Flavored pions and kaons at next-to-leading order in mixed-action staggered chiral perturbation theory
Different versions of improved staggered fermions can be used as valence quarks to reduce discretization effects in lattice QCD calculations while increasing statistics on existing staggered gauge ensembles. Such mixed-action simulations can be used to improve determinations of light quark masses, Gasser-Leutwyler couplings, decay constants, and other parameters relevant to particle phenomenology. We recall the generalization of ordinary, unmixed staggered chiral perturbation theory required to describe data from lattice calculations with a mixed action such as with HYP staggered valence quarks and asqtad sea quarks. We calculate the next-to-leading order loop diagrams contributing to the masses and decay constants of the flavored pseudo-Goldstone bosons of all tastes and here report results for the decay constants and valence-valence masses.
Flavored pions and kaons at next-to-leading order in mixed-action staggered chiral perturbation theory
SWME Collaboration††thanks: W. Lee is supported by the Creative Research Initiatives program (2013-003454) of the NRF grant funded by the Korean government (MSIP). Brookhaven National Laboratory is operated under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. J.A.B. is supported by the Basic Science Research Program (2013009149) of the National Research Foundation of Korea (NRF) funded by the Ministry of Education.
Staggered chiral perturbation theory (SChPT) has been used extensively to control extrapolations from unphysical light-quark simulation masses to physical masses and to remove dominant light quark and gluon discretization effects . Mixed-action ChPT was developed for simulations performed with Ginsparg-Wilson valence quarks and (less computationally expensive) Wilson sea quarks . The formalism for staggered sea quarks and Ginsparg-Wilson valence quarks was developed in Ref. . Mixed-action ChPT for differently improved staggered fermions was introduced for calculations of the bag parameter  and the vector form factor . As a by-product, the pseudo-Goldstone boson (PGB) masses and propagators were calculated at tree level in mixed-action SChPT. While contributions to overall errors were very small, some of the low-energy couplings (LECs) of mixed-action SChPT were not well determined by the data [4, 5].
Here we present a calculation of the next-to-leading order (NLO) loop corrections to the masses and decay constants of the flavor-charged PGBs in all taste irreps. Analyses of corresponding spectrum data may improve our knowledge of LECs that are poorly determined by existing simulations. Our results can also be used to improve determinations of the light quark masses, Gasser-Leutwyler couplings, and pion and kaon decay constants. Data for such analyses could be generated with, e.g., asqtad sea quarks and valence HYP, or with HISQ sea quarks and improved staggered valence quarks that would allow for simulating relativistic bottom quarks at lattice spacings . Mixed-action SChPT results could facilitate future calculations of quantities such as and the amplitudes.
2 Mixed-action staggered chiral perturbation theory
Mixed-action theories are generalizations of partially quenched theories with different valence and sea quark actions. The symmetries relating valence and sea quarks are broken, but with differently improved versions of the same action, the symmetries of the valence (sea) sector are the same as in the unmixed theory. As for ordinary, unmixed SChPT, mixed-action SChPT is constructed in two steps. First one builds the Symanzik effective theory (SET) for the (mixed-action) lattice theory. One then maps the operators of the SET into those of ChPT .
2.1 The leading order Lagrangian
Mapping the SET Lagrangian into the chiral theory through NLO, we have
Here () has the form of the QCD action (the leading order Lagrangian of continuum ChPT), but respects the doubler symmetry, taste SU(4). As in continuum ChPT, the Lagrangian contains kinetic energy, mass, and anomaly terms. The operators in break the continuum symmetries, including those relating valence and sea quarks, to those of the mixed-action lattice theory. A subset of four-fermion operators in respects SO(4) SU(4). They map to the potential and can be obtained from those of the unmixed SET by introducing projection operators onto the valence and sea sectors and allowing the LECs to differ in the valence and sea sectors. Generically, we have
where () is a spin (taste) matrix. To construct the potential , the projection operators are conveniently included in spurions. The resulting potential is , where () contains single-(double-)trace operators that are direct generalizations of those in unmixed SChPT, and the last term is a taste-singlet potential new in the mixed-action theory, with . The operators in have independent LECs for the valence-valence, sea-sea, and valence-sea sectors. For example,
where means parity conjugate. For the unmixed case, , , and the potential reduces to that of ordinary SChPT. The full expressions for the potentials are somewhat lengthy, and we defer writing them down .
2.2 Tree-level propagators and flavored PGB masses
The potential contributes to the tree-level masses of the PGBs, which fall into irreps of SO(4). For a taste PGB composed of quarks with flavors ,
where labels the taste SO(4) irreps (pseudoscalar, axial, tensor, vector, or scalar), and is the condensate parameter. is the tree-level mass splitting, which depends on the LECs in and , as well as the sector (valence or sea) of the flavors . We have
where the splitting is if both quarks are valence (sea) quarks and otherwise; sub(super)scripts are taste indices labeling the generators of the fundamental irrep of U(4); and if the generators for and (anti)commute. The LECs come from the potential and are defined in analogy with the unmixed case in Ref. . The residual chiral symmetry in the valence-valence sector, as for the unmixed theory (equivalently, the sea-sea sector), implies particles are Goldstone bosons for , and therefore . The same is not true for the taste pseudoscalar, valence-sea PGBs, and generically, .
In the flavor-neutral sector, , the PGBs mix in the taste singlet, vector, and axial irreps. The Lagrangian mixing terms (hairpin terms) are
where are flavor indices; () is a taste index in the vector (axial) irrep; and we use an overbar (underbar) to restrict summation to the valence (sea) sector. The -term comes from the anomaly contribution. In continuum ChPT, taking at the end of the calculation decouples the . In SChPT, taking decouples the . The flavor-singlets in other taste irreps are PGBs and do not decouple . The -terms are lattice artifacts from the potential , and the couplings depend linearly on its LECs.
Although the mass splittings and hairpin couplings are different in the three sectors, we find the tree-level propagator can be written in the same form as in the unmixed case. We have
where the disconnected propagators vanish by definition in the pseudoscalar and tensor irreps, and for the singlet, vector, and axial irreps,
The index runs over the replica flavors which include the taste degrees of freedom. The index is summed over the physical sea quark flavors such as .
3 NLO loop corrections to masses
For a taste PGB composed of quarks with flavors , the mass is defined in terms of the self-energy, as in continuum ChPT. The NLO mass can be obtained by adding the NLO self-energy to the tree-level value,
consists of connected and disconnected tadpole loops with vertices from the LO Lagrangian at and analytic terms with vertices from the NLO Lagrangian at .
For a general SO(4) irrep, the calculation of the valence-valence, flavored PGB self-energies proceeds as for the unmixed case. Quark flows for the tadpoles are shown in graphs (a-f) of Fig. 1. The kinetic energy, mass, and vertices yield graphs of type (a), (c), and (d), and the taste-singlet potential vertices () yield graphs of type (a),
where is summed over ; is summed over the physical sea quarks; and is the chiral logarithm, with the scale of dimensional regularization.
Vertices from yield graphs of type (b), (e), and (f). The hairpin vertex graphs are of type (e) and (f). As in the unmixed case, they ((e) and (f)) can be combined and converted into the form of (d). In the mixed-action case, the necessary identity is
Adding the graphs and evaluating the result at , we have the NLO, one-loop contributions to the self-energies of the valence-valence, flavored PGBs,
where unless , when it vanishes, is a trace over (a product of) generators of U(4), depends on the LECs of , and
which is the generalization of the results of Ref.  to the mixed-action case. Only graphs of type (d) contribute.
For the valence-sea, flavored PGB self-energies, the calculation proceeds similarly, but there is no symmetry under , and the taste pseudoscalars are not Goldstone bosons in the chiral limit. Graphs of type (a) contribute chiral logarithms from valence-sea and sea-sea PGBs in the loop, and graphs of type (b) enter with valence-sea PGBs in the loop. The elimination of contributions of type (e) and (f) in favor of contributions of type (d) leaves leftover chiral logarithms of the valence-valence PGBs, , multiplied by combinations of hairpin couplings that vanish in the unmixed limit. For , graphs of type (a), (b), (c), and (e) contribute; the valence-sea, taste-pseudoscalars PGBs are not true Goldstone bosons in the chiral limit. However, no new loop integrals arise. The details of the calculation and the results will be presented in Ref. .
4 NLO loop corrections to decay constants
The decay constants are defined in terms of the matrix elements of the axial currents,
The NLO corrections come from wave function renormalization [graphs (a), (c), and (d) of Fig. 1], insertions of the -terms of the LO current [graphs (g), (h), and (i) of Fig. 1], and NLO analytic terms . The LO current is determined by the kinetic energy vertices of the LO Lagrangian, and is therefore the same as in unmixed SChPT. Likewise, the NLO wave function renormalization corrections are determined by self-energy contributions from tadpoles with kinetic energy vertices from the LO Lagrangian. To generalize the results of the unmixed case, we have only to replace the propagators with those of the mixed-action theory; nothing in the calculation of the relevant part of the self-energies or the current-vertex loops is sensitive to the sector of the external quarks. We have
where can take either valence () or sea () indices. This result holds for valence-valence (), sea-sea (), and valence-sea ( and ) PGBs, and has the same form as in the unmixed theory .
Using mixed-action SChPT, we have calculated the NLO loop corrections to the valence-valence and valence-sea masses and decay constants of flavored PGBs in all taste irreps. The results have been cross-checked by performing two independent calculations, and we have verified the results reduce correctly when valence and sea quark actions are the same. In the valence-valence sector, the taste pseudoscalars are Goldstone bosons in the chiral limit, at nonzero lattice spacing. The NLO analytic terms arise from tree-level contributions of the (NLO) Gasser-Leutwyler and generalized Sharpe-Van de Water Lagrangians. They have the same form as in the unmixed case, with different couplings in the valence-valence, sea-sea, and valence-sea sectors.
The NLO loop corrections to the self-energies of the valence-valence, flavored PGBs are given in Eqs. (3.0)-(3.0); those for the valence-valence and valence-sea decay constants are given in Eq. (4.0). They have the same form as the results in ordinary, unmixed SChPT. The results for the valence-sea self-energies will be included in Ref. .
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