Family Gauge Bosons with an Inverted Mass Hierarchy ∗ Yoshio Koide A,B, , Toshifumi Yamashita B

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Physics Letters B 711 (2012) 384–389

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Physics Letters B

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Family gauge bosons with an inverted mass hierarchy ∗ Yoshio Koide a,b, , Toshifumi Yamashita b

a Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan b MISC, Kyoto Sangyo University, Kyoto 603-8555, Japan

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Article history: A model that gives family gauge bosons with an inverted mass hierarchy is proposed, stimulated by Received 9 March 2012 Sumino’s cancellation mechanism for the QED radiative correction to the charged lepton masses. The Received in revised form 13 April 2012 Sumino mechanism cannot straightforwardly be applied to SUSY models because of the nonrenormaliza- Accepted 14 April 2012 tion theorem. In this Letter, an alternative model which is applicable to a SUSY model is proposed. It Available online 17 April 2012 j is essential that family gauge boson masses m(A ) in this model is given by an inverted mass hierarchy Editor: T. Yanagida √ √ i i ∝ i ∝ m(Ai ) 1/ mei,incontrasttom(Ai ) mei in the original Sumino model. Phenomenological meaning of the model is also investigated. In particular, we notice a deviation from the e–μ universality in the tau decays. ©2012ElsevierB.V.Open access under CC BY license.

2 1. Introduction pole αem(μ) 3 μ mei(μ) = m 1 − 1 + log . (1.2) ei π 4 2 mei(μ) It seems to be meaningful to consider that the flavor physics Note that the value of K is invariant under the transformation can be understood on the basis of a family symmetry [1].Regret- tably, since a constraint from the observed K 0–K¯ 0 mixing is very → + + tight, the family gauge boson masses must be super heavy, so that mei mei(1 ε0 εi), (1.3) it is hard to observe such gauge boson effects in terrestrial ex- when ε = 0, where ε0 and ε are factors which are independent periments. Recently, one positive effect of the existence of family i i of and dependent on the family-number i (i = 1, 2, 3), respectively. gauge bosons has been pointed out by Sumino [2,3]. Since we pro- If the logarithmic term in the radiative correction (1.2) due to the pose a model with a Sumino-like mechanism in the present Letter, photon can be cancelled by a some additional effect, the relation we first give a brief review of the Sumino mechanism. K pole = K (Λ) ≡ 2/3 can be satisfied (Λ is an energy scale at which We know a miraculous formula for the charged lepton masses K = 2/3isgiven). [4]: Sumino [2] has seriously taken why the mass formula K = 2/3 is so remarkably satisfied with the pole masses, and proposed a me + mμ + mτ 2 K ≡ √ √ √ = , (1.1) cancellation mechanism of the logarithmic term in the radiative 2 ( me + mμ + mτ ) 3 correction (1.2). He has assumed that the family symmetry is local, and that the logarithmic term is cancelled by that due to the fam- − which is satisfied with the order of 10 5 for the pole masses, ily gauge bosons. In the Sumino model, the left- and right-handed pole = × ± ∗ i.e. K (2/3) (0.999989 0.000014) [5]. However, in con- charged leptons eLi and eRi are assigned to 3 and 3 of a U(3) fam- ventional flavor models, “masses” do not mean “pole masses”, but ily symmetry, respectively. (A similar fermion assignment has been “running masses”. The formula (1.1) is only valid with the order proposed by Appelquist, Bai and Piai [7].) The charged lepton mass − of 10 3 for the running masses, e.g. K (μ) = (2/3) × (1.00189 ± term is generated by a would-be Yukawa interaction pole 0.00002) at μ = mZ . The deviation of K (μ) from K is caused ye j by a logarithmic term mei log(μ/mei) in the QED radiative correc- = ¯i e eT He LΦiαΦα j eR H, (1.4) tion term [6] Λ2 where i and α are indices of U(3) and O(3), respectively, H is the − Higgs scalar in the standard non-SUSY model, and L = (ν, e ).The Corresponding author at: Department of Physics, Osaka University, Toyonaka, * VEV matrix Φe is assumed as Osaka 560-0043, Japan. E-mail addresses: [email protected] (Y. Koide), √ √ [email protected] (T. Yamashita). Φe=diag(v1, v2, v3) ∝ diag( me, mμ, mτ ). (1.5)

j Then, the family gauge boson masses m(Ai ) are given by the e–μ universality in the tau lepton decays. The present data 3 allow the lowest gauge boson mass m(A3) to be of the order of 2 i ∝ e e †αi + e e †α j ∝ + a few TeV. We will also investigate possible family-number con- m A j Φiα Φ Φ jα Φ mei mej. (1.6) serving but lepton flavor violating decays of K , D and B mesons. + − − + Since in the Sumino model, the charged lepton fields (eL , eR ) are The observations of K → πμ e and B → K μ τ are within our ∗ assigned to (eL , eR ) ∼ (3, 3 ) of the U(3), the family gauge bosons reach. Thus, we can expect fruitful low energy phenomenology. j = in the off-diagonal elements Ai (i j) cannot contribute to the radiative corrections. Then, the cancellation takes place between 2. Cancellation mechanism in a SUSY model i ∝ logmei in the QED diagram and logm(Ai) logmei in the family gauge boson diagram. (Of cause, the family gauge boson coupling In a SUSY model, the contributions of the family gauge bosons 2 2 in the vertex correction diagram become vanishing, so that the constant gF must satisfy a relation gF /4 e .) As a result, we can obtain K (Λ) = K pole. original Sumino mechanism does not work. On the other hand, In Ref. [2], and also in this Letter, it is assumed that the formula those from the wave function renormalization diagram still re- (1.1) is already given at an energy scale Λ, and is not discussed main: how to derive the formula.1 2 γeij μ αF Mij Now, it is interesting to apply this Sumino mechanism to a su- δm = 2m log = m log . (2.1) ei ei 2 ei 2 (4π) Mij 2π μ persymmetric (SUSY) scenario. It should be noted that a vertex j j correction is, in general, vanishing in the SUSY scenario, so that the Sumino mechanism cannot be applied to SUSY models straight- Here γeij gives the anomalous dimension γei when summed over forwardly. In this Letter, we investigate how to restore the Sum- j, ino mechanism in such models. The essential idea in the present =−2g2 T a T a , = , (2.2) model is as follows: The cancellation in the Sumino mechanism γeij F ij ji γei γeij occurs due to the vertex correction diagram, while that in the a j present Letter does due to the wave function renormalization dia- where T a is the generator of the U(3). Therefore, in the present grams. (The details are given in Section 2.) In the original Sumino model, the values of εi defined by Eq. (1.3) are given by mechanism, the QED correction is cancelled by gL g R logm Ai ∝ F F ( i) 2 L R 2 2 −g logmei with g =−g ≡ gF , while in the present SUSY model m Mij 3 αem 2 αF F F F ε = ρ log ei + ζ log , ρ = ,ζ= , L,R 2 j i 2 2 it is done by (g ) logm(A ). For this purpose, we will assume μ μ 4 π 3 αem F i j the gauge boson masses with an inverted hierarchy m2(Ai) ∝ 1/m i ei (2.3) as we state in the next section. In the Sumino model, the gauge boson Ai couples to = j j and Mij are the family gauge boson masses Mij m(Ai ) given by Sumino j ¯ j ¯ j 1 1 J = ψ γμψLi − ψRiγμψ , (1.7) 2 ∝ + μ i L R Mij . (2.4) mei mej so that the current–current interactions inevitably cause interac- (For a model for M , see the next section.) tions which violate the individual family number N F by | N F |=2. ij | |= j = This N F 2 effects are somewhat troublesome in the low en- Note that since the gauge bosons Ai with j i can contribute ergy phenomenology. In contrast to the Sumino currents (1.7),our to the εi term, differently from the Sumino model, the QED logmei family currents are given by the canonical form term cannot be cancelled by the gauge boson terms exactly even 0 if we adjust the parameter ζ .TheratioofK (mei) to K (m ) for j j j ei ( J ) = ψ¯ γ ψ + ψ¯ γ ψ = ψ¯ jγ ψ , (1.8) = 0 + + μ i L μ Li R μ Ri μ i mei mei(1 ε0 εi ) so that the | N |=2 effects appear only through a small quark F ≡ K (mei) family mixing. R , (2.5) K (m0 ) In summary, the present model has the following characteris- ei tics compared with the Sumino model: (i) Since we can assign is, in general, dependent on the values ζ and ε0. (The value of ε0 ∼ the family multiplets as ( f L , f R ) (3, 3) of the U(3), it is easy to is practically not essential as |ε0| 1.) make the model anomaly-less. (ii) In contrast to the Sumino cur- Next, we investigate the ζ -dependence of the ratio R.Sincethe rents (1.7), we can take a canonical form of the family currents ε0 term can always be shifted by a common value, we shift the εi | |= (1.8). Therefore, the N F 2 effects appear only through a small terms as εi → εi − ε3 and ε0 → ε0 + ε3. Then, we obtain quark mixing. (iii) Family gauge bosons with the lowest and high- 2 3 1 est masses are A and A , respectively. Note that gauge bosons 1 me1 3 1 ε1 = log which can couple to the light quarks u, d and s are only Ai with ρ m2 j e3 i, j = 1, 2. Therefore, we may consider that the contributions of the 2 2 1 m 1 m 1 + me1/me2 family gauge boson exchanges with i, j = 1, 2 are reduced com- + ζ log e3 + log e2 + log , (2.6) 2 2 2 m 2 m 1 + me2/me3 pared with a conventional family gauge boson model. This means e1 e1 that we can take a lower value of m(A3) (e.g. a few TeV), and may 2 2 + 3 1 me2 1 me3 1 me1/me2 3 + − ¯ ¯ ¯ ¯ ε = log + ζ log + log , (2.7) expect to observe A → τ τ /bb/tt via bb/tt associated produc- 2 2 2 3 ρ m 2 m 1 + me1/me3 tion in the LHC. In Section 4, we will investigate a deviation from e3 e2 and ε3/ρ = 0. Here, the first terms in the parentheses in the right-hand sides represent the contributions of the diagonal gauge 1 The first attempt to understand the mass formula (1.1) from the bilinear form i fields Ai , while the succeeding terms do those of the off-diagonal of Φe has been done by assuming a U(3) family symmetry and a “nonet” ansatz for j = = Φ [8]. For more plausible derivation of the formula, see Ref. [3],wherethemodel ones Ai ( j i). As expected, by setting ζ appropriately (ζ 2), the is based on a U(9) family symmetry. diagonal gauge fields cancel the QED logarithmic terms as in the 386 Y. Koide, T. Yamashita / Physics Letters B 711 (2012) 384–389

Table 1 The ﬁelds in the present model and their quantum numbers.

c ¯ c ¯ c ¯ ¯ e Hd L LEE Φ ΦΨΨθΦ ΘA ΘB S θS

SU(2)L 2122211111111 1 11 ∗ ∗ ∗ U(3) 33 11111333318+ 11 11 ∗ ∗ ∗ ∗ U(3) 11 1 33 3 333331 1 8+ 111 U(1)S 00 −11−10 −11 1 0 0 −2 −1 −11−1 U(1)R 1101111000022 2 02

Sumino mechanism, but the off-diagonal ones make the cancella- and doublets, respectively. Then, we obtain the following effective tion incomplete, as ε2/ρ = 2log[(1 + me1/me2)/(1 + me1/me3)] superpotential 2 2(me1/me2) and ε1/ρ log(me1/me2) in this case. Interestingly, y y y eff = Hd e ¯ i α cj however, since ε2 is quite small and safely neglected, the effect W Y iΦαΦ j e Hd. (3.2) M E ML on the parameter K which has a mild dependence on me1 is ¯ ci relatively suppressed. Although the suppression is not sufficient, Note that the counterpart of the y -term Tr[ΦΦ]ie Hd is not gen- − R − 1 = O(10 4), we expect that the deviation is cancelled by erated at the tree level, and then, in great contrast to non-SUSY some other effects, such as the tau-Yukawa effect, a misarrange- models, is protected against the radiative corrections thanks to the ment of ζ for instance due to the renormalization group (RG) nonrenormalization theorem. To be complete, we should forbid a ci effects, and so on. term ie Hd and hopefully the above effective nonrenormalizable If we want more precise value of ζ at which R becomes 1, term generated already at the cutoff scale Λ. The former term can we can obtain it numerically. For convenience, we use the fol- be forbidden by a U(1)S symmetry as usual. A concrete example of lowing input values: the observed charged lepton pole masses [5] assignment of the charge and other quantum numbers (including 0 = × −3 0 = 0 = me1 0.510998910 10 GeV, me2 0.105658367 GeV, me3 those of the U(1)R symmetry that the superpotential considered 1.77682 GeV; the fine structure constant at μ = mZ , α(mZ ) = here has) are shown in Table 1. The latter term may be forbid- 1/127.916 [5].Forthetimebeing,wedonotspecifyanenergy den effectively by assuming that the cutoff scale is large enough, = scale Λ at which the formula K (Λ) 2/3 holds. Then, we find that or by replacing for instance the mass M E with a VEV of a field S = = avalueofζ which gives R 1isζ 1.752. We can check that the charged under the U(1)S symmetry. Here, we employ the former value of R − 1atζ = 1.752 is insensitive of the value ε0.Thevalue way for simplicity and consider only renormalizable terms, while ζ = 1.752 is near to a value 7/4. If we consider ζ = 7/4, we can we also introduce the field S for the later convenience. − also show that R − 1 is always smaller than 10 5 independently To avoid the massless Nambu–Goldstone mode, we assume that of the value ε0. Thus, although the present model cannot give rig- the U(1)S charge conservation is broken by a soft term orous cancellation of the logmei term, it can practically give R = 1 −5 = − 2 with an accuracy of 10 . Wbr μS SθS εμS θS , (3.3)

where S and θS are family singlets, and ε has been put in or- 3. Model der to denote that the R charge conservation in the term (3.3) is softly broken with a small extent ε. The superpotential (3.3) leads A simple way to make a model for the charged leptons to S=εμS . anomaly-less is to assign the lepton doublet and charged lep- The effective superpotential (3.2) reduces to the charged lepton ∗ ¯ ton singlet ec to 3 and 3 of the U(3) family symmetry, re- Yukawa interaction when Φ and Φ acquire VEVs. The magnitude ¯ spectively, in contrast to the Sumino model [2,3] where both of ΦΦ is set by have been assigned to 3.Ifweadoptayukawaonmodel[9], = α ¯ i − 2 the would-be Yukawa interaction for the charged lepton sec- WΦ λ1Φi ΦαθΦ λ2 S θΦ , (3.4) tor is given by (y /Λ) (Y )i ecjH . Then, however, another term e i e j d where θΦ is a family singlet ﬁeld, which leads to j y Y eci H is also allowed by the symmetry, and there are ( e/Λ)i ( e) j d λ ¯ = 2 2 2 no reasons to forbid it. (This problem always appear when we take Tr Φ Φ ε μS . (3.5) ∗ λ amodelwith ∼ 3 and ec ∼ 3 .) Therefore, in the present Letter, 1 we do not adopt such a yukawaon model. Following to the Sum- As mentioned, we do not discuss how appropriate forms (1.5) of ¯ ino model, we will consider a bilinear contribution (Φ¯ )i (Φ )α the VEVs, Φ and Φ , are obtained but just assume that a super- e α e j ¯ i ¯ i ∼ ∗ potential W K (Φ, Φ,...) leads them. Here, in addition, we assume instead of (Ye) j . To be more concrete, we assign (Φe)α (3 , 3) 2 ∗ that W K is invariant under the exchange of the U(3) and the and (Φ )α ∼ (3, 3 ) of U(3) × U(3) family symmetries. Note that e j U(3) , which exchanges and ¯ , and that the VEVs respect this Φ Φ we consider U(3) instead of O(3) in the Sumino model. This al- S2(= Z2) symmetry: lows us to take a ﬂavor basis in which Φe is diagonal. Hereafter, ¯ ¯ ¯ we simply denote Φe and Φe in the present model as Φ and Φ, Φ=Φ=vΦ Z = vΦ diag(z1, z2, z3). (3.6) respectively. Here, in the last equality, we have used the U(3) and U(3) de- We take the following would-be Yukawa interaction terms grees of freedom to diagonalize Φ, and the parameters zi are normalized as z2 + z2 + z2 = 1, without loosing generality. With = ¯ i ¯α + cα + ¯ c α cj 1 2 3 W Y yiΦα L yHdLα Hd E ye EαΦ j e this notation, from Eq. (3.5),weobtain cα ¯ c ¯α + M E E E + ML Lα L , (3.1) λ2 α v2 = ε2 μ2 . (3.7) Φ λ S = ci = cα ¯α = ¯ α ¯ α ¯ c 1 where i (νi , ei), e , Lα (Nα, Eα), E , L (N , E ) and Eα, have the electric charges (0, −1), +1, (0, −1), +1, (0, +1) and −1, c ¯ c ¯ 2 respectively. Here, (E , E ) and (L, L) are vector-like SU(2)L singlets The superpotential (3.1) does not possess this invariance. Y. Koide, T. Yamashita / Physics Letters B 711 (2012) 384–389 387

∗ Next, we investigate a superpotential for the field Ψ whose VEV is SU(2)L doublet with 3 of the family symmetry SU(3) .(How- Ψ gives an inverted mass hierarchy (2.4): ever, we do not consider the vector-like partner N¯ c unlike the case (Ec, E¯ c). Therefore, in order to make the model anomaly free, some = ¯ i α + ¯ ¯ i α j WΦΨ λA ΨαΦ j λA ΦαΨ j (ΘA )i additional fields are needed. In this Letter, we do not comment on it.) The field generates the following superpotential terms in addi- + ¯ i α + ¯ ¯ i α − j λA ΨαΦi λA ΦαΨi μA S (ΘA ) j tion to Eq. (3.1): + α ¯ i + ¯ α ¯ i β λB Φi Ψβ λB Ψi Φβ (ΘB )α cα + cα cβ yHuLα Hu N yM N (Y M )αβ N . (3.16) β + λ ΦαΨ¯ i + λ¯ Ψ αΦ¯ i − μ S (Θ ) . (3.8) B i α B i α B B β The superpotential terms (3.1) and (3.16) leads to the effective ¯ ¯ neutrino mass matrix as Eq. (3.2): Again, we impose that WΦΨ is S2 invariant, i.e. λA = λB , λA = λB , = ¯ ¯ = = λ λ , λ λ and μA μB . Then, from the F -flatness condi- yHu y − αβ j A B A B W eff = ( H )Φ¯ i Y 1 Φ¯ ( H ). (3.17) tions, we obtain ν 2 i u α M β j u yM ML =¯ = −1 Ψ Ψ vΨ Z , (3.9) Here, we consider that the VEV value Y M αβ breaks SU(3) sym-

metry at a higher scale Λ (Λ Λ). This realizes the above which satisﬁes the D-term condition too. We also obtain assumption used in Eq. (3.15), and is a reason that we will not

μS μA discuss SU(3) family symmetry gauge boson effects in the follow- vΦ vΨ = 3ε . (3.10) + + ¯ + ¯ ing low energy scale phenomenology. λA 3λA λA 3λA

Comparing with Eq. (3.7),wesee 4. Possible effects of the family gauge bosons vΦ = ∼ εk O (ε), (3.11) Since the gauge boson masses are given by Eq. (3.15),weobtain vΨ the following hierarchical structure: = + + ¯ + ¯ where k (1/3)(λ2/λ1)(λA 3λA λA 3λA )(μS /μA ).Sincethe charged lepton masses m are given by 1 ei 2g2 v2 = M2 2M2 2M2 , F Ψ 2 11 12 13 z1 M ≡ diag(m ,m ,m ) ∝ v2 Z 2 = v2 diag z2, z2, z2 , (3.12) e e1 e2 e3 Φ Φ 1 2 3 1 2 2 = 2 2 2gF vΨ M22 2M23, from Eq. (3.2), the parameter values of zi are given by z2 √ 2 mei 1 z = √ , (3.13) 2g2 v2 = M2 . (4.1) i + + F Ψ 2 33 me1 me2 me3 z3 where (me1,me2,me3) = (me,mμ,mτ ). The explicit values of zi are The family currents in the Sumino model are given by Eq. (1.7), given by while, in the present model, those are given by

i i i i (z1, z2, z3) = (0.016473, 0.23688, 0.97140). (3.14) = ¯ + ¯ +···= ¯ +··· ( Jμ) j eL γμeLj eR γμeRj e γμe j , (4.2) Thus, we can approximately estimate the family gauge boson where, for convenience, we have denoted only charged lepton sec- i masses m(A j) as follows tor explicitly. Note that the effective current–current interactions in the Sumino model induce N F = 2 interactions, while those in M2 ≡ m2 Ai = ij j the present model do not induce such N F 2 interactions. In the present model, however, since the family number is defined 1 2 † i α ¯ i ¯ † α 2 = g Ψ Ψ + Ψ Ψ + (i → j) + O ε in a diagonal basis of the charged lepton mass matrix, in general, 2 F α i α i α quark mixings appear, so that family number violating modes will = = = 1 1 1 1 be observed through U u 1 and Ud 1, where Uq (q u, d)isde- 2 2 + ∝ + gF vΨ , (3.15) † = 2 2 m m fined by U MqU Rq Dq (Dq is a diagonal matrix). For example, zi z j ei ej Lq family currents in the down-quark sector are given by if the mixing between the U(3) gauge boson and the U(3) one (d) i = ¯0i 0 + ¯0i 0 can be neglected. This happens when the latter gauge boson is Jμ j dL γμdLj dR γμdRj sufficiently heavy and we assume such a case. Namely, we assume = † i l ¯k + → another sector that breaks U(3) at a high scale which is basically U Ld k(U Ld) j dL γμdLl (L R), (4.3) decoupled from the sector we have discussed above. so that the effective Hamiltonians for semileptonic modes and In this case, in addition to the interactions in the superpoten- nonleptonic modes, H and H , are given by tial (3.1), the gauge interaction (below the U(3) breaking scale) SL NL violates the S2 symmetry assumed in W K and WΦΨ , and the RG G eff = √ij † i l ¯k ¯ j μ effects modify the S relations shown above Eq. (3.9). Neverthe- H SL U (Ud) j d γμdl e γ ei , (4.4) 2 2 d k less, amazingly, the nonrenormalization theorem protects the VEV i, j,k,l relations (3.6) and (3.9) against the RG effects, which justifies the G eff = √ij † i l † j n ¯k ¯m μ above discussion. H NL U (Ud) j U (Ud)i d γμdl d γ dn , 2 d k d m So far, we have not discussed the neutrino mass matrix, be- i, j,k,l,m,n cause the purpose of the present Letter is to discuss how to apply (4.5) the Sumino mechanism to a SUSY model. Here, we would like to √ = 2 2 2 2 give a brief comment on the neutrino mass matrix. In order to ob- respectively, where Gij/ 2 gF /2Mij z j /2vΨ , and, for simplic- c tain neutrino masses, we add, for instance, a new field N , which ity, we have put U Ld = U Rd ≡ Ud. In this section, we investigate 388 Y. Koide, T. Yamashita / Physics Letters B 711 (2012) 384–389 + + + − 1 + + possible phenomenology of the flavor violating modes, and discuss B K → π μ e 2z4r4 B K → π 0μ ν 1 2 μ the scale of the gauge bosons. 2|V us| 0 − Usually, the most strict constraint comes from the observed K - = 4.88 × 10 8r4, (4.11) ¯ 0 K mixing. However, this N F = 2 transition occurs only via the = down-quark mixing Ud 1, so that the constraint is highly depen- + − 1 1 − + dent on the quark mass matrix model. In this Letter, we do not B K 0 → π 0μ e z4r4 B K 0 → π μ ν 1 | |2 μ discuss a model about the quark mixing. Only modes that are in- 2 2 V us − dependent of the quark mixing structures are pure leptonic decays = 9.82 × 10 8r4, (4.12) ei → e j + ν¯ j + νi . Therefore, first, let us investigate these pure leptonic decays based on the present model. The effective interactions + → + − + via the family gauge boson exchanges are given by B D π μ e 4 4 1 + ¯ 0 + f (mπ /mD ) Gij z r B D → K μ ν √ ¯ ¯ μ 1 2 μ (νiγμν j) e jγ ei , (4.6) 2|V cs| f (mK /mD ) 2 − = 5.83 × 10 9r4, (4.13) against the conventional weak interactions G F ¯ μ 0 → 0 − + √ e jγμ(1 − γ5)ν j ν¯iγ (1 − γ5)ei , (4.7) B D π μ e 2 √ 1 4 4 1 0 − + f (mπ /mD ) z r B D → K μ ν = 2 2 = 2 = 1 2 μ where G F / 2 gW /8MW 1/2v W (v W 246 GeV). By using 2 2|V cs| f (mK /mD ) the Fierz transformation, we obtain effective coupling constants − = 1.03 × 10 9r4, (4.14) (for the definitions, see [10]) in the current–current interactions V = + V = S =− S = + + − + gLL 1 j, gRR 0, gLR 2 j, gRL 0, (4.8) B B → K μ τ 2 2 where (1/4)z (v /v ) , and we have considered a case that 1 + + f (mK /mB ) j j W Ψ z4r4 B B → D¯ 0τ ν 2 2 τ the observed neutrinos are Majorana types. The result (4.8) gives 2|V cb| f (mD /mB ) the decay parameters [10] η = 0, ρ = 3/4, δ = 3/4 and ξ 1−22. − j = 1.51 × 10 2r4, (4.15) Regrettably, the results for η, ρ and δ are identical with those in the standard model (SM) and the deviation of ξ from ξ SM = 1is 0 0 − + too small to observe. On the other hand, in relation to the branch- B B → K μ τ ing ratios, we predict 4 4 1 0 − + f (mK /mB ) z r B B → D τ ντ − − 1/2 2 | |2 1 + B( → ¯ ) 2 V cb f (mD /mB ) ≡ μ = τ μ νμντ f (me/mτ ) Rτ − − , (4.9) = × −2 4 1 + e B(τ → e ν¯eντ ) f (mμ/mτ ) 2.37 10 r , (4.16) where f (x) has been defined by f (x) = 1 − 8x2 + 8x6 − x8 − where r and f (x) have been defined below Eq. (4.9). (For simplic- 4 2 ity, we have used approximate relation in the limit of massless 12x ln x and f (me/mτ )/ f (mμ/mτ ) = 1.028215. Since 2 2 = × −5 2 2 2 = × −2 2 charged leptons. Therefore, the numerical results should not be e z1r /4 6.8 10 r and μ z2r /4 1.4 10 r [r ≡ v /v ], we expect a deviation R ≡ R − 1 .Present taken rigidly.) In Eq. (4.12), under the approximation of neglecting W Ψ τ τ μ 0 → − + → ± ∓ = − → − ¯ = ± CP violation, we read B(K π μ νμ) as B(K L π μ νμ) experimental values [5] B(τ μ νμντ ) (17.39 0.04)% and − + + − − − (1/2)B(K 0 → π μ ν ) + (1/2)B(K¯ 0 → π μ ν¯ ) = B(K 0 → B(τ → e ν¯eντ ) = (17.82 ± 0.04)%give μ μ − + ¯ 0 + − 0 0 + − π μ νμ) = B(K → π μ ν¯μ) (and also B(K → π μ e ) as exp = ± 0 ± ∓ Rτ 1.0017 0.0016, (4.10) B(K L → π μ e )). In the numerical results in Eqs. (4.11)–(4.16), + → 0 + = ± we have used the observed values [5] B(K π μ νμ) i.e. μ 0.0017 0.0016. This result seems to be in favor of the in- −2 ∓ ± + ¯ 0 + 3.353 × 10 , B(K L → π μ νμ) = 0.2704, B(D → K μ νμ) = verted gauge boson mass hierarchy although it is just at 1 σ level. − − + − + + 9.4×10 2, B(D0 → K μ ν ) = 3.31×10 2, B(B → D¯ 0τ ν ) = (If the gauge boson masses take a normal hierarchy, Rτ will show μ τ × −3 0 → − + = × −2 Rτ < 1.) However, if we take the central value Rτ ∼ 0.0017, it 7 10 and B(B D τ ντ ) 1.1 10 . For reference, we list ∼ −1 means r ∼ 0.35. This value corresponds to vΨ ∼ 0.7 TeV which is the predicted values for v W /vΨ 10 (and present experimental ruled out by Kaon rare decays as we will see next. At present, we upper limits [5]) as follows: ∼ −1 should not take the value (4.10) rigidly. If we speculate r 10 + + + − − − B K → π μ e ∼ 5 × 10 12 < 1.3 × 10 11 , (vΨ of a few TeV), we may expect a sizable deviation Rτ μ 2 2 ∼ −4 0 ± ∓ −11 −11 z2r /4 10 from the e–μ universality in the tau lepton decays. B K L → π μ e ∼ 1 × 10 < 7.6 × 10 , −4 We expect that the observation Rτ 10 will be accomplished + + − + − − B D → π μ e ∼ 6 × 10 13 < 3.4 × 10 5 , by a tau factory in the near future. − + − − Next, we direct our attention to family number conserved B D0 → π 0μ e ∼ 1 × 10 13 < 8.6 × 10 5 , modes in the limit of no quark mixing. Predicted values for those + + − + − − B B → K μ τ ∼ 2 × 10 6 < 7.7 × 10 5 , modes are insensitive to the explicit values of Ud and Uu as far − + − as they are not so large. In particular, we investigate rare decays of B B0 → K 0μ τ ∼ 2 × 10 6 (no data). (4.17) − − ¯ pseudo-scalar mesons 0 → 0 + ei + e j (i = j)with N F = 0, e.g. + + + − + + + − 0 0 K → π μ e , B → K τ μ , and so on. (Since our currents We also predict B(K L → π νeν¯μ) B(K L → π μe)/2. We show − ¯ ¯ are pure vectors, they cannot contribute to decays 0 → ei + e j .) the predicted branching ratios B(P → P eie j) versus v F ≡ vΨ in When we neglect the CP violation effects and the electromagnetic Fig. 1. Therefore, if vΨ is a few TeV, observations of the lepton- mass difference of pseudo-scalar mesons, we can predict the fol- flavor violating K - and B-decays with N F = 0 will be within our lowing branching ratios: reach. Y. Koide, T. Yamashita / Physics Letters B 711 (2012) 384–389 389

with Ud 1 in the diagonal basis of the charged lepton mass matrix Me , especially with (Ud)12 0. (This means that the down- quark mass matrix Md takes a similar structure except for a unit matrix term, i.e. Md kd Me + m01.)

5. Concluding remarks

In conclusion, we have proposed a family gauge boson model with inverted hierarchical masses. The model has been motivated to give a SUSY scenario of Sumino’s cancellation mechanism between the radiative correction due to the photon and that due to the family gauge bosons. As stated in the end of Section 1,the present model has many characteristics compared with the Sum- ino model: (i) It is easy to build a model that is anomaly free, since the model takes the canonical assignments of the U(3) family. (ii) The dangerous N F = 2 interactions do not appear in the limit of no quark mixing. (iii) The family gauge bosons with the inverted hierarchical masses offer a new view for the low energy → + − ≡ Fig. 1. Predicted branching ratios B(P P ei e j ) versus the VEV value v F vΨ . phenomenology. (iv) Since our model is based on a SUSY scenario, The marks • and the dashed lines denote present lower limits of the observed the VEV relations are kept (up to the SUSY breaking effects) al- branching ratios. though in this Letter we did not discuss the derivations of the relation (1.1) and so on. We may also expect to observe the lightest family gauge boson If we take the mass relation (1.1) seriously and we want to ap- 3 A3 if its mass is a few TeV. For simplicity, we neglect the up- and ply the Sumino mechanism to a SUSY scenario, the present model down-quark mixings, i.e. (u1, u2, u3) (u, c,t) and (d1, d2, d3) will be a promising model as an alternative one of the Sumino (d, s, b). The observation is practically the same as that for Z bo- model. Whether the gauge boson mass hierarchy is inverted or son (for a review, see Ref. [11]). Although in the conventional Z normal will be confirmed by observing the direction of the de- 3 model, Z couples to the fermions of all flavors, while the A3 bo- viation form the e–μ universality in the pure leptonic tau decays. + − ¯ ¯ ¯ son couples only to τ τ , ντ ντ , bb and tt, so that the branching The present experimental result, Rτ = 1.0017 ± 0.0016, is in fa- ratios are given by vor of the inverted mass hierarchy although the error is still large. Since we speculate that the lightest gauge boson mass is a few + − 1 −4 3 → ∼ 3 → ¯ ∼ 3 → ¯ TeV, we expect the deviation Rτ = Rτ − 1 ∼ 10 .Ataufactory B A3 τ τ 2B A3 ντ ντ B A3 bb 3 in the near future will confirm this deviation. In addition, some + → + + − 1 3 2 lepton flavor violating K - and B-decays, e.g. K π μ e and ∼ B A → tt¯ ∼ . (4.18) + + + − 3 3 15 B → K τ μ , will be within our reach. We also expect a direct + − ¯ observation of τ τ /bb/tt¯ decay modes while no excesses in the + − + − + − + − We may expect to observe a peak of τ τ (but no peak in e e e e /μ μ modes in the LHC and the ILC. + − → → 3 → + − and μ μ )inpp ggX A3 X τ τ X at the LHC and + − → ∗ ∗ → 3 → + − e e Z /γ A3 X τ τ X at the ILC, although these cross Acknowledgements sections of the A3 productions are small compared with that of 3 the Z production. Similar discussion can be applied to hadronic The authors would like to thank Y. Sumino for valuable and + − jets instead of τ τ . helpful conversations, and also M. Tanaka and K. Tsumura for help- Finally, we would like to comment on a constraint from the ob- ful discussions for lepton flavor violating and collider phenomenol- 0 ¯ 0 served K –K mixing. As we stated previously, contributions from ogy. One of the authors (Y.K.) is supported by JSPS (No. 21540266). exchanges of the U(3) family gauge bosons to the K 0–K¯ 0 mixing depend on the magnitudes of the family mixing Ud in the References down-quark sector. At present, we know the observed values of = † [1] T. Maehara, T. Yanagida, Prog. Theor. Phys. 60 (1978) 822; the CKM mixing V CKM Uu Ud, while we do not know the value F. Wilczek, A. Zee, Phys. Rev. Lett. 42 (1979) 421. of Ud. Tentatively, let us assume that the CKM mixing is dom- [2] Y. Sumino, Phys. Lett. B 671 (2009) 477. inantly given by the down-quark mixing, i.e. Ud V CKM. Then, [3] Y. Sumino, JHEP 0905 (2009) 075. the dominant contribution comes from the exchange of the fam- [4] Y. Koide, Lett. Nuovo Cim. 34 (1982) 201; 2 2 2 ∗ 2 = 2 2 ∗ 2 = Y. Koide, Phys. Lett. B 120 (1983) 161; ily gauge boson A2: (gF /2M22)(V usV cs) (z2/4vΨ )(V usV cs) 2 −1 × × −4 Y. Koide, Phys. Rev. D 28 (1983) 252. (vΨ ) 6.76 10 . In the present model, the CP violating ef- [5] K. 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