Logarithmic Regge Pole

This work presents the subtraction procedure and the Regge cut in the logarithmic Regge pole approach. The subtraction mechanism leads to the same asymptotic behavior as previously obtained in the non-subtraction case. The Regge cut, on the other hand, introduces a clear role to the non-leading contributions for the asymptotic behavior of the total cross section. From these results, one introduces some simple parameterization to fit the experimental data for the proton-proton and antiproton-proton total cross section above some minimum value up to the cosmic-ray. The fit parameters obtained are used to present predictions to the $\rho(s)$-parameter as well as to the elastic slope $B(s)$ at high energies.

with the rise of the total cross section given by the leading Regge pole [25,26]. The Regge theory predicts the total cross section asymptotically behaving as s α(0)−1 , as s → ∞. The FM bound predicts, however, a rising bounded by ln s 2 . The only way to ensure the validity of the Regge formalism in front of the FM bound is with a trajectory of less than 1. Nonetheless, the fitting procedures for the total cross section always produces α(0) > 1 [3,27]. The FM bound is a crucial formal result of the high energy physics and cannot be disregarded in any theoretical approach. The analyticity principle is also not be satisfied unless the trajectory of all particles lies on the Regge trajectory.
Recently, obeying the FM bound, a novel approach to the leading Regge pole was obtained by introducing a logarithmic representation for the leading Regge pole [28]. In the present work, one continues the logarithmic Regge approach introducing the subtraction and the cut problem in the logarithmic Regge framework.
The subtraction procedure, in the present formalism, cannot be used in its pure version since the fast decreasing caused by the subtraction s −1 turns the approach useless. However, as shall be seen, there is a subtle approximation allowing the use of a less restrictive version of the subtraction mechanism. This approach will lead to a modified subtraction mechanism, whose consequence is that subtraction and non-subtraction cases produce the same functional form to the asymptotic scattering amplitude. The cut, on the other hand, seems to be a result of the sub-leading contributions. Then, this mechanism may be particularly important to explain the mixed energy region where the total cross section, for example, is controlled by the pomeron and odderon exchange.
Using naive parameterizations for the proton-proton and proton-antiproton total cross section, it is possible to understand the role of the pomeron at high energies within the logarithmic Regge approach. As obtained in [28], the double pomeron picture is favored for energies above 1.0 TeV. On the other hand, as shall be seen, starting the fitting procedures taking into account energies above 25.0 GeV, then the pomeron assumes values greater than 2 suggesting the saturation of the Froissart-Martin bound. This problem can be solved by using the recent TOTEM measurement of ρ(s) [17].
Applying the derivative dispersion relation, one obtains the real part of the elastic scattering amplitude. The fitting parameters are taken from the fitting procedures, and the predictions for the ρ(s)-parameter are presented. Assuming a null subtraction constant (it interferes only in the low energy experimental data), the curves suggest a double-pomeron exchange to reproduce the ρ(s) obtained in the TOTEM Collaboration. Then, using this constraint, one keeps the double-pomeron trajectory, obtaining a general description of the total cross section obeying the FM bound, and the correct prediction for the ρ(s)-parameter at high energies. Hereafter, ρ(s) = ρ.
The paper is organized as follows. Section II presents the experimental data set and the fitting procedure. In the section III, one presents the main results of [28] and develops the subtracted case as well as the Regge cut case. Section IV presents a brief discussion about the ρ-parameter as well as predictions for the slope of the differential cross section. The critical remarks are the subject of the last section V.

II. EXPERIMENTAL DATA AND FITTING PROCEDURES
The main quantity in the forward elastic scattering is the total cross section, connected through the optical theorem with the imaginary part of the forward elastic scattering amplitude. Bearing this in mind, one considers here the experimental data for the proton-proton (pp) and proton-antiproton (pp) total cross sections, σ pp tot (s) and σ pp tot (s). As usual, t is the squared momentum transfer and s is the squared energy, both in the center-of-mass system.
These pp and pp experimental data are used to form a joint data set since the Pomeranchuk theorem asserts they tend to the same limit if s → ∞. This behavior, predicted to occur only at the asymptotic regime, seems yet to be started from energies above √ s ≥ 25.0 GeV. Figure 1 shows the experimental data used in the fitting procedures for σ pp tot (s) and for σ pp tot (s). The goal of treating both data set as only one resides on the possibility that the absence of pp experimental data, on some energy range, can be compensated by the existence of pp data in this range, and vice-versa. Moreover, there is no data selection in any set considered. The following experimental data set are used throughout this paper.
• The SET 1 is formed by the experimental data for σ pp tot (s) and σ pp tot (s) above √ s = 1.0 TeV up to the cosmic-ray data.
• The SET 2 uses the experimental data for σ pp tot (s) and σ pp tot (s) above √ s = 1.0 TeV, excluding the cosmic-ray experimental data.
• The SET 3 contains experimental data for σ pp tot (s) and σ pp tot (s) above √ s c GeV up to the cosmic-ray data.
• The SET 4 excludes the cosmic-ray data from the SET 3.
The energy cut √ s c , corresponds to the energy for which the total cross section achieves its minimum value [29].
For the SET 1 and 2, one uses √ s c = 25.0 GeV. However, as shall be seen, the emergence of a term ln ln(s/s 0 ) in the Using the derivative dispersion relations, one obtains the real part of the forward elastic scattering amplitude. Then, it is possible to present the predictions for the ρ-parameter based on the fitting results. One uses only the experimental data for ρ above 25.0 GeV. It is also shown predictions for the slope of the differential cross section.
All the experimental data were collected from Particle Data Group [30]. Moreover, σ pp tot (s) at √ s = 2.76 TeV is from [31]. Hereafter, one uses only σ tot (s) to refer to both σ pp tot (s) and σ pp tot (s) (the same for ρ).

III. THE LOGARITHMIC LEADING REGGE POLE
In the Regge theory, the scattering amplitude is written as an analytic function of the angular momentum J. This representation is formulated in the high energy limit s → ∞, and associates the asymptotic behavior of the scattering amplitude in the s-channel to the exchange of one-particle or more, represented in the t-channel by the leading Regge poles. One writes the scattering amplitude as and one also assumes that behavior of such a function, at very high energies, is given only by the absorptive part A(s, t) ≈ ImA(s, t). As well-known, in the usual Regge pole formalism, one writes the asymptotic scattering amplitude as where η = ±1 is the signature related with the crossing symmetry s ↔ u, √ s c some critical energy, and β(t) is the residue function of the pole depending only on t. Using (2), one can attain the asymptotic form of the differential cross section and adopting the normalization sσ tot (s) = ImA(s, t) ≈ A(s, t), one has The mathematical disagreement between (4) and the FM bound given below where c is some real constant, is inevitable for 1 < α(0). However, the Regge theory and the FM bound can agree with each other if it is imposed a constraint on the scattering angle as well as a mathematical approximation on the cosine series [28]. The scattering angle is restricted to the range 0 ≤ cos θ ≤ 1 for |t| < < s. In this case, one writes the following approximation for the cosine of the scattering angle [28] cos(θ) = 1 + 2t Using the asymptotic properties of the Legendre polynomial and taking into account the approximation (6), one writes Indeed, the above result can be used to write the asymptotic scattering amplitude as a logarithmic leading Regge pole respecting the FM bound. The very successful model of Donnachie and Landshoff [3], early proposed by Badatya and Patnaik [27], describes the hadronic exchanges remarkably well assuming a simple Regge pole where C(t) is a constant depending only on t. The corresponding σ tot (s) of such parameterization is written using the optical theorem as saturating the FM bound. This Regge pole corresponds to one-pole exchange, while the double-pole exchange leads to σ tot (s) ∼ ln(s/s 0 ). The intercept of the Regge pole in this model is defined as a linear function where the intercept takes the value α(0) = α P = 1.08 and the slope α = 0.25 GeV −2 .
In the language of physics, this intercept corresponds to a soft pomeron -low momentum transfer -α P ≈ 1.05 ∼ 1.08. In contrast, the hard pomeron, predicted to mediate diffractive processes -large momentum transfer -has a value α P ≈ 1.4 ∼ 1.5. This hard pomeron emerges due to the use of the Regge formalism as an analogy to explain the structure-function F 2 (x, Q 2 ) in terms of the Bjorken scale x and the photon virtuality Q 2 .
Note that both, the one-pomeron exchange in the Donnachie and Landshoff model or the double-pomeron exchange in the logarithmic Regge pole, leads to intercepts 1 [3,28]. However, the one-pomeron exchange with an intercept slightly above 1 violates the FM bound while for the double-pomeron exchange, this violation only occurs for an intercept above 2. If the intercept is equal 2, then the triple-pomeron exchange is favorable, σ tot ≈ ln(s/s c ) 2 . Neither the soft nor the hard pomeron has been discovered yet.

A. Non-subtraction case
In [28], one considers only the non-subtraction case. Then, using the optical theorem one has a simple relation for the asymptotic total cross section In the specified range for cos(θ), it respects the FM bound if α P ≤ 2, providing a physical relation between the pomeron intercept, α P , and the saturation of this bound. The soft pomeron, if it exists, is the particle allowing the maximum growth to the total cross section, obeying the FM bound. As shall be seen, the phenomenology associated with the ρ-parameter is crucial to get the correct pomeron intercept.
Using a simple parameterization for the total cross section where β and α P are free fitting parameters, one can attain the pomeron intercept. Using the SET 1, one obtains the values for the fitting parameters shown in the first line of the Table I. Figure 2a shows the curve obtained from the fitting procedure using (13). The intercept agrees with a double-pole pomeron exchange. The fitting results using the SET 2 are shown in the second line of the Table I. From the statistical point of view, the absence of the cosmic-ray data in the SET 2 practically does not alter the results. The total cross section with one subtraction can be written using the following normalization of the optical theorem sσ tot (s) = ImA(s).
This normalization implies in the logarithmic leading Regge pole First of all, one notices that to tame the fast decreasing of the total cross section entailed by the subtraction, it is necessary a α P far above from the expected saturation of the FM bound, α P → 2. Indeed, to give some physical meaning for α P is now a hard task. The effect of using s in the above result can be observed by adopting the parameterization given below The fitting results obtained by using the parameterization (16) to the SET 1 and 2 are shown in the Table II. Of course, the subtracted case cannot be used as a realistic parameterization to describe the σ tot (s). Figure 2b shows the curve obtained from the fitting procedure using (16).  (16) in the fitting procedure to the SET 1 and 2 and taking √ sc = 25.0 GeV.
However, if we release the subtraction mechanism by introducing the δ-index as a measurement of the deviation of the non-subtraction to the subtraction case, then one writes the parameterization Using the SET 1 and 2, one obtains very small values for the δ-index (near zero). In Table III are displayed the fitting parameters. The δ-index introduces an error in the fitting parameters higher than the non-subtraction case. However, the central value of each parameter is practically the same. The fitting result is shown in Figure 3a.
To circumvent the need for the use of the δ-index, one introduces the approximation [28] 1 (s/s c ) δ ≈ 1 a(ln(s/s c )) , FIG. 2: In both panels, the solid line is for the SET 1 and the dashed line is for SET 2. In panel (a) one uses the parameterization (13). In panel (b), the parameterization used is given by (16). Experimental data are from [30,31]. which is a consequence of the fact that the inequality given below holds for 0 < s c ≤ s and 0 ≤ ≤ δ ∈ R In particular, for some 0 < a ∈ R, the approximation (18) can be used, resulting in the total cross section where α P = α P − and β = c/a. The choice of a is not unique, and in general, its value depends on the energy range where the experimental data are being analyzed. However, in the fitting procedure, it is absorbed by β . Therefore, , the parameterization used is given by (31). Experimental data are from [30,31].
the expression (20) corresponds to the subtraction case and it is exactly equal to the non-subtraction one given by (13).

C. The cut case
As well-known, the singularities play a fundamental role in the determination of the divergences of the partial-wave amplitude. Neglecting the signature of the partial-wave amplitude, one may write the singularities as where Q l (z ) is the Legendre function of the second kind and discA(s, t) is the discontinuity across the l-plane cut.
The easy way to obtain singularities from (21) is to associate the discA(s, t) with the Legendre function of first kind, P αc(t) (z), where α c (t) is the cut. Then, in the simplest case, discA(s, t) ∝ P αc(t) (z).
In this picture, one uses the properties of the Legendre functions [32] revealing a pole for l = α c (t) (l ∈ R). The Watson-Sommerfeld representation for the partial-wave expansion in the complex angular plane can be written as where R p (s, t) and R c (s, t) stands for the Regge pole and the Regge cut contributions for the scattering amplitude. The logarithmic Regge pole introduced in [28] gives for the first term in the r.h.s. of (24) If a branch point with a cut is encountered during the deformation of the contour in the complex momentum angular plane, then it contributes to the asymptotic behavior of the scattering amplitude. Then, one should take into account the cut contribution. One writes As stated above, the functional form of the discontinuity is not known a priori, and there is no phenomenological approach for it. Therefore, the only way to go through this point is by using an ansatz as performed in Ref. [33].
Of course, the above result is strongly dependent on the form of discA(l, t) and there is no information about β(t). The Regge cut α c (t) is also an unknown function of t. One suppose here, that α c (t) and β(t) are a real-valued functions as well as they assume finite values at t = 0.
It is important to stress that σ tot (s), independently of the functional form of discA(l, t), should obey the FM bound. Then, the rise of (28) is controlled by ln(s/s 0 ) 2 . It is not difficult to see that (hereafter, α c (0) = α c ) 0 ≤ 1 − ln(s/s c ) αc−α P ln ln(s/s c ) −(2+β(0)) (29) implies that in the asymptotic regime s → ∞ for β(0) > −2. The inequality α c ≤ α P implies the Regge cut is bounded by the Regge pole. Therefore, the Regge cut (28) may be used to explain the experimental data behavior from the minimum value of the total cross section up to 1.0∼2.0 TeV. To take into account the Regge pole and the Regge cut contributions for the total cross section, one uses the simple parameterization σ tot (s) = β 1 ln(s/s c ) α P − β 2 ln(s/s c ) αc ln ln(s/s c ) −(2+β(0)) , to fit the SET 3 and 4. The fitting results are displayed in Table II. The Figure 3b shows the curve obtained for the parameterization (31). The role of the cut is now clear: it represents the mixed contributions below the logarithmic Regge pole dominance, above 1.0 TeV. It is important to stress that there is no constraint on the fitting parameters. Then, one allows the pomeron intercept to assume any value necessary to fit the experimental data. The result, in a first glance, shows a pomeron intercept far above from the saturation of the FM bound -a supercritical value. Then, a control mechanism should be imposed to tame the growth of the total cross section as s → ∞. This control mechanism, as shall be seen, is obtained by looking to the ρ-parameter experimental data.

IV. THE ρ-PARAMETER AND SLOPE B(S,T)
As well-known, the validity of the Cauchy theorem is crucial for the convergence of the integral dispersion relation. This kind of relationship allows the obtaining of the real part of the forward scattering amplitude from the imaginary part. First of all, one writes the scattering amplitude as the sum of the even (+) and odd (-) amplitudes as where A ± (s, t) = ReA ± (s, t) + iImA ± (s, t) are the crossing even (+) and odd (-) amplitudes. The integral form of the dispersion relations is a consequence of analyticity, unitarity, and crossing properties. In the non-subtraction case, it is simply written as (t = 0) where P is the principal value of Cauchy integral. The convergence of the above integral can be ensured by using the subtraction procedure, i.e. by rewritten the scattering amplitude as which result in the subtraction term where K is the subtraction constant. Of course, this method is valid only for a finite number of subtractions [34]. However, the integral dispersion relations are very restrictive, since to know the value of the real part to a specific value one should know the value of the imaginary part in the whole plane. Then, despite its rigorous formulation, the use of integral dispersion relations is of little interest in the Regge theory. On the other hand, the derivative dispersion relation can be used in the present case [35,36]. These derivative relations can be written in the first-order approximation for the odd and even amplitudes as [37] ReA + (s, t) Considering (13) and (31), it is possible to obtain real part of the scattering amplitude in the subtraction as well as in the Regge cut. Then, the real part of the scattering amplitude can be used to define the ρ-parameter ρ(s) = ReA(s) ImA(s) . Without loss of generality, one set K = 0, since the influence of this parameter is restricted to the low energy region, and the main interest here is to understand the asymptotic regime. Then, all parameters were previously obtained from the fits to the total cross section.
In Figure 4 one has the predictions to the ρ-parameter. The results from the logarithmic Regge pole (13) are represented by the dotted line and by the dot-dashed line, SET 1 and 2, respectively. These curves represent the pure contribution coming from the double pomeron exchange. Therefore, these results are not able to reproduce the low energy behavior of the ρ-parameter. Using the parameterization (31), the solid and dashed lines, respectively, display the Regge cut contribution obtained from the fitting procedures for the SET 3 and 4. These predictions show behavior in the high energy regime not present in the experimental data. There is a fast-rising introduced by the experimental data below √ s = 1.0 TeV, which seems not present in the experimental data above this energy since the fittings for the SET 1 and 2 shown a pomeron intercept α P ≈ 1.05.
An attempt to solve this problem can be done by using the recent experimental data for the ρ-parameter at √ s = 13.0 TeV. These experimental data suggest a double pomeron intercept taming the rise of the total cross section. Then, taking into account the above discussion, one introduces the constraint α P ≤ 1 to reduce the fast-rising of σ tot (s) above 1.0 TeV, introduced by (31). The Table V shows the fitting parameters only for the SET 3. Despite the high value to χ 2 /ndf , the fitting parameters seems to be able to reproduce the growth of σ tot (s) as s → ∞.  The Figure 5a shows the fitting results for σ tot (s) using the parameterization (31) under the double-pomeron constraint. In Figure 5b, one shows the prediction for the ρ-parameter. Despite this naive parameterization, one can observe that the only way to reproduce the high energy behavior of ρ at √ s = 13.0 TeV is assuming a double pomeron exchange in the logarithmic Regge pole representation.
A possible understanding here is: the fast rise of σ tot (s) in the logarithmic Regge approach cannot be analyzed independently of the ρ-parameter. In particular, the ρ value at √ s = 13.0 TeV is crucial to determine an upper bound to the pomeron intercept. It should be stressed that it is a feature of the Regge theory: the phenomenology.
Another physical quantity that can be predicted from the fitting procedures performed here is the (forward) slope of the elastic differential cross section dσ/dt, defined as In the present approach, using the simple asymptotic parameterization (13), for example, one obtains for the forward case Using the TOTEM result B 0 = 19.9 ± 0.3 GeV −2 at √ s 0 = 7.0 TeV [38] and α = 0.25 GeV −2 , one has the prediction for the elastic slope B(s) = 20.0 ± 0.3 GeV −2 at √ s = 13.0 TeV. As well-known, the LHC result is B(s) = 20.36 ± 0.19 GeV −2 [31], in accordance with the value predicted here. On the other hand, starting from the TOTEM result, one cannot reproduce, for example, the unexpected value for the slope encountered by E710 Collaboration, B(s) = 16.98 ± 0.25 GeV −2 [39].
It is interesting to note that it is a very hard task to obtain the slope B(s) since it is not extracted at t = 0. In general, the slope is obtained considering very small ranges of t ≈ 0, which implies Coulomb and Nuclear contributions to the scattering amplitude. For example, for the E710 Collaboration [39], the interval for t is [0.04, 0.29] GeV 2 and for the TOTEM Collaboration one has t ∈ [0.012, 0.2] GeV 2 [31]. Then, these experimental values may carry some dependence on the momentum transfer.

V. CONCLUSIONS
One obtains here the leading Regge pole with one subtraction as well as the Regge cut, both in the logarithmic representation introduced previously in [28]. The fitting procedures for the subtraction case led to unrealistic results to σ tot (s), i.e. to a decreasing faster than the rise of the experimental data, turning the subtraction a problem in the logarithmic Regge pole.
Trying to understand this problem, one introduces a small parameter, the δ-index, to measure how strong should be the subtraction in the approach performed here. The fitting procedures indicate δ-index value near to zero (a very weak dependence on the subtraction). This result allowed the use of a logarithmic representation for the subtraction. Then, the subtraction and the non-subtraction cases can be described by the same logarithmic Regge pole.
The fitting procedures considering only energies above 1.0 TeV result in a pomeron intercept compatible with the double-pomeron value, α P ≈ 1.04 ∼ 1.05. This value corroborates the Double-Logarithmic contributions, where the higher accuracy of calculations, the lower is the intercept, resulting in the best value is given by the intercept close to 1 [40,41].
The Regge cut in the original Regge formalism does not possess a clear role. However, in the present approach, the logarithmic Regge cut may represent the contributions coming from below the logarithmic Regge pole, α P , when one adopts discA(l, t) = (α c (t) − l) 1+β(t) . Then, it can be used to explain the mixed region (25.0 GeV ≤ √ s ≤ 1.0 TeV), where the odderon and the pomeron compete as the leading contribution to the total cross section.
Then, one expects here that logarithmic Regge cut can describe the total cross section for pp and pp, from the minimum of the total cross section up to 1.0 TeV. However, assuming the parameters coming from the Regge cut can act as free fitting parameters, then they cause a supercritical value to the pomeron intercept, leading to the saturation of the FM bound. Notwithstanding, the dominance of the pomeron as the leading contribution at LHC energy seems to be an experimental fact [42]. It is important to stress that the fitting procedures for the SET 1 and 2, using the parameterization (13), furnish an important constraint on the pomeron intercept α P . Moreover, the experimental data at the cosmic-ray energies seem to have a little influence on the pomeron intercept.
To solve this problem, it is necessary to use the experimental data for the ρ-parameter. In particular, one should use the experimental result at √ s = 13.0 TeV. This value seems to impose a double pomeron exchange, resulting in an intercept α P ≤ 1. When this experimental fact is used, then σ tot (s) rises below the saturation of FM bound. Of course, as stated in [17], the values for the ρ-parameter at √ s = 13.0 TeV excluded all the models classified and published by COMPETE. Therefore, the slowing down of the σ tot (s) seems to be given by the ρ-parameter at √ s = 13.0 TeV. The slope of the differential cross section can also be predicted by the logarithmic Regge pole. Using the slope obtained by the TOTEM Collaboration, one predicts B( In the logarithmic Regge pole approach presented here, the Regge cut seems to has a clear role: it is responsible by the mixed region (25.0 GeV √ s 1.0 TeV) where the total cross section can be described by the odderon and the pomeron contributions. However, this result is strongly dependent on the discontinuities of the scattering amplitude. Unfortunately, there is no theoretical nor phenomenological information about discontinuity of A(l, t).