We discuss the theory of jet events in high-energy photon-proton interactions
using a model which gives a good description of the data available on total
inelastic $\gamma p$ cross sections up to $\sqrt{s}$=210 GeV. We show how to
calculate the jet cross sections and jet multiplicities and give predictions
for these quantities for energies appropriate for experiments at the HERA $ep$
collider and for very high energy cosmic ray observations.Comment: 12 pages + 4 figs, MAD/TH/92-8, submitted to Phys. Rev. D(Rapid
  Communications), figs. available on request from honjo@wishep.bitne

Durand, L.

Gandhi, R.

Honjo, K.

Pi, H.

Sarcevic, I.

Physical Review D

English

arXiv

Durand, Loyal

Honjo, Katsuhiko

Gandhi, Raj

Pi, Hong

Sarcevic, Ina

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-ph/9212297v1  21 Dec 1992MAD/TH/92-8Jets and jet multiplicities in high energy photon-nucleoninteractionsLoyal Durand and Katsuhiko HonjoDepartment of Physics, University of Wisconsin-Madison,Madison, Wisconsin 53706Raj GandhiDepartment of Physics, Texas A&M University,College Station, TX 77843Hong PiDepartment of Theoretical Physics, University of Lund,Lund, Sweden S-22362Ina SarcevicDepartment of Physics, University of Arizona,Tucson, Arizona 85721AbstractWe discuss the theory of jet events in high-energy photon-proton interactions usinga model which gives a good description of the data available on total inelastic γp crosssections up to√s=210 GeV. We show how to calculate the jet cross sections andjet multiplicities and give predictions for these quantities for energies appropriate forexperiments at the HERA ep collider and for very high energy cosmic ray observations.PACS number(s): 13.60.Hb, 13.87.Ce, 12.40.Lk1The hadronic structure of the photon and its interaction with hadrons are cur-rently subjects of considerable interest.The ZEUS and H1 groups at the ep colliderHERA have presented preliminary results [1] on the γp photoproduction cross sectionat 210 GeV center-of-mass energy which show the rise with energy typical of otherhadronic interactions. It was shown by Gandhi and Sarcevic [2] in a simple modelthat this rise can be used to discriminate between different sets of photon structurefunctions. The hadronic behavior of the photon at very high energies (∼ 108 GeVphoton energy) is also of interest for the theory of air showers triggered by cosmicray photons [3].We have developed an improved QCD-based model of the photon interaction withnucleons [4] which gives predictions for the total inelastic γp cross section which agreewell with the HERA data. The model has the interesting feature that the informationneeded to calculate σγpinel is largely determined by high energy πN scattering. Inthis paper, we will use the model to discuss the total jet cross section σγpjet and theprobabilities for multiple-jet events, and give our predictions for those quantities atHERA energies and the energies relevant for cosmic ray experiments.We begin by sketching our model for inelastic γp scattering. The theory is de-veloped in more detail in [4]. The incoming physical photon state |γ〉phys can beexpressed as a superposition of the bare photon and the set of virtual hadronic states|m〉 which can be distinguished in the subsequent interaction, and appear with proba-bilities Pm. Accordingly, the total inelastic γp cross section is given in a semiclassicalpicture asσγpinel = (1− Phad)σdir +∑m′ Pm σmp, Phad =∑mPm ≪ 1. (1)Included among the states |m〉 are low-mass vector meson states such as the ρ, ω andφ, and complex non-resonant final states which can be described on the average in a2quark-gluon basis.Whether the photon appears as a virtual vector meson state or a high-mass non-resonant state depends on the relative transverse momentum 2p⊥0 of the quark andantiquark in the virtual transition γ → qq which initiates the hadronic interaction.We note that p⊥0 ≈ 1/r⊥, where r⊥ is the average transverse separation of the virtualquarks during the lifetime of the qq system. If r⊥ is greater than, or on the order of,the average transverse radius R⊥ ≡ 1/Q0 of a vector meson, QCD confinement effectswill clearly set in, and the qq system will most likely appear in a hadronic collisionas a light vector meson. For r⊥ < R⊥ the qq system will be smaller than a vectormeson. In this picture, the hadronic system behaves for p⊥0 < Q0 like a vector meson,and for p⊥0 > Q0, like a system of quasi-free quarks and gluons with a transversearea smaller than that of a vector meson by a factor (Q0/p⊥0)2. Its interaction crosssection will be reduced accordingly.After eikonalizing the hadronic cross sections using an impact parameter repre-sentation to account for possible multiple parton scatterings in a single γp collision,the total inelastic γp cross section is given by [4] asσγpinel = σdir + λPρ∫d2b (1− e−2Reχρp)+∑qe2qαemπ∫Q20dp2⊥0p2⊥0∫d2b (1− e−2Reχqqp). (2)Here Pρ is the probability that a photon exists in the ρ meson state, Pρ = 4παem/f 2ρ ,where f 2ρ is the γρ coupling in the vector meson dominance model. We will treatthe ρ, ω and φ as equivalent, and will use the quark model ratios for the couplingsfV ; then λ = 4/3 for equal ρp, ωp, and φp cross sections, and λ = 10/9 for completesuppression of the φ contribution at low p⊥0 [5]. The third term in Eq. (2) includes thecontributions from the excited hadronic states of the photon, which we collectivelycall qq states. The factor (e2qαem/πp2⊥0)dp2⊥0is just the differential probability of3producing a qq pair at a relative transverse momentum 2p⊥0.The real part of the eikonal function for the ρp interaction can be written in theform [6]Reχρp(b, s) = Reχρpsoft(s) +ReχρpQCD(b, s)=12Aρp(b)[σρpsoft(s) + σρpQCD(s)], (3)where Aρp(b) is the density overlap function,Aρp(b) =∫d2b′ρρ(b)ρp(|~b− ~b′|),∫d2bAρp(b) = 1, (4)and σρpsoft and σρpQCD are the soft- and hard-scattering parts of the intrinsic crosssection. σρpsoft was parametrized in [4] using a Regge-like form,σρpsoft = σ0 + σ1(s−m2p)−1/2 + σ2(s−m2p)−1, (5)while σρpQCD was identified with the inclusive parton-level cross section for ρp scatter-ing,σρpQCD =∑ij11 + δij∫10dx1∫10dx2∫p2⊥,mindp2⊥f pi (x1, p2⊥)f ρj (x2, p2⊥)dσˆijdp2⊥. (6)Using the equivalence of the ρ and π states in the quark model, we can equate theparton distribution and density profile functions for the ρ meson to the correspondingfunctions for the pion and take f ρi =fpii and ρρ(b)=ρpi(b). The parton distributions inthe pion are known reasonably well [7]. The density profile functions ρp(b) and ρpi(b)can be taken as the Fourier transforms of the electromagnetic form factors of theproton and the π meson. The density overlap function Aρp ≈ Apip is then given by [8]Aρp(b) =14πν2µ2µ2 − ν2{νbK1(νb)− 2ν2µ2 − ν2 [K0(νb)−K0(µb)]}, (7)4where Kn(x) is a hyperbolic Bessel function, and the “size” parameters of the pionand the proton are µ2 = 0.47 GeV2 and ν2 = 0.71 GeV2, respectively. The for-mer corresponds to a root-mean-square transverse radius for the pion or ρ mesonR⊥=1/Q0 ≈ 2/µ.The eikonal function for the qqp scattering terms in Eq. (2) can be expressedsimilarly as2Reχqqp = Aqqp(b, p⊥0)[σqqpsoft(s, p⊥0) + σqqpQCD(s, p⊥0)]. (8)We assume that ρqq(b) has the same form as ρρ(b) except with the size parameterµ = 2Q0 of the ρ meson replaced by 2p⊥0 [4]. With this replacement, Aqqp(b, p⊥0)and ρqq(b, p⊥0) are continuous with the corresponding functions for the ρ meson atp⊥0 = Q0, but describe a shrinking system for p⊥0 > Q0. Finally, the cross sectionsσqqpsoft(s, p⊥0) and σqqpQCD(s, p⊥p), were assumed to scale with the physical cross sectionalarea of the qq system so thatσqqpsoft(s, p⊥0) + σqqpQCD(s, p⊥0) = (Q20/p2⊥0)[σρpsoft(s) + σρpQCD(s)]. (9)This model guarantees that σsoft and σQCD are continuous at p⊥0 = Q0, and that thesoft contributions from the qq states die out as 1/p2⊥0as expected for higher twistcontributions.We have calculated the total inelastic γp cross section using Eq. (2) and theassumptions above. The soft cross section in Eq. (5) was determined [4] using thelow-energy γp data [9]. The calculation used the parton distributions of Owens [7]for the pion and those of Eichten et al. [10] for the proton. The parameter p⊥,minin Eq. (6) which determines the transverse momentum at which hard parton-levelcollisions come into play was taken from a similar fit to π±p scattering [8]. Theresults agree with the preliminary HERA data [1] and those at lower energies asshown by the top curve in Fig. 1(a). The calculated rise in σγpinel at higher energies5arises from the hard-scattering contributions, and is a prediction of the model ratherthan a fit to the HERA data. It is therefore of interest to look for other tests of themodel. The jet cross sections predicted by the model provide one such test.The total jet cross section σγpjet(s,Q2) is defined to be the part of the inelasticγp cross section which includes events with at least one semihard parton-parton (orγ-parton) scattering with a momentum transfer p2⊥≥ Q2, irrespective of any softprocesses that may occur. To find an expression for the jet cross section σmpjet inthe interaction between the proton and the hadronic state |m〉 of the photon, weuse the fact that semiclassically exp[−2ReχQCD(b, s, Q2)] can be interpreted as theprobability that there is no parton-parton scattering with p2⊥≥ Q2 in a hadroniccollision at impact parameter b. Using this observation, we can rewrite the expressionfor the total hadronic cross section σmp to separate out the jet-free part,σmphad(s) = σmpnojet(s,Q2) + σmpjet (s,Q2)=∫d2b(1− e−2ReχmpQCD − 2Reχmpsoft)=∫d2b(1− e−2Reχmpsoft′(b, s, Q2))e−2ReχmpQCD(b, s, Q2)+∫d2b(1− e−2ReχmpQCD(b, s, Q2)). (10)Hereσmpjet (s,Q2) =∫d2b(1− e−2ReχmpQCD(b, s, Q2))(11)is the total cross section for events associated with |m〉 which contain jets withp⊥ ≥ Q. The factor σγpQCD which appears in the eikonal function in Eq. (11) isnow to be evaluated using the expression in Eq. (6) with p2⊥,min replaced by Q2. Theremaining “nojet” cross section involves a modified eikonal functionReχmpsoft′(s, b, Q2) = Reχmpsoft +ReχmpQCD − ReχmpQCD(b, s, Q2), (12)6and includes contributions from “soft” jets with p2⊥,min ≤ p2⊥ ≤ Q2 as well as theusual soft term.The total jet cross section in γp scattering is the generalized sum of the crosssections σmpjet (s,Q2), weighted with the probabilities Pm, over all the possible hadronicstates of the photon, plus the very small jet contribution from σdir(s,Q2),σγpjet(s,Q2) = σdir(s,Q2) +∑mPmσmjet= σdir(s,Q2) + λPρ σρpjet(s,Q2)+∑qe2qαemπ∫Q20dp2⊥0p2⊥0σqqpjet (s,Q2, p2⊥0). (13)The jet cross sections for specific kinematic cuts, e.g., on jet angle, can be calculatedusing the same basic formulas with the cuts imposed on the integral for σγpQCD inEq. (6).We have calculated the total jet cross section σγpjet in the HERA energy range for Q= 2, 3, 4, and 5 GeV, using the same parameters as were used in [4]. Our predictionsare shown in Fig. 1 along with the calculated total inelastic γp cross section andthe experimental data [1, 9]. At√s=200 GeV, the jet cross section is predicted tobe approximately 1.3% of the total inelastic cross section for Q= 5 GeV in roughagreement with earlier predictions [2], and 25% of the total for Q= 2 GeV.The rapid growth of the jet cross section with energy is associated with the growthof the parton distribution functions in the photon and the proton at the small valuesof x which become accessible at high energies. This growth also leads to an increasingprobability of the scattering of more than one parton in a single γp collision.For thisreason, the inclusive parton-level γp cross section σγpQCD, which counts each partoncollision separately, is not the same as σγpjet. The latter has the overcounting due tomultiple independent scatterings removed in the eikonalization. We show the effectof the eikonalization up to HERA energies in Fig. 2. The effect is not very significant7for Q = 3, 4 and 5 GeV at√s= 200 GeV, but causes about a 10% reduction in thejet cross section for Q = 2 GeV.In Fig. 2 we show σγpjet for the cosmic ray energy range for the same choices of Qas in Fig. 1, together with the calculated cross sections σγpinel and σγpQCD for Q = 2GeV. At very high energies, the jet cross section clearly comprises a large part of thetotal inelastic cross section, e.g., at Eγ = 2 × 108 GeV (or√s ≈ 2 × 104 GeV) σγpjetconstitutes ≈ 75% of σγpinel for Q = 2 GeV. Even for Q = 5 GeV, σγpjet gives nearly40% of σγpinel. Also, as more of the incident partons scatter, the effect of eikonalizationbecomes quite large as shown for Q= 2 GeV.To calculate the multiplicity of jets, we note that the average number of parton-parton scatterings with p⊥ > Q in a single γp collision at impact parameter b asso-ciated with the hadronic component |m〉 of the photon isn¯m(b, s, Q2) = σmpQCD(s,Q2)Amp(b) = 2ReχmpQCD(b, s, Q2). (14)Since the parton-parton scatterings in our model are independent, the probability ofhaving n scatterings (2n jets) in such a hadronic collision has a Poisson distribution,Pmpn (b, s, Q2) =1n![n¯m(b, s, Q2)]ne−n¯m(b, s, Q2). (15)The probability Pn(s,Q2) of having n scatterings relative to all the inelastic eventsat an impact parameter b is thenPn(s,Q2) = (σγpinel)−1∑mPmσmpjet,n= (σγpinel)−1∑mPm∫d2b Pmpn (b, s, Q2)= (σγpinel)−1∑mPm 1n!∫d2b[n¯m(b, s, Q2)]ne−n¯m(b, s, Q2) (16)for n ≥ 1. P0(s,Q2) is defined to be the probability of having a soft interaction with8no associated jets,P0(s,Q2) = (σγpinel)−1∑mPm∫d2b(1− e−2Reχmpsoft′(b, s, Q2))e−n¯m(b, s, Q2). (17)Note that we have neglected the very small two-jet contribution to these expressionsfrom the direct QCD interaction of the photon with the partons in the proton.We have calculated Pn for n = 0, 1, 2 and 3 at√s= 200 GeV using Q = 2, 3,4, and 5 GeV. The results are shown in Table I. For Q = 5 GeV the probability ofhaving a 2-jet event (a single parton scattering) is about 1%; the probability for 2n-jetevents with n larger than 1 is essentially zero. For Q = 2 GeV the total probability ofhaving any hadronic jet event is approximately 27%, with the 2-jet events comprisingmore than 90 % of the total.We also investigated jet production in γp scattering at very high energies. InFig. 3 we show the distribution of Pn at√s = 2 × 104 GeV with the choice of Q= 3 GeV. It is more likely at this energy to have jets in an event than not, withan approximately 70% jet probability. While the 2-jet production is still the mostprobable, events with more than 2 jets occur with about 20% probability.It will be quite interesting to see if the predictions above are verified in futureexperiments since they connect the hadronic properties of the photon quite directlyto those of the pion. In particular, the jet structure of pion and photon-inducedreactions on protons should be quite similar.ACKNOWLEDGMENTSThis work was supported in part through U. S. Department of Energy Grants Nos. DE-AC02-76ER00881 and DE-FG02-85ER40213. One of the authors (RG) would like tothank the World Laboratory for support.9References[1] The preliminary values of σγpinel are: ZEUS Collaboration, Phys. Lett. B 293, 465(1992), σγpinel=154±16±32 µb at a average energy of 210 GeV; H1 Collaboration(reported at the XXVI International Conference on High Energy Physics byF. Eisele), σγpinel= 165±28 µb or 150±22 µb depending on the model used inthe analysis. We show a value 160±30 µb for H1 in our figures to span theexperimental and theoretical uncertainties.[2] R. Gandhi and I. Sarcevic, Phys. Rev. D 44, R10 (1991).[3] See, for example, M. Drees and F. Halzen, Phys. Rev. Lett. 61, 275 (1988); M.Drees, F. Halzen, and K. Hikasa, Phys. Rev. D 39, 1310 (1989); T.K. Gaisser,F. Halzen, T. Stanev, and E. Zas, Phys. Lett. B 243, 444 (1990).[4] K. Honjo, L. Durand, R. Gandhi, I. Sarcevic, and H. Pi, University of WisconsinReport No. MAD/TH/92-6.[5] There is evidence for considerable suppression of the cross section expected forphotoproduction of the φ relative to that for the ρ for√s ≈10 GeV. See R.M.Egloff et al., Phys. Rev. Lett. 43, 657 (1979). We therefore used λ=10/9 in thecalculations reported here. This has rather little effect on the high energy results[4].[6] L. Durand and H. Pi, Phys. Rev. Lett. 58, 303 (1987); Phys. Rev. D 40, 1436(1989).[7] J.F. Owens, Phys. Rev. D 30, 943 (1984).[8] L. Durand and H. Pi, Phys. Rev. D 43, 2125 (1991).10[9] The lower energy data shown in Fig. 1 are from D.O. Caldwell et al.,Phys. Rev. Lett. 25, 609 (1970); ibid. 40, 1222 (1978).[10] E. Eichten, K. Lane, I. Hinchliffe, and C. Quigg, Rev. Mod. Phys. 56, 579(1984).FIGURE CAPTIONSFIG. 1. (a) The calculated total jet cross sections as functions of√s and thetransverse momentum cutoff Q. The total γp inelastic cross section calculated in [4]is compared to the low-energy [9] and HERA [1] data in the upper curve. (b) Theeikonalized and inclusive jet cross sections for the same values of Q.FIG. 2. The total inelastic and jet cross sections at cosmic ray energies. σγpQCD isthe inclusive jet cross section.FIG. 3. The probability distributions for n parton-parton collisions or 2n jets ina γp collision.11Q (GeV) P0 P1 P2 P32 0.734 0.228 0.020 0.0023 0.904 0.078 0.002 0.0004 0.954 0.030 0.000 0.0005 0.971 0.013 0.000 0.000Table 1: Probabilities Pn of having n parton-parton collisions or 2n jets in a singleinelastic γp event at√s=200 GeV.12

Jets and jet multiplicities in high-energy photon-nucleon interactions

Texas A&M University

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MAD/TH/92-8
Jets and jet multiplicities in high energy photon-nucleon
interactions
Loyal Durand and Katsuhiko Honjo
Department of Physics, University of Wisconsin-Madison,
Madison, Wisconsin 53706
Raj Gandhi
Department of Physics, Texas A&M University,
College Station, TX 77843
Hong Pi
Department of Theoretical Physics, University of Lund,
Lund, Sweden S-22362
Ina Sarcevic
Department of Physics, University of Arizona,
Tucson, Arizona 85721
Abstract
We discuss the theory of jet events in high-energy photon-proton interactions using
a model which gives a good description of the data available on total inelastic γp cross
sections up to
√
s=210 GeV. We show how to calculate the jet cross sections and
jet multiplicities and give predictions for these quantities for energies appropriate for
experiments at the HERA ep collider and for very high energy cosmic ray observations.
PACS number(s): 13.60.Hb, 13.87.Ce, 12.40.Lk
1
The hadronic structure of the photon and its interaction with hadrons are cur-
rently subjects of considerable interest.The ZEUS and H1 groups at the ep collider
HERA have presented preliminary results [1] on the γp photoproduction cross section
at 210 GeV center-of-mass energy which show the rise with energy typical of other
hadronic interactions. It was shown by Gandhi and Sarcevic [2] in a simple model
that this rise can be used to discriminate between different sets of photon structure
functions. The hadronic behavior of the photon at very high energies (∼ 108 GeV
photon energy) is also of interest for the theory of air showers triggered by cosmic
ray photons [3].
We have developed an improved QCD-based model of the photon interaction with
nucleons [4] which gives predictions for the total inelastic γp cross section which agree
well with the HERA data. The model has the interesting feature that the information
needed to calculate σγpinel is largely determined by high energy πN scattering. In
this paper, we will use the model to discuss the total jet cross section σγpjet and the
probabilities for multiple-jet events, and give our predictions for those quantities at
HERA energies and the energies relevant for cosmic ray experiments.
We begin by sketching our model for inelastic γp scattering. The theory is de-
veloped in more detail in [4]. The incoming physical photon state |γ〉phys can be
expressed as a superposition of the bare photon and the set of virtual hadronic states
|m〉 which can be distinguished in the subsequent interaction, and appear with proba-
bilities Pm. Accordingly, the total inelastic γp cross section is given in a semiclassical
picture as
σγpinel = (1− Phad)σdir +
∑
m
′ Pm σmp, Phad =
∑
m
Pm ≪ 1. (1)
Included among the states |m〉 are low-mass vector meson states such as the ρ, ω and
φ, and complex non-resonant final states which can be described on the average in a
2
quark-gluon basis.
Whether the photon appears as a virtual vector meson state or a high-mass non-
resonant state depends on the relative transverse momentum 2p⊥0 of the quark and
antiquark in the virtual transition γ → qq which initiates the hadronic interaction.
We note that p⊥0 ≈ 1/r⊥, where r⊥ is the average transverse separation of the virtual
quarks during the lifetime of the qq system. If r⊥ is greater than, or on the order of,
the average transverse radius R⊥ ≡ 1/Q0 of a vector meson, QCD confinement effects
will clearly set in, and the qq system will most likely appear in a hadronic collision
as a light vector meson. For r⊥ < R⊥ the qq system will be smaller than a vector
meson. In this picture, the hadronic system behaves for p⊥0 < Q0 like a vector meson,
and for p⊥0 > Q0, like a system of quasi-free quarks and gluons with a transverse
area smaller than that of a vector meson by a factor (Q0/p⊥0)
2. Its interaction cross
section will be reduced accordingly.
After eikonalizing the hadronic cross sections using an impact parameter repre-
sentation to account for possible multiple parton scatterings in a single γp collision,
the total inelastic γp cross section is given by [4] as
σγpinel = σdir + λPρ
∫
d2b (1− e−2Reχρp)
+
∑
q
e2q
αem
π
∫
Q2
0
dp2
⊥0
p2
⊥0
∫
d2b (1− e−2Reχqqp). (2)
Here Pρ is the probability that a photon exists in the ρ meson state, Pρ = 4παem/f 2ρ ,
where f 2ρ is the γρ coupling in the vector meson dominance model. We will treat
the ρ, ω and φ as equivalent, and will use the quark model ratios for the couplings
fV ; then λ = 4/3 for equal ρp, ωp, and φp cross sections, and λ = 10/9 for complete
suppression of the φ contribution at low p⊥0 [5]. The third term in Eq. (2) includes the
contributions from the excited hadronic states of the photon, which we collectively
call qq states. The factor (e2qαem/πp
2
⊥0
)dp2
⊥0
is just the differential probability of
3
producing a qq pair at a relative transverse momentum 2p⊥0.
The real part of the eikonal function for the ρp interaction can be written in the
form [6]
Reχρp(b, s) = Reχρpsoft(s) +Reχ
ρp
QCD(b, s)
=
1
2
Aρp(b)[σρpsoft(s) + σ
ρp
QCD(s)], (3)
where Aρp(b) is the density overlap function,
Aρp(b) =
∫
d2b′ρρ(b)ρp(|~b− ~b′|),
∫
d2bAρp(b) = 1, (4)
and σρpsoft and σ
ρp
QCD are the soft- and hard-scattering parts of the intrinsic cross
section. σρpsoft was parametrized in [4] using a Regge-like form,
σρpsoft = σ0 + σ1(s−m2p)−1/2 + σ2(s−m2p)−1, (5)
while σρpQCD was identified with the inclusive parton-level cross section for ρp scatter-
ing,
σρpQCD =
∑
ij
1
1 + δij
∫
1
0
dx1
∫
1
0
dx2
∫
p2
⊥,min
dp2
⊥
f pi (x1, p
2
⊥
)f ρj (x2, p
2
⊥
)
dσˆij
dp2
⊥
. (6)
Using the equivalence of the ρ and π states in the quark model, we can equate the
parton distribution and density profile functions for the ρ meson to the corresponding
functions for the pion and take f ρi =f
pi
i and ρρ(b)=ρpi(b). The parton distributions in
the pion are known reasonably well [7]. The density profile functions ρp(b) and ρpi(b)
can be taken as the Fourier transforms of the electromagnetic form factors of the
proton and the π meson. The density overlap function Aρp ≈ Apip is then given by [8]
Aρp(b) =
1
4π
ν2µ2
µ2 − ν2
{
νbK1(νb)− 2ν
2
µ2 − ν2 [K0(νb)−K0(µb)]
}
, (7)
4
where Kn(x) is a hyperbolic Bessel function, and the “size” parameters of the pion
and the proton are µ2 = 0.47 GeV2 and ν2 = 0.71 GeV2, respectively. The for-
mer corresponds to a root-mean-square transverse radius for the pion or ρ meson
R⊥=1/Q0 ≈ 2/µ.
The eikonal function for the qqp scattering terms in Eq. (2) can be expressed
similarly as
2Reχqqp = Aqqp(b, p⊥0)[σ
qqp
soft(s, p⊥0) + σ
qqp
QCD(s, p⊥0)]. (8)
We assume that ρqq(b) has the same form as ρρ(b) except with the size parameter
µ = 2Q0 of the ρ meson replaced by 2p⊥0 [4]. With this replacement, A
qqp(b, p⊥0)
and ρqq(b, p⊥0) are continuous with the corresponding functions for the ρ meson at
p⊥0 = Q0, but describe a shrinking system for p⊥0 > Q0. Finally, the cross sections
σqqpsoft(s, p⊥0) and σ
qqp
QCD(s, p⊥p), were assumed to scale with the physical cross sectional
area of the qq system so that
σqqpsoft(s, p⊥0) + σ
qqp
QCD(s, p⊥0) = (Q
2
0
/p2
⊥0
)[σρpsoft(s) + σ
ρp
QCD(s)]. (9)
This model guarantees that σsoft and σQCD are continuous at p⊥0 = Q0, and that the
soft contributions from the qq states die out as 1/p2
⊥0
as expected for higher twist
contributions.
We have calculated the total inelastic γp cross section using Eq. (2) and the
assumptions above. The soft cross section in Eq. (5) was determined [4] using the
low-energy γp data [9]. The calculation used the parton distributions of Owens [7]
for the pion and those of Eichten et al. [10] for the proton. The parameter p⊥,min
in Eq. (6) which determines the transverse momentum at which hard parton-level
collisions come into play was taken from a similar fit to π±p scattering [8]. The
results agree with the preliminary HERA data [1] and those at lower energies as
shown by the top curve in Fig. 1(a). The calculated rise in σγpinel at higher energies
5
arises from the hard-scattering contributions, and is a prediction of the model rather
than a fit to the HERA data. It is therefore of interest to look for other tests of the
model. The jet cross sections predicted by the model provide one such test.
The total jet cross section σγpjet(s,Q
2) is defined to be the part of the inelastic
γp cross section which includes events with at least one semihard parton-parton (or
γ-parton) scattering with a momentum transfer p2
⊥
≥ Q2, irrespective of any soft
processes that may occur. To find an expression for the jet cross section σmpjet in
the interaction between the proton and the hadronic state |m〉 of the photon, we
use the fact that semiclassically exp[−2ReχQCD(b, s, Q2)] can be interpreted as the
probability that there is no parton-parton scattering with p2
⊥
≥ Q2 in a hadronic
collision at impact parameter b. Using this observation, we can rewrite the expression
for the total hadronic cross section σmp to separate out the jet-free part,
σmphad(s) = σ
mp
nojet(s,Q
2) + σmpjet (s,Q
2)
=
∫
d2b
(
1− e−2Reχ
mp
QCD − 2Reχmpsoft
)
=
∫
d2b
(
1− e−2Reχ
mp
soft
′(b, s, Q2)
)
e−2Reχ
mp
QCD(b, s, Q
2)
+
∫
d2b
(
1− e−2Reχ
mp
QCD(b, s, Q
2)
)
. (10)
Here
σmpjet (s,Q
2) =
∫
d2b
(
1− e−2Reχ
mp
QCD(b, s, Q
2)
)
(11)
is the total cross section for events associated with |m〉 which contain jets with
p⊥ ≥ Q. The factor σγpQCD which appears in the eikonal function in Eq. (11) is
now to be evaluated using the expression in Eq. (6) with p2
⊥,min replaced by Q
2. The
remaining “nojet” cross section involves a modified eikonal function
Reχmpsoft
′
(s, b, Q2) = Reχmpsoft +Reχ
mp
QCD − ReχmpQCD(b, s, Q2), (12)
6
and includes contributions from “soft” jets with p2
⊥,min ≤ p2⊥ ≤ Q2 as well as the
usual soft term.
The total jet cross section in γp scattering is the generalized sum of the cross
sections σmpjet (s,Q
2), weighted with the probabilities Pm, over all the possible hadronic
states of the photon, plus the very small jet contribution from σdir(s,Q
2),
σγpjet(s,Q
2) = σdir(s,Q
2) +
∑
m
Pmσmjet
= σdir(s,Q
2) + λPρ σρpjet(s,Q2)
+
∑
q
e2q
αem
π
∫
Q2
0
dp2
⊥0
p2
⊥0
σqqpjet (s,Q
2, p2
⊥0
). (13)
The jet cross sections for specific kinematic cuts, e.g., on jet angle, can be calculated
using the same basic formulas with the cuts imposed on the integral for σγpQCD in
Eq. (6).
We have calculated the total jet cross section σγpjet in the HERA energy range for Q
= 2, 3, 4, and 5 GeV, using the same parameters as were used in [4]. Our predictions
are shown in Fig. 1 along with the calculated total inelastic γp cross section and
the experimental data [1, 9]. At
√
s=200 GeV, the jet cross section is predicted to
be approximately 1.3% of the total inelastic cross section for Q= 5 GeV in rough
agreement with earlier predictions [2], and 25% of the total for Q= 2 GeV.
The rapid growth of the jet cross section with energy is associated with the growth
of the parton distribution functions in the photon and the proton at the small values
of x which become accessible at high energies. This growth also leads to an increasing
probability of the scattering of more than one parton in a single γp collision.For this
reason, the inclusive parton-level γp cross section σγpQCD, which counts each parton
collision separately, is not the same as σγpjet. The latter has the overcounting due to
multiple independent scatterings removed in the eikonalization. We show the effect
of the eikonalization up to HERA energies in Fig. 2. The effect is not very significant
7
for Q = 3, 4 and 5 GeV at
√
s= 200 GeV, but causes about a 10% reduction in the
jet cross section for Q = 2 GeV.
In Fig. 2 we show σγpjet for the cosmic ray energy range for the same choices of Q
as in Fig. 1, together with the calculated cross sections σγpinel and σ
γp
QCD for Q = 2
GeV. At very high energies, the jet cross section clearly comprises a large part of the
total inelastic cross section, e.g., at Eγ = 2 × 108 GeV (or
√
s ≈ 2 × 104 GeV) σγpjet
constitutes ≈ 75% of σγpinel for Q = 2 GeV. Even for Q = 5 GeV, σγpjet gives nearly
40% of σγpinel. Also, as more of the incident partons scatter, the effect of eikonalization
becomes quite large as shown for Q= 2 GeV.
To calculate the multiplicity of jets, we note that the average number of parton-
parton scatterings with p⊥ > Q in a single γp collision at impact parameter b asso-
ciated with the hadronic component |m〉 of the photon is
n¯m(b, s, Q
2) = σmpQCD(s,Q
2)Amp(b) = 2ReχmpQCD(b, s, Q
2). (14)
Since the parton-parton scatterings in our model are independent, the probability of
having n scatterings (2n jets) in such a hadronic collision has a Poisson distribution,
Pmpn (b, s, Q
2) =
1
n!
[n¯m(b, s, Q
2)]ne−n¯m(b, s, Q2). (15)
The probability Pn(s,Q
2) of having n scatterings relative to all the inelastic events
at an impact parameter b is then
Pn(s,Q
2) = (σγpinel)
−1
∑
m
Pmσmpjet,n
= (σγpinel)
−1
∑
m
Pm
∫
d2b Pmpn (b, s, Q
2)
= (σγpinel)
−1
∑
m
Pm 1
n!
∫
d2b[n¯m(b, s, Q
2)]ne−n¯m(b, s, Q2) (16)
for n ≥ 1. P0(s,Q2) is defined to be the probability of having a soft interaction with
8
no associated jets,
P0(s,Q
2) = (σγpinel)
−1
∑
m
Pm
∫
d2b
(
1− e−2Reχ
mp
soft
′(b, s, Q2)
)
e−n¯m(b, s, Q2). (17)
Note that we have neglected the very small two-jet contribution to these expressions
from the direct QCD interaction of the photon with the partons in the proton.
We have calculated Pn for n = 0, 1, 2 and 3 at
√
s= 200 GeV using Q = 2, 3,
4, and 5 GeV. The results are shown in Table I. For Q = 5 GeV the probability of
having a 2-jet event (a single parton scattering) is about 1%; the probability for 2n-jet
events with n larger than 1 is essentially zero. For Q = 2 GeV the total probability of
having any hadronic jet event is approximately 27%, with the 2-jet events comprising
more than 90 % of the total.
We also investigated jet production in γp scattering at very high energies. In
Fig. 3 we show the distribution of Pn at
√
s = 2 × 104 GeV with the choice of Q
= 3 GeV. It is more likely at this energy to have jets in an event than not, with
an approximately 70% jet probability. While the 2-jet production is still the most
probable, events with more than 2 jets occur with about 20% probability.
It will be quite interesting to see if the predictions above are verified in future
experiments since they connect the hadronic properties of the photon quite directly
to those of the pion. In particular, the jet structure of pion and photon-induced
reactions on protons should be quite similar.
ACKNOWLEDGMENTS
This work was supported in part through U. S. Department of Energy Grants Nos. DE-
AC02-76ER00881 and DE-FG02-85ER40213. One of the authors (RG) would like to
thank the World Laboratory for support.
9
References
[1] The preliminary values of σγpinel are: ZEUS Collaboration, Phys. Lett. B 293, 465
(1992), σγpinel=154±16±32 µb at a average energy of 210 GeV; H1 Collaboration
(reported at the XXVI International Conference on High Energy Physics by
F. Eisele), σγpinel= 165±28 µb or 150±22 µb depending on the model used in
the analysis. We show a value 160±30 µb for H1 in our figures to span the
experimental and theoretical uncertainties.
[2] R. Gandhi and I. Sarcevic, Phys. Rev. D 44, R10 (1991).
[3] See, for example, M. Drees and F. Halzen, Phys. Rev. Lett. 61, 275 (1988); M.
Drees, F. Halzen, and K. Hikasa, Phys. Rev. D 39, 1310 (1989); T.K. Gaisser,
F. Halzen, T. Stanev, and E. Zas, Phys. Lett. B 243, 444 (1990).
[4] K. Honjo, L. Durand, R. Gandhi, I. Sarcevic, and H. Pi, University of Wisconsin
Report No. MAD/TH/92-6.
[5] There is evidence for considerable suppression of the cross section expected for
photoproduction of the φ relative to that for the ρ for
√
s ≈10 GeV. See R.M.
Egloff et al., Phys. Rev. Lett. 43, 657 (1979). We therefore used λ=10/9 in the
calculations reported here. This has rather little effect on the high energy results
[4].
[6] L. Durand and H. Pi, Phys. Rev. Lett. 58, 303 (1987); Phys. Rev. D 40, 1436
(1989).
[7] J.F. Owens, Phys. Rev. D 30, 943 (1984).
[8] L. Durand and H. Pi, Phys. Rev. D 43, 2125 (1991).
10
[9] The lower energy data shown in Fig. 1 are from D.O. Caldwell et al.,
Phys. Rev. Lett. 25, 609 (1970); ibid. 40, 1222 (1978).
[10] E. Eichten, K. Lane, I. Hinchliffe, and C. Quigg, Rev. Mod. Phys. 56, 579
(1984).
FIGURE CAPTIONS
FIG. 1. (a) The calculated total jet cross sections as functions of
√
s and the
transverse momentum cutoff Q. The total γp inelastic cross section calculated in [4]
is compared to the low-energy [9] and HERA [1] data in the upper curve. (b) The
eikonalized and inclusive jet cross sections for the same values of Q.
FIG. 2. The total inelastic and jet cross sections at cosmic ray energies. σγpQCD is
the inclusive jet cross section.
FIG. 3. The probability distributions for n parton-parton collisions or 2n jets in
a γp collision.
11
Q (GeV) P0 P1 P2 P3
2 0.734 0.228 0.020 0.002
3 0.904 0.078 0.002 0.000
4 0.954 0.030 0.000 0.000
5 0.971 0.013 0.000 0.000
Table 1: Probabilities Pn of having n parton-parton collisions or 2n jets in a single
inelastic γp event at
√
s=200 GeV.
12


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