QUARK MIXING IN WEAK INTERACTIONS - Ling-Lie...

QUARK MIXING IN WEAKINTERACTIONS

Ling-Lie CHAU

PhysicsDepartment,BrookhavenNationalLaboratory, Upton, New York, 11973, U.S.A.

INORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)95, No. 1(1983)1—94.North-HollandPublishingCompany

QUARK MIXING IN WEAK INTERACTIONS

Ling-Lie CHAUPhysicsDepartment,BrookhavenNationalLaboratory, Upton, New York. 11973, U.S.A.

Received9 July 1982

Contents:

Introduction 3 3.2. CP violation in partial decay rates 601. The quark mixing matrix and its determination 3 3.3. The estimateof e’ 66

1.1. Introductorydiscussion 3 3.4. The estimateof neutronelectricdipole momentd0 67

1.2. Determinationof V~dand V~ — 7 3.5. CP violation from othersources 681.3. The mixing and CP violation in the K°K°system:A 3.6. Concluding remarks 71

discussionof theformalism 8 4. Quarkmixing matrix andquark masses 711.4. Determinationof V~from the box-graphcalculation 4.1. Introduction andformulation 71

of Am ande in theK°K°system 16 4.2. Parametercounting,calculability ofquarkmixingmatrix1.5. Determination of V~from other experimentalin- in termsof quark masses,andsomeexamples 72

formation 24 4.3. Relating quark mixing matrix to quark massesneces-1.6. Concludingremarks 33 sarily leadsto flavor changingneutralcouplings 75

2. Nonleptonicdecays 34 4.4. Concluding remarks 782.1. Introductoryremarks 34 5. Alternativeto thethreeleft-handeddoubletmodel 782.2. Quarkdiagramapproach 35 5.1. Alternative models 792.3. Symmetryapproach 46 5.2. Experimentalstatusof theb-particledecay 822.4. Concludingremarks 51 5.3. Concluding remarks 83

3. CP violation 52 6. Summary andoutlook 843.1. The neutral particle—antiparticlemixing and its CP Acknowledgments 85

violation effects 52 References 85

Abstract:The statusof the quark mixing in weak interaction is reviewed. The 3 x 3 quark mixing matrix for the threeleft-handeddoublet model is

analyzed using various experimentalinformation involving strange, charmed,and b-flavored particles. Its interplay with nonleptonicdecays,implication on neutralparticle—antiparticlemixingand CF violation in heavyquark systems,andthepossibleorigin of thequark mixing from quarkmassmatrix are discussed.Finally we briefly review thestatusof alternativesourcesfor CF violation, andalternativemodelsto the threeleft-handquark doublet model.

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Ling-Lie Chau, Quarkmixing in weakinteractions 3

Introduction

Thediscoveryof thequarkhierarchyandthedevelopmentof ourunderstandingof weakinteractionsinrecentyearshavetruly madea splendidpagein thehistory of particle physics.Not only doesthe quarkgenerationrepeatitself, eventhehistoryof thediscoveryof thenewquarkstatesseemsto berepeating.In1973KobayashiandMaskawaobservedthatin orderfor CP violation to berelatedto thecouplingof theintermediatebosonwith quarks,theremust exist at least threeleft-handedquark doublets,one moredoubletthanthe two left-handeddoubletmodeloriginally postulatedin 1970 by Glashow,Iliopoulos andMaiani baseduponthesoleexperimentalobservationthatKL ~ +~ isbeingextremelysmall.Amazingly,in an almostduplicativeway to the history of the discoveryof the charmedparticlesfollowing JIçli, the bquarkstateshavenowmanifestedthemselvesfollowing thediscoveryof their implicit stateYin 1977.Eventhoughthe sixth quark, the t quark,hasso far not beenobserved,the impressivedetailedstudiesof thedecaypropertiesof theb-particlesin the Y(4s) region at CESRhavedisqualifiedmostof the alternativemodelswithout the t quark.

As was observedin the early sixties, the particle statesto which the weak intermediatebosonscouple, are mixtures of the physical particle states,and the mixing is characterizedby one rotationangleO~.In the theory with threeleft-handedquark doublets,the mixing of the quark statesis given bya 3 x 3 unitary matrix characterizedby threeanglesand onephase.This phase,as originally noted byKobayashi and Maskawa, can give the CP-violation effects. In this framework there is very richinterplay betweenthe weakdecaypropertiesof theparticlesandthe CP-violation effects.

In thisreview,we presentthe statusof ourcurrentknowledgeof the quark mixing matrix, its relationwith the weak decay propertiesof particles, implications for neutral particle—antiparticlemixing andCP-violation effectsin heavy quark systems.Discussionsare also given on the attemptsto understandthe origin of thequark mixing from symmetrypropertiesin the quark massmatrix. Finally the statusofalternativesourcesfor CP violation, andthe statusof alternativemodelswithout the t quark arebrieflyreviewed. (Many of the formulationanddiscussionsgivenherefor the quarkscan alsobe appliedto theleptons.)

1. The quark mixing matrix and its determination

1.1. Introductorydiscussion

Thehistory of thedevelopmentof our understandingaboutthequark mixing in weakinteractionsis afascinatingone. In the earlysixties,the left-handedquark statesin weak interactionwere a left-handeddoublet,and a lonely stranglesinglet [1.11,andall the right-handedquarksaresinglets:

(~‘)L’ sL; u~,dR, 5R, (1.1)

whered’ is a mixed stateof the downquarkd andthe strangequark 5,

d’= VUdd+ V~~s (1.2)

whenexpressedin termsof the CabibboangleOr,, Vud = cosO~,V~.= sin 8~.Remarkablywith this single

4 Ling-Lie Chau, Quark mixing in weak interactions

angleO~,the nuclear/3-decaysand all the hyperondecayscould be fitted. The determinationin ‘74 by

M. Roos[1.2]was

VUd = cosO~= 0.9737±0.0025, (1.3)

from the 0*~~~*01nuclear/3-decaysof ‘~oand the metastable26mAl in comparisonwith the ~ decay;and

V~= sin O~= 0.230±0.003, (1.4)

from thethen availabledataof semileptonicdecaysof the hyperons.This analysisgave

VUd~2+ V~j2=1.004±0.005. (1.5)

Thus the Cabibbotheory of VUd12 + I V~2= 1 is, within the error, satisfied. —

However, in this theory thereis the strangeness-changingneutralcurrent(d~+ sd) term arisingfromd’d’ = cos2O~dd+ sin2 O~s~+ cosO~,sin O~(d~+ sd). The rate of KL—* ~ given by such a neutralstrangenesschangingcurrentis muchtoo big for the measuredbranchingratio of [1.3],

Br(KL—*~~)=(9.1±1.8)x10~. (1.6)

This led to the introduction of the charm quark to form anotherleft-handeddoublet in the pioneerpaperby Glashow—Iliopoulso—Maiani(GIM) [1.4], 1970,

/u\ fc\UR, dR, cg, SR , (1.7)

\U/L \5/L

where(d’, s’) = (d, s)VT, andVT is the transposedmatrix of

— /V~ V~~\— ( cosO~ sinO~, 1 8— \,V~d V~~)— k,—sin O~ cosO~

and all the right-handedquarksarestill singlets.The orthogonalityof the mixing matrix, (d’d’ + s’~’)=

dd + s~,guaranteesthe absenceof strangeness-flavorchangingneutralcurrentd~term. It is now historythat after a longwait sincethe original discoveriesof J [1.51and i/i [1.6],the charmparticleswere foundin 1975 [1.71,fig. 1.1. The predictionof the GIM theory that charmparticleswould predominantlydecayintostrangleparticleshasnowbeenconfirmed[1.81.Howevertheprecisevaluesof V~.andV~dareyettobedetermined(seediscussionslater in this section).

The evolution of ideas kept pace.It was noticed in ‘73 by Kobayashiand Maskawa[1.9] that thereality of the matrix in eq. (1.8) would not allow CP violation via the intermediate-bosonW~couplingin the standardSU(2)Lx U(l) theory. Based upon this purely theoretical observation they boldlyintroducedtwo more quarks,the t and the b, which form the third pair of left-hand quark doublets.Now the left-handeddoubletsbecome

Iu\ /c\ /t\

~d’)L’~5’L’~b’)L’ UR, dR, cg,5R, tR, bR (1.9)

I +

~ \ ~ N ~ /~ \..-,-.. .—.-..

\ a. /~Ct

+ -E

+

+0

-~

0

I4,

-1

a.~1~

/1].7- ‘ 1. -E I/ ‘, I ~‘I I

I ~-•‘~_~ ~- ~ ‘ F-.--..--~-.— --..

6 Ling-Lie Chau, Quark mixingin weakinteractions

where(d’, s’, b’) = (d, s, b)VT,

/V~ V~. VUb\v=( V~d V~. V~b J (1.10)

~V~d V5. V5b JAgain all the right-handedquarksare singlets.The Va’s characterizethe couplingof the quarksq, aj tothe weak intermediateboson W~.The unitarity of the V matrix, VV~= I insures the absenceofflavor-changingneutralcurrent.

In generalfor n doublets,the numberof physically significant parametersin the quark mixing matrixV is equalto the numberof parametersfor an n x n unitary matrix, n

2, minusthe arbitraryphasesofthe quark field (2n — 1), that can be removed,i.e., n2— (2n — 1), [1.101.An orthogonalmatrix can becharacterizedby n(n — 1)72 angles, thus the rest of the parameters[n2— (2n — 1)] — n(n — 1)72 =

(n — l)(n — 2)72 have to be characterizedby phases.For example,for a 2 x 2 unitary matrix, thereare22 = 4 parameters,0, a, /3, ‘y, in thefollowing mostgeneralparametrization

A = / cos0 exp(ia) sin 0 exp(—i/3) 111\—sin 0 exp(iy) cos0 expi(/3 + y — a) - a

The (2 . 2— 1) = 3 phaseparametersa, /3, y can be removedby the bi-similarity transformationof twodiagonalunitary matrices,

(1 0 ~A(e~(p(~) 0 \ ( cosO sinO 1 lIbexp i(y — a)) \, 0 exp(—if3)) — ~‘~—sin0 cos0

These transformationsby the matrices diag[1, expi(y — a)] and diag[exp(ia), exp(i/3)] correspondtochangingthe phasesof the up and down quark fields respectively(seemore discussionsin section4).What is left then is just the single angleparameter0 (sometimeswe call it the magnitudeparameterincontrastto phaseparameter)correspondingto the 2(2— 1)72 = 1 angleparameterof a 2x 2 orthogonalmatrix. Therefore, V

51 can be characterizedby an angle0~andno phase.This is the observationmadeby KobayashiandMaskawathat in the caseof two quark doubletsthereis no complexityin the mixingmatrix, thus no CP violation from the W~coupling (seemoreelaborationon this point in sections1.3and 1.4).

For the Kobayashi—Maskawa(K—M) theory n = 3, following the samereasoning,one can see that%/~‘sarecharacterizedby threeanglesandone phase[1.9].One way to parametrizethe K—M matrix is

/‘VUd V~. VUb\ f Cl S1C3 S1S5 \V( VCd ~ V~b~=(~StC2 C1C2C3 s2ssela C1C2S3+s2csesa~. (1.12)

\1/,~ V5s 1/tb/ \—SIS2 CIS2CS + c2s3e’5 c

1s253— c2c3e1BJ

It is this complexity in V that providesthe CP violation. Thus,the salient featureof the K—M model isthat the CP violation effectis tiedwith the nonvanishingof someof the matrix elementsin the third row orthird column, which means that the b and the t flavoredparticles must havepure hadronic decays.Therefore,if the CP violation is indeedcomingfrom thiscomplexityof V, or the phase8, theunitarity ofthe 2 x 2 matrix V in eq. (1.8) cannotbe true.

It is interestingto notethat this essentialandelegantgeneralizationby KobayashiandMaskawadid

Ling-Lie (‘hau, Quark mixing in weakinteractions 7

not gainmuchattentionuntil severalyearslater [1.11,1.12]. Eventuallythe bb particle Y was found in1977 [1.13],andthe b flavoredparticleshavejust recentlybeenfound [1.14,1.15]. However, so far thequark has not showedup in any of the searchesfor it. Current experimentsat PETRA [1.16]haveset a masslimit m5> 18.5 GeV. On theotherhand,aswe shall discussin moredetail in section5, all thealternativemodelsbuilt by theoristswith a left-handedb quark without the t quark havefailed theexperimentaltests.We are now living in an interestingtime of waiting. In this reviewwe shall discussthe phenomenologyof quark mixing assumingthat the t quark exists [1,171.

Discussionspresentedin this sectioncan beeasilyusedfor the mixing matrix for the leptondoublets.

1.2. Determinationof VUd and V~.

Going back to our historicaldiscussion,the theory of Glashow—Weinberg—Salam[1.18]for electro-weak interactionsin themeantimehasgainedmuchsuccess.Its renormalizabilitywasproven [1.19,1.20].Theweakneutralcurrentwhichisanaturalpartof thetheorywasobserved[1.211.With itssingleparametersin0

2~= 0.23,awide classof electroandweakinteractionscanbeexplained[1.22].In thelight of this newpossibilityandinspiredby thefirst indicationthataheavierflavoredparticlecannotbestable[1.23],plusthefact that sinceRoos’ work of ‘74 the dataon hyperonsemileptonicdecayhadbeenimproved,with R.Shrock,we decidedto re-analyzethemixing matrix [1.24].Comparingthe0~—* 0~nuclear/3 decayof 140

and26mAl with the~tdecayas shownin figs. 1.2a, 1.2b,assumingthe absenceof lepton mixing, I V~wasdetermined,

I VUdI = c1 = 0.9737±0.0025, (1.13a)

which implies

s1 = 0.2270~ (1.13b)

Here the importantpoint is to includethe weakradiativecorrectioneffectsas shown in fig. 1.3, [1.25].This effect amountedto about 10% of the valueof VUd.

I V~~!canbedeterminedfromstrangeparticledecays.Oneway is from theK~3decayandtheotheris fromthe semileptonicdecaysof the hyperons.The averagevalue of I V~8lobtainedwas

= 0.219±0.002. (1.14)

To comparedecay rates among hyperons with that of neutron, and to estimate the K-ir formfactorf~~r(0),SU(3) symmetrymustbe used.The error in I V~~Idoesnot includethe uncertaintiesdue

N N2e-

Fig. 1 .2a. Diagramfor i.c~—~v~e~0~. Fig. I .2b. Diagramfor nuclear/3 decayN: —~N

2e v~.

S Ling-Lie Chau, Quark mixingin weakinteractions

N1~

N1 N1

Fig. 1.3. Diagramof radiativecorrectionsto nuclear/Idecay.

to SU(3) breaking.The readeris referredto [1.24]for detaileddiscussionsof errorsanduncertaintiesinvolved.

Combining the resultsof eqs.(1.13, 1.14) we have,

I VUdI2 + I VUS~2= 0.996±0.005. (1.15)

The centralvalueof I VUdI2 + I VUS~is now determinedto besmallerthan that of Roos,eq. (1.5), thoughit is still consistentwith one. Fromunitarity of V, eq. (1.15) gives

VUbI2 = (s1s3)

2= 0.004±0.005. (1.16)

From V~d= c1, Vu. = s1s3, and the determinedvalues of VUd, V~.of eqs. (1.13, 1.14), the s3 can be

determined[1.12,1.24],

1s31 = 0.28i~i~~. (1.17)

Aside from the improvementfrom future moreaccuratedataof hyperonsemileptonicdecays,it is hardto improve the determinationof thesetwo numbers,mainly dueto theoreticaluncertainties.For recentdevelopmentsof hyperondecaydataseeref. [1.26].Thereforeit will be bestto obtain VUb =

5t53 from amoredirect sourceof the b decay.Thatwill be discussedlater in section1.5.

1.3. Themixingand CP violation in theK°K°system:A discussionof theformalism

Since the model is designedto provide CP violation, someof the parametersmustbe determinedfrom the CP violation.The KL, K~systemis still the only experimentallyestablishedsystemhaving CPviolation sinceits observationin 1964 [1.27].To constrainthe other two parameters~ V~dwe use the

Ling-Lie Chau, Quark mixingin weakinteractions 9

two sets of experimentalinformation, i.e., the KL, K~mass difference ~m = mL — ms and the CPviolation parameter~. Since we shall later in section 3 discussthe mixing and CP violation in otherchargezeromesonsystems,D°D°,B°B°andT~P,we shall discussthe formulationin somedetailhere.

Considerthe regenerationwave function for theK°K°system

~(t) = aK(t)IK )+ ag(t)I~0) (:;~)~ (1.18)

with the equationof motion

id4~(t)/dt=H~(t)~(M—iF/2)~(t), (1.19)

where

H — /M11 — if’11/2 M12 — i112/2 1 20

— ~M21— iF21/2 M22— i122/2 ( . a

Mi,, r’0 aretransitionmatrix elementsfrom virtual andphysicalintermediatestatesrespectivelyandcanbe complex[1.28,1.29]

M11 = mK6~+ (iIHw,~~=2Ii)+ p ~ (iiHw,As=l!A)~AIHw,~s=lIj) (1.20b)A mK A

and

I~= 2ir ~ pA~iIHw,~StIA~(AIHW,~S.i~j), (1.20c)

whereP takesthe principal value, andPA is the densityof the A state.FromCPT,

M11 = M22, I’~= F22. (1.20d)

Fromhermiticity,

M21=M~, F21=F~. (1.20e)

Now eq.(1.20a) becomes

H — / M0 — iF0/2 M12 — iF12/2 1 20f— kM ~ — iF~/2 M0 — ilo/2 ( . )

where M0 M11 = M22, F0 = F11 = F22. Note that since the IK°~and IK°) statesdo not communicatethrough stronginteractionstheir relativephaseis not specified.The relation of IK°) to ~°) is relatedthrough CP transformationup to an arbitrary phasee

2’~,

Ling-Lie (7sau, Quark mixingin weak interactions

IK°)= e2~CPIk°)= —e2~~CIk°). (1.21)

To find the physical states,we diagonalizeH andfind its eigenvaluesA±which arethe physical massesandwidths of the physicalstates

det(H—A±I)= 0 (1.22a)

whichbecomes

det(h1÷j~m— i~u/2)/2 M12— i112/2 — 1 22b

M ~ - iF~/2 (+)(~m— i~f’I2)I2 — ‘ (- )

where

— iz~F/2)/2 M~— iF0/2 — A ~

Fromeq. (1.22b),we easily obtain

[(sm — i~F/2)/2]2= (M

12—iF12/2)(M~’2—iF~2/2). (1.23)

After diagonalizingthe matrix H, we denotethe A±eigenstatesrespectivelyby

IK~)= N~[(1+ ë)IK°)+ (1 — ~)IK°)](1 .24a)

KL) = N~[(1+~)IK°)—(1 —

whereN~is a normalizationfactor

N~= [2(1 + ~I

2)]_h/2 (1.24b)

thenfrom the eigenvalueequation

H—A I /l+E\ /—(/~m—i1XF/2)/2 M,2—i[’12/2 “~j’l+ë~~ ~125+ - ~) - \ M~2-iF~2/2 -(sm - i~F/2)/2)k1- ) -

we easily obtain

1—~(t~m—i~F/2)/2M~—iF~/2 (126)I + ~ — M12 — iF12/2 — (~m— i~F/2)/2

Note that ~ is not yet a physical quantity and dependenton the phaseconventionof IK°)and 1K0).

For exampleif ~ is pure imaginary,we have(1 ±e) = [1 + j~2]l/2 e~’°,wheretan 0= eli. We can thenredefine IK°I’ = e10IK°), IK°)’= et9IK°).Then IKE) (IK°)’+ IK°)’), IKL) (~K°)’— IK°)’), which are CPeigenstates.On the otherhand,the magnitude

Ling-Lie Chau, Quark mixing in weak interactions 11

(1 — fl/(1 + C)I = _________ 1/2 (1.27)

is a physical quantity.The deviationfrom i~= 1 specifiesthe amountof CP violation. When ~� 1 theK~,KL statesarenot orthogonal,theoverlapis given by

(KSIKL) = 2ReC/(1 + let2) = (1— ~2)/(1 + ~2) (1.28)

which is independentof phaseconvention.Using eq. (1.26), we obtain

2 Re~ — 1— ~2 — 2Im(M~Fi2/2)

1+ Ic!2 — 1+ n2 — 1F

121212 + MI

2!2 + [(L~m)2+ (z~F/2)2]/4~ (.

ThereforeCP violation is indicatedby i~� 1 or Im(F~M12)� 0, independentof phaseconvention[1.30].

In order to relate C to measurablequantities,we needto consider the decay of theK°K°system.K°,K°decaymainly into two pions[1.31].We denote

((i’rzr)ioIHwIK°)= a0exp(i8o), ((ITIT)I2IHwIK°)= a2 exp(i82), (1.30a)

((irir)10IH~Ik°)= a~exp(i30), ((1T1r)I~2IHwIk°)= a~exp(i82), (1.30b)

wherethe phasesfrom strong interactionexp(i8o) and exp(i82) are factoredout explicitly so that thephasesof a0, a2 areall from weak interaction,

a0 = lao! exp(i0o), a2 la2I exp(i02). (1.30c)

Note that the strong-interactionphasesarethe samefor K°andK°dueto C, P, T invarianceof strong

interactions.Fromeq. (1.24) we obtainao,s(L)= ((n-7r)I=olHwlKs(L)) = NE[(1+ e)ao()(1— C) a~]exp(i60), (1 .31a)

a2,s(L) ((~r1?-)I=2!Hw!Ks(L))= N~[(1 + C) a2(~)(1— C) afl exp(i82) . (1.31b)

Further,from

rOn.01 = ~(~rir)i~oI(1/V3)— ~(~‘ri’r)j~,2I(V2l\/3), (1.32a)

(1T~7rI= ((ii-’n-),=ot(V21V3)+ ((lTlr)121(17\/’i), (1.32b)

we have

Aoos(L) K1T~TIHW!KS(L))= (1/\/3)aOs(L)— (V2/V3)a2S~), (I.33a)

A+_,s(L) (ir~rrlHwlKs(L)) = (V21\/i)aO,S(L)+(1/V’~)a2,S(L). (1.33b)


Oneobtains

Aoo,L/Aoo,s= (ao,L— V2 a2L)/(ao,s— V~a15). (1.34)

Dividing ao,sboth in the numeratoranddenominatorin eq. (1.34) and defining

aO,L/aI),S, E2 a2.LlaI)S, and w a2,S/aII.S , (1.35)

we obtain

= (s — \/~E2)/(1 — \r2 w) — 2e’/(l — \/2 w), (1.36)

where

(1.37)

Similarly

— A+L ao,L+(1/V2)a2,L e+ (11V2)E2 , I-= ,— = E+E/[1+(I/V2)w]. (1.38)A~,5 ao,s+(1/v2)a2,s 1+(1/v2)w

Experimentallywe know 1a2/aol is very small (seediscussionsin section2.1)

1a2/aoI 1/20 - (1.39)

When w can be neglectedcomparingto one,eqs. (1.36, 1.38) become

(1.40a)

~7±_~E+E’. (l.40b)

Noteherethattheseexpressionsfor~ ~ + -, C, C’ aregeneral,withonlythelastapproximationutilizing theexperimentalinformationeq.(1.39). Sinceboth ~ and~+ arephysicallymeasurablequantities,C, C’ aretoo andarephase-conventionindependent,

C = (~(~+2~+~.)/3, (l.41a)

C’ = (i~+~— ThHI)/

3. (l.41b)

Now coming backto discussthe exactformula, we substituteeqs. (1.31a,b)into eq. (1.35) andobtain

CRea0+ilma0 C+it0

E — Rea0+iCIma0= 1+iCt0’ wtth tonslma0/Rea0, (1.42)


(noteherethatthephysicallymeasureds equalsC in definingtheK~,KL stateseq.(l.24a)onlywhen to = 0),

CRea2+ilma2 - /Rea2\/C+it2\)expi(62—80), (1.43)C2 expi(82—60)=(~Reao)\1+1Et0/Rea0+ iC Im a0

and

Rea2+iCIma2 . /Rea2~f1+iCt2\w = exp1(62 — 6~)= I - ) exp 1(62— 6~)), (1.44)Re a0+ iC Im a0

Rea0 \1+tCt0j

where

Im a1/Rea,. (1.45)

Now C’ of eq. (1.37) can be re-expressedas

C’(1/V2)W[2C+itol _~~..~.w(1_e2) t2tIJ

— 1+iCt2 i+iet~J~1~ (l+iCt2)(l+iCt0)

i /Rea2\~(1+iCt2= \/2 ~Rea0! \1+ iCto) (1— C

2) expi(62— 6~) ~2 tO (1.46)(1 + iCt2)(1 + iCt~1)-

Note that so far no approximationhasbeenmade.Note alsothat in our formulationherethefact that C’

vanishesas a0 and a2 havethe samephases,i.e. t2— to = 0, is alwaysmanifestlyobvious [1.32].If in a phaseconvention el ~ 1, eqs. (1.42), (1.46) becomerespectively

(1.47)

_ L (Re ~l2) (t2— t0)expi(62 — 6~). (1.48)

C’ V2 Rea0If we choosethe conventionof IK°),I(°)so that a0 is real andi.e. tO = 0, theneqs.(1.47), (1.48)become

CC, (1.49)

~ (Rea2) t2expi(62—60). (1.50)

C’ \/2 Rea0

This is the result first obtainedin the elegantwork of Wu andYang [1.31].In the literature, however some other phase convention is used, for example the K—M phase

convention

K°)=CPIK°), (1.51)

which hasthe advantagethat in calculatingweak interactionamplitudes,all the complexitiesarefrom


theK—M phase6. So to makeacomparison,we shall considerthe casesthatin certainphaseconventiona

5) is complex.Sincein K~.decay

C~I, (1.52)

with very small imaginarypart of the amplitudea0, i.e.

Im a0/Rea0 = t0 ~ 1. (1.53)

Undertheseconditions,relationsamongMi,, F11 andphysicalquantitiesaresimplified.Fromeq. (1.42)wethen obtain

C~E+it0, (1.54)

with

e~1. (1.55)

Fromeq.(1.20c),F12comesfrom thephysicaldecaysof K°andK°,which isdominatedby theI = 0 statesofthe

21T system,

F12 (K°IHwl(irir)io)((inr)ioIHwl~°), (1.56)

wherethe (irir)12 andthree-pioncontributionarenegligible. So

fl2/F~= Im(a~)2/Re(a~)2—21m a

0/Rea0 = —2t0. (1.57)

From C, C ~ 1, we can also concludefrom eq. (1.26)that

M~2/M~~1. (1.58)

Thenin this approximation,

~F~=2F~, ~ (1.59)

Further,usingthe experimentalinformation

(1.60)

from eqs.(1.26, 1.57),we obtain

- I fiM~2 1ma55\

(1.61)

Substitutingeq. (1.61) into (1.54), we obtain

Ling-LieChau, Quark mixingin weakinteractions 15

C = [11(1— i)][M~2/2M~+Im a0/Reao]

[(1+ i)/2]ItM/2+ to] , (1.62a)

where

tM =M~2/M~. (1.62b)

Thesearethe approximateformulaused often in the literature.Howeverfor heavyquark systems,theapproximationsmaynot apply.

Theexperimentalbound

lE’IEI < 1/50 (1.63)

can put a limit on the phaseratios, usingeqs. (1.38), (1 .62a).Using la2laol 1/20, we obtain

IE’lCl = (1120)l(t0 — t2)/(tM/2+ t0)l <1/50

or

l(t0— t2)l(tM/2+ to)l <2/5 . (1.64)

This boundmust be satisfiedfor thosemodelsin which a phaseconventioncan be chosenso that allphasesaresmall.

Furtherfor thosemodels t2 = 0 and all otherphasessmall, e.g. in the K—M model using the phaseconventionIK°)= CPIK°)and the “Penguin”-diagramexplanationof the L~I= ~ rule, eq. (1.64) thentakesthe form

1 + tM/(2t0)l >5/2 (1.65)

which gives

tM/(2t0)>3/2, if tM/(2t0)>0 (1.66)

ltM/(2to)l >7/2, if tM/(2tO) <0. (1.67)

Anothermeasurablequantity is the lepton asymmetryfrom KL decay.Due to CP violation, KL hasunequalmixtureof K°andK°,eq. (1.24). TheK°,K°haveverydistinctsemileptonicdecays,K°—~ tv,K°—* z.’. Due to this unequalmixtureof K°,K°in KL, the decayratesof KL into ir°t i and ir

aredifferent, andcan be calculatedto be [1.28]

— F(KL —* 1r (* v) — F(KL -~ ~ ~) — 2 R 1 — 2 /~— 2~

L — F(KL~ ~ii) + F(KL~~ — ( e e)[( x ) x ~, (1.68)

where x nsA(K°—~ir~’~zi)/A(K°—*irt1.’). The phase-independentmeasurablequantity is just theonegiven in eq. (1.29).

It, Ling-Lie Chau. Quark mixing in weak interactions

1.4. Determinationof V,1 from thebox-graphcalculation of Am and C in the K°K°system

In thissectionwe shalldiscussthe contributionto neutralparticle—antiparticletransitionmatrixM12 inthestandardelectroweaktheory.Themaincontributionis from thebox-graphshownin fig. 1.4[1.331.Herewe shall adoptthe phaseconvention

IK°) = CPI1~5~), (1.51)

so that thecomplexity in M12 is solely from that of V~,.

In comparingthe calculatedM~2/M~with measuredC, we assumeto/tM -~1 in eq. (1.62a), so thatC = [(1 + i)/21M’i2/2M~. (1.69)

After calculatingM12, we fit the mixing matrix so that

M~=Am/2, (1.70)

~M~2/Am= ReC. (1.71)

Note that Im C doesnot providefurther information.The experimentalmeasurementsfor the K°systemsare [1.34,1.35]

AT = — F1 = 0.738 X iO~’~GeV, (1.72a)

Am = ms— mL= —0.352x10~4GeV. (1.72b)

Note that the experimentalmeasurementsgive AF —2Am, which we haveusedin eq. (1.60). FromKL,S —* 2ir,

77+-i = (2.274±0.022)X iO~, ~+_ = (44.6±1.2)°, (1.73a)

177001 = (2.33±0.08)x i0~, = (54.0±5.0)°, (1.73b)

which gives

Re C = (1.536±0.062)x iO~; (1.73c)

V5~s,V~5,Vj’s —

: :±twi : ~ + u,c,t ~‘N~ :Vud ,Vcd,Vtd

Fig. 1.4. Thebox diagrams for K”~K. Ourconventionis suchthatthemixingmatrixelementata q4Wvertexis Vqif theoutgoingquarkq’ is au-likequark; is V~if theoutgoingquark q is ad-like quark.

Ling-Lie Chau, Quark mixingin weak interactions 17

from KL—* ~rtp asymmetry,

~L (3.30±0.12)xi0—~, (1.74a)

Rex= (9±20)X1O~ (l.74b)

Imx=(—4±26)x103, (1.7’k)

which gives

Re C = (1.620±0.088)xi0~. (1.74d)

Comparingwith eq. (1.73c), the two experimentalresults are consistent.From theseexperimentalinformationwe obtaina complexnumberfor M

12, usingeqs. (1.70, 1.71),

= (—0.176—i0.114x102)x iO~GeV. (1.75)

Now we shall calculate M~2,M~from the box-graph of fig. 1.4 and comparethem to data eq.

(1.75) to determine all V1~,. Ever since the original analysis [1.381,though no basic change inconclusions,therehasbeen refinementof formula and extendedanalysis to heavier quark systems[1.39—1.47].In a recentwork [1.48]a thoroughanalysisof V~and its phenomenologicalimplication aredone usingwell organizedformula and all currentlyavailableexperimentalinformation. Many resultsandformulausedin this sectionarefrom ref. [1.48].

The K°*—~K°transitionmatrix M12 given by the box graphas shownin fig. 1.4 is

M12=_~m2\~( ~ A1A.~A11)At12 (1.76)0,1= u,C,t

where

A0 = V~SVld., (1.77a)

A — x, + x1 — (11/4)x,x)+ 1 fx~[1— 2x1+ (1/4)x0x1] In x, — ~ (1 77b6 (1—x1)(1—x1) (x1—x1)t. (1—x1)

2 ‘~ ‘. -

A0 = (1_x,)2 [3—~ Xo + ~ X~]+ 2[1 — 4(1—Xe)](1)2~ (1.77c)

with the definition

x, = m~/m2w, mw= [ira/(V2 GFsin20~)]1’2.

In thisformula [1.41,1.42] all theexternalmomentaassociatedwith theexternalquark linesareneglectedin comparisonwith the W-mass.For heavyquarkstateswith massscalecomparableto that of mw, exactformulaincludingtheexternalmomentamustbeused,seesection3.1.Thefactorsin M

12 multiplying At12arethe quark diagramcontributionsof the box diagram.The matrix

At12 (K I[dy1~(1— ys)s][dye~(l._ y5)S]IK) (1.78)

IS Ling-Lie Chau, Quark mixing in weakinteractions

containsthe strong interaction contributionsof the making of K°from ~dquarks. To calculatethisquantity accuratelyrequiresour fundamentalunderstandingof the nonperturbativepart of the stronginteractionor the long distanceeffect, in the currentQCD terminology[1.44].We certainlydo not havethis knowledge.The estimatebasedupon the vacuumintermediatestateinsertiongives

42./tti2vac = —3fKmK, (1.79a)

where fK l.23m,~.The valueof At12 variesas other methodsareused.We use Att2.vac as a norm for

comparison,anddefine

At52 = BAt12.vac, (l.79b)

whereB isaconstantcharacterizingthedifferencefrom thevacuum-insertioncalculationwith itsdeviationfromone.Thebagmodel[1.44]calculationisveryuncertaindependingupontheparametersused.RecentlyB wasestimatedusingPCACandSU(3)andapositivevaluedBwith avaluesmallerthanonewasobtained.We shall use

B=lorO.4. (1.80)

The rangeof the resultsshould give someindication of the uncertainty involved in the analysis.Nowwehavetwo constraintson M12,eqs.(1.70,1.71).Given5i, s3, from theresultsof section1.2thetwo

experimentalinformation Am andE should give uniquesolutionfor s2 ands~in given quadrantsof theangles.Due to the large uncertainty in s3= 0.28i~:~determinedfrom strange-particledecays, it isappropriateto considerin the discussion

s5<0.5. (1.81)

It was notedin ref. [1.48]thatthe two constraintson M12of eq. (1.76)can besolvedalgebraicallyfor two

parametersas function of the third. First we usethe unitarity condition(1.82)

in eq. (1.76) to obtain

~, G~m~Bf~mK1Vi12~ 2

12’~r

— 2A~5+ A11)(A~)2+ 2(Auc — A~

1— ~ + A55)A~A~+ (A~0— 2A05 + A15) (A~)2], (1.83)

which is a quadraticequationof A~.GivenA~= s1c1c3,we obtaintwo solutionsfor A~correspondingto

s8>0, i.e. Im A~>0; or s~<0, i.e. Im A~<0. OnceA~is solvedas afunction of s3, c2 thencanbeobtainedfrom the quadraticequation

c~(s~c~c~+ s~s~)+ c~(2s1 c1c1 ReA2 — s~s~)+ 1A212 = 0, (1.84)

which is derivedfrom


ReA2 = —s1c1c3c~— sis3s2c2ca, (1.85a)

and

Im A2 = —s1s3s2c2s~ (1.85b)

by eliminating 6. The two solutionsof c~give the two branchesin 6:

c8 = (s1c1c2c~+ ReA2)/(s1s3c2s2), (1.85c)

and

s~= Im A2/(s1s3c2s2). (1.85d)

In the literature perturbative-QCD modifications on the naive diagram have been considered[1.46,1.47]. Theymodify eq. (1.83) in the following way

~ — 2A~1+ A05—* 771(A~~— 2A~~+ A~~)— 2773(A~1— ~ — A~5+ A~~)+ 772(A15— 2A~,+ ~(1.86a)

~ — A~1— A~5+ A5~—*—773(A~~— ~ — A~,+ A~~)+772(A51— 2A~5+ ~ (1.86b)

A55—2A~,+ A~~—*772(A1,— 2A~5+ ~ (1.86c)

Note that as ~1i= 772 = = 1, the equationsreduceto the original form of eq. (1.83). These~~‘sarerather insensitiveto m1 but sensitiveto the massscaleAQCD in the perturbative-QCDcalculation.Form5< 100 GeV, A~CD= 0.1GeV

2, ~ 0.91, 772 = 0.63,773 0.31.Thesechangesturn out not to effect thefinal resultssignificantly (‘=10%), comparedto thevariation given by thechangeof B = 0.4—1.0.Besides,the validity of suchperturbative-QCDcalculationis not established[1.47].

The resultsof this analysisfrom refs. [1.38—1.42]and the recentup-to-dateanalysis[1.48]are thefollowing. Becauseof theredundancyin thecontributionof themixing anglesto theK-M matrix, onecanpick theconventionthat01, 02, 03 areall in thefirstquadrant,andlet 6 varyin all four quadrants.(Notethatevenafterselecting01, 0~,03 to bein thefirst quadrant,thereareeffectively still two possibleconventionsfor the8, which arerelatedby ±1r.Almost all authors,exceptBargeret al. of ref. [1.38]andthecurrentParticleDataBook, usethe conventionchosenhere.)

It happensthat Am and 6 do not allow 8 to be in the fourth quadrant,i.e. — ir/2 K 8<0, but 6 isallowedin thewholeregionof 0< 6 < ir, especiallyincluding6 900 Theagreementof themodelwith thesignof Re eis important.If Reewerenegative,theforbiddenregionfor 6wouldbethefirst quadrantin thephaseconventionherefor 8. For sa = 1, in order to explain the smallnessof the CP violation effect eq.(1.33),wemusthaves

2s3’= iOfl. Thesolutionfor s~ 1 is S2‘=0.2, S3 ‘=2 X i0~.We shall seein section3thatthischoicemaygivelargeCP violation effectsin thepartial-decayratedifferencein theb andt systems.

In fig. 1.5thedependenceof 52 ands3on 6 aregiven for theallowedregion of 0.35°< 6 < 179.9°.S3 de-creasesdrasticallybelow0.1 as 6 moves1°from theextremities.52 staysconstant,abouts2 ‘=0.25 in theB = 0.4 calculation and s2 ‘=0.12 in the vacuum-insertionB = 1 calculation, for most values of 6; and

20 Ling-Lie Chau Quark mixingin weakinteractions

0.6 I ______ ___________________________1.0 — t—r———i—————~——i—~i~ I I;IIJI) I iii

S203 VACUUM ‘>~:°‘.~:~‘-~0.2 - //7_<II~”=====~~=~T 6

0.l 0.5 o ~ ~

— -~ 0.4

0.0 I I II I I I III I I I - Ib) -

~ __ TTTTill‘~‘N~8)deg~8ldegl 90 :.: . -.

0.0 I I I I iii) I —.......i.._...1.__l__i t c,c~ I i~_._~ I I i ,0.1 I 10 90 0.1 I 10 90

8 (deg) 8(degree)

Fig. 1.5. Solution of s2, s3 as functionsof 8 for 3 in thefirst quadrant Fig. 1.6. Plots of I V051, VcdI, Vd/V~_jas functions of 00<3< 180°(arrowtoward right) andsecondquadrant(arrow towardleft) from the obtainedfrom thebox-graphfit to ~m. r of theK°K°system.box-graphfit to i~m,r of the K°K°system.

beginsto part as 6 moves2°towardthe extremities.We shallseethat thesetwo featuresarecommontomany mixing-matrix elements.In figs. 1.6—1.8 various V0’s andtheir ratios aregiven as function of 6.The resultsarealsosummarizedin table 1.1. Like s2 ands3 theseIV~1l’sstay ratherconstantfor a largerangeof 2°< 6 < 178°,and some I %/~begin to vary drastically only as 6 movestoward 6 <2°and6> 178°.This implies that the valueof 8 cannotbedetermineduniquelyfrom the valuesof I V~1lunless

6 <2°or 6> 178°.In the region of 2°< 6 < 178°,we need to rely upon the phase-dependentphenomenonof CF violation to determinethe angleregions.

Besidesthe region of 0.35°<6< 179.92°,thereis a small region in the third quadrantof 6 that isallowed, 180.1°< 6 < 180.25°,0.14<s2<0.38, 0.45< s3<0.5, which was first found in ref. [1.39].In fig.1.9, s2, s~andvarious Vu’s aregiven asfunction of 6 in thisregion.The resultsare alsosummarizedintable 1.1. It is interestingto notethat with all thesevariationsin 6, for s3<0.5, we alwayshave,

1s21<0.5. (1.87)

From this box-graphfit to Am and e, as shown in figs. 1.5—1.9 and table 1.1, we havelearnedthefollowing generalfeatures.The following quantitiesstayconstant,I VC~I‘=0.22, I V0d/V~~I‘=0.22, I VUbI K

0.1 for theentire allowedrangesof 6. Thereforepreciseexperimentalmeasurementsof thesequantitiesprovide a test on the validity of the box-graphcalculation. Since I VCbI, I V~,/V0b1 assumequite different


0.6 1 I I ‘‘I 1 II I I I .C — — .._. —I ..—4— 1_4_l .4.4 — — 1_... .4—_._ã_.._4_.I..J,_I_i_j— _I — )..._‘__J

0.5~ VACUUM ~ VACUUMI ili ‘(b

)

~!~h111 II

0.8 ~°-L6I at 8-0.072° Cc) 0.8

~~—~‘---—0.83 8-0.084° Id

-- Io.c I iv—t ...i. I li~i 0.0 I IrIrlIll I I ill I0.1 I 10 90 I IC 90

8(degree) 8 (degree)

Fig. 1.7. Plots of I Vb1, I V,,~,I VU,,/V~bIasfunctions of 0°<8< 180°, Fig. 1.8. Plots of I l’,bI, I ~ I V,~/V,bI as functions of 0°<8< 180°,obtainedfrom thebox-graphfit to iXm, e of the K°K°system. obtainedfrom thebox-graphfit to ~m, e of theK°K°system.

Table 1.1Valuesof V,,~determinedfrom thebox-graphfitting of tim, r of the 1(°K°system

8 0.35°<8<2°2°<oS<178°178°<8<179.92°180.1°°zS<180.25°

0.04-0.22 0.06—0.27 0.16-0.27 0.14—0.380.03—0.5 0.0—0.14 0.03—0.45 0.46-0.5

IV~dI 0.23 0.23 0.19—0.23 0.21—0.23I V~,I 0.78—0.96 0.94—0.97 0.96-0.98 0.09—0.09!V~IV~,I 0.22—0.24 0.22 0.20—0.22 0.20-0.23VbI 0.22—0.57 0.1—0.28 0.07—0.24 0.12—0.37

IV~sI 0.01—0.10 0.0-43.04 0.01—0.1 0.07—0.09Vus! VcbI 0.020.2 0.0-0.14 0.02—1.61 0.300.93

I l4dI 0.01—0.05 0.01—0.06 0.03-0.06 0.03—0.09IV4 0.22—0.57 0.09—0.27 0.22—0.05 0.10—0.38I VibI 0.82—0.97 0.97—0.99 0.98—0.99 0.92—0.99IV,~/V~5I 0.24=0.63 0.03—0.24 0.02—0.21 0.01—0.41

22 Ling-Lie Chau, Quark mixing in weakinteractions

Vcb~ I I

2\ ~

I I III~ I I

::L~ fl05~ _05 _________ 0.1 O.I~

04’— 11111 I III liii) I

s3 IvcdIvcsI Ivub/vcbl

:.:~ ~

:~L~JLT’~ ~J nil III0.05 0.1 02 0.05 0.1 0.2 0.05 0.1 0.2 0.5

5—leo’ Ideal

Fig. 1.9. Allowed valuesof S2. s3 andvarious V~,’sas functionof 8 for 8 in the third quadrant.

valuesfor 6 <2°and 8> 178°,the measurementsof thesequantitiescan provideinformation aboutthevalue of 6. The important featuresfor the region 180.1°< 6 < 180.25°are that 0.9<I ~ and 0.3<I VUb/ VCbI <0.9.Asweshallseein our laterdiscussion,thecurrentexperimentalboundson I VUbIVCbI <0.5havealreadyput someconstraintin this region.This s8 <0 region can becompletelyruled out if futureimprovedb-decaydatacan establishI VUb/VCbI <0.3. (SeeNoteaddedin proof nearthe endof page32.)

Some other interestinggeneral resultshavealso emergedfrom this analysis.The diagonalmatrixelementsof the mixing matrix arealwaysbigger thanthe off diagonalelements.With the oneexceptionof I VUb/ VCbI > 1 for the tiny region of 179°< 6 < 179.82°,the magnitudesof matrix elementsdecreaseastheymove away from thediagonal.This saysthat quarkslike to keeptheir original generationidentitythoughthereis somequark mixing amongdifferentgenerationsof quark doublets,i.e. a phenomenonofgenerationgap. In physicalterms,quarksdecayin a cascadefashion,e.g.the b particleswill prominentlydecayinto charm particles, then charm to strange.This is now supportedby experimentfrom CESR

Ling-LieChau, Quark mixingin weakinteractions 23

[1.14, 1.49]. This principle has also been utilized for the observation of the bare b states inhadronic reactions[1.15].This analysis also predicts that the t particles will decay mainly into bparticles.Howeverthis phenomenonof generationgap alsopreventsus from knowing whethertherearemoredoublets,just like the Cabibboangle fits strangeparticle very well though the third doubletcan exist.

To be specific,wealso give the mixing matrix for threespecific valuesof 6 anda commons1 = 0.23:

(1)6minimum = 0.34°,S3 0.5,

d s b

/ 0.9737 0.197 0.114 uVB.

04 = ( —0.227 0.789 — iO.29x iO~ 0.571+ iO.51 x iO~ C; (1.88a)\—o.o228 0.582— iO.29x 10_2 —0.813— iO.50x 10-2 t

_n~no _Iln0minimum — tJ.U~/ S

3 —

0.9737 0.197 0.114Vvac = 0.228 0.822 iO.25 x i0~ 0.522+iO.43 X iO~ . (1.88b)

—0.0093 0.534+ iO.61x 10_2 —0.845— iO.10x 10_i

(2) 6 = 90°,s3= 0.00135,

0.9737 0.228 0.308xi0~Vfi04 0.222 0.947iO.31X i0~ 0.12x10

2+iO.233 (1.89a)—0.053 0.227+ iO.13x 10_2 0.61x i0~— iO.972

6 = 90°,s3= 0.003,

0.9737 0.228 0.683x iO~Vvac = 0.226 0.968 — iO.33 X l0~ 0.25X 102+ iO.110 . (1.89b)

—0.025 0.107 + iO.30x 10_2 0.42x 102 — iO.994

(3) 6maximum = 179.93°,s3 = 0.5,

/ 0.9737 0.197 0.114VBO4 = ( —0.183 0.975— iO.36X i0~ —0.122+ iO.62X i0~ 1 (1.90a)

\—0.135 0.0979+ iO.49x iO~ 0.986— iO.85x iO~ ,/

2 —1100,1° —(-‘maximum — I J.~71 S3 — -

/ 0.9737 0.197 0.113Vvac ( 0.191 0.980—iO.29X i0~ —0.0632+iO.51 X i0~ . (1.90b)

\—0.124 0.0398+ iO.45 X i0~ 0.991— iO.78x iO~

24 Ling-Lie Ghau, Quarkmixing in weakinteractions

Indeedour whole analysisis basedupon the existenceof the t quark asformulatedin the model. Infigs. l.lOa, 1.lOb thes

2 vs s3 are given for two valuesof m5= 30GeV,and 100GeV. We see that for 6 inthe quadrantsI and II the changesin allowed valuesof s2, as a function of s3 are mild, but for 6 inquadrantIII thechangesarebig.Howeverasweshalldiscusslater,in section1.5.5,if 6 indeedis in quadrantIII, the t quark massis bounded20GeV< m1 <47 GeVfrom the additionalconstraintfrom KL—~~For detaileddiscussionsseeref. [1.481;further the quadrantIII solution is alreadyseverelylimited by

I V~b/VCbI from b-particledecay,section1.5.4. (SeeNote addedin proof nearthe endof page32.)

1.5. Determinationof V11 from otherexperimentalinformation

Besidesthe Am — C determinationof the mixing matrix, which is dependenton the box-diagramcalculation,it is importantto haveindependentdeterminationof ~ V~d,VCh, VUb in away similarly tothoseof ~ V~.[1.241,asemphasizedin ref. [1.50].Recentlysuchdetailedanalysishasbeencarriedout inrefs. [1.51,1.481.

1.5.1. Determinationof I V~1Ifrom semileptonicdecaysof charmedparticles

FordeterminingI V~,the inclusivesemileptonicdecayof D~or D°can be used [1.49—1.521

f(D—~e~X)~(2±1)xlOst sect. (1.91)

UsingthenaiveW-emissionquark diagramlike fig. 1.2,oneobtainsthewidth of thecharmedsemileptonicdecay,

1.0 1.0// /

8’0.4 // VACUUM /,///

0.8 - ——— m5’IQOGeV // - 0.8 - ———rn5’IOOGeV //

rn5’ 30 GeV // mt a 30 0eV //// <2

// //0.6 // - 0.6H // -

// //// /

S2 / ~2 //0.4 /// - 0.4 // ~ -

U // /flI m. //~/11 /

0.2 - / - 0.2 / / -- II /li/

.——--_I_ ‘—-.__ I

0.0 — — 0.0 ~ ~ -~- —0.0 0.2 0.4 0.6 0.8 1.0 00 0.2 0.4 0.6 0.8 1.0

S3 S3

Fig. 1.lOa. Allowed valuesof 52,s3 from theB = 0.4 calculationof the Fig. 1.lOb. Same as fig. 1.lOa except from the vacuum-insertionbox-graphfor m1 = 30 GeV, 100 GeV for 3 in theI, II, III quadrants. calculation.

Ling-Lie Chau, Quark mixing in weak interactions 25

F(c —~ se~~e)= G~m~~ I V~5I2(l— 8z

5 + 8z?— z~— 12z~ln z1), (1.92)i~=s,d

wherez~= m~/m~.Using the experimentalnumbereq. (1.91) andm~= 1.75 GeV, m~= 0.45 GeV, weobtain

I V~~I= 0.87i~ig~. . (1.93)

Howeverthecalculationisverysensitiveto themassm~,andalsothemassratio m~/m~. Moreseriouslywehaveno way to gaugehow reliablethistypeof calculation is [1.53].

Anotherway to determine V~5Iis to considerD~—*K°e~ve.Fromthe studyof the energyspectrumEe of the electron, it is expectedthat half of the D~-~ e

4X is from D~—* K°e4Pe[1.52].Justas for K~3

decayto obtain ~ [1.24],the authorsof ref. [1.51]derivedthe following results:

F(D~—* K°e~‘-‘e)’= 1.5X 1011 sec1If~’’~(0)I2I V~~I2, (1.94)

wherethe numericalfactor is mainly from the numericalintegration of such a decay.This is a vectortransitionand the Ademollo—Gatto theorem[1.28,1.29], i.e. f~’’~(O)= 1, is expectedto be operativeeventhough the symmetrybreakingcan be largehere.Using the experimentalinformation

F(D+Ke4X)=~F(D—s.e~X)=(1±0.5)x10” sect, (1.95a)

oneobtains

= 0.8±0.2, (1.95b)

which is in agreementwith that of eq. (1.93).Howeverin thisanalysisthe uncertaintiesfrom the effectof SU(3)breakingandthe effect of the ratherlargemomentumtransferinvolved arehardto estimate.

1.5.2. Determinationof I VCdI, I V~5I,and thus other ~ from v~i-productionof opposite-Signdilepton

For I VCdI and I V~5Ithe charm productionin neutrinoand antineutrinocan providethe information.The charm production in antineutrino scatteringwill produceopposite sign dileptons through thefollowing lepton—quarkreaction

~ ~ ~±~9—)( and ~ +~2—)( (1.96)

The former is proportional to I VedI, and the latter to I ~ Assuming SU(3) symmetry for the quarkstructurefunctionsin the nucleons,

~(x) = ü(x) = t~(x)= s(x), (1.97)

the reaction is proportional to I VCdI2+ I V~

5I2.For an isoscalartarget,the charmproductioncrosssection

by i with energyE is [1.541

i~N—*1aëX)= (I %/~~I2+ I V~I

2)o~(sea), (1.98)

26 Ling-Lie (‘han. Quark mixing in weak interactions

where

â(sea)= (2!1T)G~mNEJ dx J dy [x’ — m~/(2mNE)]~(x’),

with

x’’=x+m~/(2mNEy),

wheretheintegrationrangesarew/(1 — x) < y < 1,0< x< I — w, w ‘=(m ~+2mDmN)/(2mNE).Them~,m~

are the massesof the charmedquarkandthe charmedmesonD, respectively.Thereforewe can obtain

I VCSI2 + I VCdI2 = o(iN —* ~ ~ëX)/ó~(sea)

= o~N—~X)/[ó-(sea)Br(ë—s1X)1, (1.99)

whereBr(c-.~~X) is the semileptonic-decaybranchingratio of charmeddecay.Using [1.49,1.52],

Br(ë—*tX)~(9±1)%, (1.100)

andthe knownquark structurefunctions[1.55],andthe latesthighenergyi’-production of opposite-signdilepton [1.56,1.57] areobtained[1.48],

I VCSI2 + I VCdI2 = 0.49 — 1.04, (1.101)

which gives

VCbI2= 1— IV~~I2—IVCdI2 = 0.51 — 0.0, (1.102)

or I VCbI = 0.71—0.0.The variationshereincludethe effectsof usingdifferent quark structurefunctions.Note that this is consistentwith that obtainedfrom Am—C analysisin section 1.3, and the small-6solution is preferred. From this we can estimatethe lifetime of the b-particles, using the naiveW-emissiondiagram

Tb ~ 10 sec. (1.103)

In the neutrinoreactionthe opposite-signdilepton is producedas follows

v~d—~j~c—*~L[~X, i~s—*~c—~~ (1.104)

For an isoscalartarget,the charmproductioncrosssectionby v is

u(vN —~~ cX) = I VCd12ó-(valence)+ (I v~5I

2+ I VCdI2) &(sea), (1.105)

where

ó~(valence)= (2/ir) G2FmNE J dx J dy [x’ — m~I(2mNE)]v(x’),

Ling-LieChau, Quark mixing in weakinteractions 27

with

x’ =x + m~(2mNEy),

the valencequark structurefunction is

v(x)= u(x)+d(x)2~(x), (1.106)

where~(x) is definedin eq. (1.97).If we takethedifferenceof reactions(1.96)and(1.104),the I V~~Itermswill cancelandwhatremainsis the

I VCdI term, which is proportionalto the valencepart of the d quark distribution,

I VCdI2 = [~(~wN —* j.tcX) — (i~N—~~~ëX)]/ê(valence)

= [o~(~,4N—~utX)— ~N—*~t’X)]/[ó~(valence) Br(c—~t

4X)]. (1.107)

It is obtainedin refs. [1.48,1.51, 1.581,

I VCdI = 0.17—0.23. (1.108)

The variation hereincludesthe effects from usingdifferent quark structurefunctions. Combinedwith

thepreviousresultsof I VCdI2 + I VCSI2 we obtain

I VI = 0.66 — 1.04, or I V~5I~ 0.66. (1.109)

Notethat the resultfor I V~5Iis derivedbasedon theassumptionof SU(3)symmetricocean:~(x) s(x)=

ü(x) = d(x). If thestrangeoceanisappreciablysmallerthantheup anddownoceanasaresultof symmetrybreaking,the lower limit valueof I V~~Ishouldbe larger.Using eqs.(1.108) and(1.109),we obtain

IVCdI/IVCSI = 0.17 -= 0.35. (1.110)

All theseresultsareconsistentwith thoseobtainedfrom theAm — C analysisin section1.3. Thisagreementof I VCdI, I V~~Ifrom thesetwo independentanalysesis not trivial, andquite encouraging.

Using unitarity of the mixing matrix, the mixing-matrix elementsinvolving the t quark can becalculated

I VtbI = 0.67=- 1.0, (1.111)

I V151 = 0.0 —0.72, (1.112)

IVtdI=0.17—0.0, (1.113)

andgiven m,> 20 GeV, we can calculatethe lower limit of thet-particlelifetime

Tt < 1018 sec. (1.114)

We summarizethe absolutevalue of the mixing-matrix elementshere,with I VUdI, I V~takenfrom

28 Ling-Lie Chau. Quark mixingin weakinteractions

eqs. (1.13a, 1.14)

d s b

/0.9712 — 0.9762 0.217— 0.221 0.098— 0.0 \ uVabs (I~1I) ( O.17 0.23 0.66— 1.0 0.730.06 ) C. (1.115)

\ 0.17—0.0 0.72—O.O 0.67— 1.0 /

1.5.3. Restrictionof I ~ I VCdI from nonleptonicdecaysof charmedparticlesThe nonleptonicdecaysof charmedparticlesarealsodependenton the mixing matrix, thereforecan

provide information on the mixing-matrix elements.Section 2 will be devotedto the discussionofnonleptonicdecays.Here we shall pick two nice examplesof D decayingto two pseudoscalars,fromwhich limits on I VCdI, I VCdI areobtained.

AssumingSU(3)symmetry[1.59,1.60], onecan show

= ~IVCd/VCSI2 - (1.116)

HereF’ denotesreduceddecaywidth which hasthe phasespacefactor removedfrom the decaywidthF. This is an especiallynice relation since both final statesare exotic, i.e. nonresonance-forming,andthus free from strongfinal stateinteraction [1.61].From the experimentalinformation

F(D~—~ir°ir~)/F(D~—~K°~)<0.30, (1.117)

we obtain

IV~~/V~~I2<O.60,or IVCd/VCSI <0.77. (1.118)

Anotherboundon I VCd/VCSI can be derivedfrom the following experimentalinformation [1.8],

F(D°—sKK4)/F(D°—~Kir4) = 0.113±0.030, (1.119)

F(D°—~ir~ir)/F(D°—~Kir~)= 0.033±0.015. (1.120)

Theoreticalcalculationbasedupon SU(3) invariancegives [1.59,1.601,

a(D0_~*K_1i5)=\/2AVuaVcs, (1.121)

a(D°—~KK~)= \/2 AV~.V~.+ \/2 D(VUSV~.+ V~d~ (1.122)

a(D°—~~r~r~)= V2 AV~dV~d+ \/2 D(VUSV~.+ VUdVCd), (1.123)

whereA, D areindependentamplitudes.Note that if

V~5V~~+VudVcdO, (l.124a)


as in the two-doubletmodel,we shouldhave

—~ KK~)/F(D°—~ K i~) = F’(D°—* i~ i~)/F(D°-~ K irk)

= I Vus/Vuc42 = 0.056. (1.124b)

The fact that the two decayratiosof eqs.(1.119, 1.120)arenot equalis an indication within the K—Mmodel that IVUS/VUdI�—IVCd/VCSI. (For other interpretationsee discussionin section 2.2). From eqs.(1.121—1.123),we can derivea bound[1.59,1.60],

O.2’=iD�IVUS/VUd— VCd/VCSI~ZD=0.5 (1.125)

where

= I[F’(D° —~K4K)]”2 ±[F’(D° — ±)]1/2I

ZIDJ [F’(D°~K~)]’~the lower bound is easilysatisfied,howeverthe upperboundgives the interestingbound,knowing theimaginarypart of VCd/VC. is very small,

I VCd/VCSI <0.30±0.06. (1.126)

Note that the I V~5/VUdI — I V~d/V~~Iobtainedin section 1.4 from the box-graphcalculationof Am andC

stayswithin the boundof eq. (1.125).So the resultsfrom the two different analysesareconsistent.It isvery interestingto comparetheseresults,especiallyas the accuracyof the experimentaldataimproves,becausetheyaredeterminedfrom very different methods.

1.5.4. Constraintson I V~,,J,I VCbI from the b particle decay

Using the presentexperimentalboundon the b lifetime [1.62],

Tb<l.4X 1012sec, (1.127)

and usingthe naiveW-emissiondiagramoneobtainsa lower boundon 1/a,,,

I VCbI >0.032. (1.128)

It is a ratherloosebound.We certainlyneedmoreaccuratelife-time measurementof the b.

Since VUb = s1s3, for s3 <0.5,

IVUbI<0.12. (1.129)

Unlike1/ed, so far thereis no positive identification of VUb � 0. The samesign dilepton production,

~u-bX—~c~X—e~tstX, so far is inconsistentwith the model [1.631,i.e. VUb needsto be ofthe order of magnitudeten, while the b-decayinformation from CESR is consistentwith V~b= 0,[1.64].

To estimateI VUb/VCbI, two methodscan be used: (1) Study the numberof K’s in B decay,which is

30 Ling-Lie (‘han. Quark mixing in weakinteractions

now estimatedto be 1,7±0.2±0.3per b-particleproduced,where the first error is statistical,and thesecondsystematical.The K’s arefrom the cascadedecadeof b—* c—~s. This of courseis verysensitivetothe details of fragmentationof charminto K’s. Currently, the datais consistentwith I VUbI = 0 andwithan upperboundof I VUb/VCbI <50% [1.64]. (2) Study theinclusivelepton distributionfrom the b decay.The lepton momentumis slower from the semileptonicdecayof b—*c[pe than from b—*u/?p1. Thesubsequentsemileptonicdecayof c—*stv1 from b~—~ct~t’1gives an evenslower moving t. Currentlythespectrumof t fromb—* t i3~XsuggestsaneffectivehadronmassM~ 2 GeV.This isconsistentwithbdecayingexclusivelyto c, i.e. I VUbI = 0. But atthistime a smallb—~u componentcannotbeexcluded.Theboundnow is [1.64](seeNote addedin proof nearthe endof page32)

IVubI/IVChI<O.5. (1.130)

This boundcombinedwith that from CP violation analysis,fig. 1.7 limits 6 to be 6 < 178.85°.As wementionedearlier if this bounddecreasesto I VUb/ VCbI <0.3, 6 can be excludedfrom the third quadrant.Thenthe allowedregionfor 6 is 2°< 6 < 178°.We seethat so far, all theseboundsare in agreementwith%

7,1’s determinedfrom CF violation in theK°K°system.Fromthevalueof thematrixelementswecanaskwhatconstraintstheyput on themixing angles.In figs.

l.lla,b,ccontoursof constantI V~~Iareplottedin the 52—Siplanefor 6 = 0°,90°and 180°.In thesefigureseventhoughthewholerangesof 0 < s~,s-3 < 1 areshown,but only0< s2, 53<0.5 areallowedfrom thefit tohyperonsemileptonicdecays,andto ML—MS. Thebox-graph-calculationsolutionsappearasapointon theplot for agiven 6, whichareshownascirclesin thefigures.Thesolidcirclesarefrom theB = 0.4calculation,andthedashedcirclesfromthevacuuminsertioncalculationsfor 6 179°,90°and1°.Theallowedpointsforminimum6’s arealsoshownin triangles,solid triangleis for the B = 0.4 calculationat minimum 6 = 0.3°

anddashedtriangleis for thevacuuminsertioncalculationat minimum 6 = 0.9°.We seethat I V~~Ican bevery limiting on theregionsof the angles.For examplea valueof I V~~I<0.83can rule out 6 > 90°region.Aswe shall discussin section3, the CP violation effectsin the b andt systemsare sensitiveto the 6 value.Thereforea moreprecisedeterminationof Va,, will be very valuable.

In figs. l.12a,b,cconstantIV~~/V~~Icontoursaregiven. The contourtopology is very sensitiveto thevalues I V~~/~ especiallynearthe demarcationpoint I V~~/V~~I0.24. If I

1~Cd/V~~Icouldbe shownto begreaterthan 0.27, this wouldimply6-~90°astheonly solution. In figs. 1. l3a,b,ctheconstantI VChI contoursaregiven,weseethatthecurrentexperimentalboundI VChI >0.032of eq.(2.128)givesveryweaklimitationson the angles.In figs. l.l4a,b,cconstant I VUb/VChI contoursaregiven. We seethat the presentlimit ofIV~

6/V~6I<0.5of eq. (1.130)haveno limitations on the anglesat all.In fig. 1.13, the constantIVCbI contours are given, we see that the current experimentalbound

I VCbI > 0.032 of eq. (1.128) gives very weak limitations on the angles.On theotherhand,if we takethelimitation from eq. (1.38a), O.7O<IV~~I<1,we find IVCbI<O.6, which gives a lower limit of b lifetimeTh>

2.SX 1O~4sec.Fromthesefiguresweseethattheanglesandthephasecannotbewelldeterminedfrom thevaluesof I V~,I

alone.Thecurrentcrudelydeterminedvalues of I ‘~~1Ican hardly givemuch informationaboutthe angles

andphase.The box-graphcalculationfor Am andC can providemorespecificinformation,seethecirclesandtrianglesin thesefigures.Bettermeasuredvaluesof I V411 andfutureobservationof otherCP-violationeffectspredictedby themodelareneededtodeterminetheseanglesandthephase,seediscussionin section3.2.

In table 1.2 we list all the information obtainedso far on mixing anglesand matrices and theirorigins. Comparingtheseresults with those in table 1.1, we see that the IV11I so far obtainedfrom

Ling-Lie Chau. Quarkmixing in weakinteractions 31

10 I I

0 23

8m0° 0.810.87o.8.~~d~0~%:02508 IVcsI 0.950 96

0.6 06-

CdCdSi,

04 0 ~ 0.26 0.3

\\077 04- 0.35 0.40.2502 0

0.2- 0

\\o.ss0.96\\ \ _____________

083~~ I 181m900

0.8 085 08 Vcd 0.23181m90° 024 _______

IvcsI0250.6 - 0.6

CdCd (I,

0.4 0 81

I ~ ~27~0.2 > 0 0 83 0 40.2

am 8000.8 0.8

~ii~i~18002/ 0.8

0.6

095~/

0.0 I 096 08~ 00 0.25 0.27 0.3 I 0.6)Cd CdSi) 0.4/////(/// __020.2 - 0.240.0 0.2 0.4 0.6 0.8 .0 0.0 0.2 0.4 0.6 0.8 5.0

S3 so

Fig. 1.11. ConstantI V~contoursin s~—s3for 3 =0°and180°.The solid Fig. 1.12. Sameasfig. 1.11 for I V~4/V~1I.circles arefrom the B= 0.4calculationof Section 1.3; thedashedcirclesfrom thevacuum-insertionB = 1 calculationof section1.3 for S = 1°,90°.179°.The solid triangle is for theB= 0.4calculationat minimumo = 0.3°,the dashedtriangle is for the vacuum-insertionB= 1 cal-culation at minimum S = 0.9°.

variousexperimentalinformation areconsistentwith thoseobtainedfrom box-graphcalculationof theCF violation effectsand the massdifferenceof KL, K8. It is of importanceto improvethe accuracyofsuchdeterminationof the valuesof 14.

32 Ling-Lie (‘han. Quark lnIxing in weakinteractions

0 I I I ~ .0 BmO°

8 00 0.3I6’I

045’ Iy~l08 IVcbI 0.8 IVcbl

0.6 0.6

(‘4Cd 0.1 0.12Si)

0.40.20.4 (0 0.15

0.2 00.2

0.316 ‘~\ 0.23

~I ~ _____________________________.1..I 0 I 03I~

181 m 900lBl~90°

0.8- IVcbI 0.8 0~

I ~ I0.12

06- . 0.6

024Cd 0.15(I) (0 0.I04 - 0.4-

0~0.3I6 1 0.2 0.2302 00316 0.2-

~ ~ I I.C I I

8 m 800

I VC~I I Vub I0.8 - _I 0.2308 I ~ct I 0.35

0,1 0.25.00.6 -0.6 3.2

I C ~~80° I(‘1 3.2(‘4 (I)(I)

0.4 - .00.4

0

0.2 ,I 0316 0.2 - 0.35

0.0 I0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 10S

3 53

Fig. 1.13. Sameas fig. 1.11 for VChI. Fig. 1.14. Sameas fig. 1.11 for Vuh/Va,~.

All theseanalysesdiscussedin sections1.5.1—1.5.4oughtto becontinuedas moredatais availableforthe b andeventhe t particles[1.65—1.67].

Note addedin proof. It was reportedat the XXI mt. Conf. on High EnergyPhysics,July 26—31, 1982,Paris;thattheCUSBcollaborationandCLEOcollaborationatCESR,Cornellhavenarrowedthe bound:

I VUbI/I VCbI <0.2.Thisresultrulesout thess <0solutiongivenin section1.4, seefig. 1.9;andfrom fig. 1.7weseethat the solutionsnear6 180°arealsomorelimited now 6 < 179.6°,

Ling-Lie Chau, Quark mixing in weakinteractions 33

Table 1.2Summaryof IV,11 determinationfrom sourcesotherthan box-graphfittings of ~.sn.r of the

K°K°systems

(1) V~d=c~= 0.9737±0.0025 0~—a0~nuclear$ decay(2) V~,= s1c3= 0.219—0.002 K~3andsemileptonicdecayof

(2a) s~= 0.28~~ hyperons(3) (1.64< Iv~,I<1 c—asep,(3a) I V~1I= 0.8±0.2 D~—~k°e~r(4) IV~~I

2+IV~4I

2=0.49—l.04 from i(d+1)—u~cX—*~s’rX(4a) I V~bI2= 0.0 —0.51 from I V~

1I2+ I VCIJJ~+ I V~

5I2= 1

(5) I V~dI= 0.17—- 0.23 from thedifferenceoft~(d+s)-ucX--~~t”Xandti,~(d+1)-cX’~CX

(5a) I VCdI = Isic2I, s2 <0.6 using (5)

(6) I V~1I= 1.0— 0.66 from (4) and (5)

(7) IV~1I/IV~I= 0.17—0.35 from (5) and (6)(8) I VCdI/I V~j<0.4 from (5a), S3 <0.5, s~<0.6—

(9) I!44I/Lt~’~~k0.77 from 1(D i~°ir~)/V(D~-.K°ir~)<0.3andF(D~-~ 1T°1r~)/F(D~-* k°~~)= ~IV,d/ v~,I

(10) I V~5I/IV~1I<0.30±0.06 from J’(D~—* K~K)/1(D°—* K 7r~)and F(D°-~ sr)/F(D°-*Ksr~)

(11) I V,~Iin eqs. (1.111)to (1.113) from (1), (2), (3). (4), (5) andtheunitarity of theV matrix

(12) 0.06<1V~~I from r~< 1.4X 1012sec(13) IV~sI=siss<0.lI2 from s~<0.5

(14) I~”abI/I~’cbI<0.2 b—°ce~~

1.55. constraintsfrom KL-~,LL~I~L

As we mentionedbeforethesmallnessof the rateKL—~~ promptedthe introductionof the GIMmechanismandthe introductionof the charmedquark.Now we can askif it can still furtherconstrainthe parametersin the model, like the mixing-matrix elementsandthe unobservedt quark masses.

The analysisof C andAm in section 1.3hasdeterminedall the mixing matrix in terms of oneangle,say s3, for a given m5 value. Figs. 1.lOa, 1.lOb show the dependenceof the mixing angleson m~,thedependenceis ratherweak for 6 in the quadrantsI and II, but strong for 6 in the quadrantIII. Thehopeis that by studyingonemorereactionwhich alsodependson thequark mixing matrix and m5, wecan determinethe allowedvaluesof m5 andthe mixing matrix.

It turnsout, as alreadyobservedin [1.681,that KL—* /2~/ doesnot put much moreconstrainton themixing matrix for S,~>0. For 6 in the third quadrant,i.e. s5 <0, it is concludedin ref. [1.48]that m5 isbounded20 GeV< m~<47GeV. This indeedhelpsto boundthe allowedvaluesof mixing anglesin thisregion of 6 wherethe mixing anglesaremoresensitiveto m5, figs. 1.lOa, 1.lOb. For detaileddiscussionof the analysisKL ~ - andits implicationson the quark massin the region S~>0, seeref. [1.48].

1.6. Concludingremarks

We havecome a long way since the original work of Kobayashi—Maskawa.The original deter-mination of themixing matrix in fitting Am, r of theK°K°systemwith thecrudebox-graphcalculationhassurprisinglysurvivedfurther independentexperimentalconstraints.One of the striking featuresemer-ging from theseanalysesis the generation-gapphenomenon,i.e. quarksof differentgenerationsdo mix

34 Ling-Lie (‘han, Quark mixingin weakinteractions

but the mixing getsweakeras the generationgapincreases.Physicallythis gives the cascadepatternofdecay.e.g. the observeddecayof b —* c —~s, andthe predicteddecaypatternfor theyet-to-beobservedquark t—*b--*c—~s.The analysisof I’4I canbe refinedas moreexperimentalinformationaboutthe c, b,andeventhe t quarksareavailable.It will be a continuingstudy,I am sure,for yearsto come.

While we do havethe approximaterangeof valuesof thequark mixing elementsmatrix now, we stillhavevery poor information aboutthe angles 02, 03 and especiallyabout the phase6, which can beanywherein thefirst andsecondquadrants.Thesolutionsnear6 0°or 180°maybedifferentiablefrom thevaluesof I 1/~~I.But if 6 happensto fall in themiddlevaluesawayfrom 0°and180°,thevaluesof 6 cannotbedeterminedfrom I VSJI~sinceI ‘41 iscalculatedin fitting Am andC tohavequiteconstantvaluesin thatregion.Since6 is closely relatedto the CP violation phenomenology,the confirmationof the K—M modelmustcomefrom theobservationof themanyinterestingCPviolation effectsgivenby theK—M modelotherthanthe C in the K°system,e.g. C’, partial decayratesdifferencesin A —* irp, A —* ~ andotherstrangeparticledecays,seesection3. Unfortunatelythepredictionof CP-violationeffectsin theheavyquarkD°D°,B°B°,T°T°systemsfrom the particle—antiparticletransitionareextremelysmall.Exceptfor thepurposeofcheckingthe validity of themodelby the non-observationof particle—antiparticle-transitiontypeof CPviolation, wecannotusetheinformationto determinethevalueof 6. It seemsthat the only possibility todeterminethe value of 6 is the measurementof partial decay-ratedifference betweenparticle andantiparticleinto their CP-conjugatedfinal particlestates,especiallyin theheavyquark systemsof b and t,

see section3.2. It is a strong point of the model that many interestingCF-violation phenomenaarepredicted.The observationof any such CP-violation effectsis extremelyinteresting.

One naturally raises the questionwhetherthere are more quark doubletsthan three. Due to thegeneration-gapphenomenonof the mixing-matrix V, we cannotpredict whetherhigher quark doubletsexist from the deviation from unitarity of V, for the samereasonthat the Cabibboangleworked sowell that the existenceof the third quark doublet was not suspecteduntil a good physical reasonwasfound by Kobayashiand Maskawa.Then the questionis whetherthereis a compellingreasonfor theexistenceof quark doubletsbeyondthree.Recently,CP violation with the instability of the protonhavebeenreasonedto be the causeof matter—antimatterasymmetryin our universe, developedduring itsthermal unstablestateshortly after the big bang. There have been inklings of hints that the threedoublet model with the K—M CF-violation may no~be enough to accomplish the matter—antimatterasymmetry.More heavy quark doubletsmay be needed.It will be interesting to seeif such reasoningcan besubstantiatedin the future [1.69,1.70].

2. Nonleptonicdecays

2.1. Introductoryremarks

The main reasonsto discussnonleptonicdecaysin thecontext of this reviewaretwo-fold. Firstly theanalysisof nonleptonicdecayamplitudesneedsto be carriedout in conjunctionwith mixing matrixelement analysis, thus providing information about the mixing matrix elements.Secondly, the CP-violation effects can appearin nonleptonicdecaysfrom the interferencebetweenamplitudeswhichhavephasesboth from the weak and strong interactions.Therefore the quark diagramformulationdiscussedin this section will be useful for the discussionin the next section about CF violation innonleptonicdecays.In this sectionI shall give a very qualitativedescriptionof the currentstatusof ourknowledgeaboutnonleptonicdecays.

Ling-Lie (‘han. Quark mixing in weak interactions 35

Fig. 2.1. The leading-orderQCD diagramfor nonleptonicdecay. Fig. 2.2. The “Penguin” diagramfor nonleptonlcdecay.

Let usfirst briefly reviewthe historyof the K nonleptonicdecays.About 1954,it wasobservedthatthe—*

11-~~~°decayratewasgreatlysuppressedcomparedto that of K°—* ir~if,i.e.F(K~—*

ii~ir) 1/670, [2.11.Sinceonly the isospinI = 2 componentcontributesin K~ ~ i.e. Al = ~, it wasproposed[2.2] that the weak Hamiltonian H~was dominatedby I = ~ or in SU(3) classification,H~transformslike an [8] (moredetaileddiscussionsin section2.3). However,no good dynamicalreasonwasfound for this Al = ~rule. Soonafterthe weakinteractionswere realizedto be renormalizable[1.191andpresentingthe possibility for doing perturbativeQCD calculation[2.31,the W-emissionquark diagramswith the leading-ordergluon corrections(fig. 2.1) were shown to enhancethe Al = ~ contribution

12.41.But theenhancementfactoris muchsmallerthantheexperimentalnumber670.Laterit wassuggested[2.51that the so-called“Penguin” diagramof fig. 2.2 may give significant enoughenhancement[2.61.

Sinceit hastakenmorethan20 yearsto searchfor the explanationof themechanismof the K decay,itwill probablytakesometime beforewe really understandthe nonleptonicdecaysof the charmedandnew flavored particles. It is thereforefruitful also to discussthe nonleptonicdecaysin a phenomeno-logical framework.I shallbeginwith thequark-diagramapproachfor thenonleptonicdecays.Afterwardsthe experimentaldata and the developmentof the more ambitiousdynamical interpretationof theimportanceof certaindiagramswill be discussed.Later in section2.3, the charmedparticledecayswillalso bediscussedin the traditionalsymmetryapproach.

2.2. Quark diagram approach

First we find all distinctquark diagramsfor heavy-quarkdecays.Then,by a recombinationof variousfinal statequarksthe final hadronstatesare formed. The hope is that adescriptionconsistentwithexperimentaldataemerges.Onemayfind thatoneorasubsetof diagramsdominateso thatthedescriptionof the nonleptonicdecaybecomessimpleandhasfurtherpredictivepower.Thisshouldprovideafirst steptowardtheunderstandingof themechanismof thenonleptonicdecays.Theultimategoalis of courseto findadynamicalschemethatsuchgraphscanbecalculatedandprovideadescriptionof thenonleptonicdecays.In thefollowing we shall discusswhathasbeenlearnedfrom the strangeandcharmedmesondecays,andthe interestingeffectsto look for in the future.

For the meson decay there are six distinct diagrams [2.71, as shown in fig. 2.3. Diagram (~)correspondsto what is the usuallycalled externalW-emissiondiagram,while diagram(~~)correspondsto the internalW-emissiondiagram.Note that diagrams(a) and(ö) are treateddifferently. In diagram(a) the W producedquark—antiquarkpair mustcombinewith eachother to producehadronswhile indiagram (o) they must not combinewith eachother but with the other available quarksto producehadrons.Diagram(e) is thewell-known W-exchangediagram,it contributesonly to neutralmesons,asinK°,D°decays.Diagram (d) is the W-annihilationdiagram in which the initial quark and antiquark

36 Ling-Lie (‘hau. Quark mixing in weak interactions

annihilateto makea W-bosonwhich subsequentlyconvertsinto a q~jpair involving only light quarks.Diagram (e) is the I~Penguin~~diagram while diagram (~)is the side-ways“Penguin”diagram.In thediagramsof fig. 2.3 the quark lines are not joined to make specific hadrons.These diagramsgiveinclusivetotaldecayamplitudesfor heavyquark systemswhich can decayinto manyhadronicchannels.

For exclusive decayswe need to combineq~into specific hadron states.In somegraphs quark—antiquark pairs need to be created.For example in graphs (c )—( f) we needto producean extraquark—antiquarkpair in order to producea pair of light mesonsin the final state(which we assumetobe takencare of by the strongly interactinggluons). In our discussionshereit is assumedthat uu, dd ands~pairs are producedwith equalprobability from the vacuum, as implied by SU(3) symmetry.Thefinal statequarks are then allowed to hadronizeand arrangethemselvesin all possible ways. Forpseudoscalarmesonswe use the well-known SU(3) particle contentsof the variousq~system:

uU (1/\/2)ir~~+ (l/\/6)i~ts+ (‘/‘/~)x~

dd: ~(1/\/2)ir0 + (1/\/6)iics+ (l/\/3)X°,

s~: —(V2/\/3)m1+(l/\/3)x, (21)

ud:~, dü:~,

u~:K~, d~:K15

Sd: K°, sü: K -

Fig. 2.4gives the six graphsof a mesonwith heavyquark decayinginto two lighter mesons.

We would like to emphasizethat thesegraphs are symbolic and meant to have all the stronginteractions included, i.e. gluon lines are included in all possible ways. Only in specific modelcalculationswill we specify the strong interaction involved. For example in the leading-orderQCDcalculationone gluon exchangewill be specifically included. In the following our discussionon theleading-orderQCD calculation will be very qualitative. For more quantitative discussionsee theexcellentreviewsin refs. [2.8,2.91.

i:i~ L

/Fig. 2.3. The six quark diagramsfor the inclusive nonleptonicdecay of a meson.

Ling-Lie (‘han, Quarkmixingin weakinteractions 37

~ i~t Dct/

Fig. 2.4. The six quark diagramsfor a mesondecayinginto two mesons.

2.2.1. Strange mesondecays

In table2.la,bwe list the quark diagramamplitudesfor K~,K°decayinto two pionsandthreepions.Replacing1/~by V~in table 2.la,b, we obtain the correspondingdecayamplitudesfor K, ~.° intochargeconjugatedparticlesrespectively.We shall discussin section3 the implication in CF violationdueto this changeof 14 to V~in the correspondingantiparticledecayamplitudes.

Note that the absenceof the W-annihilation diagram (d) and the “Penguin” diagram (e) inA(K~—~~ is specialto the two-pion ~ final statesof the K’ decay.Amplitudes(4 and(e) docontributeto the 3i~final statesof theK~decay,asindicatedby table2.la,which howeveris inhibitedby phasespace.Amplitudes (e) and (f) intrinsically do not contribute to the K~decay.To the K°decays,into 2ir as well as 31T final states,all the amplitudesexcept (4 contribute. The one gluonapproximationfor the amplitude (.e) is the “Penguin” diagram fIg. 2.2. As mentionedin the intro-duction of this section, it has been pointed out in ref. [2.5] that the relative enhancementofA(K°—~ir~ir)comparingto A(K~—~i~ir°)is due to the dominanceof diagram(e). We shall seein thenext section that becauseof the complexity in V~.amplitude(.e) providesa weak interactionphaseinA(K°—~irtn-) andgives the CF violation effectsof a sizeableC’.

Table 2.1quark diagramamplitudesfor K decaysinto two andthreepions±

a. K~decays

+15 — (l/’v’2) V0dVJ(a+ifl

*1111 {V2}it(1/2) lV~dV~’~ia+d+~)+ V+V+j.el3~177T {\/2}tt - [V0dt/~(a-+~fd<�)+ VVcdeI

b. K°decays

7T*~T — V~4V~°,(a+6+c+e+2f)+ V~~V~i(e+2()

{V’2}tt ‘(1/2)1 V~4V~,(i~+c+e + 2/)+ V~°~V+d(e + 2/)

~1I~,(I~f) {\/3}tt . ll/(2V2)IfV~4V~~+c+~)+ VV~�j—1V~4V~.(-e+).e)+~V,°~V11~je~

tReplacing%I,~by V~one obtainsthechargeconjugateddecays.tiThe factors in thecurly bracketsare Bose-statisticsfactorsfor identical

particlessuchthatthephasespaceis alwaysH1[d3pd(2ii~]irrespectiveof whether

theparticlesare identicalor not.

38 Ling-Lie (‘hau, Quark mixing in weakinteractions

HoweverI would like to point out an importantalternative:the enhancementof A(K°—~~-~ir) overA(K~—~~r~r°) can alsobe the consequenceof the dominanceof the W-exchangediagram(e) [2.10]. Ifindeed diagram (~)dominatesover diagrams(a), (ö), (.e), (/), due to the fact that T(K~—~3~r)~’°

we also need the dominanceof the (d) amplitude for K~—~31Tdecay, see table 2.1.However such dominanceby diagrams(~)and (at) was considerednot possible in the leading-orderQCD calculation. In the perturbativeQCD scheme,diagramsin fig. 2.4 areconsideredto be the lowestorderFeynmandiagrams.In such a scheme,decaydiagrams(.c) and(1) for pseudoscalarmesonarenegligibledueto thesmallnessof the u, d quarkmasses,just like F(n~—* e~v~)is extremelysmall. As weshallseelater in thediscussionof thecharmedmesondecays,thereare indicationsfrom thedatathattheW-exchangediagram(e), andtheW-annihilationdiagram(d ) neednot besmall. If the enhancementofF(K°—~2ir) is indeeddueto the dominanceof the diagram(c), not diagram (e), the CF violation effect C’

will be muchsmallerthan that estimatedfrom consideringdiagram(~)alone.

2.2.2. CharmedmesondecaysIn tables 2.2 and 2.3 we list the charm meson lifetimes [2.11, 2.12] and decay rates into two

pseudoscalarbosons[1.8]. In table 2.4 we list the quark-diagramamplitudesfor D°,D~,F~inclusivedecays,andtable2.5forexclusivedecaysinto twopseudoscalarmesons[2.13].Replacing14 by V~in tables2.4—2.6,we obtainthe correspondingamplitudesfor D°,D, F- decays.Sincethis is still a very rapidlydevelopingfield weshalldiscussthe experimentaldataof charmeddecayfollowing ahistoricalsequence.

In April 1979 camethe first surprise[2.14]

R1 T(D°—~KK~)/F(D°—~Kir~)= 0.113±0.030, (2.2a)

R2 F(D°—~ii~ r~)/F(D°—~Ki~-’)= 0.033±0.015. (2.2b)

(Though the errors are still large and the possibility that R~= R2 may still turn out to be true, weproceedin thecontextthat R1, R2areunequal.)Thiswas asurprisebecausecalculationsweremadeonlyin the two left-handeddoublet model assumingSU(3) symmetry [2.15,2.16], and the prediction wasR~= R2= tan

20, 0.05. This can easily be seenfrom theD°decaylisted in table 2.5a, with V~.V~.=— V~dV~d.The historically interestingthing is that althoughthe Kobayashi—Maskawathreeleft-handeddoubletmodelwas proposedin 1973,no suchcalculationwasdonein that frameworkbeforethe dataofeqs. (2.2a,b).Severalexplanationshaveso far beenoffered:

(1) Ratherthanthe four-quarkmodel, oneought to generalizethe SU(3) symmetriccalculationstothe six-quark model. Then if ‘4~/~ ~ — Vçd/14~,R~and R

2 do not haveto be the sameandthe datashown in eqs.(2.2a,b)can beeasilyaccommodated[2.171.Actually theyprovidelower andupperboundson the valueof ~ V~dV~d,asdiscussedin section1.5.3. Seealsothe discussionsat the endof thissection. If V~~/V~~=—V~d/V~~,one must resortto other explanations.

(2) Theinequalityof R1andR2maybefrom SU(3)breaking[2.18],but it isdifficult togiveaquantitativeestimate.

(3) It wassuggestedthatmaybeachargedHiggs[2.19] is involvedsinceHiggsparticleprefersto coupleto heavierquarksthan the lighter ones,soas to makeR1 > R2. SuchchargedHiggs bosoncan give CFviolation effectsin K

55 systemvia the doubleHiggs-exchangebox diagram,thereforetheir couplingandmassesmaybe quite constrained[2.19].

(4) The “Penguin” diagram[2.20] maybeimportantfor the charmeddecaytoo, howeverthereis nodefiniteconclusionas to its magnitude.

Ling-Lie (‘hau. Quark ,nixing in weak interactions 39

Table 2.2Lifetime measurementsfor charmedparticles

Particle Data Experiment

I E53114-I

0° 8C72/73I N416I-I • I-I NAI8

I—-l S I--I MarkU

E531I WA58

BC72/73NAI6

I-——— I t———-l N418I I NM

I I E531

S I NAI6I——— I S I ——-I NAI8

NAI

+ I—S——I E531A~ 5 I WA58

I • —1 NAI6

I I I I

3 5 7 9 II 3 15 7 IS

Mean Lifetime110H31 ~

E531: Fermilab,r’, hybrid emulsion spectrometerexperiment.NA16: CERN, ir, p LEBC bubble chamber andEHS spectrometerexperiment.NA28: CERN, ir, BIBC bubble chamber and streamer chamber experiment.NA1: CERN, y, emulsion-fl experiment.BC72/73: SLAC, y, hybrid facility bubble chamberexperiment.Mark II: SLAC, e~e,Mark II detectorexperiment.

Table 2.3a Table 2.3bCharm D decaybranching ratios Measurements on mixing-matrix suppressed

D decays. Upper limits are at the 90%Br (1%) confidencelevel

Decaymode Mark II LGW Decaymode Branching ratios

—~Kir~ 3.0 ±0.6 2.2±0.6 r(~-~)Ir(K-sT~) 0.033±0.0 15D°—*K°ir° 2.2±1.1 — T(KK°)/I(Kir°) 0.113±0.030D~—°K°ir~ 2.3±0.7 1.5±0.6 F(1r°1r~)/F(K°7r~) <0.30

1(K°K~)/F(K°ir~) 0.25±0.15

Table 2.3c Table 2,4°Inclusivestrangeparticlebranchingratios for D decays Quark diagram amplitudes for charmed particle inclusive decayst

D°(%) D~(%) D°-+hadrons V0dV~(~x+t~+c)+ V~V0d(~+/)+V~VUS(e+/)_______________ ________________ D -+ hadrons V*d V~(a+t~)+ ~ V0d(d+~)+ V~V01(�)

Mode Mark II LGW Mark II LGW F~-* hadrons V~Vja+e~+4)+ V°~V0d(e) + V~V~,(e)

D —~KX 55 ±11 36±10 19±5 10±7 - tReplacing V~by Vj~one obtainsthe charge conjugated decays,D—°K~X 8 ±3 — 6±4 6 ±6 This comment applies to all thesetables2.4-2.7.D —* i~°x 29±11 57±18 52±18 39±39 * See remarks in ref. [2.13]for readingtables2.4—2.7 of quark

diagrams.

.11) Ling-Lie (‘hau. Quark mixingin weakinteractions

Table 2.5Quark-diagramamplitudesfor charmedmeson—~two pseudoscalarmesonsi (tt

sameas table2.1)

a. D11 decay

VudVc*s(a+c)

(l/V2) V d” (ó—c )(l/V6) V d~”(~)

k°x° (V2/V6) V,dV~’,(~+ 2�)

KK~ V,~t/~r+c+� +2/) +V

0d%7,,J(’+2/)

~rr~ V0~V~(�+2/) +V~dV~’j(a+c+�+2/)K°K° V+~V~(c+2/) + VOdV,~d(c+2/)~.1l~.1l{V2}it . (1/2) IV=Vc*s(e + 2/) + V0dV~(—ö+ C +e + 2/))~Il~() {\/2}t~. — [VV*~+ ~c+~.e+/ )+ V~dV~(~+~c

(1/V3) [V~~V~4—ó+e) + V~dV~(—c+� )](l/V6) [V,,V~(s~+2�) + V,dV~(—2c+ 2�)!

~i°x° l/(3V2)] [V~,v~(—~ —~ + 2�) + V,dVd(2ó+2e+ 2�)]xx° {V2}tt . (1/3)1V,~V~,&+e + e + 3/) + V,dV~(~++ ~ + 3/)]

K~IT V,,V~’j(a+c)(1/\/2)VO~V~d(t~—c)

K11s~1 (l/V’~)VU,Vd(~+c)

K115° (1/\~)V,,V~’d(6+e)

b. D~decay

V0dV~(a+s~)

V,,V~.,(a+�) + V,dV,~j(4+e)

— — (l/V’2)V~dV~(a+tfl

(1/V6) [V~~V~(—26+2�) + V~dV~(a+ö+ 24+2�)]

x°7T~ (l/V~)F V~SV~-~-2�) + V~dV~(a+~ + 24+ 2�)]

V~,Vs(a+d)K>r11 -(1/V’2) V~,V~(a-d)K1~iT~ (l/V6) Vu,V:d(t~4)

(l/’ui) VusVc~,j(a+d)

c. F~’decay

K°K~ V~dV~M-‘-4)0~ (V2/’f3)VudV~(a-d)

(l/V3) V~~V~(a+4)

v,.~v:~(d+�) + V0dV~(a+e)

K~-° (1/V2) FV~,V~(d+�) + V~dV~’~(—6+�)]K’~° —(1/V6)[Vo~V,~(2a+2ö+d+�)+ V~dV~l(-ö+e)]K~° (lIV3)[V,,,V~(a+6+24+2�) + V~dV~(6+4]

K°K~ V~,V~(a+()

t See footnoteat bottoimof table 2.4.

Ling-Lie (‘han, Quark mixing in weak interactions 41

Table 2.6Quark-diagramamplitudesfor charmedmeson—* pseudoscalarandvectormesonst

a. D” decay

Kp~ V0dV~.La’+c)

(1/V2) V~d~ —

i~l’pl’ (l/V2) V~dV,(r~—c)(1/V’o) V~dV~(6’— 2c + �‘)

V,V,(a+e’+�+/)V0~V~(4+c+e’+/ ) + V0dV~,~(�’+/)

1rp* V~~V~~(e’+/) + V,sV,~(a’+c÷“~/)V0, V~(c’+/) + V,,d V,~(�+/)

KOK*II V~,V~~(c+/) + V~dV~l(c’+/)(l/2)lV~~V~(e+�’+2/)

b. D’ decay

i~1Jp~

V0dV~(a+t~’)

(l/V2)[V0,V~(—�+�‘) + V~dV~j(—a—t~’—d+d’—�+

(lW2)[VV~(�’) + V,dV~(—a’—6’+d--4’+�—e’)]

V,, V,~(a’+ �‘) + V~dV~(4’+�‘)

K*K*O V,~V~(a+/) + V,d V~~(d+/)

c. F’ decay

K*OK+ V0dV/~(ö+4)KOK*+ V~dV~(r~+d’)

(l/V2) V,dV,~(d-d’)(l/V2) V0dV/~(~d+d’)(l/V6) V~dV~(2a’+d+4’)

K’p° (l/V2) [%7~,V~(d’+/) + V,d V~~(—ö’+/)](l/V2) [V~,V,~(d+.e) + V0dVc~(—6+.e)]

+ V~dV,~(a+e)

K°p+ V,,, V,~(�I’+ e’) + V,d Vc~(a’+�‘)

t Seefootnoteat bottomof table 2.4.

(5) It hasbeenpointedout that the final stateinteractioneffects[2.21,2.22] maybeimportant. It isdifficult to knowhow to incorporatesuch effects.

(6) It was alsopointedout that this maybe an effect dueto right-handcoupling [2.231.The secondsurpriseto the leading-orderOCD argumentis that [1.8,1.521,

—~ Kir~)/r(D°—* ~°ir°) = 1.6±0.9. (2.3)

This came as a surprisebecauseit was reasonedthat in the leading-orderQCD calculation,fig. 2.1, only


diagrams(a) and(~~)of fIg. 2.4 are important[2.24—2.28].All othergraphsare smallbecauseof helicityconservation,as in the suppressionof ir’- —~e’ ~‘e-

Fromtable2.5a

F(D°—~ K ~)= I Vu~V~(a+ )12, (2.4)

F(D°—~K55ir°) = ~IVUdV~(’t~—e)12 (2.5)

The leadingorderQCD implies that (e) can beneglected.The correspondencebetweenour amplitudesin table2.5 andthoseusedin the leading-orderQCD calculationsare

a =2X±~(2f±+f)/3, (2.6a)

ó =2X~(2f+—f)/3, (2.6b)

where f+ andf aredefinedin ref. [2.28].Without gluon corrections

~ a=3 , (2.7)

which is the naive result of SU(3) color counting. Leading-orderQCD calculations typically give[2.24—2.281

f=’2.19 and f+=(f~112=0.68. (2.8)

Fromeqs.(2,4-2.6,2.8) oneobtains

F(D°—~Kii-~)/F(D°—+K°i~°) 36.6, (2.9)

which is in violent disagreementwith data, eq. (2.3). Notice that this is worse than the naive colorcountingresult of eq. (2.7).

Barring the possibleeffectsdue to final stateinteraction [2.21], two typesof suggestionshavebeengiven to “explain” eq. (2.3). First suggestionis, ref. [2.29,2.30], that one abandonsthe actual leadingorder QCD resulteq. (2.8), but assumes

f~~f+, (2.lOa)

thus

(2.lOb)

which, as we shall see in section 2.3, can be a consequenceof [6*] dominance.Then one obtainsF(D°—÷K ir~)/F(D° —* K°ir°) 2, in agreementwith data eq. (2.3). Another way is to give somereasonwhy amplitude (e) in eqs. (2.4, 2.5) maybe important. If onegluon emissionis included,fig. 2.5, thenthere is no helicity conservation argument [2.311.In this calculation two more parametersareintroduced.Educatedguesseson theseparametersgive reasonablenumberscomparedto the data.

Ling-Lie(‘hau, Quark mixing in weakinteractions 43

As we see,we can from the very beginning takea naivephenomenologicalview that the exchangegraph (~)is important [2.32,1.501.

Thethird surprisecameduring the summerof 1979.The lifetimes of D’- andD°weredifferent [1.52]:

r(D~)/r(D°) >5.8 ±1.5; ~ (2.1 la)

Formorerecentdatasee table2.2 andrefs. [2.11,2.121. [Notethat the r(D°)measuredfrom the recentSLAC Hybrid Facility bubble chambermeasurement[2.12] is considerably longer than previousexperimentsand

r(D°)/r(D~)= 1.2i~i~, (2.llb)

consistentwith unity.] This is again a surprisebecausethe leading-orderQCD calculationsneglectamplitudes (c) and (4, thus in table 2.4 only inclusive amplitudes (a) and (6) contribute andr(D’-) r(D°).The two typesof reasoningoffered beforecan explain this surprise.If, a — 6 asgivenin the typeoneargument,eq. (2.10),

F(D~ j~O~.±)~ F(D°—* K~r~), (2.12)

which is also a predictionof [6*] dominance(section2.3). Howeverdatain table 2.3atells us

Br(D~—* K°1T~) Br(D°—* K irk), (2.13)

so we obtain

r(D’-) Br(D’-—*K°ir~)F(D°~_*K1T~)>i 2r(D°) Br(D°—*K~~)F(D~ K°n~)

consistentwith eq. (2.lla).Again, if we take a phenomenologicalview, we would say eq. (2.lla) implies the importanceof

diagram (c), from table 2.4. It was pointed out in ref. [2.33]that the full dominance of amplitude (c) hasalso the interesting implication that, using table 2.4, F(D°—* K°71

0)/f’(D°—* K°~°) 1/8, while theW-emissionamplitude(6) dominanceonly gives .f’(D°—* K°~

70)/r(D°—* K°~°) 1/2. Thereforeby study-

ing the relative rateof thesetwo decayswe can obtainthe relativestrengthof amplitudes(c) and (6).Of coursethe analysiswill alwaysbe complicatedby the relativephasesof the two amplitudes.Otherresultsof W-exchangedominancecan be workedout from table2.5, [2.34].

The amplitude(s~)doesnot contributeto the decaysof chargedmesonD’- andFt The correspond-ing leading-orderQCD helicity-suppressedamplitudefor D’- andF’- is the W-annihilationdiagram (�1).Unlike for diagram (c), in order to use gluon emissionfrom the initial quark line, at leasttwo gluons

Fig. 2.5. The W-exchangediagramwith onegluon emission.

44 Ling-Lie (‘hau. Quark mixingin weak interactions

mustbeemitted dueto color conservation,fig. 2.6. It is unclearwhetherthis amplitude is enhancedornot. No numericalestimateshavebeenmade.Thereare interestingimplicationsif the amplitude (4 isenhancedto a similar amountas amplitude (c). The lifetime of F~ought to be closeto that of D°,

T(D~)> r(F~) r(D11). (2.15)

In thelimit of complete(4dominancewe find from table2.5

T(F~—* K55K’-) (3/2)F(F~—* ~j11~r~) F(F’- —* K°~-~) 3T(F~ —*x°~~). (2.16)

Seetable 2.5 for manymoresuchrelations.It is interestingto notice that neither (c) nor (xl) contributeto the mixing-matrix nonsuppressed

decayof D~.Theamplitude (xl) contributesto D~decayonly with the V~dVUd,which is suppressed.Sothat even if the amplitude (xl) is as large as the amplitude (c), thelifetimeof D’- is still longer than that of D’>.The dominanceof (xl) amplitude,which has cd first annihilatedthen ud created,will also havetheinterestingprediction that the KX mode inclusive is smaller in the D’- decaythan in D°,where themixing-matrix nonsuppressed amplitude (c) alwaysproducesa strangle particle in the final state. This isinterestinglyin agreementwith experimentalindications,table2.3.Also, from table2.5bthe(xl) amplitudedominancepredicts

F(D~—* K°K~)/E(D’- j~O+)> I VCd/VCSI2. (2.17a)

This is in agreementwith datafrom table2.3b,

F(D~—*K°K~)/T(D~—*°i~~)= 0.25±0.15, (2.17b)

while

I V~d/V~~I2<0.08 (2.17c)

as determinedin section 1. The readercan readily see many such relationsusing table2.5. All thesediscussionsare very qualitative. So we see that we needmuch moredata to checkthe importanceofamplitudes(c) and(xl). Evenmoreurgently,after so muchearnesteffort, is our needto havea reliabledynamicalcalculationscheme.

Besidesphenomenologicalstudiesof amplitudes,charm decayscan alsoprovide information aboutthe mixing matrix [2.17].For example,

F(D~—* 1T511r~)/F(D’- K°i~’-) = ~IVCd/ VCSI2, (2.18a)

Fig. 2.6. TheW-annihilationdiagramwith two gluon emission.

Ling-Lie (‘hau. Quark mixing in weak interactions 45

which is a good relation sinceboth final statesare exotic andthus free of final state interaction [1.49].Fromtable2.3b we obtain

VCd/VCSI2<0.30. (2.18b)

Thiswe havealreadyusedin Section 1.From table2.5awe can alsoconstructtheinequality [2.17]

Z1D~ VusVcsV:dVcd1 ~D, (2.19a)

~ ~l+~I~‘~““~ ~ (2.19b)

where

= [f’(D°—s.K~K)]t”2 ±[F(D°—+~._~+)]I/2

LIDJ

Ans a+c, D=~

Using the experimentalnumberfrom table 2.3b we obtain

0.2~ VUSV~ ~‘~VS~ ~0.5, (2.20a)

l0~ iii I I 11111 I 1111111 I 11FII1

~I VusV~+VudVCd 0.54$

lO2.~,

/ 1v15=3OGeV

00 I liii I I I III! I I I 11111 I I I III5X1Q2 t~’ 100 101

8 (deg)

Fig. 2.7. The rangeof themagnitudeof theamplitude 1 + DIAl asfunction of S.

46 Ling-Lie (‘hau. Quarkfluxing in weakinteractions

and

0.2± I . ~ ~0.5. (2.20b)

Note that wheneverphase-spacefactors are different, all the ratios of the widths are meantfor thereducedwidths, which has phase-spacefactor removed. As we discussedin section 1.5.3, eq. (2.20a)gives the interestingbound

IVCd/VCSI <0.3. (2.20c)

Fromeq. (2.20b), given the V0 calculatedin the ~m — r analysisin section 1.4, we can give the rangerequiredof 1 + DIAl as shown in fig. 2.7. The enhancementof amplitude D over amplitude A isrequiredto be more than 102 as 6 becomesgreaterthan a few degrees.Here we see the importantinterplay betweenthe determinationof mixing matricesandamplitudes.

In table 2.6, quark diagram amplitudesfor the charmedmesonsdecayinginto a pseudoscalarand avector mesonare given. They are taken from ref. [2.35].Similar discussionslike those given in thissectionfor charmedmesondecayinginto two pseudoscalarmesonscan be made. -

2.3. Symmetryapproach

Traditionally the nonleptonicdecayshavebeen analysedin terms of the isospin SU(2), or SU(3)symmetry propertiesof the weak interactionHamiltonian H~.The current and Hamiltonian of weakinteractionsare:

J~~ (2.21)

whereV is the mixing matrix and .f’,~.= (1 + y~)y~,which shall be droppedin future equationssinceit isnot relevantfor the discussion,

= + H.C.; (2.22)

hereH.C. standsfor Hamiltonian conjugate.The SU(2)propertiesof t~s= 1 part of J~.are

J~1 = V,~Us (2*) + (1) = (2*), (2.23a)

~ = VUdÜd (2*)x (2)= (i)+ (3)~(3), (2.23b)

~ = ~ ~J~5=0+ H.C. (2*) X (3*)~H.C. = (2*) + (4) + H.C. (2.23c)

Sothe isospinof ~ can either be I = ~,(2*); or I = ~,(4); theformergives~I = ~, andthe latter= ~ in the strangenesschangingweakdecays.As we havementionedbefore F(K°—* 1T~ rr°)/F(K

51—*

Ling-Lie (‘hau, Quark mixing in weakinteractions 47

ir~if)= 1/670 is a strong supportof H~,~51~ I = ~,or z~sI= ~. This hashadmanysuccessesin K andhyperondecays.Fora nice review, seeref. [1.28].

This symmetrypropertyconsiderationH~was later extendedto SU(3):

J x [3*]~[3] [i]+[8]= ~ (2.24)

= + H.C. [8] x [8] + H.C. (2.25a)

= [1] + [8~]+ [~7]+ {[~~]+ [~]+ [8]}~.+ H.C.. (2.25b)

The representationsin the curly bracketsare antisymmetry.They do not contributeto H~,which issymmetric. The representationsin SU(3) can further be decomposedinto representationsin SU(2)accordingto their isospin andstrangenesschangingproperties

[8] = (3)~.cI + (~)~=±~ + (1)~~=~~ (2.26a)

[~7]= {(5)+(3)-i-(i)}~...~-i-{(4)+(~)h5=±t+ (3)~~~2. (2.26b)

Since the ~S= ±1part of [~7] has both representations(4), z~I= ~, and (2), ~I = ~ however the= ±1part of [8] hasonly the phenomenologicallyneeded(2), z~I= ~,it was postulatedthat [2.36]

Hw0j8], (2.27)

which hadmanysuccesses[2.37—2.39].Howeverthe lack of a consistentpicturefor thep-wavehyperondecaysis still a major difficulty.

Similar internalsymmetrystructurescan be analyzedfor charmdecays.The SU(2)propertiesof thecharmchangingcurrents:

~ = V~~es(1), (2.28a)

J~~,&,,tAs=o= VC~~d (2), (2.28b)

~ = V~dUd (2*) X (2) = (1) + (3)~(3) (2.28c)

~ = ~ + H.C. (i)x (3)+ H.C.

~ (3) + H.C., (2.38d)

= ~ H.C. ~ (2)X (3)+ H.C.

= (2) + (4) + H.C. (2.28e)

So the mixing-matrix nonsuppressed~c = z~s= 1 decayof charm hasz~sI= 1. This hastheconsequenceof [2.15—2.17,2.40—2.41]

A(D~ —* ko17~)—A(D°—*K17~)+ V2 A(D°—*j~O~.O)=0 (2.29a)

4~ Ling-Lie (‘hau. Quarkmixing in weakinteractions

and

A(F~—* 2r’-ir°) = 0, (2.29b)

which can alsobe easily seenfrom table 2.5.

The SU(3) propertiesof the charmchangingcurrents:

~ = V~ës~ [3] , (2.30a)

~ = V~ëd [3] , (2.30b)

= VUdUd ~ [s], (2.30c)

~ = JC5=~J~,~=~=0+ H.C. (2.30d)

x [3] x [8]+ H.C. = [3] + [6*] + [~j + H.C. - (2.30e)

Note that only [6*] and ft~]havethe SU(2), I = 1, (3) representation.Similarly for the mixing matrixsuppressedcharmdecayHamiltonian

= J c=t4s~OJ c=~xO+ H.C.

~ [3]x [8] + H.C. = [3] + [~ + [6*1+ H.C.

~ [3] + ft~j+ H.C. (2.30f)

where [3] containsthe SU(2), I = ~,(2) representationand ft~]containsthe SU(2), I = ~,(4) represen-tation in eq. (2.28e).

The SU(3)propertyof H~givesthe following simple relationsamongthe z~c= zXs = 1 decays

F(D°—*K°’ij)/F(D°—* K°ir°) = ~, (2.31a)

and

F(D~—* i’r°ir~)/F(D~—* Kit) = ~IV~~/~ (2.31b)

Theseandmanyotherrelationscan be obtainedfrom table2.5.As the dominanceof the SU(3) octet [8] in ~ implies the dominanceof the SU(2)

representation(2) or ~sI= ~,it would be nice to see whether the higher groupsimply an preferencebetween[6*] and[u]. With the additionalcharmquark,the symmetrygroup can beextendedto SU(4)with quarksu, d, c, s as the fundamentalrepresentation

J~4*x4i+5 ~ (2.32a)

The SU(3)decompositionof 15 accordingto L~c= 0, ±1,is

i~=~ (2.32b)


so

H~=~J~+H.C.or 15X15+H.C.= i+15+15A+20+45+45*+84+J~j.C.. (2.32c)

Amongtheseonly therepresentations20 and84 canhaveI~c= ±1,andtheirSU(3)decompositionsare:

20 [6]~..._~+[~]~c=o~ [6*] (2.33)

84 = [6*]~2+ [~]~c=±2+ {[~] + [~M]}~c=—1 + {[3*]+ [15Z~]}~...+1+{[i] + [8]+ ~ (2.34)

We see that the ~c = 0 part of 84 has both [8] and [~7], but the iXc = 0 part of 20 has thephenomenologicallyneeded[~]only. Soif weproceedasin theextensionof SU(2)to SU(3), it is naturaltopick 20 asthe dominantcomponentfor H~.Thenthecharmchangingpart of ~ to transformlike[6*]. Assuming[6*] dominance,the charmdecaywas analyzedin ref. [2.15].

Howeverright now we arenot sureif [6*] is theonly dominantrepresentation.We shall analyzethehadronicdecayaccordingto the general symmetry propertiesof H~,which has [3], [6], [~] for the1~c= ±1transition.Since charm particlesalways decayinto ordinary quark stateswhich are classifiedaccordingto SU(3), the symmetry usedin describingthesedecaysis still SU(3)even thoughSU(4) hasbeenusedfor the discussionsof thetransformationpropertiesof H~.We considerthe decayamplitude,(/t~IHw~~...±1IJe/t),of charmedpseudoscalarmesoninto two pseudoscalars.HeretheSU(3) transformationpropertiesare

or ~[3*]I (2.35a)

~ ~ (2.35b)

or (8x 8)~= (1))+ I(s)) + (,~7)). (2.35c)

Thentherearefive independentamplitudesthat can beconstructedfor (/s~!Hw,~~±1I~/1),which should

be aSU(3)singlet,

S ~I[~]I[~]), E

T =~ F

G ~ficJ[3*]I[i]). (2.36)

In table 2.7 the two pseudoscalardecaysof D~,F’-, D°are listed in terms of thesefive amplitudes,asgiven in ref. [2.42].

Comparingtables2.5 and 2.7, we note the six to five correspondencebetweenthe quark diagramamplitudesandthe symmetry amplitudes.Recentlythe correspondencebetweenquark diagramandthetensordecompositionof the symmetry amplitudeshavebeen workedout by M. Gorn [2.35]for themixing matrix nonsuppressed~. —*~• The strict SU(3)classificationgives

a6. (2.37)

Howeverin generalthereshouldbe six amplitudes.

50 Ling-Lie (‘hau, Quark mixingin weakinteractions

Table 2.7SU(3) amplitudesfor charmedmesondecayst

a. D°decays

(2T+E—S)Ku-

0 (1I\/2)(3T—E+S) v *

~0~O (1/V6)(3T-E+S) X udVc~~-°x°(21V3)(E — S)

K~K 1~(2T+E— S)+(1/2)i(3T+2G+ F—E)—~(2T+E—S)+(1/2)zl (3T+ 2G+ F—E)

~-~° {\/2}tt. (1/2)~(3T—E+ S)+(1/4)~(—7T+2G+ F— E)1(°I(~~ (1/2)~i(—T+2G—2F+2E)

~s~1I —{\/~}tt. (1I2),~(3T—E— S)+ (I/4)~(—3T+2G—F+ E)x°x° {V2}tt . (1/2)~iG~.0~II — (l/V3) .~(3T—E+ S)+ (V3/2) ~i(—2T+F— E)

V2x(S—E)+(1/’F2)i(F--E)(V2/V3) ~(E — S) — (\/3/V2) (F —

K~ir (2T+E—S)K°ur° (1/\/2)(3T—E+S) V *

K°ij (1/V6)(3T-E+S) ~K°~ (2/V3) (E — S)

V,dV~)/2

tt The factorsin thecurly bracketsareBose-statisticsfactorsfor identicalparticlessuchthat thephasespaceis alwaysfl [d3p

1(2ir)3]irrespectiveofwhether

theparticlesareidenticalor not.

b. D decays

5TVUdV,~

~r’~r (5/V2)~T —(5/V2)~iT~ —(1/\/~)~(9T+2E+2S) +(I/V~)i(—3T+E+3F)

~(4T—E—S) +(I/2)~i(2T+E+F)~ —(2/V3)~(E+S) + ~(1/V3)(E+3F)

K°ir~ (2T+E+S) ~‘lK~u-1 (1IV2)(—3T+E+S) ~X ~K’u~°(l/V6)(3T—E—S) JK~X (2/V3)(E + S)

c. F~decays

K°K~ (2T+E+S)1~*~O —(\/2/’\./3)(3T—E—S) ~x V~V~

iT~X° (2/V3) (E + S) J

~(—3T+E+S) + ~i(2T+E+3F)K*lrO (1/\[2).~(2T+E+ S) + jlI(2V2)]~(—8T+E+ 3F)K~i~°—(1/V~)~(I2T + E + S) —[l/(2V6)] ~(12T +E + 3F)K~~°(2/V3) Z(E + S) + (1/V’~i)~i(E+3F)

K°K’ 5TV,SV~,I

t Seefootnoteat bottomof table2.4.

Ling-LieChau, Quark mixing in weak interactions 51

If the [6*] is dominantto the samedegreefor charmeddecayas the [8] is for strangeparticledecay,only the amplitude S will contribute.Certainly the dataof F(D°—*KK’-)/F(D°—* ir~r) ~ I doesnotsupportthe dominanceto such a degree.The information Br(D°-~Kii-’-) = (3.0±0.6)% andBr(D~—*

K°ir’-)= (2.3±0.7)% from table 2.2 can give an estimateof SIT, assumingthe dominanceof 5: fromtables2.1 and2.5

Br(D°~Kir~) ISI2r(D°)=~~° 238

Br(D~—*k°ir~)I5TI2T(D~) 2.3’

which gives

SI2 — 3 T(D4)2 ~ 239

ISTI2 - 2.3 r(D°) (. a)

ISI/2ITH3.5—~5.5. (2.39b)

We do not havethe overwhelmingdominanceof a single amplitude as in the strangeparticle decays[1.50,2.16, 2.30].

Fromtable2.7 wecan seethat contributionsfrom the two amplitudesS and T canadequatelyfit thepresentlyknowncharm decaydata, tables2.2, 2.3. Much more dataandtheoreticalunderstandingareneededbefore a clearpicture emergesfor the nonleptonicweakdecays.


Traditionally the nonleptonicdecaysof strangleparticles are analyzedin terms of the symmetrypropertiesof theweakinteractionHamiltonianH~andtheparticlesinvolved.Exceptfor theoutstandingdifficulty with thep-wavehyperondecaysthez~I= ~rule,the [8] dominanceof theH~worksquite well.In thecharmedparticledecays,however,therehasbeenno clearindicationwhich is thedominantSU(3),SU(4) transformationpropertiesof H~.The estheticallypreferredrepresentation[6~’]for the charmchanging~ definitely doesnot dominateto thesameextentas[~]doesfor ~ Anotherform of representatione.g. [~] for ~ may beas important.

During the study of charmeddecayin the pastfew years,an alternateapproachhas developed,i.e.analyzingcharmeddecaysaccordingto quark diagrams.Variousleading-orderQCD calculationsfor thequark diagramsappearedprior to the experimentalresults. Nature certainly had some surprisesinstock!Assumingthe experimentalresultsdo not changeas errorbar decreases,the leading-orderQCDpreferredW-emissiongraphsfail to explain manycharmeddecays.The W-exchangeor W-annihilationdiagrams,which were arguedto be negligible from the perturbativeQCD point of view, are needed.This hasbeenexplainedby consideringone gluon emissionin the W-exchangediagram,or morethantwo gluon emissionsin the W-annihilationdiagram.

Using this leading-orderQCD quark diagramapproachback to the K system,it hasbeenestimatedthat the so-called“Penguin” diagram is important. It has the interestingimplication (see discussionslater in section3) that the E~~I= ~CP violation part of KL—* 2ir can be large 1/600<Is’/~l< 1/80.

Besidesproviding a framework for perturbativeQCD calculation, the quark diagramscan alsobeusedin a morephenomenologicalway to fit the data.It is interestingto note that for the mixing-matrixnonsuppresseddecaysthe W-exchangediagramcontributesonly to the neutralparticles,D°,K°decays.

52 Ling-Lie (‘hau. Quark mixingin weak interactions

The dominanceby a factorof two or threeof the W-exchangediagram over the W-emissiondiagramcan easily fit the D5 decaysinto two pseudoscalars.The dominanceof the W-exchangediagramover theW-emissiondiagramsalsogives the interestingresult of r(D°)< r(D~), since the W-exchangediagramdoesnot contributeto the D~decay.It is currently supportedby experimentalresults, table 2.2. Theimportance of the W-exchange diagram offers an alternate reason for ~I = ~ rule to the “Penguin”diagram. The W-exchangediagram alone gives e’ = 0. Therefore the magnitude of IE’/EI can test therelative importance of the W-exchangeandthe “Penguin” diagramsin the K°system.

The correspondingdiagramfor chargedparticles(D’-, F’-) decaysto theW-emissiondiagramin the D15decayis theW-annihilationdiagram.ForD~decays,it is mixing matrixsuppressed.Its dominancewill givefewernumbersof strangeparticlesin the inclusivedecayof D~.Thisseemsto be supportedby data,table2.3. The W-annihilationdiagram for F~decayis mixing matrix nonsuppressed.If its dominanceoverW-emissionis aboutthe samemagnitudeastheW-exchangediagramin the D11 decay,onewould expectr(F’-) r(D°),andtherewill be fewer numbersof strangleparticlesin the decaythanexpectedfrom theW-emissiondiagram.

Besidesthedeterminationof therelativeimportanceof thequarkdiagramamplitudesor thesymmetrypropertiesof the weakdecayHamiltonian,heavierflavored-particledecayscan alsoprovideinformationaboutother mixing matrix elements,just like hyperondecaysdeterminedthe valueof ~

Here we have given a very qualitative discussion.Much more data andideas are neededto makefurtherprogressin our understandingof the difficult andfascinatingsubjectof nonleptonicdecays.

3. CP violation

3.1. The neutral particle—antiparticle mixing and its CP violation effects

Here we shall discuss the neutral particle—antiparticlemixing and CF violation effects_of theheavy-quarksystemsD°D°,~ B~B~andT°T°.We denotethem with a generalnotation P°P°.TheK~, KL-like statesarep~,p°respectively.The formalismof CP violation dueto K°’s-*K°transitionasintroducedin section 1 can be easily usedhere. The main differencesare heaviermassesand shorterlifetimes, andmanymorefinal statesavailablein the decay.

In the caseof theK°system,having theK°massso close to the three-pionmasses,naturehaspro-vided us with a wonderfulstateto studythe CP violation properties.Due to this phasespacelimitation,KL, which mainly decaysinto 3i~,hasa much longer lifetime thanK~andcan travel afew metersbeforethey decayaway.This is the caseof maximalmixing, i.e. regardlessof whatthe combinationsof K°,K°to begin with, in l0~°sec mainly KL will remain,which is an almost equalmixture of K1~andK°.Themassdifferencez~m= mL— ms, the CP violation parametersr, r’ can be studiedfrom theKL, K~decaysseparately,andfrom interferenceeffectsin the common i~- final statesthroughthe regenerationofK~in a KL beam, or from chargeasymmetryof ~ in the decayof KL—* ir~1p,[1.28, 1.29, 3.1]. It isremarkableand yet frustratingthat so far the only CF violation observedis still the originally observedsystemof K~,KL in 1964.

The study of CF violation effects in the heavyquark systemwill becomemuch harder.Due to thehighmasses,the lifetimes of bothp~aremuchshorter,e.g. the charmedparticle lifetime is observedofthe orderof magnitude~ l0t3sec,andthe lifetimes Tb, Tt, for the b and the t particlesareexpected

to beevensmaller,Tb 10 ~ sec,r5 < 1018 sec,as estimatedin section1.5.4.In themanydecaychannels

available,thereis no phasespacelimitation for eitherp~or p°as for KL. The lifetimes for bothp~,are

Ling-Lie (‘hau, Quark mixingin weakinteractions 53

expectedto becomparable.ThereforetheCP violation propertyof theheavyquarksystemmustbestudied

via time integratedresults[3.2].

3.1.1. Formalismfor the massandwidth difference~m,aT; parametersof mixingr, F; of CP violation ~asymmetrya -

First let us summarizethe formalism,which_isexactly analogousto that for the K°K°system,eqs.(1.20—1.29).Dueto thetransitionbetweenP°.s-*P°,theoriginal p°,p°statesareno longerphysicalstates.Thephysicalstatesareobtainedafterthe diagonalizationof the effectiveHamiltonian

H— 7 Mo—iF/2 M12—i112/2 -— ~M~—iT~/2 M0—iT/2

~jMtt—iT/2+(1/2)(am—iaT/2) 0 (31 )0 M0i1/2—(1/2)(am+iS172))’ a

where

(1/2)(am— ial/2) = [(Al’12— iT12/2) (A4’~2— if~2/2]1/2 (3.lb)

and the eigenstatesare

= [2(1 + Ie~l2)]t12[(1+ ~~)jP°)±(1 — ~~)IP°)], (3.lc)

1—~ M~—iT~/2 (1/2)(am—iaF/2) . 31d1+ ~p— (1/2)(am— ial/2) — M

12 - i112/2 (.

As we discussedin section 1.3 the phase-conventionindependentphysical measureof CF violation isthe deviationfrom unity of

1 — — AA’* ‘r* j’~ t/2= Ep — 1~’112 ii l21L.— a, ~r

I Ep lVlt2tj 12

It is relatedto otherquantitiesby

1—~2Reë~ 2Im(M~T12/2) 331 + ~ — I + ~ — 1121+ IM121

2 + [(am)2+(aT/2)2]/4 ( . )

Now the time evolution for the physicalstatesp~,is very simple

p~(t))= exp(im±t— F±t/2)jp~(0)). (3.4)

The time evolutionfor the statesjq5(t = 0)~= p°),andI4.(t = 0)) = I~°)arerespectively,

= a~(t)Ip°)+ ap(t)l~°), (3.5a)

I~(t))= ã0(t) p°)+ a~(t)Ip°), (3.5b)

54 Ling-Lie (‘han, Quark mixingin weakinteractions

where

= [exp(im±t— T+tI2)~exp(im±t— F÷t12)],

(—)a ~(t) = [(1 — ë0)/(1+ e,~)][—exp(lm±t— T±t/2)~exp(imt— Tt/2)].

Clearly the ratio of the time integratedprobability of k.(t = 0)) = Ip°)stategoesinto Ip°)and I~O)is

~ J Ia6(t)I~dt/JIa~(t)J2dt=~2j (3.6a)

and

F = J Ia~(t)f2dt/J a~(t)I2dt = ~-2~ (3.6b)

where

— (~m/F)2+ [(1/2)a171]2 (3 6c2+ (am/F)2- [(1/2)aF/F]2’ ‘

am m~- m, ar 1+ - F~,F (1+ + F)/2.Therearetwo physicalsituationswhen the mixing is maximal.First, IaFI2I F, i.e. either 1±~- F or

F ~‘ 1±,like thatof theK°system.In this situationeitherp°or ~° to beginwith will quickly endin thep~or p~staterespectivelywhich hasapproximatelyequalmixture of p°and p°,thereforemaximalmixing. Second,ôm ~‘ F so that before decayingthe systemoscillatesvery quickly betweenp°andp°and appearsas an equalmixture of p°and ~°. We shall see that this happensin theB~B~system.Inboth situationsof maximal mixing

r.Fi=I. (3.7)

For small CP violation ij ‘~ I

r=F. (3.8)

For both maximalmixing andsmall CF violation

(3.9)

This mixing of p°andp°hasthe marvelousapparenteffect that p°)can havedecayproductsthat belongdistinctively to ~0) —

ForexampleD°andD°can only havethefollowing semileptonicinclusivedecaysrespectively

Ling-LieChau, Quark mixing in weak interactions

D°—* tpeX, (3.lOa)

(3.lOb)

But as time evolvesD°,~o both becomea mixed stateof D°andD°,andwe havethe apparent

“D°+ ti7~X’-”, (3.lla)~ ~ (3.llb)

where “ “ underlinethe time integratedprobability. Actually the ratio of “D°-’+tieX~”to “D°-*~ vEX” is given by the samer, F as in eqs. (3.6a) and(3.6b), respectively,

r = (“D°—*ti~X~”)/(”D°—*6-’-v~~X”), (3.12a)

F = (“D°_*t’- v~X”)/(”D°*t ieX’-”). (3.12b)

This D°,D°mixing will give the samesign dilepton productionboth in i and v reactions,contrary tothe usual oppositesign dilepton productions;and j.tj.~, ~ (~s4,u~,~) productionin /~n(/s~n)scattering[3.3]. In e~eand nucleonnucleonscatteringtherewill be samesign dileptonproduction.

For the associatedproductionof p°p°,a nicequantity to measurethe mixing is thesum of samesigndilepton production

— (p°p°)+(~°p°) — r+F — (1+~j4)~i 313r+ — ~ (p°p°)+(p°~°)+(p°p°)— r+ F+ 1 + rF — (1 + ~)(1+ ~ ( . a)

where(p°p°)= (~~0—* p°”) (“p°—* p°”),and

(~O~O)= (~pO,~ pO~)~ _~ po~)

Note that for maximalmixing

r+=~. (3.13b)

The nice quantity to measureCF violation is thefractional differenceof samesigndilepton production

)(PP)(PP)_!~_i. —4Re~~(1+I~~l2)— 77~—i 314)

a(p — ~ (p°p°)~r+ F (1 + ~0I

2)2+4(Re~)2 — ~+ I’

Of coursethe measurabilityof a(p)dependson theexistenceof appreciablemixing, i.e. r, F not too small.

For the mixing and CF violation effectsfrom otherinitial coherentstatesof p°~°,seerefs. [1.48,3.5].

3.1.2. Calculationof theparticle—antiparticlemixingand its CP violation effectsin the ~ B~B~IandT°T°systems

Now we shall proceedto estimatethe mixing and the CP violation in the heavy quark system.We

56 Ling-Lie (‘hau, Quarkfluxing in weakinteractions

needto calculatethe width F, the dispersivepart of p°~-* ~° transition,i.e. M12, as well as the absorptivepart

T’t2, eqs.(1.2Db,c). We shall assumethatall thep°~ ~° transition,the dispersivepart M12 aswell as

theabsorptivepart F12 arefrom the boxgraphsas shownin fig. 1.4 andthe total widths aregiven by theW-emissiongraph as shownin figs. 1.1, 1.2.

The transitionmassmatrix elementis

M12(p°*-* ~°)= —[G~-Bpf~mpM~v/(127T2)]~ A,

1A,A1. (3. ISa)

Here A1 = V7~~V,q for theB-like system,or A, = V01 V~’1for theD, T-like system.m~is theneutralheavymesonmass,approximatelygivenby m~ m0+ mq, wherethe subscriptQ denotestheheavyquark.Thefactorf~wouldbe the leptonicdecayconstantif the vacuuminsertionsaturatesthe intermediatestates.It is knownin theK°systemthat the valuefor B is very modeldependent.Symbolically,our B~f~heredenotesa quantity which hasall uncertaintiescombined.We use

BDf~= (0.19GeV)2, BBf~= (0.33GcV)2, and BTf~-= (1.9GeV)2 (3. I Sb)

asestimates.Thesummationin eq.(3. l5a) isoveru,c, t channelsfor Qbeingd-like; oroverd, s, b channelsfor 0 beingu-like. TheamplitudeA,

1 isessentiallythesameexpressionof eq. (1.77) in section1, aslongastheexternalquarkmassescanbe neglected.This isgenerallyvalid for theK, D, B systems.But for thecaseof the T mesonof massnearM~,we haveto replaceit by the following exactformula [3.4]

A11= 2 J da {[_xixj — ~(x1+ x1)x0a— ~(I + ~x1x1)a(l- a) ~x1x~x0(l + 2a)] In dl

d(x,x1) d(1, 1) . . 11

+(l+4xixi)a[xlln d(l,x1) +ln d(x1,1)

d(x1, x1)asx.(1 — a)+ x1a — x0a(1—a), (3.16)

with x = m~/M~~x0 = ~ We take m5, = md = 0.3 GeV, m, = 0.5GeV, mh = 4.9GeV. For ourillustrative purpose,we assumem5 = 30 6eV.

The absorptiveamplitude 112 arisesfrom cutting the box-diagramsas shown in figs. 3.la, b. Theexactexpressionof FI2 is obtainedby replacingthe logarithmsin eq. (3.16)by 71’ andintegrateoverthedomain of negativeargumentwithin the absolutesymbols:

~ ~4 ~W

(a) (b)

Fig. 3.). Diagramsfor calculating the absorptiveparts of p0*~*p°transition; (a) gives the W-emission contribution; (b) gives the W’exchangc

contribution-

Ling-Lie (‘han, Quark mixing in weakinteractions 57

F — G~Bpf~mpM~vAiAj11 — 2(x, + ~) + (x, — x)2]1/212 — 81T(l — x

1)(1 — x,) ~ x0 x~ ]

{(1 + ~x1x1)[x0— ~(x1+ xi)— ~(x,— x3)2/x

0] + ~x1x1+ ~x0(x1x1)+~x1x1x0}. (3.17)

When mq, m0~ M~,we can safely neglectW-propagatoreffects. Fromeq. (3.17) we obtain

G~B0f~m~I 2(m~+ m~) (m ~ — m~)2 TU2

F12=_~(±I6+8)~ LJLI(m±m)2+(m±m)4j[(m~+ m~)(~±~)—(mQ—mq)2~ (m~—m~)2/(mo±mq)2]. (3.18)

The choice of negative sign gives the contributionof the W-emission graph, fig. 3.la. The choice ofpositivesign in theaboveexpressioncorrespondsto theW-exchangegraph,fig. 3.lb, which is suppressedby helicity flipping for small m

1, m~.The indices i, j run throughall kinematicallyallowedchannels.From thesecalculatedM12, F12, and substitutingthem into eq. (3.Ib), we obtainthe massandwidth

differencesam, ~F for the B~,B~,1°systems.We estimatethe total width of heavy meson in the approximationof free quark decayvia the

W-emissiondiagram,i.e. Q—*q1W—*q1aJ3,andQ-*q1W—*q~[i~

F = [(3G~m~/(192ir3)]~ I V

10~2 I V~

0J2I(m

1/m0,ma/mo,m0/m0)+ ~ ~I(m1/m0,rn~/m0,0)],

(3.19)

with

(1_a)2

I(a, b, c) = 12 J (dw/w)(1 + a2— w) (w — b2— c2) [A(l, a2, w) A(w,b2, c2)]U2, (3.20)

(b±c)2

and

A(x, y, z)= x2+ y2+ z2—2xy--2xz—2yz.

The summationis over all availablequark q,, a, /3 statesandall availablelepton4’~states.To ensuretheappropriatekinematicboundariesof the phasespacein the total width calculation,we use physicalmesonmassesfor the quarks: mb = 5.26eV for the Bd meson,or mb= 5.0GeV for the B. meson,m~=1.86GeV, m~=0.5GeV,m~=md=0.3GeV.We also assumem

5=30.3GeVfor the T~,and= 31.5GeV for theT~mesons.Now substitutingam, aF into eqs.(3.6a,3.6b)and(3.14), we obtainthemixing parametersr, F, andthe

CP-violationasymmetrya(,p). Theresultsareobtainedin ref. [1.48]andshownin figs. 3.2—3.7.Notethat incalculatingthe CP-violationparameteres,,~, theexactformulaeqs. (3.ld, 3.2) oughtto be used,otherwisespurioussharppeaking-upwill appearin theCP-violatingparametersi~anda(p) in theB°B°,T°T°systemas6 90°as happenedin someof the previouscalculations[3.8].

In fig. 3.2, 1(D), aF(D), am(D) aregiven for the K—M phase8 varying in the region allowed by theK°K° fit in section 1. The reason that T(D)>>IaF(D)I, IamilD)I is becauseF(D) is given by the

58 Ling-Lie (‘hau. Quark mixing in weak interactions

mixing-matrix non-suppresseddiagram,yet aF(D), am(D) are given by the box diagrams,which arenecessarilymixing-matrixsuppressed.SinceF(D) ~‘ IaF(D)I, lam(D)I in eq. (3.6c)therefore~i~ 1, so thereis very little mixing. In physical terms,this is the situationthat the lifetime is somuchshorterthan theoscillationperiod as well as thelifetime differencethat thereis no timefor mixing. In fig. 3.3 the mixingparametersr(D) and CF violation parametera(D) aregiven.The mixing r(D) < iO~is verysmall, so thateven thefractional CP violation asymmetrya(D) can be 102, it will be hardto be observed.

In fig. 3.4 1’, aT, am are given for B~and B~.Since F(B)’s are given by the mixing-matrixsuppresseddiagrams,am(B), ~F(B) canbe comparableto 1(B). We seethat usingBBJ~ (0.32GeV)2,

lam(Bd)I > F(B~)for 8 near 180°;andam(B~)>F(B~)for all regions of 6. In fig. 3.5, r(Bd), r(B~),areshown. In the region am > F thereis largeB°B°mixing. The Bd haslargemixing for 6 180°.Thoughthe mixing is very sensitiveto thevalue of B

8f~used,the possibilityof largeB°B11mixings shouldhave

very interesting implications for high energy e’-e, ep, and pp reactions. In the same graph, theCP-violation parametersa(Bd), a(B~)are also given, which areunfortunatelyrathersmallwhen r(Bd),r(B

5) are appreciable.In figs. 3.6, 3.7 the mixing and CP-violationparametersr anda(T) for T~.T~aregiven. They areall extremelysmalldueto the reasonthat F(T)~’jam(T)j,Iar(T)I, sincethe T(T)’s aregiven by mixing-matrix nonsuppresseddiagram,yet am(T) and aF(T) are given by the mixing-matrixsuppressedbox diagramwith GIM mechanism[3.9].

It is ratherdisappointingthat the CP-violation effectsdiscussedhereare so small. This meansthat,exceptserving to confirm the K:M model, the future non-observationof CP violation from p°~ p°transition in the D°D°,B°B°,T°T°systemcannot be used to determinethe poorly known 8. It seems

I 11111111 I I IIIIJI1 I III

QH2 r(D) 10—2 ~I I ~ I I I MI

:I:T a(D)

~ (deg) B (deg)

Fig. 3.2. CalculatedT(D), BF(D)l, fim(D)~as a function of 3, with Fig. 3.3. Calculateda(D), r(D) as function of iS; for definition ofBufi~~(0.19GeV)

2,seesubsection3.1.2 for discussionsof formula a(D), r(D) seeeqs. (3.12—15). Sinceit is a log-plot, only theabsoluteand theparametersused, valuesof a(D)I aregiven, the sign is indicatedby (+) or (—).

Ling-Lie Chau,Quark mixingin weakinteractions 59

I 11111111 11111111 111111

1010

~— ~T(Bd)

IBm(Bd)I ~8r(Bs)I> _____________

1012 Iar(Bd)J I -

I I~iii~iiiili~ I liii I ii

l0~ :

0.1 I 10 90 I 10 90

8 (deg)

Fig. 3.4. Sameasfig. 3.2 for BI, B~systems.BBf~B (0.320eV)2is used.

0 ,_,~ —i-v 4-4-4I~ $ ~1Il’1i~~ r? 11I1~

r(B~)= -~-.‘~ - ~Q-3

10-I

— —

- ~- r(B~) -i02 // I0~

I I HI

~-“ T.-~.. \I ~

a(B~) -I0’~ ,l0~~- r(T5) ~

N~ ~

I I I I III I I I I I III I I I I I 017 S I I I I liii I I I I I i”t’l+’ —r TTi I I0.1 I 10 900.1 I 0 90

8 (deg)B (deg)

Fig. 3.5. Sameas fig 3.3 for BI, B~systems. Fig. 3.6. Sameas fig. 3.3 for T~systems,BTf~ (1.9 GeV)2 is used.

that to furtherconstrain6 we needto rely on the other CF-violation effects,which shall bediscussedinthe nextsection,3.2.

In figs. 3.8, 3.9 thelifetimes for D, B, T aregiven. It is interestingto notethat in this calculationthe blifetime, thoughits decayismixing suppressed,isalwaysshorterthanthec lifetime,andab lifetime longerthan ifr’3 sec is unlikely in this modelcalculation.

6)) Ling-Lie (‘han. Quarkmixing in weak interactions

IO~ IO~ 101

1012 I I IlIll) I I II MIII I I I I III 10-1210 ~ (Ta) _______________________________

—~..--

\I0_6~=~ \ ~I0’~ Tb I0’~~

‘I’ I-

~~r(Tc) / rn~

Io~ — — — — I 30GeV/ ———100GeV

/ I -

0.1 I 10 90 10-14 I I I 11111 I I I I II III I I 10 14B (deg)

8(deg)Fig. 3.7. Sameas fig. 3.3 for T~systems.

Fig. 3.8. Lifetimes 1,, r5 for charmedandb-flavoredparticles.

I I~ I I IIIIIII I ITt

m1 ~20 GeV

1018 - -

u -

m5 ~30 GeV

10I9 I I I I I III) I I I I I I I II I I I I II

0.1 I 10 90B (deg)

Fig. 3.9. Lifetime r, for m,= 20, 30 GeV.

In section3.5 we shall comparemixing andCF-violation effectsfrom othersourcesandthe resultsaresummarizedin table3.1.

3.2. CF violation in partial decayrates

BesidescontributingCF violation effectsin the massmatrix, thecomplexity in the mixing matrix canalso give rise to CF violation effectsin the partial decayratesdue to interferencebetweenthe weakinteractionamplitudesandthe strong interactionamplitudes.It was discussedin generalityquite some

Ling-Lie (‘hau. Quark mixing in weakinteractions 61

time ago by the authorsof ref. [3.2] that, though CPT predictsequaltotal decayratefor particle andantiparticle,the partial decayratesof particleandantiparticleinto CF conjugatedfinal particlescan bedifferent if CF is not invariant.The complexity of V0 in the six-quark model of K—M provides anexplicit mechanismto such differencein partial decay rates [3.10—3.141.The quark-diagramschemediscussedin section 2.2 provides an easy way to sort out the decay channelswhere particle andantiparticledecayratescan be different [3.12].We shall seethat the partial decayratesfor particleandantiparticleare allowed to be different for many channelsin the K—M schemeof CF violation. Suchdifferencesare not allowedin the super-weaktheory for CF violation [3.15].

3.2.1. ExclusivesemileptonicdecaysFromCPT the inclusivesemileptonicdecaysof a particleandits antiparticlemustbe the same,i.e.

F(D°+ tt’eX) = F(15°4 ~eX’-) - (3.21)

The K—M modelpredicts that to the leading order there are no CF violation effectsin all exclusivesemileptonicdecays.This is due to the fact that for semileptonicdecays,inclusiveor exclusive, thereisonly the W-emissiondiagram in leading order, so no sourceof interference,which is neededfor CFviolation effectsfrom the amplitudesaswe shall demonstratenext.Thispredicctionis in agreementwiththe superweak interaction for CF violation. CF violation for other sources,like Higgs and right-handedintermediatebosons,can provide CP-violation effects in exclusive semileptonicdecays, seediscussionin section3.5.

3.2.2. StrangemesondecayThe two-pion decay amplitudesof K~,K°are given in tables 2.la,b. The correspondingdecay

amplitudesof K, K°aregiven by the sameequations,exceptreplacingV0 by V~.Typically, the decayamplitudesfor particlesandantiparticleareof the form

A(K°—~ii-~-) = V~5V~dAl+ V~~V~A2, (3.22a)

A(k°—s~ = V~~V~dAl+ V~V~dA2, (3.22b)

whereA1 = a+ £ + ~ + 2/, A2 =e + 2/. The partial decayratedifferencebetweenthe two is given by

F-F Br-Br IAI2-IAI2 -

S ~ Br+Br lAl2+IAI2— 4 Im(V~.V~dV~

5V~d)Im(A1A~)= — 4(s2s3s5cic2c3)Im(A1A~’) (3 23)

- Al2+ Al2 (IAI2 + Al2) S~2,

whereF is the particularpartial decaywidth, andBr the correspondingbranchingratio. F and Br arequantitiesfor thecorrespondingantiparticle.Sincethetotal decayratesbetweenparticleandantiparticleare the same, the F in eq. (3.23) can also be the branching ratio Br. In eq. (3.23) we divide thedenominatorby s~becauseboth A!2 and Al2 havea factor of s/. Ll~is proportionalto s

2s3sa;thesamecombinationcontributesto the CF violation parameterr in KL decay.In addition to mixing anglesandphases,~i.dependscrucially on thephasesandmagnitudeof A1 andA2. ~i. is zeroif A~and A2 have

62 Ling-Lie (‘han. Quark mixing in weak interactions

the samephase.Unfortunately we do not have reliable ways to calculate A1 and A2. The presentscheme,however, providesthe information about the channelswhereparticle—antiparticledecayratescan be different. Herewe see that the decayrate K°—*~-~ir, K°—~ir~ir can bedifferent.

Note that

F(K~—~ir’~~~-°)= F(K- —~i~~~-°), (3.24)

since -°(ir’ ~~0) hasonly I = 2 state,sothereisonly onephasefor thestronginteractionamplitudeS, i.e.Im(A~A~)= 1), our quark-diagramschemegives the sameresult,seetable 2.la.

Using table2.1 gives alsothe decayamplitudesfor the three-pionfinal states.Now, otheramplitudes(4, (e), (/)~besides(~~)and (s), will alsocontributeto the K~—~31Tdecays.ThereforeK~—*~can havedifferent decayrates, so can K~—~ir~ir°ir°.This differencein the decayratesof K~—~(31T)~has been calculated within the K—M schemeof CP violation to relate to the CP violation in theK1—~2ir,usingcurrent algebratechniques[3.16],

= [Br(K~—~~r~r”)— Br(K—* ~)]/[Br(K~—~ ir~i-~i~-’)+Br(K—* --k)]

= 0.741r’l - (3.25)

where r’ is the CP violation in the ~I = ~ transition as given in eq. (1.48). We see that iK...3,. is

unfortunatelyvery small.

3.2.3. Strangebaryon decays -

It is worthwhile to emphasizethat basedon the samequark diagramanalysis,A andA (similarly .~,

can havedifferent particle—antiparticledecayrates[3.121.Fromfigs. 3.10, 3.11

A(A —* irp) = V~5V~’~A~+ VCSVC’~JA~, (3.26a)

A(A-~~p)=V~V~~AHVC*SVCdA~ (3.26b)

whereA = a.’ + ~ + c’ + e’, the sum of all four graphsin fig. 3.10, andA = e’the ~‘Penguin”diagram.Again we obtain the differencein partial decayrates,

= — Br(A —~ii-p) — Br(A —~i~’~)= 4(s2s3s~c1c2c3)Im(A’iA~*) (~27)A Br(A -~ii~p)+ Br(A-* ir~~) (IA’I

2+ IA’I2)s72

exactlythe sameform as in eq. (3.23) for strangemesondecays.Thefactor 52535a— 10~—10~,from theanalysisin section1, dependingon the t-quarkmass.This is thesamefactorasappearedin r. Howeverweexpect Im(A~A~*)I[(IA~!2+ IA’I2)s~2]-= Im(al/

2a3/2)IIal/212, which is suppressedby L~sI= ~ dominance.

Thoughthisdifferenceissmall, it is sosimpleandit isworthwhiletoembarkon asystematicsearchfor suchdifferencesin partial decayrates.

3.2.4. CharmedmesondecayUsing table 2.5 for formulae of D°,D~,F~decaying into pseudoscalarmesons,and the same

correspondingformulae for D°,D-, F with V~,replacedby V~,we can discusswhich decaychannel

Ling-Lie Chau,Quarkmixing in weakinteractions 63

* A7Vud d

S u d _____~ V~df ~ u

U ~ U ..~—1~”-’ d

$ I U

d )~ d usI U I U

$ US u $ d~ C V~d

u ,~ d u ~ uV~d(~> ~j c : ~

d d

C.’ 2’

Fig. 3.10. Quark-diagramamplitudesa’, ~‘. c’. .e’ for .4 -+ ~rp, respectively.

~ ~US ~~$IIII~T ~

-_- _

~ ~ U C ~Vud _____________

d d d

Fig. 3.11. Quark-diagramamplitudesfor A —~

can have different partial decay rates for particle and antiparticle. Notice immediately that in themixing-matrixnonsuppressedanddoubly-suppressedchannelsthepartial decayratesare thesamefor theparticle and antiparticle. For the mixing-matrix singly suppressedchannels,typically the decay am-plitudesfor the particleand antiparticleareof the following form, e.g.,

A(D~+ K°K~) = V~5V~A1+ VUd V~~A2, (3.28a)

A(D” —~K°KT)= V. Vc~A1+ Vu~dV~dA2, (3.28b)

whereA1 = a.+ c; A2 = .d + £ . For different decays,A1, A2 representthe correspondingcombinationof amplitudes(a), (i), (a), (,d), (.e) asgiven in tables2.5, 2.6. Again thepartial ratedifferencesareexactlyintheform of eq. (3.23). Thereforewe wouldexpectthatthepercentagedifferencein partialdecayrateswillbeof orderof e.

Fromtable2.5 of charmeddecayswe can find out which partial decayratescan be differentbetweenparticle andantiparticle.All, but D~—~1T~17~

0,will havedifferentparticle and antiparticle decayrates.Togive explicit examples,herewe shall list the exclusiveandinclusivedecaysof D~,F~,D°,D°which can

64 Ling-Lie (‘hau, Quarkmixing in weak interactIons

havedifferent partial decaysamongparticleandantiparticle

D~—*K°K~,~ K~K~X~.1I,i ir~X~.11,K~X~.11,~j°X~11,etc.; (3.29a)

F~—~K°ir~,K~ir°,K~j°, ~ K~X~.I),KX~1, ir~X~±1,K~X~0 (3.29b)

g}~K-K+. ~ ~ n°~°~~ andtheir inclusive states. (3.29c)

Here s denotesstrangeness.X denotesthe inclusive states.The equality

T(D~—* ii-°ir’~’)= F(D —* 17° ir), (3.29d)

becauseof the samereasonas F(K~—~ir’4i’r°)=F(K—*i,~ir°),eq. (3.24).

Herewe seea rich variety of channelswhereonecansearchfor CF-violationeffects. Needlessto sayhigh experimentalsensitivity, in the rangeof e, is neededin such searches.

3.2.5. TheB mesondecayTheB~,Br,, B~—*doublecharm ctI~particlefinalstates:Fromthe quark diagramslike thosein fig. 2.4,

the mixing matrix and amplitude dependencesof BJ—*D°D,B—~D~D”,B~—~FDcan be easilyworked out andarelisted asfollows:

A(BU —* D°D) = V~hV~’d(a+~+ )— VUdV~d(d+e), (3.30a)

A(B~—~D~D~V~uV~~(a.+~+L )+ V5hV~~’d(L), (3.3Db)

A(B~—~F

4D)= VCdVC~(a.+~+L )+ VUbV~’d(e). (3.30c)

Thedecayamplitudesfor the antiparticlestatesarethe sameexceptreplacingV0 by V~.Thereforeall

thesedecayamplitudeshavethe generalform of, e.g.,

A(BJ ,~D°D”) = VuhVu~Al+ V~bV~~A2 (3.3la)

A(B~—~D°D~) = V~bV~dAj+ V~6V~dA2. (3.31b)

The differenceof partial decayratein CF conjugateddecaysareof the form

— Br — Br — 4 Im( V~V~V~V~d)Im(AtA~’)— 4s2s1s5c1c2c3Im(A1AflB-.ce — — — —— —Br+Br 1AI

2+1A12 (lAI2+lAI2)s~2

where (IA!2 + IAl2) is proportionalto s~.Notethen that exceptfor eqs. (3.30a), the differencein partialdecayrates are from the ioop diagrams, the so-called“Penguin” diagrams.The point hereis that eventhough the factorof the anglesin the numeratorof LI B-.cc is the sameas thoseof strangeandcharmeddecays,eq. (3.23), the denominatorof ~ i.e. the rate is rathersmall too. ThereforeJB_.ct can beappreciable.This can be easilyseenif we rewrite eq. (3.32a)in the form

Ling-Lie (‘han, Quark mixing in weakinteractions 65

LI — 2(s2/s3)s8c1c2c3Im(A1A~’)BcC - ~[IAI2+ Al2](s~y2(s

3)2 (3.32b)

where

~[lAI2+ IA!2](s1Y

2(s3Y

2= etA1!

2+ lc2[cic2 + (s2/s3)c3eia]A2l

2

— 2c1c2[cic2 + (s2/s3)c3c,5] . Re(A1Afl — 2c1c2(s2/s3)c3sa . Im(A1A~’)

we seethat as (52/53) ~‘ 1,

LIB...CC s8c1c2c3, (3.33)

which can be appreciable.The difficulty lies in estimatingthe hadronicinterferenceIm(A1A~).Some oftheseeffectshavebeencalculatedusingvariousmodels[3.10—14,3.17]. Fig. 3.12showsa calculationby

J. BernabéuandC. Jarlskog[3.13,3.18] for the decayBJ~~.*Do*D,which hasthe sameamplitudeformas given by eqs. (3.32,3.33). In their calculationthey let

5a, s2/s3beindependentparameters.However,

from CF violation analysisdiscussedin section1, s,~and s2/s3arecorrelated.The circleson the curvesin fig. 3.12 are the allowedvaluesof 5~vs s2/s3from the box-graphfit to CF violation in theK°K°system[1.48].We see that the fractionaldifferencesin partial decayratesaretypically a few to twenty percent.Other decaysin this categorythat can havedifferent particle—antiparticlepartial decayratesare, e.g.,

60— -

8 300

50 — -

40- -

8o600

130- -

8=900 I20 —

8=120°0 -

III I I I Ii

I 2 3 4 5 6 7 8 9

Fig. 3.12. Percentagepartialdecay-ratedifferencei B-.D~D betweenB~-.Do*D_ and B~ D000 asafunction of s

2/s3from [3.13].Thecircles arethepointsrelatingS to 52/23given by the box-graphfit to the ~m.e in theK°K°system.]l.48].

66 Ling-Lie (‘hau, Quark mixing in weak interactions

D°D~,D~D°X~0, ~ (3.34a)

D~DX~0,~ DX~1, (3.34b)

~ F~D~X~0,FthX~

1, D~X~1,~±1. (3.34c)

TheB~,By,, B~—.ordinary (no charm)particle final states:We first list the decayamplitudesof theB~, B~,B~ to two ordinarypseudomeson(no charmparticlein the final states),

A(B1717°)~=VUbV~~(a+~+ £ +d), (3.35a)

A(B~—~17~1r’)=V~bV~(a.+~+L +/)+ V~bV~(L+/), (3.35b)

A(B~—*17K~) = VUbVi~’d(a)+ V~hV~(L). (3.35c)

We seethat the interferencecan only comefrom the loop diagrams(e) and(/). The partial decayratesare thesamefor B’~-417_170 just as eqs. (3.24, 3.29d)for K~,D~’—*~ but different for

17~17 17~17X~o,17~x1117X~O, (3.36a)

17K~, 17K~X~o,K~X=0,17X~1. (3.36b)

The differenceof partial decayrate in the CF conjugateddecaysareexactlyof the form given by eqs.(3.32a,b)

TheB~,B~,B~mesons—~single-charmparticlefinal states:Theseare thedominantmodes.Howeverjust like in the charm decay, thereis no partial decay rate difference in the mixing-matrix nonsup-pressedparticleand antiparticledecays.

UsingthesamemethodwecaneasilysortoutchannelsofB~decays,andtflavoredparticledecayswherethepartial decayratesforparticle andantiparticlearedifferent.Hopefully thishasopeneda newfruitful areaforthe searchof CF-violation effectsandtheir origins.

3.3. The estimate of E’

It was proposedin ref. [2.5], that the K decayis dominatedby the “Penguin” diagram.In thatscenarioof explainingL~I= ~dominancewith the K—M phaseconvention,a0 is complexandhasa rathersmall imaginary part,but a2 is real. This phasedifferencebetweena0 and a2 gives r’ � 0. This was thepoint madeby Gilman and Wise [3.19].Puttingin the experimentalvalues 1a2/aol 1/20, 82— 6~°°’17/4

13.201, eq. (1.48) becomes

= —i(V2/2)(1/20)t0exp(i17/4)= — —b (Tm a0/Rea~)exp(iir/4). (3.37)

Ling-Lie (‘han, Quarkmixing in weakinteractions 67

The questionnow is what the value of Im a0/Rea0 is. Using the formulation given in section2, theamplitude a0 hasthe form

a0 = VUS Vi~dAl+ V~5V~’dA2, (3.38)

and

Im a0 — s1s2s3s8c2A2 (3 39)

Rea0 — s1c1c3A1— s1c2(c1c2c3— s2s3c~)A2’ -

whereA2 can havecontributionsfrom the “Penguin” diagrams(.e) and (/) only, but A1 can havecontributionsfrom the W exchange,the W emissiondiagrams,(c) and (a.), (eq) respectively,see table2.1.

Assuming dominance from “Penguin” diagrams alone, elaboratecalculations have been madeincluding vaiousconsiderationsof perturbativeQCD effects[3.19,3.21].The valueof Im a0/Rea0, thuse’/r, can vary a greatdeal

1.2X iO’~< tm a0/Reaol <8 x i0~, (3.40)

or -

1/600<lr’/sl < 1/80, (3.41)

dependingon the approximationscheme,and the QCD scale 0.1GeV<A <0.7GeV. The smallervalues of A tend to give smallervalues of e’Ie. Combiningthis calculationof E’ with that of s in theK—M model, as given in section1, it was notedby Hagelin[3.22]that r’ ande areparallelas s8> 0, andantiparallelas sa <0, thereforethe third quadrandsolutionsof 6 given in section 1 can bedistinguishedfrom thoseof 1800>6>00 by the measurementof the sign of Ie’/El.

Actually we ought to be cautiousabout consideringthe “Penguin” diagram alone. The exchangediagrammay be as important. If it dominates,as indicatedfrom the charm nonleptonicdecay(seesection2.2.2), A1 in eq. (3.39) can be even larger, thus the value of Im a0/Rea0 can be smaller[3.23].Currently the experimentalboundson s’/rI is, [3.24]

E’/El ~ 1/50. (3.42)

Experiments[3.25—3.26]with sensitivity of measuringIr’/~!down to 0.1% level are beingcarriedout.Theobservationof s’/eJ at that level certainlycan beviewedas atriumph of theK—M schemeoverthatof super-weaktheory. A small value of E’/EI ~ i0~can still be accomodatedby the K—M scheme,howeverit can put evenmoreseverelimits on the complexHiggs couplingasthe origin of CF violation(seediscussionlater in section3.5).

3.4. The estimateofthe neutron electricdipole momentd0

Therearethreeform factorsfor the neutron[1.28]

(nIJ~m(0)In)= ü(p’)[F1(q2) y~.+ iF

2(q2)o-~,q°’+ F

3(q2) y-~q~’]u(p), (3.43)

66 Ling-Lie(‘han. Quark mixing in weak interactions

whereF~(O)= 0 the chargeform factor, F2(0) = ~ the magneticmoment and F3(0) = d~the electricdipole moment. Again the complexity in V0 can give d~of the neutron. The relevanceof the CFviolation effect of the K—M model to the neutronelectric dipole moment was first discussedin refs.[3.27].It was notedby Shabelin[3.28]that the electricdipole momentfrom a single quark is zero,seefig. 3.13awherea photonis meantto be attachedin all possibleways.Graphsincluding gluoncorrection[3.29], or graphsof inter-quarkinteractionfig. 3.13b haveto be considered[3.30].The estimatedvalueof d0 varies a greatdeal;

10 ecm < d~< i0~°ecm, (3.44)

which is way below the experimentallimit [3.31],

d~<(2.3±2.3)xlO25ecm. (3.45)

The smallnessof c/~is onesalient featureof the K—M model dueto the GIM cancellationmechanism.Recentanalysisincluding the “Penguin”diagram,fig. 3.13c havegiven values of dn close to the largervaluesof eq. (3.44), [3.32].

CF violations of othersourceslike complexHiggs coupling,left—right symmetricgaugetheories,tendto give valuesof d~closeto the experimentalboundof eq. (3.45), (for morediscussionson comparisonswith other CP violation mechanisms and references,see the next section, 3.5), therefore animprovementon the measurementof d~will be very helpful in clarifying the origin of CF violation.

3.5. CF violation from othersources

So far we havetakenthe approachassumingthat the K—M mixing with the phase6� 0 is the solesourcefor the CPviolation. It is quite successful:the parametersadequatelyfitted the~m, E of theK°K°system,and the fitted parametersgive consistent descriptionsof other experimentalmeasurements,including e’ and dI,. Thereare other possiblesourcesof CF violation which are of currentinterest. Ishall give a brief discussion.

CF violationfrom super-weakinteraction: So far all the experimentalinformationis alsoin agreementwith CF violation from super-weakinteraction.The super-weakinteraction[1.28,3.15] was designedtoexplain the CF violation in KL —~ 217 througha super-weakinteractionwith strength~

GM2 GM22 GM22 GM2~ N) 103( -~-~-~±)l(Y°• ~ (3.46)

which contributesonly to the i~S= 2, K°~ K°transitions.It hasno effect in ~S = 1 reactions.The

w+ +

d d d ) U Hw U

d d d—~ d d—— d

(a) (b) Ic)

Fig. 3.13. Diagramsof neutronelectricdipole moment. (a) from theinteraction-diagramof a singlequark: (h) from interquarkinteraction; and(c)from Penguindiagram.


observationof s’ andmanyotherinterestingCP violation effectslike partial ratedifferencesin K~—*317

andhyperonnonleptonicdecayswill eliminatethe possibility.CFviolationfrom complexHiggs: In theelectroweakunified theories,therearebasicallytwo originsof

CF violation from the Riggsfields. Oneis from the complexity in the couplingconstants,e.g. Yukawacouplingy~e/,andAq54with complexy, A. This hasbeencalled hard CF violation in theliterature.Theother is from the complexity in thevacuumexpectationvaluesof the Higgs fields, i.e. CF symmetryisbrokenspontaneously.This is alsocalled soft CF violation.

The K—M phasecan be generatedeither way. (For more discussionsof generatingthe K—M phasespontaneouslythrough the quark massmatrix, see section4.) It is known that in order to generateanontrivial phasein the K—M matrix, at leasttwo doubletsof complexHiggs fieldswith complexvacuumexpectationvaluesareneeded[3.33,3.34]. Thus after servingthe purposeof generatingW~,Z°masses,therearestill two chargezeroHiggsH?,

2andtwo chargedH~left over.Theircouplingto thefermionsarecomplexandthereforeCF violating. SinceH?,2can givetree-levelK°~ I(°mixing, the massesof 1-1°areboundedto be mH°� 100 GeV, which in turn gives very small K°—* ~ 217, andhasvery little physicaleffects.For thechargedHiggsbosonsH~,currentexperimentallimits [3.35]setmH±>10 GeV. In thiscasethecontributionfrom H~to E, E’, L~mof the K°’sarenegligible.So in thisscenario,physicaleffectsof theHiggs arenegligible.

Anotherscenarioof spontaneouslybreakingCF invarianceis thattheK—M matrixhasthephase6 = 0,

andCF violation comessolely from thecomplexitiesof theHiggsfields. SuchamodelwasconstructedbyWeinberg[3.34],with thecharacteristicsthatthereis no flavorchangingneutralcoupling[3.36],i.e. no treelevel contribution to the K°~.-* K°transition.To be specific, we shall only discussthe Weinbergmodel.(Othermodelswith CF violation from complexHiggs fields havenot beenvery thoroughly investigatedbecauseof themanyarbitrarinessinvolvedin themodels.)In this modelthe CFviolation e from K°~—* K°

transitionis from theboxdiagramsof figs. 3.14a,b, andIm a0/Rea11 is from theHiggs“Penguin”diagramoffig. 3.14c. In a recentanalysis[3.37],it is found that -

jIm a0/Reaol ~=IW’12/2~t12l,or 1t01 ~“ ltM/2I , (3.47)

which exceedsthe currentexperimentallimit eqs.(1.63—1.67).This is a very interestingresult.Howeverwe must be cautiousin concludingnow that CF violations purely from complexHiggs are ruled out.Thereareuncertaintieswhichtypicallyplaguesuchcalculations[3.38,1.33].A bettermeasurementof le’/EIwill definitely be helpful in decidingthe fate of this mechanism.

H

s, c,u ~ d s~ c,u ~d $ , (c,u~’I,d

~ H1 ~ ~ ~

(a) (b) Ic)Fig. 3.14. (a), (b) ChargedHiggs contributionto the K~i-=K°transition. (c) “Penguin’ diagram from the chargedHiggs interaction.

The mixing and CF violation for the D°D°,B°B°,T°T~systemsin this model has also beeninvestigatedin the literature [3.38—3.40].Becauseof the many parametersinvolved in the theory it ishardto reachdefinite conclusions.Resultsfrom the latestinvestigation[3.40,1.42] indicatethatboth themixing andCF violation from p°~ ~° transitionsarevery smallexceptin theT°T°systems,which is aninterestingcontrastto the resultsfrom the K—M model,see table3.1.

7(1 Ling-Lie (‘hau. Quark mixingin weak interactions

Anothercharacteristicof CF violation from complexHiggs couplingis that it gives d~bigger thanthe K—M mechanism[3.41]

l0~27ecm < d~< 1025e cm, (3.48)

whichis very closeto the experimentalbounds,eq. (3.45).Sincethe chargedHiggs bosonsmediatelike theW~,we expectthat theywill alsocontributeto the

differenceof partial decay rates for particle and antiparticle. However therehavebeen no detailedcalculations.The CF violation from complexHiggs coupling also tend to give bigger CF violation inraredecays[3.42].Fromthe interferenceof the W3°andH°~exchange,thereareCF violation effectsinexclusivesemileptonicdecays[3.43].For example the model can give the time reversal invarianceviolation in the ~ polarization from K

03. The recent experimentalbound on such time reversalinvarianceviolation [3.44]actuallyprovideslimitations on the parametersin this model.

TheK—M modelpredictszero CF violation in all semileptonicdecays,as we mentionedin subsection3.2.1.

CP violation from strong interactions:Thenthereis the CF violation from strong interactionduetothe instantonsolutions[3.451.The boundon d~gives very severelimits on the effective CF violation

Table 3.1Summaryof (‘P violation effectsfrom variousmodels

Models C c/c r a(p) .i1. d(ecm)

nonzero nonzeroonly in the only in the

superweak fitted (I K°K°system K°K°system zero

theK—M fitted in the appreciable zero for allmodel fitted l/600—l/80 K°K°system; (°H0

2)only semileptonic

mixing can in theD°D’1 decays;nonzero

he large systemfor for manyin theB9B1 8 ~0° hadronicsystemsand channelsin the B1B1systemsfor8> 177°;small fortheD11D11 andT~~T°systems

complexHiggs fitted appreciable appreciable nonzerofor(a Ia Weinberg) >1/51) only for the only for the manychannelsno flavor- closeto T0T~system: T51’11 system; including l(I_25,

changing exceed small for small for the exclusive closetoneutral experimental theD11D°, D11D°.B°B11 semileptonic experimentalcoupling bounds B11B°systems systems decays bounds

strong (‘Pviolation 1) ) (1 () I) accommodated

= parameterfor neutralparticle—antiparticlemixing. eq. (3.12).a(p)= asymmetrybetweenp°p°and p°p°due to p°p°mixing. eq. (3.14).

= partial decay rate difference between particle and its antiparticle decaying into their corresponding(‘Pconjugatedfinal states.,see eqs.(3.23. 25. 27. 32).


phasefrom the instantoneffect [3.46].It is still a hotly pursuedtopic— why the phaseis so small [3.47].The importantfeaturefrom the CF violation pointsof view is thatthe severelimitation from theboundon d~renderstotally negligiblestrong CF-violation effectsin the K11 system.Thereadersarereferredto[3,46] for detailedcalculations.

Left—right symmetric models: An alternative to the left-handed model is the left—right symmetricmodels[3.48]which providesmore of a differencein point of view than differencein interpretingthelow energyphenomenology.In its left-handedsector,the K—M schemeworksessentiallythe sameway.Due to the additional right-handedcoupling, thereis more flexibility in interpretingthe low energyphenomenology.In oneversionof the left—right symmetricmodel,when the massmatrix is Hermitian,the strong CP violation is zero for lowestorderdiagrams[3.49].Due to the existenceboth of left andright couplings,thereare CP-violation effectsin exclusivesemileptonicdecays[3.50].


We summarizethe discussionof comparisonin table 3.1. We see that the presentstatusis that theK—M mechanismcan adequatelyfit all the known phenomenaof mixing and CF violation in theK°K° system, ~m, e, and gives d~within experimentalbounds. Further, with all uncertaintiesestimatedso far the model cannotproducevaluesbeyond IE’/eI ~ 1/80, d~~ 10~~0ecm. It is thereforecrucial to havemeasurementson r’ andd~to seewhetherother CF-violationmechanismsare needed.Theprediction of extremelysmall CP-violation effectsfrom p°.s-*~°transition for all the D°D°, B°B°,

T~Psystemsis disappointing,howeverit can serveas a test of the model.The effects of the possiblelargeB°B°mixing arevery interestingto look for. Observationsof themany CF-violation effects(e.g.thepartial ratedifferenceLIA for A, A; i~for B, B, which canbesignificantif 6 90°)in thepartial decayratedifferencebetweenparticle andantiparticlepredictedby the K—M mechanismwill be very interesting[3.51].

4. Quarkmixing matrix andquark masses

4.1. Introductionandformulation

Ever sincethe observationthat the quark statesin weak interactioncurrentsaremixed statesof thephysical quark states,attemptshavebeen madeto understandthe dynamicalorigins of the mixingangles[4.1].Eversincethe observationthat the CabibboangleO~is very closeto the massratios,

& m~/mK mJm~, (4.1)

manydiscussionshaveevolvedaroundexpressingthe mixing matrix in termsof thequark masses[4.2].Recentlytherehasbeenan increaseof activitiesin such studies[4.3].

First, we demonstratehow the mixing matrix is related to the procedureof quark mass matrixdiagonalization.The massterm in the weakLagrangiancan be written as

Im = Pt,LMJ(2/3)PJ.g+ Al. M~j(1I3)flj,p+ H.C. (4.2a)

wherethe subindicesi, j denotequark flavors, p, aremixed quarkstatesof chargeQ= ~,n aremixedquark statesof chargeQ= —~or in matrix notationin the quarkflavor space,

72 Ling-Lie (‘han, Quark mixingin weak interactions

= pLM(2/3)pR+ 1iLM(— l/3)nR+ H.C., (4.2h)

where

PR=(l+ya)P, PL—P(l+Y5).

The massmatricesM(Q) arearbitrarycomplexmatrix. To find thephysicalquarkstateswe diagonalizethemassmatrix by a bi-unitary transformation:

UL(Q) M(Q) U~(Q)= I%~I(Q), with UtU = UUt = I, (4.3a)

whereM is diagonal.Tofind UR andUL. we observethatmultiplying eq. (4.3a)by its Hermitianconjugateon theleft andright, oneobtains

URMMUR°°°M. ULMMtUL=rs’~I2. (4.3b)

SinceM~Mand M11~,ItareHermitian, there are well known proceduresto find their eigenvectorsandtheir diagonalizingtransformationsUR andUL respectively.We alsosee that UR and UL are in generaldifferentunlessMtM = MIMt. Thephysical quark statesare now

uL = UL(2/3)pL, UR = UR(2/3)pR, (4.4a)

d1 = U1(— l/3)nL, dR = UR(— l/3)nR, (4.4b)

the chargedcurrent coupledto W bosonsis

= PLY

= UL UL(213)y°U~(—l/3)dL

= ULY UL(2/3)U~(—l/3)dL. (4.5)

where

UL(2/3)U~,(—1/3)=V. (4.6)

is preciselythe quark mixing matrix of eqs. (1.8), (1.10)definedpreviouslyin section 1.

4.2. Parametercounting, calculability of quark mixing matrix in terms of quark masses,and someexamples

The U(Q) matricesare determined,up to a diagonalunitary matrix, in terms of M(Q). The hope isthat if, from somegeneralprinciple in somemodels,M(Q) aregiven in termsof the quark masses,thenthe quark mixing matrix elementscan be expressedin termsof quark masses.

In the following we give the counting procedurethat provides the information of how manyparametersthe massmatrix at most can haveso that the mixing matrix can be relatedto quark masses.

Ling-Lie (‘hau. Quark mixing in weakinteractions 73

Theresultsare: for example,in the caseof 2 x 2 massmatrices,eachof themassmatricescan havetwomagnitude parametersand arbitrary number of phaseparameters.All of thesephase parameterseventually can be removed from the mixing matrix by defining the phasesof the quark fields, asdiscussedin eqs. (1.11) and leaving the mixing matrix elementsas functions of the two magnitudeparametersthat, in turn, are expressedin terms of quark masses.In the caseof 3 x 3 massmatrices,totally therearesix invariant constraintsin termsof themassesof the threecharge2/3 andthreecharge—1/3 quarks.Therefore,in order to expressthe mixing matrix in terms of quark masseseach of thecharge — 1/3 and 2/3 mass matrices togethercan have five magnitudeparametersand an arbitrarynumberof phaseparameters.Among thesephaseparameters,at most, one will appearin the mixingmatrix, the restcan be absorbedin the redefinition of the quark phases.

For a n X n Hermitian matrix MMt, thereare n real invariantswhich can be expressedin termsofthe eigenvaluesthat in this casearethe quark masses.For a 2 x 2 matrix MM~,the two invariantsarein ~m~ and m ~+ m~,i.e. the determinantandthe trace;for a3 x 3 matrix, the invariantsare in ~tn ~m ~,

~ m~+m~+m~in generalthen invariantsare

n n n—i n(n—1) n—2 n

f’Jm~,~(flm~); ~ (flm~); ...;

for a n X n Hermitian matrix there are n2 parameters,of which n(n + 1)/2 are real magnitudeparametersand n(n — 1)/2 phaseparameters.Among the n(n — 1)/2 phaseparameter(n — 1) phaseparameterscan be removedby the similarity transformationof diagonalunitary matrix, thereforeonlythe n(n — 1)12 — (n — 1) = (n — 1)(n — 2)/2 phasefactorsare left astrue phaseparameters.Thus the totalnumberof arbitrary parametersin M1~,Itis n(n + 1)/2 + (n — 1)(n — 2)/2= n2 — (n — 1) which is just theoriginal n2 parameterminusthe (n — 1) removablephaseparameters.

From the generalmethodof diagonalizinga Hermitian matrix [4.4], one can easily show that thecomplexity in the matrix elementsU(Q) mustcome from the phaseparametersin Mt. In generalthe2 x (n — l)(n — 2)/2 phaseparametersresult in V = UL(2/3) UL(— 1/3), of which only (n — 1)(n — 2)/2 aretrue quark mixing phases,as we had also countedin section 1. So the most economicalway ofparametrizing the mass matrix is to let M(2/3) Mt(2/3) and M(—1/3) Mt(—1/3) have a total of(n — 1)(n — 2)/2phaseparameters,e.g.for n = 2, no phaseparameteris needed;for n = 3 only onephaseparameteris needed,either in M(2/3) Mt(2/3) or in M(— 1/3) Mt(— 1/3). Any additionalphaseparametersmore thanthe minimal in fl (0) can eventuallybe removedfrom the mixing matrix V.

Now we seethat in orderto give mixing anglesin termsof quark masses,restrictionsmustbeput onthe massmatrix so that eachof M(Q)Mt(Q) haveonly n parameters.Of thesetotal 2n parameters,masses,(n — 1)(n — 2)/2 phaseparametersaresufficient to generateall independentphasesin the mixingmatrix. [Of course for (n — 1)(n — 2)12> 2n, i.e. n � 7 phasesin the mixing matrix are not allindependent.]For example,in the caseof n = 2, in order for the mixing matrix to be expressableintermsof masses,eachof M(Q) can havetwo magnitudeparameters,andarbitrary numbers(includingzero)of phaseparameters,all which can eventuallybe eliminatedby requiring that the 2 x 2 mixingmatrix V be real, as shown in eqs. (1.11); for n = 3, all the mixing matrix elementsof three mixinganglesandonephasecan be given in termsof masses,if therearefive magnitudeparametersandoneormorephaseparametersjointly in M(2/3) andM(—1/3). In the end,all the phaseparametersexceptforone in the mixing matrix can be removedby the quark field phaseredefinition. We see that by thissimple countingwe can easily tell, given the parametrizationof massmatrices,which model can givequark mixing anglesin termsof quark masses.


So far thereis no good guiding principle for such restrictions.Various possibilities of imposinghorizontal(flavor) discretesymmetriesor horizontalglobal gaugeinvarianceamongthequarkswith thesamechargehavebeentried. For example,in the caseof two generationsit was observedthat a massmatrix of the form [4.5]

M(Q)= (~~), (4.7)

gives

tan2& m0ImS. (4.8)

Note that usingourcountingrule tan2& can be expressedin termsof massesregardlessof whethera, b

are real or complex,sincethe phasescan be removed.In the caseof n = 3, for example,the following form of massmatricesfor both 0 = —~and~,

/0 a 0M(Q) = ( a * 0 b , a, b, c complex; (4.9)

\o b*(0 a 0= a 0 b , a, b, c complex; (4.10)

Obc(0 a em ba/c= a em b 0 , a, b, c, a real; (4.11)

ba/c 0 c e’~( 0 0 c= 0 0 b , a real; b, c complex; (4.12)

cba

havebeenusedbasedupon discretehorizontalsymmetry [4.5],andfor example

0 cOM(Q) = —c —a b , a real; b, c complex (4.13)

0 —b a

basedupon horizontalSU(2)gaugesymmetry [4.6].Sameform of the matricesis usedfor both 0 = —~

and0 = ~ quarks but the parametersare unrelated.In all theseexamplesthereare threemagnitudeparametersandmorethan onephasefactor for eachof M(— 1/3) andM(2/3), so therearemorethanthesix parametersthatcan be relatedto quark masses.Thus only threeof the four K—M parameterscan begiven in terms of masses.Also, all the mass matricesare either symmetricor Hermitian. This is a


necessaryresultof left right symmetricmodels.If we let the numberof parametersbeoneless thanthenumberof invariantconstraints,the unknownt quark masscan be relatedto knownquark masses.Forexample,the massmatrix [4.6]

/0 A 0\M(2/3)=( A 0 B , (4.14a)

\OBC

/0 AA 0 \M(—1/3) = AA 0 rB exp(i6/2) (4.14b)

0 eB exp(i8/2) AC

hasall togetherfive magnitudeparameters,A, B, C, e, A. The phaseparameter6 is a removabletype,i.e. M(—1/3)Mt(_1/3) hasa phaseparameterthat can be rotatedaway by a similar transformationof adiagonalunitary matrix D = diag(exp(iô),1, exp(i3/2)), such that D M(— 1/3)Mt(_ 1/3) D

t hasonly realmatrix elements.Thenthet quark massm

1 can be treatedasa freeparameterto be determinedwith theotherfive parametersby the six constraintequationsof the six invariants.

In grandunified modelsthe generationof leptonmassesarerelatedto thosequarks.For example,inref. [4.7],the massmatrix is of the form

/ . . 1

/ a0 exp(i40) b0 exp(—iqSo) b0 exp(iqSo)

M(Q) = a0 exp(—i40) b0 exp(i40) b0 exp(it~o) (4.14c)

Ka0 exp(—iq’0) 0 —--~—is~b0

for Q= —1, 2/3 and—1/3; where1~= ~1~1/3 = ~~2/3 = 0 the numberof free parametersareeight: a213,b213, a113, b113, a1, b1, ~, K, thus theyareoverly constrainedby the nine-massinvariantsand againthe t quark massm~can be madeinto a free parameterto be determined.

4.3. Relatingquark mixingmatrix to quark massesnecessarilyleadstoflavor changingneutral couplings

In the unified gaugetheories,the massesof the quarksare solely generatedthrough their Yukawacouplingwith the Higgs bosons(elementaryor dynamicallygenerated),andthe spontaneoussymmetrybreakingof the Riggsfields, i.e. the Riggsfieldsgaining nonzerovacuumexpectationvalue[4.8].It hasbeennoted that the calculability of mixing matrix in termsof quark massesthrough the imposition ofsymmetryin the quark flavor spaceis incompatiblewith the requirementthatthereis no flavor changingneutralcouplings,i.e. the mixing matrix becomestrivial with mixing anglesbeing0, 17, or 17/2.Thus theimposition of relatingthe mixing matrix to quark massesnecessarilyleadsto the existenceof flavorchangingneutral coupling. This conclusionwas originally reachedby [4.9]for a single Riggscoupling,and has beengeneralizedto multi-Riggs boson coupling [4.10].Herewe should sketch the essentialingredientsof the proof.

The Lagrangianof the Yukawa coupling term of the quark fields in the standardSU(2)X U(1)

76 Ling-Lie (‘han, Quark mixing in weak interactions

electroweakunified theory is

= ~t’L1( 1/3)flR~n+ *Lf’(2/3)pRX0, (4.ISa)

where

= (~). x~= (~)are the Higgs doublets,and

= (p. n)1i. The F’m(Q)’s arematricesoperatingin the flavor spaceof quarkswith charge0. Eq. (4.l5a)

can alsohe written as

pLF(—l/3)nR0~ + nLF (—1/3)nR0~+ nLF (2/3)PRX~+ pLF(2/3)pR,~ (4.15b)

The massterms ~[m of the fermions in the Lagrangianare from the spontaneousvacuumsymmetrybreaking, i.e. the nonzerovacuumexpectationvalues of the Higgs fields,

An(0~), p.~°°°)3~~l, (4.16)

= A0fiLF

0(—1/3)nR+ /InPLF(2/3)PR

ULM( 1/3)nR + PLM(2/3)pR, (4.17)

wherethe massmatricesare

M(— 1/3) A0F (—1/3), M(2/3) ~ (2/3). (4.18)

After diagonalizationof the massmatrix we obtain the physicalquarkstatesas given in eqs. (4.4a)and(4.4b).Therequirementthat thereis no flavor changingneutralcoupling amongphysicalquarksmeansthat the transformationsthat diagonalizethe massmatricesmust also diagonalizethe neutralHiggscouplings. i.e.

UL(Q)Fn(Q)U~(Q)= t~~(Q), (4.19)

wheretn(Q)’s arediagonal.As we discussedpreviously, in order to expressthe mixing matrix elementsin termsof masses,there

must be certainconstraintsput on the massmatrix. In ordernot to be spoiledby high ordereffect ofradiative corrections,it is best to imposesuch constraintsthrough symmetry transformationsof theLagrangian.For example,considersymmetry transformations{K7~R},

nL—* K~(—1/3)nL. nR —* K7~(—1/3)nR, 4~~ (4.20a)

PL~K~(2/3)pL, PR —* K7~(2/3)pR, xn -~~ (4.20b)

Note that thesearehorizontal transformations,quarksin the sameisospin multiplet shouldtransformthe sameway, thus

Ling-Lie (‘hau, Quarkmixing in weakinteractions 77

K~(—1/3) = K~(2/3)= K~. (4.20c)

The physicalquark statestransformunder {K7.R}, like

dL—*S7,(—1/3)dL, uL—*SL(2/3)uL, (4.21)

where

S7,(Q) = UL(Q) K~,UL(Q). (4.22)

Since themixing matrix V = UL(2/3) UL(—1/3), eq. (4.6), relatesthe transformationsin the 0 = ~ quarksto 0 = —~quarks,oneeasilyobtains

Vt S~(2/3)V = S~(—1/3). (4.23)

The invarianceof theLagrangian

{K~}>.~y, (4.24)

implies

K7tI~(Q)K7~= L~2~F,~(Q), (4.25)

which can be rewrittenas

K~tULULF~(Q)U~(Q)UR(Q)K~ = ~~ULUL F0(Q)U~(Q)UR(Q)

andafter re-arrangingandusingeq. (4.19), we obtain

ULK~tU~It~(Q)UR(Q) K7~U~= ~ r~(Q), (4.26)

which is, from eq. (4.22),

S~t(Q)r~(o)S~(Q)= ~ t~(Q). (4.27)

Multiplying eq. (4.27) by its Hermitianconjugate,it follows that

S~(Q)I~(Q)Ca(Q)* S~t(Q)= ~ tf3(Q) t~(Q). (4.28)

In the specialcaseof oneRiggsdoublet

= et88~, (4.29)

eq. (4.28) reducesto

76 Ling-Lie (‘hau, Quark mixing in weak interactions

Sj~(Q)t’(Q) t~(Q)S~t(Q)= r(o) r~(Q)- (4.30)

This implies that the unitary matrix S~(Q)must be diagonal. Now let’s look at eq. (4.23). Since theleft-handsideof eq. (4.23) is diagonalthe column vectorv, of V (vi, v

2,. . . v~)must be eigenvectorsofS~’i(2/3).But now S~(—1/3) is also diagonal,its eigenvectorsmust be of the form, up to a phasefactor,

1 0 0

V1(0 ). ~2(~)~~3(~)etc. (4.31)

which is uniqueif thereis no degeneracyin the eigenvaluesof S~(2/3).Theseare the physical quarkstatesin someorder.We havethus shown that the mixing matrix V hasexactly one nonzeroentry ineachcolumn andeachrow. Sucha matrix is called monomial. It is a trivial mixing matrix.

In the caseof morethanoneRiggsfield, from eq. (4.28), we can only show thatS~(Q)aremonomial,ratherthan diagonal.To prove againthatV is monomial from eq. (4.23) is moreinvolved. The readersarereferredto [4.10].

Thereforeall thosemodelswith horizontalsymmetry haveflavor changingneutral coupling eitherthrough Higgs bosonsor horizontal gauge fields. These couplings must be suppressedto the levelconsistentwith experimentaldata. In the formulation of the K—M model, the flavor changingneutralcouplingsof theZ°are suppressedequally for all flavors. Therefore the observationof abnormallyhighb-flavor changingneutralcoupling will havetheinterestingimplication thateitherthe t quarkis not there. orthere are flavor changingHiggs or horizontal gaugecouplings.As we haveemphasizedin section5, it isimportantto establishthe suppressionlevel of the b-changingneutralcoupling.


This duplication of quarks andleptonsis one of the most fascinatingphenomena.The e-~cpuzzleturns out to be only a smallpart of abigger one. Why this duplication?Will it everend?The impressivenumberof paperson the subjectshowsthe intenseintereston the subject.Someof thesemodelshavesome phenomenologicalsuccess,and some are already ruled out. There is no lack in providingmechanismsrelatingmixing matrix, or event quark mass,to the “known” quark masses.Howeverthereis a lack of compelling fundamentalreasonsor guiding principles for imposing those horizontalsymmetriesor for certainways of generatingmasses.It probablywill be difficult to understandquarkmixing beforewe understandthequark “family problem”. Doesthe answerlie in going to largergroups14.111? Maybeit is importantto studythosetheoriesnaturally havingthreefamilies of quark fieldswithspecifiedspontaneous-symmetry-breakingmechanismbuilt in. It is most interestingto seehow natureeventuallyrevealsher secrets.

5. Alternativeto thethreeleft-handeddoubletmodel

The discoveriesof the charmandthe b particlesin the pastfew yearshavecertainlymadea brilliantchapterin the developmentof particlephysics.They demonstratedthe fruitful resultsof the interplay


betweenexperimentalobservationsandelegantphysicalreasonings.Theparticleworld with threeleft-handedquark doubletsandthreeleft-handedlepton doubletswould appearratherself sufficient, simpleand elegant, though may be boring to some physicists. The electroweak unified theory forthesequarksandleptonsareanomalyfree andthusrenormalizable[5.1].The recentlydiscoveredheavylepton r~ have all the decay propertiesexpectedof them from such a theory [5.2]. However theexpectedt quark is, sofar, still out of sight up to the highestenergyof PETRAandPEP,V5 = 37 GeV[l.l6]. This makesmanytheoristslook for alternativemodels,if indeedthe t quark is not there.

5.1. Alternativemodels

Basicallytherearethreetypesof alternativesofferedin the literature:Alternative(1). The b is a left-handedsinglet. It is the CabibboWorld enlarged,comparedto eq.

(1.7),

(~), (~),~ UR, dR, cR,5R, bR. (5.1)

dL SL

Howeverin ordernot to spoil the GIM mechanism,i.e., d’d’ + s’s’ = ss+ dd, anew quantumnumberisassignedto bL [5.3].We underlinethis point by putting a wavy line under the bL. Now the b cannotdecayinto the otherquarks.But experimentallywe know that the b decays.So new leptons1~andnewgaugebosonsW with this new quantumnumbermust beintroducedin the theory,andthe b can onlydecaysemileptonicallyinto suchnew lepton

b-*qW-*q, (5.2)

whereq denotesthe light quarks,and /2 is an ordinary lepton.Note that thesemodelsfrom atheoreticalpoint of view havetwo undesirabletheoretical features:a right-handeddoublet lepton pair must beincludedin the theory to makeit anomalyfree [5.4]the flavor changingcouplingis not suppressedin a“natural” way [3.361.

Somemodelsin thisalternativehavethe bizzaresituationof having a gaugebosonW which couplesto antiquarku andr~~:

b-+ü W—~UUT~, (5.3a)

or

B—~i5T~. (5.3b)

This alternativehasalreadybeenruled out by experimentsfrom CESRand PETRA. The b particlesare observedto decaymore than85% purehadronically[1.64,5.5].

Alternative (2.a): This is the same as alternative(1) but the neutral flavor conservation is notenforcedfor the b flavor. Thereforeb can decayinto otherquarksthroughmixing [5.6].Now we have,

(~,), (c,) ,bL; uR, dR, cR, 5R, bR, (5.4)L

19) Ling-Lie (‘hau. Quark mixingin weakinteraction.s

where

d’=°V~~d+Vs+ VUhb, s’= VCdd+ V~~s+V~bb.

This is eq. (1.1) enlarged.Taking the known value of V0d= 0.973, V~.= 0.23 from section 1, and using the orthogonality,

unitarity, and the constraintof no strangenesschangingneutralcurrent

VUd~2+tVUSI2+~VUh~2=1,

VUdVCd+ Vu~Vc~+VUbVCh = 0, (5,5)

VUd V + V~d~ = 0,

one can actuallyfind solutionsfor V0. The importantpoint hereis that now thereareb~,bd neutral

flavor changingcouplings, which producesizeableBr(b—*stt) or Br(b~dtC). It was estimatedBr(b—*sI~f)~2% [5.61.Another importantconsequenceis that the sizeableb—ssi.’i decaywill causelargeenoughmissingenergyto be tested.Both thesefeatureshavenowbeenanalysedfor the b decaysatCESR[1.64,5.5], andby Mark JandPETRA[5.7].Thepredictedamountof Br(B —* t

4~tX) andmissingenergyin the decayboth exceedexperimentalobservation.The experimentalinformation aboutthe bdecaywill be summarizedat theend of this section.

This alternativesuffersfrom the sameundesirablefeatureas alternative(1): neutral-strange-changingsuppressionis not “natural”. A new right-handleptondoublet is neededto makethe theory anomalyfree.

Alternative (2.b): To makethe neutralflavor strangenesschangingsuppressionnatural,onecan putb with u and d’ to form a triplet but then anothernew charge—1/3 quark f is neededin order to formwith c ands’ anothertriplet. Now we have

/u\ /c\( d” ) , ( s” ) , (5.6)

\b’;/i. \b~I1.

whered”, s”, b~,b~can all bemixturesof d, s, b, f. Theonly requirementis orthogonality,unitarity andno neutral strangenesschangingneutral coupling. Such arrangementsare rathercommonin unifiedmodels:SU(3) x U( 1) [5.8],SU(3)Lx SU(3)R[5.9],andE6for thegrandunificationmodels[5.10].Ofcoursefor the cancellationof anomalythe leptonshaveto form triplets too.

Insofar as the decay properties are concerned,this alternative has the same two outstandingcharacteristicsas alternative(2.a): sizeableBr(b —* qt /2) andBr(b —* qiii).

Therehavebeenmanystudieson the B decayproperties,especiallythe /~.4/~ in thefinal productandtheir phenomenologicalimplications [5.11].In a remarkableanalysisby Kane and Peskin [5.12],it isshown for both alternatives(2.a) and (2.b) in which the b quark decaysvia the normalW~,Z°gaugeboson, the inclusiveneutraldilepton decaywith inclusivechargedilepton decayratio

(5.7)

Howeverthe experimentalobservationis

Ling-Lie (‘hau. Quark mixing in weak interactions SI

Br(B—t)~0.74%, (5.8a)

or equivalently

F(B —~Xt [)/f(B —* X[~ ii) <0.08, (5.8b)

not consistentwith eq. (5.7), [1.64,5.5, 5.7].

Alternative (3): The b forms aright-handeddoubletwith the right-handedcharmquark:

(~l’)L’ (‘)L’ bL; uR, dR, SR,(~) (5.9a)

where

d’ = cosOcd + sin& s= —sin & d + cos& s.

This is a phenomenologicalmodeldesignedto beconsistentwith the presentexperimentalobservationof theB decay[5.13].Sincetheb isnowputin aright-handeddoublet,thereisno mixing of b with sandd. Itavoidstheexistenceof b~orbd neutralcurrents.By construction,it couplesonlyto thecharmc.Howeveritsfull couplingstrengthto the c canbe easilychangedby mixing the b with anothercharge— 1/3 quark,i.e. ineq. (5.9a)

(~)R~ (~‘)R’fR, (5.9b)

whereb’ = b cos °b+ f sin 01,. Thereforeit cannotbe testedbasedupon its couplingstrength,like the blifetime or its rare-modedecayrates.How can we experimentallytest this model? In principle thelepton distribution from the semileptondistribution of the b is different if the coupling is (V—A) or(V+ A). Unfortunatelythe differenceis rathersmall. It needsmuchmoreaccuratedatathanpresentlyavailableto makethe distinction.See fig. 5.1 from ref. [5.14].The crucial test againstthe model is thepositiveidentificationof b—~u decaymodes.Like otheralternatives,the model is not anomalyfree,we

(a) 2~

30- 4/2~5.27 13eV. — lb)bL_SINGLET MODEL m85.2GeV

- R rnBr 5.2 13eV - 20 - ./~= 5.2713eV -

__ ~O’~O3.O

E~1GeV) Ee 1GeV)

Fig. 5.1. Single-leptonspectrafrom B decays(a)at = 5.27GeV for left-handed and right-handed modelscomparedwith electronspectra,(b) at‘s/s = 5.27GeV for bL singlet model,from refs. 15.14.1.64].

82 Ling—Lie (‘/Iau. Quark Isi.singin is’eak interact,on.s

must increasequark or lepton doublets:(I) increasetwo right-handedheavy neutral leptonsformingtwo right-handeddoubletswith eR. IUR or TR; or two completelynew heavy right-handeddoublets:(2)or increasetwo left-hand doubletsof heavy quarks (,n~1> 20 GeV); (3) one left-handed doublet ofquarksplus oneright-handeddoublet of leptons;(4) other exotic statesof quarksandleptonsso thatthe theory is anomalyfree. Estheticallytheseare not a very elegantway out. We might aswell haveaworld with a very heavy, unobservablet quark.

5.2. E.vperimenralstatusof the h-particle decay

Let’s now sum up the experimentalsituation:(1) The semilepton leptonic branching ratios of the B decay is about 8— 14%. see table 5.1

[1.62, 1.64, 5.5]. This rulesout alternative(I) typeof models.

Table 5.)Branching ratiosfor semileptonicdecayof theB-meson

Experiments Br(B cmx) Br(B —~ji vx)

CESR: CLEO 13.6±2.1±1.7% 0.0±1.3±2.1%CUSB 13.6±2.5±3.(1% —

PETRA: Marki —

(2) The single leptonenergyspectrumis consistentwith B—+Xi i3 where inx~ 1.8GeV, andgive a

limit [1.64,5.5]

Br(b—*u)/Br(b--*c)<50% . (5.10)

(3) The averagenumberof kaonsis 2.0±0.3 kaonspernonleptonicB decay,and0.8±0.07 kaonspersemileptonicB decay[1.64,5.5]. This is consistentwith b—* c.

Theresults(2) and(3) put somestrainon thealternative(2.b) typeof models,which naturally tendtohaveb—*u.

(4) The inclusiveopposite-chargedilepton decayof the B is bounded[5.5,5.7]

Br(B—*Xt~t)<0.74%, (5.lla)

JO2_

EL 1GeV)

Fig. 5.2. Single-leptonspectrafrom b decayat rest in the standardmodel with Ib—’uI/Ib--~cI= 0,0.3,0.7,1.0,~,from ref. 15.14].

Ling-Lie(‘hau, Quark mixing in weak interactions 83

Tabel 5.2Chargedenergyfractions

Date MonteCarlo

Continuum 0.625±0.003 Exotic I <0.47Y(4S) 0.618±0.003 Exotic II <0.54BB 0.601±0.01±01.0±0.02

Exotic 1: b-.q~f,b-.qlmf, b—’qtl’, like alternative(2.a), (2.b) in the text.Exotic II: b—~qqI~,qq~like alternative(1) in thetext.

or equivalently

Br(B —* Xtt)/Br(B —* X/2i) <0.08. (5.1ib)

This is not in agreementwith alternatives(2.a), (2.b) typeof models.(5) The charged(assumedto be pions) energyfraction of the B decayis about60%, seetable 5.2

[5.5].This is not consistentwith thosemodelsin alternative(1), producinga baryon in the B decay,orthosemodelsproducingtwo neutrinosin the B decay, due to b-flavor changingneutralcoupling likealternative(2.a), (2.b).

(b) The chargedenergyfraction is measuredas function of semileptonicbranchingfraction. Thisseemsto rule out B decayvia a chargedRiggsbosonemission.The Riggsbosoncouplingto fermionsisproportionalto the fermion masses,it couplespredominantlyto ri or c~

b—*uH—-*uri7 or uës. (5.12)

This gives a lepton yield different from experimentalresults, table 5.1, and give less semileptonicbranchingfraction.

5.3. Concluding remarks

It is a remarkableachievementby the experimentaliststhat, exceptthose modelsin which the bquark doesnot decayvia the normalW~,Z°gaugebosons,all the modelswithout the t quark dreamedup by the theoristshavebeenruled out. This, of course,still doesnot meanthat t quark exists. It maybe just our lack of imagination.

The questionto what degreethe neutralb-flavor changingis suppressedis still extremelyimportantto know. Sinceif our theory is composedof only left-handeddoublets,the generalizedGIM mechanismwill suppressvery much the inclusive opposite chargedilepton tt decaysof the B, just like forK°—*p~.~ andK—* 17e~e,[3.42].Someestimatesshow [5.15]

Br(B°—~r~r) 108, (5.13)

Br(B—4)t)=m i0~. (5.14)

As we mentionedin section3, if horizontalgaugeformulation is introduced,flavor changingneutralcouplingcan be present.They are mediatedby neutralRiggsbosonsor horizontalgaugebosons,andcan give much largerbranchingratiosBr(b —* /2÷tX) thanthosegiven by the K—M model, eqs. (5.13,

84 Ling-Lie C/iou. Quark Pni,sing in weak interactions

5. 14). Thereforethe observationof neutralb-flavor changingcouplingat a level abovethatpredictedbythe K—M model has the interesting implication that either the t quark is absent,or thereare neutralflavor changingHiggs fields or horizontalgaugebosons.All thesepossibilitiesareextremelyinteresting.

6. Summaryandoutlook

The threeleft-handeddoubletquark modelwith its mixing matrixof threeanglesandonephaseis SO

far consistentwith existing relevantexperimentalinformation.Thoughstill crude, aconsistentrangeofvalues of the quark mixing matrix elementshas emerged.An interestingfeatureresulting from theseanalysesis the generation-gapphenomenon:i.e. quarksof differentgenerationsdo mix but the mixinggetsweakeras the generationgapincreases.Physically this gives the cascadepattern of decays,e.g.t—~b---~c--*s.The analysisof V0 can be refined as better theoreticalunderstandingand experimentalinformationaboutthe decaysof thecharmedandb-flavoredparticlesimprove.Most important.of course,is the finding of the t quark.

While we do have the approximaterangeof values of the quark mixing matrix elements,ourknowledgeaboutthe valuesof the angles02, 03, andthe phase8 (exceptfor cos0~= 0.9737,sin 02<0.5,

sin 03 <0.5) are still ratherpoor. Betterdeterminationof 8 will require the future observationof newCP violation phenomenain the heavyquark system.Unfortunately the K—M modelseemsto predictvery little CP violation from the neutralparticle—antiparticletransition. We need to rely upon theobservationof CP-violationeffectsin the partial decayratedifferencebetweenparticleandantiparticleinto their cP-conjugatedfinal states,e.g.. :1 —~ ii-p comparedto I —* ~ B~—~~. thedifferenceofwhich can be significant if 8 900.

The distinct featuresof CP violation from the K—M model is the absenceof first-order weakCP-violation effects in semileptonicdecays,the extremesmall value of the neutron electric dipolemoment d,, < l0~°ecm, and a maximal E’/EI < 1/80, and possible large B°B° mixing. Betterexperi-mental information on thesephenomenawill help to distinguish the model from other sourcesof CPviolation.

The analysis of the quark mixing matrix is intertwined with the study of nonleptonicdecaysofparticles.Furthermorethe CP-violationeffectscan manifestin the differenceof nonleptonpartial decayratesbetweenparticle and antiparticle decayinginto their CP-conjugatefinal states.There has beensome progressin our understandingof the difficult subject of nonleptonicdecay. Unlike the over-whelming ~I = or [8] dominancein the weakdecayof strangeparticles, thereis no cleardominanceof one single group representation in the weak decay of the charmedparticles. The new de-velopmenthasbeenthe use of the quark-diagramapproach.Though quite successfulfor semileptonicdecaysleading-orderQCD calculationsof thesenonleptonic-decaydiagramshad predictedthe singledominanceof the W-emissiondiagram.It metwith greatdifficulty whencomparedwith the presentinitialdataof D decays.Otherperturbativediagramshavebeenintroducedto accommodatethedata.If a morephenomenologicalapproachis taken,inclusionof eithertheW-exchangeor theW-annihilationdiagramsinaddition to the W-emissiondiagram seemsto be adequatein fitting the presentdata.Much work, boththeoreticalandexperimental,is needed.It will be probably some time before we truly understandthedynamicsof nonleptonicdecays.The better determinationof the quark-mixingmatrix will certainly hehelpful in the studyof the weak decaydynamics.

Oneof the searchingquestionsis why the quark statesenteringinto the weakinteractionare certainmixturesof thephysical quark state.Much work hasbeendoneattemptingto understandthe origin andthe amount of quark mixing from the symmetry of the quark mass matrix. It is a beautiful idea.

Ling-Lie (‘hou. Quark mixing in weak interactions 85

However therehasbeena lack of fundamentalguiding principle in imposingsuch symmetries.Othersearchingquestionsare why the quark generationrepeats;and if the t quark exists, how many moregenerationsof quark doubletstherereally are. The generation-gapphenomenonof the quark mixingmatrixpreventsus from detectingtheexistenceof high quark doubletsfrom the deviationof unitarity ofthe presentquark mixing matrix. It was the similar situationastheleakageto the third generationwas sosmallthat thepossibleexistenceof thethird quark doubletwasnot suspecteduntil agood physicalreasonwasfoundby KobayashiandMaskawa.Do we havecompellingreasonsnow to expectmoregenerationsofquark doublets?It is interestingto seeif thereasoningthatmoreheavyquarkdoubletsthanthreemightbeneededin orderto explain the matter—antimatterasymmetryin our universecan be substantiatedin thefuture. Onecannotbut haveakeensenseof curiosityandexcitementwaitingto seewhatthe nextsecretinnaturewill be unveiled.

Acknowledgments

This reviewis a result of the many collaborationswith WY. Keung, R.E. Schrock,M.D. Tran, SB.TreimanandF. Wilczek. Discussionswith them andmanyothercolleagueshavehelpedenormouslyinclarifying andimprovingthepresentation,specialthanksareduetoM.K. Gaillard,G.L. Kane,LB. Okun,S. Pakvasa,A. Sanda,M. Schmidt,S.C.C.Ting, T.L. Trueman,S.-H. Tye andC.N. Yang, andthemanyfriendswhorespondedto my letter askingthem to bring me up to dateon theircontributionsin the field.However,theauthoris solelyresponsiblefor anymistakesin thereview.My sincerethanksalsogoesto M.Jacobfor inviting me to write this review. It hasbeena mosteducationalexperience.Finally, I cannotexpressenoughmy appreciationto Mrs. IsabellHarrity for herartistry,andunfailingsupportandpatiencein typing the manuscript.

I would like to thankMr. F.1. Olnessfor careful proofreadingof thereferencesandeliminatingmanyerrors.

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88 Ling-Lie (‘hau. Quark in ixing in weak interactions

[1.49]For scmileptonicdecay of the charmparticle D, see W. Bacinoet a)., Phys. Rev,LeU. 43 (1979) 1073.[1.501L.-L. ChauWang,QuarkFlavorMixing anditsPhysicalImplications,Proc.XX Int’l. Conf. on High EnergyPhysics,Madison,WI, 1980,p. 510: also

Proc. VIth Int’l Coni. on ExperimentalMeson Spectroscopy.BNL. 1980. p. 403.

11.511 S. Pakvasa.S.F.Tuan andJ.J. Sakurai.Phys. Rev.D23 (1981) 2799.[1.52]Seereview talk by

J. Kirkhv, Proc. 1981 Int’l. Symp.on Leptonand Photon Interactionsat High Energies.Fermilab.Aug. 23~-29.1979.‘.531 It was pointed out that gluon correctionscan be large.

N. CahibboandL. Maiani. Phys.Lett, 79B (1978) 109:N. Cahibbo,0. Corbo and L. Maiani, NucI. Phys.B155 (1979) 93;M. Suzuki.Nuc), Phys. B145 (1978)420.For quark masseffects. secJ.L. Cortes.X.Y. Pham andA. Jounsi.Phys. Rev. D25 (1982) 188.

OuangHo-Kim andX.Y. Pham.MassEffectsin QCDCorrectonsofWeakDecays.L,avalUniv. & CERN preprintRef.‘l’h .34)7.TH.Sept. 1982.But we shall seelater in section 2. that simple OCD calculationscan he quite unreliablefor weakdecay processes.

[1.54]R.M. Barnett. Phys.Rev. Lett. 36(1976)l163.Phys.Rev. D14 (1976) 70.[1.55]R. Baer, J. EngelsandB. Petersson,Z. Phys.2 (1979) 265;

V. BargerandR.J.N. Phillips, NucI. Phys. B73 (1974) 269;R. Peierls,T.L. TruemanandL.-L. Wang,Phys.Rev.D16(1977) 1397. thecoefficient 1.07for d(x) in eq. (2.2)of thislast referenceshouldread1.107.

[1.56]BNL-Columbia Collaboration.C. Baltayet a).. Phys. Rev, Lelt, 39 (1977)62.For counterexperimentdata see CDHS Collaboration, Proc. Neutrino79. Int’l Conf, on Neutrinos,Weak Interactionsand Cosmology.Bergen. 1979. eds.A. Haatuft andC. Jarlskog.Vol. 2. p. 253.

[1.571J. Knoblock. New Resultson Dimuonsfrom CDHS, Proc.1981 Int’l. Conf,on NeutrinoPhysicsandAstrophysics.Maui,Hawaii. l981, eds.R.J.Cence,E. Ma and A. Roberts.p. 421.

11.58] E.A. PaschosandU.Tdrke, Phys.Lett. I 16B (1982)360. Their resultsof V~~Iand I V~from r, 6 reactionsaredifferent from [1.48]dueto somedifference in the analysis.See[1.481for comments.

[1.59]L.-L. Chau Wang and F. Wilczek, Phys. Rev, Lett.43 (1979)816.[1.61(1M. Suzuki.Phys. Lett. 85B (1979) 91. Phys, Rev, Lett. 43(1979)818.[1.61]L.-L. ChauWang, FlavorMixing andQuarkDecay, AlP Proc. Sixth Int’l, Conf. on ExperimentalMesonSpectroscopy-l980,BNL, eds.S.U.

Chungand S.J.Lindenbaum.p. 403.[1.62]JADECollaborationatPETRA.W. Barteletal,,Phys.Lett, 1 14B (1982)71.Seethereviewtalk by J.G.Branson,Proc.1981Int’l Symp.on Lepton

and Photon Interactionsat High Energies,Bonn. Aug. 24—29. 1981, andtalk at XXI Int’l. Conf, on High EnergyPhysics,Paris,July 26—31.1982.

[1.63]V. Barger,WY. Keung and R.J.N. Phillips, Phys.Rev. D24(1981) 244.[1.64]For semileptonicdecayof theb particles, seeCESR Collaboration.

C. Bebeket al,, Phys. Rev. Lett, 46 (1981)84;K. Chadwick et a),. Phys.Rev. Lett. 46 (1981)88;B. Niczyporuket al,, Phys. Rev, Lett, 46 (1981)92;CUSBCollaboration.0. Mageraset al., Phys. Rev,Lett. 46 (1981) 1115.For theoreticaldiscussionssee 0. Altarelli et a)., Mud, Phys. B208 (1982) 365.

[1.65]For discussionson somecharmedparticledecaynot coveredin this review, seeV. Barger,J.P. Leveille and P.M. Stevensen,Phys. Rev. Lett,44 (1980)226.

[1.661For somediscussionof thesignal of r’,6-productionof theb and t particles, seeP. Phillips. NucI. Phys. B153 (1979) 475: B159 (1979)418;K. Fujikawa, Prog. Theor, Phys.61(1979)1186.

[1.67]For somediscussionsof the b decays,seeS.P. Rosen,Phys. Lett. 93B (1980) 492;D. Zeppenfeld,Z. Phys. C8(1981) 77.

[1.68]R.E. Shrockand MB. Voloshin. Phys. Lett. 87B (1979) 375.[1.691For reviewson thesubject.see talks by

G. Senjanovicand M. Turner, AlP Conf, Proc.No. 72, Particle andFields, SubseriesNo. 23, WeakInteractionsas Probesof Unification,Virginia PolytechnicInst., 1981). eds.G.B. Collins, L.N. Changand JR. Ficenec;M.K. Gaillard. Proc. XX Int’l, Conf. on High EnergyPhysics,Madison,WI, 1980, p. 517.

11.7(1] For somecritical discussionsandalternatepointsof view. seeJ. Ellis. M.K. Gaillard, DV. NanopoulousandS. Rudaz,Phys.Lett. 99B (1981) 101; CERN preprint ‘FH 3056 (1981);Nature293 (198))41;A. Masiero,RN. MohapatraandR.D. Peccei,Phys. Lett, IO8B (1982) 111.

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12.4] M.K. Gaillard and B.W. Lee.Phys. Rev.Lett. 33 (1974) 108;0. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351.They usedthe techniqueintroducedby KG. Wilson, Phys. Rev. 179 (1959) 1499.For anotherexplanationof the~I = rule in K-. 1r~T,see0. Nardulli, G. Preparataand D. Rotondi,Phys. Rev.D27 (1983)557.

[2.5]Al. Vainshtein,VI. Zakharovand M.A. Shifman, Pisma Zh. Eksp. Teor. Fiz. 22 (1975) 123 [JETPLett. 22 (1975) 55]; NucI, Phys. B120(1977) 316; Zh. Eksp. Teor. Fiz. 72 (1977) 1275 [Soy.Phys. JETP45 (1977) 670].

[2.6] Higher ordereffectswereconsideredand werefound not to affect theconclusionsof ref. [2.5],M. WiseandE.Witten,Phys.Rev,D20(1979)1216.However,whetherthe Penguindiagramcanexplainthefull ~I = rule is somewhatuncertain.In anestimateby CT. Hill and0G. Ross,Phys.Lett. 94B (1980)234, thePenguindiagramcan only accountfor 10% of theK andA decays.

[2.71Thedevelopmentof thecompletesix-quarkdiagramfor mesondecayswas first reportedin L.-L. CltauWang,FlavorMixing andCharmDecay,Proc. 1980 Guangzhou(Canton)Conf, on TheoreticalParticlePhysics,Jan.5—14, 1980, p. 1218. The importanceof theW-exchangediagramfor theD°decaywasalsodiscussed.

[2.8] LB. Okun, Leptonsand Quarks(Science,Moscow, 1981).[2.9] H. Fritzschand P. Minkowski, Phys. Reports73. No. 2 (1981) 57.

[2.1(1]For the relevanceof theW-exchangediagramin theK systemandits implicationson e’ seeL.-L. ChauWang, QuarkMixing and Decay,Proc. Int’l, Symposiumon High Energy PhysicsandQuantumField Theory, Protvino,Sept.1980.

[2.11]For arecent reviewon charmlifetimes seeL. Foa, talk, Proc. 1981 Int’l Symp.on Lepton andPhotonInteractionsat High Energies,Bonn, p. 775;G. Kalmus,Talk at XXI Int’l Conf. on High Energy Physics,July 26—31, 1982, Paris;K. Reibel,Talk at 8th Int’l Workshopon Weak InteractionsandNeutrino,Sept. 5—Il, 1982, Javea,Spain.

[2.12]The SLAC 20GeVyP experimentby theHybrid Facility PhotonCollaboration,K. Abe et al., Phys.Rev. Lett. 48(1982)1526.[2.13]Threeremarksabout thetablesfor thequark-diagramamplitudes:

(I) In thesequark diagram amplitudes,for the Penguin diagramsthe contribution from theb quark for charmedparticledecays(and thecontributionsfrom thet quarkfor strangeparticledecays)arenot explicitly writtendownsincetheycanbecombinedwith contributionsfrom thedandthes (from the u andthec) contributions.(2) If oneneedsto takeinto considerationthedifferencesdueto differentquarkintermediatestates,thePenguinamplitudes(e). (f)with differentmixing matrix can be treateddifferently.(3) Theconventionfor the quarkmixing matrixelementataqd

1W vertexis suchthat it is V, if the outgoingquarkq, is au-like quark’, is V~if theoutgoingquark q, is a d-like quark.

[2.14]Talk by V. Luth, Proc.Int’l. Symp.on Lepton and PhotonInteractionsat High Energies,Fermilab,Aug. 23—29, 1979.[2.15]R. Kingsley, S.Treiman, F. Wilczek andA. Zee,Phys.Rev. DIl (1975) 1919;

G. Altarelli, N. CabibboandL. Maiani, NucI. Phys. B88 (1975) 285;J.F. Donoghueand BR. Holstein, Phys.Rev. D12 (1975) 1454:MB. Einhorn andC. Quigg. Phys. Rev.DI2 (1975) 2015;MB. Voloshin. VI. Zakharovand L.B. Okun,ZhETP Pis,Red. 21(1975)403 [Soy.Phys. JETPLetts. 21 l~975)183].

(2.16] For commentson theSU(4)20 dominance,M. Matsuda.M. Nakagawa.K. Okaka,S. Ogawaand M. Shintmura,Prog. Theor, Phys. 58 (1977) 1502; 59 (1978)666; 59 (1978) 1396.

[2.17]L.-L. ChauWang and F. Wilczek, Phys. Rev.Lett. 43 (1979) 816;M. Suzuki,Phys. Lett, 85B (1979) 91; Phys.Rev. Lett.43(1980)818.

[2.18]V. BargerandS. Pakvasa,Phys.Rev.Lett. 43 (1979)812;Q. Ho-Kim and H.C. Lee, Phys.Lett, bOB (1981) 420.

[2.191G. Kane,SLAC Rept.No, SLAC-PUB-2326,1979(unpublished);L.F. Abbott, P. SikivieandMB. Wise, Phys. Rev.D21 (1980) 1393.

[2.20]M. Fukugita,T. Hagiwaraand Al. Sanda,Rutherford Laboratory preprint-79-((52/T.048(1979);K. Ishikawa,UCLA preprint UCLA/79/TEP/l1(1979);M. GlOck, Phys.Lett. 88B (1979) 145;L.F. Abbott, P. Sikivie and MB. Wise, Phys.Rev. D2l, (1980) 768;Al. Sanda,Phys. Rev.D22 (1980) 2814;The resultsof thecalculation arevery muchdependenton thevalueof a. used,asdiscussedin the last reference.

[2.21]D.G. Sutherland,Phys. Lett, 90B (1980) 173;iF. Donoghue and BR. Holstein. Phys. Rev. D21 (198))) 1334;H.J. Lipkin, Phys.Rev. Lctt. 44 (1980)710.

[2.221AN. Kamal and ED. Cooper, Z. Phys. C8 (1981) 67;

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[2.24]M.K. Gaillard, B.W. Lee andJ.L. Rosner.Rev.Mod. Phys.47 (1975)277.(2.25] J. Ellis. M.K. Gaillard andD.V. Nanopoulos,NucI, Phys. BlO)) (1975)313.2.26] 0. Altarelli, N. Cahibboand L. Maiani, Nucl. Phys.B88 (1975) 285: Phys. Lett. 57B (1975) 277;

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V. Barger,J.P. Leveille and P.M. Stevenson,Phys.Rev. D22 (1980) 693;M. Bander, D. SilvermanandA. Soni. Phys.Rev. Lett. 44 (1980) 7;W. Bernreuther,0. NachtmannandB. Stech,Z. Phys. C4 (1980) 257;V.L. Chernyak and A.R. Zhitnitsky. Nuel. Phys.B20l (1982) 492; with I.R. Zhitnisky, Mud, Phys.B204 (1982)477;M. Bando.T. Okazakiand K. Fujii, Z. Phvs.C 54 (1982) 17—22.

12.321 T. Hayashi, E. Kawai. M. Matsuda,S. Ogawaand S. Shige-da,Prog. Theor. Phys.47 (1972)280, 1998;M. Matsuda,M. Nakagawaand S. Ogawa.Prog. Theor. Phys.63(198)1)35):E. Ma. S. Pakvasaand W.A. Simmons,Z. PhysikCS (1980)309;S.P. Rosen,Phys. Rev.Lett. 44 (198(1)4; Phys.Lett. 89B (1980) 246,

2.33] Hi. Lipkin. Phys. Rev.Lett. 46 (1981) 1307.[2.34]For testson W-exchangeand W-annihilationdiagramsin someotherdecay modes.,see

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[2.35]M. Gorn, NucI, Phys. B)91 (1981)269; Penguinand Annihilation Diagrams:A Relationship,Univ. München preprint 1982.[2.361H. Sugawara.Prog. Theor. Phys.31(1964)2)3:

B.W. Lee. Phys.Rev.Lett. 12 (1964)83;S. ColemanandS.L. Glashow,Phys.Rev. 134B (1964)671:N. Cabibbo, Phys.Rev. Let). 12(1964)62;M. GeIl-Mann.Phys. Rev. Lett. 12 (1964) 155:S.P.Rosen,E.C.G. Sudarshanand S.Pakvasa,Phys. Rev, 146 (1966) 1118:For a review seeP. Rosenand S. Pakvasa.in: Advancesin ParticlePhysics,Vol. II.. eds. R.L. Coo) andR. Marshak(Interscience—J.Wiley.1968). p. 473.

[2.371For a detaileddiscussionon thecomparisonof the predictionsfromM = 1/2 and[8] dominance,for strangeparticleandhyperondecaysseeChapter6.3 of ref. [1.28].anda review.M.K. Gaillard, Proc.SLAC SummerInst. on Particle Phys. 1978. ed, MC. Zipf, SLAC ReportNo. 215, p. 397.

[2.381For a more recent review of the current statusand referenceson the subjectsee talk by S. Pakvasa,Proc. XX Int’l. Conf.. Madison.Wisconsin,1980, eds, L. Durandand L.G. Pondrom;andref. 8—12 therein.p. 165.

[2.39]For recent leading-orderQCD analysisof hyperon decayseeJ. Finjord. Phys.Lett. 76B (1978) 116;P. Minkowski, Phys.Lett, 88B (1979) 138;G. Galic. D. Tadic andJ. Trampetic.Mud, Phys.B158 (1979) 106; Phys. Lett, 89B (1980)249;N. Borzin andC. Schmid.Non-LeptonicDecaysof D- and K-mesons,EidgenossischeTechnischeHochschulepreprint 1980:J.F.Donoghue,E. Golowich,WA. Ponceand B.R.Holstein, Phys. Rev. D21 (1980) 186;J. Finjord and M.K. Gaillard, Phys. Rev.D22 (1980) 778;V. Abe, K. Fujii, T. Okazaki, H. Arisue, M. BandoandM. Toya, Lett. al NuovoCimento30 (1981) 393, andreferencestherein;H.C. Lee and K.B. Winterbon, CanadianJournalof Physics60 (1982) 405. The simultaneous understanding of the s-wave as well as the

p-waveis still themain difficulty. A possibleremedyfor thisdifficulty has beenrecentlypointed out byMM. Milosevic, D. Tadic andJ. Trampetic,NucI. Phys. B202 (1982)461;andB. Holdom, Another Look at HyperonDecays,StanfordU. preprint,Nov. 1981.

[2.411]P. Rosen,Phys. Rev. D22(1980) 776; D21 (1980) 2631.[2.41]T. Hayashi, T. Karino, M. MatsudaandS. Ogawa,Prog. Theor. Phys. 49 (1973) 293.12.42] C. Quigg, Z. Phys. C 4 (1980) 55.[3.1] For reviewson the (‘P violation in theK system,see

T.D. Lee andCS.Wu, Ann. Rev.NucI. Sc). 16(1966)471;P,K. Kabir, The (‘P Puzzle (AcademicPress.New York. 1968).For a morerecent review on theexperimentalstatus,seeK. Kieinknecht,in: Proc. XVII Int’l Conf. on High EnergyPhysics,London, 1974,ed,JR.Smith, p. 111-23; andAnn. Rev.Mud, Sci. 26 (1976) 1.


[3.21Ref. [1.31]andA. Palsand SB.Treiman,Phys.Rev.D12 (1975) 2744;LB. Okun,VI. ZakharovandB.M. Pontecorvo,Lett. NuovoCim. 13 (1975)218.

[3.3] F.A. Wilczek, A. Zee,R.L. KingsleyandSB.Treiman,Phys.Rey.D12 (1975) 2768.[3.4] The original exact formulaeq. (3.16) withoutignoring theexternalmomentawasgiven in ref. [1.42],andwas put in thissimplified form in ref.

[1.48].[3.5] R.L. Kingsley, Phys.Lett. 63B (1976) 329;

M. GoldhaberandJ.L. Rosner,Phys.Rev.D15 (1977) 1254;SB.Treiman andF. Wilczek, Phys. Rey.Left. 43 (1979)1059.

[3.61For earliercalculation of B°B°systemseeJ. Ellis, M.K. Gaillard, D.N. NanopoulosandS. Rudaz,NucI. Phys. B131 (1977)285;A. Ali andZ.Z. Aydin, NucI. Phys.Bl48 (1978) 165, andrefs.[3.5—3.7].

[3.7] E. Ma, WA. SimmonsandS.F. Tuan,Phys.Rev. D20 (1979) 2888.[3.8] V. Barger,W.F. Long andS. Pakvasa,Phys.Rev.D21 (1980) 174.[3.9] For morerecentcalculationsof BB systems,with estimateof QCD effectsincluded,see

J.S.HagelinandMB. Wise, NucI. Phys.B189 (1981) 87;J.S.Hagelin, Mud. Phys. B193 (1981) 123;E. Franco,M. Lusignoli andA. Pugliese,NucI. Phys.B194 (1982) 403;For somegeneralresultsseeG.G. Volkoy, A.G.Liparteliani, V.A. Monich andYu.P. Nikitin, NucI. Phys.B199 (1982)425; Soy.J. Yad, Phys.33 (1981)446;V.A. Monich, By. StruminskyandG.G. Volkov, Soy. J. Yad. Phys.34 (1981) 245.

[3.10]M. Bander, D.SilvermanandA. Soni, Phys.Rev. Lett.43 (1979) 242.[3.11]A.B. Carterand Al. Sands,Phys. Rev.Lett. 45 (1980)952; Phys. Rev.D23 (1981) 1567;

1.1. Bigi and Al. Sanda, Nucl, Phys.B193 (1981) 85.[3.12]The resultspresentedin section3.2 aremainly from L.L. ChauWang, Phenomenologyof CPViolation from theKobayashi—MaskawaModel,

AlP Conf. Proc.No. 72, ParticleandFields,SubseriesNo. 23, WeakInteractionsasProbesof Unification, Virginia PolytechnicInst. 1980, eds.GB. Collins, L.N. ChangandJR.Ficenec.

[3.13]J. Bernabeuand C. Jarlskog,Z. Phys. C Particle and Fields 8 (1981) 233.

[3.14]Ya.l. Azimoy andA.A. logansen,Yad. Fiz. 33 (1981) 388 [Soy.J. NucI. Phys. 33(1981)205];M. Voloshin and V. Zakharov,Hints on theB-mesons,DESY ReportF15-80/04,May 1980.

[3.15]L. Wolfenstein,Phys. Rev.Lett. 13 (1964) 562.[3.161C. Avilez, Phys.Rev, D23 (1981) 1124;

This technique was originally used by L.-F. Li and L. Wolfenstein, Phys. Rev. D21 (1980) 178, to calculate s~+_O

(sr~ir ir°[HwIKs)/(ir~x- IT°]HW]KL>. They obtained si+-o — (213)J ~ — 77~o[. The uncertaintyinvolyed in thecalculationwasdiscussedinthepaper.

[3.17]The authorsin ref. [3.10]consideredthe interferenceeffects in b decaybetweena W-emissiori diagramand a “Penguin” diagram with q~productionyia a gluon emission.

[3.18] In ref. [3.13]the “Penguin” diagramsarenot considered,thereforeinterferencesin somechannels,like eqs. (3.30b,c)(3.35b,c)aremissed.[3.19]F.J. Gilman andM.B. Wise, Phys.Lett. 83B (1979) 83.

[3.20]K~experimentat CERN, PSL. Rosseletet al,, Phys.Rev, D15 (1977) 574;F. Abbud,B.W. Leeand C.N. Yang, Phys.Rev. Lett. 18 (1967)980;J. Baton,0. Laurensand J. Reignier,Phys.Lett. 33B (197(1) 528;P. Estabrooksand A.D. Martin, NucI. Phys. 79B (1974) 301.

[3.21]Theoriginal estimateof e’/e wastoo big, see FJ. Gilman andM.B. Wise.Phys. Rev.D20 (1979)2392,andothercalculations,M.B. Wise andE. Witten, Phys.Rev. D20 (1979) 1216;L. Wolfenstein,Mud. Phys.B160 (1979) 501;B. GuberinaandRD. Peccei,NucI. Phys.B163 (1980) 289;C.T. Hill, Phys.Let). 97B (1980)275;V.V. Prokhorov,Yad, Fiz, 31(1979)1019[Soy.J. Mud. Phys.31(1980)527].

[3.22]J.S. Hagelin,Phys.Rev. D23 (1981) 119.[3.23]The relevance of the W-exchange diagram in K decays and its implication on ale waspointedout in L.L. Chau Wang, QuarkMixing and

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Princeton,BNL experiment,M. Banneret al., Phys. Rev.28 (1972) 1597and seeK. Kleinknecht [3.1].[3.25]Univ. Chicago,Stanford,Saclay,FNAL experiment#617, A Studyof Direct CP Violation in thedecayof theNeutralKaonvia a Precision

Measurementof sja/s~+..j,R. Bersteinet al.;Yale—Brookhaven,BNL. experiment#749, A Measurementof Milliweak CPViolation in KL — K

0 DecaysThroughtheDeterminationof a’, R.C.Larsenet al,

[3.26]See also the proposal, D. Dundy ef al.. CERM/SPSC/8l-1 lO/P174, CERN—Dortmund—Pisa—Siegren, Measurement of ~oo[2/I~+..l2.This

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[3.29]J. Ellis and M.K. Gaillard. NucI. Phys.B150 (1979) 141:E. Shahalin.Yad, Fiz. 31(198(1)1665[Soy.J. Mud. Phys. 31(1980)864].

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3.35] Mark J. Collaborationat PETRA: B. Adevaet al., to be published.[3.36]S.L. Glashowand S. Weinberg,Phys.Rev. Dl5 (1977) 1958:

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NO. Deshpande.Phys.Rev, D23 (1981)2654;J.G. Kdrner andD. McKay. DESY preprintDESY/81/034:JO. Donoghue,J. Hagelin and B.R. Holstein, Phys.Rev. D25 (1982) 195.Theseresultsdiffer from a previouscalculation by A.A. Anselmet al.. seeref. [3.41].

[3.38]D. Chang.Phys. Rev. D25 (1982) 13l8:V. Dupont and TN. Pham, Dispersive Contribution to K°—K°Transition and Higgs—Boson ExchangeModel of (‘P Violation, ÉcolePolytechniquepreprint. A498.0482,1982:J.S.Hagelin, Phys.Lett. 117B (1982)441.

[3.39]K. ShizuyaandS-H.H. Tye, Phys. Rev. D23(1981) 1613.[3.40]Ref. [1.42].This calculation shows the uncertaintiesinvolved. Using a different set of parameterswith the limitations from g-po)arizalion

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