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On the relation between the renormalized and the bare coupling constant on the lattice
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
ON THE RELATION BETWEEN THE RENORMALIZED AND THE BARE COUPLING CONSTANT ON THE LATTICE G. PARISI
Laboratori Nazionali 1NFN, Frascati, Italy Received 4 March 1980 We discuss bow to compare the lattice regularized field theory with the continuum limit: the finite renormalization of the coupling constant must be computed on the lattice at the one-loop level.
Recently some computations have been done for renormalizable asymptotically free field theories in the low bare coupling region using lattice regularization [1,2]. The theories under consideration (the non-linear) o model and gauge theories) are formally massless and scale invariant; however there is a spontaneous generation of mass: one can define an effective running coupling constant C~R(/l),/a being the renormalization point, satisfying the renormalization group equation: dO~R(,U)/d,u2 =32o~ 2 +33~3R +O(Ot4R),
32 <0.
(1)
Eq. (1) implies that any quantity p having the dimensions of a mass squared must depend on/a and ~R(tt) in the following way: P = q/12~R(bt~3/3~ exp(1/~Z~R)[1 + O(OCR)].
(2)
The quantity C o cannot be computed in perturbation theory and depends on the definition of c~R. If we use the standard parametrization of high energy physics: --1//320tR(//) -- (133f132) I11 0tR(/d) ~ ln(/a2/A2), U ~ oo,
(3)
(4)
If the same quantity is computed on the lattice we get:
p = Lpa -2 exp [1/32c~ B + (33/32)1n C~B] ,
c~B ~ 1,
ga >> 1,
a B In/.ta ,~ 1.
(6)
In this region one gets with good accuracy: 1/OlR(/a) = (I/aB)/S 2 In #2a2S,
(7)
S being a computable constant; by substituting in the previous equations we finally find for any p:
q =s G
(8)
Eq. (8) is crucial for making an absolute comparison between the lattice and the continuum theories: S can be obtained making a simple one-loop computation on the lattice. Let us compute S for the two-dimensional nonlinear o model invariant under the group O(N) defined on a quadratic lattice. Following Br6zin and Zinn-Justin [3] we introduce an N - 1 component field and we eliminate the remaining component of the field using the non-linear constraint. Using their notation the propagator of the renormalized field is
GR1092 , t, H) = (Z/Z 1 t)(p 2 + H Z 1/%/~)
eq. (2) can be written as: p = C a 2.
We can compare eqs. (2) and (5) in the region:
(5)
where c~B is the bare coupling constant and a is the lattice spacing, L o is a new constant which can be obtained by direct non-perturbative computation on the lattice.
+ loop corrections,
(9)
where t and H are the renormalized coupling constant (temperature) and magnetic field, Z and Z 1 are the field and coupling constant renormalization constants, respectively. The renormalization constants are fixed by the condition: GRI(p2, t,# 2) = t - l ( p 2 + H ) + O(p4).
(10) 133
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
On the lattice the free propagator is:
Notice that in the continum limit the exact formula is:
tGol(p, t,H) = [2 - cos(Pxa ) - cos(Pva)] 2a -2 + H.
a-l(p, t,H) = aol(p, t,H)
(11) The integration must be done in the first Brillouin zone -Tr/a <~Px ~ 7r/a, -n/a <~py <~7fla. At the one looplevel one finds:
G-l(p,t,H)=Gol(p,t,H) --~f 2
d2K ([Go(K,t,H)/t ] { ½ ( N + 1 ) + 2 a -2
B
X [2 - cos(aPx +aKx) - cos(apy +aKx)]}-I }.
d2K
- £ 2 [p2 + {(N -
11]
f K2 + H
If one simply substitutes the lattice propagator into the formulae valid in the continuum without quantizing the theory on the lattice, one would miss the factor 7r, which is very important for not too large N. From this example it should be clear that the value of S in QCD can be extracted only by doing a straightforward, but long computation.
(12) Using the asymptotic relation [4] :
References f
d2K Go(K, t,H)/t ~ rr in(Ha2~32),
B
we get:
Z 1 = 1 + t[(N - 2) ln(32//aa 2) + n] =*S -1 = 32 e x p [ n / ( N - 2)].
134
H-+0,
[1] M. Creutz, Brookhaven preprint BNL-26847 (1979~, K. Wilson, Cargese lecture notes (1979). [2] D.S. Fisher and D.R. Nelson, Phys. Rev. B16 (1977) 2300; G. Parisi, Phys. Lett. 90B (1980) 295. [3] E. Br~zin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Phys. Rev. B14 (1976) 3110. [4] D.J. Thouless and M.E. Elzain, J. Phys. Cll (1978) 3425.
PHYSICS LETTERS
5 May 1980
ON THE RELATION BETWEEN THE RENORMALIZED AND THE BARE COUPLING CONSTANT ON THE LATTICE G. PARISI
Laboratori Nazionali 1NFN, Frascati, Italy Received 4 March 1980 We discuss bow to compare the lattice regularized field theory with the continuum limit: the finite renormalization of the coupling constant must be computed on the lattice at the one-loop level.
Recently some computations have been done for renormalizable asymptotically free field theories in the low bare coupling region using lattice regularization [1,2]. The theories under consideration (the non-linear) o model and gauge theories) are formally massless and scale invariant; however there is a spontaneous generation of mass: one can define an effective running coupling constant C~R(/l),/a being the renormalization point, satisfying the renormalization group equation: dO~R(,U)/d,u2 =32o~ 2 +33~3R +O(Ot4R),
32 <0.
(1)
Eq. (1) implies that any quantity p having the dimensions of a mass squared must depend on/a and ~R(tt) in the following way: P = q/12~R(bt~3/3~ exp(1/~Z~R)[1 + O(OCR)].
(2)
The quantity C o cannot be computed in perturbation theory and depends on the definition of c~R. If we use the standard parametrization of high energy physics: --1//320tR(//) -- (133f132) I11 0tR(/d) ~ ln(/a2/A2), U ~ oo,
(3)
(4)
If the same quantity is computed on the lattice we get:
p = Lpa -2 exp [1/32c~ B + (33/32)1n C~B] ,
c~B ~ 1,
ga >> 1,
a B In/.ta ,~ 1.
(6)
In this region one gets with good accuracy: 1/OlR(/a) = (I/aB)/S 2 In #2a2S,
(7)
S being a computable constant; by substituting in the previous equations we finally find for any p:
q =s G
(8)
Eq. (8) is crucial for making an absolute comparison between the lattice and the continuum theories: S can be obtained making a simple one-loop computation on the lattice. Let us compute S for the two-dimensional nonlinear o model invariant under the group O(N) defined on a quadratic lattice. Following Br6zin and Zinn-Justin [3] we introduce an N - 1 component field and we eliminate the remaining component of the field using the non-linear constraint. Using their notation the propagator of the renormalized field is
GR1092 , t, H) = (Z/Z 1 t)(p 2 + H Z 1/%/~)
eq. (2) can be written as: p = C a 2.
We can compare eqs. (2) and (5) in the region:
(5)
where c~B is the bare coupling constant and a is the lattice spacing, L o is a new constant which can be obtained by direct non-perturbative computation on the lattice.
+ loop corrections,
(9)
where t and H are the renormalized coupling constant (temperature) and magnetic field, Z and Z 1 are the field and coupling constant renormalization constants, respectively. The renormalization constants are fixed by the condition: GRI(p2, t,# 2) = t - l ( p 2 + H ) + O(p4).
(10) 133
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
On the lattice the free propagator is:
Notice that in the continum limit the exact formula is:
tGol(p, t,H) = [2 - cos(Pxa ) - cos(Pva)] 2a -2 + H.
a-l(p, t,H) = aol(p, t,H)
(11) The integration must be done in the first Brillouin zone -Tr/a <~Px ~ 7r/a, -n/a <~py <~7fla. At the one looplevel one finds:
G-l(p,t,H)=Gol(p,t,H) --~f 2
d2K ([Go(K,t,H)/t ] { ½ ( N + 1 ) + 2 a -2
B
X [2 - cos(aPx +aKx) - cos(apy +aKx)]}-I }.
d2K
- £ 2 [p2 + {(N -
11]
f K2 + H
If one simply substitutes the lattice propagator into the formulae valid in the continuum without quantizing the theory on the lattice, one would miss the factor 7r, which is very important for not too large N. From this example it should be clear that the value of S in QCD can be extracted only by doing a straightforward, but long computation.
(12) Using the asymptotic relation [4] :
References f
d2K Go(K, t,H)/t ~ rr in(Ha2~32),
B
we get:
Z 1 = 1 + t[(N - 2) ln(32//aa 2) + n] =*S -1 = 32 e x p [ n / ( N - 2)].
134
H-+0,
[1] M. Creutz, Brookhaven preprint BNL-26847 (1979~, K. Wilson, Cargese lecture notes (1979). [2] D.S. Fisher and D.R. Nelson, Phys. Rev. B16 (1977) 2300; G. Parisi, Phys. Lett. 90B (1980) 295. [3] E. Br~zin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Phys. Rev. B14 (1976) 3110. [4] D.J. Thouless and M.E. Elzain, J. Phys. Cll (1978) 3425.