hup://www.ict.p.t.riost.o.it/"piib_offTC/97/115

Uniled Nations Educational Scientific and Cullural Organizationriii<]

Tnlernationai Atomic Energy Agency

INTERNATIONAL CENTRE- EOR THEORETICAL PHYSICS

LEADING LOGARITHMIC QCD CORRECTIONSTO THE BB -+ 77 DECAYS IN THE TWO HIGGS DOUBLET MODEL

T.M. A WPhysics Department. Girne American University. Girne, Cyprus

andInternational Centre for Theoretical Physics.Trieste. Italy,

G. Hi Her2

Deutsches Elektronen-Synchrotron DESY, Hamburg. Germany

riii<]

E.O. Titan3

Physics DeparUvKmI.. Middle; East. T<;c}iTii<:al University. Ankara., Turkey.

ABSTRACT

We calculate t.Tie l<;a.ding logarithmic QCD corrections to t.lie decay Bs —± 77 in t.Tie two Higgsdouble;!, rnodel (2HDM) including Or t.ype long dist.a.n<:e efl'ect.s and <;st.iiTia.!.e t.lie r<;st.rict.ionHof the 2IIDM parameters. tan,3 and W.-H, using the experimental data of B —> A'.,7 decayprovided by the CLEO collaboration. A lower bound for the charged Higgs mass m-H as a.function of the renormalization scale ft is given for 2IIDM model II. We further present thedependencies of the branching ratio Br(Bs —> 77) and the ratio I/L+I2/!^"!2 on m-H and ianfiincluding leading logarithmic Q(T) corrections. The dependence on the renormalization scaleis found t.o be st.rong for bo!.li ralios. An addi!.iona.l unc<;rt.ain!.y arises from !.1K; va.riat.ion of t.liepa.ra.Tnet.erN of the bound slate model, (mi, As). \\rc see. ill at. t.o look for charged Higgs effectst.he TneaHur<;Tvi<;n!. of the branching ra.t.io Br(Bs —> 77) is proTnising.

MIRAMARE TRIESTE

August 1997

'Regular Associate of the 1CTP.2E-rna.il address: ghiller@x4u2.desy.de:iE-mail address: eiltan@lieraklit.physics.nietu.edu.tr

1 Introduction

The; experimental discovery of the inclusive; and e;xe:lusive; B —y Xs^ [1] find B —> K"j [2]

decays stimulated the study of rare B meson decays as a new force. These decays take place via

flavor-changing neutral current (TCJNC) b —V s transitions, which are absent in the Standard

Model (SM) at tree level and appear only llirougli loops. Therefore;, t.Tic: st.udy of l.lu;se; ra.re

B-meson decays can provide a sensitive test of the structure of the SM at loop level and may

shed light on the Kobayashi-Cabibbo-Maskawa (CKM) matrix elements and the leptonic decay

CGTisl.anl.K of I.IK; "B-mesoTis. On l.lie oilier hand, 1.1K; rare H-meson deca.yH ar<; in a. very promising

class to search for new physics beyond the SM, like two iiiggs doublet model (2iil)M), minimal

HiipersyniTnelric model (MSSM). et.c. [3]. Currently, I.IK; main interest, is focused on such ra.re

decays G["B-ITI<;SGTIS for vvliicli I.IK; SM predicts large; bra.nching ra.t.io and which can be; measured

in the near future in the constructed B-factories. The Bx —> 77 decay belongs to this category.

]T\ t.Tie SM. I.IK; brancliing ratio of this deca.y is o['ord<;r 10~T wit.lioul. QCD CGrrect.ioiiK. Including

leading log (LLog) QCD e:orree:t.ie>nK, t.lie bra.nching ratio (Br) b —y ^77 is of t.lie Maine order of

magnitude - 1(T6 like 6 -> sl+l~ [i].

The inveKJ.igat.ion of Bs —> 77 d<;ca.y is interesting for I.IK; following reasons:

• It is well-known, that the QCD corrections to the b —> s-f decay are considerably large (see

[5] - [8] and references therein). Therefore, one can naturally expect that the situation is

t.lie same for I.IK; b —'r ^77 decay. Recently t.lie QCD corrections in t.lie LLog approximation

to this decay have been calculated and found to be large as expected [9] - [11]. Note, that

in the literature this decay without QCD corrections was analysed in the SM [12] - [13]

and in l.lie 2HDM [14].

• In Bs —Y 77 decay, the final photons can be in a CP-odd or a CP-even state. Therefore

I.THN decay allows us t.o study CP violating effects.

• From the experimental point of view, Bs —> 77 decay can easily be identified by putting

a. cut. for t.lie energy of I.IK; final pliol.ons, e.g.. t.lie energy of each phot.on is larger t.ha.n

100 McY. In this ca.se, t.vvo hard photons will easily be; de;j.ee:t.e;d in the; experinieml.N [15].

• Finally, this ele;e:a.y is a.lse) sensitive t.e> t.he physie;s be;yond t.he SM.

In an earlier analysis [11]. the Br(Bx —Y 77) in the 211DM without QCL) corrections was

found to be enhanced with respect to the SM one for some values of the parameter space, in

t.Tic: present, work, we study Bs —> 77 decay in 2HDM wit.li pertnrbative QCD corrections in

LLog approximation. In contrast to [11]. who used the constituent quark model, we impose

a model based on heavy quark effective theory for the bound state of the Bx. Further we

perform an additional analysis with t.Tie inclusion of long-distance effects t.lirougli t.lie t.ransit.ion

Bs —Y &-{ —> 77, which we call Or-type throughout this paper, see [9] for details. We find,

that the theoretical analysis is shadowed by large uncertainties due to the renormalization

scale /./ and the parameters of t.Tie bound state. The decay Bs —Y 77 is dominated by t.he

Wilson coefficient C~^ (see Section 2). which is restricted in our analysis by the B —> Xsj

brancliing rat.io provided by CLEO data [1], Br(B -+ X,-y) tx \C'fJ\2., see Section 3. Without

any improvement, from t.he theoretical side, we see that t.he only chance t.o detect, a deviation

from the SM in Bs —> 77 decay lies in a possible enhanced branching ratio, which can be at

most 1.4 • l(Tfi in the SM [9] and Br2HI}:V!(Bs -+ 77) < 2.1 • 10~(i in model 11 (for mn = 480

GeV and large lantf) resulting from our analysis, at j.i. = 2.5 GeV including t.he C^-type long

distance effects.

T'}i<; pap<;r is organized as follows: In Section 2, we give the LLog QCD corrected Hamil-

ton i an responsible for t.he h —Y .S77 decay. We further calculate; t.he CP-odd A~ and CP-even

A+ amplitudes in an approach based on heavy quark effective theory, taking the LLog QCD

corrections into account. ^T\ Section 3, we study t.he constraint analysis for t.he 2HDM parame-

ters in 11 and lantL using t.he measured data on t.he branching ratio of the B —Y Xs~f decay [1].

Section -1 is devoted to an analysis of the dependence of the ratio [A+l2/!^"!2 and the Br on

t.he parameters j.i. (scale parameter), I.anil and rnjj and our conclusions.

2 Leading logarithmic improved short-distance contri-butions in the 2HDM for the decay Bs —> 77

Before discussing the LLog QCL) corrections to the Bx —> 77 decay, we would like to remind

t.he main features of t.he models which we use in further discussions. In t.he current, literature,

mainly two types of 211DM are discussed, in the so-called model 1, the up and down quarks

get a mass via the vacuum expectation value fv.e.v) of only one iiiggs field, in model 11, the

2

up and down quarks get mass via v.e.v of the lliggs fields 11 \ and 112, respectively, where 111

(ll-z) corresponds to first (second) lliggs doublet of the 211DM. Note, that in this sense the

Higgs secl.or of model 11 coincides vvii.li the; MSSM exUmKion of the; SM. In t.Tic: 2HDM, there

exist five physical iiiggs fields, namely, two charged 11^ and three neutral iiiggs bosons. The

interaction Lagrangian of the quarks with the charged fields, which we need for the calculation

of t.Tic b —> .<*77 decay amplil.uele;, is [17]

£ = .[^[rn^mLdj - rndttifuiLdj]ViiH

+ + h.c. , (1V ^

where L and 11 denote chiral project ions L(R) = 1/2(1 =F 7,-,) and ( and £' are the ratios of t h e

t.vvG vacmnn expecta t ion values, vi anel v2 of t.Tic HiggH fi<;ldK Hi and H2. rcKp<;d.iv<;ly. V;-; arc

t.Tic e lements of j.}i<; C K M nml.rix. In rnodcl TT,

C = -±/t = -tan3=-vlfv2 , (2)

and in rnodcl 1

£> = {• = cot3 = V2/Vl. (3)

After this preliminary remark, let us discuss the LLog QCD corrections to the b —V .S77 decay

amplitudes in OK; 2HDM. The cfl'cct.ivc Hrnnill.onirni TVK;J.}IO<] is a powerful one l.o caiculaU; QCD

corrections. The procedure is to use the effective iiamiltonian obtained by integrating out the

top quark, the W^ and ll^1 bosons. In the effective theory, only the lowest (mass) dimension

operators, which arcs coTiHtrucl.ed by quark and gauges fields, are l.aken int.o account., since; liigher

dimensional operators are suppressed by factors O(mlfmf) and O(m?Jm'lv).

The LLog Q(T) corrections are done through matching the full theory with the effective

t.Tieory at t.Tic: high scale; j.i. = rn^y and then evaluating t.Tic: Wilson coeflicientH from rnyy down

to the lower scale ft — 0{mi-). in this way the LLog QCD corrections for the b —V .S77 decay in

the SM are calculated in [9] - [11].

The; effective Haniiltonian is

where; the; (.)•• are; operators given in eq. (5) and t.Tic: (,'; are; WilKon coefliciemtK re;noniialize;d at

t.Tic: scale j.i. The; coefliciemtK can be: calculated pertnrbative;ly anel the; haelronic matrix e;le;me;ntH

> can be calculated using some non-pert urbative methods.

3

The operator basis of 'Hr,fj is given as

O2 =

{(iL8'1>i(iL<x),

O« =

^ t/, (5)

where a and 3 are SU('X) colour indices and T1-"' and Q^1' are the field strength tensors of the

dc<:l.roTnagTi<;l.i<: and HJ.rong inU;ra<:l.i()TiK, respectively.

For t.Tic: r<;a.soTi given below. t.Tie TJ.og QCD corrcjcj.ions can Tx; caiculaUjd in analog l.o t.Tie

SM. In the 2IIDM. the charged lliggs fields are present and give new contributions due to the

t.Tieir exchange; diagrams. Since I.IK; inter ad. ion verlices of I.IK; charged HiggH bosons and quarks

are proportional l.o l.he ralio mqjm\y. vvher<; mq is l.he masK of l.he c^uark and rn^y is l.he masK of

the W boson, the main contribution comes from the interaction with the t-quark. We neglect

l.he contributions corning from -it. and c quarks, sincx; l.heir rnaKseK ar<; negligibly small compared

l.o rn\y. Tn t.Tnn case; l.he calculations show, l.hal. I.IK; new contributions modify only l.he Wilson

coefficients C\ and (JH of the operators 07 and 0H at m-w scale and do not bring any new

operators [14]. Therefore I.IK; operat.or basis us<;d in I.IK; 2HDM is I.IK; same; as I.IK; basis used

in the SM for the b —> s-y~/ decay.

Denoting the coefficients for the SM with (J*M(mw) and the additional charged lliggs

cont.ribut.ion with C[*(ni\y). we; have; l.he initial values [IS]:

2x

and [H], [16]

•na

6

^r y2 . - j / 3 + % 2 + 2y4 L 4 ( y - l ) ^ i n y + 24(y - I ) 3 J ' [i)

where x = m-f/m^y and y = mjfmfj. Here the parameters ( and £' are given in eqs. (2) and

(3). From eqns. (6) and (7) the init.ial values of the coefficients for l.he 2HDM are defined as:

(8)

Using the initial values of the Wilson coefficients CfIIDM, we can calculate their contribu-

tions at any lower scale as in l.he SM case. Here we would like to make l.he following remark:

Since in our case there exists a charged lliggs boson with a mass larger than m-w-, the correct

procedure to calculate the Wilson coefficients at a lower scale j.i has two stages: First, we cal-

culate l.he value at. m\y starting from rnjj and second, we evaluate l.he result, from m\y down to

a lower scale ft. We assume that the evaluation from m-H to m-w gives a negligible contribution

to the Wilson coefficients and therefore we consider only their evaluation between m-w and a.

lower scale j.i.

U'sing the effective llamiltonian in eq. (1), the amplitude for the decay Bx —Y 77 can be

written as [9] - [14]

where f^ = ^.^gF*1*. The CP-even A+ and CP-odd A~ parts can be writt.en [9] in a HQFT

inspired approach as:

^ = 7K -2 ^f T7 T - T \,- ,-• f r , ,-:f f-. •"! U O

o.tm.GF71 = - !

(10)

where Qq = ^ for r/ = u.c and (^ = —^ for r/ = d,x.h. Here, we have used the iniil.aril.y of l.h

CKM-rnalrix J2i=u,c.t.Vi*Vib = 0 and have neglected l.he conl.ribiil.ioTi dn<; l.o V^Vi,;, <£ V l̂̂ f, =

Af. The function g_ is defined as:

The parameter A., enters eq. (10) through the bound state kinematics [9]. mrh^ and m^J- are

the effective masses of the quarks in the Bx meson bound state [9],

= rn2h -

where \ 2 comes from t h e m a t r i x e lement of t h e heavy quark expansion [19]. T h e LLog Q C D -

<:orre<:l.<;d Wilson coefficients (-'i,,,6(/') [9] - [II] enl.<;r OK; rnnplil.n<]<;s in l.he coTnbiTm.i.ions

CH(P) = Cd(ft) = (CUO -

= atri + CdriNc , (13)

where; Ar,. is l.he ninnTx;r of colours (Nc = 3 for Q C D ) . Whi le (-'i,,,6(/') <•"•<! t-M<; <:o<;ffi<:ieTil.s of l.he

opera tors Oi ...fi, C ^ (fi) is t h e ''effective*' coefficient of O7 and contains renormal iza t ion scheme

<]<;p<;n<]<;nl. conl.ribnl.ionH [rorn I.IK; fonr-qmirk opera tors Oi,,,e iTI 'H^// l.o l.he effecl.ive verl.ex in

b —̂ ,S7. Tn l.he N D R KCIKJTVK;, which w<; nsc- her<;, Cr7}i{j.i) = C7(ji.) — ^Crjji.) — Ce(fi), see [IS]

for detai ls . T h e functions l(mri). J(mri) and A ( m , j come from the irreducible d iagrams wi th

an int.erTm.1 q l.ype quark propri.gril.iTig ri.n<] are defined as

'til

J{mq) = I -mi_ -Am2,

A(rnq) =

= — 2arctan(

In our numerical analysis we used the input values given in Table ( l ) .

Parameter

m,7m-b

— i

A;.

/B. "mB,

m-t

mw

m-/.

ajrnz)A2~

Value

1,1 (CeV)4.8 (CeV)1290.044.09 - 10-|:"* (GeV)0.2 (GeV)5.369 (GeV)175 (CeV) '80.26 (CeV)91.19 (CeV)0.214 (CeV)0.1170.12 (CeV2)

(14)

Table "I: Values of t.Tic: iripnl. pri.ra.TncU;rK used in OK; innncricril calciiUiLions unless ol.}i<;rvvisc

specified.

3 Constraint analysis

There is a considerable interest in the constraints of the parameter space of the 211DM. espe-

cially in model TT, since il.s HiggH secl.or (:.oin<:i<]<;s wil.li OK; niinimal snp<;rKyTnTn<;J.ric. extension

of OK; SM one. The free pa.ramet.erN of t.Tie 2HDM are t.lie masses of t.Tie cliarged and n<;iit.ral

lliggs bosons and the ratio of the v.e.v. of the two lliggs fields, denoted by t a n ^ . in our

analysis j.}i<; n<;iit.ral HiggH bosons are irrelevant., since they do not give any contribution to t.Tie

h —Y ,^77 proc<;sH. TTier<;for<; w<; consider as free parameters j.}i<; mass rnjj of j.}i<; cliarged Higgs

boson and tanil. By using existing experimental data, it is possible to find restrictions on the

parameters rnjj and l,an;3.

7

The model independent lower bound of the mass of the charged lliggs m-n > -'l-'l (VeV comes

from the non-observation of charged 11 pairs in Z decays [20]. There are no experimental upper

bounds for rnjj e.xcepl. mjj < 1 TcV t.o sat.isfy 1.1K; unitarity condition [21]. For model 11, t.Tic

constraints have already been studied. Top decays give m-H > 1-47 C/eV'' for large tan 3 [22].

The lower bound of tan 3 is found to be 0.7 due to the decay Z —> bb [23] and in addition for

large; I.anil 1.1K; rat.io Laniljmjj is resl.ricl.ed. The current, limits are Laniljmjj < 0.38 GaV~l

[21] and tanS/m-H < 0.16 (JeV'1 [25]. which come from the experimental results of branching

ratios of the decays B —> TT> and B —V XrJ>. Recently, the exclusive decay mode B —V Dru has

been sl.udied [26] for model 11 rind 1.1K; upper bound is estimal.ed as Laniljmjj < 0.06 GaV~l.

in the present work, we estimate the constraints for the 2111)M parameters using the result

coming from 1.1K; measurement, of t.Tic: deca.y B —Y Xs~f by 1.1K; CLEO collaboration [1]:

Br(B -> A s 7 ) = (2.32 ± 0.57 ± 0.35) • 10"4. (15)

To reduce t.Tic: b-quark mass dependence let us consider t.Tic: ratio

It = -* ' ""Br(B\\ZVa

I _\ I '

where g(z) is the phase space factor in semileptonic b-decay,

g{z) = 1 - Srr + Sz6 + zs - 2i-4ln z (17)

and z = rnc/rni,.

Now we: want, to discuss t.Tic: l.Tieoretical unc:.(;rl.aintic:s prc:s<;nt in 1.1K; prediction of R.

• The ratio of the (.'KM matrix elements ,\: % = 0.95 ± 0.01 has an uncertainty whichI ^ <.b

comes from the ("P violating parameter tK [8].

• The function g(z) has an uncertainty coming from the masses mi, an<i mc via the ratio

z = rn.Jrnij. HQFT provides a mass relation [27]

m-t, — m-r = (nig — m£))[l \- Of—3-)] . (18)2mHmn rtVg

vvTic:r<; Q = b and c , rnH and rnn ar<; spin avc:ra.g<;d TVK;SOTI masses, mH = 5.31 GaV and

= 1.97 (leV [27]. Here A| is the non-perturbative parameter, which characterizes the

8

average kinetic energy of the b-quark in B meson and its value is obtained by QC1) sum

rides. Using the theoretical value for A| = —(0.1 ± 0.2) (.ieV2 [28], the mass difference

and th<; error quoted are given a.K

rnh - mi: = (3.40 ± 0.03 ± 0.03) GcV. (19)

Here the first error is due to the uncertainty in A| and the second one is from higher order

correct.ioiiK. We take the central value of b-quark rna.KS a.K mi = 4.8 GcV. The uncertainty

in mi IK Ami = ±0.1 GaV. Uning t.lie HQET result, we estimate t.lie nncertainty in mc a.K

Am(7 = ±0.16 and we get the error in g(z) as Ag(z) = 0.096 and Ag(z)/g(z) = 17.8 %,

wlier<; z = rnc/rni, = 0.29 IK 1.1K; central value.

T'}i<; Br for the sernileptonic B is

Br(B -> Xctt>r) = 0.103 ± 0.01. (20)

Both the theoretical uncertainties and the experimental errors, as given in eqs. (15) and (20),

result in an uncertainty in C* . I-sing

Br.niax(B -+ A's = 3.24 • 10" 4 ,

= 1.10 • 10"" , (21)

we get a possible range for \C~-fi\ as

rjf0.1930 < 6 7 J < 0.4049 . (22)

Tn the SM and 2HDM TVIO<1<;1 TT is \Cj \ < 0, bu t in poKsiblc; <;xtensions it, can be posi t ive.

Now we present. t.Tie low<;r bounds of rnjj for differcmt valueK of t.lie scale j.i in Table (2) for

model 11. We res t r ic ted \C~ 2 '''('/•'•) I t>y using the l imits given in eq. (22). l 'br mode l 1, a.

lower bound for t.Tie Higgs boson rna.KS IK abKent.

T'IK; para.Tvi<;ter lant} ba.K boundK st.rongly depending on tli<; scale /./ and rnjj. Tn fig. (1) ,

we plot the p a r a m e t e r tanil wi th respect to ffl« for 3 different j.i scales (2.5, 5. 10) (leY in

model TL by fixing \(_J^^2IIDM\ = 0.4049. We H<X:, llial. llie dependena^ of lanti (rnn) on mn

(I.anil) TXJCOTVKJS weak for large; values of mjj (I,anj3) and tha t a decreasing j.i. scale canseK t.lie

allowed region in the tanil - m-n p lane is to be small . It is in teres t ing t h a t at j.i = 2.5 (JeV., t h e

9

480302235158

/.i [CeV]2.5510

•mw

Table; 2: The lower bounds of the Higgs mass mjj for different scales /./ in model TT.

solnt.ion for lanfi - mn exists only in t.he region 0.4047 < | C / ^ 2 H / M 7 | < 0.4049. Tli<;refore t.he

solid curve in fig. (1) is almost the allowed region for (tan,3, run)- Tor j.i = 2.5 CeV, we get an

empirical expression for the restricted region of the parameter set (tan3. •

lanfi = f'i<••!

- mp

(23)

wil.li c\ = —0.067, ('2 = 6.9 GcV1^ and inv = rnjjm,;n — c. Here t is a positive small ninnTx;r

((. -C "I ) and rnIImin =480 GcV.

i n t h e fo l lowing a n a l y s i s w e r e s t r i c t t h e coeff icient \(^7 '2 L> \ i n t h e g i v e n r e g i o n a n d s t u d y

t.Tic: rcKiill.ing rnjj. lanfi a n d KCHIC; j.i d c p c n d c n c i c j s of t.Tic: ral . io | / 1 + | 2 / | / 1 ~ | 2 a n d t.Tic: Br for t.Tio

<]<;c.rw Bs —Y 7 7 .

4 Discussion

In the rest frame of the Bx meson, the CP = —1 amplitude ,4" is proportional to the perpen-

dicular spin polarization c[ x Co. and tli<; CP = 1 amplitude A+ is proporl.ional to 1.1K; parri.ll<;l

spin polarization t\.£>,. The ratio lA+p/IA")2 is informative to search for CP violating effects

in Bs —> 77 decays and it has been studied before in the literature in the framework of the

2HDM wit.hont QCD corrccl.ioiiK [14]. Tn our analysis we nsc- \.}\rv,v, K<;I.K of parameters (m;,, As)

given in (Table (3)), which model the bound state [9]. However, we do not present the figures

for the first two. Here we analyze the LLog //. and 211DM parameters (ffl«. tan3) dependence

of t.Tic: ratio |/1 + | 2 / | / 1 ~ | 2 and present. t.Tic: results in a series of graphK (figs. 2-7).

in figs. (2) and (3) we plot the dependence of l^+p/IA") 2 on m-t/m-H for fixed tan;3 = 2 and

four different ft scales, (m-w- 10, 5, 2.5) CeV in model 11 and model i, respectively. Decreasing

t.Tic: scale j.i. weakens t.Tic: dependence of t.Tic: ratio |/1 + | 2 / | / 1 ~ | 2 on rnjj and t.Tic: c:.ontriT>ntion of t.Tic:

charged lliggs bosons to l^+l 2 / !^"! 2 gets small. The lower limit of the iiiggs mass is sensitive

10

to the scale ft and it increases with decreasing ft in model 11. (Table (2)). However, the lliggs

mass has no lower bound in model 1.

Figs. (4) and (5) show the; dependence; of |/1+|2/|--'1~|2 <>n l-o-nfi for fixe;d mn = 500 GcV.

This ratio is sensitive only to small tan 3 values. The ft scale regulates the lower limit of tan-3 for

model 1 and model 11 in an opposite way. Further, the effect of the charged iiiggs contribution

becomes weak for large; l,an;3 values.

in figs. (6) and (7) we present the ft scale dependence of lA+p/IA")2 for the SM and 2111)M

with tan-3 = 10 and two different mass values m-n = 500 C/eV'', 800 (•'eV for model 11 and model

T. respectively. We find, t.Tiat. for model 11 t.Tie snialle;r I.IK; value; of rnjj, I.IK; le;ss dependent is

the ratio on //. Model 1 does not allow us to discriminate between the SM or different values

of mjj nse<] as e;xpe;cteel (sex; e.g. fig. (5)).

set 1A., = 370 MeVmh = 5.03 CeV

set 2A, = -180 MeVmh = 4.91 CeV

set 3A., = 590 MeVm6 = 4.79 CeV

Table 3: The parameter sets of the bound state model, [rut-,. A.,).

The lowest order result of |/1 + | 2 / | / 1 ~ | 2 in as is e)btaineel by se;1.ting j.i. = m,\y and it is 0.30

in I.IK; SM for set 3. Tl. rea.e;hes 0.85 at /./ = 2.5 GaV. Varying /./ in I.IK; range; 2.5 GeV < /./ <

10.0 CeV. 1,4+IVI^"!2 is changing in the range 0.60 < l ^ + l 2 / ! ^ " ] 2 < 0.85, resulting in an

\ ^ i ' ^ ^ 1 ^Tl l ' n c ^ ^ - ^ o v v w<! giv < ! iU] example; to coiiipa.re;

|2 on the scale; j.i. a.nel the 2HDM pa.ra.mcM,e;rs: In nie)ele;l

uncertainty of ,,4+ y\A- 2\( r ccV\

the e]e;pe;ne]e;ne:e of the ra.tio

11, for mH = 500 CeV and tan3 > 2, the lowest order result of the ratio (A+l2/!^"!2 is 0.40

and it enhances up te> 0.50 with dee;re;a.sing Ianfi. Henveve;r. at /./ = 2.5 GcV the; ra.tie) reae:heK

0.86 a.ne] the inice;rta.in1.y <]uc- te> the e;x1.endeel Higgs see:te)r is we;a.ker than the OTIC due; I.o the

scale j.i. in model 1, the behaviour is the same.

This shows, that the ratio |/1 + | 2 / | / 1 ~ | 2 is quite; se;nsil.ive; te> QCD e;orrcx;tie)ns a.nel this strong

fj. depeTideTice; ma.ke;s the analysis of the; 2HDM pa.ra.mcM,e;rs rnjj a.nd tanti for the give;n ex-

perimental value of the ratio l^+l 2 / !^"! 2 difficult. However, we believe, that the strong j.i

ele;pe;nele;ne;e will be; re;dnce;d with the; ae]ditie>n e>f the; ne;xl. I.o le;ading order (NLO) e;a.le;nlal.ion,

and the; a.na.lysis OTI the pa.ra.mete;rs will be; more; reliable;. Ne>1.e. that a. similar analysis for the

decay b —> s-f is given in [6]. [8].

in addition, there is another uncertainty due to the different parameter sets (Table (3)).

Tor set 2 a n c l f o r s e t i n t n e S M a n c l i n t n e l o w e s t

order of as. Tt. follows, that t.Tic: large ;r mr, (smaller As)- ''lie large ;r l.lie ratio. We [iirthe;r sex; l.liaf,

having increased m/;, the ratio l / l+p / l^" ! 2 becomes less sensitive to the scale ft. This ratio

essentially changes when QCI) corrections are taken into account. In the lowest order of a s , A+

and A~ ele;pe;nel bol.li on the; one; pa.rt.icle; reducible par!. (TPR). proportional l.o C'7 = Cj an el

in addition ,4" contains the one particle irreducible part (1P1), proportional to C2, see eq. (10).

If we include QCD corrections to the considered ratio, the contribution of C~ dominates over

t.lie TPT see:t.e)r anel t.Tie value of t.lie rat.io increases. This Tiie;a.ns, t.liat. t.lie vahie;s of A + anel A~

come close to each other and it can be explained as a cancellation of the I P ! sector.

Now we e:ont.imie l.o analyze the; Br elispla.ye;d in a serie;s of figures 8-13. Tn figs. 8-11, we

prese;nt the; in±- anel I.anil dependencie;s of the; Br. Dee:re;a.sing mjj, the; Br increases in nie)ele;l

11, however, the behaviour is opposite in model 1. On the other hand the Br is sensitive to

HTiia.ll LantL For large valne;s of lant} , t.lie ele;pe;nele;ne;e e>f the; Br on l,an;3 be;ce)ine;s we;a.k in

model TT. Tn model T. t.lie 2HDM re;snlt almost ce)ine;iele;s wit.li t.lie SM one; since; t.lie e:liarge;d

lliggs contribution is proportional to i/(tan,3)2. Similar to the case of lA+p/ l^" ! 2 , the Br is

HJ.re)ngly dependent, on t.lie se;ale /./. sex; figs. (12-13)). Tt. is e;nha.ne;eel [or small values of/;.. For

pa.ramet.e;r set. 3. the; lenveKJ. e>rele;r re;snlt is 3.6 • 10~7 in the; SM. Tt. increases up t.e> 6.8 • 10~7

at ft = 2.5 (.t'eV. Varying j.i in the range 2.5 (jeV < j.i < 10.0 C.!eV, the Br changes between

5.0 • "I0"7 < Br < 6.9 • "I0"7, and this gives an iine;ertahit,y ABrfBr(ji = 5 Ge;V) w ±30% te>r

1.1K: SM. For set, "I (2), the; Br increases up l.o 1.7 • 10~(i (1.0 • 10~(i) and t.lie inice;rt,a.inty also

increases, as due to the scale j.i dependence, namely 39 (35) %. This behaviour of the Br results

mainly I re mi t.lie 1/Aj. ele;pe;nele;ne;e in amplitudes.

With t.lie adelition of e;xtra. Higgs contribution, t.lie inice;rt.a.inty ehie to 2TTDM pa.ramet.e;rs

m-H and tan,3 in the Br becomes large like the one coming from the scale //. Now we will give

an e;xaniple to sex; t.lie e;fle;ct e>f t.lie 2TTDM pa.ramet.e;rs on t.lie Br by che)osing se;t 3. Tn nie)ele;l

TT r«r mu = 500 GaV, \.hv. lowest, order re;sult e>f t.be Brh 5.8 • 10"7 lor lanfi > 2 (see fig. 10)

and it reaches 1.1 • 10~(:p for smaller tan,3. Tor comparison, the value in the SM is 3.6 • 10~7.

At, ft = 2.5 GcV llic Br reae;bes 6.9 • 10"7 (9.0 • 10"7) in the; SM (2HDM).. This sliows, t.liat,

t.lie Br is alse) sensitive to the; extra. Higgs contribution. For mjj = 500 GcV anel fi. = 2.5 Gf;V,

12

the Br in the 2111)M model 11 is enhanced ~ 30% compared to the SM. in model 1, there is a,

suppression due to the extra lliggs contribution compared to the SM (see fig. (11)), however,

t.Tic: Br is still sensitive j.o the scale /./ (see fig. (13)) and the 2HDM paraniet.erH.

We complete this section by taking the Or type long distance effects (LDQ7) for both the

ratio lA+l2 / !^"!2 and the Br into account. The LDQ7 contribution to the CP-odd ,4" and OP-

even A+ amplitudes lias recently been calculated wit.li the lidp of t.Tic: Vector Meson Dominance

model (VMD) [9] and it was shown, that the influence on the amplitudes is destructive. With

the addition of the LDQ7 effects, the amplitudes entering 2 and the Br are now given

A), (24)

where ASI) arc the short, d is tance arnpliLndcK we Look int.o acconnj. in OK; pr<;vioiiK K<;d.ions

(eq. (10)). The LDQ7 amplitudes AfD are defined as [9]

IT(

A\ <' <-* i V r f * V 7

- - - - - - - ;3 m ^ ' u / v ;

where /(,(0) = 0.18 (JeV is the decay constant of o meson at zero momentum. l'\(Q) is the

ext.rapolat.ed Bs —> o form factor (for det.ails Nee [9]).

in figs, (l-i - 19) we present the W.-H, tanil and ft dependencies of the ratio l^+p/IA") 2

and the Br with the addition of LDQ7 effects for set 3. Here we use l'\(Q) = 0.16 [9]. it

can be: shown, t.hat. t.hc: valn<; of l.}i<; ratio | A + \2/1A~\2 <]<;c:.rc:a.H<;s wit.li l.}i<; acldit.ion of LDQ7

effects. However, while the scale j.i is decreasing, the effect of the LDQ7 contribution on the

ratio is also decreasing, see figs. 1 and 15. On the other hand, the uncertainty

resulting from varying j.i. bet.ween 2.5 GeV < fi. < 10.0 G<;V has increased compared to the case

without the inclusion of the LDQ7 amplitudes:

]—— R: ±40% .

The: Br decreases wit.li the addit.ion of JJDQ-, effect.s. since: t.hc: <;flec:.t is d<;st.rnc:.tiv<;. The: /./ scale

uncertainty of the Br is smaller compared to the case where no LI) effect is included and we

get for the range 2.5 GeV < ft < 10.0 GeV in the SM

ABr

Br{ii = 5 CeV)

Tlie present experimental limit GTI the decay Bs —> 77 is [l:

±27% . (26)

Br{Bs -+ 77) < 1.18 • 10-" , (27)

which IK far from t.Tic: theoretical resull.s. "By varying the para.niel.ers /i.rajj. I,anj3.(mi,, As), it

is possible to enhance the Br up to 2.1(2.5) • lO"'3 in model 11 for ffl« = -180 CeV and large

tanil. where the possible maximal value in the SM is 1.1 (1.7) • 10~f), both at ft = 2.5 CeV and

for set 1. T'}i<; TnnnTx;rK in pa.rcnt.TicK<;s corrc^porul l.o j.}i<; case vvlicrc; no LDQJ <;f[<;cJ.K ar<; taken

into account.

LLog calculations show that the Br strongly depends on the scale ft. This strong dependence

will disappear with t.Tic: addition ofNLO QCD corrections. From NLO h —y ,%-f decay, t.Tic: clioice

of ft = m-bf'l in the LLog expression reproduces effectively the NLO result, so one suggests that

this may work also for the b —> .S77 decay. An additional theoretical uncertainty arises from

t.Tic: poor knowledge of the Bs bound slate effects.

We find that the Br increases with the addition of the extra iiiggs contribution and even

at. t.Tic: scale: /./ = 2.5 GaV t.TiiK value IK ~ 2 orders of magnitude smaller than t.Tic: present,

experimental upper bound. The other possibility for an enhancement, of t.Tic: Br is the exlenKion

of the iiiggs sector. This forces us to think of further models like MSSM,...etc. and TCNC

Bs —7- 77 dc:cay will be an efficient, tool lo KearcTi for n<;w pTiysicK Tx;yond t.Tic: SM.

AcknowledgementsThis work was done within the framework of the Associateship Scheme of the International

Centre [or Theorelical Physi<:n, Trieste, Italy. (T. M. A) would like to thank Prof. S. Randjbar-

Daemi for his interest and support.

References

[1] M. S. Alam et al.. CLEO Collaboration. Phys. Uev. Lett. 74 (1995) 2885.

[2] R. Ammaret al.. CLEO Collaboration. Phys. Uev. Lett 71 (1993) 674.

14

[3] J. L. llewett, in proc. of the 21s' Annual SLAC Summer institute, ed. L. l)e Porcel and C.

Dunwoode, SLAC-PU136521.

[I] A. Ali, L.T. iiandoko, C. iiiller and T. Morozumi. Phys. Rev. D 55 (1997) 4105.

[5] B. Grinsldn, R. Springer, and M. Wise, Nncl. Pliys. B339 (1990) 269: R. GrigjaniH, P.J.

O'Donnd, M. Snllicrland and H. Navdcl, Pliys. Lcl.1.. B213 (1988) 355; Pliys. Lell. R286

(1992) E, 413; C. Cella. C. Curci. C. Uicciardi and A. Yicere. Phys. Lett. B325 (1994)

227, Nncl. Pliys. R431 (1994) 417; M. Misiak, Nncl. Pliys R393 (1993) 23, Erralmn R439

(1995) 461.

[6] K. G. Chclyrkin, M. Misiak and M. Miinz, liep-pli/9612313: C. Greub, T. Hnrl.li and D.

Wylcr, Pliys. Ld.l..R 380 (1996) 385; Pliys. Rev. D 54 (1996) 3350.

[7] M. Ciuchini. E. Eranco. C. Martinelli. L. Reina and L. Silvestrini, Phys. Lett. B316

127; Nncl. Pliys. R421 (1994) 41.

[8] A. J. Buras, M. Misiak. M. Miinz and S. Pokorski, .Nucl. Phys. B424 (1991) 374.

[9] G. Hiller and E. Tllan, licp-pli/9704385, 1.O appear in Pliys. Lcl.1.. B.

[10] C. H. V. Chang, G. L, Lin and Y. P. Yao, >K:T)-T>}I/9705345.

[II] L. Lieina. C. Uiccardi and A. Soni, hep-ph/9706253.

[12] G.-L. Lin, J. Lin and Y.-P. Yao, Pliys. Rev. Lcl.1.. 64 (1990) 1498;

C.-L. Lin, J. Liu and Y.-P. Yao, Phys. Rev. D 42 (1990) 2314.

[13] H. SiniTriaand D. Wylcr, Nncl. Pliys. R 344 (1990) 283.

[11] T. M. Aliev and G. Turan. Phys. Rev. D 48 (1993) 1176.

[15] L. Reina, G. Riccardi and A. SoTii,Pliys. Ld.l. R 396 (1997) 231.

[16] B. Crinstein. M. Savage and M. B. Wise. Nucl. Phys. B 319 (1989) 271.

[17] L. E. Aboil, P. Sikivicand M. R. Wise, Pliys. Rev. D21 (1980) 1393.

15

[18] A. Ali and C. Greub, Z. Phys. C 49 (1991) 131;

A. J. Bnras et al., Nucl. Phys. B 424 (1991) 374.

[19] A. Manohar and M. B. Wise, Phys. Rev. D 49 (1991) 1310.

[20] P. Abren el. al., Z. Phys. C 64 (1994) 183;

G. Alexander el al., Phys. Lett,. R 370 (1996) 174.

[21] M. Vellnmn, Aela Phys. Polon. B 8 (1977) 475;

B. W. Lee, C. Quigg, H. B. Thacker, Phys. Rev. D 16 (1977) 253;

M. Veltman, Phys. Lett. B 70 (1977) 253.

[22] F. Abe el. al., hep-ex/9704003.

[23] A. K. Grant. Phys. Rev. D 51 (1995) 207.

[24] M. Aeeiari el. al., Pliys. LeU.. D 396 (1997) 327.

[25] ALEPH Collaboration, contributed l.o TCHEP, Warsaw, Poland, 25-31, July 1996, Pr. No:

PA 10-019.

[26] K. Kiers and A. Soni, hep-ph/9706337.

[27] M. Nenbert, Modern. Phys. A 11 (1996) 1175.

[28] A. Falk, M. Luke, M. J. Savage, Pliys. Rev D 49 (1994) 3367; J. Ctiay, H. Georgi and B.

Grinslein, Pliys. LeU B 247 (1990) 339; T. T. Bigi, N. G. Urallsev and A. T. Vainsl.ein, Pliys.

Lett B 293 (1992) 130. ibid (B 297 (1993) 477); 1. 1. Bigi et al. Phys. Uev. Lett 71

496; R. Rlok el, al, Pliys. Rev. D 49 (1994) 3356, ( E D50 (1994) 3572).

[29] R. M. Barnett et al.. Review of Particle Properties, Phys Uev D 54 (1996) 1.

[30] M. Aeeiarri el, al. (L3 Collaboration), Phys. LeU. R 363 (1995) 127.

16

V

a Oil £00 8 00 1000

Figure 1: /,an;1 as a function of tlie mass nijj for fixed C7"J = — 0.40-49 in the model II of the2HDM. Here solid curves correspond to (lit: scale /.( = 2.5 G'<;V', dashed curves to j.i = -*j G\ 1-'and small dashed curves to /.<. = 10 Gel'",

o.s

Figure 2: rnl/-m.n dependence of the ratio |A+j2/|.4 \2 for set 3 and ianfi — 2. Here, solid lines(curves) correspond 1,0 llu: SM (model II 2IIDM) at \i = raw, long dashed lines (curves) to SM(model II 211.DM) at p. --.• 1.0 GcV, medium dashed lines (curves) to SM (model II 2HDM) atfj, = 5 GeV and small dashed lines (curves) to SM (model II 2HDM) at fi = 2.5 GeV,

0.4 0 . 6

Figure 3: Same as fig 2, but in model I.

17

| A + \"2! l / i - I

Figure 4: tani3 dependence of the ra,tio | A"1" |2 /1/J. \2 for set 3 for rn,jj = 500 GeV. Here, solidlines (curves) correspond f.o the SM (model TI 2HDM) at ft = m ^ , long dashed lines (curves) toSM (model. IT 2HDM") al /./ = "10 GcV. incdium dashed lines (curves) to SM (model II 2HDM)at ft = 5 GcV and small dashed lines (curves) to SM (model [[ 2IIDM) at ft. = 2,5 GtV.

; > I -y .• A I " /

f

Figure 5: Same as fig 4. but in model I 2HDM.

18

Figure 6: The scale dependence of the ratio [A+p/IA |2 for set 3 in the SM and for 2 valuesof ran = 500. 800 GeV with lan(j = 10. Here, solid lines (curves) correspond to the SM atJJL = rnw (at arbitrary fj. scale) , long dashed lines (curves) to model II with mH = 800 GeV atfi = mw (at arbitrary fi scale)., and small dashed lines (curves) model II with ran = 500 GeVat ji ~ mw (a*: arbitrary /.t scale).

Figure 7: Same as fig 6, but for model I. All curves coincide within errors.

19

Figure 8: mv/mj./ dependence: of the branching ratio Br for set 3 with tand = 2. Here, solidlines (curves) correspond, to the SM (model 11 2HDM) at ft = m ^ , long dashed lines (curves)to the SM (model 11 211DM) at ^ = 10 GeV, medium dashed lines (curves) to the SM (modelII 2HDM) at fi ~ 5 GeV and small dashed lines (curves) to the SM (model II 2HDM) atfi = 2.5 GeV.

10"7 3-1

0 .S

Figure 9: Same as fig 8, but in model I 2HDM.

20

til

Figure 10: tanfi dependence of the Br for seL 3 with rrij-j = 500 GeV. Here, solid lines (curves)correspond to the SM (model IT 2HDM) at y. = m\v, long dashed lines (curves) to the SM(model II 21IDM) at \i = 10 GeV, medium dashed lines (curves) to the SM (model II 2HDM)at ji = 5 GeV and small dashed lines (curves) to the SM (model II 2IIDM) at JJ, = 2.5 GeV.

tan(beta)

Figure 11: Same as fig 10, but in model 1 2HDM.

21

Figure 12: The scale dependence of the Br for set 3 and for 2 values of m^ = 500, 800 (.itVwith tanQ = 10. Here, solid lines (curves) correspond to the SM at /.t = raw (at arbitrary /J.scale), dashed lines (curves) to model IT 2HDM with nin = 800 GeV at /i = mw (at arbitraryfi scale), and small dashed lines (curves) model II 2HDM with mn = 500 GeV at JJ, = raw (atarbitrary j.t scale).

(GsV)

Figure 13: Same as fig 12, but for model 1 2IIDM. All curves coincide within errors.

22

~ 2 . ' J A - | " 2

Figure 14: mtjma dependence of the ratio |A+ |2 / jA \2 for set 3 at tanfi = 10 with the additionof LDQ7 effects. Here, solid lines (curves) correspond to the SM (model II 2HDM) at fj, = mw,loTig dashed lines (curves) to the SM (model II 2HDM) at p. = 10 GeV, medium dashed lines(curves) to t.lic SM (model 11 211DM) at \i = 5 G'eV and small dashed lines (curves) to the SM(model TI 2HDVI) at fi = 2.5 GeV.

2 / A" 12 for set 3 for mH - 500 GeV with 1 IK1

correspond to the SM (model II 2HDM)Figure "15: tan3 depeviclfnce of t.Tie ratio ',Aaddition of LDQ- effeci.s. Here, solid Hues [curvesat fi = muz, long dashed lines (curves) to the SM (model II 211UM) a.t \x = 1 0 QtV.. mediumdashed lines (curves) to the SM (model 11 2HDM) at \i ~ 5 GeV and small dashed lines (curves)to the SM (model II 2HDM) at p = 2.5 GeV.

23

Figure 16: The scale dependence of the ratio |/l + |2/|/4 \'2 for set 3 and for 2 values of500, 800 GeV at tan/3 = 10 with the addition of LDo7 effects. Here., solid lines (curves)correspond to the SM at fj, = mw (at arbitrary (j, scale), dashed lines (curves) to model II2HDM with rrtfi = 800 GeV at JJ, = mw (at arbitrary \i scale), and small dashed lines (curves)to model II 2HDM with m# = 500 GeV at fj, = raw (at arbitrary \JL scale).

Figure 17: mj-mn dependence of the Br for set 3 at tan/3 = 10 including LDo7 effects. Here,solid lines (curves) correspond to the SM (model II 2HDM) at ji — raw-, long dashed lines(curves) to the SM (model 11 2HDM) at ji- = 10 GeV, medium dashed lines (curves) to the SM(model II 2HDM) at yt - 5 GeV and small dashed lines (curves) to the SM (model II 2HDM)at fi = 2.5 GtV.

24

tan(beta)

Figure 18: tand dependence of t.Tie Br for sot 3 for m^ = 500 GeV including LDQT effects.Here. Kolid lines (curves) correspond to the SM (model II 2HDM) at \i = raw, long da.shed lines(curves) to the: SM (model 11 211DM) at fj, = 10 GeV, medium dashed lines (curves) l.o 1 he SVI(model II 2IIDM) at y, = 5 GeV" and small dashed linos (curves) 1:o the SM (model II 211UM)at fi = 2.5 GeV.

Figure 19: The scale depeiidentx? of t.he Br for set 3 and for 2 values of VIH = 500. 800 (!c\at; tanfi = 10 including LT)Q1 effeclH. Here, solid lines (curves) correspond to the SM at.p, = rn\y (at arbitrary ji scale), dashed lines (curves) to model II 2HDM with ran — 800 GeVat fj, = miy(at arbitrary fj, scale) and small dashed lines (curves) to model II 2HDM with

= 500 GeV at p, = raw (a^ arbitrary pL scale).

25