View source for Shifman-Vainshtein-Zakharov sum rules

[[Image:SVZ_erice3.png|thumb|300px|right|three|When the separation between the probe color charges exceeds critical the nonlinearity becomes so strong that it makes no sense to speak about individual gluons. The color fields between the probe charges form a flux tube with transverse dimensions <math>\sim \Lambda^{-1}\ .</math>]]

The <strong><nowiki>Shifman-Vainshtein-Zakharov (SVZ) sum rules</nowiki></strong>
relate hadronic parameters, such as meson masses and coupling constants,
baryon magnetic moments, etc., to a few characteristics of the vacuum of quantum chromodynamics (QCD): gluon and quark condensates. The method is based on
Wilson's operator product expansion (OPE) which was adapted by the authors to QCD in the mid-1970s. Quark confinement is assumed rather than proved.
The SVZ method was applied,
with remarkable success, for (approximate) calculations of a large variety
of properties of all low-lying hadronic states.
In the 1980s the SVZ method was developed in various directions
shifting the emphasis from calculation of masses and
coupling constants of "classic" resonances
to such problems as magnetic moments, form
factors at intermediate momentum transfer, weak
decays, structure functions of deep inelastic
scattering at intermediate <math>x\ ,</math> heavy quarkonium systems,
and many others. At the initial stages the method was
essentially unchallenged, it had no competitors.
Crucial elements of the method laid the foundation or were incorporated in heavy quark theory, light-cone sum rules and the
recently developed holographic (AdS/QCD correspondence) framework.
Qualitative insights into the vacuum structure provided by the SVZ
method gave rise to instanton liquid and other models of the QCD vacuum.

== Introduction ==

The basic microscopic degrees of freedom of quantum
chromodynamics (QCD) are quarks and gluons. Their interaction is
described by the Lagrangian
:<math>\label{qcdlag}
\mathcal{L} = -\frac{1}{4} G^a_{\mu\nu}G^a_{\mu\nu}
+\sum_f \bar q i\not\!\!{D}q
</math>

where <math>G^a_{\mu\nu}</math> is the gluon field strength tensor,
<math>f</math> stands for flavor, and the color indices of the quark fields
<math>q</math> are suppressed. (For simplicity the quark mass
terms are omitted in the expression above).

Neither quarks nor gluons are asymptotic states showing up in the
physically observed spectrum. Experimentally
observed are hadrons — color-singlet composite states "built" of quarks and gluons.
This phenomenon, the most salient feature of non-Abelian Yang-Mills theory,
is known as quark (or color) confinement.
If one takes a heavy probe quark and an antiquark separated by a
large distance, the potential energy between them grows linearly with distance,
leading to unavoidable formation of colorless hadrons.

In order to study
hadronic properties it is convenient to start from
the empty space — the vacuum — inject there a quark-antiquark
pair, and then follow the evolution of the valence quarks injected in
the "vacuum medium." The injection is achieved by external currents.
The most popular are the vector and axial currents. Their popularity
is due to the fact that they actually exist in nature: virtual photons
and <math>W</math> bosons couple to the vector and axial quark currents.
Therefore, they are experimentally accessible in the <math>e^+e^-</math>
annihilation into hadrons or hadronic <math>\tau</math> decays.

Identification of microscopic constituents inside hadrons — quarks and gluons —
became possible because of weakness of their interactions
at short distances (large Euclidean momenta). At short distances
perturbation theory applies.
QCD explains this phenomenon by celebrated asymptotic freedom (Gross and Wilczek 1973, Politzer 1973) stating that the effective coupling constant <math>\alpha_s</math>
falls off at large (Euclidean) momenta as
:<math>\label{asymo}
\alpha_s(p^2)\approx \frac{4\pi}{b\ln (p^2/\Lambda^2)}\, ,\qquad
b=\frac{11}{3}N_c - \frac{2}{3} N_f \, ,
</math>

where <math>N_c</math> is the number of colors, <math>N_f</math> is the number of
flavors, <math>\Lambda</math> is the dynamical scale parameter of QCD,
and, finally, <math>b</math>
is the first coefficient in the Gell-Mann—Low function.
The same formula shows that the effective color
interaction becomes stronger as separation between the
color charges increases (i.e. <math>p</math> decreases).
When the distance becomes larger than some number times
<math>\Lambda^{-1}\ ,</math> and exceeds a critical one, the
"branchings" of gluons become so
intensive (<figref>SVZ_erice3.png</figref>) that it makes no sense to speak about individual
gluons.
Rather, one should phrase one's consideration in terms of the
chromoelectric and chromomagnetic fields.

In infrared-free QED, the field induced by the probe charges separated by a large
distance, is dispersed all over
space, in accordance with the Coulomb law. In QCD a specific organization of the
QCD vacuum makes such a dispersed configuration energetically
inexpedient (Nambu 1974, 't Hooft 1975, Mandelstam 1976).
Rather, the chromoelectric field between the probe color
charges squeezes itself into a sausage-like (or string-like) configuration.
The situation is reminiscent of the Meissner effect in superconductivity.
The superconducting media do not tolerate the
magnetic field. If one imposes, as an external boundary condition, a
certain flux of the magnetic field through such a medium, the
magnetic field will be squeezed into a thin tube carrying all the
magnetic flux (the Abrikosov vortices). The flux tube gives rise to a linear potential between the magnetic sources at its endpoints.

Superconductivity is
caused by condensation of the Cooper pairs — pairs of electric
charges. A phenomenologically acceptable picture of large distance
dynamics in QCD requires ''chromoelectric'' flux tubes, rather than magnetic;
hence the phenomenon is referred to as the ''dual Meissner effect''.
The dual Meissner effect was proven to take place in a (distant) relative of
QCD, <math>\,\,{\mathcal N}=2</math> supersymmetric Yang-Mills theory,
leading to linear potential and quark confinement (Seiberg and Witten 1994).
However, the spectrum and properties of
composite mesons in supersymmetric gauge theories
are different from those one observes in QCD.

== Basic idea of the SVZ method ==

There is a long way to go from the flux-tube picture (which is not yet established in QCD, 
in spite of vigorous efforts) to a quantitative framework allowing one to
''calculate'' a variety of hadronic properties for real mesons and baryons.
The basic idea lying behind the SVZ method (Shifman et al. 1979) is as follows.
In the states comprising the low-lying part of the
hadronic spectrum, both mesonic (e.g. <math>\rho</math>)
and baryonic (e.g. <math>p</math> or <math>n</math>), quarks are not asymptotically far from each other, on average. The distance between them is of the order of
<math>\Lambda^{-1}\ .</math> Under these circumstances,
the string-like long chromoelectric flux tubes have no chance to be
developed. The valence quark pair injected in the vacuum, in a sense,
perturbs it only slightly.
Then one does not need the full QCD string theory to approximately obtain
the properties of the hadronic states.
Their basic features depend on how the
valence quarks of which they are built interact with typical ''vacuum'' field fluctuations.

It was suggested (and then established) that the QCD vacuum is sufficiently characterized by
a number of condensates (Shifman 1979):
the gluon condensate <math>\langle G^2_{\mu\nu}\rangle \ ,</math> the quark condensate
<math>\langle \bar q q\rangle\ ,</math> the mixed condensate <math>\langle\bar q \sigma G q\rangle\ ,</math>
the four-quark condensate, and a few others.
Using the SVZ method one can first find the numerical values of the above
condensates and then determine, with sufficient accuracy, parameters of a large number of mesons and baryons.

Without invoking the entire infinite set of condensates one can
capture only gross features of the vacuum medium.
Correspondingly, any calculation of the hadronic parameters
based on the SVZ method is approximate in nature. The
usefulness of the method lies in its analytic capabilities:
analytic analyses become possible
in a wide range of problems from hadronic physics, including those where the answer was
not known ''a priori''. In addition to static properties of
mesons and baryons, the SVZ method was applied in problems with three-point functions,
external electromagnetic and other auxiliary fields, light-cone kinematics,
and so on. This allowed one to approximately calculate magnetic moments, form factors
at intermediate momentum transfers, weak
decays constants, structure functions of deep inelastic
scattering at intermediate <math>x\ ,</math> transition amplitudes of the <math>B\to \rho</math> type, and many other quantities.

== Wilson's OPE ==

OPE allows one to consistently separate
the large  and  short distance contributions. The former are then represented
by the vacuum condensates while the latter are accounted for in the
coefficient functions. In constructing OPE
one must introduce a somewhat artificial boundary,
<math>\mu\ ,</math> usually referred to as ''normalization point''.
All fluctuations with frequencies higher than <math>\mu</math> are supposed to
be hard and are included in the coefficient functions.
Those with frequencies lower than <math>\mu</math> are soft.
Only soft modes are to be retained in the condensates.
Thus, the separation principle of OPE is "soft versus hard."
Both, the OPE coefficients and condensates are explicitly <math>\mu</math> dependent. All
physical quantities are <math>\mu</math> independent;
the normalization point dependence of the
condensates is compensated by that of the coefficient functions.

The theoretical basis of any calculation within the SVZ method is
the operator product expansion for correlation functions of two or more
currents. Say, for two currents, in the coordinate space, one can write
:<math>\label{tee}
\Big\langle T\{ J(x)\,\,J(0)\}\Big\rangle_{x\to 0}
= \sum_n C_n(x;\mu )\langle {\mathcal O}_n(\mu )\rangle
</math>

where the sum runs over a complete set of local gauge invariant (quark-gluon) operators
with appropriate quantum numbers. The degree of locality is set by <math>\mu\ .</math>
The series on the right-hand side is infinite. The operators <math>{\mathcal O}_n(\mu )</math>
can be ordered according to their normal dimensions. The coefficient functions
<math>C_n(x;\mu )</math> are determined mainly, but not exclusively, by perturbation theory.
Without the condensates the
perturbative series cannot be consistently defined because of
ambiguities associated with the high-order terms in the perturbative series.

In many instances it is more convenient to use OPE for correlation functions in the momentum space. 
For instance, for the conserved vector current <math>\bar u\gamma^\mu d</math>
:<math>\label{Tproduct}
\Pi_{\mu\nu} = i \int {\rm e}^{iqx} d^4 x \langle 0|T\{ J_\mu (x)
J^\dagger_\nu
(0) \} |0\rangle\, ,
</math>

where <math>q</math> is the total momentum of the quark-antiquark pair
injected in the vacuum.
Due to the current conservation <math>\Pi_{\mu\nu}</math> is transversal,
:<math>\label{dfunct}
\Pi_{\mu\nu}
= (q_\mu q_\nu -q^2g_{\mu\nu})\Pi
(q^2)\,,\qquad D(Q^2) \equiv - (4\pi^2) Q^2 (d\Pi / dQ^2 )\,.
</math>

The operator product expansion takes the form
:<math>
D(Q^2) = \sum_n C_n(q;\mu )\langle {\mathcal O}_n(\mu )\rangle \,,
</math>
where, in addition to the unit operator of dimension zero, the leading
dimension-4 and 6 condensates on the right-hand side are
:<math>\label{glop}
{\mathcal O}_G = \frac{\alpha_s}{\pi} G_{\mu\nu}^aG_{\mu\nu}^a
\, ,\qquad \langle \mathcal{O}_G\rangle \sim 0.012 \,\,\mbox{GeV}^4\, ,
</math>
and
:<math>\label{op4q}
\mathcal{O}_{4q} = (\bar q_1 \Gamma_1 q_2)( \bar q_3 \Gamma_2 q_4) \,.
</math>

To the leading order in <math>1/N_c^2</math> the four-quark condensate can be factorized,
:<math>\label{vacsat}
\langle \mathcal{O}_{4q}\rangle = \langle \bar q_1 \Gamma_1
q_2\rangle
\langle \bar q_3 \Gamma_2 q_4 \rangle \, , \qquad \langle \bar q q\rangle \sim
- (250\,\,\mbox{MeV})^3
</math>

The dimension-6 operators <math>\mathcal{O}_{qG} = {ig}\, \bar q \sigma_{\mu\nu} G_{\mu\nu} q</math>
and <math>\mathcal{O}_{3G} = G^3</math> can be neglected.
The operator product expansion in its general form was engineered by
Wilson (Wilson 1969, Wilson and Kogut 1974).
Wilson's idea was adapted for the QCD environment in (Novikov et. al. 1985).

== How OPE cures convergence problems of the perturbative series ==

As know from quantum mechanics (the same is true in field theory), perturbative series are typically factorially divergent
(Gross and Neveu, 1974;
Lautrup, 1977;
't Hooft, 1977;
Parisi, 1978, 1979;
David,  1984, 1986;
Mueller, 1985;
Beneke, 1999). 
These divergences can be Borel-summable (e.g. ultraviolet renormalons) or non-summable,
as in the case of the infrared renormalons which are responsible for the leading singularity
in the Borel plane. In some theories factorial divergence of the perturbative series 
can be associated with the existence of instantons or similar quasiclassical field configurations.
Typically the corresponding singularities in the Borel plane are less important.
The renormalon example is most transparent and instructive for understanding how 
introduction of the normalization point <math>\mu</math> and appropriate condensates
solves the problem (Shifman 2013).

The infrared renormalon is obtained by collecting all  bubble 
insertions in the gluon propagator attached to the fermion lines in the basic fermion loop in
\eqref{Tproduct},
:<math>\label{ren-1} 
D \propto \,\, Q^2 \int
dk^2\frac{k^2\alpha_s(k^2)}{(k^2+Q^2)^3} \,.
</math>
The integrand in the expression above is not exact; rather, it represents an interpolation function
with correct asymptotics at small and large <math>k^2</math>, which is sufficient for 
explanation. The running gauge coupling constant  <math>\alpha_s (k^2)</math> is
:<math>\label{alph} 
\alpha_s (k^2) = \frac{\alpha_s (Q^2)}{1-\frac{\beta_0\alpha_s (Q^2)}{4\pi} 
\ln (Q^2/k^2)} </math>
where <math>\beta_0</math> is the first coefficient of the beta function.

Focussing on the domain of small <math>k^2</math> and expanding the integrand in \eqref{ren-1}
and the denominator in \eqref{alph}
one arrives at
:<math>\label{ren-2}
  D (Q^2)= \frac{1}{Q^4} \,\alpha_s \, \sum_{n=0}^\infty \left(\frac{\beta_0\alpha_s}{4\pi} 
  \right)^n  \int\,dk^2 \,k^2 \left(\ln \frac{Q^2}{k^2}
  \right)^n
 =  \frac{\alpha_s}{2}\,\sum_{n=0}^\infty  \left(\frac{\beta_0\alpha_s}{8\pi} 
  \right)^n  \int\,dy \,
    y^n\, e^{-y}\,, 
</math>
where <math> \alpha_s \equiv \alpha_s(Q^2) </math> in \eqref{ren-2} and \eqref{r2}, 
and <math> y = 2 \ln \frac{Q^2}{k^2}</math>. Formally
the <math>y</math> integral from zero to infinity is <math>n!</math>, i.e. 
the series is factorially divergent. A characteristic value of <math>k^2</math>
saturating the integral is
:<math>y\sim n   \,\,\,{\rm or}\,\,\,   k^2 \sim Q^2\,\exp \left(-\frac{n}{2}\right)  \,.</math>

Thus, if <math>Q^2</math> is fixed and <math>n</math> is sufficiently large, the factorial divergence of 
the <math>(\alpha_s)^n</math> coefficients in \eqref{ren-2} is due to small-<math>k^2</math> domain
in the integral over <math>k^2</math>.  However, at <math>k^2\sim \Lambda^2</math> all Feynman diagrams become meaningless
since they cease to properly represent non-Abelian dynamics. According to OPE,  the integral for the coefficient function must be cut off from below at <math>k^2 =\Lambda^2</math> (for illustrative purposes one can put <math>\mu=\Lambda</math>). The remainder must
be referred to the expectation value of an appropriate subleading local operator in OPE.  In other words at large <math>y</math> 
the <math>y</math> integral must be cut off at <math>y=n_*</math> where
<math>
n_* = 2\ln\frac{Q^2}{\Lambda^2}\,.
</math>

Then the factorially divergent formula for the coefficient of the unit operator
must be truncated as follows:
<math> \label{r2}
C_{\bf I} (Q^2)\to  \frac{\alpha_s}{2}\,\sum_{n=0}^\infty  \left(\frac{\beta_0\alpha_s}{8\pi} \right)^n  n! 
 \to
  \frac{\alpha_s}{2}\,\sum_{n=0}^{n_*} \left(\frac{\beta_0\alpha_s}{8\pi} \right)^n  n! \,\,+ ... \,\,.
</math>
The ellipses represent the truncated tail of the series. The tail is ''not ignored'' -- it is represented by 
the vacuum expectation value of an appropriate operator in OPE, namely, <math>G^a_{\mu\nu}G^a_{\mu\nu}</math>
in the case at hand.

It is worth noting that <math>n_*</math> is also the critical value of
the asymptotic factorial series, i.e. the value for which the accuracy of truncation is the best.
Truncation 
at <math>n=n_*</math> ensures the deviation from the exact result to be 
<math>\exp\left(-\frac{8\pi}{\beta_0\alpha_s}\right) \sim \Lambda^4
/Q^4</math>.  The <math>O(\Lambda^4
/Q^4</math>) terms are due to the infrared sensitivity to the domain 
<math>k^2\sim \Lambda^2</math>; hence they can (and must) be absorbed in the contribution of the operator <math>G^a_{\mu\nu}G^a_{\mu\nu}</math>.

== Borel transformation for OPE ==

The condensate series for <math>D(q^2)</math> is asymptotic.
Convergence is achieved by the Borel transformation which, in the case at
hand, can be performed as follows.
For the functions obeying dispersion
relations one applies the
limiting procedure
:<math>\label{borelt}
\mathcal{\hat{B}} = \lim \frac{1}{(n-1)!}(Q^2)^n \left(
-\frac{d}{dQ^2}\right)^n\, , \,\,\, Q^2\rightarrow\infty , \,\,\, n
\rightarrow\infty , \,\,\, \frac{Q^2}{n}\equiv M^2 \,\,\,
\mbox{fixed}\, .
</math>

<math>M^2</math> is called the Borel parameter.
One can show that applying <math>\mathcal{\hat{B}}</math>
to <math>(1/Q^2)^n</math> one gets
:<math>
\mathcal{\hat{B}} \left(\frac{1}{Q^2} \right)^n =
\frac{1}{(n-1)!}\left(\frac{1}{M^2} \right)^n\, ,\qquad \mathcal{\hat{B}} ( \ln Q^2 ) = -1\,,
</math>
which entails, in turn
:<math>
\mathcal{\hat{B}} \left(\frac{1}{s+Q^2} \right) = \frac{1}{M^2}\, e^{-s/M^2}\, .
</math>
The Borel-transformed dispersion relation
takes the form
:<math>\label{btdr}
\tilde{\Pi} (M^2) \equiv \{ \mathcal{\hat{B}} \Pi (Q^2) \} =
\frac{1}{\pi M^2}\, \int ds \, \mbox{Im}\,\, \Pi (s)\, e^{-s/M^2}\, .
</math>

== Sample application: SVZ for <math>\mathbf{\rho}</math> mesons ==

Analysis of the <math>\rho</math> meson channel
was among the first applications of the SVZ method which proved its
predictive power (Shifman et al. 1979). In this channel we have
:<math>\label{psim}
\begin{align}
I(M^2) &\equiv \frac{12\pi^2}{N_c}\tilde{\Pi} (M^2) = \frac{1}{M^2}\, \int \,
ds e^{-s/M^2} \rho (s)\\
&= 1 +I_{\rm pt} + I_{\rm cond}
\end{align}
</math>

where <math>\rho(s) </math> is the spectral density and
:<math>
\begin{align}
I_{\rm pt} (M^2) &= k_1 a \left( \frac{M^2}{{\rm e}^\gamma}\right) +
k_2 \left[ a \left( \frac{M^2}{{\rm e}^\gamma}\right)\right] ^2 + (k_3
+ \frac{\pi^2}{6}
k_1)\left[ a \left( \frac{M^2}{{\rm e}^\gamma}\right)\right] ^3+...\, ,\\
I_{\rm cond}(M^2)&= \frac{\pi^2}{3M^4}\, \langle\mathcal{O}_G\rangle
-
\frac{8\pi^3}{M^6}\, \left \langle\alpha_s \left( \bar
u\gamma_\alpha\gamma_5 T^a u - \bar d\gamma_\alpha\gamma_5
T^a d \right)^2 \right\rangle\\
&- \frac{16\pi^3 }{9 M^6 }\, \left \langle
\alpha_s \left( \bar u\gamma_\alpha T^a u + \bar d\gamma_\alpha
T^a d \right)\, \sum_q \left(\bar q\gamma_\alpha T^a q\right)
\right\rangle +...\, .
\end{align}
</math>
The perturbative data <math>k_{1,2,3}</math> and <math>a</math> are
:<math>
k_1= \frac{4}{9}\,,\quad k_2=0.729 k_1\,, \quad k_3=-2.03 k_1\, ;
\qquad a \equiv\frac{b}{4} \frac{\alpha_s}{\pi}\,.
</math>
To enhance the <math>\rho</math> meson contribution
in \eqref{psim} one must choose the
value of the Borel parameter <math>M^2</math> as small as possible.
However, the condensate series explodes at small <math>M^2\ .</math>
A balance is achieved in an intermediate domain called the window of stability.
The spectral density is parametrized by three parameters to be fitted,
the <math>\rho</math> meson mass <math>m_\rho\ ,</math> its coupling constant to the vector
current <math>F_\rho\ ,</math> and <math>s_0</math> marking the onset of continuum in the given channel,
:<math>\label{origmod}
\rho (s) = F_\rho^2\, \delta (s-m_\rho^2) + \theta (s-s_0)\,,
</math>

see  <figref>SVZ_erice10.png</figref>. In fact, one is interested only in the <math>\rho</math> meson characteristics
<math>m_\rho</math> and <math>F_\rho\ .</math> Since inside the window of stability the <math>\rho</math> meson
saturates the
integral in \eqref{psim} at the level <math>\sim 90 \%\ ,</math> the fact that the above
spectral density is inaccurate in the continuum is not important.
A factor of two error in the continuum (the actual error is smaller) affects
determination of the <math>\rho</math> meson parameters <math>m_\rho^2</math>
and <math>F_\rho^2</math> at the level of <math>\lesssim 10 \%</math> accuracy. It is quite clear that the truncated condensate series does not allow
one to go to the limit <math>M\rightarrow 0</math> where determination of
the <math>\rho</math> meson parameters
would be exact.
One must stay inside the window where the expansion is still under
control. Correspondingly, the hadron parameters can only be determined approximately
within the SVZ method. This is the price one has to pay for limited knowledge
of the QCD vacuum encoded in a few condensates.

[[Image:SVZ_erice10.png|thumb|300px|right|ten|The original SVZ "delta+continuum" model for the spectral density.]]
Numerically, the theoretical and phenomenological parts of \eqref{psim}
are shown in  <figref>SVZ_erice15.png</figref>.
[[Image:SVZ_erice15.png|thumb|300px|right|fifteen|<math>I</math> ''versus'' <math>M^2</math> (in GeV<math>^2</math>) in the <math>\rho</math> meson channel: confronting experiment and theory.]]
The center of the window of stability is indicated by a vertical arrow.
The <math>\rho</math> meson example is quite typical. A similar situation takes place for
the majority of low-lying hadrons: those built from the light quarks, heavy, and light and heavy.

== Exceptional cases ==

There are a few cases in which the SVZ method does not provide one with access
to the lowest-lying mesons (Novikov et. al. 1981). This happens for specific,
"nonclassical" hadrons, with a very strong coupling to the vacuum
fluctuations, which are very different from, say, the <math>\rho</math> meson or
nucleon.
The most known example of this type is the <math>0^\pm</math> glueball.
In such channels one can reverse the argument and convert the failure
of the SVZ sum rules into
success by saying that the SVZ method predicts that not all hadrons
are alike (Novikov et. al. 1981).

== References ==
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* A. Mueller, ''On the Structure of Infrared Renormalons in Physical Processes at High-Energies,'' Nucl. Phys. '''B 250''', 327 (1985). 
* Y. Nambu, ''Strings, monopoles, and gauge fields'', Phys. Rev. D '''10''', 4262 (1974).
* V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, ''Wilson's Operator Expansion: Can It Fail?" Nucl. Phys. '''B249''',  445 (1985).
* V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, ''Are All Hadrons Alike?" Nucl. Phys. '''B191''',  301 (1981).
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* N. Seiberg and E. Witten, ''Electric-magnetic duality, monopole condensation, and confinement in <math>\mathcal{N} = 2</math> supersymmetric Yang-Mills theory'', Nucl. Phys. '''B426''', 19 (1994), (E) '''B430''', 485 (1994) [hep-th/9407087].
* N. Seiberg and E. Witten, ''Monopoles, duality and chiral symmetry breaking in <math>\mathcal{N} = 2</math> supersymmetric QCD'', Nucl. Phys. '''B431''', 484 (1994) [hep-th/9408099].
* M. Shifman, A. Vainshtein and V. Zakharov, ''QCD and Resonance Physics," Nucl. Phys. '''B147'''  385; 448 (1979).
* M. Shifman, ''New and Old about Renormalons: in Memoriam Kolya Uraltsev'', arXiv:1310.1966 [hep-th]. 
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* K. Wilson and J. Kogut, ''The Renormalization group and the epsilon expansion," Phys. Reports '''12''', 75  (1974).

== Further reading ==
* Pedagogical presentation of the SVZ sum rules and related topics can be found in
** M. A. Shifman, ''Snapshots of hadrons or the story of how the vacuum medium determines the properties of the classical mesons which are produced, live and die in the QCD vacuum'', Prog. Theor. Phys. Suppl. '''131''', 1 (1998) [arXiv:hep-ph/9802214],
** M. A. Shifman, ''Quark-hadron duality'', arXiv:hep-ph/0009131, in the Boris Ioffe Festschrift ''At the Frontier of Particle Physics'' Ed. M. Shifman (World Scientific, Singapore, 2001).

* Two review papers were published in the mid-1980s.
** B.L. Ioffe, ''Acta Phys. Polon.'' '''B16''' (1985) 543;
** L.J. Reinders, H. Rubinstein, and S. Yazaki, ''Phys. Reports'' '''127''' (1985) 1

* A collection of the original papers with an extended commentary reflecting the state of the art in the late 1980s can be found in
** ''Vacuum Structure and QCD Sum Rules'', Ed. M. Shifman (North-Holland, Amsterdam, 1992).

* A brief survey of the light-cone sum rules is given in
** V. Braun, hep-ph/9801222.

* SVZ sum rules in the heavy quark theory are discussed in
** M. Neubert, ''Phys. Reports'' '''245''' (1994) 259.

* For a review of the sum rule applications in weak decays see
** A. Khodjamirian and R. Rückl, hep-ph/9801443.

* A review on heavy quark theory based on OPE is
** I. Bigi, M. Shifman and N. Uraltsev, ''Aspects of heavy quark theory'', Ann. Rev. Nucl. Part. Sci. '''47''', 591 (1997) [arXiv:hep-ph/9703290];
** M. Shifman, "Lectures on Heavy Quarks in Quantum Chromodynamics," in "ITEP Lectures on Particle Physics and Field Theory" (World Scientific, Singapore, 1999), p. 1 [hep-ph/9510377].

== External links ==
* [http://www.physics.umn.edu/people/shifman.html Author's homepage 1]
*[http://sites.google.com/site/mashifman Author's homepage 2]

== See also ==

[[Category:Physics]]
[[Category:Theoretical high energy physics]]
[[Category:Quantum chromodynamics]]
[[Category:Eponymous]]