- Email: [email protected]
Models of supersymmetric U(2)×U(1) flavor symmetry
26 November 1998
Physics Letters B 441 Ž1998. 191–197
Models of supersymmetric U ž2 / = U ž1 / flavor symmetry Galit Eyal Department of Particle Physics, Weizmann Institute of Science, RehoÕot 76100, Israel Received 5 August 1998 Editor: M. Cveticˇ
Abstract We use a UŽ2. = UŽ1. horizontal symmetry in order to construct supersymmetric models where the flavor structure of both quarks and leptons is induced naturally. The supersymmetric flavor changing neutral currents problem is solved by the degeneracy between sfermions induced by the UŽ2. symmetry. The additional UŽ1. enables the generation of mass ratios that cannot be generated by UŽ2. alone. The resulting phenomenology differs from that of models with either abelian or UŽ2. = GUT symmetries. Our models give rise to interesting neutrino spectra, which can incorporate the Super-Kamiokande results regarding atmospheric neutrinos. q 1998 Elsevier Science B.V. All rights reserved. PACS: 11.30.Hv; 12.15.Ff; 12.60.Jv; 14.60.Pq Keywords: Supersymmetry; Flavor models; Neutrino
1. Introduction Approximate horizontal symmetries, H, can naturally explain the observed flavor structure of fermions. With abelian symmetries w1–5x all mass ratios and mixing angles are explained in a straightforward way. On the other hand, with a UŽ2. symmetry w6x it is quite difficult to explain the large m trm b ratio and the different hierarchies in the down and up sectors. In order to overcome these problems, the UŽ2. symmetry is often combined with a Grand Unified Theory ŽGUT. and a very specific choice of flavon representations w7–12x. Within Supersymmetry ŽSUSY., the horizontal symmetries should also suppress new contributions to Flavor Changing Neutral Currents ŽFCNC.. In
models with a UŽ2. symmetry and generations in 2 H 1representations Žas in other models of nonabelian horizontal symmetry w13–17x., the SUSY FCNC problem is automatically solved by degeneracy between the first two sfermion generations. On the other hand, with Abelian symmetries, the simple alignment mechanism w18x does not give strong enough suppression of the FCNC. In order to solve this problem, one is usually led to rather specific H-charge assignments that yield a very precise alignment. We construct models with a UŽ2. = UŽ1. symmetry that combine the advantages of the two frameworks. The UŽ2. symmetry solves the SUSY FCNC problem without alignment and the UŽ1. symmetry accounts for the various mass ratios without invok-
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 1 5 4 - X
G. Eyal r Physics Letters B 441 (1998) 191–197
192
ing a specific GUT structure. The resulting phenomenology is different from that of either framework. We assume that at some high-energy scale the symmetry H ' UŽ2. = UŽ1. s SUŽ2. = UŽ1.1 = UŽ1. 2 is realized. The symmetry is broken in the following hierarchical way: e
X e
X
U Ž 2 . = U Ž 1 . ™ U Ž 1 . ™ 0.
Ž 1.
This is done by giving Vacuum Expectation Values ŽVEVs. to flavon fields. We quantify all the breaking parameters as powers of a small parameter l which we take to be of O Ž0.2.:
e ; l,
e X ; l2 .
Ž 2.
The three generations are in 2 H 1 representations. In difference with previous models of UŽ2., we allow the different SM representations to carry different charges under UŽ1.1 = UŽ1. 2 . On one hand, the model is not compatible with an underlying GUT, and loses some of the predictive power found in the GUT scenario. On the other hand, with a small number of flavon fields, we are able to reproduce the mass matrices of the quarks and the leptons and the CKM matrix elements, without invoking any additional different mechanisms or symmetries in order to solve specific problems. Within our framework the large ratio m trm b can be explained without imposing large tan b . Also, the m term can naturally be of order of the SUSY breaking scale, and does not need to be put by hand. Our model reproduces the following high-energy scale mass ratios:
Ž m u ;m c ;m t . ™ Ž l7 ; l4 ;1 . ,
Ž 3.
Ž m d ;m s ;m b . ™ Ž l7 ; l5 ; l3 . ,
Ž 4.
Ž m e ;mm ;mt . ™ Ž l8 ; l5 ; l3 . .
Ž 5.
The CKM elements are of the experimentally measured order of magnitude, while the KobayashiMaskawa phase, d K M , can receive any experimentally allowed value, and is not restricted to be of O Ž1.. The charges of the matter fields are chosen in
such a way that the UŽ1.X symmetry acts only on the first generation. Due to the symmetry between the first two generations, a degeneracy between the corresponding sfermions is produced. The degeneracy can be made to be very strong, but here we choose it to be of O Ž e 2 .. This degeneracy is strong enough to solve the SUSY FCNC problem, while mild enough to still allow approximate CP - a description of all CP violating phenomena with small CP violating phases w19,20x. Since we do not use the alignment mechanism, we do not necessarily reproduce some of its generic features. In particular, D m D is not close to the experimental bound. While in our model there is room for approximate CP as a solution to the SUSY CP problem, we do not treat the CP violating phases explicitly. Within this framework there is also the possibility of relaxing the SUSY CP problem by increasing the degeneracy between the first two sfermion generations. Our models allow various interesting structures of the neutrino sector. In light of the recent results announced by Super-Kamiokande w21x which, in the three generations framework, imply
D m223 ; 5 = 10y3 eV 2 , sin2 2 u 23 ; 1,
Ž 6.
we present two models for the structure of the lepton sector: Ø lepton-model I: hierarchy mn e < mnm - mnt, Ø lepton-model II: quasi-degeneracy mn e < < mnm < < < , mnt . The structure of the paper is as follows. In Section 2 we present the quark sector of the model. Here we introduce all the flavon fields used in both the quark and the lepton sectors. We study the implications of this model to FCNC processes. In Section 3 we present two extensions of the model that describe the lepton sector. Our conclusions are summarized in Section 4.
2. The quark sector The superfields of the quark and Higgs sectors of the Supersymmetric Standard Model ŽSSM. carry H-charges, shown in Table 1. There, Q i are the quark doublets, d i and u i are the down and up quark singlets, and fu and f d are the Higgs doublet fields.
G. Eyal r Physics Letters B 441 (1998) 191–197 2
matrices M˜ f . All the terms allowed by SUSY and UŽ2. = UŽ1. are assumed to appear with coefficients of O Ž1.. When a parameter appears explicitly, it is assumed to be the O Ž1. coefficient of the corresponding term. We write the effective matrices derived after rotations needed to bring the kinetic terms into their canonical form w2–4x. We get:
Table 1 H charges of the Higgs and Quark superfields Field
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
fu fd Q1 Q2
1 1
Ž0,0. Ž0,-2.
2
Ž1,4.
Q3 u1 u2
1
Ž0,0.
2
Ž3,0.
1
Ž0,0.
ž/
2
Ž3,0.
d3
1
Ž0,6.
ž / ž/
u3 d1
d2
d
M ; ² fd :
M u ; ² fu :
2 MLqL ; m2
˜
The electroweak symmetry is spontaneously broken by the VEVs of f d and fu , and we assume that tan b '
² fu : 1 ; . ² fd : l
Ž 7.
In addition we have standard model singlet superfields: two UŽ2. doublets and two UŽ2. singlets. Their H-charges are shown in Table 2. The horizontal symmetry is broken when some of the SM-singlet fields assume VEVs. We take for the VEVs: 1 M 1 M 1 M 1 M
² f 11 : ² f 12 :
X 2 s e ; l e l
Ž 8.
² f 21 : ² f 22 :
0 ; 0 ; e l
Ž 9.
ž / ž /
ž / ž / ž / ž /
193
² x 1 :; e ; l
Ž 10 .
² x 2 :; e X ; l2
Ž 11 .
where M is a scale in which the information about this breaking is communicated to the SSM. The symmetries of the model allow the choice ² f 21 :s 0 and ² f 22 :, ² x 1 :, ² x 2 : real. The additional fields will, in general, receive complex VEVs. The VEVs and charges allow us to estimate the quark mass matrices M f and the squark mass-squared
˜
eX3 e X 2e e X 2e 3
e X 2e e Xe 2 e Xe 4
e X 3e e X 2e 2 e X 2e 2
e X 2e 2 e Xe 2 e Xe 2
e Xe 5 e4 , e2
e Xe aqe 2 e2
aqe 2 e Xe e Xe 4
0
Ž 12 .
e Xe 3 e2 , 1
0
e Xe 4 e2 1
Ž 13 .
0
Ž 14 .
Ž a is, as explained above, a coefficient of O Ž1. that marks the degeneracy of the 114 and 224 entries, and m ˜ is the SUSY breaking scale., 2 MRd R ; m2
˜
˜
2 MRu R ; m2
˜
˜
aqe 2 e Xe e X 2e 3
e Xe aqe 2 e Xe 5
e X 2e 3 e Xe 5 , 1
Ž 15 .
aqe 2 e Xe e X 2e 2
e Xe aqe 2 e Xe 2
e X 2e 2 e Xe 2 , 1
Ž 16 .
0 0
2
M˜ LqR ; mM ˜ q.
Ž 17 .
Note that the ratio m trm b is explained by the horizontal symmetries, as is the difference in hierarchies between the up and the down sectors. This is in
Table 2 H charges of the SM-singlet superfields. Field
f1 s f2 s x1 x2
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
f 11 f 12
2
Ž-1,-1.
f 21 f 22
2
Ž-1,-2.
1 1
Ž0,-2. Ž-2,0.
ž / ž /
G. Eyal r Physics Letters B 441 (1998) 191–197
194
contrast to models where the structure of the mass matrices is dictated by UŽ2. alone. We can also estimate the size of the bilinear m and B terms:
m;m ˜e,
Ž 18 .
2 m12 ;m ˜ 2e .
Ž 19 .
Thus the horizontal symmetry solves the m-problem in the way suggested in w22x. From the mass matrices we can estimate the mixing angles in the CKM matrix. We find: < Vu s < ; l ,
< Vu b < ; l4 ,
< Vcb < ; l2 ,
< Vt d < ; l3 .
Ž 20 . In order to compare quark-squark-gaugino mixing with the experimental bounds presented in w23x we use the formula given in w18x: 2
†
d Mf N ; Ž VMf M˜ Mf N VNf . rm ˜2
Ž 21 .
where M, N 4 s L, R4 , and VMf are the diagonalizing matrices of M f. The dimensionless d Mq N matrices have the simple meaning of squark mass-squared matrices Žnormalized to the average squark masssquared m ˜ 2 . in the basis where gluino couplings are diagonal and quark mass matrices are diagonal. The comparison is summarized in Table 3. There, the phenomenological bounds scale like Ž mr1 ˜ TeV. 2 , and the CP violating phases are assumed to be of O Ž1.. We learn the following points from Table 3: Ø D m K receives SUSY contributions comparable to the SM ones.
Ø Contrary to UŽ2. = GUT symmetry models w10x, SUSY contributions to D m B are negligible compared to the SM ones. Ø Contrary to abelian horizontal symmetry models w2,18x, D m D is not expected to be at the experimental limit, but rather 1 y 2 orders of magnitude smaller. Ø In order not to exceed the measured value of e K , the CP violating phase contributing to d . Ž d . ImŽ d 12 L L d 12 R R should be small.
3. The lepton sector Various anomalies in neutrino experiments provide further input to flavor models w24–33x. In the following we show how either an hierarchical spectrum or quasi-degenerate neutrinos consistent with the recent measurements of atmospheric neutrinos w21x are produced naturally in an extension of our model to the lepton sector. 3.1. Model I: Hierarchy The H charges of the lepton superfields in our model I are given in Table 4. There, L i are the lepton doublets, l i and si are the charged lepton and neutrino singlets. We get:
l
M ; ² fd : Table 3 Squark mass parameters: model predictions vs. phenomenological bounds Process d .2 ReŽ d 12 LL d . Ž d . ReŽ d 12 L L d 12 R R d .2 ReŽ d 12 RR d .2 ReŽ d 13 LL d . Ž d . ReŽ d 13 L L d 13 R R d .2 ReŽ d 13 RR u .2 ReŽ d 12 LL u . Ž u . ReŽ d 12 L L d 12 R R u .2 ReŽ d 12 RR d .2 ImŽ d 12 LL d . Ž d . ImŽ d 12 L L d 12 R R d .2 ImŽ d 12 RR
DmK DmK DmK DmB DmB DmB DmD DmD DmD eK eK eK
Bound 3
l l6 y l7 l3 l2 l4 l2 l2 l4 l2 l6 l9 y l10 l6
Model
l6 l6 l6 l6 l8 l10 l6 l6 l6 l6 l6 l6
e X 3e e X 2e 2 e X 2e 3
e X 2e 2 e Xe 2 e Xe 2
e Xe 3 e2 . e2
0
Ž 22 .
The matrices MRn R and MLn R are given in their naive form before the rotations to the canonical form are made: MRn R ; ML
MLn R ;
a e X 2e a e Xe 2 c e Xe 2
a e Xe 2 be 2 ce 3
ae X 2
0
Ž a q b . e Xe
² fu : Ž a y b . e e X
0
c e Xe 2 ce 3 , e3
ae 2 de
Ž 23 . c e Xe
0
ce 2 . 0
Ž 24 .
G. Eyal r Physics Letters B 441 (1998) 191–197 Table 4 Model I: H charges of the lepton superfields Field
SUŽ2.
L1 L2
ž/
Ž1,0.
L3
1
Ž0,0.
l1
ž/
2
Ž3,6.
l3
1
Ž0,6.
ž/
2
Ž1,2.
s3
1
Ž0,3.
l2
s1 s2
For the slepton mass-squared matrices, we get
ŽUŽ1.1 ,U Ž1. 2 .
2
195
2
M˜ Ll L ; m ˜2
2
M˜ Rl R ; m ˜2
2 MRs R ; m2
˜
˜
aqe 2 e Xe e Xe 2
e Xe aqe 2 e2
e Xe 2 e2 , 1
Ž 30 .
aqe 2 e Xe e X 2e 2
e Xe aqe 2 e Xe 2
e X 2e 2 e Xe 2 , 1
Ž 31 .
aqe 2 e Xe e Xe
e Xe aqe 2 e2
e Xe e2 , 1
Ž 32 .
0
0
0
2
M˜ Ll R ; mM ˜ l. Using the see-saw mechanism, and arranging in the canonical form: MLn L ; MLn R MRn Ry1 MLn RT ;
² fu :2 e X 2eXy1 e X e X ML
e eX
e e
0
e . 1
Ž 25 .
The hierarchy of the neutrino masses in this model is: mn e < mnm - mnt .
Ž 26 .
There is no degeneracy between any of the neutrinos. Using as input the new data from Super-Kamiokande w21x, we find: mn e ; l3 mnt ; 5 = 10y4 eV, mnm ; l mnt ; 0.01 eV,
mnt ; 0.07 eV,
sin u 23 ; 1.
Ž 27 .
Ž 28 .
We also get ML ; 5 = 10 14 GeV.
The comparison between lepton-slepton-gaugino parameters, defined analogously to the definitions in the quark sector ŽEq. 21., and the experimental bounds presented in w23x, is summarized in Table 6. There, the phenomenological bounds for the process m ™ eg scale like Ž mr1 ˜ TeV. 2 , while the bound for the Electric Dipole Moment ŽEDM. of the electron scales like Ž mr1 ˜ TeV.. The CP violating phases are assumed to be of O Ž1.. The bounds appear only for processes for which the bound is F 1. The following points should be noted in Table 6: Ø The decay m ™ eg , if the slepton masses are close to m Z , is expected to be close to the experimental limit. Ø The EDM can be close to the experimental limit, if the CP violating phases are large. 3.2. Model II: Quasi-degeneracy
sin u 12 ; l , sin u 13 ; l ,
Ž 33 .
Ž 29 .
The neutrinos do not contribute significantly to the dark matter. The mass of nm together with the mixing angle sin u 12 might point at the large angle matter enhanced solution to the solar neutrino problem Žalthough the mass is a bit too large. w31,34x.
The H charges of the lepton superfields in our model II are given in Table 5. We get: M l ; ² fd :
e X 3e e X 2e 2 e X 2e 4
e X 2e 2 e Xe 2 e X 3e 2
e X 4e 2 e X 3e 2 . eX
0
Ž 34 .
The matrices MRn R and MLn R have the following structure: MRn R ; ML
ae X 4 a e X 3e 0
a e X 3e a e X 2e 2 be
0 be , 0
0
Ž 35 .
G. Eyal r Physics Letters B 441 (1998) 191–197
196
MLn R ;
ae X 4
Ž a q b . e X 3e
² fu : Ž a y b . e X 3e 0
0
0
ce . 0
a e X 2e 2 de
Ž 36 . Using the see-saw mechanism, and arranging in the canonical form, we get: MLn L ;
X4 ² fu :2 eX 3
ML
e e e Xe 2
e X 3e e X 2e 2 e
e Xe 2 . e X2 3 e e
0
Table 6 Slepton mass parameters: model predictions vs. phenomenological bounds Process
is: mn e < < mnm < , < mnt < .
Ž 38 .
The degeneracy between mnm and mnt is O Ž l5 .. Analyzing this using the new data from SuperKamiokande, we find:
sin u 12 ; l2 ,
sin u 13 - l3 ,
sin u 23 ,
2
Model II
m ™ eg m ™ eg
1 1
l l3
l3 l3
m ™ eg
l5
l6 ² f d :
l6 ² f d :
l . < ImŽ d 11 < LR
EDM
l8
m ˜ 7 ² fd :
l7 ² f d :
l
m ˜
m ˜ m ˜
The sfermion mass-matrices have the following structure: 2
M˜ Ll L ; m ˜2
2
mnm , mnt ; 3 eV,
Model I
Ž 37 .
The hierarchy of the neutrino masses in this model
mn e ; l7 mnt ,
Bound
l . < <Ž d 12 LL l . <Ž d 12 < RR l . <Ž d 12 < LR
Ž 39 . 1
'2
,
Ž 40 . 2
M˜ Rs R ; m ˜2
and ML ; 2 = 10 12 GeV.
M˜ Rl R ; m ˜2
Ž 41 .
Here the neutrinos play an important role in structure formation and contribute a significant part to the hot dark matter. The spectrum, however, does not seem to be compatible with any of the suggested solutions to the solar neutrino problem w31,34x.
aqe 2 e Xe e X 3e 2
e Xe aqe 2 e X 2e 2
e X 3e 2 e X 2e 2 , 1
Ž 42 .
aqe 2 e Xe e Xe 4
e Xe aqe 2 e X 2e 2
e Xe 4 e X 2e 2 , 1
Ž 43 .
aqe 2 e Xe e X 3e 2
e Xe aqe 2 e X 2e 2
e X 3e 2 e X 2e 2 . 1
Ž 44 .
0 0 0
The comparison between lepton-slepton-gaugino parameters and the experimental bounds presented in w23x, is summarized in Table 6. We point out that Ø If the CP violating phases are large, the EDM can be close to the experimental limit.
4. Conclusions Table 5 Model II: H charges of the lepton superfields Field
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
ž/
2
Ž3,1.
L3
1
Ž-2,1.
ž/
2
Ž1,5.
l3
1
Ž4,1.
ž/
2
Ž3,1.
s3
1
Ž-2,1.
L1 L2 l1
l2
s1 s2
Approximate flavor symmetries naturally explain the smallness and hierarchy of the flavor parameters in SUSY models, while suppressing sources for FCNC. Abelian horizontal symmetries explain the mass ratios in a straightforward way, but need to invoke an alignment mechanism through specific H charge assignments in order to suppress FCNC. Horizontal UŽ2. symmetries suppress FCNC with a built-in degeneracy between the first two sfermion generations, but need the framework of GUT in order to explain various mass ratios. In this work, we presented a hybrid model of abelian and non-abelian symmetries. The model
G. Eyal r Physics Letters B 441 (1998) 191–197
combines the characteristics of both symmetries in such a way as to produce all the required flavor parameters and suppressions naturally, with no additional ingredient. The UŽ2. symmetry allowsX for a e e hierarchical breaking UŽ2. = UŽ1. ™ UŽ1.X ™ 0 and gives the solution to the SUSY FCNC problem. The additional UŽ1. enables generation of various mass ratios and mixing parameters in a simple way. It also allows for a natural solution of the SUSY m problem. The phenomenology of the hybrid model is different than that of either abelian or non-abelian symmetry models. Unlike in usual non-abelian models, here different SM representations carry different charges so the model is not compatible with GUT. On the other hand, D m D is not close to the experimental limit, as it is in models with alignment. This framework leaves room for different possible solutions to the SUSY CP problem, including approximate CP. Different viable neutrino spectra can arise within this framework. We gave two examples both of which are compatible with the recent observations of atmospheric neutrinos by Super-Kamiokande. The first produces a hierarchy of neutrino masses, while the second produces quasi-degenerate neutrinos, that might play a significant role in cosmology. The model presented here is not unique. It intends to demonstrate how within the hybrid framework a simple model with very few flavon fields can be built, that at the same time agrees with all measured flavor parameters and suggests attractive spectra for the neutrino masses.
Acknowledgements I thank Yossi Nir for many useful discussions and comments on the manuscript.
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Physics Letters B 441 Ž1998. 191–197
Models of supersymmetric U ž2 / = U ž1 / flavor symmetry Galit Eyal Department of Particle Physics, Weizmann Institute of Science, RehoÕot 76100, Israel Received 5 August 1998 Editor: M. Cveticˇ
Abstract We use a UŽ2. = UŽ1. horizontal symmetry in order to construct supersymmetric models where the flavor structure of both quarks and leptons is induced naturally. The supersymmetric flavor changing neutral currents problem is solved by the degeneracy between sfermions induced by the UŽ2. symmetry. The additional UŽ1. enables the generation of mass ratios that cannot be generated by UŽ2. alone. The resulting phenomenology differs from that of models with either abelian or UŽ2. = GUT symmetries. Our models give rise to interesting neutrino spectra, which can incorporate the Super-Kamiokande results regarding atmospheric neutrinos. q 1998 Elsevier Science B.V. All rights reserved. PACS: 11.30.Hv; 12.15.Ff; 12.60.Jv; 14.60.Pq Keywords: Supersymmetry; Flavor models; Neutrino
1. Introduction Approximate horizontal symmetries, H, can naturally explain the observed flavor structure of fermions. With abelian symmetries w1–5x all mass ratios and mixing angles are explained in a straightforward way. On the other hand, with a UŽ2. symmetry w6x it is quite difficult to explain the large m trm b ratio and the different hierarchies in the down and up sectors. In order to overcome these problems, the UŽ2. symmetry is often combined with a Grand Unified Theory ŽGUT. and a very specific choice of flavon representations w7–12x. Within Supersymmetry ŽSUSY., the horizontal symmetries should also suppress new contributions to Flavor Changing Neutral Currents ŽFCNC.. In
models with a UŽ2. symmetry and generations in 2 H 1representations Žas in other models of nonabelian horizontal symmetry w13–17x., the SUSY FCNC problem is automatically solved by degeneracy between the first two sfermion generations. On the other hand, with Abelian symmetries, the simple alignment mechanism w18x does not give strong enough suppression of the FCNC. In order to solve this problem, one is usually led to rather specific H-charge assignments that yield a very precise alignment. We construct models with a UŽ2. = UŽ1. symmetry that combine the advantages of the two frameworks. The UŽ2. symmetry solves the SUSY FCNC problem without alignment and the UŽ1. symmetry accounts for the various mass ratios without invok-
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 1 5 4 - X
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ing a specific GUT structure. The resulting phenomenology is different from that of either framework. We assume that at some high-energy scale the symmetry H ' UŽ2. = UŽ1. s SUŽ2. = UŽ1.1 = UŽ1. 2 is realized. The symmetry is broken in the following hierarchical way: e
X e
X
U Ž 2 . = U Ž 1 . ™ U Ž 1 . ™ 0.
Ž 1.
This is done by giving Vacuum Expectation Values ŽVEVs. to flavon fields. We quantify all the breaking parameters as powers of a small parameter l which we take to be of O Ž0.2.:
e ; l,
e X ; l2 .
Ž 2.
The three generations are in 2 H 1 representations. In difference with previous models of UŽ2., we allow the different SM representations to carry different charges under UŽ1.1 = UŽ1. 2 . On one hand, the model is not compatible with an underlying GUT, and loses some of the predictive power found in the GUT scenario. On the other hand, with a small number of flavon fields, we are able to reproduce the mass matrices of the quarks and the leptons and the CKM matrix elements, without invoking any additional different mechanisms or symmetries in order to solve specific problems. Within our framework the large ratio m trm b can be explained without imposing large tan b . Also, the m term can naturally be of order of the SUSY breaking scale, and does not need to be put by hand. Our model reproduces the following high-energy scale mass ratios:
Ž m u ;m c ;m t . ™ Ž l7 ; l4 ;1 . ,
Ž 3.
Ž m d ;m s ;m b . ™ Ž l7 ; l5 ; l3 . ,
Ž 4.
Ž m e ;mm ;mt . ™ Ž l8 ; l5 ; l3 . .
Ž 5.
The CKM elements are of the experimentally measured order of magnitude, while the KobayashiMaskawa phase, d K M , can receive any experimentally allowed value, and is not restricted to be of O Ž1.. The charges of the matter fields are chosen in
such a way that the UŽ1.X symmetry acts only on the first generation. Due to the symmetry between the first two generations, a degeneracy between the corresponding sfermions is produced. The degeneracy can be made to be very strong, but here we choose it to be of O Ž e 2 .. This degeneracy is strong enough to solve the SUSY FCNC problem, while mild enough to still allow approximate CP - a description of all CP violating phenomena with small CP violating phases w19,20x. Since we do not use the alignment mechanism, we do not necessarily reproduce some of its generic features. In particular, D m D is not close to the experimental bound. While in our model there is room for approximate CP as a solution to the SUSY CP problem, we do not treat the CP violating phases explicitly. Within this framework there is also the possibility of relaxing the SUSY CP problem by increasing the degeneracy between the first two sfermion generations. Our models allow various interesting structures of the neutrino sector. In light of the recent results announced by Super-Kamiokande w21x which, in the three generations framework, imply
D m223 ; 5 = 10y3 eV 2 , sin2 2 u 23 ; 1,
Ž 6.
we present two models for the structure of the lepton sector: Ø lepton-model I: hierarchy mn e < mnm - mnt, Ø lepton-model II: quasi-degeneracy mn e < < mnm < < < , mnt . The structure of the paper is as follows. In Section 2 we present the quark sector of the model. Here we introduce all the flavon fields used in both the quark and the lepton sectors. We study the implications of this model to FCNC processes. In Section 3 we present two extensions of the model that describe the lepton sector. Our conclusions are summarized in Section 4.
2. The quark sector The superfields of the quark and Higgs sectors of the Supersymmetric Standard Model ŽSSM. carry H-charges, shown in Table 1. There, Q i are the quark doublets, d i and u i are the down and up quark singlets, and fu and f d are the Higgs doublet fields.
G. Eyal r Physics Letters B 441 (1998) 191–197 2
matrices M˜ f . All the terms allowed by SUSY and UŽ2. = UŽ1. are assumed to appear with coefficients of O Ž1.. When a parameter appears explicitly, it is assumed to be the O Ž1. coefficient of the corresponding term. We write the effective matrices derived after rotations needed to bring the kinetic terms into their canonical form w2–4x. We get:
Table 1 H charges of the Higgs and Quark superfields Field
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
fu fd Q1 Q2
1 1
Ž0,0. Ž0,-2.
2
Ž1,4.
Q3 u1 u2
1
Ž0,0.
2
Ž3,0.
1
Ž0,0.
ž/
2
Ž3,0.
d3
1
Ž0,6.
ž / ž/
u3 d1
d2
d
M ; ² fd :
M u ; ² fu :
2 MLqL ; m2
˜
The electroweak symmetry is spontaneously broken by the VEVs of f d and fu , and we assume that tan b '
² fu : 1 ; . ² fd : l
Ž 7.
In addition we have standard model singlet superfields: two UŽ2. doublets and two UŽ2. singlets. Their H-charges are shown in Table 2. The horizontal symmetry is broken when some of the SM-singlet fields assume VEVs. We take for the VEVs: 1 M 1 M 1 M 1 M
² f 11 : ² f 12 :
X 2 s e ; l e l
Ž 8.
² f 21 : ² f 22 :
0 ; 0 ; e l
Ž 9.
ž / ž /
ž / ž / ž / ž /
193
² x 1 :; e ; l
Ž 10 .
² x 2 :; e X ; l2
Ž 11 .
where M is a scale in which the information about this breaking is communicated to the SSM. The symmetries of the model allow the choice ² f 21 :s 0 and ² f 22 :, ² x 1 :, ² x 2 : real. The additional fields will, in general, receive complex VEVs. The VEVs and charges allow us to estimate the quark mass matrices M f and the squark mass-squared
˜
eX3 e X 2e e X 2e 3
e X 2e e Xe 2 e Xe 4
e X 3e e X 2e 2 e X 2e 2
e X 2e 2 e Xe 2 e Xe 2
e Xe 5 e4 , e2
e Xe aqe 2 e2
aqe 2 e Xe e Xe 4
0
Ž 12 .
e Xe 3 e2 , 1
0
e Xe 4 e2 1
Ž 13 .
0
Ž 14 .
Ž a is, as explained above, a coefficient of O Ž1. that marks the degeneracy of the 114 and 224 entries, and m ˜ is the SUSY breaking scale., 2 MRd R ; m2
˜
˜
2 MRu R ; m2
˜
˜
aqe 2 e Xe e X 2e 3
e Xe aqe 2 e Xe 5
e X 2e 3 e Xe 5 , 1
Ž 15 .
aqe 2 e Xe e X 2e 2
e Xe aqe 2 e Xe 2
e X 2e 2 e Xe 2 , 1
Ž 16 .
0 0
2
M˜ LqR ; mM ˜ q.
Ž 17 .
Note that the ratio m trm b is explained by the horizontal symmetries, as is the difference in hierarchies between the up and the down sectors. This is in
Table 2 H charges of the SM-singlet superfields. Field
f1 s f2 s x1 x2
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
f 11 f 12
2
Ž-1,-1.
f 21 f 22
2
Ž-1,-2.
1 1
Ž0,-2. Ž-2,0.
ž / ž /
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194
contrast to models where the structure of the mass matrices is dictated by UŽ2. alone. We can also estimate the size of the bilinear m and B terms:
m;m ˜e,
Ž 18 .
2 m12 ;m ˜ 2e .
Ž 19 .
Thus the horizontal symmetry solves the m-problem in the way suggested in w22x. From the mass matrices we can estimate the mixing angles in the CKM matrix. We find: < Vu s < ; l ,
< Vu b < ; l4 ,
< Vcb < ; l2 ,
< Vt d < ; l3 .
Ž 20 . In order to compare quark-squark-gaugino mixing with the experimental bounds presented in w23x we use the formula given in w18x: 2
†
d Mf N ; Ž VMf M˜ Mf N VNf . rm ˜2
Ž 21 .
where M, N 4 s L, R4 , and VMf are the diagonalizing matrices of M f. The dimensionless d Mq N matrices have the simple meaning of squark mass-squared matrices Žnormalized to the average squark masssquared m ˜ 2 . in the basis where gluino couplings are diagonal and quark mass matrices are diagonal. The comparison is summarized in Table 3. There, the phenomenological bounds scale like Ž mr1 ˜ TeV. 2 , and the CP violating phases are assumed to be of O Ž1.. We learn the following points from Table 3: Ø D m K receives SUSY contributions comparable to the SM ones.
Ø Contrary to UŽ2. = GUT symmetry models w10x, SUSY contributions to D m B are negligible compared to the SM ones. Ø Contrary to abelian horizontal symmetry models w2,18x, D m D is not expected to be at the experimental limit, but rather 1 y 2 orders of magnitude smaller. Ø In order not to exceed the measured value of e K , the CP violating phase contributing to d . Ž d . ImŽ d 12 L L d 12 R R should be small.
3. The lepton sector Various anomalies in neutrino experiments provide further input to flavor models w24–33x. In the following we show how either an hierarchical spectrum or quasi-degenerate neutrinos consistent with the recent measurements of atmospheric neutrinos w21x are produced naturally in an extension of our model to the lepton sector. 3.1. Model I: Hierarchy The H charges of the lepton superfields in our model I are given in Table 4. There, L i are the lepton doublets, l i and si are the charged lepton and neutrino singlets. We get:
l
M ; ² fd : Table 3 Squark mass parameters: model predictions vs. phenomenological bounds Process d .2 ReŽ d 12 LL d . Ž d . ReŽ d 12 L L d 12 R R d .2 ReŽ d 12 RR d .2 ReŽ d 13 LL d . Ž d . ReŽ d 13 L L d 13 R R d .2 ReŽ d 13 RR u .2 ReŽ d 12 LL u . Ž u . ReŽ d 12 L L d 12 R R u .2 ReŽ d 12 RR d .2 ImŽ d 12 LL d . Ž d . ImŽ d 12 L L d 12 R R d .2 ImŽ d 12 RR
DmK DmK DmK DmB DmB DmB DmD DmD DmD eK eK eK
Bound 3
l l6 y l7 l3 l2 l4 l2 l2 l4 l2 l6 l9 y l10 l6
Model
l6 l6 l6 l6 l8 l10 l6 l6 l6 l6 l6 l6
e X 3e e X 2e 2 e X 2e 3
e X 2e 2 e Xe 2 e Xe 2
e Xe 3 e2 . e2
0
Ž 22 .
The matrices MRn R and MLn R are given in their naive form before the rotations to the canonical form are made: MRn R ; ML
MLn R ;
a e X 2e a e Xe 2 c e Xe 2
a e Xe 2 be 2 ce 3
ae X 2
0
Ž a q b . e Xe
² fu : Ž a y b . e e X
0
c e Xe 2 ce 3 , e3
ae 2 de
Ž 23 . c e Xe
0
ce 2 . 0
Ž 24 .
G. Eyal r Physics Letters B 441 (1998) 191–197 Table 4 Model I: H charges of the lepton superfields Field
SUŽ2.
L1 L2
ž/
Ž1,0.
L3
1
Ž0,0.
l1
ž/
2
Ž3,6.
l3
1
Ž0,6.
ž/
2
Ž1,2.
s3
1
Ž0,3.
l2
s1 s2
For the slepton mass-squared matrices, we get
ŽUŽ1.1 ,U Ž1. 2 .
2
195
2
M˜ Ll L ; m ˜2
2
M˜ Rl R ; m ˜2
2 MRs R ; m2
˜
˜
aqe 2 e Xe e Xe 2
e Xe aqe 2 e2
e Xe 2 e2 , 1
Ž 30 .
aqe 2 e Xe e X 2e 2
e Xe aqe 2 e Xe 2
e X 2e 2 e Xe 2 , 1
Ž 31 .
aqe 2 e Xe e Xe
e Xe aqe 2 e2
e Xe e2 , 1
Ž 32 .
0
0
0
2
M˜ Ll R ; mM ˜ l. Using the see-saw mechanism, and arranging in the canonical form: MLn L ; MLn R MRn Ry1 MLn RT ;
² fu :2 e X 2eXy1 e X e X ML
e eX
e e
0
e . 1
Ž 25 .
The hierarchy of the neutrino masses in this model is: mn e < mnm - mnt .
Ž 26 .
There is no degeneracy between any of the neutrinos. Using as input the new data from Super-Kamiokande w21x, we find: mn e ; l3 mnt ; 5 = 10y4 eV, mnm ; l mnt ; 0.01 eV,
mnt ; 0.07 eV,
sin u 23 ; 1.
Ž 27 .
Ž 28 .
We also get ML ; 5 = 10 14 GeV.
The comparison between lepton-slepton-gaugino parameters, defined analogously to the definitions in the quark sector ŽEq. 21., and the experimental bounds presented in w23x, is summarized in Table 6. There, the phenomenological bounds for the process m ™ eg scale like Ž mr1 ˜ TeV. 2 , while the bound for the Electric Dipole Moment ŽEDM. of the electron scales like Ž mr1 ˜ TeV.. The CP violating phases are assumed to be of O Ž1.. The bounds appear only for processes for which the bound is F 1. The following points should be noted in Table 6: Ø The decay m ™ eg , if the slepton masses are close to m Z , is expected to be close to the experimental limit. Ø The EDM can be close to the experimental limit, if the CP violating phases are large. 3.2. Model II: Quasi-degeneracy
sin u 12 ; l , sin u 13 ; l ,
Ž 33 .
Ž 29 .
The neutrinos do not contribute significantly to the dark matter. The mass of nm together with the mixing angle sin u 12 might point at the large angle matter enhanced solution to the solar neutrino problem Žalthough the mass is a bit too large. w31,34x.
The H charges of the lepton superfields in our model II are given in Table 5. We get: M l ; ² fd :
e X 3e e X 2e 2 e X 2e 4
e X 2e 2 e Xe 2 e X 3e 2
e X 4e 2 e X 3e 2 . eX
0
Ž 34 .
The matrices MRn R and MLn R have the following structure: MRn R ; ML
ae X 4 a e X 3e 0
a e X 3e a e X 2e 2 be
0 be , 0
0
Ž 35 .
G. Eyal r Physics Letters B 441 (1998) 191–197
196
MLn R ;
ae X 4
Ž a q b . e X 3e
² fu : Ž a y b . e X 3e 0
0
0
ce . 0
a e X 2e 2 de
Ž 36 . Using the see-saw mechanism, and arranging in the canonical form, we get: MLn L ;
X4 ² fu :2 eX 3
ML
e e e Xe 2
e X 3e e X 2e 2 e
e Xe 2 . e X2 3 e e
0
Table 6 Slepton mass parameters: model predictions vs. phenomenological bounds Process
is: mn e < < mnm < , < mnt < .
Ž 38 .
The degeneracy between mnm and mnt is O Ž l5 .. Analyzing this using the new data from SuperKamiokande, we find:
sin u 12 ; l2 ,
sin u 13 - l3 ,
sin u 23 ,
2
Model II
m ™ eg m ™ eg
1 1
l l3
l3 l3
m ™ eg
l5
l6 ² f d :
l6 ² f d :
l . < ImŽ d 11 < LR
EDM
l8
m ˜ 7 ² fd :
l7 ² f d :
l
m ˜
m ˜ m ˜
The sfermion mass-matrices have the following structure: 2
M˜ Ll L ; m ˜2
2
mnm , mnt ; 3 eV,
Model I
Ž 37 .
The hierarchy of the neutrino masses in this model
mn e ; l7 mnt ,
Bound
l . < <Ž d 12 LL l . <Ž d 12 < RR l . <Ž d 12 < LR
Ž 39 . 1
'2
,
Ž 40 . 2
M˜ Rs R ; m ˜2
and ML ; 2 = 10 12 GeV.
M˜ Rl R ; m ˜2
Ž 41 .
Here the neutrinos play an important role in structure formation and contribute a significant part to the hot dark matter. The spectrum, however, does not seem to be compatible with any of the suggested solutions to the solar neutrino problem w31,34x.
aqe 2 e Xe e X 3e 2
e Xe aqe 2 e X 2e 2
e X 3e 2 e X 2e 2 , 1
Ž 42 .
aqe 2 e Xe e Xe 4
e Xe aqe 2 e X 2e 2
e Xe 4 e X 2e 2 , 1
Ž 43 .
aqe 2 e Xe e X 3e 2
e Xe aqe 2 e X 2e 2
e X 3e 2 e X 2e 2 . 1
Ž 44 .
0 0 0
The comparison between lepton-slepton-gaugino parameters and the experimental bounds presented in w23x, is summarized in Table 6. We point out that Ø If the CP violating phases are large, the EDM can be close to the experimental limit.
4. Conclusions Table 5 Model II: H charges of the lepton superfields Field
SUŽ2.
ŽUŽ1.1 ,U Ž1. 2 .
ž/
2
Ž3,1.
L3
1
Ž-2,1.
ž/
2
Ž1,5.
l3
1
Ž4,1.
ž/
2
Ž3,1.
s3
1
Ž-2,1.
L1 L2 l1
l2
s1 s2
Approximate flavor symmetries naturally explain the smallness and hierarchy of the flavor parameters in SUSY models, while suppressing sources for FCNC. Abelian horizontal symmetries explain the mass ratios in a straightforward way, but need to invoke an alignment mechanism through specific H charge assignments in order to suppress FCNC. Horizontal UŽ2. symmetries suppress FCNC with a built-in degeneracy between the first two sfermion generations, but need the framework of GUT in order to explain various mass ratios. In this work, we presented a hybrid model of abelian and non-abelian symmetries. The model
G. Eyal r Physics Letters B 441 (1998) 191–197
combines the characteristics of both symmetries in such a way as to produce all the required flavor parameters and suppressions naturally, with no additional ingredient. The UŽ2. symmetry allowsX for a e e hierarchical breaking UŽ2. = UŽ1. ™ UŽ1.X ™ 0 and gives the solution to the SUSY FCNC problem. The additional UŽ1. enables generation of various mass ratios and mixing parameters in a simple way. It also allows for a natural solution of the SUSY m problem. The phenomenology of the hybrid model is different than that of either abelian or non-abelian symmetry models. Unlike in usual non-abelian models, here different SM representations carry different charges so the model is not compatible with GUT. On the other hand, D m D is not close to the experimental limit, as it is in models with alignment. This framework leaves room for different possible solutions to the SUSY CP problem, including approximate CP. Different viable neutrino spectra can arise within this framework. We gave two examples both of which are compatible with the recent observations of atmospheric neutrinos by Super-Kamiokande. The first produces a hierarchy of neutrino masses, while the second produces quasi-degenerate neutrinos, that might play a significant role in cosmology. The model presented here is not unique. It intends to demonstrate how within the hybrid framework a simple model with very few flavon fields can be built, that at the same time agrees with all measured flavor parameters and suggests attractive spectra for the neutrino masses.
Acknowledgements I thank Yossi Nir for many useful discussions and comments on the manuscript.
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