Confronting Higgcision with Electric Dipole Moments

Current data on the signal strengths and angular spectrum of the 125.5 GeV Higgs boson still allow a CP-mixed state, namely, the pseudoscalar coupling to the top quark can be as sizable as the scalar coupling: $C_u^S \approx C_u^P =1/2$. CP violation can then arise and manifest in sizable electric dipole moments (EDMs). In the framework of two-Higgs-doublet models, we not only update the Higgs precision (Higgcision) study on the couplings with the most updated Higgs signal strength data, but also compute all the Higgs-mediated contributions from the 125.5 GeV Higgs boson to the EDMs, and confront the allowed parameter space against the existing constraints from the EDM measurements of Thallium, neutron, Mercury, and Thorium monoxide. We found that the combined EDM constraints restrict the pseudoscalar coupling to be less than about $10^{-2}$, unless there are contributions from other Higgs bosons, supersymmetric particles, or other exotic particles that delicately cancel the current Higgs-mediated contributions.


I. INTRODUCTION
Since the observation of a new boson at a mass around 125.5 GeV at the Large Hadron Collider (LHC) [1,2], the most urgent mission is to investigate the properties of this new boson. There have been a large number of studies or fits of the Higgs boson couplings to the standard model (SM) particles in more or less model-independent frameworks , in the two-Higgs doublet model (2HDM) frameworks [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50], and in the supersymmetric frameworks [51][52][53][54][55]. Based on a study using a generic framework for Higgs couplings to the relevant SM particles, three of us has reported [21] that the SM Higgs boson [56] provides the best fit to all the most updated Higgs data from ATLAS [57][58][59][60], CMS [61][62][63][64][65][66][67], and Tevatron [68,69]. In particular, the relative coupling to the gauge bosons is restricted to be close to the SM values with about a 15% uncertainty while the Yukawa couplings are only loosely constrained. Furthermore, the hypothesis of a pure CP-odd state for the new boson has been mostly ruled out by angular measurements [70,71]. Nevertheless, there is still a large room for the possibility of a CP-mixed state [21].
If the Higgs boson is a CP-mixed state, it can simultaneously couple to the scalar and pseudoscalar fermion bilinears as follows: where g f = gm f /2M W = m f /v with f = u, d, l denoting the up-and down-type quarks and charged leptons collectively. We will show that non-zero values of the products proportional to g S Hf f × g P Hf (′) f (′) and g P Hf f × g HV V signal CP violation as manifested in nonzero values for electric dipole moments (EDMs) 1 . The non-observation of the Thallium ( 205 Tl) [72], neutron (n) [73], Mercury ( 199 Hg) [74], and thorium monoxide (ThO) [75] EDMs provide remarkably tight bounds on CP violation. The EDM constraints in light of the recent Higgs data were studied in Ref. [76]. Strictly speaking, only the Higgs couplings to the thirdgeneration fermions such as the top and bottom quarks and tau leptons are relevant to the current Higgs data. On the other hand, the EDM experiments mainly involve the firstgeneration fermions. Therefore, it is impossible to relate the Higgs precision (Higgcision) constraints to EDMs in a completely model-independent fashion without specifying the relations among the generations, except for the Weinberg operator. In most of the models 1 Here, g HV V denotes a generic Higgs coupling to the massive vector bosons in the interaction L HV V = studied in literature, however, the Higgs couplings to the third-generation fermions are related to those of the first-generation in a model-dependent way. In this work, to be specific, we study the contributions of the observed 125.5 GeV "Higgs" boson (H) to EDMs in the framework of 2HDMs.
The paper is organized as follows. In Sec. II we briefly describe ingredients in the framework of 2HDMs we are working with and present the 2HDM Higgcision fit to the most updated Higgs data. For notation and more details of the 2HDMs we refer to Ref. [50].
Section III is devoted to the synopsis of EDMs. In Sec. IV we present our numerical results, and summarize our findings and draw conclusions in Sec. V.

II. TWO HIGGS DOUBLET MODELS
In Ref. [50], neglecting the charged Higgs contribution to the loop-induced Higgs couplings to two photons, it was shown that the Higgcision studies in 2HDM framework can be performed with a minimum of three parameters, given by where H = h i denotes the candidate of the 125.5 GeV Higgs among the three neutral Higgs bosons h 1,2,3 in 2HDMs without further specifying which one the observed one is. The mixing between the mass eigenstates h 1,2,3 and the electroweak eigenstates φ 1 , φ 2 , a is described by and tan β in the four types of 2HDMs, see Ref. [50] for details of conventions in 2HDMs.

2HDM I
Once the three parameters C S u , C P u , and C v are given, the H couplings to the SM fermions are completely determined as shown in Table I We are using the abbreviations: s β ≡ sin β, c β ≡ cos β, t β = tan β, etc, and the convention of C v > 0.
FIG. 1. The confidence-level regions of the fit to the most updated Higgs data by varying C S u , C P u , and C v in the plane of C S u vs C P u for Type I -IV. The contour regions shown are for ∆χ 2 ≤ 2.3 (red), 5.99 (green), and 11.83 (blue) above the minimum, which correspond to confidence levels of 68.3%, 95%, and 99.7%, respectively. The best-fit points are denoted by the triangles.
In Fig. 1, we show the confidence-level (CL) regions of the fit to the most updated Higgs data by varying C S u , C P u , and C v in the plane of C S u vs C P u for Type I -IV of the 2HDMs.
Comparing to Fig. 11 in Ref. [50] for the CPV3 fit, the CL regions are mildly reduced, preferring positive C S u values slightly more than the negative ones, after the inclusion of the most recent results from H → bb [59,66,67] and τ + τ − [60,65]. Meanwhile, we note that the maximal CP violation with C S u ∼ |C P u | is still possible.

III. SYNOPSIS OF EDMS
Here we closely follow the methods used in Refs. [77][78][79][80] in the calculations of the 125.5-GeV Higgs-mediated contributions to the EDMs. We start by giving the relevant interaction Lagrangian as where F µν and G a µν are the electromagnetic and strong field strengths, respectively, the T a = λ a /2 are the generators of the SU(3) C group and G µν = 1 2 ǫ µνλσ G λσ is the dual of the SU(3) c field-strength tensor G λσ .
We denote the EDM of a fermion by d E f and the chromoelectric dipole moment (CEDM) of a quark by d C q . The major Higgs-mediated contribution comes from the two-loop Barr-Zee-type diagrams, labeled as the details of which will be discussed below. For the Weinberg operator, we consider the contributions from the Higgs-mediated two-loop diagrams: where z Hq ≡ M 2 H /m 2 q with M H = 125.5 GeV and, for the loop function h(z Hq ), we refer to Ref. [81]. We note, in passing, that (d G ) H depends on the H couplings to the thirdgeneration quarks only. For the four-fermion operators, we consider the t-channel exchanges of the CP-mixed state H, which give rise to the CP-odd coefficients as follows [77]: with r xy ≡ M 2 x /M 2 y . For the loop function J(a, b) we again refer to, for example, Refs. [77,78] and references therein. The Z-boson couplings to the quarks and leptons are given by with and a Zf f = T f 3L /2 and g Z = g/c W = (e/s W )/c W . For the SM quarks and leptons, T u,ν 3L = +1/2 and T d,e 3L = −1/2. In addition to EDMs, the two-loop Higgs-mediated Barr-Zee graphs also generate CEDMs of the light quarks q l = u, d, which take the form: g P Hq l q l g S Hqq f (τ qH ) + g S Hq l q l g P Hqq g(τ qH ) .

B. Observable EDMs
In this subsection, we briefly review the dependence of the Thallium, neutron, Mercury, deuteron, Radium, and thorium-monoxide EDMs on the EDMs and/or CEDMs of quarks and leptons, and on the coefficients of the dimension-six Weinberg operator and the fourfermion operators.

Thallium EDM
The Thallium EDM receives contributions mainly from two terms [86,87]: where d E e is the electron EDM and C S is the coefficient of the CP-odd electron-nucleon interaction L C S = C Sē iγ 5 eNN, which is given by with κ ≡ N|m ss s|N /220 MeV ≃ 0.50 ± 0.25 and with x t = 1 and x b = 1 − 0.25κ.

Thorium-Monoxide EDM
Similar to the Thallium EDM, the thorium-monoxide EDM is given by [88]: Currently, the experimental constraint is given on the quantity |d ThO /F ThO |.

Neutron EDM
For the neutron EDM we take the hadronic approach with the QCD sum-rule technique.
In this approach, the neutron EDM is given by [89][90][91][92][93] where d E q and d C q should be evaluated at the electroweak (EW) scale and d G at the 1 GeV scale, for which [91]. In the numerical estimates we take the positive sign for both d n (d G ) and d n (C bd ).

Mercury EDM
Using the QCD sum rules [92,93], we estimate the Mercury EDM as where d I ,II ,III ,IV

Hg
[S] denotes the Mercury EDM induced by the Schiff moment. The parameters C P and C ′ P are the couplings of electron-nucleon interactions as in L C P = C Pē eN iγ 5 N + C ′ Pē eN iγ 5 τ 3 N and they are given by [77] In this work, we take d I Hg [S] for the Schiff-moment induced Mercury EDM, which is given by [79] whereḡ (1)

Deuteron EDM
For the deuteron EDM, we use [77,94]: In the above, d G is evaluated at the 1 GeV scale, and the coupling coefficients g d,s,b appearing in C dd,sd,bd are computed at energies 1 GeV, 1 GeV and m b , respectively. All other EDM operators are calculated at the EW scale. In the numerical estimates we take the positive sign for d G .

Radium EDM
For the EDM of 225 Ra, we use [79]: We note that theḡ (1) πN N contribution to the Radium EDM is about 200 times larger than that to the Mercury EDM d I Hg [S] [95].

IV. NUMERICAL ANALYSIS
The non-observation of EDMs for Thallium [72], neutron [73], Mercury [74], and thorium monoxide [75] constrains the CP-violating phases through with the current experimental bounds For the normalization of the deuteron and Radium EDMs, we have taken the projected experimental sensitivity [96] to be d PRJ D = 3 × 10 −27 e cm and d PRJ Ra = 1 × 10 −27 e cm, respectively. The chosen value for d PRJ Ra is near to a sensitivity which can be achieved in one day of data-taking [97].

A. (C)EDMs of quarks and leptons and d G
In this subsection, we analyze the contributions of the Higgs boson H with the mass 125.5 GeV in the 2HDM framework to • EDMs of electron and up and down quarks: • CEDMs of up and down quarks:  together with their constituent contributions, taking the benchmark point while varying C v .
In Fig. 2, we show the electron EDM as a function of C v in units of e cm with C S u = C P u = 1/2 for the types I -IV of 2HDMs. The red and blue solid lines are for (d E e ) γH and (d E e ) ZH , respectively, and the black solid lines are for the total sum. The constituent contributions from the top, bottom, tau, and W-boson loops are denoted by the dashed black, red, blue, and magenta lines, respectively. In all types of 2HDMs, we observe that for Type I,IV and II, III, respectively: see Eq. (11).
We observe that the electron EDM is overall proportional to C P u and it flips the sign according to the change in the sign of C P u . The top and W contributions have the same signs, and the top-quark contributions are independent of C v in Types I and IV. Also, note that the two top-quark contributions in Types I and IV cancel each other so that the top-quark contribution is suppressed compared to that in Types II and III. For the reference point C S u = C P u = 1/2, we show tan β and O φ 1 i / cos β as functions of C v in Fig. 3   the electron EDM case, the up-quark EDM satisfies which are independent of the 2HDM type. We find J Z W (M H ) ≃ 5.5. The top-quark contribution is negative and the W -loop contribution is positive because v Zūu > 0. One may see (d E u ) γH vanishes when the first two terms cancel and a cancellation may also occur between (d First we note that all three contributions in Types I and III are positive because v Zdd < 0.
As in the electron EDM, we find the top-quark contributions are independent of C v . In independent of the 2HDM types and of C v : see Eq. (16).
In Fig. 7, we show the absolute values of the down-quark CEDM as a function of C v in units of cm with C S u = C P u = 1/2 for Types I -IV of 2HDMs. The labeling of lines is the same as in Fig. 6. The dominant top-quark loop contributions are proportional to for Types I,III and II,IV, respectively: see Eq. (16). Therefore, in Types I and III, the top contributions are independent of C v , while in Types II and IV there is cancellation around and, accordingly, they are independent of C v .
Before closing this subsection, we offer the following comments on the sizes of (C)EDMs of the light quarks and electron, and d G .  Therefore, the most significant constraints come from the thorium-monoxide EDM through d E e , Mercury EDM through d C u,d , and neutron EDM through d G . We are going to present more details in the next subsection.

B. Observable EDMs
In this subsection, we numerically analyze the Thallium, thorium-monoxide, neutron, and Mercury EDMs together with their constituent contributions, taking the benchmark point In Fig. 9, we show the Thallium EDM normalized to the current experimental limit in Eq. (30) as functions of C v , and in Fig. 10   EDM constraints are shown to be weaker in Types II and III. It is interesting to note that the thorium-monoxide EDM even shows a sensitivity to the C S contribution. Figure 11 shows the neutron EDM (black sold lines) and its constituent contributions from d E u,d , d C u,d , d G , and the four-fermion operators as functions of C v taking C S u = C P u = 1/2. We observe |d n /d EXP n | < ∼ 10. We also observe the d C u,d (red dashed lines) and d G (blue dashed lines) contributions dominate and they have opposite signs to each other except for the regions near C v = 0 in Types II and IV. The cancellation between the d C u,d and d G contributions is most prominent at C v = 0.25 in Types II and IV, but the milder cancellation around C v = 1 is phenomenologically more important because the current Higgs data prefer the region around C v = 1. The cancellation around C v = 1 makes the neutron EDM  constraints in Types II and IV weaker than in Types I and III, as shown in Fig. 11. We note that, in Types I and III the neutron EDM also show a sensitivity to the d E u,d EDMs (black dashed lines) near C v = 1.

C. EDM Constraints
In this subsection, we present the CL regions in the C S u -C P u plane which satisfy the current Higgs-boson data and various EDM constraints.
In Fig. 13, we show the allowed regions satisfying the Higgs-boson data and the thoriummonoxide EDM constraint at 68.3% (red), 95% (green), and 99.7% (blue) CL in the plane of C S u vs C P u for Types I -IV. We recall that the CL regions before applying the EDM constraints have been shown in Fig. 1. For each allowed point in the C S u -C P u plane in Fig. 1  1 is satisfied while varying C v within the corresponding CL regions 4 . We observe that C P u = 0 is strongly constrained in Types I and IV. While in Types II and III, the constraints are weaker in the regions centered around the point C S u = 1 due to the cancellation between the top-and W -loop contributions to the dominant electron EDM: see Figs. 2 and 10. We find that |C P u | can be as large as ∼ 0.6 for Types II and III at 95% CL (green regions). Figure 14 shows the allowed regions satisfying the Higgs-boson data and the neutron EDM constraint at 68.3% (red), 95% (green), and 99.7% (blue) CL, respectively, in the plane of C S u vs C P u for Type I -IV. The allowed regions are obtained in the same way as in The same as in Fig. 1 but with the thorium-monoxide EDM constraint the case of thorium-monoxide. The neutron EDM constraint is weaker in Types II and IV due to the cancellation between the d C u,d and d G contributions around C v = 1: see Fig. 11. We find that |C P u | can be as large as ∼ 0.6 for Types II and IV at 95% CL (green regions). Figure 15 is the same as in Figures 13 and 14 but with the Mercury EDM constraint applied. In contrast to the weaker thorium-monoxide (neutron) EDM constraint in Types II and III (Types II and IV), the Mercury EDM constraint is almost equally stringent in all four types and, specifically, |C P u | is restricted to be ∼ 0.1 for Types II and IV. The combined constraint at 95% CL from all the EDMs measurements and the Higgsboson data is obtained in Fig. 16. The black regions in Fig. 16 shows the 95 % CL regions satisfying the Thallium, thorium-monoxide, neutron, and Mercury EDM constraints simultaneously, as well as the Higgs-boson data. We find that the combination of all available EDM experiments provide remarkably tight bounds on CP violation. Thus, non-zero values of C P u are stringently restricted as FIG. 14. The same as in Fig. 1 with the relaxation factor r = 10 (orange), 30 (pink), and 100 (green). The factor r,

V. CONCLUSIONS
In this work, we have updated the Higgcision constraints on the Higgs boson couplings to SM gauge bosons and fermions, and confronted the allowed parameter space in C S u , C P u , and C v against various EDM constraints from the non-observation of the Thallium ( 205 Tl), thorium-monoxide (ThO), neutron, and Mercury ( 199 Hg) EDMs, in the framework of 2HDMs. Although the Higgs boson data still allow sizable C P u , the combined EDM constraints restrict |C P u | to a very small value of ∼ 10 −2 .
We have only considered the contributions from the 125.5 GeV Higgs boson via the Higgs-mediated diagrams in this work. There could potentially be contributions from other particles of any new physics models, e.g., the heavier Higgs bosons of multi-Higgs models, supersymmetric particles, or any other exotic particles that carry CP-violating couplings. These contributions and the contributions from the 125.5 GeV Higgs boson could cancel each other in a delicate way. If we allow 1% fine tuning, the constraints on the pseudoscalar coupling C P u are relaxed and |C P u | as large as 0.5 can be allowed.
In the following we offer a few more comments before we close.  9. We find that the deuteron and Radium EDMs can be ∼ 10 times as large as the projected experimental sensitivities even when |C P u | is restricted to be smaller than about 10 −2 by the combined EDM constraints.
Note Added: After the completion of this work, we received a paper [98], which addresses the LHC Higgs and EDM constraints in Types I and II 2HDMs.