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Nuclear Physics B (Proc. Suppl.) 73 (1999) 665-667

P R O C E E D I N G S S U P P L E M E N T S

Spha le ron ra te a t h igh t e m p e r a t u r e in 1+1 d imens ions

Jan Smit and Wai Hung Tang a *

aInstitute of Theoretical Physics, University of Amsterdam Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands

We resolve the controversy in the high temperature behavior of the sphaleron rate in the abelian Higgs model in 1+1 dimensions. The T 2 behavior at intermediate lattice spacings is found to change into T 2/8 behavior in the continuum limit. The results are supported by analytic arguments that the classical approximation is good for this model.

Sphaleron physics plays an important role in theories of baryogenesis. A simple but useful model is the abelian Higgs model in 1+1 dimen- sions, given by

s = +(Vu¢)*DU¢ L4g z

+ u: l¢ l 2 +

Recall that in 1+1 dimensions v/~ and g have di- mension of mass. A toy model for the electroweak theory is obtained by coupling to fermions, such that in the quantum theory the fermion current is anomalous. Changes in fermion number are then proportional to changes in Chern-Simons number

C = ~ dxl A1,

thereby mimicking the B+L violation in the Stan- dard Model. Here we assume space to be a circle of circumference L.

The sphaleron rate (of fermion number viola- tion) F can be identified from the diffusion of Chern-Simons number

F = -tl ( [C(t)_C(O)]2) , t--~oo.

For relatively low temperatures this rate is ex- ponentially suppressed by the sphaleron barrier. Numerical simulations [1-3] aggree with analytic results [4] in this regime. At high temperature the rate is not known analytically, but expected to be un-suppressed. For temperatures larger than any

*Supported by FOM

mass scale one may naively expect on dimensional grounds

F T (1)

Such behavior was indeed found by us numeri- cally and reported at LATTICE 94 [3]. However, De Forcrand, Krasnitz and Potting [1] gave a scal- ing argument that the behavior should instead be given by

F T213 Z " (2)

This behavior (2) can also be argued for as fol- lows. The simulations use the classical approxi- mation

<[C(t) - C(0)]2>

L,Ir e -Heff(cpJr)/T [ C ( ~ ( t ) , 7'r(;~)) - C(~p, 7"r)] 2

f~,rr e-H'ff(~'r)/T

Here %0 and 71" denote generic canonical variables and %o(t) and ~r(t) are solutions of the classi- cal Hamilton equations with initial conditions 9(0) = ~P, 7r(0) = 7r. The effective hamiltonian Heft is approximated by its classical form. Rescal- ing !a -+ tavfT, 7r -+ 7rv~ produces T only in the combination AT and e2T. Since this combination has mass dimension three, the behavior (2) in the form F/L (x (AT) ~/3 appears natural.

We analyzed the quality of the classical approx- imation in perturbation theory and found the fol- lowing favorable properties (in 1+1 dimensions!) [5l:

0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(99)00037-7

666 J. Smit, W.H. Tang~Nuclear Physics B (Proc. Suppl.) 73 (1999) 665-667

1.2,T'-','--,--","-r",""r'",'-T'-,-"T'",'"'r"',''r",,, 1.2 /- .-1 I~" a ,=-8 '-I 1.0~ <>,==12 ? 1.0

0.8]- •,,,=~e J 0.8

° L 1 0.2 0.2

0 ..~..~..~..~,. 0

0 2 4 6 la9~

: o ,e=-12 _~ -,~ .# , ,= ,6

. . . . .,°,,.~.,.,o,..sL 0 2 4 6

log z

Figure 3. Same as Fig. 1 for the first excited state in the Higgs phase.

E E o ~ ¢

v v . o~

i . . . . . . . . . . . i , , ° . . . . . . L . . . . , . . . . i ~ :

0.5 .,....,.~...= . . . . . . . . 1-° o o o o .

-IDA 0 0. t -0.1 0 0.1

Figure 4. Screening mass spectra near the crossover in channels 0 ++ and 1 - - ; 0++: (~7) Higgs states, (full triangles) W-ball states (O) Higgs states with an admixture of excited gauge d.o.f.

of the phase transition. Deeper in the Higgs phase the contribution from gauge d.o.f, is expected to disappear which has been checked in our simula- tions at M~ = 100 GeV.

The spectrum change has been studied in more detail at M~/ = 100 GeV while continuously passing the crossover line (changing/3H) at fixed gauge coupling /3c = 12. We have found a be- haviour very similar to our results obtained at M~ -- 70 GeV. The similarities concern both the high temperature side of the crossover (where one expects thermodynamic properties being close to those of the symmetric phase at smaller Higgs mass) and the region very near to the crossover line on the so-called Higgs side of the crossover.

In Fig. 4 we present the spectrum of the lowest states in the 0 ++ and 1 - - channels as function of

2 2 4 m3(g3)/g 3 (or /~H) over a certain interval above and below the crossover (i.e. in temperature). Looking at the excited states in the 0 ++ channel of Figs. 4 we conclude that the scalar and gauge sector are approximately decoupled as long as one keeps away from the crossover line on the high temperature side. The mass of the lowest W-ball

state (full triangle) is roughly independent of ~H as long as one does not come too close to the crossover. Thus, on the high temperature side of the crossover, the ordering of states qualitatively resembles the spectrum at smaller values of Higgs self-coupling (at M~ = 70 GeV).

If one approaches the crossover temperature the mass of the first (Higgs-like) excitation is moving up towards the lowest W-ball state whose mass decreases. At some point we observe a grow- ing admixture to the Higgs excitation by contri- butions from Wilson loop operators. We have ex- plicitly checked that at higher ~3H (deeper in the would-be Higgs phase) the admixture from pure gauge d.o.f, disappears again from this state.

To summarise, our operator choice was the sim- plest one to incorporate the notion of spatial ex- tension. The continuum limit of the screening masses and wave functions has to be accompa- nied by correspondingly larger lattices with the same physical volume, without automatically en- larged operator basis. Therefore, in a next step of improvement of the method, smearing of the fields entering our operators and/or the construc- tion of blocked operators will become necessary for further optimising the resolving capability of the method for excited states.

R E F E R E N C E S 1. K. Kajantie et al., Phys. Rev. Lett. 77 (1996)

2887; F. Karsch et al., Nucl. Phys. (Proc. Suppl.) 53 (1997) 623; M. Giirtler, E.-M. I1- genfritz, and A. Schiller, Phys. Rev. D56 (1997) 3888.

2. Y. Aoki and Z. Fodor, these proceedings 3. M.N . Chernodub et al., hep-lat/9805016, to

appear in Phys. Lett. B; hep-lat/9807016. 4. M. Giirtler et al., Nucl. Phys. B483 (1997)

383. 5. K. Kajantie et al., Nucl. Phys. B458 (1996)

90. 6. E.-M. Ilgenfritz, A. Schiller and C. Strecha,

hep-lat/9807023. 7. O. Philipsen, M. Teper, and H. Wittig, Nucl.

Phys. B469 (1996) 445; 8528 (1998) 379. 8. M. Ltischer and U. Wolff, Nucl. Phys. B339

(1990) 222; C. R. Gattringer and C. B. Lang, Nucl. Phys. B391 (1993) 463.

J Smit, W.H. Tang~Nuclear Physics B (Proc. Suppl.) 73 (1999) 665-667 667

of g(aZT, va/T), but notice how the combina- tion a3T implies non-commutativity of the limits a --~ 0 and T -~ c¢. Unfortunately, our a t tempts to extrapolate first to zero lattice spacing failed because the resulting errors got too large. Our present analysis seems to indicate a behavior like g ( a t , y ) " - - g ( 0 , 0 ) + ~ ( X y ) 2 / 3 j c o ~ X 4 / 3 " I - " / y 4 / 3 - [ - " • " , o r

F / L = [g( O, O) + fla2]T213 + o~a4T 2 + T T -2/3 + . . . .

R E F E R E N C E S

1. P. de Forcrand, A. Krasnitz and R. Potting, Phys. Rev. 050 (1994) 6054.

2. J. Smit and W.H. Tang, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 616.

3. J. Smit and W.H. Tang, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 590.

4. A.I. Bochkarev and G.G. Tsitsishvili, Phys. Rev. D4O (1989) 1378.

5. W.H. Tang and J. Smit, hep-lat/9805001.

0.03

0.02

0.01

- I . . . . I ' ' ' ' I ' ' ' '

0 0.01 0.02 0.03 a ' 2

Figure 2. Plot of C213 a s a function of a2[/.t[ 2.

0.06

0.05

0.04

r .

0.03

0.02

0.01

0 . 0 6

0 . 0 4 r._

0 . 0 2

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ~

, , , , I , , , ~ 1 , , , , I , ,

1.5 2 2.5 output T'2/3

. . . . I ' ' ' ' I ' ' ' ' I ' ' ' " ' I ' ' t

/ J

5 l 0 1 5 2 0

o u t p u t T z

Figure 3. Top: data showing T 2/3 behavior for alp[ = 0.16. Plotted is F versus T'2/3 and a fit to Co + c2/3T '2/3. Bottom: same data plotted versus T '2. The straight line is a fit of co + c2T '2. The curved line is the T '2/3 fit of the top figure. The data lack T 2 behavior but favor T'2/3 behavior in the region a[/~[ < 0.16.