Unitarity Cuts of Integrals with Doubled Propagators

We extend the notion of generalized unitarity cuts to accommodate loop integrals with higher powers of propagators. Such integrals frequently arise in for example integration-by-parts identities, Schwinger parametrizations and Mellin-Barnes representations. The method is applied to reduction of integrals with doubled and tripled propagators and direct extract of integral coefficients at one and two loops. Our algorithm is based on degenerate multivariate residues and computational algebraic geometry.


Introduction
Perturbative scattering amplitudes of elementary particles in quantum field theories such as Quantum Chromodyanmics (QCD) are traditionally calculated by means of Feynman diagrams and rules. The Feynman approach is very intuitive, but suffers from a severe proliferation of terms and diagrams for increasing number of external particles and order in perturbation theory. Moreover, the simplicity of the underlying theory is by no means reflected by the results. Recent years have seen enourmous progress in quantitative determination of amplitudes at the one-loop level and beyond, catalyzed by the demand of precise theoretical predictions by the Large Hadron Collider (LHC) programme at CERN.
In this paper, we define generalized unitarity cuts of integrals that otherwise appear incompatible with the usual cut prescription. Naively, applying a unitarity cut to an integral with higher powers of propagators, the immediate result is singular. Nevertheless, amplitude representations that contain integrals with doubled propagators can lead to significant simplifications as argued in [53]. Moreover, integrals with repeated propagators naturally appear in many actual calculations. We explain that such cuts can be treated as degenerate multivariate residues using computational algebraic geometry. Our algorithm makes it possible to use more general integral bases for loop amplitudes. In particular, we provide examples of two-loop integral bases, whose elements contain purely scalars and yet are adaptable for unitarity purposes. What is more, the algorithm can be used to derive IBP relations for integrals with repeated propagators.
The integrand reduction of two-loop diagrams with doubled propagators has been achieved in [44], via the synthetic polynomial division method. However, the full integral reduction for integrals with doubled propagators has not been considered from the unitarity viewpoint.

Generalized Feynman Integrals
We define the generalized dimensionally regularized n-loop Feynman integral with arbitrary integer powers (also called indices) (σ 1 , . . . , σ p ) of p propagators by where the f k 's are linear polynomials with respect to inner products of the n loop momenta {ℓ i } and m external momenta {k i }. The canonical integral is recovered when all indices are set to unity. Generally speaking, the integral will have a nontrivial numerator function and is in that case referred to as a tensor integral. We can always bring the numerator into the form of additional propagators raised to negative powers. In a typical multiloop amplitude calculation, a huge number of Feynman integrals with different indices appear. A subset of the integrals can easily be reduced algebraically, e.g. by means of Gram matrix determinants. At first glance, the remaining integrals may seem irreducible and independent, but they are in fact related by integration-by-parts identities (IBP) [60]. Discarding the boundary term in D-dimensional integration, total derivatives vanish upon integration, which can be recast as linear relations among integrals with shifted exponents, The virtue of IBP relations is that within a given topology, a few integrals may be chosen as masters in the sense that all other integrals can be expressed in a basis of them. The importance is not to be underestimated. For example, the four-point massless planar triple box has several hundred renormalizable integrals which are reduced onto a linear combination of just three integrals with at most rank 1. IBP relations can be generated by public computer codes such as FIRE [61] and Reduze [62]. In practice, the production of IBP relations is quite time consuming and requires considerable amount of memory.

Direct Extraction of Integral Coefficients
We consider schematically an n-loop amplitude contribution which is denoted A n-loop . After reduction onto a basis of master integrals, the amplitude can be written where the c k 's are rational functions of external invariants. We refer to (1.5) as the master equation. For example, at one loop the integral basis is very simple and contains only boxes, triangles, bubbles and rational terms. The integral itself is calculated once and for all and therefore the problem of computing the amplitude reduces to determining the coefficients. The trick is to probe the loop integrand by applying generalized unitarity cuts on either side of the master equation. Originally, unitarity cuts were realized by replacing a set of propagators by Dirac delta functions restricting them to their mass shell. Recently, the framework of maximal unitarity at two [28,30] and three [33] loops has been developed as a continuation of the ideas of Britto, Cachazo, Feng [5] and Forde [6]. Maximal unitarity naturally deals with amplitude contributions whose factorization properties are accessible only away from the real slices of Minkowski space, for instance one-loop quadruple cuts and two-loop hepta-cuts of double boxes. Multidimensional complex contour integrals that compute multivariate residues provide the desired genelization of the localization property, for a given ξ ∈ C n . Here, Γ ǫ is a torus of real dimension n around the pole of the integrand at z = ξ. A generalized unitarity cut, even a maximal cut which puts as many propagators on-shell as possible, is typically shared among several basis integrals, hence intermediate algebra is in principle required. Instead one seeks to construct linear combinations of residues that in a certain sense are orthogonal to each other and thus project a single basis integral. In this way, the integral coefficient is expressed in terms of residues of products of tree amplitudes that arise when the loop amplitude factorizes. These combinations are subject to the consistency requirement that parity-odd integrands and total derivatives continue to vanish upon integration. The tree-level data is easily manipulated within the spinor-helicity formalism by means of for instance superspace techniques [54,55]. Direct extraction of master integral coefficients in maximal unitarity has been demonstrated for two-loop double boxes with up to four external massive or massless legs [28,31,34], at two loops with six points in N = 4 super Yang-Mills [29], for the nonplanar double box [32] and the three-loop triple box [33]. In these calculations, only basis integrals with single propagators were considered.

Multivariate Residues
To extract the integral coefficients, we need to calculate multivariate residues. In many cases, these residues can be simply evaluated by high-dimensional Cauchy's theorem and the Jacobian determinant. However, in some cases, like the unitarity cut of the triple box topology [33] or the unitarity cut of integral with doubled propagators, the residues are degenerate and have to be evaluated by algebraic geometry. In this section, we briefly review the concept and calculation of multivariate residues. Standard mathematical references include [63][64][65].
In two cases, a multivariate residue can be calculated straightforwardly, • A residue is non-degenerate, if the Jacobian at ξ is nonzero, i.e., In this case, by the multi-dimensional verion of Cauchy's theorem, the value of residue is simply [63], In this case, the n-dimensional contour in is factorized to the product of n univariate contours, Then, we can evaluate this residue by applying the univariate residue formula n times.
However, in general, a residue is neither non-degenerate nor factorizable. For example, consider a Feynman integrand with doubled (or higher-power) propagators, At a point ξ where f 1 (ξ) = · · · = f k (ξ) = 0, the Jacobian matrix is degenerate since In general, this type of residues is not factorizable. To evaluate them, we need the transformation law [63] in algebraic geometry.
Theorem 1 (Transformation law) Let {f 1 , . . . , f n } and {g 1 , . . . , g n } be two sets of holomorphic functions and g i = a ij f j , where a ij 's are holomorphic functions. Assume that for each set, the common zeros are discrete points. Let A be the matrix of a ij 's, then This theorem holds for both non-degenerate and degenerate residues. For Feynman integrals, the denominators are all polynomials. In this case, we can use the transformation law to convert a degenerate residue to a factorizable residue, via Gröbner basis. The algorithm involves the following steps [33]: 1. Calculate the Gröbner basis {g 1 , . . . , g k } of {f 1 , . . . , f n } in the DegreeLexicographic order and record the converting matrix r ij , such that g i = r ij f j .
2. For each 1 ≤ i ≤ n, calculate the Gröbner basis of {f 1 , . . . , f n } in the Lexicographic order of z i+1 ≻ · · · ≻ z n ≻ z 1 ≻ · · · z i . Pick the univariate polynomial in z i from this Gröbner basis and name it as h i .

For each
. The transformation matrix a ij = s ik r kj . By the transformation law, the degenerate residue is converted to a factorizable residue with the matrix a ij .
Finally, we have a comment on the residues from the maximal cut of integrals with doubled (or multiple) propagators. For the residue from a general integrand, to use (2.2), for each f i , we have to collect all powers of f i as one denominator. Otherwise, the common zeros of denominators are not discrete points, so the residue is not well defined.
Hence there is no ambiguity of defining denominators for the residue computation.
In our paper, we calculate the residues from the maximal cut of integrals with doubled and tripled propagators. The degenerate residues are evaluated by our Mathematica package MathematicaM2 1 , which calls Macaulay2 [66] to finish the computation of Gröbner basis. We demonstrate the multivariate residue computation explicitly by the one-loop box integral with double propagators. Then we show this method is general by two-loop examples.

Example: One-Loop Box
Consider a one-loop box integral with four massless legs k 1 , . . . , k 4 , where the denominators are, We suppress the Feynman iǫ-prescription as it is irrelevant for our purposes and assume for simplicity that all external momenta are massless and outgoing. Multiple consecutive external momenta are summed using the shorthand notation K ij = k i + · · · + k j . In the following discussion, we set D = 4. We fix a basis of the four-dimensional space time {k 1 , k 2 , k 4 , ω} where the spurious vector ω can be represented as such that ω is orthogonal to the subspace spanned by the momentum vectors. The list of irreducible scalar products (ISP) can then be chosen as The loop momentum ℓ can be parameterized as 5) and the Jacobian for this parametrization is The cut equations f 1 (ℓ) = · · · = f 4 (ℓ) = 0 have two solutions, There is only one master integral for one-loop box, the scalar integral, where · · · stands for integrals with fewer than four propagators. Localizing the contour around ξ 1 and ξ 2 , it is clear that the Jacobian of f 's in α's are nonzero. So by Cauchy's theorem in higher dimensions (2.4), the residues are Together with the spurious integral condition I 4 (1, 1, 1, 1)[ℓ·ω] ξ 1 = 0, we have the expression for the integral coefficient for the integral I 4 (σ 1 , . . . σ 4 )[N ], Now we consider integrals with doubled propagators, for example, I 4 (1, 1, 1, 2) [1]. The Jacobian of {f 1 , f 2 , f 3 , f 2 4 } in α's is zero at both ξ 1 and ξ 2 , so direct computation does not work. We can use the transformation law (2.8) to convert the denominators to a calculable form,  where M is 4 × 4 matrix and all entries are polynomials in α's, and (3.12) Hence, around either ξ 1 or ξ 2 , So the degenerate residue can be calculated by applying the univariate Cauchy's theorem four times. The explicit form of M is found by our package MathematicaM2. The residues for I 4 (1, 1, 1, 2) [1] are (3.14) So we have Similarly, using the same method, we find that, These results are consistent with the IBP relations in the D = 4 limit. For instance, from FIRE [61],

Example: Planar Double Box
We now proceed to two-loop integrals. The generalized dimensionally regularized two-loop planar double box scalar integral ( fig. 1) with arbitrary powers of propagators reads where the seven inverse propagators {f i } are given by Closed form expressions for planar double integrals can be found in [56,57]. As in the one-loop example, we choose {k 1 , k 2 , k 4 , ω} as basis of the four-dimensional space time where again the spurious vector ω can be represented as  4) and the integrand basis contains 16 spurious and 16 nonspurious elements. Whence the nine-propagator double box topology is defined by where f 8 = ℓ 1 · k 4 and f 9 = ℓ 2 · k 1 are the nonspurious ISPs. Then we have P * * 2,2 [(ℓ 1 · k 4 ) n (ℓ 1 · k 2 ) m ] = P * * 2,2 (1, . . . , 1, −n, −m) (4.6) in the notation of [27,28].

Parametrization of Hepta-Cut Solutions
In order to expose the singularity structure of the loop integrand, we adopt a particularly convenient loop momentum parametrization of previous works, see for instance [28], It is elementary to show that the Jacobians to compensate for the change of variables from loop momenta to parameter space are where χ is a frequently used ratio of Mandelstam invariants, χ = s 14 s 12 . (4.10) The zero locus of the ideal generated by the polynomials f i defines a reducible elliptic curve associated with a sixtuply pinched torus whose components are Riemann spheres, The solutions can be summarized as follows, with (α 1 , α 2 , β 1 , β 2 ) = (1, 0, 0, 1) uniformly across all branches. Also, τ is defined by

Master Integral Projectors
The hepta-cut contours are subject to consistency requirements in order to ensure that certain integral relations are preserved after pushing the real slice integrals into C 8 . It is well known that the integrand can be parametrized in terms of four irreducible products, whose powers can be derived by renormalizability conditions and then multivariate polynomial division using the Gröbner basis method. The latter has been automated in the program BasisDet [38]. Both the spurious and nonspurious part of the basis contains sixteen elements. At the level of integrated expressions, the amplitude is expressed as a linear combination of just two masters. Accordingly, we demand that all integral identities on which the reduction relies are respected. The full list of IBP identities is available in [32]. We arrange all parity-odd and IBP constraints as a 32 × 8 matrix M that acts on the residue weights Ω. The two corresponding submatrices have rank 4 and 2 respectively. We then define two master contours by   Indeed, as ǫ → 0 the tensor integral drops out and the integral with a doubled propagator and the canonical scalar integrals equate up the factors written above. Any other powers of propagators may be treated similarly. A complete list of heptacuts of planar doubled boxes with a doubled propagator is given in appendix A.

Example: Nonplanar Double Box
We define the four-point two-loop nonplanar double box integral (see fig. 3) in dimensional regularization by and adopt the convention for propagators and momentum flow of [35], All external and internal momenta are by assumption massless. We will consider fourdimensional unitarity cuts and therefore only reconstruct the master integral coefficients to leading order in the dimensional regulator. The Feynman integral itself was calculated in [58,59]. The set of vectors {k 1 , k 2 , k 3 , ω} where ω is the spurious direction forms a basis of four-dimensional momentum space. There are again four irreducible scalar products, and the minimal representation of the integrand consists of 19 spurious and 19 nonspurious monomials. Accordingly, we define the nine-propagator version of the two-loop crossed box integral by , (5.4)
The hepta-cut equations were solved using this parametrization in [32,35] and the resulting eight solutions are quoted here in table 1.

Residues of the Loop Integrand
Once the seven inverse propagators have been expanded in the loop momentum parametrization, it is straightforward to derive the hepta-cuts of the nonplanar double box scalar integral with single propagators as nondegenerate multivariate residues [32], .

A Scalar Integral Basis
The nonplanar double box amplitude contribution with four massless external legs has already been worked out in detail for an integral basis with a scalar integral and a rank 1 tensor [32,35]. However, the reduction identities of the preceding subsection suggest an equivalent integral basis in which the tensor intergal is eliminated. We thus project the amplitude onto two scalar integrals. Our master equation reads (see also fig. 4) . . , 1, 0, 0) + C 2 X * * 1,1,2 (2, 1, . . . , 1, 0, 0) + · · · (5.34) and we thus seek to determine C 1 and C 2 . It is not hard to show that the master integral projectors in this basis are The coefficients produced here agree in D = 4 with those worked out in [32,35] as can be verified using the IBP identity (5.32).

Conclusion
In this paper, we naturally generalized the maximal unitarity method to integrals with doubled propagators and provided a simple way of reducing integrals with doubled (or even higher-order) propagators onto a master integral basis. The residues of an integral with doubled propagators are degenerate, which cannot be directly calculated by Cauchy's theorem but can be evaluated by computational algebraic geometry methods (Gröbner basis). Then from the projector information, we obtain the master integral coefficients. This method has been successfully tested on several one-loop and two-loop examples.
Since the contour projector can be found by using IBPs without doubled propagators, our method implies that the complete set of IBPs (involving integrals with or without doubled propagators) can be derived from the set of IBPs without doubled propagators. Our method can also be used for converting between different integral bases.
There are several promising future directions. So far, the maximal unitarity method for two-loop and higher-loop has been tested only for diagrams in the D = 4 limit. Therefore our paper only obtained the finite part of the reduction of integrals with doubled propagators, but not the O(ǫ) contribution. We expect that the maximal unitarity method (including integrals with doubled propagators) can be generalized to D-dimensional cases by a contour integral in the extra dimension and analytic continuation in D. Moreover, the reduction algorithm for integrals with higher powers of propagators should apply seamlessly to massive external legs.
For the computational aspect, the multivariate residue calculation can be sped up by using the relation between multivariate residues and the Bezoutian matrix [65]. Then we do not need to find the Gröbner basis in the lexicographic order.

A Planar Double Box Hepta-Cuts
For the sake of completeness, we include all hepta-cuts of four-point planar double box scalar integrals with a doubled propagator. The ordering of the propagators follows that of the main text.

B Nonplanar Double Box Hepta-Cuts
We also provide explicit forms of the hepta-cuts of all four-point nonplanar double box integrals with a single doubled propagator and a scalar numerator. The overall signs are determined by consistency of relations among residues.