Anatomy of double heavy-quark initiated processes

Received: June Anatomy of double heavy-quark initiated processes Matthew Lim 0 1 2 5 6 Fabio Maltoni 0 1 2 3 6 Giovanni Ridol 0 1 2 4 6 Maria Ubiali 0 1 2 5 6 J.J. Thomson Avenue 0 1 2 6 Cambridge 0 1 2 6 U.K. 0 1 2 6 Open Access 0 1 2 6 c The Authors. 0 1 2 6 0 Via Dodecaneso 33 , 16146 Genova , Italy 1 Chemin du Cyclotron , 1348 Louvain-la-Neuve , Belgium 2 Universite Catholique de Louvain 3 Centre for Cosmology , Particle Physics and Phenomenology CP3 4 Dipartimento di Fisica, Universita di Genova & INFN , Sezione di Genova 5 Cavendish Laboratory 6 [30] E. Byckling and K. Kajantie, Particle Kinematics, University of Jyvaskyla , Jyvaskyla A number of phenomenologically relevant processes at hadron colliders, such as Higgs and Z boson production in association with b quarks, can be conveniently described as scattering of heavy quarks in the initial state. We present a detailed analysis of this class of processes, identifying the form of the leading initial-state collinear logarithms that allow the relation of calculations performed in di erent avour schemes in a simple and reliable way. This procedure makes it possible to assess the size of the logarithmically enhanced terms and the e ects of their resummation via heavy-quark parton distribution functions. As an application, we compare the production of (SM-like and heavy) scalar and vector bosons in association with b quarks at the LHC in the four- and ve- avour schemes as well as the production of a heavy Z0 in association with top quarks at a future 100 TeV hadron collider in the ve- and six- avour schemes. We nd that, in agreement with a previous analysis of single heavy-quark initiated processes, the size of the initial-state logarithms is mitigated by a kinematical suppression. The most important e ects of the resummation are a shift of the central predictions typically of about 20% at a justi ed value of the scale of each considered process and a signi cant reduction of scale variation uncertainties. NLO Computations; QCD Phenomenology 1 Introduction Di erent heavy quark schemes: analytical comparison Di erent heavy quark schemes: numerical results LHC Run II Bottom-fusion initiated Higgs production Bottom-fusion initiated Z0 production Future colliders A Cross section in the collinear limit Introduction With the imminent restart of data-taking at LHC Run II the need for accurate theoretical of the Standard Model (SM), becomes more and more pressing. The study of associated as top and bottom quarks, are among the highest priorities of the new run. In particular, b quarks play an important role in the quest for new physics as well as for precise SM measurements from both an experimental and a theoretical perspective. Firstly, they provide a very clean signature as they may easily be identi ed in a detector due to the displacement of vertices with respect to the collision point, a consequence of the b-quark long lifetime. Secondly, the relative strength of the Higgs Yukawa coupling (or possibly both in production as well as in decay. In particular, production associated with b quarks scenarios beyond the Standard Model. At hadron colliders, any process that features heavy quarks can be described according (in the case of b quarks), the heavy quark is produced in the hard scattering and arises as a massive particle in the nal state. The dependence on the heavy quark mass mb is retained nite) splittings q ! qg or of a gluon into heavy quark pairs, g ! qq. On the other hand, in the massless or ve- avour (5F) scheme (in the case of b quarks), Q mb is assumed and the heavy quark is treated on the same footing as the light quarks: it contributes to the proton wave function and enters the running of the strong coupling constant this scheme the heavy quark mass is neglected in the matrix element and the collinear In a previous work [1], we examined processes involving a single b quark in both leptonhadron and hadron-hadron collisions. It was found that, at the LHC, unless a very heavy particle is produced in the nal state, the e ects of initial-state collinear logarithms are ical and the other of kinematical nature. The rst is that the e ects of the resummation reason is that the nave scale Q that appears in the collinear logarithms turns out to be many processes involving one b quark in the initial state was performed and a substantial found within the expected uncertainties. In this work we focus on processes that can be described by two b quarks in the initial state, such as pp ! Hbb or pp ! Zbb. As already sketched in [1], the same arguments used for single heavy-quark initiated processes can be used to analyse the double heavy-quark case. One may navely expect that the resummation e ects for processes with two b quarks in the initial state can be simply obtained by \squaring", in some sense, those of processes with only one b quark. There are, however, a number of features that are particular to the double heavy-quark processes and call for a dedicated work. One is that the lowest order contribution in the 4F scheme appears for the rst time among the NNLO real corrections to the leading order 5F scheme calculation. Furthermore, due to the simplicity of the 5F description (i.e. Born amplitudes are 2 ! 1 processes), results in the 5F scheme are now 4F scheme results have become easily accessible for a wide range of nal states. In fact, it is easy to understand that a meaningful comparison between the two schemes for double heavy-quark initiated processes starts to be accurate if results are taken at NNLO for the 5F and at NLO for the 4F case. Both pp ! Hbb or pp ! Zbb have been considered in previous works. For the LHC, it was demonstrated that consistent results for both the total cross section and di erential distributions for bottom-fusion initiated Higgs production can be obtained in both schemes [2{6]. Analogous studies were performed for bottom-fusion initiated Z producisation scales associated to these processes are to be chosen smaller than the mass of the order to stabilise the perturbative series and make the four- and ve- avour predictions Working Group (HXSWG) to match the NLO 4F and NNLO 5F scheme predictions in case of bottom-fusion initiated Higgs production via the Santander interpolation [5] and via the use of consistently matched calculations [11{14]. While previous studies support a posteriori the evidence that smaller scales make the ve- avour pictures more consistent, no complete analysis of the relation of the two schemes in the case of double heavy-quark initiated processes has been provided. In particular, no analytic study of the collinear enhancement of the cross section and the kinematics of this class of processes has been performed. In this work, we ll this gap by extending our previous work to double heavy-quark production. We rst present an analytic comparison of the two schemes that allow us to unveil a clear relation between them, establish the form of the logarithmic enhancements and determine their size. We then compare the predictions for LHC phenomenology in a of order 100 TeV, a new territory far beyond the reach of the LHC would be explored. At may become of relevance in processes involving top quarks in the initial state. The structure of the work is as follows. In section 2 we examine the kinematics of 2 to 3 body scattering and calculate the phase space factor for the particular case of binitiated Higgs production | we thus derive the logarithmic contributions to the cross section which arise in a 4F scheme. We then proceed to generate kinematic distributions for the processes and use these to analyse the 4F and 5F scheme results. We conclude the section by suggesting a factorisation scale at which results from either process may be meaningfully compared. In section 3 we compare the results on total cross sections obtained in both schemes for a number of phenomenologically relevant processes at the LHC and future colliders. Finally, our conclusions are presented in section 4. Di erent heavy quark schemes: analytical comparison We start by considering Higgs boson production via bb fusion in the 4F scheme. The relevant partonic subprocess is g(p1) + g(p2) ! b(k1) + H(k) + b(k2); where the b quarks in the nal state are treated as massive objects. Since the b-quark mass mb is much smaller than the Higgs boson mass MH , we expect the cross section for the process (2.1) to be dominated by the con gurations in which the two nal-state b (qq ! bbH) that also contributes to the leading-order cross section in the 4F scheme is very of large transverse momentum b quarks in the gg channel, as compared to the dominant collinear con gurations, we will perform an approximate calculation of the cross section for the process (2.1) limiting ourselves to the dominant terms as mb ! 0. The result will while the one in eq. (2.4) corresponds to In both cases we nd k1 = (1 k2 = (1 k1 = (1 k2 = (1 s1 = s2 = An explicit calculation yields ^4F;coll(^) = ^ dz2 Pqg(z1)Pqg(z2)L(z1; ^)L(z2; ^) (z1z2 then be compared to the full leading-order 4F scheme calculation. We present here the nal result; the details of the calculation can be found in appendix A. The di erential partonic cross section can be expressed as a function of ve independent invariants, which we choose to be s^ = (p1 + p2)2; t1 = (p1 k1)2; t2 = (p2 k2)2; s1 = (k1 + k)2; s2 = (k2 + k)2: u1 = (p1 u2 = (p2 Pqg(z) is the leading-order quark-gluon Altarelli-Parisi splitting function The su x \coll" reminds us that we are neglecting less singular contributions as mb ! 0, i.e. either terms with only one collinear emission, which diverge as log mb2, or terms which are regular as mb ! 0. The con guration in eq. (2.3) is achieved for ^ = Pqg(z) = L(z; ^) = log We now observe that the leading-order partonic cross section for the process relevant for calculations in the 5F scheme, is given by [15] b(q1) + b(q2) ! H(k); ^5F(^) = s^ = (q1 + q2)2: Hence, the 4F scheme cross section in the collinear limit, eq. (2.9), can be rewritten as ^4F;coll(^) = 2 The physical interpretation of the result eq. (2.16) is straightforward: in the limit of collinear emission, the cross section for the partonic process (2.1) is simply the bb ! H probability is logarithmically divergent as mb ! 0, and this is the origin of the two factors of L(zi; ^). The arguments of the two collinear logarithms exhibit a dependence on the momentum fractions z1; z2, eq. (2.12). This dependence is subleading in the collinear limit mb ! 0 and indeed it could be neglected in this approximation; however, the class of subleading terms induced by the factor (1 the integration bounds on t1 and t2, as shown in appendix A) and therefore universal in some sense, as illustrated in ref. [1]. We also note that the arguments of the two collinear logs depend on both z1 and z2; this is to be expected, because the integration bounds on t1 and t2 are related to each other. However, in some cases (for example, if one wants to relate the scale choice to a change of factorisation scheme, as in ref. [16]) a scale choice which only depends on the kinematics of each emitting line might be desirable. We have checked that the replacement mass energy ps is given by has a moderate e ect on physical cross sections. The replacement would make the scale at which the four- and ve- avour scheme results are comparable lower by about 20/30% but does not qualitatively modify our arguments and results below. The corresponding 4F scheme physical cross section in hadron collisions at centre-of4F;coll( ) = dx2 g(x1; 2F )g(x2; 2F )^4F;coll collinear ME collinear ME and PS where g(x; 2F ) is the gluon distributon function, F is the factorisation scale, and After some (standard) manipulations, we get 4F;coll( ) = 2 Pqg(z1)L (z1; z1z2) g Pqg(z2)L (z2; z1z2) g We are now ready to assess the accuracy of the collinear approximation in the 4F scheme. We rst consider the total cross section. In table 1 we display the total 4F scheme cross section for the production of a Higgs boson at LHC 13 TeV for two values of the Higgs mass, order result; the second column contains the cross section with the squared amplitude approximated by its collinear limit, but the exact expression of the phase space measure. Finally, in the third column we give the results obtained with both the amplitude and the phase-space measure in the collinear limit, which corresponds to the expression in eq. (2.20). From table 1 we conclude that the production of large transverse momentum b quarks, correctly taken into account in the 4F scheme, amounts to an e ect of order 20% on the total cross section and tends to decrease with increasing Higgs mass. We now turn to an assessment of the numerical relevance of the subleading terms included by the de nition eq. (2.12) of the collinear logarithms. To this purpose we study the distribution of (1 contribution to the total cross section. ments of the logs. The results are displayed in gure 1 for Higgs production at the LHC at 13 TeV and for two di erent values of the Higgs boson mass. The two distributions behave in a similar way: both are strongly peaked around values smaller than 1; in particular, the 68% threshold is in both cases around 0:2. This con rms that, altough formally subleading give a sizeable A further con rmation is provided by the distributions in gure 2, where the full cross functions of the partonic centre-of-mass energy We see that the collinear cross section provides a good approximation to the full 4F scheme result. In the same picture we show the collinear cross section with the factors of L(zi; z1z2) replaced by log MmH22 (solid black b histogram). It is clear that in this case the collinear cross section substantially di ers from the exact result. 100 200 300 400 500 600 700 800 √s s^ for a Higgs of mass 125 GeV (above) and of mass 400 GeV (below). The solid line We now consider the 5F scheme, where the b quark is treated as a massless parton and collinear logarithms are resummed to all orders by the perturbative evolution of the parton distribution function. Eq. (2.14) leads to a physical cross section pp→bbH at the 13 TeV LHC, 4FS MH = 125 GeV Coll MCEol+lMCEol+lPCSo(llMPHS) √s pp→bbH at the 13 TeV LHC, 4FS MH = 400 GeV Coll MCEol+lMCEol+lPCSo(llMPHS) 5F( ) = 2 pp→ bbH at the 13 TeV LHC, mH=125 GeV pp→ bbH at the 13 TeV LHC, mH=400 GeV z1)2=^ for b-initiated Higgs production in F are set to MH . The vertical lines represent the values below which 68% and 90% of events lie. In order to make contact with the 4F scheme calculation, we observe that the b quark PDF can be expanded to rst order in s b(x; 2F ) = Lb = log + O( s2) = ~b(1)(x; 2F ) + O( s2); Correspondingly, we may de ne a truncated 5F cross section 5F;(1)( ) which contains only one power of log mb2 for each colliding b quark. This is obtained by replacing eq. (2.22) in eq. (2.21) and performing the same manipulations that led us to eq. (2.20): we get 5F;(1)( ) = 2 Pqg(y)Lb g Pqg(z)Lb g Eq. (2.24) has exactly the same structure as the 4F scheme result in the collinear Hence, it corresponds to the solid black curve in gure 2. We are therefore led to suggest that the 5F scheme results be used with a scale choice dictated by the above results, similar to that which we have illustrated in ref. [16]. Such a scale is de ned so that the two schemes give the same result: 5F;(1)( ) = is determined by arguments of the form (1 zi)2 . For p nd the following values for ~F : The explicit expression of ~F is simply obtained by equating 5F;(1)( ), eq. (2.24), which is dependence on F due to the gluon parton density is suppressed by an extra power of s Higgs mass or the Z0 mass. The size of the logarithmic terms kept explicitly in the 4F case s = 13 GeV, and mb = 4:75 GeV, we bbH; MH = 125 GeV : bbZ0; MZ0 = 91:2 GeV : bbZ0; MZ0 = 400 GeV : 0:36 MH 0:38 MZ0 0:29 MZ0 ; ttZ0; MZ0 = 1 TeV : ttZ0; MZ0 = 5 TeV : ttZ0; MZ0 = 10 TeV : 0:40 MZ0 0:21 MZ0 0:16 MZ0 : In both cases we have used the NNPDF30 lo as 0130 PDF set [17], with the appropriate the gluon PDF and for the strong coupling constant does not modify in any signi cant way the value of ~F that we obtain. This is expected given that the gluon-gluon luminosity and the dependence on s tend to compensate between numerator and denominator. We have also checked that, after the replacement in eq. (2.17), the values of ~F are typically about 20{30% smaller. We note that the scale ~F is in general remarkably smaller than the mass of the produced heavy particle. As in the case of single collinear logarithm, the reduction is more pronounced for larger values of the mass of the heavy particle compared to the available smaller than the nave choice in previous studies [3], although perhaps with a slightly larger value in the case of Higgs boson, ~ MH =3 rather than MH =4. The argument given above identi es a suitable choice for the factorisation/renormalisaterms is correctly matched in the two schemes. At this point, further di erences between the schemes can arise from the collinear resummation as achieved in the 5F scheme and from mass (power-like) terms which are present in the 4F scheme and not in the 5F one. Closely following the arguments of ref. [3], to which we refer the interested reader for more details, we now numerically quantify the e ect of the resummation. A careful study of the impact of power-like terms can be found in refs. [11{14]. These terms have been found to have an impact no stronger than a few percent. Starting from eq. (2.22), one can assess the accuracy of the O( s1) (O( s2)) approximations compared to the full b(x; 2) resummed expression. The expansion truncated at order p s, often referred to as ~b(p)(x; 2) in the literature, does not feature the full resummation of collinear logarithms, but rather it contains powers n of the collinear log with 1 adopted throughout this work) as a function of the scale 2 for various values of the momentum fraction x. Deviations from one of these curves are an indication of the size of terms of order O( sp+1) and higher, which are resummed in the QCD evolution of the bottom quark PDFs. As observed in our previous work, at LO higher-order logarithms are important and ~b(1)(x; 2) is a poor approximation of the fully resummed distribution function. In particular, it overestimates the leading-log evolution of the b PDF by 20% at very small x and it underestimates it up to 30% at intermediate values of x. On the other hand, at NLO the explicit collinear logs present in a NLO 4F scheme calculation provide a rather accurate approximation of the whole resummed result at NLL; signi cant e ects, of order up to 20%, appear predominantly at large values of x. 1The numerical computation is performed by consistently evolving s and the PDFs in the 4FS on the right-hand side of eq. (2.25) and in the 5FS on the left-hand side. At the same time we checked that the use of a 5FS evolution for s and PDFs on the right-hand side does not modify signi cantly the resulting value of ~F , as it should be, being the change of factorisation scheme a higher order e ect. are associated to the ~b and b computations respectively. the t~ and t computations respectively. A similar behaviour characterises the top-quark PDFs. In gure 4 the ratios between the truncated top-quark PDFs t~ and the evolved PDFs t(x; 2) are displayed for four di erent values of x and varying the factorization scale . We see that for the top-quark PDF at NLO, the di erence between the 2-loop approximated PDF t~(2)(x; 2) and the fully evolved PDF t(x; 2) is very small (of the order of 5%) unless very high scales and large x are involved. A comparable behaviour was observed in ref. [18]. Di erent heavy quark schemes: numerical results In this section, we consider the production of Higgs and neutral vector bosons via bb fusion at the LHC and the production of heavy vector bosons in tt collisions at a future high energy hadron collider. We compare predictions for total rates obtained at the highest available perturbative order in the 4F and 5F schemes at the LHC and in the 5F and 6F schemes at a future 100 TeV collider. Bottom-fusion initiated Higgs production Although in the SM the fully-inclusive bb ! H cross section is much smaller than the other Higgs production channels (gluon fusion, vector boson fusion, W and Z associated Higgs become a dominant production channel when couplings are enhanced with respect to the Standard Model. More speci cally, in models featuring a second Higgs doublet the rate is typically increased by a factor 1= cos2 or tan2 , with = v1=v2 being the ratio of two Higgs vacuum expectation values. Calculations for b-initiated Higgs productions have been made available by several groups. The total cross section for this process is currently known up to next-to-nextto-leading order (NNLO) in the 5F scheme [19] and up to next-to-leading order (NLO) in the 4F scheme [20, 21]. Total cross section predictions have been also obtained via and the mass e ects on the other, without double counting common terms. A rst heuristic proposal, which has been adopted for some time by the HXSWG LHC, is based on the so-called Santander matching [5] where an interpolation between results in the 4F and in the 5F schemes is obtained by means of a weighted average of the two results. Several groups have provided properly matched calculations based on a thorough quantum theory analysis, at NLO+NLL and beyond via the FONLL method [12, 14] and an e ective eld theory approach [11, 13] that yield very similar results. Fully di erential calculations in the 4F scheme up to NLO(+PS) accuracy have been recently made available [6] in MadGraph5 aMC@NLO [22] and POWHEG BOX [23] and work is in progress in the SHERPA framework [24]. These studies conclude that the b-quark kinematics. On the other hand, for inclusive observables the di erences between 4F and 5F schemes are mild if judicious choices for scales are made. The assessment of the size of such e ects and their relevance for phenomenology is the purpose of this section. We rst compare the size and the scale dependence of the 4F and 5F scheme predictions from leading-order up to the highest available perturbative order, namely NLO in the case of the 4F scheme and NNLO in the case of the 5F scheme cross sections. Results are shown in consistently with the perturbative order of the calculation, and with s5F(MZ ) = 0:118. MH = 125 GeV, mb(MH) =MH , with F = R = . Terms proportional to ybyt in the NLO 4F scheme have been neglected. Results with the running b mass computed at a xed scale MH are also shown (right plot). In the inset the ratio between the 5F NNLO prediction and the 4F scheme NLO prediction is displayed. Both the renormalisation and factorisation scales have been taken to be equal to kMH , The treatment of the Higgs Yukawa coupling to b quarks deserves some attention. Di erent settings may cause large shifts in theoretical predictions. Here we use the MS scheme; the running b Yukawa yb( ) is computed at the scale R (left plots). We have checked that computing the Yukawa at the xed value of MH does not modify our conclusions (right plots). The numerical value of mb( R) is obtained from mb(mb) by evolving pending on the scheme. The numerical value of mb(mb) is taken to be equal to the pole settings adopted in ref. [6]) and consistently with the latest recommendation of the Higgs cross section working group.2 The 4F and 5F scheme curves at leading order show an opposite behaviour: in the 4F scheme the scale dependence is driven by the running of s and therefore decreases with the scale, while in the 5F scheme case it is determined by the scale dependence of the b-quark PDF which in turn leads to an increase. The inclusion of higher orders in both calculations drastically reduces the di erences; nonetheless, it is clear from gures 5 and 6 2The pole mass value that we use in our calculation is slightly di erent from the latest recommendation mpole = 4:92 GeV as well as from the value used in the PDF set adopted in our calculation mpole = 4:18 GeV, b b however our results are not sensitive to these small variations about the current central value. MH = 400 GeV, mb(MH) order 4F scheme prediction by a large amount, about 80%. We also observe that 4F and 5F scheme predictions are closer at lower values of the scale. The scale dependence of the 4F scheme NLO calculation is approximately of the same size as that of the 5F scheme NLO calculation, while it is stronger than the scale dependence of the 5F scheme NNLO calculation, as expected, since in the latter the collinear logarithms are resummed. served in ref. [1], for heavier nal state particles di erences between schemes are enhanced. In particular, at the central scale the NNLO 5F scheme prediction exceeds the 4F scheme case by a factor of two. Also in this case, at smaller values of the scale the di erence is signi cantly reduced. This behaviour corresponds to that expected from our analysis presented in section 2. to about 30{35%, a di erence that can be accounted for by considering rst the (positive) e ects of resummation included in the 5F scheme calculation with respect to the 4F one and second the power-like quark-mass corrections that are not included in the 5F calculation and estimated to be around 2{5%, see refs. [11{13]. The e ects of the resummation are easy to quantify by establishing the range of x which gives the dominant contribution to Higgs production via bb collisions. To this purpose, we show in gure 7 the x distribution in the leading-order bottom-quark fusion Higgs production in the 5F scheme. We observe that the x distribution has its maximum around 10 2 for the Standard Model Higgs; for such values of x, the resummation of collinear logarithms is sizeable: the di erence between the fully resummed b PDF and ~b(2) becomes as large as 10 to 15% for scales between 100 and 400 GeV. Note that we expect twice the bb→ H at the 13 TeV LHC mH=400 GeV e ect of a single b quark in the case of processes with two b quarks in the initial state, which amounts to a di erence of 20{25% from resummed logarithms at O( s3) and higher between the collinear approximation of the 4F scheme calculation and the 5F scheme calculation. This expectation is con rmed by the curves in gure 8, where we plot the 5F scheme cross section at LO (left panel) and NLO (right panel) as a function of the Higgs mass in the range 100 GeV to 500 GeV, with R = with the same settings as in gure 5. In the same panel we present the cross sections with together with the relevant ratios. We observe that, for a sensible value of the factorisation and renormalisation scales, as per the one suggested in this paper ~F MH =3, the e ect of neglecting the higher order logs resummed in the b PDF evolution beyond the ones included in the second order expansion of the b PDF, ~b(2), is smaller than 20% for the SM Higgs mass and of about 30% for a heavier Higgs. Similar conclusions are drawn if the NLO cross section is considered instead, as in the right hand-side panel. If instead we had taken as the central scale choice R = order logs would appear much more signi cant. The scale dependence of the Standard Model Higgs cross section is studied in gure 9. The plots con rm the ndings that the assessment of the e ect of the higher-order logs resummed in a 5F scheme calculation strongly depends on the scale at which the process is computed and that at a scale close to ~F the e ects of higher order logs are quite moderate, while they become signi cant if the nave hard scale of the process is chosen. Bottom-fusion initiated Z0 production A similar analysis can be carried out for the case of Z production. Z-boson production in association with one or two b-jets has a very rich phenomenology. It is interesting as a at the LHC or indirectly in the W mass determination). In addition, it represents a crucial irreducible background for several Higgs production channels at the LHC. For the SM PDF ~b(p) with p = 1; 2, with = F = R = MH =3. of MH , computed either with the fully resummed b quark PDF at LL or NLL, or with the truncated pp→bbH at the 13 TeV LHC 0.2 pMpH→=b1b25HGaetVthe 13 TeV LHC pp→bbH at the 13 TeV LHC 0.2 pMpH→=b1b25HGaetVthe 13 TeV LHC (right) as a function of k = =MH , with R = F , computed either with the fully resummed b Higgs boson, Zbb production is a background to ZH associated production followed by the decay of the Higgs into a bottom-quark pair. Finally, this process is a background to searches for Higgs bosons with enhanced Hbb Yukawa coupling. Calculations for bottom-initiated Z production have been made available by several mass) in ref. [7] for exclusive 2-jet nal states. The e ect of a non-zero b quark mass was considered in later works [8, 9] where the total cross section was also given. More recently, in ref. [10] leptonic decays of the Z boson have taken into account, together with the full correlation of the nal state leptons and the parton shower and hadronisation e ects. The total cross section for Zbb in the 5F scheme has been computed at NNLO accuracy for the rst time in ref. [26]. Bottom-initiated Z production is in principle very di erent from Higgs production because the Z boson has a non-negligible coupling to the light quarks. For simplicity, we will not take these couplings into account; to avoid confusion, we refer to the Z boson that couples only with heavy quarks as Z0, even when we take its mass to be equal to 91:2 GeV as in the Standard Model. We have calculated the 5F scheme cross sections by using a private code [26], which has been cross-checked at LO and NLO against MadGraph5 aMC@NLO. The 4F scheme cross section has been computed with MadGraph5 aMC@NLO. Our settings are the same as in the Higgs production computation. We take the same value for the factorisation and renormalisation scales. Results are presented in gure 10 as functions of k = =MZ0 for MZ0 = 91:2 GeV and = MZ0 the best 5F scheme prediction exceeds the 4F scheme prediction by almost 30%, while their di erence is reduced at lower values of the scales. In this respect the behaviour of the 4F vs 5F scheme predictions re ects what we have already observed in gure 5. We note, however, that the scale dependence of the 5F scheme predictions for Zbb is quite di erent with respect to the Hbb when expansion seems to converge more quickly for higher values of = MZ0 . The not show any signi cant qualitative di erence, apart from the fact that Zbb results have in general a milder scale dependence. The di erent scale sensitivity (with R = F ) of the in a milder scale dependence of the Zbb predictions comparing to NLO curves on the right-hand side of gures 5 and 6 and at NNLO. Future colliders The perspective of a proton-proton collider at a centre-of-mass energy of 100 TeV would open up a new territory beyond the reach of the LHC. New heavy particles associated with a new physics sector may be discovered and new interactions unveiled. quarks. We therefore expect collinear enhancements in top-quark initiated processes. In ref. [18] the question of whether the top quark should be treated as an ordinary parton at high centre-of-mass energy, thereby de ning a 6FNS, is scrutinised, and the impact of resumming collinear logs of the top quark mass is assessed. This analysis is performed i taR 1.5 pp → (b –b)Z’ at the 13 TeV LHC MZ’ = 91.2 GeV 1 pMpZ’ →=4(0b0–bG)eZV’ at the 13 TeV LHC (right). Settings are speci ed in the text. in the context of charged Higgs boson production at 100 TeV. In ref. [27], the impact of resumming initial-state collinear logarithms in the associated heavy Higgs (MH > 5 TeV) and top pair production (with un-tagged top quarks) is examined and it is found to be very large at large Higgs masses. gure 11 the total cross sections for the production of a Z0 boson of mass 5F and 6F schemes as a function of the renormalisation and factorisation scales, which are identi ed and varied between 0:2MZ0 and 2MZ0 . Results are obtained by using MadGraph5 aMC@NLO for the 5F scheme and a private code for the 6F scheme. Results in orders (NLO). At NNLO the 6F-scheme cross section displays a similar scale dependence as the NLO cross section in the 5F scheme with a residual di erence of about 40% between the two best predictions in the two schemes. To further investigate these di erences, in gure 12 we plot the distribution of the fraction of momentum carried by the top quarks 0.8 pMpZ’→=1(Ttet–)VZ’ at 100 TeV colider ()pb 0.6 σ ito 0.9 a ()pb 0.0008 σ pp → (t t–)Z’ at 100 TeV colider MZ’=5TeV σp 1.5e-05 ( o itaR 1.4 MZ’=10TeV pp collider as functions of k = =MZ0 . Top mass: mt = 173 GeV. Mass of the heavy boson: of the cross sections in the 6F and 5F schemes. bb→ Z’ at a 100TeV pp collider mZ’=1 TeV mZ’=5 TeV NNPDF30 LO nf = 5 ( s(MZ ) = 0:130). are to be associated to the absence of power-like mass terms in the 6F calculation. Conclusions In this work we have considered the use of four- and ve- avour schemes in precision physics at the LHC and in the context of b-initiated Higgs and Z production. We have extended previous work done for processes involving a single b quark in the initial state to cases in which two are present. We have followed a \deconstructing" methodology where the impacts of the various sources of di erences between the schemes have been evaluated one by one. Firstly, we have obtained the form of the collinear logarithms in the four- avour scheme by performing the explicit computation of the 2 ! 3 body scattering process and studying the collinear limit using as natural variables the t-channel invariants. We have then compared the resulting expression with the corresponding cross section in the 5- avour scheme as calculated by only keeping the explicit log in the b-quark PDF, i.e. without resummation. This has allowed us to assess the analytic form and therefore the size of the collinear logarithms and to propose a simple procedure to identify the relevant scales in the processes where the results in the two schemes should be evaluated and compared. In so doing we have considered cases where power-like e ects in the mass of the heavy quarks Secondly, we have explicitly estimated the e ects of the resummation by studying the fully evolved b PDF with truncated expansions at nite order. We have then applied our general approach to the case of Higgs and Z boson production in association with b quarks at the LHC and to heavy Z0 production in association with top quarks at a future 100 TeV collider. We have found that the resummation increases the cross section in most cases by about 20% (sometimes reaching 30%) at the LHC and in general leads to a better precision. On the other hand, the 4F scheme predictions (5F scheme in the case of associated top-quark production) at NLO also display a consistent the heavy quarks in the nal state. Acknowledgments We would like to thank Stefano Forte, Paolo Nason, Alex Mitov and Davide Napoletano for many useful discussions on this topic and for comments on this work. In particular we thank Davide Napoletano for providing the code that we used to check the e ects of the inclusion of higher order logs in the NLO ve- avour scheme cross sections. We thank the Kavli Institute for Theoretical Physics in Santa Barbara for hosting the authors during the completion of this manuscript. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The work of G.R. is supported in part by an Italian PRIN2010 grant. Cross section in the collinear limit In this appendix we illustrate in some detail the calculation of the cross section for the partonic process g(p1) + g(p2) ! b(k1) + b(k2) + H(k) in the limit of collinear emission of b quarks. We choose, as independent kinematic The remaining invariants s^ = (p1 + p2)2 = 2p1p2 t1 = (p1 t2 = (p2 k1)2 = k2)2 = s1 = (k1 + k)2 = 2k1k + mb2 + M H2 s2 = (k2 + k)2 = 2k2k + mb2 + M H2 : u1 = (p1 u2 = (p2 t = (p1 u = (p2 u1 = s1 u2 = s2 s12 = (k1 + k2)2 = 2k1k2 + 2mb2 k2)2 = k1)2 = M H2 = M H2 = are related to the independent invariants by t = u = s12 = s^ s1 s2 + M H2 + 2mb2: The leading-order Feynman diagrams are shown in gure 13. The squared invariant amplitude (averaged over initial state and summed over nal state spin and colour variables) has the general structure jMj2 = G(s; s1; s2; t1; t2) The function G(s; s1; s2; t1; t2) is a polynomial in t1; t2. It can be shown on general well known that collinear singularities do not arise in interference terms among di erent amplitudes. Thus, jMj2 = explicit calculation gives Gt = Gu = 32 s2 2mb2GF M H2 p2 Pqg(z1) Pqg(z2) ; We may now integrate over cos 1 using the delta function Figure 13. Leading order diagrams for gg ! bbH. z1 = z2 = k10 = q10 = jp~1j2 + j~k1j2 2jp~1jjk1j cos 1 + t1 : q10) = and Pqg(z) is de ned in eq. (2.11). The 3-body phase-space invariant measure d 3(p1; p2; k1; k2; k) can be factorised as We now compute each factor explicitly. We have d 2(p1; k1; q1) = d 3(p1; p2; k1; k2; k) = d 2(p1; k1; q1)d 2(p2; k2; q2)d 1(q1; q2; k); q12 = t1; q22 = t2: (2 )32k10 (2 )32k20 (2 )32k0 (2 )4 (p1 + p2 1 j~k1jdj~k1jd'1 d 2(p1; k1; q1) = d 2(p2; k2; q2) = 1 j~k2jdj~k2jd'2 jp~1j2 + j~k1j2 2jp~1jjk1j cos 1 + t1 = 0: d 3(p1; p2; k1; k2; k) = d 1(q1; q2; k): It will be convenient to adopt the centre-of-mass frame, where p1 = (1; 0; 0; 1); p2 = (1; 0; 0; 1) s1 = (k + k1)2 = (p1 + p2 s2 = (k + k2)2 = (p1 + p2 k2)2 = s^ + mb2 k1)2 = s^ + mb2 with 1 a solution of and therefore In this frame and therefore axis to replace and therefore where we have de ned a1 = s2 a2 = s1 Furthermore, we may use the invariance of the cross section upon rotations about the z d'1d'2 ! 2 d'; ' = '1 d 1(q1; q2; k) = 2 d 3(p1; p2; k1; k2; k) = 256 4s^2 ds1ds2dt1dt2 d' (q1 + q2)2 It is a tedious, but straightforward, task to show that, upon integration over the azimuth ' using the delta function, this expression is the same as the one given in [30] for the three-body phase-space measure in terms of four invariants. The two invariants u1; u2 are related to independent invariants through eqs. (A.12), (A.13), which can be written mb2 = (t2 mb2 = (t1 For small m2, ti+ = mb2 + + O(m4); ti = ai + O(m4); i = 1; 2: All the ingredients to compute the total partonic cross section in the collinear limit are now available. In this limit, the relative azimuth between b and b is irrelevant, and simply provides a factor of 2 . Furthermore The bounds for t1 are easily obtained. In the centre-of-mass frame we have 1, cos 2 = 1. We get 4mb2(a1 + s^) a2 + mb2 + cos 2 (a2 + m2)2 b 4mb2(a2 + s^) : t1 = t2 = t1 = t2 = 4mb2(a1 + s^) 4mb2(a2 + s^) : = ^ s2 mb2 GF Pqg(z1) log dz2 (z1z2 Pqg(z2) log and therefore The integrals over t1; t2 are easily computed: s1 = s^z2; s2 = s^z1 ds1 ds2 = dz1 dz2: 1 + O(1) = log m2 = log + O(1) = We nd = (z1z2s^ ^4F;coll(^) = This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. [INSPIRE]. [INSPIRE]. 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