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Two-loop electroweak Sudakov logarithms for 8 0 massive fermion scattering 0 2 n a J 7 1 AnsgarDenner,BerndJantzen ] h PaulScherrerInstitut,CH-5232VilligenPSI,Switzerland p E-mail: [email protected],[email protected] - p StefanoPozzorini∗ e h Max-Planck-InstitutfürPhysik,FöhringerRing6,D-80805München,Germany [ E-mail: [email protected] 1 v 7 We study the asymptoticbehaviourof two-loopelectroweakcorrectionsat energiesQ2 ≫M2 , W 4 wherelogarithmsofthetypeln(Q2/M2 )becomedominant. Thecalculationoftheleadingand 6 W 2 next-to-leadinglogarithmictermsformasslessandmassivefermion-scatteringprocessesissum- 1. marized. Thederivationsareperformeddiagrammaticallywithinthespontaneouslybrokenelec- 0 troweak theory. We find that the soft and collinear singularities resulting from photons can be 8 0 factorizedintoaQED-liketermandthat,uptologarithmsoftheZ–Wmassratio,theeffectsof v: symmetry breaking cancel. This result supports resummation prescriptions that are based on a i symmetricSU(2)×U(1)theorymatchedwithQEDattheelectroweakscale. X r a 8thInternationalSymposiumonRadiativeCorrections October1-5,2007 Florence,Italy ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Two-loopelectroweakSudakovlogarithmsformassivefermionscattering StefanoPozzorini 1. Introduction AtTeVcolliders,theelectroweakcorrectionsarestronglyenhancedbylogarithmsoftheform a llnj Q2/M2 , with M =M ∼M and j ≤2l. These logarithmic corrections affect every re- W Z action(cid:0)that invo(cid:1)lves electroweakly interacting particles and is characterized by scattering energies Q2 ≫M2. ForQ∼1TeV their impact can reach tens of per cent at one loop and several per cent at two loops (see for instance Refs.[1,2] and references therein). Their origin is twofold. The renormalization ofUVsingularities atthescalem <M givesrisetotermsoftheformln(Q2/m 2). R∼ R In addition, the interactions of theinitial- and final-state particles withsoft and/or collinear gauge bosons give rise to mass-singular logarithms, i.e. ln(Q2/M2) terms that are formally singular in the limit M →0. Since they originate from UV, soft and collinear singularities, electroweak logarithmic corrections have universal properties that can be studied in a process-independent way andexhibitinterestinganalogieswithQCD.Atoneloop,theleadinglogarithms(LLs)andnext-to- leadinglogarithms(NLLs)factorizeandaredescribedbyageneralformulathatappliestoarbitrary Standard-Model processes [3, 4]. The properties of electroweak logarithmic corrections beyond oneloopareinvestigatedwithtwocomplementaryapproachesintheliterature: (i)Evolutionequa- tions, which are well known in QED and QCD, are applied to the electroweak theory in order to obtain thehigher-order termsthrough aresummation [5,6,7,8,9]. Inthisapproach theevolution is split into two regimes, where the electroweak interaction is described in terms of SU(2)×U(1) andU (1)symmetricgaugetheories.1 Thissplittingisassumedtocorrectlyreproduceallrelevant em implications of electroweak symmetry breaking in LL, NLL and NNLL approximation. (ii) This assumption and the resulting resummation prescriptions can be checked and improved by means of explicit diagrammatic two-loop calculations based on the (spontaneously broken) electroweak Lagrangian. So far, all existing diagrammatic results are in agreement with the resummation prescriptions, howeveruptonowonlyasmallsubset oflogarithms andprocesses hasbeencomputed explicitly. While the LLs [11] and the angular-dependent subset of the NLLs [12] have been de- rived for arbitrary processes, complete diagrammatic calculations at (or beyond) the NLL level exist only for matrix elements involving massless external fermions [13, 14,15]. Inthe literature, noexplicitNLLcalculation existsforreactions involving massivescattering particles. In the following we present a recent two-loop NLL calculation [16] for n-fermion processes f f → f ...f involving an arbitrary number of leptons and/or quarks. While all fermions with 1 2 3 n m ≪Maretreatedasmassless,thetop-quarkmasseffectsaretakenintoaccount. Forallinvariants f r =(p +p )2weassume|r |∼Q2≫M2,andthedifferencesbetweentheW,Z,Handtmassare ij i j ij taken into account. Soft and collinear singularities from massless virtual photons are regularized dimensionally and arise as e -poles in D= 4−2e dimensions. For consistency, the same power counting is applied to ln(Q2/M2) and 1/e singularities. Thus, in NLL approximation we include all e −klnj−k(Q2/M2) terms with total power j=2,1 at one loop and j =4,3 at two loops. The photonicsingularitiesarefactorizedinagauge-invariant electromagnetic term. Theremainingpart ofthecorrections –whichisfinite,gaugeinvariant, anddoesnotdependontheschemeadoptedto regularize photonic singularities – contains only ln(Q2/M2) terms. The divergences contained in theelectromagnetic termcancelifreal-photon emissionisincluded. 1Anewapproachbasedonsoft-collineareffectivetheoryhasbeendevelopedinRef.[10]. 2 Two-loopelectroweakSudakovlogarithmsformassivefermionscattering StefanoPozzorini 2. Extraction ofsoft,collinearand UVcontributions from Feynmandiagrams Letusconsiderthetwo-loopdiagramsthatproduceNLLswithinthe’tHooft–Feynmangauge. These diagrams are obtained from the LL one-loop diagrams, i.e. diagrams with external-leg ex- change of soft-collinear gauge bosons, via insertions of one-loop sub-diagrams that produce an additional logarithm. WestartwiththediagramsthatdonotcontainYukawainteractions, V2 i V2 i V2 j i i V1 V1 V1 , V1 , i , , . (2.1) V1 V2 V3 V2 k j j j j HereV =A,Z,W±, and the external lines i,j,k =1,...,n are massless and/or massive fermions. i Inthefirstfourdiagramsin(2.1)thegaugebosonV cancoupletoanexternallineortoaninternal 2 propagator. InthelattercasetheonlyrelevantregionistheonewhereV iscollineartoanexternal 2 fermionand,usingcollinearWardidentities, wefindthatthistypeofcontributions cancel[15,16]. The remaining diagrams contain a tree sub-diagram that corresponds to the original process and depends only on the external momenta.2 These diagrams are called factorizable, since their NLL contributionfactorizesintothen-fermiontreeamplitudetimesatwo-loopNLLfactor. Forinstance, thefactorizable diagramsassociated withthefourthdiagramin(2.1)are i i k V1 V1 V2 F V2 , F V2 , F V1i . (2.2) V3 V3 V3 j j j To extract the NLLsof soft/collinear origin, we exploit the fact that in the soft/collinear limit the coupling of a gauge boson V to the i-th external fermion yields a factor −2eIV(p +q)m , where i i q and p are the gauge-boson and fermion momenta, while IV is the gauge-group generator. For i i instance, forthelastdiagram in(2.2)weobtain m 4e dDq1dDq2 −8ie3g2e V¯1V¯2V¯3 gm 1m 2(q1−q2)m 3+perm. l1m 1l2m 2l3m 3 ×IV1IV2IV3M , D Z (2p )2D (q2−M2 )(q2−M2(cid:2))(q2−M2 )(l2−m2)(l2−(cid:3)m2)(l2−m2) i k j 0 1 V1 2 V2 3 V3 1 1 2 2 3 3 whereq =−q −q ,l = p −q ,l = p −q andl = p −q . Then-fermiontreeamplitudeM 3 1 2 1 i 1 2 k 2 3 j 3 0 must be regarded as a vector carrying SU(2) indices associated with the isospin of the scattering fermions, andthegenerators IV actasSU(2)matricesonthei-thfermion. i In addition to soft/collinear contributions, there are also two-loop NLL terms that arise from UV divergences in the one-loop sub-diagrams. In this case, the UV singularities are removed by means ofaminimal subtraction atthescale m 2=Q2. Asaconsequence, only those UV-divergent sub-diagrams characterized by scales of order M2 ≪Q2 produce NLL terms. In particular, non- factorizable diagrams do not produce NLL terms of UV origin. The explicit factorization of the NLL contributions of UV origin is obtained by means of projector techniques [15, 16] since the 2Inthesetreesub-diagrams, whicharedenotedwithan’F’in(2.2), theloopmomentamustbesettozero. This followsfromthecollinearWardidentities. 3 Two-loopelectroweakSudakovlogarithmsformassivefermionscattering StefanoPozzorini soft-collinear approximation isnotapplicable inthiscase. Aftertheabovementioned minimalUV subtraction,weperformafiniterenormalizationthatrestorestheon-shellnormalizationofthewave functions andtranslates thecouplings totheminimal-subtraction schemeatthescale m . R For processes involving heavy quarks, in addition to the contributions listed above also diagrams with Yukawainteractions contribute. Many ofthem aresuppressed due tothe behaviour of theinteractionsofscalarfieldsinthesoft/collinearandM→0limits. Onlythreetypesofdiagrams arenotsuppressed, F i F i i F ′ V , V , F . (2.3) V j j j Moreover itturns out that thesum ofthe NLLterms resulting from these diagrams cancels owing to global gauge invariance of the Yukawa sector [16]. As a consequence, Yukawa interactions contribute only through fermionic self-energy diagrams that appear in the renormalization of the heavy-quark wavefunctions. 3. High-energy expansion ofthe loopintegrals The loop integrals have been calculated in the Q2 ≫ M2 limit to NLL accuracy using the sector-decomposition technique [17, 18] and, alternatively, the method of expansion by regions [19,20,21,22]. 3.1 Sectordecomposition Sector decomposition is based on the Feynman-parameter representation. Exploiting general properties ofsoft,collinearandUVsingularities, onecanfactorizethesesingularities inFeynman- parameter space insuch awaythat thedivergent integrations assume astandard form. Ifallparti- cles are massless these singular integrals are trivial and produce simple e -poles [17]. In presence of massive particles, the general solution is available to NLL accuracy [18]. This permits to con- struct, in a completely automatized way, finite integral representations for the coefficients of all NLLtermsoftypee −klnj−k(Q2/M2). Theintegrands aresmoothandcaneasilybeintegrated nu- merically.3 Moreover, although these integrals do not have a standard form, it turns out that they are relatively simple. In practice they can be solved in analytic form by means of standard tricks (partial fractioning, integration byparts, etc.) combined withatableofelementary integrals. Also this part of the calculation is highly automatized, however, the set of integration rules might need tobeextended whennewdiagramsarecomputed. 3.2 Expansionbyregions Within the expansion-by-regions method [19, 20, 21, 22], the integration domain of the loop momentaisdividedintoregionscorrespondingtotheasymptoticlimitconsidered. Theintegrandis appropriately expandedineveryrelevantregion,andeachoftheexpandedtermsisintegratedover 3Thisholdsonlyintheunphysicalregionwhereallinvariantsrijarenegative,butallowsforveryusefulchecks. 4 Two-loopelectroweakSudakovlogarithmsformassivefermionscattering StefanoPozzorini thewholeintegrationdomain. EachexpandedtermhasauniqueorderinpowersofQandM,butthe on-shellmomenta p ofmassivefermionsinvolvetwoscales, p2∼M2and2p p ∼Q2. Inorderto i i i j separate these scales, themassive momentaarereparametrized intermsoflight-like momenta p˜ i,j as p = p˜ +p2/(2p˜ p˜ )p˜ . Fortheloopintegrals needed inthis calculation, thefollowing regions i i i i j j foreachloopmomentumkarerelevant: hard,softandultrasoftregions,wherek2isoftheorderQ2, M2andM4/Q2,respectively;collinear,softcollinearandultracollinearregions,wherekiscollinear to one of the external momenta and k2 is of the order M2, M4/Q2 and M6/Q4, respectively. The lattertworegionscontributeonlyiftheexternalfermionsaremassive. Theexpandedloopintegrals have been evaluated using Mellin–Barnes representations, from whichthe extraction ofNLLshas been automatized. In addition, Mellin–Barnes representations of the unexpanded integrals have beenusedtocheckthecompleteness ofthesetofregionsusedfortheexpansion. 4. Results andconclusions Therenormalized 2-loop NLLamplitude isexpressed intermsofcombinations ofSU(2)ma- tricesactingonthetree-levelamplitude. ThisexpressionissimplifiedbymeansofSU(2)algebraic identities and additional relations resulting from global gauge invariance. The final result is con- sistent withthedouble-exponentiation formula4 a n a n a n M = exp8p (cid:229) d iejmexp8p (cid:229) d isjew1+8p (cid:229) d iZjM0. (4.1) i,j=1 i,j=1 i,j=1  j6=i   j6=i  j6=i  The term d sew behaves as if all electroweak gauge bosons would have the same mass M and is ij symmetricwithrespect toSU(2)×U(1)transformations, d sew = − (cid:229) IV¯IVI(e ,M,−r )− 1 g2YiYjb(1)+g2TaTab(1) J(e ,M,m 2). (4.2) ij i j ij (4p )2 (cid:18) 1 4 1 2 i j 2 (cid:19) R V=B,Wa Here Ta andY are the generators of the SU(2)×U(1) group, g are the corresponding coupling i i 1,2 constants, andb(1) theone-loop b -function coefficients [15]. TheNLLfunctions read 1,2 2 1 3 r yki g2m2 1 I(e ,M,r)=−L2− L3e − L4e 2+ −ln − i 2 t 2L+L2e + L3e 2 (4.3) 3 4 (cid:20)2 (cid:18)Q2(cid:19) Cew 8e2M2 (cid:21)(cid:18) 3 (cid:19) i W and J(e ,M,m 2)= I(2e ,M,Q2)−(Q2/m 2)e I(e ,M,Q2) /e with L=ln(Q2/M2). In (4.3)Cew = R R i (cid:229) IV¯IV,andtheYu(cid:2)kawatermsproportional tom2 contri(cid:3)bute withfactorsyL =1,yR=2,yR=0, V i i t t,b t b for heavy quarks i =b,t with chirality k =R,L, and yk = 0 otherwise. The term d em in (4.1) i i ij depends on the g –W mass gap and contains all soft/collinear singularities due to massless photons. These singularities involve single and double e -poles. Thus the terms d sew and d Z in (4.1) ij ij need to be expanded up to O(e 2). Wefind that d em behaves as in QED [16]. Finally we find that ij all contributions depending on the Z–W mass difference can be factorized into the one-loop term d Z=−IZIZln M2/M2 2L+2L2e +L3e 2 . Apartfromtheselatterln(M2/M2 )terms,alleffects ij i j Z W Z W ofsymmetryb(cid:0)reaking(t(cid:1)er(cid:2)msproportional to(cid:3)M /M ,mixingparametersandvacuumexpectation W Z 4Inthefollowingwesetm 2 =Q2egE/(4p ). 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