# A massive variable flavour number scheme for the Drell-Yan process

### Submission summary

 As Contributors: Rhorry Gauld Preprint link: scipost_202107_00031v1 Date submitted: 2021-07-20 10:29 Submitted by: Gauld, Rhorry Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Phenomenology Approaches: Theoretical, Phenomenological

### Abstract

The prediction of differential cross-sections in hadron-hadron scattering processes is typically performed in a scheme where the heavy-flavour quarks ($c, b, t$) are treated either as massless or massive partons. In this work, a method to describe the production of colour-singlet processes which combines these two approaches is presented. The core idea is that the contribution from power corrections involving the heavy-quark mass can be numerically isolated from the rest of the massive computation. These power corrections can then be combined with a massless computation (where they are absent), enabling the construction of differential cross-section predictions in a massive variable flavour number scheme. As an example, the procedure is applied to the low-mass Drell-Yan process within the LHCb fiducial region, where predictions for the rapidity and transverse-momentum distributions of the lepton pair are provided. To validate the procedure, it is shown how the $n_f$-dependent coefficient of a massless computation can be recovered from the massless limit of the massive one. This feature is also used to differentially extract the massless ${\rm N3LO}$ coefficient of the Drell-Yan process in the gluon-fusion channel.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202107_00031v2 on 23 September 2021

Submission scipost_202107_00031v1 on 20 July 2021

## Reports on this Submission

### Anonymous Report 3 on 2021-9-16 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202107_00031v1, delivered 2021-09-16, doi: 10.21468/SciPost.Report.3537

See attachment.

### Attachment

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### Anonymous Report 1 on 2021-8-18 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202107_00031v1, delivered 2021-08-18, doi: 10.21468/SciPost.Report.3406

### Report

The paper contains the description of a new method to combine
fixed-order (FO) computations performed considering the heavy-flavour
quarks as massless or massive. In particular, the contributions from
power suppressed terms of the form (m_q/Q)^k (where m_q is the mass of
the heavy flavour and Q the typical hard scale of the process) are
consistently combined with a massless computation, thereby obtaining
differential distributions in a "massive variable flavour number"
scheme. Such scheme allows to obtain predictions where collinear logs
(log(m_q)) are resummed to all orders and exact (m_q)^k dependence is
kept to the FO accuracy.

Apart from the proposed "massive variable flavour number" scheme, the
paper also discusses how, using the formalism, one can numerically
extract unknown results for the gg-induced DY differential cross
section at order as^3.

Although I cannot claim to be a real expert on the specific topic
addressed by the author and the associated literature (i.e. I have
never specifically worked on the details and subtleties related to the
treatment of heavy flavours in the initial state and the combination
of massless with massive flavour schemes), I believe the paper
contains original results.

Such results are timely too, in the sense that, as also remarked by
the author, the issues discussed in the paper are likely to become
more and more relevant in the years to come, in the context of
precision phenomenology at the LHC.

As far as the presentation is concerned, I've found the paper
well-written and clear. Similarities with previous works are properly
commented, and technical points are discussed when needed, giving
details at a level that I find appropriate.

For the above reasons, I recommend the paper for publication in
author.

1. "Massive computation, dsigma^M" (sec. 2, page 3): isn't there also
a mass-dependent contribution at order as^2 coming from "real-virtual"
terms for the process pp->Z+1jet (gq ->Zq), where mass terms might
arise from closed massive fermionic loops ? These terms are finite (no
UV or IR divergent) and cancel for the vectorial coupling of the Z
boson to the massive fermionic line, but for the axial-vector coupling
the cancellation is exact only for doublets degenerate in mass
(e.g. (u,d) for m_u=m_d=0), whereas a leftover scaling as (m_i)^2 -
(m_j)^2 is expected to remain, m_i and m_j being the masses of the
quarks belonging to the same SU(2)_L doublet (this is discussed in
several places; see, for instance,

Radiative Corrections to the Ratio of Z and W Boson Production
Dicus, Willenbrock
Phys.Rev.D 34 (1986) 148

).

Can the author add a comment on how these terms are included (I
suppose that, for consistency, they are included) ?

2. "Logarithmic computation" (sec. 2, end of page 4): "...required to
reconstruct the zero-mass limit of the massive computation". Can the

- there are terms without logs, without mass dependence, and not
arising from nf-type contributions (i.e. terms that don't belong to
a_{0,0} in eq. 7)

- these terms are there also in the full massive computations

- eq. 3 allows one to build them and, by consequence, in the paper,
such terms live in dsigma^{ln(m)}.

Is this correct, or not? Perhaps adding a further sentence to clarify
this point could be useful.

3. page 7: the sentence where the author describes how PDF uncertainty
have been computed is not very clear, i.e. I don't understand,
operationally, what has been done. Does the author mean that, for a
given replica, dsigma_replica = dsigma_0 * K, where dsigma_0 is the
prediction at order as^2 for the central set, and K would be instead
disgmatilde_replica/dsigmatilde_0, where both the disgmatilde's have
been computed at order as?

4. eq. 7. I guess there's a typo: m^(2i) -> (m^2)^i ?

On a related note, can the author comment on why one starts with

Always on the same equation: up to which value i and j are allowed to
run in the fit ?

I understand that, physically, what matters is the log(m)
dependence. However, I assume that, also in practice, a given
dimensional scale mu must have been used to fit the coefficients,
i.e. that, in practice, the author used log(m/mu). Is this the case ?
If so, which value of mu has been used ?

### Requested changes

See report for points to be addressed.

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### Author:  Rhorry Gauld  on 2021-08-26  [id 1709]

(in reply to Report 1 on 2021-08-18)
Category:

1) How are the finite corrections from virtual heavy-quark contributions proportional to the axial-vector treated?

These contributions have been included in my calculation (Refs. [92-94] of the original manuscript). This includes both double-virtual and real-virtual contributions, for all subprocesses. Practically, I define the massive power- corrections per each-quark flavour as the difference between the contribution when the calculated with either a massive or massless quark (when fixing the mass of the other quark in the doublet). Numerically these contributions were extremely small for the results shown in this paper (which are far below mll ̄ << mZ, and are therefore strongly power suppressed). For the referees interest, in the region of mll ̄ ∼ mZ these contributions are at the per-mille level for the b-quark with mb = 4.75 GeV.

I suggest to extend the citations to include reference to the work of Dicus and Willenbrock noted by theReferee. Practically, I re-calculated the real-virtual contributions myself and used the double-virtual results from the literature I had previously cited. I have now explicitly noted the inclusion of these (axial-vector) contributions in the manuscript.

2) Clarification on the ‘constant’ terms which are built by eq. (3).

The understanding of the Referee is correct, and this is certainly a tricky point. I have defined a0,0 in such a way that it is constructed from inputs that are defined in the renormalisation scheme (for PDFs and αs) where the heavy-flavour quark is massive (then built with the massless partonic cross-section). As such, these extra constant terms (i.e. the m-independent terms at the cross-section level) are absent from a0,0. However, they are part of the massive cross-section and must be constructed somehow (i.e. through knowledge of these decoupling relations for the PDFs, αs [and m its self at higher orders] computed by others in the past).

3) Definition of the PDF uncertainties.

This (the one quoted by the referee) is the definition that has been used throughout. Including a formula in the text would be appropriate to clarify this

4) The numerical fit and power corrections.

In the fit, j may run up to n where n is the order of the coefficient (i.e. 2 for αs2 and 3 for αs3). This is guided by the fact that there can be maximally these powers of collinear logarithms present (which I believe should dictates the possible powers of j present). The integer i was allowed to run up to 2, as it was found increasing this value made little difference on the fit (i.e. these higher-power terms were largely suppressed).

The ln[μ] dependence (which is certainly there, and is required from dimensional grounds as noted by the Referee) is absorbed into the constant coefficients of the fit. e.g. b_{0,1} ln[m/μ] → a_{0,1} ln[m] + a_{0,0}. I have used a dynamical scale choice μ0 = ET,ll ̄ in the numerical computation, and so practically I see this as the most obvious way to perform the fit.

Finally, concerning the choice of linear vs quadratic power-corrections in the fit. I have used (m2)i (and have corrected the typo, thank you) for the form of the power corrections, which is based on the behaviour of the squared matrix-element and phase space corrections which both contain corrections of the m2/Q2.