Table Of ContentWIDTH DIFFERENCE IN THE D0 D0 SYSTEMa
−
ALEXEYA. PETROV
1
0
0
Department of Physics and Astronomy, The Johns Hopkins University,
2
3400 North Charles Street, Baltimore, Maryland 21218 USA
n
a and
J
9 Laboratory of Nuclear Studies, Cornell University,
2 Ithaca, New York 14853 USAb
E-mail: [email protected]
2
v
The motivation most often cited in searches for D0 −D¯0 mixing lies with the
0
possibilityof observingasignal fromnew physics whichdominates that fromthe
6 Standard Model. We discuss recent theoretical and experimental results in D0−
1 D0 mixing, including new experimental measurements from CLEO and FOCUS
9
collaborationsandtheirinterpretations.
0
0
0
1 Introduction
/
h
p Neutral meson-antimeson mixing provides important information about elec-
- troweaksymmetrybreakingandquarkdynamics. Inthatrespect,theD0 D0
p −
e systemisuniqueasitistheonlysystemthatissensitivetothedynamicsofthe
h bottom-type quarks. The D0 D0 mixing proceeds extremely slowly, which
v: in the StandardModel (SM)1,2−,3,4,5 is usually attributed to the absence ofsu-
i perheavy quarks destroying GIM cancelations. This feature makes it sensitive
X
bothtophysicsbeyondtheStandardModelandtolong-distanceQCDeffects.
r
a The low energy effect of new physics particles can be naturally written in
terms of a seriesoflocaloperatorsofincreasingdimensiongenerating∆C =2
transitions. These operators, along with the Standard Model contributions,
generate the mass and width splittings for the eigenstates of D0 D0 mixing
−
matrix defined as
D =pD0 q D¯0 , (1)
1
| 2i | i± | i
withcomplexparameterspandqdeterminedfromthephenomenological(CPT-
invariant) D0 D0 mass matrix6. It is convenient to normalize the mass and
−
aTo be published in the proceedings of 4th Workshop on Continuous Advances in QCD,
Minneapolis,Minnesota,12-14May2000.
bafterSeptember 1st2000
1
width differences to define two dimensionless variables x and y
m m Γ Γ
2 1 2 1
x − , y − . (2)
≡ Γ ≡ 2Γ
where m (Γ ) is a mass (width) of the corresponding state, D . Clearly, y
i i 1
2
is built from the decays of D into the physical states, and so it should be
dominated by the SM contributions. If CP-violation is neglected, then p = q
and D become eigenstates of CP. To set up a relevant formalism, let us
1
| 2i
recallthatinperturbationtheory,theijth elementoftheD0 D0 massmatrix
−
can be represented as
M iΓ = 1 D0 ∆C=2 D0 + 1 hDi0|HW∆C=1|IihI|HW∆C=1†|Dj0i .
− 2 2m h i|HW | ji 2m m2 E2+iǫ
(cid:20) (cid:21)ij D D I D− I
X
(3)
HerethefirsttermofEq.(3)comesfromthelocal∆C =2(boxanddipenguin)
operators. Thesecontributionsaffect∆M onlyandexpectedtobesmallinthe
Standard Model3,4,5. It is therefore natural to expect that the ∆C = 2 part
of Eq. (3) might receive contributions from the effective operators generated
by the new physics interactions. Next come the bilocal contributions which
areinducedbytheinsertionoftwoHamiltonianschangingthecharmquantum
number by one unit, i.e. built out of ∆C = 1 operators. This class of terms
contributestobothxandy andisbelievedtogivethe dominantSMcontribu-
tion to the mixing due to various nonperturbative effects. Some enhancement
due to the ∆C = 1 operators induced by new physics is also possible, but
unlikely given the strong experimental constraints provided by the data on D
meson decays. Yet, the motivation most often cited in searches for D0 D¯0
−
mixing lies with the possibility of observing a signal from new physics which
dominates that from the Standard Model. It is therefore extremely important
to estimate the Standard Model contribution to x and y.
The mass and width differences x and y can be measured in a variety
of ways, for instance in semileptonic D Klν or nonleptonic D KK or
→ →
D Kπ decays. Let us define the D meson decay amplitudes into a final
→
state f as
A f ∆C=1 D0 , A¯ f ∆C=1 D0 . (4)
f ≡h |HW | i f ≡h |HW | i
It is also useful to define the complex parameter λ :
f
q A¯
f
λ . (5)
f
≡ p A
f
2
LetusfirstconsidertheprocessesthatarerelevanttotheFOCUS7andCLEO8
experiments. Those are the doubly-Cabibbo-suppressed D0 K+π de-
−
→
cay, the singly-Cabibbo-suppressed D0 K+K decay, the Cabibbo-favored
−
→
D0 K π+ decay,andthe three CP-conjugatedecayprocesses. Letuswrite
−
→
down approximate expressions for the time-dependent decay rates that are
valid for times t < 1/Γ. We take into account the experimental information
that x, y and tanθ are small, and expand each of the rates only to the order
c
that is relevant to the CLEO and FOCUS measurements:
Γ[D0(t) K+π−] = e−Γt A¯K+π− 2 q/p2
→ | | | |
1
× |λ−K1+π−|2+[ℜ(λ−K1+π−)y+ℑ(λ−K1+π−)x]Γt+ 4(y2+x2)(Γt)2 ,
(cid:26) (cid:27)
Γ[D0(t) K−π+] = e−Γt AK−π+ 2 p/q 2
→ | | | |
1
× |λK−π+|2+[ℜ(λK−π+)y+ℑ(λK−π+)x]Γt+ 4(y2+x2)(Γt)2 , (6)
(cid:26) (cid:27)
Γ[D0(t) K+K−] = e−Γt AK+K− 2 1+[ (λK+K−)y (λK+K−)x]Γt ,
→ | | { ℜ −ℑ }
Γ[D0(t)→K+K−] = e−Γt|A¯K+K−|2 1+[ℜ(λ−K1+K−)y−ℑ(λ−K1+K−)x]Γt ,
Γ[D0(t) K−π+] = e−Γt AK−π+ 2,(cid:8)Γ[D0(t) K+π−] = e−Γt A¯K+π−(cid:9)2.
→ | | → | |
Within the Standard Model, the physics of D0 D0 mixing and of the tree
−
level decays is dominated by the first two generations and, consequently, CP
violationcanbesafelyneglected. Inall‘reasonable’extensionsoftheStandard
Model, the six decay modes of Eq. (6), are still dominated by the Standard
Model CP conserving contributions. On the other hand, there could be new
short distance, possibly CP violating contributions to the mixing amplitude
M . Allowingforonlysucheffects ofnewphysics,the pictureofCPviolation
12
is simplified since there is no direct CP violation. The effects of indirect CP
violation can be parameterized in the following way
q/p = R ,
m
| |
λ−K1+π− = √R Rm−1 e−i(δ+φ),
λK−π+ = √R Rm e−i(δ−φ), (7)
λK+K− = Rm eiφ.
−
Here R and R are real and positive dimensionless numbers. CP violation in
m
mixing is related to R = 1 while CP violation in the interference of decays
m
6
with and without mixing is related to sinφ = 0. The choice of phases and
6
signs in Eq. (7) is consistent with having the weak phase difference φ = 0
in the Standard Model and the strong phase difference δ = 0 in the SU(3)
3
limit. The weak phase φ is universal for Kπ and KK final states under our
assumption of negligible direct CP violation. We further define
x xcosδ+ysinδ,
′
≡
y′ ycosδ xsinδ. (8)
≡ −
Withthe assumptionthatthereisnodirectCPviolationinthe processesthat
westudy,andusingthe parameterizations(7)and(8),wecanrewriteEqs.(6)
as follows:
Γ[D0(t) K+π−] = e−Γt AK−π+ 2
→ | |
R2
R+√RR (y cosφ x sinφ)Γt+ m(y2+x2)(Γt)2 ,
m ′ ′
× − 4
(cid:20) (cid:21)
Γ[D0(t) K−π+] = e−Γt AK−π+ 2
→ | |
R 2
× R+√RRm−1(y′cosφ+x′sinφ)Γt+ 4m− (y2+x2)(Γt)2
(cid:20) (cid:21)
Γ[D0(t) K+K−] = e−Γt AK+K− 2[1 Rm(ycosφ xsinφ)Γt], (9)
→ | | − −
Γ[D0(t)→ K+K−] = e−Γt|AK+K−|2 1−Rm−1(ycosφ+xsinφ)Γt ,
Γ[D0(t) K−π+] = Γ[D0(t) K+π−(cid:2)] = e−Γt AK−π+ 2. (cid:3)
→ → | |
By studying various combinations of these modes we can pin down the values
of x and y in D0 D0 system.
−
2 Theoretical expectations
The leading piece of the short-distance partof the mixing amplitude is known
to be small1,2,3,4,5, but it is instructive to see why it is so. We will also
complement the discussion by including leading 1/m corrections.
c
As discussed above, the lifetime difference is associated with the long-
distance contribution to Eq. (3), i.e. the double insertion of ∆C =1 effective
Hamiltonian
G
∆C=1 = F ξ C (µ)u¯ Γ q q¯ Γµc +C (µ)u¯ Γ q q¯ Γµc (10)
HW −√2 q{ 1 α µ β β α 2 α µ α β β}
q
X
where Γ = γ (1 + γ ) and ξ = V V represents the appropriate CKM
µ µ 5 q c∗q uq
factor for ψ =d,s. C (m ) 0.514and C (m ) 1.270,as found in a NLO
1 c 2 c
≃− ≃
QCD calculation with ‘scheme-independent’ prescription. Hereafter we shall
not write the scale dependence of Wilson coefficients explicitly. The width
difference y can be written as an imaginary part of the matrix element of the
4
time-ordered product of two ∆C = 1 Hamiltonians of Eq. (10). Physically, it
is generatedby a setof on-shellintermediate states, andtherefore,constitutes
an intrinsically non-local quantity. However, in the limit m /Λ
c QCD
→ ∞
the momentum flowing through the light (s and d quark) degrees of freedom
is large and an Operator Product Expansion (OPE) can be performed. As
a result, both x and y can be represented by a series of matrix elements of
local operators of increasing dimension. In other words, if a typical hadronic
distance z 1/m , then the decay is a local process. Of course, significant
c
≫
corrections to the leading term of this series are expected, as the expansion
parameter Λ/m (Λ Λ is some hadronic parameter) is not small.
c QCD
∼
Itiswellknownthaty shouldvanishinthelimitofequalquarkmassesby
the virtue of GIM cancelationmechanism. For the DD¯ system it is equivalent
to the requirement of flavor SU(3) symmetry. The question here is by how
much SU(3) is broken. The (parametrically) leading contribution to x and y
comes from the matrix elements of operators of dimension six
O = u¯γ (1+γ )cu¯γ (1+γ )c, O =u¯(1 γ )cu¯(1 γ )c
1 µ 5 µ 5 1′ − 5 − 5
O = u¯ γ (1+γ )c u¯ γ (1+γ )c , O =u¯ (1 γ )c u¯ (1 γ )c (11)
2 i µ 5 k k µ 5 i 2′ i − 5 k k − 5 i
Using Fierz identities and performing necessary integrations we obtain
N +1 (m2 m2)2m2+m2
∆Γ(6) = c X s− d s d C2+2C C +C2N
D πN D m2 m2 2 1 2 1 C
c c c "
2(2N 1)B M2 2 N
c− D′ D C2+ − c C2N +2C C (,12)
− 1+Nc BD (mc+mu)2 (cid:18) 2 2Nc−1 1 c 1 2 (cid:19)#
(cid:0) (cid:1)
with N = 3 being the number of colors. This result was reported in 10.
c
Numerically, the effect of including QCD evolution amounts to the enhance-
ment of the box diagram estimate by approximately a factor of two. As one
can easily see, a standard box diagram contribution is recovered in the limit
C 0, C 1 where the QCD evolution is turned off
1 2
→ →
2 (m2 m2)2 5B M2
∆mbox = X s− d 1 D′ D ,
D 3π2 D m2 − 4B (m +m )2
c (cid:20) D c u (cid:21)
4 (m2 m2)2m2+m2 5B M2
∆Γbox = X s− d s d 1 D′ D , (13)
D 3π D m2 m2 − 2B (m +m )2
c c (cid:20) D c u (cid:21)
with X is given by X ξ ξ B G2M F2. Also, the B-parameters B =
D D ≡ s d D F D D D
B =1 in the usual vacuum saturation approximation to
D′
1 4F2m2
D0 O D¯0 = 1+ D DB ,
1 D
h | | i N 2m
(cid:18) c(cid:19) D
5
1 4m2 F2m2
hD0|O1′|D¯0i= − 1− 2N (m +mD )2 2Dm DBD′ , (14)
(cid:18) c(cid:19) c u D
1 4F2m2
D0 O D¯0 = 1+ D DB ,
2 D
h | | i N 2m
(cid:18) c(cid:19) D
1 1 4m2 F2m2
D0 O D¯0 = D D DB ,
h | 2′| i − N − 2 (m +m )2 2m D′
(cid:18) c (cid:19) c u D
where 2m in the denominator comes from the normalizationof meson states
D
and F is a D-mesondecay constant. It is clear fromEq. (12) that the small-
D
nessofthe leadingorderresultcomesfromthefactorof(m2 m2)2/m2 which
s− d c
represents the GIM cancelation among the intermediate s and d quark states
and from the factor (m2+m2)/m2 which represents the helicity suppression
s d c
of the intermediate state quarks. At the end, y x 0.1%.
≪ ≪
Of course, one should be concerned with the size of (parametrically sup-
pressed)correctionstoEq.(12). Thisisespeciallyimportantforthecalculation
of y because of the SU(3) and helicity suppression of the parametrically lead-
ing term. For example, perturbative QCD corrections, while suppressed by
α (m ), include the gluon emission diagrams, which do not exhibit helicity
s s
supression factors of m2.
s
Inaddition,bothSU(3)andhelicitysuppressionfactors(m2 m2)2(m2+
s− d s
m2) can be lifted at higher orders in Λ/m , which calls for a certain reorgani-
d c
zation of the operator expansion. In spite of being parametrically suppressed,
those “corrections”are in fact numerically largerthen the leading order term.
It was realized1,11 that the higher order contributions from the operators of
dimension nine and twelve that represent interactions with the background
quark condensates do exactly that.
Taking into account new operator structures generated by the renormal-
ization group running of the effective Hamiltonian from M down to m , the
W c
contribution of dimension nine operators reads
m2 m2
∆M(9) =4ξ2G2 s− dvα (N C2+2C C +C2)
D s F N m3 c 1 1 2 2
c c (
D0 (u¯Γ c)(u¯Γ c)(ψ¯Γµψ)D0 +others
α µ
× h | | i
(cid:20) (cid:21)
+2C2 D0 (u¯Γ Tac)(u¯Γ Tac)(ψ¯Γµψ)D0 + D0 (u¯Γ c)(u¯Γ Tac)(ψ¯ΓµTaψ)D0
2 h | α µ | i h | α µ | i
(cid:20)
+ D0 (u¯Γ Tac)(u¯Γ c)(ψ¯ΓµTaψ)D0 +others (15)
α µ
h | | i
(cid:21)
6
+2N C2(dabc+ifabc) D0 (u¯Γ Tac)(u¯Γ Tbc)(ψ¯ΓµTcψ)D0 +others
c 2 h | α µ | i
(cid:20) (cid:21)
+4N C C D0 (u¯Γ c)(u¯Γ Tac)(ψ¯ΓµTaψ)D0 +others
c 1 2 α µ
(cid:20)h | | i (cid:21))
Hereψ¯Γµψ =(s¯Γµs d¯Γµd)and‘others’denotesoperatorswithcyclicpermu-
−
tations of α,µ,µ indices. v represents the heavy quark velocity. Naive power
counting argument of Ref.1 implies that the U-spin violating operator ψ¯Γµψ
would scale like m Λ2 and therefore, the overall contribution to x and y is
s
multipliedbyafactorofm3,comparedtotheleadingterm,wherex m4 and
s ∼ s
y m6. In order to develop an imaginary part (and so generate y), a gluon
∼ s
correctionshouldbeconsidered. Therefore,thecontributionofdimensionnine
operators to y is suppressed by both α and phase space factors compared to
s
x, y(9) (α /16π)x(9) x(9). While it is impossible to estimate this con-
s
∼ ≪
tribution reliably (there are unknown matrix elements of 15 operators), naive
powercountingrulesimplythatitdominatestheparametricallyleadingterms
in the expansion of x1 and y11.
Thenextimportantcontributiontoyisobtainedatthenextorderin1/m
c
andisgivenbyasubsetofmatrixelementsoftheoperatorsofdimensiontwelve.
This contribution is obtained by cutting all light fermion lines and adding a
gluontotransferlargemomentum. Itisthereforerepresentedbyasetofeight-
fermion operators. While suppressed by α /m2, it again lifts another factor
s c
of m . More importantly, y(12) x(12)! This observation11 comes from the
s
∼
fact that imaginary part of the diagram that is needed for generating ∆Γ
D
can also be obtained by dressing the gluon propagator by quark and gluon
“bubbles”. The resulting α (m ) suppression is largely compensated by the
s c
“enhancement” from the QCD β function. This results in the estimate11,12
x,y 0.1%, (16)
∼
which is obtained from the naive dimensional analysis, as there are too many
unknown matrix elements for the accurate prediction to be made.
Indeed, the short-distance analysis, while systematic, is valid as long as
one believes that the charmed quark is sufficiently heavy for 1/m expansion
c
to be performed. Moreover, truly long-distance SU(3) breaking effects might
not be captured in the short distance analysis. For example, a contribution
from a light quark resonance with m m would not be captured in this
R D
≈
analysis. For a sufficiently narrow resonance, this provides a mechanism for
breaking of local quark-hadronduality13.
Analternativewayofestimatingxandy istostartfromthelongdistance
contributions generated by the intermediate hadronic states. They arise from
7
the decays to intermediate states common to both D0 and D0. Therefore, a
sumoverallpossiblen-particleintermediatestatesallowedbythe correspond-
ing quantum numbers should be taken into account in Eq. (3). In practice,
onlyafewstatesareconsidered,soonlyanorder-of-magnitudeestimateispos-
sible. Even with this restriction, it is extremely difficult to reliably determine
the total effect from a given subset of intermediate states due to the many
decay modes with unknown final state interaction (FSI) phases. In addition,
hadronicintermediatestatesinD0 D0 mixingareexpectedtooccurasSU(3)
−
flavor multiplets, so there are cancelations among different contributions to x
and y from the same multiplet. These flavor SU(3)relations canbe analyzed.
The initial D state is an SU(3) triplet, D = (D0,D+,D+), while the final
i s
state consists of a number of particles belonging to the octet representation,
π0 + η π+ K+
√2 √6
Mki = π− −√π02 + √η6 K0 . (17)
K− K¯0 − 23η
q
A set of relations for the transition amplitudes A = D0 ∆C=1 I can be
written. The effective Hamiltonian for D transitIions,h ∆|HCW=1 | (iψ¯c)(u¯ψ)
with ψ = s,d transforms as 15 6 ¯3 ¯3 under SU(H3)W. Th∼us, D and
F i
Mi should be contractedwith the⊕vec⊕tor H⊕(¯3)i (¯3 Hamiltonian), antisymmet-
k
ric (wrt upper indices) tensor H(6)ij (6 Hamiltonian), or symmetric tensor
k
H(15)ij (15 Hamiltonian). The SU(3) relations for ∆Γ follow as ∆Γ
k D D ∼
D0 ∆C=1 I I ∆C=1 D0 and are rather complicated for a generic multi-
h |HW | ih |HW | i
particle intermediate state.
Letuselaborateonthesimplestpossiblecontribution,duetointermediate
single-particle states13. These are rather simple to analyze, as the number of
such intermediate states is constrained. A contribution to y from a resonance
state R can be written as
y = 1 Im hD2|HW|RihR|HW† |D2i (D D ) . (18)
2Γm m2 m2 +iΓ m − 2 ↔ 1
(cid:12)res D R D− R R D
(cid:12) X
(cid:12)
The p(cid:12)seudoscalar0 + (scalar0++) intermediatestates haveCP = 1(CP =
−
−
+1)andcontribute(intheCP-limit)totheD (D )partoftheaboveequation.
1 2
In principle, this contribution exhibits a resonant enhancement for a narrow
resonancewithm m . Inreality,lightquarkstateswithsuchlargemasses
R D
≈
are not narrow.
In the limit of degenerate s and d quark masses the contributionfrom the
entire SU(3) multiplet would vanish, as expected from the GIM cancelation
8
mechanism. Yet, SU(3) is known to be badly broken in D-decays15,16, so a
sizable value for the width difference might not be surprising.
A setofSU(3)relationsforthe D R transitionsfollow fromthe follow-
→
ing transition amplitude
A(D R)=A D MiH(¯3)k+A D H(6)ikMl +A D H(15)ikMl (19)
→ 3 i k 6 i l k 15 i l k
A contribution of the octet of pseudoscalar single-particle intermediate states
π , K , K¯ , η (and possibly η with η η mixing angle θ ) is
H H H H H′ H− H′ H
y res =y(KH) 1y(πH) 3cos2θHy(ηH) 1sin2θHy(ηH′ ) , (20)
|octet − 4 − 4 − 4
with the mixing amplitudes induced by resonance R calculated to be
H 2 γ
yres = | R| R , (21)
−m3 Γ (1 µ )2+γ2
D − R R
where|HR|2 ≡hD0|HW|RihR|HW† |D0i,andthedimensionlessquantitiesµR ≡
m2/m2 and γ Γ /m are the reduced squared-mass and width of the
R D R ≡ R D
resonance.
No reliable information about the size of D R matrix elements is
W
h |H | i
available at the moment. A typical contribution to y from one 0 + single-
−
particle heavy intermediate state can be calculated using vacuum insertion
ansatz. This implies H 2 = µ f2m (G a f ξ /√2)2, with f being the
| R| R R D F 2 D d R
resonancedecayconstant. Makingan“educatedguess”aboutthesizeoff ,it
R
canbe shownthatatypicalcontributionfroma0 + amountstoafew 10 4
− −
×
(see Ref.13), but might be larger.
AnestimateofH fora0++ single-particleheavyintermediatestate(like
R
K (1430)orK (1940))canbeobtainedusingthesoftpiontheoremarguments
∗ ∗
ofRef.14 andmeasuredbranchingratiosforD+ Rπ+ transitions. Assuming
→
that expected corrections to the soft pion theorem are not large we derive for
R=K (1430)
∗
8πf2 (D+ K (1430)π+)Γ γ
y0++ =−tan2θC qππ B →fm∗2D ΓDD+0 (1−µRR)2+γR2 , (22)
whereq =0.368GeV isapion’smomentum, (D+ K (1430)π+) 0.023,
π ∗
B → ≃
f (K (1430) Kπ) 0.62, f = 0.13 GeV is a pion’s decay constant,
∗ π
≡ B → ≃
and Γ /Γ 0.4. This gives
D+ D0
≃
y (1430) 0.02%, (23)
| 0++ |≃
9
which is in the same ballpark as y0−+. Now, if we assume that HK∗(1430) ≃
HK∗(1940),
y (1940) 0.1%, (24)
| 0++ |≃
ItisclearfromtheEq.(20)thaty =0intheSU(3) limit,whereµ =µ ,γ =
F i 0 i
γ ,andH =H fori=π ,K ,η andsin2θ 0. Itisthereforenecessary
0 i 0 H H H H
→
to assess the pattern of SU(3)-symmetry breaking in Eq. (20). Neglecting
singlet-octet mixing and assuming that
µ = µ +δµ ,
i 0 i
γ = γ +δγ (25)
i 0 i
H = H +δH ,
i 0 i
| | | |
we obtain an estimate of y
(1 µ )2+γ2 µ (1 µ ) δµ 1δµ 3δµ
y res m3 Γ − 0 0 =2 0 − 0 K π η
− |octet× D H γ (1 µ )2+γ2 µ − 4 µ − 4 µ
| 0| 0 − 0 0 (cid:20) 0 0 0 (cid:21)
(1 µ )2 γ2 δγ 1δγ 3δγ
+ − 0 − 0 K π η(26)
(1 µ )2+γ2 γ − 4 γ − 4 γ
− 0 0 (cid:20) 0 0 0 (cid:21)
δH 1δH 3δH
K π η
+2 .
H − 4 H − 4 H
(cid:20)| 0| | 0| | 0|(cid:21)
Unfortunately, many of the parameters of Eq. (26) are not known. Yet, it’s
not unlikely that the total resonance contribution could amount to y 0.1%
≈
or so.
Let us briefly discuss a contribution from charged pseudoscalar two-body
intermediate state. It was originally considered in Refs.5,17,18 and estimated
to be potentially large,
1 d3p d3p
y2 = 2m Γ Re (2π)321E (2π)322E hD0|HW|p1,p2ihp1,p2|HW† |D0i(,27)
D pX1,p2 Z 1 2
where one must sum over all intermediate state particles p ,p , not only
1 2
groundstatemesons. Forthechargedpseudoscalarstate p ,p = K+,K ,
1 2 −
{ } { }
π+,π , K+,π , and K ,π+ . As before, the SU(3) relations among
− − −
{ } { } { }
amplitudes imply cancelations. These cancelations occur within each mul-
tiplet, however broken SU(3) assures that they are not complete. Residual
contributions from each multiplet then have to be summed up.
In some cases available experimental data can be used. For example, for
p ,p =K+K we easily obtain from Eq. (27) that y = (D0 K+K ),
1 2 − KK −
B →
10
