New physical effects on the decay B γγ in the sequential s(d) → fourth Generation model Hong Chena and Wujun Huoa,b∗ aCollege of Physics and Technology, Southwest University, Chongqing 400715, China and 1 bThe Abdus Salam International Center for Theoretical Physics, 1 0 Strada Costiera 11, 34014 Trieste, Italy. 2 n Abstract a J We study the contributions to the branching ratios of B γγ decay in the sequential fourth 4 s(d) → 2 generation model (SM4). We find that the theoretical values of the branching ratios, BR(B s(d) → ] ph γγ), including the contributions of mt′ and the new 4×4 CKM (CKM4) matrix factors, |Vt∗′sVt′b| p- and |Vt∗′dVt′b|, are much different from the minimal standard model (SM) predictions. The new e h physics effects, especially contributed from the CKM4 matrix factors, can provide more than one [ order enhancement to the SM prediction. It is shown that the decay B γγ can test the new 1 s(d) → v physics signals from SM4. 0 6 6 4 . 1 0 1 1 : v i X r a ∗ Electronic address: [email protected] 1 I. INTRODUCTION SM a very successful theory of the electroweak interactions. But it should not be the final theory. Theoretically, it has too many unknown parameters to be put by hand and can not unify the three gauge interactions. Also, SM has been faced to some troubles from the experiments. We need the new physics beyond SM. Many new physics models have been proposed to resolve the difficulties of SM and to explain the experiments. Of course, they have to be tested in many high energy experiments, such as the rare decays of mesons. The startup of the LHC opens many new frontiers in precision flavour physics. As is well known, the rare radiative decays of B mesons are particularly sensitive to the contributions from new physics. Both inclusive and exclusive processes have been researched in the last 20 years. For example, B γγ has been studied extensively in the SM[1] and new physics s(d) → scenarios[2, 3]. The present experimental limit on the decay B γγ [4, 5]is s(d) → BR(B γγ) 8.6 10−6 (90%C.L.), (1) s → ≤ × BR(B γγ) 3.2 10−7 (90%C.L.). (2) d → ≤ × Within the SM one finds[1], BR(B γγ) 1 10−6, (3) s → ≃ × BR(B γγ) 3 10−8. (4) d → ≃ × The upper bound of B γγ is about (1) larger than the SM values. We believe, with s(d) → O the continuous accumulate of the experiment data, especially in the era of LHC and ILC, these branching ratios will be more and more precise. They will leave less room for the new physics. That is to say, B γγ, is very suitable to test the new physics models. s(d) → In ref. [3], we investigated the new physical effects on B γγ in the one generation s → Technicolor model (OGTM) and got some interesting results. In this note, we consider the sequential fourth generation model[6] to estimate the possible contributions to the decay B γγ. Recently, SM4 attracts an increasing interest and seems warming up. The s(d) → electroweak precision data does not exclude completely existence of the fourth family and there are many reasons to introduce an extra generation of heavy particles[6], (for a recent brief review on the 4th generation, see [7]). Especially, LHC has the potential to discover or fully exclude existence of a fourth generation of quarks up to 1 TeV[7], even if they are 2 too heavy to observe directly they will induce a large signal in gg ZZ that will be clearly → visible at the LHC[8]. Maybe this model will be firstly tested by the early LHC data. The sequential fourth generation model is a simple and non-supersymmetric extension of the SM, which does not add any new dynamics to the SM, with an additional up-type t and ′ ′ an down-type d quarks, a heavy charged leptonτ and a heavy neutrino ν . The model retains all the properties of the SM. The t quark like the other up-type quarks contribute to the b s transition at the loop level. Due to the additional fourth generation → there will be mixing between the t quark the three down-type quarks of the standard model and the resulting mixing matrix will become a 4 4 matrix, × Vud Vus Vub Vub′   Vcd Vcs Vcb Vcb′ V =  , (5) CKM4   Vtd Vts Vtb Vtb′      Vt′d Vt′s Vt′b Vt′b′      where Vqb′ and Vt′q are the new matrix elements in the SM4. The parametrization of this unitary matrix requires six mixing angles and three phases[9]. II. BRANCHING RATIOS OF B γγ s(d) → At quark level, b sγ, b sγγ and the exclusive decays B γγ have a close relation. s → → → Up to the corrections of order 1/m2 , the effective Hamiltonian for b sγγ at scales W → µ = (m ) is identical to the one for B X γ transition [1] and takes the form b b s O → G 6 F ∗ = V V C (µ )Q +C (µ )Q +C (µ )Q , (6) Heff √2 ts tb" i b i 7γ b 7γ 8G b 8G# i=1 X here Q ...Q are the usual four-fermion operators whose explicit form is given below. The 1 6 last two operators in the Eq.(6), characteristic for this decay, are the magnetic–penguin operators. The complete list of operators is given as follows Q = (c¯ γµb )(s¯ γ c ), (7) 1 Lβ Lα Lα µ Lβ Q = (c¯ γµb )(s¯ γ c ), (8) 2 Lα Lα Lβ µ Lβ Q = (s¯ γµb ) (q¯ γ q ), (9) 3 Lα Lα Lβ µ Lβ q=u,d,s,c,b X 3 Q = (s¯ γµb ) (q¯ γ q ), (10) 4 Lα Lβ Lβ µ Lα q=u,d,s,c,b X Q = (s¯ γµb ) (q¯ γ q ), (11) 5 Lα Lα Rβ µ Rβ q=u,d,s,c,b X Q = (s¯ γµb ) (q¯ γ q ), (12) 6 Lα Lβ Rβ µ Rα q=u,d,s,c,b X Q = (e/16π2)m s¯ σµνb F , (13) 7 b L R µν Q = (g/16π2)m s¯ σµνTab Ga . (14) 8 b L R µν where α and β are color indices, α = 1,...,8 labels SU(3) generators, e and g refer to C the electromagnetic and strong coupling constants, while F and Ga denote the QED and µν µν QCD field strength tensors, respectively. It is the magnetic γ-penguin operator Q , which 7 plays the crucial role in this decay. The effective Hamiltonian for b dγγ is obtained from → Eqs.(6-14) by the replacement s d. → The Feynman diagrams that contribute to the matrix element as the following, see Fig. 1, b b b = + s s s b b + + s s FIG. 1: Examples of Feynman diagrams that contribute to the matrix element. Within the SM, at scale m , the Wilson coefficients C (m ) at the leading order (LO) W i W approximation have been given for example in [10], C (m ) = 0 (i = 1,3,4,5,6), C (m ) = 1, (15) i W 2 W 8x3 +5x2 7x 2x2 3x3 C (m ) = t t − t t − t log[x ], (16) 7 W 24(1 x )3 − 4(1 x )4 t t t − − x3 5x2 2x 3x2 C (m ) = t − t − t t log[x ], (17) 8 W 8(1 x )3 − 4(1 x )4 t t t − − where x = m2/m2 . t t W By using QCD renormalization group equations[10], it is straightforward to run Wilson coefficients C (m ) from the scale µ = (m ) down to the lower scale µ = (m ). The i W w b O O 4 leading order results for the Wilson coefficients C (µ) with µ m are of the form [10] 7 b ≈ 8 8 C (µ) = η16/23C (m )+ η14/23 η16/23 C (m )+ h ηai, (18) 7 7 W 8 W i 3 − (cid:16) (cid:17) Xi=1 where η = α (m )/α (µ), s W s a = (14/23,16/23,6/23, 12/23, (19) i − 0.4086, 0.4230, 0.8994,0.1456), − − h = (2.2996, 1.0880, 3/7, 1/14, (20) i − − − 0.6494, 0.0380, 0.0185, 0.0057). − − − − In the sequential 4th generation model, there exists an additional contribution to b sγ → ′ induced by the 4th generation up quark t, which produce the penguin diagrams, see Fig. (2), b d;s b d;s 0 W t 0 0 t t W W (cid:13);g (cid:13) ′ FIG. 2: Magnetic photon and gluon penguin diagrams with the fourth generation t quark. ′ The new Wilson coefficients contributed by t are same as their counterparts in Eq. (16) ′ and (17) except for exchanging t quark not t quark. At the mass scale of µ , the Wilson coefficients of the dipole operators are given by b ∗ Ceff(µ ) = C(SM)eff(µ )+ Vt′sVt′bC(4)eff(µ ), (21) 7,8 b 7,8 b V∗V 7,8 b ts tb ∗ where Vt′s and Vt′b are two elements of the 4×4 CKM matrix. We recall here that the CKM ∗ coefficient corresponding to the t quark contribution, i.e., V V , is factorized in the effective ts tb Hamiltonian as given in Eq. (6). To calculate B γγ , one may follow a perturbative QCD approach which includes s(d) → a proof of factorization, showing that soft gluon effects can be factorized into B meson s(d) wave function; and a systematic way of resuming large logarithms due to hard gluons with 5 energies between 1Gev and m . In order to calculate the matrix element of Eq(1) for the b B γγ , we can work in the weak binding approximation and assume that both the s(d) → b and the s(d) quarks are at rest in the B meson, and the b quarks carries most of the s(d) meson energy, and its four velocity can be treated as equal to that of B . Hence one may s(d) write b quark momentum as p = m v where is the common four velocity of b and B . We b b s(d) have 1 p k = m v k = m m = p k , b · 1 b · 1 2 b Bs(d) b · 2 p k = (p k k ) k s(d) 1 1 2 1 · − − · 1 = m (m m ) −2 Bs(d) Bs(d) − b = p k . (22) s(d) 2 · We compute the amplitude of B γγ using the following relations s(d) → ¯ 0 s¯(d)γ γ b B (P) = if P , | µ 5 | s(d) − Bs(d) µ D E ¯ 0 s¯(d)γ b B (P) = if M , (23) | 5 | s(d) Bs(d) B D E where f is the B meson decay constant. Bs,d s(d) The total amplitude is now separated into a CP-even and a CP-odd part T(B γγ) = M+F Fµν +iM−F F˜µν. (24) s(d) µν µν → We find that 4√2αG 3C M+ = Ff m V∗ V Bm K(m2)+ 7 , (25) − 9π Bs(d) Bs(d) ts(d) tb b b 8Λ¯ (cid:18) (cid:19) 4√2αG 3C M− = Ff m V∗ V m A J(m2)+m BL(m2)+ 7 , (26) 9π Bs(d) Bs(d) ts(d) tb Bs(d) q q b b 8Λ¯ ! q X with B = (3C +C )/4, Λ¯ = m m , and − 6 5 Bs(d) − b A = (C C )N +(C C ), u 3 5 c 4 6 − − 1 A = [(C C )N +(C C )], d 3 5 c 4 6 4 − − A = (C +C C )N +(C +C C ), c 1 3 5 c 2 4 6 − − 1 A = [(C +C C )N +(C +C C )], s 3 4 5 c 3 4 6 4 − − A = A . (27) b s 6 The functions J(m2), K(m2) and L(m2) are defined by J(m2) = I (m2), 11 K(m2) = 4I (m2) I (m2), 11 00 − L(m2) = I (m2). (28) 00 with 1 1−x xpyq I (m2) = dx dy . (29) pq m2 2xyk k iε Z0 Z0 − 1 · 2 − The decay width for B γγ is simply s(d) → m3 Γ(B γγ) = Bs(d)( M+ 2 + M− 2). (30) s(d) → 16π | | | | III. NUMERICAL ANALYSIS AND SUMMARY In the numerical calculations we use as input parameters α (m ) = 0.118, , α (m ) = s Z s b 0.223, m = 80.22GeV, m = 1.27GeV, m = 4.19GeV, m = 172GeV, τ = 1.49ps, W c b t Bs f = 230MeV, λ = λ = 350MeV, m = 5.37GeV, τ = 1.55ps, f = 200MeV and Bs Bs Bd Bs Bd Bd m = 5.28GeV, respectively. Bd ′ ′ For the mass limit of t, CDF gives mt′ > 256GeV for the t qW final state [11]. → The experimental upper bounds for the fourth family quark CKM matrix elements are Vt′d < 0.063, Vt′s < 0.46, Vt′b < 0.47 [12]. By taking the CKM unitarity conditions, | | | | | | ∗ ′ V V = 0, (i = u,c,t,t), and the present measurement of 3 3 CKM matrix[13], We is(d) ib × i Pobtain the bounds for the CKM4 matrix elements in SM4, |Vt∗′dVt′b| < (1.83−2.03)×10−2, (31) |Vt∗′sVt′b| < (6.97−7.75)×10−2. (32) ∗ Fig. 3a shows the dependence of BR(Bd → γγ) with the CKM4 matrix factor |Vt′dVt′b| for different values ofmt′. We cansee thatthe new physics contributions canlead toappreciable changes of the SM predictions which may be enhanced by about more than one orders of ′ magnitude in a reason- able mass range for t. The new physics effects is very sensitive to the value of |Vt∗′dVt′b| and becomes tiny as |Vt∗′dVt′b| < 0.5×10−2. It mains that Bs → γγ can give the great limit room for the CKM4 matrix elements and give the strong constrict for the contributions to CP violation in SM4. But from Fig. 3a, the new physics effects is not 7 ′ sensitive to the mass of t. This can be seen more clearly in Fig. 3b, which shows the mass ′ dependence of BR(Bd γγ) with t for different values of mt′. From Fig. 3b, we can see → ′ that the new physics effects become bigger with increasing mass of the t. This is similar to the case of top quark to the rare B meson decays in SM, which the main contributions come from the heavy quark. -7 EXP.bound(3.2 10 ) 310-7 ) γ γ →210-7 d B ( R mt′ =256GeV B mt′ =400GeV 110-7 mt′ =600GeV mt′ =800GeV mt′ =1000GeV -8 SM(3.0 10 ) 0100 110-2 210-2 (a) |Vt′dVt∗′b| 610-7 410-7 ) γ γ -7 EXP.bound(3.2 10 ) → d B( Vt′dVt∗′b =2.03 10−2 BR 210-7 ||VVtt′′ddVVtt∗∗′′bb||==11..803××110−0−22 |Vt′dVt∗′b|=0.5×10−2 | | × -8 SM(3.0 10 ) 0100 400 600 800 1000 (b) mt′[GeV] ∗ FIG. 3: The Branching ratio of Bd → γγ versus (a) CKM4 matrix factor |Vt′dVt′b| for different ′ ∗ values of mt′; (b) the mass of t for different values of |Vt′dVt′b|. Figs. 4 and show the dependence of the decay BR(B γγ) with the CKM4 matrix s → 8 -6 EXP.bound(8.6 10 ) 810-6 ) γ γ → s B ( R 410-6 mt′ =256GeV B mt′ =400GeV mt′ =600GeV mt′ =800GeV SM(1 10-6) mt′ =1000GeV 0100 410-2 810-2 (a) |Vt′sVt∗′b| 110-5 -6 EXP.bound(8.6 10 ) ) γ 810-6 γ → Bs Vt′sVt∗′b =7.75 10−2 R( |Vt′sVt∗′b|=6.97×10−2 B |Vt′sVt∗′b|=1.0 ×10−2 410-6 |Vt′sVt∗′b|=0.5×10−2 | | × -6 SM(1 10 ) 400 600 800 1000 (b) mt′[GeV] ∗ FIG. 4: The Branching ratio of Bs → γγ versus (a) CKM4 matrix factor |Vt′sVt′b| for different ′ ∗ values of mt′; (b) the mass of t for different values of |Vt′sVt′b|. ∗ factor|Vt′sVt′b| fordifferent valuesofmt′. Wecanget thesimilar analysis but thenew physics ∗ effects is much more sensitive to the value of the CKM4 matrix factor, |Vt′sVt′b|. For both of these decays, the CKM4 matrix elements provides the dominant new physics contribution. 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