High scale flavor alignment in two-Higgs doublet models and its phenomenology
PP REPARED FOR SUBMISSION TO
JHEP
High scale flavor alignment in two-Higgs doubletmodels and its phenomenology
Stefania Gori, a Howard E. Haber, b Edward Santos b a Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA b Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064
E-mail: [email protected] , [email protected] , [email protected] A BSTRACT : The most general two-Higgs doublet model (2HDM) includes potentially large sourcesof flavor changing neutral currents (FCNCs) that must be suppressed in order to achieve a phenomeno-logically viable model. The flavor alignment ansatz postulates that all Yukawa coupling matrices arediagonal when expressed in the basis of mass-eigenstate fermion fields, in which case tree-level Higgs-mediated FCNCs are eliminated. In this work, we explore models with the flavor alignment conditionimposed at a very high energy scale, which results in the generation of Higgs-mediated FCNCs viarenormalization group running from the high energy scale to the electroweak scale. Using the currentexperimental bounds on flavor changing observables, constraints are derived on the aligned 2HDMparameter space. In the favored parameter region, we analyze the implications for Higgs boson phe-nomenology. a r X i v : . [ h e p - ph ] J un ontents B s,d → µ + µ − decays 304.3 B → τ ν decays 34 With the discovery of a particle closely resembling the Standard Model (SM) Higgs boson at the LargeHadron Collider (LHC) [1–3], attention now turns to elucidating the dynamics of electroweak sym-metry breaking. Many critical question still remain unanswered. What is the origin of the electroweakscale, and what mechanism ensures its stability? In light of the existence of multiple generations offermions, are there also multiple copies of the scalar multiplets, implying the existence of additionalHiggs scalars? If yes, how are the Higgs-fermion Yukawa interactions compatible with the apparentMinimal Flavor Violation (MFV), which is responsible for suppressed flavor changing neutral currents(FCNCs)? – 1 –otivations for extending the Higgs sector beyond its minimal form have appeared often in theliterature. For example, the minimal supersymmetric extension of the Standard Model, which is in-voked to explain the stability of the electroweak symmetry breaking scale with respect to very highmass scales (such as the grand unification or Planck scales), requires a second Higgs doublet [4–7] toavoid anomalies due to the Higgsino partners of the Higgs bosons. More complicated scalar sectorsmay also be required for a realistic model of baryogenesis [8]. Finally, the metastability of the SMHiggs vacuum [9–11] can be rendered stable up to the Planck scale in models of extended Higgs sec-tors [12–19]. Even in the absence of a specific model of new physics beyond the Standard Model, anenlarged scalar sector can provide a rich phenomenology that can be probed by experimental searchesnow underway at the LHC.One of the simplest extensions of the SM Higgs sector is the two-Higgs doublet model (2HDM). In its most general form, the 2HDM is incompatible with experimental data due to the existence of un-suppressed tree-level Higgs-mediated FCNCs, in contrast to the SM where tree-level Higgs-mediatedFCNCs are absent. To see why this is so, consider the Higgs-fermion Yukawa interactions expressedin terms of interaction eigenstate fermion fields. Due to the non-zero vacuum expectation value (vev)of the neutral Higgs field, fermion mass matrices are generated. Redefining the left and right-handedfermion fields by separate unitary transformations, the fermion mass matrices are diagonalized. Inthe SM, this diagonalization procedure also diagonalizes the neutral Higgs-fermion couplings, andconsequently no tree-level Higgs-mediated FCNCs are present. In contrast, in a generic 2HDM, thediagonalization of the fermion mass matrices implies the diagonalization of one linear combination ofHiggs-fermion Yukawa coupling matrices. As a result, tree-level Higgs-mediated FCNCs remain inthe 2HDM Lagrangian when expressed in terms of mass-eigenstate fermion fields. If it were possiblein the 2HDM to realize flavor-diagonal neutral Higgs couplings at tree-level (thereby eliminating alltree-level Higgs-mediated FCNCs), then all FCNC processes arising in the model would be generatedat the loop-level, with magnitudes more easily in agreement with experimental constraints. A natural mechanism for eliminating the tree-level Higgs-mediated FCNCs was proposed byGlashow and Weinberg [22] and by Paschos [23] [GWP]. One can implement the GWP mechanismin the 2HDM by introducing a Z symmetry to eliminate half of the Higgs-fermion Yukawa couplingterms. In this case, the fermion mass matrices and the non-zero Higgs-fermion Yukawa couplingmatrices (which are consistent with the Z symmetry) are simultaneously diagonalized. Indeed, thereare a number of inequivalent implementations of the GWP mechanism, resulting in the so-calledTypes I [24, 25], and II [25, 26], and Types X and Y [27, 28] versions of the 2HDM. For a review with a comprehensive list of references, see Ref. [20]. Even in models with flavor-diagonal neutral Higgs couplings, one-loop processes mediated by the charged Higgs bosoncan generate significant FCNC effects involving third generation quarks. Such models, in order to be consistent withexperimental data, will produce constraints in the [ m H ± , tan β ] plane. The most stringent constraint of this type, obtainedin Ref. [21] in the analysis of the Type-II 2HDM prediction for b → sγ , yields m H ± (cid:38) GeV at
CL. However, if additional degrees of freedom exist at the TeV scale, then the GWP mechanism is in general not sufficientto protect the theory from FCNCs that are incompatible with the experimental data. These TeV-scale degrees of freedom,when integrated out, can generate higher-dimensional operators of the type ( c / Λ ) ¯ Q L Y (6) u U R H | H | + · · · , which breakthe proportionality relation between quark masses and effective Yukawa interactions with the neutral scalars. As a result,such models generically generate FCNC processes that are not sufficiently suppressed [29]. – 2 –nother strategy for eliminating tree-level Higgs-mediated FCNCs is by fiat. The flavor align-ment ansatz proposed in Ref. [30] asserts a proportionality between the two sets of Yukawa matrices.If this flavor-alignment condition is implemented at the electroweak scale, then the diagonalization ofthe fermion mass matrices simultaneously yields flavor-diagonal neutral Higgs couplings. Moreover,this flavor-aligned 2HDM (henceforth denoted as the A2HDM) preserves the relative hierarchy in thequark mass matrices, and provides additional sources of CP-violation in the Yukawa Lagrangian viathe introduction of three complex alignment parameters. Unfortunately, apart from the special casesenumerated in Ref. [31], there are no symmetries within the 2HDM that guarantee the stability of theflavor alignment ansatz with respect to radiative corrections. As such, flavor alignment at the elec-troweak scale must be generically regarded as an unnatural fine-tuning of the Higgs-fermion Yukawamatrix parameters. Indeed, the Types I, II, X and Y 2HDMs are the unique special cases of flavoralignment that are radiatively stable after imposing the observed fermion masses and mixing [32].In this paper, we consider the possibility that flavor alignment arises from New Physics beyondthe 2HDM. Without a specific ultraviolet completion in mind, we shall assert that flavor alignment isimposed at some high energy scale, Λ , perhaps as large as a grand unification scale or the Planck scale,where new dynamics can emerge (e.g., see Ref. [33] for a viable model). Once we impose the flavoralignment ansatz at the scale Λ , the effective field theory below this scale corresponds to a 2HDMwith both Higgs doublets coupling to up type and down type quarks and leptons. We then employrenormalization group (RG) evolution to determine the structure of the 2HDM Yukawa couplingsat the electroweak scale. For a generic flavor alignment ansatz at the scale Λ , flavor alignment inthe Higgs-fermion Yukawa couplings at the electroweak scale is violated, thereby generating Higgs-mediated FCNCs. However, these FCNCs will be of Minimal Flavor Violation [35] type and thereforemay be small enough to be consistent with experimental constraints, depending on the choice of theinitial alignment parameters at the scale Λ .We therefore examine the phenomenology of Higgs-mediated FCNCs that arise from the assump-tion of flavor alignment at some high energy scale, Λ , that, for the purpose of our analyses, is fixed tobe the Planck scale ( M P ). We note that similar work was performed in [36], where meson mixing and B decays were used to constrain the A2HDM parameter space with flavor alignment at the Planckscale. Numerical results were obtained analytically in [36], using the leading logarithmic approxima-tion. The results of this paper are first obtained in the leading log approximation, and then numericallyby evolving the full one-loop renormalization group equations (RGEs) down from the Planck scale tothe electroweak scale. In our work, we discuss the validity of the leading log approximation and ex-amine additional FCNC processes at high energy (top and Higgs decays) and at low energy ( B mesondecays) to place bounds on the A2HDM parameters.This paper is organized as follows. In section 2, we review the theoretical framework of thegeneral 2HDM. It is convenient to make use of the Higgs basis, which is unique up to a phase degree In practice, one should also append to the 2HDM some mechanism for generating neutrino masses. An example ofincorporating the effects of neutrino masses and mixing in the context of a 2HDM with flavor changing neutral Higgscouplings can be found in Ref. [34]. In this paper, we shall simply put all neutrino masses to zero for the sake of simplicity.The extension of the results of this paper to models that incorporate a mechanism for neutrino mass generation will beconsidered in a future publication. – 3 –f freedom. All physical observables must be independent of this phase. In particular, we examinein detail the structure of the Higgs-fermion Yukawa couplings and exhibit its flavor structure. Inthe formalism presented in section 2, we initially allow for the most general form of the Higgs scalarpotential and the Yukawa coupling matrices. In particular, new sources of CP-violation beyond the SMcan arise due to unremovable complex phases in both the scalar potential parameters and the Yukawacouplings. For simplicity, we subsequently choose to analyze the case of a CP-conserving Higgsscalar potential and vacuum, in which case the neutral mass-eigenstates consist of two CP-even andone CP-odd neutral Higgs bosons. We then introduce the flavor-aligned 2HDM, in which the Yukawacoupling matrices are diagonal in the basis of quark and lepton mass-eigenstates. However, alignmentis not stable under renormalization group running. Following the framework for flavor discussedabove, we impose the alignment condition at the Planck scale and then evaluate the Yukawa couplingmatrices of the Higgs basis at the electroweak scale as determined by renormalization group running,subject to the observed quark and lepton masses and the CKM mixing matrix. The renormalizationgroup running is performed numerically and checked in the leading log approximation, where simpleanalytic expressions can be obtained. In this context, a comparison with general Minimal FlavorViolating 2HDMs is performed.In section 3, we discuss the implications of high-scale flavor alignment for high energy processes.We focus on flavor-changing decays of the top quark and on the phenomenology of the heavy neutralCP-even and CP-odd Higgs bosons. In section 4, we discuss the implications of high-scale flavoralignment for low energy processes. Here we consider constraints arising from neutral meson mixingobservables and from B s → (cid:96) + (cid:96) − , which receive contributions at tree-level from neutral Higgs ex-change, and from the charged Higgs mediated B → τ ν decay. By comparing theoretical predictionsto experimental data, one can already probe certain regions of the A2HDM parameter space. Addi-tional parameter regions will be probed by future searches for heavy Higgs bosons and measurementsof B -physics observables. Conclusions of this work are presented in section 5. Finally, in Appendix Awe review the derivation of the Yukawa sector of our model in the fermion mass-eigenstate basis, andin Appendix B we exhibit the one-loop matrix Yukawa coupling RGEs used in this analysis. Consider a generic 2HDM consisting of two complex, hypercharge-one scalar doublets, Φ and Φ .The most general renormalizable scalar potential that is invariant under local SU(2) × U(1) gaugetransformations can be written as V = m Φ † Φ + m Φ † Φ − [ m Φ † Φ + h . c . ] + λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ λ (Φ † Φ )(Φ † Φ ) + (cid:110) λ (Φ † Φ ) + (cid:2) λ (Φ † Φ ) + λ (Φ † Φ ) (cid:3) Φ † Φ + h . c . (cid:111) . (2.1)The parameters of the scalar potential can be chosen so that the minimum of the scalar potentialis achieved when the neutral components of the two scalar doublet fields acquire non-zero vacuum– 4 –xpectation vales, (cid:104) Φ (cid:105) = v / √ and (cid:104) Φ (cid:105) = v / √ , where the (potentially complex) vevs satisfy v ≡ | v | + | v | (cid:39) (246 GeV) , (2.2)as required by the observed W boson mass, m W = gv . The SU(2) × U(1) gauge symmetry is thenspontaneously broken, leaving an unbroken U(1) EM gauge group.In the most general 2HDM, the fields Φ and Φ are indistinguishable. Thus, it is always possibleto define two orthonormal linear combinations of the two doublet fields without modifying any predic-tion of the model. Performing such a redefinition of fields leads to a new scalar potential with the sameform as Eq. (2.1) but with modified coefficients. This implies that the coefficients that parameterizethe scalar potential in Eq. (2.1) are not directly physical [37].To obtain a scalar potential that is more closely related to physical observables, one can introducethe so-called Higgs basis in which the redefined doublet fields (denoted below by H and H ) havethe property that H has a non-zero vev whereas H has a zero vev [37, 38]. In particular, we definethe new Higgs doublet fields: H = (cid:32) H +1 H (cid:33) ≡ v ∗ Φ + v ∗ Φ v , H = (cid:32) H +2 H (cid:33) ≡ − v Φ + v Φ v . (2.3)It follows that (cid:104) H (cid:105) = v/ √ and (cid:104) H (cid:105) = 0 . The Higgs basis is uniquely defined up to an overallrephasing, H → e iχ H (which does not alter the fact that (cid:104) H (cid:105) = 0 ). In the Higgs basis, the scalarpotential is given by [37, 38]: V = Y H † H + Y H † H + [ Y H † H + h . c . ] + Z ( H † H ) + Z ( H † H ) + Z ( H † H )( H † H )+ Z ( H † H )( H † H ) + (cid:110) Z ( H † H ) + (cid:2) Z ( H † H ) + Z ( H † H ) (cid:3) H † H + h . c . (cid:111) , (2.4)where Y , Y and Z , . . . , Z are real and uniquely defined, whereas Y , Z , Z and Z are potentiallycomplex and transform under the rephasing of H → e iχ H as [ Y , Z , Z ] → e − iχ [ Y , Z , Z ] and Z → e − iχ Z , (2.5)since V must be independent of χ . After minimizing the scalar potential, Y = − Z v , Y = − Z v . (2.6)This leaves 11 free parameters: 1 vev, 8 real parameters, Y , Z , , , , | Z , , | , and two relative phases.In the general 2HDM, the physical charged Higgs boson is the charged component of the Higgs-basis doublet H , and its mass is given by m H ± = Y + Z v . (2.7)The three physical neutral Higgs boson mass-eigenstates are determined by diagonalizing a × realsymmetric squared-mass matrix that is defined in the Higgs basis [38, 39] M = v Z Re Z − Im Z Re Z ( Z + Y /v ) − Im Z − Im Z − Im Z ( Z + Y /v ) − Re Z , (2.8)where Z ≡ Z + Z + Re Z . – 5 – q k q k c c − s − ic s s c c − is s s ic Table 1 . Invariant combinations of the neutral Higgs boson mixing angles θ and θ , where c ij ≡ cos θ ij and s ij ≡ sin θ ij . To identify the neutral Higgs mass-eigenstates, we diagonalize the squared-mass matrix M . Thediagonalization matrix is a × real orthogonal matrix that depends on three angles: θ , θ and θ .Following Ref. [39], h h h = c c − s c − c s s − c s c + s s s c c c − s s s − s s c − c s s c s c c √ H − v √ H √ H , (2.9)where the h i are the mass-eigenstate neutral Higgs fields, c ij ≡ cos θ ij and s ij ≡ sin θ ij . Under therephasing H → e iχ H , θ , θ are invariant, and θ → θ − χ . (2.10)Assuming that Z ≡ | Z | e iθ (cid:54) = 0 , it is convenient to define the invariant mixing angle, φ ≡ θ − θ . (2.11)In light of the freedom to define the mass-eigenstate Higgs fields up to an overall sign, the invariantmixing angles θ , θ and φ can be determined modulo π . By convention, we choose − π ≤ θ , θ < π , and ≤ φ < π . (2.12)The physical neutral Higgs states ( h , , ) are then given by: h k = 1 √ (cid:26) q ∗ k (cid:18) H − v √ (cid:19) + q ∗ k H e iθ + h . c . (cid:27) , (2.13)where the q k and q k are invariant combinations of θ and θ , which are exhibited in Table 1 [39].The masses of the neutral Higgs bosons h i will be denoted by m i , respectively. It is convenient todefine the physical charged Higgs states by H ± ≡ e ± iθ H ± , (2.14)so that all the Higgs mass-eigenstate fields ( h , h , h and H ± ) are invariant under H → e iχ H . If Z = 0 , then one can always rephase the Higgs basis field H such that Z is real. In this basis, the neutral Higgsboson squared-mass matrix, M , is diagonal, and the identification of the neutral Higgs boson mass-eigenstates is trivial. – 6 –lthough the explicit formulae for the neutral Higgs boson masses and mixing angles are quitecomplicated, there are numerous relations among them which take on rather simple forms. The fol-lowing results are noteworthy [39, 40]: Z v = m c c + m s c + m s , (2.15) Re( Z e − iθ ) v = c s c ( m − m ) , (2.16) Im( Z e − iθ ) v = s c ( c m + s m − m ) , (2.17) Re( Z e − iθ ) v = m ( s − c s ) + m ( c − s s ) − m c , (2.18) Im( Z e − iθ ) v = 2 s c s ( m − m ) . (2.19)We next turn to the Higgs-fermion Yukawa couplings. As reviewed in Appendix A, one startsout initially with a Lagrangian expressed in terms of the scalar doublet fields Φ i ( i = 1 , ) andinteraction–eigenstate quark and lepton fields. After electroweak symmetry breaking, one can re-express the scalar doublet fields in terms of the Higgs basis fields H and H . At the same time, onecan identify the × quark and lepton mass matrices. By redefining the left and right-handed quarkand lepton fields appropriately, the quark and lepton mass matrices are transformed into diagonalform, where the diagonal elements are real and non-negative. The resulting Higgs–fermion YukawaLagrangian is given by in Eq. (A.16) and is repeated here for the convenience of the reader [40], − L Y = U L ( κ U H † + ρ U H † ) U R − D L K † ( κ U H − + ρ U H − ) U R + U L K ( κ D † H +1 + ρ D † H +2 ) D R + D L ( κ D † H + ρ D † H ) D R + N L ( κ E † H +1 + ρ E † H +2 ) E R + E L ( κ E † H + ρ D † H ) E R + h . c ., (2.20)where U = ( u, c, t ) and D = ( d, s, b ) are the mass-eigenstate quark fields, K is the CKM mixingmatrix, N = ( ν e , ν µ , ν τ ) and E = ( e, µ, τ ) are the mass-eigenstate lepton fields, and κ and ρ are × Yukawa coupling matrices. Note that F R,L ≡ P R,L F , where F = U , D , N and E , and P R,L ≡ (1 ± γ ) are the right and left-handed projection operators, respectively. At this stage, theneutrinos are exactly massless, so we are free to define the physical left-handed neutrino fields, N L ,such that their charged current interactions are generation-diagonal. By setting H = v/ √ and H = 0 , one can relate κ U , κ D , and κ E to the diagonal (up-type anddown-type) quark and charged lepton mass matrices M U , M D , and M E , respectively, M U = v √ κ U = diag( m u , m c , m t ) , M D = v √ κ D † = diag( m d , m s , m b ) ,M E = v √ κ E † = diag( m e , m µ , m τ ) . (2.21)However, the complex matrices ρ F ( F = U, D, E ) are unconstrained. Moreover, under the rephas-ing H → e iχ H , the Yukawa matrix acquires an overall phase, ρ F → e iχ ρ F , since L Y must beindependent of χ . To incorporate the neutrino masses, one can employ a seesaw mechanism [41–45] and introduce three right-handedneutrino fields along with an explicit SU(2) × U(1) conserving mass term. See footnote 4. – 7 –o obtain the physical Yukawa couplings of the Higgs boson, one must relate the Higgs basisscalar fields to the Higgs mass-eigenstate fields. Using Eqs. (2.13) and (2.14), the Higgs–fermionYukawa couplings are given by, − L Y = U (cid:88) k =1 (cid:26) q k M U v + 1 √ (cid:104) q ∗ k e iθ ρ U P R + q k [ e iθ ρ U ] † P L (cid:105)(cid:27) U h k + D (cid:88) k =1 (cid:26) q k M D v + 1 √ (cid:104) q k [ e iθ ρ D ] † P R + q ∗ k e iθ ρ D P L (cid:105)(cid:27) Dh k + E (cid:88) k =1 (cid:26) q k M E v + 1 √ (cid:104) q k [ e iθ ρ E ] † P R + q ∗ k e iθ ρ E P L (cid:105)(cid:27) Eh k + (cid:26) U (cid:2) K [ e iθ ρ D ] † P R − [ e iθ ρ U ] † KP L (cid:3) DH + + N [ e iθ ρ E ] † P R EH + + h . c . (cid:27) . (2.22)The combinations e iθ ρ U , e iθ ρ D and e iθ ρ E that appear in these interactions are invariant underthe rephasing of H . It is convenient to rewrite the Higgs-fermion Yukawa couplings in terms of thefollowing three × hermitian matrices that are invariant with respect to the rephasing of H , ρ FR ≡ v √ M − / F (cid:26) e iθ ρ F + [ e iθ ρ F ] † (cid:27) M − / F , for F = U, D, E , (2.23) ρ FI ≡ v √ i M − / F (cid:26) e iθ ρ F − [ e iθ ρ F ] † (cid:27) M − / F , for F = U, D, E , (2.24)where the M F are the diagonal fermion mass matrices [cf. Eq. (2.21)] and the Yukawa couplingmatrices are introduced in Eq. (2.20). Then, the Yukawa couplings take the following form: − L Y = 1 v U (cid:88) k =1 M / U (cid:26) q k + Re( q k ) (cid:2) ρ UR + iγ ρ UI (cid:3) + Im( q k ) (cid:2) ρ UI − iγ ρ UR (cid:3)(cid:27) M / U U h k + 1 v D (cid:88) k =1 M / D (cid:26) q k + Re( q k ) (cid:2) ρ DR − iγ ρ DI (cid:3) + Im( q k ) (cid:2) ρ DI + iγ ρ DR (cid:3)(cid:27) M / D Dh k + 1 v E (cid:88) k =1 M / E (cid:26) q k + Re( q k ) (cid:2) ρ ER − iγ ρ EI (cid:3) + Im( q k ) (cid:2) ρ EI + iγ ρ ER (cid:3)(cid:27) M / E Eh k + √ v (cid:26) U (cid:2) KM / D ( ρ DR − iρ DI ) M / D P R − M / U ( ρ UR − iρ UI ) M / U KP L (cid:3) DH + N M / E ( ρ ER − iρ EI ) M / E P R EH + + h . c . (cid:27) , (2.25)where is the × identity matrix. The appearance of unconstrained hermitian × Yukawamatrices ρ FR,I in Eq. (2.25) indicates the presence of potential flavor-changing neutral Higgs–quarkand lepton interactions. If the off-diagonal elements of ρ FR,I are unsuppressed, they will generate tree-level Higgs-mediated FCNCs that are incompatible with the strong suppression of FCNCs observedin nature. – 8 – .2 The limit of a SM-like Higgs boson
Current LHC data suggest that the properties of the observed Higgs boson are consistent with thepredictions of the Standard Model. In this paper, we shall identify h as the SM-like Higgs boson. Inlight of the expression for the h coupling to a pair of vector bosons V V = W + W − or ZZ , g h V V g h SM V V = c c (cid:39) , where V = W or Z , (2.26)it follows that | s | , | s | (cid:28) . Thus, in the limit of a SM-like Higgs boson, Eqs. (2.16) and (2.17)yield [39]: | s | (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) Re( Z e − iθ ) v m − m (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (2.27) | s | (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) Im( Z e − iθ ) v m − m (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (2.28)In addition, Eq. (2.19) implies that one additional small quantity characterizes the limit of a SM-likeHiggs boson, | Im( Z e − iθ ) | (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) m − m ) s s v (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) Im( Z e − iθ ) v m − m (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (2.29)Moreover, in the limit of a SM-like Higgs boson, Eq. (2.18) yields m − m (cid:39) Re( Z e − iθ ) v . (2.30)As a consequence of Eqs. (2.27) and (2.28), the limit of a SM-like Higgs boson can be achievedif either | Z | (cid:28) and/or if m , m (cid:29) v . The latter corresponds to the well-known decoupling limitof the 2HDM [39, 46, 53]. In this paper, we will focus on the decoupling regime of the 2HDM toensure that h is sufficiently SM-like, in light of the current LHC Higgs data [3]. In the exact SM-Higgs boson limit, the couplings of h are precisely those of the SM Higgs boson.In this case, we can identify h as a CP-even scalar. In general, the heavier neutral Higgs bosons, h and h can be mixed CP states. The limit in which h and h are approximate eigenstates of CP isnoteworthy. This limit is achieved assuming that | s | (cid:28) | s | . That is, (cid:12)(cid:12)(cid:12)(cid:12) s s (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) m − m m − m (cid:19) Im( Z e − iθ )Re( Z e − iθ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (2.31)In the decoupling limit, the ratio of squared-mass differences in Eq. (2.31) is of O (1) . Moreover,unitarity and perturbativity constraints suggest that Re( Z e − iθ ) cannot be significantly larger than O (1) . Hence, it follows that | Im( Z e − iθ ) | (cid:28) . (2.32) In the literature, this is often referred to as the alignment limit [46–52]. We do not use this nomenclature here in orderto avoid confusion with flavor alignment, which is the focus of this paper. Note that Eq. (2.30) implies that in the decoupling limit, m (cid:29) v implies that m (cid:29) v and vice versa. – 9 –n light of Eq. (2.10), we can rephase H → e iχ H such that θ = 0 (mod π ), i.e. c = ± .Eqs. (2.29) and (2.32) then yield | Im Z | , | Im Z | (cid:28) . For simplicity in the subsequent analysis,we henceforth assume that a real Higgs basis exists in which Z and Z are simultaneously real. Inthis case, the scalar Higgs potential and the Higgs vacuum are CP-invariant, and the squared-massmatrix of the neutral Higgs bosons given in Eq. (2.8) simplifies, M = Z v Z v Z v Y + ( Z + Z + Z ) v
00 0 Y + ( Z + Z − Z ) v , (2.33)where Z and Z are real. Moreover, c = 1 and we can set θ = θ = 0 (mod π ), or equivalently e iθ = c = (cid:15) , (2.34)where (cid:15) ≡ sgn Z , in the real Higgs basis [cf. Eqs. (2.11) and (2.12)]. To maintain the reality of theHiggs basis, the only remaining freedom in defining the Higgs basis fields is the overall sign of thefield H . In particular, under H → − H , we see that Z is invariant whereas Z (and Z ) and c change sign. We immediately identify the CP-odd Higgs boson A = √ H with squared mass, m A = Y + ( Z + Z − Z ) v . (2.35)Note that the real Higgs mass-eigenstate field, A , is defined up to an overall sign change, whichcorresponds to the freedom to redefine H → − H . In contrast, the charged Higgs field H ± defined(as a matter of convenience) by Eq. (2.14) is invariant with respect to H → − H . Indeed, by usingEq. (2.34), we can now write H ± = (cid:15) H ± . In light of Eqs. (2.7) and (2.35), m H ± = m A − ( Z − Z ) v . (2.36)The upper × matrix block given in Eq. (2.33) is the CP-even Higgs squared-mass matrix, M H = (cid:32) Z v Z v Z v m A + Z v (cid:33) , (2.37)where we have used Eq. (2.35) to eliminate Y . To diagonalize M H , we define the CP-even mass-eigenstates, h and H (with m h ≤ m H ) by (cid:32) Hh (cid:33) = (cid:32) c β − α − s β − α s β − α c β − α (cid:33) (cid:32) √ H − v √ H (cid:33) , (2.38)where c β − α ≡ cos( β − α ) and s β − α ≡ sin( β − α ) are defined in terms of the angle β defined via tan β ≡ v /v , and the mixing angle α that diagonalizes the CP-even Higgs squared-mass matrixwhen expressed relative to the original basis of scalar fields, { Φ , Φ } , which is assumed here to bea real basis. Since the real Higgs mass-eigenstate fields H and h are defined up to an overall sign Given the assumption [indicated above Eq. (2.33)] that the scalar Higgs potential and the Higgs vacuum are CP-invariant, it follows that there must exist a real basis of scalar fields in which all scalar potential parameters and the vacuumexpectation values of the two neutral Higgs fields, (cid:104) Φ i (cid:105) ≡ v i / √ (for i = 1 , 2), are simultaneously real [54]. – 10 –hange, it follows that β − α is determined modulo π . To make contact with the notation of Eq. (2.9),we note that c = 1 and c = (cid:15) [cf. Eq. (2.34)]. Assuming that h is the lighter of the two neutralCP-even Higgs bosons, then Eq. (2.38) implies the following identifications: h = h , H = − (cid:15) h , A = (cid:15) h , (2.39)and c = s β − α , s = − (cid:15) c β − α . (2.40)This means that the signs of the fields H and A and the sign of c β − α all flip under the redefinition ofthe Higgs basis field H → − H .Note that ≤ s β − α ≤ in the convention specified in Eq. (2.12). Moreover, Eq. (2.16) yields s β − α c β − α = − Z v m H − m h , (2.41)and it therefore follows that ≤ s , c ≤ and c β − α Z ≤ . The decoupling limit corresponds to m H (cid:29) m h and | c β − α | (cid:28) [cf. Eq. (2.27)], in which case we can identify h as the SM-like Higgsboson and H as the heavier CP-even Higgs boson. Finally, Eqs. (2.15)–(2.19) yield Z v = m h s β − α + m H c β − α , (2.42) Z v = ( m h − m H ) s β − α c β − α , (2.43) Z v = m H s β − α + m h c β − α − m A . (2.44)In particular, m h (cid:39) Z v in the limit of a SM-like Higgs boson h . Applying Eq. (2.40) to Table 1, q = s β − α , q = (cid:15) c β − α , (2.45) q = − (cid:15) c β − α , q = s β − α , (2.46) q = 0 , q = i . (2.47)Inserting these results into the general form of the Yukawa couplings given in Eq. (2.25), we obtain thefollowing Higgs-fermion couplings in the case of a CP-conserving Higgs scalar potential and vacuum, − L Y = 1 v (cid:88) F = U,D,E F (cid:26) s β − α M F + (cid:15) c β − α M / F (cid:2) ρ FR + iε F γ ρ FI (cid:3) M / F (cid:27) F h + 1 v (cid:88) F = U,D,E F (cid:26) c β − α M F − (cid:15) s β − α M / F (cid:2) ρ FR + iε F γ ρ FI (cid:3) M / F (cid:27) F H + 1 v (cid:88) F = U,D,E F (cid:26) M / F (cid:15) (cid:0) ρ FI − iε F γ ρ FR (cid:1) M / F (cid:27) F A + √ v (cid:26) U (cid:2) KM / D ( ρ DR − iρ DI ) M / D P R − M / U ( ρ UR − iρ UI ) M / U KP L (cid:3) DH + N M / E ( ρ ER − iρ EI ) M / E P R EH + + h . c . (cid:27) , (2.48)– 11 –here we have introduced the notation, ε F = (cid:40) +1 for F = U , − for F = D, E . (2.49)Moreover, by employing Eq. (2.34) in Eqs. (2.23) and (2.24), the expressions for ρ FR and ρ FI in termsof the Higgs Yukawa coupling matrices ρ F simplify, (cid:15) M / F ρ FR M / F = v √ (cid:0) ρ F + [ ρ F ] † (cid:1) , (2.50) i(cid:15) M / F ρ FI M / F = v √ (cid:0) ρ F − [ ρ F ] † (cid:1) . (2.51)The structure of the neutral Higgs couplings given in Eq. (2.48) is easily ascertained. If ρ FI (cid:54) = 0 ,then the neutral Higgs fields will exhibit CP-violating Yukawa couplings. Moreover, the two signchoices, (cid:15) = ± are physically indistinguishable, since the sign of Z can always be flipped byredefining the Higgs basis field H → − H . Under this field redefinition, ρ F , c β − α , H and A alsoflip sign, in which case L Y is unchanged.For completeness, we briefly consider the case where h is the heavier of the two neutral CP-evenHiggs bosons. In this case, Eq. (2.38) implies the following identifications, h = (cid:15) h , H = h , A = (cid:15) h , (2.52)and c = c β − α , s = (cid:15) s β − α . (2.53)This means that the signs of the fields h and A and the sign of s β − α all flip under the redefinition ofthe Higgs basis field H → − H . Note that Eqs. (2.41)–(2.44) are still valid. Invoking the conventiongiven by Eq. (2.12) now implies that ≤ c β − α ≤ and Z s β − α ≤ . Moreover in light of Eq. (2.26),if | s β − α | (cid:28) then H is SM-like and m H (cid:39) Z v , which is achieved in the limit of | Z | (cid:28) . Nodecoupling limit is possible in this case since m h < m H = 125 GeV. Using Eq. (2.53), one cancheck that Eqs. (2.45)–(2.47) are modified by taking s β − α → c β − α and c β − α → − s β − α . As a result,Eq. (2.48) remains unchanged.So far, the parameters α and β have no separate significance. Only the combination, β − α is meaningful. Moreover the matrices ρ FR and ρ FI are generic complex matrices, which implies theexistence of tree-level Higgs-mediated flavor changing neutral currents, as well as new sources ofCP violation. However, experimental data suggest that such Higgs-mediated FCNCs must be highlysuppressed. One can eliminate these FCNCs by imposing a discrete Z symmetry Φ → Φ and Φ → − Φ on the quartic terms of the Higgs potential given in Eq. (2.1), which sets λ = λ = 0 and gives physical significance to the Φ – Φ basis choice. This in turn promotes the CP-even Higgsmixing angle α in the real Φ – Φ basis and tan β ≡ v /v to physical parameters of the model. Likewise, if Im Z (cid:54) = 0 in a basis where Z and Z are real, then the neutral Higgs fields will also possess CP-violatingtrilinear and quadralinear scalar couplings. Since the existence of a real Higgs basis implies no spontaneous nor explicit CP-violation in the scalar sector, thereexists a Φ – Φ basis in which the λ i of Eq. (2.1), v and v (and hence tan β ) are simultaneously real. – 12 –he Z symmetry can be extended to the Higgs-fermion interactions in four inequivalent ways. In thenotation of the Higgs-fermion Yukawa couplings given in Eq. (2.48), the ρ FR,I are given by
1. Type-I: For F = U, D, E , ρ FR = (cid:15) cot β and ρ FI = 0 .2. Type-II: ρ UR = (cid:15) cot β and ρ UI = 0 . For F = D, E , ρ FR = − (cid:15) tan β and ρ FI = 0 .3. Type-X: ρ ER = − (cid:15) tan β and ρ EI = 0 . For F = U, D , ρ FR = (cid:15) cot β and ρ FI = 0 .4. Type-Y: ρ DR = − (cid:15) tan β and ρ DI = 0 . For F = U, E , ρ FR = (cid:15) cot β and ρ FI = 0 .Inserting these values for the ρ FR and ρ FI into Eq. (2.48), the resulting neutral Higgs–fermion Yukawacouplings are flavor diagonal as advertised.From a purely phenomenological point of view, one can simply avoid tree-level Higgs-mediatedFCNCs by declaring that the ρ FR and ρ FI are diagonal matrices. In the simplest generalization of theType I, II, X and Y Yukawa interactions, one asserts that both the ρ FR and the ρ FI are proportionalto the identity matrix (where the constants of proportionality can depend on F ). This is called theflavor-aligned 2HDM, which we shall discuss in the next subsection. The flavor-aligned 2HDM posits that the Yukawa matrices κ F and ρ F [cf. Eq. (2.20)] are propor-tional. When written in terms of fermion mass-eigenstates, κ F = √ M F /v is diagonal. Thus inthe A2HDM, the ρ F are likewise diagonal, which implies that tree-level Higgs-mediated FCNCs areabsent. We define the alignment parameters a F via, ρ F = e − iθ a F κ F , for F = U, D, E, (2.54)where the (potentially) complex numbers a F are invariant under the rephasing of the Higgs basis field H → e iχ H . It follows from Eqs. (2.23) and (2.24) that ρ FR = (Re a F ) , ρ FI = (Im a F ) . (2.55)Inserting the above results into Eq. (2.22), the Yukawa couplings take the following form: − L Y = 1 v U (cid:88) k =1 M U (cid:26) q k + q ∗ k a U P R + q k a U ∗ P L (cid:27) U h k + 1 v D (cid:88) k =1 M D (cid:26) q k + q k a D ∗ P R + q ∗ k a D P L (cid:27) Dh k + 1 v E (cid:88) k =1 M E (cid:26) q k + q k a E ∗ P R + q ∗ k a E P L (cid:27) Eh k + √ v (cid:26) U (cid:2) a D ∗ KM D P R − a U ∗ M U KP L (cid:3) DH + + a E ∗ N M E P R EH + + h . c . (cid:27) . (2.56) As defined here, the parameter tan β flips sign under the redefinition of the Higgs basis field H → − H , in contrast tothe more common convention where tan β is positive (by redefining H → − H if necessary). With this latter definition,the two cases of (cid:15) = ± [or equivalently the two cases of sgn( s β − α c β − α ) = ∓ ] represent non-equivalent points of theType-I, II, X or Y 2HDM parameter space. However, we do not adopt this latter convention in the present work. – 13 –his form simplifies further if the neutral Higgs mass-eigenstates are also states of definite CP. In thiscase, the corresponding Yukawa couplings are given by − L Y = 1 v (cid:88) F = U,D,E
F M F (cid:26) s β − α + (cid:15) c β − α (cid:2) Re a F + i(cid:15) F Im a F γ (cid:3)(cid:27) F h + 1 v (cid:88) F = U,D,E
F M F (cid:26) c β − α − (cid:15) s β − α (cid:2) Re a F + i(cid:15) F Im a F γ (cid:3)(cid:27) F H + 1 v (cid:88) F = U,D,E
F M F (cid:26) (cid:15) (cid:2) Im a F − i(cid:15) F Re a F γ (cid:3)(cid:27) F A + √ v (cid:26) U (cid:2) a D ∗ KM D P R − a U ∗ M U KP L (cid:3) DH + + a E ∗ N M E P R EH + + h . c . (cid:27) . (2.57)As noted above Eq. (2.41), it is convenient to choose a convention in which s β − α ≥ . It then followsfrom Eq. (2.41) that (cid:15) c β − α = −| c β − α | . That is, the neutral Higgs couplings exhibited in Eq. (2.57)do not depend on the sign of c β − α (which can be flipped by redefining the overall sign of the Higgsbasis field H ). Note that in this convention, the signs of the alignment parameters a F are physical.The Type-I, II, X and Y Yukawa couplings are special cases of the A2HDM Yukawa couplings.Since the a F ( F = U, D, E ) are independent complex numbers, there is no preferred basis for thescalar fields outside of the Higgs basis. Thus, a priori, there is no separate meaning to the parameters α and β in Eq. (2.57). Nevertheless, in the special case of a CP-conserving neutral Higgs-leptoninteraction governed by Eq. (2.57) with Im a E = 0 , it is convenient to introduce the real parameter tan β via a E ≡ − (cid:15) tan β , (2.58)corresponding to a Type-II or Type-X Yukawa couplings of the charged leptons to the neutral Higgsbosons. The theoretical interpretation of tan β defined by Eq. (2.58) is as follows. It is always possibleto choose a Φ – Φ basis with the property that one of the two Higgs-lepton Yukawa coupling matricesvanishes. Namely, in the notation of Eq. (A.1), we have η E, = 0 , which means that only Φ couplesto leptons. In the case of a CP-conserving scalar Higgs potential and Higgs vacuum, we can take the Φ – Φ basis to be a real basis and identify tan β = v /v , where (cid:104) Φ i (cid:105) ≡ v i / √ (for i = 1 , 2).However, in contrast to Type-II or Type-X models, η E, = 0 does not correspond to a discrete Z symmetry of the generic A2HDM Lagrangian, since we do not require any of the Higgs-quark Yukawacoupling matrices and the scalar potential parameters λ and λ to vanish in the same Φ – Φ basis.Note that the sign of a E in Eq. (2.58) is physical since both (cid:15) and tan β flip sign under the Higgsbasis field H → − H . In contrast to the standard conventions employed in the 2HDM with Type-I,II, X or Y Yukawa couplings where tan β is defined to be positive [cf. footnote 12], we shall not adoptsuch a convention here. In practice, we will rewrite Eq. (2.58) as, a E = (cid:15) E | tan β | , (2.59)where (cid:15) E = ± correspond to physically non-equivalent points of the A2HDM parameter space.– 14 –ne theoretical liability of the A2HDM is that for generic choices of the alignment parameters a U and a D , the flavor-alignment conditions in the quark sector specified in Eq. (2.54) are not stable underthe evolution governed by the Yukawa coupling renormalization group equations. Indeed, as shownin Ref. [32], Eq. (2.54) is stable under renormalization group running if and only if the parameters a U and a D satisfy the conditions of the Type I, II, X or Y 2HDMs specified at the end of section 2.3. Inthe leptonic sector, since we ignore neutrino masses, the Higgs-lepton Yukawa couplings are flavor-diagonal at all scales. We therefore assume that ρ F (Λ) = a F κ F (Λ) , for F = U, D, (2.60)at some very high energy scale Λ (such as the grand unification (GUT) scale or the Planck scale).That is, we assume that the alignment conditions are set by some a priori unknown physics at orabove the energy scale Λ . We take the complex alignment parameters a F to be boundary conditionsfor the RGEs of the Yukawa coupling matrices, and then determine the low-energy values of theYukawa coupling matrices by numerically solving the RGEs. To ensure that the resulting low-energytheory is consistent with a SM-like Higgs boson observed at the LHC, we shall take m h = 125 GeV,and assume that the masses of H , A and H ± are all of order Λ H ≥ GeV. In this approximatedecoupling regime, | c β − α | is small enough such that the properties of h are within about 20% of theSM Higgs boson, as required by the LHC Higgs data [3]. We employ the 2HDM RGEs given inAppendix B from Λ down to Λ H , and then match onto the RGEs of the Standard Model to generatethe Higgs-fermion Yukawa couplings at the electroweak scale, which we take to be m t or m Z . Notethat the values of κ Q (Λ H ) = √ M Q (Λ H ) /v (for Q = U , D ) are determined from the known quarkmasses via Standard Model RG running.As noted above for the lepton case ( F = E ), if ρ E (Λ) is proportional to κ E (Λ) , then ρ isproportional to κ at all energy scales. Thus, we identify the leptonic alignment parameter at lowenergies by tan β . More precisely [cf. Eqs. (2.21) and (2.59)], ρ E (Λ H ) = √ (cid:15) E | tan β | M E (Λ H ) /v . (2.61)Then, M E (Λ H ) is determined by the diagonal lepton mass matrix via Standard Model RG running. To explore the Higgs-mediated FCNCs that can be generated in the A2HDM at the electroweak scale,we establish flavor-alignment at some high energy scale, Λ , as for example at the GUT or Planckscale, and run the one-loop RGEs from the high scale to the electroweak scale. Thus, we impose thefollowing boundary conditions for the running of the one-loop 2HDM Yukawa couplings, κ Q (Λ H ) = √ M Q (Λ H ) /v, (2.62) ρ Q (Λ) = a Q κ Q (Λ) , (2.63) Under the assumption of a real Higgs basis, (cid:15) = e iθ is fixed via Eq. (2.58). This factor, which appears in Eq. (2.54),can then be absorbed into the definition of a F . – 15 –here the M Q ( Q = U , D ) are the diagonal quark matrices, and Λ H is the scale of the heavierdoublet, taken to be relatively large to guarantee that we are sufficiently in the decoupling limit. Forthe lepton sector, the corresponding boundary conditions are [cf. Eq. (2.59)], κ E (Λ H ) = √ M E (Λ H ) /v, (2.64) ρ E (Λ H ) = (cid:15) E | tan β | κ E (Λ H ) . (2.65)Satisfying the two boundary conditions for the quark sector [Eqs. (2.62) and (2.63)] is not trivial,since they are imposed at opposite ends of the RG running. For example, to set flavor-alignment at thehigh energy scale, we must know the values of κ Q (Λ) . This involves running up κ Q (Λ H ) to the highscale, but since the one-loop RGEs are strongly coupled to the ρ Q matrices, we must supply valuesfor ρ Q (Λ H ) to begin the running.With no a priori knowledge of which values of ρ Q (Λ H ) lead to flavor-alignment at the highscale, we begin the iterative process by assuming flavor-alignment at Λ H via a low-scale alignmentparameter a (cid:48) Q , ρ Q (Λ H ) = a (cid:48) Q κ Q (Λ H ) . (2.66)This flavor-alignment will be broken during RGE evolution to the high scale, and a procedure isneeded to reestablish flavor-alignment at the high scale. To accomplish this, we decompose ρ Q (Λ) into parts that are aligned and misaligned with κ Q (Λ) , respectively, ρ Q (Λ) = a Q κ Q (Λ) + δρ Q , (2.67)where a Q represents the aligned part (in general, different from a (cid:48) Q ) , and δρ Q the correspondingdegree of misalignment at the high scale.To minimize the misaligned part of ρ Q (Λ) , we implement the cost function, ∆ Q ≡ (cid:88) i,j =1 | δρ Qij | = (cid:88) i,j =1 | ρ Qij (Λ) − a Q κ Qij (Λ) | , (2.68)which, once minimized, provides the optimal value of the complex parameter a Q for flavor-alignmentat the high scale, a Q ≡ (cid:80) i,j =1 κ Q ∗ ij (Λ) ρ Qij (Λ) (cid:80) i,j =1 κ Q ∗ ij (Λ) κ Qij (Λ) . (2.69)We subsequently impose flavor-alignment at the high scale using this optimized alignment parameter, ρ Q (Λ) = a Q κ Q (Λ) , (2.70)and evolve the one-loop RGEs back down to Λ H . In principle, further running of κ U and κ D below Λ H can regenerate off-diagonal terms. However, these effects are extremely small and can be ignoredin practice. At Λ H , we use (2.62) to match the boundary conditions for the 2HDM and SM. At thispoint, the matrices κ U and κ D at the scale Λ H are no longer diagonal, so we must rediagonalize κ U and κ D in analogy with Eq. (A.12) [while respectively transforming ρ U and ρ D (at the scale Λ H ) in– 16 – igure 1 . The allowed values of a U and a D consistent with the absence of Landau poles below Λ = M P are exhibited. The blue points occupy the region of the A2HDM parameter space where the prediction for allentries of the ρ Q matrices lie within a factor of 3 from the results obtained with the full running. The red pointsoccupy the region where the leading log approximation yields results quite different from the full RG running. analogy with Eq. (A.13)]. We can now evolve κ U and κ D down to the electroweak scale to checkthe accuracy of the resulting quark masses. If any of the quark masses differ from their experimentalvalues by more than 3%, we reestablish the correct quark masses at the electroweak scale, run backup to Λ H , and then rerun this procedure repeatedly until the two boundary conditions are satisfied.The result is flavor-alignment between κ Q (Λ) and ρ Q (Λ) , and a set of ρ Q matrices at the electroweakscale that provide a source of FCNCs.In our iterative procedure, we demand that all scale-dependent Yukawa couplings remain finitefrom the electroweak scale to the Planck scale (i.e., Landau poles are absent below Λ = M P ). Thisrestricts the range of the possible seed values, a (cid:48) Q , used in Eq. (2.66) to initialize the iteration. Con-sequently, the alignment parameters a U and a D cannot be too large in absolute value. Constraints onthe alignment parameters due to Landau pole considerations during one-loop RG running have beengiven in Ref [59]. In our analysis, the allowed values of a U and a D consistent with the absenceof Landau poles at all scales below Λ are exhibited in Fig. 1. Assuming Λ H = 400 GeV , theseconsiderations lead to bounds on the alignment parameters evaluated at the Planck scale, Λ = M P , | a U | (cid:46) . | a D | (cid:46) , (2.71)which are consistent with the results previously obtained in Ref [59]. Starting the RG evolution at m Z , we use a five flavor scheme to run up to m t and a six flavor scheme above m t .Running quark mass masses at m Z and m t are obtained from the RunDec Mathematica software package [55, 56], basedon quark masses provided in Ref. [57]. We fix the initial value of the top Yukawa coupling y t ( m t ) = 0 . , correspondingto an MS top quark mass of m t ( m t ) = 163 . GeV [58]. For simplicity, the effects of the lepton masses are ignored, asthese contribute very little to the running. If a Landau pole in one of the Yukawa coupling matrices arises at the scale Λ , then both the corresponding ρ Q (Λ) and κ Q (Λ) diverge, whereas their ratio, a Q , remains finite. – 17 – .6 Leading logarithm approximation In the limit of small alignment parameters, it is possible to obtain approximate analytic solutions tothe one-loop RGEs provided in Appendix B. One can express the ρ Q matrices at the low scale as ρ U (Λ H ) (cid:39) a U κ U (Λ H ) + 116 π log (cid:18) Λ H Λ (cid:19) ( D ρ U − a U D κ U ) , (2.72) ρ D (Λ H ) (cid:39) a D κ D (Λ H ) + 116 π log (cid:18) Λ H Λ (cid:19) ( D ρ D − a D D κ D ) , (2.73)where Dκ D , Dκ U , Dρ D , Dρ U are the β -functions defined in Eqs. (B.10)–(B.14) and κ U (Λ H ) and κ D (Λ H ) are proportional to the diagonal quark mass matrices, M U and M D respectively, at thescale Λ H , according to Eq. (2.21). Working to one loop order and neglecting higher order terms, it isconsistent to set ρ F = a F κ F = a F √ M F /v (for F = U, D, E ) in the corresponding β -functions, ρ U (Λ H ) ij (cid:39) a U δ ij √ M U ) jj v + ( M U ) jj √ π v log (cid:18) Λ H Λ (cid:19) (cid:26) ( a E − a U ) (cid:2) a U ( a E ) ∗ (cid:3) δ ij Tr( M E )+( a D − a U ) (cid:2) a U ( a D ) ∗ (cid:3)(cid:20) δ ij Tr( M D ) − (cid:88) k ( M D ) kk K ik K ∗ jk (cid:21)(cid:27) , (2.74) ρ D (Λ H ) ij (cid:39) a D δ ij √ M D ) ii v + ( M D ) ii √ π v log (cid:18) Λ H Λ (cid:19) (cid:26) ( a E − a D ) (cid:2) a D ( a E ) ∗ (cid:3) δ ij Tr( M E )+( a U − a D ) (cid:2) a D ( a U ) ∗ (cid:3)(cid:20) δ ij Tr( M U ) − (cid:88) k ( M U ) kk K ∗ ki K kj (cid:21)(cid:27) . (2.75)It follows that there is a large hierarchy among the several off-diagonal terms of the ρ Q matrices, (cid:12)(cid:12)(cid:12)(cid:12) ρ D (Λ H ) ij ρ D (Λ H ) ji (cid:12)(cid:12)(cid:12)(cid:12) ∼ ( M D ) ii ( M D ) jj (cid:28) , for i < j , (2.76) (cid:12)(cid:12)(cid:12)(cid:12) ρ U (Λ H ) ij ρ U (Λ H ) ji (cid:12)(cid:12)(cid:12)(cid:12) ∼ ( M U ) jj ( M U ) ii (cid:29) , for i < j. (2.77)The inequality given in Eq. (2.76) was previously noted in Ref. [36], and provides the justification forignoring ρ Dij relative to ρ Dji , for i < j . This hierarchy of Yukawa couplings is reversed for ρ Uij . Thisreversal can be traced back to the fact that ρ U is undaggered in Eq. (2.20) whereas ρ D is daggered.It is noteworthy that the leading log results for the off-diagonal terms of the ρ Q matrices obtainedin Eqs. (2.74) and (2.75) and the corresponding full numerical calculation are typically within a factorof a few. Even for small alignment parameters, there can be some small discrepancies between thetwo approaches that can be traced back to the higher order terms that were neglected in Eqs. (2.74)and (2.75). These higher order terms are not negligible due to the running performed between theelectroweak scale and the high energy scale Λ . The leading log approximation describes less and less The misalignment contributions exhibited in Eqs. (2.74) and (2.75) were computed for the first time in Ref. [60]. To make contact with the Higgs basis Yukawa couplings ∆ u and ∆ d employed by Ref. [36], we note the relations ρ U = √ u and ρ D = √ † d . – 18 –ccurately the numerical results at larger and larger alignment parameters. This is shown in Fig. 1,where the blue points correspond to the parameter regime in which the leading log approach leadsto results within a factor of 3 of the results obtained numerically for all the elements of the ρ U and ρ D matrices. In contrast, the red points correspond to the parameter regime in which the leading logapproximation leads to results quite different from what is obtained by the full running. In the quark sector of the A2HDM, only the two Yukawa coupling matrices κ U and κ D break theSU(3) Q × SU(3) U × SU(3) D global flavor symmetry of the electroweak Lagrangian involving quarks.For this reason, our model can be thought in terms of a specific realization of a Minimal FlavorViolating (MFV) 2HDM [29]. In particular, in a general 2HDM with MFV one can write the YukawaLagrangian as − L Y , MFV = ¯ Q L Y u U R H † + ¯ Q L Y † d D R H + ¯ Q L A u U R H † + ¯ Q L A † d D R H + h . c ., (2.78)with H , H the two Higgs doublets in the Higgs basis as defined in section 2 and Q L , U R , D R flavoreigenstate quarks. In general, A u , A d can be expressed by the infinite sum [35] A u = (cid:88) n ,n ,n (cid:15) un n n ( Y d Y † d ) n ( Y u Y † u ) n ( Y d Y † d ) n Y u , (2.79) A d = (cid:88) n ,n ,n (cid:15) dn n n ( Y d Y † d ) n ( Y u Y † u ) n ( Y d Y † d ) n Y d , (2.80)with generic O (1) complex coefficients (cid:15) u,dni . In order to determine the coefficients (cid:15) u,dni in the A2HDM,we rotate to the quark mass-eigenstate basis: Y u → κ U , A u → ρ U , Y d → κ D , A d → ρ D and comparewith the leading log expressions for ρ U and ρ D as reported in Eqs. (2.74) and (2.75). We find (cid:15) u = a U − π v log (cid:18) Λ H Λ (cid:19) (cid:26) a U − a D ) (cid:2) a U ( a D ) ∗ (cid:3) Tr( M D )+ ( a U − a E ) (cid:2) a U ( a E ) ∗ (cid:3) Tr( M E ) (cid:27) , (2.81) (cid:15) u = 18 π log (cid:18) Λ H Λ (cid:19) ( a U − a D ) (cid:2) a U ( a D ) ∗ (cid:3) , (2.82)and all the higher order coefficients equal to zero. The corresponding coefficients for the down sectorare obtained from these expressions with the replacement a U → a D , a D → a U , κ D → κ U . Asexpected, the leading term in Eq. (2.81) is given by the alignment parameter at the high scale a U . Thiscoefficient receives one loop corrections. The term in Eq. (2.82) generates off diagonal terms in thematrix ρ U and is one loop suppressed. For our numerical analysis, we use the procedure described in the previous section, taking the A2HDMto be in the decoupling limit, which ensures that the properties of the lightest Higgs boson, h , are– 19 –pproximately those of the observed (SM-like) Higgs boson. As stated below Eq. (2.33), we assumethat the Higgs scalar potential and the Higgs vacuum are CP-conserving. In this case, the two heavierneutral scalars, H and A , are CP-even and CP-odd mass-eigenstates, respectively. In the decouplinglimit, these two scalars are roughly degenerate in mass, i.e., m H ≈ m A ≈ Λ H (cid:29) m h . The decouplinglimit also enforces the condition | cos( β − α ) | (cid:28) , as noted below Eq. (2.41). In this paper, we shallchoose a benchmark mass of m H = 400 GeV. Noting that in the case of a SM-like Higgs boson, m h (cid:39) Z v = (125 GeV) , which implies that Z (cid:39) . , we will furthermore assume that | Z | and Z are of similar size. Indeed, Eq. (2.41) yields | cos( β − α ) | (cid:39) . for | Z | = Z . In particular,if β − α = π/ − x , with | x | (cid:28) , then values x (cid:54) = 0 imply deviations from SM behavior of thecouplings of the 125 GeV Higgs boson to fermions and gauge bosons, as well as the appearance offlavor changing neutral Higgs couplings, the largest of which is the hbs coupling.In our analysis, we allow for CP-violating effects to enter in two ways. First, CP-violatingcharged Higgs couplings to fermion pairs are generated via the appearance of the CKM matrix, K .Second, we generically allow for the possibility of complex alignment parameters a U and a D at thehigh energy scale. Via RG-running, CP-violating neutral Higgs couplings to fermion pairs will begenerated. However, this extra source of CP violation will lead to a loop-suppressed mixing of H and A that is difficult to observe due to the near mass degeneracy of these states in the decoupling limitunder consideration here. It is instructive to examine the hb ¯ b coupling, which is the Yukawa coupling that is most affected byNew Physics in our framework, and thus plays the leading role in constraining the parameter space.Following the standard notation of the ATLAS and CMS Collaborations, we denote the coupling of h to bottom quarks normalized to the SM prediction by κ b . Due to the presence of a CP-violatingcontribution to the hb ¯ b coupling when Im ρ D (cid:54) = 0 , both scalar and pseudoscalar contributions tothe hb ¯ b coupling must be considered [see Eq. (2.48)]. In the approximation where m b (cid:28) m h , onecan simply replace γ in the expression for the Yukawa coupling with ± , in which case κ b can beexpressed by the magnitude of the complex number, κ b = (cid:12)(cid:12)(cid:12)(cid:12) s β − α + vm b √ c β − α ρ D (cid:12)(cid:12)(cid:12)(cid:12) , (3.1)and compared to its ATLAS and CMS measurement, extracted from the h → b ¯ b rate. In the leadinglog approximation, Eq. (2.75) yields, κ b = (cid:12)(cid:12)(cid:12)(cid:12) s β − α + a D c β − α + 18 πv log (cid:18) Λ H Λ (cid:19) (cid:26) ( a E − a D ) (cid:2) a D ( a E ) ∗ (cid:3) Tr( M E )+( a U − a D ) (cid:2) a D ( a U ) ∗ (cid:3)(cid:20) M U ) − (cid:88) k ( M U ) kk K ∗ k K k (cid:21)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . (3.2) For m H = 400 GeV, even a value as large as | Z | = 1 , yields | cos( β − α ) | = 0 . [cf. Eq. (2.41)], which is (barely)consistent with the measured W W and ZZ couplings of the observed Higgs boson. Note that κ b should not be confused with the matrices κ F ( F = U, D, E ) defined in Eqs. (2.20) and (2.21). – 20 – igure 2 . Prediction for the SM-like Higgs coupling to bottom quarks, normalized to the SM prediction, κ b , asa function of a U and a D , having fixed | cos( β − α ) | = 0 . (left panels) and as a function of | cos( β − α ) | and a D , having fixed a U = 0 . (right panels). The top panels exhibit the leading log predictions. The dotted linecorresponds to the SM value, κ b = 1 . The gray shaded regions produce Landau poles in the Yukawa couplingsbelow M P . The pink shaded region is favored by the LHC measurements of κ b . The bottom panels show thecorresponding results obtained via scanning the parameter space and using the full RG running. In the bottomleft panel, yellow, red, green and blue colors correspond to values of κ b in the ranges < . , [0 . , . ] , [ . , and > , respectively. Here, the boldfaced number represents the σ experimental upper bound of κ b . Allpoints shown correspond to parameter regimes where Landau poles are absent [cf. Fig. 1]. In the bottom rightpanel, the gray shaded region produces Landau poles in the Yukawa couplings below M P ; the pink shadedregion contains points favored by the LHC measurements of κ b . – 21 –n Fig. 2, we show the reduced coupling, κ b , in the leading log approximation as a function of thefree parameters a U , a D and | cos( β − α ) | . We extend the plots up to | cos( β − α ) | ∼ . , consistentwith the present measurement of the Higgs couplings to W W and ZZ . The two upper panels areobtained using the leading log approximation; the two lower ones using the full RG running. We takereal values for a U , a D to present the leading log results. Generic complex coefficients are employedin parameter scans obtained with the full running. In the left upper panel, we show the reducedcoupling as a function of a U and a D , having fixed | cos( β − α ) | = 0 . . In the right upper panel,we show the reduced coupling as a function of | cos( β − α ) | and a D , having fixed a U = 0 . . In thetwo panels, we show in blue the contour κ b = 1 . , that roughly corresponds to the present σ bound,as measured by the LHC combining ATLAS and CMS Run I data [3]. The pink regions of Fig. 2illustrate that values for | cos( β − α ) | ∼ O (0 . are still allowed for sizable values of a D of O (20) .Furthermore, the shape of the constraint is quite different, if compared to the shape obtained for the (cos( β − α ) , tan β ) plane in the Type I and II 2HDM [61, 62]. The corresponding results obtainedusing the full RG running are shown in the lower panels. Note that the bounds on the parameter spaces ( | a U | , | a D | ) and ( | cos( β − α ) | , | a D | ) are slightly weaker as compared to the leading log results.It is interesting to investigate the Higgs flavor violating couplings in the regions of parameterspace favored by the LHC measurements of the SM Higgs rates. The decay to a bottom and a strangequarks is the dominant flavor violating Higgs decay in our model. However, we have checked that thecorresponding branching ratio can be at most at the few per-mille level. Numerically, this is similar tothe result for BR( h → ¯ bs + b ¯ s ) obtained by Ref. [63] in a Type I and Type II 2HDM due to chargedHiggs loop contributions to the decay amplitude. We calculate the branching ratios for the decays t → u i h ( u i = u, c ) arising from misalignmentgenerated via radiative corrections during RG running. This is in contrast to the analysis of Ref. [64]where flavor alignment is assumed to hold at the electroweak scale, in which case only charged Higgsloop diagrams contribute to the top flavor changing decays, leading to a BR( t → u i h ) that dependsstrongly on the value of the charged Higgs mass. In this subsection, we show how the charged Higgscontributions compare to the those arising in our model due to tree-level flavor changing top couplings.Following Ref. [65], we employ the leading order formulae for both t → W b and t → u i h decayrates, assuming the top quark decay width is dominated by the SM value of Γ( t → W b ) . In addition,we include the NLO QCD correction to the branching ratio, BR( t → u i h ) = cos ( β − α )( | ρ Ui | + | ρ U i | ) v m t (1 − m h /m t ) (1 − m W /m t ) (1 + 2 m W /m t ) η QCD , (3.3)where η QCD = 1 + 0 . α s ∼ . . The flavor violating branching ratios scale with the second powerof cos( β − α ) , and thus suppressed in the cos( β − α ) = 0 limit. The couplings ρ Ui and ρ U i can beeasily extracted in the leading logarithmic approximation from Eq. (2.74). Generically, the decay into Under the assumption of no decay modes of the Higgs boson beyond the SM and no non-SM particles in the loop,Ref [3] obtains κ b = 0 . +0 . − . . – 22 – igure 3 . Tree-level contributions to top flavor changing decays assuming that | cos( β − α ) | = 0 . . Top panels:we use the leading log approximation to obtain × BR( t → ch ) [left panel] and × BR( t → uh ) [rightpanel]. The gray shaded region produces Landau poles in the Yukawa couplings below M P . The pink regionis favored by the LHC measurements of κ b (see section 3.1). Bottom panels: we exhibit the correspondingresults obtained via the full RG running. Yellow, red, green and blue colors correspond to branching ratios < − , [10 − − − ] , [10 − − − ] , > − for t → ch [left panel], and to branching ratios < − , [10 − − − ] , [10 − − − ] , > − for t → hu [right panel]. a charm and a Higgs boson has a O (10 ) larger branching ratio than the decay into an up quark and aHiggs boson since in the leading logarithmic approximation, BR( t → ch )BR( t → uh ) = | ρ U | + | ρ U | | ρ U | + | ρ U | ∼ | ρ U | | ρ U | ∼ (cid:12)(cid:12)(cid:12)(cid:12) K cb K ub (cid:12)(cid:12)(cid:12)(cid:12) . (3.4)In the top panel of Fig. 3, we show the leading log results for the branching ratios, as a functionof the two alignment parameters at the high scale, a U and a D , having fixed | cos( β − α ) | = 0 . . Grayand pink shaded regions correspond to the region producing Landau poles in the Yukawa couplings– 23 –elow the Planck scale M P , and to the region favored by the LHC measurements of κ b , respectively.Branching ratios larger than ∼ − for ch and ∼ − for uh cannot be reached, while being con-sistent with Higgs coupling measurements and with the requirement of no Landau poles below M P .For comparison, we also show our results obtained scanning the parameter space and using the fullRGEs (see the bottom panels of Fig. 3). Comparing the upper and lower panels of Fig. 3, we note thatthe agreement between the prediction at leading log and the full numerical results is less accurate atlarger values of the alignment parameters | a U | and | a D | , as expected. Values of the branching ratiosas large as ∼ × − ( ∼ × − ) for t → ch ( t → uh ) can be reached, while satisfying thecondition due to the absence of Landau poles (see the blue points). However, the majority of pointswith BR ( t → ch ) (cid:38) − and BR ( t → uh ) (cid:38) − also produces too large a deviation from SMbehavior of the Higgs coupling to bottom quarks. We have checked that the largest branching ratiocompatible with Higgs data is at around − for t → ch and at around − for t → uh .These numbers should be compared with the corresponding contributions to flavor-changing topdecays from charged Higgs loop diagrams, which are present in all 2HDMs, and are generated byflavor-changing charged Higgs interactions induced by CKM mixing [64, 66]. Based on the discussionof Ref. [64], we see that in the case of light charged Higgs bosons ( m H ± (cid:46) GeV) and hH + H − couplings as large as allowed by hγγ constraints, these latter contributions can be as large as O (10 − ) and therefore comparable to those arising from the tree-level h ¯ tu i coupling induced by RG-runningin the A2HDM.When compared to the BR( t → ch ) SM ∼ × − , BR( t → uh ) SM ∼ × − , as calculatedin the SM by Refs. [67–70], the 2HDM in general and the A2HDM in particular exhibit the possibilityof a significant enhancement of the branching ratios for flavor-changing t → u i h decays. However,both tree-level flavor changing effects and loop-level effects mediated by the charged Higgs boson aregenerically too small to be probed by the LHC and future colliders.Searches for top flavor changing decays have been performed by the ATLAS and CMS collabo-ration using Run I data [71–73], and constrain the branching ratios to BR( t → u i h ) (cid:46) . (seealso [74] for a discussion of the most recent experimental results on top flavor changing decays). Pro-jections for the HL-LHC show that the bounds on the branching ratios will likely be at the − level[75, 76]. Hence, it will be very challenging to probe our model at the LHC using top flavor changingdecays. FCC estimations show that branching ratios as small as ∼ − could be probed with 10 ab − luminosity [77]. From these numbers, we can conclude that Higgs coupling measurements typicallygive (and will give) a better probe of our model, since the region of parameter space predicting moresizable top flavor violating branching ratios, also predict large and measurable effects in the Higgscoupling to bottom quarks. As pointed out in Ref. [78], the 2HDM with flavor alignment imposed at the electroweak scale predictsa rich and novel phenomenology for the heavy Higgs bosons that is strikingly different than that of the2HDM with Type I, II, X or Y Higgs-fermion Yukawa couplings. The phenomenology is even morediverse if flavor alignment is imposed at the high scale. For example, the heavy Higgs decay to quarksis flavor non-universal (i.e., the ratios, y Hd i d i /m d i and y Hu i u i /m u i are no longer independent of the– 24 –avor i ). Moreover, flavor changing heavy Higgs decays, which are generated at the loop-level due tothe quark flavor-changing charged Higgs interactions [63, 66], receive an additional contribution fromtree-level flavor-changing neutral Higgs interactions. In contrast to the flavor-changing top decaysdiscussed in the previous section, these features are not suppressed in the limit of cos( β − α ) = 0 ,where the couplings of h coincide with those of the SM Higgs boson. This is exhibited by the tree-level partial widths of the heavy Higgs bosons to up and down quarks, which are given by Γ( H → ¯ f i f i ) = 3 G F √ π m H m f i Re (cid:32) c β − α − (cid:15) s β − α ρ iif κ iif (cid:33) (cid:32) − m f i m H (cid:33) / + Im (cid:32) c β − α − (cid:15) s β − α ρ iif κ iif (cid:33) (cid:32) − m f i m H (cid:33) / , (3.5) Γ( H → ¯ f i f j ) = Γ( H → ¯ f j f i ) = 3 G F √ π m H v × s α − β ( | ρ ijf | + | ρ jif | ) × (cid:34) − (cid:18) m f i − m f j m H (cid:19) (cid:35) (cid:32) − m f i + m f j m H (cid:33) − m f i m f j m H / ( i (cid:54) = j ) . (3.6)Henceforth, we shall set cos( β − α ) = 0 , which automatically avoids constraints from the measuredHiggs boson couplings. In the leading log approximation with real values of a D and a U assumed, thesecond term of Eq. (3.5) can be neglected since Im ( ρ iif ) = 0 [cf. Eqs. (2.74) and (2.75)].In Fig. 4, we show the leading log predictions for the most interesting branching ratios ( ¯ bb , ¯ tt , τ + τ − , ¯ bs + ¯ sb ) as a function of the two alignment parameters a U and a D , where we have fixed tan β = 10 and m H = 400 GeV. In the two panels, we only show positive values of a D , since theresults are symmetric under ( a D , a U ) ↔ ( − a D , − a U ) . For the predictions of BR( H → ¯ bs + ¯ sb ) ,we do not include loop contributions involving the charged Higgs boson. These latter contributionshave been examined in Refs. [63, 66] and have been shown generically to be considerably smallerthan the corresponding tree-level flavor violating Higgs couplings. The left upper panel shows that inour model, especially at sizable values of the alignment parameters, the Type I and II 2HDM relation,BR ( H → ¯ bb ) / BR( H → τ + τ − ) = 3 m b /m τ , is violated. In particular, our model typically predicts asmaller ratio at small values of a D , and therefore the τ + τ − mode is expected to be even more sensitivethan b ¯ b relative to that of the Type I or II 2HDM. For a D (cid:38) , the hierarchy is reversed, resulting in alarger BR ( H → ¯ bb ) as compared to BR ( H → τ + τ − ) . Furthermore, the model can predict a non zerodecay rate of the heavy Higgs to a bottom and a strange quark (see the right upper panel of Fig. 4).However, the branching ratio predicted in the leading log approximation is at most of order a fewpercent at large values of a D in the regions of the parameter space without Landau poles.Note that the branching ratios into third generation quarks are different as compared to the Type II2HDM. In the latter, BR ( H → b ¯ b ) ∼ and BR ( H → t ¯ t ) ∼ , for tan β = 10 . For comparison,we present the branching ratios into t ¯ t and b ¯ b in the lower left and lower right panels of Fig. 4. Thebehavior of the two plots is similar: at small values of a U ( a D ) the t ¯ t ( b ¯ b ) branching ratio is smallerthan the one predicted by the Type II 2HDM (see the blue contours in the two plots); the branching– 25 – igure 4 . Leading log prediction for the branching ratios of the heavy Higgs boson, H , with fixed tan β = 10 , cos( β − α ) = 0 , and m H = 400 GeV. The blue contours in the upper left and lower panels represent theprediction of a Type II 2HDM. The gray shaded regions produce Landau poles below the Planck scale
Λ = M P .The blue shaded regions have already been probed by the LHC searches for heavy scalars. ratio can even vanish for particular choices of the alignment parameters a U and a D . Larger valuesof the branching ratio are predicted for sizable values a U (cid:38) . ( a D (cid:38) ). As a byproduct, theratio of branching ratios BR( H → b ¯ b ) / BR( H → t ¯ t ) differs from the predicted value of the 2HDMwith either Type I, II, III, or IV Yukawa couplings. In particular, the A2HDM generically breaksthe relation BR( H → b ¯ b ) / BR( H → t ¯ t ) (cid:39) m b tan β/m t , which is valid in the Type II 2HDMin the limit cos( β − α ) = 0 . The branching ratios of a Type II 2HDM are recovered by choosing– 26 – igure 5 . Branching ratios of the heavy Higgs boson, H , obtained by scanning the parameter space andusing the full RG running, with fixed cos( β − α ) = 0 , tan β = 10 , and m H = 400 GeV. The yellow,red, green and blue points correspond to: upper left panel, BR ( H → ¯ bb ) m τ / BR( H → τ + τ − )3 m b < , [ , , [10 , , > ; upper right panel, BR ( H → ¯ bs + b ¯ s ) < . , [0 . , . , [0 . , . , > . ;lower left panel, BR ( H → ¯ tt ) < . , [0 . , . ] , [ . , . , > . ; and lower right panel, BR ( H → ¯ bb ) < . , [0 . , . ] , [ . , . , > . . In boldface we denote the value of the branching ratios predicted by a Type II2HDM with fixed tan β = 10 . The parameter regime with | a D | > ∼ – and | a U | < ∼ . has been eliminatedafter taking into account the LHC search for heavy Higgs bosons decaying into ¯ bb [79]. a U = ± / tan β = ± / and a D = ∓ tan β = ∓ , as discussed at the end of section 2.3.The ATLAS and CMS collaborations have performed several searches for heavy Higgs bosonsdecaying into a fermion pair: ¯ bb [79], τ + τ − [80, 81], µ + µ − [82, 83], and ¯ tt [84]. In a Type II 2HDM, τ + τ − searches are the most important ones in constraining regions of parameter space at sizablevalues of tan β . Searches for ¯ bb can only set weaker bounds in that scenario. However, as discussede.g. in Ref. [85], 2HDMs with a Yukawa texture different from Type II can be best probed by ¯ bb searches. In fact, for tan β = 10 and cos( β − α ) = 0 , only the CMS search for pp → b ( b ) H, H → ¯ bb ,performed with 8 TeV data [79], can probe sizable regions of the parameter space of the A2HDM (seethe blue shaded region in Fig. 4 at large values of a D and the corresponding parameter regime ofFig. 5). In the coming years, the LHC will be able to probe complementary regions of parameterspace. In addition to the region at large values of a D best probed by ¯ bb resonance searches, the region– 27 –t small values of a U and a D will be best probed by searches for τ + τ − and µ + µ − resonances; and theregion at small values of a D , but sizable values of a U will be best probed by ¯ tt resonance searches.For comparison, we show in Fig. 5 the corresponding results obtained through the scanning ofthe parameter space and the running of the full RGEs. Qualitatively, Fig. 5 shows a similar parameterdependence as the one obtained in the leading log approximation. Numerically, some branchingratios can be quite different, especially in the regime of sizable alignment parameters. In particular,BR ( H → ¯ bs + b ¯ s ) can reach values as large as ∼ . As we discussed in section 2.7, the A2HDM is a particular type of 2HDM with Minimal FlavorViolation. As such, it predicts interesting effects in low energy flavor observables, e.g., in mesonmixing and in B meson rare decays. In this section, we shall discuss the predictions of our model forthese low energy processes and the corresponding constraints. We shall focus on those observablesthat receive tree-level Higgs contributions, with particular attention to meson mixing, B → µ + µ − ,and B → τ ν .The lepton universality ratios, BR( B → D ( ∗ ) τ − ¯ ν ) / BR( B → D ( ∗ ) (cid:96) − ν ) , for (cid:96) = e , µ , are alsonotable, especially in light of the early BaBar measurements that yield a combined . σ deviationfrom the SM predictions [86, 87]. This anomaly is not inconsistent with subsequent Belle and LHCbmeasurements, even if with a smaller significance [88–91]. Additional data are required to clarify theimplications of these measurements and to determine whether new physics beyond the SM is required.If this anomaly persists, New Physics models need (relatively large) H ± c L b R and H ± c R b L couplingsof the same order and opposite sign (with g H ± cb /m H ± ∼ / TeV ), as shown in Ref. [92]. This israther challenging to achieve in our model while being consistent with the other flavor bounds. Amore detailed examination of these channels will be left for a future study.In principle, loop induced decays (which typically include contributions from the charged Higgsboson) can also set stringent constraints on the allowed regions of the ( m H ± , tan β ) parameterplane [93]. For example, in the Type II 2HDM the charged Higgs should be heavier than 580 GeV [21]to be in agreement with b → sγ measurements (cf. footnote 2). Moreover, going beyond the TypeII 2HDM, the b → sγ bound depends not only on the charged Higgs mass, but also on the values of a U and a D , on other non-SM-like Higgs boson masses, as well as on potential contributions of NewPhysics particles in the loop. Such constraints merit further investigation. However, the analysis ofthis section focuses on parameter regimes in which tree-level Higgs-mediated FCNC effects dominateover competing one-loop contributions. For this reason, we do not consider further the constraintsfrom b → sγ (which can be avoided for sufficiently heavy Higgs masses) in this paper. Higgs mediated contributions to neutral meson mixing ( B d,s – B d,s , K – K and D – D mixing) arisein our model. Integrating out the three neutral Higgs bosons, we obtain the following dimension sixeffective Lagrangian describing B s meson mixing L eff = C (¯ b R s L ) + ˜ C (¯ b L s R ) + C (¯ b R s L )(¯ b L s R ) + h . c ., (4.1)– 28 –ith Wilson coefficients, C = ( ρ D ) (cid:18) sin ( β − α ) m H + cos ( β − α ) m h − m A (cid:19) , (4.2) ˜ C = ( ρ D ∗ ) (cid:18) sin ( β − α ) m H + cos ( β − α ) m h − m A (cid:19) , (4.3) C = ( ρ D )( ρ D ∗ )2 (cid:18) sin ( β − α ) m H + cos ( β − α ) m h + 1 m A (cid:19) , (4.4)and corresponding Wilson coefficients for B d , K , and D mixing.In the case of degenerate heavy Higgs bosons and in the limit cos( β − α ) = 0 , only C contributesto meson mixing. In this limit, we expect small Wilson coefficients at leading log, since as discussedin section 2.6, | ( ρ D ) ij | ∝ m i /v and therefore | ( ρ D ) ij | (cid:28) | ( ρ D ) ji | , for i < j . The Wilson coefficientsare also relatively small away from the exact cos( β − α ) = 0 and m A = m H limit. More specifically, C and ˜ C will be non zero, but suppressed by the combination of masses and mixing angles shownin Eqs. (4.2) and (4.3), respectively. In the following, we will show the numerical results obtained for cos( β − α ) = 0 and m A = m H = 400 GeV. However, we have checked that the constraints on theparameter space do not change considerably by taking small but non-zero values for cos( β − α ) .We apply the bounds of Ref. [94] on the C Wilson coefficient (Ref. [95] shows slightly strongerconstraints). The leading log results for B s , B d , K , and D mixing are shown in the left panel of Fig. 6. Figure 6 . Bounds from meson mixing observables. Left panel: experimentally preferred regions, as computedin our model in the leading logarithmic approximation. The dark purple region is favored by the measurementof B s mixing, the purple region by B d mixing, and the dark pink (pink) region by the phase (mass difference)of the Kaon mixing system. D meson mixing does not give any interesting bound on the parameter space andit is not shown in the figure. Right panel: the corresponding bounds from B s mixing obtained by scanningthe parameter space and using the full RG running. The yellow, red, and green points correspond to a Wilsoncoefficient of < / , [1 / , ] , > relative to the value that yields the present bound from B s mixing. – 29 –he dark purple region is favored by the measurement of B s mixing, the purple region by B d mixing,the dark pink region by CP violation in the Kaon mixing system, and the pink region by the K – K mass difference. D mixing does not give any interesting bound on the parameter space and istherefore omitted in the figure. B s mixing leads to the most stringent bound and it constrains a D tobe smaller than ∼ . at sizable values of a U . Additionally, the bound from the measurement of CPviolation in Kaon mixing (dark pink) is significantly more stringent than the bound from the massdifference of the Kaon system (in pink). This is due to the fact that the real and imaginary parts of theWilson coefficient of the Kaon system have a similar magnitude (under the assumption that a U and a D are real). In particular, the ratio of the imaginary and real parts of the Wilson coefficient is directlyrelated to the phase of the CKM matrix: Im( C K ) / Re( C K ) = Im(( K ∗ ) K ) / Re(( K ∗ ) K ) . Incontrast, the SM Wilson coefficient has an imaginary part that is much smaller than the real part.Small differences between the constraints from CP violation and the mass difference also exist in the B s and B d systems. In Fig. 6, we only show the most constraining bound in each system, i.e. the massdifference in B s mixing and the phase in B d mixing.The right panel of Fig. 6 shows the corresponding results for the B s mixing system obtained byscanning the parameter space and using the full RG-running. The points in yellow have a Wilsoncoefficient smaller than 1/3 the present bound on the Wilson coefficient; in red we present the pointswith a Wilson coefficient smaller than the present bound, and finally in green we present the pointsthat have been already probed by the measurement of the B s mixing observables. In the limit ofsizable a U (cid:38) . , we do not find points with a D (cid:38) , in rough agreement with the leading log result. B s,d → µ + µ − decays The B -meson rare decays B s,d → (cid:96) + (cid:96) − receive contributions from the exchange of the Higgs bosons H , A and h at tree-level. This is in contrast to the numerical analysis of Ref. [96], where the flavormisalignment at the electroweak scale is set to zero. The neutral Higgs exchange contributions to theleptonic decay amplitude are proportional to m (cid:96) and hence are largest in the case of B s,d → τ + τ − .However, it is more difficult to tag the τ decay to jets and leptons at the LHC and B-factory detectors,as compared to muons. For this reason, the present LHCb bounds [97], BR ( B s ( d ) → τ + τ − ) (cid:46) × − (1 . × − ) , are relatively weak as compared to the SM prediction [98], BR( B s ( d ) → τ + τ − ) SM = (7 . ± . × − (cid:0) (2 . ± . × − (cid:1) . (4.5)At sizable values of tan β , the main contributions to B s,d → µ + µ − are typically due to H and A exchange, as they are enhanced by the second power of tan β . Furthermore, in the cos( β − α ) = 0 limit, the light Higgs ( h ) contribution vanishes at tree-level. For this reason, we shall focus henceforthon the heavy Higgs contributions that are given by [99], BR( B s,d → µ + µ − )BR( B s,d → µ + µ − ) SM (cid:39) (cid:0) | S s,d | + | P s,d | (cid:1)(cid:18) y s,d Re( P s,d ) − Re( S s,d ) | S s,d | + | P s,d | (cid:19)(cid:18)
11 + y s,d (cid:19) , (4.6)where BR( B s,d → µ + µ − ) SM is the SM prediction for the branching ratio extracted from an untaggedrate. In particular, y s = (6 . ± . and y d ∼ have to be taken into account when comparing– 30 –xperimental and theoretical results, and S s,d ≡ m B s,d m µ ( C Ss,d − C (cid:48) Ss,d ) C SM s,d (cid:115) − m µ m B s,d , (4.7) P s,d ≡ m B s,d m µ ( C Ps,d − C (cid:48) Ps,d ) C SM s,d + ( C s,d − C (cid:48) s,d ) C SM s,d . (4.8)The C i are the Wilson coefficients corresponding to the Lagrangian L s = (cid:88) i ( C i O i + C (cid:48) i O (cid:48) i ) + h . c . , (4.9)with operators O ( (cid:48) ) Ss = m b m B s (¯ sP R ( L ) b )(¯ (cid:96)(cid:96) ) , (4.10) O ( (cid:48) ) Ps = m b m B s (¯ sP R ( L ) b )(¯ (cid:96)γ (cid:96) ) , (4.11) O ( (cid:48) )10 s = (¯ sγ µ P L ( R ) b )(¯ (cid:96)γ µ γ (cid:96) ) , (4.12)and the corresponding ones for the B d system. In the limit of cos( β − α ) = 0 , the Wilson coefficientsarising from heavy neutral Higgs exchange are given by C Ps = − m B s m b ρ D ∗ √ m µ v tan β m A , C Ss = − m B s m b ρ D ∗ √ m µ v tan β m H , (4.13) C (cid:48) Ps = m B s m b ρ D √ m µ v tan β m A (cid:28) C Ps , C (cid:48) Ss = − m B s m b ρ D √ m µ v tan β m H (cid:28) C Ss , (4.14)and the analogous results for the B d system. There are no tree-level New Physics contributions to the O ( (cid:48) )10 operators.If cos( β − α ) is nonvanishing, then the scalar Wilson coefficients C Ss and C (cid:48) Ss given in Eqs. (4.13)and (4.14) due to H exchange should be changed accordingly, tan β → sin( β − α ) tan β +cos( β − α ) and ρ D → ρ D sin( β − α ) . Moreover, an additional set of contributions arise due to h exchange; thecorresponding contributions are obtained from C Ss and C (cid:48) Ss given in Eqs. (4.13) and (4.14) by makingthe following replacements, tan β → sin( β − α ) − cos( β − α ) tan β , ρ D → − ρ D cos( β − α ) and m H → m h .The SM Wilson coefficient takes the form [100], C SM10 s,d = − . e π G F √ K tb K ∗ t ( s,d ) , (4.15)and the predicted branching ratios are given by BR( B s → µ + µ − ) SM = (3 . ± . × − , (4.16) BR( B d → µ + µ − ) SM = (1 . ± . × − , (4.17)as obtained in [98] with the inclusion of O ( α em ) and O ( α s ) corrections. These values are in relatively– 31 – igure 7 . Leading log prediction for the branching ratios for B s → µ + µ − (left panel) and B d → µ + µ − (right panel) relative the SM, as a function of a U and a D , with fixed tan β = 10 , cos( β − α ) = 0 , and m A = m H = 400 GeV. The regions in pink are allowed at the σ level by the present measurements. Thepurple shaded regions are anticipated by the more precise HL-LHC measurements, assuming a measured centralvalue equal to the SM prediction. The gray shaded regions produce Landau poles in the Yukawa couplingsbelow M P . good agreement with the experimental results. The combination of the LHCb and the CMS measure-ments at Run I for the B s and B d decays to muon pairs are [101]: BR( B s → µ + µ − ) = (2 . +0 . − . ) × − , (4.18) BR( B d → µ + µ − ) = (3 . +1 . − . ) × − . (4.19)Note the much larger uncertainty in the latter decay mode.The ATLAS collaboration has also reported a Run I search for B s → µ + µ − , which yieldedBR( B s → µ + µ − ) = (0 . +1 . − . ) × − [102], although this measurement is not yet competitive withEq. (4.18). Very recently, LHCb reported a new measurement for B s,d → µ + µ − using Run II data [103].Their result, BR ( B s → µ + µ − ) = (2 . ± . × − , agrees very well with the LHCb and CMScombination quoted in Eq. (4.18). In contrast, the new LHCb B d measurement is closer to the SMprediction, BR ( B d → µ + µ − ) = (1 . +1 . − . ) × − . In the following, we will compare the predictionsof the A2HDM with the LHCb and CMS combination shown in Eqs. (4.18) and (4.19). In the comingyears, the two branching ratios will be measured much more accurately by the LHC. In particular,the B s and B d branching fractions will be measured by each experiment with a precision of ∼ and ∼ at Run-III, improving to ∼ and ∼ , respectively, at the HL-LHC [104].In Fig. 7, we show the constraints from the measurement of B s → µ + µ − (left panel) and B d → – 32 – igure 8 . Leading log prediction for the branching ratios for B s → µ + µ − (left panel) and B d → µ + µ − (rightpanel) relative the SM, as a function of M (the mass of the heavy scalar and pseudoscalar) and a D . We fix tan β = 10 , a U = 0 . , and cos( β − α ) = 0 . The pink regions are the regions allowed at the σ level bythe present measurements. The purple regions are anticipated by the more precise HL-LHC measurements,assuming a measured central value equal to the SM prediction. The gray shaded regions produce Landau polesin the Yukawa couplings below M P . µ + µ − (right panel) as functions of a U and a D , with fixed tan β = 10 , (cid:15) E = +1 [see Eq. (2.59)], cos( β − α ) = 0 , and m A = m H = 400 GeV, based on the leading logarithmic approximation. Thepink shaded region denote the parameter space favored by the CMS and LHCb combined results atthe σ level, namely BR( B s → µ + µ − )BR( B s → µ + µ − ) SM ⊂ [0 . , . , BR( B d → µ + µ − )BR( B d → µ + µ − ) SM ⊂ [0 . , . . (4.20)The purple shaded region in Fig. 7 is the parameter space favored at σ by the HL-LHC measurement,assuming a measured central value equal to the SM prediction. Comparing the region in pink to theregion in purple, one can get a sense of the improvement the HL-LHC can achieve in testing ourmodel. The expected experimental error at the HL-LHC is comparable to the present theory error.For this reason, an additional improvement can be achieved via a more precise calculation of the SMprediction for the two branching ratios, with the benefit of more accurate measurements of the CKMelements that will be obtained at the LHCb and at Belle II in the coming years.The present measurement of B s → µ + µ − constrains sizable values of a U and a D in our model.The measurement of B d → µ + µ − also sets an interesting constraint at smaller values of | a D | (cf. thewhite region where | a D | ∼ and the values of | a U | are sizable), since the central value of the mea-surement is larger than the SM prediction: BR( B d → µ + µ − ) exp / BR( B d → µ + µ − ) SM ∼ . . Fixing a different sign, (cid:15) E = − , leads to the same results, with the exchange ( a U , a D ) → ( − a U , − a D ) . – 33 – igure 9 . The branching ratio for B s → µ + µ − (left panel) and for B d → µ + µ − (right panel) relative to the SM,obtained via scanning the parameter space and using the full RG running, at fixed tan β = 10 , cos( β − α ) = 0 ,and m A = m H = 400 GeV. The yellow, red, green and blue points corresponds to branching ratios normalizedto the SM prediction < . , [ . , . ] , [ . , , > . In boldface we denote the range preferred by the LHCband ATLAS measurement of B s → µ + µ − , as reported in Eq. (4.20). However, the deviation from the SM prediction is not yet statistically significant, due to the largeexperimental uncertainty. Nevertheless, a sizable suppression of the B d decay mode is presently dis-favored. As expected, the contours for BR( B s,d → µ + µ − ) / BR( B s,d → µ + µ − ) SM in the two panelsof Fig. 7 are very similar. This is due to the fact that our model is a particular type of MFV modelin the leading logarithmic approximation [cf. section 2.7]. In particular, MFV models genericallypredict BR( B d → µ + µ − ) / BR( B s → µ + µ − ) ∼ BR( B d → µ + µ − ) SM / BR( B s → µ + µ − ) SM , withcorrections arising only from m s /m b and m d /m b terms. For this reason, it is difficult in our model toenhance one decay mode, while suppressing the other.It is also interesting to investigate the bounds as a function of the heavy Higgs boson masses. InFig. 8, we show the same constraints in the ( M, a D ) plane, where M ≡ m A = m H , having fixed tan β = 10 , a U = 0 . , and cos( β − α ) = 0 . Sizable regions of parameter space are allowed, evenfor values of M as small as ∼ GeV. Finally, in Fig. 9, we show the results obtained throughscanning the parameter space and utilizing the full RG running. These plots are qualitatively similarto the contour plots of Fig. 7 obtained in the leading logarithmic approximation, although the heavyHiggs exchange contributions to the B d,s → µ + µ − decay rates computed using the full RG runningare somewhat larger (at large alignment parameters) than the corresponding leading log results. B → τ ν decays The leptonic decays B → (cid:96)ν are interesting probes of the Higgs sector of our model and particularlyof the charged Higgs couplings, since the charged Higgs boson mediates tree-level New Physics con-tributions to these decay modes. The τ channel is the only decay mode of this type observed so far.The present experimental world average is [105] BR( B → τ ν ) exp = (1 . ± . × − , (4.21) Updated results and plots available at: . – 34 – igure 10 . The ratio BR( B → τ ν ) / BR( B → τ ν ) SM at fixed tan β = 10 and m H ± = 400 GeV. Left panel:leading log predictions, where the pink region is favored by the measurement of B → τ ν . The purple region isanticipated by future measurement at Belle II, under the assumption that the central value of the measurement isgiven by the SM prediction. Right panel: result of the parameter space scan, using the full RG running. Yellow,red, green and blue points correspond to the ratios < . , [ . , . ] , [ . , , > , respectively. In boldfacewe denote the range preferred by the present world average for BR ( B → τ ν ) . and is in relatively good agreement with the SM prediction [106] BR( B → τ ν ) SM = (0 . +0 . − . ) × − . (4.22)In our model, the New Physics contribution to this decay reads BR( B → τ ν )BR( B → τ ν ) SM = (cid:12)(cid:12)(cid:12)(cid:12) m B m b m τ C ubL − C ubR C ub SM (cid:12)(cid:12)(cid:12)(cid:12) , (4.23)where we have defined the SM Wilson coefficient C ub SM = 4 G F K ub / √ and C ubR ( L ) are the Wilsoncoefficients of the O ubR ( L ) = (¯ uP R ( L ) b )(¯ τ P L ν τ ) operators. In particular [107], C ubR ( L ) = 1 m H ± Γ LR ( RL ) ub √ m τ v tan β, (4.24)with Γ LR ( RL ) ub the two charged Higgs couplings H + ¯ u L b R , H + ¯ u R b L given by Γ LRub = (cid:88) i K ui ρ D ∗ i , Γ RLub = − (cid:88) i K ∗ ib ρ U ∗ i . (4.25)This leads to the branching ratio, BR( B → τ ν )BR( B → τ ν ) SM = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m B m b v tan β √ K ub m H ± (cid:88) i (cid:2) K ui ρ D ∗ i + K ∗ ib ρ U ∗ i (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.26) Updated results and plots available at: http://ckmfitter.in2p3.fr . – 35 –n the leading logarithmic approximation, the most important contributions come from the secondterm of the above expression ( ∝ ρ D ∗ i ), as one can easily deduce from Eqs. (2.74) and (2.75).In Fig. 10, we show our numerical results as obtained using the leading log approximation (leftpanel) and the scan of the parameter space using the full RGEs, having fixed m H ± = 400 GeV and tan β = 10 . A very large region of parameter is still allowed by the measurement of B → τ ν . Inparticular, in the leading logarithmic approximation, every value | a D | (cid:46) is allowed, irrespective ofthe value of the other alignment parameter, a U . Indeed, in the pink region shown in the left panel ofFig. 10, BR( B → τ ν ) / BR( B → τ ν ) SM ⊂ [0 . , . , consistent with the current measurements.This is no longer the case when we consider the scan based on the full RG-running. In this case,a few points at large values of | a U | are excluded by the measurement of BR ( B → τ ν ) (see theblue points in the right panel of the figure). In the left panel of Fig. 10, we also exhibit the purpleshaded region of parameter space that would be favored by the future Belle II measurement, under theassumption that the central value of the measurement is given by the SM prediction for this branchingratio [cf. Eq. (4.22)]. The allowed region of parameter space is expected to shrink considerably, thanksto the anticipated accuracy of the Belle II measurement with a total error of the order of ∼ [108],leading to an allowed range, BR( B → τ ν ) / BR( B → τ ν ) SM ⊂ [0 . , . , where we have assumedno improvement in the SM prediction of this B meson decay mode. We have explored the consequences of flavor-alignment at a very high energy scale on flavor observ-ables in the two Higgs doublet Model (2HDM). Flavor alignment at the electroweak scale genericallyrequires an unnatural fine-tuning of the matrix Yukawa couplings. If flavor alignment is instead im-posed at a higher energy scale such as the Planck scale, perhaps enforced by some new dynamicsbeyond the SM, then the flavor misalignment at the electroweak scale due to RG running will gener-ate new sources of FCNCs. The resulting tree-level Higgs-mediated FCNCs are somewhat suppressedand relatively mildly constrained by experimental measurements of flavor-changing observables.We require that the alignment parameters at the high scale remain perturbative. In particular, noLandau poles are encountered during RG running. These requirements lead to an upper bound onthe values of the alignment parameters at the Planck scale. This in turn provide an upper bound onthe size of FCNCs generated at the electroweak scale. The flavor-changing observables consideredin this paper that provide the most sensitive probe of the flavor-aligned 2HDM parameter space aremeson mixing and rare B decays such as B s,d → µ + µ − and B → τ ν . We also considered con-straints from LHC searches of heavy Higgs bosons (the most important of which are searches for pp → b ( b ) H, H → ¯ bb, τ + τ − ), and measurements of the couplings of the observed (SM-like) Higgsboson. The most stringent constraint on the flavor-aligned 2HDM parameter space arises from themeasurement of the rare decay B s → µ + µ − .We investigated the predictions of the flavor-aligned 2HDM in the regions of the parameter spacenot yet probed by the measurements listed above. The top rare flavor changing decays, t → uh , t → ch , are generated at tree-level. However, once we impose constraints from Higgs couplingmeasurements, the predicted branching ratios for these neutral flavor changing top decays are beyond– 36 – igure 11 . Summary of the constraints and predictions for the heavy Higgs phenomenology, as computed in theleading log approximation. We fix cos( β − α ) = 0 , m A = m H = m H ± = 400 GeV, tan β = 10 (upper panels),and tan β = 3 (lower panels). The contours represent the ratio BR( H → b ¯ b ) m τ / [BR( H → τ + τ − )3 m b ] ,where is the Type I and Type II 2HDM prediction. The reddish-brown regions are favored by all flavorconstraints. The green region is favored by the measurement of B → τ ν . Blue-gray and tan regions are favoredby B s mixing and B s → µ + µ − , respectively. The gray shaded regions produce Landau poles in the Yukawacouplings below M P . The left and right panels represent the bounds as they are now and as projected for thecoming years, as detailed in section 4. the LHC reach. Furthermore, the model predicts a novel phenomenology for the heavy Higgs bosons.In particular, the heavy Higgs bosons can have a sizable branching ratios into a bottom and a strangequark, and the ratios, BR( H → ¯ tt ) : BR( H → ¯ bb ) : BR( H → τ + τ − ) , can be very different, ifcompared to the predictions of the more common Type I and II 2HDMs. These features are exhibitedin our summary plots in Figs. 11 and 12. – 37 – igure 12 . Result of the parameter scan using full RG running, with fixed m A = m H = m H ± = 400 GeV, cos( β − α ) = 0 , and tan β = 10 . Blue points correspond to points allowed by the measurement of B → τ ν ,but not by the measurement of B s mixing or B s → µ + µ − . Green points are allowed by the measurements of B → τ ν and of meson mixing but not by B s → µ + µ − . Red points are allowed by all constraints. The left andright panels represent the bounds as they are now and as projected for the coming years, as detailed in section 4.In the solid white region, Landau poles in the Yukawa couplings are produced below M P . In Fig. 11, we summarize the constraints on the ( a U , a D ) parameter space, with fixed tan β = 10 (upper panels) and tan β = 3 (lower panels). In both panels, we fix the values cos( β − α ) = 0 and m A = m H = m H ± = 400 GeV. The region favored by all flavor constraints is shown inreddish-brown. At sizable values of a D , the most relevant constraint comes from the measurement of B s → µ + µ − (tan region). B s meson mixing also sets an interesting bound on the parameter space(blue-gray region). It offers some complementary with B s → µ + µ − , as it does not depend on theparticular value of tan β . Moreover, it will be able to probe the small region of parameter spacewith a U > and sizable values of a D favored by the measurement of B s → µ + µ − in the case of afuture measurement with a central value in agreement with the SM prediction. The measurementof B → τ ν imposes only a relatively weak constraint on the parameter space (green region). Forvalues of tan β = 10 (or larger), in the region of parameter space favored by present and future flavorconstraints, the ratio m τ BR( H → ¯ bb ) / m b BR( H → τ + τ − ) is smaller than the ratio predictedby Type I and II 2HDM in most of the Aligned 2HDM parameter space. The parameter space issomewhat less constrained at lower values of tan β , as shown in the lower panels of Fig. 11.In Fig. 12, we present the corresponding results obtained in the numerical scan with full RGrunning, with fixed cos( β − α ) = 0 , m A = m H = m H ± = 400 GeV, and tan β = 10 . Thequalitative features of the leading log approximation continue to hold. In particular, we again see that B s → µ + µ − provides the most stringent constraint on the aligned 2HDM parameter space. Notethat in order to emphasize the comparison of the constraints obtained from the different B physicsobservables in Figs. 11 and 12, we do not include the constraints due to the LHC searches for theheavy Higgs bosons decaying into fermion pairs in these figures. As shown in Figs. 4 and 5 forthe heavy Higgs mass values quoted above, in the region of the Aligned 2HDM parameter space We use the results in [109] for the future prospects in measuring B s mixing, corresponding to the “Stage II” scenario. – 38 –onsistent with no Landau poles below M P , the current LHC limits on H and A production eliminatethe parameter regime with | a D | > ∼ – and | a U | < ∼ . .In considering the phenomenological implications of extended Higgs sectors, the most conserva-tive approach is to impose only those constraints that are required by the current experimental data. Inmost 2HDM studies in the literature, the Yukawa couplings are assumed to be of Type I, II, X or Y. Inthis paper, we have argued that the current experimental data allows for a broader approach in whichthe Yukawa couplings are approximately aligned in flavor at the electroweak scale. The resulting phe-nomenology can yield some unexpected surprises. We hope that the search strategies of future Higgsstudies at the LHC will be expanded to accommodate the broader phenomenological framework ofthe (approximately) flavor-aligned extended Higgs sector. Acknowledgments
H.E.H. gratefully acknowledges Paula Tuzon for numerous interactions during her two month longvisit to Santa Cruz in 2010–2011. Her work on the aligned 2HDM provided inspiration for thiswork. S.G. thanks Wolfgang Altmannshofer for discussions. H.E.H. and E.S. are supported in partby the U.S. Department of Energy grant number DE-SC0010107. S.G. acknowledges support fromthe University of Cincinnati. S.G. and H.E.H. are grateful to the hospitality and the inspiring workingatmosphere of both the Kavli Institute for Theoretical Physics in Santa Barbara, CA, supported in partby the National Science Foundation under Grant No. NSF PHY11-25915, and the Aspen Center forPhysics, supported by the National Science Foundation Grant No. PHY-1066293, where some of theresearch reported in this work was carried out.
A Review of the Higgs-fermion Yukawa couplings in the Higgs basis
In a general 2HDM, the Higgs fermion interactions are governed by the following interaction La-grangian: − L Y = Q L (cid:101) Φ ¯ a η U, a U R + Q L Φ a ( η D, a ) † D R + E L Φ a ( η E, a ) † E R + h . c . , (A.1)summed over a = ¯ a = 1 , , where Φ , are the Higgs doublets, (cid:101) Φ ¯ a ≡ iσ Φ ∗ ¯ a , Q L and E L arethe weak isospin quark and lepton doublets, and U R , D R , E R are weak isospin quark and leptonsinglets. Here, Q L , E L , U R , D R , E R denote the interaction basis states, which are vectors in thequark and lepton flavor spaces, and η U, a , η D, a , η E, a are × matrices in quark and lepton flavorspaces.Note that η U, a appears undaggered in Eq. (A.1), whereas the corresponding Yukawa couplingmatrices for down-type fermions ( D and E ) appear daggered. In this convention, the transformation ofthe Yukawa coupling matrices under a scalar field basis change is the same for both up-type and down-type fermions. That is, under a change of basis, Φ a → U a ¯ b Φ b (which implies that (cid:101) Φ ¯ a → (cid:101) Φ ¯ b U † b ¯ a ), the We follow the conventions of Ref. [39], in which covariance is manifest with respect to U(2) flavor transformations, Φ a → U a ¯ b Φ b [where U ∈ U(2)], by implicitly summing over barred/unbarred index pairs of the same letter. The right and left-handed fermion fields are defined as usual: ψ R,L ≡ P R,L ψ , where P R,L ≡ (1 ± γ ) . – 39 –ukawa coupling matrices transform as η Fa → U a ¯ b η Fb and η F † ¯ a → η F † ¯ b U † b ¯ a (for F = U , D and E ),which reflects the form-invariance of L Y under the basis change.The neutral Higgs states acquire vacuum expectation values, (cid:104) Φ a (cid:105) = v ˆ v a √ , (A.2)where ˆ v a ˆ v ∗ ¯ a = 1 and v = 246 GeV. It is also convenient to define ˆ w b ≡ ˆ v ∗ ¯ a (cid:15) ab , (A.3)where (cid:15) = − (cid:15) = 1 and (cid:15) = (cid:15) = 0 .Following Refs. [37, 39], we define invariant and pseudo-invariant matrix Yukawa couplings, κ F, ≡ ˆ v ∗ ¯ a η F, a , ρ F, ≡ ˆ w ∗ ¯ a η F, a , (A.4)where F = U , D or E . Inverting these equations yields η F, a = κ F, ˆ v a + ρ F, ˆ w a . (A.5)Note that under the U(2) transformation, Φ a → U a ¯ b Φ b , κ F, is invariant and ρ F, → (det U ) ρ F, . (A.6)The Higgs fields in the Higgs basis are defined by H ≡ ˆ v ∗ ¯ a Φ a and H ≡ ˆ w ∗ ¯ a Φ a , which can beinverted to yield Φ a = H ˆ v a + H w a [39]. Rewriting Eq. (A.1) in terms of the Higgs basis fields, − L Y = Q L ( (cid:101) H κ U, + (cid:101) H ρ U, ) U R + Q L ( H κ D, † + H ρ D, † ) D R + E L ( H κ E, † + H ρ E, † ) E R + h . c . (A.7)The next step is to identify the quark and lepton mass-eigenstates. This is accomplished byreplacing H → (0 , v/ √ and performing unitary transformations of the left and right-handed up-type and down-type fermion multiplets such that the resulting quark and charged lepton mass matricesare diagonal with non-negative entries. In more detail, we define: P L U = V UL P L U , P R U = V UR P R U , P L D = V DL P L D , P R D = V DR P R D ,P L E = V EL P L E , P R E = V DR P R E , P L N = V EL P L N , (A.8)and the Cabibbo-Kobayashi-Maskawa (CKM) matrix is defined as K ≡ V UL V D † L . Note that for theneutrino fields, we are free to choose V NL = V EL since neutrinos are exactly massless in this analysis. In particular, the unitary matrices V FL and V FR (for F = U , D and E ) are chosen such that M U = v √ V UL κ U, V U † R = diag( m u , m c , m t ) , (A.9) M D = v √ V DL κ D, † V D † R = diag( m d , m s , m b ) , (A.10) M E = v √ V EL κ E, † V E † R = diag( m e , m µ , m τ ) . (A.11) Here we are ignoring the right-handed neutrino sector, which gives mass to neutrinos via the seesaw mechanism. – 40 –t is convenient to define κ U = V UL κ U, V U † R , κ D = V DR κ D, V D † L , κ E = V DR κ E, V E † L , (A.12) ρ U = V UL ρ U, V U † R , ρ D = V DR ρ D, V D † L , ρ E = V DR ρ E, V E † L . (A.13)Eq. (A.6) implies that under the U(2) transformation, Φ a → U a ¯ b Φ b , κ F is invariant and ρ F → (det U ) ρ F , (A.14)for F = U , D and E . Indeed, κ F is invariant since Eqs. (A.9)–(A.11) imply that M F = v √ κ F , (A.15)which is a physical observable. The matrices ρ U , ρ D and ρ E are independent pseudo-invariant com-plex × matrices. The Higgs-fermion interactions given in Eq. (A.7) can be rewritten in terms ofthe quark and lepton mass-eigenstates, − L Y = U L ( κ U H † + ρ U H † ) U R − D L K † ( κ U H − + ρ U H − ) U R + U L K ( κ D † H +1 + ρ D † H +2 ) D R + D L ( κ D † H + ρ D † H ) D R + N L ( κ E † H +1 + ρ E † H +2 ) E R + E L ( κ E † H + ρ E † H ) E R + h . c . (A.16) B Renormalization group equations for the Yukawa matrices
We first write down the renormalization group equations (RGEs) for the Yukawa matrices η U, a , η D, a and η E, a . Defining D ≡ π µ ( d/dµ ) = 16 π ( d/dt ) , the RGEs are given by [32]: D η U, a = − (cid:0) g s + g + g (cid:48) (cid:1) η U, a + (cid:26) (cid:2) η U, a ( η U, b ) † + η D, a ( η D, b ) † (cid:3) + Tr (cid:2) η E, a ( η E, b ) † (cid:3)(cid:27) η U, b − η D, b ) † η D, a η U, b + η U, a ( η U, b ) † η U, b + ( η D, b ) † η D, b η U, a + η U, b ( η U, b ) † η U, a , (B.1) D η D, a = − (cid:0) g s + g + g (cid:48) (cid:1) η D, a + (cid:26) (cid:2) ( η D, b ) † η D, a + ( η U, b ) † η U, a (cid:3) + Tr (cid:2) ( η E, b ) † η E, a (cid:3)(cid:27) η D, b − η D, b η U, a ( η U, b ) † + η D, b ( η D, b ) † η D, a + η D, a η U, b ( η U, b ) † + η D, a ( η D, b ) † η D, b , (B.2) D η E, a = − (cid:0) g + g (cid:48) (cid:1) η E, a + (cid:26) (cid:2) ( η D, b ) † η D, a + ( η U, b ) † η U, a (cid:3) + Tr (cid:2) ( η E, b ) † η E, a (cid:3)(cid:27) η E, b + η E, b ( η E, b ) † η E, a + η E, a ( η E, b ) † η E, b . (B.3)The RGEs above are true for any basis choice. Thus, they must also be true in the Higgs basis inwhich ˆ v = (1 , and ˆ w = (0 , . In this case, we can simply choose η F, = κ F, and η F, = ρ F, toobtain the RGEs for the κ F, and ρ F, . Alternatively, we can multiply Eqs. (B.1)–(B.3) first by ˆ v ∗ a and– 41 –hen by ˆ w ∗ a . Expanding η † ¯ a , which appears on the right-hand sides of Eqs. (B.1)–(B.3), in terms of κ † and ρ † using Eq. (A.5), we again obtain the RGEs for the κ F, and ρ F, . Of course, both methods yieldthe same result, since the diagonalization matrices employed in Eqs. (A.9)–(A.11) are defined as thosethat bring the mass matrices to their diagonal form at the electroweak scale. No scale dependence isassumed in the diagonalization matrices, and as such they are not affected by the operators D . D κ U, = − (cid:0) g s + g + g (cid:48) (cid:1) κ U, + (cid:26) (cid:2) κ U, κ U, † + κ D, κ D, † (cid:3) + Tr (cid:2) κ E, κ E, † (cid:3)(cid:27) κ U, + (cid:26) (cid:2) κ U, ρ U, † + κ D, ρ D, † (cid:3) + Tr (cid:2) κ E, ρ E, † (cid:3)(cid:27) ρ U, − (cid:0) κ D, † κ D, κ U, + ρ D, † κ D, ρ U, (cid:1) + κ U, ( κ U, † κ U, + ρ U, † ρ U, ) + ( κ D, † κ D, + ρ D, † ρ D, ) κ U, + ( κ U, κ U, † + ρ U, ρ U, † ) κ U, , (B.4) D ρ U, = − (cid:0) g s + g + g (cid:48) (cid:1) ρ U, + (cid:26) (cid:2) ρ U, κ U, † + ρ D, κ D, † (cid:3) + Tr (cid:2) ρ E, κ E, † (cid:3)(cid:27) κ U, + (cid:26) (cid:2) ρ U, ρ U, † + ρ D, ρ D, † (cid:3) + Tr (cid:2) ρ E, ρ E, † (cid:3)(cid:27) ρ U, − (cid:0) κ D, † ρ D, κ U, + ρ D, † ρ D, ρ U, (cid:1) + ρ U, ( κ U, † κ U, + ρ U, † ρ U, ) + ( κ D, † κ D, + ρ D, † ρ D, ) ρ U, + ( κ U, κ U, † + ρ U, ρ U, † ) ρ U, , (B.5) D κ D, = − (cid:0) g s + g + g (cid:48) (cid:1) κ D, + (cid:26) (cid:2) κ D, † κ D, + κ U, † κ U, (cid:3) + Tr (cid:2) κ E, † κ E, ] (cid:27) κ D, + (cid:26) (cid:2) ρ D, † κ D, + ρ U, † κ U, (cid:3) + Tr (cid:2) ρ E, † κ E, ] (cid:27) ρ D, − κ D, κ U, κ U, † + ρ D, κ U, ρ U, † ) + ( κ D, κ D, † + ρ D, ρ D, † ) κ D, + κ D, ( κ U, κ U, † + ρ U, ρ U, † )+ κ D, ( κ D, † κ D, + ρ D, † ρ D, ) , (B.6) D ρ D, = − (cid:0) g s + g + g (cid:48) (cid:1) ρ D, + (cid:26) (cid:2) κ D, † ρ D, + κ U, † ρ U, (cid:3) + Tr (cid:2) κ E, † ρ E, ] (cid:27) κ D, + (cid:26) (cid:2) ρ D, † ρ D, + ρ U, † ρ U, (cid:3) + Tr (cid:2) ρ E, † ρ E, ] (cid:27) ρ D, − κ D, ρ U, κ U, † + ρ D, ρ U, ρ U, † )+ ( κ D, κ D, † + ρ D, ρ D, † ) ρ D, + ρ D, ( κ U, κ U, † + ρ U, ρ U, † )+ ρ D, ( κ D, † κ D, + ρ D, † ρ D, ) , (B.7)– 42 – κ E, = − (cid:0) g + g (cid:48) (cid:1) κ E, + (cid:26) (cid:2) κ D, † κ D, + κ U, † κ U, (cid:3) + Tr (cid:2) κ E, † κ E, (cid:3)(cid:27) κ E, + (cid:26) (cid:2) ρ D, † κ D, + ρ U, † κ U, (cid:3) + Tr (cid:2) ρ E, † κ E, (cid:3)(cid:27) ρ E, + ( κ E, κ E, † + ρ E, ρ E, † ) κ E, + κ E, ( κ E, † κ E, + ρ E, † ρ E, ) , (B.8) D ρ E, = − (cid:0) g + g (cid:48) (cid:1) ρ E, + (cid:26) (cid:2) κ D, † ρ D, + κ U, † ρ U, (cid:3) + Tr (cid:2) κ E, † ρ E, (cid:3)(cid:27) κ E, + (cid:26) (cid:2) ρ D, † ρ D, + ρ U, † ρ U, (cid:3) + Tr (cid:2) ρ E, † ρ E, (cid:3)(cid:27) ρ E, + ( κ E κ E, † + ρ E, ρ E, † ) ρ E, + ρ E, ( κ E, † κ E, + ρ E, † ρ E, ) . (B.9)Using Eqs. (A.12) and (A.13), we immediately obtain the RGEs for the κ F and ρ F . Schemati-cally, we shall write, D κ F = β κ F , D ρ F = β ρ F , (B.10)for F = U , D and E . Explicitly, the corresponding β -functions at one-loop order are given by, D κ U = − (cid:0) g s + g + g (cid:48) (cid:1) κ U + (cid:26) (cid:2) κ U κ U † + κ D κ D † (cid:3) + Tr (cid:2) κ E κ E † (cid:3)(cid:27) κ U (B.11) + (cid:26) (cid:2) κ U ρ U † + κ D ρ D † (cid:3) + Tr (cid:2) κ E ρ E † (cid:3)(cid:27) ρ U − K (cid:0) κ D † κ D K † κ U + ρ D † κ D K † ρ U (cid:1) + κ U ( κ U † κ U + ρ U † ρ U ) + K ( κ D † κ D + ρ D † ρ D ) K † κ U + ( κ U κ U † + ρ U ρ U † ) κ U , D ρ U = − (cid:0) g s + g + g (cid:48) (cid:1) ρ U + (cid:26) (cid:2) ρ U κ U † + ρ D κ D † (cid:3) + Tr (cid:2) ρ E κ E † (cid:3)(cid:27) κ U (B.12) + (cid:26) (cid:2) ρ U ρ U † + ρ D ρ D † (cid:3) + Tr (cid:2) ρ E ρ E † (cid:3)(cid:27) ρ U − K (cid:0) κ D † ρ D K † κ U + ρ D † ρ D K † ρ U (cid:1) + ρ U ( κ U † κ U + ρ U † ρ U ) + K ( κ D † κ D + ρ D † ρ D ) K † ρ U + ( κ U κ U † + ρ U ρ U † ) ρ U , D κ D = − (cid:0) g s + g + g (cid:48) (cid:1) κ D + (cid:26) (cid:2) κ D † κ D + κ U † κ U (cid:3) + Tr (cid:2) κ E † κ E ] (cid:27) κ D (B.13) + (cid:26) (cid:2) ρ D † κ D + ρ U † κ U (cid:3) + Tr (cid:2) ρ E † κ E ] (cid:27) ρ D − κ D K † κ U κ U † + ρ D K † κ U ρ U † ) K + ( κ D κ D † + ρ D ρ D † ) κ D + κ D K † ( κ U κ U † + ρ U ρ U † ) K + κ D ( κ D † κ D + ρ D † ρ D ) , – 43 – ρ D = − (cid:0) g s + g + g (cid:48) (cid:1) ρ D + (cid:26) (cid:2) κ D † ρ D + κ U † ρ U (cid:3) + Tr (cid:2) κ E † ρ E ] (cid:27) κ D (B.14) + (cid:26) (cid:2) ρ D † ρ D + ρ U † ρ U (cid:3) + Tr (cid:2) ρ E † ρ E ] (cid:27) ρ D − κ D K † ρ U κ U † + ρ D K † ρ U ρ U † ) K + ( κ D κ D † + ρ D ρ D † ) ρ D + ρ D K † ( κ U κ U † + ρ U ρ U † ) K + ρ D ( κ D † κ D + ρ D † ρ D ) , D κ E = − (cid:0) g + g (cid:48) (cid:1) κ E + (cid:26) (cid:2) κ D † κ D + κ U † κ U (cid:3) + Tr (cid:2) κ E † κ E (cid:3)(cid:27) κ E (B.15) + (cid:26) (cid:2) ρ D † κ D + ρ U † κ U (cid:3) + Tr (cid:2) ρ E † κ E (cid:3)(cid:27) ρ E + ( κ E κ E † + ρ E ρ E † ) κ E + κ E ( κ E † κ E + ρ E † ρ E ) , D ρ E = − (cid:0) g + g (cid:48) (cid:1) ρ E + (cid:26) (cid:2) κ D † ρ D + κ U † ρ U (cid:3) + Tr (cid:2) κ E † ρ E (cid:3)(cid:27) κ E (B.16) + (cid:26) (cid:2) ρ D † ρ D + ρ U † ρ U (cid:3) + Tr (cid:2) ρ E † ρ E (cid:3)(cid:27) ρ E + ( κ E κ E † + ρ E ρ E † ) ρ E + ρ E ( κ E † κ E + ρ E † ρ E ) . For the numerical analysis of the RGEs, it is convenient to define (cid:101) κ D ≡ κ D K † , (cid:101) ρ D ≡ ρ D K † , (B.17)keeping in mind that the (unitary) CKM matrix K is defined at the electroweak scale and thus is nottaken to be a running quantity. The RGEs given in Eqs. (B.11)–(B.16) can now be rewritten by taking κ D → (cid:101) κ D , ρ D → (cid:101) ρ D and K → . The advantage of the RGEs written in this latter form is that theCKM matrix K no longer appears explicitly in the differential equations, and enters only in the initialcondition of (cid:101) κ D at the low scale [cf. Eq. (2.62)], (cid:101) κ D (Λ H ) = √ M D (Λ H ) K † /v . (B.18)In particular, the high scale boundary condition given by Eq. (2.63) also applies to (cid:101) κ D and (cid:101) ρ D , i.e., (cid:101) ρ D (Λ) = a D (cid:101) κ D (Λ) . (B.19) References [1]
ATLAS
Collaboration, G. Aad et al.,
Observation of a new particle in the search for the StandardModel Higgs boson with the ATLAS detector at the LHC , Phys.Lett.
B716 (2012) 1,[ arXiv:1207.7214 ]. – 44 – CMS
Collaboration, S. Chatrchyan et al.,
Observation of a new boson at a mass of 125 GeV with theCMS experiment at the LHC , Phys.Lett.
B716 (2012) 30, [ arXiv:1207.7235 ].[3]
ATLAS, CMS
Collaboration, G. Aad et al.,
Measurements of the Higgs boson production and decayrates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC ppcollision data at √ s = 7 and 8 TeV , JHEP (2016) 045, [ arXiv:1606.02266 ].[4] P. Fayet, Supergauge Invariant Extension of the Higgs Mechanism and a Model for the electron and ItsNeutrino , Nucl. Phys.
B90 (1975) 104.[5] K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita,
Low-Energy Parameters and Particle Masses in aSupersymmetric Grand Unified Model , Prog. Theor. Phys. (1982) 1889.[6] R. A. Flores and M. Sher, Higgs Masses in the Standard, Multi-Higgs and Supersymmetric Models , Annals Phys. (1983) 95.[7] J. F. Gunion and H. E. Haber,
Higgs Bosons in Supersymmetric Models. 1. , Nucl. Phys.
B272 (1986) 1.[Erratum: ibid.,
B402 (1993) 567].[8] L. Fromme, S. J. Huber, and M. Seniuch,
Baryogenesis in the two-Higgs doublet model , JHEP (2006) 038, [ hep-ph/0605242 ].[9] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl, and M. Shaposhnikov, Higgs Boson Mass and NewPhysics , JHEP (2012) 140, [ arXiv:1205.2893 ].[10] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, et al.,
Higgs mass and vacuumstability in the Standard Model at NNLO , JHEP (2012) 098, [ arXiv:1205.6497 ].[11] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, and A. Strumia,
Investigating the near-criticality of the Higgs boson , JHEP (2013) 089, [ arXiv:1307.3536 ].[12] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, H. M. Lee, and A. Strumia, Stabilization of theElectroweak Vacuum by a Scalar Threshold Effect , JHEP (2012) 031, [ arXiv:1203.0237 ].[13] O. Lebedev, On Stability of the Electroweak Vacuum and the Higgs Portal , Eur. Phys. J.
C72 (2012)2058, [ arXiv:1203.0156 ].[14] G. M. Pruna and T. Robens,
Higgs singlet extension parameter space in the light of the LHC discovery , Phys. Rev.
D88 (2013) 115012, [ arXiv:1303.1150 ].[15] R. Costa, A. P. Morais, M. O. P. Sampaio, and R. Santos,
Two-loop stability of a complex singletextended Standard Model , Phys. Rev.
D92 (2015) 025024, [ arXiv:1411.4048 ].[16] N. Chakrabarty, U. K. Dey, and B. Mukhopadhyaya,
High-scale validity of a two-Higgs doubletscenario: a study including LHC data , JHEP (2014) 166, [ arXiv:1407.2145 ].[17] D. Das and I. Saha, Search for a stable alignment limit in two-Higgs-doublet models , Phys. Rev.
D91 (2015) 095024, [ arXiv:1503.02135 ].[18] P. Ferreira, H. E. Haber, and E. Santos,
Preserving the validity of the Two-Higgs Doublet Model up tothe Planck scale , Phys. Rev.
D92 (2015) 033003, [ arXiv:1505.04001 ].[19] D. Chowdhury and O. Eberhardt,
Global fits of the two-loop renormalized Two-Higgs-Doublet modelwith soft Z breaking , JHEP (2015) 052, [ arXiv:1503.08216 ].[20] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Theory andphenomenology of two-Higgs-doublet models , Phys. Rept. (2012) 1, [ arXiv:1106.0034 ]. – 45 –
21] M. Misiak and M. Steinhauser,
Weak Radiative Decays of the B Meson and Bounds on M H ± in theTwo-Higgs-Doublet Model , Eur. Phys. J.
C77 (2017) 201, [ arXiv:1702.04571 ].[22] S. L. Glashow and S. Weinberg,
Natural Conservation Laws for Neutral Currents , Phys.Rev.
D15 (1977) 1958.[23] E. A. Paschos,
Diagonal Neutral Currents , Phys. Rev.
D15 (1977) 1966.[24] H. E. Haber, G. L. Kane, and T. Sterling,
The Fermion Mass Scale and Possible Effects of HiggsBosons on Experimental Observables , Nucl. Phys.
B161 (1979) 493.[25] L. J. Hall and M. B. Wise,
Flavor Changing Higgs Boson Couplings , Nucl. Phys.
B187 (1981) 397.[26] J. F. Donoghue and L. F. Li,
Properties of Charged Higgs Bosons , Phys. Rev.
D19 (1979) 945.[27] V. D. Barger, J. L. Hewett, and R. J. N. Phillips,
New Constraints on the Charged Higgs Sector in TwoHiggs Doublet Models , Phys. Rev.
D41 (1990) 3421.[28] M. Aoki, S. Kanemura, K. Tsumura, and K. Yagyu,
Models of Yukawa interaction in the two Higgsdoublet model, and their collider phenomenology , Phys. Rev.
D80 (2009) 015017,[ arXiv:0902.4665 ].[29] A. J. Buras, M. V. Carlucci, S. Gori, and G. Isidori,
Higgs-mediated FCNCs: Natural FlavourConservation vs. Minimal Flavour Violation , JHEP (2010) 009, [ arXiv:1005.5310 ].[30] A. Pich and P. Tuzon, Yukawa Alignment in the Two-Higgs-Doublet Model , Phys.Rev.
D80 (2009)091702, [ arXiv:0908.1554 ].[31] F. J. Botella, G. C. Branco, A. M. Coutinho, M. N. Rebelo, and J. I. Silva-Marcos,
NaturalQuasi-Alignment with two Higgs Doublets and RGE Stability , Eur. Phys. J.
C75 (2015) 286,[ arXiv:1501.07435 ].[32] P. Ferreira, L. Lavoura, and J. P. Silva,
Renormalization-group constraints on Yukawa alignment inmulti-Higgs-doublet models , Phys.Lett.
B688 (2010) 341, [ arXiv:1001.2561 ].[33] S. Knapen and D. J. Robinson,
Disentangling Mass and Mixing Hierarchies , Phys. Rev. Lett. (2015) 161803, [ arXiv:1507.00009 ].[34] F. J. Botella, G. C. Branco, M. Nebot, and M. N. Rebelo,
Flavour Changing Higgs Couplings in aClass of Two Higgs Doublet Models , Eur. Phys. J.
C76 (2016) 161, [ arXiv:1508.05101 ].[35] G. D’Ambrosio, G. Giudice, G. Isidori, and A. Strumia,
Minimal flavor violation: An Effective fieldtheory approach , Nucl.Phys.
B645 (2002) 155, [ hep-ph/0207036 ].[36] C. B. Braeuninger, A. Ibarra, and C. Simonetto,
Radiatively induced flavour violation in the generaltwo-Higgs doublet model with Yukawa alignment , Phys.Lett.
B692 (2010) 189,[ arXiv:1005.5706 ].[37] S. Davidson and H. E. Haber,
Basis-independent methods for the two-Higgs-doublet model , Phys. Rev.
D72 (2005) 035004, [ hep-ph/0504050 ]. [Erratum: ibid.,
D72 (2005) 099902].[38] G. C. Branco, L. Lavoura, and J. P. Silva,
CP Violation , (Oxford University Press, Oxford, UK, 1999).[39] H. E. Haber and D. O’Neil,
Basis-independent methods for the two-Higgs-doublet model. II. TheSignificance of tan β , Phys. Rev.
D74 (2006) 015018, [ hep-ph/0602242 ]. [Erratum: ibid.,
D74 (2006) 059905]. – 46 –
40] H. E. Haber and D. O’Neil,
Basis-independent methods for the two-Higgs-doublet model III: TheCP-conserving limit, custodial symmetry, and the oblique parameters S, T, U , Phys.Rev.
D83 (2011)055017, [ arXiv:1011.6188 ].[41] P. Minkowski, µ → eγ at a Rate of One Out of Muon Decays? , Phys. Lett.
B67 (1977) 421.[42] M. Gell-Mann, P. Ramond, and R. Slansky,
Complex Spinors and Unified Theories , Conf. Proc.
C790927 (1979) 315–321, [ arXiv:1306.4669 ].[43] T. Yanagida,
Horizontal Symmetry and Masses of Neutrinos , Prog. Theor. Phys. (1980) 1103.[44] R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity Violation , Phys. Rev. Lett. (1980) 912.[45] R. N. Mohapatra and G. Senjanovic, Neutrino Masses and Mixings in Gauge Models with SpontaneousParity Violation , Phys. Rev.
D23 (1981) 165.[46] J. F. Gunion and H. E. Haber,
The CP conserving two Higgs doublet model: The Approach to thedecoupling limit , Phys. Rev.
D67 (2003) 075019, [ hep-ph/0207010 ].[47] N. Craig, J. Galloway, and S. Thomas,
Searching for Signs of the Second Higgs Doublet , arXiv:1305.2424 .[48] D. M. Asner et al., ILC Higgs White Paper , in
Proceedings, Community Summer Study 2013:Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013 . arXiv:1310.0763 .[49] M. Carena, I. Low, N. R. Shah, and C. E. M. Wagner, Impersonating the Standard Model Higgs Boson:Alignment without Decoupling , JHEP (2014) 015, [ arXiv:1310.2248 ].[50] H. E. Haber, The Higgs data and the Decoupling Limit , in
Proceedings, 1st Toyama InternationalWorkshop on Higgs as a Probe of New Physics 2013 (HPNP2013): Toyama, Japan, February 13-16,2013 . arXiv:1401.0152 .[51] P. S. Bhupal Dev and A. Pilaftsis, Maximally Symmetric Two Higgs Doublet Model with NaturalStandard Model Alignment , JHEP (2014) 024, [ arXiv:1408.3405 ]. [Erratum: ibid., (2015)147].[52] A. Pilaftsis, Symmetries for standard model alignment in multi-Higgs doublet models , Phys. Rev.
D93 (2016) 075012, [ arXiv:1602.02017 ].[53] H. E. Haber and Y. Nir,
Multiscalar Models With a High-energy Scale , Nucl. Phys.
B335 (1990) 363.[54] J. F. Gunion and H. E. Haber,
Conditions for CP-violation in the general two-Higgs-doublet model , Phys. Rev.
D72 (2005) 095002, [ hep-ph/0506227 ].[55] K. Chetyrkin, J. H. Kuhn, and M. Steinhauser,
RunDec: A Mathematica package for running anddecoupling of the strong coupling and quark masses , Comput.Phys.Commun. (2000) 43,[ hep-ph/0004189 ].[56] F. Herren and M. Steinhauser,
Version 3 of
RunDec and
CRunDec , arXiv:1703.03751 .[57] Particle Data Group
Collaboration, C. Patrignani et al.,
Review of Particle Physics , Chin. Phys.
C40 (2016) 100001.[58] P. Marquard, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser,
Quark Mass Relations to Four-LoopOrder in Perturbative QCD , Phys. Rev. Lett. (2015) 142002, [ arXiv:1502.01030 ]. – 47 –
59] J. Bijnens, J. Lu, and J. Rathsman,
Constraining General Two Higgs Doublet Models by the Evolutionof Yukawa Couplings , JHEP (2012) 118, [ arXiv:1111.5760 ].[60] M. Jung, A. Pich, and P. Tuzon,
Charged-Higgs phenomenology in the Aligned two-Higgs-doubletmodel , JHEP (2010) 003, [ arXiv:1006.0470 ].[61]
ATLAS
Collaboration, G. Aad et al.,
Constraints on new phenomena via Higgs boson couplings andinvisible decays with the ATLAS detector , JHEP (2015) 206, [ arXiv:1509.00672 ].[62] CMS
Collaboration,
Summary results of high mass BSM Higgs searches using CMS run-I data , Tech.Rep. CMS-PAS-HIG-16-007, 2016.[63] A. Arhrib,
Higgs bosons decay into bottom-strange in two Higgs doublets models , Phys. Lett.
B612 (2005) 263, [ hep-ph/0409218 ].[64] G. Abbas, A. Celis, X.-Q. Li, J. Lu, and A. Pich,
Flavour-changing top decays in the alignedtwo-Higgs-doublet model , JHEP (2015) 005, [ arXiv:1503.06423 ].[65] A. Greljo, J. F. Kamenik, and J. Kopp, Disentangling Flavor Violation in the Top-Higgs Sector at theLHC , JHEP (2014) 046, [ arXiv:1404.1278 ].[66] A. Arhrib,
Top and Higgs flavor changing neutral couplings in two Higgs doublets model , Phys. Rev.
D72 (2005) 075016, [ hep-ph/0510107 ].[67] G. Eilam, J. Hewett, and A. Soni,
Rare decays of the top quark in the standard and two Higgs doubletmodels , Phys.Rev.
D44 (1991) 1473.[68] B. Mele, S. Petrarca, and A. Soddu,
A New evaluation of the t → cH decay width in the standardmodel , Phys.Lett.
B435 (1998) 401, [ hep-ph/9805498 ].[69] J. Aguilar-Saavedra,
Top flavor-changing neutral interactions: Theoretical expectations andexperimental detection , Acta Phys.Polon.
B35 (2004) 2695, [ hep-ph/0409342 ].[70] C. Zhang and F. Maltoni,
Top-quark decay into Higgs boson and a light quark at next-to-leading orderin QCD , Phys.Rev.
D88 (2013) 054005, [ arXiv:1305.7386 ].[71]
ATLAS
Collaboration, G. Aad et al.,
Search for flavour-changing neutral current top quark decays t → Hq in pp collisions at √ s = 8 TeV with the ATLAS detector , JHEP (2015) 061,[ arXiv:1509.06047 ].[72] CMS
Collaboration,
Search for the Flavor-Changing Neutral Current Decay t → qH Where the HiggsDecays to bb Pairs at √ s = 8 TeV , Tech. Rep. CMS-PAS-TOP-14-020, 2015.[73]
CMS
Collaboration,
Search for top quark decays t → qH with H → γγ in pp collisions at √ s = 8 TeV , Tech. Rep. CMS-PAS-TOP-14-019, 2015.[74] S. Gori,
Three Lectures of Flavor and CP violation within and Beyond the Standard Model , in . arXiv:1610.02629 .[75] Top Quark Working Group
Collaboration, K. Agashe et al.,
Working Group Report: Top Quark , in
Community Summer Study 2013: Snowmass on the Mississippi (CSS2013) Minneapolis, MN, USA,July 29-August 6, 2013 , 2013. arXiv:1311.2028 .[76] M. Selvaggi,
Perspectives for Top quark physics at High-Luminosity LHC , PoS
TOP2015 (2016) 054,[ arXiv:1512.04807 ]. – 48 –
77] M. L. Mangano et al.,
Physics at a 100 TeV pp collider: Standard Model processes , arXiv:1607.01831 .[78] W. Altmannshofer, S. Gori, and G. D. Kribs, A Minimal Flavor Violating 2HDM at the LHC , Phys.Rev.
D86 (2012) 115009, [ arXiv:1210.2465 ].[79]
CMS
Collaboration, V. Khachatryan et al.,
Search for neutral MSSM Higgs bosons decaying into apair of bottom quarks , JHEP (2015) 071, [ arXiv:1506.08329 ].[80] ATLAS
Collaboration,
Search for Minimal Supersymmetric Standard Model Higgs Bosons
H/A in the τ τ final state in up to 13.3 fb − of pp collisions at √ s = 13 TeV with the ATLAS Detector , Tech. Rep.ATLAS-CONF-2016-085, CERN, Geneva, 2016.[81] CMS
Collaboration,
Search for a neutral MSSM Higgs boson decaying into τ τ at 13 TeV , Tech. Rep.CMS-PAS-HIG-16-006, 2016.[82]
CMS
Collaboration,
Search for a high-mass resonance decaying into a dilepton final state in 13 fb − of pp collisions at √ s = 13 TeV , Tech. Rep. CMS-PAS-EXO-16-031, 2016.[83] ATLAS
Collaboration,
Search for new high-mass resonances in the dilepton final state usingproton-proton collisions at √ s = 13 TeV with the ATLAS detector , Tech. Rep.ATLAS-CONF-2016-045, 2016.[84]
ATLAS
Collaboration,
Search for heavy Higgs bosons A/H decaying to a top-quark pair in ppcollisions at √ s = 8 TeV with the ATLAS detector , Tech. Rep. ATLAS-CONF-2016-073, 2016.[85] M. Carena, S. Gori, A. Juste, A. Menon, C. E. M. Wagner, and L.-T. Wang,
LHC Discovery Potentialfor Non-Standard Higgs Bosons in the 3b Channel , JHEP (2012) 091, [ arXiv:1203.1041 ].[86] BaBar
Collaboration, J. P. Lees et al.,
Evidence for an excess of ¯ B → D ( ∗ ) τ − ¯ ν τ decays , Phys. Rev.Lett. (2012) 101802, [ arXiv:1205.5442 ].[87]
BaBar
Collaboration, J. P. Lees et al.,
Measurement of an Excess of ¯ B → D ( ∗ ) τ − ¯ ν τ Decays andImplications for Charged Higgs Bosons , Phys. Rev.
D88 (2013), no. 7 072012, [ arXiv:1303.0571 ].[88]
LHCb
Collaboration, R. Aaij et al.,
Measurement of the ratio of branching fractions B ( ¯ B → D ∗ + τ − ¯ ν τ ) / B ( ¯ B → D ∗ + µ − ¯ ν µ ) , Phys. Rev. Lett. (2015) 111803,[ arXiv:1506.08614 ]. [Addendum: ibid. (2015) 159901].[89]
Belle
Collaboration, M. Huschle et al.,
Measurement of the branching ratio of ¯ B → D ( ∗ ) τ − ¯ ν τ relativeto ¯ B → D ( ∗ ) (cid:96) − ¯ ν (cid:96) decays with hadronic tagging at Belle , Phys. Rev.
D92 (2015) 072014,[ arXiv:1507.03233 ].[90]
Belle
Collaboration, A. Abdesselam et al.,
Measurement of the branching ratio of ¯ B → D ∗ + τ − ¯ ν τ relative to ¯ B → D ∗ + (cid:96) − ¯ ν (cid:96) decays with a semileptonic tagging method , arXiv:1603.06711 .[91] A. Abdesselam et al., Measurement of the τ lepton polarization in the decay ¯ B → D ∗ τ − ¯ ν τ , arXiv:1608.06391 .[92] M. Freytsis, Z. Ligeti, and J. T. Ruderman, Flavor models for ¯ B → D ( ∗ ) τ ¯ ν , Phys. Rev.
D92 (2015)054018, [ arXiv:1506.08896 ].[93] F. Mahmoudi and O. Stal,
Flavor constraints on the two-Higgs-doublet model with general Yukawacouplings , Phys. Rev.
D81 (2010) 035016, [ arXiv:0907.1791 ]. – 49 – Quark Flavor Physics Working Group
Collaboration, J. N. Butler et al.,
Working Group Report:Quark Flavor Physics , in
Proceedings, Community Summer Study 2013: Snowmass on the Mississippi(CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013 , 2013. arXiv:1311.1076 .[95] A. Bevan et al.,
Standard Model updates and new physics analysis with the Unitarity Triangle fit , arXiv:1411.7233 .[96] X.-Q. Li, J. Lu, and A. Pich, B s,d → (cid:96) + (cid:96) − Decays in the Aligned Two-Higgs-Doublet Model , JHEP (2014) 022, [ arXiv:1404.5865 ].[97] L. Martini,
Search for new physics in the B meson decays: B s ) → µ + µ − , Nuovo Cim.
C39 (2016)231.[98] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, and M. Steinhauser, B s,d → l + l − in theStandard Model with Reduced Theoretical Uncertainty , Phys. Rev. Lett. (2014) 101801,[ arXiv:1311.0903 ].[99] W. Altmannshofer and D. M. Straub,
Cornering New Physics in b → s Transitions , JHEP (2012)121, [ arXiv:1206.0273 ].[100] W. Altmannshofer, P. Ball, A. Bharucha, A. J. Buras, D. M. Straub, and M. Wick,
Symmetries andAsymmetries of B → K ∗ µ + µ − Decays in the Standard Model and Beyond , JHEP (2009) 019,[ arXiv:0811.1214 ].[101] LHCb, CMS
Collaboration, V. Khachatryan et al.,
Observation of the rare B s → µ + µ − decay fromthe combined analysis of CMS and LHCb data , Nature (2015) 68, [ arXiv:1411.4413 ].[102]
ATLAS
Collaboration, M. Aaboud et al.,
Study of the rare decays of B s and B into muon pairs fromdata collected during the LHC Run 1 with the ATLAS detector , Eur. Phys. J.
C76 (2016) 513,[ arXiv:1604.04263 ].[103]
LHCb
Collaboration, R. Aaij et al.,
Measurement of the B s → µ + µ − branching fraction and effectivelifetime and search for B → µ + µ − decays , Phys. Rev. Lett. (2017) 191801,[ arXiv:1703.05747 ].[104]
CMS
Collaboration,
Technical proposal for the Phase-II upgrade of the compact muon solenoid , Tech.Rep. CERN-LHCC-2015-10, 2015.[105]
Heavy Flavor Averaging Group (HFAG)
Collaboration, Y. Amhis et al.,
Averages of b -hadron, c -hadron, and τ -lepton properties as of summer 2014 , arXiv:1412.7515 .[106] CKMfitter Group
Collaboration, J. Charles, A. Hocker, H. Lacker, S. Laplace, F. R. Le Diberder,J. Malcles, J. Ocariz, M. Pivk, and L. Roos,
CP violation and the CKM matrix: Assessing the impact ofthe asymmetric B factories , Eur. Phys. J.
C41 (2005) 1, [ hep-ph/0406184 ].[107] A. Crivellin, A. Kokulu, and C. Greub,
Flavor-phenomenology of two-Higgs-doublet models withgeneric Yukawa structure , Phys. Rev.
D87 (2013) 094031, [ arXiv:1303.5877 ].[108] G. Inguglia,
Studies of dark sector and B decays involving τ at Belle and Belle II , 2017. arXiv:1701.02288 .[109] J. Charles, S. Descotes-Genon, Z. Ligeti, S. Monteil, M. Papucci, and K. Trabelsi, Future sensitivity tonew physics in B d , B s , and K mixings , Phys. Rev.
D89 (2014) 033016, [ arXiv:1309.2293 ].].