Self-consistent description of HERA data at low Q 2 and soft hadron production at LHC

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It is well known that the deep inelatic electron-proton scattering (DIS) processes at low Q 2 and small Bjorken variable x can provide important information about the nonperturbative parton (quark and gluon) structure of the proton.Detailed knowledge of the latter is necessary for any theoretical study of different high energy processes performed within the Quantum Chromodynamics (QCD).Moreover, it is necessary for future experiments planned at the Large Hadron electron Collider (LHeC) [1] and Future Circular hadron-electron Collider (FCC-he) [2], where facilities for using electron-proton center-ofmass energies √ s = 1.3 TeV and √ s = 3.5 TeV are proposed.In the region of relatively low Q 2 ≲ 4 GeV 2 and small x the conventional (collinear) QCD approach based on Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [3] has some difficulties to describe the DIS experimental data.For example, next-to-leading order (NLO) QCD corrections to the proton longitudinal structure function F L (x, Q 2 ) are large and negative, that could lead to the negative F L (x, Q 2 ) values [4,5].However, in the considered kinematical regime the perturbative expansion contains also large logarithmic terms proportional to α n s ln n 1/x.Resummation of these terms, in general, is necessary to produce realiable theoretical predictions.Such resummation can be done in the framework of high-energy [6] or k T -factorization [7] approach with using a transverse momentum dependent (TMD, or unintegrated) gluon density f g (x, k 2 T , µ 2 ) obeying the Balitsky-Fadin-Kuraev-Lipatov (BFKL) [8] or Ciafaloni-Catani-Fiorani-Marchesini (CCFM) [9] evolution equations.
As it is suggested in [7], at very low x and Q 2 the proton structure function F 2 (x, Q 2 ) can be saturated (does not depend on Q 2 ).In [10] the partial saturation of the TMD gluon density at Q 2 less than the saturation scale Q 2 s was suggested: x.Further developments on the saturation effect were done [11][12][13][14].At asympotically low x, an equivalent description of early HERA data on F 2 (x, Q 2 ) was provided by the color dipole model [11][12][13][14], where gluon saturation effects [10] important at low scales (and, of course, at low Q 2 in DIS) can be consistently taken into account.In this approach, the DIS events are considered as an interaction of the virtual photon decaying into a color dipole q q and a proton.The saturation of the photon-proton cross section σ γ * p (x, Q 2 ) as a function of the transverse distance r between q and q is suggested at large r or small Q 2 (see [13,14] for more information).
In our previous study [15] the color dipole model was used to investigate the connection between the DIS processes studied at HERA and soft hadron production at LHC1 .An analytical expression for the TMD gluon density in a proton at the low scale µ 0 (which is of order of the hadronic scale, µ 0 ∼ 1 GeV) was proposed and its saturation dynamics was discussed.Further improvements of such investigations were developed [20][21][22][23].In particular, it was shown that a number of hard LHC processes can be reasonably well described within the k T -factorization framework if the proposed TMD gluon distribution is considered as the input for subsequent (non-collinear) CCFM evolution (see, for example, [23] and references therein).In the present note we continue our study and concentrate on self-consistent simultaneous description of latest HERA data on the reduced cross section σ r (x, Q 2 ) at low Q 2 [24] (and, consequently, F 2 (x, Q 2 ) data [25,26]) and LHC data on charged hadron production at small transverse momenta p T in the mid-rapidity region [27][28][29].Note that latest HERA data [24][25][26] have not been considered yet within the developed approach.
For the reader's convenience, we recall some important formulas on transition from high to low Q 2 in DIS.As it was mentioned above, dipole formalism for calculation of the deep inelastic and related diffractive cross sections of γ * p scattering at small x was developed [11][12][13][14].For transversely (T ) and longitudinally (L) polarized photons the γ * p can be presented in the following form: where z is the quark longitudinal momentum fraction (1 − z for antiquark) with respect to photon momentum q, 2 with p being the proton momentum.The squared photon wave functions read where McDonald functions and the summation is performed over the quark flavors f .As it was originally assumed [13,14], the effective dipole cross section is saturated at large r and presented in the following (GBW) form: where overall normalization σ 0 = 29.12mb and parameters λ = 0.277, x 0 = 4.1 • 10 −5 , Q 0 = 1 GeV were found from a fit to the inclusive DIS data.The relation of the TMD gluon density in a proton f g (x, k 2 T ) to the dipole cross section σ(x, r 2 ) was calculated [14] within the two gluon exchange approximation between color dipole q q and proton debris.It has the following form: where J 0 is the Bessel function of zero order and α s = 0.2.The relations (3) and ( 4) lead to the GBW expression for the TMD gluon density in a proton: In our previous paper [23] another analytical form of the TMD gluon density was proposed: where All phenomenological parameters essential at low x, namely, c g , c 1 , c 2 and c 3 were found from the best description of recent LHC data [27][28][29] on charged soft hadron production at the mid-rapidity region within the modified QGSM approach [16,17].Detailed information about the calculations and fitting procedure can be found in [23].Note that we kept x 0 and λ parameters as they were fitted in the GBW model.The TMD gluon density in a form (6), being extended into a whole kinematical region by appying the subsequent CCFM evolution and referred as the LLM'2022 set2 , is able to reproduce the experimental data on number of processes studied at HERA and LHC colliders (see also [31,32]).
With the listed pocket formulas we can now calculate some observables.The proton structure function F 2 (x, Q 2 ) is the sum of the transverse structure function F T (x, Q 2 ) and longitudinal one F L (x, Q 2 ): where F T (x, Q 2 ) and F L (x, Q 2 ) can be calculated at low x neglecting small term ∼ mx 2 using (1): In fact, the reduced cross section σ r (x, Q 2 ) in the DIS is measured experimentally.It can be calculated as: where d 2 σ/dxdQ 2 is the double differential DIS cross section, f (y) = y 2 /(1 + (1 − y) 2 ), inelasticity y = Q 2 /(sx) and s = 4E e E p with E e and E p being the electron and proton energies, respectively.As it was mentioned above, here we try the LLM gluon to describe the latest HERA data [25,26] on the reduced cross section σ r (x, Q 2 ) at low Q 2 within the dipole approach.First of all, we find that we have to change the default values of x 0 and λ parameters involved in (3) and (6).Our calculations according to master formulas (1) and ( 2 3), where one can see the satisfactory description of the combined H1 and ZEUS data [24].The green line corresponds to our results obtained with newly fitted LLM formula (6) and gray dashed line shows results calculated within the GBW model taken with parameters determined3 by H1 [26]: x 0 = 6.0 • 10 −5 and λ = 0.256.We only increase the normalization by 10%: σ 0 = 27 mb.One can see that GBW approach results in a worse description of the data (χ 2 /n.d.f.= 4.1).
Having determined the parameters we can now easily calculate the structure function F 2 (x, Q 2 ).We show our results in comparison with low Q 2 ZEUS [25] and H1 [26] data in Fig. 4. A good agreement of LLM results with data is achieved.
In Fig. 5 we plot the effective dipole cross section σ(x, r 2 ) evaluated according to (4) as a function of r at different values of x.We find that its saturation dynamics at large r strongly depends on x and TMD gluon density in a proton.In accordance with (3), the GBW gluon density results in the saturation in the region of r s ∼ 2/R 0 (see also discussion [14]).The corresponding saturation scale at x = x 0 = 4.2•10 −5 and Q 0 = 1 GeV is Q s ≃ 2/r s ≃ 0.8 GeV.The LLM gluon TMD results in approximately the same Q s at fitted x 0 = 1.3 • 10 −11 and Q 0 = 2.2 GeV, see Fig. 5.The effective dipole cross section, σγ * p (r, Q 2 ) and gluon density f g (x, k 2 T ) do not depend on Q 2 at low Q 2 < Q 2 s , as it shown in [13,23].The data are taken from ZEUS and H1 [24].Now we turn to the production of soft charged hadrons in pp collisions at the LHC energies.These processes are very sensitive to the gluon density in a proton at low scales µ ∼ p T ≃ 1 GeV.In the calculations we employ the modified QGSM [16,17] and strictly follow the approach described earlier [23].So, the inclusive hadron spectrum at low p T and mid-rapidities splits into two pieces: the quark contribution ρ q and the gluon one ρ g : First term is calculated within the conventional QGSM [18,19] using only one-Pomeron exchange.It can be done because in the mid-rapidity and small x T = 2p T / √ s the multi-Pomeron exchanges give negligibly small contributions [17].The second one, ρ q (x, p T ), is calculated as a convolution of the "modified" gluon distribution with fragmentation function (FF) of gluons into hadrons G g→h (z, |p T |) multiplied by the inelastic pp cross section (see, for example, [23] for more details).These FFs were calculated at leading (LO) and next-to-leading (NLO) orders [35] and presented in a factorized form G g→h (z, , where I g h (|p T |) could be approximated as: Here pT = p T − zk T with p T and k T being the transverse momenta of the produced hadron h and gluon, respectively.The FFs of quarks and/or diquarks into hadrons h can be expressed in a similar way.Below we repeat the calculations [23] with the LLM gluon density and newly fitted values of x 0 , λ and Q 0 = 2.2 GeV.We find that the best description of the LHC data [27][28][29] collected at different energies ( √ s = 0.9, 2.36, 7 and 13 TeV) is achieved with slope parameters B g h = 4.35 GeV −1 , B q h = B qq h = 5.42 GeV −1 (χ 2 /n.d.f.= 3.5).All other parameters involved into the calculations and listed in [23] taken into account.are unchanged.Our results are shown in Fig. 6.One can see that good agreement with the measurements is obtained in a wide range of energies.For a comparison we also show results obtained with the GBW TMD.It results in a different fit of the fragmentation parameters B g h = 4.75 GeV −1 , B q h = B qq h = 6.5 GeV −1 , which, however, gives a worse description (χ 2 /n.d.f.= 5.9).So, it seems that the proposed expression (6) leads to better description of the considered experimental data, than the GBW model.
Finally, we have verified the sensitivity of inclusive DIS observables to the parameters of the color dipole model, namely, x 0 , λ and the initial scale Q 0 .The dipole cross section σ(x, r 2 ) is practically not sensitive to these parameters at large r > 0.5 fm or Q 2 < 1 GeV 2 .Then, we have achieved a self-consistent simultaneous description of the latest HERA data on the reduced cross section for the ep DIS and the structure function F 2 (x, Q 2 ) at low Q 2 ≤ 4.5 GeV 2 and LHC data on the charged hadron production at small transverse momenta, p T ≤ 1 GeV.In a forthcoming study, the refined LLM gluon density in a proton will be used as the initial condition for the subsequent QCD evolution and thus extented for any scales.
) show that best description (with χ 2 /n.d.f.= 2.3) of the reduced cross section measured at different energies ( √ s = 225, 251, 300 and 318 GeV) is obtained with x 0 = 1.3 • 10 −11 and λ = 0.22.Let us note that we analyzed the HERA data only in the low Q 2 region, Q 2 < 5 GeV 2 , since we set the lowest perturbative scale Q 0 = 2.2 GeV.Above this scale, the effects of the QCD evolution can play a role.Results of our fit are shown in Fig. (1)-(

Figure 1 :
Figure 1: The reduced DIS cross section a function of x at different Q 2 at √ s = 225 GeV (upper row) and √ s = 251 GeV (lower row).The solid green line corresponds to the results obtained with LLM TMD, and the dashed gray line shows GBW based results.The data are taken from ZEUS and H1 [24].

Figure 2 :
Figure 2: The reduced DIS cross section a function of x at different Q 2 at √ s = 300 GeV.The notations are the same as on Fig. 1.The data are taken from ZEUS and H1 [24].

Figure 3 :
Figure 3: The reduced DIS cross section a function of x at different Q 2 at √ s = 318 GeV.The notations are the same as on Fig. 1.The data are taken from ZEUS and H1 [24].

Figure 4 :Figure 5 :
Figure 4: The structure function F 2 as a function of x at different Q 2 .The notations of the lines are the same as on Fig. 1.The data are taken from ZEUS [25] (white circles) and H1 [26] (black circles).

Figure 6 :
Figure 6: Transverse momentum distributions of soft charged hadrons produced in pp collisions at different LHC energies in the mid-rapidity region.The notations of the lines are the same as in Fig. 1.The experimental data are from [27-29].