aa r X i v : . [ h e p - ph ] F e b Differential equations and Feynman integrals
A. V. KotikovBogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research141980 Dubna, Russia
Abstract
The role of differential equations in the process of calculating Feynman integralsis shown. An example of a diagram is given, in the calculation of which the methodof differential equations was introduced, the properties of the inverse-mass-expansioncoefficients are shown, and modern methods based on differential equations are con-sidered.
The calculation of the Feynman integrals (FIs) provides basic information both for thematrix elements of the experimentally studied processes and for the characteristics of thephysical models themselves (their renormalization, critical behavior, etc.). When studyingrenormalization and critical behavior, it is usually sufficient to restrict oneself to the limitof massless particles at which the corresponding two-point FIs are fairly simple. However,starting at 2 or 3 loop levels, there is a need to use modern methods such as Integration byParts (IBP) [1] and Gegenbauer’s polynomial method [2] (see also the recent paper [3] andreviews in [4] and [5]). Calculating massive FIs (i.e. FIs having propagators of massive particles) is a muchmore complex problem. Simple results (in the form of a product of Γ-functions) exist onlyfor simple tadpoles (see eq. (10) below), and a massive one-loop loop is already given by aone-fold integral (see eq. (15) below).It turned out, however, that massive FIs satisfy IBP procedures [1], which lead to rela-tions between FIs equivalent to the original ones, but with different powers of the propa-gators included in them, including powers equal to zero. Diagrams containing propagatorswith degrees equal to zero are equivalent to simpler diagrams obtained by canceling thesepropagators (and reducing the points they join to one point).Such relationships can be understood in two ways. First, considering them algebraically,one can understand them as connections between diagrams that are not independent andcan be reduced to a certain set of independent diagrams, which are called masters (or masterintegrals) [7].Second, propagators with powers greater than 1 can be considered as derivatives (withrespect to the corresponding mass or external momentum) from the propagator with a de-gree of 1. Thus, the relationships between the master integrals can be considered differentialequations (DEs) for these masters (see an example in Section 2), containing inhomogeneousterms, including only simpler diagrams (i.e., obtained from the original diagrams by reducing Note that multi point massless FIs are as complex as massive FIs (see, for example, Ref. [6] about therelationship between 2-point massive FIs and 3-point massless FIs). More complicated diagrams can be obtained from these tadpolesby solving successively obtained DEs with some boundary conditions. For (dimensionallyregularized) massive FIs, a good boundary condition is the limit of infinitely large mass, atwhich these diagrams usually vanish.In Section 2 we will consider a two-loop FI, the calculation of which leds to the use ofdifferential equations. The calculation of massive diagrams is given in Section 3: rules aregiven for their efficient calculation, examples of two- and three-point diagrams are considered.The recurrence relations for the coefficients of decomposition in inverse mass are considered.In Section 4 a short review of modern computing technology is given. The appendix Acontains the derivation of DEs for massive diagrams from the inhomogeneous term of theDE for the diagram considered in Section 2.
As mentioned in the introduction, integral representations for one-loop FIs (obtained, forexample, using the Feynman parameter method [9]) are hypergeometric functions and,thus, can be represented as solutions of some DEs. The importance of DEs for FIs wasrecognized long ago (see, for example, [11, 12]). However, in my opinion, the practicalapplication awaited the emergence of IBP procedure [1] for FIs and is based on the use ofIBP relations (see eqs. (16) and (17) below).IBP-based DEs appeared in the nineties in several works, studying FIs: massive two-pointones in [13, 14], massive three-point ones in [13], four-point ones in [16] and ˆ n -point ones[17, 18] (a short short overview made at that time can be found in the issue [19] dedicated tothe 70th anniversary of Academician O.S. Parasyuk, co-author of the BPHZ renormalizationprocedure [20] (see, for example, the book [21]). In the case of Refs. [13, 25], the starting point was to study a preprint of the excellentyet unpublished work [7] on the calculation of two-loop massive FIs. Despite the excellentresults, the paper itself (i.e. the conclusions of these results) turned out to be quite difficultto understand.I decided to reproduce these results using the IBP relations, which proved to be very Sometimes it is convenient to stop the considered procedure on one-loop massive FI, the calculation ofwhich can be performed, for example, by the method of Feynman parameters (see, for example, [9]. Investigations of hypergeometric functions related to the calculation of FI are recently presented [10] ascontribution to the Proceedings. The results for massless diagrams are sometimes useful to obtain in x -space (see [22, 23]). It is convenientto do this in x -space for so-called dual diagrams (see, for example, [23, 24]). A dual diagram is obtainedfrom the initial one by replacement of all p by x with the rules of correspondence between the graph andthe integral, as in a x -space. For massive two-pint and three-point diagrams the study in dual x -space donein Refs. [25] and [26], respectively. I ( q , m ) = → q , (1)having the vertical massive propagator (see eq. (9) for definitions). The diagram has left-right and top-bottom symmetries.Applying IBP relations (16) to the left triangle of the diagram I ( q , m ) in successionwith vertical and lateral distinguished lines, we get( d − I ( q , m ) = 2 " → q − → q − m → q − m → q , (2)( d − I ( q , m ) = 2 → q − q → q − m → q . (3)Taking the combination of these equations: Eq. (2) - 2( m /q ) × Eq.(3), we have( d − (cid:18) − m q (cid:19) I ( q , m ) = 2 J ( q , m ) − m (cid:18) − m q (cid:19) → q , (4)3here J ( q , m ) = 2 → q − → q − m q → q . (5)Because 1( q + m ) = − ddm q + m ) (6)the Eq. (5) can be rewritten in the form (cid:20) ( d − (cid:18) − m q (cid:19) − m (cid:18) − m q (cid:19) ddm (cid:21) I ( q , m ) = 2 J ( q , m ) , (7)i.e. the first order DE for the original diagram with the inhomogeneous term J ( q , m )containing only simpler diagrams (i.e. obtained from the original by canceling one of thepropagators) (see eq. (5)).The first diagram in the inhomogeneous term J ( q , m ) is independent of mass andtherefore can be easily calculated as a product of the Γ-functions (see eq. (11) below):2 → q = L , ( q ) L , ( q ) = 1(4 π ) d A (2 , A (1 , q − d ) . (8)where A ( α , α ) is given in eq. (13) below.Using IBP relations, for the remaining two diagrams in the inhomogeneous J ( q , m )term diagrams, one can obtain similar equations with inhomogeneous terms containing onlyeven simpler diagrams (i.e., obtained from the original by canceling two propagators). Theseresults can be found in the Appendix A. Let us briefly consider the rules for calculating diagrams having the massive propagators. Massless propagator and the propagator with mass m will be represented as1 q α = α → q , q + m ) α = mα → q , (9)where the symbol m will be omitted in the one-mass case (as in the case of I ( q , m ) in eq.(1)). Hereafter we consider only the first order DE. The consideration of the high order DEs can be found insection 7 of the review [32]. . The massive one-loop tadpole T α ,α ( m ) and the massless loop L α ,α ( q ) can becalculated exactly as combinations of the Γ-functions: T α ,α ( m ) = Z Dkk α ( k + m ) α = α α = R ( α , α ) m α + α − d/ , (10) L α ,α ( q ) = Z Dk ( q − k ) α k α = α α → q = A ( α , α ) q α + α − d/ , (11)where A ( α , α ) = a ( α ) a ( α ) a ( α + α − d/ , a ( α ) = Γ( ˜ α )Γ( α ) , ˜ α = d − α , (12) R ( α , α ) = Γ( d/ − α )Γ( α + α − d/ d/ α ) (13)and Dk = d d kπ d/ = (4 π ) d/ D E k, D E k = d d k (2 π ) d . (14)Here D E k is the usual Euclidean measure in in ( d = 4 − ε ) space. A simple loop of two massive propagators with masses m and m can be representedas hypergeometric function, which can be calculated in a general form, for example, byFeynman-parameter method (see [9]). It is very convenient, using this approach to representthe loop as an integral of a propagator with the “effective mass” µ [13, 33, 34, 35, 36, 37, 38]: Z Dk [( q − k ) + m ] α [ k + m ] α = Γ( α + α − d/ α )Γ( α ) × Z ds s α − (1 − s ) α − [ s (1 − s ) q + m s + m (1 − s )] α + α − d/ = Γ( α + α − d/ α )Γ( α ) × Z dss − ˜ α (1 − s ) − ˜ α q + µ ] α + α − d/ , (cid:18) µ = m − s + m s (cid:19) . It is useful to rewrite the equation graphically as m m α α → q = Γ( α + α − d/ α )Γ( α ) Z dss − ˜ α (1 − s ) − ˜ α µ α + α − d/ → q . (15)The rule is very convenient in the cases m = 0 and m = m , where the variable µ isequal to µ = m /s and µ = m /s (1 − s ), respectively. Such simple forms of µ gives a5ossibility to use directly an inverse-mass expansion without a usage of Mellin-Barnes rep-resentation, which is essentially more complicated procedure. For any triangle with indices α i ( i = 1 , ,
3) and masses m i there is the followingrelation, which is based on integration by parts (IBP) procedure [1, 13, 15]( d − α − α − α ) m m m α α α → q − q → q − q → q − q = α (cid:20) m m m α +1 α − α → q − q → q − q → q − q − (cid:20) ( q − q ) + m + m (cid:21) × m m m α +1 α α → q − q → q − q → q − q + α (cid:20) α ↔ α , m ↔ m (cid:21) − m α × m m m α α +1 α → q − q → q − q → q − q . (16)Eq. (16) can been obtained by introducing the factor ( ∂/∂k µ ) ( k − q ) µ to the subintegralexpression of the triangle, shown below as [ ... ], and using the integration by parts procedureas follows: d Z Dk (cid:2) ... (cid:3) = Z Dk (cid:18) ∂∂k µ ( k − q ) µ (cid:19) (cid:2) ... (cid:3) = Z Dk ∂∂k µ (cid:0) ( k − q ) µ (cid:2) ... (cid:3)(cid:1) − Z Dk ( k − q ) µ ∂∂k µ (cid:0)(cid:2) ... (cid:3)(cid:1) (17)The first term in the r.h.s. becomes to be zero because it can be represented as a surfaceintegral on the infinite surface. Evaluating the second term in the r.h.s. we reproduce Eq.(16). Note that the equation (17) can also be applied to the ˆ n -point subgraph (see, forexample, [17]).As it is possible to see from Eqs. (16) and (17) the line with the index α is distinguished.The contributions of the other lines are same. So, we will call below the line with the index α as a “distinguished line”. It is clear that a various choices of the distinguished line producedifferent types of the IBP relations. Below in the present analysis we will concentrate mostly on two-loop two-point and three-point diagrams, which can be taken from the diagram shown in Fig. 1. We will call them6 I ( q, m , ..., m ) = m m m m m → q , ˆ P ( q, m , ..., m ) = m m m m m m → q → q → q Figure 1: Two-loop two-point diagram ˆ I ( q, m , ..., m ) and three-point diagramˆ P ( q, m , ..., m ) with q = q = 0.as: ˆ I j = ˆ I ( q, m j = m = 0 , m p = 0 , p = j ) , ˆ I ij = ˆ I ( q, m i = m j = m = 0 , m p = 0 , p = i = j ) , ˆ I ijs = ˆ I ( q, m i = m j = m s = m = 0 , m p = 0 , p = i = j = s ) , ˆ I ijst = ˆ I ( q, m i = m j = m s = m t = m = 0 , M p = 0 , p = i = j = s = t ) , (18)ˆ P j = ˆ P ( q, m j = m = 0 , m p = 0 , p = j ) , ˆ P ij = ˆ P ( q, m i = m j = m = 0 , m p = 0 , p = i = j ) , ˆ P ijs = ˆ P ( q, m i = m j = m s = m = 0 , m p = 0 , p = i = j = s ) , ˆ P ijst = ˆ P ( q, m i = m j = m s = m t = m = 0 , m p = 0 , p = i = j = s = t ) . (19)Now we repeat once again the procedure of the DE method. Application of the IBPprocedure [1] to loop internal momenta leads to relations between various FIs and, therefore,to the necessity of calculating only some of them, which in a sense are independent. Theseindependent diagrams (which were chosen completely arbitrarily, of course) are called masterintegrals [7].Applying the IBP procedure [1] to the master-integrals themselves leads to DEs [13, 25]for them with the inhomogeneous terms containing less complex diagrams. Applying theIBP procedure to diagrams in inhomogeneous terms leads to new DEs for them with newinhomogeneous terms containing even more less complex diagrams ( ≡ less complex ones).By repeating the procedure several times, in the last step we can obtain inhomogeneousterms containing mainly tadpoles, which can be easily calculated in-turn.Solving the corresponding DEs in this last step, diagrams for the inhomogeneous termsof the DEs in the previous step can be reproduced. Repeating the procedure several times,me can get the results for the original Feynman diagram.Thus, the DE method procedure is well defined, but it requires a lot of manual work anda lot of time. So, the calculations [34] of each of the diagrams P and P took about amonth of work (of course, along with checking the results). It would be nice, however, totransfer some of the work to the computer. The first attempt based on the properties of theinverse mass expansion coefficients of the master integrals is presented in the next Section.A more modern and efficient technique is discussed in Section 5. Calculations of the two-point diagrams shown in Fig. 1, which do not contain elliptic struc-tures (see Fig. 2 in Ref. [35]), as well as calculations of some three-point diagrams shown In fact, the results for these two-point diagrams were found in the late eighties and early nineties, andwere planned to be published in a long paper summarizing the results done in Refs. [13, 15]. However, thispaper has not been published. These results, after verification, were published in Ref. [35].
7n Fig. 1 (see also Fig. 3 in Ref. [35]) lead to results with interesting properties of theirinverse mass expansion coefficients.
The inverse-mass expansion of two-loop two-point and three-point diagrams with onenonzero mass (massless and massive propagators are shown by dashed and solid lines, re-spectively), can be considered asFI = ˆ Nq α X n =1 C n ( ηx ) n (cid:26) F ( n ) + (cid:20) ln x F , ( n ) + 1 ε F , ( n ) (cid:21) (20)+ (cid:20) ln x F , ( n ) + 1 ε ln x F , ( n ) + 1 ε F , ( n ) + ζ (2) F , ( n ) (cid:21) + · · · (cid:27) , where x = q /m , η = 1 or − α = 1 and 2 for two-point and three-point cases,respectively. The normalization ˆ N = ( µ /m ) ε , where µ = 4 πe − γ E µ is in the standard M S -scheme and γ E is the Euler constant. Moreover, C n = ( n !) (2 n )! ≡ ˆ C n (21)for diagrams with two-massive-particle-cuts (2 m -cuts). For the diagrams with one-massive-particle-cuts ( m -cuts) C n = 1.For m -cut case, the coefficients F N,k ( n ) should have the form F N,k ( n ) ∼ S ± a,... n b , ζ ( ± a ) n b , (22)where S ± a,... ≡ S ± a,... ( j −
1) are nested sums [39]: S ± a ( j ) = j X m =1 ( − m m a , S ± a, ± b,... ( j ) = j X m =1 ( − m m a S ± b,... ( m ) , (23)and ζ ( ± a ) = S ± a ( ∞ ) and ζ ( ± a, , ± b, ... ) = S ± a, ± b,... ( ∞ ) are the Euler-Zagier constants.For 2 m -cut case, the coefficients F N,k ( n ) can be more complicated F N,k ( n ) ∼ S ± a,... n b , V a,... n b , W a,... n b , (24)where W ± a,... ≡ W ± a,... ( j −
1) and V ± a,... ≡ V ± a,... ( j −
1) with [35] W a ( j ) = j X m =1 ˆ C − m m a , W a,b,c, ··· ( j ) = j X m =1 ˆ C − m m a S b,c, ··· ( m ) , (25) V a ( j ) = j X m =1 ˆ C m m a , V a,b,c, ··· ( j ) = j X m =1 ˆ C m m a S b,c, ··· ( m ) , (26) The diagrams are complicated two-loop FIs that do not have cuts of three massive particles. thus,their results should be expressed as combinations of Polylogarithms. Note that we consider only three-pointdiagrams with independent upward momenta q and q , which satisfy the conditions q = q = 0 and( q + q ) ≡ q = 0, where q is downward momentum. in our previous papers [23, 24, 34, 35] the nested sums K a,b,... ( j ) = P jm =1 ( − m +1 m a S b,... ( m ) = − S − a,b,... ( j ) have been used together with their analytic continuations [24, 40]. ∼ V a,... and ∼ W a,... can come only in the 2 m -cut case. The origin of theappearance of these terms is the product of series (20) with the different coefficients C n = 1and C n = ˆ C n . As an example, consider two-loop two-point diagrams ˆ I and ˆ I studied in [35]ˆ I = → q , ˆ I = → q (27)where ˆ I coincides with I ( q , m ) considered in Section 2.Their results areˆ I = ˆ Nq X n =1 x n n (cid:26) ln x − n ln x + 2 ζ (2) + 4 S − + 2 2 n + 2( − ) n n (cid:27) , (28)ˆ I = − ˆ Nq X n =1 ( − x ) n n (cid:26) n + ˆ C n (cid:18) − x − W + 2 n (cid:19)(cid:27) . (29)From (28) one can see that the corresponding functions F N,k ( n ) have the form F N,k ( n ) ∼ n − N , ( N ≥ , (30)if we introduce the following complexity of the sums (Φ = ( S, V, W ))Φ ± a ∼ Φ ± a , ± a ∼ Φ ± a , ± a , ··· , ± a m ∼ ζ a ∼ n a , ( m X i =1 a i = a ) . (31)The number 3 − N determines the level of transcendentality (or complexity, or weight) ofthe coefficients F N,k ( n ). The property greatly reduces the number of the possible elementsin F N,k ( n ). The level of transcendentality decreases if we consider the singular parts ofdiagrams and/or coefficients in front of ζ -functions and of logarithm powers. Thus, findingthe parts we can predict the rest using the ansatz based on the results already obtained, butcontaining elements with a higher level of transcendentality.Other two-loop two-point integrals in [35] have similar form. They were exactly calculatedby DE method [13, 25]. Their representations in the form of Nielsen Polylogarithms [41] canbe found also in Ref. [35]. Now we consider two-loop three-point diagrams, ˆ P and ˆ P :ˆ P = → q → q → q , ˆ P = → q → q → q . P = ˆ N ( q ) X n =1 x n n (cid:26) − ζ + 2 S ζ + 6 S − S S + 4 S n − S n + 2 S n + (cid:18) − S + S − S n (cid:19) ln x + S ln x (cid:27) , (32)ˆ P = ˆ N ( q ) X n =1 ( − x ) n n ˆ C n (cid:26) ε + 2 ε (cid:18) S − W + 1 n − ln x (cid:19) − W − W , − S + S − S W + 2 S n + 2 n − (cid:18) S + 1 n (cid:19) ln x + ln x (cid:27) , (33)Now the coefficients F N,k ( n ) have the form F N,k ( n ) ∼ n − N , ( N ≥ , (34)The diagram P (and also P , P , P and P in [35]) was calculated exactly by differentialequation method [13, 25]. To find the results for P (and also all others in [35]) we haveused the knowledge of the several n terms in the inverse-mass expansion (20) (usually lessthan n = 100) and the following arguments: • If a two-loop two-point diagram with a “similar topology” (for example, I for P ,etc.) was already calculated, we should consider a similar set of basic elements forcorresponding F N,k ( n ) of two-loop three-point diagrams but with a higher level ofcomplexity. • Let the diagram under consideration contain singularities and/or powers of logarithms.Since the coefficients are very simple before the leading singularity, or the largest degreeof the logarithm, or the largest ζ -function, they can often be predicted directly fromthe first few terms of the expansion.Moreover, often we can calculate the singular part using a different technique (see [35]for extraction of ∼ W ( n ) part). Then we should expand the singular parts, find themain elements and try to use them (with the corresponding increase in the level ofcomplexity) in order to predict the regular part of the diagram. If we need to find ε -suppressed terms, we should increase the level of complexity of the correspondingbasic elements.Later, using the ansatz for F N,k ( n ) and several terms (usually less than 100) in the aboveexpression, which can be exactly calculated, we obtain a system of algebraic equations forthe parameters of the ansatz. Solving the system, we can obtain the analytical results forFIs without exact calculations. To check the results, we only need to calculate a few moreterms in the above inverse-mass expansion (20) and compare them with the predictions ofour anzatz with the fixed coefficients indicated above.Thus, the considered arguments give a possibility to find results for many complicatedtwo-loop three-point diagrams without direct calculations. Several process options have been The evaluation of inverse mass expansion coefficients is demonstrated in Ref. [36]. b = 0 in (22) was found for the eigen-values of anomalous dimensions [43] and coefficient functions [44], as well as in the next-to-leading corrections [45] to the BFKL equation [46] for N = 4 Super Yang-Mils (SYM)model. Such a strong restriction made it possible to obtain anomalous dimensions in thefirst three orders of the perturbation theory directly from the corresponding results for QCD(the ”most complicated” parts are the same in N = 4 SYM and QCD) [47, 48], as well as inthe 4th, 5th, 6th and 7th orders (see [49], [50], [51] and [52], respectively) in the algebraicBethe ansatz [53].Note that the series (28), (29) and (32) can be expressed as a combination of the Nilson[41] and Remiddi-Vermaseren [54] polylogarithms with weight 4 − N (see [35, 34]). Morecomplicated cases were examined in [55]. Coefficients of the inverse-mass-series expansions of the two-point and three-point FIs havethe structure (30) and (34) with the rule (31). Note that these conditions greatly reducethe number of possible harmonic sums. In turn, the restriction is associated with a DEspecific form for the considered FIs. The DEs can be formally represented as [56, 57] (seethe example I ( q , m ) considered in section 2) (cid:18) ( x + a ) ddx − k ( x ) ε (cid:19) FI = less complicated diagrams( ≡ FI ) , (35)with some number a and some function k ( x ). This form is generated by IBP procedure fordiagrams including an inner ˆ n -leg one-loop subgraph, which in turn contains the product k µ ...k µ m of its internal momenta k with m = ˆ n − α i = 1 + a i ε with arbitrary a i of subgraph propagators, theIBP relation (16) gives the coefficient d − α − P pi =2 α i + m ∼ ε for m = n −
3. Importantexamples of applying the rule are the diagrams ˆ I , ˆ I and ˆ P , ˆ P (for the case ˆ n = 2 andˆ n = 3) and also the diagrams in Ref. [58] (for the case ˆ n = 3 and ˆ n = 4). However, wenote that the results for the non-planar diagrams (see Fig. 3 of [35]) obey the Eq. (34) buttheir subgraphs do not comply with the above rule. The disagreements may be related tothe on-shall vertex of the subgraph, but this requires additional research.Taking the set of less complicated Feynman integrals FI as diagrams having internalˆ n -leg subgraphs, we get their result structure similar to the one given above (34), but witha lower level of complexity.So, the integrals FI should obey to the following equation (see J (1)2 ( q , m ) in AppendixA) (cid:18) ( x + a ) ddx − k ( x ) ε (cid:19) FI = less complicated diagrams( ≡ FI ) . (36)Thus, we will have the set of equations for all Feynman integrals FI n as (cid:18) ( x + a n ) ddx − k n ( x ) ε (cid:19) FI n = less n +1 complicated diagrams( ≡ FI n+1 ) , (37)11ith the last integral FI n+1 contains only tadpoles. Note that for the case n = 2 thediagrams corresponding for the example I ( q , m ), satisfy the system of equations, formallyrepresented as eq. (37). In the last decade, several popular applications of DEs have emerged, allowing the use ofcomputer resources and, thus, to obtain results for very complicated FIs.In my opinion, the most successfully used are the so-called the canonical form [59] ofDEs (and its generalizations in Refs. [60, 61]) , the method [62] of simplified DEs, and theability to use the effective mass (see eq. (15)), as well as their combinations. DEs are alsoeffectively used in calculating FIs with an elliptical structure (see [63]).
In our notation (see eqs. (35) - (37)), the canonical form [59], which was introduced byJohannes Henn in 2013 and and wildly popular now (a huge number of publications, whichsimply cannot be listed), represents a homogeneous matrix equation of the form (see alsothe review [64]) ddx c F I − ε b K ( x ) c F I = 0 , (38)for the vector c F I = FIFI /ε... FI n /ε n , where the matrix b K contains the functions k j / ( x + a j ) as its elements. The form (38) iscalled as the “canonic basic”.Note that obtaining it is far from trivial (see, for example, Appendix A for FI n=2 di-agrams). Moreover, it is not always achievable (see [60, 61]), where FIs were consideredthat are not reducible to (38)), and its obtaining is sometimes associated with a nontrivialanalysis (see Refs. [65] and [66] containing methods and criterion to obtain the equation,respectively). However, the form of (38) is very convenient as it can be easily diagonalized.Note that formally for real calculations of FI n it is convenient to replaceFI n = e FI n FI n , where the term FI n obeys the corresponding homogeneous equation (cid:18) ( x + a n ) ddx − k n ( x ) ε (cid:19) FI n = 0 , (39)The replacement simplifies the above equation (37) to the following form( x + a n ) ddx e FI n = e FI n +1 FI n +1 FI n , (40)12aving the solution e FI n ( x ) = Z x dx x + a n e FI n +1 ( x ) FI n +1 ( x )FI n ( x ) (41)Usually there are some cancellations in the ratio FI n +1 / FI n and sometimes it is equal to1. In the last case, the equation (41) coincides with the definition of Goncharov Polylogariths[67] (see also the review [68] and the references therein).Sometimes the integrand in (41) can have a quadratic form in the denominator, for ex-ample, x ± x + 1 (sign ± can change, including when passing from the Euclidean metricto the Minkowski metric). Such forms appeared in two-point FIs, ˆ I , ˆ I and ˆ I and canbe represented as Nilson three-logarithm with complicated argument, i.e. Li ( − y ), where y = ( √ x + 4 − x ) / ( √ x + 4 + x ) is so-called conformal variable, as well as in the transformin [69] of H ( − r, ... ) functions, introduced in [70], to the Remiddi-Vermaseren polylopagitms[54] of variable ∼ y where one integral representation contains the factor x ± x + 1 in thedenominator and is thus left in this form. We note that such terms come also in contributionsof the massive form factors at 3-loop order [71]. Later, the study of such integral representa-tions leads to the discovery of cyclotomic Polylogarithms (see [72] and Ref. [32] for a review). Here we will consider other methods that can be connected both with each other and withthe canonical form and its generalizations (unfortunately, we cannot pretend here to becomplete in listing all the approaches). The simplified DE approach [62] is based on violation of momentum conservationby the parameter x , with some propagator. Using the IBP relations, we can obtain set ofequations which depend on x . We can solve it with the boundary conditions at x = 0 andtake the limit x →
1. The equations in this approach are usually representable in canonicalform, which leads to very important results (see [73]). Series expansions in singular and regular fixed points [74] (see also Ref. [75] anddiscussion therein) for DE systems, which generate eq. (38), for example, as ε b K ( x ) → b K ( x ) + ε b K ( x ) . (42)The results are obtained in the form of Goncharov Polylogarithms [67] and, in some compli-cated cases, numerically. Symmetries of FIs is a general method introduced in [76] which associates withany given Feynman diagram a system of partial DEs. The method uses the same variationswhich are used in the DE method [13] and IBP technique [1], but distinguishes itself byassociating with any diagram a natural Lie group which acts on the diagram’s parameterspace. This approach was further developed and numerous diagrams have been analyzedwithin it (see the recent paper [77] and discussions and references therein). Using the effective mass (15) reduces the number of loops in the considered diagram.In the cases under consideration, two-loop diagrams were reduced to one-loop ones. Then,one-loop diagrams were easily calculated using the DE method, and the required two-loopdiagrams were presented as integrals of the obtained one-loop results (see Ref. [37]).13 .3 Elliptic structure
Recently, the scientific community has centered its attention to the study of FIs whosegeometric properties are defined by elliptic curves. We already have a lot of progress inunderstanding simplest functions beyond usual Polylogarithms, the so-called elliptic Poly-logarithms (see the recent papers [63, 78, 79, 80] and references and discussions therein).Unfortunately, this topic is beyond the scope of this consideration (discussions about ellipticPolylogarithms can be found in Ref. [63], which is a contribution to the Proceedings), butwe would like to point out only some of the integral representations that can be used inconjunction with elliptic Polylogarithms or even instead of elliptic Polylogarithms.
The effective mass form (15) turned out to be convenient for integrals containing anelliptic structure, since it allows one to represent the final result (see Ref. [37]) as anintegral containing an elliptic kernel (i.e., a root of a polynomial of the 3rd or 4th degree)and a remainder represented in the form of an ordinary (Goncharov) polylogarithms. Thisapproach can be an alternative to the introduction of elliptic Polylogarithms, which have avery complex structure (see, for example, the recent paper [81], where the study of sunsetsin special kinematics was carried out both in the form of elliptic Polylogarithms (followingRef. [82]), as well as in the form of integral representations containing an elliptic kerneland ordinary Polylogarithms. Notice, that such analysis has been done in all orders of thedimensional regulator following the corresponding results in Ref. [83]).At the end of the section, we would like to note about the recent paper [84], wherethe results for the most complex two-point single-mass diagrams containing an ellipticalstructure were obtained in the following form: using the effective mass representation, theoriginal FIs were presented as integrals of one-loop diagrams dependent on the ratio µ/m .These one-loop diagrams were considered in a generalized canonical form (42). The authorsof Ref. [84] have obtained very convenient representations for extremely complicated FIs.
In this short review, we examined the DE applicability for calculating FIs. We have con-sidered an example I ( q , m ), which led to the DE method sometime ago. The consistentapplication of IBP relations to I ( q , m ), and then to the diagrams of inhomogeneous termsthat arise each time, made it possible to obtain a DE hierarchy for increasingly simple dia-grams obtained at each step by reducing one propagator. As noted in section 3.1, the DEmethod procedure is well defined but requires a lot of manual work and a lot of time.Next, we showed an effective method restoring the exact result for two-point and three-point two-loop diagrams in terms of inverse-mass-expansion coefficients, which have a beau-tiful structure and can be predicted using the corresponding coefficients at the poles or attranscendental constants such as Euler’s ζ -functions. These predictions were verified by an-alytical calculations of the first few terms using computer programs. Thus, this method is,apparently, the first, where computer programs were used for FI calculations using differen-tial equations.We have also given a brief overview of modern popular techniques such as the ‘ canonicalform of DEs [59], the simplified DE approach [62] and the method of the effective mass (see,for example, Ref. [38]). Section 5.2 lists other popular approaches as well.14he canonical form [59] (and its generalizations [60, 61]) are probably the most commonlyused approaches (at least as a part of the calculations).The effective mass method (see [38]) allows you to actually work with diagrams thathave fewer loops than the original ones. The results for the original diagrams are obtainedin the form of integral representations, where the integrand expressions are determined bycalculating the diagrams with fewer loops. So, in Ref. [37] the two-loop diagrams with anelliptic structure were considered. The corresponding one-loop diagrams depending on theeffective mass do not have an elliptical structure . Thus, the results of the original diagramswere presented in the form of integral representations containing an elliptic kernel (i.e., a rootof a polynomial of the 3rd or 4th degree) and ordinary Polylogarithms. These representationscan be used instead of elliptic Polylogarithms, and even more complex objects than ellipticPolylogarithms (see [84] and discussions therein).Following the discussion in Section 5.3, the combined application of the effective-mass approach and generalizations of the canonical form for effective-mass-dependent diagramscan yield results for very complicated FIs. Such an analysis has already been carried out inthe recent article [84] and, in our opinion, similar calculations can be performed in the nearfuture for many complicated FIs.Author thanks Johannes Blumlein for invitation to present a contribution to the pro-ceedings of International Conference ”Antidifferentiation and the Calculation of FeynmanAmplitudes” (4-9 October 2020, Zeuthen, Germany) and Andrey Pikelner for help withaxodraw2. J ( q , m ) in eq. (5). Now consider the following diagrams I ( α )2 ( q , m ) = α → q , S ( β,α ) ( q , m ) = αβ → q . (A1)The IBP relations for the internal loop of the diagram produce two equations:( d − − α ) I ( α )2 ( q , m ) = α J ( α +1)2 ( q , m ) − m α I ( α +1)2 ( q , m ) , (A2)( d − I (1)2 ( q , m ) = T , ( m = 0) L , ( q ) − S (2 , ( q , m ) − m → q − m I (2)2 ( q , m ) , (A3)where J ( α )2 ( q , m ) = T ,α ( m ) L , ( q ) − S (1 , ( q , m ) . (A4)15e note that T , ( m = 0) = 0 in dimensional regularization and T , ( m ) L , ( q ) = 1(4 π ) d R (0 , A (1 , m − d/ q − d/ , (A5)where R ( α , α ) and A ( α , α ) are given in eqs. (13) and (12), respectively.The IBP relations for internal triangles of the diagram I (1)2 ( q , m ) produce two additionalequations: ( d − I (1)2 ( q , m ) = S (2 , ( q , m ) − J (2)2 ( q , m ) − m I (2)2 ( q , m ) − q → q , (A6)( d − → q = S (2 , ( q , m ) − T ( m = 0) L , ( q )+ m → q − q → q . (A7)Using eqs. (A3) and (A7) as the combination: 2 × (A3) + (A7) , we have(3 d − I (1)2 ( q , m ) = − m I (2)2 ( q , m ) − m → q − q → q . (A8)So, we have for the mass-dependent part of J ( q , m ) (see eq. (5)) m → q + q → q = (cid:20) d − − m ddm (cid:21) I (1)2 ( q , m ) , (A9)i.e. the mass-dependent combinations is expressed through the diagram I (1)2 ( q , m ) and itsderivative. 16sing eq. (A2), (cid:20) d − − α − m ddm (cid:21) I ( α )2 ( q , m ) = α J ( α +1)2 ( q , m ) , (A10)i.e. the diagram I ( α )2 ( q , m ) obeys the differential equation with the inhomogeneous term J ( α +1)2 ( q , m ) having very simple form: it contains only one-loop diagrams. To look it, wesee that the last term in J ( α )2 ( q , m ) (see eq. (A4)) is expressed through massive one loop M α ,α ( q , m ): M α ,α ( q , m ) = Z Dk ( q − k ) α ( k + m ) α = α α → q . (A11)Indeed, S (1 ,α ) ( q , m ) == A (1 , M − d/ ,α ( q , m ) . (A12)The one-loop diagram M − d/ ,α ( q , m ) can be evaluated by one of some effective methods,for example, by Feynman parameters.We would like to note that I (1)2 ( q , m ) satisfies eq. (A10) with α = 1 that is not is oftype of (35). But the integral I (2)2 ( q , m ) satisfies eq. (A10) with α = 2 that is of type of(35). So, it is convenient to rewrite (A9) with I (2)2 ( q , m ) in its r.h.s.: m → q + q → q = 3 d − d − J (2)2 ( q , m ) − d − d − m I (2)2 ( q , m ) . (A13)Now we should compare the IBP-based equations for J (2)2 ( q , m ) and J (3)2 ( q , m ) ob-tained in the right-hand sides of (A13) and (A10), respectively, with eq. (35). Since J (3)2 ( q , m ) = − ( d/dm ) J (2)2 ( q , m ), consider only J (2)2 ( q , m ).So, we should prepare the IBP-based equations for the massive one-loops M ε, ( q , m )and M ε, ( q , m ). Applying IBP procedure with massive distinguished line to M ε, ( q , m ),we have( − ε ) M ε, ( q , m ) = ε (cid:2) M ε, ( q , m ) − ( q + m ) M ε, ( q , m ) (cid:3) − m M ε, ( q , m ) . (A14)The corresponding applications of the IBP procedure with massless distinguished line to M ε, ( q , m ) and M ε, ( q , m ) lead to the following results:(1 − ε ) M ε, ( q , m ) = M ε, ( q , m ) − ( q + m ) M ε, ( q , m ) , (A15) − εM ε, ( q , m ) = 2 M ε, ( q , m ) − q + m ) M ε, ( q , m ) , (A16)The last equations has the following form (cid:20) − ε − ( q + m ) ddm (cid:21) M ε, ( q , m ) = − ddm M ε, ( q , m ) (A17)17utting (A15) to (A14), we have after little algebra − ε (1 − ε ) M ε, ( q , m ) = − ε (1 − ε ) ( q + m ) M ε, ( q , m ) (cid:3) − − ε ) m M ε, ( q , m ) , (A18)which transforms to (cid:20) − ε (1 − ε ) − − ε ) ddm (cid:21) M ε, ( q , m ) = − ε (1 − ε ) ( q + m ) M ε, ( q , m ) (A19)So, eqs. (A17) and (A19) can be frustrated as a system of equations having a form similarto equation (35). References [1] K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B (1981) 159; F. V. Tkachov,Phys. Lett. B (1981) 65; A. N. Vasiliev, Y. M. Pismak and Yu. R. Khonkonen,Theor. Math. Phys. (1981) 104[2] K. G. Chetyrkin et al. , Nucl. Phys. B (1980) 345; A. V. Kotikov, Phys. Lett. B (1996) 240.[3] A. V. Kotikov and S. Teber, Phys. Rev. D (2014) no.6, 065038[4] A. Grozin, In *Grozin, Andrey: Lectures on QED and QCD* 1-156 [hep-ph/0508242];A. G. Grozin, Int. J. Mod. Phys. A (2012) 1230018[5] S. Teber and A. V. Kotikov, Theor. Math. Phys. (2017) no.3, 446; A. V. Kotikovand S. Teber, Phys. Part. Nucl. (2019) no.1, 1[6] A. I. Davydychev and J. B. Tausk, Phys. Rev. D (1996) 7381[7] D. J. Broadhurst, Z. Phys. C (1990) 115.[8] B. A. Kniehl, A. F. Pikelner and O. L. Veretin, JHEP (2017) 024; A. Pikelner,Comput. Phys. Commun. (2018) 282[9] L. H. Ryder, “Quantum Field Theory”, Cambridge University Press, 1996.[10] M. Kalmykov, V. Bytev, B. A. Kniehl, S. O. Moch, B. F. L. Ward and S. A. Yost,arXiv:2012.14492 [hep-th].[11] T. Regge, Algebraic Topology Methods in the Theory of Feynman Rela- tivistic Am-plitudes, Battelle Rencontres: 1967 Lectures in Mathematics and Physics, C.M. DeWitt,J.A. Wheeler (Eds.), (W.A. Benjamin, New York, 1968), pp. 433–458.[12] Golubeva, V. A.: Some problems in the analytical theory of Feynman integrals. Russ.Math. Surv. 31 139 (1976)[13] A. V. Kotikov, Phys. Lett. B (1991) 158[14] E. Remiddi, Nuovo Cim. A (1997) 1435.1815] A. V. Kotikov, Phys. Lett. B (1991) 314[16] T. Gehrmann and E. Remiddi, Nucl. Phys. B (2000) 485[17] A. V. Kotikov, Phys. Lett. B (1991) 123; Mod. Phys. Lett. A (1991) 3133[18] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B (1994) 217[19] A. V. Kotikov, Ukr. Fiz. Zh. (Russ. Ed. ) (1992) 303.[20] N. N. Bogoliubov and O. S. Parasiuk, Acta Math. (1957) 227; K. Hepp, Commun.Math. Phys. (1966) 301; W. Zimmermann, Commun. Math. Phys. (1969) 208 [Lect.Notes Phys. (2000) 217].[21] N. N. Bogolyubov and D. V. Shirkov, Intersci. Monogr. Phys. Astron. (1959) 1.[22] D. I. Kazakov, Phys. Lett. (1983) 406; Theor. Math. Phys. (1984) 223;D. I. Kazakov, Theor. Math. Phys. (1985) 84; N. I. Usyukina, Theor. Math. Phys. (1983) 78; V. V. Belokurov and N. I. Usyukina, J. Phys. A (1983) 2811; Theor.Math. Phys. (1989) 385.[23] D. I. Kazakov and A. V. Kotikov, Theor. Math. Phys. (1988) 1264; A. V. Kotikov,Theor. Math. Phys. (1989) 134.[24] D. I. Kazakov and A. V. Kotikov, Nucl. Phys. B (1988) 721 [Nucl. Phys. B (1990) 299].[25] A. V. Kotikov, Mod. Phys. Lett. A (1991) 677[26] A. V. Kotikov, Int. J. Mod. Phys. A (1992) 1977.[27] D. I. Kazakov and A. V. Kotikov, Phys. Lett. B (1992) 171; Yad. Fiz. (1987)1767; D. I. Kazakov, A. V. Kotikov, G. Parente, O. A. Sampayo and J. Sanchez Guillen,Phys. Rev. Lett. (1990) 1535[28] F. Herzog, S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, Phys. Lett.B (2019) 436[29] J. Blumlein, S. Klein and B. Todtli, Phys. Rev. D (2009) 094010[30] A. Behring, J. Blumlein, A. De Freitas, A. von Manteuffel, K. Sch¨onwald and C. Schnei-der, arXiv:2101.05733 [hep-ph].[31] J. Ablinger, J. Blumlein, A. De Freitas, A. Goedicke, M. Saragnese, C. Schneider andK. Sch¨onwald, Nucl. Phys. B (2020) 115059[32] J. Blumlein and C. Schneider, Int. J. Mod. Phys. A (2018) no.17, 1830015[33] J. Fleischer et al. , Phys. Lett. B (1999) 169.[34] J. Fleischer et al. , Phys. Lett. B (1998) 1631935] J. Fleischer et al. , Nucl. Phys. B (1999) 343; Acta Phys. Polon. B (1998) 2611.[36] A. V. Kotikov, Particles (2020) no.2, 394[37] B. A. Kniehl et al. , Nucl. Phys. B (2006) 306; Nucl. Phys. B (2019) 114780[38] B. A. Kniehl and A. V. Kotikov, Phys. Lett. B (2006) 531; Phys. Lett. B (2012) 233.[39] J. A. M. Vermaseren, Int. J. Mod. Phys. A (1999) 2037; J. Blumlein and S. Kurth,Phys. Rev. D (1999) 014018[40] A. V. Kotikov, Phys. Atom. Nucl. (1994) 133; A. V. Kotikov and V. N. Velizhanin,hep-ph/0501274.[41] A. Devoto and D. W. Duke, Riv. Nuovo Cim. (1984) 1.[42] B. A. Kniehl et al. , Phys. Rev. Lett. (2006) 042001; Phys. Rev. D (2009) 114032;Phys. Rev. Lett. (2008) 193401; Phys. Rev. A (2009) 052501;[43] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B (2003) 19; in: Proc. of the XXXVWinter School , Repino, S’Peterburg, 2001 (hep-ph/0112346).[44] L. Bianchi et al. , Phys. Lett. B (2013) 394[45] A. V. Kotikov and L. N. Lipatov, Nucl. Phys.
B582 (2000) 19.[46] L. N. Lipatov, Sov. J. Nucl. Phys. (1976) 338; V. S. Fadin et al. , Phys. Lett. B (1975) 50; E. A. Kuraev, et al. , Sov. Phys. JETP (1976) 443; Sov. Phys. JETP (1977) 199; I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. (1978) 822; JETPLett. (1979) 355.[47] A. V. Kotikov et al. , Phys. Lett. B (2003) 114.[48] A. V. Kotikov et al. , Phys. Lett. B (2004) 521.[49] A.V. Kotikov et al. , J. Stat. Mech. (2007) P10003; Z. Bajnok, R.A. Janik, and T.Lukowski, Nucl. Phys. B (2009) 376.[50] T. Lukowski et al. , Nucl. Phys. B , 105 (2010).[51] C. Marboe et al. , JHEP (2015) 084[52] C. Marboe and V. Velizhanin, JHEP (2016) 013[53] M. Staudacher, JHEP (2005) 054; N. Beisert and M. Staudacher, Nucl. Phys. B (2005) 1.[54] E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A (2000) 725[55] A. I. Davydychev and M. Y. Kalmykov, Nucl. Phys. B (2004) 3.2056] A. V. Kotikov, In *Diakonov, D. (ed.): Subtleties in quantum field theory* 150-174[arXiv:1005.5029 [hep-th]]; Phys. Part. Nucl. (2013) 374; A. V. Kotikov and A. I. On-ishchenko, arXiv:1908.05113 [hep-th].[57] A. V. Kotikov, Theor. Math. Phys. (2013) 913; Theor. Math. Phys. (2017)no.3, 391[58] T. Gehrmann et al. , JHEP (2012) 101.[59] J. M. Henn, Phys. Rev. Lett. (2013) 251601; J. Phys. A (2015) 153001[60] L. Adams and S. Weinzierl, Phys. Lett. B (2018) 270[61] L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, J. Math. Phys. (2016) no.12,122302[62] C. G. Papadopoulos, JHEP (2014) 088[63] S. Weinzierl, arXiv:1912.02578 [hep-ph]; arXiv:2012.08429 [hep-th].[64] M. Argeri and P. Mastrolia, Int. J. Mod. Phys. A (2007) 4375[65] R. N. Lee, JHEP (2015) 108; JHEP (2018) 176; R. N. Lee and A. I. On-ishchenko, JHEP (2019) 084;[66] R. N. Lee and A. A. Pomeransky, arXiv:1707.07856 [hep-th].[67] A. B. Goncharov, math/0103059 [math.AG].[68] C. Duhr, arXiv:1411.7538 [hep-ph].[69] A. Kotikov et al. , Nucl. Phys. B (2008) 47.[70] U. Aglietti and R. Bonciani, Nucl. Phys. B (2003) 3; U. Aglietti, R. Bonciani,G. Degrassi and A. Vicini, Phys. Lett. B (2004) 432; Phys. Lett. B (2004) 57;JHEP (2007) 021[71] J. Henn, A. V. Smirnov, V. A. Smirnov, M. Steinhauser and R. N. Lee, JHEP (2017) 139; J. Ablinger, J. Blumlein, P. Marquard, N. Rana and C. Schneider, Phys. Lett.B (2018) 528; J. Ablinger, J. Blumlein, P. Marquard, N. Rana and C. Schneider,Nucl. Phys. B (2019) 253[72] J. Ablinger, J. Blumlein and C. Schneider, J. Math. Phys. (2011) 102301[73] D. D. Canko, C. G. Papadopoulos and N. Syrrakos, JHEP (2021) 199; D. D. Cankoand N. Syrrakos, arXiv:2010.06947 [hep-ph]; N. Syrrakos, arXiv:2012.10635 [hep-ph].[74] R. N. Lee, A. V. Smirnov and V. A. Smirnov, JHEP (2018) 008[75] R. N. Lee, arXiv:2012.00279 [hep-ph].[76] B. Kol, arXiv:1507.01359 [hep-th]. 2177] B. Kol, A. Schiller and R. Shir, JHEP (2021) 165[78] P. Vanhove, arXiv:1807.11466 [hep-th][79] J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, arXiv:1807.00842 [hep-th];J. Broedel and A. Kaderli, J. Phys. A (2020) no.24, 245201[80] M. Walden and S. Weinzierl, arXiv:2010.05271 [hep-ph]; L. Adams, E. Chaubey andS. Weinzierl, PoS LL (2018) 069 [arXiv:1807.03599 [hep-ph]].[81] J. Campert, F. Moriello and A. Kotikov, arXiv:2011.01904 [hep-ph].[82] J. Broedel, C. Duhr, F. Dulat and L. Tancredi, JHEP (2018) 093[83] M. Y. Kalmykov and B. A. Kniehl, Nucl. Phys. B (2009) 365[84] M. A. Bezuglov, A. I. Onishchenko and O. L. Veretin, Nucl. Phys. B963