Toward the global fit of the TMD gluon density in the proton from the LHC data

We propose a new analytical expression for the transverse-momentum-dependent (TMD, or unintegrated) gluon density in the proton. Essential phenomenological parameters are extracted from the LHC data on inclusive hadron production in $pp$ collisions at low transverse momenta $p_T\le 1\mathrm{GeV}$. The latter are described in the framework of a modified soft quark-gluon string model, where the gluonic state and nonzero transverse momentum of partons inside the proton are taken into account. To determine the parameters important at moderate and large $x$, we use the measurements of inclusive $b$-jet and Higgs boson production at the LHC as well as the latest HERA data on proton structure functions $F_2^c(x,Q^2)$ and $F_2^b(x,Q^2)$ and reduced cross sections ${\sigma }_{\mathrm{red}}^c(x,Q^2)$ and ${\sigma }_{\mathrm{red}}^b(x,Q^2)$. The Catani-Ciafaloni-Fiorani-Marchesini evolution equation is applied to extend the initial gluon distribution to the whole kinematical region. We achieve a simultaneous description of all considered processes with ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.=2.2$, thus moving toward the global fit of the TMD gluon density from collider data. The obtained TMD gluon distribution in a proton is available for public usage and implemented in the tmdlib package and Monte Carlo event generator pegasus.

Figure 1

Left panel: charged hadron spectra calculated within the modified QGSM at different energies. Experimental data are from [45, 46, 47]. Right panel: ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ dependence of our fit performed moderate- and large-$x$ values.

Figure 2

Left: the dipole cross section as a function of $r$ at different values of $x$. Right: the LLM and GBW gluon densities as functions of $k_T$ at different gluon masses. The JH’2013 set 2 [15] gluon distribution is presented for comparison.

Figure 3

The TMD gluon densities in a proton $f_g(x,{\mathbf{k}}_T^2,{\mu }^2)$ calculated as a function of the longitudinal momentum fraction $x$ at different values of transverse momentum ${\mathbf{k}}_T^2$ and hard scale ${\mu }^2$. Shaded bands represent the uncertainties of the $b_g(0)$ fitting procedure. The uncertainties connected with the fit of the $c_g$ parameter are small and therefore not shown here. Note that the gluon densities calculated at ${\mu }^2={10}^4{\mathrm{GeV}}^2$ are multiplied by a factor of 100.

Figure 4

The TMD gluon densities in a proton $f_g(x,{\mathbf{k}}_T^2,{\mu }^2)$ calculated as a function of the transverse momentum ${\mathbf{k}}_T^2$ at different values of the longitudinal momentum fraction $x$ and hard scale ${\mu }^2$. Shaded bands represent the uncertainties of the $b_g(0)$ fitting procedure. The uncertainties connected with the fit of the $c_g$ parameter are small and therefore not shown here. Note that the gluon densities calculated at ${\mu }^2={10}^4{\mathrm{GeV}}^2$ are multiplied by a factor of 100.

Figure 5

Transverse momentum distributions of inclusive $b$-jets produced in $pp$ collisions at $\sqrt{s}=7\mathrm{TeV}$ at different rapidities calculated using the CCFM-evolved TMD gluon density (24) with fitted value $b_g(0)=5.85$. Predictions obtained with the JH’2013 set 2 gluon are shown for comparison. Shaded bands represent the estimation of theoretical uncertainties of our calculations. Kinematical cuts are described in the text. Experimental data are from CMS [52].

Figure 6

Differential cross sections of inclusive Higgs boson production at $\sqrt{s}=13\mathrm{TeV}$ (in the diphoton decay mode) calculated as functions of the diphoton transverse momentum $p_T$, rapidity $y$, and photon helicity angle $|\cos {\theta }^{*}|$ (in the Collins-Soper frame). Notation of all histograms is the same as in Fig. 5. Kinematical cuts are described in the text. Experimental data are from ATLAS [53] and CMS [55].

Figure 7

Differential cross sections of inclusive Higgs boson production at $\sqrt{s}=13\mathrm{TeV}$ (in the $H\to Z{Z}^{*}\to 4l$ decay mode) calculated as functions of the Higgs transverse momentum $p_T$, rapidity $y$, leading and subleading lepton pair invariant masses $m_{12}$ and $m_{34}$, leading lepton pair scattering angle $|\cos {\theta }^{*}|$ (in the Collins-Soper frame), and first and second antilepton production angles $\cos {\theta }_1$ and $\cos {\theta }_2$. The notation of all histograms is the same as in Fig. 5. The kinematical cuts are described in the text. The experimental data are from ATLAS [54].

Figure 8

Structure functions $F_2^c(x,Q^2)$ measured at different scales calculated using the CCFM-evolved TMD gluon density (24) with fitted value $b_g(0)=5.85$. Predictions obtained with the JH’2013 set 2 gluon are shown for comparison. Shaded bands represent the estimation of theoretical uncertainties of our calculations. Experimental data are from ZEUS and H1 [57, 58, 59].

Figure 9

Structure functions $F_2^b(x,Q^2)$ measured at different scales calculated using the CCFM-evolved TMD gluon density (24) with fitted value $b_g(0)=5.85$. Predictions obtained with the JH’2013 set 2 gluon are shown for comparison. Shaded bands represent the estimation of theoretical uncertainties of our calculations. Experimental data are from ZEUS and H1 [57, 59].

Figure 10

Reduced cross sections ${\sigma }_{\mathrm{red}}^c(x,Q^2)$ measured at different scales calculated using the CCFM-evolved TMD gluon density (24) with fitted value $b_g(0)=5.85$. Predictions obtained with the JH’2013 set 2 gluon are shown for comparison. Shaded bands represent the estimation of theoretical uncertainties of our calculations. Experimental data are from ZEUS and H1 [60].

Figure 11

Structure functions ${\sigma }_{\mathrm{red}}^b(x,Q^2)$ measured at different scales calculated using the CCFM-evolved TMD gluon density (24) with fitted value $b_g(0)=5.85$. Predictions obtained with the JH’2013 set 2 gluon are shown for comparison. Shaded bands represent the estimation of theoretical uncertainties of our calculations. Experimental data are from ZEUS and H1 [60].