Volume 145B, number 3,4 PHYSICS LETTERS 20 September 1984

THE PHENOMENOLOGICAL PREDICTIONS OF FRITZSCH-TYPE QUARK MASS MATRICES

AND THE UNDERLYING STRUCTURE OF PARTICLE PHYSICS BEYOND THE STANDARD MODEL

Michael SHIN 1

L yman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 12 June 1984

The phenomenological predictions of Fritzsch-type quark mass matrices and their implications on the underlying struc- ture of particle physics beyond the standard model are considered. Using a recent analysis on the K-M mixing matrix from the b-quark lifetime etc., R b =- r(b ---, uev-")/r(b ~ cev-') is predicted to be in the range 1.44 × 10 -3 ,; R b ~; 1.87 X 10 -2 , while the present upper bound on R b is 5 X 10 -2 . For B K = 0.33, obtained by Donoghue et al. using current algebra and the measured zX/" = 3/2 contribution to k ---, ~r~r, the physical (=- constituent) mass of the t-quark (for A~--~ = 100 MeV) is predicted to be (73.5 + 8.5) GeV and so the t-quark will not be found in PETRA and LEP experiments. For the same value of B K, our predictions on the value OfRb, I Vcbl, and the CP violating phase 8' in the K-M mixing matrix are 0.79 × 10 -z ~; R b ,; 1.01 X 10 -2, 0.066 ,; IVcbl ~ 0.070, 85.6 ° ,; 8' ~ 97.4 °, and are subject to future experiments. Our predicted value of 8' (which is independent of experimental input from CP violating phenomena) is in good agreement with the CP violation strength parameter Re e and strongly supports the validity of the Fritzsch form of quark mass matrices. It is con- jeetured that the underlying structure of particle physics is such that the two fundamental phases, o and ~', that appear in the weak charged current mixing matrix, are equal to -~r/2, corresponding to maximal CP violation in the weak interaction sector. This, together with the predicted value of the t-quark mass, can be used as a guide in future model building (o = r = -n/2 can be achieved by the phases of the VEVs of complex Higgs fields). The dependence of our results on the quark mass ratios is considered and a large mass scale for light quarks (u, d, s) is shown to be inconsistent with the Fritzsch form of quark mass matrices. Measurement of the value of R b at CLEO and CUSB and improved determination of I Vcb[ will be the best tests of the validity of Fritzsch-type quark mass matrices in the near future.

One of the most puzzling aspects of the standard model [1] of particle physics at the present time is the ob-

served generation structure of quarks and leptons. Besides the observed hierarchies in the mass eigenvalues of quarks and leptons, the observed values of various mixing angles and the phase in the weak charged current mix-

ing matrix (known as the Kobayashi-Maskawa (K-M) matrix [2] in the literature) are beyond our comprehension at this time.

One thing that we can do at this time is to guess a form for the mass matrices and examine its validity by check- ing its phenomenological implications against known experimental data. Once a plausible form for the mass ma- trices is found, one can try to determine the underlying dynamical structure which produces it.

One of the most attractive forms for the mass matrices is the one considered by Fritzsch [3]. For this form of a mass matrix, only the heaviest family (generation) has a diagonal mass term. All other masses are generated by off-diagonal mixings between neighboring generations. This form for the mass matrices requires the minimum number of input parameters (6 quark masses and 2 phases) to predict the 6 quark mass eigenvalues and the 4 mix- ing angles in the K - M matrix and gives rise to the well-known relation, tan 0 c --~ (md/ms)l /2, where 0 c is the Cabibbo angle. It is the form of mass matrix with the most predictive power. In this letter, we will be concerned with this form of quark mass matrices. We shall not ask for the dynamical origin of this particular form of mass ma- trices, but assume that the underlying structure of particle physics is such that the low energy effective theory

1 Research supported in part by the National Science Foundation under Grant No. PHY82-15249.

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is given by the standard SU(3)C X SU(2)L X U(1)y theory with Fritzsch-type quark mass matrices and deduce model-independent phenomenological predictions of these quark mass matrices. To do this, we shall compare the weak charged mixing matrix (K-M matrix) produced by Fritzsch-type quark mass matrices with constraints [4] on the K-M matrix from the recently measured b-quark lifetime, the inclusive semi-leptonic branching fraction of B mesons BR(B -* eVx), the upper bound on P(b -~ ueV)/P(b -~ ceV), and the weak CP violation strength param- eter Re e.

We begin our discussion by considering the weak charged current mixing matrix V F produced by the quark mass matrices of Fritzsch form. A 3 X 3 mass matrix M F is defined to be of Fritzsch-type [3] i fM F is of the fol- lowing form,

" 0 IA le i~A 0

i¢~t M F= b4 e 0 Inle i ~ (1)

0 IBle i¢~ ICle ice

where the constants A, B, and C and the phases are different for U and D mass matrices. In such a theory, the weak charged current mixing matrix takes the form [3,5]

1 0

V F = (RFU) T 0 e ia R D F ' 0 0 e i

where R F is the matrix

m2(m3-m2)m3 ~lZ2 ( m 2 + m l ~ m 3 - m a ) ]

( ma(m3_m2 ) ~112

( ml(m2_ml)(m3+ml) ~a/2 --\(m2+ml)(m3-m2+ml)(m3-ml) ]

(2)

) l mlm3(m,+ml) ~1/,[ mlm2(m2-ml) ~1/2 /

- \ ( m 2 + m l ~ ( m 3 + m 2 i ] \(m3-ml)(m3-m2+ml)(m3+m2)] ] ¢ m2(m3+ml) ~112 [ (m2-ml)m3 ~1/2 ] ~ (m2+ml)(m3+m2)] ~ (m3-rnl)(m3+m2)] t" i (m~-ml)m2(m3-m2~) "~1/21 (m3-m2)m3(m3+ml) ~'/z:l

- \ (m2+rhl)(rn3-m2+ml)(m3+m2)] \(m3-ml)(m3-m~+ml)(m3+m2)] j (3)

This charged current mixing matrix V F is equivalent to V, with

%% %s o V=]---C S e +CeSfSrei~ ' %% +SeS#Srei& - S % e i ' , (4)

L-cecsa-sos e-i ' ---C SoSo+CoS3e-is' %C[3 which was the new parameterization of the K-M matrix introduced m our earlier work [4].

Since all the experimental constraints on the K-M matrix were reduced to constraints on O,/~, % fi' in our earlier work [4], we only need to study I(VF)usl, I(VF)eb I, I(VF)ub 1, and the CP violating phase fi' to deduce the phenomenological predictions from the form of V F given in eqs. (2), (3). Notice that in eqs. (2), (3). V F has only 3 unknown parameters (m t, o, r) while the experimental constraints on V in eq. (4) severely constrain I(VF)usl, [(VF)eb I, I(VF)ub l, and the CP violating phase 8 '. This will lead to severe constraints on the t-quark mass, and the phases a and r. The following hand-waving arguments will help the reader to understand the main results of this work. First of all, constraints on m t come from two sources. One is from I(VF)ebl. Since

I(VF)eb [ ~ [ms/m b --(me/mt)ei(r-cr)l, (5)

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mt/m c cannot be too large. Too large a value ofmt/m c will give I(VF)ebl ~ (ms/mb)l/2 (~ (1/30)1/2 "" 0.18), while the present upper bound on I(VF)ebl is 0.07 from the observed long lifetime of the b-quark. This leads to the upper bound on mt/m c. The other is from the observed value of the CP violation strength parameter Re e in K0-K, 0 system. Since Re e is proportional to I(VF)ub I I(VF)cb I sin 6'F(m2[M2w ) where F(m2/M 2) is an increas- ing function of mt, and I(VF)ubl is very small

I(VF)ub I --~ I(m d/mb)l/2(m s/mb ) + (mu/mc)1/2 [(ms/mb)l/2 e i° -- (me/mr)1/2 eir]] -,, [(m d/mb)l/2(ms/mb)

+ (mu/mc) 1/2(VF)cb i ' (6)

m t must be large to explain the strength of Re e. This leads to a lower bound on m t in the neighborhood of the upper bound on m t from I(VF)cb I. The experimental value of 8' must also be in the neighborhood of the max- imal value, zr/2, due to the smallness of I(VF)ub I and I(VF)cb I. This agrees well with the calculated value of 8', 6' ~-- o ~- zr/2 (see below). Constraints on the phases o and r come from the observed long lifetime of the b-quark and the value of the Cabibbo angle. The observed long lifetime of the b-quark demands the smallness of I(VF)cbl (I(VF)cbl < 0.07). This, in turn, requires that the phases cr and r are roughly equal to each other in order to give maximal cancellation in l(VF)cb I since (VF)cb = (ms/mb)l/2 e i° -- (ms~mr)l~2 e it. Thus we have o --~ r. To ex- plain the observed value of I(VF)usl (old Cabibbo angle), I(VF)usl ~ (rod~ms)l~2, the value of the phase o is re- quired to be in the neighborhood of -7r/2 (or 7r/2), since (VF)us ~ --(rod/ms) 1/2 + (mu/me)l/2 e i°. These two constraints completely fix the phases o and r to be o ~- r --~ -7r/2 (or 1r/2).

In order to give definite numerical values for various quantities, we need to know the values of quark mass ra- tios in (3). These quark masses which are the parameters that appear in the lagrangian, renormalized at some scale /l (say bt = 1 GeV), are commonly called the running masses of the quarks and are extensively studied and discuss- ed in ref. [6]. (The running mass is different from the physical (= constituent) mass, which is the location of the pole in the full propagator.) Here we simply quote their results. The results are

md/m u=1.76+0.13, ms/m a=19.6-+1.6,

mu(1 GeV) = (5.1 -+ 1.5) MeV, md(1 GeV) = (8.9 + 2.6) MeV, ms(1 GeV) = (175 -+ 55) MeV,

me(1 GeV) = (1.35 -+ 0.05) GeV, rob(1 GeV) = (5.3 --- 0.1) GeV, for A~--~ = 100 MeV. (7)

From this we deduce that

ms(1 GeV) mc(1 GeV) 9ga "7 +124.2 mb(1 GeV) md(1 G e V ) - 19.6 + 1.6, mu(1 GeV) = ~ . . . . -67.7, ms(1 GeV) - 30"3+10"1 -5 .2 '

me(1 GeV) = (1.35 + 0.05) GeV. (8)

The ratio of the running masses at the same scale/a is independent of the scale parameter/a and is equal to the ra- tio of current algebra masses for light quarks (the word "current algebra mass" is meaningless for heavy quarks).

Using the central values in (8)

ms/m d = 19.6, mc/m u = 264.7, mb/m s = 30.3, (9)

and me(1 GeV) = (1.35 + 0.15) GeV, we present our main results in fig. 1. In fig. 1, we have plotted the upper bound on the physical (= constituent) mass of the t-quark, mtP, for various values of A~--~. They are obtained from the upper bound on the running mass rot(1 GeV) given by the present bound on [Vcb[ (0.04 ~ I Vcbl~ 0.069 [4] ), using the relation between mtP and rot(1 GeV) shown in fig. 2. (To relate m p with mt(1 GeV) in fig. 2, we have used the two-loop renormalization group equations [7] .) In fig. 1, we also have plotted the lower bound on mtP for various values of B K. They are obtained from the strength of the CP violation parameter Re e using table 3 of our earlier work [4]. In doing so, we have used the value of [Vub [/1Veb [ determined from the values of the

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r mtP(GeV) Bk=O'33

I00[-- Bk=O,40~

90I-- Bk=0.60 ~ ~ .,~1 A~=5OMeV / \ \ \ / .b'X~-solOOMeV 8°I

70~ ~ ~ = 200MeV

,o t ,-;.o.o7

0.04 o.o5 0.06 o,o7 IVcbl

Fig. 1. Allowed region (shaded) for mc(1 GeV) = (1.35 + 0.15) GeV, ms/m d = 19.6, rns/m b = 30.3, mc/m u = 264.7 (with B K = 0.33, AI~I~ = 100 MeV).

14C

12C

1(3£

8C

6C

4C

2C

A~=5OMeV

=IOOM~

= k:~X3#de\

I I I I 20 40 60 80 K)O 120 140 160 180 mt(iGeV)

Fig. 2. Value of m~ (in GeV) as a function of mr(1 GeV) (in GeV) for various values of A~-g.

phase, o and r, using the experimental input, IVusl = 0.231 -+ 0.003 [8] and 0.04 ~< IVcbl ~ 0.069. F o r B K = 0.33, obtained by Donoghue et al. [9] ,1 using current algebra and the measured z2d = 3/2 contribution to k -+ mr, we have shaded the allowed region for A~--~ = 100 MeV in fig. 2 (A~--~ = 100 +1°0_ MeV is the present value). The al- lowed region is bounded by

65 GeV ~< m p ~< 82 GeV, 0.066 ~< I Vcbl ~< 0.07, 0.062 ~< I Vub/Vebl( = 3/7) ~< 0.07. (10)

Taking the central value of m p, m p (which is observable through the toponium threshold, mr{ = 2m p [1 + O(a2)] ) is predicted to be

mtP = (73.5 -+ 8.5)GeV. (11)

From the value of IVub/Vcbl in (22), the semi-leptonic decay width ratio R b is predicted to be

0.79 X 10 -2 ~< R b ~ P(b -+ ue~)/F (b --> ce~) ~< 1.01 X 10-2. (12)

This is to be compared with the present upper bound on R b [11], R b ~< 5 × 10 -2. (The range of R b, which is independent o f B K, is found to be 1.44 X 10 -3 ~< R b ~< 1.87 X 10-2, corresponding to the whole region in fig. 1.)

The predicted value of IVcbl in (10) is greater than 0.06, contrary to our favored value of IVebl (~< 0.06) [4] from long lifetime of the b-quark. Although we cannot exclude the possibility of I Vebl being as large as 0.07 at this time, if the upper bound on IVcbl is determined to be as small as 0.06 by improved measurements on the b- quark lifetime and the b-quark mass parameter fits from charged lepton spectrum in the future, then the value of BK, B K = 0.33, obtained by Donoghue et al., would not be consistent with Fritzsch form of quark mass matrices

~1 BK = 0.33, obtained by the authors of ref. [9] using current algebra and the measured ~ = 3/2 contribution to k ~ 7rlr, is the most reliable, model-independent value ofB K. This result was rederived by Ginsparg et al. in the context of the chiral per- turbation theory and they obtained B K "- 0.37. See Ginsparg [10].

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if CP violation is to be explained by the phase in the K-M matrix alone. B K would then have to be at least as large as 0.6 or we must have some other source of CP violation besides the CP violating phase 6 ' in the K-M ma- trix. This would lead to the implication of new physics beyond the standard model.

The CP violating phase 6' in the K-M mixing matrix (4) is easily calculated from the form of V F given in eqs. (2), (3). Using a quark phase-independent relation,

ph(V12) + ph(V23) - ph(V22) - ph(V13) = ph[(VF)12] + ph[(VF)23] - ph[(VF)22] - ph[ (VF)13] , (13)

where ph(~/ ) - phase of ~ / , etc., 6'is found to be

77.2°~<6'~<122.1 °, f o r s i n ( r - o ) < 0 ; 70.3°~<6'~<144.7 °, f o r s i n ( r - o ) > 0 . (14)

(For most values of rot(1 GeV), except for very small values of mt(1 GeV), 6' is approximately given by 6' ~-- - o from eq. (13).)

For the allowed region (shaded) in fig. 1, the corresponding values of 6' are calculated to be

85.6 ° ~< 6' ~< 97.4 °. (15)

This is in good agreement with the values of 6' calculated from the CP violating Re e,

8 '= (80 -+ 30) ° (16)

in our earlier work (see figs. 6a-6d in ref. [4] ). We do not believe that this is an accident. The value of 6' in (15) is closely related to the observed value of

Cabibbo angle and long lifetime of the b-quark and is independent of CP violation phenomenology. We believe that this is further experimental evidence that supports the validity of a Fritzsch form of quark mass matrices. The observed value of Cabibbo angle (tan0 c "" (m d/ms)l/2), the present upper bound on IVub/Vcb I (<~ 0.165),

mt P (GeV)

10(3 Bk=O.33

90 B k = 0 . 4 0 ~

coO Bk ~0'8

50 x ¢ = 0 . 0 8

\ f:ooo

~ B 3O 7 =O.09

20 ~ 0 095

I I I o.o4 0.05 o.o6 0.07 IVcbl-

Fig. 3. Allowed region (shaded) for mc(1 GeV) = (0.135 + 0.15) GoV, ms/m d = 19.6, mb/m s = 25.1, mc/m u = 197 (with B K = 0.33, A~-~ = I00 MeV).

13C

12(3

11C

10(3

9(3

80

70

60

50

,mP(GeV)

Arz-~5OMeV

I I I I 0.04 0.o5 o.o6 o.oz Iv:~ I

Fig. 4. A l l o w e d r e g i o n ( s h a d e d ) f o r mc(1 GeV) = (1.35

± 0.15) GeV, ms/m d = 19.6, mb/m s = 40.4, mc/m u = 388.9 (with B K = 0.33, A~--~ = 100 MeV).

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mtP(GeV) m P (GeV)

I1(

100

90

80

7C

8C

5¢

40

8k=0.33

B k=0"40 ~ II A~ = 50MeV

Bk=O.60 \ ~ A M - ' S = I00 MeV

, , , ° o o , 5

0.04 0.05 o.o6 0.07 I Vcbl

Fig. 5. Allowed region (shaded) for mu(1 GeV) = 4.2 MeV, rod(1 GeV) = 7.6 MeV, ms(1 GeV) = 150 MeV, me(1 GeV) = (1.35 ± 0.15) GeV, mb(1 GeV) = 4.8 GeV (with B K = 0.33, A~--~ = 100 MeV).

7C

6C

5C

4C

30

20

I0

Bk=0.4

B k =0.8 Bk=O'G I ~ \ \ i \ ,

~,=,.o \ ,~ \I/~.V' \ \ i ~ "K~..- A~= 50MeV \ \ ', ~ A W I O O M e v

, AI~i~=2OOMeV

~=0.12

I I I 0.04 o.o5 o.o8 o.o~ iv~b[

Fig. 6. Upper and lower bounds on m p for mu(1 GeV) = 8.4 MeV, md(1 GeV) = 15.2 MeV, rns(1 GeV) = 300 MeV, me(1 GeV) = (1.35 ± 0.15) GeV, mb(1 GeV) = 4.8 GeV, for various values of A~-~ and B K.

and the observed value o f the CP violation strength parameter Re e, agree too well with the predictions from the Fritzsch form o f quark mass matrices. For values o f 6' given in (15), the corresponding values o f the phases o and r are a = - ( 9 3 . 5 +6.5-~o -5.5J ' Ir - al ~< 15.2 °, and it is our conjecture that the underlying structure of particle physics is such that a = r = - lr /2 (at least in tree level), corresponding to maximal CP violation in the weak inter- action sector. We believe that the phases, o and r, that appear in (2) are more fundamental (independent of phase convention) than the phase 6 ' in the K - M matrix (4). This conjecture (a = ~- = -rr/2), together with the range o f mtP given in (10), can be used as a guide in future model building (a = r = -rr/2 can be achieved by the phases o f the VEVs o f complex Higgs fields).

Finally, in figs. 3 - 6 , we have considered different choices of quark mass ratios other than the one in (9), and shaded the allowed region. In fig. 3 (fig. 4), we used the lowest (largest) possible values of mb/m s and mc/m u in (8). In fig. 5, we used a frequently used set o f the running masses, and we have considered a possible factor o f 2 in the light quark mass scale in fig. 6. These considerations show that too large a mass scale for the light quarks (u, d, s) is not consistent with the Fritzsch form of quark mass matrices and our results in fig. 1 are fairly stable against variations in the quark mass ratios.

To summarize, we have considered the phenomenological predictions o f Fritzsch-type quark mass matrices and their implications on the structure o f particle physics beyond the standard model in this letter. The results are very striking and are well summarized in the abstract. Some o f our results presented in this letter are in agree- ment with the previous analyses [ 12] , but most o f our results (predicted range of mtP, I Vcbl, etc.) are not. Future experiments on R b (= P(b ~ ueg)/P(b --, ceV) at CLEO and CUSB, improved bounds on I Vebl from improved measurements on r B and the b-quark mass parameter fits from the charged lepton spectrum, the t-quark mass (the t-quark is predicted not to be found in PETRA and LEP experiments since the toponium threshold is predict- ed to be at least 130 GeV), and other CP violating phenomena such as B 0 - B 0 mixing, will ultimately determine the validity o f the Fritzsch form of quark mass matrices. A more detailed account of this work will be found else- where [13] .

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I would like to thank H. Georgi for helpful discussions and suggestions.

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[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [3] H. Fritzsch, Nucl. Phys. B155 (1979) 189;Phys. Lett. 73B (1978) 317;

L.F. Li, Phys. Lett. 84B (1979) 461. [4] M. Shin, Harvard University preprint, HUTP-84/A024 (1984), to be published in Nucl. Phys. [5] H. Georgi and D.V. Nanopoulos, Nuel. Phys. B155 (1979) 52. [6] J. Gasser nad H. Leutwyler, Phys. Rep. 87 (1982) 77. [7] O. Nachtmann and W. Wetzel, Nuel. Phys. B187 (1981) 333;

R. Tarrach, Nuel. Phys. B183 (1981) 384. [8] M. Bourgin et al., Z. Phys. C21 (1983) 27. [9] J.F. Donoghue et al., Phys. Lett. l19B (1982) 412.

[10] Ginsparg et al., Phys. Lett. 127B (1983) 265. [11] S. Stone, Invited talk Intern. Symp. on Lepton and photon interactions at high energies (Cornell, August 1983), CLNS 83/

583. [12] A. Davidson, Phys. Lett. 127B (1983) 412;

A. Davidson, V.P. Nait and K.C. Wall, Phys. Rev. D29 (1984) 1513; V.P. Nalr, L. Michel and K.C. Wali, Phys. Lett. 138B (1984) 128.

[13] M. Shin, to be published in Nucl. Phys.

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