Phase structure of the generalized two dimensional Yang-Mills theories on sphere

14 Pages·0.14 MB·English

by  M. Alimohammadi

Phase structure of the generalized two dimensional
Yang-Mills theories on sphere
M. Alimohammadia1, A. Tofighi b,c 2
a Department of Physics, University of Tehran, North-Kargar Ave.,
9 Tehran, Iran
9
9 b Department of Physics, University of Mazandaran, Babolsar, Iran
1
c Institute for Studies in Theoretical Physics and Mathematics,

n
a P.O.Box 19395-5746, Tehran, Iran
J

1
3

2
v
4
0
0
7
0 Abstract
8
9
Wefindageneralexpressionforthefreeenergy ofG(φ) = φ2k generalized 2DYang-
/
h
Mills theories in the strong (A > A ) region at large N. We also show that in this
t c
-
p region, the density function of Young tableau of these models is a three-cut problem.
e
h In the specific φ6 model, we show that the theory has a third order phase transition,
v: like φ2 (YM2) and φ4 models. We have some comments for k 4 cases. At the end,
≥
Xi we study the phase structure of φ2+gφ4 model for g A region.
≤ 4
r
a
1
e-mail:[email protected]
2
e-mail:[email protected]
1 Introduction
The pure 2D Yang-Mills theory (YM ) is defined by the Lagrangian tr(F2) on a compact
2
Riemann surface. In an equivalent formulation of this theory, one can use itr(BF)+tr(B2)
as the Lagrangian, where B is an auxiliary pseudo-scalar field in the adjoint representation
of the gauge group. Path integration over field B leaves an effective Lagrangian of the form
tr(F2).
Pure YM theory on a compact Riemann surface is characterized by its invariance under
2
area preserving diffeomorphism and by the fact that there are no propagating degrees of
freedom. Now it is easy to realize that these properties are not unique to itr(BF)+tr(B2)
theory, but rather are shared by a wide class of theories, called the generalized Yang-Mills
theories (gYM ). These theories are defined by replacing the tr(B2) term by an arbitrary
2
class function Λ(B) [10]. Besides the reasons which was discussed in [1] to justify the study
of gYM , we can add two other reasons. The first one is that the Wilson loop vacuum
2
expectation value of gYM obeys the famous area law behaviour, [11], and this behaviour
2
is a signal of confinement which is one of the most important unsolved problem of QCD.
Second, the existance of the third order phase transition in some of the gYM theories (one
2
casewasstudied in[5]andother examples willbestudied inthispaper), isanotherindication
for equivalence of YM and gYM as a 2D counterpart of the theory of strong interaction.
2 2
In ref.[10], the partition function of gYM has been calculated by regarding the gener-
2
alized Yang-Mills action as perturbation of topological theory at zero area. In ref.[1], the
partition function has been obtained by generalization the Migdal’s suggestion about the
local factor of plaquettes, and has been shown that this generalizatin satisfies the necessary
requirements. And in ref.[11], this function has been calculated in continuum approach and
by standard path integral method. The gYM theories can be further coupled to fermions,
2
thus obtaining QCD and generalized QCD theories [1]. Recently there has been a lot of
2 2
interest in these theories. Phase structure, string interpretation and algebraic aspects of
these theories are reviewed in [2].
In this article we explore the issue of phase structure of the gYM theories. An early
2
study of phase transition of YM in the large-N limit on the lattice [3] reveals a third order
2
phase transition. The study of pure continuum YM in the large-N limit on the sphere also
2
yields a similar result [4]. This result is obtained by calculating the free energy as a function
of the area of the sphere (A) and distinguishing between the small and large area behaviour
of this function. In [5], the authors considered gYM in the large-N limit on the sphere and
2
they found an exact expression for an arbitrary gYM theory in the weak (A < A ) region,
2 c
where A is the critical area. They also have found a third order phase transition for the
c
specific φ4 model.
In this paper we discuss the issue of phase transition for a wider class of theories. In
1
section 2 we review the derivation of the free energy and the density function in the weak
(A < A ) region. In section 3 we study the φ2k theories. First, we show that the density
c
function of these models has two maxima in the (A < A ) region, similar to φ4 model. This,
c
then enable us to use the method of ref. [5] to obtain a general expression for the free energy
for these theories in the strong (A > A ) region. In section 4 we compute the free energy
c
near the transition point for the specific φ6 model and we show this model also has a third
order pahse transition. We also make some remarks for k 4 models. And finally in section
≥
5 we study another class of models, namely the φ2 +gφ4 models. If g A then the density
≤ 4
function will have only one maximum at the origin. We show that these models also undergo
a third order phase transition in this domain.
2 Large-N behaviour of gYM at A < A
2 c
The partition function of the gYM on a sphere is [5, and references there-in]
2
Z = d2e−AΛ(r), (1)
r
r
X
where r’s label the irreducible representations of the gauge group, d is the dimension of the
r
r-th representation, A is the area of the sphere and Λ(r) is :
p a
k
Λ(r) = C (r), (2)
Nk−1 k
k=1
X
in which C is the k-th Casimir of group, and a ’s are arbitrary constants. Now consider the
k k
gauge group U(N) and parametrize its representation by n n n , where n is
1 2 N i
≥ ≥ ··· ≥
the length of the i-th row of the Young tableau. It is found that
n n
i j
d = (1+ − )
r
j i
1≤iY<j≤N −
N
C = [(n +N i)k (N i)k]. (3)
k i
− − −
i=1
X
To make the partition function (1) convergent, it is necessary that p in eq.(2) be even and
a > 0.
p
Now, follwing [4], one can write the partition function (1) at large N, as a path integral
over continuous parameters. We introduce the continuous function:
φ(x) = n(x) 1+x, (4)
− −
where
0 x := i/N 1 and n(x) := n /N. (5)
i
≤ ≤
2
The partition function (1) then becomes
Z = dφ(x)eS[φ(x)], (6)
Z 0≤x≤1
Y
where
1 1 1
S(φ) = N2 A dxG[φ(x)]+ dx dy log φ(x) φ(y) , (7)
{− | − | }
Z0 Z0 Z0
apart from an unimportant constant, and
p
G[φ] = ( 1)ka φk. (8)
k
−
k=1
X
Now we introduce the density
dx
ρ[φ(x)] = , (9)
dφ(x)
where in the cases which G is an even function, the normalization condition for ρ is
a
ρ(λ)dλ = 1. (10)
Z−a
The function ρ(z) in this case is [5]
√a2 z2 ∞ (2n 1)!!
ρ(z) = − − a2nzqg(2n+q+1)(0), (11)
π 2nn!(2n+q +1)!
n,q=0
X
where
A
g(φ) = G′(φ), (12)
2
and g(n) is the n-th derivative of g. Similarly one can express the normalization condition,
eq.(10), as
∞ (2n 1)!!
− a2ng(2n−1)(0) = 1. (13)
2nn!(2n 1)!
nX=0 −
Defining the free energy as
1
F := lnZ, (14)
−N2
then the derivative of this free energy with respect to the area of the sphere is
1 a
F′(A) = dx G[φ(x)] = dλ G(λ)ρ(λ). (15)
Z0 Z−a
The condition n n n imposes the following condition on the density ρ(λ)
1 2 N
≥ ≥ ··· ≥
ρ(λ) 1. (16)
≤
Thus, first we determine the parameter a from eq. (13), then we calculate F′(A) from eq.
(15). Note that the above solution is valid in the weak (A < A ) region, where A is the
c c
critical area. If A > A , then the condition ρ 1 is violated.
c
≤
3
3 The G(φ) = φ2k model
In order to study the behaviour of any model in the strong (A > A ) region, we need to
c
know the explicit form of the density ρ in the weak (A < A ) region. From eq. (11) we can
c
obtain ρ for any even function G(φ). However, in this section we consider a simple case,
namely the G(φ) = φ2k model with arbitrary positive integer k . The density function ρ in
the weak region is
kA k−1 (2n 1)!!
ρ(z) = √a2 z2 − a2nz2k−2n−2. (17)
π − 2nn!
n=0
X
The interesting point is that the above density function has only one minimum at z = 0,
and two maxima which are symmetric with respect to the origin. To see this, notice that
equating the derivative of ρ equal to zero, will yield z = 0, and
k−2 (2n 1)!!
f(y) = 1+ − y−(n+1) = 0, (18)
− 2n+1(n+1)!
n=0
X
where y = z2/a2. Now as all of the coefficients of y, in eq. (18), are positive, the function
f(y) is a monotonically decreasing function and has only one root,i.e. y . Next, expanding
0
ρ near the origin, we obtain
kA (2k 3)!! 2k 1
ρ(z) = − a2k(1+ − z2a−2 + ), (19)
π 2k−1(k 1)! 2(2k 3) ···
− −
thus, ρ′′(0) > 0. The origin is then a minimum. But, near the points z = a the curve
±
(±)
ρ(z) is concave downward, consequently the points z = a√y will correspond to two
0 ± 0
symmetric maxima of the density. Therefore in the strong region, all φ2k’s models are three-
cut problems. The function F′(A) for G(φ) = φ2k in the weak regoin is [5]
1
F′(A) = . (20)
w 2kA
Next, we study the strong (A > A ) region. Following [5], we use the following ansatz
c
for ρ
1, z [ b, c] [c,b] =: L′
ρ (z) = ∈ − − (21)
s
(ρ˜s(z), z [ a, b]S[ c,c] [b,a] =: L.
∈ − − −
Then, if we define the function H (z) [6] S S
s
a ρ (w)
s
H (z) := P dw , (22)
s
Z−a z −w
where, P indicates the principal value of the integral, it has the follwing expansion at large
z
1 1 a 1
H (z) = + ρ (λ)λ2dλ+ + F′(A)+ . (23)
s z z3 Z−a s ··· z2k+1 s ···
4
Hence, one can obtain F′(A) via expansion of H (z).
s s
To calculate the function H (z), we repeat the same steps which was followed in [5], and
s
using some complex analysis techniques ([7]), which yield the following result for φ2k model:
∞ b λdλ
H (z) = kAz2k−1 kAR(z) α(n,p,q)z2(k−n−p−q−2) 2R(z) . (24)
s − − (z2 λ2)R(λ)
n,Xp,q=0 Zc −
where
(2n 1)!!(2p 1)!!(2q 1)!!
α(n,p,q) = − − − a2nb2pc2q, (25)
2n+p+qn!p!q!
and
R(z) = (z2 a2)(z2 b2)(z2 c2). (26)
− − −
q
Now, we expand H (z)/R(z) and demand that it behaves like 1/z4 at large z. It can be
s
shown that the coefficients of positive powers of z in the above expansion are identically
zero. Next, we calculate the coefficients of 1/z2 this gives
∞ b λdλ
kA α(n,p,q) 2 = 0, (27)
− R(λ)
n,p,q=0 Zc
X
in which n+p+q = k 1. By setting the coefficient of 1/z4 to unity we obtain
−
∞ b λ3dλ
kA α(n,p,q) 2 = 1, (28)
− R(λ)
n,p,q=0 Zc
X
where n+p+q = k. In k = 2 case, φ4 theory, eqs.(27) and (28) reduces to
b λdλ
A(a2 +b2 +c2) = 2 , (29)
R(λ)
Zc
and
3 1 b λ3dλ
A (a4 +b4 +c4)+ (a2b2 +b2c2 +c2a2) 2 = 1, (30)
4 2 − R(λ)
(cid:20) (cid:21) Zc
which are the same equations that was obtained in [5].
We can express the action in terms of ρ (z). If we maximize this action along with the
s
normalization condition eq.(10), as a constraint, we obtain another equation. The procedure
is fully explained in [5,8]. The result is
b b b R(z)λdλ
kA α(n,p,q)z2(k−n−p−q−2)R(z)dz +2 dz P = 0. (31)
(z2 λ2)R(λ)
n,Xp,q=0Zc Zc Zc −
Note that we can determine three unknown parameters a, b, and c from equations (27), (28),
and (31).
5
To compute the function F′(A), we start from the function H (z) directly. First we
s s
expand the R(z)
a2 b2 c2 ∞
R(z) = z3 (1 )(1 )(1 ) = z3 β(n′,p′,q′)z−2(n′+p′+q′), (32)
s − z2 − z2 − z2 −
n′,p′,q′=0
X
where
(2n 3)!!(2p 3)!!(2q 3)!!
β(n,p,q) = − − − a2nb2pc2q. (33)
2n+p+qn!p!q!
Furthermore, we define ( 3)!! = 1. If we substitute the above expansion into the eq.(24),
− −
we obtain
α(n,p,q)β(n′,p′,q′)
H (z) = kAz2k−1 +kA
s z2(n+p+q−k)+2(n′+p′+q′)+1
n,p,q,n′,p′,q′=0
X
∞ β(n′,p′,q′) b λ2n+1dλ
+2 . (34)
z2(n+n′+p′+q′)−1 R(λ)
n=0n′,p′,q′=0 Zc
X X
From eq.(23) we see that the coefficient of 1/z2k+1 in the expansion of H (z) is F′(A).
s s
Therefore from the above equation we get
b λ2n+1dλ
F′(A) = kA α(n,p,q)β(n′,p′,q′)+2 β(n′,p′,q′) . (35)
s R(λ)
n,p,q,n′,p′,q′=0 n,n′,p′,q′=0 Zc
X X
In addition, in the first summation of eq. (35) we have the follwing conditions on the indices:
(n+p+q)+(n′ +p′ +q′) = 2k, (36 a)
−
and
2k 2n 2p 2q 4 0. (36 b)
− − − − ≥ −
The condition (36-a) appears due to selection of a specific power of z in the expansion, and
the condition (36-b) is due to complex integration. Furthermore in the second summation
we have the follwing condition on the indices
n+(n′ +p′ +q′) = k +1. (36 c)
−
In this way, we find the explicit relation of the free energy of φ2k’s models. For the k = 2
case, our results agree with those of ref. [5].
4 The G(φ) = φ6 model
As an application of the previous results, we will investigate carefully the phase structure
of the G(φ) = φ6 model. From eq.(17) we obtain the density ρ for this model in the weak
region, the result is
3A 3a4 a2z2
ρ(z) = ( + +z4)√a2 z2. (37)
π 8 2 −
6
From the normalization condition ( eq.(13)) we obtain a = (16/(15A))1/6. In addition from
eq.(20) F′(A) = 1/(6A). This density function has a minimum at z=0 and two maxima
w
at z(±) = ( √3+1)a/2. At A = A , the density function (37) is equal to unity at z(±).
0 ± c 0
Using this facqt, we find the critical area A
c
3125 15625√3
A = π6( ). (38)
c
10368 − 93312
For A > A the eq.(37) is not valid.
c
Next, we analyse this model in the strong (A > A ) region. The equations (27), (28),
c
and (31) in this case become
3 1 b λdλ
3A (a4 +b4 +c4)+ (a2b2 +b2c2 +c2a2) 2 = 0, (39)
8 4 − R(λ)
(cid:20) (cid:21) Zc
5 3 1
3A (a6 +b6 +c6)+ (a2b4 +a2c4 +b2a4 +b2c4 +c2a4 +c2b4)+ a2b2c2
16 16 8
h i
b λ3dλ
2 = 1, (40)
− R(λ)
Zc
and
b a2 +b2 +c2 b b R(z)λdλ
3A (z2 + )R(z)dz +2 dz P = 0. (41)
2 (z2 λ2)R(λ)
Zc Zc Zc −
To study the structure of the phase transition, we must consider the theory on the sphere
with A = A +ǫ area, where ǫ is an infinitesimal positive number. In this region, following
c
[5], we use the follwing change of variables
c = s(1 y),
−
b = s(1+y), (42)
a = s 2√3 2+e.
−
q
Note that these parameters are introduced so that at critical point, e and y are equal to zero
and s is equal to z+. Now expanding the equations (39), (40) and (41), we find
0
π 9√3 π 1 √3
As4(18 6√3) +(As4( 3)+ ( + ))e
− − ηs 2 − ηs 2 3
9As4 π 7 √3 π 9 4√3
+( ( + ))e2 +((9+3√3)As4 ( + ))y2
8 − ηs 8 2 − ηs 4 3
3As4 π 77√3 89 π 5363 129√3
+( + ( + ))ey2 +(3As4 ( + ))y4 = 0, (43)
2 ηs 12 8 − ηs 192 8
and
7
πs πs 1 √3 45√3 9
(39√3 57)As6 1 +((42 18√3)As6 + ( + ))e+(( )As6
− − − η − η 2 3 8 − 2
πs 7 √3 πs 17 3 9√3
( + ))e2 +((33+3√3)As6 (2√3+ ))y2 +(( + )As6
− η 8 2 − η 4 2 2
πs 35√3 121 πs 199√3 8267
+ ( + ))ey2 +((3√3+24)As6 ( + ))y4 = 0, (44)
η 4 8 − η 8 192
and
π 1 √3 5√3 π 7 2√3 7√3 13
3(1+√3)As5 ( + )+(( +6)As5 + ( + ))e (( + )As5
− η 2 3 2 η 6 3 − 8 8
π 5 13√3 π 91√3 211 25 15√3
+ ( + ))e2 ((2+4√3)As5 + ( + ))y2+(( + )As5
η 2 9 − η 36 48 2 2
π 635 275√3 56 32√3 π 60673√3 23353
+ ( + ))ey2 (( + )+ ( + ))y4 = 0. (45)
η 24 18 − 3 3 η 1728 384
The parameter η = 2√3 3 is used for the sake of brevity in the above formulas. We also
−
have kept terms upqto order y4 or e2 (we will show that e is of order y2). Next we obtain s
from eq.(43), the result is
π 1 12+7√3 1 1+√3 5 √3
s = ( )5( )10 1 ( )e+( + )e2
A 648 " − 8 32 12
1 √3 167 95√3 911 11√3
+( + )y2 ( + )ey2 +( + )y4 . (46)
4 6 − 96 96 192 4 #
Substituting this s in eq.(45) will give
5√3 1 15 37√3
e = ( )y2 ( + )y4. (47)
2 − 2 − 16 48
So e is of order y2. Using eq.(44), we obtain
4√3 2 317 77√3
y2 = ( )δ +( )δ2, (48)
15 − 5 360 − 150
and
11 17√3 8533√3 14929
e = ( )δ +( )δ2. (49)
5 − 15 3600 − 3600
The parameter δ is the reduced area, i.e. δ = A−Ac.
Ac
From eq. (35), we find F′(A), the result is
s
8
3A
F′(A) = 35(a12 +b12 +c12) 12(a6b6 +a6c6 +b6c6)
s 1024 −
h
+12a2b2c2(a4b2 +a4c2 +b4a2 +b4c2 +c4a2 +c4b2)+14a4b4c4
2a2b2c2(a6 +b6 +c6) 19(a8b4 +a8c4 +b8a4 +b8c4 +c8a4 +c8b4)
− −
10(a10b2 +a10c2 +b10a2 +b10c2 +c10a2 +c10b2)
−
i
5 1
+ (a8 +b8 +c8) (a6b2 +a6c2 +b6a2 +b6c2 +c6a2 +c6b2)
64 − 16
h
1 1 b λdλ
+ a2b2c2(a2 +b2 +c2) (a4b4 +a4c4 +b4c4)
16 − 32 R(λ)
iZc
1 b λ3dλ
+ a6 +b6 +c6 (a2b4 +a2c4 +b2a4 +b2c4 +c2a4 +c2b4)+2a2b2c2
8 − R(λ)
h iZc
1 b λ5dλ b λ7dλ b λ9dλ
+ a4 +b4 +c4 2(a2b2 +a2c2 +b2c2) +(a2+b2+c2) 2 .
4 − R(λ) R(λ) − R(λ)
h iZc Zc Zc
(50)
To compute F′(A), we express the parameters a, b and c in terms of δ, and after a lengthy
s
calculations eq.(50) reduces to
1 271 1289√3
F′(A) = 1+( + )δ2 + . (51)
s 6A " 10800 504000 ··· #
If we compare this with F′(A) given in the begining of this section we find
w
271 1289√3 1 A A
F′(A) F′(A) = ( + ) ( − c)2 + . (52)
s − w 64800 3024000 A A ···
c c
Therefore, we have a third order phase transition, which is the same as the ordinary YM [4]
2
and G(φ) = φ4 model [5]. In principle one could study other φ2k models in the same manner.
The k = 4 case is solvable analytically, but the expressions become too complicated. For
k 5 one can not solve the equations analytically.
≥
5 The G(φ) = φ2 + gφ4 model
So far in our study of the phase transition for gYM theories we considered only those
2
G(φ) which contained a single term. In ref.[9] the authors study the phase transition of
gYM on a closed surface of arbitrary genus and area A. In particular they investigate the
2
G(φ) = φ2 +gφ3 model. However, their treatment is mostly qualitative. In this section we
consider a simple combination of φ2 and φ4, namely we study the G(φ) = φ2 +gφ4 model.
In the weak region we can obtain the density ρ from eq.(10), the result is
A
ρ(z) = √a2 z2(1+ga2 +2gz2). (53)
π −
9