Phenomenological analysis of near threshold periodic modulations of the proton timelike form factor
aa r X i v : . [ nu c l - t h ] M a r Phenomenological analysis of near threshold periodic modulationsof the proton time-like form factor
A. Bianconi ∗ Dipartimento di Ingegneria dell’Informazione, Universit`a di Bresciaand Istituto Nazionale di Fisica Nucleare,Gruppo Collegato di Brescia, 25133 Brescia, Italy
E. Tomasi-Gustafsson † CEA,IRFU,SPhN, Saclay, 91191 Gif-sur-Yvette Cedex, France bstract We have recently highlighted the presence of a periodically oscillating 10 % modulation in theBABAR data on the proton time-like form factors, expressing the deviations from the point-likebehavior of the proton-antiproton electromagnetic current in the reaction e + + e − → ¯ p + p . Here wedeepen our previous data analysis, and confirm that in the case of several standard parametrizationsit is possible to write the form factor in the form F + F osc , where F is a parametrization expressingthe long-range trend of the form factor (for q ranging from the ¯ pp threshold to 36 GeV ), and F osc is a function of the form exp( − Bp ) cos( Cp ), where p is the relative momentum of the final¯ pp pair. Error bars allow for a clean identification of the main features of this modulation for q <
10 GeV . Assuming this oscillatory modulation to be an effect of final state interactionsbetween the forming proton and the antiproton, we propose a phenomenological model based ona double-layer imaginary optical potential. This potential is flux-absorbing when the distancebetween the proton and antiproton centers of mass is > ∼ < ∼ pp interaction is well known in thephenomenology of small-energy antiproton interactions, and is due to the annihilation of ¯ pp pairsinto multi-meson states. We interpret the flux-creating part of the potential as due to the creationof a 1 /q -ranged state when the virtual photon decays into a set of current quarks and antiquarks.This short-lived compact state may be expressed as a sum of several hadronic states including theones with large mass Q n ≫ q , that may exist for a time t ∼ / ( Q n − q ). The decay of these largemass states leads to an intermediate stage regeneration of the ¯ pp channel. PACS numbers: 12.40.Nn Regge theory, duality, absorptive/optical models 13.40.Gp Electromagnetic formfactors ∗ E-mail: [email protected] † E-mail: [email protected] . INTRODUCTION Both the annihilation reactions e + + e − → ¯ p + p, (1)¯ p + p → e + + e − , (2)have been used to extract the electromagnetic form factors (FFs) of the proton in the time-like (TL) region (for a recent review see [1] and references therein). Assuming that theinteraction occurs through one photon exchange, the annihilation cross section is expressedin terms of the FF moduli squared, as FFs are of complex nature in the explored kinematicalregion [2].The collected statistics has not permitted the individual determination of the electric( G E ) and magnetic ( G M ) FFs due to the available limited luminosity. The cross section σ of the reactions (1) and (2), allows to extract the squared modulus of an effective formfactor F p , that is in practice equivalent to the assumption G E = G M (strictly valid only atthreshold) [3]: | F p | = 3 βq σ πα (cid:18) τ (cid:19) , (3)where α = e / (4 π ), β = p − /τ , τ = q / (4 M ), q is the squared invariant mass of thecolliding pair, and M is the proton mass. The effect of the Coulomb singularity of the crosssection at the ¯ pp threshold is removed by the β factor: β → q → M , so that βσ isfinite and the effective form factor is expected to be finite at the threshold.The reactions (1) and (2) test close-distance components of the wave function of the ¯ pp system, that are supposed to be suppressed because of ¯ pp annihilation. Data on the ¯ p -nucleonand ¯ p − nucleus annihilation process at low energies (see [4–16], and the related theoreticalanalyses [17–19]) show that a proton and an antinucleon overlap little. When their surfacescome in touch, or even within a distance of 1 fm, they annihilate into other hadron states.Elastic scattering is present, but either of diffractive origin (for p ≫
100 MeV), or of refrac-tive repulsive hard-core nature (near threshold). In all cases, the wave function of the ¯ pp relative motion is estimated to be strongly suppressed at distances lower than 1 fm. On theother side, reactions (1) and (2) involve a virtual photon with center of mass (c.m.) energy √ s ≥ M ≈ r ≤ . pp system, and TL FF complement, in this respect, the informationacquired from other annihilation experiments.Until recently, uncertainties and discontinuities between data coming from different mea-surements have prevented from appreciating the continuity features of TL FF data over alarge q range. The recent results from the BABAR collaboration [20, 21], cover a q rangegoing from near the threshold to 36 GeV , with more than 30 data points only in the region q <
10 GeV .Specific features of the effective FF related to final state interactions between ¯ p and p appear when expressing | F p ( q ) | in terms of the 3-momentum of the relative motion ofthe two hadrons. This has been illustrated in a recent work [22], where we have highlightedperiodic features in a modulation of the order of 10 % , superimposed on the long-range trendof the effective FF. The precision of the available data does not forbid the interpretationwhere the oscillation pattern is attributed to independent resonant structures, as in Ref. [23].However, the underlying assumption of the present work is that the oscillations are expressionof a unique interference mechanism, affecting all the q range where the oscillations arevisible. Of course, the two points of view may coexist within a model where two or moreresonance poles are the result of a global underlying mechanism, as in [24], or within aduality framework. As observed in our final discussion, the proposed phenomenologicalinterpretation may be framed within several models, including multiple-pole ones.In the present work we first scrutinize these oscillations by expanding the data analysisof Ref. [22]. We use four different parametrizations from the literature for the so-called”background” term (the effective form factor as it appears if one neglects the small oscillatingmodulation). For each choice of the background, we fit the residual modulation, visible inthe difference between the data and the background fit. We analyze the uncertainty on theperiodical character of the oscillations, and on their possible long-range scaling behavior(Section II).Next, we present a phenomenological model for the rescattering origin of the oscillations,within a DWIA (distorted wave impulse approximation) scheme where the outgoing (orincoming) hadron waves are distorted by an optical potential (Section III). In absence ofdistorting potentials, the background form of the TL FF is recovered (Section IV). Thisanalysis shows that it is possible to reproduce most of the features of the observed oscillations4his way, but important constraints must be satisfied by the rescattering potential (SectionV). Conclusions summarize the main finding of the paper. II. ANALYSIS OF THE DATA
The effective proton FF extracted from BABAR data on e + + e − → ¯ p + p ( γ ) [20, 21], isreported in Fig. 1 (black circles) as a function of q , that is equivalent to the total energysquared s in the TL region. As it can be noticed in the insert that highlights the nearthreshold region, 4 M ≤ q ≤
10 GeV , the data show irregularities. These irregularitiesacquire a peculiar structure when q is replaced by the relative momentum of the ¯ pp systemin the rest frame of one of the hadrons [22].We introduce a function of the form F ( p ) ≡ F ( p ) + F osc ( p ) (4)where1. The 3-momentum p ( q ) ≡ √ E − M , E ≡ q / (2 M ) − M. (5)is the momentum of one of the two hadrons in the frame where the other one is atrest.2. F ( p ) (”regular background term” ) is a function expressing the regular behavior ofthe FF over a long q range.3. F osc ( p ) describes the deviation of the TL FF from the long-range regular backgroundappearing in the region 0 ≤ p < ∼ M ≤ q < ∼ q = p q .Different forms available from the literature can be used for the background term. Asmeasured by BABAR, F [ p ( q )] is slightly steeper than expected on the ground of the corre-sponding fits of the space-like (SL) FF (dipole-like shape) and of the power law correspondingto quark-counting rules [25, 26]. The recent data are best reproduced by the function F R | F R | ( q ) | = A (1 + q /m a ) [1 − q / . , A = 7 . − , m a = 14 . . (6)Other parametrizations have been proposed. The world data prior to BABAR resultswere well reproduced in the experimental papers [28] according to the function: | F S ( q ) | = A ( q ) log ( q / Λ ) , (7)where q is expressed in GeV , A = 40 GeV − and Λ = 0 .
45 GeV .The functional form of Eq. (7) is driven by the extension to the TL region of the dipolebehavior. The dipole model of the SL FFs, more precisely their ( q ) dependence, is em-pirically well known since the first elastic scattering experiments [29] and agrees with mostof the nucleon models developed during last century, as for example, the constituent quarkmodel of Ref. [30]. It is also consistent with PQCD large- q predictions [26].Based on Ref. [31], in order to avoid ghost poles in α s , the following modification wassuggested ([32]) : | F SC ( q ) | = A ( q ) (cid:2) log ( q / Λ ) + π (cid:3) . (8)In this case the best fit parameters are A = 72 GeV − and Λ = 0 .
52 GeV .In Ref. [24] a form was suggested with two poles of dynamical origin (induced by adressed electromagnetic current) | F T P ( q ) | = A (1 − q /m )(2 − q /m ) . (9)The best fit parameters are A = 1 . m = 1 . and m = 0 .
77 GeV . Theparametrizations with the best fit parameters are illustrated in Fig. 1 and summarizedin Table I. The best fit functions are then subtracted from the data, leaving a regular oscil-latory behavior, Fig. 2. It has magnitude ∼
10 % of the regular term, and is well visible overthe data uncertainties for p > F ( p ) with the 4-parameter function F osc ( p ) ≡ A exp( − Bp ) cos( Cp + D ) . (10)Let us focus on the case F = F R , Eq. (6). The corresponding difference data are plottedin the lower panel of Fig. 2a. The relative errors in the parameters C and B show that the6 [GeV q10 20 30 40 p F − − − − FIG. 1: (color online) Data on the TL proton generalized FF as a function of q , from Ref. [20, 21]together with the fits from Eq. (6) (black solid line), Eq. (7) (blue dash-dotted line), Eq. (8)(red dashed line), and Eq. (9) (Green, long-dashed line). The insert magnifies the near thresholdregion. Because of their large error bars, the points over 16 GeV do not affect fit parameters, sothat the four fits best reproduce the data in the insert, apart for the oscillations that are the focusof this work. oscillation period is better defined than the damping coefficient exp( − Bp ). Two and a halfoscillations are clearly visible over the reaction threshold, while for p > A exp( − Bp ).The parameter D defines the position of the first oscillation maximum that occurs at p = 0within the error ∆ D P/ (2 π ), where P is the oscillation period. Estimating the oscillationperiod P = 1.13 GeV, the first oscillation maximum occurs at p = 0 within an error of 0.05GeV. Five peaks (maxima and minima) are visible and the periodicity hypothesis, that isimplicit in the cos( Cp + D ) term implies that they are regularly spaced by a half-period of1.13/2 GeV. Examining Fig. 2a (lower panel) we find: • p = 0 ± .
05 GeV (from the fit error),7 − − (d)(a)(c) (b) p [GeV] p [GeV] D F p D F p FIG. 2: Referring to Eqs. (4) and (5), we report the background fits F ( p ) of the BABAR data,according to the four parametrizations a) F = F R from [27] (Eq. (6), see text or Table 1), b) F = F S from [28] (Eq. (7)), c) F = F SC from [31] (Eq. (8)), d) F = F T P from [24] (Eq. (9)),and the corresponding fits F osc ( p ) of the differences between the data and each parametrization.In all the four cases F osc has the damped oscillation form of Eq. (10), with the best-fit parametersreported in Table 1. For each insert: (top) the data of BABAR are plotted, together with theparametrization F ( p ) (blue, dashed line) and the complete fit F R ( p ) = F ( p ) + F osc ( p ) (solid blackline); (bottom) the difference of the data and the parametrization is shown, together with the fit F osc ( p ) (solid red line). • • • • ABLE I: Background fit functions from Eqs. (6, 7, 8, 9) (see Fig. Fig:WorldData), and parameters A , B , C , D (with the related χ /n.d.f.) for Eq. (10 ) fitting in each case the difference betweenthe data and the corresponding background function.Ref. Background function A ± ∆ A B ± ∆ B C ± ∆ C D ± ∆ D χ /n.d.f.[GeV] − [GeV] − [27] | F R | = A (1 + q /m a ) [1 − q / . ± ± ± ± A = 7 . − , m a = 14 . [28] | F S | = A ( q ) log ( q / Λ ) 0.05 ± ± ± ± A = 40 GeV − , Λ = .
45 GeV [31] | F SC | = A ( q ) (cid:2) log ( q / Λ ) + π (cid:3) ± ± ± ± A = 72 GeV − , Λ = 0 .
52 GeV [24] | F T P | = A (1 − q /m )(2 − q /m ) 0.1 ± ± ± ± A =1.56, m , = 1 . , .
77 GeV • ≈
15 %.This justifies the presence of the periodic term cos( Cp + D ) in the fit. Such analysis maybe repeated for the other cases in Fig. 2, with similar results.Concerning the amplitudes of the half-oscillations, each of them is about 1 / √ F osc ( p ) is about 1/2 of the previous maximum. Thismotivates the use of exp( − Bp ) in the fit, although in this case the error on this “1 / √ F ( p )[1 + ǫ cos( Cp )] with constant ǫ ≈ F osc ( p ) /F ( p ) ≈ constant, both decreasing by 1 /e in about 1.4-1.5 GeV.Since increasing relative errors hide the possible presence of the oscillations for p > p or not. Assumingthat they are, the point of view supported in our previous work [22] and in the following is9hat we are facing an interference effect between a small number of amplitudes, effectivelycompeting in the visible momentum range. These amplitudes must be few, not forming aregular continuum, otherwise they would give rise to a diffraction pattern, rather than anoscillation pattern.Note that the oscillatory behavior is present already in other invariant functions of q ,but not periodic. The relevant point is that F osc ( p ) is periodic with respect to p , not withrespect to q or q . Since p is a variable that is uniquely associated with the relative motion ofthe hadron, we associate periodicity with interactions between the forming hadrons after thevirtual photon has been converted into quarks and antiquarks, Eq. (1), or before quarks andantiquarks annihilate into a virtual photon, Eq. (2). In both cases we name ”rescattering”these interactions. III. OPTICAL MODEL
We assume that rescattering is a relatively small perturbation, and that in absence ofrescattering the effective FF would coincide exactly with F ( p ). We also assume that it ispossible to neglect the dependence of the rescattering mechanism on q .Let ~r be the space variable that is Fourier-conjugated to ~p : r is the distance between thecenters of mass of the two forming hadrons, in the frame where one is at rest. The observedbehavior is modeled via a two-stage process where: • In the e + e − → ¯ pp ”bare” process a ¯ pp pair is formed at a distance r with spacedistribution amplitude M ( r ). • Rescattering takes place between the newly formed hadrons ( p and ¯ p ) according to anoptical potential that is function of their distance r .To introduce rescattering we use the Distorted Wave Impulse Approximation (DWIA)formalism, following the scheme employed in Ref. [33]. The starting point is the Fouriertransform F ( p ) ≡ Z d ~r exp( i~p · ~r ) M ( r ) (11)where we interpret exp( i~p · ~r ) as the plane wave final state of the ¯ pp pair in their center ofmass, and M ( r ) as a matrix element describing the earlier stage of the process. Neglecting10 (fm)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) M (r) ( /f m -5 -4 -3 -2 -1 M(r)
FIG. 3: The 3-dim Fourier transform M ( r ) of F ( p ), defined in Eq. (11). rescattering, a detailed model for the formation process would lead to a matrix element ofthe form F ( p ) = < ψ f ( x , ..., x n ) ψ f ( ~r ) | T ( r, x , .., x n , x e + e − ) | ψ i ( x e + e − ) > ≡ Z d ~r ψ f ( ~r ) M ( r ) , (12)where the hard operator T is sandwiched between the initial state ψ i , that is function ofthe 4-vector x e + e − expressing the relative coordinates of the e + e − pair, and the final states ψ f that depend on the internal coordinates x , x , ...x n of the two hadrons as well as ontheir relative position ~r . The result of integrating over all variables but r , is M ( q , r ),that in general depends on q since this is a parameter of ψ i and ψ f . We assume thatthis dependence may be neglected in the range 0 < p < ψ f ( ~r ) = exp( i~p · ~r ) . (13)In the distorted DWIA formalism exp( i~p · ~r ) is substituted by a wave including the effects11f ¯ pp rescattering. We choose a simple factorized distortion D ( ~r ) ψ f ( ~r ) = D ( ~r ) exp( i~p · ~r ) , (14)where D ( ~r ) is calculated as a Glauber-like eikonal factor: D ( x, y, z ) = exp (cid:16) − ib Z ∞ z ρ ( x, y, z ′ ) dz ′ (cid:17) (15)where ˆ z // ~p and b is a complex number, whose meaning is: • Pure real b : elastic rescattering potential, that may be attractive or repulsive depend-ing on the relative sign of Re ( b ) and p z . • Pure imaginary b : imaginary potential causing flux absorption or flux creation inrescattering.Strictly speaking the product bρ ( ~r ) is not a potential. V ( r ) ≡ bpρ ( ~r ) is a true opticalpotential appearing in a linearized form of the Schrodinger equation, but for simplicity wename ”potential” the product bρ ( ~r ).The key function is the real function ρ ( r ), describing the space distribution of the strengthand the sign of the rescattering potential ( ρ may be negative). We have tested three familiesof space densities ρ ( r ) for the rescattering potential:1. Compact rescattering densities: they are decreasing functions of r , as for exampleWoods-Saxon densities. This class includes imaginary-dominated potentials that aretypical of the theory of ¯ pp low-energy interactions.2. Hollow rescattering densities: they are very small or vanishing at small r , largein a sub-range of 0.2-2 fm, and tend to zero for larger r .3. Double-layer rescattering densities: these are the combination of two potentialsof class 2 with opposite sign. So we may have a region r a < r < r b with a repulsivepotential and a region r c < r < r d with an attractive potential, or we may have tworegions, characterized by an absorbing and a generating imaginary potential.All the potentials considered here act in a range that is typical of strong interactions. Elec-trostatic potentials have not been considered, since the short-range scheme used here is notsuitable for analyzing phase shifts that develop at distances ≫ F ( p ) as F ( p ) = 1(2 π ) Z d ~r e i~p · ~r D ( ~r ) M ( r ) , (16) M ( r ) ≡ Z d ~p e − i~p · ~r F ( p ) , (17)with D ( ~r ) defined in Eq. (15). We observe that F ( p ) does not depend on the orientationof ~p because of the choice of constraining the z -axis to the direction ~p in the calculation of D ( ~r ). IV. MODEL RESULTS
After testing several configurations, the results are the following.
1) Compact potentials .Spherical homogeneous potentials (constant up to a fixed radius), Gaussian potentials andWoods-Saxon potentials do not give positive results. Neither real nor imaginary potentialshave produced periodic oscillation patterns. This is in contrast with the case of the angulardistributions of nuclear physics (see for example [33] where evident periodic patterns areobtained via pure imaginary Woods-Saxon densities). However, in those applications therelevant variable is the t − channel momentum exchanged in elastic scattering or rescattering,while here it is the relative momentum of the colliding particles, that is equivalent to an s − channel momentum.
2) Hollow potentials (not changing sign or phase in the r range of interest).We have tested simple double-step potentials (constant between two r values, zero else-where), and shifted-Gaussian potentials exp[ − ( r − r ) /σ ], with both real and imaginaryparts. Real hollow potentials produce periodical oscillations, but the oscillation period is fartoo large (2 GeV or more). In order to make it shorter, the peak value of the potential hasto be pushed to r > M ( r ) decreases by 3-4 orders of magnitude when r increasesby 1 fm (see Fig. 3). So, at distances > M ( r ) is very small, depriving of relevancethe effects of a real potential that is active in these regions. Imaginary potentials of pureabsorbing (or pure generating) class have been found to be incompatible with our startingrequirement that rescattering is a small correction. For a hollow imaginary potential toproduce oscillations with relative magnitude 10 %, one needs a strong imaginary potential,which leading effect is damping by orders of magnitude the unperturbed term.13 r (fm)0 0.5 1 1.5 2 2.5 3 (r) ρ b -8-6-4-202468 Potential functions
FIG. 4: The three double-layer potentials used for the fits reported in figures 5, 6, and 7. Dashedcurve: potential n.1 (Eq. (18)). Continuous curve: potential n.2 (Eq. (19)). Dotted curve:potential n.3 (Eq. (20)). These potentials are used in Eq. (15) to calculate the final state distortionfactor D ( x, y, z ), that leads to the fit F ( p ) through Eq. (16).
3) Double layer potentials (presenting two r -ranges where the potential phase is op-posite). Double-layer real potentials produce weaker oscillation effects than single-layer(hollow) potentials. Our best results have been obtained with double-layer imaginary po-tentials. These have been able to produce periodic oscillations with a period of 1 GeV orshorter, and of arbitrarily large amplitudes, depending on the parameters. Such potentialspresent an inner region where the ¯ pp flux is produced and an outer region where the ¯ pp fluxis absorbed. The physical origin of this class of potentials is discussed in next section.We report the results corresponding to the three different double-layer potentials illus-trated in Fig. 4. All of them are purely imaginary ( Re ( b ) ≡ otential n.1 : Multiple-step function Im ( b ) ρ ( r ) =: 0 for r < r > . − . . < r < .
7) fm;3 . . < r < .
4) fm; (18)
Potential n.2 : Potential similar to the previous one, but regular: Im ( b ) ρ ( r ) = B G ( r − r ) T ( r − r ) W ( r − r ) , (19)where • B = 7 . • G ( r − r ) = exp[ − ( r − r ) / . ] is a Gaussian with center in r = 1 . µ = 0 . • T ( r − r ) = tanh[( r − r ) / .
05] is a ”soft sign function”, that is equal to − r ≪ r fm, to +1 for r ≫ r fm, and changes smoothly sign in a range of 0.1 fm. • W ( r − r ) = 1 + 0 . r − r ) is a weight function that (slightly) increases the strengthof the external absorbing peak over the internal generating one. Potential n.3 : Sum of a positive Gaussian in the inner region and a negative one in theouter region: Im ( b ) ρ ( r ) = A + G + ( r − r + ) − A − G − ( r − r − ) , (20)with G ± ≡ exp( − [( r − r ± ) /w ], r + = 1.35 fm, r − = r + + w , and w = 0.26 fm, A + =10, A − = 8.4.This potential is very different from the previous two: the sign of the inner and outerparts are reversed (absorption inside), the average radius is smaller, the negative and positivepeaks are more distant. It is reported here to highlight some effects of the parameters ratherthan for good fit purposes.The results obtained with these potentials are presented in Figs. 5, 6, and 7. The fitswith potentials 1 and 2 reproduce satisfactorily well the data, given the simplicity of themodel. In these two cases, the data are slightly overestimated near the threshold. Thismay be attributed to the small-energy limitations of the eikonal formalism chosen here to15eproduce the wave distortion. As observed in [33] the use of this approximation withinDWIA is good when several partial waves are involved in rescattering, and definitely it doesnot apply in a regime of S-wave dominance, corresponding to p ≤
200 MeV for ¯ pp systems[17].The example with potential n.3 shows that it is possible to obtain similar qualitativeresults with opposite configurations: absorbing potential in the outer region and generatingpotential in the inner region, or viceversa. For our best fits we have preferred the firstoption, because it corresponds to the phenomenology of ¯ pp annihilation, dominated by fluxabsorption when the proton and antiproton begin to overlap.Potential n.3 allows for an easy analysis of the separate role of the two potential layers,since one may independently modify the peak strengths A + and A − . Systematic attemptsshow that it is possible to obtain periodic oscillations of pretty large amplitude by increasingboth A + and A − , at the condition that the relative effect of the absorbing and of thecreating parts of the potential is well balanced, that may be obtained by acting on the ratio A + /A − . Apart for avoiding normalization problems, an equilibrated balance between thetwo strengths is one of the keys to get remarkable and periodic oscillations.On the other side, this potential is not suitable for producing arbitrarily short oscillationperiods, because it lacks a decisive feature of potentials n.1 and 2: their steep derivative atthe point where they change sign. To obtain this feature with potential n.3, the distancebetween the two peaks must be smaller than their width. In such conditions the two Gaus-sians overlap and cancel reciprocally. We have been able to reduce the oscillation perioddown to what is visible in Fig. 7, but not further. The conclusion is that the period of theoscillations is related to a sudden transition between the flux feeding and the flux depletingregions.Another important property shared by the three double-layer potentials is to producea systematic threshold enhancement: p = 0 corresponds to an oscillation maximum, ifthe effect of the flux-creating and flux-absorbing parts of the potential are reasonably wellbalanced. This property is very stable, it is not related to a special set of parameters, anddoes not depend on the fact that the absorbing part of the potential is external (potentials n.1and n.2) or internal (potential n.3). So, the threshold enhancement is an intrinsic propertyof the imaginary double-layer model.For large r , potentials n.1 and n.2 act qualitatively as the purely absorptive potentials16sed to fit ¯ pp LEAR data [7, 17–19]. To reproduce LEAR elastic and annihilation data,phenomena taking place at small r have no relevance, since the surface interaction at r ≈ pp channel wave function from entering the r < r is essential for thecoupling of the proton-antiproton pair with a virtual photon with virtuality q > M . Thecoupled regeneration/absorption mechanism introduced here produces, for r < pp wave function is enhanced or suppressed.Let us discuss the conditions leading to observable effects. The enhancement of the crosssections at small p , where the the ¯ pp distorted wave function exp[ i~p · ~r ] D ( ~r ) approximatelyreduces to D ( ~r ) and the Fourier transform of Eq. (16) simplifies to R d rD ( ~r ) M ( r ), suggeststhat the potential enhances the wave function at small r where M ( r ) is very large (see Fig.3). This effect is not specific of double-layer potentials: for example with a spherical realattracting potential does the same. The presence of further oscillations at larger p howeversuggests that double-layer imaginary potentials create regions where the product D ( ~r ) M ( r )alternatively becomes larger and smaller, enough to ”resonate” with the Fourier transformfactor exp( i~p · ~r ) for periodic p values far from the threshold (see Fig. 3b of [22]). If tworegions with a larger and a smaller value of D ( ~r ) are present at r + and r − respectively, r + − r − must be small, or the steep r -decrease of M ( r ) will make the modulation by D ( r + )negligible with respect to D ( r − ). This may explain the relevance of having a steep potentialat the change of sign. V. HYPOTHESIS ON THE PHYSICAL ORIGIN OF ”INNER CREATIVE/OUTER ABSORPTIVE” POTENTIALS
An optical potential with an imaginary part may be justified within several theoreticalframeworks but in general, and intuitively, its origin is related to the fact that a a multi-channel process is inclusively projected onto one channel alone.For the case of interest, this is illustrated in Fig. 8. Diagram (a) is the amplitude of γ ∗ → ¯ pp within a model that does not include ¯ pp rescattering. This is supposed to lead17 (Gev/c)0 0.5 1 1.5 2 2.5 3 | F | (a) p (GeV/c)0 0.5 1 1.5 2 2.5 3 R e ( F ) and I m ( F ) -0.100.10.20.30.40.5 (b) FIG. 5: (a): Continuous curve: | F ( p ) | , obtained with the double-layer rescattering potential n.1(the multiple-step function in Fig. 4, see text), compared to the BABAR data points (full circles).(b): real (solid line) and imaginary (dashed line) parts of the model F ( p ). p(GeV)0 0.5 1 1.5 2 2.5 3 | F | (a) p(GeV)0 0.5 1 1.5 2 2.5 3 R e ( F ) and I m ( F ) -0.100.10.20.30.40.5 (b) FIG. 6: (a): Continuous curve: | F ( p ) | , obtained with the double-layer rescattering potential n.2(continuous curve in Fig. 4, see text), compared to the BABAR data points (full circles). (b): real(solid line) and imaginary (dashed line) parts of the model F ( p ). to the background regular component of the form factor, without oscillations. Diagram(b) considers the possibility that ¯ pp annihilation into a multi-meson state depletes the finalstate produced by process (a). In our formalism this finds an expression in the absorption18 (GeV)0 0.5 1 1.5 2 2.5 3 | F | (a) p(GeV)0 0.5 1 1.5 2 2.5 3 R e ( F ) and I m ( F ) -0.4-0.200.20.40.6 (b) FIG. 7: (a): Continuous curve: | F ( p ) | , obtained with the double-layer rescattering potential n.3(continuous dotted curve in Fig. 4, see text), compared to the BABAR data points (full circles).(b): real (solid line) and imaginary (dashed line) parts of F ( p ). component of the imaginary potential. The most interesting additional diagram is (c). Thesame model that in case (a) has been used to calculate the amplitude of γ ∗ → ¯ pp , is used incase (c) to calculate the amplitude of γ ∗ → ¯ hh , where h is a hadron that is different from p , for example a neutron or a higher mass baryon. Later rescattering converts this pair intoa ¯ pp state. The intermediate state is not necessarily a two-particle state. Any multi-mesonstate with the right quantum numbers may play the role of an intermediate state that islater converted into ¯ pp .According with the previous argument, the double-layer optical potential used here isnot in conflict with the existing models for the TL FF, but is rather an effective way toinclude rescattering corrections to these models. Many models for the hadron coupling tothe virtual photon have been developed and applied to the calculation of SL FFs. Someof them may be analytically continued to the TL region. This is the case for approachesbased on vector meson dominance [34, 35] and dispersion relations [36, 37]. Constituentquark models in light front dynamics may be applied [38], as well as approaches based onAdS/QCD correspondence [24]. A phenomenological picture for the full time-evolution ofthe hadronization process has been proposed in [39].Practically all these models may be the starting point for a calculation within the pro-19 ppp πππ hh pp (a)(b)(c) FIG. 8: Examples of diagrams entering the absorptive and creative parts of the potential. (a):“direct” γ ∗ → ¯ pp . The light-grey circle represents a model amplitude without contributions byrescattering. It leads to the background regular term F ( p ) or equivalently, in r − coordinate rep-resentation, to M ( r ) (see Eq. (11)). (b): The γ ∗ → ¯ pp production is followed by an annihilationprocess reducing the final ¯ pp outcome. This contributes to the absorptive part of the optical po-tential. (c): The same model previously used to calculate the direct production γ ∗ → ¯ pp is nowused to calculate γ ∗ → ¯ hh , where h is a hadron different from a proton. Rescattering converts ¯ hh into ¯ pp increasing the final output. The effect of this diagram is taken into account by the creativepart of the potential. Even other diagrams, with intermediate hadronic states more complicatedthan ¯ hh , may contribute. posed DWIA-optical potential scheme, following the prescription suggested in fig.8:Step 1) The model is used to calculate the PWIA production amplitude of the ¯ pp pair(diagram (a)), and this leads the background regular component of the effective form factor,and to its Fourier transform M ( r ) in coordinate representation (Eq. (11)).Step 2) Final state processes implying he annihilation of the ¯ pp pair into mesons areadded (diagram (b)). The relevant amplitudes of this group may be effectively summarizedin a flux-absorbing optical potential that in coordinate representation modifies the planewave of the ¯ pp channel as in Eqs. (14) and (15).Step 3) The model is used for calculating the amplitude for the production of otherhadronic states ¯ hh that are later converted into ¯ pp by rescattering (diagram (c)). Theamplitudes for the processes of this group may be effectively summarized in a flux-creating20ptical potential distorting the plane wave of the ¯ pp channel.In principle processes as in Figs. 8b and 8c are possible everywhere in a range of a few fmaround the initial virtual photon decay point. Why should the “flux enhancing” diagramslike (c) dominate the small − r regions?While the explanation of the optical potential in terms of multi-step inelastic reactionsis straightforward, for the answer to this question we may only propose an hypothesis, thatrelates the presence of the creation part of the potential to those regions where high-massvirtual intermediate states are more likely.The amplitude for the transition from ¯ pp to a state made of 3-10 mesons is not differentfrom the amplitude for the reverse process, but phase space makes the probability of theformer process larger than the probability of the latter. So, the hadronic states that maycontribute to feeding the ¯ pp channel (Fig. 8c) and not to further depleting it (Fig. 8b),are the states made by one or two heavy hadrons like N ∗ ¯ N ∗ states. Unless q >> M p , thehadrons composing these states are virtual, short-lived and slow, with few exceptions like aneutron-antineutron intermediate state. So, they play a role for small r only, since small r corresponds in the average to small times after the photon conversion into the first ¯ qq pair.On the other side, these high-mass states must be present in the state that is initiallyproduced by the decay of a virtual photon with q ≥ M p according to the statement that thisstate has space-time size of magnitude 1 /q . According to the PQCD view[26], in the SL case(elastic electron-proton scattering) the virtual photon is absorbed by a fluctuation of theproton state consisting of valence quarks grouped within a space-time region of size 1 /q . Thefact that this fluctuation exists means that in the TL case a corresponding fluctuation of the¯ pp state exists where the required number of valence quarks and antiquarks is concentratedwithin a region of space-time size 1 /q . Indeed, the Feynman diagrams describing the PQCDkernel of the process are the same in SL and TL and in these diagrams all the involvedpartons are connected by propagator lines with off-shellness of magnitude q , that obligesthem to be within a space-time distance 1 /q .If this 1 /q -sized fluctuation takes place in a ¯ pp annihilation, we may have the rare butpossible event ¯ pp → e + e − . In the reaction e + e − → ¯ pp , the path is opposite: the virtualphoton creates a 1 /q -sized fluctuation of quarks and antiquarks, that may evolve into a ¯ pp pair, but may also evolve into other hadronic states (e.g. neutron-antineutron) since alsothese states present 1 /q -sized fluctuations of their parton content.21ny configuration of a color singlet state, like the “3 quark + 3 antiquark” small-sizedstate produced by the decay of the virtual photon, may be written as a sum over physicalhadronic states with the same quantum numbers, since these states form a complete basisfor this system. However, a state with a size of magnitude 1 /q cannot be reproduced bythe sum of a small number of hadronic states since these have a typical size 1 fm. What isneeded is a set of several states which interfere destructively at distances > /q from thevirtual photon materialization point, and constructively at distances < ∼ /q , so to build awave packet of size 1 /q . Taking into account that 1 /q is also the magnitude of the lifetime ofthis fluctuation, we may estimate that the sum must include hadronic states with a spreadof magnitude q in their center of mass energy. With a virtuality that can be of the samemagnitude as q , it is evident that many of these states cannot propagate far from the virtualphoton materialization point, and this may support the dominance of the flux-enhancingterm of the optical potential at small r .As observed, this picture behind the small- r dominance of the flux-creation part of thepotential is just an educated guess, because of the difficulties in passing from qualitativeideas to a detailed model. VI. CONCLUSIONS
We have analyzed the modulation structure shown by the precise data on the TL protonform factor, recently obtained by the BABAR collaboration. First, we have repeated thedata analysis already presented in our previous work [22] for the case of four different formfactor parameterizations available in the literature. The difference between BABAR dataand the form factor parametrization is well fitted by an oscillating function of the form A exp( − Bp ) cos( Cp ), where p is the momentum of the relative motion of the ¯ pp pair. Theperiodicity of the cos( Cp ) term is verified within 15 % in a p range from zero to 2.8 GeV.The periodicity of this oscillating modulation as a function of the relative momentumof the final hadrons has been qualitatively explained in terms of rescattering between thefinal products of the reaction e + e − → ¯ pp and reproduced via an optical potential of peculiar(double spherical layer) form.An imaginary optical potential that is mainly flux-generating in a region of small distancesbetween the centers of the forming (and still overlapping) proton and antiproton, and mainly22ux-absorbing at larger distances, produces systematic oscillations of the effective protonTL form factor, consistent with the observed ones. At distances ≈ pp annihilation data, that is itdamps the ¯ pp flux by annihilating ¯ p and p into multi-meson states. A possible explanationfor the regeneration features of the potential at smaller distances could be in terms ofcoupling between the ¯ pp final channel and large-mass virtual states (like baryon-antibaryon)temporarily produced by the virtual photon. In order to reproduce the data, the transitionfrom the flux-generating to the flux-absorbing region must be sudden. A soft transitionproduces oscillations with periods longer than the observed one. With this double-layerstructure, we always find threshold enhancement of the form factors. So, within this schemethreshold enhancement and oscillations are expressions of the same phenomenon.We have tested other simpler configurations of the potential, and also real potentials witha range typical of strong interactions, but these do not seem to allow for oscillations withthe required period and strength. The proposed phenomenological scheme is compatiblewith existing theoretical models for the TL form factors, since it may be considered as arescattering correction that does not touch the core schemes of these models. [1] S. Pacetti, R. Baldini Ferroli, and E. Tomasi-Gustafsson, Phys.Rep. , 1 (2015).[2] A. Zichichi, S. Berman, N. Cabibbo, and R. Gatto, Nuovo Cim. , 170 (1962).[3] G. Bardin, G. Burgun, R. Calabrese, G. Capon, R. Carlin, et al., Nucl.Phys. B411 , 3 (1994).[4] F. Balestra et al., Nucl. Phys.
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