Double-Lepton Polarization Asymmetries in B->K_1 l^+ l^- Decay in Universal Extra Dimension Model
aa r X i v : . [ h e p - ph ] O c t Double-Lepton Polarization Asymmetries in B → K l + l − Decay in Universal Extra Dimension Model
B. B. S¸ irvanlıGazi University, Faculty of Arts and Science, Department of Physics06100, Teknikokullar Ankara, Turkey
October 31, 2018
Abstract
Double-lepton polarization asymmetries for the exclusive decay B → K l + l − in theUniversal Extra Dimension (UED) Model is studied. It is obtained that double-lepton polar-ization asymmetries are very sensitive to the UED model parameters. Experimental measure-ments of double lepton polarizations can give valuable information on the physics beyond theStandard Model (SM). PACS number(s):12.60.–i, 13.20.–v, 13.20.He1
Introduction
The rare B-meson decays pointed out by the flavor-changing neutral currents (FCNC) have beensignificant channels for acquiring knowledge on the SM parameter and analyzing the new physicspredictions. Rare B meson decays are not allowed at the tree level in the SM and seem at looplevel. By rare B decays, one generally comprehend Cabibbo-suppressed b → u transitions orflavour-changing neutral currents (FCNC) b → s or b → d . So rare decays are significant test-ing basic of the SM and take an important part in the search for new physics. The examinationsof different FCNC processes can be used to determine different fundamental parameters of SMlike elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix , various decay constants etc.Between testing SM the FCNC processes can be very important for discovering indirect effectsof possible TeV scale extensions of SM. Therefore,we examine b → q ( q = d, s ) transitions interms of an effective Hamiltonian. For observing to the new physics in these decays, there aretwo different ways. First of all, the differences in the Wilson coefficients form the ones existingin the SM. And the second one the new operator in the effective Hamiltonian which are absent inthe SM. All decay channels of B meson include many physically quantities which are very usefultesting for the SM and investigating for new physics beyond the SM. Exclusive processes such as B → K ( K ∗ ) l + l − and B → γl + l − decays [4, 5, 6, 7, 8] have been studied extensive in literature. Colangelo et al. have studied B → K ( K ∗ ) l + l − decays in framework of one Universal ExtraDimension model (ACD),proposed in the Ref. [16] and analyzed the branching ratio and forward-backward asymmetry . In meanwhile, in the Ref. [14] the single lepton polarizations is studiedfor µ for the B → K l + l − decay in UED model. The Branching ratios (BR) of the Semileptonicdecays B ( B → K ∗ l + l − ) = 7 . ± . × − [2] and B ( B → Kl + l − ) = 5 . ± . × − [1] have been measured by BELLE [1] and BaBar [2] collaborations. It is noted that the mea-surement of the polarization of the b → s decay can provide important information about moreobservables. Some of the single lepton polarization asymmetries can be too small to be observed.Since it might not provide number of observables for control the structure of the effective Hamil-tonian,we calculate to double lepton polarization for more observables [9]. Among the differentmodels of physics beyond the SM, extra dimensions is very interesting models. Since the extradimension model contain of gravity, they give to clue on the hierarchy problem and a connectionwith string theory. The model of Appelquist, Cheng and Dobrescu (ACD) [10, 11, 20] with oneuniversal extra dimension (UED), where all the SM particles can propagate in the extra dimen-sion. Compactification of the extra dimension leads to Kaluza-Klein model in the four-dimension.In the extra dimension model, we have extra free parameter is /R ,which is inverse of the com-pactification radius. With the aid of /R , we can determined all the masses of the KK particlesand their interactions with SM particles. In the meanwhile, If we have not tree level contribu-tion of KK states to the low energy processes, KK parity is conservation in ACD model at scale µ ≪ /R .In this work, we study the double-lepton polarization asymmetries for the B → K l + l − decayin the UED model. In section 2, we shortly examine ACD model. In section 3, we obtain matrixelement for the B → K l + l − decay. In section 4, Double lepton polarization for the B → K l + l − decay are calculated. Section 5 is devoted to the numerical analysis and discussion of our results.2 B → K l + l − Decay in ACD Model
Before calculation of the double lepton polarizations few words about the ACD model. Thismodel is the minimal extension of the SM to the δ dimensions. We consider simple casewhich is δ = 1 . In the universe, we have 3 space + 1 time dimensions and one possibility is thepropagation of gravity in whole ordinary plus extra dimensional universe. The five-dimensionalACD model with a single UED uses orbifold compactification, the fifth dimension y that is com-pactified in a circle of radius R , with points y = 0 and y = πR that are fixed points of theorbifolds [11, 12, 13, 14]. The Lagrangian in ACD model can be written as: L = Z d xdy {L A + L H + L F + L Y } where L A = − W MNa W aMN − B MN B MN L H = ( D M φ ) † D M φ − V ( φ ) L F = Q ( i Γ M D M ) Q + u ( i Γ M D M ) u + D ( i Γ M D M ) DL Y = −Q e Y u φ c u − Q e Y d φ D + h.c.. where M and N are the five-dimensional Lorentz indices which can run from , , , , . W aMN = ∂ M W aN − ∂ N W aM + e gε abc W bM W cN are the field strength tensor for the SU (2) L electroweakgroup, B MN = ∂ M B N − ∂ N B M are that of the U (1) group. D M = ∂ M − i e gW aM T a − i e g ′ B M Y is thecovariant derivative, where e g and e g ′ are the five-dimensional gauge couplings for the SU (2) L and U (1) groups. Γ M are five-dimensional matrices which is Γ µ = γ µ , µ = 0 , , , and Γ = iγ . F ( x t , y ) is the periodic function of y which is /R . It can be written as follow: F ( x t , y ) = F ( x t ) + + ∞ X n =1 F n ( x t , x n ) where x t = m t m w , x n = m n m w and m n = n/R . These function can be found in [10,15]. B → K l + l − Decay
At quark level, the exclusive B → K l + l − decay is described by b → sl + l − transition governedby effective Hamiltonian: H eff = − G F √ V tb V ∗ ts X i =1 C i ( µ ) O i ( µ ) (1)where O i ’s are local quark operators and C i ’s are Wilson coefficients. G F is the Fermi con-stant and V ij are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element for B → l + l − decay is obtained by b → sl + l − sandwiching transition amplitude between initial andfinal meson states. Using effective Hamiltonian the matrix element of the B → K l + l − decaywhich can be written as follows: M = G F α √ π V tb V ∗ ts ( − m b C eff siσ µν q ν (1 + γ ) blγ µ l + C eff sγ µ (1 − γ ) blγ µ l + C sγ µ (1 − γ ) blγ µ γ l ) (2)where s = q , q is the momentum transfer, q = p + p = p B − p K . Here, p , p , p B and p K are the four-momenta of the leptons, B meson and K meson respectively. Already the freequark decay amplitude M contains certain long-distance effects which usually are absorbed intoa redefinition of the Wilson coefficient. These coefficients in UED are calculated by Ref.[11] and[12] which can be written as follows, C ( µ w ) = − D ′ ( x t , /R ) ,C ( µ ) = P NDR + Y ( x t , /R ) sin θ w − Z ( x t , /R ) + P E E ( x t , /R ) ,C = − Y ( x t , /R ) sin θ w (3)where P NDR = 2 . ± . and referring to leading log approximation. Explicit expression thefunctions of the detail D ′ ( x t , /R ) , Y ( x t , /R ) and Z ( x t , /R ) are calculated in Ref.[11, 12, 16].From Eq.(2) it follows that, for obtaining matrix element for the B → K l + l − decay we needto know following matrix elements h K ( k, ε ) | ¯ sγ µ ( γ ) b | B ( p ) i and h K ( k, ε ) | ¯ siσ µν q ν b | B ( p ) i .These matrix elements in terms of form factors are parametrized as h K ( k, ε ) | ¯ sγ µ b | B ( p ) i = iε ∗ µ ( m B + m K ) V ( s ) − ( p + k ) µ ( ε ∗ .q ) V ( s ) m B + m K − q µ ( ε.q ) 2 m K s [ V ( s ) − V ( s )] , (4) h K ( k, ε ) | ¯ sγ µ γ b | B ( p ) i = 2 iǫ µναβ m B + m K ε ∗ ν p α k β A ( s ) (5) h K ( k, ε ) | ¯ siσ µν q ν b | B ( p ) i = " ( m B − m K ) ε µ − ( ε.q )( p + k ) µ F ( s )+ ( ε ∗ .q ) " q µ − sm B − m K ( p + k ) µ F ( s ) , (6) h K ( k, ε ) | ¯ siσ µν q ν γ b | B ( p ) i = − iǫ µναβ ε ∗ ν k β F ( s ) (7)where ε is the polarization vector of the K meson. The form factors entering Eq.(4) and (5) areestimated in [18, 19]. 4 ( s ) = V (0)(1 − s/m B ∗ A )(1 − s/m ′ B ∗ A ) − sm B − m K ! (8) V ( s ) = ˜ V (0)(1 − s/m B ∗ A )(1 − s/m ′ B ∗ A ) − m K m B − m K V (0)(1 − s/m B )(1 − s/m ′ B ) (9) V ( s ) = m B + m K m K V ( s ) − m B − m K m K V ( s ) (10) A ( s ) = A (0)(1 − s/m B )(1 − s/m ′ B ) (11)We can also define to the other matrix elements of the B → K l + l − decay in terms of penguinform factors. Using the Ward identities following relationship between form factors, we get F ( s ) = − ( m b − m s )( m B + m K ) 2 A ( s ) (12) F ( s ) = − ( m b + m s )( m B − m K ) V ( s ) (13) F ( s ) = 2 m K s ( m b + m s )[ V ( s ) − V ( s )] (14)In order to avoid the kinematical singularity in the matrix element at s = 0 we demand F (0) = 2 F (0) . The corresponding values at s = 0 are given by [14, 17, 18, 19], A (0) = − (0 . ± . V (0) = − (0 . ± . V (0) = − (0 . ± . V (0) = − (0 . ± . A (0) = (0 . ± . A (0) = (0 . ± . (15)Using Eq.(4),(5)(6) and (7) for the matrix element of the B → K l + l − decay we set, M = G F α √ π V tb V ∗ ts m B (" A (ˆ s ) ǫ µναβ ǫ ∗ ν p αB p βK − iB (ˆ s ) ǫ ∗ µ + iC (ˆ s )( ǫ ∗ .p B )( p B + p K ) µ + iD (ˆ s )( ǫ ∗ .p B ) q µ (¯ lγ µ l ) + " E (ˆ s ) ǫ µναβ ǫ ∗ ν p αB p βK − iF (ˆ s ) ǫ ∗ µ + iG (ˆ s )( ǫ ∗ .p B )( p B + p K ) µ + iH (ˆ s )( ǫ ∗ .p B ) q µ (¯ lγ µ γ l ) ) (16)where 5 (ˆ s ) = − A (ˆ s )1 + m K m B C eff (ˆ s ) + 2 m b m B ˆ s C eff F (ˆ s ) B (ˆ s ) = (1 + m K m B ) " C eff (ˆ s ) V (ˆ s ) + 2 m b m B ˆ s C eff (1 − m K m B ) C (ˆ s ) = 1(1 − ( m K m B ) ) ( C eff (ˆ s ) V (ˆ s ) + 2 m b m B C eff " F (ˆ s ) + 1 − ( m K m B ) ˆ s F (ˆ s ) D (ˆ s ) = 1ˆ s " C eff (ˆ s )(1 + m K m B ) V (ˆ s ) − (1 − m K m B ) V (ˆ s ) − m K m B V (ˆ s ) ! − m b m B C eff F (ˆ s ) E (ˆ s ) = − A (ˆ s )1 + m K m B C F (ˆ s ) = (1 + m K m B ) C V (ˆ s ) G (ˆ s ) = 1(1 + m K m B ) C V (ˆ s ) H (ˆ s ) = 1ˆ s " C (ˆ s )(1 + m K m B ) V (ˆ s ) − (1 − m K m B ) V (ˆ s ) − m K m B V (ˆ s ) (17)Having the explicit expression for the matrix element for the B → K l + l − decay, the nexttask is the calculation its differential decay rate. In the center of mass frame (CM) of the dileptons l + l − , where we take z = cosθ and θ is the angle between the momentum of the B meson and thatof l − , differential decay width is found to belike follows, d Γ d ˆ s ( B → K l + l − ) = G F α | V tb V ∗ ts | m B π ∆ (18)where λ = r + ( − s ) − r (1 + ˆ s ) with ˆ s = q /m B and r = m l /m B and ˆ m l = m l /m B . s = q is the dilepton invariant mass. The function ∆ is defined as follows: ∆ = 23 m B ( m B (2 ˆ m l + ˆ s ) λ | A | + 1 r ˆ s (2 ˆ m l + ˆ s )( r + ( − s ) + 2 r ( − s )) | B | + 1 r ˆ s m B (2 ˆ m l + ˆ s ) λ | C | − m B (4 ˆ m l − ˆ s ) λ | E | + 1 r ˆ s " ˆ s r + ( − s ) + 2 r ( − s ) ! + 2 ˆ m l r + ( − s ) − r (1 + 13ˆ s ! | F | + 1 r ˆ s " m B ˆ sλ + 2 ˆ m l (1 + r + 4ˆ s − s + r ( − s )) | G | + 1 r m B ˆ m l ˆ sλ | H | + 1 r ˆ s m B (2 ˆ m l + ˆ s ) r + ( − s ) − r (3 + ˆ s ) − r ( − s + ˆ s ) ! Re ( B ∗ C )+ 1 r ˆ s " m B λ m l ( − r − s ) + ˆ s ( − r + ˆ s ) ! Re ( F ∗ G ) r m B ˆ m l λRe ( F ∗ H ) − r m B ˆ m l λ ( − r ) Re ( G ∗ H ) ) (19) Now, we would like to discuss the lepton polarizations in the B → K l + l − decays. For calcu-lation of the double lepton polarization asymmetries, in the rest frame of l + l − , unit vectors s ∓ µi ( i = L, T, N ) are defined as [8, 13] s − µL = (0 , ~e − L ) = , ~p − | ~p − | ! ,s − µT = (0 , ~e − T ) = (cid:16) , ~e − L × ~e − N (cid:17) ,s − µN = (0 , ~e − N ) = , ~p K × ~p − | ~p K × ~p − | ! ,s + µL = (0 , ~e + L ) = , ~p + | ~p + | ! ,s + µT = (0 , ~e + T ) = (cid:16) , ~e + L × ~e + N (cid:17) ,s + µN = (0 , ~e + N ) = , ~p K × ~p + | ~p K × ~p + | ! . (20)where ~p ± and ~p K are the three-momenta of the leptons l + l − and K meson in the center ofmass frame (CM) of l + l − system, respectively. The longitudinal unit vector S L is boosted to theCM frame l + l − under the Lorentz transformation: ( s ∓ µL ) CM = ( | ~p ∓ | m l , E l ~p ∓ m l | ~p ∓ | ) , (21)where ~p + = − ~p − , E l and m l are the energy and mass of leptons in the CM frame, respectively.The transversal and normal unit vectors s ∓ µT , s ∓ µN are not changed under the Lorentz boost. Thedouble lepton polarization asymmetries are defined as: P ∓ i ( s ) = d Γ ds ( ~n ∓ = ~e ∓ i ) − d Γ ds ( ~n ∓ = − ~e ∓ i ) d Γ ds ( ~n ∓ = ~e ∓ i ) + d Γ ds ( ~n ∓ = − ~e ∓ i ) (22)where ~n ∓ is the unit vectors in the rest frame of the lepton. The next step, we calculateddouble-lepton polarization asymmetries which is define as P ij : P LL = 1∆ 23 m B ( m B (2 ˆ m l − ˆ s ) λ | A | + 1 r ˆ s (2 ˆ m l − ˆ s )( r + ( − s ) + 2 r ( − s )) | B | + 1 r ˆ s m B (2 ˆ m l − ˆ s ) λ | C | + 2 m B (4 ˆ m l − ˆ s ) λ | E | r ˆ s " ˆ s r + ( − s ) + 2 r ( − s ) ! − m l r + 5( − s ) + 2 r ( − s ! | F | − r ˆ s " m B ˆ sλ − m l (5 + 5 r − s + 2ˆ s − r (5 + 2ˆ s )) | G | + 1 r m B ˆ m l ˆ sλ | H | + 1 r ˆ s m B (2 ˆ m l − ˆ s ) r + ( − s ) − r (3 + ˆ s ) − r ( − s + ˆ s ) ! Re ( B ∗ C )+ 1 r ˆ s " m B λ m l ( − r + 2ˆ s ) − ˆ s ( − r + ˆ s ) ! Re ( F ∗ G ) − r m B ˆ m l λRe ( F ∗ H ) − r m B ˆ m l λ ( − r ) Re ( G ∗ H ) ) (23) P NN = 1∆ 23 m B ( m B ( − m l + ˆ s ) λ | A | + 1 r ˆ s ˆ sλ + 2 ˆ m l ( r + ( − s ) + 2 r ( − s )) ! | B | − r ˆ s m B (2 ˆ m l + ˆ s ) λ | C | + m B (4 ˆ m l − ˆ s ) λ | E | + 1 r ˆ s (2 ˆ m l + ˆ s ) λ | F | + 1 r ˆ s " m B ˆ sλ + 2 ˆ m l (1 + r + 4ˆ s − s + 2 r ( − s )) | G | + 1 r m B ˆ m l ˆ sλ | H | − r ˆ s m B (2 ˆ m l + ˆ s ) r + ( − s ) − r (3 + ˆ s ) − r ( − s + ˆ s ) ! Re ( B ∗ C )+ 1 r ˆ s " m B λ m l ( − r − s ) + ˆ s ( − r + ˆ s ) ! Re ( F ∗ G ) − r m B ˆ m l λRe ( F ∗ H ) − r m B ˆ m l λ ( − r ) Re ( G ∗ H ) ) (24) P T T = 1∆ 23 m B ( m B (4 ˆ m l + ˆ s ) λ | A | + 1 r ˆ s − ˆ sλ + 2 ˆ m l ( r + ( − s ) + 2 r ( − s )) ! | B | + 1 r ˆ s m B (2 ˆ m l − ˆ s ) λ | C | + m B (4 ˆ m l − ˆ s ) λ | E | + 1 r ˆ s λ ( − m l + ˆ s ) | F | + 1 r ˆ s " m B ˆ sλ − m l (5 + 5 r − s + 2ˆ s − r (5 + 2ˆ s )) | G | − r m B ˆ m l ˆ sλ | H | + 1 r ˆ s m B (2 ˆ m l − ˆ s ) r + ( − s ) − r (3 + ˆ s ) − r ( − s + ˆ s ) ! Re ( B ∗ C )+ 1 r ˆ s " m B λ − m l ( − r + 2ˆ s ) + ˆ s ( − r + ˆ s ) ! Re ( F ∗ G )+ 1 r m B ˆ m l λRe ( F ∗ H ) + 1 r m B ˆ m l λ ( − r ) Re ( G ∗ H ) ) (25) P LN = 1∆ 1 r √ ˆ s m B ˆ m l π √ λ " ( − r + ˆ s ) Im ( B ∗ F ) + m B λIm ( C ∗ F ) m B ( − r )( − r + ˆ s ) Im ( B ∗ G ) + m B ( − r ) λIm ( C ∗ G ) − m B ˆ s ( − r + ˆ s ) Im ( B ∗ H ) − m B ˆ sλIm ( C ∗ H ) (26) P LT = 1∆ 1 √ ˆ s m B ˆ m l πλ s − m l ˆ s − r ( − r + ˆ s ) | F | − r m B ( − r ) λ | G | + 2 m B ˆ sRe ( B ∗ E )+ 2 m B ˆ sRe ( A ∗ F ) − r m B (2 + 2 r + ˆ s − r (4 + ˆ s )) Re ( F ∗ G ) + 1 r m B ˆ s ( − r + ˆ s ) Re ( F ∗ H )+ 1 r m B ˆ sλRe ( G ∗ H ) ! (27) P T L = 1∆ 1 √ ˆ s m B ˆ m l πλ s − m l ˆ s − r ( − r + ˆ s ) | F | − r m B ( − r ) λ | G | − m B ˆ sRe ( B ∗ E ) − m B ˆ sRe ( A ∗ F ) − r m B (2 + 2 r − s + ˆ s − r (4 + ˆ s )) Re ( F ∗ G ) + 1 r m B ˆ s ( − r + ˆ s ) Re ( F ∗ H )+ 1 r m B ˆ sλRe ( G ∗ H ) ! (28) P T N = 1∆ − m B λ r s − m l ˆ s " − m B r ˆ sIm ( A ∗ E ) + Im ( B ∗ F ) + m B ( − r + ˆ s ) Im ( C ∗ F )+ ( − r + ˆ s ) Im ( B ∗ G ) + m B λIm ( C ∗ G ) ! (29) P NL = 1∆ 1 r √ ˆ s m B ˆ m l πλ " − ( − r + ˆ s ) Im ( B ∗ F ) − m B λIm ( C ∗ F ) − m B ( − r )( − r + ˆ s ) Im ( B ∗ G ) − m B ( − r ) λIm ( C ∗ G )+ m B ˆ s ( − r + ˆ s ) Im ( B ∗ H ) + m B ˆ sλIm ( C ∗ H ) (30) P NT = 1∆ 4 m B λ r s − m l ˆ s " − m B r ˆ sIm ( A ∗ E ) + Im ( B ∗ F ) + m B ( − r + ˆ s ) Im ( C ∗ F )+ ( − r + ˆ s ) Im ( B ∗ G ) + m B λIm ( C ∗ G ) ! (31) In this section, we present our numerical results on the double lepton polarization asymmetriesfor the B → K l + l − decays. First, we present the values of input parameters are: m B = 5 . GeV , m B ∗ = 5 . GeV , m K = 1 . GeV ,m b = 4 . GeV , m s = 0 . GeV , m µ = 0 . GeV , m τ = 1 . GeV , | V tb V ∗ ts | = 0 . , α − = 137 , G F = 1 . × − GeV − , τ B = 1 . × − s. (32)The B → K transition form factors are the main input parameters in performing the numericalanalysis, which are embedded into the expressions of the double-lepton polarization asymmetries.9or them we have used their expression given by Eq. (8-15). The differential decay rate for B → K l + l − can be defined in terms of integration on ˆ s , which is determined to the range of the m l ≤ s ≤ ( m B − m K ) .In Fig.1, we present the dependence of the P LL for the B → K µ + µ − decay as a function of s/m B . We see that, P LL in UED compatible with the SM result. Increasing ˆ s , P LL is moderatefor the low of ˆ s . The effect of KK contribution in the Wilson coefficient are consistent for /R =200 GeV at low value of ˆ s . /R = 200 GeV value is greater than /R = 400 GeV . In Fig.2,Double lepton longitudinal polarization asymmetries for the B → K τ + τ − decay is presented,From this figure is follows, UED model prediction coincide with the SM result. One can seethat the value of the longitudinal polarization is different in the low of ˆ s for the B → K τ + τ − decay. While /R = 200 GeV value is max in the UED model, The SM result is approximatelytwo times lower than this value. In Fig.3, For the B → K µ + µ − decay, we analysis to thenormal polarizations. We obtained good result at the /R = 200 GeV in UED model. Wecan see that the effect of extra dimension are very noticeable at the small value of ˆ s . Whenthe value of ˆ s close to . , all the value of normal polarization is coincide with each other. In ˆ s = 0 . , the value of /R = 200 GeV is five times bigger than SM result. But in Fig.4, forthe B → K τ + τ − decay, it is similar to the P LL result. In Fig.5, We examine to the transversalpolarization for the B → K µ + µ − decay. At the /R = 200 GeV value, we compared to that ofthe SM prediction P T T is larger from SM. Again, the effects of extra dimension are distinguishedat the small value of momentum transfer ˆ s where P T T is minimum. For the ˆ s = 0 . value, allpolarization values are decreases. In Fig.6, We analysis to transversal polarization as a function ofthe ˆ s for the B → K τ + τ − decay. We observe a little contributions from UED model, especiallyin the /R = 400 GeV value. But UED model is better than SM in this figure. All modelvalues come together with the SM result in the ˆ s = 0 . value. In Fig.7, we investigate P LT polarization. We see that increasing ˆ s , P LT increase until ˆ s = 0 . GeV . After this value of ˆ s two models are decrease until ˆ s = 0 . GeV . ( P LT ) UED = 2( P LT ) SM at /R = 200 GeV . SoIt is also very useful for establishing new physics. In Fig.8, We show our predictions for the P T L for B → K τ + τ − decay. We get | ( P T L ) UED | > | ( P T L ) SM | . This result can serve as a goodtest for discrimination of two models. The other polarizations for the B → K l + l − decay, wehave imaginary part and therefore there is no interference terms between SM and UED modelcontributions.In conclusion, we have studied the double-lepton polarization asymmetries in the UED model.We obtain different double-lepton polarization asymmetries which is very sensitive to the UEDmodel. It has been shown that all these physical observebles are very sensitive to the existence ofnew physics beyond SM and their experimental measurements can give valuable information onit. Acknowledgements
The author would like to thank T. M. Aliev, M. Savcı and A. Ozpineci for useful discussionsduring the course of the work. 10 P LL ( B → Κ µ + µ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 1: The dependence of the Longitudinal polarization,for B → K µ + µ − decay, as a functionof the ˆ s . P LL ( B → Κ τ + τ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 2: The dependence of the Longitudinal polarization,for B → K τ + τ − decay, as a functionof the ˆ s . . 11 P NN ( B → Κ µ + µ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 3: The dependence of the Normal polarization,for B → K µ + µ − decay, as a function ofthe ˆ s . P NN ( B → Κ τ + τ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 4: The dependence of the Normal polarization,for B → K τ + τ − decay, as a function ofthe ˆ s . 12 P TT ( B → Κ µ + µ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 5: The dependence of the Transversal polarization,for B → K µ + µ − decay, as a functionof the ˆ s . -1-0.99-0.98-0.97-0.96-0.950.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 P TT ( B → Κ τ + τ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 6: The dependence of the Transversal polarization,for B → K τ + τ − decay, as a functionof the ˆ s . 13 P L T ( B → Κ µ + µ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 7: The dependence of the P LT polarization,for B → K µ + µ − decay, as a function of the ˆ s . P T L ( B → Κ τ + τ − ) s/m B2 SM1/R=2001/R=2501/R=3001/R=400
Figure 8: The dependence of the P T L polarization,for B → K τ + τ − decay, as a function of the ˆ s . 14 eferences [1] K. Abe, et al.,Belle Collaboration Prep, hep-ex/0410006 , (2004) .[2] B. Aubert, et al.,BaBar Collaboration
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