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Production of kink­antikink pair in collisions of high­energy par ticles
Dmitry Levkov, Sergei Demidov
Institute for Nuclear Research RAS demidov@ms2.inr.ac.ru

August 22, 2009

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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Introduction

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

semiclassical parameter coupling constant ( = g )

v0

v+



S

v+ v-

0

A

x

Proper ties: LS m-1 MS m/g

2

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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Introduction

From par ticles to solitons

E 2M
S

S
LS 1/m P (E ) A · e-

A

x

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

F (E )/g

2

Drukier, Nussinov (1982) c/g
2

Banks at al. (1990) Zakharov (1991)

No reliable estimate of P (E ) so far! Aim: calculate semiclassically F (E ).
D. Levkov, S. Demidov (INR RAS) Production of kink­antikink pair August 22, 2009 3 / 12


Semiclassical description

Semiclassical description

Introducing potential barrier


S
Attraction!

A

x



Many par ticles

Not a tunneling process?

V()


S -

A

x

vS

Critical bubble Barrier top

0

-
v+
August 22, 2009 4 / 12

0

: cr. bubble



SA; Ecr.b. 2M

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair


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 theory P (E , N ) = i ,f i |PE PN S |f
Tinyakov, 1991

Projectors N1

Mueller, 1992


2

semiclassical in­states

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

N 1/g



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

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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Semiclassical description

Semiclassical description

Semiclassical method
Rubakov, Son, Tinyakov, 1992

P (E , N ) =
i ,f

^^^ i |PE PN S |f W 1/g
2

2

= [d ][d ]



eiW



Saddle­point method!

¯ ak = e a

k

Im t
T

S / = 0 R Re t

(x , t ) C

P = A · e-

F /g

2

1 g2

F (E , N ) = 2ImS [] - TE - N

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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Semiclassical description

Numerical results

Numerical solutions
V () = 1 ( + 1)2 1 - v · f ( 2
5
- 1 a

),

f (x ) = e-

x

2

1 + x3 + x

5

S
4 3 2 1 0 Kinematically forbidden

Classically allowed

g2N

= 0.4

0

2

4

6

8

10

12

gE
Production of kink­antikink pair August 22, 2009 7 / 12

2

D. Levkov, S. Demidov (INR RAS)


Semiclassical description

Numerical results

Going to E > 2M

S

E 5.48 N 2.39, = 0.4

(2MS 6.23)

E 9.06 N 2.47, = 0.4

x

6 -

x Re t Im t Re t t

6 -

Re t



0

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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Semiclassical description

Numerical results

0: thin­wall limit!
F () = F F
-1 -1

/ + F0 + O ()
E 2ES

v1-
E2 2 4ES

Im t

(E , N ) = E

2 S

- 2arcsin

-

E ES

v+

E =N=0 x R 1/ F 1/

Voloshin, Selivanov, 1986 Rubakov et al, 1991

30

·F

20 10 0 0 2

thin-wall = 0.4 = 0.2

4

2M

S

gE
2
D. Levkov, S. Demidov (INR RAS) Production of kink­antikink pair August 22, 2009 9 / 12


Semiclassical description

Numerical results

Extrapolating to N 0
5

S
4 3 2 1 0 Kinematically forbidden

Classically allowed

g2N

= 0.4

0

2

4

6

8

10

12

gE
Production of kink­antikink pair August 22, 2009 10 / 12

2

D. Levkov, S. Demidov (INR RAS)


Semiclassical description

Numerical results

Result

4.4 4.2

F

= 0 = 0.02 = 0.4

4 3.8 2M
S

7

8

9

10

g2E
D. Levkov, S. Demidov (INR RAS) Production of kink­antikink pair August 22, 2009 11 / 12


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

D. Levkov, S. Demidov (INR RAS)

Production of kink­antikink pair

August 22, 2009

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