- Email: [email protected]
Numerical simulation of the 't Hooft-Polyakov monopole in SU(2) gauge theory
Nuclear Physics B (Proc. Suppl_) 20 (1991) 221-224 North-Holland
221
N u m e r i c a l S i m u l a t i o n o f t h e 't H o o f t - P o l y a k o v M o n o p o l e in S U ( 2 ) g a u g e theory* Jan Smit and Arian J. van der Sijs Institute for Theoretical Physics, Valckenierstraat 65, 1018 Xl= Amsterdam, The Netherlands Properties of the 't Hooft-Polyakov monopole in pure SU(2) gauge theory are studied. This is of interest for an implementation of the dual superconductor hypothesis of confinement. INTRODUCTION Last year we proposed a realization I of the dual superconductor hypothesis2 of confinement in SU(2) gauge theory. Our approach is based on an effective action for "t Hooft-Polyakov like monopole configurations. This effective action is mapped on periodic lattice QED (PQED), where the dual superconductor mechanism is realized and confinement is caused by monopole condensation 3. The key point in this mapping is that the reno~ma/ized (~nning) coupling of the SU(2) effective action is mapped on the bare coupling of PQED. The critical coupling of this U(1) theory thus translates into a critical monopole size above which conden° sation occurs. The mapping to PQED furthermore allowed us to calculate a lower bound on the string tension in SU(2). We obtained V/~ ~ 45AL. This talk gives a status report of numerical simulations that we are carrying out to verify an assumption that we macle about the behavior of the 't Hooft-Polyakov monopole in the quantum theory. THE MONOPOLE The static 't Hooft-Polyakov monopole was first found in 50(3) 5auge-Higgs theory. The Lagrangian of this model is g2£=_4
1 F2 1 2 . - + 2 ( D . ~ b ) _ ~ ( ~ b =_/z~)=
1.
*Presented by A.J. van der 5ijs
(2)
=
~TH(~),
(3)
a = 1,2,3, k , 1 : 1 , 2 , 3 , in the radial ('hedgehog') gauge, with -40 ----O. The condition of finite energy requires that [¢i[ --* p so H ( ~ r ) / ~ r -~ 1 for ~
- , oo. The ~
(~-~)
of the monopole is
(,) wheres c(,',-,~/,-,,i,,) = c(2.~) is a ~ iuo-~,,'.ing function, C(0) = 1 < C(2~t) g 1.787 = C(oo). The 5 0 ( 3 ) symmetry is broken to the O(1) ~ o u p of gauge rotations around the Highs direction ¢P. The monopole has magnetic charge --1 with respect to this residual 'electromagnetic" group. The mass of the gauge bosons corresponding to the broken generators is m w = p and the Higgs mass is
For ,X = 0 an explicit solution is known 6, H(pr) pr
--
cosh IZr sinh pr
--
1---
1
pr
1 pr
~_ 2~r e-"
(5)
(~r-~oo),
sinh pr
(1)
and the monopole solution 4 is of the form (r = li[)
A~:(£,x,) = c,kt--[1 - K(~r)l,
~'(~,=,)
(6)
(~r -, oo).
In this case the Higgs field q~ is massless and the 1/pr asymptotic behavior in eqn. (5) reflects a long range attractive interaction mediated by the Higgs field, apart from the magnetic interaction of the
0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Holland)
J. Stair, A.J. van der Sijs / ~l~e 't Hooft-Poh,akov monopole in SU(2) gaege theoly
222
:.z',
1.2
I. ,
i.
H
0.6
'~ ,''"
0.4
~
H {×) z
o-,
_- ........ -
:;::
~
0.2 •
"',
00
°- o
~
~-
-~
Figure 1: H ( x ) l x and K(:c) for (a) m H 2 I m 2w -_ 0.2 and (b) r a ~ / m ~ = 20 (solid lines). The curves for the massless Higgs case are shown for comparison (dashed lines). monopole. The dashed curves in figs. la,b are plots of the functions (5,6). In the massive Higgs case (A ~ 0), where no explicit solution is known, the asymptotic behavior for not too large A is given by H(~r) /zr
K(~)
-
1 - O (e . . . . ) ,
(7)
o(~-~')
(8)
The Higgs interaction has a short range here, and the only long distance interaction is the magnetic one. Figs. la,b show plots of H ( z ) / z and K ( z ) for two values of .~, obtained by numerical minimization of the energy functional. The A = 0 monopole can be carried over to euclidean pure SU(2) gauge theory by making the replacement ¢ -~ -44 in eqn. (3). Then it satisl~es the Landau gauge condition
O~A~=O
an infrared divergence was found in a semidassi-~! ca]c'=']at~n_ .-~.~expan_.c~onarolmd the classical solution may therefore not be a self consistent approach. In ref. I we made the assumption that an effective potential for ~ is generated dynamically. This seems plausible since space-time symmetry is broken by the presence of the static monopole, and in a similar way a mass for ~ arises in high temperature QCD. As a consequence of an effective mass for .44, the "Higgs" interaction would be screened and at the quantum |evei the monopole would interact like a regular abelian magnetic monopole at large distances. This was used in the mapping to PQED in ref. 1. THE SIMULATIONS Using the lattice regularization we perform a numerical simulation to verify the above assumption. We measure {A~(~)) and from this calculate
which is equivalent to the Coulomb gauge for this static solution. The scale/z is arbitrary here because a Higgs potential is absent.
the effective functions H e f f ( z ) / x and K e f f ( x ) in order to see if an effective mass for the 'Higgs field" . ~ is generated.
THE ASSUMPTION We are interested in the effects of the quantum fluctuations on properties of the SU(2) 't HooftPolyakov monopole. Analytical approaches7,8 have not yielded explicit results, In ref. 8, for example,
The method we use is a kind of background field technique. We put one static 't HooftPolyakov monopole on an L 3 x L4 lattice and study the quantum fluctuations around it in the Landau gauge. The monopole has to be prevented from being washed out by the fluctuations by taking
J. Stair. A~I. ~
~ r Si~ / T ~ e "t lfooft-Pot~kov monopole in SU(2) gauge theory
t~xecl monop~e boundary cond;tions. The boundary conditions have to be compatible wrth the gauge choice. It is sufficient to fix the components of the gauge fie~ tangential to the boundary 9. Links lying in the spatial bo~miary are fixed using the asymptotic form of eqn~ (2-3), neglecting exponential te~ns. Hote that the values of -44 depend on p_ We take pecind~ bonndary conditions in the time direction (kleally, the time extent for the static monolmle should be infimte). As a first test we set up the ~ continuum monopate solu6on and ¢Aeck~ ~ s t a ~ under cooling. The ~ scale parameter p is chosen in the range 1 ~
223
ditions. On the other hand, the boundary condit~ns ought to be consistent with the outcome of the s~mu|ation. To c~rcumvent this problem we do the s~mulations ~ both types of boundary co~l~tions and we decide after~rds which of the results are the most seffco~siste~t. In the case of '~ ~ 0 bou~lary ~ndi~ons ~the exact asympto#~ bc4~v-
(detem~ned by the e ~ v e
~e~)
~ .0¢
k ~ n so ~i~ set the ma~/,n~udeof A~ eqm01to ~ at the boendary. Configmatio~ are gem~ated ~ h a keat~tb algp~thm a~l p ~ in the ~ ga~lp~,
tile ~
l~zuge fireedom. We ~
the e:l:pec-
a~ ~ From tbme ezpectafi~ ~l=es the effective H and K f u . ~ = e ca/,oediated,die-
f i ~ by
~,.
Tbe d ~ ~
~=~ ( ~ )
(9)
= ;-~-~=(-~(~
00)
e~rors c~sed by ~
H
and K tlrom tile link ~riables are algam .~m~all. is i l t u s t ~ in ~ . 2 ud~re the !~_o~__~ is t ~ on the dassical mmOlX~ solutim on ~ latbce.
I= ~%
. r ~ ~ ~" ~
0_7 Q~ O-% 0.4
/
'E
0-3
2'!" F~gure 2: Recovering H(z)/z (cirdes) and K ( z ) (squares) from the 104 link configuration corresponding to the damicai solution with ap = 1.0. The exam curves are shown for comparison. An example of a simulation is shown in fig_ 3. At some places there are small jumps between the
224
J. Smit, A.J. van der Sijs / T~,e 't Hooft-Polvakov mo~opote in SU(2) gauge ~]~eory
1.
\
0.9 0.8 0.7
0,6
~ ~
\ 0.6 t
°
0.5
/
0.4
/
\x =
0.3
0.
\
/
0_3 ~-
~/
="x
/
0~ 0.1
J
\
0.7 L
~o
\
/ /
=
1
2
.3
X~
~x
/
=%
~
6~oJ
="e_
o~L/I
o!,
x --~
{=j
Heff(z)/z
Figure 3: (circles) and Keff(x) (squares) for a slmutatic~ on a 104 lattice at ~ = 3.0 for ")~ = 0" (a) and '.~ ~ O' (b) boundary conditions. The classical .~ = 0 curves (dashed lines) are shown fo¢ comparison_ points. These are probably effects of the fixed boundary conditions and the cubic |attice box. We can compare the shape of the curves through the points far from the boundary of the box with the classical curves of H and ~ for both the massiess and the massive Higss case as shown in fig. 1. The test data obtained solar indicate that the method appears to work. At a |ater stage, we may be able to determine effective Higgs and gauge boson masses from the exponential decay of Hee f t and ~=ff. Another quantity to look at is the energy density. The explicit'(>. = 0) . s ~ u ~ of eqns. (2-3), with H and K given by (5-6), is selfdua] and the electric and magnetic energy densities EE and Ej~ are equal and fall off as r -c. In the massive A4 case, only E M o c t -~, and J~E falls off exponentially fast. Piaquette expectation values obtained from the simulations may be used to distinguish between the massless and massive cases. Then the vacuum fluctuations have to be substracted, so we have to do simulations with F = 0 as well to compute these fluctuations for the vacuum in the box with our special boundary conditions. ACKNOWLEDGEMENTS We would like to thank our colleagues at NIKHEF-H for generous support and computer time on the Encore NP1 computer. Simulations were also done on the Cyber 205 and Nec SX2 supercompute;s with flaancial support by the 'Stichting SURF' from the Dutch 'Nationaal Fonds ge-
bruik Supercomputers (NFS)'. This work is supported by the 'Stichting voor FundamenteeJ Onderzoek der Materie (FOM)'. REFERENCES 1. J. Sr~t and A._I. van der Sijs, Amsterdam preprint ITFA~9-17; J. Smit and A_J. van der Sijs, Nud_ Phys_ B (P,o~. Suppl.) 17 (~_~J0) SZl (Capri 1989) 2. S. Mandelstam, Phys. Rep. 23c (1976) 245; G. "t Hooft, in "High Energy Physics', Proceedings of the EPS International Conference, Palermo 1975, ed. A. Zichichi0 Editrice ComIx~toH, Bologna 1976 3. T. Banks, R. Myerson, J. Kogut, Nud. Phys. B120 (1977) 493; M.E. PesMn, Ann. Phys. 113 (1978) 122 4. G_ "t Hooft, Nucl. Phys. B79 (1974) 276; A M . Polyakov, JETP Letters 20 (1974) 194 5. F.A Bais and J. Primack, Phys. Rev. D13 (1976) 819 6. M . K .
Pra~d
and
C.M.
Sommer~d,
Phys_
Rev. Lett. 35 (1975) 760; E.B. Bogomol'nyi, Sov. J. Nud. Phys. 24 (1976) 449 7. F,A. Bais and W. Troost, Nud. Phys. B178 (1981) 125 8. V.G. Kiselev and K.G. Selivanov, Phys. Lett. 213B (1988) 165 9. S. Coleman, "The magnetic monopole fifty years later', in "The Unity of the Fundamental Interactions', ed. A. Zichichi (Plenum Press, New York, 1983)
221
N u m e r i c a l S i m u l a t i o n o f t h e 't H o o f t - P o l y a k o v M o n o p o l e in S U ( 2 ) g a u g e theory* Jan Smit and Arian J. van der Sijs Institute for Theoretical Physics, Valckenierstraat 65, 1018 Xl= Amsterdam, The Netherlands Properties of the 't Hooft-Polyakov monopole in pure SU(2) gauge theory are studied. This is of interest for an implementation of the dual superconductor hypothesis of confinement. INTRODUCTION Last year we proposed a realization I of the dual superconductor hypothesis2 of confinement in SU(2) gauge theory. Our approach is based on an effective action for "t Hooft-Polyakov like monopole configurations. This effective action is mapped on periodic lattice QED (PQED), where the dual superconductor mechanism is realized and confinement is caused by monopole condensation 3. The key point in this mapping is that the reno~ma/ized (~nning) coupling of the SU(2) effective action is mapped on the bare coupling of PQED. The critical coupling of this U(1) theory thus translates into a critical monopole size above which conden° sation occurs. The mapping to PQED furthermore allowed us to calculate a lower bound on the string tension in SU(2). We obtained V/~ ~ 45AL. This talk gives a status report of numerical simulations that we are carrying out to verify an assumption that we macle about the behavior of the 't Hooft-Polyakov monopole in the quantum theory. THE MONOPOLE The static 't Hooft-Polyakov monopole was first found in 50(3) 5auge-Higgs theory. The Lagrangian of this model is g2£=_4
1 F2 1 2 . - + 2 ( D . ~ b ) _ ~ ( ~ b =_/z~)=
1.
*Presented by A.J. van der 5ijs
(2)
=
~TH(~),
(3)
a = 1,2,3, k , 1 : 1 , 2 , 3 , in the radial ('hedgehog') gauge, with -40 ----O. The condition of finite energy requires that [¢i[ --* p so H ( ~ r ) / ~ r -~ 1 for ~
- , oo. The ~
(~-~)
of the monopole is
(,) wheres c(,',-,~/,-,,i,,) = c(2.~) is a ~ iuo-~,,'.ing function, C(0) = 1 < C(2~t) g 1.787 = C(oo). The 5 0 ( 3 ) symmetry is broken to the O(1) ~ o u p of gauge rotations around the Highs direction ¢P. The monopole has magnetic charge --1 with respect to this residual 'electromagnetic" group. The mass of the gauge bosons corresponding to the broken generators is m w = p and the Higgs mass is
For ,X = 0 an explicit solution is known 6, H(pr) pr
--
cosh IZr sinh pr
--
1---
1
pr
1 pr
~_ 2~r e-"
(5)
(~r-~oo),
sinh pr
(1)
and the monopole solution 4 is of the form (r = li[)
A~:(£,x,) = c,kt--[1 - K(~r)l,
~'(~,=,)
(6)
(~r -, oo).
In this case the Higgs field q~ is massless and the 1/pr asymptotic behavior in eqn. (5) reflects a long range attractive interaction mediated by the Higgs field, apart from the magnetic interaction of the
0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Holland)
J. Stair, A.J. van der Sijs / ~l~e 't Hooft-Poh,akov monopole in SU(2) gaege theoly
222
:.z',
1.2
I. ,
i.
H
0.6
'~ ,''"
0.4
~
H {×) z
o-,
_- ........ -
:;::
~
0.2 •
"',
00
°- o
~
~-
-~
Figure 1: H ( x ) l x and K(:c) for (a) m H 2 I m 2w -_ 0.2 and (b) r a ~ / m ~ = 20 (solid lines). The curves for the massless Higgs case are shown for comparison (dashed lines). monopole. The dashed curves in figs. la,b are plots of the functions (5,6). In the massive Higgs case (A ~ 0), where no explicit solution is known, the asymptotic behavior for not too large A is given by H(~r) /zr
K(~)
-
1 - O (e . . . . ) ,
(7)
o(~-~')
(8)
The Higgs interaction has a short range here, and the only long distance interaction is the magnetic one. Figs. la,b show plots of H ( z ) / z and K ( z ) for two values of .~, obtained by numerical minimization of the energy functional. The A = 0 monopole can be carried over to euclidean pure SU(2) gauge theory by making the replacement ¢ -~ -44 in eqn. (3). Then it satisl~es the Landau gauge condition
O~A~=O
an infrared divergence was found in a semidassi-~! ca]c'=']at~n_ .-~.~expan_.c~onarolmd the classical solution may therefore not be a self consistent approach. In ref. I we made the assumption that an effective potential for ~ is generated dynamically. This seems plausible since space-time symmetry is broken by the presence of the static monopole, and in a similar way a mass for ~ arises in high temperature QCD. As a consequence of an effective mass for .44, the "Higgs" interaction would be screened and at the quantum |evei the monopole would interact like a regular abelian magnetic monopole at large distances. This was used in the mapping to PQED in ref. 1. THE SIMULATIONS Using the lattice regularization we perform a numerical simulation to verify the above assumption. We measure {A~(~)) and from this calculate
which is equivalent to the Coulomb gauge for this static solution. The scale/z is arbitrary here because a Higgs potential is absent.
the effective functions H e f f ( z ) / x and K e f f ( x ) in order to see if an effective mass for the 'Higgs field" . ~ is generated.
THE ASSUMPTION We are interested in the effects of the quantum fluctuations on properties of the SU(2) 't HooftPolyakov monopole. Analytical approaches7,8 have not yielded explicit results, In ref. 8, for example,
The method we use is a kind of background field technique. We put one static 't HooftPolyakov monopole on an L 3 x L4 lattice and study the quantum fluctuations around it in the Landau gauge. The monopole has to be prevented from being washed out by the fluctuations by taking
J. Stair. A~I. ~
~ r Si~ / T ~ e "t lfooft-Pot~kov monopole in SU(2) gauge theory
t~xecl monop~e boundary cond;tions. The boundary conditions have to be compatible wrth the gauge choice. It is sufficient to fix the components of the gauge fie~ tangential to the boundary 9. Links lying in the spatial bo~miary are fixed using the asymptotic form of eqn~ (2-3), neglecting exponential te~ns. Hote that the values of -44 depend on p_ We take pecind~ bonndary conditions in the time direction (kleally, the time extent for the static monolmle should be infimte). As a first test we set up the ~ continuum monopate solu6on and ¢Aeck~ ~ s t a ~ under cooling. The ~ scale parameter p is chosen in the range 1 ~
223
ditions. On the other hand, the boundary condit~ns ought to be consistent with the outcome of the s~mu|ation. To c~rcumvent this problem we do the s~mulations ~ both types of boundary co~l~tions and we decide after~rds which of the results are the most seffco~siste~t. In the case of '~ ~ 0 bou~lary ~ndi~ons ~the exact asympto#~ bc4~v-
(detem~ned by the e ~ v e
~e~)
~ .0¢
k ~ n so ~i~ set the ma~/,n~udeof A~ eqm01to ~ at the boendary. Configmatio~ are gem~ated ~ h a keat~tb algp~thm a~l p ~ in the ~ ga~lp~,
tile ~
l~zuge fireedom. We ~
the e:l:pec-
a~ ~ From tbme ezpectafi~ ~l=es the effective H and K f u . ~ = e ca/,oediated,die-
f i ~ by
~,.
Tbe d ~ ~
~=~ ( ~ )
(9)
= ;-~-~=(-~(~
00)
e~rors c~sed by ~
H
and K tlrom tile link ~riables are algam .~m~all. is i l t u s t ~ in ~ . 2 ud~re the !~_o~__~ is t ~ on the dassical mmOlX~ solutim on ~ latbce.
I= ~%
. r ~ ~ ~" ~
0_7 Q~ O-% 0.4
/
'E
0-3
2'!" F~gure 2: Recovering H(z)/z (cirdes) and K ( z ) (squares) from the 104 link configuration corresponding to the damicai solution with ap = 1.0. The exam curves are shown for comparison. An example of a simulation is shown in fig_ 3. At some places there are small jumps between the
224
J. Smit, A.J. van der Sijs / T~,e 't Hooft-Polvakov mo~opote in SU(2) gauge ~]~eory
1.
\
0.9 0.8 0.7
0,6
~ ~
\ 0.6 t
°
0.5
/
0.4
/
\x =
0.3
0.
\
/
0_3 ~-
~/
="x
/
0~ 0.1
J
\
0.7 L
~o
\
/ /
=
1
2
.3
X~
~x
/
=%
~
6~oJ
="e_
o~L/I
o!,
x --~
{=j
Heff(z)/z
Figure 3: (circles) and Keff(x) (squares) for a slmutatic~ on a 104 lattice at ~ = 3.0 for ")~ = 0" (a) and '.~ ~ O' (b) boundary conditions. The classical .~ = 0 curves (dashed lines) are shown fo¢ comparison_ points. These are probably effects of the fixed boundary conditions and the cubic |attice box. We can compare the shape of the curves through the points far from the boundary of the box with the classical curves of H and ~ for both the massiess and the massive Higss case as shown in fig. 1. The test data obtained solar indicate that the method appears to work. At a |ater stage, we may be able to determine effective Higgs and gauge boson masses from the exponential decay of Hee f t and ~=ff. Another quantity to look at is the energy density. The explicit'(>. = 0) . s ~ u ~ of eqns. (2-3), with H and K given by (5-6), is selfdua] and the electric and magnetic energy densities EE and Ej~ are equal and fall off as r -c. In the massive A4 case, only E M o c t -~, and J~E falls off exponentially fast. Piaquette expectation values obtained from the simulations may be used to distinguish between the massless and massive cases. Then the vacuum fluctuations have to be substracted, so we have to do simulations with F = 0 as well to compute these fluctuations for the vacuum in the box with our special boundary conditions. ACKNOWLEDGEMENTS We would like to thank our colleagues at NIKHEF-H for generous support and computer time on the Encore NP1 computer. Simulations were also done on the Cyber 205 and Nec SX2 supercompute;s with flaancial support by the 'Stichting SURF' from the Dutch 'Nationaal Fonds ge-
bruik Supercomputers (NFS)'. This work is supported by the 'Stichting voor FundamenteeJ Onderzoek der Materie (FOM)'. REFERENCES 1. J. Sr~t and A._I. van der Sijs, Amsterdam preprint ITFA~9-17; J. Smit and A_J. van der Sijs, Nud_ Phys_ B (P,o~. Suppl.) 17 (~_~J0) SZl (Capri 1989) 2. S. Mandelstam, Phys. Rep. 23c (1976) 245; G. "t Hooft, in "High Energy Physics', Proceedings of the EPS International Conference, Palermo 1975, ed. A. Zichichi0 Editrice ComIx~toH, Bologna 1976 3. T. Banks, R. Myerson, J. Kogut, Nud. Phys. B120 (1977) 493; M.E. PesMn, Ann. Phys. 113 (1978) 122 4. G_ "t Hooft, Nucl. Phys. B79 (1974) 276; A M . Polyakov, JETP Letters 20 (1974) 194 5. F.A Bais and J. Primack, Phys. Rev. D13 (1976) 819 6. M . K .
Pra~d
and
C.M.
Sommer~d,
Phys_
Rev. Lett. 35 (1975) 760; E.B. Bogomol'nyi, Sov. J. Nud. Phys. 24 (1976) 449 7. F,A. Bais and W. Troost, Nud. Phys. B178 (1981) 125 8. V.G. Kiselev and K.G. Selivanov, Phys. Lett. 213B (1988) 165 9. S. Coleman, "The magnetic monopole fifty years later', in "The Unity of the Fundamental Interactions', ed. A. Zichichi (Plenum Press, New York, 1983)