New determination of quark mixing angles and phase

Volume 136B, number 3

PHYSICS LETTERS

1 March 1984

NEW DETERMINATION OF QUARK MIXING ANGLES AND PHASE

PHAM Xuan-Yem Laboratoire de Physique Th~orique et Hautes Energies l, Universit~ Pierre et Marie Curie (Paris V1], Tour 16-ler btage-4 place Jussieu, 75230 Paris Cedex 05, France

VU Xuan-Chi Universit$ de Picardie, UER de Sciences et Techniques de Saint-Quentin, France

Received 21 October 1983. The Kobayashi-Maskawa angles and phase are determined with accuracy using recent B meson lifetime measurements together with the CP violation parameters of the K° system. We get 0.032 < sin 02 < 0.125, 0 ~ sin Oa < 0.06,140 ° 8 < 179° (6 in the second quadrant). Some implications, namely possible large B° - B° mixing, are discussed. 1. Introduction. Within the Glashow-WeinbergSalam-Ward theory of electroweak interactions, in order to incorporate CP violation to the coupling of W intermediate boson with quarks, there must exist at least six quarks distributed into three left-handed doublets, as originally observed by Kobayashi and Maskawa (KM) ten years ago. The quarks to which the W boson couple are not mass-eigenstates but are related to them by a unitary mixing matrix V characterized by three angles 01, 02, 03 and one phase 8.

/ gtd C1

Vts

Vtb]

$1C3

$1S3

- S 1C2 C 1 C 2 C 3 --$2S3 ei8 C 1 C 2 S 3 +S2C3ei~ - S 1 S 2 C 1 S 2 C 3 + C2S3ei~

C I S 2 S 3 - C2C3ei8

where C i = cos 0 i, S i -- sin 0 i, i E 1,2, 3. Because o f the rendundancy in the contribution o f the angles to the KM matrix, the 01 , 0 2, 0 3 are chosen to be in the first quadrant (C4, S i ~> 0), but the quadrant of ~ has physical implications and cannot be fixed by convention. The charged weak current is i Laboratoire Associ~ au CNRS. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

given by

Y. = (fi, c., t-)3'.(1 - 75)V

~ .

Thanks to the extraordinary achievements by the experimentalists successively charm, beauty, W, Z were found with expected properties, and we are now in an exciting time waiting for the revelation of the t quark. In spite o f these successes, there is a natural trend to go beyond the standard model, not because just to react against Lord Kelvin's famous word: "There is nothing new to be discovered in Physics now", but because we are unsatisfactory with too many parameters and arbitrariness unexplained by the standard model. Needless to say however that in order to test the theory, it is important to determine with great accuracy its parameters, among them the KM angles and phase, to which this paper is devoted. Once these parameters are accurately fixed, many implications and quantitative predictions follow, if future experiments turn out to disagree with these predictions, new Physics beyond the standard model is mandatory. Very recent measurements [ 1 ] of the B meson lifetime provide a new light for an accurate determination of the KM angles. We remind that 01 is well known [2]: C 1 = 0.9737 + 0 . 0 0 2 5 , S 1 = 0.2270 + 0.0104 0.0110 •

(1) 209

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complicated functions, their explicit expressions were given previously [6]. We will take M b = 5 GeV, m c = 1.5 GeV, a s = 0.21 (corresponding to AQC D = 0.1 GeV) and get:

Our main results obtained are: 0.032 ~
(2)

O~
(3)

140 ° ~< 6 ~< 179 °

(6 in the second quadrant).

(4)

We remark that S 2 is strictly non-zero and we summarize here the I Vi/I matrix elements: 10.9712-0.9762

iV i~[ = 10.2143-0.2373

\0.0049-0.0297

0.217-0.221

0.9692-0.9762 0.0178-0.0661

0.-0.012

\

0.032-0.065 I 0.9978-0.9994•"

Compared to the previous results [2,3] bearing (5) large uncertainties in the t, b sectors, our results eq. (5) are drastically improved. The methods that arrive to these results and how. reliable they are will be discussed below, few implications offering tests of the model (and our results) are mentioned.

1. The angles 02 and 03. Two independent data, u e ~ / P ( b ~ c e ~ < 0.05

P(B ~ egX) = 4.31([ Vbc 12 + 2.11Vbu 12)× 10+13(s) -1 • (7) The world average [7] semi leptonic branching ratio of the B meson is known to be (1 1.6 + 0.5)%, from which we deduce r B = 2.67 (IVbc 12 +2.1[Vbu 12) -1 X 10 -15 s.

(8)

General belief attributed to Vbc and Vbu some values around the Cabibbo angle, from which r B was expected to be short ( 1 0 - 1 3 - 1 0 -14 s). The surprise then comes with two recent measurements of ~'B [ 1 ] : r B = (1.8 + 0.6-+ 0.4) × 10 -12 s

(MAC),

rB = ~-x'z-0.30tl -,+0.45 + 0 . 3 0 ) X 10 -12 s

(MARK II)

both are consistent with the Jade upper limit [8] :

the upper limit [4]

r(b ~

1 March 1984

r B < ~ I . 4 X 10 - 1 2 s . (6)

on the one hand, and the B meson lifetime [ 1] (together with its semileptonic branching ratio) on the other hand, provide us a reliable way to extract the 02 and 0 3 domains. Inclusive leptonic decays of the B mesons can be described [5] to a good approximation ,1 by that of the b quark, the rate of which (including QCD radiative corrections [6] ,2 ) is given by (in units of F 0 = G 2M5/ 192zr3):

We will use 0 . 6 × 10 -12 s < r B < 2 . 6 X

10 -12 s.

From these data together with eq. (8) and inequality (6), we obtain (the results include the uncertainties in our choice of quark mass that we let vary within 5% range) 0~< IVbul ~ 0.012, 0.032 ~ LVbc I ~< 0.065.

(9) (10)

I'(B -~ ev~X)= I'(b -* ue~) + V(b -~ ce~),

r(b -~ qe~) = FO I Vbq 12 X [f(mq/Mb) + (Ots/Tr)g(mq/Mb) ] , q stands here for up and charmed quark. f ( x ) is the phase-space suppression factor due to the q quark mass and g(x) has an integral representation of ,1 Predictions [5] for inclusive semi leptonic branching ratio o f the B m e s o n s (as well as those o f the charmed D + m e s o n and t h e r lepton) which are in perfect agreement with experiments support the reliability o f t h e quark decay m e c h a n i s m used here, namely for heavy quark system. , 2 The notation (%]~r)g(x)rO used here is denoted by r u in ref. [6], t h e symbols P l , P2, 03 o f which are replaced here by P2 = 03 = 0, O1 = x 2.

210

Since 01 is already known with accuracy, the domains of variation for 02 and 03 satisfying eqs. (9) and (10) are first plotted for all values of 5. For every fixed 8, in the (S 2, $3) plane, the constant I Vi/I contours are the ellipses, they become flattened when 8 approaches 0 ° or 180 °. A complementary constraint for S 2 and S 3 comes from the experimental information eq. (6). Indeed, since I Vbc 12 ~< S 2 + S 2 + 2S2S 3 cos 6, the inequality (6) together with eq. (7) could be written as: 2 S 3/(S 22 + S 32 + 2S2S 3 cos 6) < 0 . 4 6 ,

(11)

or

$2/S 3 > - c o s 6 + (cos26 + 1.17) 1/2 .

(12)

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S2

~

_

-

0,06

~

- -

~

=--

6

-----

"<

.

o°

= 90"

Minimum of the right-hand side

1 March 1984

The constraint (12) added to previous domains satisfying (9) and (10) implies stingent allowed domains for 02 and 03. They are plotted in figs. 1 and 2 from which we read: 0.01~
"~

0.05

"~

0~
for 6 in the 1st quadrant,

004

or

°031\

0.023~<$2~<0.125,

0~
for 6 in the 2nd quadrant.

0.01 0.02 0.03' 0.04 0.05 0.06

53

Fig. 1. Allowed domain of $2, $3 forall values o f a in the

first quadrant. Experimental constraints are 0.032 ~ I Vbc I 0.065 and l'(b --, u)/r(b --, c) < 0.05. solid line 6 ---0° , dotted line: 6 = 90°, dot-dashed line: minimum of the righthand side of eq. (12), corresponding to 6 = 0°. 0.125' 52

0.10

These results agree with the best fit of Kleinknecht and Renk [3] using different experimental inputs. By unitarity of the KM matrix (Zq I VqQ 12 = 1 for every Q), all the I Vi/I are accurately fixed and the results are shown in eq. (5).

3. The phase 6. Since 6 reflects CP violation, analysis of the CP conserving quantities (width...) alone cannot help to constraint 6, different sources o f information are therefore needed. Inevitably comes the KS0 - K O system, with their mass and width differences Am = m s - m L, A F = F S - FL and their CP impurity parameter e. Let M12 and r12 be respectively the dispersive and absorptive parts of the off-diagonal element in the particle-antiparticle mass-matrix. For the K0-K, 0 system considered here, from some experimental facts (e ~ 1, Im F12 ~ Im M12, ~ ~ - A P / 2 ) one can use the following approximate relations:

/-

0.05

..__/,>

#

2x/~e ~-- (exp'-in) Im M12/Re M12 ,

(13)

&n ~ 2 ReM12.

(14)

Experimental measurements give [2]: Re e = (1.620 -+ 0.088) X 10 - 3 hand side of Eq. 12), corresponding to ~ = goe

Am = --0.352 X 10 -14 GeV. 0.01 001

0.02 003

0.04

Q05

0.06

53

Fig. 2. Allowed domain of $2, S 3 forail values o f 6 in the

second quadrant. Same experimental constraints as in fig. 1. solid line: 6 = 180°, dotted line: 6 = 90°, dot-dashed line: minimum of the right-hand side of eq. (12), corresponding to a = 90°.

The problem arises when one would like to identify M12 with the box-graph calculation Mlb~ x , its theoretical expression is well known [2,9] :

Mb~ x -

G2M 2 u,c,t 32nZMK

xixjaijK12,

(15)

where 211

Volume 136B, n u m b e r 3

PHYSICS LETTERS

K12 = (K 0 l dT.(l - 75)sd3'v(l - 7 5 ) s l ~0) , ~'i ~isVid and Ai] a r e known mZ//M2 not reproduced here. =

functions [9] of

The first serious objection lies in setting Re M12 r. ,-box equal to g,e s~12 since there are contributions [I0] to M12 which seem to have nothing to do with the box diagrams. Secondly, the matrix K12 is a long distance nonperturbative QCD object, its "vacuum insertion" ap= 8 2 2-proximation (K12)vac + gf~mK(fK ~ 160 MeV) might be over estimated compared to recent calculations [11]. Inthe literature, onewritesK12 =B(K12)vac. Moreover the sign of B is not known, single pion insertion even reverses [12] the sign of (K12)vac. According to recent estimates [13] using hadronic spectral sum rules, IBi is however ~<1.7. Thirdly, with the identification of Im Mb~ x in eq. (13) for e, there is still an ambiguity [14,2] due to the possible non-zero value of the imaginary part of the decay amplitude K ~ 2~r where the pions are in the isospin-zero state. Keeping the objections in mind, for lack of something better we will stick to the box diagram calculation, recalling nevertheless that its prediction [ 15 ] of the charm quark mass - which turned out to be correct - might not be a pure chance. Before looking for the solutions of/5 satisfying the box-diagram expressions of e and Am, eqs. (13) and (14), a quite remarkable property is worth to point out. Following Ginsparg et al. [16], even if we do not assume any knowledge of the sign of the parameter B, we can prove however that B sin ~ is always positive*3. Therefore is ruled out the possibility B sin/5 < 0 considered by these authors, this possibility would imply m t > 60 GeV for all B meson lifetimes plotted in their figures [16]. We now come back to the fitting of e and Am. In principle, with two unknown parameters m t and B (its sign is not even known) and with only two experi-

1 March 1984

mental informations e and Am, the determination of /5 is hazardous, unless B and m t are restricted in some plausible range. Our procedure is as follows. We scan 02 and 03 in our domain previously determined, let m t vary between 30 and 100 GeV, B from 0.3 to 1.2, and we look for/5 that must satisfy eqs. (13) and (14). Since B is assumed to be positive (as in the literature), sin/5 also must be positive, the last two quadrants 3 and 4 of/5 are consequently eliminated. Our results recover those of Chau [2]. The sign of e, as well as the suppression of l~(b -~ u) are sufficient to restrict/5 in the first and second quadrants. A new thing emerges from our fitting program: the second quadrant is much preferred to than the first one, namely the region/5 near 175 ° looms up out of the rest. In the half second quadrant for/5, the fit is excellent and the solutions are stable against the variations of the parameters 02,03, mr, B. On the other hand, in some regions 0 ° < 6 < 40 ° of the first quadrant, the solutions are unstable, some experimental constraints are even not well fullfilled. We believe that

0.12~

$2

010

/,/ ~

E --

i

F

0.0~

- -

g = 179o

. . . .

~

-----

= 140 °

Minimum of the right~and side of Eq (12), corresponding to = |4G o

i

4=3 The fact comes essentially from t h e experimental information £(b ~ u ) / £ ( b ~ c) < 0.05 (the property is still true with £(b --* u ) / r ( b ~ c) < 0.01). which can be translated into our constraint eq. (12). Indeed, from the box-diagram expression o f e as given for example in t h e eq. (7) o f Ginsparg et al. [16], the coefficient o f ( m t / m c ) 2 which is CIS2C3 + C2S 3 cos 6 m u s t be positive (whatever is cos 6) because of our eq. (12), implying by then B sin 6 > 0.

212

0.01

(101

(106

S3

Fig. 3. Allowed d o m a i n o f $2, $3 for 140 ° < 6 < 179 ° representing our final results.

Volume 136B, number 3

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unless the t quark mass is known, it is unlikely that 6 could be trapped in a more restricted domain than that obtained here: 140 ° ~ / i ~ 179 °. Fig. 3 exhibits the allowed domain for $2, S 3 when 5 is inside this range. It represents the final result o f our whole analyses combining B meson decay and CP violation o f the K 0 system. From fig. 3 we get the results announced in the introduction (and abstract).. One might also put one more experimental information from K O -+/a+# - to constrain the/i region. However the price to pay for this extra input is the introduction [17] o f an unknown parameter a describing the dispersive part o f the two virtual photons amplitude K O ~ 7 u 7 u -~ # + # - . By relaxing all parameters rot, B, (~, everything could be consistent with a long B meson lifetime.

4. Some implications. (1) in the discussion on the p0, ]~0 mixing (p0, i~0 are any neutral meson), one generally introduces a physical measurable quantity R defined as R = (1 ++ + 1 - - ) / 1 + - where 1++, 1 - - are the same sign dilepton events coming from the production, subsequent mixing and decay o f P0]Ei0 pair in e+e - annihilation, while 1+ - are the opposite sign dilepton ones. R is related to a parameter A b y R = 2A/(1 + A 2) where

(/ira/r) 2 + A-

~-(/ir / D E

2 + (/ira/F) 2 /iF, F and 5m are respectively the difference, the average width and the mass difference o f p0 and P0L. The maximum mixing occurs if A--~ 1 corresponding either t o / i F "~ 2 F (case o f the K 0 system) or t o / i r a ~ F. In the first case, a pure K 0 to begin with will quickly end in the K L one which is approximately equal mixture o f K 0 and ~ 0 , therefore maximum mixing. In the second case, if 6m >> F, then the system oscillates very quickly between P and ~0 before decaying and appears as an equal mixt___ureo f them. This case might happen to the B0 and B 0 mesons. Assuming box-graph calculation, one gets:

6m ~ (G 2BfBMBm2 /61r2) Re {(Vtd Vt*b)2} and (~m/r)B o ~

11.66(Bf2B/O.1 GeV2)(mt/50 GeV)

× Re((VtdV~b)2)/[ Vhc 12 • In the second quadrant favoured domain o f / i , Re {(Vtd Vt*b)2)/I Vbc 12 could take its maximum value

1 March 1984

as large as 0.60, then /im/F could be substantially > 1 . Consequently R could be as large as 0.90, contrary to the charmed D0D 0 pair where R is found to be around 10 - 5 . These predictions agree with the recent ones

[18]. (2) A throrough analysis by Hagelin [14] shows that the relative sign o f t h e CP parameters e and e' in the K 0 system equates that o f sin/i which makes e'/e < 0 a unique characteristic o f the third quadrant for 5. Our results indicate that e' is parallel to e, a prediction that could be checked in the forthcoming experiments. (3) Having found 02 and 0 3 to be very small, it seems likely that a high mass (say greater than 40 GeV) o f the t quark is favoured. We hope that our present knowledge o f the mixing angles and phase analysed here would shed some new light on the fundamental question about quark generations and masses. [1] E. Fernandez et al., Phys. Rev. Lett. 51 (1983) 1022; N.S. Lockyer et al., Phys. Rev. Lett. 51 (1983) 1316. [2] L.L. Chau, Phys. Rep. 95 (1983) 1. [3] K. Kleinknecht and B. Renk, Z. Phys. C20 (1983) 67. [4] E. Thomdike, Proc. 18th Recontre de Moriond (March 1983). [5] See e.g., X.Y. Pham, Proc. 17th Rencontre de Moriond (March 1982). [6] Q.oHokim and X.Y. Pham, Phys. Lett. 122B (1983) 297. [7] See, e.g., S. Stone, talk Intern. Syrup on lepton-photon interactions at high energies (CorneU, August 1983). [8] W. Bartel et al., Phys. Lett. 114B (1982) 71. [9] T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297, 1772 (E); H.Y. Chen, Phys. Rev. D26 (1982) 143. [10] T.N. Truong, Phys. Rev. Lett. 17 (1956) 1102; C. Itzykson, M. Jacob and G. Mahoux, Nuovo Cimento Supp. 5 (1957) 978; L. Woffenstein, Nucl. Phys. B160 (1979) 501. [11] J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Lett. 119B (1982) 412; P. Ginsparg and M.B. Wise, HUTP/83/A027 (1983). [12] R.E. Shrock and S. Treiman, Phys. Rev. 19D (1979) 2148. [13] B. Guberina, B. Machet and E. de Rafael, CPT-83/1485 (1983). [14] J.S. Hagelin, Nucl. Phys. B193 (1981) 123. [15] A.I. Vainshtein and I.B. Khriplovich, JETP Lett. 18 (1973) 83; E. Ma, Phys. Rev. D9 (1974) 3103; M.K. Galliard and B.W. Lee, Phys. Rev. D10 (1974) 897. [16] P.H. Ginsparg, S.L. Glashow and M.B. Wise, Phys. Rev. Lett. 50 (1983) 1415. [17] L. Bergstr6m, E. Masso, P. Singer and D. Wyler, Phys. Lett. 134B (1984) 373. [18] S. Pakvasa, KEK-TH 66 (1983); E.A. Paschos, B. Stech and U. Tiirke, CERN TH 3601 (1983). 213