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Soliton-antisoliton production in par ticle collisions
Sergei Demidov, Dmitry Levkov
Institute for Nuclear Research RAS levkov@ms2.inr.ac.ru

September 10, 2010

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

September 10, 2010

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Introduction

Topological solitons (1 + 1)
V()
1 S= 2 g dxdt (µ )2 /2 - V () =c=1

g -- semiclassical parameter and coupling constant ( = g )

v0

v+



S

v+
0

A

x

vS. Demidov, D. Levkov (INR RAS)

Proper ties: LS m-1 MS m/g 2 m -- mass scale of V ()

Soliton-antisoliton production

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Introduction

From par ticles to solitons

E 2M
S

S
LS 1/m

A

x

1/E g 2 /m Exponential suppression! Coherent­state "estimate" ¯ nS MS /m 1/g 2 2|SA
¯2 nS -n ¯S 2! e

P (E ) A(E ) · e-

F (E )/g

2

Drukier, Nussinov (1982)
2



e-c/g

Banks at al. (1990) Zakharov (1991)

Unitarity arguments for multipar ticle production No reliable estimate of P (E ) so far! Aim: calculate semiclassically F (E ).
S. Demidov, D. Levkov (INR RAS) Soliton-antisoliton production

September 10, 2010

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Semiclassical description

Semiclassical description

NOT tunneling?!


S

A

x



Many par ticles Attraction!

Not a tunneling process? Introduce a potential barrier:

V()


S -

A

x

vS

Critical bubble Barrier top

0

-
v+
4 / 15

0

: cr. bubble



SA; Ecr.b. 2M

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

September 10, 2010


Semiclassical description

Semiclassical description

In­state
Rubakov, Son, Tinyakov, 1992

Not semiclassical! E m/g
2

RST conjecture: F (E ) universal Does not depend on details of the in­state Checks of universality: 2 ^^^ Field theor y P (E , N ) = i ,f i |PE PN S |f
Tinyakov, 1991

Projectors N N 1

Mueller, 1992


2

semiclassical in­states

Toy QM models
Bonini et al, 1999 Levkov et al, 2009

1/g



F (E , N ) F (E ) F (E ) = lim F (E , N )
g 2 N 0

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

September 10, 2010

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Semiclassical description

Semiclassical description

Semiclassical method
P (E , N ) =
i ,f

^^^ i |PE PN S |f
B ( i , i ) 2

2

=

d i d

f

d i [d ] e
S [ ]

Rubakov, Son, Tinyakov, 1992 i (S + B ) 2

^^ i |PE PN |i = ei

^ i | S |

f

= [d ] ei

S 1/g



Saddle­point method!

(x , t ) C
¯ ak = e a
k

Im t
T

S / = 0 R Re t P = A · e-F /g F (E , N ) = 2g 2 Im(S + B )
2

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Numerical results

E = N = 0: Bounce

vR 1/ F 1/

Im t

v+

x

Im t Re t

Voloshin, Kobzarev, Okun, 1974

(x , t )
S. Demidov, D. Levkov (INR RAS) Soliton-antisoliton production

Coleman, 1977
September 10, 2010 7 / 15


Semiclassical description

Numerical results

Numerical solutions
V ( ) = 1 ( + 1) 2
5
2

1 - v · f(

- 1 a

),

f (x ) = e-

x

2

1 + x3 + x

5

S
4 3 2 1 0 0 2 4 6 8 10 Kinematically forbidden

= 0.4
S. Demidov, D. Levkov (INR RAS)

g2N

gE
Soliton-antisoliton production September 10, 2010 8 / 15

2


Semiclassical description

Numerical results

E < 2MS , direct tunneling
E 5. 48 N 2. 39, = 0. 4 ( 2M S 6. 23)

¯ ak = e a

k

Im t

T

R Re t

x

6 -

Re t

Im t

Re t
0

t

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Numerical results

0: thin­wall limit!
F ( ) = F F
-1 -1

/ + F0 + O ( )
E 2E
S

Voloshin, Selivanov, 1986 E2 2 4ES Rubakov et al, 1991

2 (E ) = ES - 2arcsin

-

E ES

1-

15

thin-wall

·F

10

= = = =

0.1 0.2 0.3 0.4 =0

5

0 3 4 5

2MS

g2E

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Numerical results

Going to E > 2M
5

S

S
4 3 2 1 0 0 2 4 6 Kinematically forbidden

Classically allowed

g2N

8

10

= 0.4
S. Demidov, D. Levkov (INR RAS)

gE
Soliton-antisoliton production September 10, 2010 11 / 15

2


Semiclassical description

Numerical results

Going to E > 2M

S

E 5. 48 N 2. 39, = 0. 4

( 2M S 6. 23)

E 9. 06 N 2. 47, = 0. 4

x

6 -

x Re t Im t Re t t

6 -

Re t



0

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Numerical results

Classically allowed solutions
5

S
4 3 2 1 0 0 2 4 6 Kinematically forbidden

Classically allowed

g2N

8

10

= 0.4
S. Demidov, D. Levkov (INR RAS)

gE
Soliton-antisoliton production September 10, 2010 13 / 15

2


Semiclassical description

Numerical results

Result

4.4

F

4.2

= 0.04 = 0.02 = 0

4

3.8

2MS

7

8

9 gE
2

10

11

12

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Conclusions

Conclusions

Method is applicable in 2D scalar field models. The probability of SA creation in high­energy collisions is P (E ) e-
F (E )/g
2

Generalizations to other models?

S. Demidov, D. Levkov (INR RAS)

Soliton-antisoliton production

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