Causality and loop-tree duality at higher loops

We relate a $l$-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original $l$-loop graph. Causality requires that the propagators of the trees have a modified $i\delta$-prescription and we present a simple formula for the correct $i\delta$-prescription.


INTRODUCTION
Relating loop integrals to trees goes back to Feynman [1].The Feynman tree theorem allows us to relate a lloop Feynman integral with N internal propagators to a l-fold phase space integral with a number of cuts N cut on the original integrand, with N cut ranging from l to N .One could argue that a better name for this theorem would be the Feynman forest theorem, as in general (i.e. for N cut > l) the integrand corresponds to a set of trees, i.e. a forest.This is not very convenient: If more than l cuts are present, each additional cut imposes a non-trivial constraint on the phase space integration.
What we would like to have is a formula which relates a l-loop Feynman integral to a l-fold phase space integral without any additional constraints.The integrand of the phase space integral then corresponds to a tree, not a forest, and is obtained from the original integrand by exactly l cuts.For one-loop integrals this was achieved in [2].An important result of this paper was the statement that the uncut propagators have a modified iδ-prescription.The usual iδ-prescription for a Feynman propagator is where δ > 0 is an infinitesimal small quantity.The modified iδ-prescription is a consequence of causality.We call propagators with a modified iδ-prescription dual propagators.
In this letter we present the generalisation to an arbitrary loop number l.A comment is in order: A generalisation of loop-tree duality to two loops and beyond has already been considered in [3].However, the final formulae presented there are not particular elegant and involve a mixture of Feynman propagators and one-loop dual propagators.Our result is more aesthetic: All uncut propagators are dual propagators, with a simple dual iδ-prescription, which reduces in the one-loop case to the one of Catani et al. [2].A dual propagator is of the form Only the sign of the function s j (σ) is relevant.The function s j (σ) depends on the energy E j and the energies of the cut propagators E σ1 , ..., E σ l and will be given in eq. ( 27) or alternatively in eq. ( 32) below.
Let us also briefly comment on the difference of the loop-tree duality approach with the so-called Q-cut approach [4]: The latter involves propagators linear in the loop momenta, where the information due to the infinitesimal imaginary part is lost if all propagators have the same small imaginary part as in eq. ( 1).The iδ-prescription is restored by introducing different infinitesimal small imaginary parts for the internal propagators and averaging over all possible relative orderings.In our approach, only propagators quadratic in the loop momenta occur.Furthermore, the modified iδ-prescription of the dual propagators follows directly from iδ-prescription of the Feynman propagators.Our approach applies to massless and massive particles.

NOTATION
Let Γ be a Feynman graph with l loops, n external lines and N internal edges.We denote by E Γ = {e 1 , ..., e N } the set of internal edges.A spanning tree for the graph Γ is a sub-graph T of Γ, which contains all the vertices of Γ and is a connected tree graph [5].If T is a spanning tree for Γ, then it can be obtained from Γ by deleting l internal edges, say {e σ1 , ..., e σ l }.We denote by σ = {σ 1 , ..., σ l } the set of indices of the deleted edges and by C Γ the set of all such sets.Thus |C Γ | gives the number of spanning trees for the graph Γ.
Each σ ∈ C Γ defines also a cut graph T cut , obtained by cutting each of the l internal edges e σj into two halfedges.The 2l half-edges become external lines of T cut .The graph T cut is a tree graph with n + 2l external lines.
We denote the external momenta of the graph Γ by p 1 , ..., p n and the internal momenta by k 1 , ..., k N .We will assume that the internal momenta have been labelled such that the first l internal momenta k 1 , ..., k l form a basis of independent loop momenta.For each internal edge we set We will assume that D i = D j for i = j.Otherwise we consider a reduced graph Γ ′ with edge e j contracted and a higher power of the propagator associated to edge e i .
For a function f depending on a D-dimensional momentum variable k = (E, k), where the vector k is (D − 1)-dimensional, we either write f (k) or f (E, k).We would like to integrate the function f over the hyperboloid k 2 = m 2 .The quantities give the integrals over the forward hyperboloid and the backward hyperboloid, respectively.As a short hand notation we set for the integral over the forward and the backward hyperboloid.

LOOP-TREE DUALITY
Let P Γ be a polynomial in the loop momenta.We consider We split each loop integration into an integration over the energy and the spatial components of the loop momentum: We perform the energy integrations with the help of the residue theorem.Let us assume that the polynomial P Γ is such that all energy integrations over half circles at infinity vanish.This assumption is always satisfied for scalar integrals where P Γ = 1.If this assumption is not met, we may enforce it by subtracting local ultraviolet counterterms from the integrand [6,7].Let E ⊂ C l be the set of points E = (E 1 , ..., E l ), where l internal propagators go on-shell and removing the corresponding edges gives a spanning tree.We have This number is easily obtained from the number of spanning trees and the 2 l solutions per spanning tree.For generic values of p 1 , ..., p n and k 1 , ..., k l the points of E are distinct.Points in E coincide if in addition to the cut propagators one or more uncut propagators go onshell.We distinguish the cases of a pinch singularity and a non-pinch singularity.For a non-pinch singularity we may deform the integration contour for the spatial variables k 1 , ..., k l into the complex domain.The modified iδ-prescription given in eq. ( 27) tells us in which direction we should deform.This is exactly the raison d'être for the present article.For a pinch singularity we have an infrared singularity.This singularity is either regulated by dimensional regularisation or cancelled in the combination with real contributions according to the Kinoshita-Lee-Nauenberg theorem [8,9].Let σ ∈ C Γ be a set of indices defining a spanning tree.For each cut edge we choose an orientation and we may take the l independent loop momenta to be the loop momenta flowing through the edges e σ1 , ..., e σ l with the chosen orientation.Let be a solution to In total there are 2 l solutions E σ , ..., E (2 l ) σ , given by Let us denote by n σ .We set We define the local residue [10] at The integration in eq. ( 13) is around a small l-torus encircling with orientation We consider Let us make one remark: Eq.( 16) is not a global residue for the ideal D is just the sum over the 2 l local residues, without any sign factors.This sum vanishes, whereas the sum in eq. ( 16) does in general not.
Proposition 1.The expression in eq. ( 16) is independent of the orientation of the edges e σ1 , ..., e σ l .
Proof.It is sufficient to show this for the case where we change the orientation of a single edge e σa .We may use partial fraction decomposition in the variable E σa .It is sufficient to show this for the case where r + , r − and c are E σa -independent, since higher poles and polynomial terms don't contribute to the residue.(Polynomial terms are absent due to our assumption that the integration over half circles at infinity vanish.)We have 2 α=1 (−1) Now let us change the orientation.We set E ′ σa = −E σa and Then Theorem 1.With f as in eq. ( 12) we have where the contour of integration on the left-hand side is along the real axes separating the poles at + √ ... from the poles at − √ ....
Proof.We prove eq. ( 21) by induction on the number of loops l.We start with l = 1.The function f is given in this case by Closing the contour below sums up the residues in the lower complex half plane and we obtain The minus sign comes from the clockwise orientation of the contour.If on the other hand we close the contour above we pick up the residues in the upper complex half plane and we obtain Both eq. ( 23) and eq.( 24) express the original integral as a sum of residues, and so does the average of the two: This agrees with eq. ( 21) for the case l = 1.
Let us now assume that eq. ( 21) is correct for graphs with (l −1) loops.We show that eq. ( 21) holds for graphs with l loops.Consider a graph Γ with l loops and pick a cycle (i.e. a closed path) in this graph.To this cycle corresponds an energy integration and we may take the residues as we did in the one-loop case and average over the two cases obtained by closing the contour in the upper respectively lower complex half plane.Taking a single residue will put one edge of the cycle on-shell and will fix the energy of this edge.This corresponds to cutting the cycle at this edge and we obtain a graph Γ ′ with (l − 1) loops.For Γ ′ we may use the induction hypothesis and doing this for every single residue gives eq. ( 21).
Let us now specialise to the case ν 1 = ... = ν N = 1 and let us work out the iδ-prescription for the uncut propagators.
Theorem 2. Let us assume again that P Γ is a polynomial in the loop momenta such that all energy integrations over half cycles at infinity vanish.If all propagators occur to power one, we have where s j (σ) is defined as and E is defined as follows: The set σ = {σ 1 , ..., σ l } ∈ C Γ defines a tree T cut obtained from the graph Γ by cutting the internal edges C σ = {e σ1 , ..., e σ l }.Cutting in addition the edge e j ∈ E Γ \C σ will give a two-forest (T 1 , T 2 ).We orient the external momenta of T 1 such that all momenta are outgoing.Let π be the set of indices corresponding to the external edges of T 1 which come from cutting the edges C σ of the graph Γ.The set π may contain an index twice, this is the case if both half-edges of a cut edge belong to T 1 .Then define E by Proof.Theorem 2 is a specialisation of theorem 1 to the case where the integrand has only single poles.The calculation of the residues yields where we neglected on the right-hand side the infinitesimal small imaginary part.It remains to work out the sign of the imaginary part of the uncut propagators.Let us consider D j and with the notation as above the tree T 1 .The external edges of T 1 are given by e j , the set E cut = {e π1 , e π2 , ...} and possibly a subset E ext of the external edges {1, ..., n} of the original graph Γ.Energy conservation relates E j to minus the sum of the energies of all other external particles of the tree T 1 .The energies corresponding to the edges from E ext are real, the energies corresponding to the edges from E cut have an infinitesimal small imaginary part.Taylor expansion to first order in δ gives By a slight abuse of notation we denote the O(δ 0 )-term again by E πa .Thus, the replacement makes the infinitesimal imaginary part explicit.Let us now look at D j and expand to first order in δ: For the O(δ)-term we have Although we singled out the tree T 1 from the two-forest (T 1 , T 2 ) it is easily checked that the definition of s j (σ) is invariant under the exchange Theorem 2 is the main result of this letter.It allows us to express a Feynman integral with no raised propagators as a sum of phase space integrals.Each phase space integral corresponds to a spanning tree of the original graph.The integrand of each phase space integral corresponds to a cut graph, where exactly l internal propagators have been cut and the remaining (N − l) internal propagators have a modified iδ-prescription given by eq. ( 27).Theorem 2 is the specialisation of theorem 1 to Feynman integrals with no raised propagators.For Feynman integrals with raised propagators we may still use theorem 1.The only change is that the computation of the residues is more involved.Residues of Feynman integrands with raised propagators have been considered in [11].
Although we defined the modified iδ-prescription in terms of energies in a specific Lorentz frame, we may easily formulate it in a Lorentz-covariant way: Let η be a Lorentz vector with η 0 > 0 and η 2 ≥ 0. Then s j (σ) is given by Let us look at an example.Fig. 1 shows a two-loop eightpoint graph Γ.There are 35 spanning trees, and each spanning tree defines a cut graph.Fig. 2 shows an example of a cut graph corresponding to σ = {3, 9}.We also indicate in fig. 2 an orientation for the edges e 3 , e 5 , e 6 and e 9 .As an example we consider s 5 (σ) and s 6 (σ).
The sign of the imaginary part is determined by FIG. 2: A cut diagram corresponding to σ = {3, 9}.We also indicate an orientation for the edges e3, e5, e6 and e9.
There are ample applications of our result.Let us give three examples.First of all, our result is directly geared towards numerical methods for higher-order computations [2,3,6,7, and paves the way to treat twoloop amplitudes numerically in an efficient and automated way.Secondly, and in a wider context, it sheds new light on the cancellation of infrared singularities.
Our result allows to discuss the singularity structure of loop integrands in terms of on-shell tree diagrams.This will be helpful at NNLO and beyond .Thirdly and on the formal side, our approach also suggests an extension of the concept of scattering forms [64][65][66][67] from tree-level towards loops.This will be explored in a future publication.