Parton distribution functions in Monte Carlo factorisation scheme
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Abstract
A next step in development of the KrkNLO method of including complete NLO QCD corrections to hard processes in a LO partonshower Monte Carlo is presented. It consists of a generalisation of the method, previously used for the Drell–Yan process, to Higgsboson production. This extension is accompanied with the complete description of parton distribution functions in a dedicated, Monte Carlo factorisation scheme, applicable to any process of production of one or more colourneutral particles in hadron–hadron collisions.
1 Introduction
The method of including complete NLO QCD corrections to hard processes in the LO partonshower Monte Carlo (PSMC), nicknamed KrkNLO, was originally proposed in Ref. [1], where its first numerical implementation on top of a toymodel PSMC was also presented. It was restricted there to gluon emission only and was elaborated for two processes: \(Z/\gamma ^*\) production in hadron–hadron collisions, i.e. the Drell–Yan (DY) process and deep inelastic electron–hadron scattering (DIS).
In Ref. [2], the KrkNLO method was implemented for \(Z/\gamma ^*\) production process at large hadron collider (LHC) in combination with Sherpa [3] and Herwig++ [4, 5, 6] PSMCs. Many NLOclass numerical results (distributions of transverse momenta, rapidity, integrated cross sections, etc.) were presented there and comparisons of the KrkNLO predictions with those from other methods, such as MC@NLO [7] and POWHEG [8], were also performed.
The main advantage of the KrkNLO method with respect to other, older methods of matching the fixedorder NLO calculations with PSMCs (MC@NLO and POWHEG) is its simplicity. This simplicity stems from the fact that the entire NLO corrections are implemented using a simple positive multiplicative MC weight. However, in order to profit from it, one has to use in the KrkNLO method parton distribution functions (PDFs) in a special, socalled Monte Carlo (MC) factorisation scheme and PSMC has to fulfil some minimum quality criteria. Most of modern PSMCs [9, 10, 11, 12, 13] are good enough for the KrkNLO method.
Construction of PDFs in the MC factorisation scheme (FS) has evolved step by step: in Ref. [1] it was defined for gluonstrahlung only (albeit for two different processes, DY and DIS). In Ref. [2], the KrkNLO PDFs in the MC FS were defined and numerically constructed including also gluon to quark transitions/splittings, relevant for the complete NLO corrections in the DY process, which at the LO level has only quarks and antiquarks in the initial state. PDFs in the MC scheme in Ref. [2] were defined in terms of the standard \(\overline{\text {MS}}\) PDFs, and constructed numerically by transforming the \(\overline{\text {MS}}\) PDFs into MCscheme PDFs, before they were plugged into PSMC used in the KrkNLO method.
However, in Ref. [2] certain elements in the transition matrix K, transforming the \(\overline{\text {MS}}\) PDFs into the MCscheme PDFs could be omitted, because they were not relevant (i.e. of a NNLO class) for the DY process. These elements of the transition matrix have to be added for any process with initialstate gluons, such as the Higgsboson production elaborated in the present work. They will be defined and applied in the following, such that the complete transition matrix K transforming the \(\overline{\text {MS}}\) PDFs into the MCscheme PDFs will be specified for the first time. It will be argued that PDFs in such a MCscheme can serve in the KrkNLO method for any process at a hadron–hadron collider in which a colourneutral single or multiple system of heavy particles is produced. For other processes, with one or more coloured partons in the final state at LO level, the KrkNLO method with PDFs in the MC scheme may also work, but this subject is reserved for the forthcoming publications.
The MC factorisation scheme is a complete scheme, such that NLO coefficient functions for any hard process under consideration are known, hence PDFs in the MC FS can be fitted directly to experimental DIS and DY data. However, at present, we obtain them from PDFs in the \(\overline{\text {MS}}\) scheme and leave out direct fitting to data for the future developments.
On the methodological side, as seen in Refs. [1, 2], the essence of the KrkNLO method is that certain NLO correction terms in an unintegrated/exclusive form present in the \(\overline{\text {MS}}\) scheme, which are proportional to unphysical Diracdelta terms in transverse momentum of emitted real partons, are removed in the KrkNLO methodology by means of redefinition of PDFs from the \(\overline{\text {MS}}\) to MC scheme. These ‘pathological’ terms are preventing the use of a simple multiplicative MC weight for implementing NLO corrections in the \(\overline{\text {MS}}\) scheme in realemission phase space, and they complicate implementation of the MC@NLO and POWHEG methods. These peculiar terms can be determined and calculated either by means of studying the NLO corrections to hard process (coefficient functions), or, alternatively, by means of integrating softcollinear counterterms (similar to these in the Catani–Seymour method [14]), which define the MCscheme PDFs in \(d=4+2{\varepsilon }\) dimensions.^{1} We are going to calculate them using both methods, obtaining the same results.
Last but not least, the NLO calculations for the DY process of Ref. [2] were also compared with the NNLO calculations of MCFM [15], leading to the conclusion that they are closer to the latter than the results of the MC@NLO and POWHEG methods.
The outline of the paper is the following: in Sect. 2 the KrkNLO method is characterised briefly. In Sect. 3 all distributions needed for implementation of the KrkNLO method for Higgsboson production in gluon–gluon fusion are elaborated, including also many analytical crosschecks and a necessary update of the virtual corrections in softcollinear counterterms used in Ref. [2] for the \(Z/\gamma ^*\) (DY) process. Section 4 presents numerical results for PDFs in the MC scheme. Then the first numerical results for the total cross section from the KrkNLO method for the Higgs production at the LHC are shown in Sect. 5. Finally, in Sect. 6 we summarise the paper and discuss future prospects of our work. In Appendix A the formulae for the NLO coefficient functions of the DIS process in the MC scheme are provided.
2 The method
The KrkNLO method was formulated in a few variants. For instance, in the version of Ref. [1], the MC weight implementing the NLO corrections sums the contributions from all relevant partons generated in PSMC next to the hard process “democratically”, such that it works equally well for PSMCs based on angular ordering or virtuality ordering, contrary to POWHEG which requires adding extra gluons to a PSMC event. In the present work, we are going to follow the variant of KrkNLO discussed in Ref. [2], in which the NLOcorrecting MC weight uses only one parton, the one closest to the hard process in the transverse momentum, that is, the first parton generated in the backward evolution (BEV) in the PSMC algorithm with \(k_T\)ordering.
In any case, in the KrkNLO method, the entire event of PSMC is preserved and reweighted, contrary to POWHEG and MC@NLO where the parton attributed to the hard process is generated outside PSMC and, only later on, the remaining partons are provided by PSMC. Obviously, this puts certain minimum quality requirements on the PSMC: (i) the first parton in the BEV algorithm has to be generated with the distribution which has a correct soft and collinear limit and (ii) its phase space in momentum and flavour space has to be covered completely, without empty regions. Luckily, the above requirement is fulfilled by all modern PSMCs for initialstate emissions discussed in this work.
It is worth to comment in advance on the apparent use in the following of the softcollinear counterterms (dipoles) of the Catani–Seymour (CS) subtraction scheme [14]. Their role is twofold: (1) the CS dipoles serve us as a useful benchmark, as they provide a reference model for QCD distributions of real emissions featuring the exact soft and collinear limits and (2) the CS scheme helps us in a proper inclusion of the NLO virtual corrections. However, let us point out immediately an important difference between the MC and CS scheme: the CS dipoles do not include virtual corrections, while softcollinear counterterms (SCCTs) of the KrkNLO do include them, albeit not calculated from Feynman diagrams, but deduced from PDF momentum sum rules. The role of the SCCTs in the KrkNLO methodology is also much richer than that of the dipoles in the CS scheme—our SCCTs not only provide subtractions of softcollinear singularities in realemission phase space, but they are also used to define PDFs in the MC factorisation scheme. Moreover, their sums are required to coincide with the corresponding sums of realparton distributions in PSMC.^{2}
3 Higgs production in gluon–gluon fusion
In the following we are going to collect all distributions needed for implementation of the KrkNLO method for the gluonfusion Higgs production in hadron–hadron collisions. Elements of the matrix transforming PDFs from the \(\overline{\text {MS}}\) to MC scheme will also be obtained as a byproduct.
3.1 CS dipoles and MC matrix elements
In the following formulation of the KrkNLOmethod components, the CS dipoles will serve us as useful auxiliary objects. They are formed by an initialstate (onshell) emitter a from one hadron and a spectator parton b from another hadron,^{3} see Fig. 5. Following closely the notation of the CS work [14], the emitter a splits into an offshell \(\widetilde{ac}=\bar{b}\) entering into the hard process and an emitted parton c. The CS dipoles \(\mathfrak {D}^{(ac,b)}\) relevant for processes of our interest are proportional to \(\bar{P}_{ \widetilde{ac} ,a}\), the DGLAP kernel for the \( a\rightarrow \widetilde{ac}\) splitting.^{4}
For the processes of the annihilation \(a\bar{a} \rightarrow X\) at the LO level, such as the Higgs production and the DY process, in each NLO channel \(ab \rightarrow cX \) we must have \( \widetilde{ac} =\bar{b}\) in the NLO splitting. In other words, the NLO splitting in the annihilation processes is fully determined by a and b.^{5} The above rules are illustrated in Fig. 5 and possible indices are listed in Table 1 for the emission from the incoming line a.^{6}
List of indices labelling the CS or MC softcollinear counterterms for all the NLO channels (except the \(q\bar{q}\) channel in Higgs production) of annihilation processes. Indices (a, b) denote initial partons (channel), while (ac, b) are labelling the CS/MC counterterms, with a being an emitter and b a spectator
\(a+b\rightarrow c+H\)  \(a+b \rightarrow c+Z/\gamma \)  

(a, b)  (ac, b)  (a, b)  (ac, b) 
(g, g)  (gg, g)  \((q,\bar{q})\)  \((qg,\bar{q})\) 
(q, g)  (qq, g)  \((\bar{q},q)\)  \((\bar{q}g,q)\) 
\((\bar{q},g)\)  \((\bar{q}\bar{q},g)\)  (g, q)  (gq, q) 
\((g,\bar{q})\)  \((g\bar{q},\bar{q})\) 
 (A)For the \(g + g \rightarrow H + g\) channel a typical/representative distribution of PSMC, summing the emissions from both incoming gluons, iswhere the \(g\rightarrow g\) splitting function is given by$$\begin{aligned} \mathcal{M}_{gg\rightarrow Hg}^{\text {MC}}^2 = 8\pi \alpha _s\,\mu ^{2\epsilon }\,\frac{1}{Q^2}\, \frac{1}{\alpha \beta }\,(1z)\hat{P}_{gg}(z;\epsilon ) \mathcal{M}_{gg\rightarrow H}^{\text {LO}}^2,\nonumber \\ \end{aligned}$$(3.12)It is equal to the sum of two CS dipoles \( \mathcal{M}_{gg\rightarrow Hg}^{\text {MC}}^2 = \mathfrak {D}^{(gg,g)} _{(1)}+ \mathfrak {D}^{(gg,g)}_{(2)}\), where$$\begin{aligned} \hat{P}_{gg}(z;\epsilon )= & {} 2C_A\left[ \frac{z}{1z} + \frac{1z}{z} + z(1z)\right] \nonumber \\= & {} C_A\frac{1 + z^4 + (1z)^4}{z(1z)}. \end{aligned}$$(3.13)with soft partition functions \( \frac{\alpha }{\alpha +\beta }\) and \(\frac{\beta }{\alpha +\beta }\) separating the soft singularity evenly between two incoming emitters. Indices (1) and (2) are used to distinguish the above two dipoles.$$\begin{aligned}&\mathfrak {D}^{(gg,g)}_{(1)}= \frac{\alpha }{\alpha +\beta } \mathcal{M}_{gg\rightarrow Hg}^{\text {MC}}^2, \quad \mathfrak {D}^{(gg,g)}_{(2)}= \frac{\beta }{\alpha +\beta } \mathcal{M}_{gg\rightarrow Hg}^{\text {MC}}^2,\nonumber \\ \end{aligned}$$(3.14)
 (B)For the \(g + q \rightarrow H + q\) channel we have (with a single softcollinear pole the soft partition functions are not needed):where the \(q\rightarrow g\) splitting function reads$$\begin{aligned}&\mathcal{M}_{gq\rightarrow Hq}^{\text {MC}}^2 = \mathfrak {D}^{(qq,g)}_{(1)} =8\pi \alpha _s\,\mu ^{2\epsilon }\,\frac{1}{Q^2}\, \frac{1}{\alpha }\,\hat{P}_{gq}(z;\epsilon ) \mathcal{M}_{gg\rightarrow H}^{\text {LO}}^2,\nonumber \\ \end{aligned}$$(3.15)$$\begin{aligned}&\hat{P}_{gq}(z;\epsilon ) = C_F\left[ \frac{1 + (1z)^2}{z} + \epsilon \, z\right] . \end{aligned}$$(3.16)
 (C)
Finally, for the \(g + \bar{q} \rightarrow H + \bar{q}\) channel, the CS dipole and MC distribution is the same as the previous one for quarks.
 (A)For the \(q + \bar{q} \rightarrow Z + g\) channel, the MC distribution readswhere$$\begin{aligned} \mathcal{M}_{q\bar{q}\rightarrow Zg}^{\text {MC}}^2&= \mathfrak {D}^{(q g ,\bar{q})}_{(1)} +\mathfrak {D}^{(\bar{q} g ,q)}_{(2)} = 8\pi \alpha _s\,\mu ^{2\epsilon }\,\frac{1}{Q^2}\,\nonumber \\&\quad \times \frac{1}{\alpha \beta }\,(1z)\hat{P}_{qq}(z;\epsilon ) \mathcal{M}_{q\bar{q}\rightarrow Z}^{\text {LO}}^2, \end{aligned}$$(3.17)and the soft partition function is used again:$$\begin{aligned} \hat{P}_{qq}(z;\epsilon ) = C_F\left[ \frac{1+z^2}{1z} + \epsilon (1z) \right] , \end{aligned}$$(3.18)$$\begin{aligned}&\mathfrak {D}^{(qg ,\bar{q})}_{(1)} = \frac{\alpha }{\alpha +\beta } \mathcal{M}_{q\bar{q}\rightarrow Zg}^{\text {MC}}^2, \quad \mathfrak {D}^{(\bar{q}g ,q)}_{(2)} = \frac{\beta }{\alpha +\beta } \mathcal{M}_{q\bar{q}\rightarrow Zg}^{\text {MC}}^2.\nonumber \\ \end{aligned}$$(3.19)
 (B)For the \(q + g \rightarrow Z + q\) channel we have (the soft partition function in the MC distribution is not necessary):where$$\begin{aligned}&\mathcal{M}_{qg\rightarrow Zq}^{\text {MC}}^2 = \mathfrak {D}^{(gq ,q)}_{(1)} = 8\pi \alpha _s\,\mu ^{2\epsilon }\,\frac{1}{Q^2}\, \frac{1}{\alpha }\,\hat{P}_{qg}(z;\epsilon ) \mathcal{M}_{q\bar{q}\rightarrow Z}^{\text {LO}}^2,\nonumber \\ \end{aligned}$$(3.20)$$\begin{aligned} \hat{P}_{qg}(z;\epsilon ) = T_R\left[ z^2 + (1z)^2 + 2\epsilon \, z(1z)\right] . \end{aligned}$$(3.21)
3.2 Integrated CS dipoles and counterterms of MC scheme
For the purpose of installing virtual parts (using PDF momentum sum rules) in the MC distributions (softcollinear counterterms) and defining the Kmatrix for transforming PDFs from the \(\overline{\text {MS}}\) to MC scheme, we need to integrate partly all distributions defined in the previous subsection, keeping the \(z=1\alpha \beta \) variable fixed.
Let us calculate all the above objects in more detail for the \(gg\rightarrow Hg\) channel and then, skipping details of analytical integration, for other channels.
3.3 \(gg \rightarrow Hg\) channel
3.4 \(gq \rightarrow Hq\) channel
3.5 Revisiting \(q\bar{q} \rightarrow Zg\) channel
4 PDFs in MC scheme
In Ref. [2], were the KrkNLO method was applied to the Drell–Yan process, it was sufficient to transform the \(\overline{\text {MS}}\) PDF of quarks and antiquarks. The difference between the \(\overline{\text {MS}}\) and MC PDFs for the gluon was an NNLO effect, and hence is beyond the claimed accuracy.
Looking at the elements of the transition matrix K in Eq. (4.3) one can see that the terms \(\sim \ln (1z)\) and \(\sim \ln z\) are absorbed in the MCscheme PDFs. As a result the NLO coefficient functions for the DY process and the Higgsboson production are much simpler than the corresponding ones in the \(\overline{\text {MS}}\) scheme, cf. Eqs. (3.36) and (3.37), (3.69) and (3.72), (3.47) and (3.48), (3.55) and (3.57). One can thus expect that higherorder QCD corrections, beyond NLO, will be smaller in the MC factorisation scheme than in the \(\overline{\text {MS}}\) scheme. In particular, the MCscheme coefficient functions are free of the socalled leading threshold corrections, \(\sim \ln (1z)/(1z)\), which are absorbed (and resummed) in the MC PDFs.

What is the purpose of MC factorisation scheme? It is defined such that the \(\Sigma (z)\delta (k_T)\) terms due to emission from initial partons disappear completely from the real NLO corrections in the exclusive/unintegrated form, even before PSMC gets involved.

Why is the above vital in the KrkNLO scheme? Without eliminating such terms it is not possible to include the NLO corrections using simple multiplicative MC weights on top of distributions generated by PSMC.

How to determine elements of the transition matrix \(K^{\text {MC}}_{ab}\)? They can be deduced from the difference of softcollinear counterterms of the MC and \(\overline{\text {MS}}\) scheme or from inspection of the NLO corrections in a few simple processes with initial quarks and gluons in the LO hard process. We have done it both ways.

Will the same PDFs in the MC scheme eliminate \(\sim \delta (k_T)\) terms for all processes? This is a question about the universality of the MC factorisation scheme. For all processes similar to the DY or Higgsproduction process, with produced colourneutral finalstate objects, the answer is positive.
Two types of MC PDFs are plotted: the complete version (red solid), where both quarks and gluons are transformed, and the “DY” version (green dashed), where the gluon is unchanged with respect to \(\overline{\text {MS}}\). As discussed earlier, these types of MC PDFs is sufficient for the Drell–Yan process and it was used in our previous work [2]. Hence, we show them here for comparison.
One can see that the differences between the MC and \(\overline{\text {MS}}\) PDFs are noticeable. In particular, the MC quarks are up to \(20\%\) smaller at low and moderate x, while they get above the \(\overline{\text {MS}}\) distributions at large x. For DY and Higgs production, the latter has consequences only at large rapidities of the bosons. At the same time, we notice that the gluon is larger in the MC scheme at low and moderate x. Hence, the changes in quarks and the gluon have a chance to compensate each other and, indeed, as we checked explicitly, the momentum sum rules (4.4) are numerically satisfied for our MC PDFs.
Other quark flavours, when transformed to the MC scheme, exhibit similar changes to those shown in Fig. 6 for the u and d quarks.
Finally, let us comment briefly on the processindependence (universality) of the MC factorisation scheme and the KrkNLO method. If we treat Eq. (4.2) as a definition of PDFs in the MC scheme, then their universality is just inherited from the \(\overline{\text {MS}}\) scheme. The universality of the KrkNLO method is more involved and it would imply that by means of adoption of these PDFs and a careful choice of the exclusive/unintegrated MC distributions for the initialstate splittings, we are able to eliminate from the NLO real corrections all terms proportional to \(\delta (\beta ) f(z)\) or \(\delta (\alpha ) f(z)\), which means that we can impose the NLO real corrections with the multiplicative MC weights in \(d=4\) dimensions on top of the PSMC distributions. We are able to state that the above is true for all annihilation process into colourneutral objects. This can be deduced from analysing the CS counterterms (which are compatible with the modern PSMCs), where both the emitter and the spectator are in the initial state. They are universal within the class of the above annihilation processes and therefore the KrkNLO method features the same property. The answer to the question whether extending this argument to other processes, with one or more coloured partons in the final state at the LO level, is not trivial and the relevant study is reserved to next dedicated publication.^{8}.
5 NLO cross sections for Higgs production in KrkNLO method
Values of the total cross section with statistical errors for the Higgsboson production in gluon–gluon fusion at NLO from the KrkNLO method compared to the results of MC@NLO
\(\sigma ^\mathrm{tot}_\text {H}\) [pb]  

MC@NLO  \(18.72 \pm 0.04\) 
KrkNLO  \(19.38 \pm 0.04\) 
We see that the two methods give slightly different (\(\sim 3.5\%\)) total cross sections, which come from formally higherorder terms, i.e. beyond the NLO approximation. The relevant distributions and detailed comparisons with MC@NLO, POWHEG and the NNLO calculations from the HNNLO program [25, 26] are presented in another publication [27].
6 Summary and outlook
In this work, we have presented all the ingredients of the KrkNLO method needed for its implementation for the Higgsboson production process in gluon–gluon fusion. In particular, the complete definitions of PDFs in the MC scheme, together with their numerical distributions, have been provided. Hence, PDFs in the MC FS can be fitted directly to experimental DIS and DY data. We have also presented the first result for the total cross section for the Higgs production. More distributions, comparisons with MC@NLO, POWHEG and the NNLO calculations are presented in a separate paper [27]. A dedicated study of the processindependence (universality) of the KrkNLO method and the MC factorisation scheme is also reserved for the future work.
The current state of NNLO+PS [28, 29, 30, 31, 32, 33, 34] represents a clear progress in matching fixedorder QCD calculations with PSMCs, however they are still limited to certain classes of observables. The other natural extension for KrkNLO is NNLO+NLOPS, where NLOPS is a PSMC that implements the NLO evolution kernels in the fully exclusive form and thus provides the full set of the softcollinear counterterms for the hard process.
Footnotes
 1.
They also form matrix elements of the Kmatrix transforming PDFs from the \(\overline{\text {MS}}\) to MC scheme.
 2.
At least for the initialstate emitters in the present work, but also in the finalstate ones in the future implementations of the KrkNLO method. In fact, SCCTs of the KrkNLO and PSMC distributions do not need to coincide exactly, but optional additional weight bringing the PSMC to SCCT distribution of the KrkNLO method has to be well behaved.
 3.
The role of the spectator is to provide for momentum and colour conservation.
 4.In the case of the emitted parton c being the gluon one gets \(\widetilde{ac}\equiv a\).
 5.
This is, of course, not true for other processes.
 6.
Rules for emissions from the second incoming line are analogous.
 7.
We employ here and in the following a shorthand notation \(\delta _{x=y}\equiv \delta (xy)\).
 8.
The analysis in Ref. [1] for the DIS process, albeit limited to the gluonstrahlung NLO subprocess, gives hope for a possible positive answer.
 9.
Let us note that a similar reweighting method for realparton radiation in the DY process was implemented some time ago in the PYTHIA PSMC algorithm in the socalled matrixelement correction mode [22]. However, it did not include the virtual NLO corrections and did not use the MC factorisation scheme, as it is in the case of the KrkNLO method.
Notes
Acknowledgements
We are grateful to Simon Platzer and Graeme Nail for the useful discussions and their help with the dipole parton shower implemented in Herwig 7. We are indebted to the Cloud Computing for Science and Economy project (CC1) at IFJ PAN (POIG 02.03.0300033/0904) in Kraków whose resources were used to carry out some of the numerical computations for this project. We also thank Mariusz Witek and Miłosz Zdybał for their help with CC1. This work was funded in part by the MCnetITN FP7 Marie Curie Initial Training Network PITNGA2012315877.
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