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A new method to compute the spectrum of low-lying states in massless asymptotically free field theories
Volume 118B, n u m b e r 4, 5, 6
PHYSICS LETTERS
9 December 1982
A NEW METHOD TO COMPUTE THE SPECTRUM OF LOW-LYING STATES IN MASSLESS ASYMPTOTICALLY FREE FIELD THEORIES ~
M. LUSCHER Institut fiir Theoretische Physik, Universitdt Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 12 July 1982
A systematic universal expansion for the masses o f the low-lying stable particles in pure Yang-Mills gauge theories and non-linear a-models is proposed.
1. The masses m i of the stable particles in (fourdimensional) pure Yang-Mills gauge theories and (two-dimensional) non-linear o-models are proportional to the A-parameter:
pling constant. Ultimately, we are interested only in the case, where M is the ordinary flat Minkowski space, but for the purpose of generating the expansion alluded to above, we shall take
m i = C/A~-~ .
M = R X S1
(1)
(4)
If the system is confined to a compact space + 1, the numbers C i become functions of the linear size L of the space. In this letter a systematic expansion of the Ci's is obtained, the expansion parameter being
Here, R refers to time and S 1 to space. The volume (circumference) of S 1 is denoted by L. Because space is compact, the momentum operator has a discrete spectrum of eigenvalues
z=moL ,
P = (27r/L)v,
m0:massgap.
(2)
The method is here explained for the simplest case, the O(n) non-linear o-model. Results and details of the computations in the Yang-Mills case will be given elsewhere. I shall first (section 2) define the expansion and then show how to use it to obtain C i at L = co (section 3). 2. The O(n) non-linear o-model describes n-component spin fields s(x), s(x) 2 = 1, on a two-dimensional space-time manifold M. The action is n
S -M ~0 -
f d2x [hll/2hUV~us • ~vs ,
(3)
where huu is the metric on M and f0 the (bare) couWork supported in part by Schweizerischer Nationalfonds.
#1 N o t s p a c e - t i m e , i.e. time remains an unrestricted real
v ~ Z.
(5)
The hamiltonian also has a discrete spectrum and its eigenstates group into irreducible multiplets of the internal symmetry group O(n). A first observation now is that the eigenvalues and eigenvectors of the hamiltonian can be computed in ordinary perturbation theory. For example, one may regularise with a lattice cutoff and then set up a straightforward hamiltonian formalism. Alternatively, one can compute suitable euclidean correlation functions using dimensional regularisation [1 ] and expand for large time separations (to avoid superficial infrared divergencies one should add compact extra dimensions to regularise [2]). It then turns out that the ground state is not degenerate and that the lowest excited states make up an O(n)-vector, i.e. these states occur as intermediate states in the spin field two-point function. They have vanishing momentum and their energy up to one-loop order is
variable.
0 031-9163/82/0000--0000/$02.75 © 1982 North-Holland
391
Volume l18B, number 4, 5, 6
m 0 = L - l { f ( n - 1)/2n + f 2 [(n - i)(n - 2)/2n 2 ] X (470 -1 [ln#2L 2 - In 47r - F ' ( I ) ] + ...}.
(6)
Here, f denotes the renormalized coupling constant in the dimensional regularisation scheme with minimal subtraction (MS) [3] and/a is the normalization mass. Eq. (6) has three remarkable features. (a) m 0 vanishes for f ~ 0. This behaviour is due to the fact that the classical ground state s(x) = constant is degenerate. (b) In the abelian case n = 2, the second term vanishes and rn 0 is exactly proportional to L -1 as it ought to be. (c) m 0 is a physical quantity and therefore renormalization group invariant:
[U3/3U + ~ ( f ) 3 / 3 f l m o = O,
9 December 1982
PItYSICS LETTERS
The coefficients a v can be computed by extending the series (6). They are pure numbers independent of/a and L, because mo, A~-g and z are renormalization group invariant. In other words, the expansion (13) is universal. 3. The problem is to compute C O = mo/A ~ in the infinite volume limit, i.e. for z -+ oo (provided m 0 does not vanish in this limit). The small z expansion (13) therefore does not seem to be very useful. That this impression is false, can nicely be demonstrated in the large n limit, which is exactly solvable. At n = 0% the mass gap m 0 is determined from the large-n saddle point condition ln(m0/A~-g ) = F(z),
z = moL ,
(15)
(7)
F(z)=.fd---/exp(-tz2)k~lexp(-k2/4t ) . ~(f) = -- ~
v=o
buf v+2
bo =(n--2)/2Trn,
'
(8)
bl = ( n - 2)/(2nn) 2 .
(9)
The coefficients b 0 and b I are taken from ref. [4]. b 2 is also known [5], but not needed here. We now eliminate the coupling constant f i n favour of the dimensionless, renormalization group invariant parameter z = tooL. To this end, note first that
(16)
0 Eq. (15) is an implicit equation for m0, which has a unique solution mo(L ) . mo(L ) is monotonically decreasing and lira m 0 = A~-g. L~ For small z, on the other hand, we have
(17)
F(z) = 7r/z + l n z - In 4~r - F'(1)
1
A~-~ = AMsexp(~ [ln 41r + I " ( 1 ) ] } ,
(10)
AMS = l a ( b o f ) - b l / b ~ e x p ( - 1 / b o f )
o~
+ ~ (-1)u{2u} ~'R(ZU+ 1)(z/4rr) 2u u=l \ P/
(18)
and thus obtain the large-n limit o f eq. (13): 0 Next, from eq. (6) we have
(I1)
×
f = z2n/(n - 1) - z 2 [2n/(n - 1)] 2 [(n - 2)/n] X (4n) -1 [lng2L 2 - ln47r - P'(1)] + ....
(12)
When inserted into eq. (11), the following formula is obtained:
mo/ A~-g = K(n)(ze~r/z)(n-1)/(n- 2) oo
v=l
g(n)
=
(47r)- 1exp [--I"(1)]
x [rr-l(n 392
2)/(n
-
1)11/(~-2)
rn 0 / A~--~ = (47r)- 1 exp [ - F ' ( 1 )] z exp Or/z)
(14)
a2=
l + v 1= ~
a 2v z 2u]/ '
(19)
2(47r)-2~'R(3 ),
a 4 = 2(4r0 -4 [3~'R(5 ) + ~'R(3)2],
etc.
(20)
{fR denotes Riemann's zeta function (ref. [6], § 9.522)}. The expansions (18) and (19) are absolutely convergent for Iz I < 2~r. We can therefore accurately calculate C O for (say)z < 4 by evaluating the series (19) keeping only the first few terms. This is shown in fig. 1. Besides the rapid convergence, the most remarkable fact this plot reveals is that Co(z ) is practically
Volume l18B, number 4, 5, 6
PHYSICS LETTERS
9 December 1982
% 3
3
~ n = 3 0
n:oo
I
I
I
I
f
I
I
I
I
2
3
4
5
I
2
3
4
5
z
z
Fig. 1. Plot ofC0(z ) = mo/A~~ versus z in the large-n limit l eq. (19)]. Curve "a" is obtained from the leading term alone, "b" includes the contribution ofa 2 and "c" also includes a4.
equal to its limiting value C0(oo ) = 1 already at z = 3. More precisely, the deviation Co(z ) 1 vanishes exp o n e n t i a l l y as z -+ oo:
mo/A~-g
= 1 + O(e-Z).
(21)
At z = 3 it is a b o u t 8% and then shrinks to about 2% at z = 4. This behaviour is n o t really surprising in view of the general experience that finite size corrections become negligible as soon as L is larger than three to four times the correlation length. The moral, which can be daawn from the large n limit is: (i) It is n o t necessary to extrapolate Co(z ) all the way up to z = o% because the limiting value C0(oo) is approached quickly. More precisely, Co(z ) at (say) z = 3 can be expected to be a good a p p r o x i m a t i o n to
c0(~). (ii) Only a few terms in the small z expansion (13) are needed to get accurate values for Co(z ) in the interval 0 < z ~< 4. We n o w apply these rules in the case n = 3. The leading order formula m0/A ~
= (87r2)- I exp [ - P ' ( 1 ) ] z 2 exp(27r/z)
(n = 3 ) ,
(22)
is plotted in fig. 2. As expected, Co(z ) is rapidly decreasing for 0 < z < 2 and then quickly becomes flat giving a value
Fig. 2. Plot of Co(z ) = mo/A~ vcrsusz forn = 3 andn = ~, keeping only the leading term in eq. (18) [cp. eq. (22)]. The curves for intermediate values ofn are sandwiched between the n = 3 and n = ~ curves.
m 0 ~ 1.7a~-g
(n = 3)
(23)
in the region z = 3 ... 4 [the result (23) m a y very well be off the true value by 20% as would be the case in the large-n limit, when only the lowest term in the z-expansion is used to estimate Co(z ) at z = 3 ... 4]. It is interesting to compare eq. (23) with the number o b t a i n e d from a Monte Carlo simulation of the lattice 0 ( 3 ) o-model [7,8]: m 0 ~ 100A L
(n = 3 ) .
(24)
Here, A L denotes the lattice A-parameter A L = a-l(bo,[o)-bl/b~exp(--1/bofo
)
x [1 + o(f0)l
(25)
and a is the lattice spacing. Noting AL/A~g =
32-1/2exp[-rr/2(n -
2)] ,
(26)
one gets from eq. (24) m 0 --~ 3.7A~-g
(n = 3 ) ,
(27)
i.e. a value, which is a b o u t twice as large as our result (23). There is no reason to be discouraged, however, because eq. (23) is only the lowest a p p r o x i m a t i o n and because the Monte Carlo result m a y very well be wrong by a factor of 2 *2
393
Volume 118B, number 4, 5, 6
PHYSICS LETTERS
4. The m e t h o d p r o p o s e d in this l e t t e r is a p p l i c a b l e t o a n u m b e r o f i n t e r e s t i n g p h y s i c a l q u a n t i t i e s , in particular, t o mass ratios a n d f o r m f a c t o r s at z e r o m o mentum. The main limitation on the accuracy that c a n be a t t a i n e d seems t o b e o n e ' s ability t o do p e r t u r b a t i o n t h e o r y . F o r e x a m p l e , t o get t h e n e x t t o l e a d i n g t e r m in eq. ( 1 3 ) , r e q u i r e s t h e c o m p u t a t i o n o f t h e t h r e e - l o o p / 3 f u n c t i o n a n d t h e finite p a r t s o f a set o f t w o - l o o p diagrams. :1:2 The Monte Carlo method involves an extrapolation of numerically computed data at fl ---n/f o "~ 1.5 up to 13 = ,o. Because the corrections to the exact result at/3 = o o are powers of/3 -1 , they are not easily seen in the small interval of j3, which is accessible to the computer. An error of a factor of 2 may therefore occur even if the data seem to scale perfectly.
394
9 December 1982
References [1] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [2] M. Lilscher, Ann. Phys. 142, to be published. [3] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [4] E. Br6zin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110. [5] E. Br6zin and S. Hikami, J. Phys. A l l (1978) 1141. [6] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [7] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [8] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. B205 [FS5] (1982) 188.
PHYSICS LETTERS
9 December 1982
A NEW METHOD TO COMPUTE THE SPECTRUM OF LOW-LYING STATES IN MASSLESS ASYMPTOTICALLY FREE FIELD THEORIES ~
M. LUSCHER Institut fiir Theoretische Physik, Universitdt Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 12 July 1982
A systematic universal expansion for the masses o f the low-lying stable particles in pure Yang-Mills gauge theories and non-linear a-models is proposed.
1. The masses m i of the stable particles in (fourdimensional) pure Yang-Mills gauge theories and (two-dimensional) non-linear o-models are proportional to the A-parameter:
pling constant. Ultimately, we are interested only in the case, where M is the ordinary flat Minkowski space, but for the purpose of generating the expansion alluded to above, we shall take
m i = C/A~-~ .
M = R X S1
(1)
(4)
If the system is confined to a compact space + 1, the numbers C i become functions of the linear size L of the space. In this letter a systematic expansion of the Ci's is obtained, the expansion parameter being
Here, R refers to time and S 1 to space. The volume (circumference) of S 1 is denoted by L. Because space is compact, the momentum operator has a discrete spectrum of eigenvalues
z=moL ,
P = (27r/L)v,
m0:massgap.
(2)
The method is here explained for the simplest case, the O(n) non-linear o-model. Results and details of the computations in the Yang-Mills case will be given elsewhere. I shall first (section 2) define the expansion and then show how to use it to obtain C i at L = co (section 3). 2. The O(n) non-linear o-model describes n-component spin fields s(x), s(x) 2 = 1, on a two-dimensional space-time manifold M. The action is n
S -M ~0 -
f d2x [hll/2hUV~us • ~vs ,
(3)
where huu is the metric on M and f0 the (bare) couWork supported in part by Schweizerischer Nationalfonds.
#1 N o t s p a c e - t i m e , i.e. time remains an unrestricted real
v ~ Z.
(5)
The hamiltonian also has a discrete spectrum and its eigenstates group into irreducible multiplets of the internal symmetry group O(n). A first observation now is that the eigenvalues and eigenvectors of the hamiltonian can be computed in ordinary perturbation theory. For example, one may regularise with a lattice cutoff and then set up a straightforward hamiltonian formalism. Alternatively, one can compute suitable euclidean correlation functions using dimensional regularisation [1 ] and expand for large time separations (to avoid superficial infrared divergencies one should add compact extra dimensions to regularise [2]). It then turns out that the ground state is not degenerate and that the lowest excited states make up an O(n)-vector, i.e. these states occur as intermediate states in the spin field two-point function. They have vanishing momentum and their energy up to one-loop order is
variable.
0 031-9163/82/0000--0000/$02.75 © 1982 North-Holland
391
Volume l18B, number 4, 5, 6
m 0 = L - l { f ( n - 1)/2n + f 2 [(n - i)(n - 2)/2n 2 ] X (470 -1 [ln#2L 2 - In 47r - F ' ( I ) ] + ...}.
(6)
Here, f denotes the renormalized coupling constant in the dimensional regularisation scheme with minimal subtraction (MS) [3] and/a is the normalization mass. Eq. (6) has three remarkable features. (a) m 0 vanishes for f ~ 0. This behaviour is due to the fact that the classical ground state s(x) = constant is degenerate. (b) In the abelian case n = 2, the second term vanishes and rn 0 is exactly proportional to L -1 as it ought to be. (c) m 0 is a physical quantity and therefore renormalization group invariant:
[U3/3U + ~ ( f ) 3 / 3 f l m o = O,
9 December 1982
PItYSICS LETTERS
The coefficients a v can be computed by extending the series (6). They are pure numbers independent of/a and L, because mo, A~-g and z are renormalization group invariant. In other words, the expansion (13) is universal. 3. The problem is to compute C O = mo/A ~ in the infinite volume limit, i.e. for z -+ oo (provided m 0 does not vanish in this limit). The small z expansion (13) therefore does not seem to be very useful. That this impression is false, can nicely be demonstrated in the large n limit, which is exactly solvable. At n = 0% the mass gap m 0 is determined from the large-n saddle point condition ln(m0/A~-g ) = F(z),
z = moL ,
(15)
(7)
F(z)=.fd---/exp(-tz2)k~lexp(-k2/4t ) . ~(f) = -- ~
v=o
buf v+2
bo =(n--2)/2Trn,
'
(8)
bl = ( n - 2)/(2nn) 2 .
(9)
The coefficients b 0 and b I are taken from ref. [4]. b 2 is also known [5], but not needed here. We now eliminate the coupling constant f i n favour of the dimensionless, renormalization group invariant parameter z = tooL. To this end, note first that
(16)
0 Eq. (15) is an implicit equation for m0, which has a unique solution mo(L ) . mo(L ) is monotonically decreasing and lira m 0 = A~-g. L~ For small z, on the other hand, we have
(17)
F(z) = 7r/z + l n z - In 4~r - F'(1)
1
A~-~ = AMsexp(~ [ln 41r + I " ( 1 ) ] } ,
(10)
AMS = l a ( b o f ) - b l / b ~ e x p ( - 1 / b o f )
o~
+ ~ (-1)u{2u} ~'R(ZU+ 1)(z/4rr) 2u u=l \ P/
(18)
and thus obtain the large-n limit o f eq. (13): 0 Next, from eq. (6) we have
(I1)
×
f = z2n/(n - 1) - z 2 [2n/(n - 1)] 2 [(n - 2)/n] X (4n) -1 [lng2L 2 - ln47r - P'(1)] + ....
(12)
When inserted into eq. (11), the following formula is obtained:
mo/ A~-g = K(n)(ze~r/z)(n-1)/(n- 2) oo
v=l
g(n)
=
(47r)- 1exp [--I"(1)]
x [rr-l(n 392
2)/(n
-
1)11/(~-2)
rn 0 / A~--~ = (47r)- 1 exp [ - F ' ( 1 )] z exp Or/z)
(14)
a2=
l + v 1= ~
a 2v z 2u]/ '
(19)
2(47r)-2~'R(3 ),
a 4 = 2(4r0 -4 [3~'R(5 ) + ~'R(3)2],
etc.
(20)
{fR denotes Riemann's zeta function (ref. [6], § 9.522)}. The expansions (18) and (19) are absolutely convergent for Iz I < 2~r. We can therefore accurately calculate C O for (say)z < 4 by evaluating the series (19) keeping only the first few terms. This is shown in fig. 1. Besides the rapid convergence, the most remarkable fact this plot reveals is that Co(z ) is practically
Volume l18B, number 4, 5, 6
PHYSICS LETTERS
9 December 1982
% 3
3
~ n = 3 0
n:oo
I
I
I
I
f
I
I
I
I
2
3
4
5
I
2
3
4
5
z
z
Fig. 1. Plot ofC0(z ) = mo/A~~ versus z in the large-n limit l eq. (19)]. Curve "a" is obtained from the leading term alone, "b" includes the contribution ofa 2 and "c" also includes a4.
equal to its limiting value C0(oo ) = 1 already at z = 3. More precisely, the deviation Co(z ) 1 vanishes exp o n e n t i a l l y as z -+ oo:
mo/A~-g
= 1 + O(e-Z).
(21)
At z = 3 it is a b o u t 8% and then shrinks to about 2% at z = 4. This behaviour is n o t really surprising in view of the general experience that finite size corrections become negligible as soon as L is larger than three to four times the correlation length. The moral, which can be daawn from the large n limit is: (i) It is n o t necessary to extrapolate Co(z ) all the way up to z = o% because the limiting value C0(oo) is approached quickly. More precisely, Co(z ) at (say) z = 3 can be expected to be a good a p p r o x i m a t i o n to
c0(~). (ii) Only a few terms in the small z expansion (13) are needed to get accurate values for Co(z ) in the interval 0 < z ~< 4. We n o w apply these rules in the case n = 3. The leading order formula m0/A ~
= (87r2)- I exp [ - P ' ( 1 ) ] z 2 exp(27r/z)
(n = 3 ) ,
(22)
is plotted in fig. 2. As expected, Co(z ) is rapidly decreasing for 0 < z < 2 and then quickly becomes flat giving a value
Fig. 2. Plot of Co(z ) = mo/A~ vcrsusz forn = 3 andn = ~, keeping only the leading term in eq. (18) [cp. eq. (22)]. The curves for intermediate values ofn are sandwiched between the n = 3 and n = ~ curves.
m 0 ~ 1.7a~-g
(n = 3)
(23)
in the region z = 3 ... 4 [the result (23) m a y very well be off the true value by 20% as would be the case in the large-n limit, when only the lowest term in the z-expansion is used to estimate Co(z ) at z = 3 ... 4]. It is interesting to compare eq. (23) with the number o b t a i n e d from a Monte Carlo simulation of the lattice 0 ( 3 ) o-model [7,8]: m 0 ~ 100A L
(n = 3 ) .
(24)
Here, A L denotes the lattice A-parameter A L = a-l(bo,[o)-bl/b~exp(--1/bofo
)
x [1 + o(f0)l
(25)
and a is the lattice spacing. Noting AL/A~g =
32-1/2exp[-rr/2(n -
2)] ,
(26)
one gets from eq. (24) m 0 --~ 3.7A~-g
(n = 3 ) ,
(27)
i.e. a value, which is a b o u t twice as large as our result (23). There is no reason to be discouraged, however, because eq. (23) is only the lowest a p p r o x i m a t i o n and because the Monte Carlo result m a y very well be wrong by a factor of 2 *2
393
Volume 118B, number 4, 5, 6
PHYSICS LETTERS
4. The m e t h o d p r o p o s e d in this l e t t e r is a p p l i c a b l e t o a n u m b e r o f i n t e r e s t i n g p h y s i c a l q u a n t i t i e s , in particular, t o mass ratios a n d f o r m f a c t o r s at z e r o m o mentum. The main limitation on the accuracy that c a n be a t t a i n e d seems t o b e o n e ' s ability t o do p e r t u r b a t i o n t h e o r y . F o r e x a m p l e , t o get t h e n e x t t o l e a d i n g t e r m in eq. ( 1 3 ) , r e q u i r e s t h e c o m p u t a t i o n o f t h e t h r e e - l o o p / 3 f u n c t i o n a n d t h e finite p a r t s o f a set o f t w o - l o o p diagrams. :1:2 The Monte Carlo method involves an extrapolation of numerically computed data at fl ---n/f o "~ 1.5 up to 13 = ,o. Because the corrections to the exact result at/3 = o o are powers of/3 -1 , they are not easily seen in the small interval of j3, which is accessible to the computer. An error of a factor of 2 may therefore occur even if the data seem to scale perfectly.
394
9 December 1982
References [1] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [2] M. Lilscher, Ann. Phys. 142, to be published. [3] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [4] E. Br6zin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110. [5] E. Br6zin and S. Hikami, J. Phys. A l l (1978) 1141. [6] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [7] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [8] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. B205 [FS5] (1982) 188.