Direct instantons and nucleon magnetic moments

Abstract

We calculate the leading direct-instanton contributions to the operator product expansion of the nucleon correlator in a magnetic background field and set up improved QCD sum rules for the nucleon magnetic moments. Remarkably, the instanton contributions are found to affect only those sum rules which had previously been considered unstable. The new sum rules show good stability and reproduce the experimental values of the nucleon magnetic moments with values of χ, the quark condensate magnetic susceptibility, consistent with other estimates.

Full text

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arXiv:hep-ph/9902458v1 24 Feb 1999
Direct Instantons and Nucleon Magnetic Moments∗
M. Aw†and M.K. Banerjee
Department of Physics, University of Maryland, College Park, MD 20742-4111
H. Forkel
Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg, D-69120 Heidelberg, Germany
(February 1, 2008)
Abstract
We calculate the leading direct-instanton contributions to the operator prod-
uct expansion of the nucleon correlator in a magnetic background field and set
up improved QCD sum rules for the nucleon magnetic moments. Remarkably,
the instanton contributions are found to affect only those sum rules which had
previously been considered unstable. The new sum rules show good stabil-
ity and reproduce the experimental values of the nucleon magnetic moments
with values of χ, the quark condensate magnetic susceptibility, consistent with
other estimates.
Typeset using REVTEX
∗DOE/ER/40762-176, UMD PP# 99-088
†Present address: Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213.
E–mail: mountaga@cmu.edu
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For over two decades QCD instantons have been associated with fundamental aspects of
strong interaction physics, as, for example, with the θ-vacuum [1], the issue of strong CP
violation [1], and the anomalously large η′mass [2]. Unequivocal and quantitative evidence
for their role in hadron structure, however, turned out to be much harder to establish. This
is mainly due to the complexity involved in dealing with interacting instanton ensembles
and their coupling to other vacuum fields over large distances.
Instanton vacuum models [3] attack the first part of this problem directly, by approx-
imating the field content of the vacuum as a superposition of solely instantons and anti-
instantons. This approach has been developed for more than a decade and can describe an
impressive amount of hadron phenomenology [3]. More recently, QCD lattice simulations
began to complement such vacuum models by isolating instantons in equilibrated lattice
configurations and by studying their size distribution and their impact on hadron correla-
tors [4]. While the results obtained with different, currently developed lattice techniques
have not yet reached quantitative agreement, they do confirm the overall importance of
instantons and some bulk properties of their distribution in the vacuum.
Another approach towards linking the instanton component of the vacuum to hadron
properties has been developed over the last years by generalizing the nonperturbative op-
erator product expansion (OPE) and QCD sum rule techniques [5–7]. While its range of
applicability is more limited than that of instanton vacuum models and of lattice calcu-
lations, it avoids the need for large-scale computer simulations and takes, in contrast to
instanton models, all long-wavelength vacuum fields and also perturbative fluctuations into
account. Furthermore, the approach is largely model-independent and allows the study of
instanton effects in a fully analytic and, therefore, rather transparent fashion.
In the present paper, we adapt this approach to properties which characterize the re-
sponse of the hadronic system to a weak external field. Specifically, we calculate the direct-
instanton contributions to the nucleon correlator in the presence of a constant electromag-
netic field. With the help of background-field sum-rule techniques due to Ioffe and Smilga
[8] and Balitsky and Yung [9], we then study previously neglected instanton effects in the
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QCD sum rules for the magnetic moments of the nucleon.
As already mentioned, our work is based on the nucleon correlation function
i
?
d4xeipx?0|Tη(x)¯ η(0)|0?F= Π0(p) +
√4παΠµν(p)Fµν,(1)
in the background of a constant electromagnetic field Fµν. The interpolating fields η(x) with
proton or neutron quantum numbers [10] are composite operators of massless up and down
quark fields:
ηp(x) = [uaT(x)Cγαub(x)]γ5γαdc(x)ǫabc,ηn= ηp(u ↔ d).(2)
For the application in QCD sum rules we need a theoretical description of the correlator (1)
at momenta s = −p2≃ 1GeV2, i.e. at distances x<
The information on the magnetic moments is contained in the second term on the right-
∼0.2fm.
hand side of Eq. (1). It characterizes the linear response of the nucleon to the external field
and can be decomposed into three independent Lorentz and spinor structures:
Πµν(p) = (/ pσµν+ σµν/ p)Π1(p2) + i(γµpν− γνpµ)/ p Π2(p2) + σµνΠ3(p2).(3)
Note that the invariant amplitude Π1corresponds to the chirally-even part of the correlator,
while Π2and Π3are associated with the chirally-odd part.
The nonperturbative OPE [11,8] of the above correlator at small distances can be gener-
ated by splitting each diagram contributing to (3) in all possible ways into a hard and a soft
subgraph. The hard subgraphs contribute to the Wilson coefficients and are usually calcu-
lated perturbatively, with the integration range of each internal momentum restricted1to be
larger than the OPE scale µ ∼ 0.5GeV. The soft subgraphs correspond to hadron-channel
independent condensates, renormalized at µ. In the presence of an external electromagnetic
field the OPE (up to eight-dimensional operators) involves the additional, Lorentz-covariant
condensates
1In practice, this restriction is often unnecessary (see below).
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?0|¯ qσµνq|0? =
√4παχFµν?0|¯ qq|0?,
√4πακFµν?0|¯ qq|0?,
√4παξFµν?0|¯ qq|0?.
(4)
g?0|¯ qGµνq|0? =
g?0|¯ qγ5˜Gµνq|0? =i
(5)
2
(6)
(Gµν=1
2λaGa
µν,˜Gµν=
1
2εµνρσGρσwith ε0123= −1.) The parameters χ,κ, and ξ play the
role of generalized susceptibilities and quantify the vacuum response to weak electromagnetic
fields. The magnetic susceptibility of the quark condensate, χ, for example, originates from
the induced spin alignment of quark-antiquark pairs in the vacuum. Note also that χ is
associated with the lowest-dimensional induced condensate, which enhances its role in the
OPE and the corresponding sum rules.
The OPE of Πµν(p) up to operators of dimension eight, with perturbatively calculated
Wilson coefficients, has been obtained in Ref. [8]. An inherent assumption of this calcu-
lation - and of the QCD sum rule program in general - is that the short-distance physics
associated with fields of wavelength smaller than µ−1is predominantly perturbative. It is
well known, however, that also strong nonperturbative fields of rather small size exist in the
QCD vacuum. Instantons, i.e. the finite-action solutions of the classical, Euclidean Yang-
Mills equation [12], are paradigmatic examples of such fields and have a crucial impact on
the vacuum structure.
The nonperturbative contributions from short-wavelength fluctuations of quarks and
gluons around instantons are thus neglected in the standard treatment of the OPE. Their
relative importance, and hence the justification for approximately disregarding them, de-
pends both on the instanton size distribution in the vacuum and on the quantum numbers
of the hadronic channel under consideration. Instantons of smaller (average) size ¯ ρ are ac-
companied by fluctuations of smaller wavelength, and those contribute more strongly to the
Wilson coefficients. The hadron-channel dependence of the instanton contributions orig-
inates mainly from the chirality and spin-color coupling of the quark zero-modes in the
instanton background.
Instanton-induced effects are particularly large in the pseudoscalar-isovector and scalar-
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isoscalar channels, because there the quark zero-modes contribute with maximal strength.
As a consequence, instanton contributions dominate already at short distances in the pseu-
doscalar sum rules, and are essential for their stability [6]. In the vector and axial-vector
channels, on the other hand, zero-mode contributions are, to leading order in the instanton
density, absent.
The strength of instanton contributions to the nucleon channel lies about halfway in-
between these extreme cases. In the chirally-odd amplitudes, it was found to be roughly of
the same magnitude as that of the condensate contributions [5,7], since the spin-0 diquark
operators in the interpolating fields (2) couple strongly to instantons. The nucleon channel
is therefore well suited for studying the interplay between instantons and other vacuum fields
[7].
The calculation of the leading direct-instanton contributions to the background-field
correlator (1) proceeds essentially along the lines described in Refs. [5–7], to which we
refer for more details. Similar to the condensates, the bulk properties of the instanton size
distribution are generated by long-distance vacuum dynamics and have thus to be taken as
input for this calculation. As before [5–7], we will use the standard values [13] ¯ ρ ≃
for the average instanton size and¯R ≃ 1fm for the average separation between neighboring
(anti)instantons. The results of instanton vacuum models [3,14] confirm these scales, while
1
3fm
those from the lattice [4] are not yet fully consistent but lie in the same range (with maximal
deviations of about 50 %).
Since the average instanton size is of the order of the inverse OPE scale, ¯ ρ < µ−1≃
0.4fm, instanton corrections to the Wilson coefficients can be substantial. Moreover, since
¯ ρ−1≫ ΛQCD, these corrections are essentially semiclassical, and at the relevant distances x<
0.2fm ≪¯R multi-instanton correlations should be negligible. The instanton contributions
to the OPE coefficients can therefore be calculated in semiclassical approximation, i.e. by
∼
evaluating the correlator (1) in the background of the instanton and anti-instanton field
and by then averaging the instanton parameters over their vacuum distributions. (The
distribution of the instanton’s position and color orientation is uniform, due to translational
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