Search for new physics via photon polarization of $b \rightarrow s \gamma$

We suggest a discriminant analysis of new physics beyond the standard model through a detection of photon polarization in a radiative B meson decay. This analysis is investigated in SUSY SU(5) GUT with right-handed neutrino and left-right symmetric models. New physics search via CP asymmetry in the same process are also evaluated in each model for comparison. We show that new physics can be found via detecting 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 to the standard model (SM) Higgs boson which was discovered at the LHC experiment. This fact might imply that a scale of new physics (NP) is higher than the energy scale that the LHC can reach. Thus, it is important to consider the possibility that such exotics are hard to be directly produced 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 NP and related phenomena beyond the SM even if a new particle is impossible to be produced directly. 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 unified theory (SUSY GUT) predict relatively large branching ratios of Br(µ → eγ) ∼ O(10 −15 -10 −13 ) for the NP scale (the right-handed selectron mass) of M NP ≃ (100-300) GeV in an SU(5) case and Br(µ → eγ) ∼ O(10 −13 -10 −11 ) in an SO(10) case [1,2]. Then, Ref. [3] suggested that the measurement of the angular distribution of e with respect to the spin direction of the muon in the µ → eγ process might distinguish among several extensions of the SM if the signal could be detected. 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 adopted to the b → sγ process, i.e. one would be able to discriminate among the SM and NP such as SUSY GUT models, the left-right symmetric standard model (LRSM), and the Pati-Salam models and so on, if one could precisely determine the chirality of the final s quark. The b quark can radiative decay into the s quark in the B meson and the chirality of the s quark is almost determined as left-handed in the SM. Accordingly, if we find more right-handed s quarks in the process than ones expected in the SM, we can expect that some kind of NP must cause this phenomena.
How about the measurement of the chirality of the s quark in the b → sγ process for the NP search? One may naively think that the determination of chirality of quarks except for the top quark is impossible (the top quark can decay before the hadronization). The b → sγ process occurs 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 photon polarization. In addition, there is no parity violation in QCD, the relation between the chirality of the s quark and the photon polarization is unchanged even if the hadronization is taken into account. Therefore, one can determine the chirality of s quark in the b → sγ process from the measurement of photon polarization [4,5]. In Refs. [6,7], the authors mentioned that the higher order 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 produced from the Υ(4S) resonance. The photon polarization of the B d → X s γ decay is determined from measurements of hadronic angular distributions due to the conservation of angular momentum. The LHCb collaboration actually reported the result of observation of photon polarization by measuring the angular distribution of produced mesons in the B → Kππγ process [8]. 1 In addition to the B → Kππγ process, there is another possibility to determine the photon polarization by the B → K * l + l − process. Although there exists a box diagram in the process, the radiative decay diagram (penguin) becomes dominant (the box diagram is suppressed) in a low invariant 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 is important to discuss the possibility of the detection of the photon chirality. In this work, we will consider the b → sγ process. In particular, a ratio of the Wilson coefficients of a dipole operator and a polarization parameter of photon will be firstly evaluated at a typical point in a model of SUSY SU(5) GUT with the right-handed neutrino (N R ) and LRSM in order to clarify whether one 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 0 γ), is also a sensitive observable to NP [12]. Actually, the CP violating effects from NP can be enough larger than the SM expectation as A CP (b → sγ) ≃ −0.5%. However, this CP violation is constrained by the other experiment, e.g. the chromo electric dipole moment (CEDM) [13], and it can be negligibly smaller than the SM one when the CP violating phase depending on the b → sγ process is accidentally small. We will also evaluate the magnitude of A CP (b → sγ) in the SUSY SU(5) GUT with N R and the LRSM although the magnitude strongly depends on CP violating phases in the models. Furthermore, S CP (B → K s π 0 γ) can also become larger than the SM expectation as S CP (B → K s π 0 γ) ≃ −0.3. We will show this value is insensitive to the CP phase. Then, we will compare experimental detectability for the models of NP between the determination of photon polarization and the observation of both CP asymmetry in the b → sγ process.
We will suggest that one can discriminate NP beyond the SM by the detection of photon polarization in b → sγ process. We will point out that time-dependent CP asymmetry is the most stringent constraint in our sample model point at the moment. However, it will actually turn out that the LHCb with 2 fb −1 for the determination of photon polarization may check the existence of NP scale up to several TeV in both models.

Photon polarization
We investigate the photon polarization in the radiative rare decay, b → sγ, process in a SUSY SU(5) GUT with N R and the LRSM for the search of NP. The Wilson coefficients C 7 and C ′ 7 of the dipole operator for the b → sγ process are important for the analyses of photon polarization. The effective Hamiltonian reads with the magnetic operator, where G F is the Fermi constant, m b is the bottom quark mass, σ µν = i 2 [γ µ , γ ν ], and P R,L = 1 2 (1 ± γ 5 ) (e.g., see [14]). O ′ 7 is obtained by replacing L ↔ R in O 7 . Because left-handed s quark comes from O 7 and right-handed one comes from O ′ 7 , we might be able to determine the chirality of s quark by the difference of Wilson coefficients.
When one considers physics beyond the SM, there might be additional contributions to C 7 and C ′ 7 from NP. In these cases, we can generically describe C 7 and C ′ 7 as NP , respectively. The coefficients at the b quark mass scale µ b are given by the leading logarithmic calculations with QCD corrections to the b → sγ process, at the leading order where η = α s (m W )/α s (µ b ), α s ≡ g 2 s /(4π), 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 8 is the coefficient of chromomagnetic operator in the ∆F = 1 effective Hamiltonian, where T A are the generators of SU(3) C and O ′ 8 is also obtained by replacing L ↔ R in O 8 . And, we also describe C 8 = C SM where Ψ i are 10-dimensional multiplets, Φ i are 5-dimensional ones, N i denote the right-handed neutrino superfields, and H (H) is 5-(5-) dimensional Higgs multiplets. i and j mean the generation of the fermions, i, j = 1, 2, 3. f u , f d , and f ν are Yukawa coupling matrices for the up-type quarks, down-type quarks (charged leptons) and neutrinos, respectively. These are given by without a loss of generality, where V and U are the Cabibbo-Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, respectively. ϕ u k and ϕ d i are CP-violating phases, and f u k and f d i are Yukawa couplings of the up-and down-type quarks (charged leptons), respectively. For the neutrinos sector, the light neutrino masses are given by the seesaw where v u is the vacuum expectation value (VEV) of the up-type Higgs in H, and M N i are the mass eigenvalues of the right-handed neutrinos. Here, we assume a diagonal right-handed Majorana mass matrix M ij for simplicity.

Photon polarization in SUSY SU(5) with right-handed neutrino
We discuss C 7 and C ′ 7 , which determine the magnitude of the photon polarization, in the model of SUSY SU(5) GUT with N R . In supersymmetric models, the dominant contributions to C 7,8 and C ′ 7,8 arise from loop diagrams of the charged Higgses, charginos, and gluinos. Thus, and at the weak scale and g /m 2 where m t and M H ± are the top quark and charged Higgs masses, respectively. The contributions from the charged Higgs and chargino to C ′ 7 NP are suppressed by m s /m b . In order to see the magnitude of contributions from NP, we estimate the ratio |C ′ 7 /C 7 | at the b quark mass scale, which determines the size of polarization of photon as seen below. The value of the ratio in the model of SUSY SU(5) GUT with N R is shown by the red solid curve in Fig. 1. The black solid line indicates the SM case |C ′ 7 SM /C SM 7 | which can be approximated as One can see that |C ′ 7 /C 7 | in the SUSY SU (5) with N R case is enhanced from the SM. This means that the final state of s R in b → sγ increases compared to the SM while 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 The contours with n% correspond to case that there is n% misidentification in the C 7 measurement at experiments, i.e. the contours denote |C ′ 7 (1 + n/100)/(C 7 (1 − n/100))|. This misidentification corresponds to a mismatch in the conversion of left-handed helicity to the left-handed chirality. (The helicity is determined in experiments.) For instance, if one identifies the left-handed helicity with the left-handed chirality with 10% misidentification, the right-handed chirality is over estimated as 110% of the true value. Thus, the contours go above as n increases in Fig. 1. In the calculation, we take as a typical point where CP phases are given in the radian unit. 2 We take other values of parameters in the SM and the neutrino sector (PMNS mixing angles) as the best fit values given in [19,20]. Next, we consider the polarization parameter of photon λ γ at the b quark mass scale defined as In order to measure λ γ we need to consider a parity-odd observable in the B d → X s γ decay since the photon polarization is parity-odd. In Ref. [4,5] they proposed that λ γ can be measured 2 Note that ϕ d23 is sensitive to the CEDM. The allowed minimal and maximal values by the CEDM constraint are π and 3π/2 (or π/2), respectively. In the calculation of photon polarization, we take the minimal value. The photon polarization is not so sensitive to the value of ϕ d23 but the difference between these values appears in the calculation of direct CP asymmetry as we will show in Section 3. We have numerically checked that another phase, ϕ u23 , is not sensitive to our evaluation. from the up-down asymmetry of the photon direction relative to the Kππ decay plane in the K 1 (1400) rest frame. We show λ γ in Fig. 2 and the SM and SUSY SU(5) GUT with N R cases are depicted by the black and red curves, respectively. We also show ϕ d 23 = 3π/2 case by the red dashed curve for comparison. The SM prediction for the photon polarization still have roughly 10% error due to O 2 operator [11], then we depict it by the black dashed curve. The superKEKB with an integrated luminosity of 75 ab −1 and the LHCb with 2 fb −1 might reach the 20 (|λ γ | = 0.8|λ SM γ |) and 10% (|λ γ | = 0.9|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 corresponds to M 2 ≃ 1400 GeV and Mg ≃ 4900 GeV) in this model of SUSY SU(5) with N R .

Case of Left-right symmetric standard model 2.2.1 Model
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)Ỹ . In the model, the SM left-handed doublet fermions are SU(2) R singlets, and the right-handed fermions including the neutrinos are SU(2) R doublets and SU(2) L singlets. For the Higgs sector, the model includes a bi-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)Ỹ → SU(2) L × U(1) Y → U(1) em . The symmetry breaking can be undertaken by the VEVs of Φ, ∆ R , and ∆ L as with six real numbers κ, κ ′ , ω, v R , v L , and θ L . Regarding the magnitude of the VEVs, v R should be much larger than the electroweak (EW) scale to suppress the right-handed currents at low energy, and the EW ρ-parameter limits v L to be v L 10 GeV [26]. In this work, we take v L = 0 for simplicity, which is usually taken in literatures (e.g., [27,28]). Since the VEV of Φ leads to the standard EW symmetry breaking, we define v ≡ √ κ 2 + κ ′2 = 174 GeV, tan β LR ≡ κ/κ ′ , and The charged gauge bosons are given by the admixture of the mass eigenstates as The masses of charged gauge bosons are approximated as where g L,R are the gauge couplings of SU(2) L,R and we take g R /g L = 1 for simplicity in the numerical analysis. The mixing angle is given as sin ζ ≈ (M 2  [27,29]. In this work, we represent both the charged and neutral Higgs bosons masses as M H for simplicity. Regarding the flavor mixing matrices, we assume V = V L = V R , where V L and V R are the mixing matrices for the left-and right-handed quarks, respectively. V = V L = V R is taken in the so-called the manifest LRSM [30,31] and we also take the equality in this work. In our numerical analyses, we have three free parameters, i.e.

Photon polarization in LRSM
and C ′ 7 NP in the LRSM are generally given by when one does not assume g R /g L = 1 and V = V R , where loop functions D ′ 0 (x) and A LR (x) are given in Appendix A, and x t ≡ m 2 t /m 2 The ratio |C ′ 7 /C 7 | in the LRSM is shown by the blue curves in Fig. 3. The value 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. In  Figure 4: The polarization parameter λ γ in b → sγ in the SM and LRSM which are depicted by the black and blue curves, respectively. We also show ω = π/4 and π/2 cases by the blue dashed and dashed-dotted curves for comparison, respectively. the analysis, we take as a sample point. 4 The polarization parameter in the LRSM is shown by the blue curve in Fig. 4. The SM and LRSM cases are depicted by the black and blue curves, respectively. We also show ω = π/4 and π/2 cases by the blue dashed and dashed-dotted curves for comparison, respectively. Since M NP ≃ 3.7 TeV for 10 % deviation from the SM prediction, the future LHCb experiment with 2 fb −1 will check the scale of the right-handed gauge boson up to 3.7 TeV in this case. The dependence of photon polarization on the CP phase is not strong.

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 of the process in both models of SUSY SU(5) GUT with N R and LRSM. The asymmetry is given by which of course strongly depends on the CP-phases in the model where C 2 is the coefficient of the operator O 2 = (c α γ µ P L b α )(c β γ µ P L b β ) in the effective Hamiltonian of ∆F = 2 transitions. Note that the contributions from NP to C 2 and C ′ 2 are negligibly small while C 7,8 and C ′ 7,8 include contributions from NP, i.e. C ′ 2 ≪ C 2 = C SM 2 = 1. Thus, A CP (b → sγ) is well approximated by Eq. (32). 5

SUSY SU(5) with right-handed neutrino case
We show A CP (b → sγ) in the SUSY SU (5) with N R and SM cases by red and black solid curves in Fig. 5, respectively. The experimental lower bound as −3.7 % [19] is also shown by the black dashed line. We also show ϕ d 23 = 3π/2 case by the red dashed curve for comparison. It turns out that the magnitude of the direct CP asymmetry in the case of SUSY SU(5) with N R highly depend on the phase ϕ d 23 . The current experimental lower bound does not constrain the scale of NP, M SUSY in this parameter setup. Therefore, the measurement of A CP does not currently constrain on the NP scale in the SUSY SU (5) with N R model even if one takes the maximally allowed CP phase as ϕ d 23 = 3π/2. But, the expected reach of Belle II with 50 ab −1 for A CP (B → X s+d γ) will be ±2% precision. Thus, the future determination will check NP between 450-750 GeV in this typical SUSY SU (5) with N R case.

LRSM case
For the LRSM, C NP 8 and C ′ 8 NP are  [33]. We also show the ϕ d 23 = 3π/2 case by the red dashed curve for comparison. with where ρ 8 and ρ LR are the so-called magic number which are given in the Ref. [27]. f LR (x) is also a 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/4 and π/2 cases by the blue dashed and dashed-dotted curves for comparison, respectively. One can see from Fig. 6 that the magnitude of the direct CP asymmetry in the LRSM is allowed for M NP ≥ 2 TeV for π/10. Hence, the A CP measurement does not give constraint on the existence of NP in this case at the moment. Furthermore, the future Belle II with 50 ab −1 will check the LRSM model up to 3.5TeV, 5TeV, and 5.5TeV for ω = π/10, ω = π/4, and ω = π/2, respectively. The result is really sensitive to the phase ω as with SUSY SU (5) with N R case.

Time-dependent CP asymmetry
We also evaluate the time-dependent CP asymmetry in the B → K s π 0 γ decay denoted as S CP . The definition of S CP is same in the both model: where 2β CKM ≈ 43 • is a CP phase in B → K s π 0 γ decay.  Fig. 5. We also show ω = π/4 and π/2 cases by the blue dashed and dashed-dotted curves for comparison, respectively.

SUSY SU(5) with right-handed neutrino case
We show Eq. (37) in the SUSY SU (5) with N R and SM cases by red and black solid curves in Fig. 7, respectively. The current experimental lower bound as −0.35 is also shown by the black dashed line. We also show ϕ d 23 = 3π/2 case by the red dashed curve for comparison. The current experimental bound seems to exclude up to 2 TeV of the M SUSY in Fig. 7 in this naive set up. In other words, the time-dependent CP asymmetry gives the strongest constraint within λ γ , A CP , and S CP .

LRSM case
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 set up. The figure shows that the time-dependent CP asymmetry is the strongest constraint even in the LRSM.

Comparison between Photon polarization and Measurements of CP asymmetry
As we mentioned in the Introduction, the photon polarization might become a useful way to determine NP. Actually, the ways to determine the photon chirality by measurement of angular distribution of the final state particles have been discussed in several papers [9,10,11] (See also [34]). In this paper, we have evaluated three observables: photon polarization, direct CP asymmetry, and time-dependent CP asymmetry. We give some comments on the comparison  Figure 7: The time-dependent CP asymmetry in B → K s π 0 γ in the SM and SUSY SU(5) GUT with N R , which are depicted by the black and blue solid curves. The black dashed line is the current experimental lower bound [33]. We also show ϕ d 23 = 3π/2 case by the red dashed curve for comparison.
among the results of those: • For the determination of photon polarization in the SUSY SU(5) with N R model, the future LHCb experiment with 2 fb −1 will be able to check the NP scale up to about 1700 GeV, which corresponds to M 2 ≃ 1400 GeV and Mg ≃ 4900 GeV, at the typical point of this model.
• For the measurement of A CP in the SUSY SU(5) with N R model, the experiment does not currently constrain on the NP scale even if one takes the maximally allowed CP phase as ϕ d 23 = 3π/2. The future Belle II with 50 ab −1 will check NP between 450-750 GeV.
• For the determination of photon polarization in the LRSM, the future LHCb experiment will check the NP scale up to 3.7 TeV in this case.
• For the measurement of A CP in the LRSM, the experiment constrain up to 4.7 TeV depending on the phase ω. Furthermore, the future Belle II with 50 ab −1 will check up to 5.5 TeV when ω maximize the direct CP asymmetry. On the other hand, the future experiment reach 3.5 TeV at most when ω minimize the direct CP asymmetry.
• The time-dependent CP asymmetry is the most stringent constraint in both models and this CP asymmetry does not depend on CP asymmetry parameters so much.
• Thus, we mention that there is a region where the determination of photon polarization is 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 than other observables.  Figure 8: The time-dependent CP asymmetry in B → K s π 0 γ in the SM and LRSM which are depicted by the black and blue solid curves. The black dashed line is same as in Fig. 7. We also show ω = π/4 and π/2 cases by the blue dashed and dashed-dotted curves for comparison, respectively.

Summary
One might be able to obtain the existence of NP and discriminate among the SM and NP if one can 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, there are several ways to measure the photon polarization. In addition to the determination of photon polarization, the observation of CP asymmetry in the process is still sensitive to the existence of 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 the LRSM. The ratio |C ′ 7 /C 7 |, the polarization parameter of photon, and the direct CP asymmetry in the process have been evaluated in both models. The time-dependent CP asymmetry seems to be the best way to find the NP effect. However, the combination of CP violation and photon polarization can discriminate NP beyond the SM at the end. Furthermore, there might be a region where the determination of photon polarization is more sensitive for the new physics search than that of CP asymmetry.
the Japan Society for the Promotion of Science for Young Scientists.