What Physics Does The Charged Lepton Mass Relation Tell Us?
aa r X i v : . [ h e p - ph ] N ov What Physics Does The Charged LeptonMass Relation Tell Us? Yoshio Koide
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JapanE-mail address: [email protected] §
0. Prologue
The story begins from a charged lepton mass relation[1] K ≡ m e + m µ + m τ √ m e + √ m µ + √ m τ ) = 23 . (0 . K ( m obsei ) = 23 × (0 . ± . . (0 . m predτ = 1776 .
97 MeV , (0 . m e and m µ . On the other hand, the observed mass at1982 was ( m obsτ ) old = 1784 . ± . . (0 . m obsτ was reported:( m obsτ ) new = 17776 . +0 . − . MeV . (0 . t Experimental value at 19821982 t ✲✛ Prediction t Fig.1 Predicted and experimental values of the tau mass: The experimental error bar in1992 is too small, so that we cannot denote it in the figure. A talk presented at “7th Workshop on Flavour Symmetries and Consequences in Accelerators and Cosmology”(FLASY 2018). K = 2 / K = 2 / §
1. Why is the excellent coincidence so problematic?
The charged lepton mass relation was derived based on a field theoretical model as reviewedin the next section. Therefore, we have to use the running masses for the formula K = 2 /
3, notthe pole masses. However, if we use pole masses, then, we obtain K ( m runei ) = 23 × (1 . ± . , (1 . µ = m Z ). The agreement is not so excellent.Are the present observed mass values mistaken? Is the coincidence (0.2) accidental? Will afuture experimental value be changed, and will the problem disappeared? However, such a caseis not likely.This is a serious theoretical problem. §
2. Derivation of the mass formula
Prior to review of the Sumino model, let us review the derivation[4] of the formula K = 2 / h Φ i = v diag( z , z , z ) , (2 . z + z + z = 1. 2n the model, the charged lepton mass matrix M e is given by M e = k e h Φ ih Φ i . (2 . V = µ [ΦΦ] + λ [ΦΦΦΦ] + λ ′ [Φ Φ ][Φ] , (2 . is an octet part of the nonet scalar Φ,Φ ≡ Φ −
13 [Φ] . (2 . is a unit matrix: =diag(1 , , A ] as[ A ] simply. Then, the condition ∂V /∂ Φ = 0 leads to ∂V∂
Φ = 2 (cid:0) µ + λ [ΦΦ] + λ ′ [Φ] (cid:1) Φ + λ ′ (cid:18) [ΦΦ] −
23 [Φ] (cid:19) . (2 . h Φ i as Φ simply.We want a solution Φ = , so that the coefficients of Φ and must be zero. Then, weobtain µ + λ [ΦΦ] + λ ′ [Φ] = 0 , (2 . −
23 [Φ] = 0 . (2 . K = 2 /
3. Note that Eq.(2.7) is independent of the potential parameters µ and λ .Also, recently, we have obtained another mass formula [6] κ ≡ detΦ[Φ] = √ m e m µ m τ ( √ m e + √ m µ + √ m τ ) = 12 · = 1486 , (2 . K = 2 / K ≡ [ΦΦ][Φ] = m e + m µ + m τ √ m e + √ m µ + √ m τ ) = 23 . (2 . m e , m µ , m τ ) → ( λm e , λm µ , λm τ ) . (2 .
3. Sumino mechanism
The deviation between K ( m i ( µ )) and K ( m polei ) is caused by the logarithmic term the QEDcorrection[7] m i ( µ ) = m polei ( − α ( µ ) π µ ( m polei ) !) , (3 . m i ( µ ) and m polei are running mass and pole mass, respectively. (Hereafter, for simplicity,we denote m polei as m i .) If the logarithmic term log m i is absent, then the formula K = 2 / A ji with their masses ( M ij ) ∝ ( m i + m j ). Then, the unwelcome term log( m i /µ ) inEq.(3.1) is canceled by the factor log( M ii /µ ) in the radiative mass term due to FGB. (Notethat in his model, only FGBs A ji with i = j contribute to the radiative diagram.)However, we should notice that the Sumino model has some serious shortcomings. In orderto cancel the log m i term in the QED contribution by the log M ii term in the FGB contribution,we must consider an origin of the minus sign. Sumino has assumed that the left- and right-handed charged leptons e L and e R have the same sign coupling constants e and e for photon,respectively, but the coupling constants for FGBs takes + g and − g for e L and e R , respectively.In other words, Sumino has assigned the charged leptons e L and e R to and ∗ of U(3) family,respectively. Therefore, his model is not anomaly free. Besides, in his model, unwelcome decaymodes with ∆ N family = 2 inevitably appear. §
4. Modified Sumino model
In order to avoid these defects, Yamashita and YK proposed a modified Sumino model[5]with ( e L , e R ) = ( , ) of U(3) family. In this model, the minus sign comes from the followingidea: The family gauge bosons have an inverted mass hierarchy, i.e. M ii ∝ ( m i ) − . (4 . M ii ∝ − log m i , we can obtain the minus sign for the cancellation withouttaking ( e L , e R ) = ( , ∗ ).In the modified assignment ( e L , e R ) = ( , ) with the inverted mass hierarchy (4.1), FGBwith the lowest mass is A . Note that the family number i = 1 , , i = (1 , ,
3) = ( e, µ, τ ). On the other hand, for the quark sector, we donot have any constraint from experimental observations. Both cases ( u , u , u ) = ( u, c, t ) and( u , u , u ) = ( t, c, u ) are allowed. If we choose the latter case, we can expect FGBs withconsiderably lower masses, e.g. we can suppose M ( A ) ∼ a few TeV, because the invertedfamily number assignment for quarks weakens severe constraints from K - ¯ K and D - ¯ D mixingdata, so that we can obtain considerably low FGB masses [8]. Thus, in the modified Suminomodel, we can expect fruitful phenomenology. For examples, see Ref.[9] for µ - e conversion,and see Ref.[10] for A production at LHC. However, note that in our model the transition µ → e + γ is exactly forbidden. 4 . Recent development and summary There is another effect which disturbs the K -relation: Φ ↔ Φ ≡ [Φ] / √ K and κ relations were derived from potential model under anon-SUSY scenario. Recall that there is no vertex correction in a SUSY model. Therefore, ifwe derive the relations on the basis of SUSY scenario, then the problem will disappear. Veryrecently, we succeeded to re-derive the K and κ relations on the basis of SUSY scenario [12]. .Thus, we can understand why the K - and κ -relations can keep the original forms.In conclusion, we have discussed why the K relation is so beautifully satisfied by the polemasses, not the running masses. Now we can understand the reason according to the Sumino’sidea and the modified Sumino model. References [1] Y. Koide, Lett. Nuovo Cim. (1982) 201; Phys. Lett. B (1983) 161; Phys. Rev. D (1983) 252.[2] C. Patrignani et al. (Particle Data Group), Chinese Physics C, (2016) 100001.[3] Y. Sumino, Phys. Lett. B (2009) 477; JHEP (2009) 075.[4] Y. Koide, Mod. Phys. Lett. A (1990) 2319.[5] Y. Koide and T. Yamashita, Phys. Lett. B (2012) 384.[6] Y. Koide, Phys. Lett. B (2018) 131.[7] H. Arason, et al. , Phys. Rev. D (1992) 3945.[8] Y. Koide, Phys. Lett. B736