Remarks on an anomalous triple gauge boson couplings

We address the effect of an anomalous triple gauge boson couplings on a physical observable for the electroweak sector of the Standard Model, when the $SU(2)_{L}\otimes U(1)_{Y}$ symmetry is spontaneously broken by the Higgs mechanism to $U(1)_{em}$. Our calculation is done within the framework of the gauge-invariant, but path-dependent variable formalism which is alternative to the Wilson loop approach. Our result shows that the interaction energy is the sum of a Yukawa and a linear potential, leading to the confinement of static probe charges. The point we wish to emphasize, however, is that the anomalous triple gauge boson couplings ($Z \gamma \gamma$) contributes to the confinement for distances on the intranuclear scale.


I. INTRODUCTION
It is well known that the Standard Model (SM) is an active field of research, based on the gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y , which has been successful in describing many of the particle physics phenomena. However, despite this great success, the SM must be extended to explain some aspects that need to be understood. Probably the most striking examples are: dark matter candidates, nonzero neutrino masses, baryon asymmetry of the Universe and origin of the electroweak scale. These puzzling facts have led to an increasing interest in physics beyond the SM (BSM). Mention should be made, at this point, to the Higgs boson discovered at the LHC [1,2] which clearly corroborated the electroweak symmetry breaking. In other words, the SU (2) L ⊗U (1) Y symmetry is spontaneously broken by the Higgs mechanism [3][4][5] to U (1) em . However a full understanding of this mechanism from first principles still remain elusive.
It is also known that considerable attention has been paid to the investigation of anomalous triple gauge boson couplings of the electroweak sector [6][7][8][9][10][11][12][13]. The interest in studying these couplings is mainly due to the possibility of providing a better understanding of the electroweak symmetry breaking mechanism and test the predictions in collider experiments. It is believed that the presence of these couplings can give important hints of new physics beyond the SM. In this connection, it may be recalled that the ZZ and Zγ production are the foremost processes where the triple couplings between the usual photon and Z boson (Zγγ and ZγZ) can be studied. This can be achieved by adding higher dimension effective operators to the Lagrangian of the SM. For example, in this * Electronic address: patricio.gaete@usm.cl † Electronic address: helayel@cbpf.br ‡ Electronic address: leoopr@cbpf.br work we will consider an interaction term of the form [6] where h γ 3 appears in the Zγ production process. Motivated by these observations and given experimental data on ZZ and Zγ production, it is desirable to have some additional understanding of the physical consequences presented by these anomalous triple gauge boson couplings of the electroweak sector. Of particular concern to us is the effect of the interaction term on a physical observable. To do this, we will work out the static potential for the theory under consideration by using the gauge-invariant but path-dependent variables formalism [14,15]. According this formalism, the interaction energy between two static charges is obtained once a judicious identification of the physical degrees of freedom is made. It also provides an alternative technique for determining the static potential for a gauge theory. Interestingly enough, the static potential profile contains a linear term, leading to the confinement of static probe charges.

II. INTERACTION ENERGY
As already expressed, the gauge theory we are considering describes the interaction between the familiar massless U (1) em photon with the massive vector Z-field via a new coupling. In this case, the corresponding Lagrangian density takes the form: where m Z is the mass for the gauge boson Z,F µν = 1 2 ε µνλρ F λρ , and χ = − e m 2 Z h γ 3 represents the coupling constant.
It is of interest also to notice that if we consider the foregoing model in the limit of a very heavy Z-field (and we are bound to energies much below m Z ) we are allowed to integrate over Z µ , which, then, yields an effective theory for the A µ -field. This can be readily accomplished by means of the path integral formulation of the generating functional associated to Eq. (2). Once this is done, we find that the effective theory can be brought to the form: where ∆ ≡ ∂ µ ∂ µ . It may be remarked in passing that the above Lagrangian density is a theory with non-local time derivatives. However, we stress that this paper is aimed at studying the static potential, so that ∆ can be replaced by −∇ 2 . For notational convenience we have maintained ∆, but it should be borne in mind that this paper essentially deals with the static case. We next observe that, in order to study quantum properties of the electromagnetic field in the presence of external electric and magnetic fields, we should split the A µ -field as the sum of a classical background, A µ , and a small quantum fluctuation, a µ , namely: Making use of this expression, we find that Eq. (3), up to quadratic terms in the fluctuations, reduces to where f µν = ∂ µ a ν − ∂ ν a µ . It should, however, be noted here that by defining v αβ = 1 2 ε αβµν F µν , the foregoing equation becomes And, finally, by considering the v 0i = 0 and v ij = 0 case (referred to as the magnetic one in what follows), we readily deduce that We are now in a position to evaluate the corresponding interaction energy in the case under consideration. To do that we now carry out a Hamiltonian analysis of this theory.
Let us start by observing that the canonical Hamiltonian is given by where .
Time conservation of the primary constraint, Π 0 , yields a secondary constraint. The secondary constraint is therefore the usual Gauss's law, Γ 1 ≡ ∂ i Π i = 0, and together displays the first-class structure of the theory. The extended (first-class) Hamiltonian that generates the time evolution of the dynamical variables then reads where c 0 (x) and c 1 (x) are arbitrary functions of space and time. It may be noted here that Π 0 = 0 for all time andȦ 0 (x) = [A 0 (x) , H] = c 0 (x), which is completely arbitrary. We may accordingly discard A 0 and Π 0 . It is of interest also to notice that it is redundant to retain the term containing A 0 because it can be absorbed by redefining the function c 1 (x). From the above, the extended Hamiltonian is then where and We can at this stage impose a gauge fixing condition that together with the first class constraint, Γ 1 (x), the full set of constraints become second class. We consequently choose the gauge fixing condition as [16]: where λ (0 ≤ λ ≤ 1) is the parameter describing the space-like straight path x i = ζ i + λ (x − ζ) i , and ζ is a fixed point (reference point). In passing we observe that there is no essential loss of generality if we restrict our considerations to ζ i = 0. Hence, we readly find that the only non-vanishing equal-time Dirac bracket reads We now pass on to the calculation of the interaction energy. To do this we compute the expectation value of the energy operator H in the physical state |Φ . We next observe that the physical state |Φ can be written as where the line integral is along a spacelike path on a fixed time slice, q is the fermionic charge and |0 is the physical vacuum state.
It is also important to observe that taking the above Hamiltonian structure into account, one encounters Thus, we obtain for H Φ the expression where H 0 = 0| H |0 , whereas the H (1) 0 term is given by Now making use of equation (14) and following our earlier procedure, we find that the potential for two opposite charges, located at y and y ′ , takes the form , |y − y ′ | ≡ L and Λ is a cutoff. This result explicitly shows the effect of including the interaction term (1) in the model under consideration. In fact, we see that the static potential profile displays the conventional screening part, encoded in the Yukawa potential, and the linear confining potential. Before to proceed further, we would like to illustrate how to give a meaning to the cutoff Λ. To this end one recalls that that our effective model for the electromagnetic field is an effective description that arises after integrating over the Z µ -field, whose excitation is massive. Accordingly, l Z = 1 mZ , the Compton wavelength of this excitation defines a correlation distance. Thus, physics at distances of the order or lower than 1/m Z must take into account a microscopic description of the Z-fields. To be more precise, if we work with energies of the order or higher than m Z , our effective description with the integrated effects of Z is no longer sensible. In view of the foregoing remark, we can identify Λ with m Z . Thus, finally we end up with the following static potential profile: From equation (18) it follows that the corresponding interparticle force reads With the foregoing information provided by the interparticle one-photon-exchange potential and the corresponding force, we can proceed to present and discuss some estimates that confirm the consistency of our result. We first notice the appearance of two length scales in the potential V , namely, l M = 1/M , and the Z's Compton wavelength, l Z = 1/m Z (∼ 2 × 10 −18 m); let us not forget the relationship l M = √ α l z > l z , since α > 1. By adopting the current parametrization given in the literature [12,13], our effective coupling in the anomalous Zγγvertex, χ, is actually given by χ = − eh γ 3 /m 2 z , where h γ 3 can be extracted from Zγ-producting processes. According to the results in the paper of Ref. [13], h γ 3 is estimated to be of order 10 −3 .
Since α = 1 + 2χ 2 B 2 (we are here taking v = B, the external magnetic field), even for the strongest magnetic field that may induce instabilities in the electroweak vacuum, which is estimated to be of the order of 10 19 T , the quantity α = 1+2×10 −8 . Higher values of α are possible only for magnetic fields that produce high instabilities in the Higgs vacuum and therefore the electroweak scenario is no longer consistent.
By inspecting the expressions for both the interparticle potential and force, and assuming a magnetic field of 10 19 T, we get that the Yukawa-type force dominates over the confining component up to distances L = 1.33 l Z ∼ 2.66 × 10 −18 m. For L > 1.33 l Z , the (constant) confining force is the dominant one. We have estimated the total (attractive) force for L = 1.33 l Z ; the result is F = 2.58 × 10 6 N . For distances L of the order 10 l Z and larger, the Yukawa-like force becomes negligeable and the constant confining force converges to 1.28 × 10 6 N . Just for the sake of comparison, let us recall that, by considering the purely classical Coulombian scenario for a quark-antiquark pair, the attractive force would be of O 10 7 N . The force we have attained in our effective approach is one order of magnitude weaker because the interaction under consideration is shielded by the effect of the mass of the Z-particle.
This result is particularly interesting if we consider the interaction between a quark antiquark pair. The distance 2.66 × 10 −18 m is around three orders of magnitude lower than the typical nucleous radius and our estimate indicates that the anomalous parity-preserving Zγγ-vertex we are investigating yields an effective electroweak confining force that contributes to the (much stronger) colour-confining force. But, what we wish to highlight is that this anomalous vertex enhances confinement.
Finally, we also draw attention to the fact that in the v i0 = 0 and v ij = 0 case, equation (5) can be brought to the form Proceeding in the same way as was done for the magnetic case, we find that the static potential turns out to be equation (18), in this case α = 1 − χ 2 2 E 2 and M 2 = This result shows that, in the situation of an external electric field, the previous results for a magnetic field hold through, the difference, however, lies in the expression for α, which is now no longer bigger than 1. Contrary, now, α < 1. It might happen that α = 0 if E = 2χ −2 . By taking the value for χ we have considered in the magnetic case, E turns out to be of order 10 8 GeV 2 , in international units E ∼ 10 31 V m −1 , which is a huge electric field, 13 orders of magnitude above the critical Schwinger field, namely, 10 18 V m −1 . Actually, electric fields such that eE ∼ m 2 e has enough energy to decay and produce the formation of an e − e + -pair. But, this would not be compatible with our physical scenario of an interparticle potential. The creation of e − e + -pairs spoils our potential approximation. Electric fields such that eE ∼ m 2 e are just of the order of the critical Schwinger electric field. So, we are bound to consider external fields below 10 18 V m −1 . In such a case, α < 1 + O(10 −28 ). So, for the sake of our estimations, we can still undertake that, as in the case of external magnetic fields, the presence of external electric fields weaker than the Schwinger electric field, which prevents from creating particle-antiparticle pairs, the anomalous tri-vertex we are investigating keeps on enhancing confinement.

III. FINAL REMARKS
In summary, using the gauge-invariant but pathdependent formalism, we have computed the interaction energy when an anomalous triple gauge boson couplings is taken into account in the electroweak sector of the Standard Model, and the SU (2) L ⊗ U (1) Y symmetry is spontaneously broken by the Higgs mechanism to U (1) em . Once again, a correct identification of physical degrees of freedom has been crucial for understanding the physics hidden in gauge theories. Interestingly, it was shown that the interaction energy is the sum of a Yukawa and a linear potential, leading to the confinement of static probe charges.