A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry

Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales, may have complicated symbol structures. We show that the dual conformal symmetry sheds light on the alphabet and symbol structures of multi-loop Feynman integrals. In this paper, first, as a cutting-edge example, we derive the two-loop four-external-mass Feynman integrals with uniform transcendental (UT) weights, based on the latest developments on UT integrals. Then we show that all the symbol letters can be nicely obtained from those of closely-related dual conformal integrals, by sending a dual point to infinity. Certain properties of the symbol such as first two entries and extended Steinmann relations are also studied from analogous properties of dual conformal integrals.


Introduction
With the start of the run III stage of the Large Hadron Collider (LHC), demands for the high-order perturbative computations of the precision physics get higher. For the highorder perturbation computations in quantum field theory, multiloop Feynman integrals are the key objects which are difficult to reduce or evaluate. Frequently multiloop Feynman integrals with multiple scales (Mandelstam variables or mass parameters) appear in high-order perturbative computations. These integrals, even if they can be expressed in term of polylogarithm functions, their symbol structure [1] would be very complicated: the alphabet (the list of symbol letters) may be very long and contains complicated square roots of the kinematics variables; The adjacency condition of letters in the symbols can be intricate. On the other hand, the information on the alphabet and symbol of multi-loop Feynman integrals are invaluable. It enables the simplification algorithm of polylogarithm functions [1][2][3], the bootstrap of supersymmetric Yang-Mills amplitudes [4][5][6][7][8], the bootstrap of Feynman integrals [9], the finite-field interpolation of the canonical differential equation [10] and a lot of other important applications. Therefore, it is of great interests to study the alphabet and symbol structures of multi-loop Feynman integrals.
In the literature there are many interesting studies of the theoretical properties of alphabet and symbol structures, for example, the so-called first entry-condition [11], the relation between symbols and Landau Singularity [12,13], the relation between the second entry condition and the Steinmann relation [7,14], and the final entry condition [15]. The cluster algebra structure was applied for the computation of the symbol structure in N = 4 SYM theory [16,17]. Recently, the ref. [18][19][20] studied multi-loop Feynman integrals' alphabet from the cluster algebra structure. Besides these exciting developments, the multi-scale multi-loop Feynman integrals' alphabet and symbol structures still have a lot of mysteries. In this paper, we try to illustrate the alphabet and symbol structures for Feynman integrals with uniform transcendental (UT) weight [21,22], from the viewpoint of dual conformal symmetry [23][24][25][26].
The method [21,22] of applying UT Feynman integrals and the corresponding differential equation is currently a standard way of evaluating Feynman integrals analytically. In this paper, we derive a cutting-edge example of the UT Feynman integral basis for the twoloop double box diagrams with four different massive external legs, and then show that its alphabet and symbol structure can be well understood from the dual conformal properties. This family is particularly interesting because it is a sub-family of the two-loop eight-point massless diagrams, and also a sub-family of the two-loop five-point diagrams with three massive external legs. So, this family is important both for formal theories like N = 4 SYM and also the phenomenology. It contains two Mandelstam variables as well as four mass parameters, thus the kinematics is very complicated. It is not an easy task to find the corresponding UT basis, so we turn to the latest UT basis determination methods [27,28] with the leading singularity analysis in the Baikov representation. The complete UT basis for this family is found, and the canonical differential equation is calculated analytically via the finite-field method [10]. From the canonical differential equation, we obtained the full alphabet. Furthermore, the symbols of these UT integrals are calculated. As expected for other UT integrals with multiple scales, the alphabet is long and contains many square roots in the kinematic variables. The symbol structure for this family is also complicated. Therefore, in this paper, we propose to analyze the symbology of these Feynman integrals, namely the alphabet and properties of the symbols, from certain dual conformal invariant (DCI) Feynman integrals [29] by sending a dual point to infinity. Similar considerations have appeared in [18] for one-mass five-point process, whose symbology is closely related to that of eight-point DCI integrals. The symbology of DCI integrals is much better understood by explicit computations [30][31][32][33][34][35] as well as various other "predictions", such as Landau analysis [12,13,36,37], cluster algebras [18][19][20]38] and very recently analysis based on twistor geometries known as Schubert problems [39]. Moreover, properties of the symbols such as conditions on first two entries and (extended) Steinmann relations have also been studied more extensively for DCI integrals (see [40] and references therein). By sending a generic dual point, which are not null separated from adjacent points, to infinity, dual conformal symmetry is broken and we obtain non-DCI integrals if the latter are finite. There is strong evidence [18] that such limits are still relevant for IR divergent integrals in dimensional regularization, thus we find it very rewarding to study the implications of DCI results for general, non-DCI integrals.
For our integrals with four external masses, it is necessary to consider fully massive DCI pentagon kinematics, which depends on five cross-ratios and reduces to four-mass (non-DCI) kinematics by sending any dual point to infinity. In such limits, the five crossratios reduce to ratios of kinematic invariants, s, t, m 2 1 , · · · , m 2 4 . Remarkably we find that all the 68 letters can be explained from symbol letters of related DCI integrals. Out of all 12 non-trivial rational letters, 11 of them turn out to be (the square of) leading singularities associated with one-and two-loop DCI integrals, including four-mass boxes and doubleboxes (the last one is a Gram determinant which can be obtained from following odd letters). The remaining 50 letters are parity odd with respect to the 11 square roots, which fall into two classes. There are 16 of them that has only one square root, which can all be identified with algebraic letters of four-mass boxes and two-loop generalizations; for the 34 "mixed" odd letters that depends on two square roots, we find their origin by studying twistor geometries of corresponding one-and two-loop DCI integrals. The appearance of these letters in the matrix of canonical DE also exhibit nice patterns. Moreover, we also find that, to all orders in , the first two entries of the symbols of such integrals can always be written as linear combinations of five four-mass box functions (and log log terms) which nicely explain the first-two entry conditions. Relatedly, we have also checked that extended Steinmann relations are satisfied by all the integrals, which follows from the canonical DE.
This paper is organized as: in section 2 we review the UT integrals, canonical differential equation and symbol. In section 3, we introduce the main example of two-loop double box diagram with four different external mass, derive the UT basis, canonical differential equation and the symbols. In section 4, we will first review basics of DCI integrals, focusing on one-and two-loop integrals which depend on fully-massive pentagon kinematics; we then explain that by taking any dual point to infinity, letters of DCI integrals nicely become those of double-box etc., including all square roots and odd letters; finally, we will comment on how certain properties of the symbols can be explained from DCI considerations. In section 5, we summarize this paper and provide an outlook for the future application of the DCI properties on the multi-loop UT integrals. More technical details are given in the appendices of this paper.
2 A review of differential equations, UT integrals and symbol

The canonical differential equation
Differential equation method is one of the most popular method of evaluating Feynman integrals [41][42][43]. The differential equation of a Feynman integral basis I, via the integrationby-parts reduction, reads Li n (x) n H(a 1 , · · · , a n ; x) n G(a 1 , · · · , a n ; x) n Table 1: transcendental weight of some functions where x i 's are kinematic variables and A i 's are matrices for the differential equations. Usually, solving the differential equation in (2.1) is difficult. A better choice of the integral basis will make it dramatically easier. One of the best choices of integral basis consists of integrals with uniformal transcendental (UT) weights [21]. These UT integrals have the following form in terms of expansion as, where I (n) is a pure weight-n transcendental function, i.e. T (I (n) ) = n and T ( ∂ ∂x I (n) ) = n − 1, where T stands for the transcendental weight of a function. The transcendental weight of some common functions are shown in Table 1, where Li n (x), H({a 1 , . . . a n }, x) and G({a 1 , . . . a n }, x) are, respectively, the weight n classical, harmonic and Goncharov polylogarithm functions .
The differential equations for a basis formed by UT integrals (2.2) are called canonical differential equations. For the matrices of canonical differential equations, the parameter factorizes out as [21], This is called the canonical differential equation. Consequently, we can solve the differential equations order by order in expansion as, From the integrability condition, the canonical differential equation matrices for UT basis satisfies, From the second property, we know that, where the matrixÃ takes the form as where a i 's are matrices of rational numbers, and W i 's are algebraic functions of x i 's, called symbol letters. With the form of (2.7), the canonical equation (2.3) can be solved analytically with the information of the boundary condition. On the other hand, one can easily derive the symbol [1] of the solutions of the differential equation.
The symbol S of a function is defined as, and where R and R i are rational functions and S[R 1 , · · · , R n ] ⊗ S[R] def = S[R 1 , · · · , R n , R]. With these, one can immediately derive the symbols of the solutions from the canonical differential equations as (2.11)
Here we briefly introduce the leading singularity method in Baikov representation [62][63][64]. For a Feynman integral defined as in the Baikov representation, it is (2.13) In this representation, E and L are the number of independent external momenta and loop momenta respectively, C L E is a constant irrelevant to kinematic variables, and U and P are Gram diterminants as 14) The idea of using Baikov leading singularity method to determine UT integrals is based on an conjecture [50] that an integral with constant leading singularity should be a UT integral. In the Baikov representation, the leading singularity of an integral can be derived using the cut [65][66][67], which is to replace the integration by taking residues at z i → 0, schematically as (2.16) For multi-loop integrals, a more practical way to derive the leading singularities is the loop-by-loop Baikov representations [67,68]. Take the double box with 4 external masses diagram (to be introduced in section 3.2 in detail) as an example. The propagator and kinematic information are given in (3.1) and (3.9). Consider the scalar integral in the top sector in (3.10), which is one can compute the leading singularity from full Baikov representation introduced in (2.13). In comparison, a more computationally economical way is to derive the Baikov representation loop by loop: First we treat l 1 as loop momentum and l 2 , p 1 , p 2 , and p 4 as external momenta, so that (2.18) Here, E = 3, L = 1, D = 4, C is some constant irrelevant to kinematics and Then the leading singularity for the "left" loop integral is where With the definition D 9 = (l 2 − p 1 ) 2 , we can similarly derive the leading singularity for the "right" loop as where Requiring the leading singularity of this integral to be constant, we guess that sr 1 J 1 is a UT integral candidate. Later, from the differential equations, we verify that it indeed has the uniform transcendental weights.

Feynman integrals with four different external masses
In this paper, we study the one and two-loop Feynman integrals with four different external masses. These integrals are interesting because they appear in the eight-point massless scattering processes which is the focus of the amplitude study of formal theories, and also in the five-point scattering processes with three external masses. These integrals have rich structures related to dual conformal invariance. The kinematic condition for these integrals is with p 1 + p 2 + p 3 + p 4 = 0.

One loop UT integral basis
The propagators of one-loop Feynman integral family with four different external masses are:  There are 11 master integrals after the IBP reduction.

(3.3)
These master integrals are almost UT integrals up to some rational function factors in kinematic variables. It is not difficult to determine these factor by looking at the differential equation, then the UT integral basis of this four masses box diagram family is:

Two loop UT integral basis
The propagators of two-loop Feynman integral family with four different external masses are: (3.9) The top sector, (1, 1, 1, 1, 1, 1, 1, 0, 0) for this family, is a double box diagram shown in  From the stand IBP reduction procedure, we see that the number of master integrals for this family is 74. The diagrams for all master integrals are given in the appendix A. Some of the master integrals in the family were calculated in [69].
2. Apply the Baikov leading singularity method to determine the UT integrals in the sectors with 7, 6 and 5 propagators.
• (Top sector) The top sector is (1, 1, 1, 1, 1, 1, 1, 0, 0) and there are 4 master integrals in this sector. From the experience between UT integrals and the UV finiteness condition, we tend to use as UT integral candidates. If necessary, some subsector integrals would also be used.
From the Baikov leading singularity example (2.22) in the previous section, we estimate that sr 1 G 1,1,1,1,1,1,1,0,0 is a UT integral. To make a UT integral candidate from G 1,1,1,1,1,1,1,−1,0 , based on the experience in [27], we can consider the Baikov integral with the sub-maximal cut. Given G 1,1,1,1,1,1,1,−1,0 , the ansatz for a UT integral can be set as, We use right-to-left Baikov representation on this ansatz to simplify the calculation of leading singularities. Require the leading singularities to be rational numbers, and then the coefficients f 1 ∼ f 8 are fixed up to some constants: c 1 ∼ c 6 are rational numbers, and c 1 = 0 since this ansatz is from G 1,1,1,1,1,1,1,−1,0 . We can set c 2 ∼ c 6 to 0, and c 1 = 1 for the simplicity. Finally, we get a UT integral candidate: With the leading singularity analysis in the Baikov representation, it is also easy to see that has constant leading singularities, from the maximal cut of the Baikov representation. However, later on, from the differential equation computation, we seeĨ 4 itself is not a UT integral. To upgradeĨ 4 to a UT integral, subsector integrals should be added. It is a nontrivial computation to determine those subsector integrals, so we keepĨ 4 at this stage and later describe how to get subsector integrals.
We remark that the D = 4 leading singularity analysis of integrals in this sector was calculated in the ref. [72].
3. For sectors with 3 or 4 propagators, it is relatively easy to find the UT integral candidates. We apply the strategy from [21,22]  Here our strategy in the transformation, is to set s, t, m 1 , m 2 , m 3 and m 4 as some random integer values while keep analytically. Then we can identify the entries of the differential equation matrix which are not proportional to . Explicitly, such entries are located in the row and columns of the new differential equation corresponding toĨ 4 andĨ 33 . This indicates thatĨ 4 andĨ 33 should be upgraded to UT real integrals.
We observe that entries in the DE matrix's 33-rd row, which correspond toĨ 33 , are either proportional to or has the form a + b . For the latter form, a simple transformation like that in [57], can remove all the 0 terms in the 33rd row. Here x 1 , . . . , x 6 stand for the variables s, t, m 1 , m 2 , m 3 and m 4 . A

(n)
i,j is the differential equation matrix element for the derivative in x n . To apply this transformation, we need to first evaluate A (n) i,j analytically for i = 33. The resulting integral, is indeed a UT integral.
The upgrading ofĨ 4 is more involved. The DE matrix's 4-th row contains several entries which are not linear in , so a transformation like (3.55) is not enough to make the DE canonical. Here our strategy is an induction over the number of mass parameters. We take the limit when some or all of the mass parameter to zero, and reduceĨ 4 to the known UT integral basis [21,27,73] in this limit. The reduction coefficients are not all rational constants, so we can by-hand add some integrals tõ I 4 to make the coefficients to be constants, in the partially massless or completely massless limit. After this, the resulting integral should be "closer" to a UT integral and indeed the corresponding DE matrix row is completely linear in . Then we use a similarity transformation like (3.55) again, to make the whole differential equation matrix proportional to .
The computer readable UT integrals definitions, as well as the corresponding definition of the square roots, can be found in the "output" folder of the auxiliary files, named "dbox4m UT.txt" and "rootdef.txt", respectively.

Canonical differential equations and the alphabet
WithĨ 4 andĨ 33 upgraded, numerically, we see that the differential equation is canonical.
Then the next computation is to get the analytic canonical differential equation. Here we apply the finite-field package FiniteFlow [10] for this computation, since it is well known that the canonical DE usually has much simpler coefficients than the ordinary DE and the finite field reconstruction is extremely efficient. For this family, the canonical differential equation reconstruction with FiniteFlow only takes about 14 seconds on a workstation with 50 cores. The analytic expressions of the differential equations can be found in the "output" folder of the auxiliary files, named " UTDE.txt", which is a list of 6 differential equations, with respect to the kinematic variables s, t, and m 2 1 ∼ m 2 4 , respectively. Schematically, the canonical differential equations for the UT basis can be written as  The odd letters that contain one square root are, where (3.61) The odd letters that contain 2 square roots are ) ) The definition of the polynomials f 35 ∼ f 68 is in the appendix B.
The computer readable results of the symbol letters andÃ can be found in the "output" folder of the auxiliary files, named "letterdef.txt" and "Atilde.txt", respectively.

Symbol structures
The result of the canonical differential equations (3.58) are derived in the last subsection. The integration to get the analytic expressions for the UT integrals is not easy due to the boundary condition. However, deriving the corresponding symbol letters are relatively easier, which can already help us to acquire interesting properties of the analytic expressions.The integrals' symbols can be easily derived from (3.58), using (2.11), or equivalently, the recursion with the lowest order boundary condition Having the specific results for the constant matrices a i defined in (3.58), in order to derive the symbols for the integrals, we also need the results for I i . These are the coefficients at the order −2L , being rational constants, where L is the loop order.
The coefficients I With results of I (0) i and a i defined in (3.58), we can use (3.71) and (3.72) to derive the symbols of the UT integrals and higher orders. Using this method, we derived the symbol letters of the UT integrals up to the weight-4 order. The result can be found in "output" folder of the auxiliary files, named "UTSymbols.txt", where "UTSymbol[k]" stands for I (k) i . From the results, we have observed the following phenomena: Firstly, the complexity of the symbol expressions goes rapidly up while the order increases. At the weight-1 order , the symbols are quite simple. There are only 28 integrals having non-zero symbols at this order with totally 64 terms. For example, we have S(I  These integral also have the zero symbol at the weight-3 order. At the the weight-4 order, all integrals have nonzero symbols. Note that, (3.77) is a subset of (3.76). Thirdly, letters cannot appear at arbitrary positions of the symbols. To state this, we the following terminology: The symbols of UT integrals consists of terms proportional to some S[W i 1 , W i 2 , · · ·], of which we call W i 1 the first letter (or letter at the first entry), W i 2 the second letter (or letter at second entry) and so on. As we can see from the results, among the 68 symbol letters, only W 1 ∼ W 6 can appear at the first entries, which are exactly the 6 kinematic variables s, t, m 2 1 , m 2 2 , m 2 3 and m 2 4 . The letters appearing at the second entries are W 1 ∼ W 6 and W 21 ∼ W 30 .
Moreover, some pairs of letters can never be next to each other in the symbols. We can explain this from (3.71). For constant matrices a i and a j , if a i a j = a j a i = 0, then according to (3.71), W i and W j cannot appear at adjacent entries, which means, symbols of the UT integrals contain no term like S[· · · , W i , W j , · · ·] or S[· · · , W j , W i , · · ·], at all orders of . Then we call that the adjacency between letters W i and W j is forbidden. Among all possible adjacencies of 68 letters, which are 68 × 69/2 = 2346 in total, only 919 of them are allowed and the rest 1427 are forbidden. The explicit allowed and forbidden letter pairs can be found in "output" folder of the auxiliary files, named "AdjLetters.txt" and "NonAdjLetters.txt", respectively.

Check with known dual conformal invariant integral result
Some of the symbol derived from canonical differential equations in section 3.4 can be verified with results from the dual conformal invariance (DCI) point of view. We take I 1 (3.10), which is the scalar double-box integral (proportional to G[1, 1, 1, 1, 1, 1, 1, 0, 0]) as an example. In the ref. [20], the symbol of this integral is derived from the DCI analysis, which is given as Thus, we are able to derive the symbol expressions for f (2) in terms of z andz and use the relations above to rewrite it in terms of s, t, and m 1 ∼ m 4 . Before this, we need to mention that the DCI result (3.78) is derived at dimension d = 4 and this integral is UV and IR finite. From the expression (3.78), we can derive S(f (2) ). In the right hand side of (3.78), the presenting transcendental functions are logarithm functions log(z) and poly logarithm functions Li m (z). The corresponding symbols are To derive the symbol S(f (2) ) from the symbols of above functions, we may need the following properties of symbols: For two functions f 1 and f 2 whose symbols are

(3.86)
This result can also be found in "output" folder of the auxiliary files, named "f2Symbol.txt". In order to compare with the symbol from DCI with that from the canonical differential equations of the section 3.4, we consider the relations between the two languages introduced in (3.79) and (3.80). After some simple calculations, we get the relations of the corresponding symbols, After the variable replacement using (3.87) to (3.89), S(f (2) ) is written in a summation of 474 terms of symbols formed by letters W 1 ∼ W 6 , W 25 and W 30 . This expression is identical to the symbol of I (4) In this section, we move to the study of the alphabet as well as certain properties of the symbols from viewpoint of DCI integrals. We will first give a quick review of DCI integrals, focusing on those relevant for our studies.

Review of relevant DCI integrals
Recall that for a n-point massless planar integral, we can first introduce dual points {x i } such that x i+1 − x i = p i with x n+1 = x 1 to make the momentum conservation manifest, and then the integral only depends on planar variables, which are also the first entries: It is convenient to further introduce momentum twistors Z i ∈ P 3 such that each dual point x i is associated with a bi-twistor Z i−1 ∧ Z i and planar variables become where ijkl := det(Z i Z j Z k Z l ) and I ∞ is the infinity bi-twistor. In terms of momentum twistors, the massless conditions (x i+1 − x i ) 2 = p 2 i = 0 are automatically solved, and conformal transformations of dual momenta become SL(4) linear transformations of {Z i } together with rescaling Z i → t i Z i . We say that an integral is dual conformal invariant if it only depends on cross-ratios of planar variables (or four-brackets i − 1 i j − 1 j ) and do not depend on the infinity bitwistor I ∞ .
Note that for a general planar integral with m (possibly massive) legs, the kinematics can be described by a subset of n ≥ m dual points as above, by identifying each massive momentum with two (or more) massless legs [18]. For example, a one-mass triangle kinematics depends on 3 of 4 dual points, e.g. x 2 , x 3 , x 4 (but not x 1 ), and a four-mass box kinematics depends on 4 of 8 dual points, e.g. x 2 , x 4 , x 6 , x 8 . Moreover, if we start with a DCI integral which has at least one dual point which is not null separated from adjacent ones, we can arrive at a non-DCI integral by simply sending it to infinity, or identifying this bi-twistor with I ∞ . As explained in [19], a general DCI kinematics with n−2m massless legs and m massive legs has dimension 3n−2m−15 1 . By sending a point (between two massive legs) to infinity, this is exactly the dimension of a non-DCI kinematics with m−1 massive legs and n−2m massless ones: we have 3 degree of freedoms for each massless leg, and 4 for each massive one, minus the dimension of Poincare group −10 and an overall scaling, 3 × (n−2m) + 4 × (m−1) − 10 − 1.
For our purpose, we start with a DCI kinematics that depends on five generic dual points (fully massive pentagon) with n = 10, m = 5: where each DCI four-mass box integral gives We have seen that I 4m only depends two DCI variables and it involves a square root, Note that their {u, v} are related to the five cross-ratios (4.1) as respectively. In order to arrive at non-DCI kinematics, one can send any one of the five dual points to infinity and identify the resulting massive corners with four massive legs of fourmass non-DCI kinematics. As shown in the following subsection, this limit induces a map between cross-ratios (4.1) and kinematics variables {s, t, m 2 i } i=1,···,4 , and reveals connections between DCI and non-DCI letters for one-loop integrals. In fact, this is nothing but the well-known example that in such limits, a DCI four-box becomes a (non-DCI) massive triangle integral (literally the box with a point at infinity).
Similarly, we can move to two loops, and the prototype of DCI integrals are 10-point double-box of the form: Note that this DCI integral has not been computed directly, though a similar 9-point double box (with leg 10 removed) has been computed. However, alphabets of such integrals can be predicted by considering twistor geometries associated with leading singularities (or the Schubert problems) as first proposed by N. Arkani-Hamed and worked out for numerous one-and two-loop examples in [39]. We will not review the details of such a method, but just to outline the basic idea as follows.
The Schubert problem concerns geometric configurations of intersecting lines in momentum twistor space, which are associated with maximal cuts, or leading singularities of (four-dimensional) integrals [29]. By representing each loop and external dual point as a line in twistor space, each cut propagator corresponds to intersecting two lines, and it is a beautiful problem for determining the intersections on internal or external lines. Whenever there are at least four points on such a line, the only DCI quantities one can write down are cross-ratios of such points. Remarkably, as shown in [39], such cross-ratios lead to symbol letters of DCI integrals! The classical example include A 3 , E 6 alphabets for n = 6, 7 [16] and the 9 + 9 algebraic letters for n = 8 two-loop integrals [76].
It is straightforward to apply Schubert-problem-based method to 10-point double box (similar computation has been done for 9-point case in [39]), as we will demonstrate soon. Here we first give the result for its leading singularity, which is proportional to (inverse of) As we will see, by sending certain dual point x i to infinity, this gives remaining square roots for non-DCI integrals, in addition to those from one-loop leading singularities.

Even letters
To reduce the DCI kinematics to non-DCI four-mass kinematics, we take the non-DCI limit x 2 → ∞ in the DCI fully massive pentagon, then we get a non-DCI box x 10 x 8 x 6 x 4 x 8 x 6 x 4 , and we can represent massive legs by P 1 = x 10,4 , P 2 = x 4,6 , P 3 = x 6,8 and P 4 = x 8, 10 . In this limit, finite planar variables become and we can also see that so ∆ 24810 becomes r 3 up to a factor m 2 1 ! Similarly, the square roots of five four-mass boxes in eq.(4.2) give {r 1 , r 2 , r 3 , r 4 , r 5 } (defined in eq.(3.8)) in this limit respectively.
Note that we can also set another x i to ∞ with certain identification of massive legs, then the same four-mass square root may correspond to another one-loop square root. For example, sending x 8 → ∞, ∆ 24810 gives r 5 by identifying P 1 = x 6,10 , P 2 = x 10,2 , P 3 = x 2,4 and P 4 = x 4,6 . In addition, under any identification ∆ 24810 always gives r 1 when sending x 6 → ∞. The same four-mass square root may also correspond to different forms of the same square root under different non-DCI limits. For example, if we take x 4 → ∞ and set P 1 = x 10,2 , P 2 = x 2,6 , P 3 = x 6,8 , P 4 = x 8,10 , then (4.6) Two limits x 2 → ∞ and x 4 → ∞ coincidentally give the same square root r 3 since the four-mass box I 4m (2,4,8,10) becomes the same triangle x 10 x 8 x 4 .
which reflect the fact that there are different choices to take limits for the one-loop case. Note that no matter how we write it, B i is always a monomial which is positive when all kinematic variables are positive. Furthermore, since the DCI square root is positive for any positive DCI kinematic point, these square roots stay positive in non-DCI limit for such positive regions.
We have one more comment on the W 18 and its positivity. Note that there exists a DCI letter whose non-DCI limits are always proportional to W 18 , independent of the dual point x i we send to infinity. For example, the DCI letter becomes when sending x 2 → ∞ or x 4 → ∞ respectively, and similar for the other 3 non-DCI limits. The crucial point is thatW is positive definite in the positive region G + (4, 10), as can be checked by any positive parameterization [77]. It is believed thatW is a letter of 10-point double-box integral.
Now we turn to the other 34 odd letters W 35 , . . . , W 68 involving two square roots, which take the form a + r i r j a − r i r j with a being generally involved polynomials in kinematic variables, and we call them mixed algebraic letters to distinguish them from those letters with only one square root. Mixed algebraic letters only show up as two-loop letters. We can generate them from the approach of Schubert problems and the alphabet of 10-point double-box integral above.
In the rest of this subsection, we list all the configurations we use to construct 34 mixed algebraic letters. We first classify 34 mixed algebraic letters according to different square roots, e.g. letters with only one-loop square roots and letters with various two-loop roots. For each group we fix a non-DCI limit and the corresponding identification of external legs, and present one possible choice {X 1 , X 2 , X 3 , X 4 } to construct each individual letter. We stick to the cross-ratio (X 1 ,X 3 )(X 2 ,X 4 ) (X 1 ,X 4 )(X 2 ,X 3 ) from each configuration and consider its non-DCI limit in each case. Note that the configurations we present here are not unique and these mixed algebraic letters can be generated by many different choices.
On the other hand, if we keep x 6 → ∞ but change the identification to P 4 = x 10,2 , P 1 = x 2,4 , P 2 = x 4,8 and P 3 = x 8,10 (five square roots of DCI four-mass boxes (4.1) give {r 5 , r 3 , r 1 , r 4 , r 2 } respectively), we will get the four letters with r 7 from the four configurations instead.
Note that the appearance of algebraic letters with two-loop square roots has certain pattern. For instance, on the 17th row of the DE (I 17 = r 6 G 0,1,1,0,1,1,1,0,0 , which is finite), there are five off-diagonal elements which are just the five odd letters {W 19 , W 37 , W 41 , W 55 , W 56 } with r 6 . Furthermore, these are exactly the last entries of I 17 at O( )! Similar phenomenon applies to double-triangle I 32 with r 7 and box-triangles I 5 , I 6 , I 7 , I 8 with r 8 , · · · , r 11 , respectively: at leading order, there are exactly 5 last entries, namely those 5 odd letters containing r i , for each of these integrals.
Last but not least, we note that the product of numerator and denominator of each mixed algebraic letter is always proportional to the last rational letter, which was shown to be the Gram determinant, and other factors of this product are just kinematic variables m 2 1 , . . . , m 2 4 , s, t. For example, the product of numerator and denominator of W 36 is (f 36 + r 2 r 5 )(f 36 − r 2 r 5 ) = 4m 2 2 W 18 . Therefore, any mixed odd letters are positive in the positive region.

Properties of the symbols
In this section, we investigate some analytic structures of these integrals from their symbols, which can be nicely understood from DCI integrals. From canonical differential equations dI i = j d log A ij I j , we can construct symbols of integrals in the basis by where I The starting point is to understand branch points, or the first entries of the symbol, and corresponding physical discontinuities. For any planar integral, branch points must correspond to planar variables s i,...,k := (p i + p i+1 + · · · + p k ) 2 = 0. In our cases, this means that first entries of I a should be Then we consider double discontinuities. The corresponding physical constrains are Steinmann relations, which say that the double discontinuities taken in overlapping channels of any planar integrals should vanish. In our cases, the only overlapping channels are s = 0 and t = 0. We consider the combination such that I (w) i can be written as p,q X p,q ⊗Y i,p,q where Y i,p,q are symbols of weight-(w −2), then Disc s=0 Disc t=0 (I The double discontinuity of X ac is just Disc s=0 Disc t=0 X ac = b Disc s=0 (log A ab ) Disc t=0 (log A bc ), (4.21) the product of two discontinuity matrix. We check that both Disc s=0 Disc t=0 X ac and Disc t=0 Disc s=0 X ac vanish. Steinmann relations constrain first-two entries of the symbol of I (w) a , we could further consider the same constrains on any adjacent entries of the symbol, which are known as the extended Steinmann relations. For the symbol of I i 1 A i 2 i 1 ⊗ · · · ⊗ A i k i k−1 ⊗ A i k+1 i k ⊗ · · · ⊗ A aiw Disc s=0 Disc t=0 X i k+1 ,i k−1 = 0 (4.22) and i 1 ,..., i k ...,iw I (0) i 1 A i 2 i 1 ⊗ · · · ⊗ A i k i k−1 ⊗ A i k+1 i k ⊗ · · · ⊗ A aiw Disc t=0 Disc s=0 X i k+1 ,i k−1 = 0, (4.23) for any k and the only overlapping channels s = 0 and t = 0. Therefore, Steinmann relations for X also guarantee the extended Steinmann relations of I (w) a . In [40], it's conjectured from first entry condition and Steinmann relations that the first-two entries of DCI integrals can only be linear combination of one-loop box functions and some trivial log log functions. For any order of of I (w) a , we also find that the first-two entries of I i can only be linear combinations of {log(s) 2 , log(t) 2 , log(m 2 i ) log(s), log(m 2 i ) log(t), log(m 2 i ) log(m 2 j ), F (z k ,z k )} i,j=1,...,4,k=1,...5 , (4.24) where F (z k ,z k ) are (normalized) four-mass box function whose symbol is Therefore, the first two entries are also box functions (after taking one dual point to infinity) except log log functions.

Summary and Outlook
In this paper, we provide insights for multi-loop Feynman integrals and their symbology from the viewpoint of dual conformal symmetry. The symbols of UT integrals, in principle, can be derived from the differential equations of the UT integral basis. As shown in this paper, the symbols of some integrals can be very complicated. Also, the symbols are with many "mysterious" structures. We also studied the symbology form DCI point of view. By sending certain dual point to infinity in DCI integrals, non-DCI finite integrals are acquired. Meanwhile, the corresponding symbol structures can also be analyzed. This gives us an insight about some properties of the symbols, before we comply the canonical differential equation computation, in future studies for new Feynman integrals. To explain the above specifically, in this paper, we studied a cutting-edge example, the two-loop double box diagram with 4 different external masses. We determine its UT integral basis mainly using loop-by-loop analysis of leading singularity in Baikov representation. The analytic derivation of corresponding differential equations is nontrivial and we use the package FiniteFlow based on the finite-field method for this computation.
From the differential equations, we derived the symbols of the UT integrals from order −4 to 0 . The symbols consist of 68 letters. Among the letters, there are 18 even letters being rational functions, 16 odd letters containing one square root each, and 34 odd letters containing 2 square roots each. The symbols for the 74 integrals are derived using the differential equations. We found their properties that: 1) The letters that can appear at the first and the second entries of the symbols are limited. 2) A large number of the possible letter pairs that can appear at two adjacent entries are forbidden, read from the letter structure of differential equations.
What we find particularly remarkable is that all symbol letters of these integrals have clear origin from symbol letters of related dual conformal integrals, which have been studied extensively and exhibit rich mathematical structures. Both even and odd letters are nicely obtained by taking a dual point to infinity, of the letters for one-and two-loop DCI integrals with generic pentagon kinematics. In particular, not only do the square roots correspond to one-and two-loop leading singularities, but we determine all complicated mixed odd letters from associated twistor geometries, which are also "last entries" for corresponding finite, two-loop integrals. We find it satisfying that one can obtain all letters by considering mathematical structure of DCI integrals without actually computing them, and we leave a more systematic study of this phenomenon to a future work. Last but not least, we find that important properties of DCI integrals nicely carry over to UT integrals we consider in this paper: the first two entries come from one-loop DCI box functions, and extended Steinmann relations further constrain any two adjacent entries to any order in . It would be highly desirable to understand further how structures of their symbols follow from properties of these DCI integrals.