D ec 2 01 9 NLO QCD Corrections to Inclusive Charmonium and B c Meson Production in W + Decays

We calculate the next-to-leading order (NLO) quantum chromodynamics (QCD) corrections to inclusive processes W+ → J/ψ(ηc) + c + s̄ + X and W+ → Bc(B c ) + b + s̄ + X in the framework of nonrelativistic QCD (NRQCD) factorization formalism. Result indicates that the NLO corrections are significant, and the uncertainties in theoretical predictions with NLO corrections are greatly reduced. The charmonium and Bc meson yielding rates at the Large Hadron Collider (LHC) are given. PACS numbers: 12.38.Bx, 12.39.Jh, 13.38.Be, 14.40.Pq † chenziqiang13@mails.ucas.ac.cn ‡ yanghao174@mails.ucas.ac.cn ∗ qiaocf@ucas.ac.cn, corresponding author 1


I. INTRODUCTION
In the standard model (SM), the W boson mass is generated through the electroweak spontaneous breaking mechanism. Precise measurement of W boson mass and its decay width turns out to be a unique test of the SM and hence a probe for new physics. At the Large Hadron Collider (LHC), a huge number of W bosons are produced and recorded, which enables the research on W physics feasible and meaningful.
Heavy quarkonium and as well B c meson production keeps on being an interesting and hot topic to study in high energy physics for decades, which may enrich our knowledge on the properties of quarkonium and the nature of perturbative QCD. Note, hereafter for simplicity the B c respresents for both scalar B c and vector B * c unless specifically mentioned. Based on the nonrelativistic QCD (NRQCD) factorization formalism [1], direct hadroproduction of quarkonium and B c meson was studied extensively [2][3][4][5][6][7][8][9][10][11][12].
In addition to the direct production, indirect production also stands as an independent and important source for those double-heavy measons. The quarkonium and B c meson production through top quark [13] and Z 0 decays [14][15][16] had been investigated at up to the next-to-leading order (NLO) accuracy. For indirect quarkonium and B c in W decays, the leading order (LO) analyses were performed in Refs. [17,18]. It turned out that the theoretical uncertainties at LO are very large, which suggests, and was partly confirmed, that the higher order QCD corrections in charmonium and B c productions are usually very important, even crucial sometimes, for the sake of phenomenological use. To this end, we calculate in this work the NLO QCD corrections to the inclusive charmonium and B c production in W + decays.
The rest of the paper is organized as follows. In section II we present the LO calculation of W + decay to charmonium and B c mesons. In section III, some technical details in the calculation of NLO corrections are given. In section IV, the numerical evaluation for concerned processes is performed at NLO QCD accuracy. The last section is remained for summary and conclusions.

II. THE LO DECAY WIDTH
At the LO in α s , inclusive charmonium and B c meson production through W + decays are described by the processes as shown in Fig.1. The initial and final state particles are on the mass shell: We also introduce the Mandelstam variables: The CKM-suppressed processes, such as W + → B c (B * c )+c+c, are not included in our calculation. The amplitudes of these processes are suppressed at least by a Wolfenstein parameter λ. Taking λ ∼ α s (2m c ), the suppress factor for decay width can be estimated as O(α 2 s ), which means that the contribution from these processes are less significant than the NLO corrections.
At the leading order of the relative velocity expansion, it is legitimate to take p c = pc, m H = 2m c for charmonium production and p c = mc m b pb, m H = m b + m c for B c production. The spin projection operator has the form where ǫ( 1 S 0 ) = γ 5 and ǫ( 3 S 1 ) = ǫ /. The decay width at LO reads: Here, sums over the polarizations and colors of the initial and final particles, 1

III. THE NLO CORRECTIONS
At the NLO, the W + boson decay to charmonium and B c (B * c ) meson include the virtual and real corrections of W + → J/ψ(η c )+c+s and W + → B c (B * c )+b+s processes. For η c production, new subprocess W + → η c + u +d + g should also be included. In the computation of NLO corrections, the conventional dimensional regularization with D = 4 − 2ǫ is adopted to regularize the ultraviolet (UV) and infrared (IR) divergences.
The method proposed in [19,20] is used to deal with the D dimensions γ 5 trace.
In the calculation, the package FeynArts [21] is used to generate Feynman diagrams; FeynCalc [22,23] and FORM [24,25] are used to perform algebraic calculation; FIRE [26,27] is employed to reduce the Feynman integrals into the master integrals (A 0 , B 0 , C 0 , D 0 ); With the help of Ref. [28] and Package-X [29], the master integrals are calculated analytically, and the results are checked by LoopTools [30]; The numerical phase space integration is performed by CUBA [31].
Here, Re(M virtual M * born ) contains both UV and IR singularities. Since we set p c = pc and p c = mc m b pb before the calculation of Feynman integrals, the Coulomb singularity are not expected to appear in our calculation [32].
The UV singularities are removed by renormalization. For the renormalization of heavy quark field (Z 2 ), heavy quark mass (Z m ) and light quark field (Z l ), we take the on-shell (OS) scheme; for the renormalization of gluon filed (Z 3 ) and strong coupling constant (Z g ), the modified minimal-subtraction (MS) schemes are used: Here, µ is the renormalization scale, γ E is the Euler's constant; In virtual corrections, the IR singularities arise when the gluon connecting two on shell partons is soft or collinear to finals quark. Due to p c = pc or p c = mc m b pb, parts of IR singularities cancel each other [1]. The remaining are canceled by the real corrections according to the Kinoshita-Lee-Nauenberg theorem [33,34].

B. Real corrections
Typical Feynman diagrams in real corrections are shown in Fig.3. In the calculation of the real corrections, the phase space slicing method [35] is adopted to separate the IR singularities. By introducing soft cut δ s and collinear cut δ c , the phase space can be separated into three regions: • Hard non-collinear: Here, M sg = (p s +p g ) 2 is the invariant mass ofs and g system. Then the real corrections can be written as where the superscripts "S", "HC", "HNC" represent the "soft", "hard collinear", "hard non-collinear" region respectively.
According to Ref. [35], the contributions from soft part and hard collinear part reads In the case of hard non-collinear part, the decay width reads where the four-body phase space dPS 4 with cut can be written as with where , and Θ(x) is the unit step function which return 1 when x > 0 and 0 for other case. After summing up these three parts, their dependence on technical cut are eliminated as expected.
For the subprocess W + → η c + u +d + g, there are 4 diagrams, as shown in Fig.4.
The IR singularities are eliminated after summing all the amplitude square parts. The decay width can be calculated directly in 4 dimensions as Γ HNC real .

IV. NUMERICAL RESULTS
For the numerical calculation, following input parameters are used m W = 80.399GeV, m c = 1.5 ± 0.1GeV, m b = 4.9 ± 0.2GeV, α = 1/137.065, Here, θ W is the Weinberg angle. The J/ψ wave function at the origin is extracted from its leptonic width: with Γ(J/ψ → e + e − ) = 5.55 keV [36]. The B c wave function is estimated by using the Buchmueller-Tye potential [37]. The two-loop strong coupling of is employed in the NLO calculation, in which, L = ln(µ 2 /Λ 2 QCD ), We take n f = 4, Λ QCD = 292 MeV for J/ψ and η c production; n f = 5, Λ QCD = 210 MeV for B c and B * c production.
The NLO decay width can be expressed as where C H = 25/6 for W + → J/ψ(η c ) + c +s, C H = 0 for W + → η c + u +d + g,  The decay widths are as presented in Tab The energy distribution of charmonium and B c (B * c ) meson are shown in Fig.6. It can be seen from Fig.6(b) that the η c production rate are largely enhanced at small energy region. This enhancement comes from the diagrams similar to Fig.4, except u andd are replaced by c ands. The contribution from the gluon propagator can be estimated as: Here, the B * c feed-down to B c is taken into account. In experiment, the B c meson can be fully reconstructed through B c → J/ψπ + decay, whose branching fraction is about 0.5% [40]. According to [36], the branching ratio B r (J/ψ → l + l − (l = e, µ)) = 12%, B r (η c → pp) = 0.15%, then the numbers of Numerical calculation shows that the NLO corrections are significant, and the uncertainties in theoretical predictions with NLO corrections are greatly reduced. Since B * c alomst all decays to B c , assuming B c is reconstructed through B c → J/ψπ + , J/ψ is reconstructed through J/ψ → l + l − (l = e, µ), η c is reconstructed through η c → pp, the numbers of J/ψ, η c and B c meson candidates per year may reach (1.72 ∼ 5.82) × 10 5 , (2.46 ∼ 11.8) × 10 3 and 80 ∼ 175 respectively at the LHC 2017 luminosity.
Note added: when this work was finished and the manuscript was finalizing, there appears a study on the web about the B c (B * c ) meson production in W + decay with the NLO QCD corrections [41]. We numerically compared our results with that paper, and find that by taking the same inputs we can reproduce the TableI results there 1 .