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

Abstract

The power suppressed amplitude which describes the Pauli ($\sigma ^{\mu \nu }$) coupling in the $J/\psi \rightarrow p\bar{p}$ decay is calculated within the effective field theory framework. It is shown that at the leading-order approximation this contribution is factorisable and the overlap with the hadronic final state can be described by collinear matrix elements. The obtained contribution depends on the nucleon light-cone distribution amplitudes of twist-3 and twist-4. This result is used for a qualitative phenomenological analysis of existing data for $J/\psi \rightarrow p\bar{p}$ decay: branching ratio and the angular distribution in the cross section $e^+e^-\rightarrow J/\psi \rightarrow p\bar{p}$. It is found that the power corrections provide a large numerical effect.

Appendices

Appendix

Long distance matrix elements

Here we provide a brief summary of the required nonperturbative matrix elements and LCDAs. For the heavy quark sector we only need the NRQCD matrix element

$$\begin{aligned} \langle 0\vert \chi _{\omega }^{\dag }(0)\gamma ^{\mu }\psi _{\omega }(0)\vert P\rangle =\epsilon _{\psi }^{\mu }~f_{\psi }. \end{aligned}$$

(A.1)

The operator in (A.1) is constructed from the quark $\psi _{\omega }$ and antiquark $\chi _{\omega }^{\dag }$ four-component spinor fields satisfying , . The coupling $f_\psi$ is related with the radial wave function at the origin

$$\begin{aligned} f_{\psi }=\sqrt{2M_{J/\psi }}\sqrt{\frac{3}{2\pi }}~R_{10}(0). \end{aligned}$$

(A.2)

The value $R_{10}(0)$ is well known from various potential models, for instance for the Buchmuller–Tye potential [35]

$$\begin{aligned} \left| R_{10}(0)\right| ^{2}\simeq 0.81\text {GeV}^{3}. \end{aligned}$$

(A.3)

One can also estimate this coupling from $J/\psi \rightarrow e^{+}e^{-}$ decay using the well known formula for the leptonic width

$$\begin{aligned} \Gamma [J/\psi \rightarrow e^{+}e^{-}]=\frac{16}{9}\frac{\alpha _{em}^{2}}{M_{\psi }^{2}}\left| R_{10}(0)\right| ^{2}\left( 1-\frac{ 16}{3}\frac{\alpha _{s}}{\pi }\right) . \end{aligned}$$

(A.4)

This gives ( Br$[J/\psi \rightarrow e^{+}e^{-}]=5.97\%,~\alpha _{s}=0.3,~\alpha _{em}=1/130)$

$$\begin{aligned} \left| R_{10}(0)\right| ^{2}\simeq 0.76~\text {GeV}^{3}, \end{aligned}$$

(A.5)

which is quite close to the value (A.3).

The nucleon matrix elements are more complicated. In the definitions given below we use kinematics and notations introduced in the Sect. 2. For simplicity, in this Appendix we consider the matrix elements only for the nucleon state and we also imply the light-cone gauge

$$\begin{aligned} n\cdot A^{(\bar{n})}(x)=0, \end{aligned}$$

(A.6)

in order to simplify the formulas.

The twist-3 DAs are defined as (i, j, k are the colour indices)

(A.7)

where

$$\begin{aligned} \text {FT}\left[ F(y_{i})\right]= & {} \int Dy_{i}~e^{-iy_{1}k_{-}^{\prime }z_{1+}/2-iy_{2}k_{-}^{\prime }z_{2+}/2-iy_{3}k_{-}^{\prime }z_{3+}/2}\nonumber \\&\quad F(y_{1,}y_{2},y_{3}), \end{aligned}$$

(A.8)

with

$$\begin{aligned} Dy_{i}=dy_{1}dy_{2}dy_{3}\delta (1-y_{1}-y_{2}-y_{3}). \end{aligned}$$

(A.9)

We also explicitly write the large component of the nucleon spinor

(A.10)

Three DAs $V_{1}$, $A_{1}$ and $T_{1}$ can be combined into the one twist-3 DA $\varphi _{3}$ as

$$\begin{aligned} V_{1}(x_{1},x_{2},x_{3})&=f_{N}\frac{1}{2}\left[ \varphi _{3}(x_{1},x_{2},x_{3})+\varphi _{3}(x_{2},x_{1},x_{3})\right] \nonumber \\ \end{aligned}$$

(A.11)

$$\begin{aligned} A_{1}(x_{1},x_{2},x_{3})&=f_{N}\frac{1}{2}\left[ \varphi _{3}(x_{2},x_{1},x_{3})-\varphi _{3}(x_{2},x_{1},x_{3})\right] \nonumber \\ \end{aligned}$$

(A.12)

$$\begin{aligned} T_{1}(x_{1},x_{2},x_{3})&=f_{N}\frac{1}{2}\left[ \varphi _{3}(x_{1},x_{3},x_{2})+\varphi _{3}(x_{2},x_{3},x_{1})\right] . \nonumber \\ \end{aligned}$$

(A.13)

The twist-4 LCDAs are defined as

(A.14)

where

$$\begin{aligned} \sigma _{-+}=\sigma _{\mu \nu }\bar{n}^{\mu }n^{\nu }. \end{aligned}$$

(A.15)

These LCDAs can be written in terms of three twist-4 LCDAs in the following way

$$\begin{aligned} V_{2}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}\left[ \Phi _{4}(x_{1},x_{2},x_{3})+\Phi _{4}(x_{2},x_{1},x_{3})\right] , \nonumber \\\end{aligned}$$

(A.16)

$$\begin{aligned} V_{3}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}\left[ \Psi _{4}(x_{1},x_{2},x_{3})+\Psi _{4}(x_{2},x_{1},x_{3})\right] , \nonumber \\ \end{aligned}$$

(A.17)

$$\begin{aligned} A_{2}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}\left[ \Phi _{4}(x_{2},x_{1},x_{3})-\Phi _{4}(x_{1},x_{2},x_{3})\right] , \nonumber \\\end{aligned}$$

(A.18)

$$\begin{aligned} A_{3}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}\left[ \Psi _{4}(x_{2},x_{1},x_{3})-\Psi _{4}(x_{1},x_{2},x_{3})\right] , \nonumber \\\end{aligned}$$

(A.19)

$$\begin{aligned} T_{3}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}[ \Xi _{4}(x_{1},x_{2},x_{3})+\Psi _{4}(x_{3},x_{1},x_{2})\nonumber \\&+\Phi _{4}(x_{2},x_{3},x_{1})] +(x_{1}\leftrightarrow x_{2}), \end{aligned}$$

(A.20)

$$\begin{aligned} S_{1}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}[ \Xi _{4}(x_{1},x_{2},x_{3})+\Psi _{4}(x_{3},x_{1},x_{2})\nonumber \\&+\Phi _{4}(x_{2},x_{3},x_{1})] -(x_{1}\leftrightarrow x_{2}), \end{aligned}$$

(A.21)

$$\begin{aligned} T_{7}(x_{1},x_{2},x_{3})= & {} \frac{1}{4}[ -\Xi _{4}(x_{1},x_{2},x_{3})+\Psi _{4}(x_{3},x_{1},x_{2})\nonumber \\&+\Phi _{4}(x_{2},x_{3},x_{1})] +(x_{1}\leftrightarrow x_{2}), \end{aligned}$$

(A.22)

$$\begin{aligned} P_{1}(x_{1},x_{2},x_{3})= & {} -\frac{1}{4}[ -\Xi _{4}(x_{1},x_{2},x_{3})+\Psi _{4}(x_{3},x_{1},x_{2})\nonumber \\&+\Phi _{4}(x_{2},x_{3},x_{1})] +(x_{1}\leftrightarrow x_{2}). \end{aligned}$$

(A.23)

In our calculation we use the matrix elements of twist-4 operators constructed from the large collinear components $\chi _{\bar{n}}$ (32) and their derivative $\partial _{\bot }\chi _{\bar{n}}$, see Eq. (39). In order to find expressions for these matrix elements we need to consider off light-cone correlators. The chiral even correlators have already been considered in Ref. [22]. Consider, for simplicity, the vector projection. The corresponding correlator reads

$$\begin{aligned}&-\langle 0\vert \varepsilon ^{ijk}u^{i}(z_{1})C\gamma ^{\alpha }u^{j}(z_{2})d_{\sigma }^{k}(z_{3})\vert k\rangle \nonumber \\&\quad =k^{\alpha } \left[ \gamma _{5}N\right] _{\sigma }\text {FT}\left[ V_{1}\right] +m_{N}\left[ \gamma ^{\alpha }\gamma _{5}N\right] _{\sigma }\text {FT}\left[ V_{3}\right] \nonumber \\&\qquad +\,m_{N}ik^{\alpha }\left( z_{1\beta }~\text {FT}\left[ \mathcal {V}_{1}\right] +z_{2\beta }~\text {FT}\left[ \mathcal {V}_{2}\right] \right. \nonumber \\&\qquad \left. +\,z_{3\beta }~\text {FT} \left[ \mathcal {V}_{3}\right] \right) \left[ \gamma ^{\beta }\gamma _{5}N \right] _{\sigma }. \end{aligned}$$

(A.24)

By calligraphic letters we denote the auxiliary LCDAs, which can be rewritten in terms of defined above in Eq. (A.14) twist-4 LCDAs. The explicit expressions will be given below. Performing expansion of the operator in the lhs (A.24) according to formulas (21), expanding on the rhs the coordinates $z_{i}\simeq (z_{i}\bar{n} )n/2+z_{i\bot }$ in $z_{i\bot }$ and comparing the linear in $z_{i\bot }$ contributions one finds ( in this section we denote $\xi _{\bar{n}}(x)\equiv \xi (x)$ in order to simplify notations)

(A.25)

(A.26)

For the axial projection one has

$$\begin{aligned}&-\left\langle 0\left| \varepsilon ^{ijk}u^{i}(z_{1})C\gamma ^{\alpha }\gamma _{5}u^{j}(z_{2})d_{\sigma }^{k}(z_{3})\right| k\right\rangle \nonumber \\&\quad =k^{\alpha }\left[ N\right] _{\sigma }\text {FT}\left[ A_{1}\right] +m_{N} \left[ \gamma ^{\alpha }N\right] _{\sigma }\text {FT}\left[ A_{3}\right] \nonumber \\&\qquad +\,m_{N}ik^{\alpha }\left( z_{1\beta }~\text {FT}\left[ \mathcal {A}_{1}\right] +z_{2\beta }~\text {FT}\left[ \mathcal {A}_{2}\right] +z_{3\beta }~\text {FT} \left[ \mathcal {A}_{3}\right] \right) \left[ \gamma ^{\beta }N\right] _{\sigma }, \end{aligned}$$

(A.27)

The expansion around the light-cone direction gives

(A.28)

(A.29)

We also need to consider the chiral-odd correlator

$$\begin{aligned}&-\langle 0\vert \varepsilon ^{ijk}u_{\alpha }^{i}(z_1)C\sigma ^{\mu \nu }u_{\beta }^{j}(z_2)d_{\sigma }^{k}(z_3)\vert k\rangle \nonumber \\&\quad =~ip^{\nu }\left[ \gamma ^{\mu }\gamma _{5}N\right] \text {FT}\left[ T_{1}\right] +~\frac{1}{2}m_{N}~\left[ \sigma ^{\mu \nu }\gamma _{5}N\right] \text {FT}\left[ T_{7}\right] \nonumber \\&\qquad +\,\left( z_1-z_3\right) ^{\mu }p^{\nu }~m_{N}\left[ \gamma _{5}N\right] \text {FT }\left[ \mathcal {T}_{21}\right] \nonumber \\&\qquad +~\left( z_2-z_3\right) ^{\mu }p^{\nu }~m_{N}\left[ \gamma _{5}N\right] \text { FT}\left[ \mathcal {T}_{22}\right] \nonumber \\&\qquad +~m_{N}~ip^{\nu }(z_1-z_3)_{\beta }\left[ \sigma ^{\mu \beta }\gamma _{5}N\right] \text {FT}\left[ \mathcal {T}_{41}\right] \nonumber \\&\qquad +~m_{N}~ip^{\nu }(z_2-z_3)_{\beta }\left[ \sigma ^{\mu \beta }\gamma _{5}N\right] \text {FT}\left[ \mathcal {T}_{42}\right] \nonumber \\&\qquad -(\mu \leftrightarrow \nu ). \end{aligned}$$

(A.30)

This equation yields

(A.31)

(A.32)

The LCDAs which are denoted by calligraphic letters can be rewritten in terms of the light-cone LCDAs which are defined by the light-cone matrix element (A.14). For the LCDAs $\mathcal {V}_{1,2}$ and $\mathcal {A}_{1,2}$ such expressions are already derived in Ref. [22]. We also recalculated these relations and find the same expressions. They read

$$\begin{aligned}&\mathcal {V}_{1}(x_{i})+\mathcal {V}_{3}(x_{i})+\mathcal {V}_{3}(x_{i})=0, \end{aligned}$$

(A.33)

$$\begin{aligned}&4\mathcal {V}_{k}(x_{i})=x_{3}V_{2}(x_{i})+(-1)^{k}\{ (x_{1}-x_{2})V_{3}(x_{i})\nonumber \\&\quad -\,x_{3}A_{2}(x_{i})+\bar{x}_{3}A_{3}(x_{i})\} . \end{aligned}$$

(A.34)

$$\begin{aligned}&\mathcal {A}_{1}(x_{i})+\mathcal {A}_{3}(x_{i})+\mathcal {A}_{3}(x_{i})=0, \end{aligned}$$

(A.35)

$$\begin{aligned}&4\mathcal {A}_{k}(x_{i})=-x_{3}A_{2}(x_{i})+(-1)^{k}\{ (x_{1}-x_{2})A_{3}(x_{i})\nonumber \\&\quad +\,x_{3}V_{2}(x_{i})+\bar{x}_{3}V_{3}(x_{i})\} . \end{aligned}$$

(A.36)

Notice that

$$\begin{aligned} \mathcal {V}_{2}(x_{1},x_{2},x_{3})= & {} \mathcal {V}_{1}(x_{2},x_{1},x_{3}),\nonumber \\ \mathcal {A}_{2}(x_{1},x_{2},x_{3})= & {} -~\mathcal {A}_{1}(x_{2},x_{1},x_{3}), \end{aligned}$$

(A.37)

which follows from

$$\begin{aligned} V_{i}(2,1,3)=V_{i}(1,2,3),~ A_{i}(2,1,3)=-A_{i}(1,2,3). \end{aligned}$$

(A.38)

The similar relations for the chiral-odd LCDAs $\mathcal {T}_{ij}$ have not yet been considered. Our calculations yield (see the details below)

$$\begin{aligned}&\mathcal {T}_{21}(x_{i})-\mathcal {T}_{41}(x_{i}) = \frac{x_{1}}{2}\left( T_{3}+T_{7}+S_{1}-P_{1}\right) (x_{i}) \end{aligned}$$

(A.39)

$$\begin{aligned}&\quad =\frac{x_{1}}{2}~\left[ (V_{3}-A_{3})(3,1,2)+(V_{2}-A_{2})(2,3,1)\right] , \nonumber \\\end{aligned}$$

(A.40)

$$\begin{aligned}&\mathcal {T}_{22}(x_{i})-\mathcal {T}_{42}(x_{i})=\frac{x_{2}}{2}\left( T_{3}+T_{7}-S_{1}+P_{1}\right) (x_{i}) \end{aligned}$$

(A.41)

$$\begin{aligned}&\quad =\frac{x_{2}}{2}~\left[ (V_{3}-A_{3})(3,2,1)+(V_{2}-A_{2})(1,3,2)\right] , \nonumber \\\end{aligned}$$

(A.42)

$$\begin{aligned}&\mathcal {T}_{41}+\mathcal {T}_{21}=\frac{x_{1}}{2}\left( T_{3}-T_{7}+P_{1}+S_{1}\right) =\frac{x_{1}}{2}\Xi _{4}(1,2,3), \nonumber \\\end{aligned}$$

(A.43)

$$\begin{aligned}&\mathcal {T}_{42}{+}\mathcal {T}_{22}{=}\frac{x_{2}}{2}\left( T_{3}{-}T_{7}-S_{1}{-}P_{1}\right) {=}\frac{x_{2}}{2}\Xi _{4}(2,1,3), \end{aligned}$$

(A.44)

where it was used that

$$\begin{aligned} T_{i}(2,1,3)= & {} T_{i}(1,2,3),~ S_{1}(2,1,3)=-S_{1}(1,2,3),\nonumber \\ P_{1}(2,1,3)= & {} -~P_{1}(1,2,3). \end{aligned}$$

(A.45)

Consider the derivation of Eqs. (A.40)–(A.44). Let us introduce two twist-4 light-cone operators defined as

(A.46)

(A.47)

where the projectors and are used in order to decompose collinear fields into large and small components, respectively

(A.48)

Using Eq. (34) we rewrite the first operator as

$$\begin{aligned} O_{1}=-T_{1}^{\mu \nu \alpha \lambda }\left[ \left( in\partial \right) ^{-1}\partial _{\bot \alpha }\xi (x_{-})\right] C\sigma ^{+\lambda }\xi (y_{-})\left[ \xi (z_{-})\right] _{\sigma }. \end{aligned}$$

(A.49)

where

$$\begin{aligned} T_{1}^{\mu \nu \alpha \lambda }=\left\{ g_{\bot }^{\alpha \nu }g_{\bot }^{\lambda \mu }~+\frac{1}{2}~\bar{n}^{\mu }n^{\nu }g_{\bot }^{\lambda \alpha }~-(\mu \leftrightarrow \nu )\right\} . \end{aligned}$$

(A.50)

Taking the matrix element with the help of Eq. (A.25) one obtains

$$\begin{aligned} -\langle 0\vert O_{1}\vert k\rangle= & {} ~i\bar{n}^{\mu }n^{\nu }\frac{m_{N}}{2}\left[ \gamma _{5}N_{\bar{n}}\right] _{\sigma }~ \text {FT}\left[ \frac{1}{x_{1}}\mathcal {T}_{21}\right] \nonumber \\&-\,m_{N}\frac{1}{2}\left[ \sigma _{\bot \bot }^{\mu \nu }\gamma _{5}N_{\bar{n}}\right] _{\sigma }~ \text {FT}\left[ \frac{1}{x_{1}}\mathcal {T}_{41}\right] -(\mu \leftrightarrow \nu ). \nonumber \\ \end{aligned}$$

(A.51)

On the other hand rewriting the operator (A.46) with the basic Dirac structures one finds

(A.52)

Taking the matrix element in the rhs of this equation with the help of Eq. (A.14) and comparing with the Eq. (A.51) one obtains

$$\begin{aligned} \mathcal {T}_{21}-\mathcal {T}_{41}= & {} \frac{x_{1}}{2}\left( T_{3}+T_{7}+S_{1}-P_{1}\right) ,\nonumber \\ \mathcal {T}_{21}+\mathcal {T}_{41}= & {} \frac{x_{1}}{2}\left( T_{3}-T_{7}+P_{1}+S_{1}\right) . \end{aligned}$$

(A.53)

The similar consideration for the operator $O_{2}$ in Eq. (A.47) gives

$$\begin{aligned} \mathcal {T}_{22}-\mathcal {T}_{42}= & {} \frac{x_{2}}{2}\left( T_{3}+T_{7}-S_{1}+P_{1}\right) ,\nonumber \\ \mathcal {T}_{42}+\mathcal {T}_{22}= & {} \frac{x_{2}}{2}\left( T_{3}-T_{7}-S_{1}-P_{1}\right) . \end{aligned}$$

(A.54)

The cancellation of the ultrasoft gluon contributions

Here we briefly discuss the ultrasoft gluon limit. The contribution of the sum of diagrams as in Fig. 1 can be written as

$$\begin{aligned} iM=\frac{f_{\psi }}{m_{Q}^{2}}\frac{f_{N}\lambda _{1}}{m_{Q}^{4}}\frac{m_{N} }{m_{Q}}~J,~ \end{aligned}$$

(B.1)

where the dimensionless collinear convolution integral can be schematically written as

(B.2)

For simplicity, we do not show the various indices. Notice also that the colour factors for all diagrams are the same. In Eq. (B.2) we have three gluon propagators

$$\begin{aligned} \Delta _{gi}=\frac{(-i)}{(k_{i}+k_{i}^{\prime })^{2}}\simeq \frac{(-i)}{2(kk^{\prime })}\frac{1}{x_{i}y_{i}}. \end{aligned}$$

(B.3)

The function $\hat{T}_{3g\rightarrow p\bar{p}}$ describes the contribution from the light-quark vertices and from the projections of the nucleon matrix elements. The function $D(k_{i}^{\prime },k_{j})$ describes the heavy quark lines with the quark–gluon vertices.

The scaling behaviour of the contribution in Eq. (B.1) is given by the following factors

$$\begin{aligned} \frac{f_{\psi }}{m_{Q}^{2}}\sim v^{3},~~\frac{f_{N}~\lambda _{1}}{m_{Q}^{4}} \frac{m_{N}}{m_{Q}}\sim \left( \frac{\Lambda }{m_{Q}}\right) ^{5}\sim \lambda ^{10}. \end{aligned}$$

(B.4)

The collinear integral is defined to be of order one: $J\sim$ $v^{0}$. Therefore the ultrasoft region in J must give the contribution of order one.

Consider the ultrasoft gluon limit

$$\begin{aligned} p_{g}=k_{1}+k_{1}^{\prime }\sim m_{Q}v^{2}, \end{aligned}$$

(B.5)

that gives the counting for the small momentum fractions $x_{1}\sim y_{1}\sim v^{2}$. Such limit corresponds to the contribution from the endpoint region

(B.6)

where the cut-off $\eta$ can be understood as a factorisation scale separating the hard and ultrasoft domains. Our task is to estimate the scale behaviour of $J_{us}$. For that we need to expand the integrand with respect to small fractions $x_{1}\sim y_{1}\sim v^{2}$.

The light-quark part $\hat{T}_{3g\rightarrow p\bar{p}}(x_{i},y_{i})$ includes the twist-4 nucleon LCDAs $\mathcal {V}_{i}(x_{i}),~\mathcal {A}_{i}(x_{i})$ and $\mathcal {T}_{ij}(x_{i})$ and twist-3 $\varphi _{3}(y_{i})$. These functions must be also expanded with respect to the small fractions. We assume that the expansions of the LCDAs can provide only the positive powers of the small fractions. Therefore, it is quite reasonable here to consider only the asymptotic terms, which gives the contributions with the minimal powers of all fractions. Since one of our LCDAs is of twist-3, we immediately find that

$$\begin{aligned} \hat{T}_{3g\rightarrow p\bar{p}}\simeq \varphi _{3}^{as}(y_{i}) \hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}} (x_{2}) \sim y_{1}y_{2}y_{3} \hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}} (x_{2}), \end{aligned}$$

(B.7)

where we assume that $x_{3}\simeq 1-x_{2}$. The asymptotic expressions for the twist-4 LCDAs can be easily obtained from the formulas in Appendix A. One finds

$$\begin{aligned} \mathcal {V}_{i}(x_{i})\sim x_{1}x_{2}x_{3},~\mathcal {A}_{i}(x_{i})\sim x_{1}x_{2}x_{3},~\mathcal {T}_{ij}(x_{i})\sim x_{1}x_{2}x_{3}. \end{aligned}$$

(B.8)

The factor $\hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}}$ has the following schematic structure

$$\begin{aligned} \hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}}(x_{2})\simeq \sum _{i}\frac{1}{x_{i}} X_{i}(x_{i})+X_{i}(x_{i})\frac{\partial }{\partial k_{\bot i}}, \end{aligned}$$

(B.9)

where $X_{i}$ denote one of twist-4 DAs in Eq. (B.8). The powers $1/x_{i}$ originate from the inverse derivatives $(in\partial )^{-1}$ in the twist-4 operator in Eq. (33). From Eqs. (B.8) and (B.9) can be also seen that the terms with transverse derivatives are always suppressed by factor $v^{2}$ comparing to terms with $1/x_i$ and therefore can be neglected. Then one finds that $\hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}}(x_{2})\sim \mathcal {O}(v^{0})$ which gives

$$\begin{aligned} \hat{T}_{3g\rightarrow p\bar{p}}\sim y_{1}y_{2}\bar{y}_{2}~\hat{T}^{\text {tw4}}_{3g\rightarrow p\bar{p}}(x_{2})\sim \mathcal {O}(v^{2}). \end{aligned}$$

(B.10)

Consider now the sum of the heavy quark subdiagrams $D(k_{i}^{\prime },k_{j})$. Performing expansions with respect to small fractions $x_{1}$ and $y_{1}$ one obtains that the most singular terms appear from the diagrams describing the attachments of the ultrasoft gluon to external vertices on the heavy quark line. It is convenient to divide such diagrams into two groups: the soft gluon vertex is associated with the external heavy quark or with the external heavy antiquark. Then the sum of all relevant diagrams reads

(B.11)

where$~D_{h}^{\mu _{2}\mu _{3}}$ describes the sum of the subdiagrams with the hard gluons. The expansion with respect to the small fractions yields

(B.12)

(B.13)

We see that each separate term in $D_{us}$ has the contribution of order $v^{-2}$ due to the factor $1/(x_{1}+y_{1})$ but these terms cancel in the sum. Substituting (B.10) and (B.12) in (B.6) one obtains

$$\begin{aligned} J_{us}\sim \int _{0}^{\eta }\frac{dx_{1}}{x_{1}}\int _{0}^{\eta }dy_{1}\times \mathcal {O}(v^{0})\sim v^{2}, \end{aligned}$$

(B.14)

Hence the contribution of the ultrasoft region is power suppressed.

The same conclusion is also true for other regions where $x_{i}\sim y_{i}\sim v^{2}$. This result is in agreement with the Coulomb limit described by the potential NRQCD [36,37,38,39,40,41]. In this case the ultrasoft gluon vertices are suppressed by the small velocity v.