On the $b-$ Quark Kinetic Energy in the $Λ_b$

16 Pages·0.22 MB·

by  P. Colangelo

BARI-TH/95-219
UTS-DFT-95-13
December 1995
On the b Quark Kinetic Energy in the Λ
b
P. Colangelo a, C.A. Dominguez b, G. Nardulli a,c and N. Paver d
6
a Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy
9
9
b Physics Department, University of Capetown, South Africa
1

n c Dipartimento di Fisica, Universita´ di Bari, Italy
a
J d Dipartimento di Fisica Teorica, Universita´ di Trieste, and

1
1 Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy

2
v
4
3
3 Abstract
2
1
5
9 We consider the problem of determining the matrix element of the heavy
/
h quark kinetic energy operator between Λb baryonic states by three point
K
p functionQCDsumrulesintheHeavy QuarkEffectiveTheory. Wediscussthe
-
p main features and the uncertainties associated to such an approach. Numeri-
e cally, we find the result: = (0.6 0.1) GeV2. Within the uncertainty
h hKiΛb − ±
: of the method, such result indicates a value of the heavy quark kinetic energy
v
i in the baryon similar in size to that in the meson.
X
r
a
PACS:12.39.Hg, 12.38.Lg
Typeset using REVTEX
1
I. INTRODUCTION
Importantprogressesinthetheoreticaldescriptionofhadronscontainingoneheavyquark
have been allowed by the development of the Heavy Quark Effective Theory (HQET) [1].
Based on the spin-flavor symmetry of QCD, valid in the limit of infinite heavy quark mass
m , this framework provides a systematic expansion of heavy hadron spectra and transition
Q
amplitudes in terms of the leading contribution (for m ), plus corrections decreasing
Q
→ ∞
as powers of 1/m . Being derived from ‘first principles’, i.e. from general properties of
Q
QCD, such an approach has the advantage of being model-independent to a large extent,
and predictive at the same time. For this reason, it has been extensively applied to a variety
of aspects concerning the dynamics of mesons and baryons made of both heavy and light
quarks.
Due to the flavor-spin symmetry connecting matrix elements for different heavy flavors
and angular momenta, the leading order description is simple and intuitive, and involves
a few nonperturbative quantities in terms of which a large number of processes can be de-
scribed. This leading order seems to be consistent with the main features of heavy hadron
physics as they are presently observed. However, although being suppressed by the large
heavy quark mass, the next-to-leading order corrections proportional to 1/m are also ex-
Q
pected to play a significant role. In the framework of HQET, they are limited in number,
and can be systematically identified and related to nonperturbative parameters having a
clear physical meaning. A noticeable example can be found in the effective QCD heavy
quark Lagrangian which, up to order 1/m , can be written as
Q
1 1 1
¯
= h i(v D)h + + +O( ). (1.1)
L v · v 2m K 2m S m2
Q Q Q
Here h is the effective field for a heavy quark with velocity v, related to the Dirac field Q
v
by
1+/v
h (x) = eimQv·x Q(x), (1.2)
v
2
is the nonrelativistic kinetic energy operator defined by
K
= h¯ (iD⊥)2h (1.3)
v v
K
where (D⊥)2 = D Dµ (v D)2, with Dµ the QCD covariant derivative, and is the Pauli
µ
− · S
term responsible of the spin splitting among the states belonging to the same heavy hadron
multiplet in the m limit:
Q
→ ∞
1 α (m ) 3/β0
= s Q h¯ σµνgG h . (1.4)
v µν v
S 2 αs(µ) !
Appropriately normalized matrix elements of the operators (1.3) and (1.4) account for the
next-to-leading corrections to the heavy hadron mass spectra:
1 c
H
m = m +∆ , (1.5)
H Q H H
− 2m hKi − 2m hSi
Q Q
2
where ∆ represents the binding energy of light quarks in the static approximation and c is
H
a numerical coefficient. At the same time, they are involved as ‘preasymptotic’ corrections
in the 1/m expansion of inclusive heavy hadron decay rates based on parton model ideas
Q
and on the operator product expansion [2,3]. For example, the inclusive H f weak decay
→
rate can be written as:
G2m5 1
Γ(H f) = F Q V 2 cf 1+ hKiH +cfhSiH +O( ) , (1.6)
→ 192π3 | CKM| " 3 2m2 ! 5 2m2 m3 #
Q Q Q
where cf and cf are calculable short-distance coefficients that depend on the process under
3 5
consideration.
The matrix elements of and are nonperturbative parameters that should be either
K S
determined phenomenologically from experimental data or estimated using a nonperturba-
tive theoretical approach. In the case of heavy mesons (P ), information on both and
Q hKiPQ
is available. Indeed, the chromomagnetic operator matrix element can be determined
hSiPQ
from the vector-pseudoscalar meson mass difference, indicating
3
= m2 m2 0.37GeV2, (1.7)
hSiPQ 4 VQ − PQ ≃
(cid:16) (cid:17)
and it has also been theoretically estimated by QCD sum rules giving a result in agreement
with this value [4]. Concerning the kinetic energy operator matrix element, general argu-
ments imply the inequality [5] giving 0.37GeV2 using (1.7), and
|hKiPQ| ≥ hSiPQ |hKiPQ| ≥
the most recent theoretical estimate using QCD sum rules indicates [4] 1
= (0.60 0.10)GeV2 . (1.8)
hKiPQ − ±
Finally, the first calculation with lattice QCD gives the inequality < 1 GeV2, in
|hKiPQ|
agreement with the above numbers [7].
Ontheotherhand,noanalogousinformationisavailableforthebaryonicmatrixelements
of and . In fact, the heavy baryon mass splittings relevant to the determination of the
S K
chromomagnetic expectation value are not measured yet; moreover, general lower bounds
on the kinetic energy have not been worked out in this case. To leading order in 1/m , the
Q
heavy quark kinetic energy inside the baryon can be connected to that in the meson and to
the heavy quark masses by the relation (which assumes that the charm mass m is heavy
c
enough for the expansion up to 1/m to be meaningful) [8]:
c
m m
b c
hKiPQ −hKiΛQ ≃ 2(m m ) [(mB +3mB∗ −4mΛb)−(mD +3mD∗ −4mΛc)]. (1.9)
b c
−
Using present data, Eq. (1.9) would give 0.07 0.20 GeV2, where the
hKiPQ − hKiΛQ ≃ ±
error mainly comes from the error on the the measured m , that is not small enough:
Λb
m = 5641 50 MeV [9] . Arguments have been given in favour of the near equality of the
Λb ±
kinetic energy operator matrix elements between meson and baryon states [10], which could
1For earlier estimates see [6].
3
also be expected on intuitive grounds, but some quantitative estimates are needed. Here,
we would like to consider in this regard the simplest case, represented by the Λ baryon,
b
where the chromomagnetic term vanishes in the infinite heavy quark mass limit = 0
hSiΛb
as the light diquark system carries no spin (the same is true for the Ξ ), and attempt an
Q
assessment of the heavy quark kinetic energy. Specifically, we discuss the possibility of such
an estimate in the framework of QCD sum rules, where the calculation has also been done
for the meson, so that it is of interest to compare the two cases.
II. HEAVY BARYON CURRENT AND TWO-POINT FUNCTION
A basic element in the application of QCD sum rules to baryons is the definition of a
suitable baryon interpolating effective field in terms of quark fields, and the corresponding
vacuum-to-baryon matrix element. For the case of heavy baryons this problem has been
discussed previously in Refs. [11]- [14], and we outline here this analysis for the sake of
completeness.
The general expression of the interpolating field for a baryon state with one heavy quark
and two light quarks can be written as
J(ℓ) = ǫαβγ(qTαCΓ(ℓ)τqβ)Γ˜(hγ) (2.1)
D 1 2 v D
where T means transpose, α, β and γ are color indices, D is a Dirac index, C is the charge
conjugation matrix, q and q are light quark field operators, and τ is a flavor matrix. For
1 2
the case of the spin 1/2 Λ of interest here, the matrices τ and Γ˜ are given by:
b
1
τ = (δ δ δ δ ), Γ˜ = 1, (2.2)
ij 1i 2j 2i 1j
√2 −
whereas two choices are possible for the matrix Γ(ℓ):
Γ(1) = γ , Γ(2) = γ γ . (2.3)
5 5 0
Such choices correspond to spin-parity JP = 0+ for the light quark pair, as implied by
the predicted spectroscopy of baryons containing one heavy quark in the infinite mass limit
[15,16].
Consequently, the general Λ interpolating field is the linear combination
b
J = J(1) +b J(2), (2.4)
depending on the, a priori unspecified, parameter b.
Together with the baryonic interpolating currents, we need for our estimates the values
of the vacuum-to-baryon couplings (ℓ = 1,2):
(ℓ) (ℓ)
< 0 J Λ (v) >= f (h ) . (2.5)
| D | b Λb v D
In the framework of QCD sum rules these constants can be obtained from the two-point
correlators:
Π(ℓ,ℓ′)(k) = i dx < 0 T(J(ℓ)(0)J¯(ℓ′)(x) 0 > eikx (2.6)
CD | C D |
Z
4
where, accordingtothefamiliarrulesoftheheavyquarktheory, k istheresidualmomentum,
corresponding to the decomposition pµ = m vµ + kµ of the heavy baryon momentum; we
b
parametrize it as
kµ = ω vµ. (2.7)
Taking Eqs. (1.2) and (2.7) into account, we can write the LHS of Eq. (2.6) as
Π(ℓ,ℓ′)(k) = (1+/v) Π(ℓ,ℓ′)(ω). (2.8)
CD CD
The scalar functions Π(ℓ,ℓ′) satisfy dispersion relations of the form:
1 ρ(ℓ,ℓ′)(Ω)
Π(ℓ,ℓ′)(ω) = dΩ + subtractions, (2.9)
π Ω ω
Z −
where ρ represents the spectral function: ρ(ℓ,ℓ′)(Ω) = ImΠ(ℓ,ℓ′)(Ω). The explicit form of the
polynomial representing the subtractions is not important because in the sequel we shall
apply the borelized version of the sum rule, and the Borel transform of any polynomial
vanishes.
Following the QCD sum rules method [17], ρ(ℓ,ℓ′)(Ω) can be computed in two ways: either
by saturating the dispersion relation (2.9) by physical hadronic states or by means of the
operator product expansion (OPE), which is allowed insofar k2 in (2.6) is large and negative.
Performing the OPE on Π(ℓ,ℓ′) and taking into account the operators of dimension d = 0
(the perturbative term), d = 3 and d = 5, by a straightforward calculation in the heavy
quark theory one finds the spectral functions:
Ω5
ρ(ℓ,ℓ)(Ω) = ρ(0)(Ω) = (2.10)
20π3
′
for ℓ = ℓ (‘diagonal’ sum rules), and
2 q¯q Ω2 2m2 q¯q 1 1
ρ(1,2)(Ω)+ρ(2,1)(Ω) = ρ(3)(Ω)+ρ(5)(Ω) = h i 0h i + (2.11)
− π − π −8 16
(cid:18) (cid:19)
′
forℓ = ℓ (‘nondiagonal’sum rules), asobtained previously [12–14]. InEqs. (2.10)and(2.11)
6
ρ(0,3,5) denote, respectively, the leading perturbative term in the short distance expansion
and the d = 3 and d = 5 contributions. Moreover, q¯q is the quark condensate ( q¯q =
h i h i
(230MeV)3 at the scale of 1GeV) and m is connected to the quark-gluon condensate
0
−
by the relation q¯g σ Gq = m2 q¯q (and m2 = 0.8GeV2 [17]). The separation of the
h s · i 0h i 0
d = 5 contribution in two contributions reflects the existence of two independent sources
originating such a term, namely the nonlocal quark condensate [18]
x2m2
q¯(x)q(0) = Φ(x2) q¯q 1+ 0 q¯q (2.12)
h i h i ≃ 16 !h i
and the diagram where the gluon non perturbative field is emitted from the perturbative
light quark line (in the adopted Fock-Schwinger gauge, no such gluon can be emitted from
the heavy quark leg in the infinite heavy quark mass limit).
5
Actually, in previous calculations also the d = 4 gluon condensate α G2 and the higher
s
h i
dimension operators with d > 5 have been included. The possibility of potentially non-
negligible higher order terms is briefly discussed at the end of this Section. On the other
hand, the gluon condensate contribution would be rather easy to include both in Eq. (2.10)
and in the analogous three-point functions considered in the next Section to evaluate the
kinetic energy. In principle, its presence accounts for the low momentum part of the gluon
exchange diagram of order α . In the baryon case the high momentum component, i.e. the
s
α correction to the perturbative diagram, requires a three-loop calculation, which has not
s
been attempted yet. Consequently, we do not include the gluon condensate as it would only
partially represent the gluon exchange, and we limit to the analysis of both two-point and
three-point correlators with the operators having d = 0, 3 and 5.
Turning to the representation of spectral functions in (2.9) in terms of physical hadronic
states, we write
ρ(ℓ,ℓ′)(Ω) = ρ(ℓ,ℓ′)(Ω)+ρ(ℓ,ℓ′)(Ω), (2.13)
had res cont
where ρ(ℓ,ℓ′)(Ω) contains the contribution of the low-lying baryon state, in this case the Λ ,
res b
(ℓ,ℓ′)
and ρ represents the contribution of higher mass states. Assuming, as usual, duality
cont
(ℓ,ℓ′)
between partons and resonances, ρ (Ω) can be taken as equal to the operator product
cont
expansion ρ(ℓ,ℓ′)(Ω) of Eqs. (2.10) and (2.11) for Ω > ω , with ω a continuum threshold,
c c
and vanishing elsewhere. As for ρ(ℓ,ℓ′)(Ω), they are expressed as:
res
π[f(ℓ)]2
ρ(ℓ,ℓ)(Ω) = Λb δ(Ω ∆) (2.14)
res 2 −
(1) (2)
πf f
ρ(1,2)(Ω)+ρ(2,1)(Ω) = Λb Λb δ(Ω ∆) (2.15)
res res 2 −
(ℓ)
where f have been defined in Eq. (2.5) and ∆ = m m in the limit m (see
Λb Λb − b b → ∞
Eq. (1.5)). Equating the two representations of the two-point correlators, and taking the
Borel transform in the variable ω of both sides, one obtains the set of sum rules for the
vacuum-to-baryon couplings of the interpolating fields:
[f(1)]2 = [f(2)]2 = 2 dΩ e−Ω−E∆ρ(0)(Ω) (2.16)
Λb Λb π
Z
f(1)f(2) = 1 dΩ e−Ω−E∆[ρ(3)(Ω)+ρ(5)(Ω)]. (2.17)
Λb Λb π
Z
One can notice that, due to the structure of Eq. (2.10), at the chosen order of approximation
in the operator product expansion f(1) f(2) = f from the ‘diagonal’ sum rules.2 This
| Λb | ≃ | Λb | Λb
2As it will be seen in the next Section, this does no longer occur in the case of the three-point
function, in the sense that somehow different results for the kinetic energy are obtained from the
two diagonal sum rules.
6
result should be confirmed by the non-diagonal sum rule, which can also be used to fix the
relative sign between the two constants.
In order to extract useful information from the sum rules, we have to fix the parameters
∆, ω and the allowed range for the Borel parameter E (the so-called ‘duality’ region).
c
Indeed, in order to obtain a prediction for ∆, we differentiate (2.16) ( or (2.17) ) in the
variable 1/E and take the ratio with (2.16) (or (2.17) ) itself, thus obtaining a new sum rule
where only ∆ (and not f ) appears.
Λb
The results for f and ∆ as a function of the Borel variable E, and for ω in the range
Λb c
1.1 1.3GeV, are shown in Fig. 1. In particular, in Fig. 1a we depict the result for the
−
diagonal correlators (2.16) both concerning the coupling f and ∆, in Fig. 1b the outcome
Λb
of the nondiagonal correlator (2.17). In Fig. 1c we display the result from the sum rule for
the correlator of the current J in Eq.(2.4):
1f2 (1+b)2e−∆/E = ωc dω e−ω/E 1+b2ω5 2b < qq >(ω2 m20) (2.18)
2 Λb 20π4 − π2 − 16
Z0 h i
with the choice b = 1, that we shall justify in the next Section.
As one can see, stability is always reached for values of E from 0.3GeV on, which at first
sight would be a welcomed property. However, the familiar criterion that the continuum
should not be too large with respect to the ground state baryon contribution, favoring low
values of E, rather strongly reduces the ‘working’ region to a range in the neighborhood of
E = 0.4 0.5 GeV. In the next Section this effect of the continuum will be found to be
−
even more significant in the kinetic energy calculation using three-point sum rules. Clearly,
it originates from the high power dependence of the spectral functions, that is expected
on dimensional grounds. In this regard, considering the structure of the operator product
expansion in Eqs. (2.10) and (2.11), it should be noticed that the suppression of dimen-
sional condensates is provided by smaller powers of the integration variable Ω, compared to
the leading perturbative term, rather than by large mass denominators characterizing the
applications of the QCD sum rule method with finite heavy quark mass.
Coming tothenumericalresults, itisworthobserving thatthediagonalandnon-diagonal
correlators, together with the correlator of the current J in (2.4), provide us with the same
mass parameter ∆ :
Λb
∆ = 0.9 0.1 GeV . (2.19)
Λb ±
The uncertainty on ∆ in (2.19) comes from the variation of the continuum threshold and
Λb
from changing the Borel parameter E in the range E = 0.3 0.4 GeV in Fig.1a and c,
−
and E = 0.3 0.5 GeV in Fig.1b. Using the result of the analysis for the mesonic system
−
∆ 0.5 GeV [4] we get for Λ the mass M = M +∆ ∆ 5.7 GeV, to be compared
B ≃ b Λb B Λb− B ≃
with the experimental measurement quoted above. The difference of 200 MeV between the
mass of the Λ and the continuum threshold is of reasonable size as it nearly corresponds to
b
the mass of a pair of pions.
As for the coupling f , the diagonal and non-diagonal sum rules in (2.16) and (2.17)
Λb
give rather different results:
f = (2.0 0.5) 10−2 GeV3 (2.20)
Λb ± ×
f = (3.5 0.5) 10−2 GeV3 (2.21)
Λb ± ×
7
respectively. The different results can be put in better agreement using a smaller value for
the quark condensate, as advocated in [12]. However, from the correlator of the current J,
with b = 1, we obtain:
f = (2.9 0.5) 10−2 GeV3 , (2.22)
Λb ± ×
that is the result we shall use in the next Section.
As far as higher dimensional contributions to (2.20)-(2.22)areconcerned, it was observed
in [12,14] that the most important one should be the four-quark d = 6 operator contributing
to the diagonal sum rule (2.10) and to the sum rule (2.18). The evaluation of the relevant
diagram is straightforward and leads to a term proportional to < q¯(x)q(0)q¯(x)q(0) >. The
numerical analysis can be done by making the assumption of factorization of the nonlocal
four-quarkcondensate and, using this hypothesis, we would obtaina correction to (2.20)and
(2.22) whithin the quoted uncertainty. However, the accuracy of this assumption is difficult
to assess due to the fact that the numerical result would strongly depend on the explicit
model for the nonlocal condensate and, as a matter of fact, the inclusion of this term has
been questioned, e.g. in ref. [14]. For this reason, following [14], we prefer to omit this term,
with the understanding that this neglect will be reflected in an additional uncertainty in the
final result.
III. THE THREE-POINT FUNCTION AND
hKiΛB
To apply the formalism of QCD sum rules to the kinetic energy operator , we consider
K
the three-point correlators:
Σ(ℓ,ℓ′)(k,k′) = (1+/v) Σ(ℓ,ℓ′)(ω,ω′) = i2 dxdy < 0 T(J(ℓ)(y) (0)J¯(ℓ′)(x) 0 > e−iky+ik′x,
CD CD | C K D |
Z
(3.1)
′ ′
where(l,l ) = 1,2havethesamemeaningasinEq.(2.6),andk,k aretheresidualmomenta,
corresponding to the decomposition of the heavy baryon momentum pµ = m vµ+kµ, p′µ =
b
m vµ +k′µ. Analogously to Eq. (2.7), we have taken:
b
kµ = ω vµ, k′µ = ω′ vµ. (3.2)
The scalar function Σ(ℓ,ℓ′) satisfies a double dispersion relation in the variables ω,ω′:
σ(ℓ,ℓ′)(Ω,Ω′)
Σ(ℓ,ℓ′)(ω,ω′) = dΩ dΩ′ + subtractions (3.3)
(Ω ω)(Ω′ ω′)
Z − −
where, as for the two-point function, the explicit form of the subtraction term is not impor-
tant since it will be eliminated by the double Borel transform of the sum rule.
For both k2 and k′2 large and negative, so that short distances x and y should dominate,
we perform the operator product expansion on the RHS of Eq. (3.1) taking into account
only the operators of dimension d = 0 (the perturbative term), d = 3 and d = 5, analogously
to Sec. 2, and neglecting the d = 4 gluon condensate with the same motivation given there.
8
In this approximation we obtain for the various spectral functions the following expressions,
applying the Cutkosky rules in momentum space:
3Ω7
σ(1,1)(Ω,Ω′) = δ(Ω Ω′) (3.4)
−140π4 −
5Ω7
σ(2,2)(Ω,Ω′) = δ(Ω Ω′) (3.5)
−140π4 −
2 < qq > 2m2 < qq > 3 3
σ(1,2)(Ω)+σ(2,1)(Ω) = Ω4 + 0 Ω2 + δ(Ω Ω′). (3.6)
" π2 π2 (cid:18)−8 16(cid:19)# −
Thus, σ(1,1) and σ(2,2) only contain the leading perturbative term in the short distance
expansion, and d = 3 and d = 5 condensates do not contribute to the operator product
expansion of these “diagonal’ correlators. Also the d = 6 four-quark operator does not
contribute in this case. On the other hand, the ‘nondiagonal’ spectral function σ(1,2)(Ω) +
σ(2,1)(Ω) is determined by the d = 3 q¯q vacuum condensate and by the d = 5 quark-gluon
h i
condensate. Similar to Eq. (2.11), for the latter condensate the two separate contributions
in (3.6) arise from the non local contribution of Eq. (2.12) and from the diagram where the
non perturbative low-frequency gluon is emitted from the perturbatively propagating light
quark.
The hadronic side of the sum rule is written as
σ(ℓ,ℓ′)(Ω,Ω′) = σ(ℓ,ℓ′)(Ω,Ω′)+σ(ℓ,ℓ′)(Ω,Ω′) , (3.7)
had res cont
where σ(ℓ,ℓ′)(Ω,Ω′) contains the contribution of the Λ ground state and σ(ℓ,ℓ′) contains the
res b cont
(ℓ,ℓ′)
contribution of higher mass states. By applying the duality principle, σ is assumed equal
cont
to the operator product expansion σ(ℓ,ℓ′)(Ω,Ω′) of Eqs. (3.6)-(3.6) in the domain Ω > ω and
c
′
Ω > ω , and vanishing outside this domain. Owing to the δ-function behavior, the details
c
′
of the integration domain far from the diagonal Ω = Ω should not be important.
The pole contribution is as follows:
[f(ℓ)]2
σ(ℓ,ℓ)(Ω,Ω′) = Λb hKiΛb , (3.8)
res 2(Ω ∆)(Ω′ ∆)
− −
(1) (2)
f f
σ(1,2)(Ω,Ω′)+σ(2,1)(Ω,Ω′) = Λb Λb hKiΛb , (3.9)
res res 2(Ω ∆)(Ω′ ∆)
− −
where the matrix element is defined as:
hKiΛb
= Λ (v) Λ (v) = Λ (v) h¯ (iD⊥)2h Λ (v) . (3.10)
hKiΛb h b |K| b i h b | v v| b i
′
Sum rules for are obtained by taking double Borel transforms in the variables Ω,Ω
hKiΛb
of both Σ and Σ , giving :
OPE res
1 1 1 1
(E) = e−ω/E, (E) = e−∆/E, (3.11)
B ω Ω E B ∆ Ω E
− −
9
′
and similar for Ω, and by assuming that one can equate them for some range of the Borel
′
parameters E , E . The symmetry of the spectral functions in Ω, Ω suggests the choice
1 2
E = E and, moreover, we take E = E = 2E with E the Borel parameter of the sum rule
1 2 1 2
for the two-point correlator, by analogy with the calculation of the Isgur-Wise form factor,
both for baryons [13] and mesons [19], where the normalization at zero recoil requires such a
relation. Finally, we assume the same continuum threshold ω as in the two-point function
c
sum rule, namely ω = 1.1, 1.2 and 1.3 GeV2.
c
(ℓ)
In this way, eliminating the couplings f by means of the two-point sum rules in
Λb
Eqs. (2.16) and (2.17), we obtain for the particular choices of the baryonic interpolating
field the following sum rules for the matrix element :
hKiΛb
3 I
7 ′
= (ℓ = ℓ = 1) (3.12)
hKiΛb −7 I
5
5 I
7 ′
= (ℓ = ℓ = 2) (3.13)
hKiΛb −7 I
5
I 3 m2 I
= 4 − 16 0 2 (ℓ = ℓ′) (3.14)
hKiΛb −I 1 m2 I 6
2 − 16 0 0
where
In = ωcdω ωn e−Eω . (3.15)
Z0
In these relations, the dependence on ∆ has disappeared, and in principle the analysis could
nowfollowthefamiliarprocedure, checking theexistence ofarangeintheBorelparameterE
where the results for are stable or at least smoothly dependent on E, and at the same
hKiΛb
timetheoperatorproductexpansiondisplaysaconvergentstructurewithhigherstatesgiving
smaller contributions. However, in the present case the two ‘diagonal’ sum rules (3.12) and
(3.13) do not give the same result for , at least in the approximation of the operator
hKiΛb
product expansion considered here, in contrast with the two-point function sum rules in
(2.16). Thus, the separation of ‘diagonal’ and ‘nondiagonal’ sum rules is no longer useful,
and we have to use the complete three-point correlator with the full Λ interpolating field
b
of Eq. (2.4), where the a priori unknown parameter b appears. Clearly, the corresponding
sum rule
1 f2 (1+b)2e−∆/E = ωc dωe−ω/E 3+5b2ω7 + 2b < qq >ω2(ω2 3m20) (3.16)
2hKiΛb Λb Z0 h− 140π4 π2 − 16 i
would give results depending on the parameter b, whose value must be determined somehow.
In order to get a reasonable criterion to fix the value of b, we recall that, in principle,
the three-point function sum rule can also be used to determine the parameter ∆, by taking
the derivative in 1/E of (3.16). We can then compare such determination with that given
by the two-point function sum rule (2.18) and assume, as an ‘optimal choice’, the value
of b for which the two determinations are closer. In Fig. 2 we depict the ratio of the two
determinations versus b and the Borel parameter E. As one can see, for the considered
values of E such a ratio is close to unity for b = 1, and therefore we shall assume this value
of b for the subsequent analysis of the sum rule for .
hKiΛb
10