# Mass Generation in Continuum Gauge Theory in Covariant Abelian Gauges

###### Abstract

The local action of an gauge theory in general covariant Abelian gauges and the associated equivariant BRST symmetry that guarantees the perturbative renormalizability of the model are given. I show that a global SL(2,R) symmetry of the model is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling and leads to propagators that are finite at Euclidean momenta for all elementary fields except the Abelian “photon”. The Goldstone states form a BRST-quartet. The mechanism eliminates the non-abelian infrared divergences in the perturbative high-temperature expansion of the free energy.

###### pacs:

11.25.Db,11.10.Jj,11.10.Wx^{†}

^{†}preprint: NYU-TH/99/08/30

The existence of a quark gluon plasma phase in which quarks and gluons are weakly interacting degrees of freedom at temperatures is suggested by the renormalization group flow of the effective QCD coupling constant and the analogy with the plasma phase of QED. Lattice simulations indicate that this phase transition is first order and for pure occurs at [1]. Although QCD is asymptotically free, the perturbative analysis of the high-temperature phase is plagued by infrared (IR) divergences[2]. The best one can presently achieve perturbatively at high temperatures is a resummation of the infrared-safe contributions[3]. The situation is somewhat embarrassing, since one naively might hope that an asymptotically free theory allows for an accurate perturbative description of the high temperature phase. The IR-problem encountered in the perturbative high temperature expansion in fact is part of the more general problem of defining a non-abelian gauge-fixed theory on a compact Euclidean space-time without boundaries, such as a hypertorus. It was shown[4] that normalizable zero-modes of the ghosts cause the partition function to vanish in conventional covariant gauges. An equivariant BRST construction was used to eliminate these ghost zero-modes associated with global gauge invariance at the expense of a non-local quartic ghost interaction. It was seen that this interaction leads to ghost-antighost condensation at arbitrarily small coupling[4, 5]. Subsequently, the equivariant gauge-fixing procedure was successfully used to reduce the structure group of an lattice gauge theory (LGT) to a physically equivalent Abelian LGT with a structure group[6]. The equivariant BRST symmetry of the partially gauge-fixed LGT was proven to be valid also non-perturbatively and the associated quartic ghost interaction leads to ghost-antighost condensation in this case too. The starting point of this investigation is a transcription of this partially gauge-fixed SU(2)-LGT to the continuum using the equivariant BRST algebra. A partially gauge-fixed LGT together with a corresponding critical continuum model that are defined by the same BRST algebra is perhaps of some interest.

We will see that a certain SL(2,R) symmetry associated with this gauge fixing is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling in the continuum model and leads to propagators that are regular at Euclidean momenta for all fields except the Abelian “photon”. The corresponding massless Goldstone states form a BRST-quartet and do not contribute to physical quantities such as the free energy. Screening “masses” in a certain sense thus arise naturally in an gauge theory in covariant Abelian gauges. They cure the IR-problem of the perturbative sceleton expansion for the free energy.

The critical continuum action of the lattice model[6] in Euclidean space is uniquely specified by the BRST algebra, the field content and power counting. Decomposing the non-abelian SU(2) connection in terms of two real vector-bosons (or one complex one) and a U(1)-connection (the “photon” of the model), the loop expansion is defined by the Lagrangian,

(1) |

Here is the usual -invariant Lagrangian
written in terms of the vector bosons and the
photon^{1}^{1}1Latin
indices take values in only, Einstein’s summation
convention applies and , vanishing
otherwise. All results are in the renormalization
scheme.,

(2) |

with

(3) | |||||

partially gauge-fixes to the maximal Abelian subgroup of in a covariant manner,

(4) |

with

(5) |

Note that is invariant under -gauge transformations and under an on-shell BRST symmetry and anti-BRST symmetry , whose action on the fields is

(6) |

with an obvious extension to include matter fields. Contrary to most other proposals for mass generation[7, 8] the BRST algebra Eq. (6) closes on-shell on the set of -invariant functionals: on functionals that depend only on and the matter fields, for instance effects an infinitesimal U(1)-transformation with the parameter . The algebra Eq. (6) thus defines an equivariant cohomology and ensures the perturbative renormalizability and unitarity [6, 9] of the model. Note that the physical sector comprises states created by composite operators of and the matter fields in the equivariant cohomology of (or ). They are BRST closed and -invariant.

The corresponding equivariant (anti-) BRST symmetry of the LGT is valid also non-perturbatively and it was shown[6] that expectation values of physical observables of the U(1)-LGT are the same as those of the original -LGT for any . Note that formally setting and solving the constraint as in[10] is not the same as taking the limit . The reason is inherently non-perturbative and nicely exhibited by the lattice calculation[6]: without the quartic ghost interaction, Gribov copies of a configuration conspire to give vanishing expectation values for all physical observables. No matter how small, the quartic ghost interaction is required to have a normalizable partition function and expectation values of physical observables that are identical with those of the original SU(2)-LGT.

in Eq. (1) has been added “by hand” to fix the remaining gauge invariance and define the perturbative series of the continuum model unambiguously. I will assume a conventional covariant gauge-fixing term,

(7) |

However, none of the following conclusions depend on the gauge-fixing of the Abelian subgroup – they in particular do not depend on .

The Lagrangian Eq. (1) also is invariant under a global bosonic SL(2,R) symmetry generated by

(8) |

and the ghost number . This SL(2,R) symmetry is is also realized in the lattice regularized model[6] and not anomalous. The conserved currents corresponding to are U(1)-invariant and BRST, respectively anti-BRST exact,

(9) |

I will argue that this global symmetry of the
model is spontaneously broken to the noncompact abelian subgroup
generated by the ghost number . Because the
currents Eq. (9) are (anti)-BRST exact, a
spontaneously broken SL(2,R) symmetry is accompanied by a BRST-quartet
of massless Goldstone states with ghost numbers and . They are
U(1)-invariant , , and bound states.
It is important to note that BRST quartets do not
contribute to physical quantities[11] such as the
free energy^{2}^{2}2This is analogous to the decoupling of the
Goldstone quartets of the weak interaction in
gauges[11].. The spontaneous symmetry
breaking in this sense is similar to a (dynamical) Higgs mechanism in
the adjoint.

An order parameter for the spontaneous breaking of the SL(2,R) symmetry is

(10) |

To perturbatively investigate the consequences of , the quartic ghost interaction in Eq. (4) is linearized using an auxiliary scalar field of canonical dimension two. Adding the quadratic term

(11) |

to the Lagrangian of Eq. (1), the tree level quartic ghost interaction
vanishes at
and is then formally of , proportional to the difference
of the renormalization constants of the two
couplings^{3}^{3}3The discrete symmetry relating and also ensures that only mixes with ..

We shall see that the perturbative expansion about a non-trivial solution to the gap equation

(12) |

is much better behaved in the infrared. Note that Eq. (12) is -invariant and therefore does not depend on the gauge-fixing Eq. (7). Let us for the moment assume that a unique non-trivial solution to Eq. (12) exists in some gauge ; we return to this conjecture below. The consequences for the IR-behavior of the model are dramatic. Defining the quantum part of the auxiliary scalar by

(13) |

the momentum representation of the Euclidean ghost propagator at tree level becomes

(14) |

Feynman’s parameterization of this propagator allows an evaluation of loop integrals using dimensional regularization that is only slightly more complicated than usual. More importantly, the ghost propagator is regular at Euclidean momenta when . Its complex conjugate poles at can furthermore not be interpreted as due to asymptotic ghost states[12].

When , the -boson is massless only at tree level and (see Fig. 1) acquires the finite mass at one loop,

(15) |

Fig. 1. The finite one-loop contribution to the mass.

Technically, the one-loop contribution is finite because the integral in Eq. (15) involves only the -part of the ghost propagator Eq. (14). Since , the -dependence of the loop integral is IR- and UV-finite. The quadratic UV-divergence of the subtraction at is canceled by the other, -independent, quadratically divergent one-loop contributions – (in dimensional regularization this scale-invariant integral vanishes by itself). furthermore is positive due to the overall minus sign of the ghost loop. The sign of is crucial, for it indicates that the model is stable and (as far as the loop expansion is concerned) does not develop tachyonic poles at Euclidean for . Conceptually, the local mass term proportional to is finite due to the BRST symmetry Eq. (6), which excludes a mass counter-term. The latter argument implies that contributions to are finite to all orders of the loop expansion.

Fig. 2. to order . If the model is stable at , the 1PI 2-point function of the scalar must not vanish at Euclidean either. To order , is given by the term that arises from Eq. (11) upon substitution of Eq. (13) and the one-(ghost)-loop contribution shown in Fig.2. Since a non-trivial solution to the gap equation Eq. (12) relates to a loop integral of zeroth order in the coupling, we may use Eq. (12) to lowest order to obtain a “tree-level” expression for of order . Evaluating the loop integrals, one obtains the real, positive and monotonic function

(16) | |||||

to order establishes the perturbative stability of a non-trivial solution to Eq. (12) and the fact that this solution is a minimum of the effective potential.

An expansion about a solution to the gap equation thus has lowest order propagators that are regular at Euclidean momenta for all the elementary fields except the photon (if all the matter fields are massive). The polarization of the photon vanishes at due to the -symmetry – regardless of the value of . Taking into account that the massless Goldstone quartet associated with this symmetry breaking decouples from physical quantities, the situation for is thus rather similar to QED with an unorthodox massive matter content (extending the notion of “massive matter” to include ghosts and other unphysical fields). But the perturbative expansion of the free energy of QED does not suffer from IR-divergences if all the matter fields are massive[2]. Since all the loop integrals of the sceleton expansion of the free energy are infrared finite at a non-trivial solution to Eq. (12), a perturbative evaluation of the free energy of this asymptotically free model is feasible and becomes accurate at high temperatures.

To complete the argument, we have to solve Eq. (12) for small coupling. To lowest order in the loop expansion, the relation between the renormalized couplings , the renormalization point and an expectation value implied by Eq. (12) is

(17) |

The anomalous dimension of the expectation value is simultaneously found to be

(18) |

where is the lowest order coefficient of the -function of this model ( with quark flavors in the fundamental representation as matter).

Using the relation between , and the asymptotic scale parameter , we may rewrite Eq. (17) as

(19) |

Apart from an anomalous dimension, the non-trivial solution at sufficiently small coupling is thus proportional to the physical scale in the particular gauge . The anomalous dimension in Eq. (18) is furthermore of order at . The terms of order in Eq. (19) thus also vanish in this particular gauge and higher order corrections to the asymptotic value of at small are analytic in . In the gauge , one can expand the model about

(20) |

and determine the corrections in Eq. (20) order by order in the loop expansion of the gap equation Eq. (12). At the lowest order solution Eq. (20) remains accurate to order at any finite order of the loop expansion. This does not imply that other gauges are any less physical, but it does single out as the gauge in which a perturbative evaluation of the gap equation Eq. (12) is consistent at sufficiently small values of . (In QED the hydrogen spectrum to lowest order is most readily obtained in Coulomb gauge, although it evidently does not depend on the chosen gauge. In the present case asymptotic freedom allows us to determine an optimal gauge for solving the gap equation.)

At the one-loop level, Eq. (12) has a unique non-trivial solution in any gauge and we know from of Eq. (16) that it corresponds to a minimum of the one-loop action. In the limit at finite coupling, the non-trivial one-loop solution Eq. (17) coincides with the trivial one. On the other hand, some of the couplings in the non-linear gauge-fixing become large in this limit, invalidating the perturbative analysis.

To gain some insight into the highly singular behavior of the model when , I calculated the divergent part of the self-energy to one loop. The corresponding anomalous dimensions and of the vector boson and the gauge parameter are

(21) |

Gauge dependent interaction terms proportional to at one loop thus lead to a term of order in the longitudinal part of the self-energy only. The transverse part of the self-energy is regular in the limit . Taking to vanish thus is rather tricky: Eq. (Mass Generation in Continuum Gauge Theory in Covariant Abelian Gauges) implies that the longitudinal part of the -propagator at one loop is proportional to at large momenta and no longer vanishes in this limit. Higher order loop corrections similarly contribute to the longitudinal propagator as . does not depend on the gauge parameter at one loop, due to an Abelian Ward identity that also gives the QED-like relation[10] between the renormalization constants of the photon, of the coupling and of the gauge parameter .

The anomalous dimension of the gauge parameter at sufficiently small is negative for positive values of when . With , the effective gauge parameter tends to decrease at higher renormalization scales and direct integration of Eq. (Mass Generation in Continuum Gauge Theory in Covariant Abelian Gauges) gives a vanishing at a finite value of the coupling . As already noted above, the loop expansion, however, is valid only if and . But Eq. (Mass Generation in Continuum Gauge Theory in Covariant Abelian Gauges) does show that there is no finite UV fixed point for the gauge parameter and that effectively vanishes at least as fast as as for any gauge at finite . Eq. (20) nevertheless is the asymptotic solution to Eq. (12) in the sense that it is valid at arbitrary small coupling if one chooses the gauge at that coupling to be . This is compatible with the asymptotic vanishing of the effective gauge parameter only if higher order corrections lead to an anomalous dimension that effectively remains of order even as . Since the gauge sector becomes strongly coupled when is , and the loop expansion does not give the correct behavior of Eq. (18) in this limit, this is at least conceivable.

Let me finally say that the non-trivial solution to the gap equation apparently persists to arbitrarily high temperatures. The (unique) non-trivial solution at is a consequence of the scale anomaly[5] and the Goldstone quartet of the spontaneously broken SL(2,R) symmetry does not contribute to the free energy. The renormalization point dependence of Eq. (17) and the associated UV-divergence of the loop integral are an indication of this. The character of the solution to Eq. (12) does, however, change dramatically with temperature[13]. At low temperatures deviates only marginally from Eq. (20), whereas at high temperatures .

I would like to thank D. Kabat, D. Zwanziger and R. Alkofer for suggestions, L. Spruch for his continuing support, and L. Baulieu for encouragement.

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