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The vicinity of the phase transition in the lattice Weinberg – Salam Model and Nambu monopoles. M. Zubkov ITEP Moscow 2010 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; - PowerPoint PPT Presentation
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M. ZubkovM. Zubkov
ITEP Moscow 2010ITEP Moscow 2010
1.1. B. L. G. Bakker, A. I. Veselov, M. A. Zubkov, J. Phys. G: Nucl. Part. B. L. G. Bakker, A. I. Veselov, M. A. Zubkov, J. Phys. G: Nucl. Part. Phys. 36 (2009) 075008;Phys. 36 (2009) 075008;
2.2. A.I.Veselov, M.A.Zubkov, JHEP 0812:109,2008;A.I.Veselov, M.A.Zubkov, JHEP 0812:109,2008;3.3. A.I. Veselov, M.A. Zubkov, proceedings of LATTICE2009;A.I. Veselov, M.A. Zubkov, proceedings of LATTICE2009;4.4. M.A.Zubkov, arXiv:0909.4106 M.A.Zubkov, arXiv:0909.4106 Phys.Lett.B684:141-146,2010Phys.Lett.B684:141-146,2010
The vicinity of the phase transition in The vicinity of the phase transition in the lattice Weinberg – Salam Model the lattice Weinberg – Salam Model
and Nambu monopolesand Nambu monopoles
22
AbstractAbstract
The lattice Weinberg - Salam model without The lattice Weinberg - Salam model without fermions is investigated numericallyfermions is investigated numerically for for realistic realistic choice of bare coupling constants correspondent choice of bare coupling constants correspondent to theto the value of the Higgs mass value of the Higgs mass . . On On thethe phase diagram there exists the vicinity of the phase diagram there exists the vicinity of the phase transition between thephase transition between the physical Higgs physical Higgs phase and the unphysical symmetric phase, phase and the unphysical symmetric phase, where the fluctuationswhere the fluctuations of the scalar field become of the scalar field become strong. In this region Nambu monopoles are strong. In this region Nambu monopoles are dense anddense and the perturbation expansion around the perturbation expansion around trivial vacuum cannot be applied. Out of thistrivial vacuum cannot be applied. Out of this region the ultraviolet cutoff cannot exceed the region the ultraviolet cutoff cannot exceed the value around 1 Tev. Within the fluctuational value around 1 Tev. Within the fluctuational region the maximalregion the maximalvalue of the cutoff isvalue of the cutoff is(The data is obtained on the lattice (The data is obtained on the lattice ) )
Tevac 4,1/
GevM H 300~
24203
33
2,1,
)1();2(
UeSUU i
Fields1. Lattice gauge fields (defined on links)
2. Fundamental Higgs field (defined on sites)
))1|(||(|Re
)cos1(tan
1)Re
2
11(
222
2
xsites
xlinks
yi
xyx
plaquettesplaquette
Wplaquette
xyeU
UTrS
Lattice action
Another form:
2
)|~|
~|
~|(|
~~|
)cos1(tan
1)Re
2
11(
4222
2
xsites
xlinks
yi
xyx
plaquettesplaquette
Wplaquette
xyeU
UTrS
2/~ )/)12(4(22 2/4
~
44
Transition surface
lines of constant physics
Phase diagram at constant (U(1) transition is omitted)
WPhysical phase
Unphysical phase
c2/4
~
~2
2
2
W
H
M
MTree level estimates: )tan1(
tan2
2
W
W
55
One loop weak coupling expansion: bare and are increased when the Ultraviolet cutoff is increased along the line of constant physics
log63
4
3
22
8
1
)(
1
)(
1222
hg nn
gg
log69
20
8
1
)(
1
)(
1222
hg nn
gg
log8
8
)(~1
)(~1
2
N
~
ZW
H
Z
W
H
MMM
M
MM
log8
84)(2
1)(
~
2
2
2
2
2
Along the line of constant physics ifwe neglect gauge loop corrections to
128/1)( ZM
ZZ MM
log
6
1
6
1
3
22
8
1
)(4
1)(
2 o
W 30
ZZ
W
H
MMM
M
log)(3
12
2
22
11
4
1
gg
2
~~
22
2
W
H
M
M
~
66
25.0sin30 2 Wo
W Realistic value of Weinberg angle
4
1
)tan1(
tan2
2
W
W
The fine structure constant
The majority of the results were obtained on the lattices 16123
The results were checked on the lattices 43 16;2420
77
Unphysical phase
phase diagram
line of constant renormalized
Physical phase
)128/1~;(15
)(1
Condensation of Nambu monopoles
4.0
constedrenormaliz
)128/1~;(
35,1
at
Tevc
const
)0(25.0
88
phase diagram
)128/1~;(
35,1
at
Tevc
99
approximates V(R) better than the lattice Coulomb potential
The renormalized fine structure constant
0 3222123
2sin
2sin
2sin
)(
)()(
3
p
Rip
R
pppe
LRU
constRURV
Cxy
iC
xyeW 2Re
4
2e
)]1([
][limlog)(
TRW
TRWRV
T
Right – handed lepton Wilson loop
constR
RV R
)(The simple fit
1010
The potential 12
R/1
009.0
oW 30
1683
V
277.0
]4[
]3[log)(
RW
RWRV
1111
The potential 15
oW 30
416
V
1
1212
Renormalized fine structure constant 12
009.0
oW 30
1683
edrenormaliz/1
16123
1313
GevM physZ 91
GevMa unitslattice
Z
9111
|)|(|| 0000 yxLMyxM
yxyx
latZ
latZ eeZZ
Z – boson mass in lattice units:
Evaluation of lattice spacing
00 , yx
]sin[arg 11 UZ
GevMa unitslattice
Z
2801
(the sum is over “space” coordinates of the Z boson field) are imaginary “time” coordinates
1414
Unphysical phase
Ultraviolet cutoff along the line of constant renormalized
Physical phase
9.0c
Tevc 35,1
Condensation of Nambu monopoles
128/1edrenormaliz
2.1
Tev1
1515
12oW 30
in lattice unitsFit for R = 1,2,3,4,5,6,7,8
ZM
16123
Tev4.1
GevM H 2700
416
|)|(|| 0000 yxLMyxM
yxyx
latZ
latZ eeZZ
009.0
Phase transition
Tev1
24203
Phys.Lett.B684:141-146,2010
1616
15oW 30
in lattice unitsFit for R = 1,2,3,4,5,6,7,8
ZM
Tev4.1
GevMH 800
1683
|)|(|| 0000 yxLMyxM
yxyx
latZ
latZ eeZZ
The results yet have not been checked on the larger lattices
Phase transition
1717
12
|)|(||20000 yxLMyxM
yxxyx
latH
latH eeHHH
Higgs boson mass in lattice units
Higgs boson mass in physical units:
00 , yx
y
xyx ZH 2
009.0
(the sum is over “space” coordinates of the Z boson field) are imaginary “time” coordinates
GevM H 70265
29.0
1818
Phase transition at 12
c
oW 30
1683
100/1edrenormaliz
GevM H 2700
GevM H 40300
GevM H 70265
Tev4.1
Tev1
1919
Transition surface
lines of constant physics
Phase diagram at constantWPhysical phase
Unphysical phase
Tevc 35,1
2/4~
~
Tevc 4.1
GevM H 300
GevM H 800
Effective constraint Effective constraint potentialpotentialo
W 30 12 009.0
29.0279.0
273.0c
)()0( min VVH
min
Potential barrier HightPotential barrier Hight
oW 30 12 009.0
278.02 c
273.0c
)()0( min VVH
)()( minmin VVH fluct
Minimum of the potential at Minimum of the potential at
oW 30 12 009.0
278.02 c
273.0c
min
Tevc 4,1 Tevc 12
2323
21332
2
3
;);(sin22
sin4
)2(,,2
iAAWBAZBABA
NdxdxA
suAAZNNdxZdxA
Wem
Wem
ii
LL
L
0
v
Standard Model Standard Model
GevR
TevM
200
1
1
NAMBU MONOPOLES (unitary gauge)
Z string
NAMBU MONOPOLE
NAMBU MONOPOLE
0
2424
ZdZd
eUZ
Z
i
2mod][2
1
]arg[*
2mod][2
1 *
Zddj ZZ
NAMBU MONOPOLE WORLDLINE
Worldsheet of Z – string on the lattice
AdAd
ZA
A
2mod][2
1
2mod]2[*
2mod][2
1 *
Addj AA
ZA jj
2525
009.012oW 30
Nambu monopole densitySusceptibility
22xx HH
1683
Phase transition
16123
2626
15oW 30
Nambu monopole densitySusceptibility22
xx HH
416
Phase transition
2727
c
C
c
c
Nambu monopoles
Nambu monopoles
C
Percolation Transition
Line of constant renormalized fine structure constant
Ultraviolet cutoffc
C a
2828
15oW 30
Excess of plaquette action near monopoles
Excess of link actionnear monopoles
416
Phase transition
2929
Transition surface
lines of constant physics
Phase diagram at constantWPhysical phase
Unphysical phase
Tevc 35,1
2/4~
~
Tevc 4.1
GevM H 300
GevM H 800
3030
Previous investigations of SU(2) Gauge - Higgs model
))1|(||(|Re
Re2
11
222
xsites
xlinks
yxyx
plaquettesplaquette
U
UTrS
Lattice action
2
25.0sin30 2 Wo
W At realistic value of Weinberg angle
4
1
)tan1(
tan2
2
W
W
The fine structure constant is
110
1For we have 8
3131
Cutoff (in Gev) in selected SU(2) Higgs Model studies at PublicationPublication
Joachim Hein (DESY), Jochen Heitger, Phys.Lett. B385 (1996) 242-248
1616 345345
F. Csikor,Z. Fodor,J. Hein,A. Jaster,I. Montvay Nucl.Phys.B474(1996)421
3434 880880
F. Csikor, Z. Fodor, J. Hein, J. Heitger, Phys.Lett. B357 (1995) 156-162
3535 440440
Z.Fodor,J.Hein,K.Jansen,A.Jaster,I.Montvay Nucl.Phys.B439(1995)147
4848 880880
F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Phys.Lett. B334 (1994) 405-411
5050 600600
F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Nucl.Phys.Proc.Suppl. 42 (1995) 569-574
5050 880880
Y. Aoki, F. Csikor, Z. Fodor, A. Ukawa Phys.Rev. D60(1999) 013001
8585 820820
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W.Langguth, I.Montvay,P.Weisz Nucl.Phys.B277:11,1986.
480480 12601260
W. Langguth, I. Montvay (DESY) Z.Phys.C36:725,1987 720720 14801480
Anna Hasenfratz, Thomas Neuhaus, Nucl.Phys.B297:205,1988
720720 14801480
110/1
ac /GevM H ,
3232
ConclusionsConclusionsWe demonstrate that there exists the We demonstrate that there exists the
fluctuationalfluctuationalregion on the phase diagram of the lattice region on the phase diagram of the lattice
Weinberg –Weinberg –Salam model. This region is situated in the Salam model. This region is situated in the
vicinity ofvicinity ofthe phase transition between the physical Higgs the phase transition between the physical Higgs
phasephaseand the unphysical symmetricand the unphysical symmetricphase of the model. phase of the model. In this region the fluctuations In this region the fluctuations of the scalar field becomeof the scalar field becomestrong and the perturbation strong and the perturbation expansion around trivialexpansion around trivialvacuum cannot be applied. vacuum cannot be applied.
Tev4,1max GevM H 300
16123
??
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