Following a review of the dual description of the non-linear sigma model we
investigate the one-loop quadratic divergences. We use the covariant background
field method for the general case and apply the results to the important
example of $SU(2)$.Comment: 9 pages, LaTeX2e, no figures, uses axodraw package for Feynman
  diagrams. Axodraw available via anonymous ftp,
  ftp://ftp.nikhef.nl/pub/form/axodraw/export DVI version of preprint available
  via anonymous ftp or WWW;
  ftp://hep2.phys.soton.ac.uk/pub/preprints/rdsquad.dvi
  http://wwwhep.phys.soton.ac.uk/hepftp/pub/preprints/rdsquad.dv

Simmons, R. D.

English

arXiv

Following a review of the dual description of the non-linear sigma model we investigate the one-loop quadratic divergences. We use the covariant background field method for the general case and apply the results to the important example of SU(2)

Simmons, R D

CERN Document Server

arXiv:hep-th/9507120   22 Jul 1995
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(
2
)
1. Introduction
Following the recent dramatic work of Seiberg and Witten [1] in which a version of Olive-
Montonen duality [2] was found within the context of N=2 supersymmetry, there has been
strong motivation to investigate other possible dual models. One upshot of duality is the
inversion of various coupling terms which may appear in a Lagrangian with the eect that
in some sense the strong and weak coupling regimes of a theory become interchanged.
The non-linear sigma model has been known for some time to adequately describe the
dynamics of pions [3] (and mesons in general) at energies which are small compared with
the inverse connement radius of QCD. The non-linear sigma model is inherently versatile
and nds applications to a broad range of physics; in particular it has much to say about
the symmetry breaking sector of the Standard Model and in this context has been much
investigated by Dobado and Herrero [4] and Donoghue and Ramirez [5]. With the promise
of interchanging high and low energy regimes, a study of the dual description of the non-linear
sigma model could be particularly interesting phenomenologically.
Following the work of Buscher [6] and others [7] an algorithm exists for the construction
of the dual of the non-linear sigma model in any number of dimensions. The algorithm
consists of gauging a symmetry (an isometry) of the action by introducing non-propagating
gauge elds whose eld strength is constrained to vanish by means of a Lagrange multiplier.
Integrating over the gauge elds instead yields the dual theory where now the Lagrange
multiplier has been promoted to a full dynamical eld.
In an earlier paper [8] we applied this algorithm to a general four-dimensional non-linear
sigma model and found that the Lagrange multiplier was constrained to be a rank two
antisymmetric tensor eld, the dual model being composed of two real scalar elds (the
pions 

) and the rank two tensor eld which was interpreted as the dual eld associated
with 
3
. The interesting points of comparison between the original model and the dual model
are expected to appear in the renormalisation behaviour, and it is this which we now begin
to investigate by looking at the one-loop quadratic divergences. This should be regarded as
nothing more than just a rst step towards a more general study. In particular we shall have
nothing to say about the logarithmic contributions for various reasons which shall become
apparent as we proceed.
We begin by briey reviewing the dual model.
1
2. A Dual Description of the Non-Linear Sigma Model
in Four-Dimensional Spacetime
We take a target space whose coordinates, 
a
, we split as 
a
= (
i
; ) and, without loss of
generality, dene a four-dimensional non-linear sigma model on this space as
S (; ) =
Z
d
4
x

1
2
G () @

@

 +G
i
() @


i
@

 +
1
2
g
ij
() @


i
@


j

: (1)
This action is invariant under the global transformation  ! +  and the duality emerges
upon minimally gauging this global symmetry and adding a Lagrange multiplier term con-
straining the gauge eld to be pure gauge [7]. We therefore consider
S (; ) =
Z
d
4
x

1
2
G ()D

D

 +G
i
() @


i
D

 +
1
2
g
ij
() @


i
@


j
 
1
2




F


(2)
where D

 = @

 + A

and F

= @

A

  @

A

. 

is the rank two antisymmetric tensor
eld referred to in the introduction and appears here as a Lagrange multiplier enforcing the
pure gauge condition. As hinted earlier, the dual model appears upon integrating out the
gauge eld and promoting the Lagrange multiplier to be a fully dynamical gauge eld. After
performing this manipulation we obtain [8]
S (; ) =
Z
d
4
x

1
2
G
ij
@


i
@


j
 
1
2G


@





0

0

0
@

0


0

0
+
1
G


@



G
i
@


i

(3)
where G
ij
is given from the tensors appearing in (1) by
G
ij
= g
ij
 
1
G
G
i
G
j
: (4)
The dual action is therefore seen to be related to (1) via the interchange


@



! G
i
@


i
(5)
which is in the spirit of the duality encountered in electromagnetism between the electric
and magnetic elds. The apparent problem associated with the available degrees of freedom
(six) of the rank two antisymmetric eld is discussed in detail in [8] and [9] where the
gauge freedom is noted to allow just one physical degree of freedom - consistent with its
interpretation as the dual of a real scalar eld.
In [8] we studied some phenomenology of this model and found that it reproduced known
scattering amplitudes for the charged pions as derived from the original model. Here we
consider the more general problem of the divergent behaviour of the model.
2
3. One-Loop Quadratic Divergences of the Dual Model
Having arrived at the result (3) for the dual model with the explicit inversion of various
coupling terms, we now reparameterise the action such that all coupling terms are upstairs.
This slightly eccentric decision is made purely for aesthetic reasons to make the following
formulae easier to read.
We rewrite the Lagrangian (3) in the general form
L =
1
2
G
ij
@


i
@


j
+ 

G
i
@


i
@

A

 
1
2




0

0

0
G@

A

@

0
A

0

0
: (6)
To investigate the quadratic divergences we rst expand the action using the background
eld method. This manifestly covariant [10] procedure is a well known computational tool in
quantum eld theories which allows us to compute radiative corrections whilst maintaining
manifestly the symmetries of the theory under consideration [11]. The denition of a quan-
tum eld which translates as a vector, and the use of Riemann normal coordinates, leads to
an expansion in which each term depends only on the tensors on the manifold. This geomet-
ric property of the expansion implies a corresponding geometric property of the counterterms
obtained by calculating loop diagrams with the covariant background eld vertices.
Using the background eld method we expand the Lagrange density as
L[] = L[
cl
] + DLj

cl
+
1
2
DDLj

cl
+ ::: (7)
with the action of the covariant derivatives being given by [12]
D@


i
= r


i
Dr


i
= R
i
jkl
@


l

j

k
DG = r
i
G
i
DG
i
= r
j
G
i

j
Dr
k
r
j
G
i

k
: (8)
Here 
i
denote the quantum uctuations of the 
i
elds whilst for the rank two antisymmetric
tensor eld, A

, the quantum uctuations are denoted 

and the action of the covariant
derivatives is
D@

A

= @



(9)
Upon expanding the Lagrange density (6) and inserting the expressions (8) where appro-
priate, we arrive at a propagator term for the 
i
elds of the form
G
ij
r


i
r


j
: (10)
This can be manipulated into a useable form by moving to the tangent space via the following
denitions [13]

a
= e
a
i

i

i
= E
i
a

a
; (11)
3
allowing us to write
G
ij
r


i
r


j
! e
a
i
e
b
j

ab
r


i
r


j
= 
ab
r


a
r


b
(12)
where use has been made of the relations G
ij
= e
a
i
e
b
j

ab
and r

e
a
i
= 0. Having made the
shift to the tangent space our covariant derivative is now dened via the spin connection,
!
a
ib
,
r


a
= @


a
+ !
a
ib

b
@


i
: (13)
Applying this transformation to all 
i
terms we parameterise the Lagrangian for the quantum
elds as
L =
1
2
DDS =
1
2

ab
@


a
@


b
 
1
2




0

0

0
@

e


@

0
e


0

0
+ A
ab

a

b
+B

ab

a
@


b
+ C

a

a
@

e


+D

a

a
e


+ E

a
@


a
e


+ F

e


e


+G

e


@

e


(14)
where, to get the kinetic term for the 

elds into a convenient form, we have introduced
e


=
p
G

with the suitably covariantised derivative
@



!
1
p
G
@

e


 
1
2
1
G
p
G
r
m
G@


m
e


: (15)
Using this parameterisation we obtain
2A
ab
= @


i
@


j


cd
!
c
ia
!
d
jb
+R
jmki
E
m
a
E
k
b

+ 

@

A


r
k
r
j
G
i
E
j
a
E
k
b
+G
m
R
m
jki
E
j
a
E
k
b
+ 2r
j
G
m
!
c
ib
E
m
c
E
j
a

 
1
2




0

0

0
@

A

@

0
A

0

0
r
j
r
i
GE
i
a
E
j
b
B

ac
=


cb
!
b
ia
@


i
+ 

@

A

r
j
G
i
E
i
c
E
j
a

C

a
=
1
p
G



2r
[j
G
i]
+
1
2G
G
j
r
i
G

E
j
a
@


i
  

0

0

0
@

0
A

0

0
r
i
GE
i
a

D

a
=  
r
j
G@


j
2G
p
G


h
r
m
G
i
E
m
a
+G
m
!
d
ia
E
m
d

@


i
  

0

0

0
@

0
A

0

0
r
i
GE
i
a
i
E

a
=  
1
2
1
G
p
G


@


i
G
j
r
i
GE
j
a
F

0

0
=  
1
8G
2




0

0

0
@


i
@

0

j
r
i
Gr
j
G
G

0

0

=
1
2G




0

0

0
@

0

i
r
i
G : (16)
4
Using cut-o regularisation at the scale , the Feynman diagrams which contribute  
2
divergences are (dashed lines = , coiled lines = ):
F
G G
C C C E E E
A
B B
Evaluating these diagrams we nally obtain the one-loop eective Lagrangian
L
e
=
1
2
@


i
@


j
G
R
ij
+ 

@

A

@


i
G
R
i
 
1
2




0

0

0
@

A

@

0
A

0

0
G
R
(17)
with the renormalised tensors
G
R
ij
= G
ij
+ J

R
ij
 
6
G
G
mn
r
[n
G
i]
r
[m
G
j]

G
R
i
= G
i
 
J
2

G
mn
r
n
r
m
G
i
 G
j
R
ji
 
3
G
G
mn
r
m
Gr
[n
G
i]

G
R
= G +
J
2

G
mn

3
G
r
m
G r
m

r
n
G r
j
G
i
r
m
G
n

G
mj
G
in
 G
jn
G
im


(18)
and
J =
Z

0
d
4
k
1
k
2
=  
1
16
2

2
(19)
where we have Wick rotated and d
4
k =
1
2
k
2
sin
2
 sindk
2
dd is the integration measure in
Euclidean space.
5
This is our main result and represents the general one-loop corrected eective Lagrangian
for the Abelian dualed sigma model.
1
Having derived the dual version of the non-linear
sigma model and the very general renormalisation to one-loop, we now specialise to the
important case of SU(2).
4. Application to SU(2)
We take as our starting point the non-linear sigma model parameterised by the matrix eld
U(x) belonging to the quotient space SU(2)
L

 SU(2)
R
=SU(2)
L+R
,
U(x) = exp (i:=) ; (20)
where 
a
, a = 1; 2; 3, are the Goldstone bosons associated with the symmetry breaking
SU(2)
L

 SU(2)
R
! SU(2)
L+R
, 
a
are the 2  2 Pauli matrices and  some energy scale.
(When applied to the symmetry breaking sector of the Standard Model  is xed at the
scale of the Higgs VEV = 246GeV).
Under an SU(2)
L

 SU(2)
R
transformation the matrix U transforms as U ! LUR
y
and
the SU(2) invariant sigma model can be written
L =

2
4
Tr@

U@

U
 1
: (21)
To obtain the dual version of this model we must rst massage (21) into the form (1) which
we achieve by separating out one eld - the eld which shall carry the Abelian symmetry.
When this is done (see [8] for denitions) we recover the form of (1) with
g
ij
= Æ
ij
sin
2




2
+
1


2

2
"
1 
sin
2




2
#

i

j
G
i
= 
ij
1


j
sin
2




2
G = 1: (22)
Once in this form we use the rules given in (3) and (4) to write down the dual verison of the
SU(2) non-linear sigma model to O(p
2
)
L =  
1
2


@





0

0

0
@

0


0

0
+
1



@




ij

j
@


i
sin
2




2
+
1
2
@


i
@


i
sin
2




2

1   sin
2



+
1
2
1

2


2
"
1  
sin
2




2
+
sin
4




2
#
(:@

)
2
;
(23)
1
We are only interested here in divergences  
2
and ignore the logarithmic contributions which can
only be sensibly interpreted in the full momentum expansion of which the sigma model just described is the
O(p
2
) leading term.
6
where in the language of (6) we have
G
ij
= Æ
ij
sin
2




2
h
1  sin
2


i
+

i

j

2


2
"
1 
sin
2




2
+
sin
4




2
#
G
i
= 
ij
1


j
sin
2




2
G = 1: (24)
We can now immediately use (17) to investigate the  
2
divergences appearing at the
one-loop level in the dual SU(2) non-linear sigma model. Given the relations in (24) and
inserting into (18) we arrive at the surprisingly compact expression
L =
1
2
@


i
@


j
G
ij
(1   2J) 
1
2




0

0

0
@



@

0


0

0
(1   2J)
+ 

@



@


i
G
i
: (25)
This is the one-loop corrected eectiveSU(2) Lagrangian for the Abelian dualed sigmamodel
and can be compared with the results presented in [14] for the conventional non-linear sigma
model.
5. Conclusions
We have presented a completely general discussion of the quadratic renormalisation be-
haviour of the dual non-linear sigma model at the one loop level and applied it to the
important phenomenological case of SU(2). This is the rst step in a general study of the
divergent behaviour of the dual non-linear sigma model where we hope to nd a formalism
which will facilitate a study of the infrared behaviour (xed points) which in itself will enable
a detailed comparison between this model and the standard non-linear sigma model.
The next stage in the programme will be to extend the investigations to accommodate
the logarithmic divergences. Here there is a considerable increase in the number of Feynman
diagrams which we need to consider to one loop ( 20) and in addition the infrared singu-
larities which emerge as the rank two antisymmetric tensor elds go soft in certain diagrams
demand careful attention. It is hoped to pursue this line of investigation in a subsequent
paper.
Acknowledgements
The author gratefully acknowledges N. Mohammedi for his guidance throughout the duration
of this study, and R.T. Moss who checked some of the results presented here. This work was
supported in part by PPARC.
7
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8


One-Loop Quadratic Divergences of the Dual Non-Linear Sigma Model in Four-Dimensional Spacetime

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