Zee A. Quantum Field Theory in a Nutshell
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162 | III. Renormalization and Gauge Invariance
for large k and thus the integral diverges logarithmically as
d
4
k/k
4
. (The ordinary integral
∞
dr r
n
diverges linearly for n = 0, quadratically for n = 1, and so on, and
∞
dr/r
diverges logarithmically.) Since this divergence is associated with large values of k it is
known as an ultraviolet divergence.
To see how to deal with this apparent infinity, we have to distinguish between two con-
ceptually separate issues, associated with the terrible names “regularization” and “renor-
malization” for historical reasons.
Parametrization of ignorance
Suppose we are studying quantum electrodynamics instead of this artificial ϕ
4
theory. It
would be utterly unreasonable to insist that the theory of an electron interacting with a
photon would hold to arbitrarily high energies. At the very least, with increasingly higher
energies other particles come in, and eventually electrodynamics becomes merely part of
a larger electroweak theory. Indeed, these days it is thought that as we go to higher and
higher energies the whole edifice of quantum field theory will ultimately turn out to be an
approximation to a theory whose identity we don’t yet know, but probably a string theory
according to some physicists.
The modern view is that quantum field theory should be regarded as an effective low
energy theory, valid up to some energy (or momentum in a Lorentz invariant theory) scale
. We can imagine living in a universe described by our toy ϕ
4
theory. As physicists in
this universe explore physics to higher and higher momentum scales they will eventually
discover that their universe is a mattress constructed out of mass points and springs. The
scale is roughly the inverse of the lattice spacing.
When I teach quantum field theory, I like to write “Ignorance is no shame” on the
blackboard for emphasis when I get to this point. Every physical theory should have a
domain of validity beyond which we are ignorant of the physics. Indeed were this not true
physics would not have been able to progress. It is a good thing that Feynman, Schwinger,
Tomonaga, and others who developed quantum electrodynamics did not have to know
about the charm quark for example.
I emphasize that should be thought of as physical, parametrizing our threshold of
ignorance, and not as a mathematical construct.
1
Indeed, physically sensible quantum
field theories should all come with an implicit . If anyone tries to sell you a field theory
claiming that it holds up to arbitrarily high energies, you should check to see if he sold
used cars for a living. (As I wrote this, a colleague who is an editor of Physical Review Letters
told me that he worked as a garbage collector during high school vacations, adding jokingly
that this experience prepared him well for his present position.)
1
We saw a particularly vivid example of this in chapter I.8. When we define a conducting plate as a surface
on which a tangential electric field vanishes, we are ignorant of the physics of the electrons rushing about to
counter any such imposed field. At extremely high frequencies, the electrons can’t rush about fast enough and
new physics comes in, namely that high frequency modes do not see the plates. In calculating the Casimir force
we parametrize our ignorance with a ∼
−1
.
III.1. Cutting Off Our Ignorance | 163
Figure III.1.1
Thus, in evaluating (1) we should integrate only up to , known as a cutoff. We
literally cut off the momentum integration (fig. III.1.1).
2
The integral is said to have been
“regularized.”
Since my philosophy in this book is to emphasize the conceptual rather than the
computational, I will not actually do the integral but merely note that it is equal to
2iC log(
2
/K
2
) where C is some numerical constant that you can compute if you want
(see appendix 1 to this chapter). For the sake of simplicity I also assumed that m
2
<< K
2
so that we could neglect m
2
in the integrand. It is convenient to use the kinematic variables
s ≡ K
2
=(k
1
+k
2
)
2
, t ≡(k
1
−k
3
)
2
, and u ≡ (k
1
−k
4
)
2
introduced in chapter II.6. (Writing
out the k
j
’s explicitly in the center-of-mass frame, you see that s, t , and u are related to
rather mundane quantities such as the center-of-mass energy and the scattering angle.)
After all this, the meson-meson scattering amplitude reads
M =−iλ + iCλ
2
[log
2
s
+ log
2
t
+ log
2
u
] +O(λ
3
) (2)
This much is easy enough to understand. After regularization, we speak of cutoff-
dependent quantities instead of divergent quantities, and M depends logarithmically on
the cutoff.
2
A. Zee, Einstein’s Universe, p. 204. Cartooning schools apparently teach that physicists in general, and
quantum field theorists in particular, all wear lab coats.
164 | III. Renormalization and Gauge Invariance
What is actually measured
Now that we have dealt with regularization, let us turn to renormalization, a terrible word
because it somehow implies we are doing normalization again when in fact we haven’t
yet.
The key here is to imagine what we would tell an experimentalist about to measure
meson-meson scattering. We tell her (or him if you insist) that we need a cutoff and she
is not bothered at all; to an experimentalist it makes perfect sense that any given theory
has a finite domain of validity.
Our calculation is supposed to tell her how the scattering will depend on the center-of-
mass energy and the scattering angle. So we show her the expression in (2). She points to
λ and exclaims, “What in the world is that?”
We answer, “The coupling constant,” but she says, “What do you mean, coupling
constant, it’s just a Greek letter!”
A confused student, Confusio, who has been listening in, pipes up, “Why the fuss? I
have been studying physics for years and years, and the teachers have shown us lots of
equations with Latin and Greek letters, for example, Hooke’s law F =−kx, and nobody
gets upset about k being just a Latin letter.”
Smart Experimentalist: “But that is because if you give me a spring I can go out and
measure k. That’s the whole point! Mr. Egghead Theorist here has to tell me how to
measure this λ.”
Woah, that is a darn smart experimentalist. We now have to think more carefully what
a coupling constant really means. Think about α, the coupling constant of quantum elec-
trodynamics. Well, it is the coefficient of 1/r in Coulomb’s law. Fine, Monsieur Coulomb
measured it using metallic balls or something. But a modern experimentalist could just
as well have measured α by scattering an electron at such and such an energy and at
such and such a scattering angle off a proton. We explain all this to our experimentalist
friend.
SE, nodding, agrees: “Oh yes, recently my colleague so and so measured the coupling for
meson-meson interaction by scattering one meson off another at such and such an energy
and at such and such a scattering angle, which correspond to your variables s, t , and u
having values s
0
, t
0
, and u
0
. But what does the coupling constant my colleague measured,
let us call it λ
P
, with the subscript meaning “physical,” have to do with your theoretical λ,
which, as far as I am concerned, is just a Greek letter in something you call a Lagrangian!”
Confusio, “Hey, if she’s going to worry about small lambda, I am going to worry about
big lambda. How do I know how big the domain of validity is?”
SE: “Confusio, you are not as dumb as you look! Mr. Egghead Theorist, if I use your
formula (2), what is the precise value of that I am supposed to plug in? Does it depend
on your mood, Mr. Theorist? If you wake up feeling optimistic, do you use 2 instead of
? And if your girl friend left you, you use
1
2
?”
We assert, “Ha, we know the answer to that one. Look at (2): M is supposed to be an
actual scattering amplitude and should not depend on . If someone wants to change
III.1. Cutting Off Our Ignorance | 165
we just shift λ in such a way so that M does not change. In fact, a couple lines of arithmetic
will show you precisely what dλ/d has to be (see exercise III.1.3).”
SE: “Okay, so λ is secretly a function of . Your notation is lousy.”
We admit, “Exactly, this bad notation has confused generations of physicists.”
SE: “I am still waiting to hear how the λ
P
my experimental colleague measured is related
to your λ.”
We say, “Aha, that’s easy. Just look at (2), which is repeated here for clarity and for your
reading convenience:
M =−iλ + iCλ
2
log
2
s
+ log
2
t
+ log
2
u
+ O(λ
3
) (3)
According to our theory, λ
P
is given by
−iλ
P
=−iλ + iCλ
2
log
2
s
0
+ log
2
t
0
+ log
2
u
0
+ O(λ
3
) (4)
To show you clearly what is involved, let us denote the sum of logarithms in the square
bracket in (3) and in (4) by L and by L
0
, respectively, so that we can write (3) and (4) more
compactly as
M =−iλ + iCλ
2
L + O(λ
3
) (5)
and
−iλ
P
=−iλ + iCλ
2
L
0
+ O(λ
3
) (6)
That is how λ
P
and λ are related.”
SE: “If you give me the scattering amplitude expressed in terms of the physical coupling
λ
P
then it’s of use to me, but it’s not of use in terms of λ. I understand what λ
P
is, but
not λ.”
We answer: “Fine, it just takes two lines of algebra to eliminate λ in favor of λ
P
. Big deal.
Solving (6) for λ gives
−iλ =−iλ
P
− iCλ
2
L
0
+ O(λ
3
) =−iλ
P
− iCλ
2
P
L
0
+ O(λ
3
P
) (7)
The second equality is allowed to the order of approximation indicated. Now plug this
into (5)
M =−iλ + iCλ
2
L + O(λ
3
) =−iλ
P
− iCλ
2
P
L
0
+ iCλ
2
P
L + O(λ
3
P
) (8)
Please check that all manipulations are legitimate up to the order of approximation indi-
cated.”
The “miracle”
Lo and behold! The miracle of renormalization!
Now in the scattering amplitude M we have the combination L − L
0
= [log(s
0
/s) +
log(t
0
/t) + log(u
0
/u)]. In other words, the scattering amplitude comes out as
M =−iλ
P
+ iCλ
2
P
log
s
0
s
+ log
t
0
t
+ log
u
0
u
+ O(λ
3
P
) (9)
166 | III. Renormalization and Gauge Invariance
We announce triumphantly to our experimentalist friend that when the scattering
amplitude is expressed in terms of the physical coupling constant λ
P
as she had wanted,
the cutoff disappear completely!
The answer should always be in terms of physically measurable quantities
The lesson here is that we should express physical quantities not in terms of “fictitious”
theoretical quantities such as λ, but in terms of physically measurable quantities such as
λ
P
.
By the way, in the literature, λ
P
is often denoted by λ
R
and for historical reasons called the
“renormalized coupling constant.” I think that the physics of “renormalization” is much
clearer with the alternative term “physical coupling constant,” hence the subscript P .We
never did have a “normalized coupling constant.”
Suddenly Confusio pipes up again; we have almost forgotten him!
Confusio: “You started out with an M in (2) with two unphysical quantities λ and ,
and their “unphysicalness” sort of cancel each other out.”
SE: “Yeah, it is reminiscent of what distinguishes the good theorists from the bad ones.
The good ones always make an even number of sign errors, and the bad ones always make
an odd number.”
Integrating over only the slow modes
In the path integral formulation, the scattering amplitude M discussed here is obtained
by evaluating the integral (chapter I.7)
Dϕ ϕ(x
1
)ϕ(x
2
)ϕ(x
3
)ϕ(x
4
)e
i
d
d
x{
1
2
[(∂ϕ)
2
−m
2
ϕ
2
]−
λ
4!
ϕ
4
}
The regularization used here corresponds roughly to restricting ourselves, in the integral
Dϕ, to integrating over only those field configurations ϕ(x) whose Fourier transform
ϕ(k) vanishes for k
>
∼
. In other words, the fields corresponding to the internal lines in
the Feynman diagrams in fig. (I.7.10) are not allowed to fluctuate too energetically. We will
come back to this path integral formulation later when we discuss the renormalization
group.
Alternative lifestyles
I might also mention that there are a number of alternative ways of regularizing Feyn-
man diagrams, each with advantages and disadvantages that make them suitable for some
calculations but not others. The regularization used here, known as Pauli-Villars, has the
advantage of being physically transparent. Another often used regularization is known as
III.1. Cutting Off Our Ignorance | 167
dimensional regularization. We pretend that we are calculating in d-dimensional space-
time. After the Feynman integral has been beaten down to a suitable form, we do an analytic
continuation in d and set d =4 at the end of the day. The cutoff dependences of various
integrals now show up as poles as we let d → 4. Just as the cutoff disappears when
the scattering amplitude is expressed in terms of the physical coupling constant λ
P
, in di-
mensional regularization the scattering amplitude expressed in terms of λ
P
is free of poles.
While dimensional regularization proves to be useful in certain contexts as I will note in a
later chapter, it is considerably more abstract and formal than Pauli-Villars regularization.
Each to his or her own taste when it comes to regularizing.
Since the emphasis in this book is on the conceptual rather than the computational, I
won’t discuss other regularization schemes but will merely sketch how Pauli-Villars and
dimensional regularizations work in two appendixes to this chapter.
Appendix 1: Pauli-Villars regularization
The important message of this chapter is the conceptual point that when physical amplitudes are expressed in
term of physical coupling constants the cutoff dependence disappears. The actual calculation of the Feynman
integral is unimportant. But I will show you how to do the integral just in case you would like to do Feynman
integrals for a living.
Let us start with the convergent integral
d
4
k
(2π)
4
1
(k
2
− c
2
+ iε)
3
=
−i
32π
2
c
2
(10)
The dependence on c
2
follows from dimensional analysis. The overall factor is calculated in appendix D.
Applying the identity (D.15)
1
xy
=
1
0
dα
1
[αx + (1 − α)y]
2
(11)
to (1) we have
M =
1
2
(−iλ)
2
i
2
d
4
k
(2π)
4
1
0
dα
1
D
with
D = [α(K − k)
2
+ (1 − α)k
2
− m
2
+ iε]
2
= [(k − αK)
2
+ α(1 − α)K
2
− m
2
+ iε]
2
Shift the integration variable k →k + αK and we meet the integral
[d
4
k/(2π)
4
][1/(k
2
− c
2
+ iε)
2
], where
c
2
= m
2
− α(1 −α)k
2
. Pauli-Villars proposed replacing it by
d
4
k
(2π)
4
1
(k
2
− c
2
+ iε)
2
−
1
(k
2
−
2
+ iε)
2
(12)
with
2
c
2
.Fork much smaller than the added second term in the integrand is of order
−4
and is negligible
compared to the first term since is much larger than c.Fork much larger than , the two terms almost cancel
and the integrand vanishes rapidly with increasing k, effectively cutting off the integral.
Upon differentiating (12) with respect to c
2
and using (10) we deduce that (12) must be equal to
(i/16π
2
) log(
2
/c
2
). Thus, the integral
d
4
k
(2π)
4
1
(k
2
− c
2
+ iε)
2
=
i
16π
2
log
2
c
2
(13)
is indeed logarithmically dependent on the cutoff, as anticipated in the text.
168 | III. Renormalization and Gauge Invariance
For what it is worth, we obtain
M =
iλ
2
32π
2
1
0
dα log
2
m
2
− α(1 − α)K
2
− iε
(14)
Appendix 2: Dimensional regularization
The basic idea behind dimensional regularization is very simple. When we reach
I =
[d
4
k/(2π)
4
][1/(k
2
− c
2
+ iε)
2
] we rotate to Euclidean space and generalize to d dimensions (see appen-
dix D):
I(d)= i
d
d
E
k
(2π)
d
1
(k
2
+ c
2
)
2
= i
2π
d/2
(d/2)
1
(2π)
d
∞
0
dk k
d−1
1
(k
2
+ c
2
)
2
As I said, I don’t want to get bogged down in computation in this book, but we’ve got to do what we’ve got to do.
Changing the integration variable by setting k
2
+ c
2
= c
2
/x we find
∞
0
dk k
d−1
1
(k
2
+ c
2
)
2
=
1
2
c
d−4
1
0
dx(1 − x)
d/2−1
x
1−d/2
,
which we are supposed to recognize as the integral representation of the beta function. After the dust settles, we
obtain
i
d
d
E
k
(2π)
d
1
(k
2
+ c
2
)
2
= i
1
(4π)
d/2
4 − d
2
c
d−4
(15)
As d → 4, the right-hand side becomes
i
1
(4π)
2
2
4 − d
− log c
2
+ log(4 π) − γ +O(d − 4)
where γ = 0.577
...
denotes the Euler-Mascheroni constant.
Comparing with (13) we see that log
2
in Pauli-Villars regularization has been effectively replaced by the pole
2/(4 − d). As noted in the text, when physical quantities are expressed in terms of physical coupling constants,
all such poles cancel.
Exercises
III.1.1 Work through the manipulations leading to (9) without referring to the text.
III.1.2 Regard (1) as an analytic function of K
2
. Show that it has a cut extending from 4m
2
to infinity. [Hint: If
you can’t extract this result directly from (1) look at (14). An extensive discussion of this exercise will be
given in chapter III.8.]
III.1.3 Change to e
ε
. Show that for M not to change, to the order indicated λ must change by δλ =
6εCλ
2
+ O(λ
3
), that is,
dλ
d
= 6Cλ
2
+ O(λ
3
)
III.2 Renormalizable versus Nonrenormalizable
Old view versus new view
We learned that if we were to write the meson meson scattering amplitude in terms
of a physically measured coupling constant λ
P
, the dependence on the cutoff would
disappear (at least to order λ
2
P
). Were we lucky or what?
Well, it turns out that there are quantum field theories in which this would happen and
that there are quantum field theories in which this would not happen, which gives us a
binary classification of quantum field theories. Again, for historical reasons, the former
are known as “renormalizable theories” and are considered “nice.” The latter are known
as “nonrenormalizable theories,” evoking fear and loathing in theoretical physicists.
Actually, with the new view of field theories as effective low energy theories to some
underlying theory, physicists now look upon nonrenormalizable theories in a much more
sympathetic light than a generation ago. I hope to make all these remarks clear in this and
a later chapter.
High school dimensional analysis
Let us begin with some high school dimensional analysis. In natural units in which = 1
and c = 1, length and time have the same dimension, the inverse of the dimension of mass
(and of energy and momentum). Particle physicists tend to count dimension in terms of
mass as they are used to thinking of energy scales. Condensed matter physicists, on the
other hand, usually speak of length scales. Thus, a given field operator has (equal and)
opposite dimensions in particle physics and in condensed matter physics. We will use the
convention of the particle physicists.
Since the action S ≡
d
4
xL appears in the path integral as e
iS
, it is clearly dimension-
less, thus implying that the Lagrangian (Lagrangian density, strictly speaking) L has the
same dimension as the 4th power of a mass. We will use the notation [L] = 4 to indicate
170 | III. Renormalization and Gauge Invariance
that L has dimension 4. In this notation [x] =−1 and [∂] = 1. Consider the scalar field
theory L =
1
2
[(∂ϕ)
2
− m
2
ϕ
2
] − λϕ
4
. For the term (∂ϕ)
2
to have dimension 4, we see that
[ϕ] =1 (since 2(1 +[ϕ]) =4). This then implies that [λ] =0, that is, the coupling λ is dimen-
sionless. The rule is simply that for each term in L, the dimensions of the various pieces,
including the coupling constant and mass, have to add up to 4 (thus, e.g., [λ] + 4[ϕ] = 4).
How about the fermion field ψ ? Applying this rule to the Lagrangian L =
¯
ψiγ
μ
∂
μ
ψ +
...
we see that [ψ] =
3
2
. (Henceforth we will suppress the
...
; it is understood that we
are looking at a piece of the Lagrangian. Furthermore, since we are doing dimensional
analysis we will often suppress various irrelevant factors, such as numerical factors and
the gamma matrices in the Fermi interaction that we will come to presently.) Looking at
the coupling fϕ
¯
ψψ we see that the Yukawa coupling f is dimensionless. In contrast, in
the theory of the weak interaction with L = G
¯
ψψ
¯
ψψ we see that the Fermi coupling G
has dimension −2 (since −2 + 4(
3
2
) = 4; got that?).
From the Maxwell Lagrangian −
1
4
F
μν
F
μν
we see that [A
μ
] = 1 and hence A
μ
has the
same dimension as ∂
μ
: The vector field has the same dimension as the scalar field. The
electromagnetic coupling eA
μ
¯
ψγ
μ
ψ tells us that e is dimensionless, which we can also
deduce from Coulomb’s law written in natural units V(r)= α/r , with the fine structure
constant α = e
2
/4π .
Scattering amplitude blows up
We are now ready for a heuristic argument regarding the nonrenormalizability of a theory.
Consider Fermi’s theory of the weak interaction. Imagine calculating the amplitude M
for a four-fermion interaction, say neutrino-neutrino scattering at an energy much smaller
than . In lowest order, M ∼G. Let us try to write down the amplitude to the next order:
M ∼ G + G
2
(?), where we will try to guess what (?) is. Since all masses and energies
are by definition small compared to the cutoff , we can simply set them equal to zero.
Since [G] =−2, by high school dimensional analysis the unknown factor (?) must have
dimension +2. The only possibility for (?) is
2
. Hence, the amplitude to the next order
must have the form M ∼ G + G
2
2
. We can also check this conclusion by looking at the
Feynman diagram in figure III.2.1: Indeed it goes as G
2
d
4
p(1/p)(1/p) ∼ G
2
2
.
Without a cutoff on the theory, or equivalently with =∞, theorists realized that the
theory was sick: Infinity was the predicted value for a physical quantity. Fermi’s weak
interaction theory was said to be nonrenormalizable. Furthermore, if we try to calculate to
higher order, each power of G is accompanied by another factor of
2
.
In desperation, some theorists advocated abandoning quantum field theory altogether.
Others expended an enormous amount of effort trying to “cure” weak interaction theory.
For instance, one approach was to speculate that the series (with coefficients suppressed)
M ∼ G[1 + G
2
+ (G
2
)
2
+ (G
2
)
4
+
...
] summed to Gf (G
2
), where the unknown
function f might have the property that f(∞) was finite. In hindsight, we now know that
this is not a fruitful approach.
III.2. Renormalization Issues | 171
ν
ν
ν
ν
ν
ν
Figure III.2.1
Instead, what happened was that toward the late 1960s S. Glashow, A. Salam, and S.
Weinberg, building on the efforts of many others, succeeded in constructing an elec-
troweak theory unifying the electromagnetic and weak interactions, as I will discuss in
chapter VII.2. Fermi’s weak interaction theory emerges within electroweak theory as a
low energy effective theory.
Fermi’s theory cried out
In modern terms, we think of the cutoff as really being there and we hear the cutoff
dependence of the four-fermion interaction amplitude M ∼G +G
2
2
as the sound of the
theory crying out that something dramatic has to happen at the energy scale ∼ (1/G)
1
2
.
The second term in the perturbation series becomes comparable to the first, so at the very
least perturbation theory fails.
Here is another way of making the same point. Suppose that we don’t know anything
about cutoff and all that. With G having mass dimension −2, just by high school dimen-
sional analysis we see that the neutrino-neutrino scattering amplitude at center-of-mass
energy E has to go as M ∼G +G
2
E
2
+
...
. When E reaches the scale ∼ (1/G)
1
2
the am-
plitude reaches order unity and some new physics must take over just because the cross
section is going to violate the unitarity bound from basic quantum mechanics. (Remember
phase shift and all that?)
In fact, what that something is goes back to Yukawa, who at the same time that he
suggested the meson theory for the nuclear forces also suggested that an intermediate
vector boson could account for the Fermi theory of the weak interaction. (In the 1930s the
distinction between the strong and the weak interactions was far from clear.) Schematically,
consider a theory of a vector boson of mass M coupled to a fermion field via a dimensionless
coupling constant g:
L =
¯
ψ(iγ
μ
∂
μ
− m)ψ −
1
4
F
μν
F
μν
+ M
2
A
μ
A
μ
+ gA
μ
¯
ψγ
μ
ψ (1)