# Pauli–Villars regularization elucidated in Bopp–Podolsky’s generalized electrodynamics

## Abstract

We discuss an inherent Pauli–Villars regularization in Bopp–Podolsky’s generalized electrodynamics. Introducing gauge-fixing terms for Bopp–Podolsky’s generalized electrodynamic action, we realize a unique feature for the corresponding photon propagator with a built-in Pauli–Villars regularization independent of the gauge choice made in Maxwell’s usual electromagnetism. According to our realization, the length dimensional parameter *a* associated with Bopp–Podolsky’s higher order derivatives corresponds to the inverse of the Pauli–Villars regularization mass scale \(\Lambda \), i.e. \(a = 1/\Lambda \). Solving explicitly the classical static Bopp–Podolsky’s equations of motion for a specific charge distribution, we explore the physical meaning of the parameter *a* in terms of the size of the charge distribution. As an offspring of the generalized photon propagator analysis, we also discuss our findings regarding on the issue of the two-term vs. three-term photon propagator in light-front dynamics.

## 1 Introduction

Quantum electrodynamics (QED) may be regarded as a prototype of quantum field theories with well-established renormalization program which effectively regulates the infinities present in the local gauge field theory. Due to the infinities that cannot be gotten around, e.g. radiative corrections in QED, one needs to treat and tame such infinities taking a certain regularization procedure with the renormalization condition for physical amplitudes. The very impressive agreement between high precision measurements in accelerators and the predictions of quantum field theory in the presence of radiative corrections is the key for the indication of successful renormalization program. Phenomenological success of atomic model appears ultimately backed up by the successful QED renormalization program.

Historically, the problem of infinities first arose in the classical electrodynamics of point particles in the 19th and early 20th century. The well-known example is the mass of electron including the electromagnetic mass \(m_{\mathrm{em}}\) due to its own electrostatic field given by \(m_{em}=\frac{e^2}{8\pi r_e}\) with the charge *e* and the radius \(r_e\) of the electron, which becomes an infinity as \(r_e \rightarrow 0\). It may not be an overstatement that the early work of Lorentz and Abraham [1, 2, 3] including the bare mass of the spherical shell as well as \(m_{\mathrm{em}}\) to take a consistent point limit provided the inspiration for later development of the renormalization program in QED and other local field theories. Modifying the concept of point charge to an extended charge distribution lends also the physical meaning of charge renormalization as the charge screening due to the Dirac vacuum in QED.

In the same vein, Bopp [4] and Podolsky [5] attempted to remove infinities inherent in the usual treatment of point charges introducing higher order derivatives in the Lagrangian of electrodynamics while maintaining the equations of motion still linear in the fields and preserving gauge invariance. In particular, Podolsky discussed the classical aspects of his model presenting the equations of motion, energy-momentum tensor and plane wave field solutions [5]. Traditionally, however, it has become the case to view the model due to Bopp and Podolsky (“BP model”) as a mechanism to describe massive photons without breaking gauge invariance as the propagating modes of the model comprise both massless photons as well as massive ones. In this work, we demonstrate that the BP model solution for a point charge in electrodynamics corresponds to an ordinary electrodynamic solution for a specific charge distribution. Motivated by this possible reinterpretation of BP model solution in electrodynamics, we further elucidate the BP model as a natural way of providing Pauli–Villars (PV) regularization [6] in ordinary QED. Similar finding of the charge distribution was made by Kvasnica in 1960 [7] and the connection of the BP model with the PV regularization has been discussed, while the mass introduced in the BP model was taken there as the physical mass to be compared with the results of Hoffstadter’s scattering experiments [8]. In this work, we take it here as the ultraviolet PV regularization parameter which cannot be measured but be combined with another unmeasurable quantity, i.e., bare mass, to redefine the physically measurable renormalized mass. The PV regularization scheme has been around since 1949 [6] and early attempts to connect it with higher-order derivative models can be found in the works by Slavnov [9, 10]. The equivalence between the PV regularization and the higher-order derivative regularization was also discussed in the work of Stoilov dealing with spinors only [11].

To discuss our work, it may be worthwhile to make a brief historical remark on previous works on the BP model. A few years later after the introduction of the BP Lagrangian, Podolsky and Kikuchi [12] went through the actual quantization of the model. They claimed that the usual quantization methods of the time could not be directly applied to the BP model of generalized electrodynamics due to the presence of higher order derivatives in the Lagrangian and therefore they needed to introduce extra auxiliary fields. The quantization was performed in an extended phase space and, to take gauge symmetry issues into account, a generalization of the Stueckelberg formalism was used. A review of BP model’s original results by Podolsky and Schwed can be found in Ref. [13]. Some forty years later, the BP model was revisited when Galvão and Pimentel [14] first carefully analyzed its structure of constraints performing the instant-form canonical quantization with Dirac Brackets. In terms of the canonical Dirac-Bergmann formalism [15, 16, 17], BP model has three first-class constraints generating gauge symmetries [14]. The canonical quantization was performed after gauge fixing and promotion of Dirac Brackets into operator commutators. It is worth noting though that in Ref. [14] the term of BP model’s with higher order derivatives was considered with the opposite sign. In fact, the electrodynamics of BP model with the opposite sign was then further investigated in a series of papers [14, 18, 19], leading to the discussion of tachionic propagating modes for the photon. With the advent of modern and more powerful quantization methods, Barcelos-Neto, Galvão and Natividade [18] performed the Batalin-Fradkin-Vilkovisky (BFV) quantization using two slightly different gauges, namely the usual Lorenz gauge and the other called *generalized Lorenz gauge*. More recently, Bufalo and Pimentel [20] extended the BFV analysis including matter fields. From the symplectic quantization point of view, it is also worth mentioning that an interesting duality connection between the BP model and the massive Proca model has been investigated in Ref. [21]. Most recently, the classical generalized wave equation for the BP model has been studied including the retarded Green functions for different spatial dimensions as well as the retarded and generalized Liénard-Wiechert potentials [22]. Further recent discussions on the BP model can be found in Refs. [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

In the case of the Lorenz gauge, a natural gauge-fixing term originally introduced by Podolsky and Kikuchi [12] which considerably simplifies the calculation in the quantization process has long been passed without notice in the literature and has been only recently rescued by Bufalo, Pimentel and Soto [27]. The role of this term in obtaining a simple generalized photon propagator in a straightforward manner cannot be overemphasized. In particular, we show that this term permits a nice factorization of the generalized photon propagator in all gauges analyzed in this work. Thus, the BP model parameter dependent part of the propagator appears only as a global multiplicative factor turning its mathematical structure easier to analyze and interpret as a way of introducing Pauli–Villars regularization. We discuss that it is possible to split the propagator as a sum of two parts consisting of massive and massless modes which we interpret as the Pauli–Villars regularization.

Furthermore, to our knowledge neither the axial nor the light-front gauges have been discussed so far by the functional path-integral quantization point of view in the context of BP model in the literature. The canonical structure of BP model’s generalized electrodynamics on the null-plane has been recently analyzed by Bertin, Pimentel and Zambrano [24] where the light-front Hamiltonian form evolution is considered. In Ref. [24], after unraveling the constraint structure in phase space, the generalized radiation gauge on the null-plane is adopted. In the present work, we follow a different approach considering the theory defined by the Lagrangian in the configuration space and introducing the gauge-fixing conditions in the integration measure of the generating functional via the Faddeev-Popov procedure generalized to the BP model case. The quantization is then performed in a covariant way and for instance the generalized Lorenz gauge can be achieved. The axial-gauges are obtained along the same lines, the breaking of relativistic covariance being only perceived by a particular choice of the axial direction vector \(n_\mu \). In addition, we have the opportunity to extend the ideas introduced in Refs. [34, 35] concerning the adoption of two simultaneous gauge-fixing conditions leading to a so-called doubly transverse photon propagator in the light-front gauge. We show in this work explicitly how to handle the corresponding calculations in the BP model case.

Our work is organized as follows. In Sect. 2, we define our notation and conventions, review some basic facts about the BP model as a gauge field theory and physically interpret some of its classical properties. In particular, we demonstrate that the BP model solution for a point charge in electrodynamics corresponds to an ordinary electrodynamic solution for a specific charge distribution. This motivates us in Sect. 3 to elucidate the BP model as a natural way of providing Pauli–Villars regularization in ordinary QED. We discuss in Sect. 3 the covariant Lorenz gauge fixing obtaining the corresponding photon propagator and point out the necessity of the natural gauge-fixing term in the gauge fixing action. This term permits a natural factorization of the generalized photon propagator in all gauges analyzed in Sect. 3. The BP model parameter dependent part of the propagator appears only as a global multiplicative factor turning its mathematical structure easier to analyze and interpret as a way of introducing Pauli–Villars regularization. In Sect. 4, we provide an example of the BP model application discussing the second-order correction to the electron self-energy and show explicitly the consistency with the Pauli–Villars regularized result. We close in Sect. 5 with some final comments and concluding remarks. In Appendix A, we summarize the derivation of Eq. (31) used in Sect. 2.

## 2 Bopp–Podolsky’s generalized electrostatics

^{1}

*a*is a real number with physical dimension of length or inverse mass, known as Bopp–Podolsky’s parameter [4, 5]. We use Minkowski’s coordinates with metric signature diag(\(\eta ^{\mu \nu }\)) \(=(+1,-1,-1,-1)\) and the integration measure \(d^4x\) in Eq. (1) runs throughout all space-time coordinates \(x^\mu \). It is clear that BP’s classical action \(S_0\) is a natural higher derivatives Lorentz covariant generalization of ordinary Maxwell’s electromagnetism – the latter being recovered for \(a=0\). Although with a different notation, we adopt the same original Bopp–Podolsky’s [4, 5, 12, 13] choice for the second term in Eq. (1). This choice is important if one wishes to interpret the extra degrees of freedom of BP model as physical massive excitations. Although the case of negative sign for the second term in \(S_0\) above was considered in [14, 18, 19] describing tachionic mass excitations for the gauge field, we are not going to discuss it here but rather maintain the implementation consistent with the usual causality.

Note that in Eq. (1), the short-hand \(F_{\mu \nu }\equiv \partial _\mu A_\nu - \partial _\nu A_\mu \) stands for the ordinary electromagnetic field strength tensor which is naturally invariant under the gauge group *U*(1). That means the BP extra *a*-dependent term does not spoil the original gauge invariance of the action \(S_0\) and, particularly, the propagator of the gauge field is not well defined before gauge fixing. We shall address the gauge fixing issue in Sect. 3 where the Lorenz and axial type gauge fixings will be discussed. In the following we briefly review a few immediate properties and consequences of action given by Eq. (1) and consider the static case obtaining the BP version of Poisson’s equation as well as its general solution. For a point charge delta distribution, the BP model leads to a everywhere finite potential – we shall show that it is possible to generate this very same potential within the scope of ordinary electrodynamics using a suitable charge distribution.

### 2.1 Field equations of motion and general static solution

*a*-dependent fourth-order differential operator \({{\mathbb {P}}}_a\) as

*a*goes to zero, i.e. \(a\rightarrow 0\), as expected.

*a*, the BP potential \(\phi _{P,a}(r)\) in Eq. (18) remains finite in the limit \(r\rightarrow 0\) approaching to the finite value

and reproduces back the Coulomb’s characteristic 1 / *r* behavior for large values of *r* compared to *a*. In Fig. 1, we plot BP’s potential as a function of *r* as well as its two constituent parts Coulomb’s and minus Yukawa’s for the numerical value \(a=0.5\) in the same unit as the measured distance variable *r*. Here, it may be worth noticing the important fact that Coulomb’s potential doesn’t have any length scale parameter while the BP’s potential has a natural length scale provided by the real parameter *a*. What follows in the next subsection is that this parameter *a* introduced in the BP model action given by Eq. (1) can be equivalently reinterpreted as the length scale of a specific charge distribution removing the fiasco of divergence for a point charge particle in ordinary local gauge electrodynamics. In particular, we demonstrate that the BP model solution for a point charge corresponds to an ordinary electrodynamic solution for a specific charge distribution. This motivates us in the subsequent section, Sect. 3, to elucidate the BP model as a natural way of providing Pauli–Villars regularization in ordinary QED.

### 2.2 The BP potential from ordinary electrostatics

*a*-parameter improves the model convergence properties by means of the higher-order derivatives term. As a matter of fact we have just seen Coulomb’s potential given by Eq. (16) gets smoothed out becoming finite at the critical point \({\mathbf {r}}={\mathbf {r}}_0\) when we generalize Poisson’s equation including the convergence parameter

*a*into the PBP equation given by Eqs. (12) or (14). In other words, in the BP model, Eq. (16) generalizes to Eq. (18). However, we show below that it is also possible to obtain the same effect within the realm of ordinary electrostatics by using a specific charge distribution. Indeed, let’s take a small positive length dimension parameter

*b*and consider the normalized charge distribution

*a*-dependent term in obtaining Eq. (27) as we used only the standard ordinary electrodynamics. Nevertheless, it represents the very BP potential with

*b*playing the role of the previous BP

*a*-parameter.

*a*,

*b*)-dependent potential which we denote here by \(\psi (a,b,r)\). This sort of double BP potential can be explicitly calculated using the general solution given by (21) as

*r*close to zero, we get a finite result without any divergence:

*a*and

*b*and reveals the equivalence of the result under the exchange of the role between the two parameters

*a*and

*b*which have been introduced originally with seemingly different physical motivation or physical meaning. The result of the ordinary electrostatics, i.e. \(a=0\), for a specific charge charge distribution given by Eq. (23) with \(b \ne 0\) is completely equivalent to the result of the BP electrostatics with a length scale parameter \(a \ne 0\) for a local point charge, i.e. \(b=0\). Thus, it allows the exchange of the role between the length scale parameter

*a*introduced in the BP model for a point charge and the length scale parameter

*b*for a specific charge distribution given by Eq. (23) in the ordinary electrostatics. To the extent that modifying the concept of point charge to an extended charge distribution provides the physical meaning of charge renormalization as the charge screening due to the Dirac vacuum in QED, our finding here motivates us to reinterpret the BP’s generalized electrodynamic action given by Eq. (1) as a natural way of providing Pauli–Villars regularization in ordinary QED. In the next section, we address this possible reinterpretation by looking into the details of the gauge fixing in the BP model with its functional quantization.

## 3 Gauge fixing and functional quantization

As already stated in the last paragraph of our Introduction, Sect. 1, due to gauge invariance, a direct propagator for the gauge field in BP model is ill-defined. That happens because we are working with a constrained system and to preserve explicit covariance we use more field variables than degrees of freedom [30]. In order to proceed with the quantization of the model, similarly to ordinary electrodynamics, we must choose a specific gauge suitable for perturbative calculations. In the following subsections, we show how to achieve the generalized Lorenz and axial gauges performing the functional quantization of BP model. In both cases, we shall obtain the Green functions generating functional by means of a suitable generalization of the Faddeev-Popov method.

### 3.1 Generalized Lorenz gauge

*a*-parameter, the gauge-fixing given by Eq. (34) is allowed to depend on the additional free real gauge parameter \(\xi \). For the particular case \(\xi =1\) the necessity of this natural term can already be seen in the original papers of Podolsky, Kikuchi and Schwed [12, 13]. Although this isolated term cannot be found directly in Ref. [12], a careful reading shows that it is in fact inserted and summed up in their so called

*modified Lagrangian*up to total divergences. However, the second part of Eq. (34) has been tacitly omitted in the more modern treatments of BP’s model and only recently has it been reintroduced in BP’s context by Bufalo, Pimentel and Soto [27].

^{2}

*N*is a normalization constant and \(\Delta _{FP}\) represents the determinant which arises from the Jacobian of a gauge transformation in the condition given by Eq. (44), that is,

*B*, we can rewrite the generating functional, after a convenient redefinition of the normalization factor, as

*B*field finally leads to

We have explicitly shown how the condition given by Eq. (44) leads through the Faddeev-Popov procedure to the gauge fixing term given by Eq. (34) and calculated the corresponding propagator for BP’s generalized electrodynamics. In the next subsection, we turn our attention to the axial and light-front gauges.

### 3.2 Axial and light-front gauges

*B*is the Nakanishi-Lautrup field. Paralleling the Lorenz gauge case, here we also have the BRS symmetry

*r*, those two conditions together imply

## 4 Application – the electron self-energy

*k*. This effectively illustrates how BP formulation parallels with the PV regularization.

*u*(

*p*) denotes a plane wave solution to Dirac’s equation and

*k*this term vanishes after the momentum integration in Eq. (75). By using standard gamma matrix properties, Eq. (76) can be further simplified to

*u*(

*p*), as necessary for plugging into Eq. (74), we may use Dirac’s equation to get

*x*and write

*k*, and using again Dirac’s equation we may write

*u*(

*p*) in the limit \(D\rightarrow 4\).

*u*(

*p*) on the right, as demanded by Eq. (74), and using once more Dirac’s equation the identity given by Eq. (94) leads to

*k*, amounts to zero after momentum integration. Even if we use the three-term propagator given by Eq. (67), the last term in Eq. (67) is canceled by the instantaneous interaction in the light-front dynamics [45, 46] and thus the result is identical to Eq. (93). It shows the gauge independence of the invariant amplitude given by Eq. (74) and illustrates how BP formulation parallels with the PV regularization.

However, we note that the BP parameter \(a=1/\Lambda \) is not a physically measurable quantity but corresponds to the UV cutoff parameter which is combined with another unmeasurable quantity, i.e., the bare mass, to yield the physically measurable renormalized mass of the electron. The necessity of mass renormalization already occurs in classical electrodynamics as discussed in Sect. 1 and Sect. 2. For a classical electron of radius \(r_e\), the electromagnetic mass \(\frac{e^2}{8\pi r_e}\) becomes infinite as \(r_e \rightarrow 0\). Although this divergence is also true in QED, its degree of divergence is logarithmic with the UV cutoff parameter \(\Lambda \) as shown in Eq. (93) in contrast to the linear divergence of the classical self-energy correction as \(r_e \rightarrow 0\). This weakening of the divergence in QED is a consequence of the non-trivial Dirac vacuum in cooperation with the UV cutoff parameter on par with the BP mass parameter.

## 5 Conclusion and discussion

In this work, we demonstrated that the BP model solution for a point charge in classical electrodynamics corresponds to an ordinary classical electrodynamic solution for a specific charge distribution given by Eq. (23). Motivated by this possible reinterpretation of BP model solution in classical electrodynamics, we further elucidate the BP model as a natural way of providing Pauli–Villars regularization in ordinary QED. We note that the weakening of the divergence in QED as logarithmic, in contrast to the classical linear divergence, is a consequence of the non-trivial Dirac vacuum in cooperation with the UV cutoff parameter which is combined with another unmeasurable quantity, i.e., the bare mass, to yield the physically measurable renormalized mass of the electron. The BP parameter corresponding to the UV cutoff parameter is thus as unmeasurable as the bare mass but essential to regulate the loop divergence in QED and renormalize the mass of the electron as the physically measurable quantity.

*a*and the UV cutoff parameter \(\Lambda \) in the scheme of PV regularization, i.e., \(a=1/\Lambda \). We have shown that the different gauge fixings considered do not invalidate this appealing correspondence.

## Footnotes

- 1.While Podolsky introduced the Lagrangian density corresponding to the action given in Eq. (1) in Ref. [5], Bopp actually worked with a slightly different version in Ref. [4], namelyHowever, it can be checked that these two versions due to Bopp and Podolsky are in fact equivalent to each other and lead to the same field equations of motion.$$\begin{aligned} {{{\mathcal {L}}}}_B= -\frac{1}{4} \left[ F_{\mu \nu }F^{\mu \nu } -a^2\partial _\rho F^{\mu \nu }\partial ^\rho F_{\mu \nu } \right] \,. \end{aligned}$$
- 2.

## Notes

### Acknowledgements

This work was supported by the U.S. Department of Energy (Grant No. DE-FG02-03ER41260). A.T.S. wishes to thank the kind hospitality of Physics Department, North Carolina State University, Raleigh, NC and acknowledges research grant in the earlier part of this work from Fapesp 2014/20892-2. J.H.O.S. thanks for the hospitality of North Carolina State University, Raleigh, NC which provided facilities for the completion of this work and thanks the financial support of FAPESB-PIE0013-2016, CNPq-315519/2018-5 and CAPES.

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