Solitons and instantons in an effective model of CP violation

Transcription

1 Solitons and instantons in an effective model of CP violation by N. Chandra, M. B. Paranjape, R. Srivastava GPP, Université de Montréal, Montreal and Indian Institute of Science, Bangalore arxiv:

2 Solitons and instantons in models with CPviolation Solitons are stable localized solutions of the classical equations of motion of a field theory. They represent particle-like states of the full quantum field theory. They are usually static or stationary, in some Lorentz frame. The are usually non-perturbative in the coupling constants of the theory. Instantons are solutions of the Euclideanized field equations. They are associated with tunnelling transitions and decays of meta-stable states.

3 Instantons contribute to the Euclidean path integral Z E [J] = 1 N Dφ e S E[φ]+ Jφ = h e Ĥ+ R ˆJ 'i Stationary action implies the classical equations of motion S E =0 ) (x) = 0 (x)

4 Then the path integral can be approximated by the saddle point method Z E [J] =e S E( 0 ) R D e R S E (1 + ) Often this gives a non-perturbative contribution to various matrix elements, describing tunnelling transition and decays.

5 CP violation in the standard model occurs because of the weak interactions. L = Liai µ ( µ A ij µ ) Lja + Rai µ ( µ ) Rb + LiaM ia,b Rb + h.c. Lorentz invariance implies, because of the CPT theorem, that the theory must also be time reversal violating. This requires a complex Lagrangian, real Lagrangians cannot violate time reversal. It requires at least three flavours of quarks and leptons since two flavours means the symmetry is SU() which is real.

6 Effective models of CP violating decays The kind of processes we are interested in have one scalar particle decaying into many lighter scalars. B! KK For example is a much studied CP violating decay available at LHC. An effective theory containing such a decay is given by the Lagrangian L =1/ P 5 i=1 (@ µ i@ µ i m i i i) ijklm µ i@ µ l@ m

7 In fact the interaction term is not CP violating in this case as the pseudo-scalar fields are CP odd. But it is CP odd if the fields are normal scalar. For the pseudo-scalar fields it mediates anomalous processes such as: KK!

8 Equations of motion and energy The corresponding equations of motion µ i + m i i +5/ ijklm µ l@ m =0 The energy density is given by: T 0 0 = 1 h i i + 0 i 0 i + m i i i

9 Field ansätze in the 3+1 or 4 dimensional theory We cannot find exact or even approximate solutions in the 3+1 or 4 dimensional theories. However we do find field configurations which seem to have interesting properties. We speculate that there will exist long lived or quasi-stable configurations in the field theories corresponding to our ansätze, as in the lower dimensional analogous models we can actually find exact solutions.

10 Consider a configuration parametrized as: 1 + i = f(r)e i!t ( 3, 4, 5) =g(r)ˆr(, ) The corresponding equations of motion are simply two coupled non-linear differential equations for f(r) and g(r) :! f (1/r )(r f 0 ) 0 + m f +60!g 0 g f/r = 0 (1/r )(r g 0 ) 0 60!g f 0 f/r = 0

11 Z 1+1 dimensional analogous model Here we simply mimic the higher apple dimensional model. We get: L L = 1 (@µ i )(@ µ i) m ( 1 + )+ ijk µ i(@ µ j )(@ k ) apple The corresponding equations of motion µ i + m i i ummation over ). 3 ijk µ (@ µ j )(@ k )=0

12 Z Field ansatz in Z 1+1 dimensions We take the following field configuration: 1 = f(x)cos!(t t 0 ) = f(x)sin!(t t 0 ) 3 = g(x) which gives the field equations: (m! )f f 00 3!fg 0 =0 g 00 +3!ff 0 =0

13 The second equation has an immediate first integral: g 0 = 3!f A The integration constant is fixed by insisting the energy is finite. "(x) = 1 f 0 + g 0 +(m +! )f Thus f,g,f 0,g 0! 0,r!1 which implies the integration constant is zero.

14 Then inserting into the first equation gives: f ! f 3 (m! )f =0 By magic this is the the non-linear Schrödinger equation, which is integrable. f 00 = du(f) df The equation can be rewritten as: U(f) = 9! coordinate" 8 f 4 m! f of a particle" wit

15 This is equivalent to particle motion in the potential with t! x There are two cases: m Ω 0 U f O f

16 and the opposite: m Ω 0 U f A O D B C f

17 . The -fields become q In the first, there is obviously no solution except the trivial solution f(x) =0 but in the latter, the equation integrates easily as f = p m! 3! yielding, after one integration, 33) andintegratinggives g = p m! 3! apple hph i sech m! (x x m ) hp i tanh m! (x x m ) + g m

18 h h i i The solution in terms h of the original fields i then is: 1 = p m! 3! = p m! 3! h i hp i sech m! (x x m ) cos!(t t 0 ) h hp i sechh m! (x x m i) sin!(t t 0 ) p 3 = p m! hp i tanh h m! i 3! (x x m ) + g m The energy can be calculated analytically, we get: being arbitrary constrained only by. While E = 8mp m! 9! i

19 ! The graph of the energy is: E Ω m Ω O m

20 The charge for the unbroken U(1) subgroup, 1! 0 1 =cos 1 sin 1 for infinitesimal! 0 =sin 1 +cos 1 + for infinitesimal 3! 0 3 = 3 The density is given by j 0 (x) = charge Z apple 1 3 ( 1 + ) 0 0 3( 1 + )

21 The corresponding charge is: Q = 8p m! 9! Comparing with the energy: E = 8mp m! 9! which can be solved as 1+ (m! ) 3!! = 64m4 + p (64m 4 ) +4(64m 6 )81 4 E 81 4 E.

22 In the large energy limit we get! 8m3 9 E 1+o m E Q = p m 8 m! 3! 9! p m 8m 3! 9! E m! 1+o m 3/ Thus E m p Q/ /3 a 1+1 dimensional - Which is strange, usually E Q 1/. -balls.

23 For small energy limit we have p m!! = which then gives which means p m!!! 9 E 8m m 1 n express the charge in terms of and,weget Q = E m E 18m + indicating stability even for small charges 64m E E<mQ

24 Generalizations of the Lagrangian Since can add a potential to the Lagrangian tanh +sech =1 +γ ( φ i φ i v ) ] With the appropriate choice we v = m ω 3λω the configuration remains a solution.

25 L E = 1 Instantons We obtain the equations for instantons by analytically continuing t! i The Lagrangian becomes: [ η µν E ( µφ i )( ν φ i )+m i φ i iλϵ µν E ϵ ijkφ i ( µ φ j )( ν φ k ) ] The novel aspect of this Lagrangian is that it is complex. Instantons are classical solutions of the corresponding equations of motion.

26 The solutions exist and in principle mediate tunnelling and decay in the theory. We can only find analytic solutions for the massless limit, but then they generalize to solutions with a quartic potential that is not of the symmetry breaking type. The novel thing is the solution exists only for complex field configurations. The equations of motion are µ µ φ i m i φ i + 3iλ ϵ µν ϵ ijk ( µ φ j )( ν φ k ) = 0

27 We start with the ansatz φ 1 (r, θ) = if(r) cos ωθ φ (r, θ) = if(r) sinωθ φ 3 (r, θ) = ig(r) where all of the fields are imaginary! r f + rf (m r + ω ) f +3ωλrfg = 0 r g + rg m 3r g 3ωλrff = 0 For the massless limit, we can integrate these equations analytically.

28 We can rewrite the second equation as (rg ) = 3 ωλ(f ) which integrates as g = 3 r ωλf + c 1 r ( ) replacing in the first equation with the constant taken to be zero, we get r f + rf ω f + 9 ω λ f 3 = 0

29 This equation can also be integrated as r f f + r (f ) = ω f f 9 ω λ f f 3 ( ) r (f ) = ( ω f 9 8 ω λ f 4 ) which yields f f ( λ f ) 1 = ± ω r

30 ( ) which then integrates to ie. 1 ln then 1 ( λ f ) 1 1+ ( λ f ) 1 ( ( ( ) ) ) 4 f = 4 3λ = ± ω ln c 3 r (c 3 r) ±ω ((c 3 r) ±ω +1) g = 4 3λ 1 ((c 3 r) ω +1) + c 4

31 For the fields we have φ 1 (r, θ) = 4i (c 3 r) ω 3λ ((c 3 r) ω cos ωθ +1) φ (r, θ) = 4i (c 3 r) ω 3λ ((c 3 r) ω sin ωθ +1) φ 3 (r, θ) = 4i 1 3λ ((c 3 r) ω +1) + c 4 The action is given by S E = 1 0 rdr π 0 dθ [ ( φi r ) + 1 r ( φi θ ) iλϵ ijk φ i r ( φj r )( φk θ ) ] which is easily calculated exactly

32 We get We can add interaction terms since ] the solution satisfies φ 1 + φ + φ 3 + v = with we get c 4 = i 3λ S E = 8! 7. 9 (( 3 ) +1) 3 (( 3 ) +1) [ 16 9λ 8ic ] 4 1 3λ ((c 3 r) ω +1) +(c 4 + v ) φ 1 + φ + φ 3 + v = v = c 4 = 4 9λ, This gives the additional Lagrangian ( density φi φ i + v ) L E = γ 0

33 Conclusions We have found exact solitons and instantons in a 1+1 dimensional analog effective model of CP violations The solitons corresponds particle states, the instantons presumably mediate quantum tunnelling. The solutions are of the Q-ball type, and hence we believe they are stable. We do not understand what tunnelling is being described. 3+1 dimensional effective theories may contain domain walls corresponding to our solitons and thin wall instanton configurations. The import to particle physics or early universe might be important, as such solutions usually contribute to nonperturbative CP violating processes in the field theory.