ρ parameter and H 0 → ℓ i ℓ j in models with TeV sterile neutrinos

ρparameter and H0→liljin models with TeV sterile neutrinos G. Hernández-Tom´e , J. I. Illana , and M. Masip CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, E18071 Granada, Spain (Received 4 September 2020; accepted 2 December 2020; published 21 December 2020) The presence of massive sterile neutrinos Nmixed with the active ones induces flavor violating processes in the charged lepton sector at the loop level. In particular, the amplitude of H0→¯ liljis expected to be proportional to the product of heavy-light Yukawa couplings yiyj¼2sνisνjm2 N=v2, where sνi;j express the heavy-light neutrino mixings. Here, we revisit these Higgs decays in the most generic extension of the neutrino sector, focusing on large values of yi. We show that decoupling effects and a cancellation between the two dominant contributions to these processes makes the amplitude about 100 times smaller than anticipated. We find that perturbative values of yigiving an acceptable contribution to the ρparameter imply BðH0→¯ liljÞ<10−8for any lepton flavors, a rate that is not accessible at current colliders. DOI: 10.1103/PhysRevD.102.113006 I. INTRODUCTION The nature of the neutrino masses remains as one of the most intriguing questions in particle physics. Neutrinos are different from the other fermions in that the SUð2ÞLsinglet required to give them an electroweak (EW) mass is not protected by chirality. The possible mass of this singlet will then define a new scale that, if very large, would explain the tiny value of the neutrino masses (mν<1eV) deduced from flavor oscillations. Indeed, the so-called type-I seesaw mechanism provides a minimal and very appealing way to complete the lepton sector of the standard model (SM). There are, however, other nonminimal possibilities that may be considered as well. Notice that gauge singlets, if present, can have any mass. From a phenomenological point of view, the origin of their interactions are arbitrary Yukawa couplings that mix them with the active neutrinos, so they could be very weakly coupled to matter and thus, easily avoid all experimental bounds. From a model building point of view, they appear naturally in extensions of the SM with a cutoff much lower than the seesaw scale. This is the case, for example, in little Higgs models [1–3], TeV gravity models [4,5], or composite Higgs models [6], where neutrino masses must be explained relying on physics at or below the TeV scale. In the end, it is the data on neutrino oscillations and charged-lepton flavor physics what decides about the motivation for these sterile neutrinos. The appearance of non-EW terms in the extended neutrino mass matrix and the different gauge charges of active and sterile neutrinos will imply that the rotation defining the mass eigenstates does not diagonalize, respectively, the Higgs nor the Zcoupling to the neutrinos. At the loop level, these flavor-changing neutral currents (FCNC), and also the charged currents coupled to the Wboson, induce flavor violating processes involving the charged leptons (CLFV) [7–15]. Here, we will be interested in these processes. In particular, we will study the CLFV decays H0→¯ liljin the presence of the generic heavy sterile neutrinos that appear in the context of low-scale seesaw models. These decay channels are currently searched at the LHC; at 95% C.L., ATLAS [16] and CMS [17,18] find BðH0→μeÞ<6.1×10−5ðATLASÞ;3.5×10−4ðCMSÞ; BðH0→τeÞ<2.8×10−3ðATLASÞ;6.1×10−3ðCMSÞ; BðH0→τμÞ<4.7×10−3ðATLASÞ;2.5×10−3ðCMSÞ; ð1Þ where H0→liljstands for H0→¯ lilj;li¯ lj. Our objective is to establish the maximum rate for these processes that could possibly be caused by the heavy sterile neutrinos. Previous literature reports approximate results [19–21] or detailed computations [22–25] in the context of inverse seesaw models for neutrino masses. Here, we will introduce a minimal setup [26] that contains just two heavy neutrinos but that is able to capture all the flavor effects relevant in Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 102, 113006 (2020) 2470-0010=2020=102(11)=113006(9) 113006-1 Published by the American Physical Society these processes. The simplicity of the parametrization lets us understand the limit with large (top-quark-like) Yukawa couplings for the singlets, where one may expect branching ratios near the current bounds. We show that the contribution from such couplings to the ρparameter may be acceptable (actually, we find remarkable that Δρfrom the singlet fermions may have any sign), but that the appearance of a cancellation and of decoupling effects push the decay modes well below these bounds. II. THE SETUP Flavor oscillation experiments are able to access the tiny value of the neutrino masses by combining two very different scales, L−1Eν≈Δm2 ν. In CLFVexperiments, however, the lowest available scale is ml, so these experiments are not sensitive to mν. Any observable effects will then depend on the possibly much larger masses of additional fermion singlets that mix with the active flavors. It turns out that to capture all the CLFV effects in a consistent way. it will suffice to consider two massive two-spinors that may be defining a single Dirac fermion or two Majorana fields of different mass. Although these singlets will not be responsible for the masses of the active neutrinos, the key point is that all the extra ingredients required to complete the neutrino sector will have no effect on CLFV observables. Let us be more specific (see [26] for details). Consider five Majorana (self-conjugate) fields χi¼χLi þðχLiÞc whose left-handed component χLi includes the three active neutrinos (i¼1, 2, 3) plus two sterile spinors of opposite lepton number (i¼4, 5). We will assume that in the basis of the charged-lepton mass eigenstates, the only new terms in the Lagrangian are −L⊃X 3 i¼1 yi ˜ Φ†¯ χ5PLLiþM¯ χ5PLχ4þ1 2μ¯ χ5PLχ5þH:c:ð2Þ Once the SM Higgs doublet Φgets a vacuum expectation value (v.e.v.) ( ˜ Φ¼iσ2Φ), the Majorana mass matrix for the five flavors reads, M¼ 0 B B B B B B @ 0000m1 0000m2 0000m3 0000M m1m2m3Mμ 1 C C C C C C A :ð3Þ Notice that we have ordered the fields according to the lepton number (L) of their left-handed component— positive for the first four neutrinos—that miand Mare Dirac masses—entries m0 iin the fourth row or column would break L—and that μ, a Majorana mass term for the neutrino with LðχL5Þ¼−1, is the only source of L breaking in this matrix.1Its diagonalization yields two states N1;2of mass: mN1¼1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðm2 1þm2 2þm2 3þM2Þþμ2 q−μ; mN2¼1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðm2 1þm2 2þm2 3þM2Þþμ2 qþμ;ð4Þ plus three massless neutrinos νi. It is straightforward to find that these three neutrinos have a component along the (two-dim) sterile flavor space (a heavy-light mixing), sνi¼mi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mN1mN2 p:ð5Þ For μ¼0, the two massive modes will define a Dirac field (mN1¼mN2); in this case, a small entry μ0in position M44 would give a mass mν≈μ0ðm=MÞ2to one of the standard neutrinos, as proposed in inverse-seesaw models [27,28]. In the opposite limit, if M¼0and μ→1010 GeV, the configuration describes a type-I seesaw mechanism, with one of the active neutrinos massive, mN1≈ ðm2 1þm2 2þm2 3Þ=μ, while the second singlet (χ4) is massless but decoupled. For μin the TeV range, as long as M>10 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1þm2 2þm2 3 p(i.e., the mixings are below 0.1), the model may be viable. At any rate, Mis a rank-2 matrix with three zero mass eigenvalues. As we argued above, the extra spinors and couplings required to generate light neutrino masses will have no effect on CLFV observables. In particular, the so-called TeV type-I seesaw models [7] can be obtained by adding a third singlet with an OðTeVÞ Majorana mass Λ; in a certain basis, all these models are reduced to the texture, M0¼ 0 B B B B B B B B @ 000·m1· 000·m2· 000·m3· ····M· m1m2m3Mμ0 ····0Λ 1 C C C C C C C C A ;ð6Þ where the dots indicate very small entries that are necessary to generate standard neutrino masses and light-light mixings but have no effect on the heavy-light mixings: Any OðGeVÞterm there would increase the rank 3 of this 6×6 matrix and imply a nonacceptable mass spectrum. Notice also that the third singlet does not introduce significant heavy-light mixings. Therefore, the five mass parameters in 1Notice that in inverse seesaw models, the usual ordering of the two massive neutrinos is the opposite, i.e., first the neutrino with LðχLÞ¼−1. This ordering would imply the exchange of the fourth and fifth columns or rows in our matrix M. HERNÁNDEZ-TOM´ E, ILLANA, and MASIP PHYS. REV. D 102, 113006 (2020) 113006-2 M(or two heavy masses plus three heavy-light mixings) are enough to describe all CLFV effects caused by heavy Dirac or Majorana singlets mixed with the three active families. One should also stress, however, that if μ≠0, the matrix above is not stable under radiative corrections [29]: The breaking of lepton number will contribute to all the entries in Mat the loop level, which would give mass to a linear combination of the three νi. If this breaking is small, the mass will be acceptable (i.e., below 1 eV), but if μis large, the model will require a fine-tuned cancellation of these loop contributions. In summary, the texture that we propose in Eq. (3) must be understood as approximate and established at the loop level where we work. Despite the finetune that this involves, we will consider TeV values of μin order to understand the genuine Majorana effects on CLFV observables and on the contribution to the ρparameter from heavy singlets. III. LARGE YUKAWA COUPLINGS AND Δρ The Yukawa couplings yiin Eq. (2) are the origin of any interactions of the heavy singlets, and the rate of H0→ ¯ liljwill certainly grow with them. In our model, their relation with the masses and mixings is yi¼ffiffiffi 2 pmi v¼ffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mN1mN2 p vsνi:ð7Þ The expression above shows that, for a fixed value of the mixings consistent with current constraints, large singlet masses will probe large values of yi. These couplings, however, break the custodial symmetry of the SM and will contribute to the ρparameter (or to the Peskin-Takeuchi parameter T¼ðρ−1Þ=α). These oblique corrections can be easily obtained from the contribution of the heavy neutrinos to the gauge boson self-energies at q2¼0, Δρ¼ðΠWWÞN1;2 M2 W −ðΠZZÞN1;2 M2 Z ;ð8Þ and they are constrained to be jΔρj≲0.0005 [30].Atone loop and neglecting charged lepton masses, we find (see the couplings to gauge and Goldstone bosons in Appendix A), Δρ¼g2 32π2M2 WX 3 k¼1 s2 νk2m2 N1m2 N2 ðmN1þmN2Þ2 ×3−2m2 N1þm2 N2−mN1mN2 m2 N2−m2 N1 ln mN2 mN1:ð9Þ This result presents some interesting features. Let us assume, for simplicity, mixing with just ντand consider first the case with a Dirac singlet field (μ¼0). The contribution is then obtained from Eq. (9) by taking the limit mN1;m N2 →mN: Δρ¼g2 64π2M2 W s4 ντm2 N¼g2 64π2M2 W s2 ντy3 v ffiffiffi 2 p2 :ð10Þ If we compare this with the correction from the top quark, Δρt¼3g2 64π2M2 Wyt v ffiffiffi 2 p2 ≃0.009;ð11Þ we see an extra suppression by a decoupling factor of s2 ν. Obviously, if the heavy neutrino were a sequential doublet with a purely EW mass, this suppression would be absent; in this case, the contribution should be canceled by restoring the custodial symmetry with a very similar Yukawa coupling of the charged lepton in the same doublet. But here, for sντ<0.1and y3<ffiffiffiffiffiffi 4π p,wehave that Δρ<0.00038 is within the experimental bounds. Another interesting limit goes in the opposite direction: A Majorana mass μmuch larger than M, and then mN2≫mN1. It is easy to see that if m2 N2>30m2 N1ðor μ>2.1MÞ;ð12Þ the second term in Eq. (9) dominates and the contribution to Δρis negative, something remarkable as multiplets of nondegenerate Dirac fermions always give Δρ>0. For sντ<0.1and y3<ffiffiffiffiffiffi 4π p, we obtain −Δρ<0.00012. The correction for a type-I seesaw mechanism (M¼0, μ≫1TeV) is just Δρ≈− g2 32π2M2 W m2 ντ2ln μ mντ −3;ð13Þ with mντ¼y2 3v2=ð2μÞ. Our results for Δρfrom TeV fermion singlets are consistent with the generic ones in [31]. IV. H0→¯ lilj The one-loop amplitude for H0→¯ liljis mediated in the Feynman-’t Hooft gauge by the diagrams in Fig. 1.One can see that all these diagrams are proportional to yiyjyli¼2ffiffiffi 2 psνisνj mN1mN2mli v3;ð14Þ where liabove refers to the heavier final lepton. In addition, diagrams Wχχ,χWG, and Wχare proportional to g2,χWW is proportional to g4, and χGG to the Higgs quartic coupling λ. Of course, each diagram will also depend on the mass and spin of the particles inside the loop, but one may expect that Gχχ and Gχdominate with a contribution of order M≈yiyjyli=ð16π2Þ. This estimate coincides with what is expected using an effective field theory approach (see Ref. [20]). ρPARAMETER AND H0→lilj…PHYS. REV. D 102, 113006 (2020) 113006-3 Using this estimate, we can deduce the maximum branching ratio in Higgs decays by comparing with BðH0→¯ bbÞ≃0.6. For the decay H0→τe, we expect BðH0→τeÞ¼BðH0→¯ bbÞ2ΓðH0→¯τeÞ ΓðH0→¯ bbÞ ≈BðH0→¯ bbÞ2 3y3y1yτ yb16π22 :ð15Þ Taking yi<ffiffiffiffiffiffi 4π p, this gives BðH0→τeÞ<4×10−4,a value that could be accessible once the LHC reaches its highest luminosity. However, a precise calculation will show that this is not the case. First of all, although their sum is finite, the diagrams Gχχ and Gχare both divergent. In addition, there is a value of the heavy neutrino mass that exactly cancels the sum of both contributions. For mN1¼mN2, this is ˜ mN≈0.57 MH ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 νeþs2 νμþs2 ντ q:ð16Þ Finally, at masses of the heavy neutrinos above ˜ mN, there are decoupling effects, like the extra factor of s2 ντin Δρ found in the previous section. Let us be more definite. We write the decay amplitude, MðH0→¯τeÞ¼¯ uðp2Þfτe v½mτPRþmePLvðp1Þ;ð17Þ and will give the results in terms of mN1and the ratio, r≡m2 N2 m2 N1 ≥1:ð18Þ Constraints from flavor-diagonal processes [32–35], together with Bðμ→eγÞ≈3α 8πs2 νμs2 νe<4.2×10−13;ð19Þ imply [26] FIG. 1. Diagrams contributing to H0→¯ ll0in the Feynman-’t Hooft gauge for ml0¼0. Diagram Wχis proportional to m3 land will be neglected. FIG. 2. Contribution to jfτejfrom the dominant diagrams Gχχ þGχfor fixed (maximal) mixings and different heavy neutrino masses (notice that Yukawa couplings grow with the mass). UV divergences cancel in Gχχ þGχ. The blue line (r¼1) shows the heavy Dirac case, whereas the red line (r¼100) corresponds to two Majoranas with mN2¼10mN1. We have included the estimate of jfτejin Eq. (15) for r¼1(gray dots) as well as the contribution from a massive neutrino with an active left-handed component (red dashes). HERNÁNDEZ-TOM´ E, ILLANA, and MASIP PHYS. REV. D 102, 113006 (2020) 113006-4 smax νe¼0.05;s max νμ¼4.5×10−4;s max ντ¼0.075:ð20Þ In Fig. 2, we plot the contribution to jfτejfrom the Gχχ þGχfor these maximal mixings, and r¼1, 100. We see that it grows with the heavy-light Yukawas [with the size anticipated below Eq. (14)]; then, there appears the cancellation at ˜ mNdiscussed above, and finally, the amplitude reaches a regime where it grows again with the Yukawas but is suppressed by a (decoupling) factor of s2 νeþs2 νμþs2 ντ≈0.01 for maximal mixings. This suppression is consistent with the results obtained in [24] using the mass insertion approximation in the region where Yukawa couplings become dominant (y>g,λ). The curves in Fig. 2finish at yi¼ffiffiffiffiffiffi 4π p, which implies mN1¼ 8.2ð2.6ÞTeV for r¼1ð100Þ. The plot also shows that Majorana effects (r¼1gives a Dirac heavy neutrino) do not change the qualitative behavior of the amplitude and are not able to increase the maximum value of jfτej. In the same plot, we have included the amplitude for a heavy neutrino in a SUð2ÞLdoublet:2a Dirac field with an active left-handed component. Such a neutrino does not decouple for large values of mN, which is purely EW; the plot reveals that, in this case, Gχχ þGχfollows the scaling in Eq. (15) for all values of the heavy neutrino mass. The origin of the suppression proportional to the squared mixings with the heavy singlets is the flavor-changing vertex Hχiχj, which would be flavor diagonal if the neutrinos were active (see Appendix A). In Fig. 3, we plot the modulus of each contribution and of the sum of all diagrams for r¼1, 100. We have considered mN1from 10 GeV to its maximum perturbative value just to illustrate the behavior of each contribution, although our analysis focuses on large neutrino masses.3 We see that the dominant contribution comes from χGG except at very large Yukawa couplings, i.e., maximal mixings and heavy neutrino masses above 2 TeV, when diagrams Gχχ þGχtake the lead despite the decoupling factor, yielding a maximum value that is 2 orders of magnitude smaller than the naive guess given before. In Appendix B, we present expressions for the form factors and give further details of our computation. V. SUMMARY AND DISCUSSION Vectorlike fermions at the TeV scale are a possibility with interesting phenomenological consequences. If they are quarks or charged leptons that mix with the active families, their different EW numbers will induce tree-level FCNCs that are very constrained experimentally. If they are neutrinos, however, collider effects appear at the loop level and the bounds are weaker. Here, we have focused on CLFV decays of the Higgs boson. These processes have been studied by several groups, with results that sometimes appear as contradictory. In this work, we have proposed a setup with two sterile fields that captures all flavor effects and lets us understand the results in a simple way. The model reveals, for example, that in generic low-scale seesaw models, Majorana singlets with TeV mass and unsuppressed mixings with the active neutrinos are indeed possible, although they require a fine-tuned cancellation of loop corrections so that the observed neutrinos have sub-eV masses (notice that inverse seesaw models, the heavy neutrinos, are quasi-Dirac) or that large values of the heavy-light Yukawa couplings in these models have an impact on Δρfor large enough heavy-light mixings. FIG. 3. Contribution to jfτejfrom the different diagrams in Fig. 1for fixed maximal mixings and r¼1, 100. The thick line corresponds to the sum of all the diagrams. All amplitudes are real except for Gχχ þGχand Wχχ, which have an imaginary part for mN1<M H=2. The real part of the amplitudes are positive except for Wχχ in the whole mass interval and Gχχ þGχ, which changes sign from negative to positive at intermediate masses, producing a drop. 2This case requires a charged lepton of similar mass to cancel Δρas well as extra EW fermions to cancel anomalies (e.g., to complete the whole sequential fourth family) that are excluded by the LHC. 3For mN≤mZ, current data from colliders set stringent direct limits on the active-sterile mixing from gauge boson and Higgs decays [15,36,37]. ρPARAMETER AND H0→lilj…PHYS. REV. D 102, 113006 (2020) 113006-5 Our analysis shows that the Higgs decay modes H0→ ¯ liljare not accessible at colliders. The rate of these decays is expected to grow with the Yukawa couplings that mix active and sterile neutrinos, but a cancellation of different contributions and decoupling effects proportional to the sum of squared mixings damp the final result. These two features are clearly shown in Fig. 2. We see that for a fixed mixing and a relatively light neutrino mass, the amplitude grows with the Yukawa couplings (which are proportional to the mass) as expected, until the scale in Eq. (16) where the dominant amplitude goes to zero and changes sign. At heavier neutrino masses, the amplitude grows again with the couplings; however, all but a component of order ðs2 νeþs2 νμþs2 ντÞ1=2is decoupled: The amplitude M≈ yiyjyli=ð16π2Þat low singlet masses becomes of order ðs2 νeþs2 νμþs2 ντÞyiyjyli=ð16π2Þin this decoupled regime. As a consequence, we find that the largest branching ratio consistent with the maximal mixings summarized in Eq. (20) would correspond to the channel H0→τeand is BðH0→τeÞ<1.4×10−8:ð21Þ We conclude that the observation of CLFV in Higgs decays at the LHC would involve a different type of new physics. ACKNOWLEDGMENTS We would like to thank F. del Águila, G. López-Castro, P. Roig, and J. Santiago for helpful discussions. This work was supported in part by the Spanish Ministry of Science, Innovation and Universities (FPA2016-78220-C3, PID2019–107844GB-C21/AEI/10.13039/501100011033), and by Junta de Andalucía (FQM 101, SOMM17/6104/ UGR, P18-FR-1962, P18-FR-5057). The work of G. H. T. has been funded by CONACYT of Mexico through the program “Estancias postdoctorales en el extranjero 2019-2020”. APPENDIX A: FLAVOR-CHANGING VERTICES AND MIXING MATRICES The neutrino mass eigenstates come from the interaction eigenstates by the replacement, χLi →X 5 j¼1 Uν ijχLj;ðA1Þ where Uνis the unitary matrix diagonalizing M(3) into real and positive mass eigenvalues. The Lagrangian for charge-current interactions reads, LW¼g ffiffiffi 2 pW− μX 3 i¼1X 5 j¼1 Bij ¯ liγμPLχjþH:c:; ðA2Þ where we have used the convention Dμ¼∂μ−ig ˜ Wμfor the covariant derivative, and Bij ¼X 3 k¼1 δikUν kj ðA3Þ is a rectangular 3×5mixing matrix. In the Feynman- ’t Hooft gauge, one also needs LG¼−g ffiffiffi 2 pMW G−X 3 i¼1X 5 j¼1 Bij ¯ liðmliPL−mχjPRÞχj þH:c:; ðA4Þ where Gis the charged would-be-Goldstone field. The matrix Uνintroduces tree-level flavor-changing interactions with the Zand the Higgs field: LZ¼g 4cW ZμX 5 i;j¼1 ¯ χiγμðCijPL−C ijPRÞχj;ðA5Þ LH¼−g 4MW HX 5 i;j¼1 ¯ χi½ðmχiCij þmχjC ijÞPL þðmχiC ij þmχjCijÞPRχj;ðA6Þ where Cij ¼X 3 k¼1ðUν kiÞUν kj:ðA7Þ A symmetry factor of 2 must be added in the Feynman rule for vertices including two (self-conjugate) Majorana fermions [38,39]: ðA8Þ ðA9Þ One can recover the case of active Dirac neutrinos by replacing Cij →δij,C ij →0. HERNÁNDEZ-TOM´ E, ILLANA, and MASIP PHYS. REV. D 102, 113006 (2020) 113006-6 The mixing matrix elements involving heavy neutrinos can be expressed in terms of heavy-light mixings and the squared mass ratio r¼m2 N2=m2 N1as BkN1¼−ir1 4 ffiffiffiffiffiffiffiffiffiffiffiffi 1þr1 2 psνk;B kN2¼1 ffiffiffiffiffiffiffiffiffiffiffiffi 1þr1 2 psνk;ðA10Þ CN1N1¼r1 2 1þr1 2X 3 k¼1 s2 νk;C N2N2¼1 1þr1 2X 3 k¼1 s2 νk; CN1N2¼−CN2N1¼ir1 4 1þr1 2X 3 k¼1 s2 νk:ðA11Þ APPENDIX B: FORM FACTORS The form factors fll0receive contributions from the oneloop diagrams of Fig. 1in the Feynman-’t Hooft gauge. Neglecting charged lepton masses, we find: fll0 Wχχ ¼g2 16π2X 5 i;j¼1 B liBl0jfCij ffiffiffiffiffiffiffiffi xixj p½c0þ2c1 þC ij½xjc0þðxiþxjÞc1g;ðB1Þ fll0 χWW ¼g2 16π2X 5 i¼1 B liBl0i½−2¯ c1;ðB2Þ fll0 Gχχ ¼g2 16π2X 5 i;j¼1 B liBl0j ×Cij ffiffiffiffiffiffiffiffi xixj p1 4−2c00 þ1 2ðxiþxjÞc1þ1 2xQc12 þC ijxj1 4−2c00 þxic1þ1 2xQc12;ðB3Þ fll0 χGG ¼g2 16π2X 5 i¼1 B liBl0i−1 2xHxið¯ c0þ¯ c1Þ;ðB4Þ fll0 χWG ¼g2 16π2X 5 i¼1 B liBl0i ×1 4−2¯ c00 −1 2xið¯ c0þ2¯ c1Þþ1 2xQð2¯ c1þ¯ c12Þ; ðB5Þ fll0 Wχ¼0;ðB6Þ fll0 Gχ¼g2 16π2X 5 i¼1 B liBl0i 1 2xib0;ðB7Þ where we have introduced the following dimensionless functions in terms of the standard Passarino-Veltman loop functions [40]: b0ðxiÞ≡B0ð0; M2 W;x iM2 WÞ¼B0ð0; xiM2 W;M 2 WÞ;ðB8Þ c00ðxi;x jÞ≡C00ð0;Q 2;0; M2 W;x iM2 W;x jM2 WÞ;ðB9Þ cf0;1;12gðxi;x jÞ ≡M2 WCf0;1;12gð0;Q 2;0; M2 W;x iM2 W;x jM2 WÞ;ðB10Þ ¯ c00ðxiÞ≡C00ð0;Q 2;0; xiM2 W;M 2 W;M 2 WÞ;ðB11Þ ¯ cf0;1;12gðxiÞ ≡M2 WCf0;1;12gð0;Q 2;0; xiM2 W;M2 W;M 2 WÞ;ðB12Þ with xi≡m2 χi=M2 W,xQ≡Q2=M2 W,xH≡M2 H=M2 W≈2.4, and Q2¼M2 Hfor an on-shell Higgs. We use the conventions of [41]. The functions b0,c00, and ¯ c00 are ultraviolet divergent but, thanks to relations between B and Cmatrix elements [26], the divergences in fll0 Gχχ and fll0 Gχcancel each other, and fll0 χWG is finite when summing over all neutrino states. The other diagrams are finite. It turns out convenient to cast the contributions to the form factor (B1)–(B7) into mixing-independent functions F,G,H: fll0¼g2 16π2X 5 i;j¼1 B liBl0i ×½δijFðxiÞþCijGðxi;x jÞþC ijHðxi;x jÞ:ðB13Þ In this way, the form factor can be expressed in terms of massive neutrinos only [26] as, fll0¼g2 16π2X 2 i;j¼1 B lNiBl0Njfδij½FðxNiÞ−Fð0Þþδij½GðxNi;0ÞþGð0;x NjÞ−2Gð0;0Þ þδij½HðxNi;0ÞþHð0;x NjÞ−2Hð0;0ÞþCNiNj½GðxNi;x NjÞ−GðxNi;0Þ−Gð0;x NjÞþGð0;0Þ þC NiNj½HðxNi;x NjÞ−HðxNi;0Þ−Hð0;x NjÞþHð0;0Þg;ðB14Þ ρPARAMETER AND H0→lilj…PHYS. REV. D 102, 113006 (2020) 113006-7 where, in our particular case, GðNi;0Þ¼Gð0;NiÞ¼Gð0;0Þ¼0;ðB15Þ HðNi;0Þ¼Hð0;0Þ¼0:ðB16Þ Then, substituting (A10) and (A11), the form factor has two terms, fll0¼g2 16π2sνlsνl0fð1ÞþX 3 k¼1 s2 νkfð2Þ;ðB17Þ where fð1Þand fð2Þdo not depend on mixings. Only diagrams containing the flavor-changing vertex Hχiχj contribute to fð2Þ, but we treat Gχχ and Gχtogether since they cancel the ultraviolet divergences of each other, present in the part fð1Þ. In the case of diagrams Gχχ þGχ, the part fð2Þ, subdominant at low neutrino masses in any case, cancels fð1Þat some point and, for large neutrino masses, becomes the dominant contribution despite the s2 νk suppression (see Fig. 2). This is because it keeps growing like mN1mN2. 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