Improved analysis of the rare decay processes of 【-逻*辑*与-】Lambda;
本站小编 Free考研考试/2022-01-02
Ren-hao Deng, Yong-lu Liu*, Ming-qiu HuangCollege of Art and Science, National University of Defense Technology, Hunan 410073, China
First author contact:*please provide Email for the corresponding author. Received:2021-04-14Revised:2021-05-11Accepted:2021-06-11Online:2021-08-16
Abstract It is noted that in the new Particle Data Group (PDG) version the rare decays of the ${{\rm{\Lambda }}}_{b}$ baryon have been revised with more accuracy. The new results show that most of the existing theoretical results on the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ are larger than those of experiments. With the improved higher-order light-cone distribution amplitudes of the Λ baryon, we reanalyze the process in the framework of light-cone quantum chromodynamics sum rules and the branching ratio is estimated to be ${\rm{Br}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma )=({7.38}_{-0.39}^{+0.40})\times {10}^{-6}$, which is consistent with the new experimental result. Furthermore, another process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}$ ${l}^{+}{l}^{-}$ is also analyzed in the same frame. The final branching ratio is calculated to be ${\rm{Br}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=1.20\,\times \,{10}^{-6}$, which is in good accordance with the data from the PDG and other theoretical predictions. Keywords：light-cone distribution amplitude;rare decay;light-cone QCD sum rules
PDF (615KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Ren-hao Deng, Yong-lu Liu, Ming-qiu Huang. Improved analysis of the rare decay processes of Λb. Communications in Theoretical Physics, 2021, 73(10): 105201- doi:10.1088/1572-9494/ac0a70
1. Introduction
The flavor changed process of the heavy hadron is a pure platform for studying information on the internal dynamics of the hadron, which is important for investigations of the properties of the strong interaction both at quark level and hadron level. The rare decay of the heavy b-hadron, which is forbidden at the tree level in the standard model (SM), is related to the flavor changing neutral current (FCNC). The loop of a virtual quark and a boson component dominates the process, so we can analyze the element of the Cabibbo–Kobayashi–Maskawa (CKM) matrix about Vts and Vtb [1] more accurately, which may provide information on new physics beyond the SM.
In the past 30 years, many experimental and theoretical efforts have been devoted to this topic, especially in the sector of meson decays [2–7]. The rare radiative B meson decay was seen for the first time early in 1993 by CLEO. With the observation of the process $B\to {K}^{* }\gamma $, a lot of experimental data have been accumulated and increasingly exact measurements of inclusive and exclusive branches have been reported [8–10]. Theoretically, different methods have been used to investigate the processes of the $b\to s\gamma $ section within and outside the SM [11, 12]. The theoretical predicted decay rates of the inclusive processes are well in agreement with experiments. However, the exclusive process of B-meson rare decay needs to be improved, since it is associated with nonperturbative contributions.
In comparison with processes of the B-meson, the rare decay of the b-baryon has received little attention due to its complex quark contents, and the experimental data are also not sufficient for further theoretical study [1]. Simultaneously, the process of b-baryon decay can help us learn more about the quark information inside hadrons, such as processes of hadronization and the hadron spin polarization. The hadronization process of b-quark supplies a new window for studying of the exclusive rare radiative ${{\rm{\Lambda }}}_{b}$ decays, which have been employed in many processes with the quark model [13–16], quantum chromodynamics (QCD) sum rules [17] and the QCD perturbative method [18]. Nevertheless, there are many differences in theory, even in order of magnitude. The processes of the rare decays ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ and ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}$ ${l}^{+}{l}^{-}$ have been studied with light-cone QCD sum rules (LCSR) [19–22]. Nevertheless, recent experimental data show that most of the existing theoretical results are much larger than those expected. Therefore, it is instructive to reanalyze these processes.
In the LCSR method the most likely error may come from the higher-order corrections of the light-cone distribution amplitudes (LCDAs). In previous work [23–32], the LCDAs of Λ have been renewed considering corrections from higher-order conformal spin expansion. This indicates that this correction may provide a contribution to the processes. Another uncertainty originates from the interpolating current of the baryon, which has been shown in the calculations of electromagnetic form factors [33–35]. The improvement of the current work is to use the renewed LCDAs to reanalyze the form factors related to the two rare decay processes. It is expected to give more reliable predictions which can be compared with the new experiments.
The presentation is organized as follows. Section 2 is devoted to introducing the theoretical framework of the decay processes. In section 3 we give form factors with LCSR in next to leading order (NLO) of the LCDAs using two types of interpolating currents. The numerical analysis is presented in section 4, in which the dipole formula is used to fit the form factors from LCSR. Finally, the summary and conclusion are presented in section 5.
2. Theoretical framework
In the SM, the effective Hamiltonian of the processes ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ and ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ at $\mu =O({m}_{b})$ [36, 37] are expressed as$\begin{eqnarray}\begin{array}{l}{H}_{{\rm{eff}}}(b\to s\gamma )\\ \quad =-4\displaystyle \frac{{G}_{F}}{\sqrt{2}}{V}_{{ts}}^{* }{V}_{{tb}}{C}_{7}(\mu )\displaystyle \frac{e}{16{\pi }^{2}}\bar{s}{\sigma }_{\mu \nu }({g}_{V}+{\gamma }_{5}{g}_{A}){{bF}}^{\mu \nu },\end{array}\end{eqnarray}$and$\begin{eqnarray}\begin{array}{l}{H}_{\,{\rm{eff}}}(b\to {{sl}}^{+}{l}^{-})=\displaystyle \frac{{G}_{F}{V}_{{ts}}^{* }{V}_{{tb}}{\alpha }_{{\rm{em}}}}{2\sqrt{2}\pi }\\ \left\{-{C}_{7}^{\,{\rm{eff}}}\displaystyle \frac{2}{{q}^{2}}\bar{s}\,{\rm{i}}{\sigma }_{\mu \nu }{q}^{\nu }({m}_{b}R+{m}_{s}L)b\bar{l}{\gamma }^{\mu }l\right.\\ \left.+{C}_{9}^{\,{\rm{eff}}}\bar{s}{\gamma }_{\mu }{Lb}\bar{l}{\gamma }^{\mu }l+{C}_{10}^{\,{\rm{eff}}}\bar{s}{\gamma }_{\mu }{Lb}\bar{l}{\gamma }^{\mu }{\gamma }_{5}l\right\},\end{array}\end{eqnarray}$where GF is the Fermi coupling constant, ${g}_{V}=1+{m}_{s}/{m}_{b}$ , ${g}_{A}=1-{m}_{s}/{m}_{b}$, and ${C}_{7},{C}_{7}^{\,{\rm{eff}}},{C}_{9}^{\,{\rm{eff}}},{C}_{10}^{\,{\rm{eff}}}$ are Wilson coefficients at μ.
The hadronic parts of the processes are dominated by the nonperturbative effects at the quark level, which can be parameterized with the form factors defined by the matrix elements of the interaction currents between the initial and final states:$\begin{eqnarray}\begin{array}{l}\langle {{\rm{\Lambda }}}_{b}({P}^{{\prime} })| \bar{b}{\sigma }_{\mu \nu }{q}^{\mu }s| {\rm{\Lambda }}(P)\rangle \\ ={\overline{{\rm{\Lambda }}}}_{b}({P}^{{\prime} })({f}_{1}{\gamma }_{\nu }+{f}_{2}\,{\rm{i}}{\sigma }_{\mu \nu }{q}^{\mu }+{f}_{3}{q}_{\nu }){\rm{\Lambda }}(P)\\ \langle {{\rm{\Lambda }}}_{b}({P}^{{\prime} })| \bar{b}{\sigma }_{\mu \nu }{q}^{\mu }{\gamma }_{5}s| {\rm{\Lambda }}(P)\rangle \\ ={\overline{{\rm{\Lambda }}}}_{b}({P}^{{\prime} })({g}_{1}{\gamma }_{\nu }+{g}_{2}\,{\rm{i}}{\sigma }_{\mu \nu }{q}^{\mu }+{g}_{3}{q}_{\nu }){\rm{\Lambda }}(P)\\ \langle {{\rm{\Lambda }}}_{b}({P}^{{\prime} })| \bar{b}{\gamma }_{\mu }s| {\rm{\Lambda }}(P)\rangle \\ ={\overline{{\rm{\Lambda }}}}_{b}({P}^{{\prime} })({F}_{1}{\gamma }_{\nu }+{F}_{2}\,{\rm{i}}{\sigma }_{\mu \nu }{q}^{\mu }+{F}_{3}{q}_{\nu }){\rm{\Lambda }}(P)\\ \langle {{\rm{\Lambda }}}_{b}({P}^{{\prime} })| \bar{b}{\gamma }_{\mu }{\gamma }_{5}s| {\rm{\Lambda }}(P)\rangle \\ ={\overline{{\rm{\Lambda }}}}_{b}({P}^{{\prime} })({G}_{1}{\gamma }_{\nu }+{G}_{2}\,{\rm{i}}{\sigma }_{\mu \nu }{q}^{\mu }+{G}_{3}{q}_{\nu }){\rm{\Lambda }}(P),\end{array}\end{eqnarray}$where ${{\rm{\Lambda }}}_{b}$ and Λ are the spinors of the ${{\rm{\Lambda }}}_{b}$ and Λ baryons respectively, ${P}^{{\prime} }=P+q$ is the momentum of ${{\rm{\Lambda }}}_{b}$, and q is the momentum transfer.
The final states of ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ contain a real photon, which makes the sector of hadron parameterization simpler:$\begin{eqnarray}\begin{array}{l}\langle {{\rm{\Lambda }}}_{b}(P,s)| \bar{b}{\sigma }_{\mu \nu }{q}^{\nu }({g}_{V}+{\gamma }_{5}{g}_{A})s| {\rm{\Lambda }}({P}^{{\prime} },s)\rangle \\ ={\overline{{\rm{\Lambda }}}}_{b}(P,s){\sigma }_{\mu \nu }{q}^{\nu }({g}_{V}{f}_{2}+{\gamma }_{5}{g}_{A}{g}_{2}){\rm{\Lambda }}({P}^{{\prime} },s).\end{array}\end{eqnarray}$The decay rate of of this process is expressed with the form factors as$\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma )=\displaystyle \frac{{G}_{F}^{2}| {V}_{{tb}}{V}_{{ts}}^{* }{| }^{2}{\alpha }_{\,{\rm{em}}}| {C}_{7}{| }^{2}{m}_{b}^{2}}{32{\pi }^{4}}\\ {\left(\displaystyle \frac{{M}_{{{\rm{\Lambda }}}_{b}}^{2}-{M}_{{\rm{\Lambda }}}^{2}}{{M}_{{{\rm{\Lambda }}}_{b}}}\right)}^{3}({g}_{V}^{2}{f}_{2}^{2}+{g}_{A}^{2}{g}_{2}^{2}).\end{array}\end{eqnarray}$
For the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}$ ${l}^{+}{l}^{-}$, the differential decay width on q2 is expressed with form factors as follows [21]:$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{d{\rm{\Gamma }}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})}{{{dq}}^{2}}=\displaystyle \frac{{\left({G}_{F}| {V}_{{tb}}| | {V}_{{ts}}| {\alpha }_{{\rm{em}}}\right)}^{2}}{3\times {2}^{10}{\pi }^{5}{M}_{{{\rm{\Lambda }}}_{b}}}\\ {\left[{\left(\displaystyle \frac{{q}^{2}}{{M}_{{{\rm{\Lambda }}}_{b}}^{2}}-(1+{r}^{2})\right)}^{2}-4{r}^{2}\right]}^{\tfrac{1}{2}}\\ \times \left\{-\displaystyle \frac{4{C}_{7}{C}_{7}^{* }}{{q}^{4}}\{({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r+1\right)}^{2})[({q}^{2}+2{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2}){g}_{2}^{2}{q}^{2}\right.\\ +6{M}_{{{\rm{\Lambda }}}_{b}}(r-1){g}_{1}{g}_{2}{q}^{2}+(2{q}^{2}+{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2}){g}_{1}^{2}]{\left({m}_{b}-{m}_{s}\right)}^{2}\\ +[-{\left({r}^{2}-1\right)}^{2}{M}_{{{\rm{\Lambda }}}_{b}}^{4}-{q}^{2}({r}^{2}-6r+1){M}_{{{\rm{\Lambda }}}_{b}}^{2}+2{q}^{4}]{f}_{1}^{2}\\ {\left({m}_{b}+{m}_{s}\right)}^{2}\\ +{q}^{2}[-2{\left({r}^{2}-1\right)}^{2}{M}_{{{\rm{\Lambda }}}_{b}}^{4}+{q}^{2}({r}^{2}+6r+1){M}_{{{\rm{\Lambda }}}_{b}}^{2}+{q}^{4}]{f}_{2}^{2}\\ {\left({m}_{b}+{m}_{s}\right)}^{2}\\ -6{M}_{{{\rm{\Lambda }}}_{b}}{q}^{2}({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2})(r+1){f}_{1}{f}_{2}{\left({m}_{b}+{m}_{s}\right)}^{2}\}\\ +\displaystyle \frac{2}{{q}^{2}}({C}_{9}{C}_{7}^{* }+{C}_{7}{C}_{9}^{* })\{({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2})\\ \left[({q}^{2}+2{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r+1\right)}^{2}){f}_{2}{F}_{2}\right.\\ \left.-3{M}_{{{\rm{\Lambda }}}_{b}}(r+1){f}_{2}{F}_{1}\right]({m}_{b}+{m}_{s}){q}^{2}+({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r+1\right)}^{2})\\ \times [3{M}_{{{\rm{\Lambda }}}_{b}}(r-1){q}^{2}({g}_{2}{G}_{1}+{g}_{1}{G}_{2})\\ +({q}^{2}+2{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2}){g}_{2}{G}_{2}{q}^{2}\\ +(2{q}^{2}+{M}_{{{\rm{\Lambda }}}_{b}}^{2}{g}_{1}{G}_{1})]\\ ({m}_{b}-{m}_{s})+({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2})\\ \times [(2{q}^{2}+{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r+1\right)}^{2}){f}_{1}{F}_{1}-3{M}_{{{\rm{\Lambda }}}_{b}}{q}^{2}(r+1){f}_{1}{F}_{2}]\\ ({m}_{b}+{m}_{s})\}\\ -({C}_{9}{C}_{9}^{* }+{C}_{10}{C}_{10}^{* })\{[-2{\left({r}^{2}-1\right)}^{2}{M}_{{{\rm{\Lambda }}}_{b}}^{4}\\ +{q}^{2}({r}^{2}+6r+1){M}_{{{\rm{\Lambda }}}_{b}}^{2}+{q}^{4}]{F}_{2}^{2}{q}^{2}\\ -6{M}_{{{\rm{\Lambda }}}_{b}}({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2})(r+1){F}_{1}{F}_{2}{q}^{2}\\ +[-{\left({r}^{2}-1\right)}^{2}{M}_{{{\rm{\Lambda }}}_{b}}^{4}-{q}^{2}({r}^{2}-6r+1){M}_{{{\rm{\Lambda }}}_{b}}^{2}+2{q}^{4}]{F}_{1}^{2}\\ +({q}^{2}-{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r+1\right)}^{2})[({q}^{2}-2{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2}){G}_{2}^{2}{q}^{2}\\ +6{M}_{{{\rm{\Lambda }}}_{b}}(r-1){G}_{1}{G}_{2}{q}^{2}\\ \left.+(2{q}^{2}+{M}_{{{\rm{\Lambda }}}_{b}}^{2}{\left(r-1\right)}^{2}){G}_{1}^{2}]\right\},\end{array}\end{eqnarray}$where $r={M}_{{\rm{\Lambda }}}/{M}_{{{\rm{\Lambda }}}_{b}}$ is the ratio of masses of baryons Λ and ${{\rm{\Lambda }}}_{b}$, and the form factors F3 and G3 do not appear here because the vector current is conserved.
In order to calculate these decay rates, we need to evaluate all the form factors that appear in the equation. Due to the nonperturbative properties of QCD at hadron level, we should use some nonperturbative methods when calculating these parameters. Herein, the LCSR are adopted to calculate various form factors.
3. Light-cone sum rules of the form factors
The main idea of the QCD sum rule method is to calculate the correlation function from both the hadron level and the quark level. However, the description of the hadron with quarks relies on the model of how we construct the interaction between them. Usually the interpolating current are used to couple to the hadron. Nevertheless, the choice of the interpolating current is not usually definite, especially for multi-quark states.
The criterion of the choice about the interpolating current is that it couples strongly enough to the hadron. In the following analysis of the LCSR, we use CZ-type and Ioffe-type currents respectively for a comparison [38–40]. The CZ current is$\begin{eqnarray}{j}_{{\rm{\Lambda }}}(x)={\epsilon }_{{ijk}}({u}^{i}C{\gamma }_{5}{/}\hspace{-5pt} {z}{d}^{j}){/}\hspace{-5pt} {z}{b}^{k},\end{eqnarray}$and the Ioffe current [41–43, 40] is$\begin{eqnarray}{\tilde{j}}_{{\rm{\Lambda }}}(x)={\epsilon }_{{ijk}}({u}^{i}C{\gamma }_{5}{\gamma }_{\mu }{d}^{j}){\gamma }^{\mu }{b}^{k},\end{eqnarray}$where C is the charge conjugation matrix, $i,j,k$ are the color indices, and the auxiliary four-vector ${z}^{\mu }$ is a light-cone vector which satisfies the condition ${z}^{2}=0$. The coupling coefficients of interpolating currents to the hadron are defined by the matrix elements of the currents between the vacuum and ${{\rm{\Lambda }}}_{b}$ baryon state$\begin{eqnarray}\langle 0| {j}_{{\rm{\Lambda }}}| {{\rm{\Lambda }}}_{b}({P}^{{\prime} },s)\rangle ={f}_{{{\rm{\Lambda }}}_{b}}z\cdot P{/}\hspace{-5pt} {z}{\rm{\Lambda }}({P}^{{\prime} },s),\end{eqnarray}$and$\begin{eqnarray}\langle 0| \tilde{{j}_{{\rm{\Lambda }}}}| {{\rm{\Lambda }}}_{b}({P}^{{\prime} },s)\rangle ={\lambda }_{1b}{M}_{{{\rm{\Lambda }}}_{b}}{{\rm{\Lambda }}}_{b}({P}^{{\prime} },s),\end{eqnarray}$where ${P}^{{\prime} }$ is the momentum of ${{\rm{\Lambda }}}_{b}$ and s is its spin.
The LCSR start from the correction function$\begin{eqnarray}{T}_{\mu }={\rm{i}}\int \,{{\rm{d}}}^{4}x\,{{\rm{e}}}^{\,{\rm{i}}{qx}}\langle 0| {j}_{{\rm{\Lambda }}}(0){j}_{\mu }^{\dagger }(x)| {\rm{\Lambda }}(P,s)\rangle ,\end{eqnarray}$where q is the momentum transfer.
The hadron part of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ can be parameterized as the form factors defined by the matrix elements of the interaction current ${j}_{\mu }={\overline{s}}_{\alpha }{\sigma }_{\mu \nu }{{Rq}}^{\nu }{b}_{\alpha }$ between the ${{\rm{\Lambda }}}_{b}$ state and Λ state:$\begin{eqnarray}\begin{array}{l}\langle {{\rm{\Lambda }}}_{b}({P}^{{\prime} },{s}^{{\prime} })| {\left({\overline{s}}_{\alpha }{\sigma }_{\mu \nu }{{Rq}}^{\nu }{b}_{\alpha }\right)}^{\dagger }| {\rm{\Lambda }}(P,s)\rangle \\ =\displaystyle \frac{1}{2}{\overline{{\rm{\Lambda }}}}_{b}({P}^{{\prime} },{s}^{{\prime} }){\sigma }_{\mu \nu }{q}^{\nu }({f}_{2}-{g}_{2}{\gamma }_{5}){\rm{\Lambda }}(P,s).\end{array}\end{eqnarray}$By interpolating a complete set of intermediate states with the same quantum numbers as the initial state in the correction function, we obtain the hadronic representation$\begin{eqnarray}{z}^{\mu }{T}_{\mu }={\rm{i}}\displaystyle \frac{{f}_{{{\rm{\Lambda }}}_{b}}}{{M}_{{{\rm{\Lambda }}}_{b}}^{2}}{\left(z\cdot {P}^{{\prime} }\right)}^{2}({f}_{2}-{g}_{2}{\gamma }_{5}){/}\hspace{-5pt} {z}{/}\hspace{-5pt} {q}{\rm{\Lambda }}(P,s)+......,\end{eqnarray}$in which ${P}^{{\prime} }=P-q$, and the dots represent contributions from higher resonance and continuum states. Herein the light-cone vector ${z}^{\mu }$ is used to contract the correction function and simplify the Lorentz structure in LCSR, so that contribution terms on the light-cone dominate in the processes [20, 44].
On the other hand, the correction function can be expanded on the light-cone. We can contract the b quark to calculate the correlation function at the quark level to the lead-order of ${\alpha }_{s}$ as:$\begin{eqnarray}\begin{array}{l}{z}^{\mu }{T}_{\mu }=-\displaystyle \frac{1}{2}\displaystyle \int \,{{\rm{d}}}^{4}x\displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}\,{{\rm{e}}}^{\,{\rm{i}}(q+k)\cdot x}\displaystyle \frac{{z}^{\mu }{q}^{\nu }}{{k}^{2}-{m}_{b}^{2}}\\ \{{m}_{b}{\left(C{\gamma }_{5}{/}\hspace{-5pt} {z}\right)}_{\alpha \beta }{\left({/}\hspace{-5pt} {z}{\sigma }_{\mu \nu }(1-{\gamma }_{5})\right)}_{\delta \omega }\\ +{\left(C{\gamma }_{5}{/}\hspace{-5pt} {z}\right)}_{\alpha \beta }{\left({/}\hspace{-5pt} {z}{/}\hspace{-5pt} {k}{\sigma }_{\mu \nu }(1-{\gamma }_{5})\right)}_{\delta \omega }\}\\ \langle 0| {\varepsilon }_{{ijk}}{u}_{\alpha }^{i}(0){d}_{\beta }^{j}(0){s}_{\gamma }^{k}(0)| {\rm{\Lambda }}(P,s)\rangle .\end{array}\end{eqnarray}$The nonperturbation effects can be calculated by the construction of nonlocal matrix elements between the vacuum and the Λ baryon state, which are usually parameterized as LCDAs.
With the standard procedure of LCSR, the two sides are matched, and the higher resonance contributions on the hadronic side can be expressed by the quark level with the assumption of quark–hadron duality. Hereafter, the Borel transformation is applied to suppress contributions from both the higher resonance and higher twist LCDAs. We arrive at the sum rules of form factors f1, f2, F1 and F2, which is shown explicitly in appendix A. As for form factors g1, g2, G1 and G2, we can obtain them from the relations ${g}_{1}=-{f}_{1}$, ${g}_{2}={f}_{2}$, ${G}_{1}=-{F}_{1}$ and ${G}_{2}={F}_{2}$.
A similar result can be obtained with the replacement of the interpolating current by the Ioffe-type one ${\tilde{j}}_{{\rm{\Lambda }}}$. The sum rules are shown in appendix A.
4. Numerical analysis
Before numerical analysis, some inputs need to be determined. In order to reduce the uncertainty from the sum rules itself, we choose values of the coupling constants as obtained from QCD sum rules. The results are ${f}_{{{\rm{\Lambda }}}_{b}}=(3.86\,\pm 0.13)\times {10}^{-3}\,{{\rm{GeV}}}^{2}$ and ${\lambda }_{1b}=(0.027\pm 0.03)\,{{\rm{GeV}}}^{2}$ [34]. Other parameters can be chosen from the Particle Data Group (PDG) as ${M}_{{{\rm{\Lambda }}}_{b}}=5.62\,{\rm{GeV}}$, ${M}_{{\rm{\Lambda }}}=1.116\,{\rm{GeV}}$, and the b quark mass is ${m}_{b}=4.7\,{\rm{GeV}}$ [19, 21]. The distribution amplitudes are the most important input parameters in LCSR. Hereafter, we use the renewed results which have considered effects from higher-order conformal spin expansion [23, 29, 45]. For the completeness of the text, we list the LCDAs in appendix B.
The most important auxiliary parameters in LCSR are the threshold s0 and the Borel mass MB2. Since the threshold is the parameter that is chosen to ensure that the integration of the spectral density upon it is equivalent to contributions from higher resonances, it is usually connected with the first excited state with the same quantum numbers as the ground state. Herein we choose the threshold in the region $39\,{{\rm{GeV}}}^{2}\leqslant {s}_{0}\leqslant 41\,{{\rm{GeV}}}^{2}$. Meanwhile, the choice of the Borel parameter should meet the requirement that both higher twist contributions and higher resonance contributions are suppressed simultaneously. With the simulation, we acquire the appropriate range of MB2 as $18\,{{\rm{GeV}}}^{2}\leqslant {M}_{B}^{2}\leqslant 22\,{{\rm{GeV}}}^{2}$, which is the Borel window adopted in the next analysis. It is noticed that the Borel window is different from that in the previous work, $10\,{{\rm{GeV}}}^{2}\leqslant {M}_{B}^{2}\leqslant 15\,{{\rm{GeV}}}^{2}$ [21]. The reason lies in the fact that the convergence of higher twist make it more reasonable to choose the present region, which comes from influences of higher-order corrections of LCDAs.
With the above choices of parameters, we obtain the form factor ${f}_{2}(0)$ of CZ-type current and form factors ${f}_{2}(0)$ and ${g}_{2}(0)$ of Ioffe-type current. We plot the form factor ${f}_{2}(0)$ with MB2 in figure 1 as an example. It is seen that in the region the sum rules are not sensitive to the parameter MB2. We can choose a special value ${M}_{B}^{2}=20\,{{\rm{GeV}}}^{2}$ in the Borel parameter window, in which the higher resonance contribution can be suppressed under 30%. The dependence of the form factors with the momentum transfer is plotted in figure 2 for CZ-type current and figure 3 for Ioffe-type current.
Figure 1.
New window|Download| PPT slide Figure 1.Dependence of form factors ${f}_{2}(0)$ on the Borel parameter at ${q}^{2}=0\,{{\rm{GeV}}}^{2}$.
Since form factors from the LCSR are only applicable in the middle momentum transfer, we need to extrapolate the results to the whole dynamical region to obtain the transition information. Herein we use the dipole formula to fit the form factors$\begin{eqnarray}{f}_{i}({q}^{2})=\displaystyle \frac{{f}_{i}(0)}{1+{a}_{1}{q}^{2}/{M}_{{{\rm{\Lambda }}}_{b}}^{2}+{a}_{2}{q}^{4}/{M}_{{{\rm{\Lambda }}}_{b}}^{4}},\end{eqnarray}$where a1, a2 and ${f}_{i}(0)$ have been listed in table 1. It is noted that there are relations ${g}_{1}=-{f}_{1}$, ${g}_{2}={f}_{2}$, ${G}_{1}=-{F}_{1}$ and ${G}_{2}={F}_{2}$ for CZ-type current. The same results of parameters a1, a2, and ${f}_{i}(0)$ for the Ioffe-type current are listed in table 2. In the numerical analysis only uncertainties from different choices of the threshold s0 and the Borel parameter are considered.
Table 1. Table 1.Fitted parameters of form factors by the formula (15) with CZ-type current.
With form factors fitted by the dipole formula to the whole dynamical region, the decay widths of the processes ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ and ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ can be calculated directly in aid of the formula (5) and (6). The decay width of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ is determined by the form factors f2 and g2 at the zero momentum transfer ${q}^{2}=0\,{{\rm{GeV}}}^{2}$. The form factors are estimated with ${s}_{0}=39\sim 41\,{{\rm{GeV}}}^{2}$ and ${M}_{B}^{2}=18\,\sim 22\,{{\rm{GeV}}}^{2}$ as$\begin{eqnarray}{f}_{2}(0)={g}_{2}(0)={0.584237}_{-0.010449}^{+0.010481},\end{eqnarray}$for the CZ-type current and$\begin{eqnarray}{f}_{2}(0)={g}_{2}(0)={0.0999807}_{-0.0026958}^{+0.002882},\end{eqnarray}$for the Ioffe-type current.
In analysis, we use the Fermi coupling constant ${G}_{F}=1.1166\times {10}^{-5}\,{{\rm{GeV}}}^{2}$ and the fine-structure constant ${\alpha }_{s}=1/137$. In addition, we use the CKM matrix elements $| {V}_{{tb}}| =0.999172$ and $| {V}_{{ts}}| =0.03978$ [1] and Wilson coefficient ${C}_{7}(\mu ={m}_{b})=-0.31$ [17, 46, 47] in the SM. The mass of the s-quark is ${m}_{s}=0.15{\rm{GeV}}$. The relationships ${g}_{V}=1+{m}_{s}/{m}_{b}$ and ${g}_{A}=1-{m}_{s}/{m}_{b}$ hold in the SM. Putting these coefficients into the formula (6), we obtain the decay width of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ in two different interpolating currents:$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{{\rm{CZ}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma ) & = & ({1.148}_{-0.041}^{+0.042})\times {10}^{-16}\ {\rm{GeV}},\\ {{\rm{\Gamma }}}_{{\rm{Ioffe}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma ) & = & ({3.30}_{-0.17}^{+0.18})\times {10}^{-18}\ {\rm{GeV}}.\end{array}\end{eqnarray}$In calculations we use the lifetime of ${{\rm{\Lambda }}}_{b}$ as $\tau =1.471\,\times {10}^{-12}{\rm{s}}$ [1]. Therefore, the corresponding branch ratio of the decay mode is obtained as$\begin{eqnarray}\begin{array}{rcl}{{\rm{Br}}}_{{\rm{CZ}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma ) & = & ({2.57}_{-0.09}^{+0.09})\times {10}^{-4},\\ {{\rm{Br}}}_{{\rm{Ioffe}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma ) & = & ({7.38}_{-0.39}^{+0.40})\times {10}^{-6}.\end{array}\end{eqnarray}$
It is found that the interpolating current of the ${{\rm{\Lambda }}}_{b}$ baryon can generate a fairly large difference when we estimate the branch ratio of the process. Compared with the experiments, it shows that the CZ-type current does not match well with the data, while results from the Ioffe-type are perfectly consistent with those from the PDG [1]. The results are listed in table 3 with other methods and experimental data.
Table 3. Table 3.Branch ratio of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ with different models.
The decay width and branch ratio of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ can be calculated with a similar procedure. The Wilson coefficients are set as$\begin{eqnarray}\begin{array}{rcl}{C}_{7}^{{\rm{eff}}}(\mu ={m}_{b}) & = & -0.31,\\ {C}_{9}^{{\rm{eff}}}(\mu ={m}_{b}) & = & 4.344,\\ {C}_{10}^{{\rm{eff}}}(\mu ={m}_{b}) & = & -4.669.\end{array}\end{eqnarray}$In order to give the differential decay width, we need to extrapolate the form factors to the whole dynamic region. We use the fitted formula (15) to estimate the decay width with formula (6). It is noted that the differential decay width on the momentum transfer has a pole at the point ${q}^{2}=0\,{{\rm{GeV}}}^{2}$, so results from the fit formula (15) are not applicable at the point. In the analysis, the whole dynamic region is chosen as $0.1\,{{\rm{GeV}}}^{2}\lt {q}^{2}\lt 20.286\,{{\rm{GeV}}}^{2}$. The differential decay width is shown in figure 4.
Figure 4.
New window|Download| PPT slide Figure 4.Process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ decay rate relate with the transfer momentum q2 in (a) CZ-type current and (b) Ioffe-type current.
The total decay rate can be obtained by integrating the derivative one in the dynamic region $0.1\,{{\rm{GeV}}}^{2}\,\lt {q}^{2}\lt 20.286\,{{\rm{GeV}}}^{2}$. In the analysis we ignore contributions from the region $0\sim 0.1\,{{\rm{GeV}}}^{2}$ because of the factors $1/{q}^{2}$ and $1/{q}^{4}$. The final result with CZ-type current is$\begin{eqnarray}{{\rm{\Gamma }}}_{\,{\rm{CZ}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=2.06\times {10}^{-18}{\rm{GeV}},\end{eqnarray}$and that with Ioffe-type current is$\begin{eqnarray}{{\rm{\Gamma }}}_{\,{\rm{Ioffe}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=5.38\times {10}^{-19}{\rm{GeV}}.\end{eqnarray}$In table 4, we list the corresponding branching ratios from various models and the experiment. It can be seen from the table that estimation from the Ioffe-type current matches better with the data from the PDG in comparison with other theoretical predictions. Another conclusion is that the choice of the interpolating current may affect the result to some extent and the Ioffe-type current is a more suitable choice for the processes of the description.
Table 4. Table 4.Branch ratios of the decay process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ from different models.
The rare decay processes ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ and ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-}$ are reanalyzed in the framework of LCSR with renewed distribution amplitudes of the Λ baryon. Two different interpolating currents are adopted to investigate the decay modes. The form factors defined by the matrix elements of the interaction currents between the initial and final states are obtained with LCSR respectively. To obtain complete information on the process, the dipole formulae are used to fit the form factors and then are extrapolated to the whole dynamical region. The final results for the decay width are ${{\rm{\Gamma }}}_{\,{\rm{CZ}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma )=({1.148}_{-0.041}^{+0.042})\times {10}^{-16}\,{\rm{GeV}}$ and ${{\rm{\Gamma }}}_{\,{\rm{CZ}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=2.06\, \times \,{10}^{-18}\,{\rm{GeV}}$ for the CZ-type current, and ${{\rm{\Gamma }}}_{\,{\rm{Ioffe}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma )=$ $({3.30}_{-0.17}^{+0.18})\times {10}^{-18}\,{\rm{GeV}}$ and ${{\rm{\Gamma }}}_{\,{\rm{Ioffe}}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=5.38\times {10}^{-19}\,{\rm{GeV}}$ for the Ioffe-type current.
The results show that the renewed distribution amplitudes may affect the results to some extent. In comparison with previous analysis, the predictions are much more compatible with the new data from the PDG. It can also be seen from the results that the Ioffe-type current is a more suitable choice in the description of the rare decay processes of the ${{\rm{\Lambda }}}_{b}$ baryon. The CZ-type current is used to eliminate contributions that are not important on the light-cone, while results from the Ioffe-type current may contain more complex structures and contain more detailed information, so Ioffe-type current may give relatively more reliable predictions. Therefore, our predictions for the branching ratios are ${\rm{Br}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma )=({7.38}_{-0.39}^{+0.40})\,\times {10}^{-6}$ and ${\rm{Br}}({{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}{l}^{+}{l}^{-})=1.20\,\times \,{10}^{-6}$. It is noted that the estimation of the process ${{\rm{\Lambda }}}_{b}\to {\rm{\Lambda }}\gamma $ is about one order lower than previous theoretical predictions from various models, and the result is well consistent with the renewed data from the PDG.
Appendix A. LCSR of the form factors
The LCSR of the form factors relating to the processes have been obtained in the previous work [21]. Here we only present the results in this appendix for the completeness of the paper.
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