Open loop amplitudes and causality to all orders and powers from the loop-tree duality

Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes are the main bottleneck in perturbative quantum field theory. The loop-tree duality opens multiloop scattering amplitudes to non-disjoint tree dual amplitudes by introducing as many on-shell conditions on the internal propagators as independent loop momenta, and is realized by modifying the usual infinitesimal imaginary prescription of Feynman propagators. Remarkably, non-causal singularities of the unintegrated amplitudes are explicitly cancelled in the dual representation, while the causal and anomalous threshold, soft and collinear singular structures emerge clearly in a compact region of the loop three-momenta, enabling a simultaneous computation with the extra emission real matrix elements through suitable momentum mappings. Based on the original formulation of the loop-tree duality, we present in this letter very compact and definite dual representations of a series of multiloop topologies with arbitrary powers of the Feynman propagators. These expressions are sufficient to describe any scattering amplitude up to three-loops, and their clear recurrence structure allows to conjecture other topologies with more complex combinatorics. Causal and infrared singularities are also manifestly characterized in these expressions.

LTD opens any loop diagram to a forest (a sum) of nondisjoint trees by introducing as many on-shell conditions on the internal loop propagators as the number of loops, and is realized by modifying the usual infinitesimal imaginary prescription of the Feynman propagators. The new propagators with modified prescription are called dual propagators. LTD at higher orders proceeds iteratively, or in words of Feynman [23,24], by opening the loops in succession. While the position of the poles of Feynman propagators in the complex plane is well defined, i.e. the positive (negative) energy modes feature a negative (positive) imaginary component due to the momentum independent +ı0 imaginary prescription, the dual prescription of dual propagators is momentum dependent. Therefore, after applying LTD to the first loop, the position of the poles in the subsequent loop momenta moves up and down on the real axis. The solution found in Ref. [4,5] was to reshuffle the imaginary components of the dual propagators by using a general identity that relates dual with Feynman propagators in such a way that propagators entering the second and successive applications of LTD are Feynman propagators only. This procedure requires to reverse the momentum flow of a few subsets of propagators in order to keep a coherent momentum flow in each LTD round. Recent papers have proposed alternative dual representations [19][20][21][22]. In Ref. [19,20], an average of all the possible momentum flows is proposed, which requires a cumbersome calculation of symmetry factors. In Ref. [21,22], the Cauchy residue theorem is applied iteratively by keeping track of the actual position of the poles in the complex plane. The procedure requires to close the Cauchy contours at infinity from either below or above the real axis in order to cancel the de-arXiv:2001.03564v1 [hep-ph] 10 Jan 2020 pendence on the position of the poles. In this letter, we follow a new strategy to generalize the LTD representation of loop scattering amplitudes to all orders, including also those with arbitrary powers of the original Feynman propagators. As in the original representation [4,5,13,14], we reverse sets of internal momenta whenever it is necessary to keep a coherent momentum flow, and we close the Cauchy contours always in the lower complex half plane. Causality [6,15,[25][26][27][28][29][30] is also used as a powerful guide to select which kind of dual contributions are endorsed, and then construct suitable ansätze that are proven by induction. This procedure allows to obtain explicit and very compact analytic expressions of the LTD representation for a series of loop topologies to all orders.

LOOP-TREE DUALITY TO ALL ORDERS AND POWERS
The internal propagators of any multiloop scattering amplitude can be classified into different sets or loop lines, each set collecting all the propagators that depend on the same single loop momentum or a linear combination of them. To simplify the notation, we denote by s the set of all the internal propagators that depend on the loop momentum s , with q is = s +k is their momenta, i s ∈ s, and k is a linear combination of external momenta, {p j } N . The usual Feynman propagator of one single internal loop particle is where q (+) with q is,0 and q is the time and spacial components of the momentum q is , respectively, m is its mass, and ı0 the usual Feynman's infinitesimal imaginary prescription. We extend this definition to encode in a compact way the product of the Feynman propagators of one set or the union of several sets Here, we contemplate the general case where the Feynman propagators are raised to an arbitrary power. Still, multiple powers will appear only implicitly in the following. A typical L-loop scattering amplitude is expressed as (4) in the Feynman representation, i.e. as an integral in the Minkowski space of the L-loop momenta over the product of Feynman propagators and the numerator N ({ i } L , {p j } N ), which is given by the Feynman rules of the theory. The integration measure reads [31,32], with d the number of space-time dimensions.
In the next sections, we will derive the LTD representation of the multiloop scattering amplitude in Eq. (4), and will present explicit expressions for several general topologies to all orders and arbitrary powers of the Feynman propagators.
Beyond one-loop, any loop subtopology involves at least two loop lines that depend on the same loop momentum. We define the dual function as where G F (s, t) represents the product of the Feynman propagators that belong to the two sets s and t. Each of the Feynman propagators can be raised to an arbitrary power. Notice that in Eq. (5) only the propagators that belong to the set s are set consecutively on-shell, and the Cauchy contour is closed always from below the real axis. For single power propagators and s = t, Eq. (5) provides the usual dual function at one loop [3] If some of the Feynman propagators are raised to multiple powers, then Eq. (5) leads to heavier expressions [5] but the location of the poles in the complex plane is the same as in the single power case.
Then, we construct the nested residue involving several sets of momenta In Eqs. (5) and (7), we can also introduce numerators and define the corresponding unintegrated open dual amplitudes A (L) D (1, . . . , r; n) by replacing the Feynman propagators by the integrand of Eq. (4) (see e.g. Ref. [13] at two-loops). Also, the energy component of the loop momenta can be replaced by the scalar product η · q ir , with η a future-like vector, to generalize the nested residue to an arbitrary coordinate system as in the original formulation of LTD [3]. With this compact notation, we express very easily the dual representation of benchmark multiloop scattering amplitude topologies to all orders. The causal behavior of Eq. (8) is also clear and manifest. The dual representation in Eq. (8) becomes singular when one or more off-shell propagators eventually become on-shell and generate a disjoint tree dual subamplitude. If these propagators belong to a set where there is already one on-shell propagator then the discussion reduces to the one-loop case [6]. We do not comment further on this case. The interesting case occurs when the propagator becoming singular involves the set with all the propagators off-shell [15]. For example, the first element of the sum in Eq. (8) features all the propagators in the set 1 off-shell. One of those propagators might become onshell, and there are two potential solutions, one with positive energy and another with negative energy, depending on the magnitude and direction of the external momenta [6,15]. The solution with negative energy represents a singular configuration where there is at least one on-shell propagator in each set. Therefore, the amplitude splits into two disjoint trees, with the momenta over the causal on-shell cut pointing to the same direction. Abusing notation: The on-shell singular solution with positive energy, however, is locally entangled with the next term such that the full LTD representation remains non-singular in this configuration  (1, 2, 3, . . . , n) . These local cancellations also occurs with multiple power propagators. They are the known dual cancellations of unphysical or non-causal singularities [6,[13][14][15] and they are essential to support that all the causal and anomalous thresholds as well as infrared singularities are restricted to a compact region of the loop three-momenta. Causality determines that the only surviving singularities fall on ellipsoid surfaces in the loop three-momenta space [7,8,22]. These causal singularities collapse to finite segments for massless particles leading to infrared singularities and are bounded by the magnitude of the external momenta, thus enabling the simultaneous generation with the tree contributions describing real emissions of extra radiation through suitable mappings of momenta, as defined in four-dimensional unsubstraction (FDU) [9][10][11]. A similar situation happens with the last term of the sum in Eq. (8) that features a potential causal singularity when all the on-shell momenta are aligned in the opposite direction over the causal on-shell cut, A j,0 ). The most notable property of this expression is that it is explicitly free of unphysical singularities, and the causal singularities occur, as expected, when either λ + 1,n or λ − 1,n vanishes, depending on the sign of the energy component of k n , in the loop threemomenta region where the on-shell energies are bounded, q (+) i,0 < |k 0,n |. This property also holds for powered propagators, with then λ + 1,n and λ − 1,n raised to specific powers.

NEXT-TO-MAXIMAL LOOP TOPOLOGY
The next multiloop topology in complexity, see Fig. 2, contains one extra set of momenta, denoted by 12, that depends on the sum of two loop momenta, q i12 = − 1 − 2 + k i12 . We call it next-to-maximal loop topology (NMLT). This topology appears for the first time at three loops, i.e. L = n − 1 with n ≥ 4, and its LTD representation is given by the compact and factorized expression The first term on the r.h.s. of Eq. (12) represents a convolution of the two-loop subtopology involving the sets (1,2,12) with the rest of the amplitude, such that two propagators in this subtopology are set simultaneously on-shell. In the second term on the r.h.s. of Eq. (12), the set 12 remains off-shell while there are on-shell propagators in either 1 or 2, and all the inverted sets from 3 to n contain on-shell propagators. For example, at three loops (n = 4), these convolutions are interpreted as MLT (1, 2, 12) ⊗ A and In total, the number of terms generated by Eq. (12) is 3n − 4.
Causal thresholds and infrared singularities are then determined by the singular structure of the A MLT (1,2,12) subtopology, and by the singular configurations that split the NMLT topology into two disjoint tree amplitudes with all the on-shell momenta aligned over the causal cut. Again, the singular surfaces in the loop three-momenta space are limited by the external momenta, and all the non-causal singular configurations that arise in individual contributions undergo dual cancellations.
The last term on the r.h.s. of Eq. (16) is fixed by the condition that the sets (2, 3, 23) can not generate a disjoint subtree. The second term in the r.h.s. of Eq. (15) contains a twoloop subtopology made of five sets of momenta, A MLT (1 ∪ 23, 2, 3 ∪ 12), which is dualized through Eq. (8). The total number of terms generated by Eq. (15) is 8(n − 2). As for the NMLT, the causal singularities of the N 2 MLT topology are determined by its subtopologies and by the singular configurations that split the open dual amplitude into disjoint trees with all the on-shell momenta aligned over the causal cut. Any other singular configuration is entangled among dual amplitudes and cancels. Finally, let us comment on more complex topologies at higher orders. Consider for example the multiloop topology made of one MLT and two two-loop NMLT subtopologies. This case appears for the first time at four loops. This topology is open into non-disjoint trees by leaving three loop sets off-shell and by introducing on-shell conditions in the others under certain conditions; either one off-shell set in each subtopology, or two in one NMLT subtopology and one in the other with on-shell propagators in all the sets of the MLT subtopology. Once the loop amplitude is open into trees, the singular causal structure is determined by the causal singularities of its subtopologies, and all the entangled non-causal singularities of the forest cancel.

CONCLUSIONS
We have reformulated the loop-tree duality at higher orders and have obtained very compact open-into-tree analytical representations of selected loop topologies to all orders.
These loop-tree dual representations exhibit a factorized cascade form in terms of simpler subtopologies. Since this factorized structure is imposed by the opening into non-disjoint trees and by causality, we conjecture that it holds to all loop orders and topologies. Remarkably, specific multiloop configurations are described by extremely compact dual representations which are, moreover, free of unphysical singularities. We have tested this property with several topologies. Therefore, we also conjecture that analytic dual representations in terms of only causal denominators are always plausible.
The explicit expressions presented in this letter are sufficient to describe any scattering amplitude up to three loops. Other topologies that appear for the first time at four loops and beyond have been anticipated, and will be presented in a forthcoming publication. This reformulation allows for a direct and efficient application to physical scattering processes, and is also advantageous to unveil formal aspects of multiloop scattering amplitudes.