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Experimental determination of weak mixing angles in the six quark scheme
Volume 130B, number 6
PttYSICS LETTERS
3 November 1983
EXPERIMENTAL DETERMINATION OF WEAK MIXING ANGLES IN THE SIX QUARK SCHEME K. KLEINKNECHT 1 and B. RENK Institut fffir Physik der Universit2tt Dortmund, Dortmund, Fed. Rep. Germany
Received 16 August 1983
Experimental results on weak decays of hyperons and B mesons, on neutrino production of charm quarks and on the B lifetime are used to obtain, in a combined fit, values for the three mixing angles in the Kobayashi-Maskawa scheme: sin 0 t = 0.231 ± 0.003, 0.015 < sin 02 < 0.09 and sin 03 < 0.04. In the Maiani parametrization, the angles obtained are: sin 0 = 0.231 +-0.003, Isin3'1 = 0.052_+°.b°l~and sin/3 < 0.008.
1. I n t r o d u c t i o n
The six quark mixing scheme proposed by Kobayashi and Maskawa [1 ] serves as a useful parametrization o f the connection between generations o f quarks. The elementary Uik o f the quark mixing matrix (i = u, c, t; k = d, s, b) are parametrized in terms o f three angles 0 1 , 0 2 and 03 and one phase 6, possibly related to C P violation (table 1). If C P violation is due to quark mixing, then this phase 6 is related to the parameter e describing the admixture o f wrong C P parity in the long- and shortlived neutral K meson states, measured to be e = (2.28 -+ 0.05) × 10 - 3 X e x p ( i n / 4 ) [2]. An approximate relation derived by Pakvasa and Sugawara [3] is lel = i(mt - mc)/rhcl × sin20 2 tan0 3 sin fi/(2x/~ cos01) where m t and m c are the top - and charm - quark masses. An alternative parametrization o f the matrix U h a s been given by Maiani [4], in terms o f angles O, 7, and/3 (see table 2). Experimentally, information on the weak mixing
Table 2 Parametrization of the quark mixing matrix. Maiani parametrization [4 ].
1 Also at CERN, Geneva, Switzerland.
Table 1 Parametrization of the quark mixing matrix. KobayashiMaskawa parametrization [1 ]. Cl
I
~
U = --SlC 2 ~--SI$2
81C3
S183
+ei6s2S3 c l e 2 s 3 ei6s2c3 eifc2s3 ClS2S 3 + ei6c2c3
CLC2C3 CLS2C3 --
0.031-9163/83/0000--0000/$ 03.00 © 1983 North-Holland
I CBCo U =-
-s,ycos[3e~
C~SO - sOc.~ ' c3,cO-sTs[3soei6
sl3 .,] .~,
- s[3c3,c 0 + s~/s0 e -i~ - c~s(3so - s3,C0 e -~o
sTc~3e16
c 3, c#
J
angles comes from measurements o f weak decays of light and heavy quarks and from neutrino production o f charm quarks as observed in dimuon events, as summarized in previous papers [ 5 - 8 ] . Recently, new results on the B meson lifetime [9,10] and on hyperon semileptonic decays [11] have been obtained. In this paper, we summarize the impact o f all these constraints on the weak mixing angles. We first go through the constraints on the coupling parameters Uik , a n d then proceed to derive bounds on the mixing angles. 2. C o n s t r a i n t s o n m a t r i x e l e m e n t s 2.1. L i g h t q u a r k c o u p l i n g s C o u p l i n g Uud. This coupling parameter has been
determined from a comparison o f measured rates o f nuclear beta decays with that o f muon decay. Two different evaluations of this quantity have been made and their results are Uud = 0.9730 + 0.0024 [6] and Uud = 0.9737 -+0.0025 [12]. Combining these two, one obtains Uud = 0.9733 -+ 0 . 0 0 2 4 .
(1) 459
Volume 130B, number 6
PHYSICS LETTERS
Coupling Uus. In a series of experiments, the WA2 collaboration has studied five different hyperon semileptonic decays, i.e. the electronic weak decays 2 ne~, 2;- ~ A e - b , E - -+ A e - ~ , E-- ~ ~ 0 e - ~ and A p e - ~ . Including radiative corrections and using in addit.ion the neutron lifetime [ 13], this experiment gives [11] Uus = 0.231 -+ 0.003.
(2)
This represents a substantial improvement over former analyses [6,12].
O)
from the CDHS collaboration [14]. This value is obtained by measuring the charm production by neutrinos and by antineutrinos from nuclei and by subtracting out the contribution of charm production from strange quarks, leaving the one from d quarks. Coupling Ucs. In charged current reactions this coupling appears always together with the strange-sea structure function xs(x) or its integral S = fx s(x)dx, where we use the following notation: s(x) is the quark density distribution of strange quarks in the proton in the Bjorken scaling variable x, u (x) and d (x) are the distributions of up and down quarks, with U = fxu(x)dx andD = fx d(x)dx, and similarly for antiquarks. The quantity measured is IUcs 12. 2S [5]. In the absence of an independent determination of S, only the upper limit for 2S given by SU(3) symmetry, 2S ~< U + D, and a corresponding lower limit on [Ucsl can be obtained. The product I Ucsl 2 • 2S can be extracted in three ways from the neutrino and antineutrino dimuon production data [8]. We use here the results from the x distribution of the neutrino dimuons [8,14] I Ucs 12/U2d = (6.26 + 0.73) (1 + ~*)]~,
(4)
and the one from the cross-sections of neutrino- and antineutrino4nduced dimuon production using the semileptonic branching ratio of D mesons: IUcsl 2 = (0.41 +-0.09) (1 +
a*)/a,
(5)
where a = 2S/(U + D) is the ratio of m o m e n t u m fractions carried by strange and nonstrange sea quarks in the nucleon, and a* = 2S(U2s + U2cs/rs)/(U+ D) is the same ratio modified by the threshold suppression 460
factor r s for the charm quark mass, which is r s = 1.5 for the experiment considered [14,15].
2.3. Bottom-quark couplings Ratio IUub I/IUcbI. We use the value I f u b I/IUcbl < 0.2
(90% C L ) ,
(6)
obtained by the CLEO [16] and CUSB [17] collaborations at the Cornell storage ring CESR. Coupling Uub. From the unitarity relation I Uub 12 = 1 - IUudl 2 - IUusl 2 and eqs. (1) and (2), we obtain IUub 12 = --0.0006 -+ 0.0060.
2.2. Charm-quark couplings. Coupling Ucd. We take the value Ucd = 0.24 _+0.03,
3 November 1983
(7)
B Lifetime. The lifetime r B o f b flavoured hadrons, apart from phase-space factors, depends on the magnitude of the coupling Ucb and Uub. In fact [18] rB = 0.93 ×
lO--14s/(2.751Ucb12+7.71Uub12).
(8)
The previous upper limit obtained by JADE collaboration [19], r B < 1.4 × 10 -12 s at 95% CL, is significantly improved by the recent measurements of the Mark II and MAC collaborations [9,10]. Using their high precision vertex chambers and tagging b decays by their semileptonic decays into muons or electrons they obtain rB=
+4.9 -+ 2.6) X 10 -13 s (10.4_32
r B = (18-+6 +4) X 10-13 s
[10],
[9],
(9)
which in turn gives the following constraint (iUc b [2 + 2.8 IUub 12)0.5 = a n~9 +0.009 . . . . . -0.007 "
(10)
2.4. Combined fit Using the constraints eqs. ( 1 ) - ( 7 ) and (9), we obtain for a minimum ×2 = 1.2/5 DF the values sin 01 = 0.231 +- 0.003, a -- 2S/(U + D) = 0.49 -+ 0.07 and values of sin'02 and sin 03 with the error contours in the (sin 02, sin 03) plane given in fig. 1. These contours vary slightly with the value o f the phase 5. From the resulting error contours, we obtain a finite value of 0.015 < sin 02 < 0.09 and an upper limit for sin 03 < 0.04. We conclude from this analysis that the second mixing angle 02 is smaller than the first one, 01, i.e. sin 02/sin 01 < 0.4. The third angle, 03, is still compatible with zero, with the upper limit sin 03 < 0.04 at the 67% CL. This pattern of decreasing mix-
Volume 130B, number 6
PHYSICS LETTERS r
/
1
/
,/
/
J..
/
/ /
/
/¢"
/ / /
/
/ /
/
cF ¢:
2Or
/
I"
\-A-2.
/
0.05
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I0
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0 I0 -
/
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015
/
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3 November 1983
0 6=90 o ~ _ _ _ ~
]
J
0.05
mL=lO_OGeV
± _ _
0.10 sin [ 3 / t a n 13
0.15
6=10 ° -0.05
0 5
010
015 Sin 0 3
Fig. 1. Error contours in the (sin 02, sin 03)-plane (sold lines) for three values of the phase 8 (10 ° , 90 ° and 170°). Also shown (dashed lines) are results from a phenomenological analysis [20] of the K L - K S mass difference and the CP violation parameter e, for two values of the top quark mass, mt. Along these curves, the phase 8 varies from 0 ° at sin 03 = 1 to 90 ° at sin03 = 0 on the lower branch, and from 90 ° at sin03 = 0 to 180 ° at sin03 = 1 on the upper branch.
ing angles means that weak transitions b e t w e e n m e m bers o f different quark families are suppressed more for heavy quarks t h a n for light ones. On the o t h e r h a n d , the curves given in fig. 1 f r o m a p h e n o m e n o l o g i c a l analysis [20] o f the parameters o f t h e neutral kaon system can be c o m p a r e d w i t h the e x p e r i m e n t a l l y allowed range o f angles. Along these curves the phase 8 varies f r o m 0 to n/2 o n the l o w e r branch as sin0 3 goes f r o m 1 to 0, and f r o m 7r/2 to 7r o n the u p p e r b r a n c h as sin 0 3 goes f r o m 0 to 1. The constraint sin 03 < 0.04 can be used to set the limit 2 ° <." 8 < 178 ° . A d e t e r m i n a t i o n o f 8 requires specific e x p e r i m e n t s on CP ~iolation in the K--O-K 0 or B-'0-B0 systems. Analogously for the Maiani p a r a m e t r i z a t i o n [4], the error c o n t o u r s in the plane o f the parameters sin 7 (corresponding a p p r o x i m a t e l y to sin 0 2) and sin/3/tan0 (corresponding to sin 0 3) are given in fig. 2. Here error c o n t o u r s at t h e one ( l o ) and t w o (20) standard
(I
-0.10
1
1
I
Fig. 2. Error contours in the plane of mixing parameters sin 3, and sin t3/tan 0 in the Maiani parametrization. One standard deviation (1 a) and two st. dev. (2or) contours are given.
deviation level are given. These c o n t o u r s are nearly ind e p e n d e n t o f the phase angle 8' in the Maiani parametrization. The values for the angles are sin 0 = 0.231 -+ +0 015 0.003, Isin 71 = 0 . 0 5 2 _ 0 ~()10 and sin/3 < 0.0077. F r o m the range o f values o f the mixing angles in either par a m e t r i z a t i o n , the values o f the K o b a y a s h i - M a s k a w a matrix elements can be o b t a i n e d . These are given in table 3. It is evident that the error on these matrix
Table 3 Elements of quark mixing matrix Uik from fit of experimental constraints (1 std. dev. range).
u c t
d
s
b
0.9723-0.9737 0.228-0.234 0.003-0.016
0.228-0.234 0.9704-0.9726 0.041-0.066
0.000-0.008 0.042-0.067 0.9977-0.9991
461
Volume 130B, number 6
PHYSICS LETTERS
Table 1 Elements of quark mixing matrix Uik from experimental constraints if number of quark flavours is larger than 6. 0.9709 0.9757 0.21-0.27 0.00-0.12
0.228-0.234 0.78-1.00 0.00-0.58
0.000-0.013 0.042-0.067 0.000-0.999
e l e m e n t s f r o m the c o m m o n fit is, for m o s t o f t h e m , m u c h smaller than the one o b t a i n e d f r o m individual e x p e r i m e n t a l b o u n d s o n one m a t r i x e l e m e n t . It also appears f r o m the values in table 3 that, apart f r o m the diagonal elements, and the three off-diagonal elements observed directly up to n o w , (Uus, Ucs and Ucb), there is o n l y one o t h e r non
References [1 ] M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 4 9 (1973) 652.
462
3 November 1983
[2] K. Kleinknecht, Ann. Rev. Nucl. Sci. 26 (1976) 1. [3] S. Pakvasa and H. Sugawara, Phys. Rev. D 14 (1976) 305. [4] L. Maiani, Proc. Intern. Symp. on Lepton and photon interactions at high energies (Hamburg, 1977) p. 877. [5] K. Kleinknecht, Proc. 12th Intern. Neutrino Conf. (Balatonf'ured, 1982) (Central Res. Inst. Physics, Budapest. 1982), Vol. 1,p. 115. [6] E.A. Paschos and U. Tilrke, Plays. Lett. 116B (1982) 360. [7] S. Pakvasa, Proc. 21st Intern. Conf. on High energy physics (Paris, 1982), J. Phys. 43, Suppl. 12 (1982) p. C3234. [8] K. Kleinknecht and B. Renk,Z. Phys. C15 (1982) 19; preprint Univ. Dortmund 83-276, to be published in Z. Phys. C. [9] MAC Collab., paper given at EPS Europhysics Conf. on High energy physics (Brighton, July 1983) by N. Chadwick. [10] Mark I1 Collab., paper given at EPS Europhysics Conf. on High energy physics (Brighton, July 1983) by G. Hanson. [ 11 ] M. Bourquin et al., WA 2 Collab., preprint July 1983 ; J .M. Gaillard, private communication. [12] R.E. Shrock and L.L. Wang, Phys. Rev. Lett. 41 (1978) 1692;42 (1979) 1589. [13] C.H. Christensen et al., Phys. Rev. D5 (1972) 1628; J. Byrne et al., Phys. Lett. 92B (1980) 274. [14] H. Abramowicz et al., Z. Phys. C 15 (1982) 19. [15] H.G.J. de Groot et al., Z. Phys. C1 (1979) 143. [16] B. Gittelman, CLEO Collab., Proc. 21st Intern. Conf. on High energy physics (Paris, 1982), J. Phys. 43, Suppl. 12 (1982) p. C3-110. [17] P. Franzini, CUSB Collab., Proc. 21st Intern. Conf. on High energy physics (Paris, 1982), J. Phys. 43, Suppl. 12 (1982) p. C3-114. [181 M.K. Gaillard and L. Maiani, Proc. Summer Institute on Quarks and leptons, Carg~se 1979 (Plenum, New York, 1980) p. 433. [19] W. Bartel et al., Phys. Lett. 114B (1982) 71. [20] L.L. Chau, W.-Y. Keung and M.D. Tran, BNL preprint (Aug. 1982); L.L. Chau, Phys. Rep. 95 (1983) 1.
PttYSICS LETTERS
3 November 1983
EXPERIMENTAL DETERMINATION OF WEAK MIXING ANGLES IN THE SIX QUARK SCHEME K. KLEINKNECHT 1 and B. RENK Institut fffir Physik der Universit2tt Dortmund, Dortmund, Fed. Rep. Germany
Received 16 August 1983
Experimental results on weak decays of hyperons and B mesons, on neutrino production of charm quarks and on the B lifetime are used to obtain, in a combined fit, values for the three mixing angles in the Kobayashi-Maskawa scheme: sin 0 t = 0.231 ± 0.003, 0.015 < sin 02 < 0.09 and sin 03 < 0.04. In the Maiani parametrization, the angles obtained are: sin 0 = 0.231 +-0.003, Isin3'1 = 0.052_+°.b°l~and sin/3 < 0.008.
1. I n t r o d u c t i o n
The six quark mixing scheme proposed by Kobayashi and Maskawa [1 ] serves as a useful parametrization o f the connection between generations o f quarks. The elementary Uik o f the quark mixing matrix (i = u, c, t; k = d, s, b) are parametrized in terms o f three angles 0 1 , 0 2 and 03 and one phase 6, possibly related to C P violation (table 1). If C P violation is due to quark mixing, then this phase 6 is related to the parameter e describing the admixture o f wrong C P parity in the long- and shortlived neutral K meson states, measured to be e = (2.28 -+ 0.05) × 10 - 3 X e x p ( i n / 4 ) [2]. An approximate relation derived by Pakvasa and Sugawara [3] is lel = i(mt - mc)/rhcl × sin20 2 tan0 3 sin fi/(2x/~ cos01) where m t and m c are the top - and charm - quark masses. An alternative parametrization o f the matrix U h a s been given by Maiani [4], in terms o f angles O, 7, and/3 (see table 2). Experimentally, information on the weak mixing
Table 2 Parametrization of the quark mixing matrix. Maiani parametrization [4 ].
1 Also at CERN, Geneva, Switzerland.
Table 1 Parametrization of the quark mixing matrix. KobayashiMaskawa parametrization [1 ]. Cl
I
~
U = --SlC 2 ~--SI$2
81C3
S183
+ei6s2S3 c l e 2 s 3 ei6s2c3 eifc2s3 ClS2S 3 + ei6c2c3
CLC2C3 CLS2C3 --
0.031-9163/83/0000--0000/$ 03.00 © 1983 North-Holland
I CBCo U =-
-s,ycos[3e~
C~SO - sOc.~ ' c3,cO-sTs[3soei6
sl3 .,] .~,
- s[3c3,c 0 + s~/s0 e -i~ - c~s(3so - s3,C0 e -~o
sTc~3e16
c 3, c#
J
angles comes from measurements o f weak decays of light and heavy quarks and from neutrino production o f charm quarks as observed in dimuon events, as summarized in previous papers [ 5 - 8 ] . Recently, new results on the B meson lifetime [9,10] and on hyperon semileptonic decays [11] have been obtained. In this paper, we summarize the impact o f all these constraints on the weak mixing angles. We first go through the constraints on the coupling parameters Uik , a n d then proceed to derive bounds on the mixing angles. 2. C o n s t r a i n t s o n m a t r i x e l e m e n t s 2.1. L i g h t q u a r k c o u p l i n g s C o u p l i n g Uud. This coupling parameter has been
determined from a comparison o f measured rates o f nuclear beta decays with that o f muon decay. Two different evaluations of this quantity have been made and their results are Uud = 0.9730 + 0.0024 [6] and Uud = 0.9737 -+0.0025 [12]. Combining these two, one obtains Uud = 0.9733 -+ 0 . 0 0 2 4 .
(1) 459
Volume 130B, number 6
PHYSICS LETTERS
Coupling Uus. In a series of experiments, the WA2 collaboration has studied five different hyperon semileptonic decays, i.e. the electronic weak decays 2 ne~, 2;- ~ A e - b , E - -+ A e - ~ , E-- ~ ~ 0 e - ~ and A p e - ~ . Including radiative corrections and using in addit.ion the neutron lifetime [ 13], this experiment gives [11] Uus = 0.231 -+ 0.003.
(2)
This represents a substantial improvement over former analyses [6,12].
O)
from the CDHS collaboration [14]. This value is obtained by measuring the charm production by neutrinos and by antineutrinos from nuclei and by subtracting out the contribution of charm production from strange quarks, leaving the one from d quarks. Coupling Ucs. In charged current reactions this coupling appears always together with the strange-sea structure function xs(x) or its integral S = fx s(x)dx, where we use the following notation: s(x) is the quark density distribution of strange quarks in the proton in the Bjorken scaling variable x, u (x) and d (x) are the distributions of up and down quarks, with U = fxu(x)dx andD = fx d(x)dx, and similarly for antiquarks. The quantity measured is IUcs 12. 2S [5]. In the absence of an independent determination of S, only the upper limit for 2S given by SU(3) symmetry, 2S ~< U + D, and a corresponding lower limit on [Ucsl can be obtained. The product I Ucsl 2 • 2S can be extracted in three ways from the neutrino and antineutrino dimuon production data [8]. We use here the results from the x distribution of the neutrino dimuons [8,14] I Ucs 12/U2d = (6.26 + 0.73) (1 + ~*)]~,
(4)
and the one from the cross-sections of neutrino- and antineutrino4nduced dimuon production using the semileptonic branching ratio of D mesons: IUcsl 2 = (0.41 +-0.09) (1 +
a*)/a,
(5)
where a = 2S/(U + D) is the ratio of m o m e n t u m fractions carried by strange and nonstrange sea quarks in the nucleon, and a* = 2S(U2s + U2cs/rs)/(U+ D) is the same ratio modified by the threshold suppression 460
factor r s for the charm quark mass, which is r s = 1.5 for the experiment considered [14,15].
2.3. Bottom-quark couplings Ratio IUub I/IUcbI. We use the value I f u b I/IUcbl < 0.2
(90% C L ) ,
(6)
obtained by the CLEO [16] and CUSB [17] collaborations at the Cornell storage ring CESR. Coupling Uub. From the unitarity relation I Uub 12 = 1 - IUudl 2 - IUusl 2 and eqs. (1) and (2), we obtain IUub 12 = --0.0006 -+ 0.0060.
2.2. Charm-quark couplings. Coupling Ucd. We take the value Ucd = 0.24 _+0.03,
3 November 1983
(7)
B Lifetime. The lifetime r B o f b flavoured hadrons, apart from phase-space factors, depends on the magnitude of the coupling Ucb and Uub. In fact [18] rB = 0.93 ×
lO--14s/(2.751Ucb12+7.71Uub12).
(8)
The previous upper limit obtained by JADE collaboration [19], r B < 1.4 × 10 -12 s at 95% CL, is significantly improved by the recent measurements of the Mark II and MAC collaborations [9,10]. Using their high precision vertex chambers and tagging b decays by their semileptonic decays into muons or electrons they obtain rB=
+4.9 -+ 2.6) X 10 -13 s (10.4_32
r B = (18-+6 +4) X 10-13 s
[10],
[9],
(9)
which in turn gives the following constraint (iUc b [2 + 2.8 IUub 12)0.5 = a n~9 +0.009 . . . . . -0.007 "
(10)
2.4. Combined fit Using the constraints eqs. ( 1 ) - ( 7 ) and (9), we obtain for a minimum ×2 = 1.2/5 DF the values sin 01 = 0.231 +- 0.003, a -- 2S/(U + D) = 0.49 -+ 0.07 and values of sin'02 and sin 03 with the error contours in the (sin 02, sin 03) plane given in fig. 1. These contours vary slightly with the value o f the phase 5. From the resulting error contours, we obtain a finite value of 0.015 < sin 02 < 0.09 and an upper limit for sin 03 < 0.04. We conclude from this analysis that the second mixing angle 02 is smaller than the first one, 01, i.e. sin 02/sin 01 < 0.4. The third angle, 03, is still compatible with zero, with the upper limit sin 03 < 0.04 at the 67% CL. This pattern of decreasing mix-
Volume 130B, number 6
PHYSICS LETTERS r
/
1
/
,/
/
J..
/
/ /
/
/¢"
/ / /
/
/ /
/
cF ¢:
2Or
/
I"
\-A-2.
/
0.05
"
-~.rnt= 305eV
~
~ " ~
I0
/
/
0.05
0
/
/
/
1"
0 I0 -
/
/
:
015
/
I
/
3 November 1983
0 6=90 o ~ _ _ _ ~
]
J
0.05
mL=lO_OGeV
± _ _
0.10 sin [ 3 / t a n 13
0.15
6=10 ° -0.05
0 5
010
015 Sin 0 3
Fig. 1. Error contours in the (sin 02, sin 03)-plane (sold lines) for three values of the phase 8 (10 ° , 90 ° and 170°). Also shown (dashed lines) are results from a phenomenological analysis [20] of the K L - K S mass difference and the CP violation parameter e, for two values of the top quark mass, mt. Along these curves, the phase 8 varies from 0 ° at sin 03 = 1 to 90 ° at sin03 = 0 on the lower branch, and from 90 ° at sin03 = 0 to 180 ° at sin03 = 1 on the upper branch.
ing angles means that weak transitions b e t w e e n m e m bers o f different quark families are suppressed more for heavy quarks t h a n for light ones. On the o t h e r h a n d , the curves given in fig. 1 f r o m a p h e n o m e n o l o g i c a l analysis [20] o f the parameters o f t h e neutral kaon system can be c o m p a r e d w i t h the e x p e r i m e n t a l l y allowed range o f angles. Along these curves the phase 8 varies f r o m 0 to n/2 o n the l o w e r branch as sin0 3 goes f r o m 1 to 0, and f r o m 7r/2 to 7r o n the u p p e r b r a n c h as sin 0 3 goes f r o m 0 to 1. The constraint sin 03 < 0.04 can be used to set the limit 2 ° <." 8 < 178 ° . A d e t e r m i n a t i o n o f 8 requires specific e x p e r i m e n t s on CP ~iolation in the K--O-K 0 or B-'0-B0 systems. Analogously for the Maiani p a r a m e t r i z a t i o n [4], the error c o n t o u r s in the plane o f the parameters sin 7 (corresponding a p p r o x i m a t e l y to sin 0 2) and sin/3/tan0 (corresponding to sin 0 3) are given in fig. 2. Here error c o n t o u r s at t h e one ( l o ) and t w o (20) standard
(I
-0.10
1
1
I
Fig. 2. Error contours in the plane of mixing parameters sin 3, and sin t3/tan 0 in the Maiani parametrization. One standard deviation (1 a) and two st. dev. (2or) contours are given.
deviation level are given. These c o n t o u r s are nearly ind e p e n d e n t o f the phase angle 8' in the Maiani parametrization. The values for the angles are sin 0 = 0.231 -+ +0 015 0.003, Isin 71 = 0 . 0 5 2 _ 0 ~()10 and sin/3 < 0.0077. F r o m the range o f values o f the mixing angles in either par a m e t r i z a t i o n , the values o f the K o b a y a s h i - M a s k a w a matrix elements can be o b t a i n e d . These are given in table 3. It is evident that the error on these matrix
Table 3 Elements of quark mixing matrix Uik from fit of experimental constraints (1 std. dev. range).
u c t
d
s
b
0.9723-0.9737 0.228-0.234 0.003-0.016
0.228-0.234 0.9704-0.9726 0.041-0.066
0.000-0.008 0.042-0.067 0.9977-0.9991
461
Volume 130B, number 6
PHYSICS LETTERS
Table 1 Elements of quark mixing matrix Uik from experimental constraints if number of quark flavours is larger than 6. 0.9709 0.9757 0.21-0.27 0.00-0.12
0.228-0.234 0.78-1.00 0.00-0.58
0.000-0.013 0.042-0.067 0.000-0.999
e l e m e n t s f r o m the c o m m o n fit is, for m o s t o f t h e m , m u c h smaller than the one o b t a i n e d f r o m individual e x p e r i m e n t a l b o u n d s o n one m a t r i x e l e m e n t . It also appears f r o m the values in table 3 that, apart f r o m the diagonal elements, and the three off-diagonal elements observed directly up to n o w , (Uus, Ucs and Ucb), there is o n l y one o t h e r non
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462
3 November 1983
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