Charged Lepton Mass Spectrum and Supersymmetric Yukawaon Model

On the basis of the so-called supersymmetric yukawaon model, it is investigated what form of the superpotential W can lead to the observed charged lepton mass spectrum. A simple form of W can speculate reasonable values of K(\mu)=(m_e +m_\mu + m_\tau)/(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2 and \kappa(\mu)=\sqrt{m_e m_\mu m_\tau}/ (\sqrt{m_e} + \sqrt{m_\mu}+ \sqrt{m_\tau})^3 at an energy scale \mu=\Lambda.


Introduction
If quarks and leptons are fundamental entities in the nature, their mass spectra will obey laws of the nature which have simple forms. Especially, since the charged lepton masses are almost independent of gluon corrections, it will be comparatively easy to identify such laws of the nature. In reference to this problem, for example, we know that an empirical formula [1] m e + m µ + m τ = 2 3 ( √ m e + √ m µ + √ m τ ) 2 , (1.1) is well-satisfied by the observed charged lepton masses ("pole" mass values). However, usually, masses which we deal with in a conventional mass matrix model are "running" masses m i (µ), not pole masses m pole i . It is also well known that the running masses do not satisfy the relation (1.1). In this paper, we will consider that the relation (1.1) is merely an approximate relation even for the pole masses, so that we will not attempt to derive the relation (1.1) as an exact relation. Instead, we will discuss running masses m i (µ) at a high energy scale µ = Λ. It is beyond the scope of the present work to explain validity of the mass formula (1.1) with the current experimental accuracy of order 10 −5 ; rather we aim at accuracy of order ε ∼ 10 −3 .
In the present paper, we investigate the charged lepton mass spectrum based on the running masses. Then, for investigating the mass spectrum, it is useful to use the following two quantities under the one-loop approximation [3]. When we denote K(µ) and κ(µ) as K(µ) = K pole (1 + ε K (µ)), κ(µ) = κ pole (1 + ε κ (µ)), (1.5) where K pole and κ pole are defined by pole masses correspondingly to K(µ) and κ(µ), respectively, we obtain the deviations ε K (µ) and ε κ (µ) as follows: and the values ε K (µ) and ε κ (µ) are independent of µ up to the order of α. The numerical results are as follows: However, the values (1.10) and (1.11) are pure electromagnetic corrections based on Eq.(1.4), and there are corrections from another diagrams, which are dependent on models and slightly show µ-dependence. We show a typical case in a SUSY model with tan β = 10 in Table 1, where the input charged lepton mass values are quoted from Ref. [4]. As seen in Table 1, the quantities K(µ) and κ(µ) are almost insensitive to the energy scale µ. From the results of Table 1, we take (1.12) as values which should be explained in this paper. In reference to this topic, recently, Sumino [5] has proposed an interesting idea that a flavor gauge symmetry induces an effect which exactly cancels the deviation of ε K (µ) at any energy scale µ. However, differently from Sumino's idea, in this paper, we adopt the standpoint that K pole = 2/3 is accidental. We consider that our goal is not K pole = 2/3, but K(Λ) = 2 3 (1 + ε(Λ)) with ε(Λ) ∼ 10 −3 from the beginning. (ε K (µ) is deviation of K(µ) from K pole due to running masses, while ε(Λ) is deviation of K(Λ) from the value 2/3 in the present model.) In this paper, we try to understand the values (1.12) from a new approach to a mass matrix model, the so-called "yukawaon model" [6,7]: We regard the Yukawa coupling constants Y f as "effective" coupling constants Y ef f f in an effective theory, and we consider that Y ef f f originate in vacuum expectation values (VEVs) of new gauge singlet scalars Y f , i.e.
where Λ is a scale of the effective theory which is valid at µ ≤ Λ, and we assume Y f ∼ Λ. We refer the fields Y f as "yukawaons" hereafter. Note that the effective coupling constants Y ef f f evolve as in the standard SUSY model below the scale Λ, since a flavor symmetry is completely broken at the high energy scale µ = Λ (but the supersymmetry is still unbroken at Λ > µ > Λ SU SY ∼ 10 3 GeV).
In the yukawaon model, the Higgs scalars are the same as ones in a conventional model, i.e. we consider only two Higgs scalars H u and H d as an origin of the masses (not as an origin of the mass spectra). For example, we assume an O(3) flavor symmetry [8] and we consider that the yukawaons Y f are (3 × 3) S = 1 + 5 of O(3) F . Then, the would-be Yukawa interactions are given by where q and ℓ are SU(2) L doublet fields, and f c (f = u, d, e, ν) are SU(2) L singlet fields. Here, in order to distinguish each Y f from others, we have assigned The VEVs of yukawaons are obtained from supersymmetric (SUSY) vacuum conditions for a superpotential W . In the charged lepton sector, we assume where the fields Φ e and Θ e have the U(1) X charges 1 2 x e and −x e , respectively. Here and hereafter, for convenience, we denote a trace Tr[A] of a matrix A as [A] concisely. A SUSY vacuum condition ∂W/∂Θ e = 0 leads to a bilinear mass relation (1.16) Therefore, the mass spectrum of the charged leptons are given by and We refer the field Φ e as an "ur-yukawaon". A case with K = 2/3 (an earlier attempt to give K = 2/3 has been done in Ref. [9]) is known as the charged lepton mass formula (1.1) which is excellently satisfied by the observed charged lepton masses (pole masses). However, note that masses which we deal with in the present paper are running masses, so that our goal is not The term W Φ in the superpotential (1.15) has been introduced in order to fix a VEV spectrum of Φ e . The explicit form is given in the next section. The purpose of the present paper is heuristically to find a form of W Φ which can simultaneously give reasonable values of K(Λ) and κ(Λ), and it is not to investigate an origin of such the form of W Φ . The investigation of the theoretical origin will be our next step of the investigations.

VEV structure of the ur-yukawaon
In this paper, we consider a model with explicit U(1) X symmetry breakings of the order ε SB which is negligibly small. A prototype of such a model has been discussed in Ref. [2]. We know that, in order to give K = 2/3 , it is essential to bring interaction terms with a traceless partΦ e ≡ Φ e − 1 3 [Φ e ] into the potential [7]. In the present model, we also consider a more general form of W Φ , (2.2) (Hereafter, we will drop the index "e", because we only deal with fields in the charged lepton sector in this paper.) The expression (2.1) is the most general form of ΦΦY and ΦΦΦ terms, because we know the following identical equations: and an equation with Y → Φ in Eq.(2.3). We consider that the parameters λ e , λ A , λ B , µ e and ε SB in the superpotential are "analogue" parameters which we can adjust continuously, while we assume that α i (Λ) and β i (Λ) should be integers. We consider that the relations for K(Λ) and κ(Λ) are independent of these analogue parameters, and those should be described only by the "digital" parameters α i and β i . Hereafter, we will simply denote K(Λ), κ(Λ), . . . , as K, κ, · · · , because we discuss only the relations among quantities at µ = Λ. The terms given in Eq.(2.1) explicitly break not only the U(1) X symmetry but also the R symmetry, because we assign the R charges of Φ, Y and Θ to 0, 0 and 2, respectively. We assume that the value ε SB is negligibly small. Therefore, the present model cannot break the supersymmetry spontaneously [10]. For the moment, we consider that the supersymmetry is unbroken, at least, in the yukawaon sector.
The VEV of the ur-yukawaon Φ is obtained from SUSY vacuum conditions as we show later. Then, we will obtain a cubic equation for eigenvalues v i of Φ : which necessarily and sufficiently determines the eigenvalues of Φ . The coefficients c a (a = 0, 1, 2, 3), in general, have the following relations: Therefore, we can express the quantities K and κ as follows: where Then, from a SUSY vacuum condition ∂W/∂Y = 0, we obtain From a SUSY vacuum condition ∂W/∂Φ = 0, we also obtain 14) where ξ is defined by , (2.17) and it is an "analogue" parameter at present. Note that the eigenvalues of Φ are independent of the analogue parameters except for ξ (however, we will show later that ξ becomes a digital one). By using a general formula for any 3 × 3 matrix i.e.
On the other hand, as seen in (2.6), in order to obtain a solution with [Φ] = 0, from the relations (2.14), the parameter ξ must satisfy Therefore, the parameter ξ is not an adjustable parameters, and, now, it is expressed by the digital parameters α i and β i . When we define a deviation of ε from the ideal value K = 2/3 as (2.28) Also, from Eq.(2.20), the deviation ε is expressed as In this section, we investigate a model with a simpler form i.e. with a case α 2 = β 2 = 0 in Eq.(2.1). Here, we assume that α 1 , α 3 , β 1 and β 3 are positive integers. (Note that this ansatz is only valid at µ = Λ.) In the present model, we essentially have two parameters α 3 /α 1 and β 3 /β 1 for the two observables K(Λ) and κ(Λ), so that we cannot predict K and κ definitely. For large values α i and β i , the parameters α 3 /α 1 and β 3 /β 1 approximately behave as if they are continuous parameters. In order to exclude such the cases from consideration, we assume that α i and β i are not so large, e.g. they should be smaller than about 50. (When the parameters α 1 and α 3 are given by α 1 = kn 1 and α 3 = kn 3 (k, n 1 and n 3 are positive integers), we redefine α 1 and α 3 as α 1 = n 1 and α 3 = n 3 . We also do such a re-definition for the parameters β 1 and β 3 .) In the present model (3.1), since β 2 = 0, the factor σ defined in (2.27) becomes σ = 0, so that the quantity κ is expressed simply as Since ε ∼ 10 −3 , the value κ is almost determined by the parameter α 3 /α 1 : Since we search a case κ = 0, we consider α 1 = 0. Since ∂κ/∂α 1 > 0, we can regard the value of α 1 as α 1 = 1, which gives a next smaller value of κ following to κ = 0 at α 1 = 0. Then, by neglecting the factor 2ωε 10 −2 , the expression (3.2) prdicts .

Concluding remarks
First, we would like to re-emphasize that the quantities K(µ) and κ(µ), which are defined by Eqs.(1.2) and (1.3), respectively, are useful for investigating the charged lepton mass spectrum, because they are almost independent of the energy scale µ of the running masses.
The present model is based on an effective theory which is valid at µ ≤ Λ. We have assumed that there is such the energy scale Λ at which the superpotential takes a simple form. A naive estimate of Λ leads to m ν ∼ H 0 Exactly speaking, a threshold effect at µ ∼ Λ is also important in the present model. The singlet and 5-plet parts of the effective Yukawa interactions will separately receive such corrections in the present effective theory. However, in order to estimate the effect exactly, we must know a full theory at µ > Λ. Since we do not know such the full theory at present, in this paper, we are obliged to put an ansatz that the threshold effect is negligibly small. Our goal is to simultaneously understand the both values of K(Λ) and κ(Λ) at µ = Λ ∼ 10 14−16 GeV which are given in Table 1, even at the cost of such the accuracy in the observed fact K pole ≃ 2/3.
The predicted values (3.8) and (3.10) are only valid within an approximation where we neglect such the threshold effect. In fact, the predicted value (3.8) at µ = Λ does not lead to K pole = 2/3 in the exact meaning.
The superpotential terms W Φ given in Eq.(3.1) spontaneously break the U(1) X symmetry, so that pseudo Goldstone bosons appear. An earlier estimate of such light yukawaons based on a prototype model of the present W Φ is found in Ref. [2]. The conclusions in Ref. [2] that we have three massless scalars and three pseudo Goldstone bosons with masses of the order of ε SB Λ will be unchanged in the present model. For more details of physical meaning of the light bosons, we will discuss elsewhere.
In this paper, we have considered that the observed fact K pole = 2/3 is accidental, and the model should give K(Λ) = 2 3 (1 + ε(Λ)) with ε(Λ) ∼ 10 −3 from the beginning. However, the present model have failed to give such excellent coincidences about K(Λ) and κ(Λ) as the well-known relation K pole = 2/3. In contrast to the present idea, recently, Sumino [5] has proposed a model which gives K pole = 2/3 exactly. This idea is based on a point of view that the relation (1.1) for the pole masses is exactly valid, so that his standpoint is in opposite direction against the present approach. If the charged leptons are truly fundamental entities, the laws of the nature for the charged leptons should be beautifully simple, so that it seems to be likely that K pole = 2/3 is exact as pointed out by Sumino. Even if Sumino's idea is true, the present approach based on the supersymmetric yukawaon model will still be promising for unified understanding of K(Λ) and κ(Λ), because the cubic equation (2.5) can completely determine the eigenvalues of Φ (i.e. K(Λ) and κ(Λ) simultaneously). We hope that a simple requirement for W Φ leads to simultaneous understanding of K(Λ) and κ(Λ) along a line suggested by a yukawaon model.