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Chiral Symmetry Chiral symmetry Andrea Beraudo INFN - Sezione di Torino Ph.D. lectures, AA 2015-'16, Torino Andrea Beraudo Chiral symmetry Introduction The confinement/deconfinement phase transition is accompained by the breaking/restoration of chiral symmetry. Aim of this lecture is to discuss the role of two global symmetries of the QCD lagrangian (isospin and chiral symmetry) in understanding the hadron spectrum the origin of the baryonic mass of our universe Andrea Beraudo Chiral symmetry Outline Derivation of the QCD Lagrangian (local gauge symmetry) Reminder: global symmetries and conserved currents in QFT The QCD lagrangian and its global symmetries The QCD-vacuum and chiral symmetry breaking An explicit example: the linear sigma model Chiral symmetry restoration at finite temperature Andrea Beraudo Chiral symmetry The QCD Lagrangian: construction Let us start from the free quark Lagrangian (diagonal in flavor!) free = q (x)[i@= mf ]qf (x): Lq f − The quark field is actually a vector in color space (Nc =3): T e:g: for an up quark u (x) = [ur (x); ug (x); ub(x)] The free quark Lagrangian is invariant under global SU(3) (i.e. V yV =1 and det(V )=1) color transformations, namely: q(x) V q(x) and q(x) q(x) V y; −! −! with V = exp [iθata] and [ta; tb] = i f abc tc (a=1;::: N2 1): c − f abc : real, antisymmetric structure constants of the su(3) algebra. We want to build a lagrangian invariant under local color transformations: q(x) V (x) q(x) q(x) q(x) V y(x); −! −! where now V (x) = exp [iθa(x)ta]. Andrea Beraudo Chiral symmetry free Due to the derivative term, q is not invariant under local SU(Nc ) transformations: L free 0free = free + q(x)V y(x)[i@=V (x)] q(x) (1) Lq −! L q Lq a a The solution is to couple the quarks to the gauge field Aµ Aµt through the covariant derivative ≡ @µ µ(x) @µ igAµ(x); −! D ≡ − getting: free q = q(x)[i =(x) m]q(x) = + gq(x)A=(x)q(x): L D − Lq The transformation of Aµ under local SU(Nc ) must be such to compensate the extra term in Eq. (1): 0 y i y Aµ A = VAµV (@µV )V : −! µ − g Exercise: verify that q is now invariant under local SU(Nc ) transformations. In particular:L y µq V µq = µ V µV (2) D −! D )D −! D We must now construct the lagrangian for the gauge-field Aµ Andrea Beraudo Chiral symmetry Remember the (U(1) invariant) QED lagrangian of the e.m. field QED 1 µν = Fµν F with Fµν = @µAν @ν Aµ: Lgauge −4 − The field-strength Fµν can be expressed through the covariant derivative i µ @µ + ieAµ Fµν = − [ µ; ν ] D ≡ −! e D D The generalization to QCD is now straightforward: i Fµν = [ µ; ν ] Fµν = @µAν @ν Aµ ig [Aµ; Aν ] : g D D −! − − a a a abc b c F = @µA @ν A + gf A A (verify!) µν ν − µ µ ν From the transformation of the covariant derivative in Eq. (2) one has y Fµν VFµν V ; not invariant! −! so that the proper SU(Nc )-invariant generation of the QED lagrangian is QCD 1 µν 1 a µν a = Tr(Fµν F ) = F F Lgauge −2 −4 µν where we have used Tr(tatb) = (1=2)δab. Andrea Beraudo Chiral symmetry The QCD Lagrangian and Feynman rules The final form of the QCD Lagrangian is then QCD X 1 a µν a = q [i = mf ]qf F F ; L f D− − 4 µν f leading to the following Feynman rules (ex: derive them!) µ µν j i i(pµγ + m) b,ν a,µ i( g + ...) δij δab − p2 m2 + iǫ k2 + iǫ p − k a p1 a, µ j c, ρ p2 a i p3 b, ν igtij g f abc[gµν(p p )ρ + gνρ(p p )µ + gρµ(p p )ν] 1 − 2 2 − 3 3 − 1 Andrea Beraudo Chiral symmetry Global symmetries and conserved currents Consider a Lagrangian invariant under the infinitesimal transformation φ φ + δφ −! i.e. @ @ 0= (φ + δφ) (φ) = Lδφ + L δ(@µφ) L − L @φ @(@µφ) @ @ @ = Lδφ + @µ L δφ @µ L δφ @φ @(@µφ) − @(@µφ) Employing the Euler-Lagrange equations one gets then the conserved current (at the classical level! quantum anomalies may appear) @ Z dQ j µ L δφ = Q d 3x j 0(x); with = 0 ≡ @(@µφ) ) ≡ dt a a In case of invariance under a continuous group φ~ e−iθ t φ~ one has a conseved current for each generator of the group,−! i.e. a a µ,a @ a φi φi i θ tij φj = j = i L tij φj −! − ) − @(@µφi ) Andrea Beraudo Chiral symmetry Partially conserved currents It may happen that the Lagrangian contains a term 1 which breaks explicitly the symmetry, but is very small compared toL the others: 0 + 1; with 1 0 L ≡ L L L L An example is represented by the mass term for the light quarks in the QCD Lagrangian, for which mlight ΛQCD. In this case the concept of a partially conserved current (PCC) is still very useful to understand qualitative features of the spectrum of the theory. Under the field transformation φ φ+δφ one has +δ 1, hence −! L−!L L @ @µ L δφ = δ 1 @(@µφ) L | {z } PCC Andrea Beraudo Chiral symmetry The massless QCD lagrangian: global symmetries The matter part of the massless QCD Lagrangian can be written as QCD = q [i = ]q = q [i = ]qR + q [i = ]qL: Lq D R D L D 1 γ5 In the above q [u; d; (s)]T and q ± q. ≡ R=L ≡ 2 The above Lagrangian is clearly invariant under the global symmetry UR (Nf ) UL(Nf ) = UR (1) SUR (Nf ) UL(1) SUL(Nf ) × × × × This corresponds to the following rotations in flavour space a a a a −iαR −iθ t −iα −iθ t qR e e R qR ; qL e L e L qL −! a a −! a a iαR iθ t iα iθ t q q e e R ; qL q e L e L R −! R −! L a 2 In the above t 's (a = 1; :::Nf 1) are the generators of the SU(Nf ) − a a group and all the parameters αR ; αL; θR ; θL are independent. One gets the classical conserved currents µ µ µ,a µ a jR=L = qR=Lγ qR=L; jR=L = qR=Lγ t qR=L a a µ,a µ a For Nf =2, t =τ =2 (Pauli matrices) and jR=L = qR=Lγ (τ =2)qR=L Andrea Beraudo Chiral symmetry The massless QCD lagrangian: global symmetries (II) It is convenient to combine the L/R currents into vector and axial ones V µ j µ + j µ = qγµq; ≡ R L Aµ j µ j µ = qγµγ5q; ≡ R − L V µ,a j µ,a + j µ,a = qγµ(τ a=2)q; ≡ R L Aµ,a j µ,a j µ,a = qγµγ5(τ a=2)q ≡ R − L a a Going from Nf =2 to Nf =3 one simply replaces( τ =2) (λ =2) (Gell-Man matrices in flavour space). −! They are associated to the symmetry group UV (1) SUV (Nf ) UA(1) SUA(Nf ) × × × We will comment on the role of each of these classical symmetries. Andrea Beraudo Chiral symmetry UV (1) symmetry It corresponds to the invariance for q e−iαq; q qeiα −! −! rotating by the same angle R and L components. It is associated to the conservation of the baryon number Z B (1=3)QV = (1=3) d 3x qy(x)q(x) ≡ Z 3 y y = (1=3) d x [qR (x)qR (x) + qL(x)qL(x)]; i.e. the net number of quarks (right-handed plus left-handed). Baryon number is exactly conserved in QCD. This does not mean that it is exactly conserved in nature, as suggested by the matter-antimatter asymmetry in our universe Andrea Beraudo Chiral symmetry UA(1) symmetry It corresponds to the invariance (of the massless Lagrangian) for 5 5 q e−iαγ q; q qe−iαγ (since γµ; γ5 = 0) −! −! f g 5 rotating by opposite angles R and L components (γ qR=L = qR=L). It would be associated to the conservation of the axial charge± Z Z 3 y 5 3 y y QA = d x q (x)γ q(x) = d x [q (x)qR (x) q (x)qL(x)]; R − L i.e. to the number of right-handed minus left-handed quarks. However, although being a symmetry of the classical QCD action, UA(1) is not a symmetry of the theory, being broken by quantum fluctuations: 2 µ g 1 αβµν a a @µA = Nf F F − 16π2 2 µν αβ 2 g αβ;a a Nf Fe F ≡ − 16π2 αβ Quantum anomalies of axial currents at the basis of the π0 γγ decay ! and of the η0 being much heavier than the other pseudoscalar mesons Andrea Beraudo Chiral symmetry SUV (Nf ) symmetry It corresponds to the invariance for a a a a q e−iθ t q; q qeiθ t ; (ta = τ a=2 or λa=2) −! −! rotating by the same angles θa's R and L components. It leads to the conservation (we focus on Nf =2) of the Isospin charges Z τ a Qa d 3x qy(x) q(x); V ≡ 2 which play the role of generators of the Isospin rotations. It is a symmetry of the Lagrangian and of the theory: QCD vacuum (and spectrum) invariant under Isospin transformations (more in the following) −iθaQa a e V 0 = 0 Q 0 = 0 j i j i , V j i Isospin charges annihilate the vacuum! Andrea Beraudo Chiral symmetry SUA(Nf ) symmetry It corresponds to the invariance for a a 5 a a 5 q e−iθ t γ q; q qe−iθ t γ ; (ta = τ a=2 or λa=2) −! −! rotating by opposite angles R and L components.
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