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Exact evaluation of the mass gap in the O(N) non-linear sigma model
Volume 153B, number 6
PHYSICS LETTERS
11 April 1985
EXACT EVALUATION OF THE MASS GAP IN T H E O ( N ) N O N - L I N E A R S I G M A M O D E L F. G L I O Z Z I lstituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, ltaly
Received 24 January 1985 When the Li~scher nonlocal quantum charges are transcribed in a lattice hamiltonian formalism, they become unusually manageable. Their conservation induces an exact expression for the mass of the low-lying vector multiplet of the theory. Its value in units A pv (Pauli-Villars scale) reads simply m = exp[1/(N- 2)]Apv.
The quantum O(N) nonlinear sigma model in two dimensions is interesting because it shares with QCD a number of basic properties such as renormalizability, asymptotic freedom and dimensional transmutation [1,2]. Moreover for infinite N the model can be solved exactly; for finite N the scattering matrix can be calculated explicitly [3,4] once the spectrum of the theory is known. There has been increasing evidence [5, 6] that this spectrum consists of a massive O(N) vector multiplet which provides the mass gap of the theory. The purpose of this paper is to calculate the exact value o f this mass. Comparison to previous numerical estimates is summarized in table 1. Objects of our analysis are the nonlocal conserved quantum charges built up by LiJscher, who showed that they imply, besides the absence of particle production and the factorization equations for the twoparticle S-matrix, a relation between the mass of the vector multiplet and the scale controlling the short distance behaviour o f current products. The methods used by Liischer in his construction are however non-perturbative, therefore cannot be applied to relate these physical quantities to the A scale set up by the perturbative renormalization group approach. In this paper we describe instead a procedure which makes such a kind o f connection possible. Notice that the nonlocal charges involve current products at the same point, therefore one has to apply a limiting procedure through the introduction o f a 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
space cutoff [4] in order to have well-defined, ultraviolet fmite objects. Then one comes naturally to the idea of identifying this cutoff with the spacing of a lattice regularization, so the cutoff becomes, through the renormalization group equations, a known function of the coupling constant of the theory. The hamiltonian approach offers the further advantage of allowing an explicit, straightforward evaluation of the commutator between the nonlocal charges and the hamiltonian. Now the request of conservation in the weak coupling limit implies an equation for the mass gap in lattice spacing units as a function of the coupling constant, having the universal one-loop functional form dictated by asymptotic freedom. At this point it is a simple matter to draw out the exact value of the mass in units Apv. Let us consider the standard hamiltonian of the O(N) sigma model [7,8] :
x/~g x-, H = "-~--- x/-~ [L2 - (2/g 2) n x " nx+l] ,
(1)
where the integer x labels the lattice sites, a is the spatial lattice spacing and g is the coupling constant; the "speed of light" ~ is a parameter needed to restore Lorentz invariance at each perturbative order, nxa (a = 1 ..... N) is a commuting unit vector and finally L 2 is the quadratic Casimir operator of O(AD, made with its inffmitesimal generators L a b = - L ha, normalized in a standard way: c = l. ~ x , y ( ~ ac n xb -- ~bcna), [L xab , n~,]
[nxa, nby] = 0, (2a) 403
Volume 153B, number 6
L 2x= {LxbCL~ .
PHYSICS LETTERS (2b)
It is convenient to consider the local 0 ( N ) currents, defined by
11 April 1985
The right-hand side of (6a) yields, after normal ordering,
[Qab, H] = i ~
:Aab(x):
x
jgb(x) = (l/a) L ab , j~b (x ) = ( l /ag) (nax_ l nbx - nbx_,l nxa) ,
(3)
which fulfill the following conservation law:
- i [iSb (x ), H] = (X/'~/ a) vj~b (x ) ,
(4)
with Vf(x) = f ( x + 1) - f ( x ) . In trams of these currents it is easy to write the lattice analog of the Liischer nonlocal charge:
Qab = ~
[a2/f (x)"jobc(x)+ Zaj~b (X)],
(5)
x
where/7(x) = Ey=l [.f(x + y ) - f ( x - y ) ] , and Z is a renormalization constant that will be determined perturbatively as a function o f g by charge conservation. The evaluation of the commutator [Qab, H] is the only long calculation of this paper, but it is completely straightforward. The result can be written in terms of anticommutators of currents as follows
[Qab, HI = i ~
[Aab(x) + Z a ( d / d t ) ]~b(x)] ,
(6a)
x
1
"aC
(x),lO, b c (x)+lO. b c ( x - 1 ) }
- (]lbe(x), j~C(x) + j~C(x -- 1)}).
(6b)
The explicit form of i(d/dt) j~b(x) = [i~b(x), H] is not needed here. An important feature of this relation is that the commutator is entirely expressed in terms of currents. This circumstance suggests to apply a procedure already used by Liischer for the integrand of the nonlocal charge, namely to take the t -+ oo limit of eq. (6) and express the O(N) currents in terms of the asymptotic fields ~0aut(X) which describe the massive multiplet of the theory. In such a limit the currents/~b(x)become:
•ab _ a b b a l0 (X)- ~0out(X) at~0out(X) -- ~0out(X) 3t~0out(X),
(7a)
404
~Obut(X -- 1) ~Oaout(X)] .
Z = ( N - 2) G(0; m) X/~ •
(7b)
(8)
(9)
We try now to evaluate Z as a function o f g by using the ordinary perturbative expansion. For this purpose it is convenient to make the change of variables
a - x/g rrax ,
H x --
a =/=N •
o x = (1 - g~2)1/2,
(10)
where the ~xa are the N - 1 massless fields creating (or annihilating) the associated Goldstone particles out of the perturbative (false) vacuum. In attempting to evaluate directly the right-hand side of eq. (6a), one has to face a serious difficulty, because eq. (6) includes, for any choice of the O(N) indices, operators, like L Na, rotating the degenerate vacuum. Therefore they are not expressible in terms of creation and annihilation operators. However in the hamiltonian lattice formalism - and this is a crucial difference compared to continuous regularizations - there are, as we shall see in a moment, operator identities which provide us with a simple way out. The mentioned identities are expressed by the vanishing of the local composite operator Wxabc defined by
..brca ~crab -- nv wabc"x ~ naLxbc + nx ~x + ,,x~,x
]~b (X) = (1/a) [~0aut(X -- 1) ~Oobut(X) --
dga.~tj~b(x),
where we have dropped total divergences of the form ]g Vf(x); G(x; m) is the free lattice propagator of a scalar particle of mass m as defined in ref. [8], where m is the mass gap of the theory. According to the short-distance expansion of current products worked out by Lfischer [4], the particular combination of currents appearing in G :Aab(x) : does not include local operators of dimension less than three. It follows that the charge Qab is conserved in the continuum limit a -~ 0, provided that the coefficient of ( d / d 0 j~b(x) vanishes:
nN-
where we have put, for notational convenience
Aab(x) = ~X/-~a(~/1
+i[Z-(N-2)G(0;m)V~]v-~
(1 1)
These constraints are a direct consequence of the fact that, in deriving the hamiltonian H within the transfer.
Volume 153B, number 6
PHYSICS LETTERS
matrix approach, the O(N) generators appear in the purely orbital representation
L ab = inbx ~/an a - in a ~/~n b .
f o r a , b 4 : N , in terms o f the 7rx fields, and we obtain to lowest order
(12)
Alternatively, one can realize that the Wx abc are essentially P a u l i - L u b a n s k i operators o f the EN Lie alc Then the square modgebra generated by L ab and n x. ulus W2, beir~g a Casimir operator o f such an algebra, commutes with H and hence defines an inffmite set o f constants of m o t i o n - one for each site - having no analog in the continuous theory. It is then a simple matter to conclude that, in order to have states connected to the continuum physics, one has to put W2 = 0, implying the constraints (11). Using these in eq. (6) a simple calculation shows that
[Qab, H] = i ~
[(X/~/2ag){n x "nx+l, ~TLaxb)
[Qab, H] = i ~
[(x/-~/2ag) : {nx" nx+l, v L a b ) :
X
+ (Z - 1/g - X/~ VG(y; e) [y=0,e_~0) × a ( d / d t ) j~b(x)] .
(14)
In the continuum limit, dropping again total divergences, the normal ordered product becomes proportional to the local operator of dimension three :(n x • a2nx) a l L xab.., while the other term is manifestly infrared finite, because [8] X/-~V G 0 ' ; e) [y=0,e__,0 ~ - 1 / T r ,
(15)
Thus the nonlocal charge Qab is conserved in the continuum limit, provided that
X
+ (Z - l/g) a ( d / d t ) ]~b(x)] .
11 April 1985
(13)
Now the right-hand side can be entirely rewritten,
Z = 1 / g - t/Tr.
(16)
This equation, when combined with eq. (9), yields
Table 1 Mass gap in units of Apv compared to previous estimates. Method
0(3)
0(4)
0(6)
O(10)
exact results
2.718...
1.648...
1.284...
1.133...
3.4
1.8
1.3
1.2
[ 8]
standard action
3.2
1.4
1.2
improved action
0.8
1.9 2.0 1.1 1.2
1.2
1.1
[9] [101 [11] [12]
lattice
strong coupling
hamiltonian lattice euclidean lattice
Monte Carlo simulations
standard action standard action and renormalisation group approach improved actions
f'mite volume approach
one-loop mass formula two-loop mass formula
tree level IA one-loop IA block spin RG IA
Ref.
4.8 -z-0.2 3.5 -v 0.2
[ 13] [ 14]
4.1 ~ 1.2 4.2 ~ 0.4
[15] [161
3.0 • 0.1 1.3 3.3 ~ 0.2
[ 17] [ 18] [19]
1.6
1.6
1.5
1.4
[20]
at stationary point
2.4
1.9
1.7
1.5
[21]
at stationary point and scattering matrix
2.1
[ 21 ]
405
Volume 153B, number 6
ma = 8 e x p [ 2 / ( N - 2)] e x p [ - 2 z r / ( N - 2 ) g ] ,
PHYSICS LETTERS (17)
where we have used the known asymptotic expansion
11 April 1985
bative constraint (11) b l must be zero. This is the reason why the pre-exponential term is absent in eq. (17). Note that the A ratios are independent o f b 1.
[8] x / ~ G ( 0 ; m)
1
, ln(8/ma) + O(ma) mayO ~
References (18)
Eq. (17) has the universal one-loop form expected for a physical, renormalization invariant mass which reads in units o f AH o f the present regularization m = 8 e x p [ 2 / ( N - 2)] A n .
(19)
The ratio A H / A p v with the mass scale o f the P a u l i Villars regularization has been already calculated [8] : A H / A p v = ~exp [ - 1 / ( N - 2)] .
(20)
Then we have our final result m = e x p [ 1 / ( N - 2)] A p v .
(21)
Note that for infinite N this formula reproduces the well-known result m = A p v . Table 1 shows that the true mass gap values calculated in this paper lie between the previous numerical estimates given b y the standard and the improved action [ 8 - 2 1 ], with no particular preference for the latter. This fact suggests that it is a bit too optimistic to believe that the present Monte Carlo simulations have attained the region of asymptotic scaling. Clearly, the present m e t h o d can be applied to any theory having nonlocal conservation laws, like for instance the SU(N) X SU(Ar) chiral model. I plan to come back to this subject in the future. I would like to thank S. Sciuto for helful conversations.
Note added. Recently Davis and Nahm [22] have p o i n t e d out that the S-matrix o f the O(N) sigma model is free o f infrared divergences only in regularization schemes where the second coefficient b l of the beta function vanishes. Since the schemes where Qab is well defined (see the remarks after eq. (10)) allow to build up the S-matrix [4], one concludes that in the hamiltonian lattice regularization with the nonpertur-
406
[1] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [2] E. Brezin and J. Zinn-Justin, Phys. Rev. D14 (1976) 3110. [3] A.B. Zamolodchikov and A.P. Zamolodchikov, Nucl. Phys. B133 (1978) 525. [4] M. Liiseher, Nucl. Phys. B135 (1978) 1. [5] A.M. Polyakov and P.B. Wiegrnann, Phys. Lett. J31B (1983) 121. [6] P.B. Wiegmann, Nordita preprint 84/32 (1984). [7] C. Hamer, J. Kogut and L. Susskind, Phys. Rev. LetL 41 (1978) 1337; Phys. Rev. D19 (1979) 3091. [8] J. Shigemitsu and J.B. Kogut, Nucl. Phys. B190 [FS3] (1981) 365. [9] R. Musto, F. Nicodemi, R. Pettorino and A. Clarizia, Nucl. Phys. B210 [FS6] (1982) 263. [10] M. Caselle, F. Gliozzi and R. Megna, Phys. Lett. 109B (1982) 303. [11] A. Clarizia, G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Phys. Lett. 148B (1984) 323. [12] M. Caselle, F. Gliozzi, R. Megna, F. Ravanini and S. Sciuto, Phys. Lett. 130B (1983) 81. [13] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. B205 [FS5] (1982) 188. [14] M. Fukugita and Y. Oyanagi, Phys. Lett. 123B (1983) 71. [ 15] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [16] A. Hasenfratz and A. Margaritis, Phys. Lett. 148B (1984) 129. [17] M. Falcioni, G. MartineUi, M.L. Paciello, G. Parisi and B. Taglienfi, Nucl. Phys. B225 [FS9] (1983) 313. [18] B. Berg, S. Meyer, I. Montvay and K. Symanzik, Phys. Lett. 126B (1983) 467; B. Berg, S. Meyer and I. Montvay, Nucl. Phys. B235 [FSll] (1984) 149; B. Berg, Z. Phys. C20 (1983) 243. [19] A. Hasenfratz and A. Margaritis, Phys. Lett. 133B (1983) 211. [20] M, Liischer, Phys. Lett. l18B (1982) 391;and Lectures NATO Advanced Study Institute (Carg~se, 1983), DESY preprint 83-116. [21] E.G. Floratos and D. Petcher, Bern preprint BUTP 29•84 (1984). [22] A.C. Davis and W. Nahm, Cambridge preprint DAMTP 84-24 (1984).
PHYSICS LETTERS
11 April 1985
EXACT EVALUATION OF THE MASS GAP IN T H E O ( N ) N O N - L I N E A R S I G M A M O D E L F. G L I O Z Z I lstituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, ltaly
Received 24 January 1985 When the Li~scher nonlocal quantum charges are transcribed in a lattice hamiltonian formalism, they become unusually manageable. Their conservation induces an exact expression for the mass of the low-lying vector multiplet of the theory. Its value in units A pv (Pauli-Villars scale) reads simply m = exp[1/(N- 2)]Apv.
The quantum O(N) nonlinear sigma model in two dimensions is interesting because it shares with QCD a number of basic properties such as renormalizability, asymptotic freedom and dimensional transmutation [1,2]. Moreover for infinite N the model can be solved exactly; for finite N the scattering matrix can be calculated explicitly [3,4] once the spectrum of the theory is known. There has been increasing evidence [5, 6] that this spectrum consists of a massive O(N) vector multiplet which provides the mass gap of the theory. The purpose of this paper is to calculate the exact value o f this mass. Comparison to previous numerical estimates is summarized in table 1. Objects of our analysis are the nonlocal conserved quantum charges built up by LiJscher, who showed that they imply, besides the absence of particle production and the factorization equations for the twoparticle S-matrix, a relation between the mass of the vector multiplet and the scale controlling the short distance behaviour o f current products. The methods used by Liischer in his construction are however non-perturbative, therefore cannot be applied to relate these physical quantities to the A scale set up by the perturbative renormalization group approach. In this paper we describe instead a procedure which makes such a kind o f connection possible. Notice that the nonlocal charges involve current products at the same point, therefore one has to apply a limiting procedure through the introduction o f a 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
space cutoff [4] in order to have well-defined, ultraviolet fmite objects. Then one comes naturally to the idea of identifying this cutoff with the spacing of a lattice regularization, so the cutoff becomes, through the renormalization group equations, a known function of the coupling constant of the theory. The hamiltonian approach offers the further advantage of allowing an explicit, straightforward evaluation of the commutator between the nonlocal charges and the hamiltonian. Now the request of conservation in the weak coupling limit implies an equation for the mass gap in lattice spacing units as a function of the coupling constant, having the universal one-loop functional form dictated by asymptotic freedom. At this point it is a simple matter to draw out the exact value of the mass in units Apv. Let us consider the standard hamiltonian of the O(N) sigma model [7,8] :
x/~g x-, H = "-~--- x/-~ [L2 - (2/g 2) n x " nx+l] ,
(1)
where the integer x labels the lattice sites, a is the spatial lattice spacing and g is the coupling constant; the "speed of light" ~ is a parameter needed to restore Lorentz invariance at each perturbative order, nxa (a = 1 ..... N) is a commuting unit vector and finally L 2 is the quadratic Casimir operator of O(AD, made with its inffmitesimal generators L a b = - L ha, normalized in a standard way: c = l. ~ x , y ( ~ ac n xb -- ~bcna), [L xab , n~,]
[nxa, nby] = 0, (2a) 403
Volume 153B, number 6
L 2x= {LxbCL~ .
PHYSICS LETTERS (2b)
It is convenient to consider the local 0 ( N ) currents, defined by
11 April 1985
The right-hand side of (6a) yields, after normal ordering,
[Qab, H] = i ~
:Aab(x):
x
jgb(x) = (l/a) L ab , j~b (x ) = ( l /ag) (nax_ l nbx - nbx_,l nxa) ,
(3)
which fulfill the following conservation law:
- i [iSb (x ), H] = (X/'~/ a) vj~b (x ) ,
(4)
with Vf(x) = f ( x + 1) - f ( x ) . In trams of these currents it is easy to write the lattice analog of the Liischer nonlocal charge:
Qab = ~
[a2/f (x)"jobc(x)+ Zaj~b (X)],
(5)
x
where/7(x) = Ey=l [.f(x + y ) - f ( x - y ) ] , and Z is a renormalization constant that will be determined perturbatively as a function o f g by charge conservation. The evaluation of the commutator [Qab, H] is the only long calculation of this paper, but it is completely straightforward. The result can be written in terms of anticommutators of currents as follows
[Qab, HI = i ~
[Aab(x) + Z a ( d / d t ) ]~b(x)] ,
(6a)
x
1
"aC
(x),lO, b c (x)+lO. b c ( x - 1 ) }
- (]lbe(x), j~C(x) + j~C(x -- 1)}).
(6b)
The explicit form of i(d/dt) j~b(x) = [i~b(x), H] is not needed here. An important feature of this relation is that the commutator is entirely expressed in terms of currents. This circumstance suggests to apply a procedure already used by Liischer for the integrand of the nonlocal charge, namely to take the t -+ oo limit of eq. (6) and express the O(N) currents in terms of the asymptotic fields ~0aut(X) which describe the massive multiplet of the theory. In such a limit the currents/~b(x)become:
•ab _ a b b a l0 (X)- ~0out(X) at~0out(X) -- ~0out(X) 3t~0out(X),
(7a)
404
~Obut(X -- 1) ~Oaout(X)] .
Z = ( N - 2) G(0; m) X/~ •
(7b)
(8)
(9)
We try now to evaluate Z as a function o f g by using the ordinary perturbative expansion. For this purpose it is convenient to make the change of variables
a - x/g rrax ,
H x --
a =/=N •
o x = (1 - g~2)1/2,
(10)
where the ~xa are the N - 1 massless fields creating (or annihilating) the associated Goldstone particles out of the perturbative (false) vacuum. In attempting to evaluate directly the right-hand side of eq. (6a), one has to face a serious difficulty, because eq. (6) includes, for any choice of the O(N) indices, operators, like L Na, rotating the degenerate vacuum. Therefore they are not expressible in terms of creation and annihilation operators. However in the hamiltonian lattice formalism - and this is a crucial difference compared to continuous regularizations - there are, as we shall see in a moment, operator identities which provide us with a simple way out. The mentioned identities are expressed by the vanishing of the local composite operator Wxabc defined by
..brca ~crab -- nv wabc"x ~ naLxbc + nx ~x + ,,x~,x
]~b (X) = (1/a) [~0aut(X -- 1) ~Oobut(X) --
dga.~tj~b(x),
where we have dropped total divergences of the form ]g Vf(x); G(x; m) is the free lattice propagator of a scalar particle of mass m as defined in ref. [8], where m is the mass gap of the theory. According to the short-distance expansion of current products worked out by Lfischer [4], the particular combination of currents appearing in G :Aab(x) : does not include local operators of dimension less than three. It follows that the charge Qab is conserved in the continuum limit a -~ 0, provided that the coefficient of ( d / d 0 j~b(x) vanishes:
nN-
where we have put, for notational convenience
Aab(x) = ~X/-~a(~/1
+i[Z-(N-2)G(0;m)V~]v-~
(1 1)
These constraints are a direct consequence of the fact that, in deriving the hamiltonian H within the transfer.
Volume 153B, number 6
PHYSICS LETTERS
matrix approach, the O(N) generators appear in the purely orbital representation
L ab = inbx ~/an a - in a ~/~n b .
f o r a , b 4 : N , in terms o f the 7rx fields, and we obtain to lowest order
(12)
Alternatively, one can realize that the Wx abc are essentially P a u l i - L u b a n s k i operators o f the EN Lie alc Then the square modgebra generated by L ab and n x. ulus W2, beir~g a Casimir operator o f such an algebra, commutes with H and hence defines an inffmite set o f constants of m o t i o n - one for each site - having no analog in the continuous theory. It is then a simple matter to conclude that, in order to have states connected to the continuum physics, one has to put W2 = 0, implying the constraints (11). Using these in eq. (6) a simple calculation shows that
[Qab, H] = i ~
[(X/~/2ag){n x "nx+l, ~TLaxb)
[Qab, H] = i ~
[(x/-~/2ag) : {nx" nx+l, v L a b ) :
X
+ (Z - 1/g - X/~ VG(y; e) [y=0,e_~0) × a ( d / d t ) j~b(x)] .
(14)
In the continuum limit, dropping again total divergences, the normal ordered product becomes proportional to the local operator of dimension three :(n x • a2nx) a l L xab.., while the other term is manifestly infrared finite, because [8] X/-~V G 0 ' ; e) [y=0,e__,0 ~ - 1 / T r ,
(15)
Thus the nonlocal charge Qab is conserved in the continuum limit, provided that
X
+ (Z - l/g) a ( d / d t ) ]~b(x)] .
11 April 1985
(13)
Now the right-hand side can be entirely rewritten,
Z = 1 / g - t/Tr.
(16)
This equation, when combined with eq. (9), yields
Table 1 Mass gap in units of Apv compared to previous estimates. Method
0(3)
0(4)
0(6)
O(10)
exact results
2.718...
1.648...
1.284...
1.133...
3.4
1.8
1.3
1.2
[ 8]
standard action
3.2
1.4
1.2
improved action
0.8
1.9 2.0 1.1 1.2
1.2
1.1
[9] [101 [11] [12]
lattice
strong coupling
hamiltonian lattice euclidean lattice
Monte Carlo simulations
standard action standard action and renormalisation group approach improved actions
f'mite volume approach
one-loop mass formula two-loop mass formula
tree level IA one-loop IA block spin RG IA
Ref.
4.8 -z-0.2 3.5 -v 0.2
[ 13] [ 14]
4.1 ~ 1.2 4.2 ~ 0.4
[15] [161
3.0 • 0.1 1.3 3.3 ~ 0.2
[ 17] [ 18] [19]
1.6
1.6
1.5
1.4
[20]
at stationary point
2.4
1.9
1.7
1.5
[21]
at stationary point and scattering matrix
2.1
[ 21 ]
405
Volume 153B, number 6
ma = 8 e x p [ 2 / ( N - 2)] e x p [ - 2 z r / ( N - 2 ) g ] ,
PHYSICS LETTERS (17)
where we have used the known asymptotic expansion
11 April 1985
bative constraint (11) b l must be zero. This is the reason why the pre-exponential term is absent in eq. (17). Note that the A ratios are independent o f b 1.
[8] x / ~ G ( 0 ; m)
1
, ln(8/ma) + O(ma) mayO ~
References (18)
Eq. (17) has the universal one-loop form expected for a physical, renormalization invariant mass which reads in units o f AH o f the present regularization m = 8 e x p [ 2 / ( N - 2)] A n .
(19)
The ratio A H / A p v with the mass scale o f the P a u l i Villars regularization has been already calculated [8] : A H / A p v = ~exp [ - 1 / ( N - 2)] .
(20)
Then we have our final result m = e x p [ 1 / ( N - 2)] A p v .
(21)
Note that for infinite N this formula reproduces the well-known result m = A p v . Table 1 shows that the true mass gap values calculated in this paper lie between the previous numerical estimates given b y the standard and the improved action [ 8 - 2 1 ], with no particular preference for the latter. This fact suggests that it is a bit too optimistic to believe that the present Monte Carlo simulations have attained the region of asymptotic scaling. Clearly, the present m e t h o d can be applied to any theory having nonlocal conservation laws, like for instance the SU(N) X SU(Ar) chiral model. I plan to come back to this subject in the future. I would like to thank S. Sciuto for helful conversations.
Note added. Recently Davis and Nahm [22] have p o i n t e d out that the S-matrix o f the O(N) sigma model is free o f infrared divergences only in regularization schemes where the second coefficient b l of the beta function vanishes. Since the schemes where Qab is well defined (see the remarks after eq. (10)) allow to build up the S-matrix [4], one concludes that in the hamiltonian lattice regularization with the nonpertur-
406
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