QCD sum-rules are related to an integral of a hadronic spectral function, and
hence must satisfy integral inequalities which follow from positivity of the
spectral function. Development of these Holder inequalities and their
application to the Laplace sum-rule for pions lead to a lower bound on the
average of the non-strange 2 GeV light-quark masses in the MS-bar scheme.Comment: latex2e, 8 pages. Write-up of talk presented at MRST 200

Steele, T. G.

English

arXiv

T. G. Steele

Crossref

QCD sum-rule bounds on the light quark masses

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QCD Sum-Rule Bounds on the Light Quark Masses
T.G. Steele∗
Department of Physics and Engineering Physics
University of Saskatchewan
Saskatoon, Saskatchewan, S7N 5E2
Canada
November 8, 2018
Abstract
QCD sum-rules are related to an integral of a hadronic spectral function, and hence must
satisfy integral inequalities which follow from positivity of the spectral function. Development
of these Hölder inequalities and their application to the Laplace sum-rule for pions lead to a
lower bound on the average of the non-strange 2GeV light-quark masses in the MS scheme.
The light quark masses are fundamental parameters of QCD, and determination of their values is
of importance for high-precision QCD phenomenology and lattice simulations involving dynamical
quarks. In this paper the development of Hölder inequalities for QCD Laplace sum-rules [1] is
briefly reviewed. These techniques are then used to obtain bounds on the non-strange (current)
quark masses mn = (mu +md) /2 evaluated at 2GeV in the MS scheme, updating and extending
the Hölder inequality results of ref. [2].
Although it is possible to obtain quark mass ratios in various contexts [3], the only methods
which have been able to determine the absolute non-strange quark mass scales are the lattice (see
[4] for recent results with two dynamical flavours) and QCD sum-rules [2, 5, 6, 7, 8, 9].1
In sum-rule and lattice approaches, the pseudoscalar or scalar channels are used since they have
the strongest dependence on the quark masses. This is exemplified by the correlation function
Π5 (Q
2) of renormalization-group (RG) invariant pseudoscalar currents with quantum numbers of
the pion:
Π5
(
Q2
)
= i
∫
d4x eiq·x 〈O|T [J5(x)J5(0)] |O〉 (1)
J5(x) =
1√
2
(mu +md)
[
ū(x)iγ5u(x)− d̄(x)iγ5d(x)
]
. (2)
∗email: Tom.Steele@usask.ca
1An overview of selected lattice and sum-rule results for both non-strange and strange masses can be found in
[10].
1
The Laplace sum-rule is obtained by by applying the Borel transform operator [11] B̂
B̂ ≡ lim
N, Q2→∞
Q2/N≡M2
(−Q2)N
Γ(N)
(
d
dQ2
)N
(3)
to the dispersion relation for Π5 (Q
2)
Π5
(
Q2
)
= a + bQ2 +
Q4
π
∞
∫
4m2π
ρ
5
(t)
t2 (t+Q2)
dt , (4)
where ρ
5
(t) is the hadronic spectral function appropriate to the pion quantum numbers, and the
quantities a and b represent subtraction constants. The resulting Laplace sum-rule relating the
theoretically-determined quantity
R5
(
M2
)
= M2B̂
[
Π5
(
Q2
)]
(5)
to phenomenology is
R5
(
M2
)
=
1
π
∞
∫
4m2π
ρ
5
(t) exp
(
− t
M2
)
dt . (6)
Perturbative contributions to R5 (M
2) are known up to four-loop order in the MS [5, 12].
Infinite correlation-length vacuum effects in R5 (M
2) are represented by the (non-perturbative)
QCD condensate contributions [5, 11, 13]. In addition to the QCD condensate contributions the
pseudoscalar (and scalar) correlation functions are sensitive to finite correlation-length vacuum
effects described by direct instantons [14] in the instanton liquid model [15]. Combining all these
results, the total result for R5 (M
2) to leading order in the light-quark masses is [2]
R5
(
M2
)
=
3m2nM
4
8π2
(
1 + 4.821098
α
π
+ 21.97646
(α
π
)2
+ 53.14179
(α
π
)3
)
+m2n
(
−〈mq̄q〉+ 1
8π
〈αG2〉+ π〈O6〉
4M2
)
+m2n
3ρ2cM
6
8π2
e−ρ
2
cM
2/2
[
K0
(
ρ2cM
2/2
)
+K1
(
ρ2cM
2/2
)]
, (7)
where α and mn = (mu +md) /2 are the MS running coupling and quark masses at the scale M ,
and ρc = 1/(600MeV) represents the instanton size in the instanton liquid model [15]. SU(2)
symmetry has been used for the dimension-four quark condensates (i.e. (mu +md)〈ūu〉+ d̄d〉) ≡
4m〈q̄q〉), and 〈O6〉 denotes the dimension six quark condensates
〈O6〉 ≡ αs
[
(2〈ūσµνγ5T auūσµνγ5T au〉+ u → d)− 4〈ūσµνγ5T aud̄σµνγ5T ad〉
+
2
3
〈
(
ūγµT
au+ d̄γµT
ad
)
∑
u,d,s
q̄γµT aq〉
]
(8)
2
The vacuum saturation hypothesis [11] will be used as a reference value for 〈O6〉
〈O6〉 = fvs
448
27
α〈q̄qq̄q〉 = fvs3× 10−3GeV6 (9)
where fvs = 1 for exact vacuum saturation. Larger values of effective dimension-six operators
found in [16, 17] imply that fvs could be as large as fvs = 2. The quark condensate is determined
by the GMOR (PCAC) relation
(mu +md) 〈ūu+ d̄d〉 = 4m〈q̄q〉 = −2f 2πm2π (10)
where fπ = 93MeV. A recent determination of the gluon condensate 〈αG2〉 will be used: [18]
〈αG2〉 = (0.07± 0.01) GeV4 . (11)
However, it should be noted that there is some discrepancy between [18] and the smaller value
〈αG2〉 = (0.047± 0.014) GeV4 found in [17].
Note that all the theoretical contributions in (7) are proportional to m2n, demonstrating that
the quark mass sets the scale of the pseudoscalar channel. This dependence on the quark mass
can be singled out as follows:
R5
(
M2
)
= [mn(M)]
2 G5
(
M2
)
(12)
where G5 is independent of mn and is trivially extractable from (7). Higher-loop perturbative
contributions in (7) are thus significant since they can effectively enhance the quark mass with
increasing loop order.
Determinations of the non-strange quark mass mn using the sum-rule (6) require input of a
phenomenological model for the spectral function ρ
5
(t). The mass mn can then be determined
by fitting to find the best agreement between the phenomenological model and the theoretical
prediction respectively appearing on the right- and left-hand sides of (6). For example, the simple
resonance(s) plus continuum model
1
π
ρ
5
(t) = 2f 2πm
4
π
[
δ
(
t−m2π
)
+
F 2
Π
M4
Π
f 2πm
4
π
δ
(
t−M2
Π
)
]
+Θ (t− s0)
1
π
ρQCD(t) (13)
represents the pion pole (mπ), a narrow-width approximation to the pion excitation (MΠ) such
as the Π(1300), and a QCD continuum above the continuum threshold t = s0. Of course more
detailed phenomenological models can be considered which take into account possible width effects
for the pion excitation, further resonances, resonance(s) enhancement of the 3π continuum etc.
This leads to significant model dependence which partially accounts for the spread of theoretical
estimates in [8, 9]. Since the common phenomenological portion of all these models is the pion
pole, it is valuable to extract quark mass bounds which only rely upon the input of the pion pole
on the phenomenological side of (6).
The existence of such bounds is easily seen by separating the pion pole out from ρ
5
(t), in which
case (6) becomes
[mn(M)]
2G5
(
M2
)
= 2f 2πm
4
π +
1
π
∞
∫
9m2π
ρ
5
(t) exp
(
− t
M2
)
dt . (14)
3
Since ρ
5
(t) ≥ 0 in the integral appearing on the right-hand side of (14), a bound on the quark
mass is obtained:
mn(M) ≥
√
2f 2πm
4
π
G5 (M2)
. (15)
Analysis of these bounds following from simple positivity of the “residual” portion on the right-
hand side of (14) was studied in [5, 6].
Improvements upon the positivity bound of (15) are achieved by developing more stringent
inequalities based on the positivity of ρ
5
(t). Since ρ
5
(t) ≥ 0, the right-hand (phenomenological)
side of (6) must satisfy integral inequalities over a measure dµ = ρ
5
(t) dt. In particular, Hölder’s
inequality over a measure dµ is
∣
∣
∣
∣
∫ t2
t1
f(t)g(t)dµ
∣
∣
∣
∣
≤
(
∫ t2
t1
∣
∣f(t)
∣
∣
p
dµ
)
1
p
(
∫ t2
t1
∣
∣g(t)
∣
∣
q
dµ
)
1
q
,
1
p
+
1
q
= 1 ; p, q ≥ 1 , (16)
which for p = q = 2 reduces to the familiar Schwarz inequality, implying that the Hölder inequality
is a more general constraint. The Hölder inequality can be applied to Laplace sum-rules by
identifying dµ = ρ
5
(t) dt, τ = 1/M2 and defining
S5 (τ) =
1
π
∞
∫
µth
ρ
5
(t)e−tτ dt (17)
where µth will later be identified with 9m
2
π. Suitable choices of f(t) and g(t) in the Hölder
inequality (16) yield the following inequality for S5(t) [1]:
S5 (τ + (1− ω)δτ) ≤ [S5 (τ)]ω [S5 (τ + δτ)]1−ω , ∀ 0 ≤ ω ≤ 1 . (18)
In practical applications of this inequality, δτ ≤ 0.1GeV−2 is used, in which case this inequality
analysis becomes local (depending only on the Borel scale M and not on δτ) [1, 2].
To employ the Hölder inequality (18) we separate out the pion pole by setting µth = 9m
2
π in
(17).
S5
(
M2
)
= R5
(
M2
)
− 2f 2πm4π =
∞
∫
9m2π
ρ
5
(t)e−tτ dt (19)
which has a right-hand side in the standard form (17) for applying the Hölder inequality. Note
that simple positivity of ρ
5
(t) gives the inequality
S5
(
M2
)
≥ 0 (20)
which simply rephrases (15). Lower bounds on the quark mass mn can now be obtained by finding
the minimum value of mn for which the Hölder inequality (18) is satisfied. Introducing further
4
phenomenological contributions (e.g. three-pion continuum) give a slightly larger mass bound as
will be discussed later. However, if only the pion pole is separated out, then the analysis is not
subject to uncertainties introduced by the phenomenological model.
Although the details are still a matter of dispute, the overall validity of QCD predictions at
the tau mass is evidenced by the analysis of the tau hadronic width, hadronic contributions to
αEM (MZ) and the muon anomalous magnetic moment [19], so we impose the inequality (18) at
the tau mass scale M = Mτ = 1.77GeV. This also has the advantage of minimizing perturbative
uncertainties in the running of α and mn, since the the PDG reference scale for the light-quark
masses is at 2GeV [20], in close proximity to Mτ , and the result αs (Mτ ) = 0.33 ± 0.02 [21]
can thus be used to its maximum advantage. For the remaining small energy range in which
the running of α and mn is needed, the four-loop β-function [22] and four-loop anomalous mass
dimension [23] with three active flavours are used, appropriate to the analysis of [21]. This use of
the 2GeV reference scale for mn combined with input of α (Mτ ) improves upon the perturbative
uncertainties in [2] which employed α (MZ) and a 1GeV mn reference scale which necessitated
matching through the (uncertain) b and c flavour thresholds.
Further theoretical uncertainties devolve from the QCD condensates as given in (11) and
(9) with 1 ≤ fvs ≤ 2, along with a 15% uncertainty in the instanton liquid parameter ρc [15].
The effect of higher-loop perturbative contributions to R5 (M
2) on the resulting mn bounds is
estimated using an asymptotically-improved Padé estimate [24] of the five-loop term, introducing
a 138 (α/π)4 correction into (7). Finally, we allow for the possibility that the overall scale of the
instanton is 50% uncertain.
The resulting Hölder inequality bound on the 2.0GeV MS quark masses, updating the analysis
of [2], is
mn(2GeV) =
1
2
[mu(2GeV) +md(2GeV)] ≥ 2.1MeV (21)
This final result is identical to previous bounds on mn(1GeV) [2] after conversion to 2GeV by
the PDG [20], indicative of the consistency of perturbative inputs used in the two analyses. The
theoretical uncertainties in the quark mass bound (21) from the QCD parameters and (estimated)
higher-order perturbative effects are less than 10%, and the result (21) is the absolute lowest
bound resulting from the uncertainty analysis. The dominant sources of uncertainty are α (Mτ )
and potential higher-loop corrections. The instanton size ρc is the major source of non-perturbative
uncertainty, but its effect is smaller than the perturbative sources of uncertainty.
Compared with the positivity inequality (20), as first used to obtain quark mass bounds from
QCD sum-rules [5, 6], the Hölder inequality leads to quark mass bounds 50% larger for identical
theoretical and phenomenological inputs at M = Mτ , demonstrating that the Hölder inequality
provides stringent constraints on the quark mass.
Finally, we discuss the effects of extending the resonance model to include the 3π continuum
5
calculated using lowest-order chiral perturbation theory [9]
1
π
ρ
5
(t) = 2f 2πm
4
π
[
δ
(
t−m2π
)
+Θ
(
t− 9m2π
)
ρ3π(t)
t
18 (16π2f 2π)
2
]
(22)
ρ3π(t) =
(
√
t−mπ)
2
∫
4m2π
du
t
√
λ
(
1,
u
t
,
m2π
t
)
√
1− 4m
2
π
u
{
5+
+
1
2 (t−m2π)2
[
4
3
(
t− 3
(
u−m2π
))2
+
8
3
λ
(
t, u,m2π
)
(
1− 4m
2
π
u
)
+ 10m4π
]
+
1
t−m2π
[
3
(
u−m2π
)
− t+ 10m2π
]
}
(23)
λ(x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2xz (24)
which becomes ρ3π(t) → 3 in the limit mπ → 0 . Inclusion of the 3π continuum (23) is still likely to
underestimate the total spectral function since more complicated models of the spectral function
involve resonance enhancement of this 3π continuum [9]. If this limiting form is used up to a cutoff
of 1GeV, then the resulting Hölder inequality quark mass bounds are raised by approximately
10%, and a 14% effect is observed if the cutoff is moved to infinity.2 Working with the full form
(23) complicates the numerical analysis, but the following simple form (with t in GeV units) is
easily verified to be a bound on the 3π continuum in the region below 1GeV.
ρ3π(t) ≥
4
3
[
(√
t−mπ
)2
− 4m2π
]
(25)
This approximate form of the 3π continuum again raises the resulting quark mass bounds by
approximately 10%.
This paper is dedicated to the memory of Roger Migneron. Many thanks to Vic Elias, Gerry
McKeon, and Voldoya Miransky for their efforts in organizing MRST 2001, which resulted in an
enjoyable and interesting conference.
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8


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