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.Comment: 3 pages, uuencoded compressed postscript file, contribution to the
  Conference Proceedings "LATTICE 94", Bielefeld- German

Bochkarev

de Forcrand

Grigoriev

J. Smit

Krasnitz

Smit

W.H. Tang

English

arXiv

Smit, J.

Tang, W.H.

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs modelSmit, J.; Tang, W.H.Publication date1995Published inNuclear Physics B-Proceedings SupplementLink to publicationCitation for published version (APA):Smit, J., & Tang, W. H. (1995). Sphaleron transition rate at high temperature in the (1+1)-dabelian Higgs model. Nuclear Physics B-Proceedings Supplement, 42, 590-592.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:15 Apr 2023~ LIIII I ~_,I " |  '~ -" I't,'t [1, '!  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. Tang * Inst i tute  of Theoret ical  Physics, Valckenierstraat  65, 1018 XE Amste rdam,  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  The rate F of baryon  number  violation in the S tandard  Model is difficult to calculate at finite t empera ture  T. Therefore, the classical approxi- ma t ion  was proposed which allows for making  a numerical  est imate.  It was tested in the 1+1 di- mensional  abelian Higgs model  [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 di- mensionless coupling, ~ = g2/)t, g is the gauge coupling, )~ is the quart ic  Higgs self coupling, Es = (2/3)v 2 rn¢ is the classical sphaleron energy, and f ( ( )  is calculated in [5]. This formula  is ex- pected to be valid at t empera tures  me << T << Es which is confirmed by numerical  s imulat ions [4,3]. At  high tempera tures  the rate is still unknown even for this rather  simple model.  From dimen- sional a rguments  one expects tha t  the rate be- haves as T 2, F ---- K(~, v) (v -2T)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 tempera tu re  one expects high occu- pat ion numbers  for quan ta  with energies much less than T and for their contr ibut ion a classi- cal approx imat ion  seems reasonable. Suppose we *Speaker at the conference - 2  0920-5632195/$09.50© 1995 Elsevier Science B.V. All rights reserved. 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. The diagonal line represents the analyt ic  formula  (1). The  other two lines are fits to the form co+c2 T '2. The upper  da ta  are for a '  = 0.32, the lower da ta  for a r = 0.16. derive an effective action in which the spatial  mo- men ta  are l imited to IPl < A. Then  the effective energies will be restricted by A which suggests a classical approximat ion  based on this effective ac- tion for A << T. Such an effective action S might  be defined (in a generic field theory with generic fields ~) as follows, [ exp [,~(~o, A)] = B A ( ~ , ~ ) e x p  [S(~o)] . (3) Here BA is a blocking funct ion such tha t  f D~ BA(~3, 9~) = 1, e.g. J Smit, VKH. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592 591 0.5 0 .45  ~ 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 tempera ture  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 model  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 approximat ion  to such an effective ac- tion (we omi t t ed  the ' ba r '  on the fields, A = 7r/a). For convenience we make a scale t rans format ion  to dimensionless vari- ables [2,3], a = a'/vv/~, t = t'/vx/'~, d = v¢', ? A# = vv/'-~A,, rn¢ = v/~vrn '¢,  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 obtained in our simulat ion,  which was carried out  in the same - 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  Figure 3. F versus T '2 for a' = 0.16. The  l i n e  represents the f i t  F = 0.00335-I- 0.00512 T '2. way as in [3]. The  classical approximat ion  is con- sistent with the semiclassical calculation in the semiclassical region [3,4]. We see also an inter- mediate  region which is a lmost  flat. This region appears to correspond to the m i n i m u m  of ([¢,[2) in fig. 2. Figs. 3 and 4 show more  detailed da ta  at high temperature .  We see here a T '2 behavior  which starts  already just  beyond the m i n i m u m  of ([¢,12). According to fig 1, the rate at high tem- perature  depends on the lattice spacing a ' .  W i t h  a gauge invariant  lattice act ion 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 approximat ion  appears  to hold for a '2 < 0.1. We therefore fit the da ta  in figs. 3 and 4 to the following ansatz,  F = coo + co2a '2 + (c2o + c22a '2)T '2 a2m2¢ = COO n L c 0 2 - -  2 a2m2 \ + 2  c2o+c22~- -~ )  (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 tha t  the v -4  T 2 behavior  in eq. 2 is confirmed (contradict ing [6]), with 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 .20a  Sm~, a t e = 4 .  (8) The coefficient of T 2 depends strongly on the lat- tice 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 .00347+0.192a  ~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

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs modelSmit, J.; Tang, W.H.Publication date1995Published inNuclear Physics B-Proceedings SupplementLink to publicationCitation for published version (APA):Smit, J., & Tang, W. H. (1995). Sphaleron transition rate at high temperature in the (1+1)-dabelian Higgs model. Nuclear Physics B-Proceedings Supplement, 42, 590-592.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:26 Jul 2022~ LIIII I ~_,I " |  '~ -" I't,'t [1, '!  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. Tang * Inst i tute  of Theoret ical  Physics, Valckenierstraat  65, 1018 XE Amste rdam,  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  The rate F of baryon  number  violation in the S tandard  Model is difficult to calculate at finite t empera ture  T. Therefore, the classical approxi- ma t ion  was proposed which allows for making  a numerical  est imate.  It was tested in the 1+1 di- mensional  abelian Higgs model  [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 di- mensionless coupling, ~ = g2/)t, g is the gauge coupling, )~ is the quart ic  Higgs self coupling, Es = (2/3)v 2 rn¢ is the classical sphaleron energy, and f ( ( )  is calculated in [5]. This formula  is ex- pected to be valid at t empera tures  me << T << Es which is confirmed by numerical  s imulat ions [4,3]. At  high tempera tures  the rate is still unknown even for this rather  simple model.  From dimen- sional a rguments  one expects tha t  the rate be- haves as T 2, F ---- K(~, v) (v -2T)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 tempera tu re  one expects high occu- pat ion numbers  for quan ta  with energies much less than T and for their contr ibut ion a classi- cal approx imat ion  seems reasonable. Suppose we *Speaker at the conference - 2  0920-5632195/$09.50© 1995 Elsevier Science B.V. All rights reserved. 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. The diagonal line represents the analyt ic  formula  (1). The  other two lines are fits to the form co+c2 T '2. The upper  da ta  are for a '  = 0.32, the lower da ta  for a r = 0.16. derive an effective action in which the spatial  mo- men ta  are l imited to IPl < A. Then  the effective energies will be restricted by A which suggests a classical approximat ion  based on this effective ac- tion for A << T. Such an effective action S might  be defined (in a generic field theory with generic fields ~) as follows, [ exp [,~(~o, A)] = B A ( ~ , ~ ) e x p  [S(~o)] . (3) Here BA is a blocking funct ion such tha t  f D~ BA(~3, 9~) = 1, e.g. J Smit, VKH. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592 591 0.5 0 .45  ~ 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 tempera ture  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 model  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 approximat ion  to such an effective ac- tion (we omi t t ed  the ' ba r '  on the fields, A = 7r/a). For convenience we make a scale t rans format ion  to dimensionless vari- ables [2,3], a = a'/vv/~, t = t'/vx/'~, d = v¢', ? A# = vv/'-~A,, rn¢ = v/~vrn '¢,  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 obtained in our simulat ion,  which was carried out  in the same - 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  Figure 3. F versus T '2 for a' = 0.16. The  l i n e  represents the f i t  F = 0.00335-I- 0.00512 T '2. way as in [3]. The  classical approximat ion  is con- sistent with the semiclassical calculation in the semiclassical region [3,4]. We see also an inter- mediate  region which is a lmost  flat. This region appears to correspond to the m i n i m u m  of ([¢,[2) in fig. 2. Figs. 3 and 4 show more  detailed da ta  at high temperature .  We see here a T '2 behavior  which starts  already just  beyond the m i n i m u m  of ([¢,12). According to fig 1, the rate at high tem- perature  depends on the lattice spacing a ' .  W i t h  a gauge invariant  lattice act ion 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 approximat ion  appears  to hold for a '2 < 0.1. We therefore fit the da ta  in figs. 3 and 4 to the following ansatz,  F = coo + co2a '2 + (c2o + c22a '2)T '2 a2m2¢ = COO n L c 0 2 - -  2 a2m2 \ + 2  c2o+c22~- -~ )  (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 tha t  the v -4  T 2 behavior  in eq. 2 is confirmed (contradict ing [6]), with 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 .20a  Sm~, a t e = 4 .  (8) The coefficient of T 2 depends strongly on the lat- tice 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 .00347+0.192a  ~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. 

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

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Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs model
Smit, J.; Tang, W.H.
Published in:
Nuclear Physics B-Proceedings Supplement
Link to publication
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.
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),
other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating
your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask
the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,
The Netherlands. You will be contacted as soon as possible.
Download date: 09 Feb 2021
~ LIIII I ~_,I " |  '~ -" I't,'t [1, '!  
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. Tang * 
Inst i tute  of Theoret ical  Physics, Valckenierstraat  65, 1018 XE Amste rdam,  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  
The rate F of baryon  number  violation in the 
S tandard  Model is difficult to calculate at finite 
t empera ture  T. Therefore, the classical approxi- 
ma t ion  was proposed which allows for making  a 
numerical  est imate.  It was tested in the 1+1 di- 
mensional  abelian Higgs model  [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 di- 
mensionless coupling, ~ = g2/)t, g is the gauge 
coupling, )~ is the quart ic  Higgs self coupling, 
Es = (2/3)v 2 rn¢ is the classical sphaleron energy, 
and f ( ( )  is calculated in [5]. This formula  is ex- 
pected to be valid at t empera tures  me << T << Es 
which is confirmed by numerical  s imulat ions [4,3]. 
At  high tempera tures  the rate is still unknown 
even for this rather  simple model.  From dimen- 
sional a rguments  one expects tha t  the rate be- 
haves as T 2, 
F ---- K(~, v) (v -2T)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 tempera tu re  one expects high occu- 
pat ion numbers  for quan ta  with energies much 
less than T and for their contr ibut ion a classi- 
cal approx imat ion  seems reasonable. Suppose we 
*Speaker at the conference 
- 2  
0920-5632195/$09.50© 1995 Elsevier Science B.V. All rights reserved. 
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. The diagonal line represents the analyt ic  
formula  (1). The  other two lines are fits to the 
form co+c2 T '2. The upper  da ta  are for a '  = 0.32, 
the lower da ta  for a r = 0.16. 
derive an effective action in which the spatial  mo- 
men ta  are l imited to IPl < A. Then  the effective 
energies will be restricted by A which suggests a 
classical approximat ion  based on this effective ac- 
tion for A << T. Such an effective action S might  
be defined (in a generic field theory with generic 
fields ~) as follows, 
[ 
exp [,~(~o, A)] = B A ( ~ , ~ ) e x p  [S(~o)] . (3) 
Here BA is a blocking funct ion such tha t  
f D~ BA(~3, 9~) = 1, e.g. 
J Smit, VKH. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592 591 
0.5 
0 .45  
~ 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 tempera ture  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 model  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 approximat ion  to such an effective ac- 
tion (we omi t t ed  the ' ba r '  on the fields, 
A = 7r/a). For convenience we make 
a scale t rans format ion  to dimensionless vari- 
ables [2,3], a = a'/vv/~, t = t'/vx/'~, d = v¢', 
? A# = vv/'-~A,, rn¢ = v/~vrn '¢,  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 obtained in 
our simulat ion,  which was carried out  in the same 
- 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  
Figure 3. F versus T '2 for a' = 0.16. The  l i n e  
represents the f i t  F = 0.00335-I- 0.00512 T '2. 
way as in [3]. The  classical approximat ion  is con- 
sistent with the semiclassical calculation in the 
semiclassical region [3,4]. We see also an inter- 
mediate  region which is a lmost  flat. This region 
appears to correspond to the m i n i m u m  of ([¢,[2) 
in fig. 2. Figs. 3 and 4 show more  detailed da ta  
at high temperature .  We see here a T '2 behavior  
which starts  already just  beyond the m i n i m u m  of 
([¢,12). According to fig 1, the rate at high tem- 
perature  depends on the lattice spacing a ' .  W i t h  
a gauge invariant  lattice act ion 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 approximat ion  
appears  to hold for a '2 < 0.1. We therefore fit the 
da ta  in figs. 3 and 4 to the following ansatz,  
F = coo + co2a '2 + (c2o + c22a '2)T '2 
a2m2¢ 
= COO n L c 0 2 - -  
2 
a2m2 \ 
+ 2  c2o+c22~- -~ )  (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 tha t  the v -4  T 2 behavior  in eq. 2 
is confirmed (contradict ing [6]), with 
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 .20a  Sm~, a t e = 4 .  (8) 
The coefficient of T 2 depends strongly on the lat- 
tice 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 .00347+0.192a  ~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. 


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Sphaleron transition rate at high temperature in the (1+1)-d abelian Higgs model
Smit, J.; Tang, W.H.
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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Download date: 05 Feb 2021
~ LIIII I ~_,I " |  '~ -" I't,'t [1, '!  
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. Tang * 
Inst i tute  of Theoret ical  Physics, Valckenierstraat  65, 1018 XE Amste rdam,  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  
The rate F of baryon  number  violation in the 
S tandard  Model is difficult to calculate at finite 
t empera ture  T. Therefore, the classical approxi- 
ma t ion  was proposed which allows for making  a 
numerical  est imate.  It was tested in the 1+1 di- 
mensional  abelian Higgs model  [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 di- 
mensionless coupling, ~ = g2/)t, g is the gauge 
coupling, )~ is the quart ic  Higgs self coupling, 
Es = (2/3)v 2 rn¢ is the classical sphaleron energy, 
and f ( ( )  is calculated in [5]. This formula  is ex- 
pected to be valid at t empera tures  me << T << Es 
which is confirmed by numerical  s imulat ions [4,3]. 
At  high tempera tures  the rate is still unknown 
even for this rather  simple model.  From dimen- 
sional a rguments  one expects tha t  the rate be- 
haves as T 2, 
F ---- K(~, v) (v -2T)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 tempera tu re  one expects high occu- 
pat ion numbers  for quan ta  with energies much 
less than T and for their contr ibut ion a classi- 
cal approx imat ion  seems reasonable. Suppose we 
*Speaker at the conference 
- 2  
0920-5632195/$09.50© 1995 Elsevier Science B.V. All rights reserved. 
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. The diagonal line represents the analyt ic  
formula  (1). The  other two lines are fits to the 
form co+c2 T '2. The upper  da ta  are for a '  = 0.32, 
the lower da ta  for a r = 0.16. 
derive an effective action in which the spatial  mo- 
men ta  are l imited to IPl < A. Then  the effective 
energies will be restricted by A which suggests a 
classical approximat ion  based on this effective ac- 
tion for A << T. Such an effective action S might  
be defined (in a generic field theory with generic 
fields ~) as follows, 
[ 
exp [,~(~o, A)] = B A ( ~ , ~ ) e x p  [S(~o)] . (3) 
Here BA is a blocking funct ion such tha t  
f D~ BA(~3, 9~) = 1, e.g. 
J Smit, VKH. Tang~Nuclear Physics B (Proc. Suppl.) 42 (1995) 590-592 591 
0.5 
0 .45  
~ 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 tempera ture  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 model  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 approximat ion  to such an effective ac- 
tion (we omi t t ed  the ' ba r '  on the fields, 
A = 7r/a). For convenience we make 
a scale t rans format ion  to dimensionless vari- 
ables [2,3], a = a'/vv/~, t = t'/vx/'~, d = v¢', 
? A# = vv/'-~A,, rn¢ = v/~vrn '¢,  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 obtained in 
our simulat ion,  which was carried out  in the same 
- 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  
Figure 3. F versus T '2 for a' = 0.16. The  l i n e  
represents the f i t  F = 0.00335-I- 0.00512 T '2. 
way as in [3]. The  classical approximat ion  is con- 
sistent with the semiclassical calculation in the 
semiclassical region [3,4]. We see also an inter- 
mediate  region which is a lmost  flat. This region 
appears to correspond to the m i n i m u m  of ([¢,[2) 
in fig. 2. Figs. 3 and 4 show more  detailed da ta  
at high temperature .  We see here a T '2 behavior  
which starts  already just  beyond the m i n i m u m  of 
([¢,12). According to fig 1, the rate at high tem- 
perature  depends on the lattice spacing a ' .  W i t h  
a gauge invariant  lattice act ion 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 approximat ion  
appears  to hold for a '2 < 0.1. We therefore fit the 
da ta  in figs. 3 and 4 to the following ansatz,  
F = coo + co2a '2 + (c2o + c22a '2)T '2 
a2m2¢ 
= COO n L c 0 2 - -  
2 
a2m2 \ 
+ 2  c2o+c22~- -~ )  (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 tha t  the v -4  T 2 behavior  in eq. 2 
is confirmed (contradict ing [6]), with 
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 .20a  Sm~, a t e = 4 .  (8) 
The coefficient of T 2 depends strongly on the lat- 
tice 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 .00347+0.192a  ~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. 


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

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