Bottom-flavored inclusive emissions in the variable-flavor number scheme: A high-energy analysis

We propose the inclusive semi-hard production, in proton-proton collisions, of two bottom-flavored hadrons, as well as of a single bottom-flavored hadron accompanied by a light jet, as novel channels for the manifestation of stabilization effects of the high-energy resummation under next-to-leading-order corrections. Our formalism relies on a hybrid high-energy and collinear factorization, where the BFKL resummation of leading and next-to-leading energy logarithms is used together with collinear factorization. We present results for cross sections and azimuthal correlations differential in rapidity, which are widely recognized as standard observables where to hunt for distinctive signals of the BFKL dynamics. We propose the study of double differential distributions in the transverse momenta of final-state particles as a common basis to investigate the interplay of different kinds of resummation mechanisms.


I. INTRODUCTORY REMARKS
Heavy-flavor physics is unanimously recognized as one of the most fertile grounds where to investigate modern particle physics. On one Special attention is deserved by the production in hadronic collisions of the heaviest quark species that can hadronize, the bottom one. The standard collinear description of the bb production at next-to-leading order (NLO) was set up long time ago [1][2][3], but only recently fully differential distributions were investigated with nextto-NLO perturbative accuracy [4]. is adequate (for more details see, e.g., Ref. [5] and references therein). In the FFNS only light- be resummed to all orders [6,7]. In the latter case, the so-called zero-mass variable-flavor num-  [11][12][13][14][15], and for a detailed discussion we refer the reader to Ref. [16] (see also Refs. [8,9]).
In Refs. [17,18] inclusive bottom-jet emissions in central-rapidity regions were investigated under the hypothesis of t-channel exchanges of gluon and quark Reggeons at high energies.
These studies were subsequently extended to bottomed bound states [19][20][21]. In Ref. [22] the kinematic correlations of lepton pairs from semileptonic decays of charmed and bottomed mesons were discussed. In Ref. [23] the weight of double-parton scattering effects was assessed in the hadroproduction of a D 0 B + system and of two B + mesons at the LHC.
The B-meson VFNS collinear fragmentation function (FF) was first extracted at NLO in Ref. [24] from a fit to e + e − data elaborated by the CERN LEP1 Collaboration. Then, the parametrization obtained in Ref. [25] via a fit to CERN-LEP1 and SLAC-SLC data was used to calculate the NLO cross section for the inclusive production of B mesons in pp collisions and in the GM-VFNS, namely by taking into account finite-mass effects of the bottom quark [26].
In Ref. [27] the hadroproduction of bottomflavored hadrons (B mesons and Λ b baryons, comprehensively indicated as b-hadrons) was investigated at LHC energies and compared with CMS and LHCb data. This study was performed under the assumption that a unique FF can be adopted to describe the fragmentation of partons to all b-hadrons species. Thus, the FF set for a given species could be obtained from the global one by simply multiplying the latter by a branching fragmentation fraction, which does not depend on energy. Analyses done by the Heavy Flavor Averaging Group (HFAG) [28] have shown how the universality assumption on the branching fraction is violated by LEP and Tevatron data for Λ b emissions, while its safety is corroborated for B-meson detections. A recent study on transverse-momentum distributions for the inclusive Λ b production at CMS and LHCb [29] has pointed out that the branchingfraction picture needs to be improved in the large p T -regime, and future data with reduced experimental uncertainties are expected to better clarify the situation.
The NLO fragmentation of c andb quarks to B ( * ) c mesons was studied in Ref. [30], while the first determination of a next-to-NLO b-hadron FF via a fit to e + e − annihilation data from CERN LEP1 and SLAC SLC was presented in Ref. [31]. Energy and angular distributions for b-hadrons production from semi-leptonic decays of top quarks were analyzed in Refs. [32,33].
Apart from direct-production channels, bquark emissions are employed to identify top particles and to study their properties. Thus, the bquark fragmentation mechanisms is expected to have a relevant phenomenological impact on top physics. The same formalism can be applied in electroweak precision studies to describe photon radiation from massive charged fermions, such as a Higgs-boson detection via the bb decays [34].
The role of the b-quark in the associated production of a lepton pair was discussed in Refs. [35][36][37]. The treatment of electroweak radiation from heavy fermions in the context of W -boson production with Monte Carlo generators was extensively investigated in Ref. [38].
The picture described above is still incomplete if we approach particular kinematic regions where the perturbative series is poorly convergent. A prominent example is represented by the Sudakov region, where the ratio x S between the transverse momentum of the detected particle and center-of-mass energy approaches one. Here, soft-gluon radiation produces contributions proportional to α n s ln m (1 − x S )/(1 − x S ), with m ≤ 2n − 1, which must be resummed [6,39]. This is equivalent to saying that the "true" expansion parameter is α s ln 2 (1 − x S ) instead of α s .
A similar issue arises when one approaches the so-called semi-hard region of QCD (see Section II A for further details), namely where the scale hierarchy s {Q 2 } Λ 2 QCD (s is the squared center-of-mass energy, {Q 2 } one or a set of squared hard scales given by the process kinematics, and Λ QCD the QCD mass scale) stringently holds 1 . The possibility of entering this two-scale regime via the heavy-flavor production was highlighted many years ago, when the socalled high-energy factorization (HEF) was proposed [40][41][42]. Here, m b plays the role of hard scale.
In this paper we investigate the inclusive semihard emission at the LHC of a b-hadron accompanied by another b-hadron or by a light jet, as a testfield for the manifestation of imprints of the QCD high-energy dynamics. We build predictions for distributions differential in rapidity, azimuthal angles and observed transverse momenta, calculated at the hand of a hybrid factorization that combines the Balitsky-Fadin-Kuraev-Lipatov (BFKL) resummation [43][44][45][46] of leading and next-to-leading energy logarithms with collinear PDFs and FFs. We hunt for stabilizing effects of the high-energy series under higher-order corrections and energy-scale variation, that, if confirmed, would pave the way toward prospective studies where the use of our hybrid factorization could serve as an important tool to improve precision calculations of observables sensitive to bottom-flavored bound-state emissions.

PRODUCTION
In this Section we give theoretical key ingredients to build our observables. After a brief overview on recent progresses in the phenomenology of the semi-hard sector (Section II A), we provide with analytic expressions of azimuthal-angle coefficients for our processes (see Fig. 1 heavy-light dijet system [111,112] and forward Drell-Yan dilepton production with a possible backward-jet tag [113]. On the other side, the study of single forward emissions offers us the possibility to probe the proton content via an unintegrated gluon distribution (UGD), whose evolution in the struckgluon longitudinal fraction x is driven by BFKL.
Probe candidates of the UGD are: the exclusive light vector-meson electroproduction [114][115][116][117][118][119][120][121][122][123], the exclusive quarkonium photoproduction [124][125][126], and the inclusive tag of Drell-Yan pairs in forward directions [127][128][129][130]. The information on the gluon content at small-x embodied in the UGD turned out to be relevant in the improvement of the collinear description via a first determination of small-x resummed PDFs [131][132][133], as well as in a model calculation of small- x transverse-momentum-dependent gluon densities (TMDs) [134][135][136]. Studies on the interplay between BFKL dynamics and TMD factorization were recently made in Refs. [137,138]. Mueller-Navelet reaction, cannot be studied at "natural" scales [86,88,139]. A general procedure that allows us to "optimize" scales in semihard final states was built up in Ref. [89]. It relies on the so-called Brodsky-Lepage-Mackenzie (BLM) method [140][141][142][143], which prescribes that the optimal scale value is the one that cancels the non-conformal β 0 -terms in the considered observable. Although the application of the BLM method led to a significant improvement of the agreement between predictions for azimuthal correlations of the two Mueller-Navelet jets and CMS data [144], the scale values found, much higher than the natural ones, generally bring to a substantial reduction of cross sections (observed for the first time in inclusive light charged dihadron emissions [96,98]). This issue clearly hampers any possibility of doing precision studies.
First, successful attempts at gaining stability of BFKL observables under higher-order corrections at natural scales were made via the analysis of semi-hard states featuring the detection of objects with large transverse masses, such as Higgs bosons [109] and heavy-flavored jets [111]. However, due to the lack of a NLO calculation for the corresponding impact factors (as mentioned before, the NLO Higgs impact factor was calculated quite recently in the large top-mass limit only), these reactions were studied with partial NLA accuracy. The first evidence of stabilizing effects in semi-hard processes studied at NLA came out in a recent study on inclusive Λ c emissions [145].
It was highlighted how the peculiar behavior of VFNS FFs depicting the baryon production at large transverse momenta [146] acts as a fair stabilizer of high-energy predictions for observables sensitive to double Λ c final states, while a partial stabilization was found in the production of a Λ c particle plus a light-flavored jet. Further studies on other channels featuring the tag of heavier hadron species are thus needed to corroborate the statement that the heavy-flavor production is a suitable testing ground for the manifestation of the aforementioned stabilizing effects.
In this work we consider the inclusive semihard production of two b-hadrons of a b-hadron plus jet system, where p(P 1,2 ) stands for an initial proton with momenta P 1,2 , H b (p i , y i ) for a generic bottomflavored hadron 2 with momentum p i and rapidity y i , and X contains all the undetected products of the reaction. The semi-hard configuration is realized when the two detected objects possess large transverse masses, m 1,2⊥ Λ QCD , with m 1,2⊥ = | p 1,2 | 2 + m 2 1,2 , and p 1,2 their transverse momenta. A large rapidity separation, ∆Y = y 1 − y 2 , is required in order to consider our reactions as diffractive ones 3 . We will let observed transverse momenta in ranges sufficiently 2 In our analysis we are inclusive on the production of all species of b-hadrons whose lowest Fock state contains either a b orb quark, but not both. Therefore, bottomed quarkonia are not considered. Furthermore, we ignore B c mesons since their production rate is estimated to be at most 0.1% of b-hadrons (see, e.g., Refs. [148,149]). Our choice is in line with the b-hadron FF determination of Ref. [27]. 3 The use of "diffractive" for our inclusive process is justified because the undetected hadronic activity is concentrated in the central region and summed over, thus leading, via the optical theorem, to differential cross sections (in the kinematic variable of the colliding particles' fragmentation regions) which take the same form as in truly diffractive processes, where there is no activity at all in the central region.
large to ensure the validity of a VFNS description.

B. High-energy resummed cross section
Final-state configurations that distinguish the two processes under consideration are schematically represented in Fig. 1, where a bhadron (p 1 , y 1 ) is emitted along with another bhadron or a jet (p 2 , y 2 ), featuring a large rapidity separation, ∆Y , together with an undetected system of hadrons. For the sake of definiteness, we will consider the case where the rapidity of the first detected final-state object, y 1 , is larger than the second one, y 2 , so that ∆Y is always positive, and the first object is forward while the second is backward.
The colliding protons' momenta P 1 and P 2 are taken as Sudakov basis vectors satisfying P 2 1 = P 2 2 = 0 and 2(P 1 · P 2 ) = s, so that the fourmomenta of detected objects can be decomposed as In the large-rapidity limit, the outgoing particle longitudinal momentum fractions, x 1,2 , are connected to the respective rapidities through the relation y 1,2 = ± 1 2 ln , so that one has dy 1,2 = ± dx 1,2 x 1,2 , and ∆Y = y 1 − y 2 = ln x 1 x 2 s | p 1 || p 2 | , where the spatial part of the four-vector In Eqs. (3) and (4) the r, s indices specify the parton types (quarks q = u, d, s, c, b; antiquarks q =ū,d,s,c,b; or gluon g), f r,s (x, µ F ) and Contrariwise to the pure collinear treatment, we build the cross section in hybrid factorization, where the high-energy dynamics is genuinely provided by the BFKL approach, and collinear ingredients are then embodied. We decompose the cross section as a Fourier sum of azimuthal-angle coefficients, C n , in the following way where ϕ = φ 1 − φ 2 − π, with φ 1,2 the outgoing particle azimuthal angles. In the NLA accuracy and in the MS renormalization scheme [150] the ϕ-summed cross section, C 0 , and the other coefficients, C n>0 , are given by (for details on the derivation see, e.g., Ref. [58] and in Ref. [89]) is the LO BFKL characteristic function, whilē χ(n, ν), calculated in Ref. [151] (see also Ref. [152]), is the NLO correction to the BFKL kernel χ(n, ν) = − 1 4 with and Then, c 1,2 (n, ν) are the LO forward/backward objects impact factors in the (n, ν)-representation, whose compact expression for both the H b particle and the jet reads where and the f (ν) function is defined as The remaining objects are the NLO corrections to impact factor in the Mellin representation (also known as (ν, n)-representation), c (1) i (n, ν, | p i |, x i , s 0 ). As for the H b NLO impact factor, we rely on a light-hadron calculation, done in Ref. [61]. This choice is consistent with our VFNS treatment, provided that energy scales at work are much larger than the bottom mass (see Section III). Our selection for the light-jet NLO impact factor is discussed in Section II C.
The way our hybrid factorization is realized fairly emerges from Eqs. We employ NLA expressions given in this Section at the natural energy scales given by with We introduce here also the MOM renormalization scheme [153][154][155], because this is the scheme in which the BLM procedure is developed (see with and where C A ≡ N c is the color factor associated to a gluon emission from a gluon, then we have whereas gluon and lighter quark (including c) FFs are generated through DGLAP evolution and vanish at µ F = µ 0 . Following Ref. [ When the H b + jet production channel is considered at NLA, a choice for the jet reconstruction algorithm, that enters the definition of the NLO jet impact factor, has to be made. The most popular classes of jet-selection functions are the κ ⊥ sequential-clustering [162] and the cone-type algorithms [163]. A simpler version, infrared-safe up to NLO perturbative accuracy and suited to numerical computations was derived in Ref. [59] in the so-called "small-cone" approximation (SCA) [164,165], namely for a small-jet cone aperture in the rapidity-azimuthal Working in the MOM renormalization scheme, in which the BLM procedure is natively implemented, the optimal scale for a given azimuthal coefficient, C n , is the value of µ R that satisfies the condition where dΦ(y 1,2 , | p 1,2 |, ∆Y ) stands for the final-state differential phase space (see Section III), and We remark that Eq.
In particular, we replace the analytic expres- A. ∆Y -distribution The first observable under investigation is the cross section differential in the rapidity interval, also known as ∆Y -distribution or simply C 0 . Its expression can be obtained by integrating the C 0 azimuthal coefficient (see Eq. (6)) over trans-verse momenta and rapidities of the two final-state particles, and keeping ∆Y fixed The light-flavored jet is always tagged in its typical CMS ranges [144], i.e. |y J | < 4.

B. Azimuthal correlations
Analogously to C 0 (see Eq. (27)), we define the phase-space integrated higher azimuthal coefficients, C n>0 . Thus, we can study their ratios The R n0 ratios have a straightforward physical interpretation, being the azimuthal-correlation moments cos nϕ , while the ones without zero indices represent ratios of correlations, that were originally proposed in Refs. [169,170]. We investigate the behavior of the R nm moments as functions of ∆Y and in the kinematic ranges defined in Section III A.  Recently, TM-resummed predictions were proposed for the hadroproduction of inclusive paired systems, as photon [184][185][186] and Higgs [187] pairs. The first joint resummation of TM logarithms coming from the emission of two distinct particles was considered in Ref. [188], where the concurrent measurement of the Higgs and the leading-jet transverse momenta in hadronic Higgs-boson emissions was studied up to the next-to-next-to-leading-logarithmic order via the RadISH code [189]. Those studies were then extended to TM-resummed differential observables for color-singlet channels, as the fully leptonic W + W − production at the LHC [190].
The double-differential spectrum on transverse momentum and azimuthal angle for weak gaugeboson production (W ± or Z 0 ) was recently investigated in the TM context via a soft-collinear effective theory approach [191].
becomes more and more difficult to produce a b-flavored bound state than a light jet when the transverse momentum grows.

IV. CONCLUSIONS AND OUTLOOK
We   azimuthal-angle correlations. The possibility to study azimuthal moments at natural scales also when jet emissions are allowed is a novel feature which corroborates the statement, already made in the case of Λ c production channels [145], that heavy-flavored emissions of bound states act as fair stabilizers of the high-energy series.
The next part of our program on semi-hard phenomenology relies on a two-fold strategy.
First, we plan to compare observables sensitive to heavy-flavor production in regimes where either the VFNS or the FFNS scheme is relevant, and possibly do a match between the two descriptions. The inclusion of quarkonium production channels, as done in Ref. [73], will certainly enrich our phenomenology.
Then, we project an extension of our studies on heavy flavor by considering wider kinematic ranges, as the ones reachable at the EIC [192,193], NICA-SPD [194,195], HL-LHC [196], and the Forward Physics Facility (FPF) [197]. Here, the stability of our predictions motivates our interest in (i ) proposing the hybrid high-energy and collinear factorization as an additional tool to improve the fixed-order description, and (ii ) evolving our formalism into a multi-lateral approach that embody different resummations.
We believe that the study of more and more exclusive observables, such as the double differential transverse-momentum distributions proposed in this work, goes along these directions.

Acknowledgements
We thank the Authors of Refs. [25,27] for allowing us to link native KKSS07 FF routines to the JETHAD code [139]. We thank V. Bertone and G. Bozzi for insightful discussions.