Instanton Effects on the Role of the Low-Energy Theorem for the Scalar Gluonic Correlation Function

22 Pages·0.43 MB·English

by  D. Harnett

Instanton Effects on the Role of the Low-Energy Theorem for the Scalar
Gluonic Correlation Function
D. Harnett, T.G. Steele ∗
3
Department of Physics and Engineering Physics
0
0 University of Saskatchewan
2 Saskatoon, Saskatchewan S7N 5E2, Canada.

n
a V. Elias †
J
Department of Applied Mathematics
5
University of Western Ontario

3 London, Ontario N6A 5B7, Canada.
v
9
February 1, 2008
4
0
7
0
0 Abstract
0 Instanton contributions to the Laplace sum-rules for correlation functions of scalar gluonic currents are
/ calculated. The role of the constant low-energy theorem term, whose substantial contribution is unique to the
h
p leadingLaplacesum-ruleL−1,isshowntobediminishedbyinstantoncontributions,significantlyincreasingthe
- resulting mass bounds for the ground state of scalar gluonium and improving compatibility with results from
p
higher-weight sum-rules.
e
h
:
v 1 Introduction
i
X
In the chiral limit of n quarks, the low-energy theorem (LET) for scalar gluonic correlation functions is [1]
r f
a
8π
Π(0)= lim Π Q2 = J , (1)
Q→0 β0 h i
(cid:0) (cid:1)
where
Π Q2 =i d4xeiq·x OT [J(x)J(0)] O , Q2 = q2 >0 (2)
h | | i −
Z
(cid:0) (cid:1) π2
J(x)= β(α)Ga (x)Ga (x) (3)
−αβ µν µν
0
d α(ν) α 2 α 3
β(α)=ν2 = β β +... (4)
dν2 π − 0 π − 1 π
(cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17)
11 1 51 19
β = n , β = n . (5)
0 f 1 f
4 − 6 8 − 24
The current J(x) is renormalization group (RG) invariant for massless quarks [2], and its normalization has been
chosen so that to lowest order in α
β α β α
J(x)=αGa (x)Ga (x) 1+ 1 + α2 αG2(x) 1+ 1 + α2 . (6)
µν µν β π O ≡ β π O
(cid:20) 0 (cid:21) (cid:20) 0 (cid:21)
∗email: [email protected] (cid:0) (cid:1) (cid:0) (cid:1)
†email: [email protected]
1
Most applications of dispersion relations in sum-rules are designed to remove dependence on low-energy sub-
traction constants. However, knowledge of the LET for gluonic correlation functions permits the possibility of
sum-rules that contain explicit dependence on the LET subtraction constant Π(0). For example, the dispersion
relation appropriate to the asymptotic (perturbative) behaviour of the correlation function (2) is [3]
∞
1 1 ρ(t)
Π Q2 =Π(0)+Q2Π′(0)+ Q4Π′′(0) Q6 dt . (7)
2 − π t3(t+Q2)
Z
(cid:0) (cid:1) t0
where ρ(t) is the hadronic spectralfunction with physicalthreshold t appropriate to the quantum numbers of the
0
current used to construct the correlationfunction.
Unfortunately, direct application of the dispersion relation (7) is not possible because the theoretical (pertur-
bative) calculation of Π Q2 contains a field-theoretical divergence proportional to Q4. A related problem is the
significant contribution of excited states and the QCD continuum to the integral of ρ(t) in (7). Enhancement
(cid:0) (cid:1)
of the lowest-lying resonance contribution in applications to light hadronic systems requires greater high-energy
suppression of this integral.
The established technique for dealing with these issues is the Laplace sum-rules [4]. A family of Laplace
sum-rules can be obtained from the dispersion relation (7) through the Borel transform operator Bˆ,
Q2 N d N
Bˆ lim − (8)
≡NN,/QQ22→≡∞τ (cid:0)Γ(N(cid:1)) (cid:18)dQ2(cid:19)
which has the following useful properties in the construction of the Laplace sum-rules:
Bˆ a +a Q2+...a Q2m =0 , (m finite) (9)
0 1 m
Q2n
Bˆ(cid:2) =τ( 1)ntne−(cid:3)tτ , n=0, 1, 2,... (n finite) (10)
t+Q2 −
(cid:20) (cid:21)
The theoretically-determined quantity
1
(τ) Bˆ ( 1)kQ2kΠ Q2 , (11)
k
L ≡ τ −
h (cid:0) (cid:1)i
leads to the following family of Laplacesum-rules, after applicationofBˆ to the dispersionrelation(7)weightedby
the appropriate power of Q2:
1 1
(τ) = Bˆ ( 1)kQ2kΠ(0)+( 1)kQ2k+2Π′(0)+( 1)k Q2k+4Π′′(0)
k
L τ − − − 2
(cid:20) (cid:21)
∞
1 1 Q2k+6
dt Bˆ ( 1)k ρ(t) (12)
−π τt3 − (t+Q2)
Z (cid:20) (cid:21)
t0
Therearesomeimportantconstraintsonkthatwillleadtosum-ruleswithpredictivepower. Sincetheperturbative
predictionofΠ Q2 containsdivergentconstants multiplied byQ4, the sum-rules (τ) where this contributionis
k
absent require k 2. However, the low-energy constants Π′(0) and Π′′(0) are nLot determined by the LET [i.e.
only the quanti(cid:0)ty≥Π(cid:1)−(0) appears in (1)]. Hence the sum-rules (τ) which will be independent of Π′(0) and Π′′(0)
k
L
mustsatisfyk 1,andonlythek = 1sum-rulewillcontaindependenceontheLET-determinedquantityΠ(0):
≥− −
∞
1 1
−1(τ)= Π(0)+ dt e−tτρ(t) (13)
L − π t
Z
t0
∞
1
(τ)= dttke−tτρ(t) , k > 1 (14)
k
L π −
Z
t0
2
The “resonance(s) plus continuum” model is used to represent the hadronic physics phenomenology contained
inρ(t)in(13–14)[4]. Inthismodel,hadronicphysicsis(locally)dualtothetheoreticalQCDpredictionforenergies
above the continuum threshold t=s :
0
ρ(t) θ(s t)ρhad(t)+θ(t s )ImΠQCD(t) (15)
0 0
≡ − −
The contribution of the QCD continuum to the sum-rules is denoted by
∞
1
c (τ,s )= dttke−tτImΠQCD(t) . (16)
k 0
π
Z
s0
SincethecontinuumcontributionisdeterminedbyQCD,itisusuallycombinedwiththetheoreticalquantity (τ)
k
L
(τ,s ) (τ) c (τ,s ) , (17)
k 0 k k 0
S ≡L −
resulting in the following Laplace sum-rules relating QCD to hadronic physics phenomenology:
s0
1 1
−1(τ,s0)= Π(0)+ dt e−tτρhad(t) (18)
S − π t
Z
t0
s0
1
(τ,s )= dttke−tτρhad(t) , k > 1 (19)
k 0
S π −
Z
t0
The property
lim c (τ,s )=0 (20)
s0→∞ k 0
implies that the sum-rules (17) and (14) are identical in the s limit.
0
→∞
lim (τ,s )= (τ) (21)
s0→∞Sk 0 Lk
The only appearance of the Π(0) term is in the k = 1 sum-rule, and as first noted in [5], this LET term
−
comprises a significant contribution in the k = 1 sum-rule. From the significance of this scale-independent term
−
one can ascertain the important qualitative role of the LET in sum-rule phenomenology. To see this role, we first
model the hadronic contributions ρhad(t) using the narrow resonance approximation
1
ρhad(t)= F2m2δ t m2 , (22)
π r r − r
r
X (cid:0) (cid:1)
wherethesumoverr representsasumoversub-continuumresonancesofmassm . ThequantityF isthecoupling
r r
strengthoftheresonancetothevacuumthroughthegluoniccurrentJ(0),sothesum-ruleforscalargluoniccurrents
probes scalar gluonium states. In the narrow-width approximation the Laplace sum-rules (18–19) become
S−1(τ,s0)+Π(0)= Fr2e−m2rτ (23)
r
X
Sk(τ,s0)= Fr2m2rk+2e−m2rτ , k >−1 . (24)
r
X
Thusifthe (constant)LETtermisasignificantcontributiononthe theoreticalside of(23),thenthe left-handside
of (23) will exhibit reduced τ dependence relative to other theoreticalcontributions. To reproduce this diminished
τ dependence, the phenomenological (i.e. right-hand) side must contain a light resonance with a coupling larger
3
than or comparable to the heavier resonances. By contrast, the absence of the Π(0) (constant) term in k > 1
−
sum-rules leads to stronger τ dependence which is balanced on the phenomenological side by suppression of the
lightestresonancesviatheadditionalpowersofm2r occurringin(24). ThusifΠ(0)isfoundtodominateS−1(τ,s0),
then one would expect qualitatively different results from analysis of the k = 1 and k> 1 sum-rules.
− −
Suchdistinctconclusionsdrawnfromdifferentsum-rulescanbelegitimate. Inthepseudoscalarquarksector,the
lowestsum-ruleisdominatedbythepion,andthelowmassofthepionisevidentfromtheminimalτ dependencein
thelowestsum-rule. Bycontrast,thefirstsubsequentsum-rulehasanimportantcontributionfromtheΠ(1300)[6],
asthepioncontributionissuppressedbyitslowmass,resultinginthesignificantτ dependenceofthenext-to-lowest
sum-rule.
In the absence of instantons [7], explicit sum-rule analyses of scalar gluonium [3, 5, 8] uphold the above
generalization—those which include the k = 1 sum-rule find a light (less than or on the order of the ρ mass)
−
gluonium state, and those which omit the k = 1 sum-rule find a state with a mass greater than 1GeV. The
−
prediction of a light gluonium state would have interesting phenomenological consequences as a state which could
be identified with the f (400 1200)/σ meson [9]. However, a detailed treatment of instanton contributions is
0
−
essentialinassessingthe viability ofsucha two-resonance-scalescenario(as evident in the sum-rule analysisof the
qq¯pseudoscalar channel [6]) in the scalar gluonium channel.
Seminalworkby Shuryak[10] in the instantonliquid model[11]has indicated how an asymptotic (ρ /√τ 1)
c
≫
expression for the instanton contribution to the k = 1 sum-rule may serve to compensate for that sum-rule’s
−
LETcomponentandbringthepredictedscalargluoniummassinlinewithsubsequentlatticeestimates( 1.6GeV
∼
[13]). Recent work by Forkel [12] has addressed in detail instanton effects on scalar gluonium mass predictions
from higher-weight sum-rules and has also corroborated lattice estimates. However, the overall consistency of the
k = 1 sum-rule, which is sensitive to the low-energy theorem term, and k 0 sum-rules, which are not, has not
− ≥
been addressed quantitatively.
InSection 2, we explicitly calculate the instantoncontributions to Laplacesum-rules ofscalargluonic currents.
We pay particular attention to the k = 1 sum-rule and demonstrate that instanton contributions partially cancel
−
againsttheLETconstantΠ(0)andservetoappreciablydiminishitsdominanceofthisleadingordersum-rule. The
phenomenological implications of this partial cancellation are investigated in Section 3, and a discussion relating
our work to other analyses of instanton effects in the scalar gluonium channel is presented in Section 4.
2 Instanton Effects in the Laplace Sum-Rules
Thefield-theoretical(QCD)calculationofΠ Q2 consistsofperturbative(logarithmic)correctionsknowntothree-
looporder(MSscheme)inthechirallimitofn =3masslessquarks[14],QCDvacuumeffectsofinfinitecorrelation
(cid:0)f (cid:1)
lengthparameterizedbythepower-lawcontributionsfromtheQCDvacuumcondensates[5,15],1 andQCDvacuum
effects of finite correlationlength devolving from instantons [16]
Π Q2 =Πpert Q2 +Πcond Q2 +Πinst Q2 , (25)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
with
1. ... the perturbative contribution (ignoring divergent terms proportional to Q4) given by
Q2 Q2 Q2
Πpert Q2 =Q4log a +a log +a log2 (26)
ν2 0 1 ν2 2 ν2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)α 2 659α α 2 α 3 9 α α 4
a = 2 1+ +247.480 , a =2 +65.781 , a = 10.1250
0 1 2
− π 36 π π π 4 π − π
(cid:16) (cid:17) (cid:20) (cid:16) (cid:17) (cid:21) (cid:16) (cid:17) (cid:20) (cid:21) (cid:16) (cid:17)
1Thecalculationofone-loopcontributions proportionaltohJiin[15]havebeenextended non-triviallytonf =3fromnf =0,and
theoperatorbasishasbeenchangedfrom αG2 tohJi.
(cid:10) (cid:11)
4
2. ... the condensate contributions given by
Q2 1 1
Πcond Q2 = b +b log J +c +d (27)
0 1 ν2 h i 0Q2 hO6i 0Q4 hO8i
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) α(cid:1) 175α α 2 α 2 α
b =4π 1+ , b = 9π , c =8π2 , d =8π2 (28)
0 1 0 0
π 36 π − π π π
(cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17)
= gf Ga Gb Gc , =14 αf Ga Gb 2 αf Ga Gb 2 (29)
hO6i abc µν νρ ρµ hO8i abc µρ νρ − abc µν ρλ
(cid:10) (cid:11) D(cid:0) (cid:1) E D(cid:0) (cid:1) E
3. ... and the instanton contribution given by
2
Πinst Q2 =32π2Q4 ρ4 K ρ Q2 dn(ρ) , (30)
2
(cid:0) (cid:1) Z h (cid:16) p (cid:17)i
where K (x) represents a modified Bessel function [17].
2
Thestrongcouplingconstantαisunderstoodtobetherunningcouplingatthe renormalizationscaleν,andrenor-
malization group improvement of the Laplace sum-rules implies that ν2 = 1/τ [18]. The instanton contributions
represent a calculation with non-interacting instantons of size ρ, with subsequent integration over the instanton
density distribution n(ρ). 2 The theoretical contributions to the Laplace sum-rules corresponding to (25) are
(τ)= pert(τ)+ cond(τ)+ inst(τ) . (31)
Lk Lk Lk Lk
An alternative to the direct calculation of the Laplace sum-rules through the definition of Bˆ in (8) is obtained
through an identity relating the Borel and Laplace transform [19]
∞
f Q2 = dτF(τ)e−Q2τ [F(τ)] = 1Bˆ f Q2 =F(τ)= −1 f Q2 (32)
≡L ⇒ τ L
Z
(cid:0) (cid:1) 0 (cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:3)
b+i∞
−1 f Q2 = 1 f Q2 eQ2τ dQ2 (33)
L 2πi
(cid:2) (cid:0) (cid:1)(cid:3) b−Zi∞ (cid:0) (cid:1)
where the realparameterb in the definition (33) ofthe inverseLaplace transformmust be chosenso that f Q2 is
analytictotherightofthecontourofintegrationinthecomplexplane. Usingtheresult(32),theLaplacesum-rules
(cid:0) (cid:1)
(11) can be obtained from an inverse Laplace transform of the theoretically-determined correlation function:
(τ)= −1 ( 1)kQ2kΠ Q2 . (34)
k
L L −
h (cid:0) (cid:1)i
In the complex Q2 plane where the inverse Laplace transform (33) is calculated, the QCD expression (25) for
the correlation function Π Q2 is analytic apart from a branch point at Q2 = 0 with a branch cut extending to
infinity along the negative-real-Q2 axis. Consequently, analyticity to the right of the contour in (33) implies that
(cid:0) (cid:1)
b>0. Consider the contour C(R) in Figure 1; Π Q2 is analytic within and on C(R) and so with z =Q2
1 (cid:0) (cid:1)
0= ( z)kezτΠ(z)dz , (35)
2πi −
I
C(R)
which leads to
b+iR
1 1 1
( z)kezτΠ(z)dz = ( z)kezτΠ(z)dz ( z)kezτΠ(z)dz . (36)
2πi − −2πi − − 2πi −
b−ZiR Γ1+.Z..+Γ4 Γc+ZΓǫ
2Afactorof2toincludethesumofinstantonandanti-instanton contributionshasbeenincludedin(25).
5
Taking the limit as R , which requires use of the asymptotic behaviour of the modified Bessel function [17]
→∞
π π
K (z) e−z ; z 1 , arg(z) (37)
2
∼ 2z | |≫ | |≤ 2
r
theindividualintegralsoverΓ arefoundtovanish,resultinginthefollowingexpressionfortheLaplacesum-rule.
1...4
∞ π
1 1
(τ)= tke−tτ Π te−iπ Π teiπ dt+ ( 1)kexp ǫeiθτ ǫk+1ei(k+1)θΠ ǫeiθ dθ (38)
k
L 2πi − 2π −
Zǫ (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) −Zπ (cid:0) (cid:1) (cid:0) (cid:1)
Perturbative and QCD condensate contributions to the Laplace sum-rules are well known [3, 5, 8], and serve
as a consistency check for the conventions used to determine the instanton contribution through (38). Keeping in
mind the k 1 constraint established previously, we see that the perturbative contributions to the θ integral in
≥−
(38) are zero in the limit as ǫ 0, leaving only the anticipated integral of the discontinuity across the branch cut
→
[i.e. ImΠpert(t)] to determine the following perturbative contributions to the Laplace sum-rule.
∞
t t
pert(τ)= tk+2e−tτ a 2a log +a π2 3log2 dt (39)
Lk − 0− 1 ν2 2 − ν2
Z (cid:20) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21)
0
The QCDcondensateterms proportionalto b ,c andd inthe correlationfunction Π(z)donot havea branch
0 0 0
discontinuity, so their contribution to the Laplace sum-rule arises solely from the contour Γ (represented by the
ǫ
term in (38) with the θ integral), and can be evaluated using the result
1 ezτ 0 , n=0, 1, 2,...
−2πiZ zn dz =( (τnn−−11)! , n=−1, 2,−3,... (40)
Γǫ
TheQCDcondensatetermproportionaltob requiresamorecarefultreatment. IfΠ(z)isreplacedwithlog z/ν2
1
in (38) then we find
(cid:0) (cid:1)
b+i∞ ∞ π
1 z 1 ǫ
( z)kezτlog dz = tke−tτdt+ ( 1)kexp ǫeiθτ ǫk+1ei(k+1)θ log +iθ dθ (41)
2πi − ν2 − 2π − ν2
b−Zi∞ (cid:16) (cid:17) Zǫ −Zπ (cid:0) (cid:1) (cid:16) (cid:16) (cid:17) (cid:17)
The last term in this equation will be zero in the ǫ 0 limit except when k = 1. Similarly, the t integral is well
→ −
defined in the ǫ 0 limit except when k= 1. With ν2 =1/τ, and with evaluation of the ǫ 0 limit [which, for
→ − →
k = 1, involves cancellation between the two integrals in (41)] we find
−
b+i∞ ∞
1 z tke−tτdt , k > 1
( z)kezτ log dz = − − (42)
2πi − ν2  0
b−Zi∞ (cid:16) (cid:17)  γER , k =−1
where γ 0.5772 is Euler’s constant.It is easily verified that equations (42), (40), and (39) lead to the known
E ≈
results [3, 5, 8] for the non-instanton contributions to the Laplace sum-rules for scalar gluonic currents.
To evaluate the instanton contributions to the Laplace sum-rule, we must calculate the following integral:
b+i∞ ∞
1 ( z)kezτz2 K ρ√z 2dz = 1 tk+2e−tτ K ρ√te−iπ/2 2 K ρ√teiπ/2 2 dt
2 2 2
2πi − 2πi −
b−Zi∞ (cid:2) (cid:0) (cid:1)(cid:3) Zǫ (cid:20)h (cid:16) (cid:17)i h (cid:16) (cid:17)i (cid:21)
π
1 2
+ ( 1)kexp ǫeiθτ ǫk+3ei(k+3)θ K ρ√ǫeiθ/2 dθ (43)
2
2π −
−Zπ (cid:0) (cid:1) h (cid:16) (cid:17)i
6
Simplification of (43) requires the following properties of the modified Bessel function K (z) [17]
2
2
K (z) , z 0 (44)
2 ∼ z2 →
iπH(1) zeiπ/2 , π <arg(z) π
K (z)= − 2 2 − ≤ 2 (45)
2
( iπH(2) z(cid:0)e−iπ/2(cid:1) , π <arg(z) π
2 2 −2 ≤
where H(1)(z) = J (z)+iY (z(cid:0)) and H(cid:1)(2)(z) = J (z) iY (z). The asymptotic behaviour (44) implies that the θ
2 2 2 2 2 − 2
integralof(43)willbezerointheǫ 0limitfork > 1andtheidentity(45)allowsevaluationofthediscontinuity
→ −
in the t integral of (43), leading to the following instanton contribution to the Laplace sum-rules:
∞
inst(τ)= 16π3 dn(ρ)ρ4 tJ ρ√t Y ρ√t e−tτdt 128π2 dn(ρ) (46)
L−1 − 2 2 −
Z Z0 (cid:16) (cid:17) (cid:16) (cid:17) Z
= 64π2 dn(ρ) ae−a (1+a)aK (a)+(2+2a+a2)K (a)
0 1
−
Z
∞ (cid:2) (cid:3)
inst(τ)= 16π3 dn(ρ)ρ4 tk+2J ρ√t Y ρ√t e−tτdt , k > 1 (47)
Lk − 2 2 −
Z Z0 (cid:16) (cid:17) (cid:16) (cid:17)
a4e−a
128π2 dn(ρ) [2aK (a)+(1+2a)K (a)] , k =0
ρ2 0 1
= Z
 256π2 dn(ρ) a5ρe4−a (9−4a)aK0(a)+(3+7a−4a2)K1(a) , k=1
Z
wherea ρ2/(2τ)andwhereKn is the m(cid:2)odifiedBesselfunction ofthe firstkindof(cid:3)ordern(c.f. [17]). Observethe
≡
symmetrybetween(46)and(47)brokenbythe term 128π2 dn(ρ)appearingin(46)—atermwhichcorresponds
−
to the second integral on the right-hand side of (43) and which is nonzero only for k = 1. Conversely, we note
R −
that naively substituting k = 1 into (47) leads to an incorrect expression for the instanton contribution to the
−
leading order sum-rule. This asymmetric role of the instanton contributions to k = 1 and k > 1 sum-rules is
− −
also a property of the LET as illustrated in (13,14).
AsdiscussedinSection1,wewishtodeterminewhethertheleadingordersum-rule −1 mightsupporttheexis-
L
tence of a lowest-lying resonance whose presence is mass-suppressedin subsequent higher order k > 1 sum-rules.
−
Such is indeed the case, for example, for the pion within sum-rules based on a pseudoscalar qγ q current. Corre-
5
spondingly, one might anticipate the identification of the lowest-lying scalar gluonium state with a 500–600MeV
σ (i.e. the lower-mass range of the f (400 1200)) resonance [9] whose contribution to higher order sum-rules is
0
−
suppressed by additional factors of m2 [i.e. the additional factors of m2 in (24)], a scenario analogous to the m2
σ r π
suppression of pion contributions to the pseudoscalar qγ q sum-rules [6]. As already noted in Section 1, the LET
5
constant in the absence of instantons supports this scenario for a sub-GeV scalar glueball.
However, the instanton contribution to the k = 1 sum-rule is opposite in sign and comparable in magnitude
−
to the LET subtraction constant Π(0), thereby ameliorating this term’s dominance of the lowest order sum-rule.
For example, the contribution of instanton and LET terms to the k = 1 sum rule in the dilute instanton liquid
−
(DIL) model [11],
1
dn(ρ)=n δ(ρ ρ )dρ ; n =8 10−4GeV4 , ρ = (48)
c c c c
− × 600MeV
renders trivial the remaining integrations in (46–47). If we approximate J by αG2 and employ a recently
h i
determined value of the gluon condensate [20]
(cid:10) (cid:11)
αG2 =(0.07 0.01)GeV4 (49)
±
we ob(cid:10)tain v(cid:11)ia (1) and (5) an nf =3 estimate of the LET subtraction constant:
32π
Π(0) αG2 (0.78 0.11) GeV4 . (50)
≈ 9 ≈ ±
(cid:10) (cid:11)
7
In Figure 2, we use (46) and the central value of (50) [with n and ρ given in (48)] to plot
c c
Π(0)+ inst(τ)
L−1 (51)
Π(0)
as a function τ. We note that, as anticipated, instanton effects do indeed significantly reduce the impact of the
LET on the k = 1 sum-rule: anywherefrom 20–65%for τ rangingbetween 0.6 GeV−2 and 1.0 GeV−2. Recalling
−
that the dominance of the LET over −1 is responsible for the discrepancy in gluonium mass scales in the analysis
S
of the k = 1 and k > 1 sum-rules, we see that suppression of the LET by instanton effects could reconcile this
− −
discrepancy, a possibility which is investigated further in the next section.
3 Phenomenological Impact of Instanton Effects in the Laplace Sum-
Rules
Ratios of Laplace sum-rules provide a simple technique for extracting the mass of the lightest (narrow) resonance
probed by the sum-rules. If only the lightest resonance (of mass m) is included in (23) and (24), then for the first
few sum-rules we see that
(τ,s )
S1 0 =m2 (52)
(τ,s )
0 0
S
(τ,s )
S0 0 =m2 . (53)
−1(τ,s0)+Π(0)
S
This method of predicting the mass m requires optimization of s to minimize the τ dependence that can occur in
0
the sum-rule ratios. However, a qualitative analysis which avoids these optimization issues occurs in the s
0
→ ∞
limit where bounds on the mass m can also be obtained. These bounds originate from inequalities satisfied on the
hadronic physics side of the sum-rule because of the positivity of ρhad(t). For example,
s0 s0 s0
1 1 1
dtte−tτρhad(t)= dt (t s +s )e−tτρhad(t) s dte−tτρhad(t)
0 0 0
π π − ≤ π
Z Z Z
t0 t0 t0
= (τ,s ) s (τ,s ) . (54)
1 0 0 0 0
⇒S ≤ S
Furthermore, positivity of ImΠQCD(t) leads to an inequality for the continuum.
∞ ∞ ∞
1 1 1
dtte−tτImΠQCD(t)= dt (t s +s )e−tτImΠQCD(t) s dte−tτImΠQCD(t)
0 0 0
π π − ≥ π
Z Z Z
s0 s0 s0
= c (τ,s ) s c (τ,s ) (55)
1 0 0 0 0
⇒ ≥
These inequalities can be extended to include the k = 1 sum-rules and continuum.
−
0(τ,s0) s0[ −1(τ,s0)+Π(0)] (56)
S ≤ S
c0(τ,s0) s0c−1(τ,s0) (57)
≥
We then see that
L1(τ) = S1(τ,s0)+c1(τ,s0) = S1(τ,s0) 1+ Sc11((ττ,,ss00)) S1(τ,s0) =m2 (58)
L0(τ) S0(τ,s0)+c0(τ,s0) S0(τ,s0)1+ Sc00((ττ,,ss00))≥ S0(τ,s0)
 
where the final inequality of (58) follows from c / s c / c / via (55) and (54). Similarly, we find from
1 1 0 0 1 0 0
S ≥ S ≥ S
(56) and (57) that
(τ) (τ,s )+c (τ,s ) (τ,s )
L0 = S0 0 0 0 S0 0 =m2 . (59)
−1(τ)+Π(0) −1(τ,s0)+Π(0)+c−1(τ,s0) ≥ −1(τ,s0)+Π(0)
L S S
8
Thus the ratios of the s limit of the sum-rules provide bounds on the mass in this single narrow resonance
0
→∞
approximation. Extending the analysis to many narrow resonances alters (52–53) so that the sum-rule ratios are
anupper boundonthe lightestresonance,upholding the bounds (58–59)onthe mass m2 ofthe lightestresonance.
(τ)
L1 m2 (60)
(τ) ≥
0
L
(τ)
L0 m2 (61)
−1(τ)+Π(0) ≥
L
Thesum-ruleboundsin(60–61)cannowbeemployedtodeterminethephenomenologicalimpactoftheinstanton
contributions on the sum-rule estimates of the lightest gluonium state, and to assess whether the suppression of
the LET contribution by the instanton effects is sufficient to reduce the discrepancy between sum-rule analyses
containing or omitting the k = 1 sum-rule. Collecting results from equations (27,38–40,46,47) the first few
−
sum-rules (τ) are
k
L
1 π2 τ2
L−1(τ)= τ2 −a0+a1(−2+2γE)+a2 2 +6γE −3γE2 +(−b0+b1γE)hJi−c0τhO6i−d0 2 hO8i
(cid:20) (cid:18) (cid:19)(cid:21)
64π2 dn(ρ) ae−a (1+a)aK (a)+(2+2a+a2)K (a) (62)
0 1
−
Z
1 (cid:2) b (cid:3)
(τ)= 2a +a ( 6+4γ )+a π2 6+18γ 6γ2 1 J +c +d τ
L0 τ3 − 0 1 − E 2 − E − E − τ h i 0hO6i 0 hO8i
(cid:2) a4e−a (cid:0) (cid:1)(cid:3)
+128π2 dn(ρ) [2aK (a)+(1+2a)K (a)] (63)
ρ2 0 1
Z
1 b
(τ)= 6a +a ( 22+12γ )+a 3π2 36+66γ 18γ2 1 J d
L1 τ4 − 0 1 − E 2 − E − E − τ2 h i− 0hO8i
(cid:2) a5e−a (cid:0) (cid:1)(cid:3)
+256π2 dn(ρ) (9 4a)aK (a)+(3+7a 4a2)K (a) . (64)
ρ4 − 0 − 1
Z
(cid:2) (cid:3)
Renormalization-groupimprovement has been achieved by setting ν2 =1/τ in the correlation function and in the
(three-loop, n =3, MS) running coupling α:
f
α (ν) 1 β¯ logL 1
s = 1 + β¯2 log2L logL 1 +β¯ (65)
π β0L − (β0L)2 (β0L)3 1 − − 2
ν2 (cid:2)β (cid:0) 9 (cid:1) (cid:3) 3863
L=log , β¯ = i , β = , β =4 , β = (66)
Λ2 i β 0 4 1 2 384
(cid:18) (cid:19) 0
with Λ 300MeV for three active flavours,consistent with current estimates of α (M ) [21, 22] and matching
MS ≈ s τ
conditions through the charm threshold [23].
The nonperturbative QCD parameters are needed for further analysis of the sum-rules. We employ the DIL
model[11]parameterssummarizedin(48),aswellasvacuumsaturationforthedimension-8gluoncondensate[5,24]
=14 αf Ga Gb 2 αf Ga Gb 2 = 9 αG2 2 (67)
hO8i abc µρ νρ − abc µν ρλ 16
D(cid:0) (cid:1) E D(cid:0) (cid:1) E (cid:0)(cid:10) (cid:11)(cid:1)
and instanton estimates of the dimension-six condensate [1, 4]
= gf Ga Gb Gc = 0.27GeV2 αG2 . (68)
hO6i abc µν νρ ρµ
Finally,again(cid:10)usingtheapproxim(cid:11)ati(cid:0)on J = α(cid:1)G(cid:10)2 an(cid:11)dthecentralgluoncondensatevalue[see(49)]fromreference
h i
[20],wefindthattheroleofinstantoncontributionstothesum-rules(62–64)isasillustratedinFigures3,4,and5.
(cid:10) (cid:11)
Inparticular,we see thatthe instantoncontributionsdiminish −1(τ)+Π(0). The LETtermΠ(0), whichleadsto
L
theasymptoticflatteningof −1(τ)+Π(0)atavaluesubstantiallydifferentfromzerowheninstantonsareabsent,is
L
9
clearlysuppressedbyinstantoneffectsinthelargeτ region. Asnotedearlier,suchflatteningovertheτ 1.0GeV−2
≤
regionwouldbe indicativevia (23) ofasub-GeVlowest-lyingresonance(i.e., m2τ 1 overthe physicallyrelevant
r ≪
region of τ) Instanton effects no only undo this flattening, but also increase (τ) and alter the shape of (τ).
0 1
L L
The corresponding effects of instantons on the sum-rule ratios (60–61) is shown in Figures 6 and 7. As expected
from the instanton’s impact of lowering −1(τ) and elevating 0(τ), the ratio 0(τ)/[ −1(τ)+Π(0)] is increased
L L L L
substantiallybyinclusionofinstantoneffects,increasingthecorrespondingupperboundonthemassofthelightest
gluonium state. Instanton effects also serve to lower the ratio (τ)/ (τ), decreasing the corresponding upper
1 0
L L
bound on the mass of the lightest gluonium state.
Figures 8 and 9 summarize the ratio (mass bound) analysis in the presence and in the absence of instanton
effects. Itisevidentthatinstantoneffectsleadtoasubstantialincreaseinthemassboundonthelightestgluonium
state, but other importantfeatures emerge. For example, the instanton suppressionof the LET term Π(0) reduces
the discrepancy between the ratios including or omitting the k = 1 sum-rule. Furthermore, a τ-minimum
−
stability plateau crucial for establishing a credible upper mass bound is seen to occur at reasonable energy scales
(1/√τ 1.0GeV) only when instanton effects are included. The ratios with instanton effects included (see Figure
≤
8) are remarkably flat, suggesting that the mass bounds could be close to the mass prediction that would be
obtained from a full sum-rule analysis incorporating the QCD continuum (i.e. s < ) in the phenomenological
0
∞
model.
4 Discussion
We have calculated the instanton contribution to the Laplace sum-rules of scalar gluonium and demonstrated
explicitly how,forthe lowestorderk = 1sum-rule,this instantoncontributioncancelspartofthedominantLET
−
constant.
AsnotedintheIntroduction,adiscrepancybetweenthelowestlyingstatesevidentfromthelowestandfromthe
next-to-lowestLaplacesum-rulesmaybeindicativeoftwodistinctstates. Suchisfoundtobethecase,forexample,
in the pseudoscalar channel in which the pion dominates the leading (k = 0) Laplace sum rule, but the Π(1300)
resonanceis found to dominate the next-to-leading (k =1)Laplace sum rule, because of a mass-suppressionof the
lowest-lying(pion)stateinthelattersumrule[6]. Moreover,analysesofthescalargluoniumchannelintheabsence
of explicit instanton contributions seem to exhibit a similar discrepancy between leading k = 1 and non-leading
−
k > 1 sum-rules [3, 5, 8].
−
Prior QCD sum-rule analyses of the instanton contribution to the scalar gluonium channel have focused either
on the k = 1 sum rule exclusively [10] or the LET-insensitive k > 1 sum-rules [12]. Although these two
− −
analyses (which are separatedby almosttwo decades) are both indicative of a lowest lying-resonancemass near or
above 1.4GeV, their input content (parameter values and levels of perturbation theory) are necessarily different,
suggesting the need for a single consistent treatment of leading k = 1 and non-leading k > 1 Laplace sum-rules
− −
inthe scalargluoniumchannelthat isinclusive ofinstantoneffects. We haveshownherethatcarefulconsideration
of the contribution arising from instantons within the k = 1 sum rule in this channel leads to consistency with
−
higher sum-rules in the estimation of lowest-lying resonance masses in the scalar gluonium channel.3
Correspondence with the prior treatment of the k = 1 sum rule [10] can be obtained by examining the
−
instanton contribution (46) to this sum rule in the high-energy limit of small τ. This contribution is obtained
3NotealsothatwehaveincorporatedthesignificantNNLOperturbativecorrections,asopposedtoLOcorrectionsin[10]andNLO
correctionsin[12].
10