Download 1 Axial Vector Current Anomaly in Electrodynamics By regularizing

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Axial Vector Current Anomaly in Electrodynamics
By regularizing potential infrared and gauge divergences in the Schwinger model, and examining the normal
modes of the fermion and photon fields, we have seen that the axial vector current is not conserved:
e
∂µj5µ = µν Fµν .
2π
Fluctuations in the vector potential can create a fermion anti-fermion pair with opposite chiralities out of
the vacuum. An adiabatic change ∆a1 = 2π/(eL) leaves the fermion spectrum unchanged.
Unlike the Fermi sea in a non-relativistic solid-state system, the relativistic Dirac sea has an infinite number
of filled energy levels. This ultraviolet divergence allows for a change in N+ − N−, which would not be
possible if the number of energy levels were finite.
The axial vector current in Quantum Electrodynamics with massless photons and massive electrons in 4-d
is conserved at the classical level in the limit of zero electron mass. This Noether symmetry is broken when
the theory is quantized, which results in the famous Adler-Bell-Jackiw anomaly:
e2 µνκλ
Fµν Fκλ .
=−
16π 2
It is not possible to eliminate all of the gauge degrees of freedom except for one mode because real photons
have an infinite number of transverse degrees of freedom. Different methods must be used to study the
anomaly in d = 4.
∂µj5µ
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2 OPERATOR EQUATIONS AND CURRENT CONSERVATION
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Operator Equations and Current Conservation
The equations of motion for the fermion field operators can be derived from the Lagrange density
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L = − Fµν F µν − ψ̄γ µ(i∂µ − eAµ)ψ − mψ̄ψ ,
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where m is the mass of the fermion, by formally using the classical Euler-Lagrange equations of motion.
Given a Lagrangian L(q, q̇, t), the Euler-Lagrange equations for a classical degree of freedom q are
∂L
d ∂L
−
=0.
∂q dt ∂ q̇
For a classical field with Lagrange density L(φ, ∂µφ, xµ), these equations generalize to
∂L
∂L
− ∂µ
=0.
∂φ
∂µφ
The fermion fields ψ and ψ̄ = ψ †γ 0 are independent degrees of freedom. The Euler-Lagrange equation for
ψ̄ gives
∂L
= γ µ(i∂µ − eAµ)ψ − mψ = 0 ,
∂ ψ̄
which is Dirac’s equation for the field ψ. Using (γ µ)† = γ 0γ µγ 0, the equation for ψ̄ is
(i∂µ + eAµ)ψ̄γ µ + mψ̄ = 0 .
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2 OPERATOR EQUATIONS AND CURRENT CONSERVATION
The field equations can be used to compute the divergence of the vector and axial vector currents
j5µ = ψ̄γ µγ5ψ .
j µ = ψ̄γ µψ ,
The divergence of the vector current is
∂µj µ = ∂µ(ψ̄γ µψ) = (∂µψ̄)γ µψ + ψ̄γ µ(∂µψ) = ieAµψ̄γ µψ + imψ̄ψ − ieψ̄γ µAµψ − imψ̄ψ = 0 .
The divergence of the axial vector current is
∂µj5µ =
=
=
=
∂µ(ψ̄γ µγ5ψ)
(∂µψ̄)γ µγ5ψ − ψ̄γ5γ µ(∂µψ)
ieAµψ̄γ µγ5ψ + imψ̄γ5ψ + ieψ̄γ5γ µAµψ + imψ̄γ5ψ
2imψ̄γ5ψ ,
where the anticommutation relation {γ µ, γ5} = 0 has been used.
The formal operator equations predict that the vector current is strictly conserved. The divergence of the
axial vector current is proportional to the fermion mass
∂µj5µ = 2im ψ̄ψ ,
and the current is therefore conserved in the case of massless fermions.
The vector current is conserved in massless QED, but the axial vector current is not! What went wrong
with this derivation?
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3 ULTRAVIOLET DIVERGENCES OF OPERATOR PRODUCTS
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Ultraviolet Divergences of Operator Products
The origin of this problem lies in the singular nature of quantum fields.
The types of classical fields that are useful in physics are generally smoothly varying continuous functions.
If a classical function develops a singularity, it is usually isolated and physical quantities are not singular.
Quantum fields can be much more singular than classical fields because the Uncertaintly Principle allows a
very wide range of quantum fluctuations to take place. These fluctuations could not occur in the classical
limit without violating conservation of energy or other conservation laws.
Consider an expansion of a classical field in normal modes
Z
φ(x) = dk ak eikx .
The mode amplitudes ak will tend rapidly to zero for large |k| if the field is smoothly varying. The field is
a normal function and products of the field and its derivatives are well defined.
The situation is very different for quantum field opeators. The quantity ak in the mode expansion is now an
operator that creates or destroys normal mode quanta. The number of quanta that can be created is not
restricted by classical conservation laws. In principle, a state can exist for a tiny instant of time in which all
of the modes are excited. The field operator then behaves like
Z
φ(x) ∼ dk eikx = 2πδ(x) .
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4 POINT SPLITTING REGLARIZATION
Mathematically, a quantum field must be considered to be a generalized function or distribution, like the
Dirac delta function and its derivatives.
Distributions and their linear combinations are generally well defined. Physical quantites are computed by
integrating them with smooth functions, which smear their singular behavior.
A severe problem arises when generalized functions are combined nonlinearly. In the example above
Z ∞
dk ,
|φ(x)|2 ∼ δ(x)δ(x) = δ(x)
−∞
which is ultraviolet divergent and cannot be smeared with any smooth function.
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Point Splitting Reglarization
Products of field operators at different points in space-time are generally less singular than local products at
the same space-time point. The point-splitting regularization method replaces local products with separated
products. The limit of zero separation is taken at the end of the calculation.
Separating the points in a local product can break formal symmetries satisfied by the original product. The
most important of these is Poincaré invariance. The separation of any pair of space-time points is a new
and arbitrary vector zµ which had no place in the original theory.
Another very important symmetry is CPT which interchanges particles and antiparticles.
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5 GAUGE INVARIANCE AND WILSON LINES
As a first attempt at point-splitting, consider
z µ
z
→ ψ̄ x +
γ γ5ψ x −
.
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Note that ψ̄ and ψ are treated as symmetrically as possible by splitting their arguments symmetrically in
opposite directions.
j5µ(x)
To avoid any spurious dependence on the direction in space-time of z µ, the limit z → 0 must be taken such
that tensor products of z depend only on the invariant metric tensor η µν . Vectors in z µ must vanish
lim z µ = 0 ,
z→0
and second order tensors tend to the limit
z µz ν η µν
lim
=
.
z→0 z 2
d
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Gauge Invariance and Wilson Lines
Point-splitting spoils the local gauge invariance of operator products. Under the local gauge rotation
ψ(x) → eiα(x)ψ(x) ,
the split operator product changes by an z-dependent phase:
z µ
z
z µ
z i[α(x−z/2)−α(x+z/2)]
ψ̄ x +
γ γ5ψ x −
→ ψ̄ x +
γ γ5 ψ x −
e
.
2
2
2
2
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5 GAUGE INVARIANCE AND WILSON LINES
Wilson invented a simple way of restoring local gauge invariance by inserting a “Wilson line” operator
between ψ̄ and ψ:
"
#
Z x+z/2
exp −ie
dx0λAλ(x0) .
x−z/2
Here, the integral in the exponent is the line integral along any continuous path between the two points.
Under the local gauge transformation
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Aλ(x0) → Aλ(x0) − ∂λ0 α(x0) ,
e
the line integral in the exponent gets multiplied by the phase factor
ei[α(x+z/2)−α(x−z/2)] ,
which exactly cancels the phase difference between ψ and ψ̄.
The operator
j5µ(x)
→ ψ̄ x +
z
2
"
Z
µ
x+z/2
γ γ5 exp −ie
x−z/2
is now ultraviolet finite and gauge invariant.
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#
z
dx Aλ(x ) ψ x −
,
2
0λ
0
6 VACUUM STATE AND NORMAL ORDERING
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Vacuum State and Normal Ordering
A fundamental pathology of field theories with relativistic fermions is the continuum of negative energy
states. To quantize such theories, the fermion field operator is expanded in normal modes
XZ
1 1
d−1
−ik·x
†
ik·x
√
d k
bσ (k)uσ (k)e
+ dσ (k)vσ (k)e
,
ψ(x) =
(2π)d−1
2E
k
1
σ=± 2
where the σ is a spin eigenvalue and u and v are positive and negative energy spinors, respectively, and are
normalized so that
X
X
uσ (k)ūσ (k) =
vσ (k)v̄σ (k) = γ µkµ .
σ
σ
The operator b annihilates fermion modes, and the operator d† creates antifermion modes. The field operator
ψ̄ creates fermions and annihilates antifermions.
Dirac’s prescription of filling the negative energy sea is implemented in quantum field theory by defining
operators bilinear in the fermion fields to have zero expectation value in the vacuum state, which is defined
as the state for which
b|0i = d|0i = 0 = h0|b† = h0|d† .
Therefore, bilinear operators like j5µ must be “normal ordered” with creation operators to the left of annihilation operators.
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7 DIVERGENCE OF THE AXIAL VECTOR CURRENT
The anti-commutation relations must be taken into account in the normal-ordering process:
{bσ (k), b†σ0 (k 0)} = {dσ (k), d†σ0 (k 0)} = δσ,σ0 (2π)d−1(2Ek )δ d−1(k − k 0) ,
and all other anti-commutators vanishing.
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Divergence of the Axial Vector Current
The divergence of the axial vector current in the Schwinger model can now be evaluated using the Dirac
equation for the fermion fields
∂µψ̄(x + z/2) = ieψ̄(x + z/2)Aµ(x + z/2) ,
∂µψ(x − z/2) = −ieAµ(x − z/2)ψ(x − z/2) ,
and by expanding Aµ(x ± z/2) and the Wilson line contributions in powers of z
zλ
Aµ(x ± z/2) = Aµ(x) ± ∂λAµ(x) + O(z 2) ,
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"
#
Z
x+z/2
dx0λAλ(x0) = 1 − iez λAλ(x) + O(z 2) ,
exp −ie
x−z/2
"
Z
#
x+z/2
dx0λAλ(x0) = −iez λ∂µAλ(x) + O(z 2) .
∂µ exp −ie
x−z/2
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7 DIVERGENCE OF THE AXIAL VECTOR CURRENT
The sum of the three contributions from ψ̄, ψ and the Wilson line is
∂µj5µ → −ieψ̄(x + z/2)γ µγ5ψ(x − z/2)z λ(∂µAλ − ∂λAµ) + O(z 2) .
Before taking the limit z → 0, the fermion operators must be normal ordered. Using the normal mode
expansions for the fermion fields, it is straightforward to show that
Z
i
i µν zν
ν µ
2 kν ik·z
h0|ψ̄(x + z/2)γ µγ5ψ(x − z/2)|0i =
Tr[γ
γ
γ
]
d
k
e
=
.
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(2π)2
k2
π z2
Taking the limit of the regularized and normal-ordered divergence
e
∂µj5µ − h0|∂µj5µ|0i = µν Fµν .
2π
This derivation of the anomaly can easily be generalized to massless QED in 4-d. The operator product is
more singular than in 2-d and the calculation of its vaccum expectation value is more complicated. The
final result is
e2 µνκλ
µ
µ
∂µj5 − h0|∂µj5 |0i = −
Fµν Fκλ .
16π 2
Homework Problem
Fill in the missing steps in the derivation of the Schwinger model anomaly in the last section.
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