Testing a non-perturbative mechanism for elementary fermion mass generation: lattice setup

In this contribution we lay down a lattice setup that allows for the non-perturbative study of a field theoretical model where a SU(2) fermion doublet, subjected to non-Abelian gauge interactions, is also coupled to a complex scalar field doublet via a Yukawa and an"irrelevant"Wilson-like term. Using naive fermions in quenched approximation and based on the renormalized Ward identities induced by purely fermionic chiral transformations, lattice observables are discussed that enable: a) in the Wigner phase, the determinations of the critical Yukawa coupling value where the purely fermionic chiral transformation become a symmetry up to lattice artifacts; b) in the Nambu-Goldstone phase of the resulting critical theory, a stringent test of the actual generation of a fermion mass term of non-perturbative origin. A soft twisted fermion mass term is introduced to circumvent the problem of exceptional configurations, and observables are then calculated in the limit of vanishing twisted mass.


Introduction
In [1] a new non-perturbative (NP) mechanism for elementary particle mass generation was conjectured. Existence and main properties of this phenomenon can be tested in the toy model described by where b −1 = Λ UV is the UV-cutoff. The Lagrangian (1) describes a SU(2) fermion doublet subjected to non-Abelian gauge interaction and coupled to a complex scalar field via Wilson-like (eq. (4)) and Yukawa (eq. (5)) terms. For short we use a compact SU(2)-like notation where Ψ L = (u L d L ) T and Ψ R = (u R d R ) T are fermion iso-doublets and Φ is a 2 × 2 matrix with Φ = (φ, −iτ 2 φ * ) and φ an isodoublet of complex scalar fields. The term V(Φ) in eq. (3) is the standard quartic scalar potential where the (bare) parameters λ 0 and µ 2 0 control the self-interaction and the mass of the scalar field. In the equations above we have introduced the covariant derivatives where A a µ is the gluon field (a = 1, 2, . . . , N 2 c − 1) with field strengt F a µν . . The model (1) is powercounting renormalizable (as LQCD is) with counter-terms constrained by the exact symmetries of the Lagrangian. Besides Lorentz, gauge and C, P, T , CPF 2 symmetries (see Appendix B of [1]), L toy is invariant under the following (global) transformations χ L and χ R χ L/R : which forbid power divergent fermion mass terms. The d = 4 Yukawa term L Yuk and the Wilson-like d = 6 operator L Wil , which for dimensional reasons enters in the Lagrangian multiplied by b 2 , break explicitly chiral transformationsχ L andχ R . To study possible enhancement ofχ L symmetry (by parity the same will hold also forχ R ) we consider the bare Schwinger Dyson Equation (SDE) where∆ i LÔ (0) is the variation ofÔ(0) underχ L and the associated non-conserved currents arẽ where Z ∂J andη are functions of the dimensionless bare parameters entering (1) and hence depend on the subtracted scalar squared mass µ 2 [1]. Thus we write Z ∂J = Z ∂J (η; g 2 s , ρ, λ 0 ) andη =η(η; g 2 s , ρ, λ 0 ). Ellipses in the r.h.s. of eq. (11) denote possible NP contributions to operator mixing, the possible occurrence of which is a key point that will be discussed below. Plugging (11) into (10) we get implying restoration of the fermionicχ L ⊗χ R symmetries up to O(b 2 ) UV cutoff effects.

Mass generation mechanism in the critical model (Nambu-Goldstone phase)
The physics of the model (1) at the critical value η cr crucially depends on whether the parameter µ 2 0 is such that V(Φ) has a unique minimum (Wigner phase of the χ L symmetry, µ 2 sub > 0) or whether V(Φ) develops the typical "mexican hat" shape (Nambu-Goldstone phase µ 2 sub < 0). Here µ 2 sub = µ 2 0 − µ cr , with µ 2 cr being the phase transition point. In the Wigner phase no NP terms (i.e. ellipses) are expected to occur in the mixing pattern of eq. (11) and the transformationsχ L leads to eq. (12) without the ellipses [1].
In the Nambu-Goldstone phase a non-perturbative term is expected/conjectured [1] to appear in the mixing pattern of eqs. (11) leading to a WTI of the form U is a dimensionless non-analytic function of Φ that has the same transformation properties as the latter under χ L × χ R and is well defined only if Φ = v 0. Occurrence of the c 1 Λ s term in the (13) implies the presence of c 1 Λ sΨ Ψ term in Γ loc NG , the local effective action in the NG phase. This term describe NP breaking ofχ L ⊗χ R and in particular gives fermions a mass c 1 Λ s . It does not stem from the Yukawa term and, interestingly, can give a natural (in the sense of 't Hooft [2]) understanding of the fermion mass hierarchy problem (see discussion in [1]). An idea of how the mechanism works can be obtained from a perturbative expansion where Feynman diagrams are evaluated with the Lagrangian (1) augmented by few extra terms representing the expected O(b 2 ) NP effective vertices [1], as those shown in fig. 1. These vertices can be inserted together with O(b 2 ) vertices coming from the term (4) in diagrams like the ones depicted in fig. 2, giving rise to finite self-energy contributions. It is worth noticing that if the mechanism we have conjectured really exists it will generate a NP mass term for the fermions even in the quenched approximation where the vertices (b) and (c) of fig. 1, and thus the two rightmost diagrams of fig. 2, are still present.

Lattice quenched study of L toy : regularization and renormalization
Numerical simulations of lattice models with gauge, fermions and scalars are not common and technically challenging 2 . In this first numerical study of the model (1) we can limit ourselves to a quenchedfermion simulation of the lattice model specified below. In fact in quenched approximation the gauge and the scalar fields can be updated independently of each other. The lattice regularized action 3 we consider reads 2 To our knowledge what we presented here is the first numerical study of a model with fermions, scalars and non-Abelian gauge fields in the strong interaction regime. 3 For a presentation of preliminary numerical results see [3] 4 where Φ = ϕ 0 11 + iϕ j τ j is a matrix-valued field and tr [Φ † Φ] = ϕ 2 0 + ϕ 2 with F(x) ≡ [ϕ 0 11 + iγ 5 τ j ϕ j ](x), the fermionic SU(2) doublet Ψ T = (u, d) and the lattice derivatives defined as The Wilson-like term does not remove the doublers because it involves the scalar field Φ and it has dimension six. This makes no harm in this quenched study aimed at testing whether the mass generation mechanism occurs at all. The analysis done in [4], [5] and [6] for staggered fermions can be used to analyze the fermions in the Lagrangian (15): first we rewrite the action in terms of the field 3 3 γ x 4 4 , then we perform a second change of variables where y runs over the coarse lattice x µ = 2y µ + ξ µ , ξ µ = 0, 1 and U(2y, 2y + ξ) is the average of link products along the shortest paths from y to y + ξ. With these changes of variables the action becomes where F (y) = ϕ 0 (2y)(1 1 ⊗ 11) + s B iτ i ϕ i (2y)(γ 5 ⊗ t 5 ), s B = ±1 and t µ = γ * µ are the taste matrices. The action (23) is diagonal in taste and replicas B = 1, 2, 3, 4 indices up to O(b 2 ); it describes 32 fermions species namely 4 replicas of 4 tastes of the SU(2) doublet q T = (u, d). The quark bilinears in the Ψ basis have well defined quantum numbers in the classical continuum limit once expressed in the q B basis. For example the point-split vector current once summed over the hypercube coordinate ξ and expressed in the q B basis becomes One can prove that loop effects do not generate d ≤ 4 operators besides F µν F µν , ∂ µ Φ † ∂ µ Φ, q B (γ µ ⊗ 11)∇q B , Φ † Φ, (Φ † Φ) 2 and ηq B (y)F B (y)q B (y) which are all present in the action (15). A way of seeing this is based on "spectrum doubling symmetry" [7] Ψ where H is an ordered set of four-vectors indices H ≡ {µ 1 , ..., µ h }, (µ 1 < µ 2 < ... < µ h ). For 0 ≤ h ≤ 4 there are 16 four-vectors π H with π H,µ = π if µ ∈ H or π H,µ = 0 otherwise and 16 matrices M H ≡ (iγ 5 γ µ 1 )...(iγ 5 γ µ h ). This is an exact symmetry of S lat , thus also of the effective action Γ lat [U, Φ, Ψ]. Now in order to respect the spectrum doubling symmetry Γ lat can only have terms with symmetric covariant derivatives ∇ µ acting on Ψ. Close to the continuum limit among the local terms of Γ lat only the fermion kinetic termΨ ∇Ψ and Yukawa term ηΨΦΨ are relevant. As a consequence we find that η cr , the critical value of η, is well defined (even in the presence of fermion doubling), unique and independent of the subtracted scalar squared mass µ 2 sub (thus equal for the Wigner phase and the Nambu-Goldstone phase).
Since we are doing a quenched study of the model (15) exceptional configurations of the gauges fileds and the scalars with small eigenvalues of D lat can occur in the Monte Carlo sampling leading to small eigenvalues of D lat . In order to get control over exceptional configurations we add a twisted mass term in the action S toy+tm at the price of introducing a soft (hence harmless) breaking of χ L,R (and χ L,R when restored).

Strategy of numerical study
To study whether the NP mechanism occurs we consider the renormlize axial χ SDE (see eq. 13 ) with δ ph,NG = 0, 1 for the NG and Wigner phase respectively, the current and the densities In the Wigner phase (δ ph,NG = 0) one can determine η cr by studying the SDE (28) for various η values. In NG phase the SDE (28) at η = η cr takes the form of aχ WTI with NP breaking term up to O(b 2 ) that we shall neglect from now on Expanding U around the vacuum, U = 11 + i τ· ϕ v + O( σ 2 v 2 , π 2 v 2 ), we get the corresponding expansion for P ± P ± = Ψ L U, χ invariance implies that P ± has the same renormalization constant as P ± = Ψγ 5 τ ± Ψ which we call Z P . Thus a renormalized measure of the effective NP χ breaking is given by the dimensionful quantity 2m ren AWI ≡ Z ∂Ã Z P 0|∂ µ J A ± µ |M PS ± 0|P ± |M PS ± = C 1,ren Λ s (1 + ...) C 1,ren = C 1 Z P