Symmetries_1.pdf

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ymmetries

in partic le physics

Symmetries: Spins and their addition The eightfold way revisitedDiscrete symmetries:

Charge conjugation, parity and time reversal



Symmetries pin-1/ systems

Some generalities Spin-1/ systems are often studied in physics!"#ample: electron and its spin, isospin, !!!

Spin-statistics theorem suggests that such systemsare fermionic in nature!

$nteresting in the conte#t of this lecture:%asic &uilding &loc's of matter ()uar's * leptons+are spin-1/!

Simple representation:



Symmetries,dding two spin-1/

ften, it is important to add spins"#amples: &ound states of spin-1/ fermions, spin-

or&it coupling, etc!!

$f two spin-1/ systems are added, the followingo&jects can emerge:

.aively, they have spin 1, , or -1, respectively!

%ut: .eed to distinguish total spin s and its

projection onto the 0measurement a#is s2 (here, 2 has &een chosen for simplicity+



Symmetries,dding two spin-1/

Then the truly relevant states are for s31 (triplet+

and for s3 (singlet+

.ote: the triplets are symmetric, the singlet isanti-symmetric! Catchy:



SymmetriesCle&sch-4ordan coeff ic ients

The coefficients in front of the new statescan &e calculated (or loo'ed up+! They go underthe name of Cle&sch-4ordan coefficients!

5ormally spea'ing, they are defined as follows:

indicating that two spin systems s1 and s areadded to form a new spin system with total spin

s! &viously, it is not only the total spin of eachsystem that counts here, &ut also its orientation!This is typically indicated through a 0magnetic)uantum num&er, m, replacing s

2

in the literature!



SymmetriesCle&sch-4ordan coeff ic ients

Special cases::5or s3:

Cle&sch-4ordan ta&les for two cases:(a s)uare-root over each coefficient is implied+



Symmetries5rom spin to isospin

6ho carries isospin7

8emem&er 9eisen&ergs proposal: p and n are just two manifestations of the same

particle, the nucleon! $dentify them with theisospin-up and isospin-down states of the nucleon:

Catch: $sospin conserved in strong interactions; 6ill dwell on that a &it: <lay with pions, nucleonsand Deltas!

(.ote: =ultiplicity in each multiplet: $>1+!




Dynamical implications: %ound states (deuteron+

dd two nucleons: can have isosinglet and isotriplet!

.o pp, nn-&ound states deuteron 3 isosinglet ;;;

Consider processes (> their isospin amplitudes, &elow+

1 1




6ho carries isospin7

.ucleons in isospin notation:

<ions in isospin notation:

Deltas in isospin notation:



Symmetries$ sospin and scattering ampl itudes

?se isospin for pion-nucleon scattering amplitudes

"lastic processes

Charge e#change processes





8emem&er: pions and nucleons are isopin-1 and 1/!The total isospin is either 1/ or @/ and thus, there

are only two independent amplitudes: and?se Cle&sch-4ordan coefficients:





Then: 8eactions (a+ and (f+ are pure @/:

ther reactions are mi#tures (coefficients given &ythe Cle&sch-4ordans+, e!g!





Therefore, the cross sections &ehave li'e

t a c!m! energy of 1@ =eA there is a dramatic&ump in the pion-nucleon scattering cross section,first discovered &y 5ermi in 1B1! There, the pion

and nucleon form a short-lived resonance state, theΔ which we 'now to carry $ 3 @/!




t c!m! energies aroundthe Δ-mass, one cane#pect that ,and therefore, there

"#perimentally, it issimpler to com&ine (c+

and (j+, leading to



Symmetries$ sospin and 4-parity

<ions and isospin: 4-parity 9ow does this wor' for the mesons (the pions+ 7

<ions 3 &ound states of a )uar' and an anti)uar',

so naively: 0ust add the isospins li'e the spins! %ut: 8ules of spin addition not sufficient! 9ow to0&ar a spin7 <ro&lem: want to preserve somesymmetries li'e charge conjugation under 0&arring!

4-parity (a group-theory construct+ demands:

conserved )uantum num&er in strong interactions!



Symmetries$ sospin and 4-parity

<ions and isospin: 4-parityltogether: The pion (isospin31+ multiplet reads

The une#pected minus-sign in the neutral pion(compare with spin+ is due to the 4-parity actingon the )uar's and anti-)uar's (the former have

positive, the latter negative 4-parity+!



The eightfold way, revisited ome ?(@+ relations

6hy S?(@+7$n isospin, there are two )uar's related &ysymmetry, and

The group related to this is the spin group, orS?(+! $ts generators are the <auli matrices,

The pions can &e identified with σ@ and the two

linear com&inations (of definite charge+




6hy S?(@+75or three states , similarly related througha symmetry, one could thin' a&out the group S?(@+!

$ts generators are the 4ell-=ann matrices!$n S?(@+, the mesons can &e connected to suita&lelinear com&inations of the 4ell-=ann matrices (seene#t slide+

.ote: ECDs gauge group is also S?(@+! differentiate&etween S?(@+ of flavour (up, down, strange+ andS?(@+ of colour (red, green, &lue+, although grouptheory is the same;




The 4ell-=ann matrices




Singlet-octet mi#ing.ote: $n the meson sector, also a 0singlet meson &itcontri&utes, with a wave function of the form

$t could &e realised through a unit matri#!

Typically there is a mi#ing with octet wavefunctions, most nota&ly e#amples are the andthe mi#ing in the pseudoscalar and vectormultiplet! So, typically, there are nine mesons perS?(@+-multiplet!

Th i htf ld i it d



The eightfold way, revisitedThe pseudoscalar mesons

Here, the spins anti-align

Th i htf ld i it d



The eightfold way, revisitedThe vector mesons

Here, the spins align

Th h f ld d



The eightfold way, revisited,n alternative loo'

"#ample for colour:%asic e)uation:

This should e#plain the

eight gluons and thea&sence of the ninth(the singlet+ one!

Similar for the mesons,

just replace colour withflavour!

D



Discrete symmetries: ome e#amples

Charge conjugation, parity, time reversal$n addition to continuous symmetries, which canreflect properties of space-time (li'e, e!g! under

rotations, &oosts, etc!+ or of dynamics (gaugesymmetries+, there may also &e discrete symmetries!

=ost important e#amples:Charge conjugation: FF charges (electric, colour, etc!+

are inverted! perator for that:<arity: =ove from a left-handed coordinate system to aright-handed one (mirror+! 8ealised through

Time reversal: $nvert time a#is, operator:

D



Discrete symmetries:<arity and time reversal

The operators of the discrete symmetries related tospace time, and , are )uite o&vious when actingon four-vectors:

.ote: lthough at first it loo's similar, the parity

operator is not the metric; (position of the indices+!

Di t t i



Discrete symmetries:<arity

<arity violation?ntil the 1Bs people &elieved that parity(symmetry under 0mirroring+ was conserved, and

there have &een many tests in electromagnetic andstrong interactions &ut none in wea' interactions!

This led Fee and Gang to propose an e#periment,which was later that year carries out &y C!S!6u!

$n this e#periment radioactive HCo was carefullyaligned such that its spin would point into thedirection of the positive 2-a#is!

Di t t i




<arity violationThen the Co&alt undergoes a β-decay, emitting anelectron and an anti-neutrino! 6u found that most

of the electrons would &e emitted into the positive2-direction!

$f the process is 0mirrored, the spin of the nucleus points along the negative 2-a#is, &ut the electrons

would still &e emitted into the positive 2-direction!This different &ehaviour is called a#ial-vector andvector for spin and momentum (and respectivesimilar )uantities+!

Di t t i




9elicity, chirality, and all that.ote: 8eflections turn left-handed coordinate systemsinto right-handed ones and vice versa, this affects

spins etc!=ore physical definition: Define handedness as spinwith respect to the a#is of motion (technicallyspea'ing, this is helicity+! 5or massive particles this is

not Forent2-invariant, &ut for massless ones it is!Therefore, helicity is a meaningful, fi#ed property ofmassless particles, called chirality!

Di t t i




9elicity, chirality, and all that$n the Standard =odel:FF ."?T8$.S 8" F"5T-9.D"D

This can &e seen &y considering pion decays intomuon > antineutrino! $n the rest frame of the pion,the muon and neutrino come out &ac'-to-&ac' andthe spins have to add to (since the pion has

spin-+! Therefore, the handedness (spin-direction+ ofthe anti-neutrino e)uals the handedness of the muon!

$n this e#periment, up to now, muons with only onehelicity/handedness have &een found!

Disc t sy t i s:



Discrete symmetries:C< violation

Charge conjugation, parity, time reversal5or a long time, it was thought that FF laws ofnature on the particle level are invariant under each

of these three symmetries!%ut: 6hile this is true for E"D and ECD, thewea' interactions proved to &e ma#imally parityviolating (only left-handed neutrinos+;;; $n addition, the

wea' interactions show small violation of thecom&ined operation! 6hen this was discovered,it came as a shoc'! Today we 'now that this isdue to the comple# phases in the CI= matri#!

Discrete symmetries:




=i#ing in the system of the neutral mesons prime e#ample for violation is in the systemof the neutral mesons, li'e the neutral 'aons!

$n terms of flavour, there are two eigenstates,and ! 9owever, e#perimentally,two states with wildly different lifetime areo&served: and , which predominantly

decay into two or three pions, respectively!Therefore they have different C< eigenvalues!





=i#ing in the system of the neutral mesons (contd+These C< eigenstates are nearly perfect mi#tures ofthe flavour eigenstates!

ε is related to the amount of C<-violation in the'aon system (the pro&! for decays into the 0wrongnum&er of pions+!





=i#ing in the system of the neutral mesons (contd+ε is related to the 9amiltonian of the 'aon system:

The off-diagonal elements are given &y amplitudesfor the transition &etween the two 'aons:

The amplitude is proportional to a product of fourCI= matri# elements of the form

This allows for comple# values in the 9amiltonianmatri# C< violation!




Discrete symmetries:The C<T-theorem

Charge conjugation, parity, time reversalDespite of C< violation, up to now, no violation ofthe com&ined version of all three discrete symmetrieshas &een found! So C<T seems to &e a truesymmetry of the world!

?ltimately, this allows for causal structures of thetheory as realised up to now!

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