Operator Relations for SU(3) Breaking Contributions to K and K* Distribution Amplitudes

0.24 MB·English

by  Patricia Ball

IPPP/05/76
DCPT/05/152
Operator Relations for SU(3) Breaking
Contributions to K and K∗ Distribution Amplitudes
6
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0 Patricia Ball∗ and Roman Zwicky†
2

n
a IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK
J

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v
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0
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p
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p Abstract:
e
h
: We derive constraints on the asymmetry a1 of the momentum fractions carried by quark
v
i and antiquark in K and K∗ mesons in leading twist. These constraints follow from exact
X
operator identities and relate a to SU(3) breaking quark-antiquark-gluon matrix elements
r 1
a which we determine from QCD sum rules. Comparing our results to determinations of a
1
from QCD sum rules based on correlation functions of quark currents, we find that, for
ak(K∗) the central values agree well and come with moderate errors, whereas for a (K) and
1 1
a⊥(K∗) the results from operator relations are consistent with those from quark current
1
sum rules, but come with larger uncertainties. The consistency of results confirms that the
QCD sum rule method is indeed suitable for the calculation of a . We conclude that the
1
presently most accurate predictions for a come from the direct determination from QCD
1
sum rules based on correlation functions of quark currents and are given by:
a (K) = 0.06±0.03, ak(K∗) = 0.03±0.02, a⊥(K∗) = 0.04±0.03.
1 1 1
∗[email protected]
†[email protected]
1 Introduction
Hadronic light-cone distribution amplitudes (DAs) of leading twist have been attracting
considerable interest in the context of B physics. They enter the amplitudes of QCD
processes that can be described in collinear factorisation, which include, to leading order
in an expansion in 1/m , a large class of nonleptonic B decays [1], such as B → ππ,KK.
b
DAs are also an essential ingredient in the calculation of weak decay form factors such
as B → π,ρ,K,K∗ from QCD sum rules on the light-cone [2]. These decays, and their
CP asymmetries, are currently being studied at the B factories BaBar and Belle and are
expected to yield essential information about the pattern of CP violation and potential
sources of flavour violation beyond the SM.
One particular problem in this context is the size of SU(3) breaking corrections to
K and K∗ DAs, which has been studied in a number of recent papers [3, 4, 5, 6]. The
DAsthemselves aredefined asmatrixelements ofquark-antiquarkgauge-invariant nonlocal
operators on the light-cone. To leading-twist accuracy, there are three such DAs for K and
K∗ (z2 = 0):
1
h0|q¯(z)z/γ [z,−z]s(−z)|K(q)i = if (qz) dueiξ(qz)φ (u),
5 K K
Z0
1
h0|q¯(z)z/[z,−z]s(−z)|K∗(q,λ)i = (e(λ)z)fkm dueiξ(qz)φk (u),
K K∗ K
Z0
1
h0|q¯(z)σ [z,−z]s(−z)|K∗(q,λ)i = i(e(λ)q −e(λ)q )f⊥(µ) dueiξ(qz)φ⊥(u), (1)
µν µ ν ν µ K K
Z0
with the Wilson-line
1
[z,−z] = Pexp ig dαzµA (αz)
µ
(cid:20) Z−1 (cid:21)
inserted between quark fields to render the matrix elements gauge-invariant. In the above
(λ)
definitions, e is the polarisation vector of a vector meson with polarisation λ; there are
ν
two leading-twist DAs for vector mesons, φk and φ⊥, corresponding to longitudinal and
K K
transverse polarisation, respectively. The integration variable u is the (longitudinal) meson
momentum fraction carried by the quark, u¯ ≡ 1−u the momentum fraction carried by the
(k,⊥)
antiquark and ξ = u−u¯. The decay constants f are defined in the usual way by the
K
local limit of Eqs. (1) and chosen in such a way that
1
duφ(u) = 1. (2)
Z0
All three distributions φ ,φk ,φ⊥ can be expanded in Gegenbauer polynomials C3/2,
K K K n
φ(u) = 6uu¯ 1+ a C3/2(2u−1) , (3)
n n
!
n≥1
X
where the a are hadronic parameters, the so-called Gegenbauer moments.
n
1
The most relevant quantities characterising SU(3) breaking of these DAs are the decay
constants f and f⊥,k, and a (K) and a⊥,k(K∗), which can be expressed in terms of the
K K 1 1
DAs as
5 1
a (K) = du(u−u¯)φ (u) (4)
1 K
3
Z0
and correspondingly for ak,⊥(K∗). a describes the difference of the average longitudinal
1 1
momenta of the quark and antiquark in the two-particle Fock-state component of the
meson, aquantity thatvanishesforparticleswithequal-massquarks(particleswithdefinite
G-parity). The decay constants f and fk can be extracted from experiment; f⊥ has been
K K K
calculated from both lattice [7] and QCD sum rules, e.g. Ref. [6]. In this paper we focus on
the determination of a : no lattice calculation of this quantity has been attempted yet, so
1
essentially allavailableinformationona comesfromQCDsumrulecalculations. a canbe
1 1
calculatedeither directlyfromthecorrelationfunctionoftwo quarkcurrents[3,4,6,8,9]or
from operator identities relating it to certain quark-quark-gluon matrix elements, denoted
κ , which are calculated from QCD sum rules themselves [5]. In a previous paper, Ref. [6],
4
we have obtained the following results from the first method, at the scale of 1 GeV:
a (K)BZ = 0.050±0.025, ak(K∗)BZ = 0.025±0.015, a⊥(K∗)BZ = 0.04±0.03, (5)
1 1 1
whereas Braun and Lenz found the following results from operator identities [5]:
a (K)BL = 0.10±0.12, ak(K∗)BL = 0.10±0.07. (6)
1 1
These results were obtained to first order in m and neglecting explicit terms in m2 and
s s
m in the operator identities. Numerically, however, these terms are not negligible: the
q
O(m2) correction shifts a (K) by +0.17 and ak(K∗) by +0.08 for our central value of m .
s 1 1 s
Corrections in m are small for ak(K∗), but chirally enhanced for a (K) and shift a (K)
q 1 1 1
by +0.04 for our central value of m . A consistent inclusion of O(m ) effects requires the
q q,s
calculation of these terms also for κ . In the present paper, we present such a calculation
4
andimprovethesumrulesforκ derived inRef.[5]bytheinclusionofalldominanttermsto
4
O(m2) and O(m2), which include in particular two-loopperturbative and gluon-condensate
q s
contributions. The perturbative contributions come with large coefficients and prove to be
very relevant numerically. We then construct several sum rules for κ which differ by the
4
chirality structure of the involved currents and the spin-parity assignment of the hadronic
states coupling to them. We provide criteria that allow one to identify the sum rules most
suitable for the calculation of κ and obtain the corresponding numerical results, including
4
a careful analysis of the theoretical uncertainty of κ and the corresponding values of a .
4 1
One important finding of our paper is that the results of these calculations agree, within
errors, with those from the quark current sum rules, which shows that the application of
the QCD sum rule method to the calcualation of a yields mutual consistent results. It is
1
this consistency that strengthens our confidence in the validity of the results for a .
1
Our paper is organised as follows: in Sec. 2 we derive the operator relations for a ,
1
in Sec. 3 we obtain numerical results for the corresponding matrix elements and compare
with the results of Ref. [6]. In Sec. 4 we summarise and conclude. The paper also contains
2
two appendices giving explicit expressions for all relevant correlation functions and Borel
transforms.
a
2 Exact Identities for
1
In Ref. [5], the following relations were obtained for a (K) and ak(K∗):
1 1
9 m −m m2 −m2
s q s q
a (K) = − +4 −8κ (K), (7)
5 1 m +m m2 4
s q K
3 f⊥ m −m m2 −m2
ak(K∗) = − K s q +2 s q −4κk(K∗), (8)
5 1 fKk mK∗ m2K∗ 4
where κ (K) and κk(K∗) are twist-4 quark-quark-gluon matrix elements defined by
4 4
h0|q¯(gG )iγµγ s|K(q)i = iq f m2 κ (K), (9)
αµ 5 α K K 4
h0|q¯(gG )iγµs|K∗(q,λ)i = e(λ)fkm3 κk(K∗). (10)
αµ α K K∗ 4
κ (K) and κk(K∗) vanish for m → m due to G-parity. The special structure of (7) allows
4 4 s q
one to determine the value of κ (K) to leading order in m for m → 0 [5],
4 s q
1
κ (K) = − , (11)
4
8
which is a consequence of the conservation of the axial current in the chiral limit.
The above relations were derived from the analysis of matrix elements of the local
↔ → ←
operators (D=D − D)
1 ↔ 1 ↔ 1 ↔
O(5) = q¯γ (γ )i D s+ q¯γ (γ )i D s− g q¯i(γ )D/ s, (12)
µν 2 µ 5 ν 2 ν 5 µ 4 µν 5
whose divergence can be expressed in terms of bilinear quark operators. In this section,
we rederive these relations in a different way and obtain a new relation for a⊥(K∗).
1
The starting point for our analysis are the exact nonlocal operator relations [10, 11]
∂ 1
q¯(x)γ (γ )s(−x) = −i dvvq¯(x)x gGαµ(vx)γ (γ )s(−x)
µ 5 α µ 5
∂x
µ Z−1
−(m ±m )q¯(x)i(γ )s(−x), (13)
s q 5
1
∂µ{q¯(x)γ (γ )s(−x)} = −i dvq¯(x)x Gαµ(vx)γ (γ )s(−x)
µ 5 α µ 5
Z−1
−(m ∓m )q¯(x)i(γ )s(−x), (14)
q s 5
where the total translation ∂ is defined as
µ
∂
∂ {q¯(x)Γs(−x)} ≡ {q¯(x+y)[x+y,−x+y]Γs(−x+y)} . (15)
µ
∂y
µ (cid:12)y→0
(cid:12)
(cid:12)
3 (cid:12)
The corresponding nonlocal matrix elements are, for K and K∗ (x2 6= 0):
k
1
h0|q¯(x)γ γ s(−x)|K(q)i = if q dueiξqx φ (u)+O(x2)
µ 5 K µ K
Z0
(cid:2) (cid:3)
i 1 1
+ f m2 x dueiξqx g (u)−φ (u)+O(x2) ,(16)
2 K K qx µ K K
Z0
(cid:2) (cid:3)
f m2 1
h0|q¯(x)iγ s(−x)|K(q)i = K K dueiξqx φp (u)+O(x2) , (17)
5 m +m K
s q Z0
(cid:0) (cid:1)
e(λ)x 1
h0|q¯(x)γ s(−x)|K∗(q,λ)i = fkm q dueiξqx φk (u)+O(x2)
µ K K∗ qx µ K
( Z0 h i
e(λ)x 1
+ e(λ) −q dueiξqx gv (u)+O(x2)
µ µ qx K
(cid:18) (cid:19)Z0
(cid:0) (cid:1)
1 e(λ)x 1
− x m2 dueiξqx g(3)(u)+φk (u)−2gv (u)+O(x2) . (18)
2 µ(qx)2 K∗ K K K
Z0 )
h i
In the above definitions, φ and φk are the leading-twist DAs of K and K∗, respectively;
K K k
all other functions are higher-twist DAs and have been extensively discussed in Refs. [10,
11, 12, 13].
a (K), the quantity we are interested in, is related to the first moment of φ :
1 K
5
a (K) = MφK
1 3 1
with Mf ≡ 1du(u−u¯)f(u) being the first moment of the DA f(u). Taking the matrix
1 0
elements of (13) and (14) for K and expanding to leading order in x2 and next-to-leading
R
order in qx, one obtains the exact relations
m −m
MφK −2MgK = − s q,
1 1 m +m
s q
1 MφK +MgK = −2κ (K)+MφpK, (19)
2 1 1 4 1
(cid:16) (cid:17)
from which one can determine MφK once either MφpK or MgK are known. g is a twist-4
1 1 1 K
DA and MgK contains quark-quark-gluon matrix elements itself, cf. Refs. [11, 13], whereas
1
φp is twist-3 and MφpK is completely determined in terms of the twist-2 DA φ and
K 1 K
φp
mass corrections. M K can be obtained from a second set of nonlocal operator relations
1
involving tensor currents q¯(x)σ γ s(−x) or, equivalently, from the recursion relations for
µν 5
the moments of φp given in Ref. [13]:
K
φp m2s −m2q
M K = .
1 m2
K
4
Solving (19) for a (K), we then rederive
1
9 m −m m2 −m2
s q s q
a (K) = − +4 −8κ (K), (20)
5 1 m +m m2 4
s q K
which confirms the result obtained in Ref. [5]. Note that the first term on the right-hand
side is rather sensitive to the value of m and the second one to that of m .
q s
For K∗, the same method yields the equations
k
φk g(3) gv
M K +M K = 2M K,
1 1 1
MφkK −MgK(3) = −2 fK⊥ ms −mq +2 m2s −m2q −4κk(K∗). (21)
1 1 fKk mK∗ m2K∗ 4
(3)
Again, g isatwist-4 DAwhose first moment isnot known fromanyindependent analysis,
K
whereas MgKv , the first moment of the twist-3 DA gv , can be read off Eq. (4.6) in Ref. [12]:
1 K
2MgKv = MφkK + fK⊥ ms −mq. (22)
1 1 fk mK∗
K
We can then solve (21) for ak(K∗) and obtain
1
3 f⊥ m −m m2 −m2
ak(K∗) = − K s q +2 s q −4κk(K∗), (23)
5 1 fKk mK∗ m2K∗ 4
which agrees with Eq. (8), the result obtained in Ref. [5].
Let us now apply the same method to chiral-odd operators, with the aim of obtaining
an analogous new expression for a⊥(K∗). The relevant nonlocal operator relations are
1
∂
q¯(x)σ s(−x) = −i∂ q¯(x)s(−x)+(m −m )q¯(x)γ s(−x)
µν ν s q ν
∂x
µ
1 1
+ dvq¯(x)gx Gα (vx)s(−x)−i dvvq¯(x)gx Gαµ(vx)σ s(−x),
α ν α µν
Z−1 Z−1
∂
∂µ{q¯(x)σ s(−x)} = −i q¯(x)s(−x)−(m +m )q¯(x)γ s(−x)
µν s q ν
∂x
ν
1 1
+ dvvq¯(x)gx Gα (vx)s(−x)−i dvq¯(x)gx Gαµ(vx)σ s(−x).
α ν α µν
Z−1 Z−1 (24)
These relations were first derived, without the terms in m ±m , in Ref. [10]; the terms in
s q
the quark masses are new.
The relevant K∗ matrix elements are given by [10]:
1
h0|q¯(x)σ s(−x)|K∗(q,λ)i = if⊥ (e(λ)q −e(λ)q ) dueiξqx φ⊥(u)+O(x2)
µν K µ ν ν µ K
" Z0 " #
5
e(λ)x 1 1 1
+(q x −q x ) m2 dueiξqx ht (u)− φ⊥(u)− h(3)(u)+O(x2)
µ ν ν µ (qx)2 K∗ K 2 K 2 K
Z0 (cid:20) (cid:21)
1 m2 1
+ (e(λ)x −e(λ)x ) K∗ dueiξqx h(3)(u)−φ⊥(u)+O(x2) , (25)
2 µ ν ν µ qx K K
Z0 (cid:16) (cid:17)(cid:21)
h0|q¯(x)s(−x)|K∗(q,λ)i =
m +m 1
= −i f⊥ −fk s q e(λ)x m2 dueiξqx hs (u)+O(x2) ,(26)
K K m K∗ K
(cid:18) K∗ (cid:19) Z0
(cid:0) (cid:1) (cid:0) (cid:1)
where, again, φ⊥ is the leading-twist DA of the transversely polarised K∗ and hs,t and h(3)
K K K
are higher-twist DAs. In addition, we also need the following quark-quark-gluon matrix
element:
h0|q¯(gG µ)σ s|K∗(q,λ)i =
α βµ
1
= f⊥m2 κ⊥(K∗)(e(λ)q +e(λ)q )+κ⊥(K∗)(e(λ)q −e(λ)q ) . (27)
K K∗ 2 3 α β β α 4 α β β α
(cid:26) (cid:27)
Here κ⊥(K∗) is a twist-3 matrix element, κ⊥(K∗) is twist-4; both are O(m −m ) due to
3 4 s q
G-parity.1 Taking matrix elements of (24), one obtains expressions in q , e(λ) and x . To
ν ν ν
twist-4 accuracy only the former two are relevant and yield a set of four linear equations
for the four first moments of gv , hs , ht and h(3):
K K K K
−(κ⊥(K∗)−2κ⊥(K∗))+δ MgKv +MhsK = 1 Mh(K3) + 1 Mφ⊥K ,
3 4 + 1 1 2 1 2 1
κ⊥(K∗)+2κ⊥(K∗)+δ MgKv −MhsK −δ MφkK = 1 Mh(K3) −MhtK + 1 Mφ⊥K ,
3 4 + 1 1 + 1 2 1 1 2 1
h(3) φ⊥
3M K − M K = 2δ ,
1 1 −
h3 ht φ⊥
M K −2M K +M K = 0 (28)
1 1 1
with δ = fKk ms±mq. The solution of that system implies
± f⊥ m∗
K K
δ MgKv = 1 δ + 1 δ MφkK + 1 Mφ⊥K −2κ⊥(K∗),
+ 1 6 − 2 + 1 3 1 4
which must agree with MgKv as given in Eq. (22). Solving for a⊥(K∗), one finds
1 1
3 fk m −m 3 m2 −m2
a⊥(K∗) = − K s q + s q +6κ⊥(K∗), (29)
5 1 f⊥ 2m 2 m2 4
K K∗ K∗
which is the wanted new relation for a⊥(K∗). Note that in all three relations (7), (8) and
1
(29) κ enters multiplied with a large numerical factor which implies that the theoretical
4
uncertainty of the resulting values of a will be much larger than that of κ itself.
1 4
1The normalisation of κ⊥(K∗) is chosen in such a way that DαT(α) = κ⊥(K∗) for the twist-3 DA
3 3
T(α) defined in Ref. [12].
R
6
hq¯qi=(−0.24±0.01)3GeV3 hs¯si=(1−δ )hq¯qi
3
hq¯σgGqi=m2hq¯qi hs¯σgGsi=(1−δ )hq¯σgGqi
0 5
α
s G2 =(0.012±0.003)GeV4
π
D E
m2 = (0.8±0.1)GeV2, δ = 0.2±0.2, δ = 0.2±0.2
0 3 5
m (2GeV) = (100±20)MeV ←→ m (1GeV) = (137±27)MeV
s s
m (µ) = m (µ)/R, R = 24.4±1.5
q s
α (m ) = 0.1187±0.002 ←→ α (1GeV) = 0.534+0.064
s Z s −0.052
k
f = (0.160±0.002)GeV, f = (0.217±0.005)GeV
K K
f⊥ = (0.185±0.010)GeV
K
Table 1: Input parameters for sum rules at the renormalisation scale µ = 1GeV. The value
of m is obtained from unquenched lattice calculations with n = 2 flavours as summarised
s f
in [14], which agrees with the results from QCD sum rule calculations [15]. m is taken
q
from chiral perturbation theory [16].2 α (m ) is the PDG average [18], f and fk are
s Z K K
known from experiment and f⊥ has been determined in Refs. [6, 7]. The errors of quark
K
masses and condensates are treated as correlated, see text.
3 QCD Sum Rules for κ , κk and κ⊥
4 4 4
Inorder to obtainnumerical predictions fora fromtherelations derived inthelast section,
1
one needs to know the values of the κ matrix elements. κ (K) and κk(K∗) have been
4 4 4
calculated in Ref. [5] from QCD sum rules to leading order in SU(3) breaking parameters
with the following results:
κ (K)BL = −0.11±0.03, κk(K∗)BL = −0.050±0.010, (30)
4 4
which, using the relations (7) and (8), letting m = 0 and neglecting the terms in m2
q s
translates into [5]
a (K)BL = 0.10±0.12, ak(K∗)BL = 0.10±0.07. (31)
1 1
All these results refer to a renormalisation scale of 1 GeV.
In this section we present QCD sum rules for κ (K) and κk(K∗) which are accurate to
4 4
NLO in SU(3) breaking and also a new sum rule for κ⊥(K∗) to the same accuracy. For all
4
sum rules we include O(m ) effects. The sum rules are of the generic form
q
κ4(K)fK2mnKe−m2K/M2 +contribution from higher mass states = BM2ΠG, (32)
2
m has also been determined from lattice calculations. The most recent papers on this topic are
q
Refs.[17]. Thecentralvalueofm /m determinedinthefirstofthesepaperswithn =2runningflavours
s q f
and nonperturbative renormalisation agrees with the result from chiral perturbation theory, whereas the
result of the second, obtainedwith n =3 and perturbative (two-loop)renormalisation,is a bit lower. As
f
the field appearsto developrapidly,we refrainfromtakingeither side andstaywith the resultfromchiral
perturbation theory.
7
and correspondingly for K∗. Π are correlation functions of type
G
Π (q) = i d4yeiqyh0|T[q¯(gG )Γµs](y)[s¯Γ q](0)|0i
G αµ 1 2
Z
with suitably chosen Dirac structures Γµ and Γ ; explicit expressions for all relevant Π
1 2 G
are given in App. A. B Π is the Borel transform of Π , M2 the Borel parameter and
M2 G G
n is either 2 or 4. In order to separate the ground state from higher mass contributions,
one usually models the latter, using global quark hadron duality, by an integral over the
perturbative spectral density:
∞ 1
contribution from higher mass states ≈ e−s/M2 ImΠ (s); (33)
G
π
Zs0
the parameter s is called continuum threshold. The input parameters for the QCD sum
0
rules are collected in Tab. 1.
All κ parameters can actually be determined from more than one sum rule derived
4
from various Π which can be characterised by the following features:
G
• the currents can have the same or different chirality, which results in chiral-even and
chiral-odd sum rules, respectively;
• the hadronic states saturating Π can have unique spin-parity or come with different
G
parity (e.g. 0− and 1+), which results in pure-parity and mixed-parity sum rules,
respectively.
Note that all chiral-odd sum rules are also pure-parity.
In chiral-odd sum rules the quark condensates always appear in the combination hq¯qi−
hs¯si = δ hq¯qi and hq¯σgGqi − hs¯σgGsi = δ hq¯σgGqi, which induces a large dependence
3 5
on the only poorly constrained parameters δ and also increases the impact of the gluon
3,5
condensate contribution which is equally poorly known. We therefore decide to drop all
chiral-odd sum rules and only use chiral-even ones.
As for mixed and pure-parity sum rules, they come with different mass dimensions:
n = 2 in (32) for mixed-parity vs. n = 4 for pure-parity sum rules. It is an important
result of this paper that the continuum contributions to the mixed-parity sum rules, for
typical Borel parameters M2 around 1.7GeV2, are small and below 10% for all three κ .
4
Pure-paritysumrules, ontheotherhand, havealargecontinuumcontributionaround30%.
There are two reasons for this result: first, the additional power of m2 in pure-parity sum
K
rules counteracts the exponential suppression of the continuum contribution. Second, the
contributions of particles with different parity have different sign: it was already found in
Ref. [5] that κ (K) and κk(K ) have opposite sign; we find that the same applies to κk(K∗)
4 4 1 4
and the corresponding κ (K∗) of the lowest scalar resonance, and ditto to κ⊥(K∗) and the
4 0 4
coupling κ⊥(K ) of the axial vector K meson. These results suggest that the κ matrix
4 1 1 4
elements of opposite-parity mesons have generically different signs and tend to cancel each
other in mixed-parity sum rules, which results in a small continuum contribution. From
a more formal point of view it is rather obvious from the definitions Eqs. (9), (10) and
8
(27) that the sign of κ changes under a parity transformation,3 which is in line with our
4
findings.
The mixed-parity sum rules for K and K∗ do involve the three spin-parity systems
(0−,1+), (1−,0+) and (1−,1+). Note that for all of them the “wrong”-parity ground state
(e.g. the scalar K∗(1430)) and the first orbital excitation of the “right”-parity state (e.g.
0
the vector K∗(1410)) have nearly equal mass, which makes the cancellation very effective.
We conclude that mixed-parity sum rules are more reliable than pure-parity ones and, as
a consequence, will not consider the latter in this paper. In view of the cancellation of
contributions of different sign we also decide to include explicitly only the lowest-mass
ground state in the mixed-parity sum rules, which differs from the procedure adopted by
the authors of Ref. [5].
Let us now turn to the question how to choose the Borel parameter M2 and the contin-
uum threshold s , the internal sum rule parameters. As mentioned before, the dependence
0
of the sum rules on s is weak and so we simply use the same values of s as for the
0 0
quark current sum rules, i.e. s (K) = (1.1 ± 0.3)GeV2, sk(K∗) = (1.7 ± 0.3)GeV2 and
0 0
s⊥(K∗) = (1.3±0.3)GeV2 [6]. The small dependence on s also allows one to use slightly
0 0
higher values of M2 than the usual 1 to 2GeV2, which improves the convergence of the
operator product expansion of the correlation functions and reduces the variation of the
sum rule with M2. We choose M2 = (1.6±0.4)GeV2 for K and M2 = (1.8±0.4)GeV2
for K∗.
After this general discussion of the choice of sum rules and parameters let us now turn
to the three κ parameters in turn.
4
κ K
3.1 4( )
(a)
The mixed-parity sum rule for κ (K) is obtained from the correlation function Π in
4 G,2
App. A, Eq. (A.6), and given by
α s0 s
fK2m2Kκ4(K)e−m2K/M2 = 72πs3 (m2s −m2q) dse−s/M2 10ln µ2 −25
Z0 (cid:18) (cid:19)
2 α 1 M2 ∞ ds
+ s (m hq¯qi−m hs¯si) − +γ −ln + e−s/M2
9 π s q 3 E µ2 s
(cid:26) Zs0 (cid:27)
10 α 1
s
+ (m hs¯si−m hq¯qi)+ (m hs¯σgGsi−m hq¯σgGqi)
9 π s q 6M2 s q
m2 −m2 α 1 M2 ∞ ds
+ s q s G2 1− ln −γ +1 −M2 e−s/M2
6M2 π 2 µ2 E 2s2
D E(cid:26) (cid:18) (cid:19) Zs0 (cid:27)
8πα
+ s [hq¯qi2 −hs¯si2]. (34)
27M2
This sum rule includes all relevant contributions up to dimension six. Numerically, all
dominant contributions have the same sign, with the largest one from hs¯si, followed by the
3In QCD parity is not a symmetry of the hadronic spectrum because the U(1)A-symmetry is broken.
9