New $N_f=2$ Pseudofermion Action for Monte-Carlo Simulation of Lattice Field Theory with Domain-Wall Fermions

We construct a novel $ N_f = 2 $ pseudofermion action for Monte-Carlo simulation of lattice gauge theory with domain-wall fermions (DWF), of which the effective four-dimensional lattice Dirac operator is equal to the overlap-Dirac operator with the argument of the sign function equal to $ H = c \gamma_5 D_w (1 + d D_w)^{-1} $, where $ c $ and $ d $ are parameters, and $D_w$ is the standard Wilson-Dirac operator plus a negative parameter $-m_0 \; (0<m_0<2)$. This new action is particularly useful for the challenging simulations of lattice gauge theories with large $N_f = 2n $ DWF, on the large lattices, and in the strong-coupling regime.


I. INTRODUCTION
In lattice gauge theory with N f dynamical domain-wall fermions (DWF) [1], one often considers the cases containing two fermions with degenerate masses. For example, N f = 2, N f = 2 + 1, and N f = 2 + 1 + 1 QCD, in which the masses of u and d quarks are degenerate, i.e., the theory in the isospin symmetry limit. For hybrid Monte-Carlo (HMC) simulations [2] of these theories, we have been using the N f = 2 pseudofermion action as given in Ref. [3].
Recently, for studies relating to the beyond Standard Model with composite Higgs boson, one investigates whether a lattice gauge theory with a large number of massless fermions can possess an infrared conformal fixed-point, e.g., the SU(3) lattice gauge theory with N f = 10 massless fermions, in which the pseudofermion action can be written as the sum of five identical N f = 2 massless pseudofermion actions. However, the HMC simulation based on the traditional N f = 2 pseudofermion action turns out to be rather time-consuming for large lattices at strong couplings. In this paper, we construct a new N f = 2 pseudofermion action for all variants of DWF, which is more efficient than the traditional N f = 2 pseudofermion action for the HMC simulation of the theories with large N f = 2n massless DWF, especially for large lattices in the strong-coupling regime, as first reported in Ref. [4]. In this paper, we demonstrate that, for the SU(3) lattice gauge theory with N f = 10 massless Möbius DWF on the 32 4 lattice at 6/g 2 0 = 5.70, the efficiency (speed × acceptance rate) of the HMC simulation with the new action is about 1.2 times of its counterpart with the traditional action.
In general, the 5-dimensional lattice Dirac operator of any kind of DWF with infinite extent in the fifth dimension (N s = ∞) gives the effective 4-dimensional lattice Dirac operator equal to where m q is the bare quark mass, and D w is the standard Wilson Dirac operator plus a negative parameter −m 0 (0 < m 0 < 2).
Here c and d are parameters to specify the type of DWF. Setting m q = 0, c = 1 and d = 0, (1) becomes the massless overlap-Dirac operator [5,6], In other words, the effective 4-dimensional lattice Dirac operator (1) of any kind of DWF can be regarded as the generalized overlap-Dirac operator with the argument of the sign function equal to H (2). In the massless limit m q = 0, (1) satisfies the Ginsparg-Wilson relation [7] where the chiral symmetry is broken by a contact term, i.e., the exact chiral symmetry at finite lattice spacing.
On the 5-dimensional lattice with size (N 3 x × N t × N s ), the lattice fermion operator of all variants of DWF [8][9][10][11] can be written as where s and s ′ are the indices in the fifth dimension, x and x ′ denote the lattice sites on the 4-dimensional lattice, and the Dirac and color indices have been suppressed. Here D w is the standard Wilson Dirac operator plus a negative parameter −m 0 (0 < m 0 < 2), where U µ (x) denotes the link variable pointing from x to x+μ. The operator L is independent of the gauge field, and it can be written as where Note that the matrices L ± satisfy L T ± = L ∓ , and R 5 L ± R 5 = L ∓ , where R 5 is the reflection operator in the fifth dimension, with elements (R 5 ) ss ′ = δ s ′ ,Ns+1−s . Thus R 5 L ± is real and symmetric.
Using the lattice DWF operator (4), and including the Pauli-Villars fields with bare mass m P V = 1/r = 2m 0 (1 − dm 0 ), the pseudofermion action for all variants of DWF can be written as where φ and φ † are complex scalar fields carrying the same quantum numbers (color, spin) of the fermion fields. Integrating the pseudofermion fields in the fermionic partition function gives the fermion determinant of the effective 4-dimensional lattice Dirac operator at finite Note that the pseudofermion action (7) for N f = 1 DWF cannot be used for HMC sim- is not positive-definite and Hermitian. A positive-definite, Hermitian, and exact pseudofermion action for N f = 1 DWF has been constructed in Ref. [12]. For N f = 2, it is straightforward to construct a positive-definite and Hermition pseudofermion action from (7), However, this N f = 2 pseudofermion action is not efficient for the HMC simulation.
The outline of this paper is as follows. In section 2, we derive the traditional N f = 2 pseudofermion action for DWF, which we have been using for the simulations of N f = 2 and N f = 2 + 1 + 1 lattice QCD. Even though the derivation of this N f = 2 pseudofermion action has been given in Ref. [13], we present a different derivation here, mainly for defining our notations in this paper. In section 3, we construct the new N f = 2 pseudofermion action for all variants of DWF. In section 4, we perform numerical simulations to compare the HMC efficiencies of the new and the old actions. In section 5, we conclude with some remarks.

II. THE TRADITIONAL N f = 2 PSEUDOFERMION ACTION
Using ρ s = cω s + d, and σ s = cω s − d, (4) can be rewritten as where L is defined in (5) and (6).
for the HMC simulation. Moreover, L and ω = diag(ω 1 , · · · , ω Ns ) are independent of the gauge field, we can drop the factor [cω 1/2 (1 from ω −1/2 D(m q )ω 1/2 , and obtain the re-scaled DWF operator for HMC, Here the dependence on m q has been shown explicitly in L ± , M ± , and N ± .
Next, we perform the even-odd preconditioning on D T (m q ). This is essential for halving the memory footprint as well as lowering the condition number of the conjugate gradient solver for the fermion force. With even-odd preconditioning on the 4-dimensional space-time lattice, (8) can be written as where Using the Schur decomposition, (12) becomes where Since det D T = det M −2 5 · det C, and M 5 does not depend on the gauge field, we can just use C in the Monte Carlo simulation. After including the Pauli-Villars fields with mass m P V = 2m 0 (1 − dm 0 ), we obtain the N f = 2 pseudofermion action for all variants of DWF, which has been used for the HMC simulations of N f = 2, and N f = 2 + 1 + 1 lattice QCD with the optimal DWF [14][15][16].
Then using the Sherman-Morrison formula, we obtain where Now using the optimal ω which is invariant under R 5 (i.e., R 5 ωR 5 = ω) [17], defining v ± ≡ R 5 A −1 ± u, and putting (18) into (10), we obtain where we have used Since A −1 ± is an lower/upper triangular matrix, we can solve for v ± (vectors in the fifth dimensional space) with the following recursion relation, where α s = 1/(c + dω −1 s ) and β s = −c + dω −1 s . Then we obtain where Q s ≡ α s+1 β s+1 ...α N s β N s . Using (19), we obtain where Putting (20) into (17) and using the identity det(1 + AB) = det(1 + BA), (17) becomes Note that K(m q ) is an operator on the 4-dimensional space, similar to the positive-definite Hermitian operators H 1 and H 2 in the exact pseudofermion action for one-flavor DWF [12]. However, K(m q ) is not Hermitian. For the N f = 2 pseudofermion action, it can be constructed as where φ and φ † are pseudofermion fields on the 4-dimensional lattice. This is the main result of this paper.

IV. NUMERICAL RESULTS
To where φ i and φ † i are pseudofermion fields, m P V = m 0 (2 − m 0 ) = 0.36 is the mass of the Pauli-Villars fields, and S g (U) is the Wilson plaquette gauge action Similarly, using the new N f = 2 pseudofermion action (22), the partition function of the SU(3) lattice gauge theory with N f = 10 massless fermions can be written as In both cases, the boundary conditions of the gauge field are periodic in all directions, while the boundary conditions of the pseudofermion fields are antiperiodic in all directions. In the molecular dynamics, we use the Omelyan integrator [18], and the Sexton-Weingarten multiple-time scale method [19]. Moreover, we introduce auxiliary heavy fermion fields (with mass m H a = 0.1) for the mass-preconditioning, similar to the case of Wilson fermion [20].  Table I. In Fig. 1 is estimated to be about 1.2 times of its counterpart with the traditional action. In the following, we present more details of the simulations.
In Fig. 2, we plot the maximum force (averaged over all links) among all momentum updates in each trajectory, for the gauge field, the heavy fermion field, and the light fermion field respectively. Here only the fermion forces corresponding to the first pair of pseudofermions are plotted, i.e., for i = 1 in (23)  of (a) with the traditional action, as summaried in Table I. By measuring the expectation value of ∆H, we can obtain the theoretical estimate of the acceptance rate for HMC, P acc = erfc ∆H /2 [21], which can be compared with the measured acceptance rate. As shown in Table I, the measured acceptance rate is consistent with the theoretical acceptance rate  (23) and (24). Finally, it is interesting to find out what is the renormalized coupling of this SU (3) gauge theory with N f = 10 massless Möbius DWF at 6/g 2 0 = 5.70. Performing the Wilson flow [23,24] with the small number (∼ 80) of thermalized gauge configurations generated in this test, we obtain an estimate of the renormalized coupling g 2 c (L, a) ∼ 4.5 in the finite-volume gradient flow scheme [25] with c = √ 8t/L = 0.3, which is much less than the largest coupling g 2 c ∼ 7.0 studied in Ref. [4].

V. CONCLUDING REMARK
To summarize, we have constructed a new N f = 2 pseudofermion action for lattice gauge theory with DWF. Moreover, we demonstrate that for the SU(3) gauge theory with N f = 10 massless Möbius DWF, the efficiency (speed × acceptance rate) of the HMC simulation with the new action is about 1.2 times of its counterpart with the traditional action. We expect that the gain of using the new N f = 2 pseudofermion action would become higher for more challenging simulations, i.e., lattice gauge theories with larger N f = 2n (e.g., N f = 12) massless DWF, on larger lattices (e.g., 64 4 ), and in the stronger coupling regime. Note that for the SU(3) lattice gauge theory with N f = 10 massless optimal DWF, one of us (TWC) encountered great difficulties (low acceptance rate and/or long simulation time) in the HMC simulation with the traditional N f = 2 pseudofermion action, on the 32 4 lattice at β = 6/g 2 0 = 6.45 [26]. The difficulties were circumvented by switching to the new N f = 2 pseudofermion action, and obtaining g 2 c (L, a) ∼ 8.6 in the finite-volume gradient flow scheme with c = √ 8t/L = 0.3. Nevertheless, a detailed study to compare the HMC efficiencies and characteristics of this theory with the new and the traditional actions is beyond the scope of this paper, since a single GPU (e.g., Nvidia GTX-TITAN) would take more than one year to generate 200-300 trajectories in the HMC with the traditional N f = 2 pseudofermion action.
In general, there is no guarantee that the new action would outperform the traditional action for any lattice field theory. In practice, one needs to perform numerical experiments to find out which action has higher HMC efficiency for the theory in question, which also depends on the computational platform and the algorithm implementation. Most importantly, now we have a new pseudofermion action to tackle the challenging simulations of lattice gauge thoeries with large N f = 2n massless domain-wall fermions.