Hierarchical structure of fermion masses and mixings

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

HIERARCHICAL STRUCTURE OF FERMION MASSES AND MIXINGS Johan B I J N E N S J and C h r i s t o f W E T T E R I C H Deutsches Elektronen-Synchrotron DESK D-2000 Hamburg, Fed. Rep. Germany

Received 17 August 1987

We examine patterns where ratios of the fermion masses and the W-boson mass (x,= m j m w ) are proportional to powers of a small parameter 2 (x, = c,2 r,). For a simple estimate of the uncertainty in the coefficients ci we determine the allowed values of P, and the corresponding range of 2. Using this information we search for realistic patterns in a large class of anomaly-free SU (3) × SU(2 ) × U ( 1) × U ( 1) models where ,l is related to a symmetry breaking scale and the P, follow from the quantum numbers. No realistic model is found. In contrast, realistic mass patterns can he induced from an anomalous U(1 ) symmetry.

It has been p r o p o s e d [ 1 ] that small quantities appearing in the f e r m i o n mass matrices c o r r e s p o n d to different powers o f a small p a r a m e t e r 2. Models have been constructed where all small mixing angles and small mass ratios x ~ = m J m w can be u n d e r s t o o d in terms o f a symmetry, x~ = c , 2 e' [2]. The p a r a m e t e r it is a ratio o f s y m m e t r y breaking scales and the various powers o f 2 follow from the q u a n t u m numbers u n d e r this symmetry. N o small quantities besides it are needed. In particular all the dimensionless couplings (Yukawa, gauge and scalar) are supposed to be o f the same o r d e r o f magnitude. First we discuss in what sense 2 and P, d e t e r m i n e the various quantities. Then we give an a p p r o x i m a t e diagonalization o f the ferrnion mass matrices and use this to estimate the uncertainty in c,. This inform a t i o n together with the experiental values o f the f e r m i o n masses and mixings then fix the allowed regions o f it and powers Pg. A typical Yukawa coupling o f the o r d e r o f the weak gauge coupling leads to a fermion mass of o r d e r row. We write the d i m e n s i o n less mass ratios and the mixing angles as

and j. We now want to fix 2 and P~, P~j from the x~ and 0~j. This is course depends on the allowed range o f values for the ci and c~j. These quantities cannot be u n d e r s t o o d purely in terms o f s y m m e t r y and their values d e p e n d on specific details o f a model. F o r the models considered in ref. [2] these coefficients are given by ratios o f dimensionless coupling constants. In the context o f higher d i m e n s i o n a l unification they correspond to generalized C l e b s c h - G o r d a n coefficients [ 3 ]. In a d d i t i o n the c, often have several contributions. The n u m b e r o f contributions typically increases with a higher power P,. We therefore expect a larger uncertainty for the smaller quantities, in particular for the first-generation masses. We will take the c, to be equal to one within a multiplicative uncertainty Ai, which reflects our lack o f knowledge o f the details o f a model.

x, = m , / m w = ciite'

( 1)

x 2 /,J, <~2 P' <~A,x + •

0ij =cij2 P'' •

(2)

In this letter we will take for the masses o f the third generation a s t a n d a r d uncertainty A = 2. The uncertainty for the other xi, 00 is taken as , ~ , A and ,,fn~A with n, discussed below. The powers P, and the coefficients c, come from a diagonalization o f the fermion mass matrices. We will

In (2) 0,j is the mixing angle between generation i Present address: Sektion Physik, Universit~it Mfinchen, Theresienstr. 37, D-8000 Munich 2, Fed. Rep. Germany.

1/& <~c,~A, .

(3)

So if x, .+ and x , - are the experimental u p p e r and lower b o u n d for x, the allowed values for it for a given P, are those that satisfy

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

(4)

525

Volume 199, number 4

PHYSICS LETTERS B

perform this diagonalization explicitly. The elements of the up quark mass matrix M u are given by

uo=ct/2U"mw .

(5)

Here i labels the species of right-handed quarks u7 and j stands for the generation of left-handed quarks U,. We assume the matrix to be properly ordered so that u33 is the largest element, i.e. the mass of the top quark rn, We are only interested in the power of 2 and neglect unnatural cancellations. This allows us to use the observed smallness of the mixings with the third generation to perform a simplified diagonalization of Mu. We first rotate the elements/,/13 and u23 to zero. The 33 element of the resulting matrix v0 determines the top quark m a s s ( m t = v 3 3 ~ u 3 3 ) . The other matrix elements induced by this rotation are of order ~ vll =Ull +

~31Ul3 mt

+

U21 b/23U13

rn~

'

/312 ~-'=b/12-t- U32UI3 "t- U22U23UI3

mt2

m t

1321 ~--'~/21 "j-

U31 ~23

"dr

m t

,

UI1U23U13

m~

'

U22 =U22"t - U32/'/23 "~- UI2U23UI3 m t

rn~

'

(6) (7)

(8)

(9)

U21 U23 UllUI3 + - - , m~ D'/~

(10)

V32 =U32 ,9ff U22U23 + UI2Ul~3

(11)

v31 =u31 +

m~

31 December 1987

small relative size of v, for i, j = 1, 2. We neglect them and consider only the remaining 2 × 2 matrix for the lower generations. Up to negligible corrections u U this matrix is given by v o ( i , j = 1, 2). This ~013023 is easily diagonalized and one obtains me=u22+

U32 U23

m,

UI2b/23Ul 3

+ -

m~

OU2 = U2~I-t- U31U23 "Jr 1/11/223b/~3

rnc

rnc mt

mcmY

1/31 U13 mu ~ U l l --~ _ _

m,

(

+ 1~u12 +

me\

u32u,3"~{ rn,

+

u31u23]

~ L/21

rn,

I/21 //23 U~3

+

Ull Ul~3

01u3= u3~ + _ _ + _ _ m~ m~ m?

02u3= -u32 + _U22U~3 _ + _UI2UI~3 _

m?

m,~

'

(12) (13)

This, of course, again induces elements in the top quark column (u~3, u23). They are, however, suppressed by the smallness of the angles 0,3 and the ~ Remember that we only determine the order of magnitude, not the exact value.

526

/

(16)

rn~

We have neglected terms which are proportional to other terms up to a factor of order one or smaller. The diagonalization of MD is similar. The final mixing angles are a combination from Mu and MD. 0,j = 0 ~ +00r) .

(17)

For the lepton mass matrix nothing is known about mixing angles. We nevertheless adopt the same procedure and take care of the large mixing case by considering the additional contributions to the effective 2 X 2 matrix in the second step. From (12)-(16) we can easily compute the powers P;, P0 in terms of U;j, D o and L o like (18)

Pb = D 3 3 ,

m,

(15)

'

mt

Next we rotate away the elements v3~ and v32. This defines the contributions from Mv to the mixing angles with the third generation: U21U2~3

(14)

'

Ps =min ( D22, D32 -t-D23 - 0 3 3 D12 q-D23 -t-D13 - 2D33) .

,

(19)

For the uncertainty factors we choose n; as the number of undetermined matrix elements on the righthand side of the corresponding formulae ( 12 ) - ( 16 ). Here the contributions involving more than one factor of the heaviest mass are denoted with an asterisk and are not counted in the uncertainty since they are important only under relatively rare circumstances. For example, from (8) one obtains ns= 3, no=4. (We note that m , in contrast with all other mass values should be treated as an unknown matrix element.) The n; derived from (12)-(16) are given in table 1.

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

Table 1 Quantity

Experimental value

n~

Y7 -Y,+

m, mb m~ m~ m~ m~ mo m~ m~ 023 0~3 0~_,

>~41 GeV 4.0 _+ 0.1 GeV 1784.2 _+ 3.2 MeV 0.88 _+ 0.03 GeV 105 _+ 35 MeV 105.695 MeV 3.1 _+ 0.9 MeV 5.4 _+ 1.6 MeV 0.511003 MeV 0.039-0.050 0.0-0.008 0.219-0.225

1 1 1 4 3 3 12 9 9 3 3 7

0.1 l-oo 0.010-0.042 0.011-0.043 0.011-0.019 0.00011-0.0025 0.00037-0.0044 1.6)< 10-6-0.00014 3.4>( 10 6-0.00022 1.0×10 ~'-3.7)<10.5 O.Ol 1-0.17 0.0-0.027 0.039-1.19

This simple counting rule for the uncertainty can be m o t i v a t e d by the following reasoning: F o r two matrix elements with uncertainty factors A~, A2, the uncertainty o f the p r o d u c t ( o r ratio) is a p p r o x i m a t e l y A I2 = x / ~ + Az: if the two Ai are treated as statistically i n d e p e n d e n t errors. The error o f a sum or difference cannot be so easily e s t i m a t e d but a square root addition Zll+2=x/~2 +A~ reflects at least some qualitative features. O u r rule for the error then follows if all m a t r i x elements have the same uncertainty factor A and all terms in ( 1 2 ) - ( 1 6 ) contribute equally. One m a y argue that often not all contributions to a given q u a n t i t y are i m p o r t a n t and therefore the uncertainty for the lower generations is smaller. On the other h a n d the uncertainty o f a given matrix element also tends to increase with the power o f 2 since usually m o r e ratios o f dimensionless couplings are involved (see refs. [2,4] for examples.) N o m o r e accurate estimate o f the uncertainty involved seems possible without using m o r e detailed i n f o r m a t i o n about specific models. O u r simple estimate should be regarded as an e d u c a t e d guess which qualitatively reproduces the increase o f uncertainty for the lower generations. We now turn to the d e t e r m i n a t i o n o f the allowed regions in 2 and the corresponding Pi. We assume first that the rough equality o f Yukawa couplings holds at some large scale M = 10 t7 GeV. The generation symmetry is spontaneously b r o k e n somewhat below this scale. We have to correct for the different scale dependence o f lepton, quark a n d the W - b o s o n masses according to the different r e n o r m a l i z a t i o n group equations o f the corresponding dimensionless cou-

plings. The relevant multiplicative factors in the oneloop a p p r o x i m a t i o n for a small top mass (mt~< 100 G e V ) for the rescaling from I00 GeV to 1017 GeV are 0.76 for the leptons, 0.32 for m , mc, mu, 0.33 for mb, ms, mo and 0.79 for mw ~2. A s t a n d a r d uncertainty A = 2 allows for factors o f four in (corrected) masses to be explained by differences in ClebschG o r d a n coefficients. The regions for the different quantities are given a p p r o x i m a t e l y by

Y7 = 0 . 4 3 x , /x/~A<~2 p' <<.y+ = 0.43

x+,f~M

(20)

for the quarks, and

y;- = x . / ~ < ~ , ~ ", ~ y ?

= x?~/~,J

(21)

for the leptons. The values y,-+ are shown in table 1. Q u a r k masses are taken from ref. [ 5 ] except for the recent UA 1 lower b o u n d on the top quark mass [ 6 ]. The running quark masses at # = 100 GeV are quoted (neglecting electroweak effects). Values for the mixing angles are taken from ref. [7] and the lepton masses from the particle data b o o k [8]. We use a value o f 81.5 G e V for rnw. The allowed values for 2 for the different quantities in terms o f the P, are plotted in fig. 1. The allowed regions o f 2 can be d i v i d e d according to Pb equal to 1, 2, 3 and Pc 1 or 2. There is no solution ~2 For a 160 GeV top mass a further multiplicative factor of 1.3 for rn, 1.1 for rnb, 1.15 for me, mu and 1.2 for m~, md and the lepton masses is needed. 527

Volume 199, number 4 0.01

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0.025

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PHYSICS LETTERS B 0.05

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0.15 X 025

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renormalization group corrections for the fermion masses. This scenario is more relevant for composite models. Yukawa couplings here are a consequence of strong interactions between bound states. We took this into account by replacing mw in (1) by the vacuum expectation value v = 175 GeV. The resulting values for 2 and P, can be found in table 3. In models with a generation symmetry broken somewhat below the unification scale the powers Pi can be computed in terms of the generation q u a n t u m numbers [2]. We can use the results in table 2 to decide if a given set quantum numbers leads to a realistic fermion mass pattern. We have investigated a three-parameter (m, p, r) set of anomaly-free U (1)generation symmetries. These models can all be obtained from compactification of a six-dimensional SO(12) model [9]. The quark and lepton charges are obtained from a linear combination of the U (1)1 subgroup of a generation group SU (2) ~ and another abelian symmetry U(1 )q:

?-

i

I

1 e]s

7

,

l l l

II

------4--

6

i

Q=Q, +rQq .

8 -g

The q u a n t u m numbers of the fermions under S U ( 2 ) ] M U ( 1 ) q are

t

1

[½(3+P)]~/2+[½(3-P)]_,/2,

q:

1

uC: 1

-!1l l a

Ii

1i

de:

Il i

,

[ ½(3-p+2rn)],/2

+ [½(3+p--2m)]_,/2,

[2 i r I "-

[½(3-p-2m)]

~/2

+ [½(3+p+2rn)]-,/2,

i i

L:

I

Iii

15[

?z

Fig. 1. The allowed regions for 2 in terms of the power P, for all masses and mixing angles for the unification scenario. for 2 ~<0.019 and we do not consider 2 i> 0.2 5 because the distinction between differences in c, and different powers of 2 disappears. We have subdivided the region for Pb = 2 (III and IV in fig. 1 ). The allowed values o f P, for the other quantities are given in table 2. The SU (5) example discussed in ref. [2 ] corresponds to case II. The above regions are those relevant for generation symmetries broken at a large scale. For comparison we have done a similar analysis without 528

(22)

i

i

e2s

£

31 December 1987

[½ ( 3 - 3p)],/2 + [ 1 ( 3 + 3 p ) ] _ , / 2 ,

eC:

[½(3+3p-2m)],/2

+ [½(3-3p+Zm)]-,/2

•

(23)

The standard notation is used for the S U ( 3 ) × S U ( 2 ) XU(1 ) y representation. The number in brackets is the SU (2), representation and the subscript the U(1)q quantum number. A negative n u m b e r in brackets means a mirror particle in the conjugate representation under SU (3) × S U ( 2 ) X U(1 ) y × U ( 1 )q whose SU(2)I representation is given by the absolute value of the number in brackets. The mirror particles acquire a mass from spontaneous breaking of the U(1 ) generation symmetry. We eliminate the supermassive quark-mirror pairs, taking

PHYSICS LETTERS B

Volume 199, number 4

31 December 1987

Table 2 Scenario

2

P,

Pb

P,

P.

P~

P~

Pu

Pd

P~

/°23

Pt 3

PI 2

I II 111 IV V

0.019 0.033-0.042 0.10-0.14 0.14-0.20 0.22-0.25

0 0 0,1 0,1 0,1

1 1 2 2 3

1 1 2 2 3

1 2 2,3 3,4 3,4

2 2 3,4 4,5 4-6

2 2 3,4 3,4 4,5

3 3,4 4-6 5-8 6-9

3 3,4 4-6 5-7 6-8

3 3,4 5-7 6-8 7-9

1 1 1,2 1,2 2,3

>~I >/2 />2 >--2 >/3

0 0,1 0,1 0-2 0-2

scenario I:

into account the mixing with light fermions according to the algorithm for mass m a t r i x diagonalization discussed in detail in section 3 o f ref. [4]. This leaves us then with three generations o f light fermions which are linear c o m b i n a t i o n s o f those in (23). We then allow for an a r b i t r a r y charge o f the " l e a d i n g " weak Higgs doublet [2] u n d e r the extra U(1 ) and search for a realistic set o f resulting Pi. These are given by the difference o f the fermion bilinear q u a n t u m numbers and the Higgs ones [2]. We have p e r f o r m e d a c o m p u t e r i z e d scan for p = l, 3, 5, m = - 5, - 4,..., 5 and r = - 11/2, - 9 / 2 ..... 11/2. (This leads to integer differences o f the U(1 ) charge between fermion bilinears.) We found no realistic mass patterns corresponding to cases I - V o f table 2. This d e m o n s t r a t e s how difficult it is to reproduce realistic masses from higher d i m e n s i o n a l field or string theories. (These theories generically fulfil our assumption o f dimensionless couplings all o f the same o r d e r o f m a g n i t u d e so that the structure o f mass matrices should be explained by symmetries.) A realistic f e r m i o n mass p a t t e r n is therefore a very restrictive p h e n o m e n o l o g i c a l criterion for an acceptable g r o u n d states in such theories. F o r a r b i t r a r y generation symmetries it is in general possible to find q u a n t u m n u m b e r s to r e p r o d u c e all the different scenarios discussed here. A rather complete list for scenario II can be f o u n d in ref. [ 2 ]. We list here possible sets o f q u a n t u m n u m b e r s for the different fermions u n d e r an extra U(1 ) that lead to each o f our scenarios:

q (1, 1 , 0 ) , u c (2, 0 , 0 ) , & (2, 1, 1), L(l,l,0),e

c (2,1,1),

scenario II: q(2,1,0),u

c(2,1,0),&

L(2,1,0),e

(2,1,1),

c (2,1,1),

scenario III, IV: q(3,2,0),u

c (3,1,0),d c (3,2,2),

L (3, 2, 0), e c ( 4 , 2 , 2 ) , scenario V: q(4,3, 1),u c (3,1,0),d c (3,2,2), L(4,2,1),e

c (4,2,2).

In each o f these cases the Higgs doublet has zero charge u n d e r the extra U(1 ). Very similar solutions exist for the c o m p o s i t e case. As an example we assume all c i = 1 for the scenarios III, IV m e n t i o n e d above. The following relations a n d mass values for 2 = 1/6 are obtained:

012 = 2 =

1/6 ,

023 =022 = 0 . 0 2 8

(24) ,

(25)

0~3 =032 = 0 . 0 0 5 ,

(26)

mr --Oz3mw = 2 . 3 G e V ,

(27)

Table 3 Scenario

2

Pt

P~

P~

Pc

Ps

P~

Po

Pd

Pe

P-,3

P13

PI2

I II III

0.015-0.020 0.12-0.15 0.17-0.22

0 0,1 0,1

1 2 2

1 2 3

l 2,3 3,4

2 3,4 4,5

2 4 4,5

2,3 5,6 5,6,7

2,3 4,5,6 5,6,7

3 6,7 7,8,9

1 1,2 2

>~1 ~>3 >/3

0 0,1 0,1 529

Volume 199, number 4

PHYSICS LETTERS B

m b =2.3 m, =5.3 GeV,

(28)

m~ = 0 1 2 m b = 8 8 0 M e V ,

(29)

m~ = m2c/mb = 145 M e V ,

(30)

m, =023m ~=63 MeV,

(31)

mu,e = 022 ms = 4 M e V ,

(32)

mc =O{2m~ = 0 . 3 M e V .

(33)

C o m p a r i s o n w i t h the m a s s v a l u e s in table 1 s h o w s surprisingly g o o d a g r e e m e n t d e m o n s t r a t i n g t h a t o u r a p p r o a c h can also w o r k m u c h s m a l l e r u n c e r t a i n t y factors. In this p a r t i c u l a r m o d e l t h e t o p q u a r k mass is large. T a k i n g for u ~ the charges (3, 1, 1) i n s t e a d o f (3, 1, 0) w o u l d l e a d to mflmc=mb/ms, m , = 3 2 GeV.

References [1] S. Dimopoulos, Phys. Lett. B 129 (1983) 417;

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31 December 1987

S. Dimopoulos and H. Georgi, Phys. Lett. B 140 (1984) 67; L. Wolfenstein, Phys. Rev. Lett. 51 (1984) 1945; J. Bagger and S. Dimopoulos, Nucl. Phys. B 244 (1984) 247; J. Bagger, S. Dimopoulos, H. Georgi and S. Raby, 5th Workshop on Grand unification (Providence, RI.), in: Proc. Providence grand unification (1984) p. 95; J. Bijnens and C. Wetterich, Phys. Lett. B 176 (1986) 431. [2] J. Bijnens and C. Wenerich, Nucl. Phys. B 283 (1987) 237. [3] C. Wetterich, Nucl. Phys. B 26l (1985) 461; A. Strominger and E. Witten, Commun. Math. Phys. 101 (1985) 341. [4] J. Bijnens and C. Wetterich, Nucl. Phys. B 292 (1987) 443. [ 5 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [6] S. Geer, talk EPS high energy physics Conf. (Uppsala, Sweden, June 1987). [ 7 ] K. Kleinknecht, in: Proc. Intern. Symp. on Production and decay of heavy hadrons (Heidelberg, May 1986), eds. K.R. Scubert and R. Waldi. [ 8 ] Particle Data Group, Review of particle properties, Phys. Lett. B 170 (1986) 1. [9] C. Wetterich, Nucl. Phys. B 260 (1985) 402; B 279 (1987) 711.