A Projectors

Running Couplings in Quantum Theory of Gravity Coupled with Gauge Fields

Abstract

In this paper we study the coupled system of non-abelian gauge fields with higher-derivative gravity. Charge renormalization is investigated in this coupled system. It is found that the leading term in the gauge coupling beta function comes due to interaction of gauge fields with gravitons. This is shown to be a universal quantity in the sense that it doesn’t depend on the gauge coupling and the gauge group, but may depend on the other couplings of the action (gravitational and matter). The coupled system is studied at one-loop. It is found that the leading term of gauge beta function is zero at one-loop in four dimensions. The effect of gauge fields on the running of gravitational couplings is investigated. The coupled system of gauge field with higher-derivative gravity is shown to satisfy unitarity when quantum corrections are taken in to account. Moreover, it is found that Newton constant goes to zero at short distances. In this renormalizable and unitary theory of gauge field coupled with higher-derivative gravity, the leading term of the gauge beta function, found to be universal for all gauge groups, is further studied in more detail by isolating it in the context of abelian gauge theories coupled with gravity in four dimensions. Using self-duality of abelian gauge theories in four dimensions, this term of the gauge beta function is shown to be zero to all loops. This is found to be independent of the gravity action, regularization scheme and gauge fixing condition. An explicit one-loop computation for arbitrary gravity action further demonstrates the vanishing of this term in the gauge beta function in four dimensions, independent of the regularization scheme and gauge fixing condition. Consequences of this are discussed.

I Introduction

Matter that has been observed so far can be very well described theoretically using the standard model of particle physics, which has been very accurately tested in accelerator experiments. With the discovery of a scalar particle at 125 GeV (expected so far to be Higgs particle), the standard model of particle physics is seen as an ultimate theory of nature which is capable of describing it to very high energies, may be all the way up to Planck scale (1); (2); (3). It theoretically describes three of the four known forces of nature: Electromagnetic, Weak and Strong forces. These forces are described using the gauge fields whose quanta acts as force carriers between matter. The gauge fields describing the electromagnetic force is abelian in nature, while the gauge fields describing the weak and strong forces are non-abelian in nature. All gauge fields couple with matter but in case of non-abelian gauge theories, the fields also have self coupling. The strength of these couplings are described using parameters, which in a quantum theory changes with energy scale (a generic feature of any quantum field theory), meaning that at different energy scales the strength of interaction is different, unlike in classical theories where strength of interactions remain fixed. Witnessing such theoretically predicted running of coupling parameters experimentally further justifies the methods and tools used to study quantum field theories. This running of gauge couplings can be extrapolated to very high energies as standard model of particle physics remains well defined all the way up to Planck scale. It is found that for non-abelian gauge couplings the running is such that the coupling parameter tends to zero at high enough energies (called asymptotic freedom), while for abelian gauge coupling the running leads to a singularity at a particular energy at which the coupling blows up (Landau singularity).

Theoretically these phenomenas can be studied in perturbation theory using Feynman path-integrals, which are sum over the phase-space configuration - each configuration being weighed by a phase. Perturbative quantum field theories can be studied very accurately using the path-integral techniques as long as the running coupling parameter remains small, in which case the observables obtained can be reliably estimated. In Standard Model (SM), perturbation theory is well defined all the way up to Planck scale, as the Landau singularity of the abelian gauge coupling (which is a source of worry) occurs way beyond the Planck scale (1); (2); (3). On the other hand non-abelian gauge coupling witnesses asymptotic freedom in the high energy regime thereby guaranteeing the validity of perturbation theory. The problem with abelian gauge coupling is evaded in Grand Unified Theories (GUT) by considering a bigger symmetry group of the matter fields (4); (5); (6). In these scenarios there is enhanced symmetry at high energy scales greater than GeV, which breaks down at GeV to the known standard model symmetry group. This scenario overcomes the problem of Landau singularity in a very elegant way by unifying all the standard model gauge couplings at GeV (4); (5); (6). However in the absence of such enhanced symmetry groups (as there is no experimental signature so far confirming this possibility), problem of Landau singularity for abelian gauge coupling persist and it appears as a blot on the beautiful theory of standard model. Moreover in the absence of GUT scenarios, standard model ultimately enters a non-perturbative domain, as the abelian gauge coupling becomes large near the Landau pole signaling the breakdown of the perturbation theory. GUT scenarios may seem elegant and beautiful solution to overcome the problem, but it not a necessity. In its absence the validity of the SM is limited and the perturbative path-integral is well defined only up to finite albeit larger than Plank energy scale.

Gravitational attractive force between two particles (charged or uncharged) becomes comparable to the electromagnetic, weak or strong force at the Planck scale. So far in studying SM, we have ignored gravitational interactions all the way up to Planck scale. This is justified as gravitational force is quite weak compared to other three forces at scales below Planck energy. But certainly it should not be ignored near or at Planck scale, where gravitational effects are significantly important and would contribute equally to any of the quantum effects taking place at that energy scale. Also, the standard model experiments that are performed today are local events, where background curvature is irrelevant, as its effects are negligible. However parts of background spacetime, where curvature blows up, curvature effects become large and cannot be ignored. These occur near the singularities like the one in black holes. Places like these with large curvature, leads to particle creation. This is analogous to the Schwinger mechanism (7) witnessed when strong electric field results in particle pair creation. A correct theoretical description of this is achieved when the full SM coupled with Einstein-Hilbert gravity is studied. Studying quantum matter fields on curved background provides an insight in to how the the standard model phenomenas will get corrected when background curvature is taken in to account? Due to back reaction such effects will also modify the background gravitational field and leads to higher-derivative type gravitational interactions, signaling the non-renormalizabilty of the SM on a curved background. Still, such studies are important in their own light. In a complete quantum picture it is required that all fields present in the theory be quantized. This when attempted over the coupled system of SM with Einstein-Hilbert gravity leads to plague of ultraviolet divergences, where at each order of perturbation theory new counter-terms appear which cannot be absorbed in the previous ones, indicating one to conclude that the coupled system is non-renormalizable. However, it is not a completely devastating situation, as one can still study in this coupled system how the low energy physics gets modified due to quantum nature of the gravitational field and whether such corrections will have any observable consequences? This is the attitude taken towards these studies which goes under the name of Effective Field Theory (8); (9); (10). However such effects cannot be trusted at very high energies (like Planck scale), where a knowledge of sensible quantum gravity is required.

Running of standard model couplings is one place where effects are quantum gravity are likely to affect their behavior, because the presence of new degree of freedom in the system modifies the behavior of running couplings, as the virtual particles of these new species are generated whose contributions are not negligible near Planck energy scale. In case of gauge theories, such contributions are expected to arise and may modify the high energy behavior of running gauge couplings namely asymptotic freedom and Landau singularity. This effect if present will be hardly noticeable at low energies due to Planck mass suppression, but would certainly affect the delicate unification of the gauge couplings at GUT scale in the supersymmetric GUT theories (4); (5); (6). Such unifications being very delicate and sensitive to input parameters, are very easily affected due to any minor modification to the gauge beta functions coming due to quantum gravity effects. If the presence of virtual graviton were to spoil the unification in such a manner, then it is very disturbing as the unification of Standard model gauge couplings is a necessary prediction coming out from any realistic GUT theories. In the absence of GUT (which is not a necessity to solve the problems of SM), the running of abelian gauge coupling can’t be extrapolated to very high energies as it runs in to problem (Landau singularity). On the other hand in the case of non-abelian gauge couplings the running can be extrapolated to arbitrarily high energies. It is difficult to conceive that such runnings remains unaffected even at very high energies for example Planck scale, where quantum gravity effects are important. It therefore becomes crucial to examine and study how such runnings gets altered under the influence of quantum gravity effects?

Effects of quantum gravity on the running of standard model couplings have been studied in the past. This has been mostly done in the context of standard model of particle physics coupled with Einstein-Hilbert (EH) gravity (11); (12); (13); (14); (15); (16); (17). This is a non-renormalizable system, thus any computation that has been done in this context is studied within the framework of Effective field theory (8); (9); (10). Effects of quantum gravity on the renormalization of charge were first discussed in (12); (15); (16); (17) within perturbation theory using dimensional regularization scheme (18), and concluded that at one-loop there are no quantum gravity correction to the beta function of the gauge couplings. This result was often suspected to be a consequence of the massless nature of graviton and gluons/photons, and the way dimensional regularization handles the quadratic divergences. This problem was re-examined by using a momentum cutoff with a type of gauge fixing condition at one-loop in (19). They came to the conclusion that the beta function of the gauge coupling gets a nonzero quantum gravity correction signaling asymptotic freedom of all gauge couplings even those which hits Landau singularity at high energies. This also showed that the coupling vanishes as power law as opposed to logarithmic well before Planck energies. The authors used momentum cutoff as it is able to see quadratic divergences present in the theory, which dimensional regularization misses in four dimensions.

However this result came under criticism and was re-studied by several authors using various ways in different gauge choices or regularization scheme and refuted the result of (19). The investigations which find that there is no quantum gravity correction to charge renormalization are the following: using momentum cutoff and a harmonic type gauge fixing condition, quadratic divergences were studied in (20); using dimensional regularization with a gauge independent formulation of effective action in (22); (21); in (23) the problem was studied using feynman diagram technique within both momentum and dimensional regularization scheme; using functional renormalization group by choosing a symmetry preserving gauge condition and a regulator for cutting off modes under the functional trace in (30). The literature which finds a non-zero quantum gravity correction to the running of gauge coupling are the following: using loop-regularization in (24); in the presence of cosmological constant using gauge independent formulation of effective action (Vilkowisky-DeWitt technique) in (22); (25); (26) it was found that a nonzero contribution is achieved which is proportional to cosmological constant; using Vilkowisky-DeWitt technique quadratic divergences were studied in (27); (28); using functional renormalization group equation in (29). In (19); (20); (21); (23); (24); (25); (26); (27); (28) the study of the charge renormalization was done in the context of EH gravity in the spirit of effective field theory (8); (9); (10), as the coupled system is non-renormalizable. Any results obtained from this can only be trusted at low energies as EH gravity is a low energy limit of any fundamental theory of quantum gravity. In (29); (30) however Functional renormalization group has been used to study the problem in the spirit of asymptotic safety scenario (31); (32); (33); (34); (35).

In all these cases however it is not possible to give a proper meaning to the quantum corrections to the couplings, as in these cases the theory is non-renormalizable and has quadratic divergences (36). These ambiguities don’t occur in systems which are free of quadratic divergences and are renormalizable. One such system is fourth order higher-derivative gravity which is renormalizable to all loops (37), and has recently been shown to be unitary (38); (39). The first signature which provided the motivation for studying higher-derivative gravity came when quantum matter fields were studied on a curved background. It was realized that at one-loop four kind of divergences appear (43): , , and , where is the Ricci tensor of the background metric and is the corresponding Ricci scalar. This not only showed that the coupled system is non-renormalizable but also gave an important hint that perhaps considering a quantum theory of fourth order higher-derivative gravity coupled with matter will be more reasonable system to study as it may turn out to be completely renormalizable to all loops. However historically things went around a slightly different direction. A quantum theory of Einstein-Hilbert gravity was first studied. This was also necessary, after all its a theory of classical gravity and describes a large number of phenomenas very accurately. It was found in an one-loop study that the quantum theory of pure EH gravity is renormalizable on-shell (11). In the same paper it was also noticed that the theory is non-renormalizable even at one-loop when matter is included. This was further confirmed in (12); (13); (14); (15); (16); (17). This was somewhat expected as the gravitational coupling parameter, Newton’s constant has negative mass dimensions, but a thorough study was needed to explicitly check the renormalizability of the system. A two loop analysis of EH gravity assured the non-renormalizability of the theory (44); (45); (46). Such discouragements gave boost as to modify the gravity action in order to have a better ultraviolet behavior of theory. Indeed it was found in (37) that when EH-gravity action is augmented with fourth order higher-derivative terms, then the action is perturbatively renormalizable to all loops in four spacetime dimensions. The higher derivative gravity action that was considered in (37), in arbitrary space-time dimensions is given by,

(1)

where is the Newton’s constant, has dimensions of while is dimensionless. Here the action is written in -spacetime dimensions, for . The most general fourth order derivative action that can be written includes term: Ricci scalar , Weyl-square term , Ricci scalar square term and Gauss-Bonnet term. However for the dimensions under consideration, the Gauss-Bonnet term is topological and is not relevant in perturbative studies. This implies that one can re-express the as a combination of and , thereby allowing one to rewrite as a combination of and (modulo factor of spacetime dimensions ), which is the second term in eq. (1). The -dependent coefficient in front of term is chosen for later convenience, as in the propagator it produces pole at mass .

The action given in eq. (1), although is renormalizable in four spacetime dimensions to all loops, but suffers from a serious problem of unitarity (37). This can be seen by writing the propagator of the theory. In the Landau gauge the metric propagator of the theory consist of three kind of terms: there is a massive spin-2 propagation with a pole at mass ; there is a massive scalar propagation with a pole at mass ; and the usual massless propagator for the graviton. The massive spin-2 propagator has five degrees of freedom and is called -mode, while the massive scalar propagation has one degree of freedom and is called ‘Riccion’. The massless graviton has two degrees of freedom. The propagator of the -mode has a negative residue and is responsible for breaking the unitarity of theory (47); (48).

Recently, higher-derivative gravity has been investigated in four dimensions (38); (39). There using the one-loop beta function of the gravitational couplings, it is shown that in a certain domain of coupling parameters space the mass of the spin-2 mode (which has negative norm) runs in such a way so that it is always above the energy scale, as a result the propagator of -mode never witnesses the pole. This has useful consequences: one, there is never enough energy to create this particle, so it never goes on-shell; second when it appears off-shell it doesn’t contribute to the imaginary part of scattering amplitude (Cutkosky cut). In this way the theory remains unitary once the quantum corrections are taken into account, a feature which is absent in classical theory and at tree level. Furthermore, the authors of (38); (39) also found that remains small for all energies and hence in the perturbative domain. It vanishes at some finite energy albeit larger than the Planck energy. Therefore in this context the question of charge renormalization of gauge theories is raised within the perturbation theory to all orders in the Feynman loop expansion. The coupled system of higher-derivative gravity with the gauge fields is also renormalizable to all loops (40); (41). This being free of quadratic divergences evades the criticism raised in (36). Recently gauge fields coupled with higher-derivative gravity has been studied in (42), where it was found that the leading contribution to the gauge coupling beta function in Feynman perturbation theory comes entirely due to quantum gravity effects and is seen to vanish to all loops. In (42) we wrote that the proof holds only on-shell, however the proof is valid even off-shell. Here in this paper we study the coupled system of gauge field with higher-derivative gravity in more detail. We study the unitarity of the coupled system and investigate the vanishing of the leading quantum gravity contribution in the gauge beta function to all loops off-shell.

In section II we discuss the generic structure of the gauge coupling beta function for any gauge field coupled with gravity. In section III, we build up the formalism and do a one-loop computation for the quantum gravity correction to the gauge beta function. Here we show that ’’ term is zero to one-loop in higher derivative gravity, thereby computing the finite terms at one-loop for the abelian gauge fields. In section IV, we study the gauge contribution to the beta function of gravitational couplings and investigate the behavior of running gravitational couplings and unitarity of the coupled system. In section V we use symmetry arguments to study the ’’ term of the gauge beta function to all loops. In section VI, we do one-loop computation to compute the quantum gravity contribution to gauge beta function for arbitrary gravity action. We conclude in section VII, with a discussion and implication of results.

Ii Gauge Beta Function Structure

Generically, in Feynman path-integral study of gauge field coupled with gravity, the beta function of the gauge coupling in perturbation theory is lead by a term called “” to all loops. This is the dominant contribution for small gauge couplings and is universally the same for all gauge theories coupled with gravity, including abelian gauge theories, but depends on parameters present in pure gravity sector (in the presence of matter fields, this term also depends upon the information content of the matter sector), but it is independent of the gauge coupling. Being the dominant term for small gauge coupling, it can potentially overpower the asymptotic freedom of non-abelian gauge couplings, as was also demonstrated in (19); (27); (28). In the following we define the ’’ term more precisely, and show in section V that it vanishes to all loops.

Figure 1: Propagator of gauge field and vertices involving interaction of gauge field (wavy line) with the metric fluctuations (double line). The propagator (first graph) goes like , while the vertices goes like .

The non-abelian gauge field action is given by,

(2)

where is the -gauge coupling, , is the gauge vector potential and are the structure constants. Defining quantum theory using the Feynman path-integral for the coupled action of gravity and gauge field given by eq. (1 and 2), we use background field method to do the gauge fixing. In our case where the gravity action is renormalizable and unitary, such a background gauge fixing guarantees a gauge invariant effective action in dimensional regularization scheme (18); (49); (50); (51); (52). The running of gauge coupling constant satisfies the following generic equation,

(3)

where , the function ’’ is independent of (the gauge coupling), but depends on the other couplings of the action (gravitational and matter sector couplings). It dependence on the gauge group comes implicitly through the dependence of other coupling on the number of generators of the group. The ’’-term on the other hand depends on all the couplings present in the theory and the gauge group. The ‘dots’ in the definition of ’’ and ’’ indicate the dependence on the matter couplings. For small coupling this definition of the beta function is particularly useful as by construction is a regular function of at in Feynman perturbation theory.

Figure 2: One loop contribution to the running of gauge coupling. The Feynman diagrams (a) and (b) represent the contribution to the running of gauge coupling due to self interaction of the gauge fields (only for non-abelian gauge theories). The diagrams (c) and (d) are one-loop quantum gravity contribution to the gauge coupling (for all gauge theories). Here black dots represent the bare vertices, while wavy lines attached to circle with a cross represent external legs.

From the gauge field action given in eq. (2), we can apply the background field formalism to obtain the propagator of the fluctuating gauge field and the various vertices involving interaction of metric fluctuation with both the background gauge field and the gauge field fluctuation (written in appendix B). As this formalism is explicitly gauge invariant in the background field by construction, therefore the gauge coupling and the wave-function renormalization are the same in this approach. This also means that in this picture one would only have to consider the two-point green function of the background field (51); (52). We therefore consider vertices with at most two background gauge field lines. The set of vertices and propagator for the gauge field are depicted in the Fig. 1. Counting the powers of , it is easy to realize that the propagator for the gauge fluctuation field is proportional to , while the background gauge field line carries no power of . All the vertices are proportional to . From the vertices in Fig. 1, we note that any vertex which involve only two gauge field line (either background or fluctuation field) depicts interactions where gravity couples with gauge field purely due to its energy, while any vertex involving more than two gauge field line (either background or fluctuation field) are interaction where there is also charge interactions.

In this formalism at one-loop there are four diagrams that give contribution to the running of gauge coupling: the first two diagrams come because of self interaction between gluons and are only present for non-abelian gauge fields, the other two diagrams give quantum gravity contribution to the running gauge coupling, as shown in Fig. 2. The first two diagrams are only present for non-abelian gauge fields, while the other two diagrams are present for all gauge fields. Studying the dependence of the diagrams we notice that both the one-loop diagrams that give quantum gravity contribution to the running of gauge coupling are proportional to , thereby contributing to ’’ term alone in eq. (3).

Figure 3: Various Feynman diagrams in the perturbation theory of gauge field coupled with gravity. First column contains diagrams contributing to ’’-term, while second column contain diagrams contributing to ’’-term. All diagrams containing 3- or 4-gluon vertex give contributions to ’’-term of the beta function. Matter loops attached to gluon line contributes to ’-term, while the one attached to the graviton line contributes to ’’-term. There are two kinds of ghost: gravitational ghost (which interact with only background and fluctuating metric field) and gauge field ghosts (which interact with background metric, background and fluctuating gauge field line). Gravity ghost contribute to ’’-term, while gauge field ghost contribute to ’’-term.

Now we study the nature of ’’-term to all loops. Here the task is simplified by being in background field formalism where one only has to consider two point correlation function of the background gauge field. We notice that any diagram involving a vertex with three- or four-gluon exchange will only contribute to the ’’-term of the gauge beta function. This is easily verified by counting the powers of . For example an inclusion of 3- or 4-gluon vertex in any diagram contributing to the ’’-term (meaning that it goes like ), will result in an increase in the power of by at least one factor of , meaning such diagram will eventually contribute to ’’-term. This observations can be more clearly checked from the Fig. 3, where appearance of 3- or 4- gluon vertex in a diagram shows that it gives contribution to ’’-term. So effectively one can ignore these kind of diagrams, this is equivalent to ignoring the three and four gluon terms in the expansion of , which is like considering abelian gauge fields. Diagrams arising from these terms will necessarily contribute to ’’ term alone, and will be same that is obtained in gauge field case.

Regarding matter it should be noticed that their interaction with the gauge fields (both background and fluctuation) arise via the kinetic term of the matter fields, where gauge covariant derivative is introduced in order to preserve the gauge invariance of the matter action. Such interaction terms don’t carry any power of . Matter fields also interact with metric fluctuation. Due to non-linearity of the gravity, such interaction terms are infinite in number. Matter field contribute to the gauge beta function through the loops. When matter field loop is attached to the gauge field line, then it contributes to the ’’-term, however when it is attached to the internal graviton line, then it contributes to the ’’-term. For example any diagram contributing to the ’’-term will still contribute to ’’-term when matter loop is attached or inserted in a graviton line. This is because vertices involving interaction of matter with graviton don’t carry any power of . But on other hand when matter loop is inserted/attached to gluon line then the power of increase by one. This is because although the new vertices added don’t carry any power of , but insertion of such loops splits the gluon propagator in two, thereby increasing the power of by one. Such modified diagrams will then contribute to ’’-term. This can be further verified by examining the two-loop graphs given in Fig. 3.

The case of ghost field is a bit different. There are two kind of ghost: gravitational and gauge ghosts. These two ghosts are not independent and mix with each other. This is due to the fact that the gauge field undergoes both deffeomorphism and gauge transformation. This mixing looks problematic but is actually innocuous, as it never contributes to any feynman diagram. This will become more clear later in the section of gauge fixing and ghosts III.1. As a result the gravity and gauge field ghost become independent. Gravity ghosts only interact either with the background metric or with the metric fluctuation field, as a result of which their presence doesn’t alter the power of of the diagram. Gauge field ghost on the other hand interacts with the background metric, background gauge field and the gauge fluctuation field. All these interactions don’t carry any powers of . This means that when ghost loops are inserted in the gauge field line of the diagrams, they tend to increase the power of by one. Thus any diagram which was originally contributing to ’’-term alone, after the inclusion of the gauge field ghost loop will contribute to ’’-term. This can be further verified by examining the two loop graphs involving the ghost loops depicted in Fig. 3.

These arguments tell that gauge field ghosts when present in the diagrams will always contribute to the ’’-term of the gauge beta function. On the other hand in the case of matter fields, when the loop is attached to the graviton line then it contributes to ’’-term, while when it is attached to the gluon line, it contributes to the ’’-term. Focusing just on the diagrams contributing to ’’-term alone, is equivalent to considering parts of matter action which doesn’t depend on the gauge field. This is same as saying that one is considering uncharged fields. Thus to sum up, the kind of terms of the total matter action (gauge field action plus the matter action) that contribute to the ’’-term alone to all loops are: for gauge field action, terms which don’t have 3- or 4-gluon interactions; for the matter action, terms which are independent of the gauge field (same as considering uncharged field). In Fig. 3, although we have demonstrated only up to two loops, but this argument is valid to all loops as discussed in previous paragraphs. However ’’ term is independent of the gauge coupling , but can depend upon the gauge group (number of generators of gauge group) via the gravitational and matter couplings dependence on the gauge group. The universality is a manifestation of the fact that the metric fluctuations interact universally to all gauge fields via its energy.

The formal solution of eq. (3) can be written as,

(4)

It is evident that for small the running of depends more dramatically on the sign of . If is negative then diverges as for large . If is positive, then for large , vanishes faster than as considered in (19); (27); (28). If , then the standard behavior of the running of gauge coupling qualitatively holds. The above equation is valid to any order in the loop expansion. So the qualitative behavior of namely asymptotic freedom can remain unaltered if to all loops. This was examined and studied in (42) and found that to all loops off-shell (mistakenly written on-shell, but proof is valid off-shell), while explicitly showing that it is zero at one-loop off-shell (independent of gravity action, regularization scheme and gauge fixing condition). Here we study the coupled system in more detail addressing the ’’-term to all loops off-shell and will also study the unitarity of the coupled system.

Iii Effective Action

In this section we study the coupling of higher derivative gravity with gauge field, to compute the quantum gravity correction to the running of gauge couplings. To begin with we consider the path-integral of the coupled gravity and gauge action given in eq. (1 and 2) respectively. This is given by,

(5)

where is the higher derivative gravity action given in eq. (1), is the gauge field action given in eq. (2).

We study the deffeomorphism invariant gravity action and gauge invariant gauge field action using background field method (51); (52). This has an advantage as by construction it preserves the background gauge field invariance of the effective action. In this formalism the quantum fields of gravity and gauge theory are decomposed into a background and a fluctuation. Keeping the background fixed, the invariance of the fluctuation field of the full action is broken by constraining the fluctuation fields. This procedure results in the generation of certain auxiliary fields called ghosts. The overall resulting action after applying the constraint in the presence of ghost still possess invariance over the background metric and the background gauge field, thereby producing a background gauge invariant effective action.

Writing the quantum metric as , where is some arbitrary fixed background and is the metric fluctuation, we expand the full action in powers of . As the background metric is fixed, therefore the path-integration measure over the quantum metric gets replaced by the measure over the fluctuation field . Integrating over the fluctuation field completely also implies that they will only appear as virtual particles inside the loop and never as external legs. Effective action computed after the path-integration over the fluctuation field being completely deffeomorphism invariant in the background metric, allows one to choose a particular background metric in order to simplify the computation. In particular choosing (while still keeping generic), allows one to use the methodology and formalism of the flat spacetime quantum field theory. In this way of working it is possible to attribute particle notions to the fields and , where will behave as a virtual particle, while will act as a external particle corresponding to , exactly as is the case in flat spacetime QFT for usual scalar or spinor fields. Attributing particle notions to the fluctuation field and allows one to also consider scattering matrix amplitudes. Looking from a different angle it is quickly realized that expanding the gravity action around a flat spacetime and calling the perturbations around it to be , one obtains a highly nonlinear gauge theory in the field . Treating this gauge theory along the lines of background field method, where now the quantum field is written as , allows one to quickly see that this is a flat spacetime QFT with as an external leg and as the internal line. Integrating over the fluctuation field , gives one an effective action as a functional of field, which is the effective action for an arbitrary background expanded about flat spacetime.

In this language one can setup Feynman perturbation theory by expanding the original action of gravity and matter in powers of and . This series of terms will contain propagator of the field and various vertices involving interactions of external field with internal field line . This is the expanded bare action of the theory and carries infinite number of vertices due to nonlinear nature of gravitational field. However, as we still have background gauge invariance, therefore one only needs to do the expansion of the bare action only up to certain finite order in . This means that for studying the behavior of term linear in Ricci scalar curvature under quantum corrections, it is sufficient to expand the bare action up to linear order in , and for studying the behavior of terms quadratic in curvature it is sufficient to expand the bare action up to quadratic in . Such expansion provide all the vertices that will be relevant for study of these kind of terms. This is the privilege of background field method.

For a one-loop computation on the other hand it is sufficient to expand the bare action up to second order in . This although finite in the order of , but is infinite in order of . The restriction on the order of comes from the requirement of the kind of term that is to be investigated in the effective action. For example probing the issues related to cosmological constant, it is sufficient to put i.e. truncating the series at zeroth order in . Similarly exploring the Ricci scalar term of the effective action demands to consider term up to linear in and so forth. Investigating quantum gravity contribution to charge renormalization will demand accordingly to consider terms up to zeroth order in , i.e. putting .

iii.1 Gauge Fixing and Ghosts

The path integration over the gauge and gravitational fields is not well defined. This is due to fact that for certain field configurations the integrand of the functional integral is unity, thereby implying a diverging path integral. This is because gauge and deffeomorphism transformation allows to choose field configurations for which the action is zero. Besides this, the measure of the functional integral over the gauge field is also ill defined, due to over counting of gauge orbits. Both these problems are evaded by constraining both the gauge and metric field thereby breaking the gauge and deffeomorphism invariance of the path integral. This procedure of breaking the invariance of the path integral is echoed by giving rise to ghosts, which is elegantly studied through the methodology of Faddeev-Popov (54).

However, in this style of obtaining the effective action it is difficult to see the invariance, once path integration is performed. This is overcome by using background field method, which explicitly assures the invariance of effective action in the background fields. In this picture the path integral is over the fluctuation fields and . Gauge fixing of the path integral is done in such a way so that it breaks the invariance over the fluctuation fields, while still preserving the residual invariance over the background fields. In the following we will discuss in detail how the background gauge fixing is done for this coupled system of gauge and gravity fields.

The coupled action of gauge and gravity field being deffeomorphism invariant in the field variables implies, that for an arbitrary vector field , the action should be invariant under the following transformation of the metric field variable,

(6)

where is the Lie derivative of the quantum metric along the vector field . This metric field when decomposed in to a background () and fluctuation (), allows one to obtain the transformation of the tensor field while keeping the background fixed. This will imply the following transformation of .

(7)

where is the covariant derivative whose connection is constructed using the background metric. This is the full transformation of the tensor field . Ignoring terms which are linear in allows one to investigate only one-loop effects, while they are kept in the complete analysis involving higher-loop studies.

The gauge field action given in eq. (2) on the other hand has two kind of invariances: deffeomorphism invariance and local gauge invariance. For an arbitrary vector field and a color vector field , gauge field action is invariant under the following transformation of the gauge field .

(8)

Decomposing the gauge field in to a background and a fluctuation , implies the following transformation of the fluctuating field while keeping the background fixed.

(9)

where the first four terms on the rhs of the equality denote the transformation of the fluctuation field under the deffeomorphism, while the last three terms denote the transformation under the gauge group (for abelian gauge fields, the terms proportional to structure constants will be absent). This is the complete transformation of the fluctuation field , under which the gauge field action will be invariant. This invariance of the action under the transformation of the fluctuation field is broken by choosing an appropriate gauge fixing condition for breaking the deffoemorphism and gauge invariance of the action. Choosing a gauge fixing action which is invariant under the background gauge transformation but breaks the invariance under the transformation of the fluctuation field, further assures the preservation of background gauge invariance of the effective action.

The gauge fixing action chosen for fixing the invariance under the transformation of the metric fluctuation field is given by,

(10)

where and are gauge parameters, while is either a constant or some differential operator depending upon the gravity theory under consideration. In the case of higher-derivative gravity like the one described by action in eq. (1), we consider higher-derivative type gauge fixing by taking , where . Similarly the fluctuating gauge field is constrained by choosing the following gauge fixing action,

(11)

where is a gauge parameter. Here D is the covariant derivative constructed with background metric and the background gauge field. Its action on the fluctuation gauge field is given by,

(12)

The ghost action for these gauge fixing conditions can be obtained by using the Faddeev-Popov trick (54). Calling in general the gauge fixing condition for gravitational field to be , while the gauge fixing condition for the fluctuating gauge field to be (in the present case we have and ), we introduce them in the path integral of the fluctuating fields by multiplying it with unity in the following form.

(13)

where and are the gauge transformed and respectively. If did not contain any derivatives, then this determinant would be trivial, however it will not be so if contains derivative operators, which is the case in higher-derivative gravity. The original path integral (without gauge fixing) being invariant under transformation eq. (7 and 9) of the fluctuation fields and , implies that the integration variable and can be changed to transformed fields and . This transformation don’t give rise to any non-trivial jacobians in the path-integral measure of the gauge and gravity field. Writing the measure over and , as measure over the transformation variables and , introduces a non-trivial jacobian in the path integral. This is the Faddeev-Popov determinant and is worked out as follows.

(14)

This means that the measure transforms as,

(15)

In the background field formalism, this jacobian consist of background covariant derivative, background and fluctuation fields. As the determinant is independent of the transformation parameters and , therefore it can be taken out of the functional integral over and . Changing the integration variable from and to and respectively, and ignoring the infinite constant generated by integrating over and , gives us the gauge fixed path integral including the determinants.

These functional determinants can be exponentiated by making use of appropriate auxiliary fields. Writing the functional determinant as a product of two determinants , allows us to combine the former with the Faddeev-Popov determinant in eq. (15), which is then exponentiated by making use of anti-commuting auxiliary fields, while the later determinant is exponentiated by making use of commuting auxiliary fields. The former auxiliary fields are known as Feddeev-Popov ghosts, while the auxiliary field in the later case is known as Neilsen-Kallosh ghosts (55); (56). The path integral of the full ghost sector is given by,

(16)

where and are anti-commuting fields arising from the gauge fixing in the gravitational sector, while and are anti-commuting fields arising from the gauge fixing in the gauge field sector and is the commuting ghost arising due to fact that contains derivatives. It is crucial to note here that in the coupled system of gauge field with gravity, there will always be mixing between the gravitational and gauge ghosts. The mixing term will arise, if we choose a gauge fixing condition for the gravitational field such that it also contain terms involving gauge fluctuation field . In the absence of such term, this mixing term will be always zero. However, the mixing term will be always non-zero, as the gauge fluctuation field undergoes both deffeomorphism and gauge transformation.

In the case when and are given as in eq. (10) and (11) respectively, the Faddev-Popov ghost action is given by,

(17)

where,

(18)
(19)
(20)

It should be noted that for the kind of gauge fixing condition considered here in eq. (10) and (11), only the mixing term is non-zero, while the other mixing term is zero. This means that there is no interaction between gravitational ghost and gauge ghost . This has an important consequence. This will imply that in this kind of gauge fixing, there will not be any closed loop Feynman diagram with one line in the loop to be gravitational ghost, while the other gauge ghost. Therefore, the presence of mixing term becomes completely innocuous. Had the other mixing term in the Faddeev-Popov determinant been nonzero, then we will have interaction term in the ghost action involving and . In such cases there will graphs involving loop with one gravitational ghost and one gauge ghost.

iii.2 Gravitational Field Propagator

The propagator of the fluctuating quantum field is given is obtained by expanding the bare gravitational action eq. (1) up to second order in around flat spacetime background. This is given by,

(21)

By making use of the complete set of orthogonal projectors in flat space for the symmetric rank-2 tensor field (see appendix A), this can be re-written in momentum space in term of projectors,

(22)

The gauge fixing action eq. (10), with flat spacetime as the background can also be written in the language of projectors as follows,

(23)

Writing the gauge fixing action in terms of projectors is advantageous. It shows that it only give contribution to the longitudinal modes of the field, and not to the transverse spin-2 projector. This is expected as the spin-2 component is gauge invariant under the diffeomorphism transformation of the fluctuation field . Also it should be noticed that the components corresponding to each of the projector is of type. This is due to fact that contains derivatives. Had we chosen to be proportional to constant, then we would have got coefficients corresponding to each of projector in gauge fixing action to be of type.

When gauge parameters take arbitrary values then this will over fix the gauge redundancies, and there will be longitudinal terms in the gravity propagator. Such terms may not give any contribution to one-loop diagrams (except for quadratic divergent ones) but at higher loops, contribution from them is unavoidable. However the -matrix constructed will be gauge-invariant. Its only in Landau gauge (, and ) that we don’t have any longitudinal modes in the propagator (and don’t have to worry about ghosts) describing physical degrees of freedom correctly. The gravity propagator in this gauge is given by,

(24)

where is the inverse propagator for the field including the gauge fixing and is symmetric in and . In co-ordinate space this can be written as,

(25)

The graviton propagator written in eq. (24) can be analyzed further by doing partial fraction decomposition. This enumerates the various propagating modes of the theory. This is given by,

(26)

We note that the first term is the usual massless graviton with two degrees of freedom, the second term is a massive scalar with mass and has one degree of freedom (we call it ‘Riccion’), while the third term is a massive spin-2 mode of mass with five degrees of freedom (we call it -mode). This mode has negative residue and breaks unitarity at tree level. However when quantum corrections are taken into account, it is shown that the mass of this mode runs in such a way so that it is always above the running energy scale (38); (39).

Having obtained the propagator for the gravitational field, we move on to study the gauge field action and its coupling with the gravitational field . In the next subsection we will build up the formalism to compute the quantum gravity contribution to the gauge field action.

iii.3 Formalism

Under the decomposition of gauge field in to background and fluctuation as , the expansion of field strength tensor is given by, , where and .

To compute the one-loop quantum gravity contribution to the running of gauge coupling, it is sufficient to expand the gauge field action up to quadratic in and around the flat metric. This will give rise to various vertices that will be relevant for the one-loop computation and the gauge field propagator. This second variation is given by,

(27)

The gauge fixing action for the gauge field given in eq. (11) is written for arbitrary background metric. This is crucial so as to have diffeomorphism invariance for the effective action obtained by integrating out the quantum fluctuations. However for investigating issues related to charge renormalization (divergent part of the effective action), it is sufficient to consider the flat background. This is justified further by writing the arbitrary background metric as , and expanding the gauge fixing action in powers of . The leading term of this series is the same as is obtained when background is flat, and is the only relevant term required for studying divergent contributions which are zeroth order in curvature of background metric. This gauge fixing action when added to the second variation of the gauge field action, gives the gauge fixed propagator and the vertices, which have been written in the appendix B.

Having obtained the expansion of the bare action of gravity and gauge field up to second order in fields (which is all that is needed in the one-loop computation), we start by considering the path-integral over the fluctuation fields. The zeroth order term being independent of the fluctuation fields, can be taken out of the path-integral. The linear term can be removed by redefining the fluctuation fields, which only give rise to a trivial Jacobian from the functional measure. The quadratic piece can now be tackled easily by clubbing the two fields to form a multiplet . This allows one to express the second variation in a more compact notation. The residual path-integral along with the source terms has the following form,

(28)

where is the source multiplet which couples with the fluctuation fields , and

(29)

From the generating functional we define the one-particle-irreducible (1PI) generating functional , where and is the expectation value of field. The 1PI generating functional is also the effective action. Performing the integration one obtains,

(30)

For zero source, the above equation gives the expression for 1-loop effective action ,

(31)

where the first two terms correspond to tree level diagrams, while the last term contains 1-loop quantum corrections. Our task in the following is to compute the divergent terms present in which are proportional to . There are several ways of obtaining it, but here we will do it by expanding the in a power series, and studying terms which will contain divergent contribution proportional to . In our case writing (where the first term contains only the propagator of the theory and the second terms contains the interactions), we have,