Distinguishing the right-handed up/charm quarks from top quark via discrete symmetries in the standard model extensions
aa r X i v : . [ h e p - ph ] S e p Distinguishing the right-handed up/charm quarks from top quarkvia discrete symmetries in the standard model extensions
Chao-Shang Huang, Tianjun Li,
1, 2
Xiao-Chuan Wang, and Xiao-Hong Wu
3, 1 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P. R. China School of Physical Electronics, University of Electronic Scienceand Technology of China, Chengdu 610054, P. R. China Institute of Modern Physics, East China University of Science and Technology,130 Meilong Road, Shanghai 200237, P. R. China (Dated: September 9, 2014)
Abstract
We propose a class of the two Higgs doublet Standard models (SMs) with a SM singlet and a classof supersymmetric SMs with two pairs of Higgs doublets, where the right-handed up/charm quarksand the right-handed top quark have different quantum numbers under extra discrete symmetries.Thus, the right-handed up and charm quarks couple to one Higgs doublet field, while the right-handed top quark couples to another Higgs doublet. The quark CKM mixings can be generatedfrom the down-type quark sector. As one of phenomenological consequences in our models, weexplore whether one can accommodate the observed direct CP asymmetry difference in singlyCabibbo-suppressed D decays. We show that it is possible to explain the measured values of CPviolation under relevant experimental constraints.
PACS numbers: 11.25.Mj, 12.10.Kt, 12.10.-g . INTRODUCTION Experimental data from the ATLAS [1, 2], CMS [3, 4], D0 and CDF [5] Collaborationshave confirmed the existence of the Standard Model (SM) Higgs boson. However, the quarkCKM mixing phase is not enough to explain the baryon asymmetry in the Universe andgives the contributions to electric dipole moments (EDMs) of electron and neutron muchsmaller than the experimental limits. Therefore, one needs new sources of CP violation,which has been one of the main motivations to search for new theoretical models beyondthe SM for a long time.The minimal extension of the SM is to enlarge the Higgs sector [6]. It has been shown thatthe two-Higgs-doublet models (2HDMs) naturally accommodates the electroweak precisiontests, giving rise at the same time to many interesting phenomenological effects [7]. Fora recent review on two-Higgs-doublet SMs, please see [8]. The generic scalar spectrumof the two-Higgs-doublet models consists of three neutral Higgs bosons and one chargedHiggs boson pair. The direct searches for additional scalar particles at the LHC or indirectsearches via precision flavor experiments will therefore continue being an important task inthe following years.In this paper, we will propose a class of the two Higgs doublet SMs with a SM singlet anda class of the supersymmetric SMs with two pairs of Higgs doublets, where the right-handedup/charm quarks and right-handed top quark have different quantum numbers under extradiscrete symmetries. Therefore, the right-handed up and charm quarks couple to one Higgsdoublet field, while the right-handed top quark couples to another Higgs doublet due toadditional discrete symmetries. All the down-type quarks couple to the same Higgs doublet,and all the charged leptons couple to the same Higgs doublet. Also, the quark CKM mixingscan be generated from the down-type quark sector. In particular, the first two-generationup-type quarks can have relatively large Yukawa couplings. As one of the phenomenologicalconsequences of our models we explore if one can accommodate the experimental measure-ment of direct CP asymmetry difference in singly Cabibbo-suppressed D decays.The CP asymmetry difference in D → K + K − and D → π + π − decays has been mea-sured by the LHCb Collaboration [9]. Combined with the results from the CDF [10],Belle [11], and previous BaBar [12] Collaborations, the Heavy Flavor Averaging Group yieldsa world average of the difference of direct CP asymmetry in D → K + K − and D → π + π − A CP = ( − . ± . A CP = ( − . ± . ± . . f b − data at 7 TeV have ∆ A CP = (0 . ± . ± . A CP = (+0 . ± . ± . f b − data, which have an oppositesign compared to the pion-tagged results. In combination, the current world-averaged directcharm meson CP violation is ∆ A CP = (0 . ± . A CP = A CP ( K + K − ) − A C P ( π + π − ) = − . × − , which is lower than theLHCb and CDF data. Based on the topological diagram approach for tree-level amplitudesand QCD factorization for a crude estimation of perturbative penguin amplitudes, Cheng andChiang [19] showed that the CP asymmetry difference ∆ A CP is of order − (0 . ∼ . . Even with the maximal magnitude of QCD-penguin exchange amplitude | P E | ∼ T ( T is the tree-level amplitude) and a maximal strong phase relative to T , one can only get∆ A CP = − .
25% which is still lower than the current world average. The SU (3) effects havealso been studied [20–24]. For the recent discussions on the subjects, please see Ref. [25].While the experiment is still not conclusive, there are some attempts to estimate the effectsfrom new physics models, e.g., fourth generation [26], left-right model [27], diquark [28],supersymmetry [29, 30], Randall-Sundrum model [31], compositeness [32, 33], minimal flavorviolation [34], other new physics models [35], and a χ analysis of different measurements inthe charm system [36].We calculate the direct CP asymmetry difference in charm meson decays with experi-mental constraints satisfied in our models in the paper. The new feature of our work is thatwe consider the contributions from Higgs penguin induced operators, and the mixing effectof Higgs penguin induced operator O into chromomagnetic operator O g at charm mass m c scale. We find that it is possible to explain the measured values of CP violation underrelevant experimental constraints.This paper is organized as follows. We present a class of two-Higgs-doublet SMs and aclass of the supersymmetric SMs in Sections II and III. The effective Lagrangian of c → u transition, relevant Wilson coefficients, direct CP asymmetry in charm meson decays, and3 c = 2 and ∆ c = 1 constraints are given in Section IV. We conclude in Section V. II. NONSUPERSYMMETRIC SMS
We consider the two-Higgs-doublet Standard Models [6]. First, let us explain the con-vention. We denote the left-handed quark doublets, the right-handed up-type quarks, theright-handed down-type quarks, the left-handed lepton doublets, and the right-handed lep-tons as q i , u i , d i , l i , and e i , respectively, where i = 1 , ,
3. In addition, we introduce twopairs of the Higgs doublets as φ and φ , and a SM singlet Higgs field S . Following thecommon convention, we assume that the U (1) Y charges for both φ and φ are +1.Without loss of generality, we assume that φ couples to the right-handed up and charmquarks, while φ couples to the right-handed top quark. We classify the models as follows • Model I: both the down-type quarks and the charged leptons couple to φ . • Model II: the down-type quarks couple to φ while the charged letpons couple to φ . • Model III: the charged letpons couple to φ while the down-type quarks couple to φ . • Model IV: both the down-type quarks and charged leptons couple to φ .To avoid the flavour changing neutral current (FCNC) constraints [37], we introduce a Z symmetry. Under this Z symmetry, the quark doublets, the up-type quarks, the Higgsfields, and the singlet transform as follows q i ↔ q i , u k ↔ u k , t ↔ ωt , φ ↔ φ , φ ↔ ωφ , S ↔ ωS , (1)where ω = 1, i = 1 , ,
3, and k = 1 ,
2. The transformation properties for down-typequarks, lepton doublets, and charged leptons will be given later for each model. By the way,to escape the FCNC constraints in the nonsupersymmetric SMs, we just need to consider Z symmetry, i.e. , we change each “ ω ” and “ ω ” into the “ − ” sign in our transformationequations. To match the supersymmetric SMs, we consider the Z symmetry in this paper.4 . Model I Under this Z symmetry, the down-type quarks, the lepton doublets, and the chargedleptons transform as follows d i ↔ ω d i , l i ↔ l i , e i ↔ ω e i . (2)Then, the SM fermion Yukawa Lagrangian is − L = y uki u k q i φ + y ti tq i φ + y dij d i q j ˜ φ + y eij e i l j ˜ φ + H . C . , (3)where y uij , y dij and y eij are Yukawa couplings, and ˜ φ i = iσ φ ∗ i . Here, σ is the second Paulimatrix. In particular, to avoid the FCNC constraints [37], we assume that the Yukawacouplings y u , y u , y t and y t are relatively small. It is clear that in the limit y u = y u = y t = y t = 0, there is no FCNC effect. Moreover, the quark CKM mixings are generatedfrom the down-type quark sector. Let us definetan β ≡ < φ >< φ > . (4)At large tan β , the Higgs fields with dominant components from φ will have large Yukawacouplings with the first two-generation up-type quarks.The most general renormalizable Higgs potential at tree level, which is invariant underthe SU (2) L × U (1) Y gauge symmetry and the Z symmetry, is V = λ φ † φ ) + λ φ † φ ) + λ S S † S ) + λ φ † φ )( φ † φ ) + λ φ † φ )( φ † φ )+ λ S S † S )( φ † φ ) + λ S S † S )( φ † φ ) + h ASφ † φ + H . C . i − m φ † φ − m φ † φ − m S S † S , (5)where λ i , λ S , λ S , and λ S are dimensionless parameters, m , m , and m S are mass pa-rameters, and A is a mass dimension-one parameter which is similar to the supersymmetrybreaking trilinear soft term. λ i for i = 1 , , , λ S , λ S , λ S , m , m and m S are real, while A is complex. In addition, the term λ ( φ † φ ) and its Hermitian conjugate, are forbiddenby discrete Z symmetry. Also, the terms λ ( φ † φ )( φ † φ ) and λ ′ ( φ † φ )( φ † φ ), as well astheir Hermitian conjugates, which will induce the FCNC processes [37], are forbidden in ourmodel, too. Interestingly, our model can be consistent with the constraints from the CPviolation and FCNC processes even if A is not real [38–41].5or simplicity, we assume that the up-type quark Yukawa matrix is diagonal, and thenthere are no tree-level FCNC processes. Also, we assume that A is relatively small, andthe vacuum expectation value (VEV) of S is much larger than the VEVs of φ and φ , forexample, h S i ≃ S and φ i are small and can be neglected.The Lagrangian of relevance for our discussion of direct CP violation in charm meson decayscan be written as −L = gm u k m W c α c β Hu k u k − gm u k m W s α c β hu k u k + gm t m W s α s β Htt + gm t m W c α s β htt − gm d j m W s α s β Hd j d j − gm d j m W c α s β hd j d j + i gm u k m W t β Au k γ u k + i gm t m W ct β Atγ t + i gm d j m W ct β Ad j γ d j + gm u k m W V kj t β H + u k P L d j − gm d j m W V kj ct β H + u k P R d j − gm t m W V j ct β H + tP L d j − gm d j m W V j ct β H + tP R d j + ... , where s α = sin α , c α = cos α , s β = sin β , c β = cos β , t β = tan β , and ct β = cot β , with α being the mixing angle between the real components of φ and φ . B. Model II
Under this Z symmetry, the down-type quarks, lepton doublets, and charged leptonstransform as follows d i ↔ d i , l i ↔ l i , e i ↔ ω e i . (6)So the SM fermion Yukawa Lagrangian is − L = y uki u k q i φ + y ti tq i φ + y dij d i q j ˜ φ + y eij e i l j ˜ φ + H . C . . (7)Similar to Model I, we assume that the Yukawa couplings y u , y u , y t and y t are relativelysmall. The most general renormalizable Higgs potential at tree level, which is invariantunder the SU (2) L × U (1) Y gauge symmetry and the Z symmetry, is the same as that inEq. (5) in Model I. At large tan β , the Higgs fields with dominant components from φ will have large Yukawa couplings with the first two-generation up-type quarks, and all thedown-type quarks. 6ith the same assumptions as in Model I, the Lagrangian of relevance for our discussioncan be written as −L = gm u k m W c α c β Hu k u k − gm u k m W s α c β hu k u k + gm t m W s α s β Htt + gm t m W c α s β htt − gm d j m W c α c β Hd j d j + gm d j m W s α c β hd j d j + i gm u k m W t β Au k γ u k + i gm t m W ct β Atγ t − i gm d j m W t β Ad j γ d j + gm u k m W V kj t β H + u k P L d j + gm d j m W V kj t β H + u k P R d j − gm t m W V j ct β H + tP L d j + gm d j m W V j t β H + tP R d j + ... . C. Model III
Under this Z symmetry, the down-type quarks, the lepton doublets, and the chargedleptons transform as follows d i ↔ ω d i , l i ↔ l i , e i ↔ e i . (8)So the SM fermion Yukawa Lagrangian is − L = y uki u k q i φ + y ti tq i φ + y dij d i q j ˜ φ + y eij e i l j ˜ φ + H . C . . (9)At large tan β , the Higgs fields with dominant components from φ will have large Yukawacouplings with the first two-generation up-type quarks, and all the charged leptons. Therest discussion is similar to those in Models I and II. D. Model IV
Under this Z symmetry, the down-type quarks, the lepton doublets, and the chargedleptons transform as follows d i ↔ d i , l i ↔ l i , e i ↔ e i . (10)Then, the SM fermion Yukawa Lagrangian is − L = y uki u k q i φ + y ti tq i φ + y dij d i q j ˜ φ + y eij e i l j ˜ φ + H . C . . (11)At large tan β , the Higgs fields with dominant components from φ will have large Yukawacouplings with the first two-generation up-type quarks, all the down-type quarks, and allthe charged leptons. The rest discussion is similar to those in Models I and II.7 II. SUPERSYMMETRIC STANDARD MODELS
First, let us explain the convention. We denote the chiral superfields for the quarkdoublets, the right-handed up-type quarks, the right-handed down-type quarks, the leptondoublets, and the right-handed charged leptons as Q i , U ci , D ci , L i , and E ci , respectively,where i = 1 , ,
3. We also introduce two pairs of Higgs doublets ( H u , H d ), and ( H ′ u , H ′ d ).In addition, we introduce three SM singlet Higgs fields S , S ′ and T .Without loss of generality, we assume that H u couples to the right-handed up and charmquarks, H ′ u couples to the right-handed top quark, and H d couples to the right-handeddown-type quarks. We classify the models as follows • Model A: H ′ d couples to the charged letpons. • Model B: H d couples to the charged letpons.To solve the µ problem, we consider a Z × Z ′ discrete symmetry. Under the Z symmetry,the SM quarks, the Higgs fields, and the singlet fields transform as follows Q i ↔ ωQ i , U ck ↔ ωU ck , T c ↔ ω T c , D ci ↔ ωD ci ,H u,d ↔ ωH u,d , H ′ u,d ↔ H ′ u,d , S ↔ ωS , S ′ ↔ S ′ , T ↔ ω T , (12)where ω = 1. And under the Z ′ symmetry, the SM quarks, the Higgs fields, and the singletfields transform as below Q i ↔ Q i , U ci ↔ U ci , T c ↔ ω ′ T c , D ci ↔ D ci ,H u,d ↔ H u,d , H ′ u,d ↔ ω ′ H ′ u,d , S ↔ S , S ′ ↔ ω ′ S ′ , T ↔ ω ′ T , (13)where ω ′ = 1. A. Model A
Under the Z × Z ′ symmetry, the lepton doublets and the charged leptons, respectively,transform as follows L i ↔ L i , E ci ↔ E ci ,L i ↔ ω ′ L i , E ci ↔ ω ′ E ci . (14)8hen, the SM fermion Yukawa Lagrangian is W Yukawa = y uik Q i H u U ck + y ti Q i T c H ′ u + y dij Q i H d D cj + y eij L i H ′ d E cj + λ SH d H u + λ S ′ H ′ d H ′ u + λ T H d H ′ u + λ T H ′ d H u + λ SS ′ T + κ S + κ S ′ + κ T , (15)where y uik , y ti , y dij , y eij , λ i , and κ i are Yukawa couplings. To avoid the FCNC constraints,we assume that the Yukawa couplings y u , y u , y t and y t are relatively small, similar to thenonsupersymmetric models. In our model, we definetan β ≡ < H d >< H u > , (16)which is different from the traditional minimal supersymmetric standard model. The VEVof H u can be much smaller than that of H d , since H ′ u couples to the top quark, i.e. , thecharm Yukawa coupling can be order 1. Note that the VEV of H d can be about one orderlarger that that of H ′ d , and we obtain that the Yukawa couplings of down-type quarks canbe about one order smaller than those of charged leptons compared to the SM. B. Model B
Under the Z × Z ′ symmetry, the lepton doublets and the charged leptons, respectively,transform as follows L i ↔ ωL i , E ci ↔ ωE ci ,L i ↔ L i , E ci ↔ E ci . (17)Then, the SM fermion Yukawa Lagrangian is W Yukawa = y uik Q i H u U ck + y ti Q i T c H ′ u + y dij Q i H d D cj + y eij L i H d E cj + λ SH d H u + λ S ′ H ′ d H ′ u + λ T H d H ′ u + λ T H ′ d H u + λ SS ′ T + κ S + κ S ′ + κ T . (18)To avoid the FCNC constraints, similar to Model A, we assume that the Yukawa couplings y u , y u , y t and y t are relatively small. 9 V. EFFECTIVE HAMILTONIAN AND DIRECT CP ASYMMETRIES IN D ME-SON DECAYS
The effective Hamiltonian for the c → u transition can be written as H eff ∆ C =1 = G F √ (cid:26) X p = d,s λ p ( C p O p + C p O p )+ λ b X i =3 C i O i + C γ O γ + C g O g + X i =11 X q = u,d,s,c C qi O qi (cid:27) , (19)with λ p = V ∗ cp V up ( p = d, s ) and λ b = V ∗ cb V ub .The complete list of operators is given as follows O p = (¯ up ) V − A (¯ pc ) V − A ,O p = (¯ u α p β ) V − A (¯ p β c α ) V − A ,O = (¯ uc ) V − A X q (¯ qq ) V − A ,O = (¯ u α c β ) V − A X q (¯ q β q α ) V − A ,O = (¯ uc ) V − A X q (¯ qq ) V + A ,O = (¯ u α c β ) V − A X q (¯ q β q α ) V + A ,O γ = e π m c [¯ uσ µν (1 + γ ) c ] F µν ,O g = g s π m c [¯ uσ µν T a (1 + γ ) c ] G µνa ,O q = (¯ uc ) S + P (¯ qq ) S − P ,O q = (¯ u α c β ) S + P (¯ q β q α ) S − P ,O q = (¯ uc ) S + P (¯ qq ) S + P ,O q = (¯ u α c β ) S + P (¯ q β q α ) S + P ,O q = [¯ uσ µν (1 + γ ) c ][¯ qσ µν (1 + γ ) q ] ,O q = [¯ u α σ µν (1 + γ ) c β ][¯ q β σ µν (1 + γ ) q α ] , (20)with V ± A = γ µ (1 ± γ ) and S ± P = (1 ± γ ).The direct CP asymmetry of D → K + K − can be written as a K + K − = 2Im (cid:18) λ b λ s R sK, SM (cid:19) + 2Im (cid:18) λ b λ s R sK, NP (cid:19) , (21)10here R sK, SM = a SM4 + r χ a SM6 a , R sK, NP = 1 a (cid:18) a NP4 − a s + r χ ( a NP6 + 14 a s + 3 a s ) (cid:19) , (22)where maximal strong phase is assumed, and only weak phase is included in the aboveequation. The a i coefficients are estimated in naive factorization a NP4 = 3 a NP6 = − C F α s πN C C NP8 g ,a s = C s + C s /N C ,a s = C s + C s /N C ,a s = C s + C s /N C , (23)where the Wilson coefficients C g, , , , , , are evaluated at charm quark mass m c scale.For the direct CP asymmetry of D → π + π − , the upper index s should be replaced with d .In the flavor SU (3) limit, we have a π + π − ≃ − a K + K − .The Wilson coefficients can be evolved from W boson mass m w scale to m c scale throughthe intermediate bottom quark mass scale m b [42]. The main contribution in our case is C g ( m c ), which can be written as [43–46] C g ( m c ) ≃ . C g ( m w ) − . C ( m w ) + 0 . C c ( m w ) . (24)The direct CP asymmetry in the decays D → K + K − and D → π + π − can be estimatedas ∆ a CP = a K + K − − a π + π − ≃ [ − . C NP8 g ( m w ) + 0 . C ( m w )] × . (25)For ∆ a CP ∼ . C NP8 g ( m w ) ∼
10, or C ( m w ) ∼ C c , as [46] C c = e π ( C cQ − C cQ ) , C c = e π ( C cQ + C cQ ) . (26)To follow, we will calculate C cQ , and C g at m w scale in Models I, II, and A.The contributions to C g from charged Higgs boson exchanges are C g = − cot β D ( x H ± ) − E ( x H ± ) (27)11n Model I, and C g = t β (cid:20) − D ( x H ± ) − E ( x H ± ) (cid:21) (28)in Model II, with x H ± = m b /m H ± . The one-loop functions D and E are defined in Ref. [47].In our calculations, we work in the limit of vanishing light quark masses, m u = m d = m s = 0. The Wilson coefficients C Q , at the leading order of O (tan β ) in Model I are C cQ = − m b m c m w s w c β (cid:18) c α m H + s α m h (cid:19)(cid:20) f b ( x H ± ) − f b ( x W ) (cid:21) − m c t β s w c β (cid:18) c α m H s β − α + s α m h c β − α (cid:19) f c ( x W , x H ± )+ m c t β m H ± s w | V cb | f d ( x W , x W , x H ± ) ,C cQ = m b m c t β m w s w m A (cid:20) f b ( x H ± ) − f b ( x W ) (cid:21) + 38 m c t β s w m A f c ( x W , x H ± ) − m c t β m H ± s w | V cb | f d ( x W , x W , x H ± ) . (29)where the one-loop functions f b ,c ,d are defined in Ref. [48].The Wilson coefficients C Q , at the leading order of O (tan β ) in Model II are C cQ = − m b m c m w s w t β c β (cid:18) c α m H + s α m h (cid:19) f b ( x H ± ) − m b m c m w s w t β c β (cid:18) c α m H + s α m h (cid:19)(cid:20) f c ( x H ± ) + m b m H ± f c ( x H ± ) (cid:21) + m b m c m w s w m H ± t β | V cb | f d ( x H ± ) ,C cQ = m b m c m w s w t β m A f b ( x H ± )+ m b m c m w s w t β m A (cid:20) f c ( x H ± ) + m b m H ± f c ( x H ± ) (cid:21) . (30)The leading contributions to the Wilson coefficients C g at the order of O (tan β ), and C Q , at the order of O (tan β ) in Model A from gluino exchanges are C g = − λ b g s g m W m g (cid:20) F ( x ˜ g ) δ LL + F ′ ( x ˜ g ) δ LR δ LR ∗ − m ˜ g m c F ( x ˜ g ) δ LR − m ˜ g m c F ′ ( x ˜ g ) δ LL δ LR (cid:21) ,C cQ = 43 λ b g s g s w m c m ˜ g c β (cid:18) c α m H + s α m h (cid:19) f ′ b ( x ˜ g ) δ LL δ LR , cQ = − λ b g s g s w m c m ˜ g m A t β f ′ b ( x ˜ g ) δ LL δ LR , (31)where the one-loop functions are defined in Ref. [46].The Higgs sector is subject to strong constraints from both the Higgs coupling mea-surements [57], and the direct heavier Higgs searches at LHC, in particular, pp → Φ → τ + τ − [58, 59], pp → Φ → µ + µ − [60] and pp → b Φ → bbb [61] channels, with Φ as the neutralHiggs boson. The implications of the Higgs coupling measurements are studied in Refs. [62]and [63] with direct heavier Higgs searches within the 2HDMs. Besides the up and charmquark Yukawa couplings, the other Higgs couplings in Model I are the same as in 2HDM 1,and in Model II are the same as in 2HDM 4 [62]. We note that the constraints in the β andcos( β − α ) plane are much looser in Model I than those in Model II, while the latter aretightly around the alignment limit α = β − π/
2. In the numerical calculations, we considerthe large tan β case. The direct heavier Higgs production channels through τ τ and µµ aresuppressed by sin α from Yukawa couplings in both Models I and II, while the bb channelis suppressed by sin α in Model I, and enhanced by tan β in Model II.For numerical estimations, we choose the following parameters in the Higgs sector forModel I: t β = 50, s α = − . m h = 126 GeV, m H = 180 GeV, m A = 220 GeV, and m H ± = 250 GeV. In Model II, the measurement of Br( B → X s γ ) puts a stringent bound onthe lower limit of the mass of the charged Higgs, m H ± ≥
380 GeV at 95% C.L. [64]. Witha heavy charged Higgs pair, the Higgs sector quickly approaches the decoupling limits. Fornumerical studies, we choose the following parameters for Model II: t β = 10, s α = − . m h = 126GeV, m H ≃ m A ≃ m H ± = 380GeV. In the supersymmetric version Model A, theYukawa couplings are similiar to those in Model I. We also take the supersymmetric scale m ˜ g = m ˜ q = 2 TeV [57].The charged Higgs contributions can be calculated as C H ± g ≃ − . × − in Model I, and C H ± g ≃ − .
047 in Model II. The contributions to C ( m w ) are suppressed in both Models Iand II, where we have C c ( m w ) ∼ − . × − in Model I, and C c ( m w ) ∼ − . × − inModel II. Therefore, due to the experimental constraints, the charged Higgs contributionscannot accommodate the direct CP measurement of charm decays.In Model A, for double insertion of ( δ LL δ LR ), we have C ˜ g g ∼ . × ( δ LL δ LR )10 − and C c ∼− . × ( δ LL δ LR )10 − from gluino exchange. For ( δ LL δ LR ) at the order of 10 − , we can have both C g at the order of 10 and C c of order 1, which are possible to accommodate the direct CP13easurement of charm decays.The constraint from the D − ¯ D system can be found in Ref. [49]. The nonvanishingWilson coefficients z i ( i = 1 , , ...
5) are z = g π Λ m W | λ b | m c m W x W [ I ( x W , x W /x H ± ) − I ( x W , x W /x H ± )] (32)at the leading order of O ( t β ) in Model I, and z = g π Λ m W | λ b | t β x W [ 14 I ( x W , x W /x H ± ) + m c m W I ( x W , x W /x H ± )] (33)at the leading order of O ( t β ) in Model II. The loop functions I , , are defined in Ref. [50].We can calculate z for the above parameters, z ≃ − . × − in Model I, and z ≃ . × − ( t β ) in Model II, which are below the experimental limits.In Model A, we obtain the gluino contributions z = − α s
216 ( δ LL ) [66 ˜ f ( m q /m g ) + 24 f ( m q /m g )] , ˜ z = − α s
216 ( δ LL δ LR ) f ( x ) , (34)for Λ NP = m ˜ g , where the functions f and ˜ f are given in Ref. [65], and f is defined asfollows f ( x ) = 60 x (5 + x ) ln( x ) − x − x + 300 x − x + 25 x − x − . The leading order contributions from ( δ LL ) are included in z . In the numerical estimations,we take δ LR = ( m c A c − m c µ tan β ) /m q ≃ − m c µ tan β/m q ≃ − .
015 (with µ ∼ . m ˜ q , m c ∼ . m ˜ q scale), and δ LL ≃ . z ≃ . × − ( δ LL . ) , and ˜ z ≃ − . × − ( δ LL δ LR − ) , which are belowthe limits from the constraints of the D − ¯ D system. However, due to the SU (2) gaugeinvariance, the left-left up-type squark matrix is related to the down-type one. And we have δ LL ≃ .
067 for down-type squarks, which does not satisfy the constraints from kaon systemfor the imaginary part Im( δ LL ) ≤ .
023 with the supersymmetry scale at 2 TeV [51]. Oneway out is to consider the contributions of chirally opposite operators. We can get similiarresults if the above δ LL is replaced with δ RR ∼ . δ LR with δ LR ∗ ∼ − . B ( D → µ + µ − ) < . . × − at 90% (95%) C.L. [52]and B ( D + → π + µ + µ − ) < . . × − at 90% (95%) C.L. [53]. The experimental boundon radiative charm decay is B ( D → γγ ) < . × − at 90% C.L. from the BABARCollaboration [55], and B ( D → γγ ) < . × − at 90% C.L. from BESIII [56].The corresponding Wilson coefficients are C γ = G ( x H ± ) + 16 cot βA ( x H ± ) (35)in Model I, C γ = t β [ G ( x H ± ) + 16 A ( x H ± )] (36)in Model II, and C = − − s W s W cot β x W B ( x H ± ) + cot βx H ± F ( x H ± ) ,C = − s W cot β x W B ( x H ± ) (37)in Model I, while replacing cot β with t β in Model II. The functions A, B, G, and F for the c → u transitions are defined as A ( x ) = − x
12 ( 5 − x − x (1 − x ) + 6 x (1 − x ) ln x (1 − x ) ) ,B ( x ) = − x − x + ln x (1 − x ) ) ,F ( x ) = 11 − x + 40 x − x ) + 2 − x + 3 x − x ) ,G ( x ) = − x − x ) − (1 − x ) ln x (1 − x ) ) , (38)which differ from the ones in Ref. [54] for the b → s transitions.The leading order contributions to the Wilson coefficients C γ, , at the order of O (tan β )in Model A from gluino exchanges are C γ = 272 λ b g s g m W m g (cid:20) F ( x ˜ g ) δ LL + F ′ ( x ˜ g ) δ LR δ LR ∗ − m ˜ g m c F ( x ˜ g ) δ LR − m ˜ g m c F ′ ( x ˜ g ) δ LL δ LR (cid:21) ,C = 472 λ b g s g m W m g (cid:20) f ′ ( x ˜ g ) δ LL + f ′′ ( x ˜ g ) δ LR δ LR ∗ (cid:21) , − λ b s W g s g ( − s W ) (cid:20) − f (1) c ( x ˜ g ) δ LR δ LR ∗ + f (2) c ( x ˜ g ) δ LL + f (3) c ( x ˜ g ) δ LR δ LR ∗ (cid:21) ,C = − λ b s W g s g (cid:20) − f (1) c ( x ˜ g ) δ LR δ LR ∗ + f (2) c ( x ˜ g ) δ LL + f (3) c ( x ˜ g ) δ LR δ LR ∗ (cid:21) , (39)15here the one-loop functions are defined as follows: f ′ ( x ) = x ∂f ( x ) ∂x , f ′′ ( x ) = x ∂ f ( x ) ∂x , f (1) c ( x ) = x ∂ f c ( x,y ) ∂x∂y | y − >x , f (2) c ( x ) = x ∂f c ( x,x ) ∂x , f (3) c ( x ) = x ∂ f c ( x,x ) ∂x , and F , F ′ , and f c are defined in Ref. [46].In Model I, the short distance (SD) contribution from the charged Higgs exchange isnegligible, B ( D → γγ ) ∼ − . In Model II, the contribution can be estimated as B ( D → γγ ) = 2 . × − . In Model A with a double insertion of ( δ LL δ LR ), we have C ˜ g γ ∼ − . × ( δ LL δ LR )10 − from gluino exchange. The SD contribution can be estimated as B ( D → γγ ) =5 . × − . In all three models, we have B ( D → µ + µ − ) and B ( D + → π + µ + µ − ) far belowthe current experimental bounds. V. CONCLUSION
We proposed a class of the two-Higgs-doublet SMs with a SM singlet and a class ofsupersymmetric SMs with two pairs of Higgs doublets, where the right-handed up/charmquarks and the right-handed top quark have different quantum numbers under extra discretesymmetries. So the right-handed up and charm quarks couple to one Higgs doublet field,while the right-handed top quark couples to another Higgs doublet. We have studied thedirect CP asymmetries in charm hadronic decays in Models I, II and A. We found thatthe large direct CP asymmetry difference cannot be accommodated within Model I and IIwith the contributions of charged Higgs bosons. In Model A, we can accommodate theexperimental measurement of direct CP asymmetry with both O g and O operators, whilethe constraints from the ∆ c = 2 and ∆ c = 1 processes are satisfied.We leave the detailed studies on phenomenological consequences of our models to thefuture. Acknowledgments
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