The vicinity of the phase transition in the
lattice Weinberg – Salam Model
and Nambu monopoles
M. Zubkov
ITEP Moscow 2010
1.
2.
3.
4.
B. L. G. Bakker, A. I. Veselov, M. A. Zubkov, J. Phys. G: Nucl. Part. Phys.
36 (2009) 075008;
A.I.Veselov, M.A.Zubkov, JHEP 0812:109,2008;
A.I. Veselov, M.A. Zubkov, proceedings of LATTICE2009;
M.A.Zubkov, arXiv:0909.4106 Phys.Lett.B684:141-146,2010
Abstract
The lattice Weinberg - Salam model without fermions
is investigated numerically for realistic choice of bare
coupling constants correspondent to the value of the
Higgs mass M H ~ 300 Gev . On the phase diagram
there exists the vicinity of the phase transition
between the physical Higgs phase and the unphysical
symmetric phase, where the fluctuations of the scalar
field become strong. In this region Nambu monopoles
are dense and the perturbation expansion around
trivial vacuum cannot be applied. Out of this region
the ultraviolet cutoff cannot exceed the value around
1 Tev. Within the fluctuational region the maximal
value of the cutoff is  c   / a  1,4 Tev
(The data is obtained on the lattice 203  24 )
2
Fields
1. Lattice gauge fields (defined on links)
U  SU ( 2); ei  U (1)
2. Fundamental Higgs field (defined on sites)
 ,   1,2

  2
Lattice action


1
1
S    (1  Re Tr U plaquette) 
(1  cos  plaquette)
2
2
tan W
plaquettes 

   Re  xU xy e
links
i xy
 y   (|  x |   (|  x |2 1) 2 )
2
sites
Another form:


1
1
S    (1  Re Tr U plaquette) 
(1  cos  plaquette) 
2
2
tan W
plaquettes 

~
~ 2 ~ ~
i ~
  |  x  U xy e xy  y |2   (  2 |  x |   |  x |4 )
links
~
   / 2
sites
 2  2(4  (2  1) /  )
~
  4 /  2
3
Phase diagram at constant W

(U(1) transition is omitted)
lines of constant physics
Physical phase
Transition surface
c
~
2
  4 / 
Unphysical phase
M 2H
~

2


Tree level estimates: 2
M W
tan 2 W

 (1  tan 2 W )

4
1
1
1  22 4ng nh 

1
1
1




log




g 2 (  ) g 2 () 8 2  3
3
6
 g 2 ( ) g 2 () 8 2
1
1
N 8



log
~
~

 (  )  ( ) 8 2
~
M

~
 2  
M
2
2
H
2
W
Along the line of constant physics if
we neglect gauge loop corrections to
~

 20ng nh 



log


9
6



1
1
1
 2 2
4 g
g
M 2H

1  2 

M W
3 ( M Z ) log
MZ
M 2H
2
~
M
W
 () 
1
4  8 M 2H


log
2
2
2( M Z )
8
M W
MZ
1
1
 ( ) 
 2
4( M Z ) 8

 22 1 1 
    log
MZ
 3 6 6
~
 (M Z )  1 / 128
W  30 o
One loop weak coupling expansion: bare  and  are
increased when the Ultraviolet cutoff  is increased
along the line of constant physics
5
Realistic value of Weinberg angle
W  30o  sin 2 W  0.25
The fine structure constant
tan 2 W
1
 

2
 (1  tan W )
4
The majority of the results were obtained on the lattices
The results were checked on the lattices
123  16
203  24;164
6
phase diagram

  const
line of constant renormalized
 renormalized  const
Physical phase
 c  1,35 Tev
  1 (   )
(at   ; ~ 1 / 128)
  0.25 (  0)
Condensation of
Nambu monopoles
Unphysical phase

  0.4

  15 (  ;  ~ 1 / 128)
7
phase diagram
 
 c  1,35 Tev
(at   ; ~ 1 / 128)
8
The renormalized fine structure constant
Right – handed lepton Wilson
loop
WC  Re  e
e2

4
2 i xy
xyC
W [R  T ]
V ( R )  log lim
T  W [ R  (T  1)]
The simple fit
V ( R)  
R
R
 const
V ( R )   R U ( R )  const
approximates V(R) better
than the lattice Coulomb
potential
U ( R) 

L3


p 0
sin 2
eip3R
p1
p
p
 sin 2 2  sin 2 3
2
2
2
9
The potential
83  16
W [ R  3]
V ( R)  log
W [ R  4]
  12 W
 30 o
  0.009
  0.277
V
1/ R
10
The potential
  15 W
 30 o
 
164
 1
V
11
Renormalized fine structure constant
  12 W
 30 o
  0.009
1 /  renormalized
83  16
12 16
3

12
Evaluation of lattice spacing
MZ
phys
1
1
  
91 Gev
lattice units
a MZ
 91 Gev


a

1
MZ
lattice units
280 Gev
Z – boson mass in lattice units:
Z  sin[arg U   ]
11
  Z

xy

x
Z

y
e
 M Z lat|x0  y0 |
e
 M Z lat ( L |x0  y0 |)
(the sum is over “space” coordinates of the Z boson field)
x0 , y0 are imaginary “time” coordinates
13
Ultraviolet cutoff along the line of constant renormalized

 renormalized  1 / 128
 
 c  1,35 Tev
Condensation
of Nambu
monopoles
1 Tev
Unphysical phase

Physical phase
 c  0.9
1.2
14
MZ
  0.009
  12
W  30 o
in lattice units
Fit for R = 1,2,3,4,5,6,7,8
  Z

xy

x
Z

y
e
 M Z lat|x0  y0 |
e
 M Z lat ( L |x0  y0 |)
123  16
  1 Tev
Phys.Lett.B684:141-146,2010
Phase transition
16
4
  1.4 Tev
20  24
3
M  270 Gev
0
H

15
W  30 o
MZ
in lattice units
Fit for R = 1,2,3,4,5,6,7,8
 
  15
  Z

xy

x
Z

y
e
 M Z lat|x0  y0 |
e
 M Z lat ( L |x0  y0 |)
The results yet have not been checked on the larger lattices
8  16
3
  1.4 Tev
Phase transition
M H  800 Gev

16
Higgs boson mass in lattice units
H x   Z xy
y
2
 H H  H   e
xy  x y x 
2
 M H lat|x0  y0 |
e
 M H lat ( L |x0  y0 |)
(the sum is over “space” coordinates of the Z boson field)
x0 , y0 are imaginary “time” coordinates
  0.29
  12   0.009
Higgs boson mass in physical units:
M H  265  70 Gev
17
Phase transition at
  12 W
 30 o
83  16
 renormalized  1 / 100

  1.4 Tev
M H  300  40 Gev
M
0
H
 270 Gev
  1 Tev
M H  265  70 Gev
c
18
Phase diagram at constant W

lines of constant physics
M H  300 Gev
M H  800 Gev
Physical phase
 c  1,35 Tev
 c  1.4 Tev
~
 
~
2
  4 / 
Transition surface
Unphysical phase

19
Effective constraint potential
W  30 o
  12
  0.009
H  V (0)  V ( min )
  0.29
  0.279
 c  0.273
min
Potential barrier Hight H  V (0)  V ( min )
W  30 o
  12
  0.009
 c  0.273  c 2  0.278
H fluct  V ( min   )  V ( min )

Minimum of the potential at
W  30 o
  12
min
  0.009
 c  0.273  c 2  0.278
 
 c  1,4 Tev
 c 2  1 Tev

NAMBU MONOPOLES
(unitary gauge)
Standard Model
L
NAMBU
MONOPOLE
M  1 Tev
R 
0
NAMBU
MONOPOLE
Z string
v 
 
0 
1
200 Gev


i
A
dx

Z
dx

2

N
,
N

Z
,
A

A

 su ( 2)

i
 3  

L
L

2

A
dx

dx

4

N
sin
W
 em  


Aem  2 B  2 sin 2 W ( A3  B ); Z  A3  B; W   A1  iA2
23
Worldsheet of Z – string on the lattice
Z  arg[  U ei ]
A  [ Z  2 ] mod 2
*

1
1
[d Z ] mod 2  d Z   A  [d A] mod 2  d A
Z 
2
2
*

NAMBU MONOPOLE WORLDLINE
1
jZ    Z 
d [d Z ] mod 2 
2
*
1
j A    A  
d [d A] mod 2 
2
*
j A  jZ
24
  12   0.009
W  30 o
Susceptibility
H  Hx
2
x
Nambu monopole density
2
Phase transition
83  16
123  16


25
  15
W  30 o
Susceptibility
H  Hx
2
x
164
 
Nambu monopole density
2
Phase transition


26
Percolation Transition
Nambu monopoles
Nambu monopoles
  C
  c
 
c

c
C
Line of constant renormalized fine structure constant
Ultraviolet cutoff
C 

ac
27
W  30 o
  15
Excess of link action
near monopoles
 
Excess of plaquette
action near monopoles
Phase transition
164


28
Phase diagram at constant W

lines of constant physics
M H  300 Gev
M H  800 Gev
Physical phase
 c  1,35 Tev
 c  1.4 Tev
~
 
~
2
  4 / 
Transition surface
Unphysical phase

29
Previous investigations of SU(2) Gauge - Higgs model
Lattice action
 1

S    1  Re Tr U plaquette
2

plaquettes 
  2
   Re  xU xy  y   (|  x |   (|  x |2 1) 2 )
2
links
sites
At realistic value of Weinberg angle
W  30o  sin 2 W  0.25
The fine structure constant is
tan 2 W
1
 

2
 (1  tan W )
4
For
 8
we have
1

110
30
Cutoff (in Gev) in selected SU(2) Higgs Model studies at   1/ 110
M H , Gev  c   / a
Publication
Joachim Hein (DESY), Jochen Heitger, Phys.Lett. B385 (1996) 16
242-248
345
F. Csikor,Z. Fodor,J. Hein,A. Jaster,I. Montvay
Nucl.Phys.B474(1996)421
34
880
F. Csikor, Z. Fodor, J. Hein, J. Heitger, Phys.Lett. B357 (1995)
156-162
35
440
Z.Fodor,J.Hein,K.Jansen,A.Jaster,I.Montvay
Nucl.Phys.B439(1995)147
48
880
F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay
Phys.Lett. B334 (1994) 405-411
50
600
F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay
Nucl.Phys.Proc.Suppl. 42 (1995) 569-574
50
880
Y. Aoki, F. Csikor, Z. Fodor, A. Ukawa Phys.Rev. D60(1999)
013001
85
820
Y. Aoki Phys.Rev. D56 (1997) 3860-3865
108
940
W.Langguth, I.Montvay,P.Weisz Nucl.Phys.B277:11,1986.
480
1260
W. Langguth, I. Montvay (DESY) Z.Phys.C36:725,1987
720
1480
Anna Hasenfratz, Thomas Neuhaus, Nucl.Phys.B297:205,1988 720
31
1480
Conclusions
We demonstrate that there exists the fluctuational
region on the phase diagram of the lattice Weinberg –
Salam model. This region is situated in the vicinity of
the phase transition between the physical Higgs phase
and the unphysical symmetric
phase of the model.
In this region the fluctuations
of the scalar field become
strong and the perturbation
expansion around trivial
?
vacuum cannot be applied.
M H  300 Gev
12  16
3
 max  1,4 Tev
32