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Operator product expansion and non-perturbative renormalization
UCLEAR PHYSIC...
ELSEVIER
Nuclear Physics B (Proe. Suppl.) 73 (1999) 273-275
PROCEEDINGS SUPPLEMENTS
Operator product expansion and non-perturbative renormalization Sergio Caracciolo a, Andrea Montanari a, and Andrea Pelissetto b ~Scuola Normale Superiore and INFN, Sezione di Pisa, 1-56100 Pisa, ITALIA bDipartimento di Fisica and INFN, Universitk degli Studi di Pisa, 1-56100 Pisa, ITALIA It has been recently proposed to use the operator product expansion to evaluate the expectation values of renormalized operators without the need of a direct computation of the relevant renormalization constants. We test the viability of this idea in the two-dimensional non-linear a-model discussing the non-perturbative renormalization of the energy-momentum tensor.
The Operator Product Expansion (OPE) consists in rewriting, for y --+ x,
d(x)B(y) --+ E W~AB (x - y)C(y). c
(1)
Here A, B, and C are operators; wAB(x -- y) are c-number functions (Wilson coefficients), singular when Ix - y] ~ 0, that can be computed, for example, in perturbation theory. Dimensional analysis tells us that the leading contribution for y -+ x is due to the operators C's in Eq. (1) that have the lowest dimension. Notice that the OPE is an operator relation and therefore, for any matrix element (¢lA(x)B(y)]¢), the coefficients WcAB(x- y) are independent of the states ]¢} and I¢}. This also means that if one wants to use the OPE, one should go through all the subtleties in the definition of the operators in quantum field theory (regularization, renormMization,... ). The authors of reference [1] have proposed to consider, for lattice applications, the OPE in the particular case in which A and B are the components of a conserved current J r ' This case is particularly simple because the components of J r do not need to be renormalized. This means that
[Jr(x)J~(O)]MS (#) = [Jr(x)Jv(O)]L ( l / a ) ,
(2)
where the currents are expressed in terms of the bare fields and the coupling constant on the lattice is related to the coupling constant g in the continuum at the scale # by a renormalization factor
o,(1/a'b =- Z L (o(u'~ pa) a(a"
(3)
Therefore, the lattice computation of (¢lJr(x)J~(0)l¢) provides the same expectation value as in the continuum, say in the MS-scheme, at the scale p. One can then use Eq. (1) to estimate the matrix elements of the operators C's that dominate the r.h.s, of (1) (in the continuum at the scale #), bypassing all the troubles of lattice perturbation theory (large deviations from 1 of the Z's, large corrections, improvements, ... ). The recipe is the following. One evaluates on the lattice the matrix elements (¢lJr(x)Jv(O)l¢) in the range 1 < Ixl < ~ (where ~ is as usual the correlation length) and then uses the OPE in the continuum (at the scale it) to fit the (now) numbers (¢1C(0)1¢}. One needs the Wilson coefficients W~V(x). The idea is to use their expression computed in perturbation theory in the continuum. Assuming C(O) to be multiplicatively renormalized, one can then use <¢1C(0)1¢> = Zc (gL(1/a),pa)(¢16L(0)1¢>.
(4)
In this way, by measuring (¢1CL(0)14) on the lattice, we obtain a non-perturbative determination of Zc. The discussion in [1] ends with a question: is it feasible? T h a t is, does it exist, in practice, a numerically accessible window for Ixl where, for a truncation of the sum over C, the OPE applies accurately enough to extract the expectation vaiues of the C's? To perform a check we need a case in which we already know ~he matrix elements <¢JC(0)14>, so
(~20-5632/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved. PU g0¢~20-5632(98)00570-2
S. Caracciolo et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 273-275
274
that we can verify the correctness of our results. Our example, given the experience that we have accumulated along the past years, will refer to the O(N) e-model in two dimensions with lattice action 1 2 S = 2gL ] ~ (0,~)~ (5)
where p is the momentum, so that, in a strip of size L, the expectation value between states of momentum p is
(PITMS(O)Ip) _ P (pip) L
(12)
The Wilson coefficient is given by
-Vd~
with ~ 6 S N - I and (O~f)~ = f~+~ -f~.
The
W~°(r)
=
fields are related to their continuum version by (7~ = x/-~-er(x) .
(6)
All the renormalization factors Z i have an expansion of the form co
l
Z L (g, p, a) = E g' E /=0
c~) lnn p a .
(7)
n=0
For example Z L and Z L are known up to four loops [2]. For our purposes we made use of the nonperturbative determination of Z L given in [3] for the case N -- 3. The Noether currents related to the O(N)invariance, which is preserved also on the lattice, are
•a,b
1
a
b
b
a
We have performed a Monte Carlo simulation on a lattice of size L x T = 128 × 256 with periodic boundary conditions at gL 1 = 1.54, which corresponds to ~ = 13.632(6). We measure: a) the correlation function in the one-particle sector with momentum p
C(p, 20
Z(p) e_2t~(p);
b) the correlation function of the out-of-diagonal entry of the lattice EMT with two one-particle operators
_
(9)
1 1E((SVro,o.-~oao,o)(a_t,~:.at,y)) gL L x , y × eip(~:-y) Z(p) (piT L (0)[p) e-2t~(P); 2w(p) (p[p)
(15)
here we used the symmetric definition of the derivative
g2
Til(~)- jo(O) = --~jl(rcosO, rsinO) .3o(0,0)
= [ o ~ . oo~,]Ms (o) w~°(~) + o(~ ~)
(10)
where T Ms is the continuum energy-momentum tensor (EMT), which is exactly conserved, a property that is not shared by its lattice counterpart. It follows that
f T g S ( x , t) dx = p,
(14)
2~(p)
In the continuum, using the OPE and averaging over a circle, we obtain
= eTgs(o) W~°(r) + 0(, .~)
1 E ( t r _ t , x " o.t,y)eiP(x_y ) -~ x,y
a,b
=g 2
=
Tlo(p, 2t) =
and their singlet product is
j . ( ~ ) • j~(0) = )-'~ y~,' ° ~'~" ° ~'o' ( )y,,' ().
1+5(N-2) s-----V- a (13) N-2 4~r g (7 + In r2~2r 2) + O(g2).
(11)
(a~'f)x -- fx+~, -- fx-~,, (16) 2 c) the correlation between the products of two currents in the singlet sector and two one-particle operators
~:.~ (z; p, 20 =
= l~((j.,0. L
~-,
J~,,)(~-,,~" ~,,y)>e 'p(~-y)
x,y
Z(p) (PlJ,,o'J~,,~lP) e_2t,~(p). 2~(p) (pip)
(17)
S. Caracciolo et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 273-275
2.0
AAAAAAAA AA mmmmmnun• i 0o°°°°0
-5.0e-05
-1.5e-04
275
0 p=2~% • p=a~/L A p~6n/L
1.5
1.0
© p=2n/L • p--4~/L A p=67r2L
-2.5e-04
-3"5e-04 0
10
5
0.5
15
0.0
0
z
5
10
15
Z
Figure 1. Angular average g~01(z; p, 20)/2. The abscissa of coincident points have been splitted. Notice that the rightmost points are not a faithful angular average because 27 has been measured only on a square.
Figure 2. Angular average g2~ol(z;p, 20)/2 divided by the theoretical prediction: g. (Wilson coefficient) .(energy-momentum tensor). Here in the Wilson function the first two leading logs have been summed up.
Moreover we need a lattice version of the angular average
dow seems to extend even at a distance of order ~/2 which is presumably only a lucky accident. We also looked at the OPE directly on the lattice, i.e. using the Wilson coefficient computed in lattice perturbation theory resumming the first two leading logs, the expectation value T10(P, 2t), and the non-perturbative renormalization constant of the lattice EMT. In this case we do not observe a reasonable window where the OPE is verified. We wish to thank G. Martinelli and G.C. Rossi for useful discussions on this work.
f(r)
--
1 g(r) Ef(z)
Z(z)
z
N(r) = E Z ( z ) z
=-(z) = 0(,zl-r+l) o(r+
1 _ Izl) (18)
Let us now discuss the Monte Carlo data. In Fig. 1 we plot the angular average g2L~Ol(z;p,20)/2 for various distances z and various momenta p = n2ze/L for n = 1,2,3. In Fig. 2 we plot the left-hand side of (10) divided by gTMS(o)wl°(r) renormalized at the scale ~-1 = 2.8519a. Here in the Wilson function the first two leading logs have been summed up. We find a large window of distances for which the OPE is verified, giving the correct matrix element independently of the external states. This win-
REFERENCES 1. C. Dawson, G. Martinelli, G. C. Rossi, C. T. Sachrajda, S. Sharpe, M. Talevi and M. Testa, Nucl. Phys. B514 (1998) 313. 2. S. Caracciolo and A. Pelissetto, Nucl. Phys. B455 (1995) 619 3. M. Lfischer, P. Weisz and U. Wolff, Nucl. Phys. B359 (1991) 221.
ELSEVIER
Nuclear Physics B (Proe. Suppl.) 73 (1999) 273-275
PROCEEDINGS SUPPLEMENTS
Operator product expansion and non-perturbative renormalization Sergio Caracciolo a, Andrea Montanari a, and Andrea Pelissetto b ~Scuola Normale Superiore and INFN, Sezione di Pisa, 1-56100 Pisa, ITALIA bDipartimento di Fisica and INFN, Universitk degli Studi di Pisa, 1-56100 Pisa, ITALIA It has been recently proposed to use the operator product expansion to evaluate the expectation values of renormalized operators without the need of a direct computation of the relevant renormalization constants. We test the viability of this idea in the two-dimensional non-linear a-model discussing the non-perturbative renormalization of the energy-momentum tensor.
The Operator Product Expansion (OPE) consists in rewriting, for y --+ x,
d(x)B(y) --+ E W~AB (x - y)C(y). c
(1)
Here A, B, and C are operators; wAB(x -- y) are c-number functions (Wilson coefficients), singular when Ix - y] ~ 0, that can be computed, for example, in perturbation theory. Dimensional analysis tells us that the leading contribution for y -+ x is due to the operators C's in Eq. (1) that have the lowest dimension. Notice that the OPE is an operator relation and therefore, for any matrix element (¢lA(x)B(y)]¢), the coefficients WcAB(x- y) are independent of the states ]¢} and I¢}. This also means that if one wants to use the OPE, one should go through all the subtleties in the definition of the operators in quantum field theory (regularization, renormMization,... ). The authors of reference [1] have proposed to consider, for lattice applications, the OPE in the particular case in which A and B are the components of a conserved current J r ' This case is particularly simple because the components of J r do not need to be renormalized. This means that
[Jr(x)J~(O)]MS (#) = [Jr(x)Jv(O)]L ( l / a ) ,
(2)
where the currents are expressed in terms of the bare fields and the coupling constant on the lattice is related to the coupling constant g in the continuum at the scale # by a renormalization factor
o,(1/a'b =- Z L (o(u'~ pa) a(a"
(3)
Therefore, the lattice computation of (¢lJr(x)J~(0)l¢) provides the same expectation value as in the continuum, say in the MS-scheme, at the scale p. One can then use Eq. (1) to estimate the matrix elements of the operators C's that dominate the r.h.s, of (1) (in the continuum at the scale #), bypassing all the troubles of lattice perturbation theory (large deviations from 1 of the Z's, large corrections, improvements, ... ). The recipe is the following. One evaluates on the lattice the matrix elements (¢lJr(x)Jv(O)l¢) in the range 1 < Ixl < ~ (where ~ is as usual the correlation length) and then uses the OPE in the continuum (at the scale it) to fit the (now) numbers (¢1C(0)1¢}. One needs the Wilson coefficients W~V(x). The idea is to use their expression computed in perturbation theory in the continuum. Assuming C(O) to be multiplicatively renormalized, one can then use <¢1C(0)1¢> = Zc (gL(1/a),pa)(¢16L(0)1¢>.
(4)
In this way, by measuring (¢1CL(0)14) on the lattice, we obtain a non-perturbative determination of Zc. The discussion in [1] ends with a question: is it feasible? T h a t is, does it exist, in practice, a numerically accessible window for Ixl where, for a truncation of the sum over C, the OPE applies accurately enough to extract the expectation vaiues of the C's? To perform a check we need a case in which we already know ~he matrix elements <¢JC(0)14>, so
(~20-5632/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved. PU g0¢~20-5632(98)00570-2
S. Caracciolo et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 273-275
274
that we can verify the correctness of our results. Our example, given the experience that we have accumulated along the past years, will refer to the O(N) e-model in two dimensions with lattice action 1 2 S = 2gL ] ~ (0,~)~ (5)
where p is the momentum, so that, in a strip of size L, the expectation value between states of momentum p is
(PITMS(O)Ip) _ P (pip) L
(12)
The Wilson coefficient is given by
-Vd~
with ~ 6 S N - I and (O~f)~ = f~+~ -f~.
The
W~°(r)
=
fields are related to their continuum version by (7~ = x/-~-er(x) .
(6)
All the renormalization factors Z i have an expansion of the form co
l
Z L (g, p, a) = E g' E /=0
c~) lnn p a .
(7)
n=0
For example Z L and Z L are known up to four loops [2]. For our purposes we made use of the nonperturbative determination of Z L given in [3] for the case N -- 3. The Noether currents related to the O(N)invariance, which is preserved also on the lattice, are
•a,b
1
a
b
b
a
We have performed a Monte Carlo simulation on a lattice of size L x T = 128 × 256 with periodic boundary conditions at gL 1 = 1.54, which corresponds to ~ = 13.632(6). We measure: a) the correlation function in the one-particle sector with momentum p
C(p, 20
Z(p) e_2t~(p);
b) the correlation function of the out-of-diagonal entry of the lattice EMT with two one-particle operators
_
(9)
1 1E((SVro,o.-~oao,o)(a_t,~:.at,y)) gL L x , y × eip(~:-y) Z(p) (piT L (0)[p) e-2t~(P); 2w(p) (p[p)
(15)
here we used the symmetric definition of the derivative
g2
Til(~)- jo(O) = --~jl(rcosO, rsinO) .3o(0,0)
= [ o ~ . oo~,]Ms (o) w~°(~) + o(~ ~)
(10)
where T Ms is the continuum energy-momentum tensor (EMT), which is exactly conserved, a property that is not shared by its lattice counterpart. It follows that
f T g S ( x , t) dx = p,
(14)
2~(p)
In the continuum, using the OPE and averaging over a circle, we obtain
= eTgs(o) W~°(r) + 0(, .~)
1 E ( t r _ t , x " o.t,y)eiP(x_y ) -~ x,y
a,b
=g 2
=
Tlo(p, 2t) =
and their singlet product is
j . ( ~ ) • j~(0) = )-'~ y~,' ° ~'~" ° ~'o' ( )y,,' ().
1+5(N-2) s-----V- a (13) N-2 4~r g (7 + In r2~2r 2) + O(g2).
(11)
(a~'f)x -- fx+~, -- fx-~,, (16) 2 c) the correlation between the products of two currents in the singlet sector and two one-particle operators
~:.~ (z; p, 20 =
= l~((j.,0. L
~-,
J~,,)(~-,,~" ~,,y)>e 'p(~-y)
x,y
Z(p) (PlJ,,o'J~,,~lP) e_2t,~(p). 2~(p) (pip)
(17)
S. Caracciolo et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 273-275
2.0
AAAAAAAA AA mmmmmnun• i 0o°°°°0
-5.0e-05
-1.5e-04
275
0 p=2~% • p=a~/L A p~6n/L
1.5
1.0
© p=2n/L • p--4~/L A p=67r2L
-2.5e-04
-3"5e-04 0
10
5
0.5
15
0.0
0
z
5
10
15
Z
Figure 1. Angular average g~01(z; p, 20)/2. The abscissa of coincident points have been splitted. Notice that the rightmost points are not a faithful angular average because 27 has been measured only on a square.
Figure 2. Angular average g2~ol(z;p, 20)/2 divided by the theoretical prediction: g. (Wilson coefficient) .(energy-momentum tensor). Here in the Wilson function the first two leading logs have been summed up.
Moreover we need a lattice version of the angular average
dow seems to extend even at a distance of order ~/2 which is presumably only a lucky accident. We also looked at the OPE directly on the lattice, i.e. using the Wilson coefficient computed in lattice perturbation theory resumming the first two leading logs, the expectation value T10(P, 2t), and the non-perturbative renormalization constant of the lattice EMT. In this case we do not observe a reasonable window where the OPE is verified. We wish to thank G. Martinelli and G.C. Rossi for useful discussions on this work.
f(r)
--
1 g(r) Ef(z)
Z(z)
z
N(r) = E Z ( z ) z
=-(z) = 0(,zl-r+l) o(r+
1 _ Izl) (18)
Let us now discuss the Monte Carlo data. In Fig. 1 we plot the angular average g2L~Ol(z;p,20)/2 for various distances z and various momenta p = n2ze/L for n = 1,2,3. In Fig. 2 we plot the left-hand side of (10) divided by gTMS(o)wl°(r) renormalized at the scale ~-1 = 2.8519a. Here in the Wilson function the first two leading logs have been summed up. We find a large window of distances for which the OPE is verified, giving the correct matrix element independently of the external states. This win-
REFERENCES 1. C. Dawson, G. Martinelli, G. C. Rossi, C. T. Sachrajda, S. Sharpe, M. Talevi and M. Testa, Nucl. Phys. B514 (1998) 313. 2. S. Caracciolo and A. Pelissetto, Nucl. Phys. B455 (1995) 619 3. M. Lfischer, P. Weisz and U. Wolff, Nucl. Phys. B359 (1991) 221.