CPV Phenomenology of Flavor Conserving Two Higgs Doublet Models
AACFI-T14-06CALT-68-2881
CPV Phenomenology of Flavor Conserving Two Higgs Doublet Models
Satoru Inoue , Michael J. Ramsey-Musolf , , and Yue Zhang Amherst Center for Fundamental InteractionsDepartment of Physics,University of Massachusetts AmherstAmherst, MA 01003 USA California Institute of TechnologyPasadena, CA 91125 USA (Dated: August 1, 2014)We analyze the constraints on CP-violating, flavor conserving Two Higgs Doublet Models(2HDMs) implied by measurements of Higgs boson properties at the Large Hadron Collider (LHC)and by the non-observation of permanent electric dipole moments (EDMs) of molecules, atoms andthe neutron. We find that the LHC and EDM constraints are largely complementary, with theLHC studies constraining the mixing between the neutral CP-even states and EDMs probing theeffect of mixing between the CP-even and CP-odd scalars. The presently most stringent constraintsare implied by the non-observation of the ThO molecule EDM signal. Future improvements in thesensitivity of neutron and diamagnetic atom EDM searches could yield competitive or even moresevere constraints. We analyze the quantitative impact of hadronic and nuclear theory uncertaintieson the interpretation of the latter systems and conclude that these uncertainties cloud the impactof projected improvements in the corresponding experimental sensitivities.
I. INTRODUCTION
With the discovery of a 125 GeV boson at the Large Hadron Collider, exploration of the dynamics of electroweaksymmetry-breaking (EWSB) is front and center in particle physics. Although the properties of the new boson thus faragree with expectations for the Standard Model Higgs boson, it is possible that it constitutes but one state of a richerscalar sector. Perhaps, the most widely-studied extended scalar sector is the two-Higgs doublet model (2HDM). Thecollider and low-energy phenomenology of the 2HDM have been extensively analyzed over the years [1, 2], while oneversion of this paradigm appears in an equally important scenario for physics beyond the Standard Model (BSM),the minimal supersymmetric Standard Model.One of the more interesting features of the 2HDM is the presence of new sources of CP-violation (CPV) beyondthat of the Standard Model Cabibbo-Kobayashi-Maskawa (CKM) matrix and the QCD “ θ -term”. It is well-knownthat BSM CPV is required to account for the observed excess of matter over anti-matter in the present Universe. Ifrealized in nature, the 2HDM may provide the necessary CPV and enable the generation of the matter-antimatterasymmetry during the era of EWSB [3–5], and possibly the co-generation of both baryonic and dark matter in theuniverse [6]. If so, then the 2HDM CPV may have observable signatures in laboratory tests. At the energy frontier,CPV correlations associated with the production and decay of the lightest neutral scalar may be accessible at theLHC and/or a future high intensity e + e − collider. At the low-energy “intensity frontier”, searches for the permanentelectric dipole moments (EDMs) of atoms, molecules and nucleons provide a powerful indirect probe [7–10]. Indeed,EDM searches are entering a new era of sensitivity, with the recent report by the ACME collaboration of a tentimes tighter limit on the electron EDM [11] , representing a harbinger of even more powerful probes in the future.Searches for the permanent EDM of the neutron are underway at a variety of laboratories around the world, withgoals of one-to-two orders of magnitude improvements in sensitivity. Similarly, experiments are underway to carry outimproved and/or new searches for the EDMs of Mercury, Xenon, and Radium, while longer term efforts to developstorage ring probes of the proton and light nuclei EDMs are being pursued [12].With this context in mind, it is timely to investigate the present and prospective probes of CPV in the 2HDM. Inwhat follows, we report on such a study, focusing on the implications of present and prospective EDM searches whiletaking into account the LHC constraints on the properties of the 125 GeV boson. For concreteness, we consider a The experiment actually constrains the ThO molecule response to an external electric field . In general, the ThO response is dominatedby two operators including the electron EDM and an electron-quark interaction. For the 2HDM, the electron EDM gives by far thelarger contribution. a r X i v : . [ h e p - ph ] J u l Z -symmetric variant of the 2HDM that evades potentially problematic flavor changing neutral currents (FCNCs) butstill allows for new CPV associated with the scalar potential, accommodates the present 125 GeV boson properties,and retains a rich phenomenology for future studies.Under these assumptions, we explore scenarios wherein there exists only one physical CPV phase associated withthe scalar potential. The resulting scalar spectrum contains three neutral states that are CPV mixtures of neutralscalar and pseudo-scalars and one pair of charged scalars. The scalar sector is then characterized by nine-independentparameters that can be related to the parameters in the potential using the conditions for EWSB. We take theseparameters to be the four scalar masses; the mixing angle α b that governs the CP-odd admixture of the 125 GeVscalar; the combined vacuum expectation value (vev) of the two neutral scalars, v = 246 GeV; the conventional 2HDMmixing angles β and α ; and a parameter ν (also defined below) that characterizes the degree to which the heavierstates decouple from the low-energy effective theory, leaving 125 GeV boson as the only accessible state. Note that α b → δ → α b encodes the effects of the solephysical phase in the scalar potential.From our analysis, we find that • Fits to the properties of the observed 125 GeV boson generally favor scenarios in which α ≈ β − π/ • For fixed values of the scalar masses, null results for EDMs yield constraints in the sin α b –tan β plane. • The assumption that loops involving the lightest neutral scalar are dominant over those involving the remaining2HDM scalar states does not hold in general. In the electron EDM case, for example, loops involving the heavierstates may yield the largest contribution for moderate-to-large tan β . In short, the “light Higgs effective theory”is not necessarily effective in this context. • At present the ThO result yields the strongest constraints on the CPV parameter space. Future EDM searchescould significantly extend this reach, particularly for the type II 2HDM. An order-of-magnitude improvementin the sensitivity of the neutron and
Hg EDM searches would probe regions presently allowed by the ThOlimit. A factor of 100 more sensitive neutron EDM search would go well beyond the present constraints, and inthe event of a null result, would restrict | sin α b | to less than a few × − . Successful completion of the Argonne Ra EDM search at its design sensitivity would reach well beyond the ACME constraints as well as the possibleten times better neutron and
Hg search for the type II case, though a 100 times more sensitive neutron EDMsearch would surpass the radium reach. For the type I model, the ACME constraints would survive even thefuture neutron, mercury, and radium experiments. Thus, a non-zero result for any of the latter searches wouldindicate the presence of a type II rather than a type I 2HDM. • Determination of the diamagnetic atom (
Hg,
Ra, etc ) and neutron EDM sensitivities is subject to consid-erable hadronic and nuclear many-body uncertainties. Those associated with the interpretation of the paramag-netic system (ThO) results are less significant. Consequently, the aforementioned statements about the relativesensitivities of future searches are provisional. Definitive conclusions will require substantial improvements inhadronic and nuclear many-body computations.We organize the discussion of the analysis leading to these observations as follows. In Section II we analyzethe general features of the Z -symmetric 2HDM, including the constraints of the EWSB conditions, the choice ofindependent parameters, and the structure of the interactions. Section III gives the relationship of the independentparameters and interactions to the observables of interest, including the Higgs boson event rates at the LHC (III A),the low-scale effective operators that ultimately induce EDMs of various light quark and lepton systems and theirrenormalization group evolution (III B), the sensitivity of these systems to the effective operators and the correspondingEDM constraints (III C). Section IV gives the resulting constraints on the relevant parameter space. In particularwe call the reader’s attention to Figures 6 and 10. The former gives the present constraints from ThO, the neutron,and Hg in the sin α b –tan β plane, including the hadronic and nuclear theory uncertainties. The latter shows theprospective impact of future neutron, Hg, and
Ra searches in comparison with the present ThO constraints. Inthis section, we also present an “anatomy” of the electron, neutron and diamagnetic EDMs in terms of the 2HDMdegrees of freedom as well as the various low-energy effective operators. We summarize our conclusions in Section V.Expressions for the effective operator Wilson coefficients are given in an Appendix.
II. 2HDM FRAMEWORKA. Scalar potential
In this work, we consider the flavor-conserving 2HDM in order to avoid problematic flavor-changing neutral currents(FCNCs). As observed by Glashow and Weinberg (GW) [13], one may avoid tree-level FCNCs if diagonalization of thefermion mass matrices leads to flavor diagonal Yukawa interactions. One approach to realizing this requirement is toimpose a Z symmetry on the scalar potential together with an appropriate extension to the Yukawa interactions (seebelow). In this scenario, however, one obtains no sources of CPV beyond the SM CKM complex phase. Consequently,we introduce a soft Z -breaking term that yields non-vanishing CPV terms in the scalar sector [17].To that end, we choose a scalar field basis in which the two Higgs doublets φ , are oppositely charged under thethe Z symmetry: φ → − φ and φ → φ , (1)though this symmetry will in general have a different expression in another basis obtained by the transformation φ j = U jk φ (cid:48) k . For example, taking U = 1 √ (cid:18) − (cid:19) , (2)the transformation (1) corresponds to φ (cid:48) ↔ φ (cid:48) . (3)We then take the Higgs potential to have the form V = λ φ † φ ) + λ φ † φ ) + λ ( φ † φ )( φ † φ ) + λ ( φ † φ )( φ † φ ) + 12 (cid:104) λ ( φ † φ ) + h . c . (cid:105) − (cid:110) m ( φ † φ ) + (cid:104) m ( φ † φ ) + h . c . (cid:105) + m ( φ † φ ) (cid:111) . (4)The complex coefficients in the potential are m and λ . In general, the presence of the φ † φ term, in conjunctionwith the Z -conserving quartic interactions, will induce other Z -breaking quartic operators at one-loop order. Simplepower counting implies that the responding coefficients are finite with magnitude proportional to m λ k / (16 π ). Giventhe 1 / π suppression, we will restrict our attention to the tree-level Z -breaking bilinear term.It is instructive to identify the CPV complex phases that are invariant under a rephasing of the scalar fields. Tothat end, we perform an SU(2) L × U(1) Y transformation to a basis where the vacuum expectation value (vev) of theneutral component of φ is real while that associated with the neutral component of φ is in general complex: φ = (cid:18) H +11 √ ( v + H + iA ) (cid:19) , φ = (cid:18) H +21 √ ( v + H + iA ) (cid:19) , (5)where v = (cid:112) | v | + | v | = 246 GeV, v = v ∗ and v = | v | e iξ . It is apparent that in general ξ denotes the relativephase of v and v . Under the global rephasing transformation φ j = e iθ j φ (cid:48) j , (6)the couplings m and λ can be redefined to absorb the global phases( m ) (cid:48) = e i ( θ − θ ) m , λ (cid:48) = e i ( θ − θ ) λ , (7)so that the form of the potential is unchanged. It is then straightforward to observe that there exist two rephasinginvariant complex phases: δ = Arg (cid:2) λ ∗ ( m ) (cid:3) ,δ = Arg (cid:2) λ ∗ ( m ) v v ∗ (cid:3) . (8) Another approach is to have 2HDM at the electroweak scale without the Z symmetry is to assume minimal flavor violation, flavoralignment or other variants. We do not discuss this possibility, but refer to [14–16] for recent phenomenological studies. For future purposes, we emphasize that the value of ξ is not invariant.Denoting tan β = | v | / | v | , the minimization conditions in the H k and A k directions give us the relations m = λ v cos β + ( λ + λ ) v sin β − Re( m e iξ ) tan β + Re( λ e iξ ) v sin β , (9) m = λ v sin β + ( λ + λ ) v cos β − Re( m e iξ ) cot β + Re( λ e iξ ) v cos β , (10)Im( m e iξ ) = v sin β cos β Im( λ e iξ ) . (11)From the last equation, it is clear that the phase ξ can be solved for given the complex parameters m and λ . It isuseful, however, to express this condition in terms of the δ k : | m | sin( δ − δ ) = | λ v v | sin(2 δ − δ ) . (12)In the limit that the δ k are small but non-vanishing that will be appropriate for our later phenomenological discussion,Eq. (12) then implies δ ≈ − (cid:12)(cid:12)(cid:12) λ v v m (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) λ v v m (cid:12)(cid:12)(cid:12) δ , (13)so that there exists only one independent CPV phase in the theory after EWSB.A special case arises when δ = 0. In this case, Eq. (12) implies that | m | sin( δ ) = | λ v v | sin(2 δ ) , (14)or cos δ = 12 (cid:12)(cid:12)(cid:12)(cid:12) m λ v v (cid:12)(cid:12)(cid:12)(cid:12) . (15)When the right-hand side is less than 1, δ has solutions two solutions of equal magnitude and opposite sign, corre-sponding to the presence of spontaneous CPV (SCPV) [18, 19]: δ = ± arccos (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) m λ v v (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = ± (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) m λ v cos β sin β (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (16)To the extent that the vacua associated with the two opposite sign solutions are degenerate, one would expect theexistence of cosmological domains [20] associated with these two vacua. Persistence of the corresponding domain wallsto late cosmic times is inconsistent with the observed homogeneity of structure and isotropy of the cosmic microwavebackground. Consequently, parameter choices leading to δ = 0 but δ (cid:54) = 0 should be avoided. In practice, we willscan over model parameters when analyzing the EDM and LHC constraints. As a check, we have performed a scanwith 10 points and find less than ten that give δ = 0. We are, thus, confident that the general features of ourphenomenological analysis are consistent with the absence of problematic SCPV domains.Henceforth, for simplicity, we utilize the rephasing invariance of the δ k and work in a basis where ξ = 0. In thisbasis, the phases of m and λ are redefined and related by Eq. (11). As we discuss below, we will trade the resultingdependence of observables on δ [and δ via δ in Eq. (13)] for one independent angle in the transformation thatdiagonalizes the neutral scalar mass matrix. B. Scalar spectrum
After EWSB, the diagonalization of the 2 × H + = − sin βH +1 + cos βH +2 , G + = cos βH +1 + sin βH +2 , (17)The charged scalar has a mass m H + = 12 (2 ν − λ − Re λ ) v , ν ≡ Re m csc β sec β v . (18)For the neutral Higgs sector, the mixing between CP odd components yields the Goldstone G and an orthogonalcombination A , where A = − sin βA + cos βA , G = cos βA + sin βA . (19)In the presence of explicit CP violation, A is not yet a mass eigenstate. It will further mix with the CP eveneigenstates H , H . The 3 × { H , H , A } is M = v λ c β + νs β ( λ − ν ) c β s β − Im λ s β ( λ − ν ) c β s β λ s β + νc β − Im λ c β − Im λ s β − Im λ c β − Re λ + ν , (20)where λ = λ + λ + Re λ . We define an orthogonal rotation matrix R to diagonalize the above mass matrix, with R M R T = diag( m h , m h , m h ). Generally, the matrix R can be parametrized as [21, 22] R = R ( α c ) R ( α b ) R ( α + π/
2) = − s α c α b c α c α b s α b s α s α b s α c − c α c α c − s α c α c − c α s α b s α c c α b s α c s α s α b c α c + c α s α c s α s α c − c α s α b c α c c α b c α c . (21)Both α b and α c are CP violating mixing angles in the Higgs sector that depend implicitly on δ . In this convention,the significance of the α and β angles are the same as in the minimal supersymmetric Standard Model, and theinteractions with quarks of lightest Higgs state, h , depends only on one CP violating angle α b . The mass and CPeigenstates are related via ( H , H , A ) = ( h , h , h ) R . As we discuss below, α c is determined once α b , α , β , andthe neutral scalar masses are specified. We will, thus, utilize α b rather than δ to characterize the effects of CPV inthe potential. C. Interactions
For phenomenological analysis, we are interested in interactions of the scalar sector with the other SM particles.After EWSB, the couplings of the neutral scalars with fermions and gauge bosons can be parametrized generallyas [23] L = − m f v h i (cid:0) c f,i ¯ f f + ˜ c f,i ¯ f iγ f (cid:1) + a i h i (cid:18) m W v W µ W µ + m Z v Z µ Z µ (cid:19) , (22)where i = 1 , , Z symmetry of the scalar potential by making the following assignments to the fermions: Q L → Q L u R → u R d R → d R , Type I , (23) Q L → Q L u R → u R d R → − d R , Type II . (24)One may make similar assignments for the leptons. The resulting Yukawa interactions before EWSB are L I = − Y U Q L ( iτ ) φ ∗ u R − Y D Q L φ d R + h . c . , (25) L II = − Y U Q L ( iτ ) φ ∗ u R − Y D Q L φ d R + h . c . . (26)Note that L I,II satisfy the GW criterion for the absence of tree-level FCNCs.For each of the two types of models, we solve for c f , ˜ c f , a in terms of β and the orthogonal matrix R : c t,i c b,i ˜ c t,i ˜ c b,i a i Type I R i / sin β R i / sin β − R i cot β R i cot β R i sin β + R i cos β Type II R i / sin β R i / cos β − R i cot β − R i tan β R i sin β + R i cos β (27)where all the up (down) type fermions have the universal rescaled couplings as the top (bottom) quark apart fromthe overall factor of the quark mass.The charged Higgs-fermion interactions are, respectively, L ¯ ff (cid:48) H ± = (cid:40) V ij cot β ¯ u i [ m u i (1 − γ ) + m d j (1 + γ )] d j H + + h . c . type I V ij ¯ u i [ m u i cot β (1 − γ ) − m d j tan β (1 + γ )] d j H + + h . c . type II (28)where V stands for the CKM matrix for quark mixings.The trilinear interactions between charged and neutral scalars, relevant for the scalar sector contribution to EDMs,are of the form L H ± = − ¯ λ i vh i H + H − , (29)where h i and H ± are mass eigenstates, and¯ λ i = R i · (cid:0) λ cos β + ( λ − λ − Re λ ) sin β (cid:1) cos β + R i · (cid:0) λ sin β + ( λ − λ − Re λ ) cos β (cid:1) sin β + R i · Im λ sin β cos β . (30)We do not write down the corresponding quartic terms as they are not needed for our phenomenological analysis. D. Phenomenological parameters
From the Higgs potential Eq. (4), it is possible to solve for the Higgs doublet VEVs as well as the scalar massesand mixing angles, which are more directly related to observation. Since the aim of this work is to arrive at thephenomenological constraints on the parameter space of 2HDM, it is useful to translate these constraints into thoseon phenomenologically-relevant parameters. The latter set includes the masses, mixing angles, and the parameter ν introduced in Eq. (18) and whose significance we discuss below. The following table summarizes two sets of parameters(all real). Potential parameters Phenomenological parameters λ , λ , λ , λ , Re λ , Im λ v , tan β , ν , α , α b , α c m , m , Re m , Im m m h , m h , m h , m H + (31)Each set has 10 parameters, and it would appear to be possible solve one set of parameters from the other. However,the minimization conditions for the A k in Eq. (11) imply that there exists only one independent CPV phase andhence, that the CPV mixing angles α b and α c are not independent. As we show below, one may solve for α c ( α b ) interms of α b ( α c ), the physical neutral scalar masses, α and β .Two additional remarks are in order. First, the phenomenological significance of the parameter ν is that it controlsthe mass scale of the second Higgs doublet. In the decoupling limit wherein one reverts to the SM, one has ν (cid:29) H ± , h ≈ A and the linear combination h ≈ sin βH − cos βH also decouplewith an approximately common mass ν . The resulting low energy theory contains only one CP even scalar h , whichis the SM Higgs boson. In the same decoupling limit, we also have α b,c → α → β − π/
2. Away from the decouplinglimit, both doublets are at the electroweak scale, and we have to treat ν as an independent input parameter.Second, it is useful to consider the CP conserving limit, with a real Higgs potential, i.e., Im λ = 0 and Im m = 0.In absence of SCPV, ξ = 0, and the matrix (20) is block diagonalized with vanishing M and M elements. In thisregime, one has α b = α c = 0, and the independent parameters becomePotential parameters Phenomenological parameters (no CPV) λ , λ , λ , λ , λ v , tan β , αm , m , m m h , m h , m h , m H + (32)Although there exist eight potential parameters in this case, the condition of no SCPV reduces the number ofindependent parameters to seven, which one may choose to be those in the right hand column of the table.For the general scenario that allows for CPV, it is useful to write down the relationships between the phenomeno- (cid:45) (cid:45) Ν t a n Β FIG. 1: Theoretical constraints on the ν − tan β parameter space. logical parameters and those in the potential:tan β = ( m h − m h ) cos α c sin α c + ( m h − m h sin α c − m h cos α c ) tan α sin α b ( m h − m h ) tan α cos α c sin α c − ( m h − m h sin α c − m h cos α c ) sin α b , (33) λ = m h sin α cos α b + m h R + m h R v cos β − ν tan β , (34) λ = m h cos α cos α b + m h R + m h R v sin β − ν cot β , (35)Re λ = ν − m h sin α b + cos α b ( m h sin α c + m h cos α c ) v , (36) λ = 2 ν − Re λ − m H + v , (37) λ = ν − m h sin α cos α cos α b − m h R R − m h R R v sin β cos β − λ − Re λ , (38)Im λ = 2 cos α b (cid:2) ( m h − m h ) cos α sin α c cos α c + ( m h − m h sin α c − m h cos α c ) sin α sin α b (cid:3) v sin β . (39)Note that Eq. (33) implies that α b , α c , α , β and the neutral scalar masses are not all independent, as advertised.The remaining equations (34–39), together with the minimization conditions (9–11), can be used to solve for the 9independent phenomenological parameters in Eq. (31).In order to make the presence of only one independent CPV phase apparent, we chose to eliminate one of the twoCPV mixing angles ( α c ) in terms of the other parameters, including the other CPV mixing angle ( α b ) that vanishesin the CP-conserving 2HDM and the remaining parameters that survive in the absence of CPV.
1. Parameter ranges
Theoretical constraints on the parameter space follow from requirements of stability of the electroweak vacuum andperturbativity [18]. While the latter is not an absolute requirement for the validity of the theory, our phenomeno-logical study relies on perturbative computations of observables, so we restrict our attention to the domain of na¨ıve Α b (cid:61) (cid:45) (cid:45) (cid:45) Β Α c FIG. 2: Solutions for α + c (blue), α − c (magenta) as a function of tan β using Eq. (43) for fixed α b = 0 .
02. The other parametersare m H + = 420 GeV, m h = 400 GeV, m h = 450 GeV and ν = 1 . perturbativity, expressed in terms of the quartic couplings:0 < λ < π, < λ < π, λ > − (cid:112) λ λ , λ + λ − | λ | > − (cid:112) λ λ . (40)Using Eqs. (34-39) we translate these conditions into constraints on the phenomenological parameters. To illustrate,we take h to be the 125 GeV Higgs boson discovered at the LHC. For the ranges of other parameters we allow m h , m h ∈ [125 , α, α b ∈ [ − π/ , π/
2] (notice α c is not independent as discussed above). The resultingregion consistent with the conditions (40) in the ν − tan β plane is shown in Fig. 1.As discussed in the fit to LHC Higgs data in Section III A below, especially in the type-II 2HDM, the fit to theLHC data on the Higgs boson production and decay rates points to a strong correlation between the angles α and β ,with α ≈ β − π/ α b , tan β, m H + , m h , m h , m h , ν . (41)When either of the CPV mixing angles α b or α c is fixed, the other can be obtained from Eq. (33): α b = arcsin (cid:34) ( m h − m h ) sin 2 α c tan 2 β m h − m h sin α c − m h cos α c ) (cid:35) , (42)or alternatively, α ± c = arctan ( m − m ) tan 2 β ± (cid:113) ( m − m ) tan β − α b ( m − m )( m − m )2( m − m ) sin α b . (43)For given α b , there are two solutions for α c . We find that they satisfy the relation tan α + c tan α − c =( m h − m h ) / ( m h − m h ), which is approximately 1, in the limit m h (cid:28) m h ≈ m h . Fig. 2 illustrates the aboverelations between α b and α c , for a set of sample parameters. One can see on the right panel that the expression for α ± c contains a discontinuity at tan β = 0 (for fixed color, blue or magenta). For our phenomenological studies, we choose α + c for tan β < α − c for tan β >
1, in order to avoid this discontinuity. Physically, our choice corresponds to h being mostly CP-odd; α − c for tan β < α + c for tan β > h is mostly CP-odd. Wehave observed that the choice between α + c and α − c does not make a qualitative difference in our conclusions discussedbelow. III. OBSERVABLESA. Event rates of all Higgs decay channels at LHC
In this work, we assume the light neutral scalar from the 2HDM is the 125 GeV Higgs boson discovered at theLHC. In the presence of CPV interactions in Eq. (22), the Higgs production and decay rates are modified as follows, σ gg → h σ SM gg → h = Γ h → gg Γ SM h → gg ≈ (1 . c t − . c b ) + (1 . c t − . c b ) (1 . − . , (44)Γ h → γγ Γ SM h → γγ ≈ (0 . c t − . a ) + (0 . c t ) (0 . − . , (45) σ V V → h σ SM V V → h = σ V ∗ → V h σ SM V ∗ → V h = Γ h → W W Γ SM h → W W = Γ h → ZZ Γ SM h → ZZ = a , (46)Γ h → b ¯ b Γ SM h → b ¯ b = Γ h → τ + τ − Γ SM h → τ + τ − ≈ c b + ˜ c b . (47)The modified event rates will be constrained by the inclusive data in Higgs decay channels, which are summarizedin Table. I. Since we are interested only in the couplings of h , all the above couplings a , c f , ˜ c f can be expressed interms of only three parameters β , α and α b , where α b is the CPV mixing angle. In the presence of CP violation, theglobal fit to the combined results measured by the ATLAS [24] and CMS [25] collaborations have been performed inseveral previous works [4, 26–29]. In this work, we follow the same parametrization as [4] and present the results forboth type-I and II 2HDMs. γγ W W ZZ V bb τ τ ATLAS 1 . ± . . ± . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . . ± . B. T- and P-violating effective operators, RG running and matching
We now turn to the low-energy sector, focusing on the EDMs of the neutron, neutral atoms, and molecules thatpresently yield the most stringent constraints on flavor-diagonal CPV. The relevant time reversal- and parity-violating(TVPV) effective operators for our study are the elementary fermion EDMs, the quark chromo-EDMs (CEDM), andthe Weinberg three gluon (or gluon CEDM) operators. The corresponding effective Lagrangian valid below theelectroweak scale is L eff = − i (cid:88) f d f f σ µν γ f F µν − i (cid:88) q ˜ d q qσ µν γ T a qG aµν + d w f abc (cid:15) µνρσ G aµλ G b λν G cρσ (48) ≡ i (cid:88) f δ f Λ m f e ¯ f σ µν γ f F µν + i (cid:88) q ˜ δ q Λ m q g s ¯ qσ µν γ T a qG aµν + C ˜ G g s f abc (cid:15) µνρσ G aµλ G b λν G cρσ , (49)where we take the convention (cid:15) = +1, and in the first line we have expressed the effective operators in terms ofdimensional coefficients, while in the second we have followed Ref. [7] and re-written them in terms of the dimensionlessquantities δ f , ˜ δ q , and C ˜ G , the scale of physics beyond the Standard Model Λ, and the fermion masses. Doing so isconsistent with the approach of a low-energy effective field theory, wherein one makes the relevant scales and theirhierarchy explicit. On general grounds, one then expects the remaining dimensionless Wilson coefficients to becomparable in magnitude, all other considerations being equal. We note that in the versions of the 2HDM considered Technically speaking, the paramagnetic molecule response to the external field. δ q , quark CEDM, ˜ δ q , and gluon CEDM are related to the usual definitions by [7] δ f ≡ − Λ d γf eQ q m q , ˜ δ q ≡ − Λ d Gq m q , C ˜ G = Λ d w g s . (50)Henceforth, we set Λ = v . The dominant contributions to these coefficients arise at two-loop level at the 2HDM scaleΛ ∼ v , and have been summarized in the Appendix A.We also take account of the TVPV four-quark operators that have an impact in the renormalization group (RG)running. They arise in the 2HDM model at tree level from neutral Higgs exchange, as shown in Fig. 3. We areparticularly interested in the operators containing the bottom quark, whose coefficients are enhanced when tan β islarge, thereby making a significant contribution in certain cases. Those involving only light quarks are suppressed byproducts of their small Yukawa couplings and are, therefore, neglected here. The operators under consideration are L q eff = C b Λ (¯ bb )(¯ biγ b ) + ˜ C bq Λ (¯ bb )(¯ qiγ q ) + ˜ C qb Λ (¯ qq )(¯ biγ b ) , (51)where q = u, d . At the 2HDM scale Λ = v , these coefficients are C b (Λ) = (cid:88) i m b m h i c b,i ˜ c q,i , ˜ C bq (Λ) = (cid:88) i m b m q m h i ˜ c b,i c q,i , ˜ C qb (Λ) = (cid:88) i m b m q m h i c b,i ˜ c b,i . (52) q, b q, bb bh i FIG. 3: Tree level contribution to the P and T-odd four quark operators.
In order to calculate the neutron and atomic EDMs, we take account of the renormalization group running effectdue to one-loop QCD corrections. The Wilson coefficients in Eq. (49) are evolved from Λ down to the GeV scale,based on the RG equations (RGE) [30–32] dd ln µ (cid:18) δ q Q q , ˜ δ q , − C ˜ G (cid:19) = (cid:18) δ q Q q , ˜ δ q , − C ˜ G (cid:19) · α S π C F − C F C F − N
00 2
N N + 2 n f + β , (53)where q = u, d, b , N = 3, C F = ( N − / (2 N ) = 4 / β = (11 N − n f ) / m b , we use the n f = 5 version of the above RGE. In addition, there are contributionsthrough mixing from the four-quark operators in Eq. (51). In particular, the coefficient C b mixes with, and contributesto, the b-quark CEDM, and captures the leading logarithmic terms of the one-loop result [33]. The coefficients ˜ C bq ,˜ C qb also contribute to the light quark CEDM through RGE operator mixing as discussed in detail in [31]. Thisreproduces the leading logarithmic terms in the Barr-Zee type contribution to the CEDM with a b-quark in theupper loop. In this calculation, we keep only the leading logarithmic terms that make additional contributions to theCEDMs of bottom and light quarks at the matching scale µ = m b :∆˜ δ b ( m b ) ≈ π C b (Λ) log Λ m b , (54)∆˜ δ q ( m b ) ≈ g s π m b m q ( ˜ C bq + ˜ C qb ) (cid:18) log Λ m b (cid:19) . (55)1At the same scale, the bottom quark is integrated out, and its CEDM makes a shift to the Weinberg operator [31, 34],∆ C ˜ G ( m b ) = α S ( m b )12 π ˜ δ b ( m b ) , (56)where the b-quark CEDM ˜ δ b ( m b ) at the m b scale includes both the top quark contribution Eq. (A2) which evolvesunder the RGE, and the shift (54).After taking these renormalization effects, the coefficients δ q , ˜ δ Gq and C ˜ G are further evolved down to the GeV scaleaccording to RGE with 4 or 3 flavors, for the interval above or below the charm quark mass scale, respectively. C. Current and future EDM constraints
We now analyze the constraints implied by EDM search null results considering, in turn, paramagnetic atoms andmolecules, the neutron, and diamagnetic atoms. Within the context of the flavor conserving 2HDMs, constraintsfrom the paramagnetic systems translate into limits on the electron EDM ( d e ), while in a model-independent analysisparamagnetic results bound a linear combination of d e and a dimension-six semileptonic interaction. The neutronEDM ( d n ) is sensitive primarily to the quark EDM and CEDMs as well as the CPV three gluon operator, while theCPV four-(light)quark operator contributions are subdominant in the 2HDM context. For the diamagnetic systems,such as Hg, the quark EDM contribution is in general relatively suppressed, as is a dimension six semileptonictensor interaction that can be more important in other contexts apart from the 2HDM. The quark CEDM and threegluon operators, thus, are the most significant for the diamagnetic systems in the 2HDM.
Electron EDM.
Currently, the electron EDM is most strongly constrained by the ACME experiment [11], whichsearched for an energy shift of ThO molecules due to an external electric field. The external field induces the spinof the unpaired electron to lie along the intermolecular axis, sampling the large internal electric field associated withthe polar molecule. The measured energy shift is also sensitive to the TVPV electron-nucleon interaction L eff eN = − G F √ C (0) S ¯ eiγ e ¯ N N + · · · , (57)where the “+ · · · ” in Eq. (57) denote subleading semileptonic interactions and where the leading term arises from thefour fermion operators [7] (cid:104) Im C ledq (¯ eiγ e )( ¯ dd ) − Im C (1) lequ (¯ eiγ e )(¯ uu ) (cid:105) / (2 v ) . (58)In the 2HDMs considered in this work, these four-fermion operators are obtained by integrating out the neutral Higgsbosons at tree-level, C (0) S = − g (0) s (cid:16) Im C ledq − Im C (1) lequ (cid:17) = − g (0) s (cid:88) i =1 m e m h i ( m d ˜ c e,i c d,i + m u ˜ c e,i c u,i ) , (59)where g (0) s is the isoscalar nucleon scalar density form factor at zero momentum transfer (also known as the “ σ -term”).The ACME experiment gives the constraint [11] (cid:12)(cid:12)(cid:12) E eff d e + W S C (0) S (cid:12)(cid:12)(cid:12) < . × − eV , (60)where the effective field experienced by the unpaired electron is E eff = 84 GV / cm and where W S = 1 . × − eV.Since C (0) S is proportional to the product of the electron mass m e and light quark masses m u,d one may safely neglectthe semileptonic interaction and translate Eq. (60) into a bound on d e , or equivalently, δ e . Neutron EDM.
The dependence of the neutron EDM on the leading nonleptoic CPV operators in the 2HDM isgiven by [7] d n = (cid:0) eζ un δ u + eζ dn δ d (cid:1) + (cid:16) e ˜ ζ un ˜ δ u + e ˜ ζ dn ˜ δ d (cid:17) + β Gn C ˜ G , (61)2where we have set Λ = v as indicated earlier and where the central values [7] for the hadronic matrix elements are ζ un = 0 . × − , ζ dn = − . × − , ˜ ζ un = 0 . × − , ˜ ζ dn = 1 . × − and β Gn = 2 × − e cm. The experimentalupper bound on neutron EDM is [35] d n < . × − e cm . (62) Diamagnetic Atom EDMs.
At present, the most stringent EDM limit has been obtained on the
Hg atom(see below). Efforts are underway to increase the sensitivity of this EDM search while other groups are pursuingsearches for the EDMs of other diamagnetic atoms, including
Ra and
Xe (for a discussion, see e.g.
Ref. [12]).In what follows, we will consider the present
Hg constraint as well as the prospective impact of future
Hg and
Ra searches. Diamagnetic atom EDMs arise primarily from their nuclear Schiff moments and a tensor semileptonicinteraction. In the 2HDMs, the latter is suppressed by the same light fermion Yukawa factors that suppress C (0) S . TheSchiff moment is generated by long-range, pion-exchange mediated P- and T-violating nucleon-nucleon interactions,where one vertex involves the P- and T-conserving strong πN N coupling and the second consists of TVPV interaction: L TVPV πNN = ¯ N (cid:104) ¯ g (0) π (cid:126)τ · (cid:126)π + ¯ g (1) π π + ¯ g (2) π (2 τ π − (cid:126)τ · (cid:126)π ) (cid:105) N , (63)where the terms on the right hand side correspond to isoscalar, isovector, and isotensor channels, respectively. Ingeneral, the isotensor coupling ¯ g (2) π is suppressed with respect to the other two [7], so we include only the latter inour analysis. Denoting the nuclear Schiff moment as S , one has [7], d Hg = κ S S ≈ κ S m N g A F π (cid:16) a ¯ g (0) π + a ¯ g (1) π (cid:17) , (64)where g A ≈ .
26 and F π = 186 MeV. For Hg, we take the central values for the nuclear matrix elements fromRef. [7]: a = 0 . e fm , a = ± . e fm , while the atomic sensitivity coefficient is κ S = − . × − fm − [36] .For Radium, the central values are a = − . e fm , a = 6 . e fm , and κ S = − . × − fm − [36].At the hadronic level, the TVPV coefficients ¯ g (0 , π arise from quark CEDMs and the Weinberg operators [7],¯ g (0) π = ˜ η (0) (˜ δ Gu + ˜ δ Gd ) + γ ˜ G (0) C ˜ G , ¯ g (1) π = ˜ η (1) (˜ δ Gu − ˜ δ Gd ) + γ ˜ G (1) C ˜ G , (65)where the hadronic matrix elements are ˜ η (0) = − × − , ˜ η (1) = − × − , γ ˜ G (0) ≈ γ ˜ G (1) = 2 × − . There is also acontribution to ¯ g ( i ) π from the PV four quark operators, which we find to be unimportant for the 2HDMs here.The current experimental upper bound on Mercury EDM is [37] d Hg < . × − e cm , (66)and we will use a conservative future sensitivity for the Radium EDM [12] d Ra < − e cm . (67) IV. RESULTSA. LHC Higgs Data Constraint
We perform a global fit to the Higgs data given in Table I, where all the measured event rates are normalized tothe SM predictions, and are of the form µ f ± σ f . The employed χ is defined as χ = (cid:88) f (cid:2) σ h Br h → f / ( σ SM h Br SM h → f ) − µ f (cid:3) σ f , (68) We note that Eq. (5.181) of Ref. [7] omitted the minus sign on this value of κ S . SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Α b Combined, tan
Β(cid:61) SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Combined, tan
Β(cid:61) SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Combined, tan
Β(cid:61) SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Α b Combined, tan
Β(cid:61) SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Combined, tan
Β(cid:61) SM (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Combined, tan
Β(cid:61) FIG. 4: Global fit to the LHC Higgs data on the event rates given in Table I, for different values of tan β . First row : type-Imodel;
Second row : type-II model. The other parameters are chosen to be α c = 0 . m H + = 420 GeV, m h = 400 GeV and m h = 450 GeV. The 1 , , σ regions are in green, yellow, gray, and the best fit points are in blue. where the sum goes through all the Higgs decay channels. The results are shown in Fig. 4 in the two dimensional α b − α plane, for fixed values of tan β = 0 . , ,
5, respectively. The 1 , , σ regions are in green, yellow, gray, and theblue points represent where the best fit is found.Clearly, the SM limit always gives a good fit, well within 1 σ . However, we find for low values of tan β (cid:46)
1, non-zeroCP violation is preferred over the SM. This is mainly driven by the excess in the diphoton channel which still persistsat ATLAS. We also find when tan β ∼ α ≈ β − π/
2, the largest possible α b is allowed. In this case, the Higgscouplings defined in Eq. (15) are | c t | ≈ | c b | ≈ | a | ≈ cos α b , and | ˜ c t | ≈ | ˜ c b | ≈ sin α b . The resulting χ is a function ofcos α b , which when α b →
0, approaches χ for the SM case. Because the cosine function is rather “flat” around theorigin, there is substantial room for α b to deviate from zero, while χ − χ still remains small. The physical effect=of non-zero CP violation is to enhance the rate for h → γγ and suppress the rate for V bb channel. At the same time,the Higgs total width is also slightly reduced. At large tan β , We find the effects of CPV are being enhanced, thusthe good fit region shrinks with the increasing tan β . These facts suggest the best place to look for sizable possibleCP violation angle is when tan β ≈ α , while theconstraints on the CP violating angle α b are generally relatively weaker. As we discuss below, EDM searches aremore sensitive to the non-zero α b , but less sensitive to α . In short, the Higgs studies and EDM searches providecomplementary probes of the type-I and II 2HDMs.4 Total g HWH g h ZHZh1 2 5 10 20 5010 - - - - - - tan b d e H ec m L Total g H WH g hZH Zh1 2 5 10 20 5010 - - - - - tan b d e H ec m L FIG. 5: The anatomy of various contributions to the electron EDM in flavor conserving 2HDMs .
Left : type-I model;
Right :type-II model. We plot the absolute values, so the dip in the curves implies a sign change. Parameters are chosen to be α = β − π/ α b = 0 . × − , m H + = 420 GeV, m h = 400 GeV, m h = 450 GeV and ν = 1 .
0. The parameter α c is notindependent and is obtained using Eq. (43), and we note the two solutions α ± c give very similar results here. B. Electron EDM Constraint
Drawing on the Wilson coefficients computed in Appendix A; the hadronic, nuclear and atomic computationssummarized in Section III C; and the present and prospective EDM search sensitivities, we present numerical resultsfor the EDM constraints on the CPV parameter space. We give resulting constraints in the tan β vs. sin α b (Fig. 6)as well as a breakdown, or anatomy, of the various contributions and their RG evolution as a function of tan β (Figs.5, 7, 8, and 9). We also discuss the implications of the constraints for the 2HDMs and the impact of the varioushadronic and nuclear uncertainties.We first consider the electron EDM, whose current limit is set by the ACME collaboration, d e < . × − e cmat 95% confidence level. Strictly speaking, ACME sets a bound on the linear combination of d e and a CP odd fourfermion operator [see Eq.(60)]. In practice, we note the four fermion contribution is always subdominant in the flavorconserving 2HDMs.In Fig. 5, we plot the anatomy of various contributions [see Eq. (A12) in the appendix] to the electron EDM asfunctions of tan β . Here we have fixed the CPV angle α b = 0 . × − , then α c is determined using Eq. (43). We notethe two solutions α ± c give very similar plots. We fix the other parameters to be m H + = 420 GeV, m h = 400 GeV, m h = 450 GeV, ν = 1 . α = β − π/ We find the dominant contributions are always from hγγ or Hγγ ( H = h , h ) diagrams at low or high tan β regimes, respectively. Around tan β ≈ −
20, they have comparablemagnitudes and opposite signs, leading to a cancellation in the total d e . This cancellation leads to a sign change in d e but since we plot | d e | it appears as a spike going toward zero. The H ± W ∓ γ contribution can give as large as20% corrections at large tan β . The hZγ or HZγ contribution are always subdominant, because they are accidentallysuppressed by the small
Zee vector coupling, proportional to (1 − θ W ).The first column of Fig. 6 shows the ACME experimental constraint, where the blue region is excluded. Here, inorder to compare with the fit to LHC Higgs data results in Fig. 4, we have made the plots in the sin α b –tan β plane.Again, for given α b = 0 . × − there are two solutions for α c from Eq. (43): α ± c . We find that all the EDM constraintsfor both choices give very similar results. For the type II 2HDM, there are two cancellation regions in tan β . The onenear tan β ∼ hγγ type diagrams. The second region is near tan β ≈ −
20, is due to the cancellation between hγγ and We have checked that our choice of masses is consistent with constraints from electroweak oblique parameters. General expressions foroblique parameters in 2HDM were given by Grimus et al. [38]. The charged Higgs mass is chosen to be >
380 GeV in order to satisfy B → X s γ bounds [39]. We note that the reported 3.4 σ deviation of the rate for ¯ B → D ( ∗ ) τ − ¯ ν τ [40] from the SM prediction cannot be accommodated bythe 2HDM scenario analyzed here. For the parameter space we consider, the discrepancy does not increase, though introduction ofadditional interactions would be needed to account for the difference. A study of the possibilities goes beyond the scope of the presentstudy. FIG. 6: Current constraints from the electron EDM (left), neutron EDM (middle) and
Hg EDM (right).
First row : type-Imodel;
Second row : type-II model. In all the plots, we have imposed the condition that α = β − π/
2. The other parametersare chosen to be m H + = 420 GeV, m h = 400 GeV, m h = 450 GeV and ν = 1 .
0. Again, α c is a dependent parametersolved using Eq. (43). The purple region is theoretically not accessible because Eq. (43) does not have a real solution. Forthe neutron and Mercury EDMs, theoretical uncertainties from hadronic and nuclear matrix elements are reflected by differentcurves. For the neutron EDM, we vary one of the most important hadronic matrix elements: ˜ ζ dn = 1 . × − (solid, centralvalue), 0 . × − (dot-dashed) and 4 . × − (dashed). For the Mercury EDM, we take different sets of nuclear matrixelement values: a = 0 . , a = 0 .
02 (solid, central value). a = 0 . , a = 0 .
09 (long-dashed), a = 0 . , a = − .
03 (dashed), a = 0 . , a = 0 .
02 (dotted) and a = 0 . , a = 0 .
02 (dot-dashed).
Hγγ contributions. As we will show below, these cancellation regions can be closed when the neutron and mercuryEDM limits are taken into account. A generic feature is that for growing tan β , the EDM constraints become weakerin the type-I 2HDM, but become stronger in the type-II 2HDM, which can be understood from the tan β dependencesin Eq. (27). C. Ineffectiveness of a Light-Higgs-Only Theory
From the discussion of electron EDM, we have learned that the heavy Higgs contributions via
Hγγ and H ± W ∓ γ diagrams make non-negligible contributions to the total EDM. They can even be dominant at large tan β (cid:38)
20. Thisexample illustrates the ineffectiveness of the “light Higgs effective theory”, often performed as model independentanalyses, which include the CPV effects only from the lightest Higgs (mass 125 GeV). The key point is that a CPviolating Higgs sector usually contains more than one scalar at the electroweak scale, and all of them have CPVinteractions in general. The total contribution therefore includes CPV effects from not only CP even-odd neutralscalar mixings, but also the CPV neutral-charged scalar interactions from the Higgs potential. This is necessarilymodel dependent. In this work, we have included the complete contributions to EDMs in the flavor-conserving (type-Iand type-II) 2HDMs .6
TotalEDM CEDM3G1 2 5 10 20 5010 - - - - - - tan b d n H ec m L Total
EDMCEDM 3G1 2 5 10 20 5010 - - - - - - tan b d n H ec m L FIG. 7: The anatomy of various contributions to the neutron EDM in flavor conserving 2HDMs.
Left : type-I model;
Right :type-II model. We plot the absolute values, so the dip in the curves implies a sign change. The model parameters used are thesame as Fig. 5.
D. Neutron EDM Constraint
Next, we consider the neutron EDM, whose current bound is | d n | < . × − e cm. In Fig. 7, we plot the anatomyof neutron EDM, this time in terms of the various dimension-six operator contributions. The parameters are fixedas in Fig. 5, and the contributions to neutron EDM from light quark EDMs, CEDMs, and the Weinberg three-gluonoperator are shown as functions of tan β . The plot shows that in the type-II model, the quark CEDM contributionsto neutron EDM are larger than that those from quark EDMs. In type-I, these two contributions are similar in size.In both cases, the effect of the Weinberg operator is smaller. Also in both types, EDM and CEDM contributions havethe opposite sign, and total neutron EDM tends to be suppressed as a result. However, these statements depend onthe hadronic matrix elements being close to their current best value. Total u - C E D M - CEDM1 2 5 10 20 5010 - - - - - - - tan b g p NN H L Total u - CEDM3Gd - CEDM4 - quark1 2 5 10 20 5010 - - - - - tan b g p NN H L FIG. 8: The anatomy of various contributions to the ¯ g (0) πNN for atomic EDMs in flavor conserving 2HDMs. Left : type-I model;
Right : type-II model. We plot the absolute values, so the dip in the curves implies a sign change. The model parameters usedare the same as Fig. 5.
The second column of Fig. 6 shows the bounds in the sin α b –tan β plane. The green regions are excluded, for threedifferent choices of the hadronic matrix elements. Specifically, the down quark CEDM matrix element, ˜ ζ dn , takes thevalues 1 . × − (solid), 0 . × − (dot-dashed), and 4 . × − (dashed). This matrix element has a large impact,because the down quark CEDM is the largest Wilson coefficient for most values of tan β . In type-II model, in themost sensitive case with largest matrix element ˜ ζ dn = 4 . × − , α b is constrained to be of order 0.1 or smaller. In theleast sensitive case (˜ ζ dn = 0 . × − ), no part of the sin α b –tan β plane is excluded (with α = β − π/ dé u d u d d dé d C G é - - - - - tan b » W il s on c o e ff i c i e n t s » UV scale dé u d u d d dé d C G é - - - - - tan b W il s on c o e ff i c i e n t s IR scale dé u d u d d dé d C G é - - - - - tan b » W il s on c o e ff i c i e n t s » UV scale dé u d u d d dé d C G é - - - - - tan b » W il s on c o e ff i c i e n t s » UV scale
FIG. 9: The Wilson coefficients at the 2HDM scale (left) and the GeV scale (right).
First row : type-I model;
Second row :type-II model. We plot the absolute values, so the dip in the curves implies a sign change. The differences reflect the effects ofleading-order QCD corrections in the RG running. The model parameters used are the same as Fig. 5. contributions . E. Mercury EDM Constraint
We now come to the
Hg EDM limit of | d Hg | < . × − e cm. The anatomy of the isoscalar π NN coupling¯ g (0) πNN is shown in Fig. 8. Parameters are chosen as in Figs. 5 and 7, and contributions from up and down CEDM,Weinberg operator are plotted as functions of tan β . The four-quark operator involving the up and down quarks alsoadds to g (0) πNN , but this effect is negligible as shown in the type-II plot. In the type-I model, this effect is even smaller.The down quark CEDM gives the largest contribution to the π NN coupling in the type-II model, but the Weinbergoperator can be important for large tan β . In type-I, the up CEDM consistently makes up the largest bulk of g (0) πNN .The parameter space excluded by the Hg result is plotted in the right column of Fig. 6. As for the eEDM andnEDM plots, α is assumed to have the SM value of β − π/
2. The general shape of the excluded region is similar to theother two experiments, with weakest limits on α b near tan β = 1. As in the neutron case, our limits depend heavilyon the value of hadronic and nuclear matrix elements. For illustration, several choices of the nuclear matrix elements a and a are shown. The current best values of a = 0 . e fm and a = 0 . e fm are represented by the solid line.In general, larger absolute values for the matrix elements imply stronger bounds, as expected, and the locations ofcancellation regions are sensitive to the ratio between a and a . In the more sensitive cases, α b can be constrainedto 0.1 or smaller at all values of tan β . This cancellation depends crucially on the relative signs of the hadronic matrix elements; see below F. Hadronic and nuclear uncertainties
We have noted in the discussion of neutron and
Hg constraints the effects of hadronic and nuclear matrix elements.Currently, calculations of these matrix elements are riddled with large uncertainties [7]. For the magnitudes of thematrix elements, there is guidance from naive dimensional analysis, which takes into account the chiral structures ofthe operators in question. However, the precise value of matrix elements involving quark CEDMs and the Weinbergthree-gluon operator are only known to about an order of magnitude, and dimensional analysis does not tell us thesigns of the matrix elements. We highlight two places where these uncertainties can change our results. • In Figs. 7 and 8, we see that the Weinberg three-gluon operator is always subdominant as a contribution to theneutron and mercury EDMs. It is possible, though, that the actual matrix element may be an order of magnitudelarger than the current best value. Then, the Weinberg operator would make the largest contribution to theneutron and mercury EDMs at large tan β in the type-II model. • In the left panel of Fig. 7, the quark EDM and CEDM contributions to nEDM in the type-I model are shown tobe nearly equal, but with opposite signs, suppressing the total neutron EDM in the type-I model. If overall signof the CEDM matrix element is opposite to that used here, the two effects would add constructively, makingthe neutron EDM limit much stronger.In the absence of hadronic and nuclear matrix element uncertainties, improvements in neutron and diamagneticatom searches will make them competitive with present ThO result when in constraining CPV in 2HDM. At present,however, theoretical uncertainties are significant, making it difficult to draw firm quantitative conclusions regardingthe impact of the present and prospective neutron and diamagnetic EDM results.
G. QCD Running
Fig. 9 illustrates the differences in the Wilson coefficients between the UV (weak) scale and the IR (hadronic)scale. For the type-I model (top panels), there is no dramatic difference between magnitude of the coefficients otherthan a slight enhancement in quark EDMs and CEDMs. For the type-II model (bottom panels), however, there is asignificant difference in the tan β dependence and the relative magnitudes of the coefficients. Part of this differenceis explained by diagrams involving b quarks, which is not yet integrated out at the UV scale. Specifically, much ofthe growing behaviors at large tan β for the d-quark CEDM and the Weinberg operator are accounted for by simplyadding the b-quark diagrams. However, the growth of the u-quark EDM and u-quark CEDM at large tan β , and thesimilarity of EDM and CEDM for each quark at all values of tan β are mainly due to the operator mixing in QCD.For this reason, we observe that QCD running must be taken into account when making quantitative claims abouthadronic sources of CP violation. H. Combined EDM Constraints: Present and Future
We summarize the combined EDM constraints of the electron (blue region), neutron (green),
Hg (red) and
Ra(yellow) in Fig. 10. For the neutron and atomic EDMs, we use the central values for the hadronic and nuclear matrixelements. The first column shows the present constraints. We find that the bound on electron EDM from ACMEexperiment is presently by far the strongest, except for the cancellation region, which is closed by the mercury andneutron EDM bounds. The current constraints are roughly, α b , α c (cid:46) . β in type-II 2HDM. For the type-Imodel, the constraints are α b (cid:46) .
1, while α c can still be order one for tan β (cid:38)
5. Therefore, it could be easier tosearch for CPV effects related to the heavy scalars if they are discovered in the future collider experiments.The second column of Fig. 10 shows the future constraints if the neutron and Mercury EDM experiments improvethe current sensitivities by a factor of 10. We have also shown the future constraints in blue dashed curves if electronEDM is improved by another order of magnitude. The last column shows the situation when the neutron EDM limitis improved by a factor 100, which is the goal for the experiment planned for the Fundamental Neutron PhysicsBeamline at the Oak Ridge National Laboratory Spallation Neutron Source. We find for type-II 2HDM, the futureneutron EDM experiments can improve the current limit on the CPV angles α b , α c by one order of magnitude. Theradium EDM also has the prospect to give a comparable limit.Finally, we comment on the effects of changing the masses of the heavy scalars. While we have only presented theresults for the case when the extra scalars have masses of order ∼
400 GeV, we have performed the same analyseswith different masses of up to ∼
500 GeV. The constraints on the CPV angle α b become slightly weaker with heaviernew scalars, but we find that there is no qualitative difference to our conclusions.9 FIG. 10: Current and prospective future constraints from electron EDM (blue), neutron EDM (green), Mercury EDM (red) andRadium (yellow) in flavor conserving 2HDMs.
First row : type-I model;
Second row : type-II model. The model parametersused are the same as Fig. 6. Central values of the hadronic and nuclear matrix elements are used.
Left : Combined currentlimits.
Middle : combined future limits if the Mercury and neutron EDMs are both improved by one order of magnitude. Alsoshown are the future constraints if electron EDM is improved by another order of magnitude (in blue dashed curves).
Right :combined future limits if the Mercury and neutron EDMs are improved by one and two orders of magnitude, respectively.
V. SUMMARY
The nature of CPV beyond the Standard Model remains a question at the forefront of fundamental physics. Thecosmic matter-antimatter asymmetry strongly implies that such BSM CPV should exist, but the associated mass scaleand dynamics remain unknown. With the observation of the 125 GeV boson at the LHC, it is particularly interestingto ask whether the scalar sector of the larger framework containing the SM admits new sources of CPV and, if so,whether their effects are experimentally accessible. In this study, we have explored this question in the context offlavor conserving 2HDMs, allowing for a new source of CPV in the scalar potential. The present constraints on thistype of CPV are generally weaker than for scenarios where the BSM directly enters the couplings to SM fermions, asthe associated contributions to electric dipole moments generically first appear at two-loop order. In this context, wefind that present EDM limits are complementary to scalar sector constraints from LHC results, as the latter generallyconstrain the CP-conserving sector of the type-I and type-II models, whereas EDMs probe the CPV parameter space.Moreover, despite the additional loop suppression, the present ThO,
Hg, and neutron EDM search constraints arequite severe, limiting | sin α b | to ∼ .
01 or smaller for most values of tan β .The next generation of EDM searches could extend the present reach by an order of magnitude or more and couldallow one to distinguish between the type-I and type-II models. In particular, a non-zero neutron or diamagneticatom EDM result would likely point to the type-II model, as even the present ThO limit precludes an observableeffect in the type-I scenario given the planned sensitivity of the neutron and diamagnetic atom searches. Furthermore,0it appears that a combination of searches using different systems would be needed to achieve a comprehensive probeof the relevant parameter space in the type-II model. We emphasize, however, that these expectations are somewhatprovisional, given the present substantial uncertainties associated with computations of the hadronic and nuclearmatrix elements that we have quantified in this study. Achieving more robust computations would be particularlywelcome, especially given the role of neutron and diamagnetic atom EDM searches in probing the type-II flavorconserving 2HDM. Acknowledgements
We thank J. DeVries and W. Dekens for helpful discussion of the effective operator anomalousdimension matrix. The work is partially supported by the Gordon and Betty Moore Foundation through Grant No. 776to the Caltech Moore Center for Theoretical Cosmology and Physics and by DOE Grant DE-FG02-92ER40701 (YZ)and DE-SC0011095 (SI and MJRM). The work of YZ is also supported by a DOE Early Career Award under GrantNo. DE-SC0010255.
Appendix A: Wilson coefficients of P and T-odd operators at the 2HDM scale
In this appendix, we give the results of all the Wilson coefficients by integrating out the heavy particles at the2HDM scale, Λ ∼ M Z . At this scale, the bottom quark is still light, and we discuss the matching conditions at the m b scale in subsection III B. The total contributions to the Weinberg, CEDM and EDM operators are Eq. (A1), (A2)and (A12) respectively. h i gg gt f f fg tg h i f f fγ t , H + W + , G + γ , Z h i FIG. 11:
Left : two-loop contribution to the Weinberg operator with CPV neutral Higgs mixings.
Middle : quark CEDM fromBarr-Zee type diagrams with gh i exchange and CPV neutral Higgs mixings. The conjugate diagrams are not shown. Right :quark or lepton EDM from Barr-Zee type diagrams with γh i or Zh i exchange and CPV neutral Higgs mixings.
1. Two-loop Weinberg Operator
From Ref. [42], the contribution to the d = 6 Weinberg operator arises from the top loop, as shown in the left panelof Fig. 11, which gives C ˜ G (Λ) ≡ ( C ˜ G ) t = − g s π (cid:88) i =1 h ( m t /m h i ) c t,i ˜ c t,i , (A1)where the function h ( x ) can be found in the Appendix B.
2. Two-loop Barr-Zee type contributions to CEDMs
For light fermions, the dominant contributions to their EDMs and CEDMs come from the two-loop Barr-Zee typediagrams [43], as shown in the middle and right panels of Fig. 11.1For the CEDM, the top quark in the upper (shaded) loop is first integrated out to obtain the h i GG or h i G ˜ G operators, which then contribute to the CEDM operators [44],˜ δ q (Λ) ≡ (cid:16) ˜ δ q (cid:17) hggt = − g s π (cid:88) i =1 (cid:2) f ( z it ) c t,i ˜ c q,i + g ( z it )˜ c t,i c q,i (cid:3) , (A2)where q = u, d, b , and z it = m f /m h i . The two-loop functions f ( x ) , g ( x ) can be found in the Appendix B.
3. Two-loop Barr-Zee type contributions to EDMs: diagrams with H γγ and H Zγ The corresponding Barr-Zee type EDMs for light fermions are obtained with operators h i F F or h i F ˜ F from theupper (shaded) loop. See the right panel of Fig. 11. The contribution from top quark is( δ f ) hγγt = − N c Q f Q t e π (cid:88) i =1 (cid:2) f ( z it ) c t,i ˜ c f,i + g ( z it )˜ c t,i c f,i (cid:3) . (A3)Here the external fermions relevant for our calculations are f = u, d, e . The analog contribution is to replace thephoton propagator with that of the Z -boson. It is worth noting that only the vector current of the Z ¯ f f couplingenters in the final EDM, which is( δ f ) hZγt = − N c Q f g VZ ¯ ff g VZ ¯ tt π (cid:88) i =1 (cid:104) ˜ f ( z it , m t /M Z ) c t,i ˜ c f,i + ˜ g ( z it , m t /M Z )˜ c t,i c f,i (cid:105) , (A4)with g Vf ¯ fZ = g ( T f − Q f sin θ W ) / (2 cos θ W ). The loop function ˜ f ( z, x ) and ˜ g ( z, x ) can be found in the Appendix B.The corresponding bottom quark loop contribution is properly taken into account in sec. III B.In the right panel of Fig. 11, the particles in the upper (shaded) loop can also be the W -boson and its Goldstoneboson. The gauge invariant contributions have been obtained in [45, 46],( δ f ) hγγW = Q f e π (cid:88) i =1 (cid:20)(cid:18) z iw (cid:19) f ( z iw ) + (cid:18) − z iw (cid:19) g ( z iw ) (cid:21) a i ˜ c f,i , (A5)( δ f ) hZγW = g VZ ¯ ff g ZW W π (cid:88) i =1 (cid:20)(cid:18) − sec θ W + 2 − sec θ W z iw (cid:19) ˜ f ( z iw , cos θ W )+ (cid:18) − θ W − − sec θ W z iw (cid:19) ˜ g ( z iw , cos θ W ) (cid:21) a i ˜ c f i , (A6)where z iw = M W /m h i and g W W Z /e = cot θ W .Similarly, the physical charged scalar can also run in the loop of Fig. 11. This is similar to the squark contributiondiscussed in the supersymmetric framework [47]. With the couplings defined in Eq. (30), its contributions to EDMare ( δ f ) hγγH + = Q f e π (cid:18) vm H + (cid:19) (cid:88) i =1 (cid:2) f ( z iH ) − g ( z iH ) (cid:3) ¯ λ i ˜ c f,i , (A7)( δ f ) hZγH + = g VZ ¯ ff g ZH + H − π (cid:18) vm H + (cid:19) (cid:88) i =1 (cid:104) ˜ f ( z iH , m H + /M Z ) − ˜ g ( z iH , m H + /M Z ) (cid:105) ¯ λ i ˜ c f,i , (A8)with z iH = m H + /m h and g ZH + H − /e = cot θ W (1 − tan θ W ) /
4. Two-loop Barr-Zee type contributions to EDMs: diagrams with H ± W ∓ γ The left panel of Fig. 12 represents the contribution where the upper loop yields an H ± W ∓ γ operator. Thiscontribution has not been included in the 2HDM calculations until very recently [46] (see [48] for the counterpart insupersymmetric models).Here we would like to stress that it arises from the only source of CP violation in the Higgs potential. In theeffective theory language, the possible gauge invariant operators for the upper (shaded) loop include C ij φ † i σ a W aµν φ j B µν , ˜ C ij φ † i σ a W aµν φ j ˜ B µν , (A9)where C ij , ˜ C ij are the Wilson coefficients. Because of the CP properties, fermion EDMs are proportional to theimaginary part of C ij or the real part of ˜ C ij . Here we argue that in the flavor conserving 2HDMs discussed in thiswork, only the scalar loop could contribute to C and eventually to EDMs. A representative diagram is shown inthe right panel of Fig. 12. It is proportional toIm( λ m ∗ v ∗ v ) = − (cid:12)(cid:12) λ m v v (cid:12)(cid:12) sin δ . (A10)Using the relation in Eq. (13), the above quantity is indeed related to the unique CPV source in the model.The fermionic loops do not contribute because the physical charge Higgs and quark couplings have the structureproportional to the corresponding CKM element. As a result, the coefficients C ij are purely real and ˜ C ij are purelyimaginary. They contribute to magnetic dipole moments instead of EDMs. f f ! fγ H /H + W ± H ∓ H +2 H +2 W + H +1 H H H γ FIG. 12:
Left : quark or lepton EDM from W ± H ∓ exchange and CPV Higgs interactions. Right : a scalar loop contributionto φ † σ a W aµν φ B µν effective operator, which then contributes to EDM as the upper loop of the left panel. The gauge invariant contributions to EDM from this class of diagrams have been calculated recently in [46],( δ f ) HW γH = 1512 π s f (cid:88) i (cid:20) e θ W I ( m h i , m H + ) a i ˜ c f,i − I ( m h i , m H + )¯ λ i ˜ c f,i (cid:21) , (A11)where the functions I , ( m , m ) are given in the Appendix B. The coefficient s f = − s f = +1 for down-type quarks and charged leptons.To summarize, the total contribution to fermion EDM is the sum of Eqs (A3,A4,A5,A6,A7,A8,A11), δ f (Λ) ≡ ( δ f ) hγγt + ( δ f ) hZγt + ( δ f ) hγγW + ( δ f ) hZγW + ( δ f ) hγγH + + ( δ f ) hZγH + + ( δ f ) HW γH . (A12)3 Appendix B: Loop functions
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