Diffraction and an infrared finite gluon propagator

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by  E. G. S. Luna

Diffraction and an infrared finite gluon propagator
E.G.S. Luna1,2
1Instituto de F´ısica Te´orica, UNESP, S˜ao Paulo State University, 01405-900, S˜ao Paulo, SP, Brazil
2Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas, 13083-970, Campinas, SP, Brazil
We discuss some phenomenological applications of an infrared finitegluon propagator character-
ized by a dynamically generated gluon mass. In particular we compute the effect of the dynamical
gluon mass on pp and p¯p diffractive scattering. Wealso show how the data on γp photoproduction
and hadronic γγ reactions can be derived from the pp and p¯p forward scattering amplitudes by
assuming vector meson dominance and the additivequark model.
I. A QCD-INSPIRED EIKONAL MODEL WITH
A GLUON DYNAMICAL MASS
7
0
0 Nowadays,severalstudies supportthe hypothesisthat
2 the gluonmay developadynamicalmass[1,2]. This dy-
n namicalgluonmass,intrinsicallyrelatedtoaninfraredfi-
a nitegluonpropagator[3],andwhoseexistenceisstrongly
J supported by recent QCD lattice simulations [4]), has

6 beenadoptedinmanyphenomenologicalstudies[5,6,7].
2 Hence it is natural to correlate the arbitrary mass scale

thatappearsinQCD-inspiredmodelswiththedynamical
2
gluonone,obtainedbyCornwall[1]bymeansofthepinch
v
techniqueinordertoderiveagaugeinvariantSchwinger-
9
4 Dyson equation for the gluon propagator. This connec-
1 tion can be done building a QCD-based eikonal model
9 wheretheonsetofthedominanceofgluonsintheinterac-
0 tionofhigh-energyhadronsismanagedbythedynamical
6
gluon mass scale.
0 FIG.1: Theχ2/DOF asafunctionofdynamicalgluonmass
Aconsistentcalculationofhigh-energyhadron-hadron
h/ cross sections compatible with unitarity constraints can mg.
p be automatically satisfied by use of an eikonalizedtreat-
-
p ment of the semihard parton processes. In an eikonal
e representation, the total cross sections is given by difference between pp and p¯p channels, is parametrized
h
as
: ∞
iv σtot(s)=4πZ bdb[1−e−χI(b,s)cosχR(b,s)], (1) χ−(b,s)=C−Σmg eiπ/4W(b;µ−), (3)
X 0 √s
ar where s is the square of the total center-of-mass en- where mg is the dynamical gluon mass and the parame-
ergy and χ(b,s) is a complex eikonal function: χ(b,s)= − −
tersC andµ areconstantstobefitted. ThefactorΣis
χ (b,s)+iχ (b,s). In terms of the proton-proton (pp) definedasΣ 9πα¯2(0)/m2,withthedynamicalcoupling
anRdantiprotoIn-proton(p¯p)scatterings,thiscombination ≡ s g
constant α¯ set at its frozen infrared value. The eikonal
reads χpp¯pp(b,s) = χ+(b,s)±χ−(b,s). Following the Ref. functions χsqq(b,s) and χqg(b,s), needed to describe the
[8], we write the even eikonalas the sum of gluon-gluon,
lower-energyforwarddata,aresimplyparametrizedwith
quark-gluon, and quark-quark contributions: terms dictated by the Regge phenomenology [8]; the
gluon term χ (b,s), that dominates the asymptotic be-
χ+(b,s) = χ (b,s)+χ (b,s)+χ (b,s) gg
qq qg gg havior of hadron-hadron total cross sections, is written
= i[σqq(s)W(b;µqq)+σqg(s)W(b;µqg) as χgg(b,s) σgg(s)W(b;µgg), where
≡
+σ (s)W(b;µ )]. (2)
gg gg 1
′
σ (s)=C dτF (τ)σˆ (sˆ). (4)
Here W(b;µ)is the overlapfunction atimpact param- gg Z4m2/s gg gg
g
eterspaceandσ (s)aretheelementarysubprocesscross
ij
sections of colliding quarks and gluons (i,j = q,g). The Here F (τ) [g g](τ) is the convoluted structure
gg
≡ ⊗′
overlapfunctionisassociatedwiththeFouriertransform function for pair gg, C is a normalization constant and
ofadipoleformfactor,W(b;µ)=(µ2/96π)(µb)3K (µb), σˆ (sˆ) is the subprocess cross section, calculated using a
3 gg
where K (x) is the modified Bessel function of second proceduredictatedbythedynamicalperturbationtheory
3
−
kind. The odd eikonal χ (b,s), that accounts for the [9], where amplitudes that do not vanish to all orders of
2
FIG.2: Totalcrosssectionforpp(solidcurve)andp¯p(dashed FIG.3: Gluon-gluontotalcrosssection. Thedynamicalgluon
′
curve) scattering. mass scale and the parameter C were set to mg =400 MeV
and to C′ =(12.097±0.962)×10−3, respectively.
perturbation theory are given by their free-field values,
whereas amplitudes that vanish in all orders in pertur- σγγ for the production of hadrons in the interaction of
bation theory as exp( 1/4παs) areretained atlowest one and two real photons, respectively, are expected to
∝ −
order. Only recently the physicalmeaning ofthe param- be dominatedby interactionswhere the photonhas fluc-
′
eter C has become fully [7]: it is a normalization factor tuated into a hadronic state. Therefore measuring the
that appears in the gluon distribution function (at small energy dependence of photon-induced processes should
x and low Q2) after the resummation of soft emission in improveourunderstanding ofthe hadronicnature ofthe
the leading ln(1/x) approximation of QCD, photon as well as the universal high energy behavior of
total hadronic cross sections.
(1 x)5
g(x)=C′ − , (5) Howeverthe comparisonof the experimentaldata and
xJ
thetheoreticalpredictionmaypresentsomesubtletiesde-
whereJ controlstheasymptoticbehaviorofσ (s). The pending on the Monte Carlo model used to analyze the
tot
resultsofglobalfits to allhigh-energyforwardppandp¯p data. For example, the γγ cross sections are extracted
scattering data above √s = 10 GeV and to the elastic from a measurement of hadron production in e+e− pro-
differential scattering cross section for p¯p at √s = 1.8 cesses and are strongly dependent upon the acceptance
TeV are shown in Figs. 1 e 2. The Figure 1 enables us correctionstobeemployed. Thesecorrectionsareinturn
to estimate a dynamicalgluonmass m 400+350 MeV. sensitive to the Monte Carlo models used in the simula-
g ≈ −100
More details of the fit results can be seen in Ref. [8]. tion of the different components of an event, and this
Theresultsofthefitstoσ forbothppandp¯pchannels generalprocedureproducesuncertainties inthe determi-
tot
are displayed in Fig. 2 in the case of a dynamical gluon nation of σγγ [13]. This clearly implies that any phe-
mass m =400 MeV, which is the preferred value for pp nomenologicalanalysishas totake properlyinto account
g
and p¯p scattering. The σ cross section, calculated via thediscrepanciesamongσγγ dataobtainedfromdifferent
gg
expression (4), is shown in Fig. 3. Monte Carlo generators. Therefore we performed global
fits consideringseparatelydata ofthe L3 [14] andOPAL
[15] collaborations obtained through the PYTHIA [16]
II. PHOTON-PROTON AND and PHOJET [17] codes, defining two data sets as
PHOTON-PHOTON REACTIONS
SET I: σγp and σγγ data (√s ,W 10 GeV),
PYT γp γγ ≥
Early modeling of hadron-hadron,photon-hadron and SET II: σγp and σPγγHO data (√sγp,Wγγ ≥10 GeV),
photon-photoncrosssections within Regge theory shows
a energy dependence similar to the ones of nucleon- where σγγ (σγγ ) correspond to the data of γγ total
PYT PHO
nucleon [10, 11, 12]. This universal behavior, appropri- hadronic cross section obtained via the PYTHIA (PHO-
ately scaled in order to take into account the differences JET) generator.
between hadrons and the photon, can be understood as The even and odd amplitudes for γp scattering can
follows: at high center-of-mass energies the total pho- be obtained after the substitutions σij (2/3)σij and
→
toproduction σγp and the total hadronic cross section µ 3/2µ in the eikonals (2) and (3) [13], where
ij ij
→
p
3
FIG. 4: σγp and σγγ cross sections corresponding to the case where P−1 varies with the energy (solid curves). The dashed
curves correspond to the case using a constant value of P−1. The curvehsadin (a) and (b) [(c) and (d)] are related to the SET I
had
[SET II].
χγp¯ = χ+ χ−. Assuming vector meson dominance Toextendthemodeltotheγγ channelwejustperform
γp γp ± γp
(VMD), the γp total cross section is given by the substitutions σ (4/9)σ and µ (3/2)µ in
ij ij ij ij
→ →
the evenpartofthe eikonal(2). Thecalculationleadsto
∞
the following eikonalized total γγ hadronic cross section
σγp(s)=4πPhγapdZ bdb[1−e−χγIp(b,s)cosχγRp(b,s)], ∞
where Pγp is the p0robability that the photon interacts σγγ(s)=4πNPhγaγdZ0 bdb[1−e−χγIγ(b,s)cosχγRγ(b,s)],
had
asahadron. InthesimplestVMDformulationthisprob- wherePγγ =P2 andN isanormalizationfactorwhich
ability is expected to be of (α ): had had
em takes into account the uncertainty in the extrapolation
O
to realphotons (Q =Q =0)of the hadroniccross sec-
4πα 1 1 2
Phγapd =Phad = f2em ∼ 249, tionσγγ(Wγγ,Q21,Q22)[13]. Withthe eikonalparameters
V=Xρ,ω,φ V of the QCD eikonal model fixed by the pp and p¯p data,
we have performed all calculations of photoproduction
where ρ, ω and φ are vector mesons. However, there are and photon-photon scattering [13]. We have assumed a
expected contributions to P other than ρ, ω, φ, as for phenomenological expression for P , implying that it
had had
example,ofheaviervectormesonsandcontinuumstates. increaseslogarithmicallywith the squareofthe centerof
Moreover, the probability P may also depend on the mass energy: P =a+bln(s). The total cross section
had had
energy,whichisapossibilitythatweexploreinthiswork. curves are depicted in Figure 3, where Figs. 3(a) and
4
3(b) [3(c) and 3(d)] are related to the SET I [SET II]. calculationsofstronglyinteractingprocesses. Thisresult
The results depicted in the Figures 3(c) and 3(d) show corroboratestheoretical analysis taking into account the
that the shape and normalization of the curves are in possibility of dynamical mass generation and show that,
good agreement with the data deconvoluted with PHO- in principle, a dynamical nonperturbative gluon propa-
JET[13]. ThecalculationsusingaconstantvalueofP gator may be used in calculations as if it were a usual
had
(that does not depend on the energy s) are represented (derived from Feynman rules) gluon propagator.
by the dashed curves. These global results indicate that With the help of vector meson dominance and the ad-
a energydependence of Phad is favoredby the photopro- ditive quark model, the QCD model can successfully de-
duction and photon-photon scattering data. scribethedataofthetotalphotoproductionγpandtotal
hadronic γγ cross sections. We have assumed that P
had
has a logarithmic increase with s. This choice leads to
III. CONCLUSIONS a improvement of the global fits, i. e. the logarithmic
increase of P with s is quite favoredby the data. No-
had
In this work we have investigated the influence of an tice that the data of σγγ above √s 100 GeV can
PYT ∼
infrareddynamicalgluonmassscaleinthe calculationof hardly be described by the QCD model. Assuming the
pp, p¯p, γp and γγ total cross sections through a QCD- correctnessof the model we couldsaythat the PHOJET
inspired eikonal model. By means of the dynamical per- generator is more appropriate to obtain the σγγ data
turbationtheory(DPT)wehavecomputedthe treelevel above √s 100 GeV. This conclusion is supported by
∼
gg gg cross section taking into account the dynami- the recent result that the factorization relation does not
→
cal gluon mass. The connection between the subprocess depend on the assumption of an additive quark model,
crosssectionσˆ (sˆ)andthesetotalcrosssectionsismade butmoreontheopacityoftheeikonalbeingindependent
gg
viaaQCD-inspiredeikonalmodelwheretheonsetofthe of the nature of the reaction [18].
dominance of gluons in the interaction of high energy Acknowledgments: I am pleased to dedicate this pa-
hadrons is managed by the dynamical gluon mass scale. per to Prof. Yogiro Hama, on the occasion of his 70th
Bymeansofaglobalfittotheforwardppandp¯pscatter- birthday. I am grateful to the editors of the Braz. J.
ing data and to dσp¯p/dt data at √s=1.8 TeV, we have Phys. who gave me the opportunity of contributing to
determined the best phenomenological value of the dy- the volume in his honor, and to M.J. Menon and A.A.
namical gluon mass, namely m 400+350 MeV. Inter- Nataleforusefulcomments. Thisresearchwassupported
g ≈ −100
estinglyenough,thisvalueisofthesameorderofmagni- by the Conselho Nacionalde Desenvolvimento Cient´ıfico
tudeasthevaluem 500 200MeV,obtainedinother e Tecnol´ogico-CNPq under contract 151360/2004-9.
g
≈ ±
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