A Calculation of Expectation Values

Gaussian-Perturbative Calculations with a Homogeneous External Source

Abstract

We derive the equation of the critical curve and calculate the renormalized masses of the -symmetric model in the presence of a homogeneous external source. We do this using the Gaussian-Perturbative approximation on finite lattices and explicitly taking the continuum limit. No disabling divergences are found in the final results, and no renormalization is necessary. We show that the results give a complete description of the critical behavior of the model and of the phenomenon of spontaneous symmetry breaking, at the quantum-field-theoretical level.

We show that the renormalized masses depend on the external source, and point out the consequences of that fact for the design of computer simulations of the model. We point out a simple but interesting consequence of the results, regarding the role of the model in the Standard Model of high-energy particle physics. Using the experimentally known values of the mass and of the expectation value of the Higgs field, we determine uniquely the values of the bare dimensionless parameters and of the model, which turn out to be small numbers, significantly less that one.

1 Introduction

Years ago we introduced a calculational technique that was quite successful in describing the critical behavior of the -symmetric Euclidean model in spacetime dimensions [1]. Some quantities were calculated in and compared to the results of computer simulations, yielding surprisingly good results, and describing reliably the most important qualitative aspects of the model. The observables calculated where the expectation value of the field, which is the order parameter of the critical transition of the model and describes the phenomenon of spontaneous symmetry breaking, and the two-point function, from which one can get the renormalized masses and hence the correlation lengths, in both phases of the model.

Although inspired by and superficially similar to perturbation theory, the technique can handle a phenomenon such as spontaneous symmetry breaking, which is usually considered to be out of reach for plain perturbation theory. The innovative and essential aspect of the technique is the use of certain self-consistency conditions within a framework similar to that of perturbation theory. The technique would be better described as a Gaussian approximation rather than a perturbative expansion. As such, it is able to produce good predictions for the one-point and two-point observables, since these are the moments present in the Gaussian distribution, but should not be expected to go much further than that. For lack of a better name, we shall refer to it as the Gaussian-Perturbative approximation.

The important role that the four-component model plays in the Standard Model of high-energy particle physics makes it certainly interesting to learn more about it. In this paper we extend the Gaussian-Perturbative technique introduced in [1] to the same model in the presence of external sources. These external sources are not thought of merely as analytical devices used to extract the Green’s functions from the functional generators of the model, and to be put to zero afterwards. Instead, they are thought of as actual physical sources of particles in the model. One important objective is to determine how the introduction of the external sources affects the values of the renormalized masses in either phase of the model.

These are analytical calculations performed on the Euclidean lattice, which therefore allow us to discuss, and to explicitly take, specific continuum limits in the quantum theory. As we will see, there is no need for perturbative renormalization, or for any regulation mechanism other than the lattice where the model is defined. All calculations on finite lattices are ordinary straightforward manipulations. Although there are some quantities that do diverge in the continuum limit, they all cancel off from the observables before the limit is taken.

2 The Model

Let us start by giving the definition of the model, in the classical and quantum domains, and then quickly reviewing the Gaussian-Perturbative approximation. Consider then the Euclidean quantum field theories of an -symmetric set of scalar fields defined within a periodical cubic box of side in dimensions by the classical action

where . This is the usual form of the -symmetric model in the classical continuum, with an external source , which by assumption is a constant. The vector notation is shorthand for

and the dot-product notation represents the scalar product of vectors in the internal space, that is a sum over ,

In this action the quantity is a homogeneous external source associated with the field component. Its introduction breaks the symmetry, of course, and causes the generation of a non-zero expectation value for the field component.

In order to use the definition of the quantum theory on a cubical lattice of size with sites along each direction, with lattice spacing , we consider the corresponding lattice action

where all quantities are now dimensionless, defined by the appropriate scalings,

(1)

In order for the model to be stable we must have and, in addition to this, if then we must also have . Up to this point there are no further constraints on the real parameters and .

When possible, the summations are notated, in the subscript, by the variable which is being summed over, and, in the superscript, by the number of terms in the sum. The integer coordinates are taken to vary as symmetrically as possible around the origin , that is we have with certain values of and that depend on the parity of ,

for odd , and

for even , in either case for all values of .

In this paper we will perform the calculations of the critical line and of the renormalized masses in a situation in which we have, in terms of the dimensionfull field , for ,

and, for ,

where is a constant with the physical dimensions of the field . In terms of the dimensionless field we have for the only non-trivial condition

where the dimensionless constant is given by .

We will consider continuum limits in which we have both and . In order to do this we will choose to make increase as and decrease as , so that we still have . The calculations on finite lattices will be performed with periodical boundary conditions, with the understanding that at the end of the day such a limit is to be taken.

Observe that we are specifying the value of the expectation value of rather than the value of the corresponding external source . What we are doing here is to assume that there is some external source present such that we have the expectation value specified. It follows that one of the expected results of our calculations is the determination, at least implicitly, of the form of the external source in terms of .

Our first calculational task in preparation for the Gaussian-Perturbative calculations is to rewrite the action in terms of a shifted field, which has a null expectation value. We thus define a new field variable such that

so that we have for all , with . We must now determine the form of the action in terms of . If we write each term of the action in terms of the shifted field we get

We will now eliminate all field-independent terms, since they correspond to constant factors that cancel off in the ratios of functional integrals which give the expectation values of the observables. Doing this we get the equivalent action

where except for the kinetic part the terms have been ordered by increasing powers of the field.

The last task we have to perform, in preparation for the Gaussian-Perturbative calculations, is the separation of the action in two parts. Since the symmetry is broken by the introduction of the external sources, besides the fact that depending on the values of the parameters and it might be spontaneously broken as well, this separation involves two new mass parameters, for , and for . Note that an symmetry subgroup is left over after the symmetry breakdown. We therefore adopt for the Gaussian part of the action

(2)

where there are no constraints on the parameters introduced other than and . Note that, despite the way in which this is written, we do in fact have here just an mass term for each field component , for , and an mass term for the field component . It follows that the non-Gaussian part of the action is

(3)

which has its terms now written strictly in the order of increasing powers of the field.

Let us end this section by recalling the calculational techniques that will be involved. Given an arbitrary observable its expectation value is defined by

which is a function of , where denotes the flat measure and hence integrals from to over all the field components at all sites. The expectation values of the model are obtained for , and the corresponding expectation values in the Gaussian measure of are those obtained for . The Gaussian-Perturbative approximation consists of the expansion of the right-hand side in powers of to some finite order, around the point , and the application of the resulting expression at . The first-order Gaussian-Perturbative approximation of the expectation value of the observable is given by

where the subscript indicates the expectation values in the measure of . These expectation values are most easily calculated in momentum space, where they involve only uncoupled Gaussian integrals. Therefore, let us also recall here the transformations to and from the momentum space representation of the model. We have for the field and its Fourier transform

where the sums over are taken in as symmetric a way as possible around , just as we did for . In other words, we have with the same values of and , depending on the parity of , that were used for and ,

for odd , and

for even , in either case for all values of . The orthogonality and completeness relations of the Fourier base are given by

A typical Gaussian expectation value in momentum space, and possibly the most fundamental one, is given for a generic field component by

where is either or , depending on the field component involved, and where are the eigenvalues of the discrete Laplacian on the lattice, which are given by

This and several other expectation values, Gaussian integration formulas and lattice sums can be found in Appendix B.

3 Calculations

We are now ready for the Gaussian-Perturbative calculations. We start with the calculations involving the critical behavior of the model, so that we may determine and characterize its two phases. It is important to observe that the phase structure of the model must be established right at the beginning, because everything else has to be discussed in terms of it.

3.1 The Critical Line

We will now calculate the Gaussian-Perturbative approximation for the particular observable , at some arbitrary point . If we write the observable in terms of the shifted field we get

In order to get the equation of the critical line we impose, in a self-consistent way, that we in fact have

which is the same as stating that

In the first-order Gaussian-Perturbative approximation this becomes

Since we have , because this observable is field-odd and the Gaussian action is field-even, we get for the critical line the simple equation, known as the tadpole equation,

The expectation value shown here is calculated in Appendix A, given in Equation (A.4), and the result is

The parameter cancels off from our equation, and thus we are left with the result

(4)

in which we now isolated on the left-hand side the term with the external source. This gives the general relation between and at each point of the parameter space of the model. As we shall see later, from this result we can determine the critical behavior of the model and derive the equation of the critical line.

The quantity is the width or variance of the local distribution of values of the field components , with , in the measure of ,

as one can see in Appendix B, Equation (B.6), and has the following interesting properties, so long as . First, it is independent of the position , as translation invariance would require. Second, for it has a finite and non-zero limit, so long as with a finite value of in the limit. Finally, the value of in the limit does not depend on the value of in that same limit. Analogously, the quantity is associated to the remaining field component and to the mass parameter , and has these same properties. In fact, and have exactly the same value in the limit.

3.2 The Transversal Propagator

We will now calculate the expectation value of the observable

which has the same form for all components of the field except . We call this the transversal propagator because it belongs to the field components which are orthogonal to the direction of the external source in the internal space. In this section we will assume that , in fact we will make . The observable will be taken at two arbitrary points and . The first-order Gaussian-Perturbative approximation for this observable gives

where is the two-point function with mass parameter . We must calculate the two expectation values which appear in this formula. The calculation of the first one is done in Appendix A, given in Equation (A.5), and results in

The second expectation value is also calculated in Appendix A, given in Equation (A.6), and the result is

The factor in front of can now be verified to be exactly equal to , and therefore this whole part cancels off from our observable. We may now write for the difference of expectation values that appears in it,

Finally, we can write the complete result,

where we wrote in terms of its Fourier transform.

In principle we could have used any positive value of for this calculation, but now a particular choice comes to our attention. We see from the structure of this propagator that we can make equal to the transversal renormalized mass parameter by choosing it so that the numerator of the second fraction vanishes. In this way we get a very simple propagator, with a simple pole in the complex plane, in which the parameter appears now in the role of the renormalized mass parameter,

Observe that to this order the propagator is, in fact, the propagator of the free theory. This is a self-consistent way to choose the parameter , and is equivalent to the determination of the transversal renormalized mass. This choice is equivalent to requiring that the mass parameter of the Gaussian measure being used for the approximation of the expectation values be the same as the renormalized mass parameter of the original quantum model. It gives the result

(5)

This result for is valid for a constant but possibly non-zero external source, in both phases of the model, where is the mass associated to the field components , for .

3.3 The Longitudinal Propagator

We will now complete our calculations with the expectation value of the observable

We call this the longitudinal propagator because it belongs to the field component which is in the direction of the external source in the internal space. Once more the observable will be taken at two arbitrary points and . The first-order Gaussian-Perturbative approximation for this observable gives

where is the two-point function with mass parameter . We must now calculate the two expectation values which appear in this formula. The first expectation value is the same we had before for the transversal propagator, and from Equation (A.5) we have,

The second expectation value is calculated in Appendix A, given in Equation (A.7),and the result is

The factor in front of can now be verified to be exactly equal to , and therefore once again this whole part cancels off from our observable. We may now write for the difference of expectation values that appears in it,