Structure of Forward pp and Elastic Amplitudes at Low Energies
Exact analytical forms of solutions for Dispersion Relations for Amplitudes and Dispersion Relations for Slopes are applied in the analysis of pp and scattering data in the forward range at energies below . As inputs for the energy dependence of the imaginary part, use is made of analytic form for the total cross sections and for parameters of the dependence of the imaginary parts, with exponential and linear factors. A structure for the dependence of the real amplitude is written, with slopes and a linear factor that allows compatibility of the data with the predictions from dispersion relations for the derivatives of the real amplitude at the origin. A very precise description is made of all data, with regular energy dependence of all quantities. It is shown that a revision of previous calculations of total cross sections, slopes and parameters in the literatures is necessary, and stressed that only determinations based on data covering sufficient range using appropriate forms of amplitudes can be considered as valid.
In the scattering theory in quantum mechanics, the elastic differential cross sections are written in terms of a complex amplitude with independent imaginary and real parts, that are fiunctions of two variables (spin effects neglected). In the analysis of observables , besides the nuclear amplitude, account is taken for the contribution from the real Coulomb interaction. This is very basic and obvious, but we show in the present work that this structure is not usually obeyed in the treatments of the pp and pp̄ systems, where is written without due account for the properties of the amplitudes. We give a treatment of these elastic processes using theoretical constraints and appropriate forms of the input quantities, arriving at realistic amplitudes to connect measurements and theoretical dynamical models.
Determinations of , and other parameters of pp and pp̄ forward elastic scattering are not direct experimental measurements. Rather, they result from model-dependent analytical limiting procedures, performed with forms assumed for the imaginary and real parts of the complex elastic amplitude. The work done in the laboratory consists in measuring values of the number of event rates in intervals . With attention given to fluxes and densities (we are only concerned with unpolarized beams and targets), tables of distributions in differential cross sections are produced. We stress that the identification of the amplitudes and their parameters requires use of proper theoretical framework.
The differential cross section is written as a sum of absolute values
and the disentanglement required for the determination of the amplitudes and is not at all trivial. Help is brought from the interference of nuclear and Coulomb interactions and from dispersion relations connecting real and imaginary parts through general principles of causality and analyticity.
Besides the entanglement to be resolved, we have that production rates are not obtainable directly at the origin , or even very close to it, but rather in sets of points of an interval. The determination of , , slopes and other quantities require extrapolation of data in a range, using analytical expressions, and the results obviously depend on their forms. The mathematical structures of the amplitudes are mounted using parameters that must be found in confront with the observed distribution in . Regularity in the behaviour of all quantities with the energy is important consideration to obtain sensible descriptions of the elastic processes. Experiments at different energies must be analysed globally, since separate fitting procedures may lead to values that are useless as a step for the phenomenology of the area.
The range of the data at given energy must be sufficient for representation through assumed analytical forms. In the low energy range, up to , often these conditions are not satisfied, even suffering insecure normalization in the measurements of , and compilations of published values for typical parameters result scattered in plots, without coherence and regularity. We propose an investigation of this energy range, with emphasis on the identification of the amplitudes, searching to build a bridge between measurements and mathematical description, necessary to guide models of the dynamics of the processes.
In the interval of from 30 to 60 GeV, pp and pp̄ from ISR/Cern and Fermilab data cover large range with good precision, showing fast increase in , a forward peak and a marked dip in . These measurements led to the establishment of the successful Regge phenomenology Regge (); DL (), based on the exchange of particles (pomerons, reggeons) in the channel. Several theoretical models were developed to describe dynamically this region of data in the and variables models ().
Above , experiments data_HE () have large energy gaps, passing fast by SPS/CERN, Fermilab and reaching the TeV range of LHC data_LHC (). Ingredients of QCD dynamics enter with less or more detail in the interpretation of these data SVM (); us_LHC (). According to QCD expectations, as the energy increases the response of the gluon density in the hadrons increases and the hadronic interaction becomes determined by the vacuum structure Tel-Aviv (); CGC (). The interpretation of the forward scattering parameters in the LHC experiments at 7, 8 and 13 TeV is not trivial, and ambiguities and possible discrepancies are not clarified LHC_2017 (). The potential of crucial information REAL_PART () in the real part of the forward amplitude at high energies requires that doubts in the analysis of the data be properly solved.
In this paper we analyse forward pp, pp̄ data with from to , using forms for real and imaginary scattering amplitudes restricted to the forward regime and exploring fully the theoretical resources and constraints of dispersion relations treated exactly, in order to extract pure information on the forward quantities, as much as possible independently of peculiar microscopic models. This is the most difficult range of data for the analysis, for both reasons of insufficiency in the data and sophistication of the mathematical solutions of dispersion relations at low energies. Anyhow, we believe that in this sector we can learn about determination of amplitude parameters, and hope that this technical knowledge may be useful in the present difficulties encountered in the analysis of the recent LHC experiments.
We propose a treatment of pp and pp̄ forward elastic scattering analysing all data of differential cross sections that seem qualified (namely covering necessary range with regularity) for the extraction of the real and imaginary parts of the complex amplitude. We use the simplest and realistically possible analytical forms, treating coherently the Coulomb interference and the Coulomb phase, we use dispersion relations for the amplitudes (DRA) and for their slopes (DRS) EF2007 () with exact solutions for the Principal Value integrals, obtaining coherent energy dependence of all quantities. DRA and DRS predict algebraic values for the real amplitudes and their derivatives at , and our aim is to have sound proposals for the energy dependence of other parameters in order to reduce flexibility and choices by fittings, and produce a complete coherent description of all data in pp and pp̄ unpolarized elastic scattering . To eliminate fluctuations that are not meaningful, we account for normalization uncertainties (systematic errors), investigating a normalization factor in each experiment that adjust the total cross section to the parametrized prediction. These factors are always very close to 1.
The plots and numbers presented in the Review of Particle Properties PDG () of the Particle Data Group for forward scattering parameters in the low energy range, taken from the experimental papers, are scattered and misleading. The results of our work for all data that we analyse are regularly distributed, presented in numbers and very precise plots of in Sec.III, showing a way for rationalization of the phenomenological knowledge. However, the proposed solutions are not meant to be conclusive, unique or fully convincing. Alternatives are possible and may be looked for.
In elastic pp and pp̄ scattering in the forward direction, the dependences of the amplitudes are mainly characterized by exponential forms, with slopes and that are essentially independent quantities, essentially not equal to each other: the real and imaginary amplitudes do not run parallel along the axis. We take special care in the investigation of the behavior of the real part, that has structure deviating from a pure exponential form for values included in the forward range.
Once the total cross sections for pp and pp̄ are parameterized in the energy, the usual Dispersion Relations for the Amplitudes (DRA) determine the real amplitudes at the origin (namely the parameters). Similarly, if the derivatives of the imaginary parts of pp and pp̄ at are given as functions of the energy, the derivatives of the real parts at are determined by the Dispersion Relations for Slopes (DRS). At low energies it is essential that in both DRA and DRS calculations the exact solutions Exact () be used.
The imaginary part is positive at and decreases with an exponential form, which must be multiplied by a proper factor pointing to a zero, so that the well known dip may be created in . Actually, in our analysis the dip is located outside the examined range, but a linear factor pointing to a distant zero has influence in the shape of the imaginary amplitude and its extrapolation for to use the optical theorem. In the real part the effect of the structure ( dependence) beyond the exponential slope is present in the small region, and is essential in DRA and DRS for the determination of the parameter , and here also a factor (linear, in our case) must be introduced.
The solutions for the forward amplitudes can be obtained with high accuracy, minimum freedom of parameters, and with remarkable simplicity and regularity in the energy dependence of all quantities.
Trusting to propose a realistic assumption, we write for the pp or pp̄ elastic differential cross sections
where and we call attention for the different values expected for the slopes and of the imaginary and real amplitudes and introduce factors with linear dependence in each amplitude. This expression is applied for pp an pp̄, and the energy dependent quantities , are specific for each case.
In a given normalization we write for the real and imaginary nuclear (upper label ) amplitudes
The optical theorem is implicit in Eq.(4). At , we have the usual definition of the parameter
In Eq.(2), is the fine-structure constant, is the Coulomb phase and is related with the proton form factor
for the ppp collisions, where
In the present work we follow the usual belief that the phase of the Coulomb-Nuclear interference is based on the superposition of amplitudes in the eikonal formalism phase (). In Appendix A we present the calculation of the Coulomb phase adequate for the amplitudes (3) and (4).
The expressions for the derivatives of the amplitudes are
The combinations of parameters
entering respectively as input and output in DRS, are directly related with data, and are crucial for the determination of , , .
It must be noted that the usual direct evaluation of the exponential behavior in using a straight line for the measurements, actually informs the combined average
In a complete analysis of data all quantities in this expression, and not only the average slope in , must be determined.
The forms written above for and are representations valid for small , of amplitudes for the full range, studied in several models models (); us_LHC () at ISR/Cern and higher energies that stress the peculiar properties of the real part of the elastic amplitude with common features of strong slope and a zero for small . In the low energy region here studied, there are not sufficient data for large , and the analysis is restricted to the forward forms of Eqs.(3,4), showing that all quantities (for pp and pp̄) in these expressions are necessary and sufficient for the description of data obeying constraints from DRA and DRS.
At very low energies, namely below GeV the description of elastic processes, are influenced by details of quark-quark and quark-antiquark interactions, with account for specific intermediate states, as for example in a framework of partial waves very_low_energies (). The measurement of polarized amplitudes polarization (), not considered here, depend on the precise values of non-polarized quantities, as we obtain in the present work. In the energy range of our study, gluonic interactions are present, with global dynamics that is describable by simple analytical forms in the variables .
With total cross sections, imaginary slopes and the linear terms written as analytical forms with powers and logarithms in the energy, both DRA and DRS require evaluation of Principal Value (PV) integrals with the generic structure
In recent studies, we have obtained the analytic exact solution for these integrals in terms of the Lerch’s Transcendents Exact (), and these solutions are applied in the present work, with demonstration that they are of fundamental importance, particularly in the low-energy range.
The mathematical formalism of our work is presented in Sec. II and the energy dependent inputs of the imaginary parts are written, with forms that are shown to be valid up to LHC energies and also predict correctly the integrated elastic cross sections.
In Appendix B we present in explicit form the calculation of dispersion relations for the amplitudes (DRA) and for their derivatives (DRS) with the exact solutions in terms of Lerch’s Transcendents.
In Appendix C we present alternative equivalent formalism for the total cross section in the language of pomeron and reggeon trajectories.
With established energy dependent inputs (pp,pp̄), (pp,pp̄), (pp,pp̄) we give in Sec.III precise description of the forward range of elastic pp and scattering, with the essential identification and separation of the real and imaginary amplitudes, with coherence and regularity in the energy dependence all quantities.
In Sec.IV we present conclusions and summarize achievements of our effort.
Ii Formalism and Inputs for Dispersion Relations
ii.1 Inputs for Imaginary Part of Elastic Amplitude
In this section we introduce the forms of the imaginary part of the elastic amplitudes, and explain the determination of their parameters. We stress that we only use qualified data on that may be considered as able to allow reliable analysis in terms of amplitudes written in the analytical forms of Eqs.(3,4). The method of construction of our proposal is iteractive, with inputs and outputs nourishing each other. In a first free analysis, we obtain values of parameters for , and , while the quantities of the real part are left free. The extracted values are put in regular behaviour with the energy, leading to analytical forms with terms of powers and logarithms as described below. Adopting these representations for the imaginary amplitude, we use exact forms of dispersion relations for amplitudes (DRA) and for slopes (DRS) to obtain the quantities of the real part, and then we review the imaginary amplitude.
In the reported experiments at low energies the momentum in the lab system is more often used, while at high energies the use of the the center of mass energy is more common. For pp and scattering the connection with the lab energy
where is the p/p̄ mass. To work with the dispersion relations, the most useful quantity is the dimensionless ratio
Approximate relations that are often used at high energies are obviously , , .
The usual parametrizations PDG () for the total cross sections of the pp and pp̄ interactions has the forms
with parameters constants given in milibarns, in , while are dimensionless. The upper and lower indices , refer to pp and pp̄ scattering respectively. The representation is considered to be adequate for all energies and is based on a large number of values of (pp, pp̄) found in experimental papers. We have the radical claim that these values often are not of good precision, for several reasons, and in general because the optical theorem is not applied to well identified imaginary amplitudes.
This form of amplitude is based on the Pomeron/Reggeon dynamics assumed for the strong interactions DL (), and refers only to purely elastic processes. Contributions of diffractive nature, as first studied by Gribov and later formulated by Good and Walker GW () are not included in this framework. Single diffractive, double diffractive and truly inelastic processes have not been measured in the energy range of our study, while theoretical Tel-Aviv () work based on the gluonic dynamics of QCD and measurements start at energies of the ISR/Cern and Fermilab experiments data_HE (), namely . We show indeed that our treatment describes well the integrated elastic and total cross sections at these higher energies.
Dispersion relations are defined with respect to the lab system energy, and, for low energies, terms like and appear preventing to obtain closed exact forms. We then obtain a representation for the total cross sections in terms of dimensionless variables , , with , writing
where . With this choice, the parameters , , , remain the same. and numerical values are the same, given in Table 1. With
we also need
For use as inputs in dispersion relations DRS we write the slopes and in terms of the variable as
In Eq.(4) we have in addition to the slope of the imaginary part, , a term which is linear on dependence. The inputs for both pp and pp̄ are determined by a controlled analysis. The result is that the difference between the values of for pp and pp̄ are not important, and we assume for both the form
The numerical values of the input parameters are given in Table 1 and in Fig.2 we show the quantities , and of Eqs.(19,21,22) with the points obtained in the examination of the data described in Sec.III. In the inset plots with the variable we show that the extrapolations up to LHC energies (7 and 8 TeV) are compatible with the predicted results us_LHC (); LHC_2017 ().
The integrated elastic cross section of the imaginary part is given by
Plots of and the ratio with as function of the energy are given in Fig.3. We recall that this expression gives the ratio of the purely elastic processes. The remaining part of the ratio gives diffractive plus inelastic processes. We also mark in the figure the experimental values of in the ISR range ( 30 to 60 GeV) data_HE () and our published calculations for 1.8 GeV and LHC energies us_LHC ().
are given by
The profile corresponding to the imaginary forward amplitude is dominant over the real part for low values. However from GeV ( fm) the real part can be dominant and this effect is more pronounced due to the presence of the parameter in Eq.(25).
In Appendix C alternative forms are written for the total cross section, or , in terms of power instead of logarithm as in Donnachie-Landshoff formalism, with all accuracy.
ii.2 Dispersion Relations for Amplitudes and Slopes
The well known dispersion relations for pp and pp̄ elastic scattering are written in terms of even and odd dimensionless amplitudes,
With , the even and odd combinations of amplitudes are related to the pp and pp̄ systems through
The optical theorem informs the normalization of the amplitudes by
and similarly for pp̄.
With the exponential and linear factors in the imaginary parts, we write the inputs
As explained in Sec. I, the real parts are written with exponential and linear factors
and similarly for pp̄.
The parameters are then obtained from
Thus the parameter of the real part is defined by
and similarly for pp̄.
The derivatives of the real amplitude at are written
and similarly for pp̄. These quantities are determined by DRS, that give the even and odd combinations
The quantities and are combinations of amplitude parameters as in Eq.(11).
Taking t=0 we obtain the Dispersion Relations for the Amplitudes (DRA)
The expressions in terms of PV integrals are given in Appendix B.
With t=0, these equations become the Dispersion Relations for Slopes (DRS) that we may write
where we have introduced the parameterization of the real amplitudes.
In Eqs.(II.2) and (II.2) the terms and given by Eq.(10) keep analytical form similar to that of given by Eq.(21), since the parametrization of is linear in . The presence of the quantity as an input in dispersion relations for slopes does not change the algebra of Eqs.(77), (78) (which were first written Exact () assuming ) and the contribution of can be given by the change of the parameters
Introducing analytical expressions for the terms in the imaginary parts, we fall in Principal Value integrations of the form (68) that we can solve exactly Exact (). The input forms in Eqs.(19), (21) and (22) taken into the expressions from DRA and DRS, with numbers given in Sec.III lead to values for and the coefficients of the derivatives of the real parts at the origin for pp and pp̄. In the low energy end, namely with up to 30 GeV, it is essential to use the exact solutions for the PV integrals that appear in DRA and DRS in the calculations of and . Illustrating plots are given in Appendix B.
From comparison of the results with the data, the value of the separation constant that appear in the expressions of DRA is determined. We obtain the interval
In the examples and plots of the present paper we use the value .
Given the , and inputs, the quantities and are determined by DRA and DRS. Since is a combination of and , they must be determined by the data. We obtain that presents very regular energy dependence for both pp and pp̄ systems. We introduce the forms
ii.3 Output Quantities and Plots
We must compare the results that are given in Sec.III with the predictions from dispersion relations for the quantities of the real part , and for the pp and pp̄ systems. Fig.4 shows the energy dependence of the predictions from DRA and DRS for and for the derivative coefficient .
In Fig.5(a) we show the energy dependence obtained for the parameter for pp and in Fig.5 (b) we form the quocient using as points values shown in the large table with fit results, and the lines are calculated with Eqs.(50,51) and the analytical results for and (pp and pp̄) from DRA and DRS. Note that the points where pass by zero must coincide with the zero of the sum . This condition results naturally in our solution.
We thus have a closed coherent determination of all quantities describing forward scattering, with no free local parameter.
Iii Analysis of data in forward direction
The data data_analysed () presented in Tables 3, 4 cover ranges accessible for the analysis, that requires a regular set of points with . Ideally, it would be nice to have good quality data from the very low and going up to 0.1 , but this is not always available in the low energy range.
The analysis covers all data of elastic pp and pp̄ scattering in the energy range of from 3 to 30 GeV. These data have been treated along the history with incomplete theoretical expressions for . In this energy range it is believed that the dynamics of forward scattering is mainly determined by gluonic interactions resulting in smooth energy dependence of all parameters. On the other hand, at very low energies below the direct quark-quark and quark-antiquark interactions may be more important. This seems to be particularly visible in the pp̄ case.
Above , ISR/Cern and Fermilab data, covering very wide ranges, deserves to be studied with analytical forms including the whole range. There are many models models (); us_LHC () for this purpose, and the forward scattering forms here studied are part of these full- descriptions.
An important feature of our analysis is the absorption of normalization errors that accompany the determinations of . Even when these normalization undeterminecies are small , their influence in the parameters is large. The experimental papers use criteria for normalization that we do not consider legal or correct, as ignoring existence of real part, ignoring realistic dependences in the amplitudes, and comparison with other experiments. Some experiments report values using data that are not qualified for the analysis (as insufficient range). These difficulties lead to fluctuations in values of parameters that do not represent physical effects and do not allow a regular global description.
After a smooth description has been achieved and parameters of Eq.(19) are determined, the values of are imposed at each energy. Thus we introduce a constant normalization factor for each data set, chosen so that the total cross section equals the value determined given by Eq. (19). We thus write
The value of for the dataset of each experiment is given in Tables 3, 4. Thus the central values of are assumed and is determined. The error bars in and other quantities represent sensitivities of the fit to each parameter individually, without freedom for correlations.
The resulting suggested parameter values from our analysis are collected in Tables 3, 4. The input energies for the data are written primarily in terms of , as has been more usual in the presentation of data in this range, but the table also includes values. In some cases, mainly at very low energies, where data are not rich, we combine information from different experiments of same or nearby energies, in the same numerical treatment and in plots. We observe good matching of data sets.
We show examples of the treatment of the data in many plots. More detailed information is given in the figure captions. The log horizontal scale helps to expand and exhibit the small behavior, and it is remarkable that often the descriptions work very well up to , beyond the strictly forward range that determines parameters. We interpret that this is so thanks to appropriate form assumed for the amplitudes, and to the control established by DRA and DRS.
Ranges where pass by zero are particularly delicate. The experiments at = 23.542 and 23.882 GeV ( = 294.4 and 303.1 GeV) give an example in which there is discrepancy in literature for the sign. Our treatment solves the discrepancy, namely we show that passes by zero in this region, and this is valid for both experiments. Parameters are in Table 3.
For pp̄ in the range of our analysis is always small and the data are poor. We are then strongly dependent on the predictions from DRA and DRS.
We inspect and analyse the data using a Cern Minuit program for the determination of , nominally with 6 parameters. We use the form for the total pp and cross sections in Eq.(19), and iteratively with observations of the behaviour of and , and use of dispersion relations DRA and DRS as guides for the real parts. Simultaneously we obtain a value for the subtraction constant . Determinations of , and the normalization factor are made simultaneously. , are well represented by the simple analytical forms of Eqs.(21,22) and Table 1. Once the solution for each dataset is obtained, the error bars are obtained relaxing the value of each quantity in the fitting code, so that they represent the sensitivity of the value, but in general do not include correlations.
Details of the datasets and of the calculation of the analytical representations are given in the figure captions.
iii.1 pp data analysed