CP Violation: Past, Present, and Future
Jonathan L.Rosner
EnrioFermiInstituteandDepartment ofPhysis
UniversityofChiago,Chiago,IL60637USA
Reeivedon3January,2001
We disussthe historyof CPviolation and itsmanifestationsinkaon physis, itsexplanation in
termsofphasesoftheCabibbo-Kobayashi-Maskawamatrixdesribingharge-hangingweakquark
transitions,preditions for experimentsinvolving B mesons, and thelight itan shedonphysis
beyondtheStandardModel.
I Introdution
CPsymmetryanditsviolationareimportantguidesto
fundamentalquarkpropertiesandtotheunderstanding
ofthematter-antimatterasymmetryoftheUniverse. In
thisreview,anupdatedversionofonepresentedearlier
in the year [1℄, we desribe past, present, and future
aspets of CP violation studies. After an illustration
of fundamental isrete symmetriesin Maxwell's
equa-tions (Se. II), we reall the historyof CP violation's
disovery[2℄ in the deaysof neutralkaons (Se. III).
The produt CPT sofar seems to be onserved, asis
expetedin loal Lorentz-invariantquantum eld
the-ories [3℄. Wethen disusstheeletroweaktheory's
ex-planationof CPviolation[4℄in termsofphasesof the
Cabibbo-Kobayashi-Maskawa(CKM) [4, 5℄ matrix in
Se.IV),andmentionsomepresenttestsofthistheory
with kaons(Se.V)B mesons(Se.VI), andharmed
partiles(Se.VII). ThefutureofCPviolationstudies
(Se. VIII) is veryrih, with awide varietyof
experi-ments relevanttophysisbeyondtheStandard Model
andthebaryonasymmetryoftheUniverse.
II Disrete symmetries in
Maxwell's equations
The behavior of the Maxwell equations under the
disrete symmetries P (parity), T (time reversal), C
(hargeonjugation),andCPTissummarizedinTable
I.Eahtermbehavesasshown.
UnderP,wehave
E(x;t) ! E( x;t); B(x;t)!B( x;t); (1)
r ! r; j(x;t)! j( x;t): (2)
Eletri elds hange in sign while magneti elds do
not,andurrentshangein diretion. UnderT,
E(x;t) ! E(x; t); B(x;t)! B(x; t); (3)
Magneti elds hange in sign while eletri elds do
not,sinediretionsofurrentsarereversed. UnderC,
E(x;t) ! E(x;t); B(x;t)! B(x;t); (5)
(x;t) ! (x;t); j(x;t)! j(x;t): (6)
Both eletri and magneti elds hange sign, sine
theirsouresandjhangesign. Finally, underCPT,
spaeand timeare inverted but eletriand magneti
eldsretaintheirsigns:
E(x;t)!E( x; t); B(x;t)=B( x; t): (7)
AfundamentaltermintheLagrangianbehavingas
EB,whileLorentzovariant,wouldviolatePandT.
Suhatermseemstobestronglysuppressed,inviewof
thesmallvalueoftheneutroneletridipole moment.
Its absene is a mystery, but several possible reasons
havebeenproposed(see,e.g.,[6℄).
TABLEI.BehaviorofMaxwell'sequationsunder
disretesymmetries.
Equation P T C CPT
rE=4 + +
rB=0
rB 1
E
t =
4
j
rE+ 1
B
t
=0 + +
III CP symmetry for kaons
Some neutral partiles, suh as the photon, the
neu-tralpion,andtheZ 0
not. The neutral kaon K 0
, disovered in 1946 [7℄ in
osmi radiation, was assigned a \strangeness"
quan-tumnumberS=1inthelassiationshemeof
Gell-Mann and Nishijima [8℄ in order to explainits strong
produtionandweakdeay. Produtionwouldonserve
strangeness,whiletheweakerdeayproesswouldnot.
For this sheme to make sense it was then neessary
thattherealsoexistananti-kaon,theK 0
,withS= 1.
As Gell-Mann desribed this sheme at a seminar
at the Universityof Chiago, EnrioFermi askedhim
what distinguished theK 0
from theK 0
ifbothould
deayto , asseemed tobeobserved. This question
led Gell-Mann and Pais[9℄ to propose that the states
ofdenitemassandlifetime were
K
1 =
K 0
+K 0
p
2
(C =+); (8)
K
2 =
K 0
K 0
p
2
(C = ); (9)
(10)
withtheK
1
allowedbyCinvariane(thenthoughtto
beapropertyofweakinterations)todeayto and
the K
2
forbidden to deay to . The K
2
would be
allowed to deay only to three-body nal states suh
as +
0
and thus would have a muh longer
life-time. It was looked for and found in 1956 [10℄. The
disoverythat theweakinterations violatedC andP
but apparentlypreservedtheprodutCP[11℄ ledto a
reastingoftheaboveargumentthroughthe
identia-tionCP(K
1
)=+(K
1
),CP(K
2
)= (K
2 ).
The K
1 {K
2
system an be illustrated using a
de-generate two-state example suh as a pair of oupled
pendula [12℄ or the rst exitations of a drum head.
Thereisnowaytodistinguishbetweenthebasisstates
illustratedinFig. 1(a),in whih thenodallinesareat
anglesof45 Æ
withrespettothehorizontal,andthose
inFig. 1(b),in whihtheyarehorizontalandvertial.
Figure1. Basis statesforrstexitationsof adrumhead.
(a)Nodallinesat45 Æ
withrespettohorizontal;(b)
hor-Ifaylandsonthedrum-headatthepointmarked
\", thebasis (b) orresponds to eigenstates. Oneof
the modes ouplesto the y; the other doesn't. The
basis in (a) is likethat of (K 0
;K 0
), while that in (b)
islikethatof(K
1 ;K
2
). Neutralkaonsareproduedas
in (a), while they deayas in (b), with they
analo-goustothe state. Theshort-livedstate(K
1 ,inthis
CP-onserving approximation) has a lifetime of 0.089
ns, whilethelong-livedstate('K
2
)lives600times
aslong,for52ns.
In1964Christenson, Cronin,Fith, and Turlay [2℄
found that indeed onein about500long-lived neutral
kaons did deayto +
, and one in about1000
de-ayedto 0
0
. Thestatesofdenitemassandlifetime
ouldthenbewritten, approximately,as
K
S
(\short")'K
1 +K
2 ;
K
L
(\long")'K
2 +K
1
; (11)
withaparameterwhosemagnitudewasabout210 3
andwhosephasewasabout45 Æ
. Sinethestatesof
def-inite massandlifetime were nolongerCPeigenstates,
CPhad tobeviolatedsomewhere. However,for many
yearswastheonlyparameterdesribingCPviolation.
Oneouldmeasureitsmagnitudeandphasemoreand
morepreisely(inludinglearningaboutRe()through
a study of harge asymmetries in K
L
!
l
), but
its origin remained a mystery. One viable theory
in-ludeda\superweak"one[13℄whihpostulatedanew
interation mixing K 0
=ds andK 0
=s
dbut with no
otheronsequenes.
KobayashiandMaskawaoeredanewopportunity
to desribe CP violation by boldly postulating three
quarkfamilies[4℄whenharm(the lastmemberof the
seondfamily)hadnotyetevenbeenrmlyestablished.
In the diagram of Fig. 2 desribing the seond-order
weak transition ds ! s
d through intermediate states
involvingpairsofquarksi;j=u;;twithharges2/3,
the phases of omplexweak ouplingsan have
phys-ial eets. As longas there are at least three quark
families, one annot redene quark phasesso that all
suhouplingsarereal,andoneangenerateanonzero
valueof.
Figure 2. Box diagram desribing the seond-order weak
mixingofaK 0
=dswithaK 0
=s
d. Thereisanother
dia-gramwithvertialW +
Thetime-dependeneofthetwo-omponentK 0
and
K 0
systemisgovernedbya22massmatrixM[14℄:
i
t 2
4 K
0
K 0
3
5
=M 2
4 K
0
K 0
3
5
; (12)
where M = M i =2, and M and are Hermitian
matries. Theeigenstates(11)thenorrespondto the
eigenvalues
S;L =m
S;L i
S;L
=2,with
' Im(
12
=2)+iImM
12
S
L
: (13)
UsingdataandthemagnitudeofCKMmatrixelements
one an show [14℄ that the seond term dominates.
Sine the mass dierene m
L m
S
and width
dier-ene
S
L
arenearlyequal,thephaseof
L
S is
about=4,sothatthephaseofisalso=4(mod ).
Itiseasyto modeltheCP-onservingneutralkaon
systemintable-topsystemswithtwodegeneratestates
[12℄. The demonstrationof CP violation requires
sys-tems that emulate Im(M
12
)6=0orIm(
12
)6=0. One
an ouple two idential resonant iruits
\diretion-ally" to eah other (see Fig. 3sothat theenergy fed
from iruit1 to iruit2 diersfrom that fed in the
reversediretion[15℄. Devieswiththispropertyutilize
Faraday rotationof theplane of polarization of
radio-frequenywaves;somereferenesmaybefoundin[16℄.
This asymmetriouplingalsoisinherentin the
equa-tionsofmotionofaspherial(or\onial")pendulum
in arotatingoordinate system [17℄, so that the
Fou-aultpendulumisademonstration(thoughperhapsnot
\table-top")ofCPviolation. Aballrollingwithvisous
damping in a rotating vase of elliptial ross setion
holdsmorepromiseforalaboratorysetting[16℄. Inall
suhasestheCP-violatingeetisimposed\fromthe
outside," leaving open the question of whether some
\new physis" isgoverningthe orrespondingeet in
partilephysis.
Figure3. Coupled\tank"iruitsillustratingtheK 0
K 0
system. TheouplingimpedaneZmustbeasymmetrito
emulateCPviolation.
IV Kobayashi-Maskawa theory
of CP violation
The interations of quarks with W
bosons are
de-sribedby
L
int =
g
p
2 [
U 0
L
W (+)
D
0
L
+H::℄; (14)
wheretheprimedquarksare \weakeigenstates":
U 0
2
6
6
4 u
0
0
t 0
3
7
7
5 ; D
0
2
6
6
4 d
0
s 0
b 0
3
7
7
5
: (15)
Intheweak-eigenstatebasis,themassterminthe
La-grangian,
L
m = [
U 0
R M
U U
0
L +
D 0
L M
D D
0
L
+H:: ℄; (16)
will involvea general33matrixM, whih requires
separateleftandrightunitarytransformations
R y
Q M
Q L
Q =
Q
(17)
toobtainadiagonal matrix
Q
withnon-negative
en-tries. Ifwedeneunprimed(mass)eigenstatesby
Q 0
L =L
Q Q
L ; Q
0
R =R
Q Q
R
(Q=U; D); (18)
theinteration Lagrangianmaybeexpressedas
L
int =
g
p
2 [
U
L
W (+)
VD
L
+H:: ℄; (19)
whereV L y
U L
D
is theCabibbo-Kobayashi-Maskawa
(CKM) matrix. As a result of its unitarity, V y
V =
VV y
=1,theZqqouplingsin theeletroweaktheory
are avor-diagonal. Sine it ontains no information
aboutR
U or R
D
, V providesonly partial information
aboutM
Q .
Fornu-typequarksandnd-typequarks,V isnn.
Sineitisunitary,itanbedesribedbynreal
param-eters. Relativequarkphasesaountfor2n 1ofthese,
leavingn 2
(2n 1)=(n 1) 2
physialparameters. Of
these,n(n 1)=2(thenumberofindependentrotations
in n dimensions) orrespond to angles, while therest,
(n 1)(n 2)=2,orrespondtophases.
Forn=2,wehaveone angle and nophases. The
matrix V then an always be hosen as orthogonal
[5, 18℄. For n = 3, we have three angles and one
phase, whih in general annot be eliminated by
ar-bitrary hoies of phases in the quark elds. It was
thisphasethat motivatedKobayashiand Maskawa[4℄
to introdue athird quarkdoublet in 1973when only
twowereknown. (Thebottomquarkwasdisoveredin
1977 [19℄, and thetop in 1994 [20℄.) The
Kobayashi-MaskawatheoryprovidesapotentialsoureofCP
substantial CP asymmetries in the deays of mesons
ontaining b quarks. The pattern of harge-hanging
weaktransitionsamong quarksisdepitedinFig. 4.
Figure 4. Pattern of harge-hanging weak transitions
amongquarks. Solidlines: relativestrength1;dashedlines:
relative strength 0.22; dot-dashed lines: relative strength
0.04; dotted lines: relative strength 0:01. Breadths of
levelsdenoteestimatederrorsinquarkmasses.
A onvenient parametrization of the CKM matrix
utilizesahierarhy[21℄wherebymagnitudesofelements
areapproximatelypowersofsin
'0:22,where
istheGell-Mann{Levy{Cabibboangle[5,22℄desribing
strangepartiledeays. The matrixmaybeexpressed
as
V = 2
6
6
4 1
2
2
A
3
( i)
1
2
2
A 2
A 3
(1 i) A
2
1
3
7
7
5 ;
(20)
whererowsdenoteu; ; tandolumnsdenoted; s; b.
We learn jV
b
j = A 2
' 0:0410:003 from the
dominant deays of b quarks, whih are to harmed
quarks [23, 24℄. Smaller errors are quoted in most
reviews [25℄ whih take dierent views of the
domi-nantly theoretial soures of error. As an indiation
that this number is still in some ux we note a new
measurementjV
b
j=0:0460:004bytheCLEOgroup
[26℄.) Similarly, we shalltake from harmess b deays
jV
ub =V
b
j=0:0900:025=( 2
+ 2
) 1=2
[27℄, leading
to 2
+ 2
= 0:410:11, whereas smaller errors are
quotedbymostauthors.
Figure 5. Unitarity triangle for CKM elements. Here
+i=V
ub =A
3
;1 i=Vtd=A 3
.
AsaresultoftheunitarityoftheCKMmatrix,the
quantitiesV
=A 3
=+i,V
td =A
3
=1 i,and
1form atrianglein the (;)plane (Fig. 5). Westill
do not have satisfatory limits on the angle of this
\unitarity triangle." Further information omes from
thefollowingonstraints:
1. Mixingof neutralB mesonsisdominatedbytop
quarkontributionsto graphssuh asFig. 2butwith
external quarksd
b for B 0
or s
b for B
s
. For example,
themasssplitting in thenonstrangeneutralB system
is
m
d
=0:4870:014ps 1
f 2
B B
B jV
td j
2
; (21)
where f
B
is the B meson deay onstant and B
B =
O(1)is the\vauumsaturationfator,"desribingthe
degree to whih graphs suh as Fig. 2 desribe the
mixing. Reentestimates[28℄givef
B p
B
B
=23040
MeV.Consequently,onends[23℄j1 ij=0:87
0:21. Neutral strangeB mesons are haraterized by
[29℄
m
s f
2
B
s B
Bs jV
ts j
2
>15ps 1
: (22)
Sine jV
ts
j ' jV
b
j is approximately known, this
information mainly serves to onstrain the produt
f
Bs p
B
Bs
and,giveninformationontheratioofstrange
andnonstrangeonstants[30℄,thevalueofjV
td
j,leading
toj1 ij<1:01. Thelargetopmass,m
t
=1745
GeV[31℄,isruialforthesemixings tobesolarge.
2. CP-violating K 0
{K 0
mixing through the box
graphsofFig. 2aountsfortheparameter[31℄
=(2:2710 3
)e i43:3
Æ
ImM
12 f
2
K B
K Im(V
2
td );
(23)
leadingtoaonstraint[23,24℄
(1 +0:39)=0:350:12 : (24)
Herewehaveusedf
K
=161MeVandB
K
=0:870:13
[32℄. If top quarkswere fully dominant the left-hand
sideofthis equationwouldbejust(1 ). Theterm
0.39in braketsisaorretionduetoharmedquarks.
Figure6. Regionof(;)speiedbyonstraintsonCKM
matrixparameters.Solidsemiirlesdenotelimitsbasedon
jVub=Vbj=0:0900:025;dashedarsdenotelimits0:66
j1 ij1:08basedonB 0
{B 0
mixing;dot-dashedar
denotes limit j1 ij<1:01 based onB
s {B
s
mixing;
dottedlinesdenotelimits(1 +0:39)=0:350:12based
onCP-violatingK 0
{K 0
mixing. Rays:1limitsonsin2
(seeSe.VI).Theplottedpointat(;)'(0:20;0:28)lies
The onstraints are plotted on the (;) plane in
Fig. 6. Also shown are the1 bounds on sin2, to
bedisussedpresently,fromanaverage0:490:23[24℄
of OPAL, ALEPH, CDF, BaBar, and BELLE values.
Theallowedregionislargerthanthatfavoredbymany
otheranalyses[25℄.
V The CKM matrixand
predi-tions for kaon physis
V.1 K
S;L
! rates
Ifwedene
+
A(K
L !
+
)
A(K
S !
+
) ;
00
A(K
L !
0
0
)
A(K
S !
0
0
) ;
(25)
the possibility of dierent CP-violating eets in
statesof isospinI
=2andI
=0[33℄ givesriseto
a parameter 0
suh that
+
= + 0
,
00
= 2
0
.
Thefollowingratioofratiosthenandierfromunity:
R (K
L !
+
)
(K
S !
+
) =
(K
L !
0
0
)
(K
S !
0
0
)
=1+6Re
0
: (26)
Theratio 0
=isexpetedtobeapproximatelyrealina
CPT-invarianttheory[14℄. AkeypreditionoftheKM
theory is that 0
=should beanumber oforder 10 3
.
TwotypesofamplitudesontributetoK!deays.
1. Tree amplitudes, involvingthequarksubproess
s !uud, havebothI =1=2 andI =3=2
ompo-nentsandthusontributetobothI
=0andI
=2
states. Ina standardonvention[21℄, tree amplitudes
ontain no weak phases, sine they involve the CKM
elementsV
ud andV
us .
2. Penguinamplitudes, involvingthequark
subpro-esss!dwithanintermediatelooponsistingofaW
bosonandthequarksu; ; t,andinteratingwiththe
rest of the system through one or more gluons, have
only I = 1=2 and thus an only ontribute to the
I
=0state. Thetopquarkin theloopgivesriseto
aweakphasethroughtheCKMelementV
td .
A relative weak phase of I
= 0 and I
= 2
states is thus generated in the KM theory, leadingto
0
=6=0. Eletroweakpenguinamplitudes,inwhihthe
gluononneting the s ! d subproess to therest of
the diagram is replaed by aphoton or Z 0
, an have
both I = 1=2 and I = 3=2 omponents and tend
toredue thepreditedvalueof 0
=. Onerangeof
es-timates [34℄ nds abroad and somewhat asymmetri
probability distribution extendingfrom slightlybelow
zero to above 210 3
. Others (see artiles in [35℄)
permitslightlyhighervalues.
ReentexperimentsonRe( 0
=)[36, 37,38,39℄ are
summarized in TableII.(The error in the average
in-ludesasalefator[31℄of1.86.)Themagnitudeof 0
=
is onsistent with estimates based on the
Kobayashi-Maskawa theory. The qualitative agreement is
sat-isfatory, given that we still annot aount reliably
for the large enhanement of I = 1=2 amplitudes
withrespettoI=3=2amplitudesinCP-onserving
K!deays. MoredataareexpetedfromtheF
er-milab and CERN experiments, reduing the eventual
statistialerroron 0
=toapartin 10 4
.
V.2 K!l +
l information
1. The deay K +
! +
involvesloop diagrams
involving V
td
and a small harm orretion in suh a
waythattheombinationj1:4 ijisonstrained,
withapreditedbranhingratiooforder
B(K +
! +
)'10 10
j1:4 i
1:4
2
; (27)
orfortherangepermittedinFig.6,abranhingratioof
about(0:80:2)10 10
[40℄. Additionalunertainties
areassoiated withm
[41℄and jV
b
j. A measurement
ofB(K +
! +
) to 10%will helpto onstrain(;)
moretightlythanin Fig. 6orwill expose
inonsisten-iesinourpresentpitureofCPviolation.
UptonowtheBrookhavenE787Collaborationsees
onlyoneK +
! +
eventwithnegligiblebakground
[42℄,orrespondingto
B(K +
! +
) =(1:5 +3:4
1:2 )10
10
: (28)
Moredata areexpeted from thenal analysis of this
experiment, aswellas from a future versionwith
im-provedsensitivity.
TABLE II.Reentexperimental valuesforRe( 0
=).
Experiment Referene Value(10 4
)
2
FermilabE731 [36℄ 7:45:9 3.97
CERN NA31 [37℄ 23:06:5 0.35
FermilabE832 [38℄ 28:04:1 4.65
CERN NA48 [39℄ 14:04:3 1.44
Average 19:24:6
P
2. The deays K
L
!
0
l +
l should be
dom-inated by CP-violating ontributions, both indiret
()anddiret,with aCP-onserving\ontaminant"
from K
L !
0
!
0
l +
l . The diret ontribution
probes theparameter. Eah ontribution (inluding
theCP-onservingone)isexpeted toorrespondto a
0
e +
e branhing ratio of a few parts in 10 12
.
How-ever,K
L !
0
e +
e maybelimitedbybakgroundsin
the e +
e nal stateassoiated with radiation of a
photon in K
L ! e
+
e from oneof the leptons [43℄.
Presentexperimentalupperlimits(90%.l.) [44℄are
B(K
L !
0
e +
e )<5:110 10
;
B(K
L !
0
+
)<3:810 10
; (29)
still signiantly abovemost theoretial expetations.
(See, however,[45℄.)
3. The deay K
L
!
0
should be due
en-tirely to CP violation, and provides a lean probe of
. Its branhing ratio, proportional to A 4
2
, is
ex-peted to be about 310 11
. The best urrent
ex-perimental upperlimit(90% .l.) forthis proess [46℄
isB(K
L !
0
) <5:910 7
,severalorders of
mag-nitudeabovetheexpetedvalue.
V.3 Other rare kaon deays
1. The deay K
L
!
+
e +
e involvesthree
in-dependent momenta in the nal state and thus oers
theopportunitytoobserveaT-oddobservablethrough
aharateristidistributionintheanglebetweenthe
+
and e +
e planes. A CP-orT-violatingangular
asymmetryin this proess has reently been reported
[47,48℄.
2. The deay K
L
!
+
has been studied
withsuÆientlyhigh statististo permit agreatly
im-provedmeasurementof thevirtual-photonformfator
in K
L !
[49℄. Thismeasurementis usefulin
esti-matingthe long-distaneontribution to thereal part
ofthe amplitudein K
L !
()
()
!
+
, whih in
turnallowsonetolimittheshort-distaneontribution
toK
L !
+
.
V.4IstheCKMpitureofCPviolationorret?
The KM theory is omfortable with the observed
range of 0
=, and its predition for B(K +
!
+
)
is onsistent with the oneevent seen so far. Further
antiipated tests are the measurement through the
deayK
L !
0
(see below), andthe searh forCP
violationinhyperondeays,whihisalreadyunderway
[50,51℄. Onealsolooks forwardtoarihset ofeets
in deaysofpartilesontainingb quarks,partiularly
B mesons. We now desribethe experiments andthe
eetstheyareexpetedto see.
VI CP violation in B deays
VI.1 Currentand planned experiments
Asymmetri e +
e ollisions are being studied at
\B fatories": thePEP-II mahine at SLAC with the
BaBardetetor,andtheKEK-BolliderinJapanwith
the Belle detetor. ByJuly 2000, these detetors had
aumulated about 14 and 6 fb 1
of data at the
en-ergy ofthe (4S)resonane,whih deaysalmost
ex-lusivelytoB
B[52,53℄. AsofSeptember,2000,PEP-II
andKEK-Bwereprovidingabout150and100pb 1
per
dayto theirrespetivedetetors.
Further data on e +
e ollisionsat the (4S) will
beprovidedbytheCornellEletronStorageRingwith
theupgradedCLEO-IIIdetetor. TheHERA-b
exper-iment at DESY in Hamburg hopes to study b quark
prodution viatheollisionsof 920GeV protonswith
axedtarget. TheCDFandD0detetorsatFermilab
willdevoteasigniantpartoftheirprogramatRunII
of theTevatron to B physis. Oneanexpet further
resultsonB physisfromthegeneral-purposeLHC
de-tetorsATLASand CMS,andthedediateddetetors
at LHC-batCERNand BTeVat Fermilab.
VI.2 Types ofCP violation
Inontrasttoneutralkaons,whosemasseigenstates
dier in lifetime by nearly a fator of 600, the
or-responding B 0
{B 0
mass eigenstates are predited to
dier in lifetime by at most 10{20% for strange B's
[54, 55℄, and muh less for nonstrangeB's. Thus,
in-steadofmasseigenstateslikeK
L
,twomaintypesofB
deays are of interest: deays to CP eigenstates, and
\self-tagging"deays. Bothhavetheiradvantagesand
disadvantages.
1. Deays to CP eigenstates f = CP(f)
uti-lize interferene between diret deays B 0
! f or
B 0
! f and the orresponding paths involving
mix-ing: B 0
! B 0
! f orB 0
! B 0
! f. Final states
suh as f = J= K
S
provide examples in whih one
quark subproess is dominant. In this ase one
mea-suressin2withnegligibleorretions. Forf = +
,
onewouldmeasuresin2onlyifthediretdeaywere
dominatedbya\tree"amplitude(thequarksubproess
b!uud). With ontamination from thepenguin
sub-proessb!dexpetedtobeabout30%in amplitude,
one must measure deays to other states(suh as
0
and 0
0
)tosortoutamplitudes[56℄. Indeays
toCPeigenstates,onemustdeterminetheavorofthe
deayingB attimeofprodution.
2. \Self-tagging"deays involvenal statesf suh
as K +
whih an be distinguished from their
CP-onjugates
f. A CP-violating rate asymmetry arises
whentwoweakamplitudesa
i
withweakphases
i and
strongphasesÆ
i
(i=1;2)interfere:
A(B!f)=a
1 e
i(+
1 +Æ
1 )
+a
2 e
i(+
2 +Æ
2 )
;
A(
B!
f)=a e i(
1 +Æ
1 )
+a e i(
2 +Æ
2 )
The weak phase hanges sign under CP-onjugation,
while thestrong phasedoesnot. Therateasymmetry
isthen
A(f)
(f) (
f)
(f)+ (
f)
=
2a
1 a
2 sin(
1
2 )sin(Æ
1 Æ
2 )
a 2
1 +a
2
2 +2a
1 a
2 os(
1
2 )os(Æ
1 Æ
2 )
: (31)
The two amplitudes must have dierent weak and
strong phasesin orderforarateasymmetryto be
ob-servable. The CKM theory predits the weak phases,
but no reliable estimates of strong phases exist. We
shallnote somewaystoavoidthisproblem.
VI.3 Deays to CP eigenstates
Theinterferenebetweendiretandmixingtermsin
B deaystoCPeigenstatesmodulatestheexponential
deay(see,e.g.,[57℄):
d (t)
dt e
t
(1Im
0
sinmt); (32)
wheretheuppersignreferstoB 0
deaysandthelower
to B 0
deays. m is the mass splitting, and
0
ex-presses the interferene of deay and mixing
ampli-tudes. For f = J= K
S ,
0
= e
2i
, while for
f = +
,
0 ' e
2i
only to the extent that
pen-guin amplitudes anbe negleted in omparison with
the dominant tree ontribution. The time integralof
themodulationtermis
Z
1
0 dte
t
sinmt= 1 x
1+x 2
1
1
2
; (33)
where x m= . This expression is maximum for
x = 1, and 96% of maximum for the observed value
x'0:76.
The CDF Collaboration [58℄ \tags" neutral B
mesons at the time of their prodution and measures
thedeayrateasymmetryinB 0
(B 0
)!J= K
S . This
asymmetryarisesfromthephase2haraterizingthe
twopowersofV
td
intheB 0
{B 0
mixingamplitude. The
taggingmethods are oftwomain types. In
\opposite-side"methods,sinestronginterationsprodueb and
bin pairs,onelearnstheinitialavorofadeayingB
from the \other"b-ontaininghadron produed in
as-soiationwith it. \Same-side"methods [59℄utilizethe
fat that aB 0
tendsto beassoiatedmorefrequently
witha +
, andaB 0
with a , somewherenearbyin
phasespae.
Eletron-positron ollisions provide B mesons in
pairs at the .m. energy of the (4S) resonane,
just above threshold, in states of negative
harge-onjugation eigenvalue. It then beomes neessary to
distinguishthevertiesofthedeayingandtaggingB's
from one another when studying CP eigenstates. If t
andt 0
denotethe deayand taggingpropertimes,the
asymmetryfordeaytoaCPeigenstatewillbe
propor-tionaltosinm(t t 0
),whihvanisheswhenintegrated
overall times (see, e.g., [24℄ or[62℄). The BaBar and
BELLEresults were obtainedusing asymmetri e +
e
ollisions,withtypialvertexseparationsofabout250
m and 200 m. PEP-II, onstruted in the ring of
theoldPEPmahine,ollides9GeVeletronswith2.7
GeVpositrons,whileKEK-B,onstrutedinthe
TRIS-TAN tunnel, ollides8.5 GeV eletronswith 3.5 GeV
positrons. In symmetri ollisions the (4S) is
pro-duedatrestandtheproperpathlengthofadeaying
B isonlyabout30m.
BothBaBarandBELLEusedtagsbasedonleptons
andkaonsfrom B deays. BaBaralso usedtwoneural
netmethods. Thesamplesreported by thesummerof
2000[52,53℄areshowninTableIII.
The CDF result and ones from OPAL [60℄ and
ALEPH[61℄utilizingB'sproduedinthedeaysofthe
Z 0
are ompared with those from BaBar and BELLE
in TableIV. Theaverage [24℄orrespondsto the 1
raysplotted inFig. 6. Thereisnoontradition(yet!)
with the allowed region, but we look forward eagerly
toreduederrorsfromBaBarandBELLE.Newresults
areduetobepresentedin Februaryof 2001.
TABLEIII.Samples reportedinJuly2000byBaBarandBELLECollaborations
relevanttomeasurementofsin2.
Collab. Finalstate Number No.tagged
BaBar J= K
S
!J= +
121 85(50B
0
,35B 0
)
J= K
S
!J= 0
0
19 12(7B 0
,5B 0
)
0
K
S
!J= +
28 23(13B
0
,10B 0
)
Total 168 120
BELLE CP-oddmodes 92 52(40J= K
S )
J= K
L
102 42
J= 0
10 4
VI.4 \Self-tagging"deays
Atypial\self-tagging"modesuitableforthestudy
of \diret" CP violation is B 0
! K +
. The tree
amplitude [Fig. 7(a)℄ involves the quark subproess
b!suuwithCKMfatorV
ub V
us
(weakphase). The
penguinamplitude[Fig. 7(b)℄
b!swithintermediate
u; ; t quarkshasCKM fatorV tb V ts or V b V s (weak
phase or 0), depending on how the unitarity of the
CKMmatrixisused. Therelativeweakphasebetween
thetreeand penguin amplitudesthus isnon-zero,and
diretCPviolationanariseiftherelativestrongphase
Æ
T Æ
P
alsois non-zero. Theinterpretationof arate
dierene (B 0
!K +
)6= (B 0
!K
+
)requires
independentinformationonÆ
T Æ P .
b
d
W
s
u
u
d
(a)
b
d
g
u, c, t
W
s
u
u
d
(b)
Figure 7. Contributions to B 0
! K +
. (a)
Color-favored \tree"amplitude V
ub
Vus; (b)\penguin"
ampli-tudeV
tb Vts.
TABLEIV.Valuesofsin2impliedbyreent
measure-mentsoftheCP-violatingasymmetryinB 0
!J= K
S .
Experiment Value
OPAL[60℄ 3:2 +1:8
2:0 0:5
CDF[58℄ 0:79 +0:41
0:44
ALEPH[61℄ 0:84 +0:82
1:04 0:16
BaBar[52℄ 0:120:370:09
BELLE[53℄ 0:45
+0:43+0:07
0:44 0:09
Average 0:490:23
IfonemeasuresbothaCP-violatingasymmetryand
arateratiosuhas (B!K )= (B !K )or (B ! K 0 )= (B ! K
), one an eliminate
thestrongphasediereneandsolvefor [63,64,65℄.
Onemust deal with eletroweak penguins (whih also
aeted theinterpretation of 0
=). Oneproposal (see
the rst of Refs. [63℄) to extrat from the rates for
B + !( 0 K + ; + K 0 ; + 0
) andtheharge-onjugate
proesseswasawedby the negletof these
ontribu-tions,whihareimportant[66℄. However,theyanbe
alulated [65℄, so that measurementsof therates for
theseproessesanyieldusefulinformationon.
Aneessaryonditionfortheobservabilityofdiret
CP asymmetries basedonthe interfereneof two
am-plitudes, oneweaker thanthe other, is that onemust
beableto detetproessesatthe leveloftheabsolute
squareoftheweakeramplitude[67℄. Lettheweakphase
benear=2(the mostfavorablease). Thentherate
asymmetryAin Eq.(31)hasmagnitude
jAj=O 2A 1 A 2 A 2 1 +A 2 2 ' 2A 2 A 1 forA 2 A 1 : (34)
Dene a rate based on the square of eah amplitude:
N
i
=onst:jA
i j
2
. ThenjAj'2 p N 2 =N 1 .
ThestatistialerrorinAisbasedonthetotal
num-ber of events. For A
2
A
1
, one has ÆA ' 1= p
N
1 .
Then thesignianeof theasymmetry(innumberof
standarddeviations)is
A ÆA O(2 p N 2 ): (35)
Thus(asidefromthefatorof2)onemustbeabletosee
thesquareoftheweakeramplitudeatasigniantlevel
in order to see asigniant asymmetrydue to A
1 {A
2
interferene.
In searhing for diret CP asymmetries one thus
onsiders B deays with at least two amplitudes
hav-ing an expeted weak phase dierene, with a large
enoughratethat thesmalleramplitudealonewouldbe
detetable,and withagood hane forastrongphase
dierene.
Manybranhing ratiosfor harmlessB deaysare
one to several parts in 10 5
. Ratesassoiatedwith the
subdominantamplitudesareexpetedtobe 2
'1=20
ofthese. Thuswhensensitivitiestobranhingratiosof
afew parts in 10 7
are reahed, searhes fordiret CP
asymmetrieswill takeongreatsigniane.
Two proesses whose rates favor a weak phase
exeeding 90 Æ are B 0 ! + and B 0 ! K +
[65, 68, 69℄, whih favor destrutive and onstrutive
tree-penguin interferene, respetively. A t to these
and other proesses in the seond of Refs. [69℄ nds
=(114 +24
23 )
Æ
,grazingtheallowedregionofFig. 6but
inonsistent withsomemorerestritivets[25℄. Sine
the upper bound on is set by the limit on B
s {B
s
mixing,m
s
>15ps 1
,suhmixingshould bevisible
soon. There isahintofasignalat 17ps 1
[29℄.
TheTevatronandtheLHCwillproduemany
neu-tral B's deaying to +
, K
, and K +
K [70℄.
Eahofthese hannelshaspartiularadvantages.
1. The deays B 0
! K +
K and B
s
!
+
shouldbesuppressedunlessthesenalstatesare\fed"
byresatteringfrom otherhannels[71℄.
2. The deays B 0 ! + and B s !K + K an
yield viatime-dependenemeasurements[72℄.
3. A reent proposal for measuring [73℄ utilizes
thedeaysB 0 !K + , B + !K 0 + ,B s !K + ,
andtheorrespondingharge-onjugateproesses. The
B 0 ! K + and B s ! K +
peaks are well
sep-arated from one another and from B 0 ! + and B s !K +
K kinematially[70℄.
Theproposal of Ref. [73℄ is based on the
strong phaseÆ. ThedeaysB
!K
areexpeted
to be dominated by the penguin amplitude (there is
no tree ontribution exept through resatteringfrom
other nal states), so this hannel is notexpeted to
displayanyCP-violatingasymmetries. Thepredition
(B +
!K 0
+
)= (B !K 0
)thuswillhekthe
assumption that resattering eets an be negleted.
A typial amplitude is given by A(B 0
! K +
) =
[P +Te i(+Æ)
℄, where the signs are assoiated with
phaseonventionsforstates[74℄. Dening
R
A
0
(B
0
!K +
) (B 0
!K
+
)
2 (B +
!K 0
+
)
; (36)
R
s
A
s
(B
s
!K
+
) (B
s !K
+
)
2 (B +
!K 0
+
)
; (37)
andrT=P, ~
V
us =V
ud
,onends
R=1+r 2
+2rosÆos;
R
s =
~
2
+(r= ~
) 2
2rosÆos; (38)
A
0
= A
s
= 2rsinsinÆ: (39)
ThesumofRandR
s
allowsonetodeterminer. Using
R ,r,andA
0
,oneansolveforbothÆand. The
pre-dition A
s
= A
0
heks theavorSU(3) assumption
onwhihthese relationsarebased. An errorof10 Æ
on
seemsfeasiblewithforthomingTevatrondata.
ReentupperlimitsonCP-violatingasymmetriesin
B deaystolight-quarksystems[75℄,denedas
A
CP
(B!
f) (B!f)
(B!
f)+ (B!f)
; (40)
areshowninTableV.Nosigniantasymmetrieshave
beenseen,but sensitivitiesadequatetohekthe
max-imum predited values [76℄ jA K
+
CP
j 1=3 are being
approahed.
TABLE V. CP-violating asymmetries in deays of B
mesonstolightquarks.
Mode Signalevents A
CP
K +
80
+12
11
0:040:16
K +
0
42:1 +10:9
9:9
0:290:23
K
S
+
25:2 +6:4
5:6
+0:180:24
K +
0
100 +13
12
+0:030:12
! +
28:5 +8:2
7:3
0:340:25
VII The role of harm
VII.1 Mixing and CP violation
Thedominantdeaymodesoftheneutralharmed
0
0
strangeness, respetively, and not to CP eigenstates.
Thus D 0
{D 0
mixing induedby shared nal states is
expeted to be small. Short-distane ontributions to
mixingalsoareexpetedtobesmall. Thus,inontrast
totheaseofneutralkaonsandBmesons,oneexpets
small masssplittings, m= 1, and, in ontrast to
neutral kaons, also small width dierenes. The
de-greetowhihanellations amongontributionsof
in-termediate states suh as +
, K +
K , and K
to mixing suppress suh eets further is amatter of
debate[77℄. Ifanyratediereneisexpeted,itwould
be in thediretion favoring a slightly greater rate for
theCP-evenmasseigenstate.
CP violationin theharm setoris expeted tobe
smallintheStandardModel. Itisalsoeasytolookfor,
sineD mesons are easier to produe than B mesons
andtheStandardModelbakgroundislow.
Reent interestingstudies of mixing by the CLEO
[78℄and FOCUS[79℄ Collaborationshintatthe
possi-bilityof non-zero valuesof m, , orboth, but are
notyet statistially ompelling. Noevidene for
mix-ingis found by theFermilab E791Collaboration[80℄.
It may be neessary to invoke large nal-state phase
dierenesinordertoreoniletheCLEOandFOCUS
results [81℄. No CP-violating asymmetries have been
seen in harmed meson deays at the level of several
perent[80,82℄.
VII.2Spetrosopy
A wide variety of exited qq and q states are
aessible at CLEO and FOCUS. The qq states are
providing unique insights into baryon spetrosopy
[83, 84, 85℄, while the q states [86, 87℄, are
impor-tant soures of information about the orresponding
bq states, useful for \same-side"tagging of neutral B
mesons.
VIII The future
VIII.1Envisioned measurements
Future CP studies involvea broadprogram of
ex-perimentswithkaons,harmedandBmesons,and
neu-trinos.
1. Rare kaon deays: Measurement of the
branh-ing ratio for K
L
!
0
at the required sensitivity
(B ' 310 3
) is foreseen at Brookhaven National
Laboratory[88℄andtheFermilabMainInjetor[89℄. A
Fermilab proposal [90℄ seeks to aquire enoughevents
ofK +
! +
tomeasure jV
td
jtoapreisionof10%.
2. Charmedmesons: Whilegreatstrideshavebeen
taken in the measurementof mass and lifetime
dier-enesforCPeigenstatesoftheneutralharmedmesons
D 0
, [78, 79℄, it would be worth while to follow up
studies mayalso havemore to sayaboutmixing,
life-timedierenes,andCPviolationforharmedmesons.
3. B prodution insymmetri e +
e ollisions:
Al-thoughasymmetrie +
e ollidersarenowtakingdata
atanimpressiverate,theCLEOCollaborationis
on-tinuingwithanativeprogram. Itwillbeabletoprobe
harmlessB deaysdown to branhingratios of10 6
.
ItmaybeabletodetettheelusiveB 0
! 0
0
mode,
whoseratewillhelp pindownthepenguin amplitude's
ontribution and permit adetermination of the CKM
phase[56℄. Othernalstatesofgreatinterestatthis
levelinludeVPandVV,whereP;V denotelight
pseu-dosalarand vetormesons. A useful probeof
resat-tering eets[71℄ isthedeayB 0
!K +
K . This
de-ayisexpetedtohaveabranhingratioofonlyafew
partsin10 8
ifresatteringisunimportant,butouldbe
enhaned by morethan an order of magnitude in the
presene of resattering from other hannels. A
hal-lengingbut ruialhannel isB +
! +
, whoserate
will provide information on the ombination f
B jV
b j.
Rare deayssuh asB !X` +
` andB ! Xwill
probetheeetsofnewpartilesinloops.
4.Bprodutioninasymmetrie +
e ollisions:The
BaBar and Belle detetors have made a start at the
measurement of sin2 in B 0
! J= K
S
. The moving
enter-of-mass failitates both avor tagging and
im-provementofsignalwithrespettobakground. These
mahines will makepossiblea hostof time-dependent
studiesin suh deaysasB !, B !K,et., and
theirimpressiveluminositieswilleventuallyadd
signif-iantlytotheworld'stallyofdetetedB's.
5. Hadroni B prodution: The strange B's
an-not be produed at the (4S) whih will dominate
theattentionofe +
e olliders forsomeyearstoome.
Hadroni reations at high energies will produe
o-pious b's inorporated into nonstrange, strange, and
harmed mesons,and baryons. A measurementofthe
strange-B mixing parameterm
s
is likely to bemade
soon. B
s
deaysprovidevaluableinformationonCKM
phasesandCPviolation,asinB
s !K
+
K [72℄. The
width dierene expeted between the CP-even and
CP-odd eigenstates of the B
s
system [54, 55℄ should
bevisiblein thenextroundof experiments.
6. Neutrinostudies: Themagnitudesandphasesin
theCKMmatrixareonnetedwiththequarkmasses
themselves,whosepatternwewillnotunderstanduntil
wehavemappedoutasimilar patternfortheleptons.
Wewilllearnmuhaboutneutrinomassesandmixings
fromforthomingexperimentsattheSudburyNeutrino
Observatory[91℄, Borexino[92℄, K2K[93℄, and F
ermi-lab(BooNEandMINOS)[94℄.
VIII.2 Alikely parameter spae
OurknowledgeoftheCabibbo-Kobayashi-Maskawa
is likely to improve over the next few years [95, 96℄.
With sin(2)measured in B 0
!J= K
S
deays toan
1
errors onjV
ub =V
b
j redued to 10%,strange-B mixing
bounded by x
s
= m
s =
s
> 20 (the present bound
isalreadybetterthanthis!),andB(B +
! +
)
mea-suredto20%(givingf
B jV
ub j,orjV
ub =V
td
jwhen
om-binedwithB 0
{B 0
mixing),one ndstheresultshown
in Fig. 8.
Figure8. Plotin(;)ofantiipatedonstraintsonCKM
parametersintheyear2003. Solidurves:jV
ub =V
b
j;dashed
lines: onstraint on jV
ub =V
td
j by ombining measurement
of B(B +
! +
) withB 0
{B 0
mixing; dottedlines:
on-straint due to K (CP-violating K 0
{K 0
mixing);
dash-dotted line: limit due to xs; solid rays: measurement of
sin2to0:06.
Thenarrowrangeof(;)inreasesthehanethat
anynon-standardphysiswillshowupasa
ontradi-tionamong variousmeasurements,mostlikelyby
pro-viding additional ontributions to B 0
{B 0
mixing [97℄
butpossiblydiretlyaeting deays[98℄.
VIII.3 Baryon number of the Universe
The number of baryons in the Universe is muh
larger than theorresponding number of antibaryons.
Sakharovproposed[99℄threerequirementsforthis
pre-ponderane of matter over antimatter: (1) an epoh
in whih theUniverse wasnotin thermalequilibrium,
(2)aninterationviolatingbaryonnumber,and(3)CP
(andC)violation. Theobservedbaryonasymmetryis
notexplaineddiretlybytheCPviolationintheCKM
matrix; the eets are too small, requiring some new
physis. Twoexamplesarethefollowing:
1. Supersymmetry, in whih eah partile of spin
J hasa\superpartner"ofspin J1=2,aords many
opportunitiesfor introduing newCP-violatingphases
andinterationswhihouldaetpartile-antipartile
mixing [100℄.
2. Neutrino masses at the sub-eV level an signal
largeright-handedneutrino Majorana masses,
exeed-ing 10 11
GeV [101℄. Lepton number (L), violated by
suh masses, an be reproessed into baryon number
(B)byB Lonservinginterationsattheeletroweak
sale [102℄. New CP-violating interations must exist
at thehigh mass sale iflepton number is to be
be important to understand the leptoni analogue of
theCKMmatrix!
VIII.4 Surprises ahead?
The CKM theory of CP violation in neutral kaon
deays has passed a ruial test. The parameter 0
=
is nonzero, and has the expeted order of magnitude.
Tests using B mesons, inluding the observation of a
dierene in rates between B 0
! J= K
S and B
0
!
J= K
S
, are just aroundthe orner. Progressin
\tag-ging" neutralB's and rih information from
measure-mentsofmanyBdeayrateswillroundoutthepiture.
IfB deaysdonotprovideaonsistentsetofCKM
phasesin thenextfewyears,wewillre-examine other
proposed soures of CP violation. Most of these, in
ontrastto theCKMtheory,preditneutronand
ele-trondipole momentsveryloseto theirpresent
exper-imental upper limits. If, however, the CKM piture
remainsself-onsistent,weshouldask abouttheorigin
of theCKM phasesandtheassoiatedquarkand
lep-ton masses. It is probably time to start antiipating
thispossibility,giventheresilieneoftheCKMpiture
sineitwasrstproposednearly30yearsago.
I amlookingforwardto asurprisesuh asone
en-ounteredmanyyearsagowhenexploringasmallave
in Pennsylvania. We had entered it in the afternoon
and thought we had seen all its rooms, when I ame
upon another hamberwith ghostly stalatites
silhou-etted against the darknessbehind them. A breeze of
warmairsignaledthat I wasatually looking outside,
withthe\stalatites"thefaintlyglowingnightsky,and
thedarkspaestheshadowsofpinetrees. Suha
\per-eption shift" does not ome often, but is a welome
soureofwonder.
Aknowledgments
Itisapleasure tothankAdrianoNataleand
Roge-rioRosenfeldfortheinvitationto attendtheBrazilian
MeetingofPartilesandFieldsandfortheirwonderful
hospitalityinSaoPauloandSaoLoureno,andPatrik
Roudeau for onstrutive orrespondene. This work
wassupportedinpartbytheUnitedStatesDepartment
of EnergyunderGrantNo.DEFG0290ER40560.
Referenes
[1℄ J.L. Rosner,inPartile Physisand Cosmology: 2nd
Tropial Workshop, edited by Jose F. Nieves (New
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