The decay of $\Lambda _b\rightarrow p~K^-$ in QCD factorization approach

Abstract

With only the tree-level operator, the decay of $\Lambda _b\rightarrow pK$ is predicted to be one order smaller than the experimental data. The QCD penguin effects should be taken into account. In this paper, we explore the one-loop QCD corrections to the decay of $\Lambda _b\rightarrow pK$ within the framework of QCD factorization approach. For the baryon system, the diquark approximation is adopted. The transition hadronic matrix elements between $\Lambda _b$ and p are calculated in the light-front quark model. The branching ratio of $\Lambda _b\rightarrow pK$ is predicted to be about $4.85\times 10^{-6}$, which is consistent with experimental data $(4.9\pm 0.9)\times 10^{-6}$. The CP violation is about 5 % in theory.

1 Introduction

The weak decays of the heavy baryon $\Lambda _b$ provide an ideal place to extract information as regards the Cabibbo–Kobayashi–Maskawa (CKM) parameters and explore the mechanism of CP violation complementary to the B meson system. For the non-leptonic processes, the strong interaction dynamics is very complicated. Thus, these processes are also good probes to test different QCD models and factorization approaches. In early work of [1, 2], the weak decay of $\Lambda _b$ to $\Lambda _c$ and light baryons (p, $\Lambda$) are systematically studied. The hadronic transition matrix elements parameterized by form factors are calculated by use of a light-front quark model (LFQM) [3–15]. Since there are three valence quarks in a baryon, the quark–diquark picture was employed for simplification. It is found that the diquark approximation not only greatly simplifies the calculations, but also it gives good theoretical predictions.

With a simple factorization hypothesis, many non-leptonic processes of $\Lambda _b$ to a light baryon and a meson are calculated in [1]. The theory predictions of branching ratios are well consistent with the experiment data except one process of $\Lambda _b\rightarrow p~K^-$. The theoretical result is $Br(\Lambda _b\rightarrow p~K^-)=2.58\times 10^{-7}$, which is one order smaller than the data $(4.9\pm 0.9)\times 10^{-6}$ [16]. What is the reason? In fact, the physics reason had been discussed in [1]. The calculations are performed at the tree level. In most cases, the tree operator contribution is dominant. However, for the $\Lambda _b\rightarrow p~K^-$ process, the tree-level contribution is suppressed by the CKM matrix elements $V_{ub}V^{*}_{us}$. For the penguin diagram, the main contribution comes from the loop where the top quark is the dominant intermediate fermion. The CKM entry would be $V_{tb}V^{*}_{ts}$, which is almost 50 times larger than $V_{ub}V^{*}_{us}$. Thus even though there is a loop suppression of order $\alpha _s/4\pi$, it is compensated by the much larger CKM parameter, so the contributions from penguin diagrams are dominant. The effects of the QCD penguin have been displayed in $B\rightarrow \pi K$ processes. For example, the process of $B^0\rightarrow K^+\pi ^-$ is QCD penguin dominated and its branching ratio is $(1.94\pm 0.06)\times 10^{-5}$, while for a tree dominated process $B^0\rightarrow \pi ^+\pi ^-$ with $Br(B^0\rightarrow \pi ^+\pi ^-)=(5.15\pm 0.22)\times 10^{-6}$, the ratio is a factor of three smaller than that of $B^0\rightarrow K^+\pi ^-$.

Using the method of perturbative QCD (pQCD), $\Lambda _b\rightarrow p~K^{-}$ has been calculated in [17]. The result is $1.82\times 10^{-6}$ in the conventional pQCD approach and $2.02\times 10^{-6}$ in the hybrid pQCD approach. We can see that it is smaller than a half of the experimental data $(4.9\pm 0.9)\times 10^{-6}$. In this paper, we will study the QCD corrections in the decay $\Lambda _b\rightarrow p~K^{-}$ at one-loop order within the framework of QCD factorization approach [18–21]. This factorization approach provides a systematic method to treat the non-factorizable QCD effects. It has widely been applied to many B meson non-leptonic processes. We will employ this approach into the heavy baryon decays, the $\Lambda _b\rightarrow p~K^{-}$ process in this study.

The paper is organized as follows: In Sect. 2, we list the effective Hamiltonian for the transition $\Lambda _b\rightarrow p~K^-$, give the QCD factorization approach to $\Lambda _b\rightarrow p~K^-$, the decay rate, and then we discuss CP asymmetry and the relation to decay of $\bar{B^0}\rightarrow K^- \pi ^+$. In Sect. 3, we will give the numerical calculations. In Sect. 4, a discussion and conclusion is provided.

2 The decay $\Lambda _b\rightarrow p\, K^-$

2.1 Effective Hamiltonian for $\Lambda _b\rightarrow p\, K^-$

In the decay $\Lambda _b\rightarrow p\, K^-$, the initial $\Lambda _b$ and final p are baryons with three valence quarks. When the diquark picture is employed, i.e. the inner quark structure of $\Lambda _b$ is b[ud] and p is u[ud] where [ud] is a scalar diquark in this case and acting as a spectator. The effective Hamiltonian $H_{\mathrm{eff}}$ for $b\rightarrow s$ transitions can be written

$$\begin{aligned}&{\mathcal {H}}_{\mathrm{eff}}=\frac{G_F}{\sqrt{2}}\sum \limits _{q=u,c}V_{qb}V^{*}_{qs}\nonumber \\&\quad \times \left( C_1O^{q}_1+ C_2O^{q}_2+\sum \limits ^{10}_{i=3}C_{i}O_{i}+C_{7\gamma }O_{7\gamma }+C_{8g}O_{8g}\right) ,\nonumber \\ \end{aligned}$$

(1)

where $C_i$ are the Wilson coefficients evaluated at the renormalization scale $\mu$; the current–current operators $O_1^{u}$ and $O_2^{u}$ read

$$\begin{aligned}&O^{u}_1=\bar{s}_{\alpha }\gamma ^{\mu }L u_{\alpha }\cdot \bar{u}_{\beta }\gamma _{\mu }L b_{\beta },\quad O^{u}_2=\bar{s}_{\alpha }\gamma ^{\mu }L u_{\beta }\cdot \bar{u}_{\beta }\gamma _{\mu }L b_{\alpha }.\nonumber \\ \end{aligned}$$

(2)

The usual tree-level W-exchange contribution in the effective theory corresponding to $O_1$ and $O_2$ emerges due to the QCD corrections. The QCD penguin operators $O_3$–$O_6$ are

$$\begin{aligned}&O_3=\bar{s}_{\alpha }\gamma ^{\mu }L b_{\alpha }\cdot \sum \nolimits _{q'}\bar{q}'_{\beta }\gamma _{\mu }Lq'_{\beta },~~~~~\nonumber \\&O_4=\bar{s}_{\alpha } \gamma ^{\mu }L b_{\beta }\cdot \sum \nolimits _{q'}\bar{q}'_{\beta }\gamma _{\mu }L q'_{\alpha },\nonumber \\&O_5=\bar{s}_{\alpha }\gamma ^{\mu }L b_{\alpha }\cdot \sum \nolimits _{q'}\bar{q}'_{\beta }\gamma _{\mu }R q'_{\beta }, ~~~~~\nonumber \\&O_6=\bar{s}_{\alpha }\gamma ^{\mu }L b_{\beta }\cdot \sum \nolimits _{q'}\bar{q}'_{\beta }\gamma _{\mu }R q'_{\alpha }. \end{aligned}$$

(3)

They contribute in order $\alpha _s$ through the initial values of the Wilson coefficients at $\mu \approx M_W$ [22] and operator mixing due to the QCD correction [23–26]. Some operators $O_7,\ldots ,O_{10}$, which arise from the electroweak-penguin diagrams, are

$$\begin{aligned}&O_7=\frac{3}{2} \bar{s}_{\alpha }\gamma ^{\mu }L b_{\alpha }\cdot \sum \nolimits _{q'}e_{q'}\bar{q}'_{\beta }\gamma _{\mu }R q'_{\beta }, ~~~~~\nonumber \\&O_8=\frac{3}{2} \bar{s}_{\alpha }\gamma ^{\mu }L b_{\beta }\cdot \sum \nolimits _{q'}e_{q'}\bar{q}'_{\beta }\gamma _{\mu }R q'_{\alpha },\nonumber \\&O_9=\frac{3}{2} \bar{s}_{\alpha }\gamma ^{\mu }L b_{\alpha }\cdot \sum \nolimits _{q'}e_{q'}\bar{q}'_{\beta }\gamma _{\mu }L q'_{\beta }, ~~~~~\nonumber \\&O_{10}=\frac{3}{2} \bar{s}_{\alpha }\gamma ^{\mu }L b_{\beta }\cdot \sum \nolimits _{q'}e_{q'}\bar{q}'_{\beta }\gamma _{\mu }L q'_{\alpha }. \end{aligned}$$

(4)

Here $\alpha$ and $\beta$ are the SU(3) color indices. There are still two operators,

$$\begin{aligned}&O_{7\gamma }=\frac{-e}{8\pi ^2}m_b\bar{s}\sigma _{\mu \nu }(1+\gamma _5)F^{\mu \nu }b, \nonumber \\&O_{8g}=\frac{-g_s}{8\pi ^2}m_b\bar{s}\sigma ^{\mu \nu }RG^{\mu \nu }b. \end{aligned}$$

(5)

$O_{7\gamma }$ and $O_{8g}$ are the electromagnetic, chromomagnetic dipole operators and $G^{\mu \nu }$ denotes the gluonic field strength tensor. In the above equations, L and R are the left- and right-handed projection operators with $L=1-\gamma _5$ and $R=1+\gamma _5$, respectively. The sum over $q'$ runs over the quark fields that are active at the scale $\mu =O(m_b)$, i.e. $q'={u,d,s,c,b}$. A difficult problem is how to calculate the hadronic matrix elements of the local effective operators.

Fig. 1

Order $\alpha _s$ corrections to the hard-scattering kernels $T^{1}$ (first two rows) and $T^{2}$ (last row)

2.2 $\Lambda _b\rightarrow p ~K^-$ in QCD factorization approach

The naive factorization neglects the strong interactions between the final K meson and two baryons. It is necessary to consider the non-factorizable contributions. There are several approaches which are beyond the naive factorization. In this study, we use the method called the QCD factorization approach [18–21]. The QCD factorization proves that, in the heavy quark limit, the decay amplitude can be factorized into a product of a hard-scattering kernel and a non-perturbative part. The K meson and proton are both light hadron and energetic. The interaction between them should be caused by a large momentum transfer. Although the proof is given for the B meson case, it would be valid for the baryon system, too. Since we adopt the diquark approximation, the complications caused by more valence quarks nearly vanish. The diquark, as a whole, seems to be a light quark (it should be noted that the diquark in our case is a scalar, while the quark is a fermion). Thus, we assume that QCD factorization can be applicable to $\Lambda _b\rightarrow p\, K^-$.

The diagram for the $\alpha _s$ order QCD corrections to $\Lambda _b\rightarrow p\, K^-$ is plotted in Fig. 1. The first two rows represents one-loop vertex corrections and $\alpha _s$ corrections to electromagnetic, chromomagnetic dipole operators. The last row represents the hard spectator scattering. At present, we do not know the wave function for a baryon with a quark and a diquark. One may use a meson like wave function, but a quantity like the decay constant is unknown. Thus, we will neglect the hard spectator contributions. After this simplification, the decay amplitude of $\Lambda _b\rightarrow p\, K^-$ can be written

$$\begin{aligned} \langle p\, K^-|O_i|\Lambda _b\rangle =F^{\Lambda _b\rightarrow p}~T^{1}_{i}~*~f_K\Phi _K. \end{aligned}$$

(6)

Here, $F^{\Lambda _b\rightarrow p}$ represents the $\Lambda _b\rightarrow p$ form factors which will be defined below; $*$ represents a convolution in the light-cone momentum fraction space; $T^{1}_{i}$ represents the four-quark hard-scattering kernel; $\Phi _K$ represents the kaon meson light-cone wave function.

In QCD factorization, the amplitude $\Lambda _b\rightarrow p\, K^-$ is obtained:

$$\begin{aligned} {\mathcal {M}}= & {} \frac{G_F}{\sqrt{2}}\left\{ V_{ub}V_{us}^{*}a_1+{V_{qb}V_{qs}^{*}} \left[ a_4^q+a_{10}^q+R\left( a_6^q+a_8^q\right) \right] \right\} \nonumber \\&\times \,\langle p\mid \bar{u}\gamma _{\mu }L b\mid \Lambda _b\rangle \langle K^-\mid \bar{s}\gamma ^{\mu }L u\mid 0\rangle . \end{aligned}$$

(7)

Here, a summation over $q= u,c$ is implicit. The $a_i$ are written as

$$\begin{aligned} a_1= & {} C_1+\frac{C_2}{N_c}\left[ 1+\frac{C_F\alpha _s}{4\pi }V_K\right] ,\nonumber \\ a_4^q= & {} C_4+\frac{C_3}{N_c}\left[ 1+\frac{C_F\alpha _s}{4\pi }V_K\right] +\frac{C_F\alpha _s}{4\pi } \frac{P^q_{K,2}}{N_c},\nonumber \\ a_6^q= & {} C_6+\frac{C_5}{N_c}\left( 1-6\frac{C_F\alpha _s}{4\pi }\right) +\frac{C_F\alpha _s}{4\pi } \frac{P^q_{K,3}}{N_c},\nonumber \\ a_8^q= & {} C_8+\frac{C_7}{N_c}\left( 1-6\frac{C_F\alpha _s}{4\pi }\right) +\frac{\alpha }{9\pi } \frac{P^{q,EW}_{K,3}}{N_c},\nonumber \\ a_{10}^q= & {} C_{10}+\frac{C_9}{N_c}\left[ 1+\frac{C_F\alpha _s}{4\pi }V_K\right] +\frac{\alpha }{9\pi } \frac{P^{q,EW}_{K,2}}{N_c}, \end{aligned}$$

(8)

where $C_i\equiv C_i(\mu )$, $\alpha _s\equiv \alpha _s(\mu )$, $C_F=(N^2_c-1)/(2N_c)$, and $N_c=3$. The quantities $V_K$, $P^q_{K,2}$, $P^q_{K,3}$, $P^{q,EW}_{K,2}$, and $P^{q,EW}_{K,3}$ are hadronic parameters that contain all non-perturbative dynamics. Their expressions are given in [20]. These quantities consist of convolutions of hard-scattering kernels with meson distribution amplitudes. The term $V_K$ denotes the vertex corrections, $P^q_{K,2}$ and $P^q_{K,3}$ denote QCD penguin corrections and the contributions from the dipole operators. For penguin terms, the subscript 2 or 3 indicates the twist of the corresponding projections.

2.3 The decay rate

In Eq. (7), the first factor $\langle p | J_\mu |\Lambda _b\rangle$ in the hadronic matrix element is parameterized by form factors. The calculations of these non-perturbative form factors is one essential task of hadron physics. The form factors for the weak transition $\Lambda _b\rightarrow p$ are defined in the standard way by

$$\begin{aligned}&\langle p(P') \mid \bar{u}\gamma _{\mu } (1-\gamma _{5})b \mid \Lambda _{b}(P) \rangle \nonumber \\&\quad = \bar{u}_{p}(P') \left[ \gamma _{\mu } f_{1}(q^{2}) +i\sigma _{\mu \nu } \frac{ q^{\nu }}{M_{\Lambda _{b}}}f_{2}(q^{2})\right. \nonumber \\&\qquad \left. +\,\frac{q_{\mu }}{M_{\Lambda _{b}}} f_{3}(q^{2}) \right] u_{\Lambda _{b}}(P) \nonumber \\&\qquad -\,\bar{u}_{p}(P')\left[ \gamma _{\mu } g_{1}(q^{2}) +i\sigma _{\mu \nu } \frac{q^{\nu }}{M_{\Lambda _{b}}}g_{2}(q^{2})\right. \nonumber \\&\qquad \left. +\,\frac{q_{\mu }}{M_{\Lambda _{b}}}g_{3}(q^{2}) \right] \gamma _{5} u_{\Lambda _{b}}(P). \end{aligned}$$

(9)

The second factor of matrix element in Eq. (7) defines the decay constants as follows:

$$\begin{aligned} \langle K^-(P)|A_{\mu }|0\rangle= & {} f_K P_{\mu }. \end{aligned}$$

(10)

In the above definition, we omit a factor $(-i)$ for the pseudoscalar meson decay constant for simplification.

Substituting the expressions of $\langle K^-\mid \bar{s}\gamma ^{\mu }(1-\gamma ^5)u\mid 0 \rangle$ and $\langle p\mid \bar{u}\gamma _{\mu }(1-\gamma ^5)b\mid \Lambda _b\rangle$ one can obtain the decay amplitude of $\Lambda _b\rightarrow p~ K^-$ as

$$\begin{aligned} {\mathcal {M}}(\Lambda _b\rightarrow p~K^-)=\bar{u}_p(A+B\gamma _5)u_{\Lambda _b}, \end{aligned}$$

(11)

with

$$\begin{aligned}&A=\lambda f_K(M_{\Lambda _b}-M_p)f_1(M_K^2),\\&B=\lambda f_K(M_{\Lambda _b}+M_p)g_1(M_K^2), \end{aligned}$$

where

$$\begin{aligned} \lambda =\frac{G_F}{\sqrt{2}}\left\{ V_{ub}V_{us}^{*}a_1+{V_{qb}V_{qs}^{*}} \left[ a_4^q+a_{10}^q+R\left( a_6^q+a_8^q\right) \right] \right\} . \end{aligned}$$

Then we get the decay rate of $\Lambda _b\rightarrow p~K^-$

$$\begin{aligned} \Gamma =\frac{p_c}{8\pi }\left[ \frac{(M_{\Lambda _b}+M_p)^2-M_K^2}{M^2_{\Lambda _b}}\mid A\mid ^2 \right. \nonumber \\ \left. +\frac{(M_{\Lambda _b}-M_p)^2-M_K^2}{M^2_{\Lambda _b}}\mid B\mid ^2\right] , \end{aligned}$$

(12)

where $p_c$ is the proton momentum in the rest frame of $\Lambda _b$.

2.4 CP asymmetry and relation to decay of $\bar{B^0}\rightarrow \pi ^+ K^-$

The CP violation is defined in the same way as in the PDG book [16]:

$$\begin{aligned} A_{CP}\equiv \frac{Br(\Lambda _b^0\rightarrow p~ K^-)-Br(\bar{\Lambda _b^0}\rightarrow \bar{p}~ K^+)}{Br(\Lambda _b^0\rightarrow p~ K^-)+Br(\bar{\Lambda _b^0}\rightarrow \bar{p}~ K^+)}. \end{aligned}$$

(13)

At the quark level, the CP violation is represented by b quark decay minus $\bar{b}$ quark. The similar definition of CP violation for meson is

$$\begin{aligned} A_{CP}\equiv \frac{Br(\bar{B}^0\rightarrow f)-Br(B^0\rightarrow \bar{f})}{Br(\bar{B}^0\rightarrow f)+Br(B^0\rightarrow \bar{f})}. \end{aligned}$$

(14)

Under the diquark approximation, the baryon is similar to a meson. In fact, at the quark level, $\Lambda _b\rightarrow p K^-$ has the same sub-processes $b\rightarrow su\bar{u}$ as that in $\bar{B^0}\rightarrow \pi ^+ K^-$. The amplitude of $\bar{B^0}\rightarrow \pi ^+ K^-$ is

$$\begin{aligned}&M(\bar{B}^0\rightarrow \pi ^+ K^-)= \frac{G_F}{\sqrt{2}}\{V_{ub}V_{us}^{*}a_1 \nonumber \\&\qquad +\,{V_{qb}V_{qs}^{*}} [a_4^q+a_{10}^q+R(a_6^q+a_8^q)]\}\nonumber \\&\qquad \times \,\langle \pi ^+\mid \bar{u}\gamma _{\mu }L b\mid \bar{B}^0\rangle \langle K^-\mid \bar{s}\gamma ^{\mu }L u\mid 0\rangle . \end{aligned}$$

(15)

Comparing it to Eq. (7), we can obtain a relation between the baryon and meson processes,

$$\begin{aligned} Br(\Lambda _b\rightarrow p~K^-)= & {} Br^\mathrm{Exp}(\bar{B}^0\rightarrow \pi ^+ K^- )\nonumber \\&\times \,\frac{Br^\mathrm{tree}(\Lambda _b\rightarrow p~ K^-)}{Br^\mathrm{tree}(\bar{B}^0\rightarrow \pi ^+ K^- )} . \end{aligned}$$

(16)

Here the “tree” represents the branching ratio with only the tree operator contribution. This relation will be used to estimate the branching ratio of $\Lambda _b\rightarrow p~K^-$ from the meson process $\bar{B}^0\rightarrow K^- \pi ^+$. As regards the CP violation: under the above assumption, $A_{CP}$ in the two processes should be equal.

3 Numerical results

First of all, we list some parameters used in the numerical calculations. The input parameters are taken from [16] and previous work. We have

$$\begin{aligned}&m_u=0.3~{\text {GeV}},\quad m_s=0.45~{\text {GeV}}, \quad m_c =1.3~{\text {GeV}},\\&m_K=0.4937~{\text {GeV}},\quad m_b=4.4~{\text {GeV}}, \quad m_{[ud]} =0.5~{\text {GeV}},\\&M_{\Lambda _b}=5.619~{\text {GeV}},\quad M_p=0.938~{\text {GeV}}, \quad m_B=5.280~{\text {GeV}},\\&m_{\pi }=0.1396~{\text {GeV}},\quad f_K =0.160~{\text {GeV}} \quad F_0^{B\rightarrow \pi }(0) =0.3. \end{aligned}$$

The above quark masses of u,  d are the constituting masses which are used in the LFQM. For the current quark masses, $m_u=2.3~\text {MeV}$ and $m_s=95~\text {MeV}$.

Following [1], we recalculate the from factors of $\Lambda _b\rightarrow p$ in the LFQM. The form factors at different $q^2$ are parametrized in a three-parameter form as

$$\begin{aligned} F(q^2)=\frac{F(0)}{\left( 1-\frac{q^2}{M_{\Lambda _b}^2}\right) \left[ 1-a\left( \frac{q^2}{M_{\Lambda _b}^2}\right) +b\left( \frac{q^2}{M_{\Lambda _b}^2}\right) ^2\right] }, \end{aligned}$$

(17)

where the fitted values of a, b, and F(0) are given in Table 1. Our results reproduce those given in [1].

Table 1 The value of a, b, and F(0)

Table 2 The Wilson coefficients $C_i$ in LO

Table 3 The numerical values of $a_i$ in QCD factorization

Table 4 The branching ratios of $\Lambda _b\rightarrow p~K^-$

For the Wilson coefficients $C_i$, we use the leading order (LO) results as given in [20] and list them in Table 2. As for the CKM matrix elements, we adopt the Wolfenstein parametrization beyond the LO from [27]:

$$\begin{aligned}&V_{ud}=1-\frac{1}{2}\lambda ^2-\frac{1}{8}\lambda _4+{{\mathcal {O}}}(\lambda ^6),\quad V_{us}=\lambda +{\mathcal {O}}(\lambda ^7),\nonumber \\&V_{ub}=A\lambda (\rho -i\eta ),\nonumber \\&V_{cd}=-\lambda +\frac{1}{2}A^2\lambda ^5[1-2(\rho +i\eta )]+{\mathcal {O}}(\lambda ^7),\nonumber \\&V_{cs}=1-\frac{1}{2}\lambda ^2-\frac{1}{8}\lambda ^4(1+4A^2)+{\mathcal {O}}(\lambda ^6),\nonumber \\&V_{cb}=A\lambda ^2+{\mathcal {O}}(\lambda ^8),\nonumber \\&V_{td}=A\lambda ^3\left[ 1-(\rho +i\eta )\left( 1-\frac{1}{2}\lambda ^2\right) \right] +{\mathcal {O}}(\lambda ^7),\nonumber \\&V_{ts}=-A\lambda ^2+\frac{1}{2}A(1-2\rho )\lambda ^4-i\eta A\lambda ^4+{\mathcal {O}}(\lambda ^6),\nonumber \\&V_{tb}=1-\frac{1}{2}A^2\lambda ^4+{\mathcal {O}}(\lambda ^6). \end{aligned}$$

(18)

Here, we take the values $A=0.822$, $\lambda =0.22535$, $\rho =0.155$, and $\eta =0.358$.

By use of the above input parameters, we can get the Wilson coefficients $a_i$, relevant to the process of $\Lambda _b\rightarrow p~K^-$ with the $\alpha _s$ order QCD corrections. The numerical results are given in Table 3. Considering the theoretical uncertainties, our results are consistent with those in [20]. The small difference can be ascribed to the input parameters and the hard spectator contributions we neglected. Although the scale $\mu$ dependence of the Wilson coefficients $a_i$ is reduced compared to the LO ones, there is still an effect which is not negligible. This dependence implies the importance of higher order effects.

Now, we can obtain the branching ratio of $\Lambda _b\rightarrow p~K^-$. The predicted results at a different scale $\mu$ are listed in Table 4. As discussed above, the results have an non-negligible dependence on the choice of scale $\mu$. The higher the scale is, the lower the prediction is. The result at $\mu = m_b/2$ gives a prediction of $4.85\times 10^{-6}$, which is very well in agreement with the recent LHCb data $(4.9\pm 0.9)\times 10^{-6}$. The good coincidence indicates that $\mu =m_b/2$ is more appropriate. From a phenomenological point of view, $m_b$ is the largest scale in the b quark decay subprocess and each quark in the final hadrons does not carry the total momentum. The momentum transfer between different quarks should be smaller than $m_b$. So the choice of $\mu$ at $\mu =m_b/2$ is more reasonable than at $m_b$.

Under the assumption of neglecting the strong interactions with the spectator quark (diquark for the baryon), $\Lambda _b\rightarrow p~K^-$ contains the same strong dynamics with $\bar{B}^0\rightarrow \pi ^+ K^-$. We can use the data of the meson process to extract the strong interaction information. The advantage of this method is that the scale $\mu$ dependence is eliminated and the theoretical uncertainties of the QCD factorization approach are reduced by experiment. By the aid of the experimental data of $Br(\bar{B}^0\rightarrow \pi ^+ K^-)$ and Eq. (16), we estimate the decay rate: $Br(\Lambda _b^0\rightarrow p~K^-)=4.82\times 10^{-6}$. It coincides with the experimental measurement very well.

The results of CP violation are displayed in Table 5. Contrary to the branching ratio, the numerical results of CP violation of $\Lambda _b \rightarrow p~K^-$ become smaller as the scale $\mu$ decreases. At scale $\mu =m_b/2$, the CP violation is about 5 %. The experimental data from LHCb is $0.37\pm 0.17\pm 0.03$. The central value is several times larger than theory prediction. Because the experimental error is still large, it is too early to draw a conclusion as regards whether the theory coincides with the experiment or not. It is interesting and necessary to compare the CP violation to the meson case. The data of CP violation in $\bar{B^0} \rightarrow \pi ^+ K^-$ is also provided in Table 5 for comparison. The value is $-0.080\pm 0.007\pm 0.003$ with a negative sign. In our calculations under the diquark approximation, the CP violation of $\bar{B}^0 \rightarrow \pi ^+ K^-$ and $\Lambda _b \rightarrow p~ K^-$ should be equal. However, we see that the experimental data for the two processes are quite different, especially the sign is opposite. In fact, the CP violation for the process of $\bar{B}^0 \rightarrow \pi ^+ K^-$ in the QCD factorization approach has been a challenging problem for a long time. The theoretical prediction is not only inconsistent with the experiment data but also is wrong in sign.

Table 5 The CP violation $A_{CP}(\Lambda _b \rightarrow p~K^-$)

4 Discussion and conclusion

The weak decay of $\Lambda _b$ contains fruitful information of the strong interaction and provides an important probe to test different theoretical approaches. In this work, we extend the QCD factorization approach to the heavy baryon decays, in particular the process of $\Lambda _b^0\rightarrow p~K^-$. The previous literature considers only the tree diagram contribution and the theoretical result is one order smaller than the experiment. The $\Lambda _b^0\rightarrow p~K^-$ is a type of $b\rightarrow s$ transition for which the QCD penguin diagram contribution is more important than the tree diagram because of the CKM parameter enhancement. The QCD correction is calculated to $\alpha _s$ order and the Wilson coefficients at different renormalization scales are given. For the baryon, the diquark approximation is applied. The $\Lambda _b\rightarrow p$ form factors are calculated in the light-front quark model. The branching ratio of $\Lambda _b^0\rightarrow p~K^-$ is predicted to be $4.85\times 10^{-6}$ at scale $\mu =m_b/2$. The theory coincides with the experimental data $(4.9\pm 0.9)\times 10^{-6}$ very well.

From the coincidence of theory and experiment, we can obtain some conclusions as follows: (1) The perturbative contribution is dominant. The success provides confidence in the applicability of the QCD factorization method to the more complicated heavy baryon processes. (2) The choice of $\mu =m_b/2$ is appropriate. Because the largest scale is $m_b$ in the b quark decay subprocess, the real momentum transfer cannot reach $m_b$ and should be smaller. (3) The diquark ansatz works very well. The diquark approximation not only leads to a clear physics picture but also to a great simplification in the numerical calculations. From this study and the previous literature on heavy baryon decays, we may say that the diquark is really a working ansatz.

The main theory uncertainties come from several origins: the choice of scale $\mu$, the $\Lambda \rightarrow p$ form factors, the neglected hard spectator interaction and the non-perturbative power corrections. The problem of scale $\mu$ has been discussed in the article. Its scale dependence is not negligible. The higher loop corrections may help to reduce the dependence but are usually difficult to be realized. Although the $\Lambda \rightarrow p$ form factors depend on the model calculations, the reliability can be fitted by experiment. In [1], one shows that our calculated $\Lambda \rightarrow p$ form factors give a good prediction for $\Lambda _b\rightarrow p~ \pi ^-$: the theoretical result of the branching ratio is $3.15\times 10^{-6}$ and the experimental data is $(3.5\pm 0.6(\mathrm{stat})\pm 0.9(\mathrm{syst}))\times 10^{-6}$. Thus, the model-dependent form factors do not cause large theoretical uncertainties.

For the meson case, the hard spectator scattering contributes a leading power correction. It modifies the Wilson coefficient $a_5$ largely. For the coefficients $a_4$ and $a_6$, relevant to this study, the hard spectator correction is either numerically small (about 10 %) or absent. Thus, for the baryon case, the contribution from the hard spectator interaction is small. The weak annihilation contribution is power suppressed. At the realistic $m_b$ scale, it is necessary to consider its effect. The estimation of it suffers from the problem of the end-point singularity. According to the analysis in [20], the numerical values of annihilation correction is less than 25 % compared to the leading power term. Even so, the correction has included the chiral enhanced twist-3 contribution. For the baryon case, we might expect a similar small or even smaller weak annihilation contribution, because no such chiral enhancement exists for the baryon of proton.

The higher power correction is usually difficult to calculate. The consistence of the theory at $\mu =m_b/2$ with the data indicates that the non-perturbative power correction is less important and the perturbative contribution is dominant. One can use the data from $\bar{B^0}\rightarrow \pi ^+K^-$ to reduce the theory uncertainties in the QCD factorization approach. In this way, we obtain the decay rate with $Br(\Lambda _b^0\rightarrow p~K^-)=4.82\times 10^{-6}$, which coincides with the experiment very well.

CP violation provides us with a very different physics picture. Under the diquark approximation and neglecting the spectator interactions, the theory predicts CP violation at the level of about 5 % for both the baryon process $\Lambda _b^0\rightarrow p~K^-$ and the meson case $\bar{B^0}\rightarrow \pi ^+K^-$. The origin of the strong phase in the QCD factorization approach comes from the quark loop in the vertex corrections. For the meson case, the theory result is positive. But the experiment data is negative, about $-10$ %. This obvious inconsistence implies the importance of non-perturbative corrections for CP violation. For the baryon $\Lambda _b^0\rightarrow p~K^-$, the experiment data gives a very large result: $0.37\pm 0.17\pm 0.03$. In the QCD factorization approach, the perturbative contribution cannot reach 10 %. Because it is quite difficult to estimate the non-perturbative corrections, the prediction of CP violation in theory is a challenging research. We hope the future LHCb data can provide us with a more precise measurement of CP violation in $\Lambda _b^0\rightarrow p~K^-$ to improve the development of theory.