W Boson Inclusive Decays to Quarkonium and B_{c}^{(*)} Meson at the LHC

# W Boson Inclusive Decays to Quarkonium and B(∗)c Meson at the LHC

YuQuan Road 19A, Beijing 100049, China
Theoretical Physics Center for Science Facilities (TPCSF), CAS
YuQuan Road 19B, Beijing 100049, China
June 14, 2011
###### Abstract

In this paper, the production rates of quarkonia , , , and mesons through boson decay at the LHC are calculated, at the leading order in both the QCD coupling and in , the typical velocity of the heavy quark inside of mesons. It shows that a sizable number of quarkonia and mesons from boson decay will be produced at the LHC. Comparison with the predictions by using quark fragmentation mechanism is also discussed. Results show that, for the charmonium production through decay, the difference between predictions by the fragmentation mechanism and complete leading order calculation is around , and it is insensitive to the uncertainties of theoretical parameters; however, for the bottomonium and productions, the difference cannot be ignored as the fragmentation mechanism is less applicable here due to the relatively large ratio .

PACS number(s): 13.38.Be, 13.85.Ni, 14.40.Pq, 12.38.Bx.

## I Introduction

As the only charged gauge boson in the Standard Model (SM) whose mass is generated through the electroweak symmetry spontaneous breaking (EWSB), the precise measurements on boson’s mass and decay width offer a way to distinguish the different EWSB mechanisms, and effectively put a stringent constraints on the Higgs boson mass and New Physics beyond the SMPhy (). Furthermore, the study of the productions of bosons can be a probe for searching New Physics since the possible new particles may decay to bosons. By reason of that, recently, there have appeared several papers discussing “W boson physics”, at Tevatron, ILC and LHC, respectively Tev ().

Before the LHC era, only two rare decays: and , were experimentally studied, and the respective upper limits for their branching fractions are and Pdg ()555Recently, the CDF collaboration just set a new confidence level upper limit on the relative branching fraction at which is a factor of 10 improvement over the previous limit Aaltonen:2011nn ().. The LHC has been running for two years. Due to its high luminosity, the huge number of bosons are and will be produced. According to the estimations in Phy (); Coo (); Jon (), the production cross section of boson is around nb at the LHC with the center energy 7 TeV or 14 TeV. It is expected that hundreds of million bosons will be produced per year at the LHC with the conservative expectation of the luminosity (which is twp orders of magnitude smaller than the desired LHC luminosity ). This makes the LHC also a -boson factory which facilitates the precise experimental study on boson physics, especially boson rare decays.

In this article, we will discuss a class of boson inclusive decays to the S-wave quarkonium and meson, such as , , , , and . Because boson is so heavy and participates merely the electroweak interactions, the study on boson inclusive decays to quarkonia and mesons666As inFra (); Cha (); Wu (), we treat as a non-relativistic bound state of and in this paper., offers another great place for the precise test of the perturbative quantum chromodynamics (PQCD) and the quarkonium production mechanism. As a quarkonium is presumed to be a non-relativistic bound state of heavy quark and anti-quark, a wonderful theoretical tool to deal with the processes involving quarkonium is non-relativistic quantum chromodynamics (NRQCD), in which the low-energy interactions are organized by the expansion in , the typical relative velocity of the heavy quark and anti-quark inside of quarkoniumBod (). The inclusive production cross sections can be written as the product of the perturbatively calculable short-distance coefficients and the non-perturbative NRQCD matrix elements. At the leading order in , the only non-perturbative NRQCD matrix element of the S-wave quarkonium is proportional to the Schrödinger wave function at the origin squared.

On the other hand, since the boson mass is much greater than the heavy quark masses , the parton fragmentation may dominate the boson inclusive decays to quarkonia and mesons. The pioneer works in this field were done two decades ago, the universal fragmentation functions for different quarkonium and meson were given in Fra (); Bra (); Braa (). By these universal fragmentation functions, the direct production rate of quarkonium and meson at large transverse momentum in any high-energy process can be approximated to be the product of the parton-level production rate and the universal fragmentation probability, and the corresponding computation is much easier than that done by within the NRQCD framework. In Ref.Fra (); Bra (); Bar (), for the inclusive decays to the S-wave charmonium, the authors did find great agreement between the results obtained by the parton fragmentation approximation and the complete leading order PQCD calculation.

However, in some processes, the fragmentation may not be the leading process, but main process when a certain condition of the fragmentation mechanism is not sufficiently fulfilled. In Ref.Qiao () it was found that the fragmentation contribution is important but not dominant for top quark decays into quarkonium. Thus, it is worth examining if the fragmentation mechanism works in boson inclusive decays to quarkonium and meson.

In the following sections, we present the complete leading order calculation of boson decays to quarkonium , , , and mesons; then we compare our results with that obtained by the fragmentation approximation; finally we discuss the theoretical uncertainties in our calculation.

## Ii Formalism

Some typical Feynman diagrams for quarkonia and mesons hadronic production through decay are shown in Fig. 1. We will calculate them at the leading order in and . In NRQCD, a non-relativistic bound state of heavy quarks and is considered as an expansion of a series Fock states. The leading Fock state for the S-wave hadron is constructed as

 |H[2s+1SJ](p,λ)⟩ (1) = √2mH∑i,j,λ1λ2δij√2NcC(J,λ;12,λ1,12,λ2) ×∫d3pQ√2EQ2EQ′~Ψ2s+1SJ(pQ)|Qi,λ1(pQ)¯Q′j,λ2(pQ′)⟩,

where denotes the mass of hadron and all the states are relativistically normalized. Here is momentum of hadron, stand for the color indices which run from with for QCD, and is the C-G coefficient with and being the third components of spin indices. The non-perturbative parameters is the Schrödinger wave functions in momentum space. Thus, to project out the amplitude for the S-wave hadronic states from the complete patron level amplitude at the leading order of , practically we do the following replacements for the heavy quark and -anti-quark spinors

 vi(pQ′)¯uj(pQ) → Ψ1S0(0)2√mQ+mQ′γ5(p/+mQ+mQ′)δij√Nc, (2) v(pQ′)¯u(pQ) → Ψ3S1(0)2√mQ+mQ′ϵ/∗(p/+mQ+mQ′)δij√Nc. (3)

Here is the polarization vector of state, is the Schrödinger wave function at the origin. At the heavy quark limit, we set . Throughout the paper, we adopt the relation since we are doing the LO calculation in . Obviously, we have for quarkonia, and , for .

### ii.1 W+→ηc(or J/ψ)+c+¯s

There are two Feynman diagrams for boson decay into a S-wave charmonium state associated with and . Implementing the Feynman rules of the Standard Model and projectors in Eq.(2), the amplitude for is

 M=16π3gαsVcs2√2Ψηc(0)2√6mc(A1+A2), (4)

where

 A1 = ¯u(p6)γαγ5(2mc+p/)ϵ/W(1−γ5) (5) (−p/3−p/5−p/6)(p3+p5+p6)2γαv(p5)1(p3+p6)2, A2 = ¯u(p6)γαγ5(2mc+p/)γα(mc+p/3+p/4+p/6)(p3+p4+p6)2−m2cϵ/W (6) (1−γ5)v(p5)1(p3+p6)2,

with , and being the momenta of meson, quark and quark, respectively; and the momenta of quark and quark in meson; the CKM matrix element; with the Weinberg angle and the unit electro-charge; and are the electromagnetic and the strong coupling constants, respectively; the polarization vector of boson. In this paper, we set , and quarks massless, and ignore the dependence on the relative momentum between quark and anti-quark in meson, and set . The spin-averaged partial decay width of reads

 dΓ = 128m3wπ3∑spins|M|2ds1ds2, (7)

where is the mass of boson, and . The explicit analytic expressions for the square of the amplitude for is given in Appendix A.

For the complete leading order calculation of , we just repeat the calculation that had been done in Ref.Bar (). We find that there is a misprint in the term in given in Appendix B of Bar (), which should be corrected to .

### ii.2 W+→ηb(or Υ)+X

is the leading order process of bottomonium production through decay in the expansion of . However, such processes are the CKM-suppressed with the corresponding CKM factor in the Wolfenstein parameterization where and . Numerically, the Wolfenstein parameter . Hence, we will consider some processes that are higher order in expansion of but lower order in expansion of the Wolfenstein parameter . For instance, the process depicted by the last diagram in Fig. 1 is associated with the factor which is numerically comparable to the factor accompanied with the process . Similarly, we also consider for decays to . Its amplitude is proportional to , which is numerically comparable to the factor accompanied with .

The calculation of production is similar to the production described in the previous subsection. Here we just show the results for the production of . The complete amplitude for decay to with and is

 M=16π3gαsVcb2√2ΨΥ(0)2√6mb(A1+A2), (8)

with

 A1 = ¯u(p5)γα(mc+p/3+p/5+p/6)(p3+p5+p6)2−m2cϵ/W(1−γ5)ϵ/∗Υ (9) (2mb+p/)γαv(p6)1(p3+p6)2, A2 = ¯u(p5)ϵ/W(1−γ5)(mb−p/3−p/4−p/6)(p3+p4+p6)2−m2bγαϵ/∗Υ (10) (2mb+p/)γαv(p6)1(p3+p6)2,

where , and are the momenta of , quark and quark, respectively; and are the momenta of quark and quark in meson; is the polarization vector of boson. The explicit expression for the square of amplitude is given in Appendix B.

Because of the CKM suppression, the electroweak contributions depicted in Fig. 2 is required for the production of as we argued above777However, for that of , the electroweak contribution can be ignored in LO, which is two orders of magnitude less than its QCD contribution.. Taking for example, the amplitude for this process is written as

 M=√3ge2VudΨΥ(0)2√6mb(A1+A2+A3), (11)

with

 A1 = Tr[γαϵ/∗Υ(2mb+p/)]9√2(p3+p4)2 (12) ×¯u(p6)γαp/3+p/4+p/6(p3+p4+p6)2ϵ/W(1−γ5)v(p5), A2 = −Tr[γαϵ/∗Υ(2mb+p/)]18√2(p3+p4)2 (13) ×¯u(p6)ϵ/W(1−γ5)(−p/3−p/4−p/5)(p3+p4+p5)2γαv(p5), A3 = −Tr[γαϵ/∗Υ(2mb+p/)]¯u(p6)γν(1−γ5)ϵμWv(p5)6√2(p3+p4)2((p5+p6)2−m2w) (14) ×((p5+p6−p)μgαν−(2p5+2p6+p)αgμν +(p5+p6+2p)νgαμ),

where , and are the momenta of , quark and quark, respectively; and are the momenta of quark and quark in meson. Here again we set in the calculation.

dominates the decay . Its four-body decay amplitude is quite complicated, so we do not present the complete analytic expressions here but the numerical results in the next section. The potential infrared-divergences arising from the regions where the momentum of the gluon becomes soft vanish due to the “color-transparency”, and the potential collinear-divergences arising from the region where the momentum of the gluon becomes collinear to the and -quark (or quark) are regulated by the quark mass. Thus, the decay width of is infrared-safe.

### ii.3 W+→B(∗)+c+b+¯s and W+→B(∗)+c+c+¯c

There are two processes, namely and , for bound states production in decay at the leading order. and can also contribute, however their contributions vanish at the large limit, and thus are almost two and three orders of magnitude lower than and numerically.

For , there are two Feynman diagrams and the corresponding amplitude is

 M=16π3gαsVcs2√2ΨBc(0)2√3(mc+mb)(A1+A2), (15)

with

 A1 = 1(p3+p6)2¯u(p6)γαγ5(mc+mb+p/)ϵ/W(1−γ5) (16) (−p/3−p/5−p/6)(p3+p5+p6)2γαv(p5), A2 = 1(p3+p6)2¯u(p6)γαγ5(mc+mb+p/)γα (17) (mc+p/3+p/4+p/6)(p3+p4+p6)2−m2cϵ/W(1−γ5)v(p5).

Here , , , and are, respectively, the momenta of , quark and quark in meson, quark and quark. The corresponding amplitude at the leading order for is

 M=16π3gαsVcb2√2ΨBc(0)2√3(mc+mb)(A1+A2), (18)

with

 A1 = ¯u(p6)γα(mc+p/4+p/5+p/6)(p4+p5+p6)2−m2cϵ/W(1−γ5) (19) γ5(mc+mb+p/)γαv(p5)1(p4+p5)2, A2 = ¯u(p6)ϵ/W(1−γ5)(mb−p/3−p/4−p/5)(p3+p4+p5)2−m2bγα (20) γ5(mc+mb+p/)γαv(p5)1(p4+p5)2,

where , , , and are, respectively, the momenta of , quark and quark in meson, quark and quark.

Similarly, we can obtain the amplitudes for production. In Appendix C, we detail the square of the amplitudes for both and .

## Iii Numerical Results and Analysis

### iii.1 Input parameters

Now we can employ the above formalism to evaluate the decay widths of boson to quarkonia and mesons. We adopt the mass parameters for quark, quark and boson as follows

 mc=1.50GeV,  mb=4.90GeV  and  mw=80.4GeV.

The Schrödinger wave functions at the origin for and are determined through their leptonic decay widths , at the leading order in both and , which is Bod (),

 |Ψψ(Υ)(0)|2=m2ψ(Υ)Γee16πα2e2c(b).

Taking , , and , we obtain

 |Ψψ(0)|2=0.0447 GeV3  and  |ΨΥ(0)|2=0.403 GeV3.

In principle, we can extract from the decay width of . However, since the experimental data uncertainty is still large, which is Pdg (), the corresponding extraction of is not applicable in our numerical analysis. Instead, we set , which is the consequence of the heavy quark spin symmetry in NRQCD at leading order Bod (). Similarly, we adopt .

For , we apply given in Wu () by using the Buchmüller-Tye potential Eichten (). In addition, for consistency the leading order running is adopted, i.e.

 αs(μ)=4π(11−2nf/3)ln(μ2/Λ2QCD),

where we take and as in Wu (). We choose the typical renormalization scale for charmonium and production, and for bottomonium production, correspondingly and .

### iii.2 Decay widths and feasibility at LHC

By the input parameters given above, we list the partial widths of decays to the S-wave quarkonia and mesons in Table 1. Employing these partial widths, one can estimate the event numbers of quarkonium production through decays at the LHC. Considering that the LHC runs at the center-of mass energy with the luminosity , and the cross section at the LHC Jon (), the number of events per year is expected to be . Then we present the event rates of decays to quarkonia or in Table 2.

The most readily identifiable quarkonia are and , because their leptonic decays have clear signals and relatively large branching fractions. With , and , we list the expected di-leptonic signals from and decays in Table 3. For experimental detection for such decays at the LHC, one has to look for the relevant events with . According to Gao (), the efficiency of reconstruction of from its dimuon decay channel is around for collision at LHC experiments. However, to tag whether such event is really from decay, one must reconstruct from event. Therefore, the additional information about relevant QCD background of such events could be crucial as well.

The most promising decay channels for reconstruction are , and etc, which branching ratios are around a few percent. Thus, it should be possible to observe decays to at the LHC. Certainly, such measurements also suffers the difficulty of reconstruction of as mentioned above.

is the most promising channels to identify at colliders. However, considering their small branching fractions and the efficiency of the event reconstruction, the measurements on decays to at the LHC would be quite difficult. With the similar reasoning, it would be barely possible to measure decays to at the LHC due to the lower branching fraction and the experimental difficulty for identifying .

### iii.3 Comparisons with the fragmentation mechanism

Since , one can argue that the fragmentation mechanism may be the dominant contribution to the inclusive decay rate of the into the quarkonium and meson which survives in the limit . In the fragmentation mechanism, the hadron with energy is produced by the fragment of a type parton with energy ( is the longitudinal momentum fraction of relative to type parton) directly from the decay of boson. The possibility of is presumed to be described by the universal fragmentation function . Putting all the possible parton fragmentation together, the differential decay width of inclusive production can be written as

 dΓ(W+→H(E)+X) (21) = ∑i∫10dzd^Γ(W+→i(E/z)+X,μ)Di→H(z,μ) +O(mHmw),

where is the parton level differential decay width, and the sum is over the parton type and the hadron ’s longitudinal momentum fraction . At the leading order of , does not depend on the longitudinal momentum fraction . Thus, the total decay width turns to be

 Γ(W+→H(E)+X)=∑i^Γ(W+→i(E/z)+X)Pi→H, (22) Pi→H≡∫10dzDi→H(z,μ), (23)

where is the so-called fragmentation possibility.

After paying the price of power corrections, Eq.(21) has a number of advantages in calculation. The parton level differential decay width is easy to be calculated and dependent only on the typical short-distance scale . The fragmentation functions are universal and dependent on the hadronization scale, and therefore, they can be either extracted from the experimental measurements or calculated from certain phenomenological models. Fortunately, the fragmentation functions for the S-wave quarkonium and meson can be calculated perturbatively Bra (); Braa (), together with all the non-perturbative hadronization effects being parameterized into the Schrödinger wave functions of the mesons at origin at the non-relativistic limit.

In calculation we take the following fragmentation probabilities from Bra (); Braa ():

 P(c→ψ)=32α2s(2mc)|Ψψ(0)|227m3c(118930−57ln2), (24)
 P(c→ηc)=32α2s(2mc)|Ψηc(0)|227m3c(77330−37ln2), (25)
 P(¯b→B+c)=8α2s(2mc)|ΨBc(0)|227m3cf(mcmb+mc), (26)
 P(¯b→B∗+c)=8α2s(2mc)|ΨB∗c(0)|227m3cg(mcmb+mc), (27)

where the function and are

 f(r) = 8+13r+228r2−212r3+53r415(1−r)5 (28) +r(1+8r+r2−6r3+2r4)(1−r)6ln(r),
 g(r) = 24+109r−126r2+174r3+89r415(1−r)5 (29) +r(7−4r+3r2+10r3+2r4)(1−r)6ln(r).

and can be obtained from (23) and (24) by substituting the mass for . And can be also got from (25) and (26) by interchanging and .