Heavy quarkonium production and electromagnetic fragmentation in exclusive rare W-boson decays

1. Introduction

Since the discovery of weak intermediate boson W, its exclusive decays have been concerned by experimental physicists. Particularly, it is of interest to investigate exclusive W-boson decays into hadronic final states, in order both to deepen our understanding of this fundamental gauge boson and to explore quantum chromodynamics (QCD)^1,2,3,4 describing strong interactions. However, so far no evidence for such decay modes has been obtained experimentally.⁵ On the other hand, from a theoretical perspective, the two-body hadronic radiative decays $W^{\pm }\to M^{\pm }\gamma$ with M denoting a pseudoscalar or vector meson have been studied in the past literature,^6,7,8,9 and a detailed systematical analysis presented by the authors of Ref. 8 has shown that their branching ratios are maximally around $10^{-8}$ or even smaller. This means that it is really challenging to search for these rare processes.

Very recently, three-body decays $W\to V\ell {\bar{\nu }}_{\ell }$ have been analyzed in Ref. 10, where V denotes the neutral vector particle including light $\rho$, $\omega$, and $\varphi$ mesons or heavy quarkonium $J/\mathrm{\Psi }$ etc, and $\ell$ is the lepton with $\ell =e$ or $\mu$. One can easily find that, in the standard model (SM), the dominant contributions to these decays come from $W\to {\gamma }^{*}\ell {\bar{\nu }}_{\ell }$, followed by ${\gamma }^{*}\to \bar{q}q\to V$, and their branching fractions have been predicted around $10^{-6}\sim 10^{-7}$, which are surprisingly larger than those of $W^{\pm }\to M^{\pm }\gamma$ decays. This may indicate that these three-body rare W decays would be more promising candidates in future experimental facilities, where large amount of W events can be produced.

One would naively expect that the decay rate of $W\to V\ell {\bar{\nu }}_{\ell }$ should be smaller than that of $W\to M\gamma$ since the former is suppressed by a power of ${\alpha }_{\mathrm{\text{em}}}$, where ${\alpha }_{\mathrm{\text{em}}}$ is the electromagnetic coupling constant, compared with the latter one. However, this expectation turns out to be not correct upon closer examination.¹⁰ Actually, a similar story also happens for exclusive rare Z-boson decays.^11,12,13 In particular, it has been shown in Ref. 11 that the dominant contribution to $Z\to V{\ell }^{+}{\ell }^{-}$ comes from $Z\to {\gamma }^{*}{\ell }^{+}{\ell }^{-}$ with the subsequent transition ${\gamma }^{*}\to V$, while the radiative decays $Z\to V\gamma$ are quite suppressed. As pointed out in Ref. 12, the unexpectedly large $Z\to J/\mathrm{\Psi }{\ell }^{+}{\ell }^{-}$ decay rate can be explained by an electromagnetic fragmentation contribution which is not suppressed by a factor $m_J^2/m_Z^2$, and in the fragmentation limit $m_Z\to \infty$ with the ratio of the quarkonium energy $E_J/m_Z$ fixed, the differential rate will factor into electromagnetic decay rates and individual fragmentation functions. The main purpose of the present paper is devoted to the analysis of electromagnetic fragmentation contributions in $W\to V\ell {\bar{\nu }}_{\ell }$ decays in an analogous manner. Here we will focus on heavy vector quarkonium final states including charmonium and bottomonium resonances, since they are in general reconstructed via leptonic decays, which may provide easily identifiable signatures in collider experiments.

The paper is organized as follows. In the next section, we update the calculation of the decay rates of $W\to V\ell {\bar{\nu }}_{\ell }$ for heavy vector quarkonium final states, and examine the differential rates by taking the fragmentation limit directly. The detail analysis of fragmentation contributions to the decays is presented in section 3. Finally, we give our conclusions in section 4.

2. Decay Rates

The decay rate of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ has been calculated in Ref. 10, and in the SM the leading-order Feynman diagrams have been shown in Fig. 1. We first take $J/\mathrm{\Psi }$ production as an example, and the present discussion can be easily extended to the case of other heavy vector quarkonia. Thus from Fig. 1, it is straightforward to get the decay amplitude as

where p, q, $k_1$, and $k_2$ represent the momenta of $W^{-}$, $J/\mathrm{\Psi }$, ${\ell }^{-}$, and ${\bar{\nu }}_{\ell }$, respectively. $k=k_1+k_2$, $Q_c=2/3$ is the electric charge of charm quark, and g is the weak SU(2)_L coupling constant. $f_J$ is the decay constant of $J/\mathrm{\Psi }$, which is defined by Here ${ϵ}_{\mu }^{*}$ is polarization vector of $J/\mathrm{\Psi }$. One can notice that, by using Eq. (2), we have already fulfilled the hadronization of the current $\bar{c}{\gamma }_{\mu }c$ to produce $J/\mathrm{\Psi }$ meson, and most nonperturbative strong interaction effects are confined to this matrix element. However, this is not the usual case. Generally speaking, one should utilize some approaches, such as the color evaporation model,^14,15 the color singlet model,^16,17,18 and the nonrelativistic QCD,¹⁹ to describe heavy quarkonium production.

Squaring the amplitude (1), summing or averaging the polarizations of final or initial particles, and neglecting lepton masses, the differential decay rate of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ can be easily calculated.¹⁰ Its expression is thus given by

where $\lambda =m_J^2/m_W^2$, and and Note that in Ref. 10 there is a factor 2 missed in deriving Eqs. (4), (5), and (6). The dimensionless kinematical variables of the differential rate (3) are defined as and the phase space is given by From the pure leptonic decays of W-boson and $J/\mathrm{\Psi }$, we have and Therefore, we obtain with After completing the integral ${ℐ}$ numerically, and using the experimental data of ${ℬ}(W\to \ell {\bar{\nu }}_{\ell })$ and $\mathrm{\Gamma }(J/\mathrm{\Psi }\to e^{+}{e}^{-})$ given by Particle Data Group,⁵ one can get where the error of our prediction is due to the uncertainties from the experimental data of $J/\mathrm{\Psi }$ and W-boson leptonic decays.

This branching ratio is substantially larger than those of the radiative $W^{\pm }\to M^{\pm }\gamma$ decays, which have been predicted around $10^{-8}$ or even smaller, and explicitly, it has been shown in Ref. 8 that

While from Eq. (3) one will get which contains a large factor $1/\lambda =m_W^2/m_J^2$. It is expected that this factor could overwhelm the extra power of ${\alpha }_{\mathrm{\text{em}}}$, which may thus enhance the decay rate of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ transitions.

Similarly, one can calculate the branching ratios for other heavy vector quarkonia production in rare W-boson decays, which read

It is found that, due to large Upsilon masses and small values of $\mathrm{\Gamma }(\mathrm{\Upsilon }\to e^{+}{e}^{-})$’s, transition rates for bottomonium final states are obviously suppressed.

Note that, in the rest frame of W-boson, $x=2E_J/m_W$. Therefore, the differential decay rate with respect to $J/\mathrm{\Psi }$ energy $E_J$ can be directly obtained by integrating over y in Eq. (3), which has been analyzed numerically in Ref. 10 already. On the other hand, in the limit $m_W\to \infty$ with $E_J/m_W$ fixed, the differential rate would have a concise analytic form. Interestingly, by taking this limit, Eq. (3) will be simplified as

where the ellipsis denotes the terms which will vanish by completing the integration over y and taking $m_W\to \infty$. We keep the second term in the parenthesis since integrating $1/{(1-x-y)}^2$ over y will generate the contribution $O(1/\lambda )$. Thus, after performing the integration over y in Eq. (20), we get This result will be compared to the one by the fragmentation calculation in the next section.

3. Fragmentation Contributions

Now let us study electromagnetic fragmentation contributions to $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ decays. Fragmentation is the process in which an elementary particle or a parton with high momentum splits into a collinear hadron, and the probability for the process can be described by a universal fragmentation function $D(z,{\mu }^2)$ for a parton with invariant mass less than $\mu$ to split into the hadron with longitudinal momentum fraction z. Particularly, as pointed out in Refs. 20 and 21, the direct production of charmonium state in Z-boson decays will be dominated by the fragmentation transition since in which the c and $\bar{c}$ that form the quarkonium are produced in a region of size $O(1/m_c)$, and the rate is only suppressed by a factor of $1/m_c^2$, not by the factor $1/m_Z^2$ for the short-distance transition in Z decays. This is the similar story for our present case. The charmonium $J/\mathrm{\Psi }$ production in $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ decays through the fragmentation mechanism, as shown in Fig. 1, occurs at the region of size $O(1/m_J)$. Therefore, the appearance of the large enhancement factor $m_W^2/m_J^2$ in Eq. (15) can be naturally expected.

The general form of the fragmentation contribution to the differential decay rate for the production of $J/\mathrm{\Psi }$ in Z-boson decays has already been given by the authors of Ref. 20, which was developed in the context of QCD, and the partons include quarks and gluons. This form has been further generalized to the electromagnetic transitions in Ref. 12, in which the partons are photons and charged leptons. In particular, at the leading order in ${\alpha }_{\mathrm{\text{em}}}$ the fragmentation contribution to the energy spectrum of $Z\to J/\mathrm{\Psi }{\ell }^{+}{\ell }^{-}$ can be written as¹²

where $D_{\ell \to J}(z,{\mu }^2)$ is the lepton fragmentation function. The function $P_{\gamma \to J}$ is the probability for a photon to split into a $J/\mathrm{\Psi }$, which, at lowest order, is associated with the photon fragmentation function as with z denoting the longitudinal momentum fraction of the $J/\mathrm{\Psi }$ relative to the photon. The formula (22) explicitly shows that the differential decay rate of $Z\to J/\mathrm{\Psi }{\ell }^{+}{\ell }^{-}$ can be factorized into subprocess decay rates and universal fragmentation functions with $\mu$ acting as the factorization scale. It is believe that the dependence on the arbitrary scale $\mu$ will cancel in final results.

Analogously, for $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ decays, one can straightforwardly obtain a similar fragmentation formula as (22), which reads

The physical meaning of (24) is that, for the first term on the right side, the $W^{-}$ decays into the lepton pair ${\ell }^{-}{\bar{\nu }}_{\ell }$ with energies $m_W/2$ on a distance scale of order $1/m_W$, and the charged lepton then splits into a collinear $J/\mathrm{\Psi }$ and ${\ell }^{-}$ with a separation of order $1/m_J$; for the second term, the $W^{-}$ decays into two leptons and a photon with energy $E_J$ on a distance scale of order $1/m_W$, and subsequently the photon fragments into a $J/\mathrm{\Psi }$ in a region with size of order $1/m_J$. Since in $W^{-}$ decays, only ${\ell }^{-}$ is the fragmenting lepton; while in Z decays, both ${\ell }^{-}$ and ${\ell }^{+}$ can fragment. This is the reason for a factor of 2 in the first term on the right side of the formula (24).

It is obvious that, to further carry out the present analysis, we shall proceed to evaluate the fragmentation function $D_{\ell \to J}(z,{\mu }^2)$, the fragmentation probability $P_{\gamma \to J}$, and the subprocess rates of W decays. Actually, two universal fragmentation functions are independent of the subprocess that creates the fragmenting particle, which have been given by the author of Ref. 12. Furthermore, $\mathrm{\Gamma }(W^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell })$ is already in Eq. (9), what we need do next is therefore to calculate the subprocess rate for $W^{-}\to \gamma (E_J){\ell }^{-}{\bar{\nu }}_{\ell }$ decays.

Before going ahead, first one should try to identify the regions of the $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ phase space, namely, to decide which region is for the lepton fragmentation and which is for the photon fragmentation. As shown in Eq. (24), two terms on the right side are separately dependent of the factorization scale $\mu$, and definitely this scale dependence will cancel between them. By varying the scale $\mu$, some contributions from the lepton fragmentation and the photon fragmentation will transform into each other. This implies that one can make arbitrary choices on the appropriate phase space cutoff. In the present paper, we will do the same choice as that in Ref. 12 by introducing a cutoff on the invariant mass s of the fragmenting lepton ${\ell }^{-}$ and $J/\mathrm{\Psi }$ system. Thus the lepton fragmentation contribution is defined by imposing a cutoff as $s<{\mu }^2$ in the phase space, and the photon fragmentation contribution is considered to be from the remaining region. Of course, this cutoff will translate into a limit on the phase space of the photon spectrum in the following calculation of the subprocess $W^{-}\to \gamma (E_J){\ell }^{-}{\bar{\nu }}_{\ell }$.

The lowest-order decay amplitude for $W^{-}\to \gamma {\ell }^{-}{\bar{\nu }}_{\ell }$ can be derived from Fig. 2. Thus the differential decay rate for the process is

where the kinematical variables $x=2E_{\gamma }/m_W$ and $y=2E_{\ell }/m_W$ defined in the rest frame of W-boson. As discussed above, a cutoff has been introduced to identify the regions of the phase space for the photon fragmentation and the lepton fragmentation, respectively. This leads to the limits on x and y as After integrating over y and taking the limit ${\mu }^2\ll m_W^2$, one can get the photon energy spectrum for $W^{-}\to \gamma {\ell }^{-}{\bar{\nu }}_{\ell }$, which is given by

On the other hand, the universal fragmentation functions can be directly obtained from Ref. 12, which, in our present notations, read

and

Now by inserting Eqs. (27), (28), and (29) into the formula (24), we can get the lepton fragmentation contribution

with $x=2E_J/m_W$, and the photon fragmentation contribution Combining these two terms, the total electromagnetic fragmentation contribution to the differential decay rate for $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ is

It is found that the $\mu$ dependence cancels exactly, and this result is in agreement with Eq. (21), which is reduced from the full calculation in the fragmentation limit. In Fig. 3, we have plotted the $J/\mathrm{\Psi }$ energy distribution of the full calculation from Eq. (3) and the fragmentation calculation of Eq. (32), respectively. Interestingly, the fragmentation contribution can give a result very close to the full calculation.

Finally, let us examine the scale dependence of the lepton fragmentation contribution in Eq. (30) and the photon fragmentation contribution Eq. (31), respectively. As shown in Fig. 4, at the scale $\mu =2m_J$, the photon fragmentation contribution is dominant and the lepton fragmentation part is quite small; while at the scale $\mu =4m_J$, both of them are of the same order. It is clear that, by varying the scale $\mu$, these two kinds of contributions will be dramatically changed individually, although their sum is scale independent.

One can also extend the present analysis to the case of other heavy quarkonium final states ${\mathrm{\Psi }}^{\prime }(2S)$, $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ etc.

4. Conclusions

We have investigated exclusive rare decays $W^{-}\to V{\ell }^{-}{\bar{\nu }}_{\ell }$ with V denoting a heavy vector quarkonium. In the SM, the leading order contributions to these processes come from $W\to {\gamma }^{*}\ell {\bar{\nu }}_{\ell }$, followed by ${\gamma }^{*}\to V$. One may use the measured widths of $\mathrm{\Gamma }(V\to e^{+}{e}^{-})$ given by Particle Data Group⁵ to determine the branching fractions of these transitions. The approach used in this work can, in a large extent, avoid the contamination of nonperturbative QCD effects. Our theoretical predictions on branching ratios for charmonium final states can be up to $10^{-7}$, while they are about $10^{-8}$ or smaller for bottomonium final states. In particular, ${ℬ}(W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell })=8.5\times 10^{-7}$, which is surprisingly larger than those of two-body hadronic radiative decays $W^{\pm }\to M^{\pm }\gamma$.⁸ Experimentally, the heavy quarkonium like $J/\psi$ is in general reconstructed via leptonic decays, which can provide easily detectable clear signal. Meanwhile, exclusive decays of W-boson into hadronic final states have never been observed so far. Therefore, our present study shows that the three-body rare W decays, such as $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$, could be very promising in the future experiments, especially in some facilities with large amount of W events produced. For instance, in the high-luminosity Large Hadron Collider, a huge number of W-bosons, approximately ${𝒪}(10^{11})$, will be accumulated.⁸

The second motivation of the present paper is to show that, the unexpectedly large decay rate of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ can be understood by a fragmentation mechanism. We have analyzed this process in the fragmentation limit $m_W\to \infty$ with $E_J/m_W$ fixed, and the differential decay rate can factor into the subprocess rates $\mathrm{\Gamma }(W^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell })$ and $\mathrm{\Gamma }(W^{-}\to \gamma {\ell }^{-}{\bar{\nu }}_{\ell })$, convoluted with the universal lepton fragmentation and the photon fragmentation functions. In order to calculate these fragmentation contributions, a factorization scale, acting as a cutoff on the phase space of the decay, should be introduced. Our calculation explicitly shows that the fragmentation approximation could lead to a result in agreement with the full calculation. Moreover, the scale dependence of the lepton fragmentation contribution and the photon fragmentation contribution will cancel each other, although these two contributions separately depend on the arbitrary scale dramatically.

Furthermore, according to the fragmentation mechanism, the $J/\mathrm{\Psi }$ production in the exclusive rare W-boson decay occurs in a region with size of order $1/m_J$ instead of a region with size of order $1/m_W$. This will give rise to a large factor $m_W^2/m_J^2$ in the decay rate, which could counteract the suppression of ${\alpha }_{\mathrm{\text{em}}}$. Actually, this mechanism also functions for the processes containing hadronic final particles only, such as the $Z\to J/\mathrm{\Psi }J/\mathrm{\Psi }$ decay.^13,22 It has been pointed out in Ref. 13, due to appearance of the similar large factor $m_Z^2/m_J^2$, the contribution to the decay rate from the electromagnetic dynamics can be even much larger than that from QCD.

Acknowledgments

I am greatly indebted to the editors of this memorial book for giving me the opportunity to write the present paper, dedicated to the memory of a great physicist, Professor Harald Fritzsch (1943–2022). The work was supported in part by the National Natural Science Foundation of China under Grants No. 11575175 and No. 12247103, and by National Research and Development Program of China under Contract No. 2020YFA0406400.