High scale flavor alignment in two-Higgs doublet models and its phenomenology

Journal of High Energy Physics, Jun 2017

The most general two-Higgs doublet model (2HDM) includes potentially large sources of flavor changing neutral currents (FCNCs) that must be suppressed in order to achieve a phenomenologically viable model. The flavor alignment ansatz postulates that all Yukawa coupling matrices are diagonal 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 condition imposed at a very high energy scale, which results in the generation of Higgs-mediated FCNCs via renormalization group running from the high energy scale to the electroweak scale. Using the current experimental bounds on flavor changing observables, constraints are derived on the aligned 2HDM parameter space. In the favored parameter region, we analyze the implications for Higgs boson phenomenology.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29110.pdf

High scale flavor alignment in two-Higgs doublet models and its phenomenology

JHE High scale avor alignment in two-Higgs doublet Stefania Gori 0 1 3 Howard E. Haber 0 1 2 Edward Santos 0 1 2 0 1156 High Street, Santa Cruz, CA 95064 , U.S.A 1 345 Clifton Court , Cincinnati, OH 45221 , U.S.A 2 Santa Cruz Institute for Particle Physics, University of California , USA 3 Department of Physics, University of Cincinnati The most general two-Higgs doublet model (2HDM) includes potentially large sources of avor changing neutral currents (FCNCs) that must be suppressed in order to achieve a phenomenologically viable model. The avor alignment ansatz postulates that all Yukawa coupling matrices are diagonal when expressed in the basis of mass-eigenstate fermion elds, in which case tree-level Higgs-mediated FCNCs are eliminated. In this work, we explore models with the avor alignment condition imposed at a very high energy scale, which results in the generation of Higgs-mediated FCNCs via renormalization group running from the high energy scale to the electroweak scale. Using the current experimental bounds on avor changing observables, constraints are derived on the aligned 2HDM parameter space. In the favored parameter region, we analyze the implications for Higgs boson phenomenology. Beyond Standard Model; Heavy Quark Physics; Higgs Physics - HJEP06(217) 1 Introduction 2 The avor-aligned 2HDM at the Large Hadron Collider (LHC) [1{3], attention now turns to elucidating the dynamics of electroweak symmetry breaking. Many critical question still remain unanswered. What is the origin of the electroweak scale, and what mechanism ensures its stability? In light of the existence of multiple generations of fermions, are there also multiple copies of the scalar multiplets, implying the existence of additional Higgs scalars? If yes, how are the Higgs-fermion Yukawa interactions compatible with the apparent Minimal Flavor Violation (MFV), which is responsible for suppressed avor changing neutral currents (FCNCs)? Motivations for extending the Higgs sector beyond its minimal form have appeared often in the literature. For example, the minimal supersymmetric extension of the Standard { 1 { Model, which is invoked to explain the stability of the electroweak symmetry breaking scale with respect to very high mass scales (such as the grand uni cation or Planck scales), requires a second Higgs doublet [4{7] to avoid anomalies due to the Higgsino partners of the Higgs bosons. More complicated scalar sectors may also be required for a realistic model of baryogenesis [8]. Finally, the metastability of the SM Higgs vacuum [9{11] can be rendered stable up to the Planck scale in models of extended Higgs sectors [12{19]. Even in the absence of a speci c model of new physics beyond the Standard Model, an enlarged scalar sector can provide a rich phenomenology that can be probed by experimental searches now underway at the LHC. One of the simplest extensions of the SM Higgs sector is the two-Higgs doublet model (2HDM).1 In its most general form, the 2HDM is incompatible with experimental data due to the existence of unsuppressed tree-level Higgs-mediated FCNCs, in contrast to the SM where tree-level Higgs-mediated FCNCs are absent. To see why this is so, consider the Higgs-fermion Yukawa interactions expressed in terms of interaction eigenstate fermion elds. Due to the non-zero vacuum expectation value (vev) of the neutral Higgs fermion mass matrices are generated. Rede ning the left and right-handed fermion by separate unitary transformations, the fermion mass matrices are diagonalized. In the SM, this diagonalization procedure also diagonalizes the neutral Higgs-fermion couplings, and consequently no tree-level Higgs-mediated FCNCs are present. In contrast, in a generic 2HDM, the diagonalization of the fermion mass matrices implies the diagonalization of one linear combination of Higgs-fermion Yukawa coupling matrices. As a result, tree-level Higgs-mediated FCNCs remain in the 2HDM Lagrangian when expressed in terms of masseigenstate fermion elds. If it were possible in the 2HDM to realize avor-diagonal neutral Higgs couplings at tree-level (thereby eliminating all tree-level Higgs-mediated FCNCs), then all FCNC processes arising in the model would be generated at the loop-level, with magnitudes more easily in agreement with experimental constraints.2 eld, elds A natural mechanism for eliminating the tree-level Higgs-mediated FCNCs was proposed by Glashow and Weinberg [22] and by Paschos [23] [GWP]. One can implement the GWP mechanism in the 2HDM by introducing a Z2 symmetry to eliminate half of the Higgs-fermion Yukawa coupling terms. In this case, the fermion mass matrices and the non-zero Higgs-fermion Yukawa coupling matrices (which are consistent with the Z2 symmetry) are simultaneously diagonalized. Indeed, there are a number of inequivalent implementations of the GWP mechanism, resulting in the so-called Types I [24, 25], and II [25, 26], and Types X and Y [27, 28] versions of the 2HDM.3 1For a review with a comprehensive list of references, see ref. [20]. 2Even in models with avor-diagonal neutral Higgs couplings, one-loop processes mediated by the charged Higgs boson can generate signi cant FCNC e ects involving third generation quarks. Such models, in order to be consistent with experimental data, will produce constraints in the [mH , tan ] plane. The most avor alignment ansatz proposed in ref. [30] asserts a proportionality between the two sets of Yukawa matrices. If this avor-alignment condition is implemented at the electroweak scale, then the diagonalization of the fermion mass matrices simultaneously yields avordiagonal neutral Higgs couplings. Moreover, this avor-aligned 2HDM (henceforth denoted as the A2HDM) preserves the relative hierarchy in the quark mass matrices, and provides additional sources of CP-violation in the Yukawa Lagrangian via the introduction of three complex alignment parameters. Unfortunately, apart from the special cases enumerated in ref. [31], there are no symmetries within the 2HDM that guarantee the stability of the avor alignment ansatz with respect to radiative corrections. As such, avor alignment at the electroweak scale must be generically regarded as an unnatural ne-tuning of the Higgs-fermion Yukawa matrix parameters. Indeed, the Types I, II, X and Y 2HDMs are the unique special cases of avor alignment that are radiatively stable after imposing the observed fermion masses and mixing [32]. In this paper, we consider the possibility that avor alignment arises from New Physics beyond the 2HDM. Without a speci c ultraviolet completion in mind, we shall assert that avor alignment is imposed at some high energy scale, , perhaps as large as a grand uni cation scale or the Planck scale, where new dynamics can emerge (e.g., see ref. [33] for a viable model). Once we impose the avor alignment ansatz at the scale , the e ective eld theory below this scale corresponds to a 2HDM with both Higgs doublets coupling to up type and down type quarks and leptons.4 We then employ renormalization group (RG) evolution to determine the structure of the 2HDM Yukawa couplings at the electroweak scale. For a generic avor alignment ansatz at the scale , avor alignment in the 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 therefore may be small enough to be consistent with experimental constraints, depending on the choice of the initial alignment parameters at the scale . We therefore examine the phenomenology of Higgs-mediated FCNCs that arise from the assumption of avor alignment at some high energy scale, , that, for the purpose of our analyses, is xed to be the Planck scale (MP). We note that similar work was performed in [36], where meson mixing and B decays were used to constrain the A2HDM parameter space with avor alignment at the Planck scale. Numerical results were obtained analytically in [36], using the leading logarithmic approximation. The results of this paper are rst obtained in the leading log approximation, and then numerically by evolving the full one-loop renormalization group equations (RGEs) down from the Planck scale to the electroweak scale. In our work, we discuss the validity of the leading log approximation and examine additional FCNC processes at high energy (top and Higgs decays) and at low energy (B meson decays) to place bounds on the A2HDM parameters. 4In practice, one should also append to the 2HDM some mechanism for generating neutrino masses. An example of incorporating the e ects of neutrino masses and mixing in the context of a 2HDM with avor changing neutral Higgs couplings can be found in ref. [34]. In this paper, we shall simply put all avor structure. In the formalism presented in section 2, we initially allow for the most general form of the Higgs scalar potential and the Yukawa coupling matrices. In particular, new sources of CP-violation beyond the SM can arise due to unremovable complex phases in both the scalar potential parameters and the Yukawa couplings. For simplicity, we subsequently choose to analyze the case of a CP-conserving Higgs scalar potential and vacuum, in which case the neutral mass-eigenstates consist of 2HDM, in which the Yukawa coupling matrices are diagonal in the basis of quark and lepton mass-eigenstates. However, alignment is not stable under renormalization group running. Following the framework for avor discussed above, we impose the alignment condition at the Planck scale and then evaluate the Yukawa coupling matrices 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 renormalization group running is performed numerically and checked in the leading log approximation, where simple analytic expressions can be obtained. In this context, a comparison with general Minimal Flavor Violating 2HDMs is performed. In section 3, we discuss the implications of high-scale avor alignment for high energy processes. We focus on avor-changing decays of the top quark and on the phenomenology of the heavy neutral CP-even and CP-odd Higgs bosons. In section 4, we discuss the implications of high-scale avor alignment for low energy processes. Here we consider constraints arising from neutral meson mixing observables and from Bs ! `+` , which receive contributions at tree-level from neutral Higgs exchange, and from the charged Higgs mediated decay. By comparing theoretical predictions to experimental data, one can already probe certain regions of the A2HDM parameter space. Additional parameter regions will be probed by future searches for heavy Higgs bosons and measurements of B-physics observables. Conclusions of this work are presented in section 5. Finally, in appendix A we review the derivation of the Yukawa sector of our model in the fermion mass-eigenstate basis, and in appendix B we exhibit the one-loop matrix Yukawa coupling RGEs used in this analysis. The Theoretical framework for the 2HDM Consider a generic 2HDM consisting of two complex, hypercharge-one scalar doublets, 1 2 . The most general renormalizable scalar potential that is invariant under local SU(2) U(1) gauge transformations can be written as V = m121 y1 1+m222 y2 2 [m122 y1 2+h:c:]+ + 4( y1 2)( y2 1)+ 1 2 5( y1 2)2+ potential is achieved when the neutral components of the two scalar doublet quire non-zero vacuum expectation vales, h 10i = v1=p2 and h 20i = v2=p2, where the elds ac(potentially complex) vevs satisfy v 2 is then spontaneously broken, leaving an unbroken U(1)EM gauge group. In the most general 2HDM, the elds 1 and 2 are indistinguishable. Thus, it is HJEP06(217) always possible to de ne two orthonormal linear combinations of the two doublet elds without modifying any prediction of the model. Performing such a rede nition of elds leads to a new scalar potential with the same form as eq. (2.1) but with modi ed coe cients. This implies that the coe cients that parameterize the 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 introduce the so-called Higgs basis in which the rede ned doublet elds (denoted below by H1 and H2) have the property that H1 has a non-zero vev whereas H2 has a zero vev [37, 38]. In particular, we de ne the new Higgs doublet elds: H1 = H1+! H10 v i H2 (which does not alter the fact that hH20i = 0). In the Higgs basis, the scalar potential is given by [37, 38]: V = Y1H1yH1+Y2H2yH2+[Y3H1yH2+h:c:]+ 2 Z1(H1yH1)2+ 12 Z2(H2yH2)2+Z3(H1yH1)(H2yH2) 1 +Z4(H1yH2)(H2yH1)+ 1 2 Z5(H1yH2)2+ Z6(H1yH1)+Z7(H2yH2) H1yH2+h:c: ; where Y1, Y2 and Z1; : : : ; Z4 are real and uniquely de ned, whereas Y3, Z5, Z6 and Z7 are potentially complex and transform under the rephasing of H2 ! ei H2 as [Y3; Z6; Z7] ! e i [Y3; Z6; Z7] and Z5 ! e 2i Z5 ; since V must be independent of . After minimizing the scalar potential, Y1 = 1 2 Z1v2 ; Y3 = 1 2 Z6v2 : This leaves 11 free parameters: 1 vev, 8 real parameters, Y2, Z1;2;3;4, jZ5;6;7j, and two relative phases. In the general 2HDM, the physical charged Higgs boson is the charged component of the Higgs-basis doublet H2, and its mass is given by m2H = Y2 + 1 2 Z3v2 : { 5 { (2.4) (2.5) (2.6) (2.7) The three physical neutral Higgs boson mass-eigenstates are determined by diagonalizing 3 real symmetric squared-mass matrix that is de ned in the Higgs basis [38, 39] M 2 = v2 BB Re Z6 0 Z1 Im Z6 Re Z6 12 (Z345 + Y2=v2) s13 s12c23 c12c23 c12s13s23 s12s13s23 c13s23 c12s13c23 + s12s23 s12s13c23 c12s23CC BB where Z345 Z3 + Z4 + Re Z5. M2. The diagonalization matrix is a 3 angles: 12, 13 and 23. Following ref. [39], To identify the neutral Higgs mass-eigenstates, we diagonalize the squared-mass matrix 3 real orthogonal matrix that depends on three In light of the freedom to de ne the mass-eigenstate Higgs elds up to an overall sign, the invariant mixing angles 12, 13 and can be determined modulo . By convention, we choose The physical neutral Higgs states (h1;2;3) are then given by: 1 2 1 2 where the hi are the mass-eigenstate neutral Higgs elds, cij Under the rephasing H2 ! e 12 ; 13 are invariant, and jZ6jei 6 6= 0,5 it is convenient to de ne the invariant mixing angle, hk = p qk1 H10 + qk2H20ei 23 + h:c: ; where the qk1 and qk2 are invariant combinations of 12 and 13, which are exhibited in It is convenient to de ne the physical charged Higgs states by H e i 23 H2 ; so that all the Higgs mass-eigenstate elds (h1, h2, h3 and H ) are invariant under H2 ! ei H2. 5If Z6 = 0, then one can always rephase the Higgs basis eld H2 such that Z5 is real. In this basis, the neutral Higgs boson squared-mass matrix, M2, is diagonal, and the identi cation of the neutral Higgs boson mass-eigenstates is trivial. 23 { 6 { s12c13 s13 are quite complicated, there are numerous relations among them which take on rather simple forms. The following results are noteworthy [39, 40]: one starts out initially with a Lagrangian expressed in terms of the scalar doublet elds i (i = 1; 2) and interaction-eigenstate quark and lepton elds. After electroweak symmetry breaking, one can re-express the scalar doublet elds in terms of the Higgs basis elds H1 and H2. At the same time, one can identify the 3 3 quark and lepton mass matrices. By rede ning the left and right-handed quark and lepton elds appropriately, the quark and lepton mass matrices are transformed into diagonal form, where the diagonal elements are real and non-negative. The resulting Higgs-fermion Yukawa Lagrangian is given by in eq. (A.16) and is repeated here for the convenience of the reader [40], 3 Yukawa coupling matrices. Note that FR;L PR;LF , where F = U , D, N and E, and PR;L 5) are the right and left-handed projection operators, respectively. At this stage, the neutrinos are exactly massless, so we are free to de ne the physical left-handed neutrino elds, NL, such that their charged current interactions are generation-diagonal.6 By setting H10 = v=p2 and H20 = 0, one can relate U , D, and E to the diagonal (up-type and down-type) quark and charged lepton mass matrices MU , MD, and ME , 6To incorporate the neutrino masses, one can employ a seesaw mechanism [41{45] and introduce three right-handed neutrino elds along with an explicit SU(2) U(1) conserving mass term. See footnote 4. { 7 { MU = p ME = p v v 2 2 rephasing H2 ! e LY must be independent of . U = diag(mu ; mc ; mt) ; E y = diag(me ; m ; m ) : However, the complex matrices F (F = U; D; E) are unconstrained. Moreover, under the i H2, the Yukawa matrix acquires an overall phase, F ! e i F , since To obtain the physical Yukawa couplings of the Higgs boson, one must relate the Higgs basis scalar elds to the Higgs mass-eigenstate elds. Using eqs. (2.13) and (2.14), the Higgs-fermion Yukawa couplings are given by, couplings in terms of the following three 3 3 hermitian matrices that are invariant with respect to the rephasing of H2, F R F I v p where the MF are the diagonal fermion mass matrices [cf. eq. (2.21)] and the Yukawa coupling matrices are introduced in eq. (2.20). Then, the Yukawa couplings take the following form: LY = MU1=2 qk11 + Re(qk2) UR + i 5 I U + Im(qk2) IU U i 5 R MU1=2U hk 1 v + + + 3 U X k=1 3 D X k=1 3 E X k=1 1 v 1 v p v 2 M D1=2 qk11 + Re(qk2) R D i 5 I D + Im(qk2) ID + i 5 R D ME1=2 qk11 + Re(qk2) R E i 5 I E + Im(qk2) IE + i 5 R E 3 identity matrix. The appearance of unconstrained hermitian 3 R;I in eq. (2.25) indicates the presence of potential avor-changing neutral Higgs-quark and lepton interactions. If the o -diagonal elements of F pressed, they will generate tree-level Higgs-mediated FCNCs that are incompatible with R;I are unsupthe strong suppression of FCNCs observed in nature. with the predictions of the Standard Model. In this paper, we shall identify h1 as the SM-like Higgs boson. In light of the expression for the h1 coupling to a pair of vector HJEP06(217) bosons V V = W +W or ZZ, gh1V V ghSMV V 1. Thus, in the limit of a SM-like Higgs boson, eqs. (2.16) In addition, eq. (2.19) implies that one additional small quantity characterizes the limit of a SM-like Higgs boson, (2.26) (2.27) (2.28) (2.30) j Im(Z5e 2i 23 )j ' 2(m22 m21)s12s13 v2 ' Im(Z62e 2i 23 )v2 m23 m21 1 : (2.29) Moreover, in the limit of a SM-like Higgs boson, eq. (2.18) yields m22 m32 ' Re(Z5e 2i 23 )v2 : As a consequence of eqs. (2.27) and (2.28), the limit of a SM-like Higgs boson7 can be achieved if either jZ6j 1 and/or if m2, m3 v. The latter corresponds to the well-known decoupling limit of the 2HDM [39, 46, 53].8 In this paper, we will focus on the decoupling regime of the 2HDM to ensure that h1 is su ciently SM-like, in light of the current LHC Higgs data [3]. 2.3 Neutral scalars of de nite CP In the exact SM-Higgs boson limit, the couplings of h1 are precisely those of the SM Higgs boson. In this case, we can identify h1 as a CP-even scalar. In general, the heavier neutral Higgs bosons, h2 and h3 can be mixed CP states. The limit in which h2 and h3 7In the literature, this is often referred to as the alignment limit [46{52]. We do not use this nomenclature here in order to avoid confusion with avor alignment, which is the focus of this paper. 8Note that eq. (2.30) implies that in the decoupling limit, m2 v implies that m3 v and vice versa. { 9 { are approximate eigenstates of CP is noteworthy. This limit is achieved assuming that js13j js12j. That is, In the decoupling limit, the ratio of squared-mass di erences in eq. (2.31) is of O(1). Moreover, unitarity and perturbativity constraints suggest that Re(Z6e i 23 ) cannot be signi cantly larger than O(1). Hence, it follows that j Im(Z6e i 23 )j 1 : In light of eq. (2.10), we can rephase H2 ! ei H2 such that 23 = 0 (mod ), i.e. c23 = 1. Eqs. (2.29) and (2.32) then yield j Im Z5j ; j Im Z6j 1 : For simplicity in the subsequent analysis, we henceforth assume that a real Higgs basis exists in which Z5 and Z6 are simultaneously real. In this case, the scalar Higgs potential and the Higgs vacuum are CP-invariant, and the squared-mass matrix of the neutral Higgs bosons given in eq. (2.8) simpli es, M 2 = BBZ6v2 0 Z1v2 0 Z6v2 0 Y2 + 12 (Z3 + Z4 + Z5)v2 0 0 Y2 + 12 (Z3 + Z4 Z5)v2 1 CC ; A boson A = p 2 Im H20 with squared mass, where 6 sgn Z6, in the real Higgs basis [cf. eqs. (2.11) and (2.12)]. To maintain the reality of the Higgs basis, the only remaining freedom in de ning the Higgs basis elds is the overall sign of the eld H2. In particular, under H2 ! H2, we see that Z5 is invariant whereas Z6 (and Z7) and c23 change sign. We immediately identify the CP-odd Higgs 1 2 m2A = Y2 + (Z3 + Z4 Z5)v2 : Note that the real Higgs mass-eigenstate eld, A, is de ned up to an overall sign change, which corresponds to the freedom to rede ne H2 ! H2. In contrast, the charged Higgs eld H H2 ! de ned (as a matter of convenience) by eq. (2.14) is invariant with respect to H2. Indeed, by using eq. (2.34), we can now write H = 6H2 . In light of eqs. (2.7) and (2.35), The upper 2 2 matrix block given in eq. (2.33) is the CP-even Higgs squared-mass matrix, m2H = m2A (Z4 Z5)v2 : 1 2 M2H = Z1v2 Z6v2 Z6v2 m2A + Z5v2 ! ; (2.31) (2.32) and yields h = h1 ; H = This means that the signs of the elds H and A and the sign of c rede nition of the Higgs basis eld H2 ! H2. Note that 0 s 1 in the convention speci ed in eq. (2.12). Moreover, eq. (2.16) and it therefore follows that 0 s12 ; c12 corresponds to mH mh and jc j (2.19) yield the SM-like Higgs boson and H as the heavier CP-even Higgs boson. Finally, eqs. (2.15){ 0. The decoupling limit 1 [cf. eq. (2.27)], in which case we can identify h as s c = Z6v2 m2H 1 and c m2 h ; Z6 Z1v2 = m2h s2 Z6v2 = (m2h Z5v2 = m2H s 2 + m2H c2 m2H )s c + m2h c2 ; ; m2A : where we have used eq. (2.35) to eliminate Y2. To diagonalize M2H , we de ne the CP-even mass-eigenstates, h and H (with mh mH ) by H! h = c s s c ! p p mass matrix when expressed relative to the original basis of scalar elds, f 1 ; 2g, which is assumed here to be a real basis.9 Since the real Higgs mass-eigenstate elds H and h are de ned up to an overall sign change, it follows that is determined modulo . To make contact with the notation of eq. (2.9), we note that c13 = 1 and c23 = 6 [cf. eq. (2.34)]. Assuming that h1 is the lighter of the two neutral CP-even Higgs bosons, then eq. (2.38) implies the following identi cations: In particular, m2h ' Z1v2 in the limit of a SM-like Higgs boson h. Applying eq. (2.40) to 9Given 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 elds in which all scalar potential parameters and the vacuum expectation values of the two neutral Higgs elds, h i0i p vi= 2 (for i = 1, 2), are simultaneously real [54]. (2.38) (2.39) (2.40) (2.41) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) 1 v + + + 1 v 1 v p v 2 X U KM D1=2( RD Inserting these results into the general form of the Yukawa couplings given in eq. (2.25), we obtain the following Higgs-fermion couplings in the case of a CP-conserving Higgs scalar potential and vacuum, LY = F s MF + 6c MF1=2 FR + i"F 5 IF MF1=2 F h (2.48) (2.49) (2.50) (2.51) (2.52) (2.53) where we have introduced the notation, in terms of the Higgs Yukawa coupling matrices F simplify, Moreover, by employing eq. (2.34) in eqs. (2.23) and (2.24), the expressions for FR and IF "F = (+1 1 for F = U ; for F = D; E : 6MF1=2 FRMF1=2 = i 6MF1=2 IF MF1=2 = v v p p The structure of the neutral Higgs couplings given in eq. (2.48) is easily ascertained. If IF 6= 0, then the neutral Higgs Moreover, the two sign choices, 6 = elds will exhibit CP-violating Yukawa couplings.10 1 are physically indistinguishable, since the sign of Z6 can always be ipped by rede ning the Higgs basis eld H2 ! H2. Under this eld rede nition, F , c , H and A also ip sign, in which case LY is unchanged. For completeness, we brie y consider the case where h1 is the heavier of the two neutral CP-even Higgs bosons. In this case, eq. (2.38) implies the following identi cations, and h = 6h2 ; H = h1 ; This means that the signs of the elds h and A and the sign of s rede nition of the Higgs basis eld H2 ! H2. Note that eqs. (2.41){(2.44) are still valid. 10Likewise, if Im Z7 6= 0 in a basis where Z5 and Z6 are real, then the neutral Higgs elds will also possess CP-violating trilinear and quadralinear scalar couplings. Invoking the convention given by eq. (2.12) now implies that 0 c 1 and Z6s Moreover in light of eq. (2.26), if js is achieved in the limit of jZ6j j 1 then H is SM-like and m2H ' Z1v2, which 1. No decoupling limit is possible in this case since mh < mH = 125 GeV. Using eq. (2.53), one can check that eqs. (2.45){(2.47) are modi ed by taking s ! c and c ! s . As a result, eq. (2.48) remains unchanged. So far, the parameters and have no separate signi cance. Only the combination, is meaningful. Moreover the matrices FR and IF are generic complex matrices, which implies the existence of tree-level Higgs-mediated avor changing neutral currents, as well as new sources of CP violation. However, experimental data suggest that such Higgs-mediated FCNCs must be highly suppressed. One can eliminate these FCNCs by Higgs potential given in eq. (2.1), which sets 6 = 7 = 0 and gives physical signi cance to the 1- 2 basis choice. This in turn promotes the CP-even Higgs mixing angle in the real 1- 2 basis and tan v2=v1 to physical parameters of the model.11 The Z2 symmetry can be extended to the Higgs-fermion interactions in four inequivalent ways. In the notation of the Higgs-fermion Yukawa couplings given in eq. (2.48), the FR;I are given by12 1. Type-I: for F = U; D; E, FR = 6 cot 1 and IF = 0. 2. Type-II: UR = 6 cot 1 and IU = 0. For F = D; E, FR = 6 tan 1 and IF = 0. 3. Type-X: ER = 4. Type-Y: RD = 6 tan 1 and IE = 0. For F = U; D, FR = 6 cot 1 and IF = 0. 6 tan 1 and ID = 0. For F = U; E, FR = 6 cot 1 and IF = 0. Inserting these values for the FR and IF into eq. (2.48), the resulting neutral Higgs-fermion Yukawa couplings are avor diagonal as advertised. From a purely phenomenological point of view, one can simply avoid tree-level Higgsmediated FCNCs by declaring that the FR and IF are diagonal matrices. In the simplest generalization of the Type I, II, X and Y Yukawa interactions, one asserts that both the FR and the IF are proportional to the identity matrix (where the constants of proportionality can depend on F ). This is called the avor-aligned 2HDM, which we shall discuss in the next subsection. 2.4 The proportional. When written in terms of fermion mass-eigenstates, F = p The avor-aligned 2HDM posits that the Yukawa matrices F and F [cf. eq. (2.20)] are 2MF =v is diagonal. Thus in the A2HDM, the F are likewise diagonal, which implies that tree-level 11Since the existence of a real Higgs basis implies no spontaneous nor explicit CP-violation in the scalar sector, there exists a 1- 2 basis in which the i of eq. (2.1), v1 and v2 (and hence tan ) are simultaneously real. 12As de ned here, the parameter tan ips sign under the rede nition of the Higgs basis eld H2 ! H2, in contrast to the more common convention where tan is positive (by rede ning H2 ! H2 if necessary). With this latter de nition, the two cases of 6 = 1 [or equivalently the two cases of sgn(s c ) = represent non-equivalent points of the Type-I, II, X or Y 2HDM parameter space. However, we do not adopt this latter convention in the present work. Higgs-mediated FCNCs are absent. We de ne the alignment parameters aF via, where the (potentially) complex numbers aF are invariant under the rephasing of the Higgs basis eld H2 ! ei H2. It follows from eqs. (2.23) and (2.24) that FR = (Re aF )1 ; IF = (Im aF )1 : Inserting the above results into eq. (2.22), the Yukawa couplings take the following form: LY = MU qk1 + qk2aU PR + qk2aU PL U hk MD qk1 + qk2aD PR + qk2aDPL Dhk ME qk1 + qk2aE PR + qk2aEPL Ehk 1 v + + + 1 v + + + 3 U X k=1 3 D X k=1 3 E X k=1 1 v 1 v p v 2 X This form simpli es further if the neutral Higgs mass-eigenstates are also states of de nite CP. In this case, the corresponding Yukawa couplings are given by LY = F MF s + 6c Re aF + i F Im aF 5 F h (2.57) F MF c 6s Re aF + i F Im aF 5 F H F MF 6 Im aF i F Re aF 5 F A U a D KMDPR a U MU KPL DH+ + a E N MEPREH+ + h:c: : As noted above eq. (2.41), it is convenient to choose a convention in which s then follows from eq. (2.41) that 6c = j c j. That is, the neutral Higgs couplings exhibited in eq. (2.57) do not depend on the sign of c (which can be ipped by rede ning the overall sign of the Higgs basis eld H2). Note that in this convention, the signs of the alignment parameters aF 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 the scalar elds outside of the Higgs basis. Thus, a priori, there is no separate meaning to the parameters and in eq. (2.57). Nevertheless, in the special case of a CP-conserving neutral Higgs-lepton interaction governed by eq. (2.57) with Im aE = 0, it is convenient to introduce the real parameter tan via (2.54) (2.55) (2.56) (2.58) corresponding to a Type-II or Type-X Yukawa couplings of the charged leptons to the neutral Higgs bosons. The theoretical interpretation of tan de ned by eq. (2.58) is as follows. It is always possible to choose a 1- 2 basis with the property that one of the two Higgs-lepton Yukawa coupling matrices vanishes. Namely, in the notation of eq. (A.1), we have 2E;0 = 0, which means that only 1 couples to leptons. In the case of a CP-conserving scalar Higgs potential and Higgs vacuum, we can take the 1- 2 basis to be a real basis and identify tan or Type-X models, 2 E;0 = 0 does not correspond to a discrete Z2 symmetry of the generic vi= 2 (for i = 1, 2). However, in contrast to Type-II p A2HDM Lagrangian, since we do not require any of the Higgs-quark Yukawa coupling matrices and the scalar potential parameters 6 and 7 to vanish in the same 1- 2 basis. Note that the sign of aE in eq. (2.58) is physical since both 6 and tan ip sign under the Higgs basis eld H2 ! H2. In contrast to the standard conventions employed in the 2HDM with Type-I, II, X or Y Yukawa couplings where tan is de ned to be positive [cf. footnote 12], we shall not adopt such a convention here. In practice, we will rewrite eq. (2.58) as, where E = space. a E = Ej tan j ; (2.59) 1 correspond to physically non-equivalent points of the A2HDM parameter One theoretical liability of the A2HDM is that for generic choices of the alignment parameters aU and aD, the avor-alignment conditions in the quark sector speci ed in eq. (2.54) are not stable under the evolution governed by the Yukawa coupling renormalization group equations. Indeed, as shown in ref. [32], eq. (2.54) is stable under renormalization group running if and only if the parameters aU and aD satisfy the conditions of the Type I, II, X or Y 2HDMs speci ed at the end of section 2.3. In the leptonic sector, since we ignore neutrino masses, the Higgs-lepton Yukawa couplings are avor-diagonal at all scales. We therefore assume that13 F ( ) = a F F ( ) ; for F = U; D; (2.60) at some very high energy scale (such as the grand uni cation (GUT) scale or the Planck scale). That is, we assume that the alignment conditions are set by some a priori unknown physics at or above the energy scale . We take the complex alignment parameters aF to be boundary conditions for the RGEs of the Yukawa coupling matrices, and then determine the low-energy values of the Yukawa coupling matrices by numerically solving the RGEs. To ensure that the resulting low-energy theory is consistent with a SM-like Higgs boson observed at the LHC, we shall take mh = 125 GeV, and assume that the masses of H, A and H are all of order H 400 GeV. In this approximate decoupling regime, jc j is small enough such that the properties of h are within about 20% of the SM Higgs boson, as required by the LHC Higgs data [3]. We employ the 2HDM RGEs given in appendix B from down to H , and then match onto the RGEs of the Standard Model to generate the Higgs-fermion Yukawa couplings at the electroweak scale, which we take to be mt or mZ . Note that the values of Q( H ) = p the known quark masses via Standard Model RG running. 2MQ( H )=v (for Q = U , D) are determined from 13Under the assumption of a real Higgs basis, 6 = ei 23 is xed via eq. (2.58). This factor, which appears in eq. (2.54), can then be absorbed into the de nition of aF . Then, ME ( H ) is determined by the diagonal lepton mass matrix via Standard Model RG running. Higgs-mediated FCNCs from high scale alignment To explore the Higgs-mediated FCNCs that can be generated in the A2HDM at the electroweak scale, we establish avor-alignment at some high energy scale, , as for example at the GUT or Planck scale, and run the one-loop RGEs from the high scale to the electroweak scale. Thus, we impose the following boundary conditions for the running of the one-loop 2HDM Yukawa couplings, HJEP06(217) where the MQ (Q = U , D) are the diagonal quark matrices, and H is the scale of the heavier doublet, taken to be relatively large to guarantee that we are su ciently in the decoupling limit. For the lepton sector, the corresponding boundary conditions are [cf. eq. (2.59)], Q ( H ) = p 2MQ( H )=v; Q ( ); E E ( H ) = p 2ME ( H )=v; ( H ) = E j tan j E ( H ): As noted above for the lepton case (F = E), if E ( ) is proportional to E ( ), then is proportional to at all energy scales. Thus, we identify the leptonic alignment parameter at low energies by tan . More precisely [cf. eqs. (2.21) and (2.59)], E ( H ) = p 2 E j tan jME ( H )=v : where aQ represents the aligned part (in general, di erent from a0Q), and Q the corresponding degree of misalignment at the high scale. 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 avor-alignment at the high energy scale, we must know the values of Q( ). This involves running up Q( H ) to the high scale, but since the one-loop RGEs are strongly coupled to the Q matrices, we must supply values for Q( H ) to begin the running. With no a priori knowledge of which values of Q( H ) lead to avor-alignment at the high scale, we begin the iterative process by assuming avor-alignment at low-scale alignment parameter a0Q, This avor-alignment will be broken during RGE evolution to the high scale, and a procedure is needed to reestablish avor-alignment at the high scale. To accomplish this, we decompose Q( ) into parts that are aligned and misaligned with Q( ), respectively, Q ( H ) = a0Q Q ( H ): Q To minimize the misaligned part of Q( ), we implement the cost function, Q 3 X i;j=1 3 X i;j=1 j iQj j2 = Q j ij ( ) a Q Q ij ( )j2; which, once minimized, provides the optimal value of the complex parameter aQ for avoralignment at the high scale, (2.68) (2.69) (2.70) a Q Pi3;j=1 iQj ( ) iQj ( ) Pi3;j=1 iQj ( ) iQj ( ) : Q ( ); We subsequently impose avor-alignment at the high scale using this optimized alignment HJEP06(217) parameter, D below and evolve the one-loop RGEs back down to H . In principle, further running of U and H can regenerate o -diagonal terms. However, these e ects are extremely small and can be ignored in practice. At H , we use (2.62) to match the boundary conditions for the 2HDM and SM. At this point, the matrices longer diagonal, so we must rediagonalize respectively transforming U and D (at the scale U and D at the scale H are no U and D in analogy with eq. (A.12) [while H ) in analogy with eq. (A.13)]. We can now evolve U and D down to the electroweak scale to check the accuracy of the resulting quark masses. If any of the quark masses di er from their experimental values by more than 3%, we reestablish the correct quark masses at the electroweak scale,14 run back up to H , and then rerun this procedure repeatedly until the two boundary conditions are satis ed. The result is avor-alignment between Q( ) and Q( ), and a set of Q matrices at the electroweak scale that provide a source of FCNCs. In our iterative procedure, we demand that all scale-dependent Yukawa couplings remain below nite from the electroweak scale to the Planck scale (i.e., Landau poles are absent = MP). This restricts the range of the possible seed values, a0Q, used in eq. (2.66) to initialize the iteration. Consequently, the alignment parameters aU and aD cannot be too large in absolute value. Constraints on the alignment parameters due to Landau pole considerations during one-loop RG running have been given in ref. [59]. In our analysis, the allowed values of aU and aD consistent with the absence of Landau poles at all scales below are exhibited in gure 1.15 Assuming H = 400 GeV, these considerations lead to bounds on the alignment parameters evaluated at the Planck scale, = MP, jaU j . 0:8 and jaDj . 80 ; (2.71) which are consistent with the results previously obtained in ref. [59]. 14Starting the RG evolution at mZ, we use a ve avor scheme to run up to mt and a six avor scheme above mt. Running quark mass masses at mZ and mt are obtained from the RunDec Mathematica software package [55, 56], based on quark masses provided in ref. [57]. We x the initial value of the top Yukawa coupling yt(mt) = 0:94, corresponding to an MS top quark mass of mt(mt) = 163:64 GeV [58]. For simplicity, the e ects of the lepton masses are ignored, as these contribute very little to the running. 15If a Landau pole in one of the Yukawa coupling matrices arises at the scale , then both the corresponding ( ) diverge, whereas their ratio, aQ, remains nite. = MP are exhibited. The blue points occupy the region of the A2HDM parameter space where the prediction for all entries of the Q matrices lie within a factor of 3 from the results obtained with the full running. The red points occupy the region where the leading log approximation yields results quite di erent from the full RG running. Leading logarithm approximation In the limit of small alignment parameters, it is possible to obtain approximate analytic solutions to the one-loop RGEs provided in appendix B. One can express the Q matrices at the low scale as U D ( H ) ' a ( H ) ' a U U D D ( H ) + ( H ) + 1 1 16 2 log U ); D); ; : (2.72) (2.73) (2.74) (2.75) where D D; D U ; D D; D U are the -functions de ned in eqs. (B.10){(B.14) and U ( H ) and D( H ) are proportional to the diagonal quark mass matrices, MU and MD respectively, at the scale H , according to eq. (2.21). Working to one loop order and neglecting higher order terms, it is consistent to set F = a F F = a F p 2MF =v (for F = U; D; E) in the corresponding -functions,16 U D ( H )ij ' a ( H )ij ' a U D ij ij p2(MU )jj + p2(MD)ii + v v 4(M2U )2jvj3 log p 4(M2D2)ivi3 log p H H k k (aE aU ) 1 + aU (aE) ij Tr(ME2 ) (aE aD) 1 + aD(aE) ij Tr(ME2 ) +(aD aU ) 1 + aU (aD) 3 ij Tr(M D2) 2 X(M D2)kkKikKjk +(aU aD) 1 + aD(aU ) 3 ij Tr(MU2 ) 2 X(MU2 )kkKkiKkj 16The misalignment contributions exhibited in eqs. (2.74) and (2.75) were computed for the rst time in ref. [60]. It follows that there is a large hierarchy among the several o -diagonal terms of the matrices, D( H )ij D( H )ji U ( H )ij U ( H )ji (MD)ii (MD)jj (MU )jj (MU )ii 1; for i < j ; 1; for i < j: eq. (2.20) whereas D is daggered. The inequality given in eq. (2.76) was previously noted in ref. [36], and provides the justi cation for ignoring iDj relative to jDi, for i < j.17 This hierarchy of Yukawa couplings is reversed for iUj . This reversal can be traced back to the fact that U is undaggered in It is noteworthy that the leading log results for the o -diagonal terms of the Q matrices obtained in eqs. (2.74) and (2.75) and the corresponding full numerical calculation are typically within a factor of a few. Even for small alignment parameters, there can be some small discrepancies between the two 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 the electroweak scale and the high energy scale . The leading log approximation describes less and less accurately the numerical results at larger and larger alignment parameters. This is shown in gure 1, where the blue points correspond to the parameter regime in which the leading log approach leads to results within a factor of 3 of the results obtained numerically for all the elements of the U and D matrices. In contrast, the red points correspond to the parameter regime in which the leading log approximation leads to results quite di erent from what is obtained by the full running. 2.7 A particular type of Minimal Flavor Violation In the quark sector of the A2HDM, only the two Yukawa coupling matrices U and D break the SU(3)Q SU(3)U SU(3)D global avor symmetry of the electroweak Lagrangian involving quarks. For this reason, our model can be thought in terms of a speci c realization of a Minimal Flavor Violating (MFV) 2HDM [29]. In particular, in a general 2HDM with MFV one can write the Yukawa Lagrangian as LY;MFV = QLYuURH1y + QLYdyDRH1 + QLAuURH2y + QLAydDRH2 + h:c:; (2.78) with H1; H2 the two Higgs doublets in the Higgs basis as de ned in section 2 and QL; UR; DR avor eigenstate quarks. In general, Au; Ad can be expressed by the in nite sum [35] (2.76) (2.77) Au = Ad = X n1;n2;n3 X n1;n2;n3 n1n2n3 (YdYdy)n1 (YuYuy)n2 (YdYdy)n3 Yu; u d n1n2n3 (YdYdy)n1 (YuYuy)n2 (YdYdy)n3 Yd; (2.79) (2.80) relations U = p 2 u and D = p 2 yd. with generic O(1) complex coe cients un;id. In order to determine the coe cients un;id in the A2HDM, we rotate to the quark mass-eigenstate basis: Yu ! U ; Au ! U ; Yd ! D, 17To make contact with the Higgs basis Yukawa couplings u and d employed by ref. [36], we note the as computed in our model in the leading logarithmic approximation. The dark purple region is favored by the measurement of Bs mixing, the purple region by Bd mixing, and the dark pink (pink) region by the phase (mass di erence) of the Kaon mixing system. D meson mixing does not give any interesting bound on the parameter space and it is not shown in the gure. Right panel: the corresponding bounds from Bs mixing obtained by scanning the parameter space and using the full RG running. The yellow, red, and green points correspond to a Wilson coe cient of < 1=3; [1=3; 1]; > 1 relative to the value that yields the present bound from Bs mixing. Kaon mixing system, and the pink region by the K-K mass di erence. D mixing does not give any interesting bound on the parameter space and is therefore omitted in the gure. Bs mixing leads to the most stringent bound and it constrains aD to be smaller than 4:7 at sizable values of aU . Additionally, the bound from the measurement of CP violation in Kaon mixing (dark pink) is signi cantly more stringent than the bound from the mass di erence of the Kaon system (in pink). This is due to the fact that the real and imaginary parts of the Wilson coe cient of the Kaon system have a similar magnitude (under the assumption that aU and aD are real). In particular, the ratio of the imaginary and real parts of the Wilson coe cient is directly related to the phase of the CKM matrix: Im(C4K )=Re(C4K ) = Im((K32)2K321)=Re((K32)2K321). In contrast, the SM Wilson coe cient has an imaginary part that is much smaller than the real part. Small di erences between the constraints from CP violation and the mass di erence also exist in the Bs and Bd systems. In gure 6, we only show the most constraining bound in each system, i.e. the mass di erence in Bs mixing and the phase in Bd mixing. The right panel of gure 6 shows the corresponding results for the Bs mixing system obtained by scanning the parameter space and using the full RG-running. The points in yellow have a Wilson coe cient smaller than 1/3 the present bound on the Wilson coe cient; in red we present the points with a Wilson coe cient smaller than the present bound, and nally in green we present the points that have been already probed by the measurement of the Bs mixing observables. In the limit of sizable aU & 0:7, we do not nd points with aD & 4, in rough agreement with the leading log result. 4.2 The B-meson rare decays Bs;d ! `+` 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 avor misalignment at the electroweak scale is set to zero. The neutral Higgs exchange contributions to the leptonic decay amplitude are proportional to m` and hence are largest in the case of Bs;d ! + . However, it is more di cult to tag the 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(Bs(d) ! are relatively weak as compared to the SM prediction [98], + ) . 3 At sizable values of tan , the main contributions to Bs;d ! H and A exchange, as they are enhanced by the second power of tan . Furthermore, in the cos( ) = 0 limit, the light Higgs (h) contribution vanishes at tree-level. For this reason, we shall focus henceforth on the heavy Higgs contributions that are given by [99], BR(Bs;d ! BR(Bs;d ! + + where BR(Bs;d ! ) )SM + ' jSs;dj2 + jPs;dj2 1 + ys;d Re(Ps2;d) Re(Ss2;d) jSs;dj2 + jPs;dj2 1 1 + ys;d ; (4.6) )SM is the SM prediction for the branching ratio extracted from an untagged rate. In particular, ys = (6:1 0:7)% and yd 0 have to be taken into account when comparing experimental and theoretical results, and 0:19) + (4.5) are typically due to The Ci are the Wilson coe cients corresponding to the Lagrangian with operators Ss;d Ps;d mBs;d (CsS;d Cs0S;d) s mBs;d (CsP;d Cs0P;d) (Cs1;0d + 1 C1S0Ms;d C1S0Ms;d 4m2 m2Bs;d ; C1S0Ms;d C100 s;d) : 2m 2m i Os(0)S = Os(0)P = Ls = X(CiOi + Ci0Oi0) + h:c: ; mBs mBs mb (sPR(L)b)(``); mb (sPR(L)b)(` 5`); O1(00)s = (s PL(R)b)(` 5`); and the corresponding ones for the Bd system. In the limit of cos( coe cients arising from heavy neutral Higgs exchange are given by ) = 0, the Wilson CsP = Cs0P = mb mBs p3D2 m (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) butions to the O100 operators. ( ) and the analogous results for the Bd system. There are no tree-level New Physics contrisin( sin( If cos( ) is nonvanishing, then the scalar Wilson coe cients CsS and C0S given s in eqs. (4.13) and (4.14) due to H exchange should be changed accordingly, tan ) tan + cos( ) and D D sin( contributions arise due to h exchange; the corresponding contributions are obtained from CsS and Cs0S given in eqs. (4.13) and (4.14) by making the following replacements, tan ) cos( ) tan , D D cos( ) and mH ! mh. The SM Wilson coe cient takes the form [100], HJEP06(217) ! ! (4.15) (4.16) (4.17) (4.18) (4.19) and the predicted branching ratios are given by C1S0Ms;d = 4:1 as obtained in [98] with the inclusion of O( em) and O( s2) corrections. These values are in relatively good agreement with the experimental results. The combination of the LHCb and the CMS measurements at Run I for the Bs and Bd decays to muon pairs are [101]: BR(Bs ! + + ) = (2:8+00::76) ) = (3:9+11::64) 10 9; and Bd ! Note the much larger uncertainty in the latter decay mode. The ATLAS collaboration has also reported a Run I search for Bs ! + , which yielded BR(Bs ! + ) = (0:9+10::18) 10 9 [102], although this measurement is not yet competitive with eq. (4.18). Very recently, LHCb reported a new measurement for Bs;d ! + using Run II data [103]. Their result, BR(Bs ! + ) = (2:8 0:6) 10 9, agrees very well with the LHCb and CMS combination quoted in eq. (4.18). In contrast, the new LHCb Bd measurement is closer to the SM prediction, BR(Bd ! In the following, we will compare the predictions of the A2HDM with the LHCb and CMS combination shown in eqs. (4.18) and (4.19). In the coming years, the two branching ratios will be measured much more accurately by the LHC. In particular, the Bs and Bd branching fractions will be measured by each experiment with a precision of 13% and 48% at + ) = (1:6+10::19) 10 10. Run-III, improving to 11% and 18%, respectively, at the HL-LHC [104]. In gure 7, we show the constraints from the measurement of Bs ! (right panel) as functions of aU and aD, with xed tan + (left panel) = 10, E = +1[see ) = 0, and mA = mH = 400 GeV, based on the leading logarithmic approximation. The pink shaded region denote the parameter space favored by the CMS and LHCb combined results at the 2 level, namely BR(Bs ! BR(Bs ! + + ) )SM ) )SM [0:8; 6:6]: (4.20) 21Fixing a di erent sign, E = 1, leads to the same results, with the exchange (aU ; aD) ! ( aU ; aD). + (right panel) relative to the SM, as a function of aU and aD, with xed tan ) = 0, and mA = mH = 400 GeV. The regions in pink are allowed at the 2 level by the present measurements. The purple shaded 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 poles in the Yukawa couplings below MP. The purple shaded region in gure 7 is the parameter space favored at 2 by the HL-LHC measurement, assuming a measured central value equal to the SM prediction. Comparing the region in pink to the region in purple, one can get a sense of the improvement the HLLHC can achieve in testing our model. 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 SM prediction for the two branching ratios, with the bene t of more accurate measurements of the CKM elements that will be obtained at the LHCb and at Belle II in the coming years. The present measurement of Bs ! our model. The measurement of Bd ! values of jaDj (cf. the white region where jaDj + + constrains sizable values of aU and aD in also sets an interesting constraint at smaller 3 and the values of jaU j are sizable), + )exp=BR(Bd ! + )SM 3:7. However, the deviation from the SM prediction is since the central value of the measurement is larger than the SM prediction: BR(Bd ! not yet statistically signi cant, due to the large experimental uncertainty. Nevertheless, a sizable suppression of the Bd decay mode is presently disfavored. As expected, the contours for BR(Bs;d ! + )=BR(Bs;d ! )SM in the two panels of gure 7 are very similar. This is due to the fact that our model is a particular type of MFV model in the leading logarithmic approximation [cf. section 2.7]. In particular, MFV models generically predict )=BR(Bs ! BR(Bd ! )SM=BR(Bs ! + )SM, with corrections arising only from ms=mb and md=mb terms. For this reason, it is di cult in our model to enhance one decay mode, while suppressing the other. + ) + (right panel) relative to the SM, as a function of M (the mass of the heavy scalar and pseudoscalar) and aD. We x tan = 10, aU = 0:2, and cos( ) = 0. The pink regions are the regions allowed at the 2 level by the 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 poles in the Yukawa couplings below MP. + (left panel) and for Bd ! + (right panel) relative to the SM, obtained via scanning the parameter space and using the full RG running, at xed tan ) = 0, and mA = mH = 400 GeV. The yellow, red, green and blue points corresponds to branching ratios normalized to the SM prediction < 0:4; [0:4; 1:1]; [1:1; 10]; > 10. In boldface we denote the range preferred by the LHCb and ATLAS measurement of Bs ! + to the Bd;s ! 4.3 B ! decays It is also interesting to investigate the bounds as a function of the heavy Higgs boson masses. In gure 8, we show the same constraints in the (M; aD) plane, where M mA = mH , having xed tan ) = 0. Sizable regions of parameter space are allowed, even for values of M as small as 300 GeV. Finally, in gure 9, we show the results obtained through scanning the parameter space and utilizing the full RG running. These plots are qualitatively similar to the contour plots of gure 7 obtained in the leading logarithmic approximation, although the heavy Higgs exchange contributions decay rates computed using the full RG running are somewhat larger (at large alignment parameters) than the corresponding leading log results. HJEP06(217) The leptonic decays B ! ` are interesting probes of the Higgs sector of our model and particularly of the charged Higgs couplings, since the charged Higgs boson mediates treelevel New Physics contributions to these decay modes. The channel is the only decay mode of this type observed so far. The present experimental world average is [105]22 and is in relatively good agreement with the SM prediction [106]23 (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) where we have de ned the SM Wilson coe cient CSuMb = 4GF Kub= 2 and CRub(L) are the ub Wilson coe cients of the OR(L) = (uPR(L)b)( PL ) operators. p In particular [107], CRub(L) = 1 m2H p LR(RL) 2m ub v tan ; with LR(RL) the two charged Higgs couplings H+uLbR, H+uRbL given by ub This leads to the branching ratio, ) )SM m2B mb p v tan 2Kubm2H X i Kui 3Di + Kib i1 U 2 : 22Updated results and plots available at: http://www.slac.stanford.edu/xorg/hfag. 23Updated results and plots available at: http://ckm tter.in2p3.fr. In our model, the New Physics contribution to this decay reads BR(B ! BR(B ! ) )SM = 1 + m2B mbm Cub L CSuMb CRub 2 ; uLbR = X Kui 3Di ; i = 1 uRbL = X Kib iU1 : i panel: leading log predictions, where the pink region is favored by the measurement of B ! The purple region is anticipated by future measurement at Belle II, under the assumption that . the central value of the measurement is given 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 < 0:2; [0:79; 1:71]; [1:71; 3]; > 3, respectively. In boldface we denote the range preferred by the present world average for BR(B ! ). In the leading logarithmic approximation, the most important contributions come from the second term of the above expression (/ 3Di ), as one can easily deduce from eqs. (2.74) and (2.75). In gure 10, we show our numerical results as obtained using the leading log approximation (left panel) and the scan of the parameter space using the full RGEs, having xed mH = 400 GeV and tan = 10. A very large region of parameter is still allowed by the measurement of B ! . In particular, in the leading logarithmic apment parameter, aU . Indeed, in the pink region shown in the left panel of proximation, every value jaDj . 17 is allowed, irrespective of the value of the other aligngure 10, [0:79; 1:71], 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 jaU j are excluded by the measurement of BR(B ! (see the blue points in the right panel of the gure). In the left panel of gure 10, we also ) exhibit the purple shaded region of parameter space that would be favored by the future Belle II measurement, under the assumption that the central value of the measurement is given by the SM prediction for this branching ratio [cf. eq. (4.22)]. The allowed region of parameter space is expected to shrink considerably, thanks to the anticipated accuracy of the Belle II measurement with a total error of the order of 5% [108], leading to an allowed range, BR(B ! )=BR(B ! )SM improvement in the SM prediction of this B meson decay mode. [0:86; 1:14], where we have assumed no We have explored the consequences of avor-alignment at a very high energy scale on avor observables in the two Higgs doublet Model (2HDM). Flavor alignment at the electroweak scale generically requires an unnatural ne-tuning of the matrix Yukawa couplings. If avor alignment is instead imposed at a higher energy scale such as the Planck scale, perhaps enforced by some new dynamics beyond the SM, then the avor misalignment at the electroweak scale due to RG running will generate new sources of FCNCs. The resulting tree-level Higgs-mediated FCNCs are somewhat suppressed and relatively mildly constrained by experimental measurements of avor-changing observables. We require that the alignment parameters at the high scale remain perturbative. In particular, no Landau poles are encountered during RG running. These requirements lead to an upper bound on the values of the alignment parameters at the Planck scale. This in turn provide an upper bound on the size of FCNCs generated at the electroweak scale. The avor-changing observables considered in this paper that provide the most sensitive probe of the avor-aligned 2HDM parameter space are meson mixing and rare B decays such as Bs;d ! and B ! . We also considered constraints from LHC searches with Bs ! II" scenario. bb; + of heavy Higgs bosons (the most important of which are searches for pp ! b(b)H; H ! ), and measurements of the couplings of the observed (SM-like) Higgs boson. The most stringent constraint on the avor-aligned 2HDM parameter space arises from the measurement of the rare decay Bs ! + . We investigated the predictions of the avor-aligned 2HDM in the regions of the parameter space not yet probed by the measurements listed above. The top rare avor changing decays, t ! uh, t ! ch, are generated at tree-level. However, once we impose constraints from Higgs coupling measurements, the predicted branching ratios for these neutral avor changing top decays are beyond 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 strange quark, and the ratios, BR(H ! tt) : BR(H ! bb) : BR(H ! the predictions of the more common Type I and II 2HDMs. These features are exhibited + ), can be very di erent, if compared to in our summary plots in gures 11 and 12. tan cos( In gure 11, we summarize the constraints on the (aU ; aD) parameter space, with xed = 10 (upper panels) and tan = 3 (lower panels). In both panels, we x the values ) = 0 and mA = mH = mH = 400 GeV. The region favored by all avor constraints is shown in reddish-brown. At sizable values of aD, the most relevant constraint comes from the measurement of Bs ! + (tan region). Bs meson mixing also sets an interesting bound on the parameter space (blue-gray region). It o ers some complementary , as it does not depend on the particular value of tan . Moreover, it will be able to probe the small region of parameter space with aU > 0 and sizable values of aD favored by the measurement of Bs ! + value in agreement with the SM prediction.24 The measurement of B ! imposes only a in the case of a future measurement with a central 24We use the results in [109] for the future prospects in measuring Bs mixing, corresponding to the \Stage HJEP06(217) computed in the leading log approximation. We x cos( ) = 0, mA = mH = mH = 400 GeV, tan = 10 (upper panels), and tan = 3 (lower panels). The contours represent the ratio BR(H ! bb)m2 =[BR(H ! )3mb2], where 1 is the Type I and Type II 2HDM prediction. The reddish-brown regions are favored by all avor constraints. The green region is favored by the measurement of B ! . Blue-gray and tan regions are favored by Bs mixing and Bs ! respectively. The gray shaded regions produce Landau poles in the Yukawa couplings below MP. The left and right panels represent the bounds as they are now and as projected for the coming + 400 GeV, cos( ) = 0, and tan = 10. Blue points correspond to points allowed by the , but not by the measurement of Bs mixing or Bs ! are allowed by the measurements of B ! and of meson mixing but not by Bs ! . Green points + . Red points are allowed by all constraints. The left and right 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 MP. relatively weak constraint on the parameter space (green region). For values of tan = 10 (or larger), in the region of parameter space favored by present and future avor constraints, the ratio m2 BR(H ! bb)=3mb2 BR(H ! + ) is smaller than the ratio predicted by Type I and II 2HDM in most of the Aligned 2HDM parameter space. The parameter space is somewhat less constrained at lower values of tan , as shown in the lower panels of gure 11. In gure 12, we present the corresponding results obtained in the numerical scan with full RG running, with xed cos( ) = 0, mA = mH = mH = 400 GeV, and tan = 10. The qualitative features of the leading log approximation continue to hold. In particular, we again see that Bs ! + provides the most stringent constraint on the aligned 2HDM parameter space. Note that in order to emphasize the comparison of the constraints obtained from the di erent B physics observables in gures 11 and 12, we do not include the constraints due to the LHC searches for the heavy Higgs bosons decaying into fermion pairs in these gures. As shown in gures 4 and 5 for the heavy Higgs mass values quoted above, in the region of the Aligned 2HDM parameter space consistent with no Landau poles below MP, the current LHC limits on H and A production eliminate the parameter regime with jaDj & 30{40 and jaU j . 0:1. In considering the phenomenological implications of extended Higgs sectors, the most conservative approach is to impose only those constraints that are required by the current experimental data. In most 2HDM studies in the literature, the Yukawa couplings are assumed to be of Type I, II, X or Y. In this paper, we have argued that the current experimental data allows for a broader approach in which the Yukawa couplings are approximately aligned in avor at the electroweak scale. The resulting phenomenology can yield some unexpected surprises. We hope that the search strategies of future Higgs studies at the LHC will be expanded to accommodate the broader phenomenological framework of the (approximately) avor-aligned extended Higgs sector. Acknowledgments H.E.H. gratefully acknowledges Paula Tuzon for numerous interactions during her two month long visit to Santa Cruz in 2010{2011. Her work on the aligned 2HDM provided inspiration for this work. S.G. thanks Wolfgang Altmannshofer for discussions. H.E.H. and E.S. are supported in part by the U.S. Department of Energy grant number DESC0010107. S.G. acknowledges support from the University of Cincinnati. S.G. and H.E.H. are grateful to the hospitality and the inspiring working atmosphere of both the Kavli Institute for Theoretical Physics in Santa Barbara, CA, supported in part by the National Science Foundation under Grant No. NSF PHY11-25915, and the Aspen Center for Physics, supported by the National Science Foundation Grant No. PHY-1066293, where some of the research 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 Lagrangian:25 0 0 LY = Q0L ea aU;0 UR0 + QL a( aD;0)yDR0 + EL a( aE;0)y ER0 + h:c: ; (A.1) summed over a = a = 1; 2, where 1;2 are the Higgs doublets, ea i 2 a, Q0L and EL0 are are vectors in the quark and lepton avor spaces, and aU;0; aD;0; aE;0 are 3 the weak isospin quark and lepton doublets, and UR0 , DR0, ER0 are weak isospin quark and lepton singlets.26 Here, Q0L, EL0, UR0 , DR0, ER0 denote the interaction basis states, which 3 matrices in quark and lepton avor spaces. Note that a U;0 appears undaggered in eq. (A.1), whereas the corresponding Yukawa coupling matrices for down-type fermions (D and E) appear daggered. In this convention, the transformation of the Yukawa coupling matrices under a scalar eld basis change is the same for both up-type and down-type fermions. That is, under a change of basis, a ! Uab b (which implies that ea ! ebUbya), the Yukawa coupling matrices transform as a ! Uab bF and a F F y ! bF yUbya (for F = U , D and E), which re ects the form-invariance of LY under the basis change. The neutral Higgs states acquire vacuum expectation values, (A.2) (A.3) where v^av^a = 1 and v = 246 GeV. It is also convenient to de ne where 12 = 21 = 1 and 11 = 22 = 0. 25We follow the conventions of ref. [39], in which covariance is manifest with respect to U(2) avor transformations, a ! Uab b [where U 2 U(2)], by implicitly summing over barred/unbarred index pairs of the same letter. 26The right and left-handed fermion elds are de ned as usual: R;L PR;L , where PR;L 12 ( 1 5 ). h 0ai = p ; vv^a 2 w^b v^a ab ; Following refs. [37, 39], we de ne invariant and pseudo-invariant matrix Yukawa couplings, where F = U , D or E. Inverting these equations yields F;0 v^a aF;0 ; F;0 w^a aF;0 ; F;0 = a F;0v^a + F;0w^a : Note that under the U(2) transformation, a ! Uab b , F;0 is invariant and F;0 ! (det U ) F;0 : (A.6) The Higgs elds in the Higgs basis are de ned by H1 v^ a a and H2 w^a a, which can be inverted to yield a = H1v^a + H2wa [39]. Rewriting eq. (A.1) in terms of the Higgs basis elds, 0 LY = Q0L (He1 U;0 + He2 U;0) UR0 + QL (H1 D;0 y + H1 D;0 y) DR0 0 +EL (H1 E;0 y + H1 E;0 y) ER0 + h:c: The next step is to identify the quark and lepton mass-eigenstates. This is accomplished by replacing H1 ! (0 ; v=p2) 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 matrices are diagonal with non-negative entries. In more detail, we de ne: PLU = VLU PLU 0 ; PLE = VLEPLE0 ; PRU = VRU PRU 0 ; PRE = VRDPRE0 ; PLD = VLDPLD0 ; PLN = VLEPLN 0 ; PRD = VRDPRD0 ; and the Cabibbo-Kobayashi-Maskawa (CKM) matrix is de ned as K for the neutrino elds, we are free to choose V N = VLE since neutrinos are exactly massless L in this analysis.27 In particular, the unitary matrices VLF and VRF (for F = U , D and E) VLU V D y : Note that L are chosen such that v v MU = p2 VLU U;0V U y = diag(mu ; mc ; mt) ; R MD = p2 VLD D;0 yVRD y = diag(md ; ms ; mb) ; ME = p2 VLE E;0 yVRE y = diag(me ; m ; m ) : It is convenient to de ne U = VLU U;0VRU y ; U = VLU U;0VRU y ; D = VRD D;0V D y ; L D = VRD D;0V D y ; L E = VRD E;0V E y ; L E = VRD E;0V E y : L 27Here we are ignoring the right-handed neutrino sector, which gives mass to neutrinos via the seesaw mechanism. (A.4) (A.5) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) which is a physical observable. The matrices D and E are independent pseudoinvariant complex 3 3 matrices. The Higgs-fermion interactions given in eq. (A.7) can be rewritten in terms of the quark and lepton mass-eigenstates, LY = U L( U H10 y + U H20 y)UR +N L( E yH1+ + E yH2+)ER + EL( E yH10 + E yH20)ER + h:c: (A.16) B Renormalization group equations for the Yukawa matrices We rst write down the renormalization group equations (RGEs) for the Yukawa matrices 16 2 (d=d ) = 16 2(d=dt), the RGEs are given by [32]: aU;0, a D;0 and aE;0. De ning D D a U;0 = 8gs2 + 4 9 g2 + 17 g0 2 12 D a D;0 = 8gs2 + 4 9 g2 + + 3Tr aU;0( bU;0)y + aD;0( bD;0)y + Tr aE;0( bE;0)y U;0 b 2( bD;0)y aD;0 U;0 + aU;0( bU;0)y bU;0 + 2 ( bD;0)y bD;0 U;0 + b a 1 12 bU;0( bU;0)y aU;0 ; Eq. (A.6) implies that under the U(2) transformation, a ! Uab b F is invariant and F ! (det U ) F ; for F = U , D and E. Indeed, F is invariant since eqs. (A.9){(A.11) imply that MF = p 2 F ; U , D a E;0 = 4 9 g2 + + bE;0( bE;0)y aE;0 + 12 aE;0( bE;0)y bE;0 : + 3Tr ( bD;0)y aD;0 + ( bU;0)y aU;0 + Tr ( bE;0)y aE;0 D;0 b 2 bD;0 aU;0( bU;0)y + bD;0( bD;0)y aD;0 + 2 a 1 D;0 bU;0( bU;0)y + 12 aD;0( bD;0)y bD;0; E;0 + 3Tr ( bD;0)y aD;0 + ( bU;0)y aU;0 + Tr ( bE;0)y aE;0 a and 2 F;0 = The RGEs above are true for any basis choice. Thus, they must also be true in the Higgs basis in which v^ = (1; 0) and w^ = (0; 1). In this case, we can simply choose 1F;0 = F;0 F;0 to obtain the RGEs for the F;0 and F;0. Alternatively, we can multiply eqs. (B.1){(B.3) rst by v^a and then by w^a. Expanding ay, which appears on the right-hand sides of eqs. (B.1){(B.3), in terms of y and y using eq. (A.5), we again obtain the RGEs for the F;0 and F;0. Of course, both methods yield the same result, since the diagonalization matrices employed in eqs. (A.9){(A.11) are de ned as those that bring the mass matrices to their diagonal form at the electroweak scale. No scale dependence is assumed in the diagonalization matrices, and as such they are not a ected by the operators D. (A.14) (A.15) (B.1) (B.2) E;0 b (B.3) + U;0( U;0y U;0+ U;0y U;0)+12( D;0y D;0+ D;0y D;0) U;0 +1( U;0 U;0y+ U;0 U;0y) U;0; + 3Tr U;0 U;0y+ D;0 D;0y +Tr E;0 E;0y U;0 2 D;0y D;0 U;0+ D;0y D;0 U;0 + U;0( U;0y U;0+ U;0y U;0)+12( D;0y D;0+ D;0y D;0) U;0 +1( U;0 U;0y+ U;0 U;0y) U;0; + 3Tr U;0 U;0y+ D;0 D;0y +Tr E;0 E;0y U;0 2 D;0y D;0 U;0+ D;0y D;0 U;0 + 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0] D;0 2( D;0 U;0 U;0y + D;0 U;0 U;0y)+( D;0 D;0y+ D;0 D;0y) D;0+1 D;0( U;0 U;0y+ U;0 U;0y) +1 D;0( D;0y D;0+ D;0y D;0); 2 2 2 2 2 2 (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) +( D;0 D;0y+ D;0 D;0y) D;0+1 D;0( U;0 U;0y+ U;0 U;0y) +1 D;0( D;0y D;0+ D;0y D;0); + 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0 E;0 +( E;0 E;0y+ E;0 E;0y) E;0+1 E;0( E;0y E;0+ E;0y E;0); + 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0] D;0 2( D;0 U;0 U;0y+ D;0 U;0 U;0y) + 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0 E;0 4 4 4 4 D U = D U = D D = D D = D E = D E = 8gs2+ 49 g2+ 5 g02 9 g2+ 15 g02 4 9 g2+ 15 g02 4 4 2 2 2 2 E E U U (B.10) (B.11) (B.12) (B.15) (B.16) (B.17) Using eqs. (A.12) and (A.13), we immediately obtain the RGEs for the F and F . Schematically, we shall write, D F = F ; D F = F ; for F = U , D and E. Explicitly, the corresponding -functions at one-loop order are given by, U + 3Tr U Uy+ D Dy +Tr E Ey + 3Tr U Uy+ D Dy +Tr E Ey U 2K Dy DKy U + Dy DKy U + U ( Uy U + Uy U )+ 1 K( Dy D+ Dy D)Ky U + 1 ( U Uy+ U Uy) U ; U + 3Tr U Uy+ D Dy +Tr E Ey + 3Tr U Uy+ D Dy +Tr E Ey U 2K Dy DKy U + Dy DKy U + U ( Uy U + Uy U )+ 1 K( Dy D+ Dy D)Ky U + 1 ( U Uy+ U Uy) U ; + 3Tr Dy D+ Uy U +Tr Ey E] D 2( DKy U Uy+ DKy U Uy)K + 3Tr Dy D+ Uy U +Tr Ey E] D 2( DKy U Uy+ DKy U Uy)K +( D Dy+ D Dy) D+ 1 DKy( U Uy+ U Uy)K+ E+ 3Tr Dy D+ Uy U +Tr Ey E D+ 3Tr Dy D+ Uy U +Tr Ey E] D (B.13) D+ 3Tr Dy D+ Uy U +Tr Ey E] D (B.14) 2 2 2 2 2 2 For the numerical analysis of the RGEs, it is convenient to de ne D DKy ; D e DKy ; + 3Tr Dy D+ Uy U +Tr Ey E + 3Tr Dy D+ Uy U +Tr Ey E E+ 3Tr Dy D+ Uy U +Tr Ey E keeping in mind that the (unitary) CKM matrix K is de ned at the electroweak scale and thus is not taken to be a running quantity. The RGEs given in eqs. (B.11){(B.16) can now be rewritten by taking D ! eD, D ! e D and K ! 1. The advantage of the RGEs written in this latter form is that the CKM matrix K no longer appears explicitly in the di erential equations, and enters only in the initial condition of D at the low scale [cf. eq. (2.62)], In particular, the high scale boundary condition given by eq. (2.63) also applies to eD, i.e., Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. D p ( H ) = 2MD( H )Ky=v : eD( ) = a D D ( ) : (B.18) D and (B.19) HJEP06(217) LHC pp collision data at p [1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE]. [2] CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE]. [3] ATLAS and CMS collaborations, Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the s = 7 and 8 TeV, JHEP 08 (2016) 045 [arXiv:1606.02266] [4] P. Fayet, Supergauge invariant extension of the Higgs mechanism and a model for the electron and its neutrino, Nucl. Phys. B 90 (1975) 104 [INSPIRE]. [5] K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Low-energy parameters and particle masses in a supersymmetric grand uni ed model, Prog. Theor. Phys. 67 (1982) 1889 [6] R.A. Flores and M. Sher, Higgs masses in the standard, multi-Higgs and supersymmetric models, Annals Phys. 148 (1983) 95 [INSPIRE]. (1986) 1 [Erratum ibid. B 402 (1993) 567] [INSPIRE]. JHEP 11 (2006) 038 [hep-ph/0605242] [INSPIRE]. [7] J.F. Gunion and H.E. Haber, Higgs bosons in supersymmetric models. 1, Nucl. Phys. B 272 [8] L. Fromme, S.J. Huber and M. Seniuch, Baryogenesis in the two-Higgs doublet model, [9] F. Bezrukov, M. Yu. Kalmykov, B.A. Kniehl and M. Shaposhnikov, Higgs boson mass and new physics, JHEP 10 (2012) 140 [arXiv:1205.2893] [INSPIRE]. [10] G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 08 (2012) 098 [arXiv:1205.6497] [INSPIRE]. 089 [arXiv:1307.3536] [INSPIRE]. [11] D. Buttazzo et al., Investigating the near-criticality of the Higgs boson, JHEP 12 (2013) [12] J. Elias-Miro, J.R. Espinosa, G.F. Giudice, H.M. Lee and A. Strumia, Stabilization of the electroweak vacuum by a scalar threshold e ect, JHEP 06 (2012) 031 [arXiv:1203.0237] [13] O. Lebedev, On stability of the electroweak vacuum and the Higgs portal, Eur. Phys. J. C 72 (2012) 2058 [arXiv:1203.0156] [INSPIRE]. [14] G.M. Pruna and T. Robens, Higgs singlet extension parameter space in the light of the LHC discovery, Phys. Rev. D 88 (2013) 115012 [arXiv:1303.1150] [INSPIRE]. doublet scenario: a study including LHC data, JHEP 12 (2014) 166 [arXiv:1407.2145] [17] D. Das and I. Saha, Search for a stable alignment limit in two-Higgs-doublet models, Phys. Rev. D 91 (2015) 095024 [arXiv:1503.02135] [INSPIRE]. [18] P. Ferreira, H.E. Haber and E. Santos, Preserving the validity of the two-Higgs doublet model up to the Planck scale, Phys. Rev. D 92 (2015) 033003 [Erratum ibid. D 94 (2016) 059903] [arXiv:1505.04001] [INSPIRE]. [19] D. Chowdhury and O. Eberhardt, Global ts of the two-loop renormalized two-Higgs-doublet model with soft Z2 breaking, JHEP 11 (2015) 052 [arXiv:1503.08216] [INSPIRE]. [20] G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva, Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1 [arXiv:1106.0034] [21] M. Misiak and M. Steinhauser, Weak radiative decays of the B meson and bounds on MH in the two-Higgs-doublet model, Eur. Phys. J. C 77 (2017) 201 [arXiv:1702.04571] [22] S.L. Glashow and S. Weinberg, Natural conservation laws for neutral currents, Phys. Rev. D 15 (1977) 1958 [INSPIRE]. 397 [INSPIRE]. [23] E.A. Paschos, Diagonal neutral currents, Phys. Rev. D 15 (1977) 1966 [INSPIRE]. [24] H.E. Haber, G.L. Kane and T. Sterling, The fermion mass scale and possible e ects of Higgs bosons on experimental observables, Nucl. Phys. B 161 (1979) 493 [INSPIRE]. [25] L.J. Hall and M.B. Wise, Flavor changing Higgs-boson couplings, Nucl. Phys. B 187 (1981) [26] J.F. Donoghue and L.F. Li, Properties of charged Higgs bosons, Phys. Rev. D 19 (1979) 945 [27] V.D. Barger, J.L. Hewett and R.J.N. Phillips, New constraints on the charged Higgs sector in two Higgs doublet models, Phys. Rev. D 41 (1990) 3421 [INSPIRE]. [28] M. Aoki, S. Kanemura, K. Tsumura and K. Yagyu, Models of Yukawa interaction in the two Higgs doublet model and their collider phenomenology, Phys. Rev. D 80 (2009) 015017 [arXiv:0902.4665] [INSPIRE]. conservation vs. minimal avour violation, JHEP 10 (2010) 009 [arXiv:1005.5310] [30] A. Pich and P. Tuzon, Yukawa alignment in the two-Higgs-doublet model, Phys. Rev. D 80 (2009) 091702 [arXiv:0908.1554] [INSPIRE]. quasi-alignment with two Higgs doublets and RGE stability, Eur. Phys. J. C 75 (2015) 286 [32] P.M. Ferreira, L. Lavoura and J.P. Silva, Renormalization-group constraints on Yukawa alignment in multi-Higgs-doublet models, Phys. Lett. B 688 (2010) 341 [arXiv:1001.2561] [33] S. Knapen and D.J. Robinson, Disentangling mass and mixing hierarchies, Phys. Rev. Lett. 115 (2015) 161803 [arXiv:1507.00009] [INSPIRE]. [34] F.J. Botella, G.C. Branco, M. Nebot and M.N. Rebelo, Flavour changing Higgs couplings in a class of two Higgs doublet models, Eur. Phys. J. C 76 (2016) 161 [arXiv:1508.05101] [35] G. D'Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal avor violation: an e ective eld theory approach, Nucl. Phys. B 645 (2002) 155 [hep-ph/0207036] [INSPIRE]. [36] C.B. Braeuninger, A. Ibarra and C. Simonetto, Radiatively induced avour violation in the general two-Higgs doublet model with Yukawa alignment, Phys. Lett. B 692 (2010) 189 [arXiv:1005.5706] [INSPIRE]. U.K., (1999) [INSPIRE]. [37] S. Davidson and H.E. Haber, Basis-independent methods for the two-Higgs-doublet model, Phys. Rev. D 72 (2005) 035004 [Erratum ibid. D 72 (2005) 099902] [hep-ph/0504050] [38] G.C. Branco, L. Lavoura and J.P. Silva, CP violation, Oxford University Press, Oxford [39] H.E. Haber and D. O'Neil, Basis-independent methods for the two-Higgs-doublet model II. The signi cance of tan , Phys. Rev. D 74 (2006) 015018 [Erratum ibid. D 74 (2006) 059905] [hep-ph/0602242] [INSPIRE]. [40] H.E. Haber and D. O'Neil, Basis-independent methods for the two-Higgs-doublet model III. The CP-conserving limit, custodial symmetry and the oblique parameters S, T , U , Phys. Rev. D 83 (2011) 055017 [arXiv:1011.6188] [INSPIRE]. 421 [INSPIRE]. ! e at a rate of one out of 109 muon decays?, Phys. Lett. B 67 (1977) 1103 [INSPIRE]. Rev. Lett. 44 (1980) 912 [INSPIRE]. [42] M. Gell-Mann, P. Ramond and R. Slansky, Complex spinors and uni ed theories, Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE]. [43] T. Yanagida, Horizontal symmetry and masses of neutrinos, Prog. Theor. Phys. 64 (1980) [44] R.N. Mohapatra and G. Senjanovic, Neutrino mass and spontaneous parity violation, Phys. [45] R.N. Mohapatra and G. Senjanovic, Neutrino masses and mixings in gauge models with spontaneous parity violation, Phys. Rev. D 23 (1981) 165 [INSPIRE]. [47] N. Craig, J. Galloway and S. Thomas, Searching for signs of the second Higgs doublet, boson: alignment without decoupling, JHEP 04 (2014) 015 [arXiv:1310.2248] [INSPIRE]. [50] H.E. Haber, The Higgs data and the decoupling limit, in Proceedings, 1st Toyama International Workshop on Higgs as a Probe of New Physics 2013 (HPNP2013), Toyama [51] P.S. Bhupal Dev and A. Pilaftsis, Maximally symmetric two Higgs doublet model with natural Standard Model alignment, JHEP 12 (2014) 024 [Erratum ibid. 11 (2015) 147] [52] A. Pilaftsis, Symmetries for Standard Model alignment in multi-Higgs doublet models, Phys. [arXiv:1408.3405] [INSPIRE]. Rev. D 93 (2016) 075012 [arXiv:1602.02017] [INSPIRE]. (1990) 363 [INSPIRE]. [53] H.E. Haber and Y. Nir, Multiscalar models with a high-energy scale, Nucl. Phys. B 335 [54] J.F. Gunion and H.E. Haber, Conditions for CP-violation in the general two-Higgs-doublet model, Phys. Rev. D 72 (2005) 095002 [hep-ph/0506227] [INSPIRE]. [55] K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE]. [56] F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, arXiv:1703.03751 Chin. Phys. C 40 (2016) 100001 [INSPIRE]. [57] Particle Data Group collaboration, C. Patrignani et al., Review of particle physics, [58] P. Marquard, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark mass relations to four-loop order in perturbative QCD, Phys. Rev. Lett. 114 (2015) 142002 [arXiv:1502.01030] [INSPIRE]. [59] J. Bijnens, J. Lu and J. Rathsman, Constraining general two Higgs doublet models by the evolution of Yukawa couplings, JHEP 05 (2012) 118 [arXiv:1111.5760] [INSPIRE]. [60] M. Jung, A. Pich and P. Tuzon, Charged-Higgs phenomenology in the aligned two-Higgs-doublet model, JHEP 11 (2010) 003 [arXiv:1006.0470] [INSPIRE]. [61] ATLAS collaboration, Constraints on new phenomena via Higgs boson couplings and invisible decays with the ATLAS detector, JHEP 11 (2015) 206 [arXiv:1509.00672] [62] CMS collaboration, Summary results of high mass BSM Higgs searches using CMS run-I data, CMS-PAS-HIG-16-007, CERN, Geneva Switzerland, (2016). [63] A. Arhrib, Higgs bosons decay into bottom-strange in two Higgs doublets models, Phys. Lett. B 612 (2005) 263 [hep-ph/0409218] [INSPIRE]. [64] G. Abbas, A. Celis, X.-Q. Li, J. Lu and A. Pich, Flavour-changing top decays in the aligned two-Higgs-doublet model, JHEP 06 (2015) 005 [arXiv:1503.06423] [INSPIRE]. [65] A. Greljo, J.F. Kamenik and J. Kopp, Disentangling avor violation in the top-Higgs sector at the LHC, JHEP 07 (2014) 046 [arXiv:1404.1278] [INSPIRE]. [66] A. Arhrib, Top and Higgs avor changing neutral couplings in two Higgs doublets model, Phys. Rev. D 72 (2005) 075016 [hep-ph/0510107] [INSPIRE]. [67] G. Eilam, J.L. Hewett and A. Soni, Rare decays of the top quark in the standard and two Higgs doublet models, Phys. Rev. D 44 (1991) 1473 [INSPIRE]. [68] B. Mele, S. Petrarca and A. Soddu, A new evaluation of the t ! cH decay width in the Standard Model, Phys. Lett. B 435 (1998) 401 [hep-ph/9805498] [INSPIRE]. [69] J.A. Aguilar-Saavedra, Top avor-changing neutral interactions: theoretical expectations and experimental detection, Acta Phys. Polon. B 35 (2004) 2695 [hep-ph/0409342] in pp collisions at p [arXiv:1509.06047] [INSPIRE]. Higgs decays to bb pairs at p [70] C. Zhang and F. Maltoni, Top-quark decay into Higgs boson and a light quark at next-to-leading order in QCD, Phys. Rev. D 88 (2013) 054005 [arXiv:1305.7386] [71] ATLAS collaboration, Search for avour-changing neutral current top quark decays t ! Hq s = 8 TeV with the ATLAS detector, JHEP 12 (2015) 061 [72] CMS collaboration, Search for the avor-changing neutral current decay t ! qH where the s = 8 TeV, CMS-PAS-TOP-14-020, CERN, Geneva p [73] CMS collaboration, Search for top quark decays t ! qH with H ! in pp collisions at s = 8 TeV, CMS-PAS-TOP-14-019, CERN, Geneva Switzerland, (2014). [74] S. Gori, Three lectures of avor and CP-violation within and beyond the Standard Model, in 2015 European School of High-Energy Physics (ESHEP 2015), Bansko Bulgaria, 2{15 September 2015 [arXiv:1610.02629] [INSPIRE]. [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 U.S.A., 29 July{6 August 2013 [arXiv:1311.2028] [INSPIRE]. [76] M. Selvaggi, Perspectives for top quark physics at high-luminosity LHC, PoS(TOP2015)054 [arXiv:1512.04807] [INSPIRE]. arXiv:1607.01831 [INSPIRE]. [77] M.L. Mangano et al., Physics at a 100 TeV pp collider: Standard Model processes, [78] W. Altmannshofer, S. Gori and G.D. Kribs, A minimal avor violating 2HDM at the LHC, Phys. Rev. D 86 (2012) 115009 [arXiv:1210.2465] [INSPIRE]. quarks, JHEP 11 (2015) 071 [arXiv:1506.08329] [INSPIRE]. [79] CMS collaboration, Search for neutral MSSM Higgs bosons decaying into a pair of bottom [80] ATLAS collaboration, Search for minimal supersymmetric Standard Model Higgs bosons H=A in the nal state in up to 13:3 fb 1 of pp collisions at p s = 13 TeV with the ATLAS detector, ATLAS-CONF-2016-085, CERN, Geneva Switzerland, (2016). 13 fb 1 of pp collisions at p Switzerland, (2016). pp collisions at p Geneva Switzerland, (2016). [81] CMS collaboration, Search for a neutral MSSM Higgs boson decaying into CMS-PAS-HIG-16-006, CERN, Geneva Switzerland, (2016). [82] CMS collaboration, Search for a high-mass resonance decaying into a dilepton nal state in s = 13 TeV, CMS-PAS-EXO-16-031, CERN, Geneva proton-proton collisions at p s = 13 TeV with the ATLAS detector, ATLAS-CONF-2016-045, CERN, Geneva Switzerland, (2016). [83] ATLAS collaboration, Search for new high-mass resonances in the dilepton nal state using [84] ATLAS collaboration, Search for heavy Higgs bosons A=H decaying to a top-quark pair in s = 8 TeV with the ATLAS detector, ATLAS-CONF-2016-073, CERN, [85] M. Carena, S. Gori, A. Juste, A. Menon, C.E.M. Wagner and L.-T. Wang, LHC discovery potential for non-standard Higgs bosons in the 3b channel, JHEP 07 (2012) 091 [arXiv:1203.1041] [INSPIRE]. [86] BaBar collaboration, J.P. Lees et al., Evidence for an excess of B ! D( ) Phys. Rev. Lett. 109 (2012) 101802 [arXiv:1205.5442] [INSPIRE]. [87] BaBar collaboration, J.P. Lees et al., Measurement of an excess of B ! D( ) and implications for charged Higgs bosons, Phys. Rev. D 88 (2013) 072012 decays, [arXiv:1303.0571] [INSPIRE]. [88] LHCb collaboration, Measurement of the ratio of branching fractions B(B0 ! D + )=B(B0 ! D + ibid. 115 (2015) 159901] [arXiv:1506.08614] [INSPIRE]. ), Phys. Rev. Lett. 115 (2015) 111803 [Addendum [89] Belle collaboration, M. Huschle et al., Measurement of the branching ratio of B ! D( ) relative to B ! D( )` D 92 (2015) 072014 [arXiv:1507.03233] [INSPIRE]. ` decays with hadronic tagging at Belle, Phys. Rev. [90] Belle collaboration, A. Abdesselam et al., Measurement of the branching ratio of B0 ! D + relative to B0 ! D +` ` decays with a semileptonic tagging method, arXiv:1603.06711 [INSPIRE]. [91] A. Abdesselam et al., Measurement of the lepton polarization in the decay B ! D , arXiv:1608.06391 [INSPIRE]. (2015) 054018 [arXiv:1506.08896] [INSPIRE]. [92] M. Freytsis, Z. Ligeti and J.T. Ruderman, Flavor models for B ! D( ) , Phys. Rev. D 92 [93] F. Mahmoudi and O. Stal, Flavor constraints on the two-Higgs-doublet model with general Yukawa couplings, Phys. Rev. D 81 (2010) 035016 [arXiv:0907.1791] [INSPIRE]. [94] Quark Flavor Physics Working Group collaboration, J.N. Butler et al., Working group report: quark avor physics, arXiv:1311.1076 [INSPIRE]. [95] A. Bevan et al., Standard Model updates and new physics analysis with the unitarity triangle t, arXiv:1411.7233 [INSPIRE]. [96] X.-Q. Li, J. Lu and A. Pich, Bs0;d ! `+` JHEP 06 (2014) 022 [arXiv:1404.5865] [INSPIRE]. decays in the aligned two-Higgs-doublet model, [97] L. Martini, Search for new physics in the B meson decays: B(0s) ! , Nuovo Cim. C 39 (2016) 231 [INSPIRE]. Bs;d ! `+` in the Standard Model with reduced theoretical uncertainty, Phys. Rev. Lett. 112 (2014) 101801 [arXiv:1311.0903] [INSPIRE]. (2012) 121 [arXiv:1206.0273] [INSPIRE]. Symmetries and asymmetries of B ! K JHEP 01 (2009) 019 [arXiv:0811.1214] [INSPIRE]. decays in the Standard Model and beyond, decay from the combined analysis of CMS and LHCb data, Nature 522 (2015) 68 [arXiv:1411.4413] collected during the LHC run 1 with the ATLAS detector, Eur. Phys. J. C 76 (2016) 513 branching fraction and e ective decays, Phys. Rev. Lett. 118 (2017) 191801 CERN-LHCC-2015-010, CERN, Geneva Switzerland, (2015). b-hadron, c-hadron and -lepton properties as of summer 2014, arXiv:1412.7515 [INSPIRE]. [hep-ph/0406184] [INSPIRE]. at Belle and Belle II, PoS(ICHEP2016)131 [arXiv:1701.02288] [INSPIRE]. [arXiv:1309.2293] [INSPIRE]. [15] R. Costa , A.P. Morais , M.O.P. Sampaio and R. Santos , Two-loop stability of a complex singlet extended Standard Model , Phys. Rev. D 92 ( 2015 ) 025024 [arXiv: 1411 .4048] [16] N. Chakrabarty , U.K. Dey and B. Mukhopadhyaya , High-scale validity of a two- Higgs [31] F.J. Botella , G.C. Branco , A.M. Coutinho , M.N. Rebelo and J.I. Silva-Marcos, Natural the decoupling limit , Phys. Rev. D 67 ( 2003 ) 075019 [ hep -ph/0207010] [INSPIRE]. [48] D.M. Asner et al., ILC Higgs white paper , in Proceedings, Community Summer Study 2013 : Snowmass on the Mississippi (CSS2013), Minneapolis MN U.S.A ., 29 July{6 August 2013 [49] M. Carena , I. Low, N.R. Shah and C.E.M. Wagner , Impersonating the Standard Model Higgs [98] C. Bobeth , M. Gorbahn , T. Hermann, M. Misiak , E. Stamou and M. Steinhauser , [99] W. Altmannshofer and D.M. Straub , Cornering new physics in b ! s transitions , JHEP 08 [100] W. Altmannshofer , P. Ball , A. Bharucha , A.J. Buras , D.M. Straub and M. Wick , [107] A. Crivellin , A. Kokulu and C. Greub , Flavor-phenomenology of two-Higgs-doublet models


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29110.pdf

Stefania Gori, Howard E. Haber, Edward Santos. High scale flavor alignment in two-Higgs doublet models and its phenomenology, Journal of High Energy Physics, 2017, 1-52, DOI: 10.1007/JHEP06(2017)110