EDMs vs. CPV in B_{s,d} mixing in two Higgs doublet models with MFV

We analyze the correlations between electric dipole moments (EDMs) of the neutron and heavy atoms and CP violation in B_{s,d} mixing in two Higgs doublet models respecting the Minimal Flavour Violation hypothesis, with flavour-blind CP-violating (CPV) phases. In particular, we consider the case of flavour-blind CPV phases from i) the Yukawa interactions and ii) the Higgs potential. We show that in both cases the upper bounds on the above EDMs do not forbid sizable non-standard CPV effects in B_s mixing. However, if a large CPV phase in B_s mixing will be confirmed, this will imply EDMs very close to their present experimental bounds, within the reach of the next generation of experiments, as well as BR(B_{s,d}->mu^+ mu^-) typically largely enhanced over its SM expectation. The two flavour-blind CPV mechanisms can be distinguished through the correlation between S_psi K_S and S_psi phi that is strikingly different if only one of them is relevant. Which of these two CPV mechanisms dominates depends on the precise values of S_psi phi and S_psi K_S, as well as on the CKM phase (as determined by tree-level processes). Current data seems to show a mild preference for a hybrid scenario where both these mechanisms are at work.


INTRODUCTION
In the last few years, the two B factories have established that flavor-changing and CPV processes of B d mesons are well described by the Standard Model (SM) up to an accuracy of (10 − 20)%. This observation, together with the good agreement between data and SM expectations in the kaon system, implies tight constraints on flavor-changing phenomena beyond the SM and a potential problem for a natural solution of the hierarchy problem, that calls for new physics (NP) not far from the electroweak scale [1].
An elegant way to solve this problem is provided by the Minimal Flavor Violation (MFV) hypothesis [2] (see also [3,4]), where flavor-changing transitions in the quark sector are entirely controlled by the two quark Yukawa couplings. Despite apparently being quite restrictive, the MFV hypothesis does not forbid sizable deviations from the SM in specific channels. This is particularly true in models with two or more Higgs doublets, because of the possibility to change the relative normalization of the two Yukawa couplings [2,[5][6][7][8][9].
As recently shown in [15], the general formulation of the MFV hypothesis with flavour-blind CPV phases applied to two Higgs doublet models (2HDMs) is very effective in suppressing flavour-changing neutral-currents (FCNCs) to a level consistent with experiments, leaving open the possibility of sizable non-standard effects also in CPV observables. In what follows, we will call this framework 2HDM MFV with the "bar" indicating flavour-blind CPV phases.
As discussed in [15], the 2HDM MFV can accommodate a large CPV phase in B s mixing, as hinted by CDF and D0 data [16][17][18], while ameliorating simulta-neously the observed anomaly in the relation between ǫ K and S ψKS [19,20].
On general grounds, it is natural to expect that flavour-blind CP phases contribute also to CPV flavourconserving processes, such as the EDMs. Indeed, the choice adopted in [2] to assume the Yukawa couplings as the unique breaking terms of both the flavour symmetry and the CP symmetry, was motivated by possibly too large effects in EDMs with generic flavour-blind CPV phases. This potential problem has indeed been confirmed by the recent model-independent analysis in [21].
In this Letter we address the role of EDMs, and their correlation with CPV effects in B s mixing, in the 2HDM MFV . Following the recent analysis in [15], we focus on the contributions to these observables generated by the integration of the Higgs fields only, assuming a high suppression scale for effective operators not induced by the Higgs exchange.
We analyse in particular two sources of CPV phases: flavour-blind phases in i) the Yukawa interactions and in ii) the Higgs potential. Flavour-blind phases in the Yukawa interactions, together with the assumption of small SU (2) L breaking in the heavy Higgs sector, can lead to large corrections to S ψφ . In this case the new CPV effects in B d mixing are suppressed by a factor of m d /m s , compared to B s mixing, and go in the right direction to ameliorate the prediction of S ψKS [15]. Flavour-blind phases of the type ii) can also affect S ψφ , provided the SU (2) L breaking in the heavy Higgs sector is not negligible. However, in this case the correction in B s and B d mixing is universal and the magnitude of the effect in S ψφ is bounded by the limited amount of NP allowed in S ψKS [10,20,22]. Which of the two flavour-blind CPV mechanisms dominates depends on the value of S ψφ , which is still affected by a sizable experimental error, and also by the precise amount of NP allowed in S ψKS .
We show that in both cases the upper bounds on the neutron and heavy atom EDMs do not forbid sizable non-standard CPV effects in B s mixing. Interestingly enough, in both cases sizable CPV effects in B s mixing imply lower bounds for these EDMs within the future experimental resolutions. Moreover, in both cases BR(B s → µ + µ − ) and BR(B d → µ + µ − ) are typically largely enhanced over their SM expectations.
Our paper is organized as follows. In Section 2 we recall briefly the most important ingredients of the 2HDM MFV , concentrating in particular on the two new sources of CPV in question. In Sections 3,4, we exploit the sensitivity of the EDMs and B s,d −B s,d mixing, respectively, to these new CPV phases. The numerical analysis of various correlations that have been advertised in the abstract is performed in Section 5. Here we demonstrate that the current data seems to show a mild preference for a hybrid scenario where both new mechanisms of CPV are at work.

THE 2HDM MFV
In the following, we consider a 2HDM supplemented by the MFV hypothesis, where the Yukawa matrices are the only sources of breaking of the SU (3) q flavour group, but they are not the only allowed sources of CP violation [10,12,13].
The most general renormalizable and gauge-invariant Yukawa interactions in a 2HDM are where the Higgs fields H 1,2 have hypercharges Y = ±1/2, H c 1(2) = −iτ 2 H * 1 (2) and the X i are 3 × 3 matrices with a generic flavour structure. We also assume real vevs for the two fields, 2 and, for later purpose, we define The general structure implied by the MFV hypothesis for the renormalizable Yukawa couplings X di and X ui is a polynomial expansions in terms of the two (left- [2,15]: where the ǫ (′) i are complex parameters. We work under the assumption ǫ (′) i ≪ 1, as expected by an approximate U (1) PQ symmetry that forbids non-vanishing X u1 and X d2 at the tree level. We also assume negligible violations of the U (1) PQ symmetry in the lepton Yukawa couplings.
After diagonalising quark mass terms and rotating the Higgs fields such that only one doublet has a nonvanishing vev, the interaction of down-type quarks with the neutral Higgs fields assumes the form being the CP-even (CP-odd) heavy Higges, if the Higgs potential is CP-invariant. The flavour structure of the Z d couplings, which play a key role in our analysis, is where V is the physical CKM matrix, ∆ ≡ diag(0, 0, 1) and λ u,d are the diagonal up Yukawa couplings in the limit ǫ (′) i → 0 (see [2,15] for notations). As explicitly given in [2,15], the a i are flavour-blind coefficients depending on the ǫ i , on t β , and on the overall normalization of the Yukawa couplings. Even if ǫ where the flavour structure of the C H + R,L is In analogy to the a i , also the b i and b ′ i coefficients are flavour-blind, naturally of O(1), and possibly complex.
Another source of CP violation in the 2HDM MFV , that is relevant for our analysis, arises from the Higgs potential [23][24][25] (see also [26]). We recall that the most general 2HDM potential that is renormalizable and gauge invariant is [27] All the parameters must be real with the exception of b and λ 5,6,7 . Exploiting the freedom to change the relative phase between H 1 and H 2 , we can cancel the phase of b and λ 6,7 relative to λ 5 . Moreover, the coefficients λ 6,7 can be set to zero imposing a discrete Z 2 symmetry that is only softly broken by the terms proportional to b and λ 5 .
In order to simplify the discussion, without loosing generality as far as the CP properties are concerned, we set λ 3 = λ 4 = λ 6 = λ 7 = 0 and choose the basis where only λ 5 is complex. The resulting spectrum contains a charged Higgs, with the mass and three neutral Higgses with masses where in Eqs. (9) we have assumed t β ≫ 1. Notice the approximate degeneracy of the charged Higgs and the two neutrals of mass M 2 and M 3 in the limit λ 5 → 0.
In the absence of CP violation, the physical Higgs eigenstates are given by the two CP-even fields h, H and by the CP-odd field A. In the presence of CP violation, h, H, A are mixed and the mass eigenstates are not anymore CP eigenstates. Still, it is convenient to write the Higgs potential in terms of the fields h, H, A. It turns out that where Ah and AH read Notice that in the large t β regime the mixing Ah is negligible, in contrast to AH . Moreover, if AH ≪ M 2 A,H , as it happens in the so-called decoupling regime, we can still treat the fields h, H, A as approximate masseigenstates and the mixing AH can be parameterized as an effective mass insertion in the scalar propagator.

ELECTRIC DIPOLE MOMENTS
Among the various atomic and hadronic EDMs, the Thallium, neutron and Mercury EDMs represent the most sensitive probes of CP violating effects (see table I). The effective CP-odd Lagrangian describing the quark (C)EDMs, that is relevant for our analysis, reads f stands for the quark (C)EDMs while C f f ′ is the coefficient of the CP-odd four fermion interactions.
The thallium EDM (d Tl ) can be estimated as [31][32][33] where d e is the electron EDM while C S stems from the CP-odd four fermion interactions and reads [33] C with κ ≃ 0.5 ± 0.25 [34]. The neutron EDM d n can be estimated from the naive quark model as d n ≈ 4 f are evaluated at 1 GeV by means of QCD renormalization [38], starting from the corresponding values at the electroweak scale. The alternative estimate we use in our numerical analysis is the one obtained from QCD sum rules [33,[35][36][37], which leads to for the Mercury EDM [33,37]. In these numerical formulae the d (c) f are evaluated at 1 GeV. In the following, we provide the expressions for d (c) f and C f f ′ at the electroweak scale in the context of a 2HDM, starting from the general formulae of Ref. [39,40] obtained in the context of Supersymmetry.
The CP violating effects arising from CP-odd CPeven scalar mixing and their impact on the EDMs were studied previously in the context of 2HDMs with spontaneous breaking of CP in refs. [23][24][25]. For Im  where we have defined Im The explicit expressions for the ω f f ′ relevant for our analysis are Im Im Im where we have defined Concerning the quark (C)EDMs, they are induced already at the one-loop level by means of the exchange of the charged-Higgs boson and top quark [40], as it is shown in Fig. 1.
In our scenario, their explicit expressions read where x tH = m 2 t /M 2 H ± and the expressions for the loop functions F 7,8 are given in [40].
Notice that the above effects are CKM suppressed by the factor |V td | 2 ≈ 10 −4 . In particular, even in the most favourable case where m t = M H ± and Im , leading to predictions for d n and d Hg well under control (though observable with improved experimental resolutions). Therefore, since the b i 's do not enter the predictions for B s,d mixings, we neglect the above one-loop effects to d n and d Hg in our numerical analysis.
As a result, two loop contributions might compete or even dominate over one loop-effects provided they overcome the strong CKM suppression.
Indeed, this is the case for the two-loop Barr-Zee contributions [24] to the fermionic (C)EDMs (see Fig. 3) that read where q ℓ is the electric charge of the fermion ℓ, τ q = m 2 q /M 2 A and f (τ ), g(τ ) are the two-loop Barr-Zee functions defined in [24,39]. In analogy to the down-type ω f f ′ in Eqs. (20)-(23), we have defined Im To have an idea of where we stand, let us evaluate separately the contributions to the physical EDMs arising from the fermionic (C)EDMs and from the four-fermion operator.
Assuming the example where m A = 500 GeV and t β = 10, the electron EDM contribution to d Tl reads  Finally, let us consider d Tl as induced by the four- and similarly d Hg (C S ) ≈ −10 −4 ×d Tl (C S ).
Once the Wilson coefficients of these operators C i (µ H ) are calculated at a high energy scale, where heavy degrees of freedom are integrated out, their values at scales O(µ B ) are obtained by the standard techniques [41]. The resulting low energy effective Hamiltonian then reads where i = 1, 2, and a = LR, SLL. The off-diagonal element in B meson mixing are given by Lower: Bs → µ + µ − vs. S ψφ . Red dots fulfill the EDM constraints while the black ones do not. In both plots we have assumed (|a|, |a0|, |a1|, |a2|) < 2, λ5 = 0, and 0 < (φ a , φa 0 , φa 1 , φa 2 ) < 2π.
where P a i (B q ) collect all RG effects from scales below µ H as well as hadronic matrix elements obtained by lattice methods. Updating the results of Ref. [41], it turns out that P LR 2 (B q ) ≈ 3.4 and P SLL 1 (B q ) ≈ −1.4 for µ H = 246 GeV.
Introducing the notation the B s,d mass differences and the CP asymmetries S ψKS and S ψφ are FIG. 5: Correlation between S ψφ and S ψK S with CPV sources in the Yukawa couplings and Higgs potential switched on separately (upper plot) or simultaneously (lower plot). In all plots we have assumed (|a|, |a0|, |a1|, |a2|) < 2.

correlated model-independently with S ψφ as [44]
(for an alternative model-independent formula, see [45]) where |Re(Γ s 12 /M s 12 ) SM | = (2.6 ± 1.0) × 10 −3 [43]. Assuming the a i in Eq. (3) as complex parameters and integrating out the neutral Higgs fields we have for the Wilson coefficients of the dominant operators where, in the decoupling regime (v 2 /m 2 A ≪ 1), and for large tan β values, in agreement with [26]. As can be seen, the A-H mixing, related to the SU (2) L breaking in the heavy Higgs sector, removes the zero in F − providing at the same time a possible source of CP violation independent from the a i . The mass splitting between A and H is constrained by the requirement of perturbative unitarity [46][47][48], vacuum stability [49] and by precision electroweak observables (the oblique-corrections). However, a (10−15)% splitting for M A around 500 GeV fulfills all these constraints. A close inspection of Eq. (47) reveals that: 1. the contribution of C LR 2 to B q mixing is proportional to m b m q , hence, only B s mixing can be affected in a sizable way by Q LR 2 (as in the scenario considered in [15]); 2. the effect of C SLL 1 to B s and B d mixings is the same, implying a common NP phase ϕ B d = ϕ Bs ; as a result, the limited room for NP allowed in S ψKS forbids large effects in S ψφ and A s SL coming from this operator [10,20,22]; contributes to CP violation only if a 1 = a 2 , which requires effective operators with high powers of Yukawa insertions; these operators are naturally suppressed in explicit (perturbative) models, such as supersymmetric models [50,51]. However, this suppression can be removed in a general nonsupersymmetric 2HDM with MFV.

1
is complex already at the leading order in the Yukawa insertions; however, it vanishes in the exact SU (2) limit. While in a generic 2HDM rather sizable SU (2) breaking effects (say at the 10% level) are generally allowed, in supersymmetric models such effects are loop-induced and therefore too small to provide any observable effect [50,51]. This has also been reemphasized recently in [22].
In order to understand which are the regions of the parameter space where S ψφ obtains sizable (nonstandard) values, let us first focus on the effects from C LR 2 , setting m A = 500 GeV and t β = 10. In this case it turns out that [15] S ψφ ≈ 0.15 × Im [(a 0 + a 1 )(a ⋆ 0 + a ⋆ 2 )] + (S ψφ ) SM . (49) Similarly, the SM prediction for BR(B q → µ + µ − ) is modified according to [2,15] Br where with Br(B d → µ + µ − ) SM = (1.0 ± 0.1) × 10 −10 and Br(B s → µ + µ − ) SM = (3.2 ± 0.2) × 10 −9 . Therefore, S ψφ can be large, even for moderate values of t β and relatively heavy Higgs masses, provided order one NP phases and sizable PQ-symmetry breaking sources, i.e. if Im[(a 0 + a 1 )(a ⋆ 0 + a ⋆ 2 )] ≈ 1 are present. These conditions generally imply large NP effects for BR(B q → µ + µ − ), as already observed in [15] and as clearly shown by Eqs. (50), (51). As we will show later, also for the EDMs of physical systems like the Thallium (Tl), Mercury (Hg) and the neutron EDMs large NP effects are expected. In particular, in the 2HDM MFV , large non-standard values for S ψφ imply lower bounds for the above EDMs in the reach of the expected future experimental resolutions.
As far as NP effects induced by C SLL 1 are concerned, we notice that it is easy to generate a large common phase ϕ B d = ϕ Bs provided the mass splitting between A and H is around the 10% level (or above). In this case the limits on NP effects in S ψKS set the bound S ψφ < ∼ 0.2 [10,20,22]. As shown in Eq. (47), C SLL 1 dominates over C LR 2 if we allow only flavour-blind phases in the Higgs potential (Imλ 5 = 0) and we allow a sufficient SU (2) L breaking in the heavy Higgs sector.
Finally, let us mention that there are also charged Higgs contributions to M q 12 that are sensitive to new flavour blind CP sources. These effects have recently been analyzed in [52], in the context of the 2HDM proposed in [53]. They arise from the one loop exchange of H + − H + and H + − W + and lead to where D 0 is the standard four-point functions normalized to D 0 (1, 1, 1) = −1/6. In the framework we are considering, with large/moderate t β and small/moderate U (1) PQ breaking, (C SLL 1 ) H + is always very suppressed and does not provide visible effects in physical observables. However, this is not the case in the framework analyzed in Ref [52], where the charged-Higgs contribution to B s,d -B s,d mixing can be sizable in specific regions of the parameter space (corresponding to large violations of the U (1) PQ symmetry).

NUMERICAL ANALYSIS
Having introduced the two sets of observables, namely EDMs and ∆F = 2 CPV asymmetries, we are ready to analyse their correlations in the 2HDM MFV . As outlined in the introduction, we analyse separately the cases of flavour-blind phases from i) the Yukawa interactions and ii) the Higgs potential.
Starting with case i), in the upper plot of Fig. 4 we report the expectations for the EDMs of the Thallium d T l (red dots), neutron d n (black dots), and Mercury d Hg (green dots) as a function of the phase in the B s mixing, described by the asymmetry S ψφ . The plots have been obtained by means of the following scan: (|a|, |a 0 |, |a 1 |, |a 2 |) < 2, 0 < (φ a , φ a0 , φ a1 , φ a2 ) < 2π, tan β < 60, λ 5 = 0, M H ± < 1.5 TeV and setting the hadronic parameter κ [34] entering the CP-odd four fermion coefficients of Eqs. (15), (17) to its central value κ = 0.5. Moreover, we have imposed the further condition |a 2 | < (|a|, |a 0 |, |a 1 |), since a 2 is generated only beyond the leading order in the MFV expansion in terms of the spurions Y u Y † u and Y d Y † d , in contrast to a, a 0 , a 1 . As can be seen, the current constraints from the EDMs still allow values of |S ψφ | larger than 0.5, com-patible with the highest values of the B s mixing phase reported by the Tevatron experiments. Yet, sizable non standard values for S ψφ unambiguously imply lower bounds for the above EDMs within the reach of the expected future experimental resolutions. Similarly, large values for S ψφ typically imply a BR(B s → µ + µ − ) departing from the SM prediction, as already observed in [15] and as clearly shown in the lower plot in Fig. 4.
From these plots we conclude that improvements of the experimental lower bounds for the Thallium, neutron and Mercury EDMs, as well as for BR(B s,d → µ + µ − ), together with a more accurate measurement of S ψφ , would provide a powerful tool in the attempt to test or to falsify this NP model.
In Fig. 5, we show the predictions for S ψKS vs. S ψφ where the allowed ranges for NP effects in S ψKS have been obtained combining the SM prediction sin(2β) tree = 0.734 ± 0.038 [42] with the experimental result S exp ψKS = 0.672 ± 0.023 [54]. In the upper plot, we switch on the CPV phases of the Yukawa couplings and the Higgs potential separately, to monitor their individual effect. In the former case, we employ the same scan as in Fig. 4, while in the latter case we make the scan (|a|, |a 0 |, |a 1 |, |a 2 |) < 2, (φ a , φ a0 , φ a1 , φ a2 ) = 0, |λ 5 | = 0.1 and 0 < φ λ5 < 2π. Viceversa, in the lower plot of Fig. 5, we consider an hybrid scenario, where CPV phases of both type i) and type ii) are switched on simultaneously. In both plots, red dots fulfill the EDM constraints while the black ones do not.
The correlation of the EDMs vs. S ψφ in the case ii), where the flavour blind CPV phases originates only from the Higgs potential, does not show appreciable differences with respect to Fig. 4. However, since in this cases the the effects on B s,d mixing are universal, here the larger values of S ψφ are not allowed by the constraints from B d mixing. This is clearly illustrated in the upper plot of Fig. 5.
Finally, Fig. 6 shows the correlation between A s SL /A s SL (SM) vs. S ψφ for the hybrid scenario, performing the same scan of Fig. 5. Red dots fulfill the EDM constraints while the black ones do not. Green dots further satisfy the constraint from S ψKS at the 95% CL.. As can be seen, in this scenario sizable/large effects for B d /B s mixings can be naturally accounted for, even though not necessarily in a correlated manner. According to the recent model-independent fit of the two mixing phases performed in [55], this hybrid scenario is particularly welcome by present data. Indeed the best fit of current data is obtained for a ϕ B d /ϕ Bs ratio which is in between the ϕ B d ≈ (m d /m s )ϕ Bs of the scenario i) and the ϕ B d ≈ ϕ Bs of the scenario ii).
Finally, as demonstrated in Fig.3 of [15] for large values of S ψφ a unique positive shift in ε K is implied within the model considered, bringing the theory closer to the data. This agreement is further improved through the recently calculated NNLO QCD corrections to ε K [56]. In the hybrid scenario, a value for ε K within 1 σ of current data can be obtained even for values of S ψφ ≈ 0.25.

DISCUSSION AND CONCLUSIONS
The remarkable agreement of flavour data with the SM predictions in the K and B d systems highly constrains the flavour structure of any TeV-scale UV completion of the SM. In this respect the MFV hypothesis [2], where flavour-changing phenomena are entirely controlled by the CKM matrix, represents one of the most natural and elegant explanations for such an impressive agreement. However, the MFV principle does not forbid in itself the presence of new flavour blind CP violating phases in addition to the unique phase of the CKM matrix [10,12,13].
From a phenomenological point of view, it is natural to expect that such flavour-blind phases, if present, will contribute not only to flavour-changing CPV processes, such as CP-violation in B s,d mixing, but also to flavour-conserving CPV processes such as the EDMs. Indeed the close connection within a MFV framework between CP violation in flavour physics and EDMs has already been analyzed in the context of supersymmetric extensions of the SM [50,51], and by means of a general effective theory approach [21].
In this Letter we have analyzed these connections within the 2HDM respecting the MFV hypothesis recently discussed in [15]. We have shown that the two classes of flavour-blind phases present in this model, those in the Yukawa couplings and those in the Higgs potential, provide potentially large CP violating effects in B s,d mixing while being compatible with the EDM bounds of the neutron and heavy atoms. In both cases sizable CPV effects in B s mixing imply lower bounds for the above EDMs within the future experimental resolutions. Moreover, in both cases BR(B s,d → µ + µ − ) are typically largely enhanced over their SM expectations.
What distinguishes the different flavour-blind CPV mechanisms is the correlation between S ψKS and, correspondingly, the maximal effect expected in S ψφ . Very large values of S ψφ are possible only via the mechanism pointed out in [15], which requires flavour-blind phases in the Yukawa interactions and small SU (2) L breaking in the heavy Higgs sector (or the dominance of the effective operator Q LR 2 ). On the contrary, flavour-blind phases only in the Higgs potential leads to universal effects in B s and B d mixing, which are strongly constrained by the measurement of S ψKS . Which of the two flavour-blind CPV mechanisms dominates depends on the precise values of S ψφ and S ψKS , as well as on the CKM phase (as determined by tree-level processes). Current data seems to show a mild preference for an hybrid scenario where both these mechanisms are at work.