Erratum to: A study of power suppressed contributions in $J/\psi \rightarrow p{\bar{p}}$ decay

Abstract

Few typos and two mistakes has been found in the text of the published version of the paper. Unfortunately these errors provide a strong impact on the numerical results and therefore some of the discussed observations has been changed. The updated text includes the corrected formulas, updated numerical results and conclusions. It is found that the computed power suppressed contribution provides important numerical effect for the observable $\alpha$ but the absolute value of the power suppressed amplitude is considerably smaller than one obtained before. The obtained numerical estimates are more consistent with the effective field theory power counting and provide a better agreement with the existing data.

1 Erratum to: Eur. Phys. J. A https://doi.org/10.1140/epja/s10050-020-00064-5

2 Calculation of the amplitude $A_{2}$

In this section it is important to mention that correct matching also includes the contributions from the expansion of the nucleon spinors.

In order to perform matching let us write the expansion for the amplitudes as

$$\begin{aligned}&A_1=A_1^{(0)}+{\mathcal {O}}(1/m_Q^2), \quad A_2=A_2^{(1)}+{\mathcal {O}}(1/m_Q^2), \nonumber \\&A_2^{(1)}/A_1^{(0)}\sim {\mathcal {O}}(1/m_Q^2). \end{aligned}$$

(1)

In order to expand the nucleon spinors in the definition of the amplitude in Eq. (9) with respect to powers of $1/M_\psi$. we use the equations of the motion

(2)

and perform decompositions of the spinors on large and small components

(3)

Using these definitions and performing the expansion of the amplitude $M[p{\bar{p}}\rightarrow J/\psi ]$ one finds

$$\begin{aligned}&M[p{\bar{p}}\rightarrow J/\psi ]= (\epsilon _{\psi } )_\sigma {{\bar{V}}}_{n}\gamma _{\bot }^\sigma {N}_{{\bar{n}}} ~ A^{(0)}_{1} \nonumber \\&\quad +\frac{1}{4m_{N}} \left\{ \left( \epsilon _{\psi }\cdot n\right) k_{-}^{\prime }-k_{+}\left( \epsilon _{\psi }\cdot {\bar{n}}\right) \right\} ~{{\bar{V}}}_{n}{N}_{{\bar{n}}} \end{aligned}$$

(4)

$$\begin{aligned}&\quad \times \left[ ~A^{(1)}_{2}+\frac{4m_{N}^{2}}{ k_{+}k_{-}^{\prime } }A^{(0)}_{1}\right] , \end{aligned}$$

(5)

where we only keep the leading-order contributions in front of the nucleon bispinors. This expression shows that the coefficient in front of chiral odd combination ${{\bar{V}}}_{n}{N}_{{\bar{n}}}$ includes the spin flip amplitude $A^{(1)}_{2}$ and the kinematical power correction $\sim A^{(0)}_{1}$. On the other hand, the expression for this coefficient can be computed in the EFT framework.

Therefore instead of Eq. (55) one obtains

$$\begin{aligned} A^{(0)}_{2}=\frac{m_{N}^{2}}{M_{\psi }^{2}}\frac{f_{\psi }}{~m_{Q}^{2}}\frac{ ~f_{N}^{~2}}{m_{Q}^{4}}~(\pi \alpha _{s})^{3}\frac{10}{81}~J-\frac{4m_{N}^{2}}{ M_\psi ^2}A^{(0)}_{1} , \end{aligned}$$

(6)

where the dimensionless convolution integral J reads

$$\begin{aligned} J= & {} \frac{1}{4}\left( J_{1}[V_{1},{\mathcal {V}} _{i}]+J_{2}[A_{1},{\mathcal {V}}_{i}]\nonumber \right. \\&\left. +J_{3}[V_{1},{\mathcal {A}}_{i}]+J_{4}[A_{1}, {\mathcal {A}}_{i}]+ J_{5}[T_{1} ,{\mathcal {T}}_{ij}]\right) . \end{aligned}$$

(7)

Some of the expressions for the hard coefficients $\{K_{n},L_{n},N_{n}\}$ have also been corrected, namely

$$\begin{aligned} K_{2}({\mathcal {V}}_{i}; x_{i},y_{i})&={\mathcal {V}}_{1}(x_{i})\left( \frac{x_{3}-y_{3}}{x_{1}} +2x_{1}-2y_{1}+2y_{3}\right) \nonumber \\&\quad +{\mathcal {V}}_{2}(x_{i})~2\left( x_{1}-y_{1}+\frac{x_{1}x_{3}}{x_{2}} \right) , \end{aligned}$$

(8)

$$\begin{aligned} L_{2}({\mathcal {V}}_{i}; x_{i},y_{i})&={\mathcal {V}}_{1}(x_{i})\left( \frac{x_{2}-y_{2}}{x_{1}} +2x_{2}\right) \nonumber \\&\quad +{\mathcal {V}}_{2}(x_{i})\left( \frac{y_{1}-x_{1}}{ x_{2}}-2x_{1}\right) , \end{aligned}$$

(9)

$$\begin{aligned} N_{2}({\mathcal {V}}_{i}; x_{i},y_{i})&={\mathcal {V}}_{1}(x_{i})2 \left( y_2-x_2 -\frac{x_{2}x_{3}}{x_{1}} \right) \nonumber \\&\quad +{\mathcal {V}}_{2}(x_{i})\left( \frac{y_{3}-x_{3}}{x_{2}}-2x_2+2y_2 -2y_{3}\right) , \end{aligned}$$

(10)

$$\begin{aligned} K_{3}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i})\left( \frac{y_{3}-x_{3}}{x_{1}}-2x_{1}+2y_{1}-2y_{3}\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i})~2\left( y_{1}-x_{1}-\frac{x_{1}x_{3}}{x_{2}}\right) , \end{aligned}$$

(11)

$$\begin{aligned} L_{3}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i})\left( \frac{y_{2}-x_{2}}{x_{1}}-2x_2\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i})\left( \frac{x_{1}-y_{1}}{x_{2}}+2x_1\right) , \end{aligned}$$

(12)

$$\begin{aligned} N_{3}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i}) 2\left( x_2-y_2+\frac{x_{2}x_{3}}{x_{1}}\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i})\left( \frac{x_{3}-y_{3}}{x_{2}}+2x_2-2y_2 +2y_{3}\right) , \end{aligned}$$

(13)

$$\begin{aligned} K_{4}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i}) \left( \frac{x_{3}-y_{3}}{x_{1}}+2\frac{y_{1}-x_{1}}{x_{3}}+2y_{3}-2y_{1}-2x_{1}\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i}) 2\left( \frac{y_{1}-x_{1}}{x_{3}}-\frac{x_{1}x_{3}}{x_{2}}-x_{1}-y_{1}\right) , \end{aligned}$$

(14)

$$\begin{aligned} L_{4}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i}) \left( \frac{x_{2}-y_{2}}{x_{1}}+4\frac{x_{1}x_{2}}{x_{3}}+2x_{2}\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i}) \left( \frac{x_{1}-y_{1}}{x_{2}}+4\frac{x_{1}x_{2}}{x_{3}}+2x_{1}\right) , \end{aligned}$$

(15)

$$\begin{aligned} N_{4}({\mathcal {A}}_{i}; x_{i},y_{i})&={\mathcal {A}}_{1}(x_{i}) ~2\left( \frac{y_{2}-x_{2}}{x_{3}}-\frac{x_{2}x_{3}}{x_{1}}-y_{2}-x_2\right) \nonumber \\&\quad +{\mathcal {A}}_{2}(x_{i}) \left( \frac{x_{3}-y_{3}}{x_{2}}+2\frac{y_{2}-x_{2}}{x_{3}}+2y_{3}-2y_{2}-2x_2\right) . \end{aligned}$$

(16)

3 Phenomenology

The expression for the decay width reads

$$\begin{aligned} \Gamma [J/\psi \rightarrow p{\bar{p}}]=\frac{M_{\psi }\beta }{12\pi } \left( \left| {\mathcal {G}}_{M}\right| ^{2}+\frac{2m_{N}^{2}}{M_{\psi }^{2}}\left| {\mathcal {G}}_{E}\right| ^{2}\right) , \end{aligned}$$

(17)

where we introduced the helicity amplitudes

$$\begin{aligned} {\mathcal {G}}_{M}=A_{1}+A_{2}, \quad {\mathcal {G}}_{E}=A_{1}+\frac{M_{\psi }^{2}}{4m_{N}^{2}}A_{2}. \end{aligned}$$

(18)

Using Eqs. (12) and (6) one finds

$$\begin{aligned} {\mathcal {G}}_{M}\simeq A^{(0)}_{1}=A_0 I_0, \quad {\mathcal {G}}_{E}=A_{0}~\frac{1}{4} J. \end{aligned}$$

(19)

where we introduced the convenient normalisation factor

$$\begin{aligned} A_{0}=\frac{f_{\psi }}{~m_{c}^{2}}\frac{~f_{N}^{~2}}{m_{c}^{4}}~(\pi \alpha _{s})^{3}\frac{10}{81}. \end{aligned}$$

(20)

The four-dimensional convolution integrals can be easily computed numerically. The explicit results for different convolution integrals are presented in Appendix C.

The values of the corresponding parameters are given in the Table 1. Performing the required calculations we obtain the following results.

Table 1 The LCDA parameters for the different models at $\mu ^{2}=4~\text {GeV}^{2}$

ABO-model. Let us consider first the ratio ${\mathcal {G}}_E / {\mathcal {G}}_M$, which is less sensitive to the value of the factorisation scale $\mu$. We obtain

$$\begin{aligned} \frac{ {\mathcal {G}}_E }{ {\mathcal {G}}_M }=\frac{J}{4 I_0} =0.67 ,\quad (\text {or }\alpha =0.71). \end{aligned}$$

(21)

This result is valid for all values of $\mu$ as described above. The obtained value is about 20% smaller than the experimental value and can be accepted as a reasonable leading-order approximation. The dominant numerical effect in ${\mathcal {G}}_{E}$ is provided by the chiral-odd integral $J_5$ (7) or more explicitly: by the asymptotic term in WW contribution and by the genuine twist-4 contribution $\sim \lambda _1\eta _{11}$, see more details in (8). Neglecting the contributions of the integrals $J_{1,2,3,4}$ and the momenta $\varphi _{ij}$ in (8) one finds

$$\begin{aligned} \left. \frac{J}{4 I_0}\right| _{J_{1,2,3,4}=0,\, \varphi _{ij}=0}=0.23-0.41\frac{\lambda _1\eta _{11}}{f_N}=0.558, \end{aligned}$$

(22)

which must be compared with the exact value in Eq.(21). This is a good illustration of the important role, which is provided by the 3-quark configuration with the orbital momentum $L_q=1$. The effect of the amplitude ${\mathcal {G}}_E$ in the branching ratio is small because it is suppressed by the factor $2m^2_N/M^2_\psi$, see Eq. (17). For the different choices of the factorisation scale $\mu ^2=2m_c^2-1.5$ GeV$^2$ we obtain

$$\begin{aligned} 10^{3}\text {Br}[J/\psi \rightarrow p{\bar{p}}]\simeq 0.47-1.43, \end{aligned}$$

(23)

while the experimental branching ratio reads [16]

$$\begin{aligned} 10^{3}\text {Br}[J/\psi \rightarrow p{\bar{p}}]_{\text {exp}}\simeq 2.112\pm 0.004. \end{aligned}$$

(24)

Hence we observe, that a reliable description of the branching in this model can only be obtained for the relatively small values of the renormalisation scale $\mu$.

Lattice model. In this case the coupling $f_{N}$ is smaller, see Table 1 and genuine twist-4 parameters $\eta _{ij}$ are unknown. If one sets them to zero $\eta _{10}=0$ and $\eta _{11}=0$, one gets very small value for the ratio

$$\begin{aligned} \frac{ {\mathcal {G}}_E }{ {\mathcal {G}}_M }=\frac{J}{4 I_0} =0.10,\quad (\alpha =0.99). \end{aligned}$$

(25)

Because the normalisation constant $f_N$ is also smaller, one also obtains much smaller branching ratio

$$\begin{aligned} 10^{3}\text {Br}[J/\psi \rightarrow p{\bar{p}}]= 0.08-0.22, \end{aligned}$$

(26)

The value of the ratio $\alpha$ can be easily improved taking into account the higher order twist-4 term with $\eta _{11}$. For instance, taking $\eta _{11}=0.09$ yields

$$\begin{aligned} \frac{ {\mathcal {G}}_E }{ {\mathcal {G}}_M }=\frac{J}{4 I_0} =0.77,\quad (\alpha =0.64). \end{aligned}$$

(27)

On the other hand, it is very difficult to improve the value of the branching fraction. Hence, we can conclude that for such a small $f_N$ it is very likely that a large effect from various corrections or an effect from another decay mechanism is relevant.

COZ+ model. In this model it is very important to take into account the higher coefficients $\varphi _{2i}$, which provide a strong numerical effect. Corresponding LCDA $\varphi _3$ provides sufficiently large value for the leading-twist integral $I_{0}$. This results in the large value of the amplitude ${\mathcal {G}}_M$ and reduces the value of the ratio

$$\begin{aligned} \frac{ {\mathcal {G}}_E }{ {\mathcal {G}}_M }=0.30 ,\quad (\alpha =0.93). \end{aligned}$$

(28)

Taking $\lambda _1=0$ one obtains even smaller value ${\mathcal {G}}_E /{\mathcal {G}}_M =0.20$. At the same time the value of the branching ratio is large

$$\begin{aligned} 10^{3}\text {Br}[J/\psi \rightarrow p{\bar{p}}] \simeq 4.3-16.1, \end{aligned}$$

(29)

where, remind, we assume the variation of the scale $\mu ^2=1.5-2m_c^2$. It seems, that the leading-twist contribution in this case is overestimated that strongly reduces the value ${\mathcal {G}}_E / {\mathcal {G}}_M$.

A more accurate phenomenological consideration must also include the electromagnetic contribution, which describes the subprocess $J/\psi \rightarrow \gamma ^*\rightarrow p {{\bar{p}}}$. The main conclusion, which follows from the present calculations, is a qualitative estimate of the possible effect from the helicity flip amplitudes ${\mathcal {G}}_E$. We find that the obtained result for the helicity flip amplitude ${\mathcal {G}}_E$ is quite sensitive to the genuine twist-4 nucleon DAs and can be quite large comparing with the well known leading-twist contribution in ${\mathcal {G}}_{M}$. However the contribution of ${\mathcal {G}}_E$ to the width is numerically reduced by the power $m_N^2/M_\psi ^2$ in Eq. (17) and therefore does not provide a strong numerical impact on the value of the branching fraction. Qualitatively this provides a reliable description of the data using the models of DA motivated by the light-cone QCD sum rules [22]. Such a picture suggests that the twist-four LCDAs, describing the three quarks with the orbital angular momentum $L=1$, are very important for the description of the angular behaviour of the cross section $e^+e^-\rightarrow J/\psi \rightarrow p{{\bar{p}}}$ .

4 Discussion

The different models of nucleon DAs have been used in order to compare obtained results with the existing data. The best description is obtained with the set of DAs obtained from light-cone QCD sum rules in Ref. [22]. In this case we find a reliable estimate for the ratio ${\mathcal {G}}_{E}/{\mathcal {G}}_{M}$, which describes the angular behaviour of the cross section. It is found that the Fock component of the nucleon wave function associated with the three quarks in P-wave state provides an important effect in the description of this observable. This allows one to conclude that quarkonia decays into baryon–antibaryon provide us an interesting and important insight about baryon wave functions.

The obtained description of the decay width allows one to obtain an acceptable estimate only taking the relatively low value of the renormalisation scale $\sim 1.5$ GeV$^2$. This observation opens questions about the size of the next-to-leading and power corrections to the amplitude $A_1$. For better understanding of the decay mechanism such corrections must be computed.

The lattice data [24] suggests relatively small value of the non-perturbative normalisation constant $f_N$, which makes the description of the width very problematical, because the amplitude ${\mathcal {G}}_{M}$ turns out to be very small. Therefore if such value of the $f_N$ is correct, one must expect a large contribution from other decay mechanism. The COZ model [25] works quite differently, it provides a large value of the width but on the other hand gives very small value of the ratio ${\mathcal {G}}_{E}/{\mathcal {G}}_{M}$. Most likely, this indicates that the value of the amplitude ${\mathcal {G}}_{M}$ in this case is somewhat overestimated.

Appendices

Appendix

C The convolution integrals

Here we present the results for the convolution integrals. All the relevant moments $\varphi _{ij}$, $\lambda _{1}$ and $\eta _{10,11}$ are multiplicatively renormalisable and their scale dependence are not shown for simplicity. After integrations over the momentum fractions the following expressions have been obtained

$$\begin{aligned}&I_{0}=120^2 \{ 0.1054+0.2282\varphi _{11}+0.4846\varphi _{20}\nonumber \\&\quad -0.0165\varphi _{22} +0.3958\varphi _{10}^{2}+0.6273\varphi _{10}\varphi _{21} \nonumber \\&\quad +0.370~\varphi _{11}^{2} +0.5413\varphi _{11}\varphi _{20} -0.0147\varphi _{11}\varphi _{22}\nonumber \\&\quad +0.6371\varphi _{20}^2-0.0456\varphi _{20}\varphi _{22} \nonumber \\&\quad +0.2283\varphi _{21}^{2} -0.0004\varphi _{22}^{2} \}, \end{aligned}$$

(C.1)

$$\begin{aligned}&J_{1}[V_{1},{\mathcal {V}}_{i}] + J_{3}[V_{1},{\mathcal {A}}_{i}]=120\{ P_0[\varphi _{ij}]+ \lambda _{1}/ f_{N } P_1[\varphi _{ij},\eta _{1k}] \},\nonumber \\ \end{aligned}$$

(C.2)

where

$$\begin{aligned}&P_0[\varphi _{ij}]= 23.6524 -31.2198 \varphi _{10} -16.9824 \varphi _{11} \nonumber \\&\quad +91.7657 \varphi _ {20} -25.9159 \varphi _{21} -9.1100\varphi _{22} \nonumber \\&\quad +89.8059\varphi _{10}^2 -64.5699 \varphi _{10} \varphi _{11} -24.9722 \varphi _{10} \varphi _{20}\nonumber \\&\quad +75.5738 \varphi _{10} \varphi _{21} -32.4481 \varphi _{10} \varphi _{22} \nonumber \\&\quad +11.5448 \varphi _{11}^2 -96.2095 \varphi _{11} \varphi _{20} +87.5753 \varphi _{11} \varphi _{21}\nonumber \\&\quad +9.6300 \varphi _{11} \varphi _{22} +61.6547 \varphi _{20}^2 \nonumber \\&\quad -7.3150 \varphi _{20} \varphi _{21} -21.9330\varphi _{20} \varphi _{22} \nonumber \\&\quad +12.6407 \varphi _{21}^2 +11.2552 \varphi _{21} \varphi _{22} + 0.1585 \varphi _{22}^2, \end{aligned}$$

(3)

$$\begin{aligned}&P_1[\varphi _{ij},\eta _{1k}]= 57.5592 \varphi _{10} -19.1864\varphi _{11} \nonumber \\&\quad -2.45327\varphi _{20} +36.799 \varphi _{21} -4.9061\varphi _{22} \nonumber \\&\quad +\eta _{11} (-41.3263 +31.0521\varphi _{10} -10.3507 \varphi _{11}\nonumber \\&\quad -91.8392 \varphi _{20} +23.7966 \varphi _{21} +9.72038\varphi _{22}) \nonumber \\&\quad +\eta _{10} ( -149.845\varphi _{10} +49.9484 \varphi _{11} \nonumber \\&\quad +1.2259 \varphi _{20} -18.3879 \varphi _{21} + 2.4517 \varphi _{22}). \end{aligned}$$

(4)

$$\begin{aligned}&J_{2}[A_{1},{\mathcal {V}}_{i}]+J_{4}[A_{1},{\mathcal {A}}_{i}]=120\{(\varphi _{10}+\varphi _{11}) \tilde{P}_0[\varphi _{ij},\eta _{1k}]\nonumber \\&\quad +(\varphi _{20}+5\varphi _{21}+2\varphi _{22}) {{\tilde{P}}}_1[\varphi _{ij},\eta _{1k}]\}, \end{aligned}$$

(5)

where

$$\begin{aligned} {{\tilde{P}}}_0[\varphi _{ij},\eta _{1k}]= & {} 31.2198 +29.9353 \varphi _{10} +34.6345 \varphi _{11} \nonumber \\&+4.6600\varphi _{20} -46.0060 \varphi _{21} -8.1761 \varphi _{22} \nonumber \\&+\lambda _1/f_N ( 19.1864 -49.9484\eta _{10} -31.0521\eta _{11} ), \end{aligned}$$

(6)

$$\begin{aligned} {{\tilde{P}}}_1[\varphi _{ij},\eta _{1k}]= & {} 6.0598 +7.5620 \varphi _{10} -14.9913\varphi _{11} \nonumber \\&+4.7182 \varphi _{20} +1.6796 \varphi _{21} -1.1247 \varphi _{22} \nonumber \\&+\lambda _1/f_N ( 2.4533 -1.2259 \eta _{10} -4.7593\eta _{11}). \end{aligned}$$

(7)

$$\begin{aligned} J_{5}[T_{1},{\mathcal {T}}_{ij}]= & {} 120\{ P_{t0}[\varphi _{ij}]+ \lambda _1 \eta _{11} /f_N P_{t1}[\varphi _{ij}] \}, \end{aligned}$$

(8)

where

$$\begin{aligned} P_{t0}[\varphi _{ij}]= & {} 47.3049 +28.4748 \varphi _{11}+195.6510\varphi _{20}\nonumber \\&+8.7661 \varphi _{21}+6.0192\varphi _{22} \nonumber \\&+92.3587\varphi _{11}^2 -213.08156\varphi _{11} \varphi _{20}\nonumber \\&-66.7746 \varphi _{11} \varphi _{21}-57.0375\varphi _{11} \varphi _{22} \nonumber \\&+132.7459\varphi _{20}^2+26.8001\varphi _{20} \varphi _{21}\nonumber \\&-27.2425 \varphi _{20} \varphi _{22}-0.2418 \varphi _{21}\varphi _{22} \nonumber \\&-4.1821\varphi _{22}^2, \end{aligned}$$

(9)

$$\begin{aligned} P_{t1}[\varphi _{ij}]= & {} -82.6526-82.8057 \varphi _{11}-193.1971\varphi _{20}+0.4035 \varphi _{22}.\nonumber \\ \end{aligned}$$

(10)