# Pauli–Zeldovich cancellation of the vacuum energy divergences, auxiliary fields and supersymmetry

## Abstract

We have considered the Pauli–Zeldovich mechanism for the cancellation of the ultraviolet divergences in vacuum energy. This mechanism arises because bosons and fermions give contributions of the opposite signs. In contrast with the preceding papers devoted to this topic wherein mainly free fields were studied, here we have taken their interactions into account to the lowest order of perturbation theory. We have constructed some simple toy models having particles with spin 0 and spin 1 / 2, where masses of the particles are equal while the interactions can be quite non-trivial.

## 1 Introduction

Many years ago Pauli [1] suggested that the vacuum (zero-point) energies of all existing fermions and bosons compensate each other. This possibility is based on the fact that vacuum energy of fermions has a negative sign whereas that of bosons has a positive one. As is well known, such a cancellation indeed takes place in supersymmetric models (see e.g. [2]). Subsequently in a series of papers Zeldovich [3, 4] related vacuum energy to the cosmological constant. However rather than eliminating divergences through the boson-fermion cancellation, he suggested the Pauli-Villars regularization of all divergences by introducing a number of massive regulator fields. Covariant regularization of all contributions then leads to finite values for both the energy density \(\varepsilon \) and (negative) pressure *p* corresponding to a cosmological constant, i.e. connected by the equation of state \(p =-\varepsilon \).

In our preceding paper [5] we examined the conditions for the cancellation of the ultraviolet divergences of the vacuum energy to the leading order in \(\hbar \), i.e. by considering free theories and neglecting interactions. Such conditions are reduced to some sum rules involving the masses of particles present in the model. We formulated these conditions not only for the Minkowski spacetime, but also for the de Sitter one. In the latter case, the radius of the de Sitter universe also enters into the mass sum rules. In paper [6] we applied such considerations to observed particles of the Standard Model (SM) and also studied the finite part of vacuum energy. This last contribution should be very small, so as to obtain a result compatible with the observed value of the cosmological constant (almost zero with respect to SM particle masses). We showed [6] that it was impossible to construct a minimal extension of the SM by finding a set of boson fields which, besides canceling ultraviolet divergences, could compensate residual huge contribution of known fermion and boson fields of the Standard Model to the finite part of the vacuum energy density.

On the other hand, we found that addition of at least one massive fermion field was sufficient for the existence of a suitable set of boson fields which would permit such cancellations and obtained their allowed mass intervals. On examining one of the simplest SM extensions satisfying the constraints, we found that the mass range of the lightest massive boson was compatible with the Higgs mass bounds which were known at the time of the publication of the paper [6]. As is well known, later the Higgs boson was discovered at the LHC [7, 8]. For some time it appeared that there might exist an the observed diphoton excess at 750 GeV [9]. This excess, had it been confirmed, could be interpreted as an indication for the existence of a new heavy elementary or composite particle with a mass of the order of 750 GeV. Later this phenomenon disappeared, nonetheless inspiring in the meanwhile quite a few theoretical works. In particular, we also studied in our preprint how the presence of such a new particle could be included into our scheme of the cancellation of ultraviolet divergences of vacuum energy [10].

In our preceding papers [5, 6, 10] we studied only free theories without interactions. It is also interesting to take interactions into account, at least to the lowest order of perturbation theory. This is not easy, and in the present paper we shall concentrate on the construction of relatively simple toy models where the “Pauli–Zeldovich cancellation” of ultraviolet divergences still takes place.

We wish to emphasize that the approach employed in the present paper represents a whole direction in quantum field theory which goes well beyond effective low energy field theory and, although based on some hypothesis, has not been proven to be wrong. We also wish to mention the paper by Ossola and Sirlin [11], where contributions of fundamental particles to the vacuum energy density were discussed with a special attention to relations between different regularization schemes and to the appearance of power divergences in different contexts. Other related approaches are presented in Refs. [12, 13, 14].

In the recent paper [15], it was noticed that under certain circumstances (in particular, but not limited to finite QFTs), the Pauli cancellation mechanism would survive the introduction of particle interactions. It was pointed out there that for the mass sum rules to be valid at different mass scales, it is necessary to impose some relations on mass runnings with energy. Thus, the corresponding relations between anomalous mass dimensions were formulated [15]. However, concrete examples were not constructed.

In the present paper we discuss some relatively simple examples of models where the Pauli–Zeldovich cancellation takes place to the first order of perturbation theory. Being inspired by the famous supersymmetric Wess–Zumino model [16], we consider models with spinor, scalar and pseudoscalar fields only. We hope to treat vector (gauge) fields in future works. The models which we discuss are not supersymmetric, but they have one important feature which makes them akin to supersymmetric models: the number of the fermion and boson degrees of freedom in them is the same. That implies an unexpected feature: the necessity to take so called auxiliary fields into account. Such fields are necessary in the supersymmetric models, because they allow one to formulate supersymmetry transformations in a coherent way.

But their role is even more ubiquitous. To conserve supersymmetry, it is necessary to have the balance between fermion and boson degrees of freedom not only on shell, but also off shell. However, the number of degrees of freedom of a spinor field doubles when it is off shell. For example, a Majorana spinor has two complex components, i.e. four degrees of freedom off shell. When we require the satisfaction of the first-order Dirac equation, the number of degrees of freedom becomes equal to two. Thus, for example, in the Wess–Zumino [16] model one has two fermion degrees of freedom of the Majorana spinor and two boson degrees of freedom associated with the scalar and pseudoscalar fields. Off shell the number of fermion degrees of freedom becomes equal to four, while the role of two additional boson fields is played by two auxiliary fields which become in a sense independent off shell. If we consider non-supersymmetric models with the Pauli–Zeldovich mechanism of cancellation of ultraviolet divergences for vacuum energy in the presence of interactions, then the number of the boson and fermion degrees of freedom should be equal not only on shell, but also off shell. This means that we should introduce auxiliary fields. Further, when we consider a model with interactions, we should not only take into account running of masses of the fields, but also consider cancellations of contributions coming from the potential terms in the Lagrangians. It is there that the introduction of the auxiliary fields becomes very convenient. Fortunately, we shall see that, at least in the considered class of spinor-scalar models, the introduction of auxiliary fields is equivalent to a simple rule for the calculation of some contribution to the scalar fields self-interaction. Here we can add that, in principle, one can perform all calculations and show that in the formalism where auxiliary fields are excluded, vacuum energy in the supersymmetric models is equal to zero. However, in this case there are no separate cancellations of the potential energy and of the kinetic energy between bosons and fermions. Thus, verification of the analogous cancellation in non-supersymmetric models becomes more complicated. Hence, it is better to implement rather simple rules, equivalent to the explicit introduction of auxiliary fields, which will be used in the present paper.

Here we present a model consisting of a Majorana fermion and two scalar fields with the same mass and with different kinds of interactions, and we show that for such a model, one can find a family of coupling constants such that the Pauli–Zeldovich mechanism for the cancellation still works. Then we find an analogous family of models with a Majorana fermion, a scalar field and a pseudoscalar field. Obviously, the Wess–Zumino model belongs to this family. We also discuss briefly models where particles with different masses are present.

The structure of the paper is as follows: in the second section we briefly discuss the mass sum rules for theories without interactions; in the third section we formulate rules for the conservation of the mass sum rules when interactions are switched on. In Sect. 4 we discuss the vacuum expectation values of the potential terms and the role of auxiliary fields. In Sect. 5 we present a model with one Majorana field and two scalar fields. In the sixth section we consider a model with one Majorana field, one scalar field and one pseudoscalar field. Section 7 is devoted to the discussion of models with non-degenerate masses, the last section contains some concluding remarks.

## 2 Vacuum energy and the balance between the fermion and boson fields

*m*, then \(\omega =\sqrt{k^2c^2 + m^2c^4}\), where

*k*is the wave number. In the following we shall set \(\hbar =1\) and \(c=1\). The energy density of vacuum energy of a scalar field treated as free oscillators with all possible momenta is given by the divergent integral [3]:

*S*,

*V*and

*F*denote scalar, massive vector and massive spinor Majorana fields respectively (for Dirac fields it is sufficient to put a 4 instead of 2 on the right-hand sides of Eqs. (5) and (6)). For the case in which the conditions (4), (5) and (6) are satisfied, the remaining finite part of the vacuum energy density is equal to

## 3 Running masses and anomalous mass dimensions

We shall here derive the expressions for these anomalous mass dimensions. Generally the technique of such calculations was developed many years ago [19, 20, 21, 22, 23, 24, 25]. However, for convenience and completeness we shall perform all the derivations from the start. On considering our toy models with degenerate masses, we shall not really use them explicitly. It will be enough to study shifts of masses induced by radiative corrections for different fields present in the models under consideration. However, when one considers models where particles with different masses are present, the formulas given in this section become necessary.

Our treatment of the anomalous mass dimensions in the presence of quadratic divergences is based on the approach presented in paper [26], which in turn uses the version of renormalization group formalism connected with dimensional regularization [27].

*M*and a scalar field with a mass

*m*.

*d*is the dimensionality of the spacetime such that

*B*and \(\varGamma \) functions

*M*is

*M*through the relation

*M*. Let us remember that when we use the dimensional regularization, the renormalized quantities depend on the renormalization mass parameter \(\mu \). At the same time the bare quantities depend on the regularization parameter \(\varepsilon \), but do not depend on the renormalization mass parameter \(\mu \). Thus, we can write down a general equation

*g*, Eq. (40) becomes

*g*:

*d*-dimensional spacetime. In this case, the formula (49) is replaced by

*L*is the number of loops. Let us again consider a diagram with \(E_F=0, E_B=2\). This diagram, which is quadratically divergent at \(d=4\) becomes logarithmically divergent (\(\omega (G)=0\)) at \(d = 4-\frac{2}{L}\). That means that the quadratic divergence is represented as a pole of the quantity

## 4 Contribution of potential terms into the vacuum energy and the auxiliary fields

*V*represents potential terms. For the term

*I*is defined as

*T*-exponent. This contribution (for the case of a Majorana spinor) is equal to

*M*is the fermion mass and

*m*is the scalar mass. A simple calculation shows that for the case of the Wess–Zumino model, when \(m=M\) and there are well-known relations between the coupling constants [16], the quartic divergences present in the contributions (76) and (79) do not cancel each other (we shall present detailed calculations in the next section). Namely, the contribution of the spinors is twice that of the scalars. The reason for this mismatch was already discussed in the Introduction. The point is that the number of fermion degrees of freedom is doubled off shell. To compensate this effect, we should introduce the auxiliary scalar fields as is done in supersymmetric models. A simple example shows that this exactly gives the doubling of the leading contribution to vacuum energy. Indeed, let us consider a model with the Lagrangian

*F*by means to the equation of motion

## 5 A model with one Majorana and two scalar fields

*A*and

*B*. All the fields have the same mass

*m*and the interaction is given by

*A*and

*B*should be cancelled to avoid the necessity of introducing linear in fields terms into the Lagrangian. The tadpole for the field

*A*arises due to the contraction of this field with the vertices \(A^3, AB^2\) and \(\bar{\psi }\psi A\). All these contributions are proportional to the integral (77). The corresponding combinatorial factors are \(3mh_1\), for \(A^3,mh^4\) for \(AB^2\) and \(-4mg_1\) for the vertex \(\bar{\psi }\psi A\). The last contribution arises due to the trace of the fermion propagator which is proportional to the mass

*m*. Thus, the cancellation of the tadpole diagram for

*A*requires

*B*requires

*A*obtains the contributions from the vertex \(A^4\), from the vertex \(A^2B^2\) and from the pair of vertexes \(\bar{\psi }\psi A,A^3,A^2B\) and \(AB^2\). The contributions of two quartic vertexes are both proportional to the integral

*I*. The corresponding coefficients are \(12\lambda _1\) and \(2\lambda _3\). The contribution of the fermion loop is

*I*should be canceled because such divergences do not arise in the self-energy correction to the fermion propagator. Thus, we have

*A*in the one-loop approximation has the form

*A*. The analogous shift for the second scalar field is

*A*and

*B*. The lower bound of the scalar field potential exists and is determined by the quartic terms with positive constants \(\lambda _1,\lambda _2\) and \(\lambda _3\).

## 6 Model with a Majorana field, a scalar field and a pseudoscalar field

*B*is a pseudoscalar. In this case

*A*which coincides with that given by Eq. (89). The conditions for the cancellation of quadratic divergences in the propagators of the scalar and pseudoscalar fields are also the same (94) and (95). However, the shifts of the mass squared for the fields

*A*,

*B*and \(\psi \) are different. They are proportional to

We have seen that for the case with one Majorana field, one scalar and one pseudoscalar we have less freedom in the choice of the coupling constants than in the case of two scalar fields and one Majorana field, but this choice is still broader than that in the Wess–Zumino model.

## 7 Models with non-degenerate masses

It is interesting to find toy models with masses which are not degenerate. In this case it is necessary to consider at least four boson and four fermion degrees of freedom [5]. The simplest models of this kind are those which include a certain number of “triplets” of the types described in two preceding sections, i.e. with degenerate masses inside any triplet and with coupling constants (again, describing interactions within a triplet) which satisfy the relations obtained in the Sects. 5 and 6. Naturally, in this case, if there are no interactions between the fields belonging to different triplets, then the Pauli–Zeldovich mechanism does work. If we introduce interactions between different triplets with different masses, then the coupling constants should satisfy some constraints.

*A*and

*B*and the Majorana spinor \(\psi \) belong to the first triplet, while the scalar fields

*C*and

*D*and the Majorana spinor \(\chi \) belong to the second triplet.

It is now possible to find relations which constrain the choice of the coupling constants given in the interaction Hamiltonian (120). However, this task is rather cumbersome and we shall postpone it and the construction of other models for future work [29].

## 8 Concluding remarks

In this paper we have studied the Pauli–Zeldovich mechanism for the cancellation of ultraviolet divergences in vacuum energy which is associated with the fact that bosons and fermions produce contributions to it having opposite signs. In contrast with the preceding papers devoted to this topic where only free fields were considered, here we have taken interactions up to the lowest order of perturbation theory into account. We have constructed a number simple toy models having particles with spin 0 and spin 1 / 2, wherein masses of the particles are equal while interactions can be quite non-trivial. To make calculations simpler and more transparent, it was found useful to introduce some auxiliary fields. It appears that the presence of these fields is equivalent to the modification of some contributions of the physical fields to the vacuum expectation of the potential energy. We hope to construct more complicated models including particles with different masses and in the presence of vector fields in future work [29].

## Notes

### Acknowledgements

We are grateful to A. O. Barvinsky, F. Bastianelli and O. V. Teryaev for useful discussions.

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