Perfect Lattice Actions with and without Chiral Symmetry

TL;DR

This paper constructs perfect lattice actions for fermions and gauge fields using continuum blocking, analyzing the effects of chiral symmetry preservation or breaking on locality and gauge invariance, with applications to QCD.

Contribution

It introduces a method to derive perfect lattice actions via continuum blocking, exploring the impact of chiral symmetry on locality and gauge invariance, and proposes an optimized RG transformation for QCD.

Findings

Chirally symmetric actions are nonlocal but preserve chiral symmetry.

Breaking chiral symmetry yields local actions with explicit symmetry breaking.

The spectrum matches the continuum, and anomalies are correctly reproduced.

Abstract

We use perturbation theory to construct perfect lattice actions for fermions and gauge fields by blocking directly from the continuum. When one uses a renormalization group transformation that preserves chiral symmetry the resulting lattice action for massless fermions is chirally symmetric but nonlocal. When the renormalization group transformation breaks chiral symmetry, the lattice action becomes local but chiral symmetry is explicitly broken. In particular, starting with a chiral gauge theory in the continuum one either obtains a lattice theory which is gauge invariant but nonlocal, or a local theory with explicitly broken gauge invariance. In both cases the spectrum of the lattice theory is identical with the one of the continuum and the anomaly is correctly reproduced. We also apply our techniques to vector-like theories. In particular we propose a new renormalization group transformation for QCD and we optimize its parameters for locality of the perfect action.

Equations

$${\bar{\Psi }}_x={\int }_{c_x}{d}^{d}y\bar{\psi }(y),{\Psi }_x={\int }_{c_x}{d}^{d}y\psi (y).$$

$$\Psi (p)=\sum _{l\in {\mathsf{Z}\mathsf{Z}}^d}\psi (p+2\pi l)\Pi (p+2\pi l),$$

$$\Pi (p)=\prod _{\mu =1}^d\frac{{\hat{p}}_{\mu }}{p_{\mu }},{\hat{p}}_{\mu }=2\sin (p_{\mu }/2).$$

$$\exp (-S[\bar{\Psi },\Psi ])=\int \mathcal{D}\bar{\psi }\mathcal{D}\psi \mathcal{D}\bar{\eta }\mathcal{D}\eta$$

$$\displaystyle\times\exp\Big{\{}-\frac{1}{(2\pi)^{d}}\int d^{d}p\ \bar{\psi}(-p)\delta^{f}(p)^{-1}\psi(p)\Big{\}}$$

$$\displaystyle\times\exp\Big{\{}\frac{1}{(2\pi)^{d}}\int_{B}d^{d}p\Big{[}a\bar{\eta}(-p)\eta(p)$$

$$\times [\bar{\Psi }(-p)-\sum _{l\in {\mathsf{Z}\mathsf{Z}}^d}\bar{\psi }(-p-2\pi l)\Pi (p+2\pi l)]\eta (p)$$

$$\displaystyle+\bar{\eta}(-p)[\Psi(p)-\sum_{l\in{\sf Z\!\!\!Z}^{d}}\psi(p+2\pi l)\Pi(p+2\pi l)]\Big{]}\Big{\}}.$$

$$s[\bar{\psi },\psi ]=\frac{1}{(2\pi {)}^d}\int d^{d}p\bar{\psi }(-p){\delta }^f(p{)}^{-1}\psi (p),$$

$${\delta }^f(p)=(i{\gamma }_{\mu }{p}_{\mu }+m{)}^{-1},$$

$$S[\bar{\Psi },\Psi ]$$

$${\Delta }^f(p)$$

$${\bar{\psi }}_c(-p)=\bar{\Psi }(-p){\Delta }^f(p{)}^{-1}\Pi (p){\delta }^f(p),$$

$${\Delta }^f(p)=\frac{1}{m}-\frac{2}{m^2}[\coth (m/2)-i\cot (p/2){]}^{-1}+a.$$

$$\Delta^{f}(p)=\Big{(}\frac{\hat{m}}{m}\Big{)}^{2}(i\sin p+\hat{m}+\frac{1}{2}\hat{p}^{2})^{-1}.$$

$$\Psi (\vec{p}{)}_{x_d}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }d{p}_d\Psi (p)\exp (i{p}_d{x}_d),$$

$$⟨\bar{\Psi }(-\vec{p}{)}_0\Psi (\vec{p}{)}_{x_d}⟩=$$

$$\frac{1}{2\pi }{\int }_{-\infty }^{\infty }d{p}_d\sum _{\vec{l}\in {\mathsf{Z}\mathsf{Z}}^{d-1}}\frac{m}{(\vec{p}+2\pi \vec{l}{)}^2+p_d^2+m^2}$$

$$\displaystyle\times\prod_{j=1}^{d-1}\Big{(}\frac{\hat{p}_{j}}{p_{j}+2\pi l_{j}}\Big{)}^{2}\Big{(}\frac{\hat{p}_{d}}{p_{d}}\Big{)}^{2}\exp(ip_{d}x_{d})+a\delta_{x_{d},0}=$$

$$\sum _{\vec{l}\in {\mathsf{Z}\mathsf{Z}}^{d-1}}C(\vec{p}+2\pi \vec{l})\exp (-E(\vec{p}+2\pi \vec{l})x_d)+a{\delta }_{x_d,0}.$$

$$E(\vec{p}+2\pi \vec{l}{)}^2=-p_d^2=(\vec{p}+2\pi \vec{l}{)}^2+m^2.$$

$$A_{\mu ,x}$$

$$A_{\mu }(p)=\sum _{l\in {\mathsf{Z}\mathsf{Z}}^d}{a}_{\mu }(p+2\pi l){\Pi }_{\mu }(p+2\pi l)(-1{)}^{l_{\mu }},$$

$${\Pi }_{\mu }(p)=\frac{{\hat{p}}_{\mu }}{p_{\mu }}\Pi (p).$$

$$s[a]=\frac{1}{(2\pi {)}^d}\int d^{d}p\frac{1}{2}{a}_{\mu }(-p){\delta }^g(p{)}^{-1}{a}_{\mu }(p),$$

$${\delta }^g(p)=p^{-2},$$

$$\exp (-S[A])=\int \mathcal{D}a\mathcal{D}D$$

$$\displaystyle\times\exp\Big{\{}-\frac{1}{(2\pi)^{d}}\int d^{d}p\Big{[}\frac{1}{2}a_{\mu}(-p)\delta^{g}(p)^{-1}a_{\mu}(p)$$

$$\displaystyle-iD_{\mu}(-p)a_{\mu}(p)\Pi_{\mu}(p)\Big{]}\Big{\}}$$

$$\displaystyle\times\exp\Big{\{}-\frac{1}{(2\pi)^{d}}\int_{B}d^{d}p\Big{[}iD_{\mu}(-p)A_{\mu}(p)$$

$$\displaystyle+\frac{1}{2}D_{\mu}(-p)(\alpha+\gamma\hat{p}_{\mu}^{2})D_{\mu}(p)\Big{]}\Big{\}}.$$

$$a_{\mu c}(p)={\delta }^g(p){\Pi }_{\mu }(p){\Delta }_{\mu }^g(p{)}^{-1}{A}_{\mu }(p).$$

$$\sum _{\mu }{\hat{p}}_{\mu }{\Delta }_{\mu }^g(p{)}^{-1}{A}_{\mu }(p)=0.$$

$$S[A]=\frac{1}{(2\pi {)}^d}{\int }_B{d}^{d}p\frac{1}{2}{A}_{\mu }(-p){\Delta }_{\mu }^g(p{)}^{-1}{A}_{\mu }(p),$$

$${\Delta }_{\mu }^g(p)=\sum _{l\in {\mathsf{Z}\mathsf{Z}}^d}(p+2\pi l{)}^{-2}{\Pi }_{\mu }(p+2\pi l{)}^2$$

$$+\alpha +\gamma {\hat{p}}_{\mu }^2.$$

$$S[F]$$

$$\rho (p{)}^{-1}$$

$$v[\bar{\psi },\psi ,a]=\frac{e}{(2\pi {)}^{2d}}\int d^{d}p{d}^{d}q$$

$$\times {\bar{\psi }}^i(-p)i{\gamma }_{\mu }\frac{1}{2}(1-{\gamma }_5)a_{\mu }^a(p-q){\lambda }_{ij}^a{\psi }^j(q).$$

Full text

1 Introduction

When a field theory is regularized on the lattice it is unavoidable that some symmetries of the continuum get broken explicitly. For example, one gives up Poincaré symmetry and expects that it is restored in the continuum limit. Chiral symmetry poses subtle problems in lattice theories due to the presence of anomalies. Based on mild assumptions the Nielsen-Ninomiya theorem implies that a chirally invariant lattice theory with a local action leads to fermion doubling. Doubling can be avoided by breaking chiral symmetry explicitly. In a vector-like theory such as QCD one expects that chiral symmetry is restored in the continuum limit. In a chiral gauge theory, on the other hand, breaking chiral symmetry destroys gauge invariance. On the lattice one usually tries to keep gauge invariance intact. If one breaks it explicitly, restoring it in the continuum limit requires fine tuning of counter term couplings which is complicated in a numerical simulation. An alternative is to give up locality. Also this causes problems in simulations.

Perfect actions are by definition free of cut-off effects. They are located on a renormalized trajectory that intersects the critical surface in a fixed point of a renormalization group transformation [1]. Still, a perfect action is a lattice object, and for example Poincaré symmetry can not be manifest. However, the symmetry violation does not affect the spectrum of the theory, which is identical with the one of the continuum. This is because the symmetry breaking is entirely due to the renormalization group transformation.

Here we construct perfect actions for fermions and gauge fields. Like Poincaré symmetry the chiral symmetry of the fermions may not be manifest in the action, but it is still present in the spectrum of the theory. This is the case when one uses a renormalization group transformation that breaks chiral symmetry explicitly. The resulting fixed point action is local. In a chiral gauge theory gauge invariance is then not manifest in the action, although it is still present in the spectrum. Alternatively one may choose a renormalization group transformation that preserves chiral symmetry. Then gauge invariance in a chiral gauge theory is manifest in the action, but the lattice theory becomes nonlocal. Again, by construction the nonlocality does not affect the physical content of the theory. Here nonlocality arises naturally by integrating out degrees of freedom at short distances.

We use a renormalization group transformation that blocks directly from the continuum [2]. Perfect actions for fermions and gauge fields are derived in perturbation theory and the perfect fermion-gauge vertex is obtained to leading order in the gauge coupling. This leads to a perturbative definition of lattice chiral gauge theories. At present it is unclear how our construction may be extended nonperturbatively. Our results also apply to vector-like theories. For example, we show explicitly that the correct axial anomaly is reproduced in the Schwinger model [3].

The analytic calculations presented here are a step towards the perfect action of full QCD. Then one demands a very local action, which implies non-manifest chiral symmetry. We optimize the parameters of our renormalization group transformation such that the perfect action becomes ultralocal for free fermions in one dimension and for free gauge fields in two dimensions. We also propose a new block factor 2 renormalization group transformation for QCD that reduces to blocking from the continuum for weak fields. This transformation shall be used later in a nonperturbative determination of a perfect QCD action.

We follow the concept to approximate perfect actions proposed in [4]. Work on the perfect action for pure $SU(3)$ gauge theory was presented in [5]. For a different approach to damp cut-off artifacts in the lattice action, which has attracted much attention at this conference, see [6].

2 Blocking free fermions from the continuum

We consider a fermion field $\bar{\psi},\psi$ in the continuum and construct a lattice fermion field by averaging the continuum field over unit hypercubes $c_{x}$ centered at the points $x$ of a $d$-dimensional Euclidean lattice

[TABLE]

In momentum space the lattice fermion field is given by

[TABLE]

Note that the momentum of the lattice field is restricted to the Brillouin zone $B=]-\pi,\pi]^{d}$. Due to the summation over the integer vector $l$ the lattice field is periodic over $B$. Now the continuum degrees of freedom are integrated and one arrives at an effective action for the lattice variables

[TABLE]

Here $s[\bar{\psi},\psi]$ is the continuum action for free fermions. In momentum space it takes the form

[TABLE]

where $\delta^{f}(p)$ is the free fermion propagator in the continuum. The fields $\bar{\eta}$ and $\eta$ are auxiliary lattice Grassmann variables. For $a=0$ this is a $\delta$-function renormalization group transformation which is chirally invariant. Its fixed point (at $m=0$) describes free lattice chiral fermions [7]. The corresponding fixed point action, however, is nonlocal. This is consistent with the Nielsen-Ninomiya theorem. For $a>0$ the renormalization group transformation breaks chiral symmetry explicitly. However, as we will show, the chiral breaking introduced in the renormalization group step does not affect the spectrum of the theory.

Performing the integrations in eq.(2) results in

[TABLE]

Here $\Delta^{f}(p)$ is the lattice fermion propagator. The lattice action describes a complete renormalized trajectory parametrized by the mass $m$. At $m=0$ the trajectory intersects the critical surface in the fixed point.

Performing the Gaussian integral is equivalent to solving a classical equation of motion for the continuum quark field. The solution of this equation is

[TABLE]

These relations between the continuum and lattice fermion fields will be useful when we determine the perfect vertex function.

The introduction of nonzero $a$ leads to a local perfect action with explicitly broken chiral symmetry even at $m=0$. We now vary $a$ such that the action for 1-dimensional configurations becomes ultralocal. For $d=1$ the fermion propagator takes the form

[TABLE]

Choosing $a=(\hat{m}-m)/m^{2}$, where $\hat{m}=e^{m}-1$, the propagator reduces to

[TABLE]

This corresponds to the standard nearest neighbor Wilson fermion action. It turns out that the perfect action is still very local even in $d=4$ [8].

Now we consider the spectrum of the fermionic lattice theory. The fermionic operator with definite spatial momentum $\vec{p}\in]-\pi,\pi]^{d-1}$,

[TABLE]

creates a fermion at Euclidean time $x_{d}$. Its correlation function is given by

[TABLE]

The sum over spatial $\vec{l}$ corresponds to infinitely many poles of the integrand, and thus to infinitely many states that contribute an exponential to the 2-point function. The energies of these states are given by the location of the poles, i.e. $E(\vec{p}+2\pi\vec{l})=-ip_{d}$ with

[TABLE]

This is exactly the energy of a particle with mass $m$ and momentum $\vec{p}+2\pi\vec{l}$, i.e. the spectrum of the lattice theory is identical with the one of the continuum — cut-off effects are completely eliminated. Therefore the continuous Poincaré symmetry is restored in the spectrum.

3 Blocking free gauge fields from the continuum

In this section we apply the technique of blocking from the continuum to free gauge fields $a_{\mu}$. We define a noncompact lattice gauge field

[TABLE]

where $x$ now refers to the center of the link, i.e. $x_{\mu}$ is a half-integer. This construction is gauge covariant, i.e. when one performs a gauge transformation ${}^{\varphi}\!a_{\mu}(x)=a_{\mu}(x)-\partial_{\mu}\varphi(x)$ in the continuum, it induces a lattice gauge transformation ${}^{\Phi}\!A_{\mu,x}=A_{\mu,x}-\Phi_{x+\hat{\mu}/2}+\Phi_{x-\hat{\mu}/2}$. The lattice gauge transformation is given by $\Phi_{x}=\int_{c_{x}}d^{d}y\ \varphi(y)$. In momentum space the lattice gauge field reads

[TABLE]

Because the link centers have a half-integer component $x_{\mu}$, the gauge field is antiperiodic over the Brillouin zone in the $\mu$-direction. For technical reasons we work in the Landau gauge $\sum_{\mu}p_{\mu}a_{\mu}(p)=0$. Then the continuum action takes the form

[TABLE]

where $\delta^{g}(p)$ is the free gluon propagator in the Landau gauge. As in the fermionic case we want to optimize the renormalization group transformation for locality of the fixed point action. To this end we introduce an auxiliary vector field $D_{\mu}$, analogous to the auxiliary fermion field $\eta$. For $D_{\mu}$ we introduce a smearing term $\frac{1}{2}D_{\mu}(\alpha+\gamma\hat{p}_{\mu}^{2})D_{\mu}$. This allows us to optimize the parameters in the renormalization group transformation such that the fixed point action becomes ultralocal for 2-d configurations. The parameter $\alpha$ alone is not sufficient for this purpose. This renormalization group transformation takes the form

[TABLE]

Performing the Gaussian integral for $a_{\mu}$ is again equivalent to solving a classical equation of motion which yields

[TABLE]

Since $a_{\mu}$ is in the Landau gauge, using $p_{\mu}\Pi_{\mu}(p)=\hat{p}_{\mu}\Pi(p)$ one obtains

[TABLE]

We call this the ‘fixed point lattice Landau gauge’ condition. The fixed point action in this gauge takes the form

[TABLE]

Here $\Delta^{g}_{\mu}(p)$ is the lattice gauge field propagator in the fixed point lattice Landau gauge.

Again we are looking for the optimally local fixed point action. We turn to $d=2$ and work with field strength variables $F$ that live on the plaquettes. After some manipulations one obtains

[TABLE]

For $\alpha=1/6$ and $\gamma=-1/72$ one finds $\rho(p)=1$. This corresponds to the standard plaquette action which is ultralocal. Hence in $d=2$ the optimal choice for the smearing term is $\frac{1}{12}D_{\mu}(1-\frac{1}{12}\hat{p}_{\mu}^{2})D_{\mu}$. We use the same parameters in $d=4$ and still find a very local action [8].

4 The perfect fermion-gauge vertex

Now we switch on interactions between fermions and $SU(N)$ gauge fields to leading order in the gauge coupling $e$. The continuum fermion field is denoted by $\bar{\psi}^{i}$, $\psi^{i}$ with $i\in\{1,2,...,N\}$ and the continuum gauge field is denoted by $a^{a}_{\mu}$ with $a\in\{1,2,...,N^{2}-1\}$. In momentum space the vertex of a left-handed charged fermion coupled to the gauge field takes the form

[TABLE]

Again we integrate out the continuum fields and derive the perfect vertex of the lattice theory. Before we can do so we need to define the renormalization group transformation up to $O(e)$. This can be done in many ways, and we do not want to make a specific choice yet. In general, for the fermions one can write

[TABLE]

Here $K_{\mu}(p,q)$ is a regular kernel that specifies the renormalization group transformation. It describes how the continuum quark field in a hypercube $c_{x}$ is transported to the lattice point $x$.

Compared to the free fermion calculation of section 2 we now have an extra term

[TABLE]

in the renormalization group transformation and hence in the exponential of eq.(2). The additional terms $v[\bar{\psi},\psi,a]$ and $T[\bar{\psi},\psi,a]$ modify the exponent $E[\bar{\psi},\psi]+E[a]$ of the free theory to $E[\bar{\psi},\psi,a]=E[\bar{\psi},\psi]+E[a]+v[\bar{\psi},\psi,a]+T[\bar{\psi},\psi,a]$. Since the additional terms are $O(e)$ they change the equations of motion (and hence their solutions) to $\bar{\psi}_{c}+\delta\bar{\psi}_{c}$ and $\psi_{c}+\delta\psi_{c}$ and $a_{c}+\delta a_{c}$, where $\delta\bar{\psi}_{c}$, $\delta\psi_{c}$ and $\delta a_{c}$ are of $O(e)$. Inserting the new solution into the new exponent yields

[TABLE]

The essential observation is that $\bar{\psi}_{c}$, $\psi_{c}$ and $a_{c}$ are solutions of classical equations of motion, i.e. extrema of the corresponding exponents. This implies $E[\bar{\psi}_{c}+\delta\bar{\psi}_{c},\psi_{c}+\delta\psi_{c}]=E[\bar{\psi}_{c},\psi_{c}]+O(e^{2})$ and $E[a_{c}+\delta a_{c}]=E[a_{c}]+O(e^{2})$. Therefore we read off the perfect vertex

[TABLE]

Using eqs.(6) and (16) one identifies the vertex function as

[TABLE]

This expression is in the fixed point lattice Landau gauge. It contains products of up to six $\gamma$-matrices. These may be decomposed into 1, $\gamma_{\mu}$, $\sigma_{\mu\nu}$, $\gamma_{5}$ and $\gamma_{\mu}\gamma_{5}$. The last two do not occur in QCD because then parity is conserved. The $\sigma_{\mu\nu}$-term represents a perfect ‘clover’ action.

When the renormalization group transformation is chirally symmetric (i.e. when $a=0$) the lattice propagator $\Delta^{f}$ is proportional to $\gamma_{\mu}$. Then the resulting perfect lattice vertex has the same Dirac structure as the continuum vertex. Hence only the left-handed lattice fermion couples to the lattice gauge field, and gauge invariance is manifest in the vertex function. For $a>0$, on the other hand, also the right-handed lattice fermion couples, because now $a$ enters as an additional term in $\Delta^{f}$. Then gauge invariance is explicitly broken in the lattice action. Still it is guaranteed that the theory is physically equivalent to the chiral gauge theory in the continuum, because it was obtained analytically by a renormalization group transformation. In particular the spectrum of the lattice theory does respect gauge invariance.

5 The exact axial current and the anomaly in the Schwinger model

In this section we consider the $d=2$ Schwinger model with an abelian gauge field $a_{\mu}$. We construct an exact lattice axial current and show that it has the correct anomaly [3]. For $a=0$ the renormalization group transformation and hence the perfect action is chirally invariant. In contrast to the continuum the fermionic measure on the lattice is also chirally invariant. This enables the formal construction of a conserved axial Noether current $\tilde{J}^{5}_{\mu}$, and one may argue that the anomaly vanishes on the lattice. In fact, assuming a general form of $\Delta^{f}$ and $V_{\mu}$, Pelissetto showed that $\langle\hat{p}_{\mu}\tilde{J}^{5}_{\mu}(p)\rangle_{A}=0$ in an arbitrary background lattice gauge field $A_{\mu}$, and he concluded that the anomaly is cancelled by spurious ghost states [9]. At $a=0$, however, the axial current $\tilde{J}^{5}_{\mu}$ is nonlocal, and there is no evidence for it being related to the local current

[TABLE]

in the continuum, which has the anomaly

[TABLE]

In fact, even in the continuum one can formally construct a gauge invariant axial current

[TABLE]

which is anomaly free but nonlocal. Hence one must decide carefully what lattice current to consider.

We now construct a lattice axial current that is related to the continuum current $j^{5}_{\mu}$ by the renormalization group transformation. A natural definition is to consider the total flux of the continuum current through a $(d-1)$-dimensional face $f_{\mu,x}$ that separates two hypercubes $c_{x-\hat{\mu}/2}$ and $c_{x+\hat{\mu}/2}$. The corresponding lattice current $J^{5}_{\mu,x}$ associated with the link connecting the lattice points $x-\hat{\mu}/2$ and $x+\hat{\mu}/2$ is given by

[TABLE]

We call this the ‘exact lattice axial current’. Its standard lattice divergence is by Gauss’ law identical with the divergence of the continuum current integrated over a hypercube $c_{x}$

[TABLE]

In momentum space it reads

[TABLE]

We couple the current $J^{5}_{\mu}$ to an external lattice axial current $J^{e}_{\mu}$ by adding a term $(2\pi)^{-2}\int_{B}d^{2}p\ J^{e}_{\mu}(-p)J^{5}_{\mu}(p)$ to the continuum action and again we integrate out the continuum fields. Taking a functional derivative of the lattice action with respect to $J^{e}_{\mu}(-p)$ one identifies

[TABLE]

For reasons of space the $O(e)$ term can not be displayed here.

To show that the current has the correct anomaly we compute $\langle J_{\mu}^{5}(p)\rangle_{A}$ by integrating out the lattice fermions

[TABLE]

We insert the continuum quantity $\langle j_{\mu}^{5}(p)\rangle_{a}=(e/\pi)(p_{\mu}/p^{2})\epsilon_{\nu\rho}p_{\nu}a_{\rho}(p)$, which is known e.g. from Pauli-Villars regularization. Integrating out also the continuum gauge field $a_{\mu}$ we arrive at

[TABLE]

Using $\hat{p}_{\mu}\Pi_{\neg\mu}(p+2\pi l)(-1)^{l_{\mu}}=(p_{\mu}+2\pi l_{\mu})\Pi(p+2\pi l)$, for the lattice divergence of the axial current one obtains

[TABLE]

Finally we also construct the ‘exact lattice topological charge density’

[TABLE]

by integrating the continuum charge density over a hypercube $c_{x}$. In momentum space this implies

[TABLE]

Using eq.(16) one sees that the exact lattice axial current given in eq.(35) coincides with the topological charge density. In coordinate space this implies

[TABLE]

which is the desired anomaly equation on the lattice.

6 A nonperturbative version of the renormalization group

transformation

To derive a perfect action for QCD beyond perturbation theory we want to follow the lines of [4, 5] and work with a renormalization group transformation with a blocking factor 2. Our goal is to formulate a nonperturbative version of the renormalization group transformation whose perfect action reduces to the one of blocking from the continuum at least for weak fields. Then we know that the perfect action is extremely local (in lower dimensions even ultralocal), and therefore well suited for numerical simulations.

Here we use compact link variables $U_{\mu,x}\in SU(N)$, where $x_{\mu}$ is now an integer. We rescale $eA_{\mu}\to A_{\mu}$. Then the parallel transporters are related to the gauge potential by $U_{\mu,x}=\exp(iA^{a}_{\mu,x+\hat{\mu}/2}\lambda^{a})$. Under a gauge transformation $g$ the link variables transform as

[TABLE]

In the renormalization group transformation we couple a blocked link $U^{\prime}_{\mu,x^{\prime}}$ on the coarse lattice to an average of $2^{d}$ parallel transporters on the fine lattice. The blocked link connects the centers of two neighboring $2^{d}$ blocks on the fine lattice. For each point $x\in x^{\prime}$ in the first block we construct a parallel transporter as a product of two consecutive links starting at $x$ and ending at the corresponding lattice point $x+2\hat{\mu}$ in the second block. Then we couple the parallel transporters to the blocked link $U^{\prime}_{\mu,x^{\prime}}$ by constructing a traceless anti-Hermitean matrix

[TABLE]

for each blocked link. A gauge transformation $g^{\prime}$ on the coarse lattice

[TABLE]

induces a gauge transformation $g_{x}=g^{\prime}_{x^{\prime}}$ for $x\in x^{\prime}$ on the fine lattice. Hence the anti-Hermitean matrix transforms as

[TABLE]

The field $B_{\mu}$ is a charged vector in the adjoint representation, not a lattice gauge field. The renormalization group transformation contains the covariant second derivative

[TABLE]

of $B_{\mu}$. Under a gauge transformation $g^{\prime}$ it transforms as

[TABLE]

The effective action for the coarse variables now takes the form

[TABLE]

The functional $f[U]$ ensures the proper normalization of the partition function and is given by

[TABLE]

The parameters

[TABLE]

give an optimally local fixed point action.

It may seem that the above transformation is not gauge covariant, because the fine lattice quark field $\Psi_{x}$ is not properly parallel transported to the block center $x^{\prime}$. However, using gauge invariance of the fine action and of the measure one realizes that the coarse action is indeed gauge invariant. The above transformation corresponds to $K_{\mu}=0$.

In the classical limit $e\rightarrow 0$ the full renormalization group transformation reduces to the quadratic problem of section 3 even for arbitrarily strong abelian gauge fields. For 2-d configurations the fixed point action is ultralocal at the quadratic level. Hence for any 2-d abelian gauge field the single plaquette Manton action is classically perfect.

7 Conclusions

We have used the technique of blocking from the continuum to derive perfect actions for fermions and gauge fields. Depending on the choice of the renormalization group transformation the action is chirally invariant and nonlocal, or local and chirally variant. In a chiral gauge theory this corresponds to lattice formulations with unbroken or explicitly broken gauge invariance. Since we always keep contact with the continuum theory in both cases it is guaranteed that the spectrum of the lattice theory is identical with the one of the continuum.

Also the axial anomaly is correctly reproduced as we have shown in the Schwinger model. By construction our method of blocking from the continuum extends to all orders of perturbation theory. Our construction does, however, not yet provide a nonperturbative definition of lattice chiral fermions. Still, we think that our analytic perturbative study provides insight into this problem, and may inspire nonperturbative extensions.

The other aspect of our analysis concerns the perfect action for QCD. Our perturbative results have led to an optimization of the parameters in the renormalization group transformation. The perfect vertex function will be important in the nonperturbative numerical determination of the perfect action that is presently in progress [10]. This would apply to full QCD with light quarks. The perturbatively perfect action for quarks and gluons presented here may already be useful for simulations of heavy quarks, where the leading cut-off effects are proportional to the fermion mass. With our action these effects are eliminated up to $O(e)$.

We are indebted to R. Brower, T. DeGrand, P. Hasenfratz, A. Kronfeld, H. Neuberger, F. Niedermayer and Y. Shamir for many stimulating discussions.