The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $\varepsilon$, where the $\varepsilon^0$-part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter $\varepsilon$. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems.

In this letter we report on a more involved computation. We consider the planar double box integral relevant to top-pair production with a closed top loop. This integral enters the next-to-next-to-leading order (NNLO) contribution for the process pp → tt. Up to now, it is not known analytically. The existing NNLO calculation for this process uses numerical approximations for this integral [59,60]. Our inability to compute this integral analytically has been a show-stopper for further progress on the analytical side. In this letter we show how to compute analytically this integral. Our methods are applicable to similar problems.
The planar double box integral depends on two scales (for example s/m 2 and t/m 2 , where s and t are the usual Mandelstam variables and m the mass of the heavy par-ticle). It involves the sunrise graph as a sub-topology. Therefore, we do not expect this integral to evaluate to multiple polylogarithms. Phrased differently, we expect to see elliptic generalisations of multiple polylogarithms. An obvious question is: Which elliptic curve? To some surprise, there is not a single elliptic curve associated to this integral, but three different ones. We show in this letter how to extract the elliptic curves from the maximal cuts of the (sub-) topologies. From these elliptic curves we obtain their periods.
In the next step we bring the system of differential equations to a form linear in ε, where the ε 0 -part is strictly lower triangular. We introduce kinematic variables x and y, which rationalise the square roots in the polylogarithmic case (i.e. for t = m 2 ). The transformation of the basis of master integrals is not rational in x and y, however we find a transformation which is rational in x, y, the periods of the three elliptic curves and their y-derivatives. Note that a system of differential equations linear in ε, where the ε 0 -part is strictly lower triangular, can easily transformed to an ε-form (i.e. without any ε 0 -part) by introducing primitives for the terms occurring in the ε 0 -part. Both systems are equivalent and both are easily solved order by order in the dimensional regularisation parameter ε. For the case at hand the required primitives are usually transcendental functions. We prefer to work with a system linear in ε, where in the transformation matrix only the periods and their derivatives occur as transcendental functions.
There are two interesting cases, where the solution for the Feynman integrals simplify: for t = m 2 the solution degenerates to multiple polylogarithms, for s = ∞ the solution degenerates to iterated integrals of modular forms for Γ 1 (6).

THE INTEGRAL
We consider the planar double box integral shown in fig. (1). This integral is relevant to the NNLO corrections for tt-production at the LHC. In fig. (1) the solid lines correspond to propagators with a mass m, while dashed lines correspond to massless propagators. All external momenta are out-going and on-shell. The Mandelstam variables are defined as usual We are interested in the dimensional regulated integral where γ E denotes the Euler-Mascheroni constant, D = 4 − 2ε denotes the dimension of space-time and the propagators are given by This integral has a Laurent expansion in ε: In this letter we present a method to systematically compute the j-th term of the ε-expansion. The result is expressed in terms of iterated integrals [61]. If ω 1 , ..., ω k are differential 1-forms on a manifold M and γ : [0, 1] → M a path, we write for the pull-back of ω j to the interval [0, 1] The iterated integral is then defined by Multiple polylogarithms are iterated integrals, where all differential one-forms are of the form If f (τ ) is a modular form, we simply write with a slight abuse of notation f instead of 2πif dτ in the arguments of iterated integrals.

THE KINEMATIC VARIABLES FOR THE MULTIPLE POLYLOGARITHMS
The Feynman integral is a function of two kinematic ratios, say s/m 2 and t/m 2 . A significant fraction of the sub-topologies depends only on s/m 2 , but not on t/m 2 . These integrals are expressible in terms of multiple polylogarithms and their system of differential equations can be transformed to an ε-form. This introduces square roots, which are absorbed by a change of kinematic variables. The square root −s(4m 2 − s) is typical for massive Feynman integrals, however there are also sub-topologies, which lead to the square root −s(−4m 2 − s) (note the sign in front of 4m 2 ). An example is shown in fig.(2). A transformation, which absorbs both square roots simultaneously is given by This defines the variables x and y. The variable y is not needed for integrals depending only on s/m 2 . For the integrals depending only on s/m 2 we introduce five differential one-forms Then all sub-topologies, which depend only on s/m 2 , can be expressed as iterated integrals with letters given by these five differential one-forms. From eq. (9) it is clear that they are expressible in terms of multiple polylogarithms.

ELLIPTIC CURVES
Let us consider an elliptic curve defined by the quartic equation We set and define the modulus and the complementary modulus Our standard choice for the periods is where K(x) denotes the complete elliptic integral of the first kind. For the double box integral we have to consider three elliptic curves E (a) , E (b) and E (c) , which occur for the first time in the three Feynman graphs shown in fig. (3). The equations of the elliptic curves are extracted from the maximal cuts of these Feynman integrals [62][63][64][65][66][67][68][69], specifically from the maximal cuts of We find for all three curves They differ in the values for the roots z 2 and z 3 . We have It is easily checked by computing the j-invariants that the three curves are not isomorphic. However, the curves E (b) and E (c) degenerate to curve E (a) in the limit s → ∞. Associated to the curve E (a) are modular forms of Γ 1 (6). We set Relevant to the problem is the set {g 2,0 , g 2,1 , g 2,9 , g 3,1 , p 3,0 , g 4,0 , g 4,1 , g 4,9 , p 4,0 , p 4,1 } . (17) These are modular forms of Γ 1 (6) in the variable τ 6 = ψ

MASTER INTEGRALS AND DIFFERENTIAL EQUATIONS
In order to derive the system of differential equations we first used Reduze [70], Kira [71] and Fire [72] for the integral reductions. Taking trivial symmetry relations into account, all programs give 45 master integrals. However, the reductions disagree for the three most complicated topologies. For a given set of master integrals I we obtain the system of differential equations In general, this system is not yet linear in ε, but it should satisfy the integrability condition At first sight, the results of two of three programs above fail the integrability check. Still, all three programs correctly implement the Laporta algorithm [73]. However, the Laporta algorithm does not guarantee that all relations among the Feynman integrals are found. Here, we have an example where one additional relation exists in the sub-topology shown in fig. (4). This additional relation reduces the number of master integrals in this topology from 5 to 4. Imposing this relation, the results from Reduze, Kira and Fire agree and the integrability condition is satisfied. In addition, we verified numerically the first few terms in the ε-expansion of this relation.
In this letter we are interested in the integral I 1111111 . With the help of the methods from [9] we may decouple two integrals in the top topology. Thus we have to consider a system of 42 master integrals for I 1111111 .
Under a change of basis the differential equations transform into The main result of this letter is that there exists a transformation U , such that and A (0) is strictly lower triangular (i.e. A (0) ij = 0 for j ≥ i). The system of differential equations is linear in ε and easily solved order by order in ε in terms of iterated integrals. The transformation matrix is rational in We constructed this matrix by analysing the Picard-Fuchs operators in the diagonal blocks [9] and by using a slightly modified version of the algorithm of Meyer [74,75] for the non-diagonal blocks. To give an example, the three master integrals in the topology I ν1ν2ν3ν400ν7 can be taken as where R 25,24 and R 26,24 are rational functions in (x, y), D − denotes the dimension shift operator D → D − 2 and W (b) y the Wronskian As in the sunrise sector [40], one integral is divided by a period (J 24 ), while a second integral is given as a derivative plus additional terms (J 26 ). This pattern applies to all elliptic sectors. The matrix A (0) in eq. (22) vanishes for x = 0 or y = 1. The occurrence of ε 0 -terms in the differential equations is expected from the study of the sunrise integral with unequal masses [29,33]. For y = 1 the entries of A (1) reduce to the differential one-forms of eq. (9), for x = 0 they reduce to the modular forms of eq. (17). The solution reduces therefore to multiple polylogarithms for y = 1 and to iterated integrals of modular forms for x = 0. Albeit the transformation U significantly simplifies the system of differential equations, the length of the solution still exceeds the format of this letter. The detailed solution will be given in a longer forthcoming publication [76].

CONCLUSIONS
In this letter we analysed the planar double box integral relevant to top pair production with a closed top loop. This integral depends on two scales and involves several elliptic sub-sectors. This integral has not been known analytically and impedes further progress on the analytic computation of higher-loop Feynman integrals with massive particles. In this letter we reported that we may transform the system of differential equations to a form linear in ε, where the ε 0 -term is strictly lowertriangular. With such a linear form the solution in terms of iterated integrals is immediate. Our techniques open the door for more complicated Feynman integrals.