The flavor-locked flavorful two Higgs doublet model

Journal of High Energy Physics, Mar 2018

Wolfgang Altmannshofer, Stefania Gori, Dean J. Robinson, Douglas Tuckler

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The flavor-locked flavorful two Higgs doublet model

HJE avor-locked avorful two Higgs doublet model Wolfgang Altmannshofer 0 1 Stefania Gori 0 1 Dean J. Robinson 0 1 Douglas Tuckler 0 1 0 Cincinnati , Ohio 45221 , U.S.A 1 Department of Physics, University of Cincinnati We propose a new framework to generate the Standard Model (SM) quark avor hierarchies in the context of two Higgs doublet models (2HDM). The ` avorful' 2HDM couples the SM-like Higgs doublet exclusively to the third quark generation, while the rst two generations couple exclusively to an additional source of electroweak symmetry breaking, potentially generating striking collider signatures. We synthesize the avorful 2HDM with the ` avor-locking' mechanism, that dynamically generates large quark mass hierarchies through a avor-blind portal to distinct avon and hierarchon sectors: dynamical alignment of the avons allows a unique hierarchon to control the respective quark masses. We further develop the theoretical construction of this mechanism, and show that in the context of a avorful 2HDM-type setup, it can automatically achieve realistic avor structures: the CKM matrix is automatically hierarchical with jVcbj and jVubj generically of the observed size. Exotic contributions to meson oscillation observables may also be generated, that may accommodate current data mildly better than the SM itself. Beyond Standard Model; Higgs Physics; Quark Masses and SM Parameters - The 1 Introduction 3.1 3.2 3.3 3.4 4.1 4.2 4.3 2 Review of the avorful 2HDM 3 Flavor-locking with one and two Higgs bosons Yukawa portal General avon potential and vacuum Flavor-locked Yukawas Two-Higgs avor-locking 4 Flavor violation and phenomenology Physical parameters CKM phenomenology Constraints from meson mixing 5 Conclusion and outlook A Analysis of the general avon potential A.1 General avon potential A.2 A.3 Mixing terms: single avon generation Mixing terms: arbitrary avon generations A.4 Local minimum analysis A.5 Two-Higgs alignment conditions B Flavor basis for the F2HDM Yukawa texture to top quarks has been found to be SM-like with approximately 15% uncertainty [1]. More recently, analyses of 36 fb 1 of Run II LHC data have provided evidence for the decay of the Higgs boson into a pair of b quarks with a branching fraction consistent with the SM expectation [2, 3]. Taken together, these results imply that the main origin of the masses of the weak gauge bosons and third generation fermions is the vacuum expectation value (vev) of the 125 GeV SM-like Higgs. { 1 { However, it is not known whether the vacuum of the SM Higgs eld is (solely) responsible for the generation of all the elementary fermion masses. So far, the h ! branching fraction is bounded by a factor of 2:6 above the SM prediction [ 4, 5 ]. With 300 fb 1 of data, the SM partial width for this decay mode will be accessible at LHC, and it could be measured with a precision of 8% at the High-Luminosity LHC (HL-LHC) [6{8]. The h ! cc rate is more di cult to access at the LHC. At present, the most stringent bound arises from the ATLAS search for Zh; h ! cc, exploiting new c-tagging techniques, and only probes the branching fraction down to 110 times the SM expectation [9]. Studies of future prospects for the HL-LHC have shown that LHCb may be able to set a stronger bound on the hcc coupling, at the level of 4 times the SM expectation [10]. The charm coupling may be determined more precisely at future colliders, such as e+e machines [11], as well as proton-electron colliders [12]. Finally, because of their tiny values, the SM Higgs couplings to the other light quarks, as well as the electron, are even more challenging to measure and will likely remain out of reach for the foreseeable future [13{22]. Signals that would provide immediate evidence for a beyond SM Higgs sector, such as h ! t ! ch, have branching fractions that are constrained to be less than few avor violation, such as h ! or b ! s and large branching ratios for t ! ch, as well as heavy Higgs or pseudoscalar decays H=A ! cc, tc, , and charged Higgs decays H ! bc, sc, . A di erent approach to resolving the SM mass hierarchy puzzle can be achieved with a dynamical alignment mechanism [33] | we refer to it as ` avor-locking' | in which the quark (or lepton) Yukawas are generated by the vacuum of a general avon potential, that introduces a single avon eld and a single `hierarchon' operator for each quark avor. (A detailed review follows below; see also refs. [34, 35] for a related, but intrinsically di erent approach, as well as refs. [36{39].) In this vacuum, the up- and down-type sets of avons are dynamically locked into an aligned, rank-1 con guration in the mass basis, so that each SM quark mass is controlled by a unique avon. Horizontal symmetries between the hierarchon and avon sectors in turn allow each quark mass to be dynamically set by a unique hierarchon vev. This results in a avor blind mass generation mechanism | the quarks themselves carry no avor symmetry beyond the usual U(3)Q;U;D | so that the quark mass hierarchy can be generated independently from the CKM quark mixing hierarchy, by physics that operates at scales generically di erent to | i.e. lower than | the scale of the avon e ective eld theory. In a minimal set up that features only a single SM-type Higgs, however, the CKM mixing matrix is an arbitrary unitary matrix, so that the quark mixing hierarchy itself remains unexplained. { 2 { In this work we synthesize these two approaches to the avor puzzle with the following observation: a dynamical realization of an F2HDM-type avor structure can be generated by applying the avor-locking mechanism to its Yukawas. Or alternatively: in a avorlocking scheme for the generation of the quark mass hierarchy, introducing a second Higgs doublet with F2HDM-type couplings generically produces quark mixing hierarchies of the desired size. In particular, we show that in such a setup, the 1{3 and 2{3 quark mixings are automatically produced at the observed order, without the introduction of tunings. The avor structure of this theory generically leads to tree-level contributions from heavy Higgs exchange to meson mixing observables, that vanish in the heavy Higgs in nite mass limit. However, for heavy Higgs masses at collider-accessible scales, we show these contributions may be consistent with current data, and in some cases may accommodate the current data mildly better than the SM. This paper is structured as follows. In section 2 we brie y review the general properties of the F2HDM and its avor structure. In section 3 we develop the avor-locking mechanism for F2DHM-type theories, including a review of the minimal single Higgs version. We then proceed to explore the generic avor structure of the avor-locked F2HDM in section 4, discussing both the generation of the CKM mixing hierarchies and constraints from meson mixing. We conclude in section 5. Technical details concerning the analysis of the avon potential are given in appendices. 2 Review of the avorful 2HDM The F2HDM, as introduced in refs. [27, 32], is a 2HDM in which one Higgs doublet predominantly gives mass to the third generation of quarks and leptons, while the second Higgs doublet is responsible for the masses of the rst and second generation of SM fermions, as well as for quark mixing. The most general Yukawa Lagrangian of two Higgs doublets with hypercharge +1=2 can be written as LY = X hYiuJ (QiLH~1URJ ) + Yi0Ju(QiLH~2URJ )i + X hY d (QiLH1DRJb) + Y 0d(QiLH2DRJb)i i;Jb iJb i;J i;Jb + X hYi`Jb(LiLH1ERJb) + Y 0`(LiLH2ERJb)i + h.c. ; iJb with two Higgs doublets H1 and H2 coupling to the left-handed and right-handed quarks (QL, UR, DR) and leptons (LL and ER), and H~ H . The indices i = 1; 2; 3 and J; Jb = 1; 2; 3 label the three generations of SU(2) doublet and singlet elds, respectively. We focus on quark Yukawas hereafter, but the general results of this discussion apply equally to the lepton Yukawas in eq. (2.1). The two Higgs doublets decompose in the usual way 0 0 p (v sin p (v cos 2 2 G+ sin H+ cos + h cos + H sin + iG0 sin G+ cos + H+ sin h sin + H cos + iG0 cos iA cos )A ; + iA sin )A ; { 3 { iJb 1 1 (2.1) (2.2) (2.3) where v = 246 GeV is the vacuum expectation value of the SM Higgs, G0 and G are the Goldstone bosons that provide the longitudinal components for the Z and W bosons, h and H are physical scalar Higgs bosons, A is a physical pseudoscalar Higgs boson, and H are physical charged Higgs bosons. The angle parametrizes diagonalization of the scalar Higgs mass matrix and tan is the ratio of the vacuum expectation values of H1 and H2. The scalar h is identi ed with the 125 GeV Higgs boson. The overall mass scale of the is a free parameter. The mass splitting among them is at In refs. [27, 32] the following textures of the two sets of Yukawa couplings Y and Y 0 `heavy' Higgs bosons H; A; H most of order O(v2=mH;A;H ). were chosen, (2.4a) (2.4b) HJEP03(218)9 (2.5a) (2.5b) Y 0u Y 0d Y 0u Y 0d v cos p p 2 2 v cos v cos p p 2 2 v cos 0mu mu mu 1 mu mc mc 0md Bmd ms md ms ms 3mb1 2mbC ; A ms 0mu 0md v sin p p 2 2 v sin v sin p p 2 2 v sin 0 B where each entry in the Y 0u; Y 0d Yukawas is multiplied by a generic O(1) coe cient. This structure naturally produces the observed quark masses as well as CKM mixing angles. In this work, we will focus on the dynamical generation of Yukawas of a similar form, with the schematic structure in which Uu;d and Vu;d are unitary matrices. These Yukawas will similarly produce the observed quark mass hierarchies and CKM mixing (see section 4 below), and the collider phenomenology of both Yukawa structures is expected to manifest in the same set of signatures. The F2HDM setup exhibits a very distinct phenomenology, that di ers signi cantly from 2HDMs with natural avor conservation, avor alignment, or minimal avor violation [40{44]. The couplings of the 125 GeV Higgs are modi ed in a avor non-universal eration are approximately SM like, the couplings to the rst and second generation can still deviate from SM expectations by an O(1) factor. Also, the heavy Higgs bosons H, A, and involving second generation quarks can be relevant and sometimes even dominant [32]. One important aspect of the Yukawa structures in eqs. (2.4) and eqs. (2.5) is that they imply tree-level avor changing neutral Higgs couplings. The avor-violating couplings of the 125 GeV Higgs vanish in the decoupling/alignment limit, i.e. for cos( ) = 0. However, avor-violating couplings of the heavy Higgs bosons persist in this limit and they are proportional to tan . Therefore, for large tan and heavy Higgs boson masses below the TeV scale, avor violating processes, such as meson mixing, constrain the F2HDM parameter space. Note that the rank-1 nature of the third generation Yukawas, Y , preserves a U(2)5 avor symmetry acting on the rst and second generation of fermions. This anism that realizes the avor structure in eqs. (2.4) or (2.5) has not been explicitly constructed so far. We now discuss how the avor structure (2.5) can be dynamically generated by the avor-locking mechanism, and, conversely, how a F2HDM-type theory permits the avor-locking mechanism to generate realistic avor phenomenology. (Alternatively, one may also attempt to generate the Yukawas (2.4) with horizontal symmetries directly on the SM quarks. We do not follow this approach here.) We rst review the minimal single Higgs doublet version of the avor-locking mechanism, followed by the generalization to a theory with two Higgs doublets in section 3.4. As we will discuss, while in the presence of only one SM-like Higgs doublet, the predicted quark mixing angles are generically of O(1), introducing a second Higgs doublet leads to a theory with suppressed jVcbj and jVubj. 3.1 Yukawa portal The underlying premise of the avor-locking mechanism [33] is that the Yukawas arise from a three-way portal between the SM elds (the quarks QL; UR; DR and the Higgs H), a set of ` avon' elds, , and a set of `hierarchon' operators, s: LY i QL iJ s F H H~ URJ + QiL biJb sb HDRJb : F H (3.1) The 's are bifundamentals of the appropriate U(3)Q = u; c; t and b = d; s; b, denote an arbitrary transformation property under a symmetry or set of symmetries, G and Gb, that enforces the structure of eq. (3.1). In the original avor-locking study [33], G a set of discrete Zqpq or U(1)q `quark avor number' symmetries, for q = d; s; b; u; c; t. Here, Gb was chosen to be 1We always distinguish down-type indices from up-type indices with a hat, and similarly for down-type versus up-type avon couplings and operators. { 5 { b b we similarly choose each avon ( ) to be charged under a gauged U(1) (U(1) ), but assert a S3 permutation symmetry among the up (down) avons and the corresponding b U(1) (U(1) ) gauge bosons, xing the gauge couplings g = g (g b = gb). Compared to the analysis of ref. [33] the permutation symmetry produces a convenient, higher symmetry for the avon potential, such that con gurations with the structure of eqs. (2.5) can be shown to be at its global minimum, as we will discuss in the next subsection. Note that the SM elds are not charged under the G Gb symmetry. The hierarchons s should be thought of as some set of scalar operators that eventually obtain hierarchical vevs, that break the S3 symmetries in the up and down sectors. This hierarchy will be responsible for the quark mass hierarchy, independently from any avor structure. It should be emphasized that the operators s and s do not carry the quark U(3)Q avor symmetries, i.e., they do not carry avor indices i; J; Jb. Moreover, H need not be the same as the avon scale F, and can generically be much lower. (This could permit, in principle, collider-accessible hierarchon phenomenology, b depending on the UV completion of the hierarchon sector, though we shall not consider such possibilities in this work.) In the remainder of this section, we present the general avor structures that this type of portal dynamically produces. Details of this analysis, including the identi cation of global or local minima of the avon potential, and the algebraic structure of the associated vacua, are presented in appendix A. The spontaneous breaking of continuous symmetries by the avon vacuum can result in a large number of Goldstone bosons. We assume that mechanisms are at work that remove the Goldstone bosons from the IR. 3.2 General avon potential and vacuum for this theory then transform as To generalize beyond the three avors of the SM, we contemplate a theory of N avors of up and down type quarks each, QiL; URJ ; DRJb with i; J; Jb = 1; : : : ; N , charged under the symmetry U(N )Q U(N )U U(N )D. We introduce n N pairs of avons b = 1; : : : ; n, that generate Yukawa couplings to the quarks as in eq. (3.1). The avons b , with HJEP03(218)9 N N We suppress hereafter the U(N )Q U(N )U;D indices, keeping in mind that matrix products only take the form y or y , and correspondingly in the down sector. Up-down matrix products can only take the form y b or y b , but not b y nor b y . The most general, renormalizable and CP conserving potential for the avons can then be written in the form V = X V1f + X V2f + X V1bf + X V2bf b + X Vmibx : < b<b ; b b { 6 { (3.2) (3.3) Here, the single and pairwise eld potentials are V1f = 1 Tr V2f = 3 Tr and similarly for V1bf and V2bf b, hatting all coe cients (the labeling and notation follows the choices of ref. [33]). Note that the pairwise potentials respect the U(1) and U(1) symmetries. The mixed potential is b The Sn symmetry ensures that all potential coe cients are the same for all elds ; ; ; b singly and pairwise. All i and i coe cients, as well as r and rb, are real and are chosen b A detailed analysis of the global minimum of this potential is provided in appendix A. One nds that, provided 6;2 b the potential has a global minimum if and only if the avons have the vacuum con guration 0 r C V y ; . . . CA Vb y ; b h 2i = U B r 0 C V y ; b . . AC Vb y ; r . : : : : : : with U , V , Ub , Vb unitary matrices | crucially, the matrices U , V (Ub , Vb ) are the same for all ( ) | and the CKM mixing matrix has the form Vckm = U yUb = 0 Vn n block CKM rotations are at directions of the global minimum, and therefore Vn and VN n may be any arbitrary unitary submatrices with generically O(1) entries. We refer to the con guration in eqs. (3.8) and (3.9) as being ` avor-locked'. { 7 { Flavor locking ensures that the Yukawa portal in (3.1) becomes, in the n = N = 3 case sb= H 1 A ss= H CHDR ; (3.10) under a suitable unitary rede nition of the QL; UR and DR elds. From these expressions, taking the natural choice r; rb hs i, that generates the quark mass hierarchies, i.e. hs i = H F, it is clear that it is the physics of the hierarchon vev's, y , the quark Yukawa for avor . This physics may operate at scales vastly di erent to the avor breaking scale, F. In eq. (3.10) the CKM matrix Vckm is an arbitrary 3 3 unitary matrix. One might wonder if additional terms in the avon potential of (3.3) can destabilize the vacuum identi ed above. In particular, avon-hierarchon couplings of the form ]sy s ) may be present, which can produce (mixed) mass terms that disrupt the Vmix (V2f) vacuum once the hierarchons, s , obtain vev's. Mixed mass terms may disrupt the alignment between the di erent h i, while additional mass terms induce splittings in the radial mode masses, so that the block CKM rotations are no longer at directions of the vacuum. In the UV theory, the operator product of two hierarchons with two avons may, however, be vanishingly small, e.g. if the hierarchons are composite operators in di erent sectors. Nonetheless, such terms are necessarily generated radiatively by the Yukawa portal (3.1). One may construct UV completions in which this occurs rst at the two-loop level, with the (mixed) mass contributions being log-divergent. For example, let us consider a theory containing a avored fermion i and a scalar , with interactions iJ iURJ + i QL i + y s H~ ; with m F and m H. This produces the Yukawa portal (3.1) via sα ˜ H λα Φα χα ¯ QL UR m2 2 y y 2F (16 2)2 log( H= F) r2 ; H { 8 { As hs i= H y , the quark Yukawa for avor , the corresponding (mixed) mass term for the avons is generated at two-loops by mirroring the diagram in (3.12). One nds (3.11) (3.12) (3.13) once again taking the natural choice r F. A suitable hierarchy between H and combined with the two-loop suppression, renders these terms arbitrarily small. Hence one may safely neglect these terms. Motivated by the avorful 2HDM, now we turn to consider a Yukawa potential with two Higgs elds: one that couples to the third generation, and one to the rst two generations. That is, in which we have suppressed the quark avor indices. With reference to the UV completion (3.11), one can imagine that this generational structure comes about as a consequence of t, st, and H1 belonging to a di erent UV sector (or brane) than c;u, sc;u, and H2, so that terms of the form tstH~2 or c;usc;uH~1 are heavily suppressed in the e ective eld theory. Similarly, one can also generate this structure via adding an additional symmetry to sc;u, ss;d and H2 such that sc;uH~2 and sd;sH2 are singlets. Such tstH~2 or c;usc;uH~1 terms (symmetries) will, ultimately, be generated (softly broken) via the 2H1H2y or 5H1H2yH1H2y terms in the Higgs potential, at least one of which is necessary to avoid a massless Goldstone boson. While these terms modify the phenomenology of both the Higgs bosons [32], corrections to the structure of the avon potential may arise only at high loop order (as discussed shortly below). The generational structure implies that cross-terms between the third and rst two generations in the avon potential (3.3) now vanish, and that the S3 avon-hierarchon symmetry has been replaced with a Z2 for just the two light generations. That is, the coe cients of the heavy and light avon potentials are no longer related, and the heavylight potentials V2tf , V2bfb, Vmt bix, Vmb ix vanish, for own, independent, and suppressed coe cients, identical for = c; u and b = s; d (or they obtain their = c; u and b = s; d). One then also expects the rotation matrices entering in the vacuum con guration of the avons of the rst two generations to be di erent from those of the third, breaking the heavy-light alignment conditions. Put a di erent way, we may write the full potential in the form V = V ;h + V ;l (3.15) in which the `h' and `l' pieces of the potential each have the form of the full potential (3.3), but for one heavy and two light generations, respectively. With reference to the UV completion (3.11), terms for a heavy-light mixing potential are generated radiatively by the 2H1H2y or 5H1H2yH1H2y portals combined with the Yukawas (3.14) only at the ve-loop level, along with 4 = 4F or 25 4 v = 4F factors, respectively. As such, the induced Goldstone mass may be at the electroweak scale or higher, while the induced terms for the heavy-light mixing potential remain negligible. The potentials V ;h and V ;l each have a N = 3 avor-locked vacuum, with generation number n = 1 and n = 2, respectively. Provided the conditions (3.7) are satis ed for each { 9 { potential, this leads to the vacuum structure CA Vty ; CA Vbby ; 0 0 0 0 0 0 1 1 A b CA Vb y ; r r 0 r b 0 0 0 0 1 1 A CA Vb y : We call this a `1 + 2' avor-locked vacuum. Note that the rotation matrices for the third generation quarks (Ut; Vt; Ubb; Vbb) di er in general from the corresponding rotations for the rst and second generation quarks. For the 1 + 2 avor-locked structure (3.16), the CKM structure of the global minimum in eq. (3.9) enforces U yUb and UtyUbb to each be 2 1 block unitary, i.e. U yUb = V2 1 ! ; UtyUbb = W2 ! 1 ; where V2 and W2 are 2 2 unitary matrices (see appendix A.3). The 2 1 block unitarity permits one to rotate away the tb unitary matrices, so that the Yukawa potential (3.14) (3.16) (3.17) attains the form 1 A C H~1 + U B r A 1 0 zb 3 5 C V yH~27 UR A C H1 + U V2 1 ! 0 B (3.18) with z rede ned to absorb the other unitary matrices, such that eq. (3.17) is still satis ed, and we = hs i= H and z b = hsbi= H. The unitary matrices U , V and Vb have been have written Ub = U diagfV2; 1g accordingly. Matching the structure of eq. (2.5), eq. (3.18) is the key result of this section: the dynamical generation of hierarchical aligned third generation Yukawas, and hierarchical aligned rst two generation Yukawas. An additional feature, not present in eq. (2.5), is that the up- and down-type light Yukawas are aligned up to an overall mixing angle on the left. The mixing angle is a at direction of the avon potential and therefore generically of O(1). 4 Flavor violation and phenomenology We now turn to examine the phenomenology of avor-violating processes generated by the Yukawa structure in eq. (3.18). If one treats the SM as a UV complete theory, then the quark sector alone naively features multiple tunings towards the in nitesimal: ve for the masses of all quarks except the top, and two for the small size of jVcbj and jVubj. In the minimal or F2HDM-type avor-locking scenarios, the quark mass hierarchies no longer require such tunings, as they can be generated dynamically by hs i. We show below that the structure of eq. (3.18) also characteristically produces 1{3 and 2{3 quark generation mixing comparable to the observed size of jVcbj and jVubj, without requiring ad hoc suppression of the underlying parameters. In this sense of counting tunings, the avor-locked F2HDM is a more natural theory of avor than the SM. Additionally, for the avor structure (3.18), the heavy Higgs bosons may remain light enough to be accessible to colliders, i.e. with a mass of a few hundred GeV, while not introducing unacceptably large tree-level contributions to meson mixing observables. In some regions of parameter space, these additional contributions better accommodate the current data than the SM. We explore the nature of such contributions below. Starting from the general structure of eq. (3.18), which has already selected the direction of the H1-generated component of the third generation, the Q, U and D quarks have a maximal U(2)3 U(1) avor symmetry, which breaks to baryon number. This corresponds to 3 real and 9 imaginary broken generators. The up-type Yukawa in eq. (3.18) has a total of 3 + 3 + 3 = 9 real parameters (zt;u;c, and the SO(3) rotations of U and V ) and 6+6 2 2 = 8 imaginary parameters (the phases of U and V , less the phases commuted or annihilated by the rank-2 diagonal matrix). The down-type Yukawa, excluding parameters already contained in U , has 3 + 1 + 3 = 7 real parameters (zb;d;s, and the SO(2) and SO(3) rotations of V2 and Vb , respectively) and 3 + 6 2 1 = 6 imaginary parameters (the phases of V2 and Vb , less the phases commuted or annihilated by the rank-2 diagonal matrix). This counting implies that the total number of physical parameters is 9 + 8 + 7 + 6 corresponding to 6 masses, 7 angles and 5 phases. To see this explicitly, we write a general 3 3 unitary matrix in the canonical form + QL b 6B t 0 0 B zdei d with RU rotation matrices in the 3 3 avor space, and 12; 13; 23 and ; 1;2;4;5;6 generic angles and phases, respectively. Here the indices of the angles label the 2 2 rotations. After rede ning several phases, we obtain the parametrization zcei c C RVy (#23)RVy (#13; 0)H~27 UR 0ei 1 U = B 0ei 4 ei 6 1 C ; A C H1 + RU ( 13; 0)RU ( 23) B@ A zsei s C RVyb (#b23)RVyb (#b13; 0)H27 DR : A 5 0 zuei u 0ei m 1 0 3 0 1 C A 1 A R( ) 1 ! 1 1 1 0 (4.1) 3 5 (4.2) There is a avor basis in which the above parametrization reproduces the F2HDM textures shown in (2.4), with coe cients that depend on the several angles ; #; #b. In appendix B we show explicitly how to rotate into this avor basis. and Vb . The quark mixing matrix of the full theory, however, is no longer a at direction: it is lifted by the 1+2 avor-locked structure to an O(1) 2 1 block form with all other entries suppressed by small ratios of quark masses. Diagonalizing the quark mass matrices resulting from (4.2), one nds the following schematic predictions for the CKM matrix elements O(ms=mb)CA ; 1 1 O(md=mb) O(ms=mb) where is the rotation angle in the V2 matrix (see eq. (4.2)), that is a priori a free parameter of O(1). This structure suggests that the observed CKM hierarchies can be accommodated: the 1{3 and 2{3 mixing elements are automatically suppressed at a level that resembles the experimental values. In the decoupling/alignment limit cos( ) = 0, avor-violating processes from heavy Higgs exchange vanish in the large mH;A limit. However, from eqs. (4.2) and (4.3) it is not obvious whether the avor structure of the 1 + 2 avor-locked con guration reduces to the SM in an appropriate limit. As a demonstration that the 1+2 avor-locked con guration is compatible with data, we heuristically identi ed the following example input parameters, r v1 z t F F zb b p r v1 p 2 ' 173 GeV ; 2 ' 4:8 GeV ; r v2 zc zs b p r v2 F F p 2 ' 1:9 GeV ; 2 ' 240 MeV ; r v2 zu zd b p r v2 F F p 2 ' 7 MeV ; 2 ' 21 MeV ; 13 ' m = 0, where we have de ned the two vevs, v1 v cos and v2 u; c ; m are set to zero for simplicity, as they have negligible impact on all the observables that we are considering. (The phases u; c enter in D0{D0 mixing, but, as we will discuss in section 4.3, they are only very weakly constrained.) This parameter set leads to the theoretical predictions shown in table 1 for the six quark masses and a set of ve CKM elements. We compare these predictions to data for the quark masses and CKM parameters, shown in table 1. To be self-consistent, we use data only from processes that are insensitive to heavy Higgs exchange, i.e. processes that are tree-level in the SM. (Since we are ultimately interested in considering the phenomenology of collider-accessible heavy Higgs bosons, loop-level processes in the SM will receive corrections from heavy Higgs exchanges, but measurements of tree-level processes will be insensitive to these e ects.) To reproduce (4.3) (4.4a) (4.4b) v sin . mt mb mc ms mu md 173:5 1:5 GeV 4:8 1:7 100 2:0 5:0 0:5 GeV 0:2 GeV 10 MeV 2:0 MeV 5:0 MeV Benchmark ' 173 GeV ' 4:8 GeV ' 1:7 GeV ' 100 MeV ' 2 MeV ' 5 MeV 0:225 0:023 jVusj jVcdj jVcbj (40:5 4:1) 10 3 jVubj (4:1 0:4) values correspond to the measured quark masses [45] and CKM parameters [46, 47]. All CKM parameters and the b, c, and s quark masses are assigned 10% uncertainties. In the case of the top mass we use a 1.5 GeV uncertainty, while for the up and down masses we use 100% uncertainties. Also shown are predictions corresponding to the benchmark point (4.4). the Cabibbo angle C ' 0:22506 needs to be constrained accordingly to a narrow O(1) range. Since we require only a mixing matrix with canonical entries of the same characteristic size as observed in Nature, we do not insist on such a narrow range for . Similarly, for comparison of the theoretical predictions to data, instead of using the experimental uncertainties of the observables (which in some cases are measured with remarkable precision), we choose 10% uncertainties for all CKM parameters and the bottom, charm, and strange masses. In the case of the top mass we chose a 1.5 GeV uncertainty, while for the up and down masses we use 100% uncertainties. Using these values, the theoretical predictions for the benchmark point (4.4) are in excellent agreement with the observed quark masses and CKM parameters. To quantify the \goodness" of the benchmark or other points in the parameter space, we construct a 2-like function, Xt2ree, for the six quark masses and CKM elements measured from tree-level processes, Xt2ree = X i=u;c;t;d;s;b " (miFL mi)2 # ( mi )2 + X i=us;cd;cb;ub " (jVijFL jVij)2 # ( Vi )2 + ( FL ) 2 ( )2 : (4.5) where the `FL' superscript denotes the theory prediction at a given point in the avorlocked theory parameter space (4.2), and we treat the uncertainties as uncorrelated. While such a X2 function implies a well-de ned p-value for a goodness-of- t of the quoted data to a given theory point, one cannot construct from X2 a sense of the probability for a given theory to produce the observed avor data and hierarchies. Instead, the X2 function allows us only to understand whether or not the avor-locked con guration results generically in a avor structure that agrees with observation at the level of tens of percent. In gure 1 we show the Xt2ree behavior of the avor model on various two-dimensional parametric slices in the neighborhood of the benchmark point (4.4), which is denoted by the white circle. That is, in each plot, all the theory parameters are xed to the benchmark values in eqs. (4.4), except for the two parameters corresponding to the plot axes. The number of degrees of freedom (dof) in the Xt2ree statistic is then 11 2 = 9. The contours parameter space in the neighborhood of the benchmark point (4.4). Contour values are labeled in black; the benchmark point (4.4) is shown by the white circle. show regions of Xt2ree=dof that lead to an overall good agreement between the observed quark masses and CKM parameters and those predicted in the model. As can be seen from the plots in gure 1, there are extended regions of parameter space where there is fairly good agreement between the theory predictions and the measured quark masses and CKM parameters. In particular, O(1) variations of the mixing angles 13; 23; #13; #23; #b13; #b23 around the benchmark point are possible, without worsening the agreement substantially. Only the angle that sets the Cabibbo angle is strongly constrained and has to be set to a narrow range by hand. This behavior should be contrasted to the SM, for which two CKM mixing angles | i.e. the suppressed 1{3 and 2{3 mixings | have to be tuned small. 4.3 As mentioned above and in section 2, the neutral Higgs bosons of the F2HDM setup generically have avor violating couplings. In particular, their tree-level exchange will contribute to meson oscillations. For kaon oscillations the corresponding new physics (NP) contribution to the mixing amplitude is given by The m0 parameters are the o -diagonal entries of the contribution to the down quark mass matrix from the H2 doublet in the quark mass eigenstate basis, and are fully determined by the parameters entering the 1 + 2 avor-locked Yukawas (4.2). The NP mixing amplitude also depends on the heavy Higgs masses mH and mA, the ratio of the two Higgs vacuum expectation values tan and the scalar mixing angle . As additional parametric input in eq. (4.6), we have the kaon decay constant fK ' 155:4 MeV [48]. The bag parameters B2 ' 0:46, B3 ' 0:79, B4 ' 0:78 are evaluated at the scale K = 3 GeV and are taken from ref. [49] (see also refs. [50, 51]). The parameters i encode renormalization group running e ects. From 1-loop RGEs we nd 2 ' 0:68 ; 3 ' 0:03 ; 4 = 1 : (4.7) The relevant observables that are measured in the neutral kaon system are the mass di erence MK and the CP violating parameter K . The experimental results and the corresponding SM predictions and uncertainties are collected in table 2. In terms of the NP mixing amplitude, these observables are given by MK = M KSM + 2Re(M1N2P) ; K = SKM + p 2 Im(M1N2P) MK : (4.8) In the expression for K we use = 0:94 [52] and the measured value of MK shown in table 2. In the case of neutral B meson oscillations, we nd it convenient to normalize the NP mixing amplitude directly to the SM amplitude. For Bs mixing we nd s 2 m2H 1 m2A (m0bs )2 + (m0sb)2 # mb2(VtbVts)2 : (4.9) + c 2 m2 h 1 m2A + A completely analogous expression holds for Bd oscillations. The SM loop function S0 ' 2:3, and the i factors contain QCD running as well as ratios of hadronic matrix elements. At SM Prediction the NP contributions corresponding to the benchmark point (4.4). MK and its uncertainty refers to the short distance contribution. To account for 0:5) in our numerical analysis. Also shown are where the rst (second) value corresponds to Bs (Bd) mixing. To obtain these values we used bag parameters from ref. [56] (see also ref. [55]). The meson oscillation frequencies and the phases of the mixing amplitudes are given by Ms = Md = MsSM MdSM 1 + 1 + M1N2P M1S2M ; M1N2P M1S2M ; s = 2 s + Arg 1 + d = 2 + Arg 1 + M1N2P M1S2M M1N2P M1S2M : ; (4.11) (4.12) The experimental results and the corresponding SM predictions and uncertainties for the observables are collected in table 2. Note that the NP contributions to the kaon and B meson mixing amplitudes (4.6) and (4.9) vanish in the decoupling limit cos( ) = 0, mA; mH ! 1. The NP e ects in D0{D0 oscillations are suppressed by the tiny up quark mass. We have explicitly checked that D0{D0 oscillations do not lead to relevant constraints. In the case that the heavy Higgs masses are below the TeV scale, the NP e ects in the mixing observables do not vanish, and we proceed to investigate the size of such e ects. For the following numerical study, we will set the heavy Higgs masses to a benchmark value, mH = mA = 500 GeV. We use a moderate value of tan = 5, and work in the alignment limit = =2. For the benchmark parameters in eq. (4.4), we show the NP contributions to meson mixing observables in the last column of table 2. For the benchmark point, the NP contributions are in most cases within the combined experimental and SM uncertainties. Similar to eq. (4.5), we construct a Xl2oop function, that compares the NP contributions to the di erence of the data and SM predictions, for the three mass di erences and Ms, as well as the CP violating observables K , d, and s. That is, Xl2oop = X i=K;d;s " ( ( MiNP i M exp-SM)2 # Miexp )2 + ( MSM )2 i + X i=d;s " ( iNP i i exp-SM)2 # i + ( NKP K ( exp )2 + ( SM )2 ( exp )2 + ( SM )2 MK , Md, K exp-SM)2 ; K (4.13) (- ) (- ) ( ) ( ) ( ) ( ) θ θ ( ) ( ) ( ) (- ) (- ) θ HJEP03(218)9 ψ (- ) ( ) ( ) ( ) ( )  ϑ ψ parameter space in the neighborhood of the benchmark point (4.4). Contour values are labeled in black; we also show the values for Xl2oop Xl2oop(SM) in parentheses. The benchmark point (4.4) is shown by the white circle. The contours from gure 1 are shown by the dotted lines with the corresponding contours labeled in gray. where the superscript `exp-SM' indicates that we are using the di erence of the measured values and the SM predictions given in table 2. Figure 2 shows the Xl2oop=dof behavior of the avor model on various two-dimensional parametric slices in the neighborhood of the benchmark point (4.4). As for gure 1, on each slice all theory parameters are xed to the benchmark values (4.4), except for the two parameters corresponding to the plot axes. The number of degrees of freedom in the Xl2oop statistic is then 6 2 = 4. Note that the SM predictions and experimental results for meson mixing observables from table 2 show slight tensions [55, 57, 58], as indicated by the non-negligible SM contribution to the Xl2oop function, Xl2oop(SM) ' 10:8. We observe that ranges of model parameters exist for which X2 is mildly better than in the SM: at our benchmark Xl2oop Xl2oop(SM) ' 3:7. (Identifying all regions of parameter space of our framework that can address existing tensions in meson observables is left for future studies.) Moreover, comparing with the contours obtained from the Xt2ree=dof function (dotted lines), we nd that extended regions of parameter space exist where CKM elements and masses as well as meson mixing observables are described in a satisfactory way. 5 Conclusion and outlook We have presented a new framework to address the SM avor puzzle, synthesizing the structure of the ` avorful' 2HDM with the ` avor-locking' mechanism. This mechanism makes use of distinct avon and hierarchon sectors to dynamically generate arbitrary quark mass hierarchies, without assigning additional symmetries to the quark elds themselves. In this paper, we have shown that with suitable symmetry assignments in the avon and hierarchon sectors, the global minimum of the general renormalizable avon potential can be identi ed with a ` avor-locked' con guration: an aligned, rank-1 con guration for each avon, and arbitrary (block) unitary misalignment between the up and down quark Yukawas, so that a unique hierarchon vev controls each quark mass. In the presence of only one SM-like Higgs doublet, this leads to quark mixing angles that are generically O(1). Introducing instead a avorful 2HDM Higgs sector | two Higgs doublets, such that one Higgs couples only to the third generation, while the other couples to the rst two generations | leads to a 1+2 avor-locked theory. We nd that quark avor mixing in this theory is naturally hierarchical too, once one requires that the dynamicallygenerated quark masses are themselves hierarchical | the light quark masses need not be tuned in this theory, being generated instead by the avor-blind avor-locking portal to the hierarchon sector | and the mixing is generically of the observed size. The collider phenomenology of this theory is quite rich if the additional Higgs bosons are light, with testable signatures at the LHC or HL-LHC. For an example benchmark point in the theory parameter space, we showed that this ` avor-locked avorful 2HDM' model does not require signi cant tunings in order to reproduce the observed mass, CKM and meson mixing data. In particular, O(1) variations in model parameters do not substantially or rapidly vary the agreement with the order of the observed CKM matrix, or, in other words, the hierarchical quark mixing is stable over O(1) variations in the parameters of the theory. By contrast, the SM features naively seven tunings: the ve lighter quark masses, and the mixing angles 23 and 13 in the standard CKM parametrization, that produce small jVcbj and jVubj, respectively. The reduced amount of tuning of the quark masses and CKM mixing in the avorlocked avorful 2HDM does not come at the price of large NP contributions to meson mixing, even if the additional neutral Higgs bosons are light: O(1) variation of the avor parameters does not lead to a signi cant deviation in meson mixing observables for heavy Higgs boson masses at around the electro-weak scale (e.g. mA mH 500 GeV) and (e.g. tan 5), and may in fact better accommodate current meson mixing data than the SM itself. Further exploration of the avor phenomenology of this theory is left for future studies. It is straightforward to extend this framework to the charged lepton sector. Possible ways to reproduce a realistic normal or inverted neutrino spectrum and the large neutrino mixing angles will be discussed elsewhere. Acknowledgments We thank Simon Knapen for helpful conversations and for comments on the manuscript. WA and SG thank the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support during parts of this work. The work of WA, SG and DR was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. The research of WA is supported by the National Science Foundation under Grant No. PHY-1720252. SG is supported by a National Science Foundation CAREER Grant No. PHY-1654502. We acknowledge nancial support by the University of Cincinnati. A Analysis of the general avon potential In this appendix we determine the global minimum of the avon potential (3.3). A.1 General avon potential The single and pairwise eld potentials (3.4), (3.5) are manifestly positive semide nite. Noting that the 6 terms can be written in the form Tr ([ y ]y y ) and Tr ([ y and moreover that Tr [AyA] = P zero | of V1f; V2f is attained if and only if ij jAij j2 = 0 if and only if A = 0, the global minimum | 1. Tr [h ih y i] = r2, These algebraic conditions are equivalent to the set h i being simultaneously real diagonalizable with disjoint unit rank spectra. That is, h 1i = U diagfr; 0; 0; : : :g V y ; h 2i = U diagf0; r; 0; : : :g V y ; : : : ; (A.2) with U , V generic unitary matrices, the same for all , that are at directions of the global minimum, and r real. A similar analysis follows immediately for the down-type potentials, so that h b1i = Ub diagfrb; 0; 0; : : :g Vb y ; h b2i = Ub diagf0; rb; 0; : : :g Vb y ; : : : : (A.3) We refer to this type of aligned structure as ` avor-locked'. (It is possible to switch the rank-1 structure for degeneracy by setting 2 < 0 [33], though we do not consider this possibility in this work.) A.2 Mixing terms: single avon generation The rst, 1, term of the mixed potential (3.6) manifestly respects the vacuum of V1f and V2f. It follows from the Cauchy-Schwarz inequality and positive semide niteness of y , that Tr y Tr y b b Tr y b b y : Hence for the case of n = 1 generations of avons, the 2 term and full potential is immediately positive semide nite, with global minimum at V = 0. Based on the avorlocked con gurations in eqs. (A.2) and (A.3), in which we have momentarily restored the U(N )U U(N )D indices and Vckm = U yUb is the unitary CKM matrix. Without loss of generality, we can choose the non-zero eigenvalues of the single up and down avon being in the rst diagonal entry, at the avor-locked con guration. One then obtains for the n = 1 mixed potential (A.4) (A.5) (A.6) (A.7) This vanishes if and only if Vckm is 1 (N 1) block unitary, i.e. Vmix = 2r2rb2h i 1 : Vckm = 1 0 ; in which VN 1 is an N 1 N 1 unitary submatrix (as in eq. (3.9)). Therefore, the potential has a global minimum if and only if the avons lie in the avor-locked con guration, with a block-unitary mixing matrix. A.3 Mixing terms: arbitrary avon generations For the general case that N n 1, the 2 term is not positive de nite by itself. The full potential may, however, be reorganized into the form V = X U1f + X U2f + X U1bf + X U2bfb + U m0ix + X Umibx : (A.8) in which the pure up-type potentials U1f = 1 Tr U2f = 3 Tr y y y < r 2 Tr y 2 + y + b 2 + 2 6;2 2 rb2 2 r2 b r2 b<b Tr y Tr b y 2 2 Tr y y ; y y ; (A.9) and similarly for the down-type potentials, exchanging all unhatted and hatted couplings. The two mixed potentials 2n 2 r2rb2 Tr X b y =r2 br2 b b Tr Hence each term of the full potential is now positive semide nite, provided 6;2 2rb2=r2 ; b6;2 b and 2=(2n) : (A.12) HJEP03(218)9 We write the avor-locked con guration in the ordered form of eqs. (A.2) and (A.3), so that the rst n eigenvalues of h i are non-zero. At the avor-locked con guration, the mixed potential becomes ; 1 2 b b b y =r2 : Unitarity ensures that so that on the avor-locked contour the mixing terms and hence full potential is minimized, with V = 0, if and only if Vckm is n (N n) block unitary. I.e. X Vmibx = ; b 2r2rb2 X h Vckbm 2 ; b 1=ni = 0 : n X ;b=1 Vckbm 2 n ; Vckm = U yUb = 0 Vn ! ; with Vk a k k unitary matrix. Note that the n or N n block CKM rotations are at directions of the global minimum, and therefore Vn and VN n may be any arbitrary unitary submatrices with generically O(1) entries. We often refer to eq. (A.15) in combination with eqs. (A.2) and (A.3) as the ` avor-locked' con guration, too. A.4 Local minimum analysis So far we have shown that under the conditions (A.12) the global minimum of the potential is V = 0 and it is realized if and only if the avons are in the avor-locked con guration. One may also explore the weaker condition that the avor-locked con guration is only a local minimum of the potential, by applying the general perturbations h i ! h i + X ; and h bi ! h bi + Xb : To this end, it is convenient to de ne 1 h H = r2 h iXy + X h y ii ; P = r2 1 X h ih y i ; Pb = r2 h bih ybi ; 1 X b b (A.10) (A.11) (A.13) (A.14) (A.15) (A.16) (A.17) holds (cf. (A.12)). The vacuum con guration in (3.8) is then a local minimum of the avon potential. More generically, one may also re-organize the potential, such that V = U10f + X U1f + X U2f + X U1bf + X U2bfb + U m0ix + X Umibx : 2r2rb2 2 y y =r2 Tr X b Tr 2 1 ! 2 rb2 ; 2 r2 y br2 b b 2 ; 2r2 r2b ; perturbation of the mixing terms, one nds to O( 2), Observe H is Hermitian and Tr [P ] = n. One may show that Tr [P H ] = Tr [H ], and, as a consequence of the block unitarity (A.15), that further Tr [PbH ] = Tr [H ]. Under which is positive semide nite, provided the condition 2n 2 r2rb2 X ; Tr H b 2 Tr H b 2 ; b<b b r2 Tr y Tr br2 b b b b b y =r2 2 : X H b b !) 2 rb2 ; 2n r2 Tr y + 4 Tr y y ; Tr X r2 y 3 (1 ! 0 ; 6;2 2n r2 U1f = U2f = X Tr y Tr y 6;2 in which we have de ned, for an arbitrary real coe cient, !, and analogously in the down sector for the and b pieces. The mixing terms are given by This time, under perturbations of the avor-locked con guration, one nds U m0ix = 2 2r2rb2 Tr X H P n Tr X H X H b b which is positive semide nite. Hence, no matter the form of the 1 term, a local minimum can also be achieved for the case that and similarly for the hatted couplings. but neither h tyih c;ui nor h t ih cy;ui need to vanish, and similarly for the down-type avons. The potentials V ;h and V ;l then each have a N = 3 avor-locked vacuum, with generation number n = 1 and n = 2, respectively. This leads immediately to the vacuum in eqs. (3.16) and (3.17). B Flavor basis for the F2HDM Yukawa texture Starting from the general parametrization of the avor-locked Yukawas in (4.2) we perform the following quark eld rotations in avor space UL ! UUL UL ; DL ! UDL DL ; UR ! UUR UR ; DR ! UDR DR ; (B.1) where the Ui are 2 1 block unitary matrices 0 cos UL sin UL 0 sin UL cos UL 0C ; 0 cos UR sin UR 0 sin UR cos UR 0C ; 1 A 1 A 1 1 0 0 0 cos DL ei DL sin DL 0 sin DL ei DL cos DL 0C ; 1 1 A 0 cos DR sin DR 0 sin DR cos DR 0C : 0 0 0 1 1 A The rotation angels and the phase are chosen such that Two-Higgs alignment conditions The Two-Higgs potential (3.15) is equivalent to the general potential (3.3), but with the t{c, t{u and b{d, b{s cross-terms e ectively vanishing. The vacuum for V1f + V2f then has the structure (A.25) tan UL = sin 13 tan 23 ; tan UR = sin #13 tan #23 ; tan DR = sin #b13 tan #b23 ; tan DL = sin 13 tan 23 cos DL tan tan DL = tan sin m sin 23 tan 13 tan cos m : cos 23 cos 13 cos( m + DL ) ; In this avor basis the Yukawas in (4.2) reproduce the F2HDM textures from eq. (2.4) with coe cients that depend on the several angles 13; 23; #13; #23; #b13; #b23; and phases d; s; u; c; m. Open Access. This article is distributed under the terms of the Creative Commons 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. (2017) 024 [arXiv:1708.03299] [INSPIRE]. arXiv:1709.07497 [INSPIRE]. 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Wolfgang Altmannshofer, Stefania Gori, Dean J. Robinson, Douglas Tuckler. The flavor-locked flavorful two Higgs doublet model, Journal of High Energy Physics, 2018, 129, DOI: 10.1007/JHEP03(2018)129