Neutral pion photoproduction at high energies
A. Sibirtsev, J. Haidenbauer, S. Krewald, U.-G. Meißner, A.W. Thomas
aa r X i v : . [ h e p - ph ] M a r EPJ manuscript No. (will be inserted by the editor)
FZJ-IKP-TH-2009-4,JLAB-THY-09-935
Neutral pion photoproduction at high energies
A. Sibirtsev , , J. Haidenbauer , , S. Krewald , , U.-G. Meißner , , and A.W. Thomas , Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie) und Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn,Germany Excited Baryon Analysis Center (EBAC), Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Institute of Advanced Simulations, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Theory Center, Thomas Jefferson National Accelerator Facility, 12000 Jefferson Ave., Newport News, Virginia 23606, USA College of William and Mary, Williamsburg, VA 23187, USAReceived: date / Revised version: date
Abstract.
A Regge model with absorptive corrections is employed in a global analysis of the world data on the reac-tions γp → π p and γn → π n for photon energies from 3 to 18 GeV. In this region resonance contributions are expectedto be negligible so that the available experimental information on differential cross sections and single- and doublepolarization observables at − t ≤ GeV allows us to determine the non-resonant part of the reaction amplitude reliably.The model amplitude is then used to predict observables for photon energies below GeV. A detailed comparison withrecent data from the CLAS and CB-ELSA Collaborations in that energy region is presented. Furthermore, the prospectsfor determining the π radiative decay width via the Primakoff effect from the reaction γp → π p are explored. PACS.
Recently we have completed [1] a systematic analysis of pos-itive and negative pion photoproduction at invariant collisionenergies √ s> ρ , b , and a trajectories and pionexchange. Free parameters of the amplitudes were fixed in aglobal fit of the world data on differential cross section andsingle polarization observables available at high energies, i.e. at √ s> ρ , ω , and b exchanges. Again, the free parameters of the model arefixed in a global fit to high energy data. Specifically, in our fitwe include data on γp → π p differential cross sections and sin-gle and double polarization observables available in the energyrange 3 ≤ E γ ≤
18 GeV but with the restriction − t ≤ GeV .Those data were obtained around or before 1980. After thatwe proceed to analyse data on neutral pion photoproductionon the proton collected recently by the CLAS Collaboration atJLab [2] and by the CB-ELSA Collaboration in Bonn [3,4] inthe energy region 2 ≤ E γ ≤ π radiative decay width and proportional to the charge ofthe target Z . The π → γγ decay amplitude is related to sym-metry breaking through the axial anomaly and reveals one ofthe fundamental properties of QCD [6,7,8,9,10,11,12,13].The fact that the differential cross section due to the Pri-makoff amplitude is proportional to Z initiated strong activi-ties in the determination of the π -meson radiative decay frommeasurements with nuclear targets [14,15,16,17,18,19]. In-deed, most of the results [15,16,17,18] on the neutral pionlifetime, given by the PDG [20], were obtained by utilizing A. Sibirtsev et. al : Neutral pion photoproduction at high energies the Primakoff effect. A very precise experiment (PrimEx) onthe determination of the π → γγ decay width from π -mesonphotoproduction on nuclear targets is presently performed atJLab [21].It was argued [14,21,22], however, that pion absorptionon nuclei as well as the interference between the one-photonexchange and the nuclear amplitude may complicate the dataanalysis and significantly affect the accuracy of the results ob-tained on the π -meson lifetime. While the one photon ex-change amplitude is explicitly given by theory, the evaluationof the nuclear amplitudes requires a precise knowledge of theelementary amplitude on a nucleon and the spectral functionof the target nucleus. Although the Z -argument clearly favorsnuclear targets, this advantage might be completely counter-balanced by the benefit of the π -meson photoproduction onthe proton where one has a much better handle on the hadronicpart of the production amplitude. Therefore, in the present pa-per we will re-examine [23] the prospects for determining the π radiative decay width from neutral pion photoproduction ona proton target.The paper is organized as follows. In Sect. 2 we formulatethe reaction amplitudes. The parameters of the global fit to highenergy data are given in Sect. 3. The analysis of neutral pionphotoproduction at high and low energies is given in Sects. 4and 5, respectively. Results for the ratio of the γn → π n and γp → π p differential cross section and the total cross sectionfor the neutral pion photoproduction from the proton are pre-sented in Sect. 6. In this section we also examine the energydependence of the data for fixed four-momentum transfer. InSect. 7 we discuss the Primakoff effect on the proton target andexplore the prospects for future experiments. The paper endswith a short Summary. Guided by our previous analysis of charged pion photoproduc-tion [1] we use a gauge invariant Regge model which com-bines the Regge pole and cut amplitudes for ρ , ω and b ex-changes. At high energies the interactions before and after thebasic Regge pole exchange mechanisms are essentially elas-tic or diffractive scattering described by Pomeron exchange.Such a scenario can be related to the distorted wave approxi-mation and provides a well defined formulation [24,25,26,27,28,29] for constructing Regge cut amplitudes. This approach,which can also be derived in an eikonal formalism [30] with s -channel unitarity [31], is followed in our work. Detailed dis-cussions about the non-diffractive multiple scattering correc-tions involving intermediate states which differ from the initialand final states and the relevant Reggeon unitarity equations aregiven in Refs. [24,32,33,34]. For simplicity we do not considerthese much more involved mechanisms which would increasesignificantly the number of parameters to be fitted.We use the t -channel parity conserving helicity amplitudes F i ( i =1 , ..., ). Here F and F are the natural and unnaturalspin-parity t -channel amplitudes to all orders in s , respectively. F and F are the natural and unnatural t -channel amplitudesto leading order in s . Each Regge pole helicity amplitude is parameterized as F ( s, t ) = πβ ( t ) 1+ S exp[ − iπα ( t )]sin[ πα ( t )] Γ [ α ( t )] (cid:20) ss (cid:21) α ( t ) − , (1)where s is the invariant collision energy squared, t is the squaredfour-momentum transfer and s =1 GeV is a scaling parame-ter that allows us to define a dimensionless amplitude. Further-more, β ( t ) is a residue function, S is the signature factor and α ( t ) is the Regge trajectory.From Eq. (1), we see that the factor sin[ πα ( t )] would gen-erate poles at t ≤ when α ( t ) assumes the values , − , . . . .The function Γ [ α ( t )] is introduced to suppress those poles thatlie in the scattering region because Γ [ α ( t )] Γ [1 − α ( t )] = π sin[ πα ( t )] . (2)The structure of the vertex function β ( t ) of Eq. (1) is de-fined by the quantum numbers of the particles at the interactionvertex, similar to the usual particle exchange Feyman diagram.Both natural and unnatural parity particles can be exchangedin the t -channel. The naturalness N for natural ( N =+1 ) andunnatural ( N = − ) parity exchanges is defined as N = +1 if P = ( − J , N = − P = ( − J +1 , (3)where P and J are the parity and spin of the particle, respec-tively. Furthermore, in Regge theory each exchange is denotedby a signature factor S = ± defined as [24,35,36] S = P × N = ( − J , (4)which enters Eq. (1).To proceed further we should specify the trajectories thatcontribute to neutral pion photoproduction. Note that the mech-anisms for charged and neutral pion photoproduction are differ-ent. Indeed, for charged pion photoproduction pion exchangedominates at small − t while ω -exchange is forbidden altogether.For π -meson photoproduction just the opposite is the case.One of the significant differences between the data on neu-tral and charged pion photoproduction is the presence of thedip in the γp → π p differential cross sections at the squaredfour-momentum transfer t ≃ –0.5 GeV . A similar dip is alsoobserved in other reactions. For instance, in the π − p → π n re-action such a dip results from the ρ -exchange amplitude and itsposition is related to the ρ -trajectory [37].Therefore, one expects that ρ -exchange might dominate the γp → π p reaction too. Indeed, the square of the amplitude ofEq. (1) for the ρ -exchange is proportional to | F ( s, t ) | ∝ − cos[ πα ( t )] , (5)which has a zero at t ≃ -0.6 GeV , when taking the ρ -trajectoryas obtained recently in a fit [37] to the available data for the π − p → π n reaction. Note that the additional contributions tothe reaction amplitude ( ω , b ) can move this zero closer to ex-perimentally observed value.In the γp → π p reaction there is no difference between the ρ and ω -exchanges if their trajectories are the same. Thus in . Sibirtsev et. al : Neutral pion photoproduction at high energies 3 Table 1.
Correspondence between t -channel pole exchanges and thehelicity amplitudes F i ( i =1 ÷ ). Here P is parity, J the spin, I theisospin, G the G -parity, N the naturalness and S the signature factor. F i P J I G
N S
Exchange F -1 1 1 +1 +1 -1 ρF -1 1 0 +1 +1 -1 ωF +1 1 1 +1 -1 -1 b F -1 1 1 +1 +1 -1 ρF -1 1 0 +1 +1 -1 ω some previous studies [38] both contributions were consideredas just one exchange amplitude. However, in our study we treat ρ and ω exchanges separately, because any possible differencein the amplitudes might play a role in describing observables [39,40,41]. For instance it could allow us to fix the ratio of the γn → π n and γp → π p differential cross sections. Note thatin these two reactions the contributions from the isovector ex-change enter with a different sign, whereas the isoscalar ex-change is the same in both cases.The contributions of the ρ and ω -exchanges to the reactionamplitudes F i are indicated in Table 1 together with the rele-vant quantum numbers. Both ρ and ω have natural parity andcontribute to F and F . It was argued [40] that if there areno other contributions one would expect that the photon asym-metry Σ would be predominantly +1. (However, note that Σ vanishes at forward and backward directions [42].) This can beeasily understood when considering the relations between theobservables and the t -channel parity conserving helicity am-plitudes given in the Appendix A. Experimental data [43,44]available at high energies shows that the asymmetry indeedis consistent with +1 for t values above -0.4 and below -1.1.For -1 0, based on the relations given in Eqs. (17) and (18)of Appendix A. We will discuss this point below.The trajectories are taken in the following linear form, α ( t )= α + α ′ t , (6)where the intercept and slope for the ρ and b trajectories aretaken over from analyses of other reactions [1,24,37,48]. Ex-plicitly we have for the ρ and b trajectories α ρ = 0 . 53 + 0 . t , Table 2. Parameterization of the residue functions β ( t ) for the am-plitudes F i , ( i =1 , , ). Here c ij is the coupling constant where thedouble index refers to the amplitude i and the type of exchange j , asspecified in the Table. β ( t ) Exchange j Pole amplitudes F c ρ F c ω F c t b F c t ρ F c t ω F c exp[ d t ] ρ F c exp[ d t ] ω F c exp[ d t ] b F c t exp[ d t ] ρ F c t exp[ d t ] ω F c t exp[ d t ] b F c t exp[ d t ] ρ F c t exp[ d t ] ω F c t exp[ d t ] b α b = 0 . 51 + 0 . t . (7)The ω -trajectory was parameterized by Eq. (6) with the slope α ′ =0.8 GeV − , i. e. the same value as for other trajectories.The parameter α for ω -exchange was fixed by a fit to the data.The residue functions β ( t ) used in our analysis are com-piled in Table 2. They are similar to the ones used in some ofthe previous analyses [39,49]. These residues are slightly dif-ferent from those applied in our study of charged pion photo-production [1] because here we use explicitely the Γ functionin the amplitude parameterization of Eq. (1).In defining the Regge cut amplitudes we use the followingparameterization based on the absorption model [35,39,50,51,52] F ( s, t )= π β ( t )log ( s/s ) 1+ S exp[ − iπα c ( t )]sin[ πα c ( t )] Γ [ α c ( t )] (cid:20) ss (cid:21) α c ( t ) − , (8)with the trajectories defined by α c = α + α ′ α ′ P tα ′ + α ′ P , (9)where α and α ′ are taken from the pole trajectory given byEqs. (6) and (7), and α ′ P =0 . GeV − is the slope of the Pomerontrajectory. The residue functions β ( t ) of Eq. (8) are given in Ta-ble 2.In the very forward direction of the γp → π p reaction thereis an interference of the F amplitude with the one-photon ex-change amplitude, which is known as Primakoff effect [5]. Thiseffect allows to determine the radiative decay width of the π -meson. However, the experimental resolution of the γp → π p A. Sibirtsev et. al : Neutral pion photoproduction at high energies Table 3. Parameters of the model. Here c ij is the coupling constant forthe i -th amplitude and the type j of exchange, d j is a cut-off parameterfor the Regge cut amplitude. j c ij d j i =1 i =2 i =3 − . – − . –2 . - − . -3 - . – –4 − − . − . − . . . − . data available presently is insufficient to resolve the one-photonexchange amplitude. Thus, we omitted the interference region, i. e. | t | < . GeV from the fit in order to fix the F ampli-tude. But we add the one-photon exchange amplitude lateronand compare the results with the γp → π p differential crosssections available at very forward direction.The relations between the observables analyzed in our studyand the t -channel helicity amplitudes are summarized in Ap-pendix A. The relation between the F i , the s -channel helic-ity amplitudes and the invariant amplitudes are given in Ap-pendix B. The resulting parameters of the model are listed in Table 3. Theachieved χ /dof amounts to 1.4. We find that there are someinconsistencies between data from different experiments. Thus,it is not possible to improve the confidence level of our globalanalysis unless these inconsistent data are removed from thedata base. However, it is difficult to specify sensible criteria forpruning the data base.The intercept of the ω -trajectory at t =0 obtained from thefit is α ω =0.641 ± ρ -trajectory. The coupling constants listed in Table 3show that in case of the cut amplitudes there is some compensa-tion between the ρ , ω and b -exchange contributions. However,tiny differences in the trajectories are reflected in very differ-ent couplings for the cut amplitudes. Furthermore, we find thatthe solution is very sensitive to the differential cross section inthe vicinity of the dip. Indeed the dip structure results from thepole amplitudes. Since there are many data available around thedip, i.e. at t ≃ -0.5 GeV , the parameters are well constrained bythese data and the solution turns out to be stable.In order to avoid any dependence of the fit on the startingvalues of the parameters we have used the random walk methodto construct the initial parameter vector and we have repeatedthe minimization procedure. This allows us to practically ex-clude that we obtain just a local minimum. Furthermore, an ad-ditional examination is has been done by exploring the resultsfor the parameters correlation matrix in order to find out howunique the found minimum is. Fig. 1. Differential cross section for γp → π p as a function of − t atdifferent photon energies E γ or invariant collision energies √ s . Thedata are taken from Refs. [53] (filled inverse triangles), [54] (opentriangles), [55,56] (filled squares), [57,44] (open circles), [58] (opensquares) and [59] (crosses). The solid lines show the results of ourmodel calculation. Our results for the γp → π p differential cross sections at pho-ton energies above 3 GeV are presented in Fig. 1. The modelreproduces the data quite well. As we discussed previously thedata indeed suggest a minimum or shoulder around the value t = − . GeV , which was not observed in the differentialcross sections for the γp → π + n and γn → π − p reactions. Fur-thermore, the dip becomes more pronounced with increasingphoton energy.In Fig. 2 we display the data on the polarized photon asym-metry available for the reaction γp → π p at photon energiesabove 3 GeV. The data [57,44] at the energies E γ =4, 6, and . Sibirtsev et. al : Neutral pion photoproduction at high energies 5 Fig. 2. Polarized photon asymmetry for γp → π p as a function of − t at different photon energies E γ or invariant collision energies √ s . Thedata are taken from Refs. [43] (filled triangles) and [57,44] (open cir-cles). The solid lines show the results of our model calculation. Fig. 3. Target asymmetry for γp → π p as a function of − t at photonenergy E γ =4 GeV or invariant collision energy √ s =2.9 GeV. The tri-angles are data from Ref. [46], while the squares are from Ref. [45].The solid lines show the results of our model calculation. 10 GeV are sufficiently precise and clearly indicate a dip around t ≃ -0.5 GeV . The Regge calculations reproduce the experi-mental results resonably well.Next we take a look at the data available for the target ( T )and recoil ( P ) asymmetries. Fig. 3 shows experimental resultson the target asymmetry of the γp → π p reaction at a photonenergy of 4 GeV. The data [45,46] at the same energy are fromindependent measurements. The solid lines are the result of theRegge calculations. They are in agreement with the data withinexperimental uncertainties. Fig. 4. a) Recoil asymmetry for γp → π p as a function of − t . Opencircles are the data [47] obtained at photon energies from 3 to 7 GeV.The lines show the Regge calculations at photon energies 3 and 7 GeV.b) Illustration of the Worden inequality given by Eq. (10). Closed cir-cles are the difference between P taken from experiment [47] and P given by the Regge calculation. The solid line is − Σ with the polar-ized photon asymmetry taken from the calculations. The dashed lineindicates the case | P − T | =0 . The open circles in Fig. 4a) are experimental results for therecoil asymmetry in the reaction γp → π p for incident photonenergies between 3 and 7 GeV [47]. These data allow us to ex-amine whether the target and recoil asymmetries are different.This issue was considered in Ref. [47] via a direct comparisonof experimental results [45,46,47] available for the target andrecoil asymmetries. It was argued that T = P , so that there mustbe higher order contributions to the γp → π p reaction. Indeedfollowing Eqs. (17) and (18) the amplitude F is not negligiblein such a case.However, as remarked in Ref. [47], the measurements ofthe target and recoil asymmetries were done at different ener-gies and P is averaged over photon energies ranging from 3 to7 GeV. In particular, it was emphasized that without real calcu-lations any conclusions remain quite speculative. Now we caninvestigate this issue in more detail. The two lines in Fig. 4a)show the Regge results for E γ =3 GeV and E γ =7 GeV. Indeedthe asymmetries P obtained at the two energies are slightly dif-ferent. The calculations reproduce the data fairly well and fromthat we conclude that there are no solid arguments to claim that T = P at high energies and thus to speculate about any signifi-cance of the F amplitude.A further examination of F can by done by applying theWorden inequality [29] given by | P − T | ≤ − Σ (10)and shown in the Fig. 4b). Here the closed circles indicate thedifference between the experimental results [47] for P and thecalculation for T . The solid line is the difference − Σ with thepolarized photon asymmetry taken from the calculation. Notethat in this case the Regge results are in good agreement withthe data on the T and Σ asymmetries. Fig. 4 illustrates that the A. Sibirtsev et. al : Neutral pion photoproduction at high energies inequality [29] given by Eq. (10) is satisfied within the experi-mental uncertainties. In this section we compare our predictions with older data forenergies below 3 GeV but also with the most recent experimen-tal results [2,3] for differential cross sections collected by theCLAS Collaboration at JLab and by CB-ELSA in Bonn. Bothlatter experiments cover the energy range up to E γ ≃ GeV.As pointed out in Ref. [2] the JLab results disagree with theCB-ELSA measurements at forward angles. It will be inter-esting to inspect the observed discrepancy with regard to thepredictions by the Regge model. One knows from the case ofcharged pion photoproduction say, that the Regge phenomenol-ogy works well for forward angles even down to E γ ≃ GeV.As stressed in many studies [24,52,29], the Regge theoryis phenomenological in nature. There is no solid theoreticalderivation that allows us to establish explicitly the ranges of t and s where this formalism is applicable. Since there are sev-eral well-known nucleon resonances [20] in the energy rangeup to √ s ≃ . GeV, identified in partial wave analyses [60,61,62,63,64] of pion-nucleon scattering, we expect that deviationsof our predictions from the data will start to show up for ener-gies from E γ ≃ GeV downwards. But it will be interesting tosee whether and in which observables such discrepancies in-deed occur.We also present results utilizing the amplitudes from thepartial wave analysis (PWA) of the GWU Group [65,66,67,68], which was recently extended up to √ s ≃ γp → π p at photon energies from 2.02 to 3 GeV. This energy region cor-responds to invariant collision energies of 2.16 ≤√ s ≤ √ s = √ s = Fig. 5. Differential cross section for γp → π p as a function of − t atdifferent photon energies E γ or invariant collision energies √ s . Thedata are taken from Refs. [2] (filled triangles), [3] (filled circles), [54](open triangles) and [55,56] (filled squares). The solid lines show theresults of our model calculation. The dashed lines are the results basedon the GWU PWA [69]. which it was fitted) very well up to photon energies of 2.55GeV or invariant energies of √ s =2.37 GeV. Also, for invari-ant collisions energies around 2.34 ≃ ≤ − t ≤ . It is important tonote that our model does not include any resonance contribu-tion. The GWU PWA indicates the presence of the G (2400) ∆ -resonance at the upper end of the fit to πN elastic scatteringdata [68]. But it remains unclear whether this resonance alsohas a noticable impact on their results for neutral pion photo-production.The polarized photon asymmetry, measured [70] at photonenergies from 2.1 to 2.75 GeV, is shown in Fig. 6. Unfortu- . Sibirtsev et. al : Neutral pion photoproduction at high energies 7 Fig. 6. Polarized photon asymmetry for γp → π p as a function of − t at different photon energies E γ or invariant collision energies √ s . Thedata are taken from Refs. [70]. The solid lines show the results of ourmodel calculation. The dashed lines are the results based on the GWUPWA [69]. nately, the experimental results are afflicted by large uncertain-ties and, therefore, do not allow us to draw any more quantita-tive conclusions on the reliability of the Regge predictions. Butthe results are roughly in line with the data over the whole en-ergy region. The GWU PWA is in reasonable agreement withthe data at 2.1 GeV, but develops a qualitatively different be-havior with increasing energy. The data seem to indicate thepresence of a dip at around t ≃ -0.8 GeV at almost all shownenergies. The Regge model produces such a dip, but near thevalue t ≃ -0.5 GeV , while the results from GWU PWA exhibita dip structure only at the lowest energy considered.Fig. 7 shows target and recoil asymmetries in the γp → π p reaction, measured [70] at photon energies of 2 and 2.1 GeV.The data on the target asymmetry have quite small uncertainties Fig. 7. Target (filled squares) and recoil (open squares) asymmetriesfor γp → π p as a function of − t at different photon energies E γ orinvariant collision energies √ s . The data are taken from Refs. [70].The solid lines show the results of our model calculation. The dashedlines are the results based on the GWU PWA [69]. Fig. 8. Double polarization parameter G for γp → π p as a function of − t given at different photon energies E γ . The circles are results fromRef. [45], while the squares show the data from Ref. [74]. The solidlines show the results of our model calculation. The dashed lines arethe results based on the GWU PWA [69]. and exhibit a significant variation with four-momentum trans-fer squared. Apparently, at these energies T = P . The Reggecalculations reproduce the recoil and target asymmetry roughly,but only at very forward angles. On the other hand, the GWUPWA describes T as well as P fairly well over the considered t range. A. Sibirtsev et. al : Neutral pion photoproduction at high energies Fig. 9. Double polarization parameter H for γp → π p reaction at dif-ferent photon energies E γ . The data are from Ref. [74]. The solid linesshow the results of our model calculation. The dashed lines are the re-sults based on the GWU PWA [69]. Some comments with regard to the observed difference be-tween T and P at those energies seem to be in order. In the caseof t -channel non-resonant contributions the F helicity ampli-tude is given by higher order corrections and thus we neglect it,as was discussed previously [1]. But even if F does not vanishits general influence on the various observables is expected [24,71] to be small. Note that in approaches based on an effec-tive Lagrangian, which are commonly used at low energies, thecontribution from vector-meson exchanges to pseudoscalar me-son photoproduction also results in a vanishing invariant ampli-tude A and, thus, following Eq. (22), one would expect F =0 .However, at the same time resonances can contribute [72,73] tothe amplitude F so that the difference between T and R mightbe explained in a natural way.For the γp → π p reaction there are also data [74] for thedouble polarization parameters G and H . These data are im-portant for fixing the sign of the amplitude F , as is obviousfrom Eqs. (19) and (20). Since there are no data for the dou-ble polarization parameters at higher energies we cannot deter-mine the sign of the F amplitude within our fitting procedure.Therefore, we decided to use the data at low energies to fix thatambiguity for the parameters corresponding to the F ampli-tude, listed in the Table 2.Fig. 8 shows the double polarization parameter, G , mea-sured at photon energies from 2 to 2.3 GeV, which correspondto invariant energies of 2.15 to 2.28 GeV. The solid lines arethe results obtained from our Regge model. They are only veryqualitatively in line with the data. A similar conclusion mightalso be drawn when comparing the PWA results with the mea-surements.Fig. 9 shows the double polarization parameter, H , mea-sured for photon energies between 2 and 2.3 GeV. The solidlines are the results obtained from the Regge model. Again, Fig. 10. The ratio of the differential cross sections for π -photoproduction on neutrons and protons as a function of − t for dif-ferent photon energies E γ . The data are from Refs. [78] (circles), [79](triangles) and [80] (squares). The lines show the results of our modelcalculation. these are qualitatively in line with the experiment. It is interest-ing that the PWA predicts large negative values for H that varystrongly with the four-momentum momentum squared, whilethe Regge model predicts H ≃ . π photoproduction on neutrons and protons Information about the relative contributions of the isovector ρ (and b ) exchange and the isoscalar ω exchange amplitudes canbe obtained by comparing the differential cross sections for π -meson photoproduction on neutrons and on protons [25,75,76,77]. For the proton target the total reaction amplitude is givenby the sum of the isovector and isoscalar contributions, whilefor the neutron target it is given by their difference.In Fig. 10 we present the ratio of the differential cross sec-tions for π -photoproduction on neutrons and protons for dif-ferent photon energies. The available data demonstrate that theratio R =1 , which contradicts a statement given in Ref. [40].Note that the precise measurement [78] at E γ =4 GeV indicatesthat the ratio depends considerably on t .Fig. 11 shows the total cross section for the γp → π p reac-tion as a function of the invariant collision energy. The experi-mental results were obtained [3] by integration over the angulardistributions and involve an extrapolation into the forward andbackward regions using the results from the isobar model ofAnisovich et al. [81]. The solid line is the Regge result, ob-tained by integration of the calculated differential cross sectionover the range | t |≤ . . Sibirtsev et. al : Neutral pion photoproduction at high energies 9 Fig. 11. Total cross section for γp → π p . The circles are the experi-mental results taken from Ref. [3]. The lines show the results of ourmodel calculation. Fig. 12. Differential cross section for γp → π p as a function of − t at E γ = At first sight it looks as if the Regge model would over-estimate the integrated cross section significantly at the higherenergies, i.e. at energies where it is actually expected to agreewith the data. However, the amplitude that is used for obtainingthose cross sections in [3] has some shortcomings in the for-ward direction, as one can see in Fig. 12. Specifically, it failsbadly to describe the data at very small angles [54], whereasthe Regge model reproduces even those data rather well. Con-sequently, a determination of the total cross section that utilizesthat amplitude for extrapolating to forward angles will neces-sarily underestimate the “real” value. On the contrary, basedon the quality of our fit to the small-angle data at 3 GeV, oneexpects that the predictions of the Regge model for the inte-grated cross section should be very realistic. It is interesting tosee that the results of the isobar model [81] and of our Reggefit practically coincide in the range < − t< GeV . To complete our analysis of the data we take a look at the en-ergy dependence of the γp → π p differential cross sections atfixed four-momentum transfer squared t . This allows us to shedlight on the applicability of the Regge phenomenology with re-spect to the s - as well as the t dependence. It also facilitates theinspection as to whether potential discrepancies between thecalculations and data exhibit any systematic features.Fig. 13 shows the data considered in the present analy-sis. Here the differential cross sections are multiplied by thesquared invariant collision energy s . We multiply with this fac-tor because the high energy limit of the cross section at small − t , as given by the Regge formalism, is proportional to /s .Therefore, at high energies and small − t we expect that sdσ/dt approaches a constant value. For ease of comparison we scalethe data and the curves by powers of 10.The solid lines in Fig. 13 are the results of our Regge model.At very small | t | the Regge model reproduces the data ratherwell, even down to energies of √ s ≃ GeV. In general the dataseem to agree with the high energy limit as given by the Reggephenomenology from energies of √ s = . ≤| t |≤ GeV . Below this energy regionthe data show sizeable variations with regard to the predictionsof the Regge model. Specifically, for low energies and larger | t | our model results deviate systematically from the data and thediscrepancy increases with increasing squared four-momentumtransfer.Furthermore, as the squared four-momentum increases theenergy dependence of the data and the Regge calculations be-comes steeper. Although we fit the data in the range t ≥− t = − | t | . As we showed inRef. [1], the data on charged pion photoproduction at energies √ s ≥ . GeV and at large four-momentum transfer squared arepractically independent of t and are in line with the Dimen-sional Counting Rule [82,83]. According to the DCR for theinvariant amplitude M the energy dependence of the differen-tial cross-section is given as dσdt = | M | F ( t )16 π ( s − m N ) = cs − ( n i − n f − F ( t )16 πs ∝ cs − F ( t ) , (11)in the limit m N ≪ s , where m N is the mass of the nucleon.Here c is a normalization constant, while n i and n f are thetotal number of elementary fields in the initial and final states,respectively. For single pion photoproduction n i =4 and n f =5.Furthermore, F ( t ) is a form-factor, which does not depend onthe energy s but accounts for the t dependence of the hadronicwave functions and partonic scattering. We found that, withinthe experimental uncertainties, the data on π + and π − -mesonphotoproduction indicate that F ( t ) is almost constant, i.e. doesnot depend on the squared four momentum. Moreover, it turnsout that both negative and positive pion photoproduction can bedescribed with the same normalization c =11 mb · GeV , whenassuming that F ( t )= √ s ≥ et. al : Neutral pion photoproduction at high energies Fig. 13. Differential cross section for γp → π p as a function of the invariant collision energy, at selected fixed values of t . The symbols showthe data considered in the present paper. The solid lines are the results of our model calculation. The dashed lines are the results obtained withEq. (11). The data and lines are scaled with powers of 10. good agreement with the ansatz based on the DCR. However,note that at least at t = − the Regge calculation repro-duces the data better than the DCR. The Primakoff effect [5] has not only been observed in neu-tral pion photoproduction on nuclei but also in the γp → π p reaction [54]. This effect dominates the reaction cross sectionat low momentum transfer and can be used for the determina-tion of the π → γγ decay width. But there is an interferenceof the F amplitude with the Primakoff (one photon exchange)amplitude. Thus, the determination of the π radiative decayrequires a precise knowledge of the hadronic part of the am-plitude as well as accurate data. Unfortunately, the availableexperimental results [54] on the differential cross section in thenear forward direction are afflicted by considerable uncertain-ties, as is illustrated in Fig. 14. Here the dotted lines show theresult of our Regge model without one photon exchange, whichreproduces the data at angles above 10 ◦ , say, rather well. The solid lines in Fig. 14 are results obtained with the Pri-makoff amplitude included, where the latter is given by F P = 8 m p t s πΓ ( π → γγ ) m π F D ( t ) = √ Γ ˆ F P ( t ) . (12)Here m p and m π are the proton and pion mass, respectively, Γ is the π → γγ decay width and F D is the Dirac form factor ofthe proton. For the latter we adopt the parameterization givenin Ref. [84], F D ( t ) = 4 m p − . t m p − t − t/t ) (13)with t =0 . GeV , which is derived under the assumptionsthat the Dirac form factor F D of the neutron and the isoscalarPauli form factor vanish and that a dipole form is satisfactoryfor G M ≈ µG E ( t ) , cf. [84]. There are slight deviations fromthe dipole form in the region − t< we are concernedwith here, cf. for example Ref. [85], but we neglect those in thepresent exploratory study. The amplitude of Eq. (12) is added tothe helicity amplitude F of our Regge model. We show results . Sibirtsev et. al : Neutral pion photoproduction at high energies 11 Fig. 14. Differential cross section for γp → π p as a function of theangle θ in the cm system shown for different photon energies E γ .The data are taken from Ref. [54]. The dotted lines show the Reggecalculations without one photon exchange, while the solid lines arethe results obtained with inclusion of the one photon exchange. Thedash-dotted lines are results for the one photon exchange alone. based on Γ ( π → γγ ) =8.4 eV, i.e. the value that is given by thePDG as the average π -meson lifetime. The calculations arenot folded with the pertinent angular resolution function [54],because this quantity is is not available to us for the particularexperiment in question.Fig. 14 illustrates impressively the consequences of the Pri-makoff effect. As demonstrated in the preceeding sections, the γp → π p reaction amplitude can be well fixed by the huge setof data available at larger angles θ> o and at different photonenergies. This ensures that the hadronic contribution to the pho-toproduction amplitude is known quite precisely when extrap-olating to forward angles. Apparently, the situation is differentfor measurements on nuclear targets. In that case the reactionsat large angles are entirely dominated by incoherent photopro-duction and it is very difficult to fix the coherent nuclear am-plitude, which contributes at forward angles [86,87,88,89,90].Thus, π -meson photoproduction on the proton could offer apromising alternative for the determination of the π radiativedecay width. We should emphasize, however, that for the reac-tion on the proton the relative phase between the hadronic partand the Primakoff amplitude is also an unknown quantity. Inthe present exploratory calculation we have simply added thelatter as given in Eq. (12) to our Regge amplitude. But in a con-crete application to experimental data one needs to determinethis phase together with the π radiative decay width by a fit todifferential cross sections at forward angles.For completeness we also show the γp → π p differentialcross section at forward angles for the photon energy E γ =2 GeV,cf. Fig. 15. Here the squares are data from Ref. [54], trianglesare the results from the CLAS experiment [2] and circles aredata from the CB-ELSA Collaboration [3]. Unfortunately, the Fig. 15. Differential cross section for γp → π p as a function of the an-gle θ in the cm system shown for photon energy E γ =2 GeV. The dataare taken from Refs. [54] (squares), [2] (triangles) and [3] (circles).The dotted line shows the Regge calculations without one photon ex-change, while the solid line is the results obtained with inclusion ofone photon exchange. The dash-dotted line is the result for one pho-ton exchange alone. The dashed line indicates the results based on theGWU PWA [69]. latter recent measurements [2,3] do not cover the region of veryforward angles. The dotted line in Fig. 15 is the result of theRegge model alone while the solid line was obtained with in-clusion of the Primakoff amplitude. Our model reproduces thedata at forward angles surprisingly well, but it underestimatesthe experimental results at θ> o . The dashed line indicatesresults based on the current solution of the GWU PWA [68], Fig. 16. Differential cross section for γp → π p as a function of theangle θ in the cm system shown for different photon energies E γ .Same description of curves as in Fig. 14.2 A. Sibirtsev et. al : Neutral pion photoproduction at high energies Fig. 17. Differential cross section for γp → π p at E γ =5.8 GeV di-vided by dσ P /dt (Eq. (14)). The data are from Refs. [54]. Same de-scription of curves as in Fig. 14. which describes the data at θ ≥ o but does not reproduce the γp → π p differential cross section at forward angles.Fig. 15 illustrates an interesting feature. The excellent agree-ment of our Regge calculation with the available data at for-ward angles could be an indication that even at such low ener-gies the forward photoproduction is still dominated by t -channelcontributions. If so then there would be indeed very good con-ditions for determining the π -meson lifetime from measure-ments of neutral pion photoproduction at photon energies around E γ ≃ GeV that are accessible presently at Jlab and ELSA.Predictions for higher energies are presented in Fig. 16.Experiments in this energy region will become feasibly onceJLab’s 12 GeV Upgrade Project will be completed.Finally, let us provide another view on the present situa-tion. In Fig. 17 we show again the available cross-section dataat 5.8 GeV [54], but divide the data and the curves by the contri-bution of the pure Primakoff amplitude with a normalization sothat the result at zero angle coincides with the π decay width,i.e. we divide by dσ P dt = 132 π " t | ˆ F P | ( t − m p ) . (14)Note that a logarithmic scale is used for the ordinate. On thisplot one can see down to which angles the results are still dom-inated by the hadronic amplitude. It is obvious that an appro-priate set of data points below 3 degrees, say, and with highprecision would allow an extrapolation to zero degrees and,thus, a determination of Γ ( π → γγ ) . The presently availabledata are too sparse and too inaccurate for performing such anextrapolation reliably. In the present paper we performed a global analysis of theworld data on the reactions γp → π p and γn → π n for photonenergies from 3 to 18 GeV within the Regge approach. In this region resonance contributions are expected to be negligibleso that the available experimental information on differentialcross sections and single- and double polarization observablesat − t ≤ GeV allows us to determine the reaction amplitudereliably. The Regge model was constructed by taking into ac-count both pole and cut exchange t -channel helicity amplitudesand includes the ρ , ω and b trajectories. The model parameterssuch as the helicity couplings were fixed by a fit to the availabledata in the considered E γ and t range.An excellent overall description of the available data wasachieved, indicating that for the energy and t range in questionsingle pion photoproduction is indeed dominated by nonreso-nant contributions. The model amplitude was then used to pre-dict observables for photon energies below GeV. A detailedcomparison with recent data from the CLAS (JLab) and CB-ELSA (Bonn) Collaborations in that energy region was pre-sented. It turned out that the resulting differential cross sectionsfor γp → π p were still in reasonable agreement with those newdata down to E γ ≈ − t ≤ GeV , while the veryforward data were reproduced even down to photon energies aslow as 2 GeV.Since our Regge amplitude works so well for forward an-gles, even at very low energies, we utilized it to explore theprospects for determining the π radiative decay width via thePrimakoff effect from the reaction γp → π p . Those calcula-tions indicate that corresponding measurements on a proton tar-get could be indeed promising. But, evidently, the precision towhich the decay width can be determined will depend cruciallyon the number of data points that one can collect at very smallangles and on the accuracy and the angular resolution one canachieve. Acknowledgements We acknowledge fruitful discussions with A. Bernstein, W. Chen,M. Dugger, L. Gan, H. Gao, A. Gasparian, J. Goity, I. Strakovsky,and U. Thoma. This work is partially supported by the HelmholtzAssociation through funds provided to the virtual institute “Spinand strong QCD” (VH-VI-231), by the European Community-Research Infrastructure Integrating Activity “Study of StronglyInteracting Matter” (acronym HadronPhysics2, Grant Agree-ment no. 227431) under the Seventh Framework Programme ofEU, and by DFG (SFB/TR 16, “Subnuclear Structure of Mat-ter”). This work was also supported in part by U.S. DOE Con-tract No. DE-AC05-06OR23177, under which Jefferson Sci-ence Associates, LLC, operates Jefferson Lab. A.S. acknowl-edges support by the JLab grant SURA-06-C0452 and the COSYFFE grant No. 41760632 (COSY-085). . Sibirtsev et. al : Neutral pion photoproduction at high energies 13 Utilizing the relations of Ref. [91] the γN → πN observablesanalysed in our study are given in terms of the amplitudes F i ( i =1 , ..., by [38] dσdt = 132 π (cid:20) t | F | − | F | ( t − m N ) + | F | − t | F | (cid:21) , (15) dσdt Σ = 132 π (cid:20) t | F | − | F | ( t − m N ) − | F | + t | F | (cid:21) , (16) dσdt T = √− t π Im (cid:20) − F F ∗ ( t − m N ) + F F ∗ (cid:21) , (17) dσdt P = √− t π Im (cid:20) − F F ∗ ( t − m N ) − F F ∗ (cid:21) , (18) dσdt G = Im [ tF ( F ∗ − m N F ∗ )+ F ( tF ∗ − m N F ∗ )]16 π ( t − m N ) , (19) dσdt H = √− t Im [ F ( F ∗ − m N F ∗ )+ F ( tF ∗ − m N F ∗ )]16 π ( t − m N ) . (20)In order to account for the correct behavior at very small angles t should be replaced by t − t min in the above formulae, where t min = − ( m π / E γ ) . 10 Appendix B Here we provide the relation between the t -channel helicityamplitudes F i and the s -channel helicity amplitudes S , S , N and D . Following Wiik’s abbreviations [92], S and S aresingle spin-flip amplitudes, N is the spin non-flip and D is thedouble spin-flip amplitude, respectively. The asymptotic cross-ing relation, which is useful for the analytical evaluation of thehelicity amplitudes, is given by F F F F = − √ π √− t m N √− t −√− t m N √− t √− t t m N √− t − m N √− t t − S NDS . (21)Note that Eq. (21) is appropriate only at s ≫ t , since it doesnot account for higher order corrections that are proportionalto t/ m N . The amplitudes F i are related to the usual CGLNinvariant amplitudes A i [93] by F = − A + 2 m N A ,F = A + tA ,F = 2 m N A − tA ,F = A . (22)Expressions for the experimental observables in terms of theamplitudes A i are listed, for instance, in Ref. [94]. The oftenused multipole amplitudes can be constructed from the helicityamplitudes using the relations given in Refs. [95,96,97]. References 1. A. Sibirtsev, J. Haidenbauer, S. Krewald, T.S.H. Lee, Ulf-G.Meißner and A.W. Thomas, Eur. Phys. J. 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