Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs model

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Smit, J.; Tang, W.H.

Publication date

1995

Published in

Nuclear Physics B-Proceedings Supplement

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Citation for published version (APA):

Smit, J., & Tang, W. H. (1995). Sphaleron transition rate at high temperature in the (1+1)-d

abelian Higgs model. Nuclear Physics B-Proceedings Supplement, 42, 590-592.

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P R O C E E D I N G S

S U P P L E M E N T S

H.SI-:VlFR Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592

Sphaleron transition rate at high temperature in the 1+1 D abelian

Higgs model

J. Smit and W . H . T a n g *

I n s t i t u t e of T h e o r e t i c a l Physics, Valckenierstraat 65, 1018 XE A m s t e r d a m , the Netherlands

New results for the rate are presented using the canonical ensemble in the classical approximation on a spatial lattice. We find that the rate at high temperatures is proportional to T 2, and strongly dependent on the lattice spacing a. We conclude that a better effective action is needed for the classical approximation.

1. I n t r o d u c t i o n

T h e rate F of b a r y o n n u m b e r violation in the S t a n d a r d Model is difficult to calculate at finite t e m p e r a t u r e T. Therefore, the classical approxi- m a t i o n was proposed which allows for m a k i n g a numerical estimate. It was tested in the 1+1 dimensional abelian Higgs m o d e l [1-4], for which a semiclassical calculation of the rate is available,

F(m¢,(,~

v) --- m 2 ~ -

Here L is the one dimensional volume,

rn¢

is the Higgs mass, v - 2 =

2,~/rn2~

is the effective dimensionless coupling, ~ =

g2/)t, g

is the gauge coupling, )~ is the quartic Higgs self coupling, Es = ( 2 / 3 ) v 2 rn¢ is the classical sphaleron energy, and f ( ( ) is calculated in [5]. This f o r m u l a is ex- pected to be valid at t e m p e r a t u r e s

me

<< T << Es which is confirmed by numerical simulations [4,3]. At high t e m p e r a t u r e s the rate is still u n k n o w n even for this r a t h e r simple model. From dimensional a r g u m e n t s one expects t h a t the rate be- haves as T 2,

F ---- K(~, v) ( v - 2 T ) 2 for T >> Es. (2)

2. C l a s s i c a l a p p r o x i m a t i o n

At high t e m p e r a t u r e one expects high occu- p a t i o n n u m b e r s for q u a n t a with energies m u c h less t h a n T and for their c o n t r i b u t i o n a classical a p p r o x i m a t i o n seems reasonable. Suppose we *Speaker at the conference

- 2

SSDI

0920-5632(95)00320-7 4 _= 8 - 8 10 0 i I i t . _ \ ' I ' i i I ] i i 10 i i i I i 5 #'

Figure 1. Results for ln F = l n ( r / m ~ L ) for = 4. T h e diagonal line represents the analytic f o r m u l a (1). T h e other two lines are fits to the f o r m

co+c2 T '2.

T h e u p p e r d a t a are for a' = 0.32, the lower d a t a for a r = 0.16.

derive an effective action in which the spatial m o - m e n t a are limited to IPl < A. T h e n the effective energies will be restricted by A which suggests a classical a p p r o x i m a t i o n based on this effective action for A << T. Such an effective action S m i g h t be defined (in a generic field t h e o r y with generic fields ~) as follows,

[

exp [,~(~o, A)] = B A ( ~ , ~ ) e x p [S(~o)] . (3) Here BA is a blocking f u n c t i o n such t h a t f D ~ BA(~3, 9~) = 1, e.g.

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J Smit, VKH. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592

591 0 . 5 0 . 4 5 ~ 0 . 4 0 . 3 5 0 . 3 ' ' I ' ' ' I ' ' ' I ' ' I i ' ' I ' $ t , ~ I , , , I , , , I 2 4 6 , , I , , , I , ~ 8 10

Figure 2.

(1¢'?)

versus inverse t e m p e r a t u r e fl'. BA(qS, ~) = H

5[~(x, t ) - /

dp ~(p,

t)eiP~].(4)

(~,t) IPI<A

In this spirit we interpret the effective action for the abelian Higgs m o d e l on a lattice with lattice spacing a, = rt

[ 1-'~-- (A°n+l~

0tAln) 2

( 5 ) 1 a2

[exp(-iaAl~)¢,~+l -

6,~12

+ l(ot - iAo..)¢,~12 - .~ (l¢.12 - ~ ) ] ,

as an a p p r o x i m a t i o n to such an effective action (we o m i t t e d the ' b a r ' on the fields,

A

=

7r/a).

For convenience we m a k e

a scale t r a n s f o r m a t i o n to dimensionless vari- ables [2,3], a =

a'/vv/~, t = t'/vx/'~, d = v¢',

?

A# = vv/'-~A,,

rn¢ = v / ~ v r n ' ¢ , H = v/~v 3 H ' ,

T = v ~ v 3 T', /~' = 1/T',

where H is the hamil-

tonian. 3. R e s u l t s

Fig. 1 gives an overview of the rate o b t a i n e d in our simulation, which was carried o u t in the s a m e

- I ' ' ' ' I ' ' ' ' I ' ' ' ' I 0 . 0 2 0 . 0 1 5 r. 0 . 0 1 - 0 . 0 0 5 , , , , I , , , , I , , , , I , , 0 1 2 3 T , Z F i g u r e 3. F versus T '2 for a' = 0.16. T h e l i n e represents the fit F = 0.00335-I- 0.00512 T '2.

way as in [3]. T h e classical a p p r o x i m a t i o n is con- sistent with the semiclassical calculation in the semiclassical region [3,4]. We see also an inter- m e d i a t e region which is a l m o s t flat. This region appears to correspond to the m i n i m u m of ([¢,[2) in fig. 2. Figs. 3 a n d 4 show m o r e detailed d a t a at high t e m p e r a t u r e . We see here a T '2 behavior which starts already j u s t b e y o n d the m i n i m u m of ([¢,12). A c c o r d i n g to fig 1, the rate at high tem- p e r a t u r e depends on the lattice spacing a'. W i t h a gauge invariant lattice action for bosonic fields one expects the lattice spacing dependence to be like a series in a '2. This is confirmed in figs. 5 and 6 for two values of T ~. A linear a '2 a p p r o x i m a t i o n a p p e a r s to hold for a '2 < 0.1. We therefore fit the d a t a in figs. 3 and 4 to the following ansatz, F = coo + co2a '2 + (c2o + c 2 2 a ' 2 ) T

'2

a2m2¢

= COO n L c 0 2 - - 2

a2m2 \

+ 2

c 2 o + c 2 2 ~ - - ~ ) (v-2T)2

₍₆₎

and find coo = (3.1 + 0.1)10-3, co2 = (9.6 + 1.1)10 -3, c22 = 0.20-4- 0.02, e2o = ( - 7 -4- 25)10 - 5 . (7)

From this we see t h a t the v - 4 T 2 b e h a v i o r in eq. 2 is confirmed ( c o n t r a d i c t i n g [6]), with

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592 J. Smit, W.H. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592 0.1 I . . . . I . . . . I . . . . I . . . . I ' ' L o 0 . 0 6 0 . 0 4 0 . 0 2 0 I , , , , L , , , ~ t , , , , I , , , , I , , 0 1 2 3 4 T ' ~

Figure 4. F versus T ~2 for a ~ -- 0.32 and the fit F = 0.00409 + 0.0207 T '2.

~ ( ~ , v ) ~ 0 . 2 0 a Sm~, a t e = 4 . (8) The coefficient of T 2 depends strongly on the lattice scale. Therefore, it is absolutely necessary to understand the a dependence of the coefficients in the effective action before we can say to have calculated ~.

A c k n o w l e d g e m e n t s : We would like to thank A.I. Bochkarev and A. Krasnitz for useful discus- sions. This work is financially supported by the Stichting voor Fundamenteel Onderzoek der Ma- retie (FOM) and the Stichting Nationale Com- puter Faciliteiten (NCF).

R E F E R E N C E S

1. D. Yu. Grigoriev, V.A. Rubakov and M.E. Shaposhnikov, Nucl. Phys. B326 (1989) 737.

2. A.I. Bochkarev and Ph. de Forcrand, Phys. Rev. D44 (1991) 519.

3. J. Smit and W.H. Tang, Nuclear Physics B (Proc. Suppl.) 34 (1994) 616.

4. A. Krasnitz and R. Potting, Phys. Lett. B318 (1993) 492.

5. A.I. Bochkarev and Tsitsishvili, Phys. Rev. D40 (1989) 1378.

6. Ph. de Forcrand, A. Krasnitz and R. Potting, IPS Research Report No. 94-09 (1994).

] ' ' ' ' I ' ' ' ' I 't' 0 . 1 L~ 0.05 0 I , , ~ , I , , , , I , , , 0 0.1 0.2 o.w~

Figure 5. F versus a/2 for T ~ = 4/3. The line represents the fit F = 0.00307 + 0.387a ~2.

0 . 1 5 0 . 1 I ' ' ' ' I ' ' ' ' I ' ' ' ' ' ' ' ' I ' ' ' ' O.0~ . / Z h o . . . ...

f;>(

o os ~";/"

. J

0 0.1 0.2 0.3 0.4 0.5

Figure 6. F versus a '2 for T ' = l, and the fits f - - 0 . 0 0 3 4 7 + 0 . 1 9 2 a ~2 to the seven lowest data points, g = 0.00373 + 0.171a ~ to the five lowest data points and h = 0.00367+ 0.169a ~2 + 0.270a ~4 to all data. The insert zooms in on the region a ~2 ~ 0.1.

Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs model - 13603y