Soft gluons and superleading logarithms in QCD

0.35 MB·English

by  J. R. Forshaw

MAN/HEP/2009/3
CERN-TH/2009-012
Soft gluons and superleading logarithms in QCD
9
0 J.R. Forshawa∗, M.H. Seymourab
0
aSchool of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
2

n bTheoretical Physics Group, CERN, CH-1211,Geneva 23, Switzerland.
a
J
After a brief introduction to the physics of soft gluons in QCD we present a surprising prediction. Dijet
0 production in hadron-hadron collisions provides the paradigm, i.e. h1+h2 →jj+X. In particular, we look at
2
thecasewherethereisarestrictionplacedontheemissionofanyfurtherjetsintheregioninbetweentheprimary

] (highest pT) dijets. Logarithms in the ratio of the jet scale to the veto scale can be summed to all orders in the
h strong coupling. Surprisingly, factorization of collinear emissions fails at scales above the veto scale and triggers
p the appearance of double logarithms in the hard sub-process. The effect appears first at fourth order relative to
-
theleading order prediction and is subleading in the numberof colours.
p
e
h
[

1 1. Introducing soft gluons and coherence
v
7 Given a particular short distance process in
3 QCD (“hard scattering”) we can ask how it will
0 be dressed with additional radiation. A priori,
3
the question may not be accessible to pertur-
.
1 bative QCD because hadronization effects could
0 wrecktheunderlyingpartoniccorrelations. How-
9
ever, experiment reveals that the hadronization
0
process is gentle and we are in business. Our Figure 1. Soft gluon emission (wavy line) off a
:
v attention should focus on the most important four-parton hard scattering.
i
X emissions and these involve either soft gluons or
collinear branchings, or both. By important, we
r
a meanthatthesuppressionprovidedbythestrong
wherethevarioussymbolsaredefined(forn=4)
coupling is compensated by a large logarithm,
in Fig. 1. The blob in the figure represents a
which emerges when the angle between two par-
generic short-distance process and we need only
tonsbecomessmallorwhentheenergyofagluon
everconsidersoftgluonemissionsoffexternallegs
becomes small.
since they are incapable of putting the internal
We’ll start with a very brief review of soft glu-
hard propagators on-shell2. The virtual correc-
ons. The Feynman rule for the emission of a soft
tions factorize similarly, and we will have more
gluon off any fast parton is, to good approxima-
to say about them shortly. All looks pretty sim-
tion, proportional to the four-momentum of the
ple, but there is a major obstacle preventing au-
fast parton. This simplification means that we
tomated soft gluon calculations in generic pro-
canfactorizethecross-sectionforann+1parton
cesses: the colour factor C is very difficult to
process (i.e. with one soft gluon and n partons ij
keep track of. Think of emitting many soft glu-
participating in the hard scatter) as
ons. The leading logarithmic behaviour nests,
α dE dΩ p ·p which means that each emission sees an effective
dσ =dσ s C E2 i j (1)
n+1 n ij
2π E 2π ij pi·q pj ·q 2Since we use Feynman gauge throughout, we can also
X
neglect self-energy type diagrams, both in the real and
∗Talk presented at the workshop “New Trends in HERA virtual contributions, i.e. the sum in Eq. (1) runs over
Physics”,RingbergCastle,Tegernsee,5–10October2008. i6=j.
1
2 J.R Forshaw and M.H. Seymour
short-distanceprocessinvolvingallpreviousemis- Fig. 2 is a diagrammatic statement of the
sionsandthecorrespondingcolourstructurelives Bloch-Nordsieck Theorem: Summing over cuts,
in some large representation of SU(3). This ac- the real and virtual contributions exactly cancel
tually is no problem for theorists who can calcu- inthesoftgluonapproximation. Ifrealemissions
late in a colour basis independent way [1,2] but areforbiddenforsomereason(orcontributetoan
it is a problem when it comes to putting a num- observablewithanythingotherthanunitweight)
ber on a cross-section. To date, this problem has then a miscancellationis induced and that leaves
not been solved for n > 3. Monte Carlo genera- behindalogarithminthevolumeV ofthephase-
tors, like HERWIG and PYTHIA, duck the issue space into which realemissionis suppressed. Ex-
by working in the largeN approximation. They amples of observables affected in this way are
c
also exploit the fact that, after integrating over event shapes (e.g. thrust where V =1−T), pro-
azimuth in Eq.(1), soft emissions take place into ductionnearthreshold(V =1−M2/s),Drell-Yan
cones of successively smaller angles as one moves at low p (V = p2/s), deep inelastic scattering
T T
away from the hard scatter. The major part of at large x (V = 1−x) and “gaps between jets”.
this talk will be concernedwith soft gluonsbut a Our attention now turns to the latter process: it
word first on collinear emissions. is the simplest process involving four partons in
Collinear partonic evolution is easier to deal the hard scatter. Specifically, one is interested in
with. Thecolourstructuresimplifiesandpartonic a finalstatecontainingtwohighp jets. The ob-
T
evolutionreducestoaclassicalbranchingprocess: servablecross-sectionis definedbyimposing that
it is as if particles are emitted off the parton to there be no additional jets in the rapidity region
whichtheyarecollinear. Hence(afterintegrating between the two hard jets that have p > Q .
T 0
over azimuthal angle) We emphasise that Q defines the gap in an ex-
0
perimentally well defined manner [3,4]. For this
αs dq2 observable V = Q0/Q where Q is the pT of the
dσn+1 =dσn2π q2 dz Pba(z) (2) hard jets.
ab
X
for a → bc, where parton b carries a fraction z
of parton a’s momentum, and P (z) is the cor-
ba
responding splitting function.
The large N approximation for soft emissions
c
permitsonetocombinethemwithcollinearemis-
sions in a single parton shower, successive emis-
sions being ordered in angle. Our task is to go
beyond the leading N approximationand gain a
c
better understanding of soft gluon physics.
Figure 3. Coherence.
QCDcoherenceunderpinstheangularordering
of soft gluon emissions and can be used to argue
that a wide-angle soft gluon emitted off a bunch
Figure 2. Soft gluons cancel in a sum over cuts. of collinear partons can always be re-attached to
Soft gluons and superleading logarithms in QCD 3
theprimaryhardpartonthatdefinesthedirection region is cancelled by real emissions according to
of the bunch. It is a statement at the amplitude Bloch-Nordsieck. Werefertothesoftgluonsthat
levelfor the virtualsoftgluoncorrections1 andis generate this real part as eikonal gluons. The
illustrated in Fig. 3. It is understood that there imaginary part is more subtle: it arises as a re-
are any number of additional partons exiting the sult of Coulomb gluon exchange and has noth-
hard-scatter blob (not shown) and that the soft ing to cancel against (it makes no sense to speak
gluon re-attaches to one of them. The crucial of the rapidity of these gluons). To pick up all
point to note is that coherencedefined in this di- of the leading logarithms to all orders in α one
s
agrammaticsense breaksdownfor the imaginary mightthinkthatweneedonlyiteratetheprocess
part of the amplitude. It is spoilt by virtual soft ofaddingasoftgluonbetweenthejets,witheach
gluonsthatputon-shellthepartonstowhichthey successiveemissionoccurringatmuchsmallerk
T
attach. SuchgluonsarevariouslycalledCoulomb thantheonebefore2. Itisasifallprioremissions
or Glauber gluons in the literature and they are are sitting in the short-distance blob in Fig. 4.
the bane of those wanting to prove QCD factor- The net effect is an amplitude
ization theorems. Assuming coherence leads us
to conclude that the gaps-between-jets observ- αs Q dkt
A = A exp − C (Y +ρ )
0 F jet
able should contain only single logarithms, i.e. π ZQ0 kt !
terms ∼ αnlnn(Q/Q ), since it is inclusive over
s 0 2α Q dk
the collinear regions. × exp + s tC iπ , (3)
F
π k
Z0 t !
where ρ is some function dependent upon the
jet
jetalgorithm(wehaveignoredtherunningofthe
coupling but it is easy enough to restore it). The
colour structure in this two-jet example is simple
(the qq¯must be in a colour singlet) and as a re-
sult the Coulomb gluon (iπ) term generates an
unimportant phase.
The storyis muchthe samefor three andfour-
Figure4. Thesoftgluondressingofatwo-parton parton processes. In hadron-hadron collisions
process. we need to compute the virtual corrections illus-
trated in Fig. 5. These are the leading diagrams
at large Y and there are two others (where the
soft gluon links partons 1 and 3 or partons 2 and
4),whichcontributeonlytothedependenceupon
2. Pre 2001: Exponentiation
ρ . The only subtlety arises because the colour
jet
We start with the simplest example: e+e → structure is now not so simple. For quark-quark
−
qq¯ with a restriction p < Q on the emission scattering one can think of projecting the ampli-
T 0
of additional jets into a region of rapidity Y tude onto a specific basis and the soft gluons can
somewhere in between the primary jets. Naively, generate mixing. The resultis an amplitude that
we need only compute the diagram shown in goes as
Fig.4,wheretheblobrepresenttheshortdistance
physics. Thesoftgluonshouldbeintegratedover A=exp −2αs Q dktΓ A . (4)
the region into which real emissions are forbid- π k 0
ZQ0 t !
den. In particular, the real part of this one-loop
amplitude should be integrated over rapidities in In the t-channel singlet-octet basis (e.g. the top
the gap and k > Q since the complementary
T 0 2We shall see in the next section that this does not sum
1andthecross-sectionlevelforrealemissions. alloftheleadinglogarithms.
4 J.R Forshaw and M.H. Seymour
pi1 pm1−k p3=pk1−Q pi1 p1m−k kp3 asitnglalertgeexYchaanndgefowrilslterivnegnetnutalvlyetdoso,mwinhaetree.colour
k k
j n l j n l
p2 p2+k p4 = p2+Q p2 p2+Q−k p4 3. 2001-2006: Non-global logarithms
(a) (b) The exponentiation of soft gluons is not the
whole story. In 2001,Dasgupta & Salam [8,9] re-
pi1 pm1−Q+k kp3 pi1 p1n−Q−k kp3 alized that a whole tower of leading logarithms
k k was being neglected. Fig. 6 illustrates the prob-
lem. The toptwopanesillustratethe real-virtual
j n l j m
p2 p2+Q−k p4 p2 p2−k l p4
(c) (d)
Figure5. Thesoftgluondressingofafour-parton
process.
left entry generates pure singlet evolution):
N2 1ρ √N2 1iπ
Γ= 4N− jet 2N−
√N2N2−1iπ −N1iπ+ N2Y + N42N−1ρjet !
(5)
Figure 6. The origin of non-global logarithms.
and σ = A A. Now the Coulomb gluons don’t
†
only lead to a phase: witness the iπ terms in the
evolution matrix. The inclusive nature of the re-
gionk <Q doeshowevermeanthattheycancel cancellation of Bloch-Nordsieck and the bottom
t 0
in that region since the evolution is determined two illustrate the problem. The blue (darker)
by a purely imaginary matrix, i.e. ℑmΓ. Hence blob indicates a soft gluon that is emitted out-
the universal lower cut-off of Q . The calcula- side of the gap but with p > Q . In the left
0 T 0
tion of these Sudakov logarithms in all 2 → 2 picture, it receives a virtual correction from the
sub-processes can be found in [5,6]. red gluon, which necessarily has a lower p but
T
Althoughwehavebeenfocussingourattention is still above Q . Crucially,this redgluoncan be
0
on gaps-between-jets, it is worth recalling that in the gap region. The right picture represents
the formalismcarriesoveralmostwithoutchange therealemissionthatwouldcanceltheaforemen-
to the case when a colour singlet particle(s) is tionedvirtualcorrectioninasufficientlyinclusive
producedinbetweenthejets(e.g. aHiggsboson) observable. However, the real emission diagram
since the soft gluon evolution is blind to it. The is forbidden by the gap definition (since the red
only change occurs in the jet function because gluon is “in the gap and has p > Q ”), which
T 0
the final state jets now recoil against the Higgs means that the virtual correction on the left has
[7]. NoticethatanystudiesofajetvetoinHiggs- nothing to cancel against. Clearly the argument
plus-two-jetsthatareperformedusingthegeneral applies to any number of real emissions outside
purposeMonteCarlosnecessarilymisstheeffects ofthegapandinsteadofbeingabletofocus only
of Coulomb gluons and hence any singlet-octet on virtual corrections to the primary hard scat-
mixing. This is a particularly serious deficiency ter we now have to contemplate amplitudes with
Soft gluons and superleading logarithms in QCD 5
arbitrary numbers of real emissions and the vir-
tual corrections to them. That is clearly a tall
order and, to date, these leading “non-global”
logarithms have only been summed up to all or-
ders within the leading N approximation and
c
in the case of two-parton observables.3 Physi-
cal insight into this effect can be gained by not-
Figure 7. Five-parton evolution.
ing that the all-orders result is essentially that,
by requiring the gap region to be free of radia-
tion above Q , a region outside it must also be
0
depopulated, named the buffer zone in [9]. A matrixΓ,followedbytheemissionofarealgluon
very interesting feature worth mentioning is that (denoted by D in the figure) and this in turn is
the equation that sums these logarithms [11] is followed by a period of five-partonevolution(de-
identicalinformtotheBalitsky-Kovchegovequa- noted by Λ). Kyrieleis & Seymour computed D
tion [12,13] in small-x physics [14,15]. Is this andΛforfour-quarkamplitudes[17]andSjo¨dahl
coincidence or does it reflect a deeper connec- recentlycomputedtheremainingfivepartonevo-
tion between high energy scattering amplitudes lution matrices in a specific colour basis [18]. It
and jet physics? The physics behind non-global is also possible to write down the evolution ma-
logarithms is almost embarrasingly simple: the trix for a general n-parton amplitude in a basis-
observable is obviously sensitive to the fact that independent manner [1,2]. The result is (with a
emissionsoutsideofthegapcannotemitbackinto particular choice of the overallphase):
the gap.
1 1
Γ = YT2+iπT ·T + ρ T2
2 t 1 2 4 jet,i i
4. Post 2006: Super-leading logarithms? i F
X∈
1
Asafirststeptounderstandingnon-globallog- + λjet,ij Ti·Tj
2
arithmswedecidedtocomputethetoweroflead- (i<Xj)∈L
ingnon-globallogarithmsthatariseasaresultof 1
+ λ T ·T . (6)
onegluonsittingoutsidethegap[16]. Onemight 2 jet,ij i j
viewthisasthefirst4 terminanexpansioninthe (i<Xj)∈R
numberofout-of-gapgluonsandithasthevirtue T isthecolourchargeofpartoni(partons1and
i
that we can go ahead and calculate the cross-
2 are incoming and the others are outgoing) and
section without making the large Nc approxima- T is the net charge exchanged in the t-channel,
t
tion. To make progresswe need two new ingredi- i.e. T = T = − T and the sums
t i L i i R i
ents: (i)weneedtoknowhowtoemitareal(soft) are over all inc∈oming and ou∈tgoing partons lying
gluon off the four-parton amplitude; (ii) we need P P
on either the left or right side of the gap. Notice
to be able to dress the resulting five-parton am-
that the terms involving ρ are Abelian since
jet,i
plitudewithavirtualsoftgluon. Fig.7illustrates T2 =C 1 or C 1 depending upon whether par-
i F A
schematicallyhowthefive-partonmatrixelement
ton i is a quark/antiquark or gluon. λ is a
jet,ij
is computed. Assuming thatsuccessiveemissions
second jet function.
occuratprogressivelylowerkT,startingfromthe NowΓhasaveryimportantproperty: itissafe
hard scatter at scale Q (on the right in the fig-
against final state collinear singularities. More
ure), the general amplitude contains first a pe-
specifically, if any two or more partons in the fi-
riod of virtual evolution, which is carried out by
nal state become collinear with each other, the
successiveoperationsofthefour-partonevolution
soft gluon evolution of the system is identical to
the evolutionofthe systeminwhichthecollinear
3Estimateshavebeenmadefor2→2processes in[10].
4Orsecond,ifwecounttheglobal(exponentiating) series partons are replaced by a single parton with the
asthecontributionfrom“zerogluonsoutsideofthegap”. same total colour charge. That is, if k and l are
6 J.R Forshaw and M.H. Seymour
the collinear partons, then Γ depends only upon gluon;afterintegratingoverrapidityandazimuth
T +T and not upon the T or T separately. it is equal to ℜeΓ using Eq. (5).
k l k l
A proof can be found in [2] and results in the Rather than numerically evaluate σ , we shall
1
factorization expressed in the upper diagram of compute it in the particular case that the out-
Fig. 3. This is to be contrasted with the initial of-gap gluon is collinear to one of the incoming
statecollinearlimit,i.e. inwhichoneormoreout- partons. Naively, the integral over the rapidity
going partons becomes collinear with one of the of the out-of-gap gluon yields a divergence (it
incomingpartons. Theproofclearlybreaksdown canhavearbitrarilylargerapidity). Traditionally
for the imaginary part because of the Coulomb that would not pose any problem because colour
gluon termiπT ·T , whichdepends only on the coherence(andthe“plusprescription”)wouldsay
1 2
coloursofthe initialstatepartonsandnotonthe that Ω +Ω = 0 in this limit, however in light
R V
sumofthecolourchargesofthecollinearpartons. of our previous discussion we are interested to
As a result the factorization is broken, as illus- seehowthe“coherencebreaking”Coulombgluon
tratedinthelowerpartofFig.3. Itisthisbreak- contribution works out. We need to treat the
downofnaivecoherencethatleadsdirectlytothe collinear region with a little more care than we
appearanceofsuper-leadinglogarithmsinthecal- have done. In particular we must go beyond the
culation of the gaps-between-jets cross-section. soft gluon approximation in order to account for
For one gluon outside of the gap the cross- the fact that a gluon of k > Q cannot have
T 0
section is thus infinite rapidity and finite energy. We should in-
steadtreatthecollinearregionusingthecollinear
2α Q dk dydφ
σ =− s T Ω +Ω , (7) (but not soft) approximation. The integral over
1 R V
π k 2π
ZQ0 T Zout (cid:16) (cid:17) rapidity ought then to be replaced by
where the integralis overthe phase-spacefor the
out-of-gap gluon and dσ
dy →
dyd2k
2α Q dk Z T(cid:12)soft
ΩV = A†0exp − πs kT′ Γ† ouytmax (cid:12)(cid:12)
ZQ0 T′ ! dσ (cid:12) ∞ dσ
dy + dy .
exp −2παs kT dkkT′ Γ γ Z dyd2kT(cid:12)(cid:12)soft ymZax dyd2kT(cid:12)(cid:12)collinear
ZQ0 T′ ! (cid:12)(cid:12) (cid:12)(cid:12) (10)
2α Q dk
exp − s T′ Γ A +c.c., (8)
0
π k
ZkT T′ ! Inthisequationymax dividestheregionsinwhich
2α Q dk the soft and collinear approximations are used
ΩR = A†0exp − πs kT′ Γ† D†µ and the dependence on it will cancel in the sum.
ZkT T′ ! Now we know that
2α kT dk
exp − s T′ Λ†
π k
ZQ0 T′ ! ∞ dσ
dy =
exp −2αs kT dkT′ Λ Dµ ymZax dyd2kT(cid:12)(cid:12)collinear
π ZQ0 kT′ ! ∞ dσ(cid:12)(cid:12) dσ
2α Q dk dy R + V ,
exp − πs ZkT kT′T′ Γ!A0. (9) ymZax (cid:18)dyd2kT(cid:12)(cid:12)(cid:12)collinear dyd2kT(cid:12)(cid:12)(cid:12)collinear(cid:19)
(cid:12) (cid:12) (11)
ForΩ ,thishasthestructureadvertisedinFig.7.
R
Ω involvesvirtualcorrectionstothefour-parton
V
matrixelementandγ addsthevirtualout-of-gap wherethecontributionduetorealgluonemission
Soft gluons and superleading logarithms in QCD 7
is function by choosing the factorisation scale to
equal the jet scale Q.
∞ dσ
R The miscancellationtherefore induces an addi-
dy =
Z dyd2kT(cid:12)collinear tional contribution of the form
ymax (cid:12)
(cid:12)
1 δ (cid:12)
− 1 1+z2 q(x/z,µ2)
dz A
2 1−z q(x,µ2) R
Z (cid:18) (cid:19)
0
1−δ 1 1+z2 q(x/z,µ2) 1−δ 1 1+z2
= dz2 1−z q(x,µ2) −1 AR dz2 1−z (AR+AV) = (15)
Z0 (cid:18) (cid:19)(cid:18) (cid:19) Z0 (cid:18) (cid:19)
1 δ 1
− 11+z2 ln (A +A )+subleading
+ dz AR (12) δ R V
2 1−z (cid:18) (cid:19)
Z0 ∆y Q
≈ −y + +ln (A +A ).
max R V
and the contribution due to virtual gluon emis- (cid:18) 2 (cid:18)kT(cid:19)(cid:19)
sion is
∞ dσ
V
dy =
dyd2k
Z T(cid:12)collinear
ymax (cid:12)
(cid:12) The y dependence cancels with that coming
(cid:12)1 δ max
− 1 1+z2 from the soft contribution in Eq. (10) leaving
dz A . (13)
2 1−z V onlythe logarithm. Theleadingeffectoftreating
Z (cid:18) (cid:19)
0 properly the collinear region is therefore simply
In Eq. (12), q(x,µ2) is the parton distribution to introduce an effective upper limit to the inte-
function for a quark in a hadron at scale µ2 and gration over rapidity in Eq. (7), i.e.
momentum fraction x. The factors A and A
R V
containthe z independent factors which describe
the soft gluon evolution and the upper limit on
the z integralis fixed since we require y >y 5:
max
δ ≈ kT exp y − ∆y . (14) Q dkT dy →
Q (cid:18) max 2 (cid:19) ZQ0 kT Zout
We are prepared for the fact that A +A 6= 0 Q dkT ymax Q
R V dy+(−y +ln )
max
(and we shall see it explicitly in Eq. (22)) but ZQ0 kT Z kT !
if it were the case that A +A = 0 then the
R V 1 Q
virtual emission contribution would cancel iden- = ln2 . (16)
2 Q
0
tically with the corresponding term in the real
emission contribution leaving behind a term reg-
ularised by the ‘plus prescription’ (since we can
safely take δ → 0 in the first term of Eq. (12)).
Thistermcouldthenbe absorbedinto the evolu-
tion of the incoming quark parton distribution In this way the super-leading logarithmemerges.
We see that the non-collinear region is sub-
5Theapproximationarisessinceweassumeforsimplicity
leading and hence use the fact that, in the
that∆y islargeandδ issmall. Thisapproximation does
collinear limit, the evolution matrices simplify so
not affect the leading behaviour and can easily be made
exactifnecessary. that (neglecting an overall Abelian factor) the
8 J.R Forshaw and M.H. Seymour
cross-section reduces to [2] vanish identically:
2αs Q dkT Q ( ) =t21−ta1†ta1 =0. (18)
σ1 = − 2ln 0
π k k
ZQ0 T (cid:18) T(cid:19)
′
*mt210e(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−e−2απ2sαπsRQkRT0kQTdkkdT′kT′kT′T((1212YYt2tt−2t−iπiπt1t·1t·2t)2) ( )1 =−2παs ZQk0T dkkT′T′ (t21Yt2t −ta1†YT2t t(a11)9)
(
is also zero because
′
e−2απs RQkT0 dkkT′T(12Yt2t+iπt1·t2) T2ta =tat2. (20)
−ta1†e−2απs RQkT0 dkkT′T′ (12YT2t−iπT1·T2) Extpa1ndin1g tto order α2 yields
′ s
e−2απs RQkT0 dkkT′T(12YT2t+iπT1·T2)ta1 iπY 2α kT dk 2
) = − s T′
2 π k
e−2απs RkQT dkkT′T′ (12Yt2t+iπt1·t2) m0 . (17) ( )2 (cid:18) (cid:19) ZQ0 T′ !
(cid:12)(cid:12) + t21 t2t ,t1·t2 −ta1† T2t ,T1·T2 ta1 .
(cid:12) ( )
(cid:12)
(cid:12) (cid:2) (cid:3) (cid:2) (cid:3) (21)
We have changed to a bra-ket notation where
|m i denotes the lowest order matrix element: Note that this result comes only from the case
0
it better suits the basis independent language wherethereisoneCoulombgluonandoneeikonal
wherein we are to think of the colour charges gluon either side of the cut. Now, this term is
as objects that map vectors in m-parton colour not zero but the correspondingmatrix element is
space to vectors in (m+1)-parton colour space. zero, i.e. hm0|{ }2|m0i=0.
Consequently, we have used lower (t ) and up- At the next order we obtain a non-zero result:
i
per (Ti) case symbols to represent the charge of Yπ2 2α kT dk 3
parton i in the four and five parton systems re- ≡ − − s T′
spectively. Notethatitisthenon-commutativity ( )3 6 π ZQ0 kT′ !
of T2 and T ·T (and similarly t2 and t ·t )
t 1 2 t 1 2
that prevents this expression from cancelling to t2 t2,t ·t ,t ·t
1 t 1 2 1 2
zero: if they commuted then the two exponen- (
h(cid:2) (cid:3) i
tials could be combined, all T ·T and t ·t
1 2 1 2
dependence would cancel, t1 could be commuted −ta1† T2t ,T1·T2 ,T1·T2 ta1 . (22)
through T2t and the real and virtual parts would h(cid:2) (cid:3) i )
be identical. For fewer than four external par- Substituting back into Eq. (17), evaluating the
tons colour conservation means that we can al- colour matrix element and performing the trans-
ways write T1 ·T2 ∝ 1 and hence the Coulomb verse momentum integrals, we obtain a contri-
gluons only generate an unimportant phase. bution to the first super-leading logarithm from
We can get a better idea of what is going on if configurations in which the out-of-gap gluon is
we expand the {...} in Eq. (17) order by order in hardest of
α . By setting the exponentials to unity outside
s 2α 4 Q N2−2
this bracket we are assuming that the out-of-gap σ =−σ s ln5 π2Y
1,hardest 0
gluon has the largest kT (we shall look at the (cid:18) π (cid:19) (cid:18)Q0(cid:19) 240
other possibilities shortly). The first two orders (23)
Soft gluons and superleading logarithms in QCD 9
in the case of qq → qq. The other subprocesses N andappearsfirstatfourthorderinthestrong
c
leadtodifferentcolourmatrixelementsandhence coupling relative to the lowest order. The im-
a different coefficient of the form (aN2+b). The plications for the gaps-between-jets cross-section
explicit results are presented in [2]. are clear: collinear logarithms can be summed
To complete the calculation of the order α4 intothe partondensityfunctionsonlyuptoscale
s
super-leading logarithm we need to compute the Q and the logarithms in Q/Q from further
0 0
contribution arising from the case where there is collinear evolution must be handled seperately.
just one virtual emission of higher k than the Moreover, since we now have a source of double
T
out-of-gap emission. Now we use the expression logarithms, the calculation of the single logarith-
for { } derived in Eq. (21) in conjunction with mic series necessarily requires knowledge of the
2
the orderα expansionofthe exponentialfactors two-loop evolution matrices [19,20].
s
that lie outside of the main bracket in Eq. (17). Some questions still remain open however:
Note that this is the only remaining contribution the structure of higher order super-leading log-
to the lowestorder super-leading logarithmsince arithms; how widespread they are in other ob-
all lower order expansions of the main bracket in servables for hadron collisions; and whether they
Eq.(17)(i.e. { } and{ } ) vanish. The resultis canbe reorganizedand resummedor removedby
1 0
a suitable redefinitionofobservablesor ofincom-
σ =
1,second hardest ing partonic states.
−
2α 4 Q N2−1 We close by noting that further simplifications
−σ s ln5 π2Y .
0 π Q 120 are possible. In particular, the commutators be-
(cid:18) (cid:19) (cid:18) 0(cid:19)
tween gluon exchanges from an external leg in
(24)
differentorderscanbewrittenasemissionoffthe
At order α4 relative to the lowest order result, exchanged Coulomb gluons. The colour matrix
s
the total super-leading contribution to the gaps- elements can then be shown to be identical to
between-jets cross-section for qq → qq arising those illustratedinFig.10. Once againitis clear
fromonesoftgluonemissionoutsideofthegapis that the super-leading logarithms arise as a re-
therefore sult of the non-Abelian nature of the Coulomb
gluoninteraction. Wealsoseethatthecoefficient
2α 4 Q 3N2−4
σ =−σ s ln5 π2Y . of the superleading logarithm is independent of
1 0
π Q 240
(cid:18) (cid:19) (cid:18) 0(cid:19) whether the out-of-gap gluon is collinear to par-
(25) ton 1 or 2, because the result is invariant under
interchangeoftheparticletypesintheupperand
Ofthe other sub-processes,the resultfor qg →
lowerloops. Colourstructureslikethesearerem-
qg is worth singling out because the size of the
iniscent of small-x physics.
super-leading log does not depend upon whether
the out-of-gap gluon is collinear to the quark or
to the gluon. Acknowledgements
We thank Stefano Catani, Mrinal Dasgupta,
5. Outlook
James Keates, Simone Marzani, Malin Sjo¨dahl
The super-leading logarithms constitute a sur- andGeorgeStermanforinterestingdiscussionsof
prisingbreakdownofcollinearfactorizationin an theseandrelatedtopics. Thanksalsotothework-
observablethatsumsinclusivelyoverthecollinear shop organizers, both for inviting JRF to deliver
regions: the ‘plus prescription’ fails to oper- this talk and for their very generous hospitality.
ate and double (soft-collinear) logarithms make
their appearance. The effect is due to Coulomb
REFERENCES
(Glauber) gluon exchange and it arises only in
observableswithatleastfourcolouredpartonsin 1. R. Bonciani, S. Catani, M. L. Mangano and
the hard scatter, is non-Abelian, sub-leading in P. Nason, Phys. Lett. B 575 (2003) 268
10 J.R Forshaw and M.H. Seymour
[arXiv:hep-ph/0307035].
2. J. R. Forshaw, A. Kyrieleis and
M. H. Seymour, JHEP 0809 (2008) 128
[arXiv:0808.1269[hep-ph]].
3. C.Adloffetal.[H1Collaboration],Eur.Phys.
J. C 24 (2002) 517 [arXiv:hep-ex/0203011].
4. S. Chekanov et al. [ZEUS Collaboration],
arXiv:hep-ex/0612008.
5. G. Oderda and G. Sterman, Phys.Rev. Lett.
81 (1998) 3591 [arXiv:hep-ph/9806530]. (a) (b)
6. C. F. Berger, T. Kucs and G. Ster-
man, Phys. Rev. D 65 (2002) 094031
[arXiv:hep-ph/0110004].
7. J. R. Forshaw and M. Sjo¨dahl, JHEP 0709
(2007) 119 [arXiv:0705.1504[hep-ph]].
8. M. Dasgupta and G. P. Salam, Phys. Lett. B
512 (2001) 323 [arXiv:hep-ph/0104277].
(c) (d)
9. M. Dasgupta and G. P. Salam, JHEP 0203
(2002) 017 [arXiv:hep-ph/0203009].
10. R. B. Appleby and M. H. Seymour, JHEP
0309 (2003) 056 [arXiv:hep-ph/0308086].
11. A. Banfi, G. Marchesini and G. Smye, JHEP
0208 (2002) 006 [arXiv:hep-ph/0206076].
12. I. Balitsky, Nucl. Phys. B 463 (1996) 99
[arXiv:hep-ph/9509348]. (e) (f)
13. Y. V. Kovchegov, Phys. Rev. D 60, 034008
(1999) [arXiv:hep-ph/9901281].
14. G.MarchesiniandA.H.Mueller,Phys.Lett.
B 575 (2003) 37 [arXiv:hep-ph/0308284].
15. H. Weigert, Nucl. Phys. B 685 (2004) 321
[arXiv:hep-ph/0312050].
16. J. R. Forshaw, A. Kyrieleis and
(g) (h)
M. H. Seymour, JHEP 0608 (2006) 059
[arXiv:hep-ph/0604094].
17. A.KyrieleisandM.H.Seymour,JHEP0601
(2006) 085 [arXiv:hep-ph/0510089]. Figure 8. The relevant Feynman diagrams in the
18. M. Sjo¨dahl, JHEP 0812, 083 (2008) case that the out-of-gap (dotted) gluon is the
[arXiv:0807.0555[hep-ph]]. hardest gluon. The dashed lines indicate soft
19. S. Mert Aybat, L. J. Dixon and G. Ster- (eikonal and Coulomb) gluons. Each subfigure
man, Phys. Rev. Lett. 97, 072001 (2006) represents three Feynman diagrams, correspond-
[arXiv:hep-ph/0606254]. ing to the three different ways of attaching the
20. S. Mert Aybat, L. J. Dixon and G. Ster- out-of-gapgluon. Indiagrams(e)and(f)thesoft
man, Phys. Rev. D 74 (2006) 074004 gluon to the right of the cut should only be inte-
[arXiv:hep-ph/0607309]. grated over the region in which it has transverse
momentum less than the out-of-gap gluon.