Effective actions and Schwinger
proper time
33
We have mentioned effective actions a few times already. For example, the effective action
for the 4-Fermi theory is derived from the Standard Model by integrating out the W and Z
bosons. It is an effective action since it is valid only in some regime, in this case for energies
less than mW . More generally, an effective action is one that gives the same results as a
given action but has different degrees of freedom. For the 4-Fermi theory, the effective
action does not have the W and Z bosons. In this chapter we will develop powerful tools to
calculate effective actions more generally. We will discuss three ways to calculate effective
actions: through matching (or the operator product expansion), through field-dependent
expectation values using Schwinger proper time, and with functional determinants coming
from Feynman path integrals.
The first step is to define what we mean by an effective action. The term effective action,
denoted by Γ, generally refers to a functional of fields (like any action) defined to give the
same Green’s functions and S-matrix elements
R 4 as a given action S, which is often called the
action for the full theory. We write Γ = d x Leff (x), where Leff is called the effective
Lagrangian. Differences between Γ and S include that Γ often has fewer fields, is nonrenormalizable, and only has a limited range of validity. When a field is in the full theory
but not in the effective action, we say it has been integrated out.
The advantage of using effective actions over full theory actions is that by focusing only
on the relevant degrees of freedom for a given problem calculations are often easier. For
example, in Section 31.3 we saw that in the 4-Fermi theory large logarithmic corrections
¯ decays of the form αn lnn mW could be summed to all orders in perturbation
to b → cdu
s
mb
theory. The analogous calculation in the full Standard Model would have been a nightmare.
The effective action we will focus on for the majority of this chapter is the one arising
from integrating out a fermion of mass m in QED. We can define this effective action
Γ[Aµ ] by
Z
DA exp(iΓ[Aµ ]) ≡
Z
Z
1 2
/ −m ψ .
+ ψ̄ iD
DA Dψ̄ Dψ exp i d4 x − Fµν
4
(33.1)
When Aµ corresponds to a constant electromagnetic field, Leff [A] is called the Euler–
Heisenberg Lagrangian. The Euler–Heisenberg Lagrangian is amazing: it gives us the
QED β-function, Schwinger pair creation, scalar and pseudoscalar decay rates, the chiral
anomaly, and the low-energy limit for scattering n photons, including the light-bylight scattering cross section. As we will see, the Euler–Heisenberg Lagrangian can be
calculated to all orders in αe using techniques from non-relativistic quantum mechanics.
703
704
Effective actions and Schwinger proper time
33.1 Effective actions from matching
So far, we have only discussed how effective actions can be calculated through matching.
This approach requires that matrix elements of states agree in the full and effective theories.
For example, in the 4-Fermi theory, we asked that
(33.2)
hΩ|T ψ̄ψ ψ̄ψ |ΩiS = hΩ|T ψ̄ψ ψ̄ψ |ΩiΓ ,
where the subscript on the correlation function indicates the action used to calculate it.
P
Writing the effective Lagrangian as a sum over operators Leff (x) =
Ci Oi (x) we were
able to determine the Wilson coefficients Ci by asking that Eq. (33.2) hold order-by-order
in perturbation theory. One-loop matching in the 4-Fermi theory was discussed in Section 31.3. Other examples of matching that we considered include the Chiral Lagrangian
(Section 28.2.2) and deep inelastic scattering (Section 32.4).
In the 4-Fermi theory and for deep inelastic scattering, we matched by expanding prop1
1
or p2 +Q
agators p2 −m
2
2 respectively (see Eqs. (32.70) and (32.71)). The reason one can
W
expand propagators to derive an effective Lagrangian is because when a scale such as mW
or Q is taken large, the propagator can only propagate over a small distance. In terms of
Feynman diagrams, we expand an exchange graph in a set of local interactions:
ψ1
ψ3
ψ2
ψ4
→
ψ1
ψ3
ψ2
ψ4
.
(33.3)
To see how this works in position space, consider matching a Yukawa theory with a massive
scalar,
1
/ − φ( + m2 )φ + λφψ̄ψ,
LY = iψ̄ ∂ψ
(33.4)
2
to an effective Lagrangian Leff which lacks that scalar and is useful for energies much
less than m. For large m, fluctuations of φ around its classical configuration are highly
suppressed. Thus, to leading order we can assume φ satisfies its classical equations of
λ
motion, φ = +m
2 ψ̄ψ, and that loops of φ are small corrections. Plugging the classical
solution back into the Lagrangian gives
1
λ2
ψ̄ψ
ψ̄ψ.
(33.5)
2
+ m2
In this way Leff is guaranteed to give the same correlation functions as LY but has no φ
field in it. As long as m is larger than typical momentum scales, we can also Taylor expand
this non-local effective Lagrangian in a series of local operators:
/ +
Leff = iψ̄ ∂ψ
λ2
λ2
ψ̄ψ
ψ̄ψ
−
ψ̄ψψ̄ψ + · · · .
(33.6)
2m2
2m4
If φ were the W and Z, this would give the 4-Fermi theory supplemented by additional
2
operators that have effects suppressed by powers of mE2 at low energy.
/ +
Leff = iψ̄ ∂ψ
W
33.2 Effective actions from Schwinger proper time
705
Setting φ to its classical equations of motion amounts to taking the steepest descent
approximation in the path integral. To integrate out φ to all orders, we have to perform the
path integral exactly. Thus, we can define the effective action as
Z
Z
Z
Z
Dψ̄ Dψ exp i d4 x Leff [ψ, ψ̄] = Dφ Dψ̄ Dψ exp i d4 x LY [φ, ψ, ψ̄] ,
(33.7)
which connects back to the definition given in Eq. (33.1).
33.2 Effective actions from Schwinger
proper time
The next method we discuss for computing effective actions is through Schwinger proper
time. The idea here is to evaluate the propagator for the particle we want to integrate out
as a functional of the other fields. Pictorially, we can write this as
GA (x, y) =
+
+
+ ··· .
(33.8)
Then, when we integrate out the field, we will generate an infinite set of interactions among
the other fields.
The key to Schwinger’s proper-time formalism is the mathematical identity
Z ∞
i
ds eis(A+iε) ,
(33.9)
=
A + iε
0
which holds for A ∈ R and ε > 0 (see Appendix B). This lets us write the Feynman
propagator for a scalar as
Z
d4 p ip(x−y)
i
DF (x, y) =
e
4
2
(2π)
p − m2 + iε
Z
Z
4
2
2
d p ip(x−y) ∞
ds eis (p −m +iε) .
(33.10)
=
e
4
(2π)
0
The integral over d4 p is Gaussian and can be done exactly using Eq. (14.7) with A =
−2isg µν , giving
Z ∞
2
2
+sm
−iεs
ds −i (x−y)
−i
4s
e
,
(33.11)
DF (x, y) =
16π 2 0 s2
which is an occasionally useful representation of the propagator. For m = 0 it provides a
shortcut to the position-space Feynman propagator DF (x, y) = − 4π1 2 (x−y)12 −iε .
An alternative to performing the integral over p directly is first to introduce a one-particle
Hilbert space spanned by |xi, as in non-relativistic quantum mechanics. This lets us write
hp|xi = eipx . Then, from Eq. (33.10) we get
Effective actions and Schwinger proper time
706
DF (x, y) =
Z
d4 p
hy|pi
(2π)4
Z
∞
ds eis(p
0
2
−m2 +iε)
hp|xi.
(33.12)
The analogy with quantum mechanics can be taken even further. Introduce momentum
2
operators p̂µ with p̂µ |piR = pµ |pi and define Ĥ = −p̂2 . Then eisp hp|xi = hp|e−isĤ |xi.
−4
This lets us use (2π)
d4 p |pihp| = 1 in Eq. (33.12) to get
Z ∞
Z ∞
2
2
ds e−sε e−ism hy; 0|x; si, (33.13)
ds e−sε e−ism hy|e−isĤ |xi ≡
DF (x, y) =
0
0
−isĤ
where |x; si ≡ e
|xi. In the second step, we have interpreted Ĥ as a Hamiltonian and
s as a time variable known as Schwinger proper time.1 Schwinger proper time gives an
intuitive interpretation of a propagator:
A propagator is the amplitude for a particle to propagate from x to y in proper time s,
integrated over s.
One has to be careful interpreting Ĥ however, since it conventionally includes only the p
dependence and not the m dependence (as Ĥ = m2 − p̂2 would).
We can go even further into quantum mechanics by defining the Green’s function as an
operator matrix element. Define the Green’s function operator for a massive scalar as
Ĝ ≡
i
.
p̂2 − m2 + iε
(33.14)
Then the Feynman propagator is
Z
Z
d4 p
i
i
d4 p ip(x−y)
e
=
hy|pihp| 2
|xi
DF (x, y) =
4
2
2
(2π)
p − m + iε
(2π)4
p̂ − m2 + iε
= hy|Ĝ|xi.
(33.15)
Or we can go directly to proper time, without ever introducing the p integral, through
Eq. (33.9):
Z ∞
2
ds e−sε e−ism hy|e−iĤs |xi,
(33.16)
DF (x, y) = hy|Ĝ|xi =
0
2
where Ĥ = −p̂ as before.
By the way, when you have two propagators, as in a loop, the relevant identity is
Z ∞ Z ∞
1
dt eisA+itB
(33.17)
ds
=−
AB
0
0
(the iε factors are implicit). If we then write s = xτ and t = (1 − x)τ , so that s and t are
the fractions x and (1 − x) of the total proper time τ , this becomes
Z 1
Z 1 Z ∞
1
1
iτ (xA+(1−x)B)
dx
τ dτ e
=
dx
=−
,
(33.18)
AB
[Ax
+
B
(1 − x)]2
0
0
0
1
To understand why s is called a proper time, recall from relativity that proper time s is defined by the differential ds2 = gµν dxµ dxν . Since Ĥ = −gµν p̂µ p̂ν , it naturally generates translations in proper time through
∂
∂
g µν ∂x
µ ∂xν .
33.2 Effective actions from Schwinger proper time
707
which is a Feynman parameter integral. Thus, in a loop, each particle has its own proper
time, s or t, which denote how long each particle has taken to get around its part of the
s
is how far one particle is behind the other one.
loop. Then the Feynman parameter x = s+t
33.2.1 Background fields
Now suppose a field φ interacts with a photon field, through the usual scalar QED
Lagrangian:
1 2
− φ⋆ D2 + m2 φ,
(33.19)
L = − Fµν
4
with Dµ = ∂µ + ieAµ . As a step towards calculating the Euler–Heisenberg Lagrangian,
we will need the scalar propagator in the presence of a fixed external Aµ field. We write
hA| · · · |Ai instead of hΩ| · · · |Ωi when matrix elements are taken in the presence of an
external field rather than the vacuum. Thus, the propagator in the presence of an external
field Aµ is written as
GA (x, y) = hA|T {φ(y)φ⋆ (x)}|Ai.
(33.20)
Using operator notation, we use ∂µ → −ip̂µ to define
ĜA =
i
.
(p̂ − eA(x̂))2 − m2 + iε
(33.21)
This equation illustrates an advantage of the quantum mechanics operator formalism over
Feynman diagrams: we can work in position and momentum space at the same time,
through operators such as p̂ − eA(x̂).
Then, as in Eq. (33.15), we have
Z
2
i
|xi
=
ds e−sε e−ism hy|e−iĤs |xi,
GA (x, y) = hy|ĜA |xi = hy|
(p̂ − eA(x̂))2 − m2 + iε
(33.22)
where now
Ĥ = −(p̂ − eA(x̂))2 .
(33.23)
So we get the same formula as for the free theory, but with a different Hamiltonian. The
interpretation of Eq. (33.22) is that GA (x, y) describes the evolution of φ from x to y in
time s, including all possible interactions with a field Aµ over all possible times s. This is
shown diagrammatically in Eq. (33.8).
For a spinor, we want to evaluate
GA (x, y) = hA|T {ψ(y)ψ̄(x)}|Ai.
(33.24)
First, recall from Eq. (10.106) that
e
2
/ = Dµ2 + Fµν σ µν .
D
2
(33.25)
We used this identity in Chapter 10 to show that Dirac spinors satisfy the Klein–Gordon
equation with an additional magnetic moment term. Here, the Fµν σ µν term will again
Effective actions and Schwinger proper time
708
produce the differences between the scalar and Dirac spinor cases of quantities we
calculated. Then, in momentum space, we have
e
2
/
(33.26)
(/̂
p − eA(x̂))
= (p̂ − eA(x̂))2 − Fµν (x̂)σ µν .
2
This identity lets us write the spinor Green’s function operator as
i
/
p
/̂ − eA(x̂)
− m + iε
i
/
,
= p
/̂ − eA(x̂)
+m
e
2
(p̂ − eA(x̂)) − 2 Fµν (x̂)σ µν − m2 + iε
ĜA =
(33.27)
and so the Dirac propagator is
Z ∞
2
i
/
ds e−sε e−ism hy|(/̂
p − eA(x̂)
+ m)e−iĤs |xi
|xi =
/ − m + iε
p
/̂ − eA
0
(33.28)
as before, but now with
e
(33.29)
Ĥ = −(p̂µ − eAµ (x̂))2 + Fµν (x̂)σ µν .
2
Note that there is no Dirac trace here, since the Green’s function is a matrix in spinor space.
GA (x, y) = hy|
33.2.2 Field-dependent expectation values
To connect to effective actions, recall from Section 33.1 that to integrate out a field at treelevel we set it equal to its equations of motion. Another way to phrase this procedure is that
we set the field equal to a configuration for which the Lagrangian has a minimum. Now,
classically, we can always expect to find the field at the minimum. So the minimum can be
thought of as a classical expectation. The generalization to the quantum theory is to replace
a field by its quantum vacuum expectation value:
φ → hΩ|φ|Ωi .
(33.30)
The classical and quantum expectation values agree at tree-level, but can be different when
loops or non-perturbative effects are included. We will consider how the vacuum can be
destabilized by quantum effects in Chapter 34. Our focus here is not on the expectation value in the vacuum, but in the presence of a fixed electromagnetic field. Thus, in
a background field, we can integrate out φ by replacing φ → hA|φ|Ai.
Let us go straight to the fermion case. The Lagrangian is
1 2
L = − Fµν
(33.31)
+ ψ̄(i∂/ − m)ψ − eAµ ψ̄γ µ ψ.
4
We now want to replace this by the effective Lagrangian where the current that Aµ couples to is replaced by its expectation value in the given fixed configuration, which we are
denoting as Aµ :
1 2
µ
− eAµ JA
,
(33.32)
Leff = − Fµν
4
33.2 Effective actions from Schwinger proper time
709
where
µ
JA
≡ hA|ψ̄(x)γ µ ψ(x)|Ai.
(33.33)
This is not a vacuum matrix element, but a matrix element in the presence of a given state
|Ai.
µ
Now we can calculate JA
using Schwinger proper time. First note that A = 0 is the
µ
vacuum, so J0 should reduce to the propagator G(x, y) with x = y when the field is
turned off. Indeed, being explicit about the spin indices
h
i
µ
µ
J0µ (x) = hΩ|ψ̄α̇ (x)γα̇α
ψα (x)|Ωi = −Tr hΩ|ψα (x)ψ̄α̇ (x)γα̇β
|Ωi ≡ −Tr hx|Ĝγ µ |xi.
(33.34)
The third form is meant to indicate that the trace of the matrix ψ ψ̄γ µ αβ is being taken.
In the presence of a non-zero A field, we just have to replace this by the propagator in the
Aµ background:
µ
JA
(x) = −Tr hx|ĜA γ µ |xi,
where ĜA is the Green’s function in Eq. (33.27). So,
Z ∞
2
µ
/ + m)e−iĤs |xi
ds e−sε e−ism hx|γ µ(p
JA
= −Tr
/ − eA
Z ∞0
i
h
µν
2
2
e
/ i((p̂−eA) − 2 σµν F )s |xi,
ds e−sε e−ism hx|Tr γ µ(p
=−
/ − eA)e
(33.35)
(33.36)
0
where we have used that Tr of an odd number of γ-matrices is zero. Next, note that the
current is itself a variation:
Z ∞
i
ds −sε −ism2 h
i ∂
µ
e e
Tr hx|e−iĤs |xi .
(33.37)
JA = −
2e ∂Aµ 0 s
Integrating both sides with respect to Aµ and using Eq. (33.32) gives
1 2
i
Leff (x) = − Fµν
(x) +
4
2
Z
∞
0
i
ds −sε −ism2 h
e e
Tr hx|e−iĤs |xi ,
s
(33.38)
which is only a function of the background field Aµ . For a spinor, Ĥ is given in Eq. (33.29).
For a complex scalar, the effective Lagrangian has a similar form:
Z ∞
1 2
ds −sε −ism2
Leff (x) = − Fµν
(x) − i
e e
hx|e−iĤs |xi,
(33.39)
4
s
0
with Ĥ = −(p̂ − eA(x̂))2 as in Eq. (33.23). The scalar case is actually more difficult
to derive than the spinor case using Schwinger’s method because of the A2µ φ⋆ φ term in
the scalar QED Lagrangian. We produce this Lagrangian using Feynman path integrals in
Eq. (33.52) below.
710
Effective actions and Schwinger proper time
33.2.3 Interpretation and cross check
Up to an extra factor of 1s , the proper-time integral in Eq. (33.39) looks just like hx|ĜA |yi
in Eq. (33.22) with x = y. This is easy to understand: the effective action sums closed
loops, where the particle propagates back to where it started after some proper time s.
That is, it is an integral over hx; 0|x; si. In terms of Feynman diagrams, the effective action
includes all diagrams with any number of external photons and one closed fermion loop:
1 2
Leff = − Fµν
+
4
+
+
+
+ ··· .
(33.40)
The physical interpretation of the expectation value hx|e−iĤs |xi = hx; 0|x; si in
Eq. (33.38) is therefore that it is the amplitude for a particle to go around a loop in proper
time s based on evolution with the Hamiltonian Ĥ.
Note that the first diagram in Eq. (33.40) does not involve any photons at all, thus it
should represent the vacuum energy of the system. This provides a nice consistency check.
Setting A = 0, to get just the first diagram, the effective action becomes (in the complex
scalar case)
Z
Z ∞
2
ds −sε −ism2
4
Γ[0] = −i d x
e e
hx|eip̂ s |xi.
(33.41)
s
0
R d4 k
Inserting 1 = (2π)
4 |kihk| we find
Γ[0] = −iV T
Z
∞
0
ds
s
Z
h
i
d4 k
2
~k 2 − m2 + iε)s ,
exp
i(k
−
0
(2π)4
(33.42)
where V T is the volume of space-time. It is convenient to remove this factor by writing
Γ[0] = −(V T )Veff with V eff an effective potential energy density, which in this case is just
a constant.
The integral over proper time is divergent from the s ∼ 0 region, corresponding to where
the loop has zero proper length. However, Schwinger proper time conveniently gives us a
Lorentz-invariant and gauge-invariant way to regulate such divergences: cut off the integral
for s > s0 . To evaluate V eff , we Wick rotate k0 → ik0 and can integrate over the imaginary
axis. This gives
Z
Z
Z ∞
h
i
dk 0
d3 k
ds
2
~k 2 + m2 )s
exp
−i(k
+
V eff = −
0
s
(2π)3
2π
s0
Z ∞
Z
h
i
3
ds
d k
1
~k 2 + m2 )s ,
=− √
exp
−(
(33.43)
(2π)3 s0 s3/2
2 π
where we have replaced s → −is in the second step. Then we find
q
Z
1
d3 k
~k 2 + m2 + O(√s0 ) .
−
+
V eff =
√
(2π)3
πs0
(33.44)
33.3 Effective actions from Feynman path integrals
711
1
The − √πs
is a divergent constant, corresponding to an extrinsic cutoff-dependent vacuum
0
energy. This can be
p removed with a vacuum energy counterterm. The important term is in
the integral over ~k 2 + m2 = ωk , which counts the ground-state energies of the modes.
It was this sum, not the constant, that led to the Casimir force discussed in Chapter 15.
Note that we get ωk instead of 12 ωk since this is the effective action for a complex scalar
that has twice the energy of a real scalar. For a Dirac fermion, the calculation is identical,
since Ĥ = −p̂2 in both cases when A = 0. The only difference is that the Dirac trace and
− 12 in Eq. (33.38) give a factor of 4(− 12 ) = −2 compared to the scalar case in Eq. (33.39).
The minus sign is consistent with a fermion loop and the factor of 2 is consistent with a
Dirac spinor having twice the number of degrees of freedom of a complex scalar. These
are the same results we found in Section 12.5 by computing the energy density from the
energy-momentum tensor. One consequence is that in a theory with a Weyl fermion and
a complex scalar of the same mass, such as in theories with supersymmetry, the vacuum
energy is zero.
33.3 Effective actions from Feynman
path integrals
An alternative approach to calculating the effective action is based on the Feynman path
integral. Here we want to integrate over some fields by performing the path integral. For
scalar QED, integrating out the scalar means
Z
Z
Z
1 2
⋆
2
2
⋆
4
DA exp(iΓ[A]) = DA Dφ Dφ exp i d x − Fµν − φ (D + m )φ .
4
(33.45)
In this case, since the original action is quadratic in φ, we can evaluate the path integral
exactly. We will ignore the iε in this section for simplicity.
Recall the general formula from Problem 14.1:
Z
Z
1
⋆
4
⋆
Dφ Dφ exp i d x(φ M φ + JM ) = N
exp(iJM −1 J),
(33.46)
det M
where N is some (infinite) normalization constant. Thus, for the scalar QED Lagrangian
we find
Z
Z
Z
1 2
1
DA exp(iΓ[A]) = N DA exp i d4 x − Fµν
. (33.47)
4
det(−D2 − m2 )
This equation will be satisfied if
Z
1
1 2
=N
.
exp iΓ[A] + i d4 x Fµν
4
det(−D2 − m2 )
(33.48)
Effective actions and Schwinger proper time
712
To make this notation somewhat less opaque, we can turn this mysterious determinant into
a sum by noting that
Z
1 2
− ln N = − ln det(−D2 − m2 ) = −tr ln(−D2 − m2 ) .
iΓ[A] + i d4 x Fµν
4
(33.49)
The trace is a sum over eigenvalues, in this case, eigenvalues of − ln −D2 − m2 . One
can either evaluate this trace in momentum space, as will be discussed in Chapter 34, or
in position space, as we discuss here. The beautiful thing about a trace is that it is basis
independent. So we can just evaluate the sum on position eigenstates. That is, using the
quantum mechanics notation from Section 33.2 we have
Z
i 2
2
2
4
(33.50)
iΓ[A] = d x − Fµν − hx| ln(−D − m )|xi + ln N .
4
To connect to Schwinger proper time, take a derivative with respect to m2 and introduce a
Schwinger parameter. Then,
Z ∞
2
d
1
2
2
ds e−ism hx|e−iĤs |xi,
hx|
ln(−D
−
m
)|xi
=
−hx|
|xi
=
i
dm2
−D2 − m2
0
(33.51)
2
with Ĥ = − (p̂ − eA(x̂)) as in Eq. (33.23). Integrating over m2 and restoring the iε,
which we have been ignoring in this section, gives
Z ∞
ds −sε −ism2
1 2
e e
hx|e−iĤs |xi + const,
(33.52)
Leff (x) = − Fµν − i
4
s
0
where the integration constant and ln N have been combined. Physics is unaffected by
these constants, and indeed we will exploit the fact that Leff can be shifted by a constant to
remove infinities when Leff is renormalized.
33.3.1 Fermions
For fermions, we need to evaluate
Z
Z
/ − m ψ = N det(iD
/ − m).
Dψ Dψ̄ exp i d4 x ψ̄ iD
(33.53)
1 2
/ − m)) + const,
+ Tr tr(ln(iD
d4 x − Fµν
4
(33.54)
Thus,
iΓ[A] = i
Z
where Tr indicates a Dirac trace and tr is the normal integral over xµ or pµ . The effective
Lagrangian is then
1 2
/ − m | x + const.
Leff (x) = − Fµν
− iTr hx| ln iD
4
(33.55)
33.4 Euler–Heisenberg Lagrangian
713
As before, we take a derivative with respect to m2 :
"
#
/ +m
d
i
iD
1
i
Leff (x) =
Trhx|
|xi = Tr hx|
|xi
2
2
dm2
2m
2
/ − m2
/ − m2
−D
−D
Z
h
i
2
2
1 ∞
ds e−ism Tr hx|e−iD/ s |xi ,
=
2 0
(33.56)
where we have used in the second step that the trace of an odd number of γ-matrices is 0.
Integrating over m2 gives
Z
i
2
i ∞ ds −ism2 h
1 2
e
Tr hx|e−iD/ s |xi + const.
(33.57)
Leff (x) = − Fµν +
4
2 0 s
Using Eq. (33.25), we then get
Z
i
2
µν
e
1 2
i ∞ ds −ism2 h
Leff (x) = − Fµν +
e
Tr hx|ei[(p̂−eA(x̂)) − 2 Fµν σ ]s |xi + const,
4
2 0 s
(33.58)
which agrees with Eq. (33.38).
Another way to obtain this result is to observe that
/ − m |xi = Trhx| ln(−iD
/ − m) |xi .
Trhx| ln iD
(33.59)
So averaging the two gives
/ − m) |xi =
Trhx| ln(iD
1
2
/ − m2 |xi.
Trhx| ln(−D
2
We can write this in terms of Schwinger parameters using the identity
Z ∞
ds isA
= − ln(A) − ln s0 + finite,
e
s
s0
(33.60)
(33.61)
which holds as s0 → 0. This lets us write Eq. (33.54) with Eq. (33.60) as Eq. (33.58).
33.4 Euler–Heisenberg Lagrangian
Now we are ready to do some physics! We will calculate the effective action for the case of
a constant background electromagnetic field Fµν (which is not the same as constant Aµ ).
From Eq. (33.38) we need to evaluate hx|e−iĤs |xi, where Ĥ = −(p̂−eA(x̂))2 + 21 σµν F µν
2
in the spinor case and Ĥ = − (p̂ − eA(x̂)) for scalars. There are a number of ways to
evaluate this trace. The quickest way is to work in basis |ψn i of eigenstates of Ĥ. Then we
can use
Z
Z
X
d4 xhx|e−iĤs |xi = d4 x
hx|ψn ihψn |e−iĤs |xi
=
Z
n
d4 x
X
n
2
|ψn (x)| e−iEn s .
(33.62)
Effective actions and Schwinger proper time
714
Thus, we just have to sum e−iEn s over all the eigenvalues En of Ĥ. In this way, we
reduce the problem to non-relativistic quantum mechanics. An alternative, somewhat more
general, approach is discussed in Appendix 33.A.
We are interested in constant Fµν . For a constant magnetic field in the ẑ direction, we
can take Ay = B x̂ and so the Hamiltonian becomes
Ĥ =
h
−p̂2t
+
p̂2x
+
p̂2z
+ (p̂y − eB x̂)
2
i
0
,
σz
σz
× 14×4 − eB
0
(33.63)
with the eBσz term being the spin–magnetic moment interaction coming from σµν F µν .
Ĥ has eigenstates for any values of pt , py and pz . Writing
py ipt t−ipy y−ipz z
e
(33.64)
ψnpt ,py ,pz = χn x −
eB
2
reduces the problem to finding the eigenstates of p̂2x + (eB x̂) , which is just the nonrelativistic harmonic oscillator Hamiltonian. The result is that χn are the harmonic
oscillator wavefunctions and n takes discrete values, corresponding to the Landau levels
of a non-relativistic electron in a magnetic field. The energies are therefore
Enpt ,py ,pz ,λ = −p2t + p2z + eB(2n + 1) − 2eBλ,
(33.65)
where λ = ± 21 comes from spin being up or down in the z direction.
From Eq. (33.62), we then get
Z
d4 xhx|e−iĤs |xi = 2
Z
d4 x
∞
py 2
dpt dpy dpz X X )
χn (x −
3
(2π)
eB
1
n=0
λ=± 2
×e
i(p2t −p2z )s −ies(2n+1)B 2ieBλs
e
e
,
(33.66)
where the 2 comes from Ĥ being block diagonal. To evaluate these sums and integrals, we
put the system in a Euclidean box of size L. Then the dt, dy, and dz integrals give a factor
of L3 . The dx integral just gives 1, since the wavefunctions are normalized. Because the
p
wavefunctions depend on x − eBy , unless py < LeB, the wavefunctions will shift out of
the box; so the py integral gives a factor of eBL. We then have
Z
4
d xhx|e
−iĤs
|xi = 2
X
λ=± 12
= −2iL4
eBL4
e2iseBλ
(2π)3
Z
∞
−∞
eB 1 cos(esB)
.
8π 2 s sin(esB)
2
2
dpz dpt ei(pt −pz )s
∞
X
e−ies(2n+1)B
n=0
(33.67)
This has no position dependence, since B is constant. It corresponds to an effective
Lagrangian as in Eq. (33.38) of the form
Z
1 2
eB ∞ ds −sε −ism2 1 cos(esB)
LEH = − Fµν + 2
e e
.
(33.68)
4
8π 0 s
s sin(esB)
33.4 Euler–Heisenberg Lagrangian
715
~ → iE.
~ The general
The calculation for a constant electric field is the same, but with B
Lorentz-invariant expression for the effective Lagrangian for any constant Fµν can be
written as
Z ∞
ds −sε −ism2 Re cos(esX) µν
e2
1 2
e e
F F̃µν ,
(33.69)
LEH = − Fµν +
2
4
32π 0 s
Im cos(esX)
where X is a scalar function of the electric and magnetic fields defined by
r
q
i
1 2
~ + iE)
~ 2,
Fµν − F µν F̃µν = (B
X≡
2
2
(33.70)
~ and general
with F̃ µν = 21 εµναβ Fαβ . You are encouraged to check the constant E
expression in Problem 33.1. Taking s → −is we find
Z ∞
ds isε −sm2 Re cosh(esX) µν
e2
1 2
+
e e
F F̃µν .
(33.71)
LEH = − Fµν
2
4
32π 0 s
Im cosh(esX)
In this form, the Lagrangian is more obviously real (except possibly near singularities as
discussed in Section 33.4.3).
Finally, the Lagrangian should be renormalized. We use minimal subtraction. Expanding
the integrand perturbatively in e, we find
7 µν
4
2 2
e2 s 2
Re cosh(esX) µν
2 2
2
(Fµν ) + (F F̃µν ) + · · · .
F F̃µν = − 2 2 − Fµν +
Im cosh(esX)
e s
3
45
4
(33.72)
The leading two terms result in a UV divergence from the small proper-time region of the
ds integral. These divergences can be regulated in a Lorentz-invariant and gauge-invariant
way by simply cutting off s > s0 . The required counterterms are a constant and a renormal2
ization of the leading Fµν
term. Thus, we remove the infinities with minimal subtraction,
giving
LEH
1 2
e2
= − Fµν
+
4
32π 2
Z
∞
0
ds isε −sm2 Re cosh(esX) µν
4
2 2
e e
F F̃µν + 2 2 + Fµν .
s
Im cosh(esX)
e s
3
(33.73)
This is the Euler–Heisenberg Lagrangian. It is the renormalized effective action arising from integrating out a massive fermion for constant Fµν . It is worth emphasizing that
this effective Lagrangian is non-perturbative in e. It encodes an infinite number of 1-loop
diagrams, as in Eq. (33.40), and a tremendous amount of physics. We will go through a
number of applications below.
In Appendix 33.A, we derive this Lagrangian more slowly, using Schwinger’s original
method. The basic idea is to calculate hy|e−iĤs |xi = hy; 0|x; si by solving the differential
equation
i∂shy; 0|x; si = i∂s hy; 0|e−iĤs |x; 0i = hy; 0|Ĥ|x; si.
(33.74)
d µ
d µ
x̂ = i [Ĥ, x̂µ ] and ds
p̂ = i [Ĥ, p̂µ ] are used to
The Heisenberg equations of motion ds
µ
µ
get an explicit form for x̂ (s) and p̂ (s) and therefore Ĥ(s). This method of calculation
716
Effective actions and Schwinger proper time
produces the full Green’s function G(x, y) = hy; 0|x; si , which is more generally useful
than the effective action alone. For x = y, which is relevant for the effective action, the
differential equation reduces to (cf. Eq. (33.A.150)):
e
i
(33.75)
i∂s hx; 0|x; si = −tr eF coth(esF) + σF hx; 0|x; si ,
2
2
where F = Fµν and σ = σµν are matrices. The solution with appropriate boundary
conditions is
es
−i 1
1
sinh esF
µν
− i σµν F
hx; 0|x; si =
exp − tr ln
16π 2 s2
2
esF
2
µν
2
F F̃µν
es
e
µν
exp
−i
σ
F
.
(33.76)
= −i
µν
64π 2 Im cos(esX)
2
Again, this can be checked by differentiation. For a constant magnetic field, this is
equivalent to Eq. (33.67).
The Euler–Heisenberg Lagrangian was first calculated by Heisenberg and his student
Hans Euler by finding exact solutions to the Dirac equation in a constant Fµν background
[Euler and Heisenberg, 1936]. Our derivation of it, particularly the one in Appendix 33.A,
is due to Schwinger [Schwinger, 1951].
33.4.1 Vacuum polarization
Expanding the unrenormalized Euler–Heisenberg Lagrangian, as in Eq. (33.72), we found
two divergent terms which were removed with counterterms in Eq. (33.73). If we do not
include these counterterms, the expansion gives
Z ∞
ds isε −sm2 1
1 2
e2
1 2
+ Fµν + finite.
e e
(33.77)
LEH = − Fµν − 2
4
8π 0 s
e2 s 2
6
The first term in brackets is constant. It gives the vacuum energy density, as discussed in
2
Section 33.2.3. The second term looks just like the tree-level QED kinetic term, − 41 Fµν
.
Keeping only this term (before renormalization), we have
Z ∞
ds isε −sm2
1 2 e2
1 2
e e
.
(33.78)
LEH = − Fµν − Fµν 2
4
6
8π 0 s
This is UV divergent, from the s ∼ 0 region. Regulating with a Lorentz-invariant UV
cutoff s0 , we find
Z ∞
ds isε −sm2
1 2
e2
e e
LEH = − Fµν 1 +
4
12π 2 s0 s
e2
1 2
2
ln
s
m
+
const
.
(33.79)
1−
= − Fµν
0
4
12π 2
This logarithmic dependence on the cutoff is exactly what we found from computing the
full vacuum polarization graph in QED. As discussed in Chapter 23, UV divergences determine RGEs, and this one determines the leading order β-function coefficient. We can read
33.4 Euler–Heisenberg Lagrangian
717
off from the coefficient of the logarithm in Eq. (33.79) (as discussed in Chapter 23), that
the β-function in QED at 1-loop is
β(e) =
e3
,
12π 2
(33.80)
which agrees with Eq. (16.73) (or Eq. (23.29)).
33.4.2 Light-by-light scattering
The original motivation of Heisenberg and Euler was to calculate the rate for photons to
scatter off other photons. This problem was suggested to them by Otto Halpern and is
sometimes called Halpern scattering. The relevant Feynman diagram is
p3
p1
iM =
.
(33.81)
p4
p2
This is a difficult loop to compute directly, even with today’s technology, much less with
what Euler and Heisenberg knew in 1936. We can get the answer (in the limit of lowfrequency light ω ≪ m) directly from the Euler–Heisenberg hLagrangian. The relevant
i
2
term is the one to fourth order in e, which has the form α90 m14 (F 2 )2 + 74 (F F̃ )2 . This
term was computed first in a paper by Euler and Kockel [Euler and Kockel, 1935]. Using
it for light-by-light scattering corresponds to a tree-level Feynman diagram of the form
p3
p1
iM =
p4
.
(33.82)
p2
Note that our effective Lagrangian is only valid when ∂µ Fαβ = 0; thus we will only get
p2
the result to leading order in m
2 . From the experimental point of view, this is enough, since
light-by-light scattering of real on-shell photons has not yet been experimentally observed,
at any frequency.
The matrix element is
α2 1 n 1 1
3 3⋆
4 4⋆
4 4⋆
(pµ ǫν − p1ν ǫ1µ )(p2µ ǫ2ν − p2ν ǫ2µ )(p3α ǫ3⋆
β − pβ ǫα )(pα ǫβ − pβ ǫα )
90 m4
7 µναβ 1 1
3 3⋆
4 4⋆
4 4⋆
ε
(pµ ǫν − p1ν ǫ1µ )(p2α ǫ2β − p2β ǫ2α ) × εµναβ (p3µ ǫ3⋆
+
ν − pν ǫµ )(pα ǫβ − pβ ǫα )
16
o
+permutations .
M=
(33.83)
Effective actions and Schwinger proper time
718
Summing over final polarizations and averaging over initial polarizations, the result is
1 α4 1
1X 2
M =
2224(s2 t2 + s2 u2 + t2 u2 ),
4
4 902 m8
(33.84)
pols.
which leads to a cross section
σ tot =
ω6
973
α4 8 .
10125π m
(33.85)
This is the correct low-energy limit of the exact light-by-light scattering diagram. The exact
result from the 1-loop graphs can be found in [Berestetsky et al., 1982].
33.4.3 Schwinger pair production
Notice that the effective Lagrangian in Eq. (33.73) has singularities for certain values of
the electromagnetic field. To see where the singularities are, we first consider the case with
~ and E
~ parallel. Then,
B
2
~2 − E
~ 2) = 2 B2 − E2 ,
Fµν
= 2(B
(33.86)
~ and B = |B|,
~ and
where E = |E|
~ ·B
~ = −4EB,
F µν F̃µν = −4E
(33.87)
and then, from Eq. (33.70),
X2 =
1 2
(F − iF µν F̃µν ) = (B + iE)2 .
2 µν
(33.88)
Then the Euler–Heisenberg Lagrangian in Eq. (33.73) simplifies to
LEH =
1
E2 − B2
2
Z ∞
e2
1
ds iεs −m2 s
B2 − E2
− 2
.
EB cot(esE) coth(esB) − 2 2 −
e e
8π 0 s
e s
3
(33.89)
Since coth(x) has no poles for x > 0, the singularities are all associated with constant
~ and
electric fields. Thus, we take the limit B → 0, in which case the fact that we took E
~
B parallel is immaterial. From Eq. (33.89) we find
Z ∞
ds iεs −sm2
1
1
1
2
.
(33.90)
eEs
cot(eEs)
−
1
+
e
e
(esE)
LEH = E 2 − 2
2
8π 0 s3
3
In this form, we can see that the Euler–Heisenberg Lagrangian has poles for real E when
nπ
s is equal to sn = eE
for n = 1, 2, . . . As we will now see, these poles indicate that
strong electric fields can create electron–positron pairs, a process known as Schwinger
pair production (although it was predicted first by Euler and Heisenberg).
How can electrons and positrons be produced from the Euler–Heisenberg Lagrangian,
which has no electron field in it? They cannot. However, in a unitary quantum field theory,
33.4 Euler–Heisenberg Lagrangian
719
forward scattering rates are related to the sum over real production rates via the optical
theorem. Recall from Section 24.1 that by the optical theorem (see Eq. (24.11))
1X X
dΠLIPS |M(A → X)|2 .
(33.91)
ImM(A → A) =
2
X
We can apply this theorem to QED in the situation where |Ai corresponds to a coherent
collection of photons describing a large electric field. In QED, the sum over states |Xi
includes states with on-shell electrons and positrons. Since QED is unitary, the optical
theorem holds. In the Euler–Heisenberg Lagrangian the states |Ai are the same states as
in QED. Thus, if the calculation of LEH has been done correctly, the left-hand side of
Eq. (33.91) should be unchanged, as one would expect from a matching calculation. The
right-hand side of Eq. (33.91), on the other hand, cannot be the same as in full QED, since
QED has electrons in it and the Euler–Heisenberg theory does not. Thus, what would be a
unitary process in full QED now appears as a non-unitary process in the effective theory.
Unfortunately, it is not easy to use Eq. (33.91) to calculate the pair-production rate, since
one would have to sum over an infinite number of multi-particle states.
There is a nice shortcut, due to Schwinger, for evaluating the total pair-production rate.
If there were no pair production, then the electric field state |Ai would be constant in time.
Thus hA|S|Ai = 1 where S is the S-matrix. Since in this case the action is constant, S =
2
2
eiΓ . Therefore, |hA| eiΓ |Ai| = eiΓ measures the probability for something other than
2
A to be produced. In other words, eiΓ gives the probability that no pairs are produced
over the time T and volume V of the experiment. We then have
iΓ 2
e = eiΓ e−iΓ⋆ = ei(Γ−Γ⋆ ) = e−2Im[Γ] = e−2V T ImLEH ,
(33.92)
where in the last step we use that, for given background fields, the Euler–Heisenberg
Lagrangian is just a number. Thus 2ImLEH is the probability, per unit time and volume,
that any number of pairs are created. This is the continuum field version of the optical
theorem relation ImM(A → A) = mA Γ tot , where Γ tot is the total decay rate of a single
particle of mass mA .
In order to calculate ImLEH we note that the integrand in Eq. (33.71) has poles at
π
n. There is no pole at s = 0, as can be seen from expanding the integrand at
sn = eE
small s. The imaginary part of this expression can be calculated using contour integration
(Problem 33.3). The result is that2
∞
∞
αE 2 X 1
−nπm2
1 X 1 −m2 sn
.
(33.93)
= 2
e
exp
2Im(Leff ) =
4π n=1 s2n
π n=1 n2
eE
Performing this sum, we find
πm2 αE 2
Γ E → e+ e− pairs = 2 Li2 e− eE ,
π
(33.94)
with Li2 (x) the dilogarithm function. This is the rate for Schwinger pair production in an
external electric field.
2
This sum also has an interpretation as a sum over instantons (see for example [Kim and Page, 2002]).
Effective actions and Schwinger proper time
720
m2
The rate for pair production is negligible until E & E critical = ee ≈ 1018 volts/meter,
which is an enormous field. As of this writing, Schwinger pair production in QED has still
not been observed, since it is extremely difficult to get such fields in the lab. One might
imagine, however, that such strong fields might be produced close to a particle with a very
e
large charge, such as an atomic nucleus. The field around a nucleus is E ∼ 4πr
2 Z. Now,
the Euler–Heisenberg Lagrangian is only valid for fields that have wavelengths greater
than m1e , so the best we can say is that pair production would begin for Z large enough that
E critical ∼
e
Z,
4π (m−2
e )
which gives Z =
4π 2
e2
=
1
α
∼ 137. This result is sometimes invoked
to explain why the periodic table has less than 137 elements!3
33.4.4 Connection to perturbation theory
It is informative to consider which of the predictions we have derived from LEH are
equivalent to perturbative calculations in QED, and which are not.
2
We found that the Schwinger pair-production rate depended on exp(− πm
eE ). This dependence on e indicates that pair production is a non-perturbative effect – you would never see
pair production from constant electric fields at any fixed order in perturbative QED. Of
course, you can get pair production in perturbation theory. But this would involve photon modes of frequencies larger than m. More precisely, one can show that [Itzykson and
Zuber, 1980]
s
Z
h
i2
4m2
2m2
α
4
2
2
2
+ −
~ )
1− 2
1+ 2
, (33.95)
d qθ(q − 4m ) E(q
Γ(E → e e ) =
3
q
q
~ is constant. The Schwinger pair-production rate is one of the
which vanishes when E
very few analytic non-perturbative calculations in quantum field theory that give physical
predictions.
Other results, such as the rate for light-by-light scattering, could be calculated in perturbative QED. Nevertheless, the Euler–Heisenberg Lagrangian efficiently encodes the
result of many loop calculations all at once. It is worth discussing exactly what graphs
are included in the Euler–Heisenberg Lagrangian, since this understanding will apply to
similar effective actions in other contexts.
Recall our expression for the effective Lagrangian where the fermion is integrated out,
Eq. (33.38),
Z
2
i
ds −ism2
1 2
+
e
hx|e−iD/ s |xi.
(33.96)
Leff [A] = − Fµν
4
2
s
We have not assumed Fµν is constant at this point, and in fact this effective action is exact.
That is, since the Lagrangian was quadratic in ψ, this is a formal expression for the result
of evaluating the path integral of ψ completely. It does, however, correspond to only 1-loop
3
This result actually follows more simply from dimensional analysis. The ground state of a hydrogen-like atom
has energy E0 ∼ −Z 2 α2 me . To get pair production, a nucleus has to be able to capture an electron from the
1
vacuum, emitting a positron into the continuum, so E0 . −me giving Z & α
, up to order 1 factors, which
we cannot get by dimensional analysis.
33.4 Euler–Heisenberg Lagrangian
721
graphs, those in Eq. (33.40), since there is only a single propagator going from x back to x
in proper time s. But how can this expression be exact if it does not include higher loops?
Are graphs such as
or
(33.97)
which have internal photon and/or fermion loops, included or not?
To answer this question, first recall that in the calculation of the effective action, and in
the formal exact expression Eq. (33.38), the photon propagator plays no role. In fact, if we
dropped the photon kinetic term from the original action, the only change in the effective
2
term would be missing. Thus, neither of the graphs above
action would be that the − 41 Fµν
are included in the effective action calculation, since both involve the photon propagator.
On the other hand, since nothing is thrown out (assuming the effective action Γ[A] is known
exactly), any physical effect associated with these graphs must be reproducible within the
effective theory. For example, these graphs in full QED contribute to the QED β-function,
which has physical effects. The way the effective theory reproduces the physics of these
loops is with its own loops involving effective vertices. Basically, the fermion loops are
computed first, treating the photon lines as external, which generates new vertices. Then
the photon lines coming off these vertices are sewn together in a loop amplitude using the
photon propagator in the effective theory.
For example, to reproduce the physics of the first graph in Eq. (33.97), the relevant
effective vertex can be determined by cutting through the intermediate photon and then
contracting the fermion loop to a point:
−→
.
(33.98)
The second graph in Eq. (33.97) involves this vertex, associated with the inner fermion
loop, and a 6-point vertex associated with the outer fermion loop. The physics of the
diagrams in Eq. (33.97) are then reproduced by connecting the legs in these effective
vertices:
and
.
(33.99)
These graphs would reproduce the complete result from the graphs in Eq. (33.97), but we
need the full Leff[A] to compute them.
Effective actions and Schwinger proper time
722
In the Euler–Heisenberg Lagrangian, we took Fµν constant. Thus, the full physics of the
loops in Eq. (33.97) is not reproduced by the Euler–Heisenberg Lagrangian alone. Only if
we had the full effective Lagrangian, by evaluating Γ[A] exactly, which would supplement
the Euler–Heisenberg Lagrangian with additional terms depending on ∂µ Fαβ (and give
corrections at higher order in α to the terms without derivatives), would the full theory be
reproduced. This exact Γ[A] is not known.
Even at energies above me , the exact effective Lagrangian can be used. The electron
still shows up as a pole in the scattering amplitude, as is clear already from Schwinger
pair production in the constant Fµν approximation. Thus, one can treat the electron like a
bound state and calculate S-matrix elements for it. Of course, this is a terribly inefficient
way to calculate electron production and scattering, since we already know the full theory.
It is more efficient to use the UV completion of Γ, namely QED, which has a Lagrangian
that is local and real.
33.5 Coupling to other currents
The effective action from integrating out ψ can be generalized to the case where ψ couples
to other things besides Aµ . In this way, we can calculate things such as the π 0 → γγ rate,
where π 0 is the neutral pion from QCD (see Chapter 28).
When ψ couples to things other than Aµ , the effective Lagrangian has more terms. Say
we had
1
1
L = ψ̄(i∂/ − m)ψ − φ( + m2φ )φ − π( + m2π )π − eAµ ψ̄γ µ ψ + λφψ̄ψ + igπ ψ̄γ 5 ψ,
2
2
(33.100)
which has a scalar φ and a pseudoscalar π in addition to the external field Aµ . When we
integrate out ψ, the effective Lagrangian (without ψ) will just contain the other fields coupled to the expectation value of the various ψ bilinears in the background electromagnetic
field, as in Section 33.2.2. That is,
1
1
µ
+ λφJφ + igπJπ , (33.101)
Leff [A, φ, π] = − φ( + m2φ )φ − π( + m2π )π − eAµ JA
2
2
where
µ
JA
= hA|ψ̄γ µ ψ|Ai,
Jπ = hA|ψ̄γ 5 ψ|Ai.
Jφ = hA|ψ̄ψ|Ai,
(33.102)
We sometimes call these field-dependent expectation values classical currents, since they
are just classical functionals of background Aµ (x) fields. The calculation of these classical
currents corresponds to the evaluation of Feynman diagrams such as
Jφ =
+
+
+
+ ··· .
(33.103)
33.5 Coupling to other currents
Here, the ⊗ refers to insertions of the external current in the original theory, corresponding
to an interaction with the scalar. The photon lines are the background electromagnetic
fields.
For the scalar current,
h
i
Jφ = hA|ψ̄(x)ψ(x)|Ai = −Tr hx|ĜA |xi
Z ∞
2
/ 2s
/ + m)ei(/̂p−eA)
= −Tr
ds e−ism hx|(/̂
p − eA
|xi
Z0 ∞
2
ds e−ism hx|e−iĤs |xi.
(33.104)
= −4m
0
∂
Leff [A], with Leff [A] in Eq. (33.38), a result that is useful
You may notice that Jφ = − ∂m
and not surprising, since the φψ̄ψ interaction and the mass term mψ̄ψ have the same form.
For the pseudoscalar current,
h
i
Jπ = hA|ψ̄(x)γ 5 ψ(x)|Ai = −Tr hx|ĜA γ 5 |xi
Z ∞
/ 2s 5
−ism2
i(p
−eA)
/
/
ds e
hx|(p
γ |xi
= −Tr
/ − eA + m)e
0
Z ∞
h
i
2
ds e−ism Tr hx|γ 5 e−iĤs |xi .
(33.105)
= −m
0
This current does not have a simple relation to Leff [A], but as we will see, is not hard to
compute.
33.5.1 Currents at low energy
∂
Since the scalar current is Jφ = − ∂m
Leff [A], for the case of constant electromagnetic
fields, we can read the answer from the Euler–Heisenberg Lagrangian, although additional
counterterms may be required. We find (hiding the counterterms)
Z ∞
e2 ∂
ds −m2 s Re cosh(esX) µν
Jφ = −
e
F F̃µν
32π 2 ∂m 0 s
Im cosh(esX)
Z
∞
ds −m2 s 1
e2 ∂
1 2
=
+ Fµν + · · ·
e
8π 2 ∂m 0 s
e2 s 2
6
Z
∞
2
1
1 2
e
−m2 s
dse
+ Fµν + · · · .
(33.106)
= − 2m
4π
e2 s 2
6
0
The first term is infinite and can be removed with a renormalization of the bare term Λ3 φ
in the Lagrangian. The second term is finite and gives
α 1
2
Jφ = −
Fµν
+ ··· ,
(33.107)
6π m
where the · · · are higher order in e.
For the pseudoscalar, we need
Z ∞
2
ds e−ism Tr[γ5 hx|e−iĤs |xi].
(33.108)
Jπ = −m
0
723
Effective actions and Schwinger proper time
724
Now, from Eq. (33.76),
hx|e−iĤs |xi = hx; 0|x; si = −i
and so
ie2 m
Jπ =
64π 2
Z
∞
es
e2
F µν F̃µν
µν
.
exp
−i
σ
F
µν
64π 2 Im cos(esX)
2
ds e−ism
0
2
µν
e
F µν F̃µν
Tr[γ5 e−i 2 σµν F s ].
Im cos(esX)
(33.109)
(33.110)
Since Tr[γ5 ] = Tr[σµν γ5 ] = 0, only terms with σµν to an even power will survive. Using
2
+ 2iγ5 F µν F̃µν we get
(σµν F µν )2 = 2Fµν
e
Tr[γ5 e−i 2 σµν F
µν
s
] = −4iIm cos(esX).
(33.111)
And thus,
Jπ =
e2 m
16π 2
Z
∞
0
2
ds e−ism F µν F̃µν = −i
α
F µν F̃µν .
4πm
(33.112)
Plugging Jφ and Jπ and the Euler–Heisenberg Lagrangian into Eq. (33.101) gives
1
λ α 2
Leff [A, φ, π] = LEH [A] − φ( + m2φ )φ + φ − Fµν
+ ···
2
m
6π
α g
1
πF µν F̃µν . (33.113)
− π( + m2π )π +
2
4π m
Note that the π coupling has just one term. The decay rates predicted from this effective
Lagrangian are
Γ(φ → γγ) =
α2 2 m3φ
λ
,
144π 3 m2
(33.114)
α2 2 m3π
g
.
(33.115)
64π 3 m2
Not surprisingly, the pseudoscalar rate agrees exactly with Eq. (30.11). In this method of
calculation, however, we gain additional insight into the associated anomaly.
Γ(π → γγ) =
33.5.2 Chiral anomaly
Connecting the π → γγ rate to an anomalous symmetry is straightforward in the effective
action language. Recall that the QED Lagrangian,
/ ψ − mψ̄ψ,
L = ψ̄ i∂/ − eA
(33.116)
is invariant under a vector symmetry, ψ → eiα ψ, and, in the limit m → 0, under a chiral
symmetry, ψ → eiγ5 ψ. The associated Noether currents are J µ = ψ̄γ µ ψ and J µ5 =
ψ̄γ µ γ 5 ψ. By the equations of motion, the axial current satisfies
∂µ J µ5 = 2imψ̄γ 5 ψ.
(33.117)
So the amount by which the axial current is not conserved is proportional to the fermion
mass.
33.6 Semi-classical and non-relativistic limits
725
5
Now, we already calculated the expectation
the background electro of ψ̄γ αψ inµν
value
5 magnetic field. In Eq. (33.112) we found A ψ̄γ ψ A = i 4πm F F̃µν . This is consistent
with Eq. (33.117) only if
α
A∂µ J µ5 A = − F µν F̃µν ,
(33.118)
2π
which agrees with Eq. (30.22).
33.6 Semi-classical and non-relativistic limits
The Schwinger proper-time method is not only useful for calculating loops using quantum
mechanics, it also gives a new perspective on the semi-classical and non-relativistic limits
of quantum field theory. In particular, it illustrates where the particles are hiding in the
path integral. As we will see, Schwinger proper time lets us derive one-particle quantum
mechanics as the low-energy limit of quantum field theory.
To begin, we return to the expression for the Green’s function we derived above for a
scalar particle in a background electromagnetic field, Eq. (33.22):
Z ∞
2
ds e−ism hy|e−iĤs |xi,
(33.119)
GA (x, y) = hA|T {φ(x)φ(y)}|Ai =
0
2
with Ĥ = −(p̂ − eA(x̂)) . This operator Ĥ is the Hamiltonian in a one-particle quantum
mechanical system that generates translations in Schwinger proper time s. The function GA (x, y) is computed for constant electromagnetic fields in Appendix 33.A. In this
section, we rewrite GA (x, y) in terms of a quantum mechanical path integral.
In quantum mechanics, the path integral gives the amplitude for a particle to propagate
from xµ to y µ in time s (see Section 14.2.2):
Z
Z z(s)=y
−iĤs
Dz(τ ) exp(i dτ L(z, ż)),
(33.120)
hy|e
|xi =
z(0)=x
where L = p̂x̂˙ − Ĥ is the Legendre transform of the Hamiltonian. We would like to work
out this Lagrangian in the case of a scalar in an electromagnetic field.
To simplify things, we first write Ĥ = −Π̂2 , where Π̂µ = p̂µ − eAµ (x̂). The Heisenberg
equations of motion for translation in s are
dx̂µ
= i[Ĥ, x̂µ ] = i[−Π̂2 , x̂µ ] = 2Π̂µ ,
x̂˙ µ ≡
ds
(33.121)
where [Π̂µ , x̂ν ] = [p̂µ , x̂ν ] = ig µν has been used in the last step. So,
µ 2
∂ Ĥ
dx̂µ
dx̂
L = p̂µ µ − Ĥ = −Π̂2 − 2eAµ Πµ = −
,
− eAµ
∂p
2ds
ds
(33.122)
giving
hy|e−iĤs |xi =
Z
z(s)=y
z(0)=x
Dz(τ ) exp −i
Z
s
0
dz µ
dτ
2dτ
2
− ie
Z
Aµ (z)dz µ
!
, (33.123)
Effective actions and Schwinger proper time
726
with the integral over Aµ a line integral along the path z(s). So the Green’s function is
!
Z s µ 2
Z
Z z(s)=y
Z ∞
dz
µ
−ism2
dτ
Dz(τ ) exp −i
− ie Aµ (z)dz .
ds e
GA (x, y) =
2dτ
0
z(0)=x
0
(33.124)
This is an exact formal expression, only useful to the extent that we can solve for z(τ ).
This world-line formulation was derived by a different method by Feynman [Feynman,
1950], although it had little application for many years. Interest in this approach was
revived by Polyakov [Polyakov, 1981] in the context of string theory, and by Bern and
Kosower [Bern and Kosower, 1992] who used it to develop an efficient way to compute
loop diagrams in QCD.
33.6.1 Semi-classical limit
In the limit that a particle is very massive, loops involving that particle are suppressed.
Thus, it should be possible to treat a massive particle classically and the radiation it
produces quantum mechanically.
To take the large mass limit, we first rescale s → ms2 and τ → mτ2 . This gives
GA (x, y) =
1
m2
Z
∞
0
ds e−is
Z
z( ms2 )=y
z(0)=x
× exp −i
Z
Dz(τ )
"
s
dτ m
0
2
dz µ
2dτ
2 #
− ie
Z
Aµ(z)dz
µ
!
.
(33.125)
µ
2
Now we see that, for large m, the m2 ( dz
2dτ ) term completely dominates the path integral.
Moreover, as m → ∞, the action is dominated by the point of stationary phase, which is
also the classical free-particle solution:
z µ (τ ) = xµ + v µ τ,
µ
(33.126)
µ
where v µ = y −x
is the particle’s velocity. So we get, rescaling s → sm2 back again,
s
and plugging in the stationary phase solution,
Z s
Z ∞
(y − x)2
µ
2
dτ Aµ (z(τ ))
. (33.127)
+ ev
ds exp −i sm +
GA (x, y) =
4s
0
0
The first two terms in the exponent are independent of e and represent propagation of
a free particle, similar to Eq. (33.11). The next term is equivalent to adding a term to the
Lagrangian L = −eAµ Jcµ , where Jcµ is the source current from a classical massive particle
moving at constant velocity:
Jcµ(x) = v µ δ(x − vτ ).
(33.128)
In words, a heavy particle produces a gauge potential Aµ as if it is moving at a constant
velocity.
33.6 Semi-classical and non-relativistic limits
727
This is the semi-classical limit. When a particle is heavy, the quantum field theory can
be approximated by treating that particle as a classical source, but treating everything else
quantum mechanically. You can study the fermion case in Problem 33.4.
33.6.2 Non-relativistic limit
In the non-relativistic limit, not only is the particle’s mass assumed to be larger than the
energy of typical photons, but the particle’s velocity is also assumed to be much less than
the speed of light. Define ∆t = y 0 − x0 and ∆x = |~y − ~x|. A particle moving slowly from
xµ to y µ has ∆t ≫ ∆x.
Separating out the time component, the 2-point function in Eq. (33.124) becomes
GA (x, y) =
Z
∞
ds
0
Z
z(s)=y
Dz 0 (τ )D~z(τ )
z(0)=x
× exp −i
Z
s
dτ
0
"
dz 0
2dτ
2
−
d~z
2dτ
2
#
+ m2 − ie
Z
Aµ (z)dz µ
!
.
(33.129)
The classical path that minimizes the action, from the large m limit, has
z 0 (τ ) = x0 +
∆t
τ.
s
(33.130)
We want to treat this time evolution classically, and leave the rest of the field fluctuations
dz 0 2
) and m2 are large, the
quantum mechanical. However, we can see that since both ( 2dτ
∆t
∆t
stationary phase will have 2s ∼ m and so s ∼ 2m . That is, the integral is dominated by
∆t
the region near z 0 = x0 + 2mτ and s = 2m
. To leading order in the expansion of s and
0
z around their stationary-phase points, we then find
GA (x, y) =
Z
∆t
)=y
z( 2m
z(0)=x
D~z(τ ) exp i
Now we change variables to τ =
GA (x, y) =
Z
t
2m
∆t
2m
0
dτ
"
d~z
2dτ
2
− 2m
2
#
− ie
Z
Aµ (z)dz
µ
!
.
(33.131)
to find
z(∆t)=y
z(0)=x
Z
D~z(t) exp i
Z
∆t
0
#
!
2
Z
d~z
1
µ
− m − ie Aµ (z)dz
.
dt m
2
dt
"
(33.132)
This result is exactly the path integral expression in non-relativistic, first-quantized quantum mechanics with a potential V = m. We have just derived that the non-relativistic limit
of quantum field theory is quantum mechanics!
Effective actions and Schwinger proper time
728
33.A Schwinger’s method
In this appendix, we explicitly calculate the 1-loop effective action for constant background
electromagnetic fields Fµν using Schwinger’s original method [Schwinger, 1951]. This is
an alternative way to calculate the Euler–Heisenberg Lagrangian than the sum over Landau
levels method discussed in Section 33.4. This method, although a bit longer, is appealing
because it avoids having to regulate the system in a box. It also produces a general expression for the propagator GA (x, y) of a particle in a constant background electromagnetic
field.
Our starting point is the formula for the effective action in Eq. (33.38):
Z
i
i ∞ ds −ism2 h
1 2
e
Tr hx|e−iĤs |xi ,
(33.A.133)
Leff (x) = − Fµν (x) +
4
2 0 s
with Ĥ = −(p̂µ − eAµ (x̂))2 + 2e Fµν (x̂)σ µν . We have dropped the ε term, since we will
not need it with this method. Here Aµ (x̂) is to be thought of as a classical gauge field
configuration with position replaced by the operator x̂. We would like to calculate Leff (x)
when Fµν (x̂) = (∂µ Aν − ∂ν Aµ )(x̂) is constant. We begin by calculating hy|e−iĤs |xi.
Once this is known, we will set y = x and integrate over s to get Leff .
33.A.1 Proper-time propagation
States such as |xi are eigenstates of an operator x̂µ in a first-quantized Hilbert space. The
operators x̂µ are Schrödinger-picture operators. They are related to Heisenberg-picture
operators by x̂µ (s) = eiĤs x̂µ e−iĤs . Using the definition |x; si ≡ e−iĤs |xi we find
i∂s hy; 0|x; si = i∂s hy|e−iĤs |xi = hy|e−iĤs Ĥ|xi.
(33.A.134)
hy|e−iĤs x̂µ (s) = hy|x̂µ e−iĤs = y µ hy|e−iĤs ,
(33.A.135)
x̂µ (0)|x; 0i = x̂µ |x; 0i = xµ |x; 0i.
(33.A.136)
Now,
and
Thus, if we can write Ĥ in terms of x̂(0) and x̂(s) we can turn Eq. (33.A.134) into an
ordinary differential equation whose solution gives hy; 0|x; si.
In quantum mechanics, the position and momentum operators satisfy [x̂, p̂] = i. In our
4D first-quantized setup we generalize this to
[x̂µ (s), p̂ν (s)] = −ig µν ,
(33.A.137)
with the commutation applying at the same proper time s. To simplify the form of the
Hamiltonian, we introduce the operator Π̂µ = p̂µ − eAµ (x̂). Then, assuming Fµν is
constant, we get
h
i
x̂µ (s), Π̂ν (s) = −ig µν ,
(33.A.138)
33.A Schwinger’s method
[Π̂µ (s), Π̂ν (s)] = −ieF µν .
729
(33.A.139)
In terms of Π̂µ , the Hamiltonian is
e
/̂ 2 = −Π̂µ (s)Π̂µ (s) + Fµν σ µν .
(33.A.140)
Ĥ(s) = −Π
2
For simplicity, we will drop circumflexes on operators from now on. As a notational convenience, we will also replace µ and ν indices with boldface type. So the vectors xµ and Πµ
are written as x and Π, respectively, and the matrices F µν and σ µν are written as F and σ
respectively. Then tr(σF) = σνµ F µν = −σµν F µν , with tr(· · · ) referring to a trace over
µ and ν indices in this context.
In this notation, the evolution of Πµ (s) generated by the Hamiltonian H(s) through the
Heisenberg equations of motion becomes
dΠ
= i[Ĥ, Π] = 2eF · Π,
(33.A.141)
ds
where we have used that since F is constant it commutes with all operators, including Π.
This equation is solved by Π(s) = e2esF Π(0). Similarly,
dx
= i[Ĥ, x] = 2Π,
ds
(33.A.142)
which gives
sinh(esF)
· Π(0).
(33.A.143)
seF
This solution is easy to check by differentiating. In the limit A → 0, Π → p and this
becomes x(s) = x(0) + 2sp(0), which is consistent with the eigenstates of x(s) being
those which evolve into position xµ after a time s.
Thus we have
eF
Π(0) = e−esF
· [x(s) − x(0)] ,
(33.A.144)
2 sinh(esF)
eF
· [x(s) − x(0)] .
(33.A.145)
Π(s) = eesF
2 sinh(esF)
x(s) = x(0) + 2seesF
The Hamiltonian then becomes
e
e
Ĥ = −Π(s) · Π(s) − tr(σF) = − [x(s) − x(0)] K[x(s) − x(0)] − tr(σF),
2
2
(33.A.146)
2 2
e F
with K ≡ 4 sinh2 (eFs) . Note that Kµν = Kνµ .
To evaluate hy|e−iĤs Ĥ|xi in Eq. (33.A.134) using Ĥ, it is helpful first to rewrite Ĥ so
that x(s) is on the left and x(0) is on the right. This is not hard:
Π(s)·Π(s) = x(s)Kx(s)−2x(s)Kx(0)+x(0)Kx(0)+Kµν [xµ (s), xν (0)]. (33.A.147)
Now,
sinh(esF)
· Π(0)
Kµν [xµ (s), xν (0)] = −tr K x(0), x(0) + 2eesF
eF
i
= tr[eF + eF coth(esF)].
(33.A.148)
2
Effective actions and Schwinger proper time
730
So, since tr[F] = 0, we have
e
i
Ĥ = −x(s)Kx(s) + 2x(s)Kx(0) − x(0)Kx(0) − tr[eF coth(esF)] − tr(σF) .
2
2
(33.A.149)
In this canonical form, Ĥ can be evaluated in position eigenstates.
Equation (33.A.134) becomes
i∂shy; 0|x; si = − (y − x)
e2 F 2
(y − x)
4 sinh2 (esF)
e
i
+ tr[eF coth(esF)] + tr(σF) hy; 0|x; si,
2
2
(33.A.150)
where x = xµ and y = y µ are position vectors, not operators anymore. This is just a
differential equation. The general solution is
eF
coth(esF)(y − x)
hy; 0|x; si = C(x, y) exp i(y − x)
4
es
1
sinh(esF)
+ i tr(σF)
(33.A.151)
− tr ln
2
eF
2
This can be checked by differentiation and holds for any C(x, y).
To determine C(x, y), we use the additional information that
∂
i
− eA hy; 0|x; si = hy; 0|e−iĤs Π(0)|x; si
∂x
eF
= e−esF
(y − x)hy; 0|x; si,
2 sinh(esF)
and similarly
∂
eF
−i
− eA hy; 0|x; si = eesF
(y − x) hy; 0|x; si .
∂y
2 sinh(esF)
Plugging in our general solution, we find
∂
e
i
− eA − F(x − y) C(x, y) = 0,
∂x
2
and
e
∂
− eA − F(x − y) C(x, y) = 0.
−i
∂y
2
(33.A.152)
(33.A.153)
(33.A.154)
(33.A.155)
The solution is
Z
C(x, y) = C exp ie
y
x
1
dz µ Aµ (z) + Fµν (z ν − y ν ) .
2
(33.A.156)
This line integral is independent of path since the integrand has zero curl. The constant C
can be fixed by demanding that the result reduce to the free theory as A → 0. The final
result is
33.A Schwinger’s method
731
Z y
1
−i
µ
ν
ν
dz Aµ (z) + Fµν (z − y )
hy; 0|x; si =
exp ie
16π 2 s2
2
x
eF
es
1
sinh(esF)
× exp i(y − x)
,
coth(esF)(y − x) + i tr(σF) − tr ln
4
2
2
esF
(33.A.157)
which is manifestly gauge invariant. Taking A → 0 reproduces Eq. (33.11), which confirms
the normalization.
Equation (33.A.157) is more generally useful than just for the calculation of the Euler–
Heisenberg Lagrangian. The special case when x = y is quoted in Eq. (33.76) and used
for the calculation of the π 0 → γγ rate in Section 33.5.1.
33.A.2 Effective Lagrangian
Now that we have the proper-time Hamiltonian, we are a small step away from the Euler–
Heisenberg Lagrangian. We need to calculate
Z
o
i ∞ ds −ism2 n
1 2
(x) +
e
Tr hx|e−iĤs |xi
LEH (x) = − Fµν
4
2
s
Z ∞ 0
1 2
1
1
es
1
sinh(esF)
2
= − Fµν (x)+
,
ds 3 exp −ism + i tr(σF) − tr ln
Tr
4
32π 2
s
2
2
esF
0
(33.A.158)
where Tr is the Dirac trace and tr contracts µ and ν as above.
Now, recall from Eq. (30.65) that
2
e = 8(F − iγ5 G),
[tr(σF)] = −2tr F2 − 2iγ5 tr(FF)
(33.A.159)
where F̃ µν ≡ 12 εµναβ Fαβ and
1 2
1 ~2 ~2
Fµν = (B
− E ),
4
2
1
~ · B.
~
G ≡ − F µν F̃µν = E
4
F≡
(33.A.160)
(33.A.161)
Then, since γ5 has eigenvalues ±1, the Dirac eigenvalues of Tr(σF) are
p
λσF
= ± 8(F ± iG),
i
(33.A.162)
= 4Re cos[esX] ,
(33.A.163)
with all four sign combinations possible. So,
i
h p
i
h p
i
h es
Tr ei 2 tr(σF) = 2 cos es 2(F + iG) + 2 cos es 2(F − iG)
where
X≡
r
p
i
1 2
Fµν − F µν F̃µν = 2 (F + iG) =
2
2
q
~ + iE)
~ 2.
(B
(33.A.164)
Effective actions and Schwinger proper time
732
Next we need
p
1
sinh(eFs)
= ln λ1 λ2 λ3 λ4 ,
tr ln
2
esF
(33.A.165)
where λi are the four eigenvalues of sinh(eFs)
. These eigenvalues are determined from the
esF
eigenvalues of a constant Fµν , which are (see Problem 33.5)
i
√
i h√
√
λF
=
±
F
+
iG
±
F
−
iG
,
(33.A.166)
i
2
with all four possible sign choices. After some simplification the result is
(es)2 G
sinh(eFs)
1
=−
.
exp − tr ln
2
esF
Im cos(esX)
Putting everything together, we find
Z ∞
2 Re cos(esX)
1 2
e2
1
LEH (x) = − Fµν
+
F µν F̃µν ,
ds e−im s
4
32π 2 0
s
Im cos(esX)
(33.A.167)
(33.A.168)
which is the final answer for the unrenormalized Euler–Heisenberg effective Lagrangian,
in agreement with Eq. (33.71).
Problems
33.1 Complete the calculation of the Euler–Heisenberg Lagrangian using Landau levels
in an arbitrary Fµν . Show that for an electric field B → iE is justified. Also show
that the result for a general electromagnetic field is given by Eq. (33.71).
33.2 Calculate light-by-light scattering using helicity spinors.
33.3 Calculate the contour integral to derive the pair-production rate
R ∞Eq. (33.94) from
Eq. (33.93). It is helpful to first expand the integration limits to −∞ ds, then deform
the contour to pick up the poles.
33.4 Repeat the analysis in Section 33.6.1 for a fermion. Show that in the non-relativistic
limit, the spin is irrelevant.
33.5 Show that the eigenvalues of Fµν are given by Eq. (33.A.166).