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Page 1: Introduction to Group Theory and Its Representations in Particle Physics

Introduction to group theory and its

representations and some

applications to particle physics

Valery Zamiralov

D�V�Skobeltsyn Institute of Nuclear Physics� Moscow� RUSSIA

September ��� ����

Page 2: Introduction to Group Theory and Its Representations in Particle Physics

Contents

� Introduction into the group theory ���� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Groups and algebras� Basic notions� � � � � � � � � � � ���� Representations of the Lie groups and algebras � � � � ���� Unitary unimodular group SU��� � � � � � � � � � � � � ����� SU�� as a spinor group � � � � � � � � � � � � � � � � � � � ���� Isospin group SU��I � � � � � � � � � � � � � � � � � � � � � ����� Unitary symmetry group SU��� � � � � � � � � � � � � � ��

� Unitary symmetry and quarks ����� Eightfold way� Mass formulae in SU��� � � � � � � � � � ��

����� Baryon and meson unitary multiplets � � � � � � � � � � ������� Mass formulae for octet of pseudoscalar mesons � � � ������� Mass formulae for the baryon octet JP � �

�� � � � � �

����� Nonet of the vector meson and mass formulae � � � � ������� Decuplet of baryon resonances with JP � �

�and its

mass formulae � � � � � � � � � � � � � � � � � � � � � � � ��� Praparticles and hypothesis of quarks � � � � � � � � � � ����� Quark model� Mesons in quark model� � � � � � � � � � ����� Charm and its arrival in particle physics� � � � � � � � ����� The �fth quark� Beauty� � � � � � � � � � � � � � � � � � � ���� Truth or top � � � � � � � � � � � � � � � � � � � � � � � � � ����� Baryons in quark model � � � � � � � � � � � � � � � � � � ��

� Currents in unitary symmetry and quark models ���� Electromagnetic current in the models of unitary sym�

metry and of quarks � � � � � � � � � � � � � � � � � � � � � ��

Page 3: Introduction to Group Theory and Its Representations in Particle Physics

����� On magnetic moments of baryons � � � � � � � � � � � � ������� Radiative decays of vector mesons � � � � � � � � � � � ������ Leptonic decays of vector mesons V � l�l� � � � � � � ��

��� Photon as a gauge �eld � � � � � � � � � � � � � � � � � � ���� � meson as a gauge �eld � � � � � � � � � � � � � � � � � � ���� Vector and axial�vector currents in unitary symmetry

and quark model � � � � � � � � � � � � � � � � � � � � � � ������ General remarks on weak interaction � � � � � � � � � � ������ Weak currents in unitary symmetry model � � � � � � � ������ Weak currents in a quark model � � � � � � � � � � � � � �

� Introduction to Salam�Weinberg�Glashow model � ��� Neutral weak currents � � � � � � � � � � � � � � � � � � � ��

����� GIM model � � � � � � � � � � � � � � � � � � � � � � � � ������� Construction of the Salam�Weinberg model � � � � � � � ������� Six quark model and CKM matrix � � � � � � � � � � � ��

��� Vector bosons W and Y as gauge �elds � � � � � � � � � ����� About Higgs mechanism � � � � � � � � � � � � � � � � � � �

Colour and gluons ����� Colour and its appearence in particle physics � � � � � ��

����� Gluon as a gauge �eld � � � � � � � � � � � � � � � � � ������� Simple examples with coloured quarks � � � � � � � � � �

� Conclusion ��

Page 4: Introduction to Group Theory and Its Representations in Particle Physics

Chapter �

Introduction into the group

theory

��� Introduction

In these lectures on group theory and its application to the particle physicsthere have been considered problems of classi�cation of the particles alongrepresentations of the unitary groups� calculations of various characteristicsof hadrons� have been studied in some detail quark model� First chapter arededicated to short introduction into the group theory and theory of grouprepresentations� In some datails there are given unitary groups SU�� andSU�� which play eminent role in modern particle physics� Indeed SU��group is aa group of spin and isospin transforation as well as the base ofgroup of gauge transformations of the electroweak interactons SU���U��of the Salam� Weinberg model� The group SU�� from the other side isthe base of the model of unitary symmetry with three �avours as well asthe group of colour that is on it stays the whole edi�ce of the quantumchromodynamics�

In order to aknowledge a reader on simple examples with the group theoryformalism mass formulae for elementary particles would be analyzed in de�tail� Other important examples would be calculations of magnetic momentsand axial�vector weak coupling constants in unitary symmetry and quarkmodels� Formulae for electromagnetic and weak currents are given for bothmodels and problem of neutral currents is given in some detail� Electroweak

Page 5: Introduction to Group Theory and Its Representations in Particle Physics

current of the Glashow�Salam�Weinberg model has been constructed� Thenotion of colour has been introduced and simple examples with it are given�Introduction of vector bosons as gauge �elds are explained� Author wouldtry to write lectures in such a way as to enable an eventual reader to performcalculations of many properties of the elementary particles by oneself�

��� Groups and algebras� Basic notions�

De�nition of a groupLet be a set of elements G g�� g�� ���� gn �� with the following prop�

erties��� There is a multiplication law gigj � gl� and if gi� gj � G� then

gigj � gl � G� i� j� l � �� �� ���� n��� There is an associative law gi�gjgl � �gigjgl��� There exists a unit element e� egi � gi� i � �� �� ���� n��� There exists an inverse element g��i � g��i gi � e� i � �� �� ���� n�Then on the set G exists the group of elements g�� g�� ���� gn �As a simple example let us consider rotations on the plane� Let us de�ne

a set � of rotations on angles ��Let us check the group properties for the elements of this set��� Multiplication law is just a summation of angles� �� � �� � �� � ���� Associative law is written as ��� � �� � �� � �� � ��� � ����� The unit element is the rotation on the angle �����n��� The inverse element is the rotation on the angle ������n�Thus rotations on the plane about some axis perpendicular to this plane

form the group�Let us consider rotations of the coordinate axes x� y� z� which de�ne De�

cartes coordinate system in the ��dimensional space at the angle �� on theplane xy around the axis z�

�B� x�

y�

z�

�CA �

�B� cos�� sin�� ��sin�� cos�� �� � �

�CA�B� x

yz

�CA � R����

�B� x

yz

�CA ����

Let � be an in�nitesimal rotation� Let is expand the rotation matrix R���

Page 6: Introduction to Group Theory and Its Representations in Particle Physics

in the Taylor series and take only terms linear in ��

R��� � R��� �dR�

d� �����O��� � � � iA���O���� ����

where R��� is a unit matrix and it is introduced the matrix

A� � �idR�

d� ����

�B� � �i �

i � �� � �

�CA ����

which we name �generator of the rotation around the �rd axis� �or z�axis�Let us choose � � ��n then the rotation on the angle �� could be obtainedby n�times application of the operator R���� and in the limit we have

R���� � limn���R����n�

n � limn���� � iA���n�

n � eiA���� ����

Let us consider rotations around the axis U�

�B� x�

y�

z�

�CA �

�B� cos�� � sin��

� � ��sin�� � cos��

�CA�B� x

yz

�CA � R����

�B� x

yz

�CA � ����

where a generator of the rotation around the axis U is introduced�

A� � �idR�

d� ����

�B� � � �i� � �i � �

�CA�B� x

yz

�CA � ���

Repeat it for the axis x�

�B� x�

y�

z�

�CA �

�B� � � �� cos�� sin��� �sin�� cos��

�CA�B� x

yz

�CA � R����

�B� x

yz

�CA � ����

where a generator of the rotation around the axis xU is introduced�

A� � �idR�

d� ����

�B� � � �� � �i� i �

�CA ����

Page 7: Introduction to Group Theory and Its Representations in Particle Physics

Now we can write in the ��dimensional space rotation of the Descartes coor�dinate system on the arbitrary angles� for example�

R����R����R���� � eiA���eiA���eiA���

However� usually one de�nes rotation in the ��space in some other way� name�ly by using Euler angles�

R�� �� � � eiA��eiA��eiA��

functions��Usually Cabibbo�Kobayashi�Maskawa matrix is chosen as VCKM�

�B� c��c�� s��c�� s��e

�i���

�s��c�� � c��s��s��ei��� c��c�� � s��s��s��e

i��� s��c��s��s�� � c��c��s��e

i��� �c��s�� � s��c��s��ei��� c��c��

�CA �

Here cij � cos�ij � sij � sin�ij� �i� j � �� �� �� while �ij � generalized Cabibboangles� How to construct it using Eqs������ ���� ��� and setting �� � ��

Generators Al� l � �� �� �� satisfy commutation relations

Ai �Aj �Aj �Ai � �Ai� Aj� � i�ijkAk� i� j� k � �� �� ��

where �ijk is absolutely antisymmetric tensor of the �rd rang� Note thatmatrices Al� l � �� �� �� are antisymmetric� while matrices Rl are orthogonal�that is� RT

i Rj � ij� where index T means �transposition�� Rotations couldbe completely de�ned by generators Al� l � �� �� ��� In other words� thegroup of ��dimensional rotations �as well as any continuous Lie group upto discrete transformations could be characterized by its algebra � thatis by de�nition of generators Al� l � �� �� ��� its linear combinations andcommutation relations�

De�nition of algebraL is the Lie algebra on the �eld of the real numbers if��i� L is a linear space over k �for x � L the law of multipli�

cation to numbers from the set K is de�ned�� �ii� For x� y �L thecommutator is de�ned as �x� y�� and �x� y� has the following prop�erties� �x� y� � �x� y�� �x� y� � �x� y� at � K and �x� � x�� y� ��x�� y� � �x�� y��

�x� y� � y�� � �x� y�� � �x� y�� for all x� y �L��x� x� � � for all x� y �L���x� y�z� � ��y� z�x� � ��z� x�y� � � �Jacobi identity��

Page 8: Introduction to Group Theory and Its Representations in Particle Physics

��� Representations of the Lie groups and

algebras

Before a discussion of representations we should introduce two notions� iso�morphism and homomorphism�

De�nition of isomorphism and homomorphismLet be given two groups� G and G��Mapping f of the group G into the group G� is called isomorphism or

homomorphis

If f�g�g� � f�g�f�g� for any g�g� � G�

This means that if f maps g� into g�� and g� W g��� then f also maps g�g� into

g��g���However if f�e maps e into a unit element in G�� the inverse in general

is not true� namely� e� from G� is mapped by the inverse transformation f��

into f���e�� named the core �orr nucleus of the homeomorphism�If the core of the homeomorphism is e from G such one�to�one homeo�

morphism is named isomorphism�De�nition of the representationLet be given the group G and some linear space L� Representa�

tion of the group G in L we call mapping T � which to every elementg in the group G put in correspondence linear operator T �g in thespace L in such a way that the following conditions are ful�lled�

��� T �g�g� � T �g�T �g� for all g�� g� � G���� T �e � �� where � is a unit operator in L�The set of the operators T �g is homeomorphic to the group G�Linear space L is called the representation space� and operators T �g are

called representation operators� and they map one�to�one L on L� Because ofthat the property �� means that the representation of the group G into L isthe homeomorphism of the group G into the G� � group of all linear operatorsin W L� with one�to�one correspondence mapping of L in L� If the space Lis �nite�dimensional its dimension is called dimension of the representationT and named as nT � In this case choosing in the space L a basis e�� e�� ���� en�

Page 9: Introduction to Group Theory and Its Representations in Particle Physics

it is possible to de�ne operators T �g by matrices of the order n�

t�g �

�BBB�

t�� t�� ��� t�nt�� t�� ��� t�n��� ��� ��� ���tn� tn� ��� tnn

�CCCA �

T �gek �X

tij�gej� t�e � �� t�g�g� � t�g�t�g��

The matrix t�g is called a representation matrix T� If the group G itselfconsists from the matrices of the �xed order� then one of the simple represen�tations is obtained at T �g � g � identical or� better� adjoint representation�

Such adjoint representation has been already considered by us above andis the set of the orthogonal ��� matrices of the group of rotations O�� inthe ��dimensional space� Instead the set of antisymmetrical matrices Ai� i ��� �� � forms adjoint representation of the corresponding Lie agebra� It isobvious that upon constructing all the represetations of the given Lie algebrawe indeed construct all the represetations of the corresponding Lie group �upto discrete transformations�

By the transformations of similarity PODOBIQ T ��g � A��T �gA it ispossible to obtain from T �g representation T ��g � g which is equivalent toit but� say� more suitable �for example� representation matrix can be obtainedin almost diagonal form�

Let us de�ne a sum of representations T �g � T��g � T��g and say thata representation is irreducible if it cannot be written as such a sum � For theLie group representations is de�nition is su�ciently correct�

For search and classi�cation of the irreducible representation �IR Schurr�slemma plays an important role�

Schurr�s lemma� Let be given two IR�s� t��g and t��g� of the group G�Any matrix w � such that wt��g � t��gB for all g � G either is equal to ��if t��g� and t��g are not equivalent� or KRATNA is proportional to the unitmatrix �I�

Therefore if B �� �I exists which commutes with all matrices of the givenrepresentation T �g it means that this T �g is reducible� Really�� if T �g isreducible and has the form

T �g � T��g � T��g �

�T��g �� T��g

��

Page 10: Introduction to Group Theory and Its Representations in Particle Physics

then

B �

���I

� �� ��I

��� �I

and �T �g� B� � ��For the group of rotations O�� it is seen that if �Ai� B� � �� i � �� �� � then

�Ri� B� � ���i�e�� for us it is su�cient to �nd a matrix B commuting with allthe generators of the given representation� while eigenvalues of such matrixoperator B can be used for classi�cations of the irreducible representations�IR�s� This is valid for any Lie group and its algebra�

So� we would like to �nd all the irreducible representations of �nite dimen�sion of the group of the ��dimensional rotations� which can be be reduced tosearching of all the sets of hermitian matrices J����� satisfying commutationrelations

�Ji� Jj� � i�ijkJk�

There is only one bilinear invariant constructed from generators of the algebra�of the group� �J� � J�

� � J�� � J�

� � for which � �J�� Ji� � �� i � �� �� �� So IR�s

can be characterized by the index j related to the eigenvalue of the operator�J��In order to go further let us return for a moment to the de�nition of the

representation� Operators T �g act in the linear n�dimensional space Ln andcould be realized by n�nmatrices where n is the dimension of the irreduciblerepresentation� In this linear space n�dimensional vectors �v are de�ned andany vector can be written as a linear combination of n arbitrarily chosenlinear independent vectors �ei� �v �

Pni�� vi�ei� In other words� the space Ln

is spanned on the n linear independent vectors �ei forming basis in Ln� Forexample� for the rotation group O�� any ��vector can be de�nned� as wehave already seen by the basic vectors

ex �

�B� ���

�CA � ey �

�B� ���

�CA � ez �

�B� ���

�CA

as �x �

�B� x

yz

�CA or �x � xex�yey�zez� And the ��dimensional representation �

adjoint in this case is realized by the matrices Ri� i � �� �� �� We shall writeit now in a di�erent way�

Page 11: Introduction to Group Theory and Its Representations in Particle Physics

Our problem is to �nd matrices Ji of a dimension n in the basis of nlinear independent vectrs� and we know� �rst� commutation reltions �Ji� Jj� �

i�ijkJk� and� the second� that IR�s can be characterized by �J�� Besides� it ispossible to perform similarity transformation of the Eqs������������� in sucha way that one of the matrices� say J�� becomes diagonal� Then its diagonalelements would be eigenvalues of new basical vectors�

U ��p�

�B� � � �i

� � � i ��i � �

�CA � U�� �

�p�

�B� � � i� � � i �i � �

�CA ���

UA�U�� � �

�B� � � �� � �� � ��

�CA

U�A� �A�U�� � �

�B� � �i �

i � �i� i �

�CA

U�A� �A�U�� � �

�B� � � �� � �� � �

�CA

In the case of the ��dimensional representation usually one chooses

J� ��

�B�

�p� �p

� �p�

�p� �

�CA �����

J� ��

�B� � �ip� �

ip� � �ip�� i

p� �

�CA �����

J� �

�B� � � �� � �� � ��

�CA � �����

Let us choose basic vectors as

j� � � ���B� ���

�CA � j� � ��

�B� ���

�CA � j�� � ��

�B� ���

�CA �

��

Page 12: Introduction to Group Theory and Its Representations in Particle Physics

andJ�j� � � �� �j� � � �� J�j� � �� �j� � � ��

J�j�� � �� �j� � � � �

In the theory of angular momentum these quantities form basis of the rep�resentation with the full angular momentum equal to �� But they could beidenti�ed with ��vector in any space� even hypothetical one� For example�going a little ahead� note that triplet of ��mesons in isotopic space could beplaced into these basic vectors�

��� ��� �� � j�� �� j� � � �� j�� �� j� � � �

Let us also write in some details matrices for J � �� i�e�� for the representationof the dimension n � �J � � � ��

J� �

�BBBBBBBB�

� � � � �

� �q�� � �

�q�� �

q�� �

� �q�� � �

� � � � �

�CCCCCCCCA

�����

J� �

�BBBBBBBB�

� �i � � �

i � �iq�� � �

� iq�� � �i

q�� �

� � iq�� � �i

� � � i �

�CCCCCCCCA

�����

J� �

�BBBBBB�

� � � � �� � � � �� � � � �� � � �� �� � � � ��

�CCCCCCA

�����

As it should be these matrices satisfy commutation relations �Ji� Jj� � i�ijkJk� i� j� k ��� �� �� i�e�� they realize representation of the dimension � of the Lie algebracorresponding to the rotation group O���

��

Page 13: Introduction to Group Theory and Its Representations in Particle Physics

Basic vectors can be chosen as�

j� � � ��

�BBBBBB�

�����

�CCCCCCA� j� � � ��

�BBBBBB�

�����

�CCCCCCA� j� � ��

�BBBBBB�

�����

�CCCCCCA�

j� � � ��

�BBBBBB�

�����

�CCCCCCA� j�� � ��

�BBBBBB�

�����

�CCCCCCA�

J�j���� �� ��j���� �� J�j���� �� ��j���� �� J�j�� � �� �j�� � �� J�j���� �� ��j���� �� J�j���� �� ��j���� � �

Now let us formally put J � �� although strictly speaking we couldnot do it� The obtained matrices up to a factor �� are well known Paulimatrices�

J� ��

�� �� �

��

J� ��

�� �ii �

J� ��

�� �� ��

These matrices act in a linear space spanned on two basic ��dimensionalvectors �

��

��

���

��

There appears a real possibility to describe states with spin �or isospin���� But in a correct way it would be possible only in the framework ofanother group which contains all the representations of the rotation groupO�� plus PL�S representations corresponding to states with half�integer spin�or isospin� for mathematical group it is all the same� This group is SU���

��

Page 14: Introduction to Group Theory and Its Representations in Particle Physics

��� Unitary unimodular group SU���

Now after learning a little the group of ��dimensional rotations in whichdimension of the minimal nontrivial representation is � let us consider morecomplex group where there is a representation of the dimension �� For thispurpose let us take a set of � � � unitary unimodular U �i�e�� U yU � ��detU � �� Such matrix U can be written as

U � eikak�

�k� k � �� �� � being hermitian matrices� �yk � �k� chosen in the form of Pauli

matrices

�� �

�� �� �

��

�� �

�� �ii �

�� �

�� �� ��

��

and ak� k � �� �� � are arbitrary real numbers� The matrices U form a groupwith the usual multiplying law for matrices and realize identical �adjointrepresenation of the dimension � with two basic ��dimensional vectors�

Instead Pauli matrices have the same commutation realtions as the gen�erators of the rotation group O��� Let us try to relate these matrices witha usual ��dimensional vector �x � �x�� x�� x�� For this purpose to any vector�x let us attribute SOPOSTAWIM a quantity X � ���x� �

Xab �

�x� x� � ix�

x� � ix� �x�

�� a� b � �� �� ����

Its determinant is detXab � ��x�� that is it de�nes square of the vector length�

Taking the set of unitary unimodular matrices U � U yU � �� detU � � in ��dimensional space� let us de�ne

X � � U yXU�

and detX � � det�U yXU � detX � ��x�� We conclude that transformationsU leave invariant the vector length and therefore corresponds to rotations in

��

Page 15: Introduction to Group Theory and Its Representations in Particle Physics

the ��dimensional space� and note that �U correspond to the same rotation�Corresponding algebra SU�� is given by hermitian matrices �k� k � �� �� ��with the commutation relations

��i� �j� � �i�ijk�k

where U � eikak �And in the same way as in the group of ��rotations O�� the representation

of lowest dimension � is given by three independent basis vectors�for example� x�y�z� in SU�� �� dimensional representation is given by two independentbasic spinors q�� � �� � which could be chosen as

q� �

���

�� q�

���

��

The direct product of two spinors q� and q� can be expanded into the sumof two irreducible representations �IR�s just by symmetrizing and antisym�metrizing the product�

q� � q� ��

�fq�q� � q�q�g� �

��q�q� � q�q�� � T f��g� T �����

Symmetric tensor of the �nd rank has dimension dnSS � n�n � �� and forn � � d�SS � � which is seen from its matrix representation�

T f��g �

�T�� T��T�� T��

and we have taken into account that Tf��g � Tf��g�Antisymmetric tensor of the �nd rank has dimension dnAA � n�n � ��

and for n � � d�AA � � which is also seen from its matrix representation�

T ���� �

�� T��

�T�� �

and we have taken into account that T���� � �T���� and T���� � �T���� � ��Or instead in values of IR dimensions�

�� � � � � ��

��

Page 16: Introduction to Group Theory and Its Representations in Particle Physics

According to this result absolutely antisymmetric tensor of the �nd rank ������� � ���� � � transforms also as a singlet of the group SU�� and we canuse it to contract SU�� indices if needed� This tensor also serves to upriseand lower indices of spinors and tensors in the SU���

����� � u���u������ �

��� � u��u����� � u��u

����� � �u

��u

�� � u��u

����� � DetU��� � ���

as DetU � �� �The same for ����

��� SU ��� as a spinor group

Associating q� with the spin functions of the entities of spin ���� q� � j iand q� � j i being basis spinors with ���� and ���� spin projections�correspondingly� �baryons of spin ��� and quarks as we shall see later wecan form symmetric tensor T f��g with three components

T f��g � q�q� � j i�

T f��g ��p��q�q� � q�q� � �p

�j� � i�

T f��g � q�q� � j i�and we have introduced �

p� to normalize this component to unity�

Similarly for antisymmetric tensor associating again q� with the spinfunctions of the entities of spin ��� let us write the only component of asinglet as

T ���� ��p��q�q� � q�q� � �p

�j� � i� �����

and we have introduced �p� to normalize this component to unity�

Let us for example form the product of the spinor q� and its conjugatespinor q� whose basic vectors could be taken as two rows �� � and �� ��Now expansion into the sum of the IR�s could be made by subtraction of atrace �remind that Pauli matrices are traceless

q� � q� � �q�q� � �

� ��q

�q� ��

� ��q

�q� � T �� � ��I�

��

Page 17: Introduction to Group Theory and Its Representations in Particle Physics

where T �� is a traceless tensor of the dimension dV � �n

��� corresponding tothe vector representation of the group SU�� having at n � � the dimension�� I being a unit matrix corresponding to the unit �or scalar IR� Or insteadin values of IR dimensions�

�� � � � � ��

The group SU�� is so little that its representations T f��g and T �� corresponds

to the same IR of dimension � while T ���� corresponds to scalar IR togetherwith ��I� For n �� � this is not the case as we shall see later�

One more example of expansion of the product of two IR�s is given bythe product

T �ij� � qk �

��

��qiqjqk � qjqiqk � qiqkqj � qjqkqi � qkqjqi � qkqiqj � �����

��qiqjqk � qjqiqk � qiqkqj � qjqkqi � qkqjqi � qkqiqj �

� T �ikj� � T �ik�j

or in terms of dimensions�

n�n� �� � n �n�n� � �n � �

�n�n� � �

��

For n � � antisymmetric tensor of the �rd rank is identically zero� So weare left with the mixed�symmetry tensor T �ik�j of the dimension � for n � ��that is� which describes spin ��� state� It can be contracted with the theabsolutely anisymmetric tensor of the �nd rank eik to give

eikT�ik�j � tjA�

and tjA is just the IR corresponding to one of two possible constructions ofspin ��� state of three ��� states �the two of them being antisymmetrized�The state with the sz � ��� is just

t�A ��p��q�q� � q�q�q� � �p

�j � i� ����

�Here q� �� q� ��

Page 18: Introduction to Group Theory and Its Representations in Particle Physics

The last example would be to form a spinor IR from the product of thesymmetric tensor T fikg and a spinor qj�

T fijg � qk �

��

��qiqjqk � qjqiqk � qiqkqj � qjqkqi � qkqjqi � qkqiqj � �����

��qiqjqk � qjqiqk � qiqkqj � qjqkqi � qkqjqi � qkqiqj �

� T fikjg � T fikgj

or in terms of dimensions�

n�n� �� � n �n�n� � �n� �

�n�n� � �

��

Symmetric tensor of the �th rank with the dimension � describes the stateof spin S����� ��S����� Instead tensor of mixed symmetry describes stateof spin ��� made of three spins ����

eijTfikgj � T k

S �

T jS is just the IR corresponding to the �nd possible construction of spin ���state of three ��� states �with two of them being symmetrized� The statewith the sz � ��� is just

T �S �

�p�e���q

�q�q� � e���q�q� � q�q�q� � �����

� �pj� � � i�

��

Page 19: Introduction to Group Theory and Its Representations in Particle Physics

�� Isospin group SU ���I

Let us consider one of the important applications of the group theory andof its representations in physics of elementary particles� We would discussclassi�cation of the elementary particles with the help of group theory� Asa simple example let us consider proton and neutron� It is known for yearsthat proton and neutron have quasi equal masses and similar properties asto strong �or nuclear interactions� That�s why Heisenberg suggested to con�sider them one state� But for this purpose one should �nd the group withthe �lowest nontrivial representation of the dimension �� Let us try �withHeisenberg to apply here the formalism of the group SU�� which has aswe have seen ��dimensional spinor as a basis of representation� Let us intro�duce now a group of isotopic transformations SU��I � Now de�ne nucleonas a state with the isotopic spin I � �� with two projections � proton withI� � ��� and neutron with I� � ��� in this imagined �isotopic space�practically in full analogy with introduction of spin in a usual space� Usuallybasis of the ��dimensional representation of the group SU��I is written asa isotopic spinor �isospinor

N �

�pn

��

what means that proton and neutron are de�ned as

p �

���

�� n �

���

��

Representation of the dimension � is realized by Pauli matrices �� � �k� k ��� �� � � instead of symbols �i� i � �� �� � which we reserve for spin ��� in usualspace� Note that isotopic operator �� � ����� � i�� transforms neutroninto proton � while �� � ������i�� instead transforms proton into neutron�

It is known also isodoublet of cascade hyperons of spin ��� ���� andmasses � ���� MeV� It is also known isodoublet of strange mesons of spin �K��� and masses � � � MeV and antidublet of its antiparticles �K����

And in what way to describe particles with the isospin I � �� Say� tripletof ��mesons ��� ��� �� of spin zero and negative parity �pseudoscalar mesonswith masses m��� � �� � ��� � �� ���� MeV� m��� � ���� �� � �� ���MeV and practically similar properties as to strong interactions�

��

Page 20: Introduction to Group Theory and Its Representations in Particle Physics

In the group of �isotopic rotations it would be possible to de�ne isotopicvector �� � ���� ��� �� as a basis �where real pseudoscalar �elds ���� are relat�ed to charged pions �� by formula �� � �� � i��� and �� � ��� generatorsAl� l � �� �� �� as the algebra representation and matrices Rl� l � �� �� � as thegroup representation with angles �k de�ned in isotopic space� Upon usingresults of the previous section we can attribute to isotopic triplet of the real�elds �� � ���� ��� �� in the group SU��I the basis of the form

�ab �

���

p� ��� � i��

p�

��� � i��p� ���

p�

��� �p

��� ��

�� � �p���

��

where charged pions are described by complex �elds �� � ��� � i��p��

So� pions can be given in isotopic formalism as ��dimensional matrices�

�� �

�� �� �

�� �� �

�� �� �

�� �� �

� �p�

� � �p�

��

which form basis of the representation of the dimension � whereas the rep�resentation itself is given by the unitary unimodular matrices � � � U�

In a similar way it is possible to describe particles of any spin with theisospin I � �� Among meson one should remember isotriplet of the vector�spin � mesons ���� with masses � �� MeV�

� �p��� ��

�� � �p���

�� �����

Among particles with half�integer spin note� for example� isotriplet of strangehyperons found in early ��s with the spin ��� and masses ��� � MeV �� �

which can be writen in the SU�� basis as

� �p� � �

� � �p� �

�� � �����

Representation of dimension � is given by the same matrices U �Let us record once more that experimentally isotopic spin I is de�ned as

a number of particles N � ��I�� similar in their properties� that is� havingthe same spin� similar masses �equal at the level of percents and practicallyidentical along strong interactions� For example� at the mass close to ����

Page 21: Introduction to Group Theory and Its Representations in Particle Physics

MeV it was found only one particle of spin ��� with strangeness S��� � it ishyperon ! with zero electric charge and mass ����� ���� �� MeV� Naturally�isospin zero was ascribed to this particle� In the same way isospin zero wasascribed to pseudoscalar meson ������

It is known also triplet of baryon resonances with the spin ���� strangnessS��� and masses M� ������� � ����� � � �� � m�w� M� ������� ������ � � �� � m�w� M� ������� � ����� � � �� � m�w� �resonances areelementary particles decaying due to strong interactions and because of thathaving very short times of life� one upon a time the question whether theyare �elementary� was discussed intensively ��������� � !�������� � �"or ��������� � ���� � �" �one can �nd instead symbol Y �

� ����� forthis resonance�

It is known only one state with isotopic spin I � �� �that is on exper�iment it were found four practically identical states with di�erent charges� a quartet of nucleon resonances of spin J � �� #�������� #�������#������� #������� decaying into nucleon and pion �measured mass di�er�enceM� �M�� ������� MeV� � We can use also another symbolN�������There are also heavier �replics� of this isotopic quartet with higher spins�

In the system �������� it was found only two resonances with spin ����not measured yet ����� with masses � ���� MeV� so they were put intoisodublet with the isospin I � ���

Isotopic formalism allows not only to classify practically the whole set ofstrongly interacting particles �hadrons in economic way in isotopic multipletbut also to relate various decay and scattering amplitudes for particles insidethe same isotopic multiplet�

We shall not discuss these relations in detail as they are part of therelations appearing in the framework of higher symmetries which we start toconsider below�

At the end let us remind Gell�Mann$Nishijima relation between the par�ticle charge Q� �rd component of the isospin I� and hypercharge Y � S�B�S being strangeness� B being baryon number ��� for baryons� �� for an�tibaryons� � for mesons�

Q � I� ��

�Y�

As Q is just the integral over �th component of electromagnetic current� itmeans that the electromagnetic current is just a superposition of the �rd com�ponent of isovector current and of the hypercharge current which is isoscalar�

��

Page 22: Introduction to Group Theory and Its Representations in Particle Physics

�� Unitary symmetry group SU���

Let us take now more complex Lie group� namely group of ��dimensionalunitary unimodular matrices which has played and is playing in modernparticle physics a magni�cient role� This group has already � parameters��An arbitrary complex ��� matrix depends on �� real parameters� unitaritycondition cuts them to and unimodularity cuts one more parameter�

Transition to ��parameter group SU�� could be done straightforwardlyfrom ��parameter group SU�� upon changing ��dimensional unitary uni�modular matrices U to the ��dimensional ones and to the correspondingalgebra by changing Pauli matrices �k� k � �� �� � to Gell�Mann matrices��� � �� ��� ��

�� �

�B� � � �� � �� � �

�CA �� �

�B� � �i �

i � �� � �

�CA �����

�� �

�B� � � �� �� �� � �

�CA � �

�B� � � �� � �� � �

�CA �����

�� �

�B� � � �i� � �i � �

�CA �� �

�B� � � �� � �� � �

�CA ����

� �

�B� � � �� � �i� i �

�CA �� �

�p�

�B� � � �� � �� � ��

�CA �����

��

��i��

��j� � ifijk

��k

f��� � �� f� ���� f��� � ��

�� f�� �

��� f�� �

��� f�� �

��� f�� � ��

�� f�� �p

�� � f� � �

p�� �

�In the same way being patient one can construct algebra representationof the dimension n for any unitary group SU�n of �nite n� These matricesrealize ��dimensional representation of the algebra of the group SU�� withthe basis spinors �

B� ���

�CA ��B� ���

�CA ��B� ���

�CA �

��

Page 23: Introduction to Group Theory and Its Representations in Particle Physics

Representation of the dimension � is given by the matrices � � � in thelinear space spanned over basis spinors

x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA� x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA� x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA�x �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA�

x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA� x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA�x �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA� x� �

�BBBBBBBBBBBBB�

��������

�CCCCCCCCCCCCCA�

But similar to the case of SU�� where any ��vector can be written as atraceless matrix � � �� also here any ��vector in SU�� X � �x�� ���� x� canbe put into the form of the � � � matrix�

X�� �

�p�

X�

k���kxk � �����

�p�

�BB�

x� ��p�x� x� � ix� x � ix�

x� � ix� �x� � �p�x� x� � ix

x � ix� x� � ix � �p�x�

�CCA �

In the left upper angle we see immediately previous expression ���� fromSU���

The direct product of two spinors q� and q� can be expanded exactlyat the same manner as in the case of SU�� �but now � � � �� �� � intothe sum of two irreducible representations �IR�s just by symmetrizing andantisymmetrizing the product�

q� � q� ��

�fq�q� � q�q�g� �

��q�q� � q�q�� � T f��g� T ����� ����

��

Page 24: Introduction to Group Theory and Its Representations in Particle Physics

Symmetric tensor of the �nd rank has dimension dnS � n�n � �� and forn � � d�S � which is seen from its matrix representation�

T f��g �

�B� T �� T �� T ��

T �� T �� T ��

T �� T �� T ��

�CA

and we have taken into account that T fikg � T fkig � T ik �i �� k� i� k � �� �� ��Antisymmetric tensor of the �nd rank has dimension dnA � n�n � ��

and for n � � d�A � � which is also seen from its matrix representation�

T ���� �

�B� � t�� t��

�t�� � t��

�t�� �t�� �

�CA

and we have taken into account that T �ik� � �T �ki� � tik �i �� k� i� k � �� �� �and T ���� � T ���� � T ���� � ��

In terms of dimensions it would be

n� n � n�n � ��jSS � n�n � ��jAA �����

or for n � � � � � � � ���Let us for example form the product of the spinor q� and its conjugate

spinor q� whose basic vectors could be taken as three rows �� � �� �� ��and �� � �� Now expansion into the sum of the IR�s could be made bysubtraction of a trace �remind that Gell�Mann matrices are traceless

q� � q� � �q�q� � �

n ��q

�q� ��

n ��q

�q� � T �� � ��I�

where T �� is a traceless tensor of the dimension dV � �n

��� corresponding tothe vector representation of the group SU�� having at n � � the dimension�� I being a unit matrix corresponding to the unit �or scalar IR� In termsof dimensions it would be n� �n � �n� � � � �n or for n � � � � �� � � � ��

At this point we �nish for a moment with a group formalism and make atransition to the problem of classi�cation of particles along the representationof the group SU�� and to some consequencies of it�

��

Page 25: Introduction to Group Theory and Its Representations in Particle Physics

Chapter �

Unitary symmetry and quarks

��� Eightfold way� Mass formulae in SU ����

����� Baryon and meson unitary multiplets

Let us return to baryons ��� and mesons ��� As we remember thereare � particles in each class� � baryons� � isodublets of nucleon �protonand neutron and cascade hyperons ����� isotriplet of �hyperons and isos�inglet !� and � mesons� isotriplet �� two isodublets of strange K�mesonsand isosinglet �� Let us try to write baryons B���� as a ��vector of re�

al �elds B � �B�� ���� B�� �� � N� N�� B�� B � B�� where � ��B�� B�� B��� �� �� �� Then the basis vector of the ��dimensional representation couldbe written in the matrix form�

B�� �

�p�

X�

k���kBk �

�p�

�BB� � �

�p�B� � � i � N � iN�

� � i � � � ��p�B� B� � iB

N � iN� B� � iB � �p�B�

�CCA � ����

�BB�

�p� � � �p

�!� � p

� � �p� � � �p

�!� n

�� �� � �p�!�

�CCA �

��

Page 26: Introduction to Group Theory and Its Representations in Particle Physics

At the left upper angle of the matrix we see the previous espression �����from theory of isotopic group SU��� In a similar way pseudoscalar mesonsyield �� � matrix

P �� �

�BB�

�p��� � �p

�� �� K�

�� � �p��� � �p

�� K�

K� �K� � �p��

�CCA � ����

Thus one can see that the classii�cation proves to be very impressive� in�stead of � particular particles we have now only � unitary multiplets� Butwhat corollaries would be� The most important one is a deduction of massformulae� that is� for the �rst time it has been succeded in relating amongthemselves of the masses of di�erent elementary particles of the same spin�

����� Mass formulae for octet of pseudoscalar mesons

As it is known� mass term in the Lagrangian for the pseudoscalar meson�eld described by the wave function r has the form quadratic in mass �toassure that Lagrange�Euler equation for the free point�like meson would yieldGordon equation where meson mass enters quadratically

LPm � m�

PP��

and for octet of such mesons with all the masses equal �degenerated�

LPm � m�

PP�� P

��

�note that over repeated indices there is a sum� while m � ��� MeV�mK �� � MeV� m� � ��� MeV� Gell�Mann proposed to refute principle that La�grangian should be scalar of the symmetry group of strong intereractions�here unitary group SU��� and instead introduce symmetry breaking butin such a way as to conserve isotopic spin and strangeness �or hyperchargeY � S�B where B is the baryon charge equal to zero for mesons� For thispurpose the symmetry breaking term should have zero values of isospin andhypercharge� Gell�Mann proposed a simple solution of the problem� that is�the mass term should transform as the ���component of the octet formedby product of two meson octets� �Note that in either meson or baryon octet

��

Page 27: Introduction to Group Theory and Its Representations in Particle Physics

���component of the matrix has zero values of isospin and hypercharge Firstit is necessary to extract octet from the product of two octets entering La�grangian� It is natural to proceed contracting the product P �

� P�� along upper

and sub indices as P �� P

�� or P

��P

�� and subtract the trace to obtain regular

octets

M�� � P �

� P�� �

�P �� P

��

N�� � P �

�P�� �

�P �� P

��

M�� � �� N�

� � � �over repeated indices there is a sum� Components ��of the octets M�

� I N�� would serve us as terms which break symmetry in

the mass part of the Lagrangian LPm� One should only take into account that

in the meson octet there are both particles and antiparticles� Therefore inorder to assure equal masses for particles and antiparticles� both symmetrybreaking terms should enter Lagrangian with qual coe�cients� As a resultmass term of the Lagrangian can be written in the form

LPm � m�

PP�� P

�� �m�

�P �M�� �N�

� �

� m��PP

�� P

�� �m�

�P �P��P

�� � P �

� P���

Taking together coe�cients in front of similar bilinear combinations of thepseudoscalar �elds we obtain

m� � m�

�P � m�K � m�

�P �m��P � m�

� � m��P �

�m�

�P �

wherefrom the relation follows immediately

�m�K � �m

�� �m�

� � � �� ��� � �� �� �� � �� ���g�w��The agreement proves to be impressive taking into account clearness andsimplicity of the formalism used�

����� Mass formulae for the baryon octet JP �

Mass term of a baryon B S JP � ��

�in the Lagrangian is usually linear in

mass �to assure that Lagrange�Euler equation of the full Lagrangian for free

Page 28: Introduction to Group Theory and Its Representations in Particle Physics

point�like baryon would be Dirac equation where baryon mass enter linearly

LBm � mB

�BB�

For the baryon octet B�� with the degenerated �all equal masses the corre�

sponding part of the Lagrangian yields

LBm � mB

�B��B

���

But real masse are not degenerated at all� mN � ��� m� � �� �� m� ����� m� � ���� �in MeV � Also here Gell�Mann proposed to introducemass breaking through breaking in a de�nite way a symmetry of the La�grangian�

LBm � m�

�B��B

�� �m�

�B��B

�� �m�

�B��B

���

Note that here there are two terms with the ���component as generally speak�ing m� �� m�� �While mesons and antimesons are in the same octet� baryonsand antibaryons forms two di�erent octets Then for particular baryons wehave�

p � B��� n � B�

� mp � mn � m� �m�

� � B��� � � B�

� m���� � m�

� �p!� � B�

� � m� � m� ��

��m� �m��

�� � B��� �� � B�

� m���� � m� �m�

The famous Gell�Mann�Okubo mass relation follows immediately�

��mN �m� � m� � �m�� ���� � �����

�Values at the left�hand side �LHS and right�hand side �RHS are givenin MeV� The agreement with experiment is outstanding which has been astimul to further application of the unitary Lie groups in particle physics�

����� Nonet of the vector meson and mass formulae

Mass formula for the vector meson is the same as that for pseudoscalar ones�in this model unitary space do not depends on spin indices �

��

Page 29: Introduction to Group Theory and Its Representations in Particle Physics

But number of vector mesons instead of � is � therefore we apply thisformula taking for granted that it is valid here� to �nd mass of the isoscalarvector meson �� of the octet�

m�����

���m�

K� �m���

wherefrom m�� � �� MeV� But there is no such isoscalar vector meson ofthis mass� Instead there are a meson � with the mass m� � ��� MeV anda meson � with the mass m � ���� Mev� Okubo was forced to introducenonet of vector mesons as a direct sum of the octet and the singlet

V �� �

�BB�

�p��� � �p

��� �� K��

�� � �p��� � �p

��� K��

K�� �K�� �

�CCA�

�BB�

�p��� � �

� �p��� �

� � �p���

�CCA ����

which we write going a little forward as

V �� �

�BB�

�p��� � �p

�� �� K��

�� � �p��� � �p

�� K��

K�� �K�� �

�CCA ����

���

���q

��

q��

�q

��

q��

�A� ��

��

��

whereq

�� is essentially cos�� � �� being the angle of ideal mixing of the

octet and singlet states with I � �� S � �� Let us stress once more thatintroduction of this angle was caused by discrepancy of the mass formula forvector mesons with experiment�

%Nowadays �� & there is no serious theoretical basis to treat V� �vectornonet$V�Z� as to main quantity never extracting from it �� �our ��$V�Z�as SpV� so Okubo assertion should be seen as curious but not very profoundobservation�% S�Gasiorowicz� %Elementary particles physics%� John Wiley 'Sons� Inc� NY�London�Sydney� � ��

We will see a little further that the Okubo assertion is not only curiousbut also very profound�

��

Page 30: Introduction to Group Theory and Its Representations in Particle Physics

����� Decuplet of baryon resonances with JP �

and its mass formulae

Up to � � nine baryon resonances with JP � ��

�were established� isoquartet

� with I � �� #������ N��� isotriplet ������ � !��� ��� isodublet

������� � � � �� But in SU�� there is representation of the dimension��� an analogue to IR of the dimension � in SU�� � with I � �

� which

is given by symmetric tensor of the �rd rank �what with symmetric tensor

describing the spin state JP � ��

�yields symmetric wave function for the

particle of half�integer spin&&&� It was absent a state with strangeness ��which we denote as (� With this state decuplet can be written as�

#���� #�

��� #���� #��

���

����� ����� �����

������ ������

� (���� �

Mass term of the Lagrangian for resonance decuplet B���� following Gell�Mann hypothesis about octet character of symmetry breaking of the La�grangian can be written in a rather simple way�

LB�

M� �M���B����B

��� �M���B����B

����

Really from unitary wave functions of the decuplet of baryon resonancesB���� and corresponding antidecuplet �B���� due to symmetry of indices it ispossible to constract an octet in a unique way� The result is�

M �M��

M�� �M�� �M�

M�� �M�� � �M

��

M�� �M�� � �M

��

Mass formula of this kind is named equidistant� It is valid with su�cientaccuracy� the step in mass scale being around ��� MeV� But in this case thepredicted state of strangeness �� and mass ����� � ��� � ��� MeV cannot

Page 31: Introduction to Group Theory and Its Representations in Particle Physics

be a resonance as the lighest two�particle state of strangeness �� would be�������K�� � with the mass ���� MeV& It means that if it exists it shouldbe a particle stable relative to strong interactions and should decay throughweak interactions in a cascade way loosing strangeness �� at each step�

This prediction is based entirely on the octet symmetry breaking of theLagrangian mass term of the baryon decuplet B�����

Particle with strangeness �� (� was found in � �� its mass turned outto be ����� � � �� � MeV coinciding exactly with SU�� prediction&

It was a triumph of unitary symmetry& The most of physicists believed init from � � on� �By the way the spin of the (� hyperon presumably equalto ��� has never been measured�

��

Page 32: Introduction to Group Theory and Its Representations in Particle Physics

��� Praparticles and hypothesis of quarks

Upon comparing isotopic and unitary symmetry of elementary particles onecan note that in the case of isotopic symmetry the lowest possible IR of thedimension � is often realized which has the basis �� �T � �� �T � alongthis representation� for example� N���K����K� transform� however at thesame time unitary multiplets of hadrons begin from the octet �analogue ofisotriplet in SU��I �

The problem arises whether in nature the lowest spinor representationsare realized� In other words whether more elementary particles exist thanhadrons discussed above�

For methodical reasons let us return into the times when people was livingin caves� used telegraph and vapor locomotives and thought that ��mesonswere bounded states of nucleons and antinucleons and try to understand inwhat way one can describe these states in isotopic space�

Let us make a product of spinors Na� �Nb� a� b � �� �� and then subrtractand add the trace �NcN

c� c � �� �� �� expanding in this way a product of twoirreducible representations �IR �two spinors into the sum of IR�s�

�Nb �Na � � �NbNa � �

� ab�NcN

c ��

� ab�NcN

c� ����

what corresponds to the expansion in terms of isospin ��� �

�� � � �� or �in

terms of IR dimensions �� � � � � �� In matrix form

��p� �n��pn

��

���pp� �nn� �np

�pn ���pp� �nn�

��

���pp � �nn� �

� ��pp � �nn�

�� ���

which we identify for the J�� state of nucleon and antinucleon spins andzero orbital angular momentum with the pion isotriplet and isosinglet ��

� �

� �p��� ��

�� � �p���

��

� �p��� �

� �p���

��

while for for the J�� state of nucleon and antinucleon spins and zero orbitalangular momentum with the ��meson isotriplet and isosinglet ��� �p

��� ��

�� � �p���

��

� �p��� �

� �p���

��

��

Page 33: Introduction to Group Theory and Its Representations in Particle Physics

This hypothesis was successfully used many times� For example� within thishypothesis the decay �� into two ��quanta was calculated via nucleon loop�

�� p

pp

The answer coincided exactly with experiment what was an astonishingachievement� Really� mass of two nucleons were so terribly larger than themass of the ���meson that it should be an enormous bounding energy betweennucleons� However the answer was obtained within assumption of quasi�freenucleons �� � � one of the achievements of Feynman diagram technique ap�plied to hadron decays�

With observation of hyperons number of fundamental baryons was in�creased suddenly� So �rst composite models arrived� Very close to model ofunitary symmetry was Sakata model with proton� neutron and ! hyperonas a fundamental triplet� But one was not able to put baryon octet intosuch model� However an idea to put something into triplet remains veryattractive�

��

Page 34: Introduction to Group Theory and Its Representations in Particle Physics

��� Quark model� Mesons in quark model�

Model of quarks was absolutely revolutionary� Gell�Mann and Zweig in� � assumed that there exist some praparticles transforming along spinorrepresentation of the dimension � of the group SU�� �and correspond�ingly antipraparticles transforming along conjugated spinor representationof the dimension � � and all the hadrons are formed from these funda�mental particles� These praparticles named quarks should be fermions �in

order to form existing baryons� and let it be fermions with JP � ��

q�� � �� �� �� q� � u� q� � d� q� � s� Note that because one needs at leastthree quarks in order to form baryon of spin ���� electric charge as well ashypercharge turn out to be DROBNYMI non�integer�&&& which presented ��years ago as open heresy and for many of us really unacceptable one�

Quarks should have the following quantum numbers�

Q I I� Y�S�B B

u ��� ��� ��� ��� ���d ���� ��� ���� ��� ���s ���� � � ���� ���

in order to assure the right quantum numbers of all � known baryons ofspin ��� ����� p�uud� n�ddu� ��uus� ��uds� ��dds� !�uds� ���ssu����ssd� In more details we shall discuss baryons a little further in anothersection in order to maintain the continuity of this talk�

First let us discuss meson states� We can try to form meson statesout of quarks in complete analogy with our previous discussion on nucleon�antinucleon states and Eqs���������

�q� � q� � ��q�q� � �

� �� �q�q

� ��

� �� �q�q

��

��u� �d� �s��B� u

ds

�CA �

�B� �uu �du �su�ud �dd �sd�us �ds �ss

�CA �

��

Page 35: Introduction to Group Theory and Its Representations in Particle Physics

�B� D�

�du �su�ud D� �sd�us �ds D�

�CA� �

���uu� �dd � �ssI� ����

where

D� � �uu� �

��qq �

���uu� �dd �

��uu� �dd � ��ss �

��

��q��q �

�p��q��q�

D� � �dd� �

��qq � ��

���uu� �dd �

��uu� �dd � ��ss �

� ����q��q �

�p��q��q�

D� � �ss� �

��qq � ��

��uu� �dd� ��ss � � �p

��q��q�

We see that the traceless matrix obtained here could be identi�ed with themeson octet JP � ���S�state� the quark structure of mesons being�

�� � ��ud� �� � � �du�

K� � ��us� K� � ��su�

�� ��p���uu� �dd�

K� � ��sd� �K� � � �ds�

� ��p��uu� �dd� ��ss�

In a similar way nonet of vector mesons Eq������ can be constructed� Butas they are in � we take straightforwardly the �rst expression in Eq�����with the spins of quarks forming J � � �always S�state of quarks�

��u� �d� �s� ��B� u

ds

�CA�

�B� �uu �du �su�ud �dd �sd�us �ds �ss

�CAJ��

�BB�

�p��� � �p

�� �� K��

�� � �p��� � �p

�� K��

K�� �K�� �

�CCA ����

��

Page 36: Introduction to Group Theory and Its Representations in Particle Physics

This construction shows immediately a particular structure of the � meson �it contains only strange quarks&&&

Immediately it becomes clear more than strange character of its decaychannels� Namely while � meson decays predominantly into � pions� the �meson practically does not decay in this way ��� � � � " although ener�getically it is very �pro�table� and� instead� decays into the pair of kaons ��� � �� �� " into the pair K�K� and ���� �� �� �" into the pair K�

LK�S�

This strange experimental fact becomes understandable if we expose quarkdiagrams at the simplest level�

s

�s

� u� d

K

�K

s

�s

u� d

u� d

u� d

Thus we have convinced ourselves that Okubo note on nonet was not onlycurious but also very profound�

��

Page 37: Introduction to Group Theory and Its Representations in Particle Physics

We see also that experimental data on mesons seem to support existenceof three quarks�

But is it possible to estimate e�ective masses of quarks� Let us assumethat ��meson is just made of two strange quarks� that is� ms � m��� � ���MeV� An e�ective mass of two light quarks let us estimate from nucleon massas mu � md � Mps � ��� MeV� These masses are called constituent ones�Now let us look how it works�

Mp�uu�d� �Mn�dd�u� � �� Mev �input�� ��exp

M��qq�s� � ���� Mev �� �� �exp

M��uds� � ����� Mev �� ����exp

M��ss�q� � ���� Mev �� ����exp

�There is one more very radical question�Whether quarks exist at all�From the very beginning this question has been the object of hot discus�

sions� Initially Gell�Mann seemed to consider quarks as some suitable math�ematical object for particle physics� Nowadays it is believed that quarks areas real as any other elementary particles� In more detail we discuss it a littlelater�

Page 38: Introduction to Group Theory and Its Representations in Particle Physics

Till Parisina�s fatal charmsAgain attracted every eye���

Lord Byron� �Parisina�

��

Page 39: Introduction to Group Theory and Its Representations in Particle Physics

��� Charm and its arrival in particle physics�

Thus� there exist three quarks&During �� years everybody was thinking in this way� But then something

unhappened happened�In november � �� on Brookehaven proton accelerator with max proton

energy �� GeV �USA and at the electron�positron rings SPEAR�SLAC�USAit was found a new particle$ J� vector meson decaying in pions in thehadron channel at surprisingly large mass ���� MeV and surprisingly longmean life and� correspondingly� small width at the level of ��� KeV althoughfor hadrons characteristic widths oscillate between ��� MeV for rho meson�� Mev for � and � MeV for � meson� Taking analogy with suppression of the��pion decay of the vector � meson which as assumed is mainly ��ss� statethe conclusion was done that the most simple solution would be hypothesisof existence of the �th quark with the new quantum number %charm%� Inthis case J������ is the ��cc vector state with hidden charm�

c

�c

J�

u� d

u� d

u� d

And what is the mass of charm quark� Let us make a bold assumptionthat as in the case of the ������ meson where mass of a meson is justdouble value of the ��constituent� strange quark mass ���� MeV � so called�constituent� masses of quarks u and d are around ��� Mev as we have seemass of the charm quark is just half of that of the J������ particle thatis around ���� MeV �more than ��� times proton mass&�

But introduction of a new quark is not so innocue� One assumes withthis hypothesis existence of the whole family of new particles as mesons with

��

Page 40: Introduction to Group Theory and Its Representations in Particle Physics

the charm with quark content ��uc� ��cu� � �dc� ��cd� with masses around�������������� MeV at least and also ��sc I ��cs with masses around�������������� MeV� As it is naturally to assume that charm is conservedin strong interaction as strangeness does� these mesons should decay due toweak interaction loosing their charm� For simplicity we assume that massesof these mesons are just sums of the corresponding quark masses�

Let us once more use analogy with the vector ������ meson main decaychannel of which is the decay K�� � �K�� � and the production of thismeson with the following decay due to this channel dominates processes atthe electron�positron rings at the total energy of ���� MeV�

If analogue of J������ with larger mass exists� namely� in the mass re�gion ��������������� MeV� in this case such meson should decay mostlyto pairs of charm mesons� But such vector meson ������ was really found atthe mass ���� MeV and the main decay channel of it is the decay to two newparticles$ pairs of charm mesons D����� �D����� or D������D������and the corresponding width is more than �� MeV&&&

c

�c

������ u� d

D

�D

New�found charm mesons decay as it was expected due to weak interactionwhat is seen from a characteristic mean life at the level of ����� � ����� s�

Unitary symmetry group for particle classi�cation grows to SU��� It isto note that it would be hardly possible to use it to construct mass formulaeas SU�� because masses are too di�erent in ��quark model� In any case aproblem needs a study�

Page 41: Introduction to Group Theory and Its Representations in Particle Physics

In the model of � quarks �� �avors as is said today mesons would trans�form along representations of the group SU�� contained in the direct productof the ��dimensional spinors � and ���� �� � � � �� � �� or

�q� � q� � ��q�q� � �

� �� �q�q

� ��

� �� �q�q

��

where now � �� � � �� �� �� ��

P �� �

�BBBB�

��� �� D����� D������ F��� D����� �p

��� � �p

�� �� K�

D������ �� � �p��� � �p

�� K�

F��� K� �K� � ��p�

�CCCCA ���

V �� �

�BBBB�

������ D������� D������� F �������D������� �p

��� � �p

�� �� K�

D������� �� � �p��� � �p

�� K��

F ������� K�� �K�� �

�CCCCA �����

��� The �fth quark� Beauty�

Per la bellezza delle donnemolti sono periti��� Eccl�Sacra Bibbia

Because of beauty of womenmany men have perished��� Eccl� Bible

Victorious trend for simplicity became even more clear after another im�portant discouvery� when in � �� a narrow vector resonance was found atthe mass around �� GeV� )��S� ���* ��Kev� everybody decided thatthere was nothing to think about� it should be just state of two new� �thquarks of the type �ss or �cc� This new quark was named b from beauty orbottom and was ascribed the mass around � GeV �M��� a half of the massof a new meson with the hidden %beauty% )��S� �� � ��bb�

��

Page 42: Introduction to Group Theory and Its Representations in Particle Physics

b

�b

)

u� d

u� d

u� d

Immediately searches for next excited states was begun which should sim�ilarly to ������ have had essentially wider width and decay into mesons withthe quantum number of %beauty%� Surely it was found� It was )��S�������* �� MeV� It decays almost fully to meson pair �BB and these mesons B havemass ���� � ����� � ��� MeVw and mean life around ��� ns�

b

�b

)������ u� d

B

�B

��

Page 43: Introduction to Group Theory and Its Representations in Particle Physics

�� Truth or top�

Truth is not to be sought in the good fortune of anyparticular conjuncture of time� which is uncertain� but in the light of natureand experience� which is eternal� Francis Bacon

But it was not all the story as to this time there were already leptonse�� �e���� ��� ��� �� �and antileptons and only � quarks� c� u with theelectric charge ����e and d� s� b with the electric charge ����e� Leavingapart theoretical foundations �though they are and serious we see that asymmetry between quarks and leptons and between the quarks of di�erentcharge is broken� Does it mean that there exist one more� th quark withthe new ��avour� named %truth% or %top% �

For �� years its existence was taking for granted by almost all the physi�cists although there were also many attempts to construct models withoutthe th quark� Only in � t�quark was discouvered at the mass close tothe nucleus of �Lu� �� mt ���GeV� We have up to now rather little tosay about it and particles containing t�quark but the assertion that t�quarkseems to be uncapable to form particles�

��

Page 44: Introduction to Group Theory and Its Representations in Particle Physics

�� Baryons in quark model

Up to now we have considered in some details mesonic states in frameworksof unitary symmetry model and quark model up to � quark �avours� Let usreturn now to SU�� and ���avour quark model with quarks q� � u� q� �d� q� � s and let us construct baryons in this model� One needs at leastthree quarks to form baryons so let us make a product of three ��spinors ofSU�� and search for octet in the expansion of the triple product of the IR�s������ � ���������� As it could be seen it is even two octet IR�s in thisproduct so we can proceed to construct baryon octet of quarks� Expandingof the product of three ��spinors into the sum of IR�s is more complicate thanfor the meson case� We should symmetrize and antisymmetrize all indices toget the result�

q� � q� � q� �

��q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q��

��q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q��

��q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q��

��q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q� � q�q�q� �

T f���g� T f��g� � T ����� � T ������

All indices go from � to �� Symmetrical tensor of the �rd rank has the dimen�sion NSSS

n � �n���n���n and for n�� NSSS� � ��� Antisymmetrical ten�

sor of the �rd rank has the dimension NAAAn � �n���n���n and for n��

NAAA� � �� Tensors of mixed symmetry of the dimension Nmix

n � n�n����only for n�� �Nmix

� �� could be rewritten in a more suitable form as T �� up�

on using absolutely antisymmetric tensor �or Levi�Civita tensor of the �rdrank ���� which transforms as the singlet IR of the group SU��� Really�

������� � u���u���u

������� �

���� � u��u��u

������� u��u

��u

������ � u��u

��u

������ � u��u

��u

������

�u��u��u

������ � u��u

��u

������ �

��

Page 45: Introduction to Group Theory and Its Representations in Particle Physics

� �u��u��u

�� � u��u

��u

�� � u��u

��u

�� � u��u

��u

�� � u��u

��u

�� � u��u

��u

������ �

� DetU���� � ����

as DetU � �� �The same for ���� etc�For example� partly antisymmetric ��dimensional tensor T ����� could be

reduced toB��jAsSU��� � ����T

������

and for the proton p � B�� we would have

p�jpiAsSU��� �

p�B�

� jAsSU��� � �judui� jduui� �����

Instead the baryon octet based on partly symmetric ��dimensional tensorT f��g� could be written in terms of quark wave functions as

pB�

� jSySU��� � ����fq�� q�gq��

For a proton B�� we have

pjpiSySU��� �

pB�

�jSySU��� � �juudi � judui � jduui� �����

In order to construct fully symmetric spin�unitary spin wave function ofthe octet baryons in terms of quarks of de�nite �avour and de�nite spin pro�jection we should cure only to obtain the overall functions being symmetricunder permutations of quarks of all the �avours and of all the spin projec�tions inside the given baryon� �As to the overall asymmetry of the fermionwave function we let colour degree of freedom to assure it� Multiplyingspin wave functions of Eqs����� ����� and unitary spin wave functions ofEqs����������� one gets�

p��B�

� j� �p���B�

� jAsSU��� � tjA �B�� jSySU��� � T j

S� �����

Taking the proton B�� as an example one has

p��jpi� � j � udu� uudi � j� � i �����

j� � uud� udu� duui � j� � � i �� j�u�u�d� � u�d�u� � d�u�u�

��u�d�u� � u�u�d� � d�u�u� � �d�u�u� � u�u�d� � u�d�u�i�

��

Page 46: Introduction to Group Theory and Its Representations in Particle Physics

We have used here that

juudi � j i � ju�u�d�i

and so on�Important note� One could safely use instead of the Eq������ the shorter

version but where one already cannot change the order of spinors at all&

pjpi �

pjB�

�i� � j�u�u�d� � u�d�u� � d�u�u�i� �����

The wave function of the isosinglet ! has another structure as one can seeoneself upon calcolating

�j!i� � �pjB�

�i� � jd�s�u� � s�d�u� � u�s�d� � s�u�d�i� ����

Instead decuplet of baryon resonances T f���g with JP � ��

�would be writ�

ten in the form of a so called weight diagram �it gives a convenient gra�cimage of the SU�� IR�s on the ��parameter plane which is characterized bybasic elements� in our case the �rd projection of isospin I� as an absciss andhypercharge Y as a ordinate which for decuplet has the form of a triangle�

#� #� #� #��

�� �� ��

��� ���

(�

and has the following quark content�

ddd udd uud uuu

sdd sud suu

ssd ssu

sss

In the SU�� group all the weight diagram are either hexagones or trianglesand often are convevient in applications�

��

Page 47: Introduction to Group Theory and Its Representations in Particle Physics

For the octet the weight diagram is a hexagone with the � elements incenter�

n p

� � ��! �

�� ��

or in terms of quark content�

udd uud

sdd sud suu

ssd ssu

The discouvery of �charm� put a problem of searching of charm baryons�And indeed they were found& Now we already know !�

c ������ �� �� m�w�!�c ����� ��� �m�w� ������

c ������ ����c ����� Let us try to classify themalong the IR�s of groups SU�� and SU��� Now we make a product of three��spinors q�� � �� �� �� �� Tensor structure is the same as for SU��� Butdimensions of the IR�s are certainly others� ����� � �������������� Asymmetrical tensor of the �rd rank of the dimension NSSS

n � �n���n���nin SU�� has the dimension �� and is denoted usually as ��� Reduction ofthe IR �� in IR�s of the group SU�� has the form �� � ��� ��� ������

There is an easy way to obtain the reduction in terms of the correspondingdimensions� Really� as it is almost obvious� �� spinor of SU�� reduces toSU�� IR�s as � � �� � ��� So the product Eqs�� ���� ���� for n � �� � � � �� � would reduce as

��� � ��� ��� � �� � �� � �� � �� � �� � �� � �� � �� � �� �

� � ��� � �� � �� � ���

As antisymmetric tensor of the �nd rank T ���� of the dimension NAAn �

n�n � �� equal to � at n�� and denoted by us as should be equal to itsconjugate T���� due to the absolutely anisymmetric tensor ����� and so it hasthe following SU�� content� � �� � ���� The remaining symmetric tensorof the �nd rank T f��g of the dimension NSS

n � n�n��� equal to � at n��and denoted by us as �� would have then SU�� content �� � � ��� � ���The next step would be to construct reduction to SU�� for products �� �

Page 48: Introduction to Group Theory and Its Representations in Particle Physics

��� � �� �see Eq�� � �� at n � �� and �� � � � ��� � �� �Eq�� ���� atn � ��� Their sum would give us the answer for �� � � ��

� � � ��� � ���� ��� � �� � �� � ��� �� �� � �� � ��� � �� � �� � ��� �� ���� � �� � ��� � � � ��� � ���

As ��� is now known it easy to obtain ���

�� � �� � � � ��� � ��� � � � �� � ���Tensor calculus with q� � �a q

a � � q would give the same results�

So �� contains as a part ���plet of baryon resonances ���� As to thecharm baryons ��� there are two candidate� c����� and �c���� �JP

has not been measured� ��� is the quark model prediction Note by theway that JP of the (� which manifested triumph of SU�� has not beenmeasured since � � that is for more than �� years& Nevertheless everybodytakes for granted that its spin�parity is ����

Instead baryons ��� enter ����plet described by traceless tensor of the�rd rank of mixed symmetry B�

���� antisymmetric in two indices in squarebrackets� p

B����� � �����fq�� q�gq�

� �� �� � � � �� �� �� �� This ����plet as we have shown is reduced to the sumof the SU�� IR�s as ��� � ������������ It is convenient to choose reductionalong the multiplets with the de�nite value of charm� Into ��plet with C � �the usual baryon octet of the quarks u�d�s ���� is naturally placed� Thetriplet should contain not yet discouvered in a de�nite way doubly�charmedbaryons�

��cc ���cc

(�cc

with the quark contentccd ccu

ccs�

and� for example� the wave function of ��cc in terms of quarks readspj��cci� � j�c�c�u� � c�uc� � u�c�c�i� �����

��

Page 49: Introduction to Group Theory and Its Representations in Particle Physics

Antitriplet contains already discouvered baryons with s � �

!�c

��c ��c

with the quark contentudc

dsc usc�

and� for example� the wave function of ��c in terms of quarks reads

�j��c i� � js�c�u� � c�s�u� � u�c�s� � c�u�s�i� �����

Sextet contains discouvered baryons with s � �

�c �

c ��c

���c ���c

(�c

�Note that their quantum numbers are not yet measured& with the quarkcontent

ddc udc uuc

dsc usc

ssc

and� for example� the wave function of ���c in terms of quarks reads

p��j���c i� � j�u�s�c� � �c�u�d� ����

�u�c�s� � c�u�s� � s�c�u� � c�s�u�i�Note also that in the SU�� group absolutely antisymmetric tensor of the �thrank ������ �� �� � �� transforms as singlet IR and because of that antisym�metric tensor of the �th rank in SU�� T ������ � �� � � �� �� �� �� transformsnot as a singlet IR as in SU�� �what is proved in SU�� by reduction withthe tensor ����� �� � � � �� �� � but instead along the conjugated spinor rep�resentation �� �which also can be proved by the reduction of it with the tensor������ �� �� � � � �� �� �� ��

��

Page 50: Introduction to Group Theory and Its Representations in Particle Physics

We return here to the problem of reduction of the IR of some group intothe IR�s of minor group or� in particular� to IR�s of a production of two minorgroups� Well�known example is given by the group SU� � SU���SU��Swhich in nonrelativistic case has uni�ed unitary model group SU�� and spingroup SU��S � In the framework of SU� quarks belong to the spinor ofdimension � which in the space of SU�� � SU��S could be written as� � ��� � where the �rst symbol in brackets means ��spinor of SU�� whilethe �nd symbol just states for ��dimensional spinor of SU��� Let us try nowto form a product of ��spinor and corresponding ��antispinor and reduce itto IR�s of the product SU��� SU��S �

�� � � � ��� � �� � ���� �� ��� � � ��� � �� �� � �� �� � �� � � � � ���� � � ��� � � ��� �� � ��� ��

that is� in ��� there are exactly eight mesons of spin zero and ���� vectormesons� while there is also zero spin meson as a SU� singlet� This resultsuits nicely experimental observations for light �of quarks u� d� s� mesons� Inorder to proceed further we form product of two ��spinors �rst according toour formulae� just dividing the product into symmetric and antisymmetricIR�s of the rank ��

� � � � ��� � ��� � ��� � � ��� � � �� � �� �� � �� �� � ���� � � � �

f��� � � ���� �g�� � ���� � � ����� �����dimensions of symmetric tensors of the �nd rank being n�n � �� while ofthose antisymmetric n�n���� Now we go to product ����� which shouldresults as already we have seen in the sum of two IR�s of the �rd rank � oneof them being antisymmetric of the �rd rank �NAAA � n�n�� �n�� andthe other being of mixed symmetry �Nmix � n�n� � ���

��� � � � ��� � ��� � ���� � � ����� ��� ��� � �� �� �� � � ���� �� � � � �

� ���� � � ��� ���� � ���� � � ���� � � ��� � � ��� �� ��

Instead the product ��� � � should result as already we have seen into thesum of two IR�s of the �rd rank � one of them being symmetric of the �rd

Page 51: Introduction to Group Theory and Its Representations in Particle Physics

rank �NSSS � n�n���n�� and the other being again of mixed symmetry�Nmix � n�n� � �� and we used previous result to extract the reduction ofthe ���plet�

��� � � � �� � ��� � f��� � � ����� �g � ��� � �

� �� � ��� �� � � ���� � �� � � � �� f���� � � ���� �g�� � ���� � � ��� �����

����� � � ���� � � ��� � � ��� �� ��

We now see eminent result of SU� that is that in one IR �� there areoctet of baryons of spin �� and decuplet of baryonic resonances of spin���� Note that quark model with all the masses� magnetons etc equal justreproduces SU� model as it should be� In some sense ��quark model givesthe possibility of calculations alternative to tensor calculus of SU� group�

Some words also on reduction of IR of some group to IR�s of the sumof minor groups� We have seen an example of reduction of the IR of SU��into those of SU��� For future purposes let us consider some examples of thereduction of the IR�s of SU�� into those of the direct sum SU���SU���Here we just write ��spinor of SU�� as a direct sum� �� � ���� � � ���� ��Forming the product of two ��spinors we get�

�� � �� � ��� � ��� � ���� � � ���� �� ���� � � ���� � �

� ��� � ��� � � ���� � � ���� � � ���� �� � �f�� � � ���� � � ��� �g�� � ����� � � ����� � � ��� �����

that is� important for SU�� group IR�s of dimensions � and � have thefollowing reduction to the sum SU���SU���

�� � ���� � � ���� ��

��� � ���� � � ����� � � ��� ��

In this case it is rather easy an exercise to proceed also with tensor calculus�

��

Page 52: Introduction to Group Theory and Its Representations in Particle Physics

Chapter �

Currents in unitary symmetry

and quark models

��� Electromagnetic current in the modelsof unitary symmetry and of quarks

����� On magnetic moments of baryons

Main properties of electromagnetic interaction are assumed to be known�Electromagnetic current of baryons as well as of quarks can be written

in a similar way to electrons in the theory with the Dirac equation� only weshould account in some way for the non�point�like structure of baryons intro�ducing one more Lorentz structure and two form factors� This current canbe deduced from the interaction Lagrangian of the baryon with the electriccharge e and described by a spinor uB�p and electromagnetic �eld A��xcharacterized by its polarization vector ���

e

�MB��� q�

M�B

�uB�p��P�GE�q

��

�i����P �q����GM �q��uB�p��

� �

�+p�mBuB � �� q � p� � p�� P � p� � p�

�i��� � ���� �� ��

��

Page 53: Introduction to Group Theory and Its Representations in Particle Physics

GE being electric form factor� GE�� � ��while GM is a magnetic form factorand GM �� � �B is a total magnetic moment of the baryon in terms of propermagnetons eh�mBc� Transition to the model of unitary symmetry meansthat instead of the spinor uB written for every baryon one should now putthe whole octet B�

� �What are properties of electromagnetic current in the unitary symmetry�

Let us once more remind Gell�Mann$Nishijima relation between the particlecharge Q� �rd component of the isospin I� and hypercharge Y �

Q � I� ��

�Y�

As Q is just the integral over �th component of electromagnetic current� itmeans that the electromagnetic current is just a superposition of the �rd com�ponent of isovector current and of the hypercharge current which is isoscalar�

So it can be related to the component J��� of the octet of vector currents

J���� �More or less in the same way as mass breaking was described by the�� component of the baryonic current but without specifying its space�timeproperties� The part of the current related to the electric charge shouldassure right values of the baryon charges� Omitting for the moment space�time indices we write

eJ�� � e� �B�

�B�� � �B�

�B�� �

Here p � B�� and so on� are octet baryons with JP � �

�� One can see that

all the charges of baryons are reproduced �But the part treating magnetic moments should not coincide in form with

that for their charges� as there are anomal magnetic moments in additionto those normal ones� The total magnetic moment is a sum of these twomagnetic moments for charged baryons and just equal to anomal one for theneutral baryons�

While constructing baryon current suitable for description of the magneticmoments as a product of baryon and antibaryon octets we use the fact thatthere are possible as we already know two di�erent tensor structures �whichre�ects existence of two octets in the expansion ��� � ����������������

J�� � F � �B�

�B�� � �B�

�B�� �D� �B�

�B�� � �B�

�B���

� ��D

�B��B

�� �

��

Page 54: Introduction to Group Theory and Its Representations in Particle Physics

and trace of the current should be zero� J�� � �� � �� �� � � �� �� ��� Then

electromagnetic current related to magnetic moments � we omit space�timeindices here will have the form

J�� � F � �B�

�B�� � �B�

�B�� �D� �B�

�B�� � �B�

�B�� �

��B��B

���

wherefrom magnetic moments of the octet baryons read�

��p � F ��

�D� �� � � F �

�D�

��n � ���D� �� � � �F � �

�D�

���� � ���D� �� � �

�D�

���� � �F � �

�D ��!� � ��

�D ����

�Remind that here B�� � p� Agreement with experiment in terms of only

F and D constants proves to be rather poor� But many modern modeldeveloped for description of the baryon magnetic moments contain thesecontributions as leading ones to which there are added minor correctionsoften in the frameworks of very exquisite theories�

Let us put here experimental values of the measured magnetic momentsin nucleon magnetons�

��p � ��� � �� � � �� ��� � ��������n � ��� ��� �� � � ����� ������

���� � ������ � ���������� � ������ � ������ ��!� � ������ ���� ����

And in what way could we construct electromagnetic current of quarks� Itis readily written from Dirac electron current�

J el�m� �

��t��t�

��c��c� �

��u��u�

����d��d � �

��s��s�� �

��b��b�

��

Page 55: Introduction to Group Theory and Its Representations in Particle Physics

and we have put in square brackets electromagnetic current of the ���avormodel�

And how could we resolve problem of the baryon magnetic moments inthe framework of the quark model�

For this purpose we need explicit form of the baryon wave functions withthe given �rd projection of the spin in terms of quark wave functions also withde�nite �rd projections of the spin� These wave functions have been givenin previous lectures� We assume that in the nonrelativistic limit magneticmoment of the baryon would be a sum of magnetic moments of quarks� whileoperator of the magnetic moment of the quark q would be �q�qz �quark onwhich acts operator of the magnetic moment is denoted by �� Magneticmoment of proton is obtained as �here q� � q�� q� � q�� q � u� d� s�

�p �X

q�u�d� p�j+�q�qz jp� ��

� �u�u�d� � u�d�u� � d�u�u�j+�q�qz j�u�u�d� � u�d�u� � d�u�u� ��

Xq�u�d

� �u�u�d� � u�d�u� � d�u�u�j+�qj�u��u�d� � �u�u��d����u�u�d�� � u��d�u� � u�d

��u� � u�d�u

���

�d��u�u� � d�u��u� � d�u�u

�� ��

���u � ��u � ��d � �u � �d � �u � �d � �u � �u �

���u � ��d � �

��u � �

��d�

where we have used an assumption that two of three quarks are always spec�tators so that

� u�u�d�j+�qju��u�d� ���� u�j+�qju�� �� �u etc�

Corresponding quark diagrams could be written as �we put only some ofthem� the rest could be written in the straightforward manner�

��

Page 56: Introduction to Group Theory and Its Representations in Particle Physics

u�

u�

d�

u�

u�

d�

�u�

u�

d�

u�

u�

d�

�d�

u�

u�

d�

u�

u�

In a similar way we can calculate in NRQMmagnetic moment of neutron�

�n �X

q�u�d� n�j�q�qzjn� ��

� �d�d�u� � d�u�d� � u�d�d�j�q�qz j�d�d�u� � d�u�d� � u�d�d� ��

��d � �

��u�

magnetic moment of ��

�� � �X

q�u�s� �

� j�q�qzj �� ��

� �u�u�s� � u�s�u� � s�u�u�j�q�qz j�u�u�s� � u�s�u� � s�u�u� ��

��u � �

��s�

magnetic moment of � �

�� � �X

q�d�s� �� j�q�qz j �� ��

� �d�d�s� � d�s�d� � s�d�d�j�q�qz j�d�d�s� � d�s�d� � s�d�d� ��

��d � �

��s�

magnetic moment of �� �

���� �X

q�u�s� ���j�q�qz j��� ��

��

Page 57: Introduction to Group Theory and Its Representations in Particle Physics

� �s�s�u� � s�u�s� � u�s�s�j�q�qz j�s�s�u� � s�u�s� � u�s�s� ��

��s � �

��u�

magnetic moment of �� �

���� �X

q�d�s� ��� j�q�qz j��� ��

� �s�s�d� � s�d�s� � d�s�s�j�q�qz j�s�s�d� � s�d�s� � d�s�s� ��

��s � �

��d�

and �nally magnetic moment of ! �which we write in some details as it hasthe wave function of another type�

�� �X

q�u�d�s� !�j�q�qzj!� ��

�� u�s�d� � s�u�d� � d�s�u� � s�d�u�j�q�qzjju�s�d� � s�u�d� � d�s�u� � s�d�u� ��

�� u�s�d� � s�u�d� � d�s�u� � s�d�u�j�qju��s�d� � u�s

��d� � u�s�d

���

�s��u�d� � s�u��d� � s�u�d

��

�d��s�u� � d�s��u� � d�s�u

�� � s��d�u� � s�d

��u� � s�d�u

�� ��

���u � �s � �d � �s � �u � �d � �d � �s � �u � �s � �d � �u � �s�

In terms of these three quark magnetons it is possible to adjust magneticmoments of baryons something like up to ��" accuracy�

����� Radiative decays of vector mesons

Let us look now at radiative decays of vector meson V � P � ��

Page 58: Introduction to Group Theory and Its Representations in Particle Physics

V P

Let us write in the framework of unitary symmetry model an electromagneticcurrent describing radiative decays of vector mesons as the ���component ofthe octet made from the product of the octet of preusoscalar mesons andnonet of vector mesons�

J�� � P �

� V�� � P �

� V�� �

�SpPV �

�P �� V

�� � �P

�� V

�� � P �

� V�� � �P

�� V

�� � P �

� V�� �

�SpPV �

���p��� �

��

�p��� �

�p��� �

������ � ���� � ���� � ���

We take interaction Lagrangian describing these transitions in the form

L � gV�P�J���A��

Performing product of two matrix and extracting �� component we obtainfor amplitudes of radiative decays in unitary symmetry�

M���� ��� � ���p�

�p�� �

�gV�P� �

�gV�P� �

M��� � ��� � ��� �

�gV�P� �

�gV�P��

M��� � ��� � ��p�

�p�gV�P� � gV�P� �

As masses of � and �� mesons are close to each other we neglect a di�erencein phase space and obtain the following widths of the radiative decays�

*��� � ��� � *��� � ��� � � ��

��

Page 59: Introduction to Group Theory and Its Representations in Particle Physics

while experiment yields�

���� � ��Kev � ���� � ��Kev

Radiative decay of � meson proves to be prohibited in unitary symmetry�

*��� ��� � ��

The experiment shows very strong suppression of this decay� �� �� Kev�

����� Leptonic decays of vector mesons V � l�l�

Let us now construct in the framework of unitary symmetry model an elec�tromagnetic current describing leptonic decays of vector mesons V � l�l��

V l

�l

In sight of previous discussion it is easily to see that it would be su�cient toextract an octet in the nonet of vector mesons and then take ���component�

J�� � gV �ll�V

�� �

�V �� � gV �ll��

�p��� �

�p��� �

��p�� � �� �

� gV �ll��p��� �

�p�� � �

��

Ratio of leptonic widths is predicted in unitary symmetry as

*��� � e�e� � *��� � e�e� � *��� � e�e� � � � � ��

which agrees well with the experimental data�

� �Kev � �� Kev � �� �Kev �

Let us perform calculations of these leptonic decays in quark model uponusing quark wave functions of vector mesons For �� � �p

���uu� �dd

*��� � e� � e� �

��

Page 60: Introduction to Group Theory and Its Representations in Particle Physics

ju

�u

� e�

e��

d

�d

� e�

e�j� � �

����� ���

��� � �

��

For �� � �p���uu� �dd

*��� � e� � e� �

ju

�u

� e�

e��

d

�d

� e�

e�j� � �

� ��� � ���

��� � �

���

For � � ��ss

*��� � e� � e� �

js

�s

� e�

e�j� � ���

�� � �

����

It is seen that predictions of unitary symmetry model of the quark modelcoincide with each other and agree with the experimental data�

Page 61: Introduction to Group Theory and Its Representations in Particle Physics

��� Photon as a gauge �eld

Up to now we have treated photon at the same level as other particles that asa boson of spin � and mass zero� But it turns out that existence of the photoncan be thought as e�ect of local gauge invariance of Lagrangian describingfree �eld of a charged fermion of spin ���� let it be electron� Free motionof electron is ruled by Dirac equation ����� �me�e�x � � which could beobtained from Lagrangian

L� � ��e�x�����e�x �me��e�x�e�x�

This Lagrangian is invariant under gauge transformation

��e�x � ei��e�x�

being arbitrary real phase� Let us demand invariance of this Lagrangianunder similar but local transformation� that is when is a function of x�

��e�x � ei��x��e�x�

It is easily seen that L� is not invariant under such local gauge transformation�

L�� � ���e�x������e�x �me

���e�x��e�x �

��e�x�����e�x � i��x

�x���e�x���e�x�me

��e�x�e�x�

In order to cancel the term violating gauge invariance let us introduce somevector �eld A� with its own gauge transformation

A�� � A� � �

e

��x

�x��

and introduce also an interaction of it with electron through Lagrangian

ie ��e�x���e�xA��

e being coupling constant �or constant of interaction� But we cannot intro�duce a mass of this �eld as obviously mass term of a boson �eld in Lagrangianm�A�A

� is not invariant under chosen gauge transformation for the vector

Page 62: Introduction to Group Theory and Its Representations in Particle Physics

�eld A�� Let us now identify the �eld A� with the electromagnetic �eldand write the �nal expression of the Lagrangian invariant under local gaugetransformations of the Abelian group U��

L� � ��e�x�����e�x � ie ��e�x���e�xA�

�me��e�x�e�x� �

�F��F

���

where F�� describe free electromagnetic �eld

F�� � ��A� � ��A��

satisfying Maxwell equations

��F�� � �� ��A� � ��

The ��vector potential of the electromagnetic �eld A� � ��� �A is re�

lated to measured on experiment magnetic �H and electric �elds �E by therelations

�H � rot �A� �E � ��c

� �A

�t� grad ��

while tensor of the electromagnetic �eld F�� is written in terms of the �elds�E and �H as

F�� � ��A� � ��A� �

�BBB�

� Ex Ex Ex

�Ex � Hz �Hy

�Ey �Hz � Hx

�Ez Hy �Hx �

�CCCA �

Maxwell equations �in vacuum in the presence of charges and currents read

rot �E � ��c

� �H

�t� div �H � ��

rot �H ��

c

� �E

�t���

c�j� div �E � ����

where � is the density of electrical charge and j is the electric current den�sity� In the ��dimensional formalism Maxwell equations �in vacuum in thepresence of charges and currents can be written as

��F�� � j�� j� � ��� �j�

�������F�� � �� ��A� � ��

Page 63: Introduction to Group Theory and Its Representations in Particle Physics

��� � meson as a gauge �eld

In � �� that is more than half�century ago Yang and Mills decided to tryto obtain also � meson as a gauge �eld� The � mesons were only recentlydiscouvered and seemed to serve ideally as quants of strong interaction�

Similar to photon case let us consider Lagrangian of the free nucleon �eldwhere nucleon is just isospinor of the group SU��I of isotopic transforma�tions with two components� that is proton � chosen as a state with I������and neutron � chosen as a state with I�������

L� � ��N�x�����N�x �mN��N�x�N�x�

This Lagrangian is invariant under global gauge transformations in isotopicspace

��N �x � ei�����N�x�

where � � ��� �� � are three arbitrary real phases� Let us demand nowinvariance of the Lagrangian under similar but local gauge transformation inisotopic space when � is a function of x�

��N�x � ei���x����N�x�

However as in the previous case The L� is not invariant under such localgauge transformation�

L�� � ���N�x������N�x �mN

���N�x��N�x �

��N�x�x����N�x�

�i ��N�x�������x

�x��N�x �mN

��N�x�N�x�

In order to cancel term breaking gauge invariance let us introduce an isotripletof vector �elds ��� with the gauge transformation

�� ���� � U�����Uy � �

gNN�

�U

�x�U y

where U � ei���x��� � An interaction of this isovector vector �eld could be givenby Lagrangian

gNN���N�x��������N�x�

Page 64: Introduction to Group Theory and Its Representations in Particle Physics

gNN� being coupling constant of nucleons to � mesons� Exactly as in theprevious case we cannot introduce a mass for this �eld as in a obvious waymass term in the Lagrangian m������

� is not invariant under the chosen gaugetransformation of the �eld ���� Finally let us write Lagrangian invariant underthe local gauge transformations of the non�abelian group SU���

L � ��N�x�����N�x �mN��N�x�N�x�

gNN���N�x���� ����N �x� �

��F �� �F���

�F�� describing a free massless isovector �eld �� It is invariant under gauge

transformations U y �F ���U � �F��� Let us write a tensor of free � meson �eld

�� �F�� � �kFk�� � ,F��� ���i� �j� � �i�ijk�k� i� j� k � �� �� ��

�F k�� � ����

k� � ���

k�� �gNN�i�

kij�i��j�

or,F�� � ��� ,�� � ��,��� gNN��,��� ,�� �

and see that this expression transforms in a covariant way while gauge trans�formation is performed for the �eld ��

U y��� ,��� � ��,���U �

��� ,�� � ��,�� � �Uy��U� ,���� �U y��U� ,�� ��

U y�,���� ,����U � �,��� ,�� � �

gNN��U y��U� ,���� �

gNN��U y��U� ,����

Finally�U y �F �

��U � U y��� ,��� � ��,��� � gNN��,�

��� ,�

���U �

�� ,�� � ��,�� � gNN��,��� ,�� � � �F���

It is important to know the particular characteristic of the non�abelian vector�eld � it proves to be autointeracting� that is� in the term ����j�F �� j� of thelagrangian new terms appear which are not bilinear in �eld � �as is the casefor the abelian electromagnetic �eld� but trilinear and even quadrilinear inthe �eld �� namely� �������� and ����

���

Page 65: Introduction to Group Theory and Its Representations in Particle Physics

Later this circumstance would prove to be decisive for construction of thenon�abelian theory of strong interactions� that is of the quantum chromody�namics �QCD

The Yang�Mills formalism was generalized to SU��f where the require�ment of the local gauge�invariance of the Lagrangian describing octet baryonsled to appearance of eight massless vector bosons with the quantum numbersof vector meson octet �� known to us�

Unfortunately along this way it proved to be impossible to constructtheory of strong interactions with the vector meson as quanta of the strong�eld� But it was developed a formalism which make it possible to solve thisproblem not in the space of three �avors with the gauge group SU��f butinstead in the space of colors with the gauge group SU��C where quanta ofthe strong �eld are just massless vector bosons having new quantum number�color� named gluons�

Page 66: Introduction to Group Theory and Its Representations in Particle Physics

��� Vector and axial vector currents in uni

tary symmetry and quark model

����� General remarks on weak interaction

Now we consider application of the unitary symmetry model and quark modelto the description of weak processes between elementary particles�

Several words on weak interaction� As is well known muons� neutronsand ! hyperons decay due to weak interaction� We have mentioned muon asleptons �at least nowadays are pointlike or structureless particle due to allknow experiments and coupling constants of them with quanta of di�erent�elds act� say� in pure form not obscured by the particle structure as itis in the case of hadrons� The decay of muon to electron and two neutrinos�� � e����e��� is characterized by Fermi constant GF � ����m��

p �Neutrondecay into proton � electron and antineutrino called usually neutron �� decayis characterized practically by the same coupling constant� However ��decays of the ! hyperon either ! � p � e� � ��e or ! � p � �� � ��� arecharacterized by noticibly smaller coupling constant� The same proved tobe true for decays of nonstrange pion and strange K meson� Does it meanthat weak interaction is not universal in di�erence from electromagnetic one�It may be so� But maybe it is possible to save universality� It proved tobe possible and it was done more than �� years ago by Nicola Cabibbo byintroduction of the angle which naturally bears his name� Cabibbo angle �C�

For weak decays of hadrons it is su�cient to assume that weak interactionswithout change of strangeness are de�ned not by Fermi constant GF butinstead by GF cos�C while those with the change of strangeness are de�nedby the coupling constant GF sin�C� This hypothesis has been brilliantlycon�rmed during analysis of many weak decays of mesons and barons eitherconserving or violating strangeness� The value of Cabibbo angle is � ��o�

But what is a possible formalism to describe weak interaction� Fermi hasanswered this question half century ago�

We already know that electromagnetic interaction can be given by inter�action Lagrangian of the type current� �eld�

L � eJ��xA��x � e ���x����xA

��x�

Note that� say� electron or muon scattering on electron is the process of the

Page 67: Introduction to Group Theory and Its Representations in Particle Physics

�nd order in e� E�ectively it is possible to write it the form current�current�

L��� �e�

q�J�J

��

where q� is the square of the momentum transfer� It has turned out that weakdecays also are described by the e�ective Lagrangian of the form current�currentbut this has been been taken as the �st order expansion term in Fermi con�stant�

LW �GFp�J y�J

� �Hermitian Conjugation�

The weak current should have the form

J��x � �����xO����x � ���e�xO��e�x�

��p�xO��n�xcos�C � ��p�xO����xsin�C�

The isotopic quantum numbers of the current describing neutron � decay issimilar to that of the �� meson while the current describing � decay of ! issimilar to K� meson� Note that weak currents are charged& Since � � itis known that weak interaction does not conserve parity� This is one of thefundamental properties of weak interaction� The structure of the operatorO� for the charged weak currents has been established from analysis of themany decay angular distributions and turns out to be a linear combinationof the vector and axial�vector O� � ���� � �� what named often as �V �Aversion of Fermi theory of weak interaction�

Note that axial�vector couplings� those at ���� generally speaking arerenormalized �attain some factor not equal to � which is hardly calculableeven nowadays though there is a plenty of theories while there is no needto renormalize vector currents due to the conservation of vector current���V� � ��

But dimensional Fermi constant as could be seen comparing it with theelectromagnetic process in the �nd order could be a re�ection of existence ofvery heavy W boson �vector intermediate boson in old terminology emittedby leptons and hadrons like photon� In this case the observed processes ofdecays of muon� neutron� hyperons should be processes of the �nd order inthe dimensionless weak coupling constant gW while GF � g�W�M

�W � q��

and one can safely neglect q��

Page 68: Introduction to Group Theory and Its Representations in Particle Physics

Thus elementary act of interaction with the weak �eld might be writtennot in terms of the product current� current but instead using as a modelelectromagnetic interaction�

L �gWp��J�W

�� � J y�W

�� �

����� Weak currents in unitary symmetry model

And what are properties of weak interaction in unitary symmetry model� Asweak currents are charged they could be related to components J�

� and J�� of

the current octet J�� � Comparing octet of the weak currents with the octet of

mesons one can see that the chosen components of the current correspondsexactly � in unitary structure� not in space�time one to �� and K� mesons�Vector current is conserved as does electromagnetic current and therefore hasthe same space�time structure� Then

V ���cos�C � V �

��sin�C � � �B�� ��B

�� � �B�

���B�� cos�C

�� �B�� ��B

�� � �B�

���B�� sin�C

Here p � B�� etc are members of the baryon octet matrix J

P � ��

�� But for the

part of currents violating parity similarly to the case of magnetic momentsof baryons there are possible two tensor structures and unitary axial�vectorcurrent yields

�A��� � F � �B�

� ����B�� � �B�

�����B�� �

D� �B�� ����B

�� � �B�

�����B��

Similar form is true for A����

A�� � A�

��cos�C �A���sin�C

Finally for the neutron � decay one has

GAjn�p�e���e � �F �D�

GApn � �f � dcos�C � GA

���� � ��F �Dcos�C �

GAp� �

�p��F �Dsin�C � GA

��� ��p��F �Dsin�C �

Page 69: Introduction to Group Theory and Its Representations in Particle Physics

GAn�� � ��F �Dsin�C � GA

���� � �F �Dsin�C�

GA��� �

s�

�Dcos�C �

At F � ��� D � � �SU� � SU��f � SU��J GAjn�p�e���e ���� Ex�

perimental analysis of all the known leptonic decays of hyperons leads toF � ����� � �� ���� D � ����� � ������ which reproduces the experimentalresult jGAGV jn�p�e���e j � ���� � ������

����� Weak currents in a quark model

Let us now construct quark charged weak currents� When neutron decaysinto proton �and a pair of leptons in the quark language it means that oneof the d quarks of neutron transforms into u quark of proton while remainingtwo quarks could be seen as spectators� The corresponding weak currentyields

jd� � �u���� � ��dcos�C �

For the ! hyperon this discussion is also valid only here it is s quark of the !transforms into u quark of proton while remaining two quarks could be seenas spectators� The corresponding weak current yields

js� � �u���� � ��ssin�C�

Logically it comes the form

j� � jd� � js� � �u���� � ��dC �

where dC � dcos�C � ssin�C�Thus in the quark sector �as it says now left�handed� helicity doublet

has arrived�

�udC

�L

� �� � ��

�udC

�� In the leptonic sector of weak

interaction it is possible to put into correspondence with this doublet follow�

ing left�handed�helicity doublets�

��ee�

�L

and

�����

�L

� �Thus the whole

group theory science could be reduced to the group SU�� � Then it comesnaturally an idea about existence of weak isotopic triplet of W bosons whichinteracts in a weak way with this weak isodoublet�

L � gW�j� �W� �H�C�

Page 70: Introduction to Group Theory and Its Representations in Particle Physics

� Let us remain for a moment open a problem of renormalizibility of suchtheory with massive vector bosons�

Before we �nish with the charged currents let us calculate constant GA

or more exact the ratio GAGV for the neutron � decay in quark model�Nonrelativistic limit for the operator ������ is �z�� where �� transformsone of the d quarks of the neutron into u quark�

GnpA �� p�j��q �qzjn� ��

��

� �u�u�d� � u�d�u� � d�u�u�j���qzj�d�d�u� � d�u�d� � u�d�d� ��

� �u�u�d� � u�d�u� � d�u�u�j�u��d�u� � �d�u��u��u��u�d� � d�u�u

�� � u�u

��d� � �u�d�u

�� ��

���� �� � � �� �� � � ��

��exp�� ���� � �����

Quark model result coincides with that of the exact SU� but disaccord withthe experimental data that is why in calculations as a rule unitary model ofSU��f is used�

Page 71: Introduction to Group Theory and Its Representations in Particle Physics

Chapter �

Introduction to

Salam�Weinberg�Glashow

model

��� Neutral weak currents

We are now su�ciently safe with the charged currents �and old version is quitegood but hypothesis about weak isotriplet of W leads to neutral currents inquark sector�

jneutr��ud� ��

���uO�u� �dCO�dC �

��

���uO�u� �dO�d�cos�C

� � �sO�s�sin�C��

�dO�scos�Csin�C � �sO�dcos�Csin�C�

and� correspondingly� in lepton sector�

jneutr��lept�� ��

����eO��e � �eO�e�

��

�����O��� � ��O���

�We do not write here explicitly weak neutral operator O� as it can diverge�nally from the usual charged one ���� � ��� Up to the moment when

��

Page 72: Introduction to Group Theory and Its Representations in Particle Physics

neutral currents were discouvered experimentally presence of these currentsin theory was neither very intriguing nor very disturbing�

But when in � �� one of the most important events in physics of weakinteraction of the �nd half of the XX century happened $ neutral currentswere discouvered in the interactions of neutrino beams of the CERN machinewith the matter� it was become immediately clear the contradiction to solve�although neutral currents interacted with neutral weak boson �to be estab�lished yet in those years with more or less the same coupling as chargedcurrents did with the charged W bosons� there were no neutral strange weakcurrents which were not much suppressed by Cabibbo angle� Even more�neutral currents written above opened channel of decay of neutral K mesonsinto ���� pair with approximately the same coupling as that of the main de�cay mode of the charged K� meson � into lepton pair ������ Experimentallyit is suppressed by � orders of magnitude&&&

*�K�s � ����*�K�

s � all � ���� ���

Once more have we obtained serious troubles with the model of weak inter�action&�

In what way� clear and understandable� it is possible to save it� It turnsout to be su�cient to remind of the J� particle and its interpretation as astate with the �hidden� charm ��cc� New quark with charm would savesituation�

����� GIM model

Indeed now the number of quarks is � but in the weak isodoublet only � ofthem are in action� And if one �Glashow� Iliopoulos� Maiani assumes thatthe �th quark also forms a weak isodoublet� only with the combination of dand s quarks orthogonal to dC � dcos�C � ssin�C� namely� sC � scos�C �dsin�C� Then apart from charged currents

j� � �c���� � ��sC

neutral currents should exist of the form�

jneutr��cs� ��

���cO�c� �sCO�sC �

��

Page 73: Introduction to Group Theory and Its Representations in Particle Physics

���uO�u� �sO�s�cos�C

� � �dO�d�sin�C��

��dO�scos�Csin�C � �sO�dcos�Csin�C�

The total neutral current yields�

jneutr��ud� ��

���uO�u�

��cO�c� �dO�d � �sO�s�

There are no strangeness�changing neutral currents at all�This is so called GIMmechanism proposed in � �� by Glashow� Iliopoulos�

Maiani in order to suppress theoretically decays of neutral kaons alreadysuppressed experimentally� �For this mechanism Nobel price was given&

����� Construction of the Salam�Weinberg model

Now we should understand what is the form of the operator O�� But thisproblem is already connected with the problem of a uni�cation of weak andelectromagnetic interactions into the electroweak interaction� Indeed theform of the currents in both interactions are remarkably similar to eachother� Maybe it would be possible to attach to the neutral weak currentthe electromagnetic one� It turns out to be possible� and this is the mainachievement of the Salam�Weinberg model�

But we cannot add electromagnetic current promptly as it does not con�tain weak isospin� Instead we are free to introduce one more weak� interactingneutral boson Y� ascribing to it properties of weak isosinglet� We shall con�sider only sector of u and d quarks and put for a moment even �C � � tosimplify discussion�

L � g�

���uL��uL � �dL��dLW���

�g��a�uL��uL � b�uR��uR � c �dL��dL � q �dR��dRY� �

e��

���uL��uL � �uR��uR� �

�� �dL��dL � �dR��dR�A��

��Jneutr��ud� Z�

��

��

Page 74: Introduction to Group Theory and Its Representations in Particle Physics

Having two vector boson �elds W��� Y� we should transfer to two other bo�son �elds A�� Z�

� �one of them� namely� A� we reserve for electromagnetic�eld and take into account that in fact we do know the right form of theelectromagnetic current� It would be reasonable to choose orthogonal trans�formation from one pair of �elds to another� Let it be

W�� �gZ�

� � g�A�pg� � g��

� Y� ��g�Z�

� � gA�pg� � g��

Substituting these relations into the formula for currents we obtain in theleft�hand side �LHS of the expression for the electromagnetic current thefollowing formula

gg�pg� � g��

���

�� a�uL��uL � b�uR��uR�

�����c �dL��dL � q �dR��dRA�� � eJ emA��

wherefrom

a ��

� b �

�� c �

q � ���� e �

gg�pg� � g��

Then for the neutral current we obtain

�g� � g��pg� � g��

���uL��uL�

� �dL��dLW�� � g��pg� � g��

J em �

pg� � g��

gg�

���uL��uL � �dL��dLW���

�pg� � g��

gg

g��

g� � g��J em

Let us now introduce notations

sin�W �g�p

g� � g��� cos�W �

gpg� � g��

��

Page 75: Introduction to Group Theory and Its Representations in Particle Physics

Now neutral vector �elds are related by formula

W�� � cos�WZ�� � sin�WA�� Y� � �sin�WZ�

� � cos�WA��

Finally weak neutral current in the sector of u and d quarks reads

g

cos�W��

���uL��uL � �dL��dL� sin��WJ

em��

Now we repeat these reasonings for the sector of c and s quarks and restoreCabibbo angle arriving at the neutral weak currents in the model with ��avors�

Jneutr��GWSW �

g

cos�W��

���cL��cL � �uL��uL � �dL��dL � �sL��sL�

�sin��WJ em�� ����

Remember now that the charged current enters Lagrangian as

L �gW

�p���c���� � ��sCW

�� � �u���� � ��dCW

�� �

��sC���� � ��cW�� �

�dC���� � ��uW��

and in the �nd order of perturbation theory in ud�sector one would have

L��� ��

g�W�M�

W � q��u���� � ��dC �dC���� � ��u�H�C��

what should be compared to

Leff �GFp��u���� � ��dC �dC���� � ��u�H�C�

Upon neglecting square of momentum transfer q� in comparing to the W �boson mass one has

GFp��

g�W�M�

W

�e�

�M�W sin��W

��

Page 76: Introduction to Group Theory and Its Representations in Particle Physics

wherefrom

M�W �

p�e�

�GF

p���

�GF

� ����GeV �

that isMW � ��GeV &&&

�Nothing similar happened earlier&Measurements of the neutral weak currents give the value of Weinberg

angle as sin��W � �� ����� �� ����� But in this case the prediction becomesabsolutely de�nite� MW � ��GeV� As is known the vector intermediateboson W was discouvered at the mass ��� ����� �g�w which agree with theprediction as one must increase it by ���" due to large radiative corrections�

����� Six quark model and CKM matrix

But nowadays we have and not � quark �avors� So we have to assume thatthere is a mix not of two �avors �d and s but of all � ones �d� s� b�

�B� d�

s�

b�

�CA �

�B� Vud Vus Vub

Vcd Vcs VcbVtd Vts Vtb

�CA�B� d

sb

�CA

This very hypothesis has been proposed by Kobayashi and Maskawa in � ���The problem is to mix �avors in such a way as to guarantee disappearenceof neutral �avor�changing currents� Diagonal character of neutral current isachieved by choosing of the orthogonal matrix VCKM of the �avor transfor�mations for the quarks of the charge �����

Even more it occurs that it is now possible to introduce a phase in orderto describe violation of CP�invariance � with number of �avours less then� one can surely introduce an extra phase but it could be hidden into theirrelevant phase factor of one of the quark wave functions� Usually Cabibbo�Kobayashi�Maskawa matrix is chosen as

VCKM �

�B� c��c�� s��c�� s��e

�i���

�s��c�� � c��s��s��ei��� c��c�� � s��s��s��e

i��� s��c��s��s�� � c��c��s��e

i��� �c��s�� � s��c��s��ei��� c��c��

�CA � ����

��

Page 77: Introduction to Group Theory and Its Representations in Particle Physics

Here cij � cos�ij � sij � sin�ij� �i� j � �� �� �� while �ij � generalized Cabibboangles� At ��� � �� ��� � � one returns to the usual Cabibbo angle �s � ����Let us write matrix VCKM with the help of Eqs� ������ as

VCKM � R�����D��ei�����R�����D�e

i�����R����� �

�B� � � �� cos��� sin���� �sin��� cos���

�CA�B� e�i����� � �

� � �� � ei�����

�CA� ����

�B� cos��� � sin���

� � ��sin��� � cos���

�CA�B� ei����� � �

� � �� � e�i�����

�CA�

�B� cos��� sin��� ��sin��� cos��� �

� � �

�CA �

Matrix elements are obtained from experiments with more and more pre�cision� Dynamics of experimental progress could be seen from these twomatrices divided by �� years in time�

V������CKM �

�B� �� ��� to �� �� ����� to ����� ����� to �����

����� to ����� �� ��� to �� ��� ����� to ��������� to ���� ���� to ����� �� �� to ��

�CA �����

V������CKM �

�B� �� ��� to �� ��� ���� to ���� ����� to �����

���� to ����� �� ��� to �� �� ����� to ��������� to ����� ����� to ����� �� � to �� �

�CA �����

The charged weak current could be written as

J�W � ��u� �c� �t���� � ��VCKM

�B� d

sb

�CA

Page 78: Introduction to Group Theory and Its Representations in Particle Physics

Neutral current would be the following in the standard �quark model ofSalam�Weinberg�

gpcos�W

JNEJTR��W �

gpcos�W

��

���tL��tL � �SL��SL � �uL��uL�

�dL��dL � �sL��sL � �bL��bL� sin��WJem��

��� Vector bosons W and Y as gauge �elds

Bosons W and Y could be introduced as gauge �elds to assure renormaliza�tion of the theory of electroweak interactions� We are acquainted with themethod of construction of Lagrangians invariant under local gauge transfor�mations on examples of electromagnetic �eld and isotriplet of the massless��meson �elds�

We have introduced also the notion of weak isospin� so now we requirelocal gauge invariance of the Lagrangian of the left�handed and right�handedquark �and lepton �elds under transformations in the weak isotopic spacewith the group SU��L � SU���

But as we consider left� and right� components of quarks �and leptonsapart we put for a moment all the quark �and lepton masses equal to zero�

For our purpose it is su�cient to write an expression for one left�handedisodoublet and corresponding right�handed weak isosinglets uR� dR�

L� � �qL�x����qL�x � �uR�x����uR�x � �dR�x����dR�x

This Lagrangian is invariant under a global gauge transformation

q�L�x � ei����qL�x�

u�R�L�x � ei�R�LuR�L�x�

d�R�L�x � ei��R�LdR�L�x�

where matrices �� act in weak isotopic space and � � ��� �� ���R�L� ��R�Lare arbitrary real phases�

��

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Let us require now invariance of this Lagrangian under similar but localgauge transformations when � and �R�L� �

�R�L are functions of x� As it has

been previously L� is not invariant under such local gauge transformations�

L�� � L� � i�qL�x�������x

�x�qL�x�

�i�uR����R�x

�x�uR � i �dR��

���R�x�x�

dR

�i�uL����L�x

�x�uL � i �dL��

�� �L�x�x�

dL�

In order to cancel terms violating local gauge invariance let us introduceweak isotriplet of vector �elds �W� and also weak isosinglet Y� with the gaugetransformations

�� �W �� � U y�� �W�U � �

gW

�U

�x�U y

GDE U � ei���x��� �

Y �� � Y� � �

gY

���R � ��R � �L � ��L�x�

Interactions of these �elds with quarks could be de�ned by the Lagrangianconstructed above

L ��p�g��uL��dLW

�� �

�dL��uLW�� � g

���uL��uL � �dL��dLW���

g��a�uL��uL � b�uR��uR � c �dL��dL � q �dR��dRY� �

Thus the requirement of invariance of the Lagrangian under local gauge trans�formations along the group SU��L� SU�� yields appearence of four mass�

less vector �elds �W� Y �Earlier it has already been demonstrated in what way neutral �elds W���

Y� by an orthogonal transformation can be transformed into the �elds Z��A�� After that one needs some mechanism �called mechanism of spontaneousbreaking of gauge symmetry in order to give masses to W�� Z and to leavethe �eld A� massless� Usually it is achieved by so called Higgs mechanisn�Finally repeating discussion for all other �avours we come to the alreadyobtained formulae for the charged and neutral weak currents but already inthe gauge�invariant symmetry with spontaneous breaking of gauge symmetry�

��

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��� About Higgs mechanism

Because of short of time we could not show Higgs mechanism in detail as anaccepted way of introducing of massive vector intermediate bosons W�� Z�

into the theory of Glashow�Salam�Weinberg� We give only short introductioninto the subject�

Let us introduce �rst scalar �elds � with the Lagrangian

L � T � V � ������� ���� � ��� ���

Let us look at V just as at ordinary function of a parameter � and search forthe minimum of the potential V ���

dV ��

d�� ������ ���� � ��

We have � solutions���min � ��

����min � �s���

��

That is� with ��h� we would have two minima �or vacuum states not atthe zero point& How to understand this fact� Let us �nd some telegraphmast and cut all the cords which help to maintain it in vertical state� For awhile it happens nothing� But suddenly we would see that it is falling� Andmaybe directly to us� What should be our last thought� That this is indeeda spontaneous breaking of symmetry&

This example shows to us not only a sort of vanity of our existence butalso the way to follow in searching for non�zero masses of the weak vectorbosons W�� Z�

By introducing some scalar �eld � with nonzero vacuum expectation value�v�e�v� � � �� v it is possible to construct in a gauge�invariant way theinteraction of this scalar �eld with the vector bosons W��W �� Y having atthat moment zero masses� This interaction is bilinear in scalar �eld andbilinear in �elds W��W �� Y that is it contains terms of the kind ��jW�

� j��At this moment the whole Lagrangian is locally gauge invariant which assuresits renormalization� Changing scalar �eld � to scalar �eld � with the v�e�v�equal to zero � � �� � � �� � � �� � we break spontaneously local

Page 81: Introduction to Group Theory and Its Representations in Particle Physics

gauge invariance of the whole Lagrangian but instead we obtain terms of thekind v�jW�

� j� which are immediately associated with the mass terms of thevector bosons W�

� � A similar discussion is valid for Z boson�Now we shall show this %miracle% step by step�Firs let us introduce weak isodoublet of complex scalar �elds

L � �������� ������ ������� ����

where � is a doublet� �T � ��� �� with non�zero vacuum value� h�i � v ����

It is invariant under global gauge transformations

�� � U� � ei������

Now let us as usual require invariance of the Lagrangian under the localgauge transformations� The Lagrangian Eq����� however is invariant onlyin the part without derivatives� So let us study %kinetic% part of it� With��x dependent on x we have

���� � U���� ��U��

it becomes

��������� � �����U y�����U y����U � ��� � U �

� �����U yU���� ������U y��U��

�����U yU���� ��U y���U��

Let us introduce as already known remedymassless isotriplet of vector mesons�W with the gauge transformation already proposed

�� �W �� � U�� �WU y � �

gW���UU

y�

but with two interaction Lagrangians transforming under local gauge trans�formations as�

���� �W y�� �

�W ���

� � ��� �W y��

�W���

��U yU� �W y�U

y���U�� ��U y��� �W�UyU��

��

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����U y���U��

and���� �W �

����� � ��� �W����� ��� �W�U

y���U��

The sum of all the terms results in invariance of the new Lagrangian withtwo introduced interaction terms under the local gauge transformations�

We can do it in a shorter way by stating that

��� � ig� �W ���

� �

�U�� � ���U � igU� �W�UyU � ���UU y� �

U��� � ig� �W���

Then it is obvious that the Lagrangian

��� � ig� �W y��

���� � ig� �W��

is invariant under the local gauge transformations�In the same way but with less di�culties we can obtain the Lagrangian

invariant under the local gauge transformation of the kind

�� � ei��

that is under Abelian transformations of the type use for photon previously�

��� � ig�Y������ � ig�Y ���

But our aim is to obtain masses of the vector bosons� It is in fact alreadyachieved with terms of the kind ��jW�

� j� and ��Y �� � Now we should also

assure that our e�orts are not in vain� That is searching for weak bosonmasses we should maintain zero for that of the photon� It is su�cient topropose the Lagrangian

��� � ig� �W y� � ig�Y������ � ig� �W� � ig�Y���

where g�jW�� j���� terms would yield with � � � � v masses of W� bosons

MW � v � g� while term jgW �� � g�Y�j���� would yield mass of the Z� boson

MZ � v �qg� � g�� � v � g

pg� � g��

g�

MW

cos�W�

��

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There is a net prediction that the ratio MW MZ is equal to cos�W � Experi�mentally this ratio is �omitting errors� �� � which gives the value of Wein�berg angle as sin��W � ���� in agreement with experiments on neutrino scat�tering on protons� Due to the construction there is no term jg�W �

��gY�j����that is photon does not acquire the mass&

Talking of weak bosons and scalar Higgs mesons we omit one importantpoint that is in the previous Lagrangians dealing with fermions we shouldput all the fermion masses equal to zero& Why�

It is because we use di�erent left�hand�helicity and right�hand�helicitygauge transformations under which the mass terms are not invariant as

mq�qq � mq�qLqR �mq�qRqL�

qL ��

��� � ��q� qR �

���� ��q�

What is the remedy for fermion masses� Again we could use Higgs bosons�In fact� interaction Lagrangian of quarks �similar for leptons with the samescalar �eld �� �T � ��� �� with non�zero vacuum value� h�i � v �� �� canbe written as

Ldm � �d�qL�dR �HC � �d��uL��d � �dL�

�dR �

�d��uL��d� �dL�

�dR �md�dLdR

with md � �dv��In a similar way lepton masses are introduced�

�l���lL�

�l � �lL��lR �ml

�lLlR

with ml � �lv�So we could obtain now within Higgs mechanism all the masses of weak

bosons and of all fermions either quarks or leptons�By this note we �nish our introduction into the Salam�Weinberg model

in quark sector and begin a discussion on colour�

��

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Chapter �

Colour and gluons

��� Colour and its appearence in particle

physics

Hypothesis of colour has been the beginning of creation of the modern theoryof strong interaction that is quantum chromodynamics� We discuss in thebeginning several experimental facts which have forced physicists to acceptan idea of existence of gluons � quanta of colour �eld�

� Problem of statistics for states uuu� ddd� sss with JP � ��

As it is known fermion behaviour follows Fermi�Dirac statistics and be�cause of that a total wave function of a system describing half�integer spinshould be antisymmetric� But in quark model quarks forming resonances#�� � �uuu� #� � �ddd and the particle (� � �sss should be in sym�metric S states either in spin or isospin spaces which is prohibited by Pauliprinciple� One can obviously renounce from Fermi�Dirac statistics for quarks�introduce some kind of �parastatistics� and so on� �All this is very similar tosome kind of �parapsychology� but physicists are mostly very rational peo�ple� So it is reasonable to try to maintain fundamental views and principlesand save situation by just inventing new �colour� space external to space�timeand to unitary space �which include isotopic one� As one should antisym�metrize qqq and we have the simplest absolutely antisymmetric tensor of the�rd rank �abc which �as we already know transforms as singlet representationof the group SU�� the reduction �abcq

aqbqc would be a scalar of SU�� innew quantum number called �colour� �here a� b� c � �� �� � are colour indices

��

Page 85: Introduction to Group Theory and Its Representations in Particle Physics

and have no relation to previous unitary indices in mass formulae� currentsand so on& Thus the fermion statistics is saved and there is no new quan�tum number �like strangeness or isospin for ordinary baryons in accord withthe experimental data� But quarks become coloured and number of them istripled� Let it be as we do not observe them on experiment�

� Problem of the mean life of �� mesonWe have already mentioned that a simple model of �� meson decay based

on Feynman diagram with nucleon loop gives very good agreement with ex�periment though it looks strange� Nucleon mass squared enters the denomi�nator in the integral over the loop� Because of that taking now quark model�transfer from mN � �� �� GeV to mu � �����GeV of the constituent quarkwe would have an extra factor � ��& In other words quark model result wouldgive strong divergence with the experimental data� How is it possible to savesituation� Triplicate number of quark diagrams by introducing �colour� &&&Really as �� � the situation is saved�

� Problem with the hadronic production cross�section in e�e�

annihilationLet us consider the ratio of the hadronic production cross�section of e�e�

annihilation to the well�known cross section of the muon production in e�e�

annihilation�

R ���e�e� � hadrons

��e�e� � �����

It is seen from the Feynman diagrams in the lowest order in

e�

e�

q

�q

e�

e�

��

��

that the corresponding processes upon neglecting �hadronization� of quarksare described by similar diagrams� The di�erence lies in di�erent charges ofelectrons�positrons and quarks� In a simple quark model with the pointlikequarks the ratio R is given just by the sum of quark charges squared that isfor the energy interval of the electron�positron rings up to ��� GeV it should

��

Page 86: Introduction to Group Theory and Its Representations in Particle Physics

be R � ��������

������

�� � �

�� However experiment gives in this interval

the value around ���� As one can see� many things could be hidden into thenot so understandable �hadronization� process �we observe �nally not quarksbut hadrons&� But the simplest way to do has been again triplication of thenumber of quarks� then one obtains the needed value� � � �

�� ��

With the �discouvery� of the charmed quark we should recalculate the valueof R for energies higher then thresholds of pair productions of charm particlesthat is for energies � � GeV� R � � � ����� � ���

�� � ���

� � �

��� � ��

��

Production of the pair of ��bb� quarks should increase the value R by �� whichproves to hardly note experimentally� Experiment gives above � GeV and p toe�e� energies around ����� GeV the value ��� At higher energies in�uenceon R of the Z boson contribution �see diagram � is already seen

Z�

e�

e�

q� l

�q� �l

Thus introduction of three colours can help to escape several important andeven fundamental contradictions in particle physics�

But dynamic theory appears only there arrives quant of the �eld �gluontransferring colour from one quark to another and this quant in some wayacts on experimental detectors� Otherwise everything could be �nished atthe level of more or less good classi�cation as it succeeded with isospin andhyper charge with no corresponding quanta

There is assumption that dynamical theories are closely related to localgauge invariance of Lagrangian describing �elds and its interactions withrespect to well de�ned gauge groups�

����� Gluon as a gauge eld

Similar to cases considered above with photon and � meson let us write aLagrangian for free �elds of ��couloured quarks qa where quark qa� a � �� �� �

��

Page 87: Introduction to Group Theory and Its Representations in Particle Physics

is a ��spinor of the group SU��C in colour space�

L� � �qa�x����qa�x�mq�qa�xq

a�x�

This Lagrangian is invariant under global gauge transformation

q�a�x � ei��k�k�a

b qb�x

� where �kk � �� ����� are known to us Gell�Mann matrices but now in colourspace� Let us require invariance of the Lagrangian under similar but localgauge transformation when k are functions of x�

q�a�x � ei��k�x��k�a

b qb�x�

orq��x � U�xq�x� U�x � ei�

k�x��k�

But exactly as in previous case L� is not invariant under this local gaugetransformation

L�� � �qa�x����qa�x � �qa�x�U�x����U�x

ab �xq

b�x�

�mq�qa�xqa�x�

In order to cancel terms breaking gauge invariance let us introduce masslessvector �elds Gk

�� k � �� ����� with the gauge transformation

�kG�k � U�kGkU y � �

gs

�U

�x�U y�

Let us de�ne interaction of these �elds with quarks by Lagrangian

L � gs�qa�x�Gk��k

ab��q

b�x�

to which corresponds Feynman graphs

G�ba

q�a� q�b�

Page 88: Introduction to Group Theory and Its Representations in Particle Physics

G ��p�

�B�G� � �

p�G� G� � iG� G � iG�

G� � iG� �G� � �p�G� G� � iG

G � iG� G� � iG ��p�G�

�CA �

�B�

D� G��� G

���

G��� D� G

���

G��� G

��� D�

�CA � ����

where

D� ��

��G

��� �G��� �

�G

��� �G��� � �G����

D� � ����G

��� �G��� �

�G

��� �G��� � �G����

���G

��� �G��� � �G����

Final expression for the Lagrangian invariant under local gauge transforma�tions of the non�abelian group SU��C in colour space is�

LSU���C � �qa�x����qa�x �mq�qa�xq

a�x�

gs�qa�x�Gk��k

ab��q

b�x� �

��F k��F k

���

where F k��� k � �� �� ����� is the tensor of the free gluon �eld transformingunder local gauge transformation as

�kFk��� � U y�x�l�xF l

��U�x�

It is covariant under gauge transformations U y �F ���U � �F�� � Let us write in

some detail tensor of the free gluon �eld ���F�� � �kFk�� � ,F�� � ���i� �j� �

�i�ijk�k� i� j� k � �� �� �����

�F k�� � ���G

k� � ��G

k�� �gsifkijGi

�Gj�

or,F�� � ��� ,G� � �� ,G�� gs� ,G�� ,G� �

��

Page 89: Introduction to Group Theory and Its Representations in Particle Physics

and prove that this expression in a covariant way transforms under gaugetransformation of the �eld G�

U y��� ,G�� � �� ,G

��U �

� ��� ,G� � �� ,G� � �Uy��U� ,G��� �U y��U� ,G� ��

U y� ,G���,G�� �U � � ,G�� ,G� � �

gs�U y��U� ,G��� �

gs�U y��U� ,G� ��

FinallyU y �F �

��U � U y��� ,G�� � �� ,G

�� � gs� ,G

���,G�� �U �

� �� ,G� � �� ,G� � gs� ,G�� ,G� � � �F���

The particular property of non�Abelian vector �eld as we have already seenon the example of the � �eld is the fact that this �eld is autointeracting thatis in the Lagrangian in the free term ����j�F �� j� there are not only termsbilinear in the �eld G as it is in the case of the �Abelian electromagnetic�eld but also ��and �� linear terms in gluon �eld G of the form G�G���G�

and G��G

�� to which the following Feynman diagrams correspond�

Gi Gj

Gk

Gi

Gi

Gk

Gk

This circumstance turns to be decisive for construction of the non�Abeliantheory of strong interaction � quantum chromodynamics�

The base of it is the asymptotic freedom which can be understood fromthe behaviour of the e�ective strong coupling constant of quarks and gluonss � g�s�� for which

s�Q� � s�

� � ���NC � �nf ln�Q�!��

where Q� is momentum transfer squared� �� is a renormalization point� !is a QCD scale parameter� NC being number of colors and nf number of�avours� With Q� going to in�nity coupling constant s tends to zero& Just

��

Page 90: Introduction to Group Theory and Its Representations in Particle Physics

this property is called asymptotic freedom��Instead in QED �quantum elec�trodynamics with no colors it grows and even have a pole� But one shouldalso have in mind that in the di�erence from QED where we have two ob�servable quantities electron mass and its charge �or those of �� and � �leptons in QCD we have none� Indeed we could not measure directly either quarkmass or its coupling to gluon�

Here we shall not discuss problems of the QCD and shall give only someexamples of application of the notion of colour to observable processes�

����� Simple examples with coloured quarks

By introducing colour we have obtained possibility to predict ratios of manymodes of decays and to prove once more validity of the hypothesis on exis�tence of colour�

Let us consider decay modes of lepton � discouvered practically after J�which has the mass ����� MeV �exp�������� �������� MeV� Taking quarkmodel and assuming pointlike quarks �that is fundamental at the level ofleptons we obtain that �� lepton decays emitting � neutrino �� either tolepton �two channels� e���e or ����� or to quarks �charm quark is too heavy�strange quark contribution is suppressed by Cabibbo angle and we are leftwith u and d quarks�

W

��

��

��� e�

���� ��e W

��

��

dC

�u

From our reasoning it follows that in absence of color we have two leptonmodes and only one quark mode and partial hadron width Bh should beequal to ��� of the total width while with colour quarks we have two leptonmodes and three quark modes leading to Bh � ����

Or in other words de�nite lepton mode would be �� " in absence ofcolour and �� " with the colour� Experiment gives �� � �� � ��� � �� ����� �� � �� � " and �� � e� � ��e � �� � ���� �� � �� ��"� supportinghypothesis of � colours�

Page 91: Introduction to Group Theory and Its Representations in Particle Physics

The W decays already in three lepton pairs and two quark ones

W� ��e� ���� ���

e�� ��� ��

W� dC � sC

�u� �c

This means that in absence of colour hadron branching ratio Bh would be ��"while with three colours around "� Experiment gives Bh � ��� �� �� �"once more supporting hypothesis of ��coloured quarks�

Z boson decays already along lepton and � quark modes�

Z� e�� ��� ��

e�� ��� ��

Z� ��e� ���� ���

�e� ��� ��

Page 92: Introduction to Group Theory and Its Representations in Particle Physics

Z� u� d� s� c� b

�u� �d� �s� �c��b

Thus it is possible to predict at once that in absence of colour hadron channelshould be �������" of the total width of Z while with three colours numberof partial lepton and colour quark channels increase up to �������� andhadron channel would be ��������" � Experimentally it is � �� �����"�

Page 93: Introduction to Group Theory and Its Representations in Particle Physics

Chapter �

Conclusion

In these chosen chapters on group theory and its application to the particlephysics there have been considered problems of classi�cation of the particlesalong irreducible representations of the unitary groups� have been studiedin some detail quark model� In detail mass formulae for elementary parti�cles have been analyzed� Examples of calculations of the magnetic momentsand axial�vector weak constants have been exposed in unitary symmetry andquark model� Formulae for electromagnetic and weak currents are given forboth models and problem of neutral currents is given in some detail� Elec�troweak current of the Glashow�Salam�Weinberg model has been constructed�The notion of colour has been introduced and simple examples with it aregiven� Introduction of vector bosons as gauge �elds are explained�

Author has tried to write lectures in such a way as to give possibilityto eventual reader to evaluate by him� or herself many properties of theelementary particles�