Novel two-loop supersymmetry effects on the CP asymmetry in B -> phi K-s

arXiv:hep-ph/0403188v4 4 Mar 2005

Novel two-loop SUSY effects on CP asymmetry in B→ φK^s

C.H. Chen^a∗ and C.Q. Geng^b†

aDepartment of Physics, National Cheng-Kung University, Tainan 701, Taiwan

bDepartment of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan (Dated: February 1, 2008)

Inspired by the exotic measurements on the CP asymmetry in B → φKs, we study a new diagram in supersymmetric models which can make the difference sin 2φ^{ef f}₁ (J/ΨKs) − sin 2φ^{ef f}₁ (φKs) to be 20 − 50% after satisfying the constraint from b → sγ. We also find that the direct CP asymmetry of b → sγ could be ∼ 10% and testable at B factories.

PACS numbers: 11.30.Er, 12.60.Jv, 13.25.Hw

While enjoying the large CP asymmetry (CPA) in the decay of B → J/ΨK^sobserved by Belle [1] and Babar [2]

at the precision level, the recent data on B → π⁺π⁻ [3]

and B → φKs [4, 5] have stimulated theorists to think more about other possible CP violating phases, beside the Kobayashi-Maskawa (KM) [6] phase in the standard model (SM).

It is known that with the Wolfenstein parametriza- tion [7], the tree and penguin diagrams have the same CP phase for the inclusive processes of b → s¯cc and b → s¯ss. Thus, the time-dependent CPA, proportional to ¯Γ( ¯B → fCP) − Γ(B → fCP) with fCP being the final state and having a definite CP property, arises from the B − ¯B oscillation dictated by box diagrams, in which the source of the CP phase is from Vtd= |V^td|e^−iφ1. For the channel of fCP = J/ΨKs, the CPA is related to sin 2φ1

and the mixing-induced CP violation. If there is only the KM phase involved in the low-energy, the pure pen- guin process of B → φK^s has approximately the same value of sin 2φ1as that in the decay of B → J/ΨK^s, i.e.,

∆Sφ1 = sin 2φ1(J/ΨKs) − sin 2φ1(φKs) ≃ 0 [8].

It is usually believed that new physics could go into low-energy phenomena through loop diagrams, in which new particles appearing in the loops are integrated out and the remaining effective couplings are as functions of their masses and couplings to the conventional particles.

Since the transition of b → s¯ss is a pure quantum loop effect, one can recognize immediately that B → φK^s is a good candidate to probe new physics. Furthermore, although the tree-level contributions in b → s¯cc are over a factor of 5 larger than those of penguin diagrams in the SM [9], the penguin-type diagrams induced by new physics could be enhanced, which will clearly affect the decay of B → J/ΨKs, especially on its direct CPA.

To understand the Belle’s result of the 3.5σ difference on sin 2φ1 between J/ΨKs and φKs modes [4], vari- ous theoretical models such as those with supersymme- try (SUSY) [10, 11, 12, 13] and left-right symmetry [14]

have been investigated. In addition, the authors of Refs.

[10, 11] have tried to solve the problem of unexpected

∗Email: [email protected]

†Email: [email protected]

large branching ratios (BRs) in B → η^′K decays. How- ever, we would like to address some problems on these attempts as follows:

(a) Direct CP violation on B → J/ΨKs:

We emphasize that Belle and Babar not only measure an accurate mixing-induced CPA, but also indicate no direct CPA in B → J/ΨK^s, up to the percentage level.

Those new SM-like effective four-fermion interactions for b → s¯ss will inevitably contribute to b → s¯cc. It is also known that there exist large strong phases in the produc- tion of charmed mesons (including charmonium states) [15, 16]. Therefore, to enhance the BRs of B → η^′Ks

with large CP violating effects will make the direct CPA in B → J/ΨK^sto be over the current experimental lim- its.

(b) BRs of B → ηK and B → η^(′)K^∗:

We note that the problems for the production of η^′ in B decays depend on not only B → η^′K, but also B → ηK and B → η^(′)K^∗. From the data at Babar, we have that BR(B → ηK⁰) = (2.9 ± 1.0 ± 0.2)10⁻⁶ [17], BR(B → ηK^∗0) = (18.6 ± 2.3 ± 1.2)10⁻⁶, and BR(B → η^′K^∗0) < 7.6 × 10⁻⁶[18]. By using the pertur- bative QCD approach [19], we find that the estimating BRs of B → ηK⁰and η^′K^∗0are over the current exper- imental values, whereas it is lower for B → ηK^∗0.

It is clear that to resolve the problems we need more knowledge on η^(′)mesons as well as their relevant physics.

On the other hand, we may bypass these problems by concentrating on new physics effects which are insensi- tive to hadronic uncertainties. In this paper, we will introduce a two-loop diagram illustrated in Fig. 1, in the framework of SUSY models, resulting from dipole operators. In contrast with other mechanisms, such as those discussed in Refs. [10, 12] in which the relevant off- diagonal terms of squark-mass matrices directly involve flavor changing neutral current that couples to gluino, our two-loop effect shows how to generate the flavor changing processes naturally in the SUSY models. We will illustrate that the diagram not only contributes a sizable value for the difference of sin 2φ1between J/ΨKs

and φKs channels, but also satisfies the experimental constraints such as those from the b → sγ decay and the neutron electric dipole moment (NEDM). Since the diagram involves the couplings of the charged Higgs to squarks, we first discuss the relevant couplings in SUSY

2

b_R g˜ s_L

˜b_R H⁻ s˜L

×

˜t_R

˜t_L

g

FIG. 1: Two-loop diagram by the ˜bR− ˜sL flavor changing effect and the chromodipole operator.

models, given by [20]

LH ˜f ˜f = −(2√

2GF)^1/2 ˜VtbA˜^b_L˜t^∗_L˜bR

+ ˜V_ts^∗A˜^t_R˜t^∗_Rs˜L

H⁺+ h.c., (1)

where ˜A^b_L = mb(A^∗_btan β − µ) and ˜A^t_R = mt(Atcot β − µ^∗). Here, the definition of the angle β is followed by tan β = vu/vd with vu and vd being the vacuum ex- pectation values (VEVs) of Higgs fields Φ^u and Φ^d re- sponsible for the masses of upper and down type quarks, respectively, and µ is the mixing effects of Φ^u,d. For a large tan β case, ˜A^b_L and ˜A^t_R can be simplified as A˜^b_L ≈ m^bAbtan β and ˜A^t_R≈ −m^tµ^∗. Note that we have neglected the contribution of ˜sRbecause the correspond- ing coupling is associated with the strange-quark mass.

Moreover, in order to suppress one-loop contributions, we assume that the flavor mixing effects on the down-type squark mass matrix are small.

We remark that both flavor changing and chirality flip- ping are involved in Fig. 1, in which the charged Higgs is used to change the flavor and the mixing of ˜tLand ˜tR

to govern the chirality flipping, representing by the cross in the figure. Explicitly, as usual, the mixing terms are described by [21]

m²_U˜

LR = M²_U˜

LR− µ cot βm^U, (2) where (M²_U˜)LR represent the trilinear soft breaking ef- fects. For simplicity, we have adopted the so-called super- CKM basis, where quarks are in the mass eigenstates so that mU is the diagonal upper-type quark mass matrix [21]. To overcome the NEDM constraint, it has been proposed [22] to use hermitian Yukawa and A matrices.

The construction of a hermitian Yukawa matrix can be implemented based on some symmetries, such as the hor- izontal SU (3)H [23] and left-right [24] symmetries. As a result, the CP phases of O(1) can exist naturally even with the NEDM contributions. Moreover, it implies that the CP asymmetries in hyperon decays could reach the value of O(10⁻⁴) [25], which is testable in the experi- ment E871 at Fermilab [26]. However, in the class of

models proposed in Ref. [22], the µ parameter is real which is not favored in our following discussions. To avoid this shortcoming, we address the NEDM constraint by imposing the Yukawa and A matrices to be hermi- tian and the squark mass of the first generation to be O(10) TeV. Hence, the µ parameter is regarded as a com- plex value in our approach. Due to the hermitian prop- erty, a special relation is obtained as δ_kl^U

LR = δ_kl^U

RL

with (δ^U_kl)LR ≡ (M²U kl˜ )LR/ ˜m² = (V^U†A^U†vuV^U)kl/ ˜m², where A^U† = A^U, V^U is the mixing matrix for diago- nalizing the mass matrix of upper-type quarks and ˜m is the average squark mass in the super-KM basis. In general, the trilinear SUSY soft breaking A^Q terms are not diagonal matrices. However, due to the relation of A^Q_ij = (Y^QAˆ^Q)ij with Y^Q ( ˆA^Q) being Yukawa (A- parameter) matrices and the small effect of renormal- ization group, dominant effects of A^Q are still from the diagonal elements [28] if we take ˆA^Q to be universal and diagonal at the grand unified scale. We use AQ= A^Q_ii to simplify our estimations. Therefore, the contribution in Fig. 1 is proportional to mbmtµ^∗Abtan β(δ^t₃₃)LR. Since A^U is hermitian, A^t(δ₃₃^t )LR can be regarded as real val- ues. Hence, in our mechanism, the CP violating source is focused on the complex µ term. We note that by adopting a large tan β, the µ-dependent effect is from the vertex of the charged Higgs coupling to squarks. For convenience, we write the relationship between weak and physical eigenstates for the mixing of ˜tL and ˜tR as

t˜L

˜tR



=

 cos θt sin θt

− sin θ^t cos θt

 t˜1

t˜2



. (3)

To study Fig. 1, we start with the effective interactions for quark-gluino-squark, given by [27]

L˜g ˜qq = −√

2gs(¯sPRg˜^aT^as˜L− ¯bPLg˜^aT^a˜bR) + h.c., (4)

where the flavor mixings for squarks have been neglected.

It is interesting to note that if we use the photon in- stead of the gluon and include the emission of the photon at the charged Higgs, we find that the same mechanism could also contribute to b → sγ. Therefore, sizable val- ues for both ∆Sφ1 and the rate CPA in B → X^sγ can definitely provide a hint for new physics. The effective operators for b → sγ(g) are given by

L = GF

√2V_ts^∗Vtb(C7γ(µ) O7γ+ C8g(µ) O8g) , (5)

where O7γ = mbe/(8π²)¯sσµνF^µν(1 + γ5)b, O8g = mbgs/(8π²)¯sσµνT^aG^aµν(1 + γ5)b,

3

C7γ = − cos θtsin θt

V˜_ts^∗V˜tb

V_ts^∗Vtb

αs(mb) 8π

mt

mg˜

Abtan β µ^∗

m²_˜g P_U^γI m²_˜t m²_g˜,m²_˜b

m²_˜g,m²_H m²_g˜

! ,

I m²_q˜1 m²_g˜,m²_q˜2

m²_˜g,m²_H m²_˜g

!

= m⁴_g˜ Z 1

0

dx Z ∞

0

dQ² x (1 − x) Q²

Q²+ m²_˜g 

Q²+ m²_q˜2

2

m²_q˜1(1 − x) + m²Hx + Q²x (1 − x),

C8g = C7γ/(2NcP_U^γ) and P_U^γ = CF(QU − 1) with CF = 4/3 and QU = 2/3 being the color factor and the charge of the upper-type squark, respectively. Clearly, we obtain the unique property that the effects of electric and mag- netic dipole moments are directly related to those of chro- moelectric and chromomagnetic dipole moments, respec- tively. Before we proceed further, we have to examine whether the two-loop effects are of interest. Explicitly, we would like to check whether the value of C7γ is larger or smaller than experimental constraint 0.3 < |C7γ^eff| < 0.34 [29]. For an illustration, we set the values of parameters, by satisfying the constraints from the NEDM [30], as fol- lows: tan β ∼ m^t/mb, sin θtcos θt∼ 0.2, ˜V_ts^∗V˜tb/V_ts^∗Vtb∼ O(1), Ab/m˜g ∼ µ/m˜g ∼ 4, Arg(µ) = π/2, mH ∼ 150 GeV, mg˜∼ 1 TeV, m^˜t∼ 200 GeV and m^˜b = ms˜∼ 500 GeV, and we have |C^7γ| ∼ 0.8. If we take tan β ∼ 50, sin θtcos θt ∼ 0.35, A^b/m˜g ∼ µ/m^g˜ ∼ O(1) and the re- mains to be the same as the above choices, we obtain

|C7γ| ∼ 0.14.Furthermore, by using b-quark and sbot- tom instead of s-quark and its squark, the similar two- loop diagram could contribute to the EDM of s-quark.

It is known that the current limit of the s-quark chromo EDM is |ed^Cs|^expt< 5.8 ×10⁻²⁵e cm [31]. We now exam- ine the contribution to d^C_s in our mechanism. By using Eq. (5) and assuming ms˜= m˜b and As= Ab, we obtain

|d^Cs| ∼√

2GFV_ts²ms

8π²|ImC^8g| =GFV_ts²

√2CF

ms

8π²Im|C^7γ|. (6) Numerically, we get |ed^Cs| ∼ 2.5×10⁻²⁵|C7γ| e cm, which is below |ed^Cs|exptif |C7γ| ≤ 1. Clearly, in our mechanism it is inevitable to utilize the large tan β, Ab/m˜gand µ/m˜g

scheme and, therefore, the most strict constraint is the BR of B → X^sγ.

In order to discuss the mixing-induced CP problem in B → φKs, we write the relevant definition of the time- dependent CPA as

ACP = BR( ¯B → φKs) − BR(B → φKs) BR( ¯B → φKs) + BR(B → φKs),

= CφKscos ∆mBt + SφKssin ∆mBt,

= |λ|²− 1

|λ|²+ 1cos ∆mBt − 2Imλ

|λ|²+ 1sin ∆mBt, (7) where λ = e^{−i2φef f1 (φKs)}A( ¯B → φK^s)/A(B → φK^s) and A(B → fCP) is the decay amplitude. Since the dipole operators contributing to the nonleptonic decays belong

to next-to-leading order in αs, we can safely neglect the contributions to the decay amplitude of B → J/ΨKs. For displaying the other SUSY effects on the B − ¯B mix- ing, we use φ^{ef f}₁ instead of φ1. Hence, φ^{ef f}₁ is still deter- mined by B → J/ΨKs, exclusively. For estimating the hadronic matrix element of B → φK, we use the naive factorization, given by

hφK|O8g| ¯B, pBi ≈ −2αs

9π m²_b

q²fφmφF^BK(0)ǫ^∗· pB,(8) where F^BK(0) is the transition form factor of B → K at Q² = 0, q² is the squared momentum of the virtual gluon, ǫ, fφ and mφ correspond to the polarization vec- tor, decay constant and the mass of φ, respectively. The dominant contribution of factorization assumption is con- firmed by the PQCD approach [32] in which q²is related to the momentum fractions of quarks and convolutes with wave functions. We note that although O^7γcan also con- tribute to the decay of B → φK^s, since the coupling is electromagnetic interaction and much smaller than that of strong interaction, we neglect its contribution. Ac- cordingly, the decay amplitude for B → φK0 is written as

A( ¯B → φK⁰) = GF

√2V_ts^∗Vtb 5

X

i=3

ai−2αs

9π m²_b

q²C8g

!

×fφmφF^BK(0)ǫ^∗· pB, (9) where ai, defined in Ref. [33], stand for the effective Wilson coefficients in the SM, included from electromag- netic penguin diagrams. The value of P5

i=3ai is esti- mated to be −0.045. The parameter λ in Eq. (7) for the CPA can be simplified as λ = e^{−i2φeff1 (J/ΨKs)}e^{−i2φN ew} = e^{−i2φeff1 (φKs)} with

tan φN ew= −2αs

9π m²_b

q²

ImC8g

P5

i=3ai−^2α_9π^sm_q^22bReC8g

. (10)

To display the unique character of the two-loop di- agram, we adopt the value of C7γ such that C7γ =

−C7γ^SM ± i|ImC7γ^{ef f}| and the experimental value C7γ^{ef f} = C_7γ^SM + C7γ = ±i|ImC7γ| instead of scanning the whole parameter space. By using C_7γ^SM = −0.30 and the iden- tity C8g= −3C7γ/8, the CP violating phase from the de- cay amplitude is tan φN ew= ∓(0.18±0.01^+0.11−0.06), in which

4 the first error is from |C7γ^eff| = 0.32 ± 0.02 and the sec-

ond theoretical error arises from the uncertainty in q²= (3/8 ± 1/8)m²B. Since SφKs = sin 2φ^eff₁ (φKs), by taking sin 2φ^eff₁ (J/ΨKs) ≈ 0.74 measured by Belle and Babar, we obtain SφKs = 0.46±0.01^+0.10−0.21(0.93±0.01^+0.06−0.05) where the sign of φN ew is chosen to be negative (positive). In- terestingly, the former value is close to the central value of the Babar’s result [5]. Furthermore, we can straight- forwardly calculate the difference of the CPAs to be

∆_φ^eff₁ = sin 2φ^{ef f}₁ (J/ΨKs) − sin 2φ^{ef f}1 (φKs)

=







0.28 ± 0.01^+0.21−0.10(−),

−(0.20 ± 0.01^+0.05−0.06) (+).

(11)

We now consider the two-loop effects for the CPA in b → sγ. According to the formalism shown in Ref. [34], the rate CPA for b → sγ is given by

ACP(b → sγ) ≈ 1 100

C_7γ^eff 

21.1ImC2C_7γ^eff∗

+9.52ImC_7γ^effC_8g^eff∗+ 0.16ImC2C_8g^eff∗ , where C2 ≈ 1.11 and C8g^eff = C_8g^SM + C8g. With the same C7γ used above, we get ACP(b → sγ) ≈

±(10.5 ± 0.6)% for negative and positive signs in ImC7γ, respectively. Comparing to the recent Babar’s limit of

−0.06 < A^CP(b → sγ) < +0.11 [35], we find that only

the result with negative sign in Eq. (11) is reliable, which could be used to resolve the sign ambiguity in ImC7γ. Fi- nally, we remark that although our upper value on the CP asymmetry of b → sγ is a little bit over the Babar up- per bound, the problem can be removed by relaxing the required condition C7γ = −C7γ^SM± i|ImC7γ^{ef f}| introduced for our simplified analysis.

In summary, we have studied the novel two-loop SUSY effects on the CPAs of B → φKs and b → sγ. We have found that with large values of tan β and Ab(µ)/mg˜, the difference of sin 2φ^eff₁ between J/ΨKsand φKscan have a deviation of 20 − 50%. The main theoretical error is due to the uncertainty in q². We have also shown that the two-loop effect can give the CPA in b → sγ around +10%. It is clear that, since the two-loop contributions to the CPAs in both decay modes can be the dominant ones in the SUSY models, experimental measurements at B factories on these CPAs can determine the sizes of these novel contributions.

Note added: Our two-loop SUSY mechanism has been applied to the decay of Bs→ µ⁺µ⁻ [36].

Acknowledgments

This work is supported in part by the National Science Council of R.O.C. under Grant No. NSC-91-2112-M-001- 053, No. NSC-92-2112-M-006-026 and No. NSC-92-2112- M-007-025.

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