Mixed actions: the double pole

Maarten Golterman, Taku Izubuchi, Yigal Shamir

Cyprus 2005

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:

1) unphysical effects disappear in continuum limit2) controlled by positive powers of a3) can use EFT to investigate

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

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)

Intermediate step: Symanzik expansion

For Wilson fermions, to order a: (Sharpe & Singleton)

Pauli term breaks chiral symmetry just like mass termintroduce spurion field A just like quark mass M

then set M = m , and A = a ; example:

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

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))

Propagators

0 str(Pv) is valence-“’ ” -- sea-’ integrated out (str(str((Pv+Ps))=0)

• flavor non-diagonal sector: as usual Mvv

2 = 2B0vmv + 2W0va + 2va2 + … Mss

2 = 2B0sms + 2W0sa + 2sa2 + …

• valence flavor diagonal sector:

where R = (Mvv2 - Mss

2)/N + (vv+ ss- 2vs) a2

R non-zero even if Mvv = Mss

Choice:

either: choose Mvv such that R = 0 ,

or: choose Mvv = Mss and 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)

I = 0 scattering (two pions in a box L3)

two-pion I = 0 energy shift:

= R / (82f2) , = Mvv2 / (162f2)

B0(ML) = - 0.53 + O(1/L2)

A0(ML) = 49.59 / (ML)2 + O(1/L3)

Power counting and estimates (Mvv = Mss = M):

1) ~ M2/2 ~ aQCD (Baer et al.)

one-loop/tree-level ~ 3 (ML)3 , 2 ML

2) ~ M2/2 ~ (aQCD)2 (Aoki, 2003)

one-loop/tree-level ~ (ML)3

aQCD ~ 0.1 , aM ~ 0.2 , L/a = 32:

scaling violations of order 6% small, but not negligible

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