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

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. Previously one loop diagrams with such couplings were considered in order to constrain {\it quadratric} expressions of Higgs flavor changing couplings to quarks. In the present paper the analysis is extended to the two loop level, where there are diagrams for electric dipole moments of quarks with a flavor changing Higgs coupling to {\it first order only}. The divergent loops, due to non-renormalisabillity, are parametrized in terms of an ultraviolet cut-off $\Lambda$. I also consider QCD corrections, including the mixing with the color electric dipole moment, while the contribution from the Weinberg operator is found to be negligible. The effect of QCD corrections is to suppress the bare result. 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 {\it product} of the Higgs flavor changing coupling for $(d \rightarrow b)$-transition {\it and} the CKM element $V_{td}$. Assuming that the previous bound of the {\it 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 {\it bare} result, {\it 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.

Within the SM, the nEDM is calculated to be several orders of magnitudes below the experimental bound. Calculations of the nEDM will in general put bounds on hypothetical models BSM, and any measured nEDM significantly bigger that the SM estimate (10 −32 to 10 −31 e cm) would signal New Physics.
The SM contributions to the nEDM are well known and thoroughly explained in [1]. At a low energy scale one can construct an effective Lagrangian with all possible CP-odd operators of appropriate dimension. The QCD-odd term gives the dimension 4 operator [1,5]. The dimension 5 operator contains electric dipole moment operators as well as color electric operators of quarks. The CP-odd three-gluon Weinberg operator and four-fermion interactions are then in the class of dimension six operators [1][2][3], which are not subject of this study. The electric dipole moment of a single fermion in (2) has the form where d f is the electric dipolement of the fermion, ψ f is the fermion (quark) field, F µν is the electromagnetic field tensor, and σ µν = i[γ µ , γ ν ]/2 is the dipole operator in Dirac space. The electric dipole operator in (3) for quarks appears within the SM from three loop diagrams with double Glashow-Iliopoulos-Maiani (GIM)-cancellations and in addition a gluon exchange. These are of order α s G 2 F , and are proportional to quark masses and an imaginary CKM factor. They were found to be very small, of order 10 −34 e cm [8,9]. Still within the SM, many contributions to the nEDM due to interplay of quarks in the neutron, were studied [1,[10][11][12][13][14][15][16][17][18][19]. These mechanisms, gave results of order 10 −33 to 10 −31 e cm.
The nEDM due to EDMs of light u-and d-, and even s-quarks are given by the formula similar to a corresponding formula for the magnetic moment. In the strict valence approximation, while recent lattice calculations [20,21] give γ u = −0.22 ± 0.03 , γ d = 0.74 ± 0.07 , γ s = 0.008 ± 0.010 .
Note that there is a contribution to the nEDM from the EDM of the s-quark, with a small coefficient.
Many models BSM suggest possible new particles or new interactions inducing EDMs [1-5, 10, 22-32]. In the case of New Physics presence, flavor physics might be testable through CP-violating asymmetries in mesonic decays [23,24,33]. The properties and couplings of the physical Higgs boson (H) are still not completely known. Some authors [34][35][36][37][38][39] 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 −K, D −D , and B −Bmixings, and also from leptonic flavor changing decays like µ → e γ and τ → µ γ. In the latter case two loop diagrams of Barr-Zee type [40] were also considered [34][35][36]41].
(See also [42]). Flavor changing couplings of this type will occur if the SM Higgs have non-renormalized interactions appearing when higher mass states are integrated out. For instance, flavor changing Higgs (FCH) couplings might stem from SU(2) L × U(1) Y -invariant but non-renormalizable Lagrangian terms of dimension six.
In this paper I extend the analysis of [35,36]  In the next section (II) I will present the framework for the FCH couplings. In the sections III and IV two loop calculations for the FCH couplings will be presented. In section V the results will be discussed, and the conclusion given.

II. FLAVOR CHANGING PHYSICAL HIGGS?
Within the framework in [34][35][36][37][38][39] (see also ref. [44]) the effective interaction Lagrangian for the FC transition f 1 → f 2 due to Higgs exchange can in general be written where f 1,2 are fermion fields, H the Higgs field and Y L,R (f 1 → f 2 ) are coupling constants. The Y 's are in general thought to be complex numbers. Then, from the hermitean conjugation part we obtain for the opposite Flavor changing Higgs couplings of the type presentes in eq. (7) may occur if there are non-renormalizable Higgs type Yukawa-like interactions due to dimension six operators, as shown explicitly in [36,39] : where the generation indices i and j are understood to be summed over. Further, φ is the SM Higgs field, Q i is the left-handed SU(2) L quark doublets, and the D j 's are the righthanded SU(2) L singlet d-type quarks in a general basis. Further, Λ N P 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 .
Using the assumptions based on (7), one obtains one loop diagrams for EDMs of u-and d-quarks [35,36]. The one loop diagram in Fig. 1    Adding a soft photon to the diagram in the middle and to the right, we get four diagrams for both cases. In Fig. 3 the four diagrams obtained by adding a soft photon emission to the diagram to the right in Fig. 2  The results for the loop contributions in Fig. 3 have the form: and the diagrams with interchanged order of H and W loops, as in the middle of Fig. 2 have the form: where P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2 are projectors in Dirac space. Then the electric dipole moment is found to be: There is also a contribution to the magnetic moment (i.e the gyromagnetic quantity (g − 2)) given by 2 Re(A).
The contributions from the four diagrams (i = 1-4) in Fig. 3 and its complex conjugates can then, by using (12) be written where theê i 's are the electric charges (in units e= the proton charge) of the photon-emitting particles, i.e.ê 1,3 =ê t = +2/3,ê 2 =ê b = −1/3, andê 4 =ê W = +1. Here I have used the relations (8) and (12). The V 's are CKM matrix elements in the standard notation. The constant F 2 sets the overall scale of the EDMs obtained from our two loop diagrams: where Numerically, I find If the soft photon is emitted from the top quark after exchange of the Higgs boson, as in the third diagram from left in Fig. 3, or from the 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.
Using Feynman gauge for the W -boson, one has also to add diagrams with the unphysical Higgs φ ± (i.e. the longitudinal component of the W-boson) given by the the Lagrangian The total contribution from the third digram in Fig. 3 (including the contribution from the unphysical Higgs), is Here the UV divergence is parametrized through the quantity where Λ is the UV cut-off. Numerically, C Λ is ∼ 5.5 to ∼ 7.7 for Λ ∼ 1 to 3 TeV. Further, where u t is given in (15).
The fourth diagram in Fig. 3 with the soft photon emitted from the W -boson again contains a divergent part, and the total contribution to the fourth diagram is where Summing all contributions from diagrams in Fig. 3, we find There are in addition contributions from the same diagrams in Fig. 3, but with other quarks in the loop. If the b-quark is replaced by an s-quark, the CKM factors are two orders of magnitude smaller, and in addition Y R (d → s) has a stricter bound from K −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 tand the b-quarks interchanged. This amplitude has the same structure as in (13), and is But the u-quark EDM contributions will be neglected. First, the ordinary SM coupling of the Higgs will be proportional to 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 ∼ 10 −3 compared to the analogous d-quark contributions. Second, the absolute value of Y R (u → t) has a relaxed bound of order one, but because it comes from a similar lagrangian term as in (9) one might guess that it is of same order of magnitude as compared to the d-quark EDM contribution to the nEDM.

IV. DIAGRAMS WITH ONE FC COUPLING -AND A W W H-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 W W H-coupling ∼ 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 zero 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). The relevant piece of the SM Lagrangian for a Higgs coupling to two W -bosons is given by Using Feynman gauge for the W -boson, we must also consider Lagrangian terms for a physical Higgs coupling to a W -boson and the unphysical Higgs boson φ ± . In addition to the term for quarks coupling to φ ± in (17), there is the relevant HW φ ± -coupling obtained from the Lagrangian Because of derivative couplings, the vertices involving the unphysical Higgs φ ± will depend on the loop momenta, which might give divergent (sub-)loops. There are also W γφ ± -couplings, but they do not contribute for soft photon emission.
In the preceeding section (III) all numerically relevant diagrams were proportional to m 2 t , and even m 4 t in S 3 . In the present section the diagrams have another chiral structure, and we get diagrams ∼ m 2 t only for the case when the W -boson is replaced by an unphysical Higgs φ ± . Therefore I apriori consider all quark flavors in the loops, although it is expected that the GIM-mechanism will cancel the leading terms with light quark flavors. Contributions to the 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 be written : where F 2 is given in (14) and whereê s =ê b = −1/3 are charges for the photon-emitting quarks, and the λ's and the ξ's are CKM factors: Note that only the right-handed couplings Y R contribute to the amplitude, due to the chiral structure of the two loop diagrams. Within the Wolfenstein parametrization, λ t is of fifth order in λ = λ u , while ξ u and ξ t are only of third order. The ∆f 's in (26) are differences, due to GIM-cancellation, between loop functions f (q,q), for given flavors q = s, b andq = u, c, t.
These are functions of quark, W -boson and Higgs masses and are integrals over one Feynman parameter involving logarithmic and dilogarithmic functions, and is given in eq. (65) in the Appendix. Above , we have used the shortages and so on in a self-explanatory way.
The quantities f (q,q) are finite, and are of order 10 −1 to 1.(See Table I As the f d (q, u − c)'s are very small, and λ t is of order λ 5 , while ξ t is of order λ 3 , we find that the contribution proportional to ξ t ·∆f d (b, t−c) dominates in equation (26). Also, Y R (d → s) has a stricter bound from K − K-mixing than Y R (d → b) has from B d − B d -mixing [36].
Similar to (26), contributions to the u-quark EDM due to emission from the quark q = c, t the diagram to the left in Fig. 5 is : where the κ's and the ζ's are CKM factors: Within the Wolfenstein parametrization, κ d = −λ, κ b is of fifth order, while ζ d and ζ b are of third order in λ. Further, the ∆f u 's are defined as and so on in a self-explanatory way. For the EDMs of the u-quark , the values of the various (∆f u )'s are of order 10 −3 to 10 −7 after GIM-cancellation, and can be neglected, as seen in table II in the Appendix. Again, we neglect the u-quark EDM, for the same reasons given at the end of the preceeding section.
The diagram with soft photon emission from the quarkq = u, c, t, is shown the center of Fig. 5 (i.e Fig. 5b). In this case the loop functions are named h(q,q). Adding contributions where the W is replaced by an unphysical Higgs φ ± , we obtain divergent contributions for these loop functions. Because the W Hφ ± -vertex is momemtum dependent, the left subloop is divergent, reflecting again that the theory based on the Lagrangian in eq. (7) alone is not renormalizable.
The contribution(s) from the diagram in Fig. 5b has the same structure as in (26) and (30), but all terms are not numerically relevant. The numerically relevant term from the h(q,q)-functions is given by: where ∆h d (b, t − c) is defined similar to the ∆f 's in eq. (28). From the Appendix one finds where C Λ is given in (19) and p 2 (u) in (22).
For diagrams with a soft photon emitted from the W -boson, as shown at the right of Fig. 5 (Fig. 5c), the loop functions are named k(q,q) . Also in this case there are divergent diagrams, because the left sub-loop might be divergent for the replacement W → φ ± . After GIM-cancellation the dominant term is where from the Appendix one finds Neglecting small contributions (all except those proportional to V * td V tb ≡ ξ t ), and summing all contributions from diagrams in Fig. 5 The EDM of the u-quark is neglected due to small loop functions (-after GIM-cancellation), small CKM-factors. Moreover, the comments about the Y R 's at the end of the previous sections are also relevant here.

V. SUMMARY AND CONCLUSION
As expected, there are cases where the considered two loop diagrams for the EDMs of dand u-quarks diverges. This happens for cases in section III where the left sub-loop in Fig.   6 is involved, and for diagrams where the unphysical Higgs (φ ± ) is involved both in sect III and IV. More specific, the left diagram in Fig. 6 which looks like a vertex correction for d → 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. 6 is convergent, but if the W -boson is replaced by an unphysical Higgs φ ± , 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 q is a top quark. The dominating divergent terms in section III and IV are proportional to m 2 t (-or even m 4 t in one case in section III). It should also be noted that the first and last diagram in Fig. 5 are relevant for the EDM of the electron [45]. However, in that case the divergent terms would be proportional to powers of a tiny neutrino mass, instead of the top-quark mass. Summing all two loop contributions from section III and IV, we obtain the dominanting contribution for an EDM of the d-quark: All contributions (after GIM-cancellation) not proportional to ξ t ≡ V * td V tb are neglected, using bounds on other Y R 's [35,36], 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 sections III and IV.
I have also neglected the 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 γ s /γ d ≃ 10 −2 , such that the contribution to the result in (4) from the d s is of order 5%. Also, because we are dealing with an effective theory with limited precision, we do not consider the mixing of color electric terms into the EDM term in (3) due to renormalization effects in perturbative QCD. Thus our final result for the nEDM is simply where the lattice value of γ d is given in (6). Using the experimental bound for nEDM in (1), the result (38) of the present study gives the bound In (40) I have found a bound on the coupling Y R (d → b) comparable to, but slightly more relaxed than previous bounds [35,36]. Also, the bound found here is not directly on the absolute value, as in [35,36], but on a combination of Y R (d → b) and a CKM factor, Turned [36] is (almost) saturated, then a value for nEDM not too far below the bound in (1) can not be excluded. This might be illustrated explicitly as follows: Using (38), the lattice values in (6) and absolute value of V * td V tb from [6], one may write my result for the nEDM in the following way where I have scaled the result with the bound from [35,36]: Explicitly, the function N Λ , decudeced from (38) can be written where α = 0.31 and β = 1.59. This function is plotted as a function of Λ in Fig. 7.  ) is saturated, the plot for the function N Λ in Fig. 7 shows that when the cut-off Λ is stretched up to 25 TeV, the bound for nEDM in (1)  In conclusion, I have explored the consequenses for the nEDM of having flavor changing Higgs couplings. In the scenary of [36,39] such couplings might stem from a six dimensional non-renormalizable, SU(2) L × U(1) Y gauge-invariant Lagrangian piece proportional to the third power of the SM Higgs doublet field, as seen in eq. (9). In the present paper the analysis in [35,36] is extended to two loop case for quark EDMs generated by a flavor changing Higgs coupling Y R to first order.
I have found and calculated two loop contributions which gives a bound for the imaginary part of Y R (d → b) slightly more relaxed, but of the same order of magnitude compared to the bound found in [35,36] for the absolute value, obtained from B d −B d -mixing. Turned around, if the bound for Y R (d → b) found in [35,36] is almost saturated, a value for nEDM not far below the bound in (1) cannot be excluded.

VI. APPENDIX DETAILS FROM THE TWO OOP CALCULATIONS
To simplify calculations I use the effective quark propagator in a soft electromagnetic field F [46]: where k is the four momentum and m q the mass of the quark q. The notation {A, B} ≡ AB + BA is used. Similarly, for emission of a soft photon from a W -boson, the effective propagator is: where d − r ≡ d d r/(2π) d in d dimensions within dimensional regularization. A term with ultraviolet divergence appears if r ν is replaced by p ν in the numerator when doing loop integration in the subloop containing the integration over p. This happens for instance when the W -boson is replaced by an unphysical Higgs φ ± (the longitudinal W -components) within Feynman gauge, or if the left diagram in Fig. 6 is involved. Many loop diagrams are suppressed because of chirality (P L P R = 0), or asymmetric (odd) momentum integration like: In the first subloop two Feynman parameters x and y are used. The result from the first subloop then depends on x and y and the squared loop momentum of the second loop.
Then the second loop integration is performed. In the end the Feynman parameter y is easily integrated out, and one is left with integration over the Feynman parameter x, to be done numerically.
The quantities S i in (13) are dimensionless and given by integrals over one Feynman parameter. For the first diagram in Fig. 3 I find where the the mass ratios u t and u H are given in (15). Terms suppressed by a factor of order m 2 b /m 2 t with respect to the terms in (48) are neglected. Further, The term ut 2 (1 − x) in the numerator in (48) is due to the unphysical Higgs contribution which have to be added when the Feynman gauge for the W-propagator is used.
If the soft photon is emitted from the b-quark in the second diagram from left in Fig. 3 one obtains where C 0 and C 1 are given as in (49) and the term proportional to u t · u t /2 in the numerator is the unphysical Higgs contribution. The dilogarithmic function is in my case defined as If the soft photon is emitted from the top quark after exchange of the Higgs boson, as in the third diagram from left in Fig. 3, or from soft photon emission from the W -boson in the fourth diagram, the left sub-loop containing the Higgs boson is logarithmically divergent.
It turns out that for the third diagram in Fig. 3 the divergent part is projected out due to Dirac algebra (-technically, γ α σ µν γ α = 0 ). One is then left with the finite part S W 3 : where the function K is in general given by In eq. (52), one has : where u t and u H are given in (15), and b ≡ m 2 b /M 2 W . For the third diagram in Fig. 3, with W replaced by the unphysical Higgs φ ± the contribution is divergent, and given by Divergent parts from the first subloop enters as where R is a quantity depending on masses and Feynman paprameters. The ln(R) term results in a finite term: where B 0,1 are given in (54), and where in general: Further, there is a non-logarithimic finite term (not in ln(R) in eq. (56)): where B 0,1 are given in (54), and K is given in (53). The total S 3 : is then given in (18).
For the fourth (divergent) diagram with soft photon emission from the W -boson, I find The logarithmic (∼ ln(R)) type finite term from (56) is where B 0,1 are still given in (54). For the non-logarithmic finite term where the function K is given in (53), and where B 1,0 are given as in (54). The total S 4 : is given in (21). Note that S φ 3 is of the order m 4 t in contrast to S 1,2 , S W 3 and S 4 which are of the order m 2 t . The mathematical expression for the loop functions f (q,q) in (26)-(32) for soft photon emission from the quark q given in the left of Figure 5 is given by where u H is given in (15) and Further, In the f (q,q)'s in eq. (65) the unphysical Higgs (φ ± ) contribution(proportional to −b/4)   part h Λ , the associated finite logarithmic part h N coming from ln(R) in (56) due to the unphysical Higgs (see eq. (69), and the non-logarithmic finite part h K (see eq. (71)): The logarithmic term h N (q,q) in (68) is given by where u H is given in (15), and a, b in (66), and The finite non-logarithmic part h K is given by The term proportional to b/2 above is the finite part of the unphysical Higgs contribution.
The functions h K are given in tables III and IV.
Numerically, we find for the three numerically relevant quantities in ∆h d (b, t − c): For other cases the ∆h's are of order 10 −3 or smaller. Similar to the case with emission from quark q, the contribution involving the top quark proportional to ξ t gives the biggest contributions after GIM-cancellation, as in the preceeding section.   Now, the next step is to calculate the loop diagrams when the soft photon is emitted from a W -boson. Again, I have split the functions k in a non-logarithmic finite part k K , the divergent part k Λ due to unphysical Higgs, and the finite logarithmic part k N (associated with ln(R)-terms): For the diagram to the right in Fig where B 0,1 are given as in (70), and a and b are given in (66).
When the soft photon is emitted instead from the W -boson in the middle of diagram one obtains k(q,q) 2 = 1 (u H − a) where the function K W 2 (B) is given by .
For the diagram to the right of Fig. 5 with the W in the middle replaced by an unphysical Higgs φ ± , one obtains where k Λ is the divergent part shown in (86), and the logarithmic finite part is k N (q,q) = − b 2(u H − a) where B 0,1 are given as in (70).
With emission from the W in the middle and when the W to the right is replaced by an unphysical Higgs one obtains where where K W 2 (B) is given previously in eq (76), and where The total finite amplitude for emission from W 's (k Λ and k N excluded) is  For diagrams with a soft photon emitted from some W -boson in Fig. 5, the numerically relevant part is Numerically, we find for the parts of ∆k d (b, t − c): Other ∆k's are of order 10 −3 or smaller.