Technische Universität München

Chiral symmetry breaking in QCD
andfinite-volume effects
Bertram Klein
Technische Universität München
Renormalization Group Approach from Ultra Cold Atoms to the Hot QGP, Kyoto, August 26 20111

Technische Universität München

Hadronic matter under extreme conditions• heavy-ion collisions[STAR experiment, RHIC, BNL]

[GSI]

• hadronic matter
at finite temperature and density2Technische Universität München

QCD: chiral symmetry and confinement• SU(3) gauge theory of strongly interacting quarks andgluons���

SQCD =d x41ψ̄ (i D/ − m) ψ − Fµν F µν
4• chiral flavor symmetry: symmetry for 2 massless quarks(flavors (u, d))
SU (2)L × SU (2)R
spin
spin
momentum

momentumleft-handed quarks

right-handed quarks

ψL → UL ψLψR → UR ψR
3

Technische Universität München

Spontaneous chiral symmetry
breaking
Chiral symmetry restoration and d21• Chiral quark condensate• chiral symmetry is spontaneously broken
TrTrL
in the QCD groundstate"
1.0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.00.400.50

0.600ren

0.8

0.60.6• order parameter: chiral condensate
N!=40.468††�ψ̄ψ� = �ψLψR + ψRψL � =
� T [MeV]
00.2

150

200

250

0.70

00.8

0.0
100

0.30

300

350

400

�ψ̄ψ�(T )�ψ̄ψ�(0)s,lm→0N!=460.40.2• strictly
phase• finite q• cross

T [MeV]
450

0.0
100

150200250300• Goldstonebosons of spontaneous symmetry breaking:
FIG. 11: Renormalized Polyakov loop on lattices with temporal extent N = 4, 6 and 8 (left) and the normalized difference ofand strange
quark chiral condensates
defined in Eq. 36.
The vertical lines show thelocation of the transitionpions arelightthe(almostmassless)low-energy
degreesof temperature
determined in [22] on lattices with temporal extent N = 4 (right line) and in this analysis for N = 6 (left line).
freedom[M. Cheng et al., Phys. Rev. D 77 (2008) 014511.]
τττquark mass differs slightly from the one used in that earlier study. We therefore performed a new determination of
the transition temperature for the Nτ = 6 lattice and the parameters of the LCP used here. From the peak positionsof the disconnected parts of the light and strange quark susceptibilities we find βc (Nτ = 6) = 3.445(3). Using thevalue for r0 /a quoted for this value of the coupling in Table I we find4 Tc r0 = 0.466(6) or Tc = 196(3).
We note that the region of most rapid change in the subtracted and normalized chiral condensate, ∆l,s (T ), is ingood agreement with the region where the Polyakov loop expectation value as well as bulk thermodynamic quantities,e.g. the energy and entropy densities change most rapidly.
4• low-momentum QCD: theory of weakly interacting pionsVII.

DISCUSSION AND CONCLUSIONSTechnische Universität München

Deconfinement: Z(3) symmetry breaking• Polyakov loop acts as order parameter�1
Φ(�x ) =Tr exp i
Nc• Z(3) center symmetry

Φ → zΦ

� 1/T0�dτ A4 (�x, τ )τx��
2πnz = exp i
Nc

de-confined phase: Z(3) symmetry spontaneously broken

confined phase: Z(3) symmetric�Φ� =� 0�Φ� = 0• Polyakov loop corresponds to free energy of a free static
quark:Φ ∼ exp (−Fq /T )5

Technische Universität München
Technische Universität München

Chiral symmetry restoration and deconfinement
Chiral symmetry restoration and deconfinement
210.9• •Chiral
quark
condensate
Chiralquark
condensate
0.30
0.40
0.50
0.60
0.70

1.0

1.0
0.8

Tr0�ψ̄ψ�(T )�ψ̄ψ�(0)"s,l0.6m→0• crossoverin presenceof• strictly
speaking, nota truequarksphasetransition
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
• finite1.0quarkmass
Tr0Lren• crossover
Φ(T )

N!=4
6

0.4
0.2

0.30

1.0

0.8

0.8

0.6

0.6

0.4

"s,lT [MeV]

T [MeV]400• Polyakov loop

450

0.0
100

150

200

250

300m→∞

N!=4•
finite
quark
mass
[M.
Cheng
etal.,Phys.
Rev.
D77(2008)
014511.]6es with temporal extent N = 4, 6 and 8 (left) and the normalized difference of
8
ed in Eq. 36. The vertical lines show the location of the transition temperature• strictly
speaking,a true
tent N = 4 (rightline) and in thisanalysis for N = 6 not(left line).
T [MeV]phase transition
ed in that earlier study. We therefore performed a new determination of[M. Cheng et al., Phys. Rev. D 77 (2008) 014511.]
attice and the •parametersof the LCP used here. From the peak positions
crossover
FIG. 11: Renormalized Polyakov loop on lattices with temporal extent N = 4, 6 and
0.4

0.4

τ

0.2

τ

0.2

τ

0.0
100

150

200

250

300

350

400

450

0.0100150

strange quark susceptibilities we find βc (Nτ = 6) = 3.445(3). Using theτ4lightand
strangequarkchiralcondensates
definedinEq.
36.
The
verticallines show t
oupling in Table I we find Tc r0 = 0.466(6) or Tc = 196(3).
6hange in the subtracted and normalized chiral condensate, ∆l,s (T ), is in determined in [22] on lattices with temporal extent Nτ = 4 (right line) and in this an
olyakov loop expectation value as well as bulk thermodynamic quantities,

Technische Universität München

Phase transitions in QCD

SU (2)L × SU (2)R→ O(4)Z(3)

SU (3)L × SU (3)R[F. .Karsch and E. Laermann, arXiv:hep-lat/0305025.]7

perturbative QCD and experimentTechnische Universität München

Methods for studying QCD! Asymptotic freedom at high momemtumscales (Politzer ’73, Gross and Wilczek ’73)

• QCD: perturbative at large momentum scales,
non-perturbative at low momentum scales Q! running coupling at small momentum scales?

−→ pQCD fails (Landau-pole)! confinement, χSB(Bethke ’04)

• chiral perturbation theory: Effective Field Theory
systematic expansion of low-energy QCD• lattice QCD: simulation of the gauge
theory in finite Euclidean space-time
volume!
x1/Ta8

Technische Universität München

Methods for studying QCD• Dyson-Schwinger Equations for QCD[C. Fischer and R. Alkofer, J. Lücker, ...]• functional Renormalization Group methods– for models, including confinement [B.-J. Schaefer, T. Herbst, A. Tripolt, V. Skokov, B.
Friman, K. Redlich ...]

– for QCD [J. Braun, H. Gies, L. Haas, L. Fister, Pawlowski, ... ]• (chiral) model calculations for transition regions➞ this talk9

Technische Universität München

QCD in a finite volume• finite volume introduces an additional scale 1/L:
probes different momentum regions, like temperature T• no real phase transitions, critical long-range behavior
affected by the volume size
• quark boundary conditions: what to choose in a finite
spatial volume? periodic or anti-periodic?
finite-volume effects in the analysis?
10Technische Universität MünchenTopics• Critical scaling for the chiral phase transition
in infinite and finite volume• Phenomenological implications of finite-volume effects for
the QCD phase diagram

long-range fluctuations: Renormalization Group (RG) methods

11Technische Universität München

The quark-meson model• a model for chiral symmetry breaking• no gauge degrees�of freedomΓΛ [ψ̄, ψ, σ, �π ]=

�d4 x ψ̄(i /∂ )ψ + g ψ̄(σ + iγ5�τ · �π )ψ

�
11+ (∂µ σ)2 + (∂µ�π )2 + UΛ (σ, σ 2 + �π 2 )22• chiral symmetry breaking: SU(2) × SU(2) → SU(2)as O(4) → O(3) (meson sector) �σ� =� 0• specify effective action for the model at initial scale Λ• use functional Renormalization Group(Wetterich equation) to obtain effective action[C.Wetterich, Phys. Lett. B 301 (1993) 90.]12

Technische Universität München

Renormalization Group calculation• RG flow equation with 3d optimized cutoff function:change of effective potential with change of RG scale kk∂k Uk (φ2 )=k5�

3Eπ➞ see also talk by M. Scherer����
1
11+ nB (Eπ ) Bp (kL) ++ nB (Eσ ) Bp (kL)2Eσ 2���
2Nc Nf−1 − nF (Eq , µ) − nF (Eq , −µ) Bp (kL)Eq• volume dependence encoded in mode-counting functions• depends on choice of boundary conditions(for the spatial directions)13Technische Universität München

Critical behavior, scaling and universality• Critical point (e.g. second-order phase transition)• diverging correlation length ξ ∼ 1/Mσ → ∞• critical long-range fluctuations dominate the physics ofa system close to the critical point• universality classes: the same critical behavior isobserved in different systems with the same symmetries
and dimensionality• critical behavior is characterized by a few
critical exponents β, γ, δ, ν and scaling functions
finite volume cuts off long-range fluctuation andimpacts critical scaling behavior!
14

Technische Universität München

Scaling behavior in infinite volume• two relevant couplings: temperature T and symmetry
breaking (quark mass m) H

t = (T − Tc )/T0 ,h = H/H0• Scaling function for the order parameter M ( fπ , �ψ̄ψ� ):M (t, h = 0) = (−t)β

M (t, h) = h1/δ fM (z)M (t = 0, h) = h1/δz = t/h1/(βδ)• Scaling function for the susceptibility χσ :χσ (t, h)=∂M
1 1/δ−1(t, h) =hfχ (z)∂HH015χσ [1/GeV2 ]250000200000Technische Universität München

150000100000

Susceptibility χσ from the50000model: O(4)0-3e-005

7 Quark-Meson-Modell für unendliches Volumen-1.5e-005

0• susceptibility χσ for small values of mπ < 0.9 MeV
t = (T − T )/Tc

1.5e-005

3e-00503500000.35

3000000.3

2500000.25

χσ H0 h1−1/δχσ [1/GeV2 ](a)2000001500000.20.15

1000000.1

50000

0.050-3e-005-1.5e-005

0t = (T − Tc )/T0(a)1.5e-005

3e-005O(4)0-5-4-3-2-1012345z = t/h1/βδ(b)• rescaled susceptibility
χσ H0 h1-1/δ/δ

0.35Abbildung
7.13: (a)Sigma-Suszeptibilität als Funktion von t, Mπ bei T = 0 liegt z[P. Piasecki, J. Braun, and B. Klein, Eur. Phys.
J. C71, 1576
(2011)]
schen 0.2 MeV und 0.9 MeV; (b) Reskalierte Sigma-Suszeptibilität0.3Funktion von z Mπ bei T = 0 liegt zwischen
0.2 MeV und 0.9 M
16Das Maximum χres,max = 0.33 liegt bei z = 1.31.
0.25

fπ

0.80.60.4
0.2
0Technische Universität München

Susceptibility
χσ0 fromthe3 model:O(4)-4-3-2-11245z = t/h1/βδ(a)• rescaled susceptibility
χσ H0 h1-1/δ for realistic values of mπ

0.40.3Mπ =χσ H0 h1−1/δ

• apparent scaling for

Mπ = 0.5 MeV
75 MeVlarge quark masses• differs from actual
scaling function!
• sensitivity to quark
mass larger than in QCD

Mπ = 138 MeV
Mπ = 200 MeV

0.2

0.1

0-4-3-2-1012345z = t/h1/βδ [P. Piasecki, J. Braun, and B. Klein, Eur. Phys. J. C71, 1576 (2011)](b)

Abbildung 7.14: (a)-(b) Reskalierte Pionenzerfallskonstante und Suszeptibilität als17χ σ H 0 h1

L = 30 fm
L = ∞ fm

0.2

Technische Universität München

0.1

Infinitevolume scaling in finite volume?
0-3-2-10123z = t/h1/βδ χσ H0• rescaled susceptibility

45h1-1/δ in finite volume(b)susceptibility decrease χσ ∼ L2susceptibility increase[P. Piasecki, J. Braun, and B. Klein (2010),
0.5

L = 2 fm
L = 4 fm
L = 6 fm
L = 8 fm
L = 10 fm
L = 20 fm
L = 30 fm
L = ∞ fmχσ H0 h1−1/δ

0.4
0.3

arXiv:1008.2155]

0.2
0.1
0-5-4-3-2-1

01

2

345

• mπ = 75 MeV• deviations frominfinite-volume scaling
for L < 6 fm
• effects probablyweaker in lattice QCD

z = t/h1/βδ(c)18Technische Universität MünchenScaling behavior in finite volume• correlation length is cut off by finite volume size L• Volume size L appears as additional relevant coupling• Finite-size scaling functions with additional scaling variable• Finite-size scaling function for the order parameter:

M (t, h, L) = L−β/ν QM (z, h̄),h̄ = hLβδ/ν• Finite-size scaling function for the susceptibility:χσ (t, h, L) = Lγ/ν Qχ (z, h̄)
19Technische Universität München

8 Quark-Meson-Modellür endliches VolumenFinite size
scaling: fSusceptibilityχσ• susceptibility in finite volume
L = 2 fm

1000

L = 4 fm
L = 6 fmχσ [1/GeV2 ]L = 8 fm
L = 10 fm

100

L = 20 fm
L = 30 fm

1011e-008

1e-0071e-0061e-0050.0001h(a)20Technische Universität München

Finitesize scaling: Susceptibility χσ
8 Quark-Meson-Modell für endliches Volumen• finite-size scaled susceptibility
10
L = 2 fm
L = 4 fmχσ [1/GeV2 ]L−γ/ν

1

L [fm] Mπ [MeV] Mπ L

L = 6 fm
L = 8 fm

0.1

L = 10 fm
L = 20 fm
L = 30 fm

0.01
0.001
0.0001
1e-0051e-005 0.0001

0.001

0.01

0.1

14

139

2.82

685

2.59

860

2.43

1045

2.30

20

19

1.94

10hLβδ/ν(a)2110
L = 2 fmTechnische Universität München

QCD phase diagram
Tcurvatureκcrossover(second orderfor 2 massless quarks)

at finite chemical potential...very schematically...

first-ordertransitionµ• second-order phase transition for two flavors in the chiral limit[R. D. Pisarski and F. Wilczek, Phys. Rev. D 29 (1984) 338]

• crossover at finite quark masses for finite temperature at µ = 0• conventional expectation:
first-order phase transition with critical end point22

Technische Universität München

Curvature of the transition line
• at small baryon chemical potential µ, the phase transitionline is characterized by the curvature κTχ (L, mπ , µ)Tχ (L, mπ , µ = 0)

= 1−κ

�

µ(πTχ (L, mπ , 0))

�2+ ...

RG methods:• “sign problem” in lattice QCD:
[J. Braun, Eur. Phys. J. C64, 459 (2009)]
simulations are difficult at finite µ• curvature can be calculated in lattice QCD (imaginarychemical potential, Taylor expansion) [P. de Forcrand and O. Philipsen,Nucl. Phys. B 642 (2002) 290, JHEP 01 (2007) 077; F. Karsch et al., Nucl. Phys. Proc. Suppl. 129, 614(2004).]

differences partially due to finite-volume effects?
23Technische Universität München

Why Finite-volume effects?
• curvature depends on the sensitivity of the system on�the chemical potential∂F ��µ=∂Nq �T,Vµµ• sensitivity in turn depends on the“constituent quark mass”• constituent quark mass affected by volume!
24

Technische Universität München

Change of curvature in finite volume
0.4
0.3

m# = 138 MeV

κ(L, mπ ) − κ(∞, mπ )∆κ =κ(∞, mπ )0.2!"(L)• periodic boundary
conditions for quarks• decreasing curvature
in intermediate
volume• corresponds todecreasing pion
mass/increasing
constituent quarkmass• decreased sensitivity
to chemical potential

0.1
0-0.1
2

2.5

3

3.5
L [fm]

44.5[B.-J. Schaefer, J.Braun, B. Klein]
255

Technische Universität München

Phase diagram for QCD models in
finite volume - qualitative resultsT• qualitatively clear effects offinite volume on curvature• phase transition line tends toflatten in an intermediate
volume range• curvature increases
dramatically for very small
volumesL→∞
Lc ! L < ∞L < Lcµ26

Technische Universität München

Phase diagram for QCD models in
finite volume - first results
200

T [MeV]• potential discretizedon a mesh grid• first-order phasetransition can bedetermined• effects on criticalpoint can bedetermined

L='L = 5.0 fmL = 4.0 fm
L = 3.0 fm
L = 2.25 fm

150

100500100200µ [MeV]300[A. Tripolt, B.-J. Schaefer, J.Braun, B. Klein]
27

Technische Universität München

Effects of the quark boundary conditions• Pion mass shift in V = L3 × 1/T in quark-meson model• periodic vs. anti-periodic quark boundary conditions (b.c.)
0.60.5fπ ∼ �σ�m�ψ̄ψ�
2mπ = m 2 ∼fπ
�σ�

0.4

R[mπ(L)]�ψ̄ψ� ∼ �σ�ap bc, T = 0p bc, T = 0ap bc, T = 1/Lp bc, T = 1/L

0.3mπ(∞) = 300 MeV

0.20.1
0-0.1
11.5

22.5

[J. Braun, B. Klein, H.-J. Pirner, Phys. Rev. D72, 034017 (2005).]
L [fm][J. Luecker et al., Phys. Rev. D81, 094005 (2010); D.B. Carpenter, C.F. Baillie, Nucl. Phys. B 260, 103 (1985).]33.5

Technische Universität München

Quark contributions for a finite volume• quark momentum modes
contributing to thecondensate (and theconstituent quark mass) in alarge finite volume• zero-mode
for periodic b.c.
• no zero modefor anti-periodic b.c.

Λ
k2π
L29

Technische Universität München

Quark contributions for a finite volume• quark momentum modes
contributing to thecondensate (and theconstituent quark mass) in asmall finite volume• enhancement of the zeromode contribution ∼1/V for
periodic b. c.

Λ
k2π
L30

Technische Universität München

Comparison to Chiral Perturbation Theory• comparison of model results with anti-periodic boundary
conditions from RG to ChPT in NNLO ChPT data thanks to G. Colangelo[G. Colangelo, S. Dürr, C. Haefeli, Nucl. Phys. B 271 (2005) 136.]
NNLO, all nRGR[M!(L)]1

agreement only forthis choice ofboundary conditions!

M! = 100 MeV

0.01

M! = 300 MeV

1M! = 200 MeV
23L (fm)

4keep boundary
conditions in mind forthe finite-volume
analysis of lattice
QCD results
31Technische Universität München

Conclusions• Scaling functions from the functional renormalizationgroup for the analysis of the QCD chiral phase transition:
Results from a model for the chiral phase transition• Finite-size effects in lattice simulations can lead tosignificant deviations from expected scaling behavior• Additional finite-volume effects for curvature of transitionline is expected• choice of spatial quark boundary condition is important!
32

Technische Universität München

Thanks to my collaborators• Jens Braun, Universität Jena
• Piotr Piasecki, Technische Universität Darmstadt• Bernd-Jochen Schaefer, Universität Graz
• Arno Tripolt, Universität Graz33