Two-loop Integrand Decomposition Into Master Integrals And Surface Terms

Two-loop Integrand Decomposition Into Master Integrals And Surface Terms

Harald Ita
Institut of Physics, University of Freiburg
D–79104 Freiburg, Germany
Abstract

Loop amplitudes are conveniently expressed in terms of master integrals whose coefficients carry the process dependent information. Similarly before integration, the loop integrands may be expressed as a linear combination of propagator products with universal numerator-tensors. Such a decomposition is an important input for the numerical unitarity approach, which constructs integrand coefficients from on-shell tree amplitudes. We present a new method to organize multi-loop integrands into a direct sum of terms that integrate to zero (surface terms) and remaining master integrands. This decomposition facilitates a general, numerical unitarity approach for multi-loop amplitudes circumventing analytic integral reduction. Vanishing integrals are well known as integration-by-parts identities. Our construction can be viewed as an explicit construction of a complete set of integration-by-parts identities excluding doubled propagators. Interestingly, a class of ‘horizontal’ identities is singled out which hold as well for altered propagator powers.

pacs:
11.15.Bt, 11.25.Db, 11.55.-m, 12.38.-t, 12.38.Bx

FR-PHENO-2015-011

I Introduction

Currently the experiments at the Large Hadron Collider (LHC) are entering a new energy and luminosity regime. Further upgrades are expected for a number of years to come. The increasing amount of data will allow to zoom into known physics and extend the discovery potential for new physics. An important ingredient in this quest are precise predictions which match the measurements’ standards. Predictions for key observables will be necessary, but providing a larger set of predictions beyond this minimal set will be a clear benefit. Here we present new theoretical methods towards these latter aims.

An important input for precision predictions are first-principle computations in perturbative quantum-field theory. In recent years, significant progress has been made by the theory community in providing predictions through automated fixed-order computations including quantum corrections VBFNLO ; BlackHat ; NJet ; MadLoop ; HelacNLO ; GoSam ; OpenLoops ; Recola . These have already lead to a wide class of next-to-leading order (NLO) predictions for Standard Model processes. In addition, a number of impressive two-to-two next-to-next-to-leading order (NNLO) results 2gamNNLO ; TopNNLO ; VVNNLO ; ZgamNNLO ; VjetNNLO ; HjetNNLO and further higher-order predictions HN3LO have become available. These developments have been driven by a combination of analytic and numerical advances for computing loop integrals. At one-loop level one can highlight explicit DDReduction and implicit methods UnitarityI ; GenUnitarityI ; GenUnitarityII ; GenUnitarityIII ; OPP ; NumUnitarity ; BlackHat for reducing (tensor) loop integrals to a standard set of master integrals. Similarly, at two-loop level, explicit analytic reduction techniques for integrals IBP ; Reduce ; Air ; Fire ; LiteRed ; Laporta play an important role. Here we discuss methods which bypass analytic integral reduction and make a numerical approach to multi-loop computations possible. Such methods are at the core of the unitarity based approaches BlackHat ; MZProgram ; HelacNLO ; NJet ; GoSam ; MadLoop to NLO predictions and allowed to push towards processes with many partons W3jB ; W3jR ; W4j ; W5j ; 5Jets . We are motivated by these results to explore a numerical unitarity approach for multi-loop amplitudes.

The unitarity method UnitarityI ; GenUnitarityI ; 2LoopUnitarityQCD ; MultiLoopUnitarity has continuously provided cutting edge results for formal as well as phenomenology oriented multi-loop amplitudes (see e.g. UVGravity ; BDS ; SYMUnitarity and 2LoopUnitarityQCD ; 2LoopAmpl5g ). This method relates universal representations of amplitudes in terms of master integrals to full scattering amplitudes. By comparing the analytic structure (e.g. branch cuts) of both representations the process dependent coefficients of master integrals are obtained. In this approach, the cutting operation simplifies the loop integrals to phase-space integrals over on-shell tree-level input. On the one hand, the strength of this approach arises from efficiently dealing with physical (on-shell) building blocks. On the other hand, the unitarity approach provides an implicit integral-reduction mechanism, since by cutting one targets coefficients of master integrals very directly.

In this article we discuss a numerical variant of the unitarity approach. This approach is well developed at one-loop level OPP ; NumUnitarity ; BlackHat and we extend it to higher-loop orders. In the numerical approach, the loop integrations are delayed to the very end of the computation. First one compares the rational integrands of the Feynman amplitudes with a universal basis of loop integrands. Delaying the loop integration, however, comes with a price; in order to maintain the equality of the integrand basis and Feynman amplitudes additional terms, i.e. surface terms, have to be added to the basis. These are terms that integrate to zero eventually but are required in intermediate steps. The explicit construction of the multi-loop surface terms is the main result of this article. The importance of the surface terms becomes clear in the remaining computational steps. Once the coefficients of the integrand basis have been obtained (through solving linear equations), the loop integration is performed. In this final step surface terms can be dropped and only the master integrals have to be provided to yield the loop amplitudes. In this way the reduction of tensor integrals is accomplished by the integrand parametrization and is implicit. Given the classification of surface terms we obtain a general, i.e. process and multiplicity independent, numerical algorithm.

Even though the unitarity method operates on-shell, the surface terms have to be known off-shell. This is required, since already computed unitarity cuts have to be subtracted in cuts with fewer on-shell propagators in order to avoid double counting. However, the cut conditions can only be relaxed if we have a way to take results off-shell. The prescription to go off-shell is provided through the integrand parametrization.

A number of recent developments have advanced the unitarity method to a promising approach for automated multi-loop computations in QCD. Parametrizations of loop integrands have been developed recently MLoopParamMO ; MLoopParamBFZ ; MLoopAlgGeo , which are given in terms of a minimal basis of irreducible integrands. These parametrizations identify a subset of the terms (spurious numerators) which integrate to zero, but not all. Thus, standard reduction techniques IBP ; Reduce ; Air ; Fire ; Laporta are required to obtain a final representation in terms of master integrals. Here we put forward a different type of representation of loop integrands which is organized into surface terms and master integrals. In our approach, the integral reduction is built-in and does not have to be performed in a second step, which is important in a numerical approach. Furthermore, in analogy to the numerical one-loop unitarity approach, the integral coefficients can be computed through generalized unitarity cuts 2LoopMaxCuts by solving linear equations (e.g. via Fourier transforms). This approach does not require integration over multi-dimensional phase spaces and thus differs from the direct extraction of integral coefficients MLoopContours or possible extensions using the duality between master integrals and homology cycles of the phase spaces YZhangElliptic ; MLoopCohom ; MLoopContoursCycles . Nevertheless, the latter approaches hold the promise to be very efficient once available in a complete way.

Technically, a number of new observations lead to the present construction. First of all, we combine integration-by-parts (IBP) identities and master integrals to parametrize the loop integrands. Although this approach is natural it has not been appreciated for the numerical unitarity approach so far. For the unitarity approach it is best to focus on IBP identities that do not involve integral topologies with doubled propagators KosowerIBP . The identities are obtained from a specialized set of vector fields in loop-momentum space. We provide the explicit form of such IBP vectors. Algorithms to obtain IBP vectors have been suggested in the original literature and improved in refs. sIBP ; IBPDiffGeom . We give a complete set of (off-shell) vectors for two-loop topologies. This construction is important in order to obtain compact analytic expressions as well as numerical control throughout momentum space. The presented construction reproduces the known results at one-loop level NumUnitarity .

Moreover, we find interesting properties of the IBP vectors; using general coordinate transformations to adjust the integration variables to the integral topology, IBP vectors can be constructed explicitly with pen and paper. In fact, the IBP vectors turn out to come in two types, (complexified) rotations in momentum space, which leave propagators invariant, and scaling transformations of the propagators. For our construction the former ‘horizontal’ generators will be most important. A special example of such horizontal vectors was given already in KosowerIBP (based on Gram determinants).

Although we construct special IBP relations that do not double propagators, we obtain a much bigger set of IBP relations. In fact, once the numerators of the horizontal IBP relations are obtained, the propagator powers may be changed to give new relations.

We observe that the IBP vectors are tangent vectors to the unitarity-cut phase spaces. This property allows one to link off-shell and on-shell information on unitarity cuts; we show (see section III.7) that surface terms from special IBP relations are as well surface terms on the unitarity-cut phase spaces. Interestingly, we observe as well a Lie-algebra structure which simplifies the construction of the IBP vectors. This structure appears to be fundamental linking unitarity cuts to full amplitudes (section IV.6).

Finally, the link between off-shell and on-shell information allows one to relate master integrands and surface terms to closed and exact holomorphic forms on the phase spaces, respectively. The number of master integrals is then given through topological properties of the unitarity-cut phase spaces, that is the number of closed-modulo-exact forms. The importance of cohomology for the construction of on-shell IBP identities has been discussed recently MLoopCohom . We setup a related but simplified on-shell approach to multi-dimensional phase spaces. Although we construct off-shell surface terms, the on-shell perspective serves as valuable guidance and a cross check. In fact, the on-shell construction is simpler and its completeness can be verified in a combinatorial way. We use the on-shell approach to verify that the constructed surface terms are complete.

Here we focus on planar two-loop integrals, however, our methods can be extended to non-planar as well as a full -dimensional approach, and we suggest how the generalizations can be done. In fact, we give IBP vectors for the non-planar two-loop topologies. Furthermore, we work with generic non-vanishing internal and external masses and, thus, capturing much of the -dimensional aspects. We believe that the IBP vectors are sufficient as well for most massless integrals; this is plausible assuming that factorization limits relate this massive information to the massless one111Exceptions may appear when only a single massless leg is attached to a loop.. Finally, we suggest a geometric interpretation of the IBP vectors (section IV.5) which makes generalizations to multiple loops natural.

The article is organized as follows. We start with a heuristic formulation of the central research question in section II. In section III we present important technical background and methods. This includes general coordinate transformations of the loop momenta as well as the discussion of tensor insertions and the unitarity cuts. The off-shell construction of two-loop surface terms is presented in section IV. The reader interested in the final result should be able to read starting from sections IV.3-IV.5 which give the IBP vectors, the formula for surface terms and the vectors’ geometric interpretation. In section IV.6 we speculate about the Lie-algebra structure of the IBP vectors. In section V we present the on-shell construction of surface terms and count master integrands which serves as valuable cross checks of the off-shell approach. In section IV.1 and section V.1 one-loop examples are given in order to illustrate the off-shell and on-shell constructions, respectively. Finally, we close with a summary and a discussion of a number of interesting future directions. Technical aspects of differential calculus are discussed in an appendix.

Ii Setup and Notation

We start with an heuristic introduction of the key structures the we will be dealing with.

ii.1 Loop integrand decomposition

We consider two-loop computations with the integral topologies as shown in figure fig. 1. Integrals typically include tensor insertions which are denoted by giving,

 I[t] = ∫dDℓdD~ℓt(ℓ,~ℓ)ρ0⋯ρN−1^ρ0⋯^ρ(^N−1)~ρ0⋯~ρ(~N−1), withρi=(ℓ−qi)2−m2i,^ρi=(^ℓ−^qi)2−ˆm2i,~ρi=(~ℓ−~qi)2−~m2i.

Momentum conservation is imposed . We will work with dimensional regularization keeping the loop-momentum dimensions as free parameters. The tensor insertions are assumed to be polynomial in the loop momenta as it is the case in Feynman amplitudes.

Given a complete basis of numerator tensors with the index labeling the basis elements, one can evaluate tensor integrals by first decomposing the tensor numerator into the basis,

 t(ℓ,~ℓ) = ∑\smalli ∈ numerator basis~di~mi(ℓ,~ℓ), (II.2)

with loop-momentum independent coefficients . In a second step one has to integrate all the basis tensor insertions. To this end, typically tensor reduction techniques IBP ; Reduce ; Air ; Fire ; Laporta are used to decompose the basis of tensor integrals into a small set of independent master integrals.

Here we aim to shortcut the step of the tensor reduction by constructing a particular numerator basis. Following the strategy of one-loop computations OPP ; NumUnitarity , we decompose the numerator tensors into the tensor insertions associated with master integrals and surface terms , which integrate to zero,

 t(ℓ,~ℓ) = ∑\smalli ∈ master integralsdimi(ℓ,~ℓ)+∑\smallj ∈ surface terms^djˆmj(ℓ,~ℓ), (II.3)

with the properties,

 Ii := ∫dDℓdD~ℓmi(ℓ,~ℓ)ρ0⋯~ρ(~N−1),∫dDℓdD~ℓˆmj(ℓ,~ℓ)ρ0⋯~ρ(~N−1)=0. (II.4)

Thus, we directly obtain the decomposition of the initial tensor integral in terms of master integrals ,

 I[t]=∑\smalli ∈ master integralsdiIi, (II.5)

while the coefficients drop out of the final result. The surface terms contribute only prior to the loop integration expressing for example angular correlations.

Within the (numerical) unitarity approaches one works at the integrand level and parametrizations (II.2) of the loop integrands are required. Parametrizations have been developed in the recent years MLoopParamMO ; MLoopParamBFZ ; MLoopAlgGeo , however, a decomposition in terms of surface terms and master integrals (II.3) would be important in order to avoid the explicit tensor reduction. The construction of the surface terms has so far not been developed sufficiently and we will provide this missing piece here.

It is important to know the surface terms off-shell, that is all over momentum space and in particular away from the regions of on-shell propagators. This is required on the one hand, to ensure that they in fact integrate to zero in the full loop integrals. On the other hand, even in the unitarity approach off-shell information is required to avoid double counting. That is, given a result for a unitarity cut it has to be subtracted in cuts with fewer on-shell conditions imposed. (E.g. at one-loop results from triple cuts have to be subtracted from two-particle cuts.) This can only be done if the initial cut results can be taken off-shell in a consistent way. A priori a cut, i.e. a product of on-shell tree amplitudes, cannot be taken off-shell. However, once we have used the cut to compute coefficients of an appropriate loop integrand parametrization (using on-shell momenta), we can take the latter off shell and subtract it from daughter cuts. Steps of this kind are explicit or implicit in almost all variants of the unitarity approach, but may possibly be circumvented by introducing phase space integrals or exploiting discrete symmetries GenUnitarityIII . Thus we require an off-shell representation of the loop integrands ideally in terms of surface terms. As a final remark we add that the surface terms have to be algebraic expressions in the loop momenta times propagators. This is the case since they should represent Feynman amplitudes, which have this property.

Integral relations such as IBP identities have all the properties needed for surface terms and can be used when available. This fact is very important for a numerical unitarity approach at higher-loop order. In principle IBP relations may be obtained through standard techniques. However, for the numerical unitarity cuts it is beneficial not to consider an integral basis with doubled propagators. Thus we will construct surface terms from the specialized IBP relations first introduced in ref. KosowerIBP which initially do not include integral topologies with doubled propagators.

An automated construction of the specialized IBP relations has been given in the original article KosowerIBP and has been advanced in ref. sIBP . Here we prefer to follow an analytic approach, since we require additional control over the expressions. That is we do not only need a compact representation of the surface terms, but we will also need sufficient numerical control when solving for the integral coefficients in all regions of phase space. Nevertheless, it would be instructive to compare the approaches in detail in the future. A geometric on-shell construction of specialized IBP relations has been put forward in refs. YZhangElliptic ; YZhangElliptic2 which is, however, not suitable for our purpose, since we require the full off-shell information of the surface terms. Nevertheless, we will use a related on-shell approach for cross checks below in section V.

Important structures of the loop integrals can be made manifest by using appropriate integration variables. The aim is to change from the loop-momentum components to using the inverse propagators as integration variables DOlive . Given the missmatch in the number of propagators and loop-momentum components additional internal variables have to be introduced, which we denote by indexed ’s in the following. The loop integrals are then given by integrations over the inverse propagators in addition to an internal integration over the -coordinates,

 I[t]=∫[dρ]ρ0⋯~ρ(~N−1)×t(ρ,α)μ(ρ,α)[dα], (II.6)

where is the non-trivial integration measure from the coordinate change. The expression and denote the differentials of the integration variables, ’s and ’s respectively. The insertion is the tensor evaluated in the new coordinates.

It is instructive to consider first the integration over an internal space with the inverse propagators held fixed. From this perspective we can now use the properties of the internal space to organize the computation. To give an example, at one-loop level the internal integration is performed over spheres 222Typically these spheres are part of the complex internal spaces which are tangent bundles of the real spheres .. We can think of the function being decomposed into a linear combination of spherical harmonics. Only the constant function gives a non-vanishing integral, while the higher harmonics integrate to zero. The later can be interpreted as surface terms. Thus surface terms are identified by relating the numerator tensors to spherical harmonics. Fittingly the IBP vectors that generate surface terms turn out to be generators of rotations along the internal space directions. When acting on an insertion , generators of rotations annihilate the invariant scalar parts, and give nontrivial representation of the rotation group otherwise, that is, non-trivial spherical harmonics. A similar picture holds at higher-loop level but it is also useful to consider the integral in a more formal manner. In formal terms we may relate the task of finding non-trivial integrals to understanding the cohomology of the internal spaces, so that exact forms (total derivatives) in the internal space are related to surface terms, while closed but non-exact forms are related to the non-vanishing master integrals. We will find this perspective useful when considering generalized cuts of the loop integrals.

Not all vanishing integrals arise from surface terms of the internal space alone. For example discrete symmetries can lead to further vanishing integrals. Will not consider the role of discrete symmetries further here, but focus on the identification of surface terms in the earlier sense.

ii.2.1 Maximal cuts

Once we transform to adapted coordinates (II.6) as described above we can naturally make contact with unitarity cuts. Formally, unitarity cuts amount to replacing propagators with delta distributions, . The insertion of the delta distributions localizes the integral to vanishing inverse propagators . The Jacobian factors from the coordinate change to adapted coordinates already provides the correct measure for the remaining integrations in the internal variables. (Details about the coordinate change can be found in section III.)

In the maximal cuts of a given integral topology all independent propagators are formally replaced by delta distributions. Once we localize to vanishing inverse propagators, the loop momentum takes on-shell values. Thus the on-shell loop momenta and tensor insertions are obtained by setting all the , and to zero. The internal -variables then are the coordinates of the on-shell loop-momentum space. We will refer to this subspace as the maximal-cut phase space in the following. The maximal-cut phase space shares many properties with the surfaces of fixed propagator values allowing to infer properties of the full loop integral from on-shell information.

ii.3 Surface terms as specialized IBP identities

Surface terms can be obtained from total derivatives starting from (sufficiently regular) vector fields ,

 ∫dDℓdD~ℓ[∂μ(uμt(ℓ,~ℓ)ρ0⋯~ρ(~N−1))+~∂ν(~uνt(ℓ,~ℓ)ρ0⋯~ρ(~N−1))]=0. (II.7)

The components of the vector fields are polynomial in the loop momenta to obtain relations between Feynman integrals. Typically doubled propagators appear when the derivatives act on them.

Doubled propagators can be avoided, by a very specific choice of vector-field insertions KosowerIBP fulfilling the equations,

 (uμ∂μ+~uν~∂ν)ρi=fi(ℓ,~ℓ)ρi, (II.8)

for all inverse propagators and similarly for and with independent functions and . Due to the chain rule, the doubled propagators are canceled for such special IBP vector fields (II.8). The index on the right-hand side is not summed over. The functions are again polynomial in the basic momentum contractions. Typically it is difficult to find this kind of vector field, however, we will point out a simplified construction when adapted coordinates are used.

The relation eq. (II.8) has an interesting on-shell interpretation. When specializing to the on-shell phase spaces with , we find that the vector fields turn into tangent vectors along the maximal-cut phase spaces: since the right-hand side of eq. (II.8) vanishes, the vectors generate translations that keep the propagators fixed to zero and thus point along on-shell phase space,

 {uμ,~uν}\stackbin[\small on-shell]−−−−→phase-space tangent vector. (II.9)

This property allows one to link off-shell surface terms to on-shell ones as will be discussed in section III.7. An interpretation of IBP vectors in differential geometry was given as well in IBPDiffGeom .

Although the on-shell perspective is instructive, we eventually need surface terms that are valid off-shell. To this end we can use the adapted coordinates of the loop integration. Interestingly, the construction of specialized IBP vectors can be solved by inspection. In adapted coordinates the defining equations are,

 (ua∂a+uk∂k)ρi=ui=fiρi, ⟹ui=fiρi. (II.10)

which follows from . The labels are not summed over in the above equation. The notation is explained in more detail in section III.2 and we provide only minimal explanations here. We use the shorthand notation that the index labels the -variables. Similarly the partial derivative denotes either of . Furthermore, we suppress function arguments in the -functions; .

The form of the IBP vectors allows for a natural geometric interpretation: the components generate to horizontal transformations (with fixed ) in the transverse directions and, the -components induce local conformal transformations in the individual propagator directions.

To summarize, the specialized vector fields have the restriction that the -components are proportional to and analogously for tilde/hat-coordinates. The general form reads,

 {ua,ui}={ua,ρifi}. (II.11)

(The doubled labels are not summed over here.) The component functions are unconstrained apart from the requirement that they are algebraic in the loop momenta (see also section III.5). As we will see below, given the parametrization of the loop momenta adjusted to the integral topology, it is straight forward to write down the vector fields and consequently the surface terms.

We will often consider vector fields with and refer to them as horizontal IBP vectors. These vectors induce translations along surfaces of fixed propagator values, which justifies their name. Horizontal vectors are a natural off-shell continuation of the tangent space of the unitarity-cut phase spaces,

 phase-space tangent vector\stackbin[\small off% -shell]−−−−→{ua,0}, (II.12)

linking on-shell and off-shell data. The conversion between the two is guided by their Lie-algebra structure (see sections IV.3 and IV.2).

Iii Loop Momentum Parametrizations

We will need the explicit form of the general coordinate changes to coordinates adapted to the integral topologies. We will relate the two-loop topology to nested one-loop topologies and reuse one-loop parametrizations. The below considerations hold without restrictions to the dimensionality of the loop momentum which we often suppress. For simplicity we later focus on planar integral topologies and assume generic propagator masses and external masses in order to deal with generic structures rather than special cases.

iii.1 One-loop topologies

We introduce a particular parametrization of the loop momentum adapted to the topology of the loop diagram in fig. 2. The aim is to change coordinates from the components of the loop momentum to inverse propagators . The construction of such a coordinate transformation can be obtained from ref. NumUnitarity which we review below. We use an all-outgoing convention for the external momenta . The case of generic non-vanishing external and internal masses () is considered. Using dimensional regularization in physical dimensions it suffices to consider -gon topologies with . (This allows loop momentum dimension to exceed the physical one.) Higher polygons are reducible using Gram-determinant identities NVbasis ; IntegralsExplicit and we do not considered them explicitly.

Inverse propagators will be denoted by and are expressed in terms of the loop momentum by,

 ρi = (ℓ−qi)2−m2i=(ℓ−i∑j=1pj)2−m2i. (III.1)

We set the arbitrary constant vector to zero for simplicity. When fewer propagators than loop momentum components are present an additional set of internal (angular) coordinates is required which we denote by . The final result will be the following degrees of freedom,

 inverse propagators: ρ0,…,ρDp, transverse coordinates: α1,α2,…,αDt,

with and and one additional quadratic constraint. The form of the quadratic equation will be discussed below and is given in eq. (III.1).

Explicitly, the coordinate change to adapted coordinates is given by,

 ℓ=V(ρ)+Dt∑a=1naαa,V(ρ):=Dp∑i=1rivi, ri:=−12((ρi+m2i−q2i)−(ρi−1+m2i−1−q2i−1)),qi=i∑j=1pj. (III.2)

Here the vectors and are elements of the van Neerven-Vermaseren (NV) basis NVbasis which we introduce momentarily. This basis is adapted to the integral topology and splits momentum space into a dimensional ’physical’ space spanned by the external momenta and a dimensional ’transverse’ space. For the considered -gons we have linear independent external momenta in the set due to momentum conservation. (We will also use the convenient notation .) Using the inverse of the -dimensional Gram matrix the vectors dual to the external momenta are introduced with so that they fulfill . In transverse space an orthonormal basis is used with and . The transverse basis is not unique and may be changed by (complex) orthogonal rotations. The linear dependence of the vectors and implies as well . Explicit analytic expressions for the NV basis may also be found in ref. NumUnitarity . Particularly compact expressions for the basis decomposition can be obtained in spinor-helicity notation and is inherent in most literature considering analytic unitarity methods (see e.g. GenUnitarityIII ; BlackHat ).

The parametrization (III.1) solves the linear equations,

 (pi,ℓ)=−12((ρi+m2i−q2i)−(ρi−1+m2i−1−q2i−1))=ri, (III.3)

due to the vectors and being dual. An additional quadratic constraint equation for the internal variables is imposed to make sure that the square of the loop momentum gives the inverse propagator,

 c(ρ,α) = (ℓ2−m20)−ρ0=Dt∑a=1(αa)2+C(ρ)=0, C(ρ)=V(ρ)2−ρ0−m20,

with . With the linear and quadratic equations fulfilled this parametrization returns the correct values for all inverse propagators. Most of the physical information is condensed neatly into the quadratic equation through the Gram matrix as well as its momentum and mass dependence.

A few remarks can be added here: The equation allows one to eliminate one in terms of the inverse propagators and the remaining transverse coordinates. Explicit solutions can be obtained using for example light cone coordinates,

 α1=12(t−C(ρ)+∑Dta=3(αa)2t)α2=i2(t+C(ρ)+∑Dta=3(αa)2t),

for , and the sum-term is dropped for . Here both and were traded for a new complex coordinate . For one can solve for directly to obtain,

 α1=±i√C(ρ),

where the internal manifold degenerates to two distinct points, i.e. a zero-dimensional sphere. However, it is often useful to consider the loop momentum as a hyper surface in space, without using an explicit solution of the quadratic equation inserted.

The inverse coordinate change is given by,

 ρi(ℓ) = (ℓ−qi)2−m2i,αa(ℓ)=(na,ℓ). (III.5)

For the above loop-momentum parametrization the maximal-cut on-shell conditions are implemented by setting the inverse propagators to zero, .

iii.2 Two-loop topologies

Loop-momentum parametrizations can be obtained by decomposing multi-loop diagrams into sub diagrams which admit one-loop parametrizations. To be specific, any rung in a multi-loop diagram admits a one-loop coordinate transformation yielding sets of internal coordinates and quadratic equations. When the rungs are joined in vertices the momentum-conservation conditions impose additional linear equations adding sets of linear equations. Planar as well as non-planar multi-loop parametrizations may be obtained in this way. We will focus first on the planar diagrams.

The generic two-loop topology is displayed in fig. 1. The planar integrals are obtained by specializing to the case where no external momenta are attached to the central rung, i.e. . In an approach best adapted to planar diagrams we consider the left and right one-loop sub diagrams in the figure and ignore the central rung at first. For the left loop the following external momenta and propagators are used,

 {p1,…,pN−1,pN=−(p1+⋯+p(N−1))}, ρi=(ℓ−qi)2−m2i,i=1,…N.

The quantities for the one-loop coordinate transformation , , and are obtained as in section III.1. Analogously, for the right loop we apply the one-loop transformation with the following list of external momenta inserted,

 {~p1,…,~p~N−1,~pN=−(~p1+⋯+~p(~N−1))}, (III.6) ~ρi=(~ℓ−~qi)2−~m2i,i=1,…~N.

Now we denote the parameters and functions by , , and . It is often convenient to distinguish the vectors and derived quantities by their index only, e.g. .

With these transformations we have the loop momenta parametrized in terms of inverse propagators with and . In addition and internal coordinates and are introduced, respectively.

Explicitly the loop momenta are given by,

 ℓ = V(ρ)+Dt∑a=1αana,~ℓ=~V(~ρ)+~Dt∑~a=1~α~a~n~a, (III.7)

with and . The internal coordinates have to fulfill the conditions,

 c(ρ,α)=Dt∑a=1(αa)2+C(ρ)=0,~c(~ρ,~α)=~Dt∑~a=1(~α~a)2+~C(~ρ)=0, (III.8)

where and setting .

There is one remaining transformation required; in order to express one internal degree of freedom from in terms of the inverse propagator of the central rung we have the relation,

 ^c(ρ,α,~ρ,~α) = (ℓ+~ℓ+pb)2−^ρ0−ˆm20 (III.9) = 2(ℓ+pb,~ℓ+pb)− −^ρ0−ˆm20+ρ0+m20+~ρ0+~m20−p2b.

It will be helpful to make the dependence on the -coordinates more explicit, by inserting the form of the loop momenta. We obtain the quadratic equation,

 ^c=αa~α~aˆCa~a+αaˆCa+~α~aˆC~a+ˆC, (III.10)

with the definitions,

 ˆC~ka=2(na,~v~k),ˆCk~a=2(vk,~n~a),ˆCa~a=2(na,~n~a),ˆCk~k=2(vk,~v~k), ˆCa=ˆCa~a~α~a0+ˆC~ka(~r+~r0)~k,ˆC~a=ˆCT~aaαa0+ˆCk~a(r+r0)k, αa0=(pb,na),~α~a0=(pb,~n~a),r0,l=(pb,pl),~r0,~l=(pb,~p~l), (III.11) ˆC=−^ρ0−ˆm20+ρ0+m20+~ρ0+~m20+p2b+(rk~r~k+r0,k~r~k+rk~r0,~k)ˆCk~k,

where and depend on external kinematics and inverse propagators, while the two-index terms and depend only on the external momenta. The latter matrices quantify the alignment of the physical and transverse spaces of the respective one-loop sub diagrams.

In general, complex orthogonal transformations may be used to rotate the basis vectors of the internal spaces (acting on -labels) and transform the above constraint (III.2) to canonical form. We will discuss the relation of integral topologies and the form of these equations in more detail below.

For some topologies it is possible to align the basis for the transverse spaces of left and right loop in fig. 1. This leads to a block diagonal form of and vanishing entries in and . Similarly the ()-dimensional components of the transverse space can be aligned. Rotation symmetries in these independent parts of transverse space are then manifest and simplify the quadratic equations.

In summary, we have traded the loop momenta and for the following coordinates and conditions,

 inverse propagators: ρ0,…,ρDp,~ρ0,...,ρ~Dpand^ρ0, transverse coordinates: α1,…,αDt,~α1,…,~α~Dt, quadratic equations: c=0,~c=0and^c=0. (III.12)

The quadratic equations have to be solved for the internal coordinates . Instead of finding explicit solutions it is often useful to think of the loop-momentum space as the sub manifolds defined by the quadratic equations in the unconstrained coordinate space of the ’s and ’s.

iii.3 Non-planar parametrization

Two equivalent ways to consider non-planar topologies will be discussed. The first emphasizes the general structure of multi-loop parametrizations, the second is most convenient for two-loop topologies being an adaptation of the planar setup.

1. Generic parametrization: The non-planar two-loop topology can be viewed as multiple rungs which are joined in vertices; for our notation we refer to fig. 1 (see also later in fig. 3). The individual rungs carry the loop momenta and , respectively, which are related by momentum conservation. Each rung can be parametrized using one-loop parametrizations to give three sets of -coordinates and -coordinates constrained by three quadratic equations. Compared to the planar case we obtain as well,

 ^c(^ρ,^α)=^Dt∑a=1(^αa)2+^C(^ρ)=0, (III.13)

with all functions being natural generalizations of the ones above (III.7) with ‘tildes’ replaced by ‘hats’. In a second step momentum conservation,

 (ℓ−q0)+(~ℓ−~q0)+(^ℓ−^q0)+pb=0, (III.14)

is imposed to relate the transverse coordinates of the individual rungs. In this way one obtains additional coordinates and equations, while the concepts remain the same. Considering multi-loop topologies amounts to adding further rungs and vertices in a similar way.

2. Planar induced parametrization: alternatively we can start from a planar parametrization of the loop momenta and include additional relations to transform the transverse coordinates (-coordinates) to inverse propagators.

As far as loop momentum parametrizations are concerned rungs can be exchanged, so we can always consider the central rung to have the least amount of external momenta attached; and . Given that we can have at most eight propagators we have . Thus, compared to the planar case only one additional inverse propagator variable is required. The constrains from the central rung are explicitly given by,

 ^c(ℓ,~ℓ) = (^ℓ−^q0)2−ˆm20−^ρ0=(ℓ+~ℓ−q0−~q0+pb)2−(^ρ0+ˆm20), ^c′′(ℓ,~ℓ) = (^ℓ−^q1)2−ˆm21−^ρ1=(ℓ+~ℓ−q0−~q0+pb+^p1)2−(^ρ1+ˆm21) = ^c(ℓ,~ℓ)+(ρ0+~m20)−(ρ1+ˆm21)+2(ℓ+~ℓ−q0−~q0+pb,^p1).

While the first constraint is the one already present in the planar topologies, the second one gives one additional linear equation for the loop momenta. It is useful to introduce the simplified constraint explicitly,

 ^c′(ℓ,~ℓ) = (ρ0+~m20)−(ρ1+ˆm21)+2(ℓ+~ℓ−q0−~q0+pb,^p1). (III.15)

We will return to the non-planar cases when discussing explicit IBP vectors in sections IV.2 and IV.3.

iii.4 A useful integral classification

It will be useful to refer to individual integral topologies. In principle it is sufficient to specify the number of rung momenta , with conventions as in fig. 1 and stating which of the vectors and vanish. We use the following terminology for the topologies,

 generic: pt≠0,pb≠0 semi-generic: pt=0,pb≠0,orpt≠0,pb=0 simple: pt=0,pb=0.

These sub classes differ in the number of dependent external momenta which are present in the set .

For planar topologies we label the integral topologies by only two numbers which specify external legs .

If the number of linear independent external momenta is smaller than the physical dimension the transverse spaces of left and right loops overlap and a common transverse space can be defined. We will assume that the transverse NV-vectors are aligned whenever possible.

iii.5 Algebraic Data

Tensor insertions from Feynman rules give algebraic functions that can be obtained by contracting loop momenta with themselves or with tensors derived from external momenta. These terms are natural in canonical coordinates in momentum space. When using the adapted coordinates we have to make sure that we deal with expressions that arise from coordinate transformations of such algebraic functions.

It turns out that tensor insertions are in fact in one-to-one correspondence with polynomial expressions in the adapted coordinates. This can be shown as follows: On the one hand, adapted coordinates are conventional loop momentum contractions being inverse propagators or contractions of the form . Consequently polynomials in these coordinates are polynomial in the loop momenta. On the other hand, these functions are sufficient to represent all loop momentum contractions: given an expression we can insert the completeness relations and into the contractions with and , respectively. The resulting terms and (and similar for the tilde-coordinates) give (III.3) and respectively. Both are polynomial in ’s and ’s, so that any tensor can be expressed in terms of these coordinates.

Thus we can trade any tensor in canonical coordinates for polynomials written in terms of inverse propagators and contractions with transverse vectors,

 (tμ1...νkℓμ1...~ℓνk)⟷∏a,~a,l,~l(αa)ka(~α~a)k~a×(ρl)kl(~ρ~l)k~l. (III.16)

One may as well include and variables, which however can be converted to the above monomials. The variables denote non-negative integers.

In order to obtain algebraic surface terms through IBP identities one has to consider algebraic vector fields in momentum space. Such vectors are defined to yield algebraic functions upon taking directional derivatives of algebraic functions ,

 (uμ∂μ+~uν~∂ν)t(ℓ,~ℓ)=t′(ℓ,~ℓ). (III.17)

We may use general coordinate transformations to obtain vector fields in adapted coordinates, however, it is preferable to construct them directly using the above definition in adapted coordinates,

 (ua∂a+uk∂k)t(α,ρ)=t′(α,ρ). (III.18)

(We use the shorthand notation for partial derivatives suppressing ’hats’ and ’tildes’ as in eq. (II.11).) One can show that the components and of algebraic vector fields have to be algebraic functions by acting one-by-one on the and coordinates. There is a further condition: above we worked in the coordinate space prior to imposing the conditions . Consistent vectors have to be tangent vectors to this surface, which defines the physical momentum space. This means we have to impose the three equations,

 (ua∂a+uk∂k){c,~c,^c}=0, (III.19)

to obtain an algebraic vector field. In the non-planar case we have to include one further analogous relation for (III.15).

From these definitions it is clear that multiplying an algebraic vector field by an algebraic function yields again an algebraic vector field.

Finally, given that we deal with integration, we will use differential forms and outer derivatives. As usually, these are defined as linear functions that return the components of vector fields. The differentials,

 {dαa,dρk}

are algebraic, yielding algebraic functions when acting as linear forms on algebraic vector fields with , etc. We use again a single label which runs as well over hat and tilde variables.

The 1-forms are not independent due to the relations,

and analogously for and the relations. Wedge products can be used to generate the full set of differential forms in adapted coordinates.

iii.6 Function ring and numerator tensors

We will require a complete set of tensor insertions for a given integral topology limited only by power counting (of typical field theories). Systematic constructions of such a basis of tensor insertions can be found at one-loop level in in ref. NumUnitarity (see also RevUnitarity ) and for multi-loop topologies in refs. MLoopParamBFZ ; MLoopAlgGeo ; MLoopParamMO . We will review the construction and adjust the notation to our setup. The use of the adapted coordinates makes the construction of irreducible tensor insertions very direct, so that it can often be obtained by hand.

For a given integral topology not all tensor insertions are viewed as independent; inserting an inverse propagator allows one to cancel a propagator and leads to a reduced topology. Thus we can consider numerator tensors modulo inverse propagators. This implies that for the construction of independent numerators inverse propagators are best treated as equivalent to zero . Comparing to (III.16) we can proceed in two steps. First, we identify an over complete list of numerator tensors as the polynomials in the coordinates

 ∏a,~a(αa)ka(~α~a)k~a=∏a,~a(na,ℓ)ka(~n~a,~ℓ)k~a, (III.21)

which are written as well in tensor notation for convenience. The integers and take positive values limited above by power counting. In a second step we use the relations , which allow to turn certain polynomials into inverse propagators or monomials of lower degree, thus reducing the independent tensors insertion further.

It is important to note that these equivalence relations amount to imposing the on-shell conditions, with all inverse propagators set to zero. Thus, as far as the construction of a basis of numerator tensors is concerned, linear independent numerator tensors remain independent functions when considered on-shell on the maximal-cut phase spaces.

However, not all questions can be answered modulo lower topologies and off-shell information is important. For example, considering the tensor insertion of an inverse propagator, we have an ’uninteresting’ tensor insertion,

 t(ℓ,~ℓ)=ρk∼0 (III.22)

and even obtain zero on the maximal cut. However, considering the loop integral, we clearly obtain a scalar integral of lower topology yielding a non-vanishing result,

 ∫dDℓdD~ℓρkρ0⋯~ρ(~N−1)=∫dDℓdD~ℓ1ρ1⋯ˆρk⋯~ρ(~N−1)≠0, (III.23)

with denoting the omission of the inverse propagator in the numerator. Thus when surface terms are analyzed, we have to work off-shell, although we may obtain guidance from related on-shell questions.

iii.7 Total derivatives and master-integral count

Here we discuss the relation between computing the total derivatives and cutting all propagators of an integrand. Cutting propagators amounts to replacing the propagators with delta-distributions (). When a tensor integral is written in adapted coordinates (as done in eq. (II.6)) the operation of cutting omits all propagators as well as the integration measure and sets the inverse propagators to zero,

 ∫[dρ]ρ0⋯~ρ(~N−1)×t(ρ,α)μ(ρ,α)[dα]cut⟶∫t(0,α)μ(0,α)[dα]. (III.24)

The function is a measure factor arising from transforming canonical coordinates to the adapted coordinates and denotes the tensor insertion. Terms with some of the cut propagators missing are omitted in the cutting prescription. (One might extend such terms with the necessary inverse propagators and see it vanish when the ’s are set to zero.)

For the IBP vectors that do not double propagators we now show that taking total derivatives commutes with the cut operation. That is, propagators drop out of one class of terms in the total derivative, which leads to vanishing terms when cuts are applied,

 ∫[∂i(uiμtρ0⋯~ρ(~N−1))+∂b(ubμtρ0⋯~ρ(~N−1))][dρ,dα] (III.25) cut⟶∫∂b(ubμt)[dα].

Here we use the relation and we suppress the arguments for better readability. (In the above equation the labels run as well over hat and tilde values.)

Had we first cut and then used (the pull back of) the IBP vector to obtain a total derivative we obtained the same answer. The special form of the IBP vector fields makes this identification possible in the first place: since they are tangent vectors of the maximal-cut phase spaces they are well defined intrinsically on the phase spaces and a pull back is well defined.

It is important to mention that the above reasoning did not involve the -integration itself and is valid as well at the level of integrands (or volume forms considering their Lie-derivatives Dirschmid ).

In general the cut of a vanishing integral does not need to give a surface term on phase space, but might vanish, e.g. due to the choice of the physical integration contour.

iii.7.1 On/Off-shell map

Relating on-shell and off-shell surface terms we can exploit intrinsic properties of the on-shell phase spaces. Every off-shell total derivative from special IBP vectors gives one on the maximal cut (III.7). In formal terms, we obtain exact holomorphic forms of maximal degree for each surface term. We have already seen (section III.6) that the basis of tensor insertions gives linear independent functions on the maximal-cut spaces. By multiplying with the proper volume element we obtain a holomorphic forms of maximal degree. Given that the coefficient functions are holomorphic and the forms are of maximal degree they are closed; their outer derivative vanishes. Thus surface terms and tensors are given as intrinsic objects of the phase spaces.

Master integrands can be counted on-shell. The number of master integrands is given by the number of independent tensor insertions modulo the number of surface terms. On-shell this amounts to the number of closed modulo exact holomorphic forms, that is a topological (global) property of the phase spaces. Thus a topological property, in fact the number of half-maximal cycles, counts master integrands. We will return to counting forms in section V.

Iv Construction of off-shell surface terms

We now turn to the main result: the construction of off-shell surface terms which are a central ingredient for the numerical unitarity approach. The construction differs from the one at one-loop level OPP ; NumUnitarity ; RevUnitarity which relied on tensor algebra and symmetries of one-loop integrals. The present construction reproduces the one-loop results and applies also to multi-loop topologies. Put differently, we obtain a complete set of IBP relations which might be valuable beyond its use for the unitarity approach.

The presentation focuses here on planar two-loop topologies, but gives as well non-planar surface terms. Higher-loop generalizations should work in a similar way as we will indicate briefly (section IV.5). We consider the four-dimensional construction which yields surface terms that involve the four-dimensional part of the loop momentum. These terms are as well surface terms in dimensions. Furthermore at one-loop level the four-dimensional numerators were recycled for the -dimensional approach and we believe the same construction works here. Nevertheless, additional IBP vectors can be found beyond the four dimensional ones (see eq. (IV.11)).

The central objects are specialized IBP vectors, which upon computing divergences, are used to obtain the complete set of off-shell surface terms (see section IV.4). The master integrals are obtained as convenient tensor insertions in the complement of the surface terms. The construction works topology by topology and reuses one-loop results in sub topologies. In order to introduce the key steps we start with a one-loop example in section IV.1. Next, we turn to the two-loop problem. A generating set of IBP vectors is obtained first in adapted coordinates in section IV.2 and, finally, in canonical notation in section IV.3. The latter is the main result of this article and has a very natural geometric interpretation as we discuss in section IV.5.

The set of planar generating IBP vectors have the additional property that they leave the integration measure invariant, i.e. their divergence vanishes. This simplifies the computation of surface terms to taking directional derivatives of irreducible numerator basis. In this way an over complete set of surface terms can be obtained which are equivalent to a complete set of IBP relations excluding doubled propagators. We verify the completeness using on-shell techniques in section V.

iv.1 A one-loop example

The ingredients we need are an irreducible basis of tensor insertions, IBP vectors and the integration measure in order to compute total derivatives. It will turn out, that all IBP vectors can be generated by a set of primitive ones, which we have to consider in detail. Furthermore, it turns out that the primitive IBP vectors leave the volume element and all propagators invariant. Under these circumstances the surface terms are directly obtained by acting with IBP vectors on the irreducible tensor basis. Although these statements hold more generally, we will discuss these steps for the triangle integrals.

iv.1.1 Numerator tensors for triangles

The four degrees of freedom of the loop momentum are parametrized in adapted coordinates by three inverse propagators and two internal coordinates which are constrained by a single quadratic equation (III.1),

 c(α,ρ)=α1α1+α2α2+C(ρ)=0, (IV.1)

with given by scalar terms and inverse propagators. The coordinates are related to the loop momenta through the contractions and the definitions of the inverse propagators.

Irreducible numerators are given by polynomials in the -variables (see section III.6). For standard power-counting we should consider at most cubic powers of the loop momentum in triangle functions. Thus the tensor insertions with suffice. Out of the ten monomials, only seven are linearly independent modulo inverse propagators, as can be seen by using the quadratic equation , which relates inverse propagators and internal coordinates. The three dependent monomials are and .

It is convenient to make a linear coordinate change to the coordinates (and ), so that the constraint equation is given by,

 α+α−+C(ρ)=0. (IV.2)

The reduction to a minimal numerator basis starting from monomials then simplifies; it amounts to dropping mixed monomials in and , since these are reducible . The irreducible numerator basis is then given by the seven terms with and thus the integrands,

 ~m±k(ℓ)ρ1ρ2ρ3=(α±)kρ1ρ2ρ3=(n±