Rare $\Lambda_b\to\Lambda l^+l^-$ and $\Lambda_b\to\Lambda\gamma$ decays in the relativistic quark model

TL;DR

This paper calculates decay form factors and branching ratios for rare $ ext{Lambda}_b$ baryon decays into $ ext{Lambda}$ and lepton pairs or photons using a relativistic quark model, aligning well with recent LHCb data.

Contribution

It introduces a comprehensive relativistic quark model calculation of decay form factors, including all relativistic effects, for $ ext{Lambda}_b$ rare decays, providing new theoretical predictions.

Findings

Decay branching fractions agree with LHCb measurements.

Explicit form factor dependence on momentum transfer is determined.

Predictions for $ ext{Lambda}_b o ext{Lambda} au^+ au^-$ observables are provided.

Abstract

Rare $\Lambda_b\to\Lambda l^+l^-$ and $\Lambda_b\to\Lambda\gamma$ decays are investigated in the relativistic quark model based on the quark-diquark picture of baryons. The decay form factors are calculated with the account of all relativistic effects including relativistic transformations of baryon wave functions from rest to moving reference frame and the contribution of the intermediate negative energy states. The momentum transfer squared dependence of the form factors is explicitly determined in the whole accessible kinematical range. The calculated decay branching fractions, various forward-backward asymmetries for the rare decay $\Lambda_b\to\Lambda \mu^+\mu^-$ are found to be consistent with recent detailed measurements by the LHCb Collaboration. Predictions for the $\Lambda_b\to\Lambda \tau^+\tau^-$ decay observables are given.

Tables

Table 1. Table 1: Calculated form factors of the rare weak Λ b → Λ → subscript Λ 𝑏 Λ \Lambda_{b}\to\Lambda transition.

	$f_1^V\left(q^2\right)$	$f_2^V\left(q^2\right)$	$f_3^V\left(q^2\right)$	$f_1^A\left(q^2\right)$	$f_2^A\left(q^2\right)$	$f_3^A\left(q^2\right)$	$f_1^{TV}\left(q^2\right)$	$f_2^{TV}\left(q^2\right)$	$f_1^{TA}\left(q^2\right)$	$f_2^{TA}\left(q^2\right)$
$f\left(0\right)$	0.208	$0.032$	$0.026$	0.125	$-0.003$	$-0.083$	$-0.029$	$-0.153$	0.029	$-0.153$
$f\left(q_{\max }^2\right)$	0.777	$0.632$	$0.225$	0.487	$-0.378$	$-1.09$	$-0.584$	$-0.618$	0.254	$-0.652$
$a_0$	$0.239$	$0.195$	$0.091$	$0.196$	$-0.152$	$-0.322$	$-0.180$	$-0.190$	$0.102$	$-0.263$
$a_1$	$0.633$	$-0.765$	$-0.280$	$-0.515$	$1.03$	$1.05$	$0.540$	$0.252$	$-0.592$	$0.351$
$a_2$	$-3.34$	$0.252$	$0$	$0.899$	$-1.68$	$-0.033$	$0.505$	$-0.398$	$1.19$	$0.531$

Table 2. Table 2: Comparison of theoretical predictions for the form factors of the rare Λ b → Λ → subscript Λ 𝑏 Λ \Lambda_{b}\to\Lambda transition at maximum recoil q 2 = 0 superscript 𝑞 2 0 q^{2}=0 .

	$f_1^V\left(0\right)$	$f_2^V\left(0\right)$	$f_3^V\left(0\right)$	$f_1^A\left(0\right)$	$f_2^A\left(0\right)$	$f_3^A\left(0\right)$	$f_1^{TV}\left(0\right)$	$f_2^{TV}\left(0\right)$	$f_1^{TA}\left(0\right)$	$f_2^{TA}\left(0\right)$
this paper	0.208	0.032	0.026	0.125	$-0.003$	$-0.083$	$-0.029$	$-0.153$	0.029	$-0.153$
gikls	0.107	0.043	$0.003$	0.104	0.003	$-0.052$	$-0.043$	$-0.105$	0.003	$-0.105$
aas	0.322(112)	0.011(4)	$-0.015\left(5\right)$	0.318(110)	$0.013\left(4\right)$	$-0.014\left(5\right)$	$-0.056\left(18\right)$	$-0.295\left(105\right)$	$0.101\left(35\right)$	$-0.294\left(105\right)$

Table 3. Table 4: Comparison of theoretical predictions for baryon rare decay branching fractions ( × 10 − 6 absent superscript 10 6 \times 10^{-6} ) with available experimental data.

Decay	this paper	gikls	aas	wll	llg	mr	glch	Experiment pdg
${\Lambda }_b\to \Lambda e^{+}{e}^{-}$	1.07	1.0	4.6(1.6)		$1.21\sim 2.32$		$2.03({}_9{}^{26})$
${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$	1.05	1.0	4.0(1.2)	$6.1({}_{1.7}{}^{5.8})$	$0.53\sim 0.89$	0.70		$1.08\left(28\right)$
${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$	0.26	0.2	0.8(3)	$2.1({}_{0.6}{}^{2.3})$	$0.037\sim 0.083$	0.22

Table 4. Table 5: Predictions for the averaged rare decay asymmetries and polarization fractions.

Decay	$⟨A_{FB}^{\ell }⟩$		$⟨A_{FB}^h⟩$		$⟨A_{FB}^{h\ell }⟩$		$⟨F_L⟩$
	nonres.	res.	nonres.	res.	nonres.	res.	nonres.	res.
${\Lambda }_b\to \Lambda e^{+}{e}^{-}$	$-0.288$	$-0.294$	$-0.291$	$-0.299$	0.101	0.92	0.526	0.596
${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$	$-0.286$	$-0.266$	$-0.288$	$-0.299$	0.101	0.92	0.525	0.544
${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$	$-0.161$	$-0.127$	$-0.268$	$-0.313$	0.060	0.047	0.343	0.339

Table 5. Table 6: Comparison of theoretical predictions for the rare decay Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} observables with available experimental data in the range 15 GeV ≤ 2 q 2 ≤ 20.28 {}^{2}\leq q^{2}\leq 20.28  GeV 2 .

Observable	nonres.	res.	latt	bfd	Experiment LHCb lhcb2015
$⟨Br⟩(\times {10}^{-7})$	3.54	3.22	3.99(37)	4.5(1.2)	5.9(1.4)
$⟨A_{\mathrm{FB}}^{\ell }⟩$	$-0.40$	$-0.33$	$-0.350\left(13\right)$	$-0.29\left(5\right)$	$-0.05\left(9\right)$
$⟨A_{\mathrm{FB}}^h⟩$	$-0.29$	$-0.29$	$-0.2710\left(92\right)$	$-0.26\left(3\right)$	$-0.29\left(8\right)$
$⟨A_{\mathrm{FB}}^{\ell h}⟩$	$0.145$	$0.129$	$0.1398\left(43\right)$	$0.13{(}_3^2)$
$⟨F_L⟩$	$0.38$	$0.38$	$0.409\left(13\right)$	$0.4\left(1\right)$	$0.61{(}_{14}^{11})$

Table 6. Table 7: Comparison of theoretical predictions for the binned Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} differential branching fractions ⟨ B ​ r / d ​ q 2 ⟩ delimited-⟨⟩ 𝐵 𝑟 𝑑 superscript 𝑞 2 \langle Br/dq^{2}\rangle (in units of 10 − 7 superscript 10 7 10^{-7}  GeV -2 ) with available experimental data.

$q^2$ bin (GeV2)	nonres.	res.	latt	Experiment LHCb lhcb2015
$[0.1,2]$	0.31	0.34	0.25(23)	0.36${(}_{13}^{14})$
$[2,4]$	0.27	0.31	0.18(12)	0.11${(}_9^{12})$
$[4,6]$	0.33	0.40	0.23(11)	0.02${(}_1^9)$
$[6,8]$	0.41	0.57	0.307(94)	0.25${(}_{12}^{13})$
$[11,12.5]$	0.66	0.65		0.75${(}_{21}^{21})$
$[15,16]$	0.75	0.72	0.796(75)	1.12${(}_{30}^{30})$
$[16,18]$	0.73	0.68	0.827(76)	1.22${(}_{29}^{29})$
$[18,20]$	0.56	0.49	0.665(68)	1.24${(}_{30}^{30})$
$[1.1,6]$	0.29	0.34	0.20(12)	0.09${(}_5^6)$
$[15,20]$	0.67	0.61	0.756(70)	1.20${(}_{27}^{26})$

Table 7. Table 8: Comparison of theoretical predictions for the binned lepton forward-backward asymmetries ⟨ A F ​ B ℓ ⟩ delimited-⟨⟩ subscript superscript 𝐴 ℓ 𝐹 𝐵 \langle A^{\ell}_{FB}\rangle in Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} decay with available experimental data.

$q^2$ bin (GeV2)	nonres.	res.	latt	Experiment LHCb lhcb2015
$[0.1,2]$	0.078	0.067	0.095(15)	0.37${(}_{48}^{37})$
$[11,12.5]$	$-0.35$	$-0.35$		0.01${(}_{19}^{20})$
$[15,16]$	$-0.42$	$-0.41$	$-0.374\left(14\right)$	$-0.10{(}_{16}^{18})$
$[16,18]$	$-0.43$	$-0.36$	$-0.372\left(13\right)$	$-0.07{(}_{13}^{14})$
$[18,20]$	$-0.35$	$-0.32$	$-0.309\left(15\right)$	0.01${(}_{15}^{16})$
$[15,20]$	$-0.40$	$-0.33$	$-0.350\left(13\right)$	$-0.05{(}_{10}^{10})$

Table 8. Table 9: Comparison of theoretical predictions for the binned hadron forward-backward asymmetries ⟨ A F ​ B h ⟩ delimited-⟨⟩ subscript superscript 𝐴 ℎ 𝐹 𝐵 \langle A^{h}_{FB}\rangle in Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} decay with available experimental data.

$q^2$ bin (GeV2)	nonres.	res.	latt	Experiment LHCb lhcb2015
$[0.1,2]$	$-0.26$	$-0.26$	$-0.310\left(18\right)$	$-0.12{(}_{32}^{34})$
$[11,12.5]$	$-0.30$	$-0.30$		$-0.50{(}_4^{11})$
$[15,16]$	$-0.32$	$-0.32$	$-0.3069\left(83\right)$	$-0.19{(}_{16}^{14})$
$[16,18]$	$-0.31$	$-0.31$	$-0.2891\left(90\right)$	$-0.44{(}_6^{10})$
$[18,20]$	$-0.25$	$-0.25$	$-0.227\left(10\right)$	$-0.13{(}_{12}^{10})$
$[15,20]$	$-0.29$	$-0.29$	$-0.2710\left(92\right)$	$-0.29{(}_8^8)$

Table 9. Table 10: Comparison of theoretical predictions for the binned longitudinal polarization ⟨ F L ⟩ delimited-⟨⟩ subscript 𝐹 𝐿 \langle F_{L}\rangle in Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} decay with available experimental data.

$q^2$ bin (GeV2)	nonres.	res.	latt	Experiment LHCb lhcb2015
$[0.1,2]$	$0.63$	$0.66$	$0.465\left(84\right)$	$0.56{(}_{56}^{24})$
$[11,12.5]$	$0.51$	$0.51$		$0.40{(}_{36}^{37})$
$[15,16]$	$0.42$	$0.41$	$0.454\left(20\right)$	$0.49{(}_{30}^{30})$
$[16,18]$	$0.38$	$0.38$	$0.417\left(15\right)$	$0.68{(}_{21}^{15})$
$[18,20]$	$0.35$	$0.35$	$0.3706\left(79\right)$	$0.62{(}_{27}^{24})$
$[15,20]$	$0.38$	$0.38$	$0.409\left(13\right)$	$0.61{(}_{14}^{11})$

Table 10. Table 11: Comparison of theoretical predictions for the binned hadron-lepton forward-backward asymmetries ⟨ A F ​ B h ​ ℓ ⟩ delimited-⟨⟩ subscript superscript 𝐴 ℎ ℓ 𝐹 𝐵 \langle A^{h\ell}_{FB}\rangle in Λ b → Λ ​ μ + ​ μ − → subscript Λ 𝑏 Λ superscript 𝜇 superscript 𝜇 \Lambda_{b}\to\Lambda\mu^{+}\mu^{-} decay.

$q^2$ bin (GeV2)	nonres.	res.	latt
$[0.1,2]$	$-0.024$	$-0.021$	$-0.0302\left(51\right)$
$[2,4]$	$0.003$	$0.010$	$-0.0169\left(99\right)$
$[4,6]$	$0.039$	$0.045$	$0.021\left(13\right)$
$[6,8]$	$0.068$	$0.072$	$0.053\left(13\right)$
$[15,20]$	$0.145$	$0.129$	$0.1398\left(43\right)$

Table 11. Table 12: Comparison of theoretical predictions for the rare radiative decay branching fraction B ​ r ​ ( Λ b → Λ ​ γ ) 𝐵 𝑟 → subscript Λ 𝑏 Λ 𝛾 Br(\Lambda_{b}\to\Lambda\gamma) ( × 10 − 5 absent superscript 10 5 \times 10^{-5} ) with available experimental data.

Decay	this paper	mw	gikls	wll	glch	cdfp	Experiment pdg
${\Lambda }_b\to \Lambda \gamma$	1.0	$0.77{(}_{19}^{22})$	0.4	$0.73\left(15\right)$	$0.061{(}_{13}^{14})$	3.1(6)

Equations

$$\left(\frac{b^2(M)}{2{\mu }_R}-\frac{{\mathbf{p}}^2}{2{\mu }_R}\right){\Psi }_{{\Lambda }_Q}(\mathbf{p})=\int \frac{d^{3}q}{(2\pi {)}^3}V(\mathbf{p},\mathbf{q};M){\Psi }_{{\Lambda }_Q}(\mathbf{q}),$$

$${\mu }_R=\frac{M_{{\Lambda }_Q}^4-(m_Q^2-m_d^2{)}^2}{4M_{{\Lambda }_Q}^3},$$

$$b^2(M)=\frac{[M_{{\Lambda }_Q}^2-(m_Q+m_d{)}^2][M_{{\Lambda }_Q}^2-(m_Q-m_d{)}^2]}{4M_{{\Lambda }_Q}^2}.$$

$$V(r)=-\frac{4}{3}\frac{{\tilde{\alpha }}_s(r)}{r}+Ar+B,$$

$$V_{\mathrm{conf}}^V(r)=(1-\epsilon )(Ar+B),V_{\mathrm{conf}}^S(r)=\epsilon (Ar+B),$$

$$⟨\Lambda (P)|J_{\mu }^W|{\Lambda }_b(Q)⟩=\int \frac{d^{3}p{d}^{3}q}{(2\pi {)}^6}{\bar{\Psi }}_{\Lambda \mathbf{P}}(\mathbf{p}){\Gamma }_{\mu }(\mathbf{p},\mathbf{q}){\Psi }_{{\Lambda }_b\mathbf{Q}}(\mathbf{q}),$$

$${\Gamma }_{\mu }^{(1)}(\mathbf{p},\mathbf{q})={\psi }_d^{\ast }(p_d){\bar{u}}_s(p_s){\gamma }_{\mu }(1-{\gamma }^5)u_b(q_b){\psi }_d(q_d)(2\pi {)}^3\delta ({\mathbf{p}}_d-{\mathbf{q}}_d),$$

$${\Gamma }_{\mu }^{(2)}(\mathbf{p},\mathbf{q})$$

$$\Lambda^{(-)}(p)=\frac{\epsilon(p)-\bigl{(}m\gamma^{0}+\gamma^{0}({\bm{\gamma}{\bf p}})\bigr{)}}{2\epsilon(p)}.$$

$${\Psi }_{\Lambda \mathbf{P}}(\mathbf{p})=D_q^{1/2}(R_{L_{\mathbf{P}}}^W)D_d(R_{L_{\mathbf{P}}}^W){\Psi }_{\Lambda 0}(\mathbf{p}),$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}{\gamma }^{\mu }b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}{\gamma }^{\mu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}i{\sigma }^{\mu \nu }{q}_{\nu }b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}i{\sigma }^{\mu \nu }{q}_{\nu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}{\gamma }^{\mu }b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}{\gamma }^{\mu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}i{\sigma }^{\mu \nu }{q}_{\nu }b|{\Lambda }_b(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|\bar{s}i{\sigma }^{\mu \nu }{q}_{\nu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$f_{+}(q^2)$$

$$f_{\perp }(q^2)$$

$$f_0(q^2)$$

$$g_{+}(q^2)$$

$$g_{\perp }(q^2)$$

$$g_0(q^2)$$

$$h_{+}(q^2)$$

$$h_{\perp }(q^2)$$

$${\tilde{h}}_{+}(q^2)$$

$${\tilde{h}}_{\perp }(q^2)$$

$$F(q^2)=\frac{1}{1-\frac{q^2}{M_{\mathrm{pole}}^2}}\left\{a_0+a_{1}z(q^2)+a_2[z(q^2){]}^2\right\},$$

$$z(q^2)=\frac{\sqrt{t_{+}-q^2}-\sqrt{t_{+}-t_0}}{\sqrt{t_{+}-q^2}+\sqrt{t_{+}-t_0}}.$$

$${\mathcal{H}}_{\mathrm{eff}}=-\frac{4G_F}{\sqrt{2}}{V}_{ts}^{\ast }{V}_{tb}\sum _{i=1}^{10}{c}_i{\mathcal{O}}_i,$$

$$\mathcal{M}({\Lambda }_b\to \Lambda l^{+}{l}^{-})=\frac{G_F\alpha }{2\sqrt{2}\pi }|V_{ts}^{\ast }{V}_{tb}|\left[T_{\mu }^{(1)}(\bar{l}{\gamma }^{\mu }l)+T_{\mu }^{(2)}(\bar{l}{\gamma }^{\mu }{\gamma }_{5}l)\right],$$

$$T_{\mu }^{(1)}$$

$$T_{\mu }^{(2)}$$

$$H_{{\lambda }_{\Lambda },\lambda }^m={\epsilon }^{†\mu }(\lambda )T_{\mu }^{(m)}({\lambda }_{\Lambda }),$$

$$H_{+1/2,0}^{Vm,Am}$$

$$H_{+1/2,1}^{Vm,Am}$$

$$H_{+1/2,t}^{Vm,Am}$$

$${\mathcal{F}}_1^{V1,A1}(q^2)$$

Full text

I Introduction

In the standard model the exclusive rare weak decays of hadrons, governed by the $b\to s$ quark transitions, proceed through the flavour changing neutral currents. Therefore they provide a sensitive test of different new physics extensions (see e.g. Ref. aab and references therein). Such transitions were studied in detail both theoretically and experimentally in the $B$ and $B_{s}$ meson decays. Differential decay rates and angular distributions, various asymmetry parameters were measured with rather high accuracy bgh . However, several tensions between experimental data and the standard model predictions have been found aab ; bgh ; lhcb1 . It was argued lhcb1 that these differences could be explained by contributions from physics beyond the standard model, or by unexpectedly large hadronic effects that are not properly accounted for in the predictions. Therefore it is important to refine theoretical description of the rare $b\to s$ transitions and search for similar effects in other rare decays.

The rare semileptonic $\Lambda_{b}\to\Lambda l^{+}l^{-}$ and radiative $\Lambda_{b}\to\Lambda\gamma$ decays provide a valuable testing ground. Indeed, the first observation of the baryonic flavour changing neutral current decay $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ was reported in 2011 by the CDF Collaboration cdf . It was followed by more comprehensive and precise data from the LHCb Collaboration lhcb2013 ; lhcb2015 , which not only measured the differential branching fraction but also performed the detailed angular analysis of the $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay. Theoretical description of such decays is based on the effective Hamiltonian in which intermediate gauge bosons are integrated out. The short and long distance contributions are separated by application of the operator product expansion. The short distance effects are described by the Wilson coefficients, while the calculation of the long distance part involves consideration of the hadronic matrix elements of the corresponding weak currents between baryon states, which are usually parametrized by the set of invariant form factors. The kinematically accessible momentum transfer squared $q^{2}$ range in such decays is rather broad. However, most of the theoretical approaches available in the literature provide determination of the decay form factors only in one particular kinematical point or in the limited range. Thus light-cone QCD sum rules determine form factors at large recoil of the final hadron (near $q^{2}=0$) while the lattice QCD calculations are reliable in the small recoil region (near $q^{2}=q^{2}_{\rm max}$). Then extrapolation of the theoretical predictions to the whole kinematical range is needed which introduces additional theoretical uncertainties. Therefore, reliable determination of the $q^{2}$ dependence of the hadronic form factors in the whole kinematical range without extrapolations or model assumptions is important for increasing the reliability of theoretical predictions.

In this paper we apply the relativistic quark model mmass based on the quasipotential approach with the QCD-motivated interquark interaction to the calculation of the matrix elements of the flavour changing neutral current between baryon states. In our model baryons are considered to be the relativistic quark-diquark bound systems. Their wave functions are known from the baryon mass spectra calculations bmass . Let us note that at present the convincing evidence of the existence of diquark correlations in hadrons has been collected. Information comes from different sectors of hadron physics. Thus, in the light meson sector it has been argued for a long time jaffe that mesons forming the inverted lightest scalar nonet can be well described as tetraquarks treated as diquark-antidiquark bound states ltetr . In the heavy meson sector several charged charmonium- and bottomonium-like states were discovered hq . They should be inevitably multiquark, at least four quark — tetraquark, states. One of the most successful pictures of such tetraquark states is the diquark-antidiquark model hq ; htetr . In the baryon sector it is well known that the number of observed excited states both in the light and heavy sectors is considerably lower than the number of excited states predicted in the three-quark approach. The introduction of diquarks significantly reduces this number, since in such a picture some of degrees of freedom are frozen and thus the number of possible excitations is substantially smaller. The calculations of the heavy and strange baryon spectra bmass show that all available experimental data can be well described in the framework of the relativistic quark-diquark picture of baryons. The lattice QCD calculations indicate existence of the diquark correlations in baryons lattb . Very recently the Belle Collaboration belle published data on the production cross sections of charmed baryons in $e^{+}e^{-}$ annihilation. The observed a factor of three excess of the production cross section of $\Lambda_{c}$ states over $\Sigma_{c}$ states provides a strong support for a diquark structure in the ground state and low-lying excited $\Lambda_{c}$ baryons.

Calculating the weak current matrix elements between baryon states we systematically take into account all relativistic corrections including contributions of the intermediate negative energy states and relativistic transformations of baryon wave functions from rest to the moving reference frame using the methods previously developed for the description of the semileptonic baryon decays sllbdecay . Such an approach allows us to obtain explicitly the $q^{2}$ dependence of the form factors in the whole kinematical range. For our calculation we use the same effective Hamiltonian and Wilson coefficients as in our previous consideration of the rare $B$ and $B_{s}$ meson decays rareB .

The paper is organized as follows. In Sec. II we briefly describe our relativistic quark-diquark model of baryons, present the relevant quasipotential equation and give expressions for the weak current matrix element. The rare $b\to s$ transition form factors are calculated in Sec. III. They are expressed through the overlap integrals of the baryon wave functions. The analytic expressions for the form factors, which accurately reproduce numerical results in the whole accessible kinematical $q^{2}$ range, are presented. Comparison with the previous calculations for the form factor values at $q^{2}=0$ and $q^{2}=q^{2}_{\rm max}$ is given. Rare semileptonic $\Lambda_{b}\to\Lambda l^{+}l^{-}$ decays are considered in Sec. IV. Differential branching fractions and other angular observables are calculated and compared with available experimental data and lattice calculations. The estimates for the rare radiative $\Lambda_{b}\to\Lambda\gamma$ decay are presented in Sec. V. Our conclusions are given in Sec. VI, while the Appendix contains explicit expressions for the rare decay form factors.

II Relativistic quark-diquark model

For the calculation of the rare $\Lambda_{b}$ baryon decays we use the same relativistic quark-diquark model which was previously employed for the calculation of the baryon masses bmass and weak semileptonic decays sllbdecay . The initial $\Lambda_{b}$ and final $\Lambda$ baryons are considered as the bound states of the heavy $Q$ ($b$ or $s$) quark and light scalar $[u,d]$ diquark. They are described by the wave function $\Psi_{\Lambda_{Q}}$, which satisfy the relativistic quasipotential equation of the Schrödinger type bmass

[TABLE]

where the relativistic reduced mass and and the center-of-mass system relative momentum squared on the mass shell are given by

[TABLE]

[TABLE]

Here $M_{\Lambda_{Q}}$, $m_{Q}$ and $m_{d}$ are the $\Lambda_{Q}$ baryon mass, $Q$ quark mass and diquark $d$ mass, respectively. The quark-diquark interaction quasipotential $V({\bf p,q};M)$ (see explicit expressions in Ref. bmass ) is the relativistic generalization of the Cornell potential

[TABLE]

where the first term is the smeared Coulomb potential with $\tilde{\alpha}_{s}(r)\equiv\alpha_{s}F(r)$ and the form factor $F(r)$ takes the diquark internal structure into account bmass . The confining quark interaction was taken to be the mixture of the Lorentz-vector and scalar linearly growing with $r$ potentials

[TABLE]

with the mixing coefficient $\varepsilon$, which was set to $\varepsilon=-1$ from the consideration of meson properties mmass . The vertex of the long-range vector quark interaction contains not only the Dirac part but an additional Pauli term, thus introducing the long-range anomalous chromomagnetic quark moment $\kappa$. Its value was set to $\kappa=-1$ in our previous consideration of meson properties mmass . Such choice provides the vanishing of the long-range chromomagnetic contribution to the potential, which is proportional to $(1+\kappa)$. The constituent quark masses $m_{u,d}=0.33$ GeV, $m_{s}=0.5$ GeV, $m_{c}=1.55$ GeV, $m_{b}=4.88$ GeV and the parameters of the linear potential $A=0.18$ GeV2 and $B=-0.3$ GeV have the usual values of quark models. The mass of the scalar $[u,d]$ diquark was calculated to be $m_{d}=0.71$ GeV bmass .

The matrix element of the weak current $J^{W}$, governing the rare $b\to sl^{+}l^{-}$ transition, between baryon states in the considered approach is given by f ; sllbdecay

[TABLE]

where $\Gamma_{\mu}({\bf p},{\bf q})$ is the two-particle vertex function. It receives relativistic contributions both from the impulse approximation diagram in Fig. 1

[TABLE]

and from the diagrams with the intermediate negative-energy states in Fig. 2 which are the consequence of the projection onto the positive-energy states in the quasipotential approach

[TABLE]

Here $\psi_{d}(p)$ is the diquark wave function; ${\cal V}({\bf p})$ is the quark-diquark interaction quasipotential; ${\bf k}={\bf p}_{s}-{\bf\Delta};\ {\bf k}^{\prime}={\bf q}_{b}+{\bf\Delta};\ {\bf\Delta}={\bf P}-{\bf Q}$;  $\epsilon(p)=\sqrt{m^{2}+{\bf p}^{2}}$; and

[TABLE]

$\Psi_{\Lambda\,{\bf P}}({\bf p})$ is the baryon wave function projected onto the positive-energy states of quarks and boosted to the moving reference frame with momentum ${\bf P}$ f ; sllbdecay

[TABLE]

where $\Psi_{{\Lambda}\,{\bf 0}}$ is the baryon wave function in the rest frame, $R^{W}$ is the Wigner rotation, $L_{\bf P}$ is the Lorentz boost from the baryon rest frame to a moving one with momentum ${\bf P}$, and $D^{1/2}_{q}(R^{W})$ is the rotation matrix of the quark spin sllbdecay , while the rotation matrix for the scalar diquark spin $D_{d}(R^{W})=1$.

III Form factors of the rare $\Lambda_{b}$ baryon decays

The matrix element of the flavour changing neutral current for the rare $\Lambda_{b}\to\Lambda l^{+}l^{-}$ baryon decay can be parametrized by the following set of invariant form factors

[TABLE]

where $u_{\Lambda_{b}}(p,s)$ and $u_{\Lambda}(p^{\prime},s^{\prime})$ are Dirac spinors of the initial and final baryon; $q=p^{\prime}-p$.

The other popular parameterization is the helicity-based definition of the form factors from Ref. fy

[TABLE]

with $s_{\pm}=(M_{\Lambda_{b}}\pm M_{\Lambda})^{2}-q^{2}$.

The form factors (14) and (8) are related in the following way

[TABLE]

To find the weak decay form factors we need to calculate the matrix element of the weak current between baryon wave functions known from the mass spectra calculations. The expressions for the decay form factors $f_{i}^{V,A}$ ($i=1,2,3$) parameterizing matrix elements of the vector and axial vector weak currents between baryon states were obtained in our previous paper sllbdecay . They are given in the Appendix of Ref. sllbdecay . For the calculation of the rare baryon decays we need to extend our analysis and get expressions for the rest of form factors $f_{i}^{TV,TA}$ parameterizing matrix elements of the tensor and pseudo tensor currents. To achieve this goal we follow the approach developed in Ref. sllbdecay . Namely we use the $\delta$-function in the expression for the lowest-order vertex function $\Gamma^{(1)}$ arising in the impulse approximation (see Fig. 1) to express the current matrix element (4) as the usual overlap integral of baryon wave functions. Thus, this contribution can be calculated exactly in the whole kinematical range. On the other hand, the consideration of the vertex function $\Gamma^{(2)}$ (see Fig. 2) is more complicated, since this function takes into account contributions coming from the negative-energy parts of the quark propagators and thus explicitly depends on the quark-diquark potential, in particular, on the Lorentz-structure of the confining interaction. Taking into account that the recoil momentum of the final $\Lambda$ baryon $|{\bf\Delta}|$, in the initial $\Lambda_{b}$ baryon rest frame, is significantly larger than the relative quark momentum in the baryon almost in the whole accessible kinematical range,111The square of the momentum transfer squared to the lepton pair $q^{2}$ varies from 0 to $q^{2}_{\rm max}\approx 20$ GeV2 for the $\Lambda_{b}$ decays to $\Lambda$. we neglect small relative momentum $|{\bf p}|$ with respect to the recoil momentum $|{\bf\Delta}|$ in the energies of quarks composing the energetic final $\Lambda$ baryon and replace $\epsilon_{q}(p+\Delta)\equiv\sqrt{m_{q}^{2}+({\bf p}+{\bf\Delta})^{2}}$ with $\epsilon_{q}(\Delta)\equiv\sqrt{m_{q}^{2}+{\bf\Delta}^{2}}$. As a result we can use the quasipotential equation to take one of the integrations in the current matrix element (4) and again get the expression for the current matrix element as the usual overlap integral of baryon wave functions. It is important to point out that such an approach allows us to consistently take into account all relativistic corrections including boosts of the baryon wave functions from the rest frame to the moving one (7) and contributions of the intermediate negative-energy states. The obtained expressions for the form factors are presented in the Appendix (to simplify these expressions, as previously, we explicitly set the long-range anomalous chromomagnetic quark moment $\kappa=-1$).

Substituting the baryon wave functions, found into the calculation of their mass spectra, in the expressions for the decay form factors we calculate their values and explicitly determine their dependence on the momentum transfer squared $q^{2}$ in the whole kinematical range. We find that the weak decay form factors can be approximated with a high accuracy by the expressions:

[TABLE]

where the variable

[TABLE]

Following Ref. latt we take $t_{+}=(M_{B}+M_{K})^{2}$ and $t_{0}=q^{2}_{\rm max}=(M_{\Lambda_{b}}-M_{\Lambda})^{2}$. The pole masses have the values: $M_{\rm pole}\equiv M_{B_{s}^{*}}=5.416$ GeV for $f_{1,2}^{V}$, $f_{1,2}^{TV}$; $M_{\rm pole}\equiv M_{B_{s1}}=5.830$ GeV for $f_{1,2}^{A}$, $f_{1,2}^{TA}$; $M_{\rm pole}\equiv M_{B_{s0}}=5.833$ GeV for $f_{3}^{V}$; $M_{\rm pole}\equiv M_{B_{s}}=5.366$ GeV for $f_{3}^{A}$. The fitted values of the parameters $a_{0}$, $a_{1}$, $a_{2}$ as well as the values of form factors at maximum $q^{2}=0$ and zero recoil $q^{2}=q^{2}_{\rm max}$ are given in Table 1. The difference of the fitted form factors from the calculated ones does not exceed 0.5%. Our model form factors are plotted in Fig. 3.

In Tables 3, 3 we compare the calculated values of the form factors with predictions of other approaches gikls ; aas ; latt ; ws . The covariant constituent quark model was employed in Ref. gikls , while form factors in Ref. aas were calculated in the framework of the light-cone QCD sum rules. The helicity form factors (14) were calculated using lattice QCD with relativistic $b$ quarks in Ref. latt . QCD light-cone sum rules with the account of next-to-leading perturbative corrections were used in Ref. ws . Reasonable agreement between substantially different approaches is observed. Note that most of the previous theoretical methods determine the decay form factors in the limited range of the momentum transfer squared $q^{2}$. Thus light-cone QCD sum rules provide form factors near the maximum recoil point $q^{2}=0$, while lattice calculations are performed for small values of the recoil momentum near the point $q^{2}=q^{2}_{\rm max}$. Therefore, in such approaches, the extrapolation of form factors to the whole kinematical range is required using some phenomenological model prescriptions. The important advantage of our model is the possibility to explicitly determine the $q^{2}$ dependence of the decay form factors in the whole kinematical range, which is rather broad, without extrapolations and/or additional model assumptions. In Fig. 4 we plot the helicity form factors calculated in our model.

[FIGURE:]

Now we can use the obtained form factors for the calculation of the rare semileptonic $\Lambda_{b}\to\Lambda l^{+}l^{-}$ and rare radiative $\Lambda_{b}\to\Lambda\gamma$ decay observables.

IV Rare semileptonic $\Lambda_{b}$ baryon decays

The effective Hamiltonian for the rare $b\to s$ transitions is given by bhi

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{tj}$ are Cabibbo-Kobayashi-Maskawa matrix elements, $c_{i}$ are the Wilson coefficients and ${\cal O}_{i}$ are the standard model operators.

Then the matrix element of the $b\to sl^{+}l^{-}$ transition amplitude between baryon states is given by

[TABLE]

where

[TABLE]

$T^{(m)}$ ($m=1,2$) are expressed through the form factors and the Wilson coefficients. These amplitudes can be written in the helicity basis $\varepsilon^{\mu}(\lambda)$ as follows

[TABLE]

where $\lambda_{\Lambda}$ is the helicity of the final $\Lambda$ baryon and $\lambda=0,\pm 1,t$ correspond to longitudinal, transverse and time-like helicities, respectively.

The helicity amplitudes for weak baryon transitions induced by vector ($V$) and axial vector ($A$) currents are expressed in terms of the decay form factors gikls in the following way

[TABLE]

where the upper(lower) sign corresponds to $Vm(Am)$ and the corresponding combinations of form factors are

[TABLE]

and

[TABLE]

The amplitudes for negative values of the helicities can be obtained using the relation

[TABLE]

The total helicity amplitude for the $V-A$ current is then given by

[TABLE]

The values of the Wilson coefficients $c_{i}$ and of the effective Wilson coefficient $c_{7}^{eff}$ are taken from Ref. wc . The effective Wilson coefficient $c_{9}^{\rm eff}$ contains additional pertubative and long-distance contributions

[TABLE]

The perturbative part is equal to

[TABLE]

where

[TABLE]

The long-distance (nonperturbative) contributions are assumed to originate from the $c\bar{c}$ resonances ($J/\psi,\psi^{\prime}\dots$) and have a usual Breit-Wigner structure:

[TABLE]

We include contributions of the vector $V_{i}(1^{--})$ charmonium states: $J/\psi$, $\psi(2S)$, $\psi(3770)$, $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$, with their masses ($M_{V_{i}}$), leptonic [$\Gamma(V_{i}\to l^{+}l^{-})$] and total ($\Gamma_{V_{i}}$) decay widths taken from PDG pdg .

The differential decay rate for the rare semileptonic $\Lambda_{b}$ baryon decay to the $\Lambda$ baryon reads gikls

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{qQ}$ is the CKM matrix element, $\lambda\equiv\lambda(M_{\Lambda_{b}}^{2},M_{\Lambda}^{2},q^{2})=M_{\Lambda_{b}}^{4}+M_{\Lambda}^{4}+q^{4}-2(M_{\Lambda_{b}}^{2}M_{\Lambda}^{2}+M_{\Lambda}^{2}q^{2}+M_{\Lambda_{b}}^{2}q^{2})$,

[TABLE]

and $m_{l}$ is the lepton mass.

The lepton angle differential decay distribution is given by

[TABLE]

where $\theta$ is the angle between the $\Lambda_{b}$ baryon and the positively charged lepton in the dilepton rest frame. The lepton forward-backward asymmetry is defined by gikls

[TABLE]

The fraction of longitudinally polarized dileptons is expressed by

[TABLE]

The hadron angle differential distribution of the decay $\Lambda_{b}\to\Lambda(\to p\pi^{-})l^{+}l^{-}$ is given by

[TABLE]

where $\theta_{h}$ is the angle between the proton and the $\Lambda$ baryon in the $\Lambda_{b}$ rest frame. The hadron forward-backward asymmetry has the form gikls

[TABLE]

The other useful observable is the combined hadron-lepton forward-backward asymmetry $A_{FB}^{h\ell}$. It is proportional to the coefficient in front of the term $\cos\theta\cos\theta_{h}$ in the threefold joint angular decay distribution for the decay of the unpolarized $\Lambda_{b}$ gikls . This asymmetry is expressed by

[TABLE]

where the value of the $\Lambda\to p\pi^{-}$ decay asymmetry $\alpha_{\Lambda}$ is known from experiment pdg : $\alpha_{\Lambda}=0.642\pm 0.013$.

The average values of these quantities $\langle A_{FB}^{\ell}\rangle$, $\langle A_{FB}^{h}\rangle$, $\langle A_{FB}^{h\ell}\rangle$ and $\langle F_{L}\rangle$ should be calculated by separately integrating the numerators and denominators over $q^{2}$.

Substituting the form factors calculated in the previous section into the expressions (52)–(64) we calculate the rare $\Lambda_{b}$ decay branching fractions and asymmetry parameters. We roughly estimate theoretical uncertainties of our results, originating from the calculation of the decay form factors, to be about 10% (see discussion in Ref. sllbdecay .)

In Figs. 5–9 we plot our predictions for the differential branching ratios $dBr/dq^{2}$, lepton $A^{\ell}_{FB}(q^{2})$, hadron $A^{h}_{FB}(q^{2})$ and hadron-lepton $A^{h\ell}_{FB}(q^{2})$ forward-backward asymmetries as well as the fraction of longitudinally polarized dileptons $F_{L}(q^{2})$ for rare decays $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ and $\Lambda_{b}\to\Lambda\tau^{+}\tau^{-}$ in comparison with available experimental data lhcb2015 ; cdf . By solid (dashed) lines we plot theoretical results obtained without (with) inclusion of the long-distance contributions to the Wilson coefficients coming from the charmonium resonances. Experimental data for the $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay from the LHCb lhcb2015 and CDF cdf Collaborations are plotted by dots with solid and dashed error bars, respectively.

In Table 4 we compare different theoretical predictions gikls ; aas ; wll ; llg ; mr ; glch for the total branching fractions of rare semileptonic $\Lambda_{b}$ decays with available experimental data pdg . The presented values include results of the relativistic and nonrelativistic quark model calculations gikls ; llg ; mr as well as evaluations based on various versions of the light-cone QCD sum rules aas ; wll ; glch . At present, experimental data are available for $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay only. The values obtained in our model and Refs. gikls ; llg ; mr agree well with data, while other results are significantly larger. From Table 4 we also see that predictions for the rare $\Lambda_{b}\to\Lambda\tau^{+}\tau^{-}$ decay vary significantly. Our results are close to those from quark models Refs. gikls ; mr . On the other hand, the light-cone QCD sum rules aas ; wll predict significantly larger values while the quark model llg gives a significantly lower value. Thus experimental measurement of the rare $\Lambda_{b}\to\Lambda\tau^{+}\tau^{-}$ decay branching fraction can help to discriminate between theoretical approaches.

The predicted values of the averaged asymmetries and polarization fractions of the rare semileptonic $\Lambda_{b}$ decays are given in Table 5. They include results obtained without (nonres.) and with (res.) inclusion of the charmonium resonances in the long-distance contribution (51) to the Wilson coefficient $c_{7}^{eff}$. Table 6 contains a comparison of theoretical predictions for the rare $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay observables with the LHCb experimental data lhcb2015 in the low recoil range 15 GeV${}^{2}\leq q^{2}\leq 20.28$ GeV2. In this range the lattice results latt are the most reliable. The same interval of $q^{2}$ was chosen in Ref. bfd in order to minimize the uncertainties from quark-hadron duality violation. The authors of Ref. bfd combine predictions of the lattice QCD for the rare decay form factors at low recoil with form factor relations in heavy quark effective theory. We see that theoretical approaches give close values in this $q^{2}$ range. The measured branching fraction $\langle Br\rangle$ and longitudinal polarization $\langle F_{L}\rangle$ are somewhat higher than theoretical predictions, but agree with them within $2\sigma$. The measured hadron forward-backward asymmetry $\langle A^{h}_{FB}\rangle$ agrees well with predictions, while the experimental lepton forward-backward asymmetry $\langle A^{\ell}_{FB}\rangle$ has significantly lower absolute value.

In Tables 7–11 we present the comparison of our predictions with lattice results latt and experimental data lhcb2015 for the differential branching fractions $\langle Br/dq^{2}\rangle$, forward-backward asymmetries $\langle A^{\ell}_{FB}\rangle$, $\langle A^{h}_{FB}\rangle$, $\langle A^{hl}_{FB}\rangle$ and longitudinal polarization $\langle F_{L}\rangle$ in several $q^{2}$ bins where such data is available for the rare $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay.

V Rare radiative $\Lambda_{b}$ baryon decay

The exclusive rare radiative decay rate $\Lambda_{b}\to\Lambda\gamma$ for the emission of a real photon ($k^{2}=0$) is given by

[TABLE]

Substituting the calculated values of the form factors $f_{2}^{TV,TA}(0)$ we get the prediction for the branching fraction which is given in Table 12. In this table we also give other theoretical values mw ; gikls ; wll ; glch ; cdfp and the experimental upper limit. Our result is consistent with the values from Refs. mw ; wll , but about a factor of 2 larger than the prediction of the covariant constituent quark model gikls . The result of the light-cone QCD sum rule study glch is about an order of magnitude lower, while the value obtained within three-point QCD sum rules in the heavy quark limit cdfp is more than a factor 3 larger than other theoretical predictions. Thus the measurement of the rare radiative $\Lambda_{b}\to\Lambda\gamma$ decay branching fractions can discriminate between different approaches to the form factor calculations.

VI Conclusions

The form factors of the rare $\Lambda_{b}\to\Lambda$ baryon transitions were obtained in the framework of the relativistic quark model. Relativistic quark-diquark picture of baryons was employed. The decay form factors are expressed through the overlap integrals of the baryon wave functions. The obtained expressions take into account all relativistic effects including the transformation of the baryon wave functions from rest to the moving reference frame as well as relativistic contributions of the intermediate negative energy states. The momentum transfer squared dependence of the form factors is explicitly determined in the whole accessible kinematical range without extrapolations and additional model assumptions. The analytic expressions for the form factors which approximate numerical results with high accuracy are given in Eq. (32). Such an approach significantly improves the accuracy of theoretical predictions since most of the previous calculations determine form factors in a single kinematical point or in the limited kinematical range and then require assumptions on the form factor $q^{2}$ dependence or extrapolations to the whole $q^{2}$ range.

The branching fractions, various forward-backward asymmetries and polarization fractions for the rare semileptonic $\Lambda_{b}\to\Lambda l^{+}l^{-}$ decays were calculated in the framework of the standard model using the obtained form factors. Calculations were performed both with and without long-distance contributions to the effective Wilson coefficient $c_{7}^{eff}$ arising from the account of resonances corresponding to charmonium states. Detailed comparison of the obtained predictions with previous quark model, light-cone QCD sum rules and lattice calculations as well as experimental data for the decay $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ is given. The calculations of the decay observables are performed for several $q^{2}$ bins for which experimental data is available lhcb2015 . Good agreement of our predictions with lattice latt results is found. In general reasonable agreement of the calculated and measured observables of the $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay is achieved, however in some $q^{2}$ bins deviations are found to be about $2\sigma$. Therefore additional and more precise measurements are needed for confirming the standard model predictions or revealing possible deviations from them in rare baryon decays.

Acknowledgements.

We are grateful to A. Dolgolenko, D. Ebert, M. Ivanov, J. Körner and V. Matveev for valuable discussions and support. *

Appendix A Form factors of rear the $\Lambda_{b}\to\Lambda$ transitions

The expressions for vector and axial vector decay form factors are given in Ref. sllbdecay . The tensor and pseudo tensor decay form factors are as follows (the value of the long range anomalous chromomagnetic quark moment $\kappa=-1$).

A.1 Tensor form factors

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

A.2 Pseudo tensor form factors

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

superscripts (1) and (2) correspond to vertex functions $\Gamma^{(1)}$ and $\Gamma^{(2)}$, $S$ and $V$ correspond to the scalar and vector confining potentials, $\epsilon_{d}$ is the diquark energy,

[TABLE]

subscripts $I$ and $F$ denote the initial $\Lambda_{b}$ and final $\Lambda$ baryons.