We derive a systematic procedure to compute Green functions for the massive
Schwinger model via a perturbation expansion in the fermion mass. The known
exact solution of the massless Schwinger model is used as a starting point. We
compute the corrections to the Schwinger mass up to second order.Comment: Latex, 7 pages, no figure

't Hooft

Adam

Casher

Christoph Adam

Coleman

Crewther

Dittrich

Fröhlich

Ilieva

Jayewardena

Kogut

Leutwyler

Lowenstein

Sachs

Saravi

Schwinger

Smilga

Steele

Vary

Physics Letters B

English

arXiv

Crossref

The Schwinger mass in the massive Schwinger model

We derive a systematic procedure to compute Green functions for the massive Schwinger model via a perturbation expansion in the fermion mass. The known exact solution of the massless Schwinger model is used as a starting point. We compute the corrections to the Schwinger mass up to second order

Adam, C

CERN Document Server

BUTP-95/27
The Schwinger mass in the massive Schwinger model
Christoph Adam
Institut fur theoretische Physik, Universitat Bern
Sidlerstrae 5, CH-3012 Bern, Switzerland

)
July 19, 1995
Abstract
We derive a systematic procedure to compute Green functions for the massive Schwinger
model via a perturbation expansion in the fermion mass. The known exact solution of the
massless Schwinger model is used as a starting point.
We compute the corrections to the Schwinger mass up to second order.

)permanent address: Institut fur theoretische Physik, Universitat Wien
Boltzmanngasse 5, 1090 Wien, Austria
email address: adam@pap.univie.ac.at
11 Introduction
The massless Schwinger model is wellknown to be equivalent to a theory of a free massive
boson with Schwinger mass 
2
0
=
e
2

([1]). This massive eld is formed via the chiral anomaly
and may be interpreted as a fermion{antifermion bound state ([6], [5], [10], [9], [19]). Besides, the
massless Schwinger model shows other nontrivial features like fermion condensate, instantons
and nontrivial vacuum structure ( vacuum) ([2]{[5], [7]{[9], [11]{[13], [23]). In the massless
Schwinger model physical quantities do not depend on the vacuum angle .
The massive Schwinger model is dierent in some repects. First, it is no longer exactly
solvable ([13]{[16], [10], [9]). The physical state { the massive boson { is no longer free, and its
mass acquires corrections due to the interaction. Instanton{like gauge elds and a nontrivial
vacuum structure persist to be present, and, in addition, physical quantities now depend on
the vacuum angle . The fermion condensate, too, acquires corrections due to the fermion mass
([21]).
Here we show how to compute Green functions for the massive Schwinger model within the
Euclidean path integral formalism, using a perturbation expansion in the fermion mass. (The
existence and niteness of the mass perturbation theory for the massive Schwinger model was
proven in [17]). For this purpose we use the known exact solution of the massless Schwinger
model as a starting point. From the perturbation expansion we explicitly calculate corrections
to the Schwinger mass up to second orcer in the fermion mass m, for a general vacuum angle .
2 Exact solution of the massless case
The vacuum functional (and Green functions) of the massive Schwinger model may be inferred
from n{point functions of the massless Schwinger model via an expansion in the fermion mass.
Indeed, we may write for the Euclidean vacuum functional (k : : : instanton number)
Z(m; ) =
1
X
k= 1
e
ik
Z
k
(m) (1)
where
Z
k
(m) = N
Z
D

	D	DA

k
e
R
dx
h

	(i@= eA=
k
+m)	 
1
4
F

F

i
= N
Z
D

	D	D
k
1
X
n=0
m
n
n!
n
Y
i=1
Z
dx
i

	(x
i
)	(x
i
)
 expf
Z
dx
h

	(i@=  



@


k
)	 +
1
2

k
2
2

k
i
g (2)
(A

= 

@

 corresponding to Lorentz gauge). Therefore the perturbative computation of
Z(m; ) is traced back to the computation of scalar VEVs ( h
Q
i
S(x
i
)i
0
, S(x) 

	(x)	(x) )
for the massless Schwinger model and some space time integrations. It is useful to rewrite the
scalar densities in terms of chiral ones, S(x) = S
+
(x) + S
 
(x), S



	P

	, because then only
a denite instanton sector k = n
+
 n
 
contributes to the VEV h
Q
n
+
i=1
S
+
(x
i
)
Q
n
 
j=1
S
 
(x
j
)i
0
. A
general VEV may be computed exactly (see e.g. [7]{[9], [22], [21]),
hS
H
1
(x
1
)   S
H
n
(x
n
)i
0
=


2

n
exp
h
X
i<j
( )

i

j
4D

0
(x
i
  x
j
)
i
(3)
2where 
i
= 1 for H
i
= , D

0
is the massive scalar propagator,
D

0
(x) =  
1
2
K
0
(
0
jxj);
~
D

0
(p) =
 1
p
2
+ 
2
0
; (4)
(K
0
: : :McDonald function) and  is the fermion condensate of the massless Schwinger model,
 = h

		i
0
=
e

2

0
(5)
( : : : Euler constant). The index 0 for 
0
indicates that it is the order zero result, the index
0 for the VEVs means that they are with respect to the massless Schwinger model. From this
Z(m; ) may be computed (see [21] for details, [24] for its physical implications),
Z(m; ) = e
V (m;)
;
(m; ) = m

2
2 cos +m
2


2

2
(E cos 2 + F ) + o(m
3
) (6)
(V : : : space time volume) where
E =
Z
d
2
xE(x) 
Z
d
2
x(e
 2K
0
(
0
jxj)
  1) =  8:9139 
1

2
0
F =
Z
d
2
xF (x) 
Z
d
2
x(e
2K
0
(
0
jxj)
  1) = 9:7384 
1

2
0
(7)
and for F a subtraction of a free eld singularity is necessary.
In order to compute VEVs for the massive Schwinger model one has to insert the corre-
sponding operators into the path integral (1), (2) and divide by the vacuum functional Z(m; ):
h
^
Oi
m
=
1
Z(m; )
h
^
O
1
X
n=0
m
n
n!
n
Y
i=1
Z
dx
i

	(x
i
)	(x
i
)i
0
(8)
Via the normalization all volume factors cancel completely, as it certainly has to be (we will
explicitly see this in the computations). For the computation of VEVs of scalar (S) and chiral
(S

) densities formula (3) suces and could be used e.g. for the computation of the fermion
condensate h

		i
m
(this however may be derived at once from (6), see [21]).
For the computation of the Schwinger mass additional Green functions of the massless
Schwinger model are needed. It is wellknown that the vector current correlator is the free
massive propagator in the massless Schwinger model:
hJ

(x)J

(y)i
0
=
1



0
@

0
x


0
@

0
y
D

0
(x  y) (9)
Therefore, for a perturbative calculation we need the n{point functions
hJ

(y
2
)J

(y
1
)
n
Y
i=1
S
H
i
(x
i
)i
0
=
1



0
@

0
x


0
@

0
y




2

n
h
D

0
(y
1
  y
2
) + 4(
n
X
i=1
( )

i
D

0
(x
i
  y
2
))(
n
X
j=1
( )

j
D

0
(x
j
  y
1
))
i

e
4
P
k<l
( )

k

l
D

0
(x
k
 x
l
)
(10)
which may be computed from the Euclidean path integral or from bosonization (the latter
method is easier).
33 Computation of the Schwinger mass
For the perturbative computation of the massive propagator we simply have to insert successive
orders of equ. (10) into (8). The factor (
1



0
@

0
y
2


0
@

0
y
1
) is the same for all orders, therefore we
will ignore it in the sequel.
There are two terms from (10) to be inserted in rst order, namely S
H
; H = , and four
terms, S
H
1
S
H
2
; H
i
= , in second order. For the moment we consider the case where allH
i
= +;
the other contributions can be inferred from this one by a rearrangement of some signs. Up to
second order, we nd ( hJ

(y
2
)J

(y
1
)i
m

1



0
@

0
y
2


0
@

0
y
1
T (y
1
; y
2
) )
T (y
1
; y
2
) =
1
Z(m; )
h
D

0
(y
1
  y
2
) +m

2
VD

0
(y
1
  y
2
) + 4m

2
Z
dxD

0
(x  y
2
)D

0
(x  y
1
)+
m
2
2!


2

2
Z
dx
1
dx
2
[D

0
(y
1
  y
2
) + 4(D

0
(x
1
  y
2
) +D

0
(x
2
  y
2
))
(D

0
(x
1
  y
1
) +D

0
(x
2
  y
1
))]e
4D

0
(x
1
 x
2
)
i
(11)
Inserting equ. (6) for the vacuum functional Z(m; ) and using the perturbation formula of
second order,
a
0
+ a
1
x+ a
2
x
2
+ o(x
3
)
1 + b
1
x+ b
2
x
2
+ o(x
3
)
=
a
0
+ a
1
x+ a
2
x
2
  a
0
b
1
x  a
0
b
2
x
2
  a
1
b
1
x
2
+ a
0
b
2
1
x
2
+ o(x
3
) (12)
we arrive at
T (y
1
; y
2
) = D

0
(y
1
  y
2
) + 4m

2
Z
dxD

0
(x  y
2
)D

0
(x  y
1
)+
m
2
2!


2

2
4
Z
dx
1
dx
2
[2D

0
(x
1
  y
2
)D

0
(x
1
  y
1
) + 2D

0
(x
1
  y
1
)D

0
(x
2
  y
2
)]
(E(x
1
  x
2
) + 1)  4m
2


2

2
V
Z
dxD

0
(x  y
2
)D

0
(x  y
1
)
= D

0
(y
1
  y
2
) + 4m

2
Z
dxD

0
(x  y
2
)D

0
(x  y
1
)+
m
2


2

2
4E
Z
dxD

0
(x  y
2
)D

0
(x  y
1
)+
m
2


2

2
4
Z
dx
1
dx
2
D

0
(x
1
  y
1
)D

0
(x
2
  y
2
)(E(x
1
  x
2
) + 1) (13)
where E;E(x) are given in (7) and we used the x !  x symmetry of all occurring functions.
Observe that, as claimed, all volume factors V have dropped out.
However, expression (13) is not yet the desired result for further computation, there is still
one unwanted term. The last line of expression (13) consists of two terms, due to the factor
(E(x
1
  x
2
) + 1). The second one, proportional to 1, reads
m
2


2

2
4
Z
dx
1
D

0
(x
1
  y
1
)
Z
dx
2
D

0
(x
2
  y
2
) (14)
4and obviously does not contribute to the y
1
 y
2
correlation. It is a disconnected part stemming
from the VEVs
hJ

(y
i
)S
+
(x
j
)i
0
 

0
@

0
y
i
D

0
(x
j
  y
i
) (15)
and must be subtracted from (13).
Looking for the moment at the rst order correction only, it is very easy to nd the Schwinger
mass correction:
2
y
1
T
(1)
(y
1
; y
2
) = (y
1
  y
2
) + 
2
0
D

0
(y
1
  y
2
)+
4m

2
(D

0
(y
1
  y
2
) + 
2
0
Z
dxD

0
(x  y
2
)D

0
(x  y
1
))
 (y
1
  y
2
) + 
2
1
D

1
(y
1
  y
2
) + o(m
2
) (16)
where

2
1
= 
2
0
+ 4m

2
(17)
To obtain the second order result, we rewrite expression (13), without the disconnected part,
in momentum space and substitute all functions by their Fourier transforms (thereby the con-
volutions turn into products):
~
T (p) =
 1
p
2
+ 
2
0
+ 4m

2
1
(p
2
+ 
2
0
)
2
+ 4m
2


2

2
1
(p
2
+ 
2
0
)
2
(E +
~
E(p)) =
 1
p
2
+ 
2
0
(1  4m

2
1
p
2
+ 
2
0
  4m
2


2

2
(E +
~
E(p))
1
p
2
+ 
2
0
) =
 1
p
2
+ 
2
0
+ 4m

2
+ 4m
2
(

2
)
2
(E +
~
E(p)) + (4m

2
)
2
1
p
2
+
2
0
+ o(m
3
) (18)
Therefore, for nding the mass pole, p
2
has to obey the self consistency equation (after a
rescaling p
0
2
=
p
2

2
0
, E
0
= E(
2
0
 1) = 
2
0
E etc.)
p
0
2
=  1  4
m

0

2
0
  4
m
2

2
0


2
0

2
[E
0
+
~
E
0
(p
0
) +
4
p
0
2
+ 1
] (19)
The second order part (the term in square brackets) may be rewritten like
[  ] =
Z
d
2
x[e
 2K
0
(jxj)
  1 + e
ip
0
x
(e
 2K
0
(jxj)
  1 + 2K
0
(jxj)] =
Z
1
0
drr[2(e
 2K
0
(r)
  1) +
Z
2
0
de
ijp
0
jr cos 
(e
 2K
0
(r)
  1 + 2K
0
(r))] =
2
Z
1
0
drr[e
 2K
0
(r)
  1 + J
0
(jp
0
jr)(e
 2K
0
(r)
  1 + 2K
0
(r))] (20)
where J
0
is the Bessel function of the rst kind. This expression behaves well around jp
0
j = i
and therefore we may set jp
0
j = i because deviations from this value are of higher order in m.
Using I
0
(r) = J
0
(ir) we nd
p
0
2
=  1   4
m

0

2
0
  8
2
m
2

2
0


2
0

2
A (21)
5A :=
Z
1
0
drr[e
 2K
0
(r)
  1 + I
0
(r)(e
 2K
0
(r)
  1 + 2K
0
(r))]
A =  0:6599 (22)
This result was computed for the positive chirality densities S
+
. For the pure negative chirality
densities S
 
the result is completely identical. However in second order there are mixed products
S
+
S
 
; S
 
S
+
, too. Both of them lead to the same result, where the above integralA is substituted
by a similar expression B,
B :=
Z
1
0
drr[e
+2K
0
(r)
  1 + I
0
(r)( e
+2K
0
(r)
+ 1 + 2K
0
(r))]
B = 1:7277 (23)
In this expression the nice feature of cancellation of UV divergencies occurs. Indeed, both e
2K
0
(r)
and  I
0
(r)e
2K
0
(r)
diverge like
1
r
2
for small r (this divergency corresponds to the free fermion
eld divergency of the underlying theory), but obviously the divergencies cancel each other.
In fact, this cancellation was already observed twenty years ago in [13] within a bosonization
approach, and on general grounds it should continue to hold for higher orders.
Collecting all results and multiplying each contribution with its corresponding  factor
(S

! e
i
, see (1{3)), we nd for the Schwinger mass in second order
 p
0
2


2
2

2
0
= 1 + 8
m

0

2
0
cos  + 16
2
m
2

2
0


2
0

2
(A cos 2 + B) (24)
or, inserting all numbers (remember

2
0
=
e

4
, equ. (5))

2
2
= 
2
0
(1 + 3:5621 
m

0
cos  + 5:4807 
m
2

2
0
  2:0933 
m
2

2
0
cos 2) (25)
which is our nal result.
For the special case  = 0 our result (24) precisely coincides with the result in [25], where
the second order correction for  = 0 was computed within bosonization and using near light
cone coordinates. In the same article this result was compared with a lattice calculation ([26]),
and a good agreement is obtained within the range of the expansion parameter
m

0
where the
lattice calculations were performed.
4 Summary
We have demonstrated a general method of computing n{point functions for the massive
Schwinger model in mass perturbation theory within the Euclidean path integral formalism,
using the known exact solution of the massless Schwinger model as a starting point.
All n{point functions exist perturbatively and are nite in the innite volume limit.
Using this approach we were able to compute the mass perturbation corrections to the
Schwinger mass up to second order, and almost all calculations could be done analytically. This
feature does not persist to hold for higher orders. Already in third order an equation analogous
to (19) will be a true self consistency equation that can be solved only numerically.
For  = 0, the result (24) is well reproduced by lattice calculations ([25]).
The numerical calculations in this article were done with Mathematica 2.2.
6Acknowledgements
The author thanks H. Leutwyler for very helpful discussions and the members of the Institute
of Theoretical Physics at Bern University, where this work was done, for their hospitality.
This work was supported by a research stipendium of the University of Vienna.
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The Schwinger Mass in the Massive Schwinger Model

The Schwinger Mass in the Massive Schwinger Model

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