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Strongly coupled U(1) lattice gauge theory as a microscopic model of Yukawa theory

Dynamical chiral symmetry breaking in a strongly coupled U(1) lattice gauge model with charged fermions and scalar is investigated by numerical simulation. Several composite neutral states are observed, in particular a massive fermion. In the vicinity of the tricritical point of this model we study the effective Yukawa coupling between this fermion and the Goldstone boson. The perturbative triviality bound of Yukawa models is nearly saturated. The theory is quite similar to strongly coupled Yukawa models for sufficiently large coupling except the occurrence of an additional state -- a gauge ball of mass about half the mass of the fermion.

Dynamical chiral symmetry breaking in a strongly coupled U(1) lattice gauge model with charged fermions and scalar is investigated by numerical simulation. Several composite neutral states are observed, in particular a massive fermion. In the vicinity of the tricritical point of this model we study the effective Yukawa coupling between this fermion and the Goldstone boson. The perturbative triviality bound of Yukawa models is nearly saturated. The theory is quite similar to strongly coupled Yukawa models for sufficiently large coupling except the occurrence of an additional state -a gauge ball of mass about half the mass of the fermion.
The question whether the Higgs-Yukawa mechanism of symmetry breaking and particle mass generation in the contemporary high energy physics might be an effective theory to some more fundamental renormalizable quantum field theory with dynamical symmetry breaking has been raised many times. It is of particular interest with respect to the large top quark mass, stimulating e. g. the idea of topcolor. The pursuit of this question leads frequently to devising strongly coupled gauge theories beyond the standard model with speculative nonperturbative dynamics. It might be inspiring to have prototypes of such "microscopic" gauge models of Yukawa theory with a solid and accessible dynamical framework. Numerical lattice methods are justified for this purpose provided a lattice model can be found with required basic dynamical properties, though neglecting various phenomenological aspects. In particular, these properties should be preserved when approaching the continuum limit, i.e. the model should be nonperturbatively renormalizable.
A promising lattice model has been suggested in Ref. [1]. This "χU φ model" consists of a charged fermion field χ with strong vectorlike coupling to compact U(1) gauge field U , and a scalar field φ of the same charge. Here we summarize the results of our extensive systematic investigations of this model in four dimensions by means of numerical simulations with dynamical fermions. Though we cannot decide by currently available means whether the model is renormalizable, we find some encouraging properties. The detailed account is given in a parallel paper [2] and in some previous works [3][4][5][6].
The same model has been investigated also in lower dimensions. There is little doubt that the χU φ model defines a renormalizable quantum field theory in two di-mensions, belonging to the universality class of the chiral Gross-Neveu model [7]. Also first results in three dimensions suggest renormalizability [8].
Finding a "promising" lattice model meets two major difficulties. Strongly coupled lattice gauge theories with fermions exhibit frequently the required dynamical chiral symmetry breaking. However, this phenomenon comes mostly along with undesirable confinement of fermions acquiring mass. Furthermore, at strong coupling, it is difficult to find a second order phase transition required for an approach to continuum.
In the χU φ model, the scalar field φ helps to solve both problems. Firstly, it shields the fermion charge and gives rise to an unconfined, i.e. physical massive fermion F = φ † χ in the phase with chiral symmetry broken dynamically by the gauge interaction (Nambu phase). Secondly, the scalar suppresses this symmetry breaking and at sufficiently strong gauge coupling induces a second order transition to a chiral symmetric (Higgs) phase, thus opening a way to continuum.
We find that one particular point of the phase transition line, the tricritical point E, represents a new kind (new universality class) of dynamical symmetry breaking mechanism in strongly coupled gauge theories with fermions in four dimensions. We present some evidence that a strongly coupled effective Yukawa theory results. It describes the interaction between the composite fermion F and some χχ "mesons". At this point the model could serve as a microscopic model of Yukawa theory. But in addition, the mechanism produces also a scalar gauge ball.
To explain these findings, we now summarize the most relevant features of the model (more details can be found in [2]). The model is defined by the action Here Θ P ∈ [0, 2π) is the plaquette angle, i.e. the argument of the product of U(1) gauge field link variables U x,µ along a plaquette P . Taking Θ P = a 2 gF µν , where a is the lattice spacing, and β = 1/g 2 , one obtains for weak coupling g the continuum gauge action S U = 1 4 d 4 xF 2 µν . The staggered fermion field χ has (real) bare mass am 0 in lattice units. It leads to four fermion species in the continuum limit. Sign factors η xµ are standard for staggered fermions. The complex scalar field is constrained, |φ| = 1. As its "hopping parameter" κ increases, it drives the model into the usual Higgs phase.
We note that there is no Yukawa coupling between χ and φ, as both fields have the same charge. The model has U(1) global chiral symmetry in the limit of vanishing fermion bare mass in physical units, m 0 = 0. This is the case we are really interested in. The numerical simulations have to be carried out at nonvanishing m 0 , however, and an extrapolation to the chiral limit performed. For am 0 = 0, the chiral transformation relates the model at ±am 0 .
The phase diagram at m 0 = 0 is shown in Fig. 1. One limit case is κ = 0, corresponding to the massless compact lattice QED. Another important limit case is β = 0. Here the model can be rewritten exactly as the lattice Nambu-Jona-Lasinio (NJL) model [9] with the critical point N. The strong four-fermion coupling of that model corresponds to small κ. The Nambu phase thus connects the broken chiral symmetry phases of both these limit models. Only a part of its boundary, the NE line, represents second order phase transitions. Point E is far away from any limit case and it does not appear to be accessible by any reliable analytic method, neither on the lattice nor in continuum. It is "tricritical" because in the full parameter space (including am 0 ) there are, apart from NE, two further second order "wing" lines entering E from the positive and negative am 0 directions. The existence of a common point E of these three second order lines is neither predicted nor understood. The evidence is purely numerical, but quite strong [2]. Its position is β E = 0.62 (3), κ E = 0.32 (2).
The importance of the point E roots in the experience from statistical mechanics that a tricritical point belongs to a universality class different from that of any of the second order lines entering into it [10]. (Basic properties of tricritical points relevant in our context are summarized e.g. in [11].) The usual universality of second order lines suggests, and this is supported by our earlier investigations [3], that the whole NE line except the point E corresponds to the same continuum model as the point N, the NJL model. The gauge field is presumably auxiliary and the model is therefore of limited interest there. However, the point E is expected to be different, gauge field playing an important role.
To verify this expectation, we have investigated critical exponents and spectrum of the model in the vicinity of E. The study of exponents uses advanced techniques of statistical physics and is described in [2]. Here we only mention the found value ν t ≃ 1/3 of the correlation length tricritical exponent. It is different from ν ≃ 1/2 along the NE line confirming the particularity of E. It also differs from the prediction of the classical theory of tricritical points usually believed to hold in four dimensions [10] and predicting ν t = 1/2. This indicates that the point E is a tricritical point with important role of quantum fluctuations.
Some insight into the continuum physics, which might be obtained at the point E, is provided by the spectrum and its scaling behavior. The masses am Q are in lattice units and thus only their ratios are physical. They are determined in the Nambu phase without any gauge fixing from the correlation functions of various gauge invariant composite operators Q. In this sense the massive physical fermion F , as well as other physical states, are composite. The interaction between them is due to the Van der Waals remnant of the fundamental interactions.
The fermion mass am F > 0 decreases when the NE line is approached from the Nambu phase and there it is rather insensitive to the lattice size. It is small in the Higgs phase on finite lattices, decreasing with increasing lattice size. The data are consistent with expected vanishing of am F in the Higgs phase in the infinite volume limit.
We find several χ, χ bound states and borrow their names from QCD. One of them is the obligatory pseudoscalar Goldstone boson π with the dependence on am 0 as required by current algebra. The π-meson is mass-less in the chiral limit in the whole Nambu phase. It would get "eaten" if the global chiral symmetry of the χU φ model were gauged. This process would be identical with the standard Higgs mechanism. Thus we do not need to discuss it and mention it only for completeness.
We obtain some results for the pion decay constant. The data suggest a large ratio f π /m F increasing when the point E is approached. However, the value is sensitive to the lattice volume and we cannot yet extrapolate it to the infinite volume and continuum limit. If the ratio diverges, it would indicate the triviality of the model [12]. Its current value, f π /m F ≃ 1/3 for am F ≃ 0.4 can be considered as a lower bound.
Also the σ meson in the antifermion-fermion channel is observed. The σ mass is quite dependent both on the lattice size and bare fermion mass, preventing its prediction in the continuum limit with m 0 = 0. Strong lattice size dependence of f π and σ mass has been observed and explained by means of the Schwinger-Dyson equations in the limit case β = 0, the NJL model, and holds apparently also in the vicinity of E.
The ρ mass is insensitive to the lattice size and scales like the fermion mass, with the approximate value m ρ ≃ 2m F . We cannot distinguish between a bound state and a resonance.
Also a neutral scalar (S-boson) is seen appearing as a composite of φ † and φ and as a state of pure gauge field. In the Higgs phase it corresponds to the Higgs boson associated with the perturbative Higgs mechanism occuring in that phase at large β. In the Nambu phase it is more natural to interpret the S-boson as a scalar gauge ball. The transition between both interpretations is smooth, however, and the corresponding channels appreciably mix in the vicinity of E.
The mass am S is nonzero in all phases and goes to zero when the two wing critical lines are approached for any am 0 . For am 0 = 0, the mass am S goes to zero only at the point E. This illustrates the particular character of the tricritical point E. Whereas only one of the masses am F or am S vanishes on each of the second order lines, they both vanish at the tricritical point. As their ratio remains finite, both corresponding states are present in the continuum limit taken at this point. This is the best evidence that the physical content (universality class) of the point E differs from that of any of the adjacent second-order lines and thus does not correspond to the NJL model obtained at N.
As am S is only moderately dependent on the lattice size, we can estimate its scaling behavior when point E is approached. This is shown in Fig. 2 for κ = 0.30, which is approximately the κ-coordinate of E. The ratio m S /m F remains constant when am F decreases, as long as finite size effects are negligible. The sudden rise of the data at small am F is due to the strong finite size dependence of am F in the Higgs phase and shifts correspondingly to smaller am F when the lattice volume increases. These observations suggest that the value m S /m F ≃ 1/2 would be obtained in the continuum limit at E.
In the rest of this letter we concentrate on the renormalized Yukawa coupling y R between F and π. It is obtained from the three-point function of the corresponding composite operators and thus can be interpreted as an effective Yukawa coupling, which would describe the interaction between F and π in the regime where their composite structure can be neglected. Our measurement, performed in a similar way as in [13], is described in detail in [2,6]. In spite of the complexity of the corresponding expressions, y R is measurable with good precision. The reason is the strict locality of the used operators. An attempt to measure also an analogous coupling of the S-boson failed because S is described by extended operators.
The renormalized Yukawa coupling y R is presented in Fig. 3 (full symbols) for three different am 0 at κ = 0.30 in the Nambu phase close to E. We show it in dependence on am F and compare it with the tree level relation The agreement is so good that the open symbols representing (1) in fig. 3 are nearly invisible. The values for y R are not significantly dependent on am 0 . Therefore we expect that the value for y R in the chiral limit is rather well represented by our results at am 0 = 0.01. In Fig. 4 we show these results for substantially different gauge couplings. The results close to the point E, for β = 0.64, κ = 0.30, and β = 0.55, are very similar. The values of y R at β = 0 (NJL model) are significantly below the other data, however. This implies that the Yukawa coupling gets stronger as one moves from the NJL model to the vicinity of E by increasing β at fixed am F .
In the interval y R ≃ 2 − 5, where the Yukawa coupling is determined reliably, we compare our result with the curve of maximal renormalized coupling, resulting from the first order perturbative calculation in the Yukawa model (triviality bound): with N F = 4, the number of fermions including lattice doublers. In the lattice Yukawa models this upper bound is nearly saturated by the values of y R obtained at the maximal possible bare Yukawa coupling [14,15]. As seen in fig. 4, also in our model close to E, the data are only slightly below the curve (2). Thus in this interval, y R has the same form as that in the strongly coupled Yukawa models.
For y R < 2, the data are significantly influenced by the finite size effects, and we cannot make any conclusion. It might be that some deviation from the Yukawa theory occurs there. The fact that y R is stronger in the vicinity of E than in the NJL model suggests the question whether it vanishes in the continuum limit, am F = 0, taken there. Is it zero as in Yukawa theory, or could triviality be avoided at the tricritical point? A reliable extrapolation of y R to am F = 0 at the tricritical point of the χU φ model is a challenge for future investigations.