Influence of the Quark Boundary Conditions on the Pion Mass in - PowerPoint PPT Presentation

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 Heidelberg [ Phys. Rev. D 71 (2005) 014032 [arXiv:hep-ph/0408116].]

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

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: m π � 400 MeV [UKQCD, QCDSF, JLQCD, MILC, . . . ] overlap fermions: m π � 180 MeV [H. Neuberger, T. Draper, I. Horvath, . . . ] domain wall fermions: down to m π � 100 MeV [D. Kaplan, N. Christ, . . . ] • results require extrapolation to large volumes (and to small masses) p 2 , m 2 • chiral perturbation theory (chPT) is systematic expansion in � π : requires large scale 4 πf π and converges best for small m π , 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

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

Motivation: Lattice Results • quenched simulation, Wilson fermions, periodic b.c. in spatial directions [quenched: no pions!] • R [ m π ( L )] = m π ( L ) − m π ( ∞ ) m π ( ∞ ) • comparison to chPT result (lines) [J. Gasser, H. Leutwyler, Phys. Lett. B184 (1987) 83.] 0.06 R m π (L) = (m π (L)-m π )/m π 0.05 κ = 0.1340 0.1345 a = 0 . 079 fm 0.1350 0.04 0 . 9 fm ≤ L ≤ 2 . 5 fm 0.03 κ m π ( ∞ ) [MeV] 0.02 0 . 1340 881 0.01 0 . 1345 735 0.00 0 . 1350 559 m π L 2 3 4 5 6 7 8 9 10 11 12 [M. Guagnelli, et al. [ZeRo Collaboration], Phys. Lett. B 597 (2004) 216.] Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 5

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) 0.7 a ≈ 0 . 08 fm − 0 . 13 fm 0.6 0 . 85 fm ≤ L ≤ 2 . 04 fm m PS [GeV] κ m π ( ∞ ) [MeV] 0.5 0 . 1575 643 0 . 158 490 0.4 0 . 1665 419 1 1.5 2 2.5 L [fm] [B. Orth, T. Lippert, K. Schilling, hep-lat/0503016] Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 6

Motivation: Quarks and Boundary conditions • two flavors of dynamical staggered quarks quark boundary condi- tions enter! a = 0 . 089(3) fm 0 . 7 fm ≤ L ≤ 1 . 8 fm b.c. m π a P 0.581(22) AP 0.672(22) m q a = 0 . 01 = 8 3 × (16 × 2) V [S. Aoki, et al. , Phys. Rev. D 50 (1994) 486.] • note: chPT properly needs dynamical fermions, otherwise no pions! [cf. A. AliKhan et al. ] • q chPT vs. chPT! [S. Sharpe, C. Bernard, M. Golterman, . . . ] Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 7

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 m π • 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

Quark-Meson-Model • effective low-energy model for scales below Λ UV ≃ 1 . 5 GeV • degrees of freedom: for N f = 2 flavors of quarks ⇒ 1 + ( N f 2 − 1) = 1 + 3 mesonic fields φ = ( σ,� π ) • chirally invariant → meson potential invariant under O (4)-symmetry • action at UV-scale Λ UV with parameters { λ UV , m UV 2 , gm c } : � q + 1 � 2( ∂ µ φ ) 2 + U ( φ 2 ) � d 4 x � � Γ Λ UV [ φ ] = q ¯ / ∂ + g (( σ + m c ) + i� π · � τγ 5 ) 1 2 φ 2 + 1 U ( φ 2 ) 4 λ UV ( φ 2 ) 2 = 2 m UV • explicit chiral symmetry breaking: current quark mass in ( σ + m c ) • spontaneous symmetry breaking: σ 0 = � σ � � = 0 • heavy constituent quarks M q 2 = g 2 ( � σ � + m c ) 2 , no gauge fields and no confinement Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 9

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 = Λ UV with potential with couplings m UV 2 , λ UV • renormalization step: change scale k → k − ∆ k by small ∆ k and expand around couplings m 2 ( k ) m 2 ( k ) + δm 2 (∆ k ) → λ ( k ) → λ ( k ) + δλ (∆ k ) • find effective action Γ k − ∆ k [ φ ] at momentum scale k − ∆ k • limit ∆ k → 0: obtain differential equation k ∂ ∂k Γ k [ 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 ] = Γ k → 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

Flow Equations: Finite Volume � Truncations: expansion around constant expectation value φ ( x ) = const . • for V = L 3 × T ( T finite or T → ∞ ) replace � dp i . . . → 2 π � . . . L • choice of boundary conditions for quarks in n i spatial directions: periodic p 2 F = p 2 3 3 � 2 ap = (2 π ) 2 p = (2 π ) 2 p � n i + 1 � � p 2 , p 2 n 2 anti-periodic p 2 F = p 2 or i L 2 L 2 ap 2 i =1 i =1 � Flow equations for the meson potential to one loop ( ω p = 2 π T ( n 0 ), ω ap = 2 π T ( n 0 + 1 2 )): 2 N f � � k 6 k ∂ 4 N c N f 1 � � � π 2 , L, T ) = ∂kU k ( σ,� − k ) 3 + ap + k 2 + M q 2 p + k 2 + M m 2 ( p 2 L 3 T F + ω 2 ( p 2 p + ω 2 k ) 3 n 0 m =1 { n i } k = g 2 (( σ + m c ) 2 + � 2 π 2 ) , 2 k = [eigenvalues of U ′′ π 2 )] M q M m k ( σ,� π 2 remains invariant • m c source of explicit chiral symmetry breaking: terms in ( σ − σ 0 ( k )), but � � Ansatz for the potential with explicit chiral symmetry breaking: i + j ≤ N a ij ( k )( σ − σ 0 ( k )) i ( σ 2 + � π 2 − σ 02 ( k )) j , � π 2 ) = U k ( σ,� σ 0 ( k ) = � σ � k i,j =0 Bertram Klein, GSI Darmstadt RG meeting, TU Darmstadt, June 7, 2005 11

RG calculation: strategy � tune UV couplings to physical point in infinite volume, predict volume dependence • no prediction in infinite volume: values for m π ( ∞ ), f π ( ∞ ) are used to fix parameters m UV 2 , m c • shifts in finite volume are then a prediction infinite volume parameters: Λ UV = 1 . 5 GeV, λ UV = 60 . 0 m UV [MeV] m c [MeV] f π [MeV] m π [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