Lattice QCD with mixed action – Borici-Creutz valence quark on staggered sea

Mixed action lattice QCD with Borici-Creutz valence quarks on staggered sea is investigated. The counter terms in Borici-Creutz action are fixed nonperturbatively to restore the broken symmetries. On symmetry restoration, the usual signatures of partial quenching / unitarity violation like negative scalar correlator are observed. The size of unitarity violation due to different discretization of valence and sea quark is determined by measuring ∆mix.


Introduction
Minimally doubled fermions are promising techniques to study light quarks on lattice. They are known to preserve chiral symmetry for a degenerate quark doublet and are local. This can be helpful for N f = 2 lattice simulations, and is relatively simpler and possibly faster than Ginsparg-Wilson fermions. There are two main variants of the minimally doubled fermions -Karsten-Wilczek [1,2], Borici-Creutz [3,4]. In this work, we study mixed action lattice QCD with Borici-Cruetz valence quarks in Asqtad improved staggered sea, restricting to light quarks only.
Presently, there is a dearth of numerical studies with BC fermions and very few literature that would suggest its usefulness and performance in lattice QCD simulations. Previously, using BC fermions we have studied discrete chiral symmetry breaking in a two-dimensional Gross-Neveu model [7] and mass spectroscopy in two-dimensional field theories [8] in lattice. In this work, we extend our investigations to four-dimensional lattice. Here, we consider a mixed action approach with BC valence quarks on an Asqtad improved staggered sea. The coefficients of the counterterms of the renormalized action are fixed nonperturbatively to restore the broken hypercubic symmetry of the BC action.
At finite lattice spacing, mixed action QCD violates unitarity but it is believed to have the correct continuum limit. The scalar correlator is known to be sensitive to the unitarity violation, due to different discretizations of valence and sea quarks. After the counterterms are tuned and the breaking of hypercubic symmetries minimized, the scalar correlator shows the effect of partially quenched QCD and becomes negative for valance quark mass smaller than sea quark mass.
In the leading-order mixed action chiral perturbation theory, a lattice spacing-dependent lowenergy constant ∆ mix appears as a free parameter in the mass formula for a meson made up of one valence and one sea quark. The ∆ mix gives a measure of the unitarity violation. The ∆ mix obtained in Speaker, e-mail: sbasak@niser.ac.in this work is found to be comparable to the mixed action with domain-wall fermions on a staggered sea [9,10].

Borici-Creutz action and Point split method
The free BC action in the discretized four-dimensional space-time lattice is written as where Γ = 1 2 (γ 1 + γ 2 + γ 3 + γ 4 ), {Γ, γ µ } = 1, and we have taken lattice spacing a = 1. In the momentum space, the action turns out to be diagonal and is The zeros of the free massless Dirac operator are at (0, 0, 0, 0) and (π/2, π/2, π/2, π/2). The spinor ψ(p) contains two degenerate flavors. We can construct the fields for these two flavors at two different poles by the method of point splitting [11,12]. We define such that d(0) = Γψ(0) and d(π/2) = 0, implying d is defined as an excitation around the pole at (0, 0, 0, 0). Similarly, u(0) = ψ(π/2) is defined around the pole at (π/2, π/2, π/2, π/2). In our work we have implemented the point split fields as defined in Eq. (3), where the summation over the Lorentz index µ is carried out. The BC action (1) has a special direction in Euclidean space which is the major hypercube diagonal given by Γ. This breaks reflection symmetry of the hypercube, leading to the breaking of parity and time symmetry [5]. Because of the broken hypercubic symmetry, the counterterms are necessary for a renormalized theory. The allowed counterterms for the BC action are dimension-4 counterterm c 4 (g 0 )ψ Γ µ D µ ψ and dimension-3 counterterm ic 3 (g 0 )ψ Γ ψ [13]. Then the complete renormalized action, with the gauge interaction turned on, reads wherec 3 = c 3 − 2, γ µ = Γ − γ µ = Γγ µ Γ and g 0 is the bare coupling parameter. The coefficients of the dimension-3 and -4 counterterms, c 3 and c 4 , respectively, have been evaluated in one-loop lattice perturbation theory [13]. Mixed action study has been carried out with the above renormalized BC action for valence quarks.

Simulation details
Mixed action simulation is carried out with BC fermions on three ensembles of publicly available MILC lattices with 2+1 dynamical flavors of Asqtad improved staggered fermions [14], with a fixed ratio am l /am s = 1/5. The details of MILC configurations used in this work are listed in Table 1.
Since the measurement of a 2 ∆ mix is one of the important parts of this study, we prefer to use lattices of three different spacings in lieu of multiple am l /am s ratios (for a fixed lattice spacing) as it has been shown in multiple studies that ∆ mix does not depend significantly on the sea masses [15]. The BC valence quark propagators are constructed using a range of bare quark masses [0.0075 -0.5] which restricts the m π L around 4. The strange mass is tuned by setting the fictitious ss pseudoscalar mass to 686 MeV [16]. A similar strange mass is also obtained by using m ps /m vec = 0.673 [14] for tuning. Random wall sources are used for the generation of all quark propagators. All the fits are uncorrelated and the errors are from Jack-knife analysis.

Parity condensate
The BC fermions exhibit parity-flavor breaking, which follows from the absence of hypercubic symmetry of BC action (1). Because the CPT is conserved, the T -symmetry is also broken. The counterterms are necessary for renormalized theory and those that are allowed by the remaining symmetry are added to the action. Two such counterterms of dimension-3 and 4 are introduced. The coefficients of these operators,c 3 (g 0 ) and c 4 (g 0 ), can then be tuned to restore the desired symmetries.
The quantity we measure to determine the size of parity breaking is the parity condensate | ψ iγ 5 τ 3 ψ |, where τ 3 is the third generator of S U (2). First we study the variation of the chiral condensate as we tunec 3 keeping c 4 fixed at its perturbative value for the lattice in use. The dependence of parity condensate onc 3 for various lattices is shown in Fig.1. The minimum of the condensate for all three ensembles appears aroundc 3 = −0.5. Besides, the actual values and the minimum of parity condensate appear to have much less dependence on the lattice spacing. Next we fixc 3 to this nonperturbative value and measure parity condensate for varying c 4 . This is plotted in Fig.2. We observe very little dependence of parity condensate on c 4 for all the three lattices. Here too we find that the parity condensate is almost independent of the lattice spacing. Therefore, tuningc 3 alone is sufficient for minimizing the breaking (or restoration) of parity.

Time asymmetry
The BC action has PT -symmetry, and because of breaking of parity, we expect to see the sign of time asymmetry in the theory and lattice calculation. In lattice simulation, this time asymmetry can show up in the spectrum and manifests through nondegeneracy of forward and backward propagating meson states. The pion propagators and the corresponding effective masses for different c 4 , while keeping  all the three lattices that we used. In Fig.4

Mixed action pion mass
The pion spectrum is obtained for valence quark mass in the range of [0.0075 -0.5]. In Figs. 5(a) and (b), we plotted the pion mass m π and pion mass squared m 2 π respectively as function of bare valence quark mass m. At small quark masses, the plot deviates from straight line due to the logarithmic  Figure 5. Pion mass as a function of valence quark mass m on 20 3 × 64 lattice. In (b), the dotted curve is a fit to the mass formula in Eq. (7) correction coming from one-loop chiral perturbation theory, where B is the LEC and f is the pion decay constant. The chiral log has been studied previously for different lattice fermions in the context of mixed action in [17][18][19]. A more complete mass formula in one-loop partially quenched chiral perturbation theory is given by [18,20], where C 1 , C 1L , C 2 , C 2L are independent low energy constants. The existence of chiral log in our mixed action simulation is more prominently visible when the ratio of m 2 π /m is plotted against m.

Scalar correlator
Lattice QCD with mixed action is inevitably partially quenched and violates unitarity. The full QCD can be recovered in the continuum limit only. The scalar meson (ψ ψ) is known to be sensitive to this unitarity violation, it gives the scalar correlator a negative value when valence quark mass is less than sea quark [23]. This effect of partial quenching is shown in Fig. 7, which is observed only after the parity is restored by nonperturbative tuning ofc 3 . The errors become large for partially quenched scalar correlators, but its negative value when m val (= 0.005) < m sea (= 0.01) is observable. From the plot forc 3 = −0.5, it is evident that as one increases valence quark mass, for a fixed sea quark mass, the negative contribution reduces.

∆ mix
In mixed action calculations, we can have three different types of mesons: mesons composed of (i) two valence quarks, (ii) two sea quarks and (iii) a mix of one valence and one sea quark. Each of these undergo lattice spacing dependent mass renormalization. In the leading order, the pseudoscalar meson masses for BC valence and Asqtad sea are given by where m v 1 v 2 (m s 1 s 2 ) is the pseudoscalar meson mass made up of valance (sea) quark and antiquark while m vs is the mass of the mixed meson composed of valence and sea quarks. The degree of unitarity violation at finite lattice spacing depends on the size of ∆ mix . The a 2 ∆ t are the taste splittings of the staggered pions, where t = A, T, V, I and a 2 ∆ 5 = 0 [24] and a 2∆ mix = a 2 ∆ mix + a 2 ∆ mix , where the primed term is given in Ref. [25]. The∆ mix can be extracted from the meson spectrum