# NLO PDFs from the ABMP16 fit

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## Abstract

We perform a global fit of parton distribution functions (PDFs) together with the strong coupling constant \(\alpha _s\) and the quark masses \(m_c\), \(m_b\) and \(m_t\) at next-to-leading order (NLO) in QCD. The analysis applies the \(\overline{\mathrm {MS}}\, \) renormalization scheme for \(\alpha _s\) and all quark masses. It is performed in the fixed-flavor number scheme for \(n_f=3, 4, 5\) and uses the same data as the previous ABMP16 fit at next-to-next-to-leading order (NNLO). The new NLO PDFs complement the set of ABMP16 PDFs and are to be used consistently with NLO QCD predictions for hard scattering processes. At NLO we obtain the value \(\alpha _s^{(n_f=5)}(M_Z) = 0.1191 \pm 0.0011\) compared to \(\alpha _s^{(n_f=5)}(M_Z) = 0.1147 \pm 0.0008\) at NNLO.

Parton distribution functions (PDFs) are an indispensable ingredient in theory predictions for hadronic scattering processes within perturbative QCD. Currently, the state-of-art calculations for many standard-candle processes at the Large Hadron Collider (LHC) and elsewhere are based on the QCD corrections up to the next-to-next-to-leading order (NNLO) in the strong coupling constant \(\alpha _s\) [1]. In order to match this theoretical accuracy the PDFs and other input parameters such as \(\alpha _s\) and the quark masses \(m_c\), \(m_b\) and \(m_t\) also have to be determined at the same order of perturbation theory, that is with account of the NNLO QCD corrections. In many instances, however, the Wilson coefficient functions or hard partonic scattering cross sections are known to the next-to-leading order (NLO) only. This concerns in particular Monte-Carlo studies at the LHC. Then, to meet the consistency requirements, NLO PDFs and the respective NLO values for \(\alpha _s\) and the heavy-quark masses are to be used. The NLO fit of PDFs is therefore of immediate practical use and also provides a very good consistency check of the perturbative stability of QCD calculations.

Experiment | Process | | \(\chi ^2\) | |
---|---|---|---|---|

NLO | NNLO | |||

DIS | ||||

HERA I+II | \(e^{\pm }p \rightarrow e^{\pm } X\) | 1168 | 1528 | 1510 |

\(e^{\pm }p \rightarrow \overset{(-)}{\nu } X\) | ||||

Fixed-target (BCDMS, NMC, SLAC) | \(l^\pm p \rightarrow l^\pm X\) | 1008 | 1176 | 1145 |

DIS heavy-quark production | ||||

HERA I+II | \(e^{\pm }p \rightarrow e^{\pm }c X\) | 52 | 58 | 66\(^a\) |

H1, ZEUS | \(e^{\pm }p \rightarrow e^{\pm }b X\) | 29 | 21 | 21 |

Fixed-target (CCFR, CHORUS, NOMAD, NuTeV) | \(\overset{(-)}{\nu } N \rightarrow \mu ^{\pm } c X\) | 232 | 173 | 178 |

DY | ||||

ATLAS, CMS, LHCb | \(p p \rightarrow W^\pm X\) | 172 | 229 | 223 |

\(p p \rightarrow Z X\) | ||||

Fixed-target (FNAL-605, FNAL-866) | \(p N \rightarrow \mu ^+\mu ^- X\) | 158 | 219 | 218 |

Top-quark production | ||||

ATLAS, CMS | \(pp \rightarrow tqX \) | 10 | 5.7 | 2.3 |

CDF&DØ | \(\bar{p}p \rightarrow tb X\) | 2 | 1.9 | 1.1 |

\(\bar{p}p \rightarrow tqX \) | ||||

ATLAS, CMS | \(pp \rightarrow t\bar{t}X \) | 23 | 14 | 13 |

CDF&DØ | \(\bar{p}p \rightarrow t\bar{t}X \) | 1 | 1.4 | 0.2 |

| 2855 | 3427 | 3378 |

The values of \(\chi ^2\) obtained in the present analysis for various data sets are listed in Table 1 in comparison with the earlier ones for the NNLO ABMB16 fit. The overall quality of the data description does not change dramatically between the NLO and the NNLO versions, where the former features a somewhat bigger total value of \(\chi ^2\). Of course, the theoretical description at NNLO accuracy comes with a significantly reduced theoretical uncertainty due to variations of the factorization and renormalization scales compared to the NLO one. Nevertheless, for specific scattering reactions the NNLO corrections are crucial for the respective data sets. This holds in particular for the *c*-quark and, to a lesser extent, for *b*-quark production in DIS and for hadronic *t*-quark pair-production, which constrain the heavy-quark masses and which are fitted together with \(\alpha _s\) simultaneously with the PDFs. The theoretical description at NNLO accuracy is also essential for the parameters of the higher (dynamical) twist, which contribute additively to the leading twist. The *x*-dependent twist-four contributions to the longitudinal and transverse DIS cross sections have been determined in the NNLO version (cf. Table VIII in Ref. [4]) and their central values are kept fixed in the present analysis. Also other fitted parameters like the data set normalizations are taken over unchanged from the NNLO analysis (cf. Table I in Ref. [4]). This provides a better consistency between the PDF sets obtained with different theoretical accuracy. At the same time the uncertainties in the normalization and higher twist parameters are computed in the same way as in the NNLO fit [4], by propagation of the ones in experimental data and simultaneosly with other fit parameters in order to take into account their correlations and, therefore, provide a consistent uncertainty treatment in the NLO and NNLO fits. Therefore, the uncertainties obtained for the data normalization and the twist-four contributions at NLO are only marginally different from those reported at NNLO in Tables I and VIII in Ref. [4].

*x*and the large-

*x*region, i.e., for \(x\lesssim 10^{-4}\) and \(x\gtrsim 0.3\), respectively. In these kinematic regions for example the DIS coefficient functions receive systematically large corrections at higher orders, which need to be compensated by the gluon PDF if the fit is performed at NLO accuracy. The

*u*-quark PDF in Fig. 1b does not show any big changes, except for large \(x\gtrsim 0.6\), while the

*d*-quark PDF Fig. 1c at NLO is smaller in the entire range \(x\lesssim 10^{-1}\) and decreasing more than \(20\%\) for \(x\lesssim 10^{-4}\). A similar observation holds for the strange sea displayed in Fig. 1d, which is smaller by even \(50\%\) for \(x\lesssim 10^{-4}\) at NLO, however, the PDF uncertainties for this quantity are correspondingly larger. On the other hand, the non-strange sea in Fig. 1e does not show big relative differences between NLO and NNLO. There is only a slight decrease of the NLO result by \(5\%\) to \(10\%\) for \(x\lesssim 10^{-2}\). The small-

*x*sea iso-spin asymmetry \(\bar{d} - \bar{u}\) at NLO goes lower than the NNLO one, as can be seen from a comparison of Fig. 1b, c. This reflects the impact of the NNLO corrections on the data for Drell-Yan production, which drive this asymmetry in our fit.

The values of \(\alpha _s(M_Z)\) obtained in the NLO and NNLO variants of the ABMP16 fit with various kinematic cuts on the DIS data imposed and different modeling of the higher twist terms

Fit ansatz | \(\alpha _s(M_Z)\) | ||
---|---|---|---|

Higher twist modeling | Cuts on DIS data | NLO | NNLO |

Higher twist fitted | \(Q^2>2.5~\mathrm{GeV}^2\), \(W>1.8~\mathrm{GeV}\) | 0.1191(11) | 0.1147(8) |

\(Q^2>10~\mathrm{GeV}^2\), \(W^2>12.5~\mathrm{GeV}^2\) | 0.1212(9) | 0.1153(8) | |

Higher twist fixed at 0 | \(Q^2>15~\mathrm{GeV}^2\), \(W^2>12.5~\mathrm{GeV}^2\) | 0.1201(11) | 0.1141(10) |

\(Q^2>25~\mathrm{GeV}^2\), \(W^2>12.5~\mathrm{GeV}^2\) | 0.1208(13) | 0.1138(11) |

*Z*-boson mass \(M_Z\) obtained in the present analysis at NLO is larger than the one obtained in the NNLO variant of the ABMP16 fit [4], the relative difference amounting to about \(4\%\), which is well comparable to the estimated margin due to variations of the factorization and renormalization scales, cf. Ref. [6]. In the scheme with \(n_f=5\) light flavors we find

*W*for the invariant mass of the hadronic system, where \(W^2 = M_P^2 + Q^2 (1-x)/x\) with the proton mass \(M_P\), requires modeling of the higher-twist terms. This has been discussed extensively in the ABMP16 analyses at NNLO [4]. Following the theoretical framework there, the fitted twist-four contributions to the longitudinal and transverse DIS cross sections have been used to determine the value of \(\alpha _s^{\mathrm{NLO}}(M_Z)\) in Eq. (1). Alternatively, one can impose cuts both on \(Q^2\) and \(W^2\) to eliminate data from the kinematic regions most sensitive to the higher-twist terms. Then, the fit can be performed with all higher-twist terms set to zero and the results are shown in Table 2. These variants of the fit with substantially higher cuts on \(Q^2\) and \(W^2\) and higher-twist terms set to zero display very good stability of the value of \(\alpha _s(M_Z)\), both at NLO and NNLO, and therefore very good consistency of the chosen approach.

Finally, in one of the variants of the present analysis we impose the low cuts on \(Q^2\) and \(W^2\) from the first line of Table 2, while fitting also the twist-four contributions. This gives an improvement in the value of \(\chi ^2\) equal to 86 and \(\alpha _s^{\mathrm{NLO}}(M_Z) = 0.1227 \pm 0.0011\), which is a slightly larger value than those quoted in Table 2 for the fits with higher cuts on \(Q^2\) and \(W^2\). The magnitude of these shifts in \(\alpha _s(M_Z)\) may also be considered as an indication for the limitations of the NLO approximation.

Nevertheless, the observed difference between \(\alpha _s^{\mathrm{NLO}}(\mu )\) and \(\alpha _s^{\mathrm{NNLO}}(\mu )\) is quite essential, particularly at small scales \(\mu \), where the NLO and NNLO results differ by more than \(10\%\), as illustrated in Fig. 2 for a wide range of scales. This difference is to a great extent responsible for the perturbative stability of QCD calculations at the hard scales currently probed in scattering processes at colliders. Asymptotic freedom in QCD, i.e. stability of theoretical predictions under higher order perturbative corrections requires very large scales. On the other hand, for realistic kinematics including experiments at the LHC a consistent setting of \(\alpha _s(M_Z)\) is very important to achieve sensible theoretical predictions.^{1}

In this context it is worth to mention the conventional choice \(\alpha _s^\mathrm{NLO}(M_Z)=\alpha _s^\mathrm{NNLO}(M_Z)\), which is adopted as a part of the PDF4LHC recommendations [9] and employed in the CT14 [10] and NNPDF [11] PDF fits. Under this assumption the value of \(\alpha _s\) obtained at NLO is very close to the NNLO one in a wide range of scales, as shown in Fig. 2. As a result, such an approach has significant limitations when studying the convergence of the perturbative expansion, since the NLO predictions obtained with these PDF sets might be very similar to the NNLO ones simply due to the convention used.

*c*-quark production and

*t*-quark hadro-production. On the other hand, the data on DIS

*b*-quark production are less precise, therefore the value of \(m_b\) extracted from the fit suffers from the larger uncertainties and is less sensitive to the impact of the NNLO corrections, cf. Table 3.

The values of the *c*-, *b*- and *t*-quark masses in the \(\overline{\mathrm {MS}}\, \) scheme in units of GeV obtained in the NLO and NNLO variants of the ABMP16 fi. The quoted errors reflect the uncertainties in the analyzed data

NLO | NNLO | |
---|---|---|

\(m_c(m_c)\) [GeV] | \(1.175\pm 0.033\) | \(1.252\pm 0.018\) |

\(m_b(m_b)\) [GeV] | \(3.88\pm 0.13\) | \(3.84\pm 0.12\) |

\(m_t(m_t)\) [GeV] | \(162.1\pm 1.0\) | \(160.9\pm 1.1\) |

*x*[12] (cf. also Table B in Ref. [4] for correlations between \(\alpha _S\) and the quark masses). Indeed, the fitted value of \(m_c\) only changes by \(\pm 20\, \mathrm{MeV}\) for a variation of \(\alpha _s\) in a wide range, cf. Fig. 4. However, the NNLO corrections to the Wilson coefficents for DIS heavy-quark production still have significant impact on \(m_c\), moving it up by \(\sim 100~\mathrm{MeV}\) and reducing its uncertainty. The relatively weak correlation of \(m_c\) with \(\alpha _s\) in Fig. 4 is in contrast to the observed behavior in the MMHT14 PDF set [13], where the particular variable flavor number scheme applied causes a linear relation between \(m_c\) and \(\alpha _s\) in those fits and the charm-quark mass has been treated as a variable parameter, while the resulting values of \(\chi ^2\) when fitting data on DIS

*c*-quark production have been quantified. See Sect. 3.2 and Tables 4 and 5 in Ref. [1] for a review of the theoretical treatments of DIS

*c*-quark production used in PDF fits and Table 2 in Ref. [13] as well as Table 12 in Ref. [1] for the respective values of \(m_c\), \(\alpha _s\) and \(\chi ^2\) in the MMHT14 PDF set.

Values of the heavy-quark masses \(m_c(m_c)\), \(m_b(m_b)\) and \(m_t(m_t)\) and \(\alpha _s^{(n_f=5)}(M_Z)\) in the \(\overline{\mathrm {MS}}\, \)scheme for the PDFs ABMP16_5_nlo (0+29) with \(n_f=5\). The values for pole masses \(m_b^\mathrm{pole}\) and \(m_t^\mathrm{pole}\) in the on-shell scheme obtained using RunDec [23] are also given

PDF set | \(\alpha _s^{(n_f=5)}(M_Z)\) | \(m_c(m_c)\) [GeV] | \(m_b(m_b)\) [GeV] | \(m_b^\mathrm{pole}\) [GeV] | \(m_t(m_t)\) [GeV] | \(m_t^\mathrm{pole}\) [GeV] |
---|---|---|---|---|---|---|

0 | 0.11905 | 1.175 | 3.880 | 4.488 | 162.08 | 171.44 |

1 | 0.11905 | 1.175 | 3.880 | 4.488 | 162.08 | 171.44 |

2 | 0.11906 | 1.175 | 3.880 | 4.489 | 162.08 | 171.44 |

3 | 0.11905 | 1.175 | 3.880 | 4.488 | 162.08 | 171.44 |

4 | 0.11899 | 1.175 | 3.880 | 4.488 | 162.08 | 171.43 |

5 | 0.11898 | 1.175 | 3.880 | 4.487 | 162.08 | 171.43 |

6 | 0.11907 | 1.175 | 3.880 | 4.489 | 162.08 | 171.44 |

7 | 0.11906 | 1.175 | 3.880 | 4.489 | 162.08 | 171.44 |

8 | 0.11911 | 1.175 | 3.880 | 4.489 | 162.08 | 171.44 |

9 | 0.11858 | 1.175 | 3.880 | 4.482 | 162.08 | 171.40 |

10 | 0.11925 | 1.175 | 3.880 | 4.491 | 162.08 | 171.46 |

11 | 0.11914 | 1.175 | 3.880 | 4.490 | 162.08 | 171.45 |

12 | 0.11910 | 1.176 | 3.880 | 4.489 | 162.08 | 171.44 |

13 | 0.11904 | 1.173 | 3.880 | 4.488 | 162.08 | 171.44 |

14 | 0.11909 | 1.203 | 3.880 | 4.489 | 162.08 | 171.44 |

15 | 0.11912 | 1.170 | 3.880 | 4.489 | 162.08 | 171.44 |

16 | 0.11930 | 1.169 | 3.877 | 4.489 | 162.08 | 171.46 |

17 | 0.11897 | 1.174 | 3.754 | 4.351 | 162.08 | 171.43 |

18 | 0.11883 | 1.179 | 3.878 | 4.483 | 162.09 | 171.43 |

19 | 0.11904 | 1.175 | 3.884 | 4.493 | 162.08 | 171.44 |

20 | 0.11879 | 1.180 | 3.888 | 4.493 | 162.13 | 171.47 |

21 | 0.11901 | 1.179 | 3.872 | 4.479 | 162.06 | 171.41 |

22 | 0.11914 | 1.180 | 3.882 | 4.492 | 162.05 | 171.41 |

23 | 0.11889 | 1.169 | 3.880 | 4.486 | 162.12 | 171.47 |

24 | 0.11879 | 1.178 | 3.875 | 4.479 | 161.86 | 171.19 |

25 | 0.11980 | 1.169 | 3.881 | 4.500 | 162.97 | 171.44 |

26 | 0.11914 | 1.182 | 3.881 | 4.491 | 162.12 | 171.49 |

27 | 0.11892 | 1.171 | 3.879 | 4.486 | 161.89 | 171.23 |

28 | 0.11888 | 1.176 | 3.882 | 4.488 | 161.88 | 171.21 |

29 | 0.11936 | 1.176 | 3.870 | 4.482 | 162.34 | 171.74 |

As the heavy-quark masses \(m_c(m_c)\), \(m_b(m_b)\) and \(m_t(m_t)\) have been fitted their numerical values vary for each of the 29 PDF sets and cross section computations involving heavy quarks have to account for this. For reference we list in Table 4 the heavy-quark masses in the ABMP16 grids and the values of \(\alpha _s^{(n_f=5)}(M_Z)\). These values can also be easily retrieved within the LHAPDF library framework. The bottom- and the top-quark pole masses, \(m_b^\mathrm{pole}\) and \(m_t^\mathrm{pole}\), which are required for the on-shell scheme are also provided in Table 4. In particular, for computations with the central ABMP16 set at NLO the values \(m_b^\mathrm{pole} = 4.488~\,\mathrm {GeV}\) and \(m_t^\mathrm{pole} = 171.44~\,\mathrm {GeV}\) should be used.^{2}

Cross sections at NLO and NNLO in QCD for the Higgs boson production from gluon–gluon fusion (\(\sigma (H)\) computed in the effective theory) at \(\sqrt{s}=13\) TeV for \(m_H=125.0\) GeV with the renormalization and factorization scales set to \(\mu _r=\mu _f=m_H\) and for the top-quark pair production, \(\sigma (t{\bar{t}})\), at various center-of-mass energies of the LHC with the top-quark mass \(m_t(m_t)\) in the \(\overline{\mathrm {MS}}\, \)scheme and \(\mu _r=\mu _f=m_t(m_t)\). The values of \(\alpha _s(M_Z)\) and \(m_t(m_t)\) are order dependent, see Eq. (1) and Table 3. The errors denote the PDF and \(\alpha _s\) uncertainties

\(\sigma (H)\) [pb] at \(\sqrt{s}=13\) TeV | \(\sigma (t{\bar{t}})\) [pb] at \(\sqrt{s}=5\) TeV | \(\sigma (t{\bar{t}})\) [pb] at \(\sqrt{s}=7\) TeV | \(\sigma (t{\bar{t}})\) [pb] at \(\sqrt{s}=8\) TeV | \(\sigma (t{\bar{t}})\) [pb] at \(\sqrt{s}=13\) TeV | |
---|---|---|---|---|---|

NLO, ABMP16_5_nlo | \( 33.59 \pm 0.58 \) | \( 63.74 \pm 1.44 \) | \( 172.0 \pm 3.3 \) | \( 247.9 \pm 4.5 \) | \( 835.3 \pm 14.3 \) |

NNLO, ABMP16_5_nnlo | \( 40.20 \pm 0.63 \) | \( 63.66 \pm 1.60 \) | \( 171.8 \pm 3.4 \) | \( 247.5 \pm 4.6 \) | \( 831.4 \pm 14.5 \) |

The benchmark cross sections for the Higgs-boson and top-quark pair production at the LHC at NLO and NNLO with consistent use of the PDF sets obtained in the present analysis are given in Table 5. The quoted errors denote the PDF and \(\alpha _s\) uncertainties derived from the uncertainties in the experimental data. Thus, they are of similar size at NLO and NNLO.

The Higgs boson cross section \(\sigma (H)\) is computed in the effective theory in the limit \(m_t \rightarrow \infty \), but with full \(m_t\) dependence in the Born cross section, based on the NNLO results of Refs. [15, 16, 17]. The NLO value of \(\sigma (H)\) is about 20% smaller than the NNLO one due to missing large perturbative corrections, which are only partially compensated by a larger value of \(\alpha _s\). In the effective theory the Born cross section for \(\sigma (H)\) is proportional to \(\alpha _s^2\), so that the variant of the NLO fit with the larger value for the strong coupling, \(\alpha _s^{\mathrm{NLO}}(M_Z) = 0.1227\), gives a NLO cross section increased by 5%, i.e. \(\sigma (H)=35.2 \pm 0.58\) pb with the PDF set ABMP16free_5_nlo compared to \(\sigma (H)= 33.59 \pm 0.58\) pb with the PDF set ABMP16_5_nlo in Table 5.

The inclusive cross section \(\sigma (t{\bar{t}})\) for top-quark pair production uses Ref. [18] based on Refs. [19, 20, 21, 22]. In this case, the NLO and NNLO values of \(\sigma (t{\bar{t}})\) for the range of center-of-mass energies explored at the LHC are similar, since those data have been included in both fits and are accommodated by the corresponding changes in the value of \(\alpha _s\) and the top-quark mass \(m_t\), cf. Fig. 3 and Table 3.

In summary, we have completed the determination of the ABMP16 PDF sets at those orders of perturbation theory, which are currently of phenomenological relevance, i.e., at NLO and NNLO. Essential input in the ABMP16 analysis has been the final HERA DIS combination data from run I+II, which has consolidated the available world DIS data. In addition, several new data sets from the fixed-target DIS together with recent LHC and Tevatron data for the DY process and for the top-quark hadro-production have been used.

We have discussed the features of the NLO extraction of PDFs, which in general, have a few limitations due to lacking constraints of the higher order Wilson coefficients and we have emphasized the consistent use of PDFs and an order-dependent value of \(\alpha _s(M_Z)\), which is absolutely crucial because of correlations. The same holds, to a lesser extent, in collider processes also for the values of the heavy-quark masses used.

The ABMP16 PDFs establish the baseline for high precision analyses of LHC data from run I and run II, and the NLO variant is now available for computing cross sections of scattering processes with multi-particle final states, for which the NNLO QCD corrections will not be available in the foreseeable future, or for Monte Carlo studies. Precision analyses of LHC data, however, will always require analyses to NNLO accuracy in QCD. This will become even more important with the arrival of the data from the high luminosity runs.

## Footnotes

## Notes

### Acknowledgements

We would like to thank K. Lipka and O. Zeniaev for cross-checks of the preliminary version of the LHAPDF grids derived from this analysis. This work has been supported by Bundesministerium für Bildung und Forschung (Contract 05H15GUCC1).

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