Vector mesons $ρ, ρ'$ and $ρ''$ diffractively photo- and leptoproduced

6 Pages·0.15 MB·English

by  G. Kulzinger

HD-THEP-99-1. – To be published in Fizika B
VECTOR MESONS ρ, ρ′ AND ρ′′
DIFFRACTIVELY PHOTO- AND LEPTOPRODUCED
G. KULZINGER 1,2
Institut fu¨r Theoretische Physik der Universita¨t Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany
9
9 Intheframeworkofnon-perturbativeQCDwecalculatehigh-energydiffractivepro-
9
ductionofvectormesonsρ,ρ′ andρ′′ byrealandvirtualphotonsonanucleon. The
1
initial photon dissociates into a qq¯-dipole and transforms into a vector meson by
n
scattering off the nucleon which, for simplicity, is represented as quark-diquark.
a
J Therelevantdipole-dipolescatteringamplitudeisprovidedbythenon-perturbative

4 modelofthestochasticQCDvacuum. Thewavefunctionsresultfromconsiderations
1 in the frame of light-front dynamics; the physical ρ′- and ρ′′-mesons are assumed

to be mixed states of an active 2S-excitation and some residual rest (2D- and/or
1
v hybrid state). We obtain good agreement with the experimental data and get an
3 understanding of the markedly different π+π−-mass spectra for photoproduction
1
and e+e−-annihilation.
3
1
Keywords: non-perturbativeQCD,diffraction,photoproduction,photonwavefunc-
0
9 tion, ρ-meson, excited vector mesons, hybrid
9
/
h
1. Introduction
p
-
p
Diffractive scattering processes are characterized by small momentum transfer,
e
h t<1 GeV2, and thus governed by non-perturbative QCD. To get more insight
− ∼
: inthephysicsatworkweinvestigateexclusivevectormesonproductionbyrealand
v
i virtual photons. In this note we summarize recent results from Ref. [1] on ρ-, ρ′-
X
and ρ′′-production, see also Ref. [2]. In Refs [1, 3] we have developed a framework
r
a which we here can only flash.
We consider high-energy diffractive collision of a photon, which dissociates into
a qq¯-dipole andtransformsinto a vectormeson,with a protoninthe quark-diquark
picture, which remains intact. The scattering T-amplitude can be written as an
integralof the dipole-dipole amplitude and the correspondingwavefunctions. Inte-
grating out the proton side, we have
dzd2r
TVλ(s,t)=isZ 4π ψV†(λ)ψγ(Q2,λ)(z,r)Jp(z,r,∆T), (1)
whereV(λ)standsforthefinalvectormesonandγ(Q2,λ)fortheinitialphotonwith
definitehelicitiesλ(andvirtualityQ2);z isthelight-conemomentumfractionofthe
quark, r the transverse extension of the qq¯-dipole. The function J (z,r,∆ ) is the
p T
interactionamplitudeforadipole z,r scatteringonaprotonwithfixedmomentum
{ }
transfert= ∆2;for∆ =0duetotheopticaltheoremitisthecorrespondingtotal
− T T
coss section (see below Eq. (4)). It is calculated within non-perturbative QCD: In
1 SupportedbytheDeutscheForschungsgemeinschaft undergrantno. GRK216/1-96
2E-mail: [email protected]
1
1
0.8 D
0.6
0.4
0.2
D
1
0
-10 -5 0 5 10
Figure1: Interactionamplitude(arbitraryunits)oftwocolourdipolesasfunctionoftheir
impact (units of correlation lengths a). One large qq¯-dipole of extension 12a is fixed, the
secondsmall oneof extension1ais, averaged overallitsorientations, shifted alongon top
of the first one. For theD -tensor structure of the correlator thereare only contributions
1
whentheendpointsareclosetoeachother,whereasfortheD-structurelargecontributions
showupalsofrombetweentheendpoints. Thisistobeinterpretedasinteractionwiththe
gluonic string between thequark and antiquark.
the high-energylimit Nachtmann[4] deriveda non-perturbative formulafor dipole-
dipolescatteringwhosebasicentityisthevacuumexpectationvalueoftwolightlike
Wilson loops. This gets evaluated [5] in the model of the stochastic QCD vacuum.
2. The model of the stochastic QCD vacuum
Coming from the functional integral approach the model of the stochastic QCD
vacuum [6] assumes that the non-perturbative part of the gauge field measure, i.e.
long-rangegluonfluctuationsthatareassociatedwithanon-trivialvacuumstructure
of QCD, can be approximated by a stochastic process in the gluon field strengths
with convergentcumulant expansion. Further assuming this process to be gaussian
one arrivesat a description throughthe second cumulanthg2FA(x;x )FA′(x′;x )i
µν 0 ρσ 0
which has two Lorentz tensor structures multiplied by correlation functions D and
D , respectively. D is non-zero only in the non-abelian theory or in the abelian
1
theory with magnetic monopoles and yields linear confinement. Whereas the D -
1
structure is not confining.
The underlying mechanism of (interacting) gluonic strings also shows up in
the scattering of two colour dipoles, cf. Fig. 1, and essentially determines the T-
amplitudeiflargedipolesizesarenotsuppressedbythewavefunctions. Toconfront
with experimentthis specific-largedistance predictionwe areintended to study the
broad ρ-states and, especially, their production by broad small-Q2 photons. Before
we enter the discussion of our results, however, we have to specify these states and
have to fix their wave functions as well as that of the photon.
3. Physical states ρ, ρ′ and ρ′′
Analyzing the π+π−-invariant mass spectra for photoproduction and e+e−-anni-
hilation Donnachie and Mirzaie [7] concluded evidence for two resonances in the
2
dσ/dM(M)[µb/GeV] γp π+π−p
→
signspattern
{+,+,–}
10.
1
0.1
0.01
0.5 1 1.5 2 2.5
M[GeV]
Figure2: Massspectrumofπ+π−-photoproductionontheproton: Theinterference
in the 1.6 GeV region is constructive in contrary to the case of e+e−-annihilation
into π+π−. We display our calculation together with the experimental data [7].
1.6 GeV region whose masses are compatible with the 1−− states ρ(1450) and
ρ(1700). We make as simplest ansatz
ρ(770) = 1S , (2)
| i | i
ρ(1450) = cosθ 2S +sinθ rest ,
| i | i | i
ρ(1700) = sinθ 2S +cosθ rest ,
| i − | i | i
where rest isconsideredtohave 2D -and/orhybridcomponentswhosecouplings
| i | i
to the photon both are suppressed, see Refs. [8] and [9, 10], respectively. With
our convention of the wave functions the relative signs +, ,+ of the production
{ − }
amplitudes of the ρ-, ρ′- and ρ′′-states in e+e−-annihilation determine the mixing
angle to be in the firstquadrant; fromRef. [7] then followsθ=41◦. With this value
∼
and the branching ratios of the ρ′- and ρ′′-mesons into π+π− extracted in Ref. [7]
we calculate the photoproduction spectrum as shown in Fig. 2 with the observed
signs pattern +,+, ; for details cf. [1]. We will understand below from Fig. 3
{ −}
the signs change of the 2S-productionas due to the dominance of large dipole sizes
in photoproduction in contrary to the coupling to the electromagnetic current f
2S
being determined by small dipole sizes.
4. Light-cone wave functions
Inthehigh-energylimitthephotoncanbeidentifiedasitslowestFock,i.e. qq¯-state.
The vector meson wave function distributes this qq¯-dipole z,r , accordingly.
{ }
Photon: With mean of light-cone perturbation theory (LCPT) we get explicit
expressions for both longitudinal and transverse photons. The photon transverse
size which we will see to determine the T-amplitude is governed by the product
εr, ε= zz¯Q2+m2 and r= r. For high Q2 longitudinal photons dominate by a
| |
poweropfQ2;theirz-endpointsbeingexplicitlysuppressed,LCPTisthusapplicable.
FormoderateQ2 alsotransversephotonscontributewhichhavelargeextensionsbe-
causeendpoints arenotsuppressed. ForQ2 smallerthan1GeV2 LCPTdefinitively
breaks down. However, it was shown [11] that a quark mass phenomenologically
interpolatingbetweenazerovalenceanda220MeVconstituentmassastonishingly
well mimics chiral symmetry breaking and confinement. Our wave function is thus
given by LCPT with such a quark mass m(Q2), for details cf. Refs [1, 3].
3
Vectormesons: Thevectormesonswavefunctionsofthe1S-and2S-statesare
modelledaccordingto the photon. We only replacethe photonenergydenominator
(ε2+k2)−1 by a function of z and k for which ansa¨tze according to Wirbel and
| |
Stech [12] are made; for the ”radial” excitation we account by both a polynomial
in zz¯ and the 2S-polynomial in k2 of the transverse harmonic oscillator. The pa-
rametersarefixed bythe demands thatthe 1S-statereproducesM andf andthe
ρ ρ
2S-stateisbothnormalizedandorthogonalonthe 1S-state. Fordetails cf. Ref.[1].
5. Results
Before presenting some of our results [1] we stress that all calculated quantities are
absolute predictions. Due to the eikonal approximation applied, the cross sections
are constant with total energy s and refer to √s=20 GeV where the proton radius
is fixed. (The two parameters of the model of the stochastic QCD vacuum, the
gluoncondensatehg2FFiandthe correlationlengtha,aredeterminedbymatching
low-energy and lattice results, cf. Ref. [5].)
In Fig. 3 we display – for the transverse 2S-state, λ=T – both the functions
J(0)(z,r) := 2π dϕr J (z,r,∆ =0) (3)
p Z 2π p T
0
rψV†(λ)ψγ(Q2,λ)(r) := Z 4dπz Z 2π d2ϕπr |r|ψV†(λ)ψγ(Q2,λ)(z,r) (4)
0
whichtogether,seeEq.(1),essentiallydeterminetheleptoproductionamplitude. It
is strikingly shownhow for decreasing virtualityQ2 the outer positive regionof the
wave functions effective overlap rψV†(λ)ψγ(Q2,λ) wins over the inner negative part
due to the strong rise with r of the dipole-proton interaction amplitude J(0) which
p
itselfisaconsequenceofthestringinteractionmechanismdiscussedabove. In praxi
dipole sizes up to 2.5 fm contribute significantly to the cross section.
rψV†(T)ψγ(Q2,T)(r)[10−3 fm−1] Jp(0)(1/2,r)[mb]
2S-Transverse
100
0
Q2=20GeV2
Q2=1GeV2
-100
-200 Q2=0
0 0.5 1 1.5 2 2.5 3 r[fm]
Figure 3: Dipole-proton total cross section Jp(0) and the effective overlap rψV†(T)ψγ(Q2,T)
as function of the transverse dipole size r. The black line is the function Jp(0)(1/2,r), i.e.
the total cross section of a qq¯-dipole {z=1/2,r}, averaged over all orientations, scattering
on a proton; the grey line shows the cross section for a completely abelian, non-confining
theory. TheT-amplitudeisobtainedbyintegrationovertheproductofJp andtheoverlap
function, which essentially is the effective overlap shown for Q2=0, 1 and 20 GeV2 as
short, medium and long dashed curves, respectively.
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σ(Q2)[µb]
ρ-Transverse
1
0.1 ρ-Longitudinal
0.01
85%E665-Transverse
85%E665-Longitudinal
0.001
0.05 0.1 0.5 1 5 10.
Q2[GeV2]
σ(Q2)[µb]
2S-Longitudinal
0.1
2S-Transverse
0.01
0.001
0.05 0.1 0.5 1 5 10.
Q2[GeV2]
Figure 4: Integrated cross sections of the ρ-meson and the 2S-state as a function of Q2.
E665 [13] provides data for the ρ; the pomeron contribution which corresponds to our
calculation we roughly estimate as 85% of the measured cross section, cf. Ref. [15].
Our results for integrated elastic cross sections as functions of Q2 are given in
Fig.4. For the ρ-mesonour predictionis about20 30%below the E665-data[13].
−
However, we agree with the NMC-experiment [14] which measures some definite
superpositionoflongitudinalandtransversepolarization,seeTable3inRef.[1]. For
the 2S-state,due to the nodes ofthe wave function, we predicta markedstructure;
the explicit shape, however, strongly depends on the parametrization of the wave
functions.
InFig.5wedisplaytheratioR (Q2)oflongitudinaltotransversecosssections
LT
and find good agreement with experimental data for the ρ-state. For the 2S-state
R (Q2)=σL(Q2)/σT(Q2)
LT
10. E665
ZEUSBPC1995prel.
5 ZEUS1994prel.
H11994prel.
NMC
1
0.5
2S
0.1
0.05
ρ
0.01
0.05 0.1 0.5 1 5 Q10.2[GeV2]
Figure 5: Ratio of longitudinal to transverse integrated cross sections as function of Q2
both for theρ-meson and the 2S-state. There is only data for ρ-production [13].
5
we againpredictamarkedstructure whichis verysensitive to the node positionsin
the wave functions.
Further results refering to cross sections differential in t and the ratio of
−
2π+2π−-production via ρ′ and ρ′′ to π+π−-production via ρ are given in Ref. [1].
Ackknowledgements
The author thanks H.G. Dosch and H.J. Pirner for collaboration in the underlying
work.
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