Energy dependence of proton-proton elastic scattering at large momentum transfer

The measurements of proton-proton elastic scattering for large momentum transfer at energies in the range $\approx$ 20 to 60 GeV show a simple behaviour of form $d \sigma/dt \approx {\rm const}~|t|^{-8}$, apparently with no energy dependence. In the present work detailed analysis of the data shows a decrease of the magnitude of the tail with the energy, still with preservation of the power $|t|^{-8}$. The analysis allows the definition of a band for the energy dependence with the form of a power of the strong coupling $\alpha_S^{1.57}$. The rate of decrease describes very well the data at the distant energy $\sqrt{s}$ = 13 TeV, with reduction of the cross section by a factor 5.71. This result gives prediction for new experiments at high energies, and opens important question for theoretical investigation.


Introduction
With basis on the data of Fermilab [1] and ISR experiments [2,3], it is usually believed that the differential cross section of elastic pp and pp scattering for large momentum transfer has a form |t| −8 , with no energy dependence.The interpretation given for the |t| −8 form is the dominance of perturbative process of three-gluon exchange [4], while the magnitude has no simple interpretation, as the studied multiple exchanges predict different behaviour, both in √ s and t dependence.In the present work we treat the pp elastic data for large |t| in detail, and arrive at a slow energy dependence of the real amplitude, like 1/ log √ s, still preserving the |t| −8 behaviour.The variation of the cross section in the rather small energy range, about 20 GeV to 60 GeV, of the observed data has not received attention, but the influence is very strong at the distant TeV energies.We show that the prediction for 13 TeV , with a reduction by a factor 2.39 in the amplitude and a factor 5.71 in the cross section, is in good agreement with the data.It is observed that the energy dependence of the amplitude for large |t| scattering can be described by a power of the running strong coupling with scale √ s , α S ( √ s).This information may be useful for studies of the operating mechanisms in pp elastic scattering for large momentum transfer.
The analysis of data, with analytical description of the energy dependence of the amplitude, and confirmation of prediction for 13 TeV, are all presented in Sec. 2.
In Sec. 3 we comment on the results and on expectations.

Energy dependence of the perturbative tail
The first observation of similarity of pp scattering for large |t| at different energies was made in the comparison of data at √ s = 19.6 and 27.4 GeV in Fermilab [1].Measurements of the differential cross sections in ISR/CERN at energies from 23.5 to 62.5 GeV [2] indicated the existence of a regularity in the large |t| range, with same form const |t| −8 and same magnitude for all energies.
The analysis of the ISR measurements was extended with higher accuracy [3], and a Fermilab experiment at √ s = 27.4GeV was made in very large momentum transfers with 39 points in the range 5.5 ≤ |t| ≤ 14.2 GeV 2 [5], confirming compatibility with the |t| −8 behaviour, all apparently with the same magnitude.The data are shown in Table 1.
On the theoretical side, explanation for the 1/|t| 8 behaviour of dσ/dt in terms of a real three-gluon exchange amplitude was soon given by Donnachie and Landshoff [4].The diagram with the three quarks of each proton exchanging one gluon, assuming that each of these individual exchanges is made with the same fraction of the transferred momentum, leads to where α S is the QCD strong coupling.The insertion of the diagram in a model of multi Pomeron and Reggeon exchanges was studied by Donnachie and Landshoff [4,6,7] , and also other mechanisms were analysed, such as triple singlet exchanges, but the energy independence is not consistently explained.Also the |t|dependence expected from the coupling factor [α S (t/9)] 6 is not visible in the data.The phenomenology of pp scattering at large momentum transfer is a complicated and open question, involving QCD complexities, including the proton structure.
In general, proton-proton elastic scattering for all |t| is described with a non-perturbative complex amplitude T NP covering the forward peak and the dip-bump structure in the cross section.The large |t| range is described with an additional real term, that dominates the real and imaginary parts of T NP for large |t|.Although there are inconsistencies and may be there are other mechanisms, we define in the present work an amplitude term R ggg ∼ |t| −4 , called three-gluon exchange to follow the original suggestion of a simplified three-gluon exchange mechanism.
Characteristic features of the R ggg amplitude are that it is real, has opposite signs for pp and pp scattering, and the form 1/|t| 4 .Other possibilities of exchanges exist [6,7], but the specific property of the change of sign for pp scattering seems to be fulfilled, according to the experiment at 53 GeV [8] where both pp and pp scattering are measured in the dip-bump range, that is sensitive to the sign of R ggg .
With a separate real term of perturbative nature, called R ggg , added to the complex amplitude of nonperturbative dynamics T NP , we write for the amplitude of elastic pp and pp elastic scattering The normalization used is The nonperturbative part is given by the chosen model and includes the Coulomb interaction.It is assumed that, as occurs in several models, for large |t| the perturbative real term is dominant over the nonperturbative amplitudes so that it can be identified and studied directly with the data.On the other hand, when dσ/dt is described for all |t|, the uncontrolled increase of |t| −4 in R ggg for small |t| must be accompanied with a damping factor.In the present work we are concerned only with large |t| data, and a damping factor is not directly used.We wish to investigate a possible energy dependence of the |t| −8 tail in pp elastic scattering, and then we introduce a quantity A( √ s) to be determined, and write where the sign is minus for pp scattering.The quantity A( √ s) has units GeV 6 , and t has units GeV 2 .The differential cross section in the large |t| domain is written With R ggg (s, t) in GeV −2 units and dσ/dt in mb/ GeV 2 , we have (ℏc) 2 = 0.389379 mb GeV 2 .The form |t| −4 written for the |t| dependence in Eq.( 4) is actually confirmed as very good at each energy in the interval 23.5 to 62.5 GeV examined.This is a partial support for the original simple form assumed for the mechanism of three particle exchanges in Eq. ( 1), with absence of the factor with strong coupling, that is not required by the data.
Through a detailed examination of data in the energy range 23.5 GeV to 62.5 GeV where measurements at large |t| were made [3,5], we show that the traditionally assumed energy independence for large |t| is an approximation, while actually the magnitude of the tail decreases slowly with the scattering energy.We obtain a simple form for A( √ s) with a (log √ s) −1 dependence.
The data used in our analysis are shown in Table 1 with the fitted values of A( √ s).In Fig. 1 we plot the values of A( √ s) for the six energies investigated.In the plot we identify a band limited by regular tendencies in the decreasing values.We obtain a parameterization for the central line with A( √ s) in GeV 6 and √ s in GeV .The dashed higher and lower lines defining the uncertainty band are respectively and The log forms give small changes in the range of the data in Fig. 1, but gives remarkable differences for TeV energies.For example, for 13 TeV we obtain A( √ s) = 0.1890 ± 0.0037 GeV 6 , that is about log(13000)/ log(52.8)=2.39 times smaller than the value 0.460 (see 52.8 GeV in the Table ).For the cross section the reduction is by a factor 5.71 .
Fig. 3 shows the large |t| data at 13 TeV [9] described in solid line with the new prediction of Eq.( 6).In the second part of the same figure we compare the data for 52.8 GeV and 13 TeV showing the strong reduction in the cross sections.The attempt of direct connection of 13 TeV data with points of Faissler et al [5] at 27.4 GeV, thus without energy dependence, was highly unsatisfactory [10] .
We have a formidable prediction for the energy dependence of pp elastic scattering for large momentum transfer, covering with good accuracy the energy scale with a factor of more than 200, namely from 62.5 GeV to 13000 GeV.
It is interesting to relate the magnitude in Eq.( 6) with the QCD strong coupling.Fig. 2 shows the values of α S ( √ s) from the Particle Data Group [11] Notice that at m(Z 0 ) = 91.79GeV, α(m(Z 0 )) = 0.1179 has its best determination.This is a very convenient parameterization for α S ( √ s), as we obtain simply with A( √ s) in GeV 6 and √ s in GeV.This is a compact and precise phenomenological result of the present work.Thus we say : the energy dependence of the three-gluon exchange amplitude |t| −4 goes with a simple power 1.57 of α S ( √ s).We cannot tell whether this a simple curiosity or a useful inspiration in the search for a dynamical mechanism.

Remarks
The regularity of the form 1/|t| 8 in the differential cross section in pp elastic scattering cross section for large momentum Table 1: χ 2 (namely χ 2 /d.o.f.) values for the fittings of points of large |t| in the measurements of pp elastic scattering in ISR at energies 23.5, 30.7 , 44.7, 52.8 and 62.5 GeV [3] and for 27.439 GeV from Fermilab [5], with form R ggg = A( √ s)|t| −4 in the amplitude.This form, called "tail" is based on the idea of 3-gluon exchange perturbative contribution [4] to be added to the basis given by a nonperturbative dynamical model.Only statistical errors in the reported experimental data are considered in the calculation of χ 2 values shown in the  [1,2,4].In this rather small energy interval the tails of the |t| distributions, put together in a plot of dσ/dt, seems to be in a unique narrow band.Thus Fig. 1 of the article of Donnachie and Landshoff in 1996 [7] shows all data for all ISR energies running almost together, with the unified description dσ/dt = const/|t| 8 .This form suggested [4] a mechanism of exchange of three gluons or alternatively of three singlets, but the energy dependence and the absence of the influence of the α S (|t|/9) couplings in Eq. ( 1) are not explained.
We split the band of large |t| data separating the energies, and show that the constant is an approximation.As the form |t| −8  is well preserved at all energies, we write Eq.( 4), and obtain Eq.( 6) from the numerical values in Table 1 .The decrease with the energy is slow enough to have been overlooked in the small range of the ISR data, but the reduction in the cross section from 52.8 GeV of ISR to 13 TeV of LHC is of a factor [log(13000)/ log(52.8)] 2 = 5.71 according to Eq. ( 6).This decrease is remarkably confirmed in Fig. 3 with the plot of the data of these two energies.
It is interesting that the parameterized form of α S ( √ s) in Eq.( 9) leads to the form of Eq.( 10) for the energy dependence in terms of the running strong coupling.Hopefully this relation may inspire investigations of the underlying QCD mechanisms.

Damping factor and influence on the dip-bump region .
In calculations of dσ/dt for all |t| the term R ggg in Eq. ( 2) must be accompanied by a damping factor to avoid the uncontrolled increase for not-large |t| .We suggest a form At ISR energies the dip in dσ/dt is located at |t| ≈ 1.3 GeV 2 , thus not very distant from the range where R ggg is more influential.In the dip, both imaginary and real parts of T NP (s, t) are small, and R ggg , with appropriate damping factor, has influence on the form of the dip-bump structure.
A characteristic feature of R ggg is the opposite signs in pp and pp scattering.The non-perturbative basis is nearly identical for the pp and pp systems, so that the main difference between the two systems is due to the sign of the term R ggg .The dip in dσ/dt is formed by the relative positions of the zeros of the real and imaginary parts of the T NP amplitude.The effect of the ± sign , causing the displacement of the zero of Re[T NP ], is a flattening of the dip in pp scattering [12].This change is observed at 53 GeV [8] where both pp and pp are measured in the dip-bump region of dσ/dt.
In known models, the real part is positive for large |t|, and in pp the additional positive term for the tail interferes constructively with it, putting the real zero closer to the imaginary zero.In pp it is the contrary.Thus the term R ggg has sign +1 for pp and sign -1 for pp .
At TeV energies the dip in dσ/dt is much higher and far away, at |t| ≈ 0.5 GeV 2 , and is not influenced by the damped R ggg of Eq. (11).The pp amplitudes at TeV energies fall very rapidly, and for large |t| around 3 to 4 GeV 2 has values around those from R ggg , so that the three-gluon exchange determines values at the data.We show this for 13 TeV in Fig. 3 .

The need of a specific tail term.
The treatment of the region of large |t| in pp elastic scattering requires a specific additive complement to produce the observed tail.The first example is the model of multi-exchanges by Donnachie and Landshoff [7,13] who first introduced the term with three-gluon exchange in the amplitude.
In the work by A.A. Godizov [14], the calculation in a Reggeeikonal formalism, the calculations with a Soft Pomeron and a secondary Reggeon does not fit well the large |t| data at the ISR energies, as shown in Fig. 3 of this paper.A regular difference with respect to data exists for all the energies from 23.4 GeV to 62.5 GeV for |t| ≥ 2 GeV 2 , apparently indicating that a term like the three-gluon exchange is missing.
Similarly in the work by O. V. Selyugin [15], the calculation for 52.8 GeV shown in Fig. 3 is below the data for |t| ≥ 3 GeV 2 , also missing a contribution like |t| −8 .
In the work by Gonc ¸alves and Silva [16] with the Phillips-Barger potential model the calculations are not appropriate for the large |t| range and remain much below the data.
In a treatment of 13 TeV Totem data with Lévy imaging method [17] the evidence for a missing term to cover the large |t| range is very neat.
To understand the role of the term R ggg in the description of dσ/dt it is important that the non-perturbative model provides the complex amplitude disentangled in its real and imaginary parts, with explicit exhibition of the amplitudes.

Final Comments
The observed energy and |t| dependence in Eq.( 6) or Eq.( 10) opens questions for theoretical investigation in terms of QCD structure and interactions.The description of the universal |t| −8 behaviour as a three-gluon exchange mechanism in Eq. ( 4) omits the [α S (t/9)] 6 factor in Eq. ( 1), that does not seem visible in our phenomenology.The reason for the weakness of the |t| dependence of this factor is an important problem [7].
For high energies the predictions suggest measurements in a delicate region where cross sections have very small values.The data of pp elastic scattering for large |t| are very scarce, restricted to the 8 points in Fig. 3. Few points and large error bars do not allow to confirm surely whether the pure 1/|t| 8 dependence in the tail holds also at large energies.It may happen that we must write A( √ s, t) in Eq. ( 4) and hopefully recover Eq. ( 1).
Proton-proton elastic scattering is very important in all range from very forward to large |t|, with important questions everywhere.Expectations for detailed and precise data are put in the RUN 3 of LHC. table.