E) CONNECTION BETWEEN

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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