$\Lambda_b\rightarrow \Lambda(\to p \pi^-) \ell^+\ell^-$ as probe of CP-violating New Physics

We investigate the possible sizes of all the CP-violating asymmetries offered by the angular distribution of rare decay $\Lambda_b\rightarrow \Lambda(\to p \pi^-) \ell^+\ell^-$ in the Standard Model and new physics scenarios motivated by the recent $b\to s \ell^+\ell^-$ anomalies. We work in a model-independent effective theory framework and discuss the sensitivity of CP asymmetries to new ${O}_{9,10}$ operators and their chirality flipped counterparts. We find that the size of many of the CP asymmetries can be at the level of a few percent in new physics scenarios consistent with current $b\to s\ell^+\ell^-$ data at a level of $1\sigma$. We emphasize that measurements of these CP asymmetries can be used to discriminate different new physics scenarios in $b\to s \ell^+\ell^-$.


Introduction
A major motivation for going beyond the Standard Model (SM) is to find new sources of charge-parity violation (CPV) required for the explanation of baryonic asymmetry of the Universe (BAU).The SM possesses two CPV sources: the Cabibbo-Kobayashi-Maskawa (CKM) phase (related to weak interactions), and a strong CP phase that is severely constrained by the upper limit of the neutron electric dipole moment measurement [1].So far, all experimental observations of CPV have been in the quark sector and are consistent with the CKM mechanism.However, it is well known that the SM fails to satisfy the Sakharov's conditions [2] needed for explaining the BAU.Therefore, investigations of experimentally accessible CP-violating observables offering excellent sensitivity to physics beyond the SM are highly motivated.
At the luminosity frontier of new physics (NP) searches, the physical processes with underlying quark current b → s + − transitions have been of particular interest in recent times.The measurements of several observables related to processes B → K ( * ) µ + µ − [3][4][5][6], B s → φµ + µ − [7,8], show deviations from the SM expectation.On the other hand, the recently reported measurements of lepton flavor universality (LFU) ratios R K ( * ) = B(B → K ( * ) µ + µ − )/B(B → K ( * ) e + e − ) [9,10] by the LHCb [11,12], which are updates of previous measurements [13][14][15][16], agree with the SM.However, the latest global likelihood analyses of b → s + − that include the latest R K ( * ) measurements still show large preference for the NP hypothesis over the SM [17][18][19] #1 (also see Ref. [36], which discusses impact of long-distance contributions associated with charm loops).Although these results indicate the presence of a lepton flavor universal NP, it is also worth mentioning that there may still be sufficient room for LFU violation if the NP is associated with CP violation [37].These anomalies, collectively known as neutral-current B-anomalies, will be tested rigorously in the upcoming measurements with more data, and, if confirmed, would be indisputable evidence of NP in b → s + − transitions.
The aforementioned anomalies, if confirmed, say nothing about the CP nature of the underlying NP as the concerned b → s + − observables are CP averaged.However, b → s + − decays also offer a multitude of observables which are highly sensitive to the CP nature of NP, aided by the fact that in the SM the b → s + − transitions are doubly Cabibbo suppressed [38].Combined measurements of CP-violating and CP-conserving observables therefore provide a more powerful method to understand the CP properties of NP.In the literature, several works have investigated the CP asymmetries in B → K * µ + µ − angular distributions to probe NP [17,[38][39][40][41][42][43][44][45], and, as illustrated in Ref. [43], the CP asymmetries are capable of distinguishing between NP models that address the b → s + − anomalies.
If the observed deviation in the b → s + − is indeed due to unambiguous shortdistance NP then it would, in principle, also affect all semileptonic processes with the same underlying current.One important example is the baryonic decay Λ b → Λ(→ pπ − ) + − for an unpolarized Λ b , which is the topic of this paper.There are several benefits of studying this decay.The angular distribution of Λ b → Λ(→ pπ − ) + − , similar to its mesonic counterpart B → K * (→ Kπ) + − , offers a large number of CPconserving as well as CP-violating angular asymmetries that provide complementary information about NP in b → s + − [46].The secondary decay K * → Kπ in mesonic mode B → K * (→ Kπ) + − is a strong decay and therefore it conserves parity.On the other hand, the decay Λ → pπ − in the baryonic mode is a parity violating weak decay; this characteristic will play a key role in constructing several CP-violating asymmetries in Λ b → Λ(→ pπ − ) + − , as discussed later in this paper.Furthermore, as pointed out in Ref. [47], Λ b → Λ form factors, in comparison to B → K * , are more suitable to be computed with higher precision using lattice QCD due to the stability of Λ under strong interactions.
The first observation of Λ b → Λ(→ pπ − ) + − was reported by the CDF [48].The recent angular analysis of unpolarized #2 Λ b → Λ(→ pπ − ) + − by the LHCb indicates a branching ratio that is smaller than the SM expectation [52], a pattern also observed in the B → K ( * ) µ + µ − , B s → φµ + µ − modes.There are extensive theoretical works on the model-independent study of Λ b → Λ(→ pπ − ) + − in the SM and beyond [46,47,50,.These works mostly focus on CP-conserving angular observables.To the best of our knowledge, angular observables which discriminate the decay with its CP-conjugated mode Λb → Λ(→ pπ + ) + − and their role in the probe of NP in b → s + − have not been discussed yet.This paper investigates the prospects of CP asymmetries associated with the baryonic mode and assesses their sensitivity to CP-violating NP.We give description of all possible CP-violating asymmetries that are at disposal from Λ b → Λ(→ pπ − ) + − angular distribution.Focusing on the muonic mode, we identify the sensitivity of these asymmetries to various NP WCs and show that the measurement of CP asymmetries in Λ b → Λ(→ pπ − )µ + µ − can be used to distinguish various NP solutions that are favored by current global b → sµ + µ − data.
The paper is organized as follows.We begin the next section with a description of the model-independent effective framework relevant for the study of b → s + − transitions.In section 3 we define the full angular distribution of both Λ b → Λ(→ pπ − )µ + µ − and its CP-conjugated mode, and discuss the subtleties the CP properties of the secondary decay of Λ ( Λ) particles bring in defining the corresponding angular coefficients.In section 4 we define all the CP-violating asymmetries of the Λ b decay angular distribution and present our main numerical results.Finally, we offer our conclusions in section 5.

b → s + − effective Lagrangian
In the SM, the b → s + − decays arise at the loop level only.The Lagrangian relevant at the scale µ = m b is given by [41], where G F is the Fermi's constant, and , where V ij denotes the CKM matrix element.The WCs C i contain information about short-distance physics associated with local operators O i .For the discussion of the b → s + − transition, O 7,9,10 are the dominant operators which we list below.Denoting the chiral projectors as P L,R = (1 ∓ γ 5 )/2, one has, Their explicit form can be found, for example, in Refs.[76][77][78].These operators contribute to the b → s + − through quark loops.It is customary to include their contribution in the effective WCs C eff 7(9) of operators O 7 (9) .These effective coefficients at next-to-next-to-leading logarithmic are given as, The functions h(a, b) and F 7,9 8 (q 2 ) are given in Ref. [79], and the functions F 7,9 i,c (q 2 ) (i = 1, 2) are provided in Refs.[80,81].Note that the quark masses appearing in Eqs. ( 4) and ( 5) are in the pole scheme, and the corresponding values (m pole b = 4.74174 GeV, m pole c = 1.5953GeV) are taken from Ref. [47].The numerical values of SM Wilson coefficients contributing to b → s + − are given in Table 1.
There are (sb)( ¯ ) operators with scalar and tensor structures which may arise in NP; but scalar operators are highly constrained from B s → µ + µ − data [82,83].Furthermore, since the current global fits to b → sµ + µ − data strongly prefer NP in WCs of left-handed (axial)vector operators in Eq. ( 3), we will also neglect NP tensor operators for the simplicity of the analysis.

Angular Distribution
Assuming the Λ b to be unpolarized, the 4-fold differential angular distribution for Λ b → Λ(→ pπ − ) + − is given by [46], The distribution is completely described by four variables: the invariant lepton mass squared (q 2 ) and three Euler angles, θ , θ Λ , and φ.In the rest frame of Λ b , the daughter baryon is assumed to travel along the +z axis.The θ Λ is the angle made by the proton with the +z axis in the rest frame of the Λ, θ is the angle made by the − with respect to the +z axis in the rest frame of the lepton pair, and φ defines angle in the rest frame of Λ b between planes containing pπ − and the lepton pair.Denoting mass of Λ b , Λ, and charged lepton as m Λ b , m Λ , and m , respectively, the physical region of the decay process is defined by the following values: In Eq. ( 6), the angular coefficients K i are functions of q 2 .These are conveniently described in terms of Λ b → Λ transversity amplitudes, A L,R i (q 2 ) as follows, where the q 2 dependence of the transversity amplitudes is implied.Since we will be focusing on the muonic mode only, we have ignored the lepton mass in writing the coefficients K i above.The expressions for K i , including lepton mass effects, can be found in Ref. [71].The transversity amplitudes A L,R i (q 2 ) in terms of the Λ b → Λ form factors f i (q 2 ) (see section 4) and b → s + − Wilson coefficients are given by [71] A L,(R) A L,(R) A L,(R) where and the normalization constant N , a function of q 2 , is given by, In the expressions of K i , the parity violating decay parameter α Λ arises through the secondary decay Λ → pπ − .The corresponding hadronic matrix element is given by [46], which depends on only two parameters, ξ and ω.These can be determined from experimental data on Λ → pπ − .The decay parameter α Λ then is given by [46], where r ± = (m Λ ± m p ) 2 − m 2 π .To write down the corresponding distribution for CP-conjugated mode Λb → Λ(→ pπ + ) + − , we take the following definition for Euler angles: In the rest frame of the Λb , the Λ is assumed to travel along the +z axis, and θ Λ is the angle between Λ and the antiproton in the (pπ + ) rest frame.The lepton angle θ is the angle between Λ and −#3 in the dilepton rest frame, and φ is the angle between planes of (pπ + ) and the lepton pair.With the above convention, the decay distribution for Λb → Λ(→ pπ + ) + − is simply obtained from Eq. ( 6) after the transformation (θ Λ → θ Λ , θ = θ − π, and φ → −φ).This is equivalent to replacing the functions K i 's in Eq. ( 6) with Ki 's in the following way: K 1cc,1ss,2cc,2ss,4sc,3s → + K1cc,1ss,2cc,2ss,4sc,3s , where Ki equals K i except for the weak phases conjugated.Additionally, in all but three coefficients, K 1ss , K 1cc , and K 1c , we replace the Λ → pπ − decay parameter α Λ by ᾱΛ , which corresponds to the CP-conjugated decay Λ → pπ + .This replacement follows from the fact that under a CP transformation the Dirac-field bilinears transform as ψ1 ψ 2 Therefore, in the expression of Λ → pπ + , the hadronic matrix element obtained through a CP transformation of equation (24), the ξ term picks up a minus sign but the ω term does not.The definition of ᾱΛ , defined similarly to Eq. ( 25), then implies ᾱΛ = −α Λ under strict CP-symmetry.Experimentally measured values α Λ = 0.7519 ± 0.0036 ± 0.0024 and ᾱΛ = −0.7559± 0.0036 ± 0.0030 by BESIII collaboration [84] agree well with theory.

CP-Violating Asymmetries and Results
Let Γ(H → f ) be the decay rate of H → f (where H indicates the initial state Λ b and f indicates the final state) and Γ( H → f ) be the decay rate of the CP-conjugate mode H → f .As one needs two interfering decay amplitudes to observe a direct CPV, we write the amplitudes as A #3 Note that direction of − is taken as reference for both decay and CP-conjugated mode.This convention is similar to the one used for mesonic counterpart decay in Ref. [41].
where φ w 1,2 are the weak phases, φ s 1,2 are the strong phases (arising due to final state interactions), and |A 1,2 | are the moduli of the interfering matrix elements.The decay rate asymmetry that signals the presence of CPV is As evident from this expression, a nonvanishing CP asymmetry requires that both the relative strong and weak phases of the amplitudes must be nonvanishing.The SM has a finite strong phase emanating from the imaginary part of C eff 9 , which is generated by the qq loops (q = c, u) in the current-current operators.However, the weak phase, coming from the CKM elements in the last term of Eq. ( 5), is doubly Cabibbo suppressed and small.Therefore, the CPV in b → s + − transitions in the SM is expected to be very small.
As discussed in the Introduction, the measurements of several observables associated with the b → s + − current are in tension with the SM, and the global fits to data show large preference to NP hypothesis over the SM.Except for the direct CP asymmetry A K ( * ) CP [85] and a few angular CP-asymmetries in B → K * µ + µ − [5] and B s → φµ + µ − [7,8], most of the measured b → s + − observables, including the ones that show the tensions with the SM, are CP averaged and therefore are not sensitive to the complex phases of NP.Therefore whether the NP in question is real or complex is not clear at present.To answer this question, one needs to study CP-violating observable in b → s + − transition.To this purpose, we construct several CP-violating observables in Λ b → Λ + − decay and investigate their sensitivity to complex NP Wilson coefficients.We follow the results of Ref. [17] which performed a global fit analysis of the b → sµ + µ − data to complex Wilson coefficients assuming lepton flavor universal couplings to electrons and muons.In particular, we consider the following two LFU NP scenarios The Case I and Case II can significantly improve the theory description of the data.The Case III, although cannot explain tensions in b → s data (see Ref. [86]), we #4 Since Ref. [17] provided only 1σ range of WC values, in choosing NP benchmark for our analysis, we take median for the real part but the maximum of the 1σ range for the imaginary part to have maximum CP violation effects.
include it in our analysis to assess the sensitivity of CP asymmetries to right-handed NP #5 .
In order to make numerical predictions of observables, one also needs Λ b → Λ form factors for which we use lattice QCD results of Ref. [47].The Λ b → Λ hadronic matrix elements are parametrized in terms of ten form factors #6 f V t,0,⊥ , f A t,0,⊥ , f T 0,⊥ , f T 5 0,⊥ , which are function of q 2 .The lattice calculations are fitted to two z-parametrizations: the "nominal" fit and "higher-order" fit.Defining the parameter z(q 2 , t + ) as, where , in the so called "nominal" fit the form factor parametrization is given as, while in "higher-order" fit, the parametrization is given as, where values of the coefficients a f i and the correlations among them are taken from Ref. [47].
In our numerical analysis, apart from already discussed form factors and the Wilson coefficients, we use results of CKMfitter Group [87] for values of CKM elements, while particle masses and their lifetime values are taken from Particle Data Group [88].
With the numerical inputs at our disposal, we make bin-wise predictions of different observables that we describe in the next section.To be precise, the prediction of an observable O(q 2 ) in a given bin For an observable involving a ratio of two quantities, the binned prediction is obtained after integrating the numerator and denominator separately and then taking their ratio.The two regions of q 2 where we make the predictions are q 2 ∈ [0.1, 6] and 15,20].To avoid the charmonium resonances, we refrain from making any prediction in the 6 − 15 GeV 2 range.Our numerical determinations are subject to uncertainties coming from different inputs including the form factors. Regarding form factor uncertainties, as described #5 Combination C 9 = −C 9 involving both left-and -right current NP is also favored [17] by the current data.#6 Note that the form factors labels used in Ref. [47] are different from labels we use in this paper.These are related as f V t,0,⊥ = f 0,+,⊥ , f A t,0,⊥ = g 0,+,⊥ , f T 0,⊥ = h +,⊥ and f T 5 0,⊥ = h+,⊥ .
in Ref. [47], we use the "nominal" fit of Eq. ( 32) and for an estimation of systematic uncertainties "higher-order" fit of Eq. ( 33) is used.The total uncertainty is obtained after adding statistical and systematic uncertainties in quadrature.In the appendix A we have collected bin-wise predictions of all observables that we describe next.

CP asymmetry in decay rate
The CP asymmetry in decay rate is defined as, where dΓ/dq 2 and d Γ/dq 2 are decay rates of the decay Λ b → Λ(→ pπ − )µ + µ − and its CP-conjugate decay respectively, which in terms of angular coefficients are defined as, Figure 1: Predictions for direct CP asymmetry in decay rate (A CP ) of Λ b → Λµ + µ − .In the left plot, the theoretical uncertainties are shown only for the SM case as dark grey band.In the right plot, the width of rectangle boxes denotes q 2 -bin size, and the height of the boxes shows prediction of the observable together with corresponding (1σ) uncertainties.The same style is used in the rest of figures of this paper.
In Fig. 1, we show the A CP as function of q 2 (left plot) in the SM and NP scenarios.The corresponding binned predictions are shown in the plot to the right.We find that in the SM the A CP is, as expected, very small and is ∼ O(10 −3 ).In the NP case, we note the following: • A CP is sensitive to left-handed NP at large q 2 as seen from Fig. 1 where NP cases I and II show a large deviation from the SM.We find that in case II, which has WCs with larger imaginary part, A CP can be up to 5%.
• We find that A CP is not sensitive to right-handed currents (NP case III).
• At low q 2 , we find A CP is not sensitive to any of NP scenarios.

CP asymmetry in longitudinal polarization
We define the CP asymmetry in longitudinal polarization fraction as the difference between longitudinal polarization of Λ b → Λ(→ pπ − )µ + µ − (denoted as F L (q 2 )) and its CP-conjugate decay (denoted as FL (q 2 )), normalized by the sum of corresponding decay rates, where F L (q 2 ) and FL (q 2 ) in terms of angular coefficients are given as, Figure 2: Predictions for CP asymmetry in longitudinal polarization of Λ b → Λµ + µ − .
Since angular coefficients appearing in A f L are the same as in A CP , one expects A f L to be sensitive to NP.In Fig. 2, we show results for asymmetry A f L , and indeed we find that observable is sensitive to left-handed NP (NP cases I and II) while very weakly sensitive to right-handed NP (NP case III).The NP effects are higher in the large q 2 region and A f L can be ∼ 2%.In the low q 2 region, A f L remains well below 1%.

CP asymmetry in forward-backward asymmetries
The decay Λ b → Λ(→ pπ − )µ + µ − offers three types of forward-backward (FB) asymmetries [46]: FB asymmetry (a FB ) with respect to leptonic angle θ , FB asymmetry (a Λ FB ) with respect to hadronic angle θ Λ , and FB asymmetry (a Λ FB ) with respect to the combination of θ and θ Λ , respectively.In terms of angular coefficients, these are given as, One can then take the difference between the measurement of the FB asymmetry in the decay Λ b → Λ(→ pπ − )µ + µ − and its CP-conjugate to define new CP asymmetries.These CP asymmetries can be accessed by measuring the CP asymmetries associated with angular coefficients K 1c , K 2c , K 2ss , and K 2cc .To this end, we define the following four CP asymmetries: A j = K j (q 2 ) + Kj (q 2 ) dΓ/dq 2 + d Γ/dq 2 , for j = 2c, 2ss, 2cc.
A 1c and A 2c are equivalent to CP asymmetries in a FB and a Λ FB up to a normalization constant, while CP asymmetry in a Λ FB can be determined from combined measurements of A 2ss and A 2cc .Also, note that A 1c involves the difference of K 1c and K1c , while the other CP asymmetries involves the sum of corresponding coefficients.This is because the angular coefficients in Eqs. ( 8)-( 17) are proportional to the decay parameter α Λ (or ᾱΛ in the case of CP-conjugate mode).As discussed in the previous section, α Λ −ᾱ Λ experimentally, therefore it is the combination K j + Kj (j = 2c, 2ss, 2cc, 3s, 3sc, 4s, 4sc) that vanishes in the case of purely real WCs.
In Fig. 3, we show results for A 1c , A 2c , A 2cc , and A 2ss .We note the following: • CP asymmetry A 1c is sensitive to C 10 , while no sensitivity to vector and righthanded current is found.We find that in NP case II, which has nonzero C NP 10 , the value of A 1c can be at ∼ 1.3% level in the large q 2 region.In other NP cases considered, A 1c remain indistinguishable from the SM.
• For the asymmetry A 2c , we observe similar NP sensitivity as in A 1c except that the sign of the asymmetry is opposite to that of A 1c .We also note that A 2c is largest (∼ 1.3%) at the kinematic end point q 2 ∼ 20 GeV 2 .
• In contrast, the asymmetries A 2ss and A 2ss are sensitive to C 9 , while no sensitivity to axial vector and right-handed currents is found.Both asymmetries are found to be ∼ 1% in large q 2 region, with A 2ss being somewhat slightly larger.
In Fig. 4 we show the results for these CP asymmetries and make the following observations: • We find A 3s to be sensitive to right-handed currents (NP case III) in both low and large q 2 regions, where it can be up to ∼ 2%.It is also mildly sensitive to C 10 in the low q 2 region, but has large theoretical uncertainties, as can be seen from low q 2 bins.Another point worth mentioning is that there is zero-crossing in low q 2 region of A 3s for right-handed NP scenario (case III), while no such behavior is seen in left-handed NP scenarios.
• In case of A 3sc , we find the CP asymmetry to be sensitive to C 9 as well as righthanded NP case, with its size being ∼ 1 − 3% in low q 2 region.At large q 2 , CP asymmetry remains negligible.
• For A 4s , we find that the asymmetry is sensitive to C 10 at large q 2 .The CP asymmetry remains SM-like in cases with NP in C 9 or right-handed NP.In NP case II (which has nonzero C 10 ), A 4s can be ∼ 1%.
• On the other hand, for A 4sc we find that the observable is most sensitive to C 9 (NP cases I and II), but the asymmetry remains very small, at O(10 −3 ).

Conclusion
The experimental data on the b → s + − transitions hints towards the presence of NP that is lepton flavor universal in nature.However, the CP properties of the possible underlying NP is unknown.One concrete way to answer this question is provided by the measurements of CP-violating asymmetries associated with the b → s + − transitions.In the SM, the CPV in the b → s + − transition is very small, but can be enhanced in many NP models.Thus, measurements of sizable b → s + − CP asymmetries are highly motivated.With this in mind, in this paper, we have investigated in a model-independent fashion the prospects of probing CP-violating NP in Λ b → Λ(→ pπ − )µ + µ − decay.To this end, we list all the CP asymmetries offered by the angular distribution of an unpolarized Λ b decay Λ b → Λ(→ pπ − )µ + µ − .We then present their determinations in the SM and several NP scenarios motivated by the present global fits to b → s + − data.We find that several of the CP asymmetries, depending on their NP sensitivity and q 2 region, can be enhanced to a few percent level.More importantly, we find that the measurements of these CP asymmetries can provide new methods to not only probe but also potentially distinguish NP cases discussed in the paper.Therefore, the CP asymmetries in Λ b → Λ(→ pπ − )µ + µ − provide new avenues to cross-check the SM and, in conjunction with CP asymmetries of B → K * + − , can play a useful role in searching NP in b → s + − transitions.

C 7 (
m b ) −0.3, and C 9 (m b ) ≈ −C 10 (m b ) 4.2 in the SM.Note that the primed operators are the chirality flipped counterparts of the SM operators, and in the SM their contributions (C i ) are vanishing.The operators O 1−6 are 4-quark operators related to decays b → sqq, and O 8 is dipole operator related to radiative decay b → sγ.