Monopole production and rapid decay of gauge fields

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

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

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

4. Characteristics of the gauge fields Homogeneous in longitudinal direction width of flux tube Ensemble of Z2 vortices ? Dumitru,Nara Petreska, 2013 saturation momentum field strength ( RHIC or LHC ) The gauge fields are unstable.

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

6. Exponential growth of longitudinal pressure of fluctuations around the gauge fields in expanding glasma (τ,ηcoordinates ) 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 Nielsen and Olesen 1978, Iwazaki 2008, classical instability in SU(2) gauge theory Itakura, Fujii, 2008 （charged vector fields are fluctuations around the gauge fields） （Electromagnetic fields represent the background gauge fields） The term can be positive or negative for arbitrary magnetic field B

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

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

10. Effective Lagrangian of describing the instability under inhomogeneousmagnetic fields is given such that using effective homogeneous magnetic field small effective mass ( imaginary ) 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. 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. We show that the monopole production leads to much more rapid decay of the gauge fields than the production of Nielsen-Olesen unstable modes

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

13. We calculate the decay time of background color electric (magnetic ) fields by using the effective Lagrangianof Niesen-Olesenunstable modes (magnetic monopoles ) in expanding glasma (τ,ηcoordinates ) Note that the monopoles occupy Landau levels under background electric field, while Nielsen-Olesen modes occupyLandau 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 monopoleproductionis 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. Effective Lagrangian of magnetic monopoles in the lowest Landau level Nielsen-Olesen monopole wave functions of the lowest Landau level under

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

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

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

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

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

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

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

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

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