High Energy Behaviour of Light Meson Photoproduction
Moskov J. Amaryan, William J. Briscoe, Michael G. Ryskin, Igor I. Strakovsky
HHigh Energy Behaviour of the Light Meson Photoproduction and the ”QuarkCounting Rules”
Moskov J. Amaryan, William J. Briscoe, Michael G. Ryskin, and Igor I. Strakovsky ∗ Department of Physics, Old Dominion University, Norfolk, VA 23529, USA Institute for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052, USA Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg, 188300, Russia
We evaluated recent CLAS Collaboration measurements for the 90 ◦ meson photoproduction offthe nucleon using a tagged photon beam spanning the energy interval s = 3 −
11 GeV . The resultsare compared with the ”Quark Counting Rules” predictions. ∗ Corresponding author: [email protected] a r X i v : . [ h e p - ph ] F e b I. INTRODUCTION
Binary reactions in QCD with large momentum transfer involve quark and gluon exchanges between collidingparticles. The quark counting rule (QCR) of Brodsky and Farrar [1] and Matveev, Muradyan, and Tavkhelidze [2]has a simple recipe to predict the energy dependence of the differential cross sections of two-body reactions at largemeson production angles when t/s is finite and is kept constant. The fixed production or scattering angle behaviorfor exclusive processes is expected to be [3] dσ/dt ( s ) ∝ s − ( n − , (1)where n is the minimum number of fundamental constituents (quarks) and s , t , and u are Mandelstam variables. Ifthe photon is assumed to be one elementary field, then the prediction for a meson photoproduction is dσ/dt ( s ) ∝ s − . (2)For the hadron-proton interaction, the counting rule works well, where hadron is a pion, kaon, proton, or antiproton [4–7]. The light meson photoproduction was examined in terms of the counting rule in Refs. [8–17]. As has been observed,first of all at SLAC by Anderson et al. , the reaction γp → π + n ( s = 8 . −
15 GeV ) shows agreement with constituentcounting rules that predict the cross section should vary as s − and (n - 2) = 7 . ± . s = 6 GeV , where baryon resonances are still playing a role. .Note, however, that the Quark Counting Rules account for the minimum numbers of elementary hard processesneeded to provide a large momentum transfer to the hadron. At a very large energies, these rules are modified by theso-called Sudakov form factor [19].Indeed, it is very improbable that two ensembles of constituents can get a strong transverse kick and radiate nogluons. Of course, the probability of a new gluon emission is suppressed by the QCD coupling constant α s , butsimultaneously it can be enhanced by the square of the large logarithm - ln s . The probability not to emit anyadditional gluons is called the Sudakov form factor. Thus for a very large s , we expect that the cross section of thelarge angle hadron-hadron scattering should fall down with s faster than the QCR prediction [20, 21]. The role ofSudakov form factor in large angle pp elastic scattering was considered in Refs. [22, 23].On the other hand, it was shown in Ref. [24] that due to the point-like nature of the photon, the Sudakov formfactor is absent in the case of large angle photoproduction. Thus, photoproduction allows one to check the QCRdirectly in its original form.In the present paper, we examined how the counting rules are applicable to the lightest meson photoproduction offthe nucleon up to s = 11 GeV , where modern data are available mostly produced by the CLAS Collaboration atJefferson Laboratory.Recall that there are three options of how one can consider a photon when it interacts to nucleon: • No constituents ( n γ = 0) or dσ/dt ( s ) ∝ s − , • Photon is a point-like particle which participate the strong interaction ( n γ = 1) or dσ/dt ( s ) ∝ s − , • There is a q − ¯ q configuration which actually participates in the interaction ( n γ = 2) or dσ/dt ( s ) ∝ s − II. LIGHT MESON PHOTOPRODUCTION REACTIONS
The JLab6 era has ended at Jefferson Laboratory leaving in its wake a plethora of cross section measurements forlight meson photoproduction off the nucleon. There is a unique opportunity to bridge the resonance and high-energyregions, in particular that encompassing the region in which ”Regge” theory is applicable, and evaluate the quarkcounting rule phenomenology with differential cross sections above the ”resonance” energies (Table I).The new CLAS high statistical cross sections, for instance, obtained recently for γp → π p [17] are compared inFig. 1 (top left) with previous data from CLAS measurements [25]. At higher energies (above s ∼ ) andlarge pion production angles ( θ = 90 ◦ ) in center-of-mass (c.m.) frame, the results are consistent with the s − scalingexpected from the QCR. The black dash-dotted line on 90 ◦ is a result of the best-fit of new CLAS data only [17],performed with power function ∝ s − ( n − , leading to ( n −
2) = 6 . ± .
26 (Table II).The previous CLAS study for ρ [10] and ω [12] results in ( n −
2) = 7 . ± . . ± .
7, respectively. Mesonswere identifies via the ρ → π + π − and ω → π + π − π channels. N.B. that the database for these analyses was limited Potoproduction of K-mesons were considered in terms of QCR in Ref. [18]
TABLE I. List of CLAS light meson photoproduction measurements off the nucleon.Reaction Ref. Reaction Ref. γp → π p [17, 25] γp → K + Σ [38–40] γp → π + n [11, 26] γp → K + Λ(1450) [41] γn → π − p [11, 13, 27] γp → K + Λ(1520) [41, 42] γp → ηp [28–30] γp → K + Σ(1385) [41] γp → η (cid:48) p [29, 31–33] γp → K (892) + Λ [43] γp → ωp [12, 34] γp → K (892) + Σ [43] γp → ρ p [10] γp → K (892) Σ + [44] γp → φp [35, 36] γp → f (1285) p [32] γp → K + Λ [37–39] by s = 6 . − . and divided into 3-4 energy bins. Then the joint analysis of the CLAS [37] and SLAC [8] crosssections for for the reaction γp → K + Λ covering the range s = 4 . − . gave ( n −
2) = 7 . ± . s − . TABLE II. Power factor ( n −
2) in Eq. (1) for light meson photoproduction off the nucleon came from the CLAS Collaboration.1st column gave reactions and 4th column shown best-fit results for the energy s ranges listed in the 2nd column and | t | rangesgiven in the 3rd column. Sources are given in the 5th column. To perform the best-fit for γp → π + n , we added SLAC data [8]to JLab Hall A measurements [11, 26].Reaction s | t | (n–2) Ref.(GeV ) (GeV ) γp → π p ± γp → π + n ± γn → π − p ± γp → ηp ± γp → η (cid:48) p ± γp → ωp ± γp → φp ± γp → K + Λ 4.0– 8.0 0.3–2.9 7.28 ± γp → K + Σ ± γp → K + Λ(1520) 4.8– 9.0 0.9–3.2 6.65 ± γp → K (892) + Λ 4.2– 8.1 0.7–2.6 6.65 ± γp → K (892) + Σ ± γp → f (1285) p ± For our analysis, we included a number of light meson photoproduction data sets produced by the CLAS Collabo-ration on the proton and neutron for incident photon energies above s = 3 GeV , carried out during the past 20 years.For one particular case, the γp → π + n analysis, we included JLab Hall A [11] and SLAC [8] measurements. Theresults (Fig. 1 and Table II) are consistent with the ( n −
2) = 7 scaling expected from the QCR. Oscillations observedat low energies indicate that the QCR requires higher energies and higher | t | and | u | before it can provide a validdescription. Obviously, the extended energy range would be more definitive; our results do appear to be consistentwith this limit. The JLab12 and EIC programs are capable of providing the data needed to improve our results.Recently, the analysis of the CLAS γp → η (cid:48) p , γp → K + Λ, and γp → K + Σ [15] data covered a limited energyrange of s = 6 . − . shown that the energy behaviour of 90 ◦ cross section is dσ/dt ( s ) ∝ s − . While in thecase of γp → ηp , γp → ωp , and γp → φp , results are ( n −
2) = 12 . ± .
2, ( n −
2) = 9 . ± .
1, and ( n −
2) = 12 . ± . η → π + π − π , ω → π + π − π , and φ → K + K − , respectively.The A2 Collaboration at MAMI reported differential cross sections for the γp → ωp close to threshold [45]. Theomega-meson was identified via a radiative decay mode ω → π γ . As Figure 4 of Ref. [45] shows, there is a disagreementbetween CLAS and A2 measurements below s = 3 GeV .The difference between our analysis and analysis by Dey [15] who obtained a larger power ( n −
2) for the reactions γp → ηp , γp → ωp , and γp → φp is understandable due to different energy ranges of the data included in the fits. FIG. 1. Differential cross section of γN → MB , d σ /dt, at large meson production angle θ = 90 ◦ in c.m. as a functionof invariant energy squared, s (here M is a meson and B is a baryon). Data are γp → π p [17, 25] (blue filled circles), γp → π + n [8, 11, 26], (cyan open circles), γp → π − p [11, 27], (magenta open squares), γp → ηp [28, 29], (red open asterisks), γp → η (cid:48) p [29, 32, 33], (green open diamonds), γp → ωp [12, 34], (red open triangles), γp → φp [35], (magenta filed triangles), γp → K + Λ [37], (yellow filled diamonds), γp → K + Σ [40], (green filled squares), γp → K + Λ(1520) [42], (cyan open stars), γp → K (892) + Λ [43], (magenta filled stars), γp → K (892) + Σ [43], (yellow crosses), and γp → f (1285) p [32] (blue crosses).The black dash-dotted line is a result of the best-fit summarized in Table II. Indeed, as one can see in Fig. 1 for these reactions, there is a steeper energy dependence of the higher s part of thedistribution. For the case of the φ (and partly η ) photoproduction, this can be considered as a hint in favour ofthe noticeable role of the five quark ( uuds ¯ s ) component in the proton wave function. Having such a component, theprocess can be considered as the constituent strange quark interchange between the proton and the φ meson. However,this explanation is not directly applicable to the ω photoproduction. Of course, in the case of vector meson productionbesides the constituent quark interchange, there is a contribution caused by two gluon (Pomeron) exchange. Howeveras it is seen in Fig. 2 (Lower Panel), the absolute values of the π and ω photoproduction 90 ◦ cross sections withinthe experimental error bars are practically equal. This looks natural for the case of constituent interchange assumingthe same radial wave functions of the pion and ω -meson. That is we have to conclude that the two gluon contributionis small. The additional interesting fact is that the φ and f (1285) cross sections are close to each other. Note also that for this mechanism, we have to expect a smaller and not a larger power ( n −
2) since two gluons provide a better possibilityto balance the momenta transferred to the quarks in the final hadrons.
Recall that in the case of γp → ωp and γp → φp , both analyses (our and [15]) used the same experimental data.This indicates the necessity of more experimental data in a wider energy range to obtain a more stringent constrainton the fit parameters.Let us re-frame dσ/dt ( s ) (Fig. 1) into dσ/dt ( s ) · ( s /G ( t ) ). First, recall that the proton dipole form factor is G ( t ) = (1 + | t | / . − describes all four-momentum dependencies of both electric and magnetic form factors ofproton quite well [46]. So we expect dσ/dt ( s ) ∝ G ( t ) /s . (3)Moreover, it appears natural to introduce a similar ”infrared cutoff” at a lower s as well and to replace 1 /s by1 / ( | t | + 0 . . Thus in Fig. 2, we plot the product dσ/dt ( s ) · ( | t | + 0 . . As it is seen, now the s behaviour of thisproduct is rather flat down to s = 2 − . Additionally, Figure 2 shows that the accuracy (and dispersion) of thedata points is better seen here than on Fig. 1 and it demonstrates the possible role of the infrared cutoff ( | t | + 0 . FIG. 2. Differential cross section of the light meson production off the nucleon d σ/dt · ( | t | + 0 . at meson production angle θ = 90 ◦ in c.m. as a function of c.m. energy squared s . Upper Panel: Pseudoscalar-mesons and Lower Panel: Vector-mesonsdata. Additionally, there is a differential cross section for γp → π p to compare with γp → ωp data. The notation for thedifferent reactions is the same as in Fig. 1. In Figure 3, the differential cross section dσ/dt is plotted as a function of the t -Mandelstam at s = 8 . .The red open triangles are measured data points of the γp → ωp reaction [12], while the blue solid circles are π photoproduction data points [17]. As one can see at lower values of t , the cross section of ω photoproduction is anorder of magnitude higher than that of π photoproduction, however at higher values of t , the ω photoproduction crosssection is in general still higher but the difference is not as dramatic as at lower values of t . It has to be mentionedthat data on this figure are for all meson production angles. When only the 90 ◦ production angle data are selected,these two cross sections at higher values of s reach the same level as well as all other meson production data, exceptof the φ and the f (1285) photoprodution cross sections, which lie significantly below the other mesons plateau athigher energies.It is interesting to see that the φ and f (1285) production cross sections at higher energies and the 90 ◦ productionangle are equal to each other within statistical errors, which may indicate a common mechanism of their production.While the φ has definitely s ¯ s quark structure, the K ¯ Kπ branching ratio of f (1285) is on the order of 10% whichmeans that in average the s ¯ s component of the wave function of the f (1285) is small. However, it seems that, aspresented data may indicate, the s ¯ s component of the f (1285) wave function becomes dominant in the hard scatteringprocess, when all three Mandelstam variables s , t , and u are large. FIG. 3. Differential cross section at s = 8 . for the reaction γp → π p for | t | = 0 . − . (90 ◦ corresponds to | t | = 3 . ) shown by the blue filled circles [17] and for the reaction γp → ωp for | t | = 0 . − . (90 ◦ corresponds to | t | = 3 . ) shown by red open triangles [12]. III. SUMMARY AND CONCLUSIONS
In the present paper, we study the energy dependence of the 90 ◦ light meson photoproduction off the nucleon.We consider practically all available experimental data obtained by the CLAS Collaboration over more than the lasttwo decades and compare the results with the Quark Counting Rules predictions. We emphasize that in the case of photoproduction the QCR prediction does not affected by the Sudakov form factor. This fact allows a more directinterpretation of the observed results. FIG. 4. Simplest diagram for the large t meson photoproduction. Thanks to the point-like nature of the photon, the 90 ◦ cross section dσ/dt ∝ s − . The average value of ( n −
2) forall reactions listed in Table II is 6 . ± .
18. The explanation of the s − instead of s − or s − is: • In terms of Brodsky-Farrar [1]: ”In photoproduction amplitude, the balance between the quarks momenta wasprovided by the highly virtual quark with propagator 1 / ˆ q ∝ / √ s (Fig. 4) instead of the gluon for which thepropagator is ∝ /s .” • It terms of Matveev et al. [2]: ”In photoproduction, the incoming q ¯ q pair is produced (in the case of a largemomentum transferred) very close to the interaction point and not in advance (at a large distance) as in thevector meson dominance model. That is in the incoming state, we deal with a ”point-like” q ¯ q pair and only inthe final state we have two quarks separated by a large ( ∼ . − (cid:112) /s factor.”Let us note that the cross sections for the light meson photoproduction off the nucleon at 90 ◦ is very small (minimal)and for that reason it may cause a problem for the best-fit analysis using Eq. (2).Obviously, the JLab6 program is limited by s ≤
11 GeV . Within the JLab12 program, the approved by JLab PACproposal E12—14—005 for Hall C can extend the measurement of the γp → π p reaction up to s ≤
20 GeV [47]. ACKNOWLEDGMENTS
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