A three-body form factor at sub-leading power in the high-energy limit: planar contributions

We apply the expansion (7) into the differential equations (8) with $t=x$. We note that

That is, taking derivative with respect to $x$ always decreases the power of $x$, but may or may not decrease the power of $\log(x)$. On the other hand, the matrix elements $\mathcal{P}^{(x)}_{i,j}$ may also contain poles at $x=0$. Therefore, the differential equations with respect to $x$ lead to linear relations among the coefficients $\mathcal{C}_{i,n_{1},n_{2},n_{3}}(y,z)$. From these linear relations, we can determine a set of independent coefficients. We will refer to these independent coefficients as “master coefficients” (MCs), and express the remaining coefficients as linear combinations of the MCs.

Before solving the linear relations (there are an infinite number of them), we need to constrain the highest values of the integers $n_{1}$ and $n_{2}$ for each $i$, i.e, the maximal powers of $\epsilon$ and $x$ in the expansion of the master integral $\mathcal{I}_{i}$ (note that the maximal power of $\log(x)$ is naturally determined given the values of $n_{1}$ and $n_{2}$). The first constraint comes from the required orders in the expansion of the squared-amplitude $\mathcal{F}^{(2)}$. As we have mentioned, we truncate the expansion in $x$ up to the NLP, i.e., order $x^{1}$. In addition, we truncate the expansion in $\epsilon$ up to order $\epsilon^{0}$. These requirements impose constraints on $n_{1}$ and $n_{2}$ for each master integral $\mathcal{I}_{i}$. However, we find that these constraints are too tight for our purpose: the MCs determined from these constrained linear relations are not closed under differentiation. The reason is that there are cancellations among different topologies in a family when their contributions go into Eq. (6).

In order to obtain a closed system of differential equations, we need to slightly loosen the constraints. For that purpose, we expand the coefficients in the differential equations as well:

The lowest values of $k$ and $l$ are then crucial for the determination of the highest expansion orders. As an example, we consider the differential equations with respect to $x$. The constraints from $y$ and $z$ can be similarly studied. Plugging Eqs. (7) and (10) into (8), we find that

This is the master equation for relations among expansion coefficients. For this system to be closed, it is necessary that all coefficients contributing to a given $(n_{1},n_{2},n_{3})$ are incorporated in the above equation.

Let’s focus on the equation with $i=11$, which arises from the differential equation of $\mathcal{I}_{11}$ with respect to $x$. In this case, one has $j\in\{1,8,11\}$ on the right-hand side of Eq. (11). For the NLP accuracy of the squared amplitude, one requires both $\mathcal{I}_{11}$ and $\mathcal{I}_{1}$ to be expanded to order $x^{3}$, since the minimal power of $x$ in $\mathcal{A}_{1}$ and $\mathcal{A}_{11}$ is $-2$, which means that we need to consider the equation with $n_{2}=3$. However, the minimal value of $l$ for $i=11$, $j=1$ is $l=-2$. Therefore, the coefficient with $n_{2}-1-l=4$ will appear on the right-hand side of the equation. For this reason, we need to expand $\mathcal{I}_{1}$ up to order $x^{4}$.

Considerations similar to the above also applied to the constrains on $n_{1}$, i.e., the expansion orders in $\epsilon$. We need to perform such analysis for each sector, and update the constraints until all equations are closed. As a final outcome of such analyses, we provide the number of MCs for each integral family in Table 1.

After obtaining a system of linear equations of the coefficients, we can solve the equations to express all coefficients in terms of the MCs. Before doing that, we need to decide a set of rules for choosing the MCs. This is similar to the rules for choosing the MIs in Feynman integral reduction. In practice, this requires to determine between two coefficients which one is preferred over the other.

The choice of MCs will greatly influence the process of solving their differential equations with respect to $y$ and $z$. We would like them to have as simple analytic expressions as possible. From experience, this then corresponds to selecting coefficients with smaller $n_{1}$, smaller $n_{2}$ and larger $n_{3}$. Coefficients in lower sectors are also preferred over those in higher sectors.

After deciding the preference, we can solve the linear system using the “user-defined system” functionality of Kira. We can then construct the system of differential equations of the MCs. For that we need to re-express the derivatives of the MCs in terms of the MCs again. This step can be accomplished by Kira as well. For example, we write the differential equation of a MC with respect to $y$ as

The above equation can be provided to Kira along with other linear relations. One then obtains the left-hand side expressed in terms of MCs. Finally, we arrive at the differential equations

where $I=(i,n_{1},n_{2},n_{3})$ and $J=(j,m_{1},m_{2},m_{3})$ denote two subscript sequences that label the MCs.

We first consider the simple cases where the derivatives of a MC only depend on itself and the already-solved MCs:

where $\mathcal{G}_{y}$ and $\mathcal{G}_{z}$ come from the already-solved MCs. If the coefficients $\mathcal{A}_{y}$ and $\mathcal{A}_{z}$ are both zero, the above equations can be easily solved by direct integration. This corresponds to a strictly triangular system discussed earlier. We now consider the case where $\mathcal{A}_{y}$ and $\mathcal{A}_{z}$ are nonzero.

For convenience, we introduce the following functions:

Multiplying the MC by the two functions, the differential equations Eq. (14) are transformed into:

One can see that only already-solved MCs appear on the right-hand side of the above two equations. We would like to find a transformation function $\mathcal{T}(y,z)$ such that

This is an example of a strictly triangular system of differential equations, where the derivatives of the transformed MCs only depend on already-solved ones. Comparing Eq. (17) with Eq. (16), one can see that

where the function $f_{z}(z)$ satisfies the differential equation

and similar for $f_{y}(y)$. The fact that the right-hand side of the above equation is independent of $y$ follows from the compatibility condition of Eq. (14):

which further follows from the linear-independence of $\mathcal{C}(y,z)$ with respect to $\mathcal{G}_{y}$, $\mathcal{G}_{z}$ and their derivatives. From the above analysis, we can write

Building on this result, we get general solution to the differential equaiton (17):

where $\mathcal{C}_{y}(y)$ is a function that only depends on $y$, and similarly for $\mathcal{C}_{z}(z)$. These two functions are related by the second equal sign in the above formula, and are only fixed once the arbitrary constant terms of the indefinite integrals are chosen. Using the fact that $\mathcal{C}_{y}(y)$ is independent of $z$, we can derive the differential equation of $\mathcal{C}_{z}(z)$ with respect to $z$:

which allows us to determine the analytic expression of $\mathcal{C}_{z}$ (as well as $\mathcal{C}_{y}$) up to a constant of integration. This constant should be fixed by a boundary condition, to be discussed later.

The above procedure of transforming the differential equations to a strictly triangular form can be naturally extended to $2\times 2$ blocks and $3\times 3$ blocks, as will be discussed in the following subsections.

We now consider the $2\times 2$ blocks, where the differential equations take the form

where $t\in\{y,z\}$, $\mathcal{A}^{(t)}_{ij}$’s are rational functions of $y$ and $z$, and $\mathcal{G}^{(t)}_{i}$’s come from already-solved MCs. To solve the equations, we would like to find a linear combination $\mathcal{C}^{\prime}=b_{1}\mathcal{C}_{1}+b_{2}\mathcal{C}_{2}$ with $b_{1}$ and $b_{2}$ being rational functions of $y$ and $z$, such that the derivatives of $\mathcal{C}^{\prime}$ only depend on itself. Taking the derivatives, one has

We therefore require

Once we find a suitable pair of $b_{1}$ and $b_{2}$, the differential equation of $\mathcal{C}^{\prime}$ becomes

The above equation resembles the one studied in Section 4.1. We can employ the method there to get rid of the term $\mathcal{A}^{\prime}_{t}\mathcal{C}^{\prime}$, and effectively transform the differential equation to a strictly triangular form. After obtaining the solution of $\mathcal{C}^{\prime}$, we can use it to derive analytic expressions of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Taking $\mathcal{C}_{1}$ as an example, we have

Since $\mathcal{C}^{\prime}$ is known, the above equation can again be solved using the method of Section 4.1. The expression of $\mathcal{C}_{2}$ can then be obtained from those of $\mathcal{C}^{\prime}$ and $\mathcal{C}_{1}$.

From the above discussions, it is clear that the task boils down to finding a particular solution of $b_{1}$ and $b_{2}$ satisfying Eq. (26). Note that Eq. (26) is actually a differential equation for the ratio $b_{2}/b_{1}$, which can be rewritten as

This is the so-called Riccati equation. There are two ways to find a solution.

Similar to the $2\times 2$ cases, the differential equations for $3\times 3$ blocks have the form

The method for dealing with this system is analogous. First, we need to construct a linear combination $\mathcal{C}^{\prime}=b_{1}\mathcal{C}_{1}+b_{2}\mathcal{C}_{2}+b_{3}\mathcal{C}_{3}$, whose differential equations only involve itself. This boils down to the following equation

This equation can be rewritten as

One can observe that the two unknowns $b_{2}/b_{1}$ and $b_{3}/b_{1}$ are still coupled by the two equations, and this system is challenging to solve by brute force. Fortunately, the second method introduced in the previous subsection still works, and offers an algorithmic way to tackle this kind of problems. We make polynomial ansatz for $b_{1}$, $b_{2}$ and $b_{3}$, and solve for the coefficients from Eq. (4.3). After that, we can solve for $\mathcal{C}^{\prime}$ using the method of Section 4.1. We are then left with a $2\times 2$ system of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, which can be solved following the strategy of Section 4.2.

Let’s again demonstrate our approach using an example, where

A particular solution can be found by using the ansatz with $n=2$, and is given by

This give rise to a decoupled differential equation

We don’t go into the details of the remaining steps here.

In this work, we do not encounter blocks more than $3\times 3$. But as one can see, the method introduced above can be applied to more complicated cases as well. This proceeds in an iterative way: one first solves for a linear combination whose differential equation is decoupled, which then leads to a simpler system.

To obtain the solution to the system of differential equations, we still need to determine the boundary conditions for each coefficient. For that we use AMFlow Liu:2022chg to numerically compute the small-mass expansion of all MIs at a specific kinematic point. During this process, all variables (except $m^{2}$ but including $\epsilon$), are set to exact values (rational numbers). For each value of $\epsilon$, the output has the following structure

where $\alpha_{i}(\epsilon)$ is a linear function of $\epsilon$, $w_{ij}(\epsilon)$ depends on $\epsilon$ and the kinematic variables (which we have suppressed), and $j_{\max}$ is specified according the required expansion order (in $m^{2}$) of the MIs. By varying $\epsilon$ with fixed kinematic variables, we can reconstruct the coefficients in $\alpha_{i}(\epsilon)$, as well as the function $w_{ij}(\epsilon)$ as a power series in $\epsilon$:

To determine the coefficients $a_{ijk}$, we need to sample at least $(k_{\max}-k_{\min}+1)$ different values of $\epsilon$. Note that the precision of the reconstructed $a_{ijk}$ also depends on the number of samples. Therefore, we need to choose an appropriate number according to the required expansion order (in $\epsilon$) of the MIs and the required precision for the coefficients. In practice, this number is about $50$.

With the reconstructed coefficients, we can reexpand Eq. (41) in $\epsilon$, and obtain the small-mass expansion of the MIs at the chosen kinematic point in the form

The minimal powers of $\epsilon$ and $x$, and the maximal powers of $\log(x)$ for a given pair $(n_{1},n_{2})$ can be read off from the above expansion. Note that $f_{i,n_{1},n_{2},n_{3}}$ is just the coefficient $\mathcal{C}_{i,n_{1},n_{2},n_{3}}(y,z)$ at a particular kinematic point, and the general solution of $\mathcal{C}_{i,n_{1},n_{2},n_{3}}(y,z)$ has already been obtained from the differential equations. Such a solution can be written in terms of generalized polylogarithms (GPLs) with the help of PolyLogTools Duhr:2019tlz , up to some unknown constants. These constants can then be reconstructed as transcendental numbers by the PSLQ algorithm, using the high-precision numbers (about 100 decimal digits) of $f_{i,n_{1},n_{2},n_{3}}$. The transcendental basis is generated from the elements $\{\pi,\ln(2),\zeta_{2},\zeta_{3},\zeta_{4},\mathrm{Li}_{4}(1/2),\zeta_{5}\}$ up to weight 5. It is worth mentioning that the DEs may demand higher order coefficients in $\epsilon$, as discussed in Sec. 3.1. That’s the reason why weight-5 functions and constants appear, despite the fact that two-loop amplitudes up to order $\epsilon^{0}$ only require weight-4 at most. Note that the form of the analytic expressions can be different in different kinematic regions. We concretely work in the Euclidean region where all planar MIs have no imaginary part. The expressions in other kinematic regions can be obtained via analytic continuation. Since our results are written in terms of GPLs, their analytic structure is very well understood. The branch cuts of GPLs are uniquely fixed by their symbols, and whenever one analytic continues across a branch cut, an extra term proportional to $\pm i\pi$ is generated, where the sign of $i\pi$ depends on the direction of going across the cut. See, e.g., Goncharov:2001iea ; Vollinga:2004sn ; zbMATH05494656 for more detailed discussions.

With the analytic expressions for the MIs, we can combine them to obtain the planar contributions to the squared-amplitude $\mathcal{F}^{(2)}$ up to order $x^{1}$ and $\epsilon^{0}$. It can be written in terms of GPLs up to weight $5$, with the following symbol letters:

There are two additional letters $\{1-y,\,2-y-z\}$ appearing in the MIs. But they cancel out in the final expression for the squared-amplitude. Note that the original massive amplitude involves elliptic integrals, but the expanded amplitude can be expressed by GPLs up to the NLP. The result for the squared-amplitude is very compact, with a size of about 400 KB. We attach the expression as an electronic file to this paper.

We have performed several sanity checks on our calculation. We have applied our method to the one-loop amplitude where the complete result can be easily obtained. Upon expansion in the high-energy limit, the complete result agrees exactly with our calculation. We have also numerically computed the MIs at several different kinematic points other than the chosen boundary point, and find that the results agree with the outcomes from our analytic expressions. In particular, we have verified kinematic points beyond the Euclidean region, and find that the imaginary parts agree as well. These checks demonstrate the reliability of our method.