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

11 September 2010

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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Kink-like solitons in (1+1) scalar field theories (1 + 1)
V()
1 S= 2 g dxdt (µ )2 /2 - V () =c=1

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

v0

v+



S

v+
0

A

x

vD. Levkov, S. Demidov (INR RAS)

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

Soliton-antisoliton production

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Production of soliton-antisoliton pair

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

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Semiclassical description

Semiclassical description

HOWTO calculate: NOT tunneling?!


S

A

x



Many par ticles Attraction!

Not a tunneling process? Introduce a potential barrier:

V()


S -
bubble Barrier top

A

x
Critical

v0

-
v+
4 / 17

0

: cr. bubble



SA; Ecr.b. 2M

S

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

11 September 2010


Semiclassical description

Semiclassical description

HOWTO calculate: 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

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Semiclassical description

HOWTO calculate: Semiclassical method
P (E , N ) =
i ,f

^^^ i |PE PN S |f ^ i |S |f =

Rubakov, Son, Tinyakov, 1992 2

^^^ i |PE PN S |f =

i

^^ i |PE PN |i

d i ei
W

B (i ,i )

[d ]ei

S [ ]

P (E , N ) = [d ][d ] ei W 1/g
2



Saddle­point method!

Boundar y value problem ¯ ak = e a
k

Im t
T

S / = 0

(x , t ) C
R Re t

P = A · e-
D. Levkov, S. Demidov (INR RAS)

F /g

2

1 g2

F (E , N ) = 2ImS [] - TE - N
11 September 2010 6 / 17

Soliton-antisoliton production


Semiclassical description

Semiclassical description

Outline of the procedure
1 2 3

Fix small value Solve boundary value problem for fixed E and N and find F (E , N ) Take the limit lim F (E , N ) at nonzero
g 2 N 0

4

Take the limit 0 Star ting point, solution at E = 0 and N = 0 -- bounce

v-

Im t

v+

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

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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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 12 Kinematically forbidden

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

g2N

gE
Soliton-antisoliton production 11 September 2010 8 / 17

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

D. Levkov, S. Demidov (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

2 (E , N ) = ES - 2arcsin

-

E ES

1-

E2 2 4ES

Rubakov et al, 1991

15

thin-wall

·F

10

= = = =

0.1 0.2 0.3 0.4 =0

5

0 3 4 5

2MS

g2E

D. Levkov, S. Demidov (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

12

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

gE
Soliton-antisoliton production 11 September 2010 11 / 17

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

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Numerical results

Extrapolating to N 0
5

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

Classically allowed

g2N

8

10

12

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

gE
Soliton-antisoliton production 11 September 2010 13 / 17

2


Semiclassical description

Numerical results

Classically allowed region
1 2 3

0 (x ) =

cr .bubble

(x ) + (x )

Evolution according to classical field equations Find N in asymptotic future
S

4.4

4.2

g2N
4 3.8 6 7 8 9
2

10

gE
D. Levkov, S. Demidov (INR RAS) Soliton-antisoliton production 11 September 2010 14 / 17


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

D. Levkov, S. Demidov (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?

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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Semiclassical description

Conclusions

Limit 0
E 8. 95 N 2. 42, = 0. 02 E 9. 06 N 2. 47, = 0. 4

x

6 -

x Re t

6 -

Re t



0

D. Levkov, S. Demidov (INR RAS)

Soliton-antisoliton production

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