"The Eightfold Way Anirban Sharma March 29, 2016 1 Basics of Group theory 1.0.1 Groups and representations Group is a set G under some operation\u0000 , which obeys the following properties: \u0000 If a;b2G then c =a\u0000 b2G \u0000 For a;b;c2G;a\u0000 (b\u0000 c) = (a\u0000 b)\u0000 c \u00009 an identity element e such that e\u0000 a =a\u0000 e =a8f2G \u0000 1 \u0000 1 \u0000 1 \u0000 For every element a2G,9 an inverse elementa such thata\u0000 a =a \u0000 a =e Representation: Representation of G is a homomorphism \u0000 from the the elements of G to complex general linear vector space V . \u0000 : G!GL(V ) (1) For each g2G, \u0000 (g)2GL(V ) and satis\u0000es the following: \u0000 \u0000 (e) = 1 [1 is the identity element in the space of complex general linear groups] \u0000 1 \u0000 1 \u0000 \u0000 (g )\u0000 (g ) =\u0000 (g g ) and \u0000 (g ) =\u0000 (g) 1 2 1 2 Dimension of a representation is same as the dimension of the space on which it acts.A ~ representation is unitary if all the \u0000 (g)s are unitary.Two representations \u0000 (g) and \u0000 (g) \u0000 1 ~ are equivalent if \u0000 (g) =s \u0000 (g)S Irreducible Representation Let (\u0000;V ) be a representation of a group G. We say (\u0000;V ) is an irreducible representation if the only G-stable subspaces of V are (0) and V . 11.1 Special unitary groups 1 BASICS OF GROUP THEORY 1.1 Special unitary groups y y This is the group of all matrices U, such that UU = U U = I and determinant of U is 1.These are continuous groups. Consider the group elements g2 G depend smoothly on a set of continuous parameter \u0000 such that g(\u0000 )j = e.The linear operators of the \u0000 =0 representaton will be paramatrized as \u0000 (\u0000 )j = 1. Upon Taylor expanding \u0000 (\u0000 ) in a \u0000 =0 neighbourhood of the identity element, we get the generators of the form: @ X =\u0000 i \u0000 (\u0000 )j (2) a \u0000 =0 @\u0000 a wherea=1 to N. The number of generators depend on the number of parameters needed 2 to specify the group. SU(N) has N -1 generators. (give reference) 1.2 Lie Algebra A Lie Algebra is a vector space V over some \u0000eld F together with a binary operation,[.,.]:V*V !V called the Lie bracket that satis\u0000es the following 8x;y;z2 V and a;b2F: \u0000 Bilinearity [ax +by;z] =a[x;z] +b[y;z] (3) \u0000 Alternativity [x;x] = 0 (4) \u0000 Jacobi identity [x; [y;z]] + [y; [z;x]] + [z; [x;y]] = 0 (5) \u0000 Anticommutativity [x;y] =\u0000 [y;x] (6) 1.3 Adjoint representation This representation is generated by the structure constants [X [X ;X ]] =if [X ;X ] a b c bcd a d =\u0000 f f X (7) bcd ade e Using the above in the Jacobi identity, f f +f f +f f = 0 (8) bcd ade abd cde cad bde These structure constants can be used to generate matrices which preserves the Lie alge- bra. P 3 2 Casimir Operator:The operator T is called the casimir operator. a a=1 Cartan Subalgebras and generators A subset of commuting hermitian generators which is as large as possible is called a cartan subalgebra. The elements of the cartan subalgebra that satisfy the following are called cartan generators: y H =H and [H;H ] = 0 i i j i 22 ISOSPIN 2 Isospin The mass of neutron and proton are very close. if the charge is turned o\u0000, then they can be treated as similar particles in the context of strong interaction. Strong force is invariant if we replace proton by neutron or vice versa. The nucleon doublet is given by \u0000 \u0000 p N= n Isospin formalism is similar to that of spin. So protons and neutrons were declared to be 1 of isospin . Isospin projections increase in steps of 1. 2 2.1 Isospin symmetry Isospin also has SU(2) Lie group structure. Neutron and proton form a doublet, which forms the fundamental representation. Isospin operator has three components: I ;I ;I . x y z Isospin is described by total isospin I and I . z However isospin is an approximate symmetry. Let us consider Hamiltonian of the form: H =H + \u0000 H (9) 0 whereH commutes with the generators of isospin symmetry. Isospin are invariant under 0 strong interactions. So, H =H and \u0000 H =H +H . However its not easy to 0 strong EM weak separate \u0000 H into H and H since there is a partial uni\u0000cation. Also due to the EM weak di\u0000erence in mass of u and d quarks, slight isospin violation is there. So this violation can be clubbed under \u0000 H. But the symmetry structure will still be the same. 2.2 Representations of isospin symmetry Since this is similar to spin, it will have many representation corresponding to di\u0000erent I. The dimension of the matrix of the representation will be (2I + 1)\u0000 (2I + 1). Isospin is invariant under action of SU(2) Lie group. So the representations of isospin will be basically that of SU(2). Isospin symmetry is restricted to SU(2). It forms a sub-group of a bigger symmetry called the \u0000avour symmetry. Leptons If we turn o\u0000 the mass and charge di\u0000erence and also the weak interactions, we get similar particles which have no rest mass. This will correspond to some unitary symmetry, \u0000 \u0000 \u0000 corresponding to which there will be some transformation. Let us put in a \u0000 e doublet. Now, since there is a symmetry, two types of transformations are possible. \u0000 \u0000 i\u0000 e 0 \u0000 U = (Lepton conservation) i\u0000 0 e \u0000 \u0000 This transformation leaves invariant the product of the phase factors of \u0000 and e . The determinant of the transformation is 1. 32.2 Representations of isospin symmetry 2 ISOSPIN In\u0000nitesimal transformation of the \u0000rst kind is 1 + i(\u0000\u0000 )1 and for the second kind is P 3 T k 1 +i \u0000\u0000 . Symmetry of the second kind is known as charge independence or spin k k=1 2 symmetry. Properties of T k Tr(TT ) = 2\u0000 (10) i j ij [T;T ] = 2ie T (11) i j ijk k fT;Tg = 2\u0000 1 (12) i j ij 0 1 \u0000 \u0000 @ A Now consider the triplet, e . Same as doublet there will be two types of trans- \u0000 \u0000 formations. The transformation of the second kind belongs to SU(3), the number of 2 generators of which is = (3) \u0000 1 = 8. Therefore SU(3) can be generated by \u0000 ;\u0000 ;:::\u0000 1 2 8 which are traceless Hermitian matrices satisfying the following relations: Tr\u0000 \u0000 = 2\u0000 (13) i j ij [\u0000 ;\u0000 ] = 2if \u0000 (antisymmetric) (14) i j ijk k 4 f\u0000 ;\u0000 g = \u0000 1 + 2d \u0000 (symmetric) (15) i j ij ijk k 3 \u0000 ;\u0000 and \u0000 contain T ;T and T . \u0000 commutes with \u0000 ;\u0000 and \u0000 and also is 1 2 3 1 2 3 8 1 2 3 diagonal. \u0000 is related to strangeness, 8 \u0000 \u0000 e \u0000 4 8 p Q = \u0000 + \u0000 n (16) 3 l 2 3 3 where n = number of leptons\u0000 number of antileptons. l Once the muon mass is turned on, the symmetry corresponding to \u0000 ;\u0000 ;\u0000 ;\u0000 is de- 4 5 6 7 stroyed. Also strangeness is not conserved in weak interactions. Because of electromag- netism, the symmetry under \u0000 and \u0000 is broken. We can see only lepton number and 1 2 electric charge conservation.Now the next would be to see how the baryons and mesons respond to symmetry under spin Baryons 2 Consider the isotopic spin I. Each multiplet has (2I + 1) components.I has eigenvalue 2 I(I + 1) and also [I ;I ] = 0 where a=1,2,3. Isotopic spin group is characterized by a its generators. In\u0000nitesimal group elements in the neighbourhood of identity is of the P form 1+\u0000 d\u0000I [Identity]. I s are represented by Hermitian matrices of dimension a i a (2I + 1)\u0000 (2I + 1). I also preserves the Lie algebra structure [I;I ] =ie I (17) i j ijk k 42.2 Representations of isospin symmetry 2 ISOSPIN 1 T i I = representation I = where T s are Pauli matrices Going back to the leptonic i i 2 2 doublet, consider that the antiparticles transform similarly under unitary transformation in spin space(here we will consider spin as the same as isospin just for the sake of simplicity + of notation since the underlying algebra is the same) i.e. e and -\u0000\u0000 transforms similarly \u0000 as (\u0000;e ). From doublet representation, other representations can be found by taking + \u0000 e e +\u0000\u0000\u0000 + 1 p superposition. is a singlet representation i.e. I = 0. e \u0000 = e\u0000 (T \u0000 iT )e , \u0000 1 2 \u0000 2 2 + \u0000 e e +\u0000\u0000\u0000 1 \u0000 1 p p = e\u0000 T e and \u0000e = e\u0000 (T +iT )e forms a triplet. So all the superposition \u0000 3 \u0000 \u0000 1 2 \u0000 2 2 2 1 p can be written as e\u0000 Te \u0000 i \u0000 2 write after contacting sir] \u0000 \u0000 Let us take three leptons\u0000;e ;\u0000 and make a triplet and label them asl where\u0000 =1,2,3. \u0000 Let us suppose that they have symmetry under action of unitary spin operator F . De- \u0000ne F = \u0000 =2 i=1,2,..8 where \u0000 s are the Gell-Mann matrices. SU(2) is contained in i i i p 3 SU(3).We know that T =\u0000 for i = 1; 2; 3. Therefore F \u0000 I for i = 1; 2; 3. F = Y . i i i i 8 2 This is one of the representations. The underlying algebra is the same for all the rep- resentations, i.e, [F;F ] = if F but the properties of the trace of F ’s will vary from i j ijk k i representation to representation. However there is a di\u0000erence between F and I. Upon changing sign on \u0000 \u0000 , the doublet \u0000 of antilepton e\u0000 transforms similarly as e under I whereas l doesn’t transform as l \u0000 \u0000 \u0000 \u0000 under F. This is because of the third component of\u0000 , which has a sign opposite to that 8 of the other two. Therefore, it picks up a negative sign. Also these two are inequivalent representations. Let us consider a triplet of particles L that transform like l under F. \u0000 \u0000 \u0000 eT \u0000 e L\u0000 l i i p p Similar to , we form wherei = 1; 2;::8. Under the action of F, the states form an 2 2 irreducible representation of dimension 8. The matrices can be formed by jk F =\u0000 if (18) ijk i \u0000 L\u0000 l i p All states can be formed by linear combinations of .\u0000 represents transforms like. 2 1 + + \u0000 \u0000 \u0000 L(\u0000 \u0000 i\u0000 )l \u0000 D \u0000 (19) 1 2 2 1 \u0000 0 \u0000 \u0000 \u0000 \u0000 L(\u0000 +i\u0000 )l \u0000 D e (20) 1 2 2 0 + \u0000 1 D \u0000 \u0000 D e 0 \u0000 \u0000 \u0000 p L\u0000 l \u0000 p (21) 3 2 2 1 + \u0000 p \u0000 L(\u0000 \u0000 i\u0000 )l \u0000 S \u0000 (22) 4 5 2 1 + \u0000 \u0000 n \u0000 L(\u0000 \u0000 i\u0000 )l \u0000 S e (23) 6 7 2 1 0 + \u0000 \u0000 \u0000 \u0000 L(\u0000 +i\u0000 )l \u0000 D \u0000 (24) 6 7 2 1 \u0000 0 \u0000 \u0000 \u0000 \u0000 L(\u0000 +i\u0000 )l \u0000 D \u0000 (25) 4 5 2 0 + \u0000 + \u0000 1 (D \u0000 +D e \u0000 2S \u0000 ) \u0000 p p ^ \u0000 L\u0000 l \u0000 (26) 8 2 6 0 + \u0000 HereD ;D andS+ are the notations for the particles ofL. D andS stands for doublet and singlet respectively. 52.2 Representations of isospin symmetry 2 ISOSPIN Pseudoscalar mesons In the previous section, we have seen that Baryons transform like an octet under the \u0000 action ofF . From Clebsch Gordon decomposition of a tensor product of 3 and 3, we get \u0000 3\u0000 3 = 8\u0000 1 (27) In the context of strong coupling, we want the mesons to be invariant under the action of 5 \u0000 \u0000 F . Yukawa couplings are of the formV\u0000 g \u0000 (scalar) org i\u0000 \u0000 (pseudoscalar)[wikipedia], where\u0000 is a scalar \u0000eld and is a Dirac \u0000eld. Consider baryon \u0000elds N . Now the mesons j \u0000 will couple toN\u0000 N to have Yukawa couplings. SinceN transform like an octet, we want i \u0000 \u0000 8\u0000 8. However, 8 and 8 are equivalent representations, which implies that antibaryons transform like baryons. \u0000 \u0000 8\u0000 8 = 27\u0000 10\u0000 10\u0000 8\u0000 8\u0000 1 (28) \u0000 When mass di\u0000erences are turned on, the representations 27,10 and 10 break down. The representation 27 breaks into the following: \u0000 a singlet, a triplet and a quintet for Y=0 \u0000 For both Y=1 and Y=-1, we get a doublet and a quartet \u0000 For both Y=2 and Y=-2, we get a triplet But none of the representations is similar to that of the known mesons.The 8 here is similar to that of the baryons and this corresponds to the mesons. Thus the mesons form an octet. 0 \u0000 =\u0000 (29) 8 (\u0000 \u0000 i\u0000 ) 1 2 + \u0000 = p (30) 2 (\u0000 +i\u0000 ) 1 2 \u0000 \u0000 = p (31) 2 0 \u0000 =\u0000 (32) 3 (\u0000 \u0000 i\u0000 ) 4 5 + p K = (33) 2 (\u0000 \u0000 i\u0000 ) 6 7 0 p K = (34) 2 (\u0000 +i\u0000 ) 6 7 \u0000 0 K = p (35) 2 (\u0000 +i\u0000 ) 4 5 \u0000 K = p (36) 2 The relation between\u0000 s are preserved by the matrices ofF similar to those inN s. F is i i jk given by F =\u0000 if . The Yukawa coupling between 8 mesons and 8 baryons here will ijk i 5 \u0000 be of the form 2gNi\u0000 \u0000 \u0000N in which the following holds: i [F;\u0000 ] =\u0000 if \u0000 (37) i j ijk k \u0000 It can be seen that the decomposition of 8\u0000 8 gives us 2 8 representations. This is because there are two independent sets of 8*8 matrices. One set is of symmetric matrices while other is of antisymmetric matrices. But both satisfy the same commutation relation. 62.2 Representations of isospin symmetry 2 ISOSPIN Tensor Algebra of SU(3) Let us consider a transformation, X j 0j i a = U a; i;j = 1; 2; 3 (38) i i and v is a set of 3 complex numbers. Also U is unitary. This demands y \u0000 1 U =U \u0000 \u0000 j i\u0000 \u0000 1 )U = u j i Therefore, X X \u0000 \u0000 \u0000 j 0j \u0000 i \u0000 \u0000 1 i i \u0000 a = (U ) (a ) = (U ) (a ) i j i i Any general representation of SU(N) can be given by a tensor type (p,q) having p con- travariant and q covariant indices. i i :::i 1 2 p i i i 1 2 p T =a b :::c x y :::z (39) j j j j j :::j 1 2 q i 2 q where a,b,...,c are covariant vectors and x,y,...z are contravariant vectors. T is a multi- plication of these vectors. Now the tensor transforms as the following equation. 0 0 0 0 0 0 0 0 0 i i :::i i j i i j j i i :::i 1 2 p p \u0000 1 \u0000 1 \u0000 1 q 1 2 p 1 2 1 2 T =U U :::U (U ) (U ) :::(U ) T (40) 0 0 0 i i i j j j j j :::j j j :::j 1 2 p 1 2 q 2 q i i 2 3 Here as are vectors with complex entries, so if we permute the indices it remains the i same. Therefore the tensor is symmetric under exchange of upper or lower indices. This is an invariant. Corresponding to such a tensor, a representation can be labelled asD(p;q). D(p;q) =n (41) 1 wheren = (p + 1)(q + 1)(p +q + 2) [reference]. Heren is the dimension of the represen- 2 tation. Generalized Clebsch Gorden Coe\u0000cients: Tensor Produnct of two representations can be decomposed as X 0 0 00 00 D(p;q )\u0000 D(p ;q ) = \u0000 M(p;q)D(p;q) (42) p;q where M(p,q) is the multiplicity of the representation is the decomposition. For the Isospin group the product state 0 0 00 00 0 0 00 00 jI;I ;I ;I i =jI;IijI ;I i (43) 3 3 3 3 Any statejI;Ii can be decomposed as a sum of product states. 3 X 0 0 00 00 0 0 00 00 jI;Ii = jI;I ;I ;I ihI;I ;I ;I jI;Ii (44) 3 3 3 3 3 3 0 00 I ;I 3 3 0 00 0 00 For I =I +I and I =I +I , we get non-vanishing coe\u0000cients. The coe\u0000cients are 3 3 3 same as Clebsch Gordan coe\u0000cients for SU(2). Now, in SU(3), along with I and I , Y 3 73 EIGHTFOLD WAY and n has to be incorporated in labelling a state. Similar to SU(2) we can construct product states. 0 0 0 0 00 00 00 00 0 0 0 0 00 00 00 00 jn;I;I ;Y ;n ;I ;I ;Y i =jn;I;I ;Yijn ;I ;I ;Y i (45) 3 3 3 3 Any general state can be written as X 0 0 0 0 00 00 00 00 0 0 0 0 00 00 00 00 jn;I;I ;Yi = jn;I;I ;Y ;n ;I ;I ;Y ihn;I;I ;Y ;n ;I ;I ;Y jn;I;I ;Yi 3 3 3 3 3 3 0 00 0 00 00 I ;I ;I ;I ;Y;Y 3 3 (46) Here again we can get the tabulated coe\u0000cients.(reference). The isospin and the hypercharge has to be conserved. Therefore, 0 0 0 0 00 00 00 00 0 0 00 00 0 0 0 00 00 00 hn;I;I ;Yjn;I;I ;Y ;n ;I ;I ;Y i =\u0000 0 00hI;I jI;I ;I ;I i\u0000h n;I;Yjn;I;Y ;n ;I ;Y i 3 Y;Y +Y 3 3 3 3 3 (47) Representation SU(3) is a rank 2 group. So it will have two Casimir operators. The tensors that we have discussed are the invariants. They correspond to the irreducible representations. The various irreducible representations of the group are given below.[In the diagram pq D(p;q)\u0000 D ] \u0000 D(0; 0) = 1; D(1; 0) = 3; D(0; 1) = 3; D(1; 1) = 8 Figure 1: Representation D(p;q) 3 Eightfold Way Let us consider a transformations i i i U =\u0000 +a (48) k k k \u0000 \u0000 i @U and de\u0000ne A = j . a =0 k k i @a i In 3 space, the in\u0000nitesimal operators are given by \u0000 \u0000 1 \u0000 i i \u0000 i \u0000 A =\u0000 \u0000 \u0000 \u0000 \u0000 (49) k \u0000 k k k \u0000 3 8"