Fermion Mass Mixing in Vacuum

Renormalization group procedure for effective particles is applied to a theory of fermions that interact only through mass mixing terms in their Hamiltonian. Problems with virtual pair production in vacuum are avoided by using the front form of Hamiltonian dynamics. Masses and states of physical fermions emerge at the end of a calculation that is carried out exactly irrespective of the strength of the mass mixing terms. An a priori infinite set of renormalization group equations for all momentum modes of fermion quantum fields is reduced to just one equation for a two-by-two mass matrix. In distinction from scalars, fermions never become tachyons but appear chirally rotated when the mass mixing interaction term is sufficiently strong.


Introduction
Among the issues that need to be dealt with on the way to defining and solving relativistic quantum field theory beyond the perturbative pictures based on free propagation or weak binding of field quanta, one may distinguish the issues of vacuum and renormalization, both intricately related with the struggle to understand relativity and quantum mechanics simultaneously in one theory. Literature concerning these two issues is of enormous size and scope and this article is not to review it. Instead, this article offers a description of how the renormalization group procedure for effective particles (RGPEP) can be applied to an elementary quantum field theory for the relativistic fermions that interact through a mass mixing term of arbitrary strength [1]. The idea is to present how the RGPEP maintains symmetries of special relativity in an interacting quantum field theory and keeps the vacuum state simple.
The example is solved exactly. The vacuum issue is addressed by using the front form (FF) of dynamics instead of the familiar instant form (IF) [2]. Therefore, the vacuum divergences [3] due to fermion-anti-fermion pair creation do not occur. The remaining interactions are included in the RGPEP equations. The equations produce a relativistic spectrum of solutions in agreement with the general rules of quantum representation of symmetries of special relativity [4].
Since the calculation described here concerns basic issues and differs from the IF approach, a lot of basic details are included explicitly. First, the IF approach to the model example is reviewed, introducing useful notation in a familiar context and pointing out difficulties. Then the RGPEP approach is presented in a concise form that indicates new elements in comparison with the IF approach.

Theory of Two Kinds of Free Fermion Fields
The standard form of Lagrangian density for two kinds of free fermion fields ψ and φ with mass parameters μ and ν, respectively, is The condition of stationary action, δS = 0, where S = d 4 x L, implies equations of motion (i∂ / − μ) ψ = 0 and (i∂ / − ν) φ = 0. The corresponding canonical Hamiltonian in the IF of dynamics is where T 00 can be identified as the time-time component of the energy-momentum density tensor The integral in Eq. (2) extends over the three-dimensional volume in space-time that is defined by the condition t = 0, where t is the time co-ordinate of the inertial observer who defines the theory. As a result of direct evaluation, the IF Hamiltonian for fermion fields according to this observer has the form The corresponding FF Hamiltonian is defined differently. This will be explained later.

IF Canonical Quantum Theory for Two Kinds of Free Fermions
In order to obtain a canonical quantum theory of fermions according to the IF rules [5,6] one writes the fields ψ and φ as functions of x at t = 0 in terms of their Fourier components, where the integration over momentum p and sum over spins is denoted by the symbol and the energy is E μp = μ 2 + p 2 . Similar definitions apply to the field φ with mass ν in place of μ. The coefficients u and v in the Fourier expansions are spinors of freely moving fermions. It is important for the discussion that follows to realize that these spinors are obtained by boosting spinors [7] corresponding to fermions at rest, u μ0s and v μ0s normalized to √ 2μ and √ 2ν, respectively. The explicit formulae are where the matrix B represents the boosts, Similar definitions hold for fermions with mass ν. This observation is important because the boosting requires knowledge of the fermion mass and energy. The energy is treated in the boost matrices B as if the fermions were free. Therefore, one may expect difficulties to occur when a theory includes the interactions that influence the value of mass and alter the expression for energy. The difficulties should be expected due to any form of energy that significantly differs from the free one. The usage of spinors of freely moving fermions for construction of strongly interacting quantum field operators is unlikely to be legitimate.
In the absence of interactions, having the Fourier expansions in place, one turns the fields ψ and φ into quantum field operatorsψ andφ by introducing the anti-commutation relations [8][9][10] These are satisfied as a result of imposing anti-commutation relations on the coefficients b and d in the Fourier expansions, turning them into creation and annihilation operators. Namely, one sets for fermions of mass μ, and similar relations for fermions of mass ν. Normal-ordered quantum Hamiltonian, is then obtained in the physically right form for free fermions, i.e.,

IF Canonical Quantum Theory for Fermions Interacting Through Mass Mixing
In the IF approach, one adds interaction terms to the free Lagrangian density. In the case of mass mixing terms, starting from and following the same construction steps for quantum field operators as above, one obtains the corresponding canonical quantum Hamiltonian,Ĥ =Ĥ 0 +Ĥ I , in whicĥ Having defined the quantum field operators as in the free theory, one arrives at Physically, this result is not acceptable [3]. The interaction creates a badly divergent vacuum problem. To see the problem, consider the second term inĤ I in Eq. (16), where the spinor matrix element is and ask what the norm of state |h =ĥ|0 is. Direct evaluation yields where V denotes the volume of space at t = 0 and the approximation sign indicates that the large bracket in Eq. (18) is approximated by 2, with increasing accuracy for increasing |p|. The result is infinite.
The consequence is that one does not know how to calculate the ground state of the interacting theory, or vacuum, in terms of the ground state of a free theory. All excited states contain similar divergences as the vacuum state.
To avoid the infinities and attempt a search for the spectrum of a regulated quantum Hamiltonian, one might impose a cutoff on |p|. The price would be a violation of Lorentz symmetry, since boosts can change |p| by an arbitrary amount.
Similar interaction terms appear in QED. For example, there are terms in the QED IF Hamiltonian that are capable of creating the electron, positron, and photon out of bare vacuum and thus produce divergences in a similar pattern. These interaction terms and the resulting effects in states containing electrons and photons, are known how to handle in perturbative calculations [13]. But beyond the perturbative rules, one may question the existence of QED in the IF Schrödinger picture [3]. In the case of QCD, in the domain where perturbation theory does not apply, the role of interaction terms that alter the bare vacuum in the IF approach, is not understood yet.
In summary, the mass mixing model provides an elementary example of how the vacuum issue arises in relativistic theories of physical interest, although in the realistic theories the relevant interaction terms are much more complicated than in the model. As a consequence, the general vacuum issue remains unsolved. Concerning the physics of the vacuum in the context of comparison between the IF and FF approaches, especially in the context of QCD, see Refs. [11,12] and the rich literature quoted there.

IF Re-Quantization Approach to the Mass Mixing Interaction and Vacuum Issue
In the IF approach, when one faces the vacuum problem due to the mass mixing interaction terms, one can step back to the Lagrangian density and re-write it using new field variables. Namely, one can write the density using a 2 × 2-matrix notation, in which the fields ψ and φ form a doublet and a mass-matrix, say M, is introduced; The eigenvalues and corresponding eigenvectors of M are . One can write the doublet field Ψ in terms of the eigenvectors, Ψ = ψ 1 v 1 + ψ 2 v 2 , and use the fields as new degrees of freedom in a classical theory. In terms of the new fields, the Lagrangian density reads which is a theory of free fields corresponding to masses m 1 and m 2 . Section 3 explains that the terms of typeĥ in Eq. (17) do not appear in the corresponding IF quantum Hamiltonian once the right energies are used in the construction of quantum field operators, including boosted spinors and corresponding creation and annihilation operators.
The result is that one can avoid the vacuum issue in quantum theory when one constructs it by quantizing the field degrees of freedom that appear in Eq. (24) rather than those that appear in Eq. (14). Since one steps back from a divergent quantum theory with interaction to the initial classical one and then quantizes the classical theory again using new degrees of freedom, for which the troublesome divergence turns out to cancel out, the IF cure for the vacuum problem can be called re-quantization.
In the re-quantization, one uses E m 1 p and E m 2 p instead of E μp and E νp in constructing quantum field operatorsψ 1 and ψ 2 , with new creation and annihilation operators. The resulting IF Hamiltonian, describes free fermions of masses m 1 and m 2 as desired in a relativistic theory.
The key question one faces in the IF of Hamiltonian dynamics is how to deal with the vacuum issue in the presence of complex interactions, i.e., when one does not know what masses to use. This becomes a serious roadblock when one does not even know if such masses exist. In case of theories such as QCD, which are expected to describe confined fermions, the very existence of "right" masses for IF quantization is doubtful.

FF Canonical Quantum Theory for Fermions with Mass Mixing Interaction
Previous consideration suggests that one needs some alternative approach to the IF re-quantization in order to handle theories with complex interactions. The elementary fermion model with mass mixing interaction is used here to explain how a candidate for such alternative method, the RGPEP, avoids the vacuum problem and produces the same solution that the IF re-quantization allows one to find. The FF approach to the model does not require any re-quantization. With the preliminaries laid out above, the description of FF approach given below is considerably abbreviated.
The space-time hyperplane, which in the FF of Hamiltonian dynamics is used instead of the hyperplane t = x 0 = 0, is defined by the condition x + = 0. The notation is x ± = x 0 ± x 3 and x ⊥ = (x 1 , x 2 ), the same for all tensor indices. Other conventions used here are explained in Ref. [1].
The Euler-Lagrange equation for the doublet fermion field in the Lagrangian density of Eq. (20), namely (i∂ / − M) Ψ = 0, can be written in terms of two orthogonal components Ψ ± = Λ ± Ψ , where Λ ± = γ 0 γ ± /2 are projection matrices, as an equation of motion and a constraint, respectively, Following Refs. [14,15], one obtains the FF Hamiltonian, where the relevant component of the energy-momentum density tensor is The resulting FF quantum Hamiltonian is obtained by replacing Ψ by the quantum field operatorΨ and evaluatingP In the FF representation of γ -matrices [1], the operatorΨ + has the form whereζ andω are two-component fields (the field Ψ has eight components). The anti-commutation relations at x + = x + = 0 are set in the form The FF Fourier expansions of operatorsζ andω are independent of mass parameters. Namely, where and χ s is the two-component Pauli spinor normalized to 1 (the Pauli matrix σ 1 takes care of the required spin flip and the negative sign reflects the requirement of Fermi-Dirac statistics). The creation and annihilation operators satisfy mass-independent anti-commutation relations For comparison with the IF theory, it is important to stress that the FF Fourier expansion of fields does not require any knowledge of mass or "energy" p − of the field quanta, in stark contrast to the IF construction where one assumes a value for the mass of field quanta and change of mass requires re-quantization. No re-quantization is needed in the FF version of the interacting theory. Evaluation of the FF Hamiltonian yieldŝ where p − μ = ( p ⊥ 2 + μ 2 )/ p + and p − ν = ( p ⊥ 2 + ν 2 )/ p + . The free fermion theory, the one before the mass mixing terms are added, corresponds to the Hamiltonian denoted byP − f , which is obtained fromP − by setting m = 0.
Note that in the FF HamiltonianP − there are no terms of the typeĥ that cause trouble in the IF version of the theory. The result is thatP − |0 = 0. Therefore, one may assume that the bare vacuum |0 , which by definition is annihilated by all the annihilation operators, can represent the physical vacuum. This is precisely what happens in the model. In more complex theories, one has to find out if the interactions allow for this interpretation of |0 . For example, if very strong interactions produced eigenstates ofP − with negative eigenvalues, one could not interpret |0 with eigenvalue zero as a ground state of a physical theory.
In FF Hamiltonians, the absence of interaction terms that can alter the bare vacuum state |0 is a generic feature of all theories of physical interest. This feature is a consequence of translation invariance, which implies conservation of +-component of total momentum in the dynamics. Since all quanta included in the theory have positive p + , as indicated in the Fourier spectrum of momenta in Eq. (34), the quanta cannot carry zero momentum, which characterizes the state |0 . Thus, the translation invariant Hamiltonian cannot create quanta from or annihilate them into the bare vacuum.
The only exception to stability of |0 is its interaction with the Fourier modes with p + = 0. But for quanta with non-zero masses such modes correspond to infinite eigenvalues ofP − f and by definition these modes lie outside the cutoffs used to regulate a theory. If there is a need to include any effects due to such modes, the possibility which is not excluded in singular gauge theories, these effects should be identifiable through the cutoff dependence that appears when the needed modes are absent. However, this means that their effects can be describable using counter terms to the cutoff dependence. This is explained Ref. [11]. The cutoff dependence and counter terms can be studied using the RGPEP. The FF of dynamics thus appears to provide new ways for handling the vacuum problem that is otherwise prohibitively divergent in the IF of quantum dynamics. In the fermion model with mass mixing, no cutoff dependence arises that would require questioning |0 as a good vacuum state.
In the model example, with terms of typeĥ being absent for reasons found in all physically interesting theories with regularization, the FF Hamiltonian still contains interaction terms that mix fermions of masses μ and ν with arbitrary strength. The question is how to find the spectrum. This is done below using the RGPEP within one and the same quantum theory, no re-quantization being involved.

The RGPEP in Quantum Field Theory
General non-perturbative definition of the RGPEP in quantum field theory can be found in Ref. [16]. In its essence, the procedure amounts to transforming quantum fields from the bare ones to effective ones by a unitary rotation, ψ t = U t ψ 0 U † t . This is done by expressing the bare, point-like quanta of a local theory in the canonical Hamiltonian with counter terms by the quanta of size s = t 1/4 . As a result, one obtains a non-local theory of quanta that are called effective particles. So, for fermions , but it is written using different degrees of freedom. Below, where necessary for brevity, the annihilation operators b and d are commonly denoted by q.
The canonical theory with counter terms has a Hamiltonian of the form The effective theory Hamiltonian differs by a change of operators q 0 to q t and coefficients c 0 to c t , The goal of RGPEP is to find the coefficients c t . This is done by defining H t (q 0 ) = U † t H 0 (q 0 ) U t and solving the equation where prime denotes the derivative with respect to t and G t = −U † t U t . Knowing G t , one can find where T denotes ordering operators according to the size of effective particles. In its simplest version, the RGPEP defines G t by the formula where H f is the free part of the initial Hamiltonian and which means that H Pt differs from H t only by multiplication of the coefficients c t by a square of the total + -momentum carried by quanta involved in a term.
In summary, the simplest version of RGPEP is carried out by solving the equation with initial condition H 0 = H canonical + H CT , where H CT denotes the counter terms. The counter terms are also found using the RGPEP; one adjusts the initial conditions so that the effective theories match physical data by their solutions.

Application of the RGPEP to the Fermion Mass-Mixing Model
The starting point is the Hamiltonian of Eq. (36), which is treated as an initial condition for a solution to A great simplification occurs in the fermion model: the initial condition does not require counter terms that diverge as functions of regularization parameters when regularization is being removed. The model does not require regularization of divergences; only finite adjustments of initial masses μ and ν and mass mixing parameter m are required in order to obtain desired values of fermion masses in the resulting spectrum. By examining the RGPEP equations in the model, one finds that the solution for P − t has the form where the coefficients can be written in the forms and the initial conditions for the mass parameters are The RGPEP generator in Eq. (44) involves and hence has the form By equating coefficients in front of the same products of creation and annihilation operators on both sides of Eq. (44), one obtains three spin-independent equations for every value of momentum p, But when one inserts in this set the coefficients A t , B t and C t in their forms shown in Eq. (46), the momenta p ⊥ and p + drop out and all these RGPEP equations reduce to just three for the mass parameters, One may conclude by quoting Dirac [18], who said that the FF "offers new opportunities, while the familiar IF seems to be played out." Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.