∎

11institutetext:  William I. Fine Theoretical Physics Institute, University of
Minnesota, Minneapolis, MN 55455, USA
11email<EMAIL_ADDRESS>
# OPE-based Methods in Nonperturbative QCD

SVZ sum rules, $1/M_{Q}$ expansion, and all that

Mikhail Shifman

(FTPI-MINN-22-23, UNM-TH-4132/22)

###### Abstract

I describe the inception and development of nonperturbative OPE-based methods
in hadronic physics, such as the SVZ sum rules, inverse heavy quark mass
expansion (IHQME) and so on and related topics.

Invited contribution to the EPJC Volume celebrating 50 years of Quantum
Chromodynamics, to be published in December 2022.

††journal: Eur. Phys. J. C

##  1 Preamble

Rewind to autumn of 1971. I am a student at ITEP in Moscow, working on my
Masters degree. The famous paper of Gerhard ’t Hooft GtH1 was published in
Nuclear Physics in October, but neither myself nor anybody else in ITEP
immediately noticed this ground-breaking publication. At that time I did not
even know what Yang-Mills theories meant. Now, when I think of inception of
QCD, the memories of this paper and its sequel GtH2 (issued in December of
1971) always come to my mind. For me, psychologically this was the beginning
of the QCD era.

To give an idea of the scientific atmosphere at that time I looked through the
Proceedings of the 1972 International Conference On High-Energy Physics
HEP1972 . Theoretical talks were devoted to dual models (a precursor to string
theory), deep inelastic scattering and Bjorken scaling, current algebra,
$e^{+}e^{-}\to$ hadrons, etc. In three talks – by Zumino, Bjorken and Ben Lee
– the Weinberg-Salam model (a precursor to the present-day Standard Model) was
reviewed.111There is a curious anecdote I heard later: In December 1979, after
the Glashow-Weinberg-Salam Nobel Prize ceremony, a program was aired on
Swedish radio. At some point, Weinberg quoted a phrase from the Bible. Salam
remarked that it exists in the Quran too, to which Weinberg reacted: “Yes, but
we published it earlier!” Ben Lee was the only person to refer to ’t Hooft’s
publications GtH1 ; GtH2 . The last talk of the conference summarizing its
major topics was delivered by Murray Gell-Mann. In this talk Gell-Mann
discusses, in particular, whether quarks are physical objects or abstract
mathematical constructs. Most interesting for us is his analysis of the
$\pi^{0}\to 2\gamma$ decay. Gell-Mann notes that if quarks are fermions then
theoretically predicted amplitude is factor of 3 lower than the corresponding
experimental result, but makes no statement of the inevitability of the quark
color.222For me personally the following remark in his talk was a good lesson
for the rest of my career: “Last year the rate of $K_{L}^{0}\to\mu^{+}\mu^{-}$
decay was reported to be lower than allowed by unitarity unless fantastic
hypotheses are concocted. Now the matter has become experimentally
controversial.” Alas… concocting fantastic hypotheses was the core of my
Masters thesis.

In October 1972 I was accepted to the ITEP graduate school. My first paper on
deep inelastic scattering in the Weinberg-Salam model was completed in early
1973; simultaneously, I started studying Yang-Mills theories (in particular,
the Faddeev-Popov quantization FP ) in earnest. At the same time, somewhere
far away, behind the Iron Curtain, Callan and Gross searched for a theory with
an ultraviolet fixed point at zero. In July of 1973 Coleman and Gross
submitted to PRL a paper asserting that “no renormalizable field theory that
consisted of theories with arbitrary Yukawa, scalar or Abelian gauge
interactions could be asymptotically free” DG1 . Damn Iron Curtain! If Gross
asked anyone from the ITEP Theory Department he would have obtained the answer
right away. The above theorem was known to the ITEP theorists from the Landau
time. For brevity I will refer to it as the Landau theorem, although it was
established by his students rather than Landau himself. The general reason why
this theorem holds was also known – the Källen-Lehman (KL) representation of
the polarization operator plus unitarity.

An explanatory remark concerning the Landau theorem might be helpful here. For
asymptotic freedom to take place the first coefficient of the $\beta$ function
must be negative. The sign of the one-loop graphs which determine the coupling
constant renormalization is in one-to-one correspondence with the sign of
their imaginary parts (this is due to the dispersion KL representation for
these graphs). Unitarity implies the positivity of the imaginary parts which
inevitably leads to the positive first coefficients in the $\beta$ functions
in renormalizable four-dimensional field theories based on arbitrary Yukawa,
scalar or Abelian gauge interactions. This situation is that of the Landau
zero charge in the infrared rather than asymptotic freedom. In Yang-Mills
theories in physical ghost-free gauges some graphs have no imaginary parts
which paves the way to asymptotic freedom (see e.g. MS1 ).

In fact, it is quite incomprehensible why asymptotic freedom had not been
discovered at ITEP after ’t Hooft’s 1971 publication. In Ref. MS1 the reader
can find a narrative about this historical curiosity.

May 1973 should be viewed as the discovery of asymptoric freedom GWP . That’s
when the breakthrough papers of Gross, Wilczek and Politzer were submitted –
simultaneously – to PRL. David Gross recollects DG1 :

> We completed the calculation in a spurt of activity. At one point a sign
> error in one term convinced us that [Yang-Mills] theory was, as expected,
> non-asymptotically free. As I sat down to put it together and to write up
> our results, I caught the error. At almost the same time Politzer finished
> his calculation and we compared, through Sidney, our results. The agreement
> was satisfying.

It took a few extra months for QCD to take off as the theory of strong
interactions. The events of the summer of 1973 that led to the birth of QCD
are described by H. Leutwyler in Section 1.1 of this Volume. To my mind, the
final acceptance came with the November Revolution of 1974 – the discovery of
$J/\psi$ and its theoretical interpretation as ortho-charmonium.333I should
also mention a highly motivating argument due to S. Weinberg who proved SWe
that (in the absence of the U(1) gluon anomaly)
$m_{\eta^{\prime}}\leq\sqrt{3}m_{\pi}$. This argument seemingly was discussed
during ICHEP 74 in July 1974. In the fall of 1973 we submitted a paper BNS
explaining why the Landau theorem in four dimensions fails only in Yang-Mills
theory.

QCD and its relatives are special because QCD is the theory of nature. QCD is
strongly coupled in the infrared domain where it is impossible to treat it
quasiclassically – perturbation theory fails even qualitatively. It does not
capture drastic rearrangement of the vacuum structure related to confinement.
The Lagrangian is defined at short distances in terms of gluons and quarks,
while at large distances of the order of $\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}\Lambda^{-1}_{\rm
QCD}$ (where $\Lambda_{\rm QCD}$ is the dynamical scale of QCD, which I will
refer to as $\Lambda$ below) we deal with hadrons, e.g. pions, $\rho$ mesons,
protons, etc. Certainly, the latter are connected with quarks and gluons in a
divine way, but this connection is highly nonlinear and non-local; even now,
50 years later, the full analytic solution of QCD is absent.

Non-perturbative methods were desperately needed.

##  2 Inception of non-perturbative methods

Four years before QCD Ken Wilson published a breakthrough paper KW on the
operator product expansion (OPE) whose pivotal role in the subsequent
development of HEP theory was not fully appreciated until much later. What is
now usually referred to as Wilsonian renormalization group (RG), or Wilsonian
RG flow, grew from this paper. Wilsonian paradigm of separation of scales in
quantum theory was especially suitable for asymptotically free theories.
Wilson’s formulation makes no reference to perturbation theory, it has a
general nature and is applicable in non-perturbative regime too. The focus of
Wilson’s work was on statistical physics, where the program is also known as
the block-spin approach. Starting from microscopic degrees of freedom at the
shortest distances $a$, one “roughens” them, step by step, by constructing a
sequence of effective (composite) degrees of freedom at distances $2a$, $4a$,
$8a$, and so on. At each given step $i$ one constructs an effective
Hamiltonian, which fully accounts for dynamics at distances shorter than
$a_{i}$ in the coefficient functions.

QCD required a number of specifications and adjustments. Indeed, the UV fixed
point in QCD is at $\alpha_{s}=0$; hence, the approach to this fixed point at
short distances is very slow, logarithmic rather than power-like
characteristic for the $\alpha_{s}\neq 0$ fixed points. In fact, it is not the
critical regime at the UV fixed point per se we are interested in but rather
the regime of approach to this critical point. Moreover, it was not realized
that (in addition to the dynamical scale $\Lambda$) the heavy quarks provide
an extra scale – the heavy quark mass $m_{Q}$ – which must be included in OPE
where necessary.

Surprisingly, in high-energy physics of the 1970s the framework of OPE was
narrowed down to a very limited setting. On the theoretical side, it was
discussed almost exclusively in perturbation theory. On the practical side,
its applications were mostly narrowed down to deep inelastic scattering, where
it was customary to work in the leading-twist approximation.

The fact that the UV fixed point is at zero makes OPE both more simple and
more complicated than in the general case. On one hand, the anomalous
dimensions of all composite local operators which might be relevant in the
given problem scale only logarithmically. On the other hand, slow
(logarithmic) fall off of “tails” instead of desired power-like – makes
analytic separation of scales technically difficult.

I believe that we – Arkady Vainshtein, Valentin Zakharov and myself – were the
first to start constructing a QCD version of OPE. The first step in this
direction was undertaken in 1974 in the problem of strangeness-changing weak
decays peng (currently known as the penguin mechanism in flavor-changing
decays). A mystery of $\Delta I=\frac{1}{2}$ enhancement in $K$ decays had
been known for years (for a review see VA ). A suggestion of how one could
apply OPE to solve this puzzle was already present in Wilson’s paper KW .
Wilson naturally lacked particular details of QCD. The first attempt to
implement Wilson’s idea in QCD was made in MGBL . Although these papers were
inspirational, they missed the issue of a “new” OPE needed for QCD realities.
Seemingly, we were the first to address this challenge, more exactly two of
its features: mixed quark-gluon operators (in peng we introduced

$O_{\rm
peng}=\bar{s}_{L}\gamma^{\mu}\left({\mathcal{D}}_{\nu}G^{\mu\nu}\right)d_{L}$

which is purely $\Delta I=\frac{1}{2}$) and coefficients logarithmically
depending on the charmed (i.e. heavy at that time) quark mass. Currently,
$c,b,t$ quark masses appear in the penguin operators (illustrated in Fig. 1),
the latter two being genuinely heavy. Through equations of motion the operator
$O_{\rm peng}$ reduces to a four-quark operator but its chiral structure is
different from conventional, namely, it contains both left-handed and right-
handed quark fields since
${\mathcal{D}}_{\nu}G^{\mu\nu}\sim\sum_{q}\bar{q}\gamma^{\mu}q$. Combined with
another revolutionary finding of QCD, the extraordinary smallness of the $u$
and $d$ quark masses, $m_{u,d}\sim 5\,$MeV (see Sec.1.1.15 of this Volume),
the mixed chiral structure of the emerging four-fermion operator provides the
desired enhancement of the $\Delta I=\frac{1}{2}$ amplitude. It took us over
two years to fight a succession of referees for publication of the second
paper in Ref. peng . One after another, they would repeat that mixed-chirality
four-fermion operator in the considered theory was complete nonsense.
Currently, the penguin mechanism in flavor changing weak transitions is a
basic theoretic element for a large variety of such decays. As Vainshtein put
it VA , “Penguins spread out but have not yet landed.”

Figure 1: The penguin mechanism in flavor-changing decays. Any of three heavy
quarks $c,b$ or $t$ can appear in the loop.

Systematic studies of Wilsonian OPE in QCD can be traced back to the summer of
1977 – that is when the the gluon condensate $O_{G}$ (see Table 1) was first
introduced SVZ1 . Vacuum expectation values of other gluon and quark operators
were introduced in Ref. SVZ2 , which allowed one to analyze a large number of
vacuum two- and three-point functions, with quite nontrivial results for
masses, coupling constants, magnetic moments and other static characteristics
of practically all low-lying hadronic states of mesons and baryons. A
consistent Wilsonian approach requires an auxiliary normalization point $\mu$
which plays the role of a regulating parameter separating hard contributions
included in the coefficient functions and soft contributions residing in local
operators occurring in the expansion. The degree of locality is regulated by
the same parameter.

After setting the foundation of OPE in QCD SVZ2 we were repeatedly returning
to elaboration of various issues, in particluar, in the following works: NSVZ
,Shifm2 , and Shifm3 .

##  3 SVZ sum rules. Concepts

The 1998 review Shifm2 summarizes for the reader foundations of the Shifman-
Vainshtein-Zakharov (SVZ) sum rules in a pedagogical manner. At short
distances QCD is the the theory of quarks and gluons. Yang-Mills theory of
gluons confines. This means that if you have a heavy probe quark and an
antiquark at a large separation a flux tube with a constant tension develops
between them preventing their “individual” existence. In the absence of the
probe quarks the flux tube can form closed contours interpreted as glueballs.
This phenomenon is also referred to in the literature as the area law or the
dual Meißner effect. Until 1994 the above picture was the statement of faith.
In 1994 Seiberg and Witten found an analytic proof SW of the dual Meißner
effect in ${\mathcal{N}}=2$ super-Yang-Mills.444More exactly, confinement
through the flux tube formation was proven in the low-energy limit of this
theory upon adding a small deformation term breaking ${\mathcal{N}}=2$ down to
${\mathcal{N}}=1$.  The Seiberg-Witten solution does not apply to QCD, rather
to its distant relative. The real world QCD, with quarks, in fact has no area
law (the genuine confinement is absent) since the flux tubes break through the
quark-antiquark pair creation. Moreover, light quarks are condensed, leading
to a spontaneous breaking of chiral symmetry, a phenomenon shaping the
properties of the low-lying hadronic states, both mesonic and baryonic. The
need to analytically understand these properties from first principles led us
to the development of the SVZ method.

The quarks comprising the low-lying hadronic states, e.g. classical mesons or
baryons, are not that far from each other, on average. The distance between
them is of order of $\Lambda^{-1}$. Under the circumstances, the string-like
chromoelectric flux tubes, connecting well-separated color charges, do not
develop and details of their structure are not relevant. Furthermore, the
valence quark pair injected in the vacuum – or three quarks in the case of
baryons – perturb it only slightly. Then we do not need the full machinery of
the QCD strings 555Still unknown. to approximately describe the properties of
the low-lying states. Their basic parameters depend on how the valence quarks
of which they are built interact with typical vacuum field fluctuations.

We endowed the QCD vacuum with various condensates – approximately a half-
dozen of them – in the hope that this set would be sufficient to describe a
huge variety of the low-lying hadrons, mesons and baryons. The original set
included 666A meticulous writer would have used the notation $\left\langle
G_{\mu\nu}^{2}\right\rangle$, etc. but I will omit bra and ket symbols where
there is no menace of confusion.  the gluon condensate $G_{\mu\nu}^{2}$, the
quark condensate $\bar{q}q$, the mixed condensate
$\bar{q}\sigma^{\mu\nu}G_{\mu\nu}q$, various four-quark condensates
$\bar{q}\Gamma q\bar{q}\Gamma q$, and a few others (see Table 1). Later this
set had to be expanded to address such problems as, say, the magnetic moments
of baryons.

Figure 2: A two-point correlation function in the QCD vacuum. The left side is the OPE sum with a finite number of the lowest-dimension operators ordered according to their normal dimensions. The right side is the sum over mesons with the appropriate quantum numbers. The ground state in the given channel is singled out. The excited states are accounted for in the quasiclassical approximation. We define a positive variable $Q^{2}=-q^{2}$ and a sliding $\mu^{2}$ parameter used as a separation parameter in OPE. For better convergence a Borel transformation is applied as explained below. Table 1: The lowest-dimension operators in OPE. $\Gamma$ is a generic notation for combinations of the Dirac $\gamma$ matrices. Normal dim | 3 | 4 | 5 | 6 | 6
---|---|---|---|---|---
Operator  | $O_{q}=\bar{q}q$ | $O_{G}=G_{\mu\nu}^{2}$ | $O_{qG}=\bar{q}\sigma^{\mu\nu}G_{\mu\nu}q$ | $O_{4q}=(\bar{q}\Gamma q)^{2}$ | $O_{3G}=GGG$

Our task was to determine the regularities and parameters of the classical
mesons and baryons from a limited set of the vacuum condensates. Figure 2
graphically demonstrates the SVZ concept. On the theoretical side, an
appropriate $n$-point function is calculated as an OPE expansion truncated at
a certain order. In most problems only condensates up to dimension 6 (Table 1)
are retained. In the “experimental” part the lowest-lying meson (or baryon) is
singled out, while all higher states are represented in the quasiclassical
approximation. Acting in this way, one can determine the parameters $f_{0}$
and $m_{0}$ defined in Fig. 2 and their analogs in other problems. Of course,
without invoking the entire infinite set of condensates one can only expect to
obtain the hadronic parameters in an admittedly approximate manner.

##  4 Borelization

Figure 3: Graph showing four loops renormalizing a gluon line (represented by
the dotted line). A renormalon is the sum over $n$ of such diagrams with $n$
loops.

Analyzing the sum rules displayed in Fig. 2 we realized that their predictive
power was limited – summation on both sides of the equation does not converge
fast enough. On the right-hand side the contribution of high excitations is
too large – the lowest lying states are “screened” – because the weight
factors fall off rather slowly. Likewise, to achieve reasonable accuracy on
the left-hand side one would need to add operators other than those collected
in Table 1 of which we knew next to nothing. The Borel transform came to the
rescue.

The Borel transformation is a device well-known in mathematics. If one has a
function $f(x)$ expandable in the Taylor series, $f(x)=x\sum_{n}a_{n}x^{n}$
with the coefficients $a_{n}$ which do not fall off sufficiently fast, one can
instead introduce its Borel transform

${\mathcal{B}}f=x\sum_{n}\frac{a_{n}}{n!}x^{n}$ (4.1)

and then, if needed, reconstruct $f$.777The Borel transform is closely related
to the Laplace transform. If we apply this procedure to the sum rule in Fig. 2
we obtain for a given hadronic state $i$

$\displaystyle{\mathcal{B}}\frac{f_{i}^{2}}{m_{i}^{2}+Q^{2}}$ $\displaystyle=$
$\displaystyle{\mathcal{B}}\,\frac{f_{i}^{2}}{Q^{2}}\,\sum_{n}(-1)^{n}\left[\frac{m_{i}^{2}}{Q^{2}}\right]^{n}$
(4.2) $\displaystyle\to$
$\displaystyle\frac{f_{i}^{2}}{Q^{2}}\,\sum_{n}\frac{(-1)^{n}}{n!}\left[\frac{m_{i}^{2}}{Q^{2}}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{f_{i}^{2}}{Q^{2}}\,\exp\left(-\frac{m_{i}^{2}}{Q^{2}}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{f_{i}^{2}}{M^{2}}\left(-\frac{m_{i}^{2}}{M^{2}}\right)$

where in the final step (for historical reasons) I replaced $Q^{2}$ by a Borel
parameter $M^{2}$. If $M^{2}$ can be chosen sufficiently small, higher
excitations are exponentially suppressed.

Simultaneously, we improve the convergence of OPE on the left-hand side by
applying the same operator. If the operator $O_{n}$ has dimension $2d_{n}$
then the Borel transformation of the left hand side yields

$\sum_{n}\frac{1}{(Q^{2})^{d_{n}}}\langle
O_{n}\rangle\,\to\,\sum_{n}\frac{1}{(d_{n}-1)!}\frac{1}{(M^{2})^{d_{n}}}\langle
O_{n}\rangle\,,$ (4.3)

where I have again replaced $Q^{2}$ by the Borel parameter $M^{2}$. Since the
expansion (4.3) goes in inverse powers of $M^{2}$, it is necessary to keep
$M^{2}$ large enough. The two requirements on $M^{2}$ seem contradictory.
However, for all “typical” resonances, such as, say, $\rho$ mesons, they can
be met simultaneously SVZ2 in a certain “window.” The only exception is the
$J^{P}=0^{\pm}$ channel. There are special reasons why $0^{\pm}$ mesons are
exceptional.

##  5 Practical version of OPE

At the early stages of the SVZ program the QCD practitioners often did not
fully understood the concept of the scale separation in the Wilsonian OPE. It
was generally believed that the coefficients are fully determined by
perturbation theory while non-perturbative effects appear only in the OPE
operators.888Unfortunately, this misconception lasted through the 1980s and
was visible in the literature even in the 1990s and later. This belief led to
inconsistencies which revealed themselves e.g. in the issue of renormalons
(see below). A set of graphs represented by renormalons is constructed from a
single gluon exchange by inserting any number of loops in the gluon line like
beads in a neckless (e.g Ref. Shifm3 and references therein). Being treated
formally this contribution, shown in Fig. 3, diverges factorially at high
orders. I vividly remember that after the first seminar on SVZ in 1978 Eugene
Bogomol’nyi asked me each time we met: “Look, how can you speak of power
corrections in the $n$-point functions at large $Q^{2}$ if even the
perturbative expansion (i.e. the expansion in $1/\log(Q^{2}/\Lambda^{2})$) is
not well defined? Isn’t it inconsistent?” I must admit that at that time my
answer to Eugene was somewhat evasive.

The basic principle of Wilson’s OPE – the scale separation principle – is
“soft versus hard” rather than “perturbative versus non-perturbative.” Being
defined in this way the condensates are explicitly $\mu$ dependent. All
physical quantities are certainly $\mu$ independent; the normalization point
dependence of the condensates is compensated by that of the coefficient
functions, see Fig. 2.

The problem of renormalons disappears simultaneously with the introduction of
the normalization point $\mu$. With $\mu\gg\Lambda$, there is no factorial
divergence in high orders of perturbations theory. Renormalons conspire with
the gluon condensates to produce, taken together, a well-defined OPE. The
modern construction goes under the name of the “renormalon conspiracy”; it is
explained in detail in my review Shifm3 . I hasten to add, though, that the
renormalons acquire a life of their own in those cases in which OPE does not
exist. Qualitatively, they can shed light on scaling dimensions of non-
perturbative effects. The most clear-cut example of this type is the so-called
“pole mass of the heavy quarks” Shifm7p and its relation to a theoretically
well-defined mass parameter Shifm6 .

In some two-dimensional solvable models exact OPE can be constructed which
explicitly demonstrates the $\mu$ dependence of both the coefficient functions
and the condensates in the Wilsonian paradigm and its cancellation in the
physical quantities (for a recent study see e.g. sch ). Needless to say, if
QCD was exactly solved we would have no need in the SVZ sum rules.

We had to settle for a reasonable compromise, known as the practical version
of OPE. In the practical version we calculate the coefficient functions
perturbatively keeping a limited number of loop corrections. The condensate
series is truncated too. The condensates are not calculated from first
principles; rather a limited set is determined from independent data.

The practical version is useful in applications only provided $\mu^{2}$ can be
made small enough to ensure that the “perturbative” contributions to the
condensates are much smaller than their genuine (mostly non-perturbative)
values. At the same time, $\alpha_{s}(\mu^{2})/\pi$ must be small enough for
the expansion in the coefficients to make sense. The existence of such
“$\mu^{2}$ window” is not granted a priori and is a very fortunate feature of
QCD. We did observe this feature empirically in almost all low-lying hadrons
vol . 999Theoretical understanding of the roots of this phenomenon remains
unclear. Seemingly, it has no known analogues in two-dimensional models.  At
the same time, we identified certain exceptional channels revealing unforeseen
nuances in hadronic physics alike .

##  6 Implementation of the idea and results

After the strategic idea of quark and gluon interaction with the vacuum medium
became clear we delved into the uncharted waters of microscopic hadronic
physics. Remember, in 1977 nobody could imagine that basic hadronic parameters
for at least some hadrons could be analytically calculated, at least
approximately. As a show-case example we chose the most typical mesons, $\rho$
and $\phi$, to calculate their couplings to the electromagnetic current and
masses. The agreement of our results with experiment was better than we could
a priori expect. At first we were discouraged by a wrong sign of the gluon
condensate term in the theoretical part of the appropriate SVZ sum rule. We
suddenly understood that this sign could be compensated by the four-quark
condensate – a real breakthrough. In November of 1977 we published a short
letter SVZ1 which still missed a number of elements (e.g. Borelization)
developed and incorporated later, one by one. We worked at a feverish pace for
the entire academic year, accumulating a large number of results for the
hadronic parameters. All low-lying meson resonances built from the $u,d,s$
quarks and gluons were studied and their static properties determined from
SVZ: masses, coupling constants, charge radii, $\rho$-$\omega$ mixing, and so
on, with unprecedented success. In summer of 1978, inspired by our progress,
we prepared a number of preprints (I think, eight of them simultaneously
101010In the journal publication they were combined in three articles
occupying the whole issue of Nucl. Phys. B147,
${\mathcal{N}}^{\underline{o}}5$, see SVZ2 .) and submitted to ICHEP-78 in
Tokyo. Seemingly none of us were allowed to travel to Tokyo to present our
results.

I cannot help mentioning an incident that occurred in the spring of 1978 when
we were mostly done with this work. The episode may have been funny were it
not so nerve-wracking. When we decided that the calculational stage of the
work was over, I collected all my drafts (hundreds of sheets of paper with
derivations and math expressions), I organized them in proper order, selected
all expressions we might have needed for the final draft of the paper and the
future work, meticulously rewrote them in a voluminous notebook (remember, we
had no access to photocopying machines), destroyed the original drafts, put
the the notebook in my briefcase and went home. It was about midnight, and I
was so exhausted that I fell asleep while on the metro train. A loud voice
announcing my stop awoke me, and I jumped out of the train, leaving the
briefcase were it was, on the seat. By the time I realized what have happened
the train was gone, and gone with it forever my calculations … I have never
recovered my briefcase with the precious notebook… After a few agonizing days
it became clear that the necessary formulas and expressions had to be
recovered anew. Fortunately, Vainshtein and Zakharov had kept many of their
own derivations. Vainshtein never throws away anything as a matter of
principle. Therefore, the problem was to dig out “informative” sheets of paper
from the “noise” (this was hindered by the fact that Vainshtein was in
Novosibirsk while we were in Moscow). Part of my drafts survived in the
drawers of a huge desk that I had inherited from V. Sudakov. Better still,
many crucial calculations were discussed so many times by us, over and over
again, that I remembered them by heart. Nevertheless, I think it took a couple
of uneasy weeks to reconstruct in full the contents of the lost notebook.

The SVZ method was further developed by many followers (e.g. the so-called
light-come sum rules for form-factors), see Braun and Khodj . A broad picture
of the hadronic world was obtained by the 1980s and later Colangelo . Today
the pioneering SVZ paper is cited 6000+ times. Until 1990s, when lattice QCD
based on numeric calculations, started approaching its maturity, the SVZ
method was the main tool for analyzing static hadronic properties.

##  7 Reliability and predictive power

The SVZ method is admittedly approximate. Yet, it is not a model in the sense
that it cannot be arbitrarily bent to accommodate “wrong” data. It is
instructive to narrate here the story of an alleged discovery of an alleged
“paracharmonium” referred to as $X(2.83)$ in January of 1977 X . It was widely
believed then that $X(2.83)$ was the $0^{-}$ ground sate of $\bar{c}c$ quarks,
$\eta_{c}$. If this was the case the mass difference between $J/\psi$ and
$\eta_{c}$ would be close to 270 MeV. Shortly after, the interpretation of
$X(2.83)$ as $\eta_{c}$ was categorically ruled out by the SVZ analysis volo
which predicted that the above mass difference must be 100$\pm$ 30 MeV. Two
years later, a new experiment par negated the existence of the $X(2.83)$
state. In the very same experiment the genuine paracharmonium was observed at
$2.98\pm 0.01$ GeV, in perfect agreement with volo . For us this was a triumph
and a lesson – if one believes in a theory one should stand for it!

##  8 OPE-based construction of heavy quark mass expansion

In the 1980s and early 1990s OPE was generalized to cover theoretical studies
of mixed heavy-light hadrons, i.e. those built from light, $q$, and heavy,
$Q$, flavors. In the 1990s those who used $1/m_{Q}$ expansion in theoretical
analysis of $Q\bar{q}$ and $Qqq$ systems numbered in the hundreds. A large
range of practical physics problem related to $Q\bar{q}$ and $Qqq$ systems
were solved. Lattice analyses of such systems even now remain hindered, and in
many instances the $1/m_{Q}$ expansion remains the only reliable theoretical
method.

As I have mentioned in the second paragraph of Sec. 2, heavy quarks in QCD
introduce an extra scale, $m_{Q}$. To qualify as a heavy quark $Q$ the
corresponding mass term $m_{Q}$ must be much larger than $\Lambda_{\rm QCD}$.
The charmed quark $c$ can be considered as heavy only with some reservations
while $b$ and $t$ are bona fide heavy quarks. The hadrons composed from one
heavy quark $Q$, a light antiquark $\bar{q}$, or a “diquark” $qq$, plus a
gluon cloud (which also contains light quark-antiquark pairs) – let us call
them $H_{Q}$ – can be treated in the framework of OPE. The role of the cloud
is, of course, to keep all the above objects together, in a colorless bound
state. The light component of $H_{Q}$, its light cloud, has a complicated
structure; the soft modes of the light fields are strongly coupled and
strongly fluctuate. Basically, the only fact which we know for sure is that
the light cloud is indeed light; typical excitation frequencies are of order
of $\Lambda$. One can try to visualize the light cloud as a soft
medium.111111Hard gluons do play a role too. They have to be taken into
account in the coefficient functions as will be mentioned In Sec. 10.5. The
heavy quark $Q$ is then submerged in this medium. The latter circumstance
allows one to develop a formalism similar to SVZ in which the soft QCD vacuum
medium is replaced by that of the light cloud. As a result, an OPE-based
expansion in powers of $1/m_{Q}$ emerges (see Fig. 4). When heavy quarks are
in soft medium the heavy quark-antiquark pair creation does not occur and the
field-theoretic description of the heavy quark becomes redundant. A large
“mechanical” part in the $x$ dependence of $Q(x)$ can be a priori isolated,
$Q(x)=\exp({-im_{Q}t})\tilde{Q}({x})$. The reduced bispinor field
$\tilde{Q}(x)$ describes a residual heavy quark motion inside the soft cloud;
the heavy quark mass appears only in the form of powers of $1/m_{Q}$ (first
noted in 1982).

Figure 4:  $1/m_{Q}$ expansion for a $H_{Q}$ weak inclusive decay rate (see
Eq. (8.2)). Depicted are two operators, the leading $\bar{Q}Q$ and a
subleading $(\bar{Q}q_{3})(\bar{q}_{3}Q)$. Both are sandwiched between the
heavy hadron states $\langle H_{Q}|$ and $|H_{Q}\rangle$ and the decay rate is
determined by the imaginary part. The grey area depicts the soft quark-gluon
cloud. Adapted from Ref. Sh1 .

Comprehensive reviews on the OPE-based heavy quark theory exist Shifm6 ,ur
,Shifm7 ,hqr . There the reader will find exhaustive lists of references to
original publications. Therefore, in my presentation below I will be brief,
with a focus on a historical aspect, as I remember it, and limit myself to a
few selected references.

In the early 1980s abundant data on the meson and baryon $H_{Q}$ states
started to appear. Theoretical understanding of the total decay rates beyond
the free-quark calculations became a major goal. This challenge paved the way
to the beginning of the $1/m_{Q}$ expansion in $H_{Q}$ hadron physics in the
mid 1980s. The decay rate into an inclusive final state $f$ can be written in
terms of the imaginary part of a forward scattering operator (the so-called
transition operator) evaluated to second order in the weak interactions Sh1 ,

${\rm Im}\hat{T}(Q\\!\rightarrow\\!f\\!\rightarrow\\!Q)\\!=\\!{\rm Im}\\!\int
d^{4}x\ i\,T\left({\cal L}_{W}(x){\cal L}_{W}^{\dagger}(0)\right)\ $ (8.1)

where $T$ denotes the time ordered product and ${\cal L}_{W}$ is the relevant
weak Lagrangian at the normalization point $\mu\sim m_{Q}$. The factor
$\exp({-im_{Q}t})$ mentioned above is implicit in Eq. (8.1). Descending to
$\mu\ll m_{Q}$ one arrives at the OPE expansion

$\displaystyle\Gamma(H_{Q}\rightarrow f)=G_{F}^{2}|V_{\rm
CKM}|^{2}m_{Q}^{5}\sum_{i}\tilde{c}_{i}^{(f)}(\mu)\frac{\langle
H_{Q}|O_{i}|H_{Q}\rangle_{\mu}}{2M_{H_{Q}}}$
$\displaystyle\propto\left[c_{3}^{(f)}(\mu)\frac{\langle
H_{Q}|\bar{Q}Q|H_{Q}\rangle_{(\mu)}}{2M_{H_{Q}}}\right.$
$\displaystyle\left.+c_{5}^{(f)}(\mu)m_{Q}^{-2}\frac{\langle
H_{Q}|\bar{Q}\frac{i}{2}\sigma
GQ|H_{Q}\rangle_{(\mu)}}{2M_{H_{Q}}\;\;}\right.$
$\displaystyle\left.+\sum_{i}c_{6,i}^{(f)}(\mu)m_{Q}^{-3}\frac{\langle
H_{Q}|(\bar{Q}\Gamma_{i}q)(\bar{q}\Gamma_{i}Q)|H_{Q}\rangle_{(\mu)}}{2M_{H_{Q}}\;}\right.$
$\displaystyle\left.+{\cal O}(1/m_{Q}^{4})+...\right],$ (8.2)

where $\Gamma_{i}$ represent various combinations of the Dirac $\gamma$
matrices, see also table 1. In SVZ we dealt with the vacuum expectation values
of relevant operators while in the heavy quark physics the relevant operators
are sandwiched between $H_{Q}$ states.

##  9 Applications

The expansion (8.2) allowed us to obtain Sh1 the first quantitative
predictions for the hierarchies of the lifetimes of $Q\bar{q}$ mesons and
$Qqq$ baryons ($Q$ was either $c$ or $b$ quark) in the mid-1980s – another
spectacular success of the OPE-based methods. The dramatic story of $\eta_{c}$
narrated in Sec. 7 repeated itself. With the advancement of experiment in the
late 1990s, a drastic disagreement was allegedly detected in the ratio
$\tau(\Lambda_{b})/\tau(B_{d})_{\rm exp}=0.77\pm 0.05$ compared to the
theoretical prediction

$\tau(\Lambda_{b})/\tau(B_{d})_{\rm theor}=0.9\pm 0.03$

(e.g. Shifm6 ). In the 2010s the $\Lambda_{b}$ lifetime was remeasured
shifting the above experimental ratio up to $0.93\pm 0.05$. Hurrah!

In the mid-1980s, at the time of the initial theoretical studies of the
$H_{c}$ and $H_{b}$ lifetime hierarchies Sh1 , next to nothing was known about
heavy baryons. Since then enormous efforts were invested in improving
theoretical accuracy both in mesons and baryons in particular by including
higher-dimension operators in the inverse heavy quark mass expansion (IHQME)
and higher-order $\alpha_{s}$ terms in the OPE coefficients. The status of the
IHQME for $H_{Q}$ lifetimes as of 2014 was presented in the review
Lenz:2014jha . The advances reported there and in more recent years cover more
precise determination of the matrix elements of four-quark operators via HQET
sum rules Kirk:2017juj , calculations of the higher $\alpha_{s}$ corrections,
in particular, $\alpha_{s}^{3}$ corrections to the semileptonic $b$ quark
decay Fael:2020tow , the first determination of the Darwin coefficient for
non-leptonic decays Lenz:2020 , etc. Comparison with the current set of data
on $\tau({H_{c}})$ can be found in Rusov . In this context I should also
mention an impressive publication Gratrex:2022xpm (see also references
therein) which, in addition to a comprehensive review of the OPE-based
analysis of the $H_{c}$ lifetimes, acquaints the reader with a dramatic story
of the singly charmed baryon hierarchy. Indeed, according to PDG-2018 the
lifetime of $\Omega_{c}^{0}$ is $69\pm 12\,$fs while PDG-2020 yields
$\tau(\Omega_{c}^{0})=268\pm 24\pm\\!10\\!$ fs! The latter measurement was
reported by LHCb LHC . Moreover, the newest Belle II result belle which has
been published in August 2022, $\tau(\Omega_{c}^{0})=243\pm 48\pm 11\\!$ fs,
triumphantly confirms the LHCb data. The jump in the $\Omega_{c}^{0}$ lifetime
by a factor of 3 to 4 compared to the older measurements could happen only
because (presumably) statistical and/or systematic errors in the older
measurements were grossly underestimated. With these fresh data the observed
hierarchy of lifetimes changes – $\Omega_{c}^{0}$ from the first place (the
shortest living $H_{c}$ baryon) moves to the third.121212 It is curious to
note that 30 years ago Blok and I argued blok (Secs. 4.2 and 6) that
$\Omega_{c}^{0}$ could be the longest living singly charmed baryon due to its
$ss$ spin-1 diquark structure. The question arises whether the OPE-based
theory can explain the current experimental hierarchy
$\tau(\Xi^{0})<\tau(\Lambda_{c}^{+})<\tau(\Omega_{c}^{0})<\tau(\Xi^{+})$. In
Gratrex:2022xpm it is argued that the answer is “yes, it is possible” (see
Fig. 5 in Gratrex:2022xpm ) provided one takes into account $1/m_{c}^{4}$
contributions due to four-quark operators and $\alpha_{s}$ corrections in the
appropriate coefficient functions.131313The four-quark operators introduced in
Sh1 responsible for the Pauli interference yield corrections $O(1/m_{c}^{3}$),
see Eq. (8.2). The authors of Gratrex:2022xpm go beyond this set.

I should emphasize that the theoretical accuracy in the $H_{c}$ family is
limited by the fact that the expansion parameter $\Lambda_{\rm QCD}/m_{c}$ is
not small enough. Even including sub-leading contributions will hardly provide
us with high-precision theoretical predictions. For $H_{c}$ states IHQME at
best provides us with a semi-quantitative guide. On the other hand, in the
theory of $H_{b}$ decays one expects much better accuracy.

##  10 Around 1990s and beyond

###  10.1 Heavy quark symmetry in the limit $m_{Q}\to\infty$

The light-cloud interpretation as in Fig. 4 immediately implies that at zero
recoil the (appropriately normalized) $B\to D$ formfactors reduce to unity.
This is called the “small velocity (SV) limit theorem” NW , VS . The
above“unification” is similar to the vector charge non-renormalization theorem
at zero momentum transfer, say, for the $\bar{u}\gamma^{\mu}d$ current. The
$D$ and $B$ masses are very far from each other. One has to subtract the
mechanical part of the heavy quark mass in order to see that all dynamical
parameters are insensitive to the substitution $Q_{1}\leftrightarrow Q_{2}$ in
the limit $m_{Q_{1,2}}\rightarrow\infty$, with the SV limit ensuing at zero
recoil. Perhaps, this is the reason why it was discovered so late. The next
step was made by Isgur and Wise who generalized this symmetry off the zero-
recoil point by virtue of the Isgur-Wise function IW .

###  10.2 HQET

Heavy quark effective theory which emerged in the 1990s ee formalizes and
automates a number of aspects of the generic $1/m_{Q}$ expansion. In fact, it
immediately follows from the construction similar to (8.2). Simplified rules
of behavior proved to be very helpful for QCD practitioners in the subsequent
development of various applications. In HQET the reduced field $\tilde{Q}$ is
treated quantum-mechanically, its non-relativistic nature is built in, and the
normalization point $\mu$ is $\ll m_{Q}$ from the very beginning.141414 I
personally prefer to consider the heavy quark expansions directly in full QCD
in the framework of the Wilson OPE bypassing the intermediate stage of HQET.
Applying the Dirac equation to eliminate small (lower) components of the Dirac
spinor in favor of the large components it is easy to derive the expansion of
${\cal L}_{\rm heavy}^{0}$, up to terms $1/m_{Q}^{2}$,

$\displaystyle{\cal L}_{\rm heavy}^{0}=\bar{Q}(i\not\\!\\!D-m_{Q})Q$
$\displaystyle=\bar{Q}\frac{1+\gamma_{0}}{2}\left(1+\frac{(\bfsigma\bfpi)^{2}}{8m_{Q}^{2}}\right)\left[\pi_{0}-\frac{1}{2m_{Q}}(\bfpi\bfsigma)^{2}\;-\right.$
$\displaystyle\left.-\;\frac{1}{8m_{Q}^{2}}\,\left(-(\vec{D}\vec{E})+2\bfsigma\cdot\vec{E}\times\bfpi\right)\,\right]\left(1+\frac{(\bfsigma\bfpi)^{2}}{8m_{Q}^{2}}\right)\frac{1+\gamma_{0}}{2}\,Q$
$\displaystyle+\;{\cal O}\left(\frac{1}{m_{Q}^{3}}\right)\,,$ (10.1)

where $\bfsigma$ denote the Pauli matrices and

$({\bfpi}\bfsigma)^{2}={\bfpi}^{2}+\bfsigma\vec{B}\,,$

$\vec{E}$ and $\vec{B}$ denote the background chromoelectric and
chromomagnetic fields, respectively. Moreover, the operator $\pi_{\mu}$ is
defined through

$\displaystyle iD_{\mu}Q(x)$
$\displaystyle=e^{-im_{Q}v_{\mu}x_{\mu}}\left(m_{Q}v_{\mu}+iD_{\mu}\right)\tilde{Q}(x)$
(10.2) $\displaystyle\equiv
e^{-im_{Q}v_{\mu}x_{\mu}}\left(m_{Q}v_{\mu}+\pi_{\mu}\right)\tilde{Q}(x)$

where $v_{\mu}$ is the heavy quark four-velocity.

The set of operators presented in (10.1) plays the same basic role in
$1/m_{Q}$ expansion as the set in Table 1 in SVZ sum rules.

In the remainider of this section I will briefly mention some classic problems
with heavy quarks which were successfully solved in the given paradigm.

###  10.3 CGG/BUV theorem

Up to order $1/m_{Q}^{2}$ all inclusive decay widths of the $H_{Q}$ mesons are
universal and coincide with the parton-model result for the $Q$ decay chay ,
BUV ,

$\Gamma=\Gamma_{0}\left(1-\frac{\mu_{\pi}^{2}}{2m_{Q}^{2}}\right),\qquad\mu_{\pi}^{2}=\frac{1}{2M_{H_{Q}}}\langle
H_{Q}|\bar{Q}{\vec{\pi}}^{2}Q|H_{Q}\rangle$ (10.3)

where $\Gamma_{0}$ is the parton model result. There are no corrections
$O(1/m_{Q})$. This statement is known in the literature as the CGG/BUV
theorem.

###  10.4 Spectra and line shapes

Lepton spectra in semileptonic $H_{Q}$ decays were derived in Bigi . The
leading corrections arising at the $1/m_{Q}$ level were completely expressed
in terms of the difference in the mass of $H_{Q}$ and $Q$. Nontrivial effects
appearing at the order $1/m_{Q}^{2}$ were shown to affect mainly the endpoint
region; they are different for meson and baryon decays as well as for beauty
and charm decays.

The theory of the line shape in $H_{Q}$ decays, such as $B\rightarrow
X_{s}\gamma$ where $X_{s}$ denotes the inclusive hadronic state with the $s$
quark, resembles that of the Mössbauer effect. It is absolutely remarkable
that for 10 years there were no attempts to treat the spectra and line shapes
along essentially the same lines as it had been done in deep inelastic
scattering (DIS) in the 1970s. Realization of this fact came only in 1994;
technical implementation of the idea was carried out in JR , motion- , and
motion .

###  10.5 Hard gluons

Hard-gluon contributions special for the heavy quark theory result in powers
of the logarithms $\alpha_{s}\log\left(m_{Q}/\mu\right)$. They determine the
coefficients $c_{i}$ in Eq. (8.2) through the anomalous dimensions of the
corresponding operators. They were discovered in vsp and were called the
hybrid logarithms. In HQET they are referred to as matching logarithms.

###  10.6 In conclusion

Concluding the heavy quark portion I should add that Kolya Uraltsev
(1957-2013), one of the major contributors in heavy quark theory died in 2013
at the peak of his creative abilities (see ShifmanKolya ).

Concerning the OPE-based methods in QCD in general, I would like to make an
apology to the many authors whose works have not been directly cited. The size
limitations are severe. The appropriate references are given in the review
papers listed in the text above.

Just for the record, a couple of reviews which are tangentially connected to
the topic of the present article are given in Refs. Shifm4 and Shifm5 .

##  11 Recent developments unrelated to the OPE-based methods

Quantum field theories from the same class as QCD are now experiencing
dramatic changes and rapid advances in a deeper understanding of anomalies. I
want to mention two crucial papers: Gaiotto:2014kfa and Gaiotto:2017yup . The
latter demonstrates that at $\theta=\pi$ there is a discrete ’t Hooft anomaly
involving time reversal and the center symmetry. It follows that at
$\theta=\pi$ the vacuum cannot be a trivial non-degenerate gapped state.

## Acknowledgments

This work is supported in part by DOE grant DE-SC0011842.

I am grateful to Franz Gross, Alexander Khodjamirian, Alexander Lenz and
Blaženka Melić for very useful discussions and comments.

## References

  * (1)
  * (2) G. ’t Hooft, Renormalization of Massless Yang-Mills Theory, Nul. Phys. B33, 173 (1971).
  * (3) G. ’t Hooft, Renormalizable Lagrangian for Massive Yang-Mills Fields, Nul. Phys. B35, 167 (1971).
  * (4) Proceedings of the 16th International Conference On High-Energy Physics, 6-13 Sep 1972, Batavia, Illinois, Edited by J.D. Jackson and A. Roberts.
  * (5) L. D. Faddeev and V. N. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. B 25, 29-30 (1967). [A longer and more comprehensible version appeared in Russian as Kiev preprint ITP 67-36. In the beginning of the 1970s, it was translated in English by B. Lee (NAL-THY-57, 1972). Apparently, in GtH1 ; GtH2 ’t Hooft used the short version while I could use the longer one.]
  * (6) D. J. Gross, Twenty five years of asymptotic freedom, Nucl. Phys. B Proc. Suppl. 74, 426-446 (1999) [arXiv:hep-th/9809060].
  * (7) M. Shifman, Historical curiosity: how asymptotic freedom of the Yang-Mills theory could have been discovered three times before Gross, Wilczek, and Politzer, but was not, [arXiv:2203.12030 [physics.hist-ph]].
  * (8) D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30, 1343-1346 (1973); H. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30, 1346-1349 (1973).
  * (9) S. Weinberg, The U(1) Problem, Phys. Rev. D 11, 3583-3593 (1975).
  * (10) E. Bogomolnyi, V. Novikov, and M. Shifman, Behaviour of the physical charge at small distances in nonabelian gauge theories, Sov. J. Nucl. Phys. 20, 110 (1974), [online on https://www.academia.edu/74247005/ ].
  * (11) K. G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179, 1499-1512 (1969).
  * (12) M. Shifman, A. Vainshtein and V. Zakharov, JETP Lett. 22 55 (1975); Nucl. Phys. B120, 316 (1977); Sov. Phys. JETP 45 670, (1977).
  * (13) A. Vainshtein,How penguins started to fly, Int. J. Mod. Phys. A 14, 4705-4719 (1999) [arXiv:hep-ph/9906263].
  * (14) M. K. Gaillard and B. Lee, Phys. Rev. Lett. 33, 108 (1974); G. Altarelli and L. Miani, Phys. Lett. B52, 351 (1974).
  * (15) M. Shifman, A. Vainshtein and V. Zakharov, Gluon condensate and lepton decays of vector mesons, JETP Lett. 27, 55 (1978) [Submitted on November 9, 1977].
  * (16) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147, 385; 448; 519 (1979).
  * (17) V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, Wilson’s Operator Expansion: Can It Fail?,” Nucl. Phys. B 249, 445 (1985).
  * (18) M. Shifman, Snapshots of hadrons or the story of how the vacuum medium determines […] Prog. Theor. Phys. Suppl. 131, 1 (1998) [arXiv:hep-ph/9802214].
  * (19) M. Shifman, Resurgence, operator product expansion, and remarks on renormalons in supersymmetric Yang-Mills theory, J. Exp. Theor. Phys. 120, no.3, 386-398 (2015) [arXiv:1411.4004].
  * (20) N. Seiberg and E. Witten, Nucl. Phys. B 426, 19-52 (1994) [erratum: Nucl. Phys. B 430, 485-486 (1994)] [arXiv:hep-th/9407087]; Nucl. Phys. B 431, 484-550 (1994) [arXiv:hep-th/9408099].
  * (21) I. Bigi, M. Shifman, N. Uraltsev, and A. Vainshtein, The Pole mass of the heavy quark. Perturbation theory and beyond, Phys. Rev. D 50, 2234 (1994) [arXiv:hep-ph/9402360]; M. Beneke and V. Braun, Nucl. Phys. B 426, 301 (1994) [arXiv:hep-ph/9402364].
  * (22) I. Bigi, M. Shifman and N. Uraltsev, Aspects of heavy quark theory, Ann. Rev. Nucl. Part. Sci. 47, 591-661 (1997) [arXiv:hep-ph/9703290].
  * (23) D. Schubring, C. H. Sheu and M. Shifman, Treating divergent perturbation theory: Lessons from exactly solvable 2D models at large $N$, Phys. Rev. D 104, no.8, 085016 (2021) [arXiv:2107.11017].
  * (24) See the collection M. Shifman, Vacuum Structure and QCD Sum Rules, (North-Holland, Elsevier Science Publishers, 1992); Vacuum structure and QCD sum rules: Introduction, Int. J. Mod. Phys. A 25, 226 (2010)) [https://www.academia.edu/s/7655b6c4ec].
  * (25) V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, Are All Hadrons Alike?, Nucl. Phys. B 191, 301 (1981).
  * (26) V. M. Braun, Light cone sum rules, arXiv:hep-ph/9801222.
  * (27) A. Khodjamirian, T. Mannel and N. Offen, Form-factors from light-cone sum rules with B-meson distribution amplitudes, Phys. Rev. D 75, 054013 (2007).
  * (28) P. Colangelo and A. Khodjamirian, QCD sum rules, a modern perspective, [arXiv:hep-ph/0010175].
  * (29) W. Braunschweig et al. [DASP], Radiative Decays of the J/psi and Evidence for a New Heavy Resonance, Phys. Lett. B 67, 243-248 (1977).
  * (30) M. Shifman, A. Vainshtein, M. Voloshin and V. Zakharov, $\eta_{c}$ Puzzle in Quantum Chromodynamics, Phys. Lett. B 77, 80 (1978).
  * (31) R. Partridge et al. Observation of an $\eta_{c}$ Candidate State with Mass 2978 $\pm$ MeV, Phys. Rev. Lett. 45, 1150-1153 (1980).
  * (32) N. Uraltsev, Topics in the heavy quark expansion, [arXiv:hep-ph/0010328].
  * (33) M. Shifman, Recent progress in the heavy quark theory, [arXiv:hep-ph/9505289] (published in ITEP Lectures […], WS, 1999).
  * (34) A. Manohar and M. Wise, Heavy Quark Physics, (Cambridge University Press, 2000).
  * (35) M. Shifman and M. Voloshin, Preasymptotic Effects in Inclusive Weak Decays of Charmed Particles, Sov. J. Nucl. Phys. 41, 120 (1985); Hierarchy of Lifetimes of Charmed and Beautiful Hadrons, Sov. Phys. JETP 64, 698 (1986).
  * (36) Alexander Lenz, Lifetimes and heavy quark expansion, Int. J. Mod. Phys. A, 30, 1543005 (2015) [arXiv: 1405.3601].
  * (37) M. Kirk, A. Lenz and T. Rauh, Dimension-six matrix elements for meson mixing and lifetimes from sum rules, JHEP 12, 068 (2017) [E: JHEP 06, 162 (2020)] [arXiv:1711.02100 [hep-ph]]; D. King, A. Lenz, M. L. Piscopo, T. Rauh, A. V. Rusov and C. Vlahos, Revisiting Inclusive Decay Widths of Charmed Mesons, [arXiv:2109.13219].
  * (38) M. Fael, K. Schönwald and M. Steinhauser, Third order corrections to the semileptonic $b\to{c}$ and the muon decays, Phys. Rev. D 104, no.1, 016003 (2021) [arXiv:2011.13654].
  * (39) A. Lenz, M. L. Piscopo and A. V. Rusov, Contribution of the Darwin operator to non-leptonic decays of heavy quarks, JHEP 12, 199 (2020) [arXiv:2004.09527]; T. Mannel, D. Moreno and A. Pivovarov, Heavy quark expansion for heavy hadron lifetimes: completing the $1/{m}_{b}^{3}$ corrections, JHEP 08, 089 (2020) [arXiv:2004.09485].
  * (40) D. King, A. Lenz, M. L. Piscopo, T. Rauh, A. V. Rusov and C. Vlahos, Revisiting Inclusive Decay Widths of Charmed Mesons, [arXiv:2109.13219].
  * (41) J. Gratrex, B. Melić and I. Nišandžić, Lifetimes of singly charmed hadrons, [arXiv:2204.11935].
  * (42) LHCb collaboration, Phys. Rev. Lett. 121, 092003 (2018) [1807.02024]; Phys. Rev. D 100, 032001 (2019) [1906.08350]; Sci. Bull. 67, 479 (2022) [2109.01334].
  * (43) Belle II Collaboration: F. Abudinén, et al., ArXiv 2208.08573 [hep-ex].
  * (44) B. Blok and M.A. Shifman, Lifetimes of charmed hadrons revisited. Facts and fancy, in 3rd Workshop on the Tau-Charm Factory, [hep-ph/9311331].
  * (45) S. Nussinov and W. Wetzel, Phys. Rev. D36 130, (1987).
  * (46) M. Voloshin and M. Shifman, On Production of $D$ and $D^{*}$ Mesons in B Meson Decays, Sov. J. Nucl. Phys. 47, 511 (1988).
  * (47) N. Isgur and M. Wise, Phys. Lett. B232 (1989) 113; Phys. Lett. B237 (1990) 527.
  * (48) E. Eichten and B. R. Hill, An Effective Field Theory for the Calculation of Matrix Elements Involving Heavy Quarks, Phys. Lett. B 234, 511-516 (1990); Howard Georgi, An effective field theory for heavy quarks at low energies, Phys. Lett. B 240, 447, 1990.
  * (49) J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B247 (1990) 399\.
  * (50) I. Bigi, N. Uraltsev and A. Vainshtein, Phys. Lett. B293 (1992) 430; (E) B297 (1993) 477.
  * (51) I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, QCD predictions for lepton spectra in inclusive heavy flavor decays, Phys. Rev. Lett. 71, 496-499 (1993) [arXiv:hep-ph/9304225].
  * (52) R. Jaffe and L. Randall, Nucl. Phys. B412 (1994) 79.
  * (53) M. Neubert, Phys. Rev. D49 (1994) 3392.
  * (54) I. Bigi, M. Shifman, N.G. Uraltsev and A. Vainshtein, Int. Journ. Mod. Phys., A9 (1994) 2467.
  * (55) M. B. Voloshin and M. A. Shifman, On the annihilation constants of mesons consisting of a heavy and a light quark…, Sov. J. Nucl. Phys. 45, 292 (1987); H. D. Politzer and M. B. Wise, Phys. Lett. B 206, 681 (1988).
  * (56) M. Shifman, New and Old about Renormalons: in Memoriam Kolya Uraltsev, Int. J. Mod. Phys. A 30, no.10, 1543001 (2015) [arXiv:1310.1966].
  * (57) M. Shifman, Quark-hadron duality, in At the Frontier of Particle Physics/Handbook of QCD, Ed. M. Shifman (World Scientific, Singapore, 2001), Vol. 3, p. 1447 [arXiv:hep-ph/0009131].
  * (58) M. Shifman, QCD sum rules: The Second decade, in Proc. Conf. QCD: 20 Years Later, eds. P. Zerwas and H. Kastrup, World Scientific, 1992, Volume 2, page 77 [arXiv:hep-ph/9304253].
  * (59) D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02, 172 (2015) [arXiv:1412.5148].
  * (60) D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05, 091 (2017) [arXiv:1703.00501].
  * (61)


00footnotetext: MSC 2020: Primary: 11A99, 00A05, 11B33, 00A66; Keywords:
Visual tour, $e$, $\pi$, Euler’s constant, Euler’s limit, Geometric
progression.

# A visual tour via the Definite Integration $\int_{a}^{b}\frac{1}{x}dx$

Bikash Chakraborty  Department of Mathematics, Ramakrishna Mission Vivekananda
Centenary College, Rahara, West Bengal 700 118, India.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Geometrically, $\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve
$\frac{1}{x}$ from $a$ to $b$, where $0<a<b$, and this area gives a positive
number. Using this area argument, in this expository note, we present some
visual representations of some classical results. For examples, we demonstrate
an area argument on a generalization of Euler’s limit
$\left(\lim\limits_{n\to\infty}\left(\frac{(n+1)}{n}\right)^{n}=e\right)$.
Also, in this note, we provide an area argument of the inequality
$b^{a}<a^{b}$, where $e\leq a<b$, as well as we provide a visual
representation of an infinite geometric progression. Moreover, we prove that
the Euler’s constant $\gamma\in[\frac{1}{2},1)$ and the value of $e$ is near
to $2.7$.

Some parts of this expository article has been accepted for publication in
Resonance – Journal of Science Education, The Mathematical Gazette, and
International Journal of Mathematical Education in Science and Technology.

00footnotetext: 2010 Mathematics Subject Classification: Primary 00A05,
Secondary 00A66.

##  1\. Introduction

It is well known that the function $\phi:(0,+\infty)\to\mathbb{R}$, defined by
$\phi(x)=\frac{1}{x}$, is a monotone decreasing and continuous. Thus $\phi(x)$
is Riemann integrable on $[a,b]$ where $0<a<b$. Geometrically,
$\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve $y=\frac{1}{x}$
from $a$ to $b$. Moreover, it is useful to observe that the funtion
$f(t)=\int_{1}^{t}\frac{1}{x}dx$ is strictly increasing for $t\geq 1$.

Figure 1. $\ln b-\ln a<\frac{1}{2}\cdot(\frac{1}{a}+\frac{1}{b})\cdot(b-a).$

Let $a$ and $b$ be two positive real numbers. Then the fact
$\frac{1}{x}+\frac{x}{ab}\leq\frac{1}{a}+\frac{1}{b}$ for $a\leq x\leq b$ (as
$(x-b)(x-a)\leq 0$)is equivalent to saying that the line
$y=\frac{1}{a}+\frac{1}{b}-\frac{x}{ab}$ lies above the curve $y=\frac{1}{x}$
for $a\leq x\leq b$. Thus, Figure 1 shows that the area under the curve
$y=\frac{1}{x}$ from $a$ to $b$ is less than the area of the trapeziums
covering it, i.e.,

$\ln b-\ln a<\frac{1}{2}\cdot(\frac{1}{a}+\frac{1}{b})\cdot(b-a).$

Figure 2. $\ln b-\ln a>\frac{2}{a+b}\cdot(b-a).$

Again, the fact $\frac{1}{x}+\frac{4x}{(a+b)^{2}}\geq\frac{4}{a+b}$ for $x>0$
(follows from AM-GM inequality) is equivalent to saying that the curve
$y=\frac{1}{x}$ lies above its tangent line
$y=\frac{4}{a+b}-\frac{4x}{(a+b)^{2}}$ at the point
$(\frac{a+b}{2},\frac{2}{a+b})$.

Thus, figure 2 gives the visualization that the area under the curve
$y=\frac{1}{x}$ from $a$ to $b$ is greater than the area of the trapezium
below it, i.e.,

$\ln b-\ln a>\frac{2}{a+b}\cdot(b-a).$

##  2\. Tour-1

In a recent note of the American Mathematical Monthly, R. Farhadian ([6]) made
a beautiful generalization of Euler’s limit
$\left(\lim\limits_{n\to\infty}\left(\frac{(n+1)}{n}\right)^{n}=e\right)$ as
follows:

######  Theorem 1.

([6]) Let $A_{n}$ be a strictly increasing sequence of positive numbers
satisfying the asymptotic formula $A_{(n+1)}\sim A_{n}$, and let
$d_{n}=A_{(n+1)}-A_{n}$. Then

$\lim\limits_{n\to\infty}\left(\frac{A_{(n+1)}}{A_{n}}\right)^{\frac{A_{n}}{d_{n}}}=e.$

Now, we will provide a second proof of it, which is purely pictorial.

Figure 3.

From Figure 3, it is clear that

$\displaystyle\frac{2}{A_{n}+A_{n+1}}\cdot(A_{n+1}-A_{n})<\ln(A_{n+1})-\ln(A_{n})<\frac{1}{2}\cdot\left(\frac{1}{A_{n}}+\frac{1}{A_{n+1}}\right)\cdot(A_{n+1}-A_{n}),$

i.e.,

$\frac{2}{1+\frac{A_{n+1}}{A_{n}}}<\ln\left(\frac{A_{n+1}}{A_{n}}\right)^{\frac{A_{n}}{d_{n}}}<\frac{1}{2}\cdot\left(1+\frac{A_{n}}{A_{n+1}}\right).$

Since $A_{(n+1)}\sim A_{n}$, thus
$\lim\limits_{n\to\infty}\left(\frac{A_{(n+1)}}{A_{n}}\right)^{\frac{A_{n}}{d_{n}}}=e.$

######  Remark 1.

It is well-known that if $a_{n}$ is a sequence of positive numbers satisfying
$\lim\limits_{n\to+\infty}a_{n}=0$, then

$\lim\limits_{n\to+\infty}\left(1+a_{n}\right)^{\frac{1}{a_{n}}}=e$

.
Here, we will provide a visual proof of it.

Figure 4.

From the figure, it is clear that

$\displaystyle\frac{2}{2+a_{n}}\cdot a_{n}<\ln(1+a_{n})-\ln
1<\frac{1}{2}\cdot\left(1+\frac{1}{1+a_{n}}\right)\cdot a_{n},$

i.e.,

$\lim\limits_{n\to+\infty}\left(1+a_{n}\right)^{\frac{1}{a_{n}}}=e$

.

##  3\. Tour-2

Next, we provide a pictorial description of a geometric series

$1+\frac{1}{r}+\frac{1}{r^{2}}+\frac{1}{r^{3}}+\ldots=\frac{r}{r-1}$

when $r>1$.

Figure 5. $\int_{1}^{r}\frac{dt}{t}=\int_{\frac{1}{r}}^{1}\frac{dt}{t}$

Since $\int_{1}^{r}\frac{dx}{x}=\int_{\frac{1}{r}}^{1}\frac{dy}{y}$, thus the
area covered by the rectangle
$\\{(0,1);(1,1);(1,\frac{1}{r});(0,\frac{1}{r})\\}$ is same as the area
covered by the rectangle $\\{(1,0);(r,0);(r,\frac{1}{r});(1,\frac{1}{r})\\}$.
Thus

$\displaystyle\left(1-\frac{1}{r}\right)\cdot 1$ $\displaystyle=$
$\displaystyle\left(\frac{1}{r}-\frac{1}{r^{2}}\right)\cdot(r-1)+\left(\frac{1}{r^{2}}-\frac{1}{r^{3}}\right)\cdot(r-1)+\ldots,$
$\displaystyle=$
$\displaystyle\frac{(r-1)^{2}}{r^{2}}\cdot\left(1+\frac{1}{r}+\frac{1}{r^{2}}+\ldots\right),$

i.e.,

$\left(1+\frac{1}{r}+\frac{1}{r^{2}}+\ldots\right)=\frac{r}{r-1},$

which gives the required equality.

##  4\. Tour-3

The two constants $e$ and $\pi$ have encouraged many visual proofs of the
inequality ${\pi}^{e}<e^{\pi}$. In a recent Mathematical Intelligencer note
([2]), the author provided a visual proof of the inequality $\pi^{e}<e^{\pi}$.
However, their visual proof can be used to show the more general inequality
$b^{a}<a^{b}$, where $e\leq a<b$.

Visual Proof-1
Since $\ln a\geq 1$, thus $\frac{1}{x\ln a}\leq\frac{1}{x}$ for $x>0$. Thus
the Figure 6 shows that the area under the curve $y=\frac{1}{x\ln a}$ from $a$
to $b$ is less than the area of the rectangle PQRS, i.e.,

Figure 6. $\int\limits_{a\ln a}^{b\ln a}\frac{dx}{x}<\frac{1}{a}\cdot(b-a)\ln
a.$

$\displaystyle\frac{\ln b}{\ln a}-1=\int_{a}^{b}\frac{dx}{x\ln
a}<\frac{1}{a}(b-a)=\frac{b}{a}-1,$

i.e.,

$\displaystyle b^{a}<a^{b}.$

Visual Proof-2
Also, Figure 7 shows that the area under the curve $y=\frac{1}{x}$ from $a\ln
a$ to $b\ln a$ is less than the area of the rectangle covering it. Since
$e\leq a$, so $1\leq\ln a$, i.e., $a\leq a\ln a<b\ln a$.

Figure 7. $\int\limits_{a\ln a}^{b\ln a}\frac{dx}{x}<\frac{1}{a}\cdot(b-a)\ln
a.$

$\displaystyle\ln b-\ln a<\ln a\cdot\left(\frac{b}{a}-1\right),$

i.e.,

$\displaystyle\frac{\ln b}{\ln a}-1<\frac{b}{a}-1.$

Thus

$\displaystyle a\ln b<b\ln a,\text{~{}~{}i.e.,~{}~{}}b^{a}<a^{b}.$

######  Corollary 1.

([3]) If we take $a=e$, then $(a,0)$ and $(a\ln a,0)$ will be coincided with
$(e,0)$. Also, $(a,\frac{1}{a})$ and $(a\ln a,\frac{1}{a})$ will be coincided
with $(e,\frac{1}{e})$. Thus the figure 7 becomes :

Figure 8. $\int_{e}^{b}\frac{dx}{x}<\frac{1}{e}(b-e).$

Thus we get $\ln b-1<\frac{b}{e}-1$, i.e., $b^{e}<e^{b}$.

######  Corollary 2.

By taking $a=e$ and $b=\pi$, we get $\pi^{e}<e^{\pi}$ ([2]).

##  5\. Tour-4

Considering the definition of the number $e$ by the equation

$1=\int_{1}^{e}\frac{1}{x}dx,$

we are explaining that why the value of $e$ is near to $2.7$. Basically, we
will show that $2.7<e<2.75$.

Figure 9. $\int_{a}^{b}\frac{1}{x}dx>\frac{2(b-a)}{b+a}.$

Applying the lower bound of the integral $\int_{a}^{b}\frac{1}{x}dx$ (see,
Figure 9), we have

$\displaystyle\int_{4}^{11}\frac{1}{x}dx$ $\displaystyle=$
$\displaystyle\int_{4}^{6}\frac{1}{x}dx+\int_{6}^{9}\frac{1}{x}dx+\int_{9}^{11}\frac{1}{x}dx$
$\displaystyle>$ $\displaystyle\frac{2}{5}+\frac{2}{5}+\frac{1}{5}$
$\displaystyle=$ $\displaystyle 1,$

i.e.,

$\int_{1}^{\frac{11}{4}}\frac{1}{x}dx>\int_{1}^{e}\frac{1}{x}dx~{}~{}~{}~{}~{}\Rightarrow~{}~{}e<2.75.$

Again, applying the upper bound of the integral $\int_{a}^{b}\frac{1}{x}dx$
(see, Figure 10), we have

Figure 10.
$\int_{a}^{b}\frac{1}{x}dx<\frac{1}{2}\cdot(\frac{1}{a}+\frac{1}{b})\cdot(b-a).$

$\displaystyle\int_{10}^{27}\frac{1}{x}dx$ $\displaystyle=$
$\displaystyle\int_{10}^{12}\frac{1}{x}dx+\int_{12}^{15}\frac{1}{x}dx+\int_{15}^{18}\frac{1}{x}dx+\int_{18}^{21}\frac{1}{x}dx+\int_{21}^{24}\frac{1}{x}dx+\int_{24}^{27}\frac{1}{x}dx$
$\displaystyle<$
$\displaystyle\frac{1}{2}\cdot(\frac{2}{10}+\frac{2}{12})+\frac{1}{2}\cdot(\frac{3}{12}+\frac{3}{15})+\frac{1}{2}\cdot(\frac{3}{15}+\frac{3}{18})++\frac{1}{2}\cdot(\frac{3}{18}+\frac{3}{21})$
$\displaystyle+\frac{1}{2}\cdot(\frac{3}{21}+\frac{3}{24})+\frac{1}{2}\cdot(\frac{3}{24}+\frac{3}{27})$
$\displaystyle<$ $\displaystyle 1,$

i.e.,

$\int_{1}^{2.7}\frac{1}{x}dx<\int_{1}^{e}\frac{1}{x}dx~{}~{}~{}~{}~{}~{}\Rightarrow~{}~{}e>2.70.$

##  6\. Tour-5

The Euler’s constant is defined as

$\gamma=\lim\limits_{n\to\infty}\gamma_{n},$

where

$\gamma_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots\frac{1}{n}-\ln n.$

In this visual tour, we will show that the existence of the Euler’s constant
$\gamma$ and $\gamma\in(\frac{1}{2},1)$. Since

$1-\gamma_{n}=\ln n-\frac{1}{2}-\frac{1}{3}-\ldots\frac{1}{n},$

thus $1-\gamma_{n}$ can be described as the shaded area in the following
figure.

Figure 11. $\ln n-\frac{1}{2}-\frac{1}{3}-\ldots-\frac{1}{n}=1-\gamma_{n}$

It is seen from the figure that $\\{1-\gamma_{n}\\}$ is strictly monotone
increasing, and $1-\gamma_{n}>0.$ That is $\\{\gamma_{n}\\}$ is strictly
monotone decreasing sequence and $\gamma_{n}$ is bounded above by $1$.

Next, we define a sequence $\\{A_{n}\\}$, where

Figure 12. $A_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}-\ln(n+1)$

$A_{n}$ is described by the shaded area in the following figure. Then it is
seen that $\\{A_{n}\\}$ is strictly monotone increasing and $A_{n}>0.$ Since

$A_{n}=\gamma_{n}-\ln(n+1)+\ln n,$

thus

$\gamma_{n}>\ln(n+1)-\ln n,$

which means, by Figure 13, that

Figure 13. $\ln b-\ln a>\frac{b-a}{b}$, where $b>a>0$.

$\gamma_{n}>\ln(n+1)-\ln n>\frac{1}{n}>0,$

i.e., $\gamma_{n}$ is bounded below by $0$. Thus the Euler’s constant

$\gamma=\lim\limits_{n\to\infty}\gamma_{n},$

exist, and $\gamma\in[0,1)$.

Next, we assume that

$\Gamma_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots\frac{1}{n}-\ln(n+\frac{1}{2}).$

Thus,

$\Gamma_{n+1}-\Gamma_{n}=\frac{1}{n+1}+\ln(n+\frac{1}{2})-\ln(n+\frac{3}{2}).$

Thus by the figure 14,

Figure 14. $\ln b-\ln a<\frac{b-a}{a}$, where $b>a>0$.

$\Gamma_{n+1}-\Gamma_{n}<\frac{1}{n+1}-\frac{1}{n+\frac{1}{2}}<0,$

i.e., $\\{\Gamma_{n}\\}$ is strictly monotone decreasing sequence.

Again, Figure 14 shows that

$\displaystyle\int_{n}^{n+\frac{1}{2}}\frac{1}{x}dx<\frac{1}{2n},$

and, Figure 15 shows that

Figure 15. $\ln b-\ln a<\frac{1}{2}(\frac{1}{a}+\frac{1}{b})\cdot(b-a).$

$\displaystyle\int_{1}^{n}\frac{1}{x}dx$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\int_{i}^{i+1}\frac{1}{x}dx$ $\displaystyle<$
$\displaystyle\sum_{i=1}^{n-1}\frac{1}{2}(\frac{1}{i}+\frac{1}{i+1})$
$\displaystyle=$ $\displaystyle
1+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{n-1}+\frac{1}{2n}.$

Thus

$\displaystyle\ln\left(n+\frac{1}{2}\right)$ $\displaystyle=$
$\displaystyle\int_{1}^{n+\frac{1}{2}}\frac{1}{x}dx$ $\displaystyle=$
$\displaystyle\int_{1}^{n}\frac{1}{x}dx+\int_{n}^{n+\frac{1}{2}}\frac{1}{x}dx$
$\displaystyle<$ $\displaystyle
1+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{n-1}+\frac{1}{n},$

Thus
$\Gamma_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots\frac{1}{n}-\ln(n+\frac{1}{2})>\frac{1}{2}.$
Hence $\lim\limits_{n\to\infty}\Gamma_{n}$ exist and
$\lim\limits_{n\to\infty}\Gamma_{n}\in[\frac{1}{2},1)$.
As $\gamma_{n}-\Gamma_{n}=\ln(n+\frac{1}{2})-\ln n$, thus, applying Figures 13
and 14, we get

$\frac{1}{2n+1}<\gamma_{n}-\Gamma_{n}<\frac{1}{2n},$

i.e.,

$\gamma=\lim\limits_{n\to\infty}\gamma_{n}=\lim\limits_{n\to\infty}\Gamma_{n}~{}~{}~{}\text{and}~{}~{}~{}~{}\gamma\in[\frac{1}{2},1).$

## References

  * [1] R. G. Bartle, and D. R. Sherbert, Introduction to Real Analysis (Fourth Edition), Wiley, (2014).
  * [2] B. Chakraborty, A Visual Proof that $\pi^{e}<e^{\pi}$, Math. Intelligencer, 41(1) (2019), pp. 56.
  * [3] B. Chakraborty, A Visual Proof that $b^{e}<e^{b}$ when $b>e$, The Mathematical Gazette, To apper in 2024.
  * [4] B. Chakraborty, A Visual Proof that $b^{a}<a^{b}$ when $e\leq a<b$, International Journal of Mathematical Education in Science and Technology, doi 10.1080/0020739X.2022.2102547.
  * [5] B. Chakraborty and S. Chakraborty, A Visual Proof that $2.5<e<2.75$, Resonance – Journal of Science Education (Accepted).
  * [6] R. Farhadian, A Generalization of Euler’s Limit, The American Mathematical Monthly, Volume 129, 2022, No. 4, pp. 384.
  * [7] C. Gallant, $A^{B}>B^{A}$ for $e\leq A<B$, Math. Mag., 64(1) (1991), pp. 31.
  * [8] H. N. Gupta, A simple proof that $e<3$, Math. Gaz., 62 (1978), pp. 124.
  * [9] Nick Lord, Proof without words that $2<e<3$, Math. Gaz. 95 (2011), pp. 115-117.


# Spatial-Temporal Convolutional Attention for Mapping Functional Brain
Networks

###### Abstract

Using functional magnetic resonance imaging (fMRI) and deep learning to
explore functional brain networks (FBNs) has attracted many researchers.
However, most of these studies are still based on the temporal correlation
between the sources and voxel signals, and lack of researches on the dynamics
of brain function. Due to the widespread local correlations in the volumes,
FBNs can be generated directly in the spatial domain in a self-supervised
manner by using spatial-wise attention (SA), and the resulting FBNs has a
higher spatial similarity with templates compared to the classical method.
Therefore, we proposed a novel Spatial-Temporal Convolutional Attention (STCA)
model to discover the dynamic FBNs by using the sliding windows. To validate
the performance of the proposed method, we evaluate the approach on HCP-rest
dataset. The result indicate that STCA can be used to discover FBNs in a
dynamic way which provide a novel approach to better understand human brain.

Index Terms—  fMRI, Attention Mechanism, Functional Brain Network, Brain
Function Dynamic

##  1 Introduction

Functional magnetic resonance imaging (fMRI) is a non-invasive imaging method
to estimate brain function which indirectly detect the activities of brain
function by measuring the blood-oxygen level dependence (BOLD) [1, 2]. To
explore the functional brain networks (FBNs), various analytical approaches
have been proposed, each providing a different model for mapping brain
functional pattern. Most of the previous methods for mapping FBNs using fMRI
are based on the temporal correlation between the source and voxel signals,
such as indenpendent component analysis (ICA) [3] and sparse dictionary
learning (SDL) [4]. Many of these approaches can be viewed as a blind source
separation problem, the fMRI signals are modeled as a linear combination of
the sources and the temporal patterns [5]. Although significant results have
been achieved by these methods, the process of extracting the sources is still
in a linearity way, which makes these classical methods limited by their
shallow nature.

Recently, to overcome the shallow nature of the linear models, various of deep
learning based methods have been proposed to discover the FBNs. Most of these
methods are based on the autoencoders, they use different autoencoders to
extract the sources in an self-supervised manner, and then use the generative
linear model, such as LASSO to generate the FBNs [6, 7]. In general, these
deep learning based methods can indeed extract better encoder representations
as the sources than the classical methods, such as ICA and SDL, but still
generate FBNs in a linear and independent manner, with the sources extraction
and the FBNs generation as 2 separate steps. Generating the FBNs in such way
is time-consuming and does not fully utilize the advantages of deep learning,
and cannot directly generate the FBNs with deep learning. In fact, the FBNs
should be able to be represented at the feature extraction stage if we
directly extract the spatial features, and generating the FBNs in a linear and
independent way would require more longer source signals to increase
confidence. At the same time, there is still a lack of methods for using deep
learning to generate the dynamic FBNs. The mainstream methods usually adopt
sliding window technology to generate the dynamic FBNs, they usually set the
window size and the sliding step, and then input the data into commonly used
classical models, such as ICA [8] and SDL [9]. In addition to sliding window
approaches, there are also some methods tried to discover the spatial pattern
in dynamic brain function, such as co-activation patterns (CAPs) [10, 11, 12].
The most widely used CAP model assumes that one CAP is active at each time
point, and each CAP represents a spatial pattern of the source, usually
clustering the CAP based on spatial similarity. Such a view oversimplified
about the brain function and can only provides a limited amount of CAP.

To solve these problems, we proposed a novel deep learning based method called
Spatial-Temporal Convolutional Attention (STCA) to directly generate the high
quality dynamic FBNs by the sliding window technology. The STCA model is based
on the convolutional neural network (CNN) architecture and trained in a self-
supervised manner. The CNN architecture can make full use of the FBN’s prior
knowledge which is the widespread local correlation in the fMRI signals, it is
suitable for modeling the fMRI signals naturally. To validate the performance
of the STCA model, we perform the experiments on the HCP-rest dataset [13]
which is a resting-state fMRI (rsfMRI) dataset. The result demonstrate that
STCA can generate dynamic FBNs more better than the mainstream methods, such
as ICA and SDL, the generated FBNs have higher spatial similarity to the
templates. Source code of this work can be found in
https://github.com/SNNUBIAI/STCAE.

##  2 Methods

###  2.1 Dataset and preprocessing

In this paper, we adopt the Human Connectome Project (HCP) 900 Subjects Data
Release [13] as the experiments data. The HCP-rest is a publicly accessible
rsfMRI dataset (available on https://db.humanconnectome.org.). Each individual
contains 1200 time points, all of these data have been officially preprocessed
and registered to the MNI152 $4\times 4\times 4$ standard template space. We
masked the fMRI data and resampled it to a size of $49\times 58\times 47$.

###  2.2 Spatial-Temporal Convolutional Attention (STCA)

The Fig 1 summarizes the proposed Spatial-Temporal Convolutional Attention
(STCA) based pipeline for modeling the FBNs.

Fig. 1: The architecture of the Spatial-Temporal Convolutional Attention. The
STCA used the spatial and channel-wise attention in the spatial and temporal
of the fMRI signals to discover the FBNs. The convolutional autoencoder module
is used to guide the attention modules to focus on the activated regions in
the training stage, it is not needed in the evaluate stage. The spatial-wise
attention module’s weight map are transformed to z-score values, which
represent the degree of functional activation on space, that is, the FBNs are
derived. The results are close to the known FBNs derived by the classical
methods, such as ICA and SDL.

Before going into the architectural details, we will briefly explain the
design idea. Since the existing mainstream practice is to extract the sources
through the encoder and then generate the FBNs with a generative linear model.
Extracting sources and constructing FBN are two separate steps, the noise is
unavoidable in the process of generating the FBNs, and usually need to take
longer time points of the fMRI signals to ensure the confidence. If the deep
neural network can extract the sources, the spatial pattern of the sources
should be represented in the feature extraction step. In fact, the spatial
pattern of the sources or called the FBNs, have a local correlation in the
spatial and temporal pattern. The local correlation can be easily observed in
the activation maps shown in Fig 3, the 10 resting state network (RSN) derived
by ICA [14]. The activated voxels tend to cluster together in the spatial and
the voxels in the activation regions are also correlated with each other in
the temporal. These spatial patterns widely exists in the brain functional
activities, inspired by the application of the spatial-wise attention (SA) in
computer vision tasks [15], we tried to introduce the SA to generate the FBNs,
tried to give higher weights to the activation regions. So that the output of
the SA module can be define as the FBNs.

The goal of our task is to train a neural network:

$f:X_{dyn}\longrightarrow S_{dyn}$ (1)

where $X_{dyn}=(X_{1},...,X_{t})$ is the sequence of the fMRI volumes with $t$
time points and $S_{dyn}\in\mathbb{R}^{N\times D\times H\times W}$ is the
weight maps of the volumes $X_{dyn}$ with number of the weight maps $N$, the
depth of the weight map $D$, the height of the weight map $H$ and the width of
the weight map $W$. The STCA is an independent attention module, which cannot
directly generate the weight maps in the absence of the labeled data. It needs
to be trained in a self-supervised manner to guide the attention module to
learn the FBNs. Autoencoders are commonly used self-supervised training
methods, which compress the high-dimensional information to low-dimensional
information to extract the sources. Guided by the autoencoder, the STCA learns
to focus on the spatial patterns of the source to help the encoder generate
better encoder representations as the sources, so that the spatial pattern of
the sources can be represented at the feature extraction stage.

The STCA consists of a 3D convolutional layer, a channel-wise attention (CA)
module and a spatial-wise attention (SA) module shown in Fig 1. The
convolutional autoencoder is just to guide the SA module and does not involve
in the evaluate stage. The convolutional layer and CA module are both designed
to help extract more useful features, only the SA module is used to locate the
regions where the FBNs are. We represent the $GAP$ as global average pooling
which will transform the 3D volumes to 1D vector, $W$ as the fully connected
layer, $sigmoid$ as the activation function that can map the features to
$[0,1]$. The CA can be represented as:

$CA=sigmoid(\sigma(\sigma(GAP(f^{l}){W_{1}}){W_{2}}){W_{3}})$ (2)
$\widetilde{f}^{l}=f^{l}\circ CA$ (3)

We represent the 3D convolutional layer’s output features as
$f^{l}\in\mathbb{R}^{C\times D\times W\times H}$, $conv3D$ as the 3D
convolutional layer, $\sigma$ as the activation function $GELU$,
$\widetilde{f}^{l}$ as final output of the SA module. The SA can be
represented as:

$SA=conv3D(\sigma(conv3D(f^{l})))$ (4) $\widetilde{f}^{l}=f^{l}\circ
sigmoid(SA)$ (5)

###  2.3 Generation FBNs based on the STCA

We adopt the sliding window technique to discover the dynamic FBNs. Here, the
window size is set as 40, stride is set as 1. Like the classical method, such
as ICA and SDL, both training and testing are performed in the same
individual. When training the STCA, the loss function of the autoencoder is
mean square error (MSE), the learning rate is set as 0.0001, the optimizer is
Adam and the epoch is set as 1. We convert $SA$ to z-score, and the threshold
is set as 3.6.

##  3 Experimental Results

###  3.1 Overview of the weight maps derived by STCA

The representative weight maps derived from the STCA are shown in the upper
panel of the Fig 2, these weight maps are all convert into z-score, the
thresholded maps shown in the lower panel. It can be seen that regions
activated with high weights can be easily explained by the known functional
networks, such as visual network, default mode network, etc. This result
demonstrate that the spatial pattern of the sources can indeed be captured by
the spatial-wise attention mechanism in the feature extraction stage which
means that we provide a new paradigm for discovering the functional networks.

Fig. 2: The weight and thresholded maps derived by STCA.

To evaluate the quality of the FBNs generated by STCA, we calculated the
spatial similarity between the generated FBNs and ten resting-state templates
and compared the results with ICA and SDL. The spatial similarity is defined
by the intersection over union rate (IoU) between two FBNs $N^{(1)}$ and
$N^{(2)}$, where $n$ refers to the volume size:

$IoU(N^{(1)},N^{(2)})=\frac{\sum_{i=1}^{n}|N_{i}^{(1)}\cap
N_{i}^{(2)}|}{\sum_{i=1}^{n}|N_{i}^{(1)}\cup N_{i}^{(2)}|}$ (6)

The similarity metrics $IoU(N_{STCA},N_{template})$, $IoU($
$N_{SDL},N_{template})$ and $IoU(N_{ICA},N_{template})$ are used to evaluate
the performance of STCA. We randomly selected an individual to do the
experiments, the results of the evaluation are shown in Fig 3 and Table 1, the
10 RSN templates are in line 4. The results indicate that the proposed STCA
can generate better FBNs than ICA.

Fig. 3: Comparision with the classical methods, such as ICA and SDL. The templates are based on [14] Table 1: IoU with templates in an individual in HCP-rest dataset. Template | ICA | SDL | STCA
---|---|---|---
RSN 1 | 0.6273 | 0.6053 | 0.6505
RSN 2 | 0.4260 | 0.4554 | 0.5695
RSN 3 | 0.2414 | 0.3340 | 0.3446
RSN 4 | 0.2889 | 0.3446 | 0.4250
RSN 5 | 0.1932 | 0.2635 | 0.4474
RSN 6 | 0.0888 | 0.2279 | 0.3900
RSN 7 | 0.1179 | 0.1992 | 0.4284
RSN 8 | 0.0540 | 0.1951 | 0.3159
RSN 9 | 0.1843 | 0.1890 | 0.2628
RSN 10 | 0.2436 | 0.1920 | 0.2597

###  3.2 Dynamic FBNs in resting state fMRI

The proposed STCA can be combined with sliding window technique to discover
dynamic FBNs. Here, the window size is set as 40, stride is set as 1. We plot
some representative functional network’s dynamic change process shown in Fig
4, which shows that some FBNs’ transferring processes. We can easily find that
the RSN fade in or fade out over time.Quantitatively, we also plot the spatial
similarity change between the FBNs generated by each window and 10 RSN
templates over time, which are shown in Fig 5, we can also see the fade in and
fade out process in form of IoU values.

Fig. 4: Some representative FBN’s dynamic change process.

Fig. 5: The spatial similarity change between the FBNs generated by each
window and 10 RSN templates over time. The horizontal axis is the change of
the time window, and the vertical axis is the IoU with the RSN template.

##  4 Conclusion

The proposed STCA provides a new idea and framework to construct the FBNs via
the spatial-wise attention, and shows excellent performance on generating the
FBNs. It also provides a new tool for studying dynamic FBNs. In the future, we
plan to study brain spatial state transition based on it and perform new
experiments on task-based fMRI (tfMRI). We also plan to perform the
experiments on functional connectivity (FC) analysis and diseases
classification based on the proposed STCA.

##  5 Acknowledgments

The work was supported by the National Natural Science Foundation of China
(NSFC61976131).

## References

  * [1] Nikos K Logothetis,  “What we can do and what we cannot do with fmri,”  Nature, vol. 453, no. 7197, pp. 869–878, 2008.
  * [2] Scott A Huettel, Allen W Song, Gregory McCarthy, et al.,  Functional magnetic resonance imaging, vol. 1,  Sinauer Associates Sunderland, MA, 2004.
  * [3] M. J. Mckeown,  “Detection of consistently task-related activations in fmri data with hybrid independent component analysis,”  Neuroimage, vol. 11, no. 1, pp. 24–35, 2000.
  * [4] Jinglei Lv, Xi Jiang, Xiang Li, Dajiang Zhu, Hanbo Chen, Tuo Zhang, Shu Zhang, Xintao Hu, Junwei Han, Heng Huang, et al.,  “Sparse representation of whole-brain fmri signals for identification of functional networks,”  Medical Image Analysis, vol. 20, no. 1, pp. 112–134, 2015.
  * [5] Armin Iraji, Robyn Miller, Tulay Adali, and Vince D Calhoun,  “Space: a missing piece of the dynamic puzzle,”  Trends in Cognitive Sciences, vol. 24, no. 2, pp. 135–149, 2020\.
  * [6] Mengshen He, Xiangyu Hou, Zhenwei Wang, Zili Kang, Xin Zhang, Ning Qiang, and Bao Ge,  “Multi-head attention-based masked sequence model for mapping functional brain networks,”  in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2022, pp. 295–304.
  * [7] Lin Zhao, Zihao Wu, Haixing Dai, Zhengliang Liu, Tuo Zhang, Dajiang Zhu, and Tianming Liu,  “Embedding human brain function via transformer,”  in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2022, pp. 366–375.
  * [8] Vesa Kiviniemi, Tapani Vire, Jukka Remes, Ahmed Abou Elseoud, Tuomo Starck, Osmo Tervonen, and Juha Nikkinen,  “A sliding time-window ica reveals spatial variability of the default mode network in time,”  Brain connectivity, vol. 1, no. 4, pp. 339–347, 2011.
  * [9] Bao Ge, Huan Wang, Panpan Wang, Yin Tian, Xin Zhang, and Tianming Liu,  “Discovering and characterizing dynamic functional brain networks in task fmri,”  Brain Imaging and Behavior, vol. 14, no. 5, pp. 1660–1673, 2020\.
  * [10] Xiao Liu, Catie Chang, and Jeff H Duyn,  “Decomposition of spontaneous brain activity into distinct fmri co-activation patterns,”  Frontiers in systems neuroscience, vol. 7, pp. 101, 2013.
  * [11] Xiao Liu and Jeff H Duyn,  “Time-varying functional network information extracted from brief instances of spontaneous brain activity,”  Proceedings of the National Academy of Sciences, vol. 110, no. 11, pp. 4392–4397, 2013.
  * [12] Fikret Işik Karahanoğlu and Dimitri Van De Ville,  “Transient brain activity disentangles fmri resting-state dynamics in terms of spatially and temporally overlapping networks,”  Nature communications, vol. 6, no. 1, pp. 1–10, 2015.
  * [13] David C Van Essen, Stephen M Smith, Deanna M Barch, Timothy EJ Behrens, Essa Yacoub, Kamil Ugurbil, Wu-Minn HCP Consortium, et al.,  “The wu-minn human connectome project: an overview,”  Neuroimage, vol. 80, pp. 62–79, 2013.
  * [14] S. M. Smith, P. T. Fox, K. L. Miller, D. C. Glahn, P. M. Fox, C. E. Mackay, N. Filippini, K. E. Watkins, R. Toro, and A. R. Laird,  “Correspondence of the brain’s functional architecture during activation and rest,”  Proceedings of the National Academy of Sciences, vol. 106, no. 31, pp. 13040–13045, 2009.
  * [15] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon,  “Cbam: Convolutional block attention module,”  in Proceedings of the European conference on computer vision (ECCV), 2018, pp. 3–19.