$\Lambda _b\rightarrow p$ transition form factors in perturbative QCD

Abstract

We reanalyze the $\Lambda _b\rightarrow p$ transition form factors in the perturbative QCD (PQCD) approach by including higher-twist light-cone distribution amplitudes (LCDAs) of a $\Lambda _b$ baryon and a proton. The previous PQCD evaluation performed decades ago with only the leading-twist $\Lambda _b$ baryon and proton LCDAs gave the form factors, which are two orders of magnitude smaller than indicated by experimental data. We find that the twist-4 $\Lambda _b$ baryon LCDAs and the twist-4 and -5 proton LCDAs contribute dominantly, and the enhanced form factors become consistent with those from lattice QCD and other nonperturbative methods. The estimated branching ratios of the semileptonic decays $\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell$ and the hadronic decay $\Lambda _b\rightarrow p\pi$ are also close to the data. It implies that the b quark mass is not really heavy enough, and higher-power contributions play a crucial role, similar to the observation made in analyses of B meson transition form factors. With the formalism established in this work, we are ready to study various exclusive heavy baryon decays systematically in the PQCD approach.

1 Introduction

A lot of progresses have been made on probing exclusive b-baryon decays with large amount of data collected by LHCb in recent years. CP violation (CPV) has been established in the K, B and D meson systems, but not yet in baryon systems. Therefore, exploring baryon CPV is one of the most important missions in both experimental and theoretical flavor physics. An evidence of CPV has been attained at the confidence level of $3\sigma$ in the ${{\Lambda } ^0_b} \rightarrow p {{\pi } ^+} {{\pi } ^-} {{\pi } ^-}$ decay [1]. Though CPV is not observed in other modes [2,3,4], the experimental precision has reached the percent level, such as $A_{CP}({{\Lambda } ^0_b} \rightarrow p{{K} ^-})=(-2.0{\pm }1.3{\pm }1.9)\%$ and $A_{CP}({{\Lambda } ^0_b} \rightarrow p{{\pi } ^-})=(-3.5{\pm }1.7{\pm }2.0)\%$ [2]. The above progresses motivate theoretical investigations on baryon CPV to a similar precision. A QCD-inspired formalism is definitely required for predicting CPV in heavy baryon decays, to which relative strong phases among various amplitudes are the key ingredient. Nevertheless, such a well-developed QCD method has not been available currently.

Several potential frameworks have been proposed for studies of hadronic heavy hadron decays, which include the effective theories such as the heavy quark effective theory (HQET) [5,6,7,8] and the soft-collinear effective theory (SCET) [9, 10], the factorization approaches such as the QCD factorization (QCDF) based on the collinear factorization [11,12,13,14] and the perturbative QCD approach (PQCD) based on the $k_T$ factorization [15,16,17], and phenomenological methods such as topological diagrammatic approaches [18,19,20], final-state interactions [21,22,23] and flavor symmetry analyses [24,25,26]. The above formalisms have been applied to heavy meson decays extensively, but applications to heavy baryon decays, especially to hadronic decays, are still limited. The generalized factorization assumption has been employed to estimate branching ratios of numerous b-baryon decays [27,28,29]. As to QCD-inspired methods, the QCDF approach was applied to $\Lambda _b$ baryon decays under the diquark approximation [30], and the $\Lambda _b\rightarrow pK$ and $p\pi$ branching ratios were calculated in the PQCD approach [31], but with the results being several times smaller than experimental data.

It has been known from global fits to B meson decay data [32, 33] that nonfactorizable contributions are crucial for color-suppressed tree-dominated modes, and that the W-exchange and penguin-annihilation amplitudes generate large strong phases. The above contributions cannot be computed unambiguously in the other frameworks, such as the factorization assumption and the QCDF approach, but can be in the PQCD approach [15,16,17]. For comparisons of these theoretical methods and their phenomenological impacts, refer to [34, 35]. It is the reason why the CPV in, for instance, the $B^0\rightarrow K^+\pi ^-$ and $B^0\rightarrow \pi ^+\pi ^-$ modes [36,37,38], has been predicted successfully in [15,16,17]. In fact, the PQCD approach has demonstrated a unique power for predicting CPV in two-body hadronic B meson decays. b-baryon decays involve more W-exchange and penguin-annihilation diagrams [23, 39], whose PQCD evaluation is feasible in principle. It is our motivation to examine the applicability of the PQCD formalism to exclusive heavy baryon decays in this work.

Baryonic transitions were firstly investigated in the PQCD approach in Ref. [40], where the proton Dirac form factors at large momentum transfer were derived. This framework was then extended to studies of heavy-to-light baryonic transition form factors, which are essential inputs to exclusive processes like the semileptonic decays $\Lambda _b \rightarrow p\ell {\bar{\nu }}$ [41] and $\Lambda _b \rightarrow \Lambda _c \ell {\bar{\nu }}$ at large recoil [42, 43], the radiative decay $\Lambda _b \rightarrow \Lambda \gamma$ [44], and the two-body hadronic decays $\Lambda _b \rightarrow \Lambda J/\Psi$ [45], $\Lambda _b\rightarrow p\pi ,\ pK$ [31] and $\Lambda _b\rightarrow \Lambda _c\pi ,\Lambda _c K$ [46]. Only the leading-twist light-cone distribution amplitudes (LCDAs) were considered in the factorization formulas for all the above $\Lambda _b$ baryon decays. It was noticed that the factorizable contributions to two-body hadronic decays are unreasonably smaller than the nonfactorizable ones, and the predicted $\Lambda _b\rightarrow p K^-$ branching ratio is several times lower than the measured value [31]. In another word, the $\Lambda _b\rightarrow p$ transition form factors are down by about two orders of magnitude compared to those from nonperturbative methods in the literature: the PQCD approach gave the $\Lambda _b\rightarrow p$ form factor at the maximal recoil $f_1=(2.2^{+0.8}_{-0.5})\times 10^{-3}$ in [31] and $2.3\times 10^{-3}$ in [41], while lattice QCD yielded $f_1=0.22{\pm }0.08$ [47]. To verify the applicability of PQCD, we should first resolve the difficulty appearing in the $\Lambda _b\rightarrow p$ transition form factors.

The contribution to a heavy-to-light mesonic transition amplitude is divided into two pieces at leading power in the QCDF approach, the nonfactorziable soft form factor and the factorizable hard spectator contribution. The latter is calculated in a perturbation theory, and the former can only be handled in nonperturbative methods, such as lattice QCD [47, 48], QCD sum rules (QSR) [49], light-cone sum rules (LCSR) [50, 51], light-front quark model [52,53,54], etc.. In the baryonic case it has been proved in the SCET [55] that the leading-power contribution is completely factorizable due to the absence of an endpoint singularity in the collinear factorization. The corresponding diagrams contain at least two hard-collinear gluon exchanges, so that the leading-power contribution is suppressed by $\mathcal{O}(\alpha _s^2)$ in the strong coupling. On the contrary, a soft contribution to the baryonic form factor is power-suppressed in the SCET [55], but not down by $\alpha _s$, which turns out to be numerically important. Taking the $\Lambda _b\rightarrow \Lambda$ transition form factor $\xi _\Lambda$ at maximal recoil as an example, one got $\xi _\Lambda =-0.012^{+0.009}_{-0.023}$ from the leading-power contribution [55], and $\xi _\Lambda =0.38$ from the SCET sum rules [56]. It implies that QCD dynamics is quite different between the mesonic and baryonic decays, and that the power suppression from the b quark mass may not be effective.

The soft form factor for a heavy-to-light baryonic transition is nonfactorizable in the collinear factorization, because an endpoint singularity from small parton momentum fractions will be developed, if the soft form factor is expressed as a convolution of a hard kernel and the baryon LCDAs beyond the leading twist. The PQCD approach based on the $k_T$ factorization, in which parton transverse momenta are kept to avoid the endpoint singularity [57, 58], provides a new set of power counting rules [35, 59]. The $k_T$ resummation is demanded to organize the large logarithms owing to the introduction of the additional scales $k_T$. The resultant Sudakov factors suppress the long-distance contributions now characterized by large b with b being the variables conjugate to $k_T$. Once the endpoint singularity is smeared, the higher-twist contribution to the heavy-to-light form factor is regarded as being factorizable and calculable in the PQCD formalism. The factorizable contribution then picks up the additional piece from the higher-twist LCDAs under the Sudakov suppression, such that the previous leading-twist PQCD results can be significantly enhanced.

The above discussion suggests that the contributions from higher-twist LCDAs are factorizable in the $k_T$ factorization, and their inclusion may increase the much smaller $\Lambda _b\rightarrow p$ form factors at leading power [31, 41]. Here we will analyze these form factors in the fast recoil region by including the $\Lambda _b$ baryon LCDAs up to twist 4 and the proton LCDAs up to twist 6 in the PQCD approach. It will be shown that the $\Lambda _b\rightarrow p$ form factors become comparable to those derived from other nonperturbative methods and indicated by experimental data, when the above higher-twist LCDAs are taken into account. In particular, the convolution with the twist-4 $\Lambda _b$ baryon LCDAs and the twist-4 and -5 proton LCDAs give the dominant contributions. We then estimate the branching ratios of the semileptonic decays $\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell$ with the leptons $\ell =e,\mu ,\tau$ by extrapolating the form factors at large recoil to the whole kinematic range. The agreement of our results with data encourages the generalization of the established PQCD formalism to more complicated two-body hadronic heavy baryon decays.

The remainder of this paper is organized as follows. The PQCD framework for computing the $\Lambda _b\rightarrow p$ transition form factors is explained in Sect. 2, where the $\Lambda _b$ baryon and proton LCDAs of various twists are defined. The numerical outcomes for the $\Lambda _b\rightarrow p$ form factors and for the differential widths of the semileptonic decays $\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell$ are presented and discussed in Sect. 3. We also make the preliminary prediction for the $\Lambda _b\rightarrow p\pi$ branching ratio based on the naive factorization assumption. The last section contains the conclusion. The explicit expressions for the factorization formulas together with the hard scales involved in various diagrams are collected in the Appendix.

2 Theoretical framework

2.1 $k_T$ factorization

The $\Lambda _b \rightarrow p$ transition form factors are defined via the matrix element of the $V-A$ current [5],

$$\begin{aligned}&\langle P(p^\prime ,s^\prime ) | {\bar{u}}\gamma _\mu (1-\gamma _5)b | \Lambda _b(p,s) \rangle \nonumber \\&\quad = {\overline{N}}(p^\prime ,s^\prime ) (f_1 \gamma _\mu - i f_2 \sigma _{\mu \nu } q^\nu + f_3 q_\mu )\Lambda _b(p,s)\nonumber \\&\qquad -{\overline{N}}(p^\prime ,s^\prime ) (g_1 \gamma _\mu - i g_2 \sigma _{\mu \nu } q^\nu + g_3 q_\mu )\gamma _5\Lambda _b(p,s), \end{aligned}$$

(1)

where $\sigma _{\mu \nu }=i[\gamma _\mu ,\gamma _\nu ]/2$, and $\Lambda _b(p,s)$ ($N(p^\prime ,s^\prime )$) is the spinor of the $\Lambda _b$ baryon (proton) with the momentum p ($p^\prime$) and the spin s ($s^\prime$). The form factors $f_i$ and $g_i$ depend on the invariant mass squared of the lepton pair $q^2$ with $q_\mu =p_\mu -p^\prime _\mu$. We work in the rest frame of the $\Lambda _b$ baryon, and parameterize the $\Lambda _b$ baryon and proton momenta in the light-cone coordinates as

$$\begin{aligned} p= & {} \frac{m_{\Lambda _b}}{\sqrt{2}}(1,1,{\textbf {0}}),\nonumber \\ p^\prime= & {} \frac{m_{\Lambda _b}}{\sqrt{2}}(\eta _1,\eta _2,{\textbf {0}}), \end{aligned}$$

(2)

with the large component $\eta _1\sim \mathcal{O}(1)$ and the small component $\eta _2\sim \mathcal{O}(m_p^2/m_{\Lambda _b}^2)$, $m_{\Lambda _b}$ ($m_p$) being the $\Lambda _b$ baryon (proton) mass. Namely, the fast recoiled proton has been assumed to move approximately in the plus direction. The invariant mass squared of the lepton pair is then given, in terms of $\eta _1$ and $\eta _2$, by $q^2=m^2_{\Lambda _b}(1-\eta _1)(1-\eta _2)$.

As stated before, the transverse momenta of the valence quarks are retained in the PQCD approach based on the $k_T$ factorization. We thus choose the partonic momenta as

$$\begin{aligned} k_1&=\left( {m_{\Lambda _b}\over \sqrt{2}},{x_1m_{\Lambda _b}\over \sqrt{2}},{\textbf {k}}_{1T}\right) ,\ \nonumber \\ k_1^\prime&=\left( x_1^\prime {\eta _1m_{\Lambda _b}\over \sqrt{2}},0,{\textbf {k}}_{1T}^\prime \right) ,\nonumber \\ k_2&=(\,\,\,0,\,\,\,\,\,\,{x_2m_{\Lambda _b}\over \sqrt{2}},{\textbf {k}}_{2T}),\nonumber \\ k_2^\prime&=\left( x_2^\prime {\eta _1m_{\Lambda _b}\over \sqrt{2}},0,{\textbf {k}}_{2T}^\prime \right) ,\nonumber \\ k_3&=\left( \,\,\,0,\,\,\,\,\,\,{x_3m_{\Lambda _b}\over \sqrt{2}},{\textbf {k}}_{3T}\right) ,\nonumber \\ k_3^\prime&=\left( x_3^\prime {\eta _1m_{\Lambda _b}\over \sqrt{2}},0,{\textbf {k}}_{3T}^\prime \right) , \end{aligned}$$

(3)

where $k_1$ is the b quark momentum, $k_2$ ($k_3$) and $k_2^\prime$ ($k_3^\prime$) are the spectator u (d) quark momenta in the $\Lambda _b$ baryon and the proton, respectively, and $x_i,k_{iT}$ and $x_i^\prime ,k_{iT}^\prime$ denote the corresponding light-cone momentum fractions and the transverse momenta. Note that $x_1$ is of $\mathcal{O}(m_b^2/m_{\Lambda _b}^2)$, $m_b$ being b quark mass, in order for the b quark to be off-shell by $k_1^2\approx -k_{1T}^2$, as required by the $k_T$ factorization. The dominant plus components of $k_i^\prime$ are kept, and the minus components of $k_i$ for the soft light quarks are selected by their inner products with $k_i^\prime$, which appear in the hard kernels for the $\Lambda _b\rightarrow p$ form factors. Since the two soft quarks in the $\Lambda _b$ baryon need to turn into the energetic quarks in the proton, at least two hard gluons are exchanged as shown in the leading-order Feynman diagrams in Fig. 1. That is, the $\Lambda _b \rightarrow p$ decay amplitudes start at $\mathcal{O}(\alpha _s^2)$ in the PQCD approach.

Fig. 1

Feynman diagrams for the $\Lambda _b\rightarrow p$ transition form factors, where the black dots denote the weak interaction vertices. These diagrams are labelled by $D_1$, $D_2,\ldots$ and $D_{16}$ in sequence in the text

The $\Lambda _b\rightarrow p$ transition amplitude is formulated in the PQCD approach as [31]

$$\begin{aligned} {\mathcal {A}}= & {} \Psi _{\Lambda _b}(x_i,b_i,\mu )\otimes H(x_i,b_i,x_i^{\prime },b_i^{\prime },\mu )\nonumber \\&\otimes \Psi _{P}(x_i^{\prime },b_i^{\prime },\mu ), \end{aligned}$$

(4)

where the hard kernel H is derived from the diagrams in Fig. 1, $\Psi _{\Lambda _b}$ and $\Psi _{P}$ stand for the $\Lambda _b$ baryon and proton wave functions, respectively, and $\mu$ is the factorization scale. The symbol $\otimes$ represents the convolution in the momentum fractions and in the impact parameters $b_i$ and $b_i^{\prime }$, which are conjugated to the corresponding transverse momenta. With the parton transverse degrees of freedom being included, the above $k_T$ factorization formula holds for higher-twist contributions due to the absence of endpoint singularities. At the same time, the large double logarithms $\alpha _s\ln ^2 (m_{\Lambda _b}b)$ are produced from radiative corrections to the baryon wave functions, which must be summed to all orders in $\alpha _s$ to improve the convergence of perturbation expansion. The $k_T$ resummation is thus applied to the baryon wave functions to extract the Sudakov factors [31, 60, 61] as in its application to meson wave functions [62, 63]. The Sudakov factor, which decreases fast with b and vanishes at $b=1/\Lambda _{\mathrm{QCD}}$, $\Lambda _{\mathrm{QCD}}$ being the QCD scale, is expected to effectively suppress the long-distance contributions from the large b regions. We also implement the renormalization-group evolution in $\mu$ up to a hard scale t, which is set to the maximum of all scales involved in the factorization formula [31]. The necessary constraint on the hard scale, $t\ge 1$ GeV, is imposed, because it should not go below the initial scale, at which models for the baryon wave functions are defined.

We then arrive at an factorization formula improved by the resummation and renormalization-group evolution,

$$\begin{aligned} {\mathcal {A}}= & {} \psi _{\Lambda _b}(x_i,b_i)\otimes H(x_i,b_i,x_i^{\prime },b_i^{\prime },t)\nonumber \\&\otimes \psi _{P}(x_i^{\prime },b_i^{\prime })\exp \left[ -S_{\Lambda _b}(t)-S_p(t)\right] , \end{aligned}$$

(5)

where the $\Lambda _b$ baryon (proton) wave function $\psi _{\Lambda _b}$ ($\psi _{P}$) is obtained by factorizing the extrinsic impact-parameter dependence of the wave function $\Psi _{\Lambda _b}$ ($\Psi _{P}$) into the total exponential factor $\exp [-S_{\Lambda _b}(t)]$ ($\exp [-S_p(t)]$) [31]. The remaining impact-parameter dependencies of $\psi _{\Lambda _b}$ and $\psi _{P}$ are intrinsic. We point out that another type of double logarithms $\alpha _s\ln ^2x_i$, appearing in the hard kernel, will also become crucial as the endpoint regions dominate. The threshold resummation that sums this type of double logarithms to all orders leads to the jet function (or the threshold Sudakov factor) $S_t(x_i)$ [64, 65], which can further improve the perturbative expansion. The jet function $S_t(x_i)$, extracted from the hard kernel, is process-dependent. Because a systematic derivation of $S_t(x_i)$ for baryonic decays goes beyond the scope of this paper, we will naively set $S_t(x_i)=1$ below, and investigate its effect in the future. The higher-twist $\Lambda _b$ baryon and proton wave functions and the corresponding power-suppressed contributions will be analyzed quantitatively based on Eq. (5). The factorization formulas for the $\Lambda _b \rightarrow p$ form factors from the diagrams in Fig. 1 and the involved hard scales are provided in Appendix A. It will be seen that the predictions for these form factors as well as the relevant observables in the PQCD formalism are substantially enhanced.

2.2 Light-cone distribution amplitudes of baryons

Hadronic wave functions are universal nonperturbative inputs to a factorization theorem, which need to be specified before predictions for an exclusive QCD process are made. The explicit form of a wave function, being a three-dimension object, may be very complicated. As a usual practice, one assumes that a wave function is factorized into a product of a momentum-fraction-dependent part and a transverse-momentum-dependent part. The former corresponds to a LCDA for the collinear factorization, and the latter has been parameterized as a simple Gaussian type function. This assumption has been widely adopted in applications of the $k_T$ factorization to exclusive processes. Since the transverse-momentum, i.e., impact-parameter dependencies of the baryon wave functions $\psi _{\Lambda _b}$ and $\psi _{P}$ are still not well constrained, we will neglect them in our numerical studies.

2.2.1 $\Lambda _b$ baryon light-cone distribution amplitudes

The $\Lambda _b$ baryon LCDAs are defined by the matrix elements of nonlocal operators sandwiched between the vacuum and the $\Lambda _b$ baryon state, whose general Lorentz structures can be found in Refs. [51, 66,67,68]. We start with the momentum-space projector

$$\begin{aligned}&(Y_{\Lambda _b})_{\alpha \beta \gamma }(x_i,\mu )\nonumber \\&=\frac{1}{8N_c}\Big \{f_{\Lambda _b}^{(1)}(\mu )[M_1(x_2,x_3)\gamma _5C^T]_{\gamma \beta }\nonumber \\&+f_{\Lambda _b}^{(2)}(\mu )[M_2(x_2,x_3)\gamma _5C^T]_{\gamma \beta }\Big \}[\Lambda _b(p)]_\alpha , \end{aligned}$$

(6)

where $N_c$ is the number of colors, C represents the charge conjugation matrix, and $\Lambda _b(p)$ is the $\Lambda _b$ baryon spinor. We choose the normalization constants as $f_{\Lambda _b}^{(1)} \approx f_{\Lambda _b}^{(2)}\equiv f_{\Lambda _b}= 0.021{\pm } 0.004$ $\hbox {GeV}^3$, which are close to $f_{\Lambda _b}= 0.022{\pm } 0.001$ $\hbox {GeV}^3$ from the leading-order sum rule calculation [69], and overlap with other estimates in the literature (see [70] and references therein). Note that the normalization constants have been derived in diagonal, non-diagonal and mixed sum rules at the leading-order and next-to-leading-order levels in [69]. Because the equality $f_{\Lambda _b}^{(1)}\approx f_{\Lambda _b}^{(2)}$ assumed above is not guaranteed in non-diagonal and mixed sum rules [69], and invalidated under next-to-leading-order corrections [70], we favor the result for $f_{\Lambda _b}^{(1,2)}$ from the leading-order diagonal sum rules. The terms containing the derivatives with respect to the transverse momenta of the soft light quarks have been ignored in Eq. (6). Their contributions are expected to be tiny, similar to what was observed in the PQCD analysis of B meson transition form factors [71].

The remaining parts of the projector in Eq. (6) are expressed as

(7)

(8)

where the two light-cone vectors $n=(1,0,{\textbf {0}})$ and ${\bar{n}}=(0,1,{\textbf {0}})$ satisfy $n\cdot {\bar{n}}=1$. Various models for the $\Lambda _b$ baryon LCDAs $\psi _2$, $\psi _3^{+-}$, $\psi _3^{-+}$ and $\psi _4$ have been proposed in Refs. [66,67,68]. Viewing the obvious difference among these models, we will investigate the contributions to the $\Lambda _b\rightarrow p$ form factors from all of them for completeness:

• Gegenbauer-1 [66], which was obtained by taking into account only the leading-order perturbative contribution to the associated QCD sum rules,

  $$\begin{aligned} \psi _2(x_2,x_3)&=m_{\Lambda _b}^4x_2x_3\left[ \frac{1}{\epsilon _0^4}e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _0}\right. \nonumber \\&\left. \quad +a_2C_2^{3/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \frac{1}{\epsilon _1^4}e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _1}\right] ,\nonumber \\ \psi _3^{+-}(x_2,x_3)&=\frac{2m_{\Lambda _b}^3x_2}{\epsilon _3^3}e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _3},\nonumber \\ \psi _3^{-+}(x_2,x_3)&=\frac{2m_{\Lambda _b}^3x_3}{\epsilon _3^3}e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _3},\nonumber \\ \psi _4(x_2,x_3)&=\frac{5}{{\mathcal {N}}}m_{\Lambda _b}^2\int _{m_{\Lambda _b}(x_2+x_3)/2}^{s_0}ds e^{-s/\tau } \nonumber \\&\quad (s-m_{\Lambda _b}(x_2+x_3)/2)^3, \end{aligned}$$

  (9)

  with the Gegenbauer moment $a_2=0.333_{-0.333}^{+0.250}$, the Gegenbauer polynomial $C_2^{3/2}(x)=3(5x^2-1)/2$, the parameters $\epsilon _0=200_{-60}^{+130}$ MeV, $\epsilon _1=650_{-300}^{+650}$ MeV and $\epsilon _3=230{\pm } 60$ MeV, the Borel mass 0.4 GeV $<\tau<$ 0.8 GeV, the continuum threshold $s_0=1.2$ GeV and the constant ${\mathcal {N}}=\int _{0}^{s_0}dss^5e^{-s/\tau }$.

• Gegenbauer-2 [68], which was formulated in the heavy quark limit with the moments being derived in QCD sum rules,

  $$\begin{aligned} \psi _2(x_2,x_3)&=m_{\Lambda _b}^4x_2x_3\left( \frac{a_0^{(2)}}{{\epsilon _0^{(2)}}^4}C_0^{3/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _0^{(2)}} +\frac{a_2^{(2)}}{{\epsilon _2^{(2)}}^4}C_2^{3/2}\right. \nonumber \\&\left. \quad \times \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _2^{(2)}}\right) ,\nonumber \\ \psi _3^{+-}(x_2,x_3)&=m_{\Lambda _b}^3(x_2+x_3)\left[ \frac{a_0^{(3)}}{{\epsilon _0^{(3)}}^3}C_0^{1/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _0^{(3)}} + \frac{a_2^{(3)}}{{\epsilon _2^{(3)}}^3}C_2^{1/2}\right. \nonumber \\&\left. \quad \times \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _2^{(3)}} \right] ,\nonumber \\&\quad +m_{\Lambda _b}^3(x_2+x_3)\left[ \frac{b_1^{(3)}}{{\eta _1^{(3)}}^3}C_1^{1/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\eta _1^{(3)}}+ \frac{b_3^{(3)}}{{\eta _3^{(3)}}^3}C_2^{1/2} \right. \nonumber \\&\quad \left. \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\eta _3^{(3)}}\right] ,\nonumber \\ \psi _3^{-+}(x_2,x_3)&=m_{\Lambda _b}^3(x_2+x_3)\left[ \frac{a_0^{(3)}}{{\epsilon _0^{(3)}}^3}C_0^{1/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _0^{(3)}}+ \frac{a_2^{(3)}}{{\epsilon _2^{(3)}}^3}C_2^{1/2}\right. \nonumber \\&\left. \quad \times \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _2^{(3)}} \right] ,\nonumber \\&\quad -m_{\Lambda _b}^3(x_2+x_3)\left[ \frac{b_1^{(3)}}{{\eta _1^{(3)}}^3}C_1^{1/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\eta _1^{(3)}}+ \frac{b_3^{(3)}}{{\eta _3^{(3)}}^3}C_2^{1/2} \right. \nonumber \\&\left. \quad \times \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\eta _3^{(3)}}\right] ,\nonumber \\ \psi _4(x2,x3)&=m_{\Lambda _b}^2\left( \frac{a_0^{(4)}}{{\epsilon _0^{(4)}}^2}C_0^{1/2}\left( \frac{x_2-x_3}{x_2+x_3}\right) \right. \nonumber \\&\left. \quad \times e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _0^{(4)}} + \frac{a_2^{(4)}}{{\epsilon _2^{(4)}}^2}C_2^{1/2}\right. \nonumber \\&\left. \quad \left( \frac{x_2-x_3}{x_2+x_3}\right) e^{-m_{\Lambda _b}(x_2+x_3)/\epsilon _2^{(4)}}\right) , \end{aligned}$$

  (10)

  with the Gegenbauer polynomials $C_0^{3/2}(x)=1$, $C_0^{1/2}(x)=1$, $C_2^{1/2}(x)=(2x^2-1)/2$, and the parameters $a_0^{(2)}=1$, $a_0^{(3)}=1$, $a_0^{(4)}=1$, $a_2^{(2)}=0.391{\pm }0.279$, $a_2^{(3)}=-0.161_{-0.207}^{+0.108}$, $a_2^{(4)}=-0.541_{-0.09}^{+0.173}$, $b_1^{(3)}=1$, $b_3^{(3)}=-0.24_{-0.147}^{+0.24}$, $\epsilon _0^{(2)}=0.201_{-0.059}^{+0.143}$ GeV, $\epsilon _0^{(3)}=0.232_{-0.056}^{+0.047}$ GeV, $\epsilon _0^{(4)}=0.352_{-0.083}^{+0.067}$ GeV, $\epsilon _2^{(2)}=0.551_{-0.356}^{+\infty }$ GeV, $\epsilon _2^{(3)}=0.055_{-0.02}^{+0.01}$ GeV, $\epsilon _2^{(4)}=0.262_{-0.132}^{+0.116}$ GeV, $\eta _1^{(3)}=0.324_{-0.026}^{+0.054}$ GeV and $\eta _3^{(3)}=0.633{\pm }0.099$ GeV.

• Exponential model [67],

  $$\begin{aligned} \psi _2(x_2,x_3)&= \frac{x_2x_3}{\omega _0^4}m^4_{\Lambda _b}e^{-(x_2+x_3)m_{\Lambda _b}/\omega _0},\nonumber \\ \psi _3^{+-}(x_2,x_3)&= \frac{2x_2}{\omega _0^3}m^3_{\Lambda _b}e^{-(x_2+x_3)m_{\Lambda _b}/\omega _0},\nonumber \\ \psi _3^{-+}(x_2,x_3)&= \frac{2x_3}{\omega _0^3}m^3_{\Lambda _b}e^{-(x_2+x_3)m_{\Lambda _b}/\omega _0},\nonumber \\ \psi _4(x_2,x_3)&= \frac{1}{\omega _0^2}m_{\Lambda _b}^2e^{-(x_2+x_3)m_{\Lambda _b}/\omega _0}, \end{aligned}$$

  (11)

  where $\omega _0=0.4$ GeV measures the average energy of the two light quarks.

• Free-parton approximation [67],

  $$\begin{aligned} \psi _2(x_2,x_3)&= \frac{15x_2x_3m_{\Lambda _b}^4(2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b})}{4{\bar{\Lambda }}^5}\nonumber \\&\quad \Theta (2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b}),\nonumber \\ \psi _3^{+-}(x_2,x_3)&=\frac{15x_2m_{\Lambda _b}^3(2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b})^2}{4{\bar{\Lambda }}^5}\nonumber \\&\quad \Theta (2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b}),\nonumber \\ \psi _3^{-+}(x_2,x_3)&=\frac{15x_3m_{\Lambda _b}^3(2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b})^2}{4{\bar{\Lambda }}^5}\nonumber \\&\quad \Theta (2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b}),\nonumber \\ \psi _4(x_2,x_3)&= \frac{5m_{\Lambda _b}^2(2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b})^3}{8{\bar{\Lambda }}^5}\nonumber \\&\quad \Theta (2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b}), \end{aligned}$$

  (12)

  with the theta function $\Theta$ and the scale ${\bar{\Lambda }}\equiv (m_{\Lambda _b}-m_b)/2\approx 0.8$ GeV.

All the above models for the $\Lambda _b$ baryon LCDAs obey the normalizations

$$\begin{aligned}&\int _{0}^{1}dx_1dx_2dx_3 \delta (1-x_1-x_2-x_3)\psi _2(x_2,x_3)=1,\nonumber \\&\int _{0}^{1}dx_1dx_2dx_3 \delta (1-x_1-x_2-x_3)\nonumber \\&\quad (\psi _3^{+-}(x_2,x_3)+\psi _3^{-+}(x_2,x_3))/4=1,\nonumber \\&\int _{0}^{1}dx_1dx_2dx_3 \delta (1-x_1-x_2-x_3)\psi _4(x_2,x_3)=1. \end{aligned}$$

(13)

In order to compare our results with the previous ones, we quote the simple model for the leading-twist $\Lambda _b$ baryon LCDA proposed in Ref. [72],

(14)

(15)

whose Lorentz structure has been simplified under the Bargmann–Wigner equation [73] in the heavy quark limit, such that the spin and orbital degrees of freedom of the light quark system are decoupled. The $\mu$ dependence can be organized into the total exponential factor as stated before, $f_{\Lambda _b}^\prime$ is set to the value $4.28^{+0.75}_{-0.64}\times 10^{-3}$ $\text {GeV}^2$ the same as in the previous analysis [31], $N=6.67\times 10^{12}$ is determined by the normalization condition, the shape parameter takes the value $\beta =1.0{\pm }0.2$ GeV, and $m_l=0.3$ GeV represents the mass of the light degrees of freedom in the $\Lambda _b$ baryon.

Table 1 Twist classification of the proton LCDAs in Eq. (16)

2.2.2 Proton light-cone distribution amplitudes

The proton LCDAs with definite twists have been defined in Ref. [74], and the corresponding momentum-space projector is written as

(16)

with the proton mass $m_p=0.938$ GeV. The light-like vector P can be decomposed into

$$\begin{aligned} P_\mu =p_\mu ^\prime -\frac{1}{2}z_\mu \frac{m_p^2}{Pz}, \end{aligned}$$

(17)

where $p^\prime$ is the proton momentum, and z is a light-like vector with $z^2=0$. We have adopted the shorthand notations $\sigma _{Pz}=\sigma ^{\nu \mu }P_\nu z_\mu$, and and for the “large” and “small” components of the proton spinor ${\overline{N}}$, respectively. The symbol $\perp$ denotes the projection perpendicular to z or P, and the contraction $\gamma _\perp \gamma ^\perp$ means $\gamma _\perp \gamma ^\perp =\gamma ^\mu g_{\mu \nu }^\perp \gamma ^\nu$ with $g_{\mu \nu }^\perp =g_{\mu \nu }-(P_\mu z_\nu +z_\mu P_\nu )/Pz$. The twist classification of the LCDAs $V_i$, $A_i$, $T_i$, $S_i$ and $P_i$ is specified in Table 1, and their explicit expressions are listed below:

• Twist-3 LCDAs

  $$\begin{aligned} V_1(x_i)&=120x_1x_2x_3[\phi _3^0+\phi _3^+(1-3x_3)], \end{aligned}$$

  (18)
  $$\begin{aligned} A_1(x_i)&=120x_1x_2x_3(x_2-x_1)\phi _3^-, \end{aligned}$$

  (19)
  $$\begin{aligned} T_1(x_i)&=120x_1x_2x_3[\phi _3^0+\frac{1}{2}(\phi _3^--\phi _3^+)(1-3x_3)]. \end{aligned}$$

  (20)

• Twist-4 LCDAs

  $$\begin{aligned} V_2(x_i)&=24x_1x_2[\phi _4^0+\phi _4^+(1-5x_3)], \end{aligned}$$

  (21)
  $$\begin{aligned} V_3(x_i)&=12x_3[\psi _4^0(1-x_3)+\psi _4^-(x_1^2+x_2^2\nonumber \\&\quad -x_3(1-x_3)) +\psi _4^+(1-x_3-10x_1x_2)], \end{aligned}$$

  (22)
  $$\begin{aligned} A_2(x_i)&=24x_1x_2(x_2-x_1)\phi _4^-, \end{aligned}$$

  (23)
  $$\begin{aligned} A_3(x_i)&=12x_3(x_2-x_1)[(\psi _4^0+\psi _4^+)+\psi _4^-(1-2x_3)], \end{aligned}$$

  (24)
  $$\begin{aligned} T_2(x_i)&=24x_1x_2[\xi _4^0+\xi _4^+(1-5x_3)], \end{aligned}$$

  (25)
  $$\begin{aligned} T_3(x_i)&=6x_3[(\xi _4^0+\phi _4^0+\psi _4^0)(1-x_3)\nonumber \\&\quad +(\xi _4^-+\phi _4^--\psi _4^-)(x_1^2+x_2^2-x_3(1-x_3))\nonumber \\&\quad +(\xi _4^++\phi _4^++\psi _4^+)(1-x_3-10x_1x_2)], \end{aligned}$$

  (26)
  $$\begin{aligned} T_7(x_i)&=6x_3[(-\xi _4^0+\phi _4^0\nonumber \\&\quad +\psi _4^0)(1-x_3)+(-\xi _4^-+\phi _4^--\psi _4^-)\nonumber \\&\quad (x_1^2+x_2^2-x_3(1-x_3))\nonumber \\&\quad +(-\xi _4^++\phi _4^++\psi _4^+)(1-x_3-10x_1x_2)], \end{aligned}$$

  (27)
  $$\begin{aligned} S_1(x_i)&=6x_3(x_2-x_1)[(\xi _4^0+\phi _4^0+\psi _4^0+\xi _4^++\phi _4^+\nonumber \\&\quad +\psi _4^+) +(\xi _4^-+\phi _4^--\psi _4^-)(1-2x_3)], \end{aligned}$$

  (28)
  $$\begin{aligned} P_1(x_i)&=6x_3(x_2-x_1)[(\xi _4^0-\phi _4^0-\psi _4^0+\xi _4^+-\phi _4^+\nonumber \\&\quad -\psi _4^+) +(\xi _4^--\phi _4^-+\psi _4^-)(1-2x_3)]. \end{aligned}$$

  (29)

• Twist-5 LCDAs

  $$\begin{aligned} V_4(x_i)&=3[\psi _5^0(1-x_3)+\psi _5^-(2x_1x_2-x_3(1-x_3))\nonumber \\&\quad +\psi _5^+(1-x_3-2(x_1^2+x_2^2))], \end{aligned}$$

  (30)
  $$\begin{aligned} V_5(x_i)&=6x_3[\phi _5^0+\phi _5^+(1-2x_3)], \end{aligned}$$

  (31)
  $$\begin{aligned} A_4(x_i)&=3(x_2-x_1)[-\psi _5^0+\psi _5^-x_3+\psi _5^+(1-2x_3)], \end{aligned}$$

  (32)
  $$\begin{aligned} A_5(x_i)&=6x_3(x_2-x_1)\phi _5^-, \end{aligned}$$

  (33)
  $$\begin{aligned} T_4(x_i)&=\frac{3}{2}[(\xi _5^0+\psi _5^0+\phi _5^0)(1-x_3)\nonumber \\&\quad +(\xi _5^-+\phi _5^--\psi _5^-)(2x_1x_2-x_3(1-x_3))\nonumber \\&\quad +(\xi _5^++\phi _5^++\psi _5^+)(1-x_3-2(x_1^2+x_2^2))], \end{aligned}$$

  (34)
  $$\begin{aligned} T_5(x_i)&=6x_3[\xi _5^0+\xi _5^+(1-2x_3)], \end{aligned}$$

  (35)
  $$\begin{aligned} T_8(x_i)&=\frac{3}{2}[(\psi _5^0+\phi _5^0-\xi _5^0)(1-x_3)\nonumber \\&\quad +(\phi _5^--\phi _5^--\xi _5^-)(2x_1x_2-x_3(1-x_3))\nonumber \\&\quad +(\phi _5^++\phi _5^+-\xi _5^+)(\mu )(1-x_3-2(x_1^2+x_2^2))], \end{aligned}$$

  (36)
  $$\begin{aligned} S_2(x_i)&=\frac{3}{2}(x_2-x_1)[-(\psi _5^0+\phi _5^0+\xi _5^0)\nonumber \\&\quad +(\xi _5^-+\phi _5^--\psi _5^0)x_3+(\xi _5^++\phi _5^++\psi _5^0)(1-2x_3)], \end{aligned}$$

  (37)
  $$\begin{aligned} P_2(x_i)&=\frac{3}{2}(x_2-x_1)[(\psi _5^0+\phi _5^0-\xi _5^0)\nonumber \\&\quad +(\xi _5^--\phi _5^-+\psi _5^0)x_3+(\xi _5^+-\phi _5^+-\psi _5^0)(1-2x_3)]. \end{aligned}$$

  (38)

• Twist-6 LCDAs

  $$\begin{aligned} V_6(x_i)&=2[\phi _6^0+\phi _6^+(1-3x_3)], \end{aligned}$$

  (39)
  $$\begin{aligned} A_6(x_i)&=2(x_2-x_1)\phi _6^-, \end{aligned}$$

  (40)
  $$\begin{aligned} T_6(x_i)&=2[\phi _6^0+\frac{1}{2}(\phi _6^--\phi _6^+)(1-3x_3)], \end{aligned}$$

  (41)

with the values of the involved parameters being given in Table 2.

Table 2 Parameters in the proton LCDAs in units of $10^{-2}$ $\hbox {GeV}^2$ [74]. The accuracy of those parameters without uncertainties is of order of $50\%$

3 Numerical results

3.1 $\Lambda _b \rightarrow p$ form factors at $q^2=0$

We present the numerical results of the $\Lambda _b \rightarrow p$ transition form factors in this subsection, which include the contributions from the $\Lambda _b$ baryon LCDAs up to twist 4 and the proton LCDAs up to twist 6. Because the PQCD predictions are more reliable in the large recoil (small $q^2$) region, we evaluate the form factors at $q^2=0$ and then extrapolate them to the whole kinematic range $0\le q^2\le (m_{\Lambda _b}-m_p)^2$ in order to estimate the branching ratios of the semileptonic decays $\Lambda _b \rightarrow p\ell \nu _\ell$.

We first compare in Table 3 the contributions to the form factors from the proton LCDAs of different twists by convoluting them with the hard kernels and the leading-twist $\Lambda _b$ baryon LCDA in Eq. (15). The values denoted by $\sim 0$ are smaller than $1\times 10^{-6}$, and the entries in the last column sum up the contributions from all the proton LCDAs. The result of the form factor $f_1(0)=1.9 \times 10^{-3}$ from the leading-twist $\Lambda _b$ baryon and proton LCDAs is consistent with the previous PQCD calculations [31, 41]. It is found that the contributions from the twist-4 proton LCDAs are much larger than from the leading-twist ones, which dominate the form factors $f_1(0)$ and $g_1(0)$. The twist-5 contributions to the other four form factors are also sizeable. Table 3 indicates clearly that the higher-twist contributions significantly enhance the $\Lambda _b\rightarrow p$ form factors. The enhancement of B meson transition form factors by the higher-twist pion LCDAs was also observed [75], but not as strong as in the baryonic case.

Table 3 Form factors in units of $10^{-3}$ from the leading-twist $\Lambda _b$ baryon LCDA in Eq. (15) and the proton LCDAs of various twists

We present the contributions to the form factor $f_1(0)$ from various twist combinations of the $\Lambda _b$ baryon and proton LCDAs in Table 4. All the four models of the $\Lambda _b$ baryon LCDAs are covered. The two theoretical uncertainties in the total results are estimated from the variations of the parameters in the $\Lambda _b$ baryon LCDAs introduced in Sect. 2.2.1, and in the proton LCDAs listed in Table 2, respectively. It is noticed that the dominant contribution to $f_1(0)$ comes from the combination of the twist-4 $\Lambda _b$ baryon LCDA and the twist-5 proton LCDAs, and the leading-twist contribution is about two orders of magnitude lower than the dominant one, no matter which model of the $\Lambda _b$ baryon LCDAs is employed. It further confirms that the higher-twist contributions are crucial for the $\Lambda _b\rightarrow p$ form factors at the scale of the b quark mass. We point out that the twist-6 proton LCDAs, as combined with the first three models of the $\Lambda _b$ baryon LCDAs, do give contributions smaller than the leading-twist ones. Note that the $\Lambda _b$ baryon LCDA in Eqs. (14) and (15), despite of being classified as twist 2, contains some twist-3 components through the $m_{\Lambda _b}$ term actually (but without the mixture from the twist-4 component). Thus, the results in Table 3 should be compared to those without the twist-4 $\Lambda _b$ baryon LCDA in Table 4. It is then reasonable that the contribution from the twist-4 proton LCDAs is more important than the one from the twist-5 proton LCDAs in Table 3.

Table 4 Form factor $f_1(0)$ from various twist combinations of the $\Lambda _b$ baryon and proton LCDAs. The first (second) theoretical errors of the total results come from the variations of the relevant parameters in the $\Lambda _b$ baryon (proton) LCDAs

Table 5 Numerators $h_{ij}$ of the integrands in the factorization formula for the form factor $f_1(0)$ from the diagram $D_7$ in Fig. 1 with $i=2,3^{+-},3^{-+},4$ and $j=3,4,5,6$. An overall coefficient ${\mathcal {C}}m_{\Lambda _b}^3f_{\Lambda _b} /(64\sqrt{2}N_c^2)$ with the color factor ${\mathcal {C}}=8/3$ is implicit for the entries

Fig. 2

Dependencies of the $\Lambda _b$ baryon LCDAs $\psi _2$, $\psi _3^{+-}$, $\psi _3^{-+}$ and $\psi _4$ on the momentum fractions $x_i$ for the exponential model (top) and the free-parton model (bottom) proposed in [67]. Each point inside the triangles satisfies the relation $x_1+x_2+x_3=1$

It is seen in Table 4 that the values of $f_1(0)$ derived from the two Gegenbauer models for the $\Lambda _b$ baryon LCDAs are quite different, and those from the exponential and free-parton models are close to each other. Hence, we will focus on the latter for the numerical analysis and discussion hereafter. To understand why the power-suppressed contribution incredibly surpasses the leading-power one, we take a closer look at the behaviors of the integrands in the factorization formula for $f_1(0)$. We exhibit in Table 5 the numerators $h_{ij}$ of the integrands from the diagram $D_7$ in Fig. 1, which are proportional to the products of the $\Lambda _b$ baryon LCDAs of twists $i=2,3^{+-},3^{-+},4$ and the proton LCDAs of twists $j=3,4,5,6$. The powers in the small mass ratio $r=m_p/m_{\Lambda _b}$ manifest the $1/m_{b}$ suppression associated with the higher-twist proton LCDAs. Comparing $h_{45}$ with $h_{44}$, one finds that the former contains one more power of r, but the latter acquires an additional factor $1-x_1$. We then illustrate the dependencies of the $\Lambda _b$ baryon LCDAs of different twists on the three momentum fractions $x_{1,2,3}$ in Fig. 2 for the exponential and free-parton models. It is obvious that these two models show similar behaviors with the twist-4 LCDAs strongly peaking around $x_1\approx 1$, where the b quark carries most of the $\Lambda _b$ baryon momentum. It turns out that the factor $1-x_1$ yields a severe suppression on $h_{44}$, and $h_{45}$ contributes dominantly, although it is down by a power of $1/m_b$.

THE leading-twist LCDAs contribute to the $\Lambda _b\rightarrow p$ form factors only through the diagrams $D_1$ and $D_2$, but not through the diagram $D_7$, at the level of the theoretical accuracy in the present work. We thus compare $h_{45}$ with the leading-twist one $h_{23}$,

$$\begin{aligned} h_{23}(\mathrm{from}~ D_1)&={{\mathcal {C}}m_{\Lambda _b}^3\over 64\sqrt{2}N_c^2}f_{\Lambda _b} \psi _2 4(1-x_2) (V_1-A_1+2T_1), \end{aligned}$$

(42)

$$\begin{aligned} h_{45}(\mathrm{from}~ D_7)&={{\mathcal {C}}m_{\Lambda _b}^3\over 64\sqrt{2}N_c^2}f_{\Lambda _b}r^2 \psi _4(1-x_2^\prime )\nonumber \\&\quad (V_4-V_5+A_4+A_5+T_4+T_8+S_2-P_2), \end{aligned}$$

(43)

with the color factor ${\mathcal {C}}=8/3$. As indicated in Fig. 2, $\psi _4$ in the exponential and free-parton models exhibit the maxima near the endpoint $x_1\approx 1$, i.e., $(x_2+x_3)\approx 0$, and decrease with $x_2+x_3$ owing to the factors $e^{-(x_2+x_3)m_{\Lambda _b}/\omega _0}$ and $(2{\bar{\Lambda }}-x_2m_{\Lambda _b}-x_3m_{\Lambda _b})^3$, respectively. However, $\psi _2$, being proportional to $x_2x_3$, diminishes around the endpoint region. Figure 3 shows that the combination of the higher-twist LCDAs $V_4-V_5+A_4+A_5+T_4+T_8+S_2-P_2$ is also larger than the leading-twist one $V_1-A_1+2T_1$ in the endpoint region $x_1^\prime \approx 1$, i.e., $x_2^\prime \approx 0$. The virtual particle propagators in the diagram $D_7$, being proportional to $1/(1-x_1)$ and $1/x_2^\prime$ roughly (see the factorization formulas in Appendix A), induce further enhancement as $x_1\approx 1$ and $x_2^\prime \approx 0$. The above endpoint behaviors explain why the higher-twist LCDAs of both baryons overcome the power suppression from $1/m_b$ and remarkably increase the contributions to the form factors.

The huge enhancement caused by the higher-twist baryon LCDAs warns us to investigate how the contribution to the form factor $f_1(q^2=0)$ is distributed in the impact-parameter space. We truncate the integrations over $b_i^\prime$ on the proton side at a common upper bound $b_c<1/\Lambda _{\mathrm{QCD}}$, as done in Ref. [76]. The curves for both the exponential and free-parton models increase from 0 and become flat gradually as $b_c\rightarrow 1/\Lambda _{\mathrm{QCD}}$ in Fig. 4, implying that the Sudakov suppression on the long-distance contribution is effective enough. Since the hard t is chosen as the maximum of all the scales involved in the $\Lambda _b\rightarrow p$ transition, Fig. 4 confirms that the increase of $f_1(0)$ is not attributed to nonperturbative dynamics from $b_c\sim 1/\Lambda _{\mathrm{QCD}}$. In addition, the contributions to the $\Lambda _b\rightarrow p$ transition form factor $f_1(0)$ from different ranges of $\alpha _s/\pi$ for the exponential model are displayed in Fig. 5. It implies that most of the contribution comes from the range with $\alpha _s/\pi < 0.2$, and that the contribution to a heavy-to-light baryonic transition form factor is indeed perturbative in the PQCD approach [77].

Fig. 3

Dependencies of the leading-twist proton LCDAs $V_1-A_1+2T_1$ in Eq. (42) (left) and of the twist-5 proton LCDAs $V_4-V_5+A_4+A_5+T_4+T_8+S_2-P_2$ in Eq. (43) (right) on the momentum fractions $x_i^\prime$

Fig. 4

Dependence of the form factor $f_1(0)$ on the cutoff $b_c$

Fig. 5

Contributions to the $\Lambda _b\rightarrow p$ transition form factor $f_1(q^2=0)$ from different ranges of $\alpha _s/\pi$ for the exponential model

Table 6 Form factors $f_1(0)$, $f_2(0)$, $g_1(0)$ and $g_2(0)$ from the exponential and free-parton models of the $\Lambda _b$ baryon LCDAs obtained in this work and in the NRQM, LCSR, 3-point QSR, lattice and previous PQCD studies. The theoretical errors of our results from the variations of the relevant parameters in the $\Lambda _b$ baryon and proton LCDAs have been added in quadrature

Table 7 The same as Table 4 but for the form factor $f_2(0)$ in unit of $10^{-3}$

Table 8 The same as Table 4 but for the form factor $f_3(0)$ in unit of $10^{-3}$

Table 9 The same as Table 4 but for the form factor $g_1(0)$

Table 10 The same as Table 4 but for the form factor $g_2(0)$ in unit of $10^{-3}$

Table 11 The same as Table 4 but for form factor $g_3(0)$ in unit of $10^{-3}$

We compare our predictions for the form factors $f_1(0)$, $f_2(0)$, $g_1(0)$ and $g_2(0)$ in Table 6 with those from the nonrelativistic quark model (NRQM) [78], LCSR [50, 79, 80], relativistic quark model [81], 3-point QSR [49], lattice QCD [47] and previous PQCD studies [31]. The consistency with those from light-LCSR-${\mathcal {A}}$, light-LCSR-${\mathcal {P}}$ and lattice QCD hints that the PQCD approach is an effective framework for analyzing exclusive heavy baryon decays, once the higher-twist contributions are taken into account.

The other five form factors $f_{2,3}(0)$ and $g_{1,2,3}(0)$ are also computed using the exponential and free-parton models for the $\Lambda _b$ baryon LCDAs, and the outcomes are gathered in Tables 7, 8, 9, 10 and 11. Similarly, these form factors receive large contributions from the higher-twist LCDAs, and follow the relations postulated in HQET, $f_1\approx g_1$ and $f_2\approx f_3\approx g_2\approx g_3\approx 0$ [5, 82].

Table 12 Values of the parameters for the form factors in Eq. (44)

3.2 Semileptonic and hadronic $\Lambda _b$ baryon decays

To calculate the branching ratios of the semileptonic decays $\Lambda _b\rightarrow p\ell \nu _\ell$ with $\ell =e,\mu ,\tau$, we need the behaviors of the $\Lambda _b\rightarrow p$ transition form factors in the whole kinematic range $0\le q^2\le (m_{\Lambda _b}-m_p)^2$. The form factors are first evaluated at seven points of $q^2$ in the low $q^2$ region from 0 to $m_\tau ^2$, $m_\tau$ being the $\tau$ lepton mass, where the PQCD approach is trustworthy. We then derive the form factors in the large $q^2$ region via an extrapolation from the low-$q^2$ results, assuming the parametrizations of $F(\equiv f_i,g_i)$ in the z-series formula [83],

$$\begin{aligned} F(q^2)=\frac{F(0)}{1-q^2/m_\mathrm{pole}^2}\left\{ 1+\sum _{k=1}^{N}a_k[z^k(q^2)-z^k(0)]\right\} . \end{aligned}$$

(44)

The parameter z is defined as

$$\begin{aligned} z(q^2)=\frac{\sqrt{t_+-q^2}-\sqrt{t_+-t_0}}{\sqrt{t_+-q^2}+\sqrt{t_+-t_0}}, \end{aligned}$$

(45)

with $t_0=t_+(1-\sqrt{1-t_-/t_+})$ and $t_{\pm }=(m_{\Lambda _b}{\pm } m_p)^2$. The values of F(0) and the poles $m_{\mathrm{pole}}$ are taken from the PQCD calculation and from [84], respectively. We truncate the expansion in Eq. (44) at $N=1$ for simplicity, and fit it to the seven low-$q^2$ inputs to determine the single free parameter $a_1$.

The fit quality is measured by the goodness of fit,

$$\begin{aligned} R^2=1-\frac{\sum _{j=1}^{7}(F_j-F^\mathrm{in}_j)^2}{\sum _{j=1}^{7}(F^\mathrm{in}_j-{\bar{F}}^\mathrm{in})^2}, \end{aligned}$$

(46)

where $F^\mathrm{in}_j$ denote the seven PQCD inputs in the region $0\le q^2\le m_\tau ^2$, $F_j$ come from Eq. (44) at the same $q^2$ as $F^\mathrm{in}_j$, and ${\bar{F}}^\mathrm{in}$ is the mean value of the seven $F^\mathrm{in}_j$. We have checked that $R^2$’s in all the fits range between 0.99 and 1, reflecting the satisfactory quality. The resultant parameters for Eq. (44) are given in Table 12 with the standard errors of $a_1$ being assigned. The $q^2$ dependencies of the form factors are exhibited in Fig. 6, where the bands represent the theoretical uncertainties caused by the parameters in the baryon LCDAs and by the parameter $a_1$. The absolute values of all the form factors increase with $q^2$ as expected.

Fig. 6

$q^2$ dependencies of the form factors

We then predict the branching ratios of the semileptonic decays $\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell$ based on the form factors presented in Fig. 6. We introduce the helicity amplitudes

$$\begin{aligned} H_{\lambda _2,\lambda _W}&=H_{\lambda _2,\lambda _W}^V-H_{\lambda _2,\lambda _W}^A, \end{aligned}$$

(47)

$$\begin{aligned} H_{\lambda _2,\lambda _W}^{V(A)}&=\epsilon ^{\dagger \mu }(\lambda _W)\langle p,\lambda _2|V(A)|\Lambda _b,\lambda _1\rangle , \end{aligned}$$

(48)

where $\lambda _1$, $\lambda _2$ and $\lambda _W$ denote the helicities of the $\Lambda _b$ baryon, the proton and the off-shell $W^-$ boson, respectively, $\epsilon ^\mu$ is the polarization vector of the $W^-$ boson, and V (A) labels the vector (axial vector) current. The various helicity amplitudes are then expressed, in terms of the $\Lambda _b\rightarrow p$ form factors, as [85,86,87,88,89,90,91,92]

$$\begin{aligned} H_{1/2,0}^V&=\frac{\sqrt{(m_{\Lambda _b}-m_p)^2-q^2}}{\sqrt{q^2}}\nonumber \\&\quad \times \left[ (m_{\Lambda _b}+m_p)f_1(q^2) + q^2f_2(q^2)\right] ,\nonumber \\ H_{1/2,0}^A&=\frac{\sqrt{(m_{\Lambda _b}+m_p)^2-q^2}}{\sqrt{q^2}}\nonumber \\&\quad \times \left[ (m_{\Lambda _b}-m_p)g_1(q^2) - q^2g_2(q^2)\right] ,\nonumber \\ H_{1/2,1}^V&=\sqrt{2[(m_{\Lambda _b}-m_p)^2-q^2]}\nonumber \\&\quad \times \left[ -f_1(q^2) - (m_{\Lambda _b}+m_p)f_2(q^2)\right] ,\nonumber \\ H_{1/2,1}^A&=\sqrt{2[(m_{\Lambda _b}+m_p)^2-q^2]}\nonumber \\&\quad \times \left[ -g_1(q^2) + (m_{\Lambda _b}-m_p)g_2(q^2)\right] ,\nonumber \\ H_{1/2,t}^V&=\frac{\sqrt{(m_{\Lambda _b}+m_p)^2-q^2}}{\sqrt{q^2}}\nonumber \\&\quad \times \left[ (m_{\Lambda _b}-m_p)f_1(q^2) + q^2f_3(q^2)\right] ,\nonumber \\ H_{1/2,t}^A&=\frac{\sqrt{(m_{\Lambda _b}-m_p)^2-q^2}}{\sqrt{q^2}}\nonumber \\&\quad \times \left[ (m_{\Lambda _b}+m_p)g_1(q^2) - q^2g_3(q^2)\right] , \end{aligned}$$

(49)

where the subscript t refers to the temporal component of the helicities of the off-shell $W^-$ boson [88]. The above amplitudes obey the relations $H_{\lambda _2,\lambda _W}^V=H_{-\lambda _2,-\lambda _W}^V$ and $H_{\lambda _2,\lambda _W}^A=-H_{-\lambda _2,-\lambda _W}^A$.

Fig. 7

$q^2$ dependencies of the differential widths for the $\Lambda _b\rightarrow p \ell {\bar{\nu }}_\ell$ decays with $\ell =e,\nu ,\tau$

The differential angular distributions for the semileptonic decays $\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell$ are given by [54, 85]

$$\begin{aligned} \frac{d\Gamma (\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell )}{dq^2d\cos \theta _\ell }= & {} \frac{G_F^2|V_{ub}|^2q^2|\mathbf {p}_p|}{512\pi ^3m_{\Lambda _b}^2}\nonumber \\&\times \left( 1-\frac{m_\ell ^2}{q^2}\right) ^2\left( A_1+\frac{m_\ell ^2}{q^2}A_2\right) , \end{aligned}$$

(50)

where $m_\ell$ is the charged lepton mass, $\theta _\ell$ stands for the angle between the charged lepton and the proton in the $\Lambda _b$ baryon rest frame, and the factors $A_1$, $A_2$ and $|\mathbf {p}_p|$ are written as

$$\begin{aligned} A_1&=2\sin ^2\theta _\ell (H_{1/2,0}^2+H_{-1/2,0}^2)+(1-\cos \theta _\ell )^2H_{1/2,1}^2\nonumber \\&\quad +(1+\cos \theta _\ell )^2H_{-1/2,-1}^2, \end{aligned}$$

(51)

$$\begin{aligned} A_2&=2\cos ^2\theta _\ell (H_{1/2,0}^2+H_{-1/2,0}^2)\nonumber \\&\quad +\sin ^2\theta _\ell (H_{1/2,1}^2+H_{-1/2,-1}^2)+2(H_{1/2,t}^2+H_{-1/2,t}^2),\nonumber \\&\quad -4\cos \theta _\ell (H_{1/2,0}H_{1/2,t}+H_{-1/2,0}H_{-1/2,t}) \end{aligned}$$

(52)

$$\begin{aligned} |\vec {p}_p|&=\frac{\sqrt{m_{\Lambda _b}^4+m_p^4+q^4-2(m_{\Lambda _b}^2m_p^2+m_p^2q^2+q^2m_{\Lambda _b}^2)}}{2m_{\Lambda _b}}. \end{aligned}$$

(53)

Integrating Eq. (50) over $\cos \theta _\ell$, we derive the differential decay widths

$$\begin{aligned} \frac{d\Gamma (\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell )}{dq^2}=\int _{-1}^{1}\frac{d\Gamma (\Lambda _b\rightarrow p\ell {\bar{\nu }}_\ell )}{dq^2d\cos \theta _\ell }d\cos \theta _\ell . \end{aligned}$$

(54)

plotted in Fig. 7 for $\ell =e,\mu ,\tau$, which lead to the integrated branching ratios ${\mathcal {B}}(\Lambda _b\rightarrow p\ell {{\bar{\nu }}}_\ell )=(16{\pm }11)\times 10^{-4}$ with $\ell =e,\mu$ and ${\mathcal {B}}(\Lambda _b\rightarrow p\tau \bar{\nu }_\tau )=(11{\pm }7)\times 10^{-4}$ for the exponential model of the $\Lambda _b$ baryon LCDAs, and ${\mathcal {B}}(\Lambda _b\rightarrow p\ell \bar{\nu }_\ell )=(14{\pm }10)\times 10^{-4}$ with $\ell =e,\mu$ and ${\mathcal {B}}(\Lambda _b\rightarrow p\tau \bar{\nu }_\tau )=(10{\pm }7)\times 10^{-4}$ for the free parton model. We remind that the above results for the $\Lambda _b\rightarrow p\tau \bar{\nu }_\tau$ decay depend on the extrapolation of the form factors to the large $q^2$ region completely. The central values of our predictions are higher than the LHCb data ${\mathcal {B}}(\Lambda _b\rightarrow p\mu \bar{\nu }_\mu )=(4.1{\pm }1.0)\times 10^{-4}$ [93] due to the larger form factors obtained in the previous subsection. However, they are still compatible with each other when the substantial theoretical uncertainties are considered. Besides, we expect that our results will decrease a bit, after the threshold Sudakov factor mentioned in the Introduction and the intrinsic impact-parameter dependencies of the baryon wave functions are included.

Next we estimate the branching ratio of the two-body hadronic decay $\Lambda _b\rightarrow p M$, with M denoting a light pesudoscalar meson, in the naive factorization framework. This decay is dominated by the color-favored tree contribution, for which the naive factorization assumption is supposed to hold reasonably well. The corresponding decay amplitude is expressed as

$$\begin{aligned}&\langle p M |{\mathcal {H}}_{eff}|\Lambda _b \rangle = \frac{G_F}{\sqrt{2}}V_{ub}^{*}V_{uq}a_1(\mu )\nonumber \\&\quad \langle M|{\bar{u}}\gamma ^\mu (1-\gamma _5)q|0\rangle \langle p|{\bar{u}}\gamma _\mu (1-\gamma _5)b|\Lambda _b\rangle , \end{aligned}$$

(55)

where $G_F=1.166\times 10^{-5}$ $\hbox {GeV}^{-2}$ is the Fermi constant, $V_{ub}$ and $V_{uq}$ are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and $a_1(\mu )=C_1(\mu )+C_2(\mu )/3$ represents the Wilson coefficient with $C_1(m_b)=1.076$ and $C_2(m_b)=-0.175$ at the scale of the b quark mass $m_b=4.8$ GeV.

Inserting the definition of the pseudoscalar decay constant $f_M$,

$$\begin{aligned} \langle M(q)|{\bar{u}}\gamma ^\mu (1-\gamma _5)d|0\rangle = -if_M q^\mu , \end{aligned}$$

(56)

q being the pseudoscalar meson momentum, we decompose the $\Lambda _b\rightarrow p M$ decay amplitude into

$$\begin{aligned} \mathcal{M}(\Lambda _b\rightarrow pM)=i{\bar{N}}(\mathcal{M}_1+\mathcal{M}_2\gamma _5)\Lambda _b. \end{aligned}$$

(57)

The functions $\mathcal{M}_1$ and $\mathcal{M}_2$ are given by

$$\begin{aligned} \mathcal{M}_1&= \frac{G_F}{\sqrt{2}}V_{ub}^{*}V_{uq}a_1(\mu )f_M (m_{\Lambda _b}-m_p)f_1(m_M^2), \end{aligned}$$

(58)

$$\begin{aligned} \mathcal{M}_2&= \frac{G_F}{\sqrt{2}}V_{ub}^{*}V_{uq}a_1(\mu )f_M(m_{\Lambda _b}+m_p)g_1(m_M^2), \end{aligned}$$

(59)

with the pseudoscalar meson mass $m_M$. The $\Lambda _b\rightarrow p M$ decay width is then written as

$$\begin{aligned} \Gamma (\Lambda _b\rightarrow pM)= & {} \frac{|\overrightarrow{p_p}|}{8\pi }\left[ \frac{(m_{\Lambda _b}+m_p)^2-m_M^2}{m_{\Lambda _b}^2}|\mathcal{M}_1|^2\right. \nonumber \\&\left. +\frac{(m_{\Lambda _b}-m_p)^2-m_M^2}{m_{\Lambda _b}^2}|\mathcal{M}_2|^2\right] , \end{aligned}$$

(60)

where $|\overrightarrow{p_p}|=\sqrt{[(m_{\Lambda _b}^2-(m_M+m_p)^2][m_{\Lambda _b}^2-(m_M-m_p)^2]}/(2m_{\Lambda _b})$ is the proton momentum in the rest frame of the $\Lambda _b$ baryon.

It is straightforward to get, by employing the form factors derived in the previous subsection, the $\Lambda _b\rightarrow p\pi$ branching ratio ${\mathcal {B}}(\Lambda _b\rightarrow p\pi ^-)=(13{\pm }10)\times 10^{-6}$ from the exponential model for the $\Lambda _b$ baryon LCDAs, and ${\mathcal {B}}(\Lambda _b\rightarrow p\pi ^-)=(11{\pm }8)\times 10^{-6}$ from the free-parton model, whose large theoretical uncertainties originate from those of the form factors. These predictions are also higher than the experimental data ${\mathcal {B}}(\Lambda _b\rightarrow p\pi ^-)=(4.5{\pm }0.8)\times 10^{-6}$ [94], similar to the case of the semileptonic decays.

4 Summary

The heavy-to-light transition form factors are important ingredients for exclusive heavy hadron decays. The previous studies of the $\Lambda _b \rightarrow p$ form factors in the QCDF and PQCD approaches have manifested that the leading-power contribution is much smaller than indicated by the experimental data. In this paper we extended the PQCD analysis to the inclusion of the higher-twist baryon LCDAs. It was observed that the combination of the twist-4 $\Lambda _b$ baryon LCDAs and the twist-5 (twist-4) proton LCDAs dominates the contributions to the form factors $f_{1,2,3}$ and $g_{1,2}$ (the form factor $g_3$), and that the contributions from the twist-6 proton LCDAs are indeed suppressed. We have examined the distribution of the above dominant pieces in the impact-parameter space, and concluded that the enhancement of the form factors is not attributed to the long-distance dynamics. Our results for the form factors are close to those in other theoretical methods within errors, implying that the endpoint contributions from the higher-twist LCDAs can be handled appropriately in the PQCD formalism, and the framework established here is ready for systematic applications to various semileptonic and hadronic two-body decays of heavy baryons.

Based on the obtained form factors, we have estimated the branching ratios of the semileptonic decays $\Lambda _b\rightarrow p\ell {{\bar{\nu }}}_\ell$ and of the two-body hadronic decay $\Lambda _b \rightarrow p\pi$ under the naive factorization assumption, whose central values are higher than but still compatible with the measured ones, as the significant theoretical uncertainties are considered. It suggests that a precise knowledge of the baryon LCDAs is necessary for a rigorous comparison between theoretical predictions and experimental data. It is also urgent to derive the threshold Sudakov factor for heavy-to-light baryonic transitions, which is expected to improve the agreement with the current data by lowering our results to some extent. The intrinsic impact-parameter dependencies of the baryon wave functions can also be taken into account to achieve the same purpose. At last, the complete set of topological diagrams needs to be calculated in order to make predictions for CPV in hadroic two-body heavy baryon decays. This calculation is feasible in principle in the PQCD approach.

Table 13 Auxiliary functions A, B, C and D. An overall coefficient $m_{\Lambda _b}^2$ is implicit for the entries

Table 14 The same as Table 5 but for the diagrams $D_1, D_2, D_3, D_4$ and $D_5$ in Fig. 1

Table 15 The same as Table 5 but for the diagrams $D_6, D_7, D_8, D_9$ and $D_{10}$ in Fig. 1

Table 16 The same as Table 5 but for the diagrams $D_{11}, D_{12}, D_{13}$ and $D_{14}$ in Fig. 1

Appendix A: Factorization formulas

We list below the factorization formulas for the form factor $f_1(q^2=0)$ from the Feynman diagrams $D_i$, $i=1$-14, in Fig. 1. Those for the other five form factors can be derived in a similar way. The last two diagrams with three-gluon vertices do not contribute due to the vanishing color factors:

$$\begin{aligned} f_1^{D_1}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_1})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\quad \int b_3 db_3\int d\theta _1\int d\theta _2\exp [-S^{D_1}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(D,b_3)F_3(A,B,C,b_1^\prime ,b_2), \end{aligned}$$

(61)

$$\begin{aligned} f_1^{D_2}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_2})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\exp [-S^{D_2}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(C,b_2)F_3(A,B,D,b_1^\prime ,b_3), \end{aligned}$$

(62)

$$\begin{aligned} f_1^{D_3}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_3})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_3 db_3\nonumber \\&\quad \int b_3^\prime db_3^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_3}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_3-b_1^\prime -b_3^\prime )\nonumber \\&\quad F_1(B,b_3+b_3^\prime )F_1(C,b_1)F_1(D,b_3^\prime ), \end{aligned}$$

(63)

$$\begin{aligned} f_1^{D_4}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_4})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_3 db_3\nonumber \\&\quad \int b_3^\prime db_3^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_4}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_3-b_1^\prime -b_3^\prime )\nonumber \\&\quad F_1(B,b_3+b_3^\prime )F_1(C,b_3+b_3^\prime -b_1)F_1(D,b_3), \end{aligned}$$

(64)

$$\begin{aligned} f_1^{D_5}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_5})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\quad \int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_5}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_2-b_2^\prime )\nonumber \\&\quad F_1(B,b_2+b_2^\prime )F_1(C,b_2+b_2^\prime -b_1)F_1(D,b_2), \end{aligned}$$

(65)

$$\begin{aligned} f_1^{D_6}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_6})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\quad \int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_6}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_2-b_2^\prime )\nonumber \\&\quad F_1(B,b_2+b_2^\prime )F_1(C,b_1)F_1(D,b_2^\prime ), \end{aligned}$$

(66)

$$\begin{aligned} f_1^{D_7}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\nonumber \\&\quad \int [dx]\int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_7})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_1 db_1 \int b_2^\prime db_2^\prime \int b_3 db_3\int d\theta _1\nonumber \\&\quad \int d\theta _2\exp [-S^{D_7}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(D,b_2^\prime -b_3^\prime )F_3(A,B,C,b_2^\prime ,b_1), \end{aligned}$$

(67)

$$\begin{aligned} f_1^{D_8}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\nonumber \\&\quad \int [dx]\int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_8})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_1 db_1 \int b_2 db_2\nonumber \\&\quad \int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\exp [-S^{D_8}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(C,b_2^\prime )F_3(A,B,D,b_2+b_2^\prime ,b_1), \end{aligned}$$

(68)

$$\begin{aligned} f_1^{D_9}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_9})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_3 db_3\nonumber \\&\quad \int b_3^\prime db_3^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_9}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_3-b_3^\prime )\nonumber \\&F_1(B,b_3+b_3^\prime )F_1(C,b_3+b_3^\prime -b_1^\prime )F_1(D,b_3^\prime ), \end{aligned}$$

(69)

$$\begin{aligned} f_1^{D_{10}}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_{10}})h\frac{1}{(2\pi )^7}\int b_1db_1\nonumber \\&\quad \int b_1^\prime db_1^\prime \int b_3 db_3 \int b_3^\prime db_3^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_{10}}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_3-b_3^\prime )\nonumber \\&\quad F_1(B,b_3+b_3^\prime )F_1(C,b_1^\prime )F_1(D,b_3), \end{aligned}$$

(70)

$$\begin{aligned} f_1^{D_{11}}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\int [dx]\nonumber \\&\quad \int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_{11}})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\quad \int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_{11}}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_2-b_2^\prime )\nonumber \\&\quad F_1(B,b_2+b_2^\prime )F_1(C,b_2^\prime )F_1(D,b_2+b_2^\prime -b_1^\prime ), \end{aligned}$$

(71)

$$\begin{aligned} f_1^{D_{12}}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\nonumber \\&\quad \int [dx]\int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_{12}})h \frac{1}{(2\pi )^7}\nonumber \\&\quad \int b_1db_1\int b_1^\prime db_1^\prime \int b_2 db_2\nonumber \\&\quad \int b_2^\prime db_2^\prime \int d\theta _1\int d\theta _2\nonumber \\&\quad \int d\theta _3\exp [-S^{D_{12}}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_1(A,b_1+b_1^\prime -b_2-b_2^\prime )\nonumber \\&\quad F_1(B,b_2+b_2^\prime )F_1(C,b_1^\prime )F_1(D,b_2), \end{aligned}$$

(72)

$$\begin{aligned} f_1^{D_{13}}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\nonumber \\&\quad \int [dx]\int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_{13}})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_2 db_2 \int b_2^\prime db_2^\prime \int b_3 db_3\int d\theta _1\nonumber \\&\quad \int d\theta _2\exp [-S^{D_{13}}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_2(A,B,b_2+b_2^\prime )F_1(C,b_2^\prime )F_1(D,b_3), \end{aligned}$$

(73)

$$\begin{aligned} f_1^{D_{14}}(q^2=0)&={\mathcal {C}}M_{\Lambda _b}^3 \frac{f_{\Lambda _b}}{8N_c} \frac{1}{8\sqrt{2}N_c}\nonumber \\&\quad \int [dx]\int [dx^\prime ]16\pi ^2\alpha _s^2(t^{D_{14}})h \frac{1}{(2\pi )^5}\nonumber \\&\quad \int b_2 db_2 \int b_2^\prime db_2^\prime \int b_3 db_3\nonumber \\&\quad \int d\theta _1\int d\theta _2\exp [-S^{D_{14}}(x,x^\prime ,b,b^\prime )]\nonumber \\&\quad F_2(A,B,b_2+b_2^\prime )F_1(C,b_2)F_1(D,b_3^\prime ), \end{aligned}$$

(74)

where ${\mathcal {C}}=8/3$ is the color factor and $[dx]\equiv dx_1dx_2dx_3\delta (1-x_1-x_2-x_3)$, $[dx']$ is defined analogously, the auxiliary functions A, B, C and D in Table 13 are related to the denominators of the four propagators in each diagram, and the hard kernels h are collected in Tables 14, 15 and 16. The exponent $S^{D_i}$ is the sum of the total exponents from the $\Lambda _b$ and proton wave functions with the hard scale $t^{D_i}$ involved in the diagram $D_i$. The functions $F_1, F_2$ and $F_3$ are written, in terms of the Fourier integrals, as

$$\begin{aligned}&F_1(A,b)=\int d^2k_T\frac{e^{i{\textbf {k}}_T\cdot {\textbf {b}}}}{k^2+A}\nonumber \\&\quad =2\pi \left\{ K_0(\sqrt{A}b)\theta (A)\right. \nonumber \\&\left. \qquad +\frac{\pi i}{2}\left[ J_0(\sqrt{-A}b)+iN_0(\sqrt{-A}b)\right] \theta (-A)\right\} , \end{aligned}$$

(75)

$$\begin{aligned}&F_2(A,B,b)=\int d^2k_T\frac{e^{i{\textbf {k}}_T\cdot {\textbf {b}}}}{(k^2+A)(k^2+B)}\nonumber \\&\quad =\pi \int _{0}^{1}dz\frac{b}{\sqrt{|Z_1|}}\left\{ K_1(\sqrt{Z_1}b)\theta (Z_1)\right. \nonumber \\&\qquad \left. +\frac{\pi }{2}\left[ N_1(\sqrt{-Z_1}b)-iJ_1(\sqrt{-Z_1}b)\right] \theta (-Z_1)\right\} , \end{aligned}$$

(76)

$$\begin{aligned}&F_3(A,B,C,b_1,b_2)=\int d^2k_{1T}\nonumber \\&\qquad \times \int d^2k_{2T}\frac{e^{i({\textbf {k}}_{1T}\cdot {\textbf {b}}_1+{\textbf {k}}_{2T}\cdot {\textbf {b}}_2)}}{(k_1^2+A)(k_2^2+B)((k_1+k_2)^2+C)}\nonumber \\&\quad =\pi ^2\int _{0}^{1}\frac{dz_1dz_2}{z_1(1-z_1)}\frac{\sqrt{X_2}}{\sqrt{|Z_2|}}\nonumber \\&\qquad \times \left\{ K_1(\sqrt{X_2Z_2})\theta (Z_2)+\frac{\pi }{2}\left[ N_1(\sqrt{-X_2Z_2})\right. \right. \nonumber \\&\qquad \left. \left. -iJ_1(\sqrt{-X_2Z_2})\right] \theta (-Z_2)\right\} , \end{aligned}$$

(77)

in which $J_n$ ($N_n$) is the Bessel function of the first (second) kind, $K_n$ follows the relation

$$\begin{aligned} K_n(-iz)=\frac{\pi i}{2}e^{(in\pi )/2}\left[ J_n(z)+iN_n(z)\right] , \end{aligned}$$

(78)

and the variables $Z_1$, $Z_2$ and $X_2$ are given by

$$\begin{aligned} Z_1&=Az+B(1-z), \end{aligned}$$

(79)

$$\begin{aligned} Z_2&=A(1-z_2)\nonumber \\&\quad +\frac{z_2}{z_1(1-z_1)}\left[ B(1-z_1)+Cz_1\right] , \end{aligned}$$

(80)

$$\begin{aligned} X_2&=(b_1-z_1b_2)^2+\frac{z_1(1-z_1)}{z_2}b_2^2, \end{aligned}$$

(81)

with Feynman parameters z’s. Because A, $Z_1$, $Z_2$ and $X_2$ are all positive, the imaginary parts of the above functions $F_{1,2,3}$ do not contribute.