Scattering of fermions in the Yukawa theory coupled to Unimodular Gravity

We compute the lowest order gravitational UV divergent radiative corrections to the S matrix element of the $fermion + fermion\rightarrow fermion + fermion$ scattering process in the massive Yukawa theory, coupled either to Unimodular Gravity or to General Relativity. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contribution in Dimensional Regularization. This is a nontrivial result, since in the classical action of Unimodular Gravity coupled to the Yukawa theory, the graviton field does not couple neither to the mass operator nor to the Yukawa operator. This is unlike the General Relativity case. The agreement found points in the direction that Unimodular Gravity and General Relativity give rise to the same quantum theory when coupled to matter, as long as the Cosmological Constant vanishes. Along the way we have come across another unexpected cancellation of UV divergences for both Unimodular Gravity and General Relativity, resulting in the UV finiteness of the one-loop and $\kappa g^2$ order of the vertex involving two fermions and one graviton only.


Introduction
When quantum General Relativity is formulated as an effective quantum field theory there arises a huge disparity between the actual value of the Cosmological Constant and its theoretically expected value. Unimodular gravity supplies [1,2,3] a Wilsonian solution to this problem since the vacuum energy does not gravitate: a breach of the equivalence principle is afoot -see [4], for a nice review. In this regard, it is to be stressed that when matter is coupled to Unimodular Gravity, no term occurs in the action which couples the classical potential to the graviton field.
As classical theories, whatever the value of the Cosmological Constant, Unimodular Gravity and General Relativity are equivalent [5,6,7,8] -at least as far as the equations of motion imply [9,10,11]. Of course, General Relativity and Unimodular Gravity are not equivalent as effective quantum field theories for a nonvanishing Cosmological Constant. And yet, whether that classical equivalence survives the quantization process is still an open issue, in the case of vanishing Cosmological Constant or, to be more in harmony with Nature, for physical phenomena where the Cosmological Constant can be effectively set to zero. Several papers have been published where this quantum equivalence has been discussed: see Refs. [12,13,14,15,16,17,18,19,20,21]. However, only in three of them [19,20,22] the coupling of Unimodular Gravity with matter has been studied. The fact that in Unimodular Gravity, unlike for General Relativity, the graviton field does not couple in the classical action to operators such as such λφ 4 , the mass terms, or the Yukawa vertex gψψφ , makes one strongly doubt that the equivalence in question holds at the quantum level. Indeed, the coupling of the graviton field with the terms we have just mentioned gives rise to loop contributions which are UV divergent. Let us stress that the λφ 4 , the mass terms and the Yukawa vertex are key ingredients of the Higgs sector of the Standard Model, so to ascertain their effects when coupled to quantum gravity is not a mere academic issue.
It turns out -see Ref. [20]-that in the standard multiplicative MS scheme of dimensional regularization, the gravitational contributions to the beta functions of the quartic, λ , and Yukawa, g , couplings are not the same for Unimodular Gravity as for General Relativity. As discussed in Ref. [20], this does not necessarily imply that the UV divergent behaviour of the S matrix elements differ from one theory to the other, for the aforementioned gravitational corrections have no intrinsic physical meaning. Notice that if the discrepancy between the gravitational corrections to the beta functions we have just mentioned would necessarily imply unequal UV divergent behaviour of the S matrix elements of Unimodular Gravity and General Relativity coupled to matter, then, one would be entitled to conclude that Unimodular Gravity and General Relativity are not equivalent at the quantum level, at least when interacting with matter. However, what the discrepancy does say is that it is far from clear that Unimodular Gravity and General Relativity agree at the quantum level as they do at the classical level. It is then a pressing matter to check whether the UV divergent behaviour of the gravitational corrections to the S matrix elements of massive scalar and fermion particles for Unimodular Gravity agrees with those coming from General Relativity.
In Ref. [22], the lowest order UV divergent gravitational radiative corrections to the S matrix of the scattering of two scalar particles going into two scalar particles was computed, both in Unimodular Gravity and General Relativity coupled to the massive λφ 4 theory. The outcome was that the total contribution in the Unimodular Gravity case is the same as in the General Relativity case, although the contribution coming from each individual Feynman diagram is not the same for Unimodular Gravity as for General Relativity.
The purpose of this paper is to compute the one-loop and g 2 κ 2 order -ie, the lowest order radiative contributions-UV divergent contributions to the S matrix element of the scattering process f ermion + f ermion → f ermion + f ermion in the massive Yukawa theory coupled either to General Relativity or to Unimodular Gravity, when the Cosmological Constant is set to zero. We shall show that such UV divergent behaviour is the same in Unimodular Gravity case as in the General Relativity instance, notwithstanding the fact that this equivalence does not hold Feynman diagram by Feynman diagram. This result is not trivial since Unimodular Gravity does not couple neither to the mass terms nor the Yukawa vertex gψψφ in the classical action, besides the fact that they have different gauge symmetries. Our result provides further evidence that Unimodular Gravity and General Relativity are same quantum theory for zero Cosmological Constant.
The lay out of this paper is as follows. In section 2 we display the relevant formulae that are needed to carry out the computations in Section 3. Section 3 is devoted to the computation of the one-loop and g 2 κ 2 order, UV divergent contributions to the S matrix element of the scattering f ermion + f ermion → f ermion + f ermion . Finally, we have a section to discuss the results presented in the paper.

Yukawa theory coupled to Gravity
In this section we shall just display the classical actions of the Yukawa theory coupled to a gravitational field as described by General Relativity and Unimodular Gravity. We shall also display the graviton free propagator in each case.

Yukawa theory coupled to General Relativity
Let e µ a be the vielbein, e µ a e ν b g µν = η ab , for the Lorentzian metric g µν , η ab = (+, −, −, −) . Let γ a denote the Dirac matrices: [γ a , γ b ] = η ab . The torsion-free spin connection ω µ is defined, where µ is the covariant derivative as given by the Christoffel symbols. Let ψ , denote a spinor field in spacetime, its covariant derivative being given by The classical action of General relativity coupled to the Yukawa theory reads (2.1) where κ 2 = 32πG and R[g µν ] is the scalar curvature for the metric g µν .
Up to first order in κ , S (2.5) In (2.4), contractions are carried out with η µν .

Yukawa theory coupled to Unimodular Gravity
Letĝ µν denote the Unimodular -ie, with determinant equal to -1-metric of the n dimensional spacetime manifold. We shall assume the mostly minus signature for the metric. Then, the classical action of the Yukawa theory coupled to Unimodular Gravity reads where κ 2 = 32πG , R[ĝ µν ] is the scalar curvature for the unimodular metric,ê µ a is the vielbein, e µ aê ν bĝ µν = η ab for the metricĝ µν , η ab = (+, −, −, −) , γ a denote the Dirac matrices: [γ a , γ b ] = η ab andD µ = ∂ µ +ω µ is the Dirac operator for the torsion-free spin connection µ is the covariant derivative as given by the Christoffel symbols ofĝ µν To quantize the theory we shall follow Refs. [12,16,23] and introduce an unconstrained fictitious metric,ĝ µν , thusĝ where g is the determinant of g µν . Next, we shall express the action in (2.6) in terms of the fictitious metricĝ µν by using (2.7), then, we shall split g µν as in (2.2) and, finally, we shall define the path integral by integration over h µν and the matter fields, once an appropriate BRS invariant action has been constructed.
Since our computations will always involve the matter fieldsψ , ψ and φ , and will be of order κ 2 , we shall only need -as will become clear in the sequel-the free propagator of h µν , h µν (k)h ρσ (−k) , and the expansion of S (UG) Yukawa up to first order in κ . Using the gauge-fixing procedure discussed in Ref. [16], one obtains and T µν is given in (2.5). Again, the contractions in (2.9) are carried out with the help of η µν .
Let us point out that the term in T µν which is proportional to η µν does not actually contribute to T µνĥ µν , sinceĥ µν is traceless. In terms of Feynman diagrams, this can be stated by saying that the η µν part of T µν will never contribute to a given diagram since it will always be contracted with a free propagator involvingĥ µν . This is not what happens in the General Relativity case and makes the agreement between General Relativity coupled to the Yukawa theory and Unimodular Gravity coupled to the latter a non-trivial issue already at one-loop.
We shall use the correlation function ĥ µν (k)ĥ ρσ (−k) , which can be easily obtained from (2.8). It reads (2.10) 3 The f ermion+f ermion → f ermion+f ermion scattering at one-loop and g 2 κ 2 order The purpose of this section is to work out the one-loop, and g 2 κ 2 order, UV divergent contribution, coming from General Relativity and Unimodular Gravity, to the dimensionally regularized S matrix element of the f ermion + f ermion → f ermion + f ermion scattering process.

The General Relativity case
To work out the UV divergent contribution in question to f ermion + f ermion → f ermion + f ermion , we shall need the UV divergent contributions coming from the 1PI diagramas in Figures 1 to 4. These contributions read in the General Relativity case    To define the S matrix elements it is necessary to express the bare masses M and m in terms of the corresponding physical masses -ie, the poles of the propagators-M φ and m ψ . This is accomplished by using the following formulae where the superscript G stand for gravitational -those from General Relativity in the current subsection-contributions, given in (3.1), and the superscript N G denote the corresponding contributions in absence of gravity, whose actual vaalues are not needed in this paper.
To obtain the S matrix elements from the Green functions of the fields, it is also convenient to introduce renormalized fields φ R and ψ R , so that the Laurent expansion of their propagators, φ R (p)φ R (−p) and ψ R (p)ψ R (p) , around the mass shell read The fields ψ R and φ R are obtained from the bare fields, ψ and φ -the fields in (2.1)-by introducing the following wave function renormalizations The reader should bear in mind that in the defining Z m , Z ψ , Z M and Z φ in terms of iΓ ψψ (p, κ) and iΓ φφ (p; κ) , we have taken into account that we are working at the one-loop level.
Considering (3.1), (3.3) and the definitions in (3.2), one obtains where n = 4 + 2 , n being the dimension of spacetime in Dimensional Regularization. The bits of δZ ψ and δZ φ in (3.4) that are independent of κ are the usual ones that can be found in textbooks.
The wave function renormalizations in (3.3) and (3.4) give rise to a vertex counterterm diagrammatically represented by the diagram in Figure 5, whose value reads Using the value of iΓ (GR) ψψφ (p 1 , p 2 ; κ) , displayed in (3.1), and the counterterm in (3.5), one obtains the following expression for the UV divergent General Relativity contribution to the S matrix coming from the sum of all the diagrams in Figure 6: where Q = p 1 − p 2 . Bear in mind that the blob with slanted lines in diagrams of Figure 6 represents the sum of all the diagrams in Figure 3, i.e. iΓ (GR) ψψφ (p 1 , p 2 ; κ) . Figure 6: S matrix contributions. Figure 7: Crossing S matrix contributions.
The crossing diagrams in Figure 7 yields the following UV divergent General Relativity contribution to the S matrix: Let us denote by Box8a, Box8b and Box8c the UV divergent General Relativity UV di-vergent contributions to the S matrix coming from the diagrams a), b) an c) in Figure 8, respectively. A lengthy computation yields at the following simple expressions Figure 8: Box diagrams. Figure 9: Box diagrams (crossing).
Let us introduce now the counterterm vertex in Figure 10, which comes from renormalization produced by the constants Z ψ and Z m in Minkowski spacetime applied to the energymomentum tensor in (2.5): Figure 10: Counterterm g 2 κ .
It is clear -the gravitational field here is a mere spectator-that the counterterm vertex represented by the diagram in Figure 10 has the same value for Unimodular Gravity as for General Relativity, i.e. is given by the expressions in (3.11) upon replacing h µν withĥ µν . Next, notice that iΓ  Thus, the one loop and g 2 κ order correction to theψψĥ µν is UV finite.
Using the previous result, one concludes that the sum of the diagrams in Figure 11 contains no UV divergent pieces in the Unimodular Gravity case either. Same result for the sum of the diagrams in Figure 12.
In summary, due to the UV cancellation we have just discussed, only the sum of the diagrams in Figures 6, 7, 8 and 9 gives, in the Unimodular Gravity case, a UV divergent contribution to the S matrix element of the f ermion + f ermion → f ermion + f ermion scattering at one-loop and g 2 κ 2 order. Full agreement between Unimodular Gravity and General Relativity has been thus reached.

Summary and discussion
In this paper we have shown that, at one-loop and g 2 κ 2 order, the UV divergent contribution to the S matrix element of the f ermion + f ermion → f ermion + f ermion scattering process in the Yukawa theory coupled to Unimodular Gravity, is same as the corresponding S matrix element when Unimodular Gravity is replaced with General Relativity -see (3.10) and (3.18) and recall that the sum of the diagrams in Figures 11 and 12 carries no UV divergence. We should point out that the agreement that we have just mentioned does not hold for each individual Feynman diagram -as can be seen by comparing (3.6) with (3.14), for instance-but it unfolds upon adding the contributions coming from classes of the Feynman diagrams -see (3.10) and (3.18), which yields a result independent of the gauge parameter. Of course, the gauge symmetries of Unimodular Gravity are not the same -the gauge symmetry of Unimodular Gravity being a constrained one-as those of General Relativity, and, thus, agreement between non gauge invariant objects computed in both theories is not to be expected and it does not occur in general -see (3.6) with (3.14). Hence, that agreement is reached when gauge invariant objects are computed is a far from trivial result. This non triviality is further strengthen by the fact that in Unimodular Gravity the graviton field does not couple neither to the mass terms nor the Yukawa operator in the classical action, whereas the graviton field does couple to those operators in the General Relativity case. This coupling between the graviton field and the mass terms and the Yukawa vertex explains partially the discrepancy between iΓ (GR) ψψφ (p 1 , p 2 ; κ) and iΓ (U G) ψψφ (p 1 , p 2 ; κ) in (3.1) and (3.13). Indeed, let us give just one example: diagram d) in Figure  3 gives a non-vanishing contribution in General Relativity, which reads −i 16π 2 κ 2 g − whereas it yields a vanishing contribution in the Unimodular Gravity case, for it involves the coupling of the graviton field to the Yukawa vertex.
We have also witnessed -see (3.12) and (3.19)-a surprising cancellation of all the UV divergences contributing to the fermion-fermion-graviton vertex at one loop and g 2 κ order. This cancellation to be added to the list of unexpected UV cancellations when gravity is at work at the quantum level.
A final remark, the results presented in this paper explicitly shows that the beta function of the Yukawa coupling as computed in [26] cannot be used to draw any physically meaningful conclusion, a fact already discussed in [20]. See also [22], for a similar analysis in the λφ 4 theory.
To carry out some of the computations presented in this paper, we have used Mathematica's xAct [27] and Form [28], independently.