Dispersive determination of neutrino mass ordering

Article Content

Abstract

We argue that the mixing phenomenon of a neutral meson formed by a fictitious massive quark will disappear, if the electroweak symmetry of the Standard Model (SM) is restored at a high energy scale. This disappearance is taken as the high-energy input for the dispersion relation, which must be obeyed by the width difference between two meson mass eigenstates. The solution to the dispersion relation at low energy, i.e., in the symmetry broken phase, then connects the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements to the quark masses involved in the box diagrams responsible for meson mixing. It is demonstrated via the analysis of the D meson mixing that the typical d, s and b quark masses demand the CKM matrix elements in agreement with measured values. In particular, the known numerical relation

V_{u s} \approx \sqrt{m_{s} / m_{b}}

with the s (b) quark mass

m_{s}

(

m_{b}

) can be derived analytically from our solution. Next we apply the same formalism to the mixing of the

μ^{-} e^{+}

and

μ^{+} e^{-}

states through similar box diagrams with intermediate neutrino channels. It is shown that the neutrino masses in the normal hierarchy (NH), instead of in the inverted hierarchy or quasi-degenerate spectrum, match the observed Pontecorvo-Maki-Nakagawa-Sakata matrix elements. The lepton mixing angles larger than the quark ones are explained by means of the inequality

m_{2}^{2} / m_{3}^{2} ≫ m_{s}^{2} / m_{b}^{2}

m_{2, 3}

being the neutrino masses in the NH. At last, the solution for the

τ^{-} e^{+}

–

τ^{+} e^{-}

mixing specifies the mixing angle

θ_{23} \approx 45^{\circ}

, leading to the μ–τ reflection symmetry. Our work suggests that the fermion masses and mixing parameters are constrained dynamically, and the neutrino mass orderings can be discriminated by the internal consistency of the SM.

1. Introduction

It has been believed that the parameters in the Standard Model (SM), such as particle masses and mixing angles, are free, and have to be determined experimentally. These parameters originate from the independent elements of the Yukawa matrices, which cannot be completely constrained by symmetries [1], as the electroweak symmetry is broken. Any attempt to explain their values relies on an underlying new physics theory existent at high energy, whose low-energy behavior fixes the SM parameters. However, the above observation is made at the Lagrangian level without taking into account subtle dynamical interplay among the involved gauge and scalar sectors. We have pointed out in recent publications [2], [3] that dispersion relations, which physical observables must obey owing to analyticity, connect various interactions at different scales, and thus impose stringent constraints on the SM parameters. The SM parameters should satisfy dispersive constraints from all physical observables in principle. We have demonstrated, by considering those which provide efficient constraints [2], [3], that at least some of the SM parameters can be determined dynamically within the model itself.

A dispersion relation links the high- and low-energy properties of an observable, which is defined by a correlation function. The high-energy property, calculated perturbatively from the correlation function, is treated as an input. The low-energy property is then solved directly from the dispersion relation with the given input, which demands specific values for relevant particle masses in agreement with measured ones. We have analyzed heavy meson decay widths [2], written as absorptive pieces of hadronic matrix elements of four-quark effective operators, in this inverse-problem approach [4], [5], [6], [7]. Starting with massless final-state u and d quarks, we found that the solution for the decay

c \to d u \bar{d}

(

b \to c \bar{u} d

) with heavy-quark-expansion (HQE) inputs leads to the c (b) quark mass

m_{c} = 1.35

(

m_{b} = 4.0

) GeV. The requirement that the dispersion relation for the

c \to s u \bar{d}

(

c \to d μ^{+} ν_{μ}

b \to u τ^{-} {\bar{ν}}_{τ}

) decay yields the same heavy quark mass fixes the strange quark (muon, τ lepton) mass

m_{s} = 0.12

GeV (

m_{μ} = 0.11

GeV,

m_{τ} = 2.0

GeV). The investigation on the dispersion relation respected by the correlation function of two b-quark scalar (vector) currents, with the input from the perturbative evaluation of the b quark loop, returns the Higgs (Z) boson mass 114 (90.8) GeV [3]. A reasoning for why the analyticity can constrain the SM parameters through dispersion relations for physical observables has been provided in [3].

The successful explanation of the particle masses from 0.1 GeV up to the electroweak scale by means of the internal consistency of SM dynamics encourages us to address the fermion mixing in the same formalism. We first argue that the mixing phenomenon of a neutral meson formed by a fictitious massive quark will disappear, if the electroweak symmetry of the SM is restored at a high energy scale [8], [9]: internal particles involved in the box diagrams responsible for the mixing become massless in the symmetric phase, so the contributions from all intermediate channels cancel simply owing to the Glashow–Iliopoulos–Maiani mechanism [10]. There are new physics models in the literature, which provide a suitable framework for our discussion based on the restoration of the electroweak symmetry. For instance, the composite Higgs model proposed in [11] meets the purpose well. The electroweak group in their model is broken at a scale much lower than the condensate scale, implying the existence of a symmetry restoration scale which we refer to. The disappearance of the mixing phenomenon then takes place in the region above the restoration scale and below the condensate scale. A concrete example with the incorporation of a sequential fourth generation, that realizes the restoration of the electroweak symmetry at a high energy, can be located in [12]. We do not intend to elaborate the detail of this model here, but to establish the dispersive constraints that some SM parameters should satisfy, if the electroweak symmetry is restored.

The restoration of the electroweak symmetry is the only assumption required in our analysis. The disappearance of the mixing phenomenon is taken as the high-energy input for the dispersion relation satisfied by the width difference between two meson mass eigenstates. The solution to the dispersion relation at low energy, i.e., in the symmetry broken phase, effectively binds the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the quark masses appearing in the box diagrams. Compared to conventional applications of dispersion relations to processes involving strong interaction, the symmetric phase in the present setup corresponds to the perturbative region, where high-energy dynamics is treated as inputs, and the broken phase corresponds to the nonperturbative region, where low-energy behaviors are constrained. It will be elaborated how the typical d, s and b quark masses constrain the CKM matrix elements through the dispersive relation for the D meson mixing. The connection between the fermion flavor structure and the pattern of the Yukawa matrices, together with plausible relations among the quark masses and the mixing angles, have been speculated [13], [14]. For a recent reference in this direction based on Yukawa matrix textures, see [15]. We will derive the known empirical relation

V_{u s} \approx \sqrt{m_{s} / m_{b}}

[15] with the s (b) quark mass

m_{s}

(

m_{b}

) analytically from our solution. Namely, our work realizes the speculation in the literature, and suggests that its underlying theory is the SM itself.

We then perform the similar analysis of the mixing between the

μ^{-} e^{+}

and

μ^{+} e^{-}

states, which occurs via the box diagrams containing intermediate neutrino channels. The formulas are exactly the same as of the D meson, i.e.,

c \bar{u}

–

\bar{c} u

mixing, with the quark masses

m_{d, s, b}

being replaced by the neutrino masses

m_{1, 2, 3}

, and the CKM matrix elements by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) ones. It will be shown that the neutrino masses in the normal hierarchy (NH), instead of in the inverted hierarchy (IH) or quasi-degenerate (QD) spectrum, match the observed PMNS matrix elements. The neutrino mass ordering, whose various scenarios have not been discriminated experimentally, has remained as an unsettled issue in neutrino physics [16]. Our study provides a solid theoretical support for the NH spectrum in the viewpoint of the internal consistency of SM dynamics. The neutrino mixing angles larger than the quark ones are then accounted for naturally by the inequality

m_{2}^{2} / m_{3}^{2} ≫ m_{s}^{2} / m_{b}^{2}

for

m_{2, 3}

in the NH. We further examine the

τ^{-} e^{+}

–

τ^{+} e^{-}

mixing, and find that its solution requests the angle

θ_{23} \approx 45^{\circ}

in accordance with its measured value around the maximal mixing, explaining the μ–τ reflection symmetry [17]. It is emphasized that the above relations between the fermion masses and mixing angles are established without resorting to specific new ingredients beyond the SM (for recent endeavors on this topic, refer to [18], [19], [20], [21], [22]).

2. Formalism

Consider the mixing mechanism of a neutral meson formed by a fictitious massive quark Q in the SM. Precisely, we work on the mixing between the

Q_{L} {\bar{q}}_{L}

and

{\bar{Q}}_{L} q_{L}

states, where q is a light quark and the subscript L denotes the left handedness. The external states

Q_{L} {\bar{q}}_{L}

and

{\bar{Q}}_{L} q_{L}

are formed by the quarks in the broken phase, since they provide a large mass scale. The first emissions are thus composed of real neutral scalars or weak gauge bosons [23], and only two W boson are exchanged at leading order to induce the mixing. All internal particles become massless in the symmetric phase as the external states are heavy enough, and their contributions cancel owing to the unitarity of the CKM matrix. The vanishing of a mixing observable at high energy will be taken as an input in the dispersive analysis below. We mention an alternative setup for the same purpose,

Q_{L} {\bar{q}}_{L}

scattering into

{\bar{Q}}_{L} q_{L}

at arbitrary center-of-mass energy E. As E is high enough, all intermediate particles become massless, and the corresponding amplitude diminishes.

The dispersion relation for neutral meson mixing is quoted as [24](1)

M_{12} (s) = \frac{1}{2 π} \int d s^{'} \frac{Γ_{12} (s^{'})}{s - s^{'}},

where s is the mass squared of the quark Q [4], and the application of the principal-value prescription to the right-hand side is implicit. The proposed contour and the location of s have been described in Fig. 1 of Ref. [24]. In the above expression

M_{12} (s)

and

Γ_{12} (s)

represent the real and imaginary pieces of the box-diagram contribution, respectively, which governs the time evolution of the fictitious neutral meson. Their analytical properties can be inferred from the explicit expressions in Appendix A of Ref. [25]. Simply speaking, the box-diagram contribution has no poles but contains branch cuts along the real axis with the thresholds being specified below. The piece

Γ_{12}

is related to the width difference between the two meson mass eigenstates. As s, i.e., the scale involved in the box diagrams is large enough, the argument about the disappearance of the mixing phenomenon in the symmetric phase implies

M_{12} (s) \approx 0

. For a similar reason, the upper bound of the integration variable

s^{'}

in Eq. (1) can be set to the electroweak symmetry restoration scale.

The dispersive integral on the right-hand side of Eq. (1) receives the low-mass contribution from

Γ_{12}

, which depends on the CKM matrix elements associated with various massive intermediate quarks in the symmetry broken phase. It has been illustrated that the physical

Γ_{12}

with hadronic thresholds and the perturbative

Γ_{12}

from the box diagrams with quark-level thresholds give the same dispersive integral [24]. This equality has allowed us to solve for the physical

Γ_{12}

from the dispersion relation, which accommodates the observed large D meson mixing parameters. Here we adopt the perturbative

Γ_{12}

for the evaluation of the dispersive integral in Eq. (1). The box diagrams generate the

(V - A) (V - A)

and

(S - P) (S - P)

effective operators, which should be handled independently. We concentrate on the former contribution, which is expressed as [25], [26](2)

Γ_{12} (s) \propto \sum_{i, j} λ_{i} λ_{j} Γ_{i j} (s), Γ_{i j} (s) = \frac{1}{s^{2}} \frac{\sqrt{s^{2} - 2 s (m_{i}^{2} + m_{j}^{2}) + {(m_{i}^{2} - m_{j}^{2})}^{2}}}{(m_{W}^{2} - m_{i}^{2}) (m_{W}^{2} - m_{j}^{2})} \times {(m_{W}^{4} + \frac{m_{i}^{2} m_{j}^{2}}{4}) [2 s^{2} - 4 s (m_{i}^{2} + m_{j}^{2}) + 2 {(m_{i}^{2} - m_{j}^{2})}^{2}] + 3 m_{W}^{2} s (m_{i}^{2} + m_{j}^{2}) (m_{i}^{2} + m_{j}^{2} - s)},

with the W boson mass

m_{W}

and the intermediate quark masses

m_{i}

and

m_{j}

. The overall coefficient, including the bag parameter and the meson decay constant which are irrelevant to the reasoning below, has been suppressed. For the D meson mixing,

i, j = d, s, b

label the down-type quarks, and

λ_{i} \equiv V_{c i}^{⁎} V_{u i}

are the products of the CKM matrix elements. It will be verified that the same conclusion is drawn, when the analysis is performed based on the perturbative contribution from the

(S - P) (S - P)

operator.

It seems that the W bosons in the box diagrams for the

c \bar{u}

–

\bar{c} u

mixing can be integrated out. First, Eq. (2) has appeared in the dispersive determination of the top quark mass from the

t \bar{u}

–

\bar{t} u

mixing [3]. The implication on the present subject from this mixing will be discussed in Sec. 4, for which the W boson fields should not be integrated out. Moreover, the box-diagram contribution at high s is crucial for deriving the constraints on fermion masses and mixing angles as seen shortly, in which heavy gauge bosons ought to remain dynamical. It is thus appropriate to quote the formulas in [25], [26] directly with the

m_{W}

dependence being kept. Note that in our previous study on heavy meson decay widths [2], we focused on the explanation of the heavy quark masses below

m_{W}

, the region which prefers the employment of the effective theory with W boson fields being integrated out.

To diminish the dispersive integral in Eq. (1) for large s, some conditions must be met by the CKM matrix elements. We begin with the asymptotic behavior of Eq. (2)(3)

Γ_{i j} (s^{'}) \approx Γ_{i j}^{(1)} s^{'} + Γ_{i j}^{(0)} + \frac{Γ_{i j}^{(- 1)}}{s^{'}} + \dots,

with the coefficients(4)

Γ_{i j}^{(1)} = \frac{4 m_{W}^{4} - 6 m_{W}^{2} (m_{i}^{2} + m_{j}^{2}) + m_{i}^{2} m_{j}^{2}}{2 (m_{W}^{2} - m_{i}^{2}) (m_{W}^{2} - m_{j}^{2})}, Γ_{i j}^{(0)} = - \frac{3 (m_{i}^{2} + m_{j}^{2}) [4 m_{W}^{4} - 4 m_{W}^{2} (m_{i}^{2} + m_{j}^{2}) + m_{i}^{2} m_{j}^{2}]}{2 (m_{W}^{2} - m_{i}^{2}) (m_{W}^{2} - m_{j}^{2})}, Γ_{i j}^{(- 1)} = \frac{3 (m_{i}^{4} + m_{j}^{4}) [4 m_{W}^{4} - 2 m_{W}^{2} (m_{i}^{2} + m_{j}^{2}) + m_{i}^{2} m_{j}^{2}]}{2 (m_{W}^{2} - m_{i}^{2}) (m_{W}^{2} - m_{j}^{2})} .

Each term

Γ_{i j}^{(m)}

gives contributions scaling like

Λ^{2} / s

(m_{i}^{2} + m_{j}^{2}) Λ / s

, and

(m_{i}^{4} + m_{j}^{4}) \ln Λ / s

for

m = 1, 0

, and −1, respectively, to the dispersive integral in Eq. (1), where the variable Λ is of order of the restoration scale. Suppression on these contributions characterized by large Λ is necessary for making finite the dispersive integral, which can be achieved only by imposing(5)

\sum_{i, j} λ_{i} λ_{j} Γ_{i j}^{(m)} \approx 0, m = 1, 0, - 1,

in view of the variability of Λ. Because Λ is not infinite, the left-hand sides of the above relations need not vanish exactly, but to be tiny enough.

Once the conditions in Eq. (5) are fulfilled, we recast the dispersive integral into(6)

\int d s^{'} \frac{Γ_{12} (s^{'})}{s - s^{'}} \approx \frac{1}{s} \sum_{i, j} λ_{i} λ_{j} g_{i j},

with the factors(7)

g_{i j} \equiv \int_{t_{i j}}^{\infty} d s^{'} [Γ_{i j} (s^{'}) - Γ_{i j}^{(1)} s^{'} - Γ_{i j}^{(0)} - \frac{Γ_{i j}^{(- 1)}}{s^{'}}],

and the thresholds

t_{i j} = {(m_{i} + m_{j})}^{2}

. The approximation

1 / (s - s^{'}) \approx 1 / s

has been applied, which holds well for large s, since the integral receives contributions only from finite

s^{'}

. The integrand in the square brackets decreases like

1 / s^{' 2}

, so the upper bound of

s^{'}

in Eq. (7) can be extended to infinity safely. We place the final condition(8)

\sum_{i, j} λ_{i} λ_{j} g_{i j} \approx 0,

to ensure the almost nil dispersive integral. That is, the realization of Eqs. (5) and (8) establishes a solution to the dispersion relation in Eq. (1) at large s with

M_{12} (s) \approx 0

Some remarks are in order. One may wonder whether the divergent pieces in the dispersive integral in Eq. (1) can be removed by subtraction. As the subtraction is implemented by introducing one power of

s^{'} - s_{1}

into the denominator of the dispersive integral, a subtraction constant

M_{12} (s_{1})

appears,(9)

\frac{M_{12} (s) - M_{12} (s_{1})}{s_{1} - s} = \frac{1}{2 π} \int d s^{'} \frac{Γ_{12} (s^{'})}{(s - s^{'}) (s_{1} - s^{'})} .

s_{1}

is above Λ, like s, we will have

M_{12} (s) \approx 0

M_{12} (s_{1}) \approx 0

and

(s - s^{'}) (s_{1} - s^{'}) \approx s s_{1}

. The above expression then reduces to

\int d s^{'} Γ_{12} (s^{'}) \approx 0

, which produces the conditions identical to Eqs. (5) and (8). If

s_{1}

is below Λ, we will have

M_{12} (s) \approx 0

s_{1} - s \approx - s

and

s - s^{'} \approx s

. Equation (9) thus turns into the same form as Eq. (1) that we started with. It is more difficult to extract constraints from the dispersion relation with

s_{1} < Λ

, and solving such an integral equation is not our strategy. Namely, implementing the subtraction either leaves our formalism unchanged or complicates the analysis, and does not really tame the divergent behavior of the dispersive integral. Note that we stick to the leading-order accuracy in this work as the first attempt to extract the relations among fermion masses and mixing angles by means of analyticity. Higher-order QCD and electroweak corrections, such as those from the channels involving quark pairs through W boson decays, can be taken into account systematically in the future.

3. Quark masses and the CKM matrix

It is apparent that Eqs. (5) and (8) enforce the connections between the CKM matrix elements and the quark masses speculated in the literature, which will be confronted by the data below. With the unitarity of the CKM matrix, we rewrite these conditions for the D meson, i.e.,

c \bar{u}

–

\bar{c} u

mixing as(10)

r^{2} R_{d d}^{(m)} + 2 r R_{d s}^{(m)} + 1 \approx 0, m = 1, 0, - 1, i

with the ratios(11)

R_{d d}^{(m)} = \frac{Γ_{d d}^{(m)} - 2 Γ_{d b}^{(m)} + Γ_{b b}^{(m)}}{Γ_{s s}^{(m)} - 2 Γ_{s b}^{(m)} + Γ_{b b}^{(m)}}, R_{d s}^{(m)} = \frac{Γ_{d s}^{(m)} - Γ_{d b}^{(m)} - Γ_{s b}^{(m)} + Γ_{b b}^{(m)}}{Γ_{s s}^{(m)} - 2 Γ_{s b}^{(m)} + Γ_{b b}^{(m)}},

for

m = 1, 0, - 1

. The expression for

m = i

is similar with

g_{i j}

being substituted for

Γ_{i j}^{(m)}

in Eq. (11). We will encounter real W-boson production as the massive quark mass exceeds the thresholds

m_{W} + m_{i}

m_{W} + m_{j}

and

2 m_{W}

, whose effects are not taken into account in Eq. (2). However, these thresholds, much greater than the other scales in the box diagrams like

m_{b}

, are not expected to make an impact. For example, the region of

s^{'} > m_{W}^{2}

in Eq. (7) contributes only about

10^{- 4}

of the coefficients

R_{d d}^{(i)}

and

R_{d s}^{(i)}

The factor r is defined as the ratio of the CKM matrix elements,(12)

r = \frac{λ_{d}}{λ_{s}} = \frac{V_{c d}^{⁎} V_{u d}}{V_{c s}^{⁎} V_{u s}} \equiv u + i v,

where the real part

u \equiv Re (r)

and the imaginary part

v \equiv Im (r)

have been introduced. Equation (10) contains both the real and imaginary pieces, which can be treated separately. The imaginary pieces, simply proportional to v, do not provide nontrivial constraints. Therefore, we consider the real pieces, searching for the values of u and v that minimize the squares of these real pieces simultaneously, and then check whether the obtained u and v also diminish the imaginary pieces of the conditions. It is equivalent to eliminate the product

V_{c d}^{⁎} V_{u d}

using unitarity and to work on the ratio

V_{c b}^{⁎} V_{u b} / (V_{c s}^{⁎} V_{u s})

. We have corroborated that this option leads to the same conclusion within theoretical uncertainties.

To explain how the aforementioned minima is reached, we exhibit the dependencies of the real pieces

(u^{2} - v^{2}) R_{d d}^{(m)} + 2 u R_{d s}^{(m)} + 1

on u for

v = 0

in Fig. 1(a) with the inputs of the typical quark masses

m_{d} = 0.005

GeV,

m_{s} = 0.12

GeV, and

m_{b} = 4.0

GeV [2], and the W boson mass

m_{W} = 80.377

GeV [16]. The definition of the quark masses is not very relevant in the current leading-order formalism, and our point is to demonstrate that the correct ratio of the CKM matrix elements in Eq. (12) can be produced by the ballpark values of the quark masses. Our results mainly depend on the ratios of the quark masses as seen below, such that the renormalization-group evolution effects on the quark masses largely cancel. The distinction between the curves corresponding to

m = 0

and

m = - 1

is invisible, namely, the two conditions are equivalent basically. The two coefficients of the

r^{2}

and r terms for

m = 0

and

m = - 1

are almost identical up to corrections of

O (m_{s}^{4} / (m_{W}^{2} m_{b}^{2})) \sim 10^{- 9}

. As a contrast, the coefficients for

m = 1

differ from those for

m = - 1

O (m_{s}^{2} / m_{b}^{2}) \sim 10^{- 3}

, and from those for

m = i

10^{- 2}

. The three curves labeled by

m = 1, 0, - 1

intersect at the location on the horizontal axis,(13)

u = - \frac{(m_{W}^{2} - m_{d}^{2}) (m_{b}^{2} - m_{s}^{2})}{(m_{W}^{2} - m_{s}^{2}) (m_{b}^{2} - m_{d}^{2})}, v = 0,

where the three conditions

(u^{2} - v^{2}) R_{d d}^{(m)} + 2 u R_{d s}^{(m)} + 1 = 0

are satisfied exactly. The steeper curve from the

m = i

condition favors the intersection at a more negative u, and drags the solution to

u \approx - 1

. As v increases, the

m = 1, 0, - 1

curves move downward, while the

m = i

one is relatively stable.

Below we drop the

m = 0

condition, which is equivalent to the

m = - 1

one, and minimize the sum(14)

\sum_{m = 1, - 1, i} {[(u^{2} - v^{2}) R_{d d}^{(m)} + 2 u R_{d s}^{(m)} + 1]}^{2},

by varying the unknowns u and v. It is legitimate to define this sum, because all terms in the above expression have been made dimensionless by taking the ratios. Note that

Γ_{i j}^{(m)}

have different dimensions for different m, as indicated in Eq. (4). It is easy to find that the above sum decreases with v, and arrives at its minimum, when the intersection of the

m = i

curve with the horizontal axis goes between the

m = 1

and

m = - 1

curves as displayed in Fig. 1(b). The numerical study coincides with this picture, yielding

r = - 1.0 + 0.00062 i

. Simply speaking, the value of u (v) is mainly determined by the

m = i

condition (

m = 1, - 1

conditions). The sign of v cannot be fixed, for this unknown appears as

v^{2}

in Eq. (14). We pick up the plus sign for comparison with the data, and defer the elaboration on the choice of the opposite sign to the next section.

We estimate the theoretical uncertainties associated with the result of the ratio r. The subtraction terms in Eq. (7) just need to cancel the large

s^{'}

contribution, so the lower bounds of their integrations are allowed to vary from

t_{i j}

. We increase the lower bounds to

10 t_{i j}

gradually for the three subtraction terms, and observe that the real part u is not altered, and the imaginary part v changes by only 3%, within 0.00060 and 0.00064. The results of r are insensitive to the d quark mass

m_{d}

, which can approach zero in fact, and depend on

m_{s}

and

m_{b}

through their ratio

m_{s} / m_{b}

. The minimization always returns the value

u \approx - 1

with the uncertainty at

10^{- 5}

level under the variation of the quark masses, so we scrutinize only the dependence of v on

m_{s}

. It is found that v increases with

m_{s}

, taking the value 0.00052 (0.00074) for

m_{s} = 0.11

(0.13) GeV. Hence, we summarize our prediction as(15)

r = \frac{V_{c d}^{⁎} V_{u d}}{V_{c s}^{⁎} V_{u s}} = - 1.0 + ({6.2}_{- 1.0}^{+ 1.2}) \times 10^{- 4} i .

Inserting the central value of r into the

m = 1

condition, we get(16)

r^{2} R_{d d}^{(1)} + 2 r R_{d s}^{(1)} + 1 = (4.04 - 10.96 i) \times 10^{- 7} .

It indicates that the minimization of the real pieces in Eq. (10) also guarantees the smallness of the imaginary pieces relative to the constant unity on the left-hand side, as claimed before.

We repeat the dispersive analysis based on the box-diagram contribution associated with the

(S - P) (S - P)

operator [25], [26],(17)

Γ_{i j}^{(S - P)} (s) = \frac{1}{s^{2}} \frac{\sqrt{s^{2} - 2 s (m_{i}^{2} + m_{j}^{2}) + {(m_{i}^{2} - m_{j}^{2})}^{2}}}{(m_{W}^{2} - m_{i}^{2}) (m_{W}^{2} - m_{j}^{2})} \times (m_{W}^{4} + \frac{m_{i}^{2} m_{j}^{2}}{4}) [s^{2} + s (m_{i}^{2} + m_{j}^{2}) - 2 {(m_{i}^{2} - m_{j}^{2})}^{2}],

where the overall coefficient has been also suppressed. The three terms in its asymptotic expansion differ from those in Eq. (4). In particular, the constant term vanishes, such that the numerical handling in this case is consistent with the ignoring of the

m = 0

condition in the

(V - A) (V - A)

case. The minimization of the sum over the

m = 1, - 1, i

conditions gives

r = - 1.0 + 0.00062 i

, identical to Eq. (15) from the

(V - A) (V - A)

contribution. We have to present the values up to three digits in order to reveal the distinction, i.e.,

v = 0.000617

from

(V - A) (V - A)

and

v = 0.000616

from

(S - P) (S - P)

. The above examination confirms the consistency of our approach.

The CKM matrix is written, in the Chau-Keung (CK) parametrization [27], as(18)

V_{CKM} = (\begin{matrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{- i δ} \\ - s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i δ} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i δ} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i δ} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i δ} & c_{23} c_{13} \end{matrix}) .

Given the sines of the mixing angles

s_{12} \equiv \sin θ_{12} = 0.22500 \pm 0.00067

s_{13} \equiv \sin θ_{13} = 0.00369 \pm 0.00011

, and

s_{23} \equiv \sin θ_{23} = {0.04182}_{- 0.00074}^{+ 0.00085}

, the CP phase

δ = 1.144 \pm 0.027

[16], and the corresponding

c_{12} \equiv \cos θ_{12}

c_{13} \equiv \cos θ_{13}

, and

c_{23} \equiv \cos θ_{23}

, we have the ratio r extracted from data with(19)

u = - 1.00029 \pm 0.00002, v = 0.00064 \pm 0.00002 .

The dominant error of u (v) arises from the uncertainty of δ (

θ_{13}

). It is obvious that our determination in Eq. (15) from the typical quark masses agrees with the measured values.

Equation (16) implies that the conditions in Eq. (10) are respected to a good accuracy, as the minimum is located. We can then solve for the analytical expression of v by inserting

u = - 1

corresponding to the minimum into the

m = 1

condition,(20)

v \approx \frac{(m_{W}^{2} - m_{b}^{2}) (m_{s}^{2} - m_{d}^{2})}{(m_{W}^{2} - m_{s}^{2}) (m_{b}^{2} - m_{d}^{2})} \approx \frac{m_{s}^{2}}{m_{b}^{2}},

where the approximation is valid for the large

m_{W}

and small

m_{d}

. The expansion of the ratio

V_{c d}^{⁎} V_{u d} / (V_{c s}^{⁎} V_{u s})

in the Wolfenstein parameters λ, A, ρ and η up to

λ^{4}

[28] leads to

v = A^{2} λ^{4} η

. Equating this expression to Eq. (20), we derive the known numerical relation(21)

λ = V_{u s} \approx {(A^{2} η)}^{- 1 / 4} \sqrt{\frac{m_{s}}{m_{b}}} \approx \sqrt{\frac{m_{s}}{m_{b}}},

with

{(A^{2} η)}^{- 1 / 4} \approx 1.43 \sim O (1)

for

A \approx 0.826

and

η \approx 0.348

[16]. The above relation manifests the observation that our numerical outcomes of r mainly depend on the mass ratio

m_{s} / m_{b}

Another intriguing remark is stimulated by the application of the formalism to the case with only two generations of quarks, for which the width difference between the two meson mass eigenstates has a simple form(22)

Γ_{12} (s) \propto λ_{d}^{2} [Γ_{d d} (s) - 2 Γ_{d s} (s) + Γ_{s s} (s)] .

The dispersive analysis on heavy meson lifetimes [2] has shown that the masses of the d and s quarks cannot be degenerate. The dispersion relation in Eq. (1) with

M_{12} (s) \approx 0

at large s is realized only when

λ_{d}

diminishes, namely, only when the quark mixing tends to be absent. In other words, there should exist at least three generations of fermions in order to facilitate sizable mixing among them in the SM.

4. Neutrino mass orderings

The formulas for the

c \bar{u}

–

\bar{c} u

mixing constructed in the previous section apply to the lepton mixing straightforwardly. It is natural to investigate the mixing between the

μ^{-} e^{+}

and

μ^{+} e^{-}

states, which occurs via the same box diagrams but with intermediate neutrino channels. Therefore, the dispersive constraints similar to those on the quark masses and mixing also appear in the lepton sector. The sensitivity of the mixing angles to the mass ratio

m_{s} / m_{b}

hints that it is possible to determine the neutrino mass ratio, namely, to discriminate neutrino mass orderings by means of the PMNS matrix elements. All the steps follow with the correspondence between the quark masses

m_{d, s, b}

and the neutrino masses

m_{1, 2, 3}

, and between the ratio in Eq. (12) and the ratio of the PMNS matrix elements

r = U_{μ 1}^{⁎} U_{e 1} / (U_{μ 2}^{⁎} U_{e 2})

. Here we have assumed that neutrinos are of the Dirac type. The condition labeled by

m = 0

, equivalent to the

m = - 1

condition, can be dropped. Viewing the tiny neutrino masses, we expect more serious theoretical uncertainties inherent in the framework, and aim at order-of-magnitude estimates. The global fits of data from various groups have produced the consistent parameters involved in the PMNS matrix [29], [30], [31]. We will illustrate the numerical analysis by adopting those obtained in [30].

For the neutrino masses in the NH, we take the mass squared differences

Δ m_{21}^{2} \equiv m_{2}^{2} - m_{1}^{2} = ({7.55}_{- 0.16}^{+ 0.20}) \times 10^{- 5}

eV² and

Δ m_{32}^{2} \equiv m_{3}^{2} - m_{2}^{2} = (2.424 \pm 0.03) \times 10^{- 3}

eV² [30]. As noticed before, our results are insensitive to the lightest neutrino mass, so we choose a small

m_{1}^{2} = 10^{- 6}

eV². The values of

m_{2}^{2}

and

m_{3}^{2}

are then expressed using

Δ m_{21}^{2}

and

Δ m_{32}^{2}

. We minimize the sum of the squared deviations in Eq. (14), deriving the ratio of the PMNS matrix elements(23)

r = \frac{U_{μ 1}^{⁎} U_{e 1}}{U_{μ 2}^{⁎} U_{e 2}} \approx - 1.0 - 0.02 i,

to which the variations of

Δ m_{21}^{2}

and

Δ m_{32}^{2}

cause only 2% effects. We have selected the minus sign for the value of

Im (r)

as making comparison with the data. Viewing the larger

Im (r)

, we check the

m = 1

condition in a way similar to Eq. (16), and find that it deviates from zero at

10^{- 4}

–

10^{- 3}

level. Relative to the constant unity on the left-hand side, this deviation is acceptable.

The PMNS matrix is parametrized in the same form as Eq. (18). The mixing angles

θ_{12} = {({34.5}_{- 1.0}^{+ 1.2})}^{\circ}

θ_{13} = {({8.45}_{- 0.14}^{+ 0.16})}^{\circ}

, and

θ_{23} = {({47.7}_{- 1.7}^{+ 1.2})}^{\circ}

, and the CP phase

δ = {(218_{- 27}^{+ 38})}^{\circ}

from [30] yield the measured ratio(24)

r = - ({0.738}_{- 0.048}^{+ 0.050}) - ({0.179}_{- 0.125}^{+ 0.136}) i,

where the errors mostly come from the variation of δ. The set of fit parameters

θ_{12} = {({33.46}_{- 0.88}^{+ 0.87})}^{\circ}

θ_{13} = {({8.41}_{- 0.14}^{+ 0.18})}^{\circ}

θ_{23} = {({47.9}_{- 4.0}^{+ 1.1})}^{\circ}

, and

δ = {(238_{- 33}^{+ 41})}^{\circ}

from another group [31] leads to

r = - ({0.801}_{- 0.097}^{+ 0.219}) - ({0.265}_{- 0.145}^{+ 0.090}) i

, which overlaps with Eq. (24). The real part

Re (r)

and the lower bound of the imaginary part

Im (r)

in Eq. (24) are in the same order of magnitude as our prediction in Eq. (23). That is, the dispersive constraints hold at order-of-magnitude level in the NH case.

We employ

Δ m_{21}^{2} = ({7.55}_{- 0.16}^{+ 0.20}) \times 10^{- 5}

eV² and

Δ m_{32}^{2} = (- {2.50}_{- 0.03}^{+ 0.04}) \times 10^{- 3}

eV² for determining the ratio r in the IH case [30]. The mass of the lightest neutrino is set to

m_{3}^{2} = 10^{- 6}

eV², and

m_{1}^{2}

and

m_{2}^{2}

are retrieved from

Δ m_{21}^{2}

and

Δ m_{32}^{2}

accordingly. We predict(25)

r \approx - 1.0 - O (10^{- 5}) i,

whose real part is stable against the variations of the mass squared differences. The diminishing imaginary part, always maintaining below

10^{- 5}

, differs dramatically from the observed ratio(26)

r = - ({1.03}_{- 0.16}^{+ 0.05}) - ({0.356}_{- 0.048}^{+ 0.015}) i,

inferred by

θ_{12} = {({34.5}_{- 1.0}^{+ 1.2})}^{\circ}

θ_{13} = {({8.53}_{- 0.15}^{+ 0.14})}^{\circ}

θ_{13} = {({47.9}_{- 1.7}^{+ 1.0})}^{\circ}

, and

δ = {(281_{- 27}^{+ 23})}^{\circ}

[30]. The variations of all fit parameters contribute some portions of the errors in Eq. (26). One can make plots similar to Fig. 1, which display the dependence of each term in Eq. (14) on the real part

Re (r) \equiv u

. It is observed, for the IH, that the

m = i

curve becomes close to the

m = 1

one. Namely, we need not increase the imaginary part

Im (r) \equiv v

to minimize Eq. (14); it has reached the minimum with

Im (r) \approx 0

already.

We conclude that the IH spectrum and the corresponding PMNS matrix elements do not obey the dispersive constraints because of the apparent disagreement of

Im (r)

between Eqs. (25) and (26) even after the experimental errors are considered. The conclusion should be robust, for the inclusion of subleading electroweak corrections to the box diagrams is unlikely to change the order of magnitude of our prediction in Eq. (25). Indeed, the inverted ordering is disfavored by larger

Δ χ^{2}

of global fits as stated in [16]. Nevertheless, the closeness of the measured ratios for the NH and IH indicates that it is still challenging to discriminate these two spectra experimentally. It is thus encouraging that such discrimination can be achieved theoretically in our formalism. Since the extraction of the CP phase δ is more sensitive to the neutrino mass orderings,

δ \sim 220^{\circ}

from the NH vs

δ \sim 280^{\circ}

from the IH [16], our observation also helps pin down the value of δ.

We then test the consistency of the QD spectrum under the dispersive constraints. Taking into account the bound on the neutrino mass sum

\sum m_{ν} < 0.12

eV [16] at order of magnitude, we assign a sizable value

m_{1}^{2} = 0.01

eV² arbitrarily, and write

m_{2}^{2}

and

m_{3}^{2}

by means of

Δ m_{21}^{2}

and

Δ m_{32}^{2}

in the NH. Other choices of large

m_{1}^{2}

give the same conclusion. The minimization of the sum over the squared deviations returns the ratio(27)

r \approx - 0.97 - O (10^{- 5}) i,

whose tiny imaginary part does not fit the general feature of the measured PMNS matrix elements with

Im (r) \sim 10^{- 2}

–

10^{- 1}

. The above result can be also visualized through plots similar to Fig. 1. Since only the NH scenario passes our dispersive constraints, we examine the influence from increasing the lowest mass

m_{1}

in the NH. It reduces

Im (r)

as expected, because a larger

m_{1}^{2}

makes the NH ordering closer to the QD spectrum. As

m_{1}^{2}

reaches

1.4 \times 10^{- 5}

eV²,

Im (r)

is lowered to

10^{- 3}

, which differs from the observed value significantly. This

m_{1}^{2}

for the NH, together with

Δ m_{21}^{2}

and

Δ m_{32}^{2}

, sets an upper bound of the neutrino mass sum(28)

\sum m_{ν} < 0.082 eV,

which is a bit tighter than the bound in [16].

We are ready to elucidate the different mixing patterns between the quark and lepton sectors with the solutions at hand. The real parts of r in Eqs. (23) and (24) do not differ from −1 much, so we insert

Re (r) = - 1

into the

m = 1

condition, solving for the approximate expression of

Im (r)

, which is the same as Eq. (20) but with the replacement of

m_{s}

(

m_{b}

) by

m_{2}

(

m_{3}

). It is trivial to get, from the CK parametrization in Eq. (18) which applies to both the CKM and PMNS matrices,(29)

Im (r) \propto \frac{s_{13} s_{23}}{s_{12}} .

Here only the sines of the mixing angles are highlighted. It is clear that the much larger

θ_{13}

and

θ_{23}

in the lepton sector than in the quark sector trace back to the inequality of the mass ratios,(30)

\frac{m_{2}^{2}}{m_{3}^{2}} \approx 3.1 \times 10^{- 2} ≫ \frac{m_{s}^{2}}{m_{b}^{2}} \approx 9.0 \times 10^{- 4},

where

m_{2}^{2} / m_{3}^{2}

is evaluated according to the NH spectrum.

At last, we discuss the dispersive constraints originating from the mixing between the

τ^{-} e^{+}

and

τ^{+} e^{-}

states, which corresponds to the

t \bar{u}

–

\bar{t} u

mixing in the quark sector. It is apparent that the conditions the fermion masses and mixing parameters have to meet are exactly the same as in the

μ^{-} e^{+}

–

μ^{+} e^{-}

c \bar{u}

–

\bar{c} u

mixing owing to the identical intermediate channels in the box diagrams. We remind that the W-boson thresholds should be included in the

t \bar{u}

–

\bar{t} u

mixing, which, however, do not modify the following argument. There are only two possible nontrivial outcomes other than those presented in the previous sections. First, the products of the mixing matrix elements

λ_{i} λ_{j}

must be small, such that the conditions in Eqs. (5) and (8) hold automatically. This is the case happening to the quark sector with the small mixing angles: for instance, we have

| V_{t s}^{⁎} V_{u s} |^{2} = A^{2} λ^{6} \approx 9 \times 10^{- 5}

with the Wolfenstein parameter

λ \approx 0.225

, lower than

| V_{c s}^{⁎} V_{u s} |^{2} = λ^{2} \approx 5 \times 10^{- 2}

by three orders of magnitude. Second, the minimization for the same conditions selects

Im (r)

of the opposite sign, which, as elaborated shortly, occurs to the lepton sector with the large mixing angles. The observed ratios

U_{τ 1}^{⁎} U_{e 1} / (U_{τ 2}^{⁎} U_{e 2}) = - ({1.231}_{- 0.186}^{+ 0.078}) + ({0.204}_{- 0.138}^{+ 0.085}) i

from [30] and

- ({1.139}_{- 0.207}^{+ 0.139}) + ({0.266}_{- 0.124}^{+ 0.050}) i

from [31] conform to our postulation approximately, as they are compared with the corresponding ratios

U_{μ 1}^{⁎} U_{e 1} / (U_{μ 2}^{⁎} U_{e 2})

in and below Eq. (24).

To understand how the two ratios of the PMNS matrix elements are correlated, we inspect their explicit expressions in the CK parametrization(31)

\frac{U_{μ 1}^{⁎} U_{e 1}}{U_{μ 2}^{⁎} U_{e 2}} = - \frac{c_{12}}{s_{12}} \frac{c_{12} s_{12} (c_{23}^{2} - s_{13}^{2} s_{23}^{2}) + c_{23} s_{13} s_{23} c_{δ} (c_{12}^{2} - s_{12}^{2}) - c_{23} s_{13} s_{23} s_{δ} i}{{(c_{12} c_{23} - s_{12} s_{13} s_{23})}^{2} + 2 c_{12} c_{23} s_{12} s_{13} s_{23} (1 - c_{δ})},

(32)

\frac{U_{τ 1}^{⁎} U_{e 1}}{U_{τ 2}^{⁎} U_{e 2}} = - \frac{c_{12}}{s_{12}} \frac{c_{12} s_{12} (s_{23}^{2} - c_{23}^{2} s_{13}^{2}) - c_{23} s_{13} s_{23} c_{δ} (c_{12}^{2} - s_{12}^{2}) + c_{23} s_{13} s_{23} s_{δ} i}{{(c_{12} s_{23} + c_{23} s_{12} s_{13})}^{2} - 2 c_{12} c_{23} s_{12} s_{13} s_{23} (1 - c_{δ})},

with

c_{δ} \equiv \cos δ

and

s_{δ} \equiv \sin δ

. Note that the imaginary part of Eq. (31) is a complete expression of Eq. (29). Our solution that Eqs. (31) and (32) differ only by the signs of their imaginary parts necessitates the rough equalities of the denominators and of the real pieces in the numerators, which yield(33)

(c_{12}^{2} - s_{12}^{2} s_{13}^{2}) (c_{23}^{2} - s_{23}^{2}) - 4 c_{12} c_{23} s_{12} s_{13} s_{23} c_{δ} \approx 0,

(34)

c_{12} s_{12} (1 + s_{13}^{2}) (c_{23}^{2} - s_{23}^{2}) + 2 (c_{12}^{2} - s_{12}^{2}) c_{23} s_{13} s_{23} c_{δ} \approx 0,

respectively. The combination of the above two relations, resulting in

(c_{12}^{2} + s_{12}^{2} s_{13}^{2}) (c_{23}^{2} - s_{23}^{2}) \approx 0

, thus requires

c_{23} \approx s_{23}

, i.e.,

θ_{23} \approx 45^{\circ}

in accordance with the observed

θ_{23}

around the maximal mixing. The μ–τ reflection symmetry [17] is thus realized trivially. It is also seen that both Eqs. (33) and (34) can be fulfilled by small

s_{13}

with

θ_{23} \approx 45^{\circ}

. We mention that new physics effects can deviate

θ_{23}

from

45^{\circ}

. For instance, the inclusion of a sequential fourth generation in the similar analysis, i.e., the extension of the PMNS matrix to a

4 \times 4

one gives rise to

θ_{23} \approx 47^{\circ}

in the second octant [32].

5. Conclusion

We have deduced the constraints on the fermion masses and mixing parameters from the dispersion relations obeyed by the box-diagram contributions to the mixing of two neutral states. These dispersion relations connect the behaviors of neutral state mixing before and after the electroweak symmetry breaking. They are solved with the inputs from the disappearance of the mixing phenomenon at high energy, where the electroweak symmetry is restored. The establishment of the solutions demands several conditions, which the fermion masses and mixing parameters at low energy, i.e., in the symmetry broken phase must satisfy. Taking the D meson, i.e.,

c \bar{u}

–

\bar{c} u

mixing as an example, we have demonstrated that the typical d, s and b quark masses involved in the box diagrams specify the ratio of the CKM matrix elements

V_{c d}^{⁎} V_{u d} / (V_{c s}^{⁎} V_{u s})

in agreement with the measured value. Moreover, the imaginary part of the above ratio, as solved analytically, generates the known numerical relation

V_{u s} \approx \sqrt{m_{s} / m_{b}}

. These results provide a convincing support to our formalism, which can be refined by including subleading corrections to the box diagrams systematically. The constraints obtained in the present paper are expected to be modified by these subleading corrections, which may improve the consistency between the current predictions and the data, or lead to more precise predictions.

Repeating the same analysis on the

μ^{-} e^{+}

–

μ^{+} e^{-}

mixing and the

τ^{-} e^{+}

–

τ^{+} e^{-}

mixing, which take place via the box diagrams with intermediate neutrino channels, we have shown that the neutrino masses in the NH match the observed PMNS matrix elements at order-of-magnitude level. The orderings in the IH and QD spectra generate the imaginary parts of the ratio

U_{μ 1}^{⁎} U_{e 1} / (U_{μ 2}^{⁎} U_{e 2})

, which are unequivocally too low compared with those from global fits of the data. The leptonic CP phase δ is then likely to be in the third quadrant in favor of the NH scenario. The analytical solution for the imaginary part of

U_{μ 1}^{⁎} U_{e 1} / (U_{μ 2}^{⁎} U_{e 2})

explains the larger lepton mixing angles relative to the quark ones via the inequality

m_{2}^{2} / m_{3}^{2} ≫ m_{s}^{2} / m_{b}^{2}

for

m_{2, 3}

in the NH. Our observation that the ratios

U_{μ 1}^{⁎} U_{e 1} / (U_{μ 2}^{⁎} U_{e 2})

and

U_{τ 1}^{⁎} U_{e 1} / (U_{τ 2}^{⁎} U_{e 2})

differ only by the sign of their imaginary parts requests the maximal mixing

θ_{23} \approx 45^{\circ}

, i.e., the μ–τ reflection symmetry. The above summarizes the implications from our dispersive analysis on those unresolved issues in neutrino physics.

Combining the previous works, we conjecture that part of the flavor structures in the SM, such as the particle masses from 0.1 GeV up to the electroweak scale and the distinct quark and lepton mixing patterns, are understood through the internal consistency of SM dynamics. In other words, the scalar sector of the SM may not be completely free, but arranged properly to achieve the dynamical consistency. To maintain this attractive feature, a natural extension of the SM is to include the sequential fourth generation of fermions, since the associated parameters in the scalar sector, i.e., their masses and mixing with lighter generations can be predicted unambiguously in our formalism. The predictions for the masses

m_{b^{'}} \approx 2.7

TeV and

m_{t^{'}} \approx 200

TeV of the sequential fourth generation quarks

b^{'}

and

t^{'}

, respectively, are referred to Ref. [33]. We believe that this research direction is worth of further exploration.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

We thank Y.T. Chien, T.W. Chiu, A. Fedynitch, B.L. Hu, Y.H. Lin, M.R. Wu, and T.C. Yuan for fruitful 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.

Data availability

No data was used for the research described in the article.

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Article Content

Abstract

1. Introduction

2. Formalism

3. Quark masses and the CKM matrix

4. Neutrino mass orderings

5. Conclusion

Declaration of Competing Interest

Acknowledgement

Data availability

References