Differential equations, recurrence relations, and quadratic constraints for L -loop two-point massive tadpoles and propagators
PPrepared for submission to JHEP
Differential equations, recurrence relations, andquadratic constraints for L -loop two-point massivetadpoles and propagators. Roman N. Lee and Andrei A. Pomeransky
The Budker Institute of Nuclear Physics, 630090, Novosibirsk
E-mail: [email protected] , [email protected] Abstract:
We consider L -loop two-point tadpole (watermelon) integral with arbitrarymasses, regularized both dimensionally and analytically. We derive differential equationsystem and recurrence relations (shifts of dimension and denominator powers). Since the L -loop sunrise integral corresponds to the ( L + 1) -loop watermelon integral with one cutline, our results are equally applicable to the former. The obtained differential system has aPfaffian form and is linear in dimension and analytic regularization parameters. In generalcase, the solutions of this system can be expressed in terms of the Lauricella functions F ( L ) C with generic parameters. Therefore, as a by-product, we obtain, to our knowledge forthe first time, the Pfaffian system for F ( L ) C for arbitrary L . The obtained system has noapparent singularities. Near odd dimension and integer denominator powers the system canbe easily transformed into canonical form. Using the symmetry properties of the matrixin the right-hand side of the differential system, we obtain quadratic constraints for theexpansion of solutions near integer dimension and denominator powers. In particular, weobtain quadratic constraints for Bessel moments similar to those discovered by Broadhurstand Roberts. a r X i v : . [ h e p - ph ] A p r ontents D and µ . 96 Special cases when the system acquires a block-triangular form 12 D = 2 D = 2 D = 1
23C Examples of quadratic relations for
IKM functions. 24
Multiloop massive sunrise and watermelon integrals (see Fig. 1) are ubiquitous in variousphysical applications. Being seemingly simple, they, in fact, bring with themselves a lot ofcomplexity in any calculation. The reason is that their (cid:15) -expansion can not be expressed interms of the standard functional basis, viz., multiple polylogarithms [1, 2]. Already startingfrom two loops one observes complicated iterated integrals involving modular forms and/orelliptic curves [3, 4]. For L loops, the (cid:15) expansion of massive sunrise integrals is likely toinvolve periods of certain ( L − -dimensional Calabi-Yau varieties (cf. Refs. [5, 6]).For the general case of different masses the closed expressions for these integrals areknown since Ref. [7] (see also Ref. [8]) in terms of Lauricella functions F ( L ) C defined via mul-tiple hypergeometric series. Although these functions have a long history of investigation,starting from [9], and much is known about their properties, still many questions remainunanswered. In particular, the functions which appear in coefficients of their expansion inthe Laurent series in parameters are not well investigated. Meanwhile, these coefficientsare the most important from the point of view of multiloop calculations. From the purely– 1 –athematical point of view, among the four flavors of Lauricella functions F A , F B , F C , and F D , the functions F ( L ) C are the only ones for which the differential system in Pfaffian formis not yet known for L > .In the present paper we derive the Pfaffian system of the differential equations for the L -loop watermelon and sunrise graphs with arbitrary masses regularized both dimensionallyand analytically. Similar to Refs. [7, 8], we find the fundamental system of solutions interms of the Lauricella functions F ( L ) C with generic parameters and arguments. In this way,our results give a Pfaffian system for Lauricella F ( L ) C for all L , to the best of our knowledge,for the first time .In addition to the differential system, we also derive the operators shifting dimensionor the powers of denominators. From the point of view of F ( L ) C functions these operatorsprovide a certain form of the contiguity relations. Near D = 1 we reduce the differen-tial system to canonical form which allows one to obtain the coefficients of expansion indimension and analytic regulators in terms of the Goncharov’s polylogarithms.Last but not least, we observe that the matrix in the right-hand side of the obtaineddifferential system has special features which result in the existence of the bilinear relationsfor the solutions of the systems that differ in the sign of the right-hand side (cf. Ref. [11]).Combined with the shift operators, these relations allow us to obtain the infinite set of thequadratic relations for the coefficients of expansion in dimension and analytic regulatorsnear any integer point, including the most interesting case of even D . We consider the L -loop watermelon integral, Fig.1, with different masses and powers ofdenominators (exponents), T ( m , . . . , m L ) = 2 D − iπ D/ Γ (cid:0) D (cid:1) (cid:90) δ (cid:16)(cid:80) Ll =0 p l (cid:17) L (cid:89) l =0 (cid:34) d D p l iπ D/ m l Γ ( α l ) (cid:0) m l − p l − i (cid:1) α l (cid:35) . (2.1)Here we imply the Lorentzian signature, so that p = p − p . Note that we have chosena somewhat unconventional overall normalization factor D − Γ (cid:0) D (cid:1) (cid:81) l m l Γ ( α l ) in order tosimplify some expressions below.Before moving any further let us discuss the applicability of our present considerationto the sunrise integral S ( m , . . . , m L | q ) = 2 D − iπ D/ Γ (cid:0) D (cid:1) (cid:90) δ (cid:16) q + (cid:80) Ll =0 p k (cid:17) L (cid:89) l =0 (cid:34) d D p l iπ D/ m k Γ ( α l ) (cid:0) m l − p l − i (cid:1) α l (cid:35) . (2.2) For the case L = 2 a Pfaffian system was found in Ref. [10]. – 2 – .. L loops ... Figure 1 . L -loop two-point massive tadpole. Each line corresponds to the propagator with arbi-trary mass and power. The obvious relation between S and T is S| q =0 = T . However, we can also represent the ( L − -loop sunrise integral as an L -loop watermelon integral with one cut line: ( m ) D − S ( m , . . . , m L | m ) = 2 D − iπ D/ − Γ (cid:0) D (cid:1) × (cid:90) δ (cid:16)(cid:80) L +1 l =0 p k (cid:17) d D p iπ D/ (cid:20) m m − p − i − m m − p + i (cid:21) × L (cid:89) l =1 (cid:34) d D p l iπ D/ m k Γ ( α l ) (cid:0) m l − p l − i (cid:1) α l (cid:35) , (2.3)where m = (cid:112) q . The two terms in square brackets on the last line correspond to twodifferent deformations of the integration contour over p . As the differential system we arelooking for is not aware of the integration contours, it is clear that m D − S ( m , . . . , m L | m ) satisfies the same differential equations as T ( m , . . . , m L ) with α = 1 . Therefore, in therest of the paper we will concentrate on the watermelon integral T only.After the Wick rotation we can rewrite Eq. (2.1) as T = 2 D − π D/ Γ (cid:0) D (cid:1) (cid:90) δ (cid:16)(cid:80) Ll =0 p k (cid:17) L (cid:89) l =0 (cid:34) d D p l π D/ m k Γ ( α l ) (cid:0) p l + m l (cid:1) α l (cid:35) , (2.4)where now p = p + p . Representing the function δ ( p ) as (cid:82) d D x (2 π ) D e ipx , we obtain T = ∞ (cid:90) dx x D − L (cid:89) l =0 (cid:90) d D pπ D/ m k Γ( α k ) e ipx ( p + m k ) α k = ∞ (cid:90) dx x D − L (cid:89) l =0 P ( µ k , m k , x ) . (2.5)As we shall see immediately, the integral (cid:82) d D pπ D/ m Γ( α ) e ipx ( p + m ) α depends on D and α only viathe combination µ = D − α + 1 and we have, therefore, introduced P ( µ, m, x ) = m Γ( α ) (cid:90) d D pπ D/ e ipx ( p + m ) α = m ∞ (cid:90) dλλ α − (cid:90) d D pπ D/ e ipx − λ ( p + m )= m ∞ (cid:90) dλλ − µ − e − λm − x λ = 2 m (cid:16) x m (cid:17) − µ K µ − ( mx ) . (2.6)– 3 –ere K ν ( y ) is the Macdonald function.In order to derive the differential equations for this integral, we introduce the set of L +1 functions labeled by a binary number (a string of zeros and ones) a = a a . . . a L , T a ( D, µ , m ) = ∞ (cid:90) dx x D − t a , (2.7)where t a = L (cid:89) l =0 P a l ( µ l , m l , x ) , (2.8) P is defined in eq. (2.6), and P ( µ, m, x ) = 1 m ∂ x P ( µ, m, x ) = − x ∞ (cid:90) dλλ − µ − e − λm − x λ = − m (cid:16) x m (cid:17) − µ K µ +12 ( mx ) . (2.9)The original integral (2.4) is recovered as T = T ( D, µ , m ) . The rationale behind introduc-ing auxiliary L +1 − functions is that it turns out to be possible to derive a closed systemof linear differential equations for T a . Note that all components of T a can be expressed via T with shifted dimension and indices: T a ( D, µ ) = (cid:89) l (cid:18) − m l (cid:19) a l T ( D → D + | a | , µ l → µ l + 2 a l )= (cid:89) l (cid:18) − m l (cid:19) a l T ( D → D + | a | , α l → α l + | a | / − a l ) , (2.10)where | a | = (cid:80) l a l . It is convenient to treat the L +1 quantities V a as components of thevector V in C ⊗ . . . ⊗ C (cid:124) (cid:123)(cid:122) (cid:125) L +1 factors , where V a can be either T a or t a . In what follows, we assumethat the factors in C ⊗ . . . ⊗ C are numbered starting from zero, so that, e.g., L -th factor isthe right-most one. Besides, to lighten notation we often omit the arguments of functions.In order to derive the differential system for T a we differentiate P , ( µ, m, x ) withrespect to m and use the differential equation for K ν ( y ) . We obtain m∂ m P = mxP + µP ,m∂ m P = mxP . (2.11)In matrix notations we may rewrite this as m∂ m P = ( mxσ + µ ¯ n ) P , (2.12)where P = (cid:32) P P (cid:33) , σ = σ x = (cid:32) (cid:33) , and ¯ n = 1 − n = (1 + σ z ) = (cid:32) (cid:33) .– 4 –sing these formulae, we see that the expression for the derivative of T , in addition to T multiplied by a matrix, contains the integrals of the form (cid:82) ∞ dx x D t . Note the additionalfactor x in the integrand as compared to the definition (2.7). In order to get rid of thisfactor, we consider the identity (cid:82) dx∂ x ( x D t ) = 0 . Explicitly differentiating and using therelation x∂ x P = ( m∂ m − µ ) P = ( mxσ − µn ) P , (2.13)which, again, follows from the differential equation for K ν , we obtain ∞ (cid:90) dx x D − ( xM − W ) t = 0 . (2.14)Here M = L (cid:88) l =0 m l σ l , W = L (cid:88) l =0 µ l n l − D , (2.15)and the operators σ l and n l act in l -th factor of C ⊗ . . . ⊗ C as σ and n , respectively.Therefore, ∞ (cid:90) dx x D t = M − W T (2.16)whenever M − exists. Since the operators σ l commute with each other and their eigenvaluesare η l = ± , the eigenvalues of M are (cid:80) l η l m l . Therefore, M is not invertible iff (cid:88) l η l m l = 0 (2.17)for some choice of signs η l .Thus, using Eqs. (2.12) and (2.16) we obtain the following expression for the derivativeof T with respect to m k ∂ m l T = A l T, A l = σ l M − W + µ l ¯ n l m l . (2.18)Equivalently, the above equations can be written as dT = A T , (2.19) A = L (cid:88) l =0 A l dm l = dM M − W + L (cid:88) l =0 µ l ¯ n l dm l m l . (2.20)Remarkably, the differential form A is closed ( dA = 0 ) which can be easily proved if onetakes into account the pairwise commutativity of σ l . Then the integrability condition dA = A ∧ A requires that A ∧ A = 0 , see Appendix A for the derivation. The system(2.19) equipped with the condition dA = 0 is said to be in Pfaffian form. We observe thatall singularities of the differential form A are located on the L + 1 hyperplanes defined by m l = 0 and L hyperplanes defined by Eq. (2.17) (cf. Ref. [11]). The latter hyperplanescorrespond to vanishing of the eigenvalues of the matrix M .– 5 –et us note that the differential system (2.19) in fact splits into two separate systems,each for L functions T a with | a | being either even or odd. This due to the commutativityof A with the parity operator P = ( − (cid:80) i n i = (cid:89) i (1 − n i ) . (2.21)Of course, we are mostly interested in the subsystem for even components which involvesour original tadpole integral T . Let us now obtain the recurrence relations in α k and D . For the former we use the identities P ( µ − , m, x ) = m ∞ (cid:90) dλλ − µ e − λm − x λ = − m ( m∂ m − P ( µ, m, x ) , (3.1) P ( µ − , m, x ) = − x ∞ (cid:90) dλλ − µ e − λm − x λ = − m ∂ m P ( µ, m, x ) . (3.2)Then, using Eq. (2.18), we have T ( µ − e l ) = R l ( D, µ ) T ( µ ) , (3.3)where R l ( D, µ ) = − m l (cid:18) A l − ¯ n l m l (cid:19) = − m l (cid:18) σ l M − W + ( µ l −
1) ¯ n l m l (cid:19) (3.4)and e l = ( . . . , , l -th , , . . . ) is the vector with l -th component equal to and all othercomponents equal to zero.Let us now derive the operator which shifts the dimension. Note that the identity(2.16) defines the operator R ( D, µ ) = M − W , (3.5)which shifts D by +1 at fixed µ , i.e. T ( D + 1 , µ ) = R ( D, µ ) T ( D, µ ) . (3.6)In order to obtain the operator which shifts D at fixed α , we should first use the operators R l to shift µ → µ − and then shift the dimension by − at fixed µ using the operator R − . We obtain T ( D − , µ − e ) = R ( D, µ ) T ( D, µ ) , (3.7)where R ( D, µ ) = R − ( D − , µ − e ) R − ( D − , µ − e ) L (cid:89) l =0 R l (cid:16) D, µ − (cid:80) Lj = l +1 e j (cid:17) (3.8)is the operator which shifts D by − at fixed α . Here and below e = (cid:80) Lj =0 e j = (1 , , . . . , and the product of matrices corresponds to the multiplication from the left to the right, sothat (cid:81) Ll =0 R l = R . . . R L . – 6 – Basis of solution space
While the function T as defined in Eq. (2.7) is a specific solution of the differential system(2.19) and recurrence relations (3.3) and (3.6), it is natural to consider the linear space ofall solutions of these equations. In the present section we fix a basis of L +1 functions inthis space. In order to do this, we note that the derivation of the differential equationsand recurrence relations in Sections 2 and 3 can be explicitly repeated if some of theMacdonald functions K ν are replaced with I ± ν sin πν . More specifically, let us make the followingreplacement in the definition of t a , Eq. (2.8): K µl ± ( m l x ) → ∓ σ l π πµ l I σ l µl ± ( m l x ) , (4.1)for all l ∈ S , where S is some subset of E = { , , . . . , L } . Here each σ l can be either +1 or − . Defined in this way, the function T a formally satisfies the differential and differenceequations (2.19), (3.3), and (3.6). There may be however a problem with the convergence ofthe integral over x due to the exponential growth of the modified Bessel functions. For allmasses being positive, the convergence condition is (cid:80) l ∈ S m l < (cid:80) l ∈ E \ S m l . In particular, E \ S should not be empty. Therefore, we take E \ S = { } , i.e. S = { , . . . , L } , and definethe set of functions V ( ρ ) = ∞ (cid:90) dx x D − P ( µ , m , x ) ⊗ L (cid:79) l =1 Q ( ρ l ) ( µ l , m l , x ) , (4.2) Q ( ρ ) ( µ, m, x ) = (cid:18) Q ( ρ )0 ( µ, m, x ) Q ( ρ )1 ( µ, m, x ) (cid:19) = π ( − ρ πµ m (cid:16) x m (cid:17) − µ (cid:32) I ( − ρ µ − ( mx ) I ( − ρ µ +12 ( mx ) (cid:33) , (4.3)where ρ = ρ ρ . . . ρ L is a binary number. We assume that m l > and m > (cid:80) Ll =1 m l , sothat the integrand decays exponentially at large x . Using the identity K ν ( x ) = π ( I − ν ( x ) − I ν ( x ))2 sin πν , (4.4)we can express T , Eq. (2.7), via V ( ρ ) as T = (cid:88) ρ V ( ρ ) . (4.5)The set of functions (4.2) contains L functions numbered by the vector ρ . Meanwhile,the solution space of the differential system (2.19) is L +1 -dimensional. In order to obtainthe whole set of solutions, we use the symmetry of the differential system related to theoperator P , Eq. (2.21). Therefore, we can define functions V ( (cid:37) , ρ ) = 1 + ( − (cid:37) + | ρ | P V ( ρ ) , (cid:37) = 0 , , (4.6)which are the solutions of the differential system. Note that we have introduced the addi-tional factor ( − | ρ | for further convenience. In terms of the components we have V ( (cid:37) , ρ ) a = 1 + ( − (cid:37) + | ρ | + | a | V ( ρ ) a , (4.7)– 7 –o that the solutions with (cid:37) = | ρ | (mod 2) have nonzero components with even | a | .Together with the different choices of ρ , this gives us L +1 solutions of the differentialsystem (2.19). It is also easy to see that P R l = R l P and, therefore, V ( (cid:37) , ρ ) is also a solutionof the µ -shifting recurrences (3.3). As to the the operator R , Eq. (3.6), we have P R = − R P and therefore V ( (cid:37) , ρ ) ( D + 1) = R ( D ) V (¯ (cid:37) , ρ ) ( D ) , (4.8)where we used the notation ¯ x = 1 − x . Thus, the functions (4.6) satisfy the relations shifting D by , V ( (cid:37) , ρ ) ( D + 2) = R ( D + 1) R ( D ) V ( (cid:37) , ρ ) ( D ) , (4.9)which follow from Eq. (3.6) .Note that our basis functions V ( (cid:37) , ρ ) can be expressed via Lauricella function F ( L ) C (cf.[7, 8]). This function is defined via the hypergeometric series F ( L ) C ( b , b ; c , . . . c L ; x , . . . , x L ) = ∞ (cid:88) k =0 . . . ∞ (cid:88) k L =0 ( b ) (cid:80) k i ( b ) (cid:80) k i x k . . . x k L L ( c ) k . . . ( c L ) k L k ! . . . k L ! , (4.10)where ( a ) k = Γ( a + k ) / Γ( a ) is the Pochhammer symbol. In order to do this we use theexpansion of the modified Bessel function I ν : I ν ( x ) = ∞ (cid:88) k =0 ( x/ ν +2 k k !Γ( ν + k + 1) . (4.11)Substituting this expansion in Eq. (4.2) and taking the integral over x with the help of theidentity ∞ (cid:90) dx x β − K ν ( x ) = 2 β − Γ (cid:104) β + ν (cid:105) Γ (cid:104) β − ν (cid:105) , (4.12)we obtain V ( ρ ) a = 2 D − ( − a m − D + (cid:80) Ll =0 µ l Γ [ b ] Γ [ b ] L (cid:89) l =1 ( − a l Γ [1 − c l ] (cid:18) m l m (cid:19) c l + µ l − × F ( L ) C (cid:18) b , b ; c , . . . c L ; m m , . . . , m L m (cid:19) , (4.13)where c l = 1 + ( − ρ l µ l − ( − a l ,b = D + a L (cid:88) l =1 (cid:20) c l − µ l + 12 (cid:21) ,b = b − a + 1 − µ . (4.14) Note, that we can also construct L +1 functions which satisfy (3.6), e.g., by taking V ( ρ ) = V (0 ρ ) + V (1 ρ ) and ( V (0 ρ ) − V (1 ρ ) ) g ( D ) , where g is any antiperiodic function of D , i.e., g ( D + 1) = − g ( D ) . – 8 –he functions V ( (cid:37) , ρ ) can be obtained from Eq. (4.13) with the help of Eq. (4.7).Note that V (1 ... ... ∝ m − L − D + (cid:80) Ll =0 µ l [ L (cid:89) l =1 m l ] F C (˜ b , ˜ b ; ˜ c , . . . ˜ c L ; x , . . . x L ) , (4.15)where x l = ( m l /m ) and the parameters ˜ c l = 12 (3 − µ l ) , ˜ b = D L (cid:88) l =1 [1 − µ l ] , ˜ b = ˜ b + 1 − µ (4.16)are generic for generic µ and D . Therefore, V (1 ... ... is proportional to F ( L ) C with genericparameters. Recall that the system (2.19) splits into two subsystems, for even and oddcomponents. Thus, we see that the subsystem of L equations for even components gives aPfaffian system for Lauricella F ( L ) C . D and µ . The fundamental matrix of the differential system (2.19) can be written in the form ofpath-ordered exponent U ( D, µ , m ) = Pexp (cid:20)(cid:90) C A (cid:21) , (5.1)where C is some path in L + 1 -dimensional space of mass parameters starting from somefixed point m and ending in m . Note that A is linear homogeneous in D and µ which,for the expansion near D = µ = 0 , corresponds to a certain generalization of the canonicalform of Ref. [12] to the case of analytical regularization. Thanks to this property, theexpansion around the point D = µ = 0 has the form U ( − (cid:15), − τ , m ) = (cid:88) n (cid:62) , k (cid:62) C n, k ( m ) (cid:15) n (cid:89) l τ k l l , (5.2)where the coefficients C n, l ( m ) are the iterated integrals expressed in terms of the Gon-charov’s polylogarithms [1, 2].Let us note that the recurrence relations obtained in the previous section allow us toexpress via the same polylogarithmic functions also the expansions near any integer D andeven µ . In particular, we can do it for the expansion near any odd integer value of D andinteger exponents α .In the points where at least one of µ l is an odd number, the recurrence relations can notbe used to express the corresponding expansion via polylogarithms. However, as we shallsee soon, they can be used to obtain the quadratic relations for the expansion coefficients,see Fig. 2. These relations are closely related to the ones described in Ref. [13] in the casewhen matrix A is symmetric and proportional to (cid:15) or (cid:15) + 1 / . Note that the matrix A has singularities when either det M = 0 or m l = 0 . Naturally, the path C should not intersect the singular locus of A , and, therefore, U depends not only on the end points m and m , but also on the equivalence class of C . – 9 – µ Figure 2 . µ − µ plane at fixed integer D . The quadratic relations exist near any marked point(disk, circle or square). Solid disks correspond to the points where, in addition, the expansion canbe expressed via polylogarithms. Depending on whether D is even or odd, the disks correspond tohalf-integer or integer powers of the propagators. When D is even, integer powers of propagatorscorrespond to empty squares. In order to obtain these relations, we first note that the matrix A ( D, µ ) has the followingsymmetry A (cid:124) W = W A . (5.3)Let F ± be the two solutions of the equations dF ± = ± AF ± . (5.4)Then, from Eq. (5.3) it follows that the bilinear form F T − W F + is independent of m : d ( F (cid:124) − W F + ) = − F (cid:124) − A (cid:124) W F + + F (cid:124) − W AF + = 0 . (5.5)Now we note that A ( − D, − µ ) = − A ( D, µ ) and, therefore, F − ( D, µ ) obeys the sameequation as F + ( − D, − µ ) . So, given two solutions T , ( D, µ , m ) of Eq. (2.19), we canconstruct the conserved bilinear form T (cid:124) ( − D, − µ , m ) W ( D, µ ) T ( D, µ , m ) = C ( D, µ ) , (5.6)where C ( D, µ ) in the right-hand side denotes some quantity which, in general, dependson D and µ , but not on m .Now let T , also satisfy the recurrence relations (3.3) and (3.6). Suppose that we areinterested in the expansion near D = D (cid:63) and µ = µ (cid:63) , where D (cid:63) and µ (cid:63)l are integer. Thenwe can write T (cid:124) ( D (cid:63) + 2 (cid:15), µ (cid:63) + 2 τ , m ) BT ( D (cid:63) − (cid:15), µ (cid:63) − τ , m ) = C ( D (cid:63) − (cid:15), µ (cid:63) − τ ) , (5.7)– 10 –here C ( D, µ ) is the same function as in Eq. (5.6) and B = B ( D (cid:63) , µ (cid:63) | (cid:15), τ , m ) = L (cid:89) l =0 µ (cid:63)l (cid:89) ν =1 R (cid:124) l ( D (cid:63) + 2 (cid:15), − µ (cid:63) + 2 L (cid:88) j = l µ (cid:63)j e j − ν e l + 2 τ ) × (cid:34) D (cid:63) (cid:89) d =1 R − (cid:124) ( D (cid:63) − d + 2 (cid:15), − µ (cid:63) + 2 τ ) (cid:35) W ( D (cid:63) − (cid:15), µ (cid:63) − τ ) . (5.8)Using the commutation relations derived in the Appendix A one can show that B has thesymmetry B (cid:124) ( (cid:15), τ ) = ( − D (cid:63) + (cid:80) l µ (cid:63)l +1 B ( − (cid:15), − τ ) . (5.9)When expanded in (cid:15) and τ , the identity (5.7) provides a lot of quadratic constraints forthe expansion coefficients of the solutions near D = D (cid:63) and µ = µ (cid:63) . Equations (5.6) and(5.7) are important results of the present paper.Let us now specialize Eq. (5.6) to the case T = V ( (cid:37) , ρ ) , T = V (˜ (cid:37) , ˜ ρ ) . In order tocalculate the constant in the right-hand side, we consider the limit m l → ( l = 1 , . . . , L )at fixed m . In this limit the main contribution comes from the region m x ∼ and fromEq. (4.13) we see that V ( (cid:37) , ρ ) a ( D, µ ) = m µ − D L (cid:89) l =1 m µ l (cid:18) m l m (cid:19) µ l (1 − ρ l )+1 − δ alρl F , (5.10)where F is a function of ( m l /m ) , analytic and nonzero at the origin. The value of F atthe origin can be obtained by replacing F ( L ) C → in Eq. (4.13). Since we know that theright-hand side of Eq. (5.6) does not depend on masses, it can be nonzero only if ˜ ρ = ρ .Indeed, the power m µ l (˜ ρ l − ρ l ) l can not be canceled if ˜ ρ l (cid:54) = ρ l . Besides, the exponent − δ a l ρ l inEq. (5.10) shows that only the contribution of the components with a l = ρ l ( l = 1 , , . . . , L )may contribute to the limit m l → . Taking the integral over x , we have V ( (cid:37) , ρ ) a ρ (cid:39) δ a (cid:37) ( − a D − m µ − D L (cid:89) l =1 ( − ρ l Γ (cid:2) (1 − µ l ( − ρ l ) (cid:3) m µ l ρ l m µ l (1 − ρ l ) l × Γ (cid:104) (cid:16) D − µ a − (cid:80) Ll =1 µ l ρ l (cid:17)(cid:105) Γ (cid:104) (cid:16) D + 1 − µ ¯ a − (cid:80) Ll =1 µ l ρ l (cid:17)(cid:105) (5.11)Using this asymptotics in the left-hand side of Eq. (5.6), we fix the right-hand side ofthis identity. We find V (˜ (cid:37) ˜ ρ ) (cid:124) ( − D, − µ ) W ( D, µ ) V ( (cid:37) , ρ ) ( D, µ ) = 12 δ ˜ (cid:37) (cid:37) δ ˜ ρρ (cid:34) L (cid:89) l =1 π cos πµ l (cid:35) × π sin π (cid:16) D − µ (cid:37) − (cid:80) Ll =1 µ l ρ l (cid:17) π cos π (cid:16) D − µ ¯ (cid:37) − (cid:80) Ll =1 µ l ρ l (cid:17) (5.12)Recall that ¯ (cid:37) = 1 − (cid:37) . Let us note that in Ref. [11] some bilinear relations betweenLauricella functions F C have been derived using different methods. It would be interestingto compare them with Eq. (5.12). – 11 –umming over (cid:37) and ˜ (cid:37) and using the identity x x − y + 1sin x − y x = 2cos y (cid:20) x + 1sin( x − y ) (cid:21) , (5.13)we obtain V ( ˜ ρ ) (cid:124) ( − D, − µ ) W ( D, µ ) V ( ρ ) ( D, µ ) = δ ˜ ρρ (cid:34) L (cid:89) l =0 π cos πµ l (cid:35) (cid:88) ρ =0 , π sin π (cid:16) D − (cid:80) Ll =0 µ l ρ l (cid:17) . (5.14)Finally, summing over ρ and ˜ ρ and using Eq. (4.5) we have T (cid:124) ( − D, − µ ) W ( D, µ ) T ( D, µ ) = (cid:34) L (cid:89) l =0 π cos πµ l (cid:35) (cid:88) ρ =0 , . . . (cid:88) ρ L =0 , π sin π (cid:16) D − (cid:80) Ll =0 µ l ρ l (cid:17) . (5.15)Note that we can obtain the bilinear relation separately for P -even part T + = P T . Inorder to do this, in addition to Eq. (5.13), we also use the identity x x − y − x − y x = 2cos y [cot x − cot( x − y )] . (5.16)We obtain T (cid:124) + ( − D, − µ ) W ( D, µ ) T + ( D, µ ) = (cid:34) L (cid:89) l =0 π cos πµ l (cid:35) (cid:88) ρ =0 , . . . (cid:88) ρ L =0 , × π − (cid:80) Ll =0 ρ l cot π (cid:34) D − L (cid:88) l =0 ( µ l − ρ l (cid:35) . (5.17) Let us first consider the special case µ l = µ = D − , which corresponds to the removalof the analytical regularization. The differential system (2.19) in this case should have ablock-triangular form, corresponding to the decoupling of the trivial “clover-leaf” integralsarising from the contraction of one of L + 1 lines. In order to see this structure, it isconvenient to pass to the new functions Y ( µ, m ) = M T ( D, µ , m ) (cid:12)(cid:12)(cid:12)(cid:12) D = µ +1 µ l = µ (6.1)The differential system for these functions has the form dY = µHY , (6.2) H = ( N − dMM + L (cid:88) l =0 ¯ n l dm l m l , (6.3)– 12 –here N = (cid:80) Ll =0 n l . Note that Eq. (6.2) corresponds to ( (cid:15) + 1 / -form discussed in Ref.[13]. In particular, this form allows one to write down the (cid:15) -expansion of the solution near D = 1 in terms of Goncharov polylogarithms (see Appendix B for details). Let us observethat the equations for each of the components Y ... , Y ... , . . . , Y ... (6.4)decouple and acquire the form dY ... l ... = µ (cid:88) i (cid:54) = l dm i m i Y ... l ... (6.5)with the general solution being Y ... l ... ∝ (cid:89) i (cid:54) = l m µi . (6.6)These are just the equations for the clover-leaf integrals. One can check explicitly that Y ... , Y ... , . . . , Y ... are indeed evaluated to the corresponding expressions for the clover-leaf integrals after substitution of Eq. (2.7) into Eq. (6.1). The simplest way to check thisis to represent Y ( µ ) as Y ( µ ) = lim D → µ +1 W ( D − , µ ) R − ( D − , µ ) T ( D, µ ) = lim D → µ +1 W ( D − , µ ) T ( D − , µ ) (6.7)and to note that the corresponding elements of W are vanishing in this limit. Then, onlythe pole part of the corresponding components of T ( D − , µ ) have to be evaluated (notethat W is a diagonal matrix).Let us consider now the homogeneous differential system obtained by putting compo-nents (6.4) to zero. This is the system for the maximally cut tadpole diagram. It has thesame form (6.2), where now the action of H is restricted to the subspace with Y ... = Y ... = . . . = Y ... = 0 . (6.8)Note that in this subspace the operator N − is invertible. From now on to the endof this subsection, we assume that all operators are restricted to this subspace and that allvector functions belong to it. Then, using the property H (cid:124) ( N − − = ( N − − H , (6.9)and following the same path as in the previous section, we obtain the bilinear constraint Y (cid:124) ( − µ, m )( N − − Y ( µ, m ) = const (6.10)Let us now determine the basis of solutions in the subspace constrained by Eq. (6.8). Wedefine Y ( (cid:37) , ρ ) ( µ, m ) = M V ( (cid:37) , ρ ) ( D, µ , m ) (cid:12)(cid:12)(cid:12)(cid:12) D = µ +1 µ l = µ . (6.11)– 13 –ne can check explicitly that L +1 − L − such functions with (cid:37) (cid:54) = | ρ | satisfy the constraints(6.8) and, therefore, form the basis we are looking for .For two basis functions, Y (˜ (cid:37) , ˜ ρ ) (cid:124) and Y ( (cid:37) , ρ ) , using Eqs. (5.12) and (6.7), we have Y (˜ (cid:37) , ˜ ρ ) (cid:124) ( − µ ) [ µ ( N − − Y ( (cid:37) , ρ ) ( µ ) = 12 δ ˜ (cid:37) (cid:37) δ ˜ ρρ (cid:18) π cos πµ (cid:19) L × π sin πµ ( (cid:37) − | ρ | ) π cos πµ ( ¯ (cid:37) − | ρ | ) . (6.12)Note that the right-hand side of this identity only makes sense when (cid:37) (cid:54) = | ρ | , otherwisethe argument of sin function in the denominator becomes zero.In order to obtain the quadratic relations for the coefficients of (cid:15) -expansion near µ = µ (cid:63) for some integer µ (cid:63) , we have to substitute µ = µ (cid:63) − (cid:15) in Eq. (6.12) and use the relation Y (˜ (cid:37) , ˜ ρ ) (cid:124) ( − µ (cid:63) + 2 (cid:15) ) = Y (˜ (cid:37) , ˜ ρ ) (cid:124) ( µ (cid:63) + 2 (cid:15) ) ( − ( L +1) µ (cid:63) µ (cid:63) (cid:89) ν =1 R ( − ν + µ (cid:63) + 2 (cid:15) ) , (6.13)where R ( µ ) = R ( D, µ ) (cid:12)(cid:12) D = µµ l = µ is the operator of Eq. (3.8) restricted to the subspace (6.8).Note that the quadratic relations obtained by substituting Eq. (6.13) into Eq. (6.12) arevalid only for the solutions of homogeneous equations. A natural question arises: whetherone can obtain similar relations also for generic solutions of inhomogeneous equations?Within our approach, the answer is negative. The bilinear relations (5.6) are valid for anytwo solutions T and T even if we put D − µ l = µ . However, in order to obtain thequadratic relations near any integer point µ = µ (cid:63) (cid:62) , we have to shift the first argument inthe left factor T (cid:124) at least twice by the operator R − . Then the second operator R − ( − µ, − µ ) does not make sense (since R ( − µ, − µ ) = − µM − ( N − is not invertible). Let us now discuss the case when some lines share the same µ and m . Then we can write T = ∞ (cid:90) dx x D − K (cid:89) k =0 [ P ( µ k , m k , x )] r k , (6.14)where r k is the number of lines with mass m k and parameter µ k , so that (cid:80) k r k = L + 1 .There is a symmetry group G generated by the permutations of lines sharing the same massand parameter. This symmetry, in particular, holds for the matrix A in the right-hand sideof the differential system, which now can be written as A = dM M − W + K (cid:88) k =0 µ k r k + 2 S zk ) dm k m k (6.15) M = 2 K (cid:88) k =0 m k S xk , W = K (cid:88) k =0 µ k r k − S zk ) − D , (6.16) In particular, when | ρ | (cid:62) , the function Y ( ρ ) = M V ( ρ ) (cid:12)(cid:12) D = µ +1 µ l = µ = Y (0 , ρ ) + Y (1 , ρ ) is the homogeneoussolution. – 14 –here S xk = 12 u k + r k − (cid:88) l = u k σ xl , S zk = 12 u k + r k − (cid:88) l = u k σ zl . (6.17)Here u k = (cid:80) k − s =0 r s , and the sequence u k , u k + 1 , . . . , u k + r k − enumerates the lines in k -th group. It is remarkable that A is expressed solely via S xk and S zk which are amongthe standard generators of the Lie algebra sl (2 , C ) . We can express via the same generatorsother operators which we have used previously, in particular P = i L +1 (cid:81) Kk =0 e − iπS zk .Note that there is a subtle point here that we want to stress. When all r k are even, thematrix M obviously has zero eigenvalue and, therefore, is not invertible. Thus the matrix A is ill-defined. So, from now on we assume that at least one r k is odd, i.e., at least onegroup contains odd number of lines.The symmetry of the matrix A leads to the splitting of the system into separate sub-systems, each corresponding to a specific irreducible representation of G . Since the tadpoleintegral (6.14) is invariant under the action of this group, we are mostly interested in thesubspace spanned by the tensors T a symmetric with respect to all permutations of indicesfrom this symmetry group.Alternatively, we can reduce the number of auxiliary functions from L +1 = 2 (cid:80) k r k to (cid:81) k ( r k + 1) from the very beginning. Namely, we can introduce the functions T a = ∞ (cid:90) dx x D − (cid:89) k (cid:18) r k a k (cid:19) / [ P ( µ k , m k , x )] r k − a k [ P ( µ k , m k , x )] a k (6.18)where a = a . . . a K and a k runs from to r k , and (cid:0) nk (cid:1) = n ! k !( n − k )! is the binomial coefficient.As before, it is convenient to treat the quantities T a as components of the vector T in C r +1 ⊗ . . . ⊗ C r K +1 . Then the operators S xk and S zk act on k -th factor as usual spinoperators, i.e., the matrices with nonzero elements being ( S xk ) l,l − = ( S xk ) l − ,l = 12 (cid:112) l ( r k + 1 − l ) ( l = 1 , . . . r k ) , (6.19) ( S zk ) ll = r k / − l ( l = 0 , . . . r k ) . (6.20)Again, thanks to the parity operator the differential system consisting of (cid:81) k ( r k + 1) equa-tions splits into two subsystems corresponding to P = +1 and P = − , the first involvingcomponents with | a | = (cid:80) k a k even, the second with | a | odd. Note that since we assumethat at least one r k is odd, the numbers of even and odd components coincide and are equalto (cid:81) k ( r k + 1) .Let us now discuss the parameter and dimension shifting relations. Here we haveto take into account that shifting separately each parameter is not compatible with thepermutation symmetry. Fortunately, for the dimension shift at fixed α k it is sufficient tohave possibility to shift all parameters simultaneously. Therefore, we need to define theaction of R k = u k + r k − (cid:89) l = u k R l (cid:16) D, µ − (cid:80) u k + r k − j = l +1 e j (cid:17) (6.21)– 15 –n the symmetric subspace. After some transformations we obtain R k = (cid:18) i m k (cid:19) r k e iπS xk r k (cid:88) n =0 n ! (cid:18) µ k − m k S − k (cid:19) n (cid:89) l = − r k + n +1 R ( D + l ) , (6.22)where S − k = S xk − iS yk = S xk + [ S xk , S zk ] which is the matrix with nonzero elements being ( S − k ) l,l − = (cid:112) ( r k + 1 − l ) l ( l = 1 , . . . r k ) . (6.23)Let us now construct the basis of solutions. We remind that we restrict ourselves tothe case when at least one of r k is odd. We assume that r is odd. Similarly to Section 4,we want to replace in Eq. (6.14) some P with Q (0)0 or Q (1)0 defined in Eq. (4.3). Assumingthat m > (cid:80) Kk =1 r k m k , for convergence of the integral over x we keep [ P ( µ , m , x )] (cid:100) r / (cid:101) and replace all other P with Q (0)0 or Q (1)0 . Thus we have V ( ρ ρ ) = ∞ (cid:90) dx x D − [ P ( µ , m , x )] (cid:100) r / (cid:101) (cid:104) Q (0)0 ( µ , m , x ) (cid:105) (cid:98) r / (cid:99)− ρ (cid:104) Q (1)0 ( µ , m , x ) (cid:105) ρ × K (cid:89) k =1 (cid:104) Q (0)0 ( µ k , m k , x ) (cid:105) r k − ρ k (cid:104) Q (1)0 ( µ k , m k , x ) (cid:105) ρ k , (6.24)where ρ = 0 , , . . . , (cid:98) r / (cid:99) , ρ k = 0 , , . . . , r k ( k > ), and ρ = ρ , . . . ρ K . Note thatsince r is odd, we have (cid:100) r / (cid:101) = ( r + 1) / , (cid:98) r / (cid:99) = ( r − / . Then there are ( (cid:98) r / (cid:99) + 1) (cid:81) Kk =1 ( r k + 1) = (cid:81) Kk =0 ( r k + 1) functions V ( ρ ) . This is exactly the correctdimension of P -even part of solution space. If we modify similarly other components of T a from Eq. (6.18), we immediately see that we have exactly (cid:81) Kk =0 ( r k + 1) basis vectorsalso in P -odd part of solution space, which also coincides with the correct dimension of thissubspace. Note that the components other than T involve proper symmetrization withineach group of equivalent lines. For the sake of completeness we present here the result: V ( ρ ρ ) a = ∞ (cid:90) dx x D − (cid:88) b ,c (cid:0) (cid:100) r / (cid:101) c (cid:1)(cid:0) ρ b (cid:1)(cid:0) (cid:98) r / (cid:99)− ρ a − b − c (cid:1)(cid:0) r a (cid:1) / × (cid:18) P (cid:100) r / (cid:101)− c P c (cid:104) Q (0)0 (cid:105) (cid:98) r / (cid:99)− ρ − a + b + c (cid:104) Q (0)1 (cid:105) a − b − c (cid:104) Q (1)0 (cid:105) ρ − b (cid:104) Q (1)1 (cid:105) b (cid:19) ( µ , m , x ) × K (cid:89) k =1 (cid:88) b k (cid:0) ρ k b k (cid:1)(cid:0) r k − ρ k a k − b k (cid:1)(cid:0) r k a k (cid:1) / (cid:18)(cid:104) Q (0)0 (cid:105) r k − ρ k − a k + b k (cid:104) Q (0)1 (cid:105) a k − b k (cid:104) Q (1)0 (cid:105) ρ k − b k (cid:104) Q (1)1 (cid:105) b k (cid:19) ( µ k , m k , x ) . (6.25)Again, we have the quadratic relations of the form V (˜ ρ ˜ ρ ) (cid:124) ( − D, − µ ) W ( D, µ ) V ( ρ ρ ) ( D, µ ) = δ ˜ ρρ C ( ρ , ˜ ρ , ρ | D, µ ) . (6.26)If r = 1 , the function C ( ρ , ˜ ρ , ρ | D, µ ) in the right-hand side can be easily deduced fromEq. (5.14): C (0 , , ρ | D, µ ) = (cid:34) K (cid:89) k =0 (cid:18) π cos πµ k (cid:19) r k (cid:35) (cid:88) ρ =0 , π sin π (cid:16) D − (cid:80) Kk =0 µ k ρ k (cid:17) , ( r = 1) (6.27)– 16 –owever, it is not quite clear how to fix this function for r > , and we leave this questionfor further investigations. The reason why the case r = 1 is simple for us is because, whileconstructing the basis of solutions, we always assumed that one mass ( m ) is larger thanthe sum of all others. Therefore we did not have obligations to consider the questions of an-alytical continuation of the obtained solutions across (or around) the singular hypersurfacesdefined by Eq. (2.17). However, if r > , we would have to do it.Finally, we note that if we remove the analytical regularization, the P -even subsystemcontains K + 1 decoupled equations and the number of its homogeneous solutions becomes (cid:81) k ( r k + 1) − K − . D = 2 Let us now discuss some important features of the obtained results at D = µ +1 = 2 . Belowwe assume that L > . The matrix A in this case has the form A = dM M − ( N −
2) + L (cid:88) l =0 ¯ n l dm l m l (6.28)The second term is diagonal and it is clear that the equations for the components with | a | (cid:54) = 2 form a closed subsystem. On the other hand, if we pass to Y , Eq. (6.1), we obtainthe system dY = HY (6.29)with H defined in Eq. (6.3). In this form, the equations for Y a components with | a | = 1 decouple. Let us consider the linear transformation of column of functions T a which replaces L +1 entries T ... , T ... , . . . , T ... , T ... , and T ... with Y ... , Y ... , . . . , Y ... .It is easy to check that this transformation is non-degenerate. With the new column of func-tions ˜ T the differential system acquires the block-triangular form depicted in Fig. 3. Letus observe that the component T enters the closed system of L − ( L + 1) L/ equationsdetermined by the block B . Therefore, in a certain sense, we find that the number ofmaster integrals at D = 2 is equal to L − ( L + 1) L/ . This is in agreement with knowncases L (cid:54) , [14, 15].Let us comment about the number of master integrals at D = 2 when there are groupsof identical masses. In the notations of previous subsection, we obtain that the number ofmaster integrals is equal to (cid:81) k ( r k + 1) − ( K + 1)( K + 2) / (cid:80) k δ r k , . In particular, for L -loop sunrise graph with equal masses we have L + 2 − − L master integrals. Basis of solutions.
Constructing the basis of solution space at D = 2 appears to beextremely tricky and deserves a dedicated consideration elsewhere. Here we will only explainthe complications that arise on the way.First, one might be tempted to pass to the basis of functions V [ υ ] defined as V [ υ ] = (cid:88) ρ L (cid:89) l =1 (cid:18) − π sin π(cid:15) (cid:19) υ l (1 − ¯ ρ l υ l ) V ( ρ ) , (6.30)– 17 – igure 3 . Block-triangular structure of the matrix in the right-hand side of the differential systemfor ˜ T . The block B is a diagonal matrix corresponding to L + 1 entries of the form Y ... ... . Theblock B corresponds to L − ( L + 1) L/ entries T a ( | a | even and | a | (cid:54) = 2) . The block B is adiagonal matrix corresponding to ( L + 1)( L − / remaining entries of T a with | a | = 2 . The block B corresponds to L entries T a with odd | a | . where υ = υ υ . . . υ L is a binary number. The functions of this basis have the followinglimit of a = component: V [ υ ] (cid:15) → −→ L (cid:89) l =0 (2 m l ) ∞ (cid:90) dx xK ( m x ) L (cid:89) l =1 (cid:40) I ( m l x ) if υ l = 1 ,K ( m l x ) if υ l = 0 . (6.31)However, other components of V [ υ ] diverge in the limit (cid:15) → unless υ = 1 . . . because fothe singular asymptotics of K function at small argument.In order to get rid of the divergent overall factor in Eq. (4.2) at µ → , let us definethe functions U ( ρ ) = ( − | ρ | (cid:18) π(cid:15)π (cid:19) L +1 V ( ρ ) (cid:12)(cid:12)(cid:12) D =2 − (cid:15)µ l =1 − (cid:15) . (6.32)Then if we naively take limit (cid:15) → under the integral sign, we can come to a wrongconclusion that all U ( ρ ) tend to one and the same expression independent of ρ : U ( ρ ) (cid:15) → −→ L (cid:89) l =0 (2 m l ) ∞ (cid:90) dx x (cid:32) K ( m x ) − K ( m x ) (cid:33) ⊗ L (cid:79) l =1 (cid:32) I ( m l x ) I ( m l x ) (cid:33) . (6.33)However, Eq. (6.33) is correct only for U ( ) . The reason why it breaks down for ρ (cid:54) = is that the expansion of I − (cid:15) ( mx ) in x starts from the singular term Γ[ (cid:15) ] − ( mx/ (cid:15) − .Although this term is suppressed in (cid:15) , it may survive after the integration over x due to itssingular nature. More precisely, consider the identities (cid:90) dx K − (cid:15) ( x ) x α − = 2 α Γ (cid:2) α + (cid:15) , α − (cid:15) (cid:3) , (cid:90) dx K − (cid:15) ( x ) x α = 2 α Γ (cid:2) α + (cid:15) , α − (cid:15) + 1 (cid:3) . (6.34)– 18 –hey show that when α is close to − n ( n = 0 , , . . . ), the integral over x gives poles in (cid:15) .A thorough investigation allows us to establish that nonzero corrections to the naive limit(6.33) appear only in the components with | a | = 2 . Moreover, we can explicitly calculatethese corrections using Eqs. (6.34): (cid:104) U ( ρ ) − U ( ) (cid:105) a (cid:15) → −→ δ | a | , L (cid:89) l =0 (2 m l ) (cid:88) b,c, (cid:54) b
We thank Mikhail Kalmykov for many useful discussions and David Broadhurst for valuablecomments on the manuscript. R. Lee would like to express special thanks to the MainzInstitute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (ProjectID 39083149) for its hospitality and support. R. Lee is supported by the grant of the “Basis”foundation for theoretical physics.
A Compatibility conditions
Let us examine the compatibility conditions of Eqs. (2.19), (3.3), and (3.6). First, it iseasy to see that dA = 0 , thanks to the identity [ σ l , σ k ] = 0 . Then we have to check that [ A l , A k ] = 0 , or equivalently, A ∧ A = 0 . We have A ∧ A = M − dM W M − ∧ dM W + (cid:88) k (cid:2) µ k n k , M − dM (cid:3) W ∧ dm k m k = M − dM ∧ [ W, dM ] M − W − M − dM (cid:88) k [ µ k n k , M ] M − W ∧ dm k m k = M − dM ∧ (cid:88) k µ k dm k [ n k , σ k ] M − W − M − dM (cid:88) k µ k [ n k , σ k ] M − W ∧ dm k = 0 . (A.1)– 22 –ther conditions have the form dR l ( µ ) + R l ( µ ) A ( µ ) = A ( µ − e l ) R l ( µ ) , (A.2) R j ( µ − e l ) R l ( µ ) − R l ( µ − e j ) R j ( µ ) = 0 , (A.3) dR ( D ) + R ( D ) A ( D ) = A ( D + 1) R ( D ) , (A.4) R l ( D + 1 , µ ) R ( D, µ ) = R ( D, µ − e l ) R l ( D, µ ) . (A.5)These identities can be checked along the same lines as Eq. (A.1).Let us also write down two useful identities which allow one to check the symmetryproperty (5.9) of the matrix B in the right-hand side of the quadratic relation (5.7). Theyread W ( D ) R ( ˜ D ) = R (cid:124) ( D ) W ( ˜ D ) , (A.6) W ( D, µ − e l ) R l ( D, µ ) = − R (cid:124) l ( − D, − µ + 2 e l ) W ( D, µ ) , (A.7) W ( D − , µ − e ) R ( D, µ ) = ( − L +1 R (cid:124) (2 − D, e − µ ) W ( D, µ ) . (A.8) B Expansion near D = 1 We will consider (cid:15) -expansion of Y for D = 1 − (cid:15) . Then the (cid:15) -expansion of T can beobtained with the help of Eq. (6.1). Let us rescale m k → m k x ( k = 1 , . . . , L ) and considerthe differential equation with respect to x : dYdx = (cid:15) (cid:88) a ∈ Λ H a x − a Y , (B.1)where the set (or alphabet) Λ contains and all letters of the form a ( η ) = m (cid:80) Lk =1 η k m k , (B.2)where L -tuple { η , . . . , η L } has elements ± , and H = 2 ( N − L − n ) H a ( η ) = − N − L (cid:89) k =1 − η k σ σ k (B.3)Then we choose the fundamental matrix of solutions of Eq. (B.1) as ˜ U ( (cid:15), m , x ) = lim x → Pexp (cid:15) x (cid:90) x (cid:88) a ∈ Λ H a y − a dy x (cid:15)H . (B.4)This definition corresponds to the small- x asymptotics ˜ U ( (cid:15), m , x ) ∼ x (cid:15)H . (B.5)– 23 –he (cid:15) -expansion of the fundamental matrix has the form ˜ U ( (cid:15), m , x ) = ∞ (cid:88) n =0 (cid:15) n (cid:88) a ...a n H a . . . H a n G ( a , . . . , a n | x ) , (B.6)where the sum (cid:80) a ...a n runs over all words of length n with letters from the alphabet Λ ,and G ( a , . . . , a n | x ) denotes the Goncharov’s polylogarithms [1, 2] defined recursively by G ( a , a , . . . , a n | x ) = x (cid:90) dyy − a G ( a , . . . , a n | y ) , (B.7) G (0 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n | x ) = 1 n ! ln n x . (B.8)Let us represent the specific solution Y a ρ as Y a ρ = ˜ U ( (cid:15), m , C ( (cid:37) , ρ ) a ρ (B.9)where C ( (cid:37) , ρ ) a ρ are the integration constants. In order to find C ( (cid:37) , ρ ) a ρ , we consider the asymp-totics x → . Using Eqs. (5.11) and (6.7) we have C ( (cid:37) , ρ ) a ρ = lim x → x − (cid:15)H Y ( (cid:37) , ρ ) a ρ ( m , xm k ) = 2 − (cid:15) δ a (cid:37) ( − ¯ a × Γ [1 − (¯ a − | ρ | ) (cid:15) ] Γ (cid:2) − ( a − | ρ | ) (cid:15) (cid:3) L (cid:89) k =1 ( − ρ k Γ (cid:2) + ( − ρ k (cid:15) (cid:3) m (cid:15)ρ k m (cid:15) ¯ ρ k k . (B.10) C Examples of quadratic relations for
IKM functions.
Let us present a few examples of the quadratic relations for
IKM functions related to the cutsunrise integral. For L -loop sunrise integral the overall number of Bessel and Macdonaldfunctions in the integrand is equal to L + 2 . We use the IKM functions as defined in Eq.(6.44).For L = 2 we obtain the following nontrivial quadratic relation: IKM [ { , } m , { , } ,
1] IKM [ { , } m , { , } , − IKM [ { , } m , { , } ,
3] IKM [ { , } m , { , } , (cid:0) − m (cid:1) (1 − m ) (1 − m ) . (C.1)For L (cid:62) we obtain similar relations which are too lengthy to be presented here. Theserelations, however, simplify on the pseudo-thresholds.For L = 3 we have (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) − (cid:16) { , } , { , } , (cid:17) = 20 , (C.2)– 24 – (cid:16) { , } , { , } , (cid:17) − (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) = 20 π , (C.3) − (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) + 9IKM (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) + 9IKM (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) = 0 , (C.4)For L = 4 we have a, { , } ,
3) IKM ( b, { , } , − a, { , } ,
1) IKM ( b, { , } , − a, { , } ,
5) IKM ( b, { , } ,
1) + 12IKM ( a, { , } ,
1) IKM ( b, { , } , C ( a | b ) (C.5)with C (cid:16) { , } |{ , } (cid:17) = 55468759216 , C (cid:16) { , } |{ , } (cid:17) = 0 , (C.6) C (cid:16) { , } |{ , } (cid:17) = 5546875 π . (C.7)At m = 1 / we have one relation (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) − (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) − (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) + 4IKM (cid:16) { , } , { , } , (cid:17) IKM (cid:16) { , } , { , } , (cid:17) = 9963 π . (C.8)We note that we could have easily proceeded to more loops. References [1] A. B. Goncharov,
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