Dimensional regularization for N=1 susy sigma models and the worldline formalism

We generalize the worldline formalism to include spin 1/2 fields coupled to gravity. To this purpose we first extend dimensional regularization to supersymmetric nonlinear sigma models in one dimension. We consider a finite propagation time and find that dimensional regularization is a manifestly supersymmetric regularization scheme, since the classically supersymmetric action does not need any counterterm to preserve worldline supersymmetry. We apply this regularization scheme to the worldline description of Dirac fermions coupled to gravity. We first compute the trace anomaly of a Dirac fermion in 4 dimensions, providing an additional check on the regularization with finite propagation time. Then we come to the main topic and consider the one-loop effective action for a Dirac field in a gravitational background. We describe how to represent this effective action as a worldline path integral and compute explicitly the one- and two-point correlation functions, i.e. the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. These results are presented for the general case of a massive fermion. It is interesting to note that in the worldline formalism the coupling to gravity can be described entirely in terms of the metric, avoiding the introduction of a vielbein. Consequently, the fermion--graviton vertices are always linear in the graviton, just like the standard coupling of fermions to gauge fields.


Introduction
One dimensional supersymmetric nonlinear sigma models are useful to describe in first quantization the propagation of fermionic particles in a curved background. In fact, it is wellknown that N = 1 supersymmetric sigma models describe the worldline dynamics of a spinning particle [1]. Mastering the path integral quantization of such models provides a useful tool for treating spin 1/2 particles coupled to gravity. The purpose of this paper is twofold. We first extend dimensional regularization to supersymmetric sigma models with finite propagation (proper) time. Then, with this regularization scheme at hand, we generalize the worldline formalism to include spin 1/2 fields coupled to gravity. This extends the scalar particle case treated in [2]. The resulting Feynman rules are simpler than the standard ones obtained from the second quantized action. In particular, the fermion-graviton vertices can always be taken linear in the graviton field, a fact which seems to point once more to unexpected perturbative relations between gravity and gauge theories, as reviewed in [3].
Path integrals for supersymmetric sigma models in one dimensions were originally used for deriving formulas for index theorems and chiral anomalies [4,5,6]. However, for obtaining those results the details of how to properly define and regulate the path integrals at higher loops were not necessary. Due to the worldline supersymmetry the chiral anomalies are seen as a topological quantity, the Witten index [7], which is independent of β, the propagation time in the sigma model. Thus a semiclassical approximation (which consists in calculating a few determinants) already gives the complete results. The quantum mechanical calculation of chiral anomalies can be extended to trace anomalies [8,9]. However, in the latter case the details of how to define the path integral is essential since one-loop (in target space) trace anomalies correspond to higher-loop calculations on the worldline, namely the one-loop trace anomaly in D dimensions is given by a D 2 + 1 loop calculation on the worldline. Several regularization schemes have been developed for this purpose: mode regularization (MR) [8,9,10], time slicing (TS) [11,12], and dimensional regularization (DR) [13,14]. The DR regularization was developed after the results of [15] which dealt with nonlinear sigma model in the infinite propagation time limit 4 . The first objective of this paper is to extend dimensional regularization to include fermionic fields on the worldline and treat supersymmetric nonlinear sigma models. Worldline fermions coupled to gravity give rise to new (superficially) divergent Feynman diagrams, other than those associated to the coupling of gravity with the bosonic coordinates. Hence, one may a priori expect additional counterterms to arise. In fact, in time slicing, the inclusion of the fermionic fields brings in additional non-covariant counterterms of order β 2 : they are proportional to g µν Γ λ µρ Γ ρ νλ if one uses fermions with curved target space indices, or g µν ω µ ab ω νab if one uses fermions with flat space indices [12]. Note that such counterterms only arise at two loops, and thus they do not affect the calculation of the chiral anomalies, but should be included if one wants to check with TS that there are no higher order corrections in β [17]. We are going to show that in dimensional regularization no extra counterterms arise. This implies that dimensional regularization manifestly preserves supersymmetry. In fact the bosonic part produces a coupling to the scalar curvature with the precise coefficient required by supersymmetry. We describe how to use both flat target space indices and curved ones, for the fermionic fields. Using curved indices will bring in a new set of bosonic "ghost" fields, in the same fashion of [8,9]. Having at hand a simple and reliable regularization scheme for supersymmetric sigma models, we turn to the worldline formalism. As a warm up, we first compute the trace anomaly for a Dirac fermion in 4 dimensions. We obtain the expected result, providing a further test on our application of the DR scheme. Then we come to the core of our paper: the generalization of the worldline formalism to include spin 1/2 particles coupled to gravity. Many simplifications are known to occur in the worldline path integral formulation of quantum field theory, which for this very reason provides an efficient and alternative method for computing Feynman diagrams. This method has quite a long history, rooted in [18]. Later it was developed further by viewing it as the particle limit of string theory [19], and then discussed directly as the first quantization of point particles [20,21] (see [22] for a review and a list of references). The inclusion of background gravity was presented in [2] for the case of a scalar particle. Results obtained using string inspired rules with gravity were presented in [23,24,3].
Here we consider the case of the one-loop effective action for a Dirac fermion in a gravitational background. We describe how to represent it as a worldline path integral. We compute explicitly the one-and two-point correlation functions, i.e. the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. These results are presented for the general case of a massive fermion. In our calculations we use the DR scheme constructed in the previous sections. The other known scheme explicitly developed to include worldline fermions (time slicing [12]) can be used as well, but lack of manifest covariance makes its use more complicated. It is interesting to note that in the worldline formalism the coupling to gravity can be described entirely in terms of the metric, avoiding the introduction of a vielbein. The fermion-graviton vertices are always linear in the metric field, just like the standard coupling of fermions to gauge fields are linear in the gauge potential. This fact seems to point once more to the unexpected perturbative relations between gravity and gauge theories encoded in the so-called KLT relations [25], as reviewed in [3].
The paper is organized as follows. In section 2 we introduce dimensional regularization applied to the worldline Majorana fermions and to supersymmetric sigma models. We mainly consider antiperiodic boundary conditions (which break supersymmetry), but also briefly discuss periodic boundary conditions. In section 3 we apply DR to compute the trace anomaly of a Dirac field in 4 dimension with quantum mechanics. Then in section 4 we describe the worldline formalism with Dirac fields coupled to gravity and compute explicitly the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. This is the main section, and the reader uninterested in the details of the DR regularization scheme may jump directly to it. Section 5 contains our conclusions. Conventions and useful formulas are collected in the appendix. We work with an euclidean time both on the worldline and in target space. The latter is assumed to have even dimensions D.

Dimensional regularization with fermions
In this section we describe the dimensional regularization of fermionic path integrals obtained by extending the method presented in [13] for bosonic models. We shall discuss explicitly path integrals for Majorana fermions on a circle with antiperiodic boundary conditions (ABC), as these are the only boundary conditions that will be directly needed in the applications to trace anomalies and effective action calculations. Our strategy will be as follows: we first set up the rules of dimensional regularization for fermions following [13], then we require that a two-loop computation with DR reproduces known results, and precisely those obtained by a path integral with time slicing [12] (or equivalently by heat kernel methods [26]). This requirement plays the role of a standard (in QFT) renormalization condition, and fixes once for all the DR two-loop counterterm due to fermions. Since counterterms are due to ultraviolet effects, the infrared vacuum structure and the related boundary conditions on the fields should not matter in their evaluation. Therefore one expects that the same counterterm should apply to fermionic path integral with periodic boundary conditions (PBC) as well. No higher-loop contributions to the counterterm are expected as the model is super-renormalizable, just as in the purely bosonic case.
Let us consider the path integral quantization of the N = 1 supersymmetric model written in terms of Majorana fermions with flat target space indices where as usual we have scaled the propagation time β out of the action. The propagation time β will be considered as the expansion parameter for a perturbative evaluation (i.e. the loop counting parameter). In the action we have included: i) the bosonic a µ and fermionic b µ , c µ ghost fields which exponentiate the nontrivial path integral measure, ii) the counterterm V CT which arises in the chosen regularization scheme from the bosonic sector and which is fixed in order to produce a quantum hamiltonian without nonminimal coupling to the scalar curvature R, iii) the additional counterterm V ′ CT which may arise from the fermionic sector, and iv) the potential V = 1 8 R which is required to have a supersymmetric quantum hamiltonian as given by the square of the supersymmetry charge.
The action is classically supersymmetric if all the potential terms multiplied by β 2 are set to zero (the ghosts can be trivially eliminated by using their algebraic equations of motion).
where the trace on the left-hand side is only over the Dirac matrices, and wherê is the supersymmetric Hamiltonian of the N = 1 model (one can normalize the supersymmetric charge asQ = i √ 2 / ∇, so thatĤ =Q 2 ). Note that there is an explicit coupling to the scalar curvature in (4), thus one needs to use a potential V = 1 8 R in the action together with the time slicing counterterms V T S = − 1 8 R + 1 8 g µν Γ λ µρ Γ ρ νλ and V ′ T S = 1 16 g µν ω µ ab ω νab (see [12]; later on we will derive once more this value of V ′ T S as well). Our conventions for the curvature tensors can be found in section A.1 of the appendix. Now we want to reproduce eq. (3) in dimensional regularization with a path integral over Majorana fermions. This will unambiguously fix the additional counterterm V ′ DR due to the fermions. Note that in dimensional regularization the potential V = 1 8 R cancels exactly with the counterterm V DR = − 1 8 R coming from the bosons [13].
We focus directly on the regularization of the Feynman graphs arising in perturbation theory. To recognize how to dimensionally continue the various Feynman graphs we extend the action in (2) from 1 to d + 1 dimensions as follows where Ω = I × R d is the region of integration containing the finite interval I = [0, 1], γ α are the gamma matrices in d + 1 dimensions satisfying {γ α , γ β } = 2δ αβ , and t α ≡ (τ, t) with α = 0, 1, . . . , d and with a bold face indicating vectors in the extra d dimensions.
Here we assume that we can first continue to those Euclidean integer dimensions where Majorana fermions can be defined. The Majorana conjugate is defined byψ a = ψ T a C ± with a suitable charge conjugation matrix C ± such thatψ a γ α ψ b = −ψ b γ α ψ a . This can be achieved for example in 2 dimensions 5 . It realizes the basic requirement for the Majorana fermions of the N = 1 supersymmetric model which must have a non-vanishing coupling The actual details of how to represent C ± and the gamma matrices in d + 1 dimensions are not important, as the most important thing for the rules which define the DR scheme for fermions is to keep track how derivatives are going to be contracted in higher dimensions. Apart from the above requirements, no additional Dirac algebra on the gamma matrices γ α in d + 1 dimensions is needed. With these rules one can recognize from the action (5) the propagators and vertices in d + 1 dimensions, and thus rewrite those Feynman diagrams which are ambiguous in one dimension directly in d + 1 dimensions.
The bosonic and ghost propagators are as usual and reported in section A.2 of the appendix. The fermionic fields with ABC on the worldline, ψ a (1) = −ψ a (0), can be expanded in half-integer modes ψ a (τ ) = and have the following unregulated propagator Note that the Fourier sum defining the function ∆ AF for the antiperiodic fermions is conditionally convergent for τ = σ, and yields In Euclidean 2 dimensions one can choose γ 1 = σ 3 , γ 2 = σ 1 and C + = 1. Recall that C ± are defined by is the sign function (with the value ǫ(0) = 0, obtained by symmetrically summing the Fourier series). The function ∆ AF satisfies where δ A (τ − σ) is the Dirac's delta on functions with antiperiodic boundary conditions The dimensionally regulated propagator obtained by adding a number d of extra infinite coordinates is derived from (5) and reads where the function The latter are the basic relations which will be used in the application of DR to fermions.
They keep track of which derivative can be contracted to which vertex to produce the d + 1 delta function. This delta function is only to be used in d + 1 dimensions, as we assume that only in such a situation the regularization due to the extra dimensions is taking place 6 . By using partial integration one casts the various loop integrals in a form which can be computed by sending first d → 0. At this stage one can use γ 0 = 1, and no extra factors arise from the Dirac algebra in d + 1 dimensions. This procedure will be exemplified in the subsequent calculations. Having specified how to compute the ambiguous Feynman graphs by continuation to d + 1 dimensions the DR scheme is now complete. Now we are ready to perform the two-loop calculation in the N = 1 nonlinear sigma model using DR. The bosonic vertices together with the ghosts, V and V DR give the standard contribution, as for example in [2]. The overall normalization of the fermionic path integral gives the extra factor 2 D 2 which equals the number of components of a Dirac fermion in a target space of even dimensions D. This already produces the full expected result in (3).
Thus the sum of the additional fermion graphs arising from the cubic vertex contained in ∆S = 1 0 dτ 1 2βẋ µ ω µab ψ a ψ b and the contribution from the extra counterterm V ′ DR must vanish at two loops. The cubic vertex arise by evaluating the spin connection at the background point x 0 and reads ∆S 3 = 1 2β ω µab 1 0 dτẏ µ ψ a ψ b , where y µ denotes the quantum fluctuations around the background point x µ 0 with vanishing boundary conditions at τ = 0, 1. Using Wick contractions (see appendix A.2 for the explicit form of the bosonic propagators with vanishing Dirichlet boundary conditions) we identify the following nontrivial contribution to e −S int (other graphs vanish trivially) where dotted lines represent fermions. As usual, we denote with a left/right dot the derivative with respect to the first/second variable. Using DR this contribution is regulated by s) (note the minus sign obtained in exchanging t and s in the last propagator; it is the usual minus sign arising for fermionic loops). We can partially integrate ∂ α without picking boundary terms and obtain because ∆ AF (0) = 1 2 ǫ(0) = 0 (and γ 0 = 1 at d = 0). As this example shows, the Dirac gamma matrices in d + 1 dimensions are just a book-keeping device to keep track where one can use the Green equation (13). Actually, the vanishing of this graph is achieved already before removing the regularization d → 0 by using symmetric integration in the momentum space representation of ∆ AF (t, t).
Thus no contributions arise from the fermions at order β 2 , and this fixes This is exactly what one expects to preserve supersymmetry, as the counterterm V DR is exactly canceled by the extra potential term V = 1 8 R needed to have the correct coupling to the scalar curvature in the Hamiltonian (4). Thus dimensional regularization without any counterterm preserves the supersymmetry of the classical N = 1 action since the amount of the curvature coupling brought in by DR is of the exact amount to render the quantum HamiltonianĤ supersymmetric.
To compare with TS, we can compute again the Feynman graph (14), but now using the TS rules. According to [12] we must use that • ∆ • (τ, σ) = 1 − δ(τ, σ), and integrate the delta function even if it acts on discontinuous functions. The delta function is ineffective as ǫ(0) = 0, but the rest gives This is canceled by using an extra counterterm V ′ T S = 1 16 (ω µab ) 2 which at this order contributes with a term −βV ′ T S evaluated at the background point x 0 . Thus, as expected, we recover the counterterm V ′ T S found in [12]. To summarize, we have proven that DR extended to fermions does not require additional counterterms on top of those described in [13]. In addition, supersymmetry requires that no counterterms should be added at all to the classical sigma model action.

Periodic boundary conditions
We present here some comments on the case of Majorana fermions with PBC. The mode expansion of ψ a (τ ) has now only integer modes The zero modes ψ a 0 of the free kinetic operator (∂ τ ) are treated separately, and the unregulated propagator in the sector of periodic functions orthogonal to the zero mode reads where with δ P (τ − σ) the Dirac's delta on periodic functions. Its continuum limit can be obtained by summing up the Fourier series and reads (for (τ − σ) ∈ [−1, 1]) The dimensionally regulated propagator is instead where the function Even if one uses PBC, one does not expect additional counterterms in DR, as mentioned earlier. It could be interesting to check in DR the expected β-independence of the supertrace which computes the Witten index i.e. the chiral anomaly. This is given by the path integral with periodic boundary conditions for both bosons and fermions [4,5,6]. The treatment of the bosonic zero modes is known to be somewhat delicate as a total derivative term may appear at higher loops [2,27]. However it should be possible to do a manifestly supersymmetric computation using superfields, and one could thus check if these total derivative terms survive in the supersymmetric case and, in case they do, study their meaning 7 .

Curved indices
It is interesting to consider as well the case of fermions with curved target space indices. This should be equivalent to the case of fermions with flat target space indices: it is just a change of integration variables in the path integral. However it is an useful exercise to work out, as some formulas will become simpler. The classical N = 1 supersymmetric sigma model is now written as The fermionic term could also be written more compactly using the covariant derivative D dτ ψ ν =ψ ν +ẋ λ Γ ν λρ (x)ψ ρ . Note that the action is now expressed in terms of the metric and Christoffel connection only, and there is no need of introducing the vielbein and spin connection.
The treatment of the bosonic part goes on unchanged. For the fermionic part we can derive the correct path integral measure by taking into account the jacobian for the change of variables from the free measure with flat indices Note the inverse functional determinant appearing because of the Grassmann nature of the integration variables. This extra factor arising in the measure can be exponentiated using bosonic ghosts α µ (τ ) with the same boundary condition of the fermions (ABC or PBC) and it leads to the following extra term in the ghost action One can check that the counterterms of dimensional regularization are left unchanged. The full quantum action for the N = 1 supersymmetric sigma model now reads and appears in the path integral as It is clear that supersymmetry is not broken by the boundary conditions if one uses PBC.
Then the effects of the ghosts cancel by themselves: the ghosts have the same boundary conditions and can be eliminated altogether from the path integral One can recognize that the potential divergences arising in the bosonicẋẋ contractions are canceled by the fermionic ψψ contractions, while the remaining UV ambiguities are treated by dimensional regularization as usual. In this scheme it should be simpler for example to test that the Witten index (i.e. the gravitational contribution to the chiral anomaly for a spin 1/2 field) does not get higher order contributions in worldline loops, and is thus β independent.
If one uses ABC the ghosts have different boundary conditions. Hence their cancellation is not complete, and they must be kept in the action.

Trace anomaly for a spin 1/2 field in 4 dimensions
As a further test on the DR scheme applied to fermions, we compute the trace anomaly for a spin 1/2 fields in 4 dimension. This anomaly is given by: i) extending the formula (3) to include the three-loop correction (order β 2 inside the round bracket), ii) setting D = 4, iii) picking up the order β 0 term [8] and, iv) including an overall minus sign which takes care of the fermionic nature of the target space loop. The bosonic part has been computed already in DR using Riemann normal coordinates (see [14], use ξ = 1 4 , and recall our conventions on the scalar curvature reported in appendix A.1). Multiplied by 2 D 2 (the additional normalization due the worldline ABC Majorana fermions) it reads We have now to include the fermionic contributions. On top of Riemann normal coordinates we may use a Fock-Schwinger gauge for the spin connection ω µab (x 0 + y) = 1 2 y ν R νµab (x 0 ) + ... with y µ the Riemann normal coordinates around the background point x µ 0 . Then the leading quartic vertex S 4,f = 1 4β R µνab 1 0 dτ y µẏν ψ a ψ b which originates from the spin connection produces the following 3-loop diagram where all functions ∆ and ∆ AF are functions of τ and σ in this precise order (recall that ∆ AF is antisymmetric, ∆ AF (τ, σ) = −∆ AF (σ, τ )).
Now we regulate this graph in DR as follows (the second contribution in (34) does not need regularization and could be directly computed at d = 0, but we carry it along anyway) where we have integrated by parts the α derivative in α ∆ β , which then produces a delta function when acting on fermions ("equations of motion terms"). This delta function is integrated in d + 1 dimensions and gives a vanishing contribution since ∆ AF (0) = 0. The remaining terms are then computed at d → 0. Thus This fermionic contribution must now be added to the terms inside the round bracket of eq. (33). Setting D = 4 one recognizes the following anomaly This is the correct trace anomaly for a Dirac fermion in 4 dimensions once we include the minus sign due to the target space fermionic loop.

One-loop effective action for a Dirac field in a gravitational background
It is known that, for a wide class of field theories, the one-loop effective action and the relative N-point vertex functions can be computed using one-dimensional path integrals [21,22].
Two of us have presented in [2] the extension of this formalism to include a gravitational background, considering the simplest case of a scalar field. The extension of DR to worldline fermions allows us to do the same for a Dirac field. We will get an expression for the effective action from which we derive explicitly the one-and two-point correlation functions, namely In the following we shall discuss both cases. As we shall see, the simplest set up is to use curved indices: in this case the sigma model couples linearly to the metric fluctuations h µν = g µν − δ µν , and the effective N-point vertices for the metric are obtained by integrating over the proper time the quantum average of N graviton vertex operators.

The worldline formalism
Let us consider the one-loop effective action obtained by quantizing a Dirac field Ψ coupled to gravity through the vielbein e aµ where e = det e a µ , ω µab is the spin connection, and The effective action depends on the background vielbein field e aµ and formally reads as For a Dirac field one does not expect anomalies (the euclidean effective action is real) and one can exploit standard arguments to writē In this formula we recognize the logarithm of an operator which up the mass term is proportional to the supersymmetric hamiltonian (4). Thus, we can immediately write down a path integral representation for the effective action in terms of a proper time as The subscripts PBC and ABC remind of the boundary condition at τ = 0, 1, periodic for the bosonic coordinates x µ (τ ) and antiperiodic for the fermionic ones ψ a (τ ): these boundary conditions have to be imposed to obtain the trace in (41). We have used a rescaled proper time T = β 2 with respect to the previous sections to agree with standard normalizations used in the worldline formalism [22]. We have not added any counterterm since we are going to use dimensional regularization to compute the path integral 9 . Of course, the covariant measure in (42) contains the ghost fields where One may also compute the effective action directly as a functional of the metric. This is achieved by using the sigma model written in terms of the Majorana fermions with curved indices. The corresponding formula is Note that the covariant fermionic measure now contains the new bosonic ghost α µ The fermionic term in the action (47) may be written using the covariant derivative as g µν ψ µ D dτ ψ ν , making manifest its geometrical meaning. However, one can write the Christoffel connection directly in terms of the metric and, because of the Grassmannian nature of the fields ψ µ , the action simplifies to which shows that there is only a linear coupling to the background g µν (x). To summarize, we have two options for representing the effective action in the worldline formalism, and we will consider both of them. The next step is to discuss how to treat the boundary conditions. Due to the translational invariance of the resulting propagators, we adopt the "string inspired" option: one expands the coordinate fields with periodic boundary conditions into Fourier modes and then separates the zero mode x µ 0 = 1 0 dτ x µ (τ ) from the quantum fluctuations y µ (τ ) = x µ (τ )−x µ 0 . The latter have an invertible kinetic term and the integration over the constants zero mode x µ 0 is performed separately. For the alternative option of using Dirichlet boundary conditions, see a discussion in [2]. Other options for treating the zero modes can be found in [31,22].
These subtleties do not arise for the anticommuting variables ψ a as the boundary conditions are now antiperiodic and the kinetic term has no zero mode. All these propagators are collected in section A.3 of the appendix.
For later convenience, it is useful to introduce the following notations and the corresponding Fourier transform for the vielbein vertex functions Γ a 1 ν 1 ···a N ν N (p 1 ,...,p N ) together with the relation g µν = δ ab e a µ e b ν , one finds Following a standard technique, one can obtain the vertex functions directly in momentum space [22]. Let us describe it for the effective actionΓ[e aν ]. One considersΓ[e aν ] as a power series in c aν ≡ e aν − δ aν (note that this definition induces a relative expression for the metric: g µν = δ µν + c µν + c νµ + c aµ c a ν , where c µν = c aν δ a µ ), takes the c N term as a sum of N plane waves of given polarizations (our polarization tensors include the gravitational coupling constant) and then picks up the terms linear in each ε (i) aν : this gives directly the N-point function in momentum spaceΓ (the tilde symbol can be dropped by removing the momentum delta functions as in (53)).
In the following sections we are going to compute the one-and two-point correlation functions. We will employ the worldline "string inspired" propagators together with dimensional regularization on the worldline (and in target space).

One-and two-point functions fromΓ[e aν ]
The one-point vertex function can be depicted by the Feynman diagram of fig. 1 where the external line refers to the vielbein. It gives the Dirac particle contribution to the cosmological constant. The recipe just outlined tells that the term in the effective action linear in c aν , and with c aν expressed as a single plane wave, produces where the superscript on ω (1) µab denotes the part linear in ε aν , and round brackets around indices denote symmetrization normalized to one. It can be immediately noted that the contribution of the spin connection term vanishes, being proportional to ω µab δ ab ∆ AF (0) = 0. Therefore everything proceeds as in the scalar field case [2], and the one-point function reads Clearly it diverges for even target space dimension D and renormalization is needed. Let us now discuss the two-point vertex function. We set One sees that there are two kinds of contributions, illustrated by the Feynman graphs in figures 2 and 3, which we denote as ∆ 1Γ µναβ and ∆ 2Γ µναβ , respectively. ¡ Figure 2: One-vertex graph for graviton self-energy.
In the first one there is just one vertex. It is simple to compute it, being quite similar to the tadpole. It reads ∆ 1Γ where ω (2) µab is the part of the spin connection quadratic in the c aµ field; the prescription m.l. (multi-linear) refers to the two different polarization tensors. The contribution from the spin connection term vanishes for the same reason as before. We are then left with the bosonic contribution, which gives where, according to the notation (53), we have factored out (2π) D δ D (p 1 + p 2 ) and used p = p 1 = −p 2 .
The two-vertex graph of fig. 3 produces ¡ Figure 3: Two-vertex graph for graviton self-energy. .
Three kinds of contributions are included in the previous expression: i) the square of the bosonic part which yields a term proportional to the contribution of a scalar field (nonminimally coupled with ξ = 1/4, already computed in [2]); ii) the mixed terms of the product which are zero, again being proportional to ω µab δ ab ∆ AF (0); iii) the square of the fermionic term which contains where square brackets around indices denote antisymmetrization normalized to one. This third term produces the following contribution [ab] + iε [cd] + iε After performing Wick contractions, the second line of this expression becomes (δ ac δ bd − δ ad δ bc )(2T ) 3 where ∆ 0 (τ − σ) = ∆(τ − σ) − ∆(0), and needs worldline regularization. Following the rules of dimensional regularization we write the last line of the above expression as we would have done starting from the action in 1 + d dimensions and perform an integration by parts on the α index of the first term, as already explained in (16) and (35), to get the following result The remaining worldline integral can be computed as described in section A.4 of the appendix. Using the result into eq. (64) gives where P 2 and S 2 are defined below. Collecting all terms, we find for the two-vertex part of the self-energy Consequently, the full graviton self-energy obtained by summing (62) and (69) reads where, as in [2], we have suppressed tensor indices, and used the following basis of dimensionless tensors symmetric in each pair of indices, (µ, ν) and (α, β), and symmetric under the exchange of the two pairs along with the tensor 10R For simplicity we have introduced the manifestly transverse combinations and defined

One-and two-point functions from Γ[g µν ]
In this section we describe the calculation of the one-and two-point functions employing curved indices for the worldline Majorana fermions. As already explained in sections 2.2 and 4.1, the total action including the ghost fields is given by Clearly, there are no vertices with two or more gravitons in this picture. Using with the gravitational coupling constant included into the polarization tensors, one gets the following general expression for the N-point effective vertices where the graviton vertex operator V (i) (τ i ) is given by 11 If one is interested in odd dimensions, then there is no divergence at one loop, but the formulas should be modified by substituting 2 The explicit calculations of the one-and two-point vertex functions -depicted in figures 4 and 5, respectively -give the same results previously obtained from the coupling to the vielbein (after using relations (54) and (55)). Note that external lines in figures 4 and 5 now refer to metric fluctuations. Let us describe briefly these calculations. ¡ Figure 4: Graviton tadpole.
In the one-point function the connection term (i.e. the last term inside the round brackets of the vertex operator (79)) does not contribute, and the remaining terms lead to the same worldline integral obtained previously In fact, the propagator of the extra ghost field cancels with ∆ • AF (τ − σ) = − • ∆ AF (τ − σ) due to the fermionic propagator. This exemplifies the effect of the new ghost α µ which cancels a contraction arising from the ψ µψν term. The final answer is As one might have expected, this result is −2 D 2 times the contribution of a scalar field [2]: the minus sign is the usual one due to a fermionic loop, while 2 D 2 is the number of degrees of freedom of a Dirac fermion in even D dimensions.
Let us now look at the two-point function. It corresponds to the single diagram of fig. 5, as in this scheme all vertices are linear in the graviton field. Again one may note that the α µ ghosts cancel all Wick contractions arising from the ψψ term of the vertex operators (notice that ∆ AF ∆ • AF = −∆ AF δ A = 0). Thus, only two non-vanishing contributions survive: one from the square of the kinetic term of the bosonic sector (i.e. ∼ (ẋ 2 + a 2 + bc) 2 ); the other, transverse, from the square of the connection term. The final result is ¡ Figure 5: Graviton self-energy.
This expression satisfies the expected gravitational Ward identities (for details see section A.5 of the appendix). The graviton self-energy due to a massless fermion has been already computed in [32], and agrees with the massless limit of this general result 12 .

Conclusions
We have extended the worldline formalism to include fermionic fields coupled to gravity.
To achieve this task we have found useful to study dimensional regularization on supersymmetric worldline sigma models. We have shown that dimensional regularization preserves worldline supersymmetry in that no counterterms need to be added to the classical action to maintain supersymmetry. This is in contrast to the time slicing regularization scheme, previously used for supersymmetric sigma models, which required specific counterterms to restore supersymmetry. Of course, final physical results are independent of the regularization scheme adopted. We have applied this set up to describe quantum properties of a Dirac fermion coupled to gravity. Then, we have described the one-loop effective action for a Dirac fermion coupled to gravity in the worldline formalism, and computed the corresponding oneand two-point functions, namely the one-loop fermionic contribution to the cosmological constant and graviton self-energy. We have seen that one can use a formulation either in terms of the vielbein or in terms of the metric, the latter being much simpler as the coupling to gravity is linear (and it avoids the introduction of the local Lorentz symmetry related to a choice of the vielbein). The computations are rather simple and demonstrate the efficiency of the worldline formalism in computing Feynman graphs even in the presence of gravitational fields. Our conclusion is that one can be confident and address more complicated processes using the worldline method. In particular, mixed photon-graviton amplitudes are 12 A similar result for a massive scalar field with minimal coupling to the scalar curvature can be found in [33] and agrees with the worldline result [2]. A calculation with standard Feynman rules for the scalar has been recently reported again in [34]. It may be noticed how the worldline computation produces simpler and more compact expressions. under study [35].

A.1 Covariant derivatives and curvature tensors
The covariant derivative for a vector with curved indices is ∇ µ V ν = ∂ µ V ν + Γ ν µλ V λ , where Γ ν µλ = 1 2 g νρ (∂ µ g λρ + ∂ λ g µρ − ∂ ρ g µλ ) is the usual Christoffel connection. The corresponding curvatures are defined by The covariant derivative of a vector with flat indices is is the spin connection satisfying the "vielbein postulate" ∇ µ (Γ, ω) e a ν = 0. The corresponding curvature is These curvatures are related by

A.2 Propagators for bosons and related ghosts with vanishing Dirichlet boundary conditions
For quantum fields that vanish at τ = 0, 1, we have the following propagators with Green functions ∆ and ∆ gh satisfying vanishing Dirichlet boundary conditions where θ(τ − σ) is the standard step function and δ(τ, σ) is the Dirac's delta function which vanishes at the boundaries τ, σ = 0, 1. These functions are not translationally invariant. Their extensions to d + 1 dimensions read Note that the function ∆(t, s) satisfies the relation (Green's equation) The d → 0 limits of these propagators reproduce the unregulated expressions.
When using curved indices for the Majorana fermions ψ µ there appears an extra set of bosonic ghosts α µ . Their propagators with ABC are where ∆ Agh (τ − σ) =

A.4 Recursive formula for some worldline integrals
In the calculation of 1PI correlation functions via the worldline formalism described in sec- where we have defined It is useful for the comparison with [32] to note that lim m 2 →0 (P 2 ) x = (p 2 ) x B(x + 1, x + 1) Here we have used the hypergeometric function 2 F 1 and the Euler beta function B .

A.5 Ward identities
A test for our results on one-and two-point functions is provided by the Ward identities due to general coordinate and local Lorentz invariances. Local Lorentz symmetry δe a ν = Λ a b (x)e b ν with arbitrary antisymmetric local Λ ab (x) implies that δΓ[e] δe a ν (x) Λ a b (x)e b ν (x) = 0 (107) which shows that the induced stress tensor T µν = 1 e δΓ [e] δea µ e aν is symmetric. In particular, it leads to the following identities for the one-and two-point functions in momentum spacē It is easy to check that (59) and (70) do indeed satisfy the latter, while the former is rather straightforward (p µ = 0 due to momentum conservation). Alternatively, one can derive equivalent Ward identities for the effective action Γ expressed as a functional of the metric and obtain (equivalently, one may use relations (54-55)) p µΓ µν (p) = 0 (113) Also in this case it is simple to verify that eqs. (83) and (84) satisfy these Ward identities.