素粒子論研究 ･ 電子版 Vol. 38 (2022) No. 3

Soryushiron Kenkyu

OCHA-PP-372

Infrared Divergence and Low Energy Theorem
in Non-Abelian Gauge Theory
Akio Sugamoto
Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Abstract
This is the English translation of the Doctor thesis submitted to the Universityof Tokyo in March 1978. Its Japanese version was published in Soryushiron Kenkyu
60-2 (1979-11) pp. 47–117. An appendix is newly added, which is the excerption of a
relevant mathematical part of a report submitted to Prof. Mikio Sato in August 1976.
In the thesis, ﬁrst, the cancellation of infrared divergences is reviewed. The cancellation occurs between a virtual process with soft photon corrections and the otherprocess with real soft photon emissions. The factorization of infrared divergences isshown by using the eikonal approximation, or by the (renormalization group like) differential equation controlling the infrared divergences.
Next, two examples in QCD, the fermion (quark)-fermion (quark) scattering and thefermion (quark) gauge-boson (gluon) scattering are examined at one loop, from which
the importance of Ward-Takahashi identities becomes manifest for the cancellation to
occur. (The identities represent the group properties.)
After these preparations, the factorization of infrared divergences in QCD is provedat all orders in the perturbation theory, by full usage of the Ward-Takahashi identities.
The identites are also proved at all orders. In these proofs, the axial gauge condition
is used, which simpliﬁes the derivation of the Ward-Takahashi identities as well as theproof of the unitarity. The cancellation of infrared divergences in QCD occurs among“the gauge invariant set of graphs”, if the quantum numbers of color are averaged andsummed over the initial and ﬁnal states, respectively. In this way, the low energy
theorem by F. E. Low, for emission of one or two soft gauge bosons, are proved at allorders in the perturbation expansion.
By joining the two emitted soft gauge bosons, the diﬀerential equation controllingthe infrared divergences in QCD can be derived. In QCD, the coupling constant renormalization introduces the other infrared divergence, which is governed by the beta
function β(g), or by the ultraviolet divergences in the pure Yang-Mills theory.
In the Appendix, the cancellation of infrared divergences is examined mathematid of Mikio Sato’s microfunction.
cally in terms of the singular spectrum S.S
In the Epilogue (2022), a motivation for recently translating the thesis in English
is brieﬂy stated.

Soryushiron Kenkyu

1

Introduction and Summary

The force potential acting between quarks via strong interaction is considered to rise linearly,
when the distance between quarks becomes larger. To search for its reasoning in quantum
ﬁeld theory, we have to investigate the long distance behavior [i.e., the infrared behavior
in momentum space] of the non-Abelian gauge theory, since it describes the dynamics ofgluon ﬁelds connecting quarks. From the infrared regions the divergences called infrared
divergences appear, due to the masslessness of the gluon particle. It is considered from sometime ago, that the information on the infrared behavior of the non-Abelian gauge theory,or on the conﬁnement of quarks in other words, can be obtained from the study of infrared
divergences in the theory [1].
The Quantum Electrodynamics (QED) has infrared divergences similar to non-Abeliangauge theory (in the following we will use QCD for it, Quantum Chromodynamics), since
photon is massless. The infrared divergences in QED have been studied well for a long time[2]–[4]. We have to refer to the well-known QED and extract the special characteristics in
QCD, when we study the infrared divergences in QCD.
The results obtained by the past several years’ investigation on the infrared region of
QCD are summarized roughly as follows:A) Cancellation of infrared divergences: It is shown by the lower order graphs that, byemitting non-detectable extra gauge bosons, a physically meaningful diﬀerential crosssection has no infrared divergence [5]–[8], [12]. These works are based on the renormalization carried out at oﬀ-shell; the cancellation of infrared divergences is violated when
on-mass-shell renormalization is performed. The proof of the cancellation in general isan unsolved problem.
B) Problem of summing up infrared divergences: Renormalization group-like diﬀerential
equations are predicted in various processes at the lower orders [9]–[12], and a proof
at all orders is given [13]. Its proof is complicated compared with the simplicity of its
answer.
C) Potential between quarks: Some progress has been made in the idea of attributing theorigin of the linearly rising potential between quarks to the infrared divergences in QCD[14].
D) Low energy theorem: One of the low energy theorems has been proved at all orders of theperturbation [15]. For this proof QCD in the axial gauge has been used. Generalizationof the proof to another low energy theorem has been done and it is applied to B) Problem
of summing up infrared divergences [16].
E) Others [17], [18].
The above characteristics can be summarized in a word as follows: the infrared behavior of QCD is characterized by the coexistence of the behavior identical to QED and the
QCD-speciﬁc divergences appearing in the coeﬃcients of expanding the on-mass-shell charge
by the oﬀ-mass-shell charge.
This thesis consists of the author’s work [7] [reference paper I, classiﬁed as A) in the above
classiﬁcation], his work done in collaboration with Norio Nakagawa and Hiroaki Yamamoto

2

Soryushiron Kenkyu[15] [reference paper II, classiﬁed as D)], and its generalization afterwards [15] [classiﬁed asD) and B)].
Let us summarize the contents. Section 2 is a summary of infrared divergences in QED,in Section 3 cancellation of infrared divergences at one loop is discussed, and Section 4 gives
the proof of low energy theorem using the axial gauge condition, and its application.
Sections 2–4 are divided into a number of subsections the contents of which are explainedin the following.
Firstly, taking the electron scattering by an external ﬁeld as an example, the way to
extract infrared divergence factors at one loop is stated in Subsection 2.1. This is generalized
to all orders in Subsection 2.2, by deriving a diﬀerential equation with respect to λ, where
λ is introduced as a photon mass, acting as an infrared divergence cutoﬀ. This diﬀerential
equation is easily solved and shows that the infrared divergence factor at all orders becomes
an exponential with the one-loop result raised to its power. It is very recently that thediﬀerential equation begins to be used in the study of infrared divergences in QED [19]. In
this subsection we use a method a little diﬀerent from the usual one. Our method is the
QED version of the low energy theorem proposed by Cornwall and Tiktopoulos in the studyof infrared divergences in QCD. They assumed the low energy theorem in QCD, without
giving the proof of it at all orders. [The proof of this low energy theorem in QCD is a main
theme of Section 4.] In QED, however, this low energy theorem can be easily derived, byusing the so-called eikonal identity. Therefore, this Subsection 2.2 is the training place ofvarious techniques to be used later in Section 4. In the next Subsection 2.3, the so-called
Bloch-Nordsieck theory is reviewed; that is, the infrared divergences studied in Subsection
2.2 can be cancelled by adding the soft photon emission processes which are not separable bydetection. Here, in comparison with QCD, it is shown by the explicit estimation of Z3 at one
loop that no infrared divergence newly appears in QED via the coupling renormalization.
The next Section 3 is a summary of the reference paper I, checking the cancellation of
infrared divergences in QCD at one loop. In this section the covariant gauge QCD is used.
First, in Subsection 3.1, by examining one-loop Feynman diagrams for fermion-fermion andfermion-gauge boson scatterings, graphs having infrared divergences are picked up by power
counting. As a result, it becomes a good way to sum up a number of graphs [one graph for thefermion-fermion scattering, but three graphs for the fermion-gauge boson scattering]. The
next Subsection 3.2 is important, giving a proof of the factorization of infrared divergences(at one loop) in QCD. It is trivial that the factorization in QCD of a soft gauge boson
coupled to the external fermion is identical to that in QED, except for a color factor, but
it is also proved that the factorization works also for a soft gauge boson coupled to theexternal gauge boson, by summing up a number of Feynman graphs. This set of Feynman
graphs is that for which Ward-Takahashi identities hold, and is equal to the set of Feynman
graphs appearing in Subsection 3.1. [This recognition will be deepened in the later Section
4 in which Ward-Takahashi identities play a crucial role for the factorization of infrared
divergences.]
In the next Subsection 3.3, it is cheeked that the cancellation of infrared divergences
occurs by adding the extra-emission of soft gauge bosons, leading to the cancellation of
infrared divergences at scattering amplitudes expanded in the bare coupling. Here, it is also
shown that the cancellation of infrared divergences does not occur unless the color indices areaveraged and summed over the initial and ﬁnal states, respectively [i.e., under the condition
that color can not be observed]. Furthermore, the classiﬁcation of the divergences is done in3

Soryushiron Kenkyu

this Subsection. For the one-loop diagrams, we consider three cases in which the internal lines
are massless or not, and the infrared divergences are shown to be classiﬁed, by comparing
the analysis in the Feynman parameter space and that in the momentum space. Introducing
two cutoﬀs λ′ and η [λ′ cuts the lower end of the momentum kµ as |kµ | > λ′ [no otherinfrared divergences], and η cuts the angle θ between two gauge bosons on the mass shell,as 1 − cos θ > η], the divergences of ln λ1′ cancel in the fermion-fermion scattering, while thedivergences of ln λ1′ and ln λ1′ ln η1 [the highest infrared divergence] cancel in the fermion-gaugeboson scattering. The divergence ln η1 is not discussed here, but is partly addressed in thereference paper I. [Similar cancellation of infrared divergences are checked independently by
a number of people [5]–[7], [12]. A short summary on what was done so far for what process
is given in the Reference of the reference paper I and its Note added.]
The next Subsection 3.4 gives the infrared divergences speciﬁc to QCD appearing in the
charge renormalization. The coeﬃcients of expanding the coupling constant deﬁned on themass shell gR in the bare coupling gB , have an infrared divergence speciﬁc to QCD. This
infrared divergence is equal to the coupling constant renormalization of the pure Yang-Mills
theory (without fermions), when its ultraviolet cutoﬀ is replaced by the infrared cutoﬀ.
This is an infrared divergence speciﬁc to QCD, being controlled by β(g) of pure Yang-Mills
theory. [This quantity is pointed our by a number of people.[7], [9]–[12]] Due to this infrared
divergence, the cancelation of infrared divergences is violated up to this divergence when
the on-mass-shell renormalization is performed. On the other hand, under the oﬀ-massshell renormalization, the divergence appears in the on-mass-shell charge. This provides asuggestion towards the quark conﬁnement.[14]From Section 4, the properties of infrared divergences are examined, which hold at allorders in perturbation expansion. The main theme is the proof of the low energy theorem
in QCD at all orders in perturbation.

2

Summary of infrared divergence in QED

A number of results have been obtained so far concerning infrared divergences and thelow energy theorem in QED[2]–[4], [19]. Let us summarize some of the results that are
useful when we investigate such problems as infrared divergence, low energy theorem, andquark conﬁnement in non-Abelian gauge theories, also known as quantum chromodynamics(hereinafter abbreviated as QCD).

2.1

Factorization of infrared divergence (one loop)
For simplicity, we consider electron scattering by an external ﬁeld. Figure
2.1.1 shows a process in which an electron with momentum p is scatteredby an external ﬁeld Jρ and ﬂies away with momentum p′ . Let us begin
with one-loop graphs. The following three graphs are of interest (Figure
2.1.2).

Figure 2.1.1:

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Figure 2.1.2:
First, Figure b) is considered. The scattering amplitude Sb corresponding to this Figureb) is given byZdN ki(/p′ + k/ + m)i(/p + k/ + m)
′
µ
ū(p
)ieγ
J/
ieγ ν u(p)Dµν (k),
N
′
2
2
(2π)
(p + k) − m + iϵ (p + k)2 − m2 + iϵSb =(2.1.1)

where N (being complex in general) is the spacetime dimension and Dµν (k) represents a
photon propagator. The propagator Dµν (k) has a diﬀerent form for various gause conditions.
As an example, the forms for the covariant and axial gauge are given by


−i
kµ kν
(2.1.2)Dµν (k) = 2gµν − (1 − α) 2
(covariant gauge)k + iϵk + iϵandkµ n ν + n µ kν
−i
kµ kν
2
2
gµν −
+ (αk + n )Dµν (k) = 2. (axial gauge)k + iϵk·n(k · n)2(2.1.3)In the case of QED, the eﬀect of changing the gauge condition is exhibited only in thediﬀerence of the gauge boson propagator. However, in the case of QCD, it also aﬀects thepresence or absence of Faddeev-Popov ghost ﬁelds (see Section 4.1). Now, in Eq. (2.1.1), we
estimate the value of the integral for small kµ . First we rewritei(/p′ + k/ + m)
2p′µ + (−/p′ + m)γ µ′
× · · · ≈ ū(p )(−e)× ···ū(p )ieγ ′|kµ |≪m(p + k)2 − m2 + iϵ
2p′ · k + k 2 + iϵ
2p′µ× ū(p′ ) × · · ·(2.1.4)= (−e) ′2
2p · k + k + iϵ′µ

where the commutation relation of the Dirac matrices {γ µ , γ ν } = 2g µν is used in the secondline. This transformation takes advantage of p′ being on the mass shell. That is, thepropagator of the electron (momentum p′ + k) coupled to an on-mass-shell external line anda soft photon of momentum kµ behaves as O(1/k) in the limit of |kµ | ≪ m. Similarly,
··· ×

2pν
i(/p + k/ + m)
ν
ieγ
u(p)
≈
·
·
·
×
u(p)(−e)|kµ |≪m(p + k)2 − m2 + iϵ
2p · k + k 2 + iϵ
5(2.1.5)

Soryushiron Kenkyu

holds. Using (2.1.4) and (2.1.5), the integral (2.1.1) around |kµ | ≪ m gives
Z
dN k
2pν
2p′µ
2
Sb ≈ eDµν (k)S (0) ,(2π)N 2p′ · k + k 2 + iϵ 2p · k + k 2 + iϵ

(2.1.6)

where S (0) = ū(p′ )/J u(p) represents the amplitude at the lowest order.
Setting N = 4, we ﬁnd that Eq. (2.1.6) produces a logarithmic divergence from the integral in the region |kµ | ≪ m (its speciﬁc calculation will be described later). This logarithmic
divergence is called infrared divergence. Now, the scattering amplitudes for Figures a) andb) are both given by
1 (1)(2.1.7)
Sa = Sb = Z2 × S (0) ,
2
(1)

(1)

where Z2 is the value of Z2 factor at one loop. Since Z1 = Z2 holds in QED, this Z2 can(1)
be determined from (2.1.6) (that is, from Z1 ) asSb (p = p′ )(1)(1)
Z2 = Z1 = −
S (0)
Z
N
d k
2pµ
2pν
= −e2
Dµν (k).
(2π)N 2p · k + k 2 + iϵ 2p · k + k 2 + iϵ

(2.1.8)Thus, the infrared divergence part included in the one-loop graphs of Figures a)–c) is completely extracted as a function as


Z
1 2
2pµ
dN k
2p′µ
S a + Sb + Sc ≈ − e−2(2π)N 2p′ · k + k 2 + iϵ 2p · k + k 2 + iϵ


2p′ν
2pν
×
−
Dµν (k)S (0) .
(2.1.9)2p′ · k + k 2 + iϵ 2p · k + k 2 + iϵ
When the propagators (2.1.2), (2.1.3) for various gauge conditions are substituted into thisformula, the term proportional to kµ , kν is dropped in both cases, and the formula yields thefollowing value independent of the gauge conditions (since k 2 in the denominator in (2.1.9)
is negligible):
S (1) ≡ Sa + Sb + Sc

2
Z
1 2
dN k
2p′µ
2pµ
−i
−
S (0) .
≈− e
N
′
2
2
2
2
(2π)
2p · k + k + iϵ 2p · k + k + iϵ k + iϵ

(2.1.10)

We have seen above how the infrared divergence factor (the coeﬃcient of S (0) in Eq. (2.1.10))
for one loop is decomposed.
Next, we speciﬁcally calculate the value of this infrared divergence factor. Various methods can be conceived as the regularization of infrared divergence.
1) N -dimensional method: The dimensionality of space is analytically continued to a complex number N with Re N > 4. The divergent quantity is extracted as a pole N 1−4 in thelimit N → 4.
2) Give the photon a virtual mass λ: The infrared divergence is extracted as ln λ1 in thelimit λ → 0.

6

Soryushiron Kenkyu

3) Keep the external line oﬀ the mass shell, i.e., p2 = p′2 = m2 − δ 2 : The infrared divergenceappears as ln 1δ in the limit δ → 0.
4) Cut the lower end of the momentum integral, i.e., set the domain of integration to|kµ | > λ′ : The infrared divergence is extracted as ln λ1′ in the limit λ′ → 0.
In general, infrared divergence arises when the condition of the external line being on themass shell is satisﬁed simultaneously with the condition of the photon being massless (refer
to one-loop examples). Therefore, the infrared divergence can be removed by violating oneof these two conditions. The regularization 2) is designed to violate the latter, while theregularization 3) the former.
Using the regularization 1), the factor corresponding to Sb yields
Z
dN k
2p′µ
2pν
−igµν
2
,
Ib ≡ eN′22(2π) 2p · k + k + iϵ 2p · k + k + iϵ k 2 + iϵ
and introducing Feynman parameters α, β, γ into respective factors of the integrand,
Z 1Zdα dβ dγ δ(1 − α − β − γ)dN k2′
Ib = −e 4(p · p ) 2!
N(2π) 0 [k 2 + 2(αp′ + βp) · k + iϵ]3ZZ 1dα dβ dγ δ(1 − α − β − γ)dN k2′
= −e 4(p · p ) 2!
(k-integral shifted)(2π)N 0 [k 2 − (αp′ + βp)2 + iϵ]3

Z 1
N
e2
dα dβ dγ δ(1 − α − β − γ)′
. (k-integral performed)=−4(p · p ) Γ 3 −
N
2
(4π)
2
[(αp′ + βp)2 − iϵ]3− 20(2.1.11)
The above calculation is based on the Feynman parameter formula and the formula
Z
1
dN k
1
(−1)m Γ(m − N2 )=i.
(2.1.12)N2N/2m(2π) [k − V (α) + iϵ](4π)Γ(m) [V (α) − iϵ]m− N2
In (2.1.11), Γ(3 − N2 ) −−−→ 1, which does not produce a pole of N 1−4 . This shows thatN →4

this integral Ib has no ultraviolet (UV) divergence [30]. Infrared (IR) divergence arises fromthe integral of the Feynman parameters, since in (2.1.11), the denominator of the integral
becomes zero for α = β = 0 and contributes to divergence [31]. Now we change the variables
as follows:
α = xy,β = (1 − x)y,γ = 1 − y.

(2.1.13)

Then α = β = 0 corresponds to y = 0 (the Jacobian being y), andZ 1 Z 1
e2
N
1
′
2 N
−3
2
Ib = −
4(p
·
p
)
Γ
3
−
dx
y
dy
(y
)
N .
(4π)2
2
[(xp′ + (1 − x)p)2 − iϵ]3− 200The y-integral produces a divergence from around y ≈ 0:
Z 1
1  N −4 11y N −5 dy =y=. (where Re N > 4)
0
N −4
N −4
0
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Soryushiron Kenkyu

This is a pole at N = 4, corresponding to infrared divergence. Thus
Z 1e2
14(p · p′ )Ib = −dx(4π)2 N − 4 0(xp′ + (1 − x)p)2 − iϵp1 − 4m2 /t − 1
4 − 8m2 /t
e2
1
p
p
ln
=+
(4π)2 N − 4 1 − 4m2 /t
1 − 4m2 /t + 1(2.1.14)

Using this equation and (2.1.8), the infrared divergence factor for Figures a) and c) yields
1e2
1
Ia = Ic = − Ib (p = p′ , i.e., t = 0) = 2.
22(4π) N − 4(2.1.15)

Thus the infrared divergence parts at one loop are grouped into the following form: QED1
1(1)
2[S ]IR = e
B
× S (0) ,(2.1.16)
N −4
m2
where

B

1
m2

QED

1
=
(4π)2

!
p
1 − 4m2 /t − 1
4 − 8m2 /t
ln p
.
4+ p
1 − 4m2 /t
1 − 4m2 /t + 1(2.1.17)

The replacement rule for other regularizations of infrared divergence is given by11
1
1
ln , 2 ln , ln ′ ↔
λ
δ
λ
N −4

2.2

(Re N > 4).

(2.1.18)

Factorization of infrared divergence in terms of a diﬀerential
equation (all order)

In this subsection, the result for one loop in the previous subsection is generalized to allorders. That is, we prove that the strongest infrared divergences included in the scattering
amplitude of the process of Figure 2.1.1 are grouped into
[S]IR = e[I

(1) ]

IR

× S (0)(2.2.1)(in the form of an exponential with the one-loop value [I (1) ]IR raised to its power) at allorders. In this case the exponent in (2.2.1) is the one-loop IR factor
 ′µ  ′ν

Z
1 2
dN k
p
pµ
p
pν
−igµν
(1)
[I ]IR = − eR
−
−
N
′
′
2
2
(2π)
p ·k p·k
p · k p · k k − λ2 + iϵ
 QEDm
1
2,(2.2.2)= eR ln B
λm2
1/2

where the renormalized charge eR (eR = eZ3 ) appears due to photon radiative correction.
Instead of showing (2.2.1), what happens if we derive a diﬀerential equation that it satisﬁes?
As is easily understood, the equation satisﬁed by [S]IR is given by QED∂
1
2
λ [S]IR = −eR B[S]IR .
(2.2.3)∂λm2
8

Soryushiron Kenkyu∂
We derive this equation below. The action of λ ∂λinduces mass insertion in the photon
propagator Dµν (k) (in the following the Feynman gauge is taken):(2.2.4)This represents the identity
λ

∂
−igµν
−igρν
−igµρ
= 2
2iλ2 22
2
2∂λ k − λ + iϵk − λ + iϵk − λ2 + iϵ

(2.2.5)in a graphical form.
(In this paper,is used for the photon propagator,
for the propagator of gauge bosons, andfor the propagator of Faddeev-Popov ghost ﬁelds.)
The bare propagator becomes a dressed propagator D̃µν (k) when all the graphs are
summed. Thus the λ-dependence is exhibited as the λ-dependence of D̃µν (k).
∂Application of λ ∂λto the scattering amplitude of Figure 2.1.1 results in diﬀerentiationof D̃ at every site and yields
Z
1
dN k µν∂∂K (p, p′ , k, −k)λ D̃µν (k; λ),(2.2.6)
λ S=
N
∂λ
2
(2π)
∂λ

Figure 2.2.1:

where K µν is shown by the following diagram (Figure 2.2.1), representing
all the graphs in which photons with momenta kµ , −kν (which are oﬀ
the mass shell) are emitted from any site in Figure 2.1.1. Here the righthand side of (2.2.6) includes 21 because of symmetry with respect to theexchange of (µ ↔ ν), (k ↔ −k). Now IR divergence can be determinedby examining the behavior of K µν (p, p′ , k, −k) around kµ → 0 and the∂D̃µν (k; λ). First, with regard to K µν , we prove
behavior of λ ∂λ  ′ν

 ′µ
pµ
pν
p
p
µν
′
2
K (p, p , k, l) ≈ +e−−
× S. (2.2.7)
kµ ,lν →0
p′ · k p · k
p′ · l p · l

This is the low energy theorem of F. E. Low type [2], indicating that the eﬀect of soft
photons k, l can be factored out completely. An example of one loop can be easily seenin Eq. (2.1.9). (A substantially identical formula can also be derived in QCD, but thederivation will be performed in Subsection 4.5.) First we show that photons kµ and lν
coupled to other than the incoming and outgoing electron paths, do not contribute to(2.2.7). These photons occur from the internal fermion loop, as shown in Figure 2.2.2.
The reason that the photons occurring from the internal fermion loop do not contribute to(2.2.7) is shown using the Ward identity (WI).

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Soryushiron Kenkyu

That is, emission of kµ from any site of the internal fermion yields

Figure 2.2.2:(2.2.8)where the arrow ⇐ represents the following diﬀerential operation [32].

(2.2.9)In (2.2.8) this diﬀerentiation is a diﬀerentiation for the loop momentum lµ and must be
performed before loop integration, because in the transformation of (2.2.8) the Ward identity,
represented in our notation as(2.2.10a)or in mathematical expression as
←−−
←−−
i
i
i
i
∂
∂
e
··· = ···ieγµ··· + ···e
··· ,
···
p/ − m ∂pµp/ − mp/ − m∂pµ p/ − m(2.2.10b)

is used in the integrand. The arrow ⇐ points to the part to be diﬀerentiated. The reason that
(2.2.8) vanishes is that it yields a surface term of the integral. Now we speciﬁcally calculate
this electron loop. All the electron propagators are grouped by the Feynman parameter
formula asZX
d N l ∂ l ν1 · · · l ν m
ν1 ...νmA(2π)N ∂lµ (l2 − c + iϵ)nm
nZ dN l X
X
lν1 · · · lνi−1 gµνi lνi+1 · · · lνm
ν1 ...νm=
AN(2π) i=1(l2 − c + iϵ)nZ
dN ll µ × l ν1 l ν2 · · · l νm o
+
(−2)n
,
(2.2.11)(2π)N(l2 − c + iϵ)n+1

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Soryushiron Kenkyu

where the domain of integration is shifted beforehand. It can be seen that the ﬁrst andsecond terms of the right hand side cancel out each other upon symmetric integration. The
formula necessary for it isZ
dN l lν1 · · · lνm f (l2 ) = cm × {sum of independent tensors of (gν1 ν2 · · · gνm−1 νm ) type}Z
m
× dN l (l2 ) 2 f (l2 ),(2.2.12)
(

where
cm =

0,
(m: odd)
1/N (N + 2) · · · (N + m − 2). (m: even)

Another necessary formula is
Z



1
dN l
(l2 )m(−1)n+m Γ n − m − N2 Γ N2 + m

=i
N .
N
N
2
n
N/2
(2π) (l − c + iϵ)(4π)
Γ(n)Γ 2(c − iϵ)n−m− 2(2.2.13)

Thus we have conﬁrmed above that emission of soft photons occurring from the internal
fermion loop is negligible.
Therefore we may limit the following discussion to only the case where kµ , lν are emitted
from the incoming and outgoing electron paths. This case can be addressed using thefollowing formula, called the eikonal identity [2].

(2.2.14)(2.2.15)

As described above, the symbol × here indicates that p′ and p are on the mass shell. As tothe proof of the identity, it may be suﬃcient to prove (2.2.14) only. Since k, l1 , . . . , ln aresoft, the approximation used in (2.2.4) can be used, and the left hand side of (2.2.14) yields

11

Soryushiron Kenkyu

the following expression except for (−e)n+1 p′µ p′µ1 · · · p′µn ū(p′ ):111

p′ · k p′ · (k + l1 ) p′ · (k + l1 + l2 ):::::p′ · l11p′ · (k + l

1
1 + l2 + · · · + ln )1
1 + l2 + · · · + ln )
1
1
1
1
+ ′
··· ′
+ ···
′
′
p · l1 p · (l1 + l2 ) p · (k + l1 + l2 )p · (k + l1 + l2 + · · · + ln )
1
1
1
1
1
··· ′
+ ′
.
′
′
′
p · l1 p · (l1 + l2 ) p · (l1 + l2 + l3 )p · (l1 + l2 + · · · + ln ) p · (k + l1 + l2 + · · · + ln )(2.2.16)

+

1

···

′
′
1 p · (k + l1 ) p · (k + l1 + l2 )···p′ · (k + l

First, the ﬁrst and second terms of Eq. (2.2.16) are summed and reduced to a common
denominator:
1
1
1
1
··· ′
.
′
′
′
p · l1 :::::
p · k p · (k + l1 + l2 )p · (k + l1 + l2 + · · · + ln )This is added to the third term of Eq. (2.2.16) and reduced to a common denominator:
11p′ · l1p′ · (l1′
1 + l2 ) p · k

:::::

···

p′ · (k + l

1.
1 + l2 + · · · + ln )This is added to the fourth term of Eq. (2.2.16), . . . and the same operation is repeated,
ﬁnally yielding the following factor:
1
1
1
1
·
·
·
.
p′ · l1 p′ · (l1 + l2 )p′ · (l1 + l2 + · · · + ln ) :::::
p′ · kThis factor corresponds to the right hand side of Eq. (2.2.14). Thus (2.2.14) has been proved.
(In this above proof, it should be noted how the factor p′1·k moves.) (2.2.15) is proved
similarly. This eikonal identity can be used to derive the following:(2.2.17)p′pν
(For instance, the term of −e2 p′µ·k p·lcorresponds to all the graphs in which a soft photon of
momentum l is emitted from the right electron path and a soft photon of momentum k isemitted from the left electron path.) Thus Eq. (2.2.7) has been proved.

12

Soryushiron Kenkyu∂
Next, λ ∂λD̃µν (k, λ) is simply evaluated. The dressed propagator D̃µν is related to theself-energy Πµν of a proper photon as follows:Πµν (k) = (k 2 gµν − kµ kν )Π(k 2 ),−igµν
1D̃µν (k) = 2
×+ a term proportional to kµ kν .
2
k − λ + iϵ 1 + Π(k 2 )(2.2.18)(2.2.19)∂
When λ ∂λD̃µν (k, λ) is substituted into the original equation (2.2.6), contribution to IR
divergence arises around kµ → 0. It is known that in QED, in general, Π(0) has no IR1
divergence, that is, there is no infrared divergence in Z3 = 1+Π(0)[2], [19]. Now we speciﬁcallycalculate Π(k 2 ) at one loop to show that Π(0) has no IR divergence:Π(1) (k 2 )(gµν k 2 − kµ kν )Z1dN li(/l + k/ + m)i(/l + m)= Trieγν
ieγµ 2N
2i(2π)
l − m + iϵ(l + k)2 − m2 + iϵZ 1α(1 − α)
e2
N
2
2N
dα
= (gµν k − kµ kν ) ×
Γ 2−(2.2.20)
N .
N/2
(4π)
2
[m2 − α(1 − α)k 2 − iϵ]2− 20Here the pole of N = 4 existing in Γ 2 − N2 is an UV pole (since it is a pole existingirrespective of whether the external line p is on the mass shell or oﬀ the mass shell). The
limit kµ → 0 can be taken in the remaining parameter integral to give


N
1
1
e2
(1)
Γ
2
−
2N
Π (0) =
N(4π)N/22(m2 )2− 2 6
e2 8 1(pole of UV divergence).
∼(2.2.21)N →4 (4π)2 3 4 − N
Thus we have speciﬁcally conﬁrmed that Π(1) (0) has no IR divergence. In the following we
treat Π(1) (0), and hence Z3 , as a ﬁnite quantity with regard to infrared divergence.
∂D̃µν (k) is replaced by
Returning to Eq. (2.2.6), ﬁrst, (2.2.7) is used at kµ → 0, and λ ∂λ−igµν∂
λ ∂λ k2 −λ2 +iϵ Z3 . Then the strongest infrared divergence of S yields  ′ν

Z
1
pµ
pν
∂
dN k 2 p′µ
p
∂
−igµν
λ [S]IR = −e−−[S]IR × λ
Z3
N
′
′
2
∂λ
2
(2π)
p ·k p·k
p ·l p·l∂λ k − λ2 + iϵ
 QED
1∂m
2= e Z3 λln B× [S]IR∂λ λm2
 QED12= −e Z3 B[S]IR(2.2.22)m2
where the one-loop result is used here. The diﬀerential equation thus determined is QED∂
1
2
λ [S]IR = − e Z3 B.
(2.2.23)
|{z}
∂λ
m2
e2R

This is easily solved to givem[S]IR = exp e2 Z3 ln B|{z} λ
e2R

13



1
m2

QED


 (0)
S

(2.2.24)

Soryushiron Kenkyu

where the initial condition is taken as[S]IR (λ = m) = S (0) .
e2 Z3 is a renormalized charge e2R . In the case of QED, however, Z3 has no IR divergence.
Thus, as far as infrared divergence is concerned, e (bare charge) and eR (observed charge)
are interchangeable. This concludes the proof of the factorization of infrared divergence atall orders.

2.3

Cancellation of infrared divergences

The scattering amplitude S considered up to the previous subsection includes infrared divergence as seen in (2.2.24). If we calculate the scattering cross section while keeping theinfrared divergence as it is, we would have ∞ or zero at λ → 0, which is a meaningless result. However, our actual observation necessarily involves energy resolution ∆E(̸= 0). Thus,
there may be several soft photons that carry away energy within the unobservable range.
Therefore it can be said that a scattering cross section calculated including the process ofemitting several unobserved soft photons is physically meaningful.
First, using the eikonal identity ((2.2.14) and (2.2.15)) derived in the previous subsection,
we derive an amplitude SR (n) (the subscript R refers to real emission) for emission of soft
photons with small momenta k1 , k2 , . . . , kn . The amplitude for no emission of soft photons(S up to the previous subsection) is denoted SV (V refers to virtual photon) from thissubsection. Then we have

(2.3.1)
where ϵ1 , . . . , ϵn are polarization vectors for n soft photons, and Z3 indicates the eﬀectof radiative correction for soft photons of external lines. It is obvious from (2.2.8) of theprevious subsection that soft photons do not contribute to (2.3.1) in the case of emission
from the internal fermion loop. The scattering cross section can be made from (2.3.1) as
√

dσ ≈

X 1
n

X Z δ

n! pol. sum

Z δ

···
0

0

dN −1 k1p(2π)N −1 2 |k1 |2 + λ2

Z δdN −1 knp
× |SR (n)|2 ,
N
−1
2
2
(2π)
2 |kn | + λ0

dN −1 k2p(2π)N −1 2 |k2 |2 + λ2
(2.3.2)where 1/n! is the Bose factor resulting from n particles being identical, and the k1 , . . . , kn integrations are performed in the domain of |ki | < δ (i = 1, . . . , n) for suﬃciently small δ.

14

Soryushiron Kenkyu

Using (2.3.1) in (2.3.2), we have
!
Z δ
2
N −1
X p′ · ϵ
p
·
ϵ
d
k
p
× |SV |2 .
−dσ ≈ exp +e2 Z3
N −1 2|k|2 + λ2 pol. sum p′ · k p · k
0 (2π)(2.3.3)

Substituting further the result (2.2.2) of the previous subsection into SV , we have
!
Z δ
2
N −1
X p′ · ϵ
p
·
ϵ
d
k
p
−
dσ ≈ exp +e2 Z3
N −1 2|k|2 + λ2 pol. sum p′ · k p · k
0 (2π)
 ′µ
  ′ν


Z
dN k
1 2
p
pµ
pν
−igµν
p
× exp − e Z3−−
+ c.c. × dσ (0) ,2(2π)N p′ · k p · k
p′ · k p · k k 2 − λ2 + iϵ
(2.3.4)
where dσ (0) = |S (0) |2 is the diﬀerential cross section at the lowest order.
Reviewing the meaning of (2.3.4), the ﬁrst exponential factor corresponds to the eﬀect ofsoft photon emission, and the second exponential factor corresponds to the contribution of
infrared divergence from virtual photons. We will show below that the factor associated with
soft photon emission also includes infrared divergence, which just cancels out the infrared
divergence associated with virtual photons, and that the physical cross section dσ is a ﬁnite
quantity without infrared divergence. To this end it is suﬃcient to show
Z δ
2
X p′ · ϵdN −1 kp·ϵp−
N −1 2|k|2 + λ2 pol. sum p′ · k p · k
0 (2π)
 Z
 ′µ  ′ν

pµ
pν
1
dN k
p
p
−igµν
−
−
+ c.c.
−
2(2π)N p′ · k p · k
p′ · k p · k k 2 − λ2 + iϵ
= No IR divergence.
(2.3.5)This Eq. (2.3.5) states only the cancellation of infrared divergence in one-loop approximation.
That is, in the case of QED, the cancellation of infrared divergence at all orders is ultimately
reduced to the cancellation of infrared divergence at one loop.
Now we show (2.3.5). First, with regard to the sum in polarization vectors, we note thatthe following holds:
X(2.3.6)ϵµ ϵ∗ν = −gµν + kµ k̄ν + k̄µ kν ,
pol.

where
k̄µ ≡

1
(k0 , −k).
2|k|2

Then the ﬁrst term of (2.3.5) yields:
 ′µ  ′ν
Z δ
dN −1 k
p
pµ
p
pν
p
−
−
(−igµν ).
N −1 2p′ · k p · k|k|2 + λ2 p′ · k p · k
0 (2π)(2.3.7)On the other hand, k 0 -integration is performed in the second term of (2.3.5). When the
2
 ′µ
p
pµ
0
k -integration is performed, there is concern at the positions of poles of p′ ·k − p·k . Thus
returning to the beginning and recovering iϵ (see (2.1.10)), the second term of (2.3.5) yields:



Z
dN k
p′µ
pµ
pν
−igµν
1
p′ν
−
−
−
.
(2.3.8)2(2π)N p′ · k + iϵ p · k + iϵp′ · k + iϵ p · k + iϵ k 2 − λ2 + iϵ15

Soryushiron Kenkyu

Figure 2.3.1: Positions of poles
1
1
The positions of poles are depicted as follows. The poles from p′ ·k+iϵ, p·k+iϵ
are both in the0lower half plane. Taking the k -integration path so as to enclose the upper half plane, we
have
 ′µ
  ′ν
Z
dN −1 k
p
1
p
pµ
pν
p
(2.3.8) = −
−
−
gµν
√
2
p′ · k p · k(2π)N −1 (−2) |k|2 + λ2 p′ · k p · k
k0 =− |k|2 +λ2 ′µ  ′ν
Z1
dN −1 k
p
pµ
p
pν
p
,
=
−
−
gµν
√
2
p′ · k p · k(2π)N −1 2 |k|2 + λ2 p′ · k p · k
k0 = |k|2 +λ2(2.3.9)where the transformation k → −k is performed in the last step of Eq. (2.3.9). Since theinfrared divergence of (2.3.8) arises from the suﬃciently soft part |k| < δ, the following
equation ﬁnally holds:(2.3.7) + (2.3.9) + c.c. of (2.3.9) = No IR divergence.

(2.3.10)

That is, we have shown (2.3.5). Therefore, from (2.3.4), we have shown[dσ]IR = 0.

(2.3.11)

The foregoing is the proof of cancellation of infrared divergences at all orders in QED [33].
Without this cancellation of infrared divergences, it cannot be said that a particle is physically observable in ﬁeld theory. Therefore infrared divergence remains (previous subsection)in the state of the electron not accompanied by soft photons. This is a state that is not
physically observable (conﬁned state). However, cancellation of infrared divergences holds inthe state of the electron accompanied by soft photons within the allowable range of energy
resolution (the state of the electron dressed with soft photons). Thus it can be said that thisis a physically observable state.
Therefore, when we show quark conﬁnement in QCD, we must ﬁrst examine the problem
of infrared divergence cancellation. This is the theme of the next Section 3.

3

Cancellation of infrared divergences in QCD (oneloop)

In the previous section (Section 2) we reviewed the theory of infrared divergence in QED.
From this section we examine how the theory is generalized and what diﬀerence appears in16

Soryushiron Kenkyu

the case of QCD with a focus on the author’s work. First in this section we discuss fermionfermion scattering (quark-quark scattering) and fermion-gauge boson scattering (quark-gluon
scattering) in one-loop approximation. (Quark scattering at one loop by an external ﬁeld
as seen in Subsection 2.1 is too simple to grasp the characteristics of QCD. Investigationof this reaction requires calculation at two loops at the minimum [5].) In this section we
use the covariant gauge QCD. The main theme is the problem of cancellation of infrared
divergences, summarizing Reference paper I.

3.1

Classiﬁcation of Feynman graphs (one loop)

The covariant gauge QCD is expressed by an eﬀective Lagrangian density Leﬀ including
ghosts:1 a a,µν1
Leﬀ = ψ̄(i∇/ − m)ψ − FµνF−(∂µ Aaµ )2 + g ψ̄γµ ta ψAaµ
42α
− ∂ µ ca† (∂µ ca + gf abc Abµ cc ).

(3.1.1)a
For the group SU (N ), the symbols used in (3.1.1) are given as follows. Fµν
is deﬁned asaFµν= ∂µ Aaν − ∂ν Aaµ − gf abc Abµ Acν .

(3.1.2)

ta is the representation matrix for fermions, usually an N -dimensional representation for thefundamental representation of SU (N ). The commutation relation is given by[ta , tb ] = if abc tc ,(3.1.3)where f abc is called the structure constant. ca is the Faddeev-Popov ghost ﬁeld, and α is thegauge parameter.
In the case of fermion-fermion scattering, relevant Feynman graphs are as follows. Herethe two fermions are diﬀerent fermions, and the kinematics is taken as shown in Figure 3.1.2:
where {p, p′ , q, q ′ } represent four-momenta of fermions, {i, j, l, m} represent
color indices of fermions, and k, a, ϵ represent four-momentum, color index,
and polarization vector for a soft gauge boson (gluon) emitted additionally.
We will limit the following analysis to non-forward scattering (pµ ̸=′pµ ) [34]. In this case it turns out that the following three graphs do notcontribute to the analysis of infrared divergence. In Figure a), we examine
the integral around kµ ≈ 0. The term in which the numerator is O(1)has the strongest contribution to infrared divergence. This term has no
Figure 3.1.2:
infrared divergence, since it yields, in view of p′2 = m2 , lµ ̸= 0:
Z
Z
1
1
1
1
4
∼ d4 k 3 .
(3.1.4)
dk ′
2
2
2
2
(p + k) − m k (k + l)k
Figure c) has no infrared divergence for the same reason as Figure a). Also, Figure b) cannot
have infrared divergence for lµ ̸= 0 [35].
However, the infrared divergence included in the Z1 and Z3 factors, which will be calculated later (Subsection 3.4), correspond to the infrared divergence included in Figures a)–c)in the case of forward scattering. Since two gauge bosons in graphs 8 and 9 of Figure 3.1.1
cannot simultaneously be soft (being simultaneously soft correspond to forward scattering),
infrared divergence occurs when one of them (S1 or S2 ) is soft.
Next, Feynman graphs related to fermion-gauge boson scattering are as follows.

17

Soryushiron Kenkyu

Figure 3.1.1: fermion-fermion scattering

Figure 3.1.5:

The kinematics is taken as shown in Figure 3.1.5, where {q, q ′ }, {a, b},{ϵ′ , ϵ′′ } represent four-momenta, color indices, and polarization vectors ofhard gauge bosons, and k, c, ϵ represent those of a soft gauge boson.
Here, non-forward scattering is considered. Furthermore, the initialstate and the ﬁnal state must include at least one hard gauge boson.
Thus, it turns out that in the graphs of Figure 3.1.4, although there aremany gauge bosons that can be soft, two or more gauge bosons cannot be
simultaneously soft in one graph. For instance, in graph 17 of Figure 3.1.4,

′
Gauge bosons S1 and S2 are soft ⇒ External line (momentum q ) is soft,
Gauge bosons S2 and S3 are soft ⇒ External line (momentum q) is soft,
Gauge bosons S1 and S3 are soft ⇒ Forward scattering.
Thus, these three cases are excluded, and only one of S1 , S2 and S3 can be soft. Considering
similarly for other graphs, it turns out that each graph can include only one soft gaugeboson.
Next, some gauge bosons can be soft but do not contribute to infrared divergence, suchas gauge bosons denoted t in graph 11 and denoted t1 , t2 in graph 19. For instance, if t1

18

Soryushiron Kenkyu

Figure 3.1.3:
is soft, the term in which the numerator is O(1) has the strongest divergence in the loop
integral of graph 19. However, the integral yields
Z
1
1
1
d4 k 2′2
k (k + q ) (k + q ′ − q)2
Z
Z
1 1
1
1
4
4
∼′2
dk 2
∼
d
k
,
(3.1.5)
k:soft,q =0
k 2k · q (q ′ − q)2k3
and does not contribute to infrared divergence. The same also applies to the other examples.
It should be remarked a little about the graphs omitted from Figure 3.1.4. For instance,
the following graph exists. Graph a) has no contribution for the same reason as a) of Figure
3.1.3. Also, graph c) immediately proves to have no contribution by power counting. Forgraph b), calculating the numerator as O(1), it has the same structure as graph 19 of Figure
3.1.5. Thus it may be considered that infrared divergence occurs when the gauge boson
denoted S is soft. However, the numerator is actually not O(1), and hence the graph does
not contribute to infrared divergence. Let the momentum of the ghost denoted S be kµ .
Then the relevant part yields
Z(k · ϵ′ )(k + q) · ϵ
4(3.1.6)Graph b) ≈ d k 2
k (k + q)2 (k + q ′ )2Z
(k · ϵ′ )≈ d4 k 2. : no infrared divergence(3.1.7)k (2kq)(2kq ′ )That is, the numerator being O(k) is eﬀective [36].
In view of the above reasoning, below we consider the infrared divergence in the casewhere the gauge boson denoted S in Figure 3.1.4 is soft. However, we consider all the selfenergy parts of the gauge boson in graphs 16 and 17, irrespective of soft or hard (for details,
see Reference paper I).
Now the three graphs at the lowest order are grouped and represented by2

asa single graph. Then the above graphs are classiﬁed into the following. That is, they areclassiﬁed into graphs 1–4 for correction of an external line, the graphs in which diﬀerent

19

Soryushiron Kenkyu

Figure 3.1.4: fermion-gauge boson scatteringexternal lines are connected by a soft gauge boson, and the graphs in which a soft gaugeboson is emitted from an external line. For instance, taking graph 18 of Figure 3.1.4, we
illustrate below how the graphs in which S1 –S3 are soft are grouped into a particular graphin Figure 3.1.7. Fermion-fermion scattering can also be classiﬁed similarly. In this case,there is only one graph at the lowest order. Thus the above classiﬁcation diagram Figure
3.17 is trivial. In fermion-gauge boson scattering, grouping the three graphs at the lowest
order gives a clear view.

20

Soryushiron Kenkyu

Figure 3.1.6:

Figure 3.1.7:

3.2

Factorization of infrared divergences in QCD (one loop)

On the basis of the graph classiﬁcation in the previous subsection, we prove in this subsection
the following factorization rule.

(3.2.1)(3.2.2)In the above equation, the symbol × attached to an external line represents on-mass-shell(and additionally transverse for gauge bosons). The ± sign in (3.2.1) assumes + for outgoing
G in Eq. (3.2.2) represents the sum of
fermions and − for incoming fermions. The symbol ○all the graphs at the same order (the set of gauge-invariant graphs). Thus (3.2.2) indicatesthat factorization holds only after several graphs are summed up. It is for the purpose ofusing (3.2.2) that the graphs at the lowest order are grouped in the classiﬁcation of graphs in21

Soryushiron Kenkyu

Figure 3.1.8: illustration
the last part of the previous subsection. There is no need to prove (3.2.1) because it agrees
with Eqs. (2.1.4) and (2.1.5) in §2 except for the color factor ta . [In the subsequent section
4, this eq. (3.2.1) is extended to all orders.] It seems necessary, however, to prove (3.2.2)
since it appears in QCD for the ﬁrst time. Let us rewrite the left hand side of (3.2.2) [36].

(3.2.3)In the term proportional to (1 − α) in this equation, multiplying the three-point vertex by(q + k)λ using the general formula(3.2.4)(this is a very important formula, which is a starting point of the general Ward identity)
yields the following.

(3.2.5)

In combination with the denominator (q + k)2 ≈ 2q · k, it is found that the term proportional
to (1 − α) behaves as O(k) relative to the gλλ′ term and can be omitted in the soft k limit.

22

Soryushiron Kenkyu

Then the term proportional to gλλ′ is rewritten, and the most contributing factor under soft
k is given by−i
gf bac [2qµ ϵλ′ (q) − qλ′ ϵµ (q)].
(3.2.6)
2q · k + iϵ
If the second term of this expression vanishes, only the ﬁrst term remains and agrees with
Eq. (3.2.2). The second term of (3.2.6) vanishes because substituting the sum of all theG yields the following Ward-Takahashi identity.
graphs at the same order into ○,(3.2.7)(3.2.8)Here it is assumed that the remaining external lines are all on the mass shell (and transverse
for gauge bosons). In the present consideration, the general formula is not needed, but it isG =only necessary to apply a special case of (3.2.7) with ○2(the lowest order of

fermion-gauge boson scattering) given as follows.

,(3.2.9)(3.2.10)

If these hold, the second term of (3.2.6) is dropped, completing the proof of the factorization
rule (3.2.2). Here (3.2.9) and (3.2.10) can be directly checked as
LHS of (3.2.9) = (tb ta − ta tb + itc f abc ) × ig 2 [ū(p′ )/ϵ u(p)]= 0,
LHS of (3.2.10) = (t t − t t + it f
= 0.
b a

a b

c abc

(3.2.11)′) × ig [ū(p )/q u(p)]2(3.2.12)

The three graphs correspond to the respective terms of the group commutator
tb ta − ta tb + itc f abc = 0.

23

(3.2.13)

Soryushiron Kenkyu

3.3

Cancellation of infrared divergences (in terms of bare coupling
constant)

Cancellation of infrared divergences will now be shown at one loop using the factorization
rule (3.2.1) and (3.2.2) given in the previous subsection. However, in the case of QCD, it
is necessary to clarify which coupling constant is used for expansion when the cancellation
occurs. The coupling constant in this subsection is the bare coupling constant (denoted
by gB ). First, the scattering amplitude SV without extra emission of soft gauge bosons is
expanded in terms of bare coupling constant as
SV = S2 (gB )2 + S4 (gB )4 + · · · .

(3.3.1)The scattering amplitude SB (in the thesis, SB or SR is used for the amplitude; B means
Bremmsstrahlung and R means Real emission) with extra emission of soft gauge bosons issimilarly expanded in terms of bare coupling constant as
SB = S3 (gB )3 + · · · .

(3.3.2)The scattering cross section formed from SV is given bydσV = |S2 |2 (gB )4 + (S2∗ S4 + c.c.)(gB )6 + · · · ,(3.3.3)and the scattering cross section formed from SB is given bydσB = |S3 |2 (gB )6 + · · · .

(3.3.4)dσV + dσB

(3.3.5)The sum of themis a physically meaningful cross section (see §2.3).
Since |S2 |2 includes no infrared divergence, the (gB )6 term
(S2∗ S4 + c.c.)IR + |S3 |2IR

(3.3.6)is examined below. Let us consider the following example.

(3.3.7)

Here the Feynman gauge is used.

24

Soryushiron Kenkyu

The factorization rule (3.2.1) and (3.2.2) can be used to rewrite the ﬁrst term of (3.3.7)
as follows.

(3.3.8)On the other hand, using the same rule, the second term of (3.3.7) yields the following.

(3.3.9)Here the N -dimensional method is used for the regularization of infrared divergences (theother regularization methods can also be applied in just the same way). The sum of polarization vectors can be based on Eq. (2.3.6), similarly to QED. It is shown later
Xthat thekµ k̄ν + k̄µ kν term does not contribute [(3.3.12)], which allows the replacement
ϵµ ϵ∗ν →pol−gµν . Performing k 0 integration in (3.3.8) in the same way as QED yields
Z
p′µ
−q ν
−igµν
dN k
(−g
)
(−g
)
B
B
N
′
(2π)
p · k + iϵ(−q) · (−k) + iϵ k 2 + iϵ
Z
p′µ
−qµ
dN −1 k
(−g
)
(−g
)
,
=
B
B
(2π)N −1 (−2)|k|p′ · k + iϵ(−q) · (−k) + iϵ

(3.3.10)which has the sign opposite to the k integral of (3.3.9). Therefore cancellation of (3.3.8) and(3.3.9) requires equality of the following color factors.

(3.3.11)Equality of these two factors is achieved by summation over the color index j of the fermion
in the ﬁnal state. Similar cancellation of infrared divergences holds for graphs 5–10 in Figure

25

Soryushiron Kenkyu

3.1.7 where diﬀerent external lines are connected by soft gauge bosons. This cancellation
requires summation over color indices i, j, a, b in the initial and ﬁnal states.
That is, cancellation of infrared divergences occurs for the cross section obtained byaveraging the colors in the initial state and summing over the colors in the ﬁnal state, under
the assumption that the color index is not subjected to our observation. This is characteristic
to QCD.
Let us brieﬂy comment on that the sum of polarization vectors can be replaced by −gµν .
This is manifect from the Ward-Takahashi identity which implies that the sum of externallines of all the possible gauge bosons’ emission, being multiplied by kµ , vanishes.
That is,(3.3.12)This is a Ward-(Takahashi) identity in case of soft momentum emission. [Here, factorization rules (3.2.1) and (3.2.2) have been used.] It is easy to check explicitly the vanishingof the right hand side of (3.3.12) as follows:(3.3.13)(3.3.14)(3.3.15)
which show the identities for the color factors.
About the cancellation of infrared divergences for the graphs with corrections in externallines [Figures 1–4 in Figure 3.1.7] and that of Coulomb divergences existing in Figures 7 and
8 in Figure 3.1.7, please refer to the Reference paper I. Here we omit the explanation. Next,
let us discuss what kinds of divergences were cancelled in this section.
First, the integral after using the factorization rules, becomes
Z
1
1
1
dN k
,
(3.3.16)I=N2(2π) k + iε ±2k · p + iε 2k · p′ + iε
Z
dN k
1
1
1
≈
(3.3.17)N222′(2π) k + iε (k ± p) − m + iε (k + p ) − m′ 2 + iεγβα(3.3.18)

26

Soryushiron Kenkyu′′(3.3.16) and (3.3.17) are equal when k is soft, and p2 = m2 and p 2 = m 2 hold.
For ±, + shows the cases of p going out and p′ coming in and of p coming in and p′ goingout, while − shows the case of both p and p′ are going out or coming in. From (3.3.17) let us
see how the singularities become stronger, in three cases I) m = m′ ̸= 0, II) m ̸= 0, m′ = 0,
III) m = m′ = 0. I) is the usual infrared divergences of QED, and if the divergences become
stronger for II) and III), the increased part can be classiﬁed as a new divergence inducedby the self-coupling of zero mass particles. In classifying the divergences, it is helpful tocompare a number of cases. That is, by considering various cases obtained by making mass
to be zero, external line be on-shell or oﬀ-shell, and so on, it becomes manifest that thedivergences under consideration appears in what conditions. [For example, the ultraviolet
divergences appear irrespective of these conditions such as mass is zero or not, external line
is on-shell or not, and so on. It is because the diverces are induced by the inﬁnitely large loop
momenta, which ignore mass and momentum squared.] Now, modifying (3.3.17) in terms of
Feynman parameters, and integrating over k, we have
Z 1
(−1)3
dαdβdγδ(1 − α − β − γ)
I=iΓ(3 − N/2)
(3.3.19)N/2(4π)[(αp′ ± βp)2 + iε]3−N/20[which is identical to (2.1.11).]
Here denoting the denominator as V ,′V ≡ (αp′ ± βp)2 = α(α + β)m 2 + β(α + β)m2 − αβ(p′ ∓ p)2 ,(3.3.20)

Change of variables (2.1.13) is performed, namely,
α = xy, β = (1 − x)y, γ = 1 − y, [Jacobian = y]Case I) :VI = y 2 [m2 − x(1 − x)t] Case II) : VII = y 2 (1 − x)[m2 − xt] ,Case III) : VIII = y 2 (1 − x)x[−t]where t ≡ (p′ ∓ p)2
All in the cases I), II), III), divergences arise at y = 0.
Z 1
Z 1
1
1
ydy 2 3−N/2 =dy y N −5 =[ReN > 4]
[y ]
N −4
0
0

(3.3.21)(3.3.22)

When going to II) and III), further divergences appear near x = 1 and x = 0, respectively.
These divergences are characteristic in QCD, and are called “mass singularities”, if necessary
to be speciﬁed:
Z 1
1
2
1
dx
[−(1 − x)N/2−2 ]10 ==(3.3.23)3−N/2(1 − x)N/2 − 2
N −4
0
Z 1
1
1
2
dx 3−N/2 =xN/2−2 ]10 =.
(3.3.24)(x)
N/2 − 2N −40
From this

R1
−1
1
1
Case I)[II ]IR = i (4π)
2 N −4 0 dx m2 −x(1−x)t−1121Case II)[III ]IR = i (4π)
2 N −4 × N −4 m2 −t1
2−1Case III)[IIII ]IR = i (4π)
+ N 2−4
2 N −4 ×
N −4
27




1
−t




(3.3.25)

Soryushiron Kenkyu

Maximally, the poles of 1/(N − 4)2 arise.
If the same calculation is performed in the momentum space, the meaning of the divergence can be clearer (3.3.16). [In the integral for −, the pole structure diﬀers, but thediﬀerence between + and − appears in the range of t, as is seen from the result (3.3.25)of the parametric integration. That is, the integral for − can be obtained by an analytic
continuation of the integral for +, into the region of t for −. ] Now, we estimate (3.3.16) for+: The k 0 -integration readsZ
dN −1 k1
1
1
I=i
(3.3.26)
′
N
−2
0
0
⃗
⃗
⃗
⃗
4(2π)
−2|k| 2[|k|p − k · p⃗] 2[|k|p − ⃗k · p⃗′ ]
Z
1
−1
1
=i
dN −1 k,(3.3.27)
N
−2
4(2π)
2π
2|⃗k|3 [p0 + |⃗p| cos θ][p0′ + |⃗p′ | cos θ′ ]where θ, and θ′ represent angles between ⃗k and p⃗, and ⃗k and p⃗′ , respectively.
Writing the denominator of the integral above by Ṽ ,′Ṽ = 2|⃗k|3 [p0 + |⃗p| cos θ][p0 + |⃗p′ | cos θ′ ],(3.3.28)and classifying three cases I)–III), we have′Case I) :ṼI = 2|⃗k|3 [p0 + |⃗p′ | cos θ′ ][p0 + |⃗p| cos θ], Case II) : ṼII = 2|⃗k|3 |⃗p′ |[1 + cos θ′ ][p0 + |⃗p| cos θ],Case III) : ṼIII = 2|⃗k|3 |⃗p′ |[1 + cos θ′ ]|⃗p|[1 + cos θ]. (3.3.29)′

Here we note that |⃗p| ≤ p0 , |⃗p′ | ≤ p 0 hold in general, and the equality works only for the
massless case.
Comparing eq.(3.3.29) with eq.(3.3.21), we have understood the correspondence relations;|⃗k| = ⃗0 [soft] ↔ y = 0 [α = β = 0]
′′⃗(3.3.30)cos θ = −1 [−k and p⃗ are parallel] ↔ x = 1 [β = 0],⃗cos θ = −1 [−k and p⃗ are parallel] ↔ x = 0 [α = 0]This gives the translation regulations between momentum space and parameter space. Herewe will use two cutoﬀs; introducing suﬃciently small λ′ and η, we extract the usual infrared
divergences common to I)–III) by
Z
1
1
d|⃗k|
= ln ′ + ﬁnite quantities(3.3.31)
λ|⃗k|3
λ′and extract the new infrared divergences characteristic to QCD byZ a1
1d(cos θ′ )= ln + ﬁnite quantities′
1 + cos θ
η
−1+η
Z b
1
1
d(cos θ)= ln + ﬁnite quantities
1 + cos θη−1+η
[ c.f. (3.3.23) and (3.3.24)]

28(3.3.32)(3.3.33)

Soryushiron Kenkyu

Accordingly, we haveR−11dΩ [p0′ +|⃗p′ | cos θ1′ ][p0 +|⃗p| cos θ] i (4π)
2 ln λ1−1
1
1i (4π)
2 ln λ ln η 2(p·p′ ) ,
n
o


−1
1
1
1
1
Case III)[IIII ]IR = i (4π)
ln
ln
+
ln
2
λ
η
η
2(p·p′ )Case I) [II ]IR =Case II) [III ]IR =(3.3.34)

which correspond completely to (3.3.25).
From the above discussions, the divergences which are shown to be cancelled in thissection, are as follows:◦ Fermion-fermion scattering, ln λ1 (no other divergences appear in this case),◦ Fermion-gauge-boson scattering, ln λ1 and ln λ1 ln η1 (this is the highest divergences).
Here ln η1 was not discussed; [its example can be found in the radiative corrections of theexternal gauge bosons; see (Reference paper I)]
Therefore, the discussion is complete for the cancellation of the highest divergences.

3.4

New infrared divergences for the on-mass-shell renormalization

In QED, cancellation of infrared divergences in terms of bare coupling constant implies thecancellation in terms of renormalized coupling constant [deﬁned on the mass shell]. It isbecause there appear no further infrared divergences in the coeﬃcients of the bare coupling
constant expanded in renormalized coupling constant. In other words, Z3 is infrared ﬁnite.
[Refer to the explicit demonstration at one loop in (2.2.20) and (2.2.21).]In QCD, let us expand the bare coupling gB in terms of the renormalized coupling gR .
Then, we have
−1/2
(3.4.1)gB = Z1 Z2−1 Z3 gR = gR + AgR3 + · · · ,where Z1 , Z2 , Z3 are renormalization constants deﬁned on the mass shell. They are given via
the unrenormalized vertex and propatator functions as follows:(3.4.2)(3.4.3)(3.4.4)In the actual estimation shows, [referring to Appendix B of Reference paper I, in which
how to extract the infrared divergences in the renormalization constants is quite explicitly

29

Soryushiron Kenkyu

written] that the infrared divergences at one loop are given by g 2 1 
1R(1)[Z1 ]IR =C2 (R) − C2 (G) [4 + 2(1 − α)]4π N − 42 g 2 1
3R+
C2 (G) 2(1 − α)4π N − 42 g 2 1R(1)[Z2 ]IR =C2 (R)[4 + 2(1 − α)]4π N − 4 g 2 1
10R(1)[Z3 ]IR =C2 (G)+ (1 − α) ,4π N − 43
where the symbols characteristic to QCD, C2 (R) and C2 (G) are deﬁned byXta ta = C2 (R)1 (1 is a unit matrix),(3.4.5)(3.4.6)(3.4.7)(3.4.8)(3.4.9)aX

f acd f bcd = C2 (G)δ ab ,(3.4.10)c,dwhich give C2 (R) = N2N−1 , C2 (G) = N in case of SU (N ) group. [The symbols correspondP
to angular momentum squared in case of SU (2); if taking the sum over all a as a ta ta ,it commutes with all ta s, and is proportional to a unit matrix; its coeﬃcient takes various
value depending on the representation.]
The ﬁrst and the second terms of (3.4.6) correspond, respectively to the following ﬁgures:
2

Figure 3.4.1:

Figure 3.4.2:

Especially, it is characteristic that the second term in (Figure b) is zero
for Feynman gauge (α = 1). [Writing as a reference, by jumping to a little
higher level, where it is pointed out (by Kinoshita-Ukawa [13]) that the
two-loop graphs estimated by Feynman gauge, give N 1−4 singularities forthe non-leading infrared divergences; a simple reason of this is that thepart encircled with a square in Figure 3.4.2 or (Figure b) in Figure 3.4.1,
becomes zero in the limit of kµ → 0; this is what we have mensioned above
as a characteristic feature in QCD.]
By using (3.4.6)–(3.4.8), the coeﬃcient A, connecting the bare coupling30

Soryushiron Kenkyu

with the renormalized coupling deﬁned on the mass shell, can be estimated as
1 (1)
(1)
(1)
[A]IR = [Z1 ]IR − [Z2 ]IR − [Z3 ]IR
2


1
11
1
=
C2 (G) −[where ReN > 4].
2(4π) N − 43(3.4.11)(3.4.12)This does not depend on the gauge parameter. Furthermore, it has the same form as theultraviolet divergence in QCD without fermions:


1
1
11
pure Y-M[A]UV =C2 (G) −[where ReN < 4].
(3.4.13)(4π)2 4 − N
3In other words, if the infrared divergence is regularized by introducing a mass λ to the gaugeboson, and the ultraviolet divergence is regularized by a momentum cutoﬀ Λ, we have


1
11
1
[A]IR =
(3.4.14)ln C2 (G) −(4π)2 λ3
111pure Y-M]UV =[A
ln ΛC2 (G) −.
(3.4.15)(4π)23Therefore, this infrared divergence can be said to be controlled by the βpure Y-M functionof the pure Y-M theory. Namely,β(g) ≡ λ∂∂gR = −Λ (gR )pure Y-M ≡ βpure Y-M (g)
∂λ
∂Λ


1
11
=
C2 (G) −(gR )3 + · · · .
(4π)23(3.4.16)(3.4.17)

Now, due to this new divergece, the cancellation of infrared divergences is broken in theon-mass-shell renormalization.
As is well-known, the renormalization is to store all the divergences into the redeﬁnitionof mass and charge. In the study of infrared divergences, the mass is renormalized on themass shell [δm is IR free], so that the remaining problem is the charge renormalization. If
the scattering amplitudes, SV and SB expanded in the bare coupling constant gB [see (3.3.1)
and (3.3.2)], is re-expanded in the renormalized coupling constant gR , then we have
SV =S2 (gB )2 + S4 (gB )4 + · · ·=S2 (gR )2 + [S4 + S2 × 2A](gR )4 + · · · ,SB =S3 (gB )3 + · · · = S3 (gR )3 + · · · ,(3.4.18)(3.4.19)

where (3.4.1) has been used. In this way, the coeﬃcients expanded in gR have no ultraviolet
divergences, that implies the reormalizability. However, for the infrared divergences thecancellation in Section 3.3 implies[S2∗ S4 + c.c.]IR + |S3 |2IR = 0,

(3.4.20)and hence the infrared divergence of the renormalized cross section becomes[dσ]IR = [dσV + dσB ]IR = 4Re[A]IR dσ0

31(3.4.21)

Soryushiron Kenkyu

That is, the cancellation of infrared divergences violates via [A]IR . This result is derived in
the perturbation theory at one-loop, when the on-mass-shell renormalization is adopted, thatis, considering gR [the charge deﬁned on the mass-shell] to be ﬁnite. On the other hand, if theoﬀ-mass-shell renormalization is performed, all the Z-factors are well-deﬁned and infraredﬁnite, so that the cancellation of infrared divergences holds for g(µ) [charge deﬁned oﬀ themass-shell [37], as seen from the relation (without infrared divergences) between gB andg(µ). In this latter case, [A]IR calculated in this section represents the infrared divergence
which the on-mass-shell charge has. That is,[gR ]IR = −[A]IR (gB )3 + · · · = −[A]IR [g(µ)]3 + · · · ,(3.4.22)so that gR has infrared divergence, if g(µ) is considered to be ﬁnite. [Here, [A]IR is deﬁnedby (3.4.12).]———————————————————————As a reference, we comment on the relation of infrared divergeces to quark conﬁnement.
Let us draw (3.4.1)-(3.4.4) as ﬁgures:(3.4.23)is the deﬁnition of gR . In the above calculation N dimensional method was used, but if µ[deviation from the mass shell] is used as a infrared regularization, the pole N 1−4 is replaced byln µ1 .[38] If these logarithmic infrared divergences are summed up properly to give g(µ) ∝ µ1
[on mass shell charge has a linear infrared divergence], then it can give an evidence on thelinearly rising potential between quarks [inference by Miyazawa and Cornwall [14]]. That is,(3.4.24)

where the deviations from the on-shell are taken to be (momentum transfer) 2 ,
′′
p 2 = p 2 = P 2 = P 2 = m 2 − µ2
q 2 = −µ232

Soryushiron Kenkyu

Accordingly, this inferrence may give the behavior of 1/(q 2 )2 to the gluon propagator. So
far this can not be proved rigorously, but it gives a hint to the quark conﬁnement.

4

Infrared divergence and low energy theorem in QCD(quest for the properties in all orders)

In the previous section, a number of typical properties of infrared divergences in QCD are
obtained, by the examinination at one loop level:1) Factorization rule similar to QED [(3.2.2) and (3.2.2)] exists. [Section 3.2]
2) Cancellation of infrared divergences occurs in the expansion of the bare coupling
constant [or of the coupling constant deﬁned oﬀ the mass shell].
3) The cancellation of infrared (IR) divergences is violated in the on mass shell renormalization. The amount of the violation is equal to the UV divergence of on the mass shellcharge in the oﬀ mass shell renormalization; which is controlled by the β(g) function of thepure Yang-Mills theory without fermions. [Section 3.4]
Now, in this section, we intend to discuss a number of properties which can be generalized
to all orders in QCD. This is based on the (Reference paper II) of the author performed incollaboration with Norio Nakagawa and Hiroaki Yamamoto [15], and on the development
afterwards [16].

4.1

Axial gauge QCDFirst examine QCD in the axial gauge condition [22]–[24], since it is convenient in thefollwoing discussions. The Lagrange density is given by
1 a a,µν
1FL = ψ̄(i∇/ − m)ψ − Fµν−(nµ Aa,µ )2 + g ψ̄γµ ta ψAa,µ .
42α(4.1.1)

Here, the vector nµ which ﬁxes the gauge, is chosen to be time-like (n2 > 0), and α is agauge parameter.
Then, QCD represented by (4.1.1) has a notable feature of not having Fadeev-Popov ghosts.
First we consider a simple case of α → 0. The Lagrange density for the ghost ﬁelds is, following the prescription by Fadeev and Popov [39],δ(nµ A
Lg ∝ c†a
b
δθ

a,µ

)

cc ∝ c†a nµ [∂ µ δ ac + gf abc Abµ ]cc ,(4.1.2)where θc (x) is an inﬁnitesimal gauge transformation parameter, and ca is the Fadeev-Popovghost ﬁelds. If we can show the following gauge condition holds for α → 0,nµ Ab,µ = 0,

(4.1.3)the ghost ﬁelds do not couple to gauge ﬁelds, and are excluded from the Lagrange density.
The path integral expression of the generating functional for all the Green functions(including connected and disconnected digrams), W [Jµ , η, η̄] (with Jµ , η, η̄ external souces ofgauge ﬁeld and fermions), is given by
Z
∫ 4
a a,µ
W [Jµ , η, η̄] = [dAµ ][dψ][dψ̄]ei d x{L(x)+Lg (x)+Jµ A +η̄ψ+ψ̄η} .
(4.1.4)33

Soryushiron Kenkyu

Here we take a limit of α → 0, then the α depending term becomesYY∫ 4 −1a,µ 2
ei d x 2α (nµ A ) ∝δ(nµ Aa,µ (x)),α→0x(4.1.5)agiving (4.1.4).
To show the ghost ﬁelds do not exist in case of α ̸= 0, by introducing ghost ﬁelds in theloop diagrams following Fadeev and Popov, we show that the loop integral vanishes for theghosts. First, the relevant Feynman rules for ghosts, constructed from (4.1.3), are(4.1.6)(4.1.7)

With the rules, let us integrate the ghost loop. For the ghost loop, to which gauge bosons with
momena p1 , · · · , pm come in, the numerator does not depend on the momentum. Therefore,
it is enough to consider the denominator,

Z

1
1
1
1
···
n · l n · (l + p1 ) n · (l + p1 + p2 )n · (l + p1 + · · · + pm−1 )
Z 1
Z
1
=
dα1 · · · dαm δ(1 − α1 − · · · − αm ) dN l,[n · (l + c(α))]m
0
∝

dN l

(4.1.8)where c(α) = α2 p1 + α3 (p1 + p2 ) + · · · + αm (p1 + p2 + · · · + pm−1 ). The l-integral in (4.1.8)
reads, after shifting l,
Z
Z
1
1
dN l
N
(4.1.9)d l∝=0[n · l]m(n2 )m/2(l2 )m/2
This result holds in case of using N dimensional method; in this method, ghost ﬁelds
do not couple to gauge ﬁelds, [which is based on the study by Frenkel [24]]. The otherway is to show the unitarity, as will be done in Section 4.3, by combining the Feynman
rules without ghosts obtained from (4.1.1). If this can be done, the theory is consistently
closed, without introducing the ghosts. [In the past, Feynman pointed out that Feynman
rules of the Feynman gauge violate the unitarity (in gravity and gauge theory), so that theghost ﬁelds should be introduced to recover the unitarity. [40]. On the other hand, if theunitarity is proved to hold in the axial gauge, then the ghosts are not necessary to be introduced in this gauge. ] (Refer to Section 4.3 and Reference paper II.]
Based on the above consideration, we start with the follwoing Feynman rules without ghosts.

34

Soryushiron Kenkyu

From (4.4.1) we have the following rules:(4.1.10)(4.1.11)(4.1.12)(4.1.13)(4.1.14)

The propagator of gauge boson in the axial gauge takes a complicated form (4.1.10), whichis, however, a very covenient form to develop the general theory. Here the iε-rule for 1/k · n
and 1/(k · n)2 becomes the following, in order for the unitarity to hold (see the generaltheroty in Section 4.3):


1
1
1
1
1
→ pf
≡
+
,
(4.1.15)k·n
k·n
2 k · n + iε k · n − iε


1
1
1
1
1
→ pf
≡
+
.
(4.1.16)(k · n)2(k · n)22 (k · n + iε)2 (k · n − iε)2

4.2

Diagramatical proof of Ward-Takahashi identities

Next, combining the Feynman rules given in (4.1.10)–(4.1.14) given in the previouse subsection, we will derive the Ward-Takahashi identities (W-T identities). [The identities are notdirectly related to the infrared divergences, but will be connected to the proof of unitarity
in the next subsection, and will ﬁnally lead to the cancellation of infrared divergeces.]
The proof given in the following is a remake in the axial gauge of the method which was used
by ’t Hooft in the Feynman gauge [25]. [In the axial gauge, the proof by ’t Hooft becomes
extremely simple due to the absence of ghost ﬁelds.]

35

Soryushiron Kenkyu

First we introduce some notations:(4.2.1)(4.2.2)(4.2.3)(4.2.4)Since the fermion and gauge boson have diﬀerent representations, there appears a diﬀerence
between (4.2.2) and (4.2.3), and (4.2.4). [Here f abc = −i(T c )ab , with T c the representation
matrix for gauge ﬁeld.] Following ’t Hooft, starting from tree graphs and combining of them,
we will give a proof in all orders. First, for the tree graphs, we have the following identities,
by the explicit estimation:(4.2.5)(4.2.6)(4.2.7)(4.2.8)(4.2.9)36

Soryushiron Kenkyu

Let us add a number of coments: First, (4.2.5) is a pictorial description ofiiii
igtc γµ i(p − q)µ= [−igtc ]−[−igtc ]p/ − m/q − m/q − m p/ − m(4.2.10)

Second, (4.2.6) is characteristic identity in the axial gauge [but it should be examined whether
it holds only in this gauge]. Let us compare it to the corresponding identity in the Feynman
gauge (of ’t Hooft) which is (3.2.4) used previously. Adding the gauge boson propagators,
we have−igµµ′−igν ′ ν× igf cab [(q 2 gµ′ ν ′ − qµ′ qν ′ ) − (p2 gµ′ ν ′ − pµ′ pν ′ )] × 2
2
p + iεq + iε
−ig
−i
−ig
−ig
µν
ν′ν
µν
+ 2
gf bca + (−ipµ ) 2(gf acb pν ′ ) 2= gf acb 2
q + iε p + iε
p + iεq + iε−igµµ′−i+ 2(gf bca qµ′ ) 2(−iqν )
p + iεq + iε
=(4.2.11)Since the ﬁgures correspond to the equations, one by one, so that we can skip the explanation,
but notice that the terms corresponding to Fadeev-Popov ghosts appear in the third andforth terms in the last equation or in the last ﬁgure. The last two terms, associated with
the internal ghost loops, make the W-T identities complex. In the axial gauge, however,such a complexity does not exist in (4.2.6), so that (4.2.6) completely correspons to (4.2.5)
for the fermions. The proof of (4.2.6) will not be given explicitly, but is easily understood if
factorizing the gauge boson propagator as follows:



pµ nλ
nλ pν
pµ pν
−i
λ
gν −
− iα
.
(4.2.12)gµλ −Dµν (p) = 2
p + iεp·np·n(p · n)2
Third, (4.2.7) can be understood, from (4.2.2)–(??), to represent the fundamental relationof the groupta tc − tc ta + itb f bca = 0.
(4.2.13)Forth, we omit the proof of (4.2.8) and (4.2.9), but they can be proved from the Feynman
rules and (4.2.13), by using the Jacobi identity:
f abe f ecd + f ace f edb + f ade f ebc = 0(4.2.14)

Now, let us derive the W-T identities for an arbitrary graph, by combining W-T identities
at tree level. Substituting into the following circles ○, the sum of all the connected graphs,

37

Soryushiron Kenkyu

then we have

(4.2.15)

where 21 , 16 are combinatorial coeﬃcients. Using (4.2.5)–(4.2.9) here, we have

(4.2.16)

Diﬀerentiating the line to which a dotted line attachs be external or internal, and identifying
the vertices to which the dotted line attach, then we have

(4.2.17)Here, we use (4.2.7) in the second line of the above equation, (4.2.8) in the ﬁrst term of
the third line, (4.2.9) in the second term of the same line, then the second and third lines
cancell, leading ﬁnally to(4.2.18)

These are the general W-T identities which we want to derive. They are quite similar W-T
identities in QED.
——————————————————As a reference, let us prove the W-T identities in QED, using the notations in this section.
In QED, we replace tc → 1 in (4.2.2) and (4.2.3), the identities at tree level are only
(4.2.19)(4.2.20)

38

Soryushiron Kenkyu

Using them, for a general sum of connected graphs in the circle ○, we have

(4.2.21)

From (4.2.20), we have the W-T identities in QED as follows:(4.2.22)

4.3

Proof of unitarity

Let us check the unitarity in the axial gauge QCD, using the W-T identities given in the last
subsection. The proof is a little complex, so that we restrict here to give an outline of it, butinstead add the more fundamental issues which are not written in the Reference paper II.
To begin with introduce F (x1 , · · · , xn ) which stands for, corresponds to a Feynman diagram, the products of propagators connecting the vertex points at x1 , · · · , xn , and of vertex
functions on each vertex point. [To obtain the scattering amplitude from the function F , we
cut the external lines, multiply the wave functions √1Z2 u(p, s)e−ipx , √1Z2 ū(p, s)eipx , √1Z3 εµ e−ipx
e.t.c., and integrate over x1 , · · · , xn .]For this F (x1 , · · · , xn ), the following cutting rule holds. [Cutting rule of Nishijima and
Veltman [27]]X
F (x1 , · · · , xi , · · · , xj , · · · , xn ) = 0.
(4.3.1)
all the ways to underline

Here, the meaning of underline is as follows:
1) The vertex function at the underlined point is the hermition conjugate of that without underlining. If the Lagrangian is hermitian, the vertex function has a factor i, so the underlined
vertex function becomes (−1) times the original vertex function.
2) The propagator function D(xi , xj ) changes, according to the way of underlining, as
follows:
D(xi − xj )(+) for the line (xi , xj ),D(x − x )(−) for the line (x , x ),
i
j
i
j
∗

D(xi − xj ) for the line (xi , xj ), and of courseD(x − x ) for the line (x , x ).
ijijHere, D(±) are deﬁned from the original propagator D, by
D(x) = θ(x0 )D(x)(+) + θ(−x0 )D(x)(−)

39(4.3.2)

Soryushiron KenkyuhFigure 4.3.1: The ﬁrst order (single) pole and the second order (double) pole in the propagator
The boundary conditions for the propagators [iε-rule] must satisfy[D(x)(+) ]∗ = D(x)(−)(4.3.3)

It is because, (4.3.3) is compulsory in the proof of cutting rule (4.3.1). Putting aside theproof of (4.3.1), we will show that the iε-rule for 1/(k · n) and 1/(k · n)2 can be obtained
from (4.4.3).
First, examine the relation between D(x), D(x)(+) and D(x)(−) , in the momentum space.
Propagators in the momentum space D(k)s are connected by D(x)s through the Fouriertransformation:Z
D(x) = d4 k e−ikx D(k)
Z
Z
Z
Z
3
0 −ikx3=θ(x0 )d k dk e
D(k) + θ(−x0 )d k dk 0 e−ikx D(k) . (4.3.4)
clockwise along the lower semi-circle

anti-clockwise along the uppe semi-circle

Depending on x0 > 0 or x0 < 0, the contour of the integral over k 0 should be chosen in
the lower half plane or in the upper half plane. This decomposition by x0 > 0 or x0 < 0corresponts to that in (4.3.2). Consider the following situation in which D(k) has a ﬁrst orderpole (snigle pole) 1/(k 0 − s+ + iε), and a second order pole (double pole) 1/(k 0 − d+ + iε)2
on the lower plane, while it has a ﬁrst order pole 1/(k 0 − s− − iε) and a second order pole
1/(k 0 − d− − iε)2 on the upper plane, namely,
B+ (k)
A+ (k)+ 0
D(k) = 0k − s+ + iε (k − d+ + iε)2A− (k)B− (k)+ 0+ 0
k − s− iε (k − d− + iε)240

(4.3.5)

Soryushiron Kenkyu

Then, substituting this expression into (4.3.4) yields the momentum representation D(±) (k)of D(±) (x) to be
D(+) (k) = −2πiδ(k 0 − s+ )A+ (k)+ 2πiδ ′ (k 0 − d+ )B+ (k),(4.3.6)

D (k) = 2πiδ(k − s− )A− (k)− 2πiδ ′ (k 0 − d− )B− (k),(4.3.7)

D(+) (−k)∗ = D(−) (k),(4.3.8)(−)

0

Rwhere D(±) (x) = d4 k e−ikx D(k)(±) .
Since the condition (4.3.3) in the momentum space isin order for (4.3.3) [or (4.3.8)] to hold, we can ﬁnd, by using (4.3.7), the following equations
should hold:δ(k 0 − s+ )A+ (k)∗ |k→−k = δ(k 0 − s− )A− (k),δ ′ (k 0 − d+ )B+ (k)∗ |k→−k = δ ′ (k 0 − d− )B− (k).

(4.3.9)(4.3.10)

In other words, the following relations have to hold
◦ Position of the ﬁrst order pole : s+ (⃗k)|⃗k→−⃗k = −s− (⃗k),◦ Coeﬃcient of the ﬁrst order pole : A+ (⃗k)∗ |⃗ ⃗ = A− (⃗k),(4.3.11)◦ Position of the second order pole : d+ (⃗k)|⃗k→−⃗k = d− (⃗k),◦ Coeﬃcient of the second order pole : B+ (⃗k)∗ |⃗ ⃗ = B− (⃗k).

(4.3.13)

k→−k

k→−k

(4.3.12)(4.3.14)

As is understood from the form of gauge boson’s propagator, it has a ﬁrst order pole at
k 0 = ±(|k| − iε), which represents the poles of the physical states, and satisﬁes (4.3.11). The
coeﬃcients also satisfy (4.3.12). [ The condition (4.3.3) is of course not enough to exclude
the other possibibily of k 0 = ±(|k| + iε), which is rejected by the causality.]1and the second order pole
Next, let us determine the iε-rule for the ﬁrst order pole (k·n)1.
(k·n)2
To begin with, for the ﬁrst order pole to satisfy (4.3.11), there should exist poles at⃗n0k = k·⃗± iε. The coeﬃcients of the poles satisfy (4.3.12), since we have
n0


−i
k µ n ν + n µ kν1
A+ (k)µν = A− (k)µν = 2(−1)
× .
(4.3.15)0
k + iεn2Here the last factor 12 is added so that it may reproduce the original (4.1.10), when the1iε-rule is forgotten. Thus, the k·n
is understood to be


1
1
1
1
1
→ pf
=
+
,
k·n
k·n
2 k · n + iε k · n − iε[which is (4.1.15).] In the same manner, 1/(k · n)2 should have the second order pole at⃗nk 0 = k·⃗± iε. The corresponding coeﬃcients satisfy (4.3.14), provided
n0


1
−i
2
2 kµ kνB+ (k)µν = B− (k)µν = 2(αk + n ) 0 2 × .
(4.3.16)k + iε(n )241

Soryushiron KenkyuhFigure 4.3.2: Pictorial proof of the cutting rule1
That is, (k·n)
2 in (4.1.10) must be



1
1
1
1
1
,
→ pf
=
+
(k · n)2(k · n)22 (k · n + iϵ)2 (k · n − iϵ)2
[which is the previous (4.1.16).] In this way, a general theory of determining iε-rule has been
aﬀorded by the unitarity.
Next, the proof of the cutting rule (4.3.1) [by Veltman [27] ] is recapiturated.
Let the vertex be x∗ whose time x0 is the maximum among the vertices in the righthand-side of (4.3.1). Finally, the scattering amplitude is obtained by integrating over x of
(4.3.1), so that x∗ can be one of x1 , · · · , xn in the diﬀerent integration regions.]
The left-hand-side of (4.3.1) is the sum of all the possible underlines, which can be a sumof two kinds of graphs with and without underlining for x∗ , having the common part except
for the vertex x∗ . This is depicted in Figure 4.3.2. Here, the graphs inside a squashed circle
are common about the underlining of the vertices. Comparing the ﬁrst term and the second
term in Figure (4.3.2), since x∗ has the maximum time x0∗ , the following relations hold:the ﬁrst term

the second termD(x∗ , xi )(+)
D(x∗ , xj )∗=D(x∗ , xi )=D(x∗ , xj ),(4.3.17)(4.3.18)and hence the propagators connecting to x∗ are common in the ﬁrst and the second terms.
The vertex functions at x∗ have opposite signs with or withot the underline. [Please referto the meaning of underline 1).] Thus, the ﬁrst term and the second term in Figure (4.3.2)
cancell with each other. In this way, (4.3.1) has been proved.
To derive the unitarity from this cutting rule, ﬁrst we make the scattering amplitude, by
amputating the external lines and multiplying the proper wave functions, such as √1Z2 u(p, s)e−ipx ,√1 ū(p, s)eipx , √1 εµ e−ipx , and integrating over x1 , · · · , xn . Next, understand that D(k)(+)
Z2
Z3
for the gauge boson propagator satisﬁes the following relation, under the iε-rule ﬁxed in theabove [by substituting (4.3.15) and (4.3.16) into (4.3.6) and (4.3.7), respectively],
ab
ab
Dµν
(k)(+) = Dµν(−k)(−)
k µ n ν + n µ kν
kµ kν
0
2
ab
2
2
θ(k )δ(k ) gµν −= −2πδ
+ (αk + n )k·n(k · n)2

1
1
1 ′
1
2
2
− δ(k · n) 2 (kµ nν + nµ kν ) − δ (k · n) 2 (αk + n )(kµ kν ) ,
2
k
2
k

42

(4.3.19)

Soryushiron Kenkyu[where n0 > 0.]
Then, the conditions for the cutting rule to give the unitarity for the scattering amplitude
yield the following two:
1) The terms of δ(k · n) and δ ′ (k · n) cancell in the cutting rule when applied to the scattering
amplitude, [since these terms do not correspond to the physical emission of particles.]2) Furthermore, the term of θ(k 0 )δ(k 2 ) becomes a sum of the product of physical polarization
vectors. That is, in the cutting rule of the scattering amplitude, the following substitutionis allowed to reproduce the emission process of the physical particles,XabDµν(k)(+) → +2πθ(k 0 )δ(k 2 )εµ (k)∗ εν (k)(4.3.20)
pol.

These two conditions are required to hold. The proof of the conditons is a little complecated, so that we leave it to p.12–p.15 of the Reference paper II. Then, the proof of
the unitarity has ﬁnished. Here, let us supplement the more fundamental things on thephysical polarization vectors in the axial gauge.
It is known [by Kummer et al. [23] and Frenkel [24]] that the physical polarization vectors
in the axial gauge QCD satisfynµ εa(λ)= k µ εa(λ)(λ = 1, 2)µµ(4.3.21)

Let us make this fact more familiar, by showing how to obtain the physical state condition,
starting from the Feynman graphs.
To begin with, the following things should hold, that is,1) Physical state is invariant under the time development. For this to hold, physical stateab(k), near the mass shell:
εaµ (k)phys. is the eigenvector of the dressed propatator D̃µνabD̃µν(k)εb,ν (k)phys. 2≈ Λ(k)εaν (k)phys. .

(4.3.22)k =02) Propagator of connecting physical state by physical state is equal to the free propagatorup to a factor. That is, the following should hold,iZ3
abεa,µ (k)phys. D̃µν(k)εb,ν (k)phys. 2≈ Λ(k) 2.
k =0
k + iε

(4.3.23)

Since the physical external ﬁled is √1Z3 εaµ , the propagator between the real physical states isequal to the free propagator.
Let us derive (4.3.21) from the physical state conditions 1) and 2).
ab(k). With the proper self-energy tensor ΠabFirst we construct the dressed propagator D̃µνµνab(k), we haveand bare propagator Dµν
db
ac
ab
ab
(k)Πcd(k) + D̃µλ(k) = DµνD̃µνλρ (k)Dρν (k)(4.3.24)

1
ab
ab
[D̃µν
(k)]−1 = [Dµν(k)]−1 − Πab
(k)
i µν

(4.3.25)

From this we have easilly[In the complicated gauge theories, it is better to use (4.3.25).]

43

Soryushiron Kenkyu

Now, using the Ward-Takahashi identity [(4.2.11)] for Π abµν (k), we havek µ Πabµν (k) = 0(4.3.26)[It is because the color tensor for (a, b) is not other than δab , so that it vanished as f bca δab = 0,
after multipied by f abc in the W-T identity.]In the axial gauge QCD, there exists one more vector nµ which ﬁxes the gauge, in additon
to the incoming momentum vector kµ , so that we have to decompose Πabµν (k) into two tensors,
using these two vectors, so that it satisﬁes (4.3.26):


2
2
ab
ab
(1) 2
(2) [kµ (k · n) − k nµ ][kν (k · n) − k nν ]Πµν = δΠ [k gµν − kµ kν ] + Π(4.3.27)(k · n)2[This is exactly same problem as in the deep inelastic scattering of electron and proton, in
which two form factors W1 and W2 exist and are expressed by the two momenta of protonnµ and of virtual photon kµ .] Using (4.1.10) and (4.3.27) in (4.3.25), we have obtained
−i
1
ab
D̃µν
(k) = 2k + iε 1 + Π(1)kµ n λΠ(2)n λ kν
λ
×
gµλ −
gν −
+
k·n
k·n
[1 − k 2 n2 /(k · n)2 ]Π(2) − [1 + Π(1) ]k2
22[(k · n)nµ − n kµ ][(k · n)nν − n kν ]×(k · n)4
−i
kµ kν
+ 2
αk 2
.
(4.3.28)
k + iε(k · n)2a(phys.)Using this, let us make the physical states εµso as to meet the physical state conditons
1) and 2).
First write down all the eigen-values and eigen-vectors for D̃µν (k) [up to δ ab ]. It is clearthat εµ (k)λ [λ = 1, 2] are the eigen-vectors, and the remaining eigen-vectors εµ (k)λ [λ = 0, 3]
can be obtained as linear combinations of (nν , kν ), as follows:(0)ε(0)[nν + c+ kν ]
ν = N

(4.3.29)(3)ε(3)[nν + c− kν ],ν = N(4.3.30)

where c± satisfy the following second order equation:c2 δ + c(δn2 + αk 2 )/(k · n) + α = 0,
and the solutions arec± =(4.3.31)p1[−(δn2 + αk 2 )/(k · n) ± (δn2 + αk 2 )2 /(k · n)2 − 4αδ2δ
(4.3.32)

Here, the δ is deﬁned byδ ≡ 1/1 + Π(1) − (1 − γ)Π(2)γ ≡ k 2 n2 /(k · n)244(4.3.33)(4.3.34)

Soryushiron Kenkyu

The normalization constants N (0) , N (3) are ﬁxed so as to satisfy the following normalization
condition and the completeness conditon,(j)ijg µν ε(i)µ εν = g(4.3.35)(j)gij ε(i)µ εν = gµν(4.3.36)

The eigen-values for these eigen-vectors are−ik2
× (−1)c+ δ,
k 2 + iε (k · n)21−i
Λ(1) = Λ(2) = 2,
k + iε 1 + Π(1)−ik2
Λ(3) = 2
× (−1)c− δ,
k + iε (k · n)2

Λ(0) =(4.3.37)(4.3.38)(4.3.39)

where the pole k 2 = 0 is cancelled for the unphsysical polarizations Λ(0) and Λ(3) . The(1)(2)
physical state conditon 2) of having the pole at k 2 = 0 is satisﬁed for εν and εν , so thatthese two are physical state polarizations. This is a way to construct the physical states. [Incase of the gauge parameter α → 0, the discussion is simpliﬁed ton2c+ (α = 0) = 0, c− (α = 0) = −k·n(4.3.40)

In this case our physical states reproduce those by Kummer et al. [23].
Here, we note that if εµ is an eigen-vector of D̃µν , then nµ · εµ = 0 can be derived fromk µ · εµ = 0; indeed fromΛ(nµ · εµ ) = nµ D̃µν εν = −iα1(k ν · εν ) = 0
k·n(4.3.41)

we have nµ · εµ = 0.
The unitarity proved in this way, can be used to show the cancellation of infrared divergencesin the total cross sections [41]. For example, let us consider the infrared divergences includedin the total cross section of σ(e+ e− → hadrons). It is related to the photon’s self-energyΠ(q 2 ) by unitarity, and is proved to have no infrared divergences for q 2 ̸= 0 [41], whichshows that at least the total cross section has no infrared divergences. There is, however, no
general theory exists about the cancellation of infrared divergences, when the mmomenta of
the ﬁnal quarks are ﬁxed. [ Appelquist et al. have checked explicitly at two loops for thiselectron-positron scattering process. The content of Section 3 is the check of this cancellationof infrared divergeces for the other processes.]

4.4

Generalization of F. E. Low’s low energy theorem to QCD(Part 1)

Using the axial gauge QCD studied in Section 4.1–Section 4.3, we will prove the F. E. Low”slow energy theorem (abbreviated as Low’s theorem in the following) at all orders of theperturbation. [There exists in the past the erxplicit check of it at one loop in the highenergy limit [20], but the proof at all orders is a new performance.]

45

Soryushiron Kenkyu

The process which we are going to consider is the emission of one or two soft gaugebosons in the fermion scattering process by a colorless external ﬁeld. In case of the emissionof two soft gauge bosons, the proof is restricted for the two to have the same color indices.
[In the reference paper II, only the single soft gauge boson emission process was studied.]
Let us give the Low’s theorem in ﬁgures.
That is, they are given in the axial gauge as follows:(4.4.1)(4.4.2)[The proof of (4.4.2) will be given in Section 4.5.]
The meaning of the notations is clear, but {p, p′ , k, l} are 4-momenta, {i, j, c} are color
indices, and {ε, ε′ } are polarization vectors. (4.4.1) and (4.4.2) are unrenormalized relations,being expanded in the bare coupling constant gB . The both hand sides of these equations
have infrared divergences, since the external lines are on the mass shell [and the transverse
wave for gauge bosons], which are assumed to be regularized by the N-dimensional method.
That is, these are relations which hold in the limit of kµ (and lµ ) be soft, after relularizedby N-dimensional method [or performing the analytical continuation of N to complex]. [Incase of the regularization of infrared divergence by introducing a mass λ to gauge bosons,the limit of (4.4.1) and (4.4.2) to hold is |kµ | ≪ λ ≪ m. The N dimensoinal method ismore convenient, since the ultraviolet divergences are regularized at the same time.] Z3
represents the corrections for the external line of gauge boson, being the renormalization
constant deﬁned on the mass shell, [please refer to (4.3.23) and (4.3.38).]
For the later convenience, we will write a formula of (4.4.2) after amputated the external line46

Soryushiron Kenkyu

of the gauge boson. That is,(4.4.3)[This will be proved in Section 4.5.]
Before beginning the proof, let us remind of the relations (2.3.1), (2.3.7), (2.2.17) in QED
discussed in Section 2. Note that the above mentioned (4.4.1)–(4.4.3) are the generalizationof these relations to QCD. It is however, the Low’s theorem in QED [(2.3.1), (2.3.7), (2.2.17)]
can be proved, as was explained in details in Section 2, by using the Eikonal identities(2.2.14), (2.2.15), but in QCD, these Eikonal identities can be applied directly. The Eikonal
identities hold when summing up all the possible soft photon emissions from the external
electron lines, but the same strategy does not work in QCD, in which two soft gauge bosons
emitting from the external quark line have diﬀerent color indices under the exchange of thetwo, for the non-Abelian group. In QCD, the soft gauge boson can be also emitted from theinner lines of the gauge boson. Therefore, we have to sum over all the possible soft gaugeboson emissions [not only from the external fermion lines, but also from the inner gaugeboson lines].
It is easily understood that a soft photon emission from all the possible places in QED
can be realized by diﬀerentiating the independent external momenta for fermions, as well as
the independent loop-momenta of the inner fremion loops.
As an example, let us consider the electron scattering by an external ﬁeld in Section 2.
The process with an additional photon emission with zero momentum is given by(4.4.4)Here, the arrow symbol ⇒ in QED is deﬁned by(4.4.5)[Remind of (2.2.9) and (2.2.10a) or (2.2.10b) in Section 2] The momentum p is assumed
to ﬂow along the electron line and going out to the external source, while the momentum47

Soryushiron Kenkyu

p′ enters from the source and ﬂow out to the emitted electron line. Then, the ﬁrst term
represents the emission of soft photons from the orbit of electron on the right, the second
term does from the orbit of the electron on the left, and the third term does the emission ofphoton from the inner loops of electrons.
In QCD we need to have the diﬀerential operation with color index in order to generatea gauge boson emission with zero momentum and color index c. That is, corresponding to(4.4.5), we have to introduce the following arrow symbol ⇒ in QCD. Namely,(4.4.6)(4.4.7)(4.4.8)The direction of the arrow symbol points along the part to diﬀerentiate. Now, using this arrow symbol in the following, let us generate the emission of gauge boson with zero momentum

48

Soryushiron Kenkyu

and a color c. First, for the simplest graphs, the following relations hold [42]:(4.4.9)(4.4.10)(4.4.11)(4.4.12)(4.4.13)(4.4.14)

These formulae correspond to (4.2.5)–(4.2.9) in Section 2, and can be chcked directly byusing the Feynman rules (4.1.10)–(4.1.14), the fundamental relation of the group [ta , tb ] =if abc tc , and Jacobi identities. The characteristic of the axial gauge is (4.4.11), which becomescomplicated in the other gauge, by including additional terms. [ Among (4.4.9)–(4.4.14), only
(4.4.11) depends on the gauge condition, and the others are common in any gauge. It should
be examined, however, if there are no gauge conditions other than the axial gauge which
satisﬁes (4.1.11).]
With the use of these lowest order identities (4.4.6)–(4.4.14), the following identities canbe derived for any tree graph, by acting the arrow operations (4.4.6)–(4.4.8) to one of theexternal lines:(4.4.15)

49

Soryushiron Kenkyu

The proof of it is simple.1 First, understand the characteristics of the relations (4.4.9)–(4.4.14). They are summarized to1)When the arrow symbol passes through the propagator, the gauge boson with zero
momentum and color c is emitted from the propagator. [(4.4.9)–(4.4.11)]
2) As for the vertex, the arrow symbols approach along the ﬂows of independent momenta,
join and go out to the line with a not-independent momentum. Only when a gauge boson
emission is possible from the vertex, the graph with the emission of a gauge boson with zero
momentum and color c, remains additionally. [(4.4.12)–(4.4.14)]
First we ﬁx the ﬂow of independent momenta, then there remain all the possible gaugeboson emission graphs, and the arrows are joined and gone out to the unique external linewith non-independent momentum. This is the proof of (4.4.15).
Next, we generalize (4.4.15) to the more general graphs with loops. First, write down
the result, and prove it afterwards. The general result can be depicted as follows [43]:(4.4.16)

The prove is done by induction with respect to the number of loops. First, for a tree graph(the number of loops L = 0), (4.4.16) is equal to (4.4.15) and is correct. So, assuming(4.4.16) holds for the graphs with loops less than L, consider a graph with L + 1 loops. The
left hand side of (4.4.16) for the graph with L + 1 loops, can be deformed as follows:(4.4.17)

Footnote added (2022): The left-hand-side of (4.4.15) with the arrow symbol ⇒ gives QED-like soft
photon emissions from the external lines, while the right-hand-side aﬀords all the possible soft gluon emissins.
150

Soryushiron Kenkyu

Here, we have used the following identity:(4.4.18)

which is a pictorial representation of the following equation,


∂
∂
∂
F (l, l) =+ ′ F (l, l′ )|l=l′ .
∂lµ∂lµ ∂lµ(4.4.19)

The external lines of gauge bosons with momenta l, l′ in the right hand side of (4.4.17) are
amputated. So, the last term in the right hand side of (4.4.17) represents the diﬀerentiationof the amputated propagator. Now, using the assumption of the induction in (4.4.17), we
understand(4.4.20)

which shows that the (4.4.16) is valid for the case with (L+1) loops, and the proof of (4.4.16)
for the graph with any loop has been completed. [In the above proof we add gauge boson
loops, but we can add fermion loops in the same way.]
Comparing this (4.4.16) with (4.4.1) in QED, we can understand that the contribution
from the independent loops in QCD,

(4.4.21)corresponds just to the zero momentum photon emission from the fermion loops in QED,

(4.4.22)Furthermore, (4.4.22) becomes zero due to a surface integral of the loop integral, as was
discussed in Section 2, [please refer to (2.2.8) and its proof by (2.2.11)–(2.2.13)]. In QCD

51

Soryushiron Kenkyu

if (4.4.21) is regularized by N dimensional method, it becomes zero, since (2.2.11)–(2.2.13)
hold in the same way. Namely, we have

(4.4.23)

This (4.4.23) has the important meaning.
As a simple example, let us consider a two loop graph,and take the following limit,(4.4.24)

Here, since the loop integrals in each graph are regularized by the N dimensional method,
the limit kµ → 0 can be taken continuously, as far as N is complex.
1
If the divergences, |k|, ln |k|, e.t.c., in the limit of kµ → 0 in four dimensions, appear in
the N- dimensional method as poles at N = 5, N = 4, e.t.c. in the complex N plane, after
taking the limit kµ → 0 and the integration. So, using (4.4.23), (4.4.24) implies, when the52

Soryushiron Kenkyu

limit kµ → 0 is taken while keeping N be complex,(4.4.25)1which has no poles for N ; namely, the singularities for |k|, |k|, ln |k|, e.t.c. in (4.4.24), which
may appear in the calculation in four dimensions, disappear in the N-dimensional method.
Therefore, in the limit of |kµ | ≪ m, it is possible to replace the following soft gauge boson
emission process by the soft photon emission processes.

(4.4.26)[The dotted lines in the right hand side represent the colorless photon emission with the barecoupling gB .]As is known from this example, (4.4.23) represents the cancellation mechanism among
the soft gauge boson emission graphs, with which the emission of soft gauge boson in QCD
can be reduced to soft photon emission in QED, up to a color factor [tc ]ji [ i, j are color
indices of fermions coming in and out]. (4.4.26) is one example. In the following, we willconsider this mechanism in the genaral case.
To begin with, let us prove the equality of the two renoamalizatin constats Z1 and Z2
deﬁned on the mass shell; the former Z1 is the renormalization constant for the fermionfermion-gauge boson vertex, and the latter Z2 is the wave function renormalization constantfor the fermion. This is depicted as(4.4.27)

The ﬁrst term represents the proper vertex without the lowest graph, and the second term
does the proper self-energy. Using (4.4.16) and (4.4.23), this ﬁrst term can be expressed interms of the proper self-energy Σ(p),ū(p)gtc

1 ∂× ··· ,
i ∂pµ(4.4.28)

1× ··· ,
p/ − m

(4.4.29)

and the second term can beū(p)gγµ tc

53

Soryushiron Kenkyu

The mass m in Σ(p) is renormalized on the mass shell, so that Σ(/p = m) = 0 holds, and thefollowing expansion is possible,
Σ(p) = (/p − m)Σ1 (p) + (/p − m)2 Σ2 (p) + · · ·(4.4.30)

Substitution of this expansion into (4.4.29) and (4.4.29), and comparison of the two, givethe proof of (4.4.27). Indeed, if (4.4.27) is expressed in terms of Z1 , Z2 , we have




1
1
c
c
− 1 × · · · + ū(p)igγµ t 1 −× ···0 = ū(p)igγµ t(4.4.31)
Z1
Z2
which gives Z1 = Z2 . Using this, the following replacement is allowed, for the soft bauge
boson emission from the external lines:(4.4.32)(4.4.33)√√[The · · · in the corrections of the external lines implies the multiplication of Z2 to theexternal line.]
Next, for the soft gauge boson emission from the inside of a graph, the generaalizationof the aforementioned (4.4.24) holds.
That is, the following limit of kµ → 0 can be taken continuously in the N-dimensional
regularization,(4.4.34)

54

Soryushiron Kenkyu

we obtain the following relation vanishes,(4.4.35)

Therefore, in the limit of |kµ | ≪ m, the ﬁrst term of (4.4.34) can be replaced by the soft
photon emission graphs of the second and the third terms. That is,(4.4.36)

Here, the corrections for the external lines are additionally added to (4.4.35). [Frequently
appeared notation of dotted arrow, of course, represents the emission of photon without
color index, which interacts to the external lines of fermions with the bare coupling gB .]Just to be sure, let us discuss about the limit of (4.4.34). If we take a limit kµ → 0
1, (ln |k|)m , e.t.c., appear in each term in (4.4.3).
in N = 4, the divergences, such as |k|However, before the calculation is carried out, if N is analytically continued to a complex
value, then the limit kµ → 0 can be taken continuously, but as the aftereﬀects various poles1
at N = 5, N = 4 e.t.c. arise, which correspond to the divergences |k|, (ln |k|)m , e.t.c. in
the calculation for N = 4. Therefore, the fact that the limit of (4.4.34) vanishes, due tothe cancellation mechanism of (4.4.23), without having poles in the N-dimensional method1
implies that the divergences |k|, (ln |k|)m , e.t.c. cancell in (4.4.34) in the usual calculation
for N = 4. [The property mentioned above holds always in the N-dimensional method. For
example, when some graph gives an ultraviolet divergence ln Λ in a calculation by introducing
an ultraviolet cutoﬀ Λ in N = 4 dimensions, the same calculation performed in the Ndimensional method induces that the limit of Λ → ∞ can be taken continuously, but a poleat N = 4 appears.]

55

Soryushiron Kenkyu

Now, combining (4.4.32), (4.4.33), and (4.4.36), we obtain(4.4.37)

in which the soft gauge boson emission with color has been reduced to the soft photon
emission witout color. In the right hand side, the soft photon emission processes from theleft and right fermion orbits, are summed up, so that the Eikonal identities in Section 2[(2.2.14) and (2.2.15)] can be applied, and as the result, the following identity is obtained:(4.4.38)[In the both hand sides, the corrections of the external fermion lines are taken into account.]
Adding furthermore the corrections of the external gauge boson lines, and multiplying thephysical polarization vectors, we obtain (4.4.1), one of the Low’s theorem in QCD.
Next, we considr the renormalization of this unrenormalized Low’s theorem (4.4.1). The
mass m is renormalized on the mass shell, and so examine the renormalization of the coupling constant. (]refeq4.4.1) is expanded in the bare coupling constants gB . To renormalize(]refeq4.4.), it is enough to re-expand it in the coupling constants deﬁned under various conditions. The coupling constant deﬁned on the mass shell gR and the coupling constant g(µ)
deﬁned oﬀ the mass shell, are connected to the bare coupling constant gB by the following
relations:−1/2

gB =gR Z3(4.4.39)=g(µ)Z1 (µ)Z2 (µ)−1 Z3 (µ)−1/2(4.4.40)

There is a relation of Z1 = Z2 [(4.4.27) or (4.4.31)] between the renormalization constants

56

Soryushiron Kenkyu

deﬁned on the mass shell. Oﬀ the mass shell renormaliation constants are deﬁned as follows:(4.4.41)(4.4.42)(4.4.43)+ [terms ∝ qµ qν , qµ nν + nµ qν , nµ nν ].
[Various choices for the oﬀ-mass-shell coupling constant (4.4.40) are possible. We heve toﬁnd a good choice so that the coupling constant may not depend on the gauge parameterα.] The gauge parameter is renormalized as
αB = Z3 α = Z3 (µ)α(µ)(4.4.44)

which is the same as in the covariant gauge. [Please refer to (4.3.28).] What we are discussing
is the scattering amplitudes for which they do not depend on αB and also not on the gauge
ﬁxing vector nµ , when they are expanded in gB [23], [24]; so we can ignore the αB dependence
in the following. As was pointed out in the above, an additional gauge dependence may
appear by the introduction of the oﬀ-mass-shell coupling g(µ).] The Z1 (µ), Z2 (µ, Z3 (µ) in(4.4.39) and (4.4.40) are deﬁned oﬀ the mass shell, so that thery are free from the infrared
divergences, while Z3 deﬁned on the mass shell includes infrared divergences. This is a
special property in QCD, not observed in QED. The explicit calculation [29] [by Frenkel and
Meuldermans] gives, for
1/2

g B Z3=gR (on the mass shell charge)∞X=c(n) (µ)[g(µ)]2n+1(4.4.45)n=0

the following infrared divergence exists,[c(1) (µ)]IR =11c2 (G)
2
4π(N − 4)2(4.4.46)

That is, the coeﬃcient of expanding the on-mass-shell coupling gR in terms of the oﬀ-massshell coupling g(µ), includes the infrared divergence [46]. Therefore, the Low’s theorem

57

Soryushiron Kenkyu(4.4.1) takes a diﬀerent form for the on-mass-shell renormalization [taking gR be ﬁnite], orfor the oﬀ-mass-shell renormalization [taking g(µ) be ﬁnite]. To explain this, let us express(4.4.1) formally as follows:
∞
X

(gB )

2n+1

A

(n)

1/2
= −gB Z3

n=0

 ′
 ∞
p ·ε
p·ε X
−
(gB )2n B (n)′p · k p · k n=0(4.4.47)

Then, expand it in g(µ) by using (4.4.45). [Let us include the case of on mass shell renormalization for µ = 0 in the following.]
First, introduce the renormalized amplitudes A(n) (µ) with soft photon emission, and therenormalized amplitudes B (n) (µ) without soft photon emission, both of which are connectedto the corresponding unrenormalized ones A(n) and B (n) , as follows;
∞
X
n=0
∞
X
n=0

(gB )

2n+1

A

(n)

(gB )2n B (n) ==∞X[g(µ)]2n+1 A(n) (µ)

n=0
∞
X

[g(µ)]2n B (n) (µ)(4.4.48)(4.4.49)

n=0

Substituting these into (4.4.17), we have the following renormalized Low’s theorem (4.4.1):
A

(n)

 n
 ′
p · ε X (k)p ·εc (µ)B (n−k) (µ)−(µ) = − ′p · k p · k k=0

If we take µ = 0, the renormalization becomes on the mass shell, for which
 ′

p·ε
p ·ε
(n)
−
A (0) = − ′B (n) (0)p ·k p·k(4.4.50)(4.4.51)

In summary, Low’s theorem in QCD coincides with that of QED as in (4.4.51), up to thecolor factor [tc ]ji . Applying the oﬀ mass shell renormalization, however, the Low’s theorem(4.4.50) [for µ ̸= 0], deviates from the QED type of (4.4.51) by the dependence on c(n) (µ).
The deviations c(n) (µ) are the expanson coeﬃcients of the on mass shell coupling gR in theoﬀ-mass-shell coupling g(µ), and include the infrared divergences characteristic in QCD.
This is the end of the proof of Low’s theorem in QCD in case of single soft gauge boson
emission.
The proof of Low’s theorem (4.4.2) and (4.4.3) in case of emitting two soft gauge bosons,can be done with a little generalization of the proof of (4.4.1), which will be given in thenext Subsection.

4.5

Generalization of F. E. Low’s low energy theorem in QCD(Part 2)–[Application to diﬀerential equation which controlls
infrared divergence]In this subsection,a lillte generalizing the proof of the Low’s theorem of (4.4.1) given in
the previous subsection for a single soft gauge boson emission, we will prove the Low’s
theorem of (4.4.2) and (4.4.3), wihch describes the emission of two soft gauge bosons, [witha restriction that two gauge bosons have the same colors]. Furthermore, the theorem is58

Soryushiron Kenkyu

applied to derive the renormalization group like diﬀferential equation which controles theinfrared divergences in QCD. [This is the generalization of Section 2.2 in QED to QCD.
(4.4.3) in QCD corresponds to (2.2.7) in QED.]
To begin with, by generalizing the essential formula (4.4.16) in the proof of (4.4.1) [wherein the second term, the cancellation mechanism (4.4.23) works], we will prove(4.5.1)[Here, the sum over (e, e′ ) and (l, l′ ) includes the duplication, such as (1, 2) + (1, 1) + (2, 1) +(2, 2).]
Since (4.4.16) has been already proved, to obtain (4.5.1) we have to show

(4.5.2)That is, we repeat the proof in Subsection 4.4, starting from the graph depictedin the left, having a gauge boson emission with color c, Lorentz index µ and zero
momentum.
First, for the simplest graphs, in additon to (4.4.9)–(4.4.14), the following identities are59

Soryushiron Kenkyu

needed. That is, we require additionally the following four identities:(4.5.3)(4.5.4)(4.5.5)

These identities can be proved, similar to the previous (4.4.9)–(4.4.14), by using the Feynman
rules, the fundamental relation of the group [ta , tb ] = if abc tc , and the Jacobi identity. These
identities do not hold, unless the color of the two emitting gauge bosons are equal. It should
be noted that [for example, suppose the two color indices c and d are diﬀerent with c ̸= din (4.5.3a), then there appears an additional term proportional to [tc , td ], and the (4.5.3a) is
broken].
In the same manner as in the previous subsection, using (4.4.9)–(4.4.14) and (4.5.3)–(4.5.5), we can derive(4.5.6)[This is a generalization of (4.4.15). Please remind of the proof just after (4.4.15). The
characteristics 2) mentioned there holds also for the vertex, which can emitts an additonalgauge boson with color c, Lorentz index ν, and the zero momentum.]
Next, to generalize (4.5.6) to (4.5.2), we use the induction with respect to thenumber of loops; in this case it is only to repeat the method used in Subsection
4.4 from (4.4.17) to (4.4.20), starting from the graph depicted in the right:[Therefore, we can skip to write the proof here.] Then, the proof of (4.5.3) has beenﬁnished. Here, we have used the fact that the second term in (4.5.2) has no surface term in
the N-dimensional method, which is a generalization of (4.4.23) in Subsection 4.4. This is60

Soryushiron Kenkyu

depicted in this case as(4.5.7)[Please refer to (2.2.8) and (2.2.11)–(2.2.13).] Accordingly, using the cancellation mechanismby the surface term (4.4.23), (4.5.7), we have the following formula from (4.5.2),(4.5.8)Let us apply this to our process of the fermion scattering by the colorless external source.
That is, we consider the diﬀerence of two processes in the soft momentum limit of two gaugebosons; one process is the emission of two soft gauge bosons, having the same color c with
momenta kµ and lν , while the other process is the emission of colorless two soft photons with61

Soryushiron Kenkyu

the same momenta. Pictorially, it is(4.5.9)Here, (4.5.8) has been used in the deformation of the ﬁrst line. The meaning of the limitkµ , lµ → 0 is to take the limit under the regularization by the N-dimensional method, as
was discussed in the last subsection. In (4.5.9), two cases of soft gauge boson emissions, one
from the external lines and the other from the internal lines, are not speciﬁed. For example,

62

Soryushiron Kenkyu

it includes the following two soft gauge bosons’ emission, which implies(4.5.10)[We should examine the Thomson limit in QCD, using this formula. A diﬃculty appears,
however, that a soft gauge boson can be easilly resolved into two or three soft gauge bosons
via the self-interaction between the soft gauge bosons. Therefore, it is diﬃcult to restrict
the number of soft gauge bosons to be one in the initial or in the ﬁnal state.]
By this (4.5.9), soft gauge boson emission in QCD can be reduced to the soft photon
emission as follows:(4.5.11)

Apply the Eikonal identities (2.2.14), (2.2.15) to the right hand side of this equation, we
obtain(4.5.12)[when using (2.2.14), (2.2.15), the particle with momenta l1 , l2 , · · · , ln , should be considered
as a gauge boson with a color]. In this way (4.4.3) is shown to hold in case of two softgauge bosons’ emission. Including the external corrections to the external gauge bosons,and multiplying physical polarization vectors, it becomes (4.4.2). [47]Here, we will notice by comparison that (4.4.3) obtaiend here and (2.2.7) or (2.2.17) in
Subsection 2.2 have the same form up to the color factor [tc tc ]ji . In Section 2.2, this (2.2.7)
plays an important role, which contributes to derive the diﬀerential equation (2.2.3) [or63

Soryushiron Kenkyu(2.2.23)] controling the heighest infrared divergences in all orders; this diﬀerential equation
is easily solved to give (2.2.1) [or (2.2.24)]. Therefore, using (4.4.3) in QCD, corresponding
to (2.2.7) in QED, it is easily expected to be able to derive the diﬀerential equation which
controlls the highest inftared divergences in QCD. [This derivation was given by Cornwall
and Tiktopoulos [20], but the proof of Low’s theorem in (4.4.3) was not given at that time.
On the other hand, Kinoshita and Ukawa [48] develpped a general theory to estimate thehighest logarithmic infrared divergences, and derived the above diﬀerentail equation. Theirderivation is, however, too complicated to understood easilly. The merit of the following
derivation of the diﬀerential equation is its simpleness. The new point here is that the Low’s
theorem (4.4.3) has been proved in all orders. Why the proof becomes successful largely
owes the usage of the axial gauge.]
Now, let us repeat the derivation of the diﬀerential equaion in QCD, as was done in
QED in Subsection 2.2. There, we have introduced the gauge boson mass λ as the infrared∂
regularization. Then, applying λ ∂λto the fermionic scattering amplitude S by the colorless external source, [even if it has the same notation in Subsection 2.2, but it is here theamplitude in QCD], we obtain, in the similar way to (2.2.6), the following
Z∂∂
1
dN k X µν
λ S=
Kcc (p, p′ ; k, −k)λ D̃µν (k; λ).
(4.5.13)
N
∂λ
2
(2π)
∂λ
c
µν
(p, p′ ; k, −k)[Here, N was used as the reularization of the ultraviolet divergences.] This Kcc
in QCD corresponds to Figure (4.5.1) in QED.
Two leggs of gauge bosons are connected by the dressed propagator,and have the same color c. In the following we proceed by referring to
Subsection 2.2 in QED. To examine the infrared divergences of (4.5.13),µν
we have to know the behavior of Kcc(p, p′ ; k, −k) in the limit of kµ → 0∂µν(p, p′ ; k, −k), we canand the behavior of λ ∂λD̃µν (k; λ). First, for the Kcc
use (4.4.3), the Low’s theorem for two gauge bosons’ emission with thesame color, proved in this subsection. In the present notation it reads
X
X
µν
Kcc
(p, p′ ; k, l) ≈ +(gB )2[tc tc ]Figure 4.5.1:ckµ ,lν →0

′µ


µ

µ



c

p
p
−
S
(4.5.14)′p ·l p·lP
Here, the color index c is summed over, so c tc tc = C2 (R)1 holds, [1 is a unit matrix].
[Please refer to (3.4.9) and the subsequent explanation.] This corresponds to (2.2.7). Whenusing (4.5.14) in (4.4.13), it is necessary to have an attention. It is because in the proofof (4.5.14) in this subsection, we used the N-dimensional method for both ultraviolet andinfrared regularizations. In deriving the diﬀerential equation to control the infrared divergences, we will keep the N-dimensional method for ultraviolet regularization, but thediﬀerent regularization method will be used for the infrared divergences; introduction of
non-zero mass λ to the gauge boson for the infrared regularization. Therefore, it is betterto try again the proof of (4.5.14) [or Low’s theorem (4.4.13)]. However, the diﬀerence appears only in (4.4.11) among the relations (4.4.9)–(4.4.14) for simplest graphs; for which theO(λ2 ) breaking appears in its right hand side. Namely, (4.4.11) is replaced by the following
×

p
p
−
′
p ·k p·k′µ

64

Soryushiron Kenkyu

expression:(4.5.15)cab
where Bµνλ= gf cabKνλ (q) ≡ gνλ −1∂(q 2 − λ2 + iε)2 ∂qKνλ (q),(4.5.16)µqν q λqν n λ + n ν q λ+ (αq 2 + n2 )[numerator of the propagator] (4.5.17)
q·n(q · n)2

Accordingly, each step of proof and the ﬁnal result of Low’s theorem have always O(λ2 )breakings. These breqakings do not ﬁnally remain after taking the limit of the cutoﬀ λ → 0,
so that we can ignore these breakings. This proof of Low’s theorem in this subsection
corresponds to (2.2.8)–(2.2.17) in Section 2.2.
Now, we can use (4.5.14) in (4.5.13), and obtain the following diﬀerential equation,  ′νZ δ N  ′µ
∂
1 2
d k
p
pµ
p
pν
λ
ln[S]IR = − gB C2 (R)
−
−
N
∂λ
2
p′ · k p · k p′ · k p · k
0 (2π)∂(4.5.18)× λ D̃µν (k; λ).
∂λ[Here, the part of kµ → 0 contributes to the highest infrared singularities.]
This diﬀerential equation can be solved easily, and we have[S]IR = e∫
2 C (R) δ− 21 gB
2
0

dN k
(2π)N

[

′
pµ
p µ
− p·k
p′ ·k

][

′
p ν
pν
− p·k
p′ ·k

]
×D̃µν (k;λ)× S (0) ,(4.5.19)

where S (0) = ū(p′ )/J colorless u(p). There is the arbitrariness in the initial condition. [(4.5.19)
implies that only the highest divergences are equal in both hand sides and the arbitrarinessfor the ﬁnite parts is allowed.]
In summary, the highest infrared divergences included in the fermion scattering amplitude
by a colorless source is determined, as in (4.5.18), by the behavior of the dressed gauge boson
propagator, D̃µν (k; λ). [Of cource this result holds under the axial gauge condition.]
2
Comparing with QED, the results are formally equal up to the group factor C2 (R) [ N2N−1
for SU (N )]. The essential diﬀerence exists, however, in D̃µν (k; λ). That is, in QED Z3 isinfrared ﬁnite [as in (2.2.20) and (2.2.21)], while in QCD the infrared divergence exists in
Z3 . As was shown by Frenkel and Meuldermans [29] in the one loop calculation, the infrared
divergence included in Π(1) near the mass shell, is1g2,
Π (k = 0; N )IR = − 2 c2 (G)2π(N − 4)2(1)

2(4.5.20)

while Π(2) (k 2 = 0; N )IR has no double pole at N = 4, so that it can be ignored, in comparison
with Π(1) . In our regularization adopted here, this gives
Π(1) (k 2 = 0; λ)IR = −65

 m 2
g2
c
(G)
ln
2
2π 2
λ

(4.5.21)

Soryushiron Kenkyu[Please refer to the deﬁnition of Π(1) , Π(2) in (4.3.27).] Thus, there is the infrared divergence in Z3 = 1/1 + Π(1) (on-shell), not appearing in QED. Of course, if we perform therenormalization on the mass shell, we haveD̃µν (k; λ)Z32=(gR ) D̃µν (k; λ)R ,1/2

gB2 D̃µν (k; λ) =(gB Z3 )2(4.5.22)so that divergence of Z3 in D̃µν (k; λ) is renormalized into gR . If we perform the oﬀ-mass-shell
renormalization [as in (4.4.40)–(4.4.43)], however, we have2Z1 (µ)
2
2
gB D̃µν (k; λ) = g(µ) D̃µν (k; λ, µ),(4.5.23)Z2 (µ)
so that the infrared divergence remains in D̃µν (k; λ, µ), not being renormalized into the oﬀshell coupling g(µ). [Here, D̃µν (k; λ, µ) is the dressed gauge boson propagator, renormalizd ata renormalization point µ.] Because, for µ ̸= 0, Z1 (µ), Z2 (µ), Z3 (µ) are infrared ﬁnite. Hereagain, the characteristic feature of QCD, that the infrared divergence appears in the charge
renormalization, plays the important role. In short, the problem is reduced to investigate thebehavior at kµ ∼ 0 of the dressed gauge boson propagator, D̃µν (k; λ, µ). This is the samekind of problem as that of investigating how does g(µ) behave for µ → 0 [with referring(3.4.23) and (3.4.24)], so that this will be the important target from now on.

Acknowledgements
At the end of the thesis, the author gives his sincere thanks to Dr. Ukawa for introducing
the author to this ﬁeld, and to Mr. Nakagawa and Mr. Yamamoto for their collaboration
in Reference paper II. He is grateful to thank Dr. Shimizu, Dr. Higashijima, Dr. Sato,
Dr. Arafune, Dr. Fukugita, Mr. Nishimura (A.), Dr. Nakazawa, Dr. Okamura, and Dr.
Okabayasi for their valuable discussions and advices as well as encouragements to him, since
his beginning to work in this ﬁeld. He gives his gratitude to Professor Yamaguchi and themembers of the particle physics group at University of Tokyo for their warmly watching theauthor since his entrance to the university.

5

Appendix: Application of M. Sato’s microfunction
to the cancellation problem of infrared divergences

This Appendix includes the latter half of the author’s report, which was submitted to Prof.
Mikio Sato in August (1976), to get the credit of the graduate course in physics at U. of
Tokyo. The title of the course was “Microlocal Analysis and Quantum Field Theory” [50].
The ﬁrst half of the report (Section 1 and 2) was an introductory explanation of thecancellation of infrared divergences in QED; dσV + dσB = infrared ﬁnite, where dσV is the
cross section with the virtual corrections by photon of fermion scattering by an externalsource, while dσB is the cross section associated with the emission of an additonal photons(“B” means Bremmsstrahlung), and hence this part can be omitted, except for importantsentences and equations.

66

Soryushiron Kenkyu

5.1

A claim stated in Section 1 of the report

The singular spectrum (S.S.) of a micro function gives the place where the function becomesnot microanalytic. Since the infrared divergence appearing in the scattering amplitude belongs to the S.S. of a Feynman integral (which is a micro function), the analysis by S.S.
is especially eﬀective, when the divergent part and its cancellation are dicussed without
referring to the ﬁnite part.

5.2

Deﬁnition of Ja and Jb given in Section 2 of the report

The cancellation of infrared divergences can be understood by the Cutkosky rule. For example, consider the forward amplitude, I, of the scattering of an electron by an externalsource,,(5.2.1)

where the two diﬀferent Cutkosky cuts at a and b are taken. The numerator is common toboth cases, so that we consider J, which is I up to the numerator:Z
1
1
1
J = dp′0 dq0 dk0 ′ 2· 2· 2
δ(p′ − q − k).
(5.2.2)2
2
p − m + i0 q − m + i0 k + i0
Application of Cutkosky cuts at a and b gives,Z′
Ja = dp′0 dq0 dk0 (−2πi)θ(p′0 )δ(p 2 − m2 ) ·
Z
Jb =

1
q 2 − m2 + i0·

1
k 2 + i0× δ(p′ − q − k),(5.2.3)1
dp′0 dq0 dk0 ′ 2· (−2πi)θ(q0 )δ(q 2 − m2 ) · (−2πi)θ(k0 )δ(k 2 )2
p − m + i0
× δ(p′ − q − k).
(5.2.4)

If the sum of Ja and Ja reduces the degree of singularity by one, then the infrared divergences
are cancelled between a (virtual process) and b (emission process of photon).

67

Soryushiron Kenkyu

5.3

Section 3– Application of S.S. of micro function to infrared
divergences

We will estimate the singular spectra (S.S.) of Eq.(5.2.3) and Eq.(5.2.4). Before that wesummarize the formulae for S.S [50]:2

 n
o

√ α 2
1
2
2
2
d
−1
S.S.
=
p,
d(p
−
m
)
α(p
−
m
)
=
0
,
(5.3.2)α≥0,p2 − m2 + i02o n √ α 2
2
2
2
2
2
d
S.S. θ(p0 )δ(p − m ) =p, −1 d(p − m ) α ∈ R, p − m = 0, p0 > 0 . (5.3.3)
2
The product and integral formulae are given as follows [50]: √d (f (x)g(x)) = (x, −1ξdx)| ξ = ξ1 + ξ2 s.t.
S.S.
o
√
√
d (x), (x, −1ξ2 dx) = S.S.g(x)d(x, −1ξ1 dx) = S.S.f,(5.3.4)
Z
 n
o
√
√
d
d
S.S.
f (x, y)dy = (x, −1ξdx) ∃y s.t. (x, y; −1ξdx) ∈ S.S.f (x, y) . (5.3.5)
5.3.1

d a)
S.S.(J′

If we choose the coeﬃcients of cotangent vectors in the directions of p 2 − m2 , q 2 − m2 , k 2 ,and p′ − q − k, be α2 , β2 , γ2 , and y µ , respectively, then we have
o
n
√
′
′
⃗
⃗
d
⃗
⃗
⃗ k] ,(5.3.6)
S.S.(Ja ) =p , ⃗q, k; −1[⃗udp + ⃗v d⃗q + wd′

where there exist α ∈ R, β ≥ 0, γ ≥ 0, y µ ∈ R, p 0 , q 0 , k 0 , such that the following condition
holds,α ′βγ⃗udp⃗′ + ⃗v d⃗q + wd
⃗ ⃗k = d(p 2 − m2 ) + d(q 2 − m2 ) + d(k 2 ) + ⃗y d(p⃗′ − ⃗q − ⃗k),2
2
2

(5.3.7)

which is satisﬁed, when the following relations are satisﬁed,⃗u = αp⃗′ + ⃗y , ⃗v = β⃗q − ⃗y , w⃗ = γ⃗k − ⃗y ,′0 = αp 0 + y 0 , 0 = βq 0 − y 0 , 0 = γk 0 − y 0 ,′′p 0 > 0, p 2 − m2 = 0; β(q 2 − m2 ) = 0, γ(k 2 ) = 0,(p′ − q − k)µ = 0.
2

(5.3.8)(5.3.9)(5.3.10)(5.3.11)(Footnote added in 2022) The singular spectrum (S.S.) gives the “singularity structure” of a microfunction, to which Feynman amplitudes belong. Feynman amplitude is given by a product of propagators withiϵ(= i0) rule, which means the singularities are avoided by shifting the pole positions a little in the imaginary
direction. Thus, the microfunction is deﬁned as a boundary value on the real axis of an analytic function
given in the complex plane. To manifest the shift in the complex direction of the pole, the diﬀerential(cotangent vector) giving the pole shift in the imaginary direction, is speciﬁed additionally in the deﬁnitionof S.S..
An example of the diﬀerential can be seen for the propagator as
n
o
p
p
√  p
√ α
−1 d(p2 − m2 ) = −1 α p⃗2 + m2 −d(p0 + p⃗2 + m2 ) + d(p0 − p⃗2 + m2 ) ,(5.3.1)
2which shows the directions of the shifts for the positive and negative energy poles are opposite.

68Soryushiron Kenkyu

Let us examine the conditions Eq.(5.3.8)-Eq.(5.3.11).
′
Eq.(5.3.9) appears due to the integrations over p 0 , q 0 , and k 0 . Eqs.(5.3.10)(5.3.11)
give the on shell condition, or means the singularity arises when the particle ﬂowing the
propagator becomes on the mass shell. The kinematics shows that three momenta, p′ , q, k,
can not be on the mass shell at the same time. Therefore, the following two cases remain,(Case1) : β = 0, k 2 = 0, (Case2) : q 2 − m2 = 0, γ = 0.

(5.3.12)′First we consider (Case2); from Eq.(5.3.9), y 0 = 0 and αp 0 = βq 0 = 0. Since on the′mass shell conditions tell p 0 , q 0 ̸= 0, we have
α = β = γ = y 0 = 0.

(5.3.13)This implies the corresponding cotangent vector (the direction to avoid the singularity)
vanishes,⃗udp⃗′ + ⃗v d⃗q + wd
⃗ ⃗k = ⃗0,(5.3.14)so that Ja is microanalytic in this case.
′Thus we consider (Case1); from Eq.(5.3.9), y 0 = 0 and since p 0 ̸= 0, we have
α = 0, k 0 = 0,

(5.3.15)α = β = y 0 = 0; |⃗k| = 0, ⃗u = ⃗v = ⃗0, w⃗ = γ⃗k.

(5.3.16)

where γ = 0 holds as before.
Now we have

Of course, if |⃗k| = 0, then we have w⃗ = ⃗0, and Ja becomes microanalytic. This happens
because we have not do the regularization. In reality we have to introduce the regularization(photon mass) and consider|⃗k| ∼ λ, w
⃗ = γ⃗k.
Summarizing the above consideration, we have arrived aton√d a) =
S.S.(Jp⃗′ , ⃗q, ⃗k; −1γ(⃗kd⃗k) γ > 0, |⃗k| ∼ λ, k 2 = 0, p⃗′ = ⃗q + ⃗k .

(5.3.17)(5.3.18)

Taking into account the supplementary condition ofα = β = y 0 = 0,

(5.3.19)d a ) becomes quite reasonable from the physical view point.
the result of S.S.(JWe will examine this in the following: S.S(Ja ) arises from the region of α = β = 0, γ ̸=0, |⃗k| ∼ λ, k 2 = 0. If we consider α, β, γ be “resistance” in the circuit analogy, α = β = 0
implies that the large electric currents p′ , q ﬂow along the corresponding paths, while γ ̸= 0
implies the small current |⃗k| ∼ λ ﬂows along the path, which gives the region of infrared
divergence in the limit of soft photon. That is,d a ) ⇔ Infrared divergence region of Ja .
S.S(J

69

(5.3.20)

Soryushiron Kenkyu

5.3.2

d b)
S.S.(Jd b ) can be obtained. Namely we have
In the same manner, S.S(Jon√d b) =
S.S.(Jp⃗′ , ⃗q, ⃗k; −1[⃗udp⃗′ + ⃗v d⃗q + wd
⃗ ⃗k] ,

(5.3.21)′

where there exist α ≥ 0, (β, γ, y 0 ) ∈ R, p 0 , q 0 , k 0 , such that the following conditions hold,⃗u = αp⃗′ + ⃗y , ⃗v = β⃗q − ⃗y , w⃗ = γ⃗k − ⃗y ,′00 = αp + y 0 , 0 = βq 0 − y 0 , 0 = γk 0 − y 0 ,′α(p 2 − m2 ) = 0; q 0 > 0, q 2 − m2 = 0; k 0 > 0, k 2 = 0,(p′ − q − k)µ = 0.

(5.3.22)(5.3.23)(5.3.24)(5.3.25)′

Let us examine these conditions. From Eq.(5.3.24) we have p 2 ̸= m2 , so α = 0, which
implies y 0 = 0 and β = 0 and γk 0 = 0.
In the same manner as before, if γ = 0, Jb is microanalytic, so that we consider γ ̸= 0.
Since k 0 ∼ λ and k 2 = 0, we have |⃗k| ∼ λ. Summarizing this, we have α = β = γ = 0,
yieldingon√d b) =(5.3.26)
S.S.(Jp⃗′ , ⃗q, ⃗k; −1γ(⃗kd⃗k) γ ̸= 0, |⃗k| ∼ λ, k 2 = 0, p⃗′ = ⃗q + ⃗k .
This result implies p′ and q are hard and k is soft, giving the infrared divergence of Ja .
Schematically, we haved b ) ⇔ Infrared divergence region of Jb .
S.S(J(5.3.27)d a ) and S.S.(Jd b ) overlap with each other, so that the
It is important to note that S.S.(Jsum of these two can reduce the degree of infrared divergence.
5.3.3

Sum of Ja and Jb

Instead of making a sum of Ja and Jb , consider the forward scattering amplitude J without
cut at a or b. Tnen, we have
n
o
√
d
S.S.(J)=p⃗′ , ⃗q, ⃗k; −1[⃗udp⃗′ + ⃗v d⃗q + wd
⃗ ⃗k] ,
(5.3.28)′

where there exist (α, β, γ) ≥ 0, y 0 ∈ R; p 0 , q 0 , k 0 , such that the following conditions hold,⃗u = αp⃗′ + ⃗y , ⃗v = β⃗q − ⃗y , w⃗ = γ⃗k − ⃗y ,′0 = αp 0 + y 0 , 0 = βq 0 − y 0 , 0 = γk 0 − y 0 ,′α(p 2 − m2 ) = 0; β(q 2 − m2 ) = 0; γk 2 = 0,(p′ − q − k)µ = 0.

(5.3.29)(5.3.30)(5.3.31)(5.3.32)

Let us examine these conditions. We can consider four cases.
(Case1): α = β = γ = 0 gives the zero cotangent vector, so that this case can be dropped.
′(Case2): p′ is on the mass shell, p 2 − m2 = 0, and others are oﬀ shell, q 2 − m2 ̸= 0,
k 2 ̸= 0. This case also gives the zero cotangent vector and is dropped.

70

Soryushiron Kenkyu′(Case3): q is on the mass shell, but others are oﬀ shell, p 2 − m2 ̸= 0, and k 2 ̸= 0. Thisis also irrelevant.
′(Case4): k is on the mass shell k 2 = 0, and the others are oﬀ shell, p 2 − m2 ̸= 0 and
q 2 − m2 ̸= 0. This implies α, β = 0, y 0 = 0, γk 0 = 0. Choose γ ̸= 0 and k 0 ∼ λ, we have|⃗k| ∼ λ. This can give the infrared divergence.
′(Case5): p 2 = m2 , q 2 = m2 , but k 0 ̸= 0. This implies k 0 ̸= 0, γ = 0, y 0 = 0, α = β = 0,
so that this is an irrelevant case.
′(Case6): The conditions p 2 = m2 , k 2 = 0, but q 2 ̸= m2 implies β = 0, y 0 = 0, α = 0,and γk 0 = 0, so that γ ̸= 0 and k 0 ∼ λ, and |⃗k| ∼ λ. This is a relevant case.
′(Case7): We have q 2 = m2 , k 2 = 0, and p 2 ̸= m2 . These are the same conditions to(Case6).
Since three vectors can not be on the mass shell at the same time, and hence we haveexhausted all the cases. Now the conditions lead toα = β = y 0 = 0, γ > 0,k 2 = 0, k 0 ∼ |⃗k| ∼ λ,(5.3.33)(5.3.34)

and (Case4) p′ and q are both oﬀ the mass shell, and (Case6) or (Case7) One of p′ and q
is on the mass shell, remain as the possible cases to generate infrared divergences. (Both p′
and q are oﬀ the mass shell is prohibited.) Thus, we have
n

o
√
2
′
′
⃗
⃗
⃗
⃗
⃗
d
⃗
⃗
S.S.(J) =p , ⃗q, k; −1γ(kdk) γ > 0, k = 0, |k| ∼ λ, p = ⃗q + k .
(5.3.35)
Schematically,dS.S(J)⇔ Candidate of infrared contribution to J.

(5.3.36)

By the Cutkosky rule, we have
J + J ∗ ∝ J a + Jb ,(5.3.37)

and know that the infrared singularities cancel between Ja and Jb , and that J has no infrared
singularities.
The microlocal analysis clariﬁes the position of singularities, but we need to know furthermore “Order of Singularities” at the singular position.
Physically, we know the “Ord.” of singularities by power counting with respect to thephoton momentum k, that is,Ord.[Ja ] = Ord.[Jb ] = −2,
and Ord.[J] = −1.

(5.3.38)(5.3.39)

These equations show the appearance of infrared divergences and the cancellation of them.
Next, we will give an “primitive way of power connoting”. First we introduce the following
criterions:(C1) Assign -2 to the photon propagator,(C2) Assign -1 to the internal fermion propagator which connects to the photon on themass shell and to the outgoing electron on the mass shell,(C3) Assign +1 to the integration by photon momentum.

71

Soryushiron Kenkyu

With these criterions we haveOrd.[Ja ] = Ord.[Jb ] = −2 − 1 + 1 = −2,
Ord.[Ja + Jb ] ≤ Ord.[J] = −2 + 1 = −1,(5.3.40)(5.3.41)

where the Cutkosky rule is taken into account. This reproduces the result obtained by the
explicit integration of the amplitudes.
We hope to have the technique in the micro functions to estimate the Order of Singularities (Ord.).
(Note added in 2022) Ten years later, a similar work to this Appendix appeared [51].
—————————————————–

Acknowledgements and epilogue in 2022
The author deeply thanks Shiro Komata who kindly oﬀered to translate a good part of thisthesis into English, written originally in Japanese in 1978. Without his help, the English
translation would not have been achieved.
He also thanks So Katagiri, Yoshiki Matsuoka, and Kimiko Yamashita for reading andreﬁning of the draft.

Epilogue (2022)
Even now, the interplay between the infrared (IR) behavior and the ultraviolet (UV) behavior
in some theories (especially in quantum gravity), seems not to be fully understood. During
the past 45 years, Wilson’s lattice gauge theory [52] was so successful in doing the strong
coupling expansion, which gives not only hadron masses but also scattering amplitudes
consistent to the experimental results. In this lattice gauge theory, a linearly rising potential
between quarks is given,
Vqq̄ (k) ∝g(|k|)2
1∝ 4 + · · · , or Vqq̄ (r) ∝ r + · · · ,
2
|k|
|k|

which was a mere guess in this thesis. Also the evolution equation, similar to the diﬀerential equation derived in this thesis, was developed by Dokshitzer, Gribov-Lipatov, andAltarelli-Parisi (DGLAP) [53], describing the evolution of the parton distribution functions,consistently with experiments in QCD.
This thesis claims the importance of Ward-Takahashi identities in the cancellation of
infrared as well as of mass singularities to occur, while guaranteeing the unitarity. For thecancellation to occur the gauge invariant set of graphs is essential.
To represent the set, however, Wilson operators, such as W[Path=open] andW[Path=closed] for non-Abelian gauge theory and W[Path=closed] for gravity, seem notcompletely satisfactory. The more suitable deﬁnition of variables surely exists, which ismade up of massless quarks, gluons and gravitons, moving almost in parallel with eachother. Such variables include the soft momentum cutoﬀ λ′ and the angle cutoﬀ η from theparallel direction, or the ﬁctitious mass λ introduced to quarks, gluons and gravitons. The
new variables, if denoted as
X[open; λ] and X[closed; λ],72

Soryushiron Kenkyu

are expected to follow the string dynamics, with a tension given by the cutoﬀ λ,
α′ ∝

1
.
λ2

The theories are IR ﬁnite from the beginning, due to the cancellation of IR divergences forthe gauge invariant sets, studied in this thesis, where the Ward-Takahashi identities were
fully used.
Furthermore, if the modular symmetry (τ → −1/τ ) is imposed on the partition function,then the UV ﬁniteness can also be achieved. Here, τ is an evolution parameter in thepartition function. The parameter is the ratio per unit length L of the object, of a time
diﬀerence at ﬁnite temperature T , i.e.
τ=

c∆t
ℏc
+i
.
L
kB T L

It is quite plausible that the partition function for an infrared free set or a linearly extended
object can be consistently given when the modular invariance is satisﬁed.
Recently, Ward-Takahashi identities associated with the Virasoro-like symmetry, arisingin the soft graviton limit of gravity, has been understood by A. Strominger and others [54],as the soft graviton theorem. Some 60 years ago, the symmetry was known by Bondi, vander Burg, Metzner and Sacks (BMS) [55], and the soft graviton theorem was also knownin [2], but the interrelaton between them is clariﬁed only recently. This new topics called“Celestial Holography” uses the similar terminology as in this thesis, and seems to pursuit
the similar issue mentioned in the above.
Frankly speaking, many researchers of the infrared slavery at that time, were surely
aimimg at the issue mentioned above. A reason why they (including the author) would notﬁnish the issue is the appearance of the numerically calculable lattice gauge theories. History
repeats itself.
This motivates the author for translating his old thesis recently. He hopes this English
translation contributes to the study of the issue.

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73

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74

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[25] G. ’t Hooft, Nucl. Phys. B33, 173 (1971).
[26] W. Konetschny and W. Kummer, Nucl. Phys. B108, 397 (1976).
[27] K. Nishijima, Phys. Rev. 119, 485 (1960); M. Veltman, Physica 29, 186 (1963); G. ’t
Hooft and M. Veltman, in Particle Interactions at Very High Energies, Part B, edited
by F. Halzen et al. (Plenum, New York, 1974). (The name “diagrammar” appeared.)[28] A similar remark was also made independently by K. J. Kim, Phys. Lett. 69B, 347(1977).
[29] J. Frenkel and R. Meuldermans, Phys. Lett. 65B, 64 (1976). (It seems the calculationneeds another check.)[30] Among the divergences produced by the parameter integral, it is the ultraviolet divergence (UV) that is independent on the momentum conﬁguration of the external line,
and on the mass of the particle on the internal line. This is because the external momentum conﬁgurations and the mass of the internal particles, are irrelevant, when theinternal loop momentum becomes inﬁnitely large in the UV limit.
[31] Other than the ultraviolet divergence mentioned in [30], another divergence occurs, when
the two conditions hold simultaneously; one is the external line being on the mass shell,
and the other is a particular particle being massless. This divergence is called infrared
divergence (IR), an example of which is the divergence occurring at α = β = 0. By thecircuit theory, Feynman parameter acts as a resistance against the current (momentumﬂow). Therefore α = β = 0, γ = 1 corresponds to the momentum conﬁguration, in
which p′ + k and p + k associated with α and β, respectively, being hard, and themomentum k with γ being soft.
[32] This symbol will work as a powerful tool for QCD in Section 4.
[33] However, in electron-positron pair creation and pair annihilation processes, two poles′
are split into the upper and lower half planes. Thus one of the two
at k 0 = pp′0·k and p·kp0poles contributes to infrared divergence. This contribution acts as pure imaginary for(2.3.8) and produces a pure imaginary phase ei∞ in the scattering amplitude. This iscalled the “Coulomb phase”. In the cross section the divergence vanishes since |ei∞ | = 1.
For the details, see reference [2] (in particular, the Appendix of S. Weinberg’s paper).
[34] Papers particularly discussing the forward scattering are C. T. Sachrajda, Phys. Rev.
D14, 1072 (1976); L. Matsson and R. Meuldermans, CERN prepring Ref. TH. 2334CERN (1977).
[35] In particular, for the speciﬁc form, see reference [7].
[36] The Feynman rule is shown in, for instance, Reference paper I.
[37] About the deﬁnition of g(µ), please refer for example to 3.4.23 and 4.4.40–4.4.42.

75

Soryushiron Kenkyu[38] From the experience from QED, N 1−4 ↔ 14 ln µ12 2.1.18, but the coeﬃcients in the relation
are not clear.
[39] L. D. Fadeev and V. N. Popov, Phys. Lett. 25B, 29 (1967); Understandable anduseful reference is Kiev Report No. ITP 67-36 [NAL-THY-57 (1972) english translation];Standard trextbook is E. S. Abers and B. W. Lee, Phys. Rep. 9C, 1 (1973).
[40] R. P. Feynman, Acta Physica Polonica 24, 697 (1963).
[41] This is the so-called Kinoshita Theorem: TK̇inoshita, J. Math. Phys. 3, 650 (1962);
TK̇inoshita and A. Ukawa, Phys. Rev. D13, 1573 (1976); The discussions in these
papers are done in the Feynman parameter space. The discussion in the momentum
space is E. C. Poggio and H. R. Quinn, Phys. Rev. D14, 578 (1976).
[42] It is noted that varous choices are possible for the independent momenta for the vertex.
[43] It is necessary to pay attention to the position where the arrow symbol is attached. First,
ﬁx the ﬂow of independent external momenta and independent internal loop momenta.
Then, the momentum ﬂowing a given internal line is a linearly indepemdent sum ofthese external momenta and internal loop momenta. The arrow symbol associated witha given independent momentum is attached to the lines along which the momentum
ﬂow. It is recommended to make some examples for the better understanding.
[44] It is noted that, since the external ﬁeld does not have a color, the ﬁrst and the second
terms in (4.4.36) have the same color factor [tc ]ji .
[45] It is not clear in the oﬀ-mass-shell renormalization, however, whether a relation Z1 (µ) =Z2 (µ) holds or not; which depends on how µ is chosen.
[46] The −[A]IR in Subsection 3.4 corresponds to (4.4.46) of the covariant gauge. However,there is no proper explanation of why 1/(N − 4) in the covariant gauge becomes 1/(N −4)2 in the axial gauge. Therefore, it is necessary to recheck the calculation of this[c(1) (µ)]IR .
[47] The application of oﬀ-mass-shell or on-mass-shell renormalization for (4.4.2) is the sameas the renormalization of (4.4.1) in Subsection 4.4.
Express formally (4.4.3) as
∞
X

(gB )

2n+2

D

(n)

1/2
= (gB Z3 )2

n=0

 ′
 ∞
 ′
p·ε
p ·ε p·ε X
p ·ε−−(gB )2n B (n) ,p′ · k p · k
p′ · lp · l n=0

and deﬁne the renormalized amplitudes of D(n) as D(n) (µ),
∞
X
n=0

D

(n)

=

∞
X

[g(µ)]2n+2 D(n) (µ),

n=0then the pertuabative expansion of the renormalized (4.4.3) with µ ̸= 0, is given for theoﬀ-mass-shell renoramalization by
 ′
 ∞
 ′
p·ε
p · ε p · ε X (i)p ·ε(n)(µ)−−
c (µ)c(j) (µ)B (n−i−j) ,D=p′ · k p · k
p′ · lp · l i,j≥076

Soryushiron Kenkyu

while for the on-mass-shell renormalization with µ = 0, by
 ′

 ′
p·ε
p ·ε
p ·ε p·ε(n)(0)−−
B (n) (0).
D=p′ · k p · k
p′ · lp·l[48] See the reference [13]. Let us recapitulate the formula by Kinoshita and Ukawa of
counting the highest logarithmic powers of the infrared divergences. The power of thelogarithms is less than the number of loops. Then, the graph for which the numberof the logarithmic powers is equal to the number L of loops [the highest logarithmic
power graph], is a planar graph, from each loop of which a single logarithmic powerarises. That is, from a graph G with L loops, choose a proper series of subgraphs,
S1 , S2 , · · · , SL = G, and make reduced graphs Si /Si−1 [i = 1, 2, · · · , L; where S0 is
a null set]. If each Si /Si−1 gives a logarithmic infrared divergence, then the infrared
L
divergence of the original graph G is ln λ1 . [Here, Si /Si−1 is obtained from Si by
reducing Si−1 to a point.] By simply stating, the following factorization holds in theinfrared limit:[G]IR ≈ [G/SL−1 ]IR × [SL−1 /SL−2 ]IR × · · · × [S2 /S1 ]IR[Please refer also to P. Cvitanović and T. Kinoshita Phys. Rev. D10, 3991 (1974).]
The complexity of their proof comes from the usage of the Feynman parameters. It isinteresting to perform the same thing in the momentum space; in this case the proof
seems to be extremely simple.
[49] More precisely, the integral of (4.5.16) using D̃µν (k; λ; µ), remains divergent.
[50] Mikio Sato, “Microlocal Analysis and Quantum Field Theory”, the lectures given at
University of Tokyo in 1976.
[51] T. Kawai and H. P. Stapp, “Microlocal Analysis of Infrared Singularities”, LBL-24163(1987), published in Algebraic Analysis, I, Academic Press (1988) pp.309-330.
[52] K. Wilson, Phys. Rev. D10, 2445 (1975);
About the QCD potential, see G. S. Bali, Phys. Rept. 343, 1 (2000).
[53] See for example, A. D. Martin, Acta Physica Polonica B39, 2225 (2008).
[54] T. He, V. Lysov, P. Mitra, and A. Strominger, JHEP 05, 151 (2015);
See also a good review, A.-M. Raclorin, arXiv: 2107.02075.
[55] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Proc. Roy. Soc. Lond. bf
A269, 21 (1962);
R. K. Sachs, Proc. Roy. Soc. Lond. bf A270, 103 (1962); Phys. Rev. 128 2851 (1962).

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