Electroweak Baryogenesis, Electric Dipole Moments, and Higgs Diphoton Decays

We study the viability of electroweak baryogenesis in a two Higgs doublet model scenario augmented by vector-like, electroweakly interacting fermions. Considering a limited, but illustrative region of the model parameter space, we obtain the observed cosmic baryon asymmetry while satisfying present constraints from the non-observation of the permanent electric dipole moment (EDM) of the electron and the combined ATLAS and CMS result for the Higgs boson diphoton decay rate. The observation of a non-zero electron EDM in a next generation experiment and/or the observation of an excess (over the Standard Model) of Higgs to diphoton events with the 14 TeV LHC run or a future $e^+e^-$ collider would be consistent with generation of the observed baryon asymmetry in this scenario.


I. INTRODUCTION
The Standard Model of (SM) of particle physics provides an excellent description of a wide variety of laboratory and astrophysical observations. With the discovery of the Higgslike scalar at the CERN LHC [1][2][3][4], the SM Higgs mechanism for spontaneous breaking of the gauge symmetry in the SM [5] appears to be a correct description of nature. More precise measurements of Higgs boson properties will help determine whether there exist additional degrees of freedom that participate in electroweak symmetry-breaking or otherwise generate new Higgs boson interactions. Among the most interesting observables is the rate for the Higgs to decay to two photons, a process generated at one-loop order in the SM. At present, the results for this rate obtained from the LHC are somewhat ambiguous. The associated signal strength, normalized to the SM expectation and measured by the ATLAS collaboration, is somewhat greater than unity: µ γγ = 1.55 +0. 33 −0.28 [2], whereas the CMS collaboration finds a value completely consistent with the SM: µ γγ = 0.77 ± 0.27 [4] . Combining the H → γγ, ZZ * , W W * channels ATLAS obtains the signal strength µ = 1.33 ± 0.14(sat) ± 0.15(sys) for a fixed mass hypothess corresponding to the measured value m H = 125.5 GeV. The corresponding CMS combined result is µ = 0.80 ± 0.14 for the fixed mass hypothesis m H = 125.7 GeV. It is clear that one of the key tasks for the 14 TeV run of the LHC will be to obtain more precise determinations of these rates, as they might show the evidence of Higgs boson couplings to new particles beyond those of the SM.
One motivation for the possible existence of such particles with sub-TeV scale masses is the as yet unexplained origin of the baryon asymmetry of the Universe (BAU). Combining the WMAP seven year results [6] with those from other CMB and large scale structure measurements one obtains where ρ B is the baryon number density, s is the entropy density of the Universe. The recent results obtained by the Planck satellite are consistent, giving Y B = (8.59 ± 0.11) × 10 −11 [7].
Assuming that the Universe was matter-antimatter symmetric at its birth, it is reasonable to suppose that interactions involving elementary particles generated the BAU during the subsequent cosmological evolution. To generate the observed BAU, three Sakharov criteria [8] must be satisfied in the early Universe: (1) baryon number violation; (2) C and CP violation; (3) a departure from the thermal equilibrium (assuming exact CPT invariance). These requirements are realizable, though doing so requires physics beyond the SM.
To that end, theorists have proposed a variety of baryogenesis scenarios whose realization spans the breadth of cosmic history. Electroweak baryogenesis (EWBG) [9][10][11][12][13][14][15][16] is one of the most attractive and promising such scenarios, and it is generally the most testable with a combination of searches for new degrees of freedom at the LHC and low-energy tests of CP invariance. In this respect, searches for permanent electric dipole moments (EDMs) of neutral atoms, molecules and the neutron present provide one of the most powerful probes of possible new electroweak scale CP-violating interactions [17][18][19] that may be responsible for EWBG.
In this paper, we focus on the EWBG and µ γγ implications of the most recent EDM search null result obtained by the ACME experiment using the ThO molecule, from which one may derive a limit on the electron EDM [20]: |d e | < 8.7 × 10 −29 e · cm with 90% confidence. In earlier work, the authors of Ref. [21]  fermions whose left and right-handed components transform in the same way under the 1 For other recent work investigating the interplay of EDMs, Higgs decays, and EWBG, see Ref. [22]. In that work, the authors considered a space-time varying CPV phase of the Higgs background field, a complementary approach to the one followed here.
SM gauge group). In this scenario, a strong first order phase transition is induced by the scalar potential (see, e.g., Refs. [24,25]), while new (physical) CP-violating phases enter the mass matrix of the vector-like fermions as well as the scalar potential. In what follows, we concentrate on the new CP-violation in the Higgs-vector fermion interactions 2 .
As we show below, these interactions can lead to a resonantly enhanced CP-violating source for EWBG. Because the relevant CP-violating parameter space for this relatively simple SM extension is fairly extensive, we restrict our attention to one illustrative parameter space region and demonstrate that the observed BAU can be obtained in this scenario while respecting the electron EDM constraints. A more extensive study of the parameter space will appear in a follow-up study. We also study the impact of the new fermion-scalar interactions on the Higgs diphoton rate. We find the regions favored by the observed Higgs diphoton rate and non-observation of the electron EDM overlap with regions of parameter space wherein a sizable portion of the baryon asymmetry is generated. Looking to the future, we analyze the impact of order-of-magnitude improvements in the sensitivities of both electron EDM and µ γγ probes. For the general case, the electron EDM would provide a substantially more powerful probe of the EWBG-viable parameter space. However, for scenarios where the EDM effect is suppressed (e.g., due to mixing with a SM-gauge singlet [21]), the Higgs diphoton rate may then yield an interesting sensitivity.
Our discussion of these points in the remainder of the paper is organized as follows: In Section II we describe the model in detail. Section III is devoted to a study of the EDM and modified Higgs diphoton rate. We study EWBG in Section IV. We summarize in Section VI.
A discussion of the re-phasing invariants in this scenario and their relation to the relevant couplings appears in the Appendix.

II. MODEL
We work in the Type-I 2HDM augmented by a pair vector-like fermion doublets ψ L,R , transforming as (1, 2, −1/2) and a pair vector-like fermion singlets χ L,R , transforming as 2 For other theoretical and phenomenological implications of vector-like fermions see, for example, Refs. .
(1, 1, −1). The Yukawa Lagrangian for the new fermions can be written as The mass matrix for the charged vector like fermions is then where the vacuum expectation values (VEVs) given by H i = v i , (i = 1, 2) and v 2 1 + v 2 2 = 174 GeV. Note that Eq. (3) contains a physical phase that cannot be rotated away by field redefinitions.
Although we will not consider explicit CP-violation in the scalar potential in our study, it is nevertheless useful to consider the rephasing invariants that one may construct from the parameters in Eq. (2) and the scalar potential. To that end, we follow Ref. [51] that Additional CP-violation may arise from quartic interactions, such as the Z 2 -symmetric term . As discussed in Appendix A, the rephasing invariants can be written as , θ 5 ≡ Arg(y 1 y * 2 M 2 12 ) and θ 6 ≡ Arg(y ′ 1 y ′ * 2 M 2 12 ). Including additional scalar self-interactions, term would introduce additional rephasing invariants, such as θ 7 ≡ Arg(λ 5 M 4 * 12 ). For a more detailed discussion of the CP-violating phases and rephasing invariants, see Appendix A.
In what follows, we will assume that the parameters in the scalar potential are all real and concentrate on the effects of CP-violation in the Yukawa sector (2).
To solve for the mass eigenvalues, we diagonalize the mass matrix by 2 × 2 unitary In the mass eigenbasis the mass eigenvalues can be written aŝ where The mixing angles and phases are where , θ L,R and δ L,R are the mixing angles and phases of U L,R , respectively. Notice however that U L,R are not completely determined by the They can be multiplied from the right by an arbitrary phase rotation which contains two phases that do not depend can be solely determined by the parameters in the mass matrix. As shown in Appendix A, θ L and θ R are separately rephasing invariant, while δ L and δ R are not.
We note that the mass of the neutral component of ψ is not always below that of the lighter charged state. In order to avoid the existence of a stable charged relic, it is possible to extend the model with additional electroweak singlets ξ L,R whose Yukawa interactions with the ψ L,R lead to the presence of a lightest neutral state after electroweak symmetrybreaking. Assuming the new Yukawa couplings are real, introduction of these new fields and interactions will not affect the Higgs diphoton decay rate, EDM, or EWBG. An extensive analysis of such as scenario will appear in forthcoming work [52]. For either the latter scenario or for the model considered here when the neutral states are the lightest, one could search for the vector like fermions at the LHC in the diboson plus missing energy channel. As shown, for example, in Ref. [31], the present LHC data do not preclude the existence of these fermions for masses in the several hundred GeV and above range.

III. THE HIGGS TO DIPHOTON RATE AND ELECTRON EDM
In the SM, the leading contribution to the Higgs coupling to a diphoton pair is generated by the W boson loop, which is at least four times larger than the next-to-leading contribution from the top quark loop. New charged fermions generate additional loop level contributions.
The analytical expression for the signal strength µ γγ reads Here s β = sin β with β = arctan(v 1 /v 2 ), N C = 3 is the number of the colors, Q t,χ,ψ are electric charge of the top quark and new fermion in , and the η i are couplings of new charged fermions to the SM Higgs boson. For tan β ∼ 1, global fits to the LHC Higgs boson rates imply that the [51], for the Type-I(II) 2HDM. For illustrative purposes, we will take H 0 1 to be the SM-like Higgs boson with cos α = 1. In this limit, the η i are given by We refer the read to Appendix A for details of the derivation of Eqs. (11) and (12). The explicit expressions for A 1/2 (x) and f (x) can be found in Ref. [29]. In the presence of mixing between the two neutral CP-even scalars, the RHS of Eq. (10) is multiplied by a factor of cos 2 α.
The CP-odd Yukawa couplings given in Eqs. (11) and (12) generate an elementary fermion EDM via two-loop Barr-Zee diagrams [53]. For illustrative purposes, we will work in the limit that the masses of the remaining neutral scalars (one CP-even and one CPodd) and charged scalars are sufficiently heavy that the dominant contributions arise from exchange of (a) the SM-like neutral scalar and a photon and (b) a W + W − pair. CP-violation enters the latter contribution through the relative phase of left-and right-handed charged currents. The result is well-known, and specializing to our case we arrive at the following result for the electron EDM [21]: where d and j(x, y) are given in Ref. [21]. The first term arises from W + W − -exchange and depends on Θ ≡ δ L − δ R + Arg(M ψ ) − Arg(M χ ), the relative phase between the left-and right-handed currents that is rephasing invariant as shown in Appendix A. The second term is generated by the H 0 1 γ-exchange graphs. Note that Θ → 0 in the limit that |M ψ | = |M χ |, whereas the second term remains non-zero in this regime. As we discuss in Section IV, EWBG is most viable in the regime that |M ψ | ∼ |M χ |, in which case d e will be dominated by the H 0 1 γ-exchange contribution. Apparently, Eqs. (11) and (12) are rephasing invariant and can be expressed as functions of θ j (j = 1, 3, 4, 5, 6). Since we have assumed that the mixing between the neutral Higgs fields is negligible, the phase that gives the dominant contribution to the CP-odd Hγγ coupling should be θ 1 [34]. Thus, the couplings in Eqs. (11) and (12) should govern both the magnitude of any impact on the H → γγ rate as well as d e in the |M ψ | ∼ |M χ | regime that is most relevant for EWBG. Doing so is particularly timely in light of the recent ACME result [20], from which an order of magnitude more stringent d e limit has been obtained (assuming the absence of any other CP-violating sources in the ThO molecule). As we will show in Section IV, the resulting constraints on the nevertheless leave ample room for successful EWBG.
To illustrate, we will work in a simplified region of parameter space that still allows us to assess general features of the EWBG-EDM-Higgs diphoton interplay. Specifically, we assume M χ , M ψ to be real and set y 1 = ye iδ , y ′ 1 = y where y is a real parameter. We plot in the left panel of Fig. 1 the contours of constant µ γγ in the y − δ plane, choosing tan β = 10, y 2 = 0.5, y ′ 2 = 0, M χ = 350 GeV and M ψ = 380 GeV. Clearly, the impact of this scenario on the Higgs diphoton rate is consistent with the combined ATLAS and CMS µ γγ value 1.12 ± 0.40(2σ) for a rather wide range of the y − δ parameter space. In the right panel of Fig. 1 we plot the contours of constant d e × 10 29 in the y − δ plane using the same input parameter choices. The contour line on the rightmost corresponds to the current experimental upper limit on d e obtained by ACME experiment [20] . The successive contours to the left of the exclusion line correspond respectively to d e being one and two orders of magnitude smaller than the current limit. We observe that the present d e constraints rule out most of the available parameter space at large y and δ. The diphoton decay rate displays a sensitivity only for relatively small values of the CPV phase, wherein the effect arises largely through the CP-conserving operator hF F . This feature is consistent with the general expectations based on the study of Ref. [21]. As we discuss below, future determinations of µ γγ may retain an interesting sensitivity to the new fermion masses for values of y and δ giving rise to successful EWBG, thereby complementing the information provided by d e .

IV. ELECTROWEAK BARYOGENESIS
We now proceed to study EWBG in this scenario. The three Sakharov conditions are realized in the following way. First, the two Higgs doublets potential can induce a strongly first order electroweak phase transition (EWPT) at temperatures T ∼ 100 GeV, which provides a departure from equilibrium [54,55]. During the EWPT, bubbles of broken electroweak symmetry nucleate and expand in a background of unbroken symmetry, filling the Universe to complete the phase transition. Second, the CP-violation arises from the complex phases in the couplings of the new fermions to the Higgs scalars. The phase induces CP-violating interactions at the walls of the expanding bubbles, where the Higgs vacuum expectation value is spacetime dependent, leading to the production of a CP-asymmetric charge density. This CP-asymmetry diffuses ahead of the advancing bubble and is converted into a net density of left-handed fermions, n L , through inelastic interactions in the plasma. Third, baryon number is violated by the sphaleron processes. The presence of nonzero n L biases the sphaleron processes, resulting in the production of the baryon asymmetry [56].
We ignore the wall curvature in our analysis so all relevant functions depend on the where v w is the wall velocity;z < 0, > 0 correspond to the unbroken and broken phases, respectively. Working in the closed time path formulation and under the "vev-insertion" approximation [19,[57][58][59], we compute the CP-violating source induced by the Higgs mediated processes ψ → χ → ψ, S ψ′ where n(x) = 1/exp(x) + 1 is the fermion distribution; ε χ,ψ = ω χ,ψ − iΓ χ,ψ are complex poles of the spectral function with ω 2 χ,ψ = k 2 + m 2 χ,ψ , where m χ,ψ and Γ χ,ψ are the thermal masses and thermal rates of χ and ψ, respectively. As can be seen from eqs. (A13∼A16) of Appendix A, θ i are not independent. As a result, CP-violating phases in Eqs. (14,15) can be correlated with those in Eqs. (11,12). As indicated in Section III, for illustrative purposes we assume y 1 contains the only CP phase and y ′ 2 = 0. In this case, the only nonvanishing phases are θ 1,5 , implying a non-vanishing S ψ CP (x) but zero S ψ′ CP (x). For the more general case, both CP-violating sources will contribute to the asymmetry generation. Before proceeding, we note that the vev insertion approximation used in obtaining Eqs. (14,15) is likely to lead to an overly large baryon asymmetry by at least a factor of a few, though a definitive quantitative treatment of the CPV fermion sources remains an open problem.
The results quoted here, thus, provide a conservative basis for assessing the EDM and Higgs diphoton restrictions on the EWBG-viable parameter space. For a detailed discussion of the theoretical issues associated with the computation of the CPV source terms, see Ref. [9] and references therein.
We now derive the transport equations that govern the asymmetry generation. In general, these equations depend on the densities of first and second generation left-handed quark doublets, q kL , k = 1, 2; first and second generation right-handed quarks, u R , d R , c R , and s R ; third generation left-handed quark doublets Q and right-handed singlets, T and B; the corresponding lepton densities; that for neutral scalars H; and the new fermions ψ and χ.
Since the new fermions have Dirac mass terms in Eq. (2) it makes sense to consider a single density for the Dirac fermions ψ and χ constructed from the ψ L,R and χ L,R , respectively.
Several physical considerations then allow us to reduce the number of transport equations.
Since the SM lepton Yukawa couplings are small compared to those of the third generation quarks, any reaction that converts a non-vanishing H into lepton densities will occur too slowly to have an impact on the dynamics of the plasma ahead of the advancing bubble wall.
Consequently, we may omit the SM leptons from the set of transport equations. Moreover, since all light quarks are mainly produced by strong sphaleron processes and all quarks have similar diffusion constants, baryon number conservation on time scales shorter that the inverse electroweak sphaleron rate implies the approximate constraints q 1L = q 2L = −2u R = −2d R = −2s R = −2c R = −2B = 2(Q + T ). The resulting set of transport equations can then be written as where  The transport coefficient Γ ψ can be written as: Γ ψ = 6|y| 2 I F (m ψ , m χ , m h )/T 2 , which describes the rate for the processes χ ↔ ψH to occur. We refer the reader to Ref. [60] for the general form of I F . The interaction time scale is τ ψ ≡ Γ −1 ψ . In principle, if τ ψ ≪ the diffusion time 3 τ diff , then this interaction occurs rapidly as the charge density diffuses ahead of the advancing wall, leading to chemical equilibrium. Numerically, we have τ diff ∼ 10 4 /T [61] and τ ψ ∼ 10 3 /T by taking y i ∼ 0.25, which is consistent with the µ γγ constraints indicated in Fig. 1. In this case, the new fermion Yukawa interaction is in chemical equilibrium, and the relation χ kχ − H k H − ψ k ψ = 0 is satisfied. The sum of transport equations for ψ and χ gives v w (ψ + χ) − (D ψ ψ ′′ + D χ χ ′′ ) = 0, which implies D ψ ψ = −D χ χ in the static limit [61]. Therefore, we have When top quark Yukawa interaction and strong sphaleron process are in chemical equilibrium, we have In terms of H, the left-handed fermion charge density becomes n L (z) = (5τ Q + 4τ T )H.
Since n B is determined by n L , all that remains is to solve for the Higgs charge density. The transport equations can be reduced into a single equation for H by taking the appropriate linear combination of equations: (16) + 2 × (17) + (18) + (20). Lastly, the BAU is given by GeV is the inverse washout rate for the electroweak sphaleron transitions.
The computation of n B /s relies upon many other numerical inputs; our choices are listed in Table. I. The diffusion constants were calculated in Ref [62,63] with D χ = 380 T and D ψ = 100 T . The fact that D ψ ≪ D χ enhances the left-handed lepton charge. The bubble wall velocity v w , thickness L w , profile parameters ∆β and v(T ) describe the dynamics of the expanding bubbles during the EWPT, at the temperature T . We take the Higgs profile to following Ref. [64][65][66]. The sphaleron rates are Γ ws = 6κα 4 s T and Γ ss = 6κ ′ α 4 s T 8 3 , where κ ws = 22 ± 2 and κ ss = O(1).  Fig. 1; other parameters are given in Table I. Right panel: Region consistent with observed Y B (gray region) in the M χ − M ψ plane. In each panels, the region to the left of the red dashed line is consistent with the ACME d e limit at 95% C.L., while the region surrounded to the left or above the blue dotted line is consistent with the combined ATLAS and CMS µ γγ result. We have set |y 1 | = 5/3y ′ 1 = y 2 = 0.6, y ′ 2 = 0 and δ = π/10 generating the right panel.
The contours of constant n B /s ( in units of 10 −10 ) in the y-δ plane are indicated in the left panel of Fig. 2. The initial input of the fermion mass matrix is the same as that given in the caption of Fig. 1 and other initial inputs are given in Table. I We also observe that both Higgs diphoton rate and charged lepton EDM depend nontrivially on the new fermion masses. To illustrate, we plot in the right panel of Fig but illustrative region of the model parameter space 5 , we find that it is possible for this scenario to give rise to the observed BAU while satisfying constraints from the non-observation of the electron EDM and present LHC results for the Higgs to diphoton decay rate. The complementarity of the two experimental probes considered here is most apparent in the right panel of Figure 2, where we observe their different sensitivities to the new fermion mass spectrum. Future improvements in the sensitivities of these two sets of observables will probe more deeply into the parameter space. In general, an order of magnitude improvement in d e -sensitivity would yield a considerably more comprehensive probe of the EWGB-viable parameter space considered here as compared to a factor of ten improvement in the precision of the Higgs diphoton decay rate measurement (see the right panel of Figure   3). Should additional new interactions lead to a suppression of the impact of new physics on d e , future Higgs diphoton decay studies would then yield an interesting window on the EWBG mediated by new scalar-vector like fermion interactions. More generally, the future observation of a non-zero effect in either observable could be consistent with EWBG in this scenario.

(A9)
Clearly, not all phases in Eqs. (A4 ∼ A9) are independent. Among our eight original phases, only four are physical. The following combinations are invariant combinations under the foregoing field redefinitions: In summary, the four independent rephasing invariants are Arg(y 1 y ′ 1 M * ψ M * χ ), Arg(y 2 y ′ 2 M * ψ M * χ ), Arg(y 1 y * 2 M 2 12 ) and Arg(λ 5 M 4 * 12 ). For the rephasing invariants defined in Section II, we have of which only three equations are independent. For example, one may take the rephasing invariants in this model to be θ 1,2,5,7 , with all the other rephasing invariants being constructed from these four invariants.
It is useful to show that A, B, |R| and |Q| are rephasing invariant. A direct calculation gives As a result, θ L and θ R , which only depend on A, B, |R| and |Q|, are rephasing invariant.
In contrast, δ L and δ R are not rephasing invariant, because R and Q are not rephasing invariant, as can be seen from Eqs. (A19) and (A20 ).