Exclusive semileptonic decays of \Lambda_{b}\to\Lambda l^{+}l^{-} in supersymmetric theories

Exclusive semileptonic decays of Λb→Λl+l− in supersymmetric theories

Abstract

The weak decays of () are investigated in Minimal Supersymmetric Standard Model (MSSM) and also in Supersymmetric (SUSY) SO(10) Grand Unified Models. In MSSM the special attention is paid to the Neutral Higgs Bosons (NHBs) as they make quite a large contribution in exclusive decays at large regions of parameter space of SUSY models, since part of SUSY contributions is proportional to . The analysis of decay rate, forward-backward asymmetries, lepton polarization asymmetries and the polarization asymmetries of baryon in show that the values of these physical observables are greatly modified by the effects of NHBs. In SUSY SO(10) GUT model, the new physics contribution comes from the operators which are induced by the NHBs penguins and also from the operators having chirality opposite to that of the corresponding SM operators. SUSY SO(10) effects show up only in the decay where the longitudinal and transverse lepton polarization asymmetries are deviate significantly from the SM value while the effects in the decay rate, forward-backward asymmetries and polarization asymmetries of final state baryon are very mild. The transverse lepton polarization asymmetry in is almost zero in SM and in MSSM model. However, it can reach to in SUSY SO(10) GUT model and could be seen at the future colliders; hence this asymmetry observable will provide us useful information to probe new physics and discriminate between different models.

pacs:
13.30.Ce, 14.20.Mr, 11.30.Pb

I Introduction

From last decade, rare decays induced by flavor changing neutral currents (FCNCs) have become the main focus of the studies due to the CLEO measurement of the radiative decay (1). In the standard model (SM) these decays are forbidden at tree level and can only be induced by Glashow-Iliopoulos-Maiani mechanism (2) via loop diagrams. Hence, such decays will provide helpful information about the parameters of Cabbibo-Kobayashi-Maskawa (CKM) matrix (3); (4) elements as well as various hadronic form factors. In the literature there have been intensive studies on the exclusive decays (5); (6); (7); (8); (9); (10); (11) both in the SM and beyond, where the notions and denote the pseudoscalar, vector and axial vector mesons respectively.

It is generally believed that supersymmetry (SUSY) is not only one of the strongest competitor of the SM but is also the most promising candidate of new physics. The reason is that it offers a unique scheme to embed the SM in a more fundamental theory where many theoretical problems such as gauge hierarchy, origin of mass, and Yukawa couplings can be resolved. One direct way to search for SUSY is to discover SUSY particles at high energy colliders, but unfortunately, so far no SUSY particles have been found. Another way is to search for its effects through indirect methods. The measurement of invariant mass spectrum, forward-backward asymmetry and polarization asymmetries are the suitable tools to probe new physics effects. For most of the SUSY models, the SUSY contributions to an observable appear at loop level due to the -parity conservation. Therefore, it has been realized for a long time that rare processes can be used as a good probe for the searches of SUSY, since in these processes the contributions of SUSY and SM arises at the same order in perturbation theory (12).

Motivated from the fact that in two Higgs doublet model and in other SUSY models, Neutral Higgs Bosons (NHBs) could contribute largely to the inclusive processes , as part of supersymmetric contributions is proportional to the (13). Subsequently, the physical observables, like branching ratio and forward-backward asymmetry, in the large region of parameter space in SUSY models can be quite different from that in the SM. In addition, similar effects in exclusive decay modes are also investigated (12), where the analysis of decay rates, forward-backward asymmetries and polarization asymmetries of final state lepton indicates the significant role of NHBs. It is believed that physics beyond the SM is essential to explain the problem of neutrino oscillation. To this purpose, a number of SUSY SO(10) models have been proposed in the literature (14); (15); (16); (17). One such model is the SUSY SO(10) Grand Unified Models (GUT), in which there is a complex flavor non-diagonal down-type squark mass matrix element of 2nd and 3rd generations of order one at the GUT scale (16). This can induce large flavor off-diagonal coupling such as the coupling of gluino to the quark and squark which belong to different generations. These couplings are in general complex and may contribute to the process of flavor changing neutral currents (FCNCs). The above analysis of physical observables in decay is extended in SUSY SO(10) GUT model in Ref. (18). It is believed that the effects of the counterparts of usual chromo-magnetic and electromagnetic dipole moment operators as well as semileptonic operators with opposite chirality are suppressed by in the SM, but in SUSY SO(10) GUTs their effect can be significant, since can be as large as 0.5 (18); (16). Apart from this, can induce new operators as the counterparts of usual scalar operators in SUSY models due to NHB penguins with gluino-down type squark propagator in the loop. It has been shown (18) that the forward-backward asymmetries as well as the longitudinal and transverse decay widths of decay, are sensitive to these NHBs effect in SUSY SO(10) GUT model which can be detected in the future factories.

Compared to the meson decays, the investigations of FCNC transition for bottom baryon decays are much behind because more degrees of freedom are involved in the bound state of baryon system at the quark level. From the experimental point of view, the only drawback of bottom baryon decays is that the production rate of baryon in quark hadronization is about four times less than that of the meson. Theoretically, the major interest in baryonic decays can be attributed to the fact that they can offer a unique ground to extract the helicity structure of the effective Hamiltonian for transition in the SM and beyond, which is lost in the hadronization of mesonic case. The key issue in the study of exclusive baryonic decays is to properly evaluate the hadronic matrix elements for , namely the transition form factors which are obviously governed by non-perturbative QCD dynamics. Currently, there has been some studies in the literature on transition form factors in different models including pole model (PM) (19), covariant oscillator quark model (COQM)(20), MIT bag model (BM)(21) and non-relativistic quark model (22), QCD sum rule approach (QCDSR) (23), perturbative QCD (pQCD) approach (24) and also in the light-cone sum rules approach (LCSR) (25). Using these form factors, the physical observables like decay rates, forward-backward asymmetries and polarization asymmetries of baryon as well as of the final state leptons in were studied in great details in the literature (26); (27); (28); (29); (30); (31); (32); (33). It is pointed out that these observables are very sensitive to the new physics, for instance, the polarization asymmetries of baryon in decays heavily depend on the right handed current, which is much suppressed in the SM (29).

In this paper, we will investigate the exclusive decay ( = , ) both in the Minimal Supersymmetric Standard Model (MSSM) as well as in the SUSY SO(10) GUT model (16). We evaluate the branching ratios, forward-backward asymmetries, lepton polarization asymmetries and polarization asymmetries of baryon with special emphasis on the effects of NHBs in MSSM. It is pointed out that different source of the vector current could manifest themselves in different regions of phase space. For low value of momentum transfer, the photonic penguin dominates, while the penguin and box become important towards high value of momentum transfer (12). In order to search the region of momentum transfer with large contributions from NHBs, the above decay in certain large region of parameter space has been analyzed in SuperGravity (SUGRA) and M-theory inspired models (34). We extend this analysis to the SUSY SO(10) GUT model (12), where there are some primed counterparts of the usual SM operators. For instance, the counterparts of usual operators in decay are suppressed by and consequently negligible in the SM because they have opposite chiralities. These operators are also suppressed in Minimal Flavor Violating (MFV) models (35); (36), however, in SUSY SO(10) GUT model their effects can be significant. The reason is that the flavor non-diagonal squark mass matrix elements are the free parameters and some of them have significant effects in rare decays of mesons (37). In our numerical analysis for decays, we shall use the results of the form factors calculated by LCSR approach in Ref. (25), and the values of the relevant Wilson coefficient for MSSM and SUSY SO(10) GUT models are borrowed from Ref. (12); (18). The effects of SUSY contributions to the decay rate and zero position of forward-backward asymmetry are also explored in this work. Our results show that not only the decay rates are sensitive to the NHBs contribution but the zero point of the forward-backward asymmetry also shifts remarkably. It is known that the hadronic uncertainties associated with the form factors and other input parameters have negligible effects on the lepton polarization asymmetries and polarization asymmetries of baryon in decays. We have also studied these asymmetries in the SUSY models mentioned above and found that the effects of NHBs are quite significant in some regions of parameter space of SUSY.

The paper is organized as follows. In Sec. II, we present the effective Hamiltonian for the dilepton decay . Section III contains the definitions and numbers of the form factors for the said decay using the LCSR approach. In Sec. IV we present the basic formulas of physical observables like decays rate, forward-backward asymmetries (FBAs) and polarization asymmetries of lepton and that of the baryon in . Section V is devoted to the numerical analysis of these observables and the brief summary and concluding remarks are given in Sec. VI.

Ii Effective Hamiltonian

After integrating out the heavy degrees of freedom in the full theory, the general effective Hamiltonian for in SUSY SO(10) GUT model, can be written as (18)

 Heff = −GF2√2VtbV∗ts[2∑i=1Ci(μ)Oi(μ)+10∑i=3(Ci(μ)Oi(μ)+C′i(μ)O′i(μ)) (1) +8∑i=1(CQi(μ)Qi(μ)+C′Qi(μ)Q′i(μ))],

where are the four-quark operators and are the corresponding Wilson coefficients at the energy scale (38). Using renormalization group equations to resume the QCD corrections, Wilson coefficients are evaluated at the energy scale . The theoretical uncertainties associated with the renormalization scale can be substantially reduced when the next-to-leading-logarithm corrections are included (39). The new operators come from the NHBs exchange diagrams, whose manifest forms and corresponding Wilson coefficients can be found in (40); (41). The primed operators are the counterparts of the unprimed operators, which can be obtained by flipping the chiralities in the corresponding unprimed operators. It needs to point out that these primed operators will appear only in SUSY SO(10) GUT model and are absent in SM and MSSM (12).

The explicit expressions of the operators responsible for transition are given by

 O7 = e216π2mb(¯sσμνPRb)Fμν,O′7=e216π2mb(¯sσμνPLb)Fμν O9 = e216π2(¯sγμPLb)(¯lγμl),   O′9=e216π2(¯sγμPRb)(¯lγμl) O10 = e216π2(¯sγμPLb)(¯lγμγ5l),      O′10=e216π2(¯sγμPRb)(¯lγμγ5l) Q1 = e216π2(¯sPRb)(¯ll),     Q′1=e216π2(¯sPLb)(¯ll) Q2 = e216π2(¯sPRb)(¯lγ5l),        Q′2=e216π2(¯sPLb)(¯lγ5l) (2)

with . In terms of the above Hamiltonian, the free quark decay amplitude for can be derived as (13):

 M(b → sl+l−)=−GFα√2πVtbV∗ts{Ceff9(¯sγμPLb)(¯lγμl)+C10(¯sγμPLb)(¯lγμγ5l) −2mbCeff7(¯siσμνqνsPRb)(¯lγμl)+CQ1(¯sPRb)(¯ll)+CQ2(¯sPRb)(¯lγ5l)+(Ci(mb)⟷C′i(mb))}

where and is the momentum transfer. Due to the absence of boson in the effective theory, the operator can not be induced by the insertion of four-quark operators. Therefore, the Wilson coefficient does not renormalize under QCD corrections and hence it is independent on the energy scale. Moreover, the above quark level decay amplitude can receive additional contributions from the matrix element of four-quark operators, , which are usually absorbed into the effective Wilson coefficient . To be more specific, we can decompose into the following three parts (42); (43); (44); (45); (46); (47); (48)

 Ceff9(μ)=C9(μ)+YSD(z,s′)+YLD(z,s′),

where the parameters and are defined as . describes the short-distance contributions from four-quark operators far away from the resonance regions, which can be calculated reliably in the perturbative theory. The long-distance contributions from four-quark operators near the resonance cannot be calculated from first principles of QCD and are usually parameterized in the form of a phenomenological Breit-Wigner formula making use of the vacuum saturation approximation and quark-hadron duality. The manifest expressions for and can be written as (25); (26); (27); (28); (29)

 YSD(z,s′) = h(z,s′)(3C1(μ)+C2(μ)+3C3(μ)+C4(μ)+3C5(μ)+C6(μ)) (4) −12h(1,s′)(4C3(μ)+4C4(μ)+3C5(μ)+C6(μ)) −12h(0,s′)(C3(μ)+3C4(μ))+29(3C3(μ)+C4(μ)+3C5(μ)+C6(μ)),
 YLD(z,s′) = 3α2em(3C1(μ)+C2(μ)+3C3(μ)+C4(μ)+3C5(μ)+C6(μ)) (5) ∑j=ψ,ψ′ωj(q2)kjπΓ(j→l+l−)Mjq2−M2j+iMjΓtotj,

with

 h(z,s′) = −89lnz+827+49x−29(2+x)|1−x|1/2⎧⎪⎨⎪⎩ln∣∣∣√1−x+1√1−x−1∣∣∣−iπfor x≡4z2/s′<12arctan1√x−1for x≡4z2/s′>1, h(0,s′) = 827−89lnmbμ−49lns′+49iπ. (6)

The non-factorizable effects (49); (50); (51); (56) from the charm loop can bring about further corrections to the radiative transition, which can be absorbed into the effective Wilson coefficient . Specifically, the Wilson coefficient is given by (29)

 Ceff7(μ)=C7(μ)+Cb→sγ(μ),

with

 Cb→sγ(μ) = iαs[29η14/23(G1(xt)−0.1687)−0.03C2(μ)], (7) G1(x) = x(x2−5x−2)8(x−1)3+3x2ln2x4(x−1)4, (8)

where , , is the absorptive part for the rescattering and we have dropped out the tiny contributions proportional to CKM sector . In addition, and can be obtained by replacing the unprimed Wilson coefficients with the corresponding prime ones in the above formula.

Iii Matrix elements and form factors in Light Cone Sum Rules

With the free quark decay amplitude available, we can proceed to calculate the decay amplitudes for and at hadron level, which can be obtained by sandwiching the free quark amplitudes between the initial and final baryon states. Consequently, the following four hadronic matrix elements

 ⟨Λ(P)|¯sγμb|Λb(P+q)⟩ , ⟨Λ(P)|¯sγμγ5b|Λb(P+q)⟩, ⟨Λ(P)|¯sσμνb|Λb(P+q)⟩ , ⟨Λ(P)|¯sσμνγ5b|Λb(P+q)⟩, (9)

need to be computed. Generally, the above matrix elements can be parameterized in terms of the form factors as (29); (30); (31); (32); (33):

 ⟨Λ(P)|¯sγμb|Λb(P+q)⟩ = ¯¯¯¯Λ(P)(g1γμ+g2iσμνqν+g3qμ)Λb(P+q), (10) ⟨Λ(P)|¯sγμγ5b|Λb(P+q)⟩ = ¯¯¯¯Λ(P)(G1γμ+G2iσμνqν+G3qμ)γ5Λb(P+q), (11) ⟨Λ(P)|¯sσμνb|Λb(P+q)⟩ = ¯¯¯¯Λ(P)[h1σμν−ih2(γμqν−γνqμ) (12) −ih3(γμPν−γνPμ)−ih4(Pμqν−Pνqμ)]Λb(P+q), ⟨Λ(P)|¯sσμνγ5b|Λb(P+q)⟩ = ¯¯¯¯Λ(P)[H1σμν−iH2(γμqν−γνqμ) (13) −iH3(γμPν−γνPμ)−iH4(Pμqν−Pνqμ)]γ5Λb(P+q),

where all the form factors , , and are functions of the square of momentum transfer . Contracting Eqs. (12-13) with the four momentum on both side and making use of the equations of motion

 qμ(¯ψ1γμψ2) = (m1−m2)¯ψ1ψ2 (14) qμ(¯ψ1γμγ5ψ2) = −(m1+m2)¯ψ1γ5ψ2 (15)

we have

 ⟨Λ(P)|¯siσμνqνb|Λb(P+q)⟩ = ¯¯¯¯Λ(P)(f1γμ+f2iσμνqν+f3qμ)Λb(P+q), (16) ⟨Λ(P)|¯siσμνγ5qνb|Λb(P+q)⟩ = ¯¯¯¯Λ(P)(F1γμ+F2iσμνqν+F3qμ)γ5Λb(P+q), (17)

with

 f1 = 2h2−h3+h4(mΛb+mΛ)2q2, (18) f2 = 2h1+h3(mΛ−mΛb)+h4q22, (19) f3 = mΛ−mΛbq2f1, (20) F1 = 2H2−H3+H4(mΛb−mΛ)2q2, (21) F2 = 2H1+H3(mΛ+mΛb)+H4q22, (22) F3 = mΛ+mΛbq2F1. (23)

Due to the conservation of vector current, the form factors and do not contribute to the decay amplitude of . To incorporate the NHBs effect one need to calculate the matrix elements involving the scalar and the pseudoscalar currents, which can be parameterized as

 ⟨Λ(P)|¯sb|Λb(P+q)⟩ = 1mb+ms¯¯¯¯Λ(P)[g1(mΛb−mΛ)+g3q2]Λb(P+q), (24) ⟨Λ(P)|¯sγ5b|Λb(P+q)⟩ = 1mb−ms¯¯¯¯Λ(P)[G1(mΛb+mΛ)−G3q2]γ5Λb(P+q). (25)

The various form factors and appearing in the above equations are not independent in the heavy quark limit and one can express them in terms of two independent form factors and in HQET defined by (25)

 ⟨Λ(P)|¯bΓs|Λb(P+q)⟩=¯¯¯¯Λ(P)[ξ1(q2)+v/ξ2(q2)]ΓΛb(P+q), (26)

with being an arbitrary Lorentz structure and being the four-velocity of baryon. Comparing Eqs. (10-11), (16- 17) and the Eq. (26), one can arrive at (29); (30); (31); (32); (33)

 f1 = F1=q2mΛbξ2, (27) f2 = F2=g1=G1=ξ1+mΛmΛbξ2, (28) f3 = mΛ−mΛbmΛbξ2, (29) F3 = mΛ+mΛbmΛbξ2, (30) g2 = G2=g3=G3=ξ2mΛb. (31)

Due to our poor understanding towards non-perturbative QCD dynamics, one has to rely on some approaches to calculate the form factors answering for transition. It is suggested that the soft non-perturbative contribution to the transition form factor can be calculated quantitatively in the framework of LCSR approach (57); (58); (59); (60); (61), which is a fully relativistic approach and well rooted in quantum field theory, in a systematic and almost model-independent way. As a marriage of standard QCDSR technique (62); (63); (64) and theory of hard exclusive process (65); (66); (67); (68); (69); (70); (71); (72), LCSR cure the problem of QCDSR applying to the large momentum transfer by performing the operator product expansion (OPE) in terms of twist of the relevant operators rather than their dimension (73). Therefore, the principal discrepancy between QCDSR and LCSR consists in that non-perturbative vacuum condensates representing the long-distance quark and gluon interactions in the short-distance expansion are substituted by the light cone distribution amplitudes (LCDAs) describing the distribution of longitudinal momentum carried by the valence quarks of hadronic bound system in the expansion of transverse-distance between partons in the infinite momentum frame.

Considering the distribution amplitude up to twist-6, the form factors for have been calculated in (25) to the accuracy of leading conformal spin, where the pole model was also employed to extend the results to the whole kinematical region. Specifically, the dependence of form factors on transfer momentum are parameterized as

 ξi(q2)=ξi(0)1−a1q2/m2Λb+a2q4/m4Λb, (32)

where denotes the form factors and . The numbers of parameters have been collected in Table 1.

To the leading order and leading power, the other form factors can be related to these two as

 F1(q2) = f1(q2)=q2g2(q2)=q2G2(q2), F2(q2) = f2(q2)=g1(q2)=G1(q2), (33)

where the form factors and are dropped out here due to their tiny contributions.

Iv Formula for Observables

In this section, we proceed to perform the calculations of some interesting observables in phenomenology including decay rates, forward-backward asymmetry, polarization asymmetries of final state lepton and of baryon. From Eq. (LABEL:quark-amplitude), it is straightforward to obtain the decay amplitude for as

 MΛb→Λl+l−=−GFα2√2πVtbV∗ts[T1μ(¯lγμl)+T2μ(¯lγμγ5l)+T3(¯ll)], (34)

where the auxiliary functions , and are given by

 T1μ = ¯¯¯¯Λ(P)[{γμ(g1−G1γ5)+iσμνqν(g2−G2γ5)}Ceff9 (35) +{γμ(g1+G1γ5)+iσμνqν(g2+G2γ5)}C′eff9 −2mb/s{γμ(f1+F1γ5)+iσμνqν(f2+F2γ5)}Ceff7 −2mb/s{γμ(f1−F1γ5)+iσμνqν(f2−F2γ5)}C′eff7]Λb(P+q),
 T2μ = ¯¯¯¯Λ(P)[{γμ(g1−G1γ5)+iσμνqν(g2−G2γ5)+(g3−G3γ5)qμ}C10 +{γμ(g1+G1γ5)+iσμνqν(g2+G2γ5)+(g3+G3γ5)qμ}C′10 −qμ2ml(mb+ms){(g1(mΛb−mΛ)+g3q2+G1(mΛb+mΛ)−G3q2)}CQ2 −qμ2ml(mb+ms){(g1(mΛb−mΛ)+g3q2−G1(mΛb+mΛ)+G3q2)}C′Q2]Λb(P+q),

and

 T3 = Missing or unrecognized delimiter for \right (37) +{(g1(mΛb−mΛ)+g3q2−G1(mΛb+mΛ)+G3q2)}C′Q1]Λb(P+q).

It needs to point out that the terms proportional to in do not contribute to the decay amplitude with the help of the equation of motion for lepton fields. Besides, one can also find that the above results can indeed reproduce that obtained in the SM with and .

iv.1 The differential decay rates of Λb→Λl+l−

The differential decay width of in the rest frame of baryon can be written as (74),

 dΓ(Λb→Λl+l−)ds=1(2π)3132m3Λb∫umaxumin|˜MΛb→Λl+l−|2du, (38)

where and ; , and are the four-momenta vectors of , and respectively. denotes the decay amplitude after performing the integration over the angle between the and baryon. The upper and lower limits of are given by

 umax = (E∗Λ+E∗l)2−(√E∗2Λ−m2Λ−√E∗2l−m2l)2, umin = (E∗Λ+E∗l)2−(√E∗2Λ−m2Λ+√E∗2l−m2l)2, (39)

where and are the energies of and in the rest frame of lepton pair

 E∗Λ=m2Λb−m2Λ−s2√s,E∗l=√s2. (40)

Putting everything together, we can achieve the decay rates and invariant mass distributions of with and without long distance contributions as

 dΓds = α2G2F∣∣VtbV∗ts∣∣2(umax−umin)128m3Λbπ5× {(f2ml(f2−g2((mΛ+mΛb))(CQ1C∗eff7+C∗Q1Ceff7)+ml2mb(f2