Tensor reduction of loop integrals

The computational cost associated with reducing tensor integrals to scalar integrals using the Passarino-Veltman method is dominated by the diagonalisation of large systems of equations. These systems of equations are sized according to the number of independent tensor elements that can be constructed using the metric and external momenta. In this article, we present a closed-form solution of this diagonalisation problem in arbitrary tensor integrals. We employ a basis of tensors whose building blocks are the external momentum vectors and a metric tensor transverse to the space of external momenta. The scalar integral coefficients of the basis tensors are obtained by mapping the basis elements to the elements of an orthogonaldual basis. This mapping is succinctly expressed through a formula that resembles the ordering of operators in Wick's theorem. Finally, we provide examples demonstrating the application of our tensor reduction formula to Feynman diagrams in QCD $2 \to 2$ scattering processes, specifically up to three loops.


Introduction
In gauge and gravity theories, Feynman diagrams give rise to tensor integrals.Practical computations are often organised by first projecting these tensor integrals onto scalar integrals.For the latter, powerful integration by parts methods [1][2][3] can be employed, reducing them to linear combinations of a smaller set of master integrals.
Passarino and Veltman introduced a general technique for the reduction of tensor integrals to scalar integrals in their seminal publication of Ref. [4].Their method has found countless applications since its invention.However, it involves inverting systems of equations which rapidly become intractable as the rank of the tensors increases.This poses a challenge for computations of scattering amplitudes for processes with a large number of external particles and at high loop orders.
Other methods which cast arbitrary tensor integrals as generic scalar integrals (that, in turn, need to be further reduced with integration by parts or other methods to master integrals) exploit the functional form of momentum and parametric representations [35,36] of tensor integrals.For amplitudes, methods to select external states of definite helicity can be very efficient, as it has been shown in the recent works of Ref. [37] and Refs.[38,39].For unpolarised scattering, one can avoid a reduction of tensors to scalar integrals by computing squared amplitudes summed over spins and polarisations of external particles, as they appear in cross-sections.
The Passarino-Veltman method has been superseded by all such other methods in a broad range of applications within particle physics phenomenology.For instance, as far as we are aware, no direct computation of tensor integrals in amplitudes for scattering processes involving four or more external particles has been pursued with a Passarino-Veltman reduction beyond one loop.Such challenging computations have been carried out with alternative methods (see, for example, [40][41][42][43][44][45]). However, we will show that the Passarino-Veltman method can be simplified.We believe that the improved method is competitive for cutting-edge multi-loop computations in perturbation theory.
In this article, we present a closed-form solution to the inversion problem in the Passarino-Veltman reduction.Our derivation is based on the observation that all possible tensor structures which may emerge in the tensor reduction of a general tensor integral can be identified through a properly defined ordering of operators associated with tensor indices.We project tensor integrals onto scalar integrals with a compact formula that comprises three structures: 1. Metric tensors which are transverse to external momenta.
2. Tensors which are orthogonal to products of transverse metric tensors.These have been introduced in Ref. [46] for the tensor reduction of multi-loop tadpole integrals.
3. Linear combinations of momenta that are orthogonal to the external momenta.This construction was originally developed by van Neerven and Vermaseren in Ref. [47] for the reduction of four-dimensional one-loop integrals with five or more external particles to box integrals.More recently, it has been invoked in methods for reduction to master integrals of one-loop amplitudes, see for example Refs.[14,24], and two-loop amplitudes with the method of Ref. [26].
It has also been used for the projection to helicity amplitudes in Ref. [38].
All terms of the projection are generated efficiently in a closed form, with operations analogous to contractions in applying Wick's theorem for the time-ordering of bosonic free field operators.
Our method provides an analytical solution to the large set of equations in the Passarino-Veltman system.This tensor reduction technique is universally applicable, being identical for all integrals with the same tensor rank and external momenta.Moreover, it remains independent of other integral-specific characteristics, such as the integral's topology.
Our article is organised as follows.In Section 2, we describe the relation between conventionally used tensor elements, which form a basis for the Passarino-Veltman reduction, and an ordering of operators that we attribute to the tensor indices.In Section 3, we introduce a metric that is transverse to the space of external momenta and dual elements for external momentum vectors and products of transverse metrics.With these ingredients, we build two dual bases which we use for the tensor reduction.We present our main result for the Passarino-Veltman reduction of generic tensor integrals in closed form in Section 4. In Section 5, we demonstrate the application of our formula to tensors up to rank seven, corresponding to QCD Feynman diagrams with four external partons through a perturbative order of three loops.Finally, we summarise our work and present our conclusions in Section 6.

Algebraic analogy of tensor reduction and the application of Wick's theorem in ordering operators
A general tensor loop integral of rank R in D space-time dimensions has the form f is a scalar function which depends on N p independent external momenta p i , and τ is a rank-R tensor which is a linear combination of products of the loop momenta.It can be easily seen, for example [36] in Feynman or Schwinger parameter representations, that tensor integrals can be written as a superposition of all independent rank-R tensors which can be built by multiplying external momentum vectors and/or the metric.Generically, we write 2) The tensors T µ 1 ...µ R a represent all such independent possible tensors of rank R of the metric and external momenta.The coefficients I a are scalar integrals which are amenable to further reduction with techniques such as integration by parts [1][2][3].
Expressing the tensor integral as a superposition of scalar integrals and deriving the coefficients I a is an often daunting linear algebra task.For a rank-R tensor and N p independent external momenta, tensor reduction generates terms in total, where ⌊x⌋ = max{m ∈ Z|m ≤ x} is the floor function.For example, in two-loop 2 → 2 scattering amplitudes we require R = 5, N p = 3, which results in 558 terms.For R = 6, N p = 4 in 2 → 3 two-loop processes we have 8671 terms and, by adding one more loop, for R = 8, N p = 4 we have 240809 terms.In regular Passarino-Veltman reduction this results in a system of N tensors equations that require the inversion of an N tensors × N tensors matrix in order to derive the coefficients I a .
There is an algebraic way to list all the tensor elements (products of metric tensors and external momentum vectors) which appear in the typical basis of Passarino-Veltman reduction.For concreteness, let us consider the outcome of tensor reduction of a rank-4 tensor integral.This has the form (12,34) η µ 1 µ 2 η µ 3 µ 4 + c (13,24) η µ 1 µ 3 η µ 2 µ 4 + c (14,23) We would like to generate the list of all tensors in the right-hand side of Eq. (2.4).We consider all independent vectors p µa i and we associate to their sum an index operator a which indicates the index µ a , Np i=1 This assignment defines an ordering of the indices µ a .Explicitly, We now attribute to the index operators a commutator and an ordering operation where, using the commutation of Eq. (2.8), we bring operators a i with a larger tensorindex label to the left, in front of operators with a smaller tensor-index label.For example, T The last equation is the sum of all rank-4 tensors in the list of Eq. (2.5), as it can be seen by substituting the contractions as metric tensors and index operators with the sum of momentum vectors.Generally, we can use the ordering, to sum up all the elements of the basis in which we can express a tensor integral In the following sections, inspired by the combinatorial resemblance of tensor reduction and the ordering of operators with Wick contractions, we will go one step further.We will develop a new ordering operation which will yield the full answer for tensor reduction.That is, we will obtain the sum of all the basis tensor-elements weighted with the correct scalar integral coefficients (such as the c (X) ... coefficients which appear in the right-hand side of Eq. (2.4)).

Tensor reduction and dual bases
Our starting point is to construct two complete bases of tensors, B = {T µ 1 ...µ R a } and ⟨B⟩ = {⟨T a ⟩ µ 1 ...µ R }, which are dual, satisfying the property The two bases do not need to be, and we will not take them to be, individually orthonormal.
In what follows, we will present an explicit construction of the dual bases.Assuming their existence, the reduction of tensor integrals takes a simple form.Indeed, by contracting both sides of Eq. (2.2) with ⟨T b ⟩ we can determine the scalar coefficients I b .We obtain Alternatively, we can also write Notice that on the right-hand side of Eqs.(3.2)-(3.3) the tensor indices of the integral are contracted and the tensor integral is reduced to scalar integrals.We will now build the dual bases which are needed for materialising Eq. (3.2) and Eq.(3.3).

Dual tensors for products of external momenta
Let us first start with a basis of N p external momentum vectors p µ i , i = 1 . . .N p .To find their dual, we adopt the method of van Neerven and Vermaseren in Ref. [47] (see, also, Section 3 of Ref. [24] for a elucidating variation of the formalism).We first form the symmetric N p × N p matrix of scalar products, and, with linear algebra methods, we find the inverse matrix ∆ ij , The linear combinations are dual to p i in the sense of Eq. (3.1).Indeed, We note that We can now easily extend the construction to products of momenta.For a rank-R tensor product of momentum vectors we construct its dual tensor as the product of the duals of momentum vectors which satisfies The tensor T µ 1 ...µ R p is annihilated by any other product of R vectors ⟨p i ⟩ µ than ⟨T p ⟩ µ 1 ...µ R .However, it is not annihilated by contractions with the metric tensor, such that the condition in Eq. (3.1) is not fulfilled.To resolve this issue we construct metric tensors as well as their duals, which are transverse to the external momenta.It will be useful to form a rank-2 tensor from the external momenta p i and their orthogonal momenta ⟨p i ⟩, which acts as the unity in the space of external momenta.Indeed, we have that For this reason, we will refer to u µν as the unit tensor.The natural choice for a metric transverse to external momenta is the tensor obtained by subtracting u from the metric, Explicitly, we have that In addition, the transverse metric satisfies and the dimensionality of the transverse metric is where N p is the number of independent external momenta.

Dual tensors for products of transverse metrics
While the transverse metric tensor and products of it are orthogonal to all external momenta and their duals, they are not orthogonal to other metric products.We will now construct dual elements ⟨T a ⟩ for tensors T a which are products of the transverse metric tensor, Our procedure is analogous to the method in Ref. [46] for tensor loop integrals with no external momenta.At even rank R, the set of independent tensors of the form of Eq. (3.18) contains elements.For rank two, there is only one tensor we can form, η µ 1 µ 2

⊥
. We trivially write its dual element in the orthogonal basis, satisfying At rank four, we find three independent tensors To construct the dual tensors in the orthogonal basis, we write an ansatz in which we have used the µ 1 ↔ µ 2 and µ 3 ↔ µ 4 and (µ 1 , µ 2 ) ↔ (µ 3 , µ 4 ) symmetry.
We determine the a, b coefficients from the requirements We obtain ) . (3.25) Note that the dual of a product of transverse metric tensors does not factorise Similarly, we can construct the elements of the dual basis for lengthier products.The construction for the dual basis of rank six and eight is described in Appendix A. The dual basis for the metric tensors up to rank fourteen are available in an ancillary file, where the coefficients are calculated using Form [48][49][50].These suffice, for example, for the tensors which emerge in QCD Feynman diagrams with four external partons at six loops.

Tensor reduction of a generic loop integral
We now have all ingredients to express the reduction of a generic tensor integral in a closed form.For a generic tensor integral of rank R we have constructed two bases of tensors which are } consisting of all possible products of momenta p µ i and the transverse metric tensor η µν ⊥ , • a dual basis ⟨B⟩ = {⟨T a ⟩ µ 1 ...µ R } consisting of all possible products of dual momenta ⟨p i ⟩ µ and duals of products of transverse metric tensors As an example, the basis as well as the dual basis for N p independent external momenta and tensors of rank one is given by for rank two the bases are given by whereas for rank three we get The two bases B and ⟨B⟩ for a general rank-R tensor integral satisfy the Eq.(3.1).We can then cast a tensor integral in the forms of Eq. (3.2) or Eq.(3.3) using our dual bases.Let us analyse the sum on the right-hand side of Eq. (3.3), with our initial focus on the terms that solely involve external momenta and do not include any transverse metric tensors.We have In the above, we could collect all momentum-only dependent terms in the tensor reduction of Eq. ( 3.3) into a product of unit tensors u αµ .We now turn our attention to terms with the transverse metric.These are of the form, We define a product of contractions as With the contraction symbol of Eq. (4.6), we cast the terms of Eq. (4.5) as Finally, we can sum up all terms in Eq. (3.3) compactly as an ordering operation of the unit tensors u αµ .We define an ordering symbol T as where A i = u α i µ i .For example, for a rank-4 tensor we write The tensor decomposition of Eq. ( 3.3) of a generic tensor integral is written compactly as, This is the main result of this article.
Let us remark that all multiplications in the terms of the right-hand side of Eq. (4.10) are commutative.It does not matter, for example, if we have the unit tensors with indices α 1 , µ 1 on the right and unit tensors with indices α R , µ R on the left.

Rank-2 tensor triangle integrals with two external momenta
As an instructive example, we demonstrate the steps which lead to Eq. (4.10) in the case of a rank-2 tensor integral I µν with two independent external legs p 1 , p 2 .Our basis of independent tensors consists of Correspondingly, the dual basis is given by Following (3.3), the tensor integral takes the form Collecting the momentum-dependent tensors with indices α and µ as well as β and ν allows us to rewrite the expression compactly as We can now recognise the unit tensors u αµ and u βν , defined in Eq. (3.12), for N p = 2 external momenta and write The last term corresponds to a contraction of two unit tensors where, in this example, we have The tensor integral can be written as which resembles the operation of time-ordering and Wick contractions.Using the ordering symbol defined above, we write 5 Illustrative applications Eq. ( 4.10) provides an algorithmic instruction for projecting tensor integrals to scalar integrals, in which no large matrix inversions are needed in intermediate steps.
Eq. (4.10) is a very compact expression, which includes all terms which emerge in a tensor reduction.It is useful to be able to just write down such sizable results directly, without intermittent algebraic operations.As already mentioned in Section 2, the number of possible tensors N tensors , enumerated in Eq. (2.3), rises steeply with the rank of the tensor and the number of external particles.
Still, the computational cost of merely casting the terms on the right-hand side of Eq. (4.10) is substantial.It is further increased if explicit substitutions for dual products of transverse metric tensors, for the transverse metric in terms of the Ddimensional metric and the dual momenta in terms of the original momenta are carried out analytically.
However, in realistic Feynman diagram computations, many explicit substitutions of the transverse metric η µν ⊥ in the elements of the dual basis may not be necessary.For instance, one can use directly that η µν ⊥ p iν = 0.When the transverse metric is contracted to a gamma matrix, it projects out its transverse components We can also simplify spin-chains making use of the Clifford algebra for the transverse components, which in conventional dimensional regularisation reads [51] and Also, one can make use of the property of the unit tensor, p i,µ u µν = p ν i , without expanding u µν in terms of momentum vectors.
To illustrate how Eq. (4.10) may be used and to assess its practical potential, we implemented the reduction for tensor integrals with three independent light-like external momenta.We then applied the reduction to the analytic computation of selected Feynman diagrams in 2 → 2 scattering amplitudes.We present these applications next.

Tensor reduction for integrals with three independent light-like momenta
We consider tensor integrals of generic rank R which depend on three independent light-like external momenta.We define the corresponding Mandelstam variables as The matrix of scalar products of the external momenta in our basis, Π ij = p i • p j , can be read off from Eq. (5.5), (5.6) The inverse matrix can be easily calculated (5.7) The calculation of the inverse matrix ∆ is the only process-dependent inversion which we require.Its matrix elements are the scalar products of the dual momentum vectors, ∆ ij = ⟨p i ⟩ • ⟨p j ⟩.The dual momenta, ⟨p i ⟩ , i = 1..3, can be computed from the matrix ∆ and the definition ⟨p i ⟩ = 3 j=1 ∆ ij p j .Finally, we construct the unit tensor u αµ out of the independent external momenta p 1 , p 2 , p 3 , This is the main building block for creating T R i=1 u α i µ i on the right-hand side of Eq. (4.10).
In the space of tensors which can be constructed out of metric and momenta products, the T R i=1 u α i µ i acts as a unity.We have written computer programmes which generate T R i=1 u α i µ i for a given rank R and set of independent external momenta.With explicit computations in Form we have verified that indeed for three independent external momenta and tensors up to rank five.
5.2 One-loop q q → γγ amplitude We will now apply our tensor reduction on Feynman diagrams in the scattering amplitude for quark-antiquark annihilation to a pair of real photons, q(p 1 ) + q(p 2 ) → γ(p 3 ) + γ(p 4 ) .
In parentheses we denote the momenta of the external particles, which satisfy the momentum conservation condition and Mandelstam variables as given in Eq. (5.6).All tensor integrals can be reduced to scalar integrals with the expressions for T R i=1 u α i µ i that we have built above.We start by computing the hard scattering contribution to the one-loop amplitude.We have purposefully chosen to calculate a physical amplitude, comprising of various Feynman diagrams and counterterms, in order to show the versatility of the approach in dealing with diverse diagrammatic topologies (tadpole, bubble, triangle and box in this particular example).Following Ref. [52], the process-specific finite amplitude remainder may be defined as where (5.12) The last diagram is a form-factor counterterm and removes the infrared singularities of the amplitude.The form-factor counterterm is obtained from the pictured Feynman diagram by enclosing the part of the diagram in parentheses in between spin projector factors given by Eq. (2.4) of Ref. [52] and evaluating the corresponding expression at zero gluon momentum k = 0. Diagrams with one-loop vertex and propagator corrections have ultraviolet singularities.They are subjected to an ultraviolet subtraction indicated by R k .The explicit form of the ultraviolet counterterms, is given by Eq. (6.7) and Eq.(6.15) of Ref. [52].We examine first the tensor reduction steps of the box diagram which yields tensors of the highest rank three.The diagram is written as, (5.15) In the second line of Eq. (5.13) we have already inserted the ordering operator of Eq. (4.10) which will project the loop momentum tensors to scalars.We now carry out the contractions as instructed by the ordering of the unit tensors, derived in the previous section, in the integrand.The numerator of the diagram reads (5.17) At this stage, we have projected the tensor product of loop momenta to the scalar products k 2 , k • p 1 , k • p 2 , k • p 3 .These scalar integrals can be expressed in terms of the loop momentum denominators of the diagram, The resulting scalar integrals are of the form, with integer powers i j ≤ 1.
We have already reduced the tensor integral to scalar integrals.We can now proceed with further simplifications pertinent to the spin and Lorentz structure of the diagram.We substitute into Eq.(5.16) and express the vectors of the dual momenta ⟨p ν i ⟩ in the second line of Eq. (5.16) as linear combinations of the external momenta (5.20) The vectors p 1 , p 2 , p 3 and the metric tensor are contracted with gamma matrices.We can simplify the spin chains using the Clifford algebra, in conventional dimensional regularisation, and that external states are on-shell, (5.21) The polarisation vectors are transverse to reference vectors that we choose as After these simplifications, the diagram is expressed in terms of the following minimal set of spin-chains, Our treatment of the other Feynman diagrams is analogous.Although the loop denominators for triangle and bubble graphs depend on just two or one combinations of the p 1 , p 2 , p 3 external momenta, we can decide to treat all diagrams uniformly.For example, the tensor loop integral in the second Feynman diagram of Eq. (5.12), which is a triangle, depends only on the combinations of external momenta.We could use B 2 = {q a , q b } as our basis of vectors for the tensor reduction of this particular diagram.However, it is also allowed and, perhaps, preferred to use the bigger basis B 3 = {p 1 , p 2 , p 3 } which we needed for the box diagram earlier.The only disadvantage of using the extended basis B 3 is that it results to scalar integrals TP(i 1 , i 2 , i 3 , i 4 ) with some negative powers of propagators i j .However, scalar integrals with negative propagator powers can be handled with integration by parts identities and the Laporta algorithm [3] seamlessly.In addition, beyond one loop, negative powers of propagators (or, equivalently, irreducible scalar products in the numerators) are inevitable.
We only treat ultraviolet counterterms, denoted by R k in Eq. (5.12), separately.These give rise to second rank tensor integrals of a tadpole topology.It is very easy to reduce tensors of the tadpole topology in their natural basis, consisting of metrics and no external momenta.We write (5.24) In the above, the dual of the metric is ⟨η µ 1 µ 2 ⟩ = η µ 1 µ 2 D .At this point, we have achieved our goal of reducing all tensor integrals in the amplitude to scalar integrals.For a complete analytic computation of the amplitude, one may further reduce these scalar integrals to master integrals.We reduce all scalar integrals TP(i 1 , i 2 , i 3 , i 4 ) to the one-loop box and the one-loop bubble master integrals with integration by parts identities using AIR [53], (5.25) Inserting the known master integrals (see, for example, Appendix C of Ref [54]), we obtain a finite expression in D = 4 dimensions as it is anticipated.We find 1 (s, t) 3 (s, t) 4 (s, t) (5.26) with the coefficients h i (s, t) displayed in Appendix B. As a consistency verification, we interfere the one-loop amplitude with the tree-level and obtain a result independent of our tensor reduction.5.2.1 N f contribution to the two-loop q q → γγ amplitude Our tensor reduction depends only on the rank of tensors and the external momenta of Feynman diagrams.Thus, we can use the same reduction expressions across loop orders, for tensors integrals with common external momenta and rank.For example, it is very easy to extend the previous one-loop amplitude computation, and derive similarly the N f contribution to the finite part of the two-loop q q → γγ amplitude.
We generate the N f part of the two-loop amplitude from the corresponding oneloop amplitude, by inserting a one-loop fermion loop in the gluon propagator.As an example, the box diagram in the one-loop amplitude in Eq. (5.12) with a quark self-energy inserted in the gluon propagator is displayed in Fig. 1.
We will consider the case of massless quarks in the fermion loop.As discussed in Ref. [55], we can simplify the integrand by a first tensor reduction of the fermion-loop subgraph (reducing tensors in the l momentum integration) and an elimination of terms in the integrand which cancel, due to gauge invariance, in the sum of diagrams.The integrand for the finite part of the amplitude then reads, with (5.28) We now proceed to the tensor reduction of the k-momentum integrals.The H (2,N f ) ({p 3 , ϵ 3 }) has an identical tensor numerator as H (1) ({p 3 , ϵ 3 }).Hence we reduce the tensor in H (2,N f ) to scalar products exactly as in H (1) .After the reduction to scalar integrals, we perform a further reduction to twoloop master integrals solving integration by parts identities, using AIR.The twoloop master integrals are known analytically for 2 → 2 massless QCD scattering processes [56][57][58].We have taken the master integral expressions from a computer readable input used in Ref. [54].all substitutions, we arrive at the following result for the finite N f two-loop q q → γγ amplitude, The coefficients h (2,N f ,j) i (s, t) are displayed in Appendix B and they are free of 1/ϵ poles, as it is anticipated.

A two-loop diagram with rank-5 tensors in dimensional regularisation
The previous illustrative examples are computationally simple, as they require tensor integrals of a relatively low rank, i.e. three.We would like to test the implementation of Eq. (4.10) on Feynman integrals with higher rank tensors.We find rank-5 tensors (the maximum rank for 2 → 2 QCD scattering at the two-loop order) in Feynman diagrams for the q q → γγ amplitude with seven propagators.As a representative case, we examine a Feynman diagram of a planar double-box topology together with a suitable form-factor type of counterterm [52,55,59] which removes a double soft singularity as k, l → 0, D The double-box diagram, first term on the right-hand side of Eq. (5.30), yields an 1/ϵ 4 pole.This pole is cancelled against the contribution of the form factor counterterm, which is the second term on the right-hand side of Eq. (5.30).This counterterm is again obtained by enclosing the part of the diagram in parentheses in between spin projectors and evaluating the enclosed expression at zero momenta k, l = 0. We therefore anticipate that the combination of Eq. (5.30) has a Laurent series expansion starting at the 1/ϵ 3 order.This will be a test for the correctness of our tensor reduction in this example.
The two diagrams consist of tensor integrals of rank ≤ 5 and depend on three independent light-like momenta, p 1 , p 2 , p 3 .The Mandelstam variables have been defined in Eq. (5.5) and the matrix of scalar products as well as the inverse have been defined in Eq. (5.6) and Eq.(5.7) respectively.We again use the choice of polarisation vectors defined in Eq. (5.22).We examine, first, the double-box diagram which yields tensors of the highest rank-5 tensor structures more carefully.The diagram is given by where (5.32) In the third line of Eq. (5.31) we included the ordering operator defined in Eq. (4.10), which projects the tensor integral to scalar integrals.Note that we have switched the indices in the unit tensors u µα enclosed by the ordering operator.In contrast to the one-loop case, we have chosen here to contract the loop momentum tensors with the dual metric tensors and dual external momentum vectors, whereas the non-dual transverse metric and momentum tensors are getting contracted with the gamma matrices in the spin-chains.This choice corresponds to the reduction in Eq. (3.2) instead of the previously used reduction in Eq. (3.3).Our choice is motivated by computational optimisation.Now, we can proceed directly to simplifying the spinchains without substituting the dual tensors in terms of momenta and metrics.The ordering of the five unit tensors in Eq. (5.31) leads to several terms such as a product of five unit tensors, a product of three unit tensors times one transverse metric and its dual, and so on.To enhance the performance of the code, we apply an iterative procedure to reduce the spin-chains.We explicitly substitute only one unit tensors in terms of the momenta and their dual momenta defined in Eq. (5.8) at a time.The external momenta from the unit tensor contracts with the gamma matrices in the spin-chains.This allows us to simplify the spin-chains using on-shellness of the external momenta and the Clifford algebra.We iterate this procedure, replacing one unit tensor at a time and simplifying the spin-chains, until there are no unit tensors left.This leaves the contraction of gamma matrices with the transverse metric tensors, which projects out the transverse component γ ⊥ defined in Eq. (5.1).We can make use of the transverse Clifford algebra in Eq. (5.2) and (5.3).These simplifications lead again to the minimal set of spin-chains displayed in Eq. (5.23).
As a last step have to reduce the loop momenta contracted with dual tensors to scalar integrals such that they allow for reduction to master integrals with integration by parts methods.The three scalar product structures we encounter after the insertion of the ordering operator are given by where we chose some random indices as examples.Since the metric duals are transverse to the external momenta, we can immediately rewrite the contractions with k i , only in terms of the loop momenta.We can simplify the three structures even further by invoking the dual property between the external momenta and their corresponding duals, as seen in Eq. (3.7).The three scalar product structures therefore simplify to Additionally, we include the definition of the dual transverse metrics defined in Eq. (3.20) for rank 2 and Eq.(3.25) for rank 4 and substitute the definition for the transverse metric displayed in Eq. (5.19).As a last step the dual momenta are substituted in terms of external momenta, completing the reduction of the tensor integral to scalar integrals including the scalar products In a similar fashion as in the one-loop case, the obtained scalar integrals can be reduced to master integrals with integration by parts using AIR [53].
The treatment of the counterterm diagram is analogous, where we keep all three external momenta p 1 , p 2 , p 3 as a basis for the tensor reduction.Expanding the master integrals in ϵ, the sum of diagrams is given by where we have used the minimal set of spin-chains defined in Eq. (5.23).We note that the 1/ε 4 pole cancels as expected.Additionally, we have verified the independence on tensor reduction in the interference with the tree-level, as in the one-loop case.

A three-loop diagram with rank-7 tensors in dimensional regularisation
As a last example, we apply this tensor reduction approach to one three-loop diagram for the same process (q q → γγ) considered above D This diagram is a tensor integral and has the same kinematic structure as the examples discussed above and the same choice of polarisation vectors defined in Eq. (5.22).We reduce the tensor structure analogously as in Eq. (5.31).Hence, we include the ordering operator such that the loop momenta contract with the dual momenta and metrics, whereas the non-dual momenta and metric tensors contract with gamma matrices in the spin-chain function.We take the same steps to simplify the spin-chain down to a minimal set of spin-chains seen in Eq. (5.23).We leave the scalar integrals in terms of the scalar products and was performed in TForm [49] with 16 workers in 1107.30seconds on a common desktop.Finally, we were able to confirm the independence on tensor reduction for the three-loop diagram by interfering with the tree-level, as in the previous cases.

Conclusions
In Eq. (4.10), we presented a closed-form solution to the diagonalisation problem of the Passarino-Veltman reduction method for arbitrary tensor integrals.We derived this formula by noticing a connection between the tensor elements appearing a tensor reduction and the ordering, along with associated Wick contractions, of operators that we assigned to tensor indices.This algebraic analogy of tensor reduction and ordering becomes a precise equality in Equation (4.10) when we choose an appropriate basis of tensor elements.To achieve this, we adopt a tensor basis composed of independent external momenta and metric tensors transverse to external momenta.Additionally, we construct a second dual tensor basis that is orthogonal to the first one, combining the constructions presented in Ref. [47] for momentum vectors and Ref. [46] for metric tensors.
The tensor reduction technique presented in this paper is independent of the loop order and topology of the integral.Moreover, it remains identical for all integrals with the same tensor rank and external legs.The key advantage of our method is that it avoids large matrix inversions, with the only requirement being finding the dual momenta of van Neerven and Vermaseren.This significantly reduces the computational cost of tensor reduction.
Our tensor reduction formula enables the treatment of complex Feynman diagrams, even within conventional dimensional regularisation.Extending the use of Passarino-Veltman reduction, our method becomes competitive in cutting-edge computations.In this work, we demonstrated its potential for QCD calculations by applying the method to diagrams with up to three loops and four external legs, resulting in tensors of rank seven.Throughout the article, we performed Feynman diagram computations analytically.
We are confident that Equation (4.10) can be further applied in realistic calculations, perhaps in more inventive ways.For instance, one could envisage numerical evaluations of the multiplications of elements from the dual basis and Feynman diagrams, while scalar integral coefficients can be determined using independent (analytic or numeric) methods.We eagerly anticipate future applications of our method.

A Dual transverse metrics of rank six and eight
As an ansatz for a dual transverse metric product of rank six we write (A.1) The symmetries we used are µ 1 ↔ µ 2 , µ 3 ↔ µ 4 , µ 5 ↔ µ 6 , (µ 1 , µ 2 ) ↔ (µ 3 , µ 4 ) ↔ (µ 5 , µ 6 ). (A. 2) The term proportional to a has the same structure as the dual tensor, terms proportional to b have one transverse metric tensors in common with the original tensor, terms proportional to c have none in common.The coefficients are determined using the orthogonality relations between the different metric tensor products of each type, such that we find ).As already mentioned in Section 3.2, the dual basis for the metric tensors up to rank 14 are available in ancillary file, where the coefficients are calculated using Form.

B One-loop and two-loop N f coefficients
In this Appendix, we present for completeness the components of the one-loop amplitude and the N f contribution to the two-loop amplitude which we computed in Section 5. Our results agree with an independent numerical computation by Dario Kermanschah based on the method of Ref. [60].The full two-loop amplitude for q q → γγ production has been computed analytically in Refs [54,61].The corresponding three-loop amplitude has been computed in Ref. [62].
The one-loop amplitude coefficients for Eq.(5.26) are h