Hypergeometric representation of the two-loop equal mass sunrise diagram

Physics Letters B 68 6 95 wwwelseviercom/locate/physletb Hypergeometric representation of the two-loop equal mass sunrise diagram OV Tarasov Deutsches Elektronen-Synchrotron DESY Platanenallee 6 D-578 Zeuthen Germany Received 6 April 6; accepted 4 May 6 Available online 4 May 6 Editor: GF Giudice Abstract A recurrence relation between equal mass two-loop sunrise diagrams differing in dimensionality by is derived and it s solution in terms of Gauss F and Appell s F hypergeometric functions is presented For arbitrary space time dimension d the imaginary part of the diagram on the cut is found to be the F hypergeometric function with argument proportional to the maximum of the Kibble cubic form The analytic expression for the threshold value of the diagram in terms of the hypergeometric function F of argument / isgiven 6 Elsevier BV All rights reserved Introduction The evaluation of radiative corrections for modern high precision particle physics is becoming a more and more demanding task Without inventing new mathematical methods and new computer algorithms the progress in calculating multi-loop multi-leg Feynman diagrams depending on several momentum and mass scales will be not possible An important class of radiative corrections comes from self-energy type of Feynman diagrams which also occur in evaluating vertex box and higher multileg diagrams At the two-loop level different approaches for calculating self-energy diagrams are available [] A general algorithm for reduction of propagator type of diagrams to a minimal set of master integrals was proposed in [] This recurrence relations algorithm has been implemented in computer packages in [ 5] At present the most advanced package available for calculating two-loop self-energy diagrams with arbitrary massive particles was written by Martin and Robertson [5] It includes procedures for numerical evaluation of master integrals with arbitrary masses and also a database of analytically known master integrals Integral representation for master integrals with arbitrary masses in four-dimensional space time was proposed in [6] Despite intensive efforts by many authors not all two-loop self-energy integrals with a mass are known analytically Even the imaginary part of the simplest sunrise self-energy diagram with three equal mass propagators was not known for arbitrary space time dimension d until now The two-loop sunrise integral with three equal masses was investigated in many publications [7 6] Small and large momentum expansions of this integral for arbitrary space time dimension d can be found in [] It s threshold expansion was given in [5] A numerical procedure for evaluating the sunrise integral was described in [4] The very latest effort of an analytic calculation of the diagram by using the differential equation approach [7] was undertaken in Ref [8] It is the purpose of this Letter to describe a new method and to present the analytic result for the equal mass two-loop sunrise master integral To accomplish our goal we use the method of evaluation of master integrals by dimensional recurrences proposed in [9] The application of this method to one-loop integrals was presented in [] and [] In the present publication we extend the method to two-loop integrals E-mail addresses: olegtarasov@desyde tarasov@ifhde OV Tarasov On leave of absence from JINR 498 Dubna Moscow Region Russian Federation 7-69/$ see front matter 6 Elsevier BV All rights reserved doi:6/jphysletb65

96 OV Tarasov / Physics Letters B 68 6 95 As was already discovered in the one-loop case the solutions of dimensional recurrences are combinations of hypergeometric functions The knowledge of the hypergeometric representation of an integral means that we possess the most complete mathematical information available This information can be effectively used in several respects First through analytic continuation formulae the hypergeometric functions valid in one kinematic domain can be re-expressed in a different kinematic region Second these hypergeometric functions often have integral representation themselves in which an expansion in ε 4 d/ can be made yielding expressions in logarithms dilogarithms elliptic integrals etc Since very similar hypergeometric functions come from different kind of Feynman integrals the ε expansion derived in solving one problem can be used in other applications Essential progress in the ε expansion of hypergeometric functions encountered in evaluating Feynman diagrams was achieved in [4] Third because the hypergeometric series is convergent and well behaved in a particular region of kinematical variables it can be numerically evaluated [45] In addition a hypergeometric representation allows an asymptotic expansion of the integral in terms of ratios of different Gram determinants or ratios of momentum and mass scales which can provide fast numerical convergence of the result Our Letter is organized as follows In Section we present the relevant difference equation connecting sunrise integrals with dimensionality differing by as well as the differential equation for this integral In Section the method for finding the full solution of the dimensional recurrency is elaborated Explicit expression for the sunrise integral in terms of Appell s function F and Gauss hypergeometric function F is constructed in Sections 4 and 5 the differential equation approach and the method of dimensional recurrences is compared In Appendix A some useful formulae for the hypergeometric functions F F and F are given together with their integral representations Difference and differential equations for the sunrise integral The generic two-loop self-energy type diagram in d dimensional Minkowski space with three equal mass propagators is given by the following integral: d d k d d k ν ν ν iπ d/ k m ν k k m ν k q m ν For integer values of ν j the integrals can be expressed in terms of only three basis integrals and T d m T d m d d k [iπ d ] k m Ɣ d m d The relation connecting d and d dimensional integrals ν ν ν follows from the relationship given in Ref []: J d ν ν ν ν ν ν + ν + ν + ν ν ν + ν ν + + ν ν ν ν + ν + Relation taken at ν ν ν and ν ν ν gives two equations To simplify these equations we use the recurrence relations proposed in [] From these two equations by shifting d d + two more relations follow They are used to exclude from one of the relations so that we obtain a difference equation for the master integral : J d+4 J d+ z d + d d + 4d + 4m 4 z 4z + 9z zd d 4m 8 z 9z z [ z + 7z + 8z d 4z + 9zd 48z ] m d+ Ɣ d z m 5 q The integral satisfies also a second order differential equation [] Taking the second derivative of with respect to mass gives d 6 dm dm 6 + 6 Again using the recurrence relations from [] the integrals on the rhs can be reduced to the same three basis integrals Using d 7 dm 4

OV Tarasov / Physics Letters B 68 6 95 97 from 6 we obtain: z 9zz d dz zm d 6 Ɣ d z [ 9z d 4 + zd + 8 d ]djd dz + d [ zd + 4 + d 4 ] 8 The differential equation 8 will be used in Section 4 to find the momentum dependence of arbitrary periodic constants in the solution of the difference equation 4 Solution of the dimensional recurrency Eq 4 is a second order inhomogeneous equation with polynomial coefficients in dthe full solution of this equation is given by see Ref [6] and references therein: p + w ad a + w bd b d p is a particular solution of 4 J a Jd b is a fundamental system of solutions of the associated homogeneous equation and w a d w b d are arbitrary periodic functions of d satisfying relations: w a d + w a d w b d + w b d The order of the polynomials in d of the associated homogeneous difference equation can be reduced by making the substitution 9 Ɣ d Ɣ d d Ɣ The associated homogeneous equation for J d takes the simpler form Putting 6z 7m 8 z 9z J d+4 d z 4z + 9z z 7m 4 z 9z d k ε y k ρ k J k ε we transform Eq to a standard form J d+ d d 4 6 J d Aρ y k+ + B + Ckρy k+ α + kβ + ky k A 6z 7m 8 z 9z 5 and ρ is for the time being an arbitrary constant In order to get Eq 4 into a more convenient form we will define three parameters ρ x and γ by the equations 4 B 4ε z 4z + 9z z 7 m 4 z 9z C B ε α ε β ε Aρ x x Bρ γ α + β + x Cρ x These have the solution x Cρ 9z + z q q 9m q + m ρ 7 m 4 z 9z 4A + C 4 γ Bρ + α + β + x ε z + z 7 4 and Eq 4 can accordingly be written in the form m q m q 9m q + m x xy k+ + [ xk + γ α + β + x ] y k+ α + kβ + ky k 6 7 8 9

98 OV Tarasov / Physics Letters B 68 6 95 The fundamental system of solutions of this equation consists of two hypergeometric functions [6] For example in the case when x < largeq the solutions are y k k Ɣα + kɣβ + k Ɣα + β γ + k + F α + kβ + kα + β γ + k + ; x y k Ɣα + β γ + k x k F γ α γ βγ α β + k; x Once we know the solutions of the homogeneous equation a particular solution p can be obtained by using Lagrange s method of variation of parameters Lagrange s method for finding a particular solution is well described in [6] The application of the method is straightforward but tedious Explicit result will be given in the next section It is interesting to note that the argument of the Gauss hypergeometric function is related to the maximum of the Kibble cubic form [7]: Φstu stu s + t + um m + q + m 4 m + q provided that the following condition is satisfied: s + t + u q + m The maximal value Φ max 7 q q 9m occurs at s t u q + m and we see that the kinematical variable 7 can be written as x Φstu 4 stu stu q +m This observation may be useful in finding the characteristic variable in the general mass case [8] Also one can try to apply the method described above to find the imaginary part of the sunrise integral in the general mass case in arbitrary space time dimension 4 Explicit analytic expression for To find the full solution of Eq 4 we assume that q is large The region of large momentum squared corresponds to x and therefore as a fundamental system of solutions of the homogeneous equation we take y k and y k According to and the solution of the associated homogeneous difference equation will be of the form h w z Ɣd Ɣ d d Ɣ [ d Ɣ d Ɣ d d Ɣ ρ d e iπ d F d ; ] d x ; Ɣ d + w z d [ ρ 45 Ɣ d d Ɣ F ; ] x d d ; x The arbitrary periodic functions w z and w z can be determined either from the d asymptotics or using the differential equation 8 Substituting 45 into 8 we obtain two simple equations z z + z 9z dw z + 6z 9z w z dz z + z 9z dw z + zw z 46 dz Both equations are independent of d and their solutions w z κ z + z 47 9z z w κ z z + z 9z determine the periodic functions up to integration constants κ κ which we fix from the first two terms of the large momentum expansion of presented in []: [ Ɣ + εɣ m 4ε Ɣ ε + ε zɣ + εɣ ε zε + 6Ɣ ε Ɣ ε zε The application of Lagrange s method of finding a particular solution gives p zmd 6 + z Ɣ d F d d d ; zr R ] + Oz 48 49

OV Tarasov / Physics Letters B 68 6 95 99 R 4 z + z F is the Appell function [9] defined as F αββ γγ ; xy kl α k+l β k β l x k y l γ k γ x + y < l k!l! and a n Ɣa + n/ Ɣa denotes the Pochhammer symbol Collecting all contributions setting d 4 ε applying Euler s transformation for the first F function in 45 we obtain the following solution of the difference equation 4: 6Ɣ εɣ + ε z ε z ε [ m 4ε F ; 7 z ] z Ɣ ε + z ε; + z + Ɣ + εɣ ε z ε 9z ε [ m 4ε F ; 7 z ] z Ɣ εz + z ε; + z + zm 4ε 4 + z Ɣ + εf ε ε ε; zr R This is our main result The imaginary part of on the cut comes from the first two terms in 4 The analytic continuation of the two F functions gives the relatively simple expression: Im 4zπ πm 4ε [ 9z ] ε [ 4 εɣ ε + z 8z F ; 9z ] ε; + z Ɣ At d 4 for the imaginary part this verifies the result of [4] Expanding at q 9m we reproduced the singular and finite in ε parts of several terms of the on-threshold asymptotic expansion presented in Ref [5] We found the following integral representation for Appell s F function in 4 F ε ε ε; Ɣ ε zr R Ɣ ε + z L 4zt z 4z dt [t t] ε [ ε; 4zt + z + L F ε; ] 4tz+ z L 4tz+ z + L This integral representation can be used for the ε expansion of the F function Integral representation for the F functions convenient for ε expansion is given in Appendix A Each term with the F function in 4 is a solution of the associated homogeneous differential equation 8 The term with F function is a particular solution of the differential equation 8 The analytic continuation of valid near the singular points q m 9m can be directly obtained from 4 by performing the analytic continuation of the hypergeometric functions Explicit formulae of the analytic continuations of in terms of Olsson functions [] as well as the ε expansion of the result we are planning to present in a separate publication [] Using 4 we can find the on-threshold value of the integral In formula 4 at q 9m the imaginary part of the first term cancels the imaginary part of the second term The real part coming from the F terms cancels the term with F which comes from the Appell function F at the threshold F ε ε ε; 4 4 6 ε ε F The cancellation of the F functions happens due to the fact that [ + ε ε; + ε ε; + 4Ɣ εɣ εɣ + εɣ ε F ε Ɣ εɣ5 ε ] [ + ε ε; 5 ε; ] 4 4 44 45 46 [ ε F + ε; 5 ε; ] 5 ε 5 Ɣ 4 εɣ7 4 ε ε Ɣ 4 εɣ5 ε 47

OV Tarasov / Physics Letters B 68 6 95 Adding contributions from different hypergeometric functions gives a rather simple expression for the diagram at q 9m Ɣ [ ε + ε q 9m ε ε F ε; + ε ε; ] { Ɣ + ε ε ε ε + 9 4ε + 75 8 8π } + Oε The first several terms in the ε expansion are in agreement with the result of Ref [5] 5 Conclusions Here we would like to add several remarks which underline the most important points which follow from the results of this Letter Using a new method for the first time we were able to obtain an analytic expression for the two-loop sunrise self-energy diagram with equal mass propagators Until now all attempts to find such a result with other methods failed for this integral This clearly demonstrates that the method of dimensional recurrences is a powerful tool for calculating Feynman integrals In our opinion there is a deep reason why for example with the differential equation method [7] used in [8] an explicit formula could not be found It turns out that even the associated homogeneous differential equation is rather complicated Relevant for this case is the Heun equation [] with four regular singular points located at q m 9m Since we know the explicit result and the characteristic variable 7 we can easily transform the associated homogeneous equation 8 to the hypergeometric equation However in general the reduction of the Heun equation to the hypergeometric equation is a complicated mathematical problem [] which is not completely solved until now At the same time the associated homogeneous difference equation for is rather simple and admits a reduction to a hypergeometric type of equation with linear coefficients In fact this is a rather general situation Kinematical singularities of Feynman integrals are located on complicated manifolds In the case when the differential equations are of the first order there are no problems to solve them However to solve a second or higher order differential equations in general will be a problem because of complicated structure of the kinematical singularities The location of the singularities of Feynman integrals with respect to the space time dimension d is well known This has been used for a rather evident rescaling of the integral by ratios of Ɣ functions which allowed us to significantly reduce the order of the polynomial coefficients in the difference equation as we have seen in Section Finally this simplification allowed us to obtain the explicit result We expect that a further development of the method we used in the present Letter will help to find analytic results for other more complicated Feynman integrals Acknowledgements I am very thankful to Fred Jegerlehner for useful remarks and carefully reading the manuscript This work was supported by DFG Sonderforschungsbereich Transregio 9- Appendix A The series representation for the Appell function F : F αββ γγ α k β k ; xy x k F α + kβ ; γ ; y γ k k! k is convenient for the analytic continuations and also for the evaluation of F at some particular values of their arguments The most frequently used integral representations for F is 48 A F αββ γγ ; wz ƔγƔγ ƔβƔβ Ɣγ βɣγ β For the special parameters which appear in the explicit result 4 we have: F ε ε ε; xy Ɣ εɣ ε Ɣ εɣ dudvu β v β u γ β v γ β uw vz α dudv v [u u v] ε uw vz A A

OV Tarasov / Physics Letters B 68 6 95 Furthermore we found the following specific relation between the F function and yet another Appell s function F [ F ; 7 z ] z + z ε; + z z F + ε + ε ε; 4z + z 4z z A4 To our knowledge such a relation has not been found sofar in the mathematical literature The integral representation for the F function reads F + ε + ε ε; wz Ɣ ε Ɣ Ɣ ε du u [ u wu zu ] ε and is convenient for the ε expansion This F function can be considered as a generating function of a new generalization of elliptic integrals which may appear in evaluating Feynman integrals References [] G Passarino S Uccirati Nucl Phys B 69 97; G Weiglein R Mertig R Scharf M Böhm in: D Perret-Gallix Ed New Computing Techniques in Physics Research vol World Scientific Singapore 99 p 67 [] OV Tarasov Nucl Phys B 5 997 455 [] R Mertig R Scharf Comput Phys Commun 998 65 [4] SP Martin Phys Rev D 68 75 [5] SP Martin DG Robertson Comput Phys Commun 74 6 [6] S Bauberger M Böhm Nucl Phys B 445 995 5 [7] E Mendels Nuovo Cimento A 45 978 87 [8] DJ Broadhurst Z Phys C 47 99 5 [9] C Ford I Jack DRT Jones Nucl Phys B 87 99 7; C Ford I Jack DRT Jones Nucl Phys B 54 997 55 Erratum [] AI Davydychev JB Tausk Nucl Phys B 97 99 [] DJ Broadhurst J Fleischer OV Tarasov Z Phys C 6 99 87 [] FA Berends M Buza M Böhm R Scharf Z Phys C 6 994 7 [] FA Lunev Phys Rev D 5 994 775 [4] J Fleischer F Jegerlehner OV Tarasov OL Veretin Nucl Phys B 59 999 67; J Fleischer F Jegerlehner OV Tarasov OL Veretin Nucl Phys B 57 5 Erratum [5] AI Davydychev VA Smirnov Nucl Phys B 554 999 9 [6] BA Kniehl AV Kotikov A Onishchenko O Veretin Nucl Phys B 78 6 6 [7] AV Kotikov Phys Lett B 54 99 58; AV Kotikov Mod Phys Lett A 6 99 677 [8] S Laporta E Remiddi Nucl Phys B 74 5 49 [9] OV Tarasov Phys Rev D 54 996 6479 [] OV Tarasov Nucl Phys B Proc Suppl 89 7 [] J Fleischer F Jegerlehner OV Tarasov Nucl Phys B 67 [] AI Davydychev MYu Kalmykov Nucl Phys B Proc Suppl 89 8; AI Davydychev MYu Kalmykov Nucl Phys B 65 66; AI Davydychev hep-th/998; AI Davydychev Phys Rev D 6 877; AI Davydychev MYu Kalmykov Proceedings of the Workshop CPP Tokyo Japan November KEK Proceedings November p 69 hep-th/; S Moch P Uwer S Weinzierl J Math Phys 4 6; F Jegerlehner MYu Kalmykov O Veretin Nucl Phys B 658 49; S Weinzierl J Math Phys 45 4 656; AI Davydychev MYu Kalmykov Nucl Phys B 699 4 ; MYu Kalmykov Nucl Phys B Proc Suppl 5 4 8 [] MYu Kalmykov JHEP 64 6 56 [4] S Bauberger F Berends M Böhm M Buza Nucl Phys B 44 995 8 [5] MYu Kalmykov A Sheplyakov Comput Phys Commun 7 5 45 [6] LM Milne-Thomson The Calculus of Finite Differences Chelsea New York 98 [7] TWB Kibble Phys Rev 7 96 59 [8] AI Davydychev R Delbourgo J Phys A 7 4 487 [9] P Appell J Kampé de Fériet Fonctions Hypergeometriques et Hyperspériques Gauthier Villars Paris 96 [] POM Olsson Ark Fysik 9 8 964 459; POM Olsson Ark Fysik 4 965 87 [] OV Tarasov in preparation [] A Erdèlyi et al Higher Transcendental Functions vol McGraw Hill New York 95 p 57 [] RS Maier J Differential Equations 5 7 A5