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 †V =1 and det(V )=1) color transformations, namely: q(x) V q(x) and q(x) q(x) V †, −→ −→ 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 †(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 †(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 † i † Aµ A = VAµV (∂µV )V . −→ µ − g

Exercise: verify that q is now invariant under local SU(Nc ) transformations. In particular:L † 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

† 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 q†(x)q(x) ≡ Z 3 † † = (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) −→ −→ { } 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 † 5 3 † † 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 q†(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 | i | i ⇔ V | 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. It leads to the conservation (we focus on Nf =2) of the axial charges Z τ a Qa d 3x q†(x) γ5q(x), A ≡ 2 playing the role of generators of the chiral (flavour-changing) rotations. Although being a symmetry of the Lagrangian, it is not a symmetry of the theory: QCD vacuum (and spectrum) not invariant under chiral rotations( spontaneous symmetry breaking!) −iθaQa a a a e A 0 = 0 Q 0 = 0 i.e. Q 0 Φ | i 6 | i ⇔ A| i 6 A| i ≡ | i Chiral charges create physical states: pseudoscalar mesons a a HQCD 0 = 0 and [HQCD, Q ] = 0 = H Φ = 0( Goldstone theorem) | i A ⇒ | i Pions (kaons and η) can be considered the Goldstone bosons associated to chiral-symmetry breaking: much lighter than other hadrons! Andrea Beraudo Chiral symmetry SUA(Nf ) symmetry and the QCD spectrum

According to the Goldstone theorem there is one massless boson for each generator of a broken continuous symmetry. a a λ In the case Nf =3, with generators t (a = 1, ...8), they can be ≡ 2 identified with the octet of pseudoscalar mesons

mπ 138 MeV ≈ mK 495 MeV ≈ mη 548 MeV ≈

NB due to the axial anomaly the original symmetry group is reduced

UA(Nf ) SUA(Nf ), hence the number of spontaneously broken −→ generator is 8 instead of 9: in fact mη0 = 958 MeV, much heavier than all the other pseudoscala mesons

Andrea Beraudo Chiral symmetry Quark masses and explicit symmetry-breaking

Consider the transformation of the mass term in the QCD Lagrangian

1 = qmq = q mqL q mqR L − − R − L One has, being m diag[mu, md , (ms )], ≡ qmq qmq iθaqmt aq + iθaqt amq + ... SU−→V (Nf ) − qmq qmq iθaqmt aγ5q iθaqγ5tamq + ... SU−→A(Nf ) − −

µ From ∂µj = δ 1 one gets: L µ,a a µ,a a 5 ∂µV = i q [m, t ]q and ∂µA = i q m, t γ q { } Focus on the Nf = 2 case:

as long as mu =md =0, the isospin current is exactly conserved. 6 Since mu md , isospin is an almost exact symmetry of QCD ≈ m=0 explicitly breaks chiral symmetry. However, since m ΛQCD, one6 speaks of Partially Conserved Axial Current (PCAC) 

Andrea Beraudo Chiral symmetry Transformations of mesonic currents

We now understand better the physical meaning of isospin and chiral transformation by considering their action on current operators having the quantum numbers to create/destroy different mesons: pion-like state: ~π i q ~τγ5q; sigma-like state: σ qq µ ≡ µ µ ≡ µ 5 rho-like state: ~ρ q ~τγ q; a1-like state: a~1 q ~τγ γ q ≡ ≡ Channel PS S V PV Particle π σ ρ a1 Mass (MeV) 138 500 770 1260

Pions are responsible for the long-range NN attractive interaction; Due to their large width, scalar mesons are difficult to identify. Usually one identify the sigma with the f0(500) state of the PDG, a broad 2-pion resonance, playing a major role in the nuclear binding; Vector mesons are responsible for the short-range repulsive NN interaction. In symmetric nuclear matter major role played by isoscalar ω(782) meson: ωµ q γµq. ∼ Andrea Beraudo Chiral symmetry Transformations of mesonic currents: SUV (Nf )

The transformation of the quark fields is given by a a a a q e−iθ (τ /2)q, q qeiθ (τ /2) −→ −→ From τ a, τ b = 2δab and [τ a, τ b] = 2iabc τ c , one gets: { } Isoscalar mesons: qq qq, hence σ σ −→ −→ Isovector mesons  τ b τ b  πa : i q τ aγ5q i q τ aγ5q + θb q τ aγ5 q q τ aγ5q −→ 2 − 2 = i q τ aγ5q + abc θb (i q τ c γ5q) Hence, they transform as vectors ~π ~π + θ~ ~π −→ × ~ρ µ ~ρ µ + θ~ ~ρ µ µ −→ µ ~× µ a~1 a~1 + θ a~1 −→ × Isospin rotations mix mesons belonging to the same multiplet, having the same mass. It is a symmetry of the QCD spectrum! Andrea Beraudo Chiral symmetry Transformations of mesonic currents: SUA(Nf )

The transformation of the quark fields is given by

a a 5 a a 5 q e−iθ (τ /2)γ q, q qe−iθ (τ /2)γ −→ −→ From τ a, τ b = 2δab and [τ a, τ b] = 2iabc τ c , one gets for instance { }  τ b τ b  πa : i q τ aγ5q i q τ aγ5q + θb q τ a q + q τ aq −→ 2 2 = i q τ aγ5q + θa q q

The complete set of chiral transformations is given by (verify!):

~π ~π + θ~ σ, σ σ θ~ ~π −→ −→ − · µ µ ~ µ µ µ ~ µ ~ρ ~ρ + θ a~1 , a~1 a~1 + θ ~ρ −→ × −→ × Scalar/pseudoscalar and vector/pseudovector mesons are mapped one into the other by chiral rotations. However, such a symmetry is not found

in the spectrum( mπ = mσ, mρ = ma ), it is spontaneously broken! 6 6 1 Andrea Beraudo Chiral symmetry Pion decay and PCAC

Pion decay is an electro-weak process u νµ + π _ d W+ µ+

In the Fermi theory it can be described as a point-like current-current interaction, with both vector and axial components (V A). Only the axial current has the quantum number to annihilate− a pion and the matrix element connecting the pion with the (QCD) vacuum is

a b ab −iq·x 0 A (x) π (q) = if πqµδ e h | µ | i − exp with fπ =93 MeV taken from τπ . Taking the divergence one gets:

µ a b 2 ab −iq·x 2 ab −iq·x 0 ∂ A (x) π (q) = fπq δ e = fπm δ e h | µ | i − − π µ a In case axial current were exactly conserved, ∂ Aµ =0 and hence mπ =0. Small pion mass comes from explicit symmetry breaking (mu,d =0) 6 Andrea Beraudo Chiral symmetry Spontaneous chiral-symmetry breaking: the linear σ-model

The linear σ-model is the simplest hadronic lagrangian consistent with the isospin and chiral symmetry and including the possibility of spontaneous breaking of chiral symmetry. Here we illustrate its construction. Remember the transformation of the pion and sigma fields: ~ SUV (2) : ~π ~π + θ ~π, σ σ −→ × −→ ~ ~ SUA(2) : ~π ~π + θ σ, σ σ θ ~π −→ −→ − · Hence, under infinitesimal transformations, for their squares one has:

2 2 2 2 SUV (2) : ~π ~π , σ σ −→ −→ 2 2 ~ 2 2 ~ SUA(2) : ~π ~π + 2 θ ~πσ, σ σ 2 θ ~πσ −→ · −→ − · 2 2 The combination (~π +σ ) is invariant under SUV /A(2) transformations: pion and sigma fields must enter in the Lagrangian in such a form:

φ 1 µ µ 2 2 = (∂µ~π ∂ ~π + ∂µσ ∂ σ) V (~π +σ ) Lσ−mod 2 · − where, with compact notation, φi (~π, σ)(i =1, ...4). The linear ≡ σ-model is also known, due to its symmetry, as O(4) model

Andrea Beraudo Chiral symmetry Let us now introduced also the nucleons. In order not to break chiral symmetry they must enter in the Lagrangian as massless particles: ψ,kin = iψ∂/ψ, where ψ (p, n)T Lσ−mod ≡ Under SUV /A(2) transformations the nucleon behaves as the quark field −iθa(τ a/2) iθa(τ a/2) SUV (2) : ψ e ψ, ψ ψe −→ −→ −iθa(τ a/2)γ5 −iθa(τ a/2)γ5 SUA(2) : ψ e ψ, ψ ψe −→ −→ SUV (2) SUA(2) symmetry constraints its coupling with the mesons × ψ,φ  5  = gπ ψ iγ ~π ~τ + σ ψ Lσ−mod − · Let us verify it invariance under SUV (2) transformations: ψσψ ψσψ −→ θa ψ[ iγ5~π ~τ ]ψ ψ[ iγ5~π ~τ ]ψ+i ψ[iγ5τ aτ b πb]ψ · −→ · 2 θa i ψ[iγ5τ bτ a πb]ψ + ψ iγ5(θ π) ~τ ψ − 2 × · From[ τ a, τ b]=2iabc τ c one gets ψ[ iγ5~π ~τ ]ψ ψ[ iγ5~π ~τ ]ψ · −→ · Repeat for SUA(2) transformations!

Andrea Beraudo Chiral symmetry We have seen that, although the QCD (and the effective σ-model) Lagrangian is invariant under SUA(Nf ), the spectrum is not. The potential must induce a spontaneous breaking of chiral symmetry

(a)

(y,π)

V(σ, π=0) (x,σ)

(b) (y,π)

(x,σ)

fπ σ

λ 2 V = V (~π2 +σ2) = (~π2 + σ2) f 2 4 − π Fields acquire a vacuum expectation value (VEV) minimizing the potential ~π = 0 and σ = σ0 = fπ. h i h i NB Identification of fπ with the minimum of the potential to be proven! Andrea Beraudo Chiral symmetry We can perform fluctuations around the VEV and rewrite the full Lagrangian in terms of the new field variables ~π(x) and σ(x) = fπ + s(x) i i i It is convenient to employ the field φ (~π, σ) = (~π, fπ + s) = φ0 + η , i i4 ≡ with the VEV φ δ fπ. The potential reads then 0 ≡ λ 2 λ h i V = (φi )2 f 2 = (φi + ηi )2(φj + ηj )2 2(φi + ηi )2f 2 + f 4 4 − π 4 0 0 − 0 π π Constant and linear terms in the fluctuations ηi vanish and one gets 1 λ V = (2λf 2) s2 + λf s (π2 + s2) + (π2 + s2)2 2 π π 4 Notice that:

Pions are massless: mπ = 0 (Goldstone theorem) 2 2 The σ-meson gets a mass: mσ = 2λfπ Cubic self-interaction terms among the mesons appear From the coupling with the nucleon field one gets:

ψ,kin ψ,φ  5  + = ψ [i∂/ gπfπ] ψ gπ ψ iγ ~π ~τ + s ψ Lσ−mod Lσ−mod − − · The nucleon gets its mass from chiral-symmetry breaking! Andrea Beraudo Chiral symmetry Nucleon mass: some comments

ψ,kin ψ,φ  5  + = iψ∂/ψ gπ ψ iγ ~π ~τ + σ ψ Lσ−mod Lσ−mod − ·  5  = ψ [i∂/ gπfπ] ψ gπ ψ iγ ~π ~τ + s ψ − − ·  5  = ψ [i∂/ MN ] ψ gπ ψ iγ ~π ~τ + s ψ − − ·

The model allows the nucleon to get a mass in a way consistent with the chiral-symmetry of the Lagrangian: no explicit symmetry breaking! Most of the present baryonic mass in the universe got its origin at the chiral transition, when σ = 0 σ = 0 h i −→ h i 6 One gets the Goldberger-Treiman relation gπ = MN /fπ, entailing gπ 10 (it will receive positive corrections): chiral symmetry allows one≈ to get an estimate of the (QCD!) pion-nucleon coupling, just from the nucleon mass and the pion decay-constant (e.w. process!). The pion-nucleon coupling is very large!

Andrea Beraudo Chiral symmetry Linear σ-model: the full Lagrangian

Let us collect all the terms and write the full Lagrangian:

1 µ 1 µ 1 2 2 σ−mod = ∂µ~π ∂ ~π + ∂µs ∂ s m s L 2 · 2 − 2 σ 2 2 λ 2 2 2  5  λfπ s (π + s ) (π + s ) + ψ [i∂/ MN ] ψ gπ ψ iγ ~π ~τ + s ψ − − 4 − − ·

2 2 with mσ = 2λfπ and MN = gπfπ

Andrea Beraudo Chiral symmetry Linear σ-model: the axial current

Starting from the Lagrangian written in terms of the original fields

1 µ µ λ  2 2 22 σ−mod = (∂µ~π ∂ ~π + ∂µσ ∂ σ) (~π + σ ) f L 2 · − 4 − π  5  + iψ∂/ψ gπ ψ iγ ~π ~τ + σ ψ − · and from the transformation law of the fields under SUA(2) τ a ψ ψ iγ5 θaψ ~π ~π + θσ~ σ σ θ~ ~π −→ − 2 −→ −→ − · one gets: a µ ∂ i a ¯ τ a a j = L i δΦ Aµ = ψγµγ5 ψ π ∂µσ + σ∂µπ ∂(∂µΦ ) −→ 2 − In terms of the shifted fields one has a a τ a a a A = ψγ¯ µγ5 ψ π ∂µs + fπ∂µπ + s ∂µπ , µ 2 − consistently with the pion-decay matrix element a b ab −iq·x 0 A (x) π (q) = i fπqµδ e h | µ | i − Andrea Beraudo Chiral symmetry Explicit symmetry-breaking

σ−mod must be supplemented by a small explicit symmetry breaking termL arising from the non-zero quark masses: QCD = qmq σ−mod = σ L1 − −→ L1 This corresponds to a modification of the potential

V(σ, π=0) λ 2 V = (~π2 + σ2) v 2 σ 4 − 0 −

We require σ = fπ, so to satisfy MN = gπfπ h i

∂V 2 2  = 0 v0 = fπ ∂σ −→ − λfπ σ fπ fπ

The σ-meson and pions receive a positive mass correction 2 2 2 ∂ V 2  2 ∂ V  m = = 2λf + , m = = = 0 σ ∂σ2 π f π ∂π2 f (0,fπ ) π fπ π 6 Andrea Beraudo Chiral symmetry The nucleon mass

We have imposed the Golberger-Treiman relation MN = gπfπ even in the presence of explicit symmetry breaking. However, we may ask which fraction of the nucleon mass comes from the spontaneous (v0) and from the explicit () symmetry-breaking. To linear order in  we have: 1     v0 fπ 2 MN = gπfπ = gπ v0 + 2 ≈ − 2 λfπ −→ 2λfπ Hence 2 2 2  fπmπ mπ mπ δMN = gπ = gπ = MN MN MN 2λf 2 2λf 2 m2 m2 ≈ m2  π π σ − π σ Experimentally one gets δMN 50 MeV ≈ only a small fraction of the baryonic mass of our universe is due to the Higgs mechanism (via mu,d = 0) at the electro-weak phase transistion! 6 most of the baryonic mass of our universe got its origin at the chiral QCD transition at T 150 MeV ≈ Andrea Beraudo Chiral symmetry Linear σ-model and PCAC

From the SUA(2) transformation of the symmetry-breaking term a a 1 = σ 1 + δ 1 = σ  θ π L −→ L L − µ and from ∂µj = δ 1 one gets: L µ a a 2 a ∂ A = π = fπm π , µ − − π consistently with the pion-decay matrix element µ a b 2 ab −iq·x 2 ab −iq·x 0 ∂ A (x) π (q) = fπq δ e = fπm δ e h | µ | i − − π Finally, from the link with the QCD Lagrangian σ−mod = σ QCD = qmq L1 L1 − one gets (mu md ) the Gell-Mann-Oakes-Renner (GOR) relation: ≈ m +m  σ = m qq f 2m2 = u d uu+dd h i − h i −→ π π − 2 h i establishing a link between hadronic and partonic quantities. Andrea Beraudo Chiral symmetry O(4) model at finite temperature: χ-symmetry restoration

We now wish to study the temperature evolution of the order parameter associated to chiral-symmetry breaking, i.e.

σ T (hadron level) or qq T (quark level) h i h i It is convenient to consider σ−mod as the N =4 case of the O(N) model, performing the calculationsL in the N limit. Hence: → ∞ λ 2 V = (φi )2 f 2 , with φi (~π, σ) 4 − π ≡ At finite temperature we can consider the thermal expectation value   ∂V i 2 j 2 j j = 0 (φ ) φ T fπ φ T = 0 ∂φ T −→h i − h i We can split the fields as follows: i i i φ φ T + η , where ~π T = 0, σ T σT ≡ h i h i h i ≡ i i j k One gets ( η T = η η η T = 0): h i h i i
2 j i i j i j i 2 j φ T φ T + η η T φ T + 2 η η T φ T fπ φ T = 0 h i h i h| {zi }h i |h {zi }h i − h i O(N) O(1)

Andrea Beraudo Chiral symmetry It is now convenient to perform the large-N approximation (N =4 corresponding to the physical case, with 3 pions and one σ-mesons)

i
2 j i i j 2 j φ T φ T + η η T φ T f φ T = 0 h i h i h i h i − π h i N We focus on the equation for the φ T σ T σT . One has: h i ≡ h i ≡ Z ~ 2 2 2 i i i i X dk Nk X T σT = fπ η η T with η η T = 3 i − h i h i (2π)  mi∼→0 12 i k i

i p 2 2 In the above Nk is a Bose distribution and k = k + mi . We can study two different limits:

T 0: N 1 massless pions; σ-meson (mσ T ) can be neglected → −   2   2  2 2 T 2 T σT = fπ 1 (N 1) 2 fπ 1 2 − − 12fπ N∼→4 − 4fπ

T Tc : pions and σ-meson (mσ T ) become degenerate and massless→   2   2  2 2 T 2 T σT = fπ 1 N 2 fπ 1 2 , − 12fπ N∼→4 − 3fπ

from which one gets the estimate Tc √3fπ 160 MeV. ≈ ≈ Andrea Beraudo Chiral symmetry χ-symmetry restoration: experimental studies

Signatures of chiral-symmetry restoration in the hot/dense medium produced in heavy-ion collisions studied looking at the ρ-meson spectral function. In fact

SUA(2) transformations map ρ and a1 meson into each other

µ µ ~ µ µ µ ~ µ ~ρ ~ρ + θ a~1 , a~1 a~1 + θ ~ρ , −→ × −→ × so that, if chiral-symmetry were restored, their spectral functions should merge: ρV (ω,~p) = ρA(ω,~p)

Decay-width Γ 150 MeV lifetime τ 1.3 fm/c τfireball: it decays inside the≈ medium and−→ its spectral≈ function is affected by medium-modifications In the ρ e+e− and ρ µ+µ− channels, decay products do not further interact→ with the→ medium (no colour-charge!) and carry direct information on in-medium spectral function. Furthermore, muons can be easily identified placing an absorber before the detector

Andrea Beraudo Chiral symmetry In the meddium( AA collisions) the ρ-meson spectral funtion is much broader and its strength above 1 GeV may suggest a mixing with the a1 meson.

χ-symmetry restoration: experimental studies

0.08 0.030 Total AV SF æ ALPEH 2Hn 1L data æ ALPEH 2nΠ data + Π á ALPEH 2 data 0.025 Total a1 SF Π  0.06 Total vector SF Total a1 SF Ρ SF 0.020 continuum

s ' SF

Ρ s

L continuum L s s H 0.04 H 0.015 V A Ρ Ρ 0.010 0.02 0.005 0.00 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 s HGeV2L s HGeV2L

In the vacuum( e+e− collisions) the ρ-meson spectral funtion is characterized by a sharp peak, very different from the a1-meson

Andrea Beraudo Chiral symmetry χ-symmetry restoration: experimental studies

2 3500 0.08 In-In SemiCentral 3000 all p in-med ρ æ ALPEH 2nΠ data T QGP á ALPEH 2Π data DD 0.06 Total vector SF 2500 4π mix Ρ SF Hees/Rapp sum

s Ρ' SF 2000 sum+in-med(ω+φ)

L continuum s

H 0.04 V 1500 Ρ dN/dM per 20 MeV/c 0.02 1000

500 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 s HGeV L M (GeV/c2)

In the vacuum( e+e− collisions) the ρ-meson spectral funtion is characterized by a sharp peak, very different from the a1-meson In the meddium( AA collisions) the ρ-meson spectral funtion is much broader and its strength above 1 GeV may suggest a mixing with the a1 meson.

Andrea Beraudo Chiral symmetry Bibliography

Most of the material used for this lecture can be found in: V. Koch, Aspects of Chiral Symmetry, nucl-th/9706075; For a simplified treatment of the finite-temperature part see also: J.I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications, Cambridge University Press; For a comprehensive introduction to chiral effective theories see: S. Scherer and M.R. Schindler, A Chiral Perturbation Theory Primer, hep-ph/0505265.

Andrea Beraudo Chiral symmetry