Forward scattering amplitudes in the imaginary time formalism

We study, in the imaginary time formalism, the relation between loops and on-shell forward scattering tree amplitudes in thermal field theories. This allows for an efficient evaluation, at all temperatures, of Green's functions with causal retarded boundary conditions. We present an application of this relation in quantum gravity coupled to scalar matter fields. We show that at one or two loops, involving planar diagrams, the 1PI retarded thermal Green's functions may respectively be expressed in terms of connected forward scattering tree amplitudes of one or two on-shell particles.


I. INTRODUCTION
The connections between tree amplitudes and loop integrals play an important role in quantum field theory. In a pioneering work [1], Feynman has shown that one can relate one-loop integrals in gauge theories to on-shell forward tree scattering amplitudes. Such relations are useful due to the simplicity of these amplitudes, which may be obtained by contour integrations containing the poles of the propagators. There has been a lot of effort in the literature to find out how unitarity methods, which rely on the fact that the loop amplitudes are determined by their singularities, can be used to evaluate higher order loops. These methods have been developed in a series of papers and successfully applied to loop computations in the standard model [2][3][4][5][6] When a single propagator in a one-loop graph is cut, the integrand leads to a tree-level amplitude. By developing Feynman's method, some QCD amplitudes have been computed just from single cuts. This approach has been extended, in the real time formulation at zero temperature, to higher-order loops, both for time-ordered [7][8][9][10] and for retarded boundary conditions [11].
In the present work, we consider the retarded thermal Green's functions that appear as causal response functions, which are most conveniently studied in the framework of the imaginary time formalism. In this formulation, in momentum space, a quantum field theory in 3 + 1 dimensions involves a 3-dimensional Euclidean theory with a summation over the discrete Matsubara frequencies Q 0 = 2πinT where T is the temperature [12][13][14]. This allows one to separate the usual T = 0 part from the finite temperature contribution. The main purpose of this work is to give a unified treatment of the relation between one-and two-loop graphs and on-shell forward scattering tree amplitudes, which is valid both at zero as well as at finite temperature. This method, which greatly simplifies computations at finite temperature in thermal gauge theories, has been previously used in connection with particular thermal amplitudes [15][16][17][18]. Using appropriate analytic continuations of the external energies, one can recover the corresponding results obtained in the real time formalism.
In section II we describe the forward scattering amplitude method, at one loop order, in the imaginary time formalism. An application of this approach to the calculation of the lowest order retarded Green's functions in quantum gravity at zero temperature is given in section III. The computations are done in D space-time dimensions using dimensional regularization [19]. The extension of this analysis to finite temperature is presented in section IV. A workable two loop example of this formulation in a scalar theory is examined in section V. In section VI, we summarize the results and discuss a generalization of this approach at two-loop order. Some details of the calculations are given in the Appendices.

II. THE FORWARD SCATTERING METHOD
In the imaginary time formalism, the one-loop integral associated with diagrams such as the one shown in Fig. 1 is given by [12,13] where D is the space-time dimension, Q 0 = iπ(2n + m)T are the Matsubara frequencies, and m = 0 or m = 1 for Bosons or Fermions, respectively ( α, a,s represents a sum over Lorentz, internal and spinor indices of the loop particle). The bosonic (fermionic) Matsubara summations can be done using the relation (the equivalent contours are shown in figures (2a) and (2b)) which clearly allows us to separate the T = 0 part from the finite-temperature contribution. Using Eq. (2.2), one-loop diagrams containing L internal lines, as shown in Fig. 1, can be expressed as where we have replaced Q by − Q in the second term of the integrand. In general f (Q 0 , Q, k 1 , k 2 , . . . , k L ) has the following structure where k 1 + k 2 + · · · + k L = 0, E( k i , Q) = | Q + k i | 2 + m 2 , and t(Q, k 1 , k 2 · · · k L ) is a tensor (or a scalar in the case of a scalar field theory) which is determined by the interaction vertices of the theory. Using a partial fraction decomposition like ((k 0 , k) represents any of the following possibilities: (0, 0), (k 1 0 , k 1 ), (k 1 0 + k 2 0 , k 1 + k 2 ), . . . , (−k L 0 , − k L )) the Q 0 integral can be done upon closing the integration contour on the right hand side of the complex plane and using Cauchy's integral formula (assuming that f (Q 0 , Q, k 1 , k 2 , . . . , k L ) vanishes on the curve C 2 in Fig. 2c, when C 2 is stretched to infinity).
Performing shifts in the loop momentum 1 and taking into account that the external frequencies are integer multiple of i2πT so that exp(Q 0 + i2π lT )/T = exp Q 0 /T one can show that (the overall minus sign arises from the clockwise contour on the right hand complex plane) where E Q = E(0, Q) and The tree amplitude A(k 1 , k 2 , · · · , k L ; Q) is a forward scattering amplitude which describes the scattering of an on-shell particle of momentum Q by external particles of momentum k 1 , k 2 , . . . , k L . The analytic continuation of some external energy k j 0 → k j 0 + iη j , where η j is a positive infinitesimal quantity, with all other analytic continuations involving negative infinitesimals, yields the j-th retarded amplitude. In the following, we shall not need to explicitly indicate this leg. As a simple example, we consider the self-energy in massless gφ 3 theory. In this case, the forward scattering tree amplitude is (where the on-shell condition Q 0 = E(0, Q) = | Q| is to be understood) (we have taken into account the combinatorial factor 1/2 and also added the permutation as indicated in Eq. (2.7)), so that the one-loop scalar self-energy can be written as In the Appendix A, we compute the integral in (2.9) using the retarded analytic continuation of the external energy with k 0 → k 0 + iη. This basic integral is also relevant for the self-energies of gauge fields and gravitons as we will see in the next section.

III. SCALAR LOOPS IN A GRAVITATIONAL FIELD
The action for a scalar field in curved space-time is simply For simplicity, let us set m = 0. Defining the gravitational field ϕ µν where κ = √ 16πG is related to Newton's constant G, the action becomes From (3.3) we obtain the following momentum space Feynman rules: where p 1 µ and p 2 ν are the momenta of the scalar field. In terms of the forward scattering amplitude, the graviton self-energy at zero temperature can be written as where we have used the T = 0 part of the general expression given in Eq. (2.6). Upon using the Feynman rules given in Eqs. (3.4), we obtain (we have included the combinatorial factor 1/2 and also added the permutation indicated in Eq. (2.7)). After integration, (3.7) can be expressed (by covariance) in terms of the five tensors shown in table I, so that The coefficients C i can be obtained solving the following system of five algebraic equations Using simple identities such as and setting equal to zero any dimensional regularized integrals that are independent of | Q|, then all the integrals on the right-hand side of Eq. (3.9) can be reduced to the following basic scalar integral where we have used the result given in Eq. (A8) (up to an overall factor). Solving the linear system of equations in Eq. (3.9), we obtain and .
This result satisfies the Ward identity which is stronger than the identity of the pure gravitational case in the Eq. (3.36) of [21] (see also [22]). This can be understood since there is no BRST symmetry in the scalar case. There is only the classical diffeomorphism invariance of (3.1) under

IV. GRAVITON THERMAL SELF-ENERGY
Let us now consider the thermal part of Eq. (2.6) which contains the Bose-Einstein distribution. When the external momenta k are such that k ≪ T the loop integral is dominated by the hard thermal loop region. Using expansions like to leading order in the high temperature expansion (Q = Q/| Q|). Inserting Eq. (4.2) into the thermal part of Eq. (2.6) we obtain the following expression for the thermal scalar loop contribution to the graviton self-energy  where Γ and ζ denote respectively the Gamma and Zeta functions. For D = 4, we obtain which is half the result in the case of pure gravity (A graviton loop produces a factor of two, which is associated with the extra degree of freedom of a massless spin 2 particle). At finite temperature, the tadpole diagram shown in Fig. 3a does not vanish. Taking into account the symmetry factor 1/2 and the Feynman rules in (3.4) Fig. 3b yields We can verify that the amplitudes in Eqs. (4.2) and (4.6) are related by the Ward identity Since this identity is verified at the level of the tree amplitude, the thermal self-energy and the one-graviton are also related by the same identity, namely in the hard thermal loop limit. Notice that at T = 0 Eq. (3.14) is consistent with the absence of tadpoles. In fact, it is straightforward to verify that the exact amplitude in Eq. (3.7) satisfy the same Ward identity as the hard thermal loop case, namely At T = 0, and at finite T we have where C, is a constant, u = (1, 0, 0, . . . , 0) is a rest frame four-velocity and we have used that the result is traceless, since Q 2 = 0. From the 00 component we obtain [23] For D = 4 we obtain, which is half the result obtained in pure gravity.

V. RETARDED AMPLITUDES AT TWO LOOPS
In order to study higher-order thermal loop contributions in the imaginary time formalism, we consider, for simplicity the scalar φ 4 field theory, described by the Lagrangian and examine the retarded self-energy of the scalar field which can be obtained by the prescription k 0 → k 0 + iη. The one-loop contribution can be expressed in terms of a forward scattering amplitude for a on-shell particle as discussed in Eqs. (2.6) and (2.7). In going at two loops, we note that there are two types of Feynman diagrams, as shown in the Figs. (4a) and (4b). We can evaluate the summations over the discrete frequencies Q 0 = i2πnT and q 0 = i2πsT with the help of the relation (2.2). Next, we proceed to evaluate the Q 0 and q 0 integrals by contour integrations in the right hand side of their complex planes. In the case of Fig. (4a), according to Eq. (2.6) this opens successively the Feynman propagators, and sets the Q and q particles on their mass-shell, with positive energies Q 0 = E Q and q 0 = E q . In Fig. (4a'), which has a double pole, one can do the q 0 integral by using Cauchy theorem, leading to a graph like that shown in Fig. (4a"). This diagram, which corresponds to the forward scattering amplitudes of two on-shell particles is multiplied by a phase-space integral, leading to a k-independent result given by We will next apply this method to the diagram shown in Fig. (4b), which yields a k-dependent contribution given by where Q 0 = i2πnT , q 0 = i2πsT and k 0 = i2πlT . Performing the summations with the help of Eq. (2.2) and replacing Q → − Q and q → − q in some terms, we obtain for the integrand of (5.3) the result where N (ω) denotes the statistical factor We first perform the Q 0 integration along the contour shown in Fig.(2c) by using Cauchy's theorem. Making appropriate shifts and using the fact that k 0 is an integer multiple of i2πT we obtain the following two contributions , (5.6) and This procedure opens the Q-propagator and sets the Q-particle on-shell with a positive energy Q 0 = E Q . This leads to diagrams of the form shown in Fig.(4b'), where graphs obtained by permutations of vertices as well as by the replacement Q → −Q are to be understood. We can similarly integrate over q 0 by using Cauchy's theorem. The contribution at the pole q 0 = E q coming from the equation (5.6) is The contribution coming from (5.6) at the other poles may be combined with that obtained from (5.7) at the pole q 0 = E q , to give Finally, the contribution obtained from the poles of the other propagators in (5.7) is given by where we have symmetrized under the interchange Q ↔ q.
The above expressions can be simplified by using the following identities involving the statistical factors .
In this way, the expressions given in Eqs. (5.9) and (5.10) may, respectively, be written as . (5.12) and The above procedure opens the q-propagator in the one-loop vertex correction shown in Fig.(4b') and sets the q particle on-shell with a positive energy q 0 = E q . This leads to diagrams of the form shown in Fig. (4b"), where graphs obtained by permutations and by making q ↔ −q are to be understood.
We still must take into account the contributions coming from the poles of the statistical factors N (E Q ∓ q 0 ) in the Eq. (5.7). These arise at the points q 0 = ±E Q + i2πjT , where j is an integer, along two lines parallel to the imaginary q 0 axis. Upon using Eq. (2.2) and the identities (5.11), one can similarly evaluate such contributions which should be added to those given in Eqs. (5.8), (5.12) and (5.13). Proceeding in this way, and using partial fraction decompositions like (2.5) it turns out that the complete contribution of the diagram shown in Fig. (4b) may be written in the form where E = ( k + Q + q) 2 + m 2 and The above form of the retarded self-energy is quite convenient, especially for the calculation of its imaginary part which is related to various physical processes. In particular, the imaginary part of F 0 is related to the probability that the external field creates three scalar particles at zero temperature. This can be seen from Eq. (5.16) as well as by cutting the propagator in Fig. (4b"). (Performing the Q and q integrations in Eq. (5.14), one obtains in the massless case at zero temperature the result given in the Appendix C). The result (5.14) agrees with the one obtained in the reference [24] in the imaginary time formalism, by an elaborate evaluation of the loops appearing in Fig. (4b) An alternative derivation in the real time formulation was given in Ref. [18]. In this formalism, the 2 × 2 matrix propagator is a sum of two parts, one of which represents an on-shell thermal contribution that involves a factor N (|p 0 |)δ(p 2 − m 2 ), where p is the four-momentum of the line. Such factors naturally lead to the opening of loop diagrams at non-zero temperature. However, this property does not elucidate the fact that forward scattering tree amplitudes exist as well at zero temperature.
Based on the present formulation, we may interpret the contributions coming from the thermal self-energy diagram (4b) at two loops as follows. F 0 corresponds to a forward scattering amplitude of two on-shell particles at zero temperature; The contribution linear in the statistical factor might be thought as a forward scattering amplitude of a on-shell thermal particle and of a forward scattering amplitude of a single, on-shell particle at zero temperature. Finally, the contribution quadratic in the statistical factor may be interpreted as the forward scattering amplitude of two on-shell thermal particles (for a related approach, see references [25,26]).

VI. DISCUSSION
Using the imaginary-time formalism, we have obtained the relation (2.6) between general one-loop diagrams and on-shell forward scattering tree amplitudes which holds at all temperatures T in D space-time dimensions. We note that the right hand side of Eq. (2.6) has been obtained by performing first the Q 0 integral by contour integration. This has the effect of opening one propagator in the loop which leads to a tree, on-shell forward scattering amplitude. At high temperatures, this relation provides an efficient way for computing the retarded thermal contributions in gauge theories.
As an application of Eq. (2.6), we have calculated in quantum gravity (coupled to scalar fields) the lowest orders retarded graviton amplitudes at all temperatures. At zero temperature there occurs a subtlety due to the fact that the contribution from the arcs at infinity (see Fig. 2c), which arise when first performing the Q 0 integration, is linearly divergent. However, when performing first the Q integrations, it turns out that the remaining Q 0 integration converges for D < 0 (see Appendix B), in which case the contributions from the arcs at infinity vanish. Thus, by using a consistent analytic continuation, we see that the neglect of the contributions from the arcs at infinity can be justified in dimensional regularization. At high temperatures, the leading contributions of all 1PI retarded functions are proportional to T D . This is required by the Ward identities which reflect the gauge invariance of the theory under coordinate transformations.
We have applied the above method to the evaluation, in a scalar theory, of the retarded self-energy at two loops in the imaginary time formalism. But one can show that, by using an integral reduction procedure in dimensional regularization [27], one can express any loop integral as a linear combination of scalar loop integrals. An example of this procedure in quantum gravity was given in section III. Based on these calculations, we point out some features which may hold for more generic one-particle irreducible diagrams at two loops. In this case, there are two independent internal energies which can be integrated out by contour integrations. This leads, by successive openings of the internal lines, to two cut lines which set the corresponding particles on-shell. In this way, a 1PI two-loop diagram may be opened to a connected tree-level graph. We also note here that in the imaginary time formalism, one can have at most one statistical factor for each loop (see Eq. (2.2)). However, a new feature occurs due to the presence, in intermediate calculations at two loops, of statistical factors like N (E q − E Q ) which yield non-vanishing contributions in the zero temperature limit. These add up with the usual T = 0 contributions that might be naively expected (see Eqs. (5.8), (5.9) and (5.10), respectively). This combination is necessary in order to get a result consistent with unitarity, which corresponds to a forward physical tree scattering amplitude.
Using the above properties, a simple generalization of the Eq. (2.6) (for bosonic fields) at two loop order may be written in the form +N (E Q )A 1 (k 1 , . . . , k L ; Q, q) + N (E Q )N (E q )A 2 (k 1 , . . . , k L ; Q, q) Q0=EQ;q0=Eq , (6.1) where A 0 is related to a forward scattering amplitude of two on-shell particles Q and q at zero temperature. The contribution linear in the statistical factor N (E Q ) describes a forward scattering amplitude of a thermal on-shell particle Q, and of an on-shell particle q at zero temperature. The term quadratic in the statistical factors represents a forward scattering amplitude of two on-shell thermal particles. Making in Eq. (6.1) appropriate analytic continuations of the external energies [28], one can verify that we get for planar 1PI retarded two and three point Green's functions at two loops, the same result as obtained in the real time formalism [18]. The representation given in Eq. (6.1) also holds for more generic planar amplitudes at two loops. The issue of non-planar amplitudes has been examined at zero temperature in Ref. [11]. However, a proper extension of the relation (6.1) in the case of non-planar retarded thermal Green's functions at two loops is more involved and requires further investigation.
where α and p are some integers and Q = | Q|. Performing the angular integrations, we get Integrating first over Q 0 , we see that the Q 0 integral diverges for p > 2α − 1. On the other hand, evaluating first the Q integral with the help of the formula 3.241 of [23] ∞ 0 which converges for D < 2α + 1, we obtain Here the Q 0 integration converges provided D < 2α − p. For the graviton self-energy, one has α = 2 and p = 0, . . . , 4. In this case, the Q 0 integral becomes convergent for D < 0. For example, for α = 2 and p = 4, the Q 0 integration yields the expression This result may be analytically continued to positive values of D, where it exhibits poles for D = 2n. The above relation agrees with that obtained by using Euclidean space techniques and the retarded boundary condition is selected by the prescription K 0 → K 0 + iη.