Analytic integration of real-virtual counterterms in NNLO jet cross sections II

Analy ic in eg a ion o eal- i ual coun e e ms in NNLO je c oss sec ions II This a icle has been downloaded om IOPscience. Please sc oll down o see he ull ex a icle. JHEP08(2009)079 (h p://iopscience.iop.o g/1126-6708/2009/08/079) Download de ails: IP Add ess: 128.141.30.188 The a icle was downloaded on 19/03/2010 a 10:37 Please no e ha e ms and condi ions apply. The Table o Con en s and mo e ela ed con en is a ailable Home Sea ch Collec ions Jou nals Abou Con ac us My IOPscience JHEP08(2009)079 Published by IOP Publishing o SISSA Recei ed:June 3, 2009 Re ised:July 24, 2009 Accep ed:Augus 4, 2009 Published:Augus 20, 2009 Analy ic in eg a ion o eal- i ual coun e e ms in NNLO je c oss sec ions II Paolo Bolzoni,aS en-Ola Moch,aG´abo Somogyiband Zol ´an T ´ocs´anyic aDESY, Pla anenalle 6, D-15738 Zeu hen, Ge many bIns i u e o Theo e ical Physics, Uni e si y o Z¨u ich, Win e hu e s asse 190, CH-8057 Z¨u ich, Swi ze land cUni e si y o Deb ecen and Ins i u e o Nuclea Resea ch o he Hunga ian Academy o Sciences, H-4001 Deb ecen P.O.Box 51, Hunga y E-mail: [email protected],[email p o ec ed], [email p o ec ed],[email p o ec ed] Abs ac : We p esen analy ic exp essions o all in eg als equi ed o comple e he explici e alua ion o he eal- i ual in eg a ed coun e e ms needed o de ine a ecen ly p oposed sub ac ion scheme o je c oss sec ions a nex - o-nex - o-leading o de in QCD. We use he Mellin-Ba nes ep esen a ion o hese in eg als in 4 −2ǫdimensions o ob ain he coe icien s o hei Lau en expansions a ound ǫ= 0. These coe icien s a e gi en by linea combina ions o mul idimensional Mellin-Ba nes in eg als. We compu e he coe icien s o such expansions in ǫbo h nume ically and analy ically by complex in eg a ion o e he Mellin-Ba nes con ou s. Keywo ds: Je s, QCD A Xi eP in : 0905.4390 c SISSA 2009 doi:10.1088/1126-6708/2009/08/079 JHEP08(2009)079 Con en s 1 In oduc ion 1 2 In eg als needed o he in eg a ed sub ac ion e ms 3 2.1 Basic in eg als 4 2.2 Nes ed in eg als 5 3 The me hod o Mellin-Ba nes ep esen a ions 7 4 Collinea in eg als I13 5 Nes ed collinea - ype I∗I and I∗J in eg als 15 6 Nes ed so - ype J ∗J in eg als 18 7 Nes ed so -collinea K∗J in eg al 21 8 Conclusions 21 1 In oduc ion P ecision p edic ions in pe u ba i e Quan um Ch omodynamics (QCD) a collide s de- mand calcula ing physical obse ables beyond leading o de (LO) accu acy and, in he adi ional app oach o highe o de p edic ions wi h ully di e en ial kinema ics, eal and i ual co ec ions a e sepa a ely e alua ed. In eg a ion o e he phase space hen equi es a consis en ea men o he in a ed singula i ies be o e any nume ical compu- a ion may be pe o med. A nex - o-leading o de (NLO), in a ed di e gences can be handled using a sub ac ion scheme, which exploi s he uni e sal s uc u e o he kine- ma ical singula i ies o QCD ma ix elemen s. The necessa y (p ocess-independen ) coun- e e ms egula ize he i ual co ec ions a one loop and he eal emission phase space in eg als simul aneously [1]. A nex - o-nex - o-leading o de (NNLO), he calcula ion o he adia i e co ec ions o ully di e en ial c oss sec ions is a challenging p oblem and a ious ex ensions o he sub ac ion me hod a NNLO ha e been p oposed, see e.g. e s. [2–5]. Cu en ly, he a ail- able esul s o elec on-posi on annihila ion a NNLO include o al a es [6–8] and e en shapes [9,10] o he p ocess e+e−→3 je s and a e all based on he an enna sub ac ion me hod [11–13]. On he o he hand o colo less inal s a es, such as ec o boson o Higgs boson p oduc ion a had on collide s dedica ed sub ac ion schemes a NNLO [14,15] ha e been applied. The in a ed s uc u e o sca e ing p ocesses wi h h ee o mo e colo ed pa - ons is in ol ed i calcula ed a NNLO wi h he an enna sub ac ion me hod [16] — a ac – 1 – JHEP08(2009)079 which has mo i a ed he o mula ion o al e na i e sub ac ion schemes. In pa icula , e s. [17–19] in oduce a scheme o compu ing NNLO co ec ions o QCD je c oss sec- ions o p ocesses wi hou colo ed pa ons in he ini ial s a e and an a bi a y numbe o massless pa icles (colo ed o colo less) in he inal s a e. Ve y ecen ly, ollowing he s eps o e . [17], his sub ac ion scheme has been ex ended o c oss sec ions o had on-ini ia ed p ocesses [20], al hough ye o NLO accu acy only, bu in a way which is NNLO-compa ible. Any sub ac ion scheme is o p ac ical u ili y only a e he coun e e ms o he egu- la iza ion o he eal emissions a e in eg a ed o e he phase space o he un esol ed pa ons. In he scheme o e s. [17–19] hese coun e e ms a e uni e sal (bu comple e only o p o- cesses wi hou colo ed pa icles in he ini ial s a e) and, he e o e can be compu ed once and o all. Thei knowledge is necessa y o egula ize he in a ed di e gences appea ing in he i ual co ec ions. Some o he in eg als needed explici ly in he so-called eal- i ual coun e e ms o his scheme ha e been calcula ed in e s. [21,22]. In he p esen pape we comple e his ask by compu ing all in eg als needed o he he eal- i ual coun e e ms in he sub ac ion scheme o e s. [17–19] by means o Mellin-Ba nes (MB) ep esen a ions. The use o MB in eg als when dealing wi h Feynman in eg al calculus has p o ed powe ul in he las yea s. MB in eg als we e i s applied o Feynman in eg als in e s. [23,24] and pionee ing wo k has been pe o med since hen in e s. [25–27] (see also e . [28] and e e ences he ein o many o he examples). Fo a gi en in eg al he MB ep esen a ion eplaces he powe o a sum in he in eg and by a p oduc o he indi idual e ms o he sum aised o some o he powe s. This leads hen o in eg als o e ce ain complex con ou s o Γ- unc ions. As a c ucial poin i is hen e y con enien wi h his MB ep esen a ion o esol e all singula i ies in he limi ǫ= 0 wi hin dimensional egula iza ion, d= 4 −2ǫ. In his pape , we adap he MB me hod o de i e analy ic exp essions o all in eg als appea ing in he eal- i ual coun e e ms o e s. [17–19]. Le us b ie ly discuss he me i s o he analy ic app oach o he compu a ion o he in eg a ed sub ac ion e ms. Fi s o all, in a highe -o de compu a ion, he ǫpoles o he in eg a ed sub ac ion e ms need o cancel he co esponding ǫpoles coming om he loop ma ix elemen s in he i ual co ec ions. The cancella ion o hese poles can be demon- s a ed mos con incingly once he pole s uc u e o he in eg a ed sub ac ion e ms is exhibi ed analy ically. Second, in e ms o speed and p ecision o he e alua ion, analy ic esul s a e e y as and e y accu a e compa ed o nume ical ones. Mo eo e , hey demon- s a e ha he inal esul consis s o smoo h unc ions only. Ne e heless also he nume ical e alua ion o he in eg a ed coun e e ms has i s u ili y, because i se es as an independen check. Then, he e a e indeed some cases, whe e i is e y di icul o ind he analy ic com- pu a ion o he mul i-dimensional MB in eg al and only he complex nume ical in eg a ion can be ca ied ou . In hese cases, howe e , he me hod o MB in eg als p o ides a as and eliable way o ob ain he inal esul s wi h small nume ical unce ain ies. F om a p ac ical poin o iew, he combina ion o bo h, analy ic and nume ical e alua ions o all MB in e- g als implies ha he inal esul s o he in eg a ed eal- i ual coun e e ms can be con e- nien ly gi en e.g. in he o m o in e pola ing ables which can be compu ed once and o all. This su ices o any p ac ical applica ion, because in an ac ual compu a ion he ela- i e unce ain y associa ed wi h he nume ical phase space in eg a ions is gene ally much g ea e han ha o he in eg a ed sub ac ion e ms. – 2 – JHEP08(2009)079 Q 1 b b b i b b b n nCi Q ˜ 1 b b b e i b b b en n−1⊗2 (i ) i Q 1 b b b b b b n nS Q ⊗ KQ 2 ˜ 1 b b b b b b b b b en n−1 Figu e 1. G aphical ep esen a ions o he momen um mappings and he implied phase space ac o iza ion: collinea (le ) and so momen um mapping ( igh ). The ou line o he pape is he ollowing. In sec ion 2we b ie ly e iew he phase space in eg als o he eal- i ual co ec ions a NNLO and we de ine he in eg als o he sub ac ion e ms ha we will conside in his pape . In sec ion 3we p esen a b ie explana ion o he me hod o MB ep esen a ions. We ou line he s eps o ou calcula ion and we also discuss explici ly an example o display he ypical s uc u e o he in eg als we a e in e es ed in. In sec ion 4we comple e he analy ic e alua ion o all in eg als needed o in eg a ed collinea coun e e ms. Nex , in sec ions 5–7we compu e also all di e en ypes o he nes ed in eg als. Finally in sec ion 8we p esen he conclusions o his wo k. 2 In eg als needed o he in eg a ed sub ac ion e ms The sub ac ion me hod de eloped in e s. [18,19] elies on he uni e sal so and collinea ac o iza ion p ope ies o QCD squa ed ma ix elemen s. Once he sub ac ion scheme is de ined, one has o in eg a e he sub ac ion e ms o e he ac o ized phase space o he un esol ed pa on(s). This is he con en o he p esen wo k (see also e . [21]). The e a e wo c ucial elemen s in he o mula ion o a sub ac ion scheme beyond NLO. Fi s ly, he ac o iza ion o mulae should disen angle he o e laps in so -singula ac o s and collinea singula i ies in o de o a oid mul iple sub ac ions and a simple solu ion o his p oblem has been gi en in e . [30]. Secondly, because he ac o iza ion o mulae a e alid only in he s ic so and collinea limi s, hey ha e o be ex ended o he whole phase space. Typically, his equi es a mapping o he o iginal nmomen a {p}n={p1,...,pn}in an n-pa on ma ix elemen a any o de in pe u ba ion heo y o mmomen a {˜p}m={˜p1,...,˜pm}in such a way, ha momen um conse a ion is p ese ed. He e mdeno es he numbe o ha d pa ons and n−mis he numbe o un esol ed ones. The o iginal n-pa icle phase space o o al momen um Q eads dφn(p1,...,pn;Q) = n Y i=1 ddpi (2π)d−1δ+p2 i(2π)dδ(d) Q− n X i=1 pi!,(2.1) and, o a gi en mapping, one ob ains he phase-space ac o iza ion as dφn({p}n;Q) = dφm({˜p}m;Q) [dpn−m;m({p}n−m;Q)] ,(2.2) which was i s in oduced in e . [1] in he con ex o compu ing QCD co ec ions a NLO. In his pape we a e conce ned wi h he in eg als o he singly-un esol ed coun e e ms – 3 – JHEP08(2009)079 δFunc ion g(±) I(z) 0gA1 ∓1g(±) B(1 −z)±ǫ 0g(±) C(1 −z)±ǫ2F1(±ǫ, ±ǫ, 1±ǫ, z) ±1g(±) D2F1(±ǫ, ±ǫ, 1±ǫ, 1−z) Table 1. The alues o δand g(±) I(z ) o which eq. (2.5) needs o be e alua ed. (i.e. he case m= 1), which imply wo ypes o mappings: {p}n Ci −→ {˜p}(i ) n−1={˜p1,...,˜pi ,...,˜pn},(2.3) {p}n S −→ {˜p}( ) n−1={˜p1,...,˜pn}.(2.4) In he collinea momen um mapping Ci −→in eq. (2.3) he momen a pµ iand pµ a e eplaced by a single momen um ˜pµ i and all o he momen a a e escaled, while o so - ype sub ac ions, S −→ in eq. (2.4) he momen um pµ , ha may become so , is missing om he se , and all o he momen a a e escaled and ans o med by a p ope Lo en z ans o ma ion. Bo h momen um mappings and he co esponding ac o iza ion o he phase-space measu e a e ep esen ed g aphically in igu e 1, whe e he symbol ⊗s ands o he con olu ion as implied by eq. (2.2). The in eg a ion o he singly-un esol ed sub ac ion e ms equi es h ee basic ypes o in eg als o e he co esponding ac o ized phase space, as well as i e a ions o hese (nes ed in eg als a e deno ed by a ∗). All necessa y in eg als we e de i ed in e s. [21,22]. 2.1 Basic in eg als The h ee basic in eg als a e hose used in he collinea , so and so -collinea sub ac ion coun e e ms. The collinea in eg als ha e he gene al o m Ix;ǫ, α0, d0;κ, k, δ, g(±) I=xZα0 0 dα α−1−(1+κ)ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−(1+κ)ǫ ×Z1 0 d [ (1− )]−ǫα+(1−α)x 2α+(1−α)xk+δǫ g(±) Iα+(1−α)x 2α+ (1−α)x.(2.5) These in eg als need o be known as a unc ion o x∈[0,1] in a Lau en -expansion in ǫ o k=−1,0,1,2. The necessa y alues o δand he exp essions o he unc ions g(±) Ia e gi en in able 1. He e κ= 0,1 o he i s ow and κ= 1 o all o he cases. Analy ic exp essions o all cases co esponding o he i s wo ows o able 1we e de i ed in e . [21] and con ain he i s i e e ms in he ǫ-expansion. We compu e all cases anew and p esen ou esul s explici ly in sec ion 4. The o he pa ame e s α0∈(0,1] and d0in eq. (2.5) will be speci ied in sec ion 3. Ou analy ic esul s o hese in eg als include all he coe icien s o he poles in ǫand he i s h ee e ms in he ǫ-expansion. – 4 – JHEP08(2009)079 Nex , he so sub ac ions equi e he in eg al J(Y˜ i˜ k,Q;ǫ, y0, d′ 0;κ) = −(4Y˜ i˜ k,Q)1+κǫ Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Ω(1+κǫ,1+κǫ)(cos χ) ×Zy0 0 dy y−1−2(1+κ)ǫ(1 −y)d′ 0+κǫ ,(2.6) as a unc ion o Y˜ i˜ k,Q ∈[0,1] in a Lau en expansion a ound ǫ= 0, whe e Ω(i,k)(cos χ) deno es he angula in eg al in d-dimensions Ω(i,k)(cos χ) = Z1 −1 d(cos ϑ) (sin ϑ)−2ǫZ1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ ×(1 −cos ϑ)−i(1 −cos χcos ϑ−sin χsin ϑcos ϕ)−k,(2.7) wi h cos χ= 1 −2Y˜ i˜ k,Q .(2.8) Fo he p esen pape he exac de ini ion o he kinema ic a iables xand Y˜ i˜ k,Q is unimpo an , ne e heless we ecall hei de ini ion o make hei physical meaning explici . The kinema ic a iable xis gi en by x=2˜pi ·Q Q2,(2.9) whe e ˜pi is he momen um o he pa en pa on in he (i )→i+ spli ing, which appea s on he igh hand side o eq. (2.3) abo e, while Qis he o al incoming momen um. We no e ha in he s ic collinea limi we ha e ˜pi →pi+p . The kinema ic a iable Y˜ i˜ k,Q is de ined as Y˜ i˜ k,Q =1 2 Q2(˜pi·˜pk) (˜pi·Q) (˜pk·Q)(2.10) Finally, he so -collinea sub ac ions lead o he in eg al K(ǫ, y0, d′ 0;κ) = 2 Zy0 0 dy y−(2+κ)ǫ(1 −y)d′ 0−1Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×1+ 2(1 −y) y(1−cos ϑ)1+κǫ Γ2(1 −ǫ) 2πΓ(1−2ǫ)Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ,(2.11) which does no depend on kinema ical a iables. The in eg als Jand Kin eqs. (2.6) and (2.11) ha e been compu ed in e . [21] o all ele an pa ame e s in y0,d′ 0and κ. We ha e e alua ed hese so and so -collinea in eg als, oo, and we ha e checked ha he wo esul s ag ee nume ically. We do no deal wi h he cases Jand Kin his pape . 2.2 Nes ed in eg als In a NNLO compu a ion, also i e a ions o he abo e in eg als appea . In his pape we comple e he lis o nes ed in eg als necessa y o he in eg a ed eal- i ual coun e e ms, in pa icula we co e all cases which ha e no been add essed in e . [21]. – 5 – JHEP08(2009)079 O he nes ed in eg als, which we gene ally deno e by a s a ∗, h ee a e collinea in eg als wi h one o he basic ypes in i s a gumen , I∗Ii(x;ǫ, α0, d0;k, l) = xZα0 0 dαZ1 0 d α−1−ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−ǫ ×[ (1 − )]−ǫα+(1−α)x 2α+(1−α)xk Ixα+(1−α)x(1− ) 2α+(1−α)x;ǫ, α0, d0; 0, l, 0,1,(2.12) I∗I (x;ǫ, α0, d0;k, l) = xZα0 0 dαZ1 0 d α−1−ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−ǫ ×[ (1 − )]−ǫα+ (1 −α)x 2α+ (1 −α)xk Ixα+ (1 −α)x 2α+ (1 −α)x;ǫ, α0, d0; 0, l, 0,1,(2.13) which we need o k, l =−1,0,1,2, and I∗J x;ǫ, α0, d0, y0, d′ 0;k=xZα0 0 dαZ1 0 d α−1−ǫ(1−α)2d0 −1[α+ (1 −α)x]−1−ǫ(2.14) ×[ (1 − )]−ǫα+ (1 −α)x 2α+ (1 −α)xk ×Jα(α+ (1 −α)x)(2α+ (1 −α)x)2 (α+(1−α)x )(α+(1−α)x(1− ))x2;ǫ, y0, d′ 0,0, o k=−1,0,1,2. Bo h, I∗I and I∗J a e needed as a unc ion o x∈[0,1] in an ǫ-expansion wi h Iand Jgi en in eqs. (2.5) and (2.6), espec i ely. A discussion abou he choice o he ele an pa ame e s α0,d0,y0and d′ 0is gi en a he end o sec ion 3and de ails o he compu a ion a e also gi en in sec ion 5. Th ee o he i e a ed in eg als a e de ined as so in eg als wi h o he so in eg als appea ing in he a gumen , J∗Jik(Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×Zy0 0 dy y−1−2ǫ(1 −y)d′ 0[2 −(1 + cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×J4(1 −y)Y˜ i˜ k,Q [2 −y(1 + cos ϑ)][2−y(1+cos χcos ϑ+sin χsin ϑcos ϕ)];ǫ, y0, d′ 0,0, (2.15) J∗Ji (Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ)(sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×[2 −(1 + cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×Zy0 0 dy y−1−2ǫ(1 −y)d′ 0J(1 −cos ϑ) 2−y(1 + cos ϑ);ǫ, y0, d′ 0,0,(2.16) – 6 – JHEP08(2009)079 and J∗Jk (Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ(2.17) ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×Zy0 0 dy y−1−2ǫ(1−y)d′ 0[2−(1+cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×J(1 −cos χcos ϑ−sin χsin ϑcos ϕ) 2−y(1 + cos χcos ϑ+ sin χsin ϑcos φ);ǫ, y0, d′ 0,0, wi h Jgi en in eq. (2.6). The h ee in eg als in eqs. (2.15)–(2.17) need o be calcula ed o Y˜ i˜ k,Q ∈[0,1] as expansion in ǫ. Explici esul s and de ails o he compu a ion (and alues o he pa ame e s y0and d′ 0) o hese in eg als a e p esen ed in sec ions 3and 6. The inal case is when he so in eg al appea s in he a gumen o a so -collinea one, K∗J(ǫ, y0, d′ 0) = 2 Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫZy0 0 dy y−1−2ǫ(1 −y)d′ 0−1 ×2−y(1 + cos ϑ) 1−cos ϑJ1−cos ϑ 2−y(1 + cos ϑ);ǫ, y0, d′ 0,0,(2.18) which is again independen o he kinema ics, i.e. he coe icien s o he expansion in ǫ a e pu e numbe s. De ails o he compu a ion and he pa ame e s y0and d′ 0a e gi en in sec ions 3and 7. 3 The me hod o Mellin-Ba nes ep esen a ions In his sec ion we b ie ly e iew he essen ial s eps in he de i a ion o MB ep esen a ions o he in eg als o sec ions 2.1 and 2.2. The s a ing poin is he well known basic o mula, 1 (a+b)ν=1 Γ(ν)Zq+i∞ q−i∞ dz 2πi a−ν−zbzΓ(ν+z)Γ(−z),(3.1) whe e νand qa e eal numbe s ( he case o ν= 0 is i ial) and qse s he asymp o ic posi ion o he in eg a ion con ou . The applica ion o eq. (3.1) o Feynman in eg al calculus was ini ia ed in e s. [23,24] (see also e . [28]) and is an algo i hmic p ocedu e which can be comple ely au oma ized, as e.g. in he Amb e.m package [31] in MATHEMATICA. In gene al, he con ou in eq. (3.1) is no necessa ily a s aigh line and i s s anda d de ini ion is such ha he poles o Γ(ν+z) (a z=−i−νwi h ibeing non-nega i e in ege ) a e all o he le and he poles o Γ(−z) (a non-nega i e in ege s) a e all o he igh o i . The condi ion on he poles o he Γ- unc ions can be sa is ied by such a con ou in eq. (3.1) i and only i q < 0 and ν > 0. Howe e , as a key obse a ion, e . [27] ealized s aigh -line con ou s pa allel o he imagina y axis in an algo i hmic way. I ν < 0, we s a wi h a cu ed con ou ha ul ills he condi ion on he pole and hen de o m i in o a s aigh – 7 – JHEP08(2009)079 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 -2 10-4 0 210-4 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ0) coe . o I(x, ǫ; 1,3; 1,−1,0, gC+) C- ype, κ= 1, k=−1, δ= 0 α0= 1 analy ic (A) α0= 1 nume ic (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 -1.0 102 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 6.0 102 7.0 102 -2 10-4 0 210-4 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ0) coe . o I(x, ǫ; 1,3; 1,−1,1, gD+) D- ype, κ= 1, k=−1, δ= 1 α0= 1 analy ic (A) α0= 1 nume ic (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 Figu e 3. Rep esen a i e esul s o he C- ype and D- ype in eg als. The plo s show he coe icien o he O(ǫ0) e m o k=−1 in I(x, ǫ; 1,3; 1,−1,0, g(+) C) (le ) and I(x, ǫ; 1,3; 1,−1,1, g(+) D) ( igh ) wi h d0= 3 and α0= 1. He e we in oduced he sho -hand no a ion P(m) n,k (x;a(k) m,...,a(k) 0) = Lin(1 −x) (1 −x)m m X i=0 a(k) ixi.(4.7) Acco ding o hei de ini ion, he limi o he unc ions gi en in eqs. (4.3)–(4.6) mus be ini e in x= 1 e en i some e ms a e sepa a ely di e gen . Indeed compu ing he limi a x= 1 we ind lim x→1F(x;ǫ, d0= 3,−1) = −8731 3600 −3 2ζ2,(4.8) lim x→1F(x;ǫ, d0= 3,0) = −257 60 ,(4.9) lim x→1F(x;ǫ, d0= 3,1) = −257 120 ,(4.10) lim x→1F(x;ǫ, d0= 3,2) = −1801 90 +80 3log(2) .(4.11) In igu e 3we compa e he analy ic and nume ic esul s o he ǫ0coe icien in he expansion o I(x, ǫ; 1,3; 1,−1,0, g(+) C) and I(x, ǫ; 1,3; 1,−1,1, g(+) D) o k=−1, α0= 1 and d0= 3 as ep esen a i e examples. The ag eemen be ween he wo compu a ions is excellen o he whole x- ange. The nume ic esul s ha e been ob ained using a sec o decomposi ion [40] and Mon e Ca lo in eg a ion p og am as explained in de ail in e s. [21, 22]. This shows ha he expansion coe icien s o all he collinea in eg als Iand hence also o he collinea sub ac ion e ms a e smoo h unc ions o he kinema ical a iable x. – 14 – JHEP08(2009)079 The comple e esul s o all necessa y cases (like in he la e sec ions) a e o conside - able size, such ha we shall no lis hem he e. They a e all con ained in a MATHEMATICA ile p o ided wi h he sou ces o he pape on he a chi e h p://a Xi .o g. 5 Nes ed collinea - ype I∗I and I∗J in eg als In his sec ion we discuss he analy ic compu a ion o he nes ed collinea in eg als de ined in eqs. (2.12)–(2.14). As an example we show explici ly he ully analy ic esul o he case I ∗ I (x, ǫ; 1,3; −1,2) o which we we e able o compu e he comple e pole s uc u e ana- ly ically. Choosing d0= 3 and α0= 1 we ge I∗I (x, ǫ; 1,3; −1,2) = −1 12 1 ǫ3+−2 9+1 3log(x)1 ǫ2+1 (1 −x)5−1 3ζ2−25 36 log(x) +1 3log(1−x) log(x)+ 1 3Li2(x)+1 (1−x/2)51 6log x 2+1 (1−x)4−13 36 +1 6log(x) +1/6 (1−x/2)4+1 (1−x)3−7 72 −1 18 log(x)+1/12 (1 −x/2)3+1 (1 −x)2−1 6−2 9log(x) +1/18 (1 −x/2)2+1 (1 −x)−25 72 −7 12 log(x)+1/24 (1 −x/2) +31 216 +1 6log(2) +19 9log(x) + 2 3log(1 −x) log(x)−2 3log2(x) + 2 3Li2(x)1 ǫ+ O(ǫ0).(5.1) This esul is ep esen a i e, because i s o m is ypical o all he collinea nes ed in- eg als. The plo o he O(ǫ−1) coe icien o his Lau en expansion o he in eg al I∗I (x, ǫ; 1,3; −1,2) is shown on he igh side o igu e 4 oge he wi h he compa i- son wi h he nume ical e alua ion ob ained using sec o decomposi ion and Mon e Ca lo in eg a ion. On he le side o igu e 4we plo he same coe icien o he Lau en ex- pansion o he in eg al I∗Ii(x, ǫ; 1,3; −1,2). Fo bo h cases we no e ha he ag eemen be ween he nume ical e alua ion and he analy ic esul is excellen . These plo s show also ha he coe icien s o he Lau en expansion o he nes ed collinea in eg als I∗I a e e y smoo h unc ions o x. We no e ha in he Lau en expansion o I∗I (x, ǫ; 1,3; −1,2) in eq. (5.1) he e a e some e ms ha a e di e gen in x= 1. Howe e acco ding o i s de ini ion in eq. (2.13) he limi in x= 1 mus be ini e. To e i y his is a u he check o he co ec ness o he esul . Fo he case o eq. (5.1) we ob ain ha : lim x→1I∗I (x, ǫ; 1,3; −1,2) = −1 12 1 ǫ3−2 9 1 ǫ2+3091 675 +2 3ζ2−31 6log(2)1 ǫ+ O(ǫ0).(5.2) The case o I∗I (x, ǫ; 1,3; 2,−1) is mo e di icul . Fo his in eg al we a e unable o compu e he coe icien s o he ǫpoles in a ully analy ic o m. The eason is ha in i s Mellin-Ba nes ep esen a ion also h ee- old MB in eg als a e in ol ed. Fo his case he coe icien O(ǫ−3) and O(ǫ−2) a e ully analy ic bu he coe icien o O(ǫ−1) is semi-analy ic. This las coe icien is hus w i en in e ms o an analy ic exp ession – 15 – JHEP08(2009)079 -5.0 102 -4.0 102 -3.0 102 -2.0 102 -1.0 102 0 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coe . o I∗Ii(x, ǫ; 1,3; −1,2) I∗Ii,k=−1, l= 2 α0= 1 analy ic (A) α0= 1 nume ic (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 -2.5 102 -2.0 102 -1.5 102 -1.0 102 -5.0 101 0 5.0 101 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coe . o I∗I (x, ǫ; 1,3; −1,2) I∗I ,k=−1, l= 2 α0= 1 analy ic (A) α0= 1 nume ic (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 Figu e 4. Rep esen a i e esul s o he I∗I- ype in eg als. The plo s show he coe icien o he O(ǫ−1) e m o k=−1 and l= 2 in I∗Ii(x, ǫ; 1,3; −1,2) (le ) and I∗I (x, ǫ; 1,3; −1,2) ( igh ) wi h d0= 3 and α0= 1. o which a h ee- old MB in eg al mus be added. The emaining MB in eg al can be e icien ly compu ed in MATHEMATICA by use o he package MB.m [29]. Explici ly o I∗ I (x, ǫ; 1,3; 2,−1) we ha e: I∗I (x, ǫ; 1,3; 2,−1) = −1 6 1 ǫ3+1 (1 −x/2)6−5 12 log x 2+1 (1 −x)51 6log(x) +1 (1 −x/2)5−5 12 +5 12 log x 2+1/6 (1 −x)4+5/24 (1 −x/2)4+1/12 (1 −x)3 +5/72 (1 −x/2)3+1/18 (1 −x)2+5/144 (1 −x/2)2+1/24 (1 −x)+1/48 (1 −x/2) −59 72 +1 2log(x)1 ǫ2 +1 (1 −x/2)625 24 ζ2−21 8log(2) + 5 4log2(2) + 5 3log(2) log(1 −x/2) + 21 8log(x) −5 2log(2) log(x) + 5 12 log(1 −x) log(x)−5 3log(1 −x/2) log(x) + 5 4log2(x) −5 3Li2x 2+5 12 Li2(x)+1 (1 −x)5−1 3ζ2−1 6log2(2) −1 3log(2) log(1 −x/2) +17 24 log(x) + 1 3log(2) log(x) + 1 6log(1 −x) log(x) + 1 3log(1 −x/2) log(x) −1 2log2(x) + 1 3Li2x 2+1 6Li2(x)+1 (1 −x/2)523 24 −25 24 ζ2+71 24 log(2) −5 4log2(2) −5 3log(2) log(1 −x/2) −17 8log(x) + 5 2log(2) log(x) + −5 12 log(1 −x) log(x) + 5 3log 1−x 2log(x)−5 4log2(x) + 5 3Li2x 2−5 12 Li2(x) – 16 – JHEP08(2009)079 +1 (1 −x)47 8−1 4ζ2+2 3log(2) −11 8log(x) + 1 4log(1 −x) log(x) + 1 4Li2(x) +1 (1 −x/2)4−59 48 +1 6log x 2+1 (1 −x)3−1 16 −1 12 ζ2−7 36 log(x) +1 12 log(1 −x) log(x) + 1 12 Li2(x)+1 (1 −x/2)3−29 432 −1 6log(2) + 4 9log(x) +1 (1 −x)2−1 27 +2 9log(2) −23 36 log(x)+1 (1 −x/2)2211 864 −2 9log(2)+ 7 9log(x) +1 (1−x)−31 72 −59 24 log(x)+1 (1−x/2) 139 288 −1 3log(2) + 4 3log(x)−1177 432 +7 8ζ2 −1 3log(2) + 1 6log2(2) + 1 3log(2) log 1−x 2+71 24 log(x)−1 3log(2) log(x) +1 2log(1 −x) log(x)−1 3log 1−x 2log(x)−5 6log2(x)) −1 3Li2x 2+1 2Li2(x) + MBin [x]1 ǫ+ O(ǫ0),(5.3) whe e MBin [x] is a h ee- old Mellin-Ba nes in eg al, which o his case is gi en by MBin [x] = Zq1+i∞ q1−i∞ dz1 2πi Zq2+i∞ q2−i∞ dz2 2πi Zq3+i∞ q3−i∞ dz3 2πi 2z3−1x−z1−z2−z3(5.4) ×Γ −z1,1+z1,3−z2,−2+z2,5−z1−z2−z3,−z3,2+z3, z1+z2+z3 4,4−z2!, whe e q1=q2=q3=−1/4. Simila ly o he analy ic exp ession o eq. (5.1), also in his case we ha e many e ms ha a e singula in x= 1 e en hough he ull exp ession is well de ined. Mo eo e in cases like his whe e we ha e a semi-analy ic exp ession we ind ha he analy ic pa and he emaining pa exp essed in e ms o a h ee- old MB in eg al a e sepa a ely well de ined in x= 1. In pa icula o he case o he in eg al I∗I (x, ǫ; 1,3; 2,−1) in eq. (5.3) we ob ain he ollowing limi : lim x→1I∗I (x, ǫ; 1,3; 2,−1) = −1 6 1 ǫ3+−607 60 +40 3log(2)1 ǫ2+77349 14400 +509 24 ζ2 −3571 45 log(2) + 40 3log2(2) + MBin [1]1 ǫ+ O(ǫ0),(5.5) whe e MBin [1] is gi en by MBin [1] = 0.329808.(5.6) This numbe is he esul o he MB in eg al in eq. (5.4) wi h he choice x= 1 ob ained using he MATHEMATICA package MB.m [29]. Finally we no e ha his example is ep esen a i e o a small subse o he collinea nes ed in eg als which ha e hese ea u es. They a e I∗ Ii(x, ǫ; 1,3; k, l) and I∗I (x, ǫ; 1,3; k, l) wi h k=−1,1,2 and l=−1 and I∗J(x, ǫ; 1,3,1,3; k) wi h k=−1. The esul s o he pole s uc u e o all he emaining cases o nes ed collinea in eg als a e ully analy ic. – 17 – JHEP08(2009)079 0 5.0 101 1.0 102 1.5 102 2.0 102 2.5 102 3.0 102 3.5 102 4.0 102 4.5 102 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coe . o I∗J (x, ǫ; 1,3,1,3; 0) I∗J,k= 0 α0=y0= 1 anl. (A) α0=y0= 1 num. (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0=y0= 1 Figu e 5. Rep esen a i e esul s o he I∗J- ype in eg als. The plo s show he coe icien o he O(ǫ−1) e m o k= 0 in I∗J(x, ǫ; 1,3,1,3; 0) wi h d0=d′ 0= 3 and α0=y0= 1. In able 2we lis nume ical alues o he non- i ial coe icien s o he ǫ-poles (i.e. he O(ǫ−2) and O(ǫ−1) coe icien s) o he nes ed collinea in eg als I∗I (x, ǫ; 1,3; −1,2) and I∗I (x, ǫ; 1,3; 2,−1). These numbe s ha e been ob ained using he ully analy ic exp ession in eq. (5.1) and he semi-analy ic one in eq. (5.3). Fo his las case using he de aul op ions o nume ical in eg a ion accu acy in MB.m he ela i e unce ain y is a mos o o de 10−5, o alues o xa ound one. Fo x≪1, we ind ha he analy ic pa o he ull semi-analy ic esul con ains all con ibu ions ha a e di e gen as x→0 and in ac he nume ic con ibu ion dec eases as we app oach he limi . Thus he ela i e unce ain ies become e y small. Numbe s o he O(ǫ0) coe icien o he same ep esen a i e in eg als a e lis ed in able 3. In his case hey ha e been en i ely ob ained e alua ing hei MB ep esen a ions. The ela i e unce ain ies epo ed in able 3we e ob ained wi h he nume ical in eg a ion op ion MaxPoin s se o 5 ·107in MB.m. Finally in igu e 5we plo as a u he example he ully analy ic esul o he i s o de ǫ-pole o I∗J(x, ǫ; 1,3; 0) oge he wi h he numbe s ob ained nume ically using sec o decomposi ion and Mon e Ca lo in eg a ion. As o all o he cases he ag eemen is excellen and he coe icien is gi en by a e y smoo h unc ion o x. In his sec ion as in he es o he pape he ep esen a i e plo s a e shown o he mos complica ed in eg als o which a ull analy ic esul was ob ained. 6 Nes ed so - ype J ∗J in eg als In his sec ion we discuss he analy ic compu a ion o he in eg als de ined in eqs. (2.15)–(2.17). Fo hem we we e able o compu e a ully analy ic esul o he coe icien o he Lau en expansion up o O(ǫ−2). The O(ǫ−1) coe icien is compu ed – 18 – JHEP08(2009)079 log10(x)I∗I (x, ǫ; 1,3; −1,2) I∗I (x, ǫ; 1,3; 2,−1) O(ǫ−2) an. O(ǫ−1) an. O(ǫ−2) an. O(ǫ−1) semi-an. -10. -7.89751 -374.957 -15.9061 -759.736 ±2.16974E-12 -9.66667 -7.64166 -351.104 -15.3944 -711.688 ±2.92666E-12 -9.33333 -7.38582 -328.036 -14.8828 -665.211 ±4.09941E-12 -9. -7.12998 -305.753 -14.3711 -620.305 ±6.01115E-12 -8.66667 -6.87413 -284.256 -13.8594 -576.969 ±7.79755E-12 -8.33333 -6.61829 -263.544 -13.3477 -535.205 ±1.08953E-11 -8. -6.36245 -243.618 -12.836 -495.012 ±1.54886E-11 -7.66667 -6.10661 -224.477 -12.3243 -456.389 ±1.84206E-11 -7.33333 -5.85076 -206.122 -11.8126 -419.337 ±2.91762E-11 -7. -5.59492 -188.552 -11.301 -383.857 ±3.7663E-11 -6.66667 -5.33908 -171.768 -10.7893 -349.947 ±4.44314E-11 -6.33333 -5.08324 -155.769 -10.2776 -317.608 ±6.83357E-11 -6. -4.82739 -140.556 -9.7659 -286.841 ±8.82413E-11 -5.66667 -4.57155 -126.128 -9.25423 -257.644 ±1.11892E-10 -5.33333 -4.31571 -112.485 -8.74256 -230.019 ±2.04546E-10 -5. -4.05986 -99.628 -8.2309 -203.965 ±4.4044E-10 -4.66667 -3.80402 -87.556 -7.71928 -179.482 ±9.48925E-10 -4.33333 -3.54818 -76.269 -7.20772 -156.573 ±2.0463E-9 -4. -3.29234 -65.7665 -6.69628 -135.237 ±4.4068E-9 -3.66667 -3.03649 -56.0477 -6.18508 -115.476 ±9.49612E-9 -3.33333 -2.78065 -47.1111 -5.6743 -97.2944 ±2.04532E-8 -3. -2.52481 -38.9536 -5.16432 -80.6957 ±4.39917E-8 -2.66667 -2.26896 -31.5702 -4.65576 -65.6862 ±9.46543E-8 -2.33333 -2.01312 -24.9522 -4.14969 -52.2723 ±2.03847E-7 -2. -1.75728 -19.0853 -3.64776 -40.4585 ±4.37447E-7 -1.66667 -1.50144 -13.9478 -3.15236 -30.2408 ±9.3859E-7 -1.33333 -1.24559 -9.50936 -2.66658 -21.5965 ±1.99232E-6 -1. -0.989751 -5.73082 -2.19382 -14.4712 ±4.20263E-6 -0.666667 -0.733908 -2.5675 -1.73699 -8.76877 ±8.68698E-6 -0.333333 -0.478065 0.0265877 -1.2975 -4.35339 ±1.73005E-5 0. -0.222222 2.09462 -0.874704 -1.06702 ±3.28582E-5 Table 2. Nume ical alues o he O(ǫ−2) and O(ǫ−1) coe icien s o I∗I (x, ǫ; 1,3; −1,2) (second and hi d column) and I∗I (x, ǫ; 1,3; 2,−1) (las wo columns) o a ious alues o log10(x) ( i s column). These numbe s ha e been ob ained e alua ing he ully analy ic exp ession in eq. (5.1) and he semi-analy ic one in eq. (5.3). In he las column, we also show he nume ical unce ain y as epo ed by MB.m. – 19 – JHEP08(2009)079 log10(x)I∗I (x, ǫ; 1,3; −1,2) I∗I (x, ǫ; 1,3; 2,−1) -5. -1642.9 ±1.40321 -3380.06 ±1.88480E-1 -4.66667 -1355.91 ±2.483460E-1 -2792.08 ±5.41150E-2 -4.33333 -1104.52 ±4.46988E-2 -2276.68 ±2.74382E-2 -4. -886.641 ±1.16175E-2 -1829.26 ±2.16858E-2 -3.66667 -699.73 ±8.45731E-3 -1444.98 ±1.81844E-2 -3.33333 -541.355 ±7.21400E-3 -1119.05 ±1.53613E-2 -3. -409.039 ±5.95157E-3 -846.659 ±1.28763E-2 -2.66667 -300.296 ±4.91329E-3 -622.987 ±1.08469E-2 -2.33333 -212.588 ±3.96846E-3 -443.178 ±8.21964E-3 -2. -143.342 ±3.13593E-3 -302.315 ±6.39871E-3 -1.66667 -89.9695 ±2.39827E-3 -195.382 ±4.76177E-3 -1.33333 -49.9198 ±1.77836-3 -117.264 ±3.27515E-3 -1. -20.7586 ±1.13024E-3 -62.7773 ±2.08013E-3 -0.666667 -0.267976 ±6.05459E-4 -26.8568 ±1.13023E-3 -0.333333 13.489 ±3.05477E-4 -4.81243 ±5.52177E-4 0. 22.1524 ±5.24078E-4 7.37746 ±4.201149E-4 Table 3. Nume ical alues o he O(ǫ0) coe icien o I∗I (x, ǫ; 1,3; −1,2) (second column) and I∗I (x, ǫ; 1,3; 2,−1) (las column) o a ious alues o log10(x) ( i s column). These numbe s ha e been ob ained e alua ing hei MB ep esen a ion. Aalso shown a e he nume ical unce ain ies as epo ed by MB.m. semi-analy ically simila ly o he nes ed collinea in eg al I∗I (x, ǫ; 1,3; 2,−1) discussed in sec ion 5. As a ep esen a i e example we show he s uc u e o he ully analy ic pa o he esul o he nes ed so in eg als J∗J. Fo example choosing d′ 0= 3 we ha e: J∗Jik(Y;ǫ; 1,3) = 1 ǫ4+22 3−2 log(Y)1 ǫ3+H(Y)1 ǫ2+ O(ǫ−1),(6.1) J∗Ji (Y;ǫ; 1,3) = 1 2 1 ǫ4+11 3−log(Y)1 ǫ3+533 36 −22 3log(Y) + log2(Y) +3 2Li2(1 −Y)1 ǫ2+ O(ǫ−1) (6.2) and inally J∗Jk (Y;ǫ; 1,3)= 1 ǫ4+22 3−2 log(Y)1 ǫ3+H(Y)−ζ2+1 2Li2(1−Y)1 ǫ2+O(ǫ−1).(6.3) The unc ion H(Y) which appea s in eqs. (6.1) and (6.3) is gi en by H(Y)= 497 18 −2ζ2+6−8Y 3(1−Y)2+33Y3−117Y2+126Y−44 3 (1 −Y)3log(Y) + 2 log2(Y) + 4 Li2(1 −Y). (6.4) – 20 – JHEP08(2009)079 Also o his unc ion e en i some e ms a e singula a Y= 1, we s ill ha e ha he limi is well de ined. Indeed we ind lim Y→1H(Y) = 97 3−2ζ2.(6.5) Fo hese h ee so - ype in eg als he O(ǫ−2) coe icien has been plo ed in igu e 6 using i s ully analy ic exp ession eqs. (6.1) and (6.4) and i s nume ical e alua ion ob ained using sec o decomposi ion and Mon e Ca lo in eg a ion. The ag eemen is excellen and he analy ic esul con i ms ha also he coe icien s o he Lau en expansion o he J∗J in eg als a e smoo h unc ions o Y. The numbe s in able 4ha e been ob ained e alua ing he nes ed so in eg al J∗Jik(Y;ǫ; 1,3) using he ully analy ic exp ession in eq. (6.1) o he O(ǫ−3) and O(ǫ−2) coe icien s. Fo he O(ǫ−1) coe icien a semi-analy ic exp ession in e ms o a MB in e- g al has been used and inally he ep esen a ion only in e ms o MB in eg als has been e alua ed o he O(ǫ0) coe icien . In his example he ela i e unce ain y as epo ed by MB.m wi h he de aul op ions o nume ical in eg a ion o bo h he O(ǫ−1) and O(ǫ0) coe icien s is a mos o o de 10−5. Fo he semi-analyi c esul , we see a simila phe- nomenon as in he case o he I∗I (x, ǫ; 1,3; 2,−1) in eg al: in he Y→0 limi , he analy ic pa con ains all di e gen con ibu ions while he nume ic pa dec eases. The ela i e unce ain ies hus become e y small. 7 Nes ed so -collinea K∗J in eg al In his las sec ion we discuss he pole s uc u e o he in eg al de ined in eq. (2.18). In his case he esul o he Lau en expansion is e y simple because he in eg al has no dependence on he kinema ics. The coe icien s o he poles in K∗J(ǫ, 1,3) wi h d′ 0= 3 and y0= 1 ead: K∗J(ǫ, 1,3) = −1 2 1 ǫ4−11 3 1 ǫ3−557 36 1 ǫ2+−10825 216 +5 3ζ2−3ζ31 ǫ+ O(ǫ0).(7.1) This comple es ou discussion o he analy ic compu a ion o he undamen al in eg als ha con ibu e o he singly-un esol ed coun e e ms. 8 Conclusions In his wo k we ha e comple ed he e alua ion o all in eg als needed o he compu a ion o he in eg a ed eal- i ual coun e e ms o he sub ac ion scheme o NNLO je c oss sec ions p oposed in e s. [17–19]. We ha e discussed ep esen a i e examples o all ypes o so and collinea as well as nes ed in eg als in sec ions 4–7( he comple e esul s a e con ained in a MATHEMATICA ile). These in eg als (i.e. hei Lau en expansions in ǫ o su icien dep h) ha e o be compu ed once and o all and hei knowledge is necessa y in o de o make he sub ac ion scheme an e ec i e ool. We ha e achie ed his ask by de i ing MB ep esen a ions o all in eg als unde conside a ion and, in a subsequen s ep, we ha e pe o med analy ically he summa ion o he nes ed sums o e he se ies – 21 – JHEP08(2009)079 0 2.0 102 4.0 102 6.0 102 8.0 102 1.0 103 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coe . o J∗Jik(Y, ǫ; 1,3) J∗Jik y0= 1 analy ic (A) y0= 1 nume ic (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coe . o J∗Ji (Y, ǫ; 1,3) J∗Ji y0= 1 analy ic (A) y0= 1 nume ic (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 0 2.0 102 4.0 102 6.0 102 8.0 102 1.0 103 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coe . o J∗Jk (Y, ǫ; 1,3) J∗Jk y0= 1 analy ic (A) y0= 1 nume ic (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 Figu e 6. Rep esen a i e esul s o he J∗J- ype in eg als. The plo s show he coe icien o he O(ǫ−2) e m in J∗Jik(Y, ǫ; 1,3) (le ), J∗Ji (Y, ǫ; 1,3) ( igh ) and J∗Jk (Y, ǫ; 1,3) (bo om) wi h d′ 0= 3 and y0= 1. o esidues. In some cases, his second s ep o summing he se ies has no been achie ed and we ha e eso ed o a nume ical e alua ion o he MB in eg als in he complex plane. All MB ep esen a ions o bo h he nume ical and, i a ailable, he analy ic esul s ha e been checked by an independen e alua ion o he in eg als using sec o decomposi ion as in e . [22]. We ha e shown, ha all in eg als con ibu ing o he eal- i ual coun e e ms a e smoo h unc ions. Fo p ac ical applica ions, his means ha all in eg als (in pa icula he ini e in ǫcon ibu ions) can be used in e ms o in e pola ing ables, which can be compu ed once and o all. He e we wan o s ess again ha he ables and plo s we – 22 – JHEP08(2009)079 log10(Y)J∗Jik(Y;ǫ; 1,3) O(ǫ−3) an. O(ǫ−2) an. O(ǫ−1) semi-an. O(ǫ0) MB -10. 53.385 1430.99 25680. ±1.44332E-11 347094. ±1.69388E-3 -9.66667 51.85 1350.22 23545.6 ±2.90334E-11 309328. ±1.31378E-3 -9.33333 50.3149 1271.81 21533.4 ±5.83683E-11 274744. ±1.09995E-3 -9. 48.7799 1195.75 19639.8 ±1.17829E-10 243157. ±9.10225E-4 -8.66667 47.2448 1122.05 17861.1 ±2.34891E-10 214389. ±9.47250E-4 -8.33333 45.7098 1050.7 16193.8 ±4.73302E-10 188265. ±8.07442E-4 -8. 44.1747 981.714 14634.2 ±9.39828E-10 164617. ±6.22748E-4 -7.66667 42.6396 915.081 13178.6 ±1.86537E-9 143283. ±5.50516E-4 -7.33333 41.1046 850.805 11823.6 ±3.70809E-9 124106. ±5.03212E-4 -7. 39.5695 788.886 10565.3 ±7.23854E-9 106934. ±5.38078E-4 -6.66667 38.0345 729.322 9400.38 ±1.44201E-8 91621. ±5.52685E-4 -6.33333 36.4994 672.116 8325.04 ±2.82069E-8 78027.5 ±5.08049E-4 -6. 34.9644 617.265 7335.7 ±5.49916E-8 66018.2 ±5.08676E-4 -5.66667 33.4293 564.771 6428.75 ±1.05801E-7 55463.9 ±5.09976E-4 -5.33333 31.8942 514.633 5600.58 ±2.03064E-7 46240.8 ±4.63345E-4 -5. 30.3592 466.852 4847.56 ±3.86389E-7 38230.9 ±5.01881E-4 -4.66667 28.8241 421.427 4166.08 ±7.45290E-7 31321.6 ±6.62424E-4 -4.33333 27.2891 378.358 3552.51 ±1.35799E-6 25405.6 ±6.96993E-4 -4. 25.754 337.645 3003.24 ±2.52452E-6 20381.7 ±7.53947E-4 -3.66667 24.219 299.287 2514.63 ±4.73890E-6 16153.6 ±7.61827E-4 -3.33333 22.6839 263.283 2083.06 ±8.67357E-6 12631. ±8.70475E-4 -3. 21.1488 229.632 1704.89 ±1.59261E-5 9728.88 ±8.12022E-4 -2.66667 19.6138 198.33 1376.45 ±2.79802E-5 7367.63 ±8.23619E-4 -2.33333 18.0787 169.37 1094.05 ±4.80260E-5 5473.13 ±9.09435E-4 -2. 16.5437 142.739 853.961 ±8.03383E-5 3976.65 ±1.09059E-3 -1.66667 15.0086 118.417 652.369 ±1.30870E-4 2814.76 ±1.41898E-3 -1.33333 13.4736 96.3641 485.392 ±2.06356E-4 1929.49 ±1.90893E-3 -1. 11.9385 76.5204 349.046 ±3.17701E-4 1268.34 ±2.67185E-3 -0.666667 10.4034 58.7892 239.262 ±4.48758E-4 784.581 ±3.65297E-3 -0.333333 8.86839 43.0286 151.932 ±5.96048E-4 437.509 ±4.88145E-3 0. 7.33333 29.0435 82.998 ±7.61558E-4 192.684 ±6.83182E-3 Table 4. Nume ical alues o he O(ǫ−3), O(ǫ−2), O(ǫ−1) and O(ǫ0) coe icien s o J∗Jik(x;ǫ; 1,3) o a ious alues o log10(Y). The numbe s ha e been ob ained om eq. (6.1), he semi-analy ic one o he O(ǫ−1) coe icien and MB in eg als o he O(ǫ0) coe icien . In he las wo columns, we also show he nume ical unce ain ies as epo ed by MB.m. – 23 –