# Electric dipole moment of the neutron from a flavor changing Higgs-boson

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## Abstract

I consider neutron electric dipole moment contributions induced by flavor changing Standard Model Higgs boson couplings to quarks. Such couplings might stem from non-renormalizable \(SU(2)_L \times U(1)_Y\) invariant Lagrange terms of dimension six, containing a product of three Higgs doublets. We extend previous one loop analysis to two loops. The divergent loops, due to non-renormalisabillity, are parametrized in terms of an ultraviolet cut-off \(\Lambda \). I also consider QCD corrections. Using the current experimental bound on the neutron electric dipole moment, then for cut offs from one to seven TeV, I find a constraint of order \(10^{-3}\) for the imaginary part of the *product* of the Higgs flavor changing coupling for \((d \rightarrow b)\)-transition *and* the CKM element \(V_{td}\). Assuming that the previous bound of the *absolute value* of the Higgs flavor changing coupling for \((d \rightarrow b)\)-transition obtained from \(B_d - \bar{B_d}\)-mixing is saturated, the experimental bound on the neutron electric dipole moment would be reached for the *bare* result, *if* the cut off were extended up to about ca 20 TeV. However, QCD corrections suppress this result by a factor of order ten, and keep the nEDM below the experimental bound.

## 1 Introduction

*u*- and

*d*-, and even

*s*-quarks may be given by the formula

*s*-quark, with a small coefficient.

Many models BSM suggest possible new particles and/or new interaction Lagrange terms inducing EDMs [1, 2, 3, 4, 5, 11, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In the case of New Physics (NP) presence, flavor physics might be testable through CP-violating asymmetries in mesonic decays [24, 25, 34]. The properties and couplings of the physical Higgs boson (*H*) are still not completely known. Some authors [35, 36, 37, 38, 39, 40] have suggested that the physical Higgs boson might have flavor changing couplings to fermions which might also be CP-violating. In these papers bounds on quadratic expressions of such couplings were obtained from various processes, say, like \(K-{\bar{K}}, \,D-{\bar{D}} \,\), and \( B-{\bar{B}}\) - mixings, and also from leptonic flavor changing decays like \(\mu \rightarrow e \, \gamma \) and \(\tau \rightarrow \mu \, \gamma \). In the latter case two loop diagrams of Barr-Zee type [41] were also considered [35, 36, 37, 42]. (See also [43]). Such flavor changing couplings might occur when higher mass states are integrated out. For instance, flavor changing Higgs (FCH) couplings might stem from \(SU(2)_L \times U(1)_Y\)-invariant but non-renormalizable Lagrangian terms of dimension six.

The purpose of the present paper is to extend the analysis of [36, 37] to two loop diagrams. In the one loop case one needed two FCH couplings to generate the EDM. In the two loop case it is however possible to find diagrams with the FCH coupling *to first order only*, while the rest of the couplings are ordinary SM couplings.

Some of these two loop diagrams considered here give contributions suppressed by the small mass ratio \(m_d/M_W\) for the ordinary SM Higgs coupling to fermions. (\(M_W\) denotes the mass of the *W*-boson and \(m_q\) is the mass of the quark *q*). However, if the Higgs is coupled to a top (*t*) quark one might obtain relevant non-suppressed contributions. Motivated by the result of the previous work [33], I consider such diagrams. There are additional reasons to extend the analysis in [36, 37] for nEDM to two loop level. Namely, in general, it is known that some two loop diagrams might give bigger amplitudes than one loop diagrams because of helicity flip(s) in the latter [36, 37, 42, 44]. In the present case, two loop amplitudes will be proportional to a large *ttH* coupling or a large *WWH* coupling within the SM, in contrast to the small SM Higgs couplings to light fermions. This might compensate for the two loop suppression of the diagrams. I have also adressed the issue of perturbative QCD corrections, which turn out to suppress the bare result.

In the next Sect. 2 I will present the framework for the FCH couplings. In the Sects. 3 and 4 two loop calculations for the FCH couplings will be presented. The QCD corrections are presented in Sect. 5. In Sect. 6 the results will be discussed, and the conclusion given in Sect. 7. An Appendix is given in Sect. 8.

## 2 Flavor changing physical Higgs?

*H*the physical Higgs field and \(Y_{R}(f_1 \rightarrow f_2)\)’s are coupling constants, thought to be complex numbers. Then, from the hermitean conjugation part, there will be a left-handed \(f_2 \rightarrow f_1\) coupling

*i*and

*j*are understood to be summed over the values 1, 2, 3. Further, \(\Phi \) is the SM Higgs field, \(Q_i\) is the left-handed \(SU(2)_L\) quark doublets, and the \(D_j\)’s are the right-handed \(SU(2)_L\) singlet

*d*-type quarks in a general basis. Moreover, \(\Lambda _{NP}\) is the scale where New Physics is assumed to appear. There is a similar term as in (9) for right-handed type

*u*-quarks, \(U_j\).

*u*- and

*d*-quarks [36, 37]. The one loop diagram in Fig. 1 - with FCH coupling at both vertices, puts bounds on

*quadratic*expressions of the

*Y*’s for definte choices of flavor. Note that this diagram gives a finite contribution to quark EDMs.

## 3 Diagrams with one FC coupling -and a \(t {\bar{t}} H\)-coupling

*d*-quark generated by exchange of one physical Higgs (

*H*) boson and one

*W*-boson, with a sizeable Higgs coupling \(\sim g_W \, m_t/M_W\) to a top quark and where only

*one*of the Higgs couplings are flavor changing (a soft photon is assumed to be added). The non-crossed version to the left in Fig. 2 does not give non-suppressed contributions. Taking crossed Higgs and

*W*-bosons are equivalent to the topologies in the middle and right of Fig. 2.

*A*contains coupling constants and loop functions depending on the involved masses. The diagrams with interchanged order of

*H*and

*W*loops, as in the middle of Fig. 2 have then the corresponding form:

*e*= the proton charge) of the photon-emitting particles, i.e. \({\hat{e}}_{1,3}= {\hat{e}}_t = + 2/3\), \({\hat{e}}_{2}= {\hat{e}}_b = - 1/3\), and \({\hat{e}}_{4}= {\hat{e}}_W = +1 \). Here I have used the relations (8) and (12). Note that a left-handed coupling \(Y_L(d \rightarrow b) \, P_L\) would not contribute in (13) due to wrong chirality. The

*V*’s are CKM matrix elements in the standard notation. The constant \(F_2\) sets the overall scale of the EDMs obtained from the two loop diagrams:

*W*-boson, one has also to add diagrams with the unphysical Higgs field \(\phi _\pm \) (i.e. the longitudinal component of the

*W*-boson) given by the the Lagrangian

*W*-boson, the

*t*-quark and the physical Higgs-boson

*H*are of the same order of magnitude. Therefore, because of lack of a clear mass hierarchy, it makes no sense to consider leading logarithmic approximations, in contrast to [33]. Numerically, I find

*W*-boson in the fourth diagram, the left sub-loop containing the Higgs boson is logarithmically divergent, which is not unexpected because the interaction in (9) is non-renormalizable. Each of the divergent integrals \(\sim ln(\Lambda ^2)\) are followed by finite logarithmic terms more cumbersome than for finite loop integrals, and such integrals are given by expressions like in (58), also with masses permuted for different diagrams.

*W*-boson again contains a divergent part, and the total contribution to the fourth diagram is

*b*-quark is replaced by an

*s*-quark, the CKM factors are two orders of magnitude smaller, and in addition \(Y_R(d \rightarrow s)\) has a stricter bound from \(K -{\bar{K}}\)-mixing. If the

*t*-quark is replaced by the

*u*- or

*c*-quark, the contributions are suppressed by \((m_u/m_t)^2\) and \((m_c/m_t)^2\), respectively.

There are also similar diagrams for EDM of an *u*-quark, i.e. like in Fig. 3 with the *t*- and the *b*-quarks interchanged. This amplitude has the same structure as in (13), and is proportional to the combination \(Im[Y_R(u \rightarrow t) \cdot V_{tb}^* \, V_{ub}]\). But the *u*-quark EDM contributions will be neglected. First, the ordinary SM coupling of the Higgs will be proportional to \(m_b/M_W\) instead of \(m_t/M_W\) for the *d*-quark case. Then it turns out that the prefactors \(S_i\) for *u*-quark EDM contributions are suppressed by a factor of order \((m_b/m_t)^2 \sim 10^{-3}\) compared to the analogous *d*-quark contributions. Second, even if the ratio between \(Y_R(u \rightarrow t)\) and \(Y_R(d \rightarrow b)\) would be of order \(m_t/m_b\), the *u*-quark EDM contribution to the nEDM in (4) would still be suppressed by \(|(\rho _u \cdot m_b)/(\rho _d \cdot m_t)| \, \sim 10^{-2}\) compared to the *d*-quark EDM contribution to the nEDM.

## 4 Diagrams with one FC coupling -and a *WWH*-coupling

We will now consider another class of two loop diagrams generated by FC Higgs-boson couplings. These diagrams shown in Fig. 4 have a big *WWH*-coupling \(\sim g_W \, M_W\) and *only one* FC Higgs coupling to a fermion. These two loop diagrams are divided in three types: the (a)-diagrams with Higgs exchange to the left, the (b)-diagrams with Higgs exchange in the middle, and the (c)-diagrams with Higgs exchange to the right. In the limit of small external light quark momenta, which we work, the (b)-diagrams are zero due to (odd) momentum integration, or they are suppressed by small external quark masses. The (c)-diagrams are complex conjugates of the (a)-diagrams. Soft photon emission from one of the charged particles should of course be added in Fig. 4, as seen in Fig. 5 for the (a)-diagrams. The (a) diagrams give contributions like in (10), and the (c) diagrams like in (11).

*W*-bosons is given by

*W*-boson, we must also consider Lagrangian terms for a physical Higgs coupling to a

*W*-boson and the unphysical Higgs boson \(\phi _\pm \). In addition to the term for quarks coupling to \(\phi _\pm \) in (15), there is the relevant \(H W \phi _\pm \)- coupling obtained from the Lagrangian

In the preceeding Sect. 3, for all the shown diagrams in Fig.3, the physical Higgs coupled to the top quark with strength \(\sim g_W m_t/M_W\). Also the chiral structure of the diagrams is such that these diagrams are proportional to \(m_t^2\), and even \(m_t^4\) in \(S_3\). In the present section the diagrams have a flavor blind *WWH* coupling, and have another chiral structure, and one gets diagrams \(\sim \, m_t^2\) only for the case when the *W*-boson is replaced by an unphysical Higgs \(\phi _\pm \). Therefore I have apriori considered all quark flavors in the loops, although it is expected that the GIM-mechanism will cancel the leading terms with light quark flavors, except for the difference between the *t*-quark and the *c*-quarks contribution.

*d*-quark EDM from soft photon emission from the quark \(q= s,b\) in the diagram 5a,(i.e. to the left in Fig. 5) can in the general case be written:

*W*-boson and Higgs masses. Above, I have used the shortages

*u*- and

*c*-quark contributions are very efficient, such that the differences \(\Delta f_d(s,u-c)\) and \(\Delta f_d(b,u-c)\) are of order \(10^{-3}\) to \(10^{-7}\), and can be safely neglected. Contributions with the

*t*-quark in the loops are significantly different from contributions involving the lighter quarks. Thus, the determination of

*both*the

*t*-quark

*and*the

*c*-quark contributions will be important. In this case the GIM cancellation is not efficient.

*d*-quark EDM from soft photon emission from the quark \(q= s,b\) in the diagram 5a, the dominating contribution can be written:

*b*-quark, and \(\xi _{{\tilde{q}}} = V_{td}^* V_{t b}\). There are also other contributions which are small and can be neglected.

*W*is replaced by an unphysical Higgs \(\phi _\pm \), one obtains divergent contributions for these loop functions.

*W*is replaced by the unphysical Higgs \(\phi _\pm \), and the total result from diagram 5b is

*W*-boson is shown at the right of Fig. 5 (Fig. 5c). Also in this case there are divergent diagrams, because the left sub-loop might be divergent for the replacement \(W \rightarrow \phi _{\pm }\). After GIM-cancellation the dominant term is

*u*-quark is neglected due to small loop functions (-after GIM-cancellation), and small CKM-factors. Moreover, the comments about the \(Y_R\)’s at the end of the previous sections are also relevant here.

## 5 Perturbative QCD corrections

*d*-quark:

*t*-quark scale and \(n_f=5\) below. In the present case one should do the running in four steps, from the big scale \(\mu _\Lambda \sim \Lambda \) down to the top scale \(\mu _t \sim m_t\) with \(\beta _0 = 7\), from the top scale down to the

*b*-quark scale \(m_b\) with \(\beta _0 = 23/3\), from the

*b*-quark scale down to the charm scale \(m_c\) with \(\beta _0 = 25/3\), and at last from the charm scale down to the hadronic scale \(\mu _h \sim 1\hbox { GeV}\) with \(\beta _0 =9\).

*b*-quark scale.

## 6 Summary and discussion

As expected, there are cases where the considered two loop diagrams for the EDMs of *d*- and *u*-quarks diverges. This happens for cases in Sect. 3 where the left sub-loop in Fig. 7 is involved, and for diagrams where the unphysical Higgs (\(\phi _\pm \)) is involved both in sect III and IV. More specific, the left diagram in Fig. 7 which looks like a vertex correction for \(d \rightarrow W \, + \, u, c, t\) is logarithmically divergent. Actually, this diagram generates a logarithmic divergent *right-handed current* which has no match in the SM. The diagram at the right in Fig. 7 is convergent, but if the *W*-boson is replaced by an unphysical Higgs \(\phi _\pm \), when used in two loop diagrams as in Fig. 5, we obtain logarithmic divergent diagrams due to a momentum dependent vertex, as seen from (25). These are numerically relevant if the quark \({\tilde{q}} \) is a top quark. The dominating divergent terms in Sects. 3 and 4 are proportional to \(m_t^2\) (-or even \(m_t^4\) in one case in Sect. 3). It should also be noted that the first and last diagram in Fig. 5 are relevant for the EDM of the electron [46]. However, in that case the divergent terms would be proportional to powers of a tiny neutrino mass, instead of the top-quark mass.

All contributions (after GIM-cancellation) not proportional to \( \xi _t \, \equiv \, V_{td}^* \, V_{tb}\) are neglected, using bounds on other \(Y_R\)’s [36, 37], as explaned in the preceeding sections. Also all the contributions for an EDM of the *u*-quark can be neglected, for reasons given at the end of the Sects. 3 and 4.

*s*-quark contribution \(d_s\) for the following reason: The loop functions for the

*s*-quark are numerically close to the ones for the

*d*-quark. The CKM factor is bigger, but \(\gamma _s/\gamma _d \simeq 10 ^{-2}\), such that the contribution to the result in (4) from the \(d_s\) is of order \(5 \%\). Thus our final result for the nEDM is simply

From the mathematical point of view, \(\Lambda \) is the quantity which regularise the divergent two loop diagrams, while \(\Lambda _{NP}\) in (9) is introduced as a dimensional quantity parametrising the \(Y_R\)’s and indicates the scale of new physics. But these scales are expected to be of the same order of magnitude.

*if*the bound for \(Y_R(d \rightarrow b)\) found in [37] is assumed to be saturated, then one can see how close to the experimental bound on the nEDM in (1) my value of nEDM might come. This is illustrated explicitly as follows:

Now, the *maximal value* of the parenthesis \(\{\ldots \}\) in (44) is \(= 1\). Then, *if* the bound for \(Y_R(d \rightarrow b)\) is saturated, the plot for the function \(N(\Lambda )\) in Fig. 8 shows that when the cut-off \(\Lambda \) is stretched up to 20 TeV, the bound for nEDM in (1) is reached in the bare case (upper curve), while the perturbative QCD-suppression tells us that the value of the nEDM can at maximum be of order one tenth of the experimental bound for \(\Lambda \) up to 20 TeV (lower curve). If the bound for \(|Y_R(d \rightarrow b)|\) is reduced, and also \(\Lambda \) is reduced, my value for nEDM will be accordingly smaller.

## 7 Conclusion

In conclusion, I have explored the consequenses for the nEDM of having flavor changing Higgs couplings. In the scenary of [37, 40] such couplings might stem from a six dimensional non-renormalisable, \(SU(2)_L \times U(1)_Y\) gauge-invariant Lagrangian piece proportional to the third power of the SM Higgs doublet field, as seen in Eq. (9). The considerd effective theory is non-renormalisable and some diagrams are ultraviolet divergent. I have parametrised the divergence in terms of the logarithm of the ultraviolet cut-off, of order of possible New Physics. One might think that, even if this effective theory is non-renormalisable, one might regularise it with \({\overline{MS}}\)-regularisation and add counterterms, -say like in chiral perturbation theory. However, I do not think that this will give a better description in the present case.

While previous analysis [36, 37] obtained bound(s) of *quadratic* expressions of the FCH coupling(s), in the present paper the analysis is extended to the two loop case for quark EDMs generated by a flavor changing Higgs coupling \(Y_R(d \rightarrow d)\) to *first order only*.

I have found and calculated two loop contributions which gives a bound for the imaginary part of the *product* of \(Y_R(d \rightarrow b)\) *and* the CKM entry \(V_{td}^* V_{tb}\) (where \(V_{tb}\) is known to be very close to one). This bound cannot be directly compared with the bound from [36, 37], which is on the absolute value. But even if this bound on the absolute value is saturated, and even if \(\Lambda \) is stretched up to 20 TeV, it is seen from Fig. 8 that the value of the nEDM can at maximum be of order one tenth of the present experimental bound in (1).

## Notes

### Acknowledgements

I am grateful to Svjetlana Fajfer for suggesting these calculations and for valuable discussions. Useful comments by Lluis Oliver and Ivica Picek are also acknowledged.

I am supported in part by the Norwegian research council (via the HEPP project).

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