Ultraforward production of a charmed hadron plus a Higgs boson in unpolarized proton collisions

We investigate the inclusive emission in unpolarized proton collisions of a charm-flavored hadron in association with a Higgs boson, featuring large transverse momenta and produced with a large rapidity distance. Taking advantage of a narrow timing coincidence between the ATLAS detector and the future FPF ones, we study the behavior of cross sections and azimuthal correlations for ultraforward rapidities of the detected hadron. We provide evidence that the hybrid high-energy and collinear factorization, encoding the BFKL resummation of large energy logarithms and supplemented by collinear densities and fragmentation functions, offers a fair description of this process and comes out as an important tool to deepen our understanding of strong interactions in ultraforward production regimes.


I. INTRODUCTION
The validity of Quantum Chromodynamics One of these kinematic sectors is the so-called semi-hard regime [1], which is characterized by the hierarchy of energy scales s {Q} 2 Λ 2 QCD , with s the center-of-mass energy squared and {Q} one or a set of hard scales given by the considered final states. In this regime the convergence of the perturbative expansion, truncated at a certain order in the strong coupling α s , is spoiled. This is due to the appearance of large logarithms of the form ln(s/Q 2 ), that enter the perturbative series with a power increasing with the order of α s . The most adequate and powerful mechanism to perform the allorder resummation of these energy logarithms is provided by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework [2][3][4][5]. It allows for the systematic inclusion of all terms proportional to (α s ln(s)) n , the so-called leading logarithmic approximation or LLA, and of the ones proportional to α s (α s ln(s)) n , the next-to-leading logarithmic approximation or NLA.
BFKL cross sections take the form of an elegant convolution between a process-independent Green's function, which embodies the resummation of energy logarithms, and two impact factors, depicting the fragmentation of each incoming particle to an identified outgoing one. The BFKL Green's function evolves according to an integral equation. Its kernel is known up to the next-to-leading order (NLO) in the perturbative expansion for any fixed, not growing with s, momentum transfer t and for any possible two-gluon color t-channel exchange [6][7][8][9][10][11][12]. Impact factors represent the process-dependent part of the cross section. In the context of hadron-initiated collisions with two particles emitted with a large rapidity separation (the setup matter of investigation in this article), impact factors are in turn written as a convolution between collinear parton density functions (PDFs) 1 and collinear inputs describing the production mechanisms of identified final-state hadrons, such as fragmentation functions (FFs). Therefore, in this particular case we refer to our formalism as a hybrid high-energy and collinear factorization. 2 So far, the number of impact factors calculated within NLO accuracy is quite small.
It is widely recognized that the theoretical description of semi-hard inclusive observables suffers from instabilities emerging in the BFKL series. This happens because NLA corrections both to the BFKL Green's function and impact factors turn out to be of the same size and with opposite sign of pure LLA contributions. This issue brings to large uncertainties arising from the renormalization and factorization scale choice. In particular, instabilities affecting azimuthal correlations for light jet and hadron emissions [29,31,40,42] are too strong to prevent any possibility of setting these scales at the natural values provided by kinematics, namely the observed transverse momenta. As a possible solution, it was proposed the adoption of some optimization procedures for scale fixing. The Brodsky-Lepage-Mackenzie (BLM) method [114][115][116][117] effectively helped to partially suppress these instabilities. As a side effect, however, BLM scales are definitely larger than the natural ones [61]. This translates in a loss of one or more orders of magnitude for the corresponding cross sections, and any attempt at reaching the precision level fails.
Recently, the study of semi-hard reactions featuring the emission of heavy-flavored hadrons was proposed as a new direction toward restoring the stability of the high-energy series under higher-order corrections and scale variations. The first evidence that the peculiar behavior of heavy-flavor collinear FFs [118][119][120], determined in a variable-flavor number-scheme (VFNS) framework [121,122], leads to a remarkable stabilization of the NLA series came out in the context of single-charmed [76,78] and single-bottomed [77,78] hadron studies at the LHC. This stabilizing effect emerges when a nondecreasing, smooth-behaved in energy gluon FF is convoluted with proton PDFs.
Remarkably, a strong stabilization fairly arises also when a forward quarkonium state (J/ψ or Υ) is inclusively emitted in association with a backward jet [80]. In that work the quarkonium production at large transverse momentum was described at the hand of a DGLAPevolving heavy-quark FF, modeled on the basis of the non-relativistic QCD (NRQCD) framework [123][124][125][126][127].
The emergence of such a fair stability through two independent FF descriptions, namely both via standard determinations of single-charmed hadron FFs from global fits in the collinear Figure 1: Hybrid high-energy and collinear factorization in force. Diagrammatic representation of the ultraforward inclusive hadron plus Higgs production. Red blobs depict proton collinear PDFs, green rectangles stand for charmed-hadron collinear FFs, and blue triangles denote top-quark fermion loops. The BFKL Green's function, portrayed by the yellow blob, is connected to off-shell impact factors via Reggeon waggle lines. The diagram was obtained through the JaxoDraw 2.0 package [128].
approach [118][119][120] and via heavy-quarkonium NRQCD fragmentation models [126,127], strongly corroborates our statement that intrin- More in general, the analysis proposed in Ref. [63] is one of the most recent advances in the study of Higgs differential distributions at high energies. Pioneering analyses of Higgs emissions in multi-jet events were done in Refs. [129,130]. The effect of hard-rescattering corrections from gluon ladders in the central exclusive Higgs hadroproduction was assessed in Ref. [131]. Combined BFKL and Sudakov ef-fects in the almost back-to-back production of a Higgs-jet system in proton collisions were recently investigated [132]. Ref. [133] contains the LO calculation for the doubly off-shell impact factor portraying the production of a Higgs boson from gluon-gluon fusion in the small-x limit. The small-x resummation based on the Altarelli-Ball-Forte (ABF) approach [134][135][136][137][138][139][140] was successfully employed first in the description of central Higgs production [141,142], then in rapidity [143] and momentum [144] distributions. 3 First doubly resummed predictions for central-Higgs production accounting for both the small-x and the threshold [155][156][157][158] resummations were presented in Ref. [159] (see also Ref. [160]). Small-x double-logarithmic contributions to Higgs production in the large topmass limit were considered in Ref. [161], whereas high-energy azimuthal signatures in Higgs angular distributions were investigated in Ref. [162].
In the present work we combine in our hybrid high-energy and collinear factorization the two main ingredients that have brought to the emergence of the natural stabilization of resummed semi-hard distributions. We consider the ultraforward inclusive emission of a single-charmed hadron, a Λ ± c or a D * ± meson, accompanied by a Higgs boson produced in more central regions of rapidity (see Fig. 1). We expect that distinctive signals of the mentioned natural stabi- 3 The ABF formalism relies on high-energy factorization schemes [143,[145][146][147][148][149][150][151] and has been numerically implemented in the HELL work environment [152][153][154].
lity will fairly emerge from the analysis of the rapidity and azimuthal-angle spectrum of the tagged heavy-flavor + Higgs-boson final state, thus making this reaction one of the most powerful channels to probe the high-energy QCD regime.
We add to our analysis a third, novel ingredient coming from very recent developments in the context of the Forward Physics Community. Taking advantage of forthcoming studies doable at future detectors of the planned Forward Physics Facility (FPF) [163,164], we allow the charmed particle to be detected in so far unexplored ultraforward windows of rapidity, which are definitely beyond the acceptances of current LHC ones. More in particular, we benefit from the opportunity of making FPF detectors to work in coincidence with ATLAS. According to current plans, FPF will detect decays products (mainly neutrinos) of light-or heavy-flavored hadrons, thus permitting us to reconstruct the production signal of our charmed hadron. However, the FPF will not be able to see Higgs bosons. Therefore, a very precise timing procedure will make feasible the simultaneous FPF ultraforward hadron detection and the ATLAS tag of the Higgs. Technical details on the FPF + ATLAS narrow timing coincidence are reported in Section VI E of Ref. [163].
The intersection corner between the charmflavor physics and the Higgs sector opens up fascinating windows toward frontier research lines.
A novel mechanism for the exclusive diffractive Higgs-boson production from intrinsic heavyflavor components in the proton was proposed in Refs. [165,166]. Shedding light on the intrinsic charm contribution [167][168][169][170][171], it plays a crucial role in obtaining precise determinations of proton PDFs [172,173], which in turn are needed to improve the description of Higgs production (see, e.g., Ref. [174]).
Besides a pure QCD viewpoint, Higgs-boson radiative decays to quarkonia offer us the engaging opportunity of accessing and constraining the Yukawa coupling of the charm quark [175][176][177][178].
A similar result can be achieved via the measurement of the production cross section for the Higgs in association with a charmed jet [179].
Higgs decays to charmonia at large transverse momenta can be investigated on the basis of the c-quark fragmentation approximation [180]. Observables sensitive to the emission of charmed Bmesons are thought to be quite good probes for rare Higgs decays [181][182][183].
Coming back to strong interactions in ultraforward regimes, the heavily asymmetric ranges in transverse momenta and rapidities accessible  (1) both of the two emitted objects featuring high transverse momenta, p C,H Λ QCD , and a large rapidity separation, ∆Y = y C − y H . The bound state labeled as C represents either a Λ ± c baryon or a D * ± meson, whereas H is a scalar Higgs boson. The four-momenta of the parent protons, P a,b , are taken as Sudakov vectors satisfying P 2 a,b = 0 and P a • P b = s/2, so that the finalstate transverse momenta can be decomposed in the following way with p 2 C,H⊥ = −| p C,H | 2 , and the spacial part of the four-vector P a being taken positive. Then, m H⊥ = m 2 H + | p H | 2 is the Higgs-boson transverse mass. The longitudinal momentum frac-tions, x C,H , are related to the corresponding rapidities in the center-of-mass frame via the rela- thus having dy C,H = ±dx C,H /x C,H . As for the rapidity distance, one has The cross section is given as the Fourier series of the so-called azimuthal coefficients where ϕ = ϕ C − ϕ H − π, with ϕ C,H the C-hadron and Higgs azimuthal angles. A NLA formula for the ϕ-summed cross section,Ĉ 0 , and the other coefficients,Ĉ κ>0 , readŝ for the C-hadron, and for the Higgs boson. The structure of equations presented above unambiguously shows how our hybrid high-energy and collinear factorization is built. To the high-energy convolution between the Green's function and impact factors (see Eq. (6)), the collinear convolution between the hard factor, PDFs f a (x C , µ F 1 ) for the struck parton (gluon or quark) and FFs D C a x C ξ , µ F 1 for the outgoing parton generating the C-hadron cor-responds (see Eq. (7)). The LO Higgs impact factor in the Mellin space was obtained in Eq. [63] and its expression contains the f g (x H , µ in Ref. [130] and reads with m t = 173.21 GeV the top-quark mass,

Moreover one has
We remark that our treatment for the LO Higgs vertex encodes finite top-mass contributions. The effect of taking the same calculation in the (m t → +∞) limit was gauged in Section 3.2 of Ref. [63]. A result for the Higgs NLO impact factor was recently obtained in the infinite top-mass limit in the standard BFKL approach [184] as well as within the Lipatov's highenergy effective action [185,186]. However, its numerical implementation in the Mellin space, c is not yet available. Therefore, in our study we consider a partial NLO implementation that includes some "universal" The NLO correction for the forward hadron impact factor was computed ten years ago [187].
For the sake of consistency with the treatment of the Higgs case, we take into account only the universal NLO corrections, i.e. the ones proportional to the LO hadron impact factor, thus getting c (1) wherẽ In Eqs. (11) and (12) s 0 is an energy-scale parameter, arbitrary within NLA accuracy, which we set to s 0 = | p C |m H⊥ . Furthermore, P ij (ξ) depict the DGLAP splitting kernels at LO.
As it is well-known, any variation of the renormalization (µ R 1,2,c ) and factorization (µ F 1,2 ) scales produces effects that are beyond the NLO accuracy. Therefore, in principle their choice can be arbitrary. It is convenient, however, to set these scales to the natural values provided by the process kinematics. They are generally connected with physical hard scales. From the inspection of the diagram in Fig. 1, we can distinguish two sub-processes. The first one is the production of the C-hadron in the upper fragmentation region, and the characteristic hard scale here is the | p C | transverse momentum. The second one is the emission of the Higgs boson in the lower fragmentation region, featuring m H⊥ as the typical hard scale. Therefore, we go with a multiscale strategy, namely we choose two distinct values for scales, depending on which sub-process they refer. We set µ R 1 = µ F 1 = C µ m C⊥ , with m C⊥ = m 2 C + | p C | 2 the charmed-hadron transverse mass, and µ R 2 = µ F 2 = C µ m H⊥ . Moreover, since the two sub-processes are connected via the gluon ladder (see Fig. 1), we select a third value for the renormalization scale entering the exponentiated kernel in Eq. (6), given by the geometric mean of the former ones. We thus have The C µ variation parameter will be set as explained in the next Section.

III. PHENOMENOLOGY
All the numerical studies presented in this Section were performed via the JETHAD multi-modular interface [61]. In order to gauge the sensitivity of our predictions on the scale vari- depict D * ± meson and Λ ± c baryon emissions, respectively. We remark that the use of these VFNS functions together with calculations for the hadron impact factor obtained in the lightquark limit [187] is adequate, provided that the µ F 1 scale is larger than the threshold for DGLAP evolution of the charm quark, generally set to  [192].

A. Observables and kinematics
The first class of observables included in our phenomenological study is represented by azimuthal-angle coefficients integrated over the phase space of the two outgoing particles, at fixed values of their rapidity distance ∆Y , the C κ phase-space differential coefficients being defined in Section II. Among them, C 0 is the socalled ∆Y -distribution, which corresponds to the cross section differential in ∆Y and summed over the azimuthal-angle distance ϕ.
The second class of observables matter of our investigation are the azimuthal-correlation moments, R κ0 = C κ /C 0 ≡ cos(nϕ) , built as ratios between a given C κ>0 coefficient and C 0 , and their ratios [193,194], . Therefore, we opt for a solution that stays in between the two options, namely we set 20 GeV < p H < 60 GeV. Concerning the Higgs rapidity, we allow for its detection in the ATLAS barrel, namely we set |y H | < 2.5.

B. Results
In this Section, we present our results for the observables previously introduced. In Fig. 2, the total cross section summed over azimuthal angles and differential in the final-state rapidity distance, C 0 , is shown for our two single-charmed hadron species, D * ± and Λ ± c (upper panels), and compared with the corresponding one for two lighter mesons, π ± and K ± (lower panels).
The stabilization pattern can be immediately observed from these plots. Here, NLA predictions are systematically contained inside LLA ones and the width of uncertainty bands considerably decreases when moving to the higher order. We also observe that the sensitivity of NLA predictions under scale variations lowers when charmed species are detected. This effect is due to the smoothly-behaved, non-decreasing with µ F , C-hadron gluon FFs (see Section 3.4 of Ref. [76] for technical details).
Furthermore, statistics associated with heavyparticle emissions is one to two orders of magnitude lower than that lighter-meson one, but it still remains promising. As usually observed in processes described within the hybrid factor-