One-loop off-shell amplitudes from classical equations of motion

In this letter we present a recursive method for computing one-loop off-shell amplitudes in colored quantum field theories. First, we generalize the perturbiner method by recasting the multiparticle currents as generators of off-shell tree level amplitudes. After, by taking advantage of the underlying color structure, we define a consistent sewing procedure to iteratively compute the one-loop integrands. When gauge symmetries are involved, the whole procedure is extended to multiparticle solutions involving ghosts, which can then be accounted for in the full loop computation. Since the required input here is equations of motion and gauge symmetry, our framework naturally extends to one-loop computations in certain non-Lagrangian field theories.


I. INTRODUCTION
Scattering amplitudes are central objects of study in quantum field theory. More than convenient physical observables, they are deeply rooted in the very way we intuit particle interactions. And nothing captures this statement more clearly than Feynman diagrams and their beautiful simplicity.
We soon learn, however, that Feynman diagrams are far from being the most efficient way of computing scattering amplitudes. There has been an impressive progress over the years in tree-and loop-level computations. Most of these developments involve so-called on-shell methods (see e.g. [1][2][3][4][5][6] for reviews on the always increasing number of techniques). On the other hand, off-shell methods are very scarce (see e.g. [7] and references therein). Off-shell amplitudes have in general a richer structure. Besides encoding the full on-shell information of a given process, off-shell results can be used in the computation of form factors (related to higher derivative terms in effective actions), in the study of quantum corrections of propagators and vertices of a theory, renormalization group analysis, and as building blocks for higher-loop corrections.
In between on-shell and off-shell methods, the Berends-Giele (BG) currents [8] stand out for their elegant and recursive character. Simply put, BG currents represent tree-level amplitudes with one off-shell leg, which are naturally interpreted as branches of higher-point trees. Even more interesting is the fact that the BG prescription can be used to compute quantum effects. Indeed, one-loop integrands are obtained by sewing tree-level amplitudes with two offshell legs. Following this idea, the matter contribution to one-loop amplitudes in QCD was computed in [9] by making an on-shell matter leg of the BG current off-shell. This construction, however, cannot be easily extended to gluon loops because of the gauge symmetry: lifting the on-shell condition in gauge fields is a non-trivial task. In this case, a concrete solution has only recently been found in [10], following an extensive procedure using graphic rules in pure Yang-Mills. Numerical recursions for one-loop integrands are also known in the literature, see e.g. [11,12].
We would like to offer a fresh perspective on this subject, and show how to obtain quantum off-shell currents via classical equations of motion [13]. The key ingredient is the perturbiner [15,16], a well-known method to recursively obtain Berends-Giele currents via formal multiparticle solutions of the field equations. The perturbiner method has proven to be a versatile technique to compute tree level amplitudes in different theories [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. What we propose, instead, is to view it as a generator of off-shell tree-level diagrams. This is achieved via multiparticle ansatze that solve the interacting part of the field equations, while leaving single-particle states off-shell. The standard notion of Berends-Giele currents is then generalised to a fully off-shell version. In turn, we are able to define a consistent sewing procedure to recursively generate one-loop off-shell currents, dubbed one-loop pre-integrand s.
In this letter we focus on color-ordered theories, having the bi-adjoint scalar and Yang-Mills theories as working examples. The sewing procedure has to be supplied by a cyclic completion, which is neatly implemented using the multi-index structure of the one-loop pre-integrands. It restores the cyclicity of the partial amplitude while solving the combinatorial challenge of the Feynman approach. We start by illustrating the construction in the bi-adjoint scalar theory with the derivation of the off-shell recursion and the one-loop integrands. In Yang-Mills we first perform the gauge-fixing of the action with the introduction of Fadeev-Popov ghosts. We then extend the previous analysis to ghost loop contributions, finally proposing the full n-point one-loop integrand with off-shell external gluons.
Consider a massless bi-adjoint scalar with cubic selfinteraction and classical equation of motion [31] Here we have φ = φ aã T a ⊗Tã, where a andã are adjoint indices associated with two different quadratic Lie algebras, with generators T a andTã, and We are interested in building a generator of treelevel multiparticle currents à la Berends-Giele [8], but through a modified perturbiner with off-shell external legs. To this end, we first look for a multiparticle solution to (1) of the form The sum ranges over all words P = p 1 · · · p n and Q = q 1 · · · q n of length |P | = |Q| = n, where p i and q i are single-particle labels. Furthermore, we have k P = k p1 + · · · + k pn , T aP = T a1 · · · T an , and Tã Q =Tã 1 · · ·Tã n . Inserting this back in (1) leads to k 2 p = 0 for single-particle states and the following recursion relation, where s P = k 2 P are the Mandelstam variables. The sums over P = RS and Q = T U denote deconcatenations of the word P into R and S, and the word Q into T and U , respectively. For example, for P =ijk we have (R, S) = (i, jk), (ij, k). By construction, the coefficients Φ P |Q automatically vanish unless the words P and Q are related via permutation.
The multiparticle currents given by (4) are identified with Berends-Giele currents at tree level, and can then be used to compute double-partial amplitudes [20]. On the other hand, by dropping the onshell condition for the single-particle states, equation (4) defines an off-shell recursion that only solves the equation of motion (1) at the multiparticle level. For instance, this can be expressed as where the sum is taken over the single-particle states and k 2 p = 0. It is therefore fair to say that the recursion relations in (4) solve the interacting part of the biadjoint scalar theory while leaving the singleparticle states off-shell. In other words, they can be interpreted as a generator of off-shell trees. Bearing this in mind, we may refer to the coefficients Φ P |Q as off-shell Berends-Giele double-currents.
We can then establish the connection between Φ P |Q and the off-shell scattering amplitudes in bi-adjoint scalar theory. This is realized by a direct extrapolation of the Berends-Giele prescription, such that the off-shell tree-level double-partial amplitudes are determined through the formula where the limit enforces momentum conservation.

III. BI-ADJOINT: ONE-LOOP INTEGRANDS
Off-shell trees are the building blocks of loop amplitudes via a sewing procedure. We will now show that the off-shell perturbiner expansion leads to a simple algorithm for computing one-loop integrands in the bi-adjoint scalar theory.
We start with the double-current Φ lP |lQ , in which the single-particle label l plays a special role. Using (4), such current can be explicitly expressed as We can then factor out the single-particle polarization Φ l|l on the right-hand side and recast Φ lP |lQ as where ℓ µ ≡ k µ l , and The double-current Λ P |Q (ℓ) is the fundamental ingredient for defining the one-loop integrands. It needs but a small upgrade.
In order to see this, observe that (6) and (8) yield (10) The sewing procedure Φ l|l Φ n|n → 1/ℓ 2 , with k n = −ℓ, leads to what looks like an onshell one-loop integrand I 1-loop (P |Q) ≈ Λ P |Q (ℓ). However, such an integrand is not cyclic in the words P and Q, for the singling out of the leg l has not been symmetrically done. Fortunately, the perturbiner framework enables a neat solution to this problem via a cyclic completion of the combinatorial sums defining the recursion.
We then introduce the modified double-current, such that the one-loop integrand is expressed as The words P and Q encode the (double) color ordering of the one-loop integrand. The first term inside the square brackets only yields tadpole diagrams, so it can be removed for convenience since their regularized contribution vanishes. The cyclic completion affects the remaining terms, with sums over P = [RS] and Q = [T U ] consisting of all inequivalent cyclic permutations of a given deconcatenation of P and Q. The deconcatenations of two cyclic permutations of the single-particle labels in P and Q are equivalent if they lead to the same diagram contribution on the one-loop integrand. As it turns out, they can be identified via a simple rule. 1 For P = [RS] we take the usual deconcatenation of P into the non-empty words R and S, and then add |S| − 1 cyclic permutations in P . Take, for example, the word P = 1234. The crossed permutations are not allowed. It is a straightforward exercise to verify that this expression reproduces each of the diagrams contributing to the one-loop off-shell integrand only once.
As an example, we will present the off-shell threepoint one-loop integrands. There are only two of them, namely I 1-loop ℓ (123|123) and I 1-loop ℓ (123|321). Using the rules outlined above, we obtain and with normalization Φ p|q = δ pq . These match the results obtained in e.g. [32,33].

IV. YANG-MILLS: OFF-SHELL RECURSION
The main difference between the bi-adjoint model and the Yang-Mills theory is the gauge symmetry. While at tree level, the gauge fixing procedure is trivial, the loop construction naturally involves Faddeev-Popov ghosts. The covariant gauge fixed action can then be cast as where ξ is an arbitrary parameter. The gauge field A µ and the ghost pair (b, c) are Lie algebra valued. The field strength is given by µ T a . Our next step is to build the generator of tree level multiparticle currents with off-shell external legs, just like in the bi-adjoint case. Even though ghosts do not appear in classical configurations, they will be treated for now as scalars with the wrong statistics. The equations of motion derived from (17) are given by Note that the solutions of (18) match classical Yang-Mills solutions when ∂ µ A µ = b = c = 0. Multiparticle solutions are obtained via the ansatz The currents A µ P reduce to ordinary vector polarizations ǫ µ p for one-lettered words. The multiparticle ansatz can then be plugged back in (18), leading to the following recursions, The sum over P = QRS denote deconcatenations of the word P into Q, R, and S. Following the Faddeev-Popov procedure, all physical single-particle polarizations are transversal (k p · A p = 0). Therefore, A µ P is identified with Berends-Giele currents and can then be used to compute color ordered amplitudes via the usual prescription [8]. Just like in the bi-adjoint case, we can determine a generator of off-shell trees by dropping the on-shell condition for the single-particle states, and therefore solving the equations of motion (18) only at the multiparticle level. For instance, where the sum is now restricted to single-particle states. Interestingly, this is enough to show that A µ P still satisfies the shuffle identity A µ Q¡R = 0, with Q ¡ R yielding the sum over all possible shuffles between the words Q and R. At tree level it leads to the Kleiss-Kuijf relations [34]. At one-loop, when we sew two legs and sum over specific cyclic permutations of the remaining single-particle labels, the shuffle identity indirectly leads to the Bern-Dixon-Dunbar-Kosower (BDDK) relations [35].

V. YANG-MILLS: GLUON LOOP
From now on we will work with ξ = 1, which helps to implement the recursion (20a) through a scalar-like propagator 1/s P .
Towards the one-loop construction, let us consider the wordP = lP , explicitly factorizing the singleparticle label l, with associated polarization ǫ µ l and momentum k µ l . In this case we define A lP µ = ǫ ν l J P µν , with This is but a recasting of equation (20a) that singles out the particle l, though physically meaningful: the current J P µν is the one-loop pre-integrand for a gluon loop.
Let us first examine the following object, where ǫ µ n is the polarization of an off-shell leg with momentum k µ n . The analogy with (10) is clear. The sewing procedure is simply ǫ µ l ǫ ν n → η µν /k 2 l , with k l = −k µ n , yeilding a look-alike one-loop integrand I 1-loop (P ) ≈ η µν J P µν for a single-trace color-ordered correlator. Once more, the issue with this construction is that the current J P µν is no longer cyclic in the word P , so its cyclic completion has to be introduced by hand.
As in the bi-adjoint case, we take the loop momentum to be ℓ µ ≡ k µ l , with k µ P = 0. If we explicitly remove tadpole contributions, given by the first line in (22), the one-loop integrand can be cast as with The sum over P = [QR] is analogous to the bi-adjoint case. In particular the factor of 1/2 accounts for the equivalent contributions mentioned after equation (14). The sum over P = [QRS] is simpler, related to quartic vertices in Yang-Mills: the deconcatenations of all cyclic permutations in P are inequivalent and have to be included.
Because of the color structure, as in the bi-adjoint construction, (24) reproduces each of the diagrams contributing to the one-loop off-shell integrand without repetitions. The only subtlety missed by (24) is the automorphism of the two-point integrand with P = 12 and P = 21, which has to be taken into account separately. It corresponds to the color-stripped one-loop bubble diagram with external moment k µ 2 = −k µ 1 = k µ . From (22) and (25), we obtain with (27) matching the textbook computation using Feynman diagrams. The extra factor 1/2 cancels the automorphism over counting.
We have also checked that equation (24) reproduces the three-and four-point one-loop integrands obtained using Feynman rules. These are of course only consistency checks, since our construction goes far beyond. Just like at tree level, the recursive implementation of (20a) and (22) greatly simplify the involved computational work, and can be easily implemented via commonly available software for symbolic computation.

VI. YANG-MILLS: GHOST LOOP
The steps for the definition of a ghost loop are very similar to the ones taken before. In this case, we have to consider off-shell multiparticle currents with one ghost external leg labeled by l, either b l or c l , such that b lP = b l B P and c lP = c l C P , with The currents B P and C P involve only gluons, since the ghost polarization has been explicitly stripped off. We then define the one-loop integrands where k P = 0, and the tadpole contributions have been removed. The cyclic completion is being enforced in the sums following the same recipe of the pure gluon case, yielding the full diagrammatic expansion without redundant contributions. Note that B P =C P when (k Q · A Q ) = 0, i.e., when the external gluons are on-shell. The full one-loop integrand with off-shell external gluons is where the ghost contribution appears with a minus sign (fermionic loop).
As an example, we present the one-loop gluon selfenergy, where k ≡ k 2 = −k 1 , Π µν = Π(k 2 )(k 2 η µν − k µ k ν ), and The loop integral was performed via an analytic continuation from k 2 < 0 and the dimensional regularization is implicit. The poles of Γ(1 − d/2) encode infrared as well as ultraviolet divergences.

VII. FINAL REMARKS
We have established here a robust framework for computing one-loop off-shell integrands in colorordered theories from classical equations of motion. The main ingredient in this construction is the observation that multiparticle solutions can be viewed as a generator of off-shell trees in a given field theory. Because of the color structure, it is straightforward to identify the cyclic completion necessary to obtain the full one-loop integrand.
By construction, the one-loop integrands in (11) for the bi-adjoint scalar and in (30) for Yang-Mills lead to single-trace partial amplitudes, which we denote by A n;0 (with n = |P |). Therefore, the full one-loop amplitudes in these theories are more conveniently represented by the Del Duca-Dixon-Maltoni color decomposition [36]. For instance, in Yang-Mills we have where σ is the color order (i.e., the word P in our formulas), and c n (σ) is the color basis defined by nested commutators of the group generators, with shorthand T σ = T aσ . In the sum, S n−1 denotes permutations of (n− 1) legs, and R denotes reflection. Although (33) is expanded in a single trace basis, it encodes also the double-trace (non-planar) contributions, which can be determined via the BDDK relations (see e.g. [37] for a nice summary of different color decompositions). The recursive character of the one-loop preintegrands (9), (22), and (28), represent an objective simplification over the traditional diagrammatic approach, since they can be algorithmically implemented without extra effort. More than that, their present form enable a transparent identification of the corresponding diagrams. For example, the external leg bubbles can be read off from (25) through the currents A Rµ with |R| = |P | − 1. And finally, the one-loop preintegrands can be directly used as building blocks for higher-loop integrands via different sewings, since all external legs are off-shell.
An interesting feature in our proposal is that a priori no Lagrangian is required. Therefore we can compute the one-loop off-shell scattering of field theories that are known only at the level of equations of motion. For example, six-dimensional N = (2, 0) superconformal field theory, which is supposed to describe the low energy limit of M 5-branes [38,39]. See also [40][41][42] for more details on the quantization of non-Lagrangian theories. More generally, our results can be applied to a variety of colored theories, including non-linear sigma model, Chern-Simons, super Yang-Mills, etc. For gauge theories coupled to matter, like QCD, our method becomes slightly more in-volved since we lose the rigidity of the color structure. Preliminary results on this will be reported in a different work. Perhaps more noteworthy is the fact we can generalize the results of [27] to off-shell graviton trees, including ghosts, therefore introducing a recursive tool for computing one-loop off-shell integrands in Einstein gravity [43].