We overview the general status of higher order corrections to Bhabha
scattering and review recent progress in the determination of the two-loop
virtual corrections. Quite recently, they were derived from combining a
massless calculation and contributions with electron sub-loops. For a massive
calculation, the self-energy and vertex master integrals are known, while most
of the two-loop boxes are not. We demonstrate with an example that a study of
systems of differential equations, combined with Mellin-Barnes representations
for single masters, might open a way for their systematic calculation.Comment: 8 pages, 2 figures, latex, Contribution to Proceedings of XXIX Conf.
  of Theoretical Physics: Matter to the Deepest, Ustron, Poland, 8-14 September
  2005; preprint number adde

Czakon, M.

Gluza, J.

Riemann, T.

Acta Physica Polonica B

English

arXiv

We overview the general status of higher order corrections to Bhabha scattering and review recent progress in the determination of the two-loop virtual corrections. Quite recently, they were derived from combining a
massless calculation and contributions with electron sub-loops. For a massive calculation, the self-energy and vertex master integrals are known, while most of the two-loop boxes are not. We demonstrate with an example
that a study of systems of differential equations, combined with Mellin-Barnes representations for single masters, might open a way for their systematic calculation

Czakon, Michał

Gluza, Janusz

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: On the massive two-loop corrections to Bhabha scattering  Author: Michał Czakon, Janusz Gluza, T. Riemann  Citation style: Czakon Michał, Gluza Janusz, Riemann T. (2005). On the massive two-loop corrections to Bhabha scattering. "Acta Physica Polonica B" (Vol. 36, no. 11 (2005), s. 3319-3326).   Vol. 36 (2005) ACTA PHYSICA POLONICA B No 11ON THE MASSIVE TWO-LOOP CORRECTIONSTO BHABHA SCATTERING∗ ∗∗M. Czakona,b, J. Gluzab,c, T. Riemannca Institut für Theoretische Physik und Astrophysik, Universität WürzburgAm Hubland, D-97074 Würzburg, Germanyb Institute of Physics, University of SilesiaUniversytecka 4, 40007 Katowice, Polandc Deutsches Elektronen-Synchrotron DESYPlatanenallee 6, D–15738 Zeuthen, Germany(Received November 2, 2005)We overview the general status of higher order corrections to Bhabhascattering and review recent progress in the determination of the two-loopvirtual corrections. Quite recently, they were derived from combining amassless calculation and contributions with electron sub-loops. For a mas-sive calculation, the self-energy and vertex master integrals are known,while most of the two-loop boxes are not. We demonstrate with an ex-ample that a study of systems of differential equations, combined withMellin-Barnes representations for single masters, might open a way fortheir systematic calculation.PACS numbers: 13.20.Ds, 13.66.De1. IntroductionSince Bhabha’s article [1] on the reactione+e− → e+e− (1.1)numerous studies of it were performed, triggered by better and better ex-perimentation. Bhabha scattering is of interest for several reasons. It is aclassical reaction with a clear experimental signature.∗ Presented at the XXIX International Conference of Theoretical Physics “Matter tothe Deepest”, Ustroń, Poland, September 8–14, 2005∗∗ Work supported in part by Sonderforschungsbereich/Transregio 9–03 of DFG “Com-putergestützte Theoretische Teilchenphysik”, by the Sofja Kovalevskaja Award of theAlexander von Humboldt Foundation sponsored by the German Federal Ministry ofEducation and Research, and by the Polish State Committee for Scientific Research(KBN) for the research project in 2004–2005.(3319)3320 M. Czakon, J. Gluza, T. RiemannSmall angle Bhabha scattering has a specific sensitivity to new physicsdue to its forward peaking structure, and is a prominent luminosity monitorat high energies. At LEP, the cross-section predictions were needed withan accuracy of the order of about 10−3 [2]. At the planned linear e+e−collider ILC (see the documents prepared for TESLA [3]), another order ofmagnitude (10−4) might be needed [4] and realized [5]. This accuracy is inreach of the most advanced Bhabha Monte Carlo programs BHLUMI [6, 7],NLLBHA [8,9], SAMBHA [10]. The programs contain higher order photoniccorrections, both exponentiated soft photon corrections and logarithmicallyenhanced fixed order corrections (including 2nd order). Concerning the vir-tual corrections, the logarithmically non-enhanced two-loop contributions(and also radiative one-loop contributions) were not included until recently.At small energies, large angle Bhabha scattering may be used for lumi-nosity determination. The KLOE collaboration, e.g., aims at an experimen-tal accuracy of 0.3 %, compared to a present accuracy estimate of the MonteCarlo programs (BABAYAGA [11, 12], BHAGENF [13], BHWIDE [14, 15],MCGPJ [16]) of about 0.5 % [17]. Here, the inclusion of two-loop boxes isnot needed.Many calculations of weak corrections have been performed since the firstcomplete one-loop calculation [18], and the technical accuracy is extremelyhigh nowadays. A recent one, based on DIANA [19–21] and aItalc [22],demonstrated an agreement of more than 10 digits with another calcula-tion [20]. Leading higher order weak corrections due to the top quark wereincluded in [23], where two calculations were shown to agree within few permille. It might also well come out that this accuracy (for the pure weakpart) is sufficient for applications in the foreseeable future. But, it mightwell be that a two-loop calculation of all the dominant terms of the secondorder weak corrections is of interest.The last few years brought considerable progress in the pure photonic,virtual two-loop corrections to Bhabha scattering. They were determined forthe massless case some time ago [24], and quite recently the understandingwas developed on how to transform this into the on mass shell renormal-ization scheme [25,26]. This is an absolutely needed part of the calculationin this approach since the Monte Carlo programs in use rely on a finiteelectron mass, thus regulating the collinear singularities. For this, severalinfra-red properties of the amplitudes had to exploited: exponentiation ofthe IR logarithms; factorization of the collinear logarithms into externallegs; non-renormalization of the IR exponents. The last of the three prop-erties is not fulfilled for diagrams with an internal fermion loop. For thisreason, the massless calculation has to be combined with an explicit calcula-tion of them [27–29]. In [30], the massive calculation of the Italian/Freiburggroup is summarized. So far, it is gauge invariant but not complete sinceOn the Massive Two-Loop Corrections to Bhabha Scattering 3321the massive box diagrams are an inevitable part of that program. Further,the present combination of the two existing calculations (Penin’s and Ital-ian/Freiburg approach) seems not yet to match exactly what is a physicalquantity (Bhabha scattering with several flavors); see also [31].In view of the importance of the problem and the principal interest itis quite obvious that a complete massive calculation is to be done yet. Asubstantial last missing part are the massive two-loop boxes.2. Master integralsThe two-loop vertex master integrals have been completely determinedand published by two groups in 2003 [32, 33] and 2004 [34]. For details andreferences for the self-energies see [35]. The masters are available, from bothgroups, from the internet as Form files or MATHEMATICA files [36, 37].Concerning the double boxes, the situation is less developed. A completelist of masters is known [35,36], but the evaluation is, as was expected, quitedifficult. Not only that they depend on a more complicated kinematicscompared to vertices or self-energies, namely on both s and t. Additionallyone is faced with the fact that up to six masters (for B6l3m3 1) may formsystems of differential equations, if one wants to go this way. So it may wellhappen that the use of Mellin–Barnes representations for the determinationof single masters (first successfully used for massless masters [38,39]) has tobe preferred here. In fact, the heroic efforts of Smirnov [40, 41] give stronghints to that. The two-loop Bhabha box diagrams contain Feynman integralswith seven propagators (lines) and with numerators, and their reduction toa smaller number of master integrals leads to the inclusion of masters witha smaller number of lines, and also of masters with dots (with lines raisedto a power biger than 1). The present knowledge of box masters may besummarized shortly:• Seven lines: For topology B7l4m1 (first planar) exist two masters, andB7l4m1 and B7l4m1N are known [40,41]. For topology B7l4m2 (secondplanar) we need four masters, and the basic master is known (with useof a two-fold integral in the constant term) [41]. Finally, for the threemasters of topology B7l4m3 (non-planar), only the most divergent part(∼ 1/ε2) of the basic master is determined analytically [41].• Six lines: The ten masters are not published.• Five lines: Analytical results for singularities are known, see next sec-tion.1 B6l3m3 is a master topology with six lines, three of them massive; for details of ournotations, see [35,36].3322 M. Czakon, J. Gluza, T. Riemann3. The box master integrals with five linesThe two-loop box masters depend on two variables s and t, or equiva-lently on related scaling variables x and y (the mass m = 1). There are fivemaster topologies with five internal lines, see figure 1.B5l2m1 B5l2m3B5l2m23442B5l4mB5l3mFig. 1. Two-loop MIs with five internal lines.B5l3m B5l3md1 B5l3md2B5l3md2’ B5l3md3Fig. 2. Five MIs for topology B5l3m. Masters B5l3md2’ and B5l3md2 are symmet-ric, B5l2m3d2’[s(x), t(y)]=B5l3md2[t(y), s(x)].As experience shows, some of them are already examples of quite compli-cated systems of master integrals (MIs), although two cases, namely B512m1(one MI [34, 36]) and B5l4m (two MIs [28, 34, 36]) are relatively simple.B5l2m2 is a system of three MIs, B5l2m3 is one of four MIs, while B5l3m isactually a system of even five MIs (see Fig. 2). In [35], analytical results forthe singularities of topologies B5l2m2 and B5l2m3 have been given. Herewe complete the determination of singularities of the MIs with five lines andconsider the most complex case B5l3m.On the Massive Two-Loop Corrections to Bhabha Scattering 33233.1. B5l3m: Synergetic use of Mellin–Barnes representationsand differential equationsIf we try to calculate the masters for topology B5l3m with differentialequations (DEs), we meet a system of five coupled differential equations. Infact, one of the dotted masters (B5l2m3d2) appears twice (with interchangedarguments).Fortunately, we can use another method, based on Mellin–Barnes pre-sentations, to solve some of them separately. We found it most efficient tosolve the case B5l3md2, where one eliminates two masters at once. Detailsof the calculation will be given elsewhere [42]. The result is:B5l3md2 = −1ǫ2x4(1− x2)H[0, x]+1ǫx4(1− x2)(1− y2)[2(1 + y2)H[0, x]H[0, y]− (−1 + y2)(ξ2 + 6H[−1, 0, x] − 4H[0, 0, x] − 2H[1, 0, x])]+O(1) . (3.1)Then, up to (and including) order 1/ǫ, we are left with a system of onlythree DEs. The differential equation for B5l3md3[−1] is:∂B5l3md3[−1]∂x=1 + x2x(1− x2)B5l3md3[−1] −yH[0, y](1− x2)(1− y2), (3.2)with s = −(1 − x)2/x, t = −(1 − y)2/y, and can also be solved separately.The solution is:B5l3md3[−1] = −xy(−1 + x2)(−1 + y2)H[0, x]H[0, y]. (3.3)For our notations in terms of HPLs and generalized HPLs see [43] and ref-erences therein.Finally, we have to determine the singularities of B5l3md1. They can befound from algebraic manipulations (see [44]). This relies on the fact thatthe basic integral, B5l3m, and the integrals with numerators are nonsingular.In order to use this, we replaced in the defining equations for B5l3md1 thedotted diagrams by those with numerators. The solution is:B5l3md1 =1ǫ218x(−1 + y)(1 + y)3×[(−1 + x)2y(−1 + y2 + 2yH[0, y])]+1ǫy24x(1 + x)(−1 + y)(1 + y)3)3324 M. Czakon, J. Gluza, T. Riemann×[6(−1 + x− x2 + x3)H[0, x](−1 + y2 + 2yH[0, y])−6(1 + x)(−2− 2x2 + 2y2 + 2x2y2 + yξ2 − 2xyξ2+x2yξ2 + 2(−2x− y + 2xy − x2y − 2xy2+(−1 + x)2yH[−1,−y] + 3(−1 + x)2yH[−1, y])H[0, y]−6(−1 + x)2yH[0,−1, y] − 4yH[0, 0, y]+8xyH[0, 0, y]−4x2yH[0, 0, y]+2yH[0, 1, y]−4xyH[0, 1, y]+2x2yH[0, 1, y])]+O(1), (3.4)We can easily check that the solutions (3.4) and (3.1) fulfill the original dif-ferential equations (not reproduced here for space reasons). The knowledgeof the solution (3.1), obtained by the Mellin–Barnes method, has been used.So, our combined approach is also a cross check for the analytical resultobtained with the Mellin–Barnes method. Finally, we checked that (3.4),(3.3), and (3.1) are also in agreement with a numerical approach based onsector decomposition calculations.4. SummaryWe gave an overview on status and recent progress for the higher ordercorrections for Bhabha scattering. The small angle Monte Carlo programsare approaching the accuracy standard of 10−4 once this is needed.The results of the determination of the pure constant (non-logarithmicallyenhanced) virtual two-loop corrections seem to stay perhaps negligibly small(at this accuracy level and after isolating from them not only the mass-, butalso the kinematically enhanced terms) in the kinematical region of interest,but this is not finally numerically fixed.In order to have a complete massive calculation (though in the limit ofsmall electron mass), one has to determine yet the bulk of two-loop boxes.This is underway.REFERENCES[1] H. Bhabha, Proc. Roy. Soc. A154, 195 (1936).[2] S. Jadach, hep-ph/0306083.[3] ECFA/DESY LC Physics Working Group, J. Aguilar-Saavedra et al.hep-ph/0106315, TESLA TDR, Part III: Physics at an e+e− Linear Collider,DESY 2001-011.[4] R. Hawkings, K. Mönig, Eur. Phys. J. direct C1, 8 (1999) [hep-ex/9910022].[5] H. Abramowicz et al., IEEE Trans. Nucl. Sci. 51, 1 (2004).[6] S. 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On the massive two-loop corrections to Bhabha scattering

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On the massive two-loop corrections to Bhabha scattering

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