We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of Feynman integrals, we present some selected new results for their expansions around 4 - 2ε dimensions

Luthe, Thomas

Schröder, York

Journal of Physics : Conference Series

English

arXiv

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This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 129.13.72.198This content was downloaded on 16/08/2017 at 11:50Please note that terms and conditions apply.Fun with higher-loop Feynman diagramsView the table of contents for this issue, or go to the journal homepage for more2016 J. Phys.: Conf. Ser. 762 012066(http://iopscience.iop.org/1742-6596/762/1/012066)Home Search Collections Journals About Contact us My IOPscienceYou may also be interested in:An Introduction to Quantum Theory: The Feynman path integralJ GreensiteMore Feynman errors come to lightHenning Genz, D R Lloyd and A G KleinGeometrical splitting and reduction of Feynman diagramsAndrei I DavydychevFeynman returns to centre stageDavid GoodsteinFeynman Integrals and Hypergeometric FunctionsHector Luna Garcia, Luz Maria Garcia, Ruben Mares et al.High Performance and Increased Precision Techniques for Feynman Loop IntegralsK Kato, E de Doncker, T Ishikawa et al.Alternative method of Reduction of the Feynman Diagrams to a set of Master IntegralsJulio Borja and Igor KondrashukFun with higher-loop Feynman diagramsThomas Luthe1 and York Schröder21 Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),Karlsruhe, Germany2 Grupo de Fisica de Altas Energias, Universidad del Bio-Bio, Casilla 447, Chillan, ChileE-mail: thomas.luthe@kit.edu, yschroeder@ubiobio.clAbstract. We review recent progress that we have achieved in evaluating the class of fullymassive vacuum integrals at five loops. After discussing topics that arise in classification,evaluation and algorithmic codification of this specific set of Feynman integrals, we presentsome selected new results for their expansions around 4 − 2ε dimensions.1. IntroductionIn high-energy physics experiments performed at current colliders such as the LHC, the floodof precision data requires matching theoretical efforts, in order extract the underlying event’sstructure. This is particularly relevant for Run II, where important processes are plagued withlarge QCD backgrounds, requiring high-precision theory calculations.To this end, we will showcase a few techniques and results related to investigations of thestructure of higher-loop Feynman integrals which provide one of the basic building blocks ofhigh-precision perturbative calculations within elementary particle physics. We will discuss newresults on the current (five-)loop frontier, which constitute basic building blocks for importantquantities such as anomalous dimensions in gauge field theories such as QCD.2. ClassificationLet us consider fully massive 5-loop tadpoles, in Euclidean space-time, having the same mass min all propagators 1/(q2i +m2). Due to the absence of external legs there are no further scales,such that all integrals scale trivially with the mass (which we can hence set to unity), such thatour problem corresponds to computing scale-free functions of the space-time dimension d only.The corresponding 4-loop problem has been solved already more than a decade ago in 4d [1] aswell as 3d [2].To fully define the 5-loop integral family, we need 15 propagators (lines), for which we chooseqi ∈ {k1, k2, k3, k4, k5, k13, k14, k15, k23, k24, k25, k35, k45, k124, k34} , (1)where ka...bc = ka + · · · + kb − kc. Trivalent 5-loop graphs have 12 lines, and there are fourindependent such sectors (one of them corresponding to the last diagram of Fig. 1). We labeldifferent integral sectors by their binary representation (where a 1 at position j corresponds to aline with momentum qj in the representation Eq. (1)), identify unique graphs, find all isometriesand corresponding momentum shifts, and choose the largest representative from each class.ACAT2016 IOP PublishingJournal of Physics: Conference Series 762 (2016) 012066 doi:10.1088/1742-6596/762/1/012066Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 1Figure 1. Some examples of 5-loop master integrals with ’dots’ that we choose as elements ofour basis. Each solid line corresponds to a massive propagator with unit power, while each dotrepresents an extra power. Note that we push all dots on a single line, which is the most naturalchoice with respect to the difference equation setup for their evaluation.Furthermore, using the freedom of normalization of the integral measure, we find it convenientto divide out a factor of [1-loop tadpole]#loops from each integral, achieving a convention-independent normalization of loop integrals. It is useful to recall that in 4d, the [1-looptadpole]∼ 1/ε, while it is finite in 3d.To show some 5-loop numerology: of the 215 = 32768 possible sectors that can be constructedfrom our list of momenta, 5151 do not correspond to a Feynman graph; 1941 of them are triviallyzero (in dimensional regularization, since they do not depend on one of the loop momenta), whilea further 3625 vanish non-trivially (since they can be shifted such that they become independentof one of the integration momenta); 21962 sectors can be transformed (by linear momentum shiftswith unit Jacobian) to a larger representative, while 22 others never occur in a sane Feynmandiagram labelling (since they would require a shift with Jacobian 1/2); leaving us with 48 uniqueirreducible 5-loop master sectors, plus 19 unique master sectors that correspond to products oflower-loop integrals. To compare, at 1/2/3/4-loop there were 1/1/3/10 unique sectors (plus0/1/2/6 factorized ones).While each of the 48+19 sectors contains at least one master integral (which can be chosento be the ’corner integral’ having all lines with power one), a (small) IBP reduction revealsthat some sectors contain multiple master integrals. In particular, we need in 62 (+3 factorizedones) additional masters with ’dots’, examples of which are shown in Fig. 1. Recall that at1/2/3/4-loop there were only 0/0/0/3 masters with ’dots’ [3].3. EvaluationGeneralizing our set of master integrals to carry a symbolic power x on one of their lines, wecan perform an IBP reduction on this modified set of masters and derive a hierarchical systemof difference equations for all of them. Symbolically,I(x) ≡∫1Dx1D2...DN,R∑j=0pj(x)I(x+ j) = F (x) (2)where the integer R is called order of the equation, the pj are rational functions (in twovariables, x and d), and the functions F on the right-hand sides contain integrals that are simpler(according to the ordering prescription chosen). Boundary conditions for these equations aresimple, and can be derived by shrinking a line (i.e., at x = 0) or be related to lower-loop integrals(i.e., expanding the integrand for x 1).Typically, one finally wants to extract the value for I(1). To this end, one needs to solve thedifference equation; this is possible explicitly for R = 1, while higher-order equations call for amuch more general setup [4]. Laporta’s recipe suggests to first attempt a general solution interms of factorial series, schematically:I(x) = I0(x) +R∑j=1Ij(x) , where Ij(x) = µxj∞∑s=0aj(s)Γ(x+ 1)Γ(x+ 1 + s−Kj). (3)ACAT2016 IOP PublishingJournal of Physics: Conference Series 762 (2016) 012066 doi:10.1088/1742-6596/762/1/0120662The difference equation then determines the special solution I0, fixes the constants µj , Kj(d)and implies a recurrence relation for the series coefficients aj(s) of the homogeneous solutions.Second, one needs a boundary condition for fixing, say, aj(0); it is useful to observe decoupling atlarge x, providing a particularly simple and automatable way to compute the necessary boundaryvalues:I(x) =∫k1g(k1)(k21 + 1)x⇒ I(x) ∼ (1)xx−d/2g(0) . (4)This is then to be compared with the large-x behavior of the factorial series,∑j µxjxKjaj(0).Third, having all analytic ingredients for their solution in terms of infinite sums at hand, toobtain actual results for the master integrals it is most useful to resort to a numeric treatment.This involves truncating the sum in Eq. (3) at some smax, evaluating R consecutive values ofI(x) around some large x (where the factorial series converges well), and using the differenceequation Eq. (2) to push their argument down to the required integrals, such as I(1).4. ImplementationThe above method brings with it a number of advantages: it allows for a high level of automation;works well with divergent integrals; does not rely on specific function classes; gives high-precisionresults for arbitrary ε orders; allows for expansions around any dimension; provides simple buthighly non-trivial cross-checks by putting x on different lines.However, we have found that, following the program proposed by Stefano Laporta [4], oneis faced with some problems and limitations when treating complex problems: the method isof limited use for multi-scale integrals; the complexity of coefficients in high-order equationscan grow enormously; one typically obtains recurrence relations of high orders; in numericalevaluation, one often faces instabilities of the factorial series solutions.To alleviate these problems and make feasible the computation of our set of 5-loop massivetadpoles, we had to considerably change the traditional setup. Much of our progress can beattributed to a number of fixes done to the original proposal. To name a few key ingredients [5]:we use coupled IBP equations, in order to tame the growth of complexity; we reduce (the ordersof) recurrence relations, essentially by re-using the linear solver developed for the system ofdifference equations; our code predicts instability factors, in order to assign sufficient numericalprecision.5. ResultsIn order to be specific, let us focus on results in d = 4 − 2ε here, recalling that we normalizeall our integrals with the corresponding power of the massive 1-loop tadpole. Given our high-precision numerical results, one might wonder whether it is possible to determine some of thecoefficients in analytic form. To this end, we employ the integer-relation finder PSLQ [6] togetherwith an educated guess of a basis of irrationals (such as e.g. the well-classified set of multiplezeta values and/or alternating Euler sums), to guess some of these coefficients. It helps to keepthe basis as compact as possible, so in order to absorb single powers of π as well as powers ofln 3, we definehn ≡∞∑k=0Γ(k + 1/2)Γ(k + 1)Γ(1/2)(3/4)k(2k + 1)n= n+1Fn[{1/2, .., 1/2}︸ ︷︷ ︸n+1, {3/2, .., 3/2}︸ ︷︷ ︸n, 3/4] , (5)Hn ≡ hn + h1 Coefficient[1− 3ε/2Γ(1− ε)Γ2(1− ε/2)+O(εn), ε, n− 1], (6)⇒ H1 = h1 =2π3√3, H2 = h2 −12h1 ln 3 , H3 = h3 −h18(ln2 3 + 2ζ2) , etc. (7)ACAT2016 IOP PublishingJournal of Physics: Conference Series 762 (2016) 012066 doi:10.1088/1742-6596/762/1/0120663Figure 2. Sample integrals for which results are given in Eqs. (8), (12), (13-18).The specific combinations Hn have actually been inspired by the 2-loop sunset (leftmost diagramof Fig. 2), which can be written to all orders in terms of the Hn, the first few of which read1I7.1.1 = −32− 32ε+ (9H2 − 3) ε2 + (9H2 − 18H3 − 6) ε3 + (18H2 − 18H3 + 36H4 − 12) ε4 . (8)Furthermore, in order to absorb powers of ln 2, let us define as elements of the MZV basisan ≡∞∑k=012k kn= PolyLog[n, 1/2] = Lin(12) , (9)An ≡ an + (−1)nlnn 2n!(1− n(n− 1)2ζ2ln2 2), (10)⇒ A4 = a4 − ζ2ln2 22 · 2!+ln4 24!, A5 = a5 + ζ2ln3 22 · 3!− ln5 25!, etc., (11)such that e.g. the 3-loop mercedes-type integral (second diagram of Fig. 2) readsI63.1.1 = +(0) ε0 + (0) ε1 + (−2ζ3) ε2 + (−16A4 + 27H22 +34ζ225) ε3 + . . . . (12)Our numerical approach has passed a number of cross-checks with flying colors: placing thepower x of each generalized master on topologically inequivalent lines, the values at x = 1coincide; all known results up to 4-loop level [1, 7–9] have been reproduced or improved; allfactorized sectors perfectly agree with the respective products of lower-loop integrals; and thefew previously known 5-loop coefficients [10–12] have been reproduced. Let us finally show acouple of new results on the five-loop level (for the rightmost 6 graphs shown in Fig. 2), using ourPSLQ fits for low ε-orders (the divergent ones), and truncating the constant term to 50 numericaldigits for readability:I28686.1.1 = +(−3) ε0 + (−32) ε1 + (1324) ε2 + (−12671440) ε3 + (−41933456) ε4 ++135.95072868792871461956492733702218574897992953584 ε5 + . . . , (13)I28686.1.3 = +(0) ε0 + (32) ε1 + (−12) ε2 + (−443360) ε3 + (95216) ε4 +−38.292059175062436961881799538284449799148385376441 ε5 + . . . , (14)I30862.1.1 = +(−35) ε0 + (−2710) ε1 + (−4ζ35− 42160) ε2 + (−12ζ2225+24ζ35+21124) ε3+(72ζ2225− 98ζ3 +32ζ55+1295948) ε4 ++1143.1838307558764599466030303839590323268318605888 ε5 + . . . , (15)1 Regarding our notation for integral labels: Ia,b,c refers to the master integral with binary code a as explainedin Sec. 2, having propagators with unit powers, except line number b, which carries exponent c (or c− 1 ’dots’).ACAT2016 IOP PublishingJournal of Physics: Conference Series 762 (2016) 012066 doi:10.1088/1742-6596/762/1/0120664I30862.1.2 = +(15) ε0 + (1130) ε1 + (− 130) ε2 + (−12ζ35− 9) ε3 + (−36ζ2225+548ζ315− 122915) ε4 +−102.42854342605587086319606311891941160276036031953 ε5 + . . . , (16)I30231.1.1 = +(0) ε0 + (0) ε1 + (3ζ35) ε2 + (9ζ2225+21ζ35+ 3ζ5) ε3+(−36H2ζ3 +12ζ327+63ζ2225− 21ζ235+ 27ζ3 −24ζ55) ε4 +−531.32391547725635267943444561495368318398901378435 ε5 + . . . , (17)I31420.1.1 = +(0) ε0 + (0) ε1 + (0) ε2 + (0) ε3 + (−36ζ235) ε4 ++167.81535305918474061962120601112466233675898298296 ε5 + . . . . (18)6. ConclusionsWe have studied the class of fully massive vacuum diagrams, which are essential buildingblocks for a number of phenomenological applications such as QCD thermodynamics, anomalousdimensions, or moments. A classification of all 5-loop integrals contained in this class reveals 48independent sectors, with 48+62 master integrals (the latter having ’dots’).For the evaluation, we have chosen to employ a hierarchical system of difference equations andfactorial series, amenable to automated numerical treatment. The setup required for derivingthe difference equations, the recursion relations for factorial series coefficients, as well as thenumerical solution has been implemented in the C++ code TIDE [5], essential parts of which areparallelized. We use Fermat [13] for polynomial algebra in 2 variables, and manage to employa substantial fine-tuning of the Laporta approach, without which these results could not havebeen derived on the computer hardware that we have at our disposal. At 5 loops, our code hadto deal with difference equations of up to order 20, and recurrence relations of up to order 28.To date, we have numerical results for 44 of the 48 non-factorized 5-loop master-sectors(including the ’dotted’ integrals of those sectors), with about 300 digits accuracy, for at least 10ε-orders around d = 4−2ε and d = 3−2ε. These results include all divergent integrals, bringing5-loop anomalous dimensions within reach (once the integral reduction has been performed).AcknowledgmentsThe work of T.L. has been supported in parts by DFG grants GRK 881 and SCHR 993/2, and hewishes to thank the Bielefeld Graduate School in Theoretical Sciences for facilitating finalizingthis project via a BGTS mobility grant. Y.S. acknowledges support from DFG grant SCHR993/1, FONDECYT project 1151281 and UBB project GI-152609/VC.References[1] Laporta S 2002 Phys. Lett. B549 115–122 (Preprint hep-ph/0210336)[2] Schroder Y and Vuorinen A 2003 (Preprint hep-ph/0311323)[3] Schroder Y 2003 Nucl. Phys. Proc. Suppl. 116 402–406 (Preprint hep-ph/0211288)[4] Laporta S 2000 Int. J. Mod. Phys. A15 5087–5159 (Preprint hep-ph/0102033)[5] Luthe T PhD thesis, Bielefeld University 2015[6] Ferguson H R P, Bailey D H and Arno S 1999 Math. Comp. 68 351–369[7] Czakon M 2005 Nucl. Phys. B710 485–498 (Preprint hep-ph/0411261)[8] Schroder Y and Vuorinen A 2005 JHEP 06 051 (Preprint hep-ph/0503209)[9] Chetyrkin K G, Faisst M, Sturm C and Tentyukov M 2006 Nucl. Phys. B742 208–229 (Preprint hep-ph/0601165)[10] Groote S, Korner J G and Pivovarov A A 2007 Annals Phys. 322 2374–2445 (Preprint hep-ph/0506286)[11] Broadhurst D J 1985 Phys. Lett. B164 356[12] Brown F and Schnetz O 2012 (Preprint 1208.1890)[13] Lewis R H URL http://home.bway.net/lewis/ACAT2016 IOP PublishingJournal of Physics: Conference Series 762 (2016) 012066 doi:10.1088/1742-6596/762/1/0120665

Fun with higher-loop Feynman diagrams

Thomas Luthe

York Schröder

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Fun with higher-loop Feynman diagrams

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