Mixed actions: the double pole

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

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Most serious sickness double pole
e.g. Wilson sea and GW valence add GW ghost
quarks sea quarks dont 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
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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(KNK)L ?
SU(KNK)R (M 0) (K valence quarks, N sea
quarks) (BernardMG)
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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
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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
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Lattice EFT to order a2

• start from Baer, Rupak and Shoresh (2004)
• symmetry SU(KK)L ? SU(KK)R ? SU(N)
  (GW-Wilson)
• SU(KK) ? SU(N)
  (Wilson-Wilson)
• new operators
• ?vv ?vs 0 for GW valence Wilson includes
  tmQCD
• (staggered sea see Baer et al. (2005))

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Propagators

• ?0 ??str(Pv?) is valence-? -- sea-?
  integrated out
• 
  (str(???str((PvPs)?)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

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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)
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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)

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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

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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