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The gauge invariant Nambu Jona-Lasinio model
SUP~EMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 53 (1997) 709-711
The gauge invariant Nambu Jona-Lasinio model S. Kim, ~ J. B. Kogut b and M.-P. Lombardo ~ ~Center for Theoretical Physics, Seoul National University, Seoul 151-742, Korea bphysics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801-30 CHLRZ c/o KFA, D-52425 Jiilich and DESY, D-22603 Hamburg, Germany By coupling the NJL model to gauge field it is possible to study lattice QED with a topologically trivial vacuum and to address the issue of its logarithmic triviality.
The problem in QED is to determine the critical behavior of the theory at its chiral transition and thereby determine if the theory is trivial (mean field behavior supplemented by scale breaking logarithms) or not. Unlike 'textbook' QED, previous formulations of lattice QED all have magnetic monopole excitations which percolate or condense at the theory's chiral transition [1]. Since magnetic monopoles can cause chiral symmetry breaking, these theory's are qualitatively distinct from conventional QED. Happily, by adding a four fermi interaction to the theory, the chiral and monopole transition points become distinct and the chiral transition can be studied in the theory with a topologically trivial vacuum. This is the first lattice study of QED which can address the conventional triviality question with the correct degrees of freedom only. (Of course, in the long run, theories with fundamental monopoles may be more relevant than those without!) Consider non-compact lattice QED supplemented by a four fermion term G2(¢¢) 2 , i.e. a Nambu Jona Lasinio (NJL) model with discrete chiral symmetry coupled to a U(1) gauge field. The Lagrangian for the continuum model is,
L = ¢(iTO - eTA - m)¢ + l G2((b¢)~ - ~F2 (1) The Lagrangian includes the discrete (Z2) chirally invariant four-fermion interaction (i.e. invariance under the transformation ¢ --+ 75¢), and the continuous chirally invariant electromagnetic 0920-5632(97)/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00761 -X
interaction (¢ --+ e i e ~ ¢ , where r is the appropriate flavor matrix). Both symmetries are broken by the mass term m ¢ ¢ . To formulate the model on the lattice it is convenient to add to the Lagrangian an irrelevant term--~((¢¢)o 2 , where o" is an auxil~-) iary scalar field, so that the contact four-fermion term formally disappears. The model is then discretized by using staggered fermions. The lattice Action reads:
S
=
Z Z (l,j(x)(Mxy + D::y)~,j(y) x,y
(2)
j
NL
1 T,#,P
where 1 M~y = (m ÷ ]-~ Z
~(x))~x~
(3)
1 #
In this formulation [2] a is defined on the sites of the dual lattice ~ , the symbol < x, ~ > denotes the set of the 16 lattice sites surrounding the direct site x, and the rest of the notation is standard. The Action (2) has an overall discrete symmetry under:
¢(~)
-~ (-1V1+~2+~3+~%(~)
(5)
710 ~(X)
S. Kim et al./Nuclear Physics B (Proc. Suppl.) 53 (1997) 709-711
-"}
- - ~ ( X ) ( - - 1 ) a l']-x 2 +x'3"}'~'4
(6)
-~
-z.
(7)
where ( - 1 ) xl+=2+xa+a4 is the lattice representation of 75. The resulting lattice action can be simulated directly in the chiral limit because the auxiliary scalar field ~r (essentially the chiral condensate < ¢ ¢ >) acts as a dynamical mass term for the quarks insuring that the inversion of the Dirac operator will be successful and very fast. In many cases, even if the four fermi coupling is small, the cr field will develop an expectation value due to other interactions in the model, and this will stabilize the algorithm and make the chiral limit directly accessible. In effect, this approach allows the dynamically generated 'constituent' quark mass to tune the algorithm that generated it and guide its convergence. This new approach is far more efficient and physical than the traditional lattice algorithm in which the chiral limit is singular and a nonzero bare quark mass m is a necessity. Work on lattice QCD along these lines has been presented at this conference[3], and we refer to that contribution for details on the improvement of the performance. In the exploratory study we are discussing here we used a slightly simplified form of the Hybrid Monte Carlo algorithm. Recall that the pure gauge theory piece of the action can be simulated by placing pseudo-fermion fields on even or odd sites of the lattice [4], [5]. This trick applies without error to the NJL model only at large N where the fluctuations in the ~r field are suppressed. We apply it here so that the four species model can be simulated exactly by the Hybrid Monte Carlo algorithm [4]. Since we are mostly concerned with the issue of whether gauge interactions can make the theory nontrivial, we believe that our simplification is harmless. The important point is that the lattice action and algorithm have the symmetries and degrees of freedom of interest, since we are ultimately only interested in its continuum limit and universal features. Neverthless, we are following up this study with extensive simulations which avoid the use of the even odd partitioning. Interesting limiting cases of the above Action are the zero gauge coupling model (e = 0), and
the pure QED (G = 0) limit, whose chiral phase transition is near j3~ = 1/e 2 = .204 [7], [8] for four flavors. The nature of the e = 0 limit is not known a priori, since the radius of convergence of the perturbative QED series is zero (see also the discussion in [9]). So there is no guarantee that the limit e = 0 gives the naive NJL model, and, in turn, no natural candidate for the universality class of the transition along the critical line of the gauged model. Any effective Action with the appropriate symmetry is in principle adequate (including of course the NJL effective Action itself), and we shall contrast our data with the prediction coming from various hypothesis. We scanned the 2 dimensional parameter space using the Hybrid Monte Carlo algorithm and measured the chiral condensate and monopole susceptibility as a function of ~3~ and G ~. We found that as we increased G 2 and moved off the G = 0 axis, the peak of the monopole susceptibility shifted only slightly from /3e = .204 to fie = .244. By contrast the chiral transition points shifted to larger fie and became distinct from the monopole percolation points as soon as G became nonzero. The movement of the monopole percolation peak in the (fie,G 2) plane can be understood by noticing that ~r in the Action plays the role of a site dependent mass term (Eq. 3). When the fluctuations of the g field are not important, as is the case at large G, the gauge field dynamics becomes equivalent to QED with a bare mass given by the constant cr value. So, as G 2 increases and c~ grows, the theory approaches the m --+ c~ limit of QED, i.e. quenched QED, which has a percolation transition at 3~ = .244 [10]. This result was confirmed quantitatively in the simulation. In conclusion, the chiral transition line extends from (.204, 0) to (oo, Gc(e = 0)), while the monopole percolation line extends from (.204, 0) to (.244, oo). The two transitions only coincide at the 'pure' QED point, G = 0. Thus, the gauged NJL model makes it possible to study the triviality of conventional QED (without monopoles!) along its critical line. Our numerical strategy is mainly based on the analysis of the sigma field, and of its longitudinal susceptibility. Recall that the singular piece of the longitudinal susceptibility )C diverges at the
S. Kim et al./Nuclear Physics B (Ptvc. Suppl.) 53 (1997) 709-711
critical point/3c as X+ = c+ltl -~, t =_ (j3~ - j 3 ) / ~ c , as t approaches zero from above in the broken phase, and as )~_ = c_ltl -'~ in the symmetric phase [11]. In mean field theory the universal amplitude ratio c _ / c + is 2.0 and the critical index "~ is unity. In logarithmically trivial models 7 remains unity, but the amplitudes c+ and c_ develop weak logarithmic dependences [11]. In the Z2 Ising or two component ¢4 model, c _ / c + = 2 + 4 / l n ( ~ ) , while in the Z2 NJL model, solved at large N , c _ / c + = 2 - 2 / l n ( ~ ) [6]. Fits to our data (at fixed G 2 = .85) predicted 7 = 1.0(2) with a confidence level of .49 in the broken plase and 3' -: 1.2(1) with confidence level .83 in the symmetric phase. Constrained linear fits produced the amplitude ratio c _ / c + = 3.3(3). This result is consistent, for the range of Itl shown in the figure, with the ¢4 behavior, but not with the NJL behavior. Given a relatively precise determination of the critical point /3c = .2085(5) at fixed four fermi coupling G 2 = .83, we measured the chiral condensate's response to bare fermion masses m ranging from .001 to .01 (lattice units). In mean field theory we expect the singular response o" c< m 1/'~ with the critical index (f - 3.0. In logarithmically trivial theories this scaling law is modified by logarithms : in ¢4 one expects m = a a 3 / l n ( b / ( r ) , while in the NJL model the result is m = act31n(b/~r) [6]. Again, we found that the NJL model can not accommodate the data. Finally, we considered the Equation of State in the broken phase in the chiral limit. In mean field theory one predicts 13c - j3e = ac rl/~"°g, with the critical index fl, nag = 1/2. This scaling law is also modified by logarithms: in the two component ¢4 model,/3c -/3~ = a a 2 / l n ( b / c r ) [11], and in the Z2 NJL model,/3c-ere = a c r2 ln(b/c~) [6]. Once more, we find consistency with ¢4 behavior and predict the same critical coupling/3c, within errors, found in the susceptibility calculations. Our results would give a nicely consistent picture of the triviality of QED with a Z2 chiral group. All of the data are well fit by the two component ¢4 model, rather than the large N Z2 NJL model which serves as scaffolding to the
711
gauge interactions of interest. This result should be confirmed by other algorithms. We plan to simulate the four species model without even/odd partitioning using the approximate hybrid molecular dynamics algorithm so that large N simplifications wilt not be made. This work was partially supported by NSF under grant NSF-PHY92-00148. The simulations were done on the CRAY C-90's at PSC and NERSC. The authors thank A. Kocid for calculations of the susceptibility ratios and S. Hands for conversations. REFERENCES 1. J.B. Kogut and K.C. Wang, Phys.Rev. D53, 1513 (1996). 2. S. Hands, A. Kocid and J.B. Kogut, Ann. of Phys. 224, 29 (1993) 29, and references herein. 3. J.B. Kogut and D.K. Sinclair, this volume. 4. S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195, 216 (1987). 5. S. Duane and J.B. Kogut, Phys. Rev, Lett. 55, 2774 (1985). S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D38,2245 (1988). 6. S. Kim, A. Kocid and J.B. Kogut, Nucl. Phys. B429, 407 (1994). 7. E. Dagotto, A. Koci5 and J.B. Kogut, Phys. Rev. Lett. 60,772 (1988). 8. V. Azcoiti , G. Di Carlo , A. Galante , A.F. Grillo, V. Laliena ,and C.E. Piedrafita, Phys. Lett. B353, 279 (1995), and references herein. 9. V. Azcoiti, this volume. 10. S. Hands and R. Wensley, Phys. Rev. Lett. 63, 2169 (1989). 11. C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press, 1989.)
Nuclear Physics B (Proc. Suppl.) 53 (1997) 709-711
The gauge invariant Nambu Jona-Lasinio model S. Kim, ~ J. B. Kogut b and M.-P. Lombardo ~ ~Center for Theoretical Physics, Seoul National University, Seoul 151-742, Korea bphysics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801-30 CHLRZ c/o KFA, D-52425 Jiilich and DESY, D-22603 Hamburg, Germany By coupling the NJL model to gauge field it is possible to study lattice QED with a topologically trivial vacuum and to address the issue of its logarithmic triviality.
The problem in QED is to determine the critical behavior of the theory at its chiral transition and thereby determine if the theory is trivial (mean field behavior supplemented by scale breaking logarithms) or not. Unlike 'textbook' QED, previous formulations of lattice QED all have magnetic monopole excitations which percolate or condense at the theory's chiral transition [1]. Since magnetic monopoles can cause chiral symmetry breaking, these theory's are qualitatively distinct from conventional QED. Happily, by adding a four fermi interaction to the theory, the chiral and monopole transition points become distinct and the chiral transition can be studied in the theory with a topologically trivial vacuum. This is the first lattice study of QED which can address the conventional triviality question with the correct degrees of freedom only. (Of course, in the long run, theories with fundamental monopoles may be more relevant than those without!) Consider non-compact lattice QED supplemented by a four fermion term G2(¢¢) 2 , i.e. a Nambu Jona Lasinio (NJL) model with discrete chiral symmetry coupled to a U(1) gauge field. The Lagrangian for the continuum model is,
L = ¢(iTO - eTA - m)¢ + l G2((b¢)~ - ~F2 (1) The Lagrangian includes the discrete (Z2) chirally invariant four-fermion interaction (i.e. invariance under the transformation ¢ --+ 75¢), and the continuous chirally invariant electromagnetic 0920-5632(97)/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00761 -X
interaction (¢ --+ e i e ~ ¢ , where r is the appropriate flavor matrix). Both symmetries are broken by the mass term m ¢ ¢ . To formulate the model on the lattice it is convenient to add to the Lagrangian an irrelevant term--~((¢¢)o 2 , where o" is an auxil~-) iary scalar field, so that the contact four-fermion term formally disappears. The model is then discretized by using staggered fermions. The lattice Action reads:
S
=
Z Z (l,j(x)(Mxy + D::y)~,j(y) x,y
(2)
j
NL
1 T,#,P
where 1 M~y = (m ÷ ]-~ Z
~(x))~x~
(3)
1 #
In this formulation [2] a is defined on the sites of the dual lattice ~ , the symbol < x, ~ > denotes the set of the 16 lattice sites surrounding the direct site x, and the rest of the notation is standard. The Action (2) has an overall discrete symmetry under:
¢(~)
-~ (-1V1+~2+~3+~%(~)
(5)
710 ~(X)
S. Kim et al./Nuclear Physics B (Proc. Suppl.) 53 (1997) 709-711
-"}
- - ~ ( X ) ( - - 1 ) a l']-x 2 +x'3"}'~'4
(6)
-~
-z.
(7)
where ( - 1 ) xl+=2+xa+a4 is the lattice representation of 75. The resulting lattice action can be simulated directly in the chiral limit because the auxiliary scalar field ~r (essentially the chiral condensate < ¢ ¢ >) acts as a dynamical mass term for the quarks insuring that the inversion of the Dirac operator will be successful and very fast. In many cases, even if the four fermi coupling is small, the cr field will develop an expectation value due to other interactions in the model, and this will stabilize the algorithm and make the chiral limit directly accessible. In effect, this approach allows the dynamically generated 'constituent' quark mass to tune the algorithm that generated it and guide its convergence. This new approach is far more efficient and physical than the traditional lattice algorithm in which the chiral limit is singular and a nonzero bare quark mass m is a necessity. Work on lattice QCD along these lines has been presented at this conference[3], and we refer to that contribution for details on the improvement of the performance. In the exploratory study we are discussing here we used a slightly simplified form of the Hybrid Monte Carlo algorithm. Recall that the pure gauge theory piece of the action can be simulated by placing pseudo-fermion fields on even or odd sites of the lattice [4], [5]. This trick applies without error to the NJL model only at large N where the fluctuations in the ~r field are suppressed. We apply it here so that the four species model can be simulated exactly by the Hybrid Monte Carlo algorithm [4]. Since we are mostly concerned with the issue of whether gauge interactions can make the theory nontrivial, we believe that our simplification is harmless. The important point is that the lattice action and algorithm have the symmetries and degrees of freedom of interest, since we are ultimately only interested in its continuum limit and universal features. Neverthless, we are following up this study with extensive simulations which avoid the use of the even odd partitioning. Interesting limiting cases of the above Action are the zero gauge coupling model (e = 0), and
the pure QED (G = 0) limit, whose chiral phase transition is near j3~ = 1/e 2 = .204 [7], [8] for four flavors. The nature of the e = 0 limit is not known a priori, since the radius of convergence of the perturbative QED series is zero (see also the discussion in [9]). So there is no guarantee that the limit e = 0 gives the naive NJL model, and, in turn, no natural candidate for the universality class of the transition along the critical line of the gauged model. Any effective Action with the appropriate symmetry is in principle adequate (including of course the NJL effective Action itself), and we shall contrast our data with the prediction coming from various hypothesis. We scanned the 2 dimensional parameter space using the Hybrid Monte Carlo algorithm and measured the chiral condensate and monopole susceptibility as a function of ~3~ and G ~. We found that as we increased G 2 and moved off the G = 0 axis, the peak of the monopole susceptibility shifted only slightly from /3e = .204 to fie = .244. By contrast the chiral transition points shifted to larger fie and became distinct from the monopole percolation points as soon as G became nonzero. The movement of the monopole percolation peak in the (fie,G 2) plane can be understood by noticing that ~r in the Action plays the role of a site dependent mass term (Eq. 3). When the fluctuations of the g field are not important, as is the case at large G, the gauge field dynamics becomes equivalent to QED with a bare mass given by the constant cr value. So, as G 2 increases and c~ grows, the theory approaches the m --+ c~ limit of QED, i.e. quenched QED, which has a percolation transition at 3~ = .244 [10]. This result was confirmed quantitatively in the simulation. In conclusion, the chiral transition line extends from (.204, 0) to (oo, Gc(e = 0)), while the monopole percolation line extends from (.204, 0) to (.244, oo). The two transitions only coincide at the 'pure' QED point, G = 0. Thus, the gauged NJL model makes it possible to study the triviality of conventional QED (without monopoles!) along its critical line. Our numerical strategy is mainly based on the analysis of the sigma field, and of its longitudinal susceptibility. Recall that the singular piece of the longitudinal susceptibility )C diverges at the
S. Kim et al./Nuclear Physics B (Ptvc. Suppl.) 53 (1997) 709-711
critical point/3c as X+ = c+ltl -~, t =_ (j3~ - j 3 ) / ~ c , as t approaches zero from above in the broken phase, and as )~_ = c_ltl -'~ in the symmetric phase [11]. In mean field theory the universal amplitude ratio c _ / c + is 2.0 and the critical index "~ is unity. In logarithmically trivial models 7 remains unity, but the amplitudes c+ and c_ develop weak logarithmic dependences [11]. In the Z2 Ising or two component ¢4 model, c _ / c + = 2 + 4 / l n ( ~ ) , while in the Z2 NJL model, solved at large N , c _ / c + = 2 - 2 / l n ( ~ ) [6]. Fits to our data (at fixed G 2 = .85) predicted 7 = 1.0(2) with a confidence level of .49 in the broken plase and 3' -: 1.2(1) with confidence level .83 in the symmetric phase. Constrained linear fits produced the amplitude ratio c _ / c + = 3.3(3). This result is consistent, for the range of Itl shown in the figure, with the ¢4 behavior, but not with the NJL behavior. Given a relatively precise determination of the critical point /3c = .2085(5) at fixed four fermi coupling G 2 = .83, we measured the chiral condensate's response to bare fermion masses m ranging from .001 to .01 (lattice units). In mean field theory we expect the singular response o" c< m 1/'~ with the critical index (f - 3.0. In logarithmically trivial theories this scaling law is modified by logarithms : in ¢4 one expects m = a a 3 / l n ( b / ( r ) , while in the NJL model the result is m = act31n(b/~r) [6]. Again, we found that the NJL model can not accommodate the data. Finally, we considered the Equation of State in the broken phase in the chiral limit. In mean field theory one predicts 13c - j3e = ac rl/~"°g, with the critical index fl, nag = 1/2. This scaling law is also modified by logarithms: in the two component ¢4 model,/3c -/3~ = a a 2 / l n ( b / c r ) [11], and in the Z2 NJL model,/3c-ere = a c r2 ln(b/c~) [6]. Once more, we find consistency with ¢4 behavior and predict the same critical coupling/3c, within errors, found in the susceptibility calculations. Our results would give a nicely consistent picture of the triviality of QED with a Z2 chiral group. All of the data are well fit by the two component ¢4 model, rather than the large N Z2 NJL model which serves as scaffolding to the
711
gauge interactions of interest. This result should be confirmed by other algorithms. We plan to simulate the four species model without even/odd partitioning using the approximate hybrid molecular dynamics algorithm so that large N simplifications wilt not be made. This work was partially supported by NSF under grant NSF-PHY92-00148. The simulations were done on the CRAY C-90's at PSC and NERSC. The authors thank A. Kocid for calculations of the susceptibility ratios and S. Hands for conversations. REFERENCES 1. J.B. Kogut and K.C. Wang, Phys.Rev. D53, 1513 (1996). 2. S. Hands, A. Kocid and J.B. Kogut, Ann. of Phys. 224, 29 (1993) 29, and references herein. 3. J.B. Kogut and D.K. Sinclair, this volume. 4. S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195, 216 (1987). 5. S. Duane and J.B. Kogut, Phys. Rev, Lett. 55, 2774 (1985). S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D38,2245 (1988). 6. S. Kim, A. Kocid and J.B. Kogut, Nucl. Phys. B429, 407 (1994). 7. E. Dagotto, A. Koci5 and J.B. Kogut, Phys. Rev. Lett. 60,772 (1988). 8. V. Azcoiti , G. Di Carlo , A. Galante , A.F. Grillo, V. Laliena ,and C.E. Piedrafita, Phys. Lett. B353, 279 (1995), and references herein. 9. V. Azcoiti, this volume. 10. S. Hands and R. Wensley, Phys. Rev. Lett. 63, 2169 (1989). 11. C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press, 1989.)