University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Field Theory C (215C) Spring 2013
Assignment 6 – Solutions
Posted May 30, 2013
Due 11am, Thursday, June 6, 2013
Problem Set 6
1. Matching with massive electrons. [from Iain Stewart]
Consider QED in the regime where the photon momenta q µ are much smaller than
the electron mass me . In this regime, we can integrate out the electron and write an
effective field theory involving only the photon.
(a) Calculate the QED one-loop vacuum polarization diagram using dimensional reg2
ularization, in the MS scheme. Expand Π(q 2 ) through first order in mq 2 .
e
We did this calculation in lecture and found
2 µν
δΠµν
− q µ q µ δΠ2 (q 2 )
2 (q) = q g
with
2 MS
δΠ2 (q )
2α
=
π
Z
1
dx x(1 − x) log
0
m2 − x(1 − x)q 2
µ2
.
To facilitate the expansion in q 2 , write
q2
m2 − x(1 − x)q 2
m2
= log 2 + log 1 − x(1 − x) 2 .
log
µ2
µ
µ
P
un
Taylor expanding the log log(1 − u) = − ∞
n=1 n gives
4
α
m2
α q2
q
2 MS
δΠ2 (q ) =
log 2 −
+O
2
3π
µ
15π m
m4
since
Z
0
1
1
dx x(1 − x) = ,
6
Z
0
1
dx x2 (1 − x)2 =
1
.
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(b) Write down a Lagrangian involving only the photon field operator that reproduces the first two terms in the expansion. (Hint: it should be gauge invariant
and Lorentz invariant. There is essentially (up to integration by parts) only one
addition to the Maxwell term). Use the calculation above to match between the
1
photon-only EFT and QED at µ = me , at this order in the fine structure constant
α.
The leading term is the correction to the Maxwell term. The order-q 2 bit is
produced by an operator involving two gauge fields and two extra derivatives.
c2 1
1
δL = c1 Fµν F µν + 2 Fµν ∂ρ ∂ ρ F µν + ...
4
m 4
Other gauge-invariant dimension six operators made of two As either are related
by integration by parts ( ∂ρ Fµν ∂ ρ F µν ) or break parity ( (?F )µν ∂ρ ∂ ρ F µν ) and
are therefore not produced by QED, which is parity invariant. Matching to our
expression above gives:
c1 = δΠ2 (0), c2 = m2 ∂q2 δΠ2 (0).
(c) What symmetry of QED forbids dimension-6 operators involving three field strengths?
Charge conjugation invariance acts by Fµν → −Fµν , and j µ → −j µ , where j is the
charge current. In the absence of any external charges, it acts like (−1)n where
n is the number of external photons in the amplitude. This is called ‘Furry’s
theorem’.
(d) At dimension 8, we can write down operators in the photons-only EFT which
describe light-by-light scattering. Write them down (there are two). Draw the
QED Feynman diagram which matches to these terms, and determine the number
of factors of α in their coefficients. (Don’t do the integrals.)
They are going to involve four F s and therefore go like 1/m4 . A single loop of an
electron with four photon lines coming off (a box diagram) has four vertices, which
therefore goes like e4 ∼ α2 . We must decide what to do with the indices on the
F s. Parity demands that an even number of the F s must be ?F s (can be zero).
2
µν 2
~
~
This leaves two terms, which are (Fµν (?F ) ) ∼ E · B and (Fµν F µν )2 ∼
2
(E 2 − B 2 ) . For the actual matching coefficients, see Itzykson-Zuber.
(e) [bonus] Use dimensional analysis in the low-energy EFT to estimate the size of
the γγ → γγ cross section.
The amplitude goes like
p4
A ∼ α2 4
m
with one factor of p from each F in the interaction. So the cross section is
σ(p) ∼
1
α 4 p6
2
|A|
∼
.
p2
m8
It’s pretty small.
2. Right-handed neutrinos. [from Iain Stewart, and hep-ph/0210271]
2
Consider adding a right-handed singlet (under all gauge groups) neutrino NR to the
Standard Model. It may have a majorana mass M ; and it may have a coupling gν to
leptons, so that all the dimension ≤ 4 operators are
LN = N̄R i/
∂ NR −
M c
M
N̄R NR − N̄R NRc + gν N̄R HiT Lj ij + h.c.
2
2
T
where NRc = C N̄R
is the the charge conjugate field, C = iγ2 γ0 (in the Dirac
representation), H is the Higgs doublet, L is the left-handed lepton doublet, containing
νL and eL . Take the mass M to be large compared to the electroweak scale. Integrate
out the right-handed neutrinos at tree level. [Hint: you may find it useful to work in
terms of the Majorana field
N ≡ NR + NRc
which satisfies N = N c .]
Show that the leading term in the expansion in 1/M is a dimension-5 operator made of
Standard Model fields. Explain the consequences of this operator for neutrino physics,
assuming a vacuum expectation value for the Higgs field.
In terms of N , the lagrangian is
1
∂ − M ) N + gν N̄ Hi Lj ij + gν N̄ Hi? Lcj ij .
LN = N̄ (i/
2
The equation of motion for N (from varying N̄ ) is
(i/
∂ − M ) N = −gν Hi Lj + Hi? Lcj ij
which gives
1
1
gν (Hk L` + Hk? Lc` ) k` .
LN | = − gν L̄cj Hi + L̄j Hi? ij
2
i/
∂−M
As for our discussion of W -bosons, we expand this in powers of 1/M to get a local
effective field theory. The leading term is
O(5) =
gν2 c
L̄ Hi ij L` Hk k` + h.c.
M j
Plugging in hHi =
6 0, this is a neutrino mass.
Place a bound on M assuming that the observed neutrinos have masses mν < 0.5 eV.
2
c5 v
In terms of the parameterization from lecture, mν = 2Λ
. This gives Λnew ≥ 1014 GeV
new
for c5 ∼ 1. We find Λnew /c5 ∼ M , so M ≥ 1014 GeV.
3