QCD Sum-Rule Consistency of Lowest-Lying q¯q Scalar Resonances V. Elias and AH Fariborz ...

39 Pages·1998·0.51 MB·English

QCD Sum-Rule Consistency of Lowest-Lying qq(cid:22) Scalar
Resonances
V. Elias and A. H. Fariborz(cid:3)
Department of Applied Mathematics
The University of Western Ontario
London, Ontario N6A 5B7
Canada
and
Fang Shi and T. G. Steele
Department of Physics and Engineering Physics
University of Saskatchewan
Saskatoon, Saskatchewan S7N 5C6
Canada
(cid:3) Current Address: Department of Physics, Syracuse University, Syracuse,
New York 13244-1130, USA.
1
Abstract
We investigate lowest-lying scalar mesonpropertiespredictedfrom
QCDLaplacesumrulesbaseduponisovectorandisoscalarnon-strange
q(cid:22)q currents. The hadronic content of these sum rules incorporates de-
viations from the narrow resonance approximation anticipated from
physical resonance widths. The (cid:12)eld theoretical content of these sum
rules incorporates purely-perturbative QCD contributions to three-
loop order, the direct single-instanton contribution in the instanton
liquid model, and leading contributions from QCD-vacuum conden-
sates. In the isovector channel, the results we obtain are compatible
with a (1450) being the lowest-lying qq(cid:22)resonance, and are indicative
0
of a non-qq(cid:22) interpretation for a (980). In the isoscalar channel, the
0
results we obtain are compatible with the lowest lying qq(cid:22) resonance
being f (980) or a state somewhat lighter than f (980) whose width
0 0
is less than half of its mass. The dilaton scenario for such a nar-
rower (cid:27)-resonance is discussed in detail, and is found compatible with
sum rule predictions for the resonance coupling only if the anomalous
gluon-(cid:12)eld portion of (cid:2)(cid:22) dominates the matrix element < (cid:27)j(cid:2)(cid:22)j0 >.
(cid:22) (cid:22)
A linear sigma-model interpretation of the lowest-lying resonance’s
coupling, when compared to the coupling predicted by sum rules, is
indicative of a renormalization-group invariant light-quark mass be-
tween 4 and 6 MeV.
1 Introduction: Status of the Lowest-Lying
Scalar Resonances
At present, there isa greatdealof confusionconcerning boththeidentity and
interpretation of the lowest lying I = 0 and I = 1 scalar resonances, specif-
ically the four states denoted in the Particle Data Guide by f (400-1200),
0
f (980), a (980), and a (1450). The nearness of the f (980) and a (980)
0 0 0 0 0
to a KK(cid:22) threshold has led to a widely held interpretation of these states
as isopartner KK(cid:22)-molecules [1,2,3], as opposed to light qq(cid:22)-resonances (lin-
(cid:22)
ear combinations of uu(cid:22) and dd states). However, the assumption that these
(cid:22)
states are isopartners and/or KK-molecules have both been subject to re-
cent scrutiny. In particular, Morgan and Pennington [4] have disputed the
KK(cid:22) interpretation of f (980). An analysis using the Ju¨lich model for (cid:25)(cid:25)
0
scattering [5] is compatible with a KK(cid:22) interpretation of f (980), but sees
0
2
a (980)as a dynamical threshold e(cid:11)ect, as opposed to a true resonance state.
0
An even more recent analysis of OPAL data [6] supports the consistency of
a qq(cid:22)interpretation of f (980).
0
Theory is similarly ambivalent regarding the f (980) and a (980) scalar
0 0
resonance states. A QCD sum rule analysis [7] based upon correlation-
function currents chosen to project out KK(cid:22)-molecule states concludes that
f (1500) and f (1710) are better candidates for such multiquark states than
0 0
either f (980) or a (980). Moreover, a very recent coupled channel analysis
0 0
of (cid:25)(cid:25) scattering [8] suggests that the f (980) state may really be two distinct
0
S-matrix poles (see also [4]), one corresponding to a KK(cid:22) molecule and the
other perhaps corresponding to a light qq(cid:22)state.
Indeed, the determination of the lowest lying qq(cid:22)I = 0 and I = 1 states is
of genuine value as a test of our present understanding of QCD, particularly
its nonperturbative content. Such lowest lying states, when (cid:12)rst compared
with QCD via sum rule methods [9], were necessarily found to be degen-
erate, as purely-perturbative and QCD-vacuum condensate contributions to
scalar-current correlation functions cannot distinguish between I = 0 and I
= 1 channels. Of course, this result was (cid:12)rst seen to account for the degener-
acy of f (980) and a (980) as lowest-lying qq(cid:22)isopartners [9,10]. As has been
0 0
emphasized repeatedly over the last twenty years, however, both scalar and
pseudoscalar channels exhibit signi(cid:12)cant sensitivity not only to nonpertur-
bative (cid:12)eld-theoretical e(cid:11)ects with in(cid:12)nite correlation length (QCD-vacuum
condensates), butalsotoinstantone(cid:11)ects, thenonperturbativecontentofthe
QCD vacuum characterized by (cid:12)nite correlation lengths [11,12,13,14,15,16].
The instanton component of the QCD vacuum is known to distinguish be-
tween I=0 andI=1scalar (andpseudoscalar) states[12,13]. Such a distinc-
tion is, of course, quite evident in the pseudoscalar channel’s large(cid:25)−(cid:17) mass
di(cid:11)erence, though an understanding of the pseudoscalar I = 1 channel nec-
essarily must take into account the (cid:12)rst pion-excitation state because of the
near-masslessness (and concomitantly reduced sum-rule contribution) of the
pion [17,18,19]. Similarly, the existence of instanton solutions in QCD nec-
essarily imposes the theoretical expectation that a similar split occur between
I = 0 and I = 1 qq(cid:22) scalar resonance states, with the I = 0 state substan-
tially lighter than its I = 1 isopartner [12,13]. In this regard, scalar meson
spectroscopy is a genuine test of QCD.
Recent activity [20,21,22,23] in re-analyzing old (cid:25)(cid:25) and (cid:25)N scattering
data hasled to the reinstatement of a lowest-lying I=0 scalarresonance that
3
is distinct from the f (980), conservatively labelled by the 1996 Particle Data
0
Group [1] as f (400-1200). Perhaps of equal importance, assuming a (980)
0 0
and f (980) really are either KK(cid:22) isopartners or other non-qq(cid:22)exotica, is the
0
Crystal BarrelCollaboration’srecent con(cid:12)rmationofan a (1450)I=1scalar
0
resonance state [24]. If this isovector state is identi(cid:12)ed as the lowest-lying I
= 1 qq(cid:22)-scalar resonance (lattice simulations have shown glueballs below 1600
MeV to be unlikely), it is important to determine whether the identi(cid:12)cation
of its I = 0 isopartner with any portion of the f (400-1200) mass range
0
(particularly the high end [4,25]) is compatible with the instanton-generated
mass di(cid:11)erence anticipated from QCD.
Moreover, the f (400-1200) has been widely interpreted to be the (cid:27)-
0
particle signature of chiral symmetry breaking anticipated from NambuJona-
Lasinio (NJL) dynamics [26], the linear sigma-model (L(cid:27)M) spectrum [27],
as well as in models for qq(cid:22) scattering in an instanton background [12,13]
and in one boson exchange models of the nucleon-nucleon potential [28].
Such a (cid:27)-particle, however, does not characterize the nonlinear sigma model
(NL(cid:27)M); indeed, the empirical absence of a credible (cid:27) prior to 1996 has pro-
vided impetus for the development of NL(cid:27)M ideas into a chiral perturbation
theory framework. Clearly a clari(cid:12)cation of the properties, or even the exis-
tence [29], of a light (cid:27)-resonance is required to distinguish between L(cid:27)M and
NL(cid:27)M alternatives for e(cid:11)ective theories of low-energy hadron physics.
We choose here to distinguish, somewhat arbitrarily, four di(cid:11)erent alter-
nativesforthef (400-1200)resonancethateachhavesomeempiricalsupport:
0
1. The resonance is both very light (m < 500 MeV) and very broad (Γ >
(cid:27) (cid:24) (cid:27) (cid:24)
500 MeV), as suggested by DM2 data [30] and by To¨rnqvist and Roos’s
analysisof(cid:25)(cid:25) scattering[23]. Itmustberecognized, however, thatsuch
a resonance may be generated dynamically [8] and is not necessarily a
qq(cid:22)state.
2. The resonance is (cid:27)-like in mass (500-700 MeV) but substantially nar-
rower in width (Γ < m =2), consistent with parameter ranges ex-
(cid:27) (cid:24) (cid:27)
tracted by Svec [20], S. Ishida et al [21], and Harada, Sannino, and
Schechter [22].
3. The lowest I = 0 qq(cid:22) scalar is to be identi(cid:12)ed with a qq(cid:22) pole at (or
perhaps masked by) the f (980) resonance [8], with a narrow (Γ < 150
0 (cid:27) (cid:24)
MeV) width.
4
4. The lowest I = 0 qq(cid:22) scalar resonance is a very broad structure in the
(cid:25)(cid:25) scattering amplitude characterized by a mass comparable to [4] or
substantially above [25] that of the f (980).
0
There is, of course, some blurring of the boundaries between these alter-
natives, particularly Alternatives 2 and 3. Svec [31] has recently reported
a single-pole (cid:12)t of (cid:25)-N(polarized) scattering data leading to a mass (775 (cid:6)
17 MeV) somewhat larger than NJL-L(cid:27)M expectations, and a width (147 (cid:6)
33 MeV) substantially narrower than those already quoted in support of Al-
ternative 2. Moreover, Harada, Sannino, and Schechter have demonstrated
[32] how a very light, very broad state [Alternative 1] can transmute into
a heavier, narrower state [Alternative 2] when (cid:26) exchanges are taken into
account. From a theoretical point of view, Alternatives 1 and 2 support the
existence of a (cid:27)-particle, though straightforward L(cid:27)M expectations would
favour the mass range of Alternative 2 and the broad width of Alternative 1
[33,34]. Theoretical arguments foranarrower (cid:27)-particlemorefully consistent
with Alternative 2 have been advanced through identifying the (cid:27) with the
near-Goldstone particle of dilatation symmetry in the strong coupling limit
[35,36]. Such a dilaton would be expected to have a width similar to that of
the (cid:26)-meson, corresponding to cancellation of an enhancement factor of 9/2
in a naive calculation of the width [33] against an anticipated suppression
factor 1=d2 (cid:25) 1=4, where d is the anomalous mass dimension in the strong
(cid:27) (cid:27)
coupling limit. 1
In the present manuscript, we employ QCD Laplace sum-rules as a tech-
niqueparticularlywell-suited torelatethe(cid:12)eld-theoreticalcontentofQCDto
lowest-lying resonance properties [14]. We use sum-rule methodology specif-
ically to address the following questions:
Which, if any, of the four alternatives discussed above for the lightest I =
0 qq(cid:22) scalar are supported by QCD sum rules? In particular, do QCD sum
rules rule out either of the alternatives that are consistent with an NJL/L(cid:27)M
(cid:27)-like object? If the existence of a (cid:27) is consistent with QCD sum rules, is
such a (cid:27) a broad L(cid:27)M object, or a narrower strong-coupling dilaton?
1Notethatf appearingineq. (13)of[35]shouldbeunderstoodtobe131MeV,not93
(cid:25)
MeV,sothatthe d =1predictionofΓ coincideswiththe L(cid:27)M-equivalentpredictionin
(cid:27) (cid:27)
[34]. This point has been veri(cid:12)ed through personal communciation with V. A. Miransky.
5
What is the mass range for the lightest I = 1 qq(cid:22)scalar resonance? In partic-
ular, can we rule out all but exotic interpretations of the a (980), and does
0
there exist sum-rule support for the recently con(cid:12)rmed a (1450) being the
0
lowest-lying qq(cid:22)object in this channel?
In the section that follows, we present the sum rule methodology nec-
essary to address these questions. Speci(cid:12)cally, we show how nonzero res-
onance widths can be incorporated into the hadronic contribution to QCD
Laplace sum rules, which are argued to be particularly appropriate forstudy-
ing lowest-lying resonance properties. We also demonstrate explicitly how a
lowest-lying resonance’s nonzero width elevates a sum rule determination of
that resonance’s mass.
In Section 3 we present the (cid:12)eld-theoretical content of appropriate scalar
current sum rules. We discuss the sum rule contribution arising from the 3-
loop order purely-perturbative QCD contributions to the scalar current cor-
relation function. We also present nonperturbative sum rule contributions
arising from QCD-vacuum condensates and direct single-instanton contribu-
tions to the I = 0 and I = 1 scalar current correlation functions.
In Section 4, we utilize the results of the preceding two sections to ob-
taina sum-rule determination of themasses of lowest-lying scalar resonances.
Stability curves are generated leading to estimates of such masses for a given
choice of width and the continuum threshold above which perturbative QCD
and hadronic physics are assumed to coincide. Detailed comparison is made
with earlier sum-rule generated stability curves [9], showing how the separate
incorporation of renormalization-group improvement, 3-loop perturbative ef-
fects, nonzero widths, and the contribution of instantons individually a(cid:11)ect
such curves.
In Section 5, we examine the isoscalar channel in further detail by ob-
taining values of the mass, width, continuum threshold, and coupling of the
lowest-lying qq(cid:22)resonance via a weighted least-squares (cid:12)t to the overall Borel-
parameter dependence of the (cid:12)rst Laplace sum rule. A relationship between
the anticipated resonance coupling and a phenomenologically estimable ma-
trix element is developed in Appendix A. In Section 5 this relationship is
utilized to obtain an estimate of the light quark mass. This relationship is
also used to assess the sum-rule consistency of a dilaton interpretation for
the lowest-lying resonance.
Finally in Section 6 we present our conclusions concerning the questions
6
we have raised above. We assess the compatibility of sum rule predictions
for the lowest-lying non-exotic I=1 resonance with a (980) and a (1450).
0 0
We examine in detail the four alternatives presented above for interpreting
the I = 0 scalar resonance spectrum and argue that Alternatives 1 and 4
appear to be unsupported by a sum-rule based analysis of the lowest-lying
qq(cid:22) state. We also discuss how our conclusions are a(cid:11)ected by possible sum
rule contamination from higher resonances.
7
2 Sum-Rule Methodology and Lowest-Lying
qq(cid:22)-Scalar Resonances of Nonzero Width
In the narrow resonance approximation, subcontinuum resonance contribu-
tions to the light-quark scalar-current correlation function,2
Z
(cid:5)(p2) = i d4xeip(cid:1)x < 0jTj(x)j(0)j0 >; (1)
j(x) (cid:17) [u(cid:22)(x)u(x)(cid:6)d(cid:22)(x)d(x)]=2; (2)
are proportional to a sum of delta functions:
" #
X −g
r
(Im(cid:5)(s)) = Im
res: (s−m2)+im Γ
"r r r# r
X g m Γ
r r r
=
(s−m2)2 +m2Γ2
r r r r
X
−! (cid:25)g (cid:14)(s−m2): (3)
r r
Γ ! 0 r
r
The coupling coe(cid:14)cient g is proportional to m2. However, the constant
r r
of proportionality is expected to be much larger for qq(cid:22) resonances [i.e. res-
onances that couple directly to the (cid:12)eld-theoretical operators in the scalar
current (2)] than for exotic resonances [37]. It is forprecisely this reason that
sum-rule searches for non-qq(cid:22)scalar resonance states, such as KK(cid:22) molecules
(cid:22)
[7]orglueballs[37,38],utilizecorrelationfunctionsbasedonappropriateKK-
or gluonic-currents that couple directly to such hadronic exotica.
Laplace sum rules R ((cid:28)) for the scalar current correlation function are
k
particularly sensitive to the lowest-lying qq(cid:22)resonance of a given isostructure:
Z
1 1
R ((cid:28)) (cid:17) skIm[(cid:5)(s)]e−s(cid:28)ds
k
(cid:25)
0
2We have normalized I=0 (+) and I = 1 (-) scalar currents in (2) so as to facilitate
comparison with ref. [9].
8
X
= g m2ke−m2r(cid:28)(cid:2)(s −m2)
r r 0 r
r
Z
1 1
+ skIm[((cid:5)(s)) ]e−s(cid:28)ds: (4)
pert:
(cid:25)
s0
The summation over resonances in (4) clearly follows from the (cid:12)nal line of
(3). The remaining integral in (4) reflects the anticipated duality [39] be-
tween purely-perturbative QCD and hadronic physics above some appropri-
ately chosen continuum threshold s > s . As is evident from (4), higher-mass
0
resonances areeitherabsorbed intothecontinuum (m2 > s ), orif subcontin-
r 0
uum(m2 < s ),areexponentiallysuppressed relativetolow-massresonances.
r 0
Consequently, Laplace sum rules are well-suited fordetermining properties of
the lowest-lying resonance in a given channel. The subcontinuum resonance
contribution R ((cid:28);s ) to the kth Laplace sum rule, de(cid:12)ned as
k 0
Z
1 1
R ((cid:28);s ) (cid:17) R ((cid:28))− skIm[((cid:5)(s)) ]e−s(cid:28)ds; (5)
k 0 k pert:
(cid:25)
s0
is clearly seen from (4) to satisfy the inequality
R ((cid:28);s )
k+1 0 (cid:21) m2; (6)
R ((cid:28);s ) ‘
k 0
where m is the mass of the lowest-lying resonance in a given channel. QCD
‘
sum-rule methodology for a given resonance channel generally involves ob-
taining an estimate of the mass m via minimization of the (cid:12)eld-theoretical
‘
content of the left-hand side of (6) with respect to the Borel parameter (cid:28)
[14,40]. In practice, the ratio utilized is R =R , so as to avoid methodolog-
1 0
ically inconvenient enhancement of continuum and higher-mass resonance
contributions, as well as heightened dependence on (unknown) higher di-
mensional condensates, through the factors m2k appearing in (4).
r
The (cid:12)eld theoretical content of the (cid:12)rst two Laplace sum rules R ((cid:28);s )
0;1 0
will be discussed in the section that follows. However, it is important to
recognize that (6) requires signi(cid:12)cant modi(cid:12)cation if the lowest-lying reso-
nanceisbroad. If onedoesnotinvoke thenarrowresonanceapproximation in
the (cid:12)nal line of (3), but instead assumes a Breit-Wigner (or modi(cid:12)ed Breit-
Wigner [1]) shape, one (cid:12)nds that
9
R ((cid:28);s ) (cid:24)= 1 mX2r<s0Z 1 grmrΓr ske−s(cid:28)ds
k 0 (cid:25) (s−m2)2 +m2Γ2
X r 0 r r r
= g W (m ;Γ ;(cid:28))m2ke−m2r(cid:28)(cid:2)(s −m2); (7)
r k r r r 0 r
r
with the weighting functions W demonstrably related to the narrow reso-
k
nance limit (4) via
lim W (m ;Γ ;(cid:28)) = 1: (8)
k r r
Γr!0
The net e(cid:11)ect of such weighting functions on the lowest-lying resonance con-
tribution to (6) is to replace m2 with m2W =W . If the lowest-lying reso-
‘ ‘ k+1 k
nance is the dominant subcontinuum resonance in a a given channel, then m
‘
can be extracted from the lowest-lying (‘) resonance contribution to R =R
1 0
as follows [19,41]:
!
W (m ;Γ ;(cid:28)) R ((cid:28);s )
0 ‘ ‘ 1 0 = m2: (9)
W (m ;Γ ;(cid:28)) R ((cid:28);s ) ‘
1 ‘ ‘ 0 0 ‘
For a given choice of Γ and s , one can use (cid:12)eld-theoretical expressions
‘ 0
for R ((cid:28);s ), including both perturbative QCD and nonperturbative QCD
0;1 0
contributions of in(cid:12)nite and (cid:12)nite correlation lengths [Section 3], to obtain
from (9) a self-consistent minimizing value of m . This procedure constitutes
‘
the methodological foundation for the results we obtain in Section 4.
The weighting functions W can be derived from a Breit-Wigner reso-
0;1
nance shape by expressing that shape as a Riemann sum of unit-area pulses
P centred at s = m2 [19]:
mr r
1
P [s;Γ] (cid:17) [(cid:2)(s−M2 +MΓ)−(cid:2)(s−M2 −MΓ)]; (10)
M
2MΓ
s " s #
MΓ 2 Xn n−j +f n−j +f
= lim P s; Γ ; (11)
(s−M2)2 +M2Γ2 n!1 n j −f M j −f
j=1
where f is any arbitrarily chosen constant between 0 and 1. If one approxi-
mates the resonance shape via (11) by truncating n to some (cid:12)nite number of
10