- Email: [email protected]
CP violation 1984: The KM model confronts experiment
Nuclear physics A434(1985) 525c-534~
52%
North-Holland, Amsterdam
CP VIOLATIOR 1984: THE KM HODEL CORPRORTS EKPERIHERT
Barry R. Holstein Departrnent of Physics, University of Hassachusetts,Amherst, BA 01003
Using recent experimentaldata involving the b. t quarks and the nen measurement of B+/Uoo we show that the KM picture of CP violation is barely compatiblewith these results. Slight improvementsin any of these limits could require abandonementof the KM picture as the sole source of CP nonconservation.and some of the resultant required "new physics" is discussed. 1 Although the discovery of CP violation took place two decades ago, theoreticaldiscussions on the subject have not changed much during this time. A typical talk is outlined below (I have given several such myself): a. Set up the formalism for the kaon system b) Rote that CP nonconservationhas been observed only in the kaon sector c) Show that current results are consistentwith most models d) Plead with experimentalistsfor more and better data During the two years since the last meeting, experimentalistshave been at work and have provided us with some very importantnew results. Examples are: i) the B meson lifetime:' rB
= 1.6 f 0.4 ffO.ff x lo-I2 see
+0.4 rB = 1.2 _D.36 (fO.3) x 10-l' aec
RAC BKII
(1)
ii) the t-quark mass3 = 40 GeV % 4 iii) the U /B ratio (n,,"X+7
(2)
= 1.023 f 0.031(f0.023)
(3)
iv) the branching ratio for b + u/b -w' BR(b+u)< -%0.04 BR(b+c)
(4)
I shall argue here that, by combining all these new results, we can demonstrate that the standard KM model of CP violation is at present basely compatiblewith these data and that a significant- factor of three reduction in the bu/bc ratio could essentiallyrule out this model and require the introductionof interesting"new physics." In order to illustratethis feature, we first shall go through the standard steps a) and b) quoted above. bienote that CP violation in the I0 systezacan arise from two sources from the mass matrix that mixes K*, "K* and/or directly from the decay 0375-9474/85/$03.300 Elsevier SciencePublishers (North-Holland
Physics Publishing
Division)
B.V.
B.R. Holstein / CP violation 1984
526c
amplitude . The latter arc written
AmpfKO-* (m)I)
=\
(5)
exp(i8x)
where I = 0,2 is the isospin of the final two pion state and 8, is the associated strong interactionphase shift. We
choose
a phase convention so
that real AI correspondsto CP conservation. The eigenstatesof the Ramiltonian _ 5,
Kg _ are-writtenas
KR = (Kl C &K2)Ml + [cl2
(6)
KL = (K2 + ~Kl)/41 + ItI' in terms of CP even, odd states Kl, K2. It is standard to define6
rl+-
=
ah-tHwlK~> E
&
+
E’
(7) Qoo =
~~“lH~IK~
_ c _ 2E,
ProseCPT invariance one finds7 IlnMl2 -+ 0 AUI Im A2 exp iQ*oC-E) Re A2
where
(8)
&a = "t - mR = 2 Be HI2
(9)
E = Im ho/Re A0 and Be A2 0 = -
_ 0.05
(10)
Re A0 measures the strength of the AI = 3f2 component of s
to the &I = l/2
piece as revealed in the Kg + mr decay mode. The angles 8. 8' are given by 8 = tan-l (2Amq
= 44O
e* = g2 - 6. + n/2
= 37 f so
(11)
The experimentalresult eqn. 3 then determines C'/& = -0.004 f 0.005 (*0.004)
(12)
The measurement and the asynmetry in the semileptonicdecay of the < ~ /'(K&us)
- r(K~-Wvrr+)
P(KL-qA+vlr-) + lxKg-Wvlr+)
= 2 Re c I Jr+
ICI'
(13)
B.R. Holstein / CP violation I984
d=m
= +(3.30 f 0.12)
x
527~
10-3
and of n+_. nooa In+1
= (2.27 f 0.03) x 1O-3
(14)
= (44.6 f 12.010
d+
are the only places in which CP violation has been observed. m
other tests
- in nuclear beta decay. radiative decays of nuclei, detailed balance in scattering reactions, etc. - have given negative results, as summarizedby 9 Professor Calaprice.
In order to understand the origin of CP violation without introducingany 10 really new physics, it has become standard to use the so-called KM model. Thus in the six quark model for weak and electromagneticinteractionsbased on SU(2)b x U(1) gauge theory, the charged W bosons couple to the weak current
d
(+I
J,, = &(u,
0
c. t) Yn(l + Y5NJ 8 b
I
(15)
where U is a 3 x 3 unitary matrix which arises from the diagonalizationof the q = - l/3, +2/3 quark mass matrices. By adjusting the quark field phases, U can be written in the KLIform
c1
-*lc3
-sls3
ii u = s1c2 clc2c3-s2s3e
16 clc2s3+s2c3e
i6 c1s2c3+c2s3e
i6 cls2s3-c2c3e
sls2 where
(16)
the angles 81, e2, 8, lie in the first quadrant but the
phase of the CP violating parameter 8 is not fixed. The angles ei are determined i) from nuclear beta decay" = 0.9735 f 0.0005
(17)
ii) from'&peron beta decayr2 = 0.229 f 0.003 slc3 iii) from the B meson lifetime (using mc = 1.4 GeV. mb = 4.6 GeV)13 (s$+sz+2s283c*)= 4 x 10m3 lo-l2 set B.B.(b+c) =B 832= 4 x 1O-2 lo-l2 '=' B.R.(b+u) =B
(18)
(19)
3.R. Holstein / CP violation 1984
528C
-12 se& Defining z = 10
T
and s = p co15h, s3 = p sin h the 2 B last two equations can be written as p2 _ 4 x 10-3 z B.R.(b+c) _ 4 x 1O-3 e 1 + sin 2)lcs 1 + sin 2X c6 rEbbu=O1
*
b+c
(20)
sin2 A 1 + sin Z‘Lc,cb
We may proceed further by examining theoreticalestimates of Re T2,
Im
Here there is an important short distance piece which is easily nr2. 14 calculatedusing the 2W exchange 'box" diagram Box Gp s~!&K< Re R12 =1282
2 B(nl+n,K2 9
+ 2n,K In
m, (21) 2
GP 22 2 L)(sly@%% 2~283s@~-n,+rl,K.$- + Ds In 12n2 Tnc
Box
Im W12
Here
K = s$
+ 82~3~6,
'11 - 0.7, '12- 0.6, ?I3* 0.4
(22)
are QCD enhancement factors as calculated by Gilman and Wise, and
(23) Yg>dsyu(l +yg)dlKo>
is a measure of the short distance AS = 2 operator compared to its so-called "factorization"value. The parameter B has been extracted from &I = 312
K+~T
data by use of PCAG and SIJ(3)symmetry with the result'3
B = 0.33(%50X>
(24)
where the generous uncertainty is assigned given the possibly significant errors associated with SU(3) breaking in the n-K system and the off-mass-shellextrapolationof the physical K+~TTdecay amplitude.
In principle there are in addition significant long-distancepieces, and 16 In the case Re g,, a much effort has gone into their calculation. precision estimate is impossible. However, the general concensus is that Re Hklj= 112 Am D --Re $gx Re 8%
= + Oflf
(25)
with the dominant contributioncoming from the 2r and TT,n, ?j' intermediate'states. This is confirmed empirically in that numerically one
B.R. Holstein
/ CP violation
529c
1984
finds Re l4;: _ l/2
Am B n,
(26)
Comparison with the experimentalvalue Re(l$5x+ H$x
) = l/2 Am
(27)
reveals then Re l4$
D =_=_
Box Re R12
1 - Bn
+4.6 = 3.3 -1.4
(28)
grl
BV
in general agreement with eqn. 25. In the case of Im M12. there is in principle also a long distance contribution. However. its effect is much smaller. This can be seen since s is affected not by ~41~alone but rather by the combination Im
ii12 = Im U12 + 2ERe M12
(29)
But by definition, for the 2x intermediatestate we have (2x) Im R$f"i -2ERe W12
(30)
so that Im ii:gx)=o Also, since K"+xo.~o are related to K'+mr via a simple soft pion extrapolation,it can be shown that T pn Ointermediatestate contributionsare also negligible. l7 This argument does not extend to the W(3) singlet ?j'. However, its effects can be bounded from eqn. 3 and can be 17 shown to be less than ten per cent or so. Thus we can write Im
ii12 = (Im IIFzx+2ERe $qx)(l f
(10%))
(32)
Row in the Kl4model we note that CP violation only arises from the heavy quark sector. Peverthelessan effective s+d transition is produced from the current-currentcombinations s+d+c+z
+d + gluon
+d+t+t 1 Since these interactionsare both strictly AI = l/2, we require em A2 = 0 and E = _
(2s' 0
Then from eqn. 32
.-ie' " _ z E
l IrnlI12 o Am
(34)
B.R. Holstein / CP violation 1984
53oc
Im Ml2
BOX Im y12 = E’
(35)
(1 *0(10X))
1+ >iB% Using the experimentalbound
IC’/Cl
(36)
we determine I$ ", 1111<
10%
(37)
which produces our final result: (38)
Im Gl2 = Im W~.$x(1 *C(20%))
Inserting numerical values, we find Im W12/Am = (2 ce-ie = 3.2 x 1O-3
(39)
= 828386 B (1.9 + 430(& + ~283~6)) In order to maximize the CP violation effect, we choose 1 _ n/2. (It can be shown that our numerical results are rather insensitiveto this choice as long as 70° - d - llO"). We find then an equation giving L in terms of B and x. 1.6/B = x sin 2M1.9 + 0.9s + 0.9 co8 2X 2)
(40)
This is a powerful equation of constraint on the angle k since if B, e assume their central values B = 0.33, z = l/1.4
(41)
we find 4.8 = sin 2b (1.9 + 0.6 co8 21) which cannot be satisfied for H
value of k.
(42) The best way in which to
understand the implicationsof eqn. 42 is shown in the figure below where we plot the b-u/b-c branching ratio vs. the b quark lifetime for various values of the B parameter. Assuming the allowed region r < 0.04
(43)
and
r .o.--
0.8 < T~/~O-~~ set c2.0
(44)
we see that for the allowed range of B 0.16 < B < 0.50
(45)
;o.--
only B - 0.5 is permitted, in which case r > 0.015
(46)
.04--
(47)
.02-m
If we allow say B - 0.66 we have r > 0.005 However this is only consistentwith value of Tb.
a
low
6 .a .m .w Y
531c
B.R. Holstein / CP violation 1984
Given the possible 20x or so uncertainty associated with eqn. 38, these predictions can be modified slightly, but basically this is as far as we can stretch the KU model. If CP violation is to be accounted for by the 3 generation Xl4model, then r > 0.01
(481
Thus a test of this model is to provide a firmer measurement of 'c~or to push the experimental limits on the bu/bc ratio. A value less than 0.01 would suggest the abandonment of the K?lmodel as the sole source of CP nonconserving effects and thus would signal the existance of important new physics.
It should be noted that an additional test of the validity of the IQ! picture is to seek a non-zero value for E'/E. The point here is that the Kw model does possess direct CP violation in the K+2n decay amplitude. This must show up at some level. Thus we have E'/E - o[ Am/Im"Bl2
(49)
The parameter E can be calculated semi-reliably. It was noted by Gilman and Wise'* that although CP violation arises only in direct 4-quark diagrams involvingheavy quarks (cf. eqn. 331 these interactionsgag contribute to the transitions s+d + q + q-M + gluon 19 which is directly related to so-called "penguin" diagrams. These methods yield an effective local 4-quark Hamiltonians eff BW = + Gp/fl2sls28386(Im ce)Siyu(l + Y5ldj ZqjY"(l - y5)qi
(50)
where Im cd - -0.1 is a penguin QCD enhancement factor. Calculating the 20 appropriatematrix element via the WIT bag model, we find that (again for s6 - 11 <-
-8 x 1o-4 2 sin 2A
(511
and 8 x 10-4 E-/E = 0s sin 2%.L 0.007 3.2 x 1O-3 Thus, E./C &
for B = .05
(521
be found to be non-zero if the current experimental
result - Equation 3 - is improved somewhat., This is a second unambiguous prediction of the lG4model which will soon be tested. Violation either of Equations 52 or 48 would signal the downfall of the simple KW picture and would herald the presence of new physics.
There are, of course, a number of forms which this new physics could take. We shall list four (which certainly do not exhaust the possbilities):
B.R. Holstein / CP violation 1984
532~
i) There exist more than three generations, In this situation the form of the KR matrix demanded by unitarity becomes more general than the three generation version of Equation 16. There are more mixing angles ei and additional CP violating phases di. The constraint implied by Equation 48 is obviously lost. There are astrophysicalarguments associated with the primevil He abundance which suggest that the number of such generations cannot be greater than four, but it is difficult to assess the reliabilityof this 21 suggestion. ii) Left-Right Synunetry Models. It was originally proposed by Uohapatra and Pati** that a conceivable source of CP violation was the existence of a set of right handed gauge bosons Wi w;
.
One
complementingthe usual left handed
can here choose, e.g., the left-handedweak quark couplings to be
purely real and CP conserving,while the quark couplings to 4
are in
general complex and CP violating. Since experimentalbounds force the right handed Wi
to be considerablyheavier than its left handed counterpart,a
natural mechanism for the weakness of the CP nonconserving interactionis obtained. Uore recent work has extended the original Uohapatra-Pati mode1,23 and it appears that such models can be consistentwith present phenomenology. However, they have not yet been subjected to as much theoreticalscrutiny as has KX. iii) Higgs Models. The so-called "standard"model of the electroweak interactioncontains a single Higgs doublet. The mechanism of spontaneous symmetry breaking gives them a single neutral "physical"Higgs scalar whose couplings to quarks/leptonsare flavor diagonal and proportionalto mass. Weinberg suggested,however, that in a model with three or more Higgs 24 doublets, there exist physical charged Higgs scalars as well as neutrals. The neutral Higgs couplings are required to
be
flavor diagonal. However, CP
violation can be associated with the coupling of the quarklleptonsector to the charged Higgs bosons. In such a model Deshpande and Senda have argued 25 that C'/E must be - o = 0.05 which is ruled out by experiment. However, these calculationsdid not correctly include the constraintsof chiral symmetry. When this is done, the model is barely compatiblewith the 26 present limit; however, the Higgs model also appears to be in trouble by 27 predicting too large an electric dipole moment for the neutron. iv) The Superweak Model. Ever since CP violation was first detected it has been realized that it is possible for a very weak (-lO_gL 1 AS = !@ 2 coupling between K '- K Oto be the source of CP violation. Such an interaction
would contribute
to Im q2
in first order and could be
B.R. Holstein / CP violation 1984
533c
29
produced, e.g., by a new gauge boson whose mass is
n
SW
-
~00~ WW - lo3 TeV.
Unfortunately,the extreme weakness of the associated coupling makes it all but impossibleto see effects in other than the K" -?
system. Also,
the large value of WSw implies that it would not be detectable even at the SSC. so that the validity of such a picture would have to be verified basically by ruling out everything else.
We have seen then that the currently fashionableKW model of CP violation is tightly pressed by new experimentalresults. If the latter are improved it may well be that the K.?! model will be soon ruled out as the sole source of CP noninvariance. This would not be a discouragingturn of events, however, since some sort of new physics - right-handedcurrents, a new generation,etc. - would be indicated,opening up a rich new vein for experimentaland theoretical exploration. ACKNOWLEDGRRERT It is a pleasure to acknowledge the hospitality of the Lewes Center for Physics, where this paper was written, and to thank Professor John Donoghue for many useful comments. This work was supported in part by the Wational Science Foundation. RKFERERCES 1) C.R. Christenson et al, Phys. Rev. Letters l3, 138 (1964). 2) L. Gladney. Proc. Symp. High Energy e+e- Interactions,Vanderbilt Univ. (1984). 3) C. Rubbia. CERN Colloqium (1984). 4) B. Winstein. Bull. Am. Phys. Sot. 29, 644 (1984). 51
K. Kinoshita, Proc.Proc. Symp. High Energy e+e- Interactions, Vanderbilt Univ. (1984).
61 T.T. Wu and C.R. Yang, Phys. Rev. Letters l3. 380 (19641; T.D. Lee and C.S. Wu, Ann. Rev. Nucl. Sci. l6. 471 (19661. 7) L Wolfenstein, in Theory and Phenomenologyin Particle Physics, ed. A. Zichichi. Academic Press, New York (1969). 81 Particle Data Group, Phys. Letters u,
1 (1982).
91 F. P. Calaprice. Proceedings of PANIC (1984). 101 1. Kobayashi and T. haskawa. Prog. Theo. Phys. 49. 652 (1973).
B.R. Holstein / CP violation 1984
534c
and Nuclei. Uuclear Physics with Heavy 11) D. H. Wilkinson, Synunetrics Ions and Resons, -ed.R. Balian, R. Rho and G. Ripka, Perth Holland, Amsterdam (1978). 121 J.H. Gaillard and G. Sauvage, Ann. Rev. Rucl. Part. (to be published) 13) R.B. Wise, Proc. Symposium on High Energy e+e- Interactions,Vanderbilt Univ. (1984). 14) A.J. Buras. Phys. Rev. Letters 46. 1354 (1981); L. Bergstrom, E. Rasso, P. Singer and D. Wyler, Phys. Letters m. 373 (1984). 412 15) J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Letters m. (1982); T. Applequist.J.D. Bjorken and R. Chanowitz,Phys. Rev. D7. 2225 (19731. 501 (1979); C.T. Hill, Phys. Letters m. 16) L. Wolfenstein.Rucl. Phys. m. 275 (1980); J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Letters 1358. 481 (1984). 17) J.F. Donoghue and B.R. Holstein, Phys. Rev. E. 18) F. Gilman and II.Wise, Phys. Rev. m. 19)
2088 (1984).
2392 (1979).
Shifman, A.I. Vainshtein and V.I. Zakharov, Uucl. Phys. u, 316 (1976); Zh. Rksp. Teor. Fiz. 72, 1275 (19771 (Sov. Phys. JETP 45, 670 (1977))
U.A.
201 J.F. Donoghue. E. Golowich, B.R. Holstein and W. Ponce, Phys. Rev. m, 1213 (1981). 21) G. Steigman, D.R. Schram and J.E. Gunn, Phys. Letters m. 22) R.U. llohopatraand J.C. Pati. Phys. Rev. u.
202 (1977).
566 (1975).
435 (19831; P. Herczeg, Phys. Rev. G, 23) D. Chang. Uucl. Phys. m, (1983). 24) S. Weinberg, Phys. Rev. Letters 37. 657 (1976).
200
2654 (1981); A. Sanda, Phys. Rev. m, 2647 25) U. Deshpande, Phys. Rev. m. (1981); J.P. Donoghue, J.S. Hagelin and B.R. Holstein, Phys. Rev. e. 195 (1982). 26) J.F. Donoghue et al. (in preparation). 27) G. Beall and 1. Deshpande (to be published);R.G. Deshpande.AIP Conf. Proc. Uo. 102, AIP, Uew York (1983) p. 107. 281 L. Wolfenstein,Phys. Rev. Letters l3, 562 (1964). 29) P.R. Hohapatra. J.C. Pati and L. Wolfenstein,Phys. Rev. w.
3319 (1975).
52%
North-Holland, Amsterdam
CP VIOLATIOR 1984: THE KM HODEL CORPRORTS EKPERIHERT
Barry R. Holstein Departrnent of Physics, University of Hassachusetts,Amherst, BA 01003
Using recent experimentaldata involving the b. t quarks and the nen measurement of B+/Uoo we show that the KM picture of CP violation is barely compatiblewith these results. Slight improvementsin any of these limits could require abandonementof the KM picture as the sole source of CP nonconservation.and some of the resultant required "new physics" is discussed. 1 Although the discovery of CP violation took place two decades ago, theoreticaldiscussions on the subject have not changed much during this time. A typical talk is outlined below (I have given several such myself): a. Set up the formalism for the kaon system b) Rote that CP nonconservationhas been observed only in the kaon sector c) Show that current results are consistentwith most models d) Plead with experimentalistsfor more and better data During the two years since the last meeting, experimentalistshave been at work and have provided us with some very importantnew results. Examples are: i) the B meson lifetime:' rB
= 1.6 f 0.4 ffO.ff x lo-I2 see
+0.4 rB = 1.2 _D.36 (fO.3) x 10-l' aec
RAC BKII
(1)
ii) the t-quark mass3 = 40 GeV % 4 iii) the U /B ratio (n,,"X+7
(2)
= 1.023 f 0.031(f0.023)
(3)
iv) the branching ratio for b + u/b -w' BR(b+u)< -%0.04 BR(b+c)
(4)
I shall argue here that, by combining all these new results, we can demonstrate that the standard KM model of CP violation is at present basely compatiblewith these data and that a significant- factor of three reduction in the bu/bc ratio could essentiallyrule out this model and require the introductionof interesting"new physics." In order to illustratethis feature, we first shall go through the standard steps a) and b) quoted above. bienote that CP violation in the I0 systezacan arise from two sources from the mass matrix that mixes K*, "K* and/or directly from the decay 0375-9474/85/$03.300 Elsevier SciencePublishers (North-Holland
Physics Publishing
Division)
B.V.
B.R. Holstein / CP violation 1984
526c
amplitude . The latter arc written
AmpfKO-* (m)I)
=\
(5)
exp(i8x)
where I = 0,2 is the isospin of the final two pion state and 8, is the associated strong interactionphase shift. We
choose
a phase convention so
that real AI correspondsto CP conservation. The eigenstatesof the Ramiltonian _ 5,
Kg _ are-writtenas
KR = (Kl C &K2)Ml + [cl2
(6)
KL = (K2 + ~Kl)/41 + ItI' in terms of CP even, odd states Kl, K2. It is standard to define6
rl+-
=
ah-tHwlK~> E
&
+
E’
(7) Qoo =
~~“lH~IK~
_ c _ 2E,
ProseCPT invariance one finds7 IlnMl2 -+ 0 AUI Im A2 exp iQ*oC-E) Re A2
where
(8)
&a = "t - mR = 2 Be HI2
(9)
E = Im ho/Re A0 and Be A2 0 = -
_ 0.05
(10)
Re A0 measures the strength of the AI = 3f2 component of s
to the &I = l/2
piece as revealed in the Kg + mr decay mode. The angles 8. 8' are given by 8 = tan-l (2Amq
= 44O
e* = g2 - 6. + n/2
= 37 f so
(11)
The experimentalresult eqn. 3 then determines C'/& = -0.004 f 0.005 (*0.004)
(12)
The measurement and the asynmetry in the semileptonicdecay of the < ~ /'(K&us)
- r(K~-Wvrr+)
P(KL-qA+vlr-) + lxKg-Wvlr+)
= 2 Re c I Jr+
ICI'
(13)
B.R. Holstein / CP violation I984
d=m
= +(3.30 f 0.12)
x
527~
10-3
and of n+_. nooa In+1
= (2.27 f 0.03) x 1O-3
(14)
= (44.6 f 12.010
d+
are the only places in which CP violation has been observed. m
other tests
- in nuclear beta decay. radiative decays of nuclei, detailed balance in scattering reactions, etc. - have given negative results, as summarizedby 9 Professor Calaprice.
In order to understand the origin of CP violation without introducingany 10 really new physics, it has become standard to use the so-called KM model. Thus in the six quark model for weak and electromagneticinteractionsbased on SU(2)b x U(1) gauge theory, the charged W bosons couple to the weak current
d
(+I
J,, = &(u,
0
c. t) Yn(l + Y5NJ 8 b
I
(15)
where U is a 3 x 3 unitary matrix which arises from the diagonalizationof the q = - l/3, +2/3 quark mass matrices. By adjusting the quark field phases, U can be written in the KLIform
c1
-*lc3
-sls3
ii u = s1c2 clc2c3-s2s3e
16 clc2s3+s2c3e
i6 c1s2c3+c2s3e
i6 cls2s3-c2c3e
sls2 where
(16)
the angles 81, e2, 8, lie in the first quadrant but the
phase of the CP violating parameter 8 is not fixed. The angles ei are determined i) from nuclear beta decay" = 0.9735 f 0.0005
(17)
ii) from'&peron beta decayr2 = 0.229 f 0.003 slc3 iii) from the B meson lifetime (using mc = 1.4 GeV. mb = 4.6 GeV)13 (s$+sz+2s283c*)= 4 x 10m3 lo-l2 set B.B.(b+c) =B 832= 4 x 1O-2 lo-l2 '=' B.R.(b+u) =B
(18)
(19)
3.R. Holstein / CP violation 1984
528C
-12 se& Defining z = 10
T
and s = p co15h, s3 = p sin h the 2 B last two equations can be written as p2 _ 4 x 10-3 z B.R.(b+c) _ 4 x 1O-3 e 1 + sin 2)lcs 1 + sin 2X c6 rEbbu=O1
*
b+c
(20)
sin2 A 1 + sin Z‘Lc,cb
We may proceed further by examining theoreticalestimates of Re T2,
Im
Here there is an important short distance piece which is easily nr2. 14 calculatedusing the 2W exchange 'box" diagram Box Gp s~!&K< Re R12 =1282
2 B(nl+n,K2 9
+ 2n,K In
m, (21) 2
GP 22 2 L)(sly@%% 2~283s@~-n,+rl,K.$- + Ds In 12n2 Tnc
Box
Im W12
Here
K = s$
+ 82~3~6,
'11 - 0.7, '12- 0.6, ?I3* 0.4
(22)
are QCD enhancement factors as calculated by Gilman and Wise, and
(23) Yg>dsyu(l +yg)dlKo>
is a measure of the short distance AS = 2 operator compared to its so-called "factorization"value. The parameter B has been extracted from &I = 312
K+~T
data by use of PCAG and SIJ(3)symmetry with the result'3
B = 0.33(%50X>
(24)
where the generous uncertainty is assigned given the possibly significant errors associated with SU(3) breaking in the n-K system and the off-mass-shellextrapolationof the physical K+~TTdecay amplitude.
In principle there are in addition significant long-distancepieces, and 16 In the case Re g,, a much effort has gone into their calculation. precision estimate is impossible. However, the general concensus is that Re Hklj= 112 Am D --Re $gx Re 8%
= + Oflf
(25)
with the dominant contributioncoming from the 2r and TT,n, ?j' intermediate'states. This is confirmed empirically in that numerically one
B.R. Holstein
/ CP violation
529c
1984
finds Re l4;: _ l/2
Am B n,
(26)
Comparison with the experimentalvalue Re(l$5x+ H$x
) = l/2 Am
(27)
reveals then Re l4$
D =_=_
Box Re R12
1 - Bn
+4.6 = 3.3 -1.4
(28)
grl
BV
in general agreement with eqn. 25. In the case of Im M12. there is in principle also a long distance contribution. However. its effect is much smaller. This can be seen since s is affected not by ~41~alone but rather by the combination Im
ii12 = Im U12 + 2ERe M12
(29)
But by definition, for the 2x intermediatestate we have (2x) Im R$f"i -2ERe W12
(30)
so that Im ii:gx)=o Also, since K"+xo.~o are related to K'+mr via a simple soft pion extrapolation,it can be shown that T pn Ointermediatestate contributionsare also negligible. l7 This argument does not extend to the W(3) singlet ?j'. However, its effects can be bounded from eqn. 3 and can be 17 shown to be less than ten per cent or so. Thus we can write Im
ii12 = (Im IIFzx+2ERe $qx)(l f
(10%))
(32)
Row in the Kl4model we note that CP violation only arises from the heavy quark sector. Peverthelessan effective s+d transition is produced from the current-currentcombinations s+d+c+z
+d + gluon
+d+t+t 1 Since these interactionsare both strictly AI = l/2, we require em A2 = 0 and E = _
(2s' 0
Then from eqn. 32
.-ie' " _ z E
l IrnlI12 o Am
(34)
B.R. Holstein / CP violation 1984
53oc
Im Ml2
BOX Im y12 = E’
(35)
(1 *0(10X))
1+ >iB% Using the experimentalbound
IC’/Cl
(36)
we determine I$ ", 1111<
10%
(37)
which produces our final result: (38)
Im Gl2 = Im W~.$x(1 *C(20%))
Inserting numerical values, we find Im W12/Am = (2 ce-ie = 3.2 x 1O-3
(39)
= 828386 B (1.9 + 430(& + ~283~6)) In order to maximize the CP violation effect, we choose 1 _ n/2. (It can be shown that our numerical results are rather insensitiveto this choice as long as 70° - d - llO"). We find then an equation giving L in terms of B and x. 1.6/B = x sin 2M1.9 + 0.9s + 0.9 co8 2X 2)
(40)
This is a powerful equation of constraint on the angle k since if B, e assume their central values B = 0.33, z = l/1.4
(41)
we find 4.8 = sin 2b (1.9 + 0.6 co8 21) which cannot be satisfied for H
value of k.
(42) The best way in which to
understand the implicationsof eqn. 42 is shown in the figure below where we plot the b-u/b-c branching ratio vs. the b quark lifetime for various values of the B parameter. Assuming the allowed region r < 0.04
(43)
and
r .o.--
0.8 < T~/~O-~~ set c2.0
(44)
we see that for the allowed range of B 0.16 < B < 0.50
(45)
;o.--
only B - 0.5 is permitted, in which case r > 0.015
(46)
.04--
(47)
.02-m
If we allow say B - 0.66 we have r > 0.005 However this is only consistentwith value of Tb.
a
low
6 .a .m .w Y
531c
B.R. Holstein / CP violation 1984
Given the possible 20x or so uncertainty associated with eqn. 38, these predictions can be modified slightly, but basically this is as far as we can stretch the KU model. If CP violation is to be accounted for by the 3 generation Xl4model, then r > 0.01
(481
Thus a test of this model is to provide a firmer measurement of 'c~or to push the experimental limits on the bu/bc ratio. A value less than 0.01 would suggest the abandonment of the K?lmodel as the sole source of CP nonconserving effects and thus would signal the existance of important new physics.
It should be noted that an additional test of the validity of the IQ! picture is to seek a non-zero value for E'/E. The point here is that the Kw model does possess direct CP violation in the K+2n decay amplitude. This must show up at some level. Thus we have E'/E - o[ Am/Im"Bl2
(49)
The parameter E can be calculated semi-reliably. It was noted by Gilman and Wise'* that although CP violation arises only in direct 4-quark diagrams involvingheavy quarks (cf. eqn. 331 these interactionsgag contribute to the transitions s+d + q + q-M + gluon 19 which is directly related to so-called "penguin" diagrams. These methods yield an effective local 4-quark Hamiltonians eff BW = + Gp/fl2sls28386(Im ce)Siyu(l + Y5ldj ZqjY"(l - y5)qi
(50)
where Im cd - -0.1 is a penguin QCD enhancement factor. Calculating the 20 appropriatematrix element via the WIT bag model, we find that (again for s6 - 11 <-
-8 x 1o-4 2 sin 2A
(511
and 8 x 10-4 E-/E = 0s sin 2%.L 0.007 3.2 x 1O-3 Thus, E./C &
for B = .05
(521
be found to be non-zero if the current experimental
result - Equation 3 - is improved somewhat., This is a second unambiguous prediction of the lG4model which will soon be tested. Violation either of Equations 52 or 48 would signal the downfall of the simple KW picture and would herald the presence of new physics.
There are, of course, a number of forms which this new physics could take. We shall list four (which certainly do not exhaust the possbilities):
B.R. Holstein / CP violation 1984
532~
i) There exist more than three generations, In this situation the form of the KR matrix demanded by unitarity becomes more general than the three generation version of Equation 16. There are more mixing angles ei and additional CP violating phases di. The constraint implied by Equation 48 is obviously lost. There are astrophysicalarguments associated with the primevil He abundance which suggest that the number of such generations cannot be greater than four, but it is difficult to assess the reliabilityof this 21 suggestion. ii) Left-Right Synunetry Models. It was originally proposed by Uohapatra and Pati** that a conceivable source of CP violation was the existence of a set of right handed gauge bosons Wi w;
.
One
complementingthe usual left handed
can here choose, e.g., the left-handedweak quark couplings to be
purely real and CP conserving,while the quark couplings to 4
are in
general complex and CP violating. Since experimentalbounds force the right handed Wi
to be considerablyheavier than its left handed counterpart,a
natural mechanism for the weakness of the CP nonconserving interactionis obtained. Uore recent work has extended the original Uohapatra-Pati mode1,23 and it appears that such models can be consistentwith present phenomenology. However, they have not yet been subjected to as much theoreticalscrutiny as has KX. iii) Higgs Models. The so-called "standard"model of the electroweak interactioncontains a single Higgs doublet. The mechanism of spontaneous symmetry breaking gives them a single neutral "physical"Higgs scalar whose couplings to quarks/leptonsare flavor diagonal and proportionalto mass. Weinberg suggested,however, that in a model with three or more Higgs 24 doublets, there exist physical charged Higgs scalars as well as neutrals. The neutral Higgs couplings are required to
be
flavor diagonal. However, CP
violation can be associated with the coupling of the quarklleptonsector to the charged Higgs bosons. In such a model Deshpande and Senda have argued 25 that C'/E must be - o = 0.05 which is ruled out by experiment. However, these calculationsdid not correctly include the constraintsof chiral symmetry. When this is done, the model is barely compatiblewith the 26 present limit; however, the Higgs model also appears to be in trouble by 27 predicting too large an electric dipole moment for the neutron. iv) The Superweak Model. Ever since CP violation was first detected it has been realized that it is possible for a very weak (-lO_gL 1 AS = !@ 2 coupling between K '- K Oto be the source of CP violation. Such an interaction
would contribute
to Im q2
in first order and could be
B.R. Holstein / CP violation 1984
533c
29
produced, e.g., by a new gauge boson whose mass is
n
SW
-
~00~ WW - lo3 TeV.
Unfortunately,the extreme weakness of the associated coupling makes it all but impossibleto see effects in other than the K" -?
system. Also,
the large value of WSw implies that it would not be detectable even at the SSC. so that the validity of such a picture would have to be verified basically by ruling out everything else.
We have seen then that the currently fashionableKW model of CP violation is tightly pressed by new experimentalresults. If the latter are improved it may well be that the K.?! model will be soon ruled out as the sole source of CP noninvariance. This would not be a discouragingturn of events, however, since some sort of new physics - right-handedcurrents, a new generation,etc. - would be indicated,opening up a rich new vein for experimentaland theoretical exploration. ACKNOWLEDGRRERT It is a pleasure to acknowledge the hospitality of the Lewes Center for Physics, where this paper was written, and to thank Professor John Donoghue for many useful comments. This work was supported in part by the Wational Science Foundation. RKFERERCES 1) C.R. Christenson et al, Phys. Rev. Letters l3, 138 (1964). 2) L. Gladney. Proc. Symp. High Energy e+e- Interactions,Vanderbilt Univ. (1984). 3) C. Rubbia. CERN Colloqium (1984). 4) B. Winstein. Bull. Am. Phys. Sot. 29, 644 (1984). 51
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