Radiative corrections to quark-lepton mass matrices in superstring-inspired models

Volume 188, number 1

PHYSICS LETTERSB

2 April 1987

RADIATIVE C O R R E C T I O N S T O Q U A R K - L E P T O N M A S S M A T R I C E S IN SUPERSTRING-INSPIRED M O D E L S Noriyasu O H T S U B O Kanazawa Technical College, Ishikawa 921, Japan

and Hideo M I Y A T A Kanazawa Institute of Technology, lshikawa 921, Japan

Received 15 December 1986 We examine if quark-lepton mass matrices are modified by generation changing currents in superstring-inspired models. It is shown that sufficiently large corrections to the quark mass matrices can be derived in an SU(3)c x SU(2) L x SU(2)RX U(1)L xU(1)R model with 8/> 2 and an SU(3)c xSU(2)L >(U(1)L xU(1)R model. Consistency with other experiments are investigated by means of renormalization group equations.

A great number of analyses have been performed concerning superstring theories [1,2], especially the E 8 x E~ heterotic string theories [3] as a most promising candidate for unifying all the fundamental interactions. Among the successes of these approaches is the derivation of effective theories which reasonably account for the low-energy structure of gauge groups and particles. When a six-dimensional space ( K ) is a C a l a b i - Y a u manifold with SU(3) holonomy, the remnant four-dimensional space has been found to be a Minkowski space with an N = 1 supersymmetry and a n E 6 x E 8 gauge symmetry [4]. The topology of the Calabi-Yau manifold specifies the number (ng + 3) of matter superfields and that (3) of antimatter superfields, which are assigned to the fundamental representations 27 and 27", respectively. If K is not simply connected, the E 6 gauge symmetry can break down to its subgroup which commutes with the Wilson loop U [ = P exp(ifv TaA# a d x ~) :~ 1] [5]. Some components of 3 pairs of 27 and 27* acquire superheavy masses through this breaking. It seems reasonable to regard the breaking scale M~ 1 of E6, the compactification scale M c 1 of K, the Planck scale

Mff 1 and the length scale Msxa of the string to have the same order of magnitude. Many authors [6-10] have studied which subgroups may remain unbroken in consistency with phenomenology, and which representations of 3 pairs of 27 and 27* remain massless. Although some models are able to answer the problems of proton decay, neutrino mass, SU(2)R breaking, and so on, the quark-lepton mass problems are left unanswered. Even if the quark-lepton mass matrices do not obey the relations that are expected from E 6 G U T [6], they must be restricted by some unknown relations at the tree level in order to explain the mass hierarchy and the small mixing angles. Since the phenomenological values of the masses and the mixing angles do not obey simple relations, the mass matrices at the tree level should be modified with other mechanisms. We have pointed out in a recent paper [11] [hereafter referred to as (I)] that the quark mass matrices are modified by the generation changing charged currents (GCCCs) in the SU(5) SUSY GUT. In this paper we investigate whether the effective low-energy theories of the E 8 x E~ heterotic string theory include generation changing 65

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currents (GCCs) which induce corrections to the mass matrices. When they do, we further inquire which of them can give sufficiently large corrections to the mass matrices without contradicting other phenomenological results. Let us concentrate on three models of the effective low-energy theories which are very attractive, and are realistic under certain assumptions. Two of them are invariant under SU(3)c X SU(2)L X SU(2)R X U(1)L X U(1)R [8,9] in the region between Mp and an intermediate energy M r The ones with 3 = 1 and with 3 > 2 will be denoted as model A1 and model A2, respectively. The other model, which will be denoted as model B, is invariant under SU(3)c x SU(2)L X U(1)L X U(1)R [10] in the region between M w and M e. M o d e l A 1 . The massless modes of the matter and antimatter superfields of this model consist of (3, 2, 1), (3", 1, 2),, (1, 2, 1),, (1, 1, 2),, (3, 1, 1),, (3", 1, 1)i and (1, 2, 2)i from 27i (i -- 1, 2,..., ng), (1, 2, 2)n and (1, 1, 1)H from 27H, and (1, 2; 2)~ and (1, 1, 1)~ from 27~ [8]. Here (., -, .) denotes the representations of SU(3)c , SU(2)L and SU(2)R , and the subscripts i and H are added to indicate the multiplet to which the submultiplets belong. The submultiplets (1, 1, 1)H and (1, 1, 1)~ are assumed to have the same vacuum expectation value (VEV), which is known not to break the supersymmetry with the D-terms, but to break the U(1) gauge symmetry at M I ( M w << M I << Mo) [7,9]. We arso assume that (1, 2, 2)H plays the role of the WS Higgs superfields and has VEVs ~ and v in its neutral scalar components. The Yukawa coupling of 27~ and 27 n is decomposed as

to suppress the decay. The second term gives masses to quarks and leptons, whose mass matrices satisfy the relation =

(2)

Although eq. (2) will be modified by renormalization, it is inevitable to get unrealistic relations: m d / m . = m s / m c = r n b / m t,

(3)

01 = 0 2 = 0 3 = 0 ,

(4)

without correction terms caused by GCCs. The GCCs are derived from the following Yukawa coupling terms: 27i27j27k = {(3, 2, 1)i(3", 1, 2)j +(1, 2, 1)i(1, 1, 2)j }(1, 2, 2)k + {(3", 1, 2)i(1, 1, 2)j + (3, 2, 1),(3, 2, 1)j}(3, 1, 1)k + {(3", 1, 2)/(3", 1, 2)j + ( 3 , 2 , 1),(1, 2, 1)j }(3", 1, 1)k + {(3, 1, 1),(3", 1, 2)j + ( 1 , 2, 2)i(1, 2, 2)j}(1, 1, 1)k + (rotations),

+ ((3, 2, 1)i(3", 1, 2)j +(1, 2, 1)i(1, 1, 2)j }(1, 2, 2)n (1)

The first term gives masses of O(MI) to (3, 1, 1)i, (3", 1, 1), and (1, 2, 2)v Since (3, 1, 1)/ and (3", 1, 1), have the possibility of mediating proton

(5)

where (3, 2, 1)i and (1, 2, 1)i [(3", 1, 2)i and (1, 1, 2),] are assigned to left- (right-)handed quarks and leptons, respectively. Therefore the first three terms of eq. (5) are the GCCs mediated by (1, 2, 2)k, (3, 1, 1)k and (3", 1, 1) k. The second and the third terms violate baryon number conservation. The full superpotential is assumed to be gijk27,27j27k

+ h27t] 27t] 27r~ ) F-term"

+ ( 1 , 2, 2)i(1, 2, 2)j}(1, 1, 1)n

66

decay, M I should be large enough (>t 1013 GeV)

(f/j27,27j27H +

27~27j27H = {(3, 1, 1),(3", 1, 1)j

+ (decoupling terms).

2 April 1987

(6)

Here the Yukawa terms 27i27n27 H and 27n27n27 n and the mass term 27h*27n are assumed to vanish or to be suppressed, because they could give large masses to (1, 2, 2)H and (1, 1, 1)d H. As pointed out in (I), eqs. (3) and (4) are not modified unless the supersymmetry breaks down. It has been proposed that the supersymmetry breaking is realized by a gluino condensation of

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E~ [12]. In this paper we assume that the supersymmetry is broken in this manner. Then soft breaking terms are given as (m3Z/2 (27it27/+ 27~27H + 27r~t27r~ )

+ ~m3/2 ( ~j27/27j27H

+

gijk27i27j27i~

+ h 271~ 27r~ 271~ ) } A-t. . . .

(7)

in the large-M v limit [13,14]. Here m3/2 is the gravitino mass and I~1 < 3. As derived in (I), the q u a r k - l e p t o n mass correction Amij is given as

Ami j

gikmmklgiln ~m3/2Mmn 16~r2 2 + m3/2 2 M~n g2m ~m3/2 16~r 2

M

'

(8)

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nonzero VEVs ~ and v while (1, 2, 2)H, does not, the first term of eq. (9) produces the masses of quarks and leptons and the second term, containing the GCCCs, generates the corrections to the quark masses. The mass of (1, 2, 2)w mediating the GCCs should be M n, ) o t - I M w - - - 1 0 TeV in order to suppress flavor changing neutral currents. Since this bound is near to m 3/2 = 1 TeV, we can expect desirable amounts of corrections for upquarks and down-quarks. We shall adopt m d = 8.9 MeV, m s = 175 MeV, m b = 5.3 GeV, m u --- 5.1 MeV and m c = 1.35 GeV at /~ = 1 GeV from the estimation by Gasser and Leutwyler [13]. When the mass of the t-quark is assumed to be 30 GeV, 40 GeV or 50 GeV, the Cabbibo angle is given as tan 01 -- I Amd Am s 11/2/ms = 0.12-0.37,

when the common mass M of (3, 1, 1)i, (3", 1, 1)i and (1, 2, 2)~ is much larger than rn3/2. Since the gauge hierarchy requires m3/2 = O(1 TeV) and proton stability needs M>~ 10 ~5 GeV, the mass correction (8) is much smaller than the expected value. If we can get a O(1 TeV) mass of (1, 2, 2)~, preserving the large masses of (3, 1, 1)~ and (3", 1, 1)~, then only the first term of eq. (5) is enhanced in eq. (8) and adequately large mass corrections are obtained. However, it is not known what kind of symmetry and mechanism can realize such a mass hierarchy. Even if we get this mass hierarchy, model A1 contradicts the experiments of proton decay and sinZ0w as will be explained in model A2.

Model A2. This model has the same gauge group as model A1 but 3 >~ 2. We shall assume 3 = 2 for simplicity. Two sets of 27H (27i~) are denoted as 27H and 27w, (27i~ and 27r~,), hereafter. Then the superpotential is written as ( f,v 27/27j27n + f~}27/2~27H, + (terms only with 27r~ and 27I~,) } F-t. . . .

(9)

where the second term of eq. (6) does not appear so as to relax the lower limit of M I. When it is assumed that (1, 2, 2,)n acquires

0.03-0.34

or

0.09-0.32, (10)

respectively, following the calculations used in (I). Although the other angles 02 and 03 are not determined, their smallness is well understood. If the VEVs g and o of the WS Higgs doublets belong to 27 H and 27w, respectively, the mass matrices of up-quarks and down-quarks are free from the relation (3) and are independent of each other. The q u a r k - l e p t o n mass matrices should be constrained, however, in order to explain the mass hierarchy and the small mixing angles in a natural manner. The mass corrections terms (8) might serve as connectors between the tree mass matrices with simple structures and the phenomenological mass matrices. Next, let us estimate the low-energy values of sin20w and M o by means of the renormalization group equations (RGEs). We assume that the E 6 gauge symmetry breaks down to SU(3)c × SU(2)L × SU(2) R × U(1)L × U(1) R at the energy/~ = M r , to SU(3)c x SU(2)L X U(1) at /L = MI, and furthermore to SU(3)c x U(1)e m at bt = M w. All the elements of 27/ ( i = 1 . . . . . , ng) and n H of (1, 2, 2)S , (1, 1, 1)n, (1, 2, 2)~I and (1, 1, 1)~ are supposed to contribute to the R G E s in the region M I ~
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the region M w ~< g ~ M I. We use the one-loop approximation of the coefficients of the R G E s and the step-functional approximation for passing through the threshold regions/~ - M G and g - M r Initial values of the R G E s used are a ~ - l ( M w ) = 128 and a f t ( M w ) = 9. We have studied the three cases where M I takes the values M e , ( M w M 3 ) 1/4 and ( M w M c ) :/2. In the first case the intermediate scale does not exist. The second and the third cases are the ones realized by the scheme of Dine et al. [7]. F r o m the results of the estimations listed in table 1, it can be seen that the values for nr~ >~ 2 disagree with the proton decay experiments because of the smallness of M G and also contradict the experimental values of sin20w(Mw) --- 0.23. The consistency with all of the experiments requires n H = 1 , H g ~--- 3 and M I = M G. Therefore, model A1 and model A2 are not appropriate, if the low-energy experiments are respected. Model B. There remain problems which have not been answered satisfactorily in model A1 and model A2. These are SU(2)R breaking and neutrino masses. Model B has been proposed [10] to

Table 1 Estimations by R G E s in model A2.

ng

nH

MI

MG/M w

sin20w(Mw)

a~ 1

3

1

MG

1.5×1014 5.8×101'* 2.4 × 10 x5

0.233 0.250 0.269

24.6 21.2 17.5

7.9×101l 8.6×1012 9.2 × 10 xz

0.255 0.274 0.293

23.2 19.7 16.1

6.7×1011 3 . 2 × 1 0 x: 1.6×101:

0.273 0.293 0.311

22.0 18.5 15.2

1.5 × 1014 5.8×1014 2.4 × 1015

0.233 0.250 0.269

14.2 9.0 3.4

7.9×1012 8.6 × 1012 9.2×1012

0.255 0.274 0.293

13.7 9.0 4.2

6.7 × 1011 3.2X 1011 1.6×10 H

0.273 0.293 0.311

13.3 9.0 4.9

( M w M 3 ) 1/4 (MwMG) 1/z 3

2

MG

( M w M 3 ) 1/4 (MwMG) 1/z 3

3

M~

(MwM3G) 1/4 (MwMG) 1/2 4

1

MG

( M w M 3 ) 1/4 (MwMG) 1/2 4

2

MG

( M w M 3 ) 1/4 (MwMG) 1/2 4

3

MG

( M w M 3 ) ~/4 (MwMG) H2

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2 April 1987

solve these problems by assuming discrete symmetries that forbid the Yukawa couplings of the lepton superfields to the quark superfields, and those of the right-handed neutrino superfields to the left-handed ones. There are no massless components in 8 (27 n + 27~]), so the low-energy theory is independent of 8. Therefore, this model can be applied to the theory on an orbifold [15] which has 8 = 0. The permitted terms of the Yukawa couplings under the discrete symmetries are 2727j27k = QiHjU~ ~ c + a i H j D k~ + L i H j E Kc + ~.TjN k +I21iHjN k + (rotations),

(11)

where Q = (U, D) T, U c and D e denote the quark superfields, L = (r, E ) T and E ~ denote the lepton superfields, / t and H denote the WS Higgs superfields, 7~ and T denote the triplet superfields with color quantum number and N denotes the singlet superfield. In addition, there exist right-handed neutrino superfields ~,c which are forbidden to appear in the Yukawa coupling. If at least one of N has a VEV, all of 7~, T, /q and H may get masses. Further, if one of H as well as one o f ' / ~ has a VEV, all of the quark and lepton superfields get masses in general. We shall solve the R G E s under the assumptions that E 6 breaks down to SU(3)c × SU(2)L × U(1)L X U(1)R at g = M G and further to SU(3)c × U(1)e m at /~ = Mw, and the R G E s have a contribution from ng superfields of quarks and leptons, n n pairs of H and H and n T pairs of 7~ and T in the region Mw ~I 4 are ommitted because of the large value of et~ 1, which spoils the validity of the perturbation. The generation number ng = 3 has been obtained in slightly complicated C a l a b i - Y a u manifolds [16], which cannot break, however, the E 6 t o the subgroups of model A and model B. Here, we suppose that we have been able to find a C a l a b i - Y a u manifold or an orbifold with n g -~ 3 and the subgroup of model B. The most preferable possibility n g = r/H -----r/T = 3 has suitable values of M o / M w and sinE0w . The less desirable possibility r/g = 3, n H = n T = 2 has a

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Table 2 Estimations by RGEs in model B. ng

nH

nT

Mc/M w

sinE0w

~tG 1

3

3

3 2 1 0

5.5 X 1017 1.6 x 1015 2.0 x 1013 6.7 x 1011

0.223 0.244 0.261 0.273

9.0 14.6 18.8 22.0

3 2 1 0

1.9x102° 1.1x1017 4.8x1014 7.9 x 1012

0.186 0.217 0.238 0.255

9.0 15.2 19.8 23.2

3 2 1 0

4.5 x 1023 2.2x1019 2.4x1016 1.5 x 1014

0.138 0.181 0.211 0.233

9.0 16.1 21.0 24.6

3

3

2

1

c o m f o r t a b l e v a l u e of M ~ / M w a n d a slightly too s m a l l v a l u e of sin20w . If o n e a s s u m e s m b / r n ~ ( M o ) = 1 as e x p e c t e d i n SU(5), SO(10) a n d E6 G U T , o n e will o b t a i n m J m ~ ( M w ) = 0 . 8 5 for ng = n H = n T -- 3 a n d 1.4 for n G = 3, n H = n T = 2. T h i s suggests t h a t E 6 m a s s r e l a t i o n s m u s t b e b r o k e n [6] at least for ng = nrt = n T = 3, if m o d e l - B is j u s t i f i e d as the effective l o w - e n e r g y theory. We have investigated whether GCCs modifying the q u a r k - l e p t o n m a s s m a t r i c e s c a n b e i n c o r p o r a t e d c o n s i s t e n t l y i n t o the effective l o w - e n e r g y theories of the d = 10, E s x E 8 SST. It has b e e n f o u n d t h a t m o d e l A1 w i t h S U ( 3 ) c x S U ( 2 ) L x S U ( 2 ) R X U ( I ) L X U(1)Reffective g a u g e s y m m e t r y a n d 8 = 1 c a n n o t g e n e r a t e s u f f i c i e n t l y large m a s s c o r r e c t i o n s a n d m o d e l A 2 w i t h 8 >i 2 p r e d i c t s a s m a l l M G a n d a large sin28w . O n the o t h e r h a n d , m o d e l B w i t h S U ( 3 ) c x SU(2)L X U ( I ) L X U ( 1 ) R

2 April 1987

m a y satisfy all t h e r e q u i r e m e n t s f r o m p h e n o m e n o l o g y , if u n f a v o r a b l e Y u k a w a c o u p l i n g s are forbidden. W e w o u l d like to t h a n k E. Y a m a d a for discussions a n d for r e a d i n g the m a n u s c r i p t . T h a n k s are also d u e to the o t h e r m e m b e r s of the e l e m e n t a r y p a r t i c l e g r o u p at K a n a z a w a U n i v e r s i t y . References

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