U(3) [U(1)] 3 Model and Visible Family Gauge Boson Effects

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1 Partly corrected verson of v U(3) [U(1)] 3 Model and Vsble Famly Gauge Boson Effects Yosho Kode Department of Physcs, Osaka Unversty, Toyonaka, Osaka , Japan E-mal address: kode@kuno-g.phys.sc.osaka-u.ac.jp Abstract A U(3) famly gauge boson model s nvestgated based on U(3) [U(1)] 3 gauge symmetry. In ths model, of nne U(3) famly gauge bosons A j, those wth = j are consderable heavy compared wth those wth j, so that the model can be released from severe constrants due to the observed K 0 - K 0 mxng, and so on. We speculate that the lghtest gauge boson s A 3 and ts mass s of 6 TeV. Thereby, vsble effects of the famly gauge bosons are dscussed. Especally, an observaton of µ +N e +N versus no observaton of µ e+γ wll be a promsng test for the present scenaro. PCAC numbers: Hv, , Pw, 1 Introducton It seems to be very attractve to consder famles (generatons) n quarks and leptons as a famly symmetry [1]. However, constrants from the observed pseudo-scalar-ant-pseudo-scalar (P - P ) meson mxngs (K 0 - K 0, D 0 - D 0, and so on) are too tght to allow famly gauge bosons wth lower masses, so that t s usually taken that a scale of the symmetry brakng s consderably hgh. It s taken that t s hard to observe the gauge boson effects even n the LHC era. However, f the famly gauge symmetry really exsts, t s rather lkely that the effects are certanly vsble. In 009, Sumno has proposed a new famly gauge boson model []. Quarks and leptons are assgned to trplets of a famly symmetry U(3). In the Sumno model, the famly gauge boson mass matrx s dagonal on the famly bass on whch the charged lepton mass matrx M e s dagonal. As we gve a bref revew of the Sumno model later, regrettably, the model allows effectve current-current nteracton wth N fam = (N fam s famly number), so that we cannot be released from the sever constrant due to P - P mxng. (If we take a specfc quark mxng, we can suppress the contrbuton to K 0 - K 0 mxng [3]. However, t s too artfcal.) Recently, an extended model of the Sumno model has been proposed by Yamashta and the author [4]. (Hereafter, we refer the Sumno model and the extended model as Model I and Model II, respectvely. ) In Model II, the famly gauge bosons A j of U(3) have masses wth an nverted mass herarchy. The lghtest famly gauge boson A 3 3 couples only to tau lepton, bottom and top quarks. In Model II, only gauge bosons A1 1, A and A 3 3 can contrbute to the P - P mxng through quark mxngs U u 1 and U d 1. Because of a Cabbbo suppresson factor, 1

2 for example, the domnant contrbuton to the K 0 - K 0 mxng s brought by the gauge boson A (not the lghtest gauge boson A3 3 ). Nevertheless, n Model II, too, t s hard to obtan a lowest gauge boson mass M(A 3 3 ) a few TeV [5]. Note that n Model II, the P - P mxng are caused only by the U(3) gauge bosons A 1 A and A3 3. (In Models I and II, there s no U(3)-6 scalar, so that the gauge boson mxng A j Aj s absent.) A smple way to be released from ths severe constrant due to the observed P - P mxng s to make the gauge bosons A j wth = j consderably heavy compared wth those wth j. In ths paper, we attempt to buld such a model wth heavy A, and thereby, we dscuss possble gauge boson effects. Frst, pror to gvng our model, let us gve a bref revew of a gauge boson model (Model I) proposed by Sumno [], because Model I s a startng pont of our model. The purpose of Model I was to understand why the charged lepton mass relaton [6], m e + m µ + m τ = (/3)( m e + m τ + m τ ), s well satsfed by the pole masses (not by the runnng masses). In Model I, charged lepton masses are gven by H mass = ē LΦ α Φ T αje j R + h.c., (1) where e are charged lepton felds, Φ s a scalar of (3, 3) wth U(3) O(3) famly symmetres, and H s the Hggs scalar n the standard model. Snce the O(3) symmetry s broken at a hgh energy scale µ = Λ, where we assume Λ Λ [U(3) s broken at µ = Λ], the U(3) gauge boson masses are effectvely gven by H mass = 1 g ATr[ΦΦ T AA] = 1 g A,j 1, v A j A j, () n the lmt of Λ/Λ 0, where the vacuum expectaton value (VEV) of Φ s gven by Φ α = δ α v. Therefore, the VEV of Φ s gven n terms of charged lepton masses m e as Φ = dag(v 1, v, v 3 ) dag( m e, m µ, m τ ), (3) and the gauge boson masses are gven by M j M(A j ) = k e me + m ej. A factor log m e n the QED correcton [7] s completely canceled by a factor log M n the radatve correcton due to famly gauge boson A.1 It s worthwhle notcng that, n the Sumno model, on account of the cancellaton condton (for example, see Eq.(1) later), the famly gauge couplng constants g A s not free from the electroweak gauge couplng constant g W. Recently, Yamashta and the author [4] have proposed an extended verson (Model II) of the Sumno model. In Model I, n order to realze the cancellaton, the charged lepton felds 1 Note that the charged lepton mass relaton s nvarant under the transformaton m e m e(1 + ε 0 + ε ) wth ε = 0 (ε 0 s a famly number ndependent factor). Therefore, t s essental that the factor log m e n the QED correcton [7] s canceled. Thus, n log M = log m e + log(k e), the -ndependet factor log(k e) s not essental n ths cancellaton.

3 (e L, e R ) are assgned to (3, 3 ) of U(3), so that the model s not anomaly free for U(3) symmetry, and, besdes, unwelcome current-current nteractons wth famly number change N fam = appear. In Model II, U(3) assgnment of the charged leptons s (e L, e R ) = (3, 3), and, nstead, the cancellaton s guaranteed by the famly gauge bosons wth an nverted mass herarchy: M j M (A j ) k ( m e 1 + m ej 1 ), (4).e. by log M = log m e + log(k). The nverted mass herarchy s realzed by ntroducng an addtonal scalar Ψ of (3, 3) of U(3) U(3) wth a relaton Ψ Φ 1 [Φ n Model I s reread as (3, 3 ) of U(3) U(3) n Model II]. The charged lepton masses are stll gven by the VEV Φ [we reread ΦΦ T n Eq.(1) as Φ Φ], whle the gauge boson masses are domnantly gven by Ψ Λ because of Λ Λ. In Model II, too, as well as Model I, the famly gauge boson mass matrx s dagonal on the dagonal bass of M e. Because of the U(3) assgnment of the charged leptons (e L, e R ) = (3, 3) n Model II, the famly number N f s exactly conserved n the charged lepton sector. Famly number volaton appears only n the quark sector, because quark mass matrces M u and M d are, n general, not dagonal on the dagonal bases of M e. The P - P mxngs are only caused by exchange of famly gauge bosons A j wth = j and non-vanshng quark famly mxng, U u 1 and U d 1. In Model II, for example, K 0 - K 0 mxng s domnantly caused by the gauge boson A wth a suppresson factor (U d 1 U d ) dfferently from conventonal famly gauge boson model. Besdes, snce the gauge bosons are pure vector, A cannot contrbute to P - P mxng through s-channel, and t can do only through t channel, so that a color suppresson factor 1/3 appears n the P - P mxng ampltude. Thus, n Model II, we can take a consderably lower mass value of the lghtest gauge boson A3 3. Nevertheless, n Model II, too, t s hard to obtan the lghtest gauge boson A3 3 wth a mass of a few TeV [5]. A smple way to be released from ths sever constrant due to the observed P - P mxng s to consder a model wth M M j. In order to realze such a model, n the present paper, we ntroduce three U(1) gauge symmetres whch mx wth the U(3) famly symmetry. As a result of the mxng between U(3) and [U(1)] 3, we can make A vsble effects of the famly gauge bosons A j wth j. Model bosons heavy. Thereby, we dscuss We consder gauge symmetres U(3) U(3) [U(1)] 3, whch are broken at µ = Λ, µ = Λ and µ = Λ, respectvely. The U(3) symmetry s a famly gauge symmetry of quarks and leptons. The present model (Model III) s dentcal wth Model II as far as U(3) U(3) are concerned. Correspondngly to the three U(1)s, we assume three scalars of U(3) 3 scalars (χ a, χ b, χ c ), whose VEV are gven by χ a = δ a x, χ b = δ b x, χ c = δ c x. (5) As seen n Eq.(5), the gauge bosons B a, B b and B c of gauge symmetres U(1) a, U(1) b and 3

4 U(1) c, respectvely, mx only wth A1 1, A and A 3 3. Therefore, hereafter, we denote (a, b, c) = (1,, 3) =. Scalars n the present model are summarzed n Table 1. Table 1: Scalars n the present model based on U(3) U(3) [U(1)] 3. Scalars VEV U(3) U(3) U(1) a U(1) b U(1) c Φ Φ α = δα v Ψ Ψ α = δ α u χ a χ a = δ a x 3 1 g Ba 0 0 χ b χ b = δ b x g Bb 0 χ c χ c = δc x g Bc ϕ a ϕ a = w a 1 1 g Ba 0 0 ϕ b ϕ b = w b g Bb 0 ϕ c ϕ c = w c g Bc Snce we assume Λ Λ, Λ, we obtan effectve mass terms for U(3) and [U(1)] 3 gauge bosons, A and B, H mass = g A <j (u + u j + x + x j)a j A j + g A u A A + g Bw B B + x ( ga A g B B ), (6) n the lmt of Λ Λ, Λ. The form (6) s vald only when all the U(3) gauge bosons (say, C β α ) take super heavy masses of the order Λ. For example, we can suppose U(3) 6 scalars n order to make C β α case Λ Λ, Λ, the C β α heavy. Then, C β α C α β As seen n Table 1, mxngs between A j Cβ α j mxng does not affect A mxng appears. However, snce we consder the vsbly. and B a,b,c are caused only between A j and B a,b,c wth (a, b, c) =. The mass matrx for (A, B ) s gven by wth = j ( ) M = 1 ga (u + x ) g A g B x g A g B x gb (w + x ) (7) Hereafter, we drop the ndex, because we treat only mxng between those wth component. When we defne the mxng angle θ between A and B by A = A cos θ B sn θ and B = A sn θ + B cos θ, the mxng angle θ s gven by tan θ = g A g B x g B (w + x ) g A (u + x ), (8) 4

5 and masses of A and B are gven by M (A ) = 1 [g B(w + x ) + g A(u + x )] 1 [g B (w + x ) g A (u + x )] + 8g A g B x4, (9) M (B ) = 1 [g B(w + x ) + ga(u + x )] + 1 [gb (w + x ) ga (u + x )] + 8gA g B x4. (10) Snce the gauge bosons A µ and B µ couple to U(3) currents J µ and U(1) currents j µ, respectvely, as follows: H nt = g ( ) ( ) A J µ A µ + g B j µ ga B µ = cos θ J µ g B sn θ j µ A ga µ + sn θ J µ + g B cos θ j µ B µ, when we defne effectve couplng constants of current-current nteractons by (11) the effectve couplng constants are gven by G AA = g A G BB = g B H eff = G AA J µ J µ + G BB j µ j µ + G AB J µ j µ, (1) ( cos ) θ M (A ) + sn θ M (B ) ( sn ) θ M (A ) + cos θ M (B ) G AB = g ( ) Ag B 1 sn θ cos θ M (B ) 1 M (A ) = 1 w + x w u + (w + u )x, (13) = 1 u + x w u + (w + u )x, (14) = x w u + (w + u )x. (15) Note that the effectve couplng constants G AA, G BB and G AB are only dependent on the values u, v and w, and they are ndependent of the values g A and g B. Snce quarks and leptons do not have [U(1)] 3 charges as seen n Table 1, the currents j µ do not contan quark and lepton components. Our nterest s only n the magntude of G AA. If we suppose x w, u, then we obtan G AA 1 (w + u ) w u 1 w, (16) Thus, n Model III wth w u, the current-current nteractons due to exchange of the famly gauge boson A can hghly be suppressed compared wth that of A j ( j), because the former and the latter are approxmately gven by the effectve couplng constants 1/w 5

6 and 1/(u + u j j ), respectvely. Ths s confrmed from that the mass of A M = (ga /)(u + u j ), whle the mass of (A ) s approxmately gven by ( j) s gven by M ((A ) ) g A g B gb + (w g + u ), (17) A from Eq.(9). The cancellaton mechansm proposed by Sumno [] s realzed by the famly gauge bosons (A ) n the present model as well as Model II. Therefore, we must requre a relaton w v m 1 e. In Model II, the nverted mass herarchy has been realzed Ψ Φ 1,.e. by a superpotental W = ( ) δ j µs + λφα Ψ j α Θ j, (18) based on a SUSY scenaro, where S s a U(3) U(3) snglet scalar, and Θ s a scalar wth VEV Θ = 0. (Here, the superpotental (15) has been gven by a consderably smplfed form n order to show the outlne of the model. For the full expresson, see n the reference [4].) In order to obtan a relaton w v, by followng the example of Model II, we assume a superpotental W = (µϕ a + λψ αe αa )Θ a, (19) where VEVs of E and Θ take E 1 and Θ = 0, respectvely. A SUSY vacuum condton W/ Θ = 0 leads to a VEV relaton w u v However, n order to realze m 1 e. the Sumno mechansm, the relaton M ((A ) ) m 1 e s requred. That s, not only w u m 1 e, but also g B1 = g B = g B3 must hold as seen n Eq.(17). Therefore, we assume a permutaton symmetry S 3 among the three U(1) symmetres. Then, we can obtan M ((A ) ) w u m e 1. (0) 3 Phenomenology of the famly gauge bosons 3.1 A tentatve mass value of the lghtest gauge boson A3 Let us speculate numercal values of the gauge bosons A j. As well as n Models I and II, the famly gauge couplng constant g A n Model III s not a free parameter because of the cancellaton condton g A = 3 ζ e = 3 ζ g W sn θ W, (1) where g W s the weak gauge couplng constant gven by G F / = gw /8M W, and ζ s a fne tunng parameter. In Models II and III, the parameter ζ s numercally gven by ζ = 1.75 [4]. If we suppose that the gauge boson mass of (A ) 3 s consderably low, we wll observe a volaton of the e-µ unversalty n the tau decays. From the present observed branchng ratos [8] Br(τ µ ν µ ν τ ) = (17.41 ± 0.04)% and Br(τ e ν e ν τ ) = (17.83 ± 0.04)%, we obtan 6

7 Table : Famly gauge boson masses M(A j ) ( j) and M((A ) ). The values of M j are presented n a unt of TeV. The values wth underlnes are nput values. M 3 M 31 M 1 M 33 M M the rato R Br Br(τ µ ν µ ν τ )/Br(τ e ν e ν τ ) = ± Then, the observed data gve a devaton from the e-µ unversalty [5] δ R BR (phase space factors) 1 = ± () Of course, from the value (), we cannot conclude that we found a sgnfcant dfference of the devaton from the e-µ unversalty. However, we may speculate a possblty of famly gauge bosons. We can consder that the devaton n the tau decays orgnates n exchange of gauge bosons (A ) 3 and (A ) 1 3 whch nteract as τ (A ) 3 + µ and τ (A ) e, respectvely. The present devaton δ = (.0 ± 1.6) 10 gves a famly gauge boson mass of A3 [5] by assumng an nverted mass herarchy M 3 = TeV. (3) On the other hand, from the observed K 0 - K 0 mxng and so on, we obtan a constrant [5] M 10 TeV. (4) If we take values M 3 =.6 TeV and M = 00 TeV, by usng relatons M j u + u j m 1 e + m 1 ej, M w m 1 e, (5) we obtan masses of the famly gauge bosons (A ) j as shown n Table. The values n Table are only examples n order to see the relatve ratos, and those should not be taken rgdly. 3. Vsble gauge boson effects In the present model, snce we consder that the famly gauge bosons (A ) j wth = j are suffcently heavy compared wth those wth j, those gauge bosons cannot contrbute to the P - P mxngs. (As we prevously emphaszed, n Models II and III, A j cannot contrbute to a process wth N fam =.) Also, n Model III, the famly number s exactly conserved n the charged lepton sector, so that the ratdatve charged lepton decays τ µ + γ and so on are hghly suppressed. In Model II, we have speculated [5] a volaton of the e-µ-τ unversalty n Υ decays, Υ τ + τ /µ + µ /e + e, and a drect producton of A 3 3 at LHC. However, n ths Model III, t s hard to observe those because of the large values of M as seen n Table. However, snce, for example, (A 3 ) µ can couple to tγ µ t wth up-quark mxng factor U u 33 U u 3, the decay τ µ + γ s possble through the t t loop, although t s consderably suppressed. Such a loop effect wll be dscussed elsewhere. 7

8 Nevertheless, n the present model, we can see rather frutful phenomenology, snce we do not have any constrant on the masses of A j ( j) from the observed P - P mxngs, In Model III, the famly gauge bosons nteract wth quarks and leptons as follows: [ H fam = g F (ē γ µ e j ) + ( ν γ µ ν j ) + Uk d U jl d ( d ] k γ µ d l ) + Uk u U jl u (ū kγ µ u l ) (A j )µ, (6) where U u and U d are quark mxng matrces. We can expect the followng observatons: () Devaton from the e-µ-τ unversalty n tau decays: So far, we have used an nput value estmated from the devaton from e-µ-τ unversalty n tau decays. Although the value M 3 = 6 TeV s a tentatve value, we may expect that the devaton wll soon become more accurate. () Lepton number volatng rare decays of B meson decays: For lepton-flavor volatng rare decays, bottom meson decays B + K + µ τ + and B 0 K + µ τ + become soon wthn our reach (we expect Rr 10 6 as seen n Fg.1 n Ref.[4]). Regrettably, snce M 1 40 TeV, the rare decays K + π + µ + e and K L π 0 µ ± e are nvsble as seen n Fg.1 n Ref.[4]). () µ-e converson: Most senstve test for our scenaro s to observe the so-called µ-e converson. (For a revew of the µ-e converson and more detaled calculatons, for example, see Ref.[9] and Ref.[10], respectvely.) At present, we do not know values of U q 11 U q 1 (q = u, d). Therefore, t s not practcal, at ths stage, to estmate a µ-e converson rate strctly. Instead, we roughly estmate a µ-e converson rate n the quark level as follows: ( R q σ(µ + q e + q) σ(µ + u ν µ + d) U q 11 U q 1 V ud g A g W M W M 1 ), (7) where q = u and q = d, g A /g W = (3/)ζ sn θ W = from Sumno s cancellaton relaton (1). It s lkely that U u 1 U d 1. Then, we may regard the ratos R q as R u R d, so that we can neglect contrbuton to nucleon from R u compared wth that from R d. When we suppose U11 d U 1 d / V ud 10 1, we can roughly estmate values of R d for the nput values of M 3 whch are correspondng to the values (3) speculated from τ decays. The results are lsted n Table 3. Present expermental lmt s, for nstance for Au, R(A u ) σ(µ + Au e + Au)/σ(µ capture) < [11]. The estmated values n Table 3 soon become wthn reach of our observaton, even for the case of M 1 80 TeV. If we observe µ + N e + N wthout observaton of µ e +γ, then, t wll hghly support our famly gauge boson scenaro. (v) Drect producton of the lghtest gauge boson A3 : The drect producton p + p s + X b + A 3 + X (and p + p b + X s + A 3 + X) s also expected, although the cross secton of b ( b) producton wth a large energy-momentum s qut small. If the producton of A3 s realzed, the gauge boson wll decay nto b + s, τ + µ + and ν τ + ν µ wth branchng fractons 6/9, /9 and 1/9, respectvely. The decay mode A 3 τ + µ + wll be dstnctve. However, f M 3 > 3 TeV, the observaton at the present LHC wll almost be hopeless. The value (3) s crtcal for observng of A 3. 8

9 (v) Radatve decay of hadrons: A decay mode b s + γ s also mportant to see A 3 gauge boson effects. The decay s, n fact, allowed through a t t loop, A 3 t t γ. Recently, an mportance of a rare decay mode B X s + γ has been ponted out by Buras et al. [1]. In our model, a value of the parameter U33 u U 3 u s unknown (although we assumed that t s small n the prevous tem ()), we cannot estmate t at present. Radatve decays of hadrons are not pecular n ths model, and those can be caused even n the standard model. We do not dscuss such radatve decays of hadrons n ths paper. Table 3: Rough estmate of µ-e converson rate n d quark. R d s defned by Eq.(4). Values of the gauge boson masses M j are presented n a unt of TeV. The nput values M 3 correspond to the values Eq.(0) speculated from τ decays. M 3 M 1 R d Summary In concluson, we have nvestgated a gauged U(3) [U(1)] 3 model n order to make the U(3) famly gauge bosons A j wth = j heavy compared wth those wth j. Thereby, we have speculated that the lghtest gauge boson s A3 wth a mass 6 TeV (a tentatve value determned by τ decays) [and M(A3 1) M(A 1 ) TeV]. Possble A 3 bosons effects wll be observed n a devaton from e-µ unversalty n τ decays, and n rare decays of B mesons, B + K + µ τ + and B 0 K 0 µ τ +. Possble A 1 bosons effects wll be observed n µ-e converson experments, µ + N e + N, senstvely. Also, we may expect to observe a drect producton of A 3 at LHC as p + p b + X s + A 3 + X. Acknowledgments The author thank T. Yamashta for hs valuable and helpful conversatons, and H. Yokoya for helpful comments on the drect producton of A 3 at LHC. He also thank Y. Kuno and H. Sakamoto for ther useful comments on expermental status of µ-e converson and M. Koke for hs helpful comments on estmates of µ-e converson. References [1] T. Maehara and T. Yanagda, Prog. Theor. Phys. 60 (1978) 8. For a recent work, for nstance, see A. J. Buras, M. V. Carlucc, L. Merlo and E. Stamou, JHEP 103 (01) 088. [] Y. Sumno, Phys. Lett. B 671 (009)

10 [3] Y. Kode, Y. Sumno and M. Yamanaka, Phys. Lett. B 695 (011) 79. [4] Y. Kode and T. Yamashta, Phys. Lett. B 711 (01) 384. [5] Y. Kode, Phys. Rev. D 87 (013) [6] Y. Kode, Lett. Nuovo Cm. 34 (198) 01; Phys. Lett. B 10 (1983) 161; Phys. Rev. D 8 (1983) 5. [7] H. Arason, et al., Phys. Rev. D 46 (199) [8] J. Bernger et al., Partcle Data Group, Phys. Rev. D 86 (01) [9] Y. Kuno and Y. Okada, Rev. Mod. Phys. 73 (001) 151. [10] R. Ktano, M. Koke and Y. Okada, Phys. Rev. D 66 (00) [11] W..H. Bertl et al., SINDRUM II collaboaton, Eur. Phys. J. C 47 (006) 337. [1] A. J. Buras, L. Merlo and E. Stamou, JHEP 1108 (011)