Problem Set 4
Phy 342 - Fall 2008
Assigned: Wednesday, Nov. 24
Due: Friday Dec. 12, Noon
Problem 1
Consider the chiral Lagrangian for SUL (3) × SUR (3):
L=
f2
Bf 2
Tr[∂µ Σ† ∂ µ Σ] +
Tr[m†q Σ + mq Σ† ] .
8
4
The matrix of Goldstone bosons, Σ, is
Σ = e2 iM/f
where
√
√

+
+
π 0 / 2 + η/ 6
π
K
√
√


M =
K 0√  ,
π−
−π 0 / 2 + η/ 6
K−
K̄ 0
−2η/ 6

mq = diag(mu , md , ms ) and f = fπ = 130 MeV.
a) Take the isospin limit, mu = md , and calculate the masses of π, K, and η in terms of
the quark masses and the parameter B.
b) Verify the Gell-Mann-Okubo relation: 3m2η = 4m2K − m2π .
c) Calculate m̂/ms , where m̂ = (mu +md )/2, in terms mπ and mK and obtain a numerical
estimate using mπ = 138 MeV and mK = 496 MeV.
Problem 2
µ
a) For QCD with two flavors of quarks, u and d, construct the Noether current, J3A
, for
the symmetry transformation:
δu = iγ5 α u
δd = −iγ5 α d .
(1)
µ
b) Calculate the coefficient in the J3A
anomaly
µ
∂µ J3A
= −C
e2
Fµν F̃ µν .
8π 2
c) Now consider a low energy effective theory of Goldstone bosons, π 0 and π ± . Show
that under the transformation in Eq. (1) π ± are invariant while π 0 transform by a shift
proportional to α. Hint: the effective theory is expressed in terms of fields
!
√
π +√
π0/ 2
2 iM/fπ
.
where
M=
Σ=e
π − −π 0 / 2
Under SUL (2) × SUR (2), Σ → L Σ R† . What L and R correspond to Eq. (1)?
d) Because the Noether current corresponding to Eq. (1) is nonconserved, the action must
vary as
δS =
Z
µ
d4 x α ∂µ J3A
,
under the transformation generated by Eq. (1). Use this and your result from part c) to
argue that there must be a term in the effective Lagrangian of the form
L = g π 0 Fµν F̃ µν ,
(2)
and determine g. Calculate the decay rate π 0 → γγ.
Problem 3
Show that baryon number (quark number divided by three), B, and lepton number, L,
are classically conserved but are not symmetries of the Standard Model due to anomalies.
Right down the anomaly equations for the baryon and lepton number currents. Show that
B − L has no anomaly.
Problem 4
If QCD has a nonvanishing θ term, its CP violating effects on mesons can be calculated using the Lagrangian in Problem 1. The trick is to note that under the U(1) chiral
transformations
qL i → eiαi qLi
qR i → e−iαi qRi ,
(3)
where i = u, d, s is the quark flavor, the θ parameter transforms as θ → θ − 2 i=u,d,s αi , so
the θ term can be rotated away in exchange for a complex mass matrix. The complex mass
matrix then enters the chiral Lagrangian in Eq. (1) and induces CP violating effects.
a) Show that the transformation in Eq. (3) modifies the quark mass matrix as
P
diag(mu , md , m,s ) → diag(e−2iαu mu , e−2iαd md , e−2iαs ms )
≈ diag(mu , md , m,s ) − 2i diag(mu αu , md αd , ms αs )
≡ diag(mu , md , m,s ) − iδmq .
Here we have expanded in the αi becasue we expect θ to be small.
b) Show that the modification to the chiral lagrangian in Eq. (1) is
L0 = −i
Bf 2
Tr[δmq Σ − δmq Σ† ]
4
(4)
c) Clearly we want 2 i=u,d,s αi = θ to rotate away the θ term. The remaining freedom
in choosing the αi is removed by requiring
P
δmq = λ11 ,
where 11 is the identity matrix. Why do we choose to do this? (Hint: when are there terms
linear in Goldstone bosons in Eq. (4)?) Find λ.
d) Use L0 to calculate the decay rate for η → ππ. Explain why this violates CP.