M. Zubkov ITEP Moscow B. L. G. Bakker, A. I. Veselov, M. A. Zubkov, J. Phys. G: Nucl. Part. Phys. 36 (2009) ; 2. A.I.Veselov, M.A.Zubkov, JHEP 0812:109,2008; 3. A.I. Veselov, M.A. Zubkov, proceedings of LATTICE2009; 4. M.A.Zubkov, arXiv: The vicinity of the phase transition in the lattice Weinberg – Salam Model and Nambu monopoles

2Abstract 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. 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 (The data is obtained on the lattice )

3 Fields 1.Lattice gauge fields (defined on links) 2.Fundamental Higgs field (defined on sites) Lattice action Another form:

4 Transition surface lines of constant physics Phase diagram at constant (U(1) transition is omitted) Physical phase Unphysical phase Tree level estimates:

5 One loop weak coupling expansion: bare and are increased when the Ultraviolet cutoff is increased along the line of constant physics Along the line of constant physics if we neglect gauge loop corrections to

6 Realistic value of Weinberg angle The fine structure constant The majority of the results were obtained on the lattices The results were checked on the lattices

7 Unphysical phase phase diagram line of constant renormalized Physical phase Condensation of Nambu monopoles

8 phase diagram

9 approximates V(R) better than the lattice Coulomb potential The renormalized fine structure constant Right – handed lepton Wilson loop The simple fit

10 The potential

11 The potential

12 Renormalized fine structure constant

13 Z – boson mass in lattice units: Evaluation of lattice spacing (the sum is over “space” coordinates of the Z boson field) are imaginary “time” coordinates

14 Unphysical phase Ultraviolet cutoff along the line of constant renormalized Physical phase Condensation of Nambu monopoles

15 in lattice units Fit for R = 1,2,3,4,5,6,7,8 The results yet have not been checked on the larger lattices Phase transition

16 in lattice units Fit for R = 1,2,3,4,5,6,7,8 The results yet have not been checked on the larger lattices Phase transition

17 Higgs boson mass in lattice units Higgs boson mass in physical units: (the sum is over “space” coordinates of the Z boson field) are imaginary “time” coordinates

18 Phase transition at

19 Transition surface lines of constant physics Phase diagram at constant Physical phase Unphysical phase

Effective constraint potential

Potential barrier Hight

Minimum of the potential at

23 Standard Model NAMBU MONOPOLES (unitary gauge) Z string NAMBU MONOPOLE

24 NAMBU MONOPOLE WORLDLINE Worldsheet of Z – string on the lattice

25 Nambu monopole densitySusceptibility Phase transition

26 Nambu monopole densitySusceptibility Phase transition

27 Nambu monopoles Percolation Transition Line of constant renormalized fine structure constant Ultraviolet cutoff

28 Excess of plaquette action near monopoles Excess of link action near monopoles Phase transition

29 Transition surface lines of constant physics Phase diagram at constant Physical phase Unphysical phase

30 Previous investigations of SU(2) Gauge - Higgs model Lattice action At realistic value of Weinberg angle The fine structure constant is For we have

31 Cutoff (in Gev) in selected SU(2) Higgs Model studies atPublication Joachim Hein (DESY), Jochen Heitger, Phys.Lett. B385 (1996) F. Csikor,Z. Fodor,J. Hein,A. Jaster,I. Montvay Nucl.Phys.B474(1996) F. Csikor, Z. Fodor, J. Hein, J. Heitger, Phys.Lett. B357 (1995) Z.Fodor,J.Hein,K.Jansen,A.Jaster,I.Montvay Nucl.Phys.B439(1995) F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Phys.Lett. B334 (1994) F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Nucl.Phys.Proc.Suppl. 42 (1995) Y. Aoki, F. Csikor, Z. Fodor, A. Ukawa Phys.Rev. D60(1999) Y. Aoki Phys.Rev. D56 (1997) W.Langguth, I.Montvay,P.Weisz Nucl.Phys.B277:11, W. Langguth, I. Montvay (DESY) Z.Phys.C36:725, Anna Hasenfratz, Thomas Neuhaus, Nucl.Phys.B297:205,

32Conclusions 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. ?