Mixed actions: the double pole

1. Mixed actions: the double pole Maarten Golterman, Taku Izubuchi, Yigal Shamir Cyprus 2005

2. Mixed actions: valence quarks ≠ sea quarks • very practical • field theory worries: unitarity? • similar worries exist about improved actions • and actions with GW fermions. • extend notion of universality; assume: • unphysical effects disappear in continuum limit • controlled by positive powers of a • can use EFT to investigate

3. Most serious sickness: double pole e.g. Wilson sea and GW valence: add GW ghost quarks sea quarks don’t match the valence quarks for a ≠0 double pole with residue Ra2 if also mvalence≠ msea (partial quenching) Rc1 a2 + c2(msea- mvalence)  Look at most serious consequences of double pole

4. Continuum EFT: = exp(2i/f) non-linear meson field f, B0 low-energy constants M = diag(mv,mv,…,ms,ms,…,mv,mv,…) mass matrix symmetry: SU(K+N|K)L  SU(K+N|K)R(M = 0) (K valence quarks, N sea quarks) (Bernard&MG)

5. Intermediate step: Symanzik expansion For Wilson fermions, to order a: (Sharpe & Singleton) Pauli term breaks chiral symmetry just like mass term introduce spurion field A just like quark mass M then set M = m , and A = a ; example:

6. Double Pole: Double pole comes from “super-’” terms: The (valence) super-’ field is and a term in the lagrangian c (0)2 leads to a double pole in any flavor neutral propagator of the form Note that

7. Lattice EFT to order a2: • start from Baer, Rupak and Shoresh (2004): • symmetry: SU(K|K)L SU(K|K)R SU(N) (GW-Wilson) • SU(K|K)  SU(N) (Wilson-Wilson) • new operators: vv= vs= 0 for GW valence; “Wilson” includes tmQCD (staggered sea: see Baer et al. (2005))

8. Propagators • 0 str(Pv)is valence-“’ ” -- sea-’ integrated out • (str(str((Pv+Ps))=0) • flavor non-diagonal sector: as usual • Mvv2 = 2B0vmv + 2W0va + 2va2 + … • Mss2 = 2B0sms + 2W0sa + 2sa2 + … • valence flavor diagonal sector: • where R = (Mvv2 - Mss2)/N + (vv+ ss- 2vs) a2 • R non-zero even if Mvv = Mss

9. Choice: either: choose Mvv such that R = 0 , or: choose Mvv = Mssand live with non-vanishing R. Relevant for quantities sensitive to the double pole, especially if effects are enhanced. examples: I = 0 scattering (Bernard & MG, 1996) a0 propagator (Bardeen et al., 2002) nucleon-nucleon potential (Beane and Savage, 2002)

10. I = 0  scattering (two pions in a box L3) two-pion I = 0 energy shift:  = R / (82f2) ,  = Mvv2 / (162f2) B0(ML) = - 0.53 + O(1/L2) A0(ML) = 49.59 / (ML)2 + O(1/L3)

11. Power counting and estimates (Mvv = Mss= M): 1)  ~ M2/2~ aQCD(Baer et al.)  one-loop/tree-level ~ 3(ML)3 , 2ML 2)  ~ M2/2~ (aQCD)2 (Aoki, 2003)  one-loop/tree-level ~ (ML)3 aQCD~0.1 , aM ~ 0.2 , L/a = 32: • scaling violations of order 6% small, but not negligible

12. What do we learn about mixed actions? • Assume: unphysical effects encoded in scaling violations • Important to estimate numerical size in simulations use numerical results to test assumptions • Double pole: most infrared-sensitive probe • Quantity dependent (enhancement?) • Most sensitive quantities: small, but not negligible Claude Bernard, Paulo Bedaque: thanks for discussions