Analy ic in eg a ion o eal- i ual coun e e ms in NNLO je c oss sec ions II
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JHEP08(2009)079
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Re ised:July 24, 2009
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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 –