Testing a non-perturbative mechanism for elementary fermion mass generation: numerical results

Based on a recent proposal according to which elementary particle masses could be generated by a non-perturbative dynamical phenomenon, alternative to the Higgs mechanism, we carry out lattice simulations of a model where a non-abelian strongly interacting fermion doublet is also coupled to a doublet of complex scalar fields via a Yukawa and an"irrelevant"Wilson-like term. In this pioneering study we use naive fermions and work in the quenched approximation. We present preliminary numerical results both in the Wigner and in the Nambu-Goldstone phase, focusing on the observables relevant to check the occurrence of the conjectured dynamical fermion mass generation effect in the continuum limit of the critical theory in its spontaneously broken phase.

L toy (Ψ, A, Φ) = L kin (Ψ, A, Φ) + V(Φ) + L Wil (Ψ, A, Φ) + L Yuk (Ψ, Φ) , where b −1 = Λ UV is the UV-cutoff. We denote with Ψ L = (u L d L ) T and Ψ R = (u R d R ) T the fermion iso-doublets. The Yukawa and Wilson-like terms are given by Eqs. (4) and (5), respectively. The latter is a six-dimensional operator multiplied by b 2 for dimensional reasons. The Yukawa coupling and the Wilson-like parameter are denoted by η and ρ, respectively. The scalar field Φ = (φ, −iτ 2 φ * ) is a 2 × 2 matrix with φ an iso-doublet of complex scalar fields. It obeys a quartic scalar potential denoted by the term V(Φ) of eq. (3) where µ 2 0 and λ 0 are, respectively, the (bare) values for the squared mass and the self-interaction coupling constant of the scalar field. Moreover F a µν is the field strength for the gluon field (A a µ with a = 1, 2, . . . , N 2 c − 1). Finally, the covariant derivatives are given by: A study of the unification of electroweak and strong interactions based on the above proposal has been presented in Ref. [3]. On-going work on the toy-model has been reported in Ref. [4].

Symmetries and properties of the model
The toy-model respects Lorentz, gauge, and C, P, T and CPF 2 symmetries (see Ref. [1]). Moreover it enjoys an exact symmetry under the global transformations χ L and χ R defined as: The toy-model (1), similarly to the LQCD case, is power-counting renormalizable with counter-terms constrained by the exact symmetries of the Lagrangian. In particular, thanks to the exact χ ≡ χ L ⊗ χ R symmetry, owing to the inclusion of the scalar field in the Wilson term, there is no power divergent fermion mass terms, unlike to the Wilson-LQCD case. However the pure fermionic chiral transformations,χ ≡χ L ⊗χ R , do not constitute a symmetry of L toy due to the presence of the Yukawa and Wilson terms (for non-zero values of η and ρ).
The physical implications of the toy-model depend crucially on the phase, Wigner or Nambu-Goldstone (NG), of the scalar potential V(Φ). Following the line of argument of Ref. [1] it can be shown thatχ-symmetry enhancement takes place in the Wigner phase at a critical value of the Yukawa coupling. In fact by working in a way analogous to Ref. [5] one can get the renormalised Schwinger-Dyson equation (SDE) underχ L transformations 1 : in which the operator mixing under renormalisation of the d=6 operators with the two d=4 ones has been taken into account and the current (the four-divergence of which is renormalised by Z ∂J ≡ Z ∂J (η; g 2 s , ρ, λ 0 )) is defined by: Notice that thanks to the χ-symmetry discretisation effects in Eq. (8) are of O(b 2 ) while the ellipses stand for possible contributions owing to possible NP operator mixing. The SDE of Eq. (8) becomes a WTI at a critical value of the Yukawa coupling, η = η cr (g 2 s , ρ, λ 0 ), obtained by η cr (g 2 s , ρ, λ 0 ) − η(η cr ; g 2 s , ρ, λ 0 ) = 0. In this caseχ-symmetry restoration occurs, up to discretisation effects of O(b 2 ), scalars get decoupled from quark and gluons, fermion mass is expected to vanish, and Eq. (8) becomes: In the Wigner phase no spontaneous symmetry breaking (SSB) effect takes place, so the operator mixing is expected to follow perturbation theory arguments; as a consequence there are no ellipses in Eq. (10). In the NG phase instead, aχSSB effect is expected to occur triggered by residual cutoff effects of O(b 2 ), yielding new operator mixing terms of NP nature. In that case it is conjectured that Eq. (8) takes the form: where U is a dimensionless non-analytic function of Φ given by The RGI term C 1 Λ sΨL 2 and is well defined only in the NG phase in which Φ = v 0. Λ s stands for the scale of strong interactions that in our simulation setup (see next section) is identified with Λ QCD .

Lattice simulations and results
In this preliminary numerical study of the toy-model we have performed lattice simulations in the quenched approximation, where gauge and scalar fields can be generated independently. The verification or falsification process of the NP mechanism for fermionic mass generation is totally unaffected by the present choice to carry out simulations within the (computationally cheap) quenched fermion approximation. We have employed naive Dirac fermions for which the χ L ⊗ χ R symmetry is exact. We have used the symmetric covariant derivative,∇ µ , throughout because with this choice the Wilsonlike action term has symmetry properties (see [7], sect. 2) such that, even in the presence of fermion doublers, the value of η cr is unique. In order to avoid exceptional configurations due to the possible presence of fermionic zero modes the twisted mass term, iµ QΨ γ 5 τ 3 Ψ, has been added in the lattice action (see Ref. [6]). The soft χ L ⊗ χ R symmetry breaking owing to the presence of the twisted mass  term is eliminated in the limit µ Q → 0. For full discussion of the lattice setup we refer the reader to the companion contribution at this conference [7].
In these proceedings we present a preliminary status of the simulations and analysis of the results. We have performed simulations on a lattice volume 16 3 × 40 at one value of the gauge coupling (β = 5.85) which corresponds to a lattice spacing of about a = 0.123 fm. Our lattice scale is given by r 0 = 0.5 fm determined in quenched LQCD in Refs [8] and [9].

Determination of the critical Yukawa coupling in the Wigner phase
In order to avoid unnecessary contributions in the SDEs due to the presence of the twisted mass regulator in our lattice action, we employ the vector combination of L-handed and R-handed isotriplet currents, which obeys the following renormalized SDE (for x 0): where we have defined: andJ L/R, 3 0 In the Wigner phase at η = η cr the correlation function CJD(x 0 ) ≡ x J V,3 0 (x)D S ,3 (0) is expected to vanish thanks to the restoration of theχ-symmetry. This behaviour can be noticed, as a tendency, by looking at the data in Fig. 1(a), where the correlator CJD(x 0 ) is shown for several values of η at a certain value of bµ Q = 0.0224 (in lattice units). The vanishing of lim µ Q →0 CJD(x 0 ) at η = η cr implies, in the absence of massless particles (which we explicitly check in our simulations), that all the on-shell matrix elements ofJ V,3 0 must vanish in the same limit. These remarks in turn suggest to determine η cr by looking at the renormalized SDE of vector-τ 3χ transformations, namely with kJ analytic in η at η = η cr and O(1) (see [7] about Z ∂J ). This being an operator equation (with the form of a Ward Identity at η = η cr ) that holds on-shell for arbitrary intermediate states, it looks convenient to study the ratio Indeed taking the average of r WI (x 0 ) over a x 0 -window where only few low-lying states contribute to the correlators in the ratio one gets a quantity, with reduced statistical noise and small O(b 2 Λ 2 s ) deviations from kJ(η − η cr ). In particular, if η cr is determined by imposing the condition for an appropriate time window [τ 1 , τ 2 ] kept fixed in physical units at different lattice spacings, the O(b 2 Λ 2 s ) cutoff effect in eq. (17), and the resulting one on the estimate of η cr at each β, by construction will scale nicely towards zero as b 2 → 0, thereby having no impact on the properties of the critical model that are established in the continuum limit.
The extrapolation of r [τ 1 ,τ 2 ] WI (η, µ Q ) to µ Q = 0 is easy in the Wigner phase, where absence of spontaneous symmetry breaking of χ-symmetry 3 and parity invariance entail an analytic dependence of r WI on µ 2 Q , which happens to be numerically small and comparable to the statistical errors in the explored µ Q -range (bµ Q = 0.0224, 0.0316, 0.0387).

Dynamically generated fermion mass in the NG phase
In the NG phase the χ L ⊗ χ R symmetry is broken to the χ V -symmetry. Moreover, at η cr theχ L ⊗χ R symmetry, according to our conjecture, is expected to be spontaneously broken due to O(b 2 ) effects.  In Ref. [1] it has been argued that in the NG phase the local effective action density of the model 4 reads: We also note that in the NG phase the Wilson-like term gets effectively a form analogous to the one of the Wilson term in Lattice QCD. Indeed by setting r = bvρ (with v the scalar field vev) and neglecting quantum field fluctuations the Wilson-like term in the toy model lattice action can be rewritten in the form Simulations in the NG phase are performed by employing the same values for the set of the parameters (β, λ R , ρ) and the lattice volume as in the Wigner phase. The effective quark mass (in the µ Q = 0 limit) can be read off from the axialχ WTI, e.g. where The scalar potential here, V µ 2 Φ <0 (Φ), is written in terms of the renormalised parameters µ 2 Φ andλ. In the expression (20) one could add one or more kinds of kinetic term of U that are proportional to Λ s . However, for v Λ s which is the typical regime for our mechanism these terms will be negligible. is the one-point-split current associated to the fermionic (χ) axial transformations and P ± (x) = Ψ(x)γ 5 τ 1 ± iτ 2 2 Ψ(x) is the pseudoscalar density. In Fig. 2(a) we show results for the bare quark mass (multiplied by a factor of two) in units of r 0 against the Yukawa coupling. The results have been obtained using Eq. (21) at several values of (η, µ Q ). For each value of η a linear extrapolation to µ Q = 0 has been performed. Small deviations from linearity are possible and their impact is presently under study by extra simulations at further µ Q values and more elaborate fits. At η = η cr , where the Yukawa quark mass term gets cancelled, the m WT I is expected to be equal to the conjectured quark mass of NP origin, c 1 Λ s . As it can be seen from that figure, based on our preliminary data, a rough estimate of the bare quark mass 5 in r 0 units is −2r 0 c 1 Λ s 0.06. Passing now to Fig. 2(b) where (r 0 M PS ) 2 is shown against the Yukawa coupling we notice that at η = η cr the corresponding value for the pseudoscalar mass is rather large (of about 320 MeV or larger). We would also like to draw the attention to an interesting feature which occurs at the value of the Yukawa coupling, namely η * , at which m WT I vanishes. With the help of the effective action density of Eq. (20) one can deduce that, defining m WT I ≡ (η * − η cr )v + c 1 Λ s = 0 entails η * = η cr − c 1 Λ s /v. Our data gives evidence that η * − η cr 0 which further supports the conclusion that the dynamically generated quark mass is non-zero 6 .

Summary and further developments
We have discussed a toy-model that exemplifies a novel NP mechanism proposed in Ref. [1] for dynamical fermion mass generation. The fundamental property of the mechanism consists in the enhancement of the QCD symmetries in such a way that fermion masses emerge in a natural way [11], being independent from the Yukawa interaction and the scalar field. Thanks to the NP character of the mechanism the physical implications and predictions of the associated toy-model can be tested with the help of simulations on the lattice. We have presented preliminary results based on simulations in the quenched approximation at one value of the lattice spacing. Our results for the dynamically generated effective fermion mass and the associated pseudoscalar meson mass in the NG phase, barring cutoff effects, are of O(Λ s ). Since the presentation at the conference we have performed more simulations at the present lattice spacing and improved our methods of analysis. We have also carried out simulations at a second value of the lattice spacing in order to be able to check the scaling behaviour both of the fermion mass and the pseudoscalar meson mass. All these results that show rather smooth scaling properties will be presented soon in [10].