( ) 2 = #$ 2 % 2 + #$% 3 + # 4 % 4

Transcription

1 PC 477 The Early Universe Lectures 9 & 0 One is forced to make a choice of vacuum, and the resulting phenomena is known as spontaneous symmetry breaking (SSB.. Discrete Goldstone Model L =! µ"! µ " # V (" where! "! and V (! = V ("!, i.e. Z -symmetric. Choose V (! = " ( 4! # $. there exists two minima! = ",#" which are related by Z symmetry. What we do is choose one of the vacua, and expand around it.! = " + #.! V (" = # 4 $% + % = #$ % + #$% 3 + # 4 % 4 Compare this to L =! µ"! µ " # m " " + L int :-! m " = #$ L int =!"#$ 3! " 4 $ 4 NB: if we had chosen! = "# + $ then m! would be the same, but the interaction Lagrangian would have changed slightly. We say that the symmetry is spontaneously broken by the vacuum and the resulting particle has mass!".. U( Goldstone Model L =! µ "! µ " # $ ( 4 " # %! "! ;! = " + i"! " e i#! is a global symmetry of the Lagrangian, so it is U symmetric.

2 PC 477 The Early Universe Lectures 9 & 0 This is sometimes called the Mexican Hat Potential. There exists vacua along the circle S in the complex plane!,! given by! = ", all of which are related by the action of the symmetry group! " e i#!. As before, to find the theory about the vacuum, choose one point in the vacuum and expand around it.! = " + ( # + i#! = " + "# + # ( + # L =! µ"! µ " +! µ"! µ " # $% " + O( 3 There are two particles! and!, but one is massless since there is no quadratic term for!. The choice above is not unique. More generally,! = "e i# + $ + i$ for any!.! L = " µ# " µ # + " µ# " µ # + µ ij# i # j + L int where µ ij = µ ij (! is the mass matrix. In order to find the particle masses, diagonalise µ ij and the eigenvalues would be 0,!". there are two particles, one with mass!" 3. O(N General case: L =! µ"! µ " # V (" = V (! = 0 and define the vacuum manifold Potential is invariant under G, i.e. V g! Assume that min V! and the other which is massless.

3 PC 477 The Early Universe Lectures 9 & 0 { = 0} M 0 =! :V! Now assume that G acts transitively on M 0, that is, the action of G generates the whole of M 0 from any given point. given any!,! "M 0,!" #G such that! = "!. Define the stability group of a point a!m 0 to be H a = h!g : ha = a { } Although H varies with a, it does so in a simple way due to the transitivity property. i.e. they are isomorphic. Given a!m 0, M 0 = {! a :! "G}.! a = a " H a =! H #! $ H a! H a! a =! a "!! # $H This means that! and! are in the same coset of H in G. M = G H Expansion around! = a + ", where V ( a = 0;!V (!" a = 0. where µ ij =! V!" i!" j a L =! µ"! µ " # µ ij$ i $ j + L int $ Masses of the particles are the eigenvalues of µ ij. e.g. SO N! SO( N " ; massive, N! massless. N=3, G=SO(3 SU ( 3 symmetry. Minima occurs when! = " L =! µ"! µ " # $ ( 4 " # % " = ","," 3 M 0 = {! :! = "} # S Any point on the vacuum manifold can be obtained from any other by a rotation. a = ( 0,0,! H a is the group of rotations about the nd axis (i.e. keep z fixed; can rotate around x, y freely. In general, it will be a SO( subgroup of SO( 3. G = SO( 3, H = SO( G H = SO 3 SO(! S.! = a + " 3

4 PC 477 The Early Universe Lectures 9 & 0 where a,b, c L =! µ"! µ " # $ ( a %& + & 4 =! µ"! µ " # $ (' %& + L int! =! a +! b +! 3 c form an orthogonal triad a = b = c =! massive particle, massless. L =! µ"! µ " # $% & i + L int 3.3 Symmetry Restoration at High Temperature and the finite temperature effective potential (Finite temperature = non-zero temperature The concept of Grand Unification suggests that the theory at very high temperatures is described by a simple Lie Group G and that there have been a number of Phase Transitions which lead to the symmetry being broken to the standard model. G! H! K!...! SU ( 3 " SU ( " U (! SU ( 3 " U ( The next-to last of these is the Weinberg-Salam model + QCD. The last is QCD + EM Statistical Mechanics Thermodynamic Potential:! = "T log Z = E " TS " µn where S is the entropy, µ is the chemical potential. E! TS. log Z is the grand partition function. * # log Z = ±V S d 3 %" ( E ( k k 3 log ± exp " µ '% - 0, $ (/ (! +, &% T %./ where V S is the volume, the positive case is for fermions and the negative case is for bosons. Free Energy ( µ = 0 : F = E! TS = "( µ = 0. F V S =! / d 3 k! log ± exp # " E k &, 3 + $ '. * % T ( Effective Potential The Lagrangians (e.g. Goldstone Model studied so far are for a single field corresponding to possibly a number of particles related by a symmetry. But we like to couple these particles to a thermal heat bath of particles with temperature T. Basic Concept: There exists an effective potential which encompasses the Thermal corrections. It has been shown that the computation of these thermal corrections is the same as computing the free energy. Therefore: V eff (!,T = V (! + " f n (!,T n 4

5 PC 477 The Early Universe Lectures 9 & 0 where f n (!,T = F V for the nth particle. and E k For Bosons, f n = k + m n (!. For Fermions, Hence, (!,T =!T V eff f n =! " * f n =! 7" where N = N B N f, where N B N f is the number of dof for fermions. d 3 k " log $ ± exp $ # E k '' 3 & & % T % (( # 90 T 4 + m n 4 # 70 T 4 + m n 48 T 4 + O( m n T 4 + O( m n (!,T = V (! + ( 4 m!t " # 90 NT 4 m! is the number of bosonic degrees of freedom and " + = m n (! B " f m n (! Effective Potential of Goldstone Model with U( Symmetry L =! µ "! µ " # V " with V (! = " ( 4! # $.! "! #! = ( $ + i$ L =! µ"! µ " +! µ"! µ " # $ 4 NB:! =! ( +! Eigenvalues give the masses:! V!" i!" j = # & " + " ' ( $ % m (! = " 3! # $ & " + " ' ( m (! = "! # $ # % * +, + # ij " i" j NB: If! = ", i.e. on the vacuum manifold, then m =!" and m = 0. V eff (!,T = " ( 4! # $ + " ( 4! # $ T # % 45 T 4 m eff = coefficient of! = " ( T # 6$ = " T ( # T c * + 5

6 PC 477 The Early Universe Lectures 9 & 0 where T c = 6! is the critical temperature. For T > T c, m eff ( T > 0 single minima and full U ( symmetry. For T < T c, m eff ( T < 0 spontaneous symmetry breaking, and there exists degenerate vacua. The symmetry is said to have been restored at high temperature and is broken at low temperature. 3.4 Phase Transitions 3.4. nd Order Phase Transitions Consider V eff (!,T = m eff ( T (! + " ( 4!4 m eff =! ( 6 T " 6# Sketch the curve f ( x = m eff ( T x +! 4 x4, x > 0. f '( x = m eff x +!x 3 = 0 x = 0 & x =! m eff f ' x f '' x = m eff + 3!x x = 0, f '' x x =! m eff = m eff ", f '' x If m eff > 0 T > T c If m eff < 0 T < T c x = ( 6 T c! T. =!m eff " = ( 6 T c! T., then there exists a single minimum at x = 0., then there exists a maxima at x = 0 and a minimum of The parameter grows continuously with T for T < T c towards x =!, which is characteristic of a second order phase transition. This is the only kind of phase transition we will discuss here. 6