Symbology for elliptic multiple polylogarithms and the symbol prime

Elliptic multiple polylogarithms occur in Feynman integrals and in particular in scattering amplitudes. They can be characterized by their symbol, a tensor product in the so-called symbol letters. In contrast to the non-elliptic case, the elliptic letters themselves satisfy highly non-trivial identities, which we discuss in this paper. Moreover, we introduce the symbol prime, an analog of the symbol for elliptic symbol letters, which makes these identities manifest. We demonstrate its use in two concrete examples at two-loop order: the unequal-mass sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. Finally, we also report the result of the polylogarithmic nine-point double-box integral, which arises as the soft limit of the ten-point integral.


Introduction
Scattering amplitudes as well as further quantities in Quantum Field Theory contain a rich mathematical structure, whose understanding has frequently expanded our calculational reach -benefiting both phenomenological tests of the Standard Model of Particle Physics at the LHC as well as more formal studies.
Our good understanding of MPLs is to a large extend due to the Hopf algebra structure underlying these functions [62][63][64][65][66], and in particular the symbol [63].The symbol map associates to each MPL f a simple tensor product, S(f ) = log(φ i 1 ) ⊗ • • • ⊗ log(φ i k ). 1 The entries in this tensor product, called symbol letters, are logarithms of rational or algebraic functions φ i of the kinematic invariants.Since a tensor product is easy to manipulate, and the identities of the symbol letters, log(a) + log(b) = log(ab), are well understood, the symbol provides a powerful way of finding identities between MPLs and for simplifying expressions.The symbol moreover manifests physical properties of the corresponding function.For example, the first entry of the symbol describes the discontinuities, which are heavily restricted in particular in massless theories resulting in so-called first-entry conditions [67].Moreover, discontinuities in overlapping channels are forbidden by the so-called Steinmann conditions [68,69], restricting which symbol letter in the second entry can follow a particular letter in the first entry.The symbol has made possible enormous progress for quantities consisting of MPLs, both in relation to phenomenology and more formal studies, including in particular powerful bootstrap techniques .
While the symbol for eMPLs was defined in [25,100], it has so far not been put to much use, and is still much less understood than its analog for MPLs.One reason is that the symbol letters Ω (i) for eMPLs are themselves elliptic functions of the kinematic invariants.In particular, Ω (−1) = −2πiτ occurs as a letter, where τ = ω 2 /ω 1 is the ratio of the two periods of the elliptic curve.The letters Ω (0) satisfy simple identities as the consequence of the group law on the elliptic curve.In [33], some identities for the elliptic letters Ω (i) with i = 1, 2 were observed numerically in the study of the symbol of the ten-point elliptic double-box integral.Using these identities, it was found that the elliptic letters in the first two entries simplify to logarithms, manifesting the same first-entry condition as for polylogarithmic amplitudes as well as the Steinmann conditions.Moreover, the last entries were found to be given by simple elliptic integrals Ω (0) , with Ω (2) only occurring in the third entry preceding the modular parameter τ in the last entry.
In this paper, we show how the identities observed in [33] for Ω (1) are a con-  sequence of Abel's theorem [101].Moreover, we demonstrate that the identities observed in [33] for Ω (2) are a consequence of the elliptic Bloch relation [102,103] for the elliptic dilogarithm, which generalizes the Bloch relation for the classic dilogarithm and which have also been studied in the context of finding identities between elliptic multiple polylogarithms [104,105].While the identities for Ω (1) can be reduced to three-term identities similar to log(a) + log(b) = log(ab) in the case of the logarithm, the elliptic Bloch relation, and thus the identities for Ω (2) , are five-term identities similar to the Bloch relation for the classical dilogarithm, which are made manifest only by the symbol.Thus, we introduce a symbol prime S ′ for the symbol letters Ω (2) (and similarly for Ω (n>2) ) in analogy to the symbol for MPLs and eMPLs, which makes the identities due to the elliptic Bloch relation manifest.
In general, eMPLs transform under modular transformations of τ in a complicated way [30,32], and results given in terms of eMPLs are not manifestly double periodic.However, in the examples we studied, we find that the symbol prime makes both double periodicity as well as a simple behavior under modular transformations manifest.Finally, it makes also part of the integrability conditions manifest, which result from the requirement that partial derivatives commute.
To illustrate the use of the symbol for eMPLs, the application of the identities of elliptic letters as well as the symbol prime, we study two concrete examples.The first example is the two-loop sunrise integral in two dimensions with all internal masses being unequal (see figure 1a).The second example is the ten-point two-loop doublebox integral in four dimensions with massless internal propagators (see figure 1b).In addition to the aforementioned properties and techniques, we also demonstrate how the (elliptic) symbol reduces to polylogarithmic symbol in kinematic limits where the elliptic curve degenerates.
The remainder of this paper is organized as follows.We review elliptic multiple polylogarithms in section 2. In section 3, we derive identities for elliptic symbol letters -based on Abel's addition theorem for Ω (1) 's and by introducing the symbol prime map for Ω (2) 's.We illustrate the use of these techniques for the unequal-mass sunrise integral in section 4 and for the ten-point double-box integral in section 5.In particular, we provide analytic results for the non-elliptic nine-point doublebox integral and its symbol, which result from taking the soft limit of the ten-point double-box integral.We conclude with a discussion and an outlook on open questions in section 6.In appendix A, we review the calculation of the unequal-mass sunrise integral via Feynman parameters using our conventions and notation.The details of simplifying the symbols for the sunrise integral, as well as the expressions of the functions and symbols for the ten-point elliptic integral and its soft limit, are included as ancillary files (sunrise symbol.nb,doublebox omega2 and doublebox soft). 2

Review of Elliptic Multiple Polylogarithms
Let us first review several elementary facts about elliptic multiple polylogarithms; see [19,20,25,26] for further details.We follow the notations and conventions of [33], which differ slightly from those of [19,20,25,26].

Elliptic multiple polylogarithms on the elliptic curve
In this paper, the elliptic curves C are described by monic quartic polynomials: Such elliptic curves can be birationally mapped to Weierstrass form using the rational point at (x, y) = (+∞, +∞), where (X, Y ) are related to (x, y) by On the curve C, we can introduce elliptic multiple polylogarithms E 4 , which are recursively defined as [19] 3 with E 4 (; x) = 1, where with y c = y| x=c .The definitions of ψ n (c, x) for n = ±2, ±3, . . .can be found in [19]; the kernels (2.5) are sufficient for the purpose of this paper, though.The class of elliptic multiple polylogarithms E 4 ( n 1 ... n k c 1 ... c k ; x) contains in particular all non-elliptic (Goncharov) multiple polylogarithms, defined by where G is a polylogarithm whose letters are rational functions of x and y, can be converted to E 4 functions with only the four kinds of integration kernels defined in (2.5). 4 In particular, this is the case for the (unequal-mass) sunrise integral and the ten-point double-box integral, which we will study as examples in sections 4-5.

From the elliptic curve to the torus
The functions E 4 are defined on the elliptic curve and directly arise from the Feynmanparameter representation of Feynman integrals.However, the purity of some elliptic Feynman integrals, such as integrals of the form (2.7), is not visible when expressed in terms of E 4 functions since taking the total derivative of a E 4 function does not necessarily decrease its length [25]. 5On the other hand, iterated integrals defined on the torus, such as the Γ functions we will review below, are manifestly pure and hence allow a symbol map defined via the total derivative.To connect both sides, we first need a bijection between the elliptic curve C and the torus C/Λ, where Λ is the lattice generated by the periods ω 1 and ω 2 of the elliptic curve.For an elliptic curve of the form (2.1), one can find such a map through the birationally equivalent curve in the Weierstrass normal form: first solve (x, y) The distribution of the four roots of y 2 (x) and the two integration contours γ 1 and γ − defining the period ω 1 and z − ∞ .The contour γ 2 which defines the period ω 2 runs along the real axis.in terms of (X, Y ) from (2.3), then replace X and Y with the Weierstrass elliptic function ℘(z) and its derivative ℘ ′ (z), respectively.This gives where It is easy to see that κ(0) = ∞, and hence all lattice points are mapped to the infinity point in the (x, y)-space.Furthermore, each point c in the x-space corresponds to two points (c, ±y c ) on the elliptic curve C and hence to two images on the torus C/Λ, which we denote by z ± c ; these two images satisfy since the corresponding points (X ± c , Y ± c ) and are on the same line in (X, Y )-space.The inverse map from the torus to the elliptic curve is simply given by the Abel-Jacobi map.We assume that the four roots of y 2 (x) come in complex conjugate pairs as shown in figure 2. 6 Then, the torus image z + c for any real c is given by (2.12) Hence, z + ∞ is one period of the torus, and we choose it to be ω 2 .The image z − c can be obtained by (2.10) together with z − ∞ = γ − dx/y, and the other period is ω 1 = γ 1 dx/y; see figure 2 for the definitions of the integration contours γ − and γ 1 .Due to the distribution of roots, ω 2 and iω 1 are positive reals.

Elliptic multiple polylogarithms on the torus
Due to the equivalence between the elliptic curve and the torus, another way to define elliptic multiple polylogarithms is via iterated integrals on a torus.Such iterated integrals can be formulated in several ways.In this paper, we use the so-called Γ functions [19,25], which are defined as7 with Γ(; w|τ ) = 1; we will frequently suppress the dependence on τ for ease of notation.Such an iterated integral is said to have length k and weight k n k , and in contrast to the case of MPLs both quantities are not necessarily equal.The integration kernels g (n) (w, τ ) are generated by the Eisenstein-Kronecker series where θ 1 (w|τ ) is the odd Jacobi theta function.All the integration kernels g (n) except g (0) = 1 are quasi double periodic, but meromorphic with only logarithmic poles at most [19,25].Note that the functions Γ and the integration kernels g (n) are defined on the torus with one period rescaled to be 1, and that the torus with periods (ω 1 , ω 2 ) has the two possible rescalings [1 : τ = ω 2 /ω 1 ] and [1 : τ ′ = −ω 1 /ω 2 ], which are related by the modular S-transformation τ → −τ −1 .We denote the images of c on [1 : τ = ω 2 /ω 1 ] as w ± c = z ± c /ω 1 and the images on [1 : In what follows, most of the results are written in terms of w-coordinates, but the corresponding results with w replaced by ξ also hold unless otherwise indicated.
The integration kernels ψ n can be identified as combinations of g (j) 's by matching poles on both sides.On the torus [1 : τ = ω 2 /ω 1 ], one can easily find the following relations between g (j) 's and ψ n 's, ψ −1 (c, x)dx = g (1) (w − w + c ) − g (1) (w − w − c ) + g (1) (w + c ) − g (1) (w − c ) dw , (2.16b) ) On the torus [1 : τ ′ ], the corresponding relations can be obtained by replacing w → ξ and ω 1 → −ω 2 in (2.16).Sometimes, it is more convenient to combine Γ functions into the so-called E 4 functions [26], especially if the Γ functions originally arose from an expression of E 4 functions.The elliptic multiple polylogarithms E 4 are defined in complete analogy to (2.4): and Ψ 0 (x)dx = dw, as well as analogous expressions in terms of ξ.The weight of a function E 4 ( n 1 ... n k c 1 ... c k ; x) is defined as i |n i |.

Symbol
By construction, the total derivative of Γ admits a recursive structure [25], where 0 ≡ 0 0 , w i,j ≡ w i − w j , (w 0 , w k+1 ) ≡ (w, 0), (n 0 , n k+1 ) ≡ (0, 0) as well as The forms ω (j) (w) are exact, and we can thus write them as with where η(τ ) is the Dedekind eta function and ζ j = n∈Z + n −j are the Riemann zeta values. 8The functions Ω (j) satisfy The sum representation (2.22) can be derived using the sum representation of the g (n) functions given in [25].In particular, where Ω (n) (0) vanishes for even n and is the primitive of the Eisenstein series for odd n; see [25].We will see below that the functions Ω (j) play the role of elliptic symbol letters.As can be seen from (2.22), Ω (1) has a logarithmic singularity at all lattice points, while Ω (j>1) has a logarithmic singularity at all lattice points except for the origin [25].
For a function Γ k of weight n and length k, we can define Γ Thus, it is natural to define the symbol of Note that in contrast to [25] we have included additional factors of (2πi) in the definition of the elliptic letters Ω (n) and consider the symbol of Γ k . 9This is such that the elliptic letters and symbols degenerate to logarithms and polylogarithmic symbols without additional factors of (2πi) in the limit where the elliptic curve degenerates, see sections 4-5.

Identities of Elliptic Symbol Letters and the Symbol Prime
We have briefly reviewed several elementary facts about elliptic multiple polylogarithms, and we saw that the symbol letters of elliptic multiple polylogarithms are the functions Ω (n) (w, τ ).These functions stand in the way of analyzing the elliptic symbols.For one thing, the relations among Ω (n) 's are much more complicated than the manipulation rules log a + log b = log ab for the symbol letters of multiple polylogarithms.For another, they depend on the kinematics in a rather indirect waytheir arguments w and τ are (ratios) of elliptic integrals involving kinematics.
In this section, we investigate the identities of the elliptic letter Ω (n) (w, τ ).The most trivial identities these letters satisfy are the following: Quasi periodicity : They immediately follow from (2.22).Our investigation of more non-trivial identities will be focussed on the cases n = 0, 1, 2 since -for the two examples considered in this paper, namely the sunrise integral and the double-box integral -the identities among Ω (n≤2) are sufficient to simplify the symbols after using (3.1) and (3.2).We comment on a generalization to identities among Ω (n>2) at the end of this section.
Let us start with the slightly trivial identity which is a simple consequence of applying (2.16a) the other one is obtained by further taking b → c, which yields Now one can easily see that the second equality in (2.28) is the consequence of applying (3.4) and (3.5).The two special cases are particularly useful since the letters Ω (1) on their right-hand sides always involve w ± ∞ and hence can serve as a basis.

Abel's addition theorem
Surprisingly, a very classical and powerful theorem, Abel's addition theorem [101], yields other identities for Ω (0) and Ω (1) . 10et us first spell out this theorem: Let C and C ′ be curves given by two polynomial equations where C is viewed as a fixed curve and C ′ as a variable curve with coefficients collectively denoted as {b i }.Suppose that these two curves intersect at n points (x 1 , y 1 ), ..., (x n , y n ).Let R(x, y) be a rational function defined on C. Then the following holds.
Theorem (Abel).The integral where x * is an arbitrary base point, contains at most rational functions and logarithms of {b i }.
This theorem can be proven by showing that ∂ bν I is always a rational function of {b i } for all b ν .If a symbol letter φ(u) can be expressed as u R(x, y)dx, one can try to find an addition formula for φ(u) through Abel's addition theorem.Of all applications of this theorem, we are most interested in the cases that C ′ has only two degrees of freedom and intersects C at three points.For this case, Abel's addition theorem gives where T (u, v) is an algebraic function of u and v and '• • • ' denotes simpler objects, like logarithms.An example is the addition formula for logarithms, log(x 1 )+log(x 2 ) = log(x 1 x 2 ),11 which is given by choosing These two curves intersect at the three points x 1 , x 2 , x 3 that solve and hence satisfy x 1 x 2 x 3 = 1.Now consider which satisfies since x 1 x 2 x 3 = 1.Thus, I is a constant.To fix this constant, we can pick b 1 = −3, b 2 = 3, such that x 1 = x 2 = x 3 = 1, yielding I = 0. Again using x 1 x 2 x 3 = 1, we thus have as claimed.
For the case we are most interested in, the fixed curve C is given by (2.1), and we find that a convenient choice for C ′ is (3.16) One can easily check that these two curves intersects at three point at most.Suppose that two intersection points are (x 1 , y 1 = P 4 (x 1 )) and (x 2 , y 2 = P 4 (x 2 )), then and which is the well-known group law on the elliptic curve.Furthermore, if we take Together with a little divisor theory, this gives 12 12 For any meromorphic function F on a torus, by using d log F (z) = 0 and z d log F (z) = 0, one can conclude that the number and the sum of its poles are the same as of its zeros, where poles and zeros of order n are counted n times.Now consider the function which has poles at lattice points but vanishes only at z − χ and z − ∞ , two intersection points of the curve y = −x 2 − a 3 x/2 + (a 2  3 − 4a 2 )/8 and the elliptic curve.We then obtain (3.22) by using (2.10).
Similarly, for the integral ψ −1 (c, x)dx, the same procedure gives x 3 mod Λ, a corresponding identity can be found from (3.24) using the quasi double periodicity of Ω (1) Three boundary cases of (3.24) require special care: (ii) taking x 3 → ∞ gives ) explain the subset of the identities numerically found in [33] which only involve Ω (0) 's and Ω (1) 's.
Note that the identities we presented in this subsection can be equivalently formulated in terms of divisor theory, see e.g.[104].

Elliptic Bloch relation and the symbol prime
In [33], also five identities involving Ω (2) 's were observed which are much lengthier than the other identities; each of these five identities contains at least 100 terms in the form that they were found.It turns out all these identities are consequences of the so-called elliptic Bloch relation [102,103], an elliptic generalization of the five-term identity for dilogarithms, where D(z) = ℑ(Li 2 (z)) + arg(1 − z) log |z| is the Bloch-Wigner function. 13n practice, it is difficult to simplify even expressions containing dilogarithms by using the above five-term identity directly.Instead, we introduce the symbol map [63] for polylogarithms as an assistance; we associate to each polylogarithm a tensor product whose entries satisfy simpler identities.We then exploit that the symbol of a combination of polylogarithms vanishes if that combination of polylogarithms vanishes.
A similar strategy can be used for the elliptic letters Ω (2) (w) = (2πi) −1 Γ( 2 0 ; w), although they already serve as entries of the symbol for elliptic multiple polylogarithms.Inspired by the proof of the elliptic Bloch relation for Γ( 2 0 ; w) in [104], we associate to Ω (2) (w) a rank-two tensor through the symbol prime map, where we have added a prime on "⊗" to distinguish it from the tensor product in the symbol.This map has a property similar to that of the symbol map: (3.30) for some rational coefficients c j .To show this, consider the sum j c j Γ( 1 0 0 0 ; w j ).According to (2.19), where we used that Γ( 0 0 ; w) = w, Γ( 2 0 ; w) = 2πiΩ (2) (w) and Ω (−1) = −2πiτ .If the arguments w j and coefficients c j are such that j c j Ω (2) (w j ) = 0 due to an elliptic Bloch relation, j c j Γ( 1 0 0 0 ; w j ) = 0 according to an analogous elliptic Bloch relation [104], which in turn implies that the second term on the right-hand side of (3.31), j c j S ′ (Ω (2) (w j )), drops out in the sum.In this sense, the symbol prime makes the elliptic Bloch relations manifest.
Note that we have assumed that j c j Ω (2) (w j ) = 0 vanishes due to an elliptic Bloch relation here in order to show that j c j S ′ (Ω (2) (w j )) = 0. We have indicated this in (3.30) as "⇒".However, we currently have no way of proving that all identities j c j Ω (2) (w j ) = 0 are due to an elliptic Bloch relation [102,103].This is similar to the case of dilogarithms, where (3.28) is only conjectured but not proven to generate all functional identities among dilogarithms.
The symbol map itself has a kernel, and the same is true for the symbol prime.If j Ω (2) (w j ) is in the kernel of the symbol prime, i.e. S ′ ( j Ω (2) (w j )) = 0, the first term on the right-hand side of (3.31) drops out in the sum.This implies that j Γ( 1 0 0 0 ; w j ) and thus j Ω (2) (w j ) is a function of τ only.However, not all functions only of τ are in the kernel of the symbol prime; for example, Ω (2) (τ /n) with some positive integer n only depends on τ but has a non-vanishing symbol prime. 14 One can find the action of the symbol prime map on the letters Ω (n<2) by expressing them in terms of Ω (2) using the quasi periodicity (3.2) of Ω (n) : This yields where we have moreover used quasi periodicity to simplify the entries of the symbol prime.In particular, by expressing a logarithm in terms of Ω (1) 's and Ω (0) 's either through (3.3) or (3.24), one finds Thus, for a combination of Ω (n≤2) 's and logarithms, one can compute its symbol prime.It involves only Ω (n≤1) and can thus be simplified using the techniques discussed in subsection 3.1.If the symbol prime is not zero, one may search for a 14 However, the occurrence of such a letter means that the point on the elliptic curve that corresponds to τ /n on the torus, namely κ(ω 1 τ /n), should occur in the calculation of the E 4 functions and hence is algebraic in kinematics.This is not the case for the examples of the unequal-mass sunrise integral and the ten-point double-box integral studied in sections 4-5, but it is the case for the equal -mass sunrise integral.
simpler combination of Ω (n≤2) 's and logarithms with the same symbol prime according to (3.29), (3.35) and (3.36). 15The difference of these two combinations has to be a function of τ only, and a simple expression for this function can be obtained by sending the independent w-variables in the difference to any values, say 0. In this way, we have proven the five identities involving Ω (2) 's found in [33].
The current definitions for Ω (n≤2) are sufficient for the two examples treated in this paper.For n > 2, one might similarly define the symbol prime for Ω (n) 's as due to the fact that, where Ω (n) (0) is either zero or a function only depending on τ for even or odd n, respectively.With the knowledge of the identities among Ω (n−1) , we can then find identities among Ω (n) recursively.We leave the exploration of the symbol prime for Ω (n>2) to future work.

Example I: Unequal-Mass Sunrise Integral
Two particularly interesting cases of elliptic Feynman integrals are the unequal-mass sunrise integral in two dimensions and the double-box integral in four dimensions.We will investigate these two integrals through the tools developed so far.The main focus will be on the sunrise integral treated in this section, since this integral is simple enough such that the main results can be written within a couple of lines.After applying the symbol prime map, we will see that several properties, such as double periodic invariance, modular invariance (covariance) and part of integrability are manifest.
We calculate the unequal-mass sunrise integral in terms of elliptic multiple polylogarithms E 4 in appendix A. This integral was originally calculated in terms of iterated integrals on the moduli space M 1,3 in [28].We closely follow the Feynmanparameter approach of [20] for the equal-mass case.
The resulting expression when rescaling the torus by the period ω 1 is where the periods were defined in figure 2 and T (1) is a pure combination of elliptic multiple polylogarithms of weight one and length two, where we introduced t 2 i = m 2 i /p 2 and r = −t 2 3 /t 2 1 .Note that we have included seemingly redundant factors of (2πi) in the numerator and denominator of (4.1) that ensure that the prefactor degenerates to an algebraic function in the limit where the elliptic curve degenerates, and the term in parentheses degenerates to a pure logarithm of transcendental weight two; see subsection 4.1.
In general, eMPLs transform non-trivially under the modular S-transformation τ → τ ′ = −1/τ ; for example, See [30] for the cases of iterated integrals of modular forms.The same is true for the symbol, as we will see soon.However, we will see that the application of the symbol prime map makes the behavior under the modular S-transformation manifest.

Symbol of the sunrise integral
The symbol of T (1,2) can be calculated by first rewriting E 4 's in terms of Γ's via (2.18) and then applying (2.19)-(2.26).For example, and The simplification of the symbols in this case is slightly non-trivial: it involves some non-trivial relations of Ω (1) 's, Ω (0) 's and logarithms as described in section 3.1; for instance, log log All the relations involving Ω (2) in this case are comparably trivial; they are the consequences of (3.2).We present the full simplification in the attached file sunrise symbol.nb.
The final result is where we have moreover used Similarly,

S 2πiT
(2) = log where we used (4.11) in terms of ξ-coordinates.At this point, the symbols of the sunrise integral partially show some desired properties; for example, the first entries of the first two terms in (4.10) and (4.12) indicate the physical first-entry conditions known from the massless case, and their last entries are related by simple S-transformations w → ξ.However, the first two terms on their own are neither double periodic nor integrable.
The first entries of the last terms in (4.10) and (4.12), i.e., ∂ τ T (1) and ∂ τ ′ T (2) , are rather complicated and the main obstacles to understanding the entire symbols, since it is hard to see how they render the whole symbol double periodic and integrable.
In this respect, it is instructive to consider the symbol primes of these entries: They have the following advantageous properties: 2) only by a modular Stransformation w → ξ, i.e. the symbol prime makes modular covariance manifest.
3.Moreover, the symbol prime also makes integrability with respect to τ manifest.This is slightly trivial in the case of the length-two sunrise integral, and will thus be discussed in full generality for the case of the double-box integral in section 5.1.
Finally, note that the equal-mass case can be obtained smoothly by taking t 1 = t 2 = t 3 ; we will briefly comment on this case in section 6.

Degeneration at p 2 = 0 and pseudo-thresholds
Next, let us see how the symbol of the unequal-mass sunrise integral behaves in kinematic limits where the elliptic curve degenerates.
The kinematic configurations where the elliptic curve degenerates can be easily read off from the discriminant (4.17) where t 2 i = m 2 i /p 2 as before.In particular, the sunrise integral remains finite at p 2 = 0, at the pseudo-thresholds and at the threshold p 2 = (m 1 + m 2 + m 3 ) 2 , while it diverges for m i = 0.The values Figure 3: The roots of y 2 (x) for the sunrise integral coincide in the null-momentum limit (a) and pseudo-thresholds (b).In the latter case, the position of the coinciding roots relative to −1 is shown for the case t 3 > t 1 .at p 2 = 0 and the pseudo-thresholds were given in terms of MPLs in [10].In what follows, we will show how the symbols S(2πiT (1,2) ) reproduce the corresponding symbols in these two limits. 16ull-momentum limit As p 2 → 0, t 1 , t 2 and t 3 approach infinity while their ratios remain finite.The elliptic curve degenerates in a way that r 1 → r 3 and r 2 → r 4 ; cf.figure 3a.In this case, ω 1 → ∞ since the roots pinch the corresponding integration contour γ 1 , while 1 ω 2 reproduces the same normalization factor as in [10] (up to a sign).Thus, we should expect that S(2πiT (2) ) reduces to the corresponding symbol.
Pseudo-thresholds Without loss of generality, we consider the pseudo-threshold In terms of t i , this pseudo-threshold is equal to the condition (t 1 + t 2 − t 3 ) 2 = 1.We only consider the solution t 3 = t 1 + t 2 − 1 since the treatment for the other solution is similar.At t 3 = t 1 + t 2 − 1, the roots r 1 and r 4 pinch the real axis; cf.figure 3b.Thus ω 2 diverges and we should consider the rescaling of the torus by ω 1 , T (1) .We can then close the contour γ 1 with a large semi-circle in the left half-plane and evaluate the integral via residues: 1 ω 1 reproduces the same normalization factor as in [10] at p 2 = (m 1 + m 2 − m 3 ) 2 .Thus, we should expect that S(2πiT (1) ) reduces to the corresponding symbol.At the pseudo-threshold, q = exp(2πiτ ) vanishes.Furthermore, we assume t 3 > t 1 ;17 then all E 4 ( −2 c ; x) appearing in (4.10) also vanish in this limit.Each of the three last entries in (4.10) is divergent in this limit since the roots pinch the integration contour, but all these divergences cancel in the following combinations cf. figure 3b.Correspondingly, the symbol is reduced to where each term is the symbol of Li 2 (z) − Li 2 (−z) + log(z) log (1 − z)/(1 + z) -in perfect agreement with [10].

Example II: Double-Box Integral
As a second example of Feynman integrals that evaluate to elliptic multiple polylogarithms, we consider the two-loop ten-point double-box integral: x 8 x 10 x 1 where we have introduced the dual coordinates x i defined as x i − x i+1 = p i and the notation x i,j = x i − x j .This integral is a particular component of the two-loop ten-point N 3 MHV amplitude in planar maximally supersymmetric Yang-Mills theory (N = 4 sYM theory) [110], and was recently integrated in terms of elliptic multiple polylogarithms in [33].It depends on seven dual conformal cross-ratios x 2 3,8 x 2 6,10 and contains the elliptic curve defined by where ( As shown in [33], the elliptic double-box integral can be evaluated in terms of E 4 functions whose arguments make up the set .
In the present paper, we compute I in terms of Γ functions on the torus rescaled as [1 : τ ′ = −ω 1 /ω 2 ], as well as the corresponding symbol.The computation is straightforward; however, both expressions are too lengthy to be recorded in the main text and are given in the ancillary file doublebox omega2.In what follows, we will give a schematic expression of the symbol as well as the symbol prime and comment on several interesting aspects, such as the integrability condition and the soft limit p 10 → 0.

Symbol of the double-box integral
Let us first review some results and notations from [33].After normalization by ω 1 , T (1) = I /ω 1 is a pure combination of Γ functions of length four and weight three.The corresponding symbol S(2πiT (1) ) satisfies the same physical first-entry conditions as in the MPL cases [67]; namely, the first entries are given by log u i and log v i .The first two entries are Li (5.9) The last entries of this symbol consist of seven letters of elliptic type Ω (0) .According to these last entries, the symbol can be organized as18 S(2πiT where I hex is the 6D hexagon integral (normalized to be pure), c 25 = 9,10,1,(7,8)∩ (2,3,5)   1,5,9,10 2,3,7,8 is the 25'th element of the set (5.5) and we used Ω (0) (w + ∞ ) = 2πiτ .In particular, the symbols of all weight-three functions except F τ , that are I hex , F − , etc., are polylogarithmic; their symbol entries are logarithms of kinematics.Furthermore, I hex and F − are invariant up to a sign under the two reflections where p i ≡ p i+10 , while F z 1 −1 , F z1 −1 , F −z 2 and F −z 2 form an orbit (up to a sign) under both reflections.Besides the seven elliptic last entries, elliptic letters only appear at the third entry of S(F τ ) and come in only 13 linear independent combinations of Ω (2,1,0) 's.For the full symbol alphabet and a form that is manifestly invariant under the two reflections, see [33].
In the other normalization, T = −I /ω 2 , the structure of the symbol is almost the same as (5.10),

S(2πiT
Besides the last entries, which are simple modular S-transformations of the last entries of S(2πiT (1) ), the only difference arises from the third entry of F τ ′ which consist of only Ω (2) 's.Meanwhile, the Ω (2) 's appearing in the third entry of S(F ′ τ ) also come in 13 linear independent combinations.
It is not surprising that all weight-three functions appearing in S(2πiT (1) ) except F τ are the same as in S(2πiT (2) ).Actually, for any elliptic multiple polylogarithm I given by an integral of the form (2.7), its two normalizations, T (1) = I/ω 1 and T (2) = −I/ω 2 , are related by T (2) = τ ′ T (1) .With the last entry of T (1) being of elliptic type Ω (0) , i.e., dT (1) where F i and F τ are elliptic multiple polylogarithms of lower length, a straightforward computation gives dT (2) While it is not manifest from (5.15), this particular expression for F τ ′ has length three.
Once we evaluate the symbol prime for the third entry of F τ and F ′ τ , a simple connection emerge.Both S(2πiT (1) ) and S(2πiT (2) ) have similar structures as in the case of the sunrise integral: with and a similar expression for S(2πiT (2) ) with ξ j and τ ′ in place of w j and τ .Here w j are the six last entries except τ , a ij are some algebraic functions of kinematics, f i are of the form Li 2 (1 − x ab;cd ), log x ab;cd log x a ′ b ′ ;c ′ d ′ or four-mass-box functions, and Ω i are combinations of Ω (2,1,0) that occur in the last entry of F τ associated with S(f i ).
Let us close this subsection by remarking some advantages of the form (5.16).Firstly, the symbol prime is manifestly double-periodic due to the same argument used in section 4.1: under the translation where we have used (3.36) and Ω (−1) = −Ω (0) (τ ) = −2πiτ .Secondly, this form makes a part of the integrability conditions manifest and sheds light on the (13 linearly independent) combinations of Ω (2,1,0) that occur in the last entry of F τ .The integrability condition requires which is a consequence of since (5.21) is an integrable symbol by the definition of the symbol prime.We will discuss the remaining integrability conditions in upcoming work [113].

Soft limit
In this subsection, we will consider the soft limit p 10 → 0, in which the elliptic double-box integral remains finite and becomes polylogarithmic: x 1 x 3 x 5 x 6 (5.22)In terms of momentum twistors, this limit amounts to first setting (5.23) with finite α and β, and then taking the limit ǫ → 0 [114].In this limit, the seven cross-ratios become Correspondingly, the elliptic curve (5.3) degenerates to where (5.27) In the soft limit p 10 → 0, the roots of y 2 (x) pairwise pinch the integration contours for w + z1 −1 , w + −z 2 and w + z 1 −1 , which run along the real axis.By subtracting w − ∞ and w + ∞ , respectively, we obtain integration contours that can be deformed such they are not pinched, thus resulting in finite integrals in the soft limit.polylogarithmic symbol in the soft limit with last entries (5.31)Note that in the soft limit z 1 ≡ z 1,3,5,8 and z 2 ≡ z 3,6,8,1 and the reflection symmetry R 1 is broken while R 2 survives; thus, the symbol for this nine-point double-box integrals can be expressed as where R 2 acts on the last entries via r ↔ −(1+r), z 1 ↔ z 2 and z1 ↔ z2 .Furthermore, F −z 2 | p 10 →0 and F −z 2 | p 10 →0 are related by exchanging z 2 and z2 , same as the corresponding last entries.The reason is that z and z occur symmetrically in their definition {zz = u, (1 − z)(1 − z) = v} , and thus have to occur symmetrically in the symbol as well.
The symbol alphabet of the nine-point double-box integral consists of 10 rational letters and 11 algebraic letters: 1. Rational letters: where we introduced the following notations:19 a(bc)(de)(f g) = abde acf g − acde abf g , (5.34) and (5.36) 2. Algebraic letters: We find that there are three different square roots in this alphabet; two of them are of four-mass-box type and the other, that is the square root in r and r, arises from the leading singularity of the whole Feynman diagram.Furthermore, the new type of square root only appears in the last entries.The symbol alphabet is organized such that the symbol is manifestly invariant (up to a sign) under the reflection R 2 , as well as under each of the three transformations z 1 ↔ z1 , z 2 ↔ z2 , and r ↔ r.For an analysis of these letters through Schubert problems, see [118].

Discussion and Outlook
In this paper, we have investigated various techniques for manipulating and simplifying the symbol of Feynman integrals that evaluate to elliptic multiple polylogarithms.
In particular, we studied identities between the elliptic symbol letters Ω (i) .
In contrast to ordinary multiple polylogarithms, the length of an elliptic multiple polylogarithm is not necessarily equal to its weight.A symbol letter Ω (i) , whose length is by definition one, can have weight i = 1.Identities for Ω (0) follow from the well-known group law on the elliptic curve.Moreover, we found that various identities for Ω (1) can be derived from Abel's theorem, which generalize the identity log(a) + log(b) = log(ab) in the polylogarithmic case.The higher-weight letters Ω (2)  satisfy significantly more intricate identities, closer to those of Li 2 (a) than those of log(a), which are harder to exploit in a direct fashion.We have thus introduced the symbol prime S ′ for elliptic symbol letters Ω (2) , which plays the same role the symbol S plays for Li 2 (a).We also introduced a symbol prime for Ω (i>2) but leave its exploration for future work.
We studied two concrete examples at two-loop order, namely the sunrise integral in two dimensions and the ten-point double-box integral in four dimensions.In particular, we provided proofs for the identities between symbol letters numerically found in [33].
In addition to identities between symbol letters, we also studied how the symbol behaves under kinematic limits in which the elliptic curve degenerates.We recovered the known symbols of the sunrise integral in the null-momentum limit p 2 → 0 and the pseudo threshold p 2 → (m 1 + m 2 − m 3 ) 2 [10], as well as the nine-point limit of the double box, which has not previously appeared in the literature.
The numeric values of elliptic Feynman integrals are of course independent of whether we rescale the torus by the period ω 1 or −ω 2 ; the corresponding modular parameters τ = ω 2 /ω 1 and τ ′ = −1/τ are related by a modular S-transformation.In particular, for an elliptic integral of the form G(x, y) dx/y as in (2.7), its two normalizations T (1) and T (2) , which are obtained by dividing by ω 1 and −ω 2 respectively, are simply related by T (2) = τ ′ T (1) .However, this property is not manifest when expressed in terms of elliptic multiple polylogarithms or their symbols.Instead, we found that the application of the symbol prime to the two examples in this paper yields symbols of the form ij S(f i ) ⊗ log a ij ⊗ Ω (0) (w j ) + Ω i ⊗ (2πiτ ) , (6.1) with S ′ (Ω i ) = Ω (0) (w j ) ⊗ ′ log a ij .(6.2) Not only the modular covariance is manifest in this form, but also the double-periodic invariance and the integrability conditions involving τ .However, one could not expect that the application of the symbol prime to the one-fold integral of general polylogarithms, of the form G(x, y) dx/y, yields such a structure.As a simple counter-example, consider the symbol of the integral c 0 log(x + a) dx/y, with arbitrary values of a and c as well as the elliptic curve given by (5.3), which does not follow the above structure after applying the symbol prime map.It would be very interesting to investigate why the two elliptic Feynman integrals we considered in this paper turn out to exhibit such a structure after applying the symbol prime map, and to study whether this property extends to further Feynman integrals.
Already the polylogarithmic symbol has a kernel, which is given by iπ, multiple zeta values (MZVs) and their products with MPLs.Similarly, also the symbol prime has a kernel.As discussed in section 3.2, all functions in the kernel necessarily depend only on the modular parameter τ , but not all functions that depend only on τ are in the kernel.To see that the kernel can be non-trivial, consider the symbol of the equal-mass sunrise integral, which is an iterated integral of modular forms and only depends on τ : where the E 4 are specific combinations of Ω (2) 's given in (4.11).The application of the symbol prime map to the first entry in (6.3) yields 0. We leave a comprehensive treatment of the kernel of the symbol prime map to a future study.
Another interesting problem is to lift simplified symbols to simplified functions for elliptic multiple polylogarithms.As a primary example, let us consider how to lift the simplified symbol (4.10) to a simplified function for the sunrise integral.By rewriting the logarithms in (4.10) in terms of Ω (1) 's, we find a (slightly) simpler expression for T (1) , (and a similar expression for T (2) ), in which the E 4 functions only contain c = 0, −1, ∞ but not the fourth argument r that occurs in (4.2).However, the general prescription of uplifting more complicated elliptic symbols to functions is still underexplored. 20lthough the simplified symbols of the elliptic Feynman integrals manifest some desired properties, such as double periodicity and modular covariance, after applying the symbol prime map, the singularity structures are not completely manifest.For example, the sunrise integral becomes singular at m i = 0 as well as at the threshold p 2 = (m 1 + m 2 + m 3 ) 2 , as can be seen through a Landau analysis [120].One can see the branch cut at m i = 0 explicitly from eq. (4.10) or (4.12); however, the branch cut at the threshold is not manifest from the symbol.In general, the logarithmic letter log(a) has a logarithmic singularity if a = 0 or a = ∞.In contrast, the elliptic letter Ω (1) (w) has a logarithmic singularity at all lattice points, while Ω (j≥2) (w) has a logarithmic singularity at all lattice points except for the origin.However, w is a function of the kinematics; typically, w = w + c = 1/ω 1 c −∞ dx/y, where c is an algebraic function of the kinematics.If the configuration of roots in y does not change as we vary c, w + c = 0 if c = −∞ and w + c = τ if c = +∞.However, the configuration of roots may also vary as we vary c; we leave a comprehensive analysis to a future study.
It would be interesting to apply the techniques used in this paper to bootstrap the symbol of scattering amplitudes or Feynman integrals that can be expressed in terms of elliptic multiple polylogarithms, such as the twelve-point elliptic double box.On top of the integrability condition for the final entry τ , which is made manifest by the symbol prime, this requires understanding the integrability condition for the other last entries [113].Moreover, it requires an educated guess for the alphabet of symbol letters that occur in them.For six-and seven-point amplitudes in N = 4 sYM theory, the symbol alphabet was shown to be given by cluster algebras [80,91,[121][122][123], and similar techniques have recently been extended to Feynman integrals and amplitudes [124][125][126][127] containing symbol letters that are given by logarithms of algebraic functions of the kinematics [87,95,[128][129][130][131][132][133][134].It would be interesting to extend these techniques also to the elliptic case.

Figure 1 :
Figure 1: The sunrise integral in two dimensions with unequal internal masses (a) as well as the ten-point double-box integral in four dimensions with massless internal propagators (b).

2 a,b x 2
where z a,b,c,d za,b,c,d = x a,b;c,d , (1 − z a,b,c,d )(1 − za,b,c,d ) = x d,a;b,c , with the abbreviation x a,b;c,d = x