# Strong fields and neutrino magnetic moment dynamics

###### Abstract

Interaction of magnetic moment of point particles with external electromagnetic fields experiences unresolved theoretical and experimental discrepancies. In this work we point out several issues within the relativistic quantum mechanics and the QED and we describe effects related to a new covariant classical model of magnetic moment dynamics. Using this framework we explore the invariant acceleration experienced by neutral particles coupled to the plane wave field through the magnetic moment, addressing the case of Dirac neutrinos with magnetic moment in the range of to . We show that even for the minimal predicted moment value, the neutrino interacting with present day laser systems with appropriately shaped pulses will be subject to critical acceleration facilitating radiative pair production process.

###### pacs:

13.40.Em,06.30.Ka,41.75.Jv^{†}

^{†}: Plasma Phys. Control. Fusion

Keywords: magnetic moment, laser-driven acceleration, neutrino

## 1 Introduction

The general consensus in theoretical physics is that the final word on classical Electrodynamics has not yet been said. More than hundred and fifty years have passed since its original inception by Faraday, Maxwell and many others in the 19th century, and we still face unsolved conceptual problems of a fundamental nature. One of the most prominent issues of classical Electrodynamics is the problem of radiation reaction [1, 2, 3].

Of comparable relevance is the incomplete understanding of the magnetic (Stern-Gerlach type) force, i.e. the interaction of the magnetic moment of a point particle with external EM field in both classical and quantum mechanics [4, 5]. A related experimental discrepancy exists: as of July 2017 there is a 3.5 standard deviations difference between the calculated magnetic moment of the muon based on Standard Model QFT corrections and experimental measurements [6].

We report on the recent progress in understanding the magnetic moment dynamics [7]. Here weare interested in the dynamics of a neutral particle with non-zero magnetic moment placed in the external EM field. Any new magnetic moment physics is in this situation a first order effect. As an application of these considerations we describe how Dirac neutrinos could be studied experimentally exploiting their interaction with intense laser fields. We note another effort to improve understanding of particle interaction with strong laser fields [8]. Our work can also contribute to study of the behavior of plasma influenced by external non-homogeneous fields.

Before addressing the primary contents of this report we will first consider briefly the quantum physics of the magnetic moment in section 2 clarifying how the classical and quantum physics relate. We summarize insights of Ref. [7] in section 3, and we obtain the invariant acceleration acting on any particle in the plane wave field in section 4, before describing the physics of ultrarelativistic neutrinos in interaction with the plane wave field in section 5.

## 2 Relativistic Quantum Mechanics

### 2.1 Dynamical equations

Every quantum particle should be described using three free parameters: its mass, its electric charge and its magnetic moment. However the Dirac equation reduces the number of parameters to two by predicting the magnetic moment with the gyromagnetic ration . In reality, the effective -factor is never exactly equal to two and in our effort to understand the dynamics of realistic particles we need to generalize our expressions to account for an anomalous magnetic moment with . The deviancy can be small, such as in electrons and muons due to quantum electrodynamics effects, or large, such as in protons and neutrons due to their internal structures.

The primary method of treating the anomalous magnetic dipole moment is by modification of the Dirac equation to include what is known as the Pauli term containing the anomaly deviation in the format

(1) |

The main problem with this approach is that the modified Dirac equation cannot be used to compute virtual processes since the additional so called Pauli term diverges and requires counter terms.

An alternative theoretical description of magnetic moment first “squares” the Dirac equation, resulting in a second order formulation similar to the Klein-Gordon (KG) equation for spin 0 particles supplemented with the Pauli term

(2) |

where . A solution of the Dirac equation is also a solution of this KG-Pauli Eq. (2) once is chosen. The problem with KG-Pauli is that one must carefully analyze and understand the set of solutions of the higher order equation.

The advantage of KG-Pauli Eq. (2) compared to Dirac-Pauli Eq. (1) is that we can choose an arbitrary value of the gyromagnetic factor ; if value ‘works’ so will an arbitrary value. We emphasize that these two quantum equations, the Dirac-Pauli Eq. (1) and the KG-Pauli Eq. (2) are not equivalent and result in different physical behavior. Thus experiment will determine which form corresponds to the quantum physics of e.g. bound states in hydrogen-like Coulomb potential. We will return to this matter under seperate cover.

### 2.2 Magnetic moment in QED

In principle quantum electrodynamics is formulated around a Dirac particle with with modifications arising in the context of a perturbative expansion leading to the evaluation of the actual magnetic moment i.e. of the electron in perturbative series that today requires in precision study also the consideration of strong interactions and vacuum structure. This approach masks the opportunity to use the actual particle magnetic moment for particles responsible for the vacuum properties such as is the vacuum polarization.

The study of the vacuum response to external fields has a long and distinguished history that spans over 80 years, starting with computation of the lowest order effect by Uehling [9] in 1935 and the development of the nonperturbative Euler-Heisenberg (EH) effective action characterizing all the physical phenomena present in constant fields including the decay of the field into electron positron pairs [10]. These studies introduce counter terms which served as predecessors to the full quantum field theoretical charge renormalization scheme. This effective action was revisited in a field theoretical context by Schwinger, which extended these considerations to include a demonstration of transparency of the vacuum to a single electromagnetic plane wave [11]. However, all these consideration required particles to have the Dirac value of magnetic moment .

When is introduced a modification of the analytical form of the effective EH action is discovered [12, 13] and further non-trivial modifications in the vacuum structure arise [14, 15]. A solution to the previously divergent result for effective action with was obtained [13]. Similarly the modification of the vacuum polarization was found [14]

(3) | |||||

The coefficient in Eq. (3) shows explicitly all three parameters of a particle: its magnetic moment in form of , its charge and its mass . One can easily recombine terms to show dependence on the magnetic anomaly . This form demonstrates that in perturbative QED expansion the magnetic moment dependence arises from higher order QED vacuum polarization tensor (the photon line crossing the loop) contributing. This format hides the appearance of the actual particle magnetic moment in the vacuum polarization as is seen in Eq. (3). We will return to the question how magnetic moment is renormalized under separate cover.

Once we recognize dependence of vacuum polarization on magnetic moment and the dependence of EH effective action on magnetic moment one must further revisit Schwinger‘s proof of vacuum transparency to a single plane wave for .

## 3 Magnetic moment in classical theory

There are two models which describe the magnetic moment of a point particle. The ‘Amperian’ Model approximates the particle magnetic moment by a current loop which leads to a force

(4) |

where is the magnetic moment of the particle and is magnetic field. On the other hand the ‘Gilbertian’ Model creates a magnetic dipole, consisting of two hypothetical monopoles, and leads to a different expression

(5) |

There has to be a way to reconcile these classical models and to create a covariant description of the dynamics for both particle 4-velocity and spin which would unite these two approaches. There have been efforts to do so - the first covariant model was created by Frenkel [4, 16]. This model is based on classical arguments starting with the principle of least action and couples back the spin motion of the particle with the particle motion.

Another method of approach begins with relativistic quantum Dirac theory which naturally incorporates description of the spin behavior (although strictly) and finding an appropriate classical limit should yield a full classical description of the particle behavior. The most important example of such approach is the Foldy-Wouthuysen transformation [5]

Both of these approaches predict different behavior in the external EM field and can be distinguished experimentally as was explored in the article [8]. We learn from this work that ultra-intense laser pulses are especially suitable for investigating the viability of such models.

As presented in the work [7], the spin of a particle should not be its quantum property but rather a classical characteristic similar to the particle‘s mass. Both of these are eigenvalues of Casimir operators of the Poincaré group of space-time symmetry transformations whose values describe a representation of this group for a given particle. This insight allowed us to create a new covariant description of the spin dynamics of particles [7] which has a form

(6) |

(7) | |||||

where and are arbitrary constants. We explicitly distinguish between particle charge and elementary magnetic dipole charge which is used to convert the spin of a particle to the magnetic moment as . Finally, the dual EM tensor reads , with the fully antisymmetric tensor defined as (beware of a sign if contravariant indices are used).

We see that the equation of the motion Eq. (6) of the particle depends explicitly on the spin dynamics Eq. (7) through the spin 4-vector , thus generating covariant generalization of the Lorentz force to include Stern-Gerlach force. For particles with zero magnetic moment these dynamical equations reduce to Thomas-Bargmann-Michel-Telegdi (TBMT) equations [17, 18] with . TBMT equations are widely used to model particle dynamics in external fields and yet these do not contain coupling of the spin to the particle motion.

On the other hand we can also explore the other limit: the dynamics of neutral particles with magnetic moment in external fields. Equations (6), (7) become only functions of parameter which we will further explore in section 5.

To conclude this short overview of the results obtained in Ref. [7] we note that the two forms of the force, the Amperian and the Gilbertian, were shown to be equivalent. Thus a consistent theoretical framework now exists for exploring the dynamics of a magnetic moment in external fields.

## 4 Invariant acceleration in a plane wave field

The generalized Lorentz force equation reads [7], see Eq. (6)

(8) |

Imagine a point particle with both electric charge and magnetic moment in the plane wave field given by an expression

(9) |

where is a wave vector of the plane wave, its polarization, phase, and amplitude. Just the formula for the dynamics of the 4-velocity Eq. (8) alone is sufficient to obtain an expression for invariant acceleration in the plane wave field. In this case the generalized EM tensor reads

(10) | |||||

where primes denote derivatives of the profile function with respect to its phase. If we multiply this expression with we get zero because of the identities Eq. (9). Then Eq. (8) implies that

(11) |

is an integral of motion. We can obtain the invariant acceleration by squaring the expression Eq. (8), which can be evaluated using properties Eq. (9), antisymmetric properties of , our integral of motion Eq. (11), and contraction identity

(12) |

which is a generalized Kronecker delta. The final result is

(13) |

The cross term vanishes because the force due to particle electric charge and magnetic moment are orthogonal for a plane wave field. The only unknown in this expression is the product , which is still a function of proper time.

## 5 Neutrino acceleration

In the context of magnetic dipole dynamics we are especially interested in the case of charge neutral particles in the external EM fields. The most prominent examples of such particles are neutrons and neutrinos. In the absence of the classical Lorentz force the particle dynamics is governed by spin effects and can directly be used to measure the related properties of particles.

We focus our attention on neutrinos which are very light particles and can be accelerated much easily than heavy neutrons. Interaction of neutrinos with the laser field was studied before [19] as a higher order scattering effect; in the framework we developed [7] neutrinos couple with the external field via magnetic moment directly.

We recall that by symmetry arguments only the Dirac neutrino can have a magnetic moment: in essence this is so since a Majorana neutrino is the antiparticle of itself and thus under EM interactions must be neutral(charge) and neutral(magnetic moment). A very significant effort is underway to discover the double beta-decay [20] that could demonstrate that neutrino is of the Majorana type. However, one can question if a nil result would mean that the neutrino is a Dirac neutrino [21]. We believe that the measurement of neutrino interaction with an external field via its magnetic moment would demonstrate that the neutrino is of Dirac type. Our objective in the following is to show that we not only can expect observable effects when relativistic neutrinos interact with an intense EM plane wave pulse, but that a measurement of the magnetic moment of the neutrino should be possible.

### 5.1 Magnetic moment of the neutrino

The dipole magnetic moment is a well studied electromagnetic property of the Dirac neutrino. A minimal extension of the Standard Model with non-zero Dirac neutrino masses places a lower bound on the magnetic moment of the neutrino mass eigenstate proportional to its mass and reads [22]

(14) |

where is the Bohr magneton. This value is several orders of magnitude smaller than the present experimental upper bound [23]

(15) |

### 5.2 Neutrino acceleration in the external field

Consider a beam of neutrinos with GeV. This energy of neutrinos is currently accessible, for example the OPERA experiment uses 17 GeV neutrinos produced at CERN [24]. For the rest mass of neutrinos we take eV and laser source with energy of photons eV. The de Broglie wavelength for such neutrinos compared to the wavelength of the laser light is

(16) |

where we neglected the mass of the neutrinos compared to their energy. This justifies the classical treatment because the wavelength of the 1 eV laser light is 11 orders of magnitude larger than the wavelength of the 20 GeV neutrinos, the quantum wave character of neutrinos will be invisible.

The result Eq. (13) allows us to estimate acceleration which the 20 GeV neutrino experiences in the external plane wave field. The neutrino is a neutral particle so acceleration is just the second term in Eq. (13)

(17) |

And now we can estimate individual terms. The elementary dipole charge of the neutrino can be rewritten using the neutrino magnetic moment in units of Bohr magneton as

(18) |

The product is a Doppler shifted laser frequency as seen by the neutrino being hit by the laser. In the laboratory frame and

(19) |

when the initial direction of motion of the neutrino and the direction of the plane wave propagation are parallel and opposite. The unknown product is harder to estimate and the proper treatment of this term will be presented in the paper discussing the full dynamics of neutral particles which is currently in preparation. For now we assume that during the dynamics there are long temporal periods when

(20) |

The amplitude of the laser field vector potential can be expressed in terms of the dimensionless normalized amplitude as

(21) |

Finally, we want to write the result in the units of critical acceleration for neutrino which is

(22) |

Substituting all terms Eqs. (18-22) into Eq. (17), the formula for acceleration, yields an expression

(23) |

For our 20 GeV neutrinos we see that the critical acceleration can be achieved for the whole range of our magnetic moment uncertainty with laser pulse parameters in range

(24) |

The state of the art laser systems have dimensionless normalized amplitude , Eq. (21) of the order of . If we want the product of laser parameters to have values around we need to make up the difference with an appropriate profile for the laser pulse with a high second derivative; for example laser pulses with a high contrast ratio when turning on and off. Thus this mechanism seems to allow accelerating 20 GeV neutrinos using their magnetic moment.

### 5.3 Neutrino radiation

We have shown that the ultra-relativistic neutrino can be subject to significant acceleration when considering its magnetic interaction with a specially formed laser pulse. This means that the neutrino can emit virtual quanta, here the electro-weak bosons and are relevant.

These virtual particles decay and what is observed in the experiment is the appearance of a dilepton pair of high 10-GeV energy or of lepton and hadronic shower (hadronic decay of ). Thus shooting a laser pulse onto an incoming 20 GeV neutrino beam catalyzes GeV scale particle production with momentum pointing at the source of neutrino beam. The Feynman diagrams for these two relevant cases are shown: in Figure 1; in Figure 2. There can be little doubt that if observed this neutrino-laser pulse process would demonstrate the Dirac nature of the neutrino and provide vital information about the neutrino magnetic moment and mass.

### 5.4 Neutron acceleration in the external field

Given that a neutron is about heavier compared to neutrino one cannot expect a Lorentz-factor that is anywhere near to the value that makes neutrino magnetic interactions with the external field strong. Even so we note that iThemba LABS can produce neutrons with kinetic energy of MeV [25] which corresponds to MeV. This still places their dynamics into a classical regime since for 1 eV laser photons. However, the overall acceleration effect is much smaller.

Even though the magnitude of the magnetic moment for the neutron is several orders of magnitude larger the neutrons are times heavier and in conclusion we would require the product to be as high as in order to achieve critical accelerations that allows ample radiation. On the other hand the neutron-external magnetic field interaction is appreciable and has been used to keep a neutron beam in a storage ring [26]. Thus laser-neutron interaction could become of interest in neutron beam development.

## 6 Conclusions

The novel domain of EM magnetic moment interactions in external fields which has been recently formulated also holds promise to enhance the understanding of physics of plasmas. Here we have described the nascent beginning of the exploration of applications focused on the foundational aspect of laser pulse interaction with ultrarelativistic neutrinos. As our discussion shows fascinating applications await considering that the neutrino Lorentz- factor enhances its interaction strength with the external field opening opportunity to revolutionize the study of physical properties of the neutrinos as both mass and Dirac neutrino magnetic moment maybe directly accessible to laboratory measurement.

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