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Influence of the Quark Boundary Conditions on the Pion Mass in Finite Volume [RG Seminar, TU Darmstadt, June 7, 2005] [Bertram Klein, GSI, Darmstadt] J. Braun, H.J. Pirner, [J. Braun, BK, H.J. Pirner, arXiv:hep-ph/0504127] Universit at

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Influence of the Quark Boundary Conditions on the Pion Mass in Finite Volume

[RG Seminar, TU Darmstadt, June 7, 2005] [Bertram Klein, GSI, Darmstadt]

  • J. Braun, H.J. Pirner,

Universit¨ at Heidelberg [J. Braun, BK, H.J. Pirner, arXiv:hep-ph/0504127] [ Phys. Rev. D 71 (2005) 014032 [arXiv:hep-ph/0408116].]

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Outline

  • Motivation:
  • Why do we investigate finite volume effects?
  • Why do we have to look at quark boundary conditions?
  • Model:
  • Why do we use the quark meson model?
  • Method:
  • Renormalization Group (RG) in Finite Volume
  • Results:
  • Pion mass shift in finite volume
  • Comparison to chiral Perturbation Theory (chPT)
  • Conclusions

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 2

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SLIDE 3

Motivation: Overview

  • non-perturbative methods essential for QCD: lattice gauge theory
  • currently small volumes (L 2 − 3 fm) and mass ranges of

Wilson/staggered fermions: mπ 400 MeV [UKQCD, QCDSF, JLQCD, MILC, . . . ]

  • verlap fermions:

mπ 180 MeV [H. Neuberger, T. Draper, I. Horvath, . . . ] domain wall fermions: down to mπ 100 MeV [D. Kaplan, N. Christ, . . . ]

  • results require extrapolation to large volumes (and to small masses)
  • chiral perturbation theory (chPT) is systematic expansion in

p2, m2

π:

requires large scale 4πfπ and converges best for small mπ, large L

[J. Gasser, H. Leutwyler, G. Colangelo, S. D¨ urr, C. Haefeli, . . . ]

  • renormalization group (RG) calculations can complement chPT:

expansion in 1/L is not necessary!

  • volume dependence explores scale dependence: |

p| ∼ 1/L

  • for small volumes, transition to microscopic degrees of freedom

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 3

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SLIDE 4

Motivation: Overview

  • lattice results suggest importance of quark effects in finite volume
  • visible influence of quark boundary conditions
  • lattice results seem not yet in range of chPT:

need to use actual QCD with quark degrees of freedom

  • for a first exploration, use phenomenological models instead of QCD
  • mesons and quarks, but no gauge fields as degrees of freedom
  • quark-meson-model is suitable for describing the chiral phase transition,

[e.g. B.J. Schaefer, J. Wambach, arXiv: nucl-th/0403039.]

but it has no confinement (since there are no gauge fields)

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 4

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SLIDE 5

Motivation: Lattice Results

  • quenched simulation, Wilson fermions, periodic b.c. in spatial directions [quenched: no pions!]
  • R[mπ(L)] = mπ(L)−mπ(∞)

mπ(∞)

  • comparison to chPT result (lines) [J. Gasser, H. Leutwyler, Phys. Lett. B184 (1987) 83.]

0.00 0.01 0.02 0.03 0.04 0.05 0.06 2 3 4 5 6 7 8 9 10 11 12 Rmπ(L) = (mπ(L)-mπ)/mπ mπ L κ = 0.1340 0.1345 0.1350

[M. Guagnelli, et al. [ZeRo Collaboration], Phys. Lett. B 597 (2004) 216.]

a = 0.079 fm 0.9 fm ≤ L ≤ 2.5 fm κ mπ(∞) [MeV] 0.1340 881 0.1345 735 0.1350 559

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 5

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SLIDE 6

Motivation: Lattice results compared to chPT

  • two flavors of dynamical Wilson fermions [note that our RG results apply to unquenched!]
  • comparison to (improved) chPT results (exact GL and L¨

uscher-improved)

1 1.5 2 2.5 L [fm] 0.4 0.5 0.6 0.7 mPS [GeV]

[B. Orth, T. Lippert, K. Schilling, hep-lat/0503016]

a ≈ 0.08 fm − 0.13 fm 0.85 fm ≤ L ≤ 2.04 fm κ mπ(∞) [MeV] 0.1575 643 0.158 490 0.1665 419

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 6

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Motivation: Quarks and Boundary conditions

  • two flavors of dynamical staggered quarks

[S. Aoki, et al., Phys. Rev. D 50 (1994) 486.]

quark boundary condi- tions enter! a = 0.089(3) fm 0.7 fm ≤ L ≤ 1.8 fm b.c. mπa P 0.581(22) AP 0.672(22) mqa = 0.01 V = 83 × (16 × 2)

  • note: chPT properly needs dynamical fermions, otherwise no pions! [cf. A. AliKhan et al.]
  • qchPT vs. chPT! [S. Sharpe, C. Bernard, M. Golterman, . . . ]

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 7

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SLIDE 8

Motivation

  • lattice QCD seems at present not yet in the regime

where chiral perturbation theory describes all finite volume effects (where all finite volume effects are entirely due to pions)

  • expect chPT to be quantitative description for large L, small mπ
  • for small volumes, chiral symmetry restoration becomes important
  • at corresponding large momentum scales, quark degrees of freedom become important
  • this should properly be studied in actual QCD . . .
  • . . . but for now we use a simpler phenomenological model

for an exploration

  • expect to obtain qualitative results for finite volume effects:

Can we understand why lattice results differ from chPT predictions? Can we understand how quark fields enter?

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 8

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SLIDE 9

Quark-Meson-Model

  • effective low-energy model for scales below ΛUV ≃ 1.5 GeV
  • degrees of freedom: for Nf = 2 flavors of quarks

⇒ 1 + (Nf 2 − 1) = 1 + 3 mesonic fields φ = (σ, π)

  • chirally invariant → meson potential invariant under O(4)-symmetry
  • action at UV-scale ΛUV with parameters {λUV, mUV2, gmc}:

ΓΛUV[φ] =

  • d4x
  • ¯

q

  • /

∂ + g((σ + mc) + i π · τγ5)

  • q + 1

2(∂µφ)2 + U(φ2)

  • U(φ2)

= 1 2mUV

2φ2 + 1

4λUV(φ2)2

  • explicit chiral symmetry breaking: current quark mass in (σ + mc)
  • spontaneous symmetry breaking: σ0 = σ = 0
  • heavy constituent quarks Mq2 = g2(σ + mc)2, no gauge fields and no confinement

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 9

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SLIDE 10

Renormalization Group

Goal: find effective action Γ[φ] for expectation value of fields φ Way: systematically integrate out quantum fluctuations above momentum scale k

  • start at scale k = ΛUV with potential with couplings mUV2, λUV
  • renormalization step: change scale k → k − ∆k by small ∆k and expand around couplings

m2(k) → m2(k) + δm2(∆k) λ(k) → λ(k) + δλ(∆k)

  • find effective action Γk−∆k[φ] at momentum scale k − ∆k
  • limit ∆k → 0: obtain differential equation k ∂

∂kΓk[φ] for change of action with k

⇒ “flow equation” describes change of the couplings with change of scale k

  • find “physical” effective action Γ[φ] = Γk→0[φ] by evolving to k → 0

⇒ quantum fluctuations on all scales are integrated out Tool: Proper-Time Renormalization Group (Schwinger representation of one-loop action)

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 10

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SLIDE 11

Flow Equations: Finite Volume

Truncations: expansion around constant expectation value φ(x) = const.

  • for V = L3×T (T finite or T → ∞) replace
  • choice of boundary conditions for quarks in

spatial directions: periodic p2

F = p2 p

  • r

anti-periodic p2

F = p2 ap

  • dpi . . . → 2π

L

  • ni

. . . p2

ap = (2π)2

L2

3

  • i=1
  • ni + 1

2 2 , p2

p = (2π)2

L2

3

  • i=1

n2

i

Flow equations for the meson potential to one loop (ωp = 2π

T (n0), ωap = 2π T (n0 + 1 2)):

k ∂ ∂kUk(σ, π2, L, T) = k6 L3T

  • n0
  • {ni}
  • −

4NcNf (p2

F + ω2 ap + k2 + Mq2 k)3 + Nf

2

  • m=1

1 (p2

p + ω2 p + k2 + Mm2 k)3

  • Mq

2 k = g2((σ + mc)2 +

π2), Mm

2 k = [eigenvalues of U ′′ k (σ,

π2)]

  • mc source of explicit chiral symmetry breaking: terms in (σ − σ0(k)), but

π2 remains invariant Ansatz for the potential with explicit chiral symmetry breaking: Uk(σ, π2) =

i+j≤N

  • i,j=0

aij(k)(σ − σ0(k))i(σ2 + π2 − σ02(k))j, σ0(k) = σk

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 11

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SLIDE 12

RG calculation: strategy

tune UV couplings to physical point in infinite volume, predict volume dependence

  • no prediction in infinite volume:

values for mπ(∞), fπ(∞) are used to fix parameters mUV2, mc

  • shifts in finite volume are then a prediction

infinite volume parameters: ΛUV = 1.5 GeV, λUV = 60.0 mUV [MeV] mc [MeV] fπ [MeV] mπ [MeV] 779.0 2.10 90.38 100.1 747.7 9.85 96.91 200.1 698.0 25.70 105.30 300.2

  • dependence on parameter choice is only weak [for sufficiently large volume]
  • sums are truncated in numerical evaluation

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 12

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RG results: Pion mass mπ for different boundary conditions

influence of periodic and anti-periodic boundary conditions (for fermions) Plot: R[mπ(L)] = mπ(L)−mπ(∞)

mπ(∞)

as a function of L for mπ(∞) = 100 MeV, 300 MeV

  • V = L3 × T, with T = L or T = 3

2L

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 4.5

R[mπ(L)] L [fm]

mπ(∞) = 100 MeV

a.p. b.c., T/L = 3/2

  • p. b.c., T/L = 3/2

a.p. b.c., T/L = 1/1

  • p. b.c., T/L = 1/1
  • 0.04
  • 0.02

0.02 0.04 0.06 0.08 0.1 1 1.5 2 2.5 3 3.5

R[mπ(L)] L [fm]

mπ(∞) = 300 MeV

a.p. b.c., T/L = 3/2

  • p. b.c., T/L = 3/2

a.p. b.c., T/L = 1/1

  • p. b.c., T/L = 1/1
  • pion mass in the framework of the quark meson model:

mπ2(k) = m(k)2 + λ(k) σ0(k)2 = C·mc

σ0(k)

  • scales k and L in competition, additional IR-cutoff 1/L (1/T) affects primarily fermions
  • condensation of quarks only in π

L <k<ΛUV ( π T <k<ΛUV ) Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 13

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SLIDE 14

RG results: Pion mass mπ for periodic boundary conditions

influence of side ratio (“temperature”) for periodic boundary conditions Plot: R[mπ(L)] = mπ(L)−mπ(∞)

mπ(∞)

as a function of L for mπ(∞) = 100 MeV, 300 MeV

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 4.5

R[mπ(L)] L [fm]

mπ(∞) = 100 MeV

T/L = ∞ T/L = 3/1 T/L = 3/2 T/L = 1/1

  • 0.1

0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3

R[mπ(L)] L [fm]

mπ(∞) = 300 MeV

T/L = ∞ T/L = 3/1 T/L = 3/2 T/L = 1/1

  • dependence on the box side ratio T/L (small T → large temperature) can be understood

from fermionic zero mode contribution to flow of the potential: periodic/ antiperiodic [k∂kU(σ)]F,0 = − k6 L3T · 2 · 23 · 4NcNf (k2 + (±π/T)2 +3(π/L)2 + g2 (σ + mc)2)3

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 14

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SLIDE 15

RG results compared to chPT: Pion mass mπ

Plot: R[mπ(L)] = mπ(L)−mπ(∞)

mπ(∞)

as a function of L for mπ(∞) = 100 MeV, 300 MeV

  • anti-periodic or periodic boundary conditions (for fermions in the spatial directions)
  • V = L3 × T, with T = L finite, or with T → ∞
  • for comparison: chPT [G. Colangelo, S. D¨

urr and C. Haefeli, hep-lat/0503014.]

0.01 0.1 1 1 1.5 2 2.5 3 3.5 4 4.5

R[mπ(L)] L [fm] mπ(∞) = 100 MeV

RG, a.p. b.c., T/L = ∞ chPT, nnlo, T/L = ∞ RG, a.p. b.c., T/L = 1/1 RG, p. b.c., T/L = 1/1 0.001 0.01 0.1 1 1 1.5 2 2.5 3

R[mπ(L)] L [fm] mπ(∞) = 300 MeV

RG, a.p. b.c., T/L = ∞ chPT, nnlo, T/L = ∞ RG, a.p. b.c., T/L = 1/1 RG, p. b.c., T/L = 1/1

[large L →∼ exp(−mπL)]

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 15

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SLIDE 16

Chiral Perturbation Theory

  • pion mass shift from exact one-loop calculation in chPT by J. Gasser and H. Leutwyler

[J. Gasser and H. Leutwyler, Phys. Lett. B 188 (1987) 477.]

  • new results from G. Colangelo and S. D¨

urr [G. Colangelo and S. D¨

urr, Eur. Phys. J. C 33 (2004) 543.]

using . . . L¨ uscher’s formula [M. L¨

uscher, Commun. Math. Phys. 104 (1986) 177.]

R[mπ(L)] = mπ(L) − mπ(∞) mπ(∞) = − 3 16π2 1 mπ 1 mπL ∞

−∞

dy F(iy)e−√

mπ2+y2L + O(e− ¯ mL)

  • relates finite volume mass shift to infinite volume scattering
  • F(ν) → ππ-scattering amplitude in V → ∞

= ⇒ obtain mass shift from infinite-volume result for scattering!

  • ¯

m =

  • 3

2mπ → ¯

mL provides boundary on error (error decreases with increasing mass)

  • mass shift effects included here (periodic boundary conditions for pion fields):

pions crossing the volume boundary once between interactions!

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 16

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SLIDE 17

Chiral Perturbation Theory: Results

2 2.5 3 3.5 4 L(fm) 0.0001 0.001 0.01 0.1 RM LO NLO NNLO GL (full g1) Mπ=100 MeV Mπ=300 MeV Mπ=500 MeV

[G. Colangelo and S. D¨ urr, Eur. Phys. J. C 33 (2004) 543.]

Plot: R[mπ(L)] GL: Gasser/Leutwyler exact one-loop chPT LO: L¨ uscher result with input from

  • ne-loop

infinite-volume chPT → coincide for large mπ NLO: input from two- loop chPT NNLO: input from three-loop chPT

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 17

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SLIDE 18

Chiral Perturbation Theory: Comparison to RG

  • “resummation”: include pions that cross the volume boundary n times, sum over all n → ∞

[G. Colangelo, S. D¨ urr and C. Haefeli, hep-lat/0503014.] [Thanks to G. Colangelo for the plot!]

1 2 3 4 L (fm) 0.01 1 R[Mπ(L)] LO, n=1 NLO, n=1 NNLO, n=1 LO, all n NLO, all n NNLO, all n RG Mπ = 100 MeV Mπ = 200 MeV Mπ = 300 MeV

  • corrections shift results towards larger values, surprisingly good agreement (note the slope)!
  • LEC’s unchanged from infinite volume for antiperiodic b.c.

[J. Gasser, H. Leutwyler, Nucl. Phys. B 307 (1988) 763.] Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 18

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SLIDE 19

Conclusions

  • results of RG approach to the quark-meson-model for finite volume

with finite quark mass

  • implementation of explicit chiral symmetry breaking ⇒ potential in φ = (σ,

π)

  • derived and solved flow equations in finite volume
  • results for the volume dependence of the pion mass
  • comparison to chiral perturbation theory
  • surprisingly good agreement for pion-dominated regions (for a.p. b.c., confirms G/L)
  • influence of boundary conditions: quarks important for finite volume effects,

as observed on the lattice

  • quark-meson-model provides mechanism to qualitatively understand volume dependence,

including dependence on boundary conditions

  • difference to lattice due to different cutoff, unconfined constituent quarks, gluons
  • Outlook: volume dependence of critical behavior, finite size effects

Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 19