Nuclear Physics B (Proc. Suppl_) 20 (1991) 221-224 221 North-Holland

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 ) gauge 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 hypothesis 2 of confinement in SU(2) gauge theory. Our approach is based on an effective action for "t Hooft-Polyakov like mono- pole configurations. This effective action is mapped on periodic lattice QED (PQED), where the dual superconductor mechanism is realized and confine- ment 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 ac- tion 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 sim- ulations that we are carrying out to verify an as- sumption 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 La- grangian of this model is

1 F2 1 2 g 2 £ = _ 4 . - + 2 ( D . ~ b ) _ ~ ( ~ b =_/z~)= (1)

and the monopole solution 4 is of the form (r = li[)

A~:(£, x,) = c,kt--[1 - K(~r) l , (2) 1.

*Presented by A.J. van der 5ijs

~ ' (~ ,= , ) = ~ T H ( ~ ) , (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

(,)

where s c(,' ,-,~/,-, , i , ,) = c (2 .~ ) is a ~ iuo-~,,'.- ing function, C(0) = 1 < C(2~t) g 1.787 = C(oo). The 50(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 re- spect to this residual 'electromagnetic" group. The mass of the gauge bosons corresponding to the bro- ken generators is m w = p and the Higgs mass is

For ,X = 0 an explicit solution is known 6,

H(pr) cosh IZr 1 - - ( 5 )

pr sinh pr pr

1 -- 1 - - - (~r-~oo),

p r

sinh pr (6)

~_ 2~r e - " (~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)

222 J. Stair, A.J. van der Sijs / ~l~e 't Hooft-Poh,akov monopole in SU(2) gaege theoly

1.2

I. , H

0.6 '~ ,''"

0.4 ~ ~

0.2 • "',

0 0

:.z',

i.

o-, :;::

°- o ~ ~- -~

H {×) z _- ........ -

Figure 1: H ( x ) l x and K(:c) for (a) 2 2 _ m H I m w -- 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 ) 1 - O (e . . . . ) , (7) /zr

K ( ~ ) - 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 minimiza- tion 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

which is equivalent to the Coulomb gauge for this static solution. The scale/z is arbitrary here be- cause a Higgs potential is absent.

THE ASSUMPTION We are interested in the effects of the quantum

fluctuations on properties of the SU(2) 't Hooft- Polyakov monopole. Analytical approaches 7,8 have not yielded explicit results, In ref. 8, for example,

an infrared divergence was found in a semidassi- -~! ca]c'=']at~n_ .-~.~ expan_.c~on arolmd the classical solution may therefore not be a self consistent ap- proach.

In ref. I we made the assumption that an effec- tive potential for ~ is generated dynamically. This seems plausible since space-time symmetry is bro- ken by the presence of the static monopole, and in a similar way a mass for ~ arises in high temper- ature QCD. As a consequence of an effective mass for .44, the "Higgs" interaction would be screened and at the quantum |evei the monopole would in- teract 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 assump- tion. We measure {A~(~)) and from this calculate

the effective functions He f f ( z ) / x and Kef f (x ) in order to see if an effective mass for the 'Higgs field" . ~ is generated.

The method we use is a kind of background field technique. We put one static ' t Hooft- Polyakov 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 223

t~xecl monop~e boundary cond;tions. The bound-

ary 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 us-

ing the asymptotic form of eqn~ (2-3), neglecting exponential te~ns. Hote that the values of -44 de- pend 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 ~ contin-

uum monopate solu6on and ¢Aeck~ ~ s t a ~ under cooling. The ~ scale parameter p is chosen in the range 1 ~< p-z < L/4 such that tee monopole fits easily in the lattice and disccetization errors are not too large. The relative tion effors in the energy are estimated to be of the ~ A ~ ~ _ ~.2~-,,,~H~_,,~z (~ ~ ~ h~Jce dis- tance). We use the path ordered inte~al (imple- mented by a 20-step path ordered product) to pure the link variables tirom the continuum gauge

fields. The energy of the lattice mmmlx~e solution

that is obtained after cooling is very dose to that

of the ~onfiguration before cooling, so the bound- ary conditions do indeed stabilize the monolmle on the lattice and discretDation effects are small. In order to compare the measured er~zr~ with the to- tal monopde en~gy give- by ~memon (4) ( ~ U = 1). we computed the part of the monopole en- ergy outside the lattice volume by integrating the

asymptotic energy density oc r -~ over three-spece with the tbreedimensional box excluded. The total

energy found this way is dose to the value (4).

In the quantum simulation one encountem a consistency problem in formulating the boundary

conditions because our spatial volume is a cube and not a ball. The asymptotic behavior of H / r for the

= 0 solution given by (5) has a 1 / r tail which has

to be taken into account in the boundary condition, as the boundary of the spatial cube is at varying r (we negle~ exponential terms). In contrast, the

asymptotic behavior (7) of t t / r for the ~ ~ 0 solu- tion has only negligible exponential corrections. We

want the simulation to reveal whether the quantum monopole has one or the other shape, so we should

not be prejudiced in our choice of boundary con-

ditions. On the other hand, the boundary condi- t~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~ude of A~ eqm01 to ~ 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

~,. ~= ~ ( ~ ) (9)

= ; - ~ - ~ = ( - ~ ( ~ 00) Tbe d ~ ~ 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.

0_7

Q~

O-%

0.4

0-3

2'!"

I=

~% . r ~ ~ ~" ~

/ 'E

F~gure 2: Recovering H(z)/z (cirdes) and K (z ) (squares) from the 104 link configuration corre- sponding 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

0.5

0.4

0.3

0~

0.1

0.

\

\ ~ o ~ ~ °

/ \ x =

/ = %

/ =

1 2 .3 X ~ ~ 6~oJ

0.7 L

0.6 t

0_3 ~-

o~L/I o!,

J \

\

/ \

/ ~/ = " x

/ ~ x / ="e_

x --~ {=j

Figure 3: Heff(z)/z (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 He e f t and ~=ff.

Another quantity to look at is the energy den- sity. 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 exponen- tially 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 com- pute 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 su- percompute;s with flaancial support by the 'Sticht- ing SURF' from the Dutch 'Nationaal Fonds ge-

bruik Supercomputers (NFS)'. This work is sup- ported by the 'Stichting voor FundamenteeJ On- derzoek 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', Pro- ceedings of the EPS International Conference, Palermo 1975, ed. A. Zichichi0 Editrice Com- Ix~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; AM. Polyakov, JETP Letters 20 (1974) 194

5. F.A Bais and J. Primack, Phys. Rev. D13 (1976) 819

6. M . K . P r a ~ d a n d C . M . S o m m e r ~ 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)