Some Examples of Calculation of Massless and Massive Feynman Integrals

Article Some Examples of Calculation of Massless and Massive Feynman Integrals

Anatoly V. Kotikov

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia; [email protected]

Abstract: We show some examples of calculations of massless and massive Feynman integrals.

Keywords: Feynman integrals; dimensional regularization; singularities

1. Introduction At present, the calculation of the Feynman integrals (FIs) provides basic information, both on the properties of the experimentally investigated processes and on the characteristics of the physical models under study. Calculations of the matrix elements of the cross sections of the processes under study depend on the internal properties of particles participating in the interactions, such as masses, spins, etc., and, strictly speaking, require the calculation of Feynman integrals, including those with massive propagators. Depending on the kinematics of the processes under study, the values of some masses can be neglected. Studying the characteristics of physical models (for example, critical parameters, anomalous particle sizes and operators) usually requires the calculation of massless Feynman integrals, which have a much simpler structure. This allows obtaining Citation: Kotikov, A.V. Some results for these characteristics in high orders of the perturbation theory. Examples of Calculation of Massless I would like to draw your attention to the fact that, when calculating FIs, it is rec- and Massive Feynman Integrals. ommended to use analytical methods whenever possible. The point is that the numerical Particles 2021, 4, 361–380. https:// calculation of FIs is severely limited due to the singularities arising in them, and also doi.org/10.3390/particles4030031 (especially for gauge theories) due to strong mutual cancellations between contributions from different diagrams or even between parts of the same diagram. Academic Editor: Armen Sedrakian Note that when using the dimensional regularization [1–4], i.e., when calculating

Received: 19 July 2021 the FIs for an arbitrary dimension of space, once found diagrams for some model of a Accepted: 10 August 2021 ﬁeld theory (or process) can be applied to other models (or processes), since the main Published: 14 August 2021 object of study is the so-called scalar master integrals. Consequently, the complexity of analytical calculations of FI is compensated by their versatility as applied to various

Publisher’s Note: MDPI stays neutral quantum field models. with regard to jurisdictional claims in Note also the fact that the calculation of complicated diagrams may be of some published maps and institutional affil- independent interest. For example, the use of non-trivial identities, such as the “uniqueness” iations. relation [5,6], can provide information (see [7–13]) about the properties of some integrals and series which are not yet in the reference literature. For example, the calculations of the same Feynman integral carried out in [12–14], using various methods, have made it possible to find a previously unknown relationship between hypergeometric functions with arguments 1 and 1. This relation has been neatly proven quite recently [15]. Copyright: © 2021 by the author. − Licensee MDPI, Basel, Switzerland. Recently, many powerful original methods for calculating Feynman integrals have This article is an open access article appeared (see, for example, recent reviews in [16,17]), which are often inferior in breadth of distributed under the terms and application to standard methods, such as the α-representation and the Feynman parameter conditions of the Creative Commons technique (see, for example, [18,19]), however, can significantly increase the computation Attribution (CC BY) license (https:// accuracy for a limited set of quantities (or processes). creativecommons.org/licenses/by/ 4.0/).

Particles 2021, 4, 361–380. https://doi.org/10.3390/particles4030031 https://www.mdpi.com/journal/particles Particles 2021, 1 2 Particles 2021, 1 2 Particles 2021, 4 362 This short article is devoted to the consideration of two FIs, one with massless propagators, andThis the short other article with massive is devoted propagators, to the consideration the calcula oftions two of FIs which, one just with demonstrate massless propa- the effectivenessgators,This and short the of articleother modern with is methodsdevoted massive to for propagators, the calculating consideration the Feynma calcula of twon diagrams.tions FIs, of one which with just massless demonstrate propa- theIn the effectiveness massless case, of modern we will methods consider for a single calculating 5-loop Feynma master diagramn diagrams. that contributes gators, and the other with4 massive propagators, the calculations of which just demonstrate to thetheβ effectiveness-functionIn the massless of the of modernϕ case,-model. we methods will In consider the for initial calculating a single calculations 5-loop Feynman [ master20] of diagrams. thediagram 5-loop that correction contributes 4 4 to thetoβ the-functionInβ the-function massless of the of case, theϕ -model,ϕ we-model. will the consider In results the ainitial of single four calculations 5-loop FIs were master found [20] diagram of only the 5-loop numerically. that contributes correction 4 Theirtoto analytical the theββ-function-function results of of the were theϕ4ϕ obtained-model.-model, In by the the Kazakov resultsinitial calculations of (see four [7,8 FIs,12 [, were2013]]), of butfound the they 5-loop only have correction numerically. been to publishedtheTheirβ-function analyticalwithout of any results the intermediateϕ4-model, were obtained the results. results by Moreover, of Kazakov four FIs all (see werecalculations [7 found,8,12,13 only were]), but numerically. performed they have in Their been x-space,analyticalpublished which results without can make were any them obtained intermediate difficult by Kazakov to results. understand. (see Moreover, [7,8 Recentl,12,13 all]),y,calculations but two they of the have werefour been diagrams performed published in havewithoutx been-space, exactly any which intermediate recalculated can make them results. in [16 difficult] in Moreover,p-space to understand. and all calculations are presented Recentl werey, with two performed intermediate of the four in diagramsx cal--space, culations.whichhave been Incan Section make exactly them3, recalculated we difficult provide to a in understand. neat [16] calculation in p-space Recently, for and the are two third presented of the diagram. four with diagrams intermediate have been cal- exactlyculations.In the recalculated massive In Section case, in3, [in16 we] Section in providep-space5, a we neat and considerare calculation presented the for computation with the third intermediate diagram. of one calculations. of master- In integralsSectionIn [213 the,] we contributing massive providecase, a neat to the in calculation Sectionrelationship5, for we thebetween consider third diagram.the theMS computation-mass and the of onepole-mass of master- of theintegrals HiggsIn the boson [21 massive] contributing in the case, standard in to Section the model relationship5, we in the consider limit between of the heavy the computationMS Higgs.-mass Results ofand one the of for pole-mass master- the master-integral,integralsof the Higgs [21 along] contributingboson with in the the to standard results the relationship for model other in master-i between the limitntegrals, the of heavyMS were-mass H calculatediggs. and Results the [pole-mass22] for in the the earlyofmaster-integral, the 2000s, Higgs but boson unfortunately along in thewith standard the they results have model for not other in been the master-ilimit publis ofhed.ntegrals, heavy Some Higgs. were sets calculated of Results variables for [22 the] in for integrationmaster-integral,the early 2000s, are presented alongbut unfortunately with in Appendix the results theyA for. have other not master-integrals, been published. were Some calculated sets of variables [22] in thefor early integration 2000s, are but presented unfortunately in Appendix they haveA not. been published. Some sets of variables for 2. Basic Formulas for Massless Diagrams integration are presented in AppendixA. 2.Let Basic us briefly Formulas consider for Massless the rules Diagramsfor calculation of massless diagrams. All calculations are carriedLet out us in briefly momentum consider space the rules with ford = calculation4 2ε. of massless diagrams. All calculations 2. Basic Formulas for Massless Diagrams − arePropagator carried out is represented in momentum as space with d = 4 2ε. Let us briefly consider the rules for calculation− of massless diagrams. All calculations are carriedPropagator out in is momentum represented space as with d = 4 2ε. − Propagator is represented as 1 1 = . (1) (q2)α ≡1 q2α 1 α →q= . (1) (q2)α ≡ q2α α (1) where α is called the line index. →q whereThe followingα is called formulas the line hold: index. whereA. ForTheα simpleis following called chain: the formulas line index. hold: TheA. For following simple formulas chain: hold: A. For simple chain:1 1 1 = , q2α1 q2α2 2(α1+α2)1 1 1 1 q 1 = , =2α 2α , 2(α +α ) or graphically q2α1 q2α2 q q12(qα1+2α2) q 1 2 oror graphically graphically

= , (2) α1 α2 α + α (2) →q →q= 1 2 , (2) q α1 α2 q α1 + α2 i.e., the product of propagators→ is equivalent to a new→ propagator with an index equal i.e., the product of propagators is equivalent to a new propagator with an index equal to to the sum of the indices of the original propagators. the sumi.e., of the the product indices of of propagators the original is propagators. equivalent to a new propagator with an index equal B. A simple loop can be integrated as to theB. sumA simple of the indices loop can of be the integrated original propagators. as B. A simple loop can be integrated as Dk µ2ε µ2ε Dk µ2ε = µ2ε (α , α ) , 2α 2α2ε Nd ( + 2ε) A 1 2 (q k) Dk1 k µ2 =qN2 dα1 α2 dµ/2 A(α1, α2) , Z ( )2α1 2α2 2(α1−+α2 d/2) Z− q k k = Nd q − A(α1, α2) , (q− k)2α1 k2α2 q2(α1+α2 d/2) where Z − − where ddk where Dk = ddk (3) Dk(2=π)dddk (3) Dk = (2π)d (3) (2π)d isis usual usual integration integration in in Euclidean Euclidean measure measure and and is usual integration in Euclidean measure and 1 1 a(α)aa((αβ))a(β) Γ(α˜ )Γ(α˜ ) d d N =Nd = , A(, α,Aβ()α =, β) = , a(,α)a =(α) = , α˜ ,=α˜ = α .α . (4) (4) d (4π)(d4/2π1)d/2 a(α +a(αβa+(αβd)/2a(β)d)/2) Γ(α)ΓΓ((αα˜)) 2 −2d − N = , A(α, β) = − − , a(α) = , α˜ = α . (4) d (4π)d/2 a(α + β d/2) Γ(α) 2 − − ParticlesParticles20212021, ,41 3633 Particles 2021, 1 3

ItIt is is convenient convenient to to rewrite rewrite the the equation equation graphically graphically as as It is convenient to rewrite the equation graphically as α1 α1 2ε = Nd µ A(α1, α2) . (5) 2ε α1+α2 d/2 (5) →q = Nd µ A(α1, α2) →q − . (5) α +α d/2 →q →q 1 2− α2 Therefore, all masslessα2 diagrams, which can be expressed as combinations of loops andTherefore, chains,Therefore, can all all be massless massless evaluated diagrams, diagrams, immediately, which which even can can be when be expressed expressed some ind as as combinationsicescombinations have arbitrary of of loops loops val- anduesand chains, (see chains, [23 can–33 can]). be be However,evaluated evaluated startingimmediately, immediately, already even evenwith when thewhen some two-loop some indices ind level,ices have ther have arbitrarye are arbitrary diagrams, values val- (seewhichues[23 (see– cannot33 [23]).– However,33 be]). expressed However, starting as starting combinations already already with the of with loops two-loop the and two-loop level, chain theres (see,level, are for ther diagrams, example,e are diagrams, which Figure cannot1in[which17 be]). cannot expressed For these be expressed as cases combinations there as combinations are additional of loops and of rules. loops chains and (see, chain for example,s (see, for Figure example, 1 in Figure [17]). For these cases there are additional rules. 1in[C.17]).When For these∑ αi cases= d, there there are isadditional so-called uniqueness rules. ratio [5–7] for the triangle with C. When α = d, there is so-called uniqueness ratio [5–7] for the triangle with indicesC.αi When(i = 1,2,3)∑∑ iαi = d, there is so-called uniqueness ratio [5–7] for the triangle with indices α (i = 1, 2, 3) indices iαi (i = 1,2,3) q q q q 3→− 2 3→− 2 q q q q 3→− 2 α˜ 1 3→− 2 α α 2 3 α˜ 1 α2 α3 ∑ αi=d 2ε (6) = Nd µ A(α2, α3) α˜ 3 α˜ 2 , (6) α ∑ α =d q q 1 q q i 2ε q q q q 2→ 1 1→ 3 = Nd µ A(α2, α3) 2→ 1 α˜ 3 α˜ 2 1→ 3, (6) − α − − − q q 1 q q q q q q 2→− 1 1→− 3 2→− 1 1→− 3 The results (6) can be exactly obtained in the following way: perform an inversion The results (6) can be exactly obtained in the following way: perform an inversion qi The1/qi results(i = 1,2,3 (6) can), k be1/ exactlyk in the obtained subintegral in the expression following and way: in perform the integral an inv measure.ersion q →1/q (i = 1, 2, 3), k →1/k in the subintegral expression and in the integral measure. Theiqi→ inversion1/iqi (i = keeps1,2,3) angles, k→ 1/ betweenk in the momenta. subintegral After expression the inver andsion, in the one integral propagator measure. is The→ inversion keeps angles→ between momenta. After the inversion, one propagator is cancelledThe inversion because keeps∑ α anglesi = d and between the l.h.s. momenta. becomes After to be the equal inver tosion, a loop. one propagatorEvaluating it is cancelled because ∑ α = d and the l.h.s. becomes to be equal to a loop. Evaluating it using usingcancelled the rule because (5) and∑i α returningi = d and after the it l.h.s. to the becomes initial momenta, to be equal we to recover a loop. the Evaluating rule (6). it the rule (5) and returning after it to the initial momenta, we recover the rule (6). usingD. theFor rule any (5) triangle and returning with indices after itα toi (i the= 1, initial 2, 3) theremomenta, is the we following recover relation, the rule (which6). D. For any triangle with indices α (i = 1, 2, 3) there is the following relation, which is basedD. onFor integration any triangle by with parts indices (IBP)iα procedurei (i = 1, 2, [3)6,34 there,35]. is Thisthe following paper does relation, not use which the isIBPis based based procedure on on integration integration for massless by by parts charts parts (IBP) (IBP) and, procedure therefore, procedure [6 is,34 [ not6,,3534]. pro,35 This].vided This paper here. paper does However, does not use not it the use can IBP the be procedureobtainedIBP procedure directly for massless for from massless thecharts IBP charts and, in the and, therefore, massive therefore, is case not is (see not provided pro (48vided) below) here. here. by However, However, setting all it it can masses can be be obtainedtoobtained zero. directly directly from from the the IBP IBP in in the the massive massive case case (see (see (48 (48)) below) below) by by setting setting all all masses masses toto zero. zero.E. Using equation (48) with zero masses allows you to change the indices of the lineE. diagramsE. UsingUsing by equation equation an integer. (48 (48)) with One with zerocan zero also masses masses change allows allows line you indices you to to change changeusing the the the point indices indices group of of the the of linetransformationsline diagrams diagrams by by [ an6 an,36 integer., integer.37]. The One One elements can can also also ofthe change change group line line are: indices indices (a) the usingusing transition the the point pointto moment group groupum of of transformations [6,36,37]. The elements of the group are: (a) the transition2 to momentum presentation,transformations (b) conformal [6,36,37]. The inversion elements transformation of the groupp are:p (a)′ = thep/ transitionp , (c) a special to moment seriesum of → 2 presentation,transformationspresentation, (b) (b) conformal that conformal makes inversion inversionit possible transformation transformation to make one ofp p thep ver0p=tices=p/p/p unique,p,2 (c), (c) a a special and special then series series apply of of →→ ′ transformationsthetransformations relation (6) to that that it. makes makes it it possible possible to to make make one one of of the the vertices vertices unique, unique, and and then then apply apply thethe relation relationNote that (6 ()6) to the to it. it. presence of momenta, and especially the product of momenta in the numeratorsNoteNote that that of the the propagators,presence presence of of momenta, significantly momenta, and and complicates especially especially tthehe the FI product produc resultst of and of momenta momenta requires in gener- in the the numeratorsalizationnumerators of of the of the the rules propagators, propagators, for their calculation significantly significantly (see, complicates complicates for examp thele, the FI [38 FIresults– results41]). and However, and requires requires considera- general- gener- izationtionalization of of this the of case the rules isrules forbeyond theirfor their the calculation scope calculation of (see, this (see, for work. example, for examp [38le,–41 [38]).–41 However,]). However, consideration considera- oftion this of case this is case beyond is beyond the scope the scope of this of work. this work. 3. kR′-Operation 3. kR0-Operation 3. kRCalculation′-Operation of massless diagrams is most important for calculating critical exponents of modelsCalculationCalculation and theories, of of massless massless such diagramsas diagrams anomalous is is most most dimensions important important and for forβ calc-functions. calculatingulating critical One critical of exponents the expo- most nentsimportantof models of models and recipes theories, and theories, is such the as suchBogolyubov–Parasyuk–Hepp–Zimme anomalous as anomalous dimensions dimensions and β-functions. and βrmann-functions. One (BPHZ) of the One most ofR- theoperationimportant most important [42 recipes–44], recipeswhich is the extracts is the Bogolyubov–Parasyuk–Hepp–Zimme Bogolyubov–Parasyuk–Hepp–Zimmermann all singularities of any Feynmanrmann diagram. (BPHZ) (BPHZ)Formally,RR- - operationitoperation has the form[42 [42–44–44],], which which extracts extracts all all singularities singularities of of any any Feynman Feynman diagram. diagram. Formally,Formally, it hasit has the the form form R[FI] = FI kR′[FI], (7) [ ] = − [ ] RRFI[FI] =FIFI kRkR0 ′FI[FI,], (7) (7) where the kR′-operation takes into account− all− the singularities of the subgraphs of the where the kR -operation takes into account all the singularities of the subgraphs of the consideredwhere the kR diagram,0 ′-operation except takes for into the singularities account all the of the singularities diagram itself. of the A subgraphs very important of the considered diagram, except for the singularities of the diagram itself. A very important propertyconsidered of thediagram,kR′-operations except for is the the singularities independence of ofthe the dia resultgram ofitself. its applica A verytion important to any property of the kR′-operations is the independence of the result of its application to any Particles 2021, 4 364 Particles 2021, 1 4 Particles 2021, 1 4 Particles 2021, 1 4 Particles 2021, 1 4 property of the kR0-operations is the independence of the result of its application to any [ ] FIFIdiagramdiagram (i.e., (i.e.,kRkR0′[FIFI])) from from the the external external momenta momenta and and masses masses of of this thisFIFIdiagram.diagram. This This FI diagram (i.e., kR′[FI]) from the external momenta and masses of this FI diagram. This independenceindependencediagram (i.e., is is the the basis[ basis]) from of of of of the infra-red infra-red external rearrangement rearrangement momenta and approach app massesroach of [ [4545 this]] (see (seediagram. also also [ [4646––4949 This]),]), independenceFI is thekR′ basisFI of of infra-red rearrangement approach [45] (seeFI also [46–49]), whichFIwhichindependencediagram allows allows (i.e., us us iskR to theto′[ consider considerFI basis]) from ofFI ofFI thewith infra-redwith external the the minimum rearrangementminimum momenta possibleand possible masses approach set set of of of this [ masses45 masses]FI (seediagram. and andalso externalexternal [46 This–49]), which allows us to consider FI with the minimum possible set of masses and external independencemomenta:momenta:which allows neglecting neglecting is us the to basisconsider masses masses of of and andFI infra-redwith momenta momenta the rearrangement minimum should should not not possible lead lead app to toroach theset the appearanceof appearance[45 masses] (see also and of of [ infrared46 infrared external–49]), momenta: neglecting masses and momenta should not lead to the appearance of infrared whichsingularities.singularities.momenta: allows neglecting The usThe to ability ability consider masses to to remove removeFI andwith momenta and and the change change minimum should external external not possible leadmomenta momenta to set th ofe is is appearance masseswidely widely used andused of external (see, (see,infrared for for singularities. The ability to remove and change external momenta is widely used (see, for momenta:example,example,singularities. [ [50 neglecting50]] and The and references ability references masses to remove and and momentadiscussions discussions and change should therein). therein). external not leadWe We mo show show tomenta the it it appearance inis in widely our our example example used of infrared (see, below, below, for example, [50] and references and discussions therein). We show it in our example below, singularities.wherewhereexample, we we [ will50 will The] and consider consider ability references to in in remove detail detail and the discussions the and computation computation change therein). external of of the the Wemo singularmenta singu showlar it is instructure widely structure our example used of of a(see, a 5-loop 5-loop below, for wherediagram we that will contributes consider in to thedetailβ-function the computation of the ϕ44-model. of the singular structure of a 5-loop example,diagramwhere we [that50 will] andcontributes consider references into the detail andβ-function discussions the computation of therein).the ϕ4 -model. of We the show singu itlar in structure our example of a below, 5-loop diagramTo show that contributes the applicability to the ofβ-function the -operation, of the ϕ -model. it is convenient to start with the one- wherediagramTo we show that will contributesthe consider applicability in to detail the ofβ-function thekRkR computation0′-operation, of the ϕ4 of-model. it theis convenient singular structure to start with of a the 5-loop one- loopTo and show two-loop the applicability examples. of the kR′-operation,4 it is convenient to start with the one- diagramloop andTo showthat two-loop contributes the applicability examples. to the β of-function the kR′-operation, of the ϕ -model. it is convenient to start with the one- loopOne and loop. two-loopPutting examples.α = α = 1 in Equation (5), we have loopToOne and show loop. two-loop thePutting applicability examples.α11 = α22 of= the1 inkR Equation′-operation, (5), we it is have convenient to start with the one- One loop. Putting α1 = α2 = 1 in Equation (5), we have loop andOne two-loop loop. Putting examples.α1 = α2 = 1 in Equation (5), we have One loop. Putting α1 = α2 = 1 in Equation (5), we have 2ε 2 2ε µ2ε Γ2 (1 ε) µ2ε = Nd A(1,1)µ 22εε = N4 Γ 2(1 −ε) µ 22εε ,(8) (8) = Nd A(1,1) qµ = N4 εΓ((21− 2εε)) qµ , (8) →q 2ε εΓ2(2 2ε) 2ε q = Nd A(1,1)µq 2ε = N4 Γ (1 −−ε) µq 2ε , (8) →q = N A(1,1) q = N εΓ(2−− 2ε) q , (8) → d q2ε 4 εΓ(2 −2ε) q2ε →q − where µ is the scale the MS-scheme, defined as where µ is the scale the MS-scheme, defined as where µ is the scale the MS-scheme, defined as 2ε 2ε ε where µ is the scale the MS-scheme,µ defined= µ (4π as) Γ(1 + ε) . (9) µ2ε = µ2ε (4π)ε Γ(1 + ε) . (9) µ2ε = µ2ε (4π)ε Γ(1 + ε) . (9) 2ε 2ε ε Then kR′-operation, whichµ in= theµ one-loop(4π) Γ( case1 + ε is) . equal to k-operation, because (9) no Then kR0′-operation, which in the one-loop case is equal equal to to kk-operation,-operation, because because no subgraphs,Then kR where-operation,k-operation which is an in extractionthe one-loop of the case singular is equal part, to k-operation, is the one: because no subgraphs, where′ k-operation is an extraction of the singular part, is the one: subgraphs,Then kR where′-operation,k-operation which is in an the extraction one-loop of case the singular is equal part,to k-operation, is the one: because no subgraphs, where k-operation is an extraction of the singular part, is the one: 1 k = N4 1 , (10) k " q #= N4 ε1,(10) k" →q # = N41ε , (10) k "→q #= N ε, (10) → 4 ε "→q # 2 which, of course, is q2 -independent. which, of course, is q2-independent. which,Two of loops. course,Consider isqq2-independent. the diagram Two loops. Consider2 the diagram which,TwoTwo of loops. course, loops. Consider is q -independent. the diagram Two loops. Consider the diagram 4ε 2 3 2ε 2 µ4ε N24 Γ3 (1 ε)Γ(1 + 2ε) µ2ε = N2d A(1,1) A(1,1 + ε)µ 44εε = 2 N42 Γ 3(1 −ε)Γ(21 + 2ε) µ 22εε . (11) = N 2A(1,1) A(1,1 + ε) qµ = 2ε (1N 2ε) ΓΓ(2(1−3ε)Γ((11++2εε)) qµ . (11) →q d 4ε 2ε2(1 24 2ε) Γ3(2 3ε)Γ2(1 + ε) 2ε q = Nd A(1,1) A(1,1 + ε)µq 4ε = 2N − Γ (1−−ε)Γ(12 + 2ε) µq 2ε .(11) (11) →q = N2 A(1,1) A(1,1 + ε) q = 2ε (14− 2ε) Γ(2−− 3ε)Γ (1 + ε) q . (11) → d q4ε 2ε2(1 −2ε) Γ(2 −3ε)Γ2(1 + ε) q2ε →q − − The R′-operation extracts the singularities of subgraphs. In the considered case, we The R′-operation extracts the singularities of subgraphs. In the considered case, we haveThe the singularR′-operation internal extracts loop, the which singularities singularity of subgraphs. is shown in In Equation the considered (10). Therefore, case, we haveThe theR singular0-operation internal extracts loop, the which singularities singularity of subgraphs. is shown in In Equation the considered (10). Therefore, case, we thehaveRThe′ the-operationR singular′-operation of internal the extracts considered loop, the which singularities two-loop singularity diagram of subgraphs. is has shown the folloin In Equation thewingconsidered form (10). Therefore, case, we havethe R the′-operation singular of internal the considered loop, which two-loop singularity diagram is shown has the inEquation following (10 form). Therefore, the havethe R the-operation singular of internal the considered loop, which two-loop singularity diagram is shown has thein follo Equationwing form (10). Therefore, R -operation′ of the considered two-loop diagram has the following form the0 R′-operation of the considered two-loop diagram has the following form 1 R′ = N4 1 . (12) R′ = −N4 ε1 . (12) R "→q # = →q − N ε →q . (12) "′ →q # →q 41 →q R′ " q #= q −N ε q . (12) → → − 4 ε → "→q # →q →q

Evaluating loops in the r.h.s. of (12) and taking its singular parts (by the k-operation), Evaluating loops in the r.h.s. of (12) and taking its singular parts (by the k-operation), we haveEvaluatingEvaluating loops loops in in the the r.h.s. r.h.s. of of (12 (12)) and and taking taking its its singular singular parts parts (by (by the thek-operation),-operation), we have k wewe have haveEvaluating loops in the r.h.s. of (12) and taking its singular parts (by the k-operation), we have Particles 2021, 4 365

Particles 2021, 1 5 ParticlesParticles20212021,,11 55 Particles 2021, 1 5

1 11 kR′ kR = ==k k N4 N 1 kR′′ = k − εN44 " q "" # ## " q "" −− 4qε1ε # ## kR→′"→→qq # = →k"→→qq − N→4 ε →→qq # → → − ε → (13) "→q # "→q →q # 1 1 1 1 2 22 11 11 2 22 11 11 = N4 2 1 + 5 + 2L ε 11+ 2 + L ε = N4 2 +1 ,1 (13) == NN442 2 11++ 55++22LL εε 2 2 11++ 22++LL εε == NN44 2 2 ++ ,, (13) (13) N2ε4 22ε1ε2 1 5 2L−εε −− εε12 1 2 L ε N−42ε−−22ε1ε2ε 221εε , (13) =N22ε 1 + 5 + 2L ε − ε 1 + 2 +L ε =N2− 2ε + 2ε , (13) 4 2ε2 − ε2 4 − 2ε2 2ε where wherewhere where 2 µ µµ22 L =LLln== lnln µ... (14) (14) (14) L = qln2 µ222 . (14) !qq2 !! L = ln 2 ! . (14) q ! As we can see, the r.h.s. of (13) is q2-independent.22 AsAs we we can can see, see, the the r.h.s. r.h.s. of of ( (1313)) is is qq2-independent.-independent. Fifth loops. Consider the fifth-loop diagram2 contributed to β-function of ϕ4-model.44 FifthFifthAs we loops. loops. can see,ConsiderConsider the r.h.s. the the of fifth-loop fifth-loop (13) is q -independent. diagram diagram contributed contributed to to ββ-function-function of of of ϕϕ44-model.-model. TheTheThekR′-operationkRkRFifth-operation-operation loops. ofConsider the of of diagram the the diagram diagram the has fifth-loop thehas has following the the diagram following following form contributed form form to β-function of ϕ4-model. The kR′0′-operation of the diagram has the following form The kR′-operation of the diagram has the following form

1 11 kR′ kR = ==k k N4 N 1 ,,, (15) (15)(15) kR′′ = k − εN44 A , (15) q " q "" −− 4qε1ε AA # ## kR→′ →→qq = →k"→→qq − N→4 ε →→qq # , (15) → A B → A B − ε → A B →q AA B B "→q AA B B →q BB # A B A B B which contains the diagram itself and the counter-term, corresponding the singularity of whichwhich contains contains the the diagram diagram itself itself and and the the counter-term, counter-term, corcorresponding correspondingresponding the the singularity singularity of of the internal loop. thethewhich internal internal contains loop. loop. the diagram itself and the counter-term, corresponding the singularity of Since the right-hand side in (15) is q2-independent,2 22 we can cancel the external mo- the internalSinceSince the the loop. right-hand right-hand right-hand side side side in in in (15 ( (1515) is))q is is-independent,qq2-independent,-independent, we canwe we cancancel can cancel cancel the external the the external external momenta mo- mo- menta in the considered points and transfer them2 to the points AandB: mentainmenta theSince considered in in the the the considered considered right-hand pointspoints andpoints side transfer andin and (15 transfer transfer) them is q to-independent, them them the points to to the thepoint Apoint we andss can AandB:B: AandB: cancel the external momenta in the considered points and transfer them to the points AandB:

1 11 (16) kR′ kRkR′′ = ==k kk N4 NN ... (16) (16) (16) kR′ = k − εN44 1 A A . (16) →q qq " "" q q −− →qεε qq A # ## kR′ →→q = k" → →→q N4 →→q # . (16) AAA B B B " AAA B B B→ − ε A B BB # →q A B A B →q →q B A B A B B TakingTaking the theexternal external momenta momenta in thein the points points A A and and B B as as it it was was done done in the the r.h.s, r.h.s, we we have TakingTaking the the external external momenta momenta in in the the points points A A and and B B as as it it was was done done inin the the r.h.s, r.h.s, we we havethe the following following form form for for the the considered considered diagrams diagrams havehaveTaking the the following following the external form form for for momenta the the considered considered in the points diagrams diagrams A and B as it was done in the r.h.s, we A B Ahave the followingB form for the considered2 5ε diagrams 2 4ε AA BBN AA BB µ 22 55εε C µ 22 44εε AI B 4 NN44 AI B µµ2 3 CC33 µµ2 4 II44 N4 3 II33 = N=d=CN4 AC(1,1A((+1,11,14ε+)+44εε)) µ 5NεdNN4 N AC(31,1A((+1,11,13ε++) 33εε)) µ ,4ε,, (17) (17) (17) A I4 −B ε A I3 B = Ndd C44 A(1,1 + 4εq)2 222− NddεN44 A(1,1 + 3εq)2 222 ,(17) q −− Nqεε4 d 4 qµq2 −− d 4 Cεε3 qµq2 → →→qq I4 − →ε →→qq I3 = Nd C4 A(1,1 + 4ε) q − NdN4 ε A(1,1 + 3ε) q , (17) → − ε → q2 − ε q2 →q where→ theq integrals I and I are the internal blocks, which produce the diagrams in the wherewhere the the integrals integrals4 II44 andand3 II33 areare the the internal internal blocks, blocks, which which produce produce the the diagrams diagrams in in the the r.h.s.where of (16 the) after integrals integrationI4 and ofI3 theare blocks the internal with the blocks, propagator which produce between the A and diagrams B. From in the r.h.s.r.h.s.where of of the ( (1616 integrals)) after after integration integrationI4 and I3 ofare of the the the blocks blocks internal with with blocks, the the propagator propagator which produce betwee betwee thenn diagrams A A and and B. B. inFrom From the the dimensionalr.h.s. of (16) after property, integration the integrals of the blocksI and I withcan the be propagator represented between in the following A and B. form From the thether.h.s. dimensional dimensional of (16) after property, property, integration the the integrals integrals of the4 blocksII44 andand3 withII33 cancan the be be propagator represented represented betwee in in the then following following A and B. form form From dimensional property, the integrals I4 andI4 I3 canI3 be represented in the following form the dimensional property, the integrals I4 and I3 can be represented in the following form A B A B AA BB (µ2)4ε 22 44εε AA BB (µ2)3ε 22 33εε AI B 4 44 ((µµ2))4ε AI B 3 33 ((µµ2))3ε 4 II44 = N=d=C4 4 (µ ,) , 3 II33 = N=d=C3 3 (µ ,) , (18) (18) A I4 B = NNdd CC244 1+242ε211++44ε4εε , A I3 B = NNdd CC233 1+232ε211++33ε3εε , (18) q d4(q 4)((qq(2µ))1)+4εq d(3q 3)((qq(2µ))1)+3ε → qq I4 = (q ) →, qq I3 = (q ) , (18) →→q Nd C4 2 1+4ε →→q Nd C3 2 1+3ε (18) q (q ) q (q ) wherewherewhereC4 andCC andandC→3 areCC theareare coefficient the the coefficient coefficient functions functions functions of the→ of of integrals the the integrals integralsI4 andII andandI3. II .. where C44 and C33 are the coefficient functions of the integrals I44 and I33. where C and C are the coefficient functions of the integrals I and I . Therefore,Therefore,Therefore,4 we get we we3 that get get that that 4 3 whereTherefore,C4 and C3 weare get the that coefficient functions of the integrals I4 and I3. Therefore, we get that Particles 2021, 4 366

Particles 2021, 1 6 ParticlesParticles20212021,,11 66 Therefore, we get that Particles 2021, 1 5

C3 kR′ = N4 Sing C4 A(1,1 + 4ε) CC A(1,1 + 3ε) , (19) = ( + ) − Cε33 ( + ) (19) kRkR′′ →q == NN44SingSingCC44AA((1,11,1++44εε)) AA((1,11,1++33εε)) ,, (19) (19) −− εε →→qq A B AA B B 1 kR′ = k N4 " q # " q − ε q # → where all q2-dependence→ is cancelled in the r.h.s. singular→ terms. 22 wherewhereFor all all theqq2-dependence-dependence similar diagram is is cancelled cancelled but with in in the the the index r.h.s. r.h.s. 1 singular singularε in the terms. terms. line between A and B, we have ForFor the the similar similar diagram diagram but but with with the the index index 11 −εεinin the the line line between between A A and and B, B, we we have have 1 the followingFor the similar1 diagram but with the index1 1−−1 in the line between A and B, we have = N2 1 +thethe5 + following following2L ε 1 + 2 + L ε = N2 + − , (13) 4 2ε2 the following− ε2 4 − 2ε2 2ε whereC C CC C3 kR′ = N4 Sing C4 A(1 ε,1 + 4ε)2 A(1 ε,1 + 3ε) . (20) − µ −CCε33 − kRkR′′ q == NN44SingSing CC44AA((11 Lεε=,1,1ln++44εε)) .AA((11 εε,1,1++33εε)) .. (20)(20) (14) ′ → 4 4 −− q2−− εε −− →→qq A B − !− ε − → 1 ε AA− B B 11 εε −− 2 AsWe we would can see,like tothe note r.h.s. that of (in13 this) is q diagram-independent. we can represent, as an outer line, the line β ϕ4 betweenWeWeFifth would would points loops. like like AConsider and to to note note C, that thethat that is, ﬁfth-loop in in we this this get diagram diagram diagram we we contributed can can represent, represent, to as as-functionan an outer outer line,of line, the the-model. line line betweenThebetweenkR′-operation points points A A and and of the C, C, thatdiagram that is, is, we we has get get the following form

C C C CC CC CC C C 1 1 C kR′ = k (21) kR′ = k −N114ε A , (15) kRkR′′ q →q ==kk"q − ε→q q A # (21)(21) → → −− εε →AA →→qq AA B B AA B B →→qq BB AA B B →q AA B B BB qq and, thus, →→q whichand, thus, contains the diagram itself and the counter-term, corresponding the singularity of and,and, thus, thus, the internal loop. Since the right-hand side in (15) is q2-independent, we can cancel the external momenta in the consideredC points and transfer them to the points AandB: CC C C3,1 (22) kR′ = N4 Sing C4,1 A(1,1 + 3ε) A(1,1 + 2ε) , (22) −CC3,13,1ε kRkR′′ q == NN44SingSing CC4,14,1AA((1,11,1++33εε)) AA((1,11,1++22εε)) ,, (22) (22) kR′ → N4 C4,1 A −− εε A →→qq A B − ε → 1 ε AA− B B 1 kR′ = k 11 εε N4 . (16) where C4,1 and C3,1 are−− the coefficient functions of the integralsA I4,1 and I3,1, which are q " q − ε q # → wherewherethe internalCC4,14,14,1andand blocks,CCC3,13,13,1areare whichare the the thecoefficient produce coefficient coefficient the functions functions functions diagrams→ of of of thein the→ the the integrals integrals integrals r.h.s. of(I4,1II4,1214,1and)andand afterI3,1I,I3,13,1 integrationwhich,, which which are are theare of A B 4,1 3,1 A B 4,1 B 3,1 theinternalthe internal internal blocks blocks, with blocks, blocks, which the which which propagator produce produce produce the between the thediagrams diagrams diagrams A and in the C. in in theFromthe r.h.s. r.h.s r.h.s ofthe.. (21 of( of(dimen)21 after21))sional after after integration integration integration property, of the the of of thethe blocks blocks with with the the propagator propagator between between A A and and C. C. From From the the dimen dimensionalsional property, property, the the blockstheintegrals blocks withI4,1 with theand propagator theI3,1 propagatorcan be between represented between A and in A C. the and From following C. the From dimensional form the dimen property,sional property, the integrals the integralsintegralsI4,1 andTakingI3,1II4,14,1can theandand be externalII3,13,1 representedcancan be bemomenta represented represented in the in following the in in points the the following following form A and Bform form as it was done in the r.h.s, we have the followingA form C for the considered2 4ε diagramsA C 2 3ε 4 (µ ) 3 (µ ) AAI4,1 C C = N C 22 44εε ,5ε AAI3,1 C C = N C 22 334εεε , (23) A B A B 4d4 4,1 ((µµ2 )1)+23ε 3d3 3,1 ((µµ22 )1)+2ε N4 II4,14,1 4 (q ) µ II3,13,1C3 3 (qµ ) I4 I3q I4,1 == NNddCC4,14,1 2 1++3ε,, q I3,1 == NNddCC3,13,1 2 1++2ε ,, (23)(23) → = Nd C4 A(1,1d + 4((εqq)22))11+323εε →NdN4 A(1,1 +d 3ε) ((qq222))11+22εε, (17) − ε →→qq (q ) q →→q−q ε (qq ) →q because→q they→ contain one line with the index 1→ ε. becausebecause they they contain contain one one line line with with the the index index 1 1 −εε.. wherebecauseNow the they integralswe contain considerI4 oneand the lineI diagram,3 are with the the internalsimilar index 1 to blocks,−− theε. initial which one, produce but w theith lines diagrams between in the A − r.h.s.andNow BNow of and (16 we we A) after and consider consider C, integration having the the the diagram, diagram, diagram, the of indices the similar blockssimilar similar 1 to withε to. to the Taking the the theinitial initial initial propagator as one, above, one, one, but but but the with betwee w linewith linesith betweenn lines lines Abetween and between between A B.A and From and A A C − andBandtheas and an dimensional B B external and Aand and A A andC, and one, having property, C, C, we having having have the the indices the the integrals indices followingindices1 εI 1. 14 Takingand resultsεε.. TakingI Taking3 ascan above, be as as represented above, above, the line the the between line linein the between between following A and A A C and and form as an C C − −− asexternalas an an external external one, we one, one, have we we havethe have following the the following following results results results A B 2 4ε A B 2 3ε 4 (µ ) 3 (µ ) C I4 = N C4 , I3 = N C3 , (18) d (q2)1+4ε d (q2)1+3ε D1CC ε q E q C− → → C3,1 kR′ DD11 εε E E = N4 Sing C4,1 A(1 ε,1 + 3ε) A(1 ε,1 + 2ε) . (24) D1−−ε E − −CC3,13,1ε − kRkR′′ q where− C4 and C3==areNN44 theSingSing coefficientCC4,14,1AA((11 functionsεε,1,1++33ε ofε)) the integralsAA((11 Iε4ε,1,1and++22Iε3ε).) .. (24) (24) kR′ → N4 C4,1 A −− −− εε A −− →→qq A B − − ε − → Therefore,1 ε we get that AA− B B 11 εε −− Particles 2021, 1 6

C3 kR′ = N Sing C A(1,1 + 4ε) A(1,1 + 3ε) , (19) 4 4 − ε →q A B

where all q2-dependence is cancelled in the r.h.s. singular terms. For the similar diagram but with the index 1 ε in the line between A and B, we have − the following

C C3 kR′ = N Sing C A(1 ε,1 + 4ε) A(1 ε,1 + 3ε) . (20) 4 4 − − ε − →q A B 1 ε − We would like to note that in this diagram we can represent, as an outer line, the line between points A and C, that is, we get

C C C 1 kR′ = k (21) − ε A →q →q A B A B B →q and, thus,

C C3,1 kR′ = N Sing C A(1,1 + 3ε) A(1,1 + 2ε) , (22) 4 4,1 − ε →q A B 1 ε − where C4,1 and C3,1 are the coefﬁcient functions of the integrals I4,1 and I3,1, which are the internal blocks, which produce the diagrams in the r.h.s. of(21) after integration of the blocks with the propagator between A and C. From the dimensional property, the integrals I4,1 and I3,1 can be represented in the following form

A C 2 4ε A C 2 3ε 4 (µ ) 3 (µ ) I4,1 = N C , I3,1 = N C , (23) d 4,1 (q2)1+3ε d 3,1 (q2)1+2ε →q →q because they contain one line with the index 1 ε. Particles 2021, 4 − 367 Now we consider the diagram, similar to the initial one, but with lines between A and B and A and C, having the indices 1 ε. Taking as above, the line between A and C − as an external one, we have the following results

C D1 ε E − C3,1 kR′ = N Sing C A(1 ε,1 + 3ε) A(1 ε,1 + 2ε) . (24)(24) 4 4,1 − − ε − Particles 2021, 1 →q 7 Particles 2021, 1 A B 7 1 ε Particles 2021, 1 − 7 The l.h.s. diagram, which contains two lines with the index 1 ε, can be evaluated Particles 2021, 1 − 7 exactly.The l.h.s. Indeed, diagram, we can which represent contains the two r.h.s. lines diagrams with the as index combinations 1 ε, can blocks, be evaluated containing The l.h.s. diagram, which contains two lines with the index− 1 ε, can be evaluated exactly.two Indeed,lines with we the can index represent 1 ε, the and r.h.s. some diagrams additional as line com betweenbinations D blocks, and− E: containing exactly.The Indeed, l.h.s. diagram, we can whichrepresent− contains the r.h.s. two diagrams lines with as the com indexbinations 1 ε, blocks, can be containing evaluated two lines with the index 1 ε, and some additional line between D and− E: exactly.two lines Indeed, with the we index can− represent 1 ε, and the some r.h.s. additional diagrams line as betweencombinations D and blocks, E: containing The l.h.s. diagram, which− contains two lines with the index 1 ε, can be evaluated two lines with the index 1 ε, and some additional line between D− and E: exactly. Indeed, we can represent− the r.h.s. diagrams as combinations blocks, containing twoC lines with the index 1 ε, andC some additional line betweenC D and E: D1 ε C E D− 1 ε C E DC E − − 1 1 ε kR′ D1C ε E= k D1C ε E 1 D− C E kR − = k − − ε A 1 ε →q′ D1 ε E →q D1 ε E →q D− E (25) − − −1ε A kR′ →q AC B = k →q AC B →q 1 ε C B 1 ε 1 ε − Dq 1−Aε B E Dq 1−Aε B E −1ε Dq A 1 ε B E → − 1 ε → − 1 ε → 1 ε− kR′ = k − 1 ε A− B C3,2 A− B − ε A − B = Nd Sing q C4,2 A(1,1 + 21ε) ε A(1,1+qε) , 1 ε q (25) → − − ε C3,2 → − → 1 ε = Nd Sing C4,2 AA(1,1 + B2ε) A(1,1 + ε) , A B − B (25) 1 ε −C3,2ε 1 ε = Nd Sing C4,2 A(1,1 −+ 2ε) A(1,1 + ε) , − 1 ε (25) where C4,2 and C3,2− areε the coefficient functions of the integrals I4,2 and−I3,2, which can be where C4,2 andCC3,23,2 are the coefficient functions of the integrals I4,2 and I3,2, which can be = Nd Sing Cobtained4,2 Awhere(1,1 from+C4,22ε) theand diagramsC3,2Aare(1,1 the in+ thecoefficientε) r.h.s., taking functions out the of the line integrals betweenI4,2 Dand I E.3,2 Since, which they can(25) be obtained from− theε diagrams in the r.h.s. taking out the line between D and E. Since they havewhereobtained two linesC4,2 from withand theC the3,2 diagramsare index the 1 coefficient inε, fromthe r.h.s. dimensional functions taking of out properties the the integrals line b theetweenI4,2 integralsand DI and3,2I,4,2 which E.and SinceI can3,2 they be have two lines with the index− 1 ε, from dimensional properties the integrals I and I canobtained behave imagined two from lines as the with diagrams the index in 1 theε, r.h.s. from taking dimensional out the properties line between the integralsD and E.I4,2 Since4,2 and theyI3,23,2 where C4,2 and C3,2 are the coefficient−− functions of the integrals I4,2 and I3,2, which can be havecancan be be two imagined imagined lines with as as the index 1 ε, from dimensional properties the integrals I and I obtained from the diagrams in the− r.h.s. taking out the line between D and E. Since4,2 they3,2 can beD imagined as E ε2 4ε D E 2 3ε have two lines with the index4 1 (µ,) from dimensional properties3 the(µ integrals) I4,2 and I3,2 DI4,2 = EN C4,2 − 2, 4ε DI3,2 = EN C3,2 2,3ε (26) can be imagined as d 4 2 1+(2µε ) d 3 2 1+(µε ) I4,2 (q ) I3,2 (q ) →q D E = Nd C4,2 22 14+ε2→εq, D E = Nd C3,2 22 13+ε ε ,(26) (26) 4 ((qµ)) 3 (qµ )) →q I4,2 = N C4,2 , →q I3,2 = N C3,2 , (26) D E d (q22)14+ε2ε D E d (q22)13+ε ε q 4 (µ ) q 3 (µ ) Now we→ considerI4,2 the integral I4.2. After integrating→ I3,2 of the internal loop, we have = Nd C4,2 2 1+2ε , = Nd C3,2 2 1+ε , (26) Now we consider the integral(qI4.2) . After integrating of the internal(q loop,) we have Now→q we consider the integral I4.2. After→ integratingq of the internal loop, we have Now we considerC the integral I4.2. After integrating of theC internal loop, we have D1 ε C E D1 ε C E − − Now we considerD the integral1C ε =I E4.2N.d AfterA(1,1 integrating) D of the internal1C ε loop,. E we have (27) − = N A(1,1) − . (27) →q D1 ε E d →q D1 ε E →q AC− B = N A(1,1) →q AC− B . (27) 1 ε d ε 1 ε (27) Dq 1−Aε B E Dq 1−Aε B E → − 1 ε = N A(1,1) → ε − 1 ε . (27) The vertex DAC in the r.h.s.A diagram− B is unicald and, thus, it can beA replaced− B by triangle →q 1 ε →q ε 1 ε as it was shownThe vertex in Equation DAC in the (A6). r.h.s. Therefore,− B diagram we is have unical and, thus, it caAn be− replaced B by triangle 1 ε ε 1 ε as itThe was vertex shown DAC in Equation in the r.h.s. (−6). diagram Therefore, is unicalwe have and, thus, it can be− replaced by triangle The vertex DAC in the r.h.s. diagramC is unical and, thus, itC can be replaced by triangle as itThe was vertex shown DAC in Equation in the r.h.s. (6). diagram Therefore, is unicalwe have and, thus, it can be replaced by triangle as it was shown in EquationD (6). Therefore,1 ε C E we haveDC E − 2 2ε as it was shownNd A(1,1 in Equation) D (6). Therefore,1 ε we= have E D− E (28) C− D C 2 2ε Nd A(1,1q ) q= − (28) → D1 ε E → DD 2 2ε E N A(1,1) →q AC− B = →q C B − (28) d ε 1 ε D Dq 1−Aε B E Dq 2 2εB E N A(1,1) → ε − 1 ε = → − (28) Now the triangled CBE in the r.h.s. diagramA− B is unical and,D thus, it can beB replaced by(28) vertex inNow an agreement the triangle with CBE→ (q6): in theε r.h.s. diagram1 ε is unical→q and, thus, it can be replaced by A− B B vertexNow in an the agreement triangle CBE with in (6 the):ε r.h.s. diagram1 ε is unical and, thus, it can be replaced by vertex in an agreement with (6): − Now the triangle CBE in the r.h.s. diagram is unical and, thus1, itε can be replaced by vertex in an agreementC with (6): C − C C 1 ε D2 2ε E D E ε − D− = EN A(1,1) D 1 ε (29) D C 2 2ε d D C E− ε q − = Nd A(1,1q ) 1 ε (29) → DD 2 2ε E → D D −1 ε ε →q C B − = N A(1,1) →q C B E−1 ε (29) D d D − Dq 2 2εB E Dq 1B εε → − = N A(1,1) → E − (29) Therefore, finally,D we have B d D B →q →q 1 ε Therefore, finally, we have B B− Therefore, finally, we have 1 ε C C − 1 ε D1 ε C E D C − − Therefore, finally, we have E ε (1) 1 D1 ε = ENd A(1,1) D 1 ε = Nd A(1,1) J , (30) C− D C E− ε 1 q2(ε1) 1 →q = Nd A(1,1→q ) 1 ε = Nd A(1,1) J1 , (30) D1 ε E D D −1 ε ε (1) q12ε →q AC− B = N A(1,1) →q C B E−1 ε = N A(1,1) J , (30) 1 ε d D − d 1 2ε Dq 1−Aε B E Dq 1B εε (1) q1 → − 1 ε = N A(1,1) → E − = N A(1,1) J , (30) A− B d D B d 1 q2ε →q 1 ε →q 1 ε A− B B− 1 ε − ParticlesParticles20212021,,11 77

TheThe l.h.s. l.h.s. diagram, diagram, which which contains contains two two lines lines with with the the index index 1 1 εε,, can can be be evaluated evaluated −− exactly.exactly. Indeed, Indeed, we we can can represent represent the the r.h.s. r.h.s. diagrams diagrams as as com combinationsbinations blocks, blocks, containing containing twotwo lines lines with with the the index index 1 1 εε,, and and some some additional additional line line between between D D and and E: E: −−

CC CC CC DD11 εε E E DD11 εε E E 1 DD E E −− −− 1 1 ε kRkR′′ == kk 1− ε −− εε AA − →→qq →→qq →→qq AA B B AA B B BB 1 ε 1 ε 1− ε 1− ε 11 εε − − −− CC3,23,2 ==NNd SingSing CC4,2 AA((1,11,1++22εε)) AA((1,11,1++εε)) ,, (25)(25) d 4,2 −− εε

wherewhereCC4,24,2andandCC3,23,2areare the the coefﬁcient coefﬁcient functions functions of of the the integrals integralsII4,24,2andandII3,23,2,, which which can can be be obtainedobtained from from the the diagrams diagrams in in the the r.h.s. r.h.s. taking taking out out the the line line b betweenetween D D and and E. E. Since Since they they havehave two two lines lines with with the the index index 1 1 εε,, from from dimensional dimensional properties properties the the integrals integralsII4,2 andandII3,2 −− 4,2 3,2 cancan be be imagined imagined as as

DD E E 22 44εε DD E E 22 33εε 4 ((µ )) 3 ((µ )) I4,2 4 µ I3,2 3 µ I4,2 ==NNd CC4,2 + ε ,, I3,2 ==NNd CC3,2 +ε ,, (26) (26) d 4,2((qq22))11+22ε d 3,2((qq22))11+ε →→qq →→qq

NowNow we we consider consider the the integral integralII4.24.2.. After After integrating integrating of of the the internal internal loop, loop, we we have have

CC CC DD11 εε E E DD11 εε E E −− = N A(1,1) −− . (27) = Ndd A(1,1) . (27) →→qq →→qq A B A B A B ε A B 11 εε ε 11 εε −− −− TheThe vertex vertex DAC DAC in in the the r.h.s. r.h.s. diagram diagram is is unical unical and, and, thus, thus, it it ca cann be be replaced replaced by by triangle triangle asas it it was was shown shown in in Equation Equation ( (66).). Therefore, Therefore, we we have have

CC CC D1 ε E D E D1− ε E D22 22εε E Particles 2021, 4 NNd AA((1,11,1)) − == −− (28)(28)368 d DD →→qq →→qq A B B ε A B B ε 11 εε −− NowNowNow the the the triangle triangle triangle CBE CBE CBE in in in the the the r.h.s. r.h.s. r.h.s. diagram diagram diagram is is is unical unical unical and, and, and, thus thus thus,,, it it it can can can be be be replaced replaced replaced by by by vertexvertexvertex in in in an an an agreement agreement agreement with with with ( (6 (66):):):

11 εε CC CC −− DD2 2ε E E DD εε 2− 2ε = N A(1,1) EE (29)(29) D − = Ndd A(1,1) D (29) q D q D 1 ε →→q →→q 1− ε BB BB−

Therefore,Therefore,Therefore, finally, finally, finally, we we we have have have

11 εε CC CC −− D1 ε E D ε D1− ε E D E ε ((11)) 11 − ==Nd A((1,1)) E ==Nd A((1,1))J , (30) Nd A 1,1 D Nd A 1,1 J11 22εε , (30)(30) q q D 1 ε qq Particles 2021, 1 →→q →→q 1− ε 8 AA B B BB− 11 εε −− (1) where the integral J(1) is the table one (see [16]) (Note that the results obtained in [16] are where the integral J1 is the table one (see [16]) (Note that the results obtained in [16] are a simple recalculation1 of the corresponding results [7,8,12,13] found in the x-space. For a simple recalculation of the corresponding results [7,8,12,13] found in the x-space. For recalculation, it is convenient to use the concept of the so-called dual diagrams (see, for recalculation, it is convenient to use the concept of the so-called dual diagrams (see, for ex- example, [38–41]), which are obtained from the original ones by replacing all momenta with ample, [38–41]), which are obtained from the original ones by replacing all momenta with coordinates. With this replacement, the results of the FIs themselves remain unchanged, and coordinates. With this replacement, the results of the FIs themselves remain unchanged, only their graphical representation changes. In general, dual diagrams are used [38–41]) and only their graphical representation changes. In general, dual diagrams are used [38– in the massless case, but they can also be used [51–53] for propagators with masses.): 41]) in the massless case, but they can also be used [51–53] for propagators with masses.): (1) J(1) = J1(1, 1, 1, 1, 1, 1 ε, 1 ε). Here J1 = J1(1,1,1,1,1,1 − ε,1 − ε). Here 1 − − a1 a5 a6 (µ2)3ε J (a , a , a , a , a , a , a )(q2) = = N3 C (a , a , a , a , a , a , a ) , (31) 1 1 2 3 4 5 6 7 a 4 1 1 2 3 4 5 6 7 2 a 3d/2 (31) q 2 (q ) − → a7 a3 a6

where a = ∑7 a , and where a = ∑k=1 ai, and 1 1 3 2 C1(a1, a2, a3, a4, a5, a6, a7) = B0ζ5 + B1ζ6 + B2ζ ε + O(ε ) (32) C1 a1, a2, a3, a4, a5, a6, a7 1 2ε B0ζ5 B1ζ6 B2ζ3 ε O ε (32) 1 − 2ε − with 7 a = 1 + a ε, B = 20, B = 50, B = 20 + 6 a (33) ai = 1 + aiε, B0 = 20, B1 = 50, B2 = 20 + 6 ∑ ai (33) k=4 (1) It is convenient to write C = C1(1,1,1,1,1,11, 1, 1, 1, 1, 1 ε,1, 1 ε) as 1 1 − − 2 (1) (0) 3ζ3 C 1 = C 0 1 3 ε , (34) C1 C1 5ζ − 5ζ5 !

where 10 (0) 10 3 2 C = C1(1,1,1,1,1,1,11, 1, 1, 1, 1, 1, 1) = 2ζ5 + 5ζ6 + 2ζ3 ε + O(ε ) (35) 1 1 1 2ε 5 6 3 − 22 2 2 The counter-terms II33((qq )),, I3,1(q ) and I3,2(q ) can be expressed also though J1(a1, a2, a3, a4, a5, a6, a7) in the following from: 2 2 2 I3,2(q2) = J1(1,1 ε,1 ε,1,1,1,1), I3,1(q2) = J1(1 ε,1,1,1,1,1,1), I3(q2) = J1(1,1,1,1,1,1,1) . (36) I3,2(q ) = J1(1, 1 − ε, 1 − ε, 1, 1, 1, 1), I3,1(q ) = J1(1 − ε, 1, 1, 1, 1, 1, 1), I3(q ) = J1(1, 1, 1, 1, 1, 1, 1) . (36) − −Therefore, within the accuracy− O(ε2) their coefﬁcient functions are coincide:

2 2 (0) 2 C3 = C3,1 + O(ε ) = C3,2 + O(ε ) = C1 + O(ε ) . (37)

Taking into account above relations and the results for the one-loop results A(α, β), it is possible to show that within the accuracy O(ε2) the results for four-loop coefﬁcient functions are also coincide. Indeed,

and, thus, 2 (1) 2 (0) 3ζ3 2 C4 = C1 + O(ε ) = C1 1 + ε + O(ε ) (39) 10ζ5 ! Particles 2021, 4 369

Therefore, within the accuracy O(ε2) their coefﬁcient functions are coincide:

2 2 (0) 2 C3 = C3,1 + O(ε ) = C3,2 + O(ε ) = C1 + O(ε ) . (37)

A(1, 1 + 2ε) C e( Lε) C = C 3,1 − A(1, 1 + ε) A(1 ε, 1 + 2ε) + O(ε2), 4,1 4,2 A(1 ε, 1 + 3ε) − ε A(1 ε, 1 + 3ε) − − − − h i A(1, 1 + 3ε) C e( Lε) C = C 3 − A(1, 1 + 2ε) A(1 ε, 1 + 3ε) + O(ε2) 4 4,1 A(1 ε, 1 + 4ε) − ε A(1 ε, 1 + 4ε) − − − − h i and the terms C and C are suppressed and we have ∼ 3,1 ∼ 3 2 2 C4 = C4,1 + O(ε ) = C4,2 + O(ε ) (38) Particles 2021, 1 9 ParticlesParticles 20212021,, 11 99 Particles 2021, 1 and, thus, 9 2 (1) 2 (0) 3ζ3 2 C4 = C1 + O(ε ) = C1 1 + ε + O(ε ) (39) Therefore, for the initial diagrams, shown in the10 r.h.sζ5 ! of Equation (15), using the r.h.s Therefore, for the initial diagrams, shown in the r.h.s of Equation (15),), using using the the r.h.s r.h.s of (19Therefore,), we have for the initial diagrams, shown in the r.h.s of Equation (15), using the r.h.s of (19Therefore,),), we we have have for the initial diagrams, shown in the r.h.s of Equation (15), using the r.h.s of (19),), we we have have

5ε (1) 2 C(1) µ 55εε 5 (11) 22 = N5 C µ 5ε (40) = 54 2 11(1) µ2 (40) q = N4 5ε2 (1 C2ε)(1 6ε) µq2 ! (40) →q = N45 5εε2((1 −2ε1ε)()(1 −6εε)) qq2 ! (40)(40) →q 4 5ε2(1− 2ε)(1− 6ε) q2 ! →q − − ! and and and

4ε (0) 2 C(0) µ 44εε 4 (10) 22 = N4 C µ 4ε .(41) = 4 2 11(0) µ2 q = N4 4ε2 (C1 5ε) µq2 ! .. (41) (41) →q = N44 4εε2((11−5εε)) qq2 ! . (41) →q 4 4ε2(1− 5ε) q2 ! →q − !

Therefore, for the initial combination shown in (15), we have Therefore, for the initial combination shown in (15),), we we have have Therefore, for the initial combination shown in (15), we have

(1) 5Lε (0) 4Lε C(1) e C(0) e 10 (11) 55LLεε (10) 44LLεε (42) (4π)10 kR′ = Sing C1(1)ee C1(0)ee , (42) (4π) 10 kR′ = Sing 5ε2(1 12ε)(5L1ε 6ε) − 4ε21(1 4L5εε) , (42) (4π)10 kR′ q = Sing " 22 C1 e C221 e #, (42) (4π) kR′ →q = Sing" 5εε ((1 −2εε)()(1 −6εε)) − 4εε ((1 −5εε))# , (42) →q A B " 5ε2(1− 2ε)(1− 6ε) − 4ε2(1− 5ε) # →q A B " # A B − − − (1) (0) where L is giveA in B Equation (14). Taking the results for C( ) and C( ) , given in where is give in Equation (14). Taking the results for (1(111)) and (100), given in where LL is give in Equation ( (14).). Taking the results for for CC1(1) and CC1(0),, given in whereEquationsL is (34 give) and( in35 Equation), respectively, (14). we Taking have the results for C11 and C1 , given in EquationsEquations ( (3434))) and( and(and (3535),), respectively, respectively, we we have have 1 1 Equations (34) and(35), respectively, we have

ζ 1 5 17 (4π)10 kR = ζ 5 1 5 ζ +17 ζ2 5ζ . (43) 1010 ′ ζ552 1 5 6 17 223 5 (43) (4π) kR′ = − ε − ε 2ζ6 + 5 ζ3 −5ζ5 . (43) (4π)10 kR′ q = ζ225 1 5 ζ6 + 17 ζ32 5ζ5 . (43) (4π) kR′ →q = − εε − εε 2 ζ + 5 ζ − 5ζ . (43) →q − ε2 − ε 2 6 5 3 − 5 →q 4. Calculation of Massive Feynman Integrals (Basic Formulas) 4. Calculation of Massive Feynman Integrals (Basic Formulas)s) 4. CalculationFeynman ofintegrals Massive with Feynman massive Integrals propagators (Basic are Formula significants) ly more complicated ob- Feynman integrals with massive propagators are significantlyly more more complicated complicated ob- ob- jectsFeynman compared integrals to the massless with massive case (see, propagators for example, are significant the recently review more complicated [54]). The basic ob- jectsjects compared compared to to the the massless massless case case (see, (see, for for example, example, the the re recentcent review review [ [54]).]). The The basic basic jectsrules compared for calculating to the such massless diagrams case (see,are discussed for example, in Sectio the rencent2, which review are [54 supplemented]). The basic rulesrules for for calculating calculating such such diagrams diagrams are are discussed discussed in in Sectio Section 2,, which which are are supplemented supplemented rulesby new for ones calculating containing such directly diagrams massive are discussed propagators. in Section 2, which are supplemented by new ones containing directly massive propagators. by newLet ones us containing briefly directlyconsider massive the rulespropagators. for calculating diagrams with the Let us briefly consider the rules for calculating diagrams withth the the massiveLet propagators. us briefly consider the rules for calculating diagrams with the massive propagators. massivePropagator propagators. with mass M is represented as Propagator with mass M isis represented represented as as Propagator with mass M is1 represented as M 1 = M . (44) 2 1 2 α = (q2 +1M2 )α = q Mα .. (44) (44) ((qq2 + M 2))α = →q α . (44) (q2 + M2)α →q α The following formulas hold: →q The following formulas hold: The following formulas hold: Particles 2021, 1 9

Therefore, for the initial diagrams, shown in the r.h.s of Equation (15), using the r.h.s of (19), we have

(1) 5ε C µ2 = N5 1 (40) 4 5ε2(1 2ε)(1 6ε) q2 →q − − !

and

(0) 4ε C µ2 = N4 1 . (41) 4 4ε2(1 5ε) q2 →q − !

Therefore, for the initial combination shown in (15), we have

(1) 5Lε (0) 4Lε 10 C1 e C1 e (4π) kR′ = Sing , (42) 5ε2(1 2ε)(1 6ε) − 4ε2(1 5ε) →q " − − − # A B (1) (0) where L is give in Equation (14). Taking the results for C1 and C1 , given in Equations (34) and(35), respectively, we have

10 ζ5 1 5 17 2 (4π) kR′ = ζ6 + ζ 5ζ5 . (43) Particles 2021, 4 − ε2 − ε 2 5 3 − 370 →q

4. Calculation of Massive Feynman Integrals (Basic Formulas) 4. Calculation of Massive Feynman Integrals (Basic Formulas) Feynman integrals with massive propagators are significantly more complicated ob- Feynman integrals with massive propagators are significantly more complicated jects compared to the massless case (see, for example, the recent review [54]). The basic objects compared to the massless case (see, for example, the recent review [54]). The basic rules for calculating such diagrams are discussed in Section 2, which are supplemented rules for calculating such diagrams are discussed in Section2, which are supplemented by by new ones containing directly massive propagators. new ones containing directly massive propagators. Let us briefly consider the rules for calculating diagrams with the Let us briefly consider the rules for calculating diagrams with the massive propagators. massive propagators. Propagator with mass M is represented as Propagator with mass M is represented as 1 M = . (44) Particles 2021, 1 (q2 + M2)α α 10(44) →q Particles 2021, 1 10 The following formulas hold: The following formulas hold: A. ForA. For simple simple chain chain of two of two massive massive propagators propagators with with the samethe same mas mass,s, we have we have

A. For simple chain1 of1 two massive1 1 propagators with1 1 the same mass, we have 2 2 α 2 2 =α = , 2( +2 α1 ) 1 2( +2 α2 ) 2 2 2 2 (α2 +(αα1+) α2) (q +q M M)1 (q +q M M)1 (q (+q M+ M) 1)1 2 = , or graphically 2 2 α1 2 2 α2 2 2 (α +α ) or graphically (q + M ) (q + M ) (q + M ) 1 2 or graphically M M M = , (45)(45) q α1 Mα2 M q α1 + α2M → →= , (45) i.e., the product of propagatorsα with the sameα mass M is equivalent+ to a new propagator →q 1 2 →q α1 α2 withi.e.,i.e., the the themass product productM and of of an propagators propagators index equal with with to the the the sum same same of massthe mass indicesMMisis equivalent ofequivalent the original to to a apro new newpagators. propagator propagator withwithB. Massive the the mass mass tadpoleMMandand isan an integrated index index equal equal as to to the the sum sum of of the the indices indices of of the the original original propagators. propagators. B.B. MassiveMassive tadpole tadpole is is integrated integrated as as Dk R(α1, α2) = Nd 2α1 ( 2 + 2)α2 2(α1+α2(αd,/2α) ) Z k k MDkDk M RR(−α1 , α2 ) = Nd 1 2 2α1 2 2 α2 = Nd 2(α1+α2 d/2) k 2α1((k 2++M 2))α2 M 2(α1+α2− d/2) where ZZ k k M M − Γ(d/2 α1)Γ(α1 + α2 d/2) wherewhere R(α, β) = − − . (46) Γ(d/2Γ(d/2α ))ΓΓ((αα )+ α d/2) R(α, β) = Γ(d/2− α11)Γ(2α11 + α22− d/2). (46) R(α, β) = −Γ(d/2)Γ(α ) − . (46) C. A simple loop of two massive propagatorsΓ(d/2 with)Γ(α masses22) M1 and M2 can be represented asC. hypergeometricA simple loop function, of two which massive can be propagators calculated in with a general masses form,M forand example,M can be C. A simple loop of two massive propagators with masses M1 and1 M2 can2 be repre- by Feynman-parameterrepresentedsented as hypergeometric as hypergeometric method. function, With function, this which approach, which can be canitcalculated is verybe calculatedconvenientin a general in to a form,general represent for form, example, the for loopexample,by asFeynman-parameter the integral by Feynman-parameter of the propagator method. With with method. thisthe “effective approach, With this ma it approach, isss” veryµ [55convenient– it67 is]: very to convenient represent the to represent the loop as the integral of the propagator with the “effective mass” µ [55–67]: loop as the integral of the propagatorDk with the “effective mass” µ [55–67]: (4π)d/2 × [(q k)2 + M2]α1Dk[k2 + M2]α2 (4π)dd/2/2Z − 1 Dk 2 (4π) 2 2 α1 2 α 21 α2 α 1 Γ(α + α ××Zd/2[([(q)q k1k))2++MM12]]α1[[kkds2++ s M1M22](]α12 s) 2 = 1 2 − Z − 1 − 2 − − − 1 2 2 α1 1 2 α2α 1+α d/2 ΓΓ((αα11)Γ+(αα22) d/20) [s(1 s)q + Mdss s+α1−M1(1(1 s)s)]α2−1 1 2 = Γ(α1 + α2− dZ/2) − ds1 s − 2(1−−s) − − = − 2 2 −2 α1+α2 d/2 Γ(α1)Γ(α2) 1 0 [s(1 s)q 2+ M 2s + M 2(1 s)] α1+α2 d/2 2 2 Γ(α + αΓ(α1)dΓ/2(α)2) Z 0 [s(1ds s)q + M1 s + M12 (1 s)] −− M M = 1 2 Z −− 1 2 −− , µ2 = 1 + 2 . − 11 α˜ 1 α˜ 2 2 α +α d/2 2 2 ΓΓ((αα11)Γ+(αα22) d/20) s 1 2 (1 sds) 1 [q + µ ] 1 12 1 sM 2 s M 2 = Γ(α1 + α2− dZ/2) − − ds− 1− , µ22=− M11 + !M22 . = Γ(α )Γ(−α ) 11 α˜α˜2 (1 )11 α˜α˜1 [ 2 + 2]α1+α2 d/2 , µ =1 s + s . Γ(α1 )Γ(α2 ) Z00 ss − 2 (1 ss) − 1 [qq2 +µµ2]α1+α2− d/2 1 s s ! It is useful to rewrite1 2 the equationZ − graphically−− − as − −− ! ItIt is is useful useful to to rewrite rewrite the the equation equation graphically graphically as as M1

α1 M1 µ Γ(α + α d/2) 1 ds α1 1 2 = Nd − 1 α˜ 1 α˜ . (47) Γ(α1)Γ(α2) 0 s 1 2 (1 s) 1 α +α d/2µ q Γ(α1 + α2 d/2Z ) − ds− q 1 2− → M2 − → (47) = Nd − 1 α˜ 1 α˜ . (47) Γ(α1)Γ(α2) 0 s 2 (1 s) 1 α +α d/2 →q Z − − − →q 1 2− α2 M2

α2 D. For any triangle with indices αi (i = 1, 2, 3) and masses Mi there is the following relation, which is based on integration by parts (IBP) procedure [6,34,35,57–59] D. For any triangle with indices αi (i = 1, 2, 3) and masses Mi there is the following relation, which is based on integration by parts (IBP) procedure [6,34,35,57–59] Particles 2021, 4 371

D. For any triangle with indices αi (i = 1, 2, 3) and masses Mi there is the following relation, which is based on integration by parts (IBP) procedure [6,34,35,57–59] Particles 2021, 1 11

q →q 3− 2 M2 M3 α2 α3 M1 (d 2α1 α2 α3) α − − − q q 1 q q 2→− 1 1→− 3 q →q q →q 3− 2 3− 2 M2 M3 M2 M3 α2+1 α3 α2+1 α3 (48) M1 M1 2 2 2 = α2 (q2 q1) + M1 + M2 α q q α1 1 q q− − q× q 1 q q # 2→− 1 − 1→− 3 2→− 1 1→− 3 q →q 3− 2 M2 M3 α2 α3 M1 2 +α3 α2 α3, M2 M3 2M1α1 . (48) ↔ ↔ − q× q α1+1 q q 2→− 1 1→− 3 µ EquationEquation (48) ( can48) can be obtained be obtained by byentering entering the the factor factor(∂/(∂∂/kµ∂k)()(k k q1)q )intoµ into the the trian- triangle µ − − 1 gle integrand,integrand, shown shown below below as as[...[...].]. Indeed, Indeed, using using the the procedure procedure of integration of integration by parts, by parts, we obtainwe obtain ∂ ∂ µ ∂ ∂ µ d Dk ... = Dk (k q1) µ... = Dk (k q1) ...µ d Dk ... = Dk∂kµ −(k q1) ... = ∂Dkkµ −(k q1) ... Z Z ∂kµ − Z ∂kµ − Z Z Z µ ∂ ∂ Dk (Dkk (qk1) q )µ ...... (49) (49) − − ∂1kµ ∂ Z − Z − kµ TheThe first first term term in the in the r.h.s. r.h.s. becomes becomes to tobe be zero zero because because it it can can be berepresented represented as as a a sur- surface faceintegral integral on the the infinite infinite surface. surface. Evaluating Evaluating the the second secondterm term in thein the r.h.s. r.h.s. wereproduce we reproduce EquationEquation (48). (48). As it is possible to see from Equations (48) and (49) the line with the index α1 is distin- As it is possible to see from Equations (48) and (49) the line with the index α1 is guished.distinguished. The contributions The contributions of the other of lines the other are the lines same. are theTherefore, same. Therefore, we will call we below will call the line with the index α1 as a “distinguished line”. It is clear that a various choices of the below the line with the index α1 as a “distinguished line”. It is clear that a various choices distinguishedof the distinguished line produce line different produce types different of the types IBP relatof theions. IBP relations. TheThe IBP IBPrelations relations lead lead to differential to differential equations equations [57–59 [57,68–59–72,68] for–72] the for considered the considered dia- di- gramsagrams (see an (see example an example in Section in Section5) with5) inhomogeneous with inhomogeneous terms containing terms containing simpler simpler diagrams,diagrams, i.e., diagrams i.e., diagrams containing containing fewer propagators. fewer propagators. By repeating By repeating the IBP procedure the IBP procedure sev- eralseveral times, in times, the last in the step last we step can obtainwe can an obtain inhomogeneous an inhomogeneous term containing term containing only very only simplevery diagrams simple diagrams that can be that computed can be computed using the A–C using rules the discA–Cussed rules above, discussed as well above, as as the ruleswell as discussed the rules in discussed Section 2. in By Section integrating2. By successivelyintegrating successively the terms in the all other terms inhomo- in all other geneousinhomogeneous terms, in the terms, last step in the we last can step get the we result can get for the the result original for the diagram. original diagram. E. IE. wouldI would also also like like to note to note the the importance importance of of the the inverse-mass inverse-mass expansions expansions of of mas- massive siveFIs FIs depending depending on on one one mass mass (or on two masses in the on-shalon-shalll case). The structure of of the the coefficientscoefficients of of such such expansions expansions often often has has some some universal universality,ity, preserving preserving the the complexity complexity (or (or rang)rang) of of harmonic harmonic (or (or nested) nested) sums sums [ [7373,,7474]] (a more detailed discussion can be found found in a in arecent recent review review [75 [75]).]). This This property property simplifies simplifies the the structure structure of the of the results results (and (an thed correspond-the cor- respondinging ansatz ansatz for it), for and it), also and makes also makes it possible it possible to predict to pred theict unknown the unknown terms of terms the expansion. of the expansion.This property This property is associated is associated with a specific with a specificform of formdifferential of differential equations equations (see discussions (see discussionsin [76–80 in]) [ and76–80 is]) most and successfully is most successfully used in the used so-called in the canonical so-calledapproach canonical [ a71pproach,72], which [71,72is], currently which is the currently most popular. the most popular. NoteNote also thatalso a that similar a similar property property (in reality, (in evenreality, more even stri morect) takes strict) place takes in the placeN = in4 the superN Yang–Mills= 4 super (SYM) Yang–Mills model (SYM) not only model for some not masteronly for integr someals, master but for integrals, the kernel but of the for the Balitsky–Fadin–Lipatov–Kuraev (BFKL) equation [81–88], as well as for the anomalous dimensions contributed to the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAR) equations [89–93], and for Wilson coefficient functions (see, respectively, [94–99]). This property was called [94] the principle of maximal transcendentality and allows us to obtain anomalous dimensions of Wilson operators and Wilson coefficient functions without any Particles 2021, 4 372

kernel of the Balitsky–Fadin–Lipatov–Kuraev (BFKL) equation [81–88], as well as for the anomalous dimensions contributed to the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi Particles 2021, 1 12 ParticlesParticles20212021,,11 (DGLAR) equations [89–93], and for Wilson coefficient functions (see, respectively, [94–9912]).12 This property was called [94] the principle of maximal transcendentality and allows us to obtain anomalous dimensions of Wilson operators and Wilson coefficient functions without calculations,any calculations, directly directly from from the corresponding the corresponding QCD QCD results results(if they (if they exist). exist). Moreover, Moreover, this propertythiscalculations,calculations, property (togetherdirectly (together directly with from from with the the rulesthe the corresponding corresponding rules [100 [,101100],101 for] analyticQCD for QCD analytic results results continuation) continuation)(if(if they they exist). exist). allows allows Moreover, Moreover, predicting predicting this thisthe property (together with the rules [100,101] for analytic continuation) allows predicting the propertyansatzthe ansatz [102 (together [,102103,]103 for with] finding for findingtherules the solution the [100 solution,101 of] for the of analytic corresponding the corresponding continuation) Bethe-ansatz Bethe-ansatz allows predicting[104– [104106–]106 and the] ansatz [102,103] for finding the solution of the corresponding Bethe-ansatz [104–106] and obtainingandansatz obtaining [102 results,103 results] for for finding foranomalous anomalous the solution dimensions dimensions of the in corresponding in high high orders orders o Bethe-ansatzf of perturbation perturbation[104 theory theory–106] (see, and respectively,obtainingobtaining results results [95–98 for for,,107 anomalous anomalous–111].]. Thus, dimensions dimensions the the anomalous anomalous in in high high dimensions orders orders o off perturbation perturbation of the Wilson Wilson theory theory operators operators (see, (see, respectively, [95–98,107–111]. Thus, the anomalous dimensions of the Wilson operators wererespectively, found [111 [95]–98 in, the107– seven-loop111]. Thus, approximation. the anomalous dimensions of the Wilson operators werewere found found [ [111111]] in in the the seven-loop seven-loop approximation. approximation. 5. Two-Loop on-Shall Master Integral 5. Two-Loop on-Shall Master Integral 5. Two-LoopConsider on-Shall the two-loop Master on-shall Integral master integral (with 2 = 2) Consider the two-loop on-shall master integral (with q 2 = m 2) ConsiderConsider the the two-loop two-loop on-shall on-shall master master integral integral (with (withqq2 ==−mm2)) −− m mm

I(m2, M2) = M , (50) I(m22, M22) = M , (50) I(m , M ) = →q M m , (50) →qq mm → M MM contributing to the αs-correction to the ratio between the pole and MS masses of the Higgs contributingcontributing to to the theααs-correction-correction to to the the ratio ratio between between the the pole pole and andMSMSmassesmasses of of the the Higgs Higgs bosoncontributing in the standardto the αss-correction model. to the ratio between the pole and MS masses of the Higgs bosonboson in in the the standard standard model. model. bosonExcept in the for standard special model. places, below we will not indicate the masses m and M, but will ExceptExcept for for special special places, places, below below we we will will not not indicate indicate the the masses massesm andm andM, butM, will but willuse use ratherExcept thin for and special thick places, lines for below propagators we will not with indicatem and M the, respectively. masses m and M, but will ratheruseuse rather rather thin thin thinand and thickand thick thick lines lines lines for propagators for for propagators propagators with with withm andmmandandM, respectively.MM,, respectively. respectively.2 2 Applying the IBP relation for the inner loop of the integral I(m22, M22), we have ApplyingApplying the the IBP IBP relation relation for for the the inner inner loop loop of of the the integral integralII((mm2,,MM2)),, we we have have

2 2 2 2 (d 3) I(m , M ) = 2 (4M m ) . (51) (d− 3) I(m22, M22) = − − (4M22− m22) . (51) (d 3) I(m , M ) = 22 (4M m ) .(51) −− 2 −− −− −− 22 2 22 It is possible to see that the last integral in the r.h.s. can be represented as ItIt is is possible possible to to see see that that the the last last integral integral in in the the r.h.s. r.h.s. can can beb bee represented represented as as 1 ∂ 1 ∂ I(m2, M2) (52) − 21 ∂M∂ 2 ( 22 22) 2 II(mm ,,MM ) (52)(52) −− 22 ∂∂MM2 and, thus, Equation (51) can be rewritten as the differential equation and,and, thus, thus, Equation Equation ( (5151)) can can be be rewritten rewritten as as the the differential differential equation equation 1 ∂ (4M2 m2) 1 ∂ I(m2, M2)=(d 3) I(m2, M2) + J(m2, M2) , (53) ( 22− 22)21 ∂M∂ 2 ( 22 22)=( − ) ( 22 22) + ( 22 22) (44MM mm ) 2 II(mm ,,MM ))=( = (dd 33) II(mm ,,MM ) + JJ(mm ,,MM ) ,, (53) (53) −− 22 ∂∂MM2 −− where the inhomogeneous term wherewhere the the inhomogeneous inhomogeneous term term

2 2 J(m , M ) = 2 (54) 22 22 − JJ((mm ,,MM )) = = 22 (54)(54) −− 2 (54) 22 contains only less complicated diagrams. The solution of the equation with the boundary containscontains only only less less2 complicated complicated2 diagrams. diagrams. The The solution solution of of th thee equation equation with with the the boundary boundary conditioncontains onlyT(M less2 complicated∞, m 2) = 0 diagrams.has the following The solution form of the equation with the boundary condition T((M2 → ∞, m2)) = = 0 has the following form condition T(M2 →∞, m2) = 0 has the following form T M → m 2 2ε 2 →2 2 2 (µ ) 2 2 2 µ I(M , m ) = N I(x) (µ22)22εε, J(M , m ) = N J(x) µ22 , (55) 22 22 422 ((µ22)22εε 22 22 422 ( µ22)2ε II((MM2,,mm2)) = = NN24 II((xx)) (mµ ) ,, JJ((MM2,,mm2)) = = NN24 JJ((xx)) mµ ,, (55) (55) I(M , m ) = N4 I(x) ((m22))22εε, J(M , m ) = N4 J(x) ((m22))22εε, (55) 4 (m2)2ε 4 (m2)2ε ∞ 1/2 ε z 1/2 ε 2I1(x1)dx1 (4 z) − 2J(z1)dz1 ( ) = ( ) ∞ = 1/21/2 εε z I x 4x 1 1/2− ε ∞ 22II1((xx1))dx3/2dx1 ε ((44−1/2zz)) ε − z 1/2+ε22JJ((zz1))dzdz1 , (56) I(x) =− (4x− 1)1/2 −ε x∞ (4x 1( 1) 1 =− ( z− )1/2−ε 0z ( ( 1) )(13/2 ε) , (56) I(x) = (4x 1)1/2−ε Z 2I11 x1 dx3/21− ε = 4 −1/2z − ε − Z z11/2+2ε J4 z1 zdz1 1( − ) , (56) I(x) = −(4x −1) − x (4x1− 1)3/2 ε = − −z1/2 ε 0 z1/2+ε (4− z )(3/23/2 εε) , (56) − − Zx (4x1 1)3/2−−ε − z1/2−−ε Z0 z1/21 +ε(4 z11)(3/2 −ε) − − Zx (4x1 −−1) − − z − Z0 z1 (4 −z ) − where Z − Z 1 − 1 − wherewhere M2 M2 1 x = M22, z = M22 = 1. (57) Mm2 Mm2 x1 xx == 2 ,, zz== 2 == .. (57) (57) mm2 mm2 xx Particles 2021, 4 373

where

Particles 2021, 1 2 2 13 Particles 20212021,, 11 M M 1 1313 x = , z = = . (57) Particles 2021, 1 m2 m2 x 13

5.1. J(x) 5.1. JJ((x)) 5.1. J(Thex) result for the first diagram of ( ) can be written in the following form [112,113]: 5.1. J(Thex) result for the first diagram of JJ(xx) can be written in the following form [112,113]: The result for the first diagram of JJ((x)) can be written in the following form [112,,113]:]: The result for the first diagram of J(x) can be written in the following form [112,113]: 2 2ε 2 ((µ22))22εε 1 1 1 1 zz 2 = N2 (µ ) 1 + 1 1 + ln z + 1 −z Li2(z) , (58) 2 = N244 22 22εε 2 + + lnln z+ − Li2((z)) ,, (58) (58)(58) 2 N42 ((Mµ 22)))22εε 21εε22 21εε − 12 z 1 −zz z 2 z 2 = N (M ) 2ε + 2ε − 2 + ln z + −z Li (z), (58) 4 (M2)2ε 2ε2 2ε − 2 z 2 where where wherewhere Li2((zz)) = = Li2((zz)) + +lnlnzzlnln((1 zz)) (59)(59) Li22(z) = Li22(z) + ln z ln(1 −z) (59) LiLi22((zz)) = =LiLi22((zz)) + +lnlnzzlnln((11− zz)) (59)(59) and Li2((zz)) isis dilogarithm dilogarithm [ [114]] (more (more complicated complicated functions functions−− can can be be found found in in Refs. Refs. [ [115– and Li22(z) is dilogarithm [114] (more complicated functions can be found in Refs. [115– 117and]).Li (z) is dilogarithm [114] (more complicated functions can be found in Refs. [115–117]). 117and]).]). Li22(z) is dilogarithm [114] (more complicated functions can be found in Refs. [115– TheThe result result for for off-shall off-shall one-loop one-loop can can be be presented presented in in the the for formm (by (by using, using, for for example, example, 117]).The result for off-shall one-loop can be presented in the form (by using, for example, thethe rule rule C) C) thethe rule ruleThe C) C) result for off-shall one-loop can be presented in the form (by using, for example, the rule C) N (µ22)εε 1 1 + y 1 = N44 ((µ2))ε 1 + 1 + y ln y + ε 1 ln22 y 2 ln y ln(1 + y) 2Li ( y) ζ , (60) = 4 22 εε + ln y+ εε ln2 y 2 ln y ln((1+ y)) 2Li22(( y)) ζ 22 , (60) ((1 N42εε)) ((µm22)))εε 1εε 11+ yy ln y 12 ln2 y −2 ln y ln 1 y −2Li2 −y − 2 , (60)(60) →qq = (1 −2ε) (m ) ε + 1 −y ln y + ε 2 ln y − 2 ln y ln(1 + y) − 2Li (−y) − ζ , (60) →q − 2 ε − 2 2 q (1 2ε) (m ) ε 1 y 2 − − − − → −where y is so-called − conformal variable (see Appendix A with the substitutionm22 q22), where y isis so-called so-called conformal conformal variable variable (see (see Appendix AppendixA with the substitution m2 → −q2),), wherewhichwhere isyisis very so-called so-called convenient conformal conformal in the variable variable case of (see massive (see Appendix Appendix FIs (see,A with for example, the substitution [55,56,66m,11822→]) − q22),), which isy very convenient in the case of massive FIs (see, for example, [55,,56,,66m,,118→])]) −q whichTaking is very on-shall convenient limit in (see the also case Appendix of massiveA), FIs i.e., (see, for example, [55,56,66,118→]) − whichTaking is very on-shall convenient limit in (see the also case Appendix of massiveA),), FIs i.e., i.e., (see, for example, [55,56,66,118]) TakingTaking on-shall on-shall limit limit (see (see also also Appendix AppendixA), i.e., √3 + i 1 + y √3 πi 1 2i π z = 1, y = √3 + ii , 1 + y = √3 , ln y = πii , Li ( y) = 1 ζ + 2ii Cl π i22 = 1, (61) z = 1, y = √√3 + i,, 1 + y= √√3,, ln lny = πi,, Li Li22(( y)) = = 1ζ 22+ 2iCl22 π ii2 = 1, (61) √33++ii 11+ yy ii3 −π3i 2 − −13 2 23i 2 π3 22 − zz==1,1, yy==√3 + ii,, 1 −y == i ,, ln lnyy==− 3 ,, Li Li2((−yy)) = =− 3 ζζ2++ 3 ClCl23 ii ==−1,1 , (61) (61) √√3 + i 11− y i −33 2 − −33 2 33 2 33 − 3 + i −y i − − − − where Cl−2((π/3)) isis the the Clausen Clausen function, function, we we have have where Cl22(π/3) is the Clausen function, we have wherewhere Cl Cl22((ππ/3/3))isis the the Clausen Clausen function, function, we we have have N (µ22)εε 1 N44 (µ2 )ε 1 = N4 (µ ) 1 a1 a2εε ,, (62) (62) = (1 2ε) (m222)εεε ε −a 1 −a 2ε , (62) (1 N42ε) (mµ2))ε 1ε − 1 − 2 = (1 −2ε) (m ) ε − a1 − a2ε, (62)(62) (1 − 2ε) (m2)ε ε − − − with with π 4 π π withwith aa1 = π ,, aa2 = 4 Cl2 π π ln3.ln3. (63) (63) a11 = − √π , a22 = √ Cl22 π3 − √π ln3. (63) − √π3 √443 3π − √π3 aa11== 3 ,, aa22== 3 ClCl22 3 ln3.ln 3 . (63) (63) Then, the result for the second−−√√33 diagram√√33 of J(x3)3can −− be√√3 written3 in the following form Then, the result for the second diagram of JJ((x)) can be written in the following form Then,Then, the the result result for for the the second second diagram diagram of ofJJ((xx))cancan be be written written in in the the following following form form

N22 (µ22)22εε 1 a N24 (µ2 )2ε 1 a11 = N44 (µ ) 1 a1 a2 . (64) = 2 22 2εε 2ε 22 εε 22 a2 .. (64) (64) ((1 N42εε)) ((M(2µ))ε )((m2 ))ε ε1ε2 − aεε1 − a2 (64) 2 = (1 −2ε) (M ) (m ) ε − ε − a2 . (64) 22 (1 − 2ε) (M2)ε(m2)ε ε2 − ε − 2 − Thus, for JJ((x)) we have the following result Thus,Thus, for forJJ((xx))wewe have have the the following following result result Thus, for J(x) we have the following result 1 3 1 9 1 zz 1 2 J(z) = 11 + a1 33 11 + a2 + 2a1 99 + a1 ln z + 11−z Li2(z) + 11 ln22 z . (65) JJ((z)) = = 22 + a 1 + a 2 + 2a 1 + a 1 lnlnz + − zLi2((z)) + + lnln 2z .. (65) (65) J(z) = − 21εε2 + a11− 32 1εε + a22 + 2a11− 92 + a11 ln z +1 −z−z z Li22(z) +12 ln2 z . (65) J(z) = −−22εε2+ a1 −−22 εε+ a2 + 2a1 −−22+ a1 ln z + −zz Li2(z) + 22 ln z. (65) − 2ε2 − 2ε − 2 z 2 5.2. II((x)) 5.2.5.2.II((xx)) 5.2. I(Tox) find the result for the initial diagram I(x), see Equation (56), we have to calculate To find the result for the initial diagram II((x)),, see see Equation Equation ( (56),), we we have have to to calculate calculate severalTo integrals. find the result for the initial diagram I(x), see Equation (56), we have to calculate severalseveralTo integrals. integrals.find the result for the initial diagram I(x), see Equation (56), we have to calculate severalThe integrals. first integrals, which corresponds to z-independent part of J(x), is very simple: severalThe integrals. first integrals, which corresponds to z-independent-independent part part of of JJ((x)),, is is very very simple: simple: The first integrals, which corresponds∞ to z-independent part of J(x), is very simple: 1/2 ε ∞ dx 1 1 I (x)=(4x 1)1/2 −ε dx11 = 1 . (66) II11((x)=()=(4x 1))1/2 −ε ∞ 3/2 ε = .. (66) (66) 1 − 1/2−ε xx ((4x 1 dx11))3/23/2 −εε 2((1 1 2εε)) I (x)=(4x − 1) Zx (4x11 −1) − = 2(1 −2ε) . (66) 1 − Z − 3/2−ε ( − ) − Zx (4x1 1) − 2 1 2ε Other integrals can be calculated at the accuracy− εε = 0. The integral− lnlnzz inin JJ((x))cancan Other integrals can be calculated at the accuracy ε = 0. The integral ∼ ln z in J(x) can be evaluatedOther integrals using integration can be calculated by parts at the(A similar accuracy applicatε = 0.ionion The of of integral the the integration integration∼ ln z in byJ by(x parts) partscan be evaluated using integration by parts (A similar application of the integration∼ by parts be evaluated using integration by parts (A similar application of the integration by parts Particles 2021, 4 374

The ﬁrst integrals, which corresponds to z-independent part of J(x), is very simple:

∞ 1/2 ε dx1 1 I1(x) = (4x 1) − = . (66) − x (4x 1)3/2 ε 2(1 2ε) Z 1 − − − Other integrals can be calculated at the accuracy ε = 0. The integral ln z in J(x) can ∼ be evaluated using integration by parts (A similar application of the integration by parts procedure for integral representations can be found in the recent article [119], where FIs containing elliptic structures were considered.) as

∞ 1/2 dx1 1 1 1 I2(x) = (4x 1) ln = ln I˜(x) , (67) − x (4x 1)3/2 x 2 x − Z 1 − 1 where (see AppendixA)

∞ dx (4 z)1/2 z dz ˜( ) = ( )1/2 1 = 1 I x 4x 1 1/2 −1/2 1/2 − x x1(4x1 1) z 0 z (4 x )1/2 Z − Z 1 − 1 2 t dt 1 + y = 1 = ln y (68) t 0 1 + t2 − 1 y Z 1 − and, thus, 1 1 + y I (x) = ln z + ln y . (69) 2 2 1 y − The integral ln2 z in J(x) can be evaluated using integration by parts similarly to ∼ the previous one. We have

∞ 1/2 dx1 2 1 1 2 1 I3(x) = (4x 1) ln = ln + 2I˜1(x) , (70) − x (4x 1)3/2 x 2 x Z 1 − 1 where (see AppendixA)

∞ dx (4 z)1/2 z dz ˜ ( ) = ( )1/2 1 = 1 I1 x 4x 1 1/2 ln x1 −1/2 1/2 ln z1 − x x1(4x1 1) − z 0 z (4 x )1/2 Z − Z 1 − 1 t 1 2 dt1 1 + y dy1 = 2 ln[z1(t1)] = ln[z1(y1)] . (71) − t 0 1 + t − 1 y y y Z 1 − Z 1 The last integral can be calculated using integration by parts as 1 1 dy1 dy1(1 + y1) 1 2 ln[z1(y1)] = ln y ln z ln y1 = ln y ln z + ln y + 2Li2(1 y) T1(y) , (72) − y y − y y (1 y ) 2 − ≡ Z 1 Z 1 − 1 and, thus,

1 + y 1 1 + y I˜ (x) = T (y) and I (x) = ln2 z + T (y) . (73) 1 1 y 1 3 2 1 y 1 − − The last term Li (z) in J(x) can be evaluated using integration by parts similarly to ∼ 2 the previous ones. It can be represented as

∞ 1/2 dx1 (x1 1) 1 2 + z I4(x) = (4x 1) − Li2(1/x1) = Li2(z) + I˜2(x) , (74) − x (4x 1)3/2 2 − 2z Z 1 − Particles 2021, 4 375

where

∞ 1/2 dx1 (x1 + 1/2) ∂ I˜2(x) = (4x 1) Li2(1/x1) − − x x (4x 1)1/2 ∂x Z 1 1 − 1 (4 z)1/2 z dz (2 + z ) ∂ = 1 1 ( ) −1/2 1/2 Li2 z1 (75) z 0 2z (4 z )1/2 ∂z1 Z 1 − 1 Since ∂ ln z Li (z) = (76) ∂z 2 − 1 z − we have

(4 z)1/2 z dz (2 + z ) ln z 2 t dt (1 + 3t2) ˜ ( ) = 1 1 1 = 1 1 [ ( )] I2 x −1/2 1/2 2 2 ln z1 t1 − z 0 2z (4 z )1/2 1 z1 − t 0 (1 + t )(1 3t ) Z 1 − 1 − Z 1 − 1 1 t 1 3 1 1 + y = dt1 ln[z1(t1)] = T1(y) 3I˜21(x) . (77) t 0 1 + t2 − 1 3t2 − 2 1 y − Z " 1 − 1 # −

Now, we study the term I˜21(x). Considering the integral

t 1 1 1 √3t dt1 = ln − , (78) 0 1 3t2 − 2√3 1 + √3t Z − 1 ! we see an appearance of the new variable

1 √3t ξ = − (79) 1 + √3t

Using the new variable ξ (see AppendixA), we have for I˜21(x):

1 1 dξ (1 ξ )2 I˜ (x) = 1 ln − 1 (80) 21 √ ξ 2 2 3t Zξ 1 (1 + ξ1 + ξ1) ! Since (1 ξ )2 (1 ξ )3 1 1 + ξ − 1 = − 1 and = , (81) (1 + ξ + ξ2) (1 ξ3) √3t 1 ξ 1 1 − 1 − we can evaluate the integral I˜21(x) in the following form (We see the appearance of a 2 polynomial structure (1 + ξ1 + ξ1) in the integrand (80), which leads to the appearance of the poly-logarithms with the argument ξ3 below in (82). A similar polynomial structure has already been developed, for example, in [55,56,67,120–124] and in the more complicated cases the structure leads to the appearance of the cyclotronic poly-logarithms [125–127].)

1 + ξ 1 8 1 + ξ I˜ (x) = 3Li (ξ) Li (ξ3) ζ T (ξ) (82) 21 2(1 ξ) 2 − 3 2 − 3 2 ≡ 2(1 ξ) 2 − − and, thus,

1 + y 3(1 + ξ) I˜ (x) = T (y) T (ξ) , 2 − 2(1 y) 1 − 2(1 ξ) 2 − − 2 + z 1 + y 3(1 + ξ) I (x) = Li (z) T (y) T (ξ) . (83) 4 − 4z 2 − 4(1 y) 1 − 4(1 ξ) 2 − − Particles 2021, 4 376

Thus, the initial master integral I(x) can be expressed as

1 1 5 2a 1 + y 1 I(x) = + − 1 + 19 8a 2a a ln z + ln y ln2z 2 ε2 ε − 1 − 2 − 1 1 y − 2 − 2 + z 1 + y 3(1 + ξ) + Li (z) T (y) + T (ξ) . (84) 2z 2 − 2(1 y) 1 2(1 ξ) 2 − − 6. Conclusions In this short review, we have presented the results of calculating some massless and massive Feynman integrals. In the massless case, we considered a 5-loop master diagram that contributes to the β-function of the ϕ4-model. The results for this diagram were obtained [7,8,12,13] by Dmitry Kazakov, but it published without any intermediate calculations. Our calculations are performed in detail (the other two diagrams were discussed in [16]) and the ﬁnal result coincides with that obtained by Kazakov. In the massive case, we considered the computation of one of the master integrals contributing to the relationship between the MS-mass and the pole-mass of the Higgs boson in the standard model in the limit of heavy Higgs. The results for this master integral contain dilogarithms with unusual arguments.

Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The author is grateful to Mikhail Kalmykov for collaboration in the calculation of on-shall diagrams, as well as for the correspondence. He is grateful also to Andrei Pikelner for him help with Axodraw2. Conﬂicts of Interest: The author declares no conﬂict of interest.

Appendix A Here we present the sets of the new variables that are useful for integrations in the case of massive diagrams:

z 4t2 4 8t(dt) t2 = , z = , 4 z = , (dz) = ; 4 z 1 + t2 − 1 + t2 (1 + t2)2 − 1 it 1 y 4y 2 (dy) (dt) 1 (dy) y = − , t = − , 1 + t2 = , (dt) = , = ; 1 + it i(1 + y) (1 + y)2 − i (1 + y)2 1 + t2 − 2i y 1 √3t 1 1 ξ 4t2 (1 ξ)2 (1 ξ)3 ξ = − , t = − , z = = − = − , 1 + √3t √3 1 + ξ 1 + t2 1 + ξ + ξ2 1 ξ3 − 2(dy) (dt) 1 (dξ) (dt) = , = . (A1) − √3(1 + y)2 1 3t2 − 2√3 ξ −

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