Effective field theories for QCD with rooted staggered fermions Claude Bernard, Maarten Golterman and Yigal Shamir Lattice 2007, Regensburg

Aim of this talk: For small-enough lattice spacing, a, EFTs like the Symanzik effective theory (SET) and chiral perturbation theory (ChPT) account for lattice artifacts through a systematic expansion in a  QCD Key assumption: the underlying lattice theory is local However: QCD with rooted staggered fermions (Det 1/4 (D stag )) is non-local  can the construction of a SET and staggered ChPT be extended to rooted staggered QCD? Intuitive idea: consider n r replicas, then continue n r  1/4, but dependence of EFT coefficients on n r is not known Even at a = 0.06 fm lattice artifacts (e.g., mass splittings) are significant!

Start from Shamir’s RG analysis: 1) Go to taste basis (Q is unitary): 2) Carry out n RG blocking steps (postpone integration over gauge fields): Here n r is the number of “replicas” -- take integer for now! D taste,n is staggered Dirac operator after n RG steps; are blocked gauge fields; B n is local (on coarse lattice) Boltzmann weight: So far standard RG set-up -- see Shamir for detailed discussion

3) Define split (no doublers!) and assume where a f = a is the original fine lattice spacing and a c = 2 n a f ; i.e., taste violations scale to zero in the continuum limit 4)Staggered theories with n r replicas have the same continuum limit as reweighted, taste-invariant, local theories with n s = 4n r taste-singlet fermions with Dirac operator 5)Shamir: this should also work for n r = 1/4 (i.e., n s = 1)!! (key point: scaling of  n on “rooted” ensemble; a c  QCD << 1)

Generalized theory: Replace For t = 1 and n s = 4n r this is staggered theory with n r replicas ; For t = 0 this is reweighted, local theory with n s taste-singlet fermions; For n r = any positive integer, and any t, this defines a local theory  assume that SET (and thus ChPT) exist Now take n s fixed, not equal to 4n r, then SET still exists -- think of SET as expansion in a f, with coefficients that depend on a c (need to assume this works for partially quenched theories)

Important consequences: Can expand determinant ratio in t :  n  a f  power of n r less than power of t less than or equal to power of a f Lattice: all correlation functions, expanded to a fixed order in a f are polynomial in n r, hence we may continue in n r to n s /4 !! SET: n r dependence comes from Symanzik coefficients and loops  “staggered SET with the replica rule” (set t = 1)

Comments: SET is complicated for t ≠ 1, but don’t need explicit form (lattice spacing is a c, depends also on a f ) Only staggered external legs and t = 1: 1) all staggered symmetries apply (shift, U(1)  )  form of SET is that of Lee & Sharpe (after field redefinitions) 2) if also n r = n s /4: lattice spacing is a f Expansion in hence the generalized theory has no 1/a f divergences (same continuum limit for all t !) Transition to ChPT works the same way  “SChPT with the replica rule” (Aubin & Bernard)

How does shift symmetry work in the SET? Shift symmetry is lattice translation plus phases, with generators Any representation thus takes the form with However, any continuum EFT is invariant under translations; which, for distance r, act on any continuum field as Choose r such that p  r = –p   EFT is invariant under the group  4 generated by the  

Conclusions The n s taste-singlet fermions play the role of the physical flavors; continuum limit (n  ) is independent of n r (and a “perfect” action) Turning on t “decorates” or “staggers” these fermions: n r appears; to any given order in a f correlation functions are polynomial in n r thus we may continue n r to n s /4 For n r = n s /4 with n s not a multiple of four the theory is non-local and this non-locality is reproduced by the EFT: non-local behavior is reproduced by continuing the n r dependence appearing through loops to n s /4 (see Bernard for example)  “SET/SChPT with replica rule” Some refs: RG approach: Shamir, PRD75, (2007) non-locality: Bernard, MG, Shamir, PRD73, (2006) replica-SChPT: Bernard, PRD73, (2006)