Nailing Down the KM matrix and the top quark mass

Volume 195, number 3 PHYSICS LETTERS B 10 September 1987

NAILING DOWN T H E KM MATRIX AND T H E TOP QUARK MASS

L. ANGELINI, L, NITTI, M. PELLICORO Dipartimento di Fisica, Universita" di Bari and Istituto Nazionale di Fisica Nucleate, Sezione di Bari, L 70126 Bari, Italy

and

G. PREPARATA Dipartimento di Fisica, Universita" di Milano and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, 1-20133, Milan, Italy

Received 22 May 1987

We present a determination of the KM matrix elements and a prediction of the top quark mass, based on available experimental information and the only theoretical framework, known to us, that is free of experimental and/or theoretical difficulties.

In t roduc t ion . The notion of quark-lepton univer- sality is one of the focal points of the very successful synthesis of subnuclear physics which is known as the standard model. According to quark-lepton uni- versality, to the known three generations of leptons there must correspond three generations of quarks (u, d; c, s; t, b), and this explains why finding the top quark (t) as well as determining the weak-interaction mixing among the three generations - the Kobayashi-Maskawa (KM) matrix [ 1 ] - i s consid- ered to be of paramount importance.

It is clear that in the case of quarks, different from leptons, the experimental determination of the weak interaction parameters depends on our ability to get hold of the matrix elements of current operators (expressed in terms of quarks) between physical states (described by hadrons). That this is a highly nontrivial task is witnessed by the fact that Cabibbo [2] could give substance to this bold hypothesis of weak d-s mixing only when SU(3) and current alge- bra could be employed for determining, with small uncertainties, the relevant current matrix elements. And when it gets to calculating nonleptonic matrix elements the task becomes even harder.

The history of nonleptonic physics ~ shows that

~ For a recent account see ref. [3].

finding a suitable and reliable framework for the computation of the matrix elements of products of weak current operators has kept theoreticians busy with varying degrees of success, since the beginning of the sixties. The observation, in 1968 [4], that in a world that exhibits the scaling phenomenon at short distances the weak nonleptonic hamiltonian is dom- inated by long distance physics, was at the founda- tions of a research program that attempted quite successfully to calculate nonleptonic physics from well-determined weak semileptonic matrix elements [ 3 ]. Apparently this program was superseded by the discovery of asymptotic freedom and by the claim [ 5 ] that nonleptonic physics is quite sensitive to what happens at distances O(1/mw), and that perturba- tive QCD gives a full account of such effects. How- ever, there is now evidence that these ideas meet a number of difficulties in comparing with experi- ments, and in particular in accounting for the large difference in the D°-D + lifetimes [ 3,6].

In the light of this we believe it very useful to present an analysis of the experimental inferences on both the KM matrix and the top quark mass within the theoretical framework of anisotropic chromody- namics (ACD) [7], which now appears to be subtly related to the dynamically stable realization of QCD [ 8 ]. This is what we shall do in this letter.

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As is well known, the KM matrix depends on four parameters, three angles 0i and one phase d, which controls CP violation. Of the three angles, one is the Cabibbo angle ~1 (sin 0~ = 0.221 +_ 0.002), while the other two can be related, via 0~, to the matrix elements Vub and Vt~. From the experimental side, the important information on the top mass and the CP violating phase comes from the K°-I~ ° mass difference [Sm=mL--ms=(3.521 +0.014)×10 -~5 OeV] and the magnitude and phase of the CP violating parameter e [Re e= (1.621 +0.088) × 10 -31 [91. In addition B decays, in particular the surprisingly large lifetime (rB = 1.42_+ 0.27 ps) and their lepton spectrum, will be seen to yield important constraints on the magnitudes of the matrix elements I rob I, I g~bl. Finally Ba-ga and B , - B , mixings, which are now being determined [ 10,11 ], will prove to constrain rather strongly the KlVl model, the top mass and the theoretical ideas utilized in the computation of the nonleptonic matrix elements.

But let us see how.

K°-K ° physics. The matrix element to be com- puted is the AS---2 transition shown in fig. 1. The short distance hamiltonian, inducing the AS=2 K°-I~ ° transition, is given by ~2 [ 13 ]

2 2 LrAS=2 GFMw (22rhE ~ +22tlzE2 +2)~cltr/3E3 ) **~rr - 167t2

X :dy*'(1 - 75)s dgu( 1 - ys)s: , (1)

where

ki = V* Via , (2)

El = A ~ +Ac~ -- 2A~, (3)

~2 For an extensive review see ref. [ 12].

E2 =A. . + A t / - 2Aut , (4)

E3 = A ~ +A~t - A ~ -A~t • (5)

The Au's, which are functions of ( m / M w ) 2, i=u , c, t, are given in ref. [ 12] and according to AF the QCD corrections are r/l~-0.7, ~/z~0.6, q3~-0.4 ¢3. As we have argued elsewhere [3,6,14] ACD, on the other hand, implies that all r/factors should be to a good approximation ~4 equal to 1; and this is one of the important differences with the AF approach.

In order to compute the normal product of current operators appearing in eq. (1), we use the same strat- egy that has proven successful ~5 in the calculation of the A/=3/2 amplitude in K--+2~ decay, namely we separate the disconnected amplitude (vacuum inter- mediate state) from the connected amplitude, and saturate the latter with the lowest one-particle inter- mediate states (pseudoscalars and vectors) to obtain within less than 10% the "naive" result [ 16,17]

<K ° I : a~ (1 - ~ 5 ) s a te(1 - ~5)s: IK ° ) = - 4f2~ mK, (6)

where we have fK = 0.17 GeV from K-+ gv. Thus a realistic calculation of the current-current matrix element has yielded the simple result B = 1 (within 10%), and it is difficult to envisage how any rea- sonable physics can yield a much smaller B. Notice that in the following we shall have to deal with sim- ilar matrix elements between B mesons, and there the B = 1 result is on even firmer ground.

From what we know of the couplings of the t-quark (see later) 8m=mL- -ms is sensitive only to the charmed quark mass rnc and to the Cabibbo angle,

,3 See ref. [ 12 ] and references quoted therein. ~4 Likewise the so-called "penguin" diagrams should be negligible. ~s For a thorough discussion see ref. [ 15 ].

most si~ular , short disfance-parr

W

Ko

•&S=2 ~ff

Ko K °

Fig. 1. The most singular short-distance part contributing to the K°-l( ° mixing.

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while the CP violating parameter e depends on both the full structure of the KM matrix and the top quark mass. Thus leaving m~ as a parameter and neglecting the genuine long distance contribution, the experimental value o f gin yields me= 1.55 GeV, just in the right ball park. On the other hand if we use the ACD value for m~= 1.2 GeV, we see that there is room for a long distance contribution comparable with the short distance one ~6. As for the CP violating amplitude we predict e' _~ 0, and the experimental value o f e provides a stringent constraint on the CP violating phase ~ and the top mass mr, that shall be discussed at the end.

B decay. The relevant amplitudes that ACD is capable to compute ~7 are shown in fig. 2. A careful comparison of our theory with the experimental data [ 19] on the leptonic spectrum yields

p= ] Vub/V~b l =0.21 + 0 . 0 7 , (7)

while from the measured lifetime of the B mesons we obtain

[ V~b [ =0.042_+ 0.003. (8)

B°-B ° mixing. The evaluation o f the B°-B ° mixing amplitude runs completely parallel to the calculation

~6 As advocated in ref. [ 18]. ~7 The calculations are reported in ref. [ 14]. Notice, however,

that our determinations in eqs. (7) and (8) are based on a more refined subsequent analysis that shall be reported elsewhere.

of the K°-I~ ° mixing. The effective hamiltonian can be written as (we set all ~/'s to 1) [12,13,20]

rras-2 = GvMw Z V~.q V* Vjq V~jbAjj a " t e f f 167g 2 i , j = u , c, t

X :ClTU(1 - ?'5)b Clyu(1 - ? :5 )b : ,

where q is the spectator quark (d, s) and

(B° I :QTu(1 - ~'5)b (tTu(1 - ~ 5 ) b : If3 ° )

(9)

= -~ f~qmBq . (10)

ACD determines fBq as [ 21 ]

fBa=0.15 G e V , fBs=0.18 G e V . (11)

In this way we can determine the B°-l) ° mixing parameters ra and rs ~8 again as a function o f the KM matrix elements and of mr.

Injecting now, as a final experimental information, the recent ARGUS measurement [ 11 ]

rd = 0 . 2 3 + 0 . 0 8 , (12)

we are in a position to exploit the constraint already found from Re E, together with the determination o f p and I Vcbl [see eqs. (7) and (8)] , to draw the allowed domains for mt and ~ that we report in fig. 3. In fig. 3a, we plot the allowed values o f mt ~9 as a function of p in the allowed range of eq. (8). One

~8 For a definition of mixing parameters rd and rs see ref. [20]. ~9 Realistically we have restricted our analysis to mt< 175 GeV.

d, "~

q

Fig. 2. The relevant diagrams for B decay: (a) semi-leptonic and hadronic (b~) colour enhanced, (b2) colour suppressed.

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PHYSICS LETTERS B I 0 September 1987 Volume 195, number 3

>-~160 L~

140

120

100

80

60

40

0.14

a •

I , I I I I I ,

0.16 0.18 0.2 0.22 0.24 0.26 0,28 iv=/v=l

:~ 160 (.~

1,$0

120

100

80

60

40

0.

b

I I I t I I I 0.25 0.5 0.75 1. 1.25 1,5 1.75 2.

Fig. 3. The allowed regions for the variable pairs mt-p (a), ~ -m t (b) , '~-p (c).

~/~

sees that there are two solutions, the "low" (L) solution,

mt = 7 0 + 10 GeV, (13)

and the "high" (H) solution,

mt ----- 125-t- 10 GeV. (14)

The allowed regions for fi as a function of mt are

drawn in fig. 3b, where one sees that we have the solution for the "low" mt

~/n =0.86+_0.07, (15)

while for "high" mt we have

fi/n-- 1.28 +_ 0.05. (16)

Finally fi/zt as a function of p appears in fig. 3c, showing the structure of the two solutions eqs. (15)

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PHYSICS LETTERS B 10 September 1987 Volume 195, number 3

2.

=o 1.75

1.5

1.25

1.

0.75

0.5

0.25

O. 0.14 I I 0.16 0.18

C

I I I I 0.2 0.22 0.24 0.26 0.28

iv,,./v=l

Table 1 Our calculated KM matrix elements for the ( L ) and (H) solutions.

[0.976 V(I-) = 0.221

0.018_+0.001

0.976 v(H)= 0.221

0.011 _+ 0.001

-0.221 0.975

(0.038_+ 0.003) +i(0.019 + 0.005)

-0.221 0.975 0.019 + 0.003) +i( -0.037 + 0.002)

-0-010-+0.001 [ (-0.027+0.007) +i(-0.032_+0.006)

(0.909 + 0.044) +i( - 0.403 _+ 0.095)

-0.011 -+ 0.001 ] (0.0017_+ 0.004) +i(-0.038_+0.002) (0.632 _+ 0.040) + i(0.773 _+ 0.032)

and (16) as a funct ion o f p, f rom which we conclude that no solut ion exists for

p < 0 . 1 8 . (17)

In all cases, we calculate r s - ~ 1 in agreement with the indicat ions o f U A l [ 10].

Discussion. We conclude with a discussion of the theoret ical re l iabi l i ty o f our predict ions, eqs. (13), (14), (15) and (16):

( i ) Box diagram. The operators in eqs. (1) and (9) are certainly rel iable for the CP violat ing phase and the B mass matr ix, for they involve the exchange o f high mass quarks.

( i i ) ~ / ' s~ l . The absence o f AF/shor t -d is tance effects is suppor ted [ 3] by dynamica l calculat ions o f K +--, n + n o and D + ~ hadrons, where we do not f ind any evidence ei ther o f suppression factors for K +--,n +n o or enhancing factors for D + --, hadrons.

( i i i ) B - 1 . This is defini tely suggested by a dynamical calculat ion [ 17 ] whose result agrees with the naive de te rmina t ion [ 16].

Thus we may state with confidence that, within the three-generat ion ansatz, the structure o f weak inter- act ion mixing ( in table 1 we repor t the K M matr ix for the L- and H-solut ions) and o f the quark masses are now essentially de termined.

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Nailing Down the KM matrix and the top quark mass