Axial-Vector Emitting Weak Nonleptonic Deacys of $\Omega_c^0$ Baryon

0.24 MB·English

by  Rohit Dhir

May 12, 2015
Axial-Vector Emitting
5 Weak Nonleptonic Deacys of Ω0 Baryon
1 c
0
2

y
a Rohit Dhir and C. S. Kim
M † ‡∗
Department of Physics and IPAP,

1 Yonsei University, Seoul 120-749, Korea
1

]
h
p
-
p
e
h
[

Abstract
2
v The axial-vector emitting weak hadronic decays of Ω0 baryon are investigated. After employing
9 c
5 the factorization and the pole model framework to predict their branching ratios, we derive the
2
symmetry breaking effects on axial-vector-meson-baryon couplings and effects of flavor dependence
4
0 on baryon-baryon weak transition amplitudes and, consequently, on their branching ratios. We
.
1 found that the W-exchange process contributions dominate p-wave meson emitting decays of Ω0
c
0
5 baryon.
1
:
v PACS numbers: 13.30.Eg,11.40.Dw, 14.20.Mr
i
X
r
a
∗ †[email protected], ‡[email protected]
I. INTRODUCTION
The production of heavy baryons have always posed experimental challenges and hence,
have generated much interest in their studies [1–4]. Many interesting observations by CDF,
D0, SELEX, FOCUS, Belle, BABAR, CMS, LHCb etc. [5–14] in context of mass spectrum,
lifetimes and decay rates have been made in resent years. Most recently, LHCb and CDF
collaborations [15–22] have announced more precise measurement of masses and lifetimes
of (Ξ0c, Ξ+c , Λb(∗), Ξ−b , Ξ0b, Ω−b ) baryons. Also, LHCb has now identified two new strange-
′
beauty baryonic resonances, denoted by Ξ and Ξ [23], though many of doubly and triply
b− ∗b−
heavy states are yet to be confirmed. In two body nonleptonic decay sector, first observation
of (Ω Ω0π ) decay process and measurement of CP-asymmetries for Λ pπ and
−b → c − b → −
Λ pK are reported by CDF collaboration [17, 24]. On the other hand, LHCb has
b −
→
reported first observation of Λ Λ+D and Λ J/ψpπ decays and the measurement of
b → c (−s) b → −
thedifferenceinCP-asymmetries between Λ J/ψpπ andΛ J/ψpK andmanyother
b − b −
→ →
decays involving b baryons [25–29]. However, little progress has been made in observing
−
decays of heavy charm meson. All these recent measurements have attracted, much needed,
attention to heavy baryonic sector.
On the theoretical side, various attempts had been made to investigate weak decays
of heavy baryons [30–58]. A number of methods, mainly, current algebra (CA) approach,
factorization scheme, pole model technique,non-relativistic quark model (NRQM), heavy
quark effective theory (HQET), framework based on next-to-leading order QCD improved
Hamiltonian etc., have been employed. Recent experimental developments have prompted
more theoretical efforts in b-baryon decays [58–64]. In all these works, the focus has so far
been on s-wave meson emitting decays of heavy baryons including Ω0 decays. Being heavy,
c
charmandbottombaryonscanalsoemit p-wavemesons. Inpast, thep-wave emittingdecays
of charmandbottombaryons have been studied using factorizationandpolemodel approach
[65–70]. However, p-wave emitting decays of Ω0 baryon remain untouched. The fact, that
c
Ω0 baryon is the heaviest and only doubly strange particle in charmed baryon sextet that
c
is stable against strong and electromagnetic interactions, makes it an interesting candidate
for the present analysis. Moreover, study of s-wave emitting decays of Ω0 baryon reveals
c
that nonfactorizable W-exchange terms dominate as compared to factorizable contributions
[37]. This makes study of Ω0 decays even more important to understand the mechanism
c
underlying W-exchange processes.
In our previous work [70], we have studied the scalar meson emitting decays of bottom
baryons employing the pole model. We have shown that such decays can acquire significant
pole(W-exchange) contributions tomake their branching ratioscomparable to s-wave meson
emitting decays. In the present work, we analyze the axial-vector meson emitting exclusive
nonleptonic decays of Ω0 baryon. We have already seen that for Ω0 decays the factorization
c c
contribution are small in comparison to the pole contributions in case of s-wave meson
emitting decays. Thus, the factorizable contributions to p-wave meson emitting decays
of Ω0 baryon are also expected to be suppressed. Therefore, we study weak nonleptonic
c
decays of Ω0 emitting axial-vector mesons in the factorization and the pole model approach.
c
1
We obtain the factorization contributions using the non-relativistic quark model (NRQM)
[33] and heavy quark effective theory (HQET) [47] based form factors. We employ the
effects of symmetry breaking (SB) on strong couplings that may decide crucial pole diagram
contributions [50, 71]. We use the traditional non-relativistic approach [72] to evaluate weak
matrix element to obtain flavor independent pole amplitude contributions at ground state
1+ intermediate baryon pole terms. Adding factorizable and pole contributions we predict
2
branching ratios (BRs) of Ω0 decays. Later, we employ the possible flavor dependence via
c
variationofspacial baryonwave function overlapinweak decay amplitude. Wefindthat BRs
of all the decay modes are significantly enhanced on inclusion of flavor dependent effects.
The article is organized as follows: In sec. II , we give a general framework including spec-
troscopy of axial-vector mesons, decay kinematics and effective Hamiltonian. Sec. III deals
with weak decay amplitudes both pole terms and factorization terms, weak transitions and
axial-vector meson-baryon couplings. Numerical results are given in sec IV. We summarize
our findings in last section.
II. GENERAL FRAMEWORK
A. Spectroscopy of Axial-Vector Mesons
The axial-vector meson spectroscopy has extensively been studied in literatures [73–77].
Here, we list the important facts. Spectroscopically, there are two types of axial-vector
mesons: 3P (JPC = 1++) and 1P (JPC = 1+ ). 3P and 1P states can either mix within
1 1 − 1 1
themselves or can mix with one another. Experimentally observed non-strange and un-
charmed axial-vector mesons exhibit first kind of mixing and can be identified as follows:
3P : meson 16-plet includes isovector a (1.230) 1 and four isoscalars, namely, f (1.285),
1 1 1
f (1.420)/f (1.512) and χ (3.511). The following mixing scheme has been used in
1 1′ c1
isoscalar (1++) mesons:
1
f (1.285) = (uu+dd)cosφ +(ss)sinφ ,
1 A A
√2
1
f (1.512) = (uu+dd)sinφ (ss)cosφ . (1)
1′ √2 A − A
1P : meson multiplet consists isovector b (1.229) and three isoscalars h (1.170), h (1.380)
1 1 1 ′1
and h (3.526), where spin and parities of h (3.526) and h (1.380) states are yet to be
c1 c1 ′1
confirmed, experimentally. These isoscalar (1+ ) mesons can mix in following manner:
−
1
h1(1.170) = (uu+dd)cosφA′ +(ss)sinφA′,
√2
1
h′1(1.380) = √2(uu+dd)sinφA′ −(ss)cosφA′. (2)
1
Here the quantities in brackets indicate their respective masses (in GeV).
2
The mixing angles are given by relation: φA(A′) = θ(ideal) θA(A′)(physical). The experimen-
−
tal observations predominantly favor the ideal mixing for these states i.e., φA = φA′ = 0◦.
The hidden-flavor diagonal states a (1.230) and b (1.229) cannot mix owing to C- and
1 1
G-parity considerations. However, there are no such restrictions for the states involving
strange partners namely, K1A and K1A′ of A(1++) and A′(1+−) mesons, respectively. They
mix in the following convention to generate the physical states :
K1(1.270) = K1AsinθK1 +K1A′cosθK1,
K1(1.400) = K1AcosθK1 −K1A′sinθK1. (3)
Several phenomenological analyses based on the experimental information obtained
twofold ambiguous solutions for θ i.e. 37 and 58 [73–77]. We wish to point out that
K1 ± ◦ ± ◦
the experimental measurement of the ratio of K γ production in B decays and the study of
1
charm meson decays to K (1.270)π/K (1.400)π favor negative angle solutions. Very recently
1 1
[75], it has been shown that choice of mixing angle θ is intimately related to choice of
K1
′ ′
angle for f f and h h mixing schemes. The mixing angle θ 35 is favored over
− − ′ ′ K1 ∼ ◦
55 for near ideal mixing for f f and h h . Therefore, we use θ = 37 for our
∼ ◦ − − K1 − ◦
calculation; however, we also give results on 58 for comparison.
◦
−
B. Kinematics
The matrix element for the baryon decay process e.g. B (1+, p ) B (1+, p ) +
i 2 i → f 2 f
A (1+, q) can be expressed as
k
B (p )A (q) H B (p ) = iu¯ (p )ε µ(A γ γ +A p γ +B γ +B p )u (p ), (4)
h f f k | W| i i i Bf f ∗ 1 µ 5 2 fµ 5 1 µ 2 fµ Bi i
where u are Dirac spinors for baryonic states B and B . εµ is the polarization vector of
B i f
the axial-vector meson state A . A ’s and B ’s represent parity conserving (PC) and parity
k i i
violating (PV) amplitudes, respectively. The decay width for the above process is given by
q E +m E2
Γ = µ f f 2( S 2 + P 2)+ A( S +D 2 + P 2) , (5)
8π m | | | 2| m2 | | | 1|
i h A i
where m and m are the masses of the initial and final state baryons, and q = (p p )
i f µ i f µ
−
is the four-momentum of axial-vector meson
1
q = [m2 (m m )2][m2 (m +m )2],
| µ| 2miq i − f − A i − f A
where m is the mass of emitted p-wave meson [34, 41]. The decay emplitude of the final
A
state is now an admixture of S, P and D wave angular momentum states with
q m +m
µ i f
S = A , P = B +m B ,
1 1 1 i 2
− −E (cid:18)E +m (cid:19)
A f f
q q2
µ µ
P = B , D = (A m A ),
2 1 1 i 2
E +m −E (E +m ) −
f f A f f
3
where E and E are the energies of the axial-vector meson and the daughter baryon,
A f
respectively. Furthermore, there are two independent P-wave amplitudes: one corresponds
to the singlet spin combination of the parent and daughter baryon and the other corresponds
to the triplet. The interference between S and D wave amplitudes and P-wave amplitudes
results in asymmetries for the daughter state with respect to the spin of the parent state.
The corresponding asymmetry parameter is
4m2 Re[S P ]+2E2 Re[(S +D) P ]
α = A ∗ 2 A ∗ 1 . (6)
2m2( S 2+ P 2)+E2( S +D 2 + P 2)
A | | | 2| A | | | 1|
Thus, to determine the decay rate and asymmetry parameters we require to estimate ampli-
tudes, A and B.
C. Hamiltonian
The QCD modified current current effective weak Hamiltonian consisting Cabibbo-
⊗
favored (∆C = ∆S = 1), Cabibbo-suppressed (∆C = 1,∆S = 0) and Cabibo-doubly-
− −
suppressed (∆C = ∆S = 1) modes is given by
− −
G
Heff = F V V c (u¯d)(c¯s)+c (s¯d)(u¯c) +
W √2 ud c∗s 1 2 (∆C=∆S= 1)
(cid:8) (cid:2) (cid:3) −
¯ ¯
V V c (s¯c)(u¯s) (dc)(u¯d) +c (u¯c)(s¯s) (u¯c)(dd)
ud c∗d 1{ − } 2{ − } (∆C= 1, ∆S=0) −
(cid:2) ¯ ¯ (cid:3) −
V V c (dc)(u¯s)+c (u¯c)(ds) , (7)
us c∗d 1 2 (∆C= ∆S= 1)
(cid:2) (cid:3) − − (cid:9)
where V and(q¯q ) q¯γ (1 γ )q denote theCabibbo-Kobayashi-Maskawa (CKM) matrix
ij i j i µ 5 j
≡ −
elements and the weak V-A current, respectively. We use the QCD coefficients c (µ) = 1.2,
1
c (µ) = 0.51 at µ m2 [35]. Furthermore, nonfactorizable effects may modify c and c ,
2 − ≈ c 1 2
thereby indicating that these may be treated as free parameters. The discrepancy between
theory and experiment is greatly improved in the large N limit. Interestingly, the charm
c
conserving decays of Ω0 are also possible but they are kinematically forbidden in present
c
analysis.
III. DECAY AMPLITUDES
The hadronic matrix element B A H B for the B B + A process may be
f k W i i f k
h | | i →
expressed as
B A H B + , (8)
f k W i Pole Fac.
h | | i ≡ A A
where and represent pole (W-exchange) and factorization amplitudes, respec-
Pole Fac.
A A
tively. The polediagramcontributions involving theW-exchange process areevaluated using
the pole model framework [35]. One may consider factorization as a correction to the pole
model which includes the calculation of possible pole diagrams via s , u and t channels,
− − −
where t channel virtually implicate tree-level diagrams i.e. factorizable processes. The con-
−
tribution of these terms can be summed up in terms of PC and PV amplitudes.
4
A. Pole Amplitudes
The first term, , involves the evaluation of the relevant matrix element
Pole
A
B H B = u¯ (B +γ A)u (9)
h f| | ii Bi 5 Bj
between two 1+ baryon states. A and B are s wave and p wave decay amplitudes, re-
2 − −
ceptively. A and B include the contributions of s and u channels for positive-parity
− −
intermediate baryon (JP = 1+) poles; henceforth, given by Apole and Bpole as follows:
2
g a a g
Apole = Σ BfBnAk ni + fn BnBiAk , (10)
−n (cid:20) mi mn mf mn (cid:21)
− −
g b b g
Bpole = Σ BfBnAk ni + fn BnBiAk , (11)
n (cid:20) mi +mn mf +mn (cid:21)
where g are the strong axial-vector meson-baryon coupling constants; a and b are weak
ijk ij ij
baryon-baryon matrix elements defined as
B H B = u¯ (a +γ b )u . (12)
h i| W| ji Bi ij 5 ij Bj
It is well known that the PV matrix element b vanishes for the hyperons owing to
ij
B A HPV B = 0 in the SU(3) limit. This also implies for non-leptonic charm meson
h f k| W | ii
decays that b a suppressing s-wave contributions for 1+ poles. These contributions
ij ≪ ij 2 −
are further suppressed by presence of sum of the baryon masses in the denominator. Thus,
only PC terms survive for non-leptonic decays of charm baryons. It may be noted that
the negative-parity intermediate baryon (JP = 1−) may also contribute to these processes
2
and may turn out to be important. However, evaluation of such terms require knowledge
of the axial-vector meson strong coupling constants for (1−) states. Unfortunately, no such
2
theoretical or experimental information is available in literature. Moreover, in the leading
non-relativistic approximation, one can ignore JP = 1−, 3−.... and higher resonances as
2 2
they would require at least one power of momentum in H in order to connect them with
W
the relevant ground state in the overlap integral. That means one has consider terms of the
order v/c. In the same manner, to connect radial excitations with the corresponding ground
state, one would need terms of order (v/c)2; otherwise the overlap integral would be zero due
to orthogonality of the wave functions [51]. Therefore, we have restricted our calculation to
ground state 1+ intermediate baryon pole terms to estimate the pole contributions to the
2
axial-vector meson emitting decays of charm baryons.
B. Factorizable Amplitudes
Likewise meson decays, the reduced matrix element (4) can be factorized to obtain decay
amplitudes (ignoring the scale factors) in the following form:
< A (q) A 0 >< B (p ) Vµ +Aµ B (p ) >, (13)
k µ f f i i
| | | |
5
where
< A (q) A 0 >= f m ε , (14)
k | µ| A A ∗µ
and f is the decay constant of the emitted axial-vector meson A . The baryon-baryon
A k
matrix elements of the weak currents are defined as
f f
< B (p ) V B (p ) >= u¯ (p )[f γ 2 iσ qν + 3 q ]u (p ), (15)
f f µ i i f f 1 µ µν µ i i
| | − m m
i i
and
g g
< B (p ) A B(p ) >= u¯ (p )[g γ γ 2 iσ qνγ + 3 q γ ]u (p ), (16)
f f µ i f f 1 µ 5 µν 5 µ 5 i i
| | − m m
i i
where, f and g denote the vectot and axial-vector form factors as functions of q2 [35]. The
i i
factorizable amplitudes are thus given by
G m m
Afac = FF f c m [gBi,Bf(m2) gBi,Bf(m2) i − f],
1 −√2 C A k A 1 A − 2 A m
i
G
Afac = FF f c m [2gBi,Bf(m2)/m ],
2 √2 C A k A 2 A i
G m +m
Bfac = FF f c m [fBi,Bf(m2)+fBi,Bf(m2) i f],
1 √2 C A k A 1 A 2 A m
i
G
Bfac = FF f c m [2fBi,Bf(m2)/m ], (17)
2 −√2 C A k A 2 A i
where F contains appropriate CKM factors and Clebsch-Gordan (CG) coefficients and c
C k
are QCD coefficients.
The baryon-baryon transition form factors f and g are evaluated in literature using the
i i
non-relativistic quark model (NRQM) [33] and the heavy quark effective theory (HQET)
[47]. The NRQM based form factors are calculated in Breit frame and include correction like
the q2 dependence of the form factors, the hard-gluon QCD contributions, and the wave-
function mismatch. Later, in light of the fact that the form factors for heavy baryon-baryon
transitions should also include constraints from the heavy quark symmetry, 1/m correction
Q
totheformfactorswasintroducedusingHQET. ItmaybenotedthattheΩ decaysinvolving
c
factorizable amplitudes only include Ω0 Ξ0 and Ω0 Ξ form factors which come out to
c → c → −
be equal, numerically. The evaluated form factors using NRQM [33] are given by
Ω Ξ : f (0) = 0.23, f (0) = 0.21, g (0) = 0.14, g (0) = 0.019. (18)
c 1 2 1 2
→ − −
Similarly, form factors calculation in HQET [47] yields :
Ω Ξ : f (0) = 0.34, f (0) = 0.35, g (0) = 0.10, g (0) = 0.020. (19)
c 1 2 1 2
→ − −
We wish to remark here that numerical calculations of the factorizable branching ratios we
use dipole q2 dependence following HQET constraints.
The decay constants of axial-vector mesons [73–77] used for numerical evaluations are
given by
6
f f = 0.221 GeV; f (1270) = 0.175 GeV;
a1 ≈ f1 K1
whiledecayconstantforK (1.400)maybecalculatedbyusingrelationf (1.400)/f (1.270) =
1 K1 K1
cotθ i.e.
1
f (1.400) = 0.099 GeV, for θ = 58 ; f (1.400) = 0.225 GeV, for θ = 37 .
K 1 ◦ K 1 ◦
1 − − 1 − −
It may also be noted that decay constants of axial-vector mesons are not so trivial to
understand. As these may be effected by factors like, C-parity/G-parity conservations,
mixing scheme and SU(3) breaking etc. For more details readers are referred to [73–77].
C. Weak Transitions
The flavor symmetric weak Hamiltonian [40, 45] for quark level process q +q q +q
i j l m
→
can be expressed as,
H = V V c (m )[B¯[i,j]kB H[l,m]], (20)
W ∼ il j∗m c [l,m]k [i,j]
−
where where c = c +c and the brackets, [ , ], represent the anti-symmetrization among
1 2
− [1,3]
the indices. The spurion transforms like H . We obtain the weak baryon-baryon matrix
[2,4]
elements a for CKM favored and CKM suppressed modes from the following contraction:
ij
H = a [B¯[i,j]kB H[l,m]], (21)
W ∼ W [l,m]k [i,j]
where a is weak amplitude. It is worth remarking here that the enhancement due to hard
W
gluon exchanges, coming through c , will effect weak baryon-baryon matrix element. Also,
−
we ignore the long-distance QCD effects reflected in the bound-state wave functions.
D. Axial-vector meson-baryon couplings
In SU(4), Hamiltonian representing the strong transitions is given by
1
H =√2(g +g )( B¯[a,b]dB Ac)
strong D F 2 [a,b]c d
+ √2(g g )(B¯[a,b]dB Ac), (22)
D − F [a,b]c d
where B ,B¯[a,b]dand Ac are the baryon, anti-baryon, and axial-vector meson tensors, re-
[a,b]c d
spectively and g (g ) are conventional D-(F)-type strong coupling constants [30, 33, 37].
D F
Experimentally, there are no measurements available for the axial-vector-meson-baryon
coupling constants for charm sector. Since, it is difficult to determine g directly, a
aNN
reasonable estimate could be obtained by using Goldberger-Treiman (GT) relation:
g m
A N
g = , (23)
NNπ
f
π
which relates pion-nucleon coupling g with axial-vector coupling g [78, 79]. The GT
NNπ A
relation exhibits direct relation between spontaneous chiral symmetry breaking and PCAC
7
hypothesis at SU(2) SU(2) or, with a possible extension, SU(3) SU(3) . Here, g
L R L R A
× ×
represents contribution to the dispersion relation of all the axial-vector states higher than
the pion. In light of the PCAC, the heavier axial vector states contributing to g must
A
reproduce the pion pole at q2=0. Thus, the combined contribution of all the heavier states
may be replaced by an effective pole a i.e if we assume axial-vector dominance2 [81, 82] to
get,
g m2
g A a = 8.60 (24)
NNa1 ≈ f
a1
for g = 1.26 given by β decay [79]. To proceed forward, we use QCD sum rules analysis
A
[73] based
g
D
g = 6.15 and g = 2.45 for = 2.5, (25)
D F
g
F
which in turn yields axial-vector meson-baryon strong coupling constants based on SU(4)
symmetry.
We wish to pointed out that the SU(4) symmetry is badly broken, hence it would not
be wise to use SU(4) symmetry based strong coupling constants for charm baryon decays.
Therefore, we consider the SU(4) breaking effects in strong coupling constants by using the
Coleman-Glashow null result [83]. The tadpole mechanism can generate breaking effects,
namely, the medium strong, the electromagnetic and the weak effects, that transforms like
an SU(3) octet, via a single symmetry breaking term. Thus, except for the tadpole term
the hadronic Hamiltonian remains SU(3) invariant. In SU(3) octet, the strangeness chang-
ing scalar tadple S , transforming as sixth component of symmetry breaking octet, can be
6
rotated away by unitary transformation. These strangeness changing effects produced by
S tadpole must vanish, leaving behind the electromagnetic and the weak effects. Khanna
6
and Verma [71] exploited the null result to obtain SU(3) broken baryon-baryon-pseudoscalar
couplings. After validating SU(3) case they extended their results to SU(4), where symmetry
breaking effects belong to similar regular representation 15. In SU(4), the weak interaction
Hamiltonian responsible for hadronic weak decays of charm baryons belongs to representa-
′′
tion 20 . The tadpole term of the weak Hamiltonian belongs to representation 15. In this
case the charm changing effects generated through S tadpole must vanish (for details see
9
[71]). We wish to remark here that the tadpole-type symmetry breaking effects does not in-
clude any additional parameter. The symmetry broken (SB) baryon-meson strong couplings
are calculated by
′
′ M +M 1 ′
gBB (SB) = B B gBB (Sym), (26)
A 2M α A
N P
′
where gBB (Sym) is SU(4) symmetric couplings and the ratio of mass breaking terms α is
A P
2 As a consequence of spontaneous chiral symmetry breaking, Weinberg sum rules[80] relate m and m
a1 ρ
by assuming vector and axial-vector dominance.
8
given by
δM 3m m
c c u
α = = −
P δM r8m m
s s u
−
[71]. TheobtainedvaluesofSBstrongaxial-vectormeson-baryoncoupling constantsrelevant
for our calculation have been given in table I. For heavy baryon decays, it has been observed
[84] that mass independent strong couplings lead to smaller pole contributions. It is quite
obvious that symmetry breaking will result in larger values for strong couplings as compared
to symmetric ones due to mass dependence. Consequently, higher pole contributions would
′
be expected. We also give the expressions for the gBB in terms of g and g . However,
A D F
we have given the absolute numerical values for the strong couplings where the actual sign
would depend upon the conventions used and could be determined from their expressions in
present case.
E. Baryon Matrix Element
In general, numerical evaluation of W-exchange terms (pole terms) involves weak matrix
element of the form B HPC B . Riazuddin and Fayyazuddin [72] has calculated this
h f| W | ii
matrix element for noncharmed hyperon decays in the non-relativistic limit. Following their
analysis one can obtain the matrix element for the charm baryons as a first approximation.
Though Ω is heavy and, therefore, outgoing quarks may have large momenta, we use the
c
non-relativistic approximation to get the first estimates of the baryonic matrix elements.
The matrix element for the W-exchange process (c+d s+u) can be expressed as
→
G
FV V ψ¯ (p′)γ (1 γ )γ ψ (p )ψ¯ (p′)γ (1 γ )α+ψ (p )+i j (27)
M ≈ √2 du cs u i µ − 5 i− d i s j µ − 5 i c j ↔
(cid:2) (cid:3)
where ψ’s are Dirac spinors and q = p p′ = p′ p . The operators α+ convert d u and
i− i j − j i →
γ convert c s. In the leading non-relativistic approximation, only terms corresponding
j− →
to γ0 and γiγ have non zero limits, which are then reduced to only parity conserving part
5
of . Thus, in leading non-relativistic approximation we have
M
G
PC = FV V (γ α+ +α+γ )(1 σ σ ), (28)
M √2 du cs i− j i j− − i · j
Xi>j
where S =σ /2 are Pauli spinors representing spin of ith quark. Fourier transformation of
i i
the above expression gives the parity conserving weak Hamiltonian
G
HPC = FV V α+γ (1 σ σ )δ3(r), (29)
W √2 du cs i j− − i · j
X
i=j
6
following which, we can get a reasonable estimate of these terms. One can fix the scale by
assuming the baryon overlap wave function to be flavor independent such that
ψ δ3(r) ψ ψ δ3(r) ψ , (30)
f i c f i s
h | | i ≈ h | | i
9