Phys. Rev. D 35, 3187 (1987)

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Effective potential in scalar field theory

Kerson Huang, Efstratios Manousakis, and Janos Polonyi
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
(Received 5 January 1987)

We study the lambda phi4 theory in 4 space-time dimensions in a Monte Carlo simulation on a 104 lattice, through an especially simple and accurate way to calculate the effective potential. All renormalized parameters are obtained via the effective potential and the propagator. In the continuum limit we confirm the vanishing of the renormalized self-coupling, and show that the system can exist in one of two possible phases, both having a free particle of arbitrary mass. In one phase the vacuum expectation of the field vanishes, while in the other it is nonzero. This opens the possibility that, even though the self-coupling vanishes, the field can still be used to generate masses for gauge bosons and fermions.

©1987 The American Physical Society.

PACS: 11.15.Ha 11.10.Ef


Ever since Wilson^{1}showed that in 4 space time dimensions the self coupling of the lambda phi^{4}theory vanishes, there has been extensive analytic and numerical study on the subject.^{2,3}The interest is justified by the importance of understanding this simplest of all quantum field theories, as a prelude to studying the Higgs sector in the standard model, and in grand unified theories. We offer some new insight to the problem gained through a Monte Carlo calculation of the effective potential.

Consider a one component scalar field phi (x) in an external source J(x), with classical Lagrangian density in Minkowski space given by

L(x) = cf12 ( partial phi )^{2}- cf12 r_{0}phi^{2}- case 1 over case 4 lambda_{0}phi^{4}+ J phi ,   (1)

where lambda_{0}> 0_ and - inf < r_{0}< inf_. The quantum theory is defined through the usual Euclidean path integral. An ultraviolet cutoff is introduced by discretizing Euclidean space time as a four dimensional (4D) square lattice of spacing a. Eventually the cutoff is removed by taking the continuum limit a -> 0, with the usual attendant renormalizations. The field phi and the mass gap m (invariant mass of the lowest excited state) both have dimensions of inverse length, and hence have the forms

phi = a^{-1}phi_{roman}latt , m = a^{-1}m_{roman}latt ,   (2)

where phi_{roman}latt_ is a dimensionless field (but still unrenormalized) and m_{roman}latt_ is a dimensionless mass (renormalized). The renormalized field is phi_{r}=Z^{-1/2}phi_, where Z is the wave function renormalization constant. Correspondingly Z^{-1/2}phi_{roman}latt_ is the dimensionless renormalized field. Clearly both m_{roman}latt_ and phi_{roman}latt_ must vanish in the continuum limit, but the parameter

b = langle phi_{r}rangle /m = Z^{-1/2}langle phi_{roman}latt rangle /m_{roman}latt   (3)

may be nonzero and finite. At fixed lambda_{0_}the continuum limit corresponds to the critical value of r_{0_}at which m_{roman}latt =0_. On both sides of this critical point we expect to find different phases, which correspond to continuum field theories with or without spontaneous symmetry breaking (i.e., the parameter b defined above is nonzero in one phase, but vanishes in the other). The critical value of r_{0_}is generally nonzero; it vanishes only in lowest order perturbation theory.

The mass m_{roman}latt_ can be extracted from the two particle correlation function, or propagator DELTA (x), as follows. We integrate DELTA (x) over spatial coordinates at large values of the Euclidean time x_{4_.}The result should be proportional to exp (-mx_{4})_. Setting x_{4}= a tau_, so that tau is dimensionless, we have m x_{4}= m_{roman}latt tau_. The wave function renormalization constant Z may be extracted from the Fourier transform DELTA tilde (p) of DELTA (x), as the residue of the pole at p^{2}= m^{2.}Defining DELTA tilde_{roman}latt (p) = a^{-2}DELTA tilde (p)_, we can write

Z = ^lim from {a -> 0} ^ [ m_{roman}latt ^{2}DELTA tilde_{roman}latt (0) ] .   (4)

Our renormalized self coupling constant lambda_{r_}is defined as case 1 over case 6 of the amputated one particle irreducible (1PI) 4-point function at zero momenta. (The factor case 1 over case 6 is chosen so that in lowest order perturbation theory lambda_{r}= lambda_{0_.)}In numerical calculations, we can get better accuracies by extracting it from the effective potential U( phi ), which can be calculated as follows.^{4}Introduce a constant external source J, and calculate the vacuum expectation f = langle phi rangle. By regarding J as a function of f we have the derivative of the effective potential:

U prime (f)=J(f) .   (5)

Let phi_{0_}be the vacuum expectation of the field in the absence of an external source, i.e., J( phi_{0}) =0_. The effective potential has the following expansion:

U( phi ) = ^sum from n=2 to inf ^ {G^{(n)}}over {n !} ( phi - phi_{0})^{n} ,   (6)

where G^{(n)}is the Fourier transform of the unrenormalized amputated 1PI n-point function, with all external four momenta set to zero. The renormalized version of G^{(n)}is

G_{r^{(n)}=}Z^{n/2}G^{(n)} ,   (7)

where the wave function renormalization constant Z can be obtained from (4) by noting that

J prime ( phi_{0}) = [ DELTA tilde_{roman}latt (0) ]^{-1} .   (8)

Thus we can obtain the following renormalized parameters from J(f):

lambda_{r}= Z^{2}J prime prime prime ( phi_{0}) ,   (9)

lambda_{3,r}= Z^{3/2}J prime prime ( phi_{0}) ,   (10)

m_{roman}latt ^{2}= Z J prime ( phi_{0}) .   (11)

Here lambda_{r_}is the renormalized four particle vertex, and lambda_{3,r_}the renormalized three particle vertex, which is expected to be present when phi_{0}!= 0_. The relation (11) can be used as a check on m_{roman}latt_.

From the results of Ref. 2 we can deduce that a continuum limit does not exist for any lambda_{r}> 0_. The reason is that lambda_{0_}becomes divergent at a nonzero lattice constant. This phenomenon is reminiscent of the exactly soluble Lee model,^{5}in which a ghost^{6}(bound state with negative norm) appears for all nonzero real values of the renormalized coupling constant. In fact, perturbation theory extended by the renormalization group^{7}gives a result very similar to that in the Lee model:

1 over {lambda_{0}}= 1 over {lambda_{r}}- 3 over {128 pi} ^ln ( {LAMBDA^{2}}over {m^{2}}right ) ,   (12)

where LAMBDA is the cutoff momentum and m is a mass scale. For a fixed lambda_{r}> 0_ the cutoff is bounded by a critical value LAMBDA_{c_,}at which lambda_{0_}diverges:

( LAMBDA_{c}/m)^{2}= ^exp (128 pi^{2}/3 lambda_{r}) .   (13)

Thus the cutoff can approach infinity only if lambda_{r}-> 0_. We call LAMBDA_{c_}the ``ghost point,'' by analogy with the Lee model. The same phenomenon has been conjectured in quantum electrodynamics (the Landau ghost),^{7}for which LAMBDA_{c}= m_{e}^exp (3 pi / alpha )_, where m_{e_}is the electron mass and alpha the fine structure constant. As we shall see, however, perturbation theory vastly overestimates the value of LAMBDA_{c_.}

The divergence of lambda_{0_}at the ghost point means that our theory reduces to a 4D Ising model, whose critical behavior therefore governs the continuum limit. Thus, the order parameter and the inverse correlation length have the behavior

langle phi_{r}rangle -> t^{beta}, m -> t^{nu} ,   (14)

where t = | r_{0}- r_{c}|_, r_{c_}being the critical value of r_{0_.}We can now verify that b -> t^{{beta}- nu} = const(up to possible logarithmic corrections) because in four dimensions the critical indices have the mean field values beta = nu = cf12.

The relation (12) indicates that the ghost arises from the fact that the theory is not asymptotically free. From a more elementary point of view we can qualitatively understand the vanishing of lambda_{r_}in the continuum limit by appealing to the nonrelativistic analog of the system, which is an N-particle system with repulsive delta function interactions. It is well known that such an interaction in the Schroumldinger equation in three spatial dimensions does not lead to scattering; i.e., the T matrix is identically zero.


We make Monte Carlo simulations of the field theory on a 10^{4}square lattice with periodic boundary conditions. To update a site, we first flip the sign of phi with a probability determined by a ``heat bath'' algorithm, then increment it by a random number D, with |D| < D_{0_.}The change is accepted or rejected according to a standard Metropolis algorithm. The value of D_{0_}is adjusted to give roughly a 50% acceptance rate. The purpose of the sign flip is to prevent the value of phi from being trapped in the neighborhood of one of two possible potential minima. For a fixed value of lambda_{0_}we search for the critical point by varying r_{0_.}For each lambda_{0_}and r_{0_,}the following quantities are calculated: (a) the propagator with J=0, which gives the wave function renormalization Z and the mass gap m_{roman}latt_; (b) the ensemble average f = langle phi_{roman}latt rangle_ for a range of values of J, which yields J(f) as the derivative of the effective potential. We fit J(f) to a least squares polynomial in order to extract m_{roman}latt_ and lambda_{r_.}

The computations were done on a VAX 780. Typically it took 200 sweeps to warm up the lattice. For the case J=0, 10^000 ndash 20^ 000 sweeps were needed to obtain an acceptable level of accuracy for the propagator. Generating an acceptable effective potential is easier, requiring only 2000 ndash 5000 sweeps, depending on how close we were to the critical point. The CPU time required was about 1 day for 10^000 sweeps. At this rate a calculation of lambda_{r_}via the 4-point function would be practically impossible, due to the large errors introduced by the subtractions required for connected parts. On the other hand, calculating the effective potential proves to be a very efficient and economical way to obtain both lambda_{r_}and m_{roman}latt_.

Because of the finite size of the lattice, there is no sharp phase transition, and m_{roman}latt_ never actually goes to zero. To help determine the critical point as best we could, we kept a record of the average field over the lattice, at successive sweeps. That is, we kept track of the evolution of the average field. In this manner, we could see in detail how spontaneous symmetry breaking develops when r_{0_}was varied across the critical point. Some samples are shown in Fig. 1 for the case lambda_{0}= 1000_. For r_{0}= -150_ the average field makes small fluctuations about zero. The fluctuations become more pronounced as r_{0_}is decreased toward the critical point between -160 and -165. Spontaneous symmetry breaking is already evident at -165, where the field flip flops with a period much greater than the characteristic time scale at -160. By -175 the period has become much greater than the observation time, and broken symmetry becomes manifest on a macroscopic scale.


We have carried out computations for lambda_{0}=1_, 100, 1000, inf. The first value is small enough to allow a comparison with perturbation theory, and the last limiting value corresponds to the 4D Ising model. Figure 2 shows plots of the derivative of the effective potential for lambda_{0}=1_, for a range of values of r_{0_}that includes the critical point. Figure 3 shows various parameters as functions of r_{0_,}for lambda_{0}=1_. In both figures the predictions of zero loop and one loop perturbation theories are also shown for comparison. Figures 4 and 5 give the same plots for lambda_{0}=100_, but comparison with perturbation theory for this case is inappropriate, and therefore omitted.

The data for Z in Figs. 3 and 5 are obtained from the effective potential. The corresponding data obtained from the propagator via (4) are consistent with the above in the symmetric phase, both in central values and statistical errors. In the ``broken'' phase, however, the data obtained via the propagator have such large errors as to render them useless. This demonstrates the efficacy of the effective potential method.

In the weak coupling case lambda_{0}=1_, the predictions of one loop perturbation theory compare well with the numerical results away from the critical point, but they break down near the critical point. For example, perturbation theory yields a complex effective potential, whereas the actual effective potential is always real. The one loop perturbation theory completely fails in its prediction Z=1. As we can see in the figures, Z depends on r_{0_,}and vanishes at the critical point.

From the plots in Figs. 3 and 5 we see that langle phi_{roman}latt rangle_ and m_{roman}latt_ both vanish at the critical point, but the ratio b is discontinuous, approaching zero from one side, and nonzero from the other. This defines the two phases of the system in the continuum limit, a symmetric and a symmetry broken phase. We shall return shortly for a closer look at the behavior of b in the symmetry broken phase.

In Fig. 6 we plot the renormalized coupling lambda_{r_}as a function of m_{roman}latt_, for various values of the unrenormalized coupling lambda_{0_.}The origin corresponds to the continuum limit. Separate plots are given for the two different phases corresponding to r_{0}> r_{c_}and r_{0}< r_{c_.}We see that the limiting curve with lambda_{0}= inf_ divides the plane into two parts, and no point falls to the left of it. This clearly shows the ghost point, the smallest possible m_{roman}latt_ for fixed renormalized coupling, where the bare coupling diverges. The forbidden region is presumably ghost land, to which unfortunately computers are still denied access.

Figure 7 shows results for the three particle vertex in the symmetry broken phase. This parameter is proportional to lambda_{r_}in lowest order perturbation theory, but in general it might be an independent quantity. Here we check that it indeed vanishes in the continuum limit as expected, thus explicitly demonstrating that in the continuum limit there is neither three particle nor four particle vertex.

The boundary of the forbidden region in Fig. 6 is the locus of the ghost point corresponding to various values of lambda_{r_.}These values, the closest possible approach to the continuum limit for a given lambda_{r}> 0_, are larger from those predicted by perturbation theory in (12) by orders of magnitude. That is, on the scale of Fig. 6, perturbation theory would have lambda_{r_}drop to zero precipitously near the continuum limit. Instead, our results show a gentle decrease. The reason is that the vanishing of lambda_{r_}in Fig. 6 is caused mainly by the fact that Z -> 0, whereas perturbation theory, even with the summing of all one loop graphs, gives Z=1.

In the present theory the value of the parameter b is without physical significance. Nevertheless we shall examine it in greater detail, for it may give an indication of whether the Higgs mechanism still works when the scalar field is coupled to other fields, despite the fact that lambda_{r}=0_. To this end we exhibit b in Fig. 8 as a function of m_{roman}latt_ in the symmetry broken phase, for various values of lambda_{0_.}It appears that b extrapolates to different values in the continuum limit, depending on lambda_{0_.}This would indicate that the vacuum expectation of the field sets an arbitrary mass scale independent of m.

However, the numerical data cannot rule out the possibility that b diverges in the continuum limitmdashan expected behavior if one believes in the relation b prop lambda_{r^{-1/2_}from}perturbation theory. To test this, we plot lambda^{1/2}b as a function of m_{roman}latt_ in Fig. 9, for various values of lambda_{0_.}The function is remarkably constant for lambda_{0}=1_; but for larger values of lambda_{0_}it appears to approach zero in the continuum limit. We take this to be tentative indication that b is a parameter independent of lambda_{r_}and that lambda_{0_}is not an irrelevant parameter of the theory.


The existence of a phase in the continuum limit with broken symmetry suggests that, even though the renormalized self coupling is zero, a gauge field coupled to the scalar field can acquire mass through the usual Higgs mechanism. In fact Coleman and Weinberg^{8}have verified this in perturbation theory. The answer is by no means certain, however, because the self consistency of the perturbative treatment relies on the implicit assumption lambda_{r}> 0_, which is not true.

The question of generating fermion masses remains equally open. If we couple a fermion to the scalar field through the Yukawa coupling g psi bar phi psi we might expect to generate a mass term g_{r}langle phi rangle psi bar psi_. But the renormalized Yukawa coupling g_{r_}may vanish in the continuum limit, since it is not asymptotically free. Then, again, this may be compensated by the fact that b actually diverges. On top of all the uncertainty, we must add the further caveat that the mass term above is no more than a naive expectation suggested by perturbation theory.

In closing we must draw particular attention to the fact that the resulting field theory in the continuum limit is drastically different from what we have been conditioned to expect by perturbation theory. In the phase with broken symmetry, the ``Higgs''-boson mass m is arbitrary, even though lambda_{r}= 0_. Perturbation theory would have us erroneously believe that m is proportional to lambda_{r}^{1/2_.}The lesson we learn is that it is futile to make conjectures on generating masses for gauge fields and fermions. First of all, the perturbative connection between coupling constants and masses may not hold. Second, introducing new couplings may drastically change the system. These physically relevant problems will have to be studied by nonperturbative methods and we are continuing our computational program to address them.


We thank J. Shigemitsu for pointing out an error in the original handling of the data in Fig. 6. This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under Contract No. DE AC02 76ER03069.


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