hep-ph/9707369 PDF

U C L-IP T -97-09
arXiv:hep-ph/9707369v1 17 Jul 1997
F ield T heory approach to K
system s
0
K
0
and B 0
B0
M .Beuthea; 1,G .Lopez C astroa; b and J.Pestieaua
a
Institutde Physique T heorique,U niversite catholique de Louvain,
B-1348 Louvain-la-N euve,Belgium
b
D epartam ento de F sica,C investav delIPN ,A partado Postal
14-740,C .P.07000 M exico,D .F.M exico
A bstract
Q uantum
eld theory provides a consistent fram ework to dealw ith unstable
particles. W e present here an approach based on eld theory to describe the
production and decay of unstable K
0
K
0
and B 0
B 0 m ixed system s. T he
form alism is applied to com pute the tim e evolution am plitudes of K
0
and K
0
studied in D A PH N E and C PLEA R experim ents. W e also introduce a new set
ofparam eters that describe C P violation in K !
isospin decom position ofthe decay am plitudes.
1
IISN researcherunder contract 4.4509.66
decays w ithoutrecourse to
1
Introduction
N eutral strange and beauty pseudoscalar m esons, K 0K
0
and B 0B 0, are system s of
two unstable m ixed states ofspecialinterest for the study ofweak interactions. T hey
are particularly suited to study the phenom ena of C P violation together w ith the
oscillations in their tim e-dependent decay probabilities [1,2].
T he traditional description of unstable neutral kaons is based on the W ignerW eisskopf(W W )form alism [3].In thisapproach,the tim e evolution ofdecaying states
is governed by a Schrodinger-like equation based on a non-herm itian ham iltonian [4]
that allow s particle decays. A s a result,the diagonalizing transform ations,in general,
arenotunitary,thecorresponding eigenstatesarenotorthogonaland thenorm alization
cannot be done w ithout am biguities.
Besides these unsatisfactory features ofthe W W form alism ,one faces other di culties. Projected factories ofK and B m esons [5,6]are expected to m easure the C P
violation and oscillation param eters to a higher accuracy than present experim ents.
W hile it is not clear w hether the approxim ations involved in the W W form alism are
valid for both the K and B system s,a consistent schem e is certainly required to com pute these observables to a high degree ofaccuracy.
In this paper we adopt the view that the quantum m echanicalbehavior ofa com plete processinvolving the production and decay ofunstable statescan only be consistently described in the fram ework ofquantum
eld theory [7]. In Q FT ,the S-m atrix
am plitude becom es the basic object that describes the properties of a physical process am ong particles. T his am plitude is taken between in-and out-asym ptotic states
w hich are de ned as non-interacting states (stable particles) existing far away the interaction region. T herefore,as a generalrule,unstable particles cannot be considered
as asym ptotic states.
U nder these conditions, unstable particles appear only as interm ediate states to
w hich we associate G reen functions (propagators) to describe the propagation am pli1
tudes from their production to their decay spacetim e locations. T he form of these
propagators,w hich is consistent w ith specialrelativity and causality,determ ines the
tim e evolution ofthe decay probabilities. Since Lorentz covariance is im plicit to the
eld theory approach,neither boosttransform ations nor the choice ofa speci c fram e
[8]are required to de ne the tim e param eter in the am plitude.
In this paper we w illalso address som e questions related to the usualtreatm ent of
C P violating param eters.A siswellknow n,the K 0K 0 (and B 0B 0)system requirestwo
param etersto accountforC P violation in the propagation (indirect)and decay (direct)
ofneutralkaons,usually related to two theoretically de ned com plex param eterscalled
respectively " and "0 [9].O n the one hand the description based on the W W form alism
isnotvalid beyond order" becauseoftheaforem entioned di culty in thenorm alization
ofnon-orthogonalstates. Since "0
O ("2)forthe K 0K
0
system ,itbecom es necessary
to establish a correctform alism [10]to accountconsistently forterm soforder"2.T his
isallthe m ore needed because the usualapproxim ations forneutralkaonsin the W W
form alism m ight failin the case ofB m esons [11].
O n the other hand, the reduction of observable C P violating param eters in the
K 0K
0
system to only two theoretical param eters " and "0 cannot be done w ithout
assum ing isospin sym m etry and the factorization of strong rescattering e ects [9].
T hese assum ptions are rather strong in view of the sm allness ofdirect C P violating
e ects [12,13]. In this paper,we give up the isospin decom position ofthe am plitudes,
param etrizing C P violation in term s ofthree param eters. T he rst,^,describes the
m ixing ofC P eigenstates K 1
and
00, account
K 2 (indirect C P violation),w hile the two other,
+
for the C P violating 2 decays of K 2 (direct C P violation) in our
approach.
T he paper is organized as follow s. In section 2 we discuss the diagonalization of
m ixed propagatorsin m om entum space forthe system ofunstable neutralpseudoscalar
K and B m esons. In section 3 we focus on the space-tim e representations of these
2
propagators. Section 4 is devoted to the applications of our form alism to com pute
the tim e-dependent distributions ofneutralkaon decays as adapted to C PLEA R and
D A PH N E experim ents. O ur conclusions are presented in section 5.
2
U nstable particle propagator in m om entum space
A s previously discussed,the propagator is the basic object in the S-m atrix am plitude
that describes the propagation ofan unstable state from its production at space-tim e
pointx to itsdecay atpointx0.In thissection we study them om entum space representation ofthe propagatorforthe neutralkaon system ,w hich w illbe needed to com pute
the S-m atrix am plitudes.
T he description ofthe evolution ofan unstable particle requires non perturbative
inform ation to be introduced in the bare propagator. T he fullpropagator is obtained
from a D yson sum m ation ofself-energy graphs.Since the weak interaction couplesthe
avorstatesK
0
and K 0,therenorm alized propagatorforthesetwo unstableparticlesis
a non diagonal2
2 m atrix [14].By im posing the C PT sym m etry,we can param etrize
the inverse propagatorfor unstable kaons offour-m om entum p as follow s
0
iD
1
a+ b C
d
B
1
(p2)= @
a
b
A
(1)
2
00(p );
(2a)
d
w here
p2
d
m 20
iIm
a+ b
2
00(p );
(2b)
a
00
(p2);
(2c)
b
(p2)w ith ; = 0;0 aretherenorm alized
w herem 0 istherenorm alized m assand i
com plex self-energies ofthe neutralkaon system in an obvious notation. R em ark that
the non diagonalterm s depend on the phase convention chosen for the kaons.
3
W e de ne the C P eigenbasis as
B K1 C
A
@
K2
w here S = S
1
0
1 0
1
1
p B@
2 1
1 C B K
1
0
A @
1
1
0
0
C
A
B K
K
0
0
1
0
K
C
A
S@
(3)
and C PjK 0i= jK 0i. T he corresponding inverse propagator is
0
iD
1
(p2)
1
S iD
(p2)S
1
B d+ a
1
b C
= @
A
b
d
(4)
:
a
N ow ,ifwe introduce the com plex param eter "
^ as
"
^
1+ "
^2
b
;
2a
(5)
we can diagonalize the inverse propagator as follow s
1 0
0
iD
1
(p2) =
1
1
"
^2
B
@
1 0
1 "
^ C B d+ a
A @
"
^ 1
1 "
^2
1+ "
^2
0
0
d
a
"
^2
C B
A @
1
1
"
^C
A
1
1+ "
^2
"
^
:
(6)
1
T herefore,the physicalbasisofneutralkaonsconsistsoftwo statesK L ;S ofde nite
m asses m L ;S and decay w idths
L ;S ,such
p2
dS
2
dL
p
that
m 2S + im S
m
2
L
+ im L
S
L
1 "
^2
1+ "
^2
(7a)
1 "
^2
;
a
1+ "
^2
(7b)
= d+ a
= d
T he constant w idth approxim ation w illbe justi ed in section 3. C onsistency between
Eqs.(2a)and (7)dem andsto approxim ate Im
2
00(p )by
a constantterm
m0
0.T he
propagatorD (p2) now reads
0
iD (p2)=
1
1
"
^2
B
@
1 0
1 0
1 "
^ C B dS
1
A @
"
^ 1
0
0 C B
A @
dL 1
1
1
"
^C
A
:
(8)
"
^ 1
A s already anticipated, the diagonalization of the non-herm itian m atrix given in
Eq.(4) involves a non-unitary m atrix. In order to provide a link w ith the usualform alism ,we can obtain a proper orthogonaland norm alized physicalbasis ifwe de ne
4
independent ket (in-) and bra (out-) states,respectively, as left-hand and right-hand
eigenvectors ofthe inverse propagator[4]:
0
@
p
A
jK L i
and
"
^2
1
p
A
@
A @
@
hK L j
1
1
"
^2
B
@
jK 2i
1
1 0
1
"
^ C B hK 1j C
A
A @
"
^
(9)
A
"
^ 1
0
B hK S j C
1
^ C B jK 1i C
B 1 "
1
1
0
1 0
0
1
B jK S i C
1
(10)
:
hK 2j
N otice that for an arbitrary "
^ bra states do not correspond to herm itian conjugate of
ket states.
T he quantities m S;L ;
a;b;m 0 and
0
S;L
can be m easured experim entally, w hile the param eters
can bein principlecom puted from thetheory.T herelationshipsbetween
these two sets ofparam eters are
!
1 1+ "
^2
a=
fm 2L
2 1 "
^2
m 20
b=
3
im 0
0
=
m 2S
i(m L
L
mS
S )g ;
(11a)
1 2
fm + m 2S
2 L
i(m L
L
+mS
S )g ;
(11b)
"
^
fm 2L m 2S i(m L
2
1 "
^
L
mS
S )g:
(11c)
Space-tim e evolution of resonance propagators
In thissection we are interested in the tim e dependentpropertiesofthe propagation of
unstable particles for the purposes ofstudying C P violation and the tim e oscillations
in the kaon system . W e shalltherefore focus on the properties ofthe unstable state
propagatorin con guration space.
Let us rst consider the propagatorfor a stable spin zero particle :
Z
0
F (x
x) = i
0
d4p
e ip:(x x)
:
(2 )4 p2 m 2 + i"
5
(12)
T he tim e dependence w illbecom e m anifest in the am plitude ifwe put this expression
into another form show ing a separate tim e evolution for the particle and the antiparticle. A contour integration in the com plex p0 plane gives
Z
0
F (x
w ith E
p
0
0
d3p eip~:(~x ~x)e iE (t t)
(t0 t)
3
(2 )
2E
0
0
Z
3
ip
~:(~
d p e x ~x)eiE (t t)
+
(t t0)
(2 )3
2E
x) =
(13)
p
~ 2 + m 2.
D epending ofthe speci c process,the rst(second)term in Eq. (13)w illsurvive in
the tim e-dependent am plitude and w illdescribe a particle (antiparticle) propagating
forward in tim e.
Letusnow considerthepropagatorofa spin zero resonance.T heD yson sum m ation
of self-energy graphs leads to the follow ing renorm alized propagator in m om entum
space representation :
p2
m
2
i
;
iIm (p 2)
(14)
w here i (p 2)isthe renorm alized self-energy w hose absorptive partvanishesundera
threshold p2th in the case ofonly one decay channel.
In orderto justify theconstantw idth approxim ation used in Eqs.(7),letusconsider
the 2 contribution to the kaon self-energy. A direct com putation ofIm (p 2),for a
kaon ofsquared four-m om entum s,gives
(g12 + g22=2) s sth
8
s
Im (s)=
w here sth = 4m 2 (w ith the approxim ation m
e ective couplings for K
0
!
+
;
0 0
+
1=2
(s
= m
0
sth )
(15)
) and w here g1, g2 are the
respectively.
R em arking that cutting rules give Im (s = m 2) =
m
w here
is the parti-
cle w idth in the center-of-m ass fram e and m the kaon m ass,we can w rite the above
expression as
m 2 s sth
Im (p 2)= p
s m 2 sth
6
1=2
(s
sth ):
(16)
T he in uence ofthe kaon w idth in the propagator is only felt for
p
s values near
the kaon m ass. T herefore the propagator can be greatly sim pli ed by neglecting all
but the rst term in an expansion ofIm (s) around s = m 2. M ore precisely,
Im (s)=
for m
p
x
s
1+ O
m
x
m
(17)
sth
m + x ,w ith x an arbitrary num ber such that x =m < < 1 and
w ith sth < < m .
Since
S =(m S
2m )
O (10
14
) [2],the form ofthe propagator w ith a constant
w idth
i
m + im
p2
2
(p2
p2th )
(18)
;
turns out to be an extrem ely good approxim ation for the renorm alized propagator.
T herefore,thespace-tim erepresentation ofthespin zero propagatorfortheunstable
particle can be w ritten as
Z
0
R (x
e ip:(x
m 2 + im
d4p
(2 )4 p2
x) = i
0
x)
(p2
(19)
:
p2th )
Sim ilarly asdone above forthe stable particle propagator,we would like to express
0
R (x
explicitly the tim e dependence of
x). It becom es convenient to separate the
propagatorinto two pieces :
0
R (x
x)=
(1) 0
R (x
(2) 0
R (x
x)+
(20)
x)
w ith
Z
(1) 0
R (x
x) = i
Z
(2) 0
R (x
x) = i
0
d4p
e ip:(x x)
(2 )4 p2 m 2 + im
3
dp
(2 )3
Z
p0th
p0th
(
0
dp
e
2
q
w here p0th =
p
~ 2 + p2th .
7
1
ip:(x0 x)
p2
1
m
2
p2
m 2 + im
)
(21)
q
p2th )< < 1,we can show that
U sing the condition =(m
0
(2) 0
R (x
x)
1
O@
A
q
;
p2th
m
w hich allow s to w rite
Z
0
R (x
0
e ip(x x)
d4p
(2 )4 p2 m 2 + im
x)= i
0
0
@ 1+
11
O@
AA
q
m
:
(22)
p2th
In orderto m ade explicitthe tim e dependence ofthe unstable propagatorletususe
the follow ing pole decom position
!
2
p
w here E =
2
m + im =
p
p0
im
E +
2E
!
2
im
2E
p0 + E
1+ O
!!
(23)
m2
p
~ 2 + m 2.
T herefore, by neglecting very sm allterm s of order 10
the com plex p0 plane w ith the poles located at (E
Z
0
R (x
0
14
,the contour integralin
im =2E ) gives
0
m
d3p eip~:(~x ~x )e iE (t t)
0
x) =
e 2 E (t t) (t0 t)
3
(2 )
2E
0
0
Z
m
d3p eip~:(~x ~x )e iE (t t )
0
+
e 2 E (t t ) (t t0):
3
(2 )
2E
(24)
T he interpretation is sim ilar to the one for the stable particle, except for the decay
constant
w hich expresses the unstability ofthe particle and antiparticle. T he case
ofK 0K 0 system considered in thispaperism ore involved,because the propagatorisa
2 2 m atrix. T his problem iscircum vented by perform ing the diagonalization (see Eq.
(8)) before doing the contour integration.
N otice that
= t0
t is the tim e elapsed between the production and decay lo-
cations ofthe resonance. N ote also that,contrary to non-relativistic approaches,the
factor m =E naturally appears in the exponential decay factor. T herefore, no boost
transform ationsare required to relate the proper tim e to the tim e param eterofa m oving particle. O fcourse,the exponentialdecay takesitsusualform e
fram e ofthe resonance.
8
=2
[1]in the rest
4
A pplications
In this section we com pute the fullS-m atrix am plitudes for the production and decay
of neutral kaons as studied in C PLEA R and D A PH N E experim ents. T hen, we derive the tim e evolution ofthese transition am plitudes and introduce the C P violation
param eters intrinsic to our description.
4.1
C P LE A R experim ent
A t the C PLEA R experim ent [15],K
0
and K
0
are produced at point x in the strong
interaction annihilation of pp, and subsequently decay at point x0 to
e ects of weak interactions. T he production m echanism s of K
K 0K
+
; K 0K
+
0
and K
+
0
by the
are pp !
,thus neutralkaons can be tagged by identifying the accom pa-
nying charged kaon [15]. A ftertheirproduction,K
0
(orK 0)oscillates between itstwo
com ponents K L and K S before decaying to the 2
nalstates. W e are interested in
the description ofthe tim e evolution ofthe fulldecay am plitude and its interference
phenom ena. Itisinteresting to note thatdespite the factthatcharged kaonsand pions
have sim ilarlifetim esasK L ,they can be treated asasym ptotic particlesin the present
case.
In order to relate the di erent S-m atrix am plitudes,let us rst consider the production m echanism ofK 0K 0. Since strong interactions conserve strangeness,we have
+
M (pp ! K
+
K 0) = M (pp ! K
K 0)= 0;
(25)
w hich,according to equation (3),im plies
M (pp ! K
M (pp ! K
+
+
+
K 1) = M (pp ! K
K 1) =
M (pp ! K
+
K 2)
A
(26a)
K 2)
B:
(26b)
C:
(26c)
A ssum ing C PT invariance we obtain
M (pp ! K
+
K 0) = M (pp ! K
9
+
K 0)
C ollecting allthese constraints,we get
C
A = B = p :
2
(26d)
N ow ,let us rst consider the com plete process for the production ofa K
into
0
decaying
+
p(q)+ p(q0)! K (k)+
+
(k0)+ K 0(p)! K (k)+
+
(k0)+
+
(p1)+
(p2) : (27)
T he full am plitude corresponding to this process can be w ritten (the subscript +
0
refers to the charges ofthe two pions from K
decay):
Z
0
d4x d4x0 ei(p1 + p2 ):x M (K 1 !
=
T+
+
0
K 1K
R
K 1K
R
w here
2
(x0
2
(x0
+
); M (K 2 !
)
(28)
1
0
B M (K ! K 1) C
x)@
A M (pp ! K
M (K 0 ! K 2)
+
0
K 0)ei(k+ k
x) is the propagator m atrix for the coupled K 1
q q0):x
K 2 system in
con guration space.
W ith the help ofequations (8) and (26),this gives
Z
T+
=
0
d4x d4x0 ei(p1 + p2 ):x M (K 1 !
+
); M (K 2 !
d4p
e
(2 )4
i
ip:(x0 x)
1
0
"
^2
0
A @
@
1
1 0
^ C B dS 1(p)
B 1 "
1
)
1 0
0
Z
+
0
"
^ 1
1
dL (p)
C B
A @
1
"
^C
A
"
^
1
1
0
B
@
M (K ! K 1) C p
A
0
2 A ei(k+ k
q q0):x
(29)
M (K 0 ! K 2)
Inserting equations (3) and (7),we obtain
Z
T+
d4x d4x0
= i
(
[M (K 1 !
d4p i(p1 + p2
e
(2 )4
+
+ ["
^ M (K 1 !
p):x0
)+ "
^ M (K 2 !
+
)+ M (K 2 !
10
0
ei(k+ k
A
1+ "
^
1
)] 2
2
p
m S + im S S
)
1
: (30)
)] 2
p
m 2L + im L L
q q0+ p):x
+
+
A n expression depending on the four-m om entum is obtained by perform ing allthe
integrals :
= i(2 )4
T+
(
(4)
(q+ q0
k0
k
A
M (K 1 ! +
p2)
1+ "
^
"
^+ +
+
(p1 + p2)2 m 2L + im L
)
p1
1+ + "
^
2
(p1 + p2)2 m S + im S
S
)
(31)
L
w here
+
M (K 2 !
M (K 1 !
+
)
;
)
+
(32)
is the param eter describing direct C P violation in our approach.
A nother possibility is to keep the tim e dependence,proceeding as in section 3,on
the diagonalized propagator ofEq.(30). In that case,the am plitude ofan originally
pure K
T+
0
+
state decaying into
= (2 )4
(4)
(q+ q0
(
k
reads
k0
p2)
p1
A
M (K 1 !
1+ "
^
+
)
(33)
Z
(1 +
+
+ (^
"+
m S
m S
1
1
dt
t
t
(t)+ ei(E S E )t e2 S E S
( t)
e i(E S E )t e 2 S E S
2E S
)
Z
m L
m L
1
1
dt
t
i(E L E )t
i(E L E )t 2 L E L t
2 L EL
)
e
e
(t)+ e
e
( t)
2E L
"
^)
+
q
w hereE = p01+ p02 isthetotalenergy ofthe
+
(p
~1 + ~
p2)2 + m 2S;L .
system and E S;L =
T he tim e t in eq. (33) has been de ned as the tim e elapsed from the production
to the decay locations ofK 0. T hus,the transition am plitude T (t) describing the tim e
evolution ofthe system for t> 0 is given by the integrand proportionalto (t) in eq.
(33),nam ely
T+ (t) = (2 )4
(
(4)
(q+ q0
1+ + "
^
e
2E S
k0
k
1
2
m S
S E
S
t
A
M (K 1 !
1+ "
^
"
^+ +
+
2E L
iE L t
e
Let us now consider the analogous process w here a pure K
0
duced and then decay to
+
iE S t
p2)
p1
e
,i.e. pp ! K
11
+
K
0
e
! K
+
)eiE t
+
)
1
2
m L
L E
L
t
:
(34)
state is initially pro+
. Follow ing the
sam e procedure as in the case ofK
0
production and decay,we com pute the follow ing
0
decays
k0
p1
expression for the tim e evolution ofK
T+
(t) = (2 )4
(
(4)
(q+ q0
1+ + "
^
e
2E S
k
iE S t
e
1
2
m S
S E
S
A
M (K 1 !
p2)
1 "
^
"
^+ +
e iE L t e
2E L
t
0 0
Let us notice that ifwe were interested in the
by
00
)eiE t
)
1
2
m L
L E
L
t
(35)
:
decay m ode ofneutralkaons,we
+
would have to replace in Eqs. (34) and (35) M (K 1 !
+
+
) by M (K 1 !
0 0
) and
w here
0 0
M (K 2 !
M (K 1 !
00
)
(36)
:
0 0)
U sing Eqs. (34) and (35), we can express the m easurable ratio of C P-violating
to C P-conserving decay am plitudes of K L , K S states in term s of the C P-violating
param eters proper to our approach:
M (K L !
M (K S !
+
+
"
^+
)
=
) 1+
+
+
+
"
^
;
(37)
and
00
M (K L !
M (K S !
A s is well know n, the param eters
+
0 0
)
0 0)
and
=
00
"
^+
1+
00
^
00"
:
(38)
are com m only used to express the
violation ofC P in the two pion decays ofK L (see for exam ple pages 422-425 in [2]).
N ote that the above relations between m easurable quantities and the param eters that
quantify directand indirectviolation ofC P,arederived withoutrelying on assum ptions
based on isospin sym m etry,contrary to the relations obtained for the
param eters in
term softheusualparam eters"and "0.Furtherm ore,itcan beexplicitly show n thatthe
param eters
+
and
00
are independent ofthe phase convention chosen for K 0; K 0,
w hich is not the case for "
^ and
+
; 00.
U sing theisospin sym m etry and theW u-Yang phase convention [9],we observe that
= "
^ (see R ef. [2],p.102)and the param eters
12
+
; 00
are expected to be very sm allso
thatterm sofO ( ij")can be neglected in the above equations. In thatlim it,we obtain
+
00
=
=
0
2"0:
Finally,let us m ention that Eqs. (34) and (35) reduce to the current expressions
for the tim e evolution used in the analysis ofthe C PLEA R collaboration [15],w hen
we choose the centerofm assfram e (p
~1 + ~
p2 = ~
0)ofthe two pion produced in K
0
K
0
decays.
4.2
N eutralkaon production at D A P H N E
In thissection weconsidertheoscillationsofthepairofneutralkaonsproduced in e+ e
annihilations at D A PH N E [5]. T he results obtained in the present form alism for the
K
0
K 0 system can be straightforwardly generalized to describe the sam e phenom ena
in pair production ofneutralB m esons in the (4s) region [6].
109 pairs K 0K 0/year)
N eutraland charged kaons w illbe copiously produced (
in e+ e collisionsoperating ata centerofm assenergy around the m assofthe (1020)
m eson [5]. T he
m esons produced in e+ e annihilations decay at point x into K 0K
pairs,and subsequently each neutralkaon oscillates between its K L
0
K S com ponents
before decaying to nalstates f1(p) and f2(p0) at spacetim e points y and z :
(q)! K 0K 0 ! f1(p)f2(p0)
(39)
w here q; p and p0 are the corresponding four-m om enta.
Since each nalstate can be produced by eitherK
the two am plitudes arising from the exchange of K
0
0
orK 0,we m ustadd coherently
and K
0
as interm ediate states.
Taking into account the charge conjugation properties ofthe electrom agnetic current,
the pair of neutral kaons is found to be in an antisym m etric state [16]. T hus, the
relative sign of the two contributions to
13
! f1f2 decays m ust be negative. T he
S-m atrix am plitude for the process indicated in Eq. (39) is
Z
0
d4x d4y d4z eip:y+ ip :zM ( ! K 0K 0)e
Tf1 f2 =
28
>
<
6
4 (M
>
:
8
>
<
iq:x
0
0
(K 1 ! f1); M (K 2 ! f1))
K 1K
R
2
B
(y
x)@
M (K !
M (K 0 !
0
K 1K
R
(M (K 1 ! f2); M (K 2 ! f2))
>
:
2
(z
B
x)@
M (K
0
M (K
0
19
K 1) C >=
A
>
K 2) ;
19
! K 1) C >=
A
>
! K ) ;
2
i
sam e expression w ith K 0 $ K
Letusde ne M
0
(40)
:
M (K i ! fj).W ith the help ofEqs. (3)and (8),we can reexpress
ij
the previous am plitude as
Z
Tf1 f2 =
Z
d4k d4k0 ik:(y x)
d x d yd ze
e
(2 )4 (2 )4
(
M 11 + "
^M 21 "
^M 11 + M 21 M 12 + "
^M 22
+
dS (k)
dL (k)
dS (k0)
^M
^M 11 + M 21 M 12 + "
M 11 + "
^M 21 "
dS (k)
dL (k)
dS (k0)
4
4
ip:y+ ip0:z iq:x
4
( 1)
2(1 "
^2)
"
^M 12 + M 22
dL (k0)
)
"
^M 12 + M 22
22
+
dL (k0)
ik0:(z x)
M ( ! K 0K 0)
=
(2 )4
(
(4)
(q
p
p0)M ( ! K 0K 0)
+"
^M 21
m + im S
"
^M 11 + M 21
+ 2
p
m 2L + im L
M
p2
"
^M
11
2
S
S
p02
M
L
p02
1
"
^2
1
12 + M 22
m 2L + im L L
+"
^M 22
2
m S + im S
12
)
(41)
:
S
A s in the previous subsection, the tim e evolution of the am plitude is obtained
through the insertion of the explicit tim e-dependent propagator into the am plitude
(41).T he result is
Z
Tf1 f2 =
0
)+ other term s
dtdt0 T (t;t0) (t) (t
0
( t)or ( t
)
(42)
w here t and t0 are the tim es taken by unstable kaons to propagate from the com m on
production point (x) up to their disintegration into f1 at point y and f2 at point z,
respectively.
14
T hus,the explicit tim e evolution ofthe decaying am plitude is given by
T (t;t0) = (2 )4
(4)
(q
0 t+
p0)eip
p
0
ip0 t0
(
(M
11
+"
^M
"M 12
21)(^
+M
1
1
M ( ! K 0K 0)
4E S E L
1 "
^2
22)e
iE S (p)t
1
2
m S
S E
S
t
e
iE L (p0)t0
1
2
(43)
m L
L E
L
t0
)
+ (^
"M
11
+M
21)(M 12
+"
^M
22)e
iE S (p0)t0
1
2
m S
S E
S
t0
e
iE L (p)t
1
2
m L
L E
L
t
q
w here E S;L (p)
p
~ 2 + m 2S;L .
A s we have already pointed out in the case ofthe C PLEA R experim ent,no boost
transform ations are required to adequate the tim e evolution of the decay am plitude
to a given reference fram e. O bserve that,due to the initialantisym m etrisation ofthe
K 0K
5
0
system ,T (t;t)= 0 iff1 = f2 and p = p0 as noted in R ef. [16].
D iscussion and C onclusions
In this paper we have discussed a form alism based on quantum
eld theory w hich
describes a system oftwo unstable coupled pseudoscalar particles. Properties related
to the production and decay ofthese unstable statesare consistently incorporated into
the relativistic S-m atrix am plitude w hich is the physically m eaningfull object for a
given process [7]. T herefore,this form alism does not exhibit the lim itations intrinsic
to the W igner-W eisskopf approxim ation or non-relativistic approaches that we have
discussed in the introduction ofthis paper.
W e have applied this form alism to describe the tim e evolution ofthe K
0
and K
0
decay am plitudes in C PLEA R and D A PH N E experim ents. Since Lorentz covariance
is im plicit to the eld theory approach, our results are valid in any reference fram e
contrary to the results obtained in other approaches w hich require boost transform ationsto relate the tim e param etersde ned in restand m oving fram es. Letusm ention
thatotherpapers,w hich departfrom the W W approxim ation,have appeared recently
[17,18,19,20];however,they allpresentlim itationsm ainly related to the introduction
15
ofa propertim e param eterw hich isintrinsic to non-relativistic treatm ents. N otice also
thatan interference term show ing tim e oscillations w illappear in the decay probabilitiesobtained from Eqs. (34),(35)and (43).T histerm can be converted to a evolution
in space by using the classicalform ula t= (E =jp
~j)j~
xj,w hich applies only to particles
observed in the detector and not to (o -shell) unstable K
L
and K S states.
Finally, we would like also to stress that the present form alism allow s to de ne
direct(
+
; 00)and
indirect(^
")C P-violating param etersforK decaysw ithoutrelying
on assum ptions based on isospin sym m etry. T his is particularly suitable because the
factorization ofstrong rescattering e ects m ay not be fully justi ed for the study of
C P violation in B decays.
Letusrem ark thatthepresentform alism can beextended straightforwardly to treat
the isospin violation in the pion electrom agnetic form factor in the
! resonance
region [21].
A cknow ledgem ents. W e thank Jean-M arc G erard et Jacques W eyers for useful
discussions.
R eferences
[1] See forexam ple: P.K .K abir,T he C P Puzzle (A cadem ic Press,N ew York,1968).
[2] T he Particle D ata G roup,R .M .Barnett etal,Phys.R ev.D 54 Part I(1996).
[3] E.W igner and V .F.W eisskopf,Z.Phys.63 (1930)54;ibid.65,18 (1930).
[4] R .G .Sachs,A nn.Phys.22,239 (1963).
[5] T he D A N E PhysicsH andbook Vol.I,Eds.L.M aiani,G .Pancheriand N .Paver,
(Frascati1992).
16
[6] K EK -B 1995 B factory D esign R eport K EK ;BA BA R 1995 TechnicalD esign R eport SLA C -R -95-457.
[7] M .Veltm an,Physica 29,186 (1963).
[8] K .G otfried and V .F.W eisskopf,C onceptsofParticle Physics,(O xford U niversity
Press,1984) Vol.Ip.151.
[9] See for exam ple: R .E.M arshak, R iazzuddin and C .P.R yan, T heory of W eak
Interactions in Particle Physics,(W iley-Interscience,1969).
[10] Q .W ang and A .I.Sanda,Phys.R ev.D 55,3131 (1997).
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605 (1997).
[12] E731 collaboration,L.K .G ibbonsetal.,Phys.R ev.Lett.70,1203 (1993);Phys.
R ev.D 55,6625 (1997);N A 31 collaboration,G .D .Barretal.,Phys.Lett.B 317,
233 (1993).
[13] B.W instein and L.W olfenstein,R ev.ofM od.Phys.65,1113 (1993).
[14] L.Baulieu and R .C oquereaux,A nn.Phys.140,163 (1982).
[15] C PLEA R collaboration,R .A dleretal.,Phys.Lett.B 363,243 (1995);ibid.B 363,
237 (1995).
[16] H .J.Lipkin,Phys.R ev.176,1715 (1968).
[17] H .J.Lipkin,Phys.Lett.B 348,604 (1995).
[18] B.K ayser, Proc.of the 28th International C onference on H igh Energy Physics,
(W orld Scienti c,W arsaw ,1996)p.1135;B.K ayserand L.Stodolsky,Phys.Lett.
359,343 (1995).
17
[19] J.Lowe,B.Bassallech,H .Burkhardt,A .R usek,G .J.Stephenson and T .G oldm an,Phys.Lett.B 384,288 (1996).
[20] S.Srivastava,A .W idom and E.Sassaroli,Z.Phys.C 66,601 (1995).
[21] K .M altm an,H .B.O ’C onnelland A .G .W illiam s,Phys.Lett.B 376,19 (1996);
A .G .W illiam s,H .B.O ’C onnelland A .W .T hom as,e-print hep-th/9707253.
18