Analytic results for two-loop planar master integrals for Bhabha scattering

The Bhabha scattering process, i.e., the elastic scattering of an electron-positron pair, is one of the standard candles at lepton colliders, and it will play a crucial role at future circular or linear colliders. A precise theoretical knowledge of this process, including up to next-to-next-leading order effects in the QED coupling constant, is therefore highly desirable. So far the NNLO cross section is only known in the massless limit, supplemented by the leading logarithmic finite mass effects (see, e.g., ref. Banerjee:2021mty for recent results). Complete NNLO results including the full dependence on the electron mass, however, are not yet available.

One of the main obstacles to obtain the complete NNLO results is the complexity of the two-loop integrals involved. The relevant two-loop integrals were evaluated in the small-mass limit in refs. Actis:2006dj ; Actis:2007gi ; Actis:2007pn ; Actis:2007fs ; Actis:2008br . Up to now, there are only partial results for two-loop integrals with massive fermions. Analytic results for diagrams with one-loop insertions and a closed massive fermion loop were obtained in refs. Bonciani:2003cj ; Bonciani:2004gi . First analytic results for massive two-loop double-box Bhabha diagrams were obtained in refs. Smirnov:2001cm ; Heinrich:2004iq . More attempts to evaluate two-loop Bhabha integrals can be found in refs. Czakon:2004wm ; Czakon:2005jd ; Czakon:2005gi ; Czakon:2006pa ; Czakon:2006hb .

The master integrals relevant for the calculation of the two-loop corrections to Bhabha scattering can be classified into two planar families and one non-planar one. The systematic evaluation of the integrals in the first family was started in ref. Henn:2013woa , where the master integrals for the family associated with graph (a) of fig. 1 were evaluated in terms of multiple polylogarithms Lappo:1927 ; Goncharov:1998kja ; GoncharovMixedTate (MPLs)

in the framework of the method of differential equations Kotikov:1990kg ; Remiddi:1997ny ; Gehrmann:1999as with the help of the strategy based on canonical bases Henn:2013pwa . In ref. Henn:2013woa , a solution in terms of MPLs had been provided for all integrals except one, whose analytic evaluation was hindered by the presence of the a non-rationalisable square-root in the symbol alphabet. More recently, it was shown that also this last integral can be expressed in terms of multiple polylogarithms by an integration technique based on an ansatz of MPLs with suitable arguments Heller:2019gkq .

The goal of the present paper is to analytically evaluate the master integrals for the second planar family, which is associated with graph (b) of fig. 1. In a way that is reminiscent of the first planar family of integrals, we will show that also in this case we can obtain results in terms of MPLs for all master integrals but one (see fig. 2), due to the presence of the same non-rationalisable square root found in the evaluation of the master integrals for graph (a). While it can be shown by direct integration techniques that also for this master integral a representation in terms of MPLs exist, the representation we obtained is extremely cumbersome and of no practical use. Nevertheless, it turns out that a compact result for this integral can be derived in terms of the elliptic MPLs introduced in refs. Broedel:2017kkb ; Broedel:2017siw ; Broedel:2018iwv . As a byproduct of our analysis, we will also provide a very compact analytic result for the remaining master integral in the first family in fig. 1(a) in terms of the same class of functions.¹¹1This result had first been presented by one of the authors at the Loops and Legs Conference 2018 in St. Goar, but it has never been published before.

The rest of the paper is organised as follows. In section 2 we explain the notation, present the system of canonical differential equations for the problem at hand, and introduce the alphabet needed to describe the planar family in fig. 1(b). The alphabet contains multiple square roots, and it is a priori not obvious that the differential equations can be integrated in terms of multiple polylogarithms. In section 3 we review general sufficient criteria and algorithms to solve a system of canonical differential equations in terms of MPLs or their elliptic generalisations. We then continue in section 4 to apply these ideas to our computation. We explain how our differential equations can straightforwardly be solved in terms of multiple polylogarithms for all elements of the basis of the master integrals but one, which is connected with the graph shown in fig. 2.

In section 5 we show how a convenient and compact representation for this integral can be found in terms of elliptic generalisations of MPLs. We also provide a very compact, alternative, analytic solution for one of the master integrals in the first planar family considered in refs. Henn:2013pwa ; Heller:2019gkq . Finally we draw our conclusions in section 6.

The Feynman integrals for the family of fig. 1 (b) can be organised in an integral family with nine propagators, where the first seven propagators correspond to the edges of the graph and the last two are so-called irreducible numerators,

The indices $a_{i}$, $i=1\ldots 7$ can be positive or negative integers, but we restrict our computation to $a_{8},a_{9}\leq 0$. We work in dimensional regularisation in $D=4-2\epsilon$ dimensions in order to regulate both infrared and ultraviolet divergences. The electron mass is denoted by $m$, and the external momenta $p_{i}$ are on shell, $p_{i}^{2}=m^{2}$. We introduce the usual Mandelstam variables

with $s+t+u=4m^{2}$.

Using the public codes FIRE Smirnov:2019qkx and KIRA Maierhoefer:2017hyi ; Klappert:2020nbg , we solve the integration-by-parts (IBP) identities Chetyrkin:1981qh ; Tkachov:1981wb and reveal 43 independent master integrals, which we collect into the vector $g=(g_{1},\ldots,g_{43})^{T}$. This vector satisfies a system of linear differential equations of the form

where $v=s,t,m^{2}$, $\partial_{v}=\frac{\partial}{\partial v}$ and the matrices $A_{s},A_{t},A_{m^{2}}$ are rational functions of $s,t,m^{2}$ and $\epsilon$. In the following, it will be useful to collect all the partial derivatives into a total differential, and to work with the equation

In order to evaluate the master integrals, it is convenient to search for a so-called canonical basis Henn:2013pwa , i.e., a basis of master integrals for which the matrix $dA$ on the right-hand side of eq. (4) is proportional to $\epsilon$ and its entries can all be expressed as linear combinations of total differentials of logarithms. While it has been suggested (at least conjecturally) that the study of the residues of the integrands can provide the full information to determine the elements of the canonical basis ArkaniHamed:2010gh , this analysis can become computational very expensive when square-roots are involved.²²2See ref. Henn:2020lye for an automated implementation of this approach. Therefore, we follow here a mixed approach to find a canonical basis. In order to select suitable candidates, we start by analysing only the leading singularities associated to the maximal cuts of the various integrals, and we supplement this analysis by the method of ref. Gehrmann:2014bfa . By choosing integrals with unit leading singularities at the level of the maximal cuts, one can often bring the initial differential equations into a so-called precanonical form, where the corresponding matrices depend linearly on $\epsilon$. Once this is achieved, the prescriptions of ref. Gehrmann:2014bfa can be successfully applied to arrive at a fully canonical form. In our case, the precanonical basis is composed of the $43$ Feynman integrals present on the right-hand side of the above expression for the $\epsilon$-basis. In this way we obtain the new system of differential equations

with $\bar{A}$ independent of $\epsilon$. Our canonical basis $f_{i}$ is given in appendix A. The price to pay for casting the equations in canonical form is that the new matrix $d\bar{A}$ is not a matrix of rational one-forms, but it involves four square roots:

More precisely, the matrix $\bar{A}$ takes the general form

where the $R_{i}$ are algebraic functions, referred to as letters (and their collection is called the alphabet):

As we will see, the appearance of these square roots makes the solution of the differential equation (5) in terms of MPLs highly non-trivial.

It is easy to write down a formal solution to eq. (5) as

where $\mathbb{P}\textrm{exp}$ denotes the path-ordered exponential and $f_{0}(\epsilon)$ encodes the initial condition and is related to the value of $f$ at a specific point $(s_{0},t_{0},m_{0}^{2})$. The path $\gamma$ connects the initial point $(s_{0},t_{0},m_{0}^{2})$ to the generic point $(s,t,m^{2})$.

When expanding eq. (9) in $\epsilon$, then at each order we can write $f$ in terms of Chen iterated integrals ChenSymbol , defined in the following way: consider a path $\gamma$ and a collection of one-forms $\omega_{i}$. If $t$ denotes a local coordinate on $\gamma$, we can pull each $\omega_{i}$ back to $\gamma$ and write $\gamma^{*}\omega_{i}=dt\,F_{i}(t)$. We can then define the iterated integral of a sequence $\omega_{i_{1}}\ldots\omega_{i_{n}}$ (usually referred to as a word) by

The number $n$ of integrations is called the length of the iterated integral. In general, this integral will not be homotopy-invariant, i.e., it will depend on the details of the path $\gamma$. There is a necessary and sufficient condition, called the integrability condition, for a combination of iterated integrals to be homotopy-invariant ChenSymbol . The details of this criterion are not important in the following. Here it suffices to say that it is always satisfied for the solutions in eq. (9). The corresponding Chen iterated integrals are then multi-valued functions of the end point $(s,t,m^{2})$ of the path $\gamma$, where the multi-valuedness only comes from choosing two non-homotopic paths from $(s_{0},t_{0},m_{0}^{2})$ to $(s,t,m^{2})$.

The master integrals $f(s,t,m^{2};\epsilon)$ can be expressed at every order in terms of Chen iterated integrals over words from the alphabet in eq. (8). The coefficient of $\epsilon^{n}$ in the path-ordered exponential in eq. (9) only involves iterated integrals of length $n$. Chen iterated integrals are a very general class of functions, and it can be useful to express them in terms of a class of functions that are well-studied in the literature. In the next section we discuss sufficient criteria for when Chen iterated integrals over $d\log$-forms with algebraic arguments, as the ones above, can be expressed in terms of other classes of functions.

Since the solution of differential equations in canonical form as Laurent series in dimensional regularisation leads naturally to linear combinations of Chen iterated integrals over words of $d\log$-forms, it is natural to ask when it possible to evaluate such iterated integrals in terms of other classes of functions, and if it is possible to perform this rewriting algorithmically. The advantage of this rewriting lies in the fact that these classes of functions may be well studied in the literature, and there may be established methods or computer codes for their manipulation and/or their numerical evaluation.

Let us consider Chen iterated integrals of the form $\int_{\gamma}w$, where $\gamma$ is a path from an initial point ${\bf x}_{0}=(x_{0,1},\ldots,x_{0,p})$ to the point ${\bf x}_{1}=(x_{1,1},\ldots,x_{1,p})$, and $w$ is an integrable combination of words of length $n$ of the form

We consider the initial point ${\bf x}_{0}$ fixed, and we see the integral as a multi-valued function of the end-point ${\bf x}_{1}$. The arguments of the logarithms are assumed to be algebraic functions of the variables ${\bf x}=(x_{1},\ldots,x_{p})$.

For some time, there was a folklore belief in the physics community that all such iterated integrals could be evaluated in terms of a rather simple class of iterated integrals, namely the so-called multiple polylogarithms (MPLs), defined by Lappo:1927 ; Goncharov:1998kja ; Remiddi:1999ew ; GoncharovMixedTate

where the recursion starts with $G(;z)\equiv 1$. In the special case where all the $a_{i}$’s are zero, we define

The shared belief in physics was that, whenever the $d\log$’s in eq. (11) have algebraic arguments $R_{i}$, then this integral can be written in terms of MPLs whose arguments $a_{i}$ and $z$ are algebraic functions of ${\bf x}_{0}$ and ${\bf x}_{1}$. This belief was shown to be false in ref. Brown:2020rda , where an example of an iterated integral of length two was constructed that cannot be evaluated in terms of MPLs with algebraic arguments. The result of ref. Brown:2020rda shows that the question of whether an iterated integral of $d\log$-forms with algebraic arguments can be expressed in terms of MPLs depends in general on the details of the integral, including the details of the integration contour.

In the remainder of this section we discuss some special cases of algebraic letters $R_{i}$ where the Chen iterated integrals can be evaluated in terms of other classes of special functions. The starting point is the observation that in the context of Feynman integrals, the $R_{i}$ are not generic algebraic functions, but often involve at most square roots of polynomials, i.e., they are of the form:

where the $r_{i,j}$ are rational functions, and the $q_{a}$ are polynomials. Without loss of generality, we assume the $r_{i,j}$ to be polynomials. At this point we have to make a comment: The integrand of the iterated integration, and thus the functional form of the letters $R_{i}$, is sensitive to a change of variables. In particular, one may ask if we can find a change of variables ${\bf x}=\psi({\bf y})$, with $\psi$ a rational function, such that $q_{a}(\psi({\bf y}))=u_{a}({\bf y})^{2}$ is a perfect square, for some values of $a$. This operation is known as rationalisation of square roots. Over the last few years, several algorithmic criteria have been developed to find such a parametrisation for specific Feynman integral computations, or to prove that rationalisation is instead not possible Festi1 ; Festi:2018qip ; Besier:2018jen ; Besier:2019kco ; Besier:2020klg ; Besier:2020hjf ; Festi:2021tyq . Note that rationalisability can easily be decided for a single square root of a one-variable polynomial ($p=N_{\textrm{sqrt}}=1$) based on the degree $\deg q_{1}$ of the polynomial: the square root $\sqrt{q_{1}(x_{1})}$ can be rationalised (and the corresponding function $\psi$ can be constructed explicitly) if and only if $\deg q_{1}\leq 2$. The question of the rationalisability for $p>1$ is much more involved, even in the presence of a single square root, see refs. Besier:2018jen ; Besier:2019kco ; Besier:2020klg ; Besier:2020hjf ; Festi:2021tyq .

In the following we discuss two special cases of eq. (14), in which we can express the Chen iterated integrals in terms of other classes of iterated integrals:

• •

  If $N_{\textrm{sqrt}}=0$, the Chen iterated integrals can be expressed in terms of MPLs evaluated at algebraic arguments.

• •

  If $N_{\textrm{sqrt}}=1$ and $\deg q_{1}=3$ or $4$, the Chen iterated integrals can be expressed in terms of elliptic generalisations of MPLs evaluated at algebraic arguments.

In particular, we explain how we can algorithmically rewrite all Chen iterated integrals that meet these criteria in terms of MPLs and their elliptic analogues. This algorithm is well known, but we document it here because it will be an important tool to obtain compact analytic expressions for some master integrals that contribute to Bhabha scattering at two loops. Two comments are in order: First, in the presence of (multiple) square roots, it may be possible to change variables and rationalise (some of) the roots. In this way it may possible to reduce the problem to a situation covered by the criteria above, even though the original problem did not satisfy these conditions. Second, we stress that the aforementioned conditions are only sufficient, but by no means necessary, to rewrite Chen iterated integrals in terms of (elliptic) MPLs. In particular, there are several examples of Feynman integrals whose alphabets involve non-rationalisable square roots, but nevertheless, it was possible to express them in terms of ordinary MPLs, cf.,  e.g., refs. Heller:2019gkq ; Heller:2021gun ; Kreer:2021sdt .

After the general remarks of the previous section, we go back to the explicit solution for the system in eq. (5). The standard way to rationalize the first two square roots in eq. (6) is to turn to the dimensionless Landau variables $x$ and $y$ related to the Mandelstam invariants by

In terms of these variables the Euclidean region $s,t<0$ corresponds to $0\leq x,y\leq 1$ (using the symmetry of eq. (33) under $(x,y)\to(1/x,1/y)$).

There exist different strategies to solve the differential equation in eq. (5). The first method consists in evaluating numerically the $\epsilon$-expansion of the path-ordered exponential in eq. (9). This can be achieved by application of the Frobenius method to look for power-series solutions of ordinary differential equations in the vicinity of regular singular points. The Frobenius method has been successfully used in various one-dimensional problems in the past Pozzorini:2005ff ; Aglietti:2007as ; Caffo:2008aw ; Bonciani:2018uvv , and more recently a generalisation of this method has been proposed in ref. Francesco:2019yqt to deal with complicated multidimensional problems, see for example Bonciani:2019jyb ; Abreu:2020jxa . This strategy has also been implemented into the public code code DiffExp Hidding:2020ytt .³³3A Mathematica notebook is provided as ancillary material with the arXiv submission. The input data for this code are the matrices that define the differential equations, and the boundary conditions in some limit, e.g., at some point $(x_{0},y_{0})$. The code then uses this input to evolve the solution numerically from the point $(x_{0},y_{0})$ to some generic point $(x,y)$. We fix the boundary conditions in the limit $s,t\to 0$, which corresponds to $(x_{0},y_{0})=(1,1)$ in terms of the Landau variables in eq. (33). Using the expansion by regions strategy implemented in the public code asy.m Pak:2010pt ; Jantzen:2012mw (which is now included in the code FIESTA Smirnov:2015mct ) we obtain the following leading order asymptotic behaviour in this limit

and $f_{i}\sim 0$, i.e., $f_{i}=\cal O(s,t)$, for all the other elements.

Let us emphasise that, in order to profit maximally from the automated code DiffExp, it is crucial that the input data are in an optimal form. This includes providing differential equations in canonical form and fixing the boundary conditions in a simple point ($s=t=0$, i.e., $x=y=1$). With our input, the code works very well and provides the possibility to obtain high-precision numerical results (100 digits accuracy and more), both in the Euclidean and the physical regions. In other words, DiffExp allows us to obtain high-precision numerical results for all master integrals and for all values of the input parameters. The code runs fast enough to allow its usage for practical applications. This is then of course sufficient for all phenomenological applications one has in mind. However, both from a formal and from a practical point of view, it may still be desirable in some situations to have full-fledged analytic representations of the master integrals in terms of functions that are well studied in the literature, e.g., MPLs. In the remainder of this section we explain how we can obtain analytic results for all master integrals but $f_{14}$ in terms of MPLs evaluated at algebraic arguments. The reason why $f_{14}$ is different will be discussed in section 5. Here it suffices to say that the square root $r_{u}$ only enters the differential equation for $f_{14}$. Since our strategy of solving the differential equations in terms of MPLs will be closely based on the ideas from section 3.1, it is not suprising that an additional square root implies that the sufficient condition of section 3.1 is not satisfied. We will discuss the case of $f_{14}$ in detail in section 5, and we focus for the rest of this section only on the other master integrals.

In order to solve the master integrals in terms of MPLs, we start by observing that the square root $r_{st}$ does not appear when solving the differential equations up to weight 3 for all elements but $f_{37}$, and at weight 4 for all elements but $f_{i},i\in\{35,36,37,38,39,41,43\}$. The equations can be solved by integrating first in $x$, resulting in MPLs of the form $G(\vec{c};x)$ with $c_{i}\in\{0,-1,1,-y,-1/y\}$. This allows us to fix the solution up to a function of $y$, which can be determined by substituting the results of the $x$-integration into the differential equations in $y$, checking that the variable $x$ disappears in them, and finally solving them in terms of MPLs of the form $G(\vec{c};y)$ with $c_{i}\in\{0,-1,1\}$, i.e., harmonic polylogarithms Remiddi:1999ew . At this point the solution is fixed up to a set of undetermined integration constants, which we fix using the boundary conditions in eq. (34).

To evaluate $f_{37}$ at weights 3 and 4 and $f_{i}$ with $i\in\{35,36,38,39,41,43\}$ at weight 4, we have to deal with the square root $r_{st}$. It can be rationalized by the following further change of variables

The resulting system of differential equations for 42 elements can be solved, first in $w$ and then in $y$. Solving in $w$ gives a linear combination of MPLs $G(\ldots;w)$ with the letters the alphabet $\{l^{w}_{1},\ldots,l^{w}_{15}\}$, where $l^{w}_{i},i=1,\ldots,11$, are taken from the set

and $l^{w}_{i},i=12,\ldots,15$ are the roots of the polynomial $\cal Q(y,w)$ defined by