Feynman rules for Four Bosons Electromagnetism

A whole electromagnetism carrying four electric charge messengers is studied. Based on light invariance and conservation of electric charge, it provides a fields set } , , {     V U A . Something beyond Maxwell appears. The usual photon is accomplished by others electric charge porters, which are a massive photon plus two charged photons. They carry electromagnetic processes with charge exchange 0 = Q  and 1 |= | Q  . There is still room for an electromagnetism on electric charge transmission to be understood. Through such so-called four bosons electromagnetism a new way to conduct the electric charge is proposed. It says that the electromagnetic phenomena is something more than Maxwell’s charge distribution. It establishes the presence of four fields association responsible for the electric charge transmission. It develops a quanta set which means electromagnetism based on eight messengers with spin-1 and spin-0 to be analysed. Thus given such fields collection } , , {       V U A A I one studies the corresponding propagations and interactions. Derive the corresponding Feynman rules for this electric charge transmission. The model shows itself renormalizable and unitary. New features are obtained as selfinteracting photons. The photon is no more necessarily coupled to the electric charge. A diversity of coupling constants is obtained. The electromagnetism universality is on the ubiquous photon and not on the electric charge as coupling constant.

There is still room for an electromagnetism on electric charge transmission to be understood. Through such so-called four bosons electromagnetism a new way to conduct the electric charge is proposed. It says that the electromagnetic phenomena is something more than Maxwell's charge distribution. It establishes the presence of four fields association responsible for the electric charge transmission. It develops a quanta set which means electromagnetism based on eight messengers with spin-1 and spin-0 to be analysed.
Thus given such fields collection

Introduction
The main theory explaining the electromagnetic phenomena is due to Maxwell who summarized two centuries of experimental findings into his equations. Maxwell's theory [1] works very well, as proven by experiments. But it has some limitations too. In fact, three basic considerations can be done. Firstly, the masslessness of the photon can not be proven, only a lower experimental limit can be given. Also, the superposition principle can be proven only to a certain precision. Thirdly, adopting Maxwell's theory in the quantum framework is not straightforward and is somewhat artificial.
There is still room for an electromagnetism beyond Maxwell. Currently there are 38 extended models [2]. Based on two basic electromagnetic postulates which are light invariance and electric charge conservation this work studies a model where while Maxwell focus on charges distribution its objective drives on electric charge transportation. For this, from an abelian whole model [3], a four bosons electromagnetism is developed [4]. It proposes an electric charge transmission where beyond the photon it adds three additional electromagnetic messengers.
Thus from Maxwell photon field one moves for a fields set  (2) (1) SO U  transforming under a common gauge parameter, an electric charge porter electromagnetism is derived. From [4], one gets Decomposing the kinetic term, , the corresponding antisymmetric and symmetric parts are written as where the granular fields strength are defined as follows , , , Considering that physical fields are that ones which diagonalize the transverse sector, we should not consider terms like The mass term is written as In principle the gauge-fixing should given by However, although eq. (1.9) fixes the gauge parameter it does not introduces the SO(2) invariance. For this, one has to consider the relationships  Eq. (1.10) shows that from one gauge parameter one obtains a gauge fixing term composed by four different parameters. In a similar situation with spontaneous breaking symmetry models [5], the gauge fixing term provides the model with four free parameters. Another aspect from eq. (1.10), is that differently from the standard QED, the gauge fixing condition does not necessarily takes one degree of freedom. Consequently, by associating a common symmetry for four fields embedded in the Lorentz representation ) 2 1 , 2 1 ( , one gets four particles with spin-1 plus four with spin-0.
The interaction Lagrangian is decomposed in trilinear term and quadrilinear parts: Preserving light invariance and electric charge conservation postulates, eq. (1.2) introduces through an U(1)  SO(2) symmetry, the minimal model for electric charge transmission. It introduces a richest electromagnetic world than Maxwell. Granular and collective fields, non-linearity, massive intermediary particles are among new aspects to be observed.

Perturbative Lagrangian
Our intention here is to study the quantum perspective to the involved fields and associated particles. So a next step is to organize the perturbative approach for the corresponding particles and their interactions. Taking the flavour and for the interacting sector as Thus, from eq. (1.3), the kinetic term can be decomposed in three parts The interaction term gives,   (1)    [13] [24] [13] [34] [12] (33) ( [13] [24] [13] [34] [12] (33) (12)

Feynman rules
A non-linear electromagnetism is obtained. A next step is to express the corresponding Feynman rules. The effective action of the classical field is defined by the functional Legendre transformation,

Expanding in powers of
The conventions for the momentum flows are indicated in Fig. 1.
For the vertices, the correspondence between the interacting terms that appear in the Lagrangian and the Feynman vertices is not one-to-one. The introduction of more potential fields in the same group enlarges the possibilities for playing with the Lorentz indices and also appear different possibilities for distributing the flavour indices. Thus, it appears a kind of topology of gauge invariance where a determined graph incorporates different contributions from the Lagrangian terms.
Thus three-gauge-boson proper vertices are systematized in three structures. They correspond graphs with three different fields, two fields being equal and the case where all fields interacting in a vertex are the same. Taking the Fourier transform of the action corresponding to one gets the following Feynman rule for the first case, For the case with two-equal fields, the vertex receive contributions from ten different terms. It gives, Finally, when the three involved fields are equal, one gets the expression Similarly, for the four-boson vertex which yields the following cases Four features can be taken from these Feynman rules. First, the above Feynman rules structures a new approach to a non-linear electromagnetism. The first relevant try was due to Born and Infeld in 1934 [6]. The difference here is that the non-linearity is on potential fields. Three and four vector bosons vertices are developed independently.
We should also observe that this non-linearity is not ruled by electric charge. Eqs. (3.13-3.22) are showing different coupling constants which can take any value without breaking gauge invariance. As consequence this welcomed fact release the model on facing the crucial result from electrodynamics which is the accuracy on the electron anomalous magnetic moment measurement. The possibility for a model adding new vertices to agree with the experimental value to more than 10 significant figures [7] is to be free for adjusting the coupling constants being included.
As a third aspect, the above graphs are showing on deviation from linearity in the quantum regime at tree level.
For the transversal sector, the corresponding propagators are where the corresponding residues expressions are Reading off propagators, one obtains information on mass and spin contents. Eqs. (4.5-4.7) are saying that the model carry four poles with spin-1. They are a massless photon, a massive photon with a mass where the residues are given by the following expressions  As an important result, considering that the parameters that define the transversal and longitudinal parameters are independent, one can get the solutions where the model does not depend on ghosts. This means that it is unitary.
Concluding on these two quantum numbers masses and spin, being studied through the poles and residues of propagators, we should also notice that them are explicitly showing on the antireductionistic character of the model. Their expressions are depending on the system as a whole. As result, one can control on tachyons and ghosts through free coefficients and also given their different poles and residues, the spin-1 and spin-0 quanta can be identified as different particles.

Discrete symmetries
A next aspect is to study on discrete symmetries at eq. (1.2). For this, as example, we are going to select the following possibilities: Then any composition between such possibilities or between themselves as