End-to-end probability for an interacting center
vortex worldline in Yang-Mills theory
Bruno Fernando Inchausp Teixeira
August-2011
IX Oficina de Teoria Quˆ
antica de Campos-IX OTQC
CBPF-RJ
1
Introduction
Motivations and Objective
2
Integration on an ensemble of intantons chains
End-to-end probability for an interacting center vortex worldline in Yang-Mills theory
Motivations
Try to understand better one of the existent confinament scenarios, in
particular, the dual superconductivity mechanism;
The CFN formalism allows us to represent monopoles as center vortices like
topological defects in a local color frame;
Objective
We want to determine the partition function with interactions for one center
vortex attempting to derive all the possible topological configurations involving
ensemble of instantons and anti-instantons connected for a couple of center
vortices;
Partition Function of defects
By applying of the Cho-Faddeev-Niemi decomposition in the Yang-Mills action we can to demonstrate that the partition function involving the center vortices that are coupled with monopoles is given by:
Zv ,m= Z
[Dm][Dv ]ei R
d 3 x λµdµ, (1)
where dµis on the defects and contain all the singularities and λµis a dual field associated to the charged sector. Using dµ(α)=2π g Z d σdx α µ d σδ (3) (x − xα(σ)), (2)
is easy to show that
Zv ,m= Z
[Dφ]e−SφX n
Zn, (3)
where we have defined
Zn= Z [Dm]n[Dv ]nexp " i2π g 2n X k=1 ZLk 0 ds ˙xα(k)λα(x(k)) − Sd # , (4)
Partition Function of defects and Sd= 2n X k=1 ZLk 0 ds " µ + 1 2κ˙u (k) α ˙u (k) α + φ(x (k) ) # . (5) Fenomenological Parameterization
The action of defects is purely fenomenological; The first term is related to tensile center vortices;
The second term is related to the stiffness of the center vortices (curvature energy); Finally, the last term represents the excluded volume effects.
It’s so easy to see from the partition function that exists a “fundamental block”representing an integration of the only one center vortex with fixed edges:
Q(x , x0) = Z∞ 0 dL e−µLq(x , x0, L), q(x , x0, L) = Z [Dx (s)] e− RL 0ds h 1
2κ˙uα ˙uα+φ(x(s))−i2πguα(s)λα(x(s)) i
Fenomenological Parameterization
where [Dx (s)] represents an integral over all the possibles center vortices of length L and fixed edges x and x0. We identified the action Sφas:
e−Sφ= e12 R
d 3 xd 3 x 0 φ(x )V −1 (x ,x 0 )φ(x 0 )
, (7)
and using the density ρ(x ) =P k
RLk
0 ds δ(x − x(k)(s)), this action can be integrated, and then
e− 12 R d 3 xd 3 x 0 ρ(x )V (x ,x 0 )ρ(x 0 ) , Z d3yV−1(x , y )V (y , z) = δ(x − z). (8)
If the interaction is V (x − y ) = (1/ζ)δ(x − y ) we obtain a contact interaction:
e−Sφ= e R
d 3 xζ2φ2
. (9)
Stiffness in string-like objects
It’s hard even in the non-interacting case;
Stiffness allows us to define an effective size for a monomer in polymer field theory; The end-to-end probability for a random chain is:
qN(x , x0) = N Y n=1 " Z (d3∆xn) 1 4πa2δ(|∆xn| − a) # δ(x − x0− N X n=1 ∆xn). (10)
For large N we obtain a gaussian distribution:
qN(x , x0) ≈ 3 2πNa2 !3/2 exp " −3(x − x0) 2 2Na2 # . (11)
Note that we can not apply the continuum limit in a straightforwardly form. However, if we consider the effective monomer size Na2/3 → L/α, α = 3/a
effin the stiffness case, we obtain:
q(x , x0, L) = α 2πL !3/2 e− α2L(x −x0)2= Z d3k (2π)3e − L2αk2 eik·(x −x0), (12)
Stiffness in string-like objects
and the integration in L yields:
Q(x , x0) = 2α Z d3k (2π)3 eik·(x −x0) k2+ m2 , m 2= 2αµ, (13) that satisfies (−∇2+ m2)Q(x , x0) = 2αδ(x − x0). (14) Introduction of interactions
We can write also:
Q(x , x0) = Z∞ 0 dL e−µL Z d2u 0 4π d2u 4π q(x , u, x0, u0, L). (15) A discretized version of the function q(x , u, x0, u0, L) can be obtained from a recursive relation given by:
Figure:
Interacting center vortices with fixed endpoints, orientations, and
length are associated with the weight q(x , u, x
0
, u
0
, L).
Introduction of interactions
qj +1(x , u, x0, u0) = e−ω(x,u) Z
d3x0d2u0Φ(u − u0)δ(x − x0− u∆L) qj(x0, u0, x0, u0), (16)
with the initial condition
q0(x , u, x0, u0) = δ(x − x0)δ(u − u0) e−ω(x,u). (17)
For example, to j = 0 we obtain:
q1(x , u, x0, u0) = e−[ω(x,u)+ω(x0,u0)]Φ(u − u0)δ(x − x0− u∆L), (18)
and continuing the iteration, it is not difficult to see that for N iterations we will have (defining x = xNe u = uN): qN(x , u, x0, u0) = Z d3x1d2u1. . . d3xN−1d2uN−1e− PN i =0ω(xi ,ui )× × N−1 Y j =0 Φ(uj +1− uj)δ(xj +1− xj− uj +1∆L). (19)
Introduction of interactions
Choosing a normalized angular distribution (stiffness term in the discretized version) and the interaction function: Φ(u − u0) = N e− 12κ∆L u−u0 ∆L 2 , (20) ω(x , u) = ∆L φ(x ) − i2π g u · λ(x ) , (21)
we can see that this is the discretized version for N monomers, in other words, q(x , u, x0, u0, L) = lim
N→∞qN(x , u, x0, u0), (22)
Introduction of interactions
From the recursive relation we can write:
qN+1(x , u, x0, u0) = e−ω(x,u) Z
d3x0d2u0Φ(u − u0) qN(x − u∆L, u0, x0, u0), (23)
such that for large N we expanded in the variable ∆L = NL in both sides of the last equation, and knowing that Φ(u − u0) is localized, the following difusion equation is obtained when we expand
qN(x − u∆L, u0, x0, u0) around u ≈ u0, ∂Lq(x , u, x0, u0, L) = κ 2 ∇2u− φ(x) − u · Dx q(x , u, x0, u0, L), (24)
where u · u = 1, u0· u0= 1, ~∇~u2is the Laplacian on an unitary sphere, Dx= ∇x− i2πgλ(x ), and when we take the limit ∆L → 0 we apply the following initial condition to solve this equation:
q(x , u, x0, u0, 0) = δ(x − x0)δ(u − u0). (25)
“Gymnastics”of functions
We must integrate initially, the function q(x , u, x0, u0) in the u0variable in order to obtain Q(x , x0). So,
“Gymnastics”of functions
q(x , u, x0, L) = Z d2u
0
4π q(x , u, x0, u0, L), (26)
whose integration in the equations (21) and (22) yields:
∂Lq(x , u, x0, L) = κ 2 ∇2u− φ(x) − u · Dx q(x , u, x0, L), (27) q(x , u, x0, 0) = δ(x − x0). (28)
In addition, one can integrate over the differents possible lengths
Q(x , u, x0) = Z∞
0
dL e−µLq(x , u, x0, L). (29)
“Gymnastics”of functions " κ 2∇ 2 u− φ(x) − u · Dx # Q(x , u, x0) = Z∞ 0 dL e−µL∂Lq(x , u, x0, L) = Z∞ 0 dL ∂L[e−µLq(x , u, x0, L)] + µ Z∞ 0 dL e−µLq(x , u, x0, L) = −q(x , u, x0, 0) + µQ(x , u, x0), (30) in other words, " −κ 2∇ 2 u+ φ(x ) + u · Dx+ µ # Q(x , u, x0) = δ(x − x0). (31)
Expansion in spherical harmonics
Finally we can obtain Q(x , x0) from Q(x , u, x0) making the following expansion in spherical harmonics
Q(x , u, x0) = X
l =0
Ql(x , u, x0) −→ Q(x , x0) ≡ Q0(x , x0). (32)
Expansion in spherical harmonics So, u · DxQ(x , x0, u) = X l =0 u · DxQl= X l =0 Rl, (33) where R0 = [u · DxQ1]0 R1 = [u · DxQ0+ u · DxQ2]1 R2 = [u · DxQ1+ u · DxQ3]2 . . . (34)
and when l = 0 we obtain
[φ(x ) + µ]Q0+ R0= δ(x − x0). (35) For l 6= 0 we have 1 fl(x ) Ql+ Rl= 0, fl(x ) = " φ(x ) + µ +l (l + 1)κ 2 #−1 . (36)
Expansion in spherical harmonics
It’s possible determine the R0and then we find the following differential equation for Q0(x , x0)
−1
3κD
2+ φ(x ) + µQ
0(x , x0) = δ(x − x0). (37)
The solution of this equation provides us a Green function associated to a center vortice. From this result we can to generate all the contributions of the topological defects in the generating function Znand one becomes possible derive an effective theory for this model.
