of the two-dimensional world will allow a ... where. We use the following conventions. (3) goo = 1. E 01 = 1 d2x = dx'dx' ... a1 . ..n) f j dxl... dxn exp(ixipi) S(l...n).
SIX-PUB-2407 September 1979 (T/E)
CONNECTION BETWEEN NEUTRINO AND GAUGE INVARIANCE: * A TWO-DIMENSIONAL MODEL **
R. Ferrari Stanford Linear Accelerator Center Stanford University, Stanford, California 94305
ABSTRACT We discuss electrons
a two-dimensional
and neutrinos.
for a local
conservation
quantum electrodynamics. tree
approximation
of the current
Moreover,
we show that for this
(Submitted
*
The absence of a neutrino law,
similar
We discuss
associated
between
mass allows
to gauge invariance the peculiar
gives wrong results
equation
responsible
model of interaction
with
for
fact
in that
the
the conservation
gauge transformations.
the anomaly of the axial
current
is
property.
to Nuovo Cimento Letters.)
Work supported in part by the Department of Energy under contract number DE-AC03-76SF00515. *src Supported by a NATO Fellowship. On leave of absence from Istituto di Fisica, Universitg di Pisa, Italy and CERN, Geneva, Switzerland.
I -2-
The existence questions, that
particularly
at least
to render
in nature
from the theoretical
one neutrino
observed parity
violating
important
symmetry or field
fact
the neutrino
this
will
allow
to associate
compatible
property
some local
between group invariance
and a neutrino. a state
The fact requirement
with
the Poincar; no other
is associated
with
the
is massless.
problem in a two-dimensional
electron
of view.
However, to our knowledge,
theoretical
absence of the mass of the neutrino, connection
rouses some puzzling
the minimal
processes
laws. .t
of the physical
We would like
point
is massless is just
invariance
that
of massless fermions
conservation
similar
to what happens to the
and photon.
We shall
model of interaction
The peculiarity
of neutrino-antineutrino
law to the
investigate
between an
of the two-dimensional to play the role
world
of
gauge-particle. In the tree approximation diagram rules
derived
the theory
from the Lagrangian
is given by the Feynman f density
(1)
where
We use the following goo = 1
conventions E01 = 1
(3) d2x = dx'dx'
d2)(x>
= 6(x0> 6(x1)
-3-
The regularization
and the renormalization
such a way to exert
local
procedure
will
be elected
in
gauge invariance
$J(x> -f
exp i(ACx>
4(x)
exp
- y5X(x)
) $(x)
(4) +
s,,v a?(x)
where 3,x(x)= a gauge-invariant
in order
+
and consequently
jp(x)
of the fields
and gauge-invariant
procedure,
e.g.,
split-point
for
if
we define
J in
should transform
in the definition
of J in (2) can be made
regularization
(l),
subtraction
On the other
side,
the
way to enforce
(5) for j:
is some renormalization
5 the most singular
the free propagator,
.JY,$
factor.
2):
dk+ j dk-
c Perm.
+
(14)
X
-
By using
q1
4::
...
2 ql+is
qi+i.s
(14) one can easily
show that
.
(15)
From (15) and from
. ..n)
= 0
follows
that
(1
one obtains
(3) that
for all
L?(l . ..n)
(17) From the lemma just function point
proved,
= 0 it
is given only by the chain-graphs,
function
propagator invariant j-two
momenta
point
(10) connects
in (10))
function.
where the one-loop
the one-particle-irreducible
does not contribute
Thus (11) is valid
Green j-two-
(including
Since J is built
blobs of .
way, the electron
the j-two-point
the
in a gauge-
to the divergence
for any number of loops.
of the
-7-
A further
result
can be derived
a simple use of the Ward identity
shows that
Z3 in (6) is also the wave function By power counting
neutrino.
of fields,
constant
is the disconnected
constant
part
vanishes
of the limit
the
any product
B
