University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Field Theory C (215C) Spring 2013
Assignment 5 – Solutions
Posted May 21, 2013
Due 11am, Thursday, May 30, 2013
Problem Set 5
1. All possible terms.
Perturb the gaussian fixed point of the XY model in D > 3 by a term
Z
δS6 = dD x g6 (x)(φ? φ)3
where g6 is a short-ranged coupling function. Only if you find new relevant perturbations (this does not include terms that may be absorbed in changes in the bare values
of perturbations that we have already studied) must you keep track of the numerical
coefficients.
What would you find if you included also δS2n with n > 3?
The point of this problem is to convince ourselves that we haven’t missed any marginal
or relevant operators by keeping only the quadratic and quartic perturbations of the
gaussian fixed point.
The basic point is that nothing the quantum mechanics does in perturbation theory
can overcome the scaling from the engineering dimensions. With scaling of the field
defined to keep the k 2 φ2k term fixed, we have
Z
6
−D
6 D−2
δS6 = s 2 s g6 dD x̃ φ̃? φ̃
so we must have
g̃6 = s2D−6 g6
which is not close to marginal for D ≥ 4. In D = 3 it is close and maybe we should
worry about it at the Wilson-Fisher fixed point.
To give a little bit more detail: again we can proceed by cumulant expansion, the
leading term of which is
Z
hδS6 i>,0 =
h(φ< + φ> )3 (φ< + φ> )?3 i
|k|<Λ
which is 64 terms. Of these, 1 has only external slow lines and gives directly the
tree-level self-interaction of the slow modes. One has only external fast lines and gives
1
a contribution to the cosmological constant. Many (32) vanish because they have an
odd number of external fast lines. The remaining diagrams renormalize the quartic
and quadratic amplitudes. The renormalization of the quartic term involves just one
contraction of fast modes and has the same form as the δr from δu.
2. (Upper) critical dimension. [from Kaplan, nucl-ph/0510023]
Define the ‘critical dimension’ dc for an operator to be the spacetime dimension for
which that operator is marginal. How will that operator behave in dimensions d > dc
and d < dc ? In a theory of interacting relativistic scalars, Dirac fermions, and gauge
bosons, determine the critical dimension for the following operators:
(a) A gauge coupling to either a boson or a fermion through the covariant derivative
in the kinetic term (‘minimal coupling’).
L = ψ̄ (iD
/ − m) ψ −
1 2
F
4e2
with
D
/ = γ µ (∂µ + iAµ ) ,
F = dA
tells us that
1 = [∂] = [A]
(I am using mass dimensions) and
D = [L] = −2[e] + 2[∂A] = −2[e] + 4 =⇒ [e] = −
D−4
.
2
The critical dimension is D = 4.
(b) A Yukawa interaction, φψ̄ψ.
L = ψ̄ i∂/ − m ψ + g ψ̄ψφ + (∂φ)2 + m2 φ2
D = [L] = 2[ψ] + [∂] =⇒ [ψ] =
[φ] =
D−1
.
2
D−2
2
D = [L] = [g] + 2[ψ] + φ =⇒ [g] = D − (D − 2) −
D−1
D−1
=−
+ 2.
2
2
Again D = 4.
(c) An anomalous magnetic moment coupling for a fermion g ψ̄σ µν Fµν ψ (where σ µν ≡
[γ µ , γ ν ] is the generator of spin rotations).
This one is a little tricky because our counting depends whether we normalize the
gauge by the Maxwell term or by the covariant derivative (i.e. whether we absorb
a factor of e into A) I will do the latter.
D−1
D−1
, [F ] = 2 =⇒ [g] = D − 2
− 2 = −1
2
2
It seems that this interaction is never marginal!
[ψ] =
2
(d) A four-fermion interaction ψ̄ψ
2
L = ψ̄ (i∂/ − m) ψ + g ψ̄ψ
2
D−1
D−1
=⇒ [g] = D − 4
= 2 − 2D
2
2
The critical dimension is 2.
[ψ] =
3. Non-relativistic QFT and the Schrödinger equation. [from Kaplan, nucl-ph/0510023]
We’ve discussed in lecture a Lagrangian for a non-relativistic field theory in d spatial
dimensions:
Z
∇2
g
2
d
?
?
S = d ~xdt ψ (~x, t) i∂t −
ψ(~x, t) −
(ψ (~x, t)ψ(~x, t))
.
2m
8m2
Translate this into ‘first-quantized’, single-particle language: that is, write down the
Hamiltonian H acting on the sector of the Hilbert space where there are two particles.
This sector is closed because the particle number is conserved.
Spoilers: show that this interaction is equivalent to a δ d (~x1 − ~x2 ) interaction potential
between two particles in d spatial dimensions. Relate the strength of the potential to
g by matching the Born approximation (=tree level) amplitudes.
What does the RG scaling of the interaction (it can be relevant, marginal or irrelevant,
depending on d) mean for the solution of the Schrödinger problem? What is the critical
dimension for the δ d (~r) potential?
R
This field theory has a conserved particle number, dd xψ † ψ. And the non-relativistic
field cannot create antiparticles. So we may work in a sector with a fixed number of
particles and the dynamics will never force us to leave it. For example, we may think
of the sector with two particles.
The Hamiltonian is
Z
H=
1 ~
~ + g ρ(x)2
∇π · ∇ψ
d ~x π −
2m
8m2
d
2
where the canonical field momentum is π = ∂∂L
= iψ ? , and the particle density is
tψ
ρ(x) = iπ(x)ψ(x). In the sector with two particles, this evaluates to
H=−
~2
~2
∇
∇
g d
1
− 2 +
δ (~x1 − ~x2 ) .
2m 2m 8m2
This is the problem we studied in the first problem set. The critical dimension is 2.
4. 1, 2, many.
By integrating momentum shells, derive the RG equations for the quartic and quadratic
operators of the relativistic O(N ) vector model in D dimensions,
Z
1
1 a a
a
D
a µ a
a a 2
SE [φ ] = d x
∂µ φ ∂ φ + rφ φ + u (φ φ )
.
2
2
3
Here a = 1..N .
This is mostly in my lecture notes. The only step which is not explicit is making the
RG equation for u infinitesimal. We found
2
2
Z Λ
1
1
1 2
1 2
s=e`
D
D
d¯ q
δu0 = − u0 (8N + 64)
= − u0 (8N + 64)KD Λ
2
r0 + r2 q 2
2
r0 + r2 Λ2
Λ/s
with KD ≡
ΩD−1
.
(2π)D
ũ0 ≡ u0 + `
du0
+ ... = s−3D ζ 4 (u0 + δu0 ) = s−D+4 (u0 + δu0 )
d`
So
du0
1
= (4 − D) u0 − u20 (8N + 64) KD ΛD
d`
2
4
1
r0 + r2 Λ2
2
+ O u30