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Unlike the case of the gauge action, attempts to straightforwardly discretize the fermion action quickly run into trouble. Considering, for simplicity, a single flavor, the Euclidean Dirac action can

be written as S = ψ_α(x)K_αβ(x, y)ψ_β(y), with K_αβ(x, y) = 1

2a(γ_µ)_αβ(δ_y,x+aˆµ− δ_y,x−aˆµ) + mδ_αβδ_xy (1.26) and α, β denoting spinor indices. By taking the Fourier transform of this operator and inverting in momentum space, it can be shown [4] that the lattice propagator is

ψ_α(x)ψ_β(y) = Z π/a

−π/a

d⁴p (2π)⁴

[ − iP

µγ_µpe_µ+ m]_αβ P

µep²_µ+ m² e^ip·(x−y), (1.27) with

peµ= 1

asin (apµ) . (1.28)

In the limit a → 0 we should recover the continuum Dirac propagator, but this is spoiled by the observation that peµ ≈ p_µ not only near the origin, but also when |p_µ|≈ π/a. Since this is true of any individual component of the momentum, we see that the naïve fermion action of Equation (1.26) actually describes sixteen degenerate fermion “tastes” in the continuum limit. This is known as the fermion doubling problem.

In Refs. [7, 8] Nielsen and Ninomiya provided an elegant characterization of the fermion doubling problem through a famous no-go theorem. They proved that it is not possible to construct a lattice Dirac operatorD for an even dimensional spacetime which is simultaneously:

1. Hermitian

2. Translationally invariant

3. Local, i.e. D(x, y) decays exponentially fast at large distances |x − y|  1

4. Consistent with chiral symmetry at vanishing quark mass, i.e. respecting {D, γ5} = 0 5. Free of doublers

In essence, their proof exploits the Poincaré-Hopf index theorem to demonstrate that conditions 1-4 necessarily lead to doublers for a lattice theory defined on an even-dimensional torus T^d.

A number of fermion actions are in common use in the literature, including Wilson, staggered, twisted mass, domain wall, and overlap fermions. These actions typically involve trade-offs between

violating particular conditions of the Nielsen-Ninomiya theorem, the size of lattice artifacts at finite lattice spacing, and the relative computational cost of performing a simulation. The best choice of action for a particular calculation is often highly dependent on the details of the target physics and the available computational resources. We will not attempt to provide a general overview, since reviews of each formulation can be found in the literature, but will instead focus on the domain wall fermion action used in this thesis.

Domain Wall Fermions

Domain wall fermions (DWF) avoid the Nielsen-Ninomiya no-go theorem in a particularly clever way: by adding a fictitious fifth spatial direction — conventionally labeled s, with L_s lattice sites along this direction — to sidestep the critical assumption of an even-dimensional spacetime.

Shamir and Furman [9, 10], building off of earlier work by Kaplan [11], demonstrated that effective 4D chiral fermions can be recovered at the s-boundaries of a five dimensional theory. While the DWF formalism has the nice property that it can have arbitrarily exact chiral symmetry in the limit L_s → ∞, and is empirically found to maintain excellent chiral symmetry even at modest L_s, this advantage comes at the price of an O(L_s) increase in the computational cost due to the extra dimension.

The generic domain wall-type fermion action takes the form SDWFψ, ψ, U  =X

xs

X

x⁰s⁰

ψ_xs(DDWF)_xs;x0s⁰ψ_x⁰_s⁰, (1.29)

where

(DDWF)_xs;x0s⁰ = b_s(D_W)_xx0δ_ss⁰+ δ_xx⁰δ_ss⁰ + c_s(D_W)_xx0L_ss⁰− δ_xx⁰L_ss⁰ (1.30) is the DWF Dirac operator,

(DW)_xx0 = (4 + M5) δxx⁰ −1 2

X

µ

h

(1 − γµ) Uµ(x)δx+ˆµ,x⁰ + (1 + γµ) U_µ^†(x⁰)δx−ˆµ,x⁰

i

(1.31)

is the four-dimensional Wilson Dirac operator, and

L_ss⁰ = (L₊)_ss0P_R+ (L−)_ss0P_L (1.32)

is the 5D hopping matrix, with

This construction may be regarded as a theory of L_sWilson fermions of mass −M₅that mix through the “mass” matrix L_ss⁰. The gauge field remains a four-dimensional object and is merely replicated for each s-slice. Four dimensional fermion fields q and q with mass m and definite chiralities are recovered from the five dimensional quark fields ψ and ψ at the boundaries of the fifth dimension

q_L= P_Lψ₀ q_R= P_Rψ_Ls−1

q_L= ψ₀P_R q_R= ψ_Ls−1P_L

. (1.34)

Correlation functions constructed from q and q approximate continuum QCD arbitrarily well in the simultaneous continuum and infinite volume limits.

Propagation and mixing of the light left-handed and right-handed modes through the fifth dimension is exponentially suppressed in L_s, but still nonzero when L_s is finite. In addition, the doubler states appear as heavy modes propagating in the five-dimensional bulk. It can be shown that this leads to mild chiral symmetry breaking effects, the largest of which is an additive renormalization of the bare fermion mass m → m+m_resby the residual mass (m_res) [12]. Simulating QCD with light pions forces L_s to be taken sufficiently large to keep m_res under control. In the limit L_s → ∞, however, the heavy modes propagating in the five-dimensional bulk dominate the spectrum, leading to a divergence. This divergence can be removed by introducing a heavy, Pauli-Villars regulator field: in practice one always computes a determinant ratio

det

 D(m) D(mpv)



(1.35) with m_pv  m when simulating QCD with domain wall fermions. This modification can be shown to remove the bulk divergence without affecting the desired low-energy chiral physics [13].

In addition to tuning L_s, the coefficients b_s and c_s can also be chosen to further suppress chiral symmetry breaking, at the expense of making domain wall fermions more expensive to simulate; the ability to achieve the same m_res with smaller L_s often justifies the use of these more sophisticated actions. The original Shamir DWF construction of Shamir and Furman has b_s= 1 and c_s= 0 for all s. Other variants commonly used in the literature include:

• Möbius DWF [14–16]: b_s− c_s = 1 and b_s+ c_s = α for all s, where α is a free parameter known as the Möbius scale.

• Optimal DWF [17]: b_s and c_s are real parameters constructed to minimize chiral symmetry breaking at fixed L_s.

• zMöbius DWF [18, 19]: b_s and c_s are complex parameters constructed to minimize chiral symmetry breaking at fixed L_s.

The simulations presented in this thesis make use of either the Shamir or Möbius DWF action.