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

Dear Editor, and Referees,

I would like to thank the Referees for the careful reading of the manuscript, and for providing constructive comments in various places and for making suggestions to improve the manuscript. In the following I provide a list of minor changes I have introduced to clarify various points throughout the manuscript.

With kind wishes,

Rhorry Gauld

CHANGES INTRODUCED TO ADDRESS REPORT 1
Overall, four points were raised.

1a. How are the finite corrections from virtual heavy-quark contributions proportional to the axial-vector treated?
I have extended the citations to include reference to the work of Dicus and Willenbrock noted by the Referee. 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.
``This includes those axial contributions arising due the presence of heavy-quark triangle diagrams, see for example [89--91]"

2a. Clarification on the ‘constant’ terms which are built by eq. (3).
I have added the following sentences to help clarify how these terms are constructed:
``Constructed in this way (i.e. using massless inputs) the logarithmic calculation will also contain those terms which are independent of $m$. They are generated by the constant terms contained in $\hat{A}_{ab}$ and $\Delta_{n_f}(\alphas)$---i.e. those which define the de-coupling across heavy-flavour thresholds in a variable flavour number scheme. It is necessary to account for these terms as they are part of the massive calculation (i.e. they appear on the LHS of Eq.~(2)), but are not generated when ${\rm d}\sigma^{\rm m=0,n_f}$ is computed with inputs (PDFs and $\alphas$) defined in the massive scheme (e.g. $n_f^{\rm max} = 4$ for the $b$-quark)."

3a. Definition of the PDF uncertainties.
I have altered the text to explicitly include the formula used to approximate the PDF uncertainties up to O(αs2):
``Where shown, PDF uncertainties have been obtained from individual replica predictions ($i$) calculated in the following way:
%
\begin{align}
{\rm d}\sigma_i = K \, {\rm d}\sigma_i[\mathcal{O}(\alpha_s)] \,,\qquad K = \frac{ {\rm d}\sigma_0[\mathcal{O}(\alpha_s^2)] }{ {\rm d}\sigma_0[\mathcal{O}(\alpha_s)] }\,.
\end{align}
%
That is to say that a differential K-factor is calculated for the central PDF member at $\mathcal{O}(\alpha_s^2)$, and then applied to each of the individual replica cross-sections which are computed at $\mathcal{O}(\alpha_s)$."

4a. The numerical fit and power corrections.
Several points regarding the value of the integers in the fit, the μ-dependence, and form of the ansatz of the power-corrections were raised. To address each these comments above (related to the fit), the text has been altered to:

``The form of this ansatz is motivated by the behaviour of the squared matrix-element and phase space which both contain corrections of the form $m^2/Q^2$. The integer $j$ is limited to $2(3)$ when the $\alpha_s^{2(3)}$ coefficient is fitted, and a maximum value of $i = 2$ is considered in each case. The choice for $j$ is guided by the powers of collinear logarithms which may be present at each order, whereas increasing $i$ beyond 2 had little impact on the fit. The $m$-independent constant $a_{0,0}$ is equivalent to ${\rm d}\sigma^{\rm m=0,n_f}$, while the remaining terms describe the power corrections. Fitted in this way, all $m$-independent information (such as dependence on $\mu$, which is chosen as the dynamic scale $E_{\rT,\Pll}$) is absorbed into the $a_{i,j}$ coefficients."

I have also changed the inline math to read $m^2)^i$ instead of $m^{2i} when defining the ansatz.

CHANGES INTRODUCED TO ADDRESS REPORT 3
The referee noted 5 places where improvements could be made.

1b. Comment about providing more detail on the construction of the logarithmic cross-section in eq. (3).
I have extended the discussion in Sec.~2, which now includes the explicit construction of the logarithmic cross-section up to $\mathcal{O}(\alpha_s^2)$, now appearing as Eq.~(5). The formula for $\Delta_{n_f}^{(i)}$ has also been given to second order in Eq.~(4)---it has been provided in the scheme where the heavy quark mass is defined in the on-shell scheme, consistent with the OME calculations.

2b. Comment about providing more detail on (previously eq. (4)) eq. (6), for the heavy quark mass slicing procedure.
The discussion appearing after eq. (4) has been extended. In particular, I have now introduced an explicit example on how the logarithmic cross-section is built at third-order (including the subtraction terms). I have also introduced an explicit discussion about the scheme dependence of the third-order results which addresses a comment in point 4. (see below) of the referee regarding the scheme dependence of the OMEs in ref.[24], and the reproducibility of the results.

3b. Validity of collinear factorisation with massive initial states.
I certainly agree that the concept of collinear factorisation is at stake, which is why Section 4 includes the statement “A deeper theoretical understanding of factorisation theorems for massive-initial states remains desirable today.” and reference to the previous work on violation of the standard factorisation theorem. The intention of this Section is to state that the procedure I have developed to extract differential massive power corrections is fully applicable to processes with massive initial states (and not to address the long-standing issue of factorisation itself). I have not introduced any changes in Section 4.

4b. Generality of the presentation of the formalism, and reproducibility of results.
The extended discussion introduced in Section 2 and Section 3 of the manuscript (see point 1. and 2. above, respectively) now addresses the comment raised by the referee about reproducibility of individual components of the calculation. The referees criticism of the approach taken by the NNPDF collaboration to extract an intrinsic charm quark PDF may be valid, but I do not believe such a discussion is relevant/appropriate/necessary within the context of my work. I also note that all the citations in the “Theoretical implementation” Section have been relevant for the current computation (either directly used, or used as a cross-check of various numerical/analytic results).

5b. Relevance of the phenomenology in Section 8.
I agree that Section 8 is out of context given the main results of the paper (which is theoretical/conceptual in nature and solves the issue of defining a massive variable flavour number scheme for differential collider observables). However, Section 8 demonstrates that the procedure is applicable to experimentally accessible distributions. These distributions have also been provided to the LHCb collaboration and will appear in a comparison to data in a forthcoming publication by the experiment. This Section is therefore still highly valuable, and remains unchanged in the revised version.

OTHER CHANGES
I have also introduced other small changes/corrections:
1c. I have included a correction in the legend (mis-label) of Figure 1 that had been made in the original submission
2c. An additional clarifying sentence in the “Numerical inputs” subsection has been included.
3c. I have included + signs before the positive values appearing in Table 1 and Table 2, and I have corrected the ”Order” for the second entry in Table 2 to now read $\mathcal{O}(\alpha_s^2)$. The latter change is necessary as is also contains the αs coefficient the way I have defined it.