Effect of both Z and Z'-mediated flavor-changing neutral currents on the baryonic rare decay lambd a b decays into lambda l + l −
Effect of both Z and Z ′ -mediated flavor-changing neutral currents on the baryonic rare decay −+Λ→Λ ll b S Sahoo , C K Das and L Maharana Department of Physics, National Institute of Technology, Durgapur-713 209, West Bengal, India E-mail: [email protected] Department of Physics, Trident Academy of Technology, Bhubaneswar – 751 024, Orissa, India. E-mail: [email protected] Department of Physics, Utkal University, Bhubaneswar – 751 004, Orissa, India. E-mail: [email protected]
Abstract
We study the effect of both Z and Z ′ -mediated flavor-changing neutral currents (FCNCs) on the −+ Λ→Λ ll b ( ) τµ , = l rare decay. We find the branching ratio is reasonably enhanced from its standard model value due to the effect of both Z and Z ′ -mediated FCNCs, and gives the possibility of new physics beyond the standard model. The contribution of Z ′ -boson depends upon the precise value of / Z M . . Keywords:
Z-boson; Z ′ -boson; Semileptonic baryonic decays; Heavy quark effective theory PACS Nos.: . Introduction
Rare B decays [1,2] induced by flavor-changing neutral current (FCNC) −+ → ll )( dsb transitions are very important to probe the flavor sector of the standard model (SM). In the SM, they arise from one-loop diagrams and are generally suppressed in comparison to the tree diagrams. Nevertheless, one-loop FCNC processes can be enhanced by orders of magnitude in some cases due to the presence of new physics. New physics (NP) comes into play in rare B decays in two different ways: (a) through a new contribution to the Wilson coefficients or (b) through a new structure in the effective Hamiltonian, which are both absent in the SM. Rare decays can give valuable information about the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, btstdt VVV ,, , etc and leptonic decay constants. Moreover, −+ → ll )( dsb decay is very sensitive to the new physics beyond the SM. The study of −+ → ll sb decays is one of the most reliable tests of FCNC. These decays have been studied in the SM, two-Higgs-doublet model (2HDM) and minimal supersymmetric standard model (MSSM) [3-14]. The theoretical study of the inclusive decays is easy but their experimental detection is quite difficult. For exclusive decays the situation is opposite i.e. their experimental detection is easy, but theoretical analysis is very difficult. One of the exclusive decay which is described at inclusive level by −+ → ll sb transition is the baryonic −+ Λ→Λ ll b ( ) τµ ,, e = l decay. This decay has been studied in the SM [15], in the supersymmetric model with and without R-parity [16-19], in the two-Higgs-doublet model [20] and in a model independent way [21]. In comparison with B meson decays, b Λ baryon decays contain some particular observables, involving the spin of the b quark, which are sensitive to new physics and more easily detectable [22]. Theoretically it is predicted that Z ′ -bosons are exist in grand unified theories (GUTs), superstring theories and theories with large extra dimensions but experimentally the Z ′ -boson is not discovered so far [23]. If the Z ′ -bosons couple to quarks and leptons not too weakly and if their mass is not too large, they will be produced at the Tevatron and the LHC and easily detected through their leptonic decay modes [24]. Therefore the search for these particles is a very challenging topic in experimental physics. There has been rigorous study in the Z ′ sector to understand physics beyond the SM [25-28]. It has been shown that a leptophobic Z ′ -boson can appear in E gauge models due to mixing of gauge kinetic terms [29,30]. Flavor mixing can be induced at the tree level in the up-type and/or down-type quark sector after diagonalizing their mass matrices. Mixing between ordinary and exotic left-handed quarks induces Z-mediated FCNCs. The right-handed quarks RR sd , and R b have different )1( ′ U quantum numbers than exotic R q and their mixing will induce Z ′ -mediated FCNCs [29,31,32] among the ordinary down quark types. Tree level FCNC interactions can also be induced by an additional Z ′ -boson on the up-type quark sector [33]. In the Z ′ model [34], the FCNC Zsb ′−− coupling is related to the flavor-diagonal couplings
Zqq ′ in a predictive way, which is then used to obtain upper limits on the leptonic Z ′ ll couplings. Hence, it is ossible to predict the branching ratio for −+ Λ→Λ ll b rare decay. With FCNCs, both Z and Z ′ -boson contributes at tree level, and its contribution will interfere with the SM contributions [31,32,35]. In this paper, we study −+ Λ→Λ ll b rare decay considering the effect of both Z and Z ′ -mediated FCNCs that change the effective Hamiltonian and modifies the branching ratio. This paper is organized as follows: in Section 2, we discuss the −+ Λ→Λ ll b rare decay in the standard model. In Section 3, we give a brief account of the extended quark sector model and explain why it implies FCNC at the tree level. Then, we evaluate the effective Hamiltonian for −+ Λ→Λ ll b rare decay considering the contributions from both the Z and Z ′ -bosons. In Section 4, we calculate the branching ratio for −+ Λ→Λ ll b decay. Then we discuss our results, so obtained. −+ Λ→Λ ll b Decay in the Standard Model
Let us consider the −+ Λ→Λ ll b ( ) τµ , = l rare decay process. In the standard model, this process is loop-suppressed. However, it is potentially sensitive to new physics beyond the SM. At the quark level, the −+ Λ→Λ ll b decay is described by the −+ → ll sb transition. The matrix element of the −+ → ll sb process contains terms describing the virtual effects induced by cctt , and uu loops which are proportional to ** , scbcstbt VVVV and * usbu VV respectively. From unitarity of the CKM matrix and neglecting * usbu VV in comparison to * stbt VV and * scbc VV , it is clear that the matrix element of the −+ → ll sb contains only one independent CKM factor * stbt VV . The effective Hamiltonian describing −+ Λ→Λ ll b decay process is given [2,36]: ( ) ( ) ( ) ( )( ) ,]2[2
27 5109 ll llll µµνµ µµµµ γσ γγγγγλπα − += bPppismC bPsCbPsCGH
Rbeff LLefftFeff (1) where F G is the Fermi coupling constant, α is the electromagnetic coupling constant, * stbtt VV =λ , ( ) γ±= LR P , p is the momentum transferred to the lepton pair and −+ += ppp the sum of the momenta of the + l and − l , and eff C , C and C are Wilson coefficients [37] evaluated at the b quark mass scale in the modified minimal subtraction ( ) SM scheme ( 6.4 = b m GeV). −==−=
CCC eff . (2) he coefficient eff C has a perturbative part and a resonance part which comes from the long distance effects. Therefore, we can write eff C = res CsYC )( ++ , (3) where ps = and the function )( sY is the perturbative part coming from one loop matrix elements of the four quark operators and is given by [20,38] ( ) ( ) ( ) ( ) CCsgCCCCCCsmgsY c +−+++++= ( ) ( ) ( ) CCCCCCCCsmg b +++++++− , (4) where ( ) ( ) ( )( ) ( ) −−+ − −− −+− ×−+−++−= jjjjj jjjpolebjj yyiyyy yyymmsmg θπθ , (5) with smy jj /4 = . The values of the coefficients C i in next-to-leading-logarithmic (NLL) order are taken from [37] as ,012.0,059.1,151.0 ==−= CCC ,034.0 −= C = C and 040.0 −= C . The long distance resonance effect is given as [5,8,39] ( ) ( ) iiv ivSSv ires vmvism VmvkCCCCCCC iii Γ−− →Γ×+++++= −+= ∑ ll ψψ απ , (6) where the phenomenological parameter k [9] is satisfied the relation ( ) −=+ CCk . Now the amplitude of ( ) ( ) ( ) ( ) −−++ΛΛ
Λ→Λ pppp b b ll decay can be obtained by sandwiching eff H for the −+ → ll sb transition between initial and final baryon states i.e. beff H ΛΛ . The matrix elements of the various hadronic currents between the initial b Λ and the final Λ baryon can be written as [2]: [ ] b upfpfifubs b ΛΛ ++=ΛΛ µννµµµ σγγ , [ ] b upgpgigubs b ΛΛ ++=ΛΛ µννµµµ γγσγγγγ , [ ] b upfpfifubpis TTTb ΛΛ ++=ΛΛ µννµµννµ σγσ , [ ] b upgpgigubpis TTTb ΛΛ ++=ΛΛ µννµµννµ γγσγγγσ , (7) here −+ΛΛ +=−= ppppp b is the momentum transfer, and i f and i g are the various form factors which are functions of p . The processes for the heavy to light baryonic decays such as those with Λ→Λ b have been studied based on the heavy quark effective theory (HQET) in [40]. It is found that the number of independent form factors is reduced to two in the heavy quark symmetry limit. In this limit, the matrix elements of all hadronic currents, irrespective of their Dirac structure, can be written as ( ) ( ) ( ) ( ) [ ] bb upFvpFupbsp b ΛΛΛΛ
Γ/+=ΛΓΛ , (8) where Γ is the product of Dirac matrices, bb mpv ΛΛ = / µµ is the four velocity of b Λ and F are the form factors. The relations among these two sets of form factors are given as [16-19]: FrFgffg TT +==== , b mFfgfg Λ ==== , ( ) ΛΛΛ += mmmFg bb T , ( ) ΛΛΛ −−= mmmFf bb T , pmFgf b TT Λ == , (9) where / b mmr ΛΛ = . The form factors F and F for the −+ Λ→Λ ll b decay are calculated in QCD sum rule approach combined with heavy quark symmetry in [15,16] and values are also presented in [21]. The values of form factors are 462.0 = F and 077.0 −= F . Using the recent data [16,41] we get the ratio of form factors, 04.020.0 ±−= r . From equation (9), it is clear that )( gf and ( ) TT gf are proportional to F . Hence, they are large whereas all others are small. Now using these form factors, the transition amplitude can be written as [2]: ( ) ( ) ( ) [ ] { } ( ) ( )( ) ++ ++++ +++×=Λ→Λ ΛΛ ΛΛ−+ bb uPEPDp PEPDpiPEPDu uPBPApiPBPAuGM LR LRLR LRLRtFb
33 22115 2211 µ ννµµµ ννµµµ σγγγ σγγλπα ll llll ………….(10) where the parameters i A , i B and jj ED , ( i = 1, 2 and j = 1, 2, 3) are defined as ( ) ( ) TiTibeffiieffi gfp mCgfCA +−−=
221 , ( ) ( ) ,221
TiTibeffiieffi gfp mCgfCB −−+= ( ) jjj gfCD −= , ( ) jjj gfCE += . (11) rom equation (11), it is clear that T f and T g are associated with C which is about one order magnitude smaller than C and C [equation (2)] so that their effects due to the deviation of the results in the HQET are small [18]. Differential Decay Rates and Branching Ratios:
The double partial decay rates for −+ Λ→Λ ll b ( ) τµ , = l can be obtained from the transition amplitude [equation (10)] as: ( ) ( ) zsKsrpmmGzdsdd ltF b ,ˆ,,1412ˆ λπλα −=Γ Λ , (12) where /ˆ b mss Λ = , θ cos = z , the angle between b p Λ and + p in the center of mass frame of −+ ll pair, and ( ) ( ) cabcabcbacba ++−++= λ is the usual triangle function. The function ( ) zsK , is given as ( ) ( ) ( ) ( ) sKzsKzsKzsK , ++= , (13) where ( ) sK = ( ) ( ) ( ) ( ) *133*132232342 Reˆ164ˆ1ˆ32
EDEDsrmmEDsrsmm bb ll +−−++−+ ΛΛ + ( ) ( ) Reˆ664
EDmsmrm bb l ΛΛ − + ( ) ( ) ( ) [ ] Reˆ1Reˆ264
EEDDsrEDsmrmm bb l ++−+× ΛΛ + ( ) ( ) ( ) ( ) ( )( ) ( ) { } +− +−−−++−×+ Λ ΛΛΛΛ
ReˆRe2 Reˆ1Reˆ1ˆ232
BAsmBAr BABAsrmBBAArmsrmsmm b bbbb l + ( ) ( ) { } [ ] ( ) ˆ1ˆ148 BAsrmsrmm bb l +×−−+−+ ΛΛ + ( ){ } ( ) { } [ ] ( ) ˆ1ˆˆˆ148 BAsrsmssrmm bb l +×−−+−++ ΛΛ λ ( ) ( ) { } [ ] ( ) ˆ1ˆ148 EDsrmsrmm bb l +×−−−−+− ΛΛ + × Λ ˆ8 l vsm b ( ) ( ) ( ) ( ) ( ) ( ) { } { } +−−++−−− ++−+− ΛΛ ˆ1Reˆ14 Reˆ14Reˆ8 EDsrmEDEDsr EEDDrsrEDrsm bb , (14) ( ) ( ) ( ) ( ) { } Re2Re2Re2ˆ16
EAEADBDBmEBDAvsmsK bb l −+−+−−= ΛΛ λ ) ( ) { } ReRe)1(ˆ32
EBEBDADArEBDArmvsm bb l −−++−−+ ΛΛ λ ……(15) and ( ) [ ] ˆ8 EDBAvsmsK l b +++= Λ λ [ ] EDBAvm l b +++− Λ λ , (16) where λ is the short form for ( ) sr ˆ,,1 λ . Now integrating equation (12) w. r. t. the angular dependent parameter z, we can get the expression for decay width as : ( ) ( ) ( ) +−= Γ Λ sKsKsrpmmGsdd ltF b λπ λα , (17) The limits for s are based on the kinematic phase space: ( ) ΛΛ −≤≤ mmsm b l . (18) The branching ratios can be calculated by multiplying decay width Γ with the life time of b Λ i.e. ( ) ( ) b bb B Λ−+−+
Λ→ΛΓ=Λ→Λ τ llll , (19) where b Λ τ is the life time of b Λ and ( ) −+−Λ ×= b τ s [41]. The value of the branching ratios for −+ Λ→Λ ll b decay in the standard model [2] is given as: ( ) −+ Λ→Λ µµ b B = − × , ( ) −+ Λ→Λ ττ b B = − × . (20) These branching ratios have also been calculated in an explicit supersymmetric standard model [18] as: ( ) −+ Λ→Λ µµ b B = − × , ( ) −+ Λ→Λ ττ b B = − × . (21) The Model
In extended quark sector model [35], besides the three standard generations of the quarks, there is an L SU )2( singlet of charge − . This model allows for Z-mediated FCNCs. The up quark sector interaction eigenstates are identified with mass eigenstates but down quark sector interaction eigenstates are related to the mass eigenstates by a 4 × ( ) −+ +− += µµµµ JWJWgL W int , (22) LjLiji duVJ µµ γ= − . (23) The charged-current mixing matrix V is a 3 × jiji KV = for == ji . (24) Here, V is parametrized by six real angles and three phases, instead of three angles and one phase in the original CKM matrix. The neutral-current interactions are described by ( ) µµµ θθ meWWZ JJZgL sincos −= , (25) LjLijiLqLpqp uuddUJ µµµ γδγ +−= . (26) In neutral-current mixing, the matrix for the down sector is U = V † V. Since in this case V is not unitary, ≠ U . Its nondiagonal elements do not vanish: qpqp KKU −= for qp ≠ . (27) Since the various qp U are non-vanishing, they allow for flavor-changing neutral currents that would be a signal for new physics. Now consider the −+ Λ→Λ ll b decay process in the presence of Z-mediated FCNC [35] at tree level. The sbZ FCNC coupling, which affects B-decays, is parameterized by one independent parameter bs U and this parameter is constrained by branching ratio of the process −+ → ll S XB and is found to be − ×≅ bs U [42]. The BELLE Collaboration [43] has measured the branching ratio ( ) =→ −+ ll S XBB ( ) −+− ×± . Considering the contribution of the Z-boson to −+ Λ→Λ ll b ( ) τµ , = l decay, one can write the effective Hamiltonian [2] as ( ) [ ] ( ) [ ] ll ll γγγγγ µµµ AVbsFeff
CCbsUGZH −−= , (28) where l V C and l A C are the vector and axial vector −+ ll Z couplings and are given as −=+−= ll AWV CC θ , (29) where W θ is the weak mixing (or Weinberg) angle [44]. The contributions to decay −+ Λ→Λ ll b mainly come from the Wilson coefficients C and C , and corresponding operators. In this model, the structure of the effective Hamiltonian is in the same form as hat of the SM. Hence, its effect can be evaluated by replacing the SM Wilson coefficients ( ) SMeff C and ( ) SM C by ( ) *99 stbt bsSMeffeff VVUCC απ += , ( ) *1010 stbt bsSMeff VVUCC απ −= . (30) From equation (2), it is clear that the value of the Wilson coefficients C and C are opposite to each other. Hence, the new physics contributions to C and C are opposite to each other and one will get constructive and destructive interference of SM and NP amplitudes for πθ = or zero (where θ is the relative weak phase between the SM and NP contribution in the above equation). We consider the weak phase difference to be π to get constructive interference between the SM and NP amplitudes. The updated model-independent values for these coefficients can be found in Ref. [45,46]. The branching ratios for −+ Λ→Λ ll b ( ) τµ , = l decay in the presence of Z-mediated FCNC are calculated in [2] as: ( ) Zb B −+ Λ→Λ µµ = − × , ( ) Zb B −+ Λ→Λ ττ = − × . (31) The same idea can be applied to a Z ′ -boson i.e., mixing among particles which have different Z ′ quantum numbers will induce FCNCs due to Z ′ exchange [31,32,47]. Since the Zqp U are generated by mixing that breaks weak isospin, they are expected to be at most O ( / MM ), where )( MM is typical light (heavy) fermion mass. On the other hand, the Z ′ -mediated coupling / Zqp
U can be generated via mixing of particles with same weak isospin and, so, suffer no suppression. Even though Z ′ -mediated interactions are suppressed relative to Z, these are compensated by the factor ZqpZqp UU / / ∼ ( / MM ). The flavor-changing coupling Zbs U ′ is constrained by the process νν sb → [48,49] and is found to be −′′ ×≤ zzzsb MMU . This can be turned into a bound on
Zsb U ′ if one assumes a value for Z M ′ . If we assume Zbs U ′ ~ * stbt VV , then it is possible to write bs U instead of Zbs U ′ , which gives significant contributions to the −+ Λ→Λ ll b decay process. Thus the new contributions from Z ′ -boson are exactly in the similar manner as in the Z-boson. Therefore, we write the general effective Hamiltonian [31,32] that contribute to −+ Λ→Λ ll b in the light of equation (28) as : ( ) [ ] ( ) [ ] ′−−=′ ′ ZZAVbsFeff
MMggCCbsUGZH ll ll γγγγγ µµµ , (32) where ( ) WW eg θθ cossin/ = and g ′ is the gauge coupling associated with the )1( ′ U group. The absence of the suppression in the mixing for the Z ′ -mediated FCNC can compensate for / ZZ MM ′ suppression of the Z ′ amplitude relative to the Z amplitude, which implies that the coefficients describing the Z and Z ′ flavor-changing effective interactions can be comparable in size. The net effective Hamiltonian can be written, from equation (28) and (32), as )()( ZHZHH effeffeff ′+= and ( ) [ ] ( ) [ ] ′+−−= ′ ZZAVbsFeff
MMggCCbsUGH ll ll γγγγγ µµµ , (33) and the corresponding branching ratios for the baryonic rare decays −+ Λ→Λ ll b ( ) τµ , = l are calculated in the next section. . Results and Discussions
In this section, we calculate the branching ratios for the baryonic rare decays −+ Λ→Λ ll b ( ) τµ , = l using recent data [41]: ( ) ±= µ m MeV, ( ) ±= τ m MeV, ( ) ±= e m MeV, ( ) ±= Λ m MeV, ( ) ±= Λ b m MeV, ( ) +−Λ = b τ − × s, Z M = ( ) ± GeV, ( ) −− ×±= GeVG F , = W θ and − ≅ bs U [42]. Since the Z ′ has not yet been discovered, its mass is unknown. However, the Z ′ mass is constrained by direct searches at Fermilab, weak neutral current data and precision studies at LEP and the SLC [50-53], which give a model-dependent lower bound around 500 GeV if the interaction is comparable to the other couplings of the standard model. However, the lower mass limit can be as low as 130 GeV [54] if the coupling is weak. The experimental bounds on / Z M above 1 TeV are not expected without excessive fine-tuning of supersymmetry breaking mass parameters, or unusual choices of )1( ′ U charge assignments. In a study of B meson decays with Z ′ -mediated flavor-changing neutral currents [32], they study the Z ′ -boson in the mass range of a few hundred GeV to 1 TeV. In this paper, we study the Z ′ -boson in the mass range 130 GeV – 1 TeV. In general, the value of gg / ′ is undetermined [55]. However, generically, one expects that 1/ ≈′ gg if both U(1) groups have the same origin from some grand unified theory. We take 1/ ≈′ gg in our calculations. sing the lower limit for the mass of Z ′ -boson, / Z M = 130 GeV, we get ( ) ZZb B ′+−+ Λ→Λ µµ = ( ) − ×± , ( ) ZZb B ′+−+ Λ→Λ ττ = ( ) − ×± . (34) Again using the mass of Z ′ -boson, / Z M = 1000 GeV, we get ( ) ZZb B ′+−+ Λ→Λ µµ = ( ) − ×± , ( ) ZZb B ′+−+ Λ→Λ ττ = ( ) − ×± . (35) From equation (34) and (35), it is clear that depending on the precise value of / Z M , the Z ′ -mediated FCNCs gives sizable contributions to −+ Λ→Λ ll b decay process. Our estimated branching ratios for −+ Λ→Λ ll b decay process are reasonably enhanced from its standard model value [equation (20)]. Hence, the −+ Λ→Λ ll b decay process could provide signals for new physics beyond the standard model. It is also found that the forward-backward asymmetries ( A FB ) are different from that of the standard model value due to Z-mediated FCNC [2]. We expect that the Z ′ -mediated FCNC will also change the forward-backward asymmetries values from that of the standard model value. The position of the zero value of A FB is very sensitive to the presence of new physics. These facts lead to enrichment in the phenomenology of both the Z and Z ′ -mediated FCNCs and −+ Λ→Λ ll b decay; and the physics beyond the standard model will be known after the discovery of the Z ′ -boson which is expected at the LHC. We are hopeful that the effect of Z ′ -mediated FCNCs in b Λ decays would be measured in the Tevatron. The LHCb is expected to give us more insight on the flavor structure of new physics through precise measurements of rates and CP asymmetries in rare decays. The experimental observation of the rare B-decays −+ Λ→Λ ll b would provide precision tests of the SM in the crucial and as yet untested FCNC sector of B-decays. It is also expected [18,22,56-58] that the measurements of −+ Λ→Λ ll b can serve as a promising quantity to explore new physics effects as well as to constrain the parameter space of various models beyond the SM. We have no doubt that an exciting future is ahead of us! Acknowledgments
We gratefully acknowledge helpful and enlightening discussions with Prof. T. M. Aliev, Middle East Technical University, Turkey; Dr. A. K. Giri, Punjabi University, Patiala and Dr. R. Mohanta, University of Hyderabad, India. We also thank Dr. Kartik Senapati, Cambridge University and Dr. S. C. Martha & Mr. Sudhansu Biswal, IISc. Bangalore for their help in the preparation of the manuscript. We thank the referee for suggesting valuable improvements of our manuscript. eferences Mohanta R 2005
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