Electroweak and QCD corrections to $Z$-boson production with one $b$ jet in a massive 5 Flavor Scheme

We compute the $O(\alpha_s \alpha^2)$ and $O(\alpha_s^2 \alpha)$ contributions to the production cross section of a $Z$ boson with one $b$ jet at the Large Hadron Collider (LHC), and study their phenomenological relevance for LHC physics. The accurate prediction of hadronic $Z+b$-jet production is needed to control a background that greatly affects both the measurement of Higgs-boson properties and searches of new physics at the LHC. At the same time it could enable the first precise measurement of the $b$-quark parton distribution function. In this context $b$-quark mass effects become relevant and need to be studied with care, both at the level of the hard process and at the level of the initial- and final-state parton evolution. It is the aim of this paper to explore some of these issues in the framework of a massive 5 Flavor Scheme and to assess the need for both the inclusion of electroweak corrections, in addition to QCD corrections, and $b$-quark mass effects in the prediction of total and differential cross sections for hadronic $Z+b$-jet production.


I. INTRODUCTION
The production of a Z boson with one or more b jets plays a very important role in the physics program of the Large Hadron Collider (LHC) both for direct searches of physics beyond the Standard Model (SM) and for precision measurements of SM processes that could reveal deviations induced by new physics beyond the direct reach of the LHC. Dedicated experimental studies from the LHC, which upgraded previous Tevatron results [1,2], have been published during Run I [3][4][5][6][7][8][9], and we look forward to results obtained at higher centerof-mass (c.m.) energy and with much higher statistics during Run II.
In general, signals of physics beyond the SM will likely emerge, if any, from signatures containing heavy SM particles such as Z and W bosons together with t and b quarks. In the specific case of Higgs-boson production, Z + b jets is one of the main background processes for several production modes and the theoretical accuracy of its prediction will affect the precision on measurements of Higgs-boson couplings reached at the LHC.
Besides searches for new physics, the interest in a precision measurement of Z + b-jet production is also motivated by the possibility of obtaining the first direct measurement of the b-quark parton distribution function (PDF). Indeed, if one assumes a non-zero b-quark PDF, i.e. if one works in a 5 Flavor Scheme (5FS), the production of a Z boson with one b jet proceeds dominantly via bg → Zb. If this is very natural at energy scales much larger than m b , it becomes less justified at lower energies where working in a 4 Flavor Scheme (4FS) may also be appropriate, and the channels qq → Zbb and gg → Zbb are the main channels for Z production with both one and two b jets. In the last few years a lot of theoretical activity has gone into clarifying the interplay between the two approaches [10][11][12][13][14][15][16][17][18] (for a review see also Ref. [19]), and times are now mature to develop a dedicated experimental program aimed at the measurement of the b-quark parton density via processes like Z + b jets, or γ + b jets, which provides analogous yet complementary information.
As already proven by the incredibly successful physics program of Run I of the LHC, both the intricacy of new physics searches and the challenge of SM precision measurements require the improvement of the accuracy of theoretical predictions to the percent level. If the knowledge of the first order(s) of QCD corrections was mandatory for Run I, electroweak (EW) corrections will also become important at the energies of Run II. At the same time, a more accurate assessment of the theoretical uncertainties intrinsic to sophisticated Monte Carlo tools used to match higher-order QCD/EW calculations to parton-shower generators is clearly needed. Among others, the question of properly including heavy-quark mass effects (b-quark mass effects in our case), both at the level of the hard-scattering matrix element and at the level of the PDF, should be carefully considered, in particular for bquark initiated processes. Given its relevance for the physics of Run II, Z + b-jet production offers a particularly interesting prototype case to be considered in this context.
With this in mind, we consider in this paper the production of a Z boson with one b jet as generated at Leading Order (LO) via the tree-level O(α s α) process bg → Zb. The first order of QCD corrections, i.e. the O(α s ) or Next-to-Leading Order (NLO) QCD corrections to this process, have been calculated for the first time in Ref. [20] and implemented in MCFM [21], assuming a massless b quark. To improve the precision of the NLO QCD theoretical predictions one could either add the second order (i.e. the next-to-next-to-leading order or NNLO) QCD corrections or the first order (i.e. the NLO) EW corrections. Adding NNLO QCD corrections should certainly stabilize the cross section by reducing the dependence on the renormalization and factorization scales, while adding NLO EW corrections could add a few percent to total rates and have a visible impact on high-energy tails of distributions.
NNLO QCD corrections to Z+jets have been presented in Refs. [22,23] and, with due care, they could be used in the future to extract the NNLO QCD prediction for Z + b-jet for the case of a massless b quark. Obtaining NNLO QCD corrections for a massive b quark is a more challenging endeavor and should be considered only after the inclusion of massive initial-state partons, some issues of which we discuss in this paper. NLO EW corrections to hadronic Z + j production have been presented in Refs. [24][25][26][27], and combined NLO QCD+EW corrections in Ref. [28]. The first order of EW corrections to Z + b-jet production via bg → Zb, i.e. the O(α s α 2 ) term in the perturbative expansion of the cross section for Z + b-jet production, is the main subject of this paper, where we also study the effect of considering an initial-state massive b quark and discuss the interplay with the definition of the corresponding b-quark PDF. This can be considered as the first building block of a more general program that will have to connect both 4FS and 5FS calculations, including both QCD and EW corrections. Since the EW corrections to bg → Zb represent a well defined set of corrections, in a well defined flavor scheme, this calculation allows to estimate in a consistent way the impact of EW corrections on Z + b-jet production through the process that also most affects the measurement of the b-quark PDF. Furthermore, working with the idea of implementing a 5FS calculation in a parton-shower Monte Carlo event generator, we treat the b quark as massive also in the initial state, since this is necessary in order to properly implement the backward evolution of final-state massive b quarks. Indeed, if it is customary to treat initial-state b quarks as massless (traditional 5FS), this is nevertheless just a simplification of the calculation, not a requirement [29][30][31].
More to the point, this approximation does not lend itself well to the implementation of methods, like phase-space slicing methods or subtraction methods, which evaluate higher-order real radiative corrections with the help of auxiliary terms that algorithmically approximate the real-emission contributions in the soft/collinear regions starting from the corresponding Born processes. For example, a real emission subtraction term to the real-emission process gg → Zbb, with massive b-quarks, cannot be generated in a kinematically consistent way from the corresponding Born process bg → Zb, with massless b-quarks, by convoluting with the splitting function for g → bb. Since it is, for example, at the core of the implementation of 5FS processes in Monte Carlo event generators that match NLO cross sections to parton showers, this issue has recently attracted some attention and studies aimed at introducing what has been dubbed as massive 5FS (m5FS 1 ) have appeared [32]. Hence, in view of future developments in Monte Carlo event generators, we consistently develop the calculation of both the first order of QCD and EW corrections with a massive b quark. This incidentally also implies that we extend the existing NLO QCD calculation [20] to the case of a massive b quark.
In this paper, both QCD and EW virtual corrections have been obtained through the NLOX one-loop provider [33], as well as by independent in-house codes. The corresponding real corrections have been computed, in both the massless and massive cases, via independent codes using phase-space slicing, as well as an implementation of the dipole subtraction method based on the formalism of Ref. [34], extended to QED radiation off massive quarks (in both initial and final state). More details will be presented in Section II. Having obtained both the O(α s ) and the O(α) corrections to bg → Zb including full b-quark mass effects, we can assess: 1) the impact of mass effects on the fixed-order total cross section and distributions by comparing NLO QCD cross sections with massless and massive b quarks, 1 So far we have used 5FS to denote a generic scheme with 5 active flavors. In the following we will use 5FS to denote the case in which the b quark is considered massless, and m5FS to denote the case in which the b quark is considered massive. and 2) the relative impact of QCD and EW corrections on fixed-order total cross sections and distributions by comparing the O(α 2 s α) and O(α s α 2 ) cross sections with massive b quarks. Indeed, independently of the necessity of introducing massive initial-state partons for automated implementations of NLO QCD and EW corrections, it is also clear that mass effects and EW corrections can be of the same order, although they typically affect physical observables in different kinematical regions. Even if small, they both need to be accounted for when one aims for percent-level precision predictions over a broad kinematical range.
In the remainder of this paper we systematically review in Section II the relevant technical details of the calculation and present results in Section III where we also assess the impact of such corrections and discuss the need for future improvements. Conclusions and suggestions for future developments are presented in Section IV.

II. DETAILS OF THE CALCULATION
We write the hadronic cross section, σ, for Z + b-jet production at the LHC as follows: where i and j, with i + j ≥ 2 and j ≥ 1, refer to the coupling order of the partonic cross section, and σ (i,j) denotes the term in the perturbative expansion of the cross section that is proportional to α i s α j (where α s = g 2 s /(4π) and α = e 2 /(4π), g s and e being the QCD and QED coupling constants, respectively). In a scheme with 5 active flavors (such as the 5FS or m5FS), the set of "lowest-order" contributions with i + j = 2 consists of all sets of tree-level diagrams that satisfy such relation, i.e. σ (1,1) corresponds to the tree-level contributions to bg → bZ and σ (0,2) to the tree-level contributions to bγ → bZ 2 . It is clear that the calculation of the cross section also includes the correspondingbg-initiated andbγ-initiated processes, even if it is not explicitly repeated throughout the paper. Both processes consist of s-and t-channel contributions, as illustrated in Fig. 1 for the case of bg → Zb (the corresponding diagrams for bγ → Zb are obtained by replacing a gluon with a photon in both diagrams). 2 Note that the set of all possible "lowest-order" contributions is made of contributions of different couplingpower combinations and that those also arise from different sets of initial-state particles. The same holds for the set of all possible "next-to-lowest-order" contributions.
FIG. 2: Coupling-power flow chart for Z + b-jet production at the level of the partonic cross section. The notation (i, j) corresponds to the coupling-power combination α i s α j . From left to right we have increasing i / decreasing j in steps of 1. From top to bottom the total order i + j increases in steps of 1. The upper row depicts all possible coupling-power combinations for the "lowest-order" contributions, with i+j = 2, while the lower row depicts all possible coupling-power combinations for the higher-order corrections of one order higher, with i + j = 3. Note that the (1, 2) contribution is depicted twice, as it originates from two different "lowest-order" contributions with different sets of initial-state particles.
The set of "next-to-lowest-order" contributions consists of the O(α s ) and O(α) corrections to the set of "lowest-order" contributions, as depicted in Fig. 2. The O(α 2 s α) term, σ (2,1) , corresponds to the O(α s ) corrections to σ (1,1) . It was first computed in Ref. [20] for massless b quarks, and the corresponding calculation is implemented in MCFM [21]. We have reproduced it independently in this paper, where we also extend it to the case of a massive b quark in order to consistently compare the impact of NLO QCD and EW corrections, and in order to assess non-zero b-quark mass effects by comparing massive and massless NLO QCD results.
The O(α s α 2 ) term, σ (1,2) , is also presented for the first time in this paper and is indeed the main focus of our study. Such term originates from both the O(α) corrections to σ (1,1) (bg → Zb) and the O(α s ) corrections to σ (0,2) (bγ → Zb), as depicted in Fig. 2. It is entirely dominated by the O(α) corrections to the bg → Zb tree-level process, since the cross section for bγ → Zb is much smaller due to the smallness of the photon parton density in the initial-state protons 3 . Hence in our study we will only consider the O(α) corrections to bg → Zb.
Also the O(α 3 ) term, σ (0, 3) , is entirely negligible and will not be considered here. We will thus from here on simply speak of σ (1,1) as the LO contribution, and define and of σ (2,1) and σ (1,2) as the O(α s ) and O(α) corrections, or simply NLO QCD and EW corrections, respectively, and define the corresponding NLO QCD and NLO EW cross sections In order to implement both the NLO EW and NLO QCD cross sections in the m5FS, the full b-quark mass dependence has been retained both in the hard-scattering matrix elements and in the phase-space integration. We therefore have defined the initial-state parton-level kinematics of b(p b ) + g(p g ) → Zb as follows [35]: where √ s is the hadronic c.m. energy, x 1 and x 2 are the longitudinal fractions of the corresponding hadron momenta carried by each parton. Their ranges assure compatibility with the parton-level kinematics in the hadronic c.m. frame, where we impose that m b ≤ Of course, the symmetric case in which x 1 and x 2 are exchanged is also considered, and, due to the massiveness of the initial-state b quark, care must be taken to keep the two kinematic cases separate when convoluting the partonic cross section with the PDF to obtain the hadronic cross section. In the case of bb → Zbb with an initial-state 3 Using the setup described in Section III, one finds that σ LO = α s ασ (1,1) 376 pb, while the cross section for the "lowest-order" photon-induced process is three orders of magnitude smaller, α 2 σ (0, 2) 0.1 pb, and one order of magnitude smaller than α s α 2 σ (1,2) 5 pb. massive b, the initial-state kinematic configuration becomes [35]: In the 5FS, which assumes a massless b quark, the definition of the parton-level kinematics for bg → Zb is the same as for massless partons (obtained by setting m b = 0 in Eqs. (4) and (5)).
In the following we will review the essential features of the O(α s ) and O(α) virtual and real corrections in Sections II A and II B, respectively. In particular, since considering a massive b quark consistently in all parts of our calculation introduces some novel aspects in the calculation apart from the difference in beam kinematics, we describe in detail in Section II A 2 the gluon wave-function and α s renormalization scheme and in Section II C the choice of PDF subtraction terms.

A. Virtual corrections with NLOX
The virtual O(α s ) and O(α) corrections to σ LO , contributing to σ QCD NLO and σ EW NLO respectively, have been produced with the one-loop provider NLOX [33]. They have also been crosschecked against several independent in-house codes and the one-loop provider Recola NLOX is a new program for the automated computation of one-loop QCD and EW corrections in the Standard Model. A non-public predecessor of NLOX has been available in the past, to calculate one-loop QCD corrections to selected processes [38,39]. NLOX has seen quite some progress in recent years, and most recently partook in a technical comparison on NLO EW automation [40]. The current version of the program provides fully renormalized QCD and EW one-loop corrections in the Standard Model, for all the possible QCD+EW mixed coupling-power combinations to a certain parton-level process up to one-loop accuracy, in- 4 The O(α) and O(α s ) corrections at the amplitude-squared level have been provided by NLOX and crosschecked on the phase-space point level against in-house codes and Recola. At the cross-section level NLOX and Recola have been used with a dedicated interface to the Cuba library [37], in order to confirm the virtual cross-section numbers from interfacing NLOX and the in-house codes to in-house Monte Carlo integration routines.
cluding the full mass dependence on initial-and final-state particle masses. NLOX has been interfaced to a selection of in-house Monte Carlo integration routines. A more general interface to publicly available Monte Carlo integrators and event generators is currently under development. In the following we will only give a brief description of NLOX. More details will soon be available in Ref. [33], which will go along with a public release of the code.

NLOX
NLOX is based on a traditional approach of Feynman diagrams, utilizing QGRAF [41], FORM [42], and Python to algebraically generate C++ code for the virtual QCD and EW one-loop contributions to a certain process in terms of one-loop tensor-integral coefficients. The tensor-integral coefficients are calculated recursively at runtime through standard reduction methods by the C++ library tred, an integral part of NLOX. Several reduction techniques are available to tred, many of which are found in Refs. [43,44]. The scalar one-loop integrals are evaluated by either using OneLOop [45], or QCDLoop [46,47].
In NLOX UV and IR singularities are regularized in d-dimensional regularization (with d = 4 − 2 , | | 1). UV singularities are renormalized, while IR singularities are reported in terms of the Laurent coefficients of the corresponding 1/ 2 and 1/ poles. The UV renormalization in NLOX is carried out by means of counterterm diagrams, which provides a flexible way to systematically include mass renormalization for massive propagators as well as Yukawa-type vertices.
The renormalization constants in terms of which the EW UV counterterms are formulated are derived in the on-shell renormalization scheme, as described in Ref. [48], or in the complex-mass scheme 5 , as described in Ref. [49]. As EW input scheme choices NLOX provides both the α(0) and the G µ EW input schemes [48,[50][51][52]. Per default the α(0) EW input scheme is used. The renormalization constants in terms of which the QCD UV counterterms are formulated are derived in a mixed renormalization scheme: the on-shell renormalization scheme is used for the wave-function and mass renormalization of massive quarks, while the MS renormalization scheme is used for massless quarks and gluons, where, however, in the latter case heavy-quark-loop contributions are decoupled by subtracting them at zero momentum, considering the CWZ prescription [53]. We provide a detailed description of the gluon wave-function and α s renormalization in NLOX, in general and for the special case of a consistent implementation of the m5FS, in Section II A 2.
Finally, for what pertains to IR divergences, the virtual IR poles are collected and canceled against the corresponding IR poles arising from the O(α s ) and O(α) real corrections and from the PDF subtraction terms. More details on the implementation of the PDF subtraction terms for the specific case of the bg → Zb process will be discussed in Section II C.

Gluon wave-function and α s renormalization in NLOX
The gluon wave-function renormalization constant, δZ G , that is used in NLOX is given by (dropping O( ) terms as done in all equations in this section, and setting the 't Hooft-mass in dimensional regularization equal to the renormalization scale µ r ) The coupling renormalization constant, δZ gs , that is used in NLOX is consequently given by The renormalized gluon self-energy at zero momentum is in NLOX thus given by where we keep the closed-quark-loop contributions in the analytic derivation of the unrenormalized gluon self-energy ({Q }, n hf and n lf ) separate from their corresponding counterparts ({Q}, n hf and n lf ) in the gluon wave-function renormalization constant. We see that for {Q} = {Q }, n hf = n hf , and n lf = n lf the above expression simplifies, such that the contributions of closed heavy-quark loops vanish entirely and the renormalized gluon self-energy at zero momentum is then given by Furthermore the gluon wave-function residue,R G , is in NLOX thus given by where, in order to fix the residues of the external gluon wave-functions a term (1/2)δR G {Q }, {Q} times the corresponding Born contribution has to be considered for each external gluon. We see again that for {Q} = {Q }, n hf = n hf , and n lf = n lf the above expression simplifies, such that the contributions of closed heavy-quark loops vanish entirely and δR G is then given by to the partonic m5FS cross section,σ QCD NLO , as follows: where we have used that n lf + n hf = n lf + n hf , and whereσ LO denotes the tree-level partonic cross section for bg → Zb (c.f. Eq. (2); c.f. Eqs. (20) and (21)).
The particular expressions that we use for δR G , δZ G , and δZ gs in NLOX in case of the m5FS are thereby given by Z + b-jet production depends on the experimental signature selected. An example of a set of analysis cuts in Z +j production which warrants the inclusion of real EW gauge-boson emission in addition to virtual EW corrections can be found in Ref. [55], where the well-known incomplete cancellation of EW Sudakov logarithms between these two contributions [56,57] is also discussed. Since our study is not aiming at a detailed analysis of this effect, which should only be done in collaboration with the experimentalists performing the measurement of Z + b-jet production, we do not consider the real emission of an extra Z/W/H. On the other hand, we consider the QED part of the real radiation (photon emission) which needs to be consistently included in order to cancel the IR divergences present in the virtual cross section (due to photon exchange).
The calculation of the O(α) real-photon emission cross section can be easily implemented using PSS [54], and actually involves just a subset of the singularities encountered in the QCD case for massive b quarks. With due differences, the result can be easily obtained from there. On the other hand, the QED case provides an interesting testing ground for the implementation of the dipole subtraction (DS) method with initial-state massive dipoles, whose working knowledge, in the QCD as well as QED cases, is a necessary step towards the proper implementation of processes with initial-state massive partons in NLO event generators. All dipoles for QED radiation off leptons, including initial-state massive dipoles, have been calculated in [34], where they have also been tested against phase-space slicing in specific QED processes. QED massive dipoles have also been used and tested against PSS in [58] for hadronic W +jet production. In the specific case of photon emission from the b-quark lines of bg → Zb we have implemented the dipole subtraction terms corresponding to the configurations of massive initial-state emitter and massive final-state spectator (both b quarks), and vice versa. For illustration, a summary of the comparison between the two methods is given in Table I, where we report the total NLO EW cross section calculated using either PSS or DS. In the case of PSS the result depends on the soft cutoff δ s and we report in Table I the values of σ EW NLO for four decreasing values of δ s , to show the existence of a plateau region in which σ EW NLO is actually independent of δ s . We notice that the results reported in Table I have been obtained without imposing a recombination cut, and therefore slightly differ from what is included in the final results presented in Section III. Indeed, Table I shows that results obtained using the PSS method with a soft cutoff δ s varying between 10 −3 and 10 −5 are compatible with results obtained using the DS method based on Ref. [34], to which we refer for more technical details. If the successful implementation of the PSS method depends on a careful study of the analytical dependence on the soft and collinear cutoffs, obtaining meaningful numerical results using the dipole method involves its own subtleties. Hence, the cross check between the two methods is all but trivial, and allows us to move forward to further extensions and future studies with more confidence.
We notice that initial-state massive QCD dipoles have first been studied in Ref. [59], where however very little detail is provided, while a more detailed discussion has recently appeared  in Ref. [32].

C. PDF subtraction terms
After combining real and virtual NLO QCD or NLO EW corrections, residual poles in dimensional regularization, due to collinear radiation off massless initial-state partons, are absorbed into NLO PDF via MS factorization. This is systematically achieved by defining PDF subtraction terms, aka collinear counterterms.
The corresponding treatment in the case of massive initial-state quarks needs a more detailed discussion. Indeed, in both NLO QCD and NLO EW calculations with a massive b quark, the b-quark mass acts as a physical regulator for collinear singularities in b → b(g, γ) and (g, γ) → bb splittings. In particular, radiative corrections involving a massive initialstate b quark do not contain collinear poles in dimensional regularization, but logarithms of the b-quark mass instead, which, being finite, do not necessarily need to be absorbed into the b-quark PDF. However, since at LHC energies the mass of the b quark is relatively small, these mass logarithms can be large and retaining them in the calculation of the partonic cross section, i.e. not absorbing them into the b-quark PDF, can lead to unnaturally large corrections that could eventually affect the numerical stability of the hadronic cross section.
In a fully massive calculation the numerical stability of the hadronic cross section is retained by absorbing these mass logarithms via e.g. the generalized MS scheme of Collins, aka the ACOT scheme [29,30] (see also Ref. [60]) 6 . This is achieved by defining corresponding 6 For an overview on the ACOT scheme, as well as its applicability over a range of scales and its convergence to the MS scheme in the massless limit, see, e.g., Refs. [51,[60][61][62]. For a review and a comparison of the heavy-quark schemes based on ACOT, TR, and FONLL used respectively in the CTEQ, MSTW (now PDF subtraction terms in the case of near-collinear emission off massive b quarks, quite in analogy to subtracting poles in dimensional regularization in the case of collinear emission off massless b quarks, such that MS factorization is retained in the massless limit. In obtaining the 5FS NLO QCD results presented in Section III we have used the S-ACOT scheme (or simplified ACOT scheme) [31], where the b quark is considered to be massless everywhere, except in the calculation of those partonic sub-processes that only involve b quarks in the final state. As such, in this scheme the b-quark mass is used as a regulator for the g → bb splitting, while the initial-state b → bg splitting is treated as for all massless quarks (namely, initial-state collinear poles are absorbed in the corresponding PDF, by PDF subtraction terms, in MS factorization). The S-ACOT scheme is what is usually assumed in implementing a traditional 5FS (as also done in Ref. [20]), and is also what is assumed in defining the corresponding PDF in recent CTEQ PDF sets, including the CT14qed PDF set that we use for our study 7 .
On the other hand, in obtaining NLO QCD and NLO EW results in the m5FS we need to implement a fully massive factorization scheme like the ACOT scheme, where the b quark is considered massive everywhere. In this case, mass logarithms originating from both g → bb (or γ → bb, in case initial-state photons are considered) and b → bg (or b → bγ) splittings are absorbed into the b-quark PDF, and all partonic sub-processes are calculated considering a massive b quark 8 . In accordance with this, in our m5FS calculation of O(α) and O(α s ) corrections to bg → Zb, we subtract from the partonic cross section all the mass logarithms arising from both initial-state g → bb and b → bg (or b → bγ) splittings (see Eqs. 17 and 19).
In our study we use the b-quark PDF of the CT14qed PDF set [67]. For the best accuracy, however, a dedicated PDF set should be determined in the ACOT scheme, which is beyond the scope of this study 9 .
MMHT) and NNPDF PDF sets, see, e.g., Ref. [63]. A thorough discussion of the case of charm-quark initiated processes in the FONLL scheme can be found for instance in Refs. [64][65][66]. 7 The CT14qed PDF set [67] is determined in the same CTEQ global analysis in which the CT14 PDF set [68] is determined, using the S-ACOT(χ) scheme [61], but including QED in the DGLAP evolution of the PDF. 8 Although one could use the S-ACOT scheme, which agrees with the ACOT scheme in the high-energy limit, for best accuracy and a fully consistent treatment of massive initial-state quarks, including the possibility of constraining an instrinsic b-quark parton density, the use of a scheme like the ACOT scheme should be preferred. For an overview on ACOT vs. S-ACOT see, e.g., Refs. [51,[60][61][62]. 9 The study presented in Ref. [32] pursues the direction of determining dedicated PDF sets in a fully massive As a reminder on how the PDF subtraction terms enter our calculation, in a condensed notation, the NLO cross section, using a PSS method, can be sketched as follows: where σ V denotes the set of UV renormalized virtual contributions and σ R the set of realemission contributions. In the real-emission contributions, soft and collinear photon/parton emissions are treated using a PSS method with a soft (δ s ) and a collinear (δ c ) cutoff, as defined at the beginning of Section II B, separating the real-emission contributions into soft/collinear (σ soft/coll ) and hard/non-collinear (σ hard/non−coll ) contributions. In the following, we will collect the expressions for the PDF subtraction terms (σ sub ), which we used for our calculation in the 5FS and m5FS for the various cases of initial-state splittings. We will not repeat the details of their derivation, as these can be found in the literature (see, e.g., Refs. [30,31,51,54,60,62,[69][70][71]). We will set the renormalization scale µ r equal to the factorization scale µ f , as µ r = µ f = µ (and do not distinguish between the QED and QCD factorization scales).
As discussed above, in both the 5FS and m5FS scheme, the b-quark mass appears as physical regulator of the collinear singularity in the g → bb splitting, and the corresponding subtraction term reads [30,60,62] σ g→bb whereσ LO =σ LO (x 1 , x 2 , µ) denotes the tree-level partonic cross section for bg → Zb, and the last line takes into account that an analogous term where the role of the two gluons is exchanged needs to be included. Here and in the following it is also understood that the analogous subtraction terms forbg → Zb are considered.
In contrast, the collinear b → bg (and b → bγ) initial-state splittings occurring in the O(α s ) (and O(α)) corrections to the bg-initiated processes are treated quite differently in scheme, also focusing on the definition of a m5FS.
the traditional 5FS and a m5FS. In a 5FS NLO calculation (with a massless initial-state b quark) the corresponding PDF subtraction term reads [54] S −1 σ b→b(γ,g) sub Here and in the following the last line takes into account that an analogous term where b and g come from the respectively opposite hadrons needs to be included. Note that, performing the dz integration in Eq. (18), only subleading terms in δ s remain, which vanish in the limit δ s → 0 (this is also true for Eqs. (19), (20) and (21) in the remainder of this section). On the other hand, in the m5FS (with a massive initial-state b quark) the PDF subtraction term reads [51,60,62,[69][70][71] σ b→b(γ,g) sub 1−δs Finally, in the m5FS NLO QCD calculation the subtraction term for the g → gg splitting is also affected by the b-quark mass as follows [71]: 1−δs where C g = C A = N (with N = 3), and n lf = 4 (see also Section II A 2), whereas in the 5FS the same subtraction term reduces to [54] S −1 σ g→gg 1−δs with n lf = 5.

III. NUMERICAL RESULTS
In this section we present results for total cross sections and distributions of inclusive are chosen in such a way that the value for the hadronic five-flavor contribution to the photon vacuum polarization, ∆α (5) had (M 2 Z ) = 0.027572 [73], as derived from low-energy e + e − data with the help of dispersion relations, is recovered. In the spirit of avoiding large NLO corrections, these universal corrections resulting from EW renormalization can be absorbed into the corresponding lowest-order prediction using specific EW input schemes [48,[50][51][52].
For instance, in the so-called α(M Z )-scheme the dependence on the light-quark masses, which originates from the renormalization constant for the electric charge, δZ e , is absorbed in the running of α when replacing the fine-structure constant α(0) by α(M Z ) = α(0)/(1 − ∆α) with ∆α = −4παRe(Π γ (M Z ) − Π γ (0)). Here we provide results for NLO cross sections calculated in the so-called G µ -scheme according to which in the Born-level couplings α(0) is replaced by: where ∆r [74] describes the EW one-loop corrections to muon decay and includes ∆α(M Z ) as well as corrections to the ρ parameter. For the input parameters given in Eq. (22) we find ∆r = 0.02968. Note that ∆r = 0 in the calculation of the NLO QCD cross sections.
The G µ scheme indeed lends itself better to processes that involve not only photons but also weak gauge bosons. Note that the relative EW corrections are still evaluated with α(0).
Consequently, the NLO cross section in the G µ scheme can be derived from the NLO cross section in the α(0) scheme as follows: In this way, the logarithmic dependence on the light-quark masses due to δZ e in the EW one-loop corrections is completely canceled by the corresponding contribution in ∆r.
Since we include both QCD and QED corrections, we choose to use the CT14qed PDF set [67] with α s (M Z ) = 0.118. We do not include in our study an estimate of the current PDF uncertainty, since, for b-quark initiated processes, this will become more meaningful when a complete implementation of the m5FS is included in the available PDF sets. Likewise, we do currently also not consider uncertainties from varying the b-quark mass. On the other hand, the choice of a PDF set which includes QED radiation is necessary when considering EW corrections. In both NLO and LO results we use NLO PDF and the two-loop running of α s with n lf = 5 flavors (c.f. Eq. (6)). We choose as central renormalization and factorization scale µ = µ r = µ f = M Z and quantify the uncertainty from scale variation by varying µ by a factor of two above and below the central value. We notice that the scale uncertainty through µ r variation is entirely driven by QCD effects.
Since we consider inclusive Z + b-jet production, we include Z + b + X and Z +b + X    In Table II while in the multiplicative approach the NLO QCD result is multiplied by the EW K-factor (K EW ) [75]: hence including mixed QCD×EW corrections that are of higher order. As a first estimate of the overall effect of each kind of NLO corrections and their combined effect on the LO cross section, in the second row of Table II we  With this in mind, in the following we will quantify the impact of EW corrections in terms of the following ratios: and which correspond to the additive and multiplicative approach of combining NLO QCD and EW cross sections respectively. these effects may be less pronounced when the experimental analysis requires the inclusion of real EW gauge-boson radiation. In the low-p T region the impact of EW corrections is completely within the scale uncertainty of the differential NLO QCD cross section. In the case of pseudorapidity distributions, both the additive and multiplicative combination of QCD and EW NLO corrections show that EW corrections could lead to visible deviations from the pure NLO QCD distributions. For η(Z) this happens most notably in the regions around ±3.2, while for η(b jet) effects can be seen throughout the whole η range.
Following the same logic, we quantify the effect of switching from a massless 5FS to a m5FS in terms of the ratio representing the fractional change in the NLO QCD cross sections in going from the 5FS to the m5FS. This ratio is given in the lower plots of Figs retaining the b quark initial-state mass dependence.

IV. CONCLUSIONS
In this paper we have considered the production of a Z boson with at least one b jet as a phenomenologically interesting testing ground to assess the impact of finite initial-state b-quark masses, and to evaluate the relevance of a still missing piece of NLO corrections, the combined QCD and EW NLO corrections. We have consistently implemented the initialstate massive kinematics and the corresponding matching with existing PDF, setting the stage for a consistent development of a massive 5FS, which should provide the correct interface between fixed-order NLO calculations and parton-shower Monte Carlo event generators. We have presented the first calculation of the first-order EW corrections to bg → Zb, using the NLOX one-loop provider, and implementing real photon-emission corrections using both a phase-space slicing method and dipole subtraction with massive initial-state dipoles. EW and QCD NLO corrections have been combined using both an additive and multiplicative approach.
Both mass effects and NLO EW corrections are small effects, compared to the size of NLO QCD corrections, and are mostly within the uncertainty of the NLO QCD cross section.
Still, there are clear indications of their impact, in particular on angular distributions, and in complementary kinematic regions: EW corrections mainly affect the high-p T region of both b-jet and Z boson p T , while b-quark mass effects are more visible in the low-p T regions. Interestingly, they both tend to be more visible in angular distributions, such as pseudorapidity distributions.
Given the phenomenological relevance of Z + b-jet production, both as background to Higgs-boson precision measurements and new physics searches, as well as a potential candidate for a direct precision measurement of the b-quark PDF, we should aim to reduce the theoretical uncertainty in the future. In this respect, it is clear that the inclusion of NNLO QCD corrections will greatly help to mitigate unphysical scale dependencies. At the same time, the consistent inclusion of b-quark mass effects in existing PDF sets, where the b-quark mass only plays the role of an IR regulator so far, will be needed.