In the two-Higgs-Doublet model the alignment of the Yukawa flavour matrices
of the two scalar doublets guarantees the absence of tree-level
flavour-changing neutral couplings. The resulting fermion-scalar interactions
are parameterized in terms of three complex parameters, leading to a generic
Yukawa structure which contains as particular cases all known specific
implementations of the model based on Z2 symmetries. These three complex
parameters are potential new sources of CP violation.Comment: 7 page

Pich, Antonio

Tuzon, Paula

Acta Physica Polonica B Proceedings Supplement

English

arXiv

In the two-Higgs-Doublet model the alignment of the Yukawa flavour matrices of the two scalar doublets guarantees the absence of tree-level flavour-changing neutral couplings. The resulting fermion-scalar interactions are parameterized in terms of three complex parameters, leading to a generic Yukawa structure which contains as particular cases all known specific implementations of the model based on Z2 symmetries. These three complex parameters are potential new sources of CP violation

Tuzón Marco, Paula

Pich Zardoya, Antonio

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:1001.0293v2  [hep-ph]  13 Jan 2010The Aligned Two-Higgs-Doublet Model∗Paula Tuzo´nandAntonio PichDepartament de F´ısica Teo`rica, IFIC, Universitat de Vale`ncia - CSICApt. Correus 22085, E-46071 Vale`ncia, SpainIn the two-Higgs-Doublet model the alignment of the Yukawa flavourmatrices of the two scalar doublets guarantees the absence of tree-levelflavour-changing neutral couplings. The resulting fermion-scalar interac-tions are parameterized in terms of three complex parameters ζf , leadingto a generic Yukawa structure which contains as particular cases all knownspecific implementations of the model based on Z2 symmetries. These threecomplex parameters are potential new sources of CP violation.PACS numbers: 12.60.Fr, 12.15.-y, 12.15.Ji, 14.80.Cp1. The generic two-Higgs-doublet modelThe two-Higgs-doublet model (THDM) [1, 2] is one of the most sim-plest extensions of the Standard Model (SM) and brings many interest-ing phenomenological features (new sources of CP violation, dark mat-ter candidates, axion phenomenology . . . ). It is a gauge theory with thesame fermion content as the SM and two Higgs doublets φa (a = 1, 2)with hypercharge Y = 12. The neutral components of the scalar dou-blets acquire vacuum expectation values that are, in general, complex:< 0|φTa (x)|0 >= 1√2 (0, vaeiθa). One can fix θ1 = 0 through an appropriateU(1)Y transformation, leaving the relative phase θ ≡ θ2 − θ1.A global SU(2) transformation in the scalar space (φ1, φ2) allows us todefine the so-called Higgs basis (Φ1,Φ2) where only one doublet acquires a∗ Presented at the Flavianet topical workshop on Low energy constraints on extensionsof the Standard Model, Kazimierz, Poland, 23-27 July 20092vacuum expectation value v ≡√v21 + v22 :(Φ1−Φ2)≡ 1v[v1 v2v2 −v1] (φ1e−iθφ2), (1)Φ1 =[G+1√2(v + S1 + iG0)], Φ2 =[H+1√2(S2 + iS3)]. (2)In addition to the three Goldstone fields G±(x) and G0(x), there are fivephysical degrees of freedom in the scalar sector: two charged fields H±(x)and three neutrals ϕ0i (x) = {h(x),H(x), A(x)}, which are related to the Sifields through an orthogonal transformation ϕ0i (x) = RijSj(x). The formof Rij depends on the scalar potential, which could violate CP in its mostgeneral version; in this case the resulting mass eigenstates will not have adefinite CP parity. If it is CP symmetric, the CP admixture disappears.1.1. Yukawa sectorThe most general Yukawa Lagrangian is given byLY = − Q¯′L(Γ1φ1 + Γ2φ2)d′R − Q¯′L(∆1φ˜1 +∆2φ˜2)u′R− L¯′L(Π1φ˜1 +Π2φ˜2)l′R + h.c. , (3)where Q¯′L and L¯′L are the left-handed quark and lepton doublets, respec-tively, and φ˜a(x) ≡ iτ2φ∗a the charge-conjugated scalar fields. All fermionicfields are written as NG–dimensional vectors and the couplings Γa, ∆a andΠa are NG × NG complex matrices in flavour space. Moving to the Higgsbasis, the Lagrangian readsLY = −√2v{Q¯′L(M′dΦ1 + Y′dΦ2)d′R + Q¯′L(M′uΦ˜1 + Y′uΦ˜2)u′R+ L¯′L(M′l Φ˜1 + Y′l Φ˜2)l′R + h.c.}. (4)Here, M ′f (f = u, d, l) are the non-diagonal fermion mass matrices and Y′fcontain the Yukawa couplings to the scalar doublet that doesn’t acquire avacuum expectation value. Each right-handed fermion couples to two differ-ent matrices that in general cannot be diagonalized simultaneously. In thefermion mass-eigenstate basis f(x), the M ′f matrices become the diagonalmass matrices Mf , while Y′f become Yf which remain non-diagonal and un-related to Mf . This originates dangerous flavour-changing neutral-current(FCNC) interactions, which are tightly constrained phenomenologically.3To avoid this problem one can assume that the non-diagonal Yukawacouplings are proportional to the geometric mean of the two fermion masses,gij ∝ √mimj [3]. This kind of phenomenologically viable models, knownas Type III THDM [4], can be obtained assuming particular textures ofthe Yukawa matrices [3]. Another possibility is to make the scalars heavyenough to suppress low-energy FCNC effects. However, this assumptionleads to a phenomenologically non relevant THDM.A more elegant solution is to impose a discrete Z2 symmetry to enforcethat only one scalar doublet couples to a given right-handed fermion [5]; oneof the two Yukawa matrices has to be zero. One requires the Lagrangianto remain invariant under the change φ1 → φ1, φ2 → −φ2, QL → QL,LL → LL, and appropriate transformation properties for the right-handedfermions. There are four non-equivalent choices: the Type I (only φ2 couplesto fermions) [6, 7, 8], Type II (φ1 couples to d and l and φ2 to u) [7, 9, 8],Type X or leptophilic (φ1 couples to fermions and φ2 to quarks) and TypeY (φ1 couples to d and φ2 to u and l) models [8, 10, 11, 12, 13]. The Z2symmetry can be also implemented in the Higgs basis, forcing all fermionsto couple to Φ1; this inert doublet model is a natural frame for dark matter[14, 15, 16].2. Aligned THDMA more general way to avoid tree-level FCNCs is to require the alignmentin flavour space of the Yukawa coupling matrices [17]:Γ2 = ξde−iθΓ1 , ∆2 = ξ∗ueiθ∆1 , Π2 = ξle−iθΠ1 , (5)with ξf arbitrary complex numbers. The alignment guarantees that the ma-trices Y ′f and M′f are proportional and can be simultaneously diagonalized:Yd,l = ζd,lMd,l , Yu = ζuMu , ζf ≡ ξf − tan β1 + ξf tan β, (6)where tan β ≡ v2/v1. In terms of mass-eigenstate fields, the Yukawa La-grangian takes then the form:LY = −√2vH+(x) {u¯(x) [ζdVMdPR − ζuMuV PL] d(x) + ζl ν¯(x)MlPRl(x)}− 1v∑ϕ,fϕ0i (x)yϕ0if f¯(x)MfPRf(x) + h.c. , (7)where V is the CKM matrix, PR,L ≡ 1±γ52 and yϕ0if are neutral couplings ofthe physical scalar fields [17]. Therefore, within this approach:4Table 1. Choices of ξf associated with Z2 models and corresponding values of ζf .Model (ξd, ξu, ξl) ζd ζu ζlType I (∞,∞,∞) cot β cot β cot βType II (0,∞, 0) − tan β cot β − tan βType X (∞,∞, 0) cot β cot β − tan βType Y (0,∞,∞) − tan β cot β cot βInert (tan β, tan β, tan β) 0 0 0– All scalar-fermion couplings are proportional to the correspondingfermion masses.– The neutral Yukawas are diagonal in flavour.– The only source of flavour-changing interactions is the quark-mixingmatrix V in the charged sector.– All leptonic couplings are diagonal in flavour, since there are no right-handed neutrinos in the theory.– There are only three new parameters ζf , which encode all possiblefreedom allowed by the alignment conditions. These couplings satisfyuniversality among the different generations, i.e. there is the sameuniversal coupling for all fermions with a given electric charge. Theyare also invariant under global SU(2) transformations of the scalarfields φa → φ′a = Uabφb [18], i.e. they are scalar basis independent.– The usual models based on Z2 symmetries are recovered taking theappropriated limits shown at Table 1.– The ζf can be arbitrary complex numbers. Their phases introducenew sources of CP violation without tree-level FCNCs.Lepton-flavour-violating neutral couplings are identically zero to all or-ders in perturbation theory, because of the absence of right-handed neu-trinos. So, the usually adopted Z2 symmetries are not necessary in thelepton sector, making the Type X and Y models less compelling. From aphenomenological point of view, the coupling ζl could take any value.Quantum corrections could introduce some misalignment of the quarkYukawa coupling matrices, generating small FCNCs suppressed by the cor-responding loop factors. However, the flavour mixing induced by loop cor-rections has a very characteristic structure [17] which resembles the popularMinimal Flavour Violation scenarios [19]. This restriction in the structureof the local FCNC terms is due to the following symmetry of the alignedTHDM: the Lagrangian remains invariant under flavour-dependent phase5transformations of the different fermion mass eigenstates, fi(x)→ eiαfi fi(x),provided the quark mixing matrix is transformed as Vij → eiαui Vije−αdj [17].The only possible FCNC terms are then of the type u¯i[V (Md)nV †]ijuj andd¯i[V†(Mu)mV ]jdj (n,m > 0), or similar structures with additional factorsof V and V † [17]. Structures of this type have been recently discussed inRef. [20].3. PhenomenologyOne of the most distinctive features of the THDM is the presence of acharged scalar H±. Assuming that Br(H+ → cs¯) + Br(H+ → τ+ντ ) = 1,the combined LEP data on e+e− → H+H− constrain MH± > 78.6 GeV(95% CL) [21]. The CDF [22] and D0 [23] collaborations have searched fort → H+b decays with negative results. CDF assumes Br(H+ → cs¯) = 1(ζl << ζu,d), while D0 adopts the opposite hypothesis Br(H+ → τντ ) = 1(ζl >> ζu,d). Both experiments find upper bounds on Br(t→ H+b) around0.2 (95% CL) for charged scalar masses between 60 and 155 GeV. Thisimplies |ζu| < 0.3 − 2.5, where the exact number depends on MH± .One can also extract information from the semileptonic decays of a pseu-doscalar meson P+ → l+νl, which are sensitive to H+ exchange due tothe helicity suppression of the SM amplitude: Γ(P+ij → l+νl)/Γ(P+ij →l+νl)SM = |1 − ∆ij|2, where the new physics information is encoded in∆ij = (mP±ij/MH±)2ζ∗l (ζumui + ζdmdj)/(mui + mdj). The correction ∆ijis in general complex and its real part can have either sign. To determineits size one needs to know Vij and a theoretical determination of the me-son decay constant. From the present knowledge of B+ → τ+ντ [24], weobtain |1−∆ub| = 1.32± 0.20 which implies MH±/√|Re(ζ+l ζd)| > 5.7 GeV(90% CL). With fDs = 242 ± 6 MeV [25], the recent CLEO measurementBr(D+s → τ+ντ ) = (5.62 ± 0.64)% [26] implies |1 −∆cs| = 1.07 ± 0.05 andMH±/√|Re(ζ+l ζu)| > 4.4 GeV (90% CL).FCNC processes such as b → sγ or P 0 − P¯ 0 mixing are also very sen-sitive to scalar contributions and give strong constraints on the parameterspace (MH± , ζu,d). A detailed phenomenological analysis will be presentedsomewhere else.AcknowledgementsThis work has been supported in part by the EU Contract MRTN-CT-2006-035482 (FLAVIAnet), by MICINN, Spain [grants FPA2007-60323 andConsolider-Ingenio 2010 CSD2007-00042 CPAN] and by Generalitat Valen-6ciana (PROMETEO/2008/069). P.T is indebted to the Spanish MICINNfor a FPU fellowship.REFERENCES[1] J. Gunion, H.E. Haber, G. Kane and S. Dawson, The Higgs Hunters Guide(Addison-Wesley, New York, 1990).[2] G.C. Branco, L. Lavoura and J. P. 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The Aligned two-Higgs Doublet model

Topical Workshop on Low Energy Constraints on Extensions of the Standard Model.
Kazimierz, POLAND, JUL 23-27, 2009In the two-Higgs-doublet model the alignment of the Yukawa flavour matrices of the two scalar doublets guarantees the absence of tree-level flavour-changing neutral couplings. The resulting fermion-scalar interactions are parameterized in terms of three complex parameters zeta(f), leading to a generic Yukawa structure which contains as particular cases all known specific implementations of the model based on Z(2) symmetries. These three complex parameters are potential new sources of CP violation.This work has been supported in part by the EU Contract MRTN-CT-
2006-035482 (FLAVIAnet), by MICINN, Spain [grants FPA2007-60323 and
Consolider-Ingenio 2010 CSD2007-00042 CPAN] and by Generalitat Valenciana (PROMETEO/2008/069). P.T is indebted to the Spanish MICINN
for a FPU fellowship

The Aligned two-Higgs Doublet model

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