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Pauli-Villars regularization of non-Abelian gauge theories

As an extension of earlier work on QED, we construct a BRST-invariant Lagrangian for SU(N) Yang-Mills theory with fundamental matter, regulated by the inclusion of massive Pauli-Villars (PV) gluons and PV quarks. The underlying gauge symmetry for massless PV gluons is generalized to accommodate the PV-index-changing currents that are required by the regularization. Auxiliary adjoint scalars are used, in a mechanism due to Stueckelberg, to attribute mass to the PV gluons and the PV quarks. The addition of Faddeev--Popov ghosts then establishes a residual BRST symmetry. Although there are drawbacks to the approach, in particular the computational load of a large number of PV fields and a nonlocal interaction of the ghost fields, this formulation could provide a foundation for renormalizable nonperturbative solutions of light-front QCD in an arbitrary covariant gauge.


I. INTRODUCTION
Nonperturbative Hamiltonian approaches to quantum field theories [1,2] in more than two dimensions require regularization. The standard approach to regularization of non-Abelian gauge theories is dimensional regularization [3], but this is inherently perturbative. For Abelian theories, Pauli-Villars (PV) regularization has proven useful [4,5]. However, the ordinary PV regularization of non-Abelian theories fails. Gauge invariance is violated, blocking any hope of BRST invariance, which confounds proofs of renormalizability [6]. To implement a PV regularization [7], we repair gauge invariance and then break it properly with a gauge-fixing term. Ghost fields [8] are added to build a BRST invariance.
The causes of gauge non-invariance in ordinary PV regularization are the PV gluon mass, the non-degenerate PV quark masses, and the PV-index changing currents for null interactions. A mass is acceptable for PV photons in an Abelian theory [9], but the approach used does not generalize to non-Abelian theories. Null interactions (meaning interactions that involve only null combinations of positive and negative metric fields) are necessary to provide the subtractions that regulate theory. So, how can one restore gauge invariance?
The Higgs mechanism is, of course, one possible approach, but investigations of this invariably found only null combinations of gluon fields to be massless, rather than a massless physical gluon. There is, however, another mechanism for generating masses [10][11][12], by coupling the PV gluons and PV quarks to a PV scalar, with all removed from the spectrum in the infinite-mass limit. The mixing of currents can be overcome by extension of the gauge transformation to include mixing PV fields; the original gauge transformation is obtained when the fields are removed from the spectrum. The construction is done is such a way as to facilitate quantization in an arbitrary covariant gauge, as has been done previously for QED [5]. The remaining gauge invariance can be tested by studying the dependence of physical quantities on the gauge-fixing parameter.
In the remainder of this paper, we present the PV-regulated Lagrangian term by term, along with the extended gauge transformation and the BRST transformations. We then discuss light-front quantization in an arbitrary covariant gauge and the light-front coupledcluster method [13] for solution of the Hamiltonian eigenvalue problem in Fock space.

II. A PAULI-VILLARS REGULATED LAGRANGIAN
For the purposes of discussion, we divide the PV-regulated Lagrangian into four terms L = L massless + L gluon + L quark + L ghost . (2.1) Here L massless is a gauge-invariant Lagrangian for massless gluons and quarks, with null interactions; L gluon is the mass and gauge-fixing term for gluons and auxiliary scalars; L quark is the mass term for quarks; and L ghost is the Faddeev-Popov ghost term. The first is given by where the field tensor is 3) The indices are k for (PV) gluons and i, j for (PV) quarks, with values of zero for the physical fields. The gauge transformations of the fields are The regularization is provided by the null field combinations: Here r k and s i equal ±1 (e.g. (−1) k and (−1) i ), A µ a is Abelian, and ψ is gauge invariant. With these definitions, this term of the Lagrangian can be written as (2.7) The free terms are those for fields with metrics r k and s i , and the interaction terms involve only null fields. The four-gluon interaction is implicit in the infinite-PV-mass limit through a contraction of two three-gluon interactions.
The null field combinations provide the necessary two PV subtractions to regulate any loop. For example, consider a loop with vertices from interactions of the form gψγ µ T a A aµ ψ. Assume that there is one gluon contraction and one quark contraction to form the loop. The contraction of the kth gluon field yields the metric signature r k and contraction of the ith quark field, s i . The coupling coefficients are selected as ξ k , ξ ′ k , β i , and β ′ i . The loop contribution then contains the factors r k ξ k ξ ′ k and s i β i β ′ i . These factors provide the two subtractions, because on their own they sum to zero The gluon mass and gauge-fixing term is (2.8) The first term is gauge-invariant, and the second, gauge-fixing. The scalar fields φ ak obey the gauge transformation The line integral allows the derivative to transform as This part of the Lagrangian can be reduced to the form where the gluon mass µ k is explicit and the scalar is given a mass µ k / √ ζ and inherits the metric r k . This is a non-Abelian extension of a Stueckelberg mechanism [10][11][12].
The quark mass term is made null by the additional constraint k r k ξ 2 k µ 2 k = 0 and with µ PV ≡ max k µ k . The gauge transformation of the combination is Abelian: (2.14) To make all couplings null, we define with m PV ≡ max i m i , and impose the constraints i s i m 2 i β 2 i = 0 and i s i m i β 2 i = 0. The second constraint makes ψ and ψ mutually null, to allow couplings between these combinations, and cancels the quartic coupling term. The quark mass term then simplifies to A standard construction [14] yields the ghost term [8] For these to be (mutually) null, we require k r k µ 2 k ξ 2 k = 0 and k r k µ 4 k ξ 2 k = 0. As a summary of the various constraints, the following need to be satisfied for for the adjoint fields: and for the quark fields: For the PV masses to be chosen independently, these constraints require four PV gluons, four PV ghosts and antighosts, five PV scalars, and three PV quarks. For pure Yang-Mills theory, the number of PV scalars is four, with the k = 0 field dropped and µ 0 = 0; the fields φ a0 are used only in splitting the masses of the PV quarks. In either case, the number of PV fields translates to a large, but necessary computational load. The BRST transformations are with ǫ a real Grassmann constant, for which ǫ 2 = 0. For the various null combinations, we then find δA µ a = ǫ∂ µ c a , δ φ a = ǫµ PV c a , δc a = 0, δψ = 0, δ ψ = 0. (2.26) The full Lagrangian is invariant with respect to these transformations.
The nondynamical components of the fermion fields satisfy the constraints When multiplied by s i β i and summed over i, this becomes which is the same constraint as for a free fermion field, in any gauge. Quantization can then proceed without need of light-cone gauge.

IV. LIGHT-FRONT COUPLED-CLUSTER METHOD
Given a field-theoretic light-front Hamiltonian P − , we wish to solve the fundamental eigenvalue problem P − |ψ = M 2 +P 2 ⊥ P + |ψ . The LFCC method [13] is designed to do so in terms of a Fock-state expansion but without the usual truncation of Fock space. Instead, write the eigenstate as |ψ = √ Ze T |φ . Here Z controls the normalization, which is fixed as ψ ′ |ψ = δ(P ′ − P ). The ket |φ is the valence state, with norm φ ′ |φ = δ(P ′ − P ). The operator T contains terms that only increase particle number; it does, however, conserve the quantum numbers of the full state, such as J z , light-front momentum P , and charge. Because p + is always positive, T must include annihilation operators, and powers of T include contractions. With construction of the effective Hamiltonian P− = e −T P − e T and the definition of a projection P v onto the valence Fock sector, we have the coupled system: This is still an infinite system of equations. The LFCC approximation is to truncate T at a fixed increase in particle count, but not truncate the exponential of T . The projection (1 − P v ) is truncated to a consistent set of Fock sectors above the valence sector, enough to have a finite, but sufficient system of (nonlinear) equations for the functions in T and in the valence state. The leading approximation of T for QCD would include quark → quark + gluon, gluon → quark + antiquark, and gluon → gluon + gluon transitions.
The LFCC approach requires no Fock-space truncation and no sector dependence or spectator dependence of self-energies or bare parameters. It is systematically improvable by adding terms to T and sectors to (1 − P v ) with more constituents. Various applications have been carried out, including a heavy-fermion model [13], φ 4 theory [16], and QED [17].

V. SUMMARY
With these tools in place, the time has come for a concentrated assault on QCD. The procedure would be to invoke PV regularization and light-front quantization in an arbitrary covariant gauge with variable gauge parameter. Any (numerical) approximation [2,18,19] would then be applied to a finite theory where the continuum limit can be taken independent of any chosen renormalization scheme. Fock-space truncation and the resulting uncanceled divergences can be avoided by using the LFCC method. The best places to start would be simpler systems in quenched approximation, such as heavy-quark mesons and glueballs. The computational load and effort are likely comparable to lattice gauge theory; the large number of PV modes and the discretizations being the analogs of lattice size and spacing.