Structure of Forward pp and p\=p 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 $\rm {p \bar p}$ scattering data in the forward range at energies below $\sqrt(s)\approx 30 \GeV$. 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 $t$ dependence of the imaginary parts, with exponential and linear factors. A structure for the $t$ dependence of the real amplitude is written, with slopes $B_R$ and a linear factor $\rho-\mu_R t$ 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 $d\sigma/dt$ data, with regular energy dependence of all quantities. It is shown that a revision of previous calculations of total cross sections, slopes and $\rho$ parameters in the literatures is necessary, and stressed that only determinations based on $d\sigma/dt$ data covering sufficient $t$ range using appropriate forms of amplitudes can be considered as valid.


I. INTRODUCTION
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, which are functions of two variables s, t (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 pp systems, where dσ=dt 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 σ, ρ and other parameters of pp and pp 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 ΔN=Δt of event rates in intervals t → t þ Δt. With attention given to fluxes and densities (we are only concerned with unpolarized beams and targets), tables of t distributions in differential cross sections dσ=dt 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 T I ðs; tÞ and T R ðs; tÞ 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 t ¼ 0, or even very close to it, but rather in sets of points of an interval. The determination of σ, ρ, slopes and other quantities requires extrapolation of data in a jtj 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 t distribution in dσ=dt. Regularity in the behavior of all quantities with the energy is an important consideration to obtain sensible descriptions of the elastic processes. Experiments at different energies must be analyzed globally, since separate fitting procedures may lead to values that are useless as a step for the phenomenology of the area.
The t range of the data at given energy must be sufficient for representation through assumed analytical forms. In the low energy range, up to ffiffi ffi s p ≈ 30 GeV, often these conditions are not satisfied, even suffering insecure normalization in the measurements of dσ=dt, 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 ffiffi ffi s p from 30 to 60 GeV, pp and pp from ISR/CERN and Fermilab data cover the large t range with good precision, showing fast increase in σ, a forward peak and a marked dip in dσ=dt. These measurements led to the establishment of the successful Regge phenomenology [1,2], based on the exchange of particles (Pomerons, Reggeons) in the t channel. Several theoretical models were developed to describe dynamically this region of data in the s and t variables [3].
Above ffiffi ffi s p ¼ 60 GeV, experiments [4] have large energy gaps, passing fast by SPS/CERN, Fermilab and reaching the TeV range of LHC [5]. Ingredients of QCD dynamics enter with less or more detail in the interpretation of these data [6,7]. 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 [8,9]. 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 [10]. The potential of crucial information [11] 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 analyze forward pp, pp data with ffiffi ffi s p from ≈3 to ≈30 GeV, 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 pp forward elastic scattering analyzing all data of differential cross sections that seem qualified (namely covering necessary t 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) [12] 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 t ¼ 0, 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 pp 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 adjusts 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 [13] 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 analyze are regularly distributed, presented in numbers and very precise plots of dσ=dt 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 pp scattering in the forward direction, the t dependences of the amplitudes are mainly characterized by exponential forms, with slopes B I and B R that are essentially independent quantities, essentially not equal to each other: the real and imaginary amplitudes do not run parallel along the t axis. We take special care in the investigation of the behavior of the real part, which has structure deviating from a pure exponential form for t values included in the forward range.
Once the total cross sections for pp and pp are parametrized in the energy, the usual dispersion relations for the amplitudes (DRA) determine the real amplitudes at the origin (namely the ρ parameters). Similarly, if the derivatives of the imaginary parts of pp and pp at t ¼ 0 are given as functions of the energy, the derivatives of the real parts at t ¼ 0 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 [14] be used. The dispersion relations for the amplitudes has been effectively used in investigations of the energy dependence of total cross section and ρ parameter in pp and pp scattering, being a very important tool of control in the analysis of the data [15].
The imaginary part is positive at t ¼ 0 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 dσ=dt. Actually, in our analysis the dip is located outside the examined jtj range, but a linear factor pointing to a distant zero has influence in the shape of the imaginary amplitude and its extrapolation for t ¼ 0 to use the optical theorem. In the real part the effect of the structure (t dependence) beyond the exponential slope is present in the small jtj region, and is essential in DRA and DRS for the determination of the parameter ρ, 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 pp elastic differential cross sections dσ dt ðpp; ppÞðs; tÞ where t ≡ −jtj and we call attention for the different values expected for the slopes B I and B R of the imaginary and real amplitudes and introduce factors with linear t dependence in each amplitude. This expression is applied for pp and pp, and the energy dependent quantities σðsÞ, B I ðsÞ, μ I ðsÞ, ρðsÞ, B R ðsÞ, μ R ðsÞ are specific for each case. In a given normalization we write for the real and imaginary nuclear (upper label N) amplitudes and T N I ðs; tÞ ¼ The optical theorem is implicit in Eq. (4). At t ¼ 0, we have the usual definition of the ρ parameter remarking that the value of ρ obtained by fitting of data in a certain t range depends on the analytical forms (3) and (4) of the amplitudes. In Eq. (2), α is the fine-structure constant, Φðs; tÞ is the Coulomb phase and F C ðtÞ is related with the proton form factor for the pp=pp 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 [16]. 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 and entering respectively as input and output in DRS, are directly related with data, and are crucial for the determination of ρ, μ R , B R . It must be noted that the usual direct evaluation of the exponential behavior in dσðtÞ=dt ¼ ðdσðtÞ=dtÞðt ¼ 0Þ × expðBtÞ 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 B in dσ=dt, must be determined. The forms written above for T I ðtÞ and T R ðtÞ are representations valid for small jtj, of amplitudes for the full t range, studied in several models [3,7] 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 B R and a zero for small jtj. In the low energy region here studied, there are not sufficient data for large jtj, and the analysis is restricted to the forward forms of Eqs. (3) and (4), showing that all quantities (for pp and pp) in these expressions are necessary and sufficient for the description of data obeying constraints from DRA and DRS.
At very low energies, namely below p LAB ¼ 4 GeV, the description of elastic processes is 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 [17]. The measurement of polarized amplitudes [18], not considered here, depend on the precise values of nonpolarized 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 s, t.
With total cross sections, imaginary slopes and the linear terms μ I 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 [14], 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 A we calculate the phase of the nuclear-Coulomb interference for real amplitude of the form of Eq. (3).
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 σðpp; ppÞ, B I ðpp; ppÞ, μ I ðpp; ppÞ we give in Sec. III a precise description of the forward range of elastic pp and pp scattering, with the essential identification and separation of the real and imaginary amplitudes, with coherence and regularity in the energy dependence of all quantities.
In Sec. IV we present conclusions and summarize achievements of our effort.

II. FORMALISM AND INPUTS FOR DISPERSION RELATIONS
A. 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 dσ=dt that may be considered as able to allow reliable analysis in terms of amplitudes written in the analytical forms of Eqs. (3) and (4). The method of construction of our proposal is interactive, with inputs and outputs nourishing each other. In a first free analysis, we obtain values of parameters for σ, B I and μ I , while the quantities of the real part are left free. The extracted values are put in regular behavior 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 p LAB is more often used, while at high energies the use of the center of mass energy ffiffi ffi s p is more common. For pp and pp scattering the connection with the lab energy where m is the p=p mass. To work with the dispersion relations, the most useful quantity is the dimensionless ratio and then Approximate relations that are often used at high energies are obviously s ≈ 2mE, x ≈ s=2m 2 , x ≈ p LAB =m. The usual parametrizations [13] for the total cross sections of the pp and pp interactions have the forms with parameters P, H, R 1 , R 2 constants given in millibarns, s 0 in GeV 2 , while η 1 , η 2 are dimensionless. The upper and lower indices −,þ refer to pp and pp scattering respectively. The representation is considered to be adequate for all energies s ≥ s 0 and is based on a large number of values of σ (pp, pp) 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 [2], and refers only to purely elastic processes. Contributions of diffractive nature, as first studied by Gribov and later formulated by Good and Walker [19], 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 [8] work based on the gluonic dynamics of QCD and measurements start at energies of the ISR/CERN and Fermilab experiments [4], namely ffiffi ffi s p ⪆ 30 GeV. 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 log 2 ðE þ mÞ and ðE þ mÞ −η appear preventing to obtain closed exact forms. We then obtain a representation for the total cross sections in terms of dimensionless variables and analyze all dσ=dt data for p LAB from 4 to 500 GeV=c using the assumed structures of the amplitudes in Eqs. (3) and (4). We keep the value s 0 ¼ 16 GeV 2 suggested [13] for Eq. (18), now appearing as where m ¼ 0.93827 GeV. With this choice, the parameters P, H, R 1 , R 2 remain the same and numerical values are the same, given in Table I. With we also need ðℏcÞ 2 1 m 2 ¼ 0.4423 mb: In Fig. 1 we show the comparison between Eqs. (18) and (19) for pp written with the same parameters. The difference between the curves represents the deviation of Eq. (17) to the approximated form x ≈ s=2m 2 .
For use as inputs in dispersion relations DRS we write the slopes B pp I ðxÞ and B pp I ðxÞ in terms of the x variable as again with symmetry in the coefficients for pp and pp. As in Eqs. (18) and (19), for s=2m 2 ≫ 1 the slopes B I of Eq. (21) can be written with similar analytical forms in the variable s. In Eq. (4) we have in addition to the slope of the imaginary part, B I , a term μ I which is linear on t dependence. The inputs μ I ðxÞ for both pp and pp are determined by a controlled analysis. The result is that the difference between the values of μ I ðxÞ for pp and pp is not important, and we assume for both the form The numerical values of the input parameters are given in Table I and in Fig. 2 we show the quantities σðxÞ, B I ðxÞ and TABLE I. Parameters of total cross section, slopes and linear terms of imaginary parts in Eqs. (19), (21), and (22).    (18) and (19) using the same parameter values, showing the deviation for low energies due to kinematics. μ I ðxÞ of Eqs. (19), (21), and (22) with the points obtained in the examination of the data described in Sec. III. In the inset plots with the variable ffiffi ffi s p we show that the extrapolations up to LHC energies (7 and 8 TeV) are compatible with the predicted results [7,10].
The integrated elastic cross section of the imaginary part is given by Plots of σ elastic I and the ratio with σ as a 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 1 − σ el I =σ gives diffractive plus inelastic processes. We also mark in the figure the experimental values of σ elastic in the ISR range ( ffiffi ffi s p ≈ 30 to 60 GeV) [4] and our published calculations for 1.8 GeV and LHC energies [7]. The dimensionless Fourier transforms of the amplitude Tðs; tÞ in Eqs. (3) and (4). with respect to the momentum transferT  (19), (21), and (22), together with values obtained in the study of the data in Sec. III. The connections of the analytical forms with our previously published calculations [7] for very high energies (1.8 TeV and LHC) are shown in the insets in terms of ffiffi ffi s p .
are given bỹ The profile corresponding to the imaginary forward amplitude is dominant over the real part for low b values. However from b ≥ 14 GeV (≃2.8 fm) the real part can be dominant and this effect is more pronounced due to the presence of the μ R parameter in Eq. (25). In Appendix C alternative forms are written for the total cross section, σðxÞ or σðsÞ, in terms of power instead of logarithm as in Donnachie-Landshoff formalism, with all accuracy.

B. Dispersion relations for amplitudes and slopes
The well known dispersion relations for pp and pp elastic scattering are written in terms of even and odd dimensionless amplitudes, With x ¼ E=m, the even and odd combinations of amplitudes are related to the pp and pp systems through The optical theorem informs the normalization of the amplitudes by and similarly for pp.
With the exponential and linear factors in the imaginary parts, we write the inputs ImF pp ðx; tÞ ImF pp ðx; tÞ with functions σðxÞ, B I ðxÞ and μ I ðxÞ for pp and pp given in Eqs. (19), (21), and (22). As explained in Sec. I, the real parts are written with exponential and linear factors and similarly for pp. The ρ parameters are then obtained from with the lhs given by dispersion relations (27) and (28). Thus the ρ parameter of the real part is defined by and similarly for pp. The derivatives of the real amplitude at jtj ¼ 0 are written and similarly for pp. These quantities are determined by DRS, which give the even and odd combinations and The quantities D pp R and D pp R are combinations of amplitude parameters as in Eq. (11).
Substituting these expressions into Eqs. (27) and (28), written in terms of the dimensionless variable x, we obtain and Taking t ¼ 0 we obtain the dispersion relations for the amplitudes (DRA) and The expressions in terms of PV integrals are given in Appendix B.
To obtain DRS, we take derivatives of Eqs. (40) and (41) with respect to t, writing With t ¼ 0, these equations become the dispersion relations for slopes (DRS) that we may write where we have introduced the parametrization of the real amplitudes.
In Eqs. (47) and (48) the terms D pp I and D pp I given by Eq. (10) keep analytical form similar to that of B I ðxÞ given by Eq. (21), since the parametrization of μ I is linear in logðxÞ. The presence of the quantity μ I ðxÞ as an input in dispersion relations for slopes does not change the algebra of Eqs. (B10) and (B11) (which were first written [14] assuming μ I ¼ 0) and the contribution of μ I 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 (B1) that we can solve exactly [14]. 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 ρ and the coefficients of the derivatives of the real parts at the origin D R for pp and pp. In the low energy end, namely with p LAB 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 ρ and D R . Illustrating plots are given in Appendix B.
From comparison of the results with the ρ data, the value of the separation constant K that appears in the expressions of DRA is determined. We obtain the interval In the examples and plots of the present paper we use the value K ¼ −310.
Given the σðxÞ, B I ðxÞ and μ I inputs, the quantities ρ and D R are determined by DRA and DRS. Since D R ¼ ρB R =2 − μ R is a combination of μ R and B R , they must be determined by the data. We obtain that μ R presents very regular energy dependence for both pp and pp systems. We introduce the forms and  (50) and (51), obtained for K ¼ −310. The corresponding lines are shown in Fig. 5. The central values of c 0 and c 2 are shown with high precision to put coincident the zeros of ρ and Numerical values for the constants are given in Table II and plots are shown in Fig. 5.

C. 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 ρ, D R and μ R for the pp and pp systems. Figure 4 shows the energy dependence of the predictions from DRA and DRS for ρ and for the derivative In Fig. 5(a) we show the energy dependence obtained for the parameter μ R for pp pp and in Fig. 5(b) we form the quotient B R ¼ 2ðD R þ μ R Þ=ρ using as point values shown in the large table with fit results, and the lines are calculated with Eqs. (50) and (51) and the analytical results for ρ and D R (pp and pp) from DRA and DRS. Note that the points where ρ passes by zero must coincide with the zero of the sum D R þ μ R . 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 [20] presented in Tables III and IV cover jtj ranges accessible for the analysis, which requires a regular set of points with jtj ≤ 0.5 GeV 2 . Ideally, it would be nice to have good quality data from the very low jtj ≈ 0.001 GeV 2 and going up to 0.1 GeV 2 , but this is not always available in the low energy range.
The analysis covers all data of elastic pp and pp scattering in the energy range of ffiffi ffi s p from 3 to 30 GeV. These data have been treated along the history with incomplete theoretical expressions for dσ=dt. 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 p LAB ¼ 4 GeV the direct quark-quark and quark-antiquark interactions may be more important. This seems to be particularly visible in the pp case.
Above ffiffi ffi s p ¼ 30 GeV, ISR/CERN and Fermilab data, covering very wide t ranges, deserves to be studied with analytical forms including the whole t range. There are many models [3,7] for this purpose, and the forward scattering forms here studied are part of these full-jtj descriptions. An important feature of our analysis is the absorption of normalization errors that accompany the determinations of dσ=dt. Even when these normalization indeterminacies 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 t dependences in the amplitudes, and comparison with other experiments. Some experiments report σ values using data that are not qualified for the analysis (as insufficient t range). These difficulties lead to fluctuations in values of parameters that do not represent physical effects and do not allow a regular global description.  50) and (51) and the points obtained from the data. (b) Energy dependence of the slopes B R ðppÞ and B R ðppÞ of the real amplitudes. The lines are fully predicted by the dispersion relations for slopes DRS (that give D R ¼ ρB R =2 − μ R for pp and pp), dispersion relations for the amplitudes (that give ρ for pp and pp) and by Eqs. (50) and (51)  After a smooth description has been achieved and parameters P, H, R 1 , R 2 , η 1 , η 2 of Eq. (19) are determined, the values of σ are imposed at each energy. Thus we introduce a constant normalization factor f for each data set, chosen so that the total cross section σ equals the value determined given by Eq. (19). We thus write   The value of f for the data set of each experiment is given in Tables III and IV. Thus the central values of σ are assumed and f is determined. The error bars in σ 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 III and IV. The input energies for the data are written primarily in terms of p LAB , as has been more usual in the presentation of data in this range, but the table also includes ffiffi ffi s p 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 dσ=dt data in many plots. More detailed information is given in the figure captions. The log jtj horizontal scale helps to expand and exhibit the small jtj behavior, and it is remarkable that often the descriptions work very well up to jtj ¼ 0.2, 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 ρ passes by zero are particularly delicate. The experiments at ffiffi ffi s p ¼ 23.542 and 23.882 GeV (p LAB ¼ 294.4 and 303.1 GeV) give an example in which there is discrepancy in literature for the ρ sign. Our treatment solves the discrepancy, namely we show that ρ passes by zero in this region, and this is valid for both experiments. Parameters are in Table III.
For pp in the range of our analysis ρ is always small and the data are poor. We are then strongly dependent on the predictions from DRA and DRS.
We inspect and analyze the data using a CERN MINUIT program for the determination of χ 2 , nominally with six parameters. We use the form for the total pp and pp cross sections in Eq. (19), and iteratively with observations of the behavior of B I and μ I , and use of dispersion relations DRA and DRS as guides for the real parts. Simultaneously we obtain a value for the subtraction constant K. Determinations of B I , μ I and the normalization factor f are made simultaneously. B I , μ I are well represented by the simple analytical forms of Eqs. (21) and (22) and Table I. Once the solution for each data set 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 χ 2 value, but in general do not include correlations.
Details of the data sets and of the calculation of the analytical representations are given in the figure captions.
A. pp data analyzed Figure 6 shows data from four experiments, [20], [20], [20], and [20], that cover regularly large energy and jtj ranges. In the plots these are called Beznogikh (1973) [20], Kuznetsov (1981) [20], Fajardo (1981) [20] and Burq (1983) [20]. The parameters for the lines are given in Table III. We observe that the representations obtained by Eqs. (2)-(4) are very faithful to the data up to surprisingly large jtj values, namely above jtj ¼ 0.1 GeV 2 and up to 0.3 in some cases. The compatibility of different experiments can be noted.

Low energies: p LAB = 4-10 GeV
In the very low energy end, there are forward data from Jenni et al. [20] at p LAB ¼ 4.2, 7 and 10 GeV. At 4.2 and 7 GeV we join Ambats et al. [20] at 3.65 and 6 GeV, with factors 0.99 to compensate for the energy dependence.
At p LAB ¼ 10 GeV, to exhibit a representative recommendation we join to Jenni et al. [20] the data from Beznogikh et al. [20] at 9.43 GeV and from Brandenburg et al. [20] at 10.4 GeV, with energy adjustment factors, forming a large set with 153 data points.
In all these cases the matching of the experimental sets is very good, and we are then able to find representations of very good quality for dσ=dt in elastic pp scattering in all this energy range. The parameters are shown in Table III and the data and representative curves are shown in Fig. 7. of the Fermilab data [20] shown in Fig. 6. The parameters are given in Table III Fig. 6.

Large jtj: Behavior beyond the forward range
The situation at large jtj is shown with Ayres et al. data [20] in Fig. 9, where we study the behavior for jtj beyond 0.1 GeV 2 . These data do not cover the low-jtj range, and are plotted together with the lines representing the Beznogikh (1973) and Fajardo (1981) data at the corresponding energies. We see a remarkable matching of normalization in the points with smaller jtj and an increasing deviation as jtj increases. We here confirm that the equations for the amplitudes in forward scattering have the validity confirmed up to 0.1 GeV, where regular deviation of the data upwards may start.
Two other plots in the same Fig. 9 show the behavior of data in comparison with the analytic representations for jtj beyond the strict forward region. Details are given in the figure caption.

B. pp data analyzed
At low energies, the analysis of the pp forward scattering data is particularly difficult, because ρ is very small, and changes sign for p LAB ¼ 117 GeV ( ffiffi ffi s p near 12 GeV).

pp in the range p LAB = 4.2-10.0
The range is ffiffi ffi s p ¼ 3. 14-4.54 GeV in pp scattering is difficult to treat, because ρ is very small, possibly passing through zero in this range. In Fig. 10 we show data collected at the energies p LAB ¼ 4.2, 6.0, 8.0 and 10.0 GeV corresponding to measurements of Jenni et al. [20] with jtj starting at about 10 −3 GeV 2 . These data are not smooth and do not cover a wide t range, so that they are  [20], [20], [20] and [20]. The data are regular, cover t ranges adequate for determination of forward scattering quantities, and the representation with parameters given in Table III are extremely precise. The compatibility among the four experiments is remarkable. The parameter ρ and the combination ðρB R =2 − μ R Þ are in agreement with predictions from dispersion relations for amplitudes DRA and for their derivatives DRS. The framework is explained in the text. The zero of ρ at p LAB ¼ 277 GeV that occurs in the energy range of these data is well treated by the representations.
combined with other data in the range to build sufficient sets. The assemblage is shown in plots, with the lines representing the results of the analysis, with parameters given in Table IV. At 4.2 GeV the combination is made with the p LAB ¼ 5 GeV data of Ambats et al. [20], multiplied by 1.06 to account for the energy dependence. At 6 GeV the merging is with Braun et al. [20] at p LAB ¼ 5.7 GeV (multiplied by the energy dependence factor 0.98). At 8 GeV the combination is made with Russ et al. [20] at the same energy. At 10 GeV the set is completed with data at 10.4 GeV from Brandenburg et al. [20] with a factor 1.0117. The energy factors are based on the squared ratio of the total cross section given by the parametrized σ input. We thus obtain values for parameters that give reliable proposals in a region of energies that is very difficult, because ρ is very small. Our representations with Eqs. (2)-(4) and parameters given in Table IV give correct descriptions of these data. The dispersion relations predictions for ρ and for the derivative combination ρB R =2 − μ R are satisfied.

pp data in the mid-jtj range for p LAB = 16-50 GeV
The data in Fig. 11, with p LAB ¼ 16-50 GeV, cover only the mid-jtj range. It is interesting to observe in general that the representation of data using the forward scattering expression for dσ=dt, in principle is only valid up to jtj ¼ 0.1 GeV 2 , in many cases remain good up to 0.5.
According to Ayres [20], Akerlof data [20], also at midjtj, have a normalization problem, and then are not included in our report in Table IV.

pp in the range p LAB = 70-200 GeV
This is a sensitive range for the determination of ρ, due to the change of sign. In pp ρ is small and negative at low energies up to about p LAB ¼ 116 GeV. The data by Ayres [20] at p LAB ¼ 50, 70, 100, 140 and 175 GeV are regular and with small error bars but are restricted to jtj values that are not small enough for determination of forward scattering parameters, and we then form combinations with other sets of data, as shown in Fig. 12. At p LAB ¼ 70 and 175 GeV these data can be combined with the data from Fajardo et al. [20] to produce sets that cover both small and mid-jtj values. Some parts of Fajardo data [20] in the higher jtj part are excluded, giving place to Ayres data [20], which are more regular. It is remarkable that the data from the two experiments match very well, and a common solution seems to be adequate. At p LAB ¼ 70 GeV the data give ρ compatible with zero, and in Table IV we choose the value given by the DRA prediction. This is a choice, and the value and sign have the corresponding uncertainty. GeV respectively with large jtj are incorporated in the sets, after multiplication by the same factor 0.99 accounting for the energy dependence. At p LAB ¼ 10 we plot together in a single set data from three different experiments [20], [20], and [20] at 9.43, 10 and 10.4 GeV respectively, introducing conversion factor 0.97 at 10.24 GeV to account for energy dependence. For 9.43 the factor is nearly 1, and is ignored. In all cases the connection of the data sets is remarkable. The data at 9.43 GeV are also plotted in a pile of the same experiment [20] in Fig. 6. The parameters are given in Table III.  [20] and [20] in the higher energy part of our analysis with parameters given in Table III. Other measurements in the same region are shown in Fig. 6.  [20]. The lines are NOT obtained with these data, but rather are taken from the description of the forward range in Fig. 6, with the parameters as given in Table III, at the corresponding energies. For the 140 GeV data there is no corresponding low-jtj experiment, and we use the line for 150 GeV [20]. The matching of points of lowest jtj at each energy is impressive, showing that different experiments succeed in the normalization of their data. At all energies there is a progressive detachment between data and predicted curves starting at, say, 0.1 GeV, showing clearly that these data by themselves cannot be used for determinations. In the second plot (b) (rhs at top) we show the connection between the solution at 199 GeV [20] and the forward part (up to 0.2 GeV 2 ) of data by Schiz et al. [20] that starts at jtj ¼ 0.02 GeV 2 . The solid line represents the curve for Kuznetsov data at 199 GeV in Fig. 6 with parameters given in Table III. The agreement of Schiz data with the curve up to jtj ≈ 0.1 GeV 2 is remarkable but it seems that improvement could be obtained with normalization. We observe that for higher jtj there appears a displacement between data and the curve, exhibiting again the limitation for the use of the forward amplitudes. For illustrative purpose, we show together Akerlof et al. points [20] added to the 200 GeV plot, demonstrating that they should not be considered in good agreement (notice that the scale is very tight), presenting strong displacement as jtj increases. In the third plot (c) (bottom) we show Apokin et al. data [20] at p LAB ¼ 42.5 and 52.2 GeV, covering smaller jtj ranges, inserted together with Beznogikh [20] and Kuznetsov [20] data respectively. We observe compatibility (the 42.5 data are corrected with factor 1.02 in the plot, to reduce to energy 40.62 GeV). The lines of the solution for Apokin [20] are shown in dashed form, while the Beznogikh and Kuznetsov lines are solid. There is reasonable compatibility, but it is clear that the measurements reaching smaller jtj values are more qualified for parameter determination.
We observe in Table IV that at p LAB ¼ 125 GeV, ρ is positive, confirming the change of sign that occurs at p LAB ¼ 117 GeV.
At all energies of the range, the suggested solutions in Table IV give very good descriptions, as indicated by the low χ 2 values. We are able to follow the DRA and DRS predictions for the value of the amplitude and for its derivative at the origin. It is interesting to observe that often the representation of data using the forward scattering expression for dσ=dt, in principle only valid up to jtj ¼ 0.1 GeV 2 , in many cases remains good up to 0.2 GeV 2 . This is seen also in both Figs. 11 and 12. We observe that in this range the numbers reported in experimental papers for the ρ parameter are very irregular and with contradictions.   [20] together with data from Ambats [20] at 5 GeV (multiplied by 1.06 to account for energy dependence). Data from Jenni et al. [20] at 6 GeV, and from Braun et al. [20] for p LAB ¼ 5.7 GeV (multiplied by 0.98). Data from Jenni et al. [20] and from Russ et al. [20] at 8 GeV, and data from Jenni et al. [20] at 10 GeV and from Brandenburg et al. [20] at 10.4 GeV (multiplied by 1.0117). Visually (up to jtj ≈ GeV 2 ) and in the calculations the matching of data is impressive. Parameter values for each line are given in Table IV.  Russ et al. [20], Antipov et al. [20], Ayres et al. [20], Batyunia et al. [20] at p LAB ¼ 16, 22.4, 25.2, 40.1 and 50 GeV, described with the parameter values of Table IV, presenting regular energy dependence. It is remarkable that the lines reproduce well the data up to 0.5 GeV 2 . Notice that the original Antipov data at 40.1 GeV have been remeasured and published with changes in 1977 [20], and a factor 1.15 is used to adjust the old to the new published data.  [20], and from Ayres et al. [20] described by the parameters given in

IV. CONCLUDING REMARKS
In this paper we propose a revision of the existing information on forward parameters in pp and pp scattering, giving coherent and precise description of all elastic data in the energy range from p LAB ≈ 4 to 500 GeV, namely ffiffi ffi s p ≈ ¼ 3 to 30 GeV.
We treat the fundamental problem of the identification of the real and imaginary parts of the complex amplitude in dσ=dt, using Eqs. (3) and (4) and exact solutions of dispersion relations for amplitudes (DRA) and for slopes (DRS). The Coulomb interference is included with correct expression for the phase in Eq. (2), which depends on the actual analytical form of the real amplitude, and is studied in Appendix A. We do not perform open fittings, but rather rationalize descriptions, in an interplay between data and mathematical representations, building a bridge for the construction of full dynamical models.
The analysis of the dσ=dt data leads to the parameter values presented in Tables III and IV, given in Sec. III. Determination of parameters is guided by dispersion relations and criteria of regularity in the energy dependence, in general achieving low χ 2 values, with visual performances exhibited in plots in Sec. III. The description is coherent and theoretically respectable and formally simple.
A global analysis in the low energy range must be restricted to the forward range, because there are not enough data for a full-t study of the amplitudes. Since the imaginary amplitudes dominate at small jtj, our precision work has particular importance as an investigation of the structure of the real parts. At low energies the use of dispersion relations requires the use of exact solutions for principal value integrals, as we prove in Appendix B.
For a precise treatment of the data, the t dependence of the amplitudes cannot be limited to exponential forms. It is known that the imaginary part must point to a zero because there is a dip in dσ=dt shown when data have more extended jtj. Also the real part must show a zero when ρ is positive, determined by Martin's theorem [21]. We we thus are led to the amplitudes written in Eqs. (3) and (4). To deal with this complexity without ambiguities in the analysis of data, we make use of the dispersion relations for slopes (DRS), which determine the derivatives of the real part when the derivatives of the imaginary parts are given as a function of the energy.
The proposals for σ, B I and μ I (for pp and for pp) in Eqs. (19), (21) and ((22), obtained from the data, are basic in our work as inputs for DRA and DRS. Figure 2 shows that the analytical forms have correct extensions up to the very distant LHC energies.
Starting with the unitarized form for σðxÞ originated in Pomeron/Reggeon phenomenology [22], and complemented with the jtj structure of the imaginary part, the description of the dσ=dt data shows that all information in elastic scattering at low energies can be represented by simple expressions for the amplitudes. We believe that the simple forms of energy dependence are a consequence of the very complex gluonic dynamics of QCD. Local effects that may be meaningful microscopically are absorbed in the complexity and do not become visible in elastic data.
In Appendix C it is shown that the log-squared form of the total cross section in Eq. (19) is shown to be equivalent to a simple-pole-Pomeron/Reggeon form, with powers instead of logarithms, and thus without limitation for the analytical solutions of PV integrals.
As an important test for the amplitudes, Fig. 3 in Sec. II shows that the integrated elastic cross section σ elastic very well reproduces the data at higher energies (ISR, Fermilab, LHC) [4,5]. The ratio σ elastic =σ total is small, corresponding to a large fraction of diffractive and inelastic contributions, which have not been measured in the low energy range and are not in our scope. The single and double diffractive processes, measured at ISR/CERN, Fermilab and LHC [4,5], are theoretically studied in microscopic terms based on multi-Pomeron/Reggeon ideas [8] and on gluonic dynamics of the color glass condensate [9].
The analysis of the pp data at low energies is particularly delicate, because ρ is very small and has change of sign. Most pp data do not cover jtj ranges sufficient for determination of the amplitude parameters without support of DRA, DRS and search for regular energy dependence. We include as much pp information as possible, forming combinations of different experiments at nearby energies to have sets suitable for analysis, arriving at reliable solutions.  [20] and from Amos et al. [20] described by the parameters given in  [20], and also in the CERN Library. Figure 4 shows that ρ and the derivative D R ¼ρB R =2−μ R follow DRA and DRS predictions. Figure 5 shows the proposed decomposition of D R , with μ R given by Eqs. (50) and (51), and shows the calculated B R , compared to values obtained from the data. Everything seems satisfactory, including the behavior as high energies, where these quantities appear [7,10]. Negative μ R predicts the zero of the real amplitude in a region where ρ is positive, as at high energies [21], with the real zero approaching jtj ¼ 0 with increasing energy. Of course alternative solutions can be obtained for μ R and B R , maintaining the combination Eq. (11) as fixed by DRS.
This extensive work is an effort to place order in the multitude of determinations of pp and pp forward scattering parameters at low energies. The treatment is as simple as possible, without relation to any specific model based on microscopic ingredients, and identifies the real and imaginary parts of the complex elastic amplitude. We hope it can be considered as necessary and useful.

ACKNOWLEDGMENTS
The authors wish to thank the Brazilian Agencies CNPq and CAPES and Gobierno de Aragón (Spain) (Project No. E24/1) for financial support.

APPENDIX A: COULOMB PHASE
We here present the Coulomb interference phase ϕ used in the phenomenology. This phase was studied by several authors [16].
The expression for the phase depends on the analytical structure of the real and imaginary parts of the nuclear amplitude and also on the Coulomb amplitude and electromagnetic form factor. The real and imaginary nuclear amplitudes are given by Eqs. (3) and (4) respectively and the Coulomb amplitude is given by Eq. (6) with the proton form factor (7). To simplify calculations, we may alternatively use the exponential representation for the form factor with Λ 2 ¼ 0.71 GeV 2 , trusting that differences in results for the phases with different form factors are unimportant, as confirmed by Cahn [16]. The Coulomb-nuclear-interference amplitude is given by with normalization defined by The correspondence between the dimensionless nuclear amplitude F N and the phenomenological T N is given by We start from the expression for the phase with the signs ∓ corresponding to the pp=pp systems. As a generalization with respect to Cahn's calculation, we take for the nuclear amplitude the same expressions in Eqs. (3) and (4). Then we need to evaluate integrals where we have used the definition with β ¼ R, I. The results of the integrations (k ¼ 0, 1) are where γ ¼ 0.5772 is the Euler Gamma constant. The phase is then written with real and imaginary parts respectively and Equations (A10) and (A11) are our final results for the phase calculated with form factors, in a generalization of the work by Cahn [16], assuming more complete structures for the real and imaginary parts of the scattering amplitude. It may be of practical usefulness to define and and then write and It must be observed that in these expressions b R , b I and −t have compatible units, as GeV −2 and GeV 2 . The result is simple: in the real part the t dependence is purely linear in logð−tÞ, the imaginary part is very small constant, and there is no explicit energy dependence.
In the simplified case μ R ¼ μ we obtain Cahn's original form.
Obviously this alternative representation can be written in terms of the variable x for use in dispersion relations.