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Gauge-theory and chiral theory on a momentum lattice
Nuclear Physics B (Proc. Suppl.) 20 (1991) 755-757 North-Holland
755
GAUGE-THEORY AND CHIP~.L THEORY ON A MOMENTUM LATTICE
D. BdrubC, H. KrOgeP, R. Lafrance ~, S. Lantagne~, L. Marleau ~, K.J.M. Moriarty 2 and J. Potvinz3. D6partement de Physique, Universit6 Lava/, Qu6bec, P.Q., GIK 7P4, Canada 2 Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, N.Sc. B3H 3J5, Canada Department of Physics, Boston University, Boston, MA 02215, USA We discuss a proposal how to put gauge theories arid chiral theo6es on a momentum lattice in a noncompact formulation. We present numerical results for m R and Z R of ~ e ~'-model and for the action of U(1) and SU(2) gauge theory. 1. MOTIVATION There are serval reasons to consider lattice field theory formulated on a momentum lattice. (a) Dynamical observables like the S-matrix, or decay amplitudes are expressed in k-~ace. (b) Physical obsev;ab!es lake mR, ZR, gR, etc. can be defined at k=0. It may be that small k-lattices are sufficient to compute those observables. (c) Foarier acceleration at the critical point can be implemented in k-space straight forwardly. There are reasons to implement gauge theories in particular on a momentum lattice. (tl) The k-lattice formulation is non-compact. It allows an independent check o f confinement. (b) There is no fermion doubling. (c) The classical action conserves gauge symmetry and chiral symmetry in a weak sense, which opens the avenue to treat chiral fermions on a latdce. However, there is a draw back, of cause. The formulation is non-local, which means an increased numerical effort. By using fast Fourier transformation, the effort to compute the action (without fermions) goes like V log V on the klattice, compared to V on the k-lattice. In order to demonstrate the feasibility of doing numerical simulations on the k-lattice, we consider the one-component ~4-model in the symmetric phase. We have computed m R (Fig. 1) and Z R (Fig. 2) and compared it with the analytical results by Liischer and Weisz [1]. One observes that the Langevin algorithm on the k-lattice workes well, as long as the bare parameters Zo, m, ~, g. are positive. However, when approaching the critical point, mo2<0 which makes the results less reliable at k=0.
k [J:Itrice -
i
Z
e2 006 ~
O1 "E"
Fig. i Renormalized mass rn~ as a fune6on of v---(K~- -~)/r~.. The full line gives the scaling behavior, fated to the dam of ReL [I]. 2. NON-COMPACT FORMULATION OF GAUGE THEORY. A proposition -has been made in Ref. [2-5] how m pat gauge theories in a non-compact formulation on a k-lattice. It has the following prope~es: (a) Existence of a symmetry group, Co) Weak ga=ge end chiral invariance of the chiral action. (c) Absence of fermion doubling. (d) Correct continuum limit of quantum expectation values from lattice perturbation theory. Let us discuss the topic (d). Firstly, we have considered in QED the vacuum polarisation [4], starting from k-lattice perturbation theory and then going to the continuum limit. We have considered the diagram to 1-1oop order. The intrimsic k-lattice regulator is a cut-off Ik~l < A.
0920-5632/91/$3.50 © Elsevier Sdence Publishers B.V. (North-Holland)
D. Brrub6 et al. ~Gauge-theory and chiral theory
756
ZR
influe,~ces strongly the values of the scattering matrix.
5ci 4C
50
k- Loilice
70°
• L~Jscher + Weisz 20
t
o 54 a 74
I0
0.02 0 0 4
006
008
010
-c
Fig. 2 Wave function renormalization Z~. However, this is not sufficient, as such'a cut-off in ordinary continuum perturbation theory is known to )field a result which is not gauge invariant. If we employ in addition a Pauli-Villars regularisation and then let A --~ ,,% we obtain the standard gauge invariant result [4]. Similarly, we have computed the chiral triangle anomaly and also find the standard result [5].
3OO 2O0 ooo
i --
~&o
2o~o
Number
of
~--~ooo noises
Fig. 3 Action Re(F~(k) F~(-k)) for U(1)-model in 13=-4. The dashed live gives the exact solution.
NUMERICAL RESULTS We have done some numerical simulations on the k-lattice by computing the expectation value of the action using stochastic quantizatinn and solving the Langevin equation for the U(1)-model in D=4. We have displayed in Fig. [3] Re(F~(k) F~(-k)) as a function of the number of Gaussian noises. For the SU(2)-medel in 13=-2, we have computed the action on a small lattice (32) in the weak coupling regime (13---400) and compared with stochastic perturbation theory (Fig. [4]). The results are in agreement within the statistical errors. The effects of confinement on scattering processes can also be studied using the k-lattice in order to obtain the scattering matrix for the collision of various hadron-like states such as "glueballs" and "mesons". This formulation is fully gauge invariant and uses Gauss' law to define the quantum numbers of the hadrons. One of the several interesting aspects revealed by such a study is the relation between the range of the "conf'mement" interaction and the scattering matrix itself. In the strong coupling regime of compact QED, the interaction naturally yields quark confinement. Moreover, the range between two hadron-like states depends crucially on the proper inclusion of the first few orders of perturbation theory used in computing the matrix elements of the Hamiltonian, and therefore
8~
a ~2
Number
of noises
Fig. 4 Action for SU(2) model on a 32 lattice. The dashed line corresponds to stochastic perturbation theory. ACKNOWLEDGEMENTS We would like to thank John Buchanan, Terence R.B. Donahoe and Don Cameron of the Government of Nova Scotia for their continued interest, support and encouragement and grant support; HNSX Supercomputers, Inc., for HNSX Supercomputer Fellowship Awards [Grants 90HSFA01 and 90HSFA01]; James R. Berrett, Samuel W. Adams
D. B~rub~ et al./ Gauge-theory and chiral theory
and Dana Hoffman for their conti~.uexi interest, encouragement, and support and access to the NEC SX-2 in the Houston Advanced Research Center in the Wood/ands, Texas; the Natural Sciences and Engineering Research Council of Canada [Grant Nos. NSERC A8420 and NSERC A9030] for financial support; the Atlantic Canada Opportunities Agency [AAP Project No. 2060-339, 905 and AAP Project No. 2060-343, 394] for f'maz~ial support; Pietou County Economie Development Fund [Grant 90PCEDFOI] for financial support; and the Canada/Nova Scotia Technology Transfer and Industrial Innovation Agreement [Grant Nos. 87110I, 881"F1101, 89"IT!10l, 9ffFITI01, 9ffITlI02, 90TTII03 and 90TTII04] for fur~er financial support.
757
REFERENCES [1] M. Ltisher and P. Weisz, Nuci. Phys. B 290 [FS 201 (1987) 25. [2] D. B~rut~ and H. KrOger, Phys. Lett. B238 (1990) 348, [3] D. B6rub~, H. Kr6ger, 1L Lafi"ance and L Marleau, Universit~ Laval, preprint LAVALPHY-90/1. [4I D. B~'ub~, H. Kr0ger, R. Lafi'ance and L. Marleau, Universit~ Laval, preprint LAVA[,-
PHY-9014. [5] D. B~rul~ I~ Ka0ger, R. Lafrance and L Marleau, Universit6 Lavat, preprint LAVALPHY-90/5o
755
GAUGE-THEORY AND CHIP~.L THEORY ON A MOMENTUM LATTICE
D. BdrubC, H. KrOgeP, R. Lafrance ~, S. Lantagne~, L. Marleau ~, K.J.M. Moriarty 2 and J. Potvinz3. D6partement de Physique, Universit6 Lava/, Qu6bec, P.Q., GIK 7P4, Canada 2 Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, N.Sc. B3H 3J5, Canada Department of Physics, Boston University, Boston, MA 02215, USA We discuss a proposal how to put gauge theories arid chiral theo6es on a momentum lattice in a noncompact formulation. We present numerical results for m R and Z R of ~ e ~'-model and for the action of U(1) and SU(2) gauge theory. 1. MOTIVATION There are serval reasons to consider lattice field theory formulated on a momentum lattice. (a) Dynamical observables like the S-matrix, or decay amplitudes are expressed in k-~ace. (b) Physical obsev;ab!es lake mR, ZR, gR, etc. can be defined at k=0. It may be that small k-lattices are sufficient to compute those observables. (c) Foarier acceleration at the critical point can be implemented in k-space straight forwardly. There are reasons to implement gauge theories in particular on a momentum lattice. (tl) The k-lattice formulation is non-compact. It allows an independent check o f confinement. (b) There is no fermion doubling. (c) The classical action conserves gauge symmetry and chiral symmetry in a weak sense, which opens the avenue to treat chiral fermions on a latdce. However, there is a draw back, of cause. The formulation is non-local, which means an increased numerical effort. By using fast Fourier transformation, the effort to compute the action (without fermions) goes like V log V on the klattice, compared to V on the k-lattice. In order to demonstrate the feasibility of doing numerical simulations on the k-lattice, we consider the one-component ~4-model in the symmetric phase. We have computed m R (Fig. 1) and Z R (Fig. 2) and compared it with the analytical results by Liischer and Weisz [1]. One observes that the Langevin algorithm on the k-lattice workes well, as long as the bare parameters Zo, m, ~, g. are positive. However, when approaching the critical point, mo2<0 which makes the results less reliable at k=0.
k [J:Itrice -
i
Z
e2 006 ~
O1 "E"
Fig. i Renormalized mass rn~ as a fune6on of v---(K~- -~)/r~.. The full line gives the scaling behavior, fated to the dam of ReL [I]. 2. NON-COMPACT FORMULATION OF GAUGE THEORY. A proposition -has been made in Ref. [2-5] how m pat gauge theories in a non-compact formulation on a k-lattice. It has the following prope~es: (a) Existence of a symmetry group, Co) Weak ga=ge end chiral invariance of the chiral action. (c) Absence of fermion doubling. (d) Correct continuum limit of quantum expectation values from lattice perturbation theory. Let us discuss the topic (d). Firstly, we have considered in QED the vacuum polarisation [4], starting from k-lattice perturbation theory and then going to the continuum limit. We have considered the diagram to 1-1oop order. The intrimsic k-lattice regulator is a cut-off Ik~l < A.
0920-5632/91/$3.50 © Elsevier Sdence Publishers B.V. (North-Holland)
D. Brrub6 et al. ~Gauge-theory and chiral theory
756
ZR
influe,~ces strongly the values of the scattering matrix.
5ci 4C
50
k- Loilice
70°
• L~Jscher + Weisz 20
t
o 54 a 74
I0
0.02 0 0 4
006
008
010
-c
Fig. 2 Wave function renormalization Z~. However, this is not sufficient, as such'a cut-off in ordinary continuum perturbation theory is known to )field a result which is not gauge invariant. If we employ in addition a Pauli-Villars regularisation and then let A --~ ,,% we obtain the standard gauge invariant result [4]. Similarly, we have computed the chiral triangle anomaly and also find the standard result [5].
3OO 2O0 ooo
i --
~&o
2o~o
Number
of
~--~ooo noises
Fig. 3 Action Re(F~(k) F~(-k)) for U(1)-model in 13=-4. The dashed live gives the exact solution.
NUMERICAL RESULTS We have done some numerical simulations on the k-lattice by computing the expectation value of the action using stochastic quantizatinn and solving the Langevin equation for the U(1)-model in D=4. We have displayed in Fig. [3] Re(F~(k) F~(-k)) as a function of the number of Gaussian noises. For the SU(2)-medel in 13=-2, we have computed the action on a small lattice (32) in the weak coupling regime (13---400) and compared with stochastic perturbation theory (Fig. [4]). The results are in agreement within the statistical errors. The effects of confinement on scattering processes can also be studied using the k-lattice in order to obtain the scattering matrix for the collision of various hadron-like states such as "glueballs" and "mesons". This formulation is fully gauge invariant and uses Gauss' law to define the quantum numbers of the hadrons. One of the several interesting aspects revealed by such a study is the relation between the range of the "conf'mement" interaction and the scattering matrix itself. In the strong coupling regime of compact QED, the interaction naturally yields quark confinement. Moreover, the range between two hadron-like states depends crucially on the proper inclusion of the first few orders of perturbation theory used in computing the matrix elements of the Hamiltonian, and therefore
8~
a ~2
Number
of noises
Fig. 4 Action for SU(2) model on a 32 lattice. The dashed line corresponds to stochastic perturbation theory. ACKNOWLEDGEMENTS We would like to thank John Buchanan, Terence R.B. Donahoe and Don Cameron of the Government of Nova Scotia for their continued interest, support and encouragement and grant support; HNSX Supercomputers, Inc., for HNSX Supercomputer Fellowship Awards [Grants 90HSFA01 and 90HSFA01]; James R. Berrett, Samuel W. Adams
D. B~rub~ et al./ Gauge-theory and chiral theory
and Dana Hoffman for their conti~.uexi interest, encouragement, and support and access to the NEC SX-2 in the Houston Advanced Research Center in the Wood/ands, Texas; the Natural Sciences and Engineering Research Council of Canada [Grant Nos. NSERC A8420 and NSERC A9030] for financial support; the Atlantic Canada Opportunities Agency [AAP Project No. 2060-339, 905 and AAP Project No. 2060-343, 394] for f'maz~ial support; Pietou County Economie Development Fund [Grant 90PCEDFOI] for financial support; and the Canada/Nova Scotia Technology Transfer and Industrial Innovation Agreement [Grant Nos. 87110I, 881"F1101, 89"IT!10l, 9ffFITI01, 9ffITlI02, 90TTII03 and 90TTII04] for fur~er financial support.
757
REFERENCES [1] M. Ltisher and P. Weisz, Nuci. Phys. B 290 [FS 201 (1987) 25. [2] D. B~rut~ and H. KrOger, Phys. Lett. B238 (1990) 348, [3] D. B6rub~, H. Kr6ger, 1L Lafi"ance and L Marleau, Universit~ Laval, preprint LAVALPHY-90/1. [4I D. B~'ub~, H. Kr0ger, R. Lafi'ance and L. Marleau, Universit~ Laval, preprint LAVA[,-
PHY-9014. [5] D. B~rul~ I~ Ka0ger, R. Lafrance and L Marleau, Universit6 Lavat, preprint LAVALPHY-90/5o