Multiple Hard Partonic Collisions with Correlations in Proton-Proton Scattering

0.32 MB·English

by  T. C. Rogers

Multiple Hard Partonic Collisions with Correlations in Proton-Proton Scattering
T. C. Rogers
Department of Physics and Astronomy,
Vrije Universiteit Amsterdam,
NL-1081 HV Amsterdam, The Netherlands
M. Strikman
Department of Physics,
Pennsylvania State University,
University Park, PA 16802, USA
We propose a simple method for incorporating correlations into the impact parameter space
0 description of multiple (semi-)hard partonic collisions in high energy hadron-hadron scattering.
1 The perturbative QCD input is the standard factorization theorem for inclusive dijet production
0 with a lower cutoff on transverse momentum. The width of the transverse distribution of hard
2 partons is fixed by parameterizations of the two-gluon form factor. We then reconstruct the hard
contribution to the total inelastic profile function and obtain corrections due to correlations to the
n
morecommonlyusedeikonaldescription. Estimatesofthesizeofdoublecorrelationcorrectionsare
a
based on the rate of double collisions measured at theTevatron. Wefind that, if typical values for
J
the lower transverse momentum cutoff are used in the calculation of the inclusive hard dijet cross
5
section,thenthecorrelationcorrectionsarenecessaryformaintainingconsistencywithexpectations
2
for thetotal inelastic proton-proton cross section at LHC energies.

] Keywords: perturbativeQCD,unitarization,multiplecollisions
h
p
-
p I. INTRODUCTION approaches see, e.g. [7, 8], and references therein.) The
e hardcontributioniscalculatedusingthewell-knownper-
h Models of multiple partonic collisions in high energy turbative QCD (pQCD) leading twist factorization for-
[
hadron-hadron scattering are important for simulations mula for the inclusive dijet production cross section, in-

2 of complex events in upcoming experiments like those volvingaconvolutionofthestandardpartondistribution
v at the LHC or in high energy cosmic ray air showers. functions (PDFs) with a partonic cross section. An im-
1 Furthermore, measurements of the rate of multiple par- mediate complication is that, in order for perturbation
5 toniccollisionscanbeusedtotestcurrentmodelsofpro- theory to be applicable, the relative transverse momen-
2
0 ton structure. Already, measurements at accessible en- tumofthe producedjetsmustbelargerthansomemini-
. ergies [1–5] yield a much smaller effective cross section mumcutoffscalepct. Thecutoffshouldbechosensmallso
8 σ than what is naively expected if partons are homo- as to maximize the range of the perturbative expression,
0 eff
geneouslydistributedoverthetransverseareaofthepro- butstilllargeenoughforperturbativemethodstoberea-
9
0 ton. (Figure 1 shows a schematic depiction of a double sonable. For describing events with transverse momen-
v: partonic collision.) The definition of σeff is Atumprelescssripthtiaonnpfoct,rnmoantpcehrintugrbthaetivheigmheatnhdodloswarterannesevdeerdse.
i
X σ2 momentum behavioris necessaryfor a complete descrip-
σeff =m 2 (1) tion over the full range of transverse momentum.
r 2σ
4
a The inclusive pQCD harddijet crosssectionis numer-
where σ is the inclusive cross section for a single par- ically very sensitive to the precise choice of pc. This has
2 t
tonic collision (resulting in a dijet), σ is the inclusive a tendency to lead to substantial variations between dif-
4
crosssectionfor a double collision,and m is a symmetry ferent model predictions of the minijet cross section at
factor that depends on whether the partons are identi- high energies. The question of what values of pc are ap-
t
cal. New measurements of multiple collisions are cur- propriate continues to be discussed in current research
rentlybeing proposedfor the LHC [6]. Hence, novelnew on the development of models and simulations (see, for
phenomena involving multiple hard partonic collisions, example, recent discussions in [8–11]).
which will enhance understanding of proton structure, An additional complication is that a description is
canbe expectedinthenextgenerationofexperimentsat needed for the distribution of hard partons in impact
the high energy frontier. parameter space. In the past, it has usually been mod-
However,multipleinteractionsinvolveacomplexinter- eled or assumed to be equal to the electromagnetic form
play of soft, hard, and semihard physics, so a complete factor of the nucleon. Also, in a number of models, soft
description using purely perturbative techniques is not and hard partonic interactions are incorporated into a
possible. Insimulationsofcomplexhadronicfinalstates, singleeikonalpicture. Inthisway,bothsoftandhardin-
methods are needed for combining hard and soft colli- teractions are included in a way that respects s-channel
sions in a consistent way. (For an overview of current unitarity. Data for the total and inelastic pp cross sec-
2
FIG. 1: Schematic depiction of a double hard collision. A disconnected pair of partons from each proton collide to produce a
pair of high transverse momentum dijets.
tions can then be used to fit parameters such as pc and the contribution to the total inelastic profile function
t
the width of the distribution of hard partons in impact from the production of hard dijet pairs becomes larger
parameterspace. However,differentchoicesforthesepa- than the total inelastic profile function itself. Although
rameters can provide equally good fits to the total cross the inclusive dijet cross section is unitarized in the ba-
section at accessible energies while leading to very dif- sic eikonal description, it nevertheless grows too rapidly
ferent extrapolations at high energies (see, for example, withenergy. Aswewillargue,thisproblemismostlikely
Ref. [12]). Therefore, it is important to make use of any a symptom of the common assumption that partons are
experimental or observational information that can nar- uncorrelated.
row the range of allowed parameters and falsify some of The study of correlations in multiple hard collisions
the models currently in use. is already an active area of research [19–22]. Imple-
It is nowadays possible to use pQCD to obtain direct mentationsofDokshitzer-Gribov-Lipatov-Alterelli-Parisi
experimental informationabout the transversespreadof (DGLAP) evolution in multiparton distribution func-
hard partons in the proton via parametrizations of the tionssuggestthatcorrelationsareindeedsignificant[21].
generalized parton distributions (GPDs). (See Ref. [13] CorrelationsmayalsobeinducedbyevolutioninBjorken
andreferencesthereinforareviewofthephenomenology x [23] in the very high energy limit. However,numerical
of GPDs.) The gluon GPD of the proton, for example, estimates in Ref. [18] suggest that hard unitarity (satu-
canbeextractedfrommeasurementsofthetdependence ration) effects contribute to only a small fraction of the
in deep inelastic production of light vector mesons or total inclusive dijet cross section, even at LHC energies.
photoproduction of heavy vector mesons [14, 15]. Then, Moreover,theinconsistenciesencounteredinRef.[18]oc-
sincethe GPDis auniversalobject[16], itcanbe reused curevenatlargeimpactparameters,&1.0fm,andvalues
in other processes. In particular, it can be used in the of pc that are not particularly small, e.g. pc 2.5 GeV.
t t ≈
description of the impact parameter dependence of hard Intheinstantonliquidmodel(seeRef.[24]andreferences
collisionsinhadron-hadronscattering. Assuch,weadopt therein) one can expect a strongcorrelationbetween the
thepointofviewinthispaperthattheimpactparameter quark and gluon fields in the constituent quarks which
descriptionofa hardcollisionis notanadjustable model will result in correlations at all impact parameters. It
parameter, but rather is fixed by fits to the gluon GPD. may be somewhat diluted at large b due to the contri-
An accurate description of the impact parameter de- bution from the 4qq¯component. On the other hand for
partons at large enough distances from the center it is
pendence is important because it allows for a measure
of how close the hard interaction is to the unitarity (or likelythatanotherpartonshouldbepresentatadistance
comparable to the confinement scale, leading to correla-
“saturation”) limit. Furthermore, the proximity to the
unitaritylimitisrelatedtothenumberofhightransverse tions of qq¯pairs at relative distances 0.5 fm that are
≤
rather small which could easily compensate for the dilu-
momentum jets that are produced, and hence to the use
of centrality as a dijet trigger [17]. However, as was il- tion effect. Therefore, it is likely that nonperturbative
correlations, unrelated to the overall impact parameter,
lustrated in Ref. [18], using GPDs to describe the im-
pact parameter dependence of hard collisions within the play a role in determining the size of the dijet contri-
bution to the total inelastic cross section. Heuristically,
approximation where partons are not correlated in the
transverse plane (which is implemented in the simplest one can imagine scattering at large impact parameters
as the scattering of pion clouds. The nonperturbative
eikonaldescription) leads to inconsistencies with general
expectations for the extrapolation of the total inelastic dynamics responsible for binding the qq¯pairs introduces
potentially large correlations.
profile function to high energies, unless a large value is
usedforthetransversemomentumcutoffpc. Specifically, Collinear constituents of the incoming protons are ex-
t
3
p1 p3
pected, onaverage,tohaveaspacetimeseparationofor-
der 1/Λ . Because ofthe breakdownof asymptotic
QCD
∼
freedomatthese scales,itshouldbeanticipatedthatthe
constituentsofthe nucleonaresubjectto strongnonper-
turbative correlations. In other words, one should not
expect the multiparton distribution functions to be sim-
ply related to products of the single parton distribution
function at any scale. p2 p4
As a specific illustration, let us consider the double
FIG. 2: Momentum labels for pp scattering.
parton event in Fig. 1. If f (x ,x ) is the distribution
N 1 2
function for parton pairs with momentum fractions x
1
and x inside proton N, and f (x ) and f (x ) are the
2 N 1 N 2 outlined in Ref. [18] for dealing with multiple hard colli-
standard PDFs, then an often made ansatz is
sions in terms of inclusive dijet cross sections, applying
?? theresulttothespecialcaseofuncorrelatedscatteringin
f (x ,x ) f (x )f (x ). (2)
N 1 2 ≈ N 1 N 2 Sec. IV. In Sec. V these steps are extended to allow for
In some treatments of correlations this relation follows impact parameter independent correlations. In Sec. VI
after an integration over impact parameters, even if the we use estimates of correlations based on Tevatron mea-
factorizationansatzisbrokeninimpactparameterspace. surementsofσ tocalculatethehardcontributiontothe
eff
ThequestionmarksinEq.(2)indicatethatwewillques- inelastic profile function at LHC energies, and we com-
tion the validity of this assumption. Given the strong pare with standard extrapolations of the total inelastic
binding between the constituent partons in the proton, profilefunction. We speculate onprospects forincluding
the approximation in Eq. (2) is probably very rough at impactparameterdependenceofcorrelationsinSec.VII.
any scale. We will argue that deviations can have an We close in Sec. VIII with a conclusion and a discussion
important effect on extrapolations to high energies. of the main results.
For a large cutoff pc, the inclusive dijet cross section
t
gives a small fraction of the total inelastic cross section,
and there is no conflict with s-channel unitarity. What II. IMPACT PARAMETER REPRESENTATION
is needed at smaller pc is a method for organizing corre-
t
lation corrections which does not rely on the commonly A. S-Channel Unitarity
used assumption that correlations can be neglected.
To summarize, the aim of this paper is to set up a
We work in impact parameter space, defining the pro-
methodfororganizingcorrectionstotheuncorrelatedap-
file function,
proach at small pc. We will use this to simultaneously
t
incorporate the following information into a description 1
Γ(s,b) d2qeiq·bA(s,t), (3)
of multiple hard partonic collision: ≡ 2is(2π)2 Z
Theimpactparameterdistributionofhardpartons
• HereA(s,t)istheamplitudeforelasticppscattering(see
obtained from measurements of the gluon GPD.
Fig. 2), and s and t are the usual Mandelstam variables,
The width of the distribution of hard partons is
s = (p + p )2 and t = (p p )2. We work in the
not a fixable parameter in our approach. 1 2 1 − 3
high energy limit, s >> t where one may approximate
An estimate of the size of double parton correla- t q2 (see e.g. Ref. −[25] for a review of kinematics
• ≈ −
tions obtained from measurements σ . In this inthehighenergylimit). Inthetwo-dimensionalFourier
eff
paper, we will assume that these correlations are transformtocoordinatespace,bistheimpactparameter.
roughly independent of impact parameter. Unitarity and analyticity allow the total, elastic and
inelastic cross sections to be calculated in terms of the
Fromthis information,wewillreconstructthe harddijet
profile function via the familiar relations:
contribution to the total inelastic profile function. Com-
pared with the uncorrelated expression, the result ob-
tained with correlationswill be shown to exhibit greater σtot(s) = 2 d2bReΓ(s,b), (4)
Z
consistency with common high energy extrapolations of
the total inelastic cross section, even with a relatively σ (s) = d2b Γ(s,b)2, (5)
el
small and fixed value for pc in the perturbative inclusive Z | |
t
dijetcalculation. Thus,includingcorrelationsinthisway σ (s) = d2b 2ReΓ(s,b) Γ(s,b)2 . (6)
mayprovideanaturalresolutiontotheconsistencyprob- inel Z (cid:16) −| | (cid:17)
lems encounteredinRef.[18]atlargeimpactparameters
and fixed pc. WerefertotheintegrandofEq.(6)astheinelasticprofile
t
function,
The paperisorganizedasfollows: InSec.II wereview
thebasicsetupfordiscussinghighenergycollisionsinim-
pactparameterspace,andinSec.IIIwereviewthesteps Γinel(s,b) 2ReΓ(s,b) Γ(s,b)2 . (7)
≡(cid:16) −| | (cid:17)
4
Inthe veryhighenergylimit, itis appropriateto neglect where g(x;µ) is the standard integrated gluon distribu-
the imaginary part of the amplitude. Then unitarity re- tion function evaluated at hard scale µ. The two-gluon
quires form factor obeys the condition,
Γinel(s,b),Γ(s,b) 1. (8) F (x,t=0;µ)=1 (12)
g
≤
Theprofilefunctionforthetotalproton-protoncrosssec- so Eq. (11) reduces to the standard PDF in the limit of
tionanditsextrapolationtohighenergieshasbeenstud- t 0. The parameter mg(x,µ) in Eq. (10) determines
→
ied extensively overthe lastfew decades. Our main con- the width of the peak around t=0. Following Ref. [17],
cern in this paper is whether common extrapolations of we allow it to have x and µ dependence to account for
Eq. (7) are consistent with extrapolations of the total evolution in the hard scale µ and diffusion at small x.
hard contribution. The Fourier transform of the two-gluon form factor into
the transverse plain is
B. Inclusive Dijet Cross Section (x,b;µ)= d2∆ F (x,t;µ)e−i∆t·b, t ∆2.
Fg Z t g ≈− t
(13)
Hard collisions are described by the leading twist Again,the approximationt ∆ is justified so long as
t
pQCD expression for the inclusive dijet cross section, ≈−
s >> t. Using the dipole form in Eq. (10) one finds
−
explicitly
K
σinc (s;pc)= dx dx dp2
pQCD t i,Xj,k,l 1+δkl Z 1 2Z t× g(x,b;µ)= mg(x;µ)3bK1(mg(x;µ)b). (14)
F 4π
dσˆ
× dijp→2kl fi/p1(x1;pt)fj/p2(x2;pt)θ(pt−pct). (9) Inthis paper, Kn forintegerndenotesamodifiedBessel
t function of the second kind. The overlap function is de-
The collision is between parton types i and j inside pro- fined as
tonsp andp respectively,andthepartonichardscatter-
1 2
ing cross section dσˆij→kl/dp2t is calculated at tree level. P2(b,x1,x2;µ)=
We have explicitly included a K factor and a symme-
d2b′ (x , b′ ;µ) (x , b b′ ;µ). (15)
try factor 1/(1+δ ). The f (x ;p ) and f (x ;p ) g 1 g 2
kl i/p1 1 t j/p2 2 t Z F | | F | − |
aretheusualintegratedpartondistributionfunctionsfor
Using Eq. (14) yields
partons with longitudinal momentum fractions x and
1
x2, evaluated at a hard scale equal to the dijet trans- m2(x;pc) m (x;pc)b 3
verse momentum. The lower bound pct is in principle P2(s,b;pct)= g12π t (cid:18) g 2 t (cid:19) K3(mg(x;pct)b).
arbitrary, but it should be chosen large enough for it
(16)
to be a reasonable hard scale. It is not clear exactly
Here we have made the usual approximation, x x
whatis the minimum pct thatcanbe used, but Relativis- x 2pc/√s. Amoredetailedtreatmentshould1ta≈kei2nt≈o
tic Heavy Ion Collider (RHIC) data for pion production ≡ t
accounttheseparateintegrationsoverx andx —there
suggest that perturbation theory is still reliable for pi- 1 2
is not, in general, a one-to-one mapping between values
ons with pc & 1 GeV [26] for forward production where
t ofpc andx (2). Fornow,wementionthatdirectnumeri-
background from soft physics is small. (Note here that t 1
calcalculationsverifythatthisapproximationintroduces
for suchkinematics p of the progenitorquark is close to
t less than 10% error in the essential region of integration
p of the pion.)
t for the cross section. Note that P (s,b;pc) is normalized
Information about the impact parameter distribution 2 t
to unity,
of hardpartons in the protonis obtained from the gluon
GPD which is parametrized in experimental measure-
d2bP (s,b;pc)=1. (17)
ments of the t dependence in exclusive heavy vector me- Z 2 t
son photoproduction and light vector meson electropro-
CombiningEqs.(10-17)withEq.(9)allowsthe inclusive
duction at small x. For the t dependence of the differen-
dijet cross section to be written in the form
tial exclusive vector meson production cross section, we
use the dipole parametrization, σinc (s;pc)= d2bχ (s,b;pc) (18)
pQCD t Z 2 t
1
F (x,t;µ)= . (10)
g 2 where
1 t
(cid:16) − mg(x,µ)2(cid:17) χ (s,b;pc)=σinc (s;pc)P (s,b;pc). (19)
2 t pQCD t 2 t
ThebasicFeynmandiagramcontributingtothisreaction
We will refer to χ (s,b;pc) as the impact parameter de-
involves the exchange of two gluons in the t channel, so 2 t
pendent inclusive cross section [44]. Using the GPD
we refer to it as the two-gluon form factor. The gluon
to write the inclusive dijet cross section in the form of
GPD g(x,t;µ) is then
Eq. (18) enables one to analyze the contribution from
g(x,t;µ)=g(x;µ)F (x,t;µ), (11) different regions of impact parameter space.
g
5
III. MULTIPLE HARD PARTONIC COLLISIONS In principle, if allthe inclusive n-dijet impact parameter
dependent cross sections χ (s,b;pc) are known, then it
2n t
Bytakingintoaccountmultiplehardscatteringevents, is possibleto obtainexactlythe dijet contributionto the
it is in principle possible to reconstruct the total hard inelasticprofilefunctionbysummingallthetermsinthe
scatteringcontributionto the totalinelastic profilefunc- last line of Eq. (24). In practice, higher orders in n need
tion using probabilistic arguments [27, 28]. In this sec- to be modeled or approximated.
tion, we briefly review the steps for constructing the di- Consistencyrequirestheharddijetcontributiontothe
jet contribution to the total inelastic profile function in total inelastic cross section to be less than the actual
terms of a series involvingthe inclusive n-dijet crosssec- total inelastic cross section so
tions.
We startbydefiningχ2n(s,b;pct)tobe the analogueof Γidniejelts(s,b;pct)≤Γiancetlual(s,b), (25)
χ (s,b;pc)forthe caseofn-dijetproduction(nhardcol-
2 t
lisions). Namely, integrating over all impact parameters where the right side is the “actual” inelastic profile
yields function, which could be obtained from either a mea-
surement or a model extrapolation. Hence, Eq. (25)
d2bχ (s,b;pc)=σinc(s;pc), (20) provides a means of checking that an expression for
Z 2n t 2n t Γinel (s,b;pc), constructed from Eq. (24), is consistent
dijets t
with other methods for obtaining the total inelastic pro-
whereσinc(s;pc)istheinclusivecrosssectionforproduc-
2n t file function. A violation of Eq. (25) means either that
ing n-dijet pairs.
the model/extrapolation is incorrect or that there is a
Next, χ˜ (s,b;pc) is defined to be the exclusive ana-
2n t problem with the χ (s,b;pc) used in Eq. (24).
logue of χ (s,b;pc). It describes the production of ex- 2n t
2n t
actly n hard collisions, differential in b. Integrating over
impact parameters gives
IV. UNCORRELATED SCATTERING
d2bχ˜ (s,b;pc)=σex(s;pc), (21)
Z 2n t 2n t The simplest and most common way to obtain an ex-
plicit expression for Γinel (s,b;pc) from Eq. (24) is to
where σex(s;pc) is the integrated cross section for pro- dijets t
2n t assume that all partonic collisions occur completely in-
ducingexactly n-dijetpairs. [Theχ˜ (s,b;pc)correspond
2n t dependently from one another. It was shownin Ref. [27]
to the “exclusive” cross sections of Ref. [20], the quotes
thattheinclusiveimpactparameterdependentcrosssec-
referring to the fact that these cross sections are still in-
tion for production of n dijets is then
clusive in soft fragments.]
Now we reconstruct the hard dijet contribution to the
1
inelastic profile function by writing down the expression χ (s,b;pc)= χ (s,b;pc)n. (26)
2n t n! 2 t
for the inclusive cross section for k-dijet production in
terms of the exclusive cross sections:
This can be inserted into the last line of Eq. (24) and
∞ summedtoreproducethe familiarunitarizedeikonal-like
n
χ2k(s,b;pct)= (cid:18)k(cid:19)χ˜2n(s,b;pct). (22) expression,
nX≥k
∞
The combinatorial factor counts all the ways an n-dijet Γinel (s,b;pc)= 1 ( 1)n−1χ (s,b;pc)n =
eventcancontributetotheinclusivek-dijetcrosssection. dijets t n! − 2 t
nX=1
Equation (22) can be inverted to obtain the exclusive
=1 exp[ χ (s,b;pc)] . (27)
impactparametercrosssections interms ofthe inclusive − − 2 t
ones,
The single, double, and triple scattering terms are rep-
∞ resented graphically in Fig. 3. (This kind of graphical
n
χ˜2k(s,b;pct)= (cid:18)k(cid:19)(−1)n−kχ2n(s,b;pct). (23) representation will be useful later for describing combi-
nX≥k natorial factors when correlations are included.) Each
circle-cross represents a hard scattering event. The un-
The total inelastic profile function is obtained by sum-
correlated assumption of Eq. (26) is symbolized by the
ming all the exclusive components. Using Eq. (23), we
absence of any lines connecting the different hard colli-
obtain
sions - each graph is simply χ (s,b) raised to the appro-
2
∞ priate power.
Γinel (s,b;pc)= χ˜ (s,b;pc)= For many practical purposes, Eq. (27) is sufficient. In
dijets t 2k t
Xk=1 general, the reconstructed profile function simply needs
∞ to reproduce the correct pQCD expression at large b
= (−1)n−1χ2n(s,b;pct). (24) wheremultiplecollisionsareveryrare,whiletheminimal
nX=1 unitarityrequirementthatΓinel .1shouldbe enforced
dijets
6
 
1  1
Γinel ∼ − +   .
dijets (cid:18) (cid:19) 2  6 
   
   
   
 
 
FIG.3: Graphicalrepresentationofthetermintheseriesforuncorrelatedscattering—thefirstthreetermsinthesecondline
of Eq. (27), assuming no correlations. Spectator partons are not shown.
at small b. That is, the basic requirements in the high Therefore, we organize the description of correlations
energy limit are, around the assumption that the effect is to introduce a
simple (impact parameter independent) rescaling from
Γinel (s,b) b→=∞ χ (s,b;pc) (28) the uncorrelated case. As a first example, we reconsider
dijets 2 t
double hard collisions. Equation (26) gives the uncorre-
b→0
Γinel (s,b) . 1. (29) lated expression
dijets
1
In the high energy limit, Γidniejelts(s,b) is expected to ap- χ4(s,b;pct)= 2χ2(s,b;pct)2 (30)
proach one at small b (black disk limit). Equation (27)
is completelysatisfactoryasfar asconditions(28,29)are which should be replaced in the correlated case by
concerned. Forlargeb,onlythefirsttermintheseriesin
Eq. (27) — i.e. single scattering — is important. How- 1
χ (s,b;pc) (1+η (s))χ (s,b;pc)2, (31)
ever, in a precise treatment, one should also account for 4 t → 2 4 2 t
potential for violations of the consistency requirement
in Eq. (25) in the high energy limit and at intermedi- whereη4(s)parametrizesthedeviationfromuncorrelated
atevalues ofbwherecorrectionsoforderχ (s,b;pc)2 are scattering. Our strategy is to estimate the size of the
2 t
non-negligible. Indeed, such consistency problems were double correlation by directly fitting Eq. (31) to experi-
found in Ref. [18]. mentaldata,giventheconstraintthatχ2(s,b;pct)isfixed
by the GPD in Eq. (19). Note that we place no condi-
tion on the b integral of Eq. (30). In particular, we do
not use the approximation in Eq. (2). In general, the
V. ORGANIZING CORRELATIONS IN
correlationcorrectionwill also depend on both pc and b.
MULTIPLE COLLISIONS t
Forour analysis,we willnotexplicitly writethe pc argu-
t
ments in Eq.(30) because we are mainly concernedwith
A. Impact Parameter Independence correlation corrections in the limited range of pc where
t
Eq. (25) becomes problematic within the usual eikonal
Deviationsfromuncorrelatedscatteringcanarisefrom picture. As we will see, neglecting the b dependence in
multiple sources. As discussed in the Sec. I, correlations η (s) will allow for a direct parametrization of the cor-
4
can be generated both in perturbative evolution equa- relation correction in terms of experimentally observed
tions and in nonperturbative models. double scattering rates. It is likely that this is a very
Correlations will also be induced by kinematical con- roughapproximation,butitwillallowforafirstestimate
straints. We will assume, however,that most active par- oftheroleofcorrelationsatlargeimpactparameters. We
tonshavesmallenoughxthattheseconstraintsareunim- also remark that the dynamics responsible for confine-
portant, at least for the first few terms in the series in ment are likely to induce large correlations regardless of
Eq. (27). For this paper, we will assume that the in- impactparameter. Wewilldiscusspossiblebdependence
comingpartonsthattakepartinmultiple hardcollisions in greater detail in Sec. VII.
movenearlyparallelwithtransversemomentumoforder In experiments the effect of double partonic collisions
1/Λ . That is, they have momentum typical for is most commonly represented by the observable,
QCD
∼
bound constituents of the incoming hadrons. In general,
if pc is allowed to be larger than a few GeV, the partons 1σinc(s;pc)2
t σ = 2 t . (32)
will undergo DGLAP evolution, and hence may include eff 2 σinc(s;pc)
4 t
partonswithlargertransversemomentum. Furthermore,
one expects significant dependence of σ on the hard In the uncorrelated case, using Eq. (30) and Eq. (19) in
eff
scale at large pc [22]. In such cases, it is possible that Eq. (32) yields
t
correlations may be understood as arising from parton
evolution. However, conflicts with Eq. (25) become less 1
σuncor = . (33)
likely at larger pc. eff d2bP (s,b;pc)2
t 2 t
R
7
Eq. (31):
1 
−
2
  1
  χ (s,b;pc)= χ (s,b;pc)2 χ (s,b;pc)
  6 t 6 2 t 2 t
  (cid:2) (cid:3)
1
(1+η (s))χ (s,b;pc)2 χ (s,b;pc). (36)
FIG. 4: Graphical representation of the extra contribution → 6 4 2 t 2 t
( 1/2)η4(s)χ2(s,b;pct)2 due to double partonic correlations (cid:2) (cid:3)
−
in Eq.(31). There is, therefore, an extra contribution equal to
η4(s)(χ (s,b))3 for each of the 3 = 3 ways a pair of
6 2 2
incoming bound partons can bec(cid:0)om(cid:1) e correlated. This is
In general, the value of σ can be fitted to experimen-
eff illustrated graphically in Fig. 5, which shows the addi-
tally measured values by changing the width or shape of
tional contributions that must be added to the n = 3
P2(s,b;pct). However,inourapproachP2(s,b;pct)isfixed terminEq.(27)/Fig.3. The expressionfor χ6(s,b;pct)is
byexperimentalmeasurementsoftheGPD,sothewidth
therefore
of P (s,b;pc) is not a free parameter.
2 t
If Eq. (31) is used in Eq. (32), one obtains for σeff in χ (s,b;pc)= 1(1+3η (s))χ (s,b;pc)3. (37)
the correlated case 6 t 6 4 2 t
1
σcor = . (34) Following this example, it is now clear how to include
eff (1+η (s)) d2bP (s,b;pc)2
4 2 t double correlation corrections in n-parton scattering. In
R an n-parton collision, there are n additional contribu-
With P (s,b;pc) fixed by the two-gluonform factor, one 2
2 t tions equal to 1η (s)χ (s,b;pc)n(cid:0).(cid:1)In terms of diagrams
can only tune Eq. (34) to the measured value of σ by n! 4 2 t
eff likeFig.5,thiscorrespondstoallthewaysthattwocolli-
adjusting η (s). Note also that, although we have as-
4 sionscanbeconnectedbyasinglezigzagline. Therefore,
sumed impact parameter independent correlations, the
toincludedoublecorrelationsinthedescriptionofthein-
relative rate of double collisions compared with single
clusive n-dijet cross section, the uncorrelated relation in
collisions, χ (s,b)/χ (s,b), still depends on impact pa-
2 4 Eq. (26) should be replaced with,
rameter.
1 n(n 1)
χ (s,b;pc)= 1+η (s) − χ (s,b;pc)n.
B. Double Partonic Correlations in Multiple 2n t n!(cid:18) 4 2 (cid:19) 2 t
Collisions (38)
Theeffectofdoublecorrelationsinann-collisionevent Using Eq. (38) in Eq. (24) and summing over all n pro-
may now be organized in a very convenient way. We duces an analytic expression for the hard dijet contribu-
start by looking at how Γinel (s,b;pc) is modified by tion to the total inelastic profile function,
dijets t
the inclusion of double correlations. In Eq. (31), η (s)
4
parametrizes the deviation of χ4(s,b) from the uncorre- Γinel(s,b;pc)=
jets t
lated case, η (s) = 0. It represents a correction to the
4 ∞ ( 1)n−1 n(n 1)
assumption in Eq. (2) that the integrated double parton − 1+η (s) − χ (s,b;pc)n
PDF is simply a product of the standard PDFs. The n! (cid:18) 4 2 (cid:19) 2 t
nX=1
additional term proportional to η (s) is represented by
4 =1 exp[ χ (s,b;pc)]
Fig.4. Thezigzaglineconnectingthetwohardcollisions − − 2 t −
may be thought of loosely as representing the effect of η4(s)χ (s,b;pc)2 exp[ χ (s,b;pc)] . (39)
summing all soft gluons exchanged between the nearly − 2 2 t − 2 t
parallel incoming and outgoing partons. We call η (s)
4
the double correlation correction factor. Note that this equation respects the basic requirements
Next, we reconsider the uncorrelated description of ofEqs.(28,29)aslongasη4(s)>0. Ifη4 isallowedtobe
triple parton scattering, graphically represented by the lessthanzero,thenthereisapotentialforEq.(39)tobe
third term in Fig. 3. With no zigzag lines, we get the greater than one for some intermediate impact parame-
naive uncorrelated contribution from Eq. (26) ters. Thethirdlineisthe standardunitarizedexpression
Eq.(27)fortheprofilefunction,familiarfromtheeikonal
1 model, while the last line is a correction due to double
χ (s,b;pc)= χ (s,b;pc)3. (35)
6 t 6 2 t correlations. Thedoublecorrelationcorrectiontermcon-
tributesapowerofχ (s,b)2 inaseriesexpansioninsmall
2
For each pair of incoming partons there is another dou- χ (s,b;pc). Therefore, it will only become important at
2 t
ble correlation correction. In other words, for each pair impact parameters which are small enough that terms
of colliding partons there is another replacement like proportional to χ (s,b;pc)2 are non-negligible.
2 t
8
 
1
 + + 
6 
 
 
 
 
 
 
FIG. 5: Graphical representation of theextra contribution to then=3 term of Eq. (27) dueto doublepartonic correlations.
C. Higher Correlations η (s)foreachnumbernofcollisions. Intheseriesrepre-
2n
sentationforΓinel(s,b;pc),acorrelationcorrectionfactor
jets t
In the last section, we assumed that only a single pair η2j(s) is accompanied by powers χ2(s,b;pct)j or higher.
of partons can become correlated. This is reasonable if Equation (41) always satisfies the basic conditions
the goal is simply to account for double correlations at Eqs. (28,29) of a unitarized profile function. [Although,
large or intermediate impact parameters where the con- depending on the signs of the η (s), it may need to be
2n
tribution from double collisions [order χ (s,b)2] is sig- checked that the profile function does not exceed unity
2
nificant, while contributions from triple collisions [order for some intermediate value of b.] Contributions from
χ (s,b)3]andhigherarenegligible. Theexplicitsumover higher n correlation corrections are suppressed by fac-
2
allcollisionsinEq.(39)is neededtoproduce ananalytic tors of (χ (s,b))n/n! and can be neglected so long as
2
expressionforthecorrectedinelasticprofilefunctionthat χ (s,b) is sufficiently smaller than 1. By truncating the
2
still satisfies Eqs. (28,29) and is less than unity for all b. series at larger n, we obtain an increasingly refined de-
In reality, there are of course corrections from triple scription of the b tail at moderate to large b. By using
and higher correlations; in our graphical representation, models of multiple collisions to obtain the η (s), or di-
2n
triple correlations are represented by zigzag lines con- rectly parametrizing the size of correlation corrections
necting three ofthe interactionpoints —see Fig.6. The from experimental data, it should therefore be possible
contribution from triple correlations becomes important to reconstruct an inelastic profile function that respects
only at order χ (s,b;pc)3. If powers of χ (s,b;pc)3 are Eq. (25).
2 t 2 t
significant, then we can iterate the steps of Sec. VB by Unfortunately, there is as yet very little direct exper-
replacing Eq. (37) with imental knowledge of η (s) for n > 2. However, in the
2n
next section we will argue that even when only double
1
χ (s,b;pc) (1+3η (s)+η (s))χ (s,b;pc)3, (40) correlations are included, the corrections are important
6 t → 6 4 6 2 t
at moderate to large impact parameters. Once data are
exactly analogous to Eq. (31) for double correlations. available, steps analogous to those in Sec. VA can be
The η6(s) parametrizes the correction from triple cor- used to parametrize η6(s). As in Ref. [6], we can define
relations in triple and higher partonic collisions. the triple effective cross section,
Takingintoaccountthecontributionfromtriplecorre-
lations to n>3 collisions,andincluding the appropriate σT 2 = 1σ2inc(s;pct)3. (42)
combinatorialfactors by counting all ways of connecting eff 6 σinc(s;pc)
(cid:0) (cid:1) 6 t
three hard collisions, we then recover Eq. (39), but with
a triple correlationcorrection term equal to Then, including up to triple correlations, we have
η66(s)χ2(s,b;pct)3 exp[−χ2(s,b;pct)]. σeTff 2 = (1+3η (s)+η (s)1) d2bP (s,b;pc)3. (43)
(cid:0) (cid:1) 4 6 2 t
Now it is a simple matter to generalize the steps from R
Sec.VBtothearbitrarycaseofncorrelationcorrections. FromEq.(43),wecancalculatethecorrectionfromdou-
The resulting general expression for the inelastic profile ble correlationsto σT with triple correlationsneglected.
eff
function is In the next section, we will find values of 1.3 or 2.1 for
η (s) at currently accessible energies and pc 2.5 GeV.
Γinel(s,b;pc)= T4hesegiveσT =12.8mband10.5mbrespetct≈ively,com-
jets t eff
pared with σT = 28.3 mb for the case with no double
=1 exp[ χ (s,b;pc)] eff
− − 2 t − correlations.
∞ (−1)nη2n(s)χ (s,b;pc)n exp[ χ (s,b;pc)]. We end this section by pointing out that, in princi-
− n! 2 t − 2 t ple, the steps leading to Eqs. (39,41) remain valid if we
nX=2
allow the η (s) to have impact parameter dependence.
(41) 2n
The hypothesis of impact parameter independent corre-
The series after the first line includes all correlationcor- lations is only needed if we wish to estimate the size of
rections. There is a new correlation correction factor correlations by using Eq. (34).
9
FIG. 6: Graphical illustration of triple correlations in triple parton scattering.
VI. NUMERICAL ESTIMATES rateofgrowthofthe radiusatsmallxandfixedp isnot
t
currently well established but will likely become clearer
A. Total Cross Sections asnew data become available. Forthe xdependence, we
will use the parametrization in Ref. [17].
Any numerical results that we obtain using Eq. (41) The 34 mb calculation obtained with the uncorrelated
should be compared with common extrapolations of the approximation should be compared with the measured
total inelastic profile function to high energies so that value of 14.5 mb [1] from the CDF collaboration taken
consistency with Eq. (25) can be verified. A standard at a center-of-mass energy of √s = 1.8 GeV. The un-
parametrization of the total profile function takes the correlatedcalculationisroughlyafactorof2.3toolarge,
form implyingthatitisunsafetoneglectcorrectionsfromcor-
relations. At a minimum one shouldkeepthe η (s) term
4
σ (s) b2
tot in Eq. (39) with a correlation correction factor η 1.3.
Γ(s,b)= exp (44) 4
≈
4πB(s) (cid:26)−2B(s)(cid:27)
It was argued in Ref. [36] that the analysis in [1] ac-
with B(s) B + α′lns. Regge theory fits give α tually overestimates σeff. If three-jet events are taken
0
0.25GeV−2≈for the rate ofgrowthofB(s). For the LH≈C into account (to make the cross section truly inclusive),
then a new estimate is σ 11 mb. This suggests
energy of √s = 14 TeV, a survey of common models eff ≈
that the correlation correction is closer to η 2.1.
and extrapolations in the literature [29–33] suggests the 4 ≈
More recent measurements from the D0/ collaboration
following as a range of reasonable parameters:
find σ =15.1 mb [5], without cuts on three-jet events.
eff
90mb . σtot(√s=14TeV). 120mb (45) So the precise size of σeff remains unclear. We remark
thatσ maydependonpc,whichmayleadtodifferences
19GeV−2 . B(√s=14TeV). 23GeV−2. (46) eff t
in measured values [22].
For example, in Ref. [34] it is found that Eq. (44) with The CDF measurements in [1] find that correlations
Γ(b = 0) = 1 and B = 21.8 GeV−2 is in very close depend weakly on x, suggesting that η4 may be roughly
agreementwiththeReggeparametrizationofRef.[30]as constant with energy. Therefore, we will test the effect
well as with the non-Gaussian model of Ref. [35]. [Some of using 1.3 . η4(√s = 14TeV) . 2.1 in the calculation
fitsputthemaximumfromthetotaluppererrorbandfor of the inelastic profile function using Eq. (39). Plots of
σ (√s = 14TeV) at around 130 mb. However, a total Eq. (39) are shown as dotted curves in Figs. 7(a-b). In
tot
cross sectionthis large wouldalso require a very large B thesecalculationsweallowmg tovaryslowlywithxand
to avoid having a profile function greater than unity at pc in accordance with the parametrization in Ref. [17].
t
small b.] Thetotalinclusivedijetcrosssectionisthesameaswhat
is used in [18], based on the CTEQ6M gluon distribu-
tion function [37] with a K factor of 1.5. With next-to-
B. Correlated vs uncorrelated partons leading-orderPDFs being used, the K factor is closer to
1.2. However the dominant contribution to final states
To calculate σ within the uncorrelated assumption, typically involves at least three jets, corresponding to
eff
we use Eq. (33), and obtain χ (s,b;pc) from the two- K = 1.5 for our calculation [38]. See also the discussion
2 t
gluon form factor, as in Eq. (19). We use m 1 GeV in Ref. [18].
g
≈
which works well for 0.03 x 0.05 and large p , InFig.7(a),wehaveusedη 1.3whileinFig.7(b)we
t 4
≤ ≤ ≈
relevant for most Tevatron data. More data on J/ψ have used η 2.1. In addition, we have tested the
4
≈
electro(photo)-production and deeply virtual Compton sensitivity to higher correlations by including terms up
scattering in this range of kinematics would be very de- to n = 4 in Eq. (41), and using the approximation η
8
≈
sirableforimprovingtheaccuracyofthedeterminationof η η η. The resulting curves are shown as dashed
6 4
the bdependence ofquarkandgluonGPDsatx 10−2. line≈s in ≡Figs. 7(a-b). The suppression at large b from
∼
The value of σ obtainedfromEq.(33) is then about double correlations could in principle be spoiled if the
eff
34 mb. At small x, the width of the χ (s,b;pc) grows triplecorrelationislargeandpositive,sowehavechecked
2 t
and results in an even larger value for σ . The precise thecasewhereη >0. Then,toavoidthepossibilitythat
eff 6
10
Γ>1 at very small b, η is made positive. tionEq.(16)obtainedfromthe two-gluonformfactorat
8
For comparison, the completely uncorrelated eikonal- √s = 14 TeV, and the overlap function from Ref. [33],
type expression, Eq. (27), is plotted as a dash-dotted also at √s=14 TeV [45]. A comparison of these curves
curve in Figs. 7(a-b). Note that there is a substantial illustratesthatmodels ofthe overlapfunctiontendto be
difference between the correlated curves and the uncor- muchnarrowerthanwhatisexpectedfromthetwo-gluon
related expression for 0.8 fm. b . 2 fm in both plots. form factor.
In all cases, the correlations result in a suppression of Because a and η are fixed by the measured value of
1 2
the totalinelasticcrosssectionfromdijets [calculatedby σ ,Eq.(47)yieldsbyconstructionthesamerateofdou-
eff
integrating Γinel (s,b) over b] by more than 15%. The ble collisions as Eq. (39). For triple and quadruple colli-
dijets
shadedregionsmarktheareacoveredbythestandardex- sionsthetwoapproachesgiveroughlysimilarrates(same
trapolations of the total inelastic profile function. They orders of magnitude). Therefore, in practical situations,
correspond to Eq. (44) with the range of parameters in usingEq.(47)withanarrowpeakmaybeaneconomical
Eqs. (45, 46). The p cutoff in all cases is fixed at the way to model the effects of correlations. However, our
t
typical value of pc 2.5 GeV. basic aim in this paper is to incorporate the maximum
t ≈
The uncorrelated curve lies entirely above the shaded amount of available experimental input into the descrip-
area for b . 1.6 fm, in violation of Eq. (25). That is, tionofhardcollisionsbyusing the factorizationtheorem
the hard contribution to the total inelastic cross section andparametrizationsoftheGPDtodescribetheoverlap
is larger than the total inelastic cross section itself for function. Using the overlap function obtained directly
much of the essential range of impact parameters. The fromthe t dependence ofthe J/ψ photoproductioncross
curves that include double or quadratic correlations ex- section requires either that a larger transverse momen-
hibit greater consistency for the full range of b for both tum cutoff (pc &3.5 GeV) is used, or that double corre-
t
Figs. 7(a) and 7(b). In the case of the moderate sized lations are incorporated by using Eq. (39) with η >0.
2n
correlationcorrectionsinFig.7(a),theeffectofincluding Otherwise, there are potential problems with the consis-
triple and higher correlation corrections is rather small tency relation Eq. (25), even for a relatively large b.
comparedwiththecasewhereonlycorrectionsfromdou- One source of uncertainty is the shape of the overlap
ble correlations are kept. Including higher correlation function P (s,b;pc). A Gaussianform, forinstance, may
2 t
corrections does seem to smooth out the shape of the be preferred to Eq. (16). Therefore, we have repeated
profile function. (We have also assumed all correlation the calculation of Fig. 7, but now with
corrections to be positive.) However, the higher correc-
tion terms only become significant at small impact pa- 1 b2
P (s,b;pc)= exp − . (48)
rameters where the profile function is already close to 2 t 2πb (cid:20)2b (cid:21)
0 0
unityanyway. Ifthecorrelationcorrectionsarelarger,as
in Fig. 7(b), then the higher n>2 correlationsare more The parameter b0 is fixed by requiring that the average
significant. b2,
Now let us consider what is needed for the radius of
tinheEhqa.r(d41o)v(erreldapucfiunngcttoiotnheifsatlalntdhaerdη2eni(kso)naarlefosremtutolaz).erIof hb2i=Z d2bb2P2(s,b;pct), (49)
a smallvalue ofpc is used to evaluate σinc (s;pc), then
t pQCD t isthesameforbothEq.(48)andEq.(16). Theresulting
fitsofthetotalcrosssectiontocurrentdatarequireavery
plots are shown in Fig. 8. The drop with b is slightly
narrow width for the overlap function [12]. In theoret-
steeper at intermediate b in Fig. 7, but otherwise the
ical calculations, a narrow overlap function is obtained,
plots are very similar. We also point out that a recent
for example, in the semiperturbative approach proposed
experimental study [39] finds good agreementbetween ρ
in [33] where the radius of the hard overlapfunction de-
andφelectroproductiondataandthedipoleformforthe
creaseswith energy. In Pythia the hardoverlapfunction
two-gluon form factor.
is modeled by the double Gaussian parametrization[8]
Anothersourceofuncertaintyisthe contributionfrom
diffractiontotheinelasticcrosssection,whichisexpected
(1 β)2 b2
P (b)= − exp − + tobemuchmoreperipheralthangenericinelasticinterac-
2 2a21 (cid:26)2a21(cid:27) tions. Thisisknownalreadyfromanalysesofthediffrac-
2β(1 β) b2 β2 b2 tive processes at lower energies [40] and should be even
+ − exp − + exp − , (47)
a2+a2 (cid:26)a2+a2(cid:27) 2a2 (cid:26)2a2(cid:27) more prominent at √s= 2TeV and above where inelas-
1 2 1 2 2 2
tic diffraction cannot occur at small impact parameters,
with a =0.4a and β =0.5 (in Tune A). The radius in and where the interaction is practically black. Inelastic
2 1
Eq. (47) does not vary with energy. We determine a by diffractionconstitutes asignificantfractionofthe inelas-
1
usingEq.(47)inEq.(33)forσ withuncorrelatedmul- tic cross section at √s = 2TeV, 25% - 30 %, and is
eff
tiple hard scattering and fixing it to the measured CDF expected to remain significant at the LHC.
value. In Fig. 9 we show Eq. (47) with a calculated us- Hence, in the region where we use the consistency re-
1
ingσ =14.5mb(Ref.[1])andσ =11mb(Ref.[36]). quirement, Eq. (25), a large fraction of Γinel is due to
eff eff
For comparison we have also plotted the overlap func- inelasticdiffraction. Atthesametimethe Tevatrondata