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Chapter 7 Unitary Symmetries and QCD As a Gauge Theory

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Chapter 7 Unitary Symmetries and QCD As a Gauge Theory Chapter 7 Unitary symmetries and QCD as a gauge theory Literature: Lipkin [23] (group theory concepts from a physicist’s point of view) • Lee [24], chapter 20 (extensive treatment of Lie groups an Lie a!gebras in the • context of ifferentia! geometry) Interactions "etween particles shou! respect some o"served symmetry% &ften, the proce- ure of postu!ating a speci(c symmetry !ea s to a uni)ue theory% *his way of approach is the one of gauge theories% The usual examp!e of a gauge theory is +,-, which correspon s to a !oca!U(1)-symmetry of the La grangian / ieqeχ(x) ψ ψ � 0 e ψ, (1%.) → A A � 0A ∂ χ(x). (1%2) µ → µ µ − µ 2e can co e this comp!icated transformation "ehavior "y rep!acing in the +,- La- grangian∂ µ "y the covariant derivativeD µ 0∂ µ 3 ieqeAµ% 7.1 IsospinSU(2) 4or this section we consi er on!y the strong interaction an ignore the electromagnetic an weak interactions. $n this regar , isobaric nuclei (with the same mass num"er A) are very simi!ar% Heisen"erg propose to interpret protons an neutrons as two states of the same o"ject / the nucleon/ . p 0ψ(x) , | � 0 � � 0 n 0ψ(x) . | � . � � .25 .26 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY 2e note the analogy to the spin forma!ism of nonrelativistic )uantum mechanics, which originated the name isospin% 1 In isospin'space, p an n can "e represented as a two-component spinor withI0 2 % p has thenI 0| 3� 1 an| �n hasI 0 1 % | � 3 2 | � 3 − 2 :ince the strong interaction is "!in to other charges (e!ectromagnetic charge, weak hy' percharge), the (strong) physics must "e the same for any !inear com"inations of p an n % $n other wor s, for, | � | � p p� 0α p 3β n , | �→| � | � | � n n� 0γ p 3δ n , | �→| � | � | � for some α,β,γ,δ C, or, ∈ ψp N 0 N � 0U N , (1%3) | � ψn → | � | � � � for some 2 2 matrixU with comp!ex entries, t he (strong) physics oes not change if we × switch from N to N � to escri"e the system% | � | � 2e remark at this point that this symmetry is on!y an approximate symmetry since it is violated "y the other interactions, an is hence not a symmetry of nature. First we require the conservation of the norm N N which we interpret as the num"er of particles !ike in )uantum mechanics% This yi�el |s,� ! N N N � N � 0 N U †U N 0 N N � | �→� | � � | | � � | � U †U0UU † 0 U U(2). (7% 4) ⇒ ⇒ ∈ 5 genera! unitary matrix has 4 rea! parameters. :ince the effect ofU an e iϕU are the same, we (x one more parameter "y imposing, etU 0! . U SU(2), (1%8) ⇒ ∈ the special unitary group in 2 imensions. *his group is a Lie group (a group which is at the same time a manifo!d)% 2e use the representation, ˆ U0e iαj Ij , (1%9) where theα j’s are arbitrary group parameters (constant, or epen ing on the spacetime coor inatex), an the I;j’s are the generators of the Lie group% 2e concentrate on in(nitesima! transformations, for whichα j .% $n this approximation we can write � U 3 iα I; . (1%1) ≈ j j 7.1. ISOSPINSU(2) .27 The two e(ning con itions ofSU(2), ,)% (1%4) an (1%8), imp!y then for the generators, ; ; Ij† 0 Ij (hermitian), (1%8) *r I;j 0 0 (traceless). (1%=) In or er for the exponentiation procedure to converge for nonin(nitesimalα j’s, the gen' erators must satisfy a comutation relation, thus e(ning the Lie algebra su(2) of the groupSU(2)% +uite in general, the commutator of two generators must "e expressi"!e as a !inear com' "ination of the other generators 1% In the case of su(2) we have, [I;i, I;j] 0 iεijkI;k, (1%.0) whereε ijk is the total!y antisymmetric tensor withε 123 0 3.% They are characteristic of the (universa! covering group of the) Lie group ("ut in epen ent of the chosen represen' tation) an calle structure constants of the Lie group% The representations can "e characterized accor ing to their tota! isospin% Consi er now I0. 2, where the generators a re given "y . I; 0 ! i 2 i with! i 0" i the @au!i spin matrices (this notation is chosen to prevent confusion with ordinary spin)/ 0 . 0 i . 0 ! 0 ! 0 ! 0 1 . 0 2 i0− 3 0 . � � � � � − � which fu!(!! ["i,"j] 0 2iεijk"k. *he action of the matrices of the representation (see ,)% (1%6)) is a non'a"elian phase transformation/ #» #» iα τ N � 0e · 2 N . | � | � 4orSU(2), there exists on!y one ia gona! matrix (!3). In genera!, forSU(N), the fo!!owing ho! s true: Rankr0N ./ *here arer simu!taneous!y iagona ! operators. • − Dimension of the Lie a!gebra#0N 2 ./ *here are# generators of the group an • − therefore# group parameters. ,% g. in the case ofSU(2) ∼0 S$(3) this means that there are three rotations/generators an three ang!es{± as} parameters. 1Since we are working in a matrix representation ofSU(2) this statement mak es sense. The difference between the abstract group and its matrix representation is often neglected. .28 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY � ���� �������� �3 �� �� Figure 1%./ The nucleons n and p form an isospin double t. | � | � Isospin particle mu!tip!ets (representations) can "e characterized "y their )uantum num' "ersI an I 3/ There are 2I 3 . states. Consi er for examp!e once again the caseI0. 2. There are two states, characterize "y theirI 3 )uantum num"er/ I0 1 ,I 0 3 1 p 2 3 2 0 | � . I0 1 ,I 0 1 n �� 2 3 − 2 �� �| �� � � � This is visua!ized in Fig% 1%1, a!ong� with the action of the operators! 0 . 2(!1 iτ2): ± ± 0 0 . 0 ! p 0 0 0 n − | � . 0 0 . | � � � � � � � ! n 0 p + | � | � ! n 0! + p 00. − | � | � This is the sma!!est non-trivia! representation of SU(2) an therefore its fun amenta! representation% 4urther examp!es for isospin mu!tip!ets are I mu!tip!ets I3 + 3 1 1 p % 2He 3 2 2 n %0 36 1 � � � � �1 � − 2 &+ 3. &0 0 &− . ++ −3 B 3 2 + 1 3 B 3 2 2 0 1 B 2 − 3 B− − 2 wherem 1232 MeD an m 93< MeV. Δ ≈ p,n ≈ 5!!I� . representations can " e o"taine from irect pro ucts out of the fun amental I0. 2 representation 2 where “2F enotes the num"er of states. In ana!ogy to the a ition of two electron spins where the ?!ebsch'Gor an ecomposition rea s rep% 1/2 ⊗ 7.1. ISOSPINSU(2) .29 rep%1/2 0 rep%0 rep%1 an where there are two states for the spin'.A2 representation, one state for the spin-0⊕ representation, an three states for the spin'. representation, we have 2 2 0 . 3 . (7%.1) ⊗ ⊕ I= 1 1 =0,1 isosinglet,I=0 isotriplet,I=1 | 2 ± 2 | However, there is an important� �� �ifference "etwee���� n isospin���� an spin mu!tip!ets. $n the !atter case, we are consi ering a "oun system an the constituents carrying the spin have the same mass. &n the other han , pions are not simp!e "oun states. Their structure wi!! "e escri"ed "y the )uark mo el% 7.1.1 Isospin invariant interactions Isospin invariant interactions can "e constructed "y choosingSU(2) invariant interaction terms �. 4or instance, consi er the Hukawa mo e!, escri"ing nucleon'pion coup!ing, whereL #» #» #» #» � 0 igNI !N &0 ig NI � !N � & � (1%.2) LπN · · which is an isovector an where the secon i entity is ue toSU(2) invariance% In(nites ' ima!!y, the transformation !ooks as fo!!ows: i #» #» N � 0UN U03 α ! (1%.3) 2 · i #» #» 1 NI � 0 NUI † U † 0 α !0U − (1%.4) − 2 · #» #» #» #» & � 0( &(03i α t . (1%.5) · #» The parameters t can "e etermine from the isospin invariance con ition in ,)% (1%12): 1 N!I jN&j 0 NUI − !iUN(ij&j. With( ij 0δ ij 3 iαk(tk)ij (cp% ,)% (1%.8)) an inserting the expressions forU an U †, this yiel s i i ! 0 α ! ! 3 α ! δ 3 iα (t ) j − 2 k k i 2 k k ij k k ij � � � � � � i 2 =τ i + 2 αk[τi, τk] + (α k) i O 2 =τ i + αk2iεiklτl + (α ) � 2 �� O k � 0! 3 iα iε ! 3! (t ) j k { jkl l i k ij} 0! 3 iα ! iε 3 (t ) j k i { jki k ij} ! =0 (t ) 0 iε . ⇒ k ij −� kij �� � .30 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY This means that the 3 3 matricest k,)0.,2,3, are given "y the struct ure constants (see ,)% (7..0))% 4or the×commutator we therefore have [t , t ] 0 ε ε 3ε ε 0ε ε 0 iε ( iε ) 0 iε (t ) (1%.9) k l ij − kim lmj lim kmj klm mij klm − mij klm m ij where the secon i entity fo!!ows using the Jaco"i i entity% This means that the matrices tk fu!(!! the Lie algebra [tk, tl] 0 iεklmtm. Thet ks form the a joint representation ofSU(2). 7.2 Quark model of hadrons It is experimental!y we!! esta"!ishe that the proton and the neutron have inner structure.
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