# Puzzles of Divergence and Renormalization
in Quantum Field Theory

Guang-jiong Ni a,b<EMAIL_ADDRESS>a Department of Physics, Portland State
University, Portland, OR97207, U. S. A.
b Department of Physics, Fudan University, Shanghai, 200433, China  Jianjun Xu
b<EMAIL_ADDRESS>b Department of Physics, Fudan University, Shanghai,
200433, China  Senyue Lou c,d<EMAIL_ADDRESS>c Department of Physics,
Shanghai Jiao Tong University, Shanghai, 200030, China
d Department of Physics, Ningbo University, Ningbo 315211, China

###### Abstract

A regularization renormalization method ($RRM$) in quantum field theory
($QFT$) is discussed with simple rules: Once a divergent integral $I$ is
encountered, we first take its derivative with respect to some mass parameter
enough times, rendering it just convergent. Then integrate it back into $I$
with some arbitrary constants appeared. Third, the renormalization is nothing
but a process of reconfirmation to fix relevant parameters (mass, charge,
etc.) by experimental data via suitable choices of these constants. Various
$QFT$ problems, including the Lamb shift, the running coupling constants in
$QED$ and $QCD$, the $\lambda\phi^{4}$ model as well as Higgs mass in the
standard model of particle physics, are discussed. Hence the calculation,
though still approximate and limited in accuracy, can be performed in an
unambiguous way with no explicit divergence, no counter term, no bare
parameter and no arbitrarily running mass scale (like the $\mu$ in $QCD$).
Keywords: renormalization, quantum field theory, Lamb shift, running coupling
constant
PACS: 03.70.+k; 11.10.-z; 11.10.Gh; 11.10.Hi

##  I Introduction

Since the establishment of quantum mechanics ($QM$) and the quantization of
electromagnetic field, the quantum electrodynamics ($QED$) in particular and
the quantum field theory ($QFT$) in general have been developed for over 80
years. A common prominent feature of $QFT$ is the emergence of the divergence
in calculations beyond the tree level. To handle these divergences, various
regularization and renormalization methods ($RRMs$) have been proposed.

Despite the great success of $QFT$, the present status of $RRMs$ remains
ambiguous to some extent. For example, in the theory of Chromodynamics ($QCD$)
for describing the strong interactions of colored quarks and gluons, a
commonly used renormalization scheme ($RS$) is the modified minimal
substraction ($\bar{MS}$) scheme (see a summary in the Review of Particle
Physics in 2008, 1 , p.157) where an arbitrary renormalization mass scale
$\mu$ is introduced (see next section). Physicists believe the fundamental
theorem of $RS$ dependence: Physical quantities, such as the cross-section
calculated to all orders in perturbation theory, should not depend on the
$RS$. However, it follows that a truncated series does exhibit $RS$
dependence. In practice, $QCD$ cross-section are known to different orders,
depending on the choice of $RS$ (and $\mu$) in different sensitive ways. We
still don’t know what is the ”best” choice for $\mu$ within a given scheme
(usually $\bar{MS}$). There is no definite answer to this question yet.

##  II What a divergence means?

Physicists often talk about different orders of a divergence, based on its
dimension with respect to mass (i.e., momentum, we use natural unit system
with $\hbar=c=1$). For example, if a Feynman diagram integral ($FDI$) in $QFT$
reads

$I=\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{K^{n}}{(K^{2}-M^{2})^{2}}$ (1)

where $K$ corresponds to the (4-dimensional)momentum of virtual particle (say,
virtual photon in $QED$) and $M$ is a mass parameter (maybe in a complex form)
characterizing the $QFT$ under consideration. Then

$n=\left\\{\begin{array}[]{l}0,\quad\text{logarithmic divergence}\\\
1,\quad\text{linear divergence}\\\ 2,\quad\text{quadratic
divergence}\end{array}\right.$ (2)

However, among these categories, only the first one is really meaningful in
mathematics. This is because the definition of a number sequence
$A_{i}\;(i=1,2,\ldots)$ having a limit being $\infty$ is as follows: Given
arbitrarily a large number $M$, one can always find such a number $N$ so that
$A_{i}>M$, when $i>N$. Here $A_{i}$ and $M$, let alone $i$ and $N$, are all
dimensionless numbers. A number $M\gg 1$ is called a large number, whereas
$\varepsilon\ll 1$ a small number.

On the other hand, the space-time coordinates $\bf x$ and $t$, mass $m$ and
momentum $p$ (or $k$) are physical quantities and each with certain dimension.
If treating them as dimensionless numbers, we will run into trouble
inevitably.

Example A: Assume that in $QFT$, a $FDI$ with linear divergence is
approximately expressed as $I\sim 10^{3}M$ with $M$ being a mass parameter in
the unit of $mg$. If $M=1\,mg$, we have $I\sim 10^{3}mg$ with $10^{3}\gg 1$
being a large number. But if changing the unit from $mg$ to $kg$, we will have
$I\sim 10^{-3}kg$ with $10^{-3}\ll 1$ being a small number. A mathematician
would ask: ”Could you still treat your $I$ as a divergent quantity?”Who could
answer his question?

Example B: In the $\bar{MS}$ scheme, the renormalization mass scale $\mu$ is
introduced as follows (see p.137 in 2 (2))

$\dfrac{\Gamma(2-d/2)}{(4\pi)^{d/2}(m^{2})^{2-d/2}}=\dfrac{1}{(4\pi)^{2}}\left[\dfrac{2}{\varepsilon}-\gamma+\ln(4\pi)-\ln(m^{2})\right]\to\dfrac{1}{(4\pi)^{2}}\left[-\ln\left(\dfrac{m^{2}}{\mu^{2}}\right)\right]$
(3)

where $m$ is some mass parameter containing in the model. Eq.(3) is derived
from the ”dimensional regularization” method. The 4-dimensional (Euclidean)
space has been analytically continued into $d$-dimensional one with
$\varepsilon\sim 4-d\sim 0$ and Gamma function
$\Gamma(2-d/2)=\Gamma(\varepsilon/2)=2/\varepsilon-\gamma$
($\gamma=0.4772\ldots$ is the Euler constant). Obviously, the left-handed-side
($LHS$) has a dimension of $m^{-\varepsilon}$, whereas in the right-handed-
side ($RHS$), the function $\ln\left(\frac{m^{2}}{\mu^{2}}\right)$ becomes
dimensionless after the $\mu$ is introduced. However, the mathematician would
feel quite uncomfortable because $\varepsilon\neq 0$. He will focus on the
middle of Eq.(3) and ask: ”Why the divergent number $2/\varepsilon$ disappears
at the $RHS$ and becomes finite? Where the term $\ln(m^{2})$ comes from? What
is its dimension?”

We physicists accept Eq.(3) since it could be derived from a ”mathematical
formula” like (see p.57 in 3 (3))

$(m^{2})^{d/2-2}=\exp\left[\left(\frac{d}{2}-2\right)\ln m^{2}\right]\simeq
1+\left(\frac{d}{2}-2\right)\ln m^{2}$ (4)

Then the mathematician would say: ”No! In the mathematical formula

$x^{y}=\exp[y\ln x]$ (5)

both $x$ and $y$ must be dimensionless numbers. So in the $LHS$ of Eq.(3), you
correctly write down:

$(4\pi)^{-d/2}=(4\pi)^{2-d/2-2}=\dfrac{1}{(4\pi)^{2}}\exp\left[(2-\frac{d}{2})\ln(4\pi)\right]\simeq\dfrac{1}{(4\pi)^{2}}\left[1+(2-\frac{d}{2})\ln(4\pi)\right]$

But Eq.(4) is wrong because $x=m^{2}$ is a physical quantity with dimension.
That’s why you got a strange result at the $RHS$ of Eq.(3)”.

Hence if insisting on mathematical rigor, we should admit that the
introduction of $\mu$ via Eq.(3) is groundless. Then a question arises: Why
the $\mu$ seems necessary in $QCD$ ?

To our understanding, the answer lies in the fact that in high energy $QCD$,
the quarks’ masses were often neglected. Therefore, in order to express the
running coupling constant ($RCC$) of strong interaction, $\alpha_{s}$, as a
function of $Q$, the 3-dimensional momentum transfer in collision, one needs
$\mu$ as shown by the solution of renormalization-group-equation ($RGE$) (see
p.532 of 2 (2) and Eq.(53) below):

$\alpha_{s}(Q)=\dfrac{\alpha_{s}(\mu)}{1+\alpha_{s}(\mu)\frac{\beta_{0}}{2\pi}\ln(Q/\mu)}$
(6)

It is interesting to solve Eq.(6) for $\alpha_{s}(\mu)$, yielding

$\alpha_{s}(\mu)=\dfrac{\alpha_{s}(Q)}{1+\alpha_{s}(Q)\frac{\beta_{0}}{2\pi}\ln(\mu/Q)}$
(7) 111see Appendix of 4 (4), where a typing error exists in the denominator
of Eq.(A.5), ”$+$” should be ”$-$”.

We see that Eqs.(6) and (7) are symmetrical with respect to mutual change of
$Q\leftrightarrow\mu$. $Q$ and $\mu$ are essentially equivalent. Why we need
both of them? The answer is: only $Q/\mu$ is capable of expressing a
dimensionless $\alpha_{s}$. However, as shown in Eq.(3), the existence of
$\mu$ is doubtful, even superfluous. Once we take the quarks’ masses into
account, there will be no need of $\mu$ at all (see section V below).

##  III Self-Energy Correction of an Electron, Lamb Shift

As is well known (see e.g., Refs.4 (4, 5, 6)), the $FDI$ of a free electron’s
self-energy at one loop ($L=1$) level of $QED$ in covariant form reads

$-i\Sigma(p)=(ie)^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{g_{\mu\nu}}{ik^{2}}\gamma^{\mu}\dfrac{i}{\not\\!\\!{p}-\not\\!\\!{k}-m}\gamma^{\nu}$
(8)

where the Bjorken-Drell metric ($\not\\!\\!{p}=\gamma^{\mu}p_{\mu}$) and
rationalized Gaussian units are adopted with electron charge $-e\,(e>0)$ and
mass $m=m_{e}$. In Eq.(8), $p$ and $k$ are momenta of electron and (virtual)
photon. After introducing the Feynman parameter $x$ and making a shift in
momentum integration: $k\to K=k-xp$, Eq.(8) is recast into

$-i\Sigma(p)=-e^{2}\int_{0}^{1}dx[-2(1-x)\not\\!\\!{p}+4m]I$ (9)

with

$I=\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{1}{(K^{2}-M^{2})^{2}},\quad
M^{2}=p^{2}x^{2}+(m^{2}-p^{2})x$ (10)

being a logarithmically divergent integral, see Eq.(1).

Note that in Eq.(10) we can change the unit of $M$ (and $K$) at our disposal
without any change in the value of $I$, which is just a ”dimensionless”,
”large” but ”uncertain” number. However, in the past, we used to pay too much
attention to its feature of being ”large”, trying to curb the divergence by
means of some regularization method, which led to complicated renormalization
schemes ($RS$).

By contrast, now we believe the more important, even essential feature of a
divergence is hiding in its ”uncertainty”. To stress this cognition, we just
use a simple trick to regulate the $I$ in Eq.(10) as follows.

To render it convergent, we perform a differentiation with respect to the
mass-square parameter $M^{2}$, yielding

$\dfrac{\partial I}{\partial
M^{2}}=2\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{1}{(K^{2}-M^{2})^{3}}=\dfrac{-i}{(4\pi)^{2}}\dfrac{1}{M^{2}}$
(11)

Then we reintegrate Eq.(11) with respect to $M^{2}$ and arrive at

$I=\dfrac{-i}{(4\pi)^{2}}(\ln
M^{2}+C_{1})=\dfrac{-i}{(4\pi)^{2}}\ln\dfrac{M^{2}}{\mu_{2}^{2}}$ (12)

where an arbitrary constant $C_{1}=-\ln\mu_{2}^{2}$ (with $\mu_{2}$ a mass
scale to be fixed later) is introduced so that the ambiguity of dimension in
the $\ln M^{2}$ term can be eliminated.

Further integration of Eq.(9) with respect to $x$ leads to
($\alpha=\frac{e^{2}}{4\pi}$)

$\begin{array}[]{l}\Sigma(p)=A+B\not\\!\\!{p},\\\\[14.22636pt]
A=\dfrac{\alpha}{\pi}m\left[2-2\ln\dfrac{m}{\mu_{2}}+\dfrac{(m^{2}-p^{2})}{p^{2}}\ln\dfrac{(m^{2}-p^{2})}{m^{2}}\right],\\\\[14.22636pt]
B=\dfrac{\alpha}{4\pi}\left\\{2\ln\dfrac{m}{\mu_{2}}-3-\dfrac{(m^{2}-p^{2})}{p^{2}}\left[1+\dfrac{(m^{2}+p^{2})}{p^{2}}\ln\dfrac{(m^{2}-p^{2})}{m^{2}}\right]\right\\}\end{array}$
(13)

Using the chain approximation, we can derive the modification on the electron
propagator as

$\dfrac{i}{\not\\!\\!{p}-m}\to\dfrac{i}{\not\\!\\!{p}-m}\dfrac{1}{1-\frac{\Sigma(p)}{\not\\!\\!{p}-m}}=\dfrac{iZ_{2}}{\not\\!\\!{p}-m_{R}}$
(14)

where

$Z_{2}=\dfrac{1}{1-B}$ (15)

is the renormalization factor for electron’s wave function and

$m_{R}=\dfrac{m+A}{1-B}$ (16)

is the renormalized mass of $m$. The increment of mass reads

$\delta m=m_{R}-m=\dfrac{A+mB}{1-B}$ (17)

In the past, many physicists viewed $\delta m$ as some real contribution of
”radiation correction”. While $m_{R}$ should be identified with the observed
mass $m_{obs}$, or physical mass $m_{e}$, the original $m$ (usually denoted by
$m_{0}$ or $m_{B}$ in the expression of Lagrangian density) was thought to be
a ”bare mass”. Both $\delta m$ and $m_{0}$ were divergent quantities. (see,
e.g., p.220 in 2 (2)).

We don’t think so. Let us read carefully the seminal paper by Bethe in 19478
(8). The theory for the hydrogenlike atom begins with a Hamiltonian in the
center-of-mass frame

$H_{0}=\dfrac{{\bf p}^{2}}{2m}+\dfrac{{\bf
p}^{2}}{2m_{N}}-\dfrac{Z\alpha}{r}=\dfrac{{\bf
p}^{2}}{2\mu}-\dfrac{Z\alpha}{r}$ (18)

Bethe pointed out that the effect of electron’s interaction with the vector
potential $\bf A$ of radiation field

$H_{int}=\dfrac{e}{mc}{\bf A}\cdot{\bf p}$ (19)

should properly be regarded as already included in the $m_{obs}$, which is
denoted by $m$ in Eqs.(18) and (19).

In our understanding on Bethe’s claim, the ”self-interaction” of electron with
radiation field is indivisible from the free electron mass $m$. In other
words, in the covariant form of $QED$, certain contributions of $FDIs$ for
”self-energy” (with Eq.(8) being merely that at $L=1$ order) at all orders (up
to $L\to\infty$) are already contained in the value of $m$. To show this
cognition, we impose the mass-shell condition $p^{2}=m^{2}$ in Eq.(17)
together with

$\delta m|_{p^{2}=m^{2}}=\dfrac{\alpha m}{4\pi}(5-6\ln\frac{m}{\mu_{2}})=0$
(20)

which in turn fixes the arbitrary constant $\mu_{2}$ to be

$\mu_{2}=me^{-5/6}$ (21)

and thus

$Z_{2}|_{p^{2}=m^{2}}=\dfrac{1}{1+\frac{\alpha}{3\pi}}\simeq
1-\dfrac{\alpha}{3\pi}$ (22)

Note that $m_{R}=m=m_{obs}=m_{e}$ with no bare mass at all and $Z_{2}$ is
fixed and finite, in sharp contrast to that in previous theories.

Our reader may wonder: ”In this case, does the calculation on $FDI$ for the
self-energy become worthless ?” The answer is ”No” due to two reasons. First,
at the $QM$ level, the parameters $m$ and $e$ in Eqs.(18) and (19) can be
regarded as well-defined. But they are not so at the level of $QED$. As
discussed before Eq.(20), the new effect of radiative corrections of $FDIs$
for self-energy is inevitably confused with that in the mass, the dividing
line between them is blurred. In some sense, the appearance of divergence in
the $FDI$ is just a warning: the new effect you want to calculate has become
entangled with the mass $m$, rendering both of them uncertain. Hence the aim
of so-called mass renormalization is nothing but a reconfirmation of $m$ as we
did in Eqs.(20)-(22), where the mass $m$ is renormalized on the mass-shell
$p^{2}=m^{2}$ with $m=m_{e}$ being fixed by the experimental value and thus
well-defined. This is one important thing we must do and at most we can do on
the mass-shell for a free electron.

Second, the increment of mass, $\delta m$, ceases to be zero once when the
electron is moving off-mass-shell ($p^{2}\neq m^{2}$). Then Eq.(17) will
provide some information about the new effect of radiation corrections. For
example, for a bound electron in a hydrogenlike atom, in Ref.7 (7), we replace
the electron mass $m=m_{e}$ by reduced mass
$\mu=\frac{m_{e}m_{N}}{m_{e}+m_{N}}$ (not to be confused with the $\mu$ in
$QCD$) and write (see also 29 (29)):

$p^{2}=\mu^{2}(1-\zeta)$ (23)

Here a dimensionless parameter $\zeta\,(>0)$ is introduced to show (on
average) how large the extent of ”off-mass-shell” is. Substitution of Eq.(23)
into Eq.(17) yielding

$\delta\mu\simeq\dfrac{\alpha\mu}{4\pi}\dfrac{(-\zeta+2\zeta\ln\zeta)}{1+\frac{\alpha}{3\pi}}$
(24)

where some terms of the order of $\zeta^{2}$ or $\zeta^{2}\ln\zeta$ are
neglected since $\zeta\ll 1$.

As a perturbative calculation at $L=1$ order, we may ascribe $\delta\mu$ to
the (minus) binding energy $B$ of electron in the Bohr theory

$\delta\mu=\varepsilon_{n}=-B=-\dfrac{Z^{2}\alpha^{2}}{2n^{2}}\mu$ (25)

Combination of Eqs.(24) and (25) gives the value of $\zeta=\zeta^{<S>}$ with
the superscript $<S>$ referring to ”self-energy (at $L=1$ order)”.

Another ”nonperturbative” method to fix the $\zeta$ in Eq.(23) is to resort to
the Virial theorem: For an electron in the Coulomb potential
$V=-\frac{Z\alpha}{r}$, its kinetic energy $T=\frac{1}{2\mu}{\bf p}^{2}$ can
be evaluated on average as

$<{\bf p}^{2}>=2\mu<T>=2\mu[-B-<V>]=2\mu B$ (26) $<p^{2}>=<E^{2}-{\bf
p}^{2}>=<(\mu-B)^{2}-{\bf p}^{2}>\simeq\mu^{2}(1-\dfrac{4B}{\mu})$ (27)

Comparing Eq.(27) with Eq.(23), we obtain

$\zeta^{<V>}=\dfrac{4B}{\mu}=\dfrac{2Z^{2}\alpha^{2}}{n^{2}}$ (28)

where the superscript $<V>$ refers to ”Virial theorem”.

In Ref.7 (7), for explaining the Lamb shift of energy levels in hydrogenlike
atoms, we find the result being expressed in terms of $\zeta$. Throughout the
entire calculation, all ultraviolet divergences are handled like that in
Eqs.(9)-(12) while the infrared divergence disappears due to the introduction
of $\zeta$. However, the formulas are still approximate and either one of
$\zeta^{<S>}$ and $\zeta^{<V>}$ is not reliable. So in the following table
$I$, not only $\zeta^{<S>}$ and $\zeta^{<V>}$, but also two kinds of
”average”, $\zeta^{<S+V>}=\frac{1}{2}(\zeta^{<S>}+\zeta^{<V>})$ and
$\zeta^{<SV>}=\sqrt{(\zeta^{<S>}\zeta^{<V>}}$ are given.

Table I. Off-mass-shell parameter $\zeta$ and $\ln\zeta$ |
---|---
$\frac{Z^{2}}{n^{2}}$ | $\zeta^{<S>}\times 10^{4}$ |  -$\ln\zeta^{<S>}$ | $\zeta^{<V>}\times 10^{6}$ |  -$\ln\zeta^{<V>}$ | $\zeta^{<S+V>}\times 10^{5}$ |  -$\ln\zeta^{<S+V>}$ | $\zeta^{<SV>}\times 10^{5}$ | $-\ln\zeta^{<SV>}$
$\frac{1}{16}$ | $1.546093458$ | $8.77461$ | $\frac{\alpha^{2}}{8}=6.6564192$ | 11.91992886 | $8.0632$ | 9.425609 | 3.2080284 | 10.34727
$\frac{1}{4}$ | $7.446539697$ | 7.20259 | $\frac{\alpha^{2}}{2}=26.6256771$ | 10.5336345 | $38.5639$ | 7.860609 | 14.0808 | 8.86816225
1 | $37.73719345$ | 5.57969 | $2\alpha^{2}=106.502$ | 9.147340142 | $194.011$ | 6.2450103 | 63.39626 | 7.36351521

There are $8$ cases discussed in 7 (7). The first one is the hydrogen atom’s
”classical Lamb shift” measured as:

$L_{H}^{exp}(2S-2P)=E_{H}(2S_{1/2})-E_{H}(2P_{1/2})=1057.845\;MHz$ (29)

Theoretically, the radiative correction (at $L=1$ order) makes the dominant
contribution, yielding:

$\begin{array}[]{l}\Delta E_{H}^{Rad<S>}(2S-2P)=1000.657\;MHz\\\ \Delta
E_{H}^{Rad<S+V>}(2S-2P)=1089.651\;MHz\\\ \Delta
E_{H}^{Rad<SV>}(2S-2P)=1226.087\;MHz\\\ \Delta
E_{H}^{Rad<V>}(2S-2P)=1451.791\;MHz\end{array}$ (30)

Taking the small contribution from the nuclear size effect into account, we
adopt the $<S+V>$ scheme to obtain

$L_{H}^{theor}(2S-2P)=1089.794\;MHz$ (31)

which is larger than the experimental value, Eq.(29), by $3\%$.

The most interesting case is the $1S-2S$ two-photon transition in hydrogen $H$
or deuterium $D$ because its natural width is so tiny ($1.3Hz$) and thus
allows precision measurement in recent years9 (9):

$\Delta E_{H}^{exp}(1S-2S)=2466061413187.34(84)\;kHz$ (32)

The isotope shift of $1S-2S$ transition between $H$ and $D$ had been measured
first by Schmidt-Kalar et al.10  and quoted in 11 (11) as:

$\Delta E_{D-H}^{exp}(2S-1S)=670994337(22)\;kHz$ (33)

Theoretically, the above accurate data cannot be explained by the original
Dirac equation with nucleus having mass $m_{N}\to\infty$. We propose a reduced
Dirac equation ($RDE$) with electron mass replaced by reduced mass for $H$ and
$D$ being respectively

$\mu_{H}=\dfrac{m_{e}m_{p}}{m_{e}+m_{p}},\quad\mu_{D}=\dfrac{m_{e}m_{d}}{m_{e}+m_{d}}$
(34)

Then theoretically, the $RDE$ predicts:

$\Delta E_{H}^{RDE}(2S-1S)=2.466067984\times 10^{15}\;Hz$ (35) $\Delta
E_{D-H}^{RDE}(2S-1S)=6.7101527879\times 10^{11}\;Hz$ (36)

which are larger than the experimental values by only $3\times 10^{-6}$ and
$3\times 10^{-5}$ respectively. Further radiative corrections on Eq.(35) will
be sensitive to the choice of schemes in Table $I$, the best one is $\Delta
E_{H}^{Theor<SV>}(2S-1S)$, deviating from the experimental data, Eq.(32), by
$-1\times 10^{-7}$ only. On the other hand, besides Eq.(36), the $\Delta
E_{D-H}^{Theor}(2S-1S)$ is influenced considerably by the nuclear size effect
and so less sensitive to the scheme choice of the smaller radiative
correction, bringing the discrepancy between theory and experimental data,
Eq.(33), from $3\times 10^{-5}$ down to $3\times 10^{-6}$ approximately.

##  IV Renormalization Group Equation ($RGE$) for $QED$ and Its Solution

In $QED$, the $FDI$ for photon self-energy (i.e., vacuum polarization) can
also be evaluated 4 (4), bringing the charge $e$ into its renormalized one:

$e^{2}\to e^{2}_{R}=Z_{3}e^{2}$ (37)
$Z_{3}=1+\dfrac{\alpha}{3\pi}\left(\ln\dfrac{m^{2}}{\mu^{2}_{3}}-\dfrac{q^{2}}{5m^{2}}+\cdots\right)$
(38)

Here $m$ is the fermion (say, electron) mass, $q$ is the momentum of photon
and $\mu_{3}$ is an arbitrary constant emerging from the treatment on the
divergence like that in Eqs.(9)-(12). Although Eq.(37) looks like that in
previous theories, it is really a new one: $e$ is the observed (physical)
charge, not a ”bare charge”, and $Z_{3}$ remains finite.

The vertex function between two fermions’ momenta $p$ and $p^{\prime}$ with
$p^{\prime}-p=q$ will give another $Z_{1}$ 4 (4, 7). Adding all the $FDIs$, we
find the renormalized charge being:

$e_{R}=\dfrac{Z_{2}}{Z_{1}}Z_{3}^{1/2}e$ (39)

But the Ward-Takahashi identity ($WTI$) implies that6 (6)

$Z_{1}=Z_{2}$ (40)

Hence Eq.(37) remains valid and

$e_{R}(Q)=e\left\\{1+\dfrac{\alpha}{2\pi}\left[\dfrac{1}{3}\ln\dfrac{m^{2}}{\mu^{2}_{3}}+\dfrac{1}{15}\dfrac{Q^{2}}{m^{2}}+\cdots\right]\right\\}$
(41)

where $Q^{2}=-q^{2}>0$, with $Q$ being the 3-dimensional momentum transfer at
fermion collision. The observed charge should be defined at $Q\to 0$ (Thomson
scattering limit):

$e_{obs}=e_{R}|_{Q=0}=e$ (42)

which dictates that

$\mu_{3}=m$ (43)

As usual, the beta function is defined as

$\beta(\alpha,Q)\equiv Q\dfrac{\partial}{\partial Q}\alpha_{R}(Q)$ (44)

From Eq.(41), it is found in 4 (4) that

$\beta(\alpha,Q)=\dfrac{2\alpha^{2}}{3\pi}-\dfrac{4\alpha^{2}m^{2}}{\pi
Q^{2}}\left[1+\dfrac{2m^{2}}{\sqrt{Q^{4}+4m^{2}Q^{2}}}\ln\dfrac{\sqrt{Q^{4}+4m^{2}Q^{2}}-Q^{2}}{\sqrt{Q^{4}+4m^{2}Q^{2}}+Q^{2}}\right]$
(45)
$\beta(\alpha,Q)=\dfrac{2\alpha^{2}}{15\pi}\dfrac{Q^{2}}{m^{2}},\quad(\dfrac{Q^{2}}{m^{2}}\ll
1)$ (46)

Evidently, the $e_{R}(Q)$ will increase with $Q$, becoming a running coupling
constant ($RCC$). To calculate it, usually a renormalization-group-equation
($RGE$) was derived for $QED$ by setting $\alpha\to\alpha_{R}(Q)$ and
$Q\to\infty$ in $\beta(\alpha,Q)$ yielding:

$Q\dfrac{d}{dQ}\alpha_{R}=\dfrac{2\alpha_{R}^{2}}{3\pi}$ (47)

with its solution

$\alpha_{R}(Q)=\dfrac{\alpha}{1-\frac{2\alpha}{3\pi}\ln\frac{Q}{m}}$ (48)

Here, the renormalization was forced to be made at $Q^{2}=m^{2}$ so that

$\alpha_{R}|_{Q^{2}=m^{2}}=\alpha$ (49)

This is inconsistent with the physical condition, Eq.(42), a defect due to
ignoring the mass $m$, which plays a dominant role at low $Q$ region as shown
by the Eq.(46). As an improvement, a more practical $RGE$ is constructed in 4
(4) by changing $\alpha$ into $\alpha_{R}(Q)$ in Eq.(45) for the entire $Q$
region and adding up contributions from $9$ elementary charged fermions
($e,\mu,\tau,u,d,s,c,b,t$). Then $\alpha_{R}$ can be numerically calculated as
a function of $\ln(Q/m_{e})$, starting from

$\alpha_{R}(Q=0)=\alpha=(137.03599)^{-1}$ (50)

and passing through another experimental data point 12 (12)

$\alpha_{R}(Q=M_{Z}=91.1880\,GeV)=(128.89)^{-1}$ (51)

In this way, after adopting three heavy quarks’ masses as
$m_{c}=1.031\,GeV,\;m_{b}=4.326\,GeV$ (see section 9.5D in 13 (13)) and
$m_{t}=175\,GeV$, three light quarks masses

$m_{u}=8\,MeV,\;m_{d}=10\,MeV,\;m_{s}=200\,MeV$ (52)

(or averaged mass for $u,d,s$ being $92\ MeV$) can be fitted as shown in Fig.1
of Ref.4 (4).

##  V $RGE$ for $QCD$, Threshold Energies of Quarks Hadronization

Different from the $RCC$ in $QED$, the $RCC$ in $QCD$, denoted by
$\alpha_{s}(Q)=\frac{1}{4\pi}g_{s}^{2}(Q)$, is much larger and decreases with
the increase of $Q$. Usually, the $RGE$ for $QCD$ reads (see p.551-552 in 2
(2))

$Q\dfrac{d}{dQ}\alpha_{s}(Q)=\beta(\alpha_{s}(Q))=-\dfrac{\beta_{0}}{2\pi}\alpha_{s}^{2}(Q)$
(53)

where

$\beta_{0}=11-\dfrac{2}{3}n_{f}$ (54)

with $n_{f}$ being the number of quarks’ flavor. Eq.(53) looks similar to
Eq.(47), but the negative sign in beta function implies the property of
asymptotic freedom in the strong interaction. Solution of Eq.(53) is give by
Eq.(6), also bears some resemblance to Eq.(48).

Let us make a comparison between Eqs.(6) and (48). Besides the difference in
sign (”$+$” versus ”$-$”) in the denominator, there is another one: The mass
scale $\mu$ remains arbitrary in $QCD$ whereas $m$ in $QED$ means the mass of
an observed charged fermion (usually electron).

As we already see, even for $QED$, one mass ($m_{e}$) is far from enough, let
alone in $QCD$, where quarks’ masses are much heavier. How can we ignore them?

Usually, to remove the arbitrary mass scale $\mu$, a parameter $\Lambda_{QCD}$
is often defined via equation

$\dfrac{\beta_{0}}{2\pi}\alpha_{s}(\mu)\ln\left(\dfrac{\mu}{\Lambda_{QCD}}\right)=1$
(55)

such that a simpler formula for $\alpha_{s}(Q)$ can be found as

$\alpha_{s}(Q)=\dfrac{2\pi}{\beta_{0}\ln(Q/\Lambda_{QCD})}$ (56)

Then the precision experimental data 1

$\alpha_{s}(M_{Z}=91.1876\,GeV)=0.1176$ (57)

serves as a substitute for $\alpha_{s}(\mu)$ in Eq.(55), yielding

$\Lambda_{QCD}=M_{Z}\exp\left[\dfrac{-2\pi}{\alpha_{s}(M_{Z})\beta_{0}}\right]=\left\\{\begin{array}[]{l}240\;MeV,\;(n_{f}=3)\\\
150\;MeV,\;(n_{f}=4)\\\ 85.8\;MeV,\;(n_{f}=5)\\\
44.2\;MeV,\;(n_{f}=6)\end{array}\right.$ (58)

It is evident from Eq.(56) that

$\alpha_{s}(\Lambda_{QCD})\to\infty$ (59)

which implies the ”infrared confinement” of quarks.

However, the value of $\Lambda_{QCD}$ sensitively depends on the flavor number
$n_{f}$ as shown by Eq.(58) but is independent of the concrete flavor of a
quark under consideration. Moreover, the divergence of $\alpha_{s}$ appears at
$\Lambda_{QCD}$. These features seem not so reasonable and are not consistent
with the experimental fact that the lighter a quark’s mass is, the lower its
”threshold energy for hadronization” will be.

The way out of above difficulties is clear. Just like in Eq.(45) for $QED$,
where rather than just the first term, all terms for $9$ fermions should be
added, now for $RGE$ in $QCD$, all masses of $6$ quarks should be preserved.
In this way, the $\alpha_{si}(Q)$ are numerically calculated for $i=u,d,s,c,b$
respectively in Ref.4 (4). Starting from $\alpha_{s}(M_{Z})=0.118$ (the common
renormalization point), their running curves (as shown by Figs.2 and 3 in 4
(4)) follow the trend of experimental data (as shown on p.158 of Ref.1 ) quite
well but separate at the low $Q$ region. Each of them rises to a maximum
$\alpha_{si}^{max}$ at $Q=\Lambda_{i}$ and then suddenly drops to zero at
$Q=0$.

For example, for $b$ quark, $\Lambda_{b}=7.04\,GeV,\;\alpha_{sb}^{max}=0.161$.
If tentatively explain $L_{b}\sim\hbar/\Lambda_{b}$ as some critical length
scale of $b\bar{b}$ pair, then $L_{b}\sim 0.02805\,fm$. In 4 (4), it is
further guessed that $E_{b}^{theor}\sim\alpha_{sb}^{max}/L_{b}\sim 1.133\,GeV$
being the order of excitation energy for breaking the binding $b\bar{b}$ pair,
i.e., the hadronization threshold energy of Upsilon $\Upsilon(b\bar{b})$
against its dissociation into two bosons. It is indeed the case found
experimentally1 :

$M(\Upsilon(4S))-M(\Upsilon)=1.12\,GeV,\quad M(\Upsilon(4S))\to
B^{+}B\;\text{or}\;B^{0}\bar{B^{0}}$ (60)

Similarly, we can estimate from Ref.4 (4) that $E_{c}^{theor}\sim 0.398\,GeV$
which also corresponds to experiments that1

$M(\psi(3770))-M(\psi(3097))=673\,MeV,\quad\psi(3770)\to
D^{+}D^{-}\;\text{or}\;D^{0}\bar{D^{0}}$ (61)

It seems that $E_{s}^{theor}\sim 90\,MeV$ and $E_{u,d}^{theor}\sim 0.4\,MeV$
are not so reliable but still reasonable.

##  VI $\lambda\phi^{4}$ Model and Higgs Mass in the Standard Model

The Lagrangian density of $\lambda\phi^{4}$ ($\phi(x)$ is a real scalar field)
model is defined by

${\cal
L}=\dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\dfrac{1}{2}\sigma\phi^{2}-\dfrac{\lambda}{4!}\phi^{4}$
(62)

The importance of this model lies in the ”wrong sign” of mass term
($\sigma=-m^{2}>0$) which leads to the spontaneous symmetry breaking ($SSB$)
at the tree level ($L=0$). The effective potential ($EP$) reads

$V_{0}(\phi)=-\dfrac{1}{2}\sigma\phi^{2}+\dfrac{\lambda}{4!}\phi^{4}$ (63)

(The subscript ”$0$” refers to $L=0$). Obviously, $V_{0}(\phi)$ has two
extremum, one is a maximum:

$\phi_{0}=0\quad\text{(symmetric phase)}$ (64)

while the other one is a minimum:

$\phi_{1}^{2}=\dfrac{6\sigma}{\lambda}\quad\text{(SSB phase)}$ (65)

At the $QFT$ level, the $EP$ evolves into

$V=V_{0}+V_{1}+\cdots$ (66)

The theory for $EP$ had been developed by various authors 14 (14, 15, 16, 17,
18), with $L=1$ contribution to $EP$ being evaluated as:

$V_{1}(\phi)=\dfrac{1}{2}\int\dfrac{d^{4}k_{E}}{(2\pi)^{4}}\ln(k_{E}^{2}-\sigma+\frac{1}{2}\lambda\phi^{2})$
(67)

This highly divergent integral (in 4-dimensional Euclidean momentum space) is
treated in Ref.19 (19) like that for Eqs.(10)-(12). First, three times of
differentiation with respect to $M^{2}=-\sigma+\frac{1}{2}\lambda\phi^{2}$ are
needed before it becomes just convergent.

$\dfrac{\partial^{3}V_{1}}{\partial(M^{2})^{3}}=\int\dfrac{d^{4}k_{E}}{(2\pi)^{4}}\dfrac{1}{(k_{E}^{2}+M^{2})^{3}}=\dfrac{1}{2(4\pi)^{2}M^{2}}$
(68)

Second, three times integration with respect to $M^{2}$ are performed,
yielding

$V_{1}(\phi)=\dfrac{1}{2(4\pi)^{2}}\left\\{\dfrac{M^{4}}{2}(\ln
M^{2}-\dfrac{1}{2})-\dfrac{1}{2}M^{4}+\dfrac{1}{2}C_{1}M^{4}+C_{2}M^{2}+C_{3}\right\\}$
(69)

As expected, three arbitrary constants $C_{1},C_{2},C_{3}$ appear. The
renormalization amounts to fix them at our disposal.

Third, like that in Eq.(3), for eliminating the ambiguity of dimension in the
first term involving $\ln M^{2}$, the only possible choice of
$C_{1}=-\ln\mu^{2}$ is fixed. Then the choice of $C_{2}=\mu^{2}=2\sigma$ and
$C_{3}=-\sigma^{2}+(4\pi)^{2}\frac{3\sigma^{2}}{\lambda}$ leads to
$V=V_{0}+V_{1}$ with its derivatives being given at the Table II 19 (19).

Table II. Effective potential of $\lambda\phi^{4}$ model with $SSB$ |
---|---
|  $SSB$ phase | symmetric phase
$\phi$ | $\phi_{1}=\sqrt{\dfrac{6\sigma}{\lambda}}$ | $\phi_{0}=0$
$V$ | $0$ | $-\dfrac{\sigma^{2}}{2(4\pi)^{2}}\left[\dfrac{15}{4}+\dfrac{1}{2}\ln 2-i\dfrac{\pi}{2}\right]+\dfrac{3}{2}\dfrac{\sigma^{2}}{\lambda}$
$\dfrac{dV}{d\phi}$ | $0$ | $0$
$\dfrac{d^{2}V}{d\phi^{2}}$ | $2\sigma$ | $-\sigma\left[1-\dfrac{\lambda}{2(4\pi)^{2}}(3+\ln 2-i\pi)\right]$
$\dfrac{d^{3}V}{d\phi^{3}}$ | $\lambda\sqrt{\dfrac{6\sigma}{\lambda}}\left[1+\dfrac{3\lambda}{2(4\pi)^{2}}\right]$ | $0$
$\dfrac{d^{4}V}{d\phi^{4}}$ | $\lambda\left[1+\dfrac{9\lambda}{2(4\pi)^{2}}\right]$ | $\lambda\left[1-\dfrac{3\lambda}{2(4\pi)^{2}}(\ln 2-i\pi)\right]$

Note that, with the above assignment of $C_{i}\,(i=1,2,3)$, both the position
of the $SSB$ phase, $\phi_{1}$, and the mass $m_{\sigma}$ excited above it
take the same expression as that at the tree level

$m_{\sigma}^{2}=\dfrac{d^{2}V}{d\phi^{2}}|_{\phi=\phi_{1}}=2\sigma$ (70)

However, the renormalization coupling constant

$\lambda_{R}=\dfrac{d^{4}V}{d\phi^{4}}|_{\phi=\phi_{1}}=\lambda\left[1+\dfrac{9\lambda}{2(4\pi)^{2}}\right]$
(71)

does receive some quantum correction on its classical value $\lambda$. Hence
it is suddenly realized that the invariant meaning of $\lambda$ in the
Lagrangian, Eq.(62), is by no means a ”coupling constant”, but the ratio of
two mass scales 21 (21).

$\lambda=3\dfrac{m_{\sigma}^{2}}{\phi_{1}^{2}}$ (72)

Two parameters, $\sigma$ and $\lambda$, together with Eqs.(70) and (72),
should all be preserved through out high loop ($L$) evaluations in
perturbation theory until $L\to\infty$, i.e., in any nonperturbative
treatment.

On the other hand, the above assignment of $C_{i}$ renders the appearance of
imaginary part in $V$ and its derivatives at the symmetric phase
($\phi_{0}=0$). It means the instability of symmetric phase at the presence of
stable $SSB$ phase.

It is interesting to see that an alternative choice of

$C_{1}=-\ln(-\sigma),\;C_{2}=-\sigma,\;C_{3}=-\dfrac{1}{4}\sigma^{2}$ (73)

would leads to the survival of $\phi_{0}=0$ as a semistable state with

$V(0)=\dfrac{dV}{d\phi}|_{\phi=0}=\dfrac{d^{3}V}{d\phi^{3}}|_{\phi=0}=0$ (74)
$\dfrac{d^{2}V}{d\phi^{2}}|_{\phi=0}=-\sigma,\;\dfrac{d^{4}V}{d\phi^{4}}|_{\phi=0}=\lambda$
(75)

whereas no real $SSB$ solution exists. Hence we see that two different choices
of $C_{i}$ lead two separable sectors in the effective potential 19 (19).

In 1989, we had estimated the upper and lower bounds of Higgs mass $M_{H}$ in
the standard model of particle physics by using a nonperturbative approach in
$QFT$ — the Gaussian effective potential ($GEP$) method, yielding20 (20)

$76\,GeV<M_{H}<170\,GeV$ (76)

Like many authors, we were bothered a lot by divergences. After a deeper
understanding on $\lambda\phi^{4}$ model19 (19), this problem was restudied in
1998 by a combination of $GEP$ with our $RRM$, yielding

$M_{H}=138\,GeV$ (77)

which is based on the following input of experimental data:

$\begin{array}[]{l}M_{W}=80.359\,GeV,\;M_{Z}=91.1884\,GeV,\\\
\alpha^{-1}=\dfrac{4\pi}{g^{2}\sin^{2}\theta_{W}}=128.89,\;\sin^{2}\theta_{W}=0.2317\end{array}$
(78)

where $\theta_{W}$ is the weak mixing (Weinberg) angle. As now the search for
Higgs particle becomes so urgent experimentally while the theoretical
estimation about its mass still remains uncertain1 , we think our method 21
(21) with its prediction, Eq.(77), deserves to be reconsidered.

##  VII Summary and Discussion

Our $RRM$ was first proposed by J. F. Yang in 199422 , then elaborated in a
series of papers since 1998 (19 (19, 21, 23, 24, 4, 7) etc.). What we have
been thinking about is: Why the divergence emerges inevitably in $QFT$? What
is the essential meaning of a regularization and renormalization procedure?

For instance, in a pioneering work to explain the Lamb shift, Welton (25 (25),
see also section 9.6B in Ref.13 (13)) encountered a divergent integral
$I=\int\frac{d\omega}{\omega}$ with $\omega$ being the (angular) frequency of
virtual photon (vacuum fluctuation). He simply set the lower and upper bounds
by $\omega_{min}\sim mZ\alpha=\frac{Z}{a}$ ($a$ is Bohr radius) and
$\omega_{max}\sim m$ respectively, arriving at
$I\simeq\ln(\frac{1}{Z\alpha})=4.92\,(Z=1)$ which leads to an estimation of
Lamb shift $L_{H}^{theor}(2S_{1/2}-2P_{1/2})\simeq 668\,MHz$. However, if
instead of Bohr radius, the lower cutoff is provided by the electron’s binding
energy, one would get $I\simeq\ln(Z\alpha)^{-2}$ and
$L_{H}^{theor}(2S_{1/2}-2P_{1/2})\simeq 1336\,MHz$ (see Eq.(30) in the Ref.26
(26)). We see the integral $I$ being a dimensionless number, not very large
($I\leq 10$), but uncertain indeed. The root cause of uncertainty lies in the
fact that a reconfirmation process of electron mass like Eq.(20) was missing.

For further clarity, a study on Lamb shift in the form of noncovariant $QED$
was performed in 24 (24) (see also Appendix 9A in 13 (13)). Beginning with
Eqs.(18) and (19) (with $m\to\mu=\frac{mm_{N}}{m+m_{N}}$), the perturbative
calculation of electron’s self-energy at second order in noncovariant form
(corresponding to the one-loop ($L=1$) order in covariant form) leads to an
energy increase of an electron with momentum $\bf p$, $\Delta E_{p}$, which
contains divergence and can be handled just like that in Eqs.(10) and (67),
yielding

$\begin{array}[]{l}\Delta E_{p}=b_{1}p^{2}+b_{2}p^{4}+\cdots\\\\[14.22636pt]
b_{1}=\dfrac{\alpha}{\pi\mu}(\dfrac{4}{3}\ln
2+\dfrac{4}{3}\ln\mu-\dfrac{4}{3}C_{1})\\\\[14.22636pt]
b_{2}=\dfrac{\alpha}{\pi\mu^{3}}(-\dfrac{2}{15})\end{array}$ (79)

The only choice of arbitrary constant $C_{1}$ is to make $b_{1}=0$ such that
the reduced mass $\mu$ in Eq.(18) can be reconfirmed. However, Eq.(19) must be
supplemented by the interaction between electron’s spin and the radiation
field

$H^{\prime}_{int}=\dfrac{ge\hbar}{4\mu
c}{\boldsymbol{\sigma}}\cdot\nabla\times{\bf A}$ (80)

where $g=2\times 1.0011596522$ is gyromagnetic ratio of electron. Similar
treatment leads to

$\begin{array}[]{l}\Delta
E^{\prime}_{p}=b^{\prime}_{0}+b^{\prime}_{1}p^{2}+b^{\prime}_{2}p^{4}+\cdots\\\\[11.38109pt]
b^{\prime}_{0}=\dfrac{g^{2}}{4}\dfrac{\alpha\mu}{\pi}[4(\ln
2+\ln\mu)-4C_{2}-\dfrac{2C_{3}}{\mu}-\dfrac{C_{4}}{\mu^{2}}]\\\\[11.38109pt]
b^{\prime}_{1}=\dfrac{g^{2}}{4}\dfrac{\alpha}{\pi\mu}(\dfrac{4}{3}\ln
2+2+\dfrac{4}{3}\ln\mu-\dfrac{4}{3}C_{2})\\\\[11.38109pt]
b^{\prime}_{2}=\dfrac{g^{2}}{4}\dfrac{\alpha}{\pi\mu^{3}}(-\dfrac{1}{15})\end{array}$
(81)

Because $\mu$ has already been reconfirmed (by $b_{1}=0$), the only choice of
arbitrary constant $C_{2}$ is to cancel the ambiguous term with
$\ln\mu,\;C_{2}=\ln\mu$, leaving a nonzero $b^{\prime}_{1}p^{2}$ and combining
with $\frac{1}{2\mu}{\bf p}^{2}$. Hence $\mu$ really acquires a modification
as

$\mu\to\mu_{obs}=\dfrac{\mu}{1+\beta},\qquad\beta=\dfrac{g^{2}\alpha}{2\pi}(\dfrac{4}{3}\ln
2+2)$ (82)

where $\mu_{obs}=\frac{m_{e}m_{N}}{m_{e}+m_{N}}$. Then constants $C_{3}$ and
$C_{4}$ must be chosen such that $b^{\prime}_{0}=0$.

Notice that, however, the spin induced interaction, Eq.(80), endows electron
with relativistic feature, creating a term ($-\frac{1}{8\mu^{3}}p^{4}$) in its
kinetic energy. Yet the modification on $\mu$ shown in Eq.(82) does induce a
corresponding change
$-\frac{1}{8}(\frac{1}{\mu_{obs}^{3}}-\frac{1}{\mu^{3}})p^{4}$, which should
be regarded as an invisible ”background” and subtracted from the $p^{4}$ term
induced by the radiative corrections. Hence the ”renormalized” $b_{2}$ should
be

$b_{2}^{R}=b_{2}+b^{\prime}_{2}+\frac{1}{8\mu^{3}}(3\beta+3\beta^{2}+\beta^{3})\simeq\dfrac{\alpha}{\pi\mu_{obs}^{3}}(1.99808)$
(83)

where only the lowest approximation is kept ($\mu\simeq\mu_{obs}$). Hence the
radiative correction on the energy level of a stationary state $|Z,n,l\rangle$
in hydrogenlike atom simply reads

$\Delta E^{rad}(Z.n,l)=\langle
Z,n,l|b_{2}^{R}p^{4}|Z,n,l\rangle=[\dfrac{8n}{3l+1}-3]\dfrac{b_{2}^{R}Z^{4}\alpha^{4}}{n^{4}}\mu_{obs}^{4}$
(84)

This contribution, together with that from the vacuum polarization (borrowed
from covariant theory) and nuclear size effect, gives a theoretical value for
the Lamb shift 24 (24)

$L_{H}^{theor}(2S_{1/2}-2P_{1/2})=1056.52\,MHz$ (85)

which is smaller than the experimental value, Eq.(29), by $0.13\%$ only.

Despite the approximation involved, the above method clearly shows that our
regularization is by noo means a trick to curb the divergence. Rather, it is a
natural way to transform a divergence into some arbitrary constants, revealing
that the essential meaning of divergence is just the ”uncertainty” in the
theory. Thus so-called renormalization turn out to be nothing but a process of
reconfirmation to fix these constants via experiments. We must reconfirm a
mass before it could be modified via radiative corrections. Either ”skipping
over the first step” or ”combining two steps into one” is not allowed.

In deeper understanding, our $RRM$ is based on a ”principle of relativity” in
epistemology27 (27): Every thing is moving and becomes recognizable only in
relationship with other things. What we can understand is either no mass scale
or two mass scales, but never one mass scale. This scenario is clearly
displayed in the Gross-Neveu model28 (28), also in the $\lambda\phi^{4}$ model
with $SSB$ as shown by Eqs.(65), (70) and (72). The vacuum expectation value
of field, $\phi_{1}$, just provides a ”mass unit” for the mass $m_{\sigma}$
excited on the vacuum with $SSB$.

Similarly, in perturbative $QFT$, we will be able to calculate various
radiative corrections on a particle only when its mass $m$ can be reconfirmed
again and again throughout any high loop ($L$) order of theory until
$L\to\infty$. Just like one has to reconfirm his plane ticket before his
departure from the airport, he must use the same name throughout his entire
jouney4 (4).

## Acknowledgements

We thank S. Q. Chen, Y. S. Duan, R. T. Fu, S. S. Feng, T. Huang, P. T. Leung,
W. F. Lu, X. T. Song, F. Wang, H. B. Wang, K. Wu, Y. L. Wu, J. Yan, G. H.
Yang, J. F. Yang and Z. X. Zhang for close collaborations and/or helpful
discussions.

## References

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11institutetext: DISI, University of Trento, Italy.
11email<EMAIL_ADDRESS>

# From Knowledge Representation to
Knowledge Organization and Back

Fausto Giunchiglia  0000-0002-5903-6150 Mayukh Bagchi  0000-0002-2946-5018

###### Abstract

Knowledge Representation (KR) and facet-analytical Knowledge Organization (KO)
have been the two most prominent methodologies of data and knowledge modelling
in the Artificial Intelligence community and the Information Science
community, respectively. KR boasts of a robust and scalable ecosystem of
technologies to support knowledge modelling while, often, underemphasizing the
quality of its models (and model-based data). KO, on the other hand, is less
technology-driven but has developed a robust framework of guiding principles
(canons) for ensuring modelling (and model-based data) quality. This paper
elucidates both the KR and facet-analytical KO methodologies in detail and
provides a functional mapping between them. Out of the mapping, the paper
proposes an integrated KO-enriched KR methodology with all the standard
components of a KR methodology plus the guiding canons of modelling quality
provided by KO. The practical benefits of the methodological integration has
been exemplified through a prominent case study of KR-based image annotation
exercise.

###### Keywords:

Knowledge Representation Knowledge Organization Model Quality Faceted Approach
Information Science Data Science AI.

##  1 Introduction

Knowledge Representation is the arena of Artificial Intelligence (AI) dealing
with _“how knowledge can be represented symbolically”_ [5] within intelligent
systems. To that end, KR encompasses a wide spectrum of advanced
_technologies_ (e.g., the Semantic Web technology stack [24]) and
_methodologies_ (e.g., [13, 23]) to facilitate generation of KR artifacts
(e.g., ontologies [29], conceptual models [30], Knowledge Graphs (KGs) [33]).
Such methodologies have been widely adopted or adapted for application
scenarios from as conventional as, e.g., data integration [28], to as
innovative as, e.g., image annotation [18], wherein, conceptual models are
designed to organize (objects in) images which are later exploited as high-
quality training data for computer vision tasks like object recognition.
However, while KR methodologies have remained highly mature in terms of
supporting technology and scalable in terms of technology-enabled services, a
_key_ criticism has been that they have traditionally underemphasized
_modelling quality_ , e.g., of ontologies or conceptual models (see, [1, 10,
46], for a few prominent studies), resulting in, often, flawed and biased
datasets designed according to such models.

Knowledge Organization, on the other hand, is the arena of Information Science
dealing with the cumulative set of activities concerning, quoting [32], the
_“description, indexing and classification”_ of information resources (e.g.,
books) in different kinds of ‘memory institutions’ (e.g., libraries). To that
end, KO encompasses a wide spectrum of modelling systems [53], e.g.,
classification schemes, taxonomies, catalogs, etc., and, different approaches
[32], e.g., enumerative approach, facet-analytical approach, etc., which
integrate different KO systems. In this paper, we concentrate exclusively on
the facet-analytical KO approach originally proposed by Ranganathan [41, 42].
Noticeably, while the activities in (faceted) KO are predominantly
intellectual in nature and less technology-driven, they have, in order to
support the aforementioned activities, developed a huge number of _guiding
principles_ for (conceptual) modelling, termed _canons_ , following which
_high-quality models_ , .e.g., conceptual hierarchies, conceptual models and
model-based _datasets_ , can be generated.

From the above overviews, we notice that both KR and KO have several
complementary strengths and weaknesses. We concentrate on the specific problem
of how the modelling quality of KR artifacts can be methodologically ensured.
One clear way, in the light of the above discussion, can be the incorporation
of the guiding principles for modelling, namely, the _canons_ , developed
within the facet-analytical approach of KO in mainstream KR. In fact, the
potentiality of such an incorporation can be extremely crucial because of
growing concerns about systematic design flaws inherent in the _quality of
data_ which are designed using KR models, e.g., data quality issues [50] in
ImageNet [9], and, additionally, the _decline in both the quality and
performance_ of resulting data-driven models, e.g., [47, 35]. In this paper,
we propose a KO-enriched KR methodology which not only includes the standard
roles and activities key to the generation of a KR model, e.g., an ontology-
driven KG, but also, most importantly, incorporates the otherwise missing
activity of generating the high-quality ontology model structuring the KG
following the guiding canons prescribed in the facet-analytical approach of
KO. It also presents brief highlights of a recent study-cum-experiment in KR-
based image annotation [19] which preliminarily validates the benefits of
implementing an adapted version of the aforementioned methodology.

The remainder of the paper is organized as follows: Section 2 and Section 3
describes the roles and activities within the different phases of a standard
KR and facet-analytical KO methodology, respectively. Section 4 maps and
integrates the methodologies in the previous two sections into a single KO-
enriched KR methodology, with special emphasis on the activity which exploits
the guiding principles (canons) to ensure model (and model-based data)
quality. Section 5 describes a case study in image annotation which
preliminarily validates the advantages of the integrated KO-enriched KR
methodology. Section 6 discusses related work and Section 7 concludes the
paper.

##  2 Knowledge Representation (KR)

Figure 1: A high-level view of the KR Methodology.

Let us first concentrate on a detailed exposition of the various phases, and
within each phase, the various roles, activities and artifacts, which together
compose to form a standard KR methodology. See Figure 1 for a high-level view
of the KR methodology. The methodology can be seen as being constituted of the
following four distinct phases (the terminologies being detailed later):

  1. 1.

The first phase initiating with guiding principles and concluding with the
generation of the Lightweight Ontology.

  2. 2.

The second phase concerning the development of the Entity Type Graph (ETG)
repository.

  3. 3.

The third phase of the methodology concentrating on how the data scientist
takes in three different inputs, namely, the Lightweight Ontology, Entity
Dataset(s) and ETGs from the ETG repository, and, suitably integrates them to
generate the Entity Graph (EG).

  4. 4.

Finally, the fourth and the final phase concentrating on the different ways in
which a user can use and exploit the EG.

The diagrammatic symbols of Figure 1 include, amongst others, the various
roles (visualized via actor icon), activities (visualized as edge labels) and
artifacts (visualized variously via other (coloured) icons). Notice also that
the numbers and lowercase alphabets which identify roles and artifacts,
respectively, in Figure 1, are employed later (in Section 4) for functional
mapping purposes. We now consider each phase of the methodology individually.

The first phase, as already noted, commences with guiding principles
articulated as the intensional definition-building paradigm of Genus-
Differentia [38] proposed by Aristotle (visualized via a cloud) over two
millennia ago. According to the paradigm, the linguistic definition of any
real-world object [25] expressed, e.g., as a noun, is formulated in terms of
two constituent definitions: Genus and Differentia. While Genus defines an _a
priori_ set of properties shared across distinct objects, e.g., the property
of being a stringed musical instrument, Differentia defines a novel set of
properties used to differentiate objects having the same Genus, e.g., the
properties of musical instruments having six strings or thirteen strings.
Therefore, as illustrated in Figure 1, the Genus-Differentia guidelines are
taken in input and adhered to by the Language Engineer to create machine-
processable language data, e.g., WordNet-like lexical-semantic hierarchies of
synsets [36] codifying word meanings in different natural languages and in
different domains [2] represented in a machine processable format, e.g.,
Lexical Markup Framework [14].

Such machine-processable language data are stored and managed as part of a
Lexical-Semantic Resource (see Figure 1) which is usually designed as a
collection of WordNet-like machine-processable lexical hierarchies and, in
some cases [20, 21], with an additional language-independent semantic layer
unifying different language-specific lexical hierarchies. Given the design of
the Lexical-Semantic Resource, the final activity of this phase shifts to the
Knowledge Engineer (see Figure 1) who has to now generate the Lightweight
Ontology (Figure 1) which is an intermediate machine-processable formal
hierarchy _“consisting of backbone taxonomies”_ [27] that are being considered
for representing knowledge in the context of a specific purpose provided by
the user. To that end, the Knowledge Engineer has to take in two important
inputs. Firstly, (s)he has to take in input the appropriate lexical-semantic
hierarchy of words in a specific language which will inform the syntax and
modelling of the taxonomical hierarchy of the Lightweight Ontology she will
generate. In addition, (s)he also takes in input the (dataset) schema of the
entities which (s)he wants to model in the Lightweight Ontology, this,
providing her with the exact way in which the subsumption hierarchy of parent
and child concepts, pertaining to the entities (and their datasets), should be
organized111thereby, significantly speeding up the process of merging the
datasets with the KR model at a later stage of the methodology..

The second phase, as illustrated in Figure 1, concerns the design and
development of a repository (exposed via a catalog) of reusable Entity Type
Graphs (ETGs) [15], wherein, ETGs are defined as machine-processable
ontological representations formalizing entity types [26] which capture the
semantics inherent in (dataset) entities. Notice the fact that, ETGs, being
ontological representations, also encode object properties (modelling how an
entity is related to other entities) and data properties (modelling the
attributes which describe an entity) as is standard to any KR model. This
repository is crucial to the methodology in all the four dimensions -
Findability, Accessibility, Interoperability, and Reusability - advanced by
the FAIR paradigm [51] of scientific data management. Especially, the
repository would facilitate not only interoperability amongst ETGs (as they
are modelled following the same technological standards) but also enable ETG-
based interoperability of data (when ETGs from the repository are reused, in
conjunction with lightweight ontology, to produce the final EG). It would also
promote the circular reuse of ETGs, for instance, the reuse of the same ETG
but for different use case scenarios of KR. A crucial observation. Notice that
while there are several technical advantages of designing an ETG repository as
elucidated above, the methodology does not specify any activity path or roles
for enforcing guiding principles which ensure the high-quality of the ETG
models constituting the repository (noticeable by the lack of any activity
preceding the ETG repository in Figure 1).

The third phase concentrates on how the Data Scientist (see Figure 1) exploits
the outputs of the previous two phases and the concrete (entity) datasets of
the use case at hand to generate the final Entity Graph (EG). An EG is a type
of Knowledge Graph (KG) which is: (i) taxonomically structured via a
lightweight ontology, and, (ii) interrelated and described with object
properties and data properties from the relevant ETG. To that end, the Data
Scientist receives in input the lightweight ontology (output of the first
phase), wherein, the use-case specific concepts are hierarchically modelled
following the appropriate lexical-semantic hierarchy. The lightweight ontology
is then grounded in the relevant classificatorily-compliant ETG from the ETG
repository (output of the second phase). This activity has three key
advantages. Firstly, the ETG endows the lightweight ontology-ETG combined KR
artifact with crucial object properties (which interrelate, at a conceptual
level, the entities it encode) and data properties (which encode the
descriptive attributes of entities). Secondly, the grounding also ensures the
completeness of the KR model in terms of grounding use-case specific concepts
(modelled bottom-up) into general (domain) concepts (modelled top-down).
Thirdly, the above grounding also ensures the fact that, later, the data
encoded via the specific lightweight ontology becomes interoperable with data
encoded via other lightweight ontologies which commit to the same ETG. Given
the lightweight ontology-ETG combined KR artifact, the Data Scientist takes in
input the entity datasets and semi-automatically maps them to the combined
artifact (see, [22], for the detailed process), one dataset at a time, to
generate the final Entity Graph (EG) which is a KG modelled by populating the
nodes of the combined KR artifact with data.

Finally, the fourth phase of the methodology concentrates on the different
ways in which different users can exploit the Entity Graph (EG). This, in
turn, informs as well as depends on the different type of basic and
specialized services which can be developed to explore the EG. A set of basic
services can include the option to download a (part of a) KG in different
formats, the option to query a KG (see, [15], for an enumeration of potential
services). Advanced services can be designed depending upon the use-case, for
instance, to exploit a generated KG for image annotation tasks. A prime
example can be of an interactive service to facilitate egocentric image
annotation (with egocentrism becoming increasingly pivotal to the computer
vision community [11]). Such a service, for instance, would allow a user to
use (a portion of) a KG schema and populate each of its node (i.e., a concept)
with (multiple) relevant images to generate a hierarchically organized image
dataset ready to be downloaded and used as training data for computer vision
models.

##  3 Knowledge Organization (KO)

Figure 2: A high-level view of the facet-analytical KO Methodology.

In sync with the previous section on the KR methodology, let us now
concentrate on a detailed exposition of the various phases which together
compose to form a standard KO methodology within the facet-analytic tradition
of KO. See Figure 2 for a high-level view of the KO methodology. The
methodology can be seen as being constituted of the following four distinct
phases (the terminologies being detailed later):

  1. 1.

The first phase initiating with the guiding principles for _Classification_
[42] and concluding with the generation of the book classification number.

  2. 2.

The second phase concerned with the guiding principles for _Cataloguing_ [41]
and concluding with the design of the catalogue code.

  3. 3.

The third phase of the methodology concentrating on how the cataloguer takes
in three different inputs, namely, the book classification number, the
physical copy of the book and the catalogue code, and, suitably integrates
them to generate the book catalogue record.

  4. 4.

Finally, the fourth and the final phase concentrating on the different ways in
which a reader visiting a library can use and exploit the book catalogue
record.

The diagrammatic symbols of Figure 2 are similar as before. We now consider
each phase of the methodology individually.

The first phase, as already noted, commences with guiding principles, termed
canons of classification, postulated by Ranaganathan [42] to oversee the
generation of high-quality classification hierarchies for any subject. Notice
that, by subject, Ranganathan meant not only the universe of _macro thought_
(e.g., disciplines such as Mathematics, Chemistry) but also, crucially, the
universe of micro thought [43] (e.g., depth classification of differential
equations). We now enumerate below an overview of the three groups of canons
of classification (detailed in [42]) which are core to the methodology
illustrated in Figure 2:

  1. 1.

Canons of Idea Plane, which focus on the modelling of concepts in a
classification hierarchy based on their perceivable properties. They include:

    1. (a)

canons about characteristics based on which concepts should be differentiated
with respect to a single level in the classification hierarchy, e.g., canon of
relevance, canon of ascertainability.

    2. (b)

canons about succession of characteristics, i.e., how differentiating
characteristics should succeed one after the other with respect to a
classification hierarchy, e.g., canon of relevant succession.

    3. (c)

canons about arrays based on which concepts, within the same horizontal level
in the classification hierarchy, should be modelled, e.g., canon of
exhaustiveness.

    4. (d)

canons about chains based on which concepts, within a single hierarchical path
in the classification hierarchy, should be modelled, e.g., canon of
modulation.

  2. 2.

Canons of Verbal Plane, which focus on the proper linguistic rendering of the
concepts modelled following the canons of the Idea Plane, e.g., canon of
reticence.

  3. 3.

Canons of Notational Plane, which focus on assigning a unique numerical
identifier for each linguistically labelled concept in the classification
hierarchy, e.g., canon of synonym and canon of homonym.

Thereafter, as illustrated in Figure 2, the canons of classification as
briefed above are taken in input by the Classificationist to design (a set of)
faceted classification schedules for either general knowledge organization
usage, e.g., the Colon Classification (CC) [44] designed and developed by
Ranganathan himself (indicated with dashed lines), or, for specialized
knowledge organization usage, e.g., Uniclass222https://www.thenbs.com/our-
tools/uniclass, specialized classification for the construction sector. Given
the design of the classification schedule(s), the final activity of this phase
shifts to the Classifier (see Figure 2) who has to now generate the unique
Book Classification Number (Figure 2), e.g., Colon Number, for the subject
matter of a book. To that end, the Classifier has to take in two important
inputs. Firstly, (s)he has to take the _a priori_ designed classification
schedules which provide him/her with the concept hierarchy (with each concept
uniquely identified via an identifier) as well as the formula (i.e., the facet
formula) in which relevant concepts should be combined to generate the book
classification number. Secondly, (s)he has to take in input the thought
content of the book to be classified. Finally, the Classifier follows
Ranganathan’s analytico-synthetic classification number generation procedure
[44] to generate the Book Classification Number which uniquely identifies its
subject matter. Notice that the above process is valid not only for books but
also for any library resource.

The second phase, as illustrated in Figure 2, concerns the design and
development of a catalogue code. It commences with the canons of cataloguing
as guiding principles postulated by Ranaganathan [41] to oversee the
generation of high-quality description of any book (or, any library resource).
Some of the canons are notable to be briefed. For example, the canon of sought
heading mandates that the metadata attributes which should be captured about a
book in a catalogue should be strictly based on the likelihood of how a user
might approach the catalogue. To that end, all unnecessary metadata should be
excluded from the catalogue record. Further, the canon of consistence
prescribes that, for a specific type of (library) resource, the set of
metadata which constitute its catalogue record should be consistent, unless
otherwise prescribed by the canon of context. Another crucial principle is
that of local variation which allows flexibility in the description of a
catalogue record if need arises due to a very typical resource specific to a
context. These canons of cataloguing, amongst many others detailed in [41],
are taken as input guidelines by the Catalogue Code Designer to finally design
a Catalogue Code \- a body of specifications on how and what metadata should
be encoded in a catalogue record for a specific bibliographic resource type.
In fact, Ranganathan himself designed and developed one such catalogue code
termed the Classified Catalogue Code (or, CCC; shown in Figure 2 via dashed
lines).

The third phase concentrates on how the Cataloguer (see Figure 2) exploits the
outputs of the previous two phases and the concrete physical copy of the
resource (e.g., book) at hand to generate the final Book Catalogue Record. A
bibliographic catalogue record encodes metadata which are termed as access
points, e.g., title, author, year of publication, subject headings, etc.,
which might be help the reader in her quest to search and identify the actual
copy of the book (or, resource). To that end, the Cataloguer receives in input
the book classification number (output of the first phase), wherein, the
subject matter of the book is modelled following the appropriate
classification schedule (and facet formula). Given the book classification
number, the procedure of Chain Indexing is performed, whereby, the
classification number is reverse-engineered through a series of specified
steps (see [41] for details) to generate subject headings (tags in modern
parlance) which can serve as access points in the catalogue record. Further,
the Cataloguer also receives two other inputs: the concrete copy of the book
which contains all its imprint details, and, the catalogue code (e.g., CCC)
which strictly specifies which and how such imprint information should be
modelled in the catalogue record. Thereafter, the Cataloguer integrates the
subject headings together with the requisite imprint attributes and the
classification number (together with the call number to identify the book’s
exact place in the shelves) to generate the final Book Catalogue Record.

Finally, the fourth phase concentrates on how a library reader can exploit the
book catalogue record. In addition to the usual ways of using the catalogue
(see, e.g., [41]), Ranganathan’s APUPA principle [45] facilitates a reader in
finding very related and somewhat related books/resources on either side of
the particular resource one is searching for). Further, a reader can use the
catalogue record of an Online Public Access Catalog (OPAC) which is enhanced
with library discovery services [6], thereby, going beyond traditional means
of library search to include web-scale exploratory search and recommendations.

##  4 From KR to KO and Back

Phase | KR | KO | KO-Enriched KR
---|---|---|---
1 |  | Language Engineer
---
(1.1)
| Classificationist
---
(1.1)
| Language Engineer
---
(1.1)
2 |  | Knowledge Engineer
---
(2.1)
| Classifier
---
(2.1)
| Knowledge Engineer
---
(2.1)
3 |  | Data Scientist
---
(1.2 + 2.2)
| Catalogue Code Designer
---
(1.2)
Cataloguer
(2.2)
| Ontology Engineer
---
(1.2 + 2.3)
Data Scientist
(2.2)
4 |  | User
---
(3)
| Reader
---
(3)
| User
---
(3)

Table 1: Functional mapping between roles of KR, KO and KO-Enriched KR.

In the previous two sections, we’ve provided a detailed elucidation of the
principle roles, activities and artifacts of both the Knowledge Representation
and the facet-analytical Knowledge Organization methodology, respectively. In
this section, before showing how to complete the loop from KR to KO and back,
we first concentrate on a functional mapping of, chiefly, the roles and the
artifacts, of the two aforementioned methodologies (see the first three
columns of Table 1). Notice two things. Firstly, via functional mapping, we
ascertain whether or not two roles or artifacts perform the same or synonymous
broad function, irrespective of differences in their syntax, semantics or
form. Secondly, in Table 1, we illustrate only the roles as they embody the
major functional differences between the two methodologies and not the
artifacts (duely elucidated in the following description) which are
functionally synonymous. We constantly refer to Table 1, and, to Figure 1 and
Figure 2 as required, in the following discussion.

At the very outset, note that we proceed the functional mapping on the basis
of the four informal phases via which we detailed each individual methodology.
We notice that, in the first phase, there is a mapping between the roles of
Language Engineer (1.1) and that of Classificationist (1.1). This is clearly
due to the fact that both these roles, based on input guiding principles,
generate lexical-semantic classification hierarchies. Next, we also note that
the artifacts that the above two roles produce, namely, the Lexical-Semantic
Resource (a) (see Figure 1) and the Classification Schedule (a) (see Figure 2)
are also in functional mapping to each other, given that both are essentially
constituted of (a set of) lexical-semantic classification hierarchies focused
on different subjects, domains, etc. Further, the roles Knowledge Engineer
(2.1) and Classifier (2.1) are also mapped to each other because their central
function is to use prescribed lexical-semantic hierarchies and classify the
subject of resources according to their relevant concept within the hierarchy.
To that end, the output artifacts they produce, i.e., Lightweight Ontology (c)
(see Figure 1) and Book Classification Number (c) (see Figure 2) also serve
the synonymous function of encoding the categorization of either the entity
dataset schema or the book subject matter.

In the second phase, the ETG Repository (b) (see Figure 1) and Catalogue Code
(b) (see Figure 2) are also functionally synonymous in the sense that both of
these artifacts endow their input artifacts with reusable specifications of
how to describe concepts via attributes within a classification hierarchy. The
first major break in functional mapping occurs with respect to the role of
Data Scientist (1.2 + 2.2). The Data Scientist role subsumes two roles,
namely, that of the Catalogue Code Designer (1.2) which exclusively deals with
the specification of the descriptive (data) attribute schema in addition to
that of the Cataloguer (2.2) whose function is to generate the book catalogue
record. Further, notice the functional synonymity between the two artifacts:
Entity Graph (d) (see Figure 1) and Book Catalog Record (d) (see Figure 2),
both of whose function is to provide the end user with an integrated view of
top-down and bottom-up knowledge. Finally, the roles of a User (3) and a
Reader (3) are also mapped as both function as end-users using the respective
output artifacts of their methodologies in different ways. Notice also the
fact that the activities across both the methodologies are also functionally
mapped (see the labelled edges on Figure 1 and Figure 2, respectively) with
the only exception of the gap prior to the ETG repository in Figure 1.

Figure 3: A high-level view of the KO-Enriched KR Methodology.

Given the functional mapping, we now concentrate on how the guiding
principles, i.e., canons, advanced by the facet-analytic KO approach can be
incorporated within the KR methodology to ultimately result in a KO-enriched
KR methodology. To that end, we concentrate on the various phases (harmonized
from both the KR and KO methodologies via the functional mapping) which
together compose to form the KO-enriched KR methodology (see Figure 3). The
methodology can be seen as being constituted of the following four distinct
phases:

  1. 1.

The first phase initiating with guiding principles and concluding with the
generation of the Lightweight Ontology.

  2. 2.

The second phase initiating with guiding principles (canons) guiding the
development of high-quality Entity Type Graphs (ETGs) within the ETG
repository.

  3. 3.

The third phase of the methodology concentrating on how the data scientist
generates the Entity Graph (EG), and,

  4. 4.

The final phase concentrating on the different ways in which a user can use
and exploit the EG.

The diagrammatic symbols of Figure 3 are similar as before. We now consider
each phase of the methodology briefly.

The first phase (see Figure 3) is exactly the same as that of the original KR
methodology (see Figure 1). Briefly, the Language Engineer designs a Lexical-
Semantic Resource composed of machine-processable language data, i.e.,
lexical-semantic hierarchies, designed by following the Genus-Differentia
guiding principle of Aristotle. These lexical-semantic hierarchies are then
used by the Knowledge Engineer in conjunction with the entity dataset schemas
to generate the Lightweight Ontology. The second phase of the methodology,
i.e., the development of the ETG repository, is markedly different from that
in Figure 1 and constitutes the core of the back loop in from KR to KO and
back. It initiates with the still ongoing work (represented via dashed line in
Figure 3) on the adaptation of Ranganathan’s guiding canons of both
classification and cataloguing (see Section 3) for the development of high-
quality ETGs. Notice that the current effort (and the final goal) is to adapt
all the relevant canons for classification (i.e., the different canons from
the Idea Plane, Verbal Plane and Notational Plane) to generate taxonomically
well-founded ETG hierarchies and the canons for cataloguing to mandate
guidelines as to how (data) properties should be modelled to describe
(conceptual) entities in such hierarchies. To that end, the methodology
introduces a new role, that of an Ontology Engineer, who will ensure the
development of the ETGs, and thereby, the ETG repository, in full conformance
with the quality guidelines prescribed by the canons of Ranganathan suitably
adapted. Notice that, with the support of Ranaganathan’s canons of
classification, the Ontology Engineer would be able to generate ETGs not only
restricted to macro-domains but also focused on fine-grained micro domain of
interests.

The third and the fourth phase of the methodology (see Figure 3) is also the
same as that of the original KR methodology (see Figure 1). Briefly, the Data
Scientist takes in input the lightweight ontology, the entity datasets and the
relevant quality-enriched ETG and suitably integrates them, in an iterative
fashion, to ultimately generate an EG. Finally, the fourth phase concentrates
on the ways in which the user can exploit the output EG. At this point, let us
recollect the example of an advanced service of egocentric image annotation
described in the fourth phase of the original KR methodology (in Section 2).
In addition to the above service, the methodological incorporation of
Ranganathan’s canons would also ensure, in a major advance from the state-of-
the-art, the generation of fine-grained classification (ETG) hierarchies
which, e.g., as evidenced from (their lack in) computer vision
literature333https://paperswithcode.com/task/fine-grained-image-
classification, is bound to deeply enrich (KR-based) dataset building and data
quality in the entire spectrum of fine-grained image classification.

Notice also, from Table 1, how the roles within the integrated KO-enriched KR
methodology functionally compares with that of the original KR and facet-
analytical KO methodology. The role of the Language Engineer (1.1), Knowledge
Engineer (2.1) and User (3) in the integrated methodology is functionally
similar to that of their counterparts in the KR and KO methodology (which have
been functionally mapped before). The key difference, however, is reflected in
the role of Ontology Engineer (1.2 + 2.3), which, builds the ETG hierarchy
adhering to the canons (a completely new role absent in the other
methodologies) and additionally specifies the data attributes within the ETG
(part of the role of Data Scientist in the KR and the role of Catalog Code
Designer in the KO methodology). The Data Scientist (2.2) in the KO-enriched
KR methodology is functionally mapped to the role of Cataloguer in the KO and
to a part of the role of Data Scientist in the KR methodology. This difference
is because the former Data Sceintist (2.2) role only integrates the previous
three outputs to generate the EG, whereas, the latter role of Data Sceintist
(1.2 + 2.2), in addition to the above function, also specifies data attributes
(performed by the new role of Ontology Engineer in the KO-enriched KR
methodology). In terms of artifacts, the KO-enriched KR methodology is,
functionally, the same as that of the KR methodology, and, thereof, to the KO
methodology. Finally, in terms of activities, the KO-enriched KR methodology
adds two extra activty flows over and above the KR methodology, namely: (i)
the first activity flow from Ranganathan and Ongoing Work to the Ontology
Engineer in terms of adapting guiding principles of CC and CCC, and (ii) the
activity flow from the Ontology Engineer to the ETG Repository in terms of
designing the repository.

##  5 Case Study

Let us now focus on how the incorporation of the guiding principles of
modelling quality (i.e., the canons) within the overall KO-enriched KR
methodology can significantly improve the quality of those data which, within
the data science research landscape, are organized and generated by exploiting
KR models. To that end, we consider the case of ImageNet [9] which is a
benchmark dataset for various computer vision tasks such as image annotation,
object detection, etc. In fact, ImageNet is also the most prominent dataset
within the data science landscape which is structured by populating the nodes
of the WordNet lexical-semantic ontology with hundreds of thousands of images
of different objects. While it has been heavily used to test and develop
multiple computer vision models (e.g., [31, 34]), a recent study-cum-
experiment [19] revealed a fundamental problem in the quality of the ImageNet
ontology model which results in systematic flaws in its data quality (see
[50]).

The study established that, in ImageNet, there was a many-to-many mapping
between, quoting [19], _“the visual information encoded in an image and the
intended semantics of the corresponding linguistic descriptions”_. In its
detailed analysis, the study found that the underlying cause was the lack of
an explicit methodology by the ImageNet creators, backed by guiding principles
for ensuring conceptual model quality, to align the way visual classification
hierarchies, e.g., of images, are modelled and the way in which lexical-
semantic hierarchies are modelled. The study, quoting a previous study [50],
also noted that this misalignment between visual and linguistic classification
led to two major categories of design flaws444Notice that the study [19] also
found an additional category of design flaw - Multi-Object Images, the root
cause of which, however, is object localization and not classification model
quality and alignment. systemic in ImageNet data annotation (and affecting all
models developed and trained on it). Firstly, there are Mislabelled Images,
wherein, there is a complete misalignment in how an image is visually
classified and annotated and how it is linguistically classified and
annotated. A famous example is that of a birthday cake labelled as an acoustic
guitar (see, [19]). Secondly, there are Single-Object Images, wherein, the
misalignment between the visual classification and linguistic classification
of an image is chiefly due to either visual polysemy or linguistic polysemy
(several examples in [19]). To that end, notice the fact that the root of all
the aforementioned design flaws in ImageNet data is the lack of modelling
quality in classification, namely, in visual and linguistic classification of
an image, and, in their alignment thereafter.

In response to the above problems, the study proposed an overall image
annotation methodology within which KO-enriched KR modelling is a key
component. Notice, however, the fact that KO-enriched KR modelling component
in the said methodology has been adapted and tuned to the needs of image
classification. To that end, the study proposed that any KR-based image
classification and annotation exercise should be comprised of the following
activities:

  * •

Firstly, Linguistic Classification, involving the lexical-semantic
hierarchical modelling of the space of linguistic labels used to annotate
images. Here the meaning of each label within the linguistic hierarchy should
be defined in terms of linguistically defined properties encoding a selected
set of visual properties, thereby, factoring in the alignment of visual and
linguistic classification at the language level. For example, the definition
of a guitar as being a “string instrument with six strings”.

  * •

Secondly, Visual Classification, involving the selection of visual properties
of an object (in an image) and annotating it with an appropriate label from
the linguistic hierarchy as defined via the linguistic classification. For
example, for the object Guitar in an image, the annotator would would not
choose the label Guitar but, rather the property that it has six strings. The
label Guitar comes for free because of how it is defined via the same property
in the linguistic classification hierarchy, thereby, eliminating any visual or
linguistic ambiguity.

It is very interesting to note that the aforementioned modelling of visual and
linguistic classification is completely guided by the same set of canons of
classification proposed by Ranganathan (briefed in Section 3) and adapted for
KO-enriched KR-based visual classification. The reader is referred to the
study in [16, 17] for details regarding how the canons were adapted for visual
classification and examples of several key modelling instances where the
canons ensured the quality of alignment between visual and linguistic
classification.

Finally, the study in [19] also performed three experiments to quantify the
improvement, if any, brought by the dataset generated following the canons-
based high quality alignment between visual and linguistic classification. To
that end, the study modelled a dataset on a selected fragment of ImageNet,
namely, the stringed musical instruments hierarchy and performed three
separate experiments on: the accuracy of computer vision (CV) models when
trained on the dataset, the improvement in the inter-annotator agreement with
respect to the visual and linguistic classification, and, the improvements (if
any) in the cost of image annotation. As detailed in [19] (see Section 5:
Evaluation), the study found significant improvements for all the above
experiments and attributed the improvement to the enrichment in classification
model quality facilitated by adherence to the canons.

##  6 Related Work

There are three important research lines within Data and Information Science,
namely, KR model quality, KR methodologies and data quality in data science
ecosystem, which are significantly related to the current work. We consider
each of the above areas briefly.

Firstly, we focus on literature analyzing the quality of classification
semantics of KR models of various kinds, e.g., ontologies, conceptual models.
The research in, e.g., [1, 40] advanced a checklist of modelling issues which
directly affect the classification semantics of ontology models, including,
amongst others, cycles in a hierarchy, creating polysemous elements, creating
synonyms as classes, using different inconsistent naming criteria. More
recently, there have also been a lot of work in uncovering _antipatterns_ \-
taxonomically inadmissible hierarchical patterns - in KR models [30, 46].
Notice that modelling quality enrichment activity path based on Ranganathan’s
canons in our proposed KO-enriched KR methodology tackles all of the
aforementioned modelling issues which chiefly impact the classification model
of the hierarchy.

Secondly, let us focus on some of the prominent state-of-the-art KR
methodologies. The highly cited methodology in [13] advanced a _“life cycle to
build ontologies based in evolving prototypes”_. The methodology proposed in
[37] offered the flexibility of choosing top-down, bottom-up or middle-out
approaches while modelling ontologies without any explicit emphasis on
modelling quality. Recently, the NeOn methodology [48] proposed reuse, re-
engineering and merging of ontological resources and the XD methodology [3]
stressed on reuse of ontology design patterns. Notice that modelling quality,
especially for the latter, is crucial for integrating the various ontology
patterns reused. To that end, differently from all above KR methodologies, the
KO-enriched KR methodology includes a dedicated activity path for ensuring the
classification modelling quality of the KR model.

Last but not the least, let us focus on data quality in data science research.
The research reported in [49] is an early effort in analyzing the issue of
data quality. Recent research, e.g., [39, 52], proposes adherence to best
practices-based methodologies in the modelling of datasets that are attentive
to limitations [4] and impact. It is also worthy to note that the
crowdsourcing community has also focused extensively on the problem of data
quality, see, e.g., [12, 7]. Some early work have also been done towards
improving the process of ensuring quality, see, e.g., [8]. In our current
work, the proposed KO-enriched KR methodology is a significant advance in the
aforementioned context of the methodological incorporation of quality best
practices in modelling.

##  7 Conclusion

To summarize, the paper provided a non-trivial functional mapping between the
components of the KR and facet-analytical KO methodologies and derived an
integrated KO-enriched KR methodology which subsumes all the components of the
KR methodology with the major addition of the KO-based canons.

## Acknowledgement

The research has received funding from JIDEP under grant number 101058732.

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