Model independent New Physics analysis in $Λ_b\toΛμ^+μ^-$ decay
Diganta Das


		
		




 Model independent New Physics analysis in Λ_b→Λμ^+μ^- decay


		




Diganta Das  

Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India


		







We study the rare Λ_b→Λμ^+μ^- decay in the Standard Model and beyond. Beyond the Standard Model we include new vector and axial-vector operators, scalar and pseudo-scalar operators, and tensor operators in the effective Hamiltonian. Working in the helicity basis and using appropriate parametrization of the Λ_b →Λ hadronic matrix elements, we give expressions of hadronic and leptonic helicity amplitudes and derive expression of double differential branching ratio with respect to dilepton invariant mass squared and cosine of lepton angle. Appropriately integrating the differential branching ratio over the lepton angle, we obtain the longitudinal polarization fraction and the leptonic forward-backward asymmetry and sequentially study the observables in the presence of the new couplings. To analyze the implications of the new vector and axial-vector couplings, we follow the current global fits to b→ sμ^+μ^- data. While the impacts of scalar couplings can be significant, exclusive B̅→ X_sμ^+μ^- data imply stringent constraints on the tensor couplings and hence the effects on Λ_b→Λμ^+μ^- are negligible.






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§ INTRODUCTION

The Standard Model (SM) of particle physics has so far been enormously successful in explaining most of the particle physics experiments. Recently however, discrepancies between SM predictions and experimental measurements have been observed in several decay modes of B mesons. For example, LHCb measurements <cit.> of the ratio of branching ratios of B̅→ K(K^∗)ℓ^+ℓ^- decays into di-muons over di-electrons deviate from the SM predictions ∼ 1 by about 3σ. Other notable deviations in the b→ sμ^+μ^- mediated transitions include the so called P_5^' anomaly between theory <cit.> and experiments <cit.> in the B̅→ K^∗μ^+μ^-, and the systematic deficit in the branching ratio of B̅_s→ϕμ^+μ^- <cit.>. In the b→ cℓν mediated transitions, ratio of B→ D^(∗)ℓν decay involving τ leptons over light leptons has been measured by BaBar <cit.>, Belle <cit.> and LHCb <cit.> which deviate from the SM predictions <cit.> by about 2-3σ with a combined deviation of ≈ 4σ. Similar deviations from the SM prediction in B^+→ J/ψℓ^+ν decay <cit.> has also been observed in the recent LHCb measurement <cit.> where the deviation from the SM value is 1.7σ. These deviations, though statistically small and unable to provide unambiguous signal of New Physics (NP) can not be completely ignored. It is rather natural to explore all possible avenues of flavor physics to draw a definite conclusion on the existence of NP.

In the past several decades, the main phenomenological focus in this context has been on exclusive and inclusive decays of B meson. The recent B physics program at LHCb involves in addition to precise measurements of B decay modes, study of baryonic decays such as b→ sℓ^+ℓ^- mediated Λ_b→Λℓ^+ℓ^-. The important difference between Λ_b baryon and the B meson decay modes is that due to the spin of the Λ_b baryon, its decay distributions are generally more involved which makes the baryonic modes phenomenologically rich. At present, data on baryonic modes are rather limited and only recently LHCb has reported a measurement of several Λ_b→Λμ^+μ^- observables where the branching ratio is found to be lower than the SM prediction at low q^2 and higher at large q^2 <cit.>. Earlier, Λ_b →Λμ^+μ^- branching ratio was measured by CDF <cit.>.

In the SM, angular distributions of Λ_b→Λℓ^+ℓ^- have been studied in <cit.> and  <cit.>. Other studies in the SM and in the context of NP have been reported in Refs. <cit.>. The Λ_b→Λμ^+μ^- angular distributions for polarized Λ_b were recently worked out in <cit.>. In Ref. <cit.> the LHCb data on Λ_b→Λμ^+μ^- decay in combination with b→ sμ^+μ^- data from B meson decays were used to perform a model-independent fit to Wilson coefficients. One of the challenging aspect of Λ_b→Λℓ^+ℓ^- decay is to estimate the hadronic Λ_b→Λ form factors. Recent progress in this respect includes the lattice QCD results at low and intermediate recoil <cit.> and the QCD sum-rules analysis of spectator-scattering corrections to form factor relations which are valid at large recoil <cit.>. Also, better theoretical understanding of light-cone distribution amplitudes of Λ_b baryon has recently been achieved in <cit.>. In the large recoil region, the form factors have been calculated in the light cone sum-rules (LCSR) <cit.>. 

In this paper, we supplement the previous studies with a model independent analysis of Λ_b→Λμ^+μ^- for the most general effective Hamiltonian involving new vector and axial vector (VA), scalar and pseudo-scalar (SP), and tensor (T) operators for unpolarized Λ_b and massless leptons. We work in the helicity basis and derive the expressions of helicity amplitudes for this Hamiltonian. We present expressions of the double differential branching ratio with respect to dilepton invariant mass squared q^2 and cosine of lepton angle θ_ℓ, and then define forward-backward asymmetry involving leptons and longitudinal polarization fraction. We use helicity parametrization of Λ_b→Λ hadronic matrix elements. At large recoil, which correspond to low q^2, the form factors are taken from recent calculations in the light cone sum-rules <cit.> and at low recoil, which correspond to large q^2, we use results from the calculations in lattice QCD <cit.>.


The paper is organized as follows. In Sec. <ref> we write down the most general effective Hamiltonian involving VA, SP and T operators. After describing the decay kinematics in Sec. <ref>, we describe the helicity formalism in Sec. <ref>. The form factors are discussed in Sec. <ref> and the model independent analysis is performed in Sec. <ref>. The results are summarized in Sec. <ref>. The paper is supplemented with a number of appendixes which contain the details of the calculations.





§ EFFECTIVE HAMILTONIAN 

The most general low energy effective Hamiltonian for rare |Δ B|=|Δ S|=1 transition involving VA, SP and T operators is


    ℋ = ℋ^ VA + ℋ^ SP + ℋ^ T  ,

where ℋ^ VA, ℋ^ SP and ℋ^ T are


    ℋ^ VA   =    -4G_F/√(2)V_tbV_ts^∗α_e/4π[_9^ effs̅γ^μ P_Lb ℓ̅γ_μℓ + _10s̅γ^μ P_L b ℓ̅γ_μγ_5 ℓ - 2m_b/q^2_7^ effs̅iq_νσ^μνP_R b ℓ̅γ_μℓ 
       +   _V s̅γ^μ P_L b ℓ̅γ_μℓ + _A s̅γ^μ P_L b ℓ̅γ_μγ_5 ℓ + ^'_V s̅γ^μ P_R b ℓ̅γ_μℓ + ^'_A s̅γ^μ P_R b ℓ̅γ_μγ_5 ℓ]  ,
    
    ℋ^ SP   =    -4G_F/√(2)V_tbV_ts^∗α_e/4π[ _S^'s̅ P_L b ℓ̅ℓ + _P^'s̅ P_L b ℓ̅γ_5 ℓ + _S s̅ P_R b ℓ̅ℓ + _P s̅ P_R b ℓ̅γ_5 ℓ]  ,
    
    ℋ^ T   =    -4G_F/√(2)V_tbV_ts^∗α_e/4π[ _T s̅σ^μν b ℓ̅σ_μνℓ +  _T5s̅σ^μν b ℓ̅σ_μνγ_5ℓ]  .


Here G_F is the Fermi-constant, α_e is the fine structure constant, V_tbV_ts^∗ are the Cabibbo-Kobayashi-Maskawa(CKM) elements, P_L,R=(1∓γ_5)/2 are the chiral projection operators, and σ_μν=i[γ_μ,γ_ν]/2. The b-quark mass multiplying the dipole operator is in the MS-mass scheme. In the SM, _V=_V^' =_A=_A^'= _S=_S^'=_P= _P^'=_T=_T5=0. The expressions of _9^ eff and other SM Wilson coefficients are given in Appendix <ref>. In this article, we neglect the mass of the leptons. 

In the Hamiltonian (<ref>), we assume only the factorizable quark-loop contributions to four-quark operators and gluonic operator which are conveniently absorbed in the Wilson coefficients ^ eff_7,9. For simplicity we ignore the non-factorizable corrections which are expected to play a non-negligible role, particularly at low q^2 <cit.>.




§ KINEMATICS OF Λ_B→Λℓ^+ℓ^- DECAY 

The momenta and spins variables for the different particles in the decay process are assigned as follows


    Λ_b(p,s_p ) →Λ(k,s_k) ℓ^+(q_1) ℓ^-(q_2)  .


Here p,k,q_1 and q_2 are the momenta of the parent baryon Λ_b, daughter baryon Λ and the negatively and positively charged leptons, respectively, and s_p,k are the projections of the baryonic spins on to the z-axis in their respective rest frames. 

For future convenience, we define the four momentum of the dilepton pair as


    q^μ = q_1^μ + q_2^μ  .

We assume that in the rest frame of Λ_b, the daughter baryon Λ travels along the positive z-direction and the lepton pair travels in the negative z-direction. So that in Λ_b rest frame (Λ_b RF) we have 


    q^μ|_Λ_b RF = (q^0, 0, 0, -|q|)  ,   k^μ|_Λ_b RF =((m_Λ_b-q^0), 0, 0, +|q|)  ,


where 


    q^0|_Λ_b RF = -+q^2/2  ,   |q||_Λ_b RF = √(λ(,,q^2))/2  ,


and we have defined λ(a,b,c)=a^2+b^2+c^2-2(ab+bc+ca).
The Λ_b→Λ(→ Nπ)ℓ^+ℓ^- decay can be described in terms of four independent kinematic variables which we choose as the dilepton invariant mass squared q^2 and three angles θ_ℓ, θ_Λ and ϕ. The angles are defined as follows: θ_ℓ is the angle made by the negatively charged lepton ℓ^- with the +z-direction in the dilepton rest frame, θ_Λ is made by N with respect to the +z-direction in the Nπ rest frame, and ϕ is the angle between the ℓ^+ℓ^- and Nπ decay planes. 






§ THE HELICITY FORMALISM 




 §.§ Decay process 

We derive the decay distribution in the helicity formalism where we restrict ourselves to unpolarized Λ_b since the polarization has been measured by LHCb to be very small <cit.>. In the rest frame of the Λ_b, the two-fold differential decay distribution in terms of q^2 and the angle θ_ℓ can be represented as



    dΓ/dq^2 dcosθ_ℓ = 1/2m^3_Λ_b2√(λ(,,q^2))/(8π)^31/2s_p+1∑_λ_1,λ_2∑_s_p,s_k|ℳ^λ_1,λ_2(s_p,s_k)|^2  ,


where λ_1,2 are the helicities of the final state leptons. The Λ_b →Λℓ^+ℓ^- helicity amplitudes ℳ^λ_1,λ_2(s_p,s_k) corresponding to the effective Hamiltonian (<ref>) consist of contributions from VA, SP and T operators


    ℳ^λ_1,λ_2(s_p,s_k) = ℳ^λ_1,λ_2_ VA(s_p,s_k) + ℳ^λ_1,λ_2_ SP(s_p,s_k) + ℳ^λ_1,λ_2_ T(s_p,s_k)  ,


which are defined as


    ℳ^λ_1,λ_2_ VA(s_p,s_k) = -G_F/√(2) V_tbV_ts^∗α_e/4π∑_λη_λ[ H^L,s_p,s_k_ VA,λ L^λ_1,λ_2_L,λ + H^R,s_p,s_k_ VA,λ L^λ_1,λ_2_R,λ]  ,
    
       ℳ^λ_1,λ_2_ SP(s_p,s_k) = -G_F/√(2) V_tbV_ts^∗α_e/4π[ H_ SP^L,s_p,s_k L^λ_1,λ_2_L + H_ SP^R,s_p,s_k L^λ_1,λ_2_R ]  ,
    
       ℳ^λ_1,λ_2_ T(s_p,s_k) = -2G_F/√(2) V_tbV_ts^∗α_e/4π∑_λ,λ^'η_λη_λ^'[ H^L,s_p,s_k_ T,λλ^' L^λ_1,λ_2_L,λλ^' + H^R,s_p,s_k_ T,λλ^' L^λ_1,λ_2_R,λλ^']  .


Here λ,λ^' are the polarization states of the virtual gauge boson that decays in to dilepton pair, η_t = 1 and η_±1,0 = -1. H^L(R) and L_L(R) are the hadronic and leptonic helicity amplitudes corresponding the left- (L) and right-handed (R) chiralities of the lepton currents. 

The hadronic helicity amplitudes are the projections of Λ_b→Λ matrix elements on the direction of the polarization of virtual gauge boson. For VA, SP and T operators these are 


    H^L(R),s_p,s_k_ VA,λ   =   ϵ̅^∗_μ(λ) ⟨Λ(k,s_k)| [ ( (_9^ eff∓_10) + (_V ∓_A) )s̅γ^μ (1-γ_5)b  
       +    (_V^'∓_A^') s̅γ^μ(1+γ_5)b - 2m_b/q^2_7^ effs̅iq_νσ^μν (1+γ_5) b ] |Λ_b(p,s_p)⟩  ,
    
    
    H^L(R),s_p,s_k_ SP   =   ⟨Λ(k,s_k) | [ (_S^'∓_P^') s̅(1-γ_5)b  
       +    (_S ∓_P) s̅(1+γ_5)b ] | Λ_b(p,s_p) ⟩  ,
    
    
    
    H^L(R),s_p,s_k_ T,λλ^'   =    iϵ̅^∗_μ(λ) ϵ̅^∗_ν(λ^') ⟨Λ(k,s_k) | s̅σ^μν b | Λ_b(p,s_p) ⟩ (_T ∓_T5)   .


Here ϵ̅_μ(λ^(')) denote the polarization vectors of the virtual gauge boson. Our choice for the polarization vectors are summarized in Appendix <ref>. Similarly, the leptonic helicity amplitudes are 


    L^λ_1,λ_2_L(R) = ⟨ℓ̅(λ_1)ℓ(λ_2) | ℓ̅ (1∓γ_5) ℓ | 0⟩  , 
    
        L^λ_1,λ_2_L(R),λ = ϵ̅^μ(λ) ⟨ℓ̅(λ_1) ℓ(λ_2) | ℓ̅γ_μ (1∓γ_5) ℓ | 0⟩  ,
    
       L^λ_1,λ_2_L(R),λλ^' = -iϵ̅^μ(λ) ϵ̅^ν(λ^') ⟨ℓ̅(λ_1) ℓ(λ_2) | ℓ̅σ_μν (1∓γ_5) ℓ | 0⟩  .


The tensor amplitudes are anti-symmetric under the exchange of λ and λ^', i.e., L^λ_1,λ_2_L(R),λ^'λ = - L^λ_1,λ_2_L(R),λλ^'.






 §.§ Hadronic Helicity amplitudes 

The Λ_b →Λ hadronic matrix elements for different operators are defined in terms of ten helicity form factors f^V_t,0,⊥, f^A_t,0,⊥, f^T_0,⊥, f^T5_0,⊥ <cit.> and spinor matrix elements. The definitions are summarized in Appendix <ref> and the spinor matrix elements for different combinations of spin orientations are worked out in Appendix <ref>. Using these results we write down the expressions of the helicity amplitudes defined in Eqs. (<ref>)-(<ref>) for different operators. For VA operators the non-vanishing amplitudes are


    H^L(R),_ VA,0   =    f^V_0( + ) √(s_-/q^2)^L(R)_ VA,+ - f^A_0( - ) √(s_+/q^2)^L(R)_ VA,- 
       +   2m_b/q^2( f^T_0 √(q^2 s_-) - f^T5_0 √(q^2 s_+))  _7^ eff  ,
    
    
    H^L(R),_ VA,0   =    f^V_0( + ) √(s_-/q^2)^L(R)_ VA,+ + f^A_0( - ) √(s_+/q^2)^L(R)_ VA,- 
       +   2m_b/q^2( f^T_0 √(q^2 s_-) + f^T5_0 √(q^2 s_+))  _7^ eff  ,
    
    
    H^L(R),_ VA,+   =    -f^V_⊥√(2s_-)^L(R)_ VA,+ + f^A_⊥√(2s_+)^L(R)_ VA,- 
       -   2m_b/q^2( f^T_⊥( + ) √(2s_-) - f^T5_⊥ ( - ) √(2s_+))_7^ eff ,   
    
    
    H^L(R),_ VA,-   =    -f^V_⊥√(2s_-)^L(R)_ VA,+ - f^A_⊥√(2s_+)^L(R)_ VA,- 
       -   2m_b/q^2( f^T_⊥( + ) √(2s_-) + f^T5_⊥ ( - ) √(2s_+))_7^ eff ,



where the variables s_± are


    s_± = (±)^2 - q^2  ,


and we have defined


    _ VA,+^L(R) = (_9^ eff∓_10)+(_V∓_A)+(_V^'∓_A^')  ,
       _ VA,-^L(R) = (_9^ eff∓_10)+(_V∓_A)-(_V^'∓_A^')  .


The non-vanishing helicity amplitudes for the SP operators are


    H^L(R),_ SP   =   ( f^A_t  + /m_b√(s_-) + f^V_t  - /m_b√(s_+))(_S^'∓_P^') 
       +   ( -f^A_t  + /m_b√(s_-) + f^V_t  - /m_b√(s_+)) (_S ∓_P) ,  
    
    
    H^L(R),_ SP   =   ( -f^A_t  + /m_b√(s_-) + f^V_t  - /m_b√(s_+))(_S^'∓_P^') 
       +   ( f^A_t  + /m_b√(s_-) + f^V_t  - /m_b√(s_+)) (_S ∓_P) ,


where we have neglected the mass of the strange quark. Finally, for the T operators the non-vanishing amplitudes are


    H^L(R),_ T,+-   =    - H^L(R)_ T,+- = -f^T5_0 √(s_+) (_T ∓_T5)  ,
    
    H^L(R),_ T,+0   =    H^L(R)_ T,-0 = f^T5_⊥ ( - ) √(2s_+/q^2) (_T ∓_T5)  ,  
    
    
    H^L(R),_ T,+t   =    H^L(R)_ T,-t = f^T5_⊥ ( + ) √(2s_-/q^2)(_T ∓_T5)  ,
    
    
    H^L(R),_ T,0t   =    H^L(R)_ T,0t = -f^T_0 √(s_-)(_T ∓_T5)  .


The tensor amplitudes are anti-symmetric under the exchange of λ and λ^', i.e., H^L(R)_ T,λ^'λ=-H^L(R)_ T,λλ^'. 






 §.§ Leptonic Helicty amplitudes 
 
Using the representations of the lepton spinors given in Appendix <ref>, we calculate the leptonic helicity amplitudes defined in Eqs. (<ref>)-(<ref>). Neglecting the mass of the leptons, the expressions of the non-vanishing amplitudes for the SP and VA operators are


    L^_L = - L^_R = 2√(q^2)  ,
    
        L^_R,+ = -L^_L,- = - √(2q^2) (1 - cosθ_ℓ)  ,
    
    
        L^_L,+ = -L^_R,- = + √(2q^2) (1 + cosθ_ℓ)  ,   L^_L,0 = L^_R,0 = 2√(q^2)sinθ_ℓ  ,


and for the tensor operators the expressions are


    L^_L,+- = L^_R,+- = -L^_L,0t = L^_R,0t = - 2 √(q^2)cosθ_ℓ  , 
    
        L^_L,+0 = L^_R,+0 = L^_L,-0 = L^_R,-0 = + √(2q^2)sinθ_ℓ  ,
    
        L^_L,+t = -L^_R,+t = -L^_L,-t = L^_R,-t = - √(2q^2)sinθ_ℓ  .



Other non-vanishing amplitudes can be obtained with the following relation L^λ_1,λ_2_L(R),λ^'λ = - L^λ_1,λ_2_L(R),λλ^'  .




 §.§ Differential Distributions and Observables

With the expressions of the helicity amplitudes in Secs. <ref> and <ref> we write down the two-fold differential branching ratio


    d^2ℬ/dq^2 dcosθ_ℓ   = N^2(q^2) ( A^ VA + A^ SP  + A^ T + A^ inter)  ,
 

where A^ VA, SP, T correspond to the contributions from VA, SP and T operators respectively and A^ inter is the interference terms between SP and T contributions. The normalization constant N(q^2) is given by



    N(q^2) = G_F V_tbV_ts^∗α_e  √(τ_Λ_bq^2√(λ(,,q^2))/2^15 m^3_Λ_bπ^5 )  .


In terms of the hadronic helicity amplitudes the expressions of A^ VA,SP,T are 


    A^ VA   = 4 (1 - ) ( |H_ VA,0^L,|^2 + |H_ VA,0^L,|^2 + |H_ VA,0^R,|^2 + |H_ VA,0^R,|^2) 
         + 2(1-)^2( | H_ VA,-^L,|^2 + | H_ VA,+^R,|^2)  
       + 2(1+)^2 ( | H_ VA,-^R,|^2 + | H_ VA,+^L,|^2| )  ,
    
    
    A^ SP    = 4 ( | H_ SP^L,|^2 + | H_ SP^L,|^2 + | H_ SP^R,|^2 + | H_ SP^R,|^2) ,
    
    
    A^ T    = 8[ ( |H_ T,-0^L,|^2 + |H_ T,-t^L,|^2 + |H_ T,+0^L,|^2 + |H_ T,+t^L,|^2 + { L ↔ R })  
        + ( 2 | H_ T,0t^L,|^2  + 2 | H_ T,0t^L,|^2  - | H_ T,-0^L,|^2 - | H_ T,+0^L,|^2 
        - | H_ T,-t^L,|^2 - | H_ T,+t^L,|^2  + 2 | H_ T,+-^L,|^2 + 2 | H_ T,+-^L,|^2 + { L ↔ R })  
        + ( 4[H_ T,+-^L, (H_ T,0t^L,)^∗] + 4[H_ T,+-^L, (H_ T,0t^L,)^∗] 
        + 2[H_ T,-t^L, (H_ T,-0^L,)^∗] - 2 [H_ T,+t^L, (H_ T,+0^L,)^∗] - { L ↔ R })  
        -  ( 2[H_ T,-t^L, (H_ T,-0^L,)^∗] -  2[H_ T,+t^L, (H_ T,+0^L,)^∗] -  { L ↔ R }) ] ,


and the interference term is


    A^ inter_ SP-T    = -16[( [H_ T,0t^L,(H_ SP^L,)^∗] + [H_T,0t^L,(H_ SP^L,)^∗] +{L ↔ R })  
       + ([H_ T,+-^L,(H_ SP^L,)^∗] + [H_ T,+-^L,(H_ SP^L,)^∗] - {L ↔ R }) ]   .



As can be seen, there are no interference terms of VA with SP and T contributions. This is because VA-SP and VA-T interferences are proportional to either lepton mass or its squared and therefore vanish in the limit of massless leptons.

By partially integrating the differential distribution (<ref>) over the angle θ_ℓ various observables can be constructed. Here we recall the definitions of the leptonic forward-backward asymmetry A^ℓ_ FB


    A^ℓ_ FB(q^2) = ∫_-1^0 dcosθ_ℓd^2ℬ/dq^2dcosθ_ℓ - ∫_0^+1 dcosθ_ℓd^2ℬ/dq^2dcosθ_ℓ/∫_-1^0 dcosθ_ℓd^2ℬ/dq^2dcosθ_ℓ + ∫_0^+1 dcosθ_ℓd^2ℬ/dq^2dcosθ_ℓ  ,


and the longitudinal polarization fraction F_L


    F_L(q^2) = ∫_-1^+1 dcosθ_ℓ (2-5cos^2θ_ℓ) d^2ℬ/dq^2dcosθ_ℓ/∫_-1^+1 dcosθ_ℓd^2ℬ/dq^2dcosθ_ℓ  .


In section <ref> we perform a model-independent analysis of these observables in the presence of the new operators.




§ Λ_B →Λ HELICITY FORM FACTORS 

There are ten q^2 dependent form factors f^V_t,0,⊥, f^A_t,0,⊥, f^T_0,⊥, f^T5_0,⊥ that parametrize the Λ_b→Λ hadronic matrix elements which are summarized in Appendix <ref>. Note that our notations of the form factors is same as in <cit.>. Somewhat different notation has been followed in <cit.> and our notation is related to their's as f^V_t,0,⊥ = f_0,+,⊥, f^A_t,0,⊥ = g_0,+,⊥, f^T_0,⊥=h_+,⊥ and f^T5_0,⊥=h̃_+,⊥. 

In the large recoil region, we use the LCSR results <cit.> for the form factors. The q^2 dependence are obtained from fits of the LCSR results to z-parametrization with following form


    f^i(q^2) = f^i(0)/1-q^2/m^2_B^∗_s( 1 + b_1^i [ z(q^2,t_+) - z(0,t_+) ]  )  ,


for the form factors f^V_0,f^V_⊥, f^T_0 and f^T_⊥,


    f^i(q^2) = f^i(0)/1-q^2/m^2_B_s( 1 + b_1^i [ z(q^2,t_+) - z(0,t_+) ]  )  ,


for the form factor f^A_t, and


    f^i(q^2) = f^i(0) ( 1 + b_1^i [ z(q^2,t_+) - z(0,t_+) ]  )  ,


for the form factors f^V_t, f^A_0,⊥, f^T5_0,⊥. Here z(q^2,t_+) is defined as


    z(q^2,t_+) = √(t_+ - q^2) - √(t_+ - t_0)/√(t_+ - q^2) + √(t_+ - t_0)  ,


where t_0 = ( - )^2 and t_+ = (m_B_s + m_π)^2. The values of the form factors at zero momentum transfer f^i(0), the fit parameters b_1^i, and the masses of B_s^(∗) are taken from <cit.>. In our numerical analysis at large recoil we restrict ourselves to q^2 ≤ 7 GeV^2 in dilepton invariant mass squared.

At low recoil region we use lattice QCD results <cit.> for the form factors. In Ref. <cit.> two z-parametrization fits, “nominal" fit and “higher-order" fit have been used to obtain the q^2 dependence. For our numerical analysis, we use their “higher-order" fit which correspond to the following z-parametrization



    f^i(q^2) = 1/1-q^2/(m^f^i_ pole)^2[ a^f^i_0 + a_1^f^i z(q^2,t^'_+) + a_2^f^i (z(q^2,t^'_+))^2 ]  ,


where t^'_+ = (m_B + m_K)^2. The fit parameters are taken from <cit.>. We note that if both the “nominal" and “higher-order" fit are used in the way mentioned in Ref. <cit.>, it would lead to different central values and errors in our results.  For numerical analysis, our choice of q^2 range for low recoil is 14.18 < q^2 < ( - )^2 GeV^2. 

With the form factors at hand, we are ready to study the angular observables in the SM. In Fig. <ref> the differential branching ratio dℬ/dq^2, leptonic forward-backward asymmetry A_ FB^ℓ and the longitudinal polarization fraction F_L are shown as function of q^2. The bands are obtained by varying the form factors and other inputs within their quoted uncertainties. A list of numerical inputs and their values are collected in table <ref>.



We have superimposed the latest LHCb data by gray crosses. In our case, the angle θ_ℓ is defined with respect to the negatively charged lepton, where as LHCb choses the positively charged lepton to define the angle θ_ℓ. This difference has been taken care of to display the LHCb data on A^ℓ_ FB.





§ MODEL INDEPENDENT ANALYSIS 

In this section we study the impacts of NP couplings on the differential branching ratio, forward-backward asymmetry and longitudinal polarization fraction. We sequentially discuss the consequences of the presence of (i) only _V,A^(') couplings, (ii) only _S,P^(') couplings, (iii) only _T,T5 couplings, and (iv) combinations of _V,A^(') _S,P^(') and _T,T5 couplings. All the NP couplings are considered real. The values of numerical inputs used in our analysis are collected in the table <ref> in Appendix <ref>.




 §.§ Only VA couplings are present 

There exist a wealth of data on b→ sμ^+μ^- transitions from various B meson decay modes some of which show deviations from the SM predictions. In the past few years a number of global fits have appeared to interpret the deviations through new VA couplings. If such new couplings are indeed present, it is worth studying the consequences on Λ_b→Λμ^+μ^- observables. We do so by first summarizing the important features when only _V,A^(') are present. In our discussions, we choose those set of couplings that have large 'pull' in the global fits to b→ sμ^+μ^- data. In this context we follow the fits presented in Ref. <cit.> where it is assumed that NP appear only in the muon modes and the electron modes are SM like. In Fig. <ref> we show plots with representative sets of couplings which are chosen so as to show the interesting variants. For all our analysis, the signs of the couplings are same as is found in global fits<cit.>. Our observations are as follows


	
  •  It was realized very early from global fits that a negative value of _V can easily alleviate the tensions in b→ sμ^+μ^- data. We find that for a negative value of _V the branching ratio is reduced at all q^2. With only positive _A, or, with _V = -_A also the branching ratio is reduced. However, for _V = _A < 0, the branching ratio increases for all q^2. 
	
  •  For a negative value of _V, the zero-crossing of leptonic forward-backward asymmetry A^ℓ_ FB is shifted to the higher values of q^2 than in SM. For _V = ±_A also, the zero-crossing points are shifted in this direction. However, if only _A is present, the zero-crossing point is SM like, which is understood from the fact that in the SM the zero-crossing point is proportional to _7^ eff/_9^ eff. 


There are other combinations of VA couplings that can be considered, such as only _V,A^' or _V^' = ±_A^'. In global fits these cases have very small `pull' value and we therefore do not discuss these further. 

We also examined four pairs of uncorrelated couplings, (_V,_A), (_V,_V^'), (_V,_A^') and (_V^',_A) having large pull values in global fits <cit.>. Here we summarize some of our observations


	
  •  We find that the branching ratio is reduced for all these cases.
	
  •  Apart from when the pair (_V^',_A) is present, the A^ℓ_ FB zero-crossing is shifted to the higher value of q^2 for all other pairs. For the case when the pair (_V^',_A) is present, the zero-crossing can shift to lower value of q^2, but the effect is negligible unless the couplings are large.
	









 §.§ Only SP couplings are present

The scalar operators are constrained from B̅_s→μ^+μ^- branching ratio. To B̅_s→μ^+μ^- decay, there is no contribution from new vector couplings _V^('), while the contributions from new axial vector couplings _A^(') are suppressed by m_μ/m_B_s. There is also no contribution from tensor operators since the matrix element ⟨ 0|s̅σ^μνb|B_s⟩ vanishes. In the presence of SP operators the of B̅_s →μ^+μ^- branching ratio is given by 


    ℬ(B̅_s→μ^+μ^-)    =   G_F^2α_e^2 m_B_s^5 f_B_s^2 τ_B_s/64π^3 |V_tbV_ts^∗|^2 √(1 - 4m_μ^2/m_B_s^2) 
       ×   {( 1 - 4m_μ^2/m_B_s^2) | _S - _S^'/m_b|^2 + | _P - _P^'/m_b + 2m_μ/m_B_s^2_10|^2 }  .


 The SM prediction for the branching ratio is ℬ^ SM(B̅_s→μ^+μ^-) = 3.53 × 10^-9 <cit.> while the experimentally measured values by CMS<cit.> and LHCb <cit.> are 3.0^+1.0_-0.9× 10^-9, (3.0 ± 0.6^+0.3_-0.2) × 10^-9, respectively.

As can be seem from Eq. (<ref>), the constraints from B̅_s→μ^+μ^- alone can be evaded due to the cancellations between _S,P and _S,P^'. The only stringent constraint that can be obtained are |_S-_S^'|≲ 0.1 and |_P-_P^'|≲ 0.3 <cit.>. Therefore we combine with B̅_s→μ^+μ^- the measurements of B̅→ X_sμ^+μ^-, which is discussed in the next section, and find that the _S,P^(') are constrained as


    _S,P^(')≡ [-4.0,4.0]  .


We use these values and our main observations are 


	
  •  Branching ratio increases in the presence of SP.
	
  •  The A^ℓ_ FB zero-crossing always remain SM like. This is understood from the fact that the A^ SP term does not have any cosθ_ℓ dependence and therefore do not contribute to the numerator of A^ℓ_ FB. This is also the reason why |A^ℓ_ FB| decreases for all q^2 as can be seen in Fig. <ref>.
	
  •  Longitudinal polarization fraction reduces.






 §.§ Only tensor couplings are present

The tensor couplings are constrained by the inclusive B̅→ X_sμ^+μ^- branching ratio. This is a theoretically clean mode for indirect searches of NP. The q^2-cut dependent branching fractions has been measured by Belle <cit.> and BaBar <cit.> and the weighted average of the results read


    ℬ(B̅→ X_sμ^+μ^-)^[1,6] GeV_ exp   =    (1.58 ± 0.37) × 10^-6 ,
    ℬ(B̅→ X_sμ^+μ^-)^q^2 > 14.2, GeV_ exp   =    (0.48 ± 0.10) × 10^-6  .


In the presence of the tensor couplings, the expression of q^2-integrated branching ratio following <cit.> is


    dℬ(B̅→ X_sμ^+μ^-)/dq^2   =   ℬ(B̅→ X_sμ^+μ^-)|_ SM + ℬ_0/2m_b^8  16M_9(q^2)(|_T|^2 + |_T5|^2)   ,


where ℬ(B̅→ X_sμ^+μ^-)|_ SM is the branching ratio in the SM. The normalization factor is 


    ℬ_0    =   3α_e^2/16π^2|V_tbV_ts^∗|^2/|V_cb|^2ℬ(B→ X_sℓν)/f(m̂_c)κ(m̂_c)  ,
    
    f(m̂_c)    =    1 - 8 m̂_c^2 + m̂_c^6 - m̂_c^8 - 24 m̂_c^4 ln(m̂_c)  ,  m̂_c = m_c/m_b
    κ(m̂_c)    =    1 - 2α_s(m_b)/3π[( π^2 - 31/4)(1 - m̂_c)^2 + 3/2]  ,


where we use ℬ(B̅→ X_cℓν) = 10.33% <cit.> and the quark masses are in the pole mass scheme. The experimental data (<ref>) imply severe constraint on _T,T5 couplings


    _T^2+_T5^2 ≲ 0.55 ,

and due to this the effects of tensor couplings on Λ_b→Λℓ^+ℓ^- are negligibly small as shown in Fig. <ref>.













 §.§ Combinations of VA, SP and T couplings

After studying the patterns of the observables when VA, SP and T couplings are present separately, we study what happens when both VA and SP couplings are present. The terms A^ VA and A^ SP both contribute in this case.  We considered all the four scalar couplings _S,P^(') simultaneously along with different combinations of VA couplings considered in Sec. <ref>. Due to the interplay between VA and SP couplings, the resultant effects on branching ratio depend on the choice of values of the couplings. In general, if _S,P^(') are large so that A^ SP dominates over A^ VA, then branching ratio increases. Presence of both _V,A^(') and _S,P^(') has interesting effects on A^ℓ_ FB. Since A^ SP do not contribute to the numerator of A^ℓ_ FB, the zero-crossings in the simultaneous presence of _V,A^(') and _S,P^(') are same as when only _V,A^(') couplings are present. But the absolute value |A^ℓ_FB| is reduced compared to when only VA couplings are present and this is shown in Fig. <ref> (see also Fig. <ref> to compare).


 
We also considered combinations of SP and T couplings. In this case in addition to A^ SP and A^ T the interference term A^ inter contribute to the differential distribution. We find that due to stringent constraints on tensor couplings, the effects of combinations of SP and T couplings are almost identical to that when only SP couplings are present.





§ SUMMARY 

In this paper we have investigated the effects of NP couplings on the b→ sμ^+μ^- mediated rare decay Λ_b→Λμ^+μ^-. The NP considered are new VA operators, SP and T operators. We work in the helicity formalism and derive the expressions of helicity amplitudes with NP operators. We give the expressions of two fold differential distribution and study the observables A^ℓ_ FB and F_L in the presence of the NP couplings. The hadronic matrix elements are defined in terms of so called helicity form factors. At the large recoil the form factors are taken from LCSR calculations while at the low recoil the form factors are taken from calculations in lattice QCD, and both are fitted to z-parametrization which give their q^2 dependence.

We sequentially discuss the implications of the NP couplings on differential branching ratio, A^ℓ_ FB and F_L. To understand the implications of the new VA couplings, we have followed the trends of these couplings suggested by the global fits to b→ sμ^+μ^- data. We find that for most of the cases that are largely favored by the global fits, the branching ratio is reduced which is somewhat preferred by the LHCb data at low q^2 but is disfavored at large q^2. However, we have also identified a case for which the branching ratio increases.  In leptonic forward-backward asymmetry, the zero-crossings shift to the higher value of q^2 for most of the cases. We have however identified scenarios where it can shit to the lower value or remain SM like.

The scalars couplings are constrained together by the B̅_s→μ^+μ^- and the inclusive B̅→ X_sμ^+μ^- branching ratios. We show that in the presence of only scalar couplings, the Λ_b→Λμ^+μ^- branching ratio increases while the zero crossing of A^ℓ_ FB remain SM like. The tensor couplings however are severely constrained by the exclusive B̅→ X_sμ^+μ^- so that the effects on Λ_b→Λμ^+μ^- are negligibly small. If VA and SP couplings are simultaneously present, the A^ℓ_ FB zero-crossing remain VA like, but |A^ℓ_ FB| is reduced compared to when only VA couplings are present.






§ ACKNOWLEDGEMENTS

	The author is supported by the DST, Govt. of India under INSPIRE Faculty Fellowship.







§ SM WILSON COEFFICIENTS 

The effective Wilson coefficient 𝒞_9^ eff read


    𝒞_9^ eff(μ) = 𝒞_9^ SM(μ) + Y^ pert(q^2)   ,


where 𝒞_9^ SM(μ) indicates the value in the SM at the scale of b-quark mass, μ_b ≈𝒪(m_b)=4.8 GeV <cit.>. The contributions from one-loop matrix elements of the four-quark operators <cit.> are contained in Y^ pert(q^2) and is given by 


    Y^ pert(q^2)    =    h(m_c,q^2) ( 3𝒞_1 + 𝒞_2 + 3𝒞_3 + 𝒞_4 + 3𝒞_5 + 𝒞_6 ) 
       -   1/2 h(m_b,q^2) (4𝒞_3 + 4𝒞_4 + 3𝒞_5 + 𝒞_6)- 1/2h(0,q^2) (𝒞_3 + 3𝒞_4) 
       +   2/9 (3𝒞_3 + 𝒞_4 + 3𝒞_5 + 𝒞_6)  ,


where the loop functions read


    h(m_q, q^2)    =    -8/9lnm_q/m_b + 8/27 + 4/9x - 2/9(2+x)|1-x|^1/2{[ ln| √(1-x) + 1/√(1-x) - 1| - iπ,             x ≡4 m_c^2/ q^2 < 1,;                                                                  ;                2 arctan1/√(x-1),             x ≡4 m_c^2/ q^2 > 1, ]. 
    
    h(0,q^2)     =   8/27 - 4/9lnq^2/m_b^2  + 4/9 iπ.


The quark masses appearing in the loop functions are in the pole mass scheme. All the values of SM Wilson coefficients including _10 and _7^ eff are taken from <cit.>. 






§ POLARIZATION CONVENTIONS 

In the Λ_b rest frame, the polarization four-vectors of the virtual gauge boson that decays to dilepton pair is 


    ϵ̅^μ(±) = 1/√(2)(0, ± 1, -i, 0)  ,  ϵ̅^μ(0) = 1/√(q^2)(|q|, 0, 0, -q^0)  ,  ϵ̅^μ(t) = 1/√(q^2)(q^0, 0, 0, -|q|)  .
 

The polarization vectors satisfy the following orthonormality and completeness relations


    ϵ̅^∗μ(n) ϵ̅_μ(n^') = g_nn^'  ,  ∑_n,n^'ϵ̅^∗μ(n) ϵ̅^ν(n^') g_nn^' = g^μν  ,   n, n^' = t, ± 1, 0  ,


where g_n,n^' =  diag(+1,-1,-1,-1) and our choice of the metric tensor is g^μν=diag(1,-1, -1,-1).

In the rest frame of the ℓ^+ℓ^- system, the transversity polarizations of virtual gauge boson remain same where as the longitudinal and time like polarizations read ϵ̅^μ(0) = (0;0,0,-1) and ϵ̅^μ(t) = (1;0,0,0).




§ Λ_B →Λ HADRONIC MATRIX ELEMENTS 

The Λ_b →Λ hadronic matrix elements are conveniently written in the helicty basis <cit.>. For the vector currents we have


    ⟨Λ(k,s_k)|s̅γ^μ b |Λ(p,s_p)⟩ =   u̅(k,s_k)[f^V_t(q^2)(-)q^μ/q^2
     +    f^V_0(q^2) +/s_+{p^μ + k^μ  - q^μ/q^2( - ) }
     +     f^V_⊥(q^2) {γ^μ - 2/s_+p^μ - 2/s_+k^μ}]u(p,s_p)  ,


where the variables s_± are defined in Eq. (<ref>). For the axial-vector currents we get


    ⟨Λ(k,s_k)|s̅γ^μγ_5 b |Λ(p,s_p)⟩ =    - u̅(k,s_k) γ_5 [ f_t^A(q^2) ( + ) q^μ/q^2
     +    f_0^A(q^2)  - /s_-{p^μ + k^μ - q^μ/q^2 ( - ) }
     +     f_⊥^A(q^2) {γ^μ + 2/s_-p^μ - 2/s_-k^μ}] u(p,s_p)  .


The matrix elements for the scalar and the pseudo-scalar currents can be obtained from Eqs. (<ref>) and (<ref>) via the equations of motion


    ⟨Λ(k,s_k)|s̅ b |Λ(p,s_p)⟩ =    f^V_t(q^2) -/m_bu̅(k,s_k)u(p,s_p)  ,
    
    ⟨Λ(k,s_k)|s̅γ_5 b |Λ(p,s_p)⟩ =    f_t^A(q^2)  + /m_bu̅(k,s_k) γ_5 u(p,s_p)  ,


where we have neglected the mass of the strange quark in the denominator. For the dipole operators we get 


    ⟨Λ |s̅i q_νσ^μνb|Λ_b⟩   =    -u̅(k,s_k)[ f^T_0(q^2) q^2/s_+(p^μ + k^μ - q^μ/q^2( - ) )  
       +    f^T_⊥ (+) ( γ^μ - 2/s_+p^μ - 2/s_+k^μ) ]u(p,s_p)  ,


and 


    ⟨Λ |s̅i q_νσ^μνγ_5 b|Λ_b⟩   =    -u̅(k,s_k)γ_5 [f^T5_0 q^2/s_-( p^μ + k^μ - q^μ/q^2( - ) )  
       +    f^T5_⊥ ( - ) ( γ^μ + 2/s_-p^μ - 2/s_-k^μ) ] u(p,s_p)  .


The hadonic matrix elements for the tensor current is given by


    ⟨Λ |s̅iσ^μνb|Λ_b⟩   =   u̅(k,s_k)[ 2 f_0^T(q^2) p^μ k^ν - p^ν k^μ/s_+ + f_⊥^T(q^2) (  m_Λ_b + m_Λ/q^2(q^μγ^ν-q^νγ^μ) 
       -    2 (1/q^2 + 1/s_+) (p^μ k^ν - p^ν k^μ) ) + f_0^T5( iσ^μν - 2/s_-[m_Λ_b(k^μγ^ν - k^νγ^μ) 
       -    m_Λ (p^μγ^ν - p^νγ^μ) + p^μ k^ν - p^ν k^μ]   ) + f_⊥^T5(q^2)  m_Λ_b - m_Λ/q^2 s_-
       ×   ( (m_Λ_b^2 - m_Λ^2 - q^2 )(γ^μ p^ν - γ^ν p^μ) - ( m_Λ_b^2 - m_Λ^2 + q^2)(γ^μ k^ν - γ^ν k^μ) 
       +    2(m_Λ_b - m_Λ )(p^μ k^ν - p^ν k^μ) ) ]u(p,s_p)  .







§ SPINOR BILINEARS 

To calculate the hadronic helicity amplitudes, we give the explicit expression of the spinor matrix elements. We follow the spinor representations given in <cit.> for these calculations. For (pseudo-)scalar operators we have the following non-vanishing elements


    u̅(k,± 1/2)u(p,± 1/2) = √(s_+)  ,
       u̅(k,± 1/2)u(p,∓ 1/2) = 0  ,
    
       u̅(k,± 1/2)γ_5 u(p,± 1/2) = ∓√(s_-)  ,
       u̅(k,± 1/2)γ_5 u(p,∓ 1/2) = 0  ,


and for (axial-)vector operators we get


    u̅(k,± 1/2) γ^μ u(p,± 1/2) = (√(s_+), 0, 0, √(s_-))  ,
       u̅(k,± 1/2) γ^μ u(p,∓ 1/2) = √(2 s_-)ϵ^μ(±)  ,
       u̅(k,± 1/2) γ^μγ_5 u(p,± 1/2) = ±(√(s_-), 0, 0, √(s_+))  ,
       u̅(k,± 1/2) γ^μγ_5 u(p,∓ 1/2) = ±√(2s_+)ϵ^μ(±)  .


The non-vanishing elements of the tensor operators are


    u̅(k,±1/2) σ^14
    σ^23 u(p,±1/2)  =  -i√(s_-)
    ±√(s_+)  ,
    
       u̅(k,± 1/2) σ^12
    σ^13
    σ^24
    σ^34 u(p,∓ 1/2)  = ∓ i√(s_-)
     -√(s_-)
    ± i √(s_+)
    √(s_+)  ,


where u̅(k,s_k)σ^μνu(p,s_p) = - u̅(k,s_k)σ^νμu(p,s_p). 








§ SPINORS IN DILEPTON REST FRAME 

To calculate the lepton helicity amplitudes, we work in the dilepton rest frame. The angle θ_ℓ is defined as the angle made the negatively charged lepton ℓ^- with the Λ_b in the dilepton rest frame. The four momentum of the charged leptons ℓ^- and ℓ^+ (here we use the subscript 2ℓ to denote this frame) respectively are



    q_-^μ|_2ℓ = (E_ℓ, |q_2ℓ|sinθ_ℓ, 0, |q_2ℓ|cosθ_ℓ)  ,
    
       q_+^μ|_2ℓ = (E_ℓ, -|q_2ℓ|sinθ_ℓ, 0, -|q_2ℓ|cosθ_ℓ)  ,


with 


    |q_2l| = β_ℓ/2√(q^2)  ,     E_ℓ = √(q^2)/2  ,    β_ℓ = √(1-4m_ℓ^2/q^2)  .



The explicit expressions of the amplitudes in Eqs. (<ref>)-(<ref>) are obtained by contractions of the currents


    u̅_ℓ^-{1,γ_μ, σ_μν} (1∓γ_5) v_ℓ^+  ,


with 1, ϵ̅^μ(λ), -iϵ̅^μ(λ)ϵ̅^ν(λ^') respectively.

Following <cit.> we give the explicit expressions of the spinors in this frame are


    u_ℓ^-(λ) = 
    [    √(E_ℓ+m_ℓ)χ^u_λ; 2 λ√(E_ℓ-m_ℓ)χ^u_λ ]  ,
    
      χ^u_+1/2 = [ cosθ_ℓ/2; sinθ_ℓ/2 ]  ,
    
    
      χ^u_-1/2 = [ -sinθ_ℓ/2;  cosθ_ℓ/2 ]  ,
    
    
        v_ℓ^+(λ) = 
    [     √(E_ℓ-m_ℓ)χ^v_-λ; -2 λ√(E_ℓ+m_ℓ)χ^v_-λ ]  ,
    
      χ^v_+1/2 = [  sinθ_ℓ/2; -cosθ_ℓ/2 ]  ,
    
    
      χ^v_-1/2 = [ cosθ_ℓ/2; sinθ_ℓ/2 ]  .






§ NUMERICAL INPUTS 


In the following table we collect the numerical values of the inputs used in the paper.

cr|cr
	inputs      values      inputs      values   
 
	α_e(m_b)      1/127.925(16) <cit.>     |V_tbV_ts^∗|      0.0401 ± 0.0010 <cit.>  
 
	m_c(MS)     1.28 GeV <cit.>          5.619 GeV <cit.> 
 
	μ_b     4.8 GeV <cit.>          1.115 GeV <cit.> 

	m_b(MS)     4.2 GeV <cit.>     τ_Λ_b     (1.470± 0.010)× 10^-12 s <cit.> 
  
	m_b( pole)     4.8 GeV <cit.>      m_B^0     5.279 GeV <cit.> 
 
	α_s(m_b)     0.214 <cit.>      m_K     0.494 GeV <cit.> 
 
	|V_cb|     (4.15 ± 0.07)× 10^-2 <cit.>        

	List of inputs and their values. 




99
	
	
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