LHC13 forward elastic scattering: Dynamical gluon mass and semihard interactions

In the context of a QCD-based model with even-under-crossing amplitude dominance at high-energies, it is shown that the $pp$ and $\bar{p}p$ elastic scattering data on $\sigma_{tot}$ and $\rho$ above 10 GeV are quite well described, especially the recent TOTEM data at 13 TeV. Specifically, we investigate the role of low-$x$ parton dynamics in dictating the high-energy behavior of forward scattering observables at LHC energies, by using a nonpertubative cutoff linked to the dynamical generation of a gluon mass. Unexpected features of the data, such as the rather small $\rho$ value at 13 TeV recently reported by the TOTEM Collaboration, are addressed using an eikonalized elastic amplitude, where unitarity and analyticity properties are readily build in. The model provides an accurate global description of $\sigma_{tot}$ and $\rho$ with pre- and post-LHC fine-tuned parton distributions, CTEQ6L and CT14, even if data at 8 and 13 TeV are not included in the dataset analyzed. These findings suggest that the low-$x$ parton dynamics, as well as the nonperturbative dynamics of QCD, play a major role in the driving mechanism behind the pre-asymptotic $\rho$ decrease at LHC energies.


I. INTRODUCTION
The elastic hadronic scattering at high energies represents a rather simple kinematic process. However, its complete dynamical description is still a fundamental problem in QCD, since the confinement phenomena precludes a pure perturbative approach. Over the past few years, the LHC has released precise measurements of elastic proton-proton scattering which has become an important guide for selecting models and theoretical approaches, looking for a better understanding of the theory of strong interactions.
Among other physical observables, two forward quantities play a fundamental role in the investigation of the elastic scattering at high energies, the total cross section and the ρ parameter, which can be expressed in terms of the scattering amplitude A(s, t) by σ tot (s) = 4πIm A(s, t = 0), where s and t are the Mandelstam variables and t = 0 indicates the forward direction. Recently, the TOTEM Collaboration has provided new experimental measurements on σ tot and ρ from LHC13, the highest energy reached in accelerators. In a first paper [1], by using as input ρ = 0.10, the measurement of the total cross section yielded σ tot = 110.6 ± 3.4 mb.
In a subsequent work [2], an independent measurement of the total cross section was reported, together with the first measurements of the ρ parameter: ρ = 0.10 ± 0.01 and ρ = 0.09 ± 0.01.
Although the values of σ tot are in consensus with the increase of previous measurements by TOTEM, the ρ values indicate a rather unexpected decrease, as compared with measurements at lower energies and predictions from the wide majority of phenomenological models. This new information has originated a series of recent papers and discussions on possible phenomenological explanations for the rather small ρ-value. The main concern in these theoretical discussions is the full understanding of the Odderon concept (a crossing odd color-singlet with at least three gluons) [3][4][5] and of the Pomeron one (a crossing even color-singlet with at least two gluons) [6,7].
In this rather intricate scenario, we present here a phenomenological description of the forward pp andpp elastic scattering data in the region 10 GeV -13 TeV. In our model the behavior of the forward quantities σ tot (s) and ρ(s), given by Eqs. (1) and (2), are expected to be asymptotically dominated by the so-called semihard interactions. This type of process originates from hard scattering of partons which carry a very small fraction of the momenta of their parent hadrons, leading to the appearance of minijets [31,32]. The latter can be viewed simply as jets with transverse energy much smaller than the total center-of-mass energy available in the hadronic collision. The energy dependence of the cross sections is driven mainly by semihard elementary processes that include at least one gluon in the initial state, since at low x they are responsible for the dominant contribution.
In our QCD-based formalism these partonic processes are written by means of the standard QCD cross sections convoluted with updated sets of partonic distribution functions. However, these processes are potentially divergent at low transferred momenta, and for this reason they must be regularized by means of some cutoff procedure. In a nonperturbative QCD context, one natural regulator was introduced by Cornwall some time ago [33], and since then has become an important feature in eikonalized models [34][35][36][37][38]. This regularization process is based on the increasing evidence that the gluon may develop a momentum-dependent mass, which introduces a natural scale able to separate the perturbative from the nonperturbative QCD region.
Thus, taking into account the possibility that the infrared properties of QCD can, in principle, generate an effective gluon mass, we explore the nonperturbative aspects of QCD in order to describe the total cross section and the ratio of the real-to-imaginary parts of the forward elastic scattering amplitude in pp andpp collisions. Most importantly, two components are considered in our eikonal representation, one associated with the semihard interactions and calculated from QCD and a second one associated with soft contributions and based on the Regge-Gribov phenomenology. Except for an odd under crossing Reggeon contribution, necessary to distinguish between pp andpp scattering at low energies, all the dominant components at high energies (soft and semihard) are associated with even under crossing contributions, namely we have Pomeron dominance and absence of Odderon.
The work is organized as follows. In Sect. II a short review on the concept of the dynamical gluon mass is presented. In Sect. III we introduce all the inputs and details concerning our QCD-based model and in Sect. IV we specify the data set and the fit procedures. In Sect. V the fit results are presented, followed by a discussion on the corresponding physical interpretations and implications. Our conclusions and final remarks are the contents of Sect. VI. The paper is complemented by four appendixes, where it is presented: details on the analytical parametrization for the partonic cross section (A), tests related to the effect of the leading soft contribution (B), energy-independent semihard form factor (C) and changes in the dataset (D).

II. THE DYNAMICAL GLUON MASS
As pointed out in the previous section, scattering amplitudes of partons in QCD contain infrared divergences. One procedure to regulate this behavior is by means of a dynamical mass generation mechanism which is based on the fact that the nonperturbative dynamics of QCD may generate an effective momentum-dependent mass M g (Q 2 ) for the gluons, while preserving the local SU (3) c invariance [39][40][41]. The dynamical mass M g (Q 2 ) introduces a natural nonperturbative scale and is linked to a finite infrared QCD effective chargeᾱ s (Q 2 ). The existence of a dynamical gluon mass is strongly supported by QCD lattice results. More specifically, lattice simulations reveal that the gluon propagator is finite in the infrared region [42][43][44][45][46][47][48][49] and this result corresponds, from the Schwinger-Dyson formalism, to a massive gluon [33,[50][51][52][53][54]. It is worth mentioning that infrared-finite QCD couplings are quite usual in the literature (for a recent review, see [55]). In addition to the evidence already mentioned in the lattice QCD, a finite infrared behavior of α s (Q 2 ) has been suggested, for example, in studies using QCD functional methods [56][57][58], and in studies of the Gribov-Zwanziger scenario [59][60][61].
Since the gluon mass generation is a purely dynamical effect, a formal continuum approach for tackling this nonperturbative phenomenon is provided by the aforementioned Schwinger-Dyson equations that govern the dynamics of all QCD Green's functions [33, 50-54, 62, 63]. These equations constitute an infinite set of coupled nonlinear integral equations and, after a proper truncation procedure, it is possible to obtain as a solution an infrared finite gluon propagator, while preserving the gauge invariance (or the BRST symmetry) in question. In this work we adopt the functional forms of M g andᾱ s obtained by Cornwall [33] via the pinch technique in order to derive a gauge invariant Schwinger-Dyson equation for the gluon propagator and the triple gluon vertex: where Λ is the QCD scale parameter, β 0 = 11−2n f /3 (n f is the number of flavors) and m g is the gluon mass scale to be phenomenologically adjusted in order to yield well founded results in strongly interacting processes. Note that the dynamical mass M 2 g (Q 2 ) vanishes in the limit Q 2 Λ 2 . It is thus evident that in this same limit the effective chargeᾱ s (Q 2 ) matches with the one-loop perturbative coupling: In the limit Q 2 → 0, in turn, the effective chargeᾱ s (Q 2 ) have an infrared fixed point, i.e. the dynamical mass tames the Landau pole. More precisely, if the relation m g /Λ > 1/2 is satisfied thenᾱ s (Q 2 ) is holomorphic (analytic) on the range 0 ≤ Q 2 ≤ Λ 2 [37]. In fact, this is the case, since the values of the ratio m g /Λ obtained phenomenologically typically lies in the interval

A. Eikonal Representation
The correct calculation of high-energy hadronic interactions must be compatible with analyticity and unitarity constraints, where the latter is satisfied simply by means of eikonalized amplitudes. We adopt the following normalization for the elastic scattering amplitude: where s is the square of the total center-of-mass energy, b is the impact parameter, q 2 t = −t is the usual Mandelstam invariant, with the complex eikonal function denoted by In this picture Γ(s, b) = 1 − e −χ(s,b) is the profile function, which, by the shadowing property, describes the absorption effects resulting from the opening of inelastic channels. In addition, in the impact parameter space and according the unitarity condition of the scattering S-matrix it may be also written as Therefore, the scattering process cannot be uniquely inelastic since the elastic amplitude receives contributions from both elastic and inelastic channels. In this representation P (s, b) = e −2χ R (s,b) can be defined as the probability that neither hadron is broken up in a collision at a given b and s. Such an absorption factor is crucial to determine rapidity gap survival probabilities in pp andpp scattering at high-energies, which in turn are crucial to disentangle inelastic diffractive (single and double) and central exclusive processes from the dominant minimumbias (non-diffractive) cross section [74,75].
Within the eikonal representation, Eq. (9), the total cross section and the ρ parameter in Eqs. (1) and (2) are given by: The eikonals for elastic pp andpp scattering are connected with crossing even (+) and odd (−) eikonals by Real and imaginary parts of the eikonals can be connected either by Derivative Dispersion Relations (DDR) [76][77][78][79][80][81] or Asymptotic Uniqueness (AU), which is based on the Phragmén-Lindelöff theorems [82,83] (see [84], appendixes B,C,D for a recent short review on these subjects). We have tested both methods and in what follows we present the results with the AU approach, also referred to as asymptotic prescriptions or real analytic amplitudes [83].

B. Semihard and Soft Contributions
The eikonal function is assumed to be the sum of the soft and the semihard (SH) parton interactions in the hadronic collision [85,86], with each one related, in the general case, to the corresponding crossing even and odd contributions: In what follows we specify the inputs for each one of the four aforementioned contributions to the eikonal.

Semihard Contributions and the Dynamical Gluon Mass
The fundamental basis of models inspired upon QCD, or also known as minijet models, is that the semihard scatterings of partons in hadrons are responsible for the observed increase of the total cross section. Here we assume a Pomeron dominance, represented by a crossing even contribution, namely we consider that the semihard odd component does not contribute with the scattering process, In respect to the even contribution, it follows from the QCD improved parton model. At leading order, this semihard eikonal can be factorized as where W SH (s, b) is the overlap density distribution of semihard parton scattering, σ QCD denotes the cross section of hard parton scattering in the region where pQCD can be safely applied, namely above the cutoff Q 2 min . We assume (as in previous studies [37]) that hard parton scattering configuration in the transverse plane of the collision (in b-space) to be given by the Fourier-Bessel transform: where with ν SH = ν SH (s) taken as an energy dependent scale of the dipole. Specifically, we assume a logarithmic dependence for ν SH , namely: where ν 1 and ν 2 are two free fit parameters and the scale √ s 0 = 5 GeV is fixed. Regarding this dependence of the form factor on the energy, though not being formally established in the context of QCD, it is truly supported by the wealth of accelerator data available (as we shall see in Section IV) and seems to us more realistic than taking a static partonic configuration in b-space. In addition, many other phenomenological models have been proposed in literature (see e.g. [87][88][89][90][91][92][93]), in which the energy dependence in form factors play a crucial role in pp andpp elastic scattering dynamics and, therefore, in accurate descriptions of the data beyond √ s ∼10 GeV. The dynamical contribution, σ QCD (s), is calculated using perturbative QCD as follows: where x 1 and x 2 are momentum fraction carried by partons in the hadrons A and B, respectively,ŝ = x 1 x 2 s, |t| ≡ Q 2 stands for Mandelstam invariants of partonparton scatterings such as e.g. gg → gg, qg → qg and gg →qq (whose partonic cross sections are given afterwards) and f i/A (x 1 , |t|), f j/B (x 2 , |t|) are the parton distribution functions (PDFs) for partons i and j.
The indexes i, j = q,q, g identify quark (anti-quark) and gluon degrees of freedom and Q 2 min represent the minimum momentum transfer scale allowing for pQCD calculations of partonic hard scattering, obeying the constraint 2Q 2 min < 2|t| <ŝ. Concerning the differential cross section at elementary level, the major contribution at high energies are the ones initiated by gluons 1 i. gluon-gluon elastic scattering, ii. quark-gluon elastic scattering, iii. gluon fusion into a quark pair, with kinematical constraints imposed and connected with the dynamical mass, namely: (i)ŝ +t +û = 4M 2 g (Q 2 ), for gluon elastic scattering (gg → gg) and (ii)ŝ +t +û = 2M 2 g (Q 2 ) + 2M 2 q (Q 2 ) for gluon fusion (gg →qq) and quark-gluon scattering qg → qg. Importantly, in what follows we assume the Cornwall's dynamical gluon mass (in Euclidean space) [33], Eq. (6), with the infrared frozen effective QCD charge, Eq. (7), to interpolate two QCD domains: (i) Q 2 ≈ 0, i.e. at infrared, where M 2 g freezes and the gluons carries an effective bare mass, , dynamical mass generation from nontrivial vacuum structure becomes unimportant and perturbative QCD limit is achieved.
As discussed in Section II, recent phenomenology and lattice studies support bare gluon masses in the range, m g : 300 − 700 MeV. Here we fix m g = 400 MeV while also accounting, for completeness, the subdominant role of dynamical quark generation at high energies. We assume, for simplicity which also recovers the bare mass m q (with m q < m g ) at infrared and reaches the massless quark limit for Q 2 m 2 q . In all calculations we take m q = 250 MeV as fixed scale. At last, as commented before, the complex eikonal χ + SH (s, b) is determined through the asymptotic even prescription s → −is. The details on this dependence and the evaluation of the real and imaginary parts of σ QCD (s) are presented and discussed in Appendix A.

Soft Contributions
The full even and odd soft contributions are based on the Regge-Gribov formalism and are constructed in accordance with Asymptotic Uniqueness (Phragmén-Lindelöff theorems). Assuming also leading even component, they are parametrized by where denote analytical even and odd cross sections and A, B, C and D are free fit parameters. Moreover, the impact parameter structure derives from bidimensional Fourier transform of dipole form factors, namely: GeV is a fixed parameter and µ + sof t a free fit parameter. As in the case of the SH form factor, the energy scale is fixed at We notice that in the Regge-Gribov context, the soft even contribution consists of a Regge pole with intercept 1/2, a critical Pomeron and a triple-pole Pomeron, both with intercept 1. The odd contribution is associated with only a Regge pole, with intercept 1/2.

IV. DATASET AND FIT PROCEDURES
In the absence of ab initio theoretical QCD arguments to determine the parameters A, B, C, D, µ + sof t , ν 1 and ν 2 , we resort to a fine-tuning fit procedure described in what follows. As we are interested in the very high-energy behavior of σ tot and ρ, we shall use only pp andpp elastic scattering data. Moreover, in order to test our QCDbased model in the t = 0 limit, we perform global fits that include exclusively forward data, as described in Section III.

A. Dataset
Our dataset is compiled from a wealth of collider data on pp andpp elastic scattering, available in the Particle Data Group (PDG) database [94] as well as in the very recent papers of LHC Collaborations such as TOTEM [1,2,95,96] and ATLAS [97,98], which span a large c.m. energy range, namely 10 GeV √ s 13 TeV. For the sake of clarity and completeness we furnish in Table  I all the recent LHC data on σ tot and ρ, still absent in the PDG2018 review.
We call attention to the fact that we do not apply to this dataset, composed of 174 data points on σp p,pp tot and ρp p,pp , any sort of selection or sieving procedure, which might introduce bias in the analysis. TABLE I. Total cross section, σtot, and ρ-parameter data recently measured by TOTEM and ATLAS Collaborations at the LHC, but not compiled in the PDG2018 review [94]. This dataset totalizes 13 new data points on pp forward elastic scattering at high energies, most of which are currently published. For completeness, we provide all the appropriate references to the data we have used in our fits in the last column.

B. Fit Procedures
To provide statistical information on fit quality, we perform a best-fit analysis, furnishing as goodness of fit parameters the chi-squared per degrees of freedom (χ 2 /ζ) and the corresponding integrated probability, P (χ 2 , ζ) [102]. Since our model is highly nonlinear, numerical data reduction is called for. Despite the limitation of treating statistical and systematical uncertainties at the same foot, we apply the χ 2 /ζ tests to our dataset with uncertainties summed in quadrature 2 . Our fits are done using the TMINUIT class of the ROOT framework [105], through the MIGRAD algorithm. While the number of calls of the MIGRAD routine may vary in the fits with PDFs CETQ6L, CT14 and MMHT, full convergence of the algorithm was always achieved. Moreover, all data reductions were performed with the interval χ 2 − χ 2 min = 8.18, which corresponds to 68.3 % of Confidence Level (1σ) [106] in our case (7 free parameters).
Furthermore, in all fits performed we set the low energy cutoff, lowing we present our results, according to the choice of three distinct PDFs: CTEQ6L [107] (pre-LHC), CT14 [108] and MMHT [109] (fine-tuned with LHC data) and setting three different high-energy cutoffs, as previously discussed. In testing different PDFs we look for a better understanding of the impact of low-x parton dynamics in defining the very high-energy behavior of σp p,pp tot and ρp p,pp . For comparison, the behavior of the gluon distribution function in each PDF set in given in Figs. 1 and 2.

V. RESULTS AND DISCUSSION
The results for the free fit parameters, using each one of the three PDFs (CTEQ6L, CT14, MMHT) and for each high-energy cutoff in the dataset ( √ s max = 13 TeV, 8 TeV and 7 TeV), are displayed in Table II, together with the statistical information on the data reductions (reduced chi square and corresponding integrated probability). The curves of σ tot (s) and ρ(s) for the three PDFs, compared with the experimental data, are shown in Figures Fig. 3, the results are in plenty agreement with all the σ tot data, independently of the PDF employed. For ρ the results with CTEQ6L and CT14 also describe quite well the TOTEM data at 13 TeV (and data at lower energies), but that is not the case with MMHT. Indeed, from Table II, in this case the integrated probability is the smallest one among the three PDFs. Notice that the result with CT14 (finetuned with LHC data) gives exactly ρ = 0.1 at 13 TeV. Despite a barely underestimation of the ρ datum from pp at 546 GeV, we conclude that our QCD-based model with CTEQ6L and CT14 provides a consistent description of the forward data in the interval 10 GeV -13 TeV, mainly a simultaneous agreement with the σ tot and ρ  [107], CT14 [108] and MMHT [109] for highenergy cutoffs √ smax = 13 TeV, 8.0 TeV and 7.0 TeV. Quality fit estimators, chi-squared per degree of freedom, χ 2 /ζ, and integrated probability, P (χ 2 ; ζ), are also furnished (where ζ specifies the number of degrees of freedom (dof) in each fit).  Table II, the integrated probability with √ s max = 7 TeV is the highest one among the three cutoffs and the corresponding predictions at higher energies indicate the decreasing in ρ(s).
These results show the powerful predictive character of the results, since the σ tot and ρ data at 13 TeV are simultaneously described in all cases, even with √ s max = 7 TeV (for PDFs CT14 and CTEQ6L) and without Odderon contribution. In addition, looking for some insights into the formalism, it may be important to notice the effects of two phenomenological inputs, one related to the soft even eikonal and the other to the semihard form factor. In the first case, χ + sof t (s, b) as given by Eq. (26), has a component which increases with the energy, namely the term with coefficient C. In the second case, the dipole form factor G SH (s, k ⊥ ; ν SH ), Eqs. (19) and (20), also depends on the energy through the logarithmic. The effect of these terms can be investigated by assuming either C = 0 or ν 2 = 0 and re-fitting the dataset. These tests are presented and discussed in Appendixes B and C.
By showing the values in the Table II, we can see that the parameter µ + sof t has, in general, the value 0.90 GeV. This restriction is due to the fact that the inverse of both µ + sof t and µ − sof t parameters characterizes the range of these soft interactions. Since the odd soft eikonal χ − sof t (s, b) is more sensitive to the longer-range ρ and ω exchanges, it is expected the inverse of the odd exchanges, (µ − sof t ) −1 , to be larger than the inverse of the even (a 2 and f 2 ) exchanges, (µ + sof t ) −1 . Thus in our analysis we impose the reasonable condition 1 < µ + sof t /µ − sof t ≤ 1.8. Indeed, in all cases the parameter µ + sof t fall within the expected range.
Next we turn the focus to the physical intepretations of our results, mainly concerning high-energy QCD dynamics.
In QCD-based (s-channel) models like ours, the driving mechanism behind the rapid rise of the total cross section is linked to the growth with energy of low-p t jets (called minijets). This idea, while proposed many years ago, remains a powerful one in the scope of models of strong interactions at high-energies, as it provides a clear connection between perturbative QCD and hadronic elastic observables, such as σ tot and ρ, in a unitarized framework.
Those minijets arise from partonic interactions (mainly gluons) carring very small momentum fraction of their parent hadrons. On the one hand, from eq. (21), we see that the smallest x scale probed by this model is which, taking Q 2 min 1 GeV 2 , yields x min ∼ 10 −10 at LHC13. On the other, it is well-known that at very lowx the PDF's diverge, as gluon emissions -which naturally occur in any partonic process at high energiesare not suppressed by DGLAP evolution at higher momentum transferred. This behaviour can be readily seen from Figure 1 and 2 where the gluon distribution function from parton distributions CT14, CTEQ6L and MMHT are displayed at the minimum scale Q min = 1.3 GeV and two higher scales, Q = 10 GeV and 100 GeV. From these plots one may notice that MMHT grows faster than CT14 and CTEQ6L, specially at low momentum scales, such as Q min = 1.3 GeV.
As matter of fact, very low-x gluons are the key ingredient to understand our results for various PDF's, as shown in Figures 3, 4 and 5. Once the QCD cross section (21) is dominated by low-x partons, and gluon iniciated processes are the leading component of this cross section, one expects the magnitude of σ SH (s) calculated with MMHT to be larger than the corresponding curves for CT14 and CTEQ6L at high energies. As we show in Figure 6 in Appendix A, that turns out to be exactly the case.  Table  II.

VI. CONCLUSIONS
In this paper we have presented recent studies of pp and pp elastic scattering within an eikonal QCD-based model, which combines the perturbative parton-model approach to model the semihard interactions among partons, with a Regge-inspired model to describe the underlying soft interactions. We present a phenomenological analysis undertaken to improve the understanding of elastic processes taking place in the LHC. We address this issue by means of a model involving only even-under-crossing amplitudes at very high energies. As a result, we see that Best fit parameters and quality estimators are given in Table  II. the QCD-based model allows us to describe successfully the forward scattering quantities σ tot and ρ from √ s = 10 GeV to 13 TeV.
Nowadays, with the recent release of LHC13 data by the TOTEM Collaboration, it seems that we have achieved a true impasse: (i) in the Regge phenomenology context, LHC13 data is interpreted as clear evidence for the Odderon discovery, in the maximal strong scenario (namely the maximal Odderon) [8,9] ; (ii) however, in other t-channel approaches, based on eikonal rescatterings, such as [17,18], a small or vanishing Odderon contribution at 13 TeV is found to be compatible with the  Table  II. real-to-imaginary ratio, ρ = 0.10 ± 0.01, measured by TOTEM; (iii) in addition, violations of t-channel unitarity have also been addressed in [110] and seems to be unavoidable if QCD interactions manifests in the strongest form; (iv) other approaches, based on s-channel unitarity, such as ours, find the LHC13 data on forward observables to be compatible with a vanishing high-energy odd under crossing amplitude. According to this picture, a detailed scrutiny on the asymptotic nature of the C-parity of the scattering amplitude continues to be a core task in physics. Hence, we devote most of this paper to analyzing forward observables in hadron-hadron collisions, bringing up information about the infrared properties of QCD by considering the possibility that the nonperturbative dynamics of QCD generate an effective charge.
Our analysis, which follows a previous short letter [111], explores in detail the various effects that could be important in the global fits, in special three major points: (i) the use of three different PDFs (CT14,CTEQ6L and MMHT), investigating not only the difference and similarities among them, but also the effect of being pre or post LHC distributions; (ii) the study of their compatibility with the LHC13 data; (iii) the descriptions and predictions provided according to three high energy cutoffs, namely √ s max = 7, 8 and 13 TeV.
On general grounds, the present results demonstrate an overall agreement of all PDFs with σ tot at 13 TeV and, apart from MMHT, an excellent agreement with ρ at the same energy. From a rigorous statistical point of view, our results show that the TOTEM measurements can be simultaneously well described by a QCD scattering amplitude dominated by only single crossing-even elastic terms. At first glance, the behavior of the ρ parameter obtained by means of the MMHT set could be regarded as a consequence of its gluon steeply-rising component, as depicted in Figs. 1 and 2. We observe that its gluon distribution function increases rapidly and becomes higher than the CTEQL and CT14 gluon distributions. Note that this rapid variation, around the initial scale Q = 1.3 GeV, occurs in the kinematic region that contributes most to the integral (21). We argue that the success of our model in describing the unexpected ρ decrease at LHC13 may be attributed to the effect of introducing infrared properties of QCD, by considering that the nonperturbative dynamics of QCD generate an effective gluon mass. Specifically, the essential inputs of our model, namely the low-x behavior of parton distribution functions and the dynamical gluon mass scale, are found to be crucial in the phenomenological description of present available data at center-of-mass energies spanning from 10 GeV to 13 TeV. This mass scale is a natural regulator for the potentially divergent partonic processes and apparently also plays an important role in the unexpected decrease of the ρ parameter at high energy. The study of infrared properties of QCD is currently a subject of intense theoretical interest. Our expectation is to improve the understanding about the influence of the dynamical-mass generation mechanism on semihard processes.
Appendix A: Parametrization for σQCD(s) One of the most important ingredient of the QCDbased model is the even-under-crossing partonic crosssection σ QCD (s), given by Eq. (21). In this appendix we present the details of the evaluation of this quantity, using three distinct PDFs: CTEQ6L [43], CT14 [44] and MMHT [45]. Some additional results are presented and discussed. The evaluation is based on the steps that follow.
First we consider the complex analytic parametrization where b 1 , ..., b 10 are free fit parameters and provides the adequate complex and even character of the analytic function through the substitution s → −is, leading to Re σ QCD (s) and Im σ QCD (s).
Next, by means of Eq. (21) and using the three distinct PDFs, we generate around 30 points for each one of these parton distributions, which are then fitted by the Re σ QCD (s), with less than 1% error. With the values of the free fit parameters determined for each PDF, the corresponding Im σ QCD (s) are evaluated.
For CTEQ6L, CT14 and MMHT we display in Table  III the best-fit parameters b i , i = 1, · · · , 10 and in Fig. 6 the dependencies of Re σ QCD (s) and Im σ QCD (s).
From the figure, we see in all cases the steep rise of the partonic cross-sections with the energy. For example at √ s = 10 TeV, most results lie around 580 mb. Notice, however, that this rise is tamed in the physical crosssections, since we have an eikonalized model. We note that among the PDFs post-LHC, MMHT led to the fastest rise of both Re σ QCD (s) and Im σ QCD (s) and CT14 led to the slowest rise. The results with CTEC6L (pre-LHC) lie between these two cases.  Table III.
The extreme fast rise of σ QCD (s) in case of MMHT, may be the responsible for the overestimation of ρ at 13 TeV, a result which is independent of the high-energy cutoff (Figures 1, 2 and 3).
Appendix B: Effect of the leading contribution in χ + sof t (s, b) One of the ingredients of the QCD-based model is the soft-even component of the eikonal, Eqs. (26) and (27), which comprise a leading Pomeron contribution given by the quadratic term in Eq. (27), with coefficient C. In order to investigate the relevance of this leading soft contribution at high energies in our global results, we present here a test in which this term is excluded. Specifically, we fix C = 0 in Eq. (27) and refit the dataset. As illustration, we consider the high-energy cutoff at 13 TeV and the three PDFs employed in this work. The results of these fits are presented in Table IV and Fig. 7.
Let us compare the results in Fig. 3 (C free fit parameter) with those in Fig. 7 (C = 0 fixed), focusing the TOTEM data at 13 TeV (inserts) in the cases of PDFs CT14 and CTEQ6L. From Fig. 3, the results for σ tot cross the middle of the lower error bar and for ρ they cross the central value of the highest measurement. On the other hand, from Fig. 7 the results for σ tot barely reach the end of the lower error bar and for ρ they cross the middle of the upper error bar.
We conclude that, although not being the leading contribution at the highest energies, the triple pole Pomeron in the soft component is important for the correct description of σ tot and ρ at 13 TeV and for an adequate fit result in statistical grounds. Although not so usual in the present phenomenological context, one of the ingredients of the QCD-based model is the energy dependence embodied in the semihard form factor, Eqs. (18) and (19). As commented in our introduction, this assumption is associated with the possibility of a broadening of the spacial gluon distribution as the energy increases. In order to investigate the relevance of this assumption in our global results, we present here a test in which this energy dependence is excluded. Specifically, we fix ν 2 = 0 in Eq. (20), so that ν SH = ν 1 and refit the data set. As illustration, we consider the highenergy cutoff at 13 TeV and the three PDFs employed in this work. The results of these fits are presented in Table  V and Fig. 8.
Let us compare the results in Fig. 3 (ν 2 free fit parameter) with those in Fig. 8 (ν 2 = 0 fixed), focusing the TOTEM data at 13 TeV (inserts) in the cases of PDFs CT14 and CTEQ6L. For ρ(s), the results with ν 2 = 0 indicate a steeper decrease at high energies, present agree-  Table IV. ment with the pp ρ data and also with thepp data at 546 GeV. However, for σ tot (s) with both PDFs the results lie far below the lower error bars.
In respect the statistical quality of the fits, comparison of Tables II (ν 2 free fit parameter) and V (ν 2 = 0 fixed) shows that the exclusion of the energy dependence results in a rather unaccepted goodness of fit since χ 2 /ζ increase to 1.3 − 1.4 and P (χ 2 ) decrease to 10 −3 − 10 −4 , at least two order of magnitude smaller.
We conclude that the broadening of the spacial gluon distribution, as provided by Eqs. (18) and (19), is an important ingredient for the adequate description of both σ tot and ρ data at the LHC energy region. Here we develop two tests on the efficiency of the QCD-based model related to two different choices of the dataset. In the first test the low-energy cutoff is lowered from 10 GeV down to 5 GeV and in the second test the ATLAS data at 7 and 8 TeV are not included in the dataset. We present the results obtained with the three PDFs and as illustration, we consider only the high-energy cutoff at 13 TeV. Since the results are similar to those presented in the main text with our standard dataset, we focus the discussion on those obtained with the PDF CT14.
Although the integrated probability decreases two order of magnitudes for √ s min = 5.0 GeV, we see that the visual description of the data is quite good and the quality of the fit is reasonable for this data set (without any sieve procedure), showing that the model can cover efficiently the whole region 5 GeV -13 TeV.

D.2: Fits without the ATLAS data
It is well known the discrepancies between the TOTEM and ATLAS data on σ tot at 7 and 8 TeV [112]. Here we present two tests with low-energy cutoffs at 10 GeV and 5 GeV, in which the ATLAS data are not included in the data set. The results are presented in Table VII, Figure  10 ( √ s min = 10 GeV) and Table VIII, Figure 11 ( √ s min  Table V = 5 GeV).
Our results with the complete dataset (ATLAS data included) are shown in Table II, Fig. 1 for √ s min = 10 GeV ( √ s max = 13 TeV, PDF CT14) and Table VIII, Fig. 9 for √ s min = 5 GeV. By comparing the results we see that, without the ATLAS data, for both cutoffs the integrated probability increases as a consequence of the aforementioned discrepancies. In particular, it is interesting to note that with the exclusion of the ATLAS data, for √ s min = 10 GeV we obtain χ 2 /ζ = 1.071 for ζ = 165, resulting in the highest integrated probability: P (χ 2 ; ζ) = 0.25.