We study the nonlinear $O(N)$ sigma model on $S^2$ with the gravitational
coupling term, by evaluating the effective potential in the large-$N$ limit. It
is shown that there is a critical curvature $R_c$ of $S^2$ for any positive
gravitational coupling constant $\xi$, and the dynamical mass generation takes
place only when $R<R_c$. The critical curvature is analytically found as a
function of $\xi$ $(>0)$, which leads us to define a function looking like a
natural generalization of Euler-Mascheroni constant.Comment: 7 pages, LaTe

Kim, K. H.

Song, Dae-Yup

English

arXiv

We study the nonlinear O(N) sigma model on S^2 with the gravitational coupling term, by evaluating the effective potential in the large-N limit. It is shown that there is a critical curvature R_c of S^2 for any positive gravitational coupling constant \xi, and the dynamical mass generation takes place only when R0), which leads us to define a function looking like a natural generalization of Euler-Mascheroni constant

Kim, K H

Song, D Y

CERN Document Server

Title: Critical curvature of large-N nonlinear O(N) sigma
model on S
2
Short title: Critical curvature on S
2
K.H. Kim and Dae-Yup Song
1
Department of Physics, Sunchon National University,
Sunchon, Chonnam 540-742, Korea
Abstract
We study the nonlinear O(N) sigma model on S
2
with the gravitational coupling
term, by evaluating the eective potential in the large-N limit. It is shown that
there is a critical curvature R
c
of S
2
for any positive gravitational coupling constant
, and the dynamical mass generation takes place only when R < R
c
. The critical
curvature is analytically found as a function of  (> 0), which leads us to dene a
function looking like a natural generalization of Euler-Mascheroni constant.
PACS numbers: 04.62.+v, 11.15.Pg
1
Electronic address: dsong@sunchon.sunchon.ac.kr
1
1 Introduction
For a model described by boson elds n
i
(n = 1; 2;    ; N) on a curved spacetime, a
gravitational interaction term 
i
Rn
i
n
i
may be added to the Lagrangian [1], where 
i
is a gravitational coupling constant and R is the scalar curvature of the spacetime (
In this paper, we consider only Euclidean spacetime. ). Since this interaction term
is a bilinear term of the elds, the change of mass through quantum loop corrections
may be of interest.
Recently one of us [2] studied the three dimensional nonlinear O(N) sigma model
on S
2
R
1
in the leading order of 1=N expansion, where S
2
is the two sphere with
the canonical metric. For the O(N) symmetry we assume that 
i
is the same  for all
i. On the three dimensional at spacetime, there exists a critical coupling constant
of the model and the dynamical mass generation takes place only when the coupling
constant is larger than the critical one (strong coupling regime) [3]. In the strong
coupling regime, we showed that there is a critical curvature (radius) of S
2
for some
value of . Though the O(N) model can be treated by the 1=N expansion in both
two and three dimensions, the properties of the models are substantially dierent.
One of them is that the dynamical mass generation takes place in two dimension no
matter how small the coupling constant is, as it should be.
In this paper, we will study the two-dimensional O(N) model [4] on S
2
with the
gravitational coupling term in the large-N limit, by evaluating the eective potential
[5]. We will show that there is a critical curvature R
c
() for every  (> 0). As 
approaches 0, the R
c
goes to innity and decreases monotonically as  increases.
By introducing a function which looks like a functional generalization of Euler-
Mascheroni constant, the R
c
can be neatly expressed by the function.
2
2 Eective potential and critical curvature
We assume that the metric of S
2
is the canonical one so that the curvature R is
2=
2
, where  is the radius of S
2
. Then the Lagrangian of our model is written as:
L =
Z
dd' 
2
sin  + (
2
sin
2
)
 1
@
'
n
i
@
'
n
i
+Rn
2
+ (n
2
 Ng
 2
) ]; (1)
where g
0
is the bare coupling constant. For the evaluation of the eective potential
we can write the Lagrangian density as
L = n
i
Dn
i
 N=g
2
0
: (2)
In (2) D is dened as 
 2
(L
2
+ 2) + , where L is the quantum mechanical an-
gular momentum operator. By performing the Gaussian functional integral [6], the
eective potential per unit volume is given as
V
N
=  

g
2
0
+
1
4
2
(Tr lnD + C); (3)
where C is a constant which may be determined by requiring V j
=0
= 0. As is
well-known, in the leading order of 1=N expansion the eective potential in (3) is
written in terms of spacetime independent :
V
N
=  

g
0
+
1
4
2
I
X
l=0
(2l + 1) ln(1 +

l(l+1)

2
+
2

2
); (4)
where we introduce an integer cut-o I for the quantum number of the operator L.
In order to compare eective potential in (4) with that on R
2
, as in the previous
paper [2], one can use the following formula:
N
X
l=0
f(l) =
1
2
f(0) +
Z
N+1
0
f(x)dx+
N
X
l=0
Z
1
0
f
0
(x+ l)(x 
1
2
)dx 
1
2
f(N + 1);
which gives
V
N
=  

g
2
0
+
1
2
Z
I
0
(x+
1
2
) ln(1 +

2
(x+
1
2
)
2
+ 2  
1
4
)dx +
1
8
2
ln(1 +

2
2
)
+
1
2
2
I
X
l=0
Z
1
0
dx(x 
1
2
)
2
6
4
ln(1 +

2
(x+l+
1
2
)
2
+2 
1
4
)
+
2(x+l+
1
2
)
2
(x+l+
1
2
)
2
+2 
1
4
+
2
 
2(x+l+
1
2
)
2
(x+l+
1
2
)
2
+2 
1
4
3
7
5
: (5)
3
Since the integrals in (5) are well dened without infra-red problem for  > 0, we
will restrict our attention to these cases. By dening a cut-o  = (I +
1
2
)= whose
dimension is that of a momentum, we can write the potential as
V
N
=  

g
2
0
+
1
2
Z

1=2
y ln(1 +

y
2
+
2 
1
4

2
)dy +
1
8
2
ln(1 +

2
2
)
+
1
2
2
1
X
l=1
Z
1
2
 
1
2
dt t
2
4
ln(1 +

2
(t+l)
2
+2 
1
4
)
+
2(t+l)
2
(t+l)
2
+2 
1
4
+
2
 
2(t+l)
2
(t+l)
2
+2 
1
4
3
5
+O(1=)
=  

g
2
0
 

4
(ln
2 + 
2

2

2
  1) +
1  4
8
2
ln(1 +

2
2
)
+
1
2
2
1
X
l=1
Z
1
2
 
1
2
dt t
2
4
ln(1 +

2
(t+l)
2
+2 
1
4
)
+
2(t+l)
2
(t+l)
2
+2 
1
4
+
2
 
2(t+l)
2
(t+l)
2
+2 
1
4
3
5
+O(1=): (6)
The eective potential in (6) once again conrms the well-known fact that topolog-
ical change of spacetime does not give rise to new counterterms, and thus we can
use the renormalization relation of the model on R
2
[5]
 
1
g
2
0
=  
1
g
2
+
1
4
ln
M
2

2
; (7)
where M is the renormalization mass. It is convenient to dene 
0

0
= M
2
e
 4=g
2
(8)
which denotes the square of the dynamically generated mass on R
2
. Making use of
the relations (7) and (8), one can nd the renormalized eective potential
V
N
=  

4
(ln
2 + 
2

0

2
  1) +
1   4
8
2
ln(1 +

2
2
)
+
1
2
2
1
X
l=1
Z 1
2
 
1
2
dt t
2
4
ln(1 +

2
(t+l)
2
+2 
1
4
)
+
2(t+l)
2
(t+l)
2
+2 
1
4
+
2
 
2(t+l)
2
(t+l)
2
+2 
1
4
3
5
: (9)
To have a better understanding of the shape of V in (9), we evaluate the rst
derivative of V with respect to :
1
N
@V
@

V
0
()
N
=  
1
4
ln
2 + 
2

0

2
+
1
8
1
2 + 
2
+
1
2
1
X
l=1
Z 1
2
 
1
2
dt t
 (t+ l)
2
+ 2  
1
4
+ 
2
[(t+ l)
2
+ 2  
1
4
+ 
2
]
2
: (10)
4
As shown in the appendix in detail, one can nd a simpler form of the rst derivative
in (10):
1
N
@V
@
=
1
4
[ln(
0

2
) +
1
2 + 
2
+ 2(2 + 
2
 
1
4
)]; (11)
where the function  is dened by
() = lim
N!1
[
N
X
n=1
n+
1
2
(n+
1
2
)
2
+ 
 
1
2
ln(N
2
+ )]: (12)
The  function looks like a generalization of Euler-Mascheroni constant

E
= lim
N!1
[
N
X
n=1
n
 1
  lnN ] = 0:5772    :
In fact, making use of the formulae e.g. in [7], one can show that (0) = 
E
 2 2 ln 2.
Furthermore, it is easy to show that () ( >  
1
4
) is a monotonically decreasing
function of , and it approaches to  1 as  goes to 1.
Now one can nd that, for a xed , V
0
() monotonically decreases to  1 as
 increases. That is, a global stationary point of the eective potential V which
denotes the dynamical mass generation appears only when V
0
(0) > 0. Therefore,
the critical radius of S
2
is given by

0

2
c
= exp( 
1
2
  2(2  
1
4
)): (13)
Or, the critical curvature is given by
R
c
= 2
0
exp(
1
2
+ 2(2  
1
4
)): (14)
The critical curvature which is1 in the limit  ! 0 decreases monotonically to 0 as
 increases to 1, and the dynamical mass generation takes place only when R < R
c
(or  > 
c
).
3 Discussion
By evaluating the eective potential (9) in the leading order of 1=N expansion, we
have shown that, the O(N) nonlinear  model with the gravitational coupling term
5
described by the Lagrangian (1), has a critical curvature ( given in (13,14) ) which
decreases as  increases.
For the conformal symmetry,  must be 
c
=
1
4
[(d 2)=(d 1)] in a d-dimensional
spacetime. That is, the conformal coupling constant 
c
is 0 in two dimension and
1
8
in three dimension. One of the common features of the O(N) nonlinear  model on
S
2
and S
2
R
1
[2] is that the dynamical mass generation takes place for any nite
R in the limit  ! 
c
while there exists a critical curvature for  > 
c
.
Acknowledgments
One of us (D.Y.S) would like to thank Jeong Hyeong Park for the discussion on the
 function. This work is supported in part by the Korea Research Foundation.
Appendix
Making use of the identity
Z
t[ (t+ l)
2
+ ]
[(t+ l)
2
+ ]
2
dt =  
l(t+ l) + 
(t+ l)
2
+ 
 
1
2
ln[(t+ l)
2
+ ]
for (t+ l)
2
+  > 0; (15)
one can nd the following equality which could be used to nd (11) and (12) from
(10);
1
X
l=1
Z
1
2
 
1
2
t[ (t+ l)
2
+ x 
1
4
]
[(t+ l)
2
+ x 
1
4
]
2
dt
=
1
4x
+
1
2
ln x+ lim
N!1
f
N 1
X
n=1
n+
1
2
(n+
1
2
)
2
+ x 
1
4
 
1
2
ln[(N +
1
2
)
2
+ x 
1
4
]g:(16)
6
References
[1] Birrell N D and Davies P C W 1982 Quantum Fields in Curved space (Cam-
bridge:Cambridge Univ. Press)
[2] Song D Y 1995 Sunchon Nat'l Univ. preprint, hep-th/9505036
[3] Aref'eva 1979 Ann. Phys. (N.Y.) 117 393
[4] Polyakov A M 1975 Phys. Lett. 59B 79
Berezin E and Zinn-Justin J 1976 Pys. Rev. B 14 3110
[5] Colman S 1985 Aspects of Symmetry (Cambridge:Cambridge Univ. Press)
[6] Jackiw R 1974 Phys. Rev. D 9 1686
[7] Whittacker E T and Watson G N 1927 A Course of Modern Analysis (Cam-
bridge:Cambridge Univ. Press)
7


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