Search for new physics via photon polarization of b→sγ
Naoyuki Haba, Hiroyuki Ishida, Tsuyoshi Nakaya, Yasuhiro Shimizu, Ryo Takahashi
aa r X i v : . [ h e p - ph ] A p r SU-HET-01-2015TU-987
Search for new physics via photon polarization of b → sγ Naoyuki Haba , Hiroyuki Ishida , Tsuyoshi Nakaya , Yasuhiro Shimizu , and Ryo Takahashi Graduate School of Science and Engineering, Shimane University,Matsue, 690-8504, Japan Department of Physics, Kyoto University,Kyoto, 606-8502, Japan Department of Physics, Tohoku University,Sendai, 980-8578 Japan Academic Support Center, Kogakuin University,Hachioji, 192-0015 Japan
Abstract
We suggest a discriminant analysis of new physics beyond the stan-dard model through a detection of photon polarization in a radiative Bmeson decay. This analysis is investigated in SUSY SU(5) GUT withright-handed neutrino and left-right symmetric models. New physicssearch via CP asymmetry in the same process are also evaluated in eachmodel for comparison. We show that new physics can be found via de-tecting the photon polarization in a parameter space of TeV energy scale.
Introduction
In this decade, we have never seen an exotic elementary particle except for the most likely tothe standard model (SM) Higgs boson which was discovered at the LHC experiment. This factmight imply that a scale of new physics (NP) is higher than the energy scale that the LHC canreach. Thus, it is important to consider the possibility that such exotics are hard to be directlyproduced by collider experiments even though the next LHC will become 14 TeV.On the other hand, indirect searches become powerful way to explore the existence of NPand related phenomena beyond the SM even if a new particle is impossible to be produceddirectly. One popular approach is flavor physics. For instance, flavor changing processes such as µ → eγ and b → sγ retain much information about NP. Models of supersymmetric grand unifiedtheory (SUSY GUT) predict relatively large branching ratios of Br( µ → eγ ) ∼ O (10 − -10 − )for the NP scale (the right-handed selectron mass) of M NP ≃ (100-300) GeV in an SU(5) caseand Br( µ → eγ ) ∼ O (10 − -10 − ) in an SO(10) case [1, 2]. Then, Ref. [3] suggested that themeasurement of the angular distribution of e with respect to the spin direction of the muon inthe µ → eγ process might distinguish among several extensions of the SM if the signal could bedetected. This implies that the precise determination of the chirality of the final e state in the µ → eγ process might become a clue to obtain the evidence of NP. This situation is adoptedto the b → sγ process, i.e. one would be able to discriminate among the SM and NP suchas SUSY GUT models, the left-right symmetric standard model (LRSM), and the Pati-Salammodels and so on, if one could precisely determine the chirality of the final s quark. The b quarkcan radiative decay into the s quark in the B meson and the chirality of the s quark is almostdetermined as left-handed in the SM. Accordingly, if we find more right-handed s quarks in theprocess than ones expected in the SM, we can expect that some kind of NP must cause thisphenomena.How about the measurement of the chirality of the s quark in the b → sγ process for the NPsearch? One may naively think that the determination of chirality of quarks except for the topquark is impossible (the top quark can decay before the hadronization). The b → sγ processoccurs through the dipole type operators, s L σ µν bF µν ( s R σ µν bF µν ) which induce left- ( right-) handed photon. The information on the chirality of the s quark is imprinted on the photonpolarization. In addition, there is no parity violation in QCD, the relation between the chiralityof the s quark and the photon polarization is unchanged even if the hadronization is taken intoaccount. Therefore, one can determine the chirality of s quark in the b → sγ process from themeasurement of photon polarization [4, 5]. In Refs. [6, 7], the authors mentioned that the higherorder correction may induce the right-handed photon even though it is in the SM.At e + e − colliders, such as Belle and BaBar, B d mesons, which are spin 0 particles, are pro-duced from the Υ(4 S ) resonance. The photon polarization of the B d → X s γ decay is determinedfrom measurements of hadronic angular distributions due to the conservation of angular momen-tum. The LHCb collaboration actually reported the result of observation of photon polarization1y measuring the angular distribution of produced mesons in the B → Kππγ process [8]. Inaddition to the B → Kππγ process, there is another possibility to determine the photon polar-ization by the B → K ∗ l + l − process. Although there exists a box diagram in the process, theradiative decay diagram (penguin) becomes dominant (the box diagram is suppressed) in a lowinvariant mass region of dileptons [9, 10] (see also [11]). The chirality of the s quark in the K ∗ meson can be lead by the chirality of photon due to the conservation of the spin. Then, it isimportant to discuss the possibility of the detection of the photon chirality. In this work, we willconsider the b → sγ process. In particular, a ratio of the Wilson coefficients of a dipole operatorand a polarization parameter of photon will be firstly evaluated at a typical point in a model ofSUSY SU(5) GUT with the right-handed neutrino ( N R ) and LRSM in order to clarify whetherone can find an evidence of NP or distinguish among the SM and NP, or not.In addition to the determination of the photon chirality, the CP asymmetry in the b → sγ process which are direct CP asymmetry, A CP ( b → sγ ) and time-dependent CP asymmetry, S CP ( B → K s π γ ), is also a sensitive observable to NP [12]. Actually, the CP violating effectsfrom NP can be enough larger than the SM expectation as A CP ( b → sγ ) ≃ − . b → sγ process is accidentally small. We will also evaluate the magnitudeof A CP ( b → sγ ) in the SUSY SU(5) GUT with N R and the LRSM although the magnitudestrongly depends on CP violating phases in the models. Furthermore, S CP ( B → K s π γ ) canalso become larger than the SM expectation as S CP ( B → K s π γ ) ≃ − .
3. We will show thisvalue is insensitive to the CP phase. Then, we will compare experimental detectability for themodels of NP between the determination of photon polarization and the observation of both CPasymmetry in the b → sγ process.We will suggest that one can discriminate NP beyond the SM by the detection of photonpolarization in b → sγ process. We will point out that time-dependent CP asymmetry is themost stringent constraint in our sample model point at the moment. However, it will actuallyturn out that the LHCb with 2 fb − for the determination of photon polarization may check theexistence of NP scale up to several TeV in both models. We investigate the photon polarization in the radiative rare decay, b → sγ , process in a SUSYSU(5) GUT with N R and the LRSM for the search of NP. The Wilson coefficients C and C ′ ofthe dipole operator for the b → sγ process are important for the analyses of photon polarization.The effective Hamiltonian reads H eff ⊃ − G F √ V tb V ∗ ts ( C O + C ′ O ′ ) , (1) This issue tells us that the theorists have to clarify the prediction of photon polarization in each model. O = e π m b (¯ sσ µν P R b ) F µν , (2)where G F is the Fermi constant, m b is the bottom quark mass, σ µν = i [ γ µ , γ ν ], and P R,L = (1 ± γ ) (e.g., see [14]). O ′ is obtained by replacing L ↔ R in O . Because left-handed s quarkcomes from O and right-handed one comes from O ′ , we might be able to determine the chiralityof s quark by the difference of Wilson coefficients.When one considers physics beyond the SM, there might be additional contributions to C and C ′ from NP. In these cases, we can generically describe C and C ′ as C = C SM7 + C NP7 and C ′ = C ′ + C ′ , respectively. The coefficients at the b quark mass scale µ b are given by theleading logarithmic calculations with QCD corrections to the b → sγ process, C ( µ b ) = η C ( m W ) + 83 ( η − η ) C ( m W ) + X i =1 h i η a i , (3) C ′ ( µ b ) = η C ′ ( m W ) + 83 ( η − η ) C ′ ( m W ) , (4)at the leading order where η = α s ( m W ) /α s ( µ b ), α s ≡ g s / (4 π ), g s is the strong coupling constant, m W is the W boson mass, and h i and a i are numerical coefficients [15, 16, 17]. C is the coefficientof chromomagnetic operator O = g s π m b (¯ sσ µν T A P R b ) G Aµν , (5)in the ∆ F = 1 effective Hamiltonian, H eff ⊃ − G F √ V tb V ∗ ts ( C O + C ′ O ′ ) , (6)where T A are the generators of SU (3) C and O ′ is also obtained by replacing L ↔ R in O . And,we also describe C = C SM8 + C NP8 and C ′ = C ′ + C ′ including contributions from NP. At first, we give a brief review of a model of SUSY SU(5) GUT with N R . In a simple SU(5)GUT model, the final s quark must have the same chirality as in the SM. When there is N R ,a neutrino Yukawa coupling induces additional flavor mixings in the right-handed down squarkwhich derives the opposite chirality of s quark. Thus, we adopt the model of SUSY SU(5) GUTwith N R . The superpotential in this model is given by W = 14 f uij Ψ i Ψ j H + √ f dij Ψ i Φ j ¯ H + f νij Φ i ¯ N j H + M ij ¯ N i ¯ N j , (7)3here Ψ i are -dimensional multiplets, Φ i are -dimensional ones, N i denote the right-handedneutrino superfields, and H ( ¯ H ) is - (¯ -) dimensional Higgs multiplets. i and j mean thegeneration of the fermions, i, j = 1 , , f u , f d , and f ν are Yukawa coupling matrices for theup-type quarks, down-type quarks (charged leptons) and neutrinos, respectively. These are givenby f uij = V ki f u k e iϕ uk V kj , (8) f dij = f d i δ ij , (9) f νij = e iϕ di U ∗ ij f ν j , (10)without a loss of generality, where V and U are the Cabibbo-Kobayashi-Maskawa (CKM) andPontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, respectively. ϕ u k and ϕ d i are CP-violatingphases, and f u k and f d i are Yukawa couplings of the up- and down-type quarks (charged lep-tons), respectively. For the neutrinos sector, the light neutrino masses are given by the seesawmechanism m ν i = f ν i v u /M N i , where v u is the vacuum expectation value (VEV) of the up-typeHiggs in H , and M N i are the mass eigenvalues of the right-handed neutrinos. Here, we assumea diagonal right-handed Majorana mass matrix M ij for simplicity. We discuss C and C ′ , which determine the magnitude of the photon polarization, in the modelof SUSY SU(5) GUT with N R . In supersymmetric models, the dominant contributions to C , and C ′ , arise from loop diagrams of the charged Higgses, charginos, and gluinos. Thus, C NP7 , = C H ± , + C ˜ χ ± , + C ˜ g , and C ′ NP7 , = C ′ H ± , + C ′ ˜ χ ± , + C ′ ˜ g , in the SUSY SU(5) model with N R where C H ± , ( C ′ H ± , ), C ˜ χ ± , ( C ′ ˜ χ ± , ), and C ˜ g , ( C ′ ˜ g , ) are the contributions to C , ( C ′ , ) from thecharged Higgses, charginos, and gluinos, respectively. These contributions are calculated as [18] C H ± = C ′ H ± ≃ (cid:18) − ǫt β ǫt β (cid:19) h ( y t ) , (11) C ˜ χ ± = 4 G F √ g ˜ m (cid:20) ( δ LLu ) V tb V ∗ ts µM ˜ m f (1)7 ( x , x µ ) + m t M W A t µ ˜ m f (2)7 ( x µ ) (cid:21) t β ǫt β , (12) C ′ χ ± = 4 G F √ g ˜ m (cid:20) ( δ RRu ) V tb V ∗ ts µM ˜ m f (1)7 ( x , x µ ) + m t M W A t µ ˜ m f (2)7 ( x µ ) (cid:21) t β ǫt β , (13) C ˜ g ± = 4 G F √ g s ˜ m (cid:20) M ˜ g m b ( δ RLd ) V tb V ∗ ts g (1)7 ( x g ) + M ˜ g µ ˜ m t β ǫt β ( δ LLd ) V tb V ∗ ts g (2)7 ( x g ) (cid:21) , (14) C ′ g ± = 4 G F √ g s ˜ m (cid:20) M ˜ g m b ( δ LRd ) V tb V ∗ ts g (2)7 ( x g ) + M ˜ g µ ∗ ˜ m t β ǫt β ( δ RRd ) V tb V ∗ ts g (1)7 ( x g ) (cid:21) , (15)4nd C H ± = C ′ H ± ≃ (cid:18) − ǫt β ǫt β (cid:19) h ( y t ) , (16) C ˜ χ ± = 4 G F √ g ˜ m (cid:20) ( δ LLu ) V tb V ∗ ts µM ˜ m f (1)8 ( x , x µ ) + m t M W A t µ ˜ m f (2)8 ( x µ ) (cid:21) t β ǫt β , (17) C ′ χ ± = 4 G F √ g ˜ m (cid:20) ( δ RRu ) V tb V ∗ ts µM ˜ m f (1)8 ( x , x µ ) + m t M W A t µ ˜ m f (2)8 ( x µ ) (cid:21) t β ǫt β , (18) C ˜ g ± = 4 G F √ g s ˜ m (cid:20) M ˜ g m b ( δ RLd ) V tb V ∗ ts g (1)8 ( x g ) + M ˜ g µ ˜ m t β ǫt β ( δ LLd ) V tb V ∗ ts g (2)8 ( x g ) (cid:21) , (19) C ′ g ± = 4 G F √ g s ˜ m (cid:20) M ˜ g m b ( δ LRd ) V tb V ∗ ts g (2)8 ( x g ) + M ˜ g µ ∗ ˜ m t β ǫt β ( δ RRd ) V tb V ∗ ts g (1)8 ( x g ) (cid:21) , (20)at the weak scale and ǫ ≃ α s / (3 π ) ∼ O (10 − ) for a degenerate SUSY spectrum, t β = tan β ≡ v u /v d , v d is the VEV of down-type Higgs, g is the SU (2) L gauge coupling constant, ˜ m is anaveraged squark mass, ( δ XYq ) ij ( q = u, d and X, Y = L, R ) are mass insertion parameters, µ isthe supersymmetric Higgs mass, M x ( x = 2 , ˜ g ) are the gaugino masses, and A t is the soft scalarscoupling for the top quark. h , , f (1 , , , and g (1 , , are loop functions, which are given in AppendixA. The mass insertion parameters are given in Appendix B. And, we define y t ≡ m t /M H ± , x ≡ | M | / ˜ m , x µ ≡ | µ | / ˜ m , and x g ≡ M g / ˜ m where m t and M H ± are the top quark andcharged Higgs masses, respectively. The contributions from the charged Higgs and chargino to C ′ are suppressed by m s /m b .In order to see the magnitude of contributions from NP, we estimate the ratio | C ′ /C | atthe b quark mass scale, which determines the size of polarization of photon as seen below. Thevalue of the ratio in the model of SUSY SU(5) GUT with N R is shown by the red solid curvein Fig. 1. The black solid line indicates the SM case | C ′ /C SM7 | which can be approximated as | C ′ /C SM7 | ≃ m s /m b . One can see that | C ′ /C | in the SUSY SU(5) with N R case is enhancedfrom the SM. This means that the final state of s R in b → sγ increases compared to the SMwhile the most of final state of b → sγ in the SM is s L due to the suppression proportional to m s /m b .In Fig. 1, the horizontal axis is a typical scale of NP, which is a SUSY breaking scale M SUSY in the SUSY SU(5) GUT with N R case. One can see that at a large limit of M SUSY , the ratio | C ′ /C | closes to the SM case, | C ′ /C | → | C ′ /C SM7 | . The contours with n % correspond tocase that there is n % misidentification in the C measurement at experiments, i.e. the contoursdenote | C ′ (1 + n/ / ( C (1 − n/ | . This misidentification corresponds to a mismatch inthe conversion of left-handed helicity to the left-handed chirality. (The helicity is determinedin experiments.) For instance, if one identifies the left-handed helicity with the left-handedchirality with 10% misidentification, the right-handed chirality is over estimated as 110% of the5 M 0 % SM 10 % SM 30 % SUSY SU H L % SUSY SU H L % SUSY SU H L %
500 1000 1500 20000.00.10.20.30.40.50.6 M SUSY @ GeV D È C ' (cid:144) C È Figure 1: The magnitude of | C ′ /C | in the SM and SUSY SU(5) GUT with N R which aredepicted by the black and red curves, respectively. The contours with n % correspond to casesthat there is n % misidentification in the C measurement at experiments, i.e. each of the contoursdenotes | C ′ (1 + n/ / ( C (1 − n/ | .true value. Thus, the contours go above as n increases in Fig. 1. In the calculation, we take t β = 10 , M SUSY = m / , m = A = A t = A b = 1 TeV , µ = 1 . m , (21) M = 0 . M SUSY , M ˜ g = 2 . M SUSY , M H c = 10 GeV , ˜ m u = q m Q L m u R , (22)˜ m d = q m Q L m d R , m Q L = m + 6 . M , m u R = m + 6 . M , (23) m d R = m + 6 . M , f ν i = 1 , ϕ u = 0 . , ϕ d = π, (24)as a typical point where CP phases are given in the radian unit. We take other values ofparameters in the SM and the neutrino sector (PMNS mixing angles) as the best fit values givenin [19, 20].Next, we consider the polarization parameter of photon λ γ at the b quark mass scale definedas λ γ ≡ Re[ C ′ /C ] + Im[ C ′ /C ] − C ′ /C ] + Im[ C ′ /C ] + 1 . (25)In order to measure λ γ we need to consider a parity-odd observable in the B d → X s γ decaysince the photon polarization is parity-odd. In Ref.[4, 5] they proposed that λ γ can be measured Note that ϕ d is sensitive to the CEDM. The allowed minimal and maximal values by the CEDM constraintare π and 3 π/ π/ ϕ d but the difference between these values appears inthe calculation of direct CP asymmetry as we will show in Section 3. We have numerically checked that anotherphase, ϕ u , is not sensitive to our evaluation. USY SU H L H j d =Π L SUSY SU H L H j d = Π (cid:144) L SMSM H % error L
500 1000 1500 2000 2500 3000 - - M SUSY @ GeV D Λ Γ Figure 2: The polarization parameter λ γ in b → sγ in the SM and SUSY SU(5) GUT with N R which are depicted by the black and red curves, respectively. We also show ϕ d = 3 π/ Kππ decay plane in the K (1400) rest frame. We show λ γ in Fig. 2 and the SM and SUSY SU(5) GUT with N R casesare depicted by the black and red curves, respectively. We also show ϕ d = 3 π/ O operator [11], then we depict it by the black dashed curve.The superKEKB with an integrated luminosity of 75 ab − and the LHCb with 2 fb − mightreach the 20 ( | λ γ | = 0 . | λ SM γ | ) and 10% ( | λ γ | = 0 . | λ SM γ | ) precision, respectively [11, 21]. Thus,the future experiment will be able to check the NP scale up to about 1700 GeV (which corre-sponds to M ≃ M ˜ g ≃ N R . Next, we consider the case in the left-right symmetric standard model (LRSM) [22, 23, 24, 25].The model is based on the gauge group SU (2) L × SU (2) R × U (1) ˜ Y . In the model, the SMleft-handed doublet fermions are SU (2) R singlets, and the right-handed fermions including theneutrinos are SU (2) R doublets and SU (2) L singlets. For the Higgs sector, the model includes abi-doublet scalar Φ under the SU (2) L × SU (2) R transformation, an SU (2) R triplet ∆ R , and an SU (2) L triplet ∆ L in order to realize a realistic symmetry breaking, SU (2) L × SU (2) R × U (1) ˜ Y → SU (2) L × U (1) Y → U (1) em . The symmetry breaking can be undertaken by the VEVs of Φ, ∆ R ,7nd ∆ L as h Φ i = (cid:18) κ κ ′ e iω (cid:19) , h ∆ R i = (cid:18) v R (cid:19) , h ∆ L i = (cid:18) v L e iθ L (cid:19) , (26)with six real numbers κ , κ ′ , ω , v R , v L , and θ L . Regarding the magnitude of the VEVs, v R shouldbe much larger than the electroweak (EW) scale to suppress the right-handed currents at lowenergy, and the EW ρ -parameter limits v L to be v L .
10 GeV [26]. In this work, we take v L = 0for simplicity, which is usually taken in literatures (e.g., [27, 28]). Since the VEV of Φ leads tothe standard EW symmetry breaking, we define v ≡ √ κ + κ ′ = 174 GeV, tan β LR ≡ κ/κ ′ , and ǫ LR ≡ v/v R .The charged gauge bosons are given by the admixture of the mass eigenstates as (cid:18) W − L W − R (cid:19) = (cid:18) cos ζ − sin ζ e iω sin ζ e iω cos ζ (cid:19) (cid:18) W − W − (cid:19) . (27)The masses of charged gauge bosons are approximated as M W ≃ g L v √ − ǫ sin β LR cos β LR ) , M W ≃ g R v R (1 + 14 ǫ ) , (28)where g L,R are the gauge couplings of SU (2) L,R and we take g R /g L = 1 for simplicity in thenumerical analysis. The mixing angle is given as sin ζ ≈ ( M W /M W ) sin 2 β LR . M W is identifiedwith M NP in the LRSM. There are also charged and heavy neutral Higgs bosons in the LRSM.And their masses are nearly the same, M H ± ≃ M H ≃ M A , where M H ± are the masses of thecharged Higgs bosons and M H ,A are the neutral Higgs bosons masses [27, 29]. In this work, werepresent both the charged and neutral Higgs bosons masses as M H for simplicity. Regardingthe flavor mixing matrices, we assume V = V L = V R , where V L and V R are the mixing matricesfor the left- and right-handed quarks, respectively. V = V L = V R is taken in the so-called themanifest LRSM [30, 31] and we also take the equality in this work. In our numerical analyses,we have three free parameters, i.e. M H , M NP = M W , and tan β LR . The value of M W is not exactly determined even if one takes g R /g L = 1 and fixes the value of M H , becausethe heavy Higgs masses depend on scalar quartic couplings, which can be in region from 0 to 4 π , and/or trilinearcouplings. Thus, one can generally take both M W and M H as free parameters in this model. M 0 % SM 10 % SM 30 % LRSM 0 % LRSM 10 % LRSM 30 % M W R @ GeV D È C ' (cid:144) C È Figure 3: The magnitude of | C ′ /C | in the SM and LRSM which are depicted by the black andblue curves, respectively. The meaning of each contour is the same as Fig. 1. C NP7 and C ′ in the LRSM are generally given by C NP7 = − sin ζ (cid:18) D ′ ( x t ) − M W M W D ′ (˜ x t ) (cid:19) + m t m b g R g L V Rtb V tb sin ζ cos ζ e iω (cid:18) A LR ( x t ) − M W M W A LR (˜ x t ) (cid:19) + m c m b g R g L V ∗ cs V Rcb V ∗ ts V tb sin ζ cos ζ e iω (cid:18) A LR ( x c ) − M W M W A LR (˜ x c ) (cid:19) + m t m b tan(2 β LR )cos(2 β LR ) e iω V Rtb V tb h ( y )+ tan(2 β LR ) A H ( x ) , (29) C ′ = g R g L V R ∗ ts V Rtb V ∗ ts V tb (cid:18) sin ζ D ′ ( x t ) + cos ζ M W M W D ′ (˜ x t ) (cid:19) + m t m b g R g L V R ∗ ts V ∗ ts sin ζ cos ζ e − iω (cid:18) A LR ( x t ) − M W M W A LR (˜ x t ) (cid:19) + m c m b g R g L V R ∗ cs V cb V ∗ ts V tb sin ζ cos ζ e − iω (cid:18) A LR ( x c ) − M W M W A LR (˜ x c ) (cid:19) + m t m b tan(2 β LR )cos(2 β LR ) e − iω V R ∗ ts V ∗ ts h ( y ) + V R ∗ ts V Rtb V ∗ ts V tb (2 β LR ) A H ( x ) , (30)when one does not assume g R /g L = 1 and V = V R , where loop functions D ′ ( x ) and A LR ( x )are given in Appendix A, and x t ≡ m t /m W , ˜ x t ≡ m t /m W , x c ≡ m c /m W , ˜ x c ≡ m c /m W ,and y ≡ m t /M H . The ratio | C ′ /C | in the LRSM is shown by the blue curves in Fig. 3. Thevalue of the ratio in the LRSM is larger than the both cases of SM and SUSY SU(5) with N R model because the right-handed current in the LRSM is more effective than those models. In9 RSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L SMSM H % error L - - - - - - - M W R @ GeV D Λ Γ Figure 4: The polarization parameter λ γ in b → sγ in the SM and LRSM which are depictedby the black and blue curves, respectively. We also show ω = π/ π/ g R g L = 1 , V = V R , tan β LR = 10 , ω = π , M H ± = 15TeV , (31)as a sample point. The polarization parameter in the LRSM is shown by the blue curve in Fig. 4. The SM andLRSM cases are depicted by the black and blue curves, respectively. We also show ω = π/ π/ M NP ≃ . − will check the scale of the right-handed gauge boson up to 3 . Next, we evaluate the CP asymmetry in b → sγ process in each model. The CP asymmetrycan be categorized into two parts: One is the direct CP asymmetry which is induced by theCP phase in the decay amplitude, and the other is the time-dependent CP asymmetry which isinduced during the meson mixing. Although there are crossing points of the NP lines with the SM prediction line in all figures afterward, theyare due to the fixing of the charged Higgs mass. .1 Direct CP asymmetry In addition to the determination of photon polarization, the observation of CP asymmetry in the b → sγ process is still sensitive to the existence of NP. Thus, we evaluate the CP asymmetry ofthe process in both models of SUSY SU(5) GUT with N R and LRSM. The asymmetry is givenby A CP ( b → sγ ) ≡ Γ( B → X ¯ s γ ) − Γ( ¯ B → X s γ )Γ( B → X ¯ s γ ) + Γ( ¯ B → X s γ ) ≃ − | C | + | C ′ | (1 .
23 Im[ C C ∗ ] − .
52 Im[ C C ∗ + C ′ C ′ ∗ ] + 0 .
10 Im[ C C ∗ ]) − . , (32)which of course strongly depends on the CP-phases in the model where C is the coefficient of theoperator O = (¯ c α γ µ P L b α )(¯ c β γ µ P L b β ) in the effective Hamiltonian of ∆ F = 2 transitions. Notethat the contributions from NP to C and C ′ are negligibly small while C , and C ′ , includecontributions from NP, i.e. C ′ ≪ C = C SM2 = 1. Thus, A CP ( b → sγ ) is well approximated byEq. (32). We show A CP ( b → sγ ) in the SUSY SU(5) with N R and SM cases by red and black solid curvesin Fig. 5, respectively. The experimental lower bound as − . ϕ d = 3 π/ N R highly depend on the phase ϕ d . The current experimental lower bound does not constrainthe scale of NP, M SUSY in this parameter setup. Therefore, the measurement of A CP does notcurrently constrain on the NP scale in the SUSY SU(5) with N R model even if one takes themaximally allowed CP phase as ϕ d = 3 π/
2. But, the expected reach of Belle II with 50 ab − for A CP ( B → X s + d γ ) will be ±
2% precision. Thus, the future determination will check NPbetween 450-750 GeV in this typical SUSY SU(5) with N R case. For the LRSM, C NP8 and C ′ are C = ρ ∆ LR C + ρ LR m c m b sin ζ cos ζ e iα V Rcb V Lcb , (33) C ′ = ρ ∆ LR C ′ + ρ LR m c m b sin ζ cos ζ e − iα V R ∗ cb V L ∗ cb , (34) There is also an error in the SM prediction but it is enough small [32]. Therefore, we neglect such correctionhere just for simplicity. USY SU H L H j d =Π L SUSY SU H L H j d = Π (cid:144) L SMEXP LB
500 1000 1500 2000 - - - - - M SUSY @ GeV D A C P H b ® s Γ L @ % D Figure 5: The direct CP asymmetry in b → sγ in the SM and SUSY SU(5) GUT with N R ,which are depicted by the black and red solid curves. The black dashed line corresponds to thecurrent experimental lower bound [33]. We also show the ϕ d = 3 π/ LR C = m t m b sin ζ cos ζ e iα V Rtb V Ltb f LR (˜ x t ) , (35)∆ LR C ′ = m t m b sin ζ cos ζ e − iα V R ∗ ts V L ∗ ts f LR (˜ x t ) , (36)where ρ and ρ LR are the so-called magic number which are given in the Ref. [27]. f LR ( x ) is alsoa loop function for the left-right symmetric model given in Appendix A.The asymmetry in the LRSM is shown by the blue curve in Fig. 6. We also show ω = π/ π/ M NP ≥ π/
10. Hence, the A CP measurement does not give constraint on the existenceof NP in this case at the moment. Furthermore, the future Belle II with 50 ab − will checkthe LRSM model up to 3 . . ω = π/ ω = π/
4, and ω = π/ ω as with SUSY SU(5) with N R case. We also evaluate the time-dependent CP asymmetry in the B → K s π γ decay denoted as S CP .The definition of S CP is same in the both model: S CP = 2 Im (cid:2) e − iβ CKM C C ′ (cid:3) | C | + | C ′ | , (37)where 2 β CKM ≈ ◦ is a CP phase in B → K s π γ decay.12 RSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L SMEXP LB - - - - - M W R @ GeV D A C P H b ® s Γ L @ % D Figure 6: The direct CP asymmetry in b → sγ in the SM and LRSM which are depicted by theblack and blue solid curves. The black dashed line is same as in Fig. 5. We also show ω = π/ π/ We show Eq. (37) in the SUSY SU(5) with N R and SM cases by red and black solid curvesin Fig. 7, respectively. The current experimental lower bound as − .
35 is also shown by theblack dashed line. We also show ϕ d = 3 π/ M SUSY in Fig. 7 in this naiveset up. In other words, the time-dependent CP asymmetry gives the strongest constraint within λ γ , A CP , and S CP . We show Eq. (37) in the LRSM and SM cases by blue and black solid curves in Fig. 8, respectively.The current experimental lower bound constrains the mass of W R up to 7 TeV in this simple setup. The figure shows that the time-dependent CP asymmetry is the strongest constraint evenin the LRSM. As we mentioned in the Introduction, the photon polarization might become a useful way todetermine NP. Actually, the ways to determine the photon chirality by measurement of angulardistribution of the final state particles have been discussed in several papers [9, 10, 11] (Seealso [34]). In this paper, we have evaluated three observables: photon polarization, direct CPasymmetry, and time-dependent CP asymmetry. We give some comments on the comparison13
USY SU H L H j d =Π L SUSY SU H L H j d = Π (cid:144) L SMEXP LB
500 1000 1500 2000 2500 3000 - - - - - M SUSY @ GeV D S C P Figure 7: The time-dependent CP asymmetry in B → K s π γ in the SM and SUSY SU(5) GUTwith N R , which are depicted by the black and blue solid curves. The black dashed line is thecurrent experimental lower bound [33]. We also show ϕ d = 3 π/ • For the determination of photon polarization in the SUSY SU(5) with N R model, thefuture LHCb experiment with 2 fb − will be able to check the NP scale up to about 1700GeV, which corresponds to M ≃ M ˜ g ≃ • For the measurement of A CP in the SUSY SU(5) with N R model, the experiment does notcurrently constrain on the NP scale even if one takes the maximally allowed CP phase as ϕ d = 3 π/
2. The future Belle II with 50 ab − will check NP between 450-750 GeV. • For the determination of photon polarization in the LRSM, the future LHCb experimentwill check the NP scale up to 3 . • For the measurement of A CP in the LRSM, the experiment constrain up to 4 . ω . Furthermore, the future Belle II with 50 ab − will check upto 5 . ω maximize the direct CP asymmetry. On the other hand, the futureexperiment reach 3 . ω minimize the direct CP asymmetry. • The time-dependent CP asymmetry is the most stringent constraint in both models andthis CP asymmetry does not depend on CP asymmetry parameters so much. • Thus, we mention that there is a region where the determination of photon polarizationis more ascendant for the NP search than that of direct CP asymmetry in both models.However, time-dependent CP asymmetry always gives us more stringent constraint thanother observables. 14
RSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L LRSM H Ω=Π (cid:144) L SMEXP LB - - - - - - - - M W R @ GeV D S C P Figure 8: The time-dependent CP asymmetry in B → K s π γ in the SM and LRSM which aredepicted by the black and blue solid curves. The black dashed line is same as in Fig. 7. Wealso show ω = π/ π/ One might be able to obtain the existence of NP and discriminate among the SM and NP if onecan precisely determine the chirality of the s quark in the b → sγ process. The chirality of s quark can be determined by measuring the polarization of photon in the process. And, thereare several ways to measure the photon polarization. In addition to the determination of photonpolarization, the observation of CP asymmetry in the process is still sensitive to the existenceof NP. Thus, simultaneous studies of photon polarization and CP asymmetry in the b → sγ process will be intriguing for the experimental search of NP.We have investigated the b → sγ process in the SUSY SU(5) GUT with N R model and theLRSM. The ratio | C ′ /C | , the polarization parameter of photon, and the direct CP asymmetryin the process have been evaluated in both models. The time-dependent CP asymmetry seemsto be the best way to find the NP effect. However, the combination of CP violation and photonpolarization can discriminate NP beyond the SM at the end. Furthermore, there might be aregion where the determination of photon polarization is more sensitive for the new physicssearch than that of CP asymmetry. Acknowledgement
We would like to thank to Y. Okada for fruitful discussion. This work is partially supportedby Scientific Grant by Ministry of Education and Science, No. 24540272 and Grant-in-Aidfor Scientific Research on Innovative Areas titled “Unification andDevelopment of the NeutrinoScience Frontier”, No. 25105001. The work of R.T. are supported by Research Fellowships of15he Japan Society for the Promotion of Science for Young Scientists.
AppendixA Loop functions
We give loop functions [18] which are utilized in our analyses: h ( x ) = − x − x − x ) − x − x − x ) log x, (38) h ( x ) = − x − x − x ) + x − x ) log x, (39) f (1)7 , ( x, y ) = 2 x − y ( f (2)7 , ( x ) − f (2)7 , ( y )) , (40) f (2)7 ( x ) = − − x − x ) − x − x − x ) log x, (41) f (2)8 ( x ) = 1 + 5 x − x ) + x (2 + x )4(1 − x ) log x, (42) f LR ( x ) = − x + x + 44(1 − x ) − x − x ) log x, (43) g (1)7 ( x ) = − x )9(1 − x ) − x (1 + x )3(1 − x ) log x, (44) g (1)8 ( x ) = 11 + x − x ) + 9 + 16 x − x − x ) log x, (45) g (2)7 ( x ) = − x + x )9(1 − x ) − x (1 + x )3(1 − x ) log x, (46) g (2)8 ( x ) = 53 + 44 x − x − x ) + 3 + 11 x + 2 x − x ) log x, (47) D ′ ( x ) = − x − x + 7 x x − + 3 x − x x − log x, (48) A LR ( x ) = − x + 31 x − x − − x − x ( x − log x, (49) A H ( x ) = 22 x − x + 25 x x − − x − x + 4 x x −
1) log x. (50)16 Mass insertion parameters
The mass insertion parameters are defined as (e.g., see [35])( δ XXq ) ij ≡ ( m q X ) ij ˜ m f , (51)( δ XYd ) ij ≡ v d ( A d − µt β ) ij ˜ m f , (52)( δ XYu ) ij ≡ v u ( A u − µ cot) ij ˜ m f , (53)with X = Y where ˜ m f ( ˜ f = ˜ u , ˜ d ) denotes up- and down-type averaged squark mass, and thenumerator of Eq. (51) can be written as( m u L ) ij ≃ − V i V ∗ j f t (4 π ) (3 m + A ) (cid:18) M M H c + log M H c M (cid:19) , (54)( m u R ) ij ≃ − e − iϕ uij V ∗ i V j f b (4 π ) (3 m + A ) log M M H c , (55)( m d L ) ij ≃ − V ∗ i V j f t (4 π ) (3 m + A ) (cid:18) M M H c + log M H c M (cid:19) , (56)( m d R ) ij ≃ − e − iϕ dij U ∗ ki V kj f ν k (4 π ) (3 m + A ) log M M H c . (57)Here, m and A are the universal scalar mass and trilinear coupling, respectively. References [1] R. Barbieri and L. J. Hall, Phys. Lett. B (1994) 212 [hep-ph/9408406].[2] R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B (1995) 219 [hep-ph/9501334].[3] Y. Kuno and Y. Okada, Phys. Rev. Lett. (1996) 434 [hep-ph/9604296].[4] M. Gronau, Y. Grossman, D. Pirjol and A. Ryd, Phys. Rev. Lett. , 051802 (2002)[hep-ph/0107254].[5] M. Gronau and D. Pirjol, Phys. Rev. D (2002) 054008 [hep-ph/0205065].[6] B. Grinstein, Y. Grossman, Z. Ligeti and D. Pirjol, Phys. Rev. D (2005) 011504[hep-ph/0412019].[7] Y. Y. Keum, M. Matsumori and A. I. Sanda, Phys. Rev. D (2005) 014013[hep-ph/0406055]. 178] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. (2014) 161801 [arXiv:1402.6852[hep-ex]].[9] D. Melikhov, N. Nikitin and S. Simula, Phys. Lett. B (1998) 381 [hep-ph/9807464].[10] C. S. Kim, Y. G. Kim, C. D. Lu and T. Morozumi, Phys. Rev. D (2000) 034013[hep-ph/0001151].[11] D. Becirevic, E. Kou, A. Le Yaouanc and A. Tayduganov, JHEP (2012) 090[arXiv:1206.1502 [hep-ph]].[12] J. M. Soares, Nucl. Phys. B (1991) 575.[13] M. Endo and N. Yokozaki, JHEP (2011) 130 [arXiv:1012.5501 [hep-ph]].[14] A. J. Buras, hep-ph/9806471.[15] A. L. Kagan and M. Neubert, Eur. Phys. J. C (1999) 5 [hep-ph/9805303].[16] M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett. B (1993)127 [hep-ph/9307364].[17] A. J. Buras, M. Misiak, M. Munz and S. Pokorski, Nucl. Phys. B (1994) 374[hep-ph/9311345].[18] W. Altmannshofer, A. J. Buras, S. Gori, P. Paradisi and D. M. Straub, Nucl. Phys. B (2010) 17 [arXiv:0909.1333 [hep-ph]].[19] K.A. Olive et al. (Particle Data Group), Chin. Phys. C, (2014) 090001.[20] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, arXiv:1409.5439 [hep-ph].[21] E. Kou, A. Le Yaouanc and A. Tayduganov, Phys. Rev. D (2011) 094007[arXiv:1011.6593 [hep-ph]].[22] J. C. Pati and A. Salam, Phys. Rev. D (1974) 275 [Erratum-ibid. D (1975) 703].[23] R. N. Mohapatra and J. C. Pati, Phys. Rev. D (1975) 566.[24] R. N. Mohapatra and J. C. Pati, Phys. Rev. D (1975) 2558.[25] G. Senjanovic and R. N. Mohapatra, Phys. Rev. D (1975) 1502.[26] U. Amaldi, A. Bohm, L. S. Durkin, P. Langacker, A. K. Mann, W. J. Marciano, A. Sirlinand H. H. Williams, Phys. Rev. D (1987) 1385.[27] M. Blanke, A. J. Buras, K. Gemmler and T. Heidsieck, JHEP (2012) 024[arXiv:1111.5014 [hep-ph]]. 1828] E. Kou, C. -D. Lu and F. -S. Yu, JHEP (2013) 102 [arXiv:1305.3173 [hep-ph]].[29] Y. Zhang, H. An, X. Ji and R. N. Mohapatra, Nucl. Phys. B (2008) 247 [arXiv:0712.4218[hep-ph]].[30] G. Senjanovic, Nucl. Phys. B (1979) 334.[31] M. A. B. Beg, R. V. Budny, R. N. Mohapatra and A. Sirlin, Phys. Rev. Lett. (1977)1252 [Erratum-ibid. (1977) 54].[32] T. Hurth, E. Lunghi and W. Porod, Nucl. Phys. B (2005) 56 [hep-ph/0312260].[33] J. N. Butler et al. [Quark Flavor Physics Working Group Collaboration], arXiv:1311.1076[hep-ex].[34] See for example: R. H. Li, C. D. Lu and W. Wang, Phys. Rev. D (2011) 034034[arXiv:1012.2129 [hep-ph]] S. J¨ager and J. Martin Camalich, JHEP (2013) 043[arXiv:1212.2263 [hep-ph]].[35] J. Hisano, M. Kakizaki, M. Nagai and Y. Shimizu, Phys. Lett. B604