On scalar and vector fields coupled to the energy-momentum tensor

On scalar and vector fields coupled to the energy-momentum tensor

Jose Beltrán Jiménez, Jose A. R. Cembranos and Jose M. Sánchez Velázquez
Abstract

We consider theories for scalar and vector fields coupled to the energy-momentum tensor. Since these fields also carry a non-trivial energy-momentum tensor, the coupling prescription generates self-interactions. In analogy with gravity theories, we built the action by means of an iterative process that leads to an infinite series, which can be resumed as the solution of a set of differential equations. We show that, in some particular cases, the equations become algebraic and that is also possible to find solutions in the form of polynomials. We briefly review the case of the scalar field that has already been studied in the literature and extend the analysis to the case of derivative (disformal) couplings. We then explore theories with vector fields, distinguishing between gauge- and non-gauge-invariant couplings. Interactions with matter are also considered, taking a scalar field as a proxy for the matter sector. We also discuss the ambiguity introduced by superpotential (boundary) terms in the definition of the energy-momentum tensor and use them to show that it is also possible to generate Galileon-like interactions with this procedure. We finally use collider and astrophysical observations to set constraints on the dimensionful coupling which characterises the phenomenology of these models.

\affiliation

Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, Madrid, E-28049, Spain.
Departamento de Física Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain.
Departamento de Física Teórica and UPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain.

\emailAdd

jose.beltran@uam.es \emailAddcembra@fis.ucm.es \emailAddjmsvelazquez@ucm.es

IFT-UAM/CSIC-18-031

1 Introduction

General Relativity (GR) is the standard framework to describe the gravitational interaction and, after more than a century since its inception, it still stands out as the most compelling candidate owed to its excellent agreement with observations on a wide regime of scales [Will:2014kxa]. From a theoretical viewpoint, GR can be regarded as the theory describing an interaction mediated by a massless spin 2 particle. The very masslessness of this particle together with explicit Lorentz invariance makes it to naturally couple to the energy-momentum tensor and, since it also carries energy-momentum, consistency dictates that it needs to present self-interactions. This requirement has sometimes led to regard gravity as a theory for a spin 2 particle that is consistently coupled to its own energy-momentum tensor so that the total energy-momentum tensor is the source of the gravitational field111Let us note however that the actual crucial requirement is to maintain the gauge symmetry at the non-linear level.. These interactions can be constructed order by order following the usual Noether procedure (see for instance [Ortin]) and one obtains an infinite series of terms. One could attempt to re-sum the series directly or to use Deser’s procedure [Deser:1969wk] of introducing auxiliary fields so that the construction of the interactions ends at the first iteration. Either way, GR arises as the full non-linear theory and the equivalence principle together with diffeomorphism symmetry come along in a natural way (see also [Padmanabhan:2004xk, Butcher:2009ta, Deser:2009fq, Barcelo:2014mua] for some recent related works discussing in detail the bootstrapping procedure).

In this work, we intend to develop a family of theories for scalar and vector fields following a similar bootstrapping approach as the one leading to GR, i.e., by prescribing a coupling to the energy-momentum tensor that remains at the full non-linear level. Unlike the case of gravity where the coupling to the energy-momentum tensor comes motivated from the requirement of maintaining gauge invariance (so it is a true consistency requirement rather than a prescription), in our case there is no necessity to have a consistent coupling to the energy-momentum tensor nor self-couplings of this form. However, the construction of theories whose interactions are universally described in terms of the energy-momentum tensor (as to fulfill some form of equivalence principle) is an alluring question in relation with gravitational phenomena. Let us remind that, starting from Newton’s law, the simplest (perhaps naive) relativistic completion is to promote the gravitational potential to a scalar field. However, the most leading order coupling of the scalar to the energy-momentum tensor is through its trace and, therefore, there is no bending of light. This is a major obstacle for this simple theory of gravity based on a scalar field since the bending of light is a paramount feature of the gravitational interaction. Nevertheless, the problem of finding a theory for a scalar field that couples in a self-consistent manner to the energy-momentum tensor is an interesting problem on its own that has already been considered in the literature [Kraichnan:1955zz, Freund:1969hh, Deser:1970zzb, Sami:2002se]. Here we will extend those results for the case of more general couplings for a scalar field (adding a shift symmetry that leads to derivative couplings) and explore the case of vector fields coupled to the energy-momentum tensor.

The paper is organised as follows: We start by briefly reviewing and re-obtaining known results for a scalar field coupled to the trace of the energy-momentum tensor. We then extend the results to incorporate a shift-symmetry for the scalar in the coupling to the energy-momentum tensor, what leads to a theory for a derivatively coupled scalar. After obtaining results in the second order formalism, we turn to discuss the construction of the full non-linear theories in the first order formalism, where the resummation procedures can be simplified. We end the scalar field case by considering couplings to matter. After working out the scalar field case, we consider theories for a vector field coupled to the energy-momentum tensor. We will devote Sec. LABEL:sec:Superpotentials to discuss the role played by superpotential terms and Sec. LABEL:Sec:EffectiveMetrics to present a procedure to obtain the interactions from a generating functional defined in terms of an effective metric. In Sec. LABEL:Sec:Phenomenology we give constraints obtained from several phenomenological probes and we conclude in Sec. LABEL:Sec:Discussion with a discussion of our results.

2 Scalar gravity

We will start our study with the simplest case of a scalar field theory that couples to the trace of the energy-momentum tensor to recover known results for scalar gravity. Then, we will extend these results to include derivative interactions that typically arise from disformal couplings. Such couplings will be the natural ones when imposing a shift symmetry for the scalar field, as usually happens for Goldstone bosons. We will also consider the problem from a first order point of view. Finally, couplings to matter, both derivative and non-derivative, will be constructed.

2.1 Self-interactions for scalar gravity

Let us begin our tour on theories coupled to the energy-momentum tensor from a scalar field and focus on the self-coupling problem neglecting other fields, i.e., we will look for consistent couplings of the scalar field to its own energy-momentum tensor. Firstly, we need to properly define our procedure. Our starting point will be the action for a free massive scalar field given by

(1)

The goal now is to add self-interactions of the scalar field through couplings to the energy-momentum tensor. This can be done in two ways, either by imposing a coupling of the scalar field to its own energy-momentum tensor at the level of the action or by imposing its energy-momentum tensor to be the source in the field equations. We will solve both cases for completeness and to show the important differences that arise in both procedures at the non-linear level. Let us start by adding an interaction of the scalar to the energy-momentum tensor of the free field in the action as follows

(2)

with some mass scale determining the strength of the interaction and the trace of the energy-momentum tensor of the free scalar field, i.e., the one associated to . We encounter here the usual ambiguity due to the different available definitions for the energy-momentum tensor that differ either by a term of the form with some super-potential antisymmetric in the first pair of indices so that it is off-shell divergenceless, or by a term proportional to the field equations (or more generally, any rank-2 tensor whose divergence vanishes on-shell). In both cases, the form of the added piece guarantees that all of the related energy-momentum tensors give the same Lorentz generators, i.e., they carry the same total energy and momentum. We will consider in more detail the role of such boundary terms in Sec. LABEL:sec:Superpotentials and, until then, we will adopt the Hilbert prescription to compute the energy-momentum tensor in terms of a functional derivative with respect to an auxiliary metric tensor as follows222This definition is not free from subtleties either since one also needs to choose a covariantisation procedure and the tensorial character of the fields. We will assume a minimal coupling prescription for the covariantisation and that all the fields keep their tensorial character as in the original Minkowski space.

(3)

where in the action we need to replace with some background (Lorentzian) metric and its determinant. This definition has the advantage of directly providing a symmetric and gauge-invariant (in case of fields with spin and/or internal gauge symmetries) energy-momentum tensor. In general, this does not happen for the canonical energy-momentum tensor obtained from Noether’s theorem, although the Belinfante-Rosenfeld procedure [BelinfanteRosenfeld] allows to correct it and transform it into one with the desired properties 333See also [Gotay&Marsden] for a method to construct an energy-momentum tensor that can be interpreted as a generalisation of the Belinfante-Rosenfeld procedure.. For the scalar field theory we are considering, the energy-momentum tensor is

(4)

which is also the one obtained as Noether current so that the above discussion is not relevant here. However, in the subsequent sections dealing with vector fields this will be important since the canonical and Hilbert energy-momentum tensors differ.

After settling the ambiguity in the energy-momentum tensor, we can now write the first order corrected action for by incorporating the coupling (2), so we obtain

(5)

As usual, when we introduce the coupling of the scalar field to the energy-momentum tensor, the new energy-momentum tensor of the whole action acquires a new contribution and, therefore, the coupling receives additional corrections that will contribute to order . The added interaction will again add a new correction that will contribute an order term and so on. This iterative process will continue indefinitely so we end up with a construction of the interactions as a perturbative expansion in powers of and, thus, we obtain an infinite series whose resummation will give the final desired action. The iterative process for the case at hand gives the following expansion for the first few terms:

(6)

It is not difficult to identify that we obtain the first terms of a geometric progression with ratios and which can then be easily resummed. One can confirm this by realising that a term gives a correction , while a term introduces a correction . Then, the resummed series will be given by

(7)

with

(8)

Technically, the geometric series only converges for444The convergence of the generated perturbative series will be a recurrent issue throughout this work. In fact, most of the obtained series will need to be interpreted as asymptotic series of the true underlying theory. We will discuss this issue in more detail in due time. , but the final result can be extended to values , barring the potential poles at and that occur for positive values of the scalar field, assuming , while for the functions are analytic. Let us also notice that, had we started with an arbitrary potential for the scalar field instead of a mass term, the corresponding final action would have resulted in a re-dressed potential with the same factor, i.e., the effect of the interactions on the potential would be and, as a particular case, if we start with a constant potential corresponding to a cosmological constant, the same re-dressing will take place so that the cosmological constant becomes a dependent quantity. In any case, we find it more natural to start with a mass term in compliance with the prescribed procedure of generating the interactions through the coupling to the energy-momentum tensor, i.e., the natural starting point is the free theory.

An alternative way to resum the series that will be very useful in the less obvious cases that we will consider later is to notice that the resulting perturbative expansion (6) allows to guess the final form of the action to be of the form (7). Then, we can impose the desired form of our interactions to the energy-momentum tensor so that the full non-linear action must satisfy

(9)

with the trace of the energy-momentum tensor of the full action, i.e.,

(10)

We have also included here an arbitrary potential for generality. Thus, we will need to have

(11)

from which we can recover the solutions for and given in (8). Notice that this method allows to obtain the final action without relying on the convergence of the perturbative series and, thus, the aforementioned extension of the resummed series is justified. As a final remark, it is not difficult to see that, had we started with a coupling to an arbitrary function of of the form , the final result would be the same with the replacement in the final form of the function and , recovering that way the results of [Sami:2002se].

We have then obtained the action for a scalar field coupled to its own energy-momentum tensor at the level of the action. However, as we mentioned above, we can alternatively impose the trace of the energy-momentum tensor to be the source of the scalar field equations, i.e., the full theory must lead to equations of motion satisfying

(12)

again with the total energy-momentum tensor of the scalar field. Before proceeding to solve this case, let us comment on some important differences with respect to the gravitational case involving a spin-2 field. The above equation is perfectly consistent at first order, i.e., we could simply add on the RHS, so we already have a consistent theory and there is no need to include higher order corrections. This is in high contrast with the construction in standard gravity where the Bianchi identities for the spin-2 field (consequence of the required gauge symmetry) are incompatible with the conservation of the energy-momentum tensor and one must add higher order corrections to have consistent equations of motion. For the scalar gravity case, although not imposed by the consistency of the equations, we can extend the construction in an analogous manner and impose that the source of the equation is not given in terms of the energy-momentum tensor of the free scalar field, but the total energy-momentum tensor. As before, we could proceed order by order to find the interactions, but we will directly resort to guess the final action to be of the form given in (7) and obtain the required form of the functions and for the field equations to be of the form given in (12). For the sake of generality, we will consider a general bare potential instead of a simple mass term. By varying (7) w.r.t. the scalar field we obtain

(13)

that must be compared with the prescribed form of the field equation

(14)

Thus, we see that the functions and must satisfy the following equations

(15)

The solution for can be straightforwardly obtained to be

(16)

where we have chosen the integration constant so that , i.e., we absorbed into the normalization of the free field. It might look surprising that the solution for in this case is related to (8) by a change of sign of . This could have been anticipated by noticing that the construction of the theory so that appears as a source of the field equations requires an extra minus sign with respect to the coupling at the level of the action to compensate for the one introduced by varying the action. Thus, the two series only differ by this extra factor in the series that results in the overall change of sign of .

From the obtained equations we see that the solution for depends on the form of the bare potential . We can solve the equation for an arbitrary potential and the solution is given by

(17)

with an integration constant that must be chosen so that . If , we need to set . Remarkably, if we take a quadratic bare potential corresponding to adding a mass for the scalar field (which is the most natural choice if we start from a free theory), the above solution reduces to . In that case, the resummed action reads

(18)

which is the action already obtained by Freund and Nambu in [Freund:1969hh], and which reduces to Nordstrøm’s theory in the massless limit. We have obtained here the solution for the more general case with an arbitrary bare potential, in which case the solution for depends on the form of the potential. Finally, if we have , the starting action contains a cosmological constant and the obtained solution for gives the scalar field re-dressing of the cosmological constant, which is obtained after setting as it corresponds to have . We will re-obtain this result in Sec. 2.4 when studying couplings to matter fields.

2.2 Derivatively coupled scalar gravity

After warming up with the simplest coupling of the scalar field to the trace of its own energy-momentum tensor, we will now look at interactions enjoying a shift symmetry, what happens for instance in models where the scalar arises as a Goldstone boson, a paradigmatic case in gravity theories being branons, that are associated to the breaking of translations in extra dimensions [DoMa]. This additional symmetry imposes that the scalar field must couple derivatively to the energy-momentum tensor and this further imposes that the leading order interaction must be quadratic in the scalar field, i.e., we will have a coupling of the form . This is also the interaction arising in theories with disformal couplings [Disformal]. Although an exact shift symmetry is only compatible with a massless scalar field, we will leave a mass term for the sake of generality (and which could arise from a softly breaking of the shift symmetry). In fact, again and for the sake of generality, we will consider a general bare potential term. Then, the action with the first order correction arising from the derivative coupling to the energy-momentum tensor in this case is given by

(19)

with some mass scale. It is interesting to notice that now the coupling is suppressed by so that the leading order interaction corresponds to a dimension 8 operator, unlike in the previous non-derivative coupling whose leading order was a dimension 5 operator. As before, the added interaction will contribute to the energy-momentum tensor so that the interaction needs to be corrected. If we proceed with this iterative process, we find the expansion

(20)

where we have defined . Again, we could obtain the general term of the generated series and eventually resum it. However, it is easier to use an Ansatz for the resummed action by noticing that, from (20), we can guess the final form of the action to be

(21)

with and some functions to be determined from our prescribed couplings. Thus, by imposing that the final action must satisfy

(22)

with the total energy-momentum tensor, we obtain the following relation:

where the prime stands for derivative w.r.t. its argument. Thus, the functions and will be determined by the following first order differential equations

(24)
(25)

We have thus reduced the problem of resuming the series to solving the above differential equations. The existence of solutions for these differential equations will guarantee the convergence (as well as the possible analytic extensions) of the perturbative series. Although not important for us here, it is possible to obtain the explicit analytic solutions as

(26)
(27)

where stands for the exponential integral function of order and we have chosen the integration constants in order to have a well-defined solution for . In principle, one might think that boundary conditions must be imposed so that . However, these boundary conditions are actually satisfied by all solutions of the above equations since they are hardwired in the own definition of the functions and through the perturbative series. The way to select the right solution is thus by imposing regularity at the origin . Even this condition is not sufficient to select one single solution and this is related to the fact that the perturbative series must be interpreted as an asymptotic expansion555In fact, the solution resembles one of the paradigmatic examples of asymptotic expansion ., rather than a proper series expansion. In fact, it is not difficult to check that the perturbative series is divergent, as it is expected for asymptotic expansions. Thus, the above solution is actually one of many different possible solutions. We will find these equations often and we will defer a more detailed discussion of some of their features to the Appendix LABEL:Appendix.

So far we have focused on the coupling , but, at this order, we can be more general and allow for another interaction of the same dimension so that the first correction becomes

(28)

where and are two arbitrary dimensionless parameters, one of which could actually be absorbed into , but we prefer to leave it explicitly to keep track of the two different interactions. The previous case then reduces to , which is special in that the coupling does not depend on the metric and, as we will see in Sec. LABEL:Sec:EffectiveMetrics, this has interesting consequences in some constructions. For this more general coupling, the perturbative series reads

(29)

To resum the series we can follow the same procedure as before using the same Ansatz for the resummed action as in (21), in which case we obtain that the following relation must hold:

(30)

Thus, the equations to be satisfied in this case are

(31)
(32)

The additional freedom to choose the relation between the two free parameters and allows now to straightforwardly obtain some particularly interesting solutions. Firstly, for , the equations become algebraic and the unique solution is given by

(33)

This particular choice of parameters that make the equations algebraic is remarkable because it precisely corresponds to coupling the energy-momentum tensor to the orthogonal projector to the gradient of the scalar field . On the other hand, we can see from the perturbative series (29) that the condition cancels all the corrections to the kinetic term and this can also be seen from the differential equations where it is apparent that, for those parameters, is the corresponding solution. Moreover, for that choice of parameters, we see from the perturbative expansion that , which can be confirmed to be the solution of the equation for with . Likewise, for , all the corrections to the potential vanish and only the kinetic term is modified. It is worth mentioning that the iterative procedure used to construct the interactions also allows to obtain polynomial solutions of arbitrarily higher order by appropriately choosing the parameters. All these interesting possibilities are explained in more detail in the Appendix LABEL:Appendix.

Finally, let us notice that a constant potential that amounts to introducing a cosmological constant in the free action leads to a re-dressing of the cosmological constant analogous to what we found above for the non-derivative coupling, but with the crucial difference that now the cosmological constant becomes kinetically re-dressed in the full theory.

In the general case we see that we obtain a particular class of K-essence theories where the -dependence is entirely given by the starting potential, but it receives a kinetic-dependent re-dressing. On the other hand, if we start with an exact shift symmetry, given that the interactions do not break it, the resulting theory reduces to a particular class of theories.

2.3 First order formalism

In the previous section we have looked at the theory for a scalar field that is derivatively coupled to its own energy-momentum tensor. The problem was reduced to solving a couple of differential equations expressed in (25). Here we will explore the same problem but from the first order formalism perspective. In the case of non-abelian gauge fields and also in the case of gravity, the first order formalism has proven to significantly simplify the problem since the iterative process ends at the first iteration [Deser:1969wk]. For non-abelian theories, the first order formalism solves the self-coupling problem in one step instead of the four iterations required in the Lagrangian formalism. In the case of gravity, the simplification is even greater since it reduces the infinite iterations of the self-coupling problem to only one. This is in fact the route used by Deser to obtain the resummed action for the self-couplings of the graviton [Deser:1969wk]. The significant simplifications in these cases encourages us to consider the construction of the theories with our prescription in the first order formalism in order to explore if analogous simplifications take place. As a matter of fact, the first order formalism for scalar gravitation was already explored in [Deser:1970zzb] for the massless theory and with a conformal coupling so that the trace of the energy-momentum tensor appears as the source of the scalar field. It was then shown the equivalence of the resulting action with Nordstrøm’s theory of gravity and the massless limit of the theory obtained by Freund and Nambu [Freund:1969hh] with the first order formalism (which we reproduced and extended above). We will use this formalism for the theories with derivative couplings to the energy-momentum tensor, what in the first order formalism means couplings to the canonical momentum.

The first thing we need to clarify is how we are going to define the theory in the first order formalism. The starting free theory for a massive scalar field can be described by the following first order action:

(34)

with the corresponding momentum in phase space. Upon variations with respect to the momentum and the scalar field we obtain the usual Hamilton equations and , which combined gives the desired equation . At the lowest order we then prescribe a coupling to the energy-momentum tensor as666Of course, we could have also added a term , but the considered coupling will be enough to show how the use of the first order formalism leads to simpler non-differential equations.

(35)

with the energy-momentum tensor corresponding to the free theory . This is the form that we will also require for the final theory replacing by the total energy-momentum tensor. Following the same reasoning as in the previous section, the final theory should admit an Ansatz of the following form:

(36)

where the Hamiltonian777Let us stress that this Hamiltonian function will not give, in general, the energy of the system, although that is the case for homogeneous configurations. will be some function of the phase space coordinates. Lorentz invariance imposes that the momentum can only enter through its norm. As in the Lagrangian formalism, the energy-momentum tensor admits several definitions that differ by a super-potential term or quantities vanishing on-shell. As before, we shall resort to the Hilbert energy-momentum tensor. In this approach, one needs to specify the tensorial character of the fields, which are usually assumed to be true tensors. In some cases, it is however more convenient to assume that some fields actually transform as tensorial densities. In the Deser construction, assuming that the graviton is a tensorial density simplifies the computations. At the classical level and on-shell, assuming different weights only results in terms that vanish on-shell in the energy-momentum tensor888If we re-define a given field with a metric-dependent change of variables (as it happens when the re-definition corresponds to a change in the tensorial weight of the field), we have the following relation for the variation of the action

Since the second term on the RHS vanishes on the field equations of we obtain that both energy-momentum tensors coincide on-shell.. In the present case, it is convenient to assume that is a tensorial density of weight 1 such that is a tensor of zero weight. The advantage of using this variable is twofold: on one hand, this tensorial weight for makes already covariant without the need to introduce the in the volume element and, consequently, this term will not contribute to the energy-momentum tensor. On the other hand, the variation of the Hamiltonian with respect to the auxiliary metric gives

(37)

which is proportional to the orthogonal projector to the momentum and, thus, although it does contribute to the energy-momentum tensor, it will not contribute to the interaction . To derive the above variation, we have taken into account that the Hamiltonian remains a scalar after the covariantisation and, because of the weight of , it will become a function of . The full energy-momentum tensor computed with the described prescription is given by

(38)

If we compare with the canonical energy-momentum tensor , we see that the difference is , which vanishes upon use of the Hamilton equation and the relation between the Hamiltonian and the Lagrangian via a Legendre transformation . The final action must therefore satisfy the relation

(39)

where we see that the chosen weight for has greatly simplified the interaction term that simply reduces to . The above relation then leads to the algebraic equation

(40)

so that the Hamiltonian of the desired action will be given by

(41)

We see that the use of the first order formalism has substantially simplified the resolution of the problem since we do not encounter differential equations. Needless to say that the solutions obtained in both first and second order formalisms are different. Expressing the theory obtained here in the second order formalism is not very illuminating so we will not give it, although it would be straightforward to do it. Let us finally notice that, if the leading order term in the Hamiltonian for the limit is assumed to be , then it is not difficult to see that the full Hamiltonian will be

(42)

i.e., the procedure simply re-dresses the seed Hamiltonian with the factor . One interesting property of the resulting theory is that the Hamiltonian density for the massless case saturates to the scale at large momenta.

2.4 Coupling to matter fields

In the previous subsections we have found the action for the self-interacting scalar field through its own energy-momentum tensor, both with ultra-local and derivative couplings. Now we will turn our analysis to the couplings of the scalar field with other matter fields following the same philosophy, i.e., the scalar will couple to the energy-momentum tensor of matter fields. For simplicity, we will only consider the case of a matter sector described by a scalar field . The derivative couplings will be the same as we will obtain for the vector field couplings to matter that will be treated in the next section so that, in order not to unnecessarily repeat the derivation, we will not give it here and discuss it in Sec. LABEL:Sec:VectorMatter. Thus, we will only deal with the conformal couplings so that our starting action for the proxy scalar field including the first order coupling to is

(43)

where is the trace of the energy-momentum tensor of the proxy field and the corresponding potential. Going on in the iterative process yields the following series for the total action

(44)

This expression has the form of a geometric series so it is straightforward to resum it yielding the final action for scalar gravity algebraically coupled to matter:

(45)

with

(46)

Not very surprisingly, we obtain the same result as for the self-couplings of the scalar field, i.e., both the kinetic and potential terms get re-dressed by the same factors as we found in Sec. 2.1 for . If the matter sector consists of a cosmological constant, which would correspond to a constant scalar field in the above solutions, the coupling procedure gives rise to an additional modification of the potential or, equivalently, the cosmological constant becomes a -dependent quantity, which is also the result that we anticipated in Sec 2.1 for a constant potential of . Some phenomenological consequences of this mechanism were explored in a cosmological context in [Sami:2002se].

As in the self-coupling case, we could have introduced the coupling so that the trace of the energy-momentum tensor appears as a source of the scalar field equations. In order to obtain the theory with the required property, we shall follow the procedure of assuming the following action for the scalar gravity field and the scalar proxy field :

(47)

with , , and some functions of that will be determined from our requirement and is some potential for the scalar . In our Ansatz we have included the self-interactions of the scalar field encoded in and . Since this sector was resolved above, we will focus here on the couplings to so that and are the functions to be determined by imposing the field equations be of the form

(48)

The trace of the total energy-momentum tensor derived from (47) is given by

(49)

and hence we obtain that the equation of motion must be of the form

(50)

On the other hand, varying (47) with respect to yields

(51)

Comparing (50) and (51) will give the equations that must be satisfied by the functions in our Ansatz for the action. The sector has already been solved in the previous subsection, so we will only pay attention to the sector now. Then, we see that the functions and must satisfy the following equations

(52)

The solution for these equations, taking into account the functional form of given in 16, is then

(53)

that coincides with the expression given in [Freund:1969hh]. We see again that, although both procedures give the same leading order coupling to matter for the scalar field, the full theory crucially depends on whether the coupling is imposed at the level of the action or the equations. If we consider again the case of a cosmological constant as the matter sector, we see that its re-dressing with the scalar field will be different in both cases. It could be interesting to explore the differences with respect to the analysis performed in [Sami:2002se], where the coupling was assumed to occur at the level of the action.

3 Vector gravity

After having revisited and extended the case of a scalar field coupled to the energy-momentum tensor, we now turn to the case of a vector field. Since vectors present a richer structure than scalars due to the possibility of having a gauge invariance or not depending on whether the vector field is massless of massive, we will distinguish between gauge invariant couplings and non-gauge invariant couplings. For the latter, the existence of a decoupling limit where the dominant interactions correspond to those of the longitudinal mode will lead to a resemblance between some of the interactions obtained here and those of the derivatively coupled scalar studied above.

3.1 Self-coupled Proca field

Analogously to the scalar field case, our starting point will be the action for a massive vector field given by the Proca action999Of course we could consider an arbitrary potential, but a mass term is the natural choice if we really assume that we start with a free theory.

(54)

where , and is the mass of the vector field. The energy-momentum tensor of this field is given by

(55)

Unlike the case of the scalar field, this energy-momentum tensor does not coincide with the canonical one obtained from Noether’s theorem and, thus, the Belinfante-Rosenfeld procedure would be needed to obtain a symmetric energy-momentum tensor, showing the importance of the choice in the definition of the energy-momentum tensor in the general case.

Along the lines of the procedure carried out in the previous sections, we will now introduce self-interactions of the vector field by coupling it to its energy-momentum tensor, so the first correction will be

(56)

with the corresponding coupling scale. In this case, the leading order interaction corresponds to a dimension 6 operator. Since this interaction will also contribute to the energy-momentum tensor, we will need to add yet another correction as in the previous cases, resulting in an infinite series in that reads:

(57)

where . Again, to resum the iterative process we will use a guessed form for the full action. The above perturbative series makes clear that the final form of the action will take the form

where the functions , and will be obtained by imposing the desired form of the interactions through the total energy-momentum tensor, i.e., we need to have

(58)

Thus, the coupling functions in the resummed action need to satisfy the following first order differential equations

(59)
(60)
(61)

These equations are of the same form as the ones obtained for the derivatively coupled scalar field and the solutions will also present similar features. For instance, the perturbative series will need to be interpreted as asymptotic expansions of the solutions of the above equations. Furthermore, the integration constants that determine the desired solution are already implemented in the equations so we need to impose regularity at the origin, but this only selects a unique solution for a given semi-axis, either or , that can then be matched to an infinite family of solutions in the complementary semi-axis (see Appendix LABEL:Appendix). The particular form of the solutions is not specially relevant for us here (although it will be relevant for practical applications), but we will only remark that they correspond to the class of theories for a vector field that is quadratic in the field strength or, in other words, in which the field strength only enters linearly in the equations.

As for the derivative couplings of the scalar field, we can be more general and allow for a coupling of the form

(62)

The iterative process in this case gives rise to

We can again use our Ansatz for the final action to obtain that the differential equations to be satisfied are

(63)
(64)
(65)

Remarkably, we see from the perturbative expansion that the case exactly cancels all the corrections to the kinetic part so that , and only the potential sector is modified. It is not difficult to check that this is indeed a solution to the above differential equations. We can also see here again that the choice , which corresponds to a coupling to the orthogonal projector to the vector field given by , reduces the equations to a set of algebraic equations whose solution is

(66)
(67)
(68)

These solutions show that and present different analytic properties depending on whether the field configuration is timelike or spacelike, but, in any case, has a pole for so that it seems reasonable to demand for this particular solution. The properties of these equations are similar to the ones we found in Sec. 2.2 for the derivatively coupled case and, in fact, the equation for here is the same as the equation for in (32), which is of course no coincidence. Thus, the more detailed discussion given in Appendix LABEL:Appendix also applies and, in particular, it will also be possible to obtain polynomial solutions by appropriately choosing the parameters, owed to the recursive procedure used to construct the interactions.

From our general solution we can also analyse what happens if our starting free theory is simply a Maxwell field, i.e., . In that case, only the terms containing and will have an effect and, in fact, they will provide the vector field with a mass around non-trivial backgrounds of