Undoubled Chiral Fermions on a Lattice

TL;DR

This paper demonstrates that a lattice formulation of an $SU_L(2) imes U_R(1)$ chiral theory can avoid fermion doubling and spontaneous symmetry breaking, maintaining chiral symmetry and producing a single chiral fermion in the continuum limit.

Contribution

It introduces a lattice approach with strong multifermion coupling that successfully decouples doublers and preserves chiral symmetry without breaking it.

Findings

Doublers are decoupled as massive Dirac fermions.

The spectator fermion remains free and decoupled.

A single chiral fermion persists in 1+1 dimensions.

Abstract

We analyze the dynamics of an $SU_L(2)\otimes U_R(1)$ chiral theory on the lattice with a strong multifermion coupling. It is shown that no spontaneous symmetry breaking occurs; the ``spectator'' fermion $\psi_R$ is a free mode; doublers are decoupled as massive Dirac fermions consistently with the chiral symmetries. In 1+1 dimension, we show that the right-handed three-fermion state disappears at the threshold and an undoubled left-handed chiral fermion remains in the continuum limit.

Equations

$$S$$

$$S_k$$

$${\psi }_R(x)\to {\psi }_R(x)+ϵ.$$

$$W(\eta )$$

$$Z(\eta )$$

$$\frac{1}{2a}\partial {\psi }_R^{\prime }+g_2⟨\Delta \left({\bar{\psi }}_L^i\cdot \Delta {\psi }_R{\psi }_L^i\right)⟩-\frac{\delta \Gamma }{\delta {\bar{\psi }}_R^{\prime }}=0.$$

$${\int }_x{e}^{-ipx}\frac{{\delta }^{(2)}\Gamma }{\delta {\psi }_R^{\prime }(x)\delta {\bar{\psi }}_R^{\prime }(0)}$$

$$\frac{1}{2}{\Sigma }^i(p)$$

$$w(p)=\sum _{\mu }(1-\cos p_{\mu }),{\Sigma }^i(0)=0.$$

$${\psi }_L^i\to (g_2{)}^{\frac{1}{4}}{\psi }_L^i;{\psi }_R\to (g_2{)}^{\frac{1}{4}}{\psi }_R.$$

$$Z=2^{4N}{\left(\det {\Delta }^2(x)\right)}^4,$$

$$S_{RL}^j(x)$$

$$S_{MR}^j(x)$$

$$S_{RL}^j(x)$$

$$S_{MR}^j(x)$$

$$S_{RL}^j(p)\equiv {\Sigma }^j(p)=0,S_{MR}^j(p)=0.$$

$${\Psi }_R^i=\frac{({\bar{\psi }}_R\cdot {\psi }_L^i){\psi }_R}{2a};{\Psi }_L^n=\frac{({\bar{\psi }}_L^i\cdot {\psi }_R){\psi }_L^i}{2a},$$

$${\Psi }_c^i=({\psi }_L^i,{\Psi }_R^i),{\Psi }_n=({\Psi }_L^n,{\psi }_R).$$

$$S_{LL}^{ij}(x)$$

$$S_{ML}^{ij}(x)$$

$$S_{MM}^{ij}(x)$$

$$S_{LL}^{ij}(x)$$

$$S_{MM}^{ij}(x)$$

$$S_{ML}^{ij}(x)$$

$$S_c^{-1}(p)=\frac{i}{a}\sum _{\mu }{\gamma }_{\mu }\sin p^{\mu }+8a{g}_2{w}^2(p).$$

$$E(P)=\pm \sqrt{{\sin }^{2}P+(8a^{2}g{w}^2(P){)}^2},$$

$$P=p_1+p_2+p_3\ll \frac{\pi }{2}.$$

$$E$$

$$E_i$$

$$E_t(P)$$

$$p_3$$

$$\delta =E_t(P)-E(P)>0.$$

$$\delta =E_t(P)-E(P)=0.$$

$$\delta (P)\to 0,$$

Full text

Since the demonstration of “no-go” theorem [2] of Nielsen and Ninomiya in 1981 the problem of chiral fermion “doubling” and “vector-like” phenomenon on a lattice still exists if one insists on preserving chiral symmetry. One of the ideas to get around this “no-go” theorem was proposed with a multifermion coupling model [3]. However, it was pointed out [5] that such a model of multifermion couplings fails to give chiral fermions in the continuum limit.

We have been studying multifermion couplings on the lattice for years [6] and we believe that such models still have a chance to work [7]. Let us consider the following fermion action of the $SU_{L}(2)\otimes U_{R}(1)$ chiral symmetries on the lattice with a multifermion coupling.

[TABLE]

where “$a$” is the lattice spacing; $\psi^{i}_{L}$ ($i=1,2$) is an $SU_{L}(2)$ doublet, $\psi_{R}$ is an $SU_{L}(2)$ singlet and both are two-component Weyl fermions. The $\psi_{R}$ is treated as a “spectator” fermion. The $\Delta(x)$ is a second order differential operator on the lattice. The multifermion coupling $g_{2}$ is a dimension-10 operator relevant only for doublers $p=\tilde{p}+\pi_{A}$, but irrelevant for the chiral fermions $p=\tilde{p}\sim 0$ of the $\psi^{i}_{L}$ and $\psi_{R}$. Action (1) preserves the global chiral symmetry $SU_{L}(2)\otimes U_{R}(1)$ and the $\psi^{i}_{L}(x)$ can be gauged to have the exact local $SU_{L}(2)$ chiral gauge symmetry. In addition, action (1) possesses a $\psi_{R}$-shift-symmetry[8],

[TABLE]

The $U_{L}(1)$ global symmetry relating to the conservation of the fermion number of the $\psi_{L}^{i}(x)$ is explicit in eq.(1).

To seek a possible segment $g_{2}\gg 1$, where an undoubled $SU_{L}(2)$-chiral fermion exists in the continuum limit, we are bound to demonstrate the following properties of the theory in this segment:

the normal mode of $\psi_{R}$ is a free mode and decoupled; 2. 2.

no spontaneous chiral symmetry breaking occurs ; 3. 3.

all doublers $p=\tilde{p}+\pi_{A}$ are bound to be massive Dirac fermions and decoupled consistently with the $SU_{L}(2)\otimes U_{R}(1)$ chiral symmetry; 4. 4.

an undoubled chiral fermion $p=\tilde{p}$ of $\psi^{i}_{L}$ exists in the low-energy spectrum.

To prove the first and second points, we use the Ward identity stemming from the $\psi_{R}$-shift-symmetry (2). Considering the generating functional $W(\eta)$ of the theory,

[TABLE]

we define the effective action $\Gamma(\psi^{\prime i}_{L},\psi_{R}^{\prime})$ as the Legendre transform of $W(\eta)$. Making the parameter $\epsilon$ in eq.(2) to be space-time dependent, and varying the generating functional (3) according to the transformation rule (2) for arbitrary $\epsilon(x)\not=0$, we arrive at the Ward identity corresponding to the $\psi_{R}$-shift-symmetry of the action (1):

[TABLE]

From this Ward identity, one can obtain all one-particle irreducible vertices $\Gamma^{(n)}_{R}$ containing at least one external $\psi_{R}$. Taking functional derivatives of eq.(4) with respect to appropriate “primed” fields and then putting external sources $\eta=0$, one can derive:

[TABLE]

where $w(p)$ is the Fourier transform of ${1\over 2}\Delta(x)$, which is the Wilson factor [9]. One has

[TABLE]

Eq.(5) indicates an absence of the wave-function renormalization of the $\psi_{R}$. Eqs.(6,7) show the vanishing of the NJL symmetry breaking [4] for $p=0$.

To adopt the technique of strong-coupling expansion in powers of ${1\over g_{2}}$ and we make a rescaling of the fermion fields

[TABLE]

For the strong coupling $g_{2}\rightarrow\infty$, the kinetic terms $S_{k}$ can be dropped and we compute the integral of $e^{-S(x)}$ in this limit:

[TABLE]

where the determinant is taken only over the lattice-space-time and “$N$” is the number of lattice sites. For the non-zero eigenvalues of the operator $\Delta^{2}(x)$, eq. (9) shows an existence of the sensible strong-coupling limit. However, as for the zero eigenvalue of the operator $\Delta(x)$, this strong-coupling limit is not analytic and the strong-coupling expansion in powers of ${1\over g_{2}}$ breaks down.

To show the second point concerning the NJL symmetry breaking, we calculate the two-point functions:

[TABLE]

At non-trivial leading order $O({1\over g_{2}})$, we get the recursion relations:

[TABLE]

These recursion relations are not valid where the operator $\Delta(x)$ has zero eigenvalues. For $p\not=0$ and $\Delta(p)=2w(p)\not=0$, the Fourier transform of these recursion relations leads to the solution ($p\not=0$),

[TABLE]

Together with eq.(7), we prove the second point.

We turn to the third point that concerns decoupling of doublers. On this extreme strong coupling ($g_{2}\gg 1$) symmetric segment, the $\psi^{i}_{L}$ and $\psi_{R}$ in (1) are bound up to form the three-fermion states[3, 7]:

[TABLE]

which carry the appropriate quantum numbers of the chiral group that accommodates $\psi^{i}_{L}$ and $\psi_{R}$. These three-fermion states are Weyl fermions and respectively pair up with the $\bar{\psi}_{R}$ and $\bar{\psi}_{L}^{i}$ to be massive, neutral $\Psi_{n}$ and charged $\Psi_{c}^{i}$ Dirac fermions,

[TABLE]

To show this phenomenon, we compute the following two-point functions,

[TABLE]

In the lowest non-trivial order $O({1\over g_{2}})$, we obtain the following recursion relations

[TABLE]

Eq.(20) suggestes that the states coupling to operators $\psi^{i}_{L}(x)$ and $\Psi^{i}_{R}(x)$ are mixed, producing a massive four-component Dirac fermion. To find the masses, we make the Fourier transform of these recursion equations for $p\not=0$ and $\Delta^{2}(p)=4w^{2}(p)\not=0$ and obtain the inverse propagators of the charged Dirac fermions,

[TABLE]

This shows that all doublers $p=\tilde{p}+\pi_{A}$ are decoupled as massive Dirac fermions consistently with the $SU_{L}(2)\otimes U_{R}(1)$ chiral symmetries.

We are left with the last point that undoubled chiral fermions $(p=\tilde{p}\sim 0)$ of $\psi^{i}_{L}$ exist in this segment. We repeat the same calculations and discussions in 1+1 dimensions, where the time direction is continuous and space is discrete. We obtain the dispersion relation corresponding to the Dirac fermion (21) for $P\not=0$,

[TABLE]

which is analytically continuous to low momentum states $P\rightarrow 0$, unless it hits the threshold. For a given total momentum $P$, we consider the system that contains three free chiral fermions, i.e. right-movers $\bar{\psi}_{R},\psi_{R}$ with momenta $p_{1}$ and $p_{2}$ and a left-mover $\psi_{L}^{i}$ with momentum $p_{3}$ ($|p_{i}|\ll{\pi\over 2},i=1,2,3$),

[TABLE]

Since NJL spontaneous symmetry breaking does not occur when $|p_{i}|\rightarrow 0$, the total energy of such a system is given by

[TABLE]

where all negative-energy states have been filled. The lowest energy (the threshold) of such a system and corresponding configuration are

[TABLE]

The three-fermion state is stable if there is a gap $\delta$ between the threshold (25) and the energy (22) of the three-fermion state, i.e.

[TABLE]

The three-fermion state disappears, when the gap $\delta$ goes to zero,

[TABLE]

Substituting eqs.(22) and (25) into eq.(27), we obtain in the continuum limit $P\rightarrow 0$, the gap

[TABLE]

where the three-fermion-state spectrum dissolves into free chiral fermion spectra. As a result, an undoubled chiral fermion exists in the continuum limit.

I thanks Profs. G. Preparata, H. B. Nielsen and E. Eichten for discussions on the last point.