1. Monopole production and rapid decay of gauge fields Aiichi Iwazaki Nishogakusha University

2. High energy heavy ion collisions Generation of color electric and magnetic fields according to a model of color glass condensate

3. thermalized quark gluon plasma Quarks and gluons are produced by the rapid decay of the gauge fields decay time < <1fm/c （～ 0.5fm/c ?) Hirano, Nara 2004 We have not yet found the rapid decay mechanism of the gauge fields. We wish to propose a rapid decay mechanism of the gauge fields. High energy density of the color gauge fields ~

4. width of flux tube Homogeneous in longitudinal direction field strength saturation momentum Ensemble of Z 2 vortices ? The gauge fields are unstable. Dumitru,Nara Petreska, 2013 ( RHIC or LHC ) Characteristics of the gauge fields

5. |A(p L =p,t)/A(p L =p,t=0| 2 J. Berges, S. Scheffer and D. Sexty, 2008 Exponential growth of the distance between nearby gauge fields at t=0 B B’ t=0 Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013) Exponential growth of fluctuations around the gauge fields

6. Exponential growth of longitudinal pressure of fluctuations around the gauge fields in expanding glasma ( τ,η coordinates ) Romatschke and Venugopalan 2006 Fukushima and Gelis 2011 It has been found that these instabilities do not lead to sufficiently rapid decay of the gauge fields for QGP to be realized within 1fm/c. They have been discussed to be Nielsen-Olesen instabilities.

7. Nielsen-Olesen instability classical instability in SU(2) gauge theory （ Electromagnetic fields represent the background gauge fields ） The term can be positive or negative for arbitrary magnetic field B Iwazaki 2008, Itakura, Fujii, 2008 （ charged vector fields are fluctuations around the gauge fields ） Nielsen and Olesen 1978,

8. Homogeneous B negative potential Nielsen-Olesen unstable modes occupying lowest Landau level Negative potential for homogeneous B -2gB Potential for inhomogeneous B growth rate Bound states in the Lowest Landau level Bound states exist with We may represent these bound states by using effective magnetic field Eq. of motion

9. Numerical results ( nonexpanding glasma ) growth rate saturation momentum A roughly estimated decay time of the background fields J. Berges, S. Scheffer and D. Sexty, 2008 Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013) When we represent the growth rate by using effective homogeneous magnetic field such as, we find

10. small effective mass ( imaginary ) Effective Lagrangian of describing the instability under inhomogeneous magnetic fields is given such that using effective homogeneous magnetic field small growth rate long decay time

11. Using the effective Lagrangian, we calculate the back reaction of the unstable modes on the background gauge fields and show how fast the fields decay. We show that the monopole production leads to much more rapid decay of the gauge fields than the production of Nielsen-Olesen unstable modes Similarly, we wish to calculate the back reaction of magnetic monopoles on the background gauge fields by using an effective Lagrangian of the monopoles. The monopoles are such objects whose condensation gives rise to “quark confinement” in QCD.

12. Effective Lagrangian of magnetic monopoles describing quark confinement larger than magnetic charge dual gauge fields ‘tHooft Mandelstam 1976 Koma, Suzuki 2003 describes dual superconductors for monopole field

13. We calculate the decay time of background color electric ( magnetic ) fields by using the effective Lagrangian of Niesen-Olesen unstable modes ( magnetic monopoles ) in expanding glasma ( τ,η coordinates ) Note that the monopoles occupy Landau levels under background electric field, while Nielsen-Olesen modes occupy Landau levels under magnetic field.

14. Our assumptions Relevant monopoles occupy only the lowest Landau level Their distribution is almost homogeneous in transverse plain so that magnetic field affected by the monopole production is almost homogeneous. The similar assumptions for Nielsen-Olesen unstable modes are adopted.

15. Effective Lagrangian of Nilesen-Olesen (N-O) unstable modes in τ,ηcoordinates Assuming homogeneous distribution of N-O modes in transverse plain, the dynamical variable is left. ( wave functions of the lowest Landau level )

16. wave functions of the lowest Landau level under Effective Lagrangian of magnetic monopoles in the lowest Landau level Nielsen-Olesen monopole

17. Equations of motion of Nielsen-Olesen modes homogeneous in We assume that background magnetic field decreases with the expansion The equations describe how the electric field decays via the production of Nielsen-Olesen unstable modes Maxwell eq.

18. initial conditions We use the initial conditions given by Dusling, Gelis, and Venugopalan Whittaker function τ→0 Without taking average of initial values after obtaining the time evolution of we take the initial value, τ→0 2011, 2012 That is, we include next to leading order of quantum effects on the evolution of the background gauge fields.

19. Positive energy solutions of the equation with the parameters, as τ→0 Whittaker function

20. τ→0 This initial condition comes from the average, with the use of the formulae, τ→0 For simplicity, we take the simple initial conditions, Similar procedures of initial conditions even in the case of the magnetic monopoles are assumed.

21. Decay of the electric field producing Nielsen-Olesen modes fm/c Decay of the magnetic field producing magnetic monopoles ten times more rapid decay We should note how the gauge field rapidly decays producing the magnetic monopoles. tentative results 0.1fm/c 1 fm/c

22. Initial amplitude of Nielsen-Olesen unstable modes Initial amplitude of magnetic monopoles The initial amplitude is 10 times larger than the amplitude of Nielsen- Olesen unstable modes

23. Pair creations of magnetic monopoles under magnetic fields by Schwinger mechnism production rate of monopoles Tanji and Itakura, 2012 The production rate of the monopoles is about 10 times larger than that of Nielsen-Olesen unstable modes. Compare the production rate of the monopoles with that of Nielsen-Olesen unstable modes production rate of N-O modes

24. conclusions We have shown that the gauge fields generated after high energy heavy ion collisions decay much more rapidly producing magnetic monopoles than Nielsen-Olesen unstable modes. Although our calculation does not properly take into account precise initial conditions so that the result is preliminary, it shows that the role of the magnetic monopoles in the realization of thermalized QGP is important.

26. time J. Berges, S. Scheffer and D. Sexty, 2008 Numerical simulations exponential growth of the fluctuations

27. Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013) Exponential growth of the distance between nearby gauge fields B B’ t=0 t Numerical simulations ( in our notations )