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

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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.

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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

??