A Monte Carlo study of the SU(3) adjoint Higgs model

Volume 138B, number 1,2,3 PHYSICS LETTERS 12 April 1984

A MONTE CARLO STUDY OF THE SU(3) ADJOINT HIGGS MODEL

Subhash GUPTA and Urs M. HELLER a The Institute for Advanced Study, Princeton, NJ 08540, USA

Received 28 November 1983 Revised manuscript received 9 January 1984

We examine the full SU(3) gauge-Higgs theory, with the Higgs fields in the adjoint representation, and allow for radial fluctuations of the Higgs fields by means of a Monte Carlo simulation. We compare the resulting phase diagram with the predictions based on a perturbative analysis of the model and find qualitative agreement. We were not able to isolate a Coleman-Weinberg-type phenomenon.

1. Introduct ion. Theories of SU(N) gauge fields interacting with Higgs fields in the adjoint representation are of importance in building models of grand unified theories [1 ] and also in cosmology [2]. The inflation- ary universe scenario is based on such Higgs theories with a particular choice of parameters such that the (effective) potential for the Higgs fields has the form of a Coleman-Weinberg potential [3]. These theories have so far only been studied within perturbation theory and renormalization group improved perturbation theory. In this paper we will report on a non-perturbative study of interacting gauge-Higgs theories put on a space - t ime lattice.

Some investigations exist for the SU(2) case, but with a Higgs fields whose length was kept fixed [4]. For the Z 2 gauge-Higgs system it has been shown [5] that allowing for radial excitations of the Higgs field may alter the phase structure. We therefore decided to perform a Monte Carlo simulation of the full gauge - Higgs system, including a {b 4 self-coupling to control the radial modes. We were interested in mapping out the phase structure and in seeing whether the perturbative predictions remain valid.

In our Monte Carlo simulation we only considered the gauge group SU(3). We review the perturbative

1 Present address: CERN, Theory Division, CH-1211 Geneva 23, Switzerland.

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

analysis of the continuum theory in the next section. This will serve us as a guide for what we have to look for in the numerical simulation. The theory on the lattice is introduced in section 3. We describe the observables measured and their significance in determining the phases of the theory. Also given there are some technical details concerning the Monte Carlo algorithm used. In section 4 we present and discuss the results. Section 5 finally contains our conclusion.

2. Perturbative analysis and expectat ions. The most general lagrangian in the euclidean continuum theory is

O = - ( 1 / 2 g 2) t r (FuvFuv ) - tr(Duq~Du~ )

- la 2 tr ~ 2 _ X(tr qb2)2 , ( 1 )

where for simplicity we have imposed a • ~ -q~ symmetry. Otherwise there would be an extra term ~'2 tr ~1,3. Note that for SU(3), which we consider here, (tr ~2)2 and tr c1,4 are not independent.

For negative/a2 and k > 0 the theory has at the tree

level a broken SU(3) gauge group. The minimum of the potential is at

tr q~2 = _/a2/23, (2)

and the direction in group space is undetermined. But, since there are infinitely many gauge-inequivalent U(1)

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X U(1) directions in SU(3), we expect the gauge group to be broken to U(1) X U(1). Inclusion of a tr q~3 term lifts the degeneracy and uniquely chooses an SU(2) × U(1) minimum. But even if such a term is not in- cluded, higher perturbative corrections might lift the degeneracy. For this we have considered the one-loop effective potential [3]. The contribution from the Higgs fields gives an effective potential that is a function of tr q)2 only. Thus in a pure Higgs theory, the degeneracy would remain. The contribution from the gauge fields on the other hand lifts the degeneracy and chooses one of the SU(2) X U(1) directions for the symmetry breaking.

Perturbatively, the coupling constants g(R 2) of the unbroken subgroup SU(2) and e R of the subgroup U(1) are supposed to be equal to the SU(3) coupling constant g(R 3) at the scale tr Cb2ac = (tr ~2). Mthough the renormalization ofg(R 2) and e R depends on XR and /12 , it is only sensitive to tr q,2, especially when ~k R is

(2) small. So, we expect that the gauge couplings gR and e R would effectively be the same as g(R3Jdefined through the beta function. Furthermore, we expect that the average SU(3) plaquette would be more ordered in the broken phase, since it has a smaller unbroken group space. This analysis assumes the decoupling of the heavy degrees of freedom, i.e. the heaw gauge and Higgs bosons, which get masses proportion- al to tr Cb2ac , have effects on scales below tr qb2ac at most of O(1/tr q52ac ).

Of particular interest is the massless theory (/a 2 = 0). At the tree level the minimum occurs at tr ~2 = 0, but the perturbative theory is sick and full of infrared problems. Coleman and Weinberg [3] have shown, to one loop, that dimensional transmutation takes place and the theory realizes in a broken SU(2) X U(1) phase. By a clever choice of the renormalization prescription, this can be arranged to all orders in perturbation theory. Since we are interested in finding signs of this Coleman-Weinberg phenomenon, let us discuss this renormalization prescription in some more detail.

The "massless" theory is really defined by

t/2 = ~2 Veff/a~b 2 [42 =0 = 0 (3)

to all orders in perturbation theory. Note that/j2R is not the position of a pole in the physical propagators. In fact, the physical mass is non-zero. (3) is a purely perturbative definition. There is no analog to it on the

lattice, since it is impossible to enforce • = 0. Further- more, the effective potential evaluated in a Monte Carlo simulation, as far as this is possible at all ,1 , would, because of its convexity property, in this case of broken symmetry by fiat in the region tr ~2

< tr qb21at minimum [71 and (3) would be trivially sat- isfied.

Also we define, perturbatively,

)t R = ~4 V/~b4ltr ~2=~z2 = 0 (4)

following an observation due to D. Politzer ,2 . Note that X R has to be defined at tr q~2 away from zero, since at tr ~2 = 0 this is infrared sick. The definition X R = 0 is most useful and appealing, as it basically cor- responds to setting Vtree level = 0 and demanding that all Higgs couplings arise due to gauge interactions. This might even be true in view of the general belief that the pure 44 theory (at least the one-component theory) be free [81.

3. The theory on a lattice. We put the continuum theory (1) on a (hypercubic) lattice in a straightfor- ward way. The partition function is defined by

z = f dUxu' x,a[I d • e S . (5)

Here Ux, u are the gauge group elements (parallel trans- porters) living on the links of the lattice and dUx, u is the invariant group measure. The hermitean traceless matrix ~x on each site is written as

di, x = q):~ xa /2 , (6)

with X a the Gell-Mann matrices. The integration over the ~x a ranges from -~" to + ~ (this measure for q~x preserves hermicity and tracelessness).

The lattice action S in (5) was taken to be

S = S G +S H ,

S G = -~13 ~ tr(Up + U~p), p

S H = 2 ~ t r ( q ~ Y U ~b + U + ) . . . . , / . t X /.~ X , / 2

X , , u

_ (y2 + 2d) ~ tr qbx2 - X ~ (tr q~x2) 2 . (7) x x

,1 For an attempt, see ref. [6]. #2 Coleman and Weinberg [3] ascribe this observation to him.

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Eqs. (7) are the simplest lattice action that reduce to f ddx 22 of eqs. (1) in the naive continuum limit. (In (7) the lattice spacing a was set to 1.)

For the Monte Carlo simulation we used a Metropolis algorithm [9] for both the gauge and Higgs field updating. The U's were updated by multi- plying with a random SU(3) matrix V from a table, U ~ U' = UV, and the change was accepted with prob- ability exp [S(U') - S (U)] . The 4a's were changed by 4 a -+ 4 a' = 4 a + 84 with 84 generated according to a gaussian distribution. Again the change was accepted with probabil i ty exp [S(4 a ) - S(4a)] . We used random sweeps through the lattice and made 5 hits per U and 4 a. The table of random SU(3)matr ices V and their inverse V -1 and the width o f the gaussian distribution for 84 were tuned to get an acceptance rate of about 30%. An iteration consisted of a sweep through the lattice updating the U's followed by a sweep updating qb's.

We did not want to fix a gauge and make gauge-de- pendent measurements. Thus we only measured gauge- invariant objects. This is enough to get all the relevant information about the realizations of the theory. For given parameters/3 = 6/g 2, X and/.t 2 were measured (tr ~2), ((tr alp2)2) and the average plaquette

E = (1 - 1Re(tr Up)). (8)

In case o f symmetry breaking it is difficult to determine from these observables the unbroken subgroup, though they give enough information to distinguish between confining and broken realizations. To determine the unbroken subgroup we looked at the eigenvalues 0¢ i of the hermitean matrices ~x (these are gauge invariant). We ordered them in decreasing order, i.e. a i >1 ~/for i < j . Then we measured the expectation values <ai ) and their fluctuations

Aoq : (<~2> _ (ai)2)1/2 .

In an SU(2) X U(1) phase, two of the eigenvalues should be equal. Since • is traceless, a 3 is determined to be - ( a l + a2)" Hence we can either have a 1 >~ o~ 2 or ~1 ~ -2a2" Note that the eigenvalues will always fluctuate somewhat and that because of our ordering

therefore always (0~ 1 ) ~> <a 2 ) or <a I ) > --2<a2). As a criterion of two eigenvalues being equal, the unbroken subgroup being SU(2) X U(1), we required that either (a 1) - (a 2) or (a I ) + 2(a 2) is of the same order as the fluctuations Aa I . In contrast, when the expectation

values (a i) and the fluctuations A%. were o f the same order, we took this as an indication that the symmetry remained unbroken (qb = 0 perturbatively).

4. Results o f the Monte Carlo simulation. We started off with some Monte Carlo runs for the pure Higgs theory (the U-matrices in eq. (7) set identically to the unit matrix). For various couplings X we scanned the system changing/l 2 from +0.4 to - 4 . 0 in steps of 0.2. We found that (tr ~2) varied smoothly and for X

0.5 and//2 ~ - 0 . 5 , (tr (b 2) followed the perturbative prediction (2) quite closely. Our measurements were not accurate enough to see whether there might be a second-order phase transition at/~2 = 0 for small enough X. However, when (tr qb2) was big enough to

indicate symmetry breaking, inspection of the eigen-

a ,8=6.0 x = o . 0 4 CZ 4

mmm w~m~m m ~m~mm ~I 5

emtmlem~mXxxxx 4 Xxxx x

° o o o o o o o oo xl~m ~ml = m e l 4 , , ~ - o ~ - ~ - . - , , . , ~ , ~ + . - . ~ , - ~ H . 2

................. t/, (12

b e = 6 . 0 X = 0 . 0 4 E

o ' o°o°o°oo°°x~X~RRx~ .4

o 3,5 x

.3

xX ,25 xxXx

o Oo~ x~OR~,~X~x xxx ~ RRx xo~o .2

, _~ _~ r . ~ -4 -5

Fig. i . A typical hysteresis run, performed at ~ = 6.0 and X = 0.04. #2 was first lowered from 0 in steps of 0.1 to -4.0 (dots) and then increased again up to 0 (crosses). Each point is the average over the last 20 of 40 iterations: (a) the eigenvalues al (positive axis) and a2 (negative axis); (b) the average plaquette, see eq. (8).

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values of ~ always showed that the broken phase was U(1) X U(1).

To determine the phase diagram of the full lattice SU(3) gauge-Higgs theory we found it most conve- nient to fix X and/3 to certain values and then run "hysteresis" runs in/l 2. A typical run is shown in fig. 1 for/~ = 6.0 and X = 0.04. The step size z%u 2 was 0.1, and the run was started at/~2 = 0. At each point we made 40 iterations and the data shown is the average over the last 20. The figure shows a clear hysteresis loop, which indicates that the system undergoes a first- order phase transition. We made some longer runs for /~2 between - 1 . 0 and - 2 . 0 starting from configura- tions which correspond to the broken and unbroken phase, respectively. From these we estimate that the transition occurs for ~t 2 ~ -1 .4 .

The resulting phase diagram in the/3-/~ 2 plane at 3 ̀= 0.04 is shown in fig. 2, and for 3 ̀= 0.004 in fig. 3. We notice that in this range of the Higgs self-coupling X there is very little dependence of the phase diagram on 3 .̀ In the broken phase we find

(tr q~2)~ _/12/23,, (9)

in agreement with the perturbative analysis. Inspection of the eigenvalues of ~I, shows that the system always breaks to an SU(2) X U(1) phase, as predicted by the one-loop effective potential. For bigger 3, the phase transition becomes weaker and eventually disappears for X ~> 0.5.

For the Higgs coupling 3, in the range 10 -6 to 0.1 the average plaquette E, eq. (8), decreases in the broken phase as compared to the unbroken one, showing that the gauge fields tend to be more ordered (closer to the unit matrix) in the broken phase. This is in agreement with the notion of the decoupling of the degrees of freedom as discussed in section 2.

L 4 6 8 40 I I I I

p2

- 2

_3[

Fig. 2. The phase diagram in the ~2-13 plane at h = 0.04.

/3 '~ 2 4 6 8 t0

I I I I I

/xz -4

- 2

Fig. 3. Same as fig. 2, but for X = 0.004.

For small 3, and/~2 near the phase transition, one expects the Coleman-Weinberg mechanism to operate (for large ~). To look for such a signal, we studied this region, X < 10 -6 , in much greater detail. We started the system in SU(2) × U(1) at small values of tr 4) 2 and at very large values of tr ~b 2 and studied the Monte Carlo development. We expected to find a slow change in t r ~b 2 (in the Monte Carlo time) finally falling into the deep minimum. We did find a slow evolution starting from small values o f t r q~2 towards larger values of tr ~2,but were unable to find a stable minimum at tr q~2 4 :0 even when we started with very large values of tr q~2. The potential seemed to be unbounded below. As far as we could see, the second minimum only existed in the range where there was no flat potential. In this region, we were only able to find the symmetry unbroken minimum, and not the Coleman- Weinberg minimum. But, this could also be due to lack of computer time available to investigate this region, rather than the unboundedness of the potential.

Most calculations were done on a 44 lattice. We did some checks on 34 and 54 lattices and found no size dependence of our results.

5. Conclusions. We have performed a Monte Carlo simulation of the SU(3) gauge theory coupled to Higgs fields in the adjoint representation. We have found the phases and symmetry breaking behavior which was expected from a perturbative analysis.

We did not find anything peculiar near the phase transition from broken to unbroken phase where one would expect the Coleman-Weinberg mechanism to operate. Of course, to really see some signals of Coleman-Weinberg phenomenon, one has to sit at the phase-transition point and study the behaviour as continuum theory is reached (i.e./~ ~ ~ limit). We were unable to do so.

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We would like to thank Herbert Hamber for giving

us his SU(3) gauge theory program. U.H. would like to

thank the theory group at Lawrence Berkeley Labora- tory for their hospitality and the use of their comput-

er facility where part of this work was done. He also

would like to express his thanks for a grant from the

Federal Republic of Germany. S.G.'s work was sup-

ported by the US Department of Energy under Grant

No. DE AC02-76ER02220.

References

[1 ] H. Georgi and S.C. Glashow, Phys. Rev. Lett. 32 (1974) 438.

[2] A.D. Linde, Phys. Lett. 106B (1982) 389; A. Albrecht and P.S. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220.

[3] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888.

[4] C.B. Lang, C. Rebbi and M. Virasoro, Phys. Lett. 104B (1981) 244; R.C. Bower et al., Phys. Rev. D25 (1982) 3319; F. Karsch, E. Seller and 1.O. Stamatescu, Ref. TH. 3612, CERN.

[5] T. Munehisa and Y. Munehisa, Phys. Lett. l16B (1982) 353; Nucl. Phys. B215 [FS7] (1983) 508.

[6] P. Ginsparg and H. Levine, Harvard Report No. HUTP-82-/ A034.

[7] T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291; Y. Fujimoto, L. O'Raifeartaigh and G. Parravicini, Nucl. Phys. B212 (1983) 268.

[8] M. Aizenmann, Phys. Rev. Lett. 47 (1981) 1; J. Fr6hlich, Nucl. Phys. B200 [FS4] (1982) 281; C. Aragao de Carvalho, C.S. Carracciolo and J. Fr~hlich, Nucl. Phys. B215 [FS7] (1983) 209, and references therein.

[9] M. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087.

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A Monte Carlo study of the SU(3) adjoint Higgs model