The (IR)relevance of the Gribov ambiguity in gauge theories with fundamental Higgs matter
Abstract
It is well accepted that dealing with the Gribov ambiguity has a major impact on correlation functions in gaugefixed Yang–Mills theories, in particular in the low momentum regime where standard perturbation theory based on the Faddeev–Popov approach fails. Recent results, derived from functional tools (Dyson–Schwinger equations or exact RG) or the effective Gribov–Zwanziger action method, pointed towards e.g. gauge boson correlation functions that are not compatible with the properties of observable degrees of freedom. Although such an observation is a welcome feature for gauge theories exhibiting confinement, it would be a discomfort for gauge theories supplemented with Higgs fields, cfr. the experimental success of the electroweak model based on a gauge group. The purpose of this short note is to assure that the effective action resolution to the Gribov ambiguity reduces to the standard Faddeev–Popov method in the perturbative regime of sufficiently small coupling/large Higgs condensate, thereby not compromising the physical particle spectrum of massive gauge bosons and a massless photon for the gaugeHiggs model. The closer the theory gets to the limit of vanishing Higgs condensate, the more the Gribov problem resurfaces with all its consequences. We give some speculations w.r.t. the Fradkin–Shenker insights about the phase diagram.
1 Motivation
The continuum path integral or canonical Hamiltonian quantization of gauge theories require the choice of a gauge fixing condition. This is perfectly well understood in the Abelian case, but in the nonAbelian case the nontrivial topology of the gauge group can spoil the simple picking up of a unique gauge field per set of gauge equivalent fields that fulfills the gauge condition. This problem was first investigated by Gribov in [1] for the Landau and Coulomb gauges, and later on many nontrivial results were more rigorously probed in the work of Zwanziger, Dell’Antonio and others [2]. Also in curved spaces, some results were booked in more recent times [3].
In a series of papers, we and others investigated the dynamical stability and further quantum alterations to the seminal results of Gribov and Zwanziger [4]. The ensuing propagators of the gauge bosons (gluons) and FaddeevPopov ghosts are in good agreement with their lattice counterparts [5]. The Gribov–Zwanziger approach is thus one example of effective functional approaches to nonAbelian gauge fixed theories, thereby supplementing other potential approaches as in [6]. The propagators proposed by us recently also found use in studies of Casimir effect [7] or the finite temperature phase diagram [8, 9].
A key observation of the nonperturbative Landau gauge gluon propagator is the violation of positivity, that is the gluon is not a particle that has a physical interpretation with associated KällénLehmann spectral integral representation with positive density. This violation of positivity is wellappreciated from simulational and analytical approaches [10], and it can be seen as a signal of confinement: gluons cannot be observed. In the Gribov–Zwanziger approach, this violation of positivity annex unphysicalness of the gluon is imminent because of the presence of complex conjugate poles in the propagator [11].
This led us to the interesting question what the consequences would be of the gauge fixing ambiguity and its GribovZwanziger resolution when the gauge theory is coupled to a Higgs sector, most notably in the case of a theory with fundamental Higgs matter, of physical relevance to the electroweak sector of the Standard model. At the energies at which the electroweak theory is probed, a valid perturbation theory with a set of 3 observable massive gauge bosons and a massless photon should be the outcome. Complex conjugate masses or a positivity violation in the gauge boson sector should thus preferably not manifest themselves. Of course, the Gribov problem is still there, irrespective whether Higgs field are present or whether the Higgs mechanism takes place. The issue of having multiple solutions to the gauge fixing condition is a classical observation, only influenced by having a nonAbelian gauge field structure. We would thus rather like to find out under which circumstances treating the Gribov copy problem becomes trivial at the quantum level, thereby not affecting the standard perturbative expansion and accompanying results on the spectrum etc.
The difference or not between a confining and Higgslike spectrum of gaugeHiggs system is a very deep question, most notable brought to attention in the seminal theoretical lattice work of Fradkin and Shenker [12]. They presented evidence, using either an area or perimeter law for the Wilson loop expectation value to label the confining and Higgs phase, that both regions are analytically connected in a (gauge coupling, Higgs ) diagram. From lattice works as [13], one also learns that the 2 regions are separated by a line corresponding to a 1st order phase transition, whereby it is crucial to mention that this line has an endpoint. The spectrum is in both phases generated from gauge invariant operators that “interpolate” analytically between massive gauge bosons, resp. mesonlike bound states in the Higgs, resp. confining phase, see for example also [14]. We refer to [12, 15] for a more detailed exposure on this.
2 Setup
In this section we define the notation and conventions used throughout the paper. The system we study is the electroweak sector of the standard model, thereby generalizing our earlier works [16, 17, 18]. In the following sections we consider in detail the possible effects of Gribov copies in such a setting, based on the original Gribov nopole analysis [1, 19]. A summary of our results can be found in section 6.
2.1 The classical action
Let us begin by recalling the action describing a Higgs field in the fundamental representation of :
(1) 
with covariant derivative
(2) 
and the vacuum expectation value (vev) of the Higgs field
(3) 
The indices refer to the fundamental representation of , and , , are the Pauli matrices. The coupling constants and refer to the groups and respectively. It will be understood that we work in the limit for simplicity, i.e. we have a Higgs field frozen at its vacuum expectation value, as in [12] and as in our previous works. In principle, in the continuum, such limit should only be taken after the Feynman diagrams up to a certain order have been computed and properly renormalized, since otherwise problems with the UV renormalization can be expected. In this work, as in its predecessors [16, 17, 18], we shall however only need a one loop computation for the ghost propagator (see further) and at this order, there are no contributions proportional to yet, so the limit is well under control for the time being. In any case, the aim of the current paper is still not that of being exhaustive on the Gribov/Higgs issue, given the mere complexity of it. The to be presented results should be interpreted as a first step towards a better understanding of the gauge fixing ambiguity in gauge theories supplemented with Higgs matter fields.
The respective field strengths are defined as
(4) 
For later usage, consider quadratic part of the action (1), which reads
(5)  
Making use of
(6) 
which has the inverse transformation
(7) 
one easily derives that is diagonal:
(8)  
2.2 Gauge fixing, the Gribov ambiguity and the restriction to the Gribov region
As we are working in a continuum setting, we need to restrict the local gauge freedom using a suitable gauge fixing. Here, we adopt the Landau gauge, i.e. we demand that all gauge fields be transverse (= vanishing divergence). The gauge fixing action can be written as
(9) 
after a suitable linear redefinition of the necessary Lagrange multipliers and antighosts. The corresponding gauge field propagators read
(10a)  
(10b) 
The indices and take the values 1 and 2 and denote the offdiagonal modes. We also introduced the transversal projector . Rewriting these propagators in terms of the physical fields , and gives
(11) 
Now, it is well known since the pioneering paper [1] that imposing the Landau gauge fixing does not completely fix the gauge freedom. Indeed, since an infinitesimal gauge transformation corresponds to a shift of the gauge fields with the covariant derivative of some function, a solution gauge equivalent to the Landau gauge —with infinitesimal gauge transformation— is possible whenever the Faddeev–Popov operator has zero modes, that is if the Jacobian of the gauge fixing has zero modes. Since the Jacobian matrix enters the Faddeev–Popov quantization procedure explicitly, it is clear that the latter is incompatible with having gauge copies.
Gribov’s original proposal was to restrict the Landau gauge configurations to the subset defined by the Faddeev–Popov operator having no zero modes. Needless to say, this at least kills off the infinitesimal gauge copies. This approach was later put on a firmer footing by Zwanziger, see [20] for a recent review with a myriad of relevant references. Related or somewhat different approaches can be found in e.g. [21, 22].
We will implement the restriction to the Gribov region in terms of the original fields and and only afterwards move to the other fields , , , and . This simplifies the calculations of the Gribov form factors, see next section. From now on, we shall be working with expression (9). Adding the gauge fixing action to the expression (1), the total action reads
(12) 
It is interesting to notice here that in general, thanks to the transversality of the Landau gauge, there will be no mixing terms between the massive gauge bosons and associated Goldstone mode fields, due to the presence, after partial integration of a factor. This is not a surprise, since after all the Landau gauge is an extremal case of the wellknown gauges which have the explicit property of killing the mixing terms between the gauge bosons and Goldstones. The diagrammatic expansion of the here studied theory should thus be well under control. We do however wish to remark that the issue of unitarity of gauge theories with massive vector particles due to a Higgs mechanism, is ideally not discussed in the Landau gauge, mainly because of the constraint issues related to setting with the help of a Lagrangian multiplier . Other gauges are better suited for this, in particular the wellknown unitary gauge. On the other hand, this gauge might provide with nice physical interpretation at the classical level, it is evenly wellknown that the UV renormalization of the unitary gauge is badly behaved, to say the least. In this work, we thus specifically choose the Landau gauge, since it is the only covariant gauge in which relatively a lot is known about the Gribov gauge fixing ambiguity, while being a welldefined gauge at the quantum level for carrying out loop computations. In addition, it also has the advantage of being under scrutiny using lattice simulational tools, see [23], allowing a comparison between different methodologies. In theory, if unitarity is established in one gauge, gauge invariance dictates it is valid in any gauge. At the quantum level, the unitarity is always better discussed using BRST tools anyhow [24, 25]. We will not dwell upon that here.
To restrict to the first Gribov region at lowest nonvanishing order, we can compute the twopoint ghost function and impose a nopole condition for the latter quantity. Indeed, the ghost propagator is nothing but the inverse of the Faddeev–Popov operator. The tree level ghost propagator reads and is thus positive. If the perturbative corrections get too large — which we typically expect in strongly interacting gauge theories to happen at lower momentum transfer — the quantum shift in the ghost selfenergy could overcome the tree level value. In other words, the ghost propagator would turn negative, indicating that we are outside of the region , defined by the ghost propagator being positive.
3 The nopole condition made explicit
In order to impose the nopole condition on the SU(2) ghost propagator, we first need a closed expression for it. The twopoint ghost correlation function is
(13) 
which can be written in matrixform as
(14) 
Here we defined the offdiagonal twopoint function
(15a)  
and the diagonal one  
(15b) 
Thus
(16) 
The quantities and turn out to be decreasing functions of , explicitly visible upon taking the average with a suitable measure, see also [1]. Thus, the nopole condition is eventually satisfied by the conditions
(17) 
where
(18a)  
and  
(18b) 
Conditions (17) imply that the ghost propagators and are always positive, thus ensuring that one remains inside the Gribov region . We employed the property
(19) 
which follows from the transversality of the gauge field, . Using rotational invariance, for a generic function , we may always use
(20) 
where, upon contracting both sides of equation (20) with ,
(21) 
3.1 Lowestorder gap equations from the nopole condition
In order to restrict to the Gribov region , we implement the nopole conditions (17) by means of suitable step functions in the functional integral, namely
(22) 
where is the quadratic part of the total action (12). Making use of the integral representation
(23) 
we get for the functional integral
(24)  
It turns out to be convenient to perform the change of variables , , leading us to
(25)  
where
(26) 
One can easily get the twopoint correlation function of the fields , and by inverting the matrices and . Thus, since is diagonal, the offdagonal boson propagator can be written as
(27) 
Similarly, inverting immediately gives the diagonal boson propagators, namely
(28a)  
(28b)  
(28c) 
With the relations (6), one obtains the propagators of , and :
(29a)  
(29b)  
(29c)  
(29d) 
In order to derive the gap equations, we first integrate out the fields to obtain
(30) 
The two determinants appearing can immediately be evaluated as
(31a)  
(31b) 
where we have used the fact that the elements of commute with each other, allowing to compute the determinant in the usual way. The partition function can be written as
(32) 
Applying the saddle point approximation, one gets
(33) 
For each condition we get a gap equations: the first one, form the derivative,
(34a)  
and the second one, from the derivative,  
(34b) 
3.2 The limit .
An important check to be done is the case where , which must recover the results of [17] with the Higgs field in the fundamental representation, obtaining a massless gauge field decoupled from the gauge sector. This decoupling can be easily seen just by setting in the propagator expressions (27) and (28):
(35a)  
(35b) 
Also, as in the last section, one should be able to write the propagators in terms of the fields , and obtaining
(36a)  
(36b) 
These propagators, (35) and (36), could also be derived by taking in the quadratic partition function, or even in the generating function (24), and following the steps of the last section.
3.3 The vacuum energy
Looking at the above propagators, beside the decoupling of the gauge field from the gauge field, one should note the likeness between the diagonal and offdiagonal propagators, though in general the two Gribov parameters, and , differ. Therefore, given the important role played by the gap equations, it seems to be worth to analyze the two gap equations (34) in the limit . In this limit, the gap equations (34) become
(37) 
It is clear that these two equations are identical. Thus, when , there is only one gap equation and, therefore, only one Gribov parameter. Consequently, the diagonal and offdiagonal propagators of (35) are identical. These results coincide with what was found in [17] in the case of Higgs field in the fundamental representation.
Furthermore, from equation (32), we get for the vacuum energy,
(38) 
which, again, is in agreement with the expression for the vacuum energy for the case of in the fundamental representation, upon redefining .
4 About and without the Gribov parameters
Before trying to solve the gap equations, it seems to be worthwhile to study what happens with and in the absence of the Gribov parameters, which will allow us to search for regions where the Gribov parameters and are unnecessary, which happens whenever and/or are less than one.
Thus, given (18) and the propagators (10) we have
(39) 
Using standard techniques, this gives
(40) 
where
(41) 
such that the offdiagonal ghost form factor can be rewritten as
(42) 
where is the Weinberg angle. With the 2nd expression of (39) and (42) we are able to identify three possible regions:

Region I, where and , meaning . In this case the Gribov parameters are both zero so that we have the massive and , and a massless photon, as in (11), in what we can call the Higgs phase.

Region II, where and , or equivalently . In this region we have while , leading to a modified propagator, and a free photon and a massive boson.

The remaining parts of parameter space, where and , or . In this regime we have both and , which modifies the , and photon propagators, as shown in equations (29). Furthermore this region will fall apart in two separate regions III and IV due to different behavior of the propagators.
5 The offdiagonal gauge bosons
Let us first look at the behavior of the offdiagonal bosons under the influence of the Gribov horizon. The propagator (29a) only contains the Gribov parameter, meaning does not need be considered here.
As found in the previous section, this is not necessary in the regime , due to the ghost form factor always being smaller than one. In this case, the offdiagonal boson propagator is simply of the massive type:
(43) 
In the case that , the relevant ghost form factor is not automatically smaller than one anymore, and the Gribov parameter becomes necessary. The value of is given by the gap equations (34b), which has exactly the same form as in the case without electromagnetic sector. Therefore the results will also be analogous. As the analysis is quite involved, we just quote the results here.
For