diffractive production: The nature of the Pomeron and a test of hard diffractive factorization

$: diffractive production: The nature of the Pomeron and a test of hard diffractive factorization$
PHYSICAL REVIEW D, VOLUME 59, 014028

AssociatedJ/c1g diffractive production: The nature of the Pomeronand a test of hard diffractive factorization

Jia-Sheng XuDepartment of Physics, Peking University, Beijing 100871, China

Hong-An PengChina Center of Advance Science and Technology (World Laboratory), Beijing 100080, China

and Department of Physics, Peking University, Beijing 100871, China~Received 2 July 1998; published 7 December 1998!

We present a study of diffractive associatedJ/c1g production at energies reached at the Fermilab Tevatronand CERN LHC based on the Ingelman-Schlein model for hard diffractive scattering and the factorizationformalism of nonrelativistic QCD for quarkonia production. We find that this process (p1 p→p1J/c1g1X) can be used to probe the gluon content of the Pomeron and test the assumption of diffractive hardscattering factorization. Using the renormalized Pomeron flux factorD.0.11(0.052) at Tevatron~LHC! en-ergy, the single diffractive associatedJ/c1g production cross section in the region of 4,PT,10 GeV,21,h,1 is found to be of the order of 3.0 pb~8.5 pb!. The ratio of single diffractive to inclusive productionis 0.50% ~0.15%! in the central region at the Tevatron~LHC! for the gluon fraction in the Pomeronf g

50.7, independent of the values of the color-octet matrix elements.@S0556-2821~99!00501-9#

PACS number~s!: 12.40.Nn, 13.85.Ni, 14.40.Gx

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I. INTRODUCTION

In the early 1960s it was realized that in high-enerstrong interactions the Regge trajectory with a vacuum qutum number, the Pomeron, plays a particular and veryportant role in soft processes, such as the energy dependof s tot(s), the behavior of the elastic differential cross sectidsel/dt at small utu, and the single and double diffractivdissociation processes in hadron-hadron collisions@1#. Afterthe emergence of QCD, physicists began to study the naof the Pomeron in the framework of QCD@2#. Since thePomeron carries the quantum number of the vacuumterms of QCD, it is a colorless entity, which leads to tdiffractive events which are characterized by a large rapidgap, a region in rapidity devoid of hadronic energy floThese distinct classes of events are observed by the Zand H1 Collaborations at the DESYep collider HERA in thedeep inelastic scattering region@3#, which offers a uniquechance to study the soft scattering process with a hard virphoton probe.

Ingelman and Schlein@4# pointed out that hard diffractivescattering processes would give new and valuable insabout the nature of the Pomeron. They assumed thatPomeron, similar to the nucleon, is composed of partomainly of gluons, and the hard diffractive scattering prcesses can be calculated in a factorized way: first,Pomeron is emitted from the diffractively scattered hadrthen one parton of the Pomeron takes part in hard subcesses. The results will be diffractively produced high-PT

jets inP2P( P),g2P collisions or large rapidity gap eventin diffractive deep inelastic scattering~DDIS!; therefore, thepartonic structure of the Pomeron could be establishedstudied experimentally. These predictions were confirmsubsequently. The UA8 Collaboration at the CERN SuProton Synchrotron~S ppS! collider with As5630 GeV

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have studied the diffractive jet distribution; they find a domnant hard partonic structure of the Pomeron; however,diffractive dijet event topology alone cannot distinguish btween a hard-quark or a hard-gluon structure in the Pome@5#. The DDIS and dijet photoproduction experimentsHERA have shed light on the partonic structure of tPomeron. Combining the measurements of the diffractstructure function in DDIS and the photoproduction jet crosections, the ZEUS Collaboration gave the first experimeevidence for the gluon content of the Pomeron and demined that the hard gluon fraction of the Pomeronf g is in therange 0.3, f g,0.8, independent of the validity of the momentum sum rule for the Pomeron and the normalizationthe flux of the Pomeron from the proton@6#. The H1 Col-laboration also determined the fraction of the momentumthe Pomeron carried by the hard gluon, which isf g;0.9 atQ254.5 GeV2 and f g;0.8 at Q2575 GeV2 @7#. The par-tonic structure of the Pomeron, based on the IngelmSchlein~IS! model for hard diffraction, has been studied bthe Collider Detector at Fermilab~CDF! Collaboration at theTevatron through diffractiveW production and dijet production @8,9#, which gives further evidence for the hard partonstructure of the Pomeron. Combining these two experimedata, the CDF Collaboration determined the hard-gluon ctent of the Pomeron to bef g50.760.2; this result is alsoindependent of Pomeron flux normalization or the validitythe momentum sum rule for the Pomeron.

However, there is an important problem, the factorizatproblem. The IS model is based on the assumption offractive hard scattering factorization. Recently, a factorition theorem has been proven by Collins@10# for the leptoninduced diffractive hard scattering processes, such as Dand diffractive direct photoproduction of jets. This factoriztion theorem justifies the analysis given by the ZEUS andCollaborations, and establishes the universality of the

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JIA-SHENG XU AND HONG-AN PENG PHYSICAL REVIEW D59 014028

fractive parton distributions for those processes to whichtheorem applies. In contrast, no factorization theorembeen established for diffractive hard scattering in hadrhadron collisions. As was known even before the advenQCD, factorization fails for hard processes in diffractihadron-hadron scattering@11#. Furthermore, in order to preserve the shapes of theM2 and t distributions in soft singlediffraction ~SD! and predict correctly the experimentally oserved SD cross section at all energies inP2 P collisions,Goulianos@12# proposed to renormalize the Pomeron fluxan energy-dependent way; this approach indicates the brdown of the triple-Regge theory for soft single diffractivexcitation, and implies that diffractive hard factorizationlikely to breakdown in hadron-hadron collisions which apears to be confirmed by CDF experiments@8,9#. At largeutu, where perturbative QCD applies to the Pomeron, itbeen proven that there is a leading twist contribution brothe factorization theorem for diffractive hard scatteringhadron-hadron collisions@13#. Evidence for this substantiacoherent perturbative contribution has been observed byUA8 experiment in diffractive jet production, in which thsquare of the proton’s four-momentum transfert is in theregion 22,t,21 GeV2 @5#. On the other hand, for thePomeron at smallutu, nonperturbative QCD dominatewhere the IS model would be appropriate. Therefore, varihard diffractive processes may be used to probe the partstructure of the Pomeron, and test the hard diffractive facization in hadron-hadron collisions@14,15#. Recently, Aleroet al. @16# extracted the parton densities of the Pomeron frthe HERA data on DDIS and diffractive photoproductionjets, and used the fitted parton densities to predict the difftive production of jets and weak bosons inPP collisions atthe Tevatron. The predicted cross sections are several thigher than the experimental data, which signals a bredown of hard scattering factorization in diffractive hadrohadron collisions.

In this paper, we will discuss another diffractive procethe associatedJ/c1g single diffractive production at largePT :

P1 P→P1J/c1g1X. ~1!

This process is of special interest because the largePT J/cproduced is easy to detect through its leptonic decay moand theJ/c ’s large PT is balanced by the associated higenergy photon. We will see through the following calcutions that although the production cross section of the aciatedJ/c1g is sensitive to the color-octet matrix elementhe ratio of the single diffractive production cross sectionthe inclusive production cross section is not so. We find tthe ratio is sensitive to the productD f g , where D is therenormalization factor of the Pomeron flux~which indicatesthe factorization broken effects! and f g is the hard gluonfraction in the Pomeron. The measurement of this procesthe Tevatron and the CERN Large Hadron Collider~LHC!would shed light on the nature of the Pomeron and testdiffractive hard scattering factorization. Furthermore, tprocess is also of interest for the study of heavy quarkonproduction mechanism.

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Our paper is organized as follows. We describe our cculation scheme in detail in Sec. II, in which, a brief intrduction of the heavy quarkonium production mechanism,hard subprocesses for associatedJ/c1g production, and asummary of the kinematics related to theJ/c PT distributionare included. Our results and discussions are given in SIII.

II. CALCULATING SCHEME

Based on the IS model for diffractive hard scattering, tassociatedJ/c1g single diffractive production process alarge PT consists of three steps~shown in Fig. 1!. First, thePomeron is emitted from the proton with a small squafour-momentum transferutu. Second, partons from the antproton and Pomeron scatter in the hard subprocessesproduce almost pointlikecc pair with largePT and an asso-ciated photon. In the third step,J/c is produced from thepointlike cc pair via soft interactions.

A. Heavy quarkonium production mechanism

Prior to 1993, the conventional wisdom for heavy quarknium production was based on the so-called color-singmodel ~CSM! @17# which assumes that the heavy quark pis produced in a color-singlet state with the right quantunumbers of the final heavy quarkonium in the hard subpcesses and on the belief that the dominant processes of csinglet cc production are at lowest order inas . However,data collected by the UA1 Collaboration@18# already con-tained indications of deviations from CSM predictions fcharmonium production. At the Tevatron, the data of promand directJ/c production are higher than expectations froCSM at largePT . The color-singlet model underestimatedirect J/c production by a factor varying from 80 to 30 fo5 GeV,PT,18 GeV @19#. For c8, the prompt data aremore than an order of magnitude higher~approximately afactor 50! than the CSM prediction@19#. The first majorconceptual advance in heavy quarkonium production wasrealization that fragmentation dominates at sufficiently laPT @20# which indicates that most charmoniums at largePT

FIG. 1. Sketch diagram for diffractive associatedJ/c1g pro-duction in the Ingelman-Schlein model for diffractive hard scatting.

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ASSOCIATEDJ/c1g DIFFRACTIVE PRODUCTION: . . . PHYSICAL REVIEW D 59 014028

are produced by the fragmentation of individual largePT

partons. The fragmentation functions are calculated in CSfor example, the fragmentation function forg→J/c is cal-culated from the parton processg→cc1gg in CSM. Includ-ing this fragmentation mechanism indeed brings the theoical predictions for promptJ/c production at the Tevatron towithin a factor of 3 of the data@21#. But the prediction forthec8 production cross section remains a factor of 30 belthe data even after including the fragmentation contribut~the c8 ‘‘surplus’’ problem!. Furthermore, the presencethe logarithmic infrared divergences in the production crsections forP-wave charmonium states and the annihilatirate forxcJ→qqg indicate that the CSM is incomplete. Athese indicate that an important production mechanismyond CSM needs to be included@22,23#. So the color-octetmechanism@24# is proposed which is based on the factoriztion formalism of nonrelativistic quantum chromodynami~NRQCD! @25,26#. Contrary to the basic assumptionCSM, the heavy quark pair in a color-octet state can bindform heavy quarkonium.

Although the color-octet fragmentation picture of heaquarkonium production@23# has provided valuable insighthe approximations that enter into fragmentation computions break down when a quarkonium’s energy becomcomparable to its mass. The fragmentation predictionscharmonium production are therefore unrealiable at lowPT .In the case ofY production, the data at the Tevatron arethe PT,15 GeV region, which significantly disagree witthe fragmentation predictions. Based upon the severalcently developed ideas in heavy quarkonium physics aboCho and Leibovich@27# identified a large class of color-octediagrams that mediate quarkonia production at all energwhich reduce to the dominant set of gluon fragmentatgraphs in the highPT limit. By fitting the data of promptJ/cand Y production at the Tevatron, numerical values for tlong distance matrix elements were extracted, which are gerally consistent with NRQCD power scaling rules@28#. Inorder to convincingly establish the color-octet machanismis important to test whether the same matrix elementsable to explain heavy quarkonim production in other higenergy processes, such as inclusiveJ/c production ine1e2

annihilation on theZ0 resonance and inelasticJ/c photopro-duction at HERA.

B. The hard subprocesses for associatedJ/c1g production

J/c is described within the NRQCD framework in termof Fock state decompositions as

uJ/c&5O~1!ucc@3S1~1!#&1O~v !ucc@3PJ

~8!#g&

1O~v2!ucc@1S0~8!#g&1O~v2!ucc@3S1

~1,8!#gg&

1O~v2!ucc@3PJ~1,8!#gg&1¯ , ~2!

where thecc pairs are indicated within the square bracketsspectroscopic notation. The pairs’ color states are indicaby singlet ~1! or octet ~8! superscripts. The color octetcc

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state can make a transition into a physicalJ/c state by softchromoelectric dipole (E1) transition~s! or chromomagneticdipole M1 transition~s!:

~cc!@2S11L j~8!#→J/c. ~3!

The NRQCD factorization scheme@25# has been establisheto systematically separate high- and low-energy scale inactions. It is based upon a bouble power series expansiothe strong interaction fine structure constantas and the smallvelocity parameterv. The production of a (cc)@2S11L j

(1,8)#pair with separation less than or of order 1/mc can becalculated perturbatively. The long-distance effectsthe produced almost pointlikecc to form the bound stateare isolated in nonperturbative matrix elements. Furthmore, NRQCD power counting rules can be exploited totermine the dominant contributions to various quarkoniuprocesses@28#. For direct J/c production, the color-octematrix elements ^0uO8

J/c@3S1#u0&, ^0uO8J/c@1S0#u0&, and

^0uO8J/c@3PJ#u0& are all scaling asmc

3vc7 . So these color-

octet contributions toJ/c production must be included foconsistency.

On the partonic level, the associatedJ/c1g production iscomposed of gluon fusion, which is sketched in Fig. 2. Tis

g1g→g1~cc!@3S1~1! ,3S1

~8!#,

g1g→g1~cc!@1S0~8! ,3PJ

~8!#. ~4!

The quark initiated subprocesses (qq channel! are stronglysuppressed and will be neglected. The color-singlet glugluon fusion contribution to associatedJ/c1g production iswell known @29#:

ds

d t~singlet!

5N1

16p s2 F s2~ s24mc2!21 t2~ t24mc

2!21u2~ u24mc2!2

~ s24mc2!2~ t24mc

2!2~ u24mc2!2 G ,

~5!

where

s5~p11p2!2, t5~p22P!2,u5~p12P!2. ~6!

The overall normalizationN1 is defined as

FIG. 2. The subprocessg11g2→g1cc@2S11LJ(1,8)#→g1J/c

in the NRQCD framework.

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3G1~J/c!, ~7!

where

G1~J/c!5^0uO1

J/c@3S1#u0&

3mc2 5

G~J/c→ j 1l 2!

~2/3!pec2a2 ~8!

andec25 2

3 .The average-squared amplitude of the subprocessg1g

→g1(cc)@3S1(8)# can be obtained from the average-squa

amplitude ofg1g→g1(cc)@3S1(1)# by taking into account

the different color factor. The result is

ds

d t@g1g→g1~cc!@3S1

~8!#→g1J/c#

51

16p s2

15

6SuM „g1g→g1~cc!@3S1

~1!#…u2

31

24mc^0uO8

J/c@3S1#u0&

51

16p s2

15

6

1

8

1

1264gs

4ec2e2mc

2

3F s2~ s24mc2!21 t2~ t24mc

2!21u2~ u24mc2!2

~ s24mc2!2~ t24mc

2!2~ u24mc2!2 G

31

24mc^0uO8

J/c@3S1#u0&. ~9!

The average-squared amplitudes of the subprocessesg1g→g1(cc)@1S0

(8)# andg1g→g1(cc)@3PJ(8)# can be found

in Ref. @30#:

ds

d t@g1g→g1~cc!@1S0

~8!#→g1J/c#

51

16p s2 SuM „g1g→g1~cc!@1S0~8!#…u2

31

8mc^0uO8

J/c@1S0#u0&,

ds

d t@g1g→g1~cc!@3PJ

~8!#→g1J/c#

51

16p s2 (J

SuM „g1g→g1~cc!@3PJ~8!#…u2

31

8mc

^0uO8J/c@3P0#u0&, ~10!

where the heavy quark spin symmetry

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is exploited.

C. The PT distribution of J/c

Now we consider thePT distribution ofJ/c produced inprocess

p~Pp!1 p~Pp!→p~Pp8!1J/c~P!1g~k!1X. ~12!

Based on the IS model for diffractive hard scattering, tdifferential cross section can be expressed as

ds5 f IP/p~j,t ! f g/IP~x1 ,Q2!

3 f g/ p~x2 ,Q2!ds

d tdjdtdx1dx2d t, ~13!

wherej is the momentum fraction of the proton carried bthe Pomeron andt5(Pp2Pp8)

2 is the square of the proton’four-momentum transfer.f IP/p(j,t) is the Pomeron flux fac-tor

f IP/p~j,t !5d2sSD/djdt

sTIPP~s8,t !

5b1

2~0!

16pj122a~ t !F2~ t !

5Kj122a~ t !F2~ t !, ~14!

where the parameters are chosen as@12#

K50.73 GeV2, a~ t !5110.11510.26 ~GeV22!t,

F2~ t !5e4.6t. ~15!

In the following calculation, we use the renormalized flufactor for the Pomeron, proposed by Goulianos@12# in orderto preserve the shapes of theM2 and t distribution in softsingle diffraction~SD! and predict correctly the observed Scross section at all energies inpp collisions:

f IP/pRN ~j,t !5D f IP/p~j,t !, ~16!

the renormalization factorD is defined as

D5minS 1,1

ND ~17!

with

N5Ejmin

jmaxdjE

2`

0

dt f IP/p~j,t !, ~18!

wherejmin5M02/s with M0

251.5 GeV2 ~effective threshold!and jmax50.1 ~coherence limit!. At the Tevatron energy(As51800 GeV),D5 1

9 .We now consider the kinematics. In thepp c.m. frame,

we can express the momenta of the incidentp,p and gluons,etc., as

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Pp5As

2~1,1,0W !,

Pp5As

2~1,21,0W !,

PIP5As

2j~1,1,0W !,

p15As

2x1j~1,1,0W !5x1PIP ,

p25As

2x2~1,21,0W !5x2Pp ,

~19!

where the first component is the energy, the second is lotudinal momentum, and the third is the transverse componof the four-momentum andx1 ,x2 are the momentum fractions of the gluons. The momenta of the outgoingJ/c aregiven by

P5~E,PL ,PW T!5~E,PT sinh h,PW T!, ~20!

wherePT is the transverse momentum ofJ/c, h is the pseu-dorapidity ofJ/c andE5Amc

21PT2 cosh2 h.

The Mandelstam variables are

s5~Pp1Pp!2,

s5~p11p2!25x1x2js,

t5~p22P!25mc22Asx2~E1PTsinh h!,

u5~p12P!25mc22Asx1j~E2PTsinh h!.

~21!

Using s1 t1u5mc2 , we have

x25Asjx1~E2PTsinh h!2mc

2

x1js2As~E1PTsinh h!. ~22!

In order to obtain the distribution in the transverse momtum PT for the process Eq.~12!, we express the differentiacross section as

ds5 f IP/pRN ~j,t ! f g/IP~x1 ,Q2! f g/ p~x2 ,Q2!

3ds

d tJS x1x2 t

x1hPTD djdtdx1dhdPT , ~23!

where the Jacobian can obtain from Eqs.~21! and ~22!,

JS x1x2 t

x1hPTD 5

2sx1x2jPT2coshh

E@x1js2As~E1PTsinh h!#. ~24!

Then thePT distribution ofJ/c is expressed as

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dPT

5Ehmin

hmaxdhE

jdw

jupdjE

x1 min

x1 maxdx1E

21

0

dt f IP/pRN ~j,t !

3 f g/IP~x1 ,Q2! f g/ p~x2 ,Q2!JS x1x2 t

x1hPTD ds

d t,

~25!

where the allowed regions ofx1 ,j are given by

x1 min5E1PTsinh h2mc

2/As

j~As2E1PTsinh h!,

jdw5E1PTsinh h2mc

2/As

As2E1PTsinh h. ~26!

In order to suppress the Reggon contributions, we setjup50.05 as usual.

III. NUMERICAL RESULTS AND DISCUSSIONS

For numerical predictions, we usemc51.5 GeV, L45235 MeV, and set the factorization scale and the renormization scale both equal to the transverse mass ofJ/c, i.e.,Q25mT

25(mc21PT

2). For the color-octet matrix element^0uO8

J/c@3S1#u0&, ^0uO8J/c@1S0#u0&, and^0uO8

J/c@3P0#u0& weuse the values determined by Beneke and Kra¨mer @31# fromfitting the directJ/c production data at the Tevatron@32#using Gluck-Reya-Vogt~GRV! leading order~LO! partondistribution functions:

^0uO8J/c@3S1#u0&51.1231022 GeV3,

^0uO8J/c@1S0#u0&1

3.5

mc2 ^0uO8

J/c@3P0#u0&53.9031022 GeV3.

~27!

Since the matrix elements ^0uO8J/c@1S0#u0& and

^0uO8J/c@3P0#u0& are not determined separately, we pres

the two extreme values allowed by Eq.~27! as

1S0 saturated case:0uO8J/c@1S0#u0&53.9031022 GeV3,

^0uO8J/c@3P0#u0&50,

3PJ saturated case:0uO8J/c@3P0#u0&51.11

31022mc2 GeV3

^0uO8J/c@1S0#u0&50. ~28!

For the parton distribution functions, we use the GRV Lgluon distribution function for the antiproton@33# and thehard gluon distribution function for the Pomeron@9#:

x fg/IP~x,Q2!5 f g6x~12x!, f g50.760.2. ~29!

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We use the central value off g above for numerical calculation. We neglect anyQ2 evolution of the gluon density of thePomeron at present stage.

With all ingredients set as above, in Fig. 3 we showPT distributionB(ds/dPT) for single diffractive associatedJ/c1g production inpp collisions atAs51.8 TeV, inte-grated over the pseudorapidity region21<h<1 ~the cen-tral region!, whereB50.0594 is theJ/c→m1m2 leptonicdecay branching ratio. The lower solid line is the colosinglet gluon-gluon fusion contribution, while the lowedashed, dotted, and dash-dot-dotted lines are1S0

(8) saturated,3PJ

(8) saturated, and3S1(8) color-octet contributions, respec

tively. For comparison, we also calculate the inclusive asciated J/c1g production p1 p→J/c1g1X in the samekinematic region, the results are shown as the upper lineFig. 3. The code for the lines is the same as the singlefractive production case. As shown in Fig. 3, the color-oc3S1

(8) contribution is strongly suppressed compared withothers over the entirePT region considered in both production cases. The1S0

(8)-saturated and3PJ(8)-saturated contribu-

tions are smaller than the singlet contribution wherePT,5 GeV. Although their contributions dominate in the higPT region, their differential cross section in the highPT re-gion is much smaller than that in lowPT region. So inte-

FIG. 3. Transverse momentum ofJ/c(PT) distributionBds/dPT , integrated over theJ/c pseudorapidity rangeuhu<1~central region!, for single diffractive~lower! and inclusive~upper!associatedJ/c1g production at the Tevatron. HereB is thebranching ratio ofJ/c→m1m2(B50.0594). The solid line is thecolor-singlet gluon-gluon fusion contribution, the dashed line rresents the1S0

(8)-saturated color-octet contribution, the dotted onethe 3PJ

(8)-saturated color-octet contribution, and the dot-dot-dasline is the 3S1

(8) color-octet contribution.

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grated over thePT region, their contributions (Bs50.08 and16 pb for 1S0

(8)-saturated contribution to SD and inclusivproduction, respectively; Bs50.06 and 12 pb for3PJ

(8)-saturated contribution to SD and inclusive productiorespectively! are smaller than the color-singlet contributio@Bs(singlet)50.10 and 20 pb for SD and inclusive prodution, respectively#. The total SD~inclusive! production crosssection atAs51.8 TeV, integrated over 4<PT<10 GeV incentral region is BsSD(cen)50.18 pb @Bs inclusive(cen)536 pb# for 1S0

(8)-saturated case,BsSD(cen)50.16 pb@Bs inclusive(cen)532 pb# for the 3PJ

(8)-saturated case. Thratio of the total SD production cross section to that ofclusive production in the central regionRcen is 0.50% forboth the 1S0

(8)-saturated case and the3PJ(8)-saturated case.

In Fig. 4, we show the PT distribution of J/c,B(ds/dPT), integrated over the pseudorapidity region24<h<22 ~the forward region!, at As51.8 TeV. The linesare the same in Fig. 3. One can find that the differential crsection is significantly smaller than that in central region.the diffractively producedJ/c should be concentrated in thcentral detectors of the collider. The same characterfound earlier by Bergeret al. @14# in the rapidity distributionof SD production of bottom and top quarks. This contrawith the naive expectation that diffractively producted sy

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d

FIG. 4. Transverse momentum ofJ/c(PT) distributionBds/dPT , integrated over theJ/c pseudorapidity range24<h<22 ~forward region!, for single diffractive~lower! and inclusive~upper! associatedJ/c1g production at the Tevatron. HereB is thebranching ratio ofJ/c→m1m2(B50.0594). The solid line is thecolor-singlet gluon-gluon fusion contribution, the dashed line reresents the1S0

(8)-saturated color-octet contribution, the dotted onethe 3PJ

(8)-saturated color-octet contribution, and the dot-dot-dasline is the 3S1


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TABLE I. The ratio R(PT) as a function of PT at the Tevatron in central region (21<h<1) Rcen(PT) and forward region (24<h<22) Rfwd(PT).

PT ~GeV! 4 5 6 7 8 9 10

Rcen(PT)(%) 0.52 0.52 0.50 0.48 0.47 0.46 0.44Rfwd(PT)(%) 0.28 0.28 0.28 0.28 0.29 0.29 0.30

ein

ean

ve

rix

ro

ol-

RA

theard

sssof

ionral

-nts

theline

tem would appear only at large rapidity. The total SD~inclu-sive! production cross section atAs51.8 TeV, integratedover 4<PT<10 GeV in the forward region isBsSD(fwd)55.731022 pb @Bs inclusive(fwd)523 pb# for 1S0

(8)

-saturated case and BsSD(fwd)55.131022 pb@Bs inclusive(fwd)520 pb# for the 3PJ

(8)-saturated case. Thratio of the total SD production cross section to that ofclusive production in the forward regionRfwd are 0.25% forboth the 1S0

(8)-saturated case and3PJ(8)-saturated case.

In Table I, we show the ratio

R~PT!5dsSD

dPTY ds inclusive

dPT~30!

at As51.8 TeV in the central and forward region, where

dsSD

dPT5

dsSD~singlet!

dPT1

dsSD~3S1~8!!

dPT1

dsSD~1S0~8!!

dPT~31!

for the 1S0(8)-saturated case and

dsSD

dPT5

dsSD~singlet!

dPT1

dsSD~3S1~8!!

dPT1

dsSD~3PJ~8!!

dPT~32!

for the 3PJ(8)-saturated case.

The ratioR(PT) for the 1S0(8)-saturated case is the sam

as that for the3PJ(8)-saturated case. From this table, we c

see thatR(PT) is almost constant forPT<6 GeV both in thecentral and forward regions.R(PT) varies slowly asPT in-creases. Because the differential cross section in the highPTregion is much smaller than in the lowPT region, integratedover the overallPT region, the ratiosRcen and Rfwd are in-sensitive to thePT smearing effects. Furthermore, we havaried the color-octet matrix elements0uO8

J/c@3S1#u0&,^0uO8

J/c@1S0#u0&13.5/mc2 ^0uO8

J/c@3P0#u0& and multipliedthem by a factor between110 and 2; the ratiosRcen andRfwd

are unvaried. This character demonstrates that the ratiosRcen

and Rfwd are insensitive to the values of color-octet matelements. The ratiosRcen andRfwd are proportional toD f g ,hence they are sensitive to the gluon fraction of the Pomef g and the renormalization factorD which indicates the fac-torization broken effects. From other single diffractive prduction experiments,f g can be determined, as the CDF Colaboration at Tevotran and the ZEUS Collaboration at HEDESY have done, the renormalization factorD can be deter-mined precisely from those ratios, andvice versa. So mea-

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n

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suring these ratios can probe the gluon density inPomeron and shed light on the test of the diffractive hscattering factoriztion theorem.

In Fig. 5, we show the PT distribution of J/c,B(ds/dPT), integrated over the pseudorapidity region21<h<1 at LHC energyAs514 TeV with the factorD50.052 calculated from Eq.~17!. The lines are the same ain Fig. 3. As expected, the increasing of the differential crosection with c.m. energy is slowed by the renormalizationthe Pomeron flux. The total SD~inclusive! production crosssection atAs514 TeV, integrated over 4<PT<10 GeV inthe central region isBsSD(cen)50.51 pb @Bs inclusive(cen)53.43102 pb# for the 1S0

(8)-saturated case andBsSD(cen)50.46 pb @Bs inclusive(cen)53.03102 pb# for the3PJ

(8)-saturated case. The ratio of the total SD productcross section to that of inclusive production in the cent

FIG. 5. Transverse momentum ofJ/c(PT) distributionBds/dPT , integrated over theJ/c pseudorapidity rangeuhu<1~central region!, for single diffractive~lower! and inclusive~upper!associatedJ/c1g production at LHC. HereB is the branchingratio of J/c→m1m2(B50.0594). The solid line is the colorsinglet gluon-gluon fusion contribution, the dashed line represethe 1S0

(8)-saturated color-octet contribution, the dotted one is3PJ

(8)-saturated color-octet contribution, and the dot-dot-dashedis the 3S1


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e

ct2ay

he

be

wo

if-nth

be

if

za-h

ionthe

o

oandthe

anns.ralda-e


regionRcen is 0.15% for both the1S0(8)-saturated case and th

3PJ(8)-saturated case.In the above calculations, we have set the values of fa

D according to Eq.~17!; it is 19 at Tevatron energy and 0.05

at LHC energy. But the value is not unique, since it mchange with different choices of the parameters such asM0and jmax in Eq. ~18!. If we use the central value ofD50.1860.04 measured by the CDF Collaboration at tTevatron, the predicted ratio in the forward regionRfwd at theTevatron, taking into account the proton or antiproton candiffractively scattered, will be 0.81%, which is close to thmeasured production rate of diffractive dijet events with tjets of ET.20 GeV, 1.8,uhu,3.5, andh1h2.0.

Experimentally, the nondiffractive background to the dfractive associatedJ/c1g production must be dropped iorder to obtain useful information about the nature ofPomeron and the factorization broken effects. This canattained by performing the rapid gap analysis as was donthe CDF diffractive dijet experiment.

In conclusion, in this paper we have shown that the dfractive associatedJ/c1g production at largePT is sensi-

hn-

B

c

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or

e

eein

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tive to the gluon content of the Pomeron and the factorition broken effects in hard diffractive scattering. Althougthe single diffractive and inclusive production cross sectis sensitive to the values of coler-octet matrix elements,ratio of single diffractive to inclusiveJ/c1g production isnot so and proportional toD f g , hence they are sensitive tthe gluon fraction of the Pomeronf g and the renormalizationfactorD which indicates the factorization broken effects. Sexperimental measurement of this ratio at the TevatronLHC can shed light on the nature of the Pomeron and testassumption of diffractive hard scattering factorization.

ACKNOWLEDGMENTS

We thank Professor Kuang-Ta Chao and Dr. Feng Yufor their reading of the manuscript and useful discussioThis work was supported in part by the National NatuScience Foundation of China, the Doctoral Program Fountion of the Institution of Higher Education of China, and thHebei Natural Province Science Foundation, China.

cl.

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diffractive production: The nature of the Pomeron and a test of hard diffractive factorization

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