Wess-Zumino term by Vacuum Overlap Formula

TL;DR

This paper investigates the lattice regularization of the Wess-Zumino term in two-dimensional SU(2) models, demonstrating correct continuum behavior perturbatively and revealing nonperturbative properties like Gaussian distribution and sharp crossover effects.

Contribution

It provides a nonperturbative analysis of the vacuum overlap formula for the Wess-Zumino term, highlighting its behavior in different coupling regimes and the impact of species doublers.

Findings

Perturbatively reproduces the Wess-Zumino term in the continuum limit.

Nonperturbatively shows a sharp Gaussian distribution in the scaling region.

Identifies a sharp crossover from strong to weak coupling regimes.

Abstract

We examine the vacuum overlap formula for the two-dimensional SU(2) Wess-Zumino term in lattice regularization. Perturbatively it reproduces the Wess-Zumino term correctly in the continuum limit and yields the IR fixed point in the beta function of the chiral model. Nonperturbatively it shows a sharp Gaussian distribution for the SU(2) chiral field configurations in the scaling region, where smooth configurations dominate even in the symmetric phase due to asymptotic freedom. Crossover is sharp from the strong coupling region where the Wess-Zumino term fluctuates hard and the species doublers' contribution is suspected to affect it.

Equations

$$\det (\mathbf{C}{)}_{\mathrm{reg}}\equiv \frac{{}_{1\mathrm{l}}⟨-|-{⟩}_U{}_U⟨-|+{⟩}_U{}_U⟨+|+{⟩}_{1\mathrm{l}}}{{|}_{1\mathrm{l}}⟨-|-{⟩}_U|{}_{1\mathrm{l}}⟨-|+{⟩}_{1\mathrm{l}}|{}_U⟨+|+{⟩}_{1\mathrm{l}}|}.$$

$$H^{\pm }(U)$$

$$C_{n\text{mathchar}28939i,m\text{mathchar}28940j}$$

$$B_{n\text{mathchar}28939i,m\text{mathchar}28940j}$$

$$\{a_{n\text{mathchar}28939i},a_{m\text{mathchar}28940j}^{†}\}$$

$$\{a_{n\text{mathchar}28939i},a_{m\text{mathchar}28940j}\}$$

$${}_U⟨-|+{⟩}_U={\det }_{K,K^{\prime }}\left(\sum _n\text{mathchar}28960{}_K^{-}(n,U{)}^{†}\text{mathchar}28960{}_{K^{\prime }}^{+}(n,U)\right),$$

$${}_U⟨\pm |\pm {⟩}_{1\mathrm{l}}={\det }_{K,K^{\prime }}\left(\sum _n\text{mathchar}28960{}_K^{\pm }(n,U{)}^{†}\text{mathchar}28960{}_{K^{\prime }}^{\pm }(n,1\mathrm{l})\right),$$

$$U_{n,\text{mathchar}28950}\to U_{n,\text{mathchar}28950}^g=g_n^{†}{U}_{n,\text{mathchar}28950}{g}_{n+\widehat{\text{mathchar}28950}}.$$

$$\text{mathchar}28960{}_K^{\pm }(n,U)\to \text{mathchar}28960{}_K^{\pm }(n,U^g)=g_n^{†}\text{mathchar}28960{}_K^{\pm }(n,U),$$

$${}_{U^g}⟨\pm |\pm {⟩}_{1\mathrm{l}}={\det }_{K,K^{\prime }}\left(\sum _n\text{mathchar}28960{}_K^{\pm }(n,U{)}^{†}{g}_n\text{mathchar}28960{}_{K^{\prime }}^{\pm }(n,1\mathrm{l})\right).$$

$${\mathcal{W}}^{\pm }(U,g)$$

$$\exp (i\text{mathchar}28928{}_{WZ}[U,g])$$

$${}_{{1\mathrm{l}}^g}⟨\pm |\pm {⟩}_{1\mathrm{l}}={\det }_{K,K^{\prime }}\left(\sum _n\text{mathchar}28960{}_K^{\pm }(n,1\mathrm{l}{)}^{†}{g}_n\text{mathchar}28960{}_{K^{\prime }}^{\pm }(n,1\mathrm{l})\right),$$

$$\exp (i\text{mathchar}28928{}_{WZ}[g])$$

$$g_n=e^{i\text{mathchar}28958{}_n}\cong 1\mathrm{l}+i\text{mathchar}28958{}_n+\frac{1}{2!}(i\text{mathchar}28958{}_n{)}^2+\cdots .$$

$$\text{mathchar}28936{}^{\pm }[g]$$

$$C_{(M)}^{\pm }(k_1,\cdots ,k_{M-1})$$

$${\int }_k$$

$$K_j$$

$$(\text{mathchar}28942g_n{)}_{ij}$$

$$O_{pq}^{\pm }$$

$$S^{\pm }(p{)}_{\text{mathchar}28939\text{mathchar}28940}$$

$$\text{mathchar}28928{}_{WZ}[g]=\mathrm{Im}\left(\text{mathchar}28936{}^{+}[g]-\text{mathchar}28936{}^{-}[g]\right).$$

$${\text{mathchar}28928{}_{WZ}[g]}^{(3)}$$

$$J^{\pm }$$

$$C_{\text{mathchar}28950}(k)=\sin k_{\text{mathchar}28950},B(k)=r\sum _{\text{mathchar}28950}\left(1-\cos k_{\text{mathchar}28950}\right).$$

$${\text{mathchar}28928{}_{WZ}[g]}^{(3)}{\cong }_{a\to 0}-\frac{i}{12\text{mathchar}28953}\int d^{2}x\text{mathchar}28943{}_{\text{mathchar}28950\text{mathchar}28951}\mathrm{tr}\left(\text{mathchar}28958(x){\partial }_{\text{mathchar}28950}\text{mathchar}28958(x){\partial }_{\text{mathchar}28951}\text{mathchar}28958(x)\right).$$

$${\text{mathchar}28928{}_{WZ}[g]}^c=\frac{1}{12\text{mathchar}28953}\int d^{3}y\text{mathchar}28943{}^{ABC}\mathrm{tr}\left(g^{-1}{\partial }_{A}g{g}^{-1}{\partial }_{B}g{g}^{-1}{\partial }_{C}g\right)$$

$$J^{+}=-J^{-}$$

$$\text{mathchar}28949{}_p$$

$$Z=\int \mathcal{D}g{e}^{-S[g]}{e}^{in\text{mathchar}28928{}_{WZ}[g]},$$

$$S[g]$$

$$g_n=g_{0n}{e}^{\text{mathchar}28949i\text{mathchar}28953{}_n},$$

$$S_{cl}[g_0]=-\sum _{n,\text{mathchar}28950}\mathrm{tr}{\left(g_{0n}^{-1}{\hat{\partial }}_{\text{mathchar}28950}{g}_{0n}\right)}^2.$$

Full text

Wess-Zumino term by Vacuum Overlap Formula

Yoshio Kikukawa*)))E-mail address: kikukawa@gauge.scphys.kyoto-u.ac.jp and Shunji Miyazaki**)))E-mail address: miyazaki@gauge.scphys.kyoto-u.ac.jp

1 Introduction

The vacuum overlap formula[1, 2, 3, 4, 5, 6] gives a well-defined method to calculate the chiral fermion determinant in lattice regularization. By this formula, one first considers two Wilson fermion fields which posses positive mass and negative mass, respectively. They are assumed to couple to the link variables of given background gauge field. We next solve the ground states of the two fields. Then the overlap of the two ground states gives the chiral determinant. There remains an ambiguity in the complex phase of the determinant. It is fixed by the convention that the inner product of each ground state with the corresponding free ground state is real and positive.

The complex phase of the vacuum overlap is able to reproduce the full complex phase of chiral determinant correctly for a smooth background gauge field. The complex phase is gauge noninvariant and its variation under a gauge transformation was shown to give the consistent anomaly.[1, 7, 8, 9] In the continuum U(1) chiral gauge theory which is defined on the two-dimensional torus, the chiral determinant with the uniform background gauge potential (toron) can be calculated exactly. The vacuum overlap reproduces this exact result.[2]

Actually, this lattice definition of the complex phase of chiral determinant is closely related to the $\mathchar 28945\relax$ invariant definition in the continuum theory.[10] The vacuum overlap formula can be derived from the $(2n+1)$ dimensional system of Wilson fermion with kink mass[11] by taking infinite limit of the extra dimension, provided that appropriate Pauli-Villars (Wilson) fields are included.[1, 12, 4] For this limiting procedure to be well-defined, it also needs that the background gauge field is topologically trivial.[4] By virtue of the Pauli-Villars fields, we may take the naive continuum limit $a\rightarrow 0$ in this formula keeping the kink and Pauli-Villars masses finite. Then it reduces to the formula which defines the $\mathchar 28945\relax$ invariant. The only technical difference is that the smooth interpolation function of two gauge fields is replaced by the sharp step function at the boundary of the wave guide of gauge interaction.

For generic gauge fields on the lattice, however, several authors[13, 12, 4, 6, 14] argued that the nature of gauge degrees of freedom may spoil the chiral structure of the formula. For the $(2n+1)$ dimensional Wilson fermion with the kink mass and the wave guide of gauge interaction,[11] it was shown[13] that the large fluctuation of gauge freedom induces another light fermion mode of opposite chirality at the boundary of the wave guide. Such a light mode could also give the contribution to the complex phase of chiral determinant.

Singular gauge transformations can also affect the complex phase of the vacuum overlap. In the continuum U(1) gauge theory which is defined on the two-dimensional torus, we can consider the singular gauge transformation which wipe out the toron into the singular potential with delta functions. Then, it was shown[6] that the gauge invariance breaks by the singular gauge transformation even in the anomaly free U(1) chiral gauge theory, provided that the background toron is large and the anti-periodic boundary condition is chosen for all fermions. The vacuum overlap reproduces the result as well. The authors of ref. [6] claimed that it is these singular gauge transformations which spoils the chiral phase when averaged on the gauge orbit. ))***) For the small toron, the chiral determinant in the continuum limit is gauge invariant even under the singular gauge transformation.[6] Choosing such small toron, the gauge integration of chiral determinant was performed.[14] The result shows that the large gauge fluctuation affects the chiral phase even for the small toron and seems to cause it to vanish.

Therefore, it is a crucial problem in this approach to control and reduce the fluctuation of gauge degree of freedom. In this paper, we examine the gauge freedom of the two-dimensional SU(2) gauge field. In order to control and reduce the fluctuation, we introduce the kinetic term for the variable of gauge freedom, which consists of the nearest neighbor coupling. We calculate the Wess-Zumino term by the vacuum overlap formula and measure its distributions. The whole system we consider here is the lattice counterpart of the SU(2) Wess-Zumino-Witten model off the critical point.

This type of kinetic term for gauge freedom was examined in the U(1) case.[15] It was shown that the reduction of the fluctuation is not sufficient near the critical point of the corresponding X-Y model. On the contrary, in the case of SU(2) chiral field considered here, the coupling in the critical region is weak because of the asymptotic freedom. It is expected that the fluctuation can be reduced in this region even thought the SU(2) chiral symmetry is realized linearly.

This paper is organized as follows. In section 2, the lattice counterpart of the two-dimensional SU(N) Wess-Zumino-Witten model is examined in the lattice perturbation theory. We show that the vacuum overlap formula reproduces the Wess-Zumino term correctly in the continuum limit. We also calculate the beta function of the model and show that it has the nontrivial IR fixed point, as in the continuum theory. In section 3, the Wess-Zumino term is examined numerically. The SU(2) chiral field is generated by the Monte Carlo method (Cluster algorithm[16]) with the action of the nearest neighbor coupling for several values of the coupling constant. With these configurations, we calculate the complex phase of the vacuum overlap and its distribution is obtained. In section 4, we give a discussion concerning the numerical estimate of observables in the Wess-Zumino-Witten model by incorporating the imaginary Wess-Zumino action.

2 Two-dimensional Wess-Zumino term on the Lattice

2.1. Vacuum overlap formula and Wess-Zumino term

The definition of the vacuum overlap formula for the lattice regularized chiral determinant in $2n$ dimension is given by [1]

[TABLE]

$|\pm\rangle_{U}$ are the vacuum state of the second quantized Hamiltonian ${H}^{\pm}$, which defined by

[TABLE]

where we set the lattice spacing $a$ to unity, $m_{0}$ is a bare mass parameter , $C$ and $B$ are the massless Dirac operator and the Wilson term respectively. $a_{n\mathchar 28939\relax i},a^{\dagger}_{n\mathchar 28939\relax i}$ are field operators satisfying the following anti-commutation relations :

[TABLE]

The vacuum overlaps, $\,{}_{U}\langle-|+\rangle_{U}\,$ etc., can be expressed by the one particle wave functions $\mathchar 28960\relax^{\pm}_{K}(n,U)$, which are the eigenfunctions of $H^{\pm}(U)$ with positive eigenvalues labeled $K$:

[TABLE]

where color and spin indices are suppressed.

The gauge transformation of the link variables is defined by

[TABLE]

where $g_{n}$ are elements of gauge group SU($N$) and $U^{g}$ denote the gauge transformation of $U$.

By this gauge transformation, $\mathchar 28960\relax^{\pm}_{K}(n,U)$ transform as

[TABLE]

Then the vacuum overlap ${}_{U}\langle-|+\rangle_{U}$ is gauge invariant but the others are gauge dependent and are expressed as

[TABLE]

The variation of the complex phase of the chiral determinant by gauge transformation gives the gauged Wess-Zumino effective action.[17] The corresponding quantity of the vacuum overlap formula is written by

[TABLE]

If all the link variables are set to be unity, it is expected that Eq. (2.13) gives the Wess-Zumino effective action (we call as Wess-Zumino term in the following). In this case the vacuum overlaps (2.11) are expressed by

[TABLE]

and the Wess-Zumino term is defined by

[TABLE]

For the continuum theory in two dimensions, it is well-known that the chiral determinant can be calculated exactly and the explicit expression can be obtained[17]. Next we verify that Eq. (2.15) has the correct continuum limit in two dimensions.

2.2. Perturbative Calculation of Wess Zumino term

First we examine (2.14) perturbatively. The group elements $g_{n}$ can be expressed as $e^{i\mathchar 28958\relax_{n}}$ by the variable $\mathchar 28958\relax_{n}$ which takes the value on the Lie algebra. We take this variables as the small quantum fluctuation from the unity (not some background) so that we can expand as follows:

[TABLE]

Then, $\mathchar 28928\relax_{\scriptscriptstyle WZ}[g]$ can also be expanded in $\mathchar 28958\relax_{n}$ and the continuum limit of the terms in the expansion series can be evaluated.

Let $\mathchar 28936\relax^{\pm}[g]$ denote the effective actions for Eq. (2.14). Expanding by $\mathchar 28958\relax_{n}$, we obtain the following result,

[TABLE]

where

[TABLE]

$H^{\pm}(p)$ is the momentum representation of Hamiltonians Eq. (2.3) in the free case and $\mathchar 28960\relax^{\pm}_{p}$ and $\mathchar 28949\relax^{\pm}_{p}$ are the eigenfunctions and the eigenvalues of them. The explicit expressions are given in the appendix A. Then the Wess-Zumino term on the lattice is given by

[TABLE]

The oder $\mathchar 28958\relax$ contribution stems from $M=1$ term, but $C^{+}_{\scriptscriptstyle(1)}$ and $C^{-}_{\scriptscriptstyle(1)}$ are equal to each other and cancel. The oder $\mathchar 28958\relax^{2}$ contribution also vanishes since $\mathchar 28958\relax^{2}$ term in $M=2$ terms are real. Therefore the leading term is the order $\mathchar 28958\relax^{3}$ and is given by the following,

[TABLE]

where the explicit expressions of $C^{\pm}_{\scriptscriptstyle(3)}$ are also given in the Appendix A. In order to take the continuum limit, we introduce physical momenta $p_{i}$ and express the dimensionless momenta $k_{i}$ as $k_{i}=p_{i}a$ with the lattice spacing $a$. Then we expands the coefficient $C^{\pm}_{\scriptscriptstyle(3)}$ in $a$ and obtain

[TABLE]

with $f^{,\,\mathchar 28950\relax}=\partial_{\mathchar 28950\relax}f$ .

For the Wilson fermion, $C$ and $B$ are given by

[TABLE]

Then the integrands of $J^{\pm}$ are complex expressions of sine and cosine and include two parameters, $m_{0}$ and $r$. It dose not seem to be able to accomplish the integrals analytically. By numerical calculation we obtained $J^{+}=\mathchar 28953\relax,J^{-}=0$ for the region of $m_{0}/r~{},0<m_{0}/r<2$. Therefore, when $a$ goes to zero Eq. (2.26) becomes as follows:

[TABLE]

This is identical with the leading term of the continuum Wess-Zumino term,

[TABLE]

in the $\mathchar 28958\relax$-expansion. ))****) For the continuum vacuum overlap formula,[1] the same calculation can be performed. We obtain the similar expansion as Eq. (2.27) in the $(k/m)$-expansion where m is Pauli-Villars mass.

$\displaystyle J^{+}=-J^{-}$ $\displaystyle=$ $\displaystyle\int d^{2}p\,\frac{m}{4\mathchar 28949\relax_{p}^{3}},$

(2.31)

$\displaystyle\mathchar 28949\relax_{p}$ $\displaystyle\equiv$ $\displaystyle[p^{2}+m^{2}]^{\frac{1}{2}}\quad.$

It is easy to carry out this integral analytically and obtain the results $J^{+}=-J^{-}=\mathchar 28953\relax/2$. It also reproduce the correct Wess-Zumino term. The fact that $J^{-}=0$ in lattice regularization is consistent with the calculation of the Chern-Simons current by Golterman et al.[19]

2.3. Wess-Zumino-Witten model on the lattice

Next we formulate a model with the Wess-Zumino term defined in the previous subsections. The partition function of the model is defined by

[TABLE]

where $S[g]$ is the action of $SU(N)$ spin model:

[TABLE]

and $n$ is an arbitrary integer. This model is considered as the lattice counterpart of the two dimensional Wess-Zumino-Witten model.[18] In order to verify that the theory have the correct continuum limit, we perform the calculation of the Callan-Symanzik $\mathchar 28940\relax$ function and show it is identical to that of the continuum model.

To evaluate the $\mathchar 28940\relax$ function, one may use the background field method, in which the field variables are separated into two parts such as

[TABLE]

with $g_{0\,n}$ a smooth background and $\mathchar 28953\relax_{n}$ as small fluctuation around $g_{0\,n}$. Then one calculates the correction to the coefficient of the functional of $g_{0\,n}$,

[TABLE]

However, in our model, it is not easy to calculate the Wess-Zumino functional defined by Eq. (2.13) keeping the nontrivial background $g_{0\,n}$. So we first expand the total action in $\mathchar 28958\relax_{n}$ by which $g_{n}$ is expressed as $e^{i\mathchar 28958\relax_{n}}$. Then we shift $\mathchar 28958\relax_{n}$ around a background $\mathchar 28958\relax_{0\,n}$ not linearly but nonlinearly by applying the Hausdorff’s formula to Eq. (2.34) as follows:

[TABLE]

In this case, the lowest order of the functional (2.35) is

[TABLE]

and it is sufficient for us to estimate the corrections to its coefficient. The one-loop order corresponds to $O(\mathchar 28958\relax_{0}^{2},\mathchar 28949\relax^{2})$. To obtain the propagator and vertices necessary to calculate the correction to this order, we first expand the actions $S[g]$ and $\mathchar 28928\relax_{\scriptscriptstyle WZ}[g]$ in $\mathchar 28958\relax_{n}$ as follow,

[TABLE]

Next we shift $\mathchar 28958\relax_{n}$ around $\mathchar 28958\relax_{0\,n}$ as mentioned above and obtain the followings.

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

The vertex functions $V$’s are listed in appendix B. Then the partition function is calculated as follows:

[TABLE]

where

[TABLE]

The contribution of the original spin model is calculated as[21]

[TABLE]

where $C_{A}$ is the seconde Casimir coefficient in the adjoint representation and $m$ is the infrared cut-off mass. On the other hand, the contribution from the Wess-Zumino term is calculated as follows. It is easy to see the second term vanishes. The first term is given by

[TABLE]

[TABLE]

Toward the continuum limit we also scale the dimensionless external momenta $k$ by $a$ and consider the momenta $ka$ near zero. Following Karsten and Smit[20], we can estimate the loop integral. The divergent contribution stems from the momenta near zero, so that we can use the continuum limit of the vertex $C(k,p)$ obtained in the previous section. It is calculated as

[TABLE]

Therefore the contribution from Wess-Zumino term is

[TABLE]

Adding two parts, we obtain

[TABLE]

where $\mathchar 28949\relax_{R}^{2}$ denotes the renormalized coupling constant given by

[TABLE]

From this, $\mathchar 28940\relax$ function is obtained as

[TABLE]

For $n>0$, the $\mathchar 28940\relax$ function has an IR-fixed point at $\mathchar 28949\relax^{2}=\frac{8\mathchar 28953\relax}{n}$ as expected (the factor 2 differ from Witten’s results[18] because of the difference of the convention). For $\mathchar 28949\relax^{2}<\frac{8\mathchar 28953\relax}{n}$, this model is asymptotically free, so that it has a renormalization group invariant scale parameter, $\mathchar 28931\relax$ given by

[TABLE]

3 Numerical Calculation of the Wess-Zumino term

In this section, we show our results of the numerical calculation of the Wess-Zumino term. We consider the SU(2) chiral field and adopt the lattice action with the nearest neighbor coupling. We generate SU(2) chiral fields by the cluster algorithm.[16] Lattice size is set to 16 x 16. For most of the values of $\mathchar 28940\relax\equiv\frac{2}{\mathchar 28949\relax^{2}}$ examined, 500 configurations are generated. For $\mathchar 28940\relax=1.1,1.5$, we generate 4000 more configurations. With these configurations, the vacuum overlaps are calculated numerically through the LU decomposition. We measure the distribution of the Wess-Zumino term by dividing the range of the value $[-\mathchar 28953\relax,\mathchar 28953\relax]$ into bins with the width 0.1. We also calculate the standard deviation $\mathchar 28955\relax_{WZ}$ of the distribution of the Wess-Zumino term for each $\mathchar 28940\relax$.

First of all, we show the behavior of the Wess-Zumino term for randomly generated configurations of SU(2) chiral field in Figure 1. ($\mathchar 28940\relax$ is set to $0.025$.) The Wess-Zumino term distributes almost uniformly in the range $[-\mathchar 28953\relax,\mathchar 28953\relax]$. In Figure 2, we show a typical configuration of the random SU(2) chiral field in momentum space. We can find that almost all modes of momenta contribute equally. The species doublers are then suspected to contribute to the complex phase.

Next we show the distributions of the Wess-Zumino term for $\mathchar 28940\relax=0.5$, $1.1$ and $1.5$ in Figure 3. We can clearly observe that for $\mathchar 28940\relax=1.1$, the evaluated values of the Wess-Zumino term start to concentrate around zero and the distribution becomes like Gaussian. When $\mathchar 28940\relax$ increases, the distribution becomes sharp Gaussian. $\mathchar 28940\relax$ dependence of the standard deviation $\mathchar 28955\relax_{WZ}$ of the Wess-Zumino term distribution is plotted in Figure 4. The onset of the Gaussian distribution is around $\mathchar 28940\relax=0.9$. Crossover is sharp from the strong coupling region where the Wess-Zumino term fluctuates hard.

In Figure 5, we show a typical configuration of the SU(2) chiral field at $\mathchar 28940\relax=1.5$ in momentum space. We can find that the modes with small momenta dominates. In this region of $\mathchar 28940\relax$, the species doublers are almost suppressed.

In Figure 6, we show the susceptibility $\mathchar 28959\relax$ of the SU(2) chiral model for $L=16$ and $L=64$. The scaling of the susceptibility starts around $\mathchar 28940\relax=0.9$. This coincides with the onset of the Gaussian distribution of the Wess-Zumino term.

In summary, the two-dimensional Wess-Zumino term defined by the vacuum overlap formula shows a sharp Gaussian distribution for the configurations of SU(2) chiral field in the scaling region. In this region, the configurations are smooth and the contribution of the species doublers is suppressed. In the strong coupling region, the Wess-Zumino term fluctuates hard and the species doublers’ contribution is suspected to affect it. However the crossover is rather sharp from the strong coupling region to the weak coupling region.

A few comments are in order for our numerical calculation. In Figure 6, we find that the finite size effect on $\mathchar 28959\relax$ is substantial for $L=16$. We also suspect the finite size effect on the Wess-Zumino term. The onset of the Gaussian distribution of the Wess-Zumino term in the scaling region is also expected for larger lattice. However there can be much finite size effect in the values of $\mathchar 28955\relax_{WZ}$. We need further investigation on large lattice.

4 Numerical Evaluation of Observable in Wess-Zumino-Witten model

In this section, we discuss the possibility of the numerical estimate of observables in the Wess-Zumino-Witten model by incorporating the Wess-Zumino action. The lattice Wess-Zumino-Witten model we are considering is defined with the complex action,

[TABLE]

where $S[g]$ and $\mathchar 28928\relax_{WZ}[g]$ are given by Eq. (2.33) and by Eq. (2.15), respectively. $n$ is the integer coupling constant of the Wess-Zumino term. In order to evaluate the observables in this model, we need to perform the following integral,

[TABLE]

In general, without the positivity of the measure, the Monte Carlo method cannot be applied. One possible way out is to incorporate the imaginary Wess-Zumino term into the observable and to perform the Monte Carlo integration only with the real part of the action. That is, we consider

[TABLE]

where $\langle\langle\,\rangle\rangle_{MC}$ denotes the Monte Carlo integration only with the real part of the action,

[TABLE]

Another possible method is to introduce the spectral functions[22] defined as follows:

[TABLE]

where the delta function is the periodic one. With these spectral functions, we can write the integrals Eq. (4.2) and Eq. (4.3) as

[TABLE]

The integration of $\mathchar 28946\relax$ is then performed as the Riemann integral.

Actually, the distribution of the Wess-Zumino term obtained in the previous section can be regarded as the spectral function, ))*****) It may be better to use the set method[23] for the precise evaluation of the spectral function. In this case, however, since the value of the Wess-Zumino term is in the compact region $[-\mathchar 28953\relax,\mathchar 28953\relax]$, we can do without it if we have enough statistics.

[TABLE]

in the discrete approximation with the normalization,

[TABLE]

We found that in the scaling region, it is given by a sharp Gaussian distribution with the width $\mathchar 28955\relax_{WZ}$ depending on $\mathchar 28940\relax$,

[TABLE]

For such a Gaussian distribution of the imaginary part of the action, it is known that the first method mentioned above may work only for small $n$.[22] To see this, we perform the following test. We assume the Gaussian distribution for the Wess-Zumino term, Eq. (4.14). We take the square of the Wess-Zumino term as an observable. Since the spectral function is obtained by the Monte Carlo method, we simulate the situation by performing the Monte Carlo integration on $\mathchar 28946\relax$. Thus we consider the following integrals for various $n$ at fixed $\mathchar 28955\relax_{WZ}$,

[TABLE]

where $\langle\langle\,\rangle\rangle_{\mathchar 28946\relax}$ denotes the Monte Carlo integration of $\mathchar 28946\relax$ with the weight $\mathchar 28954\relax(\mathchar 28946\relax)$,

[TABLE]

In Figure 7, we show the result. We found that for $\mathchar 28955\relax_{WZ}=0.5$, up to $n=3$ the numerical result reproduces the exact one within the acceptable error of three significant digits. For $\mathchar 28955\relax_{WZ}=0.2$, $n=12$.

On the other hand, it is also possible to perform the Riemann integral in Eq. (4.10) numerically for the same observable. We also found the upper bounds for $n$. In order that the numerical result reproduces the exact one within the acceptable error, we should have $n\leq 12$ for $\mathchar 28955\relax_{WZ}=0.5$ and $n\leq 30$ for $\mathchar 28955\relax_{WZ}=0.2$. We have tried several schemes of higher order integration and obtained the similar result.

Thus we obtained the upper bound for the accessible value of the coupling constant $n$ of the Wess-Zumino term for given $\mathchar 28955\relax_{WZ}$. In other word, for a given $n$, there exists the upper bound of $\mathchar 28955\relax_{WZ}$, with which the measurement of the observable can be performed by the direct Monte Carlo integration or by the Riemann integration of the spectral functions. In Figure 9, we show the upper bounds of $\mathchar 28955\relax_{WZ}$ versus the coupling constant $n$ of the Wess-Zumino term which is translated to the IR fixed point $\mathchar 28940\relax_{c}=\frac{n}{4\mathchar 28953\relax}$. From this figure, it seems possible to simulate the model across the IR fixed point for $\mathchar 28940\relax_{c}\geq 1.3$, (that is $n\geq 17$), if the spectral density method is used. On the contrary, by the direct Monte Carlo calculation, only the weak coupling region can be simulated.

Our discussion given here is based on the numerical result shown in the previous section. Since the systematic and statistical errors are suspected in the result, the numbers of the upper bounds given in this discussion should not be taken literary. We believe, however, that this discussion gives a correct qualitative estimate of the possibility.

5 Discussion and Conclusion

The vacuum overlap formula for the two-dimensional Wess-Zumino term was examined both perturbatively and nonperturbatively. We showed by the lattice perturbation theory that the formula correctly reproduces the Wess-Zumino term in the continuum limit and yields the IR fixed point in the beta function of the SU(2) chiral model. We calculated the vacuum overlaps numerically and found that the complex phase shows a sharp Gaussian distribution for the SU(2) chiral fields in the scaling region. In this region, the smooth configurations were obtained even in the symmetric phase due to asymptotic freedom. We also found that the crossover is rather sharp from the strong coupling region where the Wess-Zumino term fluctuates hard and the species doublers’ contribution is suspected to affect it.

It was also argued that, if we use the spectral density method, it seems possible to examine the region of the IR fixed point of the Wess-Zumino-Witten model numerically.

Also from this study, we found that the asymptotically free coupling for gauge degree of freedom is able to reduce the gauge fluctuation. In two dimensions, the nearest neighbor coupling of the SU(2) chiral field plays such a role. In four dimensions, it is known in the continuum theory that the four-derivative coupling which is induced from the gauge fixing term is asymptotically free.[24] It may be interesting to examine the gauge degree of freedom of the four-dimensional nonabelian gauge field with the gauge fixing term from this point of view.[25]

Acknowledgements

The authors would like to thank H. Hata for enlightening discussions about the perturbative treatment of the non-linear sigma model. They are also grateful to W. Bock for his helpful advice about the Monte Carlo simulation with the Cluster algorithm.

This work is supported by National Laboratory for High Energy Physics, as KEK Supercomputer Project (Project No. 10).

A Eigenvalues and Eigenfunctions of $H^{\pm}(p)$

In this appendix we give the detail expression of the functions used in subsection 2.1. The momentum representation of Hamiltonians and the Wave functions are

[TABLE]

For Wilson fermion ,

[TABLE]

But one may take any regularization function as $C(p)$ and $B(p)$. Imaginary part of $C_{\scriptscriptstyle(3)}^{\pm}$ are

[TABLE]

Expanding on $k_{1},k_{2}$, one obtain the expression of the integral $J^{\pm}$.

B Vertices for One-Loop Calculation

Here we give the vertex functions used in subsectection 2.3.

[TABLE]

C Convention for SU(N) Lie Algebra

[TABLE]