University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Field Theory C (215C) Spring 2015
Assignment 3 – Solutions
Posted April 21, 2015
Due 11am, Thursday, April 30, 2015
Relevant reading: Zee, Chapter IV.3. Zee, Chapter III.3 also overlaps with recent lecture
material.
Problem Set 3
1. Propagator corrections in a solvable field theory.
Consider a theory of a scalar field in D dimensions with action
S = S0 + S1
where
Z
S0 =
dD x
and
1
∂µ φ∂ µ φ − m20 φ2
2
Z
1
dD x δm2 φ2 .
2
We have artificially decomposed the mass term into two parts. We will do perturbation
theory in small δm2 , treating S1 as an ‘interaction’ term. We wish to show that the
organization of perturbation theory that we’ve seen lecture will correctly reassemble
the mass term.
S1 = −
(a) Write down all the Feynman rules for this perturbation theory.
The propagator is the usual one, with amplitude p2 −mi 2 +i . The only ‘vertex’ is a
2-point vertex, which comes with amplitude −iδm2 .
(b) Determine the 1PI two-point function in this model, defined by
X
iΣ ≡
(all 1PI diagrams with two nubbins) .
The only feynman diagrams are concatenations of free propagators and the mass
insertion. The only 1PI diagram is the single mass insertion (with two nubbins
sticking off).
iΣ = −iδm2 .
1
(c) Show that the (geometric) summation of the propagator corrections correctly
produces the propagator that you would have used had we not split up m20 + δm2 .
The geometric series for the two-point function results in
1
G(p) = G0 (p) + G0 (p)iΣG0 (p) + G0 (p)iΣG0 (p)iΣG0 (p) + ... = G0 (p)
1 − iΣG0 (p)
!
1
i
= 2
2
p − m0 + i 1 − iΣ p2 −mi 2 +i
0
i
i
= 2
.
= 2
2
2
p − m0 + Σ + i
p − m0 − δm2 + i
2. Coleman-Weinberg potential.
(a) [Zee problem IV.3.4] What set of Feynman diagrams are summed by the ColemanWeinberg calculation?
[Hint: expand the logarithm as a series in V 00 /k 2 and associate a Feynman diagram
with each term.]
Recall that this calculation was the order-~ contribution to the saddle-point expansion of
Z
2
[Dφ]ei((∂φ) −V (φ)+Jφ)
about φ = φc + ϕ. They are diagrams with a single loop of the fluctuation ϕ, no
external ϕ legs, and arbitrary numbers of φc insertions. Then one perspective is to
use the Feynman rules for the expansion φ = φc + ϕ, thinking of the potential as
a perturbation. For simplicity consider the case where V (φ) = φp is a monomial.
The CW potential comes from just one-loop diagrams, so vertices for ϕ in the
expansion of V (φ = φc + ϕ) of order higher than two do not contribute – they
would turn the one ϕ running in the loop into more than one, and then we would
need an extra loop to get rid of it.
In the figure, we show the example with V (φ) = φ4 , so that the ‘interaction
vertex’ for the fluctuations ϕ, V 00 (φc ) is quadratic in φc . This is an interaction
vertex in the same sense as in the previous problem – it’s just an insertion in
the propagator. The one-loop diagram with n insertions of the propagator has a
2
cyclic symmetry which rotates the diagram like a clock; this group has order n
and so the answer is of the form
n
Z
1 V 00 (φc )
D
.
d¯ k
n k 2 + i
(Actually the diagram also has a reflection symmetry, since ϕ is real; this produces
an extra overall factor of 12 .) A little bit more explicitly, the propagators for the
fluctuations are
−i
hϕ(k)ϕ(−k)i = 2
k
00
and the 2-point vertex is −iV (φc ). The solid lines are ϕ propagators, and the
dotted lines represent insertions of φc .
(b) [Zee problem IV.3.3] Consider a massless fermion field coupled to a scalar field φ
by a coupling gφψ̄ψ in D = 1 + 1. Show that the one loop effective potential that
results from integrating out the fluctuations of the fermion has the form
2
φ
1
2
(gφ) log
VF =
2π
M2
after adding an appropriate counterterm.
In this problem, the terms in the boson action which do not involve the fermions
just go along for the ride. The fermion path integral is explained in Zee §IV.
Notice that the fermion mass is m(φ) ≡ m + gφ. The two important differences
from the boson case are:
• the determinant of the kinetic operator ends up in the numerator
Z
dψdψ̄ eiψ̄Dψ = det D = e+tr log D
R
• the trace over single particle states, which previously was tr... = d¯D khk|...|ki
now also involves a trace over the spin space:
Z
X
tr log D = d¯D k
hk|Dαα (k)|ki.
α
It is still diagonal in momentum space. Zee explains a trick for getting rid of
the gamma matrices, which I will not repeat. One reason we know they must go
away is that they have a vector index and we are computing a rotation-invariant
quantity which doesn’t depend on any vectors. This gives
2
Z
k + m(φ)2
D
.
VF = d¯ k log
k2
As in lecture, I added a (cutoff dependent) constant to eliminate the contribution to the vacuum energy (it also makes Mathematica happier about doing the
integral). We can regulate this integral by euclidean cutoff. That gives
2
Z Λ
kE + m(φ)2
D
Λ
VF =
d¯ kE log
kE2
3
Z Λ2
2π
u + m2
1
=
du
log
(2π)2 0 2
u
1
−m2 log m2 − Λ2 log Λ2 + (m2 + Λ2 ) log(m2 + Λ2
=
2π
(1)
Don’t forget that m = m(φ) = mψ + gφ here. Interesting effects occur if we set
mψ = 0 so that setting φ = 0 makes the fermion massless. We shouldn’t think
about values of φ that are close to the cutoff, so we may assume φ Λ in which
case the 2nd and 3rd terms in (??) may be expanded as
φΛ
−Λ2 log Λ2 + (m2 + Λ2 ) log(m2 + Λ2 ' m2 1 + log Λ2 .
Since m = gφ, to get a cutoff-independent effective action for φ, we must add a
counterterm for the boson mass, δL = B(Λ)φ2 . The problem would have been
better-posed had I specified what you can measure – i.e. if I had told you what
renormalization conditions to impose – rather than the answer, which determines
a family of renormalization conditions with hindsight. A good example of a reasonable RC which gives the stated answer is analogous to our quartic condition
in four dimensions:
!
00
Veff
(φc )|φc =M 2 = µ2M .
This demands that
µ2M = 2B(Λ) + VF00 (φ = M ) = 2B − 6 − 2g 2 log(g 2 M 2 ) + 2g 2 1 + log Λ2
so
1
B(Λ) =
2
g2M 2
2
2
µM + 6 + 2g log
.
Λ2
The important leftover φ dependent terms are
Veff (φ) =
φ2
1 2 2
g φ log 2 .
2π
M
3. 2PI generating functional.
In this problem, we will consider 0 + 0-dimensional φ4 theory for definiteness, but the
discussion also works with labels. Parametrize the integral as
Z ∞
g 4 1
1 2 2
2
Z[K] ≡
dq e− 2 m q − 4! q − 2 Kq ≡ e−W [K] ;
−∞
we will regard K as a source for the two-point function.
The 2PI generating functional is defined by the Legendre transform
Γ[G] ≡ W [K] − K∂K W [K]
with K = K(G) determined by G = ∂K W [K]. Here G is the 2-point Green’s function
G ≡ hq 2 iK .
4
(a) Show from the definition that Γ satisfies the Dyson equation
1
∂G Γ[G] = − K.
2
(b) Show that the perturbative expansion of Γ can be written as
A
Γ[G] =
1
1
log G−1 + m2 G + γ2P I + const.
2
2
(The A over the equal sign means that this is a perturbative statement.) Here
γ2P I is defined as everything beyond one-loop order in the perturbation expansion.
(c) Show that −γ2P I is the sum of connected two-particle-irreducible (2PI) vacuum
graphs, where 2PI means that the diagram cannot be disconnected by cutting any
two propagators.
(d) Define the 1PI self-energy in this Euclidean setting as
X
A
−Σ =
(1PI diagrams with two nubbins) .
Show that
∂γ2P I
∂K
by examining the diagrams on the BHS.
Taking a 2PI diagram and cutting a single line produces a 1PI diagram. Differentiating with respect to K produces a sum of ways of cutting a single line.
A
Σ=
(e) Show that the expression for G that results from the Dyson equation
G−1 = K + 2∂G γ2P I
agrees with our summation of 1PI diagrams.
(f) I may add another part to this problem which shows why 2PI effective actions
are useful.
See this paper and this paper.
5