Understanding small neutrino mass and its implication
Hsiang-nan Li







Institute of Physics, Academia Sinica,
Taipei, Taiwan 115, Republic of China







We have derived previously the relations between the neutrino masses and mixing angles 
in a dispersive analysis on the mixing of neutral leptonic states. The only involved 
assumption is that the electroweak symmetry of the Standard Model (SM) is restored at 
a high energy scale in some new physics scenario, which diminishes the box diagrams 
responsible for the mixing. Here we include corrections to the analysis up to three 
loops, arising from exchanges of additional neutral and charged scalars in the electroweak 
symmetric phase. The solution to the dispersion relation for the μ^-e^+-μ^+e^- 
mixing generates a typical neutrino mass m_ν∼ O(1) eV in the SM unambiguously. The 
solution also favors the normal ordering of the neutrino masses over the inverted one, and 
links the large electroweak symmetry restoration, i.e., new physics scale to the small 
neutrino mass. 




Understanding small neutrino mass and its implication
    Hsiang-nan Li
    April 29, 2024
=====================================================







§ INTRODUCTION




It has been a long pursuit in particle physics to understand the flavor structure of the 
Standard Model (SM). We proposed recently <cit.> that 
analyticity imposes additional dynamical constraints on the SM parameters beyond the Lagrangian 
level, which can account for the mass hierarchy and the distinct mixing patterns between quarks 
and leptons. Take the mixing of the neutral leptonic states 
μ^- e^+ and μ^+ e^- as an example, where the muon is treated as being heavy with the 
invariant mass squared s, and the electron is regarded as being massless. The mixing amplitude 
governed by the leading-order (LO) box diagrams must respect a dispersion relation formulated in 
the s complex plane. It was argued that the mixing phenomenon will disappear, if the 
electroweak symmetry of the SM is restored at a high energy scale Λ 
<cit.>. The composite Higgs model described in <cit.> 
provides a suitable platform for the argumentation; the electroweak group in their model is 
broken at a scale much lower than the condensate scale, implying the existence of a symmetric 
phase which we refer to. The disappearance of the mixing at s>Λ^2 was taken as the LO 
input to the dispersion relation, and the corresponding solution at low s<Λ^2, i.e., 
in the symmetry broken phase, was found to effectively bind the neutrino masses and the 
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix elements involved in the box diagrams. 
 

Several important observations were extracted from the dispersive constraints on the 
neutrino masses and the PMNS matrix elements <cit.>. The neutrino mass ordering, 
whose various scenarios have not been discriminated experimentally, has remained unsettled in 
neutrino physics <cit.>. It was noticed that the neutrino masses in the normal ordering 
(NO), instead of in the inverted ordering (IO) or quasi-degenerate spectrum, match the 
observed PMNS matrix elements. The neutrino mixing angles larger than the quark ones were 
attributed to the inequality of the mass ratios, m_2^2/m_3^2≫ m_s^2/m_b^2, 
where m_2 (m_3) is the second-generation (third-generation) neutrino mass and m_s 
(m_b) is the s (b) quark mass. The combination with the dispersive analysis of the 
τ^-e^+-τ^+e^- mixing further demands unequivocally the mixing angle 
θ_23≈ 45^∘ in accordance with its measured value around the maximal mixing. 
We emphasize that the above observations are 
established without resorting to specific new ingredients beyond the SM (for recent endeavors 
on this topic, refer to <cit.>).
The only required assumption is the restoration of the electroweak symmetry 
at a high energy scale. The successful explanation on part of the SM flavor structure
thus also sheds light on potential new physics models.

Precisely speaking, our previous study explored the connection between the mixing angles and 
the ratios of the neutrino masses, instead of the absolute values of the neutrino masses 
<cit.>. Therefore, one of the key issues in neutrino physics, i.e., the 
smallness of the neutrino masses, has not been addressed. As a follow-up, we include 
loop corrections to the LO formalism, which arise from exchanges of additional
neutral and charged scalars in the electroweak symmetric phase.  The solution to 
the dispersion relation for the μ^-e^+-μ^+e^- mixing with the high-energy inputs
up to three loops generates a tiny neutrino mass scale in the SM via the equation

    m_ν^2√(lnm_W^2/m_ν^2)≈√(3)/128π^2m_μ^2 m_e^2/v^2,

with the W boson mass m_W, the muon (electron) mass m_μ (m_e) and the vacuum 
expectation value (VEV) v of the Higgs field. Compared to the well-known seesaw mechanism 
<cit.>, the VEV v plays the role of a heavy Majorana neutrino mass in suppressing 
neutrino masses relative to charged lepton masses. Equation (<ref>) leads to 
m_ν∼ O(1) eV unambiguously, which, as an order-of-magnitude estimate, is  
compatible with the upper bound on the neutrino mass m_ν<0.9 eV at 90% CL from the 
the endpoint spectrum of the tritium β-decay measured by the KATRIN Collaboration 
<cit.>. 

We also find from the solution that the tiny neutrino mass hints a large electroweak 
symmetry restoration Λ through the order-of-magnitude 
formula

    lnΛ/m_W∼ O(1)√(lnm_W^2/m_ν^2),

where O(1) denotes a coefficient of order of unity. In other words, the small neutrino mass 
is indeed linked to a new physics scale <cit.>. As 
mentioned before, we have demonstrated <cit.> that the neutrino masses in the NO 
satisfies better the LO dispersive constraints from the mixing of neutral leptonic states, 
compared to the IO. We examine whether this postulation sustains its validity by deriving the 
constraint under the multi-loop input,

    U^*_μ 1 U_e1(m_3-m_1)+U^*_μ 2 U_e2(m_3-m_2)=O(m_ν^2/v)
    ≈ 0,

with the PMNS matrix elements U_μ i and U_ei and the first-generation neutrino mass
m_1. It will be shown that Eq. (<ref>) confirms the preference on the fitted 
parameters associated with the NO. The discrimination is mainly due to the different CP 
phases determined for the two possible mass orderings.

The rest of the paper is organized as follows. We recapture our framework <cit.>
in Sec. II, and present the real part of the mixing amplitude in the symmetric phase and the 
imaginary part in the broken phase. The two pieces are substituted into the
dispersion relation respected by the mixing amplitude in Sec. III to construct the
solution. Equations (<ref>)-(<ref>) are established, and their physical implications
are elaborated. Section IV contains the conclusion and outlook. The real
part of the mixing amplitude is evaluated up to three loops in the Appendix.






§ FORMALISM



Consider the mixing of the neutral leptonic states ℒ^-_Lℓ^+_L and 
ℒ^+_Lℓ^-_L, where ℒ (ℓ) stands for a massive (light) charged 
lepton and L labels the left-handedness. The right-handedness will be labeled by R below. 
Before the electroweak symmetry breaking, all particles are massless, and leptons are in their 
flavor eigenstates. The mixing occurs through exchanges of charged or neutral scalars among 
leptons, whose strengths are characterize by the Yukawa couplings. The Yukawa matrix elements 
are not all independent, so the discussion of the mixing in the symmetric phase will be more 
tedious, if it is based on the Yukawa matrices. A more transparent picture is attained by 
implementing the fermion field transformations adopted in the symmetry broken phase. The 
standard unitary transformation of fermion fields from the flavor eigenstates with the 
superscript f to the mass eigenstates reads

    ν_L^(f)= U_νν_L,    ν_R^(f)= V_νν_R,    ℓ_L^(f)= U_ℓℓ_L,    ℓ_R^(f)= V_ℓℓ_R,

which diagonalizes the Yukawa matrices with the superscript d

    Y_ν^(d)=U_ν^† Y_ν V_ν,    Y_ℓ^(d)=U_ℓ^† Y_ℓ V_ℓ,

and defines the PMNS matrix

    U= U_ℓ^†  U_ν.


The relevant Lagrangian in the symmetric phase is then given by  

    Y_ν^(d)ν̅_L(-ϕ̅^0)ν_R
    +Y_ℓ^(d)ℓ̅_Lϕ^0ℓ_R
    +Y_ν^(d)ℓ̅^'_Lϕ^-ν_R
    +Y_ℓ^(d)ν̅^'_Lϕ^+ℓ_R+h.c.,

after the fermion field transformation,
where the real component of ϕ^0 in the first two terms corresponds to the SM Higgs 
boson. The imaginary component of ϕ^0 and the charged scalars ϕ^± in the last 
two terms disappear after the symmetry breaking, and turn into the longitudinal components 
of weak gauge bosons. Equation (<ref>) indicates that the Yukawa matrices have been 
diagonalized, but the charged scalar currents persist in the symmetric phase. Note that the 
left-handed charged leptons ℓ^'_L, which couple to the right-handed neutrinos 
ν_R via charged scalar currents, differ from the physical mass eigenstates 
ℓ_L. The same differentiation applies to ν^'_L and ν_L. The 
relations between them are governed by the PMNS matrix in Eq. (<ref>),  

    ℓ_L=Uℓ^'_L,    ν_L=U^†ν^'_L;

namely, the left-hand sides are the linear combinations of the right-hand sides 
weighted by the PMNS matrix elements. 

The external neutral states ℒ_L^-ℓ_L^+ and ℒ_L^+ℓ_L^- 
are formed by the charged leptons in the broken phase, so their first emissions are 
composed of real neutral scalars or weak gauge bosons. As the magnitude of the ℒ 
invariant mass exceeds the restoration scale, internal leptons become massless, and start to 
exchange imaginary neutral scalars and charged scalars. The 
ℒ^-_Lℓ^+_L-ℒ^+_Lℓ^-_L mixing must involve 
exchanges of charged scalars ϕ^± or W^± bosons in the symmetric phase. The 
Yukawa matrices associated with charged scalar emissions are also diagonal with 
Y_ℓ^(d) for the ℓ_R→ν^'_L transition and Y_ν^(d) for the 
ν_R→ℓ^'_L transition. When a right-handed neutrino ν_R emits a 
charged scalar ϕ^+ and transits to a left-handed charged lepton ℓ_L, which is one 
of the components of ℓ_L^' according to Eq. (<ref>), the emission is 
characterized by a PMNS matrix element. The same PMNS matrix element is 
associated with the W^+ boson mission in the ν_L→ℓ_L transition. This 
coincidence is expected, since a charged scalar is equivalent to the longitudinal component of 
a W boson.

The dispersion relation for the mixing amplitude Π(s)≡ M(s)-iΓ(s)/2 is quoted as 
<cit.>

    M(s)=1/2π∫^R ds'Γ(s')/s-s'
    +1/2π i∫_C_R ds'Π(s')/s'-s,

where s is the mass squared of the heavy charged lepton ℒ, 
and M(s) and Γ(s)/2 represent the real and imaginary parts of the mixing amplitude, 
respectively. The contour consists of one horizontal path below the branch cut along the 
positive real axis, another horizontal path above the branch cut, and a circle C_R of large 
radius R. The invariant mass squared s is located in a deep Euclidean region inside the
large circle, far away from the branch cut, and its magnitude is above the restoration
scale Λ^2. We decompose the mixing amplitude into a sum over various intermediate 
neutrino channels,

    Π(s)=∑_i,j=1^3λ_iλ_jΠ_ij(s)+⋯≡∑_i,j=1^3λ_iλ_j[M_ij(s)-i/2Γ_ij(s)]+⋯,
 
where λ_i≡ U^*_ℒ iV_ℓ i is the product of the PMNS matrix 
elements, and ⋯ denote the contributions proportional to higher powers of λ_i,j. 
Because an intermediate state can be identified experimentally in principle, the 
contribution from each channel with the lowest power of λ_iλ_j satisfies its own 
dispersion relation,

    M_ij(s)=1/2π∫^R ds'Γ_ij(s')/s-s'
    +1/2π i∫_C_R ds'Π_ij(s')/s'-s,

in which the lower bound of s' for the dispersive integral of Γ_ij(s') will be 
specified later.





 §.§ Real Contribution




We calculate the real part of the mixing amplitude in the symmetric phase, and collect
the detail in the Appendix. As elucidated above, the external states 
ℒ_L^-ℓ_L^+ and ℒ_L^+ℓ_L^- exchange only two W bosons 
at one-loop level as depicted by the box diagram in Fig. <ref>. The real contribution 
from the box diagrams with the massless internal neutrinos ν_iL, ν_jL and two 
massless W bosons is written as

    M_ij^(1)(s)=-1/16π^2(g/2√(2))^44/s,

which is of O(g^4), g being the weak coupling. The above expression is independent of 
intermediate states, such that the summation over all intermediate channels vanishes owing 
to the unitarity condition ∑_iU^*_ℒ iU_ℓ i=0. The absence of 
the mixing dynamics in the symmetric phase has been adopted as the input in our previous 
LO analysis <cit.>.



At two-loop level, the heavy charged lepton ℒ_L^- can first emit a Higgs 
boson, transiting to a right-handed charged lepton ℒ_R^-, which 
then emits a charged scalar ϕ^-, transiting to a left-handed neutrino ν_iL. 
At last, this ν_iL becomes the external charged lepton ℓ_L^- in the external 
state ℒ_L^+ℓ_L^- by emitting a W^+ boson as illustrated in 
Fig. <ref>. All the above internal particles are massless. The Higgs boson vertex 
carries the Yukawa coupling, i.e., the diagonal element of the Yukawa matrix in 
Eq. (<ref>), (Y_ℓ^(d))_ℒℒ/√(2)=m_ℒ/v. The 
charged scalar vertex is characterized by the PMNS matrix element U^*_ℒi and 
the Yukawa coupling (Y_ℓ^(d))_ℒℒ=√(2)m_ℒ/v 
according to the Lagrangian in Eq. (<ref>), 
and the W boson vertex contains the PMNS matrix element U_ℓ i and the weak coupling 
g. The combination of U^*_ℒi and U_ℓ i forms the PMNS factor 
λ_i. The couplings associated with the emissions from the external charged lepton 
ℓ_L^+, the internal right-handed charged lepton ℓ_R^+, and the internal 
left-handed neutrino ν_jL are assigned in a similar way, which gives rise to another PMNS 
factor λ_j. This contribution, unlike Eq. (<ref>), is counted as O(g^2) 
and the fourth power in the Yukawa couplings for charged leptons.



There are other two-loop diagrams of course. For instance, an additional photon or Z boson
can be exchanged between the charged leptons in Fig. <ref>,
or a W boson can proceed a hadronic decay, splitting into a quark-antiquark pair. These 
diagrams, being of higher orders in the electroweak coupling, are classified as radiative 
corrections to Fig. <ref>. To have them in the real part of the mixing amplitude, the 
corresponding diagrams must be added to the imaginary part for a consistent construction of
the dispersion relation. We will not consider these contributions in the present 
order-of-magnitude investigation. The boson exchanges in the sequence of W^-, ϕ^+ 
and ϕ^0, opposite to that displayed in Fig. <ref>, are allowed. The diagram 
involving a ϕ^0ϕ^0ϕ^+ϕ^- four-scalar vertex and a W boson exchange also exists 
at O(g^2). For our purpose, we compute only Fig. <ref> in the Appendix, and the result 
reads

    M_ij^(2)(s)=-(1/16π^2)^2(g/2√(2))^2
    m_ℒ^2 m_ℓ^2/v^44/s.

The above formula, vanishing the summation over all intermediate channels, implies that 
the mixing phenomenon in the symmetric phase is still off at two-loop level. 
Note that the lepton mass dependence has been introduced via the Yukawa matrix 
Y^(d)_ℓ, though all internal particles are massless in the symmetric phase. 





We then extend the evaluation to three-loop level, at which there are even more 
diagrams. The contribution from exchanges of two W bosons and two charged scalars, being 
proportional to the PMNS factor to the fourth power, should be discarded. The diagram with 
exchanges of two photons (or two Z bosons) and two W bosons stands for a higher-order 
correction in the electroweak coupling to Fig. <ref>, and is thus neglected. The 
contribution of Fig. <ref>(a) with two Higgs bosons and two charged scalars is 
proportional to O(g^0) and the eighth power in the Yukawa couplings for charged leptons,  
different from Eqs. (<ref>) and (<ref>). At the same order, a Higgs boson can attach 
to ℒ_L^- and ℓ_L^- and another can attach to ℓ_L^+ 
and ℒ_L^+; the two Higgs bosons in the above diagram can cross each other; 
the external leptonic states can interact via two Higgs bosons and a ϕ^+ϕ^+ϕ^-ϕ^- 
four-scalar vertex. Figure <ref>(b) with two W bosons and two neutral scalars
[We have assumed that neutrinos are of the Dirac type to allow the 
existence of Fig. <ref>(b)], being proportional to (g^4) and the fourth power in the 
Yukawa couplings for neutrinos, also constitutes a higher-order correction to Fig. <ref>.
Nevertheless, it produces the first piece of contributions dependent on intermediate 
neutrino masses, so that the summation over all intermediate channels does not vanish 
because of ∑_iU^*_ℒ iU_ℓ i(Y^(d)_ν)_ii^2≠0. Therefore, the 
reasoning for the disappearance of the mixing phenomenon in the symmetric phase 
<cit.> holds to an extremely high precision in view of the tininess of Y^(d)_ν. 



We estimate only the O(g^0) contribution from Fig. <ref>(a) in the Appendix for our 
order-of-magnitude analysis, obtaining  

    M_ij^(3)(s)≈1/64(1/16π^2)^3
    m_ℒ^4 m_ℓ^4/v^83/s.

The contribution from Fig. <ref>(b) is also needed for discriminating the neutrino mass 
orderings in the next section, but its explicit expression is not crucial. We summarize the 
real part of the mixing amplitude in the symmetric phase from Eqs. (<ref>), (<ref>) 
and (<ref>),

    M_ij(s)≈-1/16π^2[4(g/2√(2))^4
    +1/4π^2(g/2√(2))^2
    m_ℒ^2 m_ℓ^2/v^4
    -3/64(1/16π^2)^2m_ℒ^4 m_ℓ^4/v^8
    +O(g^4m_i^2m_j^2/v^4)]1/s,

where the last term is associated with Fig. <ref>(b). Equation (<ref>)
will be inserted into the left-hand side of the dispersion relation in Eq. (<ref>).




 §.§ Imaginary Contribution




The neutral leptonic states ℒ_L^-ℓ_L^+ and ℒ_L^+ℓ_L^- 
mix through the box diagrams with two W boson exchanges in the broken phase. To coordinate 
with the derivation of the real part M_ij(s) in the previous subsection, we sum up the 
contributions to the imaginary part Γ_ij(s) from 
all possible cuts on the internal lines in the box diagrams. This treatment differs from 
that in our earlier work <cit.>, where only the two internal neutrino 
lines were cut. It makes sense, since such an intermediate state can be identified 
experimentally in principle, and the corresponding real part vanishes at LO owing to the 
unitarity of the PMNS matrix. The box diagrams induce the (V-A)(V-A) and (S-P)(S-P)
structures, which ought to be handled separately. We concentrate on the former, which 
contributes <cit.>

    Γ_ij(s)   =   G_F^2/16π∑_k'Γ_ij^k(s),
    Γ_ij^k(s)   =   ∫_α_k^l^α_k^u dα(4m_W^4+m_i^2m_j^2)F_k(α)-4m_i^2m_j^2m_W^2+2s
    [α m_i^2+(1-α)m_j^2]m_W^2/(m_W^2-m_i^2)(m_W^2-m_j^2),
 
with the notation ∑'_k≡∑_k=1^2-∑_k=3^4 and the intermediate neutrino 
masses m_i and m_j. 
 
The functions F_k take the forms

    F_1(α)   =   α m_i^2+(1-α)m_j^2-α(1-α)s,
    
    F_2(α)   =   m_W^2-α(1-α)s,
    
    F_3(α)   =   α m_W^2+(1-α)m_j^2-α(1-α)s,
    
    F_4(α)   =   α m_i^2+(1-α)m_W^2-α(1-α)s,

whose vanishing fixes the upper and lower bounds of the integration variable α, 

    α_1^u,l   =   1/2s[s-m_i^2+m_j^2±√(s^2 - 2s(m_i^2+m_j^2)+(m_i^2-m_j^2)^2)],
    α_2^u,l   =   1/2s[s±√(s^2 - 4sm_W^2)],
    α_3^u,l   =   1/2s[s-m_W^2+m_j^2±√(s^2 - 2s(m_W^2+m_j^2)+(m_W^2-m_j^2)^2)],
    α_4^u,l   =   1/2s[s-m_i^2+m_W^2±√(s^2 - 2s(m_i^2+m_W^2)+(m_i^2-m_W^2)^2)].

The indices k=1, 2, 3 and 4 correspond to the intermediate states with two real neutrinos, 
two real W bosons, the real neutrino of mass m_i and one real W boson, and the real 
neutrino of mass m_j and one real W boson, respectively. The thresholds for
the dispersive integral in Eq. (<ref>) are then set to the values t_k=(m_i+m_j)^2, 
4m_W^2, (m_i+m_W)^2 and (m_j+m_W)^2, respectively. 


It is observed that the linear growth of Γ_ij(s) at large s <cit.> in 
the broken phase cancels among the cuts in Eq. (<ref>). The cancellation is so strong, 
that Γ_ij(s) turns out to decrease like 1/s asymptotically. It is not hard to 
realize this strong cancelation; the functions F_k in Eq. (<ref>), as well as the upper 
and lower bounds of α in Eq. (<ref>), become identical in the s →∞ limit. 
The real part M_ij(s) in Eq. (<ref>) and the aforementioned decrease of 
Γ_ij(s) at high s manifest that the mixing amplitude Π_ij(s) goes down like 
1/s. Hence, the contour integral from the big circle diminishes with the radius R, and 
the large s' region does not contribute to the dispersive integral in Eq. (<ref>). This 
fact is also reflected by the R independence of M_ij(s) on the left-hand side of 
Eq. (<ref>). Although the behavior of Γ_ij(s) in the transition region between 
the symmetry broken and restored phases is unknown, physical insights can be extracted from 
the convergent dispersive integral. First, we remove the interval above the restoration 
scale, Λ^2<s'<R, motivated by the disappearance of the R dependence. The restoration 
scale, representing an order-of-magnitude concept, can mimic the uncertainty associated with 
the transition region. The integration is then performed with the known integrand Γ_ij(s') 
in Eq. (<ref>), which is valid within the interval t_k<s'<Λ^2. 

Keeping only the terms that survive in the large Λ^2 limit, we expand the dispersive 
integral as

    1/2π∑_k'∫_t_k^Λ^2 ds'Γ_ij^k(s')/s-s'≈-G_F^2m_W^4/16π^2[1 + m_im_j/m_W^2lnΛ^2/m_W^2 - m_i^2m_j^2/m_W^4(lnm_W^2/m_im_j - 1/4)]2/s,

where the approximation 1/(s-s')≈ 1/s for s≫Λ^2 has been applied,
and those suppressed by higher powers of m_i,j/m_W have been dropped for clarity. 
To proceed further, we recast the above expression, by employing the relations
G_F=√(2)g^2/(8m_W^2) and m_W=gv/2, into

    1/2π∑_k'∫_t_k^Λ^2 ds'Γ_ij(s')/s-s'≈ 
    -1/16π^2[4(g/2√(2))^4 
    + 2(g/2√(2))^2m_im_j/v^2lnΛ^2/m_W^2 
    - m_i^2m_j^2/v^4(lnm_W^2/m_im_j - 1/4)]1/s.

It is noticed that powers of m_W have compensated some of the weak coupling, and the 
expansions on both sides of Eq. (<ref>), i.e., in Eqs. (<ref>) and (<ref>), coincide 
perfectly in powers of g. In particular, the uncertainty from the transition 
region shows up only in the O(g^2) term. Adding more photons, Z bosons and Higgs bosons 
to the box diagrams just introduces higher-order corrections to Eq. (<ref>), which will 
be ignored in the present order-of-magnitude setup, as stated in the previous subsection. 





§ SOLUTION






We are ready to acquire information on neutrino masses from Eq. (<ref>), given the real
piece M_ij(s) in the symmetric phase and the imaginary piece Γ_ij(s) in the 
broken phase. Our framework is appropriate for the mixing of heavy-light systems, 
like μ e and τ e. This claim is justified by the connections among the PMNS matrix 
elements discovered in our previous work <cit.>; the ratio 
U^*_μ 1U_e1/(U^*_μ 2U_e2) of the PMNS matrix elements for the μ e mixing and 
U^*_τ 1U_e1/(U^*_τ 2U_e2) for the τ e mixing are roughly equal, except 
the opposite signs of their imaginary parts. Such a prediction for heavy-light systems agrees 
well with the outcomes of global fits <cit.>. However, the ratio 
U^*_τ 1U_μ 1/(U^*_τ 2U_μ 2) for the τμ mixing does not follow the 
same pattern, suggesting that τμ should not be regarded as a heavy-light system. The 
distinction is natural in view of the charged lepton masses; a muon is about 200 times heavier 
than an electron, while a τ lepton is only 17 times heavier than a muon. 



We have shown in Sec. IIA that M_ij(s) in the symmetric phase is proportional to the 
masses of the external charged leptons. To probe how small a typical neutrino mass can be, 
we consider the μ e mixing, and set m_i,j in Eq. (<ref>) 
to a common mass scale m_ν. As seen shortly, the value of m_ν determined from the 
dispersion relation is of order eV, so that there is no need to differentiate which generation 
of neutrinos this typical scale is assigned to; an eV scale is much higher than the 
measured mass differences reflected by 
Δ m^2_21≡ m^2_2-m^2_1= (7.55^+0.20_-0.16)× 10^-5 eV^2 
and Δ m^2_32≡ m^2_3-m^2_2=(2.424± 0.03)× 10^-3 eV^2 
from the global fit <cit.>. We establish a solution to the dispersion relation 
by equating the terms in Eqs. (<ref>) and  (<ref>) with the same powers of the weak 
coupling g, for g can vary freely in a mathematical viewpoint. 
It is immediately found that the first, i.e., O(g^4), terms in Eqs. (<ref>) and (<ref>) 
are the same. We emphasize that this equality is insensitive to the behavior of Γ_ij(s) 
in the transition region, namely, independent of the symmetry restoration scale.






The equality of the third, i.e., O(g^0), terms in Eqs. (<ref>) and (<ref>) leads to 
Eq. (<ref>), where m_ℒ, m_ℓ and m_i,j have been replaced by the 
muon mass m_μ, the electron mass m_e, and the typical neutrino mass m_ν, respectively. 
The constant 1/4 is negligible relative to the logarithmic term ln(m_W^2/m_ν^2). 
Equation (<ref>), arising from the convergent part of the dispersive integral, does not 
depend on the restoration scale either. 
It then designates a small mass scale for neutrinos unambiguously, 

    m_ν≈ 3  eV,

with the inputs m_W≈ 80 GeV, m_μ≈ 0.1 GeV, m_e≈ 0.5 MeV and 
v≈ 250 GeV <cit.>. Note that this low mass scale m_ν∼ O(1) eV is demanded 
by the internal consistency the SM dynamics, whose origin is quite different from the seesaw 
mechanism <cit.> as stated in the Introduction. Our order-of-magnitude estimate is 
not far beyond the upper bound on the neutrino mass m_ν<0.9 eV at 90% CL measured by the 
KATRIN Collaboration <cit.>. We mention that their measurement is independent of 
cosmological models and does not rely on assumptions whether a neutrino is a Dirac or 
Majorana particle. 




Next we equate the second, i.e., O(g^2), terms in Eqs. (<ref>) and (<ref>),
reaching

    m_ν^2lnΛ/m_W≈1/16π^2m_μ^2 m_e^2/v^2,

whose combination with Eq. (<ref>) yields Eq. (<ref>). It is stressed that 
Eq. (<ref>) is independent of the masses m_ℒ and m_ℓ of the external 
charged leptons. Because of the uncertain behavior of Γ_ij(s) in the transition region, 
we have regarded the coefficient on the right-hand side of Eq. (<ref>) as being of O(1). 
Unfortunately, the value of the restoration scale Λ inferred from Eq. (<ref>) is 
sensitive to the O(1) coefficient, so no definite result can be obtained (for a very crude 
guesstimate, Eq. (<ref>) corresponds to Λ≳ O(100) TeV via Eq. (<ref>)). 
Instead, we highlight the essential indication that the large electroweak symmetry restoration, 
i.e., new physics scale is tied to the tiny neutrino mass. A precise prediction for the 
new physics scale can be achieved by exploring Γ_ij(s) in the transition region and by 
calculating exact multi-loop contributions to M_ij(s) in the symmetric phase.


At last, we sum both Eq. (<ref>) and (<ref>) over all intermediate channels, and
check whether the constraint on the PMNS matrix elements and the neutrino mass 
orderings in our previous LO analysis <cit.> is maintained. Only the 
O(m_i^2m_j^2/v^4) term in Eq. (<ref>), which depends on the intermediate neutrino 
masses, survives the summation under the unitarity of the PMNS matrix. The summation for 
Eq. (<ref>) is dominated by the second term of O(m_im_j/v^2). The elimination of 
λ_3=-λ_1-λ_2 by means of the unitarity condition produces

    lnΛ^2/m_W^2[λ_1(m_3-m_1)+ λ_2(m_3- m_2)]^2=∑_i,j
    O(m_i^2m_j^2/v^2)≈ 0,

which leads to Eq. (<ref>) trivially irrespective of the restoration scale 
Λ; the O(m_im_j) coefficient of ln(Λ^2/m_W^2) must diminish to match the 
right-hand side of O(m_i^2m_j^2/v^2). 

To facilitate the discussion, we quote the PMNS matrix in the Chau-Keung 
parametrization <cit.>,

    U_ PMNS=( [                   c_12c_13                   s_12c_13                  s_13e^-iδ; -s_12c_23-c_12s_23s_13e^iδ  c_12c_23-s_12s_23s_13e^iδ                   s_23c_13;  s_12s_23-c_12c_23s_13e^iδ -c_12s_23-s_12c_23s_13e^iδ                   c_23c_13 ]) ,
 
with the notations s_12≡sinθ_12, c_12≡cosθ_12, etc. for
the mixing angles, and the CP phase δ. The products of the PMNS matrix elements then 
read 

    λ_1=-c_12c_13(s_12c_23+c_12s_23s_13e^-iδ),    λ_2=s_12c_13(c_12c_23-s_12s_23s_13e^-iδ).

Defining r=λ_1/λ_2, we require the cancellation between the real parts of 
r(m_3-m_1) and (m_3- m_2) according to Eq. (<ref>),

    Rer(m_3-m_1)+(m_3- m_2)≈ 0,

and the small imaginary part of r(m_3-m_1), i.e., a small s_δ≡sinδ. 

It is easy to verify 
that the mixing angles θ_12= (34.5^+1.2_-1.0)^∘, 
θ_13=(8.45^+0.16_-0.14)^∘ and θ_23=(47.7^+1.2_-1.7)^∘, 
and the CP phase δ=(218^+38_-27)^∘ from the global fit 
<cit.> in the NO scenario meet better the above requirements, compared to 
θ_12=(34.5^+1.2_-1.0)^∘, θ_13=(8.53^+0.14_-0.15)^∘, 
θ_23=(47.9^+1.0_-1.7)^∘ and δ=(281^+23_-27)^∘ in the IO 
one <cit.>. We set θ_12, θ_13 and θ_23 to their
central values owing to their small errors, and vary δ within its 1-σ range.
Equation (<ref>) can be satisfied by c_δ<0 (Rer>-1) and m_3-m_1>0 
for the NO, and by c_δ>0 (Rer<-1) and m_3-m_1<0 for the IO. 
The whole range of δ for the NO is allowed, for the resultant Rer takes 
values between -0.9 and -0.7, while δ in the IO case is restricted to 
(270^∘, 304^∘). We then have s_δ≈ -(0.62^+0.35_-0.43) from the 
NO, whose magnitude could be much lower than that of s_δ∈ (-1,-0.83) from the IO.
The latter narrow range is attributed to that δ for the IO is about 
3π/2, so the amount of errors similar to the NO case causes minor variation of 
s_δ. Our solution favors negative c_δ and diminishing s_δ, i.e., 
δ∼ 180^∘, close to that derived in a seesaw model with right-handed 
neutrino masses being dynamically generated by strong horizontal gauge interactions 
<cit.>.


























§ CONCLUSION



We have proposed a viable explanation on the extremely small neutrino masses, which comes 
as a straightforward extension of our continuous efforts in understanding the SM flavor 
structure. Dispersion relations, which analytical physical observables must obey, 
bind interactions at various scales, and enforce stringent constraints on 
the SM parameters. The assumption for the existence of an electroweak symmetric phase
grants such a bond between the dynamics above and below the symmetry restoration scale. 
We have taken into account the contributions to the mixing of neutral leptonic states up to 
three loops in the symmetric phase, which arise from the exchanges of neutral and charged 
scalars and W bosons. The standard box diagrams responsible for the mixing in the 
symmetry broken phase involve the neutrino masses and the PMNS matrix elements. It was then 
demonstrated that the dispersion relation for the μ^-e^+-μ^+e^- mixing 
demands a typical neutrino mass scale m_ν at eV order in the SM unambiguously. It is 
much lower than a lepton mass because of the huge suppression factor m_ℓ/v formed by 
the lepton mass and the electroweak scale. At two-loop level, the dispersion relation hints 
that the large electroweak symmetry restoration, i.e., new physics scale is linked to the 
small neutrino mass. 

The calculation presented in the Appendix confirmed that the disappearance of the mixing 
phenomenon in the symmetric phase, argued in our previous publication based on the unitarity 
of the PMNS matrix, holds to high precision, up to corrections of O(m_ν^4/v^4). In 
other words, the first nonvanishing contribution which survives the summation over all 
intermediate channels stars at three-loop level. It is thus expected that the dispersive 
constraint on the neutrino masses and the PMNS matrix elements with the loop corrections 
still prefers the NO over the IO. The pivot for discriminating the neutrino mass orderings 
is played by the CP phase, which got distinct results in the global fits for the NO and 
the IO. Since it is challenging to determine the neutrino mass spectrum experimentally, our 
theoretical attempt should be worthwhile. The above summaries the solution to the dispersion 
relation for the mixing of heavy-light leptonic systems. We admit that our approach is 
preliminary, accurate only to orders of magnitude, and needs to be refined in the future. 
In principle, more definite predictions for the neutrino masses and for the new physics scale 
can be made by exploring the imaginary piece Γ_ij(s) in the transition region and 
by computing complete multi-loop contributions to the real piece M_ij(s) in the symmetric 
phase. 





We affirm that the current formalism for generating a small neutrino mass cannot be extended 
to the quark case, simply due to strong interaction among quarks. The strong coupling 
g_s≳ 1 is much larger than most of the quark Yukawa couplings, even greater than the 
top Yukawa coupling, such that the loop corrections discussed in this work are overwhelmed by 
QCD ones. For example, the two neutral scalars in Fig. <ref>(a) for the mixing between 
the neutral quark states cu̅ and c̅ u can be replaced by two gluons. The resultant 
contribution then dominates over the former one by a factor g_s^4 v^4/(m_c^2m_u^2)≫ 1 with 
the charm (up) quark mass m_c (m_u). It is obvious that Eq. (<ref>) is not justified. 
In this sense, the origin of the tiny neutrino 
mass traces back to the fact that a neutrino participates only weak and scalar interactions. 
Combining the present and previous observations, we conjecture that the mass hierarchy inherent 
in the SM, from the neutrino masses up to the electroweak scale, and the distinct quark and 
lepton mixing patterns, may be accommodated by means of the internal consistency of the SM 
dynamics. This understanding also points to possible new physics scenarios above the electroweak 
symmetry restoration scale, such as the composite Higgs model, which will be explored in 
forthcoming papers.







§ ACKNOWLEDGEMENT


We thank Y. Chung, X.G. He, Y.B. Hsiung, Y. Liao, J.L. Liu and M.R. Wu for illuminating 
discussions. This work was supported in part by National Science and Technology Council of 
the Republic of China under Grant No. MOST-110-2112-M-001-026-MY3.






§ MIXING AMPLITUDE UP TO THREE LOOPS IN SYMMETRIC PHASE





 §.§ One Loop


The box-diagram contribution, including both the real and imaginary parts, has been computed 
in <cit.>. We set the intermediate neutrino masses to zero, and then take the m_W→ 0 
limit to get the leading real contribution in the symmetric phase shown in Eq. (<ref>). 
The dispersion relation at O(g^4) is then verified by 
Eqs. (<ref>) and (<ref>). To reduce the load for two-loop and three-loop calculations, 
we propose a simplified scheme below. The amplitude from the box diagram in Fig. <ref> 
for the ℒ^-ℓ^+-ℒ^+ℓ^- mixing with 
massless intermediate particles is written as

    -i M=1/2(-ig/2√(2))^4∫d^4l/(2π)^4u̅_ℓγ_ν(1-γ_5)i(p̸-l̸)γ_μ(1-γ_5) u_ℒv̅_ℓγ_μ'(1-γ_5)(-i)l̸γ_ν'(1-γ_5) v_ℒ
    (-i)^2g^νν'g^μμ'/(l^2+iϵ)^2[(p-l)^2+iϵ]^2.

The PMNS matrix elements have been suppressed, the spin factor 1/2 comes from the 
definition of the neutral leptonic state, the coupling -ig/(2√(2)) is associated with 
a weak vertex, and the tensors g^νν' and g^μμ' are from the massless W 
boson propagators. Strictly speaking, the aforementioned m_W→ 0 limit urges that we 
start with a massive m_W boson propagator, 

    -ig^μμ'-l^μ l^μ'/m_W^2/l^2-m_W^2,

involving an additional longitudinal polarization. Here we employ a massless W boson propagator 
directly for simplification.

The Fierz identity is inserted into Eq. (<ref>) to factorize the lepton currents,

    M   =   -1/4(g/2√(2))^4
    u̅_ℓγ^α(1-γ_5) u_ℒv̅_ℓγ^β(1-γ_5)v_ℒi∫d^4l/(2π)^4 tr[γ_μγ_αγ^μl̸γ^νγ_βγ_ν(p̸-l̸)
    (1+γ_5)]/(l^2+iϵ)^2[(p-l)^2+iϵ]^2.

If we work on the (S-P)(S-P) structure, the term (1-γ_5)⊗(1-γ_5)
in the Fierz identity will need to be inserted too, and the derivation will be more cumbersome. 
To extract the (V-A)(V-A) current, we delete the current product
u̅_ℓγ^α(1-γ_5) u_ℒv̅_ℓγ^β(1-γ_5)v_ℒ, and
contract the remaining piece by the projector g^αβ/4. The projector should 
be (g^αβ-p^α p^β/p^2)/3 to avoid the (S-P)(S-P) component. However, we 
have confirmed that g^αβ/4 reproduces Eq. (<ref>) from Eq. (<ref>) with 
massless W boson propagators. This simpler scheme will be applied to the evaluations of 
two-loop and three-loop diagrams for an order-of-magnitude estimate. We acquire from 
Fig. <ref>

    M   =   2(g/2√(2))^4i∫d^4l/(2π)^4l·(p-l)/(l^2+iϵ)^2[(p-l)^2+iϵ]^2
    =2/16π^2(g/2√(2))^41/s,

with s=p^2, which is assumed to be in the Euclidean region. 

We have quoted the imaginary contribution of the box diagrams in the broken phase 
from <cit.> as stated in Sec. II. It is reminded that the analysis in <cit.> 
was conducted for the mixing of neutral quark states. A quark possesses degrees of freedom from 
colors, and the box diagrams with vertical W boson lines (in Fig. <ref>) and with 
horizontal W boson lines have different color factors, 1 for the former and 3 for the latter. 
Since a color factor can vary arbitrarily in a mathematical viewpoint, each box diagram 
respects its own dispersion relation. We will take advantage of this fact for the leptonic state 
mixing, combining the above two box diagrams in the way the same as in the quark case, although 
a lepton has no color charge. The contribution from the second box diagram with horizontal 
W boson lines turns out to be identical to Eq. (<ref>) but with an opposite sign. The 
combination of the two box diagrams then yields Eq. (<ref>). We do not include the diagram 
with the two W bosons in the second diagram crossing each other, because it obeys a separate 
dispersion relation with different branch cuts; the two intermediate neutrinos of unequal 
masses cannot be on shell simultaneously in this case.





 §.§ Two Loops



The two-loop diagram in Fig. <ref> contributes in the symmetric phase

    -i M   =   1/2(-im_ℒ/v)
    (-i√(2)m_ℒ/v)
    (-im_ℓ/v)(-i√(2)m_ℓ/v)
    (-ig/2√(2))^2
    ∫d^4l_1/(2π)^4∫d^4l_2/(2π)^4
          ×u̅_ℓγ_μ(1-γ_5)i(p̸-l̸_2)i(p̸-l̸_1)u_ℒv̅_ℓ(-i)l̸_1(-i)l̸_2γ_ν(1-γ_5) v_ℒ/(p-l_2)^2(p-l_1)^2l_1^2l_2^2i/l_1^2i/(l_2-l_1)^2-ig^μν/(p-l_2)^2,

where the Yukawa couplings for the neutral and charged scalar emissions have been specified 
in Sec. II. Inserting the Fierz identity to factorize the lepton currents and 
contracting the remaining part by the projector g^αβ/4, we have

    M   =   -m_ℒ^2 m_ℓ^2/v^4(g/2√(2))^2
    ∫d^4l_1/(2π)^4∫d^4l_2/(2π)^4l_1· l_2(p-l_2)·(p-l_1)/((p-l_2)^2)^2(p-l_1)^2(l_1^2)^2l_2^2(l_2-l_1)^2.

We first work out the integration over l_1 under the Feynman parametrization,

    ∫d^4l_1/(2π)^4l_1· l_2(p-l_2)·(p-l_1)/(l_1^2)^2(p-l_1)^2(l_2-l_1)^2
    =6∫_0^1 dv∫_0^1-vdu(1-u-v)∫d^4l_1/(2π)^4l_1· l_2(p-l_2)·(p-l_1)/[l_1^2-2up· l_1+us-2vl_2· l_1+vl_2^2]^4.

The variable change l_1→ l_1+up+vl_2 and the Wick rotation lead the above integral to

    iπ^2/(2π)^4∫_0^1 dv∫_0^1-vdu(1-u-v){(up+vl_2)· l_2(p-l_2)·[(1-u)p-vl_2]/[v(1-v)l_2^2-2uvp· l_2+u(1-u)s]^2.
          .
    -(p-l_2)· l_2/2[v(1-v)l_2^2-2uvp· l_2+u(1-u)s]}.


We then turn to the integration over l_2. The first piece in Eq. (<ref>) gives,
under the Feynman parametrization,

    I_1   =   ∫d^4l_2/(2π)^4(up+vl_2)· l_2(p-l_2)·[(1-u)p-vl_2]]/l_2^2((p-l_2)^2)^2[v(1-v)l_2^2-2uvp· l_2+u(1-u)s]^2
       =   ∫_0^1 dw∫_0^1-wdt24tw/v^2(1-v)^2∫d^4l_2/(2π)^4(up+vl_2)· l_2(p-l_2)·[(1-u)p-vl_2]/{l_2^2-2tp· l_2-2uwp· l_2/(1-v)+ts+u(1-u)ws/[v(1-v)]}^5.

The variable change 

    l_2→ l_2+(t+uw/1-v)p

brings Eq. (<ref>) into

    I_1
    
    
       =   iπ^2/(2π)^4∫_0^1 dw∫_0^1-wdt2tw/(1-v)^2A (B + D) + (B-C)D+ 3D^2/D^3,

with the functions

    A   =   (u/v+t+uw/1-v)(t+uw/1-v)s
    
    B   =   (1-t-uw/1-v)(1-u/v-t-uw/1-v)s
    
    C   =   (u/v+2t+2uw/1-v)
    (1+1-u/v-2t-2uw/1-v)s/4
    
    D   =   [t+u(1-u)w/v(1-v)]s-(t+uw/1-v)^2s.

We calculate the second term in Eq. (<ref>) in a similar way,

    I_2   =   1/2∫d^4l_2/(2π)^4(l_2-p)· l_2/l_2^2((p-l_2)^2)^2[v(1-v)l_2^2-2uvp· l_2+u(1-u)s]
       =   ∫_0^1 dw∫_0^1-wdt6t/2v(1-v)∫d^4l_2/(2π)^4(l_2-p)· l_2/{l_2^2-2tp· l_2-2uwp· l_2/(1-v)+ts+u(1-u)ws/[v(1-v)]}^4.

The same variable change in Eq. (<ref>) generates 

    I_2
    
    
    =iπ^2/(2π)^4∫_0^1 dw∫_0^1-wdtt/2v(1-v)2D - E/D^2,

with the function

    E=(1-t-uw/1-v)(t+uw/1-v)s.



At last, the contribution from Fig. <ref> reads 

    M   =   -m_ℒ^2 m_ℓ^2/v^4(g/2√(2))^2
    iπ^2/(2π)^4∫_0^1 dv∫_0^1-vdu(1-u-v)(I_1+I_2)
       =   m_ℒ^2 m_ℓ^2/v^4(g/2√(2))^2
    (1/16π^2)^22/s,

where the integrations over the Feynman parameters u, v, t and w are carried out 
numerically. The result from the diagram with horizontal boson lines is has the same 
expression but with an opposite sign. Following the special combination of the two 
diagrams, we arrive at Eq. (<ref>).







 §.§ Three Loops


The three-loop diagram with four vertical scalar lines in Fig. <ref>(a) contains the 
loop integral

    -i M   =   1/2(-im_ℒ/v)^2
    (-i√(2)m_ℒ/v)^2
    (-im_ℓ/v)^2(-i√(2)m_ℓ/v)^2
    ∫d^4l_1/(2π)^4∫d^4l_2/(2π)^4∫d^4l_3/(2π)^4
          ×u̅_ℓ il̸_3il̸_2i(p̸-l̸_1)u_ℒv̅_ℓ(-i)l̸_1(-i)(p̸-l̸_2)(-i)(p̸-l̸_3) v_ℒ/(p-l_1)^2(p-l_2)^2(p-l_3)^2 l_1^2l_2^2l_3^2i/l_1^2i/(p-l_1-l_2)^2i/(l_2-l_3)^2i/l_3^2.

We have arranged the loop momentum flows to facilitate the identification of the leading
regions for the integral. It is straightforward to tell that finite l_1 gives an important 
contribution, for both the infrared and ultraviolet regions are suppressed. Small l_2 
dominates, in which the integration over l_3 develops an infrared logarithmic enhancement. 
We thus neglect l_2 in the factor p-l_1-l_2, such that the integration over l_1 is 
simplified to the one for the box diagram in Fig. <ref>, and retain only the terms up to 
those linear in l_2 in the integral over l_3. An order-of-magnitude estimate serves our 
purpose, so the above approximation is appropriate, without which the handling of three-loop 
diagrams would be extremely tedious. The reasoning then motivates the separation
of Eq. (<ref>) into three pieces by inserting the Fierz identity,

    M   =   2i/8^4m_ℒ^4 m_ℓ^4/v^8u̅_ℓγ^α(1-γ_5) u_ℒv̅_ℓγ^β(1-γ_5)v_ℒ
          ×∫d^4l_2/(2π)^4 tr[ (p̸-l̸_2)
    γ^λ(1-γ_5)l̸_2γ^σ(1-γ_5)]/(p-l_2)^2l_2^2
          ×∫d^4l_1/(2π)^4 tr[l̸_1(1-γ_5)γ_σ
    (p̸-l̸_1)(1-γ_5)γ_α]/(p-l_1)^2(p-l_1-l_2)^2(l_1^2)^2
          ×∫d^4l_3/(2π)^4 tr[(p̸-l̸_3)
    (1-γ_5)γ_βl̸_3(1-γ_5)γ_λ]/(p-l_3)^2(l_3^2)^2(l_2-l_3)^2.


The integral over l_1 is denoted by

    J_1   =   ∫d^4l_1/(2π)^4 tr[l̸_1(1-γ_5)γ_σ
    (p̸-l̸_1)(1-γ_5)γ_α]/(p-l_1)^2(p-l_1-l_2)^2(l_1^2)^2≈ 2∫d^4l_1/(2π)^4 tr[l̸_1γ_σ
    (p̸-l̸_1)γ_α(1+γ_5)]/((p-l_1)^2)^2(l_1^2)^2,

in the small l_2 region. The matrix γ_5 does not contribute, because the vectors 
p and l_1 must appear in even powers after the Feynman parametrization. 
Equation (<ref>) becomes

    J_1   ≈   24∫_0^1 du u(1-u)∫d^4l_1/(2π)^44u(1-u)p_α p_σ
    -g_ασ[l^2+2u(1-u)s]/[l_1^2-u(1-u)s]^4
    =i/π^2p_α p_σ/s^2,
 
where the portion proportional to g_ασ turns out to vanish. 

The third integral is computed as

    J_3   =   ∫d^4l_3/(2π)^4 tr[(p̸-l̸_3)
    (1-γ_5)γ_βl̸_3(1-γ_5)γ_λ]/(p-l_3)^2(l_3^2)^2(l_2-l_3)^2
       =   12∫_0^1 du∫_0^1-udv(1-u-v)i∫d^4l_3/(2π)^4
          ×2(up+vl_2)_λ[(1-u)p-vl_2]_β+2(up+vl_2)_β[(1-u)p-vl_2]_λ
    -g_λβ{l_3^2+2(up+vl_2)·[(1-u)p-vl_2]}/[l_3^2-u(1-u)s-v(1-v)l_2^2+2uvp· l_2]^4,

to which the Feynman parametrization and the variable change l_3→ l_3+up+vl_2 have
been implemented. Keeping the terms up to those linear in l_2, we get

    J_3   ≈   i/4π^2∫_0^1 du∫_0^1-udv(1-u-v)2u(1-u)p_λ p_β
    -(1-2u)vl_2λp_β+(1-2u)vp_λ l_2β+g_λβ(1-2u)vp· l_2/[u(1-u)s+v(1-v)l_2^2-2uvp· l_2]^2
       ≈   i/4π^2[2p_λ p_β/s^2lns/l_2^2
    +(g_λβp· l_2-l_2λp_β+p_λ l_2β)1/s l_2^2].

To reach the last expression, we have focused on the leading region with small u and v,
i.e., on the terms linear in u and v.

We then work out the integral over l_2 in the small l_2 region,

    J_2   =   i/4π^2∫d^4l_2/(2π)^4 tr[ (p̸-l̸_2)γ^λ(1-γ_5)
    l̸_2γ^σ(1-γ_5)]/(p-l_2)^2l_2^2[2p_λ p_β/s^2lns/l_2^2
    +(g_λβp· l_2-l_2λp_β+p_λ l_2β)1/s l_2^2]
       ≈   2i/π^2∫d^4l_2/(2π)^4p^λ l_2^σ
    +l_2^λ p^σ-g^λσl_2· p/s l_2^2[2p· l_2/s2p_λ p_β/s^2lns/l_2^2
    +(g_λβp· l_2-l_2λp_β+p_λ l_2β)1/s l_2^2],

where the expansion of the denominator (p-l_2)^2+iϵ≈ p^2-2p· l_2+iϵ
in powers of l_2 has been made. The above integral reduces to 

    J_2   ≈   2i/π^2∫d^4l_2/(2π)^41/s^2[4l_2^σ p· l_2 p_β/s l_2^2lns/l_2^2
    +s l_2^σ l_2β-l_2^2p^σ p_β+p· l_2 p^σ l_2β
    +p· l_2 l_2^σ p_β-g^σ_β (p· l_2)^2/(l_2^2)^2]
       ≈   3/64π^4p^σ p_β/s.

Note that the Wick rotation of the zeroth component of l_2 introduces -i,
instead of +i, since the pole is located in the first quadrant as indicated by the
denominator p^2-2p· l_2+iϵ in Eq. (<ref>). We have restricted the range of
the integration variable to 0<l_2^2<s/2 after the Wick rotation in accordance with 
the soft approximation.

At last, we group all the ingredients together, remove the lepton currents, and contract 
the remaining part by g^αβ/4, deriving 

    M≈2i/8^4m_ℒ^4 m_ℓ^4/v^8g^αβ/4i/π^2p_α p_σ/s^23/64π^4p^σ p_β/s
    =-3/128(1/16π^2)^3
    m_ℒ^4 m_ℓ^4/v^81/s.


The combination with the contribution from the diagram with horizontal scalar lines
then produces Eq. (<ref>).






99




Li:2023ncg
H. n. Li,

[arXiv:2306.03463 [hep-ph]].


Li:2023dqi
H. n. Li,

Phys. Rev. D 107, 094007 (2023).




Li:2023yay
H. n. Li,

Phys. Rev. D 108, no.5, 054020 (2023).



Chien:2018ohd
Y. T. Chien and H. n. Li,

Phys. Rev. D 97, 053006 (2018).



Huang:2020iya
L. Huang, S. D. Lane, I. M. Lewis and Z. Liu,

Phys. Rev. D 103, 053007 (2021).



Kaplan:1983fs
D. B. Kaplan and H. Georgi,

Phys. Lett. B 136, 183 (1984).

PDG
R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022).


Alvarado:2020lcz
J. S. Alvarado and R. Martinez,

[arXiv:2007.14519 [hep-ph]].

Xu:2023kfi
G. Xu and Y. Zhang,

EPL 143, 44001 (2023).



Patel:2023qtw
A. A. Patel and T. P. Singh,

Universe 9, 440 (2023).



Bora:2023teg
H. Bora, N. K. Francis, A. Barman and B. Thapa,

Phys. Lett. B 848, 138329 (2024).



Thapa:2023fxu
B. Thapa, S. Barman, S. Bora and N. K. Francis,

JHEP 11, 154 (2023).




Chung:2023rie
Y. Chung,

[arXiv:2311.17183 [hep-ph]].


Supanyo:2023jkh
S. Supanyo, C. Hasuwannakit, S. Yoo-Kong, L. Tannukij and M. Tanasittikosol,

[arXiv:2312.16587 [hep-ph]].

Lampe:2024gqo
B. Lampe,

PoS EPS-HEP2023, 373 (2024).




Shaikh:2024ufv
A. R. Shaikh and R. Adhikari,

[arXiv:2404.11570 [hep-ph]].


Bilenky:2002aw
S. M. Bilenky, C. Giunti, J. A. Grifols and E. Masso,

Phys. Rept. 379, 69 (2003).



see
P. Minkowski, Phys. Lett. 67B, 421 (1977);
M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C790927, 315 (1979);
T. Yanagida, Conf. Proc. C7902131, 95 (1979);
R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).

Nature
The KATRIN Collaboration, 

Nat. Phys. 18, 160 (2022). 











Mohapatra:2006gs
R. N. Mohapatra and A. Y. Smirnov,

Ann. Rev. Nucl. Part. Sci. 56, 569 (2006);



R. N. Mohapatra,

Nucl. Phys. B 908, 423 (2016).


Li:2022jxc
H. n. Li,

Phys. Rev. D 107, 054023 (2023).



Li:2020xrz 
H. n. Li, H. Umeeda, F. Xu and F. S. Yu,

Phys. Lett. B 810, 135802 (2020).

Cheng 
H. Y. Cheng, Phys. Rev. D 26, 143 (1982).

BSS
A. J. Buras, W. Slominski and H. Steger,
Nucl. Phys. B245, 369 (1984).


deSalas:2017kay
P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola and J. W. F. Valle,

Phys. Lett. B 782, 633 (2018).



CK84
L. L. Chau and W. Y. Keung, Phys. Rev. Lett. 53, 1802
(1984); L. Maiani, in Proceedings of the 1977 International Symposium on Lepton and 
Photon Interactions at High Energies (DESY, Hamburgh, 1977), p. 867.










