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Constraints on the existence of chiral fermions in interacting lattice theories
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PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 590-592 North-Holland
Constraints on the existence of chiral fermions in interacting lattice theories Yigal Shamir ~* aDepartment of Physics Weizmann Institute of Science, Rehovot 76100, ISRAEL Any interacting theory defined on a regular lattice is shown to have a vector-like spectrum if the following conditions are satisfied: (a) hermiticity, (b) locality, (c) relativistic continuum limit without massless bosons, and (d) pole-free effective vertex functions for conserved currents. The proof exploits the zero frequency inverse retarded propagator of an appropriate set of interpolating fields as an effective quadratic hamiltonian, to which the Nielsen-Ninomiya theorem is applied.
1. I N T R O D U C T I O N The last year has seen a renewed interest in the long standing problem of constructing a lattice chiral gauge theory. The well known stumbling block is the doubling problem. In the continuum, currents pertaining to classical symmetries m a y develop anomalies at the q u a n t u m level. On the other hand, on the lattice a current that corresponds to an exact s y m m e t r y of the action is always conserved. The appearance of doublers resolves the conflict between the continuum and lattice theories when b o t h the continuum action and its latticized version have axial symmetries [1], albeit at the price of making the continuum limit of the lattice theory different from the initial continuum theory. Alternatively, the presence of doublers can be attributed to the fact that m o m e n t u m space for a regular lattice (the Brillouin zone) is periodic. This line of argument was studied in detail by Nielsen and Ninomiya in a famous No-Go theorem [2]. Given a free hamiltonian on a regular lattice, Nielsen and Ninomiya showed that the massless fermion spectrum is always vector-like with respect to to full set of compact global symmetries, provided the hamiltonian is sufficiently local so that the dispersion relation has a contin*This research was supported in part by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities, and by a grant from the United States - Israel Binational Science Foundation.
uous first derivative. The Nielsen-Ninomiya (NN) theorem is mathematically rigorous, but by its very nature it cannot alone answer the question "can we define a regular lattice model whose continuum limit will be a consistent chiral gauge theory?" One obvious reason is that, while hermiticity, locality etc. are needed for consistency in the continuum limit, the NN theorem provides little information as to whether limited violations of these properties, which vanish when the lattice cutoff tends to zero, can allow for the construction of a chiral gauge theory. Another limitation is that the NN theorem is formulated for free lattice hamiltonians. It therefore does not apply to models [3] in which one tries to decouple the doublers dynamically, i.e. to achieve a chiral massless spectrum by introducing additional, strong interactions which lead to a non-trivial phase diagram. Clearly, only in the perturbative region of the phase diagram can the spectrum be reliably determined by switching off all coupling constants. Our main goal in the present work is to show that the NN theorem has a natural generalization to lattice theories with some additional non-gauge interactions. The m a j o r progress is in relating the absence of chiral fermions to locality properties of the fermionic two point function. The work is still incomplete because a detailed discussion of which lattice models satisfy our locality assumptions re-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00346-W
Y. Shamir / Constraints on the existence of chiral fermions in interacting lattice theories
mains to be given. We believe that eventually this may turn out to include all lattice models with a short range hamiltonian. In the limited room allocated to each contribution in these proceedings it is impossible to give a thorough presentation of our results. What follows is an informal report on our main results [4,5], with emphasis on the physical justification for each assumption, as well as an account on questions left for future research. 2. A N O - G O
THEOREM
The setup for the theorem is an interacting hamiltonian defined on a regular space lattice. We assume a positive norm Hilbert space, thus we do not address the Rome approach which is based on gauge fixing. Any extension of the NN theorem to lattice models with additional non-gauge interactions, automatically extends its scope to include also lattice models whose action is not gauge invariant. Practically any lattice model with a gauge variant action is mathematically equal to a model with a gauge invariant action which includes an additional field, a unitary (frozen) Higgs field. The frozen Higgs has no quadratic action, and its interactions are completely determined by the gauge variant part of the original action. The strength of the Higgs couplings is typically of order unity. In what follows we assume that the action is gauge invariant, but in view of the above comments this by itself does not lead to any loss of generality. Gauge variant models may turn out to be qualitatively different only if the integration over the frozen Higgs field changes substantially the canonical structure of the second quantized fermion fields. In order to determine the fermion content of a gauge invariant theory we will switch off the gauge coupling. The physical justification for this procedure is as follows. When approaching the continuum limit in an asymptotically free gauge theory, there has to exist an intermediate scaling region in which the gauge coupling is still small, and physics is correctly described by continuum perturbation theory. In this region, apart from the common strength of the gauge coupling,
591
the vertex describing the gloun interaction with a given fermion is completely determined once the fermion is assigned to a specific representation of the global symmetry, which is left over when the gauge coupling is switched off. With the gauge coupling switched off, what we have at hand is typically an interacting model containing only fermions and (possibly) scalar fields. Our goal is to determine the massless fermion spectrum as one approaches some phase boundary from within a s y m m e t r i c phase. We do not address directly what happens as a critical line is approached from within a broken phase. There is, however, extremely convincing evidence that if the doublers' mass arises from symmetry breaking, then these particles cannot be made very heavy and they remain in the physical spectrum. Our task now reduces to determining what massless fermions (if any) are there in a given complex representation of some unbroken global symmetry. Our strategy will be as follows. We first have to pick a set of fermion operators (which in general may or may not be elementary) that create all the massless fermions in that representation out of the vacuum. These will be called the interpolating fields. Having decided on the set of interpolating fields, we will compute a specific two point function - their retarded propagator, denoted T¢(~, t), and its Fourier transform 7~(iff, a~). The latter is chosen (rather than the commonly used time ordered propagator) because of its good behaviour at space-like separations. We will then be able to determine the number of positive helicity ("right handed") and negative helicity ("left handed") fermions in this representation from the singularities of the propagator. Moreover, it follows from the existence of a spectral representation that 7~-1(/7, w = 0) is hermitian. With suitable assumptions, the effective hamiltonian 7~- 1(/7, w = 0) will satisfy all the conditions of the NN theorem, and so the spectrum will be vectorlike. In order to show that 7~-1(/7,¢0 = 0) satisfies all the conditions of the NN thereom two major ingredients are needed. The first one concerns the analytic structure of the retarded propagator for a general lattice momentum. This, in turn, is
592
Y. Shamir / Constraints on the existence of chiral fermions in interacting lattice theories
determined by the behaviour of TC(~,t) at large space-like separations. On the lattice, (anti)commutators cannot vanish identically outside the light cone, but they have to tend to zero very rapidly if the hamiltonian is local. Using the "Edge of the Wedge" theorem, we show in ref. [4] that an exponentially decreasing anticommutator gives rise to an analytic 7~(i~,w). It is easy to show that free short range hamiltonians always give rise to exponentially bounded anticommutators. We anticipate a similar behaviour for a very large class of, and perhaps for all finite range interacting hamiltonians. Work on this subject is in progress. We also showed [5] that, in fact, a power law bound is sufficient to exclude the existence of chiral fermions. In order to avoid this milder bound the hamiltonian has to be highly non-local, and Lorentz invariance, causality etc. in the continuum limit may be jeopardized. The second ingredient concerns the allowed singularities of the propagator. The analytic structure of an exponentially bounded retarded propagator allows for singularities at a finite set of critical lattice momenta denoted generalized degeneracy points for which the spectrum contains eigenstates of vanishing energy. In order to control these singularities we have to make some reasonable assumptions on the c o n t i n u u m limit of the model with the gauge interactions switched off. This limit is easy to analyze, since we want it to describe massless fermions which are essentially non-interacting. More specifically, the only interactions which are allowed in the scaling region are non-renormalizable ones. With this information on the low energy limit, it follows that even at generalized degeneracy points our effective hamiltonian 7~- 1(/7, w = 0) has a continuous first derivative. The only way 7~-1 (iff,w = 0) may not satisfy all the assumptions of the NN theorem is if there are zeros in the propagator. This may typically happen as a result of a bad choice of interpolating fields. Should this occur, we have to return to the starting point and look for a different set of interpolating fields which is free of propagator's zeros. Finally we should consider the possibility that a zero-free propagator cannot be found. Under these circumstances the spectrum can be chi-
ral. However, propagator's zeros always reappear as poles in the vertex function. In a very general framework it has been shown [6] that they contribute to the vacuum polarization as ghosts, thus rendering the coupling to gauge fields inconsistent. We comment that condition (d) in the abstract lumps together the assumptions that the propagator is zero-free, and that the symmetry is not spontaneously broken. In conclusion, generalizing the NN theorem can be a fruitful line of investigation. Such a generalization can lead to a deeper understanding of the physical reasons for species doubling, and can therefore be instrumental in looking for new models that may succeed were all existing models have failed so far. We believe that the work reported above represents an important progress on these fronts. The potential implication of the present work to the recent proposal of D. Kaplan [7] is the following. If a domain wall model can be accurately approximated by one where the extra direction is finite, than this model cannot have a chiral spectrum. Indeed, the only concrete proposal [8] not yet ruled out, does not seem to be describable in terms of a local lagrangian theory, and it remains to be seen whether it has a consistent continuum limit. REFERENCES
1. L.H. Karsten and J. Smit, Nucl. Phys. B 1 8 3 (1981) 103. 2. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 1 8 5 (1981) 20, E r r a t u m Nucl. Phys. B 1 9 5 (1982) 541; Nucl. Phys. B 1 9 3 (1981) 173. 3. J. Smit, Acta. Phys. Pol. B 1 7 (1986) 531. P. Swift, Phys. Lett. B 1 4 5 (1984) 256. E. Eichten and J. Preskill, Nucl. Phys. B 2 6 8 (1986) 179. 4. Y. Shamir, Phys. Rev. Lett. 71 (1993) 2691. 5. Y. Shamir, preprint WIS-93/57/YUL-PH. 6. M. Campostrini, G. Curci and A. Pelissetto, Phys. Lett. B 1 9 3 (1987) 279. A. Pelissetto, Ann. Phys. 182 (1988) 177. 7. D.B. Kaplan, Phys. Lett. B 2 8 8 (1992) 342. 8. R. Narayanan and H. Neuberger, preprints RU-93-25 and RU-93-34.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 590-592 North-Holland
Constraints on the existence of chiral fermions in interacting lattice theories Yigal Shamir ~* aDepartment of Physics Weizmann Institute of Science, Rehovot 76100, ISRAEL Any interacting theory defined on a regular lattice is shown to have a vector-like spectrum if the following conditions are satisfied: (a) hermiticity, (b) locality, (c) relativistic continuum limit without massless bosons, and (d) pole-free effective vertex functions for conserved currents. The proof exploits the zero frequency inverse retarded propagator of an appropriate set of interpolating fields as an effective quadratic hamiltonian, to which the Nielsen-Ninomiya theorem is applied.
1. I N T R O D U C T I O N The last year has seen a renewed interest in the long standing problem of constructing a lattice chiral gauge theory. The well known stumbling block is the doubling problem. In the continuum, currents pertaining to classical symmetries m a y develop anomalies at the q u a n t u m level. On the other hand, on the lattice a current that corresponds to an exact s y m m e t r y of the action is always conserved. The appearance of doublers resolves the conflict between the continuum and lattice theories when b o t h the continuum action and its latticized version have axial symmetries [1], albeit at the price of making the continuum limit of the lattice theory different from the initial continuum theory. Alternatively, the presence of doublers can be attributed to the fact that m o m e n t u m space for a regular lattice (the Brillouin zone) is periodic. This line of argument was studied in detail by Nielsen and Ninomiya in a famous No-Go theorem [2]. Given a free hamiltonian on a regular lattice, Nielsen and Ninomiya showed that the massless fermion spectrum is always vector-like with respect to to full set of compact global symmetries, provided the hamiltonian is sufficiently local so that the dispersion relation has a contin*This research was supported in part by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities, and by a grant from the United States - Israel Binational Science Foundation.
uous first derivative. The Nielsen-Ninomiya (NN) theorem is mathematically rigorous, but by its very nature it cannot alone answer the question "can we define a regular lattice model whose continuum limit will be a consistent chiral gauge theory?" One obvious reason is that, while hermiticity, locality etc. are needed for consistency in the continuum limit, the NN theorem provides little information as to whether limited violations of these properties, which vanish when the lattice cutoff tends to zero, can allow for the construction of a chiral gauge theory. Another limitation is that the NN theorem is formulated for free lattice hamiltonians. It therefore does not apply to models [3] in which one tries to decouple the doublers dynamically, i.e. to achieve a chiral massless spectrum by introducing additional, strong interactions which lead to a non-trivial phase diagram. Clearly, only in the perturbative region of the phase diagram can the spectrum be reliably determined by switching off all coupling constants. Our main goal in the present work is to show that the NN theorem has a natural generalization to lattice theories with some additional non-gauge interactions. The m a j o r progress is in relating the absence of chiral fermions to locality properties of the fermionic two point function. The work is still incomplete because a detailed discussion of which lattice models satisfy our locality assumptions re-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00346-W
Y. Shamir / Constraints on the existence of chiral fermions in interacting lattice theories
mains to be given. We believe that eventually this may turn out to include all lattice models with a short range hamiltonian. In the limited room allocated to each contribution in these proceedings it is impossible to give a thorough presentation of our results. What follows is an informal report on our main results [4,5], with emphasis on the physical justification for each assumption, as well as an account on questions left for future research. 2. A N O - G O
THEOREM
The setup for the theorem is an interacting hamiltonian defined on a regular space lattice. We assume a positive norm Hilbert space, thus we do not address the Rome approach which is based on gauge fixing. Any extension of the NN theorem to lattice models with additional non-gauge interactions, automatically extends its scope to include also lattice models whose action is not gauge invariant. Practically any lattice model with a gauge variant action is mathematically equal to a model with a gauge invariant action which includes an additional field, a unitary (frozen) Higgs field. The frozen Higgs has no quadratic action, and its interactions are completely determined by the gauge variant part of the original action. The strength of the Higgs couplings is typically of order unity. In what follows we assume that the action is gauge invariant, but in view of the above comments this by itself does not lead to any loss of generality. Gauge variant models may turn out to be qualitatively different only if the integration over the frozen Higgs field changes substantially the canonical structure of the second quantized fermion fields. In order to determine the fermion content of a gauge invariant theory we will switch off the gauge coupling. The physical justification for this procedure is as follows. When approaching the continuum limit in an asymptotically free gauge theory, there has to exist an intermediate scaling region in which the gauge coupling is still small, and physics is correctly described by continuum perturbation theory. In this region, apart from the common strength of the gauge coupling,
591
the vertex describing the gloun interaction with a given fermion is completely determined once the fermion is assigned to a specific representation of the global symmetry, which is left over when the gauge coupling is switched off. With the gauge coupling switched off, what we have at hand is typically an interacting model containing only fermions and (possibly) scalar fields. Our goal is to determine the massless fermion spectrum as one approaches some phase boundary from within a s y m m e t r i c phase. We do not address directly what happens as a critical line is approached from within a broken phase. There is, however, extremely convincing evidence that if the doublers' mass arises from symmetry breaking, then these particles cannot be made very heavy and they remain in the physical spectrum. Our task now reduces to determining what massless fermions (if any) are there in a given complex representation of some unbroken global symmetry. Our strategy will be as follows. We first have to pick a set of fermion operators (which in general may or may not be elementary) that create all the massless fermions in that representation out of the vacuum. These will be called the interpolating fields. Having decided on the set of interpolating fields, we will compute a specific two point function - their retarded propagator, denoted T¢(~, t), and its Fourier transform 7~(iff, a~). The latter is chosen (rather than the commonly used time ordered propagator) because of its good behaviour at space-like separations. We will then be able to determine the number of positive helicity ("right handed") and negative helicity ("left handed") fermions in this representation from the singularities of the propagator. Moreover, it follows from the existence of a spectral representation that 7~-1(/7, w = 0) is hermitian. With suitable assumptions, the effective hamiltonian 7~- 1(/7, w = 0) will satisfy all the conditions of the NN theorem, and so the spectrum will be vectorlike. In order to show that 7~-1(/7,¢0 = 0) satisfies all the conditions of the NN thereom two major ingredients are needed. The first one concerns the analytic structure of the retarded propagator for a general lattice momentum. This, in turn, is
592
Y. Shamir / Constraints on the existence of chiral fermions in interacting lattice theories
determined by the behaviour of TC(~,t) at large space-like separations. On the lattice, (anti)commutators cannot vanish identically outside the light cone, but they have to tend to zero very rapidly if the hamiltonian is local. Using the "Edge of the Wedge" theorem, we show in ref. [4] that an exponentially decreasing anticommutator gives rise to an analytic 7~(i~,w). It is easy to show that free short range hamiltonians always give rise to exponentially bounded anticommutators. We anticipate a similar behaviour for a very large class of, and perhaps for all finite range interacting hamiltonians. Work on this subject is in progress. We also showed [5] that, in fact, a power law bound is sufficient to exclude the existence of chiral fermions. In order to avoid this milder bound the hamiltonian has to be highly non-local, and Lorentz invariance, causality etc. in the continuum limit may be jeopardized. The second ingredient concerns the allowed singularities of the propagator. The analytic structure of an exponentially bounded retarded propagator allows for singularities at a finite set of critical lattice momenta denoted generalized degeneracy points for which the spectrum contains eigenstates of vanishing energy. In order to control these singularities we have to make some reasonable assumptions on the c o n t i n u u m limit of the model with the gauge interactions switched off. This limit is easy to analyze, since we want it to describe massless fermions which are essentially non-interacting. More specifically, the only interactions which are allowed in the scaling region are non-renormalizable ones. With this information on the low energy limit, it follows that even at generalized degeneracy points our effective hamiltonian 7~- 1(/7, w = 0) has a continuous first derivative. The only way 7~-1 (iff,w = 0) may not satisfy all the assumptions of the NN theorem is if there are zeros in the propagator. This may typically happen as a result of a bad choice of interpolating fields. Should this occur, we have to return to the starting point and look for a different set of interpolating fields which is free of propagator's zeros. Finally we should consider the possibility that a zero-free propagator cannot be found. Under these circumstances the spectrum can be chi-
ral. However, propagator's zeros always reappear as poles in the vertex function. In a very general framework it has been shown [6] that they contribute to the vacuum polarization as ghosts, thus rendering the coupling to gauge fields inconsistent. We comment that condition (d) in the abstract lumps together the assumptions that the propagator is zero-free, and that the symmetry is not spontaneously broken. In conclusion, generalizing the NN theorem can be a fruitful line of investigation. Such a generalization can lead to a deeper understanding of the physical reasons for species doubling, and can therefore be instrumental in looking for new models that may succeed were all existing models have failed so far. We believe that the work reported above represents an important progress on these fronts. The potential implication of the present work to the recent proposal of D. Kaplan [7] is the following. If a domain wall model can be accurately approximated by one where the extra direction is finite, than this model cannot have a chiral spectrum. Indeed, the only concrete proposal [8] not yet ruled out, does not seem to be describable in terms of a local lagrangian theory, and it remains to be seen whether it has a consistent continuum limit. REFERENCES
1. L.H. Karsten and J. Smit, Nucl. Phys. B 1 8 3 (1981) 103. 2. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 1 8 5 (1981) 20, E r r a t u m Nucl. Phys. B 1 9 5 (1982) 541; Nucl. Phys. B 1 9 3 (1981) 173. 3. J. Smit, Acta. Phys. Pol. B 1 7 (1986) 531. P. Swift, Phys. Lett. B 1 4 5 (1984) 256. E. Eichten and J. Preskill, Nucl. Phys. B 2 6 8 (1986) 179. 4. Y. Shamir, Phys. Rev. Lett. 71 (1993) 2691. 5. Y. Shamir, preprint WIS-93/57/YUL-PH. 6. M. Campostrini, G. Curci and A. Pelissetto, Phys. Lett. B 1 9 3 (1987) 279. A. Pelissetto, Ann. Phys. 182 (1988) 177. 7. D.B. Kaplan, Phys. Lett. B 2 8 8 (1992) 342. 8. R. Narayanan and H. Neuberger, preprints RU-93-25 and RU-93-34.