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A massive variable flavour number scheme for the Drell-Yan process
by R. Gauld
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): | Rhorry Gauld |
Submission information | |
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Preprint Link: | scipost_202107_00031v2 (pdf) |
Date accepted: | Nov. 24, 2021 |
Date submitted: | Sept. 23, 2021, 3:08 p.m. |
Submitted by: | Gauld, Rhorry |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
Specialties: |
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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 nf-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 N3LO coefficient of the Drell-Yan process in the gluon-fusion channel.
Author comments upon resubmission
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
List of changes
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 ˆAab and Δnf(\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 dσm=0,nf is computed with inputs (PDFs and \alphas) defined in the massive scheme (e.g. nmaxf=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:
%
dσi=Kdσi[O(αs)],K=dσ0[O(α2s)]dσ0[O(αs)].
%
That is to say that a differential K-factor is calculated for the central PDF member at O(α2s), and then applied to each of the individual replica cross-sections which are computed at O(αs)."
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 m2/Q2. The integer j is limited to 2(3) when the α2(3)s 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 a0,0 is equivalent to dσm=0,nf, while the remaining terms describe the power corrections. Fitted in this way, all m-independent information (such as dependence on μ, which is chosen as the dynamic scale E\rT,\Pll) is absorbed into the ai,j coefficients."
I have also changed the inline math to read m2)i instead of m2iwhendefiningtheansatz.CHANGESINTRODUCEDTOADDRESSREPORT3Therefereenoted5placeswhereimprovementscouldbemade.1b.Commentaboutprovidingmoredetailontheconstructionofthelogarithmiccross−sectionineq.(3).IhaveextendedthediscussioninSec. 2,whichnowincludestheexplicitconstructionofthelogarithmiccross−sectionupto\mathcal{O}(\alpha_s^2),nowappearingasEq. (5).Theformulafor\Delta_{n_f}^{(i)}hasalsobeengiventosecondorderinEq. (4)−−−ithasbeenprovidedintheschemewheretheheavyquarkmassisdefinedintheon−shellscheme,consistentwiththeOMEcalculations.2b.Commentaboutprovidingmoredetailon(previouslyeq.(4))eq.(6),fortheheavyquarkmassslicingprocedure.Thediscussionappearingaftereq.(4)hasbeenextended.Inparticular,Ihavenowintroducedanexplicitexampleonhowthelogarithmiccross−sectionisbuiltatthird−order(includingthesubtractionterms).Ihavealsointroducedanexplicitdiscussionabouttheschemedependenceofthethird−orderresultswhichaddressesacommentinpoint4.(seebelow)oftherefereeregardingtheschemedependenceoftheOMEsinref.[24],andthereproducibilityoftheresults.3b.Validityofcollinearfactorisationwithmassiveinitialstates.Icertainlyagreethattheconceptofcollinearfactorisationisatstake,whichiswhySection4includesthestatement“Adeepertheoreticalunderstandingoffactorisationtheoremsformassive−initialstatesremainsdesirabletoday.”andreferencetothepreviousworkonviolationofthestandardfactorisationtheorem.TheintentionofthisSectionistostatethattheprocedureIhavedevelopedtoextractdifferentialmassivepowercorrectionsisfullyapplicabletoprocesseswithmassiveinitialstates(andnottoaddressthelong−standingissueoffactorisationitself).IhavenotintroducedanychangesinSection4.4b.Generalityofthepresentationoftheformalism,andreproducibilityofresults.TheextendeddiscussionintroducedinSection2andSection3ofthemanuscript(seepoint1.and2.above,respectively)nowaddressesthecommentraisedbytherefereeaboutreproducibilityofindividualcomponentsofthecalculation.TherefereescriticismoftheapproachtakenbytheNNPDFcollaborationtoextractanintrinsiccharmquarkPDFmaybevalid,butIdonotbelievesuchadiscussionisrelevant/appropriate/necessarywithinthecontextofmywork.Ialsonotethatallthecitationsinthe“Theoreticalimplementation”Sectionhavebeenrelevantforthecurrentcomputation(eitherdirectlyused,orusedasacross−checkofvariousnumerical/analyticresults).5b.RelevanceofthephenomenologyinSection8.IagreethatSection8isoutofcontextgiventhemainresultsofthepaper(whichistheoretical/conceptualinnatureandsolvestheissueofdefiningamassivevariableflavournumberschemefordifferentialcolliderobservables).However,Section8demonstratesthattheprocedureisapplicabletoexperimentallyaccessibledistributions.ThesedistributionshavealsobeenprovidedtotheLHCbcollaborationandwillappearinacomparisontodatainaforthcomingpublicationbytheexperiment.ThisSectionisthereforestillhighlyvaluable,andremainsunchangedintherevisedversion.OTHERCHANGESIhavealsointroducedothersmallchanges/corrections:1c.Ihaveincludedacorrectioninthelegend(mis−label)ofFigure1thathadbeenmadeintheoriginalsubmission2c.Anadditionalclarifyingsentenceinthe“Numericalinputs”subsectionhasbeenincluded.3c.Ihaveincluded+signsbeforethepositivevaluesappearinginTable1andTable2,andIhavecorrectedthe”Order”forthesecondentryinTable2tonowread\mathcal{O}(\alpha_s^2)$. The latter change is necessary as is also contains the αs coefficient the way I have defined it.
Published as SciPost Phys. 12, 024 (2022)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2021-10-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00031v2, delivered 2021-10-17, doi: 10.21468/SciPost.Report.3691
Report
(through the SciPost "answer to question" form), and for having
taken into account my observations in the revised version of the
manuscript.
In my opinion, now the manuscript can be published in SciPost.