pQCD Physics of multiparton interactions

0.51 MB·English

by  B. Blok

pQCD physics of multiparton interactions
B. Blok,1 Yu. Dokshitzer,2 L. Frankfurt,3 and M. Strikman4
1Department of Physics, Technion—Israel Institute of Technology, 32000 Haifa, Israel∗
2Laboratory of High Energy Theoretical Physics (LPTHE), University Paris 6, Paris, France †‡
3School of Physics and Astronomy, Raymond and Beverly Sackler
Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel§
4Physics Department, Penn State University, University Park, PA, USA¶
Westudyproductionoftwopairsofjetsinhadron–hadroncollisionsinviewofextractingcontri-
bution of double hard interactions of three and four partons (3→4, 4→4). Such interactions, in
spite of being power suppressed at the level of the total cross section, become comparable with the
standard hard collisions of two partons, 2→4, in the back-to-back kinematics when the transverse
momentum imbalances of two pairing jets are relatively small.
Weexpressdifferentialandtotalcrosssectionsfortwo-dijetproductionindoublepartoncollisions
throughthegeneralizedtwo-partondistributions, GPDs[1],thatcontainlarge-distancetwo-parton
2 2
correlationsofnon-perturbativeoriginaswellassmall-distancecorrelationsduetopartonevolution.
1
We find that these large- and small-distance correlations participate in different manner in 4-jet
0
2 production,andtreatthemintheleadinglogarithmicapproximationofpQCDthatresumscollinear
logarithms in all orders.
n
Aspecialemphasisisgivento3→4doublehardinteractionprocessesthatoccurasaninterplay
a
between large- and short-distance parton correlations and were not taken into consideration by
J
approachesinspiredbythepartonmodelpicture. Wedemonstratethatthe3→4mechanism,being
0 of the same order in αs as the 4→4 process, turns out to be geometrically enhanced compared to
3
the latter and should contribute significantly to 4-jet production.
The framework developed here takes into systematic consideration perturbative Q2 evolution of
]
h 2GPDs. ItcanbeusedasabasisforfutureanalysisofNLOcorrectionstomulti-partoninteractions
p (MPI) at LHC and Tevatron colliders, in particular for improving evaluation of QCD backgrounds
- to new physics searches.
p
e
PACSnumbers:
h
Keywords:
[

2
v I. INTRODUCTION Apossibilityofadoublehardcollisionbecomesmoreim-
3 portantwithincreaseoftheenergyofthecollisionwhere
3 scattering off small x partons which have much higher
Understanding the rates and the structure of multi-
5 densities becomes possible.
jet production in hadron–hadron collisions is of primary
5
importance for new physics searches. In recent years multiparton collisions have attracted
.
6 closeattention. FollowingthepioneeringworkofRefs.[2,
Productionofhightransversemomentumjetsisahard
0 3],alargenumberofrelatedtheoreticalpapersappeared
process which implies a head-on collision of QCD par-
1
[4–8],basedonthepartonmodelandgeometricalpicture
tons — quarks and/or gluons — from the small-distance
1
in the impact parameter space. More recently, this topic
: wave functions of initial hadrons. Cross section of a
v has been intensively discussed in view of the LHC pro-
hard collision is small compared with the size of hadron,
Xi σ ∝ Q−2 (cid:28) R2 , with Q2 the scale related to trans- gram[9,10]. MonteCarloeventgeneratorsthatproduce
ar vtyeprsiceamllyomitenistathwoafodtrhpearptroondsutchedatjeetxsp,eQr2ie∼ncej⊥2a. hTahredrecfoolrlei-, mthuelotrieptliecaplaprtaopnercsoelxlipsiloonrisngarperobpeeinrtgiedseovfeldoopuebdle[1p1a–r1to3n];
distributions and discussing their QCD evolution have
sion in a given event. A large angle scattering of these
appeared [1, 14, 15].
two partons produces two (or more) final state partons
that manifest themselves as hadron large transverse mo- In our view, however, important elements of QCD
mentum jets. At the same time, one cannot exclude a that are necessary for theoretical understanding of the
possibility that more than one pair of partons happen to multiple hard interactions issue have not yet been prop-
collide in a given event, giving rise to a multi-jet event. erlytakenintoaccountbyabove-mentionedintuitiveap-
proaches.
The problem is, sort of, educational: both the proba-
bilistic picture, the MC generator technology is based
†On leave of absence: St. Petersburg Nuclear Physics Institute, upon, and the familiar Feynman diagram technique,
Gatchina,Russia when used in the momentum space, prove to be in-
∗Electronicaddress: [email protected]
adequate for careful analysis and understanding of the
‡Electronicaddress: [email protected]
physics of multiple collisions.
§Electronicaddress: [email protected]
¶Electronicaddress: [email protected] From experience gained by treating standard (single)
2
hard processes, one became used to a motto that a large (πR2 ) is often referred to in the literature as “an ef-
int
momentum transfer scale Q2 ensures the dominance of fective cross section” σ . We prefer, however, not to
eff
small distances, r2 ∼Q−2, in a process under considera- look at this quantity as a cross section, since it reflects
tion. With the multiple collisions under focus, however, transversal area of parton overlap as well as longitudinal
onehastodistinguishtwospace-timescales: thatoflocal- correlationsofthepartons. Atthesametime,ithaslittle
ization of the parton participating in a hard interaction, todowiththemeasureofthestrengthoftheinteraction,
∆r2 ∼ Q−2, and that of transverse separation, ∆ρ, be- which is what “cross section” represents.
tween the two hard collision vertices. The latter can be In a two-parton collision, scattered partons form two
large, of the order of the hadron size, even for large Q2. nearly back-to-back jets, while additional jets (should
In order to be able to trace the relative distance be- there be any) tend to be softer and to align with the di-
tweenthepartons, onehastousethemixedlongitudinal rections of initial and final partons, because of collinear
momentum–impact parameter representation which, in enhancements due to radiative nature of secondary par-
the momentum language, reduces to introduction of a tons. Suchwillbetypicalcharacteristicsofa4-jet event,
mismatch between the transverse momentum of the par- in particular. On the other hand, four jets produced as
ton in the amplitude and that of the same parton in the a result of a double hard collision of two parton pairs
amplitude conjugated. would, on the contrary, form two pairs of nearly back-to-
back jets. Thiskinematicalpreferenceisinstarkcontrast
Another unusual feature of the multiple collision anal-
with “hedgehog-like” configurations of four jets stem-
ysis that may look confusing at the first sight is the fact
ming from a single collision and can be used in order
that—evenatthetreelevel—the amplitude describing
to single out double hard collisions experimentally.
the double hard interaction process contains additional
Suchexperimentalstudieswererecentlycarriedoutby
integrations over longitudinal momentum components;
theCDFandD0collaborationswhohavestudiedproduc-
more precisely — over the difference of the (large) light-
tionofthreejets+photon[9,16–18]. Theanalysisofthe
conemomentumcomponentsofthetwopartonsoriginat-
dataperformedin[6,7]usinginformationaboutgeneral-
ing from the same incident hadron (see Section IVA).
izedpartondistribution(GPDs)obtainedfromthestudy
of hard exclusive processes at HERA has found that ob-
Inthepreviousshortpublication[1]wehaveconsidered served 3 jet+γ rates were a factor ≥2 higher than the
productionoftwopairsofnearly back-to-back jetsresult- expected rates based on a naive model that neglected
ingfromsimultaneoushardcollisionsoftwopartonsfrom correlations between partons in the transverse plane.
the wave function of one incident hadron with two par- The use of GPD allows one to incorporate such cor-
2
tons from the other hadron (“four-to-four” processes). relations and predict their Q2 evolution.
As we have shown, this necessitates introduction of a
new object — a generalized double parton distribution,
Onthetheoryside,theback-to-backenhancementhas
GPD,thatdependsonanewtransversemomentumpa-
2 been discussed, at tree level, in a number of studies
rameter∆(cid:126) conjugatetotherelativedistancebetweenthe
of various channels (see, for example, discussion of the
two partons in the hadron wave function. Generalized 2 jets+b¯b in [19] and references therein).
double parton distributions provide a natural framework
In the present paper we study perturbative radiative
forincorporatinglongitudinalandtransversecorrelations
effectsinthedifferential 4-jetdistributionintheback-to-
between partons in the hadron wave function at the Q2
0 back kinematics and derive the expression for the corre-
scale, and for tracing the perturbative Q2-evolution of
spondingcrosssectionintheleadinglogarithmiccollinear
the correlations.
approximation. It takes into account QCD evolution of
Thecorresponding4-jetcrosssectioncanbeexpressed
the generalized double parton distributions as well as ef-
in terms of GPD’s as follows
2 fects due to multiple soft gluon radiation, and turns out
tobeadirectgeneralizationoftheknown“DDTformula”
dσ(x ,x ,x ,x ) dσ13 dσ24
1 2 3 4 = for back-to-back production of two large transverse mo-
dtˆ dtˆ dtˆ dtˆ
1 2 1 2 (1) mentum particles in hadron collisions [20].
(cid:90) d2∆(cid:126) We also discuss and treat new specific correlations be-
× D (x ,x ;∆(cid:126))D (x ,x ;−∆(cid:126)).
(2π)2 a 1 2 b 3 4 tween transverse momenta of jets due to 3-parton in-
teractions producing 4 jets,“three-to-four”. Such pro-
The factor on the second line has dimension of inverse cesses are induced by perturbative splitting of a parton
area: from one of the hadrons, the offspring of which enter
double hard collision with two partons from the wave
→− →− →−
1 (cid:90) d2∆ D(x ,x ,∆)D(x ,x ,−∆) function of the second hadron. The hard scale of this
= 1 2 3 4 , (2)
πR2 (2π)2 D(x )D(x )D(x )D(x ) parton splitting is determined by transverse momentum
int 1 2 3 4
imbalances of pairs of jets, δ , δ , and exhibits specific
13 24
where D(x ) are the corresponding one-parton distribu- collinear enhancement in the kinematical region where
i
tions. The ratio of the product of two single-inclusive two jet imbalances practically compensate one another,
cross sections and the double-inclusive cross section δ(cid:48)2 =((cid:126)δ +(cid:126)δ )2 (cid:28)δ2 (cid:39)δ2 .
13 24 13 24
3
Consistently taking into account three-to-four parton introduce transverse momentum imbalance: (cid:126)j +(cid:126)j =
1⊥ 3⊥
processsolvesalongstandingproblemofdoublecounting (cid:126)δ which constitutes another hard scale: Q2 (cid:29) δ2 (cid:29)
13 1 13
in treating multi-parton interactions. Λ2 . For production of four jets in the back-to-back
QCD
kinematics, this gives four different hard scales. As we
Discussion of the 2-parton distribution has a long his-
shallseebelow,inthe3-partonscollisionsproducingfour
tory. It is commonly defined in the momentum space as
jets yet another scale enters the game: δ(cid:48)2 (cid:29)Λ2 with
a 2-particle inclusive quantity depending on two parton QCD
momenta, see [21, 22]. Being related to (the imaginary δ(cid:126)(cid:48) = (cid:126)δ13+(cid:126)δ24 — the total transverse momentum of the
partof)acertainforward scatteringamplitude, itthere- 4-jet ensemble.
fore disregards impact parameter space geometry of the In what follows we consider transverse momenta of all
interaction. Exploring properties of 2-parton distribu- four jets to be of the same order, Q2 = j2 (cid:39) j2 ∼
1⊥ 3⊥
tionssodefined,anapproachtothestudyofthemultiple j2 (cid:39) j2 . This is not necessary but helps to avoid
2⊥ 4⊥
jet production has been recently suggested in Ref. [23]. complications in the hierarchy of relevant scales.
The reason why this approach has faced difficulties, [24], Finally, let us mention that in what follows it will be
and did not solve, in our view, the problem of system- tacitlyimpliedthatfixingthesescales—fromthelargest
atic pQCD analysis of 4-jet production is clear: it did one,Q2,downtosmallerones,δandδ(cid:48),—isnotcompro-
notincorporateeffectsduetovariationsofthetransverse misedbyuncertaintiesindeterminationofthetransverse
separation between the partons — information encoded momenta of the jets.
by GPD’s but not by the 2-parton momentum distribu-
2
tions.
The GPD’swererecentlyusedinRef.[25]forintuitive B. Back-to-back kinematics
2
description of the total 4-jet production cross section.
However,thedifferentialdistributionswerenotdiscussed
Thebasic2-jetproductioncrosssectionscales,asymp-
in that paper, and not all relevant pQCD contributions
totically, as
were included, so that our results are different from the
ones obtained in [25]. dσ(2→2) α2
∝ s. (3)
dtˆ Q4
The paper is organized as follows.
In Section II we recall the main ingredients of the Accordingto(1), productionoffourjetsinsimultaneous
perturbative analysis based on selection of maximally hard collisions of four partons yields
collinearenhancedcontributionsinallorders. InSection
IIIwepresenttheevolutionequationforgeneralizedtwo- dσ(4→4) (cid:18)α2(cid:19)2 α4
parton distributions. Section IV is devoted to the per- ∝ R−2· s ∝ s , (4a)
dtˆdtˆ Q4 R2Q8
turbative analysis of small-distance correlations between 1 2
partons. The main result of the paper — the differential with R2 = 1/(cid:10)∆2(cid:11) the characteristic distance between
distributionof4-jetproductionintheback-to-backkine-
the two partons in the hadron wave function. At large
matics — is formulated in Section V, and the total cross
Q2 this cross section is parametrically smaller than that
sectionoftwo-dijetproductionisdescribedinSectionVI.
forproductionoffourwellseparatedjetswithtransverse
Conclusions and outlook are presented in Section VII.
momenta j2 ∼Q2 in a 2-parton collision:
i⊥
dσ(2→4) α4
II. PERTURBATIVE ANALYSIS ∝ s (4b)
dtˆdtˆ Q6
1 2
A. Hard scales (with transverse momenta of two out of four jets being
integrated over).
The perturbative approach implies that all hard- Qualitatively, the production mechanism (4a) can be
ness (transverse momentum) scales that characterize the labelled a “higher twist effect”. Nevertheless, it may
problem are comfortably larger than the intrinsic QCD turn out to be essential — comparable with the “lead-
scale ΛQCD: Q2i (cid:29)Λ2QCD. The process under considera- ing twist” 2 → 4, Eq. (4b) — if one looks at specific
tion may have up to five hard scales involved. kinematics of the 4-jet ensemble.
Indeed,intheleadingorderinαs,largetransversemo- Let z be the direction of colliding hadron momenta.
mentum partons are produced in pairs and have nearly Imagine that we are triggering on two jets moving along
opposite transverse momenta, setting the hard scale the x and y axes in the transverse plane, and look for
Q2 = j2 (cid:39) j2 . Within the parton model framework two accompanying jets inside some solid angles ∆Ω (cid:28)
1 1⊥ 3⊥
(neglectingfinitesmearingduetointrinsictransversemo- 4π around the −x and −y directions. The production
menta of incident partons), one has dσ ∝ δ((cid:126)j +(cid:126)j ). mechanism(4b)doesnotpopulatethisregion: thehigher
1⊥ 3⊥
Secondary QCD processes — evolution of initial parton order 2→4 QCD matrix element is enhanced when two
distributions and accompanying soft gluon radiation — finalstatepartonsbecomequasi-collinear butisperfectly
4
smooth in the back-to-back kinematics. Therefore, its D. Double Logarithmic parton form factors
contribution will be suppressed,
In the leading order in α , it suffices to have just one
(cid:18)∆Ω(cid:19)2 vs. 1 , partonpresentwith(cid:126)k⊥ =−(cid:126)sδ13inordertoassureδ13 (cid:54)=0.
4π R2Q2 At the same time, inclusive production of accompanying
partonswithtransversemomentak turnsouttobesup-
⊥
contrary to the 4→4 production mechanism (4a) which
pressed in a broad interval δ2 (cid:28) k2 (cid:28) Q2, as long as
is concentrated in this very kinematical region. 13 ⊥
one wants to preserve the collinear enhancement factor
δ2 in the jet correlation (5a).
13
This dynamical “veto” has two consequences.
C. Collinear approximation
Firstofall,itresultsinreduction ofthehardnessscale
of the parton distributions from the natural scale Q2
Thedifferential4-jetproductioncrosssectionpossesses
(scale of the parton distributions in the integrated cross
two collinear enhancements. Depending on the kinemat- section)downtotheobservation-inducedscaleδ2 (cid:28)Q2.
13
ics of the jets, they are, symbolically,
Then, it introduces double logarithmic (DL) form fac-
torsofparticipatinginitialstatepartons,sincethetrans-
α2
dσ(4→4) ∝ s d2j d2j ·dΣ, verse momentum of the jet pair can be compensated not
δ2 δ2 3⊥ 4⊥
13 24 only by a hard (energetic) parton from inside initial par-
δ2 (cid:28)Q2, δ2 (cid:28)Q2; (5a) tondistributionsbutalsobyasoftgluon whoseradiation
13 24
α2 did not affect inclusive parton distributions due to real–
dσ(3→4) ∝ δ(cid:48)2sδ2 d2j3⊥d2j4⊥·dΣ, virtual cancellation.
δ(cid:48)2 (cid:28)δ2 (cid:28)Q2, δ2 =δ2 (cid:39)δ2 . (5b) The presence of the DL form factors depending on the
13 24 logarithm of a large ratio of scales, ln(Q2/δ2), is typical
ij
Here dΣ = dΣ(tˆ,tˆ) is the cross section integrated for the so-called “semi-inclusive” processes [20, 26].
1 2
over the transverse momenta of the “backward” jets 3
Productionofmassiveleptonpairsinhadroncollisions
and 4. The integrated cross section dΣ contains the
(the Drell–Yan process) is a classical example of a two-
squared matrix element of the four-parton production
scaleproblem. Hereenterformfactorsofcollidingquarks
and is of the order of α4, cf. Eq. (4a). At the Born
s that depend on the ratio of the invariant mass q2 to the
level, the jets in pairs are exactly back-to-back, so that
transverse momentum of the lepton pair, α ln2(q2/q2),
dσ(4→4) ∝ δ2((cid:126)δ )δ2((cid:126)δ ) in (5a). To have a non-zero s ⊥
13 24 in the dominant kinematical region q2 (cid:28)q2:
value of the transverse momentum imbalance, one has ⊥
to have additional large transverse momentum parton(s) dσ dσ
produced. = tot
dq2dq2 dq2
In the second important contribution to the cross sec- ⊥
(cid:26) (cid:27)
tion, Eq. (5b), one power of the coupling emerges from × ∂ Dq(cid:0)x ,q2(cid:1)Dq¯(cid:0)x ,q2(cid:1)S2(cid:0)q2,q2(cid:1) .(6)
thesplittingofapartonfromoneoftheincidenthadrons ∂q2 a 1 ⊥ b 2 ⊥ q ⊥
⊥
into two, and the second power is due to production of
an additional final state parton with(cid:126)k =−(cid:126)δ(cid:48). Sq is the double logarithmic QCD quark form factor.
⊥
In both cases the smallness due to additional powers Sudakovquarkandgluonformfactorscanbeexpressed
of the coupling is compensated by two broad (logarith- via the exponent of the total probability of the parton
mic) integrations over transverse momentum imbalances decay in the range of virtualities (transverse momenta)
as indicated in (5). between the two hard scales:
(cid:40) (cid:90) Q2 dk2α (k2)(cid:90) 1−k/Q (cid:41)
S (Q2,κ2) = exp − s dzPq(z) , (7a)
q k2 2π q
κ2 0
S (Q2,κ2) = exp(cid:40)−(cid:90) Q2 dk2αs(k2)(cid:90) 1−k/Qdz(cid:2)zPg(z)+n Pq(z)(cid:3)(cid:41). (7b)
g k2 2π g f g
κ2 0
Here Pk(z) are the non-regularized one-loop DGLAP splitting functions (without the “+” prescription):
i
5
1+z2
Pq(z)=C , Pg(z)=Pq(1−z),
q F 1−z q q
(8)
Pq(z)=T (cid:2)z2+(1−z)2(cid:3), Pg(z)=C 1+z4+(1−z)4;
g R g A z(1−z)
theupperlimitofz-integralsproperlyregularizesthesoft of the parton system, see [27] and references therein.
gluon singularity, z → 1 (in physical terms, it can be In the present paper we concentrate on resummation
looked upon as a condition that the energy of a gluon of collinear enhanced DL and SL terms and avoid com-
should be larger than its transverse momentum, [20]). plications due to soft SL corrections. This means ig-
noring color transfer effects in hard interactions. Thus,
Thecaseofhadroninteractionsproducinglargetrans- production of four jets with large transverse momenta
versemomentumpartons (insteadofcolorlessobjectslike j⊥ ∼Qandpairimbalancesδij willbeequivalent,inour
a Drell–Yan pair or an intermediate boson) is more in- treatment,toproductionoftwocolorlessDrell–Yanpairs
volved since here the transverse momentum imbalance withinvariantmassesO(Q)andtransversemomentaδ13
may be compensated by QCD radiation from the final and δ24. Generalization of the results of [27] to the case
state partons too. of double parton scattering seems straightforward and
The azimuthal correlation between two nearly back- should be considered separately.
to-backlargetransversemomentumparticles wasconsid-
ered in [20]. An analog of the “DDT formula” has been
derived in the collinear approximation, which expression III. GENERALIZED DOUBLE PARTON
DISTRIBUTION
contained the product of four form factors, two initial
parton distributions and two fragmentation functions.
A. Geometry of GPD
2
E. Single Logarithmic soft gluon effects The 2GPD in the expression for the multiparton pro-
duction cross section has a meaning of a two body form
factor when partons 1 and 2 receive transverse momenta
Thecasewhenjets arebeingreconstructedinthefinal
∆ and −∆ leaving the hadron intact. Nonrelativistic
state is more complicated to analyze as it yields an an-
analogue of this form factor is familiar from the double
swer depending on the jet finding algorithm. The prob-
scatteringamplitudeinthemomentumspacerepresenta-
lem has been addressed by Banfi and Dasgupta in [27]
tionoftheGlaubermodel,seee.g.[28]. Recallthat[1]the
where a smart way of defining the final state jets was
scale ∆ in GPD is conjugate to the relative transverse
formulated that permitted to write down a resummed 2
distance between the two partons in the GPD in the
QCD formula for soft gluon effects in “2 partons → 2 2
impact parameter representation considered in [2, 3, 14].
jets” cross sections.
Collinear logarithms due to hard splittings of the fi-
nal state partons do not pose a problem: such secondary Two partons may originate from soft low-scale fluc-
partons populate the jets. Partons that appear as sep- tuations inside the hadron; they can also emerge from
arate out-of-jet radiation — and are relevant for trans- a perturbative splitting of a common parent parton at
verse momentum imbalance compensation — have to be relatively large momentum scales. It is clear that these
produced at sufficiently large angles with respect to the two contributions to GPD will have essentially different
2
jet axis. This is the domain of large-angle gluon radi- dependence on the parameter ∆.
ation. Production of soft gluons in-between jets is also The first contribution we will denote
logarithmically enhanced and induces single logarithmic
(SL) corrections, (cid:2)αsln(Q2/δ2)(cid:3)n, that may also be sig- [2]Dabc(x1,x2;q12,q22;∆(cid:126)), (9)
nificant and should be resummed in all orders.
Contrary to collinear enhanced effects (that drive evo- with the subscript stressing that here the partons b
[2]
lution of parton distributions and fragmentation func- and c emerge from the no-perturbative wave function of
tionsanddeterminetheSudakovformfactors),thelarge- the hadron a. It should decrease rapidly at scales larger
angle gluon radiation cannot be attributed to one or an- than a natural scale of short-range parton correlation in
other of the partons participating in the hard scattering. a hadron (this scale may be slightly different for quarks
It is coherent and depends on the kinematics and color and gluons and could in principle be significantly larger
topology of the hard parton ensemble as a whole. As a than the 1/r as there exists another non-perturbative
N
result, resummation of these SL corrections becomes a scale of the chiral symmetry breaking which maybe as
matrixproblemthatinvolvestracingvariouscolorstates large as 700 MeV).
6
GPD should rapidly decrease for ∆2 ≥ 1.5/(cid:10)r2 (cid:11) B. On the geometrical enhancement of the
2 tg
where(cid:10)r2 (cid:11)1/2 isthetransversegluonicradiusofthenu- interference effects due to pQCD correlations
tg
cleon. In the mean field approximation when the corre-
lations are neglected, it can be approximated by a fac-
torized expression [1]
By total cross section in the present context we mean
the back-to-back 4-jet cross section integrated over pair
Db,c(x ,x ;q2,q2;∆(cid:126)) = F (∆2;x ,q2)F (∆2;x ,q2)
[2] a 1 2 1 2 g 1 1 g 2 2 jet imbalances in the dominant logarithmic region δ (cid:28)
ik
× Gb(x ,q2)Gc(x ,q2), (10a) j (cid:39)j .
a 1 1 a 2 2 i⊥ k⊥
where G are the single-parton distributions and the two-
gluon form factor F can be parametrized as We start by noting that the product of two small-
distance parton fluctuations, D × D, does not con-
[1] [1]
(cid:18) ∆2 (cid:19)−2 tribute to the process we are interested in. Indeed, in
Fg(∆2) = 1+ m2(x) . (10b) this case the integral over ∆ in Eq. (1) formally diverges
g andyieldsahardscale(insteadofR−2)inthenumerator.
Thismeansasignificantcontributiontothecrosssection
Theparameterm2 isoftheorderof1GeV2 forx∼10−2
g butnottheonewearelookingfor. BelowinSectionIVB
andgraduallydropswithdecreaseofxandwithincrease
wewillexplicitlyverifythatadoublehardcollisionoftwo
of virtuality [29].
parton pairs each of which originates from perturbative
The second contribution we will denote splitting, lacks the back-to-back enhancement. In fact,
the product D × D corresponds to a one-loop cor-
[1]Dab,c(x1,x2;q12,q22;∆(cid:126)), (11) rectiontothe[1“]leadin[1g] twist”perturbativeproductionof
fourjetsinahardcollisionoftwopartons(“two-to-four”)
where the subscript stands as a reminder of the fact
[1] whose distribution is smooth in the back-to-back region
thatbandcoriginatefromperturbativesplittingofasin-
and as such gets subtracted as background.
gle parton from the hadron wave function. This contri-
bution is practically ∆-independent and should decrease
with ∆ much more slowly, due to logarithmic pQCD ef- Keeping this in mind, the back-to-back 4-jet produc-
fects. (Asteeppowerfalloffstartsonlywhen∆2 exceeds tion cross section is proportional to the inverse “interac-
the relevant hard scale, q2.) tions area” S described by the expression
1 (cid:90) d2∆ (cid:18) (cid:19)
= D (∆) D (∆) + D (∆) D (∆) + D (∆) D (∆) , (12)
S (2π)2 [2] a [2] b [2] a [1] b [1] a [2] b
where indices a, b mark two interacting nucleons. This leads to fast convergence of the integral whose median is
expression is somewhat symbolic; a careful analysis of positioned at a value as low as ∆2 ≈0.1m2.
g
the“interactionarea”willbecarriedoutbelowinSec.V
The case of the three-to-four process is different. This
(see Eq. (32)).
process corresponds, as we explained above, to interac-
The first term in Eq. (12) we will refer to as a “four-
tion of the offspring of the perturbative splitting of a
to-four”process: two partonsfrom thewavefunctions of
parton from the wave function of one hadron, with two
the hadron a interact with two partons from the hadron
partons from the non-perturbative wave function of the
b producing four jets. The second and the third terms in
second colliding hadron. On the side of D, the param-
Eq. (12) describe hard collisions of one parton from one [1]
eter∆enters“perturbativeloop”duetopartonsplitting
hadron with two partons from the second hadron. Until
and as a result the dependence of D on ∆ turns out to
recently, these “three-to-four” processes were commonly [1]
be only logarithmic, that is, parametrically much slower
ignored in the literature (see, however, [25] ). At the
than that of the non-perturbative form factor F (∆2).
same time, they turn out to be somewhat enhanced. g
Thus, the three-to-four contribution to the double inter-
Indeed, the contribution due to four-to-four processes
action cross section reduces to
to the geometrical factor Eq. (12) is given by
(cid:90) (d22π∆)2 Fg2(∆2)×Fg2(∆2)= m7π2g. (13a) (cid:90) (d22π∆)2 [1]D(x1,x2;∆)Fg2(∆2)(cid:39) [1]D(cid:12)(cid:12)∆=0(cid:90) (d22π∆)2Fg2(∆2),
(13b)
Fast decrease of the product of two squared form factors (where we have neglected the logarithmic ∆-dependence
7
of D). This corresponds to the fact that in the impact C. Perturbative QCD effects in GPD
[1] 2
parameter space, the distance between partons coming
from a perturbative splitting is much smaller than the
hadronsize,sothattheanswerisproportionaltotheden- Thus, we represent the generalized double parton dis-
sity of non-perturbative two-parton correlation at small tribution GPD as a sum of two terms:
2
distances — “in the origin”:
(cid:90) d2∆ F2(∆2) = m2g. (13c) Dab,c(x1,x2;q12,q22;∆(cid:126)) = [2]Dab,c(x1,x2;q12,q22;∆(cid:126))
(2π)2 g 3π
+ Db,c(x ,x ;q2,q2;∆(cid:126)). (15)
[1] a 1 2 1 2
Comparison with the estimate (13a) shows that the con-
tribution to the cross section of the “interference term”
1+2→4 is enhanced, relative to the 2+2→4 process,
by the factor Theterm[2]Ddescribesthedistributionoftwopartons
from the non-perturbative wave function of the hadron
7 a that are independently evolved to large perturbative
×2∼5. (14)
3 scales q2 and q2 according to the standard one-parton
1 2
evolution equation. The perturbative evolution involves
(For the case of the Gaussian form factors this enhance-
momentum scales much larger than the hadron wave
ment is 15% smaller — a factor of 4.) This estimate was functioncorrelationscale(cid:10)∆2(cid:11)∼Q2. Therefore,theevo-
0
obtained for the case when all four partons participating
lutionpracticallydoesnotaffectthe∆-dependenceofthe
in the hard collisions are gluons. A detailed numerical
two-parton spectrum. This piece of the GPD acquires
2
study of the x -dependence of the effective interaction
i but a mild additional logarithmic dependence at the tail
area S will be presented in the paper under preparation.
of the ∆-distribution in addition to a non-perturbative
So we conclude that the three-to-four processes may
power falloff (10).
provideasizablecontributiontothecrosssectionevenif
they constitute a small correction to GPD. The integral QCD evolution equation for Dbc reads
2 [2] a
Db,c(x ,x ;q2,q2;∆(cid:126)) = S (q2,Q2 )S (q2,Q2 ) Db,c(x ,x ;Q2,Q2;∆(cid:126))
[2] a 1 2 1 2 b 1 min c 2 min [2] a 1 2 0 0
+(cid:88)(cid:90) q12 dk2αs(k2)S (q2,k2)(cid:90) dz Pb(z) Db(cid:48),c(cid:16)x1,x ;k2,q2;∆(cid:126)(cid:17)
b(cid:48) Q2min k2 2π b 1 z b(cid:48) [2] a z 2 2 (16)
+(cid:88)(cid:90) q22 dk2αs(k2)S (q2,k2)(cid:90) dz Pc(z) Db,c(cid:48)(cid:16)x ,x2;q2,k2;∆(cid:126)(cid:17).
k2 2π c 2 z c(cid:48) [2] a 1 z 1
c(cid:48) Q2min
Here Pk(z) are the non-regularized one-loop DGLAP stitutes — the starting non-perturbative scale Q2 of the
i 0
splitting functions (8) and S — the double logarithmic perturbative evolution.
i
Sudakov parton form factors defined in Eq. (7).
The lower limit of the perturbative evolution in
Eq. (16), The second term in Eq. (15), [1]Dabc, represents the
small-distance correlation between the two partons that
Q2 =max(Q2,∆2) (cid:39) Q2+∆2, (17) emerge from a perturbative splitting of a common par-
min 0 0
ent parton taken from the hadron wave function. It can
is the only source of additional (logarithmic) ∆- beexpressedintermsofstandardinclusivesingle-parton
dependence. It emerges when ∆2 exceeds — and sub- distributions as follows
Db,c(x ,x ;q2,q2;∆(cid:126))= (cid:88) (cid:90) min(q12,q22) dk2αs(k2)(cid:90) dy Ga(cid:48)(y;k2,Q2)
[1] a 1 2 1 2 k2 2π y2 a 0
a(cid:48),b(cid:48),c(cid:48) Q2min (18)
×(cid:90) dz Pb(cid:48)[c(cid:48)](z) Gb (cid:18)x1;q2,k2(cid:19)Gc (cid:18) x2 ;q2,k2(cid:19).
z(1−z) a(cid:48) b(cid:48) zy 1 c(cid:48) (1−z)y 2
8
The ∆-dependence of D is very mild as it emerges enter the denominator of the amplitude in terms of the
[1]
solely from the lower limit of the logarithmic transverse Sudakov variables read
momentum integration Q2 .
min k2 =x βs−k2, k2 =−x βs−k2,
1 1 ⊥ 2 2 ⊥
with s = 2p p and k2 ≡ ((cid:126)k )2 > 0 the square of the
a b ⊥ ⊥
IV. ANALYSIS OF PERTURBATIVE two-dimensional transverse momentum vector.
TWO-PARTON CORRELATIONS
A singular contribution we are looking for originates
from the region β (cid:28)1, so that precise form of the longi-
A. 3→4 tudinal smearing does not matter and the integral yields
(cid:90) dβ 2πiN 1
Let us analyze the lowest order interaction amplitude N = .
(x βs−k2 +i(cid:15))(−x βs−k2 +i(cid:15)) (x +x )k2
showninFig.1thatproducesadoublehardcollisionand 1 ⊥ 2 ⊥ 1 2 ⊥
involves parton splitting. The numerator of the amplitude is proportional to the
first power of the transverse momentum k . As a result,
⊥
the squared amplitude (and thus the differential cross
section)acquiresthenecessaryfactor1/δ2 thatenhances
the back-to-back jet production.
B. 2→4
NowweshouldverifythatthediagramofFig.2where
both incident partons split, and their offspring engage
into double hard scattering, does not favor back-to-back
jetkinematics. Inotherwords,itdoesnotleadtoasmall
imbalance factor 1/δ2 in the differential cross section.
FIG. 1: 3→4
Weexpresspartonmomentak intermsoftheSudakov
i
decompositionusingthelight-likevectorsp ,p alongthe
a b
incident hadron momenta:
k = x p +βp +k , k (cid:39)(x −β)p ;
1 1 a b ⊥ 3 3 b
k = x p −βp −k , k (cid:39)(x +β)p ;
2 2 a b ⊥ 4 4 b
(cid:126)k = (cid:126)δ =−(cid:126)δ (δ(cid:48) ≡0); k (cid:39)(x +x )p .
⊥ 12 34 0 1 2 a
Here k , k and k are momenta of incoming (real) par-
0 3 4
tons, and k and k — virtual ones. Light-cone fractions
1 2
x , i=1,..4, are determined by jet kinematics (invariant FIG. 2: 2→4
i
masses and rapidities of jet pair). The fraction β that
measures the difference in longitudinal momenta of the
twopartonscomingfromthehadronb,isarbitrary. Fixed Sudakov decomposition:
values of the parton momenta x −β and x +β cor-
3 4
respond to the plane wave description of the scattering k1 =(x1−α)pa+βpb+k⊥(cid:48) , k3 =(x3−β)pb+αpa−k⊥;
process in which the longitudinal distance between the k =(x +α)p −βp −k(cid:48) , k =(x +β)p −αp +k ;
2 2 a b ⊥ 4 4 b a ⊥
two scatterings is arbitrary. This description does not
(cid:126)k(cid:48) −(cid:126)k =(cid:126)δ =−(cid:126)δ (δ(cid:48) ≡0);
correspond to the physical picture of the process we are ⊥ ⊥ 12 34
interested in. In order to ensure than the partons 3 and k0 (cid:39)(x1+x2)pa, k5 (cid:39)(x3+x4)pb.
4 originate form the same hadron of finite size, we have
to introduce an integration over β in the amplitude, in Thisisaloopdiagramanditcontainsexplicitintegration
the region β =O(1). over the loop momentum:
The Feynman amplitude contains the product of two (cid:90) dαdβ (cid:90) d2k d2k(cid:48)
s ⊥ ⊥δ2((cid:126)k +(cid:126)δ−(cid:126)k(cid:48) ).
virtual propagators. The virtualities k2 and the k2 that (2π)2i (2π)2 ⊥ ⊥
1 2
9
To get an enhanced contribution we have to have parton factor, ln(Q2/δ2), instead of the power back-to-back sin-
virtualities that enter the denominator of the Feynman gularity Q2/δ2 we were looking for.
amplitudetoberelativelysmall,oftheorderofδ2 (cid:28)Q2. So, the diagram Fig. 2 with double parton splitting
This implies |α|,|β| (cid:28) 1 in the essential integration re- constitutes but a negligible loop correction to the usual
gion. Adopting this approximation, we can simplify par- “hedgehog” 4-jet kinematics typical for 2→4 QCD pro-
ton propagators and reduce the longitudinal momentum cesses. Thefactthatthisloopdiagramdoesnotproduce
integrationstotheproductoftwoindependentintegrals: a pole singularity in δ2 could have been extracted, e.g.,
from numerical studies of double Z-boson production in
i (cid:90) dβ s
two-parton collisions [30] and, more generally, of multi-
s 2πi(βx1s−k⊥2 +i(cid:15))(βx2s+k⊥2 −i(cid:15)) leg parton amplitudes [31].
(cid:90) dα s The logarithmic character of this correction has been
×
2πi(αx s−k(cid:48)2+i(cid:15))(αx s+k(cid:48)2−i(cid:15)) recently confirmed by the systematic study of “box inte-
3 ⊥ 4 ⊥ grals” in [24].
i 1
= . The presence of the double parton splitting contri-
(x +x )(x +x )s k2 k(cid:48)2
1 2 3 4 ⊥ ⊥ bution of Fig. 2 is being treated in the literature as a
source of potential problem of double counting (see, e.g.,
The remaining transverse momentum integration takes
[15, 31, 32]). The present paper solves this problem.
the form
(cid:90) d2k V
⊥ (20)
(2π)2 (cid:126)k2((cid:126)k +(cid:126)δ)2 C. 3→4 with additional parton emission
⊥ ⊥
Duetogaugeinvariancethenumeratorofthediagram— Wehavetoreturnnowtothe3→4processandexam-
the “vertex factor” V — is linear in transverse momenta ine the possibility of producing an additional parton, in
of the loop partons: V ∝ kµk(cid:48)ν. Therefore, the integral collinear enhanced manner, in order to lift off the Born
⊥ ⊥
(20) produces no more than a logarithmic enhancement level kinematical constraint δ(cid:48) =0.
FIG. 3: Three-to-four amplitudes with extra emission from inside the splitting fork
Consider the diagram of Fig. 3a. The momenta of We have three virtual propagators subject to integration
quasi-real colliding partons are over β:
δ(cid:48)2
k (cid:39)(x +x +α)p ;k (cid:39)(x −β)p ,k (cid:39)(x +β+ )p ,
0 1 2 a 3 3 b 4 4 αs b k = x p +βp +δ , (21a)
1 1 a b 13
andtheradiatedon-mass-shellpartoncarriesmomentum (cid:18) δ(cid:48)2(cid:19)
k = x p − β+ p +δ , (21b)
2 2 a αs b 24
δ(cid:48)2
(cid:96)−k2 = αpa+ αspb−δ(cid:48), δ(cid:48) =δ13+δ24. (cid:96) = (x2+α)pa−βpb−δ13. (21c)
10
Closing the contour around the pole k2 + i(cid:15) = 0, we the parton 1 collides with 3, while the parton (cid:96) keeps
1
obtain βs=δ2 /x and evolving and scatters off 4 with a much larger momen-
13 1
tum transfer δ . Evolution of the parton (cid:96) in between
24
x +x +α
−(cid:96)2 =(x +α)βs+δ2 =δ2 · 1 2 , (22a) thesetwoscalesistheoriginofprobable(logarithmically
2 13 13 x1 enhanced) production of additional parton(s).
(cid:18) δ(cid:48)2(cid:19) x δ2 +x δ2 x Analogously, the diagram Fig. 3b with a parton pro-
−k2 =x β+ s+δ2 = 1 24 2 13 + 2δ(cid:48)2.(22b)
2 2 αs 24 x α ducedoffthevirtualline1contributesinthecomplemen-
1
tary kinematical region
Taken together with the residue of the β-integration,
1/x , Eq. (22b) produces the universal factor present in δ2 (cid:28) δ2 (cid:39) δ(cid:48)2. (24b)
1 24 13
all the amplitudes considered, (including the diagrams
with parton emission off the external lines, see Fig. 4 Full perturbative analysis of the production of a parton
below): from inside the “splitting fork”, Fig. 3, together with
emission off the incoming line “0” (to be treated below
P−1 =x δ2 +x δ2 + x1x2δ(cid:48)2. (23) in Sec. VB) is sketched in the Appendix.
1 24 2 13 α
We observe that both propagators (22) are enhanced in
V. DIFFERENTIAL DISTRIBUTION
the back-to-back kinematics. The amplitude of Fig. 3a
gives a double collinear enhanced contribution to the
A. 2+2→4
cross section in the region
δ2 (cid:28) δ2 (cid:39) δ(cid:48)2. (24a) Now that we know the structure of the perturbative
13 24
correctionsto GPD,Eqs.(15)–(18),weareinaposition
2
The inequality (24a) corresponds to the following phys- towritedownthegeneralizationoftheDDTformula(6)
ical picture. An incident parton k splits into k and (cid:96) for(thefirstcontributionto)thedifferentialcrosssection
√ 0 1
early, at time O(cid:0) s/δ2 (cid:1) corresponding to some com- of 4-jet production in nearly back-to-back kinematics. It
13
paratively low perturbative scale δ . At this time scale reads
13
dσ(4→4) dσ ∂ ∂ (cid:26)
π2 = part · D1,2(x ,x ;δ2 ,δ2 )× D3,4(x ,x ;δ2 ,δ2 )
d2δ d2δ dtˆ dtˆ ∂δ2 ∂δ2 [2] a 1 2 13 24 [2] b 3 4 13 24
13 24 1 2 13 24
(cid:27)
× S (cid:0)Q2,δ2 (cid:1)S (cid:0)Q2,δ2 (cid:1)×S (cid:0)Q2,δ2 (cid:1)S (cid:0)Q2,δ2 (cid:1) . (25)
1 13 3 13 2 24 4 24
Here dσ is the cross section of double hard parton B. 1+2→4
part
scattering, andS standforSudakovformfactorsoffour
i
participating partons. Sum over parton species and con-
volution over ∆(cid:126) as in Eq. (1) is implied.
The differential transverse momentum imbalance dis-
Taking derivative over the scale δ2 of the function de-
tribution due to the cross-terms D× D contains two
pending on α logδ2, produces the factor α /δ2. Differ- [2] [1]
s s pieces.
entiating the Sudakov form factor of a given parton de-
1. Two compensating partons
scribes the situation when the jet imbalance is compen-
sated by radiation of a soft gluon off this parton. Dif-
ferentiationofthepartondistributioncorrespondstothe
situation when a hard parton takes the recoil. The first one has the same structure as Eq. (25):
dσ(3→4) dσ ∂ ∂ (cid:26)
π2 1 = part · D1,2(x ,x ;δ2 ,δ2 )· D3,4(x ,x ;δ2 ,δ2 )
d2δ d2δ dtˆ dtˆ ∂δ2 ∂δ2 [1] a 1 2 13 24 [2] b 3 4 13 24
13 24 1 2 13 24
(cid:27)
× S (cid:0)Q2,δ2 (cid:1)S (cid:0)Q2,δ2 (cid:1)·S (cid:0)Q2,δ2 (cid:1)S (cid:0)Q2,δ2 (cid:1) . (26)
1 13 3 13 2 24 4 24