MPP-2020-28

Revisiting Neutrino Self-Interaction Constraints from Z and τ decays

Vedran Brdar1,a Manfred Lindner1,b Stefan Vogl2,c and Xun-Jie Xu1 d

1Max-Planck-Institut fur Kernphysik, 69117 Heidelberg, Germany2Max-Planck-Institut fur Physik, 80805 Munchen, Germany

Given the elusive nature of neutrinos, their self-interaction is particularly difficult to probe.Nevertheless, upper limits on the strength of such an interaction can be set by using data fromterrestrial experiments. In this work we focus on additional contributions to the invisibledecay width of Z boson as well as the leptonic τ decay width in the presence of a neutrinocoupling to a relatively light scalar. For invisible Z decays we derive a complete set ofconstraints by considering both three-body bremsstrahlung as well as the loop correction totwo-body decays. While the latter is usually regarded to give rather weak limits we find thatthrough the interference with the Standard Model diagram it actually yields a competitiveconstraint. As far as leptonic decays of τ are concerned, we derive a limit on neutrino self-interactions that is valid across the whole mass range of a light scalar mediator. Our boundson the neutrino self-interaction are leading for mφ & 300 MeV and interactions that preferντ . Bounds on such ν-philic scalar are particularly relevant in light of the recently proposedalleviation of the Hubble tension in the presence of such couplings.

I. INTRODUCTION

Recent studies have revealed a discrepancy between local measurements of the Hubble constant [1–3] and those obtained by analyzing the Cosmic Microwave Background (CMB) data [4] at a & 4σlevel. This has sparked an ongoing controversy in cosmology and the search for potential solutionsis currently ongoing. At the moment the origin of the Hubble tension is unclear; potential solutionsinclude, for example, early dark energy [5], light dark matter [6], majorons [7], dark matter neutrinointeractions [8], certain classes of non-Gaussian primordial fluctuations [9] or, more prosaically,underestimated systematics [10]. Most of these ideas fall clearly into the realm of cosmology andastrophysics and cannot be tested in laboratory experiments. However, it was proposed recentlythat strong neutrino self-interactions (νSI) can alleviate this tension [11]1. The preferred value ofthe interaction strength is in the ballpark of 107 ∼ 109 in units of Standard Model (SM) weakinteraction strength GF . In this regime neutrino free-streaming is suppressed at high red-shift andit is not surprising that such an interaction can have remarkable consequences for the physics ofthe early Universe.

Large νSI present a challenge from a particle physics perspective and it is expected that ter-restrial experiments can help scrutinize this option. In [13], the authors explored different optionsfor enhanced neutrino interaction. While they found that the vector forces of the aforementionedstrength are already disfavored from laboratory experiments, light (below O(102) MeV) bosonsstrongly coupled to neutrinos remain viable. The only surviving option which alleviates the Hub-ble tension is ντ -philic light scalar; this is expected since it is well known that new interactions of ντare generically the least constrained compared to other flavors. Let us note that the authors of [14]have recently reached similar conclusion by performing an analysis in the framework of effectivetheory which respects SM gauge invariance. It is therefore timely to revisit the constraints on such

a vbrdar@mpi-hd.mpg.deb lindner@mpi-hd.mpg.dec stefan.vogl@mpp.mpg.ded xunjie.xu@mpi-hd.mpg.de1 It was also noted in Ref. [11] that including the CMB polarization data in the fit tends to reduce the statistical

significance of this scenario, though an earlier study [12] found that including the polarization data increases thestatistical significance.

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interactions from particle physics processes and pay particular attention to the interactions of τneutrino.

There are numerous studies of νSI through the exchange of “light” mediators in the literature.This class of new physics was explored in meson decays [15–19], double beta decay [18, 20–26],invisible Z decays [18, 27, 28] and τ decays [16]. In addition, it has been recently pointed out thatstrong νSI can also play a relevant role in producing sterile neutrino dark matter [29, 30] as wellas testing ultralight dark matter scenarios [31, 32]. We would also like to point out further studiesinvolving cosmological [18, 33–35] as well as astrophysical (primarily Supernovae) [36–43] probes.

For a light scalar interacting with ντ many of the most sensitive probes of new physics connectedto νe and νµ are not sensitive and the two most relevant laboratory bounds arise from τ and Zdecays. The authors of Ref. [16] were the first to estimate the bound on the neutrino coupling tolight scalar by studying the former process. The reported limit only applies to a particular choiceof mφ and cannot be extrapolated to the mass range of interest easily. One of our goals in thispaper is to derive this limit as a function of scalar mass by using state of the art numerical tools.

In Ref. [18] the authors present a comprehensive analysis of constraints on light neutrinophilicscalars. What is very interesting for us, in light of couplings to ντ , is the constraint arising frominvisible Z decay, namely the process Z → ννφ, where φ is light scalar. In addition to this process,we will also consider Z invisible decay (Z → νν) via a triangle loop diagram. Naively such acontribution may appear subdominant since it contains two powers of scalar coupling to neutrinosalready at the amplitude level. However, it interferes with the SM tree level diagram and, therefore,the leading contribution is of the same order in the new physics coupling as the φ-bremsstrahlungand should be expected to give a competitive constraint.

The paper is organized as follows. In Section II we present the main results of our investigationof the new physics contribution to invisible Z decays while relegating the details of the calculationto appendices. In Section III we discuss the procedure for obtaining limits on new physics fromleptonic τ -decays. We analyze the implications of our results for the allowed interaction strengthof a neutrino-philic light mediator and comment on the implication for the proposed solution ofthe Hubble tension in Section IV. While the motivation for our study is mostly connected to τneutrino flavor, for completeness we also present limits for a νe and νµ-philic scalar as well as flavoruniversal coupling scenario. In Section V we summarize our results and present our conclusions.

II. Z DECAY

The new neutrino interactions to be considered in this work are parameterized by

L ⊃∑α, β

1

2yαβ νcα PL νβ φ+ h.c. , (1)

where να and νβ are Dirac spinors of neutrinos (νcα is the charge conjugate of να) and α and βstand for flavor indices. Furthermore, yαβ = yβα is a symmetric Yukawa matrix and φ is a scalarfield. Finally, note that the left projector PL = (1− γ5)/2 ensures that only left-handed neutrinosare involved in the interaction. This interaction term can be generated for instance in the seesawscenario; the coupling of singlet φ with right-handed neutrinos induces interaction of φ with activeneutrino states through lepton mixing [31].

In the presence of these interactions two new physics processes contribute to invisible Z decaysand we show the corresponding Feynman diagrams in Fig. 1. The loop contribution contains twoYukawa vertices being proportional to |yαβ|2 while the bremsstrahlung diagram is proportional toyαβ. Therefore, at the amplitude level, the left diagram is suppressed with respect to the rightone by a higher power of Yukawa coupling. However, the loop diagram can interfere with the SM

3

νL

νL

Z

νL

νL

φ

νL

νL

Z

νL

φ

FIG. 1. Representative Feynman diagrams contributing to invisible Z decays.

invisible Z decay and this substantially enhances its contribution to the decay width. Consequently,there is no obvious hierarchy between the two processes and both contributions to the invisiblewidth should be considered in a complete analysis.

A. Loop Contribution

Let us first consider that two neutrino species with flavors denoted by α and β (α 6= β) are coupledto φ and other couplings in Eq. (1) are absent. The result obtained for this simple case can beeasily generalized to the most general Yukawa matrix.

In the presence of a νcαPLνβφ interaction with α 6= β, there are new physics contributions toboth Z → νανα and Z → νβνβ which have identical amplitude and therefore it is enough to onlyconsider Z → νανα. The 1-loop amplitude for Z → νανα reads

iM(Z → νανα) =i|yαβ|216π2

εµ(q)u(p2)(gZγµPL)v(p1)

[1

2ε′+ log

mφ

mZ+

1 + iπ

2+O

(m2φ

m2Z

)], (2)

where gZ is the gauge coupling of Z boson to neutrinos; εµ(q), u(p2) and v(p1) denote the externallegs associated to the Z boson, neutrino and antineutrino, respectively while mφ (mZ) is the massof φ (Z). The 1-loop diagram for this process is UV divergent. We have adopted gauge invariantdimensional regularization. The typical terms appearing in such calculation are abbreviated by ε′

1

ε′≡ 1

ε− γE + log(4π) + log

µ2

m2φ

. (3)

Here, ε = (4−d)/2 with d representing the number of dimensions, while γE is the Euler-Mascheroniconstant.

The interaction in Eq. (1) can also generate other loop diagrams corresponding to neutrinoself-energy corrections to the Z → νανα process; see e.g. diagrams in Fig. 6. In the conventionalrenormalization scheme where only amputated diagrams need to be computed, such diagrams areincluded by computing the correction to the Z-ν-ν counter term caused by the wave-functionrenormalization. Without adding such a counter term, one can also compute these diagramsdirectly and add them to Eq. (2). These two methods are known to be equivalent; see Appendix Afor explicit verification.

Adding the neutrino self-energy corrections to Eq. (2), we obtain

iM(Z → νανα) =i|yαβ|216π2

εµ(q)u(p2)(gZγµPL)v(p1)

[1

ε′+ log

mφ

mZ+

3 + 2iπ

4+O

(m2φ

m2Z

)]. (4)

4

Compared to Eq. (2), the log(mφ/mZ) term is not changed.In a complete model, the UV-divergence arising from the considered diagram is expected to be

canceled by other diagrams (including counter terms). Here “complete” means not only that alloperators should be of dimension 4 or lower, but also that gauge invariance has to be respected.We would like to stress that the UV cancellation is model-dependent and, consequently, the finitepart can not be predicted fully without committing to a specific UV-completion. Nevertheless,the log(mφ/mZ) term is a generic feature and is independent of the regularization scheme. Thiscan for instance be seen by considering only the loop integral with the loop momentum runningbetween the scales of mφ and mZ which yields a result proportional to log(mφ/mZ). This impliesthat this term can be physically interpreted as the contribution of the loop momentum running inthe intermediate scale and being insensitive to the UV or IR behavior of the underlying completemodels. We refer the interested reader to appendix A where we show the cancellation explicitly ina toy model and find the behavior detailed above.

In the SM, the tree-level amplitude for Z → νανα is

iMSM(Z → νανα) = −iεµ(q)u(p2)(gZγµPL)v(p1), (5)

which leads to the decay width2 [44]

ΓSM(Z → νανα) =GFm

3Z

12√

2π. (6)

By comparing Eq. (2) and Eq. (5), we can obtain the decay width including the loop contributionwhich yields

Γnew(Z → νανα) = ΓSM(Z → νανα)

∣∣∣∣1 +|yαβ|216π2

(L+ iπ/2)

∣∣∣∣2 , with L = logmZ

mφ+

3

4. (7)

One can check that the final result for the case α = β turns out to be the same as Eq. (7)with β → α. Therefore, in the presence of the most general Yukawa matrix, one only needs toreplace |yαβ|2 with

∑β |yαβ|2 in Eq. (7). If we sum over α indices and restrict ourselves to terms

proportional to the second power of Yukawa coupling or lower we get

Γnew(Z → νν) ≡∑α

Γnew(Z → νανα) ≈ GFm3Z

12√

2π

[3 +

tr[Y Y †]

16π22L

], (8)

where Y is the 3× 3 Yukawa matrix with yαβ elements. One can also see that Eq. (8) is invariantunder ν → Uν, Y → UY U † basis transformations where U is an arbitrary unitary matrix.

B. Bremsstrahlung

The bremsstrahlung process is depicted by the right diagram in Fig. 1. Again, we first considerZ → ναφνβ with α 6= β. The decay width in case of α 6= β reads

Γnew(Z → ναφνβ) =g2Z |yαβ|2mZ

24(2π)3F , (9)

2 The relation between neutrino coupling to Z boson and the weak interaction strength GF reads g2ZmZ =√

2GFm3Z .

5

where

F ≈(

1 + 3m2φ

m2Z

)log

mZ

mφ− 17

12.

For details of the derivation including the expression with the full mφ dependence of Γnew(Z →ναφνβ) see Appx. B. The expression for the total width of the φ bremsstrahlung with the mostgeneral Yukawa couplings is given by

Γnew(φ bremsst.) =1

2

∑α 6=β

Γnew(Z → ναφνβ) +∑α

Γnew(Z → ναφνα) (10)

=1

2

∑α 6=β

Γnew(Z → ναφνβ) +1

2

∑α

Γnew(Z → ναφνβ)|β→α , (11)

where the first 1/2 factor is due to double counting of∑

α6=β, and the last 1/2 factor accounts forthe phase space of identical particles. In the last term of Eq. (11), Γnew(Z → ναφνβ) takes thesame expression as Eq. (9). Eq. (11) allows the formulation in a basis-independent form similar toEq. (8)

Γnew(φ bremsst.) =1

2

g2ZmZtr[Y Y †]

24(2π)3F. (12)

The bremsstrahlung diagram in Fig. 1 represents Z → νφ∗ν process. By flipping the arrows inthe diagram one obtains a similar diagram for Z → νφν with identical decay width.

Also note that there is the charge conjugate process Z → νφ∗ν, which has the same decay widthas Z → νφν. Therefore, upon combining all the bremsstrahlung processes we reach the expressionfor the total contribution to invisible Z decay

Γnew(φ/φ∗ bremsst.) =

√2GFm

3Ztr[Y Y †]

24(2π)3F. (13)

We would like to stress that we simulated this three-body decay numerically in CalcHEP [45] andfound excellent agreement with our analytic results.

Note that in the limit of mφ → 0, the sum of loop and bremsstrahlung contribution is divergent.In some theories such as QED, it is well known that the infrared (IR) divergence in the trianglediagram cancels the IR divergence in the bremsstrahlung diagram. But here one should not expectsuch cancellation due to the chirality-flipping feature of scalar interactions. The processes Z → ννand Z → ννφ, in the limit of mφ → 0 and zero momentum of φ, are still physically distinguishablesince ν and ν are different and the IR divergence is regulated by neutrino masses (mν). A carefultreatment of the case when mφ is comparable or lower than mν is beyond the scope of this work.Nonetheless, our results are valid in regime mφ � mν > 0 that is considered throughout this work.

III. TAU DECAY

Another relevant probe of neutrino self-interactions that is particularly relevant for ντ is the decayof τ leptons. Similar to the φ-bremsstrahlung from Z decays the basic idea here is to constrainthe scalar-neutrino coupling by investigating the impact of attaching a scalar line to the final stateneutrino line; this for instance turns the diagram for the standard three-body decay into a charged

6

ν

νµ

W±

φ

µ

ντ

τ

ν

νµW±

φ

µ

ντ

τ ν

FIG. 2. Representative Feynman diagrams contributing to τ decays due to real emission of the new scalarφ (left panel) and at one loop (right panel).

lepton (electron or muon) and a pair of neutrinos into a 4-body process containing an extra lightscalar boson in the final state. We illustrate this process in the left panel of Fig. 2.

Most τ leptons decay hadronically but with a leptonic branching ratio Brl=e,µ ≈ 34% theleptonic final states are hardly suppressed. As the leptonic channels are much cleaner we focus onthem in the following. A similar process has been considered previously in the context of a modelwith light majorons [16]. In principle the majoron limits from the literature, available for mφ = 1keV, could be used to estimate the bounds in the model under consideration here. However, as ouranalysis shows, such bound cannot simply be extrapolated to higher mφ and limits derived fromrescaling the results of [16] become unreliable in the mass range of interest here.

We supplement the interaction term in Eq. (1) to the full SM implementation provided by theFeynRules [46] team. Then we generate a UFO model [47] which allow us to simulate the processof interest with MadGraph5 aMC@NLO [48]. As a cross check we first calculate the partial width forτ− → l− νl ντ where l = µ− or e− in the SM and find good agreement with the observed values.We determine the decay rate of the process τ− → l− νl ντ φ as a function of mφ numerically andconstruct a fit functions to derive the limit on the Yukawa coupling. In principle, the rates for thedecay into electrons and muons are different due to the different masses of final state particles. Inpractice, the discrepancies are expected to be rather small due to the large hierarchy of chargedlepton masses. We find that the differences between electron and muon channels are within thenumerical uncertainties. The obtained partial width for the Yukawa coupling yττ equal to 1 isshown in Fig. 3 as a function of scalar mass, mφ. This decay rate is used for obtaining the limit aswill be demonstrated in Section IV.

Finally, we would like to comment on another channel that can be constrained from τ decays. Inthe diagram for SM process τ− → l− νl ντ , the two neutrino lines could be connected with a scalarsimilar to the loop correction to Z → νν, see the right panel of Fig. 2 for an illustrative diagram.Note, however, that in contrast to Z decays only off-diagonal components of the Yukawa lead toa contribution that can interfere with the SM amplitude. These off-diagonal couplings are alreadystrongly constrained by meson decays and, therefore, we do not consider this process further.

IV. CONSTRAINTS OF NEUTRINO INTERACTION WITH LIGHT SCALAR

With all the necessary ingredients available we can now turn to actual observables and derive limitson the parameters of the neutrino self-interaction model under consideration here. We will firstconsider the impact of the measurement of invisible Z decay before turning to τ decays.

7

10-3 10-2 10-1 1 101 102 10310-16

10-15

10-14

mϕ/MeV

De

ca

yR

ate

[Ge

V]

MadGraph5_aMC@NLO output

τ -→μ -

νμντϕ

FIG. 3. Partial width of the four-body decay τ− → µ−νµντφ for a representative Yukawa coupling yττ = 1.

Combining the results in Eqs. (8) and (13), the total Z invisible width is given by

Γnew(Z → inv.) ≈ GFm3Z

12√

2π

[3 +

tr[Y Y †]

16π22L+

tr[Y Y †]

8π2F

]. (14)

Conveniently, the experimental measurement of Z invisible width can be expressed in terms of thenumber of light neutrino species [49–51] (see also [52–56])

Nν = 2.9963± 0.0074, (15)

which means that the observed invisible width is about 2σ lower than the SM prediction. Sinceboth L and F in Eq. (14) are positive, the new physics we introduce can only enhance the Zinvisible width. To get our limits we set the confidence level to 3σ so that the exclusion bound canbe obtained by requiring

3 +tr[Y Y †]

16π22L+

tr[Y Y †]

8π2F < 2.9963 + 0.0074× 3. (16)

In the case of τ decays the situation is more subtle. Since a ντ is emitted in every τ decay acorrection to all decay modes is expected for yτα 6= 0. Naively, one could assume that the correctionof the different decay modes is very similar since a φ emitted from the ντ is only sensitive to thetotal momentum of the remaining final state. Consequently, the branching ratios remain similar tothe SM prediction while the total width/lifetime of the τ changes. In contrast, a coupling to νe or νµonly affects the partial width of the leptonic decay modes. In order to derive reliable bounds on yττand yµµ we make use of the partial width Γτµ for the three-body decay τ− → µ− ντ νµ which can bedetermined by combining the measured lifetime (290.6±1.0)×10−15s with the observed branchingratio of (17.41± 0.04)% [57]. The central value for the partial decay rate reads 3.94× 10−13 GeV.

8

101 102 103 104

mφ/MeV

10−2

10−1

100

y ee

Z → inv.

τ decay

K− → e−νφ

D− → e−νφπ− →

e− νφ

19931126

101 102 103 104

mφ/MeV

10−2

10−1

100

y µµ

Z → inv.

τ decay

K− → µ−νφ

D− → µ−νφ

100 101 102 103

mφ/MeV

10−2

10−1

100

y ττ

Z → inv.τ decay

favo

red

byC

MB

BB

N

101 102 103 104

mφ/MeV

10−2

10−1

100

y ee

=y µµ

=y ττ

Z → inv.

τ decay

K− → e−νφ

K− → µ−νφ

D− → e−νφD− → µ−νφ

π− →

e− νφ

FIG. 4. Constraints on νSI from Z invisible decay (blue) and τ decay (orange) shown together with otherknown constraints taken from Ref. [18]. For the case of ντ -philic scalar we also show the preferred region torelax the Hubble tension [13].

In order to get an estimate of the relative error on the leptonic partial width we add the relativeerrors of the lifetime and the branching ratio in quadrature and find δΓτµ/Γτµ ≈ 0.004. Therefore,we set the 3σ exclusion limit on the couplings by requiring

Γτ−→µ− ντ νµφ ≤ 3× 0.004× 3.94× 10−13 GeV. (17)

A similar procedure utilizing τ− → e− ντ νeφ leads to essentially identical results for yee sinceδΓτµ/Γτµ ≈ δΓτe/Γτe. When there are contributions to both τ− → µ− ντ νµφ and τ− → e− ντ νeφ,we combine both channels together to set our limit.

In Fig. 4, we present our results; constrains on the diagonal elements of Y are calculatedassuming the other elements of the Yukawa matrix are zero. More specifically, for the case ofnonvanishing yee, yµµ and yττ , we take tr[Y Y †] = |yee|2, |yµµ|2 and |yττ |2, respectively. For theyee = yµµ = yττ figure (lower right), we take tr[Y Y †] = |yee|2 + |yµµ|2 + |yττ |2. For flavor off-diagonal elements (yαβ with α 6= β), one can simply interpret bounds from any of these figures

as the bounds on√

tr[Y Y †] and convert it to the bounds on yαβ. For the coupling of νe, νµ aswell as flavor universal scenario (upper panels as well as lower right panel) we also superimposelimits from meson decays [18]. As can be seen these bounds are stronger than those derived in thiswork for mφ . 300 MeV. In the cosmologically most interesting case of a ντ -philic scalar (lowerleft panel) we also show the preferred region for alleviating the Hubble tension (green) as well as aconstraint from BBN [13]. While the derived laboratory constraints are certainly a relevant playerfor excluding the parameter space in yττ & 0.1 range, the viable region still remains in the range0.1 & yττ & 0.01. This points towards mφ ∼ O(10) MeV.

9

V. SUMMARY AND CONCLUSIONS

In this work we revisited constraints on neutrino self-interactions arising from a neutrino-philiclight scalar φ. The employed probes are invisible Z decays and the leptonic decay modes of the τ .For invisible Z decays we consider two contributions: one with νν in the final state where we findthat 1-loop diagram interfering with the usual SM contribution yields rather significant limit; theother, complementary, contribution to the invisible width arises from bremsstrahlung where twoneutrinos (or antineutrinos) appear in the final state alongside φ. Summing both contributions wederive bounds on the new interactions for the case where the light scalar interacts with all flavorsindividually as well as a flavor universal scenario. In addition, we derive new limit from leptonicτ decays. To the best of our knowledge these are the first results that take the dependence on theφ mass and the coupling fully into account while previous calculations in the literature only applyfor a restricted set of parameters. We provide a full picture of our results in Fig. 4 and comparethem to constraints from meson decays. Our results constitute the leading bound on scalars withmφ & 300 MeV irrespective of the preferred flavor. In the case of ντ self-interactions, which is aparticularly relevant scenario in light of recently proposed solution to the Hubble tension, theseconstraints constitute the leading laboratory limit throughout the considered mass range. However,a scalar in the mass range 10−100 MeV remains a viable option for large neutrino self-interactionsand we are not able to exclude the whole parameter space preferred by cosmology.

ACKNOWLEDGEMENTS

The authors would like to thank Ibragim Alikhanov for finding a typo in one of the equations. VBwould like to thank Sudip Jana for discussions regarding usage of Madgraph.

Appendix A: Loop calculation in a chiral U(1) toy model

In this appendix, we discuss a toy model which is complete and rather minimal containing onlya chiral fermion νL, a gauged U(1) with the gauge boson denoted as Zµ and a scalar boson φ.Although the toy model is not realistic, it illustrates how the UV cancellation works explicitly and,in addition, shows the potential difference between the results in an incomplete model with respectto the complete one.

The model is formulated by the following Lagrangian

L ⊃ νLi /DνL + |Dµφ|2 −m2φ†φ− 1

4FµνFµν −

1

2m2ZZ

µZµ −[y

2νcLφνL + h.c.

]. (A1)

Here, all terms are gauge invariant with the charge assignments νL ∼ Qν = −1 and φ ∼Qφ = +2, except for the gauge boson mass term m2

ZZµZµ, which can be easily generated by, e.g.,

introducing another scalar that has a charge of +3 and a nonzero VEV. Note that such detailsare irrelevant for our discussion below. The covariant derivatives can be explicitly expressed asDµ = ∂µ − igQZµ, where Q takes Qν or Qφ.

In Fig. 5, we present the Feynman diagrams involved in our analyses. We will show explicitlythat the UV divergent parts in these diagrams cancel each other, as long as the U(1) charge isconserved (2Qν +Qφ = 0).

First, we compute the 1PI diagram generated by the Yukawa interaction, which will only leadto renormalization of the wave function of νL. It will not lead to mass renormalization as one canexpect from the chiral symmetry, so νL remains massless after the loop corrections. The self-energy

10

pp − kp

k

p p

p1

p2

qp1 + k

p2 − k

k

p1

p1 + k

p2 − k

p2

k

q

p1

p2

q

FIG. 5. Feynman diagrams in the chiral U(1) toy model.

generated by the top left diagram in Fig. 5 reads

−iΣ(/p) =

∫4d4k

(2π)4

−iy∗2

PRi

/p− /kPL−iy

2

i

k2 −m2φ

(A2)

= |y|2I(p2)/pPL , (A3)

with

I(p2) =i

16π2

1

2ε′+ 1−

m2φ

2p2+

(p2 −m2

φ

)2

2p4log

m2φ

m2φ − p2

. (A4)

Here, we used Package-X [58] to evaluate the loop integral. When p2/m2φ is small, we have the

following expansion:

I(p2) =i

16π2

[1

2ε′+

1

4+

p2

6m2φ

+p4

24m4φ

+O(p6

m6φ

)]. (A5)

The UV divergence in the neutrino self-energy is canceled by wave function renormalization

νL → (1 + δZ)1/2νL. (A6)

The wave function renormalization generates a counter term δZνLi /DνL which then can be split totwo counter terms δZνLi/∂νL and δZνLgQνZµνL. The first term, corresponding to the top rightdiagram in Fig. 5, cancels the UV divergence in Eq. (A4); while the second term, correspondingto the bottom right diagram in Fig. 5 cancels the UV divergences of the two triangle diagrams inFig. 53.

Now, by adding the counter term iδZ/pPL to Eq. (A3) and requiring the UV cancellation, weobtain

δZ = i|y|2 I(p2)∣∣p2→0

=−|y|216π2

[1

2ε′+

1

4

]. (A7)

3 Note that in this toy model, if we are only interested in loop corrections of the Yukawa interactions to the ZνLνLvertex, then only the wave function renormalization is sufficient to remove all the UV divergences in the loopdiagrams shown in Fig. 5.

11

Next, we compute the Feynman diagrams for the Zµ → νLνL decay. The amplitudes of thethree bottom diagrams in Fig. 5 are

iM(a) =

∫d4k

(2π)4u(p2)(−iy∗)PR

i

/p2− /k (−igQνcγµ)

−i/p1

+ /kPL(−iy)v(p1)

× i

k2 −m2φ

εµ(q) , (A8)

iM(b) =

∫d4k

(2π)4u(p2)(−iy∗)PR

i

/kPL(−iy)v(p1)

× i

(p2 − k)2 −m2φ

(igQφ)(p2 − p1 − 2k)µi

(p1 + k)2 −m2φ

εµ(q) , (A9)

iM(c) = u(p2)(−igδZQνγµPL)v(p1)εµ(q) . (A10)

Note that Qνc = −Qν because instead of νL(g Qν)ZµνL, the Z-vertex should take the chargeconjugate −νcL(g Qν)Zµν

cL in the left bottom diagram. Also note that the neutrino propagators

running in the loops are related to 〈νcνc〉 instead of 〈νν〉 so when a fermion current arrow isopposite to a momentum arrow in the loops, it implies that the anti-fermion current is alignedwith the momentum arrow. Hence the numerators above /p2

− /k, /p1+ /k, and /k in Eqs. (A8) and

(A9) should be i, −i, and i respectively. After computing the loop integrals and expanding theresults in m2

φ/q2 (q2 = m2

Z), we obtain

iM(a) =i|y|2Qν16π2

εµ(q)u(p2)(gγµPL)v(p1)

[1

2ε′+

1

2log

m2φ

q2+

1 + iπ

2+O

(m2φ

q2

)], (A11)

iM(b) =i|y|2Qφ

16π2εµ(q)u(p2)(gγµPL)v(p1)

[1

2ε′+

1

2log

m2φ

q2+

3 + iπ

2+O

(m2φ

q2

)], (A12)

iM(c) =i|y|2Qν16π2

εµ(q)u(p2)(gγµPL)v(p1)

(1

2ε′+

1

4

). (A13)

Now we can clearly see that the the UV divergent parts in the above expressions cancel out if

Qν +Qφ +Qν = 0 . (A14)

This corresponds to Qφ = −2Qν , which can be understood from symmetry: νcLφνL in Eq. (A1)respects the U(1) symmetry only if Qφ = −2Qν .

Taking Qφ = −2Qν and q2 → m2Z , we get

iM(a) + iM(b) + iM(c) = − i|y|2Qν

16π2εµ(q)u(p2)(gγµPL)v(p1)

[1

2log

m2φ

m2Z

+9 + 2iπ

4+O

(m2φ

m2Z

)].

(A15)In the above calculation, we have adopted the conventional renormalization scheme which in-

volves counter terms. In such a renormalization scheme, one only needs to compute amputateddiagrams while the diagrams in Fig. 6 should not be added [59]. Actually since the counter termδZνLi /DνL is generated by the 1PI diagram, the two diagrams in Fig. 6 have already been taken intoaccount by the last diagram in Fig. 5. Nonetheless, it is still interesting to compute the diagramsin Fig. 6 to explicitly check that they give result equivalent to the counter term diagram. FromFig. 6, we write down the sum of the two amplitudes:

12

p1

p2

q k − p1

p2 − k

k

p2

q

p1

k

p1

p2

FIG. 6. Loop corrections to the external neutrino legs in the chiral U(1) toy model. These two diagramsare equivalent to the last counter term diagram in Fig. 5; see the text for more discussions.

iM′(c) =

∫d4k

(2π)4u(p2)

[i

(p2 − k)2 −m2φ

(−iy∗)PRi

/kPL(−iy)

i

/p2−m1

(−igQνγµ)

+(−igQνγµ)−i

/p1−m2

(−iy∗)PR−i/kPL(−iy)

i

(k − p1)2 −m2φ

]v(p1)εµ(q) . (A16)

Here we have inserted two masses m1 and m2 in order to treat singularities properly. At the endof the calculation we will take the zero limit for both. Performing the loop integral, we obtain

iM′(c) = u(p2)|y|2[I(p2

2)/p2PL

gQν

/p2−m1

γµ + γµgQν

/p1−m2

I(p21)/p1

PL

]v(p1)εµ(q)

= u(p2)|y|2gQν[I(p2

2)/p2

/p2γµL +m1γ

µR

p22 −m2

1

+γµL/p1

+ γµRm2

p21 −m2

2

I(p21)/p1

]v(p1)εµ(q) , (A17)

where in the second line we have moved PL to the left side of γµ and defined γµL/R ≡ γµPL/R, so

that all the other gamma matrices can either meet u(p2) or v(p1). Then using u(p2)/p2= u(p2)m2

and /p1v(p1) = −m1v(p1), we obtain

iM′(c) = u(p2)|y|2gQν[I(m2

2)m2

2γµL +m2m1γ

µR

m22 −m2

1

+−m2

1γµL + γµRm1m2

m22 −m2

1

I(m21)

]v(p1)εµ(q)

≈ u(p2)i|y|2gQν

16π2γµL

(1

2ε′+

1

4+

m42 −m4

1

6m2φ(m2

2 −m21)

)v(p1)εµ(q) , (A18)

where in the second line we have used the expansion in Eq. (A5) and ignored higher order terms.In addition, γµR terms are also ignored because they vanish in the limit of m2 → 0 and m1 → 0.Comparing Eq. (A18) with Eq. (A13), we can see that iM′(c) = iM(c) in the limit of m2 → 0 andm1 → 0. This verifies that the two diagrams in Fig. 6 are indeed equivalent to the counter termdiagram in Fig. 5.

The result in Eq. (A15) contains an IR divergence if mφ → 0. In the main text, we havediscussed that this result is only valid for mZ � mφ � mν . In the presence of nonzero mν , oneneeds to insert mν in all the neutrino propagators in Eqs. (A3), (A8), and (A9). Then following astraightforward but lengthy calculation, we obtain a result which can be written in a form similar

to Eq. (A15) with the 12 log

m2φ

m2Z

replaced by another function:

1

2log

m2φ

m2Z

→ L(mφ,mν), (A19)

13

10−11 10−10 10−9 10−8 10−7 10−6 10−5

mφ/MeV

−34

−32

−30

−28

−26

−24

−22

L(m

φ,m

ν)

L(mφ, 0) = log(mφ/mZ)

L(mφ, 0.1eV)

FIG. 7. Numerical evaluation of the L(mφ,mν) function given in Eq. (A21). When mφ is not well above

mν , the 12 log

m2φ

m2Z

in Eq. (A15) should be replaced by L(mφ, mν). The plot shows that the IR divergence in

the limit of mφ → 0 is removed when mν 6= 0.

where

L(mφ,mν) =1

4m4νm

2Z

(m2Z − 4m2

ν

) (A20)

×[−4m4

ν log

(m2φ

m2ν

)(−m2

φ

(4m2

ν +m2Z

)+m2

Z

(m2Z − 2m2

ν

)+m4

φ

)−4m4

νm2φ

(−8m4

ν +m2φ

(2m2

ν −m2Z

)+ 2m2

νm2Z

)Cνφν0

+8m4ν

(m4φ

(m2Z − 2m2

ν

)+m2

φ

(8m4

ν − 4m2νm

2Z

)+m2

νm2Z

(m2Z − 2m2

ν

))Cφνφ0

+8m6ν

(m2φ − 4m2

ν

)Λ(m2ν ,mφ,mν

)+2m4

ν

(8m4

ν +m2φ

(2m2

Z − 4m2ν

)+ 2m2

νm2Z −m4

Z

)Λ(m2Z ,mν ,mν

)−4m4

ν

(2m2

ν −m2Z

) (m2Z − 2m2

φ

)Λ(m2Z ,mφ,mφ

)−2m2

νm2Z

(m2φ − 2m2

ν

) (4m2

ν −m2Z

)Λ(m2ν ,mφ,mν

)+2m4

ν

(4m2

ν −m2Z

) (4m2

ν − 5m2Z + 2m2

φ

)−2m2

νm2Z

(m2φ − 3m2

ν

) (4m2

ν −m2Z

)+m2

Zm2φ

(m2φ − 4m2

ν

) (4m2

ν −m2Z

)log

(m2φ

m2ν

)]−(

9

4+iπ

2

). (A21)

Here Cνφν0 and Cφνφ0 involve two-dimensional integrals that have to be evaluated numerically:

Cνφν0 ≡ C0

(m2ν ,m

2ν ,m

2Z ,mν ,mφ,mν

), (A22)

Cφνφ0 ≡ C0

(m2ν ,m

2ν ,m

2Z ,mφ,mν ,mφ

), (A23)

C0 (s1, s2, s3;m2,m1,m0) ≡ limε→0+

∫ 1

0dy

∫ 1−y

0dz[s1y

2 + s2z2 + (s1 + s2 − s3) yz

14

+y(−m2

0 +m21 − s1

)+ z

(−m2

0 +m22 − s2

)+m2

0 − iε]. (A24)

The Λ functions is defined as

Λ (x, y, z) ≡log[(√

x2 − 2xy2 − 2xz2 + y4 − 2y2z2 + z4 − x+ y2 + z2)/ (2yz)

]x (x2 − 2xy2 − 2xz2 + y4 − 2y2z2 + z4)−1/2

. (A25)

In Fig. 7 we show result of numerical evaluation of L(mφ,mν). In particular, it is demonstratedthat the IR divergence in the limit of mφ → 0 is removed when mν 6= 0. It is also shown that for

mφ � mν , L(mφ,mν) expectedly converges to 12 log

m2φ

m2Z

.

Appendix B: Analytical Calculation of three-body invisible Z decay

The amplitude for the process Z(q)→ να(p1)νβ(p2)φ(k) reads

M = iε∗(q) gZ yαβ u(p2)

[γµPL( /p1 + /k)

(p1 + k)2+

( /p2 + /k)γµPL

(p2 + k)2

]v(p1) , (B1)

which leads to

|M|2 =1

3

∑polarizations

MM∗ =

=g2Z |yαβ|2

3

(2(2E2 −mZ)(2E1 −mZ)mZ(8E1E2(E1 + E2)− 12E1E2mZ +m3

Z)

m2Z(mZ − 2E1)2(mZ − 2E2)2

−

2m2φ (16E1E2(E1mZ + E2mZ − E1E2 −m2

Z) +m4Z)

m2Z(mZ − 2E1)2(mZ − 2E2)2

), (B2)

where we used the expression for the massive vector polarization sum∑polarizations

ε(q) ε∗(q) =

(−gµν +

qµ qνm2Z

), (B3)

while E1 and E2 are energies of particles with 4-momenta p1 and p2, respectively. By employingenergy conservation E1 = mZ −E2−Ek the square matrix element |M|2 can be expressed only interms of 2 energies - one of massive (Ek) and one of effectively massless (E2) final state particle.This allows for a straightforward evaluation of non-trivial three-body phase space integrals.

The differential decay rate reads

dΓ =

∫ mZ2

+−Ek+

√E2k−m2

φ

2

mZ2−Ek+

√E2k−m2

φ

2

1

16mZ(2π)4

d3|k|√E2k −m2

φEk

g2Z |yαβ|2

3f(E2, Ek) dE2, (B4)

where f(E2, Ek) is Eq. (B2) with the aforementioned substitution for E1. The integral g(Ek) =∫f(E2, Ek)dE2 can be evaluated analytically. We obtain the following result

g(Ek) =2√E2k −m2

φ(−2EkmZ − 3m2Z +m2

φ)

m2Z

+ 8Ek ArcCoth

Ek√E2k −m2

φ

. (B5)

15

After inferring d3|k|/√E2k −m2

φEk = 4πdEk, one obtains the expression for the decay rate

Γ(Z → νανβφ) =4πg2

Z |λ|2αβ16× 3mZ(2π)4

∫ (m2φ+m2

Z)/2mZ

mφ

g(Ek)dEk

=g2Z |yαβ|2mZ

24(2π)3

{(1 + 3r2

)log

(1

r

)− 17− 9r2 − 9r4 + r6

12

}, (B6)

where r = mφ/mZ . Notice that in case α = β we get an extra 1/2 factor from the phase space.

[1] A. G. Riess et al., A 2.4% Determination of the Local Value of the Hubble Constant, Astrophys. J.826 (2016), no. 1 56, [1604.01424].

[2] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Large Magellanic Cloud CepheidStandards Provide a 1% Foundation for the Determination of the Hubble Constant and StrongerEvidence for Physics beyond ΛCDM, Astrophys. J. 876 (2019), no. 1 85, [1903.07603].

[3] K. C. Wong et al., H0LiCOW XIII. A 2.4% measurement of H0 from lensed quasars: 5.3σ tensionbetween early and late-Universe probes, 1907.04869.

[4] Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters,1807.06209.

[5] V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Early Dark Energy Can Resolve TheHubble Tension, Phys. Rev. Lett. 122 (2019), no. 22 221301, [1811.04083].

[6] J. Alcaniz, N. Bernal, A. Masiero, and F. S. Queiroz, Light Dark Matter: A Common Solution to theLithium and H0 Problems, 1912.05563.

[7] M. Escudero and S. J. Witte, A CMB Search for the Neutrino Mass Mechanism and its Relation tothe H0 Tension, 1909.04044.

[8] E. Di Valentino, C. Behm, E. Hivon, and F. R. Bouchet, Reducing the H0 and σ8 tensions with DarkMatter-neutrino interactions, Phys. Rev. D97 (2018), no. 4 043513, [1710.02559].

[9] S. Adhikari and D. Huterer, Super-CMB fluctuations can resolve the Hubble tension, 1905.02278.[10] T. Shanks, L. Hogarth, and N. Metcalfe, Gaia Cepheid parallaxes and ’Local Hole’ relieve H0 tension,

Mon. Not. Roy. Astron. Soc. 484 (2019), no. 1 L64–L68, [1810.02595].[11] C. D. Kreisch, F.-Y. Cyr-Racine, and O. Dore, The Neutrino Puzzle: Anomalies, Interactions, and

Cosmological Tensions, 1902.00534.[12] L. Lancaster, F.-Y. Cyr-Racine, L. Knox, and Z. Pan, A tale of two modes: Neutrino free-streaming

in the early universe, JCAP 1707 (2017), no. 07 033, [1704.06657].[13] N. Blinov, K. J. Kelly, G. Z. Krnjaic, and S. D. McDermott, Constraining the Self-Interacting

Neutrino Interpretation of the Hubble Tension, Phys. Rev. Lett. 123 (2019), no. 19 191102,[1905.02727].

[14] K.-F. Lyu, E. Stamou, and L.-T. Wang, Self-interacting neutrinos: solution to Hubble tension versusexperimental constraints, 2004.10868.

[15] V. D. Barger, W.-Y. Keung, and S. Pakvasa, Majoron Emission by Neutrinos, Phys. Rev. D25 (1982)907.

[16] A. P. Lessa and O. L. G. Peres, Revising limits on neutrino-Majoron couplings, Phys. Rev. D75(2007) 094001, [hep-ph/0701068].

[17] P. S. Pasquini and O. L. G. Peres, Bounds on Neutrino-Scalar Yukawa Coupling, Phys. Rev. D93(2016), no. 5 053007, [1511.01811]. [Erratum: Phys. Rev.D93,no.7,079902(2016)].

[18] J. M. Berryman, A. De Gouva, K. J. Kelly, and Y. Zhang, Lepton-Number-Charged Scalars andNeutrino Beamstrahlung, Phys. Rev. D97 (2018), no. 7 075030, [1802.00009].

[19] A. de Gouva, P. S. B. Dev, B. Dutta, T. Ghosh, T. Han, and Y. Zhang, Leptonic Scalars at the LHC,1910.01132.

[20] C. P. Burgess and J. M. Cline, Majorons without Majorana masses and neutrinoless double betadecay, Phys. Lett. B298 (1993) 141–148, [hep-ph/9209299].

[21] C. P. Burgess and J. M. Cline, A New class of Majoron emitting double beta decays, Phys. Rev. D49(1994) 5925–5944, [hep-ph/9307316].

16

[22] KamLAND-Zen Collaboration, A. Gando et al., Limits on Majoron-emitting double-beta decaysof Xe-136 in the KamLAND-Zen experiment, Phys. Rev. C86 (2012) 021601, [1205.6372].

[23] M. Agostini et al., Results on ββ decay with emission of two neutrinos or Majorons in 76 Ge fromGERDA Phase I, Eur. Phys. J. C75 (2015), no. 9 416, [1501.02345].

[24] K. Blum, Y. Nir, and M. Shavit, Neutrinoless double-beta decay with massive scalar emission, Phys.Lett. B785 (2018) 354–361, [1802.08019].

[25] R. Cepedello, F. F. Deppisch, L. Gonzalez, C. Hati, and M. Hirsch, Neutrinoless Double-β Decay withNonstandard Majoron Emission, Phys. Rev. Lett. 122 (2019), no. 18 181801, [1811.00031].

[26] T. Brune and H. Paes, Massive Majorons and constraints on the Majoron-neutrino coupling, Phys.Rev. D99 (2019), no. 9 096005, [1808.08158].

[27] M. S. Bilenky, S. M. Bilenky, and A. Santamaria, Invisible width of the Z boson and ’secret’neutrino-neutrino interactions, Phys. Lett. B301 (1993) 287–291.

[28] P. Machado, Y. Perez, O. Sumensari, Z. Tabrizi, and R. Z. Funchal, On the Viability of MinimalNeutrinophilic Two-Higgs-Doublet Models, JHEP 12 (2015) 160, [1507.07550].

[29] A. De Gouvea, M. Sen, W. Tangarife, and Y. Zhang, The Dodelson-Widrow Mechanism In thePresence of Self-Interacting Neutrinos, Phys. Rev. Lett. 124 (2020), no. 8 081802, [1910.04901].

[30] K. J. Kelly, M. Sen, W. Tangarife, and Y. Zhang, Origin of Sterile Neutrino Dark Matter via VectorSecret Neutrino Interactions, 2005.03681.

[31] G. Krnjaic, P. A. Machado, and L. Necib, Distorted neutrino oscillations from time varying cosmicfields, Phys. Rev. D 97 (2018), no. 7 075017, [1705.06740].

[32] V. Brdar, J. Kopp, J. Liu, P. Prass, and X.-P. Wang, Fuzzy dark matter and nonstandard neutrinointeractions, Phys. Rev. D 97 (2018), no. 4 043001, [1705.09455].

[33] C. Boehm, M. J. Dolan, and C. McCabe, Increasing Neff with particles in thermal equilibrium withneutrinos, JCAP 1212 (2012) 027, [1207.0497].

[34] A. Kamada and H.-B. Yu, Coherent Propagation of PeV Neutrinos and the Dip in the NeutrinoSpectrum at IceCube, Phys. Rev. D92 (2015), no. 11 113004, [1504.00711].

[35] G.-y. Huang, T. Ohlsson, and S. Zhou, Observational Constraints on Secret Neutrino Interactionsfrom Big Bang Nucleosynthesis, Phys. Rev. D97 (2018), no. 7 075009, [1712.04792].

[36] K. Choi, C. W. Kim, J. Kim, and W. P. Lam, Constraints on the Majoron Interactions From theSupernova SN1987A, Phys. Rev. D37 (1988) 3225.

[37] K. Choi and A. Santamaria, Majorons and Supernova Cooling, Phys. Rev. D42 (1990) 293–306.[38] M. Kachelriess, R. Tomas, and J. W. F. Valle, Supernova bounds on Majoron emitting decays of light

neutrinos, Phys. Rev. D62 (2000) 023004, [hep-ph/0001039].[39] S. Hannestad, P. Keranen, and F. Sannino, A Supernova constraint on bulk Majorons, Phys. Rev.

D66 (2002) 045002, [hep-ph/0204231].[40] Y. Farzan, Bounds on the coupling of the Majoron to light neutrinos from supernova cooling, Phys.

Rev. D67 (2003) 073015, [hep-ph/0211375].[41] K. C. Y. Ng and J. F. Beacom, Cosmic neutrino cascades from secret neutrino interactions, Phys.

Rev. D90 (2014), no. 6 065035, [1404.2288]. [Erratum: Phys. Rev.D90,no.8,089904(2014)].[42] Y. Farzan, M. Lindner, W. Rodejohann, and X.-J. Xu, Probing neutrino coupling to a light scalar

with coherent neutrino scattering, JHEP 05 (2018) 066, [1802.05171].[43] M. Bustamante, C. A. Rosenstroem, S. Shalgar, and I. Tamborra, Bounds on secret neutrino

interactions from high-energy astrophysical neutrinos, 2001.04994.[44] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics. Oxford University

Press, 2007.[45] A. Belyaev, N. D. Christensen, and A. Pukhov, CalcHEP 3.4 for collider physics within and beyond

the Standard Model, Comput. Phys. Commun. 184 (2013) 1729–1769, [1207.6082].[46] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, FeynRules 2.0 - A complete toolbox

for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250–2300, [1310.1921].[47] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, and T. Reiter, UFO - The Universal

FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201–1214, [1108.2040].[48] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer,

P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading orderdifferential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079,[1405.0301].

17

[49] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLDElectroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., Precisionelectroweak measurements on the Z resonance, Phys. Rept. 427 (2006) 257–454, [hep-ex/0509008].

[50] G. Voutsinas, E. Perez, M. Dam, and P. Janot, Beam-beam effects on the luminosity measurement atLEP and the number of light neutrino species, Phys. Lett. B 800 (2020) 135068, [1908.01704].

[51] P. Janot and S. a. Jadach, Improved Bhabha cross section at LEP and the number of light neutrinospecies, Phys. Lett. B 803 (2020) 135319, [1912.02067].

[52] A. A. Penin, Two-loop corrections to Bhabha scattering, Phys. Rev. Lett. 95 (2005) 010408,[hep-ph/0501120].

[53] T. Becher and K. Melnikov, Two-loop QED corrections to Bhabha scattering, JHEP 06 (2007) 084,[0704.3582].

[54] R. Bonciani, A. Ferroglia, and A. Penin, Heavy-flavor contribution to Bhabha scattering, Phys. Rev.Lett. 100 (2008) 131601, [0710.4775].

[55] S. Actis, M. Czakon, J. Gluza, and T. Riemann, Virtual hadronic and leptonic contributions toBhabha scattering, Phys. Rev. Lett. 100 (2008) 131602, [0711.3847].

[56] S. Actis, M. Czakon, J. Gluza, and T. Riemann, Virtual Hadronic and Heavy-Fermion O(alpha**2)Corrections to Bhabha Scattering, Phys. Rev. D 78 (2008) 085019, [0807.4691].

[57] Particle Data Group Collaboration, M. Tanabashi et al., Review of Particle Physics, Phys. Rev.D98 (2018), no. 3 030001.

[58] H. H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loop integrals,Comput. Phys. Commun. 197 (2015) 276–290, [1503.01469].

[59] M. Peskin and D. Schroeder, An Introduction to quantum field theory, Addison-Wesley, 1995, USA.