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CU-TP-1153 FERMILAB-PUB-06-325-T The Kaon B-parameter in Mixed Action Chiral Perturbation Theory C. Aubin,1,∗ Jack Laiho,2,† and Ruth S. Van de Water2,‡ 1Department of Physics, Columbia University, New York, NY 10027 7 2Theoretical Physics Department, Fermilab, Batavia, IL 60510 0 0 (Dated: February 1, 2008) 2 n Abstract a J We calculate the kaon B-parameter, BK, in chiral perturbation theory for a partially quenched, 9 1 mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We find that 3 the resulting expression is similar to that in the continuum, and in fact has only two additional v 9 0 unknown parameters. At one-loop order, taste-symmetry violations in the staggered sea sector 0 9 only contribute to flavor-disconnected diagrams by generating an O(a2) shift to the masses of 0 6 taste-singlet sea-sea mesons. Lattice discretization errors also give rise to an analytic term which 0 / shifts the tree-level value of B by an amount of O(a2). This term, however, is not strictly due to t K a l - taste-breaking, andisthereforealsopresentintheexpressionforBK forpureG-Wlattice fermions. p e We also present a numerical study of the mixed B expression in order to demonstrate that both h K : v discretization errors and finite volume effects are small and under control on the MILC improved i X staggered lattices. r a PACS numbers: 11.15.Ha,12.39.Fe, 12.38.Gc ∗[email protected] †[email protected] ‡[email protected] 1 I. INTRODUCTION Lattice quantum chromodynamics allows nonperturbative calculations of low-energy QCD quantities from first principles. Until recently, limited computing resources and the inability to include quark loop effects have prevented lattice calculations from achieving realistic results. In the past few years, however, lattice simulations of both light meson and heavy-light meson quantities with dynamical staggered quarks have shown excellent numerical agreement with experimental results [1]. These have included both post-dictions, such as the pion and kaon decay constants [2], and predictions, as in the case of the B meson mass c [3]. Such successes demonstrate that many of the systematic uncertainties associated with lattice simulations are under control, and therefore give confidence that lattice simulations can reliably calculate quantities that cannot be accessed experimentally. One of the simplest quantities of phenomenological importance that can only be calculated using lattice QCD is the kaon B-parameter, B . Thus a precise measurement of B is an important goal for K K the lattice community.1 0 B parameterizesthehadroniccontributiontoK0−K mixing; itthereforeplaysacrucial K role in extracting information about the CKM matrix using measurements of the neutral kaon system. In particular, the size of indirect CP violation in the neutral kaon system, ǫ , when combined with a numerical value for B , places an important constraint on the K K apex of the CKM unitarity triangle [7, 8]. Because ǫ is well known experimentally [9], the K dominant source of error in this procedure is the uncertainty in the lattice determination of B . It is likely that new physics would give rise to CP-violating phases in addition to K that of the CKM matrix; such phases would manifest themselves as apparent inconsistencies among different measurements of quantities which should be identical within the standard CKM picture. Thus a precise determination of B will help to constrain physics beyond K the standard model. Because lattice simulations with staggered fermions can at present reach significantly lighter quark masses than those with other standard discretizations [10, 11], calculations of weak matrix elements using the available 2+1-flavor Asqtad staggered lattices appear promising. Unfortunately, however, one pays a significant price for the computational speed 1 Promising calculations of BK including dynamical quark effects are currently in progress using both improved staggered fermions [4, 5] and domain-wall fermions [6]. 2 of staggered simulations: each flavor of staggered quark comes in four species, or “tastes.” In the continuum limit, these species become degenerate and can be removed by taking the fourth root of the fermion determinant. In practice, however, one must take the fourth root of the quark determinant during the lattice simulation in order to remove the extra tastes; thus it is an open theoretical question whether or not one recovers QCD after taking the continuum limit of fourth-rooted lattice simulations. Recently several papers have appeared addressing the validity of the “fourth-root trick” [12, 13, 14, 15, 16]. Although the issue has not yet been resolved, there are indications that the fourth root does not introduce pathologies when taking the continuum limit of the lattice theory. We recommend a recent review by Sharpe, Ref. [17], as a clear summary of the current status of the fourth-root trick. In this paper we assume the validity of the fourth-root trick. Even working under this assumption, however, the additional tastes introduce complications to staggered lattice simulations. The degeneracy among the four tastes is broken by the nonzero lattice spacing, a,andresultsindiscretizationerrorsofO(a2). Thusonemustusefunctionalformscalculated in staggered chiral perturbation theory (SχPT), in which taste-violating effects are explicit, to correctly extrapolate staggered lattice data [18, 19, 20, 21]. Staggered chiral perturbation theory has been used to successfully fit quantities such as f and f , for which the SχPT expressions are simple and taste-symmetry breaking pri- π D marily enters through additive corrections to the pion masses inside loops [20, 22]. In the case of B , however, the additional tastes also make the matching procedure between the K lattice ∆S = 2 effective four-fermion operator and the desired continuum operator more difficult. The latticized version of the continuum B operator mixes with all other lattice K operators that are in the same representation of the staggered lattice symmetry group [23] – including those with different tastes than the valence mesons. Current staggered calculations only account for mixing with operators of the correct taste and only to 1-loop order in α ; this results in a 20% uncertainty in B [4]. In order to achieve a precise determination S K of B with staggered fermions, one must either perform the lattice-to-continuum matching K nonperturbatively using a method such as that of the Rome-Southampton group [24] or one must include the effects of the extra operators in the SχPT expression for B [25]. Thus K far the large number of staggered operators has prevented matching calculations beyond 1-loop order [26, 27], although nonperturbative matching including all relevant staggered operators is, in principle, possible. The effects of this truncated lattice-to-continuum opera- 3 tor matching can be included in an extended version of SχPT, but the resulting expression for B has many undetermined fit parameters, only a few of which are already known from K measurements ofother quantities. Therefore use ofthis expression may prove just asdifficult as implementing fully nonperturbative matching. The calculationof weak matrix elements such as B with Ginsparg-Wilson(G-W)quarks K [28], on the other hand, is theoretically much cleaner than that with staggered quarks. This is because in the massless limit, G-W quarks possess an exact chiral symmetry on the lattice and do not occur in multiple species. Although in practice, lattice simulations use approximate G-W fermions, the degree to which chiral symmetry is broken in simulations can be controlled either by the length of the fifth dimension in the case of domain-wall quarks [29, 30] or through the degree to which the overlap operator is realized in the case of overlap quarks [31, 32, 33]. Consequently, while the ∆S = 2 lattice operator still mixes with wrong-chirality operators, there are significantly fewer operators than in the staggered case, and nonperturbative renormalization can be used in the determination of B . As an K additional benefit, the approximate chiral symmetry also ensures that the appropriate chiral perturbation theory expression for use in the extrapolation of B lattice data is continuum- K like at next-to-leading order (NLO). Lattice simulations with G-W fermions, however, are 10 to 100 times more computationally expensive than those with staggered fermions with comparable masses and lattice spacing, and thus are unfortunately not yet practical for realizing light dynamical quark masses. A computationally affordable compromise is therefore to calculate correlation functions with Ginsparg-Wilson valence quarks on a background of dynamical staggered gauge con- figurations. This “mixed action” approach combines the advantages of staggered and G-W fermions while not suffering from their major disadvantages, and is well-suited to the calculation of the weak matrix element B . By using staggered sea quarks and G-W valence K quarks one can better approach the chiral regime in the sea sector while minimizing operator mixing and allowing the use of nonperturbative renormalization. Additionally, because the MILC staggered lattices with 2+1 flavors of dynamical quarks are publicly available and offer a number of quark masses and lattice spacings [10, 11], one can perform unquenched three-flavor simulations at the same cost as quenched G-W simulations. Mixed action simulations have already been successfully used to study quantities of interest to nuclear physics [34, 35], thus we expect that a similar method can be used to calculate B . K 4 One might be concerned that the use of a mixed action could introduce new theoretical complications into the lattice determination of B . Although the mixed staggered sea, G-W K valence lattice theory reduces to QCD in the continuum limit, at nonzero lattice spacing, it is manifestly unphysical in that it violates unitarity. Thus, in order to extract physical QCD quantities from mixed action simulations, it is essential that one can correctly describe and remove contributions due to unphysical mixed action effects from quantities such as B K using the appropriate lattice chiral perturbation theory. It has not been rigorously proven that the mixed action chiral perturbation theory developed in [36] is the correct chiral effective theory for mixed G-W, staggered simulations in which the fourth-root of the quark determinant is taken in the sea sector. Nevertheless, Ref. [12] showed that, given a few plausible assumptions, staggered chiral perturbation theory is the correct chiral effective theory for describing the pseudo-Goldstone boson sector of rooted staggered simulations. A similar line of reasoning should hold for mixed action chiral perturbation theory. Assuming so, mixed action lattice simulations can be used to correctly calculate quantities involving pseudo-Goldstone bosons (such as B ) in QCD. More complicated quantities, however, K should be considered on a case-by-case basis. In this paper we calculate B in χPT for a lattice theory with G-W valence quarks and K staggered sea quarks. We present results for a “1+1+1” theory in which m 6= m 6= m u d s in the sea sector, for a “2+1” theory in which m = m 6= m in the sea sector, and for u d s a “full QCD”-like expression in which we tune the valence-valence meson masses equal to the taste-singlet sea-sea meson masses. (We emphasize, however, that these expressions only truly reduce to QCD when m = m and the lattice spacing a → 0.) These sea valence expressions will be necessary for the correct chiral and continuum extrapolation of mixed- action B lattice data. We find that the expression for B in mixed action χPT has only K K two more parameters than in the continuum. The first coefficient multiplies an analytic term which shifts the tree-level value of B by an amount of O(a2); this term is also present K in the case of pure G-W lattice fermions. The second new parameter shifts the mass of the taste-singlet sea-sea meson (which only appears inside loop diagrams) by an amount of O(a2). This mass-splitting has already been separately determined, however, in the MILC spectrum calculations so we do not consider it tobe anunknown fit parameter. Therefore, in practice, the chiral and continuum extrapolation of mixed action B lattice data should be K 5 no more complicated than that of domain-wall lattice data. A numerical analysis using the taste-breaking parameters measured on the MILC coarse lattices (a ≈ 0.125 fm) shows that the size of non-analytic discretization errors should be less than a percent of the continuum value of B over the relevant extrapolation range. In addition, we find that finite volume K effects in B are also small, and are of O(1%) for the lightest pion mass on the MILC coarse K ensemble. This paper is organized as follows. We review mixed action chiral perturbation theory (MAχPT) in Sec. II. In Sec. III we calculate B to next-to-leading order in MAχPT. This K is divided into four subsections: we first present the spurion analysis necessary to map the quark-level B operatorontoanoperatorinthechiraleffective theoryinSec.IIIA, calculate K the 1-loop contributions to B at NLO in Sec. IIIB, determine the analytic contributions K to B at NLO in Sec. IIIC, and finally present the complete expressions for B at NLO K K in MAχPT in Sec. IIID. Next, in Sec. IV, we estimate the numerical size of both taste- symmetry breaking contributions and finite volume effects on the existing MILC ensembles using the resulting mixed action χPT formulae. Finally, we conclude in Sec. V. II. MIXED ACTION CHIRAL PERTURBATION THEORY In this section we review the leading-order mixed action chiral Lagrangian, first determined in Ref. [36], and discuss some of its physical consequences for the pseudo-Goldstone boson sector. We consider a partially quenched theory with N Ginsparg-Wilson valence quarks and val N staggered sea quarks. Each staggered sea quark comes in four tastes, and each G-W sea valence quark has a corresponding bosonic ghost partner to cancel its contribution to loop diagrams. For example, in the case N = 2 and N = 3 , the quark mass matrix is given val sea by M = diag(m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ), (1) u u u u d d d d s s s s x y x y sea valence ghost where we label the|dynamical quarks by u{,zd, and s and the valenc}e|qu{azrk}s b|y{xzan}d y. Near the chiral and continuum limits, the mixed action theory has an approximate SU(4N + sea N |N ) ⊗ SU(4N + N |N ) graded chiral symmetry. In analogy with QCD, we val val L sea val val R 6 assume that chiral symmetry spontaneously breaks to its vector subgroup, SSB SU(4N +N |N ) ⊗SU(4N +N |N ) −−→ SU(4N +N |N ) , (2) sea val val L sea val val R sea val val V and gives rise to (4N + 2N )2 −1 pseudo-Goldstone bosons (PGBs). These PGBs can sea val be packaged in an SU(4N +N |N ) matrix: sea val val U π+ K+ Q Q ··· ··· ux uy  π− D K0 Q Q ··· ···  dx dy  K− K0 S Q Q ··· ···   sx sy  2iΦ   Σ = exp , Φ = Q† Q† Q† X P+ R† R† , (3) f  ux dx sx xex yex  (cid:18) (cid:19)    Q† Q† Q† P− Y R† R†   uy dy sy xey yey     ··· ··· ··· Rxex Rxey X P+       ··· ··· ··· Ryex Ryey P− Y   e e    where f is normalized such that f ≈ 131MeV. The upper-left block of Φ contains sea-sea π e e PGBs, each of which comes in sixteen tastes. For example, 16 T b U = U , (4) b 2 b=1 X where the Euclidean gamma matrices T = {ξ ,iξ ξ ,iξ ξ ,ξ ,ξ } (5) b 5 µ 5 µ ν µ I are the generators of the continuum SU(4) taste symmetry (ξ is the 4×4 identity matrix). I The fields in the central block are the flavor-charged (P+ and P−) and flavor-neutral (X and Y) valence-valence PGBs, while those in the lower-right block with tildes are the analogous PGBs composed of only ghost quarks. Finally, the off-diagonal blocks contain “mixed” PGBs: those labelled by R’s are composed of one valence and one ghost quark, while those labelled by Q’s are composed of one valence and one sea quark. We do not show the mixed ghost-sea PGBs explicitly; their locations are indicated by ellipses. Under chiral symmetry transformations, Σ transforms as Σ −→ L Σ R†, L,R ∈ SU(4N +N |N ) . (6) sea val val L,R The standard mixed action chiral perturbation theory power-counting scheme is p2 /Λ2 ∼ m /Λ ∼ a2Λ2 , (7) PGB χ q QCD QCD 7 so the lowest-order, O(p2 ,m ,a2), mixed action chiral Lagrangian is PGB q f2 µf2 L¸ = Str ∂ Σ∂ Σ† − Str ΣM† +MΣ† +a2(U¸ +U¸′ +U¸ ), (8) 8 µ µ 4 S S V (cid:0) (cid:1) (cid:0) (cid:1) where Str indicates a graded supertrace over both flavor and taste indices and µ is an undetermined dimensionful parameter of O(Λ ). The leading-order expression for the QCD mass-squared of a valence-valence PGB is identical to that of the continuum because the chiral symmetry of the valence sector prevents an additive shift due to lattice spacing effects: m2 = µ(m +m ). (9) xy x y U and U′ comprise the well-known staggered potential and come from taste-symmetry S S breaking in the sea quark sector [19]. U splits the tree-level masses of the sea-sea PGBs S into degenerate groups: m2ff′,t = µ(mf +mf′)+a2∆t, (10) where ∆ is different for each of the SO(4)-tasteirreps: P, V, A, T, I. In particular, ∆ = 0 t P because the taste-pseudoscalar sea-sea PGB is a true lattice Goldstone boson. The mixed action Lagrangian contains only one new operator, and thus one new low-energy constant, as compared to the staggered chiral Lagrangian: U¸ = −C Str τ Στ Σ† , (11) V Mix 3 3 (cid:0) (cid:1) where τ = P¸ −P¸ = diag(I ⊗I ,−I ,−I ). (12) 3 sea val sea taste val val This operator links the valence and sea sectors and generates a shift in the mass-squared of a mixed valence-sea PGB: 16C m2 = µ(m +m )+a2∆ , ∆ = Mix . (13) fx f x Mix Mix f2 Althoughtheparameter∆ hasnotyet beencalculatedinmixedactionlatticesimulations, Mix it can, in principle, be determined by calculating the mass of a mixed valence-sea meson on the lattice. As in any partially quenched theory, the mixed action theory contains flavor- neutral quark-disconnected hairpin propagators which have double pole contributions. The only flavor-neutral propagators that appear in the expression for B are those with two K valence quarks; these have the following form: δ XY G (q) = +D (q), (14) XY q2 +m2 XY X 8 where 1 1 (q2 +m2 )(q2 +m2 )(q2 +m2 ) D (q) = − UI DI SI (15) XY 3(q2 +m2 )(q2 +m2 ) (q2 +m2 )(q2 +m2 ) X Y πI0 ηI is the disconnected (hairpin) contribution. Note that the sea-sea PGBs in the above expression are all taste singlets because the valence quarks do not transform under the taste symmetry. III. B AT NLO IN MIXED ACTION CHIRAL PERTURBATION THEORY K In this section we outline the calculation of B in mixed action χPT. We divide it into K four subsections. We first determine the operators that contribute to B in the mixed K action chiral effective theory using a spurion analysis in Sec. IIIA. In Sec. IIIB we outline the 1-loop calculation of B , and in Sec. IIIC we follow this up with an enumeration of the K corresponding analytic terms. Finally, the complete NLO results are presented in Sec. IIID. A. B Spurion Analysis K The spurion analysis for B in the mixed action case is similar to that in the continuum K [25, 37]; we will point out differences when they occur. In continuum QCD, B is defined as a ratio of matrix elements: K M K B ≡ . (16) K M vac The numerator in the above expression measures the hadronic contribution to neutral kaon mixing: M = hK0|O |K0i, (17) K K O = [sγ (1−γ )d][sγ (1−γ )d], (18) K µ 5 µ 5 where we have dropped color indices in O because both color contractions give rise to the K sameoperatorsinthechiraleffectivetheory. O isanelectroweak operatorwhichtransforms K as a (27 ,1 ) under the standard continuum chiral symmetry group. The denominator in L R Eq. (16) is the same matrix element as in the numerator evaluated in the vacuum saturation approximation: 8 Mvac = hK0|sγ (1−γ )d|0ih0|sγ (1−γ )d|K0i, (19) K 3 µ 5 µ 5 9 so that B is normalized to be of O(1). K In the mixed action theory, we define B in an analogous manner, except that both the K external kaons and the operator O must now be composed of valence quarks: K Olat = [yγ (1−γ )x][yγ (1−γ )x]. (20) K µ 5 µ 5 We can rewrite this operator as follows: Olat = 4[q (γ ⊗P )q ][q (γ ⊗P )q ], (21) K L µ yx L L µ yx L where the subscript “L” on the quark field indicates the left-handed projection and the matrix P is a 4N + 2N matrix in flavor space that projects out the yx component yx sea val of each quark bilinear ([P ] = δ δ for N = 3, N = 2). If we promote P to yx ij i,14 j,13 sea val yx a spurion field, F , which can transform under the mixed action chiral symmetry group K SU(4N +N |N ) ⊗SU(4N +N |N ) , then F must transform as sea val val L sea val val R K F → LF L†, (22) K K so that Eq. (21) remains invariant. One cannot build a chirally invariant operator out of Σ and the spurion field F without derivatives, but one can build two such operators at K O(p2 ): PGB Str[Σ∂ Σ†F ]Str[Σ∂ Σ†F ], (23) µ K µ K µ X Str[Σ∂ Σ†F Σ∂ Σ†F ]. (24) µ K µ K µ X It turns out however, that these two operators are equivalent when one demotes the spurion F to the matrix P . Thus we are left with a single chiral operator: K yx Oχ = Str[Σ∂ Σ†P ]Str[Σ∂ Σ†P ]. (25) K µ yx µ yx µ X This operator is identical in form to the continuum B operator [37], but Σ contains more K fields and the standard trace has been promoted to a supertrace. Because this operator is of O(p2 ), operators of O(m ) and O(a2), if present, could PGB q also potentially contribute at the same order in χPT. Recall that the quark mass matrix, when promoted to a spurion field, must transform as M → LMR†; thus we cannot form a chiral operator with a single power of M and two powers of F that is invariant under the K 10