COURSE
HEAVY QUARK PHYSICS AND CP VIOLATION
Je rey D Richman
University of California
y
Santa Barbara California U S A
y
E mail richman charm physics ucsb edu
c
Elsevier Science B V Al l rights reserved
Photograph of Lecturer
Contents
Intro duction
Roadmap and Overview of Bottom and Charm Physics
Intro duction to the Cabibb o Kobayashi Maskawa Matrix and a First Lo ok
at CP Violation
Exp erimental Challenges and Approaches in Heavy Quark Physics
Historical Persp ective Bumps in the Road and Lessons in Data Analysis
Avery short history of heavy quark physics
Bumps in the road case studies
Some rules for data analysis
Leptonic Decays
Intro duction to leptonic decays
Measurements of leptonic decays
Lattice calculations of leptonic decay constants
Semileptonic Decays
Intro duction to semileptonic decays
Dynamics of semileptonic decay
Heavy quark e ective theory and semileptonic decays
Inclusive semileptonic decay and jV j
cb
Lepton endp oint region in semileptonic B deca y and jV j
ub
Form factors and kinematic distributions for exclusive semileptonic
decay HQET predictions and the Isgur Wise function
Exclusive semileptonic decay jV j and jV j
cb ub
Hadronic Decays Lifetimes and Rare Decays
Hadronic Decays
Lifetimes
Rare decays
CP Violation and Oscillations
Intro duction to CP violation
CP violation and cosmology
CP violation in decay direct CP violation
CP violation in mixing indirect CP violation
Phenomenology of mixing
CP violation due to interference b etween mixing and decay
Acknowledgements
App endix Remarks on Hadronic Currents Form Factors and Decay Constants
App endix Remarks on CP Conjugate Amplitudes
References
Intro duction
I am delighted to present these lectures on the physics of charm and
b ottom hadrons to this enthusiastic group of graduate students and
p ostdo cs here in Les Houches Whyisheavy quark physics interesting
I hop e to show you that we can now address an extraordinarily broad
range of issues ranging from the mysteries of the Cabibb o Kobayashi
Maskawa CKM quark mixing matrix to the dynamics of the strong
and weak interactions to fascinating rare pro cesses that are sensitive
to physics beyond the standard mo del In the near future several
new and ongoing exp eriments will address the driving question of this
eld what is the origin of CP violation So far CP violation has b een
observed only as a tiny part in a thousand e ect in kaon decays There
is as yet very little empirical evidence to establish that physics within
the framework of the standard mo del namely the phase structure of
the CKM matrix is resp onsible If the CKM matrix is indeed the
source then we will observe large CP violating asymmetries in b oth
B and B meson decays If the exp ected pattern of CP violation is
s
not observed then these investigations will provide a windowinto new
physics beyond the standard mo del
I was more than a little surprised to discov er that nearly all of the
memb ers of this audience are theorists This fact is b oth intriguing and
a little daunting I have the opp ortunity to explain the challenges and
excitement of building exp eriments and p erforming new measurements
At the same time there is a risk that I will bore you with to o many
exp erimental details or fail to motivate them su ciently For some of
you learning how measurements are p erformed may b e something like
learning how your sausages are made you would rather not know I
hop e that by the end of these lectures you will at least conclude that
the sausages we make are kosher
between theory and exp eriment in heavy quark The interaction
physics has been extremely pro ductive and I will devote a signi
cant part of these lectures to a review of theoretical progress from
the perspective of an exp erimentalist I will discuss theory at a fairly
simple level mainly to obtain insights into the key physical ideas and
J D Richman
to help explain the phenomenology that app ears in exp erimental mea
surements For discussions of the more technical theoretical issues I
refer you to the lectures of other sp eakers at this scho ol for example
Buras Manohar Martinelli and Wise
Exp erimentalists need to know ab out theory for several reasons
First in planning an incisive and coherent set of measurements it
helps greatly to understand the theoretical issues It is also imp ortant
to have a sense of how reliable the predictions are and to know the
key assumptions that are really being tested In p erforming a mea
surement one m ust understand all of the kinematic distributions that
describ e the pro cesses under study since both the detection e ciency
and the ability to reject backgrounds dep end on knowledge of such
distributions
Similarly theorists should have some knowledge of the strengths and
weaknesses of di erent exp eriments as well as a sense of which kinds of
measurements are practical and which are not It is also imp ortantfor
theorists to b e aware of assumptions that exp erimentalists make and to
understand the dep endence of the measurements on these assumptions
A measured branching fraction for example may well dep end either
on a mo del for the decay distributions for the signal or on a mo del de
scribing the background comp osition For phenomenologists esp ecially
it is imp ortant to b e able to communicate with exp erimentalists since
predictions that are not clearly stated may simply b e ignored
These lectures are unashamedly p edagogical so I will not aim for
the level of impartiality that is customary in a review talk or article I
hav e made some recent attempts at such reviews My own work
in heavy quark physics has b een mainly on the CLEO exp eriment al
though more recently I have b een involved in the construction of the
BaBar detector Many of my examples are from CLEO analyses b oth
b ecause I am most familiar with them and b ecause CLEO results are
often as good as those from other exp eriments When other measure
ments are b etter than those of CLEO however I will fo cus on them
esp ecially to show the advantages of di erent metho ds
My goal then is to present a balanced and coheren t picture of b oth
exp eriment and theory The p edagogical approachgives me more free
dom to be selective to sp end more time than usual on simple physics
arguments and to present some of myown opinions ab out the strengths
and weaknesses of various measurements In the pro cess I hop e b oth
to giveyou a picture of howheavy avor physics is done and to convey
my enthusiasm for this eld
I am happy to say that these lectures are destined to become out
Heavy Quark Physics and CP Violation
of date in the relatively near future as exp eriments provide us with
a vast amount of new data We can therefore exp ect ma jor advances
in nearly all areas of bottom and charm quark physics and we need
young p eople likeyou to help explore this new territory
Roadmap and Overview of Bottom and Charm Physics
The physics of b ottom and charm hadrons is a broad sub ject and it is
useful to begin with a roadmap of the main topics and physics issues
Here is an outline of the lectures
Intro duction to the CKM matrix
Exp erimental c hallenges of b and c physics
Historical p ersp ective bumps in the road and lessons in data anal
ysis
Leptonic decays of pseudoscalar mesons
Semileptonic decays and measurements of CKM elements
Hadronic decays lifetimes and rare decays
CP violation and oscillations
All of the physical pro cesses that I discuss involve underlying weak
pro cesses mediated by the W b oson but strong and even electromag
netic interactions play a crucial role as well The thread that runs
through these lectures ties all of these sub jects to the quest to under
stand CP violation
Although this list is rather long it is far from complete For exam
ple I will say very little ab out baryons apart from lifetime measure
ments or ab out sp ectroscopy The physics of heavy quark pro duction
including Z bb and other related electroweak phenomena such as the
b forward backward asymmetry is entirely absent these topics could
easily b e the sub ject of another series of lectures
I b egin in Section with a short intro duction to the CKM matrix
and a review of its present status Because m m the b quark must
b t
decay into quarks outside its own generation As a consequence even
b c decay mo des are suppressed by jV j the dominant
cb
resulting in the long lifetimes of b hadrons of order ps In some
exp eriments these long lifetimes have made distinguishing b hadrons
from backgrounds much easier since separated decay vertices are ev
ident if the b hadron is moving rapidly B decays provide various
ern the strengths of b c ways to measure jV j and jV j which gov
cb ub
and b u transitions Furthermore the app earance of the t quark
in virtual intermediate states of B and B mixing and p enguin decay s
J D Richman
pro cesses allows one to extract jV j and jV j We will also see that
td ts
phases of the CKM elements not just their magnitudes are exp eri
mentally accessible and are asso ciated with CP violating asymmetries
For reasons that I will discuss the determination of CKM elements is
less imp ortant in charm physics Although jV j and jV j can be de
cs cd
termined from charm semileptonic decays the main fo cus has been to
infer these quantities from unitarity of the CKM matrix and then to
use measured branching fractions to test theoretical predictions for the
absolute scale of the decay rates
Section describ es the main challenges confronting exp eriments that
study b ottom and charm quark physics I compare the strengths and
weaknesses of di erent metho ds and indicate the kinds of measurements
for which each is b est suited Currently b physics exp eriments use three
di erent approaches all of which involve colliding b eams e e
S B B e e Z bb andpp bbX The S a bb bound
state just ab ove the threshold for pro ducing B B pairs has provided
the largest data samples Over ve million S B B events have
b een obtained by the CLEO exp eriment at the Cornell Electron Storage
Ring CESR this is the largest B sample in the world The CLEO
detector has b een upgraded a numb er of times CLEO I p erformed the
original observation of B mesons CLEO I I dramatically advanced the
eld with its CsI calorimeter whichprovides sup erb photon detection
in CLEO I I a silicon vertex detector was added and CLEO I I I is now
under construction It will include a new particle ID system based on a
ring imaging Cerenkov detector Two new detectors BaBar at SLAC
and Belle at KEK are now under construction they will op erate at
asymmetric energy e e machines that will give the S a boost
relativ e to the detector frame allowing for vertexing of the daughter B
mesons The ARGUS exp eriment at DESY is no longer in op eration
but it also contributed greatly to this eld
The pro cess e e Z bb has also been used with great e ec
tiveness Because the energy in this pro cess is so high a large variety
of b hadrons can be pro duced B B b baryons as well as radially
s
and orbitally excited B mesons The b hadrons are pro duced at high
far enough that high momentum p GeV c and they move
precision vertex detectors help greatly in reconstructing the decay se
quences The LEP exp eriments ALEPH DELPHI OPAL and L as
well as the SLD detector at the Stanford Linear Collider SLC have
p erformed manyinvestigations of b hadrons including a comprehensive
set of lifetime and mixing measurements More recently successful b
hadron lifetime and mixing measurements have b een p erformed by the
Heavy Quark Physics and CP Violation
CDF exp eriment at the Fermilab Tevatron pp collider Here again
high precision vertexing has b een essential
In contrast high statistics data on charm particles have been ob
tained not only by collider exp eriments such as CLEO but also by
xed target sp ectrometers mainly at Fermilab In these exp eriments
vertex detectors play a crucial role in separating the charm signals
from the light quark background These xed target exp eriments be
ginning with Fermilab E provided the rst high precision charm
lifetime measurements PresentFermilab exp eriments suchasFOCUS
E have accumulated data samples that should yield over one mil
lion reconstructed charm decays
Section surveys some of the main achievements of this eld Many
of the most interesting discoveries are quite recent and there is ev
ery exp ectation that rapid progress will continue As in any scienti c
endeavor however there have been o ccasional missteps In the spirit
of the teaching goals of this scho ol I present some examples of mea
surements that have not sto o d the test of time It is not always clear
precisely what lessons should be drawn from such mistakes but these
examples can nevertheless be instructive I list some guidelines to an
alyzing data that might help others to avoid these problems
In the next sections I discuss the decay mo des of charm and b ottom
mesons progressing from the simplest to the most complicated pro
cesses Figure shows some of the most imp ortant diagrams which
suggest the remarkable variety of phenomena we can observe
Given that the initial state is a hadronic system the simplest
of mo des from a theoretical p ersp ective are the leptonic decays
charged mesons discussed in Section In these decays Fig a the
initial quark and antiquark within the meson annihilate into a virtual
W which then pro duces a two b o dy nal state consisting of a lepton
and its antineutrino Studies of leptonic decays provide information on
meson decay constants which measure the amplitude for the quark and
antiquark to overlap Sophisticated theoretical metho ds such as lat
tice QCD are b eing develop ed to describ e the strong interaction forces
within mesons and to calculate decay constants We will see that this
information is valuable in a wide variety of situations in particular
it is needed in some cases to extract values of CKM matrix elements
In general I will not describ e the details of data analyses To give
some idea of the di culties one encounters however I have chosen
somewhat arbitrarily two measurements of D to describ e
s
more fully The approaches used in these two measurements are quite
di erent
J D Richman
Fig Diagrams for B and D decays and oscillations a Leptonic decay b Semilep
tonic decay c hadronic decay external sp ectator diagram d hadronic decay internal
sp ectator diagram e gluonic p enguin f radiative p enguin with real photon g ra
B oscillations mixing These gures do not show diative p enguin with dilepton h B
all of the p ossible decay diagrams but only the most imp ortant ones
Semileptonic decays Section are shown in Fig b These mo des
are much easier to study exp erimentally than leptonic decays mainly
b ecause the branching fractions are larger Theoretical calculations
while di cult are reliable enough that semileptonic decays provide
good measurements of jV j and jV j As for leptonic decays the am
cb ub
plitude can be factorized into leptonic and hadronic pieces However
the nal state in semileptonic decay involves at least three particles
so that q the mass squared of the virtual W is a variable The
hadronic current can b e parametrized with a small numb er of functions
of q called form factors these functions describ e the transition of the
hadronic system from the initial to the nal state The form factors
dep end on q because this quantity determines the value of the recoil
Heavy Quark Physics and CP Violation
kick given to the daughter quark of the decaying heavy quark In some
cases the form factors can b e predicted with go o d precision as in the
case B X where X denotes a charm meson Predictions based
c c
on heavy quark e ective theory HQET and heavy quark expansions
haveprovided valuable insights into the dynamics of these decays and
have led to more reliable measurements of jV j The B X de
cb u
cays where X represents a non charm nal state are used to extract
u
are much jV j The rate and kinematic distributions of these mo des
ub
more di cult to predict and they are currently a sub ject of much
exp erimental and theoretical research
Hadronic decays Section are the most complicated mo des of all
The e ects of strong interactions cannot b e isolated to a single current
and nal state interactions between the hadronic systems can be sig
ni cant Nevertheless there are sp ecial and intriguing phenomena in
hadronic decays that at least in the standard mo del cannot o ccur in
leptonic or semileptonic decays Because there can be more than one
decay diagram leading to a given nal state hadronic decays can dis
play a variety of remarkable interference e ects In some cases these
e ects give insight into the dynamics of the decay pro cess while in
others they provide a powerful prob e of CP violating phases
Figures c and d show the so called external and internal sp ecta
tor diagrams for hadronic decay To a certain extent the initial state
light quark in these decays can be regarded as a sp ectator since it
do es not play a direct role in the underlying weak pro cess Generally
sp eaking the decays that can pro ceed through the external sp ectator
diagram have rates that are larger than those which must go through
the internal sp ectator diagram In the latter case there can b e a color
mismatchbetween the quarks pro duced bytheW and the other quarks
in the system suppressing the rate Interference between the external
and internal sp ectator diagrams has extremely imp ortant implications
in D decay The interference in this case is destructive and it sub
stantially lengthens the D lifetime relativetothatofthe D We will
see that the interference has the opp osite sign in B decays but its
e ects on the B lifetime are much less signi cant Lifetime measure
ments Section have reached an impressive level of precision and
in some cases are raising some interesting theoretical questions
In Section I discuss rare B decays which have branching frac
tions of ab out or less Such decays are due to a variety of pro
cesses The sp ectator diagrams in Figs c and d can lead to rare
decays through b u transitions which are suppressed by roughly
jV j jV j relative to the dominant b c mo des Figure
ub cb
J D Richman
e shows a gluonic p enguin diagram This one lo op pro cess involves
the emission and reabsorption of the same W and it leads to an e ec
tive avor changing neutral current giving b s or b d transitions
In some cases the p enguin pro cess can interfere with a b u sp ecta
tor decay Since the CKM phases are di erent for the two diagrams
the decay rate to a particular nal state can be di erent from that
for the CP conjugate mo de in which the weak phases change sign
example one might discover di erent rates for B K and For
B K CP violation resulting from this typ e of interference is
called direct CP violation and would be extremely interesting to ob
serve Because it also dep ends on poorly understo o d strong phases
however it will b e di cult to use this e ect to extract reliable values
of the CKM phases Currently we have very little exp erimental data
on gluonic p enguins but there should be much more in the relatively
near future
Penguin diagrams can also o ccur with the emission of a real or vir
tual photon Figs f and g The former pro cess has b een measured
by CLEO for both exclusive and inclusive mo des and more recently
there have also been observations from the LEP exp eriments Radia
tive p enguin decays are sensitivetophysics b eyond the standard mo del
such as sup ersymmetry since new heavy particles can app ear in the
virtual lo ops
In Section I discuss the phenomenon of B B oscillations some
times called mixing Figure h shows two box diagrams which lead
to the sp ontaneous oscillation of a B into a B and vice versa Sim
B oscillations The oscillation ilar diagrams are resp onsible for B
s s
probability is prop ortional to exp t sin M t where
M ps is the mass di erence b etween the two neutral B mass
d
eigenstates and is their essentially equal lifetime The oscillation
frequency is su ciently rapid that an appreciable fraction
b efore they decay and the oscillation of neutral B mesons oscillate
probability p eaks at ab out four lifetimes The frequency is su ciently
slow that the oscillation of rapidly moving B mesons can be directly
followed as they propagate through space There is also great interest
in observing B oscillations but their exp ected rate is ab out twenty
s
times faster The probability to decay in the mixed state is essen
tially maximal but the structure of B oscillations is to o ne
s
for present exp eriments to resolve From these measurements one can
extract the magnitudes of V and V which app ear b ecause the dom
td ts
t contributions are due to virtual intermediate states containing inan
t and t quarks An exp erimental determination of the ratio of B to d
Heavy Quark Physics and CP Violation
fCP
Fig CP violating asymmetries result from interference e ects involving phases that
change sign under the CP op erator The weak phase of the CKM matrix has this prop erty
One way to observe CP violation is to use the interference between the direct decay
B f and the pro cess B B f The standard mo del predicts substantial
CP CP
asymmetries b etween this pro cess and that in which the initial meson is a B
B oscillation frequencies would help to reduce some of the theoretical
s
uncertainties that arise from hadronic e ects
The sub ject of CP violation in B decays will op en up dramatically
in the next few years Iintro duce the C and P op erators in Sec and
brie y discuss the relevance of CP violation to cosmology in Sec
All CP violating e ects involveinterfering amplitudes and the various
ways in which this can o ccur are examined in Sec CP violation
in decay Sec CP violation in mixing and Sec CP viola
tion in the interference between mixing and decay I have found the
sub jects of oscillations and CP violation to be quite di cult and for
this reason I include some discussion of the neutral kaon system as well
for p ersp ective Section discusses the phenomenology of mixing in
these two systems The ma jor new e orts to study CP violation in the
B system are based on the general idea illustrated in Figure which
shows that mixing can provide a second amplitude to interfere with a
decay to a particular nal state In certain cases this metho d can pro
vide information on the CKM phases with little or no uncertainty from
strong interaction e ects Exp eriments using this approach are under
construction in the United States Japan and Europ e these ma jor ef
forts will dramatically improve our knowledge of CP violation and the
CKM matrix
Bottom and charm physics involves the study of a large variety of
pro cesses which can be used to measure fundamental parameters of
the standard mo del to search for e ects of physics beyond the stan
and to prob e the dynamics of QCD and the electroweak dard mo del
interactions Although there are a great many individual physics is
sues to fo cus on it is also imp ortant to take a di erent view and to ask
J D Richman
whether we understand b ottom and charm physics in a broader sense
Do the measurements t together Are there hiding in the dense forest
of B decays unexp ected pro cesses that we are missing
Intro duction to the Cabibb o Kobayashi Maskawa Matrix
and a First Lo ok at CP Violation
In this section we examine the basic phenomenology of the CKM
matrix We will see that the CKM matrix is not simply an ad hoc
parametrization of generation changing quark transitions but is in
stead an integral part of the standard mo del that is closely connected
with the generation of quark masses We brie y review pro cesses that
have already constrained the CKM matrix and summarize the cur
rent status of these constraints Many of the quantities discussed later
in the pap er are de ned and explained in this section including the
Wolfenstein parameters A and and the angles and of
the unitarity triangle Finally I give a simple example illustrating how
CP violation can arise from the CKM matrix
The standard mo del successfully accounts for avor changing quark
transitions in terms of a V A charged weak current op erator J that
couples to the W b oson according to the interaction Lagrangian
g
y
p
L J W J W
int
where for quark transitions
X X
V d J V J u
ij j ij i
ij
i j i j
The V are the elements of the CKM matrix and the indices i and j
ij
run over the three quark generations The eld op erators u i
i
annihilate u c and t or create their antiparticles and the d annihi
j
late d s and b The op erator W annihilates a W or creates a W
the reverse is true for W Thus the amplitudes for the pro cesses in
u and u W d are prop ortional whichaW is radiated d W
i i j j
to V whereas the amplitudes for pro cess in which a W is radiated
ij
u W d and d W u are prop ortional to V
i j j i
ij
The CKM matrix V can be regarded as a rotation from the quark
mass eigenstates d s and b to a set of new states d s and b with
Heavy Quark Physics and CP Violation
diagonal couplings to u c and t The standard notation is
d V V V d
ud us ub
C B C B C B
s V V V s
A A A
cd cs cb
b V V V b
td ts tb
To a rst approximation the CKM matrix is simply the unit matrix
so that the dominant transitions are u d c s and t b In
reality none of the o diagonal elements is exactly zero leading to
generation changing transitions b etween quarks and as we will see to
the p ossibilityofa CP violating phase
The values of CKM matrix elements like fermion masses are fun
damental input parameters of the standard mo del and cannot be pre
dicted In a comprehensive theory of quark avor b eyond the stan
dard mo del these parameters would be explained in terms of other
physics or at the very least related to a smaller set of constants Nev
ertheless the standard mo del provides a key insight the values of b oth
fermion masses and CKM elements originate in the unknown couplings
of the fermions to the Higgs eld The detailed discussion of the
relationship b et ween the fermion mass matrices and the CKM matrix
is beyond the scop e of this pap er I will however brie y outline the
argument as a reminder of how closely related the masses and CKM
elements are
The form of the Yukawa terms in the Lagrangian that couple the
Higgs eld to the quarks is constrained by SU gauge invariance
L
but this condition do es not require the terms to be diagonal in quark
avor
X
j j
k
L Y u d u
Yukawa jk
L L
R
j k
j j
k
Y u d d h c
jk L L R
where i and j run over the quark generations L and R denote left and
right handed comp onents of the quark elds and Y and Y are the
jk
jk
Yukawa couplings The complex Higgs doublet undergo es sp ontaneous
symmetry breaking
p
v H x
where v is the Higgs vacuum exp ectation value and H x is the eld
corresp onding to the Higgs particle After sp ontaneous symmetry
J D Richman
breaking the Lagrangian b ecomes
X
j j
k k
p
v H x L Y u u Y d d h c
Yukawa jk
L R jk L R
j k
The terms prop ortional to v couple the left and right handed comp o
nents of the quark elds and generate the mass terms To determine
the quark mass eigenstates however one must diagonalize the mass
matrices
v v
p p
m Y m Y
jk jk
jk jk
The CKM matrix is a pro duct of unitary matrices that accomplish this
task and it is therefore unitary by construction The nine fermion
masses and four CKM elements are therefore intimately related and
together account for of the standard mo del parameters The
remaining parameters are the three SU SU U gauge
C L Y
coupling constants the Higgs mass and the Higgs vacuum exp ectation
value
The magnitudes of CKM elements are determined largely but not
exclusively from semileptonic pro cesses A high precision value of jV j
ud
is obtained by comparing the rates for sup erallowed nuclear
decays to the muon decay rate Semileptonic K and K decays
are used to extract jV j The values for jV j and jV j can be deter
us cs cd
mined using a variety of techniques The rate of charm pro duction in
neutrino interactions with valence quarks in nucleons has b een used to
determine jV j A similar determination of jV j is complicated by lack
cd cs
of knowledge regarding the p opulation of ss pairs in the nucleon sea
Alternatively jV j can be determined from the decay D Ke
cs e
The magnitudes of the CKM elements for the sector governing
transitions within the rst two generations are
jV j jV j
ud us
jV j jV j
cd cs
These values are determined without any assumptions regarding uni
tarity apoint that we will revisit b elow
element is determined from a measured decay rate When a CKM
or branching fraction an absolute scale must be set by theory If the
initial meson with quarks Qq is denoted M the branching fraction
for a semileptonic decay to a hadronic nal state X can be written
q q
B M X jV j
Qq q q q Q M
Heavy Quark Physics and CP Violation
where is the mean lifetime of M Exp erimentalists measure the
M
branching fraction and lifetime thus determining the decay rate it
falls to theorists to determine A partial exception to this argument
is the determination of jV j where the overall scale is determined from
ud
muon decay Still imp ortant theoretical corrections are required even
in this case The value for jV j given ab ove relies on a theoretical
cs
calculation of a D semileptonic transition form factor to set the scale
adopt a di erent point of For b oth V and V however one can
cd cs
view which is quite imp ortant in practice In this view we assume
that the CKM is unitary as required in the standard mo del Unitarity
of the third row and third column implies
q q q
jV j jV j jV j jV j jV j
tb td ts ub cb
Thus jV j and jV j must each be less than ab out which implies
td ts
that
jV j jV j jV j
ud cd td
jV j jV j jV j
us cs ts
In other words if we assume unitarity the known very small values
of jV j and jV j imply that jV j and jV j are essentially determined
cb ub cd cs
from jV j jV j and Eq
ud us
For charm decays the real aim has therefore not b een to determine
jV j and jV j but to use these inferred values so that the absolute scale
cd cs
of the form factors can be obtained from exp eriment This pro cedure
allows one to check complicated QCD calculations that give the form
factor normalizations This approach cannot be applied to b hadron
decays where we do not have a comparable check of the theoretical
predictions for the factor in Eq In B decays one can measure
the shapes q dep endences and relative sizes of the form factors but
the overall scale must be determined from theory
We will see in Sec that semileptonic decays of B mesons are used
to measure jV j and jV j
cb ub
jV j
ub
jV j
cb
B where the value for jV j is based on the world average value from
cb
D Table and that for jV j is obtained from the CLEO II
ub
measurements of B and B Eq Thus
jV V j
ub cb
J D Richman
Information on jV j and jV j can b e extracted from B B and B B
td ts s s
mixing and from B X pro cesses Using the B B mixing rate
s
M ps and other input parameters the PDG
B
d
quotes
jV V j
td
tb
whichshows that V is of the same order as V If we use the unitarity
td ub
and third columns of the CKM matrix condition b etween the second
V V V V V V
us cs ts
ub cb tb
and ignore the rst term compared to the other two we nd that jV j
ts
jV V V j jV j which is b orne out bytheB X measurements
cs cb s
cb tb
The current exp erimental constraint on V is very weak Measure
tb
ments of semileptonic t decays from CDF and D indicate that
jV j
tb
jV j jV j jV j
td ts tb
consistent with the exp ectation from unitarityofthe third column
Before leaving the sub ject of the magnitudes of CKM elements I
would like to point out a simple consequence of Eq for the un
certainty on a magnitude Exp erimenters generally measure branching
fractions so using this equation involves taking a square ro ot From the
simple propagation of errors this means that the relative uncertainty
on the CKM element is half of that on the branching fraction If you
are impressed by a exp erimental uncertainty on jV j for example
cb
rememb er that the branching fraction had a uncertainty
Although wehave considered the magnitudes of nine CKM elements
there are only four indep endent real parameters needed to sp ecify them
in the standard mo del In general an n n unitary matrix has n
indep endent real parameters There are n real parameters but there
are n constraints from the normalization of each column and n n
constraints real plus imaginary from the orthogonality between each
pair of columns The conditions on the rows do not add additional
U exp iH constraints One can also see this result by writing
where H is a hermitian matrix In the CKM matrix not all of these
parameters are physically meaningful however b ecause for n quark
generations n phases can be absorb ed by the freedom to select
the phases of the quark elds The number of physical parameters is
therefore n n n whichgives four parameters for three generations
Heavy Quark Physics and CP Violation
Let s examine how this quark rephasing can work We can
attach a phase factor to each of the quark op erators so that the current
op erator b ecomes
i
d
de V V V
ud us ub
C B C B
i i i i
s u c t
se V V V J ue ce te
A A
cd cs cb
i
b
be V V V
td ts tb
The u c and t phases each allow us to multiply a row of the CKM
matrix by a phase while the d s andb phases eachallowustomultiply
a column by a phase We can therefore select the u c and t quark
phases so as to make one element in each of the three rows real say
V V and V Having made the second column real we can select
us cs ts
the d s and b phases so as to make one elementineach of the rst and
third columns real say V and V Thus with six quarks we are able
ud cb
to remove ve CKM phases with n quark generations we can remove
n phases
What degrees of freedom remain in the CKM matrix We know
n unitary matrix contains n real parameters that an arbitrary n
If the CKM matrix were merely orthogonal and real there would be
n n n n n n indep endent real rotation parameters
We can summarize the results so far as
A general unitary matrix has n indep endent real parameters
Of these n n are real rotation angles
Because we can rephase the quark elds the numb er of indep endent
phase factors is therefore n n n n n n
The total number of physical parameters is n
The results for and generations are listed in Table A key
conclusion is that there must b e at least three generations in order for
the CKM matrix to contain a complex phase factor As we will see
this implies that at least three generations are required for the CKM
matrix to pro duce CP violating asymmetries
A standard parametrization for the CKM matrix used bytheParticle
Data Group is the set of angles and sp ecifying the
rotation
i
c c s c s e
C B
i i
s c c c s s s e s c c s s e V
A
i i
s s c c s e c s s c s e c c
etc The phase pro duces CP where c cos s sin
J D Richman
Table
Degrees of freedom in the CKM matrix as a function of the number n of quark
generations
n generations Total indep params Real rot angles Complex phase factors
n n n n n
violation and would not app ear if there were only two generations
This formidable lo oking matrix takes on a simpler form if we use the
fact that jV j is very small so that c is extremely close to
ub
unity We can then neglect terms prop ortional to s relative to terms
of order unity which gives
i
c s s e
B C
V s c c c s
A
i
s s c c s e c s c c
In this approximation only V and V carry phases
ub td
Empirically there is a hierarchy in the magnitudes of CKM elements
whichwehave already b egun to exploit by using the smallness of jV j
ub
This hierarchy motivates a particular expansion of the CKM matrix
rst given by Wolfenstein in the small parameter sin
C
where is the Cabibb o angle
C
V V V
ub ud us
C B
V V V V
A
cd cs cb
V V V
td ts tb
A i
B C
O
A
A
A i A
The known values of CKM elements can b e used to motivate this form
in which only four indep endent parameters remain A and
Note that to O the upp er left p ortion of the CKM
matrix the matrix asso ciated with Cabibb o rotations of the d and
s quarks is constructed to b e nearly unitary
O
q
in accord with measurements whichgive jV j jV j Of
ud us
course exact unitarity of this sector was p ostulated by Glashow Il
Heavy Quark Physics and CP Violation
iop oulos and Maiani to suppress strangeness changing neutral currents
in kaon decays Even though b c transitions involve only one genera
tion change the magnitude of V is measured to b e quite small ab out
cb
As a consequence the B meson lifetime is quite long Naively
one mighthave guessed from the magnitude of V that the amplitude
us
for any one generation transition would be of order but this is not
correct Thus it is natural to write V A where A is a constant
cb
j of order unity We also know that jV j or jV V
cb ub ub
This suggests that we write V A i where we cho ose to
ub
incorp orate the phase in this element in accord with Eq From
the unitarity of the third column the magnitude of V is equal to unity
tb
up to corrections of O The orthogonalityof the second and third
columns then gives V V A Finally orthogonalitybetween
ts cb
the third and rst columns sp eci es
A i A V O
td
or V A i This parametrization is very convenient for
td
understanding how various measurements constrain the CKM matrix
and is completely adequate for our purp oses For discussions in which
higher accuracy is required the expansion can be carried out further
as discussed by Buras and Fleischer
By applying the orthogonality condition to the rst and third
columns we can obtain a useful relation between the two smallest
elements of the CKM matrix V and V
ub td
V V V V V V
ud cd td
ub cb tb
in which each term in the sum is of order This relation which
has been emphasized by Bjorken and Chau and Keung can be
represented by a triangle in the complex plane Fig In the
parametrization given ab ove V V andV are all real and by using
cd cb tb
V V and V we obtain
ud tb cd
V V
td
ub
jV V j jV V j
cd cb cd cb
In terms of the Wolfenstein parameters given in Eq the co ordi
nates of the vertices of the triangle are and
From Eq it is also easy to see that measurements of jV j and
cb
jV j provide the constraints
ub
q
jV V j A and jV V V j
cb us cd cb
ub
Thus measurements of jV j essentially determine A The constraint
cb
from jV j de nes a circle in the plane when errors are taken ub
J D Richman
Fig The CKM unitarity triangle a shows the orthogonality condition b etween the
rst and third columns of the CKM matrix the orientation of the triangle is arbitrary
and dep ends on phase convention b shows a rescaled version of the triangle in which
each side has b een divided by the jV V j so that the base has unit length The base
cd
cb
has also b een aligned with the real axis bychoice of phase convention c shows a com
monly used lab eling resulting from the approximation V V In the Wolfenstein
ud tb
parametrization the upp er vertex of the triangle has co ordinates All CP violating
amplitudes in the standard mo del are prop ortional to the area of the triangle
into account this constraint b ecomes an annulus The equations that
describ e constraints from B B and B B mixing are given in Sec
s s
y virtual tt inter Eq Since the mixing rates are dominated b
mediate states B B measurements constrain
jV j A
td
A particular value of jV j corresp onds to a circle centered at the p oint
td
Figure shows the separate constraints on the lo cation
Heavy Quark Physics and CP Violation
of the p oint fromjV j from B X jV j from B B mix
ub u td
ing and j j from CP violation in the neutral K system This gure
K
is meant to illustrate the wayinwhich the individual constraints enter
a much more carefully constructed plot of the combined constraint
taking into account correlations is discussed below
What information do we obtain from B B mixing From Eq
s s
we see that the mixing rate determines jV j which is essentially the
ts
jV j The real value of the B B mixing measurement how same as
cb s s
ever is that it allows us to reduce the uncertainties that a ect the
determination of jV j The ratio of B B to B B mixing rates deter
td s s
mines
f B
M jV j M
B
d td B
B
d
d
d
M jV j M f B
s ts B B
s s
B
s
The ratios of the decay and bag constants in this equation are much
easier to predict than their individual values so measurements or even
limits on B B mixing are extremely valuable
s s
We digress brie y to discuss the constraint on the unitarity triangle
from To agoodapproximation
K
i
e
p
Im M
K
M
K
where a small correction term prop ortional to Re M has b een
ignored and
G M
F W
f B M V V S x V V S x M
K K cs c ts t
K cd td
V V V V S x x
cs ts c t
cd td
where x m M i c t f is the K decayconstant and B is a
i K K
i W
phenomenological bag constant parametrizing non p erturbative QCD
e ects The factors and
are relatively well known QCD corrections that dep end on
the heavy quark masses The functions S x and S x x are given
c t
by
x x x x ln x
S x
x x
x x ln x x
t t t
t
ln S x x x
c t c
x x x
c t t
where in the last expression only the linear terms in x have been
c
kept The largest uncertainty in using these relations has been in the
J D Richman
value of B However the precision and reliability of lattice QCD
K
calculations have improved signi cantly over the past few years and
the value B is commonly used
K
To derive the constraint in the plane we need to express the
CKM elements in Eq in terms of and and then take the
imaginary part The term from the tt intermediate state gives
Im V V Imf A A i g
ts
td
A
This result shows that for well away from the extraction of j j
K
dep ends on
jV j
cb
A
Thus the relative uncertainty in jV j is magni ed by as much as a
cb
factor of four when it propagates into the uncertaintyonj j The term
K
from the ct intermediate state gives Im V V V V A and is
cs ts
cd td
therefore less sensitive to jV j The evaluation of Im V V requires
cb cs
cd
use of the O correction to the imaginary part of V in other words
cd
one has to carry out the expansion of the Wolfenstein parametrization
to higher order This expansion is discussed in Buras and Fleischer
V iA this implies Im V V A with the result
cd cs
cd
With these results one can show that the measurementofj j together
K
with the value of jV j yields the hyp erb olic constraint curves shown
cb
in Fig
Avery careful analysis of the CKM constraints has b een p erformed
by Grossman Nir Plaszczynski and Schune This analysis takes
into account the non gaussian nature of the mo del dep endent errors
q
on jV j f B B and f B f B The combined constraint
ub B B K B B
s
B B
d d d
s
d
is shown in Fig a without the measured limit on B mixing and in
s
Fig b with the limit incorp orated
One can show that in order for CP violation to b e p ermitted in the
standard mo del the area of the unitarity triangle must b e nonzero In
particular V and V must b e nonzero and complex relativeto V V
ub td cd cb
More quantitatively theinterference terms that pro duce CP violation
are prop ortional to the quantity
k l J jIm V V V V j i
CP ij kl
il kj
where no sums are implied What is this equation telling us The term
is one term in the sum of terms that gives the inner pro duct z V V
ij il
Heavy Quark Physics and CP Violation
x d εΚ , Vcb η
Vub
ρ
Fig Individual constraints on the plane arising from measurements of jV j B B
ub
mixing jV j CP violation in kaon decays and jV j It is imp ortant to recognize that
td cb
the hyp erb olic curves derived from CP violation in kaon decays also dep end on jV j The
cb
bands shown corresp ond to The prop er calculation of the overlap region however
must takeinto account correlations This is done in Fig which also uses more recent
values of all input quantities From Ref
between column j and column l Similarly z V V is the complex
kl
kj
conjugate of another one of these terms Using polar co ordinates it
is easy to show that the quantity Im z z is prop ortional to the sine
of the angle between z and z in fact the result is twice the area
of the triangle of which z and z are two sides This quantity gives
a measure of CP violating interference terms that is indep endent of
the overall orientation of unitarity triangle constructed from the inner
pro duct b etween columns j and l In the Wolfenstein parametrization
J A
CP
so is required for CP violation
It is worth rememb ering that there are in fact six CKM unitarity
triangles three from inner pro ducts between the columns and three
from inner pro ducts between the rows It can be shown that all of
these triangles have the same area The three triangles corresp onding
to inner pro ducts between the columns represent the relations
V V V V V V
ud cd td
us cs ts
V V V V V V
ud cd td
ub cb tb
V V V V V V
ts cs us
tb cb ub
J D Richman
Fig Current constraints on the plane a without B B mixing information
s s
B mixing information included This analysis incorp orates non gaussian and b with B
s s
errors on various mo del dep endent quantities The allowed region shaded corresp onds
to C L Figure provided byM H Schune
We can count powers of to get a rough feeling for the app earance of
these triangles
O O O
O O O
O O O
The triangles are clearly very di erent the column column
triangle has two long sides of order and one very short side of or
der while the column column triangle is sp ecial in that
all three sides have lengths of order It is crucial however to re
member that all of these triangles have the same area and it is not
to Factors hard to verify Eq What do es this area corresp ond
like V V V V are asso ciated with interference terms between
ub tb
us ts
rst order weak amplitudes For example for B K there could
A from a b uW be a tree amplitude A V V
ub us
Heavy Quark Physics and CP Violation
W u s decay and and a penguin amplitude A V V A
tb
ts
from a b t s transition with the emission and reabsorption of a
W See Fig Equation is telling us that the pro duct of of
CKM factors for all such interference term are the same
In a measurement however it is not the size of the interference term
by itself that matters but the size of the interference term relative to
A the the rate for the pro cess For twointerfering amplitudes A and
asymmetry b etween aprocess and its CP conjugate is of the form
jA A j jA A j
A
CP
jA A j jA A j
where A and A are the amplitudes for the CP conjugate pro cess The
numerator gives the interference term while the denominator contains
the squares of the magnitudes of the individual amplitudes as well In
some sense the strategy for CP studies is to nd situations in which the
denominator is not much bigger than the numerator A This
is precisely what can happ en with interference terms based on column
column all sides of the triangle are O so the denominator
itself is prop ortional to We can therefore have asymmetries of order
unity In contrast the squashed shap e of the triangle relev ant to the
neutral kaon system means that the denominator will be large
compared to the numerator in that case
Further discussion of the connection between complex elements of
the CKM matrix and CP violation can be found at the end of this
section in Sec and in Sec
We will see that the angles and can be extracted from CP
violating asymmetries in B and B decays Using the result z z
s
z j exp i where z jz j exp i andz jz j exp i jz j j
we can relate the terms in Eq to the angles of the unitarity triangle
shown in Fig For example from the upp er left and upp er rightsides
of the triangle we have
V V jV jjV j
ud ud
ub ub
i
e
V V jV jjV j
td td
tb tb
or
V V jV jjV j V V
td td td
tb tb tb
i
e arg
V V jV jjV j V V
ud ud ud
ub ub ub
The upp er right and lower sides give
V V V V jV jjV j
cd cd cd
cb cb cb
i
e arg
V V jV jjV j V V
td td td
tb tb tb
J D Richman
Finally the upp er left and lower sides give
V V V V jV jjV j
ud ud ud
ub ub ub
i
e arg
V V jV jjV j V V
cd cd cd
cb cb cb
In our discussion of CP violation we will encounter the quantities
V V V V
td ub
tb ud
Im sin
V V V V
tb ud
td ub
V V V V
tb cd
td cb
sin Im
V V V V
cb td
cd tb
V V V V
ud cb
ub cd
Im sin
V V V V
ub cd
ud cb
In the conventional parametrization of the CKM matrix V V
ud cd
V and V are real and V so that
cb tb cb
jV j V
td td
i i i
e V jV je V jV je
td td ub
ub
V jV j
ub
ub
and
V V
ub td
sin Im
V V
ub td
V
td
sin Im
V
td
V
ub
sin Im
V
ub
We can also connect and with the Wolfenstein parameters
and From Fig we have
tan tan
tan
Iwant to brie y discuss an example of CP violation since the central
idea is quite simple and there are several interesting observations that
we can already make We will compare the decay rates for B bd
and B bd Each of these pro cesses has a contribution
from a tree diagram with amplitudes
A B aV V
T ub
ud
V A B aV
ud T ub
Heavy Quark Physics and CP Violation
For simplicity we ignore the p enguin contribution to the decay The
co e cient a contains a lot of complicated physics including nal state
interactions and it is in general a complex numb er However a is not
conjugated in going from B to B The relation between the two
amplitudes in Eq is treated much more carefully in Sec andin
the App endix If these were the only amplitudes present there would
be no CP asymmetry since the magnitudes of the amplitudes are the
trian same In fact the factor V V is just one side of the unitarity
ub
ud
gle and its orientation dep ends on the CKM phase convention You
can make a single amplitude p oint anywhere you want To pro duce
a CP asymmetry we need another amplitude to generate an interfer
ence term and a second amplitude is generated from mixing A prop er
treatment of mixing requires an analysis of the coupled B B system
and the calculation of the mass eigenstates A great simpli cation
arises from the fact that the intermediate state in mixing is dominated
by tt The total amplitudes included b oth the tree diagram and mixing
are derived in Sec and are of the form
V V
tb
td
A B V V aV V b
ud ub
ub ud
V V
td
tb
V V
td
tb
V V A B aV V b
ub ud
ud ub
V V
tb
td
In fact there can be di erent overall phases multiplying these expres
sions but they do not a ect the physical results See the discussion
in Sec and in the App endix The di erence in decay rates is pro
p ortional to
jA B j jA B j
V V
td
tb
ab V V V V
ub ub
ud ud
V V
tb
td
V V
tb
td
a b V V V V
ud ud
ub ub
V V
td
tb
V V
tb
td
ab V V V V
ud ud
ub ub
V V
td
tb
V V
td
tb
a b V V V V
ub ub
ud ud
V V
tb
td
V V
tb
td
V V Im ab Im
ud
ub
V V
td
tb
There are several observations to make regarding this result First
J D Richman
to obtain a nonzero di erence between the decay rates we require
Im ab If we write a jaj exp i and b jbj exp i then the
a b
condition Im ab jajjbj sin implies that a and b must
a b
be noncollinear in the complex plane While the phases asso ciated
with a and b do not reverse sign under the CP op eration they must
nevertheless still b e present to pro duce a CP asymmetry In Sec we
will see how the role of this strong phase is lled in the inteference
between mixing and decay Similarly the imaginary part of the CKM
factors in Eq is prop ortional to the sine of twice the angle b etween
the two complex numbers z V V and z V V The angle
ud td
ub tb
between z and z is where is shown in Fig Thus the
asymmetry b etween the rates for B and B prob es
sin You can also see explicitly in Eq that because the CKM
elements for B decay are the complex conjugates of those for B decay
the two terms prop ortional to ab do not cancel and the two terms
prop ortional to a b do not cancel except for the case in which the
unitarity triangle collapses to a straight line Thus to pro duce a CP
violating asymmetry we require the presence of b oth the CP violating
phases from the CKM elements and the non CP violating phases from
a and b
We note also that the angle between z V V and z V V is
ud td
ub tb
de ned using only the rst and third generations one do es not need
all three generations to b e present in a given process to pro duce a CP
asymmetry two complex numb ers are su cient to de ne an angle and
with three generations the angle can b e nonzero One might then won
der sp eci cally how CP asymmetries are excluded in a world with just
two generations Let s consider the analogous situation for K
decay assuming that the third generation is missing Then ignoring
p enguins as b efore the interference bet ween direct decay and mixing
followed by decay gives
V V
cs
cd
V V A K aV V b
ud us
us ud
V V
cd
cs
V V
cd
cs
V V A K aV V b
us ud
ud us
V V
cs
cd
and
jA K j jA K j
V V
cs
cd
Im ab Im V V
ud
us
V V
cd cs
Heavy Quark Physics and CP Violation
But with only two generations unitaritygives
V V V V
ud cd
us cs
Thus the CKM factor in Eq is real and there can be no CP
asymmetry Here the unitarity constraint Eq is telling us that
the two complex numb ers are collinear this is how the equations know
that there areonlytwo generations
Finally this example gives us insight as to why simple phase rota
tions of the quark elds cannot somehow be confused with CP viola
tion We have seen that when a quark eld is multiplied by a phase
factor an entire row or column of the CKM matrix is e ectively mul
tiplied by this phase factor Thus every term in the inner pro duct
between two rows or two columns is multiplied by the phase factor
and the entire unitarity triangle rotates The angles b etween the sides
of the triangle do not change and it is those angles that enter into CP
asymmetries
Exp erimental Challenges and Approaches in Heavy Quark
Physics
Exp erimentalists have used a wide variety of techniques to study the
physics of b ottom and charm hadrons The most imp ortantchallenges
are straightforward to state
How can we pro duce and detect millions of b and c hadrons
How can we distinguish events containing b and c hadrons from other
pro cesses
How can we extract the maximum information from each event
The answers to these questions are not simple In each case creativity
compromise and technological innovation come into play and there
are often many di erent approaches that can succeed This section de
scrib es the exp erimental environments and compares their advantages
tages and disadvan
Table summarizes some of the imp ortant features of current b
exp eriments and the accelerators at which they op erate Although
the states were discovered in a xed target exp eriment at Fermilab
in all successful b hadron exp eriments have been p erformed in
colliding b eam machines Currently three main approaches are used
The CLEO e e S B B e e Z bb and pp bbX
exp eriment at the Cornell Electron Storage Ring CESR has the
largest b hadron data sample with over million S B B events
J D Richman
CESR
SYNCHROTRON
WEST EAST TRANSFER TRANSFER LINE LINE
LINAC e-
e+
CHESS WEST CHESS CLEO II EAST
Positron Bunch - Clockwise
Electron Bunch - Counter Clockwise
Fig The CESR machine consists of three main parts a linear accelerator which
brings the b eams to an energy of ab out MeV a synchrotron which further accelerates
the b eams to GeV and the storage ring itself The e e collisions take place at an
interaction p oint surrounded by the CLEO detector The circumference of CESR is only
m and the ring is lo cated underneath a playing eld at Cornell University
Most of the CLEO analyses discussed in this pap er use million events
or less however since the data pro cessing is relatively slow The CESR
ring is fairly small by present standards its circumference is m
and it roughly ts under one of the playing elds at Cornell University
It now has the highest luminosity of any collider in the world L
cm s Figure shows the main features of CESR
One of the most imp ortant advances in achieving high luminosi
ties in storage rings is multibunch op eration which allows higher to
tal beam currents to be stored At present CESR op erates with
bunches of electrons and bunches of p ositrons Table shows a
dramatic increase in the number of stored bunches with two new ma
chines KEKB in Japan bunches b eam and PEP I I at SLAC
bunches b eam Two new detectors Belle at KEK and
BaBar at SLAC are now being constructed by international col
lab orations and should b egin op eration at these machines in
Heavy Quark Physics and CP Violation
Table
Comparison of existing exp eriments studying b hadron physics The pro duction cross section
at the Z is expressed as a function of R Z bb Z hadrons The number
b
of b events available at the Tevatron exp eriments is very large of order but this number
cannot be directly compared with the e e data samples b ecause it is only p ossible to
trigger on a small fraction of the decays as discussed in the text The quantity sp eci ed
for CDF is the number of b hadrons rather than the number of bb pairs The luminosity
quoted for CESR is the p eak value at the b eginning of a run while others are typical values
or averages
Machine Detectors L N bb
bb bb
bb Pro duction cm s nb
had
CESR e e CLEO I I
S B B
LEP ALEPH DELPHI R
b
e e Z bb OPAL L
SLC SLD R
b
e e Z bb
Tevatron CDF D b
pp bbX
Table
Comparison of e e accelerators now under construction that will op erate at the S
CESR is an upgrade to the existing machine These machines are exp ected to b egin
op eration in
Machine Circum E L bunches b eam sep bunch
beam
m GeV cm s beam metho d sep m
CESR mr xing
angle
KEKB mr xing
angle
PEP I I magnetic
Multibunch op eration has a numb er of complications The elds left
b ehind in the accelerator by one bunch can interact with a subsequent
bunch leading to instabilities whichmust somehowbekept under con
trol There must also b e amechanism to prevent collisions of bunches
at p oints other than at the designated interaction p oint The metho ds
for avoiding these so called parasitic collisions have a ma jor impact on
the accelerator the interaction region and the detector design
In CESR the b eams are stored within a single ring of magnets and
radio frequency cavities This is p ossible because the beams have the
same energy but are travelling in opp osite directions To avoid para
sitic collisions the b eams must b e put into slightly di erent orbits For
J D Richman
many imp ortant measurements of CP violation however it is neces
sary to have unequal beam energies so that the center of mass of the
B B system moves with resp ect to the detector and the asymmetries
can be measured as a function of the relative B decay times This is
the motivation for building the KEKB and PEP I I accelerators Ta
ble which are now under construction The need for unequal b eam
energies led to two separate rings this feature in turn allows a vast
increase in the number of bunches
Still there is the problem of howtoprevent parasitic collisions near
the interaction region in a two ring machine In PEP I I for example
the bunch separation is only m the rst parasitic crossing would
o ccur at only half this distance from the interaction point Figure
shows a top view of the PEP I I interaction region The tra jectories of
the b eams are separated by p ermanent dip ole magnets B placed very
close to the interaction point The b eam separation comes at a price
a large ux of synchrotron photons sprays o the b eams during the
b ends These photons must be intercepted by carefully placed masks
Furthermore the magnets also occupy valuable space that could be
used to add more detector elements At PEP II the forward region is
extremely valuable due to the boost degrees in the S frame
corresp onds to only degrees to the b eam axis in the detector frame
The KEKB machine uses a completely di erent approach the beams
do not collide head on but instead enter with a crossing angle of
mrad In CESR an upgrade to the symmetric energy CESR ma
chine there are relatively few bunches and a larger bunch spacing so
t to solve the problem that a crossing angle of only mrad is su cien
In these high current storage rings the quality of the vacuum es
p ecially near the interaction point is crucial Coulomb scattering or
bremsstrahlung of the b eam particles from residual gas molecules in
the b eam pip e can lead to electromagnetic showers and enormous back
grounds in the detector Thus high luminosity can be achieved but
not without great attention to its consequences for backgrounds in the
detector and for radiation damage to detector comp onents
For exp erimenters the three techniques used to pro duce b hadrons
o er di erent advantages and disadvantages A distinctive feature of
running at the S is that b ecause its mass is only ab out MeV
ab ove that for a B B or B B pair no additional pions are pro duced
nor are any other typ es of b hadrons which are to o heavy As a con
sequence in a symmetric energy machine such as CESR the energy of
each B meson is equal to the beam energy When computing the in
variant mass of a set of particles X assumed to come from the pro cess
Heavy Quark Physics and CP Violation
Support Tube Q1 Magnet
B1 Magnet
Mask HEB LEB
Septum
Fig The interaction region of the new PEP I I machine is complicated by the need to
avoid parasitic crossings of the closely spaced b eam bunches In this view from ab ove
the detector is not shown and the scale transverse to the beam direction is expanded
by a factor of ten relative to the scale along the beam direction The GeV electron
b eam HEB enters from the left while the GeV p ositron b eam LEB enters from
the right After interacting the b eams are separated by p ermanent dip ole magnets B
that start at ab out cm on either side of the interaction p oint The space on the b eam
pip e b etween the magnets is o ccupied bya ve layer silicon vertex detector not shown
The drift chamb er is lo cated outside the supp ort tub e
B X X X one can therefore imp ose the constraint E E
n B b eam
v
u
n
X
u
t
m E p
B
beam
In analyses of hadronic decays this technique gives a mass resolution of
MeV ab out a factor of ten b etter than the resolution without
M
b
the constraint see Sec Before using this constraint it is imp ortant
to verify that the measured energy of the candidate system is consistent
with the b eam energy within resolution
n
q
X
E m jp j E
b eam
The best approach is to lo ok at the m distribution b oth for events
B
with E and for sidebands with E and E For a
two body decay to low mass particles such as B m and E
B
essentially measure the di erence and sum of the momenta
q q
E p p E jp j jp j m
B
b eam beam
J D Richman
25
ϒ(1S) 20
15 ϒ(2S) Hadrons)(nb) 10
→ ϒ(3S)
- e
+ ϒ(4S) 5 (e σ
0 9.44 9.46 10.00 10.02 10.34 10.37 10.54 10.58 10.62
Mass (GeV/c2)
Fig The hadronic cross section vs center of mass energy in the energy region The
S at GeV is the third radial excitation of the ground state It has su cient
mass to decayinto B B or B B pairs which results in its larger width The continuum
events underneath the S typically havea two jet top ology which allows them to b e
distinguished from the much more isotropic distribution of tracks in S B B decays
q q
E E jpj jpj E p p m m
beam b eam
where wehave used the fact that the B mesons movevery slowly in the
S rest frame so the momentum vectors of the daughter hadrons
are nearly back to back
Figure shows the hadronic cross section in the energy region
measured by the CLEO detector at CESR The S is a rather mo d
est resonance it is much broader than the lighter s b ecause the B B
channel is op en The S cross section is nb and there is an
nb contribution to the total hadronic cross section from additional
continuum nonresonant e e q q q u d s c events Still this
gives the most favorable signal to backgound ratio of anyenvironment
and there is a p ositive side both e e cc and e e events
are interesting in themselves
Two strategies are used to deal with continuum background at
the S First many of these events can be remo ved from the
data sample with a relatively simple selection cut based on overall
eventshape whichcharacterizes the distribution of particle momentum
vectors In continuum events the particles are generally collimated
into two back to back jets in contrast the distribution of tracks in
S B B events is much more isotropic The spherical top ology
Heavy Quark Physics and CP Violation
of S events is due to the very low B momentum in the S
rest frame p MeV c Although useful for separating B B events
from continuum the spherical event shap e has a disadvantage The
decay pro ducts of the two B mesons overlap in space making it more
di cult to asso ciate the correct tracks with each other and resulting
in combinatorial backgrounds The second strategy for dealing with
the continuum background is to measure it CLEO sp ends one third of
running at a center of mass energy ab out MeV below the its time
S Each analysis p erformed at the S is also p erformed on the
continuum sample providing a direct measurement of this backgound
contribution
A particular strength of the CLEO detector shown in Fig is its
ability to measure b oth charged tracks and photons with very good
precision The momentum of charged particles is determined from the
track curvature in a T magnetic eld The high value of the mag
netic eld helps to provide go o d momentum resolution for sti tracks
from low multiplicity mo des such as B It has the disadvan
tage that low momentum tracks can curl multiple times complicating
the pattern recognition for the hits in the drift chamb er Measurements
of the soft pion from the ubiquitous D D decay can su er
from this problem The electromagnetic calorimeter constructed from
CsI crystals can detect photons down to energies of ab out
MeV with high e ciency The resolution is E for a MeV
E
photon and for a GeV photon relevant for B K a typical
mass resolution is MeV c Since electrons or p ositrons also
calorimeter also provides a pro duce electromagnetic showers the CsI
powerful to ol for electron detection and measurement This device adds
enormously to the detector s capability for reconstructing exclusive
nal states and b oth BaBar and Belle are using similar technology for
their calorimeters Figure shows a B K decay observed in the
CLEO detector
The pro cess e e Z bb has complementary advantages and
disadvantages to running at the S The b and b quarks hadronize
into a wide variety of particles The fractions for the di erent sp ecies
of hadrons are
f f
B
B
d
f
B
s
f
b baryon
Fig The CLEO I I Detector The b eam pip e which runs left to right is surrounded by
tracking chamb ers that measure the tra jectories of charged particles Outside of the drift
chamb er are time of ightscintillators and the CsI crystal calorimeter lab elled shower
detectors and a T sup erconducting coil A three layer silicon vertex detector not
shown was installed in This recent upgrade is called CLEO I I and will b e followed
by the ma jor CLEO I I I upgrade in
These particles are b oth pro duced directly from the bb pair and as
decay pro ducts of various typ es of excitations of the ground state b
hadrons Unlike the S there are accompanying lighter particles
so that the b hadrons do not have a xed momentum although there
peak around GeV c Since Z bb events are jet like there is is a
Heavy Quark Physics and CP Violation
2771196-018 Run: 47779CLEO XD Event: 16528
I + K I I o o o + I B K* B D I + I o K -
K + +
Fig The decay B K K K observed in the CLEO I I detector The
curved tracks in the drift chamb er are shown in xy pro jection p erp endicular to the b eam
axis whereas the clusters in the CsI calorimeter are shown in p ersp ective as if lo oking
into a barrel The outermost detectors whichonlyhave noise hits in this event are muon
chamb ers
very little confusion between the decay pro ducts of the two b hadrons
in contrast to the situation at the S
Although Z decays provide access to many di erent typ es of b hadrons it is not a simple matter to determine the pro duction rates of
J D Richman
individual sp ecies with high precision This intro duces an uncertainty
in the normalization of branching fractions and related quantities such
as CKM elements For that matter it has not b een fully established
empirically that B S B B B S B B al
though this equality is usually assumed and is exp ected from the nearly
identical values of the B and B masses Some S analyses in
corp orate a systematic error due to the uncertainty in the B B and
pro duction fractions In general it is always worth paying at B B
tention to how an exp eriment normalizes its branching fractions and
to understand what assumptions if any are present
Table and Table compare the relativistic b o ost factors in dif
ferent exp erimental environments The highest momentum b hadrons
with are pro duced at the Z The typical distance of the b
decay vertex from the interaction p ointis c mm Although
B
the b hadrons still decay within the b eam pip e the decayv ertex can b e
clearly resolved by extrap olating tracks measured with high precision
silicon vertex detectors which are placed just outside the beam pip e
The use of such devices has resulted in a very successful b physics pro
gram at LEP Figure shows tracks in a Z bb decay detected by the
DELPHI vertex detector Compare this to the CLEO event Fig
where the decay pro ducts of the two B mesons are not collimated into
jets The typical B decay length in CLEO is only m whichistoo
small to be resolved However vertexing of charm hadrons pro duced
in B decays can still b e very useful The asymmetric B factories give
a boost ten times larger than that at CLEO and a tenth of
the b o ost at LEP The typical B decay length is ab out mm and
vertexing will b e extremely imp ortant
The CDF exp erimentattheTevatron Fermilab has demonstrated
that certain b physics measurements can be p erformed successfully in
the hadron collider environment The pro cess pp bbX pro duces an
enormous number of b hadrons since b However most
bb
these events cannot be written to tap e b ecause bb of
inelastic
and they cannot be distinguished by the triggers which initiate
detector readout from the large number of uninteresting background
events The triggers for selecting b hadron events are based on leptons
The inclusive single lepton trigger selects b hadrons with a typical
p GeV c while the J trigger gives a typical p
T T
GeV c where p is the momentum transverse to the b eam axis
T
The b hadron p sp ectrum is peaked at the low end so the triggers
T
select only a small fraction of decays even for these mo des In spite of
these di culties there are a number of excellent b physics studies at
Heavy Quark Physics and CP Violation
DELPHI 26024 / 1730
0.0 cm 7.5 c DELPHI 26024 / 1730
0.0 cm 2.0 c
Fig A Z bb event observed in the DELPHI vertex detector The decay pro ducts
of the b hadrons are highly collimated into jets The vertex detector surrounds the b eam
pip e and by extrap olating tracks back to their intersection p oints it is p ossible to identify
decayvertices and determine their lo cations Figure used with p ermission
CDF esp ecially in the area of lifetime measurements The enormous
pro duction rate allows the use of hadronic decay mo des whichtypically
give the smallest systematic uncertainties The CDF exp eriment also
has a go o d chance to observe CP violation in the decay B J K
s
Measurements of charm hadron decays have been p erformed b oth
in e e colliding b eam exp eriments such as CLEO and at xed target
exp eriments The challenge for xed target exp eriments is to suppress
ery large background from light quark pro duction In photopro duc av
tion exp eriments where a high energy photon is incidenton a nuclear
target the charm cross section is typically b or of the to
tal cross section In hadropro duction exp eriments where a proton
J D Richman
is incident on a nuclear target the charm pro duction cross section is
higher but represents even less of the total cross section For pro
ton momenta in the range GeV c to GeV c the charm cross
section is b to b while the total cross section is times
greater To suppress the backgrounds xed target exp eriments
exploit the relative long charm hadron lifetimes s to s
which together with a large b o ost factor enable charm particles to
travel measurable distances from the primary pro duction point Sili
con vertex detectors again playakey role in distinguishing signal events
from backgrounds Examples of such xed target charm exp eriments
are E an upgraded version of the E photopro duction exp eri
ment and E hadropro duction at Fermilab The E exp eriment
shown in Fig exp ects to reconstruct over a million reconstructed
charm decays from its current data sample
Fig The FOCUS E xed target charm exp erimentat Fermilab The photon
beam enters from the left Note the planes of the silicon strip detectors in the segmented
target Figure used with p ermission
Fixed target exp eriments have not yet b een able to obtain useful
samples of b hadrons b ecause the cross sections are to o low at the
energies of presently available b eams The HERA B exp eriment now
under construction at DESY Hamburg will be able to study CP
aswell as B mixing and a variety violation in the decay B J K
s S
Heavy Quark Physics and CP Violation
of other pro cesses The exp erimenthasanovel con guration GeV
protons from the HERA proton synchrotron are incident up on a wire
target that is mounted internally to the accelerator beam pip e The
target consists of short pieces of wire z mm along the beam
direction providing a well de ned primary vertex lo cation The wires
will b e inserted into the halo of the proton b eam which extends out a
few millimeters from the b eam axis The detector itself is a meter
long sp ectrometer that covers the angular range milliradians
It will b egin op eration in the relatively near future
The longer term future will bring several additional exp eriments
measure CP violation These include the LHC with the capability to
exp eriments Atlas and CMS as well as the dedicated B physics exp eri
ment LHC B AtFermilab the BTeV exp erimentisnow b eing designed
to use forward pro duction of b hadrons at the Tevatron
Historical Persp ective Bumps in the Road and Lessons in
Data Analysis
A very short history of heavy quark physics
Table lists some of the most imp ortantachievements in b ottom and
charm quark physics I have included not only discoveries but also
imp ortant technological advances and milestones The ARGUS exp er
iment which is no longer in op eration made numerous discoveries in
cluding B B mixing The exp erimen t ran at the DORIS e e storage
ring at DESY from to accumulating ab out B B
events Theorists might argue that my selection of historical highlights
is biased toward exp erimental results In any case the developmentof
heavy quark e ective theory starting around has had an enormous
e ect on the eld
Bumps in the road case studies
Exp eriments in b ottom and charm physics have pro duced an enormous
number of measurements and it is not surprising that some of them
have not stood the test of time At the risk of o ending some of
my colleagues who I hop e will appreciate the p edagogical goals of
these measurements here I these lectures I will examine some of
have restricted myself to exp eriments studying b physics these provide
J D Richman
Table
A brief history of charm and b ottom physics
Discovery of J Bro okhaven and SLAC
Exploration of cc sp ectroscopy
theoretical studies of QCD p otential
Discovery of the D meson SLAC
Discovery of the S FNAL
Discovery of the B meson Cornell
Precision measurements of D and D in high rate
xed target charm exp eriments using silicon strip
detectors FNAL
Discovery of B B mixing DESY
First discussions of asymmetric B factories
Heavy Quark E ective Theory Isgur Wise and many others
provides new metho ds for extracting jV j
cb
Discovery of B and CERN
s
b
Discovery of B X Cornell and DESY sho wed that
u
jV j
ub
Discovery of radiative p enguin decays B K and B X
s
inclusivedecay Cornell
Discovery of B K and other rare hadronic mo des Cornell
Fixed target charm exp eriments approaching reconstructed charm
decays FNAL
CESR reaches L cm s
plenty of examples and at the end of this section I will try to draw
some lessons from them
In the CLEO collab oration published the pap er Obser
vation of Exclusive Decay Mo des of b avored Mesons Because this
study provided the rst measurement of the B meson mass it can be
regarded as the discovery of the B At the time it was not uni
versally accepted that it would even be p ossible to reconstruct such
decays so the analysis op ened the do or to exclusive B physics For
a detailed discussion of the CLEO I analysis as well as an historical
p ersp ective see Stone
There were however already two indications that b avored hadrons
were b eing pro duced at the S the large width of the S com
pared to that of the S S and S resonances and the
observation of high energy leptons asso ciated with S decays The
reconstruction of sp eci c exclusive nal states was quite di cult b oth
b ecause B branching fractions are small and because the dominant
recon mo des lead to charm mesons which themselves can be easily
structed only in a relatively small fraction of their decay mo des
The di culties were particularly formidable in the CLEO I detector
Heavy Quark Physics and CP Violation
Fig CLEO I In the original observation of exclusive B decays several decay mo des
were summed to get su cient sensitivity
which was not able to reconstruct mo des with s and which also had
poor e ciency for reconstructing low momentum charged pions In
the CLEO I analysis the only D decay mo de used was D K
which has a branching fraction of ab out In such situations when
an exp eriment is attempting to discover a new particle it is often
the case that there are to o few events in any single decay mo de to
The exp erimenters must resort to summing signals establish a signal
over avariety of nal states to obtain su cient sensitivity to claim an
observation
Figure shows the B mass distributions for four low multiplicity
mo des The B and B masses were obtained from a t in whichtheB
energy D mass and D mass were constrained to known values leading
to a B candidate mass distribution that p eaks somewhat b elow
GeV c These decays conjugate mo des are implied along with the
number of events assigned in the CLEO I analysis and the pro jected
num b ers based on the recent estimates by Stone are
i B D events pro jected
ii B D events pro jected
iii B D events pro jected
J D Richman
Table
CLEO I and current B branching fractions To make a precise comparison the CLEO I
values needed to b e renormalized to takeinto account di erent assumptions on S
D and D branching fractions The B D decayisnowknown to b e dom
inated by D this branching fraction is counted twice in the right hand column
giving what amounts to an upp er limit for the sum
Mo de CLEO I branching fraction PDG branching fraction
B D
B D
B D dominated by D
B D
Sum
iv B D events pro jected
In most cases the number of events observed in CLEO I is far higher
than the pro jected number
Table compares the CLEO I branching fractions with current
values from the Particle Data Bo ok The CLEO I measurements are
roughly a factor of higher than presentvalues Given that the CLEO
I signal was events current data indicate that there should only have
b een a few events from the mo des listed To make a precise compar
assumed values ison it is imp ortant to check for di erences between
of input branching fractions For example CLEO used B S
B B and B S B B whereas
wenow assume for b oth values In addition CLEO used the value
B D K the currentvalue is
The CLEO pap er contains an imp ortant quali cation which to some
extent explains the extremely large values measured for the branching
fractions If a B decay contains a low energy particle that escap es
detection the remaining particles from that B may still b e consistent
with the b eam energy constraint and give an acceptable t The pap er
states that this problem a ects B D and B D but
not B D and B D so only the latter two pro cesses
are used in the B mass determination CLEO obtained the masses
M MeV c
B
M MeV c
B
which can be compared with the current PDG values
M MeV c
B
MeV c M B
Heavy Quark Physics and CP Violation
The CLEO I and current values of the masses are consistent to within
ab out
It can b e very di cult to determine what problems might b e present
in a study p erformed many years ago Stone has recently exam
ined the CLEO I analysis It is likely that the di culty is precisely the
one discussed in the CLEO pap er kinematic tting to the wrong de
cay hyp othesis sweeping mo des with missed low energy particles into
the signal Although this problem was recognized it app ears that its
impact on the analysis was underestimated It is interesting that sim
ilar problems a ected the Mark III measurement of D branching frac
recognized and corrected in a reanalysis tions the problems were
of the same data
The situation in which very weak signals in several mo des are com
bined to establish a statistically signi cant signal is inherently fraught
with the p ossibility of mishap This of course is often the situa
tion in which we nd ourselves in measurements with discovery po
tential With very low statistics signals it is imp ortantto distinguish
between the statistical uncertainty on the central value of the mea
surement and the probability that the observed events is due to
background For a small number of events the relative statistical error
p
on the signal is very large N but if the background is known to
be much smaller the signi cance of the signal can still be very high
Just b ecause a signal has only a few events is not reason to dismiss
it you need to know the size of the background and its uncertainty
as well However small signals present great di culties b ecause it is
extremely easy to overlo ok or underestimate the contributions of small
background sources In addition the usual checks that one p erforms
on the signal by examining various kinematic distributions are only
marginally useful putting even more reliance on correct determination
of the background
Figure in Sec shows a recent set of mass plots for hadronic
decays from the CLEO I I The contrast with Fig is enormous The
integrated luminosity at the S used in the CLEO I I analysis is
fb compared with only pb in The CLEO I I detector
is also vastly sup erior to CLEO I B physics has come a long way but
it had to start somewhere
My second example also comes from another low statistics measure
ment in this case p erformed by the Crystal Ball exp eriment
after it was moved from the SPEAR ring at SLAC to the DORIS ring
at DESY The Crystal Ball was a spherical array of ab out NaI Tl
crystals and it had earlier p erformed detailed studies of charmonium
J D Richman
Fig Initial evidence from the Crystal Ball for the pro cess S The plots
on the left and right are for high and low multiplicity nal states resp ectively Within
each group the gures are a photon energy sp ectrum b tted photon energy sp ectrum
and c photon energy sp ectrum after background subtraction No signals were presentin
a subsequent data sample Figure used with p ermission
states by observing the sp ectrum of photons from transitions between
these states
At the summer conference at SLAC the Crystal Ball collab ora
tion presented a pap er Evidence for a NarrowMassive State in the
Radiative Decays of the Upsilon The new particle called the
was apparently pro duced in the radiative decay S It
was conceivable although considered unlikely that the could
be a non standard mo del Higgs particle There was simply no obvi
ous explanation for the The Crystal Ball detector did not have
a magnetic eld and it had only minimal charged particle tracking
detectors so the signal was established almost entirely on the basis
of the recoil photon sp ectrum rather than by reconstructing a
mass p eak Still some basic prop erties of the charged tracks and event
top ology could be determined and the events were divided into two
samples multi hadrons and low multiplicityjets
These two event samples were said to b e statistically indep endent
The evidence for the was based on p eaks in the photon energy
sp ectra in the two event samples The sp ectrum for the multi hadron
Heavy Quark Physics and CP Violation
mo de shown in Fig a had a p eak while the low multiplicity
jet sample had a p eak with a statistical signi cance of The signif
icance for the combined signal was over a frequently used threshold
for claiming a new e ect The mass and width were measured to be
M MeV c
MeV c C L
The signal was obtained in a sample of K S decays Although
the Crystal Ball collab oration presented the in conference pa
p ers they did not publish the result in a journal Instead they accu
mulated a second sample of K S decays to verify the signal In
this new sample there were events at the exp ected mass in
the high multiplicity sample and the claim was retracted The CUSB
collab oration at CESR also searched for a signal and found none
It app ears that the p eak was the combination of two things
a random statistical uctuation and enhancement of this uctuation
by the tuning of event selection cuts on the data There is very little
that we can do ab out large statistical uctuations if we lo ok at enough
histograms there are b ound to b e some The Crystal Ball adopted one
approach to the problem of tuning cuts on the data they analyzed a
second data sample using the same cuts In my view it is far preferable
to use Monte Carlo events for de ning event selection criteria a point
that I discuss in the following section
My last example is the claim for the rst observation of the exclu
sive semileptonic decay B from the ARGUS collab oration
This rare decaymodeprovides a metho d to measure the magnitude of
V one of the smallest and least well measured elements of the CKM
ub
matrix The rst evidence for b u transitions was based on CLEO
and ARGUS measurements of the inclusive lepton energy sp ectrum b e
yond the B X endp oint where X represents a nal state with
c c
charm The theoretical uncertainties asso ciated with extracting jV j
ub
interest from the endp oint measurement are large so there was great
in measuring jV j in a di erent way
ub
At the Lepton Photon Conference the ARGUS collab ora
tion presented the results of a search for B The
metho d was to select events with a mo derately high energy lepton
E GeV imp osing various kinematic requirements for consis
tency with the semileptonic decay of a B meson and then to plot the
distri mass sp ectrum of the candidates Figure shows this
bution with a p eak at the mass This signal led to the branching
J D Richman
Fig Evidence from ARGUS for B presented at the Lepton Photon
Conference The p oints with error bars are the continuum subtracted data the histogram
is the overall t and the background from b c decays is represented by the shaded
region Used with p ermission and drawn from Ref
fraction measurement
B B
One of the most di cult asp ects of the measurementwas the deter
mination of the shap e of the background under the p eak The dom
inant contribution is from semileptonic decay to charm nal states
B X but there can be smaller contributions from various
c
B X pro cesses Comp ounding this di culty is the fact that
u
the is a very broad state MeV c and with low statistics
it is di cult to distinguish it from a combinatorial backgound
that often peaks in roughly the same mass region CLEO eventually
presented a limit that excluded the ARGUS central value but a
signal was not found until signi cantly later b ecause the branching
fraction was far b elow that claimed by ARGUS The CLEO II mea
surement is
B B
where the errors are statistical systematic and estimated mo del
dep endence resp ectively Since
B B
by isospin the CLEO I I result is e ectively that of ARGUS The
the AR ARGUS result was never published in a journal Statistically
GUS measurementwas marginal some might argue that given its large
Heavy Quark Physics and CP Violation
uncertainty it was consistent with the CLEO II result However as
suming that the CLEO I I branching fraction is correct the probability
for ARGUS to observe any signal in their data sample was extremely
low One interesting asp ect of the ARGUS analysis was the rather low
value of the lepton energy cut As we will see this mo de is exp ected
from rather general considerations to have a very hard lepton energy
sp ectrum Thus a useful test of the signal would have b een to apply a
much higher value of the cut to see if the signal was enhanced relative
to the background
Some rules for data analysis
Never determine your event selection criteria using the data
sample that you will use to measure your signal It is very
easy to exploit statistical uctuations in the data sample biasing the
signal upwards Usually the b est approach is to tune cuts on a Monte
Carlo sample used solely for this purp ose If the Monte Carlo do esn t
p erfectly describ e the distributions of the signal your cuts might not
be optimal However they will not be biased which is much more
imp ortant If a data sample is used to tune cuts it should not be
used to measure the size of the signal or to determine the level of
backgound Thus the rst steps of an analysis should be make
rough estimates and map out the main issues and strategies write
event generators and simulate the analysis The rst step should not
be to lo ok at the data As for most rules there are exceptions to this
one and with extreme care one can develop event selection cuts based
on the data An interesting example is the Bro okhaven E search
for K
Don t use more cuts than y ou need A simple analysis is easier
to understand check duplicate and present
Always check to see whether your signal is robust as you
vary your cuts If there are systematic variations in the size of your
signal after background subtraction and e ciency correction then
you might not understand your background comp osition the kinematic
distributions of your signal or the detector resp onse
Lo ok at all the distributions you can think of for your signal
and compare them with what you exp ect For example angular
distributions can often be predicted they can build your con dence
by cutting or in the signal or even be used to improve its signi cance
tting
Lo ok at the distributions of the events that your cuts ex
J D Richman
clude Try to select event samples with little or no signal contribution
to check your understanding of the background
When p ossible use data rather than Monte Carlo events to
measure e ciencies and background levels This is particularly
imp ortant for quantities that dep end sensitively on the particle inter
actions with the detector resolutions misidenti cation probabilities
lepton detection e ciencies
Do not use Monte Carlo events blindly nd out where the
information came from that went into the Monte Carlo co de
The Monte Carlo maydowell in predicting the background in someone
analysis but it may never have been checked for the mo des or else s
region of phase space relevantto your analysis
Be careful not to underestimate the systematic errors as
so ciated with ignorance of the signal e ciency back
ground comp osition and background shap es For example
the signal e ciency is usually sensitive to the signal decay distribu
tions which are not always p erfectly known In some cases however
the uncertainties in these distributions are not evaluated or ev en rec
ognized
Go o d luck
Leptonic Decays
Introduction to leptonic decays
Leptonic decays provide a clean way to prob e the strong interactions
that bind the quark antiquark system in the initial state meson simply
b ecause there are no strong interactions in the nal state to complicate
the interpretation of the decay rate measurement I ll b egin by posing
afew questions related to leptonic decays
Whichcharm and b ottom mesons have the largest leptonic branching
fractions Which mo des are exp erimentally accessible now
Why are the leptonic branching fractions of charm and bottom
mesons so small compared to those of the and K
How can leptonic decays b e separated from the much larger semilep
tonic backgrounds
What are the dominant systematic errors in the b est leptonic decay
measurements
How precise are current predictions for decay constants What kinds
of theoretical uncertainties a ect these predictions
Heavy Quark Physics and CP Violation
Fig Feynman diagram for leptonic decay
What are the e ects of radiative corrections in charm and bottom
leptonic decays
Figure shows a Feynman diagram for leptonic decay The initial
meson of mass M and four momentum p contains a Qq pair that
meets at a point annihilating via a virtual W into a lepton and a
neutrino The CKM element V or V enters at the annihilation
vertex The four momentum q of the W is xed q p and q
M This situation contrasts with that in semileptonic decay where
q varies dep ending on the kinematic con guration and q can take
on a broad range of positive values In scattering pro cesses q
Exp erimentally one observes only the lepton directly in favorable
situations the four momentum of the neutrino can be inferred from
the missing momentum in the event
Because the lepton neutrino system cannot interact strongly with
the initial state the matrix element can b e factorized into the pro duct
of hadronic and leptonic currents
G
F
p
M V h jJ jM iu k v p s
where the leptonic current is written out in terms of Dirac spinors
For the case of pseudoscalar meson decay there is only one four vector
that can b e used to construct the hadronic current namely p the four
vector of the initial meson The current op erator J has a V A form
but as we show in the app endix Sec only the matrix element of
the axial vector part of the current contributes The hadronic current
can then be written as
q QjP p i if p h j
M
where f denotes the meson decay constant It is remarkable that all
M
of the e ects of the strong interactions can be parametrized in terms
of this single quantity which measures the amplitude for the Q and
q quarks to have zero separation In fact in a non relativistic meson
J D Richman
where both quarks are heavy the decay constant can be related in
a very simple way to the wave function at the origin f M
M
j j
Using Eq and Eq it is simple to calculate the leptonic decay
rate for a pseudoscalar meson ignoring radiative corrections
G m
F
M jV j f Mm
Qq qQ
M
M
where f is the decay constant V is the CKM matrix element and
M qQ
m and M are the masses of the lepton and the decaying charged meson
M resp ectively
If we start with a vector meson the hadronic current must be pro
p ortional to its p olarization vector giving
h jq QjV i f V M
M Qq V
In practice it is very di cult to observe the leptonic decayofavector
meson b ecause this weak pro cess is o verwhelmed in rate by strong or
electromagnetic decays to the pseudoscalar meson for example D
D or D D Nevertheless knowledge of both pseudoscalar and
vector meson decay constants have imp ortant applications as we will
now see
Figure shows several situations in which decay constants are in
volved In Fig a a B meson decays into a two b o dy hadronic nal
state In Sec we will analyze this pro cess in the framework of the fac
torization approximation roughly sp eaking this means that the upp er
and lower vertices can be considered separately The quark antiquark
pair pro duced by the W is created at a point and the amplitude for
the pair to pro duce a particular meson is therefore prop ortional to the
meson decay constant In contrast the pro cess by which the daughter
charm quark and the sp ectator quark form a D or D meson is quite
di erent since the quark antiquark pair is not pro duced at a p oint or
even on a distance scale that is small compared to the size of the me
son We will later use transition form factors from semileptonic decay
to analyze the physics at the lower vertex
Figure b shows an extremely imp ortant application of decay con
and B B mixing which can be used to stants the pro cesses of B B
s s
extract the magnitudes of the CKM elements V and V We will see
td ts
in Sec that the oscillation frequency is prop ortional to f Even
B
though it app ears that exchange of W b osons could lead to a spatial
separation of the b and d quarks the short range of the W means that
this separation is negligible on the scale of the size of the B meson
Heavy Quark Physics and CP Violation
(a)
B B
(b)
(c)
(d)
Fig Pro cesses in which meson decay constants are involved a hadronic decay
b B B oscillations c decay and d neutral vector meson decay to lepton pairs
The gluons shown in a are simply a reminder that pro cesses involving hadrons are very
complicated
Figure c shows the decay of a lepton Because the leptonic
current is known this pro cess can be used to determine f for kine
M
matically accessible mesons pro duced at the upp er vertex The rate
for the decay is given by
m M M
M G b jV j f
M ij
F M
m m
where b for J and b for J Decay constants
M M
for neutral vector mesons can b e determined from the electromagnetic
pro cess shown in Fig d Again the decay constant enters because
the q and q must meet at a p oint
As a result of the factor m in Eq leptonic decay of a pseu
doscalar meson is forbidden in the limit m As m increases the
leptonic rate also increases to a maximum value and then falls again
to zero as m approaches M Both vector and axial vector
couplings at a vertex favor decays in which the nal state fermion and
J D Richman
antifermion have opp osite helicities the more relativistic the fermions
the stronger the e ect hence the dep endence on m This e ect is
sometimes called helicity suppression and it is a particular consequence
of helicity conservation The e ects of helicity conservation are ubiq
uitous in particle physics we will meet them again for example in
the discussion of kinematic distributions for semileptonic decays A
well known application of Eq ischarged pion decay where
e
e
For D decay Eq gives
s
D e D D
e
s s s
To understand this e ect it is imp ortant to distinguish between
chirality and helicity The left and right handed chiral pro jections
of a spin eld are de ned by P and P where
L L R R
P and P These op erators satisfy P
L R
L
P P P and P P the usual conditions on pro jection
L R R L
R
op erators The eld op erator either destroys a fermion or creates an
do es the reverse The op erator either destroys antifermion while
L
a left handed chirality fermion or creates a right handed antifermion
Eachchiral pro jection is a mixture of b oth negative and p ositive helicity
states but in the relativistic limit the left handed chiral pro jection of
a spin particle is purely helicity while the right handed
chiral pro jection is purely helicity
It is easy to show that a vector or axial vector current involving a
fermion eld can b e decomp osed as
L R
L R
where L and R are used to indicate chiral pro jections and repre
sents either a vector or an axial vector coupling As a consequence
in a t channel pro cess a left handed chirality fermion in the initial
state couples only to a left handed fermion in the nal state and right
handed to right handed If a fermion and antifermion not necessarily
of the same typ e are either pro duced or annihilated with a vector
or axial vector coupling Eq implies that they must have opp o
site chirality In the relativistic limit these statements ab out c hirality
translate directly in statements ab out helicity Figure illustrates
these e ects of helicity conservation
Heavy Quark Physics and CP Violation
(a) (b) (c) (d)
Fig Helicity conservation in various situations In a and b helicity is conserved
in a t channel scattering pro cess a helicity fermion is scattered into a helicity
nal state and similarly for helicity The spin vector rotates along with the
momentum vector The same e ect implies that helicity fermions couple to helicity
antifermions and vice versa as shown in c and d These statements apply to
vertices with vector or axial vector couplings and are exact in the limit of zero fermion
masses
Another physical e ect is that in pro cesses like e e S or
e e Z the vector particles cannot have a spin pro jection zero
along the b eam axis In W mediated weak decays these considerations
apply but there is the additional e ect that P picks out only one of
L
the two p ossible ways to satisfy the helicity conservation constraints by
forcing the W s daughter fermion to be left handed and the daughter
antifermion to b e right handed
We now return to leptonic decay Fig shows two decay con g
urations in which the neutrino has negative helicity The lepton and
neutrino helicities are equal in Fig a and opp osite in Fig b If
the decaying meson has spin however the opp osite helicity con g
uration is forbidden by angular momentum conservation since there
would b e nonzero angular momentum along the decay axis Note that
in calculating angular momentum quantized along a two b o dy decay
axis we don t have to worry ab out orbital angular momentum since
L r p there can t be any such contribution The decay can only
pro ceed through the con guration shown in Fig a but this is helic
ity suppressed In the limit of zero lepton masses the requirements of
helicity conservation and angular momentum conservation are in direct
y is forbidden except for the mitigating e ects of con ict and the deca
radiative corrections
We can use Eq to give us an idea which leptonic decays of charm
and b ottom hadrons would be easiest to search for Note rst of all
that the dep endence on the CKM element is crucial
D jV j
cd
D jV j
cs
s
B jV j
ub
J D Richman
(a)
(b)
Fig Leptonic decay of a pseudoscalar meson a Con guration allowed by angular
momentum conservation but suppressed by the axial vector coupling b con guration
forbidden by angular momentum conservation but favored by the axial vector coupling
Table
Predictions for leptonic branching fractions of
and B under the assumption the D D
s
that f f and f are all equal to
D D B
s
MeV For comparison the measured values of
the and K leptonic branching fractions
are included
B e B B
D
D
s
B
K
The D should have the most accessible leptonic decays since they
s
are CKM favored It is unfortunate that B leptonic decay is highly
suppressed since wewould like to use it to measure f which is needed
B
for B mixing calculations It will probably b e some time b efore leptonic
B decays are observed
Table lists the predictions of Eq for the leptonic branching
fractions for the D D and B under the assumption that f
D
s
f f MeV For comparison I also list the measured leptonic
D B
s
branching fractions for the and K mesons As exp ected the mo des
D and D have the largest branching fractions the
s s
advantage of the relatively large branching fraction for D is
s
partly o set by the di culty in reconstructing the decay
Heavy Quark Physics and CP Violation
At rst glance it is startling that
B D B K
s
used in Table may not be quite Even though the assumed value f
D
s
right the dramatic di erence in the D and K leptonic branching
s
fractions will not change if we use any value within the measured or
exp ected range What happ ens to f as the meson mass increases An
M
asymptotic scaling law which can b e derived from HQET predicts
that
t for large M f M constan
M
Note that this factor app ears in the leptonic decay rate formula Eq
In contrast the total decay rate for a heavy meson is prop ortional to
M so the leptonic branching fraction b ecomes small If we compare
rates instead of branching fractions we have
K s
D s
s
D s
s
which shows that the leptonic decay rate is in fact higher for the D
s
than for the K
Measurements of leptonic decays
Imagine that you are given a large sample of events containing a mix of
charm hadrons moving at high momentum The sample also contains
a large number of events without charm particles Howwould you nd
leptonic decay a signal for D
s
The most conspicuous problem is that the only directly observed
particle from the decay is the lepton At best the neutrino can be
inferred from the missing momentum v ector assuming that the ini
tial D momentum is somehowknown and the detector is very nearly
s
hermetic If there is no such estimate of the neutrino momentum
then the D mass peak cannot be observed In this case the signal
s
must be established by some other technique Even if the neutrino
momentum can b e estimated the resolution is notasgoodasthat for
a directly measured particle The D mass peak will b e broader than
s
for a hadronic decay making it more di cult to distinguish the signal
from backgrounds Another problem is that there are plenty of leptons
around from other sources in particular from charm semileptonic de
ys Given these di culties and the small branching fractions it is ca
J D Richman
Fig Observation of D leptonic decay from the E exp erimentatFermilab The
s
left hand gure a shows the distribution of p muon momentum transverse to the D
T
s
direction for the sample of one prong kinks The dominant source of events is semileptonic
D decay but there is an excess of events at high p this excess is attributed to leptonic T
decay The right hand gure b shows the p sp ectrum for two prong vees
T
not surprising that current measurements of leptonic branching frac
tions do not have high precision Nevertheless steady progress is b eing
made and in this section we examine some of the measurements
One way to get additional information ab out the decay is to observe
not just the charged lepton but also the D itself b efore it decays
s
y would then pro duce a kink in the tra jectory of a A leptonic deca
charged track since the muon recoils against an unobserved neutrino
In most exp eriments it is not p ossible to observe the kink b ecause the
D decays b efore it ever reaches a detector Figure shows data from
s
the E exp eriment at Fermilab which had the ability to observe
such kinks The charm hadrons are pro duced by the interactions of a
high energy b eam incident on a photographic emulsion target The
target is followed by planes of silicon strip detectors and a magnetic
sp ectrometer To analyze the events the emulsion must rst be re
moved from the exp eriment It is then pro cessed and scanned for kinks
h are due to neutral particles and other top ologies such as vees whic
decaying into a pair of opp ositely charged tracks The kinks and vees
are identi ed in the emulsion while the silicon vertex detectors serve
to link the tracks found in emulsion with those in the sp ectrometer
where the momenta are measured
The observation of a kink however is not su cient to identify a
leptonic decay The semileptonic pro cess D K for example
ys from can pro duce an identical top ology To separate leptonic deca
Heavy Quark Physics and CP Violation
this background E measures the momentum p of the muon
T
p erp endicular to the parent particle s direction The distribution of
p for tracks in the kink sample is shown in Fig a The dominant
T
source of events is D K which pro duces a broad sp ectrum
in p The signal has a much narrower p distribution p eaked at
T T
ab out half the D mass this peak is largely beyond the endp oint for
s
the semileptonic decay Beyond the nominal endp oint there are
events ab ove the semileptonic background but there is still an small
contamination estimated to b e events from D
Given the small size of the signal it is imp ortant to demonstrate that
it is not simply an artifact related to the background Figure b shows
the p distribution for the sample of vees two opp ositely charged
T
tracks emerging from a common vertex p oint There is a large contri
bution from D K which pro duces a p sp ectrum similar to
T
that from D K As exp ected there is no excess beyond the
semileptonic endp ointinthis sample
In the E exp eriment it is not p ossible to determine the total
number of D decays so the leptonic signal is normalized relative
s
to the signal for D which is the b est measured D decay
s s
mo de Unfortunately this branching fraction is quite p o orly known
uncertainty propagates B D and this
s
into all D branching fractions The problem of normalizing signals is
s
quite imp ortant and we will return to it in our discussion of hadronic
decays
Currently the largest D signal is from a recent CLEO II
s
measurement which uses ab out e e cc events pro duced
p
at s M S or MeV below These events typically have
a two jet top ology and the fragmentation of the charm quark leads
not only to D and D pro duction but also to D and D Rather
s
s
than simply searching for D the CLEO analysis exploits the
s
constraints provided by the decay sequence D D D
s s s
The momentum of the neutrino is inferred somewhat crudely from
tum and it is used to reconstruct D and D the missing momen
s s
masses Because the neutrino momentum p is used to compute b oth
quantities the error on p tends to cancel when the di erence b etween
the reconstructed masses is computed
M M M
The resolution on M is reasonably go o d with MeV c
and the signal is ultimately identi ed as a peak in the M distri
J D Richman
bution This pro cedure is analogous to measurements that use the
reconstructed D D mass di erence when studying certain D decays
A second key idea in the CLEO analysis is to use events with elec
trons to measure the dominant comp onentofthebackground Because
D e is negligible due to helicity suppression we can assume
e
s
that events passing the analysis cuts but which have electrons instead
of muons must be due to background pro cesses The backgounds in
this analysis are due to three main sources
i semileptonic pro cesses primarily D K D and D
s
ii leptonic pro cesses a signal events in which the wrong photon is
selected b D decays for which the D did not come
s s
from D decay but was pro duced directly and c D
s
which is Cabibb o suppressed and
iii events in which a hadron has been misidenti ed as a muon
The semileptonic backgrounds have nearly the same rates to the e
and channels so the electron sample can be used to measure the
semileptonic contamination of the muon sample The leptonic and
misidenti cation backgrounds are very di erentinthee and samples
and must b e studied in a di erent way
The problem of hadrons being misidenti ed as muons is quite sig
ni cant and it is the source of one of the most imp ortant systematic
errors in the analysis In fact this fake rate is now thought to hav e
b een underestimated in the original CLEO I I analysis of this mo de re
sulting in a value of f that was to o large Muons in the D
D
s
s
analysis are required to have momenta p GeV c This value is
relatively high and it greatly suppresses background from semileptonic
decays while still retaining of the signal events Atsuchhighmo
menta however hadrons are more likely to p enetrate the steel absorb er
in front of the muon detectors thus app earing to be muons To esti
mate the number of such o ccurrences one can study tracks tagged
reconstructed in mass p eaks as or K mesons by virtue of being
for example from K and D D D K Such
s
pro cesses serve to identify the daughter hadrons simply by kinematics
The revised misidenti cation rates used in the new CLEO II analysis
are based on a sample of such events this study found that the
probability for a to fake a signature is
Figure shows the M mass di erence distribution for the CLEO
II measurement a b efore background subtraction and b after sub
Heavy Quark Physics and CP Violation
1631197-043 120
) ( a ) ( b ) 2 300 80 200
40 100
Events / (20 MeV c 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 2
Mass Difference M (GeV / c )
Fig CLEO I I signal for D The left hand plot a shows the M mass dif
s
ference for D candidates for muon data p oints with error bars electron data dashed
s
histogram and the additional background due to the excess of muon fakes over elec
tron fakes shaded region The solid histogram is the overall t The two background
contributions are not summed in this gure The right hand gure b shows the M dis
tribution for muon events after the backgrounds estimated from the electron sample and
from the excess of muon fakes over electron fakes have b een subtracted There remains
some combinatorial background and background from D The signal contains
events
traction of all but leptonic backgrounds which are taken in to account
in the t The background p eaks in the same region as the signal
which itself is not overwhelming Nevertheless this signal is by far
the most signi cant yet observed by any exp eriment The CLEO II
analysis gives
D
s
D
s
where the rst error is statistical and the second is systematic When
computing the actual branching fraction and f there is a large ad
D
s
ditional uncertainty from the error on the branching fraction for the
D normalization mo de as discussed ab ove Improvement in
s
the precision of this key branching fraction is extremely desirable
ts Both measurements from various exp erimen Table lists f
D
s
the published value of f and a corrected value are included The
D
s
corrected values are obtained from the CLEO II pap er in which the
results are renormalized to a common set of branching fractions The
LEP exp eriments L and OPAL have also measured f using D
D
s
s
D D The L signal is shown in Fig it has
s s
MeV while the events and gives f
D s
J D Richman
Table
from D decay The corrected values use a com Measurements of f
D
s
s
mon set of branching fractions as describ ed in the CLEO pap er The rst error
is statistical the second systematic and the third is the uncertainty due to the
error on the branching fraction for the normailizing mo de D Values of
s
obtained from D are given in the text f
D
s
s
MeV MeV Corrected f Exp eriment Observed Evts Published f
D D
s s
WA
BES same
E
CLEO I I
Table
Values of decay constants for vari
ous charged and neutral mesons f
and f are from the Particle Data
K
Bo ok while the others are from
Neub ert and Stech The decay
constants for neutral vector mesons
V are obtained from the rate for V
e e
Decay Constant Value MeV
f
f
K
f
f
K
f
a
f
f
J
Delphi signal has events including b oth D and
s
D and gives f MeV Values of decay constants
D
s
s
for other charged and neutral mesons are listed in Table
The correct calculation of an exp erimental average of the f values
D
s
requires some work due to a number of correlated systematic errors
MeV I have computed one such average f
D
s
where the rst error combines the statistical and most of the systematic
uncertainties and the second error is due to the uncertainty from the
overall normalization of D branching fractions For a recent review
s
of leptonic decay results see the pap er by Nipp e who obtains the
average value f MeV Nipp e uses B D
D
s
s
rather than the value see PDG which
Heavy Quark Physics and CP Violation
20
Data
15 Ds→τν, µν Background
10
Decays / 40 MeV 5
0 2 2.2 2.4 2.6 2.8 3
M(γDs) (GeV)
Fig L signal for D D D The p oints with error bars are the data
s s s
the hatched region represents the Monte Carlo estimate for the background and the op en
histogram shows the tted signal
is used in Ref
Searches for B B are not yet sensitive to rates at the
level predicted by the standard mo del The current limit from LEP
exp eriments is B B C L
Because helicity suppression plays suchakey role in leptonic decays
to see whether the emission of a photon can substan it is interesting
tially a ect the decay rates We might exp ect photon radiation to
mitigate the e ects of helicity suppression when e or
there should b e little overall increase in the rate for since there
is much less helicity suppression in that case Estimates of these e ects
are mo del dep endent and have b een discussed by several authors For
example Atwood Eilam and Soni nd that the dominant contri
bution comes from photon radiation from the light quark in the meson
and that B B essentially indep endent of the
sp ecies of lepton The rate is large relative to the purely leptonic
but not for rate for
B B
B B
J D Richman
B B
B B
The e ect on D decays is very small
s
B D
s
B D
s
CLEO has obtained the limits B B and
B B e b oth at C L These results demon
strate how di cult it will be to make precision measurements of lep
tonic decays but progress will b e p ossible at the new B factories
Lattice calculations of leptonic decay constants
The calculation of heavy meson decay constants using lattice QCD have
b een progressing steadily and they may have even reached the point
where exp erimentalists should start taking them seriously These nu
merical calculations of hadron prop erties are based on computer simula
tions using a discrete space time lattice Because the limiting resource
is computing p ower certain approximations are made First the space
time volume is typically divided into to spatial lattice sites and
to time intervals To contain a meson the lattice must have
a spatial length of fm or more which means that the spacing a is
typically between fm and fm For numerical reasons lattice
calculations are p erformed using imaginary time it rather than t and
Et
excited states must b e allowed to decayaway e leaving a con gu
ration that corresp onds to a su ciently pure ground state This is why
the number of time intervals on the lattice is typically larger than the
number of lattice sites along a spatial dimension Many lattice groups
calculate their results for a series of values of a and then extrap olate
to a the so called continuum limit This extrap olation is a signi
cant source of uncertainty A second limitation of lattice calculations is
that until recently they have b een p erformed in the so called quenched
approximation In this mo del of hadron dynamics the interactions o c
cur b etween valence quarks and gluons only there are no virtual quark
lo ops W e are now seeing the rst attempts to include quark lo ops and
there are some rough estimates of how big these lo op related e ects are
At the th International Conference on Heavy Flavor Physics
Claude Bernard gave a talk that clari ed many points regard
ing the history of lattice calculations He argued that there have been
three overlapping eras ancient history roughly the mid
dle ages roughly and the mo dern era which we app ear
Heavy Quark Physics and CP Violation
to just b e entering Many of the calculations p erformed in the ancient
era were simply wrong Some were a ected by p o orly understo o d lat
tice artifacts in others the ground state meson was not su ciently
isolated from its excited states in the calculation Improvements were
made during the middle ages but here again the calculations su ered
from such limitations as no extrap olation to a and no estimates of
the quenching error Calculations in the mo dern era are characterized
by detailed estimates of all uncertainties at least within the quenched
approximation The implication of all of this is that there is no point
in making enormous tables listing all of the lattice calculations since
at least by the exp erts I have many of them are known to be wrong
therefore listed some of the most recent results most of which are de
clared to b e mo dern in Bernard s talk
Ideally lattice predictions come with three errors a statistical er
ror a systematic error within the quenched approximation often dom
inated by the a extrap olation and an estimate of the quenching
error Figure shows preliminary results from a calculation of f
B
from the MILC collab oration as a function of the lattice spacing
Two p ossible p oints at a are used to estimate an error asso ciated
with the extrap olation to the continuum limit Table lists some of the
recent lattice results There seems to be a consensus that f is signif
B
icantly b elow MeV at least in the quenched approximation Two
signi cant statements can be made ab out this approximation First
lo ops are exp ected to increase f so the corrected values should be
B
somewhat higher than the quenched results Second based on the
studies of the MILC collab oration which has p erformed some calcula
tions with lo ops such e ects are not exp ected to b e dramatic p ossibly
bringing up their result by MeV which is only a shift
The lattice predictions of f are consistent within errors with the
D
s
exp erimental averages discussed in the previous section It is clear
however that b oth exp eriment and theory need to improve the preci
sion of these results
Semileptonic Decays
Introduction to semileptonic decays
Semileptonic decays due to their simplicity provide an excellent lab
oratory in which to measure jV j and jV j These pro cesses also allow
cb ub
us to study the e ects of nonp erturbative QCD interactions on the
J D Richman
Table
Some recent results on decay constants from lattice calculations The er
rors are statistical systematic within the quenched approximation and
for the MILC result an estimate of the uncertainty due to the quenched
approximation The MILC JLQCD and FNAL results are reviewed by
Bernard the Ali Khan et al results are from Ref
Group f MeV f f f MeV
B B B D
s s
MILC
JLQCD
FNAL
Ali Khan et al
Fig Lattice calculation of f from the MILC collab oration as a function of lattice
B
spacing a Two p ossible extrap olations to the continuum limit a are shown The
uncertainty on the extrap olation is based on the di erence between these two values
Figure provided by R Sugar
weak decay pro cess These goals may sound contradictory how can
we measure standard mo del parameters if complicated hadronic e ects
are present
We can untangle the e ects of strong interactions on semileptonic
decays for a numb er of reasons First it is imp ortant to recognize that
although such e ects are di cult to calculate they are isolated to the
teractions hadronic current As a consequence the e ect of strong in
can be rigorously parametrized in terms of a small number of form
Heavy Quark Physics and CP Violation
factors which are functions of the Lorentz invariant quantity q the
square of the mass of the virtual W Thus we have a solid starting
p oint for understanding how strong interactions a ect the decay Sec
ond the di culty in understanding these e ects is quite dep endent
on the decay mo de For b c decays the large masses of b oth
the b and c quarks provide the key to reliable theoretical predictions
based on heavy quark e ective theory HQET for exclusive decays
and heavy quark expansions for inclusive decays Third it has been
recognized that in di erent regions of the three b o dy phase space for
semileptonic decay the strong interactions play a very di erent role
Thus by fo cusing on the particular kinematic con gurations in which
the hadronic system is least disturb ed by the decay it has b een p ossi
ble to reduce the uncertainties asso ciated with hadronic e ects These
considerations have led to ma jor advances in the reliable determination
of jV j
cb
Heavy quark symmetry helps in two ma jor ways in b c de
cays First it relates di erent form factors to each other reducing the
numb er of indep endent functions that need to b e understo o d Second
it tells us the value normalization of the form factors when the
daughter charm hadron has zero momentum with resp ect to the par
ent b hadron the so called zero recoil con guration The zero recoil
p oint corresp onds to the maximum value of q
p p E E q p p
B D B D B D
m m
B D
where we have used the decay B D as an example In the
heavy quark symmetry limit m and m the hadronic
b c
system is undisturb ed by the replacement of one very heavy quark by
another Since the b and c quark masses are not truly in nite there are
corrections to predictions based on the heavy quark symmetry limit
but they are relatively small Heavy quark symmetry cannot give us
the actual q dependenceof the form factors sinc e this depends on the
complexities of the strong interactions The symmetry only relates the
q dependenceof the di erent form factors to each other and gives the
normalization at zero recoil The three form factors for B D
and the single form factor for B D all b ecome directly related in
the heavy quark symmetry limit to a single form factor the Isgur Wise
function
Signi cantchallenges in understanding semileptonic decays remain
however The problems are particularly di cult in b u decays
tally and While the determination of jV j is improving b oth exp erimen ub
J D Richman
theoretically there are still large uncertainties Due to the small value
of the u quark mass the zero recoil con guration in b u decays
do es not provide a solid normalization p oint Furthermore the small
mass of the daughter meson results in a large accessible range in q
over which there can be considerable variation of the form factors
Although jV j has been measured with reasonably go o d precision
cb
further improvements are highly desirable b ecause some tests of the
CKM framework are highly sensitivetothevalue of jV j For example
cb
to determine the constraint in the plane arising from measure
ments of the CP violating quantity j j measured in neutral K decays
K
one needs the quantity A jV j see Sec This dep endence on
cb
jV j means that the relative uncertaintyonjV j is magni ed byafac
cb cb
tor of four Thus aquantitative understanding of the e ects of strong
interactions on the underlying weak decay is essential for extracting
precise values of CKM elements and for testing the CKM framework
of CP violation
–
Fig Semileptonic decay of a B meson The complicated e ects of the strong in
teractions can b e rigorously parametrized bya smallnumb er of form factors whichare
functions of q the square of the mass of the virtual W The recoil velo city of the hadronic
system is directly related to q
Figure shows the diagram for the semileptonic decay of a B me
son The large number of gluons is meant to suggest the complexity
of non p erturbative strong interaction e ects Unlike leptonic decays
the quantity q the square of the mass of the virtual W is not a
of the kinematic con guration We constant but varies as a function
will see that there are twokey variables that enter into the description
of the decay q and E the lepton energy To understand semilep
tonic decays it is very often useful to analyze the dynamics at several
representativevalues of q For eachvalue we can think of the semilep
tonic decay M X as pro ceeding in two steps the two b o dy
decay M XW q and the two b o dy decay W q I
Heavy Quark Physics and CP Violation
will present a simple physical picture of semileptonic decays that is
centered on understanding the role of q the V A coupling meson
quantum numb ers and simple dynamical considerations
The discussion of semileptonic decays is divided into several sections
covering the dynamics of semileptonic decay a brief intro duction to
HQET inclusive b c and b u semileptonic decays form factor
predictions and measurements of exclusive decays To conclude this
intro duction I o er a list of questions to p onder
How do the strong interactions a ect semileptonic decay in di erent
kinematic regions What is the general behavior of the form factors
How do es this behavior di er among di erent semileptonic pro cesses
What determines the shap e of the lepton energy sp ectrum Why is
the shap e of the sp ectrum di erent for b ottom and charm decays
What determines the shap e of the observed q distribution Why is
this distribution completely di erent for B D and B D
What role do es the V A structure of the weak current play in
di erent pro cesses Do es this role dep end on the quantum numb ers of
the hadrons
What do es heavy quark e ective theory tell us ab out semileptonic
decays
If P is a pseudoscalar meson and V is the vector meson with the same
quark content what is the value of the ratio M V M
P Hint it s not generally three What determines the relative
p opulations of the helicity states of V
How are the observed decay distributions related to form factors
What have exp eriments told us ab out form factors How much do
form factors typically vary over the allowed kinematic range How do
the results compare with theoretical predictions
Given the presence of an unobserved neutrino how are semileptonic
decays identi ed
What are the best metho ds for determining jV j and jV j Which
cb ub
uncertainties dominate in each case exp erimental or theoretical
What can we learn from the semileptonic branching fraction as op
p osed to the semileptonic rate Are the values of the semileptonic
branching fractions in B and D decay understo o d
Review articles on semileptonic decays have b een written by Rich
latter article contains man and Burchat and by Neub ert The
extensive information on HQET
J D Richman
Dynamics of semileptonic decay
I will b egin with a qualitative picture of the dynamics of the semilep
tonic decay M X where M is a pseudoscalar meson a D or
B containing a heavy quark We will fo cus on decays in which the
hadronic system X is a single meson since such pro cesses dominate
the D and B semileptonic decay rates In semileptonic D decays the
known resonant exclusive mo des come fairly close to saturating the
inclusive semileptonic rate and multib o dy nonresonant nal states
have not been observed In B decays the two mo des B D and
B D account for ab out two thirds of the semileptonic rate Part
of the remainder is to due to D pro duction but there is still a little
ro om for nonresonant nal states
A powerful to ol for describing the dynamics of semileptonic decay
is the Dalitz plot which maps the probability for di erent kinematic
con gurations over the allow ed region of phase space Figure is
based on a Monte Carlo simulation of a sample of B D decays
which we have generated using HQET based form factor parameters
given by Neub ert These predicted values are quite similar to
those measured by CLEO this measurement is discussed belowin
the section on exclusive semileptonic decays In this plot each point
represents a single B D decay Because Dalitz plot variables
energies or squared masses are used phase space is uniform over
the plot so that a constant matrix element would give a uniformly
distributed set of points In this section we will analyze the physical
signi cance of di erent Dalitz plot regions and qualitatively explain
the pattern shown in Fig
For Dalitz plot variables wehavechosen E the energy of the lepton
measured in the rest frame of the initial meson M whose mass is also
lab eled M and q the variable mass squared of the W
q m p p P p M m ME
X X X
W
where P is the four momentum of M and E is the energy of X in the
X
M meson rest frame This result is simply an equation from two b o dy
decay kinematics M XW and it shows that E is linearly related
X
to q Traditionally form factors have been parametrized in terms of
q but in our discussion of HQET we will use an alternate form based
on four velo cities rather than four momenta The form factors are then
written in terms of
P E M m q p
X X
X
w v v
X X
M m m Mm
X X X
Heavy Quark Physics and CP Violation
12 ] 2 ) 8 2 GeV/c (
[ cos θ cosθ 4 2 =–1 = +1 q
0 0 0.5 1.0 1.5 2.0 2.5
3–95 E (GeV) 7713A16
Fig A Monte Carlo simulation of the Dalitz plot for the pro cess B D using
HQET based form factors from Neub ert Because Dalitz plot variables are used
equal sized areas in the allowed region corresp ond to equal amounts of available phase
space The form factors are largest at high q which increases the density of points
towards the top of the plot At a xed value of q the range in lepton energies from left
to right corresp onds to the variation of cos where is the p olar angle of the lepton in
the W rest frame from to see Fig The increase in density across the Dalitz
plot from left to right can b e traced to the cos distribution which is asymmetric due to
the V A coupling This coupling enhances the amplitude for the negative helicit y state
of the W relative to the p ositive helicity state Sp ecial cases o ccur at q where the W
max
or the D is unp olarized and at q where it is in a pure helicity zero state
This quantity is just the relativistic gamma factor of the daughter
meson in the rest frame of the parent meson and it has a minimum
value of one The literature contains a number of di erent names for
this quantity including and y
Twokey semileptonic decay con gurations are shown in Fig The
initial B meson which contains a b quark and a sp ectator quark q is
shown in Fig a while the con gurations for q q and q q
max min
are shown in Figs b and c
The maximum value of q o ccurs when E is minimized E m
X X X
the virtual W is as heavy and w E m Fig b At q
X X
max
as it can p ossibly be and there is no kinetic energy Both the W and
the daughter meson X are pro duced at rest in the parent meson rest
M m at zero recoil the that q frame Wehave seen ab ove
X max
J D Richman
b q
(a)
c l q ν 2 2 q = qmax (b)
ν c l q 2 2 q = qmin
(c)
Fig Kinematic con gurations for the semileptonic decayofaB meson a B meson
b efore decay b decay con guration for q q zero recoil where the form factors
max
are largest for pro ducing a D or D meson in the nal state and c con guration for
q q where the form factors are smallest
min
W therefore decays into a back to back lepton and neutrino each of
p
which has energy E q ignoring the lepton mass
The minimum value of q the lightest the system can be is
q m which is zero to a go o d approximation for electrons or
min
muons This value o ccurs whenever the lepton and the neutrino are
collinear Fig c The recoil energy of the hadronic system E is
X
maximized when q is minimized Eq so at q the hadronic
min
system X recoils fast in the opp osite direction The value of w corre
sp onding to q is
min
M m
X
w
max
Mm
X
Table lists the ranges in q and w foranumb er of imp ortant semilep
tonic decays For B D and B D the range is very mo dest
w For B the range is very large w
Heavy Quark Physics and CP Violation
Table
Ranges of q and w for semileptonic decays of charm and b ot
tom mesons The variable w is the relativistic factor for the
X
daughter meson X as measured in the rest frame of the par
ent meson Note that w is the factor at maximum recoil
max
velo city whichoccursatthemimimum value of q q
min
Decay q q GeV c w
max
max
D
D K
D
D K
B
B
B D
B D
and the pion b ecomes very relativistic We therefore exp ect a larger
variation of the form factors over phase space for this decay
For a given lepton energy E the allowed range of q is given by
m E
X
m q ME
E M
which de nes the b oundary of the Dalitz plot
Both q and E are exp erimentally accessible although it may not
be completely obvious how one measures q Broadly sp eaking there
are two p ossible approaches In an exclusive pro cess where X is a
particular hadronic nal state one can measure q by identifying X
or its decay pro ducts measuring the energy of X and transforming
the energy to the rest frame of the decaying particle M Equation
is then used to compute q The determination of the M rest frame
which is also needed to determine E cannot be taken for granted
b ecause there is an unobserved particle the neutrino In xed target
charm exp eriments q is determined only up to a quadratic ambiguity
b ecause the charm meson direction but not its energy is determined
by high resolution tracking detectors For B mesons pro duced in the
pro cess S B B the B is pro duced nearly at rest in the S
frame e machines the S In symmetric energy e
is at rest in the detector frame while in asymmetric energy machines
there is a boost as discussed in Sec
A second approach to measuring q can be used with hermetic de
J D Richman
tectors With su cient care the missing momentum vector of an event
can to a go o d approximation b e asso ciated with the neutrino Then
q can be calculated from the lepton and neutrino momenta This
approach su ers from p o orer resolution b oth because particles be
sides the neutrino such as K s may not be observed and because
L
in colliding b eam detectors the comp onent of the missing momentum
along the b eam direction is not always well measured Nevertheless
this pro cedure can work and in principle it can b e applied to inclusive
decays where the X system is not explicitly reconstructed
Let s now consider qualitatively how the Dalitz plot variables q and
E are related to the underlying physics of the decay Distributions
of these variables are determined by two e ects the dynamics of the
formation of the hadronic system X and the spin structure of the decay
We now analyze these e ects starting with the variable q
To compare the decay dynamics in di erent regions of q we refer
again to Fig Athighq when the lepton and the neutrino are pro
duced nearly back to back Fig b the daughter meson receives only
a very small momentum kick For pro cesses in which the daughter
quark mass accounts for most of the meson mass B D the
daughter quark itself also receives very little recoil kick at q A
max
heavy static source of color eld at the center of the meson is replaced
by a color source of a di erent avor but the color eld is not changed
Relativistic e ects which dep end on the mass of the heavy quark be
come negligible For example the color magnetic moment of the heavy
quark is prop ortional to m so only the color electric eld is im
Q
p ortant Thus the so called light degrees of freedom the sp ectator
quark and the gluons are least disturb ed at q These observations
max
are among the key ideas in HQET whichw e will discuss further in the
anaysis of exclusive semileptonic decays
In low q con gurations the daughter quark q recoiling against the
W receives a large kick and initially moves rapidly with resp ect to the
sp ectator quark q For the daughter and sp ectator quarks to form a
b ound state gluons must b e exchanged in order to transfer momentum
to the light degrees of freedom of the meson The larger the recoil the
stronger the suppression resulting from this e ect As a consequence
the q con guration is typically the least favorable for the forma
tion of a meson There are however imp ortant spin related e ects
the minimum kick This situation contrasts with that in a scattering pro cess where
o ccurs at q In a scattering pro cess this is however still the highest value of q
which is either negative or zero
Heavy Quark Physics and CP Violation
that can change this conclusion we will discuss these e ects later in
this section Theoretical calculations are usually di cult at low q
b ecause the hadronic system is highly disturb ed and it is at q
that one might exp ect signi cant pro duction of nonresonant nal states
to o ccur in analogy to QCD jets
Thus we have three key observations
For pro cesses in which both the initial and nal quarks are b oth
heavy compared with the strong interactions that bind the mesons
typi ed by the sp ectator quark and the gluons are almost
QCD
completely undisturb ed at q q At q calculations of the decay
max max
rate for b c have relatively small uncertainties to while
those for b u are still quite di cult
Even if the daughter quark is light the zero recoil con guration is
generally still the most favorable one for the formation of a low mass
meson
Decay mo des with a large accessible range in q are exp ected to
have a corresp ondingly large decrease in the amplitude as q decreases
from q to q This variation of the amplitude with q is
max min
parametrized by form factors
We can see the suppression of the rate as q decreases in Fig the
density of p oints is highest at the top of the Dalitz plot and b ecomes
signi cantly smaller toward the b ottom We note however that the
actual probability distribution of q do es not p eak at q but somewhat
max
b elow b ecause at q the amount of phase space goes to zero
max
We also need to consider the consequences of angular momentum
conserv ation which directly a ect the distributions of b oth q and
E These e ects are strongly dep endent on the spin of the daughter
meson The rst step is to show that the W behaves like a spin
particle From our discussion of helicity conservation in Sec we
know that due to the V A coupling the lepton and neutrino spins
p oint in the same direction in the W frame in the limit that the lepton
has a zero mass These e ects give an angular momentum pro jection
of J along the direction in a W decay or J
z z
along the direction in a W decay Because there is one unit
of spin angular momentum along the decay axis and orbital angular
momentum cannot cancel it L r p the W cannot behave like a
spin particle even though it is virtual This argument holds in the
limit that we can neglect the mass of the charged lepton More formally
we will see that certain form factors cannot signi cantly a ect the decay
We can therefore unless the charged lepton is heavy as for
imagine that the rst step of semileptonic decay is M XW where
J D Richman
W is a spin particle
Now if b oth M and X are pseudoscalar particles B D B
D K D then the decay cannot conserve angular
momentum unless there is one unit of orbital angular momentum to
cancel the spin angular momentum of the W The decays D KW
or B DW are therefore P wave pro cesses whereas D K W can
o ccur in an S P or D wave As a consequence the decay rate when
X is a pseudoscalar meson contains a factor jp j one power of jp j
X X
from phase space two more from the amplitude squared
As can be seen from Eq jp j at q q so the factor
X
max
suppresses the rate at high q where the large form factor values would
ordinarily lead to a large decay rate Alternatively one can easily see
that suchadecaymust b e forbidden at q b ecause the back to back
max
lepton neutrino system has one unit of angular momentum along its
line of ight and this angular momentum cannot be cancelled by the
daughter pseudoscalar meson In terms of w we have
p
w jp j m
X X
so that the jp j factor greatly increases the probability for large values
X
of w If Fig were made for B D rather than for B D
the density of p oints in the upp er region would b e signi cantly reduced
The dramatic result of the P wave decay can be seen in the CLEO
II data for B D Figure a shows the strong suppression of
the decay rate for small D momenta resulting from the p factor
D
Figure b shows the b ehavior of the decay amplitude as a function of w
after the P wave and phase space factors are divided out The mo dest
increase in the rate towards small w are due to the form factor which
describ es the nonp erturbative QCD e ects discussed in connection with
Fig In this decay the form factor e ects are nearly overwhelmed
by e ects of angular momentum conservation
Finally we note that the q distribution will b e di erent if the nal
state meson has orbital excitation under such circumstances the typical
value of q can b e pushed signi cantly lower
Having discussed the physics that controls the q distribution we
turn now to the factors that in uence the distribution of E We can
see directly from the Dalitz plot b oundary in Fig that if phase space
ould be peaked were p opulated uniformly the lepton energy sp ectrum w
toward the high end However the lepton energy sp ectrum is strongly
a ected by three asp ects of the dynamics the V A coupling
the quantum numb ers of the particle X and the distribution in q
We now consider each of these in turn
Heavy Quark Physics and CP Violation
3420597-002 2.5 ( a ) 2.0
/ 0.05) 1.5 I 1 1.0
0.5
d (ns / dw 0
( b ) 0.10
I I 0.08 cb V I I 0.06
F(w) 0.04
0.02 0 1.0 1.1 1.2 1.3 1.4 1.5 1.6
w
Fig CLEO II analysis of B D a the distribution of the rate d dw of
w E m which ranges from zero recoil to ab out maximum recoil The
D D
points with error bars are the background subtracted e ciency corrected data and the
histogram is the b est t assuming a linear dep endence of the form factor on w The P
wave nature of the decay suppresses the rate at low recoil b the w dep endence of the
amplitude after the P wave dep endence has b een divided out the remaining w dep endence
is due to the form factor whichfavors decay con gurations at lowvalues of w high q
The solid line is the b est t the dashed lines show the slop es allowed within the statistical
errors on the t The dotted line is the sup erimp osed curve from the corresp onding t in
the B D analysis
A direct consequence of the V A coupling is that the charged lep
ton and the neutrino share the available energy di erently for charm
and b ottom decays The pro cesses b c and b u pro duce c
and u quarks that are predominantly helicity in asso ciation
with a charged lepton that is almost purely helicity The
decays c s and c d also pro duce predominantly
s and d quarks but they are in asso ciation with a charged lepton that
As a result the collinear con guration in is almost purely
which the charged lepton recoils against the daughter quark and the
neutrino the con guration leading to the highest lepton energy is
allowed for b quark decays but forbidden by angular momentum con
servation for c quark decays Thus in the case of b or b decay the
lepton energy sp ectrum p eaks at a higher energy than the neutrino
sp ectrum the reverse is true for c decays
ignores the quantum numb ers of the The argument we have given
meson X but these are quite imp ortant in determining the lepton
energy sp ectrum in an exclusive decay In particular the V A e ect
J D Richman
describ ed ab ove is imp ortantwhen X is a spin particle but is masked
when X has spin These e ects are b est understo o d by relating via a
Lorentz transformation the distribution of E in the rest frame of the
parent meson M to the angular distribution of the charged lepton
in the W rest frame We de ne the angle Fig as the polar
l M θ W* l X zz
ν
Fig The p olar angle is de ned in the rest frame of the virtual W denoted here
as W in which the charged lepton and the neutrino are back to back The angle is
measured with resp ect to the axis z pointed opp osite to the momentum vector of X the
daughter meson Due to the Lorentz b o ost b etween the W and M rest frames leptons
with small values of have higher energy in the M rest frame than leptons with large
values of at xedq
angle of the lepton in the W rest frame with resp ect to the direction
of the W momentum vector in the M rest frame In the M rest frame
p p Ignoring the charged lepton mass its energy in the W
W X
p
W
q However leptons going forward rest frame is simply E
along p in the W rest frame are given a higher energy in the M
W
rest frame due to the b o ost than those going backward From the
Lorentz transformation to the M rest frame
max min max min
E E E E E cos
where
max min
M q m M jp j E
X
X
M
and
s
M q m
X
m jp j
X X
M
Heavy Quark Physics and CP Violation
As q increases the lepton energy tends to increase as well b ecause
M q m
X
max min
E E
M
The range of lepton energies observed in the M rest frame however
max min
decreases as q increases b ecause E E jp j jp j de
X W
creases as the b o ost b ecomes smaller The width of the Dalitz plot at
a xed q is just the two b o dy phase space factor for B XW q
At q q the W and the X system are each at rest in the M rest
max
frame and E is the same for all angles Both of these features the
max min
increase in E E withq and the diminishing range in lepton
energies are simply phase space e ects and can b e seen directly from
the shap e of the Dalitz plot b oundary in Fig
It is clear that to predict the observed lepton energy sp ectrum one
must understand the physics underlying the distribution of cos I will
digress to showhow easy it is to calculate angular distributions for two
body decays we will then apply the results to the pro cess W
Consider the decay A B C where J is the spin of A M is
J M
B C
its spin pro jection along an arbitrary z axis and and are the
B C
helicities of B and C resp ectively We consider nal states in which B
and C have de nite and opp osite linear momenta which we sp ecify
in the rest frame of A as well as the de nite helicities and
B C
In the helicity formalism such states are called two particle
plane wave helicity states The amplitude for this pro cess is
M hp p p p jU jA i
B f B C f C J M
op erator that propagates the initial state where the op erator U is the
though the interaction to the nal state particles Because particles B
and C have equal and opp osite momenta in the rest frame of A we
can characterize the nal state by the direction n of the decay
axis with resp ect to the z axis spin quantization axis of A by the
magnitude of either particle s momentum and by the helicities and
B
Thus suppressing jp j b ecause it is xed
C f
M h jU jA i
B C J M
The probability distribution dN d cos d is prop ortional to jMj It
is imp ortantto plot angular distributions as a function of cos and
rather than and since phase space is prop ortional to d d cos d
The use of will not give a wrong answer but it will intro duce a
physically meaningless Jacobian that will make the plots more di cult
to interpret
J D Richman
The key idea of the helicity formalism is that rotational invariance
of the helicities s p allows one to de ne a set of two particle
basis states jj m i that have de nite total angular momentum
B C
j angular momentum pro jection m and helicities and Wecan
B C
then exploit conservation of angular momentum by inserting a complete
set of these states into Eq
X
A i M h jj m ihj m jU j
J M B C B C B C
j m
X
h jj m i A
B C B C jJ mM
B C
j m
h jJ M iA
B C B C
B C
J
constant D A
M
B C
J
where and D are the rotation op erators
B C
M
J i M J
D e d
M M
The angle is the p olar angle of p with resp ect to the spin quantiza
B
tion axis of A is the azimuthal angle ab out this axis The so called
helicity amplitudes A can be di erent for each set of p ossible he
B C
licities but they contain no angular dep endence The last step in the
derivation is where most of the work is the original derivation is in
the pap er by Jacob and Wick All of the dep endence is contained in
phase factors which for a single non sequential two b o dy decay dis
app ear when the square of the amplitude is computed If the helicities
are sp eci ed in the nal state there is no coherence between their am
J
plitudes The functions d are listed in a table of the Particle
m m
Data Bo ok on the same page as the Clebsch Gordon co e cients A
j j j
m m
d d useful identityisd
m m m m m m
We will need the following d functions
cos cos
d d
sin
p
d
Let s apply these results to semileptonic decay To do this we need
to sp ecify the quantization axis for the spin of the W The W cor
resp onds to the parent particle A in the discussion ab ove This axis
is taken to be the momentum direction of the W in the B rest frame
or equivalently the direction opp osite to that of the daughter hadron
The quantum number M in the formulas ab ove then corresp onds to
Heavy Quark Physics and CP Violation
the helicity of the W and the p olar angle of the lepton is measured
in the W frame with resp ect to the direction of the W momentum in
the B frame
The dN d cos distribution is connected to b oth the V A couplings
and to the quantum numb ers of X In b or c quark semileptonic decay
the daughter quark helicity is predominantly If this quark
combines with the sp ectator quark to form a pseudoscalar meson as
K the helicity information in B D B or D
is lost since the helicity of the meson must be zero Because the
initial meson has spin zero angular momentum conservation along the
decay axis forces the W to have helicity zero as well This means
that M in Eq The helicity di erence for the system is
We therefore use d which gives
dN
sin
d cos
This result is mo del indep endent and holds for all values of q If
you think you have discovered a new signal of the form P P
where P and P are pseudoscalars it is always a good idea to check
the angular distribution of the lepton in the W rest frame
If however the daughter quark and the sp ectator form a spin
meson as in D K B D orB then the helicity
information of the daughter quark is not lost It is manifested as a
higher probability for the vector meson to have helicity than
Roughly sp eaking a rapidly recoiling daughter
quark can combine with the sp ectator quark to form a or
meson as shown in Fig This description is not appropriate when
the daughter quark is nonrelativistic or when the W is massless as
discussed b elow The probabilities for di erentvector meson helicities
also apply to the W b ecause to conserve angular momentum along the
decay axis the helicities of the W and the vector meson must be the
same
The predominance of over a ects the lepton
W W
sp ectrum di erently for B and D decays In B decay the pro cess
W pro duces a charged lepton and a
antineutrino For the angular distribution of the lepton is
W
dN d cos jd j cos
so the leptons tend to go forward in the W frame and are boosted to
higher energy in the B frame For the angular distribution W
J D Richman
l bc
W* c q
ν
Fig In B semileptonic decay the V A coupling at the b c or b u vertex
pro duces a c quark that is predominantly helicity In the simple mo del shown
here the helicity of the meson X is then determined by whether the c quark combines with
a sp ectator quark that has or If X is a spin zero meson only
sp ectator quarks can contribute If X has spin b oth helicities of the sp ectator quark
contribute leading to X helicities of and but not It is easy to see
that this V A e ect combined with overall angular momentum conservation results in
a harder energy sp ectrum for the charged lepton than for the neutrino as observed in the
rest frame of M
is
dN d cos j j cos d
so the leptons tend to go backwards in the W frame and are b o osted
to lower energy in the B frame Since tends to predominate
W
over in B decays to a spin particle D the lepton
W
sp ectrum is harder than the neutrino sp ectrum If we switched from
b oth the daughter quark and the charged a B to a B the helicities of
lepton would reverse and the angular distribution would b e the same
For D decay the c s or c d transition again pro duces a hadron
with more than but the lepton is p ositively charged
and has Again to conserve angular momentum the W
spin pro jection must cancel that of the daughter hadron so it has
more than Since the W decay pro duces a
charged lepton and neutrino the lepton angular
distribution for is
W
dN d cos jd j cos
Heavy Quark Physics and CP Violation
which leads to a softer b o osted energy sp ectrum for the lepton than
the neutrino
Finally the lepton energy sp ectrum is a ected bytheq distribution
If q is forced to b e high by the b ehavior of the form factors E will also
tend to be large see Eq Conversely in decays in which X is a
spin particle the P wave e ect suppresses high q decays softening
the lepton energy sp ectrum
The simpli ed arguments we have made do not hold at q q
max
or at q At q the daughter vector meson is at rest Its
max
helicity is therefore unde ned and b oth the vector meson and the W
are unp olarized As a result the cos distribution b ecomes uniform at
high q At small values of q the lepton and neutrino b ecome parallel
in the M rest frame and their combined spin pro jection along their
direction of motion is zero The helicity comp onents are absent
and there is no lepton forward backward asymmetry in the W frame
The helicityofthevector meson must also b e zero in this con guration
We now summarize our understanding of the Dalitz plot for B
D in Fig It is useful to refer also to Fig which shows
the contributions to the rate from the di erent helicities as a function
of w Near the top of the Dalitz plot where q is large the D is
moving very slowly and is nearly unp olarized and are
present in approximately equal amounts The distribution of cos is
then uniform b ecause the W is also unp olarized The form factors are
largest in this region accounting for the high density of points As
q decreases the comp onent of the D b egins to dominate
over the comp onent which explains the excess of points on
the right side of the Dalitz plot compared with the left side At the
lowest value of q the charged lepton and the neutrino are parallel in
the lab frame leading to maximum D recoil and b oth the D and W
are forced into a pure state There is no asymmetry in the cos
distribution at this edge of phase space is distribution at q its
dN d cos sin This e ect can b e seen in Fig in the depletion
of points at high and low lepton energies for small values of q
Heavy quark e ective theory and semileptonic decays
In the last few years a new theoretical approach known as heavy quark
e ective theory HQET has emerged for analyzing so called heavy
light mesons mesons containing one heavy and one light quark as
well as baryons containing a heavy quark and two light quarks Many
authors have contributed to the developmentofHQET whose history
J D Richman
Fig Contributions to the rate in B D from di erent D or W helicities
while maximum recoil o ccurs at w The zero recoil con guration is at w q
max
q Atintermediate values of w the V A coupling pro duces more rate into the
state than into
is traced in the extensive review by Neub ert A number of the
separate ideas underlying HQET emerged over a long p erio d and can
be found in the pap ers of among others Shuryak Nussinov and
Wetzel and Voloshin and Shifman Two pap ers by Isgur
and Wise played a ma jor role in synthesizing and extending
this development and they are among the most frequently cited pa
p ers in particle physics over the last few years This work led to a rapid
expansion in the study of HQET among the key pap ers are those of
Eichten and Hill Falk et al Grinstein and Georgi
Many reviews of this sub ject are also available such as those by
Wise Bigi Shifman and Uraltsev Grinstein Man
nel and Neub ert If you p erform a literature searc h using
SPIRES http www spires slac stanford edu FIND hep or with the
Los Alamos preprint server http xxx lanl gov you will b e rewarded
with an avalanche of pap ers by these and many other authors
A simple argument indicates that within a hadron containing a
heavy quark the heavy quark moves nonrelativistically The momen
tum of the heavy quark p must balance that of the light constituents Q
Heavy Quark Physics and CP Violation
of the hadron p
light
jp j jp j
Q light had QCD
where GeV is a typical hadronic interaction scale
had
In the proton for example we have m MeV In
had P
heavy light hadrons the typical momentum transfer b etween the light
constituents and the heavy quark is but the asso ciated velocity
had
transfer to the heavy quark is v ery small
jp j
Q had
jv j
Q
m m
Q Q
so that in the limit m the heavy quark behaves essentially
Q had
as a stationary source of a color eld The velo city of the hadron and
the heavy quark b ecome the same Furthermore the heavy quark s
spin whichinteracts with the system through a color magnetic moment
prop ortional to m also decouples from the dynamics in this limit
Q
In the heavy quark limit the actual value of the mass of the heavy
quark therefore b ecomes irrelevant The con guration of the light con
stituents of a heavy light hadron is not a ected by the replacement of
aheavy quark Q v s with another heavy quark Q v s where the
Q Q
heavy quarks have the same four velo city v but di erent avors Q
Q
and Q and di erent spins s and s Thus instead of describing the
form factor variation of the semileptonic decay M X in terms of
the square of the four momentum transfer q HQET calculations use
the square of the four velo city transfer v v v v where
v and v are the four velo cities p m of the initial and nal hadrons
We have already seen that in the rest frame of the initial hadron
v v has a simple physical interpretation
q
v v
X
X
where is the velo city of the nal state hadron Eq The quantity
X
v v which is often called w or y in the literature is dimensionless
and ranges from at minimum recoil to the maximum value
X
M m Mm It is linearly related to q by
X
X
M m q
X
w v v
Mm
X
or
q q
max
v v
Mm X
J D Richman
Since the typical mass scale for the light constituents within the hadron
is the square of the four momentum transferred to these con
had
stituents during the decaymust b e v v indep endent of the
had
heavy quark mass
The heavy quark symmetry limit provides a go o d description of a
real physical system if the light constituents have su ciently small
momenta that they cannot prob e distance scales of the order m
Q
Since the momentum transfer to the light constituents has the typical
scale v v this condition is equivalent to the statement v
had
v m If this condition is violated in a certain region of
Q had
phase space then the quark under consideration cannot be regarded
as heavy
In practice the heavy quark symmetry limit is the starting point
for an expansion in the general framework of HQET In HQET the
prop erties and decays of hadrons containing a heavy quark are analyzed
in terms of a systematic expansion in the variable E m where E
Q
can be due to a number of QCD e ects such as the kinetic energy
of the heavy quark or the chromomagnetic interaction energy At
each order in m there is an asso ciated p erturbative expansion in
Q
the strong coupling constant In contrast to calculations based
s
on hadron mo dels the HQET expansion is derived directly from the
fundamental theory of QCD Although the terms in the expansion can
be di cult to evaluate the systematic and rigorous nature of HQET
means that uncertainties are easier to identify and estimate than those
for calculations based on hadron mo dels
For the expansion to b e useful the higher order terms must b e small
at least in the regions of phase space of interest The decays b est suited
to treatment using HQET inv olve b c transitions since b oth the
initial and nal state hadrons contain a heavy quark Examples of
such decays are B D B D and Note
b c
that compared with the strange quark cannot be regarded as
had
heavy since m GeV c Thus HQET is not applicable to charm
s
semileptonic decays
Inclusive semileptonic decay and jV j
cb
In the inclusive approach to semileptonic decays one considers the sum
over all p ossible nal state hadrons ignoring the detailed breakdown
among the individual decay mo des that contribute to the semileptonic
rate Exp erimentally it is necessary to observe only the lepton elim
inating the di culty of reconstructing what are often very complex
Heavy Quark Physics and CP Violation
decay sequences of the daughter hadrons Theoretical calculations of
inclusive prop erties have certain advantages of simplicityaswell since
calculations in which the heavy quark is assumed to decay as a free par
ticle with the light quark acting merely as a sp ectator provide a go o d
starting point for predictions Recently there has b een great interest
in re ning calculations for b hadron semileptonic decays using heavy
quark expansions It has b een shown that sp ectator mo del predictions
corresp ond to the lowest order term in such expansions
The inclusive semileptonic branching fraction is de ned by
P
M X
X
B
SL
M all
where is either an electron or a muon but not b oth For precise
measurements it may be necessary to correct for the slightly smaller
phase space available in the muon mo de or for radiative e ects which
are larger in the electron mo de Because the lepton mass is large
the case is treated separately Often the branching fraction
measured by exp eriments is an average over more than one sp ecies of
heavy hadron M b ecause measurements of the lepton alone are not
su cient to distinguish b etween di erenttyp es of hadrons carrying the
same heavy quark
Inclusive measurements of leptons can provide a remarkable amount
of information
B can b e used to determine what fraction of the semileptonic rate
SL
can b e accounted for byknown exclusivechannels and thus help to de
termine whether there are additional exclusive mo des to b e discovered
An imp ortant application of B is that it can be used to compute
SL
the total semileptonic decay width B where is the
SL SL M M
measured lifetime of the hadron M In B decays one can compare the
inclusive semileptonic decay width to theoretical calculations to deter
mine jV j The measurement of jV j using this metho d has attracted
cb cb
muchinterest b ecause B and have b een measured with very go o d
SL B
precision and theorists believe that the theoretical uncertainties can
also b e made very small
The lepton energy sp ectrum contains imp ortant information ab out
which mo des contribute For example the lepton energy sp ectrum
from the set of decays B X where X is a non charm hadron
u u
extends to a higher energy than the sp ectrum from decays B X
c
where X is a charm hadron One can therefore measure the semilep
c
t region of the lepton energy sp ectrum to tonic rate in the endp oin
measure jV j ub
J D Richman
In B decays it is of interest to know if there are any large unexp ected
contributions to the hadronic rate for example from gluonic penguin
pro cesses We can use B to calculate the ratio b etween the hadronic
SL
rate and the semileptonic rate Since the semileptonic rate can be
calculated with go o d precision B e ectively provides a constrainton
SL
the hadronic rate
Separate measurements of B for Qu and Qd mesons allow one to
SL
extract the ratio of the lifetimes of the charged and neutral mesons
For example in D meson decays
D all D X D all
D all D X D all
B B
SL SL
where we have assumed that the charged and neutral D mesons have
the same semileptonic partial widths This assumption is exp ected
to be very go o d since the two mesons di er only in the isospin of the
light quark There will b e a very small di erence b ecause for example
D is not exactly the same as D
Decays to leptons are also used simply as a means to determine the
avor of the decaying heavy quark In measurements of CP violation
and mixing one uses the fact that b while b
In b hadron decay the challenge for inclusive measurements is to de
termine what part of the observed lepton momentum sp ectrum is due
to leptons from b decay primary leptons and what part is due to lep
tons from charm decay secondary leptons or other sources misidenti
ed hadrons photon conversions J decays etc A standard tech
nique which has now been surpassed is to t the observed lepton
momentum sp ectrum to a sum of the shap es exp ected for primary and
secondary decays after subtracting backgrounds from other sources
Thus a large part of the e ort and uncertainty in the analysis is in
the determination of these shap es
Figure shows the electron and muon data from CLEO I I
which has the largest event sample The muon sp ectrum cuts o b elow
GeV c due to the iron absorb er in frontofthemuon detectors but
the electron sp ectrum is measured down to GeV c Backgrounds
from continuum pro cesses photon conversions Dalitz
decays and B X decays have been subtracted The electron
sp ectrum has b een radiatively corrected according to the prescription
of Atwood and Marciano so the electron and muon data can be
directly compared
It is clear from Fig that it is not trivial to extract the contribution
Heavy Quark Physics and CP Violation
0.20
e
µ
-1 0.15 fit
b→clν
b→ulν
b→c→ylν 0.10 (4S)) x dN/dp (GeV/c) Υ 1/2N( 0.05
0.00 0 1 2 3
Lepton Momentum (GeV/c)
Fig CLEO I I inclusive single lepton analysis Although the sp ectrum can b e measured
extremely well the di culty is to extract the contributions from primary dashed curve
and secondary dot dashed curve leptons The uncertainty in the shap es of these sp ectra
pro duces the dominant systematic error in the measurement In the t shown ab ove the
shap e of the primary lepton sp ectrum is obtained from the ACCMM inclusivemodel An
alternative metho d using dileptons avoids this problem
to the sp ectrum from semileptonic B decays In fact the main limi
tation of using the inclusive single lepton sp ectrum to measure B is
SL
that it is necessary to t the sp ectrum to shap es that describ e the con
tributions from primary b and charge conjugate and secondary
b c and charge conjugate contributions The primary lepton
sp ectrum is usually describ ed by mo dels and the uncertainty in the
predicted shap es pro duces the dominant uncertainty in the measure
men t For this reason we turn to an alternative measurementtechnique
for measuring B
SL
The ARGUS collab oration intro duced a metho d based on
charge correlations in dilepton events that allows one to separate the
contributions of primary and secondary leptons without relying on
mo del dep endent shap es This technique which has now been used
byCLEO aswell substantially reduces the need for mo dels in the
determination of B
SL
The rst step is to require that one lepton the tagging lepton
have high momentum so that it must nearly always be primary For
with p GeV c only ab out of these leptons are example
secondary The lepton sp ectrum extracted is that for the other lepton
in the event for which no such cut is applied
J D Richman
Assuming that the tagging lepton is always primary a small cor
rection for secondary tags must b e made there are three p ossibilities
for the other lepton These p ossibilities and the corresp onding lepton
charge correlations are assuming no B B mixing
the other lepton is primary and is pro duced in the decay of the other
B meson resulting in opp osite sign leptons
the other lepton is secondary and is pro duced in the decay chain of
the other B meson resulting in same sign leptons or or
the other lepton is secondary and is pro duced in the decay chain
of the same B meson in the event resulting in opp osite sign leptons
Thus if one can eliminate events corresp onding to the third scenario
then a lepton with charge opp osite to that of the tag must b e primary
whereas a lepton with the same charge as the tag must be secondary
This discussion ignores mixing but the B B mixing rate is well mea
sured and it is not di cult to correct for this e ect
If primary and secondary leptons are from the decay chain of the
same B meson the third case listed ab ove there is a strong angu
lar correlation resulting from momentum conservation such that they
tend to b e in opp osite hemispheres In contrast leptons from di erent
B mesons have uncorrelated angular distributions since the two B s
are pro duced essentially at rest relative to each other By requiring
b oth leptons to be in the same hemisphere events in which the two
leptons come from the decay chain of a single B meson are e ectively
removed
With these analysis cuts a lepton whose charge is opp osite to that
of the tagging lepton must b e primary while one with the same charge
as the tagging lepton must be secondary up to the mixing correc
tion One can therefore separately measure the number of primary
and secondary leptons in each momentum bin Figure shows the
primary and secondary electron momentum sp ectra obtained from the
CLEO II analysis There is a lower cuto in the electron identi ca
tion e ciency however and a small extrap olation based on mo dels
is required to obtain the total semileptonic rate In the CLEO I I mea
surement the minimum momentum in the measurementis GeV c
and the measured branching fraction ab ove this cuto is mo del inde
p endent giving B B X p GeV c
The extrap olation to zero momentum amounts to only of
the semileptonic rate The central value of this extrap olation fraction
is based on the average of the ACCMM and the ISGW predictions
and the error is based on the di erence between these two mo dels
Heavy Quark Physics and CP Violation
Table
Measurements of the inclusive semilep
tonic branching fraction The CLEO I I
and ARGUS measurements are averaged
over the B and B while the LEP
measurements are averaged over the b
hadrons pro duced in Z bb Full ref
erences to these measurements are given
in Ref
Exp eriment B b q
ARGUS
CLEO I I
S avg
ALEPH
DELPHI
L
OPAL
Z avg
The ISGW is a mo di ed version of the ISGW mo del in which the
B D branching fraction is allowed to oat
Table lists the values of B obtained from the ARGUS CLEO
SL
II dilepton analyses as well as those measured at LEP In computing
the LEP average I assumed a common systematic of ab
solute The discrepancy between S and LEP measurements is
not very large but the LEP results have always been systematically
somewhat higher even though the presence of b baryons should very
slightly decrease the semileptonic branching fraction
We now show how jV j is extracted from the semileptonic rate fol
cb
lowing the review of Gibb ons The analysis assumes that the av
erage B lifetimes at the S are h i ps while the
B
average b hadron lifetime at the Z is h i ps In the
b
m expansion
Q
a m
s np s
c
O a
s
m m m m
b b b b
where the a term parametrizes p erturbative QCD corrections a m
np
b
parametrizes non p erturbative corrections which are small and
m G m
c F b
jV j z
cb
m b
J D Richman
The function z x is the phase space factor
z x x x x x ln x
The calculations of Ball et al and Shifman et al give
jV j ps Ball et al
cb
jV j ps Shifman et al
cb
Combined jV j ps
cb
The combined value is the average of the two predictions but the error
is the larger of the two stated errors which are essentially the same
Next we correct the measurements of B by subtracting a small
SL
amount b u transitions An inclusive prediction by Rosner
gives
V B B X
ub u
to
B B X V
c cb
where the exp erimental error has been included The nal value from
the S measurements is
jV j
cb
where the rst error is exp erimental and the second theoretical Note
that the theoretical uncertainty in the rate translates into a
alue from the uncertainty in jV j an impressive achievement The v
cb
LEP exp eriments is
jV j
cb
This value is consistent with that obtained at the S and its errors
are comparable It is not clear to me however to what extent mo del
dep endent systematic errors are included in the LEP results
Lepton endpoint region in semileptonic B decay and jV j
ub
The determination of jV j is one of the most imp ortant and challenging
ub
measurements in B physics The curve in Fig representing the
b u rate shows clearly how little these pro cesses p erturb the
total sp ectrum below the endp oint region As we have just seen in
Eq over all lepton energies simple free quark mo dels predict
that b u b c The key to measuring jV j ub
Heavy Quark Physics and CP Violation
0970995-001 0.15 1 -
0.10
0.05 dB / dp (0.1 GeV c)
0 0 123 p (GeV / c)
e
Fig CLEO I I lepton momentum sp ectrum electrons from B decays based on the
dilepton charge correlation analysis The solid p oint with error bars are the primary lepton
sp ectrum B Xe the op en circles with error bars are the secondary lepton sp ectrum
B X Ye The curves show the b est t to the mo di ed ISGW mo del whichhas
c
B D
is to take advantage of the large lepton momenta made accessible by
the small mass of the daughter u quark By working in the region at
and beyond the lepton momentum sp ectrum endp oint for B X
c
pro cesses one gains enormously in sensitivity to B X decays
u
Although the advantages of working in this endp oint region
p GeV c are decisive there are also disadvantages The ma jor
di culty is the need to convert the measured rate for this tiny p ortion
of phase space into a value of jV j This calculation can b e p erformed
ub
using either inclusive or exclusive mo dels but b oth have substantial
uncertainties
We turn now to the measurement of the rate in the endp oint region
Although the measurement is prop erly describ ed as inclusive the
analysis is rather di erent from that of the inclusive lepton sp ectrum
e The reason is that continuum pro cesses pro duce high describ ed ab ov
momentum leptons which constitute an enormous background unless
suppressed by kinematic cuts The signal e ciency of these cuts is
much more sensitive to the shap es of kinematic distributions esp ecially
to that of q than the very lo ose cuts used in the analysis of the
inclusive lepton sp ectrum This sensitivity intro duces another source
of mo del dep endence into the results beyond the overall scale
thy
The primary characteristic used to remove continuum events is
J D Richman
event top ology continuum events are usually much more jet like than
S B B events which are quite spherical In the CLEO II mea
surement which has the highest statistics the event shap e is de
scrib ed quantitatively using the variable R H H where the H are
i
Fox Wolfram moments The R variable ranges from completely
spherical to completely jetlike for the endp oint analysis CLEO re
quires R Even with continuum suppression cuts a signi cant
uum background remains Fortunately it can be directly mea contin
sured by running just below the S resonance and p erforming the
same analysis
Figure shows the CLEO lepton momentum sp ectrum including
b oth electrons and muons in the endp oint region In the upp er plot for
which the stricter cuts were used the b u yields in the two bins
are events p GeV c and
events p GeV c with a total of events
Prior to the observation of the exclusive B and B
signals these yields and those from a similar ARGUS analysis were
the only evidence that jV j is nonzero The corresp onding yields with
ub
the lo oser less mo del dep endent cuts are events and
events The data sample used for this analysis contains
S B B events so a substantial improvement will only
b e p ossible when the full CLEO data sample million events is used
The B X signal is normalized to the B X rate in
u c
the endp oint region Thus the analysis determines jV V j For the
ub cb
ACCMM inclusive mo del the value is jV V j while
ub cb
for the ISGW mo del sum of exclusive mo des the value is jV V j
ub cb
We will compare these results with those of the exclusive
measurements in the following section
Form factors and kinematic distributions for exclusive semilep
tonic decay HQET predictions and the Isgur Wise function
In this section we examine the connections between form factors and
observable kinematic distributions in exclusive semileptonic decays Al
though the formalism may seem complicated at rst the connections
turn out to be fairly simple We also write the form factors and de
cay distributions using the standard HQET formalism which while
the completely general allows one to easily identify the behavior in
heavy quark symmetry limit The relationship between various ob
servable form factors and the Isgur Wise function is also discussed All
Heavy Quark Physics and CP Violation
120
(a) 80
40
0
(b) Events/(50 MeV/c) 250
0 2.00 2.25 2.50 2.75 3.00
3–95 Lepton Momentum (GeV/c) 7713A19
Fig Inclusive end p oint sp ectrum in B decays from CLEO I I The upp er plot a is
from the analysis with tighter cuts the lower plot b is from the analysis with lo oser cuts
E ciencies in the latter analysis have less sensitivity to mo dels but the backgrounds are
larger In b oth plots the solid p oints with error bars are the data taken at the S
while the op en circles with error bars represent the continuum background level whichis
measured using data taken b elow the S The histogram shows the total background
prediction which includes b oth continuum and B X The excess of observed events
c
over this total background is interpreted as evidence for B X decays
u
of these ideas will b e used in Sec where we present the results of
measurements of exclusive semileptonic decays
The matrix element for the semileptonic decay M X can
Qq q q
be written as the pro duct ofahadronicand a leptonic current
ig
p
M M X hX jq V QjM i
Qq q q q q q Q Qq
ig
p
P q u v
where the op erator Q annihilates the quark Q or creates Q and the
W propagator is given by
i g q q M g
W
P q i
q M M
W W
The last expression for the propagator is appropriate when the energies
are much less than M We have used the form of the CKM element
W
V appropriate to the case in whichaW is emitted e g V or V
q Q cb ub
if a W were emitted as in c s or c d the form of the CKM
J D Richman
element is V e g V or V but the absolute magnitude is taken
Qq cs cd
in the end in any case We therefore have the phenomenological form
for the matrix element
G
F
p
M M X i V L H
Qq q q q Q
p
where G g M The leptonic current is exactly known
F
W
L u v
and the hadronic current is given by
H hX jq QjM i
q q Qq
We can use Lorentz invariance to construct the hadronic cur
rent from the available four vectors which are momenta and spin
p olarization v ectors The Lorentz vector or axial vector quantities thus
formed haveLorentz invariant co e cients form factors that are func
tions of q We will consider two main classes of exclusive semileptonic
decay P P where b oth P and P are pseudoscalar particles and
P V where V isavector particle
In the case of a P Q q P q q decay there are only two inde
p endentfourvectors whichwe can taketobep p and q p p For
these quantum numb ers the hadronic current H has no contribution
from the matrix element of the axial vector current as discussed in the
app endix and can be written
M m
P
hP p jV jP p i F q p p q
q
M m
P
q F q
q
where V q Q and F F so there is no singular b ehavior
at q The form factors F q and F q can b e asso ciated with the
P P
exchange of particles with quantum numb ers J and J
resp ectively Another common way to write the current is
hP p jV jP p i f q p p f q p p
where f q F q and
q
F q f q f q
M m
P
In practice these expressions for the currents simplify b ecause the
terms prop ortional to q are nearly always negligible b oth here and for
Heavy Quark Physics and CP Violation
the case P V The reason is that in the limit m q L
where L is the lepton current Thus P P is to a very good
approximation describ ed by only one form factor F q when e
or
hP p jV jP p i F q p p
The di erential decay rate for P P can now be calculated
giving
d G jV j p
q Q
F P
jf q j
dq
where Q and q are the initial and nal state quarks in the underlying
transition Q q The momentum p is the magnitude of the
P
three momentum of the nal state meson P in the rest frame of P It
is a function of q see Eq In fact the dominant q dep endence
usually arises not from the fallo of the form factor as q decreases
q We have but from the p term which enhances the rate at low
P
integrated over all decay angles to obtain Eq as we have seen in
Eq however the lepton angular distribution in the W frame is
not isotropic but is dN d cos sin The leptons tend to come
out p erp endicular to the W direction
Many di erent metho ds have b een applied to the di cult problem
of calculating the form factors Quark mo del calculations estimate me
son wave functions and use them to compute the matrix elements that
app ear in the hadronic currents These integrals are p erformed by an
alyzing the decay at a particular value of q either q orq q
max
One p ersp ective is that the hadronic system is least disturb ed at high
q so q is where the integrals are most naturally evaluated His
max
torically however the convention in charm decays has been to sp ecify
the form factors at q The variation of the form factors with q is
determined as a separate step in the calculation In fact this variation
is usually assumed to have a very simple form Because the physics
b eing describ ed is nonp erturbative none of these phenomenological
forms should be taken to o seriously at a fundamental level although
do reasonably well in describing the data One approach they often
used in the WSB and KS mo dels is called nearest p ole dom
inance which has its origin in vector dominance ideas Here the q
dep endence of a form factor f is assumed to have the form
i
f
i
f q
i n
q
m p ole
J D Richman
where n is an integer usually one for mesons The pole mass m
pole
is the mass of the lowest lying Qq meson with the quantum numb ers
appropriate to a given part of the hadronic current Thus for a D
K decay the quark transition is c s so the p ole mass for the
vector form factor is equal to m
D
s
In the last few years much e ort has gone into lattice calculations
of semileptonic form factors Some of the lattice predictions for D
semileptonic decays are discussed in the following section
The form factors used in HQET are somewhat di erent from those
describ ed ab ove b ecause in the limit of very heavy quark masses the
four velo cities of the hadrons rather than their four momenta are the
appropriate quantities to use The hadronic current for P P is
expressed in terms of HQET form factors h v v and
h v v
hP v j V jP v i
q
h i
h v v v v h v v v v Mm
P
The di erential decay rate is then
d P P G jV j
cb
F
m w m m
P P
P
dw
m m
P P
h w h w
m m
P P
We will see that in the heavy quark symmetry limit h w w
where w is the Isgur Wise function while h w
For the pro cess P Q q V q q where V is a vector meson
each term in the current must be linear in the p olarization vector
of the vector meson Actually the complex conjugate is used since
V is an outgoing particle The p olarization vector transforms as a
Lorentz axial vector under parity the time like comp onent reverses
but the space like comp onents are unchanged The simplest term in the
hadronic current is essentially just multiplied by a function A q
tribution from the which is called an axial vector form factor The con
term asso ciated with this form factor unlike others do es not go to zero
as the vector meson momentum jp j go es to zero and it dominates the
rate at high q Another term that can be written is prop ortional to
p p which transforms as a Lorentz four vector For
this term transforms into itself under parity for
it transforms into minus itself This term
The multiplies a function V q which is called a vector form factor
Heavy Quark Physics and CP Violation
most general form for the current is
i
p V q p hV p jV A jP p i
M m
V
q
p p A q M m A q
V
M m
V
q q
m q A q m q A q
V V
q q
where V q Q A q Q and
M m M m
V V
A q A q A q
m m
V V
with A A Again terms prop ortional to q only play an
imp ortant role for the case Thus P V is essentially
describ ed by three form factors A q V q and A q
i
hV p jV A jP p i p p V q
M m
V
q
M m A q p p A q
V
M m
V
The form factors A A and V are dimensionless and relatively real
since CP is conserved in these decays and there are no nal state strong
interactions Possible CP violating e ects in semileptonic B de
cays due to physics beyond the standard mo del are discussed in
We can therefore take them to b e real A q andA q can b e asso
P
ciated with the exchange of a particle with quantum numb ers J
P
whereas V q is asso ciated with J We will see later Eq
and Eq that A contributes to all three helicity comp onents of
the nal state vector meson or the W A contributes only to the
helicity zero comp onent and V q contributes only to the helicity
comp onents
In terms of HQET form factors the hadronic current is
q
h
ih v v v v hV v j V A jP v i Mm
V V
h v v v v h v v v v
A A
i
h v v v v
A
J D Richman
Heavy quark symmetry do es not predict the w or q dep endence
of the form factors but it do es relate the form factors to each other
reducing the numb er of indep endent unknown functions In the heavy
quark symmetry limit for example m m in b c
b c
h w h w h w h w w
V A A
where w the Isgur Wise function is a common form factor for de
cays in to pseudoscalar and vector mesons These states are in the same
HQET multiplet b ecause they di er only in the relative spin orienta
tions of the quarks Furthermore in the heavy quark symmetry limit
h w h w
A
The Isgur Wise function can also b e regarded as the form factor in the
heavy quark symmetry limit for the elastic scattering of the meson by
a current that gives a kick to the heavy quark The form factor gives
the amplitude that the meson remains intact through the scattering
pro cess
Nonp erturbative metho ds such as lattice QCD or QCD sum rules
are needed to calculate the long range soft gluon strong interactions
that the Isgur Wise function describ es In the key decay mo de B
D however the range of v v is fairly small to A
D
Taylor explansion of ab out w v v is exp ected to work well
over most of this region
w w O w
At zero recoil w v v the ligh t constituents of the meson
X
are essentially undisturb ed by the heavy quark decay In the heavy
quark symmetry limit there is then a complete overlap between the
initial and nal meson wave functions and the value of the Isgur Wise
function is known This result whichmust b e mo di ed to take
into account the nite quark masses is of great practical imp ortance
for measuring jV j
cb
In the short w range for B D and B D the shap e of
the Isgur Wise function is thus characterized primarily by the slop e
Although is di cult to calculate it must b e p ositive and is exp ected
to b e roughly in the range to The primary theoretical to ols used
to determine the Isgur Wise function are QCD sum rules which are
based on quark hadron duality and lattice QCD in which a computer
calculation is p erformed using a discrete space time lattice
A ma jor theoretical e ort has been undertaken to evaluate correc
tions to the heavy quark symmetry limit for various pro cesses In gen
eral the largest corrections are from terms of order m m which
b c
Heavy Quark Physics and CP Violation
can b e estimated in the framework of HQET At the zero recoil p oint
w the two form factors h and h are protected against m
A Q
corrections As w increases however all of the heavy quark symme
try relations are sub ject to signi cant symmetry breaking corrections
These results have ma jor implications for the determination of jV j
cb
since h is the only form factor governing the rate at zero recoil for
A
B D The practical application of this result is discussed in
Sec
The HQET form factors are related to the traditional form factors
F V A and A by
M m
P
RF q h w h w
M m
P
R V q h w
V
w
h w R A q
A
m
V
R A q h w h w
A A
M
where we have explicitly written m and m instead of m in order
P V
to distinguish between the di erent masses for the nal state mesons
The constants R and R are given by
p p
Mm Mm
P V
R and R
M m M m
P V
The heavy quark symmetry limit has not yet b een imp osed on Eq
it is generally valid
We can use these results to show how the traditional form factors
F V A and A are themselves related to the Isgur Wise function
in the heavy quark symmetry limit For P P we have
F q R w
and for P V we have also in the heavy quark symmetry limit
A q
q R w V q A
q
M m
V
Four indep endent kinematic variables completely describ e the
semileptonic decay P V where the vector meson decays to two
pseudoscalars V P P The four variables most commonly used are
q or w and the three angles shown in Fig The angle is measured
in the W or rest frame where the lepton and the neutrino are
J D Richman
back to back it is the p olar angle b etween the charged lepton and the
direction opp osite to that of the vector meson The angle is mea
V
sured in the rest frame of the vector meson where the pseudoscalars P
and P are back to back In this frame is the polar angle between
V
one of these mesons say P and the direction of the vector meson in
the parent meson s rest frame Although either P or P can b e chosen
for this de nition one must b e careful to use a consistentchoice for the
angle whichwe de ne to b e the azimuthal angle b etween the pro jec
tions of the momenta of the lepton and P in the plane p erp endicular
to the decay axis
l χ D θl * W D θV z B
ν π
Fig De nition of the angles and in the decay B D These angles
V
are used for any P V in which the vector meson decays into two pseudoscalars
The lepton and neutrino are drawn back to back b ecause they are shown in the W rest
frame Similarly theD and the are shown in the D rest frame The angle is thus
measured in the W rest frame while is measured in the D rest frame The azimuthal
V
angle is measured b etween the W and D decay planes In the literature the angle
is sometimes de ned as the direction b etween the charged lepton and the recoiling vector
meson measured in the rest frame
The di erential decay rate for P Qq V q q V P P can
be expressed in terms of these four kinematic variables q and
V
d P V V P P p q
V
G jV j B V P P
q Q
F
dq d cos d cos d M
V
f cos sin jH q j
V
cos sin jH q j
V
sin cos jH q j
V
sin cos sin cos cos H q H q
V V
Heavy Quark Physics and CP Violation
sin cos sin cos cos H q H q
V V
sin sin cos H q H q g
V
The amplitudes for helicities and are prop ortional to
H q H q and H q Because the parent meson has spin zero
the vector meson and the W must have the same helicity The detailed
dynamics of the hadronic current are describ ed by the variation of
these helicity amplitudes with q whichwehave not yet sp eci ed The
magnitude of the three momentum of V in the rest frame of P p
V
is a function of q see Eq The factor is equal to for B
decays and forD decays It is this factor that leads to the di erent
lepton energy distributions for b ottom and charm decays Note that
the angle is de ned with resp ect to the direction of the virtual W
in the parent rest frame this accounts for the sign di erences between
our formula and certain others in the literature which use
The joint distribution given in Eq incorp orates the V A struc
ture of the leptonic current as well as the assumption that the mass
of the charged lepton can be neglected In general there is a fourth
helicity amplitude corresp onding to the time likehelicity comp onentof
the virtual W but its contribution is negligible when the lepton mass
is small The di erential decay rate formula for nite lepton mass is
given for example in Ref Eq also assumes that the daugh
ter meson is a narrow resonance if this is not the case a Breit Wigner
line shap e should be included
The helicity amplitudes can in turn be related to the two axial
vector form factors A q and A q and the vector form factor
V q which app ear in the hadronic current Eq
p
M m q M m A q H q
V
V
m q
V
M p
V
A q
M m
V
and
Mp
V
H q M m A q V q
V
M m
V
We note that while A contributes only to H and V contributes only to
H A contributes to all three helicity amplitudes At high q small
p each of the helicity amplitudes is dominated by A
V
We can also relate the helicity amplitudes to the set of form factors
de ned in HQET Since this result is applicable mainly to the decay
J D Richman
B D we write the result with the relevant masses
s
m m
B D
H w m m w h w
B D A
q w
h i
w
R w
r
and
s
m m
B D
H w m m w h w
B D A
q w
s
p
h i
wr r w
R w
r w
m The terms w are related to q by where r m
B D
m m q
B D
w
m m
B D
The form factor ratios R w and R w are given by
h w q V q
V
R w
h w M m A q
A V
h w m M h w
A V A
R w
h w
A
q A q
M m A q
V
These results are general and are not restricted to the heavy quark
symmetry limit Using Eq and Eq or Eq and Eq
we see that in the heavy quark symmetry limit R w and R w
independent of w Recent measurements from CLEO discussed in
the following section have shown that these ratios are indeed close to
unity for B D From insp ection of Eq and Eq it is
clear that as w H and H are gov erned by h w
A
Figure shows a prediction for the B D and B D
form factors from an HQET based calculation by Neub ert All of the
form factors asso ciated with these decays are exp ected to have essen
tially the same shap e
Many exp eriments have measured ratios of the traditional form fac
tors
V q A q
r and r
V
A q A q
Heavy Quark Physics and CP Violation
V
F 1
2 f(q )A 1
A 2
Fig The q dep endence of form factors in B D and B D according to an
HQET based calculation by Neub ert The curves represent V dot dashed F dashed
f q A where f q q m m solid and A dotted From with
B D
p ermission from M Neub ert
These quantities are usually assumed to be constant in the t How
ever in the heavy quark symmetry limit it is not r and r that should
V
be approximately q indep endent but R and R For charm decays
this consideration is not esp ecially relevant since the heavy quark sym
metry limit is not exp ected to hold even approximately For B decays
however the ratios R and R are preferred
In the real world the heavy quark symmetry limit do es not apply
and it is therefore of great interest to know how large the symmetry
breaking corrections are Neub ert s mo del dep endent estimate of
these corrections for B D gives
R and R
Neub ert argues that HQET predicts unambiguously that R for B
D must be signi cantly larger than unity whereas the prediction
for R is less certain In reality b oth R and R are exp ected to
have a mild q or w dep endence
R w w w
R w w w
y of the form factors Thus symmetry breaking a ects b oth the equalit
J D Richman
at w and the equality of their slop es It is imp ortant to remember
that a typical value of w in B D is ab out so the
predicted variation of R and R with w is rather small Thus the ob
served b ehavior of the form factors should b e similar to that predicted
in the heavy quark symmetry limit Ideally we would measure the w
dep endence of all three form factors and determine whether the ratios
predicted in Eq are correct We will discuss the B D
form factor measurements in the following section
It is interesting to analyze the w or q dep endence of each term con
tributing to the di erential rate All of the w dep endence resides in the
H H terms multiplied by the factor p q which form the co e cients
i j V
of the angular terms in Eq The momentum can be expressed in
terms of w by
p
p jp j m w
V D D
Fig shows the w dep endence of these co e cients using as inputs the
form factors measured by CLEO for the decay B D Recall
from Eq that the minimum value w corresp onds to q
q where the hadronic system has zero recoil velo city in the parent
max
meson s rest frame There is no phase space for this con guration
which explains why all of the curves in Fig go to zero at w
ever as w the rates from the three p ossible helicities How
contribute equally b ecause both the daughter meson and the virtual
W are stationary in the parent rest frame As we discussed at the end
of Sec this forces these particles to b e unp olarized
As w increases we see that the H term quickly b egins to dominate
over the H term This result is exp ected from the V A nature of
the W couplings which leads to a higher probability for the vector
meson and the virtual W tohave helicity than See
e ect also pro duces a larger value for the Sec and Fig This
interference term H H than for H H at all values of q except the
endp oints The CLEO measurements discussed in Sec show clear
e ects in the correlation b etween and cos due to this interference
V
term
At the other extreme w w or q the recoil velo city
max
is maximum The lepton and neutrino momenta are parallel in the
rest frame of P and their combined spin pro jection along their direc
tion of motion is zero Hence only the H combination contributes in
this limit This p oint is also evident from insp ection of Eq and
Eq which show that p q H remains nite as q whereas
V
go to zero It is also instructive to exam and p q H b oth p q H
V V
Heavy Quark Physics and CP Violation
Fig The predicted w dep endence of di erent helicitycombinations in the decay B
jV j p q G
V
cb
F
H q H q i j D The curves show the quantities
i j ij
M
which app ear as co e cients of the angular factors in Eq Wehave assumed R
and R with a linear w dep endence slop e for h The overall scale is
A
also determined by jV j The solid lines corresp ond to the terms that contribute
cb
to the total decay rate i j The terms i j do not contribute to the total
ij ij
decay rate b ecause the angular functions that they multiply in Eq integrate to zero
as w decreases both The D helicity can only be zero at maximum recoil w
helicity and contribute but helicity dominates due to the V A coupling At
minimum D recoil all three helicities contribute equally but phase space go es to zero
ine the dep endence of Eq on the form factors themselves via the
helicity amplitudes Eq and Eq When the recoil of the
hadronic system is small w the terms prop ortional to V q and
A q can be neglected compared with those prop ortional to A q
which app ear in all the helicity amplitudes As a consequence the A
form factor dominates the rate at large values of q
J D Richman
Since the form factor A q app ears in all three helicity amplitudes
Eqs and and typically dominates the rate it is natural to
use the new variables r V A and r A A or in HQET R and
V
R de ned in Eq
It is common to rep ort the values of certain other integrated observ
ables that can be derived from the form factors These are A the
FB
forward backward asymmetry of the lepton in the W rest frame
R R
d d
d cos d cos
d cos d cos
A
R
FB
d
d cos
L T
d cos
and A which is related to the ratio of longitudinal to transverse
pol
p olarization of the vector meson
R
p q jH j dq
V L
R
A
pol
p q jH j jH j dq
V T
where
Z
G jV j q
cb
F
dq p jH q j
i V i
m
B
and
L T
Wecaneasily see which form factors these observables are sensitive
to by examining the expressions for the helicity amplitudes Eq
and Eq Since the only di erence between H and H is the
sign of the co e cient of V q it is clear that the di erence
is prop ortional to an integral over V q Thus A provides a
FB
measurement of r or R in the HQET picture In contrast the
V
ratio of longitudinal to transverse p olarization is almost completely
controlled by r or R in the HQET version Roughly sp eaking then
r or R is determined from the cos distribution and the cos vs
V V
correlation whereas r or R is determined from the cos and q
V
distributions
Exclusive semileptonic decay jV j and jV j
cb ub
The sub ject of exclusive semileptonic decays has changed dramatically
over the past several years Huge data samples at CLEO LEP and
the xed target charm exp eriments at Fermilab have led to vastly im
proved measurements of b oth branching fractions and decay distribu
tions The exp erimental sensitivity has also allowed the observation of
Heavy Quark Physics and CP Violation
CKM suppressed mo des although the statistical uncertainties are still
fairly large In the case of B mesons the observation of B
and B represents ma jor milestones in the e ort to measure
jV j Impressive theoretical progress has also been made and predic
ub
tions based on HQET lattice QCD and other metho ds have put the
sub ject on amuch more solid foundation
There is a vast literature on this sub ject For a theoretical p ersp ec
tive I particularly recommend the pap ers describing the ISGW and
ISGW mo dels whichhaveplayed a ma jor role in the development
of this sub ject Exp erimentalists have found the ISGW mo dels to be
extremely useful b ecause all of the form factors are predicted for a wide
variety of mo des Neub ert has written a comprehensive review of
HQET which contains much information regarding semileptonic de
cays Richman and Burchat have written a review of leptonic and
semileptonic decays
Table lists the branching fractions for exclusive semileptonic de
cays of D D B andB mesons Have a lo ok at this table and see
what observations you can make Here are some of my conclusions
The branching fractions for the CKM favored mo des c s and b
c are relatively large In fact B D has the largest branching
fraction of any exclusive B decay
There is a remarkable di erence between the vector to pseudoscalar
V P ratios for the CKM favored D and B decays
e e
B D K B B D
e e
e e
B D K B B D
e e
The sizes of jV j jV j and jV j jV j can be crudely estimated
cd cs ub cb
from
e
jV j B D
cd e
e
B D K jV j
e cs
e
B B jV j
e ub
e
B B D jV j
e cb
where di erences in phase space and in the form factors are ignored
The uncertainties on the branching fractions for the CKM suppressed
e and e mo des are quite large both in B and in D decays and
we do not yet have go o d measurements of the V P ratios The ratios
do not app ear to be large however See the discussion later in this
section on B for more details on this case
The only mo de so far observed to an orbitally excited state is B
D which has a substan tial branching fraction
For semileptonic decays there is a vast amount of information be
J D Richman
Table
Measured branching fractions for exclusive D and B meson decays All values
are given in and con dence levels are at The results for B D e
D and B D e assume that B D and B D
D The D is a P wave charm meson L relative orbital
angular momentum b etween the c and the light quark the orbital angular
momentum and the light quark spin S are to a very go o d approximation
q
coupled to give j L S The D is the j state with total
q
J Both states are narrow since they must decayina D wave in contrast
to the two j states which can decayinan S wave The large di erence
between the D and D branching fractions is due to All B
D
D
branching fractions in the table are from CLEO Recentworld
averages for B semileptonic branching fractions are giv en in Drell The
D K e branching fractions are taken from the Particle Data Bo ok
and the D e and D e branching fractions are myaverages based on
CLEO E E E and Mark I I I measurements
B B D D
e
K K e D e D e
e e e e
e
K K e D e D e
e e e e
D e
e
D e
e
e e e e
e e e e
e
e e e
e e e e
yond that conveyed by a table of branching fractions The additional
information is contained in the kinematic distributions and their cor
relations which we will examine in this section
Let s b egin with the decays P P where b oth P and P are
pseudoscalars From Eq we can see that the q distribution is
sensitive not only to the square of the form factor but also to the p
factor arising from the P wave decay and phase space
CLEO has observed a signal of ab out events in the decay
K e The analysis uses over two million e e cc events D
e
pro duced in the continuum The metho d is to study the decay sequence
D D D K e in which the signal app ears as a peak
e
soft
near zero in the distribution of the reconstructed mass di erence M
Heavy Quark Physics and CP Violation
M K Thus the neutrino four vector is simply M K
soft
ignored resulting in a degradation of the M resolution Nevertheless
the backgrounds are su ciently low that the q distribution is quite
clean as shown in Figure
As exp ected from the p factor there is a dramatic decrease in the
K
rate as q increases The remaining q dep endence is in fact very mild
and there is as yet very little exp erimental sensitivity to the curvature
To parametrize the q dep endence of the form factors one can use for
example an exp onential form
q q q
max
f q f q e f e
max
or a p ole form
f
f q
q M
pole
For charm decays it is traditional to quote form factors at q
whereas in b c decays where HQET is applicable it is more appro
priate to give their values at q Figure shows explicitly how the
max
q dep endence of the decay rate dep ends on p and the square of the
K
form factor
The CLEO I I data shown in Fig give
GeV
GeV M
p ole
for the exp onential and pole forms resp ectively The world average
value of M is given in the review by Ryd
p ole
M GeV
p ole
GeV The exp ected value of the p ole mass for D K is M
D
s
which is roughly consistent with the measured value The CLEO II
data together with the D lifetime and the value of jV j determined
cs
from unitarity giv e
f
where the exp onential form was used in the integration of the form
factor
The E exp eriment has su cient sensitivity in the D K
mo de to obtain a rough measurementofthe f form factor which only
enters for nonzero lepton mass With D K events the result
is
f
f
J D Richman
500
400
2 300 dN/dq 200
100
0 0.0 0.4 0.8 1.2 1.6
q 22 (GeV )
e
Fig The q distribution for D K measured by CLEO The p oints with error
bars are the data the solid curve is the t to signal plus background the dashed curve
is combinatorial background and the dotted curve is the background from D K
decays The dip near q is due to resolution e ects
Fig The q dep endence for D K is dominated bythep factor arising from the
P wave nature of the amplitude The solid line shows the full q dep endence the dashed
line shows the dep endence if the form factor were constant and the dot dashed line shows
the dep endence arising from the squared form factor only The scale of the vertical axis
is arbitrary
Heavy Quark Physics and CP Violation
which shows how di cult it is to prob e the f form factor
The imp ortance of these results is that we can determine the absolute
normalization of the form factor b ecause jV j is indep endently known
cs
from unitarity of the CKM matrix see Sec We will use f
b elow when we examine the mysteriously small value of the vector to
pseudoscalar ratio in charm semileptonic decay
The analogous pro cess in B decays is B D Before the devel
opment of HQET many p eople felt that from a theoretical standp oint
this mo de was preferable to B D for measuring jV j The
cb
reason w as that B D dep ends on only one form factor while
B D dep ends on three At that time the values of jV j were
cb
extracted from the total rate which dep ends on all of the form factors
involved in the decay Exp erimentally however larger backgrounds
make it more di cult to measure B D than B D It
app eared that the optimal mo de for theorists was di erent from the
optimal mo de for exp erimentalists a situation that o ccurs all to o of
ten
This picture changed completely with the development of HQET If
we consider B D at zero recoil only one form factor h w
A
is involved in the original basis of form factors A q is the only one
contributing at zero recoil This result can b e seen from Eq and
Eq set w and there won t be any dep endence on R w or
Furthermore R w or from Eq and Eq set p p
V D
w form factor has no m corrections at zero recoil a result the h
A Q
known as Luke s theorem In contrast B D is suppressed at
zero recoil by the p factor and there are leading m corrections
Q
D
For these reasons together with its exp erimental advantages of higher
rate and lower background B D has b ecome the preferred mo de
for extracting jV j Nevertheless we should try to measure the CKM
cb
elements in as many ways as p ossible In addition it is imp ortant to
compare the form factor slop es in B D and B D to see
if they are essentially identical as exp ected in HQET In the following
discussion I will present results for B D and B D more
or less in parallel since the form factors for these mo des are related in
HQET and b oth are used to extract jV j
cb
Figure shows the signals for B D and B D in a
CLEO II analysis This measurement used a technique in whic h
the momentum of the neutrino is estimated from the missing momen
tum in the event One can then compute E and the b eam energy
constrained B mass for each candidate as discussed in Sec Eq
and Eq From Fig it is clear that there is a lot of background
J D Richman
whichmust b e determined with great care The histograms show that
the dominantbackgrounds are quite di erentforthetwo mo des Why
The D mo de has more background from D since all D de
cays and ab out two thirds of D decays pro duce D s This problem
is therefore much less severe for the D mo de but here there is
more combinatorial background since the D is reconstructed in the
three b o dy deca y D K
3420597-001 140 120 100 80 60 40 20 0 300 250 Events / 7.5 MeV 200 150 100 50 0 5.12 5.14 5.16 5.18 5.20 5.22 5.24 5.26 5.28 5.30 M (GeV)
cand
Fig CLEO I I analysis of B D The upp er gure shows B D while the
lower one shows B D The p oints with error bars are the data the shaded region
is the background from continuum events and events with fake leptons the diagonal hatch
region is the combinatoric background the vertical hatchistheB D background
and the crosshatch region is B D and other backgrounds From Ref
Wehave already seen the mild w dep endence of the B D form
factor in Fig Exp erimentally there is very little sensitivitytoany
w dep endence beyond the linear term of the Taylor expansion of the
form factor ab out the zero recoil p oint In the HQET basis B D
dep ends on both h w and h w Eq where h w w
and h w in the heavy quark symmetry limit The combination
of these form factors that app ears in Eq can b e written as a new
function F w whichalsogoesto w intheheavy quark symmetry
D
limit
p
d B D jV j G
cb
F
m m m w F w
B D D
D
dw
Heavy Quark Physics and CP Violation
p
where p w m w For B D we can integrate Eq
D D
over all of the angular variables to obtain in the HQET form factor
basis
p
G jV j d B D
cb
F
m m m w w
B
D D
dw
w m wm m m
B D
B D
F w
D
w m m
B D
where
X
w m wm m m
B D
B D
F w jh w j jH w j
D A i
w m m
B D
i
and
h i
w
jH w j R w
r
s
h i
wr r w
R w jH w j
r w
m See Eq and Eq Recall that in the where r m
B D
heavy quark symmetry limit R w R w but that there are
corrections to this result in the real world Thus the function F w
D
has a rather complicated relationship to the individual form factors
We note that when R w R w
X
w m wm m m
B D
B D
jH w j
i
w m m
B D
i
which explains why this term is explicitly factored out in Eq
Measurements of the w distribution by itself are sensitive to F w
D
whereas measurements of the full correlated kinematic distribution can
extract information on all three form factors as we will see below
Because the hadronic system is essentially undisturb ed at zero recoil
the overlap integral of the initial and nal mesons is essentially unity
This statement corresp onds to the unit normalization of the Isgur Wise
function For nite masses the form factors at zero recoil are
computed in the framework of the HQET expansion in m
Q
F
D
F D
J D Richman
which are indeed close to unity The calculation of these values and
the understanding of their uncertainties represent a ma jor theoretical
e ort over the last few years
In this approach we need to know the rate at zero recoil w so
that Eq and Eq can b e used to extract jV j Unfortunately
cb
there is no rate at zero recoil because phase space go es to zero This
can be seen from Fig or from Fig The width of the physical
region of the Dalitz plot in Fig is equal to the momentum of the
daughter hadron in the B rest frame As we have seen B D
is further suppressed in the zero recoil region by the P wave e ect
which gives the p part of the p in Eq For these reasons
D D
exp eriments have had to measure the rate at zero recoil by p erforming
an extrap olation using the full range in w Since the shap e of F w
is not known the extrap olation intro duces an additional uncertainty
Fortunately the range in w for b oth B D and B D is
quite mo dest w see Table and the form factors are
exp ected to have a nearly linear dep endence on w over this range A
common approach is to write F w as a Taylor expansion ab out the
zero recoil p oint
h i
F w F w c w
The notation is meant to distinguish this slop e from the slop e
of the true Isgur Wise function which corresp onds to the heavy quark
symmetry limit and from the slop es of the individual form factors
For example is the slop e of the h form factor There are di er
A
A
ent expansion co e cients for F w and F w The d dw distribu
D
D
tion are usually t under various assumptions to explore the sensitivity
to the shap e of the extrap olation function Sometimes theoretical pre
dictions are also used to limit the size of the quadratic co e cient c
Figure shows the CLEO II measurements of the w distri
and B D before acceptance cor butions for B D
rections are applied The low detection e ciency for the soft from
D D strongly a ects the distribution for B D near
w Recall from Sec that due to the high magnetic eld T
such soft pions do not travel very far out into the detector but instead
spiral around multiple times in the inner layers The new silicon vertex
detector should help to signi cantly improve this e ciency By cor
recting for the detection e ciency one can extract the w dep endence
of F w as shown in Fig The extrap olations to w assuming
D
linear and quadratic forms are shown
The values of F jV j and measured by CLEO I I ARGUS and cb
Heavy Quark Physics and CP Violation
3330694-015 160 _ 0 + - _ ( a ) B D* ν
80
0 - 0 - _ ( b ) B D* ν Events / 0.08
80
0 1.00 1.10 1.20 1.30 1.40 1.50 ~
y
Fig CLEO II analysis of B D showing a the w distribution rate for
B D where the detection e ciency is poor for low values of w and b the
w distribution for B D where the e ciency is b etter at low w The variable
plotted is calledy to emphasize that there is some smearing of the measured quantity due
to resolution e ects including the unknown direction of the B meson The B momentum
is however small at the S The p oints with error bars are the data the solid line
histogram shows the b est t which takes into account acceptance corrections and the
dashed line histogram shows the contribution from backgrounds From Ref
the LEP exp eriments have b een compiled by Drell and are listed
in Table ARGUS was the rst exp eriment to use this technique
The spread in the values of is quite large and shows that wehavealot
more work to do on this mo de Still the values of jV j are reasonably
cb
consistent The separate averages computed by Drell for B D
and B D are listed Table
What measurements allow us to test the predictions of HQET With
data from semileptonic decay there are essentially two approaches
B comparison of the di erent form factors governing with D
each other and comparison of B D with B D Wecan
see from Table that the values of for the two mo des are consistent
within errors Figure shows the decay angles that together with q
J D Richman
3330994-016 40 ( a ) Linear Fit
20 3
0 x F ( y ) 10 |
cb ( b ) Quadratic Fit V |
20
0 1.00 1.10 1.20 1.30 1.40 1.50
Y
Fig CLEO I I analysis of B D showing the form factor dep endence on the
measured value of w This is not a plot of the rate but of the amplitude The data from
B D and B D havebeencombined The upp er histogram a shows
a linear t to the data while the lower histogram b shows a quadratic t The solid
lines are the b est t and the dashed lines show the curves allowed within the statistical
errors From Ref
Table
Measurements of F jV j and using B D and B D
cb
From Ref
Mo de Exp eriment F jV j
cb
B D ALEPH
DELPHI
OPAL
CLEO
ARGUS
B D ALEPH
CLEO
are used in the CLEO II measurement of the individual B
D form factors The form factors are measured by p erforming a
Heavy Quark Physics and CP Violation
Table
Average values for jV j and from B
cb
D and B D The similar val
ues observed for for the two mo des are
exp ected from HQET From Ref
Mo de jV j
cb
B D
B D
joint four dimensional maximum likeliho o d t to these variables The
analysis is p erformed using B D where D D and
D K or D K These mo des are fairly clean with a
total of signal events over ab out background events The D
signals for the two channels are shown in Fig
Fig D signals from the CLEO II measurement of the B D form fac
tors The gures show the signals in the distributions of the reconstructed D D
mass di erence M M K M K left and M M K
s s
M K right
bined B Figure shows some of the t pro jections for the com
D mo des The two upp er histograms compare the distributions of
J D Richman
cos in the lower and upp er half of the q range Although accep
V
tance e ects which are taken into account in the t gradually reduce
the e ciency as cos increases it is apparent that in the lower q
V
range there is a strong forward backward p eaking comp onent At low
q the lepton and antineutrino b ecome collinear with zero net spin
along their common direction forcing the D also to have zero helicity
In contrast This e ect pro duces a distribution dN d cos cos
V V
at very high q the D is nearly at rest and unp olarized pro ducing
a cos distribution that is uniform apart from acceptance e ects
V
The lower two histograms show distributions of the azimuthal angle
for the lower and upp er range of cos The correlation b etween these
V
histograms arises from an interference term prop ortional to the di er
ence between negative and p ositive helicity amplitudes multiplied by
the zero helicity amplitude The observed upward slop e of the rst
histogram c and the downward slop e of the second histogram d are
consistent with either V A V A or V A V A couplings
which cannot b e distinguished However the observed correlation rules
out a mixed coupling V A V A
The t determines R w and R w as well as the dimen
sionless form factor slop e of h Since the true form of h w is
A A
A
not known several di erent functions are tried in addition to a simple
linear function with slop e Table compares the measured values
A
for R and R obtained from the combined t to B D
and B D with theoretical predictions from Neub ert
Close and Wambach and the ISGW mo del Within errors
the measurements are in go o d agreement with the predictions and they
followthe exp ected pattern R and R
In summary we can say that R and R are quite consistent
with HQET predictions they are actually consistent with the heavy
e sizes quark symmetry limit R R itself as are the relativ
of form factor slop es for B D and B D With larger
data samples we can exp ect substantial improvements in these mea
surements
Weturnnow to measurements of the D K branching fraction
and form factors and to the question of the B D K B D
K ratio There has b een a long struggle b oth exp erimental and
theoretical to understand the physics of these mo des and the lowvalue
of the vector to pseudoscalar ratio The D K form factors have
b een measured by several exp eriments Fig shows the distributions
from a recent analysis from the E charm hadropro duction ex
Heavy Quark Physics and CP Violation
3281095-015 120 ( a ) ( b ) 60 80 40
40 20
0 0 -1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0 2 2 2 2 cos for q / q < 0.5 cos for q / q > 0.5 θV max θV max 100 ( c ) ( d ) 80 60
60 40 40 20 20
0 1 2 3 0 123 for cos < 0 for cos > 0
χ θV χ θV
Fig Distributions of cos and in the CLEO I I measurementofthe B D
V
form factors The histograms show a cos for the lower half of the q range b cos
V V
for the upp er half of the q range c for the lower half of the cos range and d
V
for the upp er half of the cos range The p oints with error bars are the data the
V
solid histogram is the t including backgrounds and the dashed line is the contribution
from backgrounds The interpretation of these histograms is discussed in the text From
Ref
p erimentatFermilab The event sample consists of approximately
D K e K K decays and the kinematic distributions
e
studied are analogous to those measured for B D see Fig
Given that HQET is not exp ected to be applicable to this decay it is
appropriate to t for the form factor ratios r and r Eq rather
V
than the HQET motivated ratios R and R The tradition in charm
physics is to quote r and r at q The E measurements are
V
the most precise so far giving
V
r
V
A
A
r
A
J D Richman
Table
CLEO I I measurements of form factor ratios for B
D and comparison with theoretical predictions
The t also yields the slop e of the form factor h w
A
at w The value obtained
A
for this slop e is sensitive to the shap es assumed for
the form factors but the values of R and R are
relatively insensitive
R w R w
CLEO I I
Neub ert
Close
Wambach
ISGW
Table
Comparison of form factors for D K e
e
measured by E with ISGW mo del predic
tions
A A V
E
A q A q V q
max max max
E
ISGW
Using the measured rate and the value of jV j determined from unitar
cs
ity the absolute scale of the form factors can be determined and the
values are given in Table along with predictions from the ISGW
mo del
Figure shows a comparison of the exp erimental averages for
the D K and D K form factors at q with theoretical
predictions from lattice calculations There is go o d agreementbetween
theory and exp eriment on the single form factor f for D K
The value of jV j determined from unitarity is used to set the scale In
cs
contrast the more precise predictions for the D K form factors
which is the most imp ortant one for the overall in particular A
rate tend to b e higher than what is measured We can conclude that
at some level the overall rate for D K is understo o d but that
the rate for D K is lower than exp ected It will b e interesting
to see how this puzzle is ultimately resolved
The measurement of exclusive B X decays is one of the
u
ma jor goals of B physics With improvements in lattice QCD and
Heavy Quark Physics and CP Violation
Fig E D K analysis The p oints with error bars are the data and the
histograms are the b est t predictions which incorp orate acceptance e ects The plots are
a cos for q q upp er and q q lower b cos for q q
V e
max max max
upp er and q q lower and c for cos upp er and cos
V V
max
lower The low q region has a larger fraction of helicity zero which is manifested as
a larger forward backward p eaking comp onentof K s from K K cos We also
V
see a dramatic change in the lepton energy distribution at low q the leptons tend to
come out at deg to the W direction at higher q they are more backward p eaked in
contrast to the situation in B decays where the forward p eaking leads to a hard lepton
sp ectrum The azimuthal angle shows a strong dep endence on cos Figure used with
V
p ermission
other theoretical metho ds the uncertainties in predicting the rate for
exclusive B X mo des should b ecome smaller than those for the
u
inclusive endp oint sp ectrum and hence provide a more precise value
of jV j
ub
mo del predictions For B X decays unlike B X
u c
indicate that the rate should b e distributed over many exclusivechan
nels with no dominant mo des The hadronic system X can range
u
over much of the light quark hadron sp ectrum including radially ex
cited and P wave mesons However this picture is somewhat mo di ed
for the lepton sp ectrum endp oint region where the backgrounds from
B X are suppressed and exp erimental sensitivity is b est In
c
this region theoretical mo dels indicate that a small number of exclu
sive decays are dominant B B B
B and B A key prediction which is essentially
mo del indep endent is that the lepton sp ectra for the decays to nal
states with vector mesons are exp ected to p eak at very high energy
J D Richman
∗
→ and → Semileptonic D K D K
BKS BKS LMMS LMMS ELC ELC APE APE UKQCD UKQCD WUP WUP LANL LANL Expt Expt
0.6 0.8 1. 1. 1.5 2.
f V
A A
BKS BKS LMMS LMMS ELC ELC APE APE UKQCD UKQCD WUP WUP LANL LANL Expt Expt
0.6 0.8 1. 0 0.25 0.5 0.75
Fig Comparison of exp erimental average values with lattice predictions for D K
f and D K A A and V form factors Figure provided by
J Flynn
as discussed in Sec Figure shows predictions for lepton energy
and q distributions from a calculation in which lattice results are used
to constrain a form factor mo del Figure shows the results of a
calculation by Ball of the B form factors and a com
parison with lattice results There is considerable variation of some of
the form factors over the q range esp ecially in comparison with those
D from B
The rates for B and B are connected by
Heavy Quark Physics and CP Violation
−
B → ρ l ν l
− −
−
| |
| | GeV d /dq Vub d /dE Vub 0.8 16 14 0.6 12 10 0.4 8 6 0.2 4 2 0 0
0 5 10 15 20 0 0.5 1 1.5 2 2.5
GeV V
q Ge E
−
B → π l ν l
− −
−
| |
| | GeV d /dq Vub d /dE Vub
0.4 6
0.3 4 0.2 2 0.1
0 0
0 5 10 15 20 25 0 0.5 1 1.5 2 2.5
GeV V
q Ge E
Fig Predictions for E and q distributions for B and B These
curves are based on a calculation in which lattice results are used to constrain a form factor
mo del Note the sharply p eaked lepton energy sp ectrum for B The solid curves
are the central value predictions while the dotted curves are allowed within one sigma
errors Figure provided by J Flynn
isospin symmetry since the B and B have the same space and spin
wave functions in this limit as do the and A simple way to
p
see this is to note that the avor wave function is uu dd
while that for the is ud For B decay the avor wave
function pro jection of the daughter uu system onto the therefore
p
in the amplitude whereas for B the pro jection gives
J D Richman
Fig Calculation of the form factors with t q for B by P Ball using
line cone sum rules curves and bytheUKQCD lattice collab oration p oints with error
bars The variation with q is esp ecially dramatic for the V form factor Figure provided
byP Ball
of the daughter ud system onto the gives unity Isospin symmetry
tells us that in all other resp ects the decays are the same Although the
rate for B cannot be related to these by a avor symmetry
it is exp ected in the quark mo del to b e approximately equal to that for
B Thus
B B B
Heavy Quark Physics and CP Violation
B B
In reality there may b e a small amount of isospin violation due to
mixing so these relationships may b e a ected to some extent
The CLEO II measurement of B and B is
unusual in that it relies on the very go o d solid angle coverage her
meticity of the detector to asso ciate the neutrino momentum with the
missing momentum in the event
E p jp j p
miss miss
where
X
p p
miss
Here the index runs over the charged tracks and neutral clusters de
tected in the calorimeter Although it might app ear obvious that the
neutrino momentum should b e identi ed with the missing momentum
there are a great many e ects that can contribute to missing momen
tum other than a neutrino such as undetected K s or mismeasured
L
tracks To achieve go o d resolution on the neutrino four momentum a
detailed understanding of these e ects is required as we will discuss
b elow
Using the neutrino four vector one pro ceeds as in a hadronic anal
ysis As discussed in Sec Eq and Eq we de ne E
the energy di erence b etween the candidate B decay pro ducts and the
b eam energy
E E E E E
beam
and m the b eam energy constrained mass
B
M E jp p p j
B
beam
Signal events are exp ected to have E and m p eaking at the B
B
mass The real question is what resolution can be achieved in these
quantities and at what exp ense in detection e ciency
Although the identi cation of neutrino momentum with the missing
momentum in the event has b een used in many other exp eriments in
particular at hadron colliders the CLEO II analysis is able to take
the metho d a step further by carefully selecting events such that the
neutrino momentum is measured well Events are required to have only
one lepton to ensure that only one neutrino is present and events with
large missing mass for example from undetected K s are removed
L
E MeV With these and other cuts to by requiring M
miss miss
J D Richman
suppress spurious tracks and clusters the resolution on E is MeV
ab out the resolution on the neutrino direction is ab out deg and
the resolution on the b eam energy constrained mass is MeV c
The cuts required to achieve go o d resolution result in rather low signal
e ciencies for B and for B
1850696-001
30 30 30
20 20 20
10 10 10
0 0 0 ( ) ( ) ( ) 40 40 40 Events / 100 MeV Events / (200 MeV/c) < Events > / 7.5 MeV 20 20 20
0 0 0 5.2 5.3 I 0.5 0 0.5 23 M (GeV) E (GeV) p (GeV / c) m Lepton
Fig CLEO I I analysis of exclusive B X decays The histograms show the
u
b eam energy constrained mass of the candidate B m decay pro ducts left the
E distribution middle and the lepton energy distribution right The upp er plots
are for the B channel while the lower plots are for the combined vector meson
and channels The p oints with error bars are the data after subtraction of
continuum background and fake lepton events The coarse crosshatch grey and unshaded
comp onents are b cX b H and signal comp onents resp ectively where H
u u
represents higher mass non charm states For the mo des the ne crosshatch shows
the vector cross feed and the single hatch the cross feed For the vector
mo des the ne cross hatch shows the vector cross feed and the single hatchshows
the vector vector cross feed The arrows in the lepton momentum histograms show the
cuts on lepton momentum From Ref
As in any B X analysis there are large backgrounds from
u
These and other backgrounds are incorp orated into a B X
c
joint tinthe E m plane for the ve mo des B B
B
B B and B For the and
mo des the M and M distributions are also included in
B X backgrounds except the t Because there are enormous
c
for relatively high lepton momenta leptons are required to have p
GeV c for the mo des and p GeV c for mo des
These and other cuts intro duce a mo del dep endence into the branching
fractions b ecause theoretical mo dels must be used to extrap olate the
rate over the full phase space Figure shows the pro jections of the
t onto m E and lepton momentum Small but signi cant signals B
Heavy Quark Physics and CP Violation
are evident for both B and B The charged and
neutral mo des are constrained in the t by the isospin relation given
in Eq
Figure shows the signal in the mass distribution No signif
icant signal is seen in but the e ciency is lower in this channel
Figure lists the branching fractions obtained using various mo d
els to determine the detection e ciencies The e ciencies dep end on
absolute the shapes of the distributions from the mo dels not on their
normalizations The CLEO II results can be used to test the various
predictions for the ratio B B the K orner
Schuler mo del is only consistent with the measurement at the
level and it is not used in the subsequent averages over mo dels which
give
B B
B B
where the errors are statistical systematic and the estimated mo del
dep endence resp ectively
The vector to pseudoscalar ratio can be extracted from the ts al
though the results are again mo del dep endent The value averaged over
the di erent mo dels studied is
B B
B B
where the rst error is statistical the second systematic and the third
re ects the mo del dep endence This result is an indication that the
vector to pseudoscalar ratio may be lower here than for the b c
mo des
Figure shows the values of jV j extracted from the CLEO II
ub
measurements and the various mo dels In contrast to the measurements
of branching fractions jV j do es dep end on the absolute normalization
ub
each mo del the value of jV j is of rates predicted by the mo dels For
ub
obtained by constraining the ratio to the value predicted by that
mo del A nal value for jV j is then calculated by averaging over
ub
several mo dels and explicitly incorp orating an uncertainty due to the
mo del dep endence
jV j
ub
where the uncertainties are statistical systematic and the estimated
resp ectively It is quite encouraging that this re mo del dep endence
sult is consistent with the value from the inclusive endp oint sp ectrum
J D Richman
1850696-002 Events / 40 MeV 15 50 10
40 5 0 0.7 0.8 0.9 1.0 30 3 Mass (GeV) Events / 190 MeV
20 15 Events / 190 MeV
10 10 5 0 0 0.5 1.0 1.5 0.5 1.0 1.5 o o
Mass (GeV) Mass (GeV)
Fig CLEO I I analysis of B The histograms show the reconstructed mass
distributions for left top right and for b ottom right no signal
exp ected in the M E signal bin The arrows indicate the mass ranges included in
m
the t From Ref
analysis jV j There is certainly a long waytogo
ub
in determining a precise value for jV j and with larger data samples
ub
there will be b oth detailed studies of the kinematic distributions for
B and B and measurements of additional mo des
Hadronic Decays Lifetimes and Rare Decays
Hadronic Decays
The decays of B and D mesons into nal states consisting only of
hadrons are called nonleptonic or hadronic decays I will use the latter
term even though it is considered imprecise by some p eople These
mo des are much more complicated than leptonic or semileptonic de
cays b ecause all of the particles involved can interact strongly Fur
thermore there is no rigorous way even to parametrize the e ects of
the strong interaction in terms of form factors Nev ertheless there are
many asp ects of hadronic bottom and charm decays that can be un
dersto o d and there are situations in which quantitative predictions are
p ossible
Here are some of the key questions that relate to these mo des
Can we construct any framework that allows us to think ab out
Heavy Quark Physics and CP Violation
ISGW II 2.0±0.5±0.3
WSB 1.8±0.5±0.3
Melikhov 1.8±0.4±0.3±0.2
Burdman-Kambor 1.7±0.4±0.3±0.2
Average 1.8±0.4±0.3±0.2
1234 Br(B0→π+l-ν) / 10-4
0.4 ISGW II 2.2±0.4±0.6
0.5 WSB 2.8±0.5±0.8
0.5 Melikhov 2.8±0.5±0.8±0.4
0.4 UKQCD/Stech 2.1±0.4±0.6±0.4
0.5 Average 2.5±0.4±0.7±0.5 1234
Br(B0→ρ+l-ν) / 10-4
Fig CLEO I I branching fractions for B and B The mo del
dep endence arises b ecause the detection e ciency dep ends on the predicted kinematic
distributions for lepton energy and other variables Figure provided byL Gibbons
hadronic decays Can we explain the patterns that app ear in values of
measured B branching fractions
What role do interference e ects play in hadronic decays
What role do nal state interactions play in hadronic decays
What are the inclusive prop erties of hadronic decays For example
what is the mean number of c and c quarks pro duced per B decay
do es the inclusive D momentum sp ectrum lo ok like What
How can we explain the striking observation that D D
whereas B B In the sp ectator mo del where the D
and D lifetimes are essentially determined by the decay rate of the
charm quark alone this result is quite unexp ected A related question
is whether the large lifetime ratio is due to an enhancement of the D
J D Richman
-3 |Vub| / 10
0.3 ISGW II 3.4±0.2±0.4
WSB 2.9±0.2±0.3
0.4 Melikhov 4.0±0.2±0.5±0.5
0.3 Burdman-Kambor/UKQCD 3.1±0.2±0.4±0.5 0.3 Average 3.3±0.2±0.4±0.7 Ball et al.
Khodjamirian-Rückl
Narison
APE
ELC
Demchuk et al.
Faustov et al.
Grach et al.
Lellouch
Endpoint Method 3.1±0.8
05
Fig Values of jV j extracted from the CLEO II measurements of B and
ub
B Mo del dep endence enters mainly in relating the overall decay rate to jV j
ub
but also in determining the detection e ciencies which dep end on the predicted kine
matic distributions The value of jV j extracted from the analysis of the inclusive lepton
ub
sp ectrum endp oint region is also shown and it is consistent with the results from the
exclusive analyses Figure provided by L Gibb ons
decay rate or to a suppression of the D decay rate
Some useful review articles on hadronic decays are Browder et al
Neub ert et al all of which and Neub ert and Stech and
contain extensive references The early pap er of Bauer Stech and
Wirb el was among the rst to use factorization to make systematic
predictions for two b o dy charm and b ottom decays
Let s begin by trying to make sense of some of the data on exclu
sive hadronic B decays Table lists some branching fractions for a
selected set of two b o dy mo des Most of the decays listed involve D or
Heavy Quark Physics and CP Violation
Table
Branching fractions for some hadronic B and B decays
The D and D values are from recentCLEOIImea
surements One more signi cant gure is quoted
in the references and a detailed breakdown of contribu
tions to the uncertainties is given The uncertainties are
somewhat correlated due to common D and D branching
fractions The J K branching fractions are from the
Particle Data Bo ok All con dence levels are at
This table do es not list all branching fractions with
measured non zero values in particular two b o dy decays
with other charmonium systems J S P have
c
b een observed What is the a It s a very broad
MeV isovector I state with orbital
P
excitation and J
B decay BR B decay BR
D D
D D
D D
D D
D
D
D
D
D a D a
D a D a
J K J K
J K J K
D mesons and pro ceed through the dominant b c transition Ex
amine these data make comparisons and write down any interesting
conclusions or conjectures you mighthave Note that a comparison of
B and B branching fractions is equivalent toacomparisonofdecay
rates to a good approximation b ecause the lifetimes of these parti
cles are very nearly the same You mightalso nd it useful to draw a
few Feynman diagrams but don t attempt any elab orate calculations
The CLEO I I signals for the D D and D a mo des are shown in
Fig
Here are some observations
i All of the branching fractions are less than
J D Richman
ii B X Y B X Y comparison
B B D B B D
B B D B B D
iii D D comparison
B B D B B D
B B D B B D
iv D D comparison
B B D B B D
B B D B B D
v D D comparisons
B B D B B D
B B D B B D
B B D B B D
B B D B B D
vi B B comparisons
B B D B B D
B B D B B D
B B D B B D
B B D B B D
vii The only observed B decays to neutral neutral nal states involve
cc mesons
The rst observation that all of the branching fractions are below
tells us that even many of the most natural two b o dy hadronic B
decays have small rates Of all B decays the largest branching fraction
for an exclusive mo de is B B D We can contrast this
observation with the situation in D decays where there are some mo des
D K with very substantial branching fractions such as B
and B D K Thus two b o dy
mo des are much more dominantin D decays than in B decays Wealso
Heavy Quark Physics and CP Violation
120 150 60 + − + − + − B→ D π B→ D ρ B→ D a1
80 100 40
20 40 50
0 0 0 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3
400 120 B→ D0π− B→ D0ρ− B→ D0a− 300 100 1 80 200 200 60
100 40 20 0 0 0 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3
100 ∗+ − 80 ∗+ − ∗+ − B→ D π B→ D ρ B→ D a1 75 40 50 40
25
0 0 0 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3
100 60 ∗0 − 80 ∗0 − ∗0 − B→ D π B→ D ρ B→ D a1
40 50 40 20
0 0 0
5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3 5.2 5.22 5.25 5.27 5.3
Fig CLEO I I studies of B decays to exclusive hadronic nal states The distributions
show the b eam energy constrained B mass reconstructed in various nal states with a D
or D meson and and a light meson ora
momentum sp ectrum in B decays shown in see that the inclusive D
Fig is very broad with no visible narrow peaks corresp onding to
two body pro cesses
The second observation compares B decays to neutral neutral
J D Richman
1630197-022 2400
2000
1600
1200 Events / 0.05 800
400
0 0 0.25 0.50 x = p / p
max
Fig CLEO I I measurement of the inclusive D sp ectrum in B decay The scaled
momentum variable is de ned as x p p p GeV c
max max
X Y nal states with decays to charged charged X Y nal
states It is clear that the X Y mo des are strongly suppressed none
of the B D or B D mo des have even b een observed
yet and the limits are at least an order of magnitude b elowtheX Y
branching fractions The only observed X Y mo des involve cc bound
states Contrast these results with B K B K
s s
or with B D K and B D K
Figure compares the external and internal sp ectator diagrams
for B and B decay In B decay the external sp ectator diagram
leads to the X Y nal states while the internal sp ectator diagram
leads to the X Y nal states The data therefore indicate that the
internal sp ectator diagram is highly suppressed relative to the external
sp ectator This observation makes the contrast with K and D decays
even more interesting Internal sp ectator diagrams in B decay are
often said to be color suppressed since the quarks from the W decay
which are pro duced as a color singlet system do not in general have
the correct colors to form color singlet systems with the other quarks
in the pro cess For B decay the two typ es of diagrams can lead to
the same nal state X Y so interference e ects can o ccur
The third observation notes that the D nal state do esn t get
Heavy Quark Physics and CP Violation
+
Fig Comparison of B and B external and internal sp ectator decays For B decay
the twotyp es of pro cesses lead to distinct nal states X Y and X Y which therefore
do not interfere For B decay the twotyp es of diagrams b oth lead to X Y nal states
and the decay rate is therefore sensitivetointerference e ects
any more rate than D This fact is not very surprising given that
the D is forced by angular momentum conservation to have helicity
zero so there are no additional helicities that can be p opulated In
addition the D and D masses and wave functions are quite similar
since the charm quark mass is large the spin is nearly decoupled from
the dynamics of the system Of course the D and D masses aren t
exactly the same this measures the size of a m correction to the
c
heavy quark symmetry limit
The next observation is more interesting it compares the rates for
D with D nal states Since the D state can nowhave three p ossi
ble helicity con gurations and still conserve angular momentum there
is now the mystery as to whytheD D ratios aren t larger Tocom
plicate matters a little I have listed decays of the typ e B D D
s
in Table In this case the D D mo des do have a higher rate than
s
the DD D D or DD decays This suggests that the momenta
s s s
of the nal state particles are relevant and that the vector meson he
licities in low momentum and high momentum decays are p opulated
di erently We will discuss these points below when we compare the
p olarization of the daughter vector meson in hadronic decay with that
in semileptonic decay
keeping everything In the fth observation we substitute a for a
else the same The D D ratios are around three This result
cannot b e due to the having more spin states we see the e ect even
in D D and our previous comparison of D D showed that the
extra spin states don t increase the rate bymuchanyway The e ect is
J D Richman
Table
Branching fractions for two b o dy hadronic B
and B decays to nal states with D mesons
s
The values are from the Particle Data
Bo ok www edition
B decay BR B decay BR
D D D D
s s
D D D D
s s
D D D D
s s
D D D D
s s
due in large part to the di erence in the decay constants of the and
the In the external sp ectator diagrams which give the dominant
contributions the quarks that feed into the and are pro duced at a
p oint from the W decay The amplitudes for the quarks to overlap in
space within a or are prop ortional to f and f resp ectively We
therefore exp ect the rates to b e roughly in the ratio
f
MeV
f MeV
Next we noted that all of the X Y branching fractions are larger
than the corresp onding X Y branching fractions Wehaveseenthat
X Y mo des get contributions from b oth the external and internal
sp ectator diagrams However from the fact that B X Y mo des
havesuch small branching fractions it would seem likely that a signif
icant b o ost to the X Y rate could only come from the interference
term This argument suggests that the interference is constructive in
contrast to D decays where it is destructive
Finally of the internal sp ectator pro cesses the only ones that are
observed so far involve cc systems We can tentatively attribute this
in part to the large decay constant of the J f MeV See
J
Table
Although it is not p ossible to make precise calculations for hadronic
decays the factorization hyp othesis provides a framework for under
standing many of the observed features of two b o dy mo des and even
quantitative predictions In the factorization ap for making some
proach one writes the decay amplitude as the pro duct of two currents
in analogy to semileptonic decay The e ects of strong interactions
are divided into two categories short distance hard gluon e ects
parametrized by the co e cient a for the external sp ectator diagram
and a for the internal color suppressed sp ectator diagram and
Heavy Quark Physics and CP Violation
(a) External Spectator f u π d
a b F (q 2 =m 2 ) 1 c H π q q
(b) Internal Spectator c f D u b d a2 22 FL (q =mD )
q q
Fig Sp ectator diagrams for B decays with a an external W and b an internal W
The QCD hard gluon corrections are parametrized by the co e cients a and a while
nonp erturbative e ects are describ ed in our B D example bythedecay constant f
or f and the form factor F q m or F q m
D H L
D
long distance soft gluon e ects which are parametrized by decay con
stants and form factors The elements used in the factorization de
scription are shown in Fig Final state interactions between the
daughter hadrons are ignored in the factorization picture It is how
ever p ossible to parametrize these e ects and to use data to constrain
the parameters to some extent
In Fig a the ud system from the W decay is pro duced at a
p oint so that the appropriate meson decay constant which measures
the overlap of the quark anti quark pair parametrizes the amplitude
to pro duce the meson To be de nite we take this meson to be a
pion The daughter charm quark recoiling at the lower W vertex
however must bind together with the sp ectator quark The physics of
ery similar to the hadronic transition in semileptonic this pro cess is v
decay discussed earlier and it is describ ed by the appropriate form
factor evaluated at q m The lower the q value the higher the
recoil velo city and in general the lower the amplitude for pro ducing
the meson from the daughter quark and the sp ectator quark Thus for
the external sp ectator diagram example the non p erturbative QCD
physics is describ ed by the pion decay constant f and a heavy to
q m heavy B X form factor F
c H
Consider now the internal sp ectator diagram shown in Fig b
J D Richman
Because the W has such a short range compared with the size of a
hadron the c and the u are still e ectively pro duced at the same p oint
and the relevant decay constantisf instead of f It is the daughter d
D
quark that nowmust bind with the sp ectator so there is nowa heavy
to light B X form factor which I denote by F q m
u L
D
For a typ e decays with large daughter hadron recoil velo city i e
decays at low q one might exp ect factorization to b e a go o d approxi
q pair mation from the following argument due to Bjorken The q
pro duced in the W decay is formed at a point as a color singlet and
for low q pro cesses the pair moves out of the decaying hadron at high
velo city in a collinear fashion Thus the pair lo oks like a very small
color dip ole that gradually grows to the size of a meson The pair will
not form a meson however until it moves a distance c where
h h
is a typical hadronization time in the rest frame fm c This
h
distance can be quite large for example c fm for the pion in
h
B D deca y Since this distance is much larger than the size of the
decaying meson the q q pair can escap e from the cloud of quarks and
gluons without signi cantly interacting with it
For a B decay the external and internal diagrams lead to di erent
nal states X Y and X Y whereas in B decaytwo u quarks can
be present in the nal state which means that the same X Y nal
state can be reached in two ways Thus we have
B bd X Y ja f F j
H
B bd X Y ja f F j
D L
B bu X Y ja f F a f F j
H D L
This pattern is illustrated in Fig Fig and Fig
The factorization picture can be put on a more solid foundation by
relating a and a to a set of strong interaction parameters that can b e
calculated or at least estimated from QCD Here I will give only a short
summary of the theory underlying factorization based predictions for
an excellent review see Neub ert and Stech which I will followin
the discussion b elow The task is to separate the amplitude into parts
that can be identi ed with hard gluon e ects and soft gluon
interactions which can then be describ ed in terms of decay constants
and form factors An imp ortant question is where to draw the dividing
line b etween soft and hard gluons the so called factorization scale
f
Given that decay constants and form factors describ e the interactions of
q q systems that remain b ound together the hard gluons are considered
to b e those that can rearrange the quarks into di erent hadrons This
Heavy Quark Physics and CP Violation
Fig External upp er gure and internal lower gure sp ectator diagrams for hadronic
decayofa B meson
b B –
u
Fig External upp er gure and internal lower gure sp ectator diagrams for hadronic
decayofa B meson
Fig B decay via color suppressed internal sp ectator diagram to a cc meson At
present the only observed internal sp ectator pro cesses are of this typ e
J D Richman
scale is at some level pro cess dep endent if there is a large energy release
in the decay it is harder to rearrange the quarks and the factorization
scale is higher
We now fo cus on b cu d decays following the analysis of Neu
b ert and Stech The hard gluon corrections down to a scale can be
summed using the renormalization group equation which leads to an
e ective hamiltonian
G
F
p
H b cud V V c du c b c cu db
e cb
ud
The Wilson co e cients c and c describ e the hard gluon ef
fects and at the scale m have the values
b
c m c m
b b
in next to leading order The expression cb is short hand for a V A
color singlet current The matrix elements of the four quark op erators
c b and cu db must b e somehow related to the decay constants du
and form factors It it is not obvious how this is to be done however
b ecause the latter are matrix elements of two quark op erators For
B D decay the amplitudes are written
G
F
p
A B D V V a F
cb
BD
ud
G
F
p
V V a F A B D
cb B D
ud
where F and F are the so called factorizable contributions
BD B D
F h j du j ihD j cb jB i
BD
F hD j cu j ih j db jB i
B D
of these equa The rst matrix element on the right hand side of each
tions is related to a decay constant f or f while the second is
D
related to a form factor B D or B The physics connected
with a and a is quite complicated since these co e cients are de ned
so as to makeEq exact
h i
c
BD BD
c a c
N
c
h i
c
B D B D
a c c
N c
Heavy Quark Physics and CP Violation
where N is the number of colors Non factorizable contributions
c
from currents with color singlet and o ctet structure are parametrized
by the co e cients O N and O N While j j
c
c
cannot b e neglected relative to other terms of order N Recalling
c
that O c N we have
c
a c m O N
b
c
a c m c m O N
b b
c
where N m Since cannot at present b e calculated the
c b
strategy is to use measured branching fractions to extract a and a
for as many di erent decays as p ossible and to see if consistent values
are obtained In principle is pro cess dep endent but its size and
variation are empirical questions
Using measured branching fractions for hadronic B decays one can
extract the co e cients a and a arising from hard gluon e ects This
pro cedure requires knowledge of b oth decay constants and form fac
tors As discussed earlier our knowledge of these quantities is far from
p erfect esp ecially in the case of the form factors for heavy to light
transitions Compared with the state of our understanding of hadronic
decays however form factor predictions mightbe regarded as reason
ably trustworthy but it is imp ortant to remember that there is more
uncertainty in calculations made within the factorization framework
than is often explicitly stated
One interesting wrinkle occurs in decays to nal states with two
pseudoscalars The amplitude for B D is
G
F
p
V V a h j du j ihD j cb jB i A B D
cb A V
ud
G
F
p
V V a if q
cb
ud
m m
B D
B D
q p p f F q
q
m m
B D
B D
q g F q
q
where q p p When the two currents are contracted the term
B D
prop ortional to F q vanishes giving
G
F
B D
p
q m a f F m m A B D i V V
cb
D B ud
J D Richman
The form factor that enters here is precisely the one for which there
is almost no sensitivity in semileptonic decay where its contribution
vanishes in the limit of zero lepton mass See Sec and In
practice this is not a serious problem In the present example
B D B D B D
F q m F F
B D B D
where the relation between F F is discussed after
B D
Eq We can regard F as an approximately known quan
tity given that jV j can b e determined from other decays In addition
cb
HQET can b e used to provide relations to form factors that havebeen
prob ed exp erimentally As I have already mentioned however fac
torization predictions involving heavy to light form factors do have to
make assumptions that have not b een tested by exp eriment
In B decays there is sensitivity to a a through the interference
term Some predictions from Neub ert et al are
B B D
R a a
B B D
B B D
a a R
B B D
B B D
R a a
B B D
B B D
R a a
B B D
In a recent CLEO II analysis the mo dels of Deandrea et
al and the BSWI I mo del of Neub ert et al were used to
extract a and a from the measured branching fractions Table
lists some of the predictions of these mo dels and Table lists the
values of a and a extracted from an overall t to the data Three
out features stand
Both a and a are measured with go o d precision however the errors
do not re ect the full theoretical uncertainties
a as exp ected from QCD predictions
a is substantially smaller as exp ected The sign of a a is
p ositive corresp onding to constructiveinterference b etween the exter
nal and internal sp ectator diagrams
It is imp ortant to check that the values extracted for a and a are
consistent from mo de to mo de Using B K decays one can
Heavy Quark Physics and CP Violation
Fig The ratio of a a as a function of the running coupling constant evalu
s
f
ated at the factorization scale The bands indicate the phenomenological values of a a
extracted from B D and D K decays From Neub ert Used with p ermission
extract
ja j
Similar values are obtained in Ref which also quotes
and m
b
The situation in charm decays is much more complicated The data
on charm decays show that unless nal state interactions are taken into
account factorization predictions by themselves do not yield consistent
K D K results Browder Honscheid and Pedrini use D
and other mo des to obtain
a D a D
Although the errors do not take into account all of the uncertainties
it is clear that the sign of a a is negative in D decays and p ositive
in B decays This e ect can be explained as a consequence of
the running of since the factorization scale is lower in D
s f f
than in B decays leading to a much larger value of the strong coupling
constant The relation between the running of and a a is shown
s
in Fig The presence of destructive interference together with the
predominance of two b o dy decays explains the long D lifetime
While factorization app ears to have some validity many p eople re
gard it with a great deal of skepticism given the complexity of hadronic
J D Richman
Table
Predictions for two body B decays in factorization based mo dels The pre
dictions of the mo dels were slightly mo di ed as describ ed in Ref to
takeinto account more recentvalues of the input parameters
B DecayMode B BSWI I BCDDFGN
D a a
D a a
D a a
D a a
D a a a a
a a a a D
D a a a a
D a a a a a a a a
Table
Values of a and a extracted from tting the BSWI I and CDDFGN mo d
els to CLEO I I measurements of D and D branching fractions The
values are from Ref The stated errors do not include any uncertainty
asso ciated with the form factors
Mo del ja j a a dof
CDDFGN
BSWI I
pro cesses Most of the results discussed ab ove are from two b o dy de
cays that have a large energy release or low q where wewould exp ect
factorization to work b est It is p erhaps not to o surprising that among
these mo des the values of a and a are consistent It is imp ortantto
make as many quantitative tests of factorization as p ossible and to
determine when it b egins to break down We know that in D decays
factorization without including nal state interactions is inadequate to
describ e the data One approach to testing factorization is to com
pare the decay rate for a hadronic mo de with the rate at the same value
of q for a semileptonic decay F rom the Dalitz plot for B D
shown in Fig it is clear that q m q m and q m all
a
corresp ond to the low q region of phase space where the D p olar
ization is mostly longitudinal For q m we have moved to ab out
D
s
is a the mid p osition of the q range For B D P decays where P
pseudoscalar meson one can de ne the ratio
B D P
R
P
d B D dq j
q m P
Heavy Quark Physics and CP Violation
Table
Values of a extracted from
a comparison of hadronic and
semileptonic B decays The con
sistency of these values is an in
dication of whether factorization
provides a go o d description of
the hadronic decays
Mo de a
B D
B D
B D
B D
B D a
B D D
s
f a jV j X q m
ij
P P P
The quantities X and X are for DP and D P decays resp ectively
P
P
and are equal to the squares of the ratios of form factors for hadronic
and semileptonic decay The ratio is de ned precisely in Neub ert et
al In most cases X is very close to one For B D V
P
the corresp onding ratios X and X are exactly unity decays
V
V
The ratio on the upp er line of Eq can be evaluated from ex
p eriment while the lower line f a jV j can b e calculated from
ij
p
theory the comparison giving a test of whether factorization is valid
The constant a canbe calculated from QCD as discussed ab ove
but I prefer to use Eq to extract a for di erent decay mo des and
then to check whether the resulting values are consistent with each
other and the prediction
Table shows the v alues of a for a set of B decay mo des that can
be compared with B D and B D The agreement
among these values is reasonably go o d indicating that at the lowvalues
of q fast recoil at which these hadronic decays o ccur factorization
provides a go o d description of the pro cess The data for the comparison
with B D are taken from the CLEO while the B D
comparison is made using combined CLEO and ARGUS data compiled
D D B in Browder et al The data for the are from
s
CLEO and assume f f
D D
s
s
We can also compare the p olarization of the D in the hadronic
decay B D with that in the semileptonic decay B D
J D Richman
measured at the same value of q In a semileptonic decay at low
q the lepton and antineutrino are nearly collinear so their net spin
along their direction of motion is zero Since the B has spin zero
the recoiling meson must also have helicity zero In the factorization
picture the quark antiquark pair from the W would behave like the
lepton antineutrino system so that the or D is exp ected to be
almost completely longitudinally p olarized This explains why B B
D B B D An up date of CLEO I I measurement of the
B D form factors gives For
L
q m
B D CLEO obtains avery similar
L
value It will be interesting to make this comparison at higher q for
example when there are enough events in B D D to measure the
s
angular distributions
Wehave considered the relatively small slice of B decays that corre
sp ond to two body or quasi two b o dy mo des with easily reconstructed
nal states There are many other imp ortant topics in hadronic de
cays including the observation of B D D K mo des that I
cannot cover here
I will conclude this section with a list of some inclusive prop erties
of B decays Table The information in the table can be used to
compute the average number of c or c quarks pro duced per B decay
harmonium states counttwice The value measured by CLEO c
hn i can be used along with the semileptonic branching
c
fraction to test our understanding of the hadronic rate Historically
theorists have had an extremely di cult time obtaining a value of the
semileptonic branching fraction B that is as low as the measured
SL
value see Sec Bigi et al argued that B must be greater than
SL
if measurements yielded a lower value this could b e an indica
tion either that the perturbative QCD corrections to the semileptonic
rate were surprisingly large or that there was new physics involved
This issue is related to the value of n b ecause a larger B ccX rate
c
would increase n and lower B Recent analyses of this problem have
c SL
therefore fo cussed on understanding b oth of these quantities together
For a recent review see Drell
Lifetimes
The most striking overall feature of b hadron lifetimes is how similar
they are in contrast with those for charm hadrons I will not discuss
few of the charm lifetimes in any detail but it is worth recalling a
measured values which are listed in Table Charm lifetimes lie
Heavy Quark Physics and CP Violation
Table
CLEO I I measurements of inclusive prop erties of B
meson decays
Mo de Branching Fraction
B D X
B D X
B D X
B D X
B D X B D X
B D X
s
B X
c
B X
c
B X
c
B B CharmoniaX
hn i
c
roughly in the range ps to ps even though considerable this
range is still much less than that in the kaon system where
K ps K ps
L
K ps K ps
S S
In the early days of charm physics many people exp ected D
and D to be very similar It was argued that the lifetime of the
charm quark should essentially determine the lifetime of the hadron
with the other quarks acting as sp ectators Initially the lifetime ra
tio D D was inferred from the ratio of semileptonic branching
fractions In the s high statistics xed target charm exp eriments
at Fermilab nailed down the individual lifetimes using direct decay
length measurements With the installation of a new silicon strip ver
tex detector the CLEO exp eriment should so on b e comp etitive in this
areaaswell From Table we see the uncertainties on the D lifetimes
are now quite small Although the lifetimes are roughly similar we see
that D D D This observation suggests that it is
s
the D lifetime that is in some sense anomalous as discussed in the
previous section there is also go o d evidence that destructive interfer
ence in hadronic D decays suppresses its rate The b quark however
ful ll the old exp ectation is su ciently heavy that b hadrons b egin to
that the lifetimes should be nearly equal
Lifetimes of b hadrons have b een measured by the LEP exp eriments
and SLD using Z bb and by the CDF exp eriment which uses
pp bbX Figure lists the averages computed by the LEP B
J D Richman
Table
A summary of lifetimes
for ground state charm
hadrons The values are
averages from the
Particle Data Group up
date
Particle s
D cu
D cd
cs D
s
cud
c
csu
c
csd
c
css
c
Lifetime Working Group Their averages take into account the
many sources of correlated exp erimental error such as assumed frag
mentation mo dels decay mo dels and branching fractions Individual
measurements of B and B lifetimes are reaching an impressive pre
cision with systematic errors as small as ps CDF and total
errors as small as ps DELPHI There are substantial improve
ments in the lifetime measurements for b oth the B and but here
s b
the even t samples are much smaller and more data are needed to ob
tain precise values Before considering these measurements welookat
some of the theoretical predictions
Predictions for ratios of b hadron lifetimes are based on the heavy
quark m expansion and have b een made by many theorists I
Q
will give only a couple of examples here A recent review of inclusive
b hadron predictions based on the heavy quark expansion was given
by Mannel Neub ert gives the following estimates for the
lifetime ratios
B
O m
b
B
B
s
O m
b
B
b
O m
b
B
where the estimate for B includes corrections that arise at
b
Although these ratios might app ear very close to unity order m b
Heavy Quark Physics and CP Violation
Fig Summary of b hadron lifetimes The averages currentas of Oct were
calculated by the LEP B Lifetime Working Group and takeinto account correlated
systematic errors among the measurements Figure redrawn and used with p ermission
Neub ert argues that the m corrections to B B might be
b
large due to phase space enhancement of e ects involving the sp ec
tator quark and he concludes that theoretical uncertainties allow a
lifetime ratio in the range B B This sub ject is
controversial and other theorists have placed tighter contraints on this
ratio For example Bigi concludes that the B lifetime should
be longer than the B lifetime
f B
B
B MeV
There is more consensus on B B which is exp ected to b e unity
s
up to corrections of order Theoretical estimates for are
B
b
typically in the range to
Lifetime measurements can be group ed into three broad categories
corresp onding to the use of semileptonic decays fully reconstructed
hadronic decays and inclusive metho ds such as top ological vertex
ing The main advantages of using semileptonic decays such as
fractions the presence of B D X are the large branching
the lepton and good vertex determination There are however sig
ni cant disadvantages Because there is always at least one missing
particle the neutrino it is generally not p ossible to reconstruct a B
mass peak As a consequence it can b e di cult to determine whether
J D Richman
Fig Prop er time distribution for the ALEPH measurementof B and B using
semileptonic B decay The p oints with errors show the data for a D events and
b D events which corresp ond mainly to the B and B decays The dashed curves
represent the background contribution and the solid curves show the total t
there are additional missing particles esp ecially neutrals The neutrino
and other missing particles degrade the B momentum resolution and
hence reduce the precision of the prop er decay time measurement In
addition missing charged particles hinder the clean separation of B
ws the prop er time distributions from and B samples Figure sho
ALEPH for D and D samples whichwould ideally corre
sp ond to B and B decays In reality the D sample is
pure and the D is pure so these distributions are t si
multaneously A mo del is needed to account for p ossible feed down con
tributions from various D mo des In spite of these complications
ALEPH achieves avery go o d systematic error on these measurements
ps and ps DEL
B
B
PHI rep orts a very precise result B ps
based on a study of B D X decays Their measurement uses
a metho d in whichtheD D decay is reconstructed inclusively
leading to a sample of decays
The second metho d for determining b hadron lifetimes uses fully
reconstructed exclusive hadronic decays and the analyses are usually
much simpler than those using semileptonic mo des A b hadron mass
study the lifetime p eak is observed whose sidebands can be used to
Heavy Quark Physics and CP Violation
distribution of the background Since all particles are observed the
decay vertex and the b hadron momentum are well determined and
the conversion to prop er lifetime is very straightforward The only
disadvantage is that the event samples are typically smaller than those
for semileptonic decay In hadron colliders however a su ciently large
number of B mesons is pro duced for decays to nal states with J s to
provide a comp etitive metho d for measuring lifetimes Figure shows
the reconstructed B mass from CDF for the nal states J K
J K S K and S K The upp er and lower histogram
in the gure show the data b efore and after the prop er decay length
cut c m This cut is not actually used for the lifetime t but
it demonstrates that the background is concentrated at short prop er
lifetimes Figure shows the distribution of prop er decay lengths
in the signal region upp er histogram and in the B mass sidebands
lower histogram together with the ts to the data This measurement
yields the value B ps the result for the B
lifetime is B ps In my view these very small
systematic errors which are related to the considerations discussed
ab ove demonstrate that hadronic decays are signi cantly b etter than
semileptonic mo des for lifetime measurements assuming that enough
events are available
The third metho d for measuring b hadron lifetimes makes use of
top ological vertexing in whichtheB meson charge is determined from
the total charge of the tracks asso ciated with the decay vertex The
SLD results are
B ps
B ps
The measurement errors here are comparable to those of many of the
LEP and CDF results despite a much smaller event sample
hadronic Z decays This is due not only to the inclusive nature of the
metho d but also to the advantages of the SLC accelerator with its
very small diameter b eams small radius b eampip e and electron b eam
p olarization
To compare the B and B lifetimes it is b est to calculate the
lifetime ratio for each exp eriment so that man y systematics cancel
and then to compute the average of the individual lifetime ratios The
average computed by the LEP B Lifetime Working Group is
B B
This value is certainly consistent with unity but it is not su ciently
J D Richman
Fig CDF measurementof B The upp er histogram a shows the invariant mass
distributions for B candidates in the J K J K S K and S K nal
states b shows the same distributions after the cut c m Figure used with
p ermission
precise to rule out a lifetime di erence at the level The p ositive
sign of a a however would indicate that B B As
other B decay mo des are observed it will b e interesting to see if the
interference is constructive or destructive
The uncertainties in B lifetime measurements are much larger than
s
those for the B and B due to smaller event samples and larger combi
natorial backgrounds Most of the measurements use the semileptonic
decay B D X and reconstruct the D decay in one or more
s
s s
mo des although some use an inclusive D signal or a D with an
s s
asso ciated hadron The average lifetime from measurements based on
these mo des is
B ps
s
consistent with the B and B lifetimes
Because the B and B can oscillate into one another the two
s s
mass eigenstates are exp ected to di er somewhat in their masses and
lifetimes Theoretical estimates indicate that the di erence in decay
widths may be as large as CDF has recon
structed events in the decay B J which is exp ected to b e
s
X nal state which the D predominantly CP even in contrast to s
Heavy Quark Physics and CP Violation
Fig CDF measurementof B The upp er histogram shows the prop er decay length
distribution for events in the B mass p eak the lower histogram shows the distribution for
events in the B mass sidebands Figure used with p ermission
is exp ected to b e an equal mixture of CP even and CP o dd The result
from B J B ps is not yet su ciently
s s
precise to address this issue but future measurements with more data
should b e extremely interesting
The measured values of b baryon lifetimes are systematically lower
than those for B mesons These analyses are p erformed using a vari
ety of metho ds that select di erent comp ositions of b baryons whose
pro duction fractions at the Z or at the Fermilab collider are not well
known In fact some pap ers use the symbol to denote a generic
b
b baryon but I use to denote the I bud ground state baryon
b
Measurements of the lifetime generally use correlations to se
b c
lect events from the decay X By fully reconstructing
b c
decay and using kinematic cuts one can obtain a the pK
c
high purity sample of decays Alternatively one can ob
b c
tain larger event samples by using the inclusive decay X and
c
searching for or even p combinations These metho ds however
have a lower purity of baryons since for example there can be
b
background from X
b c c
By including measurements that use and p correlations the
LEP B Lifetime Working Group obtains the average b baryon
J D Richman
ps This lowvalue is di cult to accomo date in calculations
based on the m expansion which predict B tobeinthe
Q b
range to
OPAL has measured the ratio R B b baryon
X B b baryon X which should be a good approximation
to the average b baryon semileptonic branching fraction They obtain
R which is consisten t with the exp ectation
based on the shorter b baryon lifetime and the assumption of equal
semileptonic widths of B mesons and b baryons
In spite of a tremendous e ort there remain basic questions on b
hadron lifetimes all of which call for more precision
How similar are the B B B lifetimes
s
Are the b baryon lifetimes really signi cantly lower than those of the
mesons
Do the B mass eigenstates have signi cantly di erent lifetimes
s
The CDF exp eriment should b e able to contribute much to answering
these questions as it obtains more data
Rar e decays
With B data samples of over events exp eriments havenowachieved
sensitivities to branching fractions in the range to op ening
up new typ es of pro cesses for study Several new mo des were recently
discovered by CLEO and it is likely that many more will be found in
the next few years The study of rare decays has therefore become an
exciting and rapidly developing area of B physics There are imp ortant
rare pro cesses to search for in charm decays as well Because charm
particles have Cabibb o favored c s decays however rare mo des tend
to have smaller branching fractions and it is more di cult to achievea
comparable sensitivity to exotic pro cesses than it is in B decays Lingel
Skwarnicki and Smith have written a review of rare decays from
an exp erimental p ersp ective recent theoretical reviews include those
by Buras and Fleischer and Ali
In this section we consider three typ es of rare decays
Hadronic decays resulting from b u transitions These are very
similar to the hadronic decays discussed in Sec but are highly
due to the factor jV j suppressed
ub
Hadronic decays resulting from p enguin pro cesses Sometimes called
gluonic p enguins these decays involve the emission and reabsorption of
a W resulting in a b Q s or b Q d transition
where Q is u c or t
Heavy Quark Physics and CP Violation
Radiative penguin decays These pro cesses also involve the emis
sion and reabsorption of a W but a photon is radiated from one of
the charged particles Radiative p enguins with real photons such as
B K have b een discovered The photon can also b e virtual ma
terializing into a e e or pair At present exp eriments do not
have su cient sensitivitytoobserve these decays
Many rare hadronic decays such as B and B K can
pro ceed either through a b u sp ectator diagram or through a gluonic
p enguin as shown in Fig In a gluonic p enguin a b d or b s
transition o ccurs through a virtual lo op containing a W and either a t
c oru quark with the radiation of a gluon The dominant contribution
is exp ected to arise from the t quark intermediate state but e ects from
the c quark are not necessarily negligible Buras and Fleischer
discuss this p ointindetail
The gluonic p enguins are of interest not only as one lo op pro cesses
but also b ecause they are relevanttostudies of CP violation We will
see in Sec that CP violation can o ccur in a numb er of di erentways
all of which require interference between at least two amplitudes CP
violation can result from interference between tree and penguin dia
grams for example giving a di erence in rate between B K
and B K Another metho d for studying CP violation can
be applied when b oth B and B decay into the same nal state f
Then interference between B f and B B f can pro duce a
CP asymmetry In this case it is quite undesirable to ha ve amplitudes
with di erent CKM factors contributing to the decay b ecause the mea
surement b ecomes sensitive to p o orly known strong interaction phases
The term p enguin p ollution is used to describ e this problem It is
amazing how quickly a fascinating new pro cess can b ecome a problem
for the next set of measurements
A simple argumentgives a rough idea of the p ossible relativecontri
bution of the tree and p enguin amplitudes in B and B K
Assuming that the t quark dominates the lo op the p enguin contribu
tion to B is suppressed relative to that for B K by
j O in the rate where is the sine of the Cabibb o jV V
ts td
angle The same conclusion holds when there is a c quark in the inter
mediate state Furthermore the tree contribution to B K is
suppressed relativetothatforB by a factor jV V j
us ud
in the rate By itself all this tells us is that the tree diagram con
tributes more to than to K and the p enguin contributes more to
K than to
B B K as is crudely Now supp ose that B B
J D Richman
Fig Diagrams for B and B K Both mo des have contributions from
tree b u and p enguin pro cesses The gure shows the dep endence of each amplitude
on sin as determined from the Wolfenstein parametrization of the CKM matrix
C
Simple p ower counting shows that the tree pro cess should contribute more to B
than to B K while the p enguin pro cess with intermediate t or c quarks should
contribute more to B K than to B Both the tree and p enguin
are O In a scenario where B B K B B contributions to B
we could conclude that B K was predominantly p enguin and that B
was predominantly a tree pro cess That is the upp er left and lower right diagrams
must dominate For if the B K were mainly a tree pro cess upp er right diagram
O then the B tree contribution upp er left O would b e even larger
and wewould observe B B B B K Similarly ifB were
mainly p enguin lower left O then B K p enguin contribution lower right
O would b e even larger Current data havevery large statistical uncertainties but
suggest that the p enguin amplitude may b e somewhat larger than exp ected increasing
the B K rate and pro ducing p enguin p ollution in B
indicated by exp eriment Then our CKM argument implies that B
K must b e mainly p enguin or else the K ratio would b e
larger Furthermore even if all of the B K rate were p enguin
the assumed near equality of the branching fractions implies that the
p enguin contribution to B must be fairly small since this
contribution is suppressed by in the rate relative to the penguin
contribution to K B
The dominant background in most CLEO measurements of rare
due to continuum pro cesses which can pro hadronic decay mo des is
duce high momentum tracks In the analyses great e ort is required
Heavy Quark Physics and CP Violation
0990695-018 400
300
200 Arbitrary Scale
100
0 -1.0 -0.5 0 0.5 1.0 cos
θT
Fig CLEO II distribution of the cos variable for B histogram and
T
for continuum background solid squares The signal distribution is taken from a Monte
Carlo simulation
to suppress this background Figure shows one powerful variable
that is used for this purp ose We de ne as the p olar angle b etween
T
the thrust axis of the B candidate tracks b oth charged and neutral
and the thrust axis of all the remaining tracks in the event In jetlike
continuum events these axes tend to be highly correlated resulting
in peaks at cos while for the more spherical S B B
T
events the axes are essentially uncorrelated resulting in a uniform dis
tribution
Exp erimentally it can be quite di cult to distinguish between
B and B K b ecause the daughter hadrons have
momenta p GeV c In CLEO the dE dx separation between
kaons and pions at this momentum is only Until recently b oth
e only been able to rep ort mea CLEO and the LEP exp eriments hav
surements of the sum of the two mo des Table lists a set of new
CLEO II measurements based on a sample of million B B
pairs of B B K and B KK mo des in various charge
states The B signi cance at is still not su ciently
has a signif large to claim a signal but the B K mo de
icance and has a branching fraction of Figure
shows some of the signals The quoted signi cance is based on a careful
J D Richman
Table
CLEO I I branching fractions B and theoretical predictions for B K
B and B KK N is the number of signal events The quoted
S
signi cance Sig is statistical only E is the detection e ciency The errors
on B are statistical t systematics and e ciency systematics resp ectively
The symbol h represents either a or a K The results are based on
million S B B decays From Ref whichcontains references to
the theoretical predictions Following standard practice conjugate mo des are
included that is each branching fraction is the average over the indicated
nal state and its conjugate
Mo de N Sig E B Theory B
S
K
K
K
K
K K
K K
K K
h
analysis of the joint K likeliho o d function which is shown
along with those for other mo des in Fig
The B branching fraction is of great interest b ecause this
mo de can be used to measure the CKM angle Using the measured
branching fraction for B
together with the factorization hyp othesis one can estimate
B B very close to B B K
given in Table The central value of the t to the data
B B is consistent with the factorization
mo del prediction It has been said that the B branching
fraction is turning out to b e low er than exp ected but at this p oint the
uncertainties are to o large to draw any strong conclusion
Table also shows that there is a signi cant signal for B K
or B K In Fig we see that the p enguin diagram pro duces
the appropriate set of quarks for this decay sddu while the tree dia
gram pro duces su uu Wemight conclude that there can b e absolutely
no tree contribution in which case this signal would represent the rst
unambiguous observation of a gluonic penguin pro cess In principle
system in the however nal state interactions could convert the uu
Heavy Quark Physics and CP Violation
3461197-014 8
+ I ( a ) K
4
0 + ( b ) h 0
4 Events / 2.5 MeV 0 ( c ) K0 + 4
2
0 5.20 5.22 5.24 5.26 5.28 I 0.1 0 0.1 0.2 M (GeV) E (GeV)
B
Fig CLEO I I observation of a B K b B h h is either a or
a K and c B K The solid curves show the scaled pro jections of the total
likeliho o d t while the dotted curves show the continuum background comp onentineach
channel The E distributions are calculated assuming B and the signal E
distribution for B K is centered at MeV This kinematic distribution gives
a separation b etween B K and B
tree diagram into a dd pair To the extent that nal state interactions
are likely to be small the original conclusion still holds Furthermore
the amplitude for the tree diagram is O while that for the p enguin
is O making it extremely unlikely that the tree contribution could
be signi cant
A number of other rare B decay mo des have now b een observed
The most striking are B K and B K whose branching
fractions are listed in Table Although the branching fractions
have large uncertainties their large values suggest that something in
teresting might b e going on Figure shows that there are sev eral di
agrams that may contribute to these mo des CLEO has also searched
for two b o dy B decays with or mesons The K mo des have
no tree contributions while the K mo des can have b oth penguin
and tree contributions A signi cantsignalforB K is listed in
Table
Radiative p enguins are pro cesses in which the e ective avor
changing neutral current transition is accompanied by the emission of
a photon rather than a gluon Although these are electromagnetic de
J D Richman
40 3461097-013 ( a ) 5 4
30 I 3
+ 20 2
N 1
10 X
0 1020 30 40 N I +
+I K 40 ( b ) 5 30 4
0 3 I
+ 20 2
N 1 10 X
0 10 20 30 40 N
+I K 0 20 ( c ) 16 4
3 I + 12 2 0 1 K S X N 8
+ 4 I
0 4 8 12 16 20
N +I K0K
S
Fig Likeliho o d contours for the CLEO I I B B K andB KK analyses
The b est t results are marked with an x
cays strong interaction e ects are extremely imp ortant giving a large
enhancement in the rate There has b een a heroic theoretical e ort
to calculate the standard mo del prediction to next to leading order in
QCD this work involved many groups and was completed in A
review with references is given by Buras and Fleischer At the same
time there havebeenintensive exp erimental investigations Currently
the branching fraction for the exclusive decay B K has b een mea
Heavy Quark Physics and CP Violation
t
Fig The decay B K has a p enguin contribution left The tree diagram
right do es not contain a d quark and therefore cannot directly pro duce a K In principle
nal state interactions which are exp ected to b e small could convert the uu pair pro duced
in the tree diagram into a dd pair and a K system could b e pro duced In addition
the amplitude for the tree diagram is O while that for the p enguin is O
3300198-001 + + W W I I I I b s b s + + , K , K s u * + + g I B u, c, tg B u, c, t I s + + u K , K , u u * u u
( a ) ( b )
u, c, t g u + + I
K , K I + I * W s b s I I + + b u + + g K , K* + B W B , u u u u
( c ) ( d )
Fig Diagrams that can contribute to B K a and b showtwointernal
p enguin pro cesses c shows an external sp ectator diagram and d shows a avor singlet
p enguin Constructiveinterference among these pro cesses may account for the large ob
served rate for B K
sured by CLEO and that for the inclusive b s pro cess has been
measured by both CLEO and ALEPH
Figure shows the standard mo del diagrams for B K and
B Part of the interest in these decays comes from the p ossibility
that b esides the W other charged particles such as charged Higgs
b osons could app ear in the lo op There are a host of predictions from
extended versions of the standard mo del with extra Higgs doublets
from SUSY mo dels and from mo dels with comp osite W b osons
Historically the rst signal for an electromagnetic penguin came
J D Richman
Table
CLEO I I results on selected rare hadronic B decays The symbol
h represents either a or a K The results are based on
million S B B decays These results as well as references to
the theoretical predictions are given in Ref and Ref
Mo de N E B Theory B
S
B K
B K
B K
B K
B K
B K
K B
B K
B K
B
B h
B
B K
B K
B
B
Table
CLEO I I upp er limits C L for B decays to
K and K The results are based on million
S B B decays From Ref which also con
tains references to the theoretical predictions
Mo de E B Theory B
B
B
B
B K
B K
B K
B K
B K
B K
B K
B K
from a CLEO analysis of the exclusive decay B K Why
lo ok for K rather than K The answer is simple B K violates
Heavy Quark Physics and CP Violation
Table
CLEO I I B K results N is the numb er of signal
S
events E is the e ciency and B is the branching fraction
Final state N E B
S
K K
K K
K K
K K
Combined K
conservation of angular momentum since the K cannot cancel the pho
ton s helicity and there can be no orbital angular momentum pro jec
tion along the decay axis An up dated version of the original analysis
with million S B B events was later p erformed The
result is
B B K
The signal is observed in four nal states B K with K
K or K K and B K with K K or K
K A comparison of the yields and e ciencies of these mo des is
given in Table
Fig Diagrams for the radiative p enguin decays B K left and B right
The measurement of the inclusive B X branching fraction has
s
b een p erformed by b oth CLEO and ALEPH Here X represents any
s
kinematically accessible nal state containing a strange particle It
is remarkable that such an analysis is even p ossible given the huge
number of photons from s The large value of the photon energies
a signi cant help From a theoretical GeV to GeV is
p ersp ective this measurement is much more valuable than that for
J D Richman
B K which is harder to predict since one must understand the
hadronization into a particular nal state rather than summing over all
nal states The standard mo del prediction including next to leading
order QCD corrections gives
B B X
s
The relatively small uncertainty on this quantity represents a ma jor
achievement The prediction is scaled to the measured semileptonic
rate and uses unitarityofthe CKM matrix to determine jV j
ts
Two essentially indep endent measurements w ere p erformed in
CLEO The rst analysis was based on the photon energy sp ectrum
and event shap e cuts while the second also attempted to explicitly re
construct the X system into nal states K K n where n
s
and at most one was allowed Figure shows the photon energy
sp ectrum from the second analysis which has a signi cant excess of
events over the predicted background Figure shows the distribu
tion of the X masses recoiling against the photon A p eak is observed
s
at the K mass as well as a broader bump ab ove ab out GeV where
there are many high mass strange states Together the two CLEO II
inclusive analyses give
B B X
s
This result is based on a sample of S B B events An
ALEPH analysis using Z bb events gives
B b s
Both of these results are consistent with the standard mo del prediction
they provide a powerful constraint on parameters in given ab ove and
theories beyond the standard mo del
CP Violation and Oscillations
Fundamental symmetries and symmetry breaking are central issues in
particle physics The three symmetry op erators C P and T pro duce
discrete transformations
C a a a is the antiparticle of a
P x x parityinversion of spatial co ordinates
T t t motion or time reversal
Heavy Quark Physics and CP Violation
1851194-003 200 ( a ) 150
100
50
0 ( b )
Events / 0.1 GeV 25
0
-25 1.8 2.0 2.2 2.4 2.6 2.8 E (GeV)
γ
Fig CLEO I I photon energy sp ectrum from B X analysis a shows the S
s
data solid histogram the scaled o resonance data dashed histogram and the sum of
scaled o resonance and the background from S squares with error bars b shows
the backgound subtracted data squares with error bars and the Monte Carlo prediction
for the shap e of the B X signal solid curve
s
These transformations play a sp ecial role in our understanding and
classi cation of particle interactions The famous CP T theorem states
that all relativistic quantum eld theories must p ossess symmetry un
der the combined CP T op eration so that CP violation implies a com
p ensating T violation Both e ects havenow b een observed in neutral
K decays while no CP T violation is seen
CP violation was discovered at the Bro okhaven Alternating
GradientSynchrotron AGS in well b efore there was a theoreti
cal framework let alone a prediction to guide exp erimentalists Chris
tenson Cronin Fitch and Turlay observed the decay K
L
with a branching fraction of If CP were conserved
this decay would be forbidden as discussed later in this section I
particularly recommend the articles by Cronin and Fitch in
Reviews of Modern Physics which contain their Nob el Prize lectures
Fitch notes that
Professor Cronin and I are being honored for a purely exp eri
there were no precursive mental discovery a discovery for which
indications either theoretical or exp erimental It is a discovery
J D Richman
1851194-004 20
15
10
5
Events / 0.1 GeV 0
-5
-10 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M (X ) (GeV)
s
Fig CLEO I I inclusive analysis of B X the apparent distribution of M X
s s
The squares with error bars represent the background subtracted data not corrected
for e ciency or cross feed the solid histogram shows the t to the Monte Carlo based
distributions and the dotted histogram shows the comp onent of the t from resonances
other than K
for which after more than years years there is no satisfactory
accounting
CP violation was not easy for everyone to swallow and there was a lot
of healthy skepticism regarding the measurement Cronin recalls the
International Conference on High Energy Physics at Dubna
As the session neared a close one of my Soviet colleagues sug
gested that p erhaps the e ect was due to regeneration of short
lived K mesons K in a y unfortunately trapp ed in the helium
S
bag We did a quick backoftheenvelop e estimate of the density
of the y necessary to pro duce the e ect The density required
was far in excess of uranium
The exp erimental result withsto o d all scrutiny but the source of CP
violation remained a mystery We now do have a possible explanation
for CP violation namely interference e ects that are sensitive to the
phase in the CKM matrix Sec The CKM matrix in turn originates
in the electroweak symmetry breaking physics of the Higgs sector and
ways for this it is intimately related to quark masses There are many
Heavy Quark Physics and CP Violation
CP violating phase to manifest itself exp erimentally and several ma jor
new exp eriments will b egin to test the CKM framework using B decays
These exp eriments will b e able to measure CP odd observables or CP
asymmetries which vanish if there is no CP violation In the CKM
framework the amplitudes of CP violating pro cesses are prop ortional
to the area of the unitarity triangle We have already obtained some
imp ortant constraints on these quantities by measuring the lengths
of the sides of the unitarity triangle Sec and Sec If the new
measurements are not compatible with the CKM framework they will
op en the do or to physics beyond the standard mo del
Cronin concludes his article with the statement
the e ect is telling us that at some tinylevel there is a funda
mental asymmetry b etween matter and antimatter and it is also
telling us that at some tiny level interactions will showanasym
metry under the reversal of time Weknow that improvements in
detector technology and quality of accelerators will p ermit even
more sensitive exp eriments in coming decades We are hop eful
then that at some epoch p erhaps distant this cryptic message
from nature will b e deciphered
The following sections describ e the next stage in our e ort to decipher
the message
Introduction to CP violation
CP violation is a deep and fascinating sub ject but it is not always
easy to see through the formalism to the simplicity of the underlying
physics I will try to present a coherent picture of the di erenttyp es of
CP violation and I will discuss CP violation in K decays to a certain
extent In these lectures however I will only b e able to scratch the sur
face of the sub ject There are many excellent pap ers and b o oks on CP
violation and I have listed some of them in the references Ihave found
the discussions of Buras and Fleischer Fleischer Kayser
Leader and Predazzi Nakada Nelson Nir Nir
and Quinn Bigi and Sanda and Sanda to b e particu
larly useful A careful discussion of CP violation in the neutral kaon
system is given in the b o ok on weak interactions by Marshak Riazud
din and Ryan A much more recent review of this topic is given
by Winstein and Wolfenstein
y
The parity op erator P is a unitary op erator P P that reverses
the sign of spatial co ordinates r r If a particle a has four
J D Richman
momentum p E p and helicity the transformation is
P ja p i ja p i
P
where p E p Parity reverses the momentum direction but not
the spin direction the helicity s p therefore reverses sign under
y
parity Since P is de ned such that P we have P P P
From unitarity we have
y
hajai hajP P jai ha ja i
P P P P
P P
where ja i is the parity transformed version of jai This result implies
P
that exp i Moreover we can use P to show that
P P
P
jai P jai P j a i jai
P P
P
so that and
P
P
Charge conjugation C is a unitary op erator that changes particle to
antiparticle and vice versa Its name is somewhat misleading b ecause
it reverses not only electric charge but also other quantum numb ers
such as color R R It has no e ect on momentum spin direction
or helicity
C ja p i ja p i
C
where a is the antiparticle of a and is a phase factor Like P C is
C
y
de ned such that C so that C C C
We can let C op erate twice on a particle jai and use C
jai C jai a C ja i a a jai
C C C
as a conse so that a a Since must be a phase factor
C C C
quence of unitarity this implies
i i
C C
a a e a e
C C C
If jai is an eigenstate of C a a
C C
The combined e ect of C and P is
CP ja p i ja p i
CP
where From the prop erties of C and P itiseasytoshow
CP C P
y
CP CP CP that
Supp ose that jii and jf i are arbitrary states They can be multi
particle states for example and they mayhave de nite orbital angular
Heavy Quark Physics and CP Violation
momentum rather than de nite linear momentum If CP is conserved
CP H and we have
y y
hf jH jii hj CP CP H CP CP jii
y
hf j CP H CP jii
so that if b oth jf i and jii are eigenstates of CP
hf jH jii f i hf jH jii
CP CP
This relation implies either that the CP eigenvalues are the same or
that the amplitude is zero i e the transition is forbidden
What if CP is conserved but the states are not eigenstates of CP
Let s consider the decay a b c and analyze the amplitudes for
transitions between states of de nite linear momentum and helicity
Then
hb p c p jH ja p i
b b c c a a
y y
hb p c p j CP CP H CP CP ja p i
b b c c a a
b c a hb p c p jH ja p i
CP b b c c a a
CP CP
and p E p etc The matrix element where p E p
a a a a a a
on the right hand side of Eq is the amplitude for the transition
between the CP transformed versions of the original states We see
c a CP conservation b therefore that due to the factor
CP
CP CP
do es not quite imply that the original and CP transformed amplitudes
are the same although their magnitudes are In general it is imp or
tant to keep track of whether the states are lab eled by de nite linear
momenta or orbital angular momenta It can t be b oth When we
consider a state at rest it generally means that we are discussing an
intrinsic prop ertysuchastheintrinsic parity of the state and lab els
P
such as momentum and helicity are suppressed
We will often fo cus on pro cesses involving a pseudoscalar meson P
that is distinct from its CP conjugate particle P Neither P nor P
is an eigenstate of CP A standard phase convention is to de ne
CP jP i jP i
i CP jP i jP
Although this convention is natural and can b e convenient I will try to
keep nearly all of my presentation indep endentofany particular phase
convention This approach helps to reduce confusion between what is
J D Richman
and what is not a physically meaningful phase In fact we have the
freedom to rede ne the phases such that
i i
j P i e jP i j P i e jP i
This freedom is a consequence of avor conservation in the strong and
electromagnetic interactions which are used to de ne the quantum
numb ers of P Flavor conservation means that the interactions are
invariant under global U symmetry so we can rede ne the particle
phases as long as we do it in a consistent manner
Let s consider a simple example of K decay keeping the
arbitrary phase throughout the discussion Since the system is
either or in a de nite state of orbital angular momentum
it is an eigenstate of CP with eigenvalue equal to one For the
system for example
CP P C
C
The action of CP on jK i and jK i is given by
i i
CP CP
jK i jK i CP jK i e CP jK i e
where is an arbitrary phase The factor of in the exp onentisnot
CP
in any way fundamental but corresp onds to a common usage Let s
assume that CP is conserved in K decays Then
A h jH jK i
y y
j CP CP H CP CP jK i h
i
CP
e h jH jK i
i
CP
e A
y
where wehave used and CP H CP H It is easy to
CP
show that the orthogonal states
i
CP
p
jK i jK i jK i e
CP
i
CP
p
jK i e jK i jK i
CP
are eigenstates of CP with eigenvalues and resp ectively Then
A h jH jK i
CP
i h
i
CP
p
i h jH jK ie h jH jK
h i
i
CP
p
A A e
Heavy Quark Physics and CP Violation
h i
i i
CP CP
p
A A e e
p
A
where the relation between A and A is de ned in Eq Thus
if CP is conserved the CP o dd eigenstate cannot decay into In
p erforming the corresp onding exp eriment to search for CP violation it
is crucial to b e able to de ne the typ e of kaon that is decaying We will
encounter the general issue of tagging the initial state throughout the
discussion of CP violation and mixing
CP violation and cosmology
In Sakharov noted that CP violation has an imp ortant
connection to cosmology b ecause it is one of three conditions neces
sary to generate an asymmetry between the number of baryons and
antibaryons in the universe and therefore to explain the prep onderance
of matter over antimatter that we observe to day It is considered un
likely by some theorists that sources of CP violation accessible in B
physics such as the CKM matrix are resp onsible for this e ect and
the relation of up coming exp eriments to cosmological issues may be
overstated It is nevertheless interesting to see what role CP viola
tion plays in generating a baryon numb er asymmetry and we will now
examine this issue
Consider ah yp othetical particle X whichhas decay mo des to nal
states lab eled Y B where B is the di erence b etween the baryon
i i i
numb ers of Y and X If X has no baryon numb er then B is just
i i
the baryon number of Y and anymodewith B violates baryon
i i
numb er The universe is assumed initially to b e in thermal equilibrium
at some temp erature with equal numb ers of X and X particles In
thermal equilibrium all states with the same energy must be equally
p opulated Baryon numb er violation is a necessary condition for gen
but it is not su cient Why not erating a baryon numb er asymmetry
It is easy to show that starting with equal numb ers of X and X
particles baryon number violation in X decay would be exactly cancel led
by an opposite sign baryon number violation in X decay if either CP
or C were conserved Let X and X decay with branching fractions
f B X Y B
i i i
f B X Y B
i i i
J D Richman
The net baryon number resulting from these decays is
X X
B B X Y B B B X Y B B
net i i i i i i
i i
X
f f B
i i i
i
Thus B implies that there must b e a decay mo de that has b oth
net
baryon number violation B and a di erence in the partial
i
widths f and f Baryon number violation by itself is not su cient
i i
Equality of f and f can be guaranteed either by C conservation
i i
or CP conservation either of these symmetries relates the particle
decay pro cess to the antiparticle decay pro cess For C the helicities
are unchanged while for CP they are reversed However we are simply
counting the numb er of baryons regardless of their spin states so either
C or CP invariance implies that the p opulations of Y and Y will be
i i
the same Thus to violate baryon n umber we need to violate both C
and CP If b oth symmetries are broken then the one with the smaller
symmetry violation will determine the baryon number asymmetry In
the weak interactions C is violated to a much greater extentthanCP
so if weak pro cesses were the source of the baryon numb er asymmetry
it would be the extent of CP violation that would determine the size
of the asymmetry
One more condition is necessary to generate a baryon numb er asym
metry departure from thermal equilibrium If the system is in thermal
equilibrium all states with the same energy will b e equally p opulated
so particle and antiparticle p opulations will be the same Thermal
equilibrium means that crudely sp eaking there is an in nite amount
of time at least compared to the time scales of the pro cesses under
consideration Thus we have Sakharov s famous three conditions for
generating a baryon number asymmetry in the early universe
i baryon number violation
ii C and CP violation
iii departure from thermal equilibrium
The CP violating mechanism op erative in the early universe may
well have no relation to the CP asymmetries in K or B decays As
particle physicists we shouldn t ap ologize even if it do esn t what we
observe will b e imp ortant enough It is fair to say however that stud
a necessary rst step in a ies of these pro cesses can be regarded as
Heavy Quark Physics and CP Violation
long quest to understand the broader issues and implications of CP
violation
A CP violating phase factor can manifest itself in several ways
CP violation in decay
CP violation in mixing
CP violation in the interference between mixing and decay
In the following sections we will consider each of these in turn
CP violation in decay direct CP violation
Conceptually one of the simplest ways to study CP violation is to
compare the decay rates P f with P f where P is a
pseudoscalar meson and f and f are CP conjugate nal states One
can also study direct CP violation by comparing the Dalitz plots for P
and P decays but here we will sum over all kinematic con gurations
so the particle states will not carry momentum lab els We de ne the
action of the CP op erator on the states jP i and jf i by
i P
CP jP i e jP i
i f
CP jf i e jf i
where the values of the phases are convention dep endent Given the
large number of things that P can represent I should emphasize that
exp i P is the intrinsic CP phase factor asso ciated with the pseu
doscalar particle P
We can determine a simple condition that determines whether CP
is conserved or violated in a decay pro cess Let s assume that CP is
conserved H C P We can then write the amplitude A A P
f for the P f decay as
y y
A hf jH j P i hf j CP CP H CP CP jP i
y i P f
hf j CP H CP jP i e
i P f
hf jH jP i e
i P f
A e
where
A A P f hf jH jP i
I am using the symbol hf jH jP i somewhat lo osely here Strictly sp eaking the amplitude
A P f isgiven by the matrix elementofthetransition operator and I have just written
the rst order term of the p erturbation expansion For our purp oses the distinction is
not imp ortant
J D Richman
is the amplitude for the CP conjugate pro cess Thus CP conservation
implies jA Aj j exp i P f j indep endent of phase
convention so that
A
CP violation in decay
A
If this condition is met the rates P f and P f will be
di erent which we can express as an asymmetry
P f P f
A
P f P f
In the standard mo del CP conjugate amplitudes di er from the orig
inal amplitude at most by a phase factor As a consequence if only a
single amplitude contributes to a given decay process there cannot be
an observable CP asymmetry
Wenow supp ose that there is more than one amplitude A and that
j
each amplitude has an asso ciated CP violating phase
X X
i
j
A hf jH jP i A a e
j j
j j
X
i P f i
j
P i e a e A hf jH j
j
j
This equation is similar to Eq except that there are phases that
change sign under CP These so called weak phases can be phases
from the CKM matrix but they could also be due to new physics In
general the co e cients a will b e of the form
j
i
j
a ja je
j j
where the are non CP violating phases that can arise for example
j
from strong nal state interactions The are often called strong
j
phases they do not change sign under CP Let us rst assume that
all such phases are zero Then
j
P
i
j
ja je
A
j
j
P
i
j
ja je A
j
j
P P
ja j cos i ja j sin
j j j j
j j
P P
ja j cos i ja j sin
j j j j
j j
Simply put the numerator is just the complex conjugate of the de
nominator so the magnitudes of the amplitudes are the same This
Heavy Quark Physics and CP Violation
situation is shown in Fig a for the case of two interfering ampli
tudes Thus the presence of CP violating phases even if there is more
than one amplitude is not su cient to pro duce a CP asymmetry We
now allow for the p ossibility that there are non zero strong phases
j
in the amplitudes We can rewrite our result as
P
i
j j
ja je
A
j
j
P
i
j j
A ja je
j
j
P P
ja j cos i ja j sin
j j j j j j
j j
P P
ja j sin ja j cos i
j j j j j j
j j
An alternativeway to express the result is
X
jAj jAj ja jja j sin sin
i j i j i j
i j
Usually we are just concerned with two amplitudes in which case the
asymmetry is
jAj j Aj ja jja j sin sin
A
jAj jAj ja j ja j ja jja j cos cos
Examples showing the interaction of the strong and weak phases are
shown in Fig b and Fig c
These results have imp ortant implications We can observe CP
violating e ects by comparing P f with P f only if there
are both CP violating and non CP violating phases In fact this may
well be the way in which CP violation is discovered in the B system
For example one could compare B K with B K
These decays have b oth tree and p enguin contributions whichhavedif
ferent weak and presumably di erent strong phases Unfortunately
it is not p ossible at present to calculate the strong phases Although
observation of direct CP violation in B decays would be very imp or
general to tant the value of the weak phase would be ambiguous In
obtain a large CP asymmetry we need twointerfering amplitudes that
are roughly equal in magnitude By studying the interference between
mixing and decay we can achieve this situation Furthermore the un
certainties asso ciated with strong interaction e ects can b e eliminated
if there is only one amplitude for the direct decay and one can then
extract precise CKM phase information
J D Richman
Fig Amplitudes a and a in direct CP violation The amplitudes for the conjugate
pro cess are a and a In a there is a relative weak phase between a and a but
no relative strong phase In this case the CP conjugate amplitudea a a has the
same magnitude as the original amplitude a and there is no CP asymmetry In b and
c there is b oth a relative weak and a relative strong phase b etween a and a The
magnitudes ofa and a now di er giving a CP asymmetry
An example of what can be done with direct CP violation is the
Fleischer Mannel bound which is based on an analysis
of branching fractions for the various charged and neutral B K
mo des Here I brie y summarize the argument full details and dis
cussion of the many controversial assumptions are contained in the
references We have seen in Sec that the decay B K has
b oth p enguin and tree contributions while to a go o d approximation
Heavy Quark Physics and CP Violation
B K is entirely a p enguin pro cess Furthermore the penguin
amplitudes for these pro cesses should b e essentially identical since the
corresp onding decays di er only in the isospin of the sp ectator quark
We can therefore write the decay rates as
i i
B K jA A e e j
P T
i i
B K jA A e e j
P T
K jA j B K B
P
where A and A contain no CKM phases and is the relative strong
P T
phase shift between the p enguin and tree pro cesses As we have seen
CLEO has rep orted the average branching fraction for B K
and B K We therefore compute
B K B K
jA j r cos cos r
P
where r A A Thus we can compute
T P
B K
d
R r cos cos r
B K
The minimum value of R as a function of r is attained when r
cos cos and is given by
R cos cos sin
Assuming that sin we have a constraint in the plane
sin R
Current data yield R which suggests that there maybea
useful constraint once the exp erimental uncertainties are reduced with
more data and if various theoretical issues are resolved
There has b een a tremendous exp erimental e ort to detect
direct CP violation in K decays In general a two pion system
can have isospin I or Bose symmetry however requires that
the overall wave function of the system be symmetric and since
angular momentum conservation in K requires the isospin
wave function must be symmetric As a consequence only I and
I are allowed The strong interactions conserve isospin and for
each of these channels there are separate phase shifts and due to
nal state interactions
i i
h jH jK i a e h jH jK i a e
I I
J D Richman
where are the strong phase shifts The amplitudes a and a contain
the weak phases and
i i
a ja je a ja je
Direct CP violation is then characterized by the parameter
a i
i
p
Im e
a
where There are a number of slightly di erent
in the literature You might wonder whether it is de nitions of
p ossible for the weak interactions to pro duce the pions in b oth I
and I states This is indeed p ossible for the standard sp ectator
diagram as well as certain electroweak p enguin diagrams In gluonic
p enguin diagrams only I is allowed when a uu or dd pair is
pro duced by a gluon the pair must have I In principle these
quarks can then combine with the other nal state quarks to give total
isospin I or I but Bose symmetry excludes I
Direct CP violation in K decays can b e studied by comparing the
ratios
A K A K
L L
A K A K
S S
Since and are di erent mixtures of the I and I
states comparison of and is sensitive to direct CP violation
If CP violation in the K K system is due to state mixing only the
nal state is irrelevant and Using great care to control
systematic errors two exp eriments have measured the double ratio
A K A K
L L
Re
A K A K
S S
Note that the measurement of direct CP violation is related to
Re sin sin which vanishes if either the weak
or the strong phases are equal The results are
Re NA
Re E
The NA result indicate a small amount of direct CP violation in K
decays but the uncertainties are to o large to draw a de nite conclusion
Heavy Quark Physics and CP Violation
New measurements from NA at CERN and KTeV at Fermilab should
help to clarify the situation
The calculation of is extraordinarily di cult due to the large
numb er of pro cesses involved and the presence of long distance strong
interaction e ects Predicted values are in the range to a few
times Observation of a non zero value however would establish
that the observed CP violation could not be attributed entirely to
a new S sd sd Fermi typ e interaction the sup erweak
hyp othesis which would pro duce CP violation in mixing but not in
decay
CP violation in mixing indirect CP violation
The sp ontaneous oscillation of a particle into its antiparticle often
called mixing provides a remarkable example of quantum mechanical
b ehavior Mixing do es not necessarily mean that there will b e CP vi
olation but it do es provide interfering amplitudes that can p otentially
pro duce CP violation CP violation in neutral K decays is largely if
not entirely attributable to interfering amplitudes intro duced through
the mixing pro cess Using the B B system we can extract clean infor
mation on certain CP violating phases by exploiting the interference
bet ween mixing and decay amplitudes
In this section we consider how CP violation can be manifested
in the mixing of two pseudoscalar mesons P and P which could
represent K K D D B B While large mixing e ects B or B
s s
have b een observed in b oth the K K and B B systems D D mixing
cancellations the has not been observed Due to approximate GIM
mixing rate predicted within the standard mo del is exp ected to be
quite low Furthermore D D mixing may well be sensitive to new
physics and it is a key topic for the charm physics program We will
see that in contrast B B oscillations are exp ected to b e quite rapid
s s
so that extremely good time resolution is required to observe them
In practice this implies that one needs excellent resolution on decay
length and high statistics data samples Searches for B B oscillations
s s
ve so far only pro duced limits ha
The standard formalism for mixing is based on a time dep endent
p erturbation theory analysis of a two state system jP i and jP i to
gether with a continuum of states jf i into which the particles P and
P can decay The total hamiltonian is written
H H H
w
J D Richman
where H contains the strong and electromagnetic interactions and
H contains the weak interactions that induce P P P f and
w
P f transitions The basis states jP i jP i andjf i are eigenstates
of H
H jP i m jP i
H jP i m jP i
H jf i E jf i
f
and any state in the space can be written as
X
t a t jP i b t jP i c t jf i
f
f
As the system evolves P and P decay so that probabilitymoves into
the states jf i However the standard formalism recasts Eq and
Eq into an equivalent form
a t H H a t a t
H i
H b t H b t b t
t
in which jf i do es not explicitly app ear The new e ective hamiltonian
matrix H is not hermitian since we are only considering a pro jection
onto the state space spanned by jP i and jP i
The form of H is constrained by CP T symmetry which implies
that the matrix can be written
i i
M M
H H
H
i i
H H
M M
i
M M
M M
i
M
Note that H H M M and but H H
The equality of the diagonal elements is simply the usual result that
the masses of a particle and its antiparticle are the same as are their
lifetimes The matrix M is called the mass matrix and is called the
decay matrix The p erturbation theory analysis shows that to second
order their elements are given by
X
hijH jf ihf jH jj i
w w
M m hijH jj i P
ij ij w
m E
f
f
X
hijH jf ihf jH jj i m E
ij w w f f
Heavy Quark Physics and CP Violation
where P denotes the principal value prescription for p erforming the
sum over intermediate states avoiding states for which m E The
f
term hijH jj i corresp onds to a S non standard mo del interaction
w
of the typ e discussed in connection with
I will not derive these results here but it may be useful to note
the b ehavior of the energy denominator that app ears in p erturbation
theory
m E
f
lim lim i
m E i m E m E
f f f
P i m E
f
m E
f
The principal value like the function only has meaning when used
in p erforming an integral It s not hard to convince yourself that these
results are plausible Since the real term is o dd in x m E plot it
f
it deletes the contribution at x and therefore gives the principal
part of the integral you integrate up to x and then beyond it
but don t include x The imaginary part is even in x m E
f
If you plot x you will see that it tends to x as
We will not sp eci cally use the results in Eq for calculations
but they provide insight into the origin of the terms in the mass and
decay matrices by relating them to particular classes of physical pro
cesses Recalling that P and P we note that
The diagonal elements M of the mass matrix are dominated by the
eigenvalue m of the unp erturb ed hamiltonian H which contains in
formation on the quark masses and the strong interactions that bind
the quarks into the mesons
The o diagonal elements of the mass matrix M and M are in
the standard mo del due to second order weak transitions P P and
P P resp ectively via virtual o shell intermediate states The
fact that these states must b e virtual is enforced by the principal value
prescription which means that in the sum over intermediate states jf i
those states with m E are excluded
f
to all allowed The diagonal elements of the decay matrix are due
decays P f and P f These states are on shell due to the factor
m E
f
The o diagonal elements of the decay matrix and are due
to transitions P f P and P f P where f is again an
on shell intermediate state that is one into whichbothP and P can
decay
J D Richman
The term hijH jj i is zero in the standard mo del since S
w
C or B pro cesses do not o ccur in rst order The so
called sup erweak mo del p ostulates a new interaction that p ermits such
pro cesses in rst order It has not b een ruled out exp erimentally that
suchaninteraction is in fact resp onsible for the CP violation observed
in the K K system
It is very simple to solve the eigenvalue problem for this system
The hamiltonian matrix can be written
H
H K H H
H
where is the unit matrix and K is an o diagonal matrix
Thus the eigenvectors of K will also be eigenvectors of H We write
the eigenvectors in the form
q
jP i pjP i q jP i
jpj jq j
q
jP i pjP i q jP i
jpj jq j
Solving
p p H
H q q
we nd that
H q
H H
p H
so that the eigenvalues of H are
i
H H H M
where M and are by de nition real quantities
M Re M Re H H
Im H H
Due to the square ro ot the signs of the eigenvalues do not neces
sarily map onto the same signs in the state vectors We will discuss
this p oint further below see Eq In many presentations of this
mass eigenstates are lab eled with subscripts L or H to sub ject the
indicate that one has b een chosen as the lighter and one the heavier
Heavy Quark Physics and CP Violation
mass state For now we will simply keep the subscripts according
to the sign of the co e cient of q
From Eq and Eq we see that the eigenvectors of K and
H are sp eci ed by
i
M
q H
i
p H
M
This quantity will app ear many times in our discussion of CP violation
It is imp ortant not to confuse the M and with M and ap
p earing in H However wewould exp ect that the o diagonal elements
of H determine the mass and width splittings b etween the eigenstates
and this is indeed the case From Eq we can see that
M M M Re H H
Im H H
ed then j j Furthermore we will see below that if CP is conserv
and j M j M
In Sec we will see that for the case of the B B system
H M and H M due to the complete dominance of the
tt intermediate state in all mixing transitions As a consequence
H H jM j is nearly real and there is a mass splitting M
but very little di erence in the widths
In all of this formalism it is easy to lose track of the key question
what is the condition for CP violation in mixing To nd the answer
we need rst to de ne the action of the CP op erator on jP i and jP i
We work in an arbitrary phase convention where
i
CP
jP i CP jP i e
i
CP
CP jP i e jP i
In the subspace spanned by jP i and jP i we can therefore write the
matrix representation of CP op erator as
i
CP
e
CP
i
CP
e
If CP is conserved then CP H or CP H CP H Let s
write out the matrix version of this statement Since CP commutes
with the identity matrix we need only calculate
CP K CP
i i
CP CP
H e e
i i
CP CP
H e e
J D Richman
i
i
CP
e M
i
i
CP
e M
i
M
i
M
where the last line holds only if CP H CP H Thus if
CP H then
i i
i
CP
e M M
so that q p must b e a pure phase factor directly related to the CP
phases of jP i and jP i From Eq and Eq we have in this
case
i n
CP
e
where n is an integer When we de ne our mass eigenstates in Eq
b elow we will take n even so that in the limit of no CP violation
i
CP
e
and the sign of the co e cient of will be the CP eigenvalue Equa
tion implies that
i
M
i
CP
e
i
M
The condition for CP violation can be written
i
M
q H
CP violation in mixing j j
i
p H
M
It may help to rememb er that H is simply the amplitude for a P
P transition while H is the amplitude for a P P transition If
the magnitudes of these quantities are di erent wehave an asymmetry
between the b ehavior of matter and antimatter Yet another way to
understand the condition for CP violation is to write
M jM j exp i
M
j j exp i
and then to substitute these expressions into the de nition of
Eq After some algebra one nds
i
M
q jM j e j j
im i
e e
p jM j j j jM jj j sin
M
Heavy Quark Physics and CP Violation
Fig CP violation in mixing In a the amplitudes M and are collinear so that
Im M and there is no CP asymmetry The overall orientation of M and
however dep ends on phase convention In b M and point in di erent directions
pro ducing a CP asymmetry
im
where the factor e m integer has again b een used to indicate that
there are two p ossible solutions Insp ection of this result shows that
will b e a pure phase factor if
n
M
where n is yet another integer If M and are not collinear in the
complex plane there will be CP violation in mixing But that is not
the only way to make j j If either M or we would
also have j j These conditions can be stated as
i
M
Im M jM jj j Im e
jM jj j sin
M
CP violation in mixing
This condition contains the analog of the requirement that sin
i
in the formula for a direct CP violating asymmetry Eq
j
and there must be at least two interfering amplitudes Figure
shows situations that corresp ond to CP conservation and CP violation
in mixing
The condition stated in Eq is remarkably simple given the
complexityofalltheintermediate states that can play a role in mixing
It would seem that a myriad of interference e ects could take place
even within the set of on shell or o shell transitions to pro duce a
CP asymmetry Yet Eq seems to be telling us that there is
cannot something sp ecial ab out the mass and decay matrices There
be CP asymmetry in mixing unless the magnitudes of both M and
J D Richman
are nonzero At the end of this section I will try to explain why
this is
When CP H M and point either in the same or opp osite
direction The phases of M and however dep end on convention
since
q
arg m n
CP
p
Using all of these phase relations we have
n m n
CP M
n m
CP
Thus when CP is conserved in mixing
i
n i n m n
CP
M i jM j j j e
i
i n m n n
CP
M i j j e jM j
Since H H enters directly into expressions for measurable quantities
it must be indep endent of phase convention
i
n
j j H H jM j
and the mass and width splittings are given by
M jM j
n
j j
n
The sign ambiguities and the factor p ermit either the CP even
or o dd state to b e the heavier one and either to have a longer lifetime
leading to four p ossibilities The appropriate signs can be determined
from data or in principle from a detailed theoretical calculation Since
CP violation in mixing is very small in all cases these results for the
splittings are valid to a go o d approximation
Other useful expressions related to that are valid even when CP
is not conserved are
j j Im M jM j
q
j j
p jM j j j Im M
and
Im M jpj jq j
jpj jq j
jM j j j M
Heavy Quark Physics and CP Violation
where we have used
i
j M j M
s
j j Im M jM j
Again a non zero value of Im M or j j signals CP violation
in mixing
Nowthatwehave identi ed the phase convention indep endent cri
terion for CP violation in mixing we can analyze the e ect of CP
violation on the time evolution of the P P system The normalized
eigenstates of H the mass eigenstates are
q
jP i jP i jP i
j j
q
jP i jP i jP i
j j
Actually by itself do es not completely sp ecify the eigenstates in
Eq b ecause it only contains the relative phase of q p The states
in Eq in principle could have an overall phase factor exp i
p
where is the phase of p Since there is no physics connected with
p
this overall phase factor we will set it to unity from now on This is
ention b ecause that phase not equivalenttocho osing a CP phase conv
is opp osite for jP i and jP i We also note that in many treatments
one of these mass eigenstates is selected to be the heavier than the
other and they are lab elled with subscripts H heavy or L light
An alternative formulation for the co e cients in the mass eigenstates
is sometimes used esp ecially for the neutral kaon system
h i
q
p
jK i jK i jK i
S
j j
h i
q
p
jK i jK i jK i
L
j j
where the bar over the is meant to emphasize that is a phase
convention dep endent quantity By comparing the two ways to write
the ratio q p we can relate and
J D Richman
Although the two forms have the same q p in Eq the co e cient
of jK i is not real but the overall phase freedom is used instead to set
Im q Im p
If CP is violated in mixing the mass eigenstates are not orthogonal
since
jpj jq j j j
hP jP i
jpj jq j j j
If CP H is just exp i so that these states b ecome the
CP
CP eigenstates that we constructed in Eq which in our notation
for a general pseudoscalar particle are
i
CP
p
jP i jP i e jP i
CP
i
CP
p
jP i jP i e jP i
CP
Note that if one adopts the natural convention that CP jP i jP i
and CP jP i jP i then exp i
CP
Although this relentless series of relations among di erent basis
states can b e mind numbing it s useful to express the mass eigenstates
in termsofthe CP eigenstates
i
CP
e
i
CP
q
p
e jP i jP i jP i
CP CP
i
CP
e
j j
i
CP
e
i
CP
q
p
jP i jP i jP i e
CP CP
i
CP
e
j j
You can check that when CP is conserved these formulas behave as
exp ected the mass eigenstates b ecome CP eigenstates
Because jP i and jP i are eigenstates of H their time dep endence
is extremely simple
i t
jP t i P e jP i
i
i M t
q
P e jP i jP i
j j
i t
jP t i P e jP i
i
t i M
q
jP i jP i P e
j j
Heavy Quark Physics and CP Violation
where P and P are the amplitudes at t
To determine the time dep endence of the avor eigenstates we ex
press them in terms of the mass eigenstates
q
i i jP j j jP jP i
q
j j
jP i jP i jP i
The time dep endence of states that are initially jP i or jP i is therefore
q
i i
i M t t i M
jP t i ie jP ie jP j j
q
j j
i i
i M t i M t
jP t i jP ie jP ie
Finally we obtain the time evolution of the states that are pro duced
as avor eigenstates expressed in the avor eigenstate basis
jP t i f t jP i f t jP i
f t jP i f t jP i jP t i
where
iM t t iM t t
f t e e e e
iM t t iM t t
e e e e f t
These expressions are general and can be used as a starting point to
analyze any of the neutral meson systems with oscillations We can
now calculate the mixing probabilities as a function of time
h i
t t t
Prob P at tjP at t e e e cos Mt
h i
t t t
Prob P at tjP at t j j e e e cos Mt
h i
t t t
Prob P at tjP at t e e e cos Mt
j j
h i
t t t
Prob P at tjP at t e e e cos Mt
J D Richman
where
M M M
These expressions b ehave as exp ected for t jP t i has not yet
develop ed any jP i comp onent and jP t i has not yet develop ed
any jP i comp onent We are accustomed to the survival probabilityof
a particle b eing an exp onential For j j
Prob P at tjP at t Prob P at tjP at t
h i
t t
e e
That is the total probability to survive either as a P or a P is the
sum of two exp onentials If CP is violated this probability isn t even
the sum of two exp onentials since the oscillating term is present
In the B B system there is essentially no lifetime splitting so
and the oscillating term can b e written
M t
cos Mt cos
Thus the quantity M characterizes the mixing rate when we mea
sure time in units of prop er lifetime
We can also see that if j j
Prob P P Prob P P
If j j we get less P than P b ecause
j j jH H j jhP jH jP i hP jH jP ij
The equality of the P t P and P t P probabilities is a
consequence of CP T symmetry Figure shows the behavior of the
formulas in Eq for the neutral kaon system As discussed in
Sec the lifetimes of the mass eigenstates are very di erent
L
ns and ns
s
Before leaving this section I would like to discuss why it is that the
condition for CP violation in mixing Im M is so remark
ably simple What is magic ab out the mass and decay matrices After
CP violation is generated by the interference b etween di erentam all
plitudes and there can b e many amplitudes with on shell intermediate
states why can t these interfere with each other pro ducing CP viola
tion and many amplitudes for o shell intermediate states why can t
these interfere with each other pro ducing CP violation The state
is saying that CP violation in mixing can only ment Im M
Heavy Quark Physics and CP Violation
Fig Time evolution of particles initially tagged as a K and K The probabilities for
K K and K K are equal as required by CP T symmetry For j j however
there is a di erence between K K and K K The CP asymmetry is seen
as the di erence b etween the dashed line and the dotted line and it was generated with
j j corresp onding to an asymmetry ab out ten times the actual size j j
The curves app ear to reach an essentially at asymptote this is b ecause and the
L S
lifetimes scale only extends out to twenty K
S
interference between the total amplitude via on shell originate in the
intermediate states and the total amplitude for o shell intermediate states
J D Richman
Fig CP violation in mixing The interference o ccurs b etween the set of amplitudes
with virtual intermediate states M and on shell intermediate states The factor
i exp i plays the role of the strong phase factor exp i in direct CP
i j
violation it do esn t change sign for the CP conjugate amplitude
To resolve this question consider Fig which shows how CP
violation in mixing is generated Keep in mind that we are comparing
H and H
i
H hP jH jP i hP jM jP i hP j jP i
i
M
i
H hP jH jP i hP jM jP i hP j jP i
i i
M M
Figure a shows the two amplitudes that contribute to P P
Fig b shows the analogous amplitudes that contribute to P P
Each of these amplitudes is actually a bundle of the amplitudes for
various intermediate states but by itself neither bundle can pro duce
CP violation Thus if there are only o shell intermediate states such
as tt there can be no CP violation in mixing no matter what CKM
phases are present in amplitudes b ecause CP T invariance guarantees
that M M Without an amplitude from on shell intermedi
ate states the rates for P P and P P would be equal
Similarly b ecause mixing via on shell intermediate states
alone cannot pro duce CP violation The factor of i exp i
that multiplies b oth and comes from the co e cient of the
function in Eq that de nes the on shell intermediate states It
is this i that plays the role that strong phases play in direct CP viola
same sign in the CP conjugate amplitude while b oth tion it has the
M and are complex conjugated
Heavy Quark Physics and CP Violation
Phenomenology of mixing
The formalism presented in the previous section applies equally well
to K K D D B B or B B oscillations Di erences in the de
s s
tailed physics however lead to approximations and results that are
di erent for each case In this section we examine the sp eci cs of the
phenomenology and review some of the mixing measurements
be The K K system is characterized by a huge lifetime splitting
tween the two mass eigenstates
K ns m
L
K ns cm
S
The origin of this splitting is simple CP violation in K K mixing is a
very small e ect so to a go o d approximation the physical mass eigen
state K corresp onds to the CP even eigenstate and K corresp onds to
S L
the CP o dd eigenstate Note that the mass eigenstates K K and
S
are given by Eq Wehave already seen in Eq that K K
L
the CP o dd state cannot decay into Thus the K essentially has
L
only four signi cant decay mo des
and e all of which are dramatically sup
pressed by three b o dy phase space In contrast the K has decays
s
which account for nearly all of its decay and K K
S S
rate The approximate ratio K K is a
S S
consequence of the I rule
For the neutral kaon system the mass and width splittings are com
parable
s
K K K
S S L
M M M M M s
K K
L S
where the mass di erence corresp onds to ab out eV Remember
that the subscripts here are not CP eigenvalues but the eigenvalues
the states would have if CP violation were turned o Thus
M
The lifetime splitting has imp ortant implications for exp erimen ts
since it means that any neutral kaon beam evolves over distance into
an essentially pure K beam As a consequence it is p ossible to know
L
that the particles under study are K s without identifying a correlated
L
is an eigenstate particle which would provide a tag Since the K L
J D Richman
of H it remains a K unless a K comp onent is regenerated by an
L S
interaction with matter which intro duces di erent amplitudes for the
K and K pieces of the K wave function
L
Because K sd but K sd semileptonic de
cays can b e used to monitor the K and K content of the state From
Eq
K K
L L
K K
L L
jhK jK ij jhK jK ij
L L
jhK jK ij jhK jK ij
L L
j j
hK jK i
L S
j j
This value demonstrates that CP violation is presentinK K mixing
but that it is a small e ect We can use it to measure the size of j j
j j
We can summarize this bysaying that K K K semileptonic
L
decays have slightly more p ositively charged than negatively charged
t excess of K over K leptons due to a sligh
T D Lee has presented some interesting observations regard
ing the charge asymmetry in Eq Before meeting each other the
representatives of two civilizations from di erent parts of the universe
decide that it would b e prudent to determine whether their de nitions
of matter and antimatter were the same The sp ectroscopy of their
atoms would be identical even if the atoms of one civilization were
made from antiprotons and p ositrons However they could use the
decays of the longer lived neutral kaon to determine whether direct
contact would result in their mutual annihilation From the asym
metry in Eq the civilizations could establish a sign convention
for charge the lepton that is pro duced more often could be called
p ositively charged and they could compare this charge to that of
their atomic nuclei Note that a decay that violates parity or charge
conjugation only such as would not be adequate for this
purp ose You could say that a p ositively charged muon was predomi
nantly helicity but then the civilizatons would have to nd some
way to establish a common convention for handedness A way around
this would be to communicate with p olarized light signals de nining
Heavy Quark Physics and CP Violation
a handedness convention by a direct handshake However if unp olar
ized lightwere used then an analysis based on CP violation would b e
required
Another metho d for studying mixing e ects in the K K system
has b een used by the CPLEAR exp eriment at CERN In this exp er
iment pp collisions at rest can pro duce neutral kaons in asso ciation
with charged kaons
pp at rest K K
K K
Thus by selecting these pro cesses and using their charge correlations
one can determine whether the neutral kaon is pro duced as a K or a
K
The picture for the B B system is di erent in many resp ects To
generate and hence we need mo des with large rates that b oth
B and B can decay into These supply the on shell intermediate
states in the mixing amplitude Since b c and b c usually lead to
di erent sets of quarks in B and B decay the rate into such common
mo des is exp ected to be small Both B and B can decay into uu dd
systems but these involve highly suppressed b u transitions There
are also common states with ccd d quark content such as D D and
J K b oth of these are Cabibb o suppressed and the latter is color
suppressed Finally there are common states with quark content cu dd
d In this case either the B or B decay must involve a b u or cud
transition and the other one is Cabibb o suppressed The amplitudes
for these pro cesses enter into the mixing diagram with di erent signs
so there will even be some cancellation Thus and are b oth
exp ected to be very small and in our mixing formulas we can set
This conclusion is supp orted by a direct calculation of
in the standard mo del whose result is given belo w
It is interesting that in the B B system can be signi cant be
s s
cause both B and B can decay into the Cabibb o favored nal states
s s
with quark content ccs s As a consequence the lifetime splitting be
tween the two B mass eigenstates could b e
s
The mass di erence in the B B system on the other hand is gen
erated by mixing amplitudes with o shell intermediate states In the
standard mo del these amplitudes are completely dominated bythebox
M is diagrams with tt The standard mo del prediction for
G M m B f
B B B
F W B
i B
CP
M V V S x e
tb t
td
J D Richman
where x m m f is the B meson decay constant B is the
t B B
t W
B meson bag constant parametrizes calculable QCD corrections
B
and
x ln x x x x
t t
t t t
S x
t
x x
t t
For large x S x x so
t t t
O m m
b t
M
It is therefore useful to write q p in a form appropriate for small
M
j j
i n
M
sin e
M
jM j
The mass dep endence of the term sin m m is given
M c b
in Buras and Fleischer Thus CP violation in B B mixing is
prop ortional to m m m m m m O
b t c b c t
Since the lifetime splitting is negligible b etween the neutral B mass
eigenstates the state of the B at the initial time cannot b e determined
by letting one comp onent decay away Instead the initial state must
b e tagged by identifying a particle whose avor is correlated with that
of the B We can again use semileptonic decays to determine whether
the B is a B or B at the decay time We can therefore measure the
asymmetry
B t X B t X
A
CP
B t X B t X
jhB jB t ij jhB jB t ij
jhB jB t ij jhB jB t ij
j j
j j
j j
j j j j
j j
This result has a di erent form than the asymmetry we wrote down
for the K Eq b ecause here we are not starting with a mass
L
B mixing is given to a eigenstate Because is very small for B
go o d approximation by
s
q M
p M
Heavy Quark Physics and CP Violation
which implies that j j to a go o d approximation Even though
carries a non trivial CKM phase CP violation in mixing alone has no
sensitivity to it This is b ecause so that is a pure phase
factor and the CP asymmetry in Eq vanishes We will return
to Eq in the following section when we discuss CP violation in
the interference between mixing and decay That typ e of interference
in is sensitive to the phase of Using j j and
Eq we can obtain formulas for the B B mixing probabilities as
a function of time
Prob B at tjB at t Prob B at tjB at t
t
e cos Mt
Prob B at tjB at t Prob B at tjB at t
t
e cos Mt
The rate for B B mixing is exp ected to be much higher than that
s s
for the B B system In each case M is dominated by the tt inter
mediate state so that the relevantCKM factors are
M V V M V V
d tb s tb
td ts
shows while the other factors are similar for the two cases Figure
calculated curves for B and B mixing where the current exp erimental
d s
lower limit on M is used in the latter case
s
B B B B and B The predicted values of M for both B
s d s d
mixing are given in Buras and Fleischer
q
B f
B B
jV j m m
d d
td t t B
M ps
d
MeV GeV
p
B f m m jV j
B B t t ts B
s s
M ps
s
MeV GeV
The exp erimental results on B B and B B mixing were recently
s s
reviewed bySchune who describ ed the advantages and disadvan
tages of various techniques and rep orted an average of LEP SLD and
CDF exp eriments These exp eriments measure the time dep endence
of mixing Eq using high precision vertex detectors to measure
the B decay distance which is then converted into the prop er decay
J D Richman
Fig The mixing probabilities solid lines for B and B mesons as a function of
s
d
time The dashed lines show the exp onential decays The value of m used for the B
d
is essentially the world average value the value for the B is slightly ab ove the current
s
lower limit
time using the measured B momentum The world average value for
M calculated by the LEP B Oscillations Working Group is
d
M ps
d
Measurements of M have b ecome a ma jor industry values are
d
included in this average
Measurements of B B mixing have so far only yielded the lower
s s
limit
M ps C L
s
Mixing in the B B system is su ciently slow that it is also p ossible
to measure it using time integrated rates for opp osite sign and like sign
dilepton rates in the pro cess S B B The mixing probability
can be expressed in terms of the parameter
R
P t dt
B B
R R
P t dt t dt P
B B
B B
M
M
Heavy Quark Physics and CP Violation
where P t is the probability for a meson initially tagged as a B
B B
to b e found at time t as a B This technique works well for the B B
system where M M is not to o large giving
The measurement compares the yield of like sign dilepton events which
are mainly due to mixing to the yield of opp osite sign dilepton events
which are unmixed Considerable care is required b ecause the like sign
sample has a contribution from secondary leptons from charm decays
In addition the contribution from S B B decays must be
accounted for Nevertheless this metho d has pro duced precise results
the metho d used by ARGUS for the discovery of B B and it was
mixing We will see in the following section that when a B and
B are pro duced at the S their time evolution is correlated until
one or the other particle decays their oscillations cannot pro duce a
B B or B B system
For large M the B meson oscillates many times b efore it decays
B mixing a measurement of so that approaches Thus for B
s s
the time integrated mixing rate has little sensitivity to M and it
s
is essential to use high precision vertex detectors to measure the time
dep endence of the oscillation directly The pro cess Z bb is well
suited for this purp ose due to the long b hadron decay length ab out
mm The measurement must provide three pieces of information
i The avor of the B meson when it was pro duced is obtained by
measurement of the other b hadron in the event The e ects of
mistagging are signi cant and must b e corrected for
ii The a vor of the B meson when it decays is determined from
semileptonic decay either inclusive or semi inclusive
iii To determine the prop er decay time one measures the decay length
which requires knowledge of b oth the primary Z decay vertex po
sition and the B decay vertex To convert the decay length into
the prop er decay time the B momentum must be measured Since
a semileptonic decay is used to reconstruct the B there is some
uncertainty asso ciated with the neutrino momentum
decay It is interesting to examine how the uncertainty on the prop er
time t dep ends on measured quantitities Using L p m ct it is easy
to show that the uncertainty is given by
v
u
u
M
L p
t
t
t
p p
J D Richman
where is the uncertainty on the decay length p is the momentum
L
is the uncertainty on the momentum For large t the uncertainty
p
on the momentum which gives the b o ost to the rest frame gives the
dominantcontribution
The e ects of decay length resolution and mistagging called dilu
tion are shown in Fig for the separate cases of B and B mixing
d s
The dilution is simply
N
B incorr tagged
N
B corr tagged
The e ects on B oscillation measurements are dramatic Prosp ects
s
app ear to be go o d for measurements of B mixing in hadron collider
s
exp eriments
CP violation due to interference between mixing and decay
As wehave seen CP violating e ects require interfering amplitudes in
order that the weak phases can b e manifest in the asymmetry b etween
rates As shown earlier Fig these amplitudes can be supplied by
i Direct decay B f where f is a nal state that b oth B and B
can decay into
ii B B mixing followed by B f
Both the B and B must b e able to decay into f whichcan be a CP
eigenstate f e g or D D but need not be e g D
CP
Using the gold plated decay B J K one can measure sin
S
with virtually no uncertainties due to strong interaction e ects In
other cases suchasB the situation is more complicated due
to the simultaneous presence of direct CP violation with the attendant
strong phases and CP violation due to interference between mixing
and decay As we will see this mo de will b e used to extract sin but
it remains to be seen how the uncertainties due to strong phases will
be sorted out In any case the metho d of using interference between
mixing and decay to a CP eigenstate is very powerful b ecause the
interfering amplitudes are of comparable size and the asymmetries can
be very large
We start with the general mixing formulas which we rep eat here
with P B for convenience
jB t i f t jB i f t jB i
jB t i f t jB i f t jB i
Heavy Quark Physics and CP Violation
Fig The e ects of decay length resolution and dilution mistagging on B and B
s
d
oscillation measurements Figure provided byM H Schune
where
iM t t iM t t
e e e e f t
iM t t iM t t
f t e e e e
Recall that jB t i is the state that at t is pure jB i and jB t i is
i the time evolution of the system the state that at t is pure jB
however mixes in their antiparticles The states jB t i and jB t i
are sometimes lab eled jB t i and jB t i to emphasize that they
phys phys
are not purely jB i or jB i
J D Richman
As discussed in the previous section for B B mixing we can assume
The co e cients f t and f t can then be written
iM t t iM t t
f t e e e e
i i M M t M M t t i M M t t
e e e e e
Mt
iM t t
e e cos
and
iM t t iM t t
e e e e f t
Mt
iM t t
ie e sin
where
M M M M M M
The factor of i in Eq is crucial It is asso ciated with the mixing
amplitude and plays the role of the strong phase factor i exp i
in our discussion
Recall that the subscripts onM refer to the sign of the co e cient
of jB i in Eq We now assume that M M and M M
H L
where M and M are the masses of the heavier and lighter mass
H L
M M M eigenstates resp ectively Thus M M
H L
The time dep endence of states jB t i and jB t i that at t are
jB i and jB i is therefore
Mt Mt
t iM t
jB i i sin jB i jB t i e e cos
Mt Mt
t iM t
jB t i e e jB i cos jB i i sin
Wenow let these states decayinto a common nal state jf i which
CP
we take to be a CP eigenstate with eigenvalue f The CP
Heavy Quark Physics and CP Violation
amplitude for B t f is given by
CP
t iM t
e hf jH jB t i e
CP
Mt Mt
hf jH jB i cos i hf jH jB i sin
CP CP
while that for B t f is
CP
t iM t
e hf jH jB t i e
CP
Mt Mt
hf jH jB i i sin hf jH jB i cos
CP CP
In these two equations we see explicitly the two amplitudes whose
interference can pro duce a CP asymmetry We next write these equa
tions in terms of a ratio of amplitudes
v
u
i
u
M
hf jH jB i hf jH jB i
CP CP
t
i
hf jH jB i hf jH jB i
M
CP CP
This quantity plays a key role in determining whether there is a CP
asymmetry as we will show below
The amplitude for B t f is
CP
t iM t
hf jH jB t i e e hf jH jB i
CP CP
Mt hf jH jB i Mt
CP
cos i sin
hf jH jB i
CP
and that for B t f
CP
t iM t
e hf jH jB i i hf jH jB t i e
CP CP
Mt hf jH jB i Mt
CP
sin i cos
hf jH jB i
CP
If the initial state at t is jB i the probability to pro duce jf i
CP
at time t is therefore
t
jhf j H jB t ij e jhf jH jB ij
CP CP
Mt Mt Mt Mt
cos j j sin Im cos sin
J D Richman
t
e jhf jH jB ij j j
CP
j j cos Mt Im sin Mt
while the probabilityto pro duce jf i at time t starting from jB i at
CP
time t is
t
jhf j H jB t ij e jhf jH jB ij
CP CP
Mt Mt Mt Mt
j j sin Im cos sin cos
t
jhf jH jB ij j j e
CP
j j cos Mt Im sin Mt
We now use the fact that and that see Eq
i
CP
M a V V e
tb
td
i
CP
M a V V e
td
tb
where a is a real constant Thus
s
M V V
td
tb
i
CP
e
M V V
tb
td
i B i B
M CP
e e B B mixing
where the last expression de nes the mixing phase B The rst
M
imp ortant consequence of this result is that j j j j Below we
will evaluate for various cases and show that it has no dep endence
B mixing is given by on phase convention We note that for B
s s
V V
ts
tb
i B
s
CP
e B
s
V V
tb
ts
i B i B
s s
M CP
e e B B mixing
s s
We can now calculate the time dep endent CP asymmetry
jhf jH jB t ij jhf jH jB t ij
CP CP
A t
CP
jhf jH jB t ij jhf jH jB t ij
CP CP
j j cos Mt Im sin Mt
j j
Heavy Quark Physics and CP Violation
which is one of the key results in the analysis Incidentally to com
pute the so called time integrated asymmetry you should not integrate
this expression Instead compute the asymmetry between the time
integrated rates see below
To evaluate we use
y y
hf jH jB i hf j CP CP H CP CP jB i
CP CP
y i
CP
H CP jB i f e hf j CP
CP CP
Two rather di erent typ es of situations can emerge at this point de
p ending on whether the direct decay to f is dominated by a single
CP
amplitude If this is the case then
i
D
hf jH jB i jaje
CP
y i
D
hf j CP H CP jB i jaje
CP
i i
D
where e is as usual a strong CP conserving phase factor and e
is the weak phase factor that we want to extract from the measure
ment The second of these two equations is discussed in the Ap
p endix Thus
i i
CP D
hf jH jB i f e jaje
CP CP
Putting the pieces together we have
V V
td
tb
i i B
D M D
f e f e
CP CP
V V
tb
td
where the two phase convention dep endent factors have cancelled In
this imp ortantcase a single weak phase in the decay amplitude
j j
so that the term in Eq prop ortional to cos Mt do es not con
tribute This term in e ect measures direct CP violation and would
b e presentiftwo pro cesses with di erent CKM and strong phases con
tributed signi can tly to the decay Assuming j j in Eq and
Eq we obtain the remarkably simple results
h i
t
jhf jH jB t ij e jhf jH jB ij Im sin Mt
CP CP
h i
t
jhf jH jB t ij e jhf jH jB ij Im sin Mt
CP CP
J D Richman
which give the asymmetry
A t Im sin Mt
CP
Let s continue with the assumption that only one diagram con
tributes to the direct weak decay or more precisely that all of the
contributing pro cesses have the same weak phase Since a B meson
contains a b quark the CKM factors asso ciated with tree diagrams give
V V
cb
cs
i
D
b ccs e
V V
cs
cb
V V
cb
cd
i
D
b ccd e
V V
cd
cb
V V
ub
ud
i
D
e b uu d
V V
ud
ub
Each of these quantities gives the orientation in of a quantity V V
ij
ik
in the complex plane This orientation dep ends on the CKM phase
convention but the angle between two such quantities do es not For
B we have
V V
td
tb
i
D
e
V V
tb
td
V V V V
ub td
ud tb
i
e
V V V V
tb ud
td ub
which implies that
Im sin
Now consider the D D nal state Using Eq we have
V V V V
td cb
tb cd
i
e
V V V V
tb cd
td cb
or
Im sin
If all of the CKM factors in these equations b ecome to o mesmerizing
just lo ok at the CKM triangle in Fig The pairs V V V V and
ud cd
ub cb
V V are simply the complex numb ers representing the directed sides
td
tb
of the triangle Thus their orientation in the complex plane dep ends on
the choice of CKM phase convention As discussed in Sec a di erent
choice of quark phase will result in a rotation of the triangle However
if z and z represent any two of the directed sides of the triangle
Heavy Quark Physics and CP Violation
the quantity z z jz jjz j exp i measures the convention
indep endent area see the discussion at the end of Sec Similarly
z z z z exp i gives a convention indep endent mea
sure that governs the size of the asymmetries in the case of interference
between mixing and decay
We next consider the decay B J K To a very go o d approx
S
imation we regard the K as a CP eigenstate with eigenvalue so
s
the CP eigenvalue of the nal state is CP J K Wehave to
S
b e careful b ecause B J K but B J K Toreach the com
mon nal state the K s must mix intro ducing an additional phase
di erence between the two pro cesses Since gives the ratio B B
the required additional factor is
V V
cs
cd
K
V V
cd
cs
and
V V V V V V
cb cs td
cs cd tb
V V V V V V
tb cs cd
td cb cs
i
e
so that
Im sin
The extraordinary feature of these results is that wehave b een able
to express measurable CP asymmetries directly in terms of CKM quan
tities without knowledge of hadronic matrix elements This simplicity
is compromised in the cases where additional amplitudes contribute
since a dep endence on strong phases is intro duced Most imp ortantly
from an exp erimental p ersp ective the asymmetries can b e quite large
since the angles of the unitarity triangle are not all near or and
the magnitudes of the interfering amplitudes are comparable in size
Figure shows the decay rates vs time for B mesons initially tagged
are non exp onential Eq a char as B or B The decay rates
acteristic of CP violation The gure also shows the asso ciated CP
asymmetry which has a sinusoidal time dep endence Eq
Before pro ceeding I would like to write the decay amplitudes in a
form that we can easily relate to a simple graphical analysis Again
sp ecializing to the case of a single direct decay amplitude and using
Eq Eq and Eq
i B i B
M CP
e e
J D Richman
Fig CP asymmetry for neutral B mesons resulting from the interference between
mixing and decay The upp er plot shows the decay rates as a function of prop er lifetime
for B t f and B t f where B t is tagged as a B at t and
CP CP
B t is tagged as a B at t The asymmetry is a sinusoidal function of time reaching
a maximum value equal to Im The lower plot shows the decay rates on a logarithmic
scale The main p oint is that the size of the CP asymmetries in B decays can b e very
large this is a consequence of the fact that all three sides of the corresp onding unitarity
triangle have lengths of order giving large interior angles and hence large values of
i
The value Im is a plausible value for sin Im sin
i
i
D
hf jH jB i jaje
CP
i B i
CP D
hf jH jB i f e jaje
CP CP
in Eq and Eq we have
t iM t i
D
hf jH jB t i jaje e e
CP
Mt Mt
i
M D
cos i f e sin
CP
Heavy Quark Physics and CP Violation
and
t iM t i i B
CP D
hf jH jB t i jaj f e e e e
CP CP
Mt Mt
i
M D
cos i f e sin
CP
Note that the strong phase factor exp i is common to all the am
plitudes so it factors out and has no e ect on observable quanti
ties The phase i exp i from mixing lls the role played by
the strong phase in direct CP violation it is not conjugated in go
ing from B t to B t decay The weak phase which is conjugated
is given by The complicated hadronic e ects in mix
M D
ing have neatly divided out in since mixing is completely domi
nated by o shell pro cesses and H M has the same magnitude
as H M M The two interfering amplitudes have similar
magnitudes allowing for large CP violating asymmetries Figure
shows the amplitudes schematically and Figure shows a graphical
construction of the amplitudes as a function of time
We again see that CP violation pro duces a nonexp onential decay
probability
t
jA B t f j jaj e f sin Mt sin
CP CP M D
t
jA B t f j jaj e f sin Mt sin
CP CP M D
We have already computed the CP asymmetry as a function of time
en by Note that the asymmetry b etween the time integrated rates is giv
R
jA B t f j jA B t f j
CP CP
R
A
CP
jA B t f j jA B t f j
CP CP
x
f sin
CP M D
x
f sin
CP M D
M
where x This time integrated asymmetry can be measured
in a hadron collider exp eriment such as LHC B or BTeV but not in
e e S exp eriments as we will now see
For exp eriments that pro duce B B pairs using the decay S
B B there are imp ortant tagging considerations arising from the
J D Richman
Fig Amplitudes contributing to the interference b etween mixing and decaytoaCP
eigenstate f in the B B system For simplicity most of the relative phases have
CP
b een attached to the sin Mt term The overall exp onential decay factors havebeen
omitted
C quantum number of the S Since the strong and elec
tromagnetic interactions conserve C the B B system must also have
C We can write the overall wave function of this system as
p
j t jB t pijB t pi t i
C
jB t pijB t pi
for C states Wehave sp eci ed a particular momentum direction
for the B mesons the actual wave function for the nal state is a
sup erp osition of such two particle plane wave states with a weighting
d due to helicity conservation see factor d sin no
Sec which gives the dN d cos sin angular distribution
Substituting the expressions for jB t i and jB t i from Eq we
obtain
j t t i
C
p
f t f t f t f t jB pijB pi f
f t f t f t f t jB pijB pi
f t f t f t f t jB pijB pi
f t f t f t f t jB pijB pig
Heavy Quark Physics and CP Violation
Fig Graphical analysis of the amplitudes contributing to the interference between
mixing and decayintheB B system as a function of time The amplitudes corresp ond
to those in Fig except the exp onential decay factors have b een ignored for clarity
These factors would rapidly shrink the triangles in the gure The triangles on the
left corresp ond to decays B t f while those on the right corresp ond to B t
CP
f In each case the vectors in the horizontal direction real axis corresp ond to the
CP
cos Mt term Todisplay the sin Mt term we b egin with the strong phase
i exp i asso ciated with mixing This phase factor is oriented vertically on the page
along the imaginary axis The weak phase factor exp i exp i rotates
M D
either counterclo ckwise or clo ckwise with resp ect to this axis dep ending on whether weare
considering B t or B t Note that the B t or B t amplitudes at time Mt
have di erent magnitudes as do those at Mt At Mt the
amplitudes havedi erent orientations but their magnitudes are the same
Eq We substitute the results for f t Eq and f t
which gives
t t iM t t
p
j t t i e e
C
J D Richman
M
t t jB pijB pi jB pijB pi cos
M
i sin t t jB pijB pi jB pijB pi
For t t the sine terms vanish so only jB pijB pi or
jB pijB pi terms contribute to the C wave function This
means that the two B mesons evolve coherently and until one of them
them is a B and the other one a B If one B decays decays one of
into a mo de that tags its avor as say a B at time t t then
the other one is a B at that instant t and it evolves from that time
onward as jB t t i
We can pro ject the state vector j t t i onto a state
i One B meson decays at time t into a CP eigen jf t f t
CP tag
B
state while the other decays at time t into a state that tags it as a B
The tagging nal state could b e B D or another semileptonic
mo de This calculation gives
t t
jhf t f t j H j t t ij e jA j jA j
CP TAG TAG f
B
CP
j j j j cos M t t f
Im sin M t t g
This result is essentially the same as that in Eq but the corre
lations due to tagging at the S have b een taken into account It
is evident that the corresp onding asymmetry can be either p ositiveor
negative dep ending on the sign of t t t t If the time
CP TAG
of the tag is not known relative to the time of the decay to the CP
eigenstate no asymmetry will be observed since there will be a can
cellation of the asymmetry for t t withthatfort t As
a consequence it is crucial to p erform the asymmetry measurementas
a function of the time t t
Measurements of CP violation at the S using the interference
between mixing and decaymust therefore accomplish three main tasks
Determine the avor of one of the neutral B mesons directly from
semileptonic decay its decay pro ducts e g
and B can B Reconstruct the other B meson in a state that b oth
decay into In the simplest case this state is an eigenstate of CP but
it is not essential that this b e the case
Measure the time di erence t t t between the two decays
Heavy Quark Physics and CP Violation
t=ttag t=t _ CP S B 0 B
z= c( t)
Fig CP Asymmetry measurement as a function of B vertex separation At the
PEP I I and KEKB asymmetric energy e e colliders the S is b o osted resulting in
a typical separation of the B decayvertices of z m The typical transverse
momentum of the B mesons is much smaller than their longitudinal momentum One
B is reconstructed in a state that tags its avor the other is reconstructed in a CP
eigenstate In the case shown t t Since one B meson is determined to b e a B
TAG CP
at time t t the other one is known to b e at B at that time Due to mixing this
TAG
other B meson can b e either a B or a B when it decays into the CP eigenstate f
CP
The b o ost allows the CP asymmetry to b e measured as a function of the di erence of the
two decay times This element of the measurementisessential b ecause at the S the
time integrated asymmetry vanishes
These times are actually the prop er times but the B mesons move
slowly in the S rest frame so this is not an imp ortant considera
tion
Figure shows how the measurement of the decay time di erence
t is p erformed Because the B mesons are b o osted along the z axis
the decay time di erence translates into a di erence z c t in
decay p ositions along the z b eam axis The mean separation for
is hz i m This distance scale is measurable with
silicon vertex detectors placed just outside the b eampip e as shown in
Fig view transverse to the b eam axis and Fig view along the
b eam axis
The requirement that one neutral B meson be avor tagged with
a relatively high e ciency also places demanding requirements on the
detector Leptons of su ciently high momentum from semileptonic B
y to tag decay are mostly primary and provide one straight forward wa
the avor A second metho d uses charged kaons from the B D K
decay sequence Neither of these metho ds is p erfect and there is an
asso ciated mistagging probability
One B meson must also b e reconstructed in a CP eigenstate and it
is desirable to use as many mo des as p ossible For B there
J D Richman
BaBar Silicon Vertex Tracker
Kevlar/carbon-fiber support rib Si detectors Carbon-fiber endpiece z=0
Cooling ring Carbon-fiber Upilex fanouts support cone Hybrid/readout ICs Beam pipe 30o 350 mr
e- e+
Fig BaBar silicon vertex tracker side view The approximate radii of the layers
are mm mm mm mm and mm Note the asymmetry of the detector
with resp ect to the interaction p oint The mo dules are mounted to co oling rings on the
carb on b er supp ort cones The region within the cones is o ccupied by the B dip ole
magnets shown in Fig Flexible upilex fanouts allow the hybrid readout circuits to b e
mounted at an angle with resp ect to the silicon detectors therebykeeping the readout
electronics out of the tracking volume
Detector wafer
Support ribs
Fig BaBar silicon vertex tracker cross section in xy plane
is background from B K To cleanly separate these mo des it
is very helpful to have go o d particle ID capability at high momentum
which has led to innovativenew approaches to particle ID
A useful summary of the statistical requirements of the CP asymme
try measurements using the interference b etween mixing and decayhas
b een presented by Cassel The number of S B B events
required to observe an asymmetry a in a nal state f with branching
Heavy Quark Physics and CP Violation
fraction B B B f and detection e ciency is
f f
S B s
N
B B
f B d x a S
f f tag d
where s a is the statistical signi cance S B is the ratio of the
a
number of signal to background events and f is the fraction of
B B events in S decays The dep endence on tagging arises from
the quantity
B w
tag tag tag
where B is the branching fraction for the tagging mo de is the
tag tag
e ciency for the tagging mo de w is the fraction of wrong tags to
total tags and d x is a dilution factor that accounts for the loss in
d
asymmetry due to tting the time distributions Although the CP
asymmetries in B decay can be large the mo des that are common to
B and B have small branching fractions so an enormous amount of
data will be required to obtain measurements of good precision The
next generation of exp eriments is exp ected to acquire over
S B B events p er year leading to a precision on sin of b etter
than ab out
This discussion of CP violation is far from comprehensive and an
enormous amountofwork and thought is going into guring howtoex
ploit all of the information that will b e available in forthcoming exp er
iments In particular the complications caused by penguin pro cesses
in B and the di culty of measuring sin have led to many cre
ative if not entirely successful metho ds for dealing with these issues
For a recent review see Fleischer
The program of measuring and understanding CP asymmetries in
B decays will be a ma jor part of the high energy physics e ort for at
enormously least the next decade Although B physics has matured
since the discovery of the rst B meson events in the expanse
of uncharted territory remains vast There remain key unanswered
questions in charm physics as well such as the rate for D D mixing
Heavy quark physics will o er many opp ortunities and challenges and
there will b e a great deal of excitement as the pieces of the puzzle are
collected and the overall picture emerges
Acknowledgements
It is a pleasure to thank the vast number of p eople who help ed in the
preparation of these lectures My colleagues at UCSB Claudio Cam
J D Richman
pagnari Michael Witherell Rollin Morrison Steve Giddings Harry
Nelson and Bob Sugar help ed with many valuable comments sugges
tions and ideas I would like to esp ecially thank Natalia Kuznetsova
a UCSB graduate student who did an extraordinary amount of work
checking the text and equations
Many other colleagues help ed by generously providing information
and gures I m sure I won t remember all of them but they include
Patricia Ball Elliot Blo om Tom Browder Patricia Burchat Lucia
Di Ciaccio Jonathan Flynn Lawrence Gibb ons Joseph Kroll David
Lange Matthias Neub ert Alex Nipp e Ritchie Patterson Anders Ryd
Marie H el ene Schune Jim Smith No el Stanton Sheldon Stone Ed
Thorndike Jim Wiss and Renata Zaliznyak My colleagues in CLEO
have over the years given me all sorts of ideas and information that has
made its way into this pap er Dorothy McLaren at UCSB drew many
of the original gures her hard work contributed greatly to this pap er
I would also like to thank the organizers and sta at Les Houches
in particular Ra jan Gupta for putting together a sup erb scho ol in
a b eautiful and congenial setting Finally I want to thank Patricia
Metrop olis for valuable editorial assitance and Kate Metrop olis for her
help and encouragement in countless ways and esp ecially for her b elief
that this pap er would one day b e nished Iwould like to thank Daniel
Richman for many useful conversations
App endix Remarks on Hadronic Currents Form Fac
tors and Decay Constants
In this app endix I will derive some of the results regarding hadronic
currents that are used in the text For example in leptonic decay
of a pseudoscalar particle P only the matrix element of the axial
vector part of the current op erator contributes Eq while in a
semileptonic decay P P where P and P are b oth pseudoscalars
only the matrix element of the vector part contributes Eq In
P V decay where V is a vector meson there are contributions
from b oth the vector and axial vector parts of the current Eq
We rst consider the hadronic current that enters into the semilep
tonic decay P P The S matrix elementis
Z
x W x y J y S i d xd yJ
fi
lep had
Heavy Quark Physics and CP Violation
where
ig
p
x hP p jH x jP p i V J
had
is the hadronic current W x y is the W b oson propagator and
J y is the leptonic current The hadronic current has the well known
lep
V A form
hP p jH x jP p i hP p j x jP p i x
q Q
To avoid getting hop elessly confused it is useful to remember what
the ob jects in the hadronic current actually are In particular x
Q
annihilates Q or creates Q but it also contains the spinors appropri
ate for fermions and antifermions There are also spinors asso ciated
x x has no remaining so that the pro duct with
Q q q
spinor structure it consists only of particle creation and annihilation
op erators and functions of the space time co ordinate x The standard
pro cedure is to use translational invariance to remove this co ordinate
dep endence Let P b e the four momentum op erator Then
iP x iP x
H x e H e
We next substitute this expression into Eq letting the momentum
op erators act on the states to the left and right to obtain
i p p x
jP p i hP p jH x jP p i e hP p j
Q q
When integrated along with similar terms from the leptonic current
the exp onentials b ecome part of the overall four momentum conserving
function We can nowwork with the currentevaluated at x Let
be the parity op erator We use a di erent symbol here than in the
text to avoid confusion with the momentum op erator P Since is
y
unitary We can relate the hadronic current to its parity
transformed version
hP p jH jP p i hP p j jP p i
q Q
y y
hP p j jP p i
q Q
y
P P hP p j jP p i
q Q
where P and P are the intrinsic parities of the particles P and
P resp ectively and
p E p p E p
J D Richman
p E p p E p
The particle states must be lab eled with linear momenta because we
want to parametrize the currents in terms of simple functions of these
momenta Wenow use the transformation prop erties of the vector and
axial vector currents under parity
y
a
q Q q Q
y
a
q Q q Q
where a is a matrix with the elements along the diag
onal and zeros o the diagonal The relations in Eq simply say
that under parity the spatial comp onents only of a vector change
sign while the timelike comp onent only of an axial vector quantity
changes sign Note in particular that
p a p
Using the expressions for the transformed currents in Eq we
obtain
hP p jH jP p i P P a hP p j jP p i
q Q
There are only two linearly indep endent four vectors available in the
We can therefore write current whichwecho ose to b e p p and p p
V
q p p hP p j jP p i f
q Q
V
q p p f
and
A
jP p i f q p p hP p j
Q q
A
f q p p
V A V A
The functions f q and f q must b e Lorentz invariant so they
The can dep end only on q p p as well as m and m
P P
quantity p p can b e expressed in terms of these quantities We will
A A
show that f q f q which is equivalent to the statement
that only the matrix element of the vector current contributes Using
q p p q the expression on the right hand side of Eq
can be expressed in terms of form factors
V V
jP p i f q p p f q p p hP p j
Q q
A A
f q p p f q p p
Heavy Quark Physics and CP Violation
We substitute this expression into Eq to obtain
V V
hP p jH jP p i P P a f q p p f q p p
A A
q p p q p p f f
or
V V
hP p jH jP p i P P f q p p f q p p
A A
f q p p f q p p
But the original de nition of H given in the rst line of Eq
has a V A structure so that
V V
hP p jH jP p i f q p p f q p p
A A
f q p p f q p p
By comparing Eq with Eq we conclude that
A A
q f q f
Thus the only terms that contribute are those from Eq
A similar analysis can b e used for leptonic decays Here the hadronic
current is
h jH jM p i h j jM p i
Qq q Q Qq
y y
h j jM p i
q Q Qq
y
M h j jM p i
Qq q Q Qq
where j i represents the vacuum state Since the momentum of the
meson is the only available four vector in the hadronic current wecan
write
V
h j jM p i if p
q Q Qq
M
A
h j jM p i if p
q Q Q q
M
Using the transformation prop erties of the vector and axial vector cur
rents Eq we have
h jH jM p i M a h j jM p i
Qq Qq q Q Qq
h i
V A
f f p i M a
M M
h i
V A
i f f p
M M
J D Richman
But the original form of the current can be written
i h
A V
p f h jH jM p i i f
M M
V
so wemust have f and the nonzero contribution comes from the
M
matrix elementofthe axial vector current
App endix Remarks on CP Conjugate Amplitudes
In Sec we required the relationship between the two amplitudes
hf jH jB i and hf jH jB i We can write
y y
hf jH jB i hf j CP CP H CP CP jB i
i B f y
CP CP
e hf j CP H CP jB i
where we have used
i B
CP
CP jB i e jB i
i f
CP
CP jf i e jf i
If jf i is a CP eigenstate then replace exp i f with f
CP CP
We will consider the amplitude for a single weak pro cess the
tree diagram in B or B The relevant part of the weak
hamiltonian ignoring the strong interactions is
G
F
p
H V V V V
ub d u u b ud u d b u
ud ub
where We now compute
G
F
p
V V CP H CP
ub u d b u
ud
V V
ud d u u b
ub
Several steps were skipp ed here The op erator resp onsible for B
decay must destroy a b quark so that
G
F
y
p
V V hf j jB i hf j CP H CP jB i
ub u d b u
ud
while
G
F
p
V hf j jB i hf jH jB i V
ud u d b u
ub
Heavy Quark Physics and CP Violation
Thus
y
hf j CP H CP jB i
i B f
CP CP
hf jH jB i e hf jH jB i
hf jH jB i
Using Eq and Eq in Eq we nd
V V
ub
ud
i B f
CP CP
hf jH jB i e hf jH jB i
V V
ud
ub
which relates the amplitude for a particular pro cess with that for the
CP conjugate particles
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