SLIDE 1
SU(4)-Ward identities for QCD with restored chiral symmetry
Vasily Sazonov
LPT Orsay, University of Paris Sud 11
SLIDE 2 Low mode truncation
According to the Banks-Casher relation: lim
m→0ψψ = πρ(0)
When chiral symmetry is restored (for instance at T > Tc) ψψ = 0 = ⇒ ρ(0) = 0 What if we artificially enforce ρ(0) = 0, by removing zero and near-zero modes of the Dirac operator from computations? How one can do it? [C.B. Lang, Mario Schr¨
- ck, Phys. Rev. D 84 (2011) 087704 ]:
S = SFull −
k
1 λi |λiλi| The correlators of the hadron interpolators lim
t→∞O(t)O†(0) =
0|O|nn|O†|0e−tEn are expressed as convolutions of S.
SLIDE 3
Expected degeneracies in the spectrum
Each meson is denoted as (I, JPC), with I isospin, J total spin, P parity and C charge conjugation. The left column represents the irreducible representations of the parity-chiral group, (IR, IL). Below each state its generating current is given. The scheme is adopted from [L. Ya. Glozman, M. Pak, Phys. Rev. D 92, 016001 (2015)]
SLIDE 4
Evolution of hadron masses under the low-mode truncation
[M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502]: J = 1 k is the number of removed lowest eigenmodes and σ is the corresponding energy gap (100 gauge field configurations, generated with NF = 2 dynamical overlap fermions on a 163 × 32 lattice with the spacing a ∼ 0.12 fm)
SLIDE 5
Observed degeneracies in the spectrum
Each meson is denoted as (I, JPC), with I isospin, J total spin, P parity and C charge conjugation. The left column represents the irreducible representations of the parity-chiral group, (IR, IL). Below each state its generating current is given. The scheme is adopted from [L. Ya. Glozman, M. Pak, Phys. Rev. D 92, 016001 (2015)]
SLIDE 6
Development of the topic
◮ J = 2 mesons, [M. Denissenya, L.Ya.Glozman, M.Pak, PRD
91(2015)114512]
◮ J = 1/2 baryons, [M. Denissenya, L.Ya.G, M.Pak, PRD 92
(2015) 074508]
◮ SU(4) symmetry, [L.Ya.Glozman, EPJA 51(2015)27],
[L.Ya.Glozman, M. Pak, PRD 92(2015)016001]
◮ Prediction of the SU(4) restoration at T > Tc, relation to
confinement, [L.Ya.Glozman, arXiv 1512.06703]
◮ Approximate SU(4) degeneracy, observed at T > Tc, [C.
Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, L. Ya. Glozman, S. Hashimoto, C. B. Lang, S. Prelovsek, Phys. Rev. D 96, 094501 (2017)]
SLIDE 7 SU(4)-symmetry transformations
The quark QCD Lagrangian with NF = 2 flavors in the Euclidean space is L = Ψ(1F ⊗ (γµDµ + m))Ψ , µ = 1..4 , The transformations of the SU(4) group are completely determined by generators Tl, satisfying the su(4) algebra commutation
- relations. Tl are given by 15 matrices
{(τ a ⊗ 1D), (1F ⊗ Σi), (τ a ⊗ Σi)} , a, i = 1, 2, 3 . τ a are the generators of SU(2)L × SU(2)R chiral symmetry.
SLIDE 8
The matrices Σi are Σi = {γ4, iγ5γ4, −γ5} and satisfy the su(2) algebra commutation relations. The generators Σi define chiral spin SU(2)CS group with transformations acting only in the Dirac space Ψ → Ψ′ = eiǫ(x)·(1F ⊗Σ)Ψ . The transformations of the full SU(4) group Ψ → Ψ′ = eiǫ(x)·TΨ , Ψ = uL uR dL dR mix both quark flavors and left-/right-handed components.
SLIDE 9 In the Euclidean space spinors Ψ and Ψ are completely
- independent. The action of SU(4) on the spinor Ψ is defined to be
equivalent of the SU(4) action on the Ψ = −iΨ†γ4. Generators δΨ δΨ 1F ⊗ γ4 iǫ(x)γ4Ψ −iǫ(x)Ψγ4 1F ⊗ iγ5γ4 −ǫ(x)γ5γ4Ψ −ǫ(x)Ψγ5γ4 1F ⊗ (−γ5) −iǫ(x)γ5Ψ −iǫ(x)Ψγ5 τa ⊗ 1D iǫ(x)τaΨ −iǫ(x)Ψτa τa ⊗ γ4 iǫ(x)τaγ4Ψ −iǫ(x)Ψγ4τa τa ⊗ iγ5γ4 −ǫ(x)τaγ5γ4Ψ −ǫ(x)Ψγ5γ4τa τa ⊗ (−γ5) −iǫ(x)τaγ5Ψ −iǫ(x)Ψγ5τa
Table: Infinitesimal variations of spinors Ψ and Ψ under the SU(4) transformations, the tensorial product of flavor and Dirac spaces is
- mitted to shorter notations.
SLIDE 10 Lagrangian transformations
The quark Lagrangian is not invariant under the SU(4) transformations. Generators δL = δΨ D Ψ + Ψ D δΨ 1F ⊗ γ4 i(∂µǫ(x))Ψγµγ4Ψ− 2iǫ(x)Ψγ4γi[∂i − igAi]Ψ 1F ⊗ iγ5γ4 −(∂µǫ(x))Ψγµγ5γ4Ψ − 2mǫ(x)Ψγ5γ4Ψ −2ǫ(x)Ψγiγ5γ4[∂i − igAi]Ψ 1F ⊗ (−γ5) −i(∂µǫ(x))Ψγµγ5Ψ − 2iǫ(x)mΨγ5Ψ τa ⊗ 1D i(∂µǫ(x))ΨγµτaΨ τa ⊗ γ4 i(∂µǫ(x))Ψγµγ4τaΨ − 2iǫ(x)Ψγ4γiτa[∂i − igAi]Ψ τa ⊗ iγ5γ4 −(∂µǫ(x))Ψγµγ5γ4τaΨ − 2mǫ(x)Ψγ5γ4τaΨ −2ǫ(x)Ψγiγ5γ4τa[∂i − igAi]Ψ τa ⊗ (−γ5) (∂µǫ(x))Ψγµγ5τaΨ − 2iǫ(x)mΨγ5τaΨ
Table: Variations of the quark Lagrangian under the SU(4)
- transformations. Here D = (1F ⊗ (γµDµ + m)) and in the right part of
the table the tensorial product of flavor and Dirac structures is omitted to shorter notations.
SLIDE 11 Anomalous violation of the SU(4) symmetry
Consider QCD in a finite volume with appropriate boundary
- conditions. Then, the anomaly term of the action is given by the
sum −iTrF[gF]
φ†
k(x)
where Ψ → Ψ
′ = Ψ eiǫ(x)·T ,
T = (gF ⊗ Σ) , gF = {1F, τa} and an appropriate regularization is assumed (see later). When gF = 1F, TrF[gF] = NF, alternatively, one has TrF[gF = τa] = 0
SLIDE 12 Identities
We treat terms containing ∂µǫ(x) employing periodic boundary conditions for ǫ(x) and periodic/anti-periodic boundary conditions for quark fields. The bilinear forms containing the mixing of quark flavors performed by τ a vanish after the integration over the quark fields and one ends up with the three following identities ∂µΨγµγ4ΨA + 2Ψγ4γi[∂i − igAi]ΨA = 0 , ∂µΨγµγ5γ4ΨA − 2mΨγ5γ4ΨA − 2Ψγiγ5γ4[∂i − igAi]ΨA +4i lim
M→∞
φ†
k(x)γ5γ4e−(λk/M)2φk(x) = 0 ,
∂µΨγµγ5ΨA + 2imΨγ5ΨA + 1 8π2 Tr ∗FµνFµν = 0
SLIDE 13 Spectral properties of SU(4)-Ward identities Classical part of identities
Consider the terms, related to the non-invariance of the classical action utilizing the spectral representation Ψ(x)OΨ(x)A =
φ†
n(x)Oφn(x)
1 m − iλn The summation over n is ill-defined and has to be regularized Ψ(x)OΨ(x)A = lim
M→∞
φ†
n(x)Oe−(λn/M)2φn(x)
1 m − iλn Then, we split the sum over n as
+
+
SLIDE 14 The anti-commutation of γ5 with matrices γµ ensures that iγµDµ(γ5φn) = −λn(γ5φn) ≡ λ−nφ−n and φ−n = γ5φn ⇒ φ†
−n = φ† nγ5
Now we join the sums over the positive and negative eigenvalues Ψ(x)OΨ(x)A = lim
M→∞ λn=0
φ†
n(x)Oφn(x) 1
m +
n(x)Oe−(λn/M)2φn(x)
1 m − iλn +φ†
n(x)γ5Oe−(λn/M)2γ5φn(x)
1 m + iλn
SLIDE 15 Operators from Ward identities O = {γ4γi[∂i − igAi], γiγ5γ4[∂i − igAi]} commute with γ5 [O, γ5] = 0 Then, Ψ(x)OΨ(x)A = lim
M→∞ λn=0
φ†
n(x)Oφn(x) 1
m +
φ†
n(x)Oe−(λn/M)2φn(x)
2 m λ2
n + m2
- Therefore, in the zero mass limit the dominant contributions to the
expectation values Ψ(x)OΨ(x)A come from zero modes of the Dirac operator.
SLIDE 16 Anomalous part
◮ Axial anomaly = n+ − n−, consequently vanishes in a
presence of a spectral gap
◮ (1F ⊗ (iγ5γ4)) - related anomaly
φ†
k(x)γ5γ4e−(λk/M)2φk(x) =
φ†
k(x)γ5γ4e−(λk/M)2φk(x) +
k(x)γ5γ4e−(λk/M)2φk(x) + φ† −k(x)γ5γ4e−(λk/M)2φ−k(x)
φ†
k(x)γ5γ4e−(λk/M)2φk(x) +
φ†
k(x){γ5, γ4}e−(λk/M)2φk(x) =
φ†
k(x)γ5γ4e−(λk/M)2φk(x)
hence, it is also inconsistent with the spectral gap
SLIDE 17 Banks-Casher relation, T > TC, SU(4)
lim
m→0ψψ = πρ(0) .
Consequently, above Tc, when ψψ = 0
◮ ρ(0) = 0 ◮ ρ(λ) = |λ|α, α > 2, [S. Aoki, H. Fukaya, Y. Taniguchi, Phys.
- Rev. D 86, 114512 (2012)]
◮ or ρ(λ) = 0, for λ < λc ◮ The latter means that the pion, sigma, delta, and eta(-prime)
meson correlators are all identical, realizing the U(1)A symmetry, [T. D. Cohen, Phys. Rev. D 54, 1867 (1996)].
SLIDE 18
Conclusions
◮ Ward identities provide the most natural framework for
describing quantum symmetries.
◮ Existence/emergence/restoration of symmetries is realized in
the simplicity of corresponding Ward identities.
◮ The re-parametrization identities for SU(4) transformations
become Noether conservation laws if the spectrum of the Dirac operator contains a gap at low modes.
◮ The latter can be the case at sufficiently high temperatures. ◮ All derivations are equally applicable to interacting and free
quarks, consequently, there is no chance to judge existence of confinement on the basis of the possible emergence of the SU(4)-symmetry.
