The Chiral Dirac Determinant According to the Overlap Formalism

TL;DR

This paper evaluates the chiral Dirac determinant using the overlap formalism, demonstrating its consistency with continuum results for gauge fields, thereby validating the formalism's effectiveness.

Contribution

It provides the first detailed comparison showing the overlap formalism reproduces continuum chiral Dirac determinants for gauge backgrounds.

Findings

Real and imaginary parts match continuum results

Overlap formalism passes a key theoretical test

Supports use of overlap formalism in chiral gauge theories

Abstract

The chiral Dirac determinant is calculated using the overlap formalism of Narayanan and Neuberger. We compare the real and imaginary parts of the determinant with the continuum result for perturbative gauge field backgrounds and show that they are identical. Thus we find that the overlap formalism passes a crucial test.

Equations

$$H_{\pm}=\int d^{4}x\psi^{\dagger}\left(x\right)\gamma_{5}\left(\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }\pm\Lambda\right)\psi\left(x\right)$$

$$\mbox{det}[{{1}\over{2}}(1+\gamma_{5})(\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} })]=\lim_{\Lambda\rightarrow\infty}\frac{\langle+\mid A+\rangle\langle A+\mid A-\rangle\langle A-\mid-\rangle}{\langle+\mid-\rangle\mid\langle+\mid A+\rangle\langle-\mid A-\rangle\mid}$$

$$\displaystyle{\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}\left(A\right)\phi_{j}\left(A\right)$$

$$\displaystyle{\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}\left(A\right)\gamma_{5}\phi_{j}\left(A\right)$$

$$h_{\pm}=\gamma_{5}\left(\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }\pm\Lambda\right)$$

$$v_{\pm ,j}\left(A\right)$$

$$u_{\pm ,j}\left(A\right)$$

$$h_{\pm }{v}_{\pm ,j}$$

$$h_{\pm }{u}_{\pm ,j}$$

$$\left({\varphi }_j\left(A\right),{\varphi }_k\left(A\right)\right)=({\gamma }_5{\varphi }_j\left(A\right),{\gamma }_5{\varphi }_k\left(A\right))={\delta }_{jk},$$

$$({\varphi }_j\left(A\right),{\gamma }_5{\varphi }_k\left(A\right))=0.$$

$$⟨A+\mid A-⟩$$

$$M_{jk}$$

$$\text{mbox}detM$$

$$\underset{\Lambda \to \infty }{\lim }\frac{⟨A+\mid A-⟩}{⟨+\mid -⟩}$$

$$=$$

$$h_{\pm }{\psi }_{L,j}$$

$$h_{\pm }{\psi }_{R,j}$$

$$\text{mbox}Im\ln \text{mbox}det=\pi {\eta }_1\left(0\right)-2\pi Q_5^0\left({\hat{A}}_{t,1}\right).$$

$$\pi\frac{d\eta_{u}\left(0\right)}{du}=2\pi\frac{d}{du}Q_{5}^{0}+\frac{1}{2}\mbox{tr}\gamma_{5}\left(\frac{d{\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}\left(A_{u}\right)}{du}\right){\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}^{-1}\left(A_{u}\right)\left(1-{\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}^{2}\left(A_{u}\right)/M^{2}\right)^{-1/2}$$

$$\displaystyle\lim_{M\rightarrow\infty}\frac{1}{2}\mbox{tr}\gamma_{5}\left(\frac{d{\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}\left(A_{u}\right)}{du}\right){\parbox[b]{10.00002pt}{$D$}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{$/$} }}^{-1}\left(A_{u}\right)\!\!\!\!\!$$

$$\frac{d}{du}\left(\pi {\eta }_u\left(0\right)-2\pi Q_5^0\left({\hat{A}}_{t,u}\right)\right)$$

$$\text{mbox}Im\ln ⟨+\mid A+⟩⟨A-\mid -⟩=\text{mbox}Im\left(\ln \text{mbox}det{M}_{+}+\ln \text{mbox}det{M}_{-}\right).$$

$$⟨+\mid A+⟩$$

$$M_{+jk}$$

$$⟨A-\mid -⟩$$

$$M_{-jk}$$

$$M_{+jk}$$

$$M_{-jk}$$

$$\frac{d}{du}\ln \text{mbox}det{M}_{\pm u}$$

$$\frac{d}{du}\ln ⟨+\mid A+⟩⟨A-\mid -⟩$$

$$\lim_{\Lambda\rightarrow\infty}\frac{\langle+\mid A+\rangle\langle A+\mid A-\rangle\langle A-\mid-\rangle}{\langle+\mid-\rangle\mid\langle+\mid A+\rangle\langle-\mid A-\rangle\mid}=\parbox[b]{62.59605pt}{ \raisebox{-11.38109pt}{${}^{\sqrt{\left|\frac{\mbox{det}\parbox[b]{5.0pt}{$D$}\parbox[b]{4.00002pt}{ \raisebox{0.43054pt}{$/$} }\left(A\right)}{\mbox{det}\parbox[b]{5.0pt}{$\partial$}\parbox[b]{4.00002pt}{ \raisebox{0.43054pt}{$/$} }}\right|}}$}}e^{i\left(\pi\eta_{1}\left(0\right)-2\pi Q_{5}^{0}\left({\hat{A}}_{t,1}\right)\right)}.$$

Full text

Nordita 95/80

November, 1995

Lattice regularization of field theories is the only known non-perturbative regularization available to us. Chiral gauge theories have eluded a non-perturbative regularization for reasons summarized by the Nielsen-Ninomiya [1, 2] theorem which states that there exists no discretization of the chiral Dirac operator which simultaneously preserves a number of desirable physical properties. This is an unfortunate state of affairs since, at least at low-energies, the Weinberg-Salam model describes the physics of the world we live in, and this model involves chiral couplings of fermions to gauge fields. Recently attempts have been made to evade the theorem of Nielsen and Ninomiya in various ways (see [2, 3] for a recent review of progress in this direction.) We will be concerned with the approach of Narayanan and Neuberger[4] who, inspired by an idea of Kaplan’s[5], have proposed a new way of calculating chiral quantities on the lattice. They evade the Nielsen-Ninomiya theorem by studying an auxiliary problem in one dimension higher. Quantities in this auxiliary problem can then be related to the lower dimensional theory by taking certain limits. Thus the conclusions of the no-go theorem are avoided by formulating a problem which ostensibly has nothing to do with the original problem and in fact is formulated in odd dimensions where chirality is not an applicable concept.

In this letter we calculate the determinant of the chiral Dirac operator in $4$ dimensions using the recipe of Narayanan and Neuberger which has been dubbed the “overlap formalism”. First we evaluate the modulus of the determinant and show that it reproduces correctly the continuum result. We then evaluate the phase of the determinant and compare our result with the continuum result of Alvarez-Gaume, Della Pietra and Della Pietra[6] (similar expressions have also been derived by [7, 8]. We find that the results of the two approaches are identical. While calculating the imaginary part of the effective action we always work with perturbative gauge fields since this is the assumption under which the continuum results have been obtained. Our results confirm that at least the chiral determinant can be defined on the lattice using the overlap formalism.

The overlap formalism has been applied by a number of authors to various problems involving chiral and non-chiral fermions. Randjbar-Daemi and Strathdee have calculated chiral anomalies in 2 and 4 dimensions, the gravitational anomaly in 2 dimensions and the vacuum polarization in 4 dimensions [9, 10, 13, 12]. They have also calculated the two point functions for chiral fermions [9] and verified anomaly cancellation in the standard model using this formalism. Randjbar-Daemi and Fosco calculated the determinant of the chiral Dirac operator in a constant background gauge field with non-trivial holonomy on the two dimensional torus, verifying that the continuum result is reproduced including the holomorphic anomaly [11]. Narayanan and Neuberger have applied their formalism to the twisted chiral Dirac operator and confirmed numerically the continuum result [14]. Narayanan, Neuberger and Vranas [15] applied the overlap formalism to the Schwinger model and obtained results consistent with the continuum exact solution. This last piece of work is particularly interesting in that the gauge fields involved in that calculation are topologically non-trivial and therefore involve zero modes of the Dirac operator. While work on the present project was in progress we received [16] where the phase of the chiral determinant is calculated for domain wall fermions. The authors suggest that their results also apply to the overlap formalism.

The overlap formalism expresses chiral quantities in terms of certain objects in an auxiliary problem in one dimension higher. Specifically, one considers two five dimensional Hamiltonians (we use the simpler notation of Randjbar-Daemi and Strathdee developed in [9] and [13])

[TABLE]

where $\Lambda>0$ is a mass for the five dimensional fermions. Notice that the two Hamiltonians differ only by the sign of the mass term. The Dirac vacua for the two Hamiltonians are denoted by $\mid A\pm>$ which are Slater determinants of the non-positive eigenvalue states of the first quantized Dirac Hamiltonians. The overlap formalism states that the chiral Dirac determinant is given by the following expression:

[TABLE]

where $\mid\pm>$ are the Dirac vacua of the problem with vanishing gauge fields.

Before proceeding to the calculation we comment on the regularization procedure. We assume that the five dimensional overlap problem can be regularized on the lattice. We will never explicitly define a lattice regularization but will assume its existence and other formal properties of the lattice regularized operator which must be identical with the continuum problem. The Dirac operator as we will use it will always be a finite dimensional matrix and we will take 111 For our calculation of the modulus of the determinant it is sufficient to take $\Lambda\gg||A||$

$\Lambda\gg 2\pi/a+||A||$ where $||A||$ is the supremum of the expectation value of the gauge field and $a$ is the lattice spacing, ensuring that the eigenvalues of the Dirac operator are small compared to the mass $\Lambda$. There are two large scales in the problem: $\Lambda$ and the inverse of the lattice spacing $1/a$. Both are large but the relevant limit to reproduce the continuum result is the one in which $\Lambda\gg 1/a$. However, one could imagine various limits controlled by the dimensionless parameter $a\Lambda$. We will have nothing to say about this but we hope to return to this issue in the future (see, however, the discussion in [13]222 After submitting this work we received a paper by Randjbar-Daemi and Strathdee where this question is taken up (eprint archive: hep-th/9512112)). It would be very interesting to characterize any gauge non-invariant terms, which we neglect in the present work, “suppressed” by $a\Lambda$ and others suppressed by $a$ and $1/\Lambda$ . This may shed light on the continuum limit of the lattice regularized overlap formalism.

The four dimensional Dirac operator anti-commutes with $\gamma_{5}$, this allows one to pair the non-zero eigenmodes of the operator as follows:

[TABLE]

We adopt the notation that the $\phi_{j}$ are positive eigenvalue modes of -i{\parbox[b]{10.00002pt}{D}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{/} }} and $\epsilon_{j}$ denotes a positive eigenvalue. We assume first that there are no zero modes. Then the eigenstates $\left\{\phi_{j},\gamma_{5}\phi_{j}\right\}$ of the Dirac operator form a complete basis. We will always assume that $\phi_{j}\left(0\right)$ and $\phi_{j}\left(A\right)$ are smoothly related to each other by an interpolating gauge field $A_{t}$ between the gauge configurations [math] and $A$ such that the Dirac operator does not develop a zero mode anywhere along the interpolation. The first quantized hamiltonians:

[TABLE]

commute with {\parbox[b]{10.00002pt}{D}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{/} }}^{2} and allow one to diagonalize $h_{\pm}$ and {\parbox[b]{10.00002pt}{D}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{/} }}^{2} simultaneously. We can express the eigenstates of $h_{\pm}$ as linear combinations of a positive eigenstate of

$D$

$/$

and its negative eigenvalue pair. This has the virtue that the eigenstates of $h_{\pm}$ will be linear combinations of $\phi_{j},\gamma_{5}\phi_{j}$ and the $\Lambda$ dependence will occur only in the coefficients multiplying the eigenstates of

$D$

$/$

. Using the usual notation denoting positive (negative) eigenstates of the hamiltonians $h_{\pm}$ as $u_{\pm}$($v_{\pm}$) we get the following eigenstates:

[TABLE]

These wave functions satisfy:

[TABLE]

The Dirac vacua are then given by the Slater determinants of the negative energy states. We have normalized the above wave functions in such a way that if we have an inner product such that

[TABLE]

then the $v_{\pm},u_{\pm}$ are orthonormal with respect to the same inner product.

Now we will calculate $\langle A+\mid A-\rangle$. The states $\mid A\pm>$ are Slater determinants of the $v_{\pm}$, assuming for the moment that

$D$

$/$

has no zero modes,

[TABLE]

We can now evaluate $M_{jk}$ using equations ( 6) and ( 8). We find

[TABLE]

Thus

[TABLE]

The last equality follows from recalling that the $\epsilon_{j}$ are the positive eigenvalues of

$D$

$/$

.

Now we would like to show that if

$D$

$/$

has zero modes then $\langle A+\mid A-\rangle=0$. To demonstrate this we divide the zero modes of

$D$

$/$

into positive ($L$) and negative ($R$) “chirality” (i.e. with respect to $\gamma_{5}$) modes. Then denoting by $\psi_{L(R),j}$ the zero modes of

$D$

$/$

we see that they are eigenstates 333Note that the index $j$ does not necessarily run over the same number of values for the left and right handed wave functions, this number can be different if the gauge field carries a non-zero instanton number. of $h_{\pm}$:

[TABLE]

Therefore, the right handed zero modes will be in the $\mid A+\rangle$ vacuum but will not appear in $\mid A-\rangle$. The converse is true for the left handed zero modes. Using the orthogonality of the left and right handed modes then proves that $\langle A+\mid A-\rangle=0$. Of course, if the number of left and right handed zero modes is not the same then $\langle A+\mid A-\rangle=0$ for an additional reason than the one just stated, namely, there is a mismatch in the number of states in the two vacua.

So far we have evaluated $\langle A+\mid A-\rangle/\langle+\mid-\rangle$ for arbitrary gauge configurations. Since the remaining part of the Dirac determinant ( 2) is a phase while $\langle A+\mid A-\rangle/\langle+\mid-\rangle$ is a real non-negative number we have evaluated the magnitude of the chiral Dirac determinant and found that it is precisely as it should be. We turn now to the phase of the Dirac determinant which is, in a sense, at the heart of the matter since all the information about chirality is stored in this phase. The magnitude is merely the square root of the full Dirac determinant with vector couplings.

We are interested in calculating the phase of the determinant in background configurations for which there are continuum results available for comparison. The phase of the determinant was calculated by Alvarez-Gaume, Della-Pietra and Della-Pietra for perturbative background gauge fields for which there are no zero modes of the Dirac operator. They found that the phase can be written as:

[TABLE]

Where $Q_{5}^{0}\left[{\hat{A}}_{t,1}\right]$ is the five dimensional Chern-Simons form and ${\hat{A}}_{t,u}$ is a two parameter extension of the four dimensional gauge field such that ${\hat{A}}_{t,u}=0$ for $-\infty\leq t\leq-T$, ${\hat{A}}_{t,u}$ smoothly interpolates between [math] and $A_{u}$ for $-T<t<T$ and finally ${\hat{A}}_{t,u}=A_{u}$ for $T<t<\infty$. While $A_{u}$ is an interpolating field between [math] (for $u=0$) and $A$ (for $u=1$). $A$ is the four dimensional gauge field appearing in the Dirac operator whose determinant we wish to calculate. The object $\eta_{u}\left(0\right)$ is the so-called eta-invariant associated with the five dimensional Dirac operator H_{u}=i\gamma_{5}\partial_{t}+i{\parbox[b]{10.00002pt}{D}\parbox[b]{8.00003pt}{ \raisebox{0.86108pt}{/} }}\left({\hat{A}_{t,u}}\right). Crucial to the derivation of this result is the identity derived in [6]:

[TABLE]

The $M$ appearing on the right hand side is a Pauli-Villars mass regularizing the expression, and a limit where it is taken to infinity is implicit. If one takes this limit the last term on the right hand side becomes:

[TABLE]

This expression, without the Pauli-Villars mass, has appeared in the physics literature previously in a paper by Niemi and Semenoff [7]. Alternatively, one could define a lattice regularization of the operators appearing in the trace and then remove the Pauli-Villars regulator. In either case we need to take this limit to be able to compare with our calculation which has no Pauli-Villars regulator but a lattice regulator instead. Finally, we can evaluate the trace over the basis $\left\{\phi_{j}\left(A_{u}\right),\gamma_{5}\phi_{j}\left(A_{u}\right)\right\}$ to get

[TABLE]

The phase of the determinant in the overlap formalism is given by

[TABLE]

Where

[TABLE]

and

[TABLE]

We now take the limit $\Lambda\rightarrow\infty$ and evaluate the determinants in that limit.

[TABLE]

To compare the overlap method with the continuum method we consider a field $A_{u}$ interpolating between the configurations $A_{0}=0$ and $A_{1}=A$. Using the completeness of $\left\{\phi_{j}(0),\gamma_{5}\phi_{j}\left(0\right)\right\}$ It is easy to check that $M_{\pm}$ are unitary matrices in the limit $\Lambda\rightarrow\infty$. Thus:

[TABLE]

We arrive finally at the expression:

[TABLE]

Comparing with equation (17) we see that the continuum result for the phase of the chiral determinant is reproduced by the overlap formalism. Together with equation (12) we have

[TABLE]

We conclude with a few comments. The overlap recipe for the chiral Dirac determinant has passed an important test by reproducing the continuum result. What is most satisfying about this result is that while in the continuum the imaginary part of the effective action naively vanishes but is produced due to the regularization of the determinant and survives the limit in which the regulator is removed, in the overlap formalism one can reproduce this result independently of the specifics of the regularization procedure and any delicate limits. This is a consequence of the fact that we are always working with a parity non-invariant system which has a non-vanishing imaginary part independently of the lattice regulator. In our derivation we have neglected terms of order $\epsilon/\Lambda$ where $\epsilon$ is a typical eigenvalue of the Dirac operator. It would be interesting to keep these terms and to see how various limiting procedures in which $a\rightarrow 0$ and $\Lambda\rightarrow\infty$ can affect continuum limits. In any problem where more than one large scale is available one must eventually address the question of their relative size.

Acknowledgements We would like to thank B. Kileng, A. Krasnitz, L. Kärkkäinen, and especially A. Kronfeld for discussions. We are grateful to M. Schmaltz for pointing out some misprints in an earlier version of this article.