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The Schwinger model and sum rules
Volume 132B, number 4,5,6
PHYSICS LETTERS
1 December 1983
THE SCHWINGER MODEL AND SUM RULES A.A. PIVOVAROV, N.N. TAVKHELIDZE and V.F. TOKAREV Institute for Nuclear Research of the Academy o f Sciences of the USSR, Moscow 11 7312, USSR Received 24 May 1983 Revised manuscript received 23 August 1983
It is shown with the example of the two dimensional strictly solvable model that the knowledge of the asymptotic behaviour of the correlation function of two interpolating currents at the great euclidean momenta does not make it possNle to recover the parameters of the resonance on the basis of using the standard sum rules. This is due to the fact that the resonance lies out of the asymptotic freedom region. The use of the sum rules in x-space, however, allows us to fix resonance parameters reliably.
1. Introduction. As is well known the perturbation theory calculations within the QCD framework are reliable only in the deep euclidean region [ 1 - 4 ] . One o f the methods of getting information on the low energetic behaviour o f the theory is the use of one or another of the sum rules. The finite energy sum rules [ 5 - 7 ] , being the mathematical expression of the local q u a r k - h a d r o n duality idea, allow us to get interesting information about the radial excitations of many resonances [8] which is fairly reliable at the perturbation theory level. To study the lowest lying resonance we have to go behind the limits of the perturbation theory and to take into account the nonperturbative effects which determine the physics of great distances. In refs. [9] the authors suggested a method of taking into account the so-called power corrections to the Green function of the different currents on the basis'of using the Wilson expansion. The use of this technique allowed them to show in many cases that arising power corrections really define the parameters of the physical spectrum in the lowenergy region with reasonable assumptions about its qualitative character. However the motivation of the technique used is in general of an heuristic character and so it is attractive to check this method on the example of the strictly solvable model of field theory.
402
In the present paper we show that this method is a reliable way to determine the resonance parameters if the resonance lies in the asymptotic freedom region only. This means that the power series expansion of the corresponding correlator converges rather well in the neighbourhood of the resonance. Such a situation really takes place in the configuration space of the model and it is not so in the m o m e n t u m space. The absence o f the hard correlation between scales of the asymptotic freedom o f the Green functions in x- and p-spaces is quite unexpected. 2. The model. As a model for our research we choose the two-dimensional massless electrodynamics [10] which has many attractive features which make it resemble QCD. The model is asymptotically free and the confinement phase is realized in it explicitly [11,12]. The lagrangian of this model has the form .(2 = - ¼ ( F u r ) 2 + ~-(i~ + e~4)~b,
(I)
where Fur = 3 uA v - 8vAu is the electromagnetic field strength tensor, ff is the massless spinor field in two-dimensional s p a c e - t i m e . This model has been very well studied. There is the operator solution expressing the interacting fields q) and A u through the free (pseudo) scalar field ~ with the mass m = e/x/7, the massless scalar field r~ and the massless spinor field
~0
0.03 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 ] $ 03.00 © 1983 North-Holland
Volume 132B, number 4,5,6
PHYSICS LETTERS
p0. =
if(x) = :exp l i v e r s (Y.(x) + r/(x))] "4; ° (x),
A , ( x ) - (x/~/e)%~O v (2(x) + rl(x)), e O1 = - e 10 = 1.
(2)
In the unitary gauge the spectrum consists of the massive (pseudo) scalar field 2; and the fields r/and qj0 generate the complex structure of the ground state [11,12]. We investigate the physical spectrum in the channel with quantum numbers JP = 0 - . The interpolating current for the corresponding meson has the form J5(x) = i~75 ~(x) and the use of the operator solution (2) leads to the representation /'5 (x) = (m7/2rr): sin(Zx/~y-(x)) :,
(3)
where the normal ordering is taken with respect to the vacuum of the free field £, 7 = exp C, C is the Euler constant, C = 0.577 .... The exact expression for the correlator of the currents]5(x ) has the form
n(x) =
1 December 1983
= (m272]4rr2)sh[(4rc/i)A(x, m2)],
(4)
+ m2],/2.
(7 con'd)
The power series expansion of the function (4) in the short distance limit has the form (in euclidean space-time) ll(x) = (m 2/27r2x 2) { 1 - z(L x - 2)
+ (z2/2!)(L2x - ~ L x + ~ -- 274) -(z3/3!)(L3x -'£~xlS/2 +(s9_5_ + 674)Lx --~-317 _ 1274)
+ (z4/4!)[L 4 -llL3x __ \(635972
--
423,4)Lx
+ (~ -+
102116
- 123,4)L 2 --
303,4 ] + O(z5LSx) ),
(8) where z = (½rex) 2 ,
L x = ln(y2z).
It is easy to get the corresponding expansion in momentum space which is true in the great momentum limit (remind once more that we work in euclidean space-time) ~(p2) =
f dxil(x)eipX = ~1 {ln(uZ/pZ) + m2 /p2
where A(x, m 2) is the propagator of the free scalar field of the mass m,
+ (m2/p2)2(Lp - ¼)
A(x, m 2) = (i/2rOKo(m(-x 2 + i0) 1/2)
+ (m2/p2)3(2L 2 + 2Lp + 474 --T)35
and Ko(z ) is the McDonald function of the zero order. For the two point function (4) the K/~llen-Lehmann representation takes place
+ (mZ/pZ)4 [6L3 _ ~p33/2 _ (1~__~3 + 3674)Lp
n(x) = 1 ;
p(s) l A ( x , s ) d s ,
(5)
0 here O(s) is the spectral density, the explicit expression of which is [13]
O(s) = FS(s - m 2) + F ~
_ _ 2 2 k ~22k+1(s), k=l (2k + 1)!
(6)
where F = 2m272 = 6.34 ... m 2 , and ~2n is the n-particle phase volume
i=t 2p 0 5(pl + "" + Pn - q),
(7)
41s
6974 - 24f(3)] +
O(L4p/plO)},
(9)
where Lp = ln(m2/p2),/a is the renormalization mass, ~'(3) = 1.202 .... Note that having calculated the Fourier transformation of the expression (8) we omit the infinitly differentiable terms like X 2n which lead to the contact terms of the form ([]p)n6(p) in the momentum space and can be omitted in the great momentum limit. It is seen from (9) that the expansion parameter is the effective coupling constant ap = m2/p 2 which tends to the zero in the high energy limit. From (9) and (5) the asymptotic value for p(s) follows paS(s) = 1. Having used the representation (7) we found the explicit formula for P3(S) which gives p(9m 2) = 1.92 .... p(25m 2) = 0.89 .... 403
3. The method. Now we shall try to calculate the resonance parameters F and m in the formula (6) b y the sum rule method. We write the sum rules in the form, suggested b y ITEP group (9) d._[s= / } M f pexp(s)e-s/p2 112
2
[2n~(p2) ] _= II(M2) ' (10)
where II~M2) is the Borel transformation of the function 2n II(p 2) from (9) we get II(M 2) = 1 + M - 2 - - M - 4 ( L M
+ 0.673)
+ M - 6 ( L 2 + 0.846 L M + 17.7) - M-8(L3M + 1.02 L 2 - 68.5 L m + 36.9) +O(L4/MIO),
1 December 1983
PHYSICS LETTERS
Volume 132B, number 4,5,6
LM=lnM
2.
(11)
Here and in the following we suppose m 2 = e2/n =1. As peXP(s) in formula (10) we chose the fit in the form peXP(s) = f R 8(s - m2R) + O(s - So),
(12)
which takes into account the existence of the resonance and the beginning of the asymptotic continuum at some energy s o . We know that in the exact spectral density (6) the continuum starts from the three particle threshold (3m) 2 = 9 but in the formula (12) we can (and even must) use some "effective threshold" allowing to optimize the continuum contribution to the correlator (11) rather than the true value o f the threshold energy. This freedom of the choice o f s 0 arises due to rough fitting the form of the exact p(s) when s > (3m) 2 and can partially compensate for the roughness o f this approximation. The relations (10), (12) give
8 m R / m R = ~1- ( g 2 /m 2R - 1 ) ( S f l ' / f l ' + 6fl/fl),
(15)
where (l = dI'l(M2)/dM2 . It is seen that when M 2 ~ m 2 the accuracy of the resonance mass determination is proportional to one o f the calculation of the functions l'l and (I' but when M 2 >> m ~ the error increases considerably due to the presence of the great factor M 2/m2R . (When M 2 = m 2 the function H' must be equal to zero - in real cases because of error 61:I' it is only in the vicinity of zero and one has to calculate accurately the uncertainty %). So the reliable determination of the resonance mass is possible if we can work in the region of small values o f M 2 , namely, M 2 --~ m 2 , otherwise the error in the mass determination will be very great. The expansion (11) however does not work at this point I1(1) = 1 + 1 - 0.673 + 17.7 - 36.9 + ... In fig. 1 we plot m R versusM 2 for several values o f s 0 and in different orders of the expansion parameter O~M = m 2 / M 2 . It happens to be impossible to calculate the resonance parameters b y the standard sum rule technique taking into account the expansion o f the exact function H(M 2) to fourth order on a M only. Note that in the expansion (9) we neglect the contact terms which can be essential in the resonance region that can be easily seen from the expansion (8) if we take it at the point (mx) 2 = 1. It is worth noticing that the representation (5), written in the form
mR y ~
II(M2) - e x p ( - s o / M 2) = (I(M 2)
11/i/1"j = (FR/M2)exp(-m2/M
2)
~ ......
(13)
and the resonance mass is m 2 - M 2 (d/dM 2 )(M 2 I'I(M2))
F~
(14)
,
l](M 2 ) In the linear approximation we find the guaranteed accuracy of the determination o f this parameter from eq. (14) 404
,
,rt
.
J
,
[ .
,
4r~
.
~
30
M~
Fig. 1. m R versus M 2 . 2 ( 3 , 4 ) . So = 6, 2 n d ( 3 r d , 4 t h ) a p p r o x i m a t i o n o n c~M. - - (-. -) - so = 9(4,5), 4th approximation on aM"
Volume 132B, number 4,5,6
PHYSICS LETTERS
n(x) = ~ J p(s)Ko(xx/s)ds, 4rr2 ~)
(16)
is the set of sum rules depending on the parameter x. From this point x will define the length of the euclidean vector x u. Note that the continuum in p(s) is pressed due to the exponential decreasing of function Ko(XX/~) in the limit s -+ oo Having substituted the fit (12) in the sum rules (16) we obtain
FR½X2Ko(mR x) = 2n2x 2 H(x) - X/~OXgl(xX/So) =f(x),
(17)
where K t ( z ) is the McDonald function of the first order. After differentiation by x and excluding F R we get the equation for the m R determination
E - I ( m R x) = 2 - xf'/f, E(z) =Ko(z)/zKI(Z),
f ' ( x ) - df/dx.
The relation,between the accuracy of the calculation of the functions f(x) and f'(x) and one of the parameter m R analogous to (15) is
8 m R/m R = P(mRxXSf'/f + 8f/f), P(z) = [ 2 E ( z ) - 1]E(z)/[1 -K2(z)/K2(z)].
(18)
Eq. (18) means that a greater accuracy of the deter-
1 December 1983
mination of m R will be obtained if one works in the region of large x since IP(z)l-~ 1 when z-~ oo and [P(z)I -~ oo when z -+ 0. In fig. 2 we plot the value of m R as a function of 1/x 2 (that analogous to M 2 in p-space). When 1Ix 2 is small (~1) the continuum contribution is about 10% only, P(1) ~- 0.6, and the results depend on the accuracy of the expansion of the exact function If(x). We see that the successive approximations on a x = m2x 2 converge rather well (m R) = 1.84, m ~ ) = 0.82, m ~ ) = 1.00) for the optimal choice of continuum threshold sO = 6. The fourth approximation allows us to fix the resonance parameter with great accuracy m R = 1.00, F R = 6.18. In the region of large 1/x 2 the continuum contribution is not small but it is correct enough due to the optimal choice ofs 0 . So, one can think that the difference between the contribution of the truth continuum and that of effective one is much smaller than the contribution itself (for instance if the continuum amounts 50% of the magnitude of the function f ( x ) it doesn't mean that all this contribution is wrong). Really when 1/x 2 = 30 the continuum contribution is about 80% but the parameters of the resonance are m R = 1.13, F R = 6.78 and we find still reasonable agreement with the exact values. This fact reflects the successful chocie o f s 0. For s O = 9 we have m R -- 1.42, F R = 9.53 and for sO = 4.5 - m R = 0.9, F R = 5.36. Note that in this point there exists almost no difference between the exact function [I(x) and its approximations. Note the function P(z) belongs to the interval 0.6 < P < 5.7 when 1 < 1/x z < 30, while the maximum of analogous function in p-space in eq. (15) can reach the value 15. Thus we may say that inx-space does exist the large and well expressed region of stability of the resonance parameters relative to changing sum rules parameter x. In this region the successive approximations converge rapidly and the influence of the continuum is small. In p-space it is not so.
L 5
10
~
~
.
.
.
30 .
-
~/X a
Fig. 2. m R v e r s u s 1 / x 2. 2 ( 3 , 4 ) . So = 6, 2 n d ( 3 r d , 4 t h ) a p p r o x i m a t i o n o n c~x . - - (-. -) - so = 9(4,5) 4th approximation o n a x.
The authors are indebted to K.G. Chetyrkin, N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, M.E. Shaposhnikov, A.N. Tavkhelidze, F.V. Tkachov for useful discussions. We are grateful to the referee for careful reading of the manuscript and valuable comments. 405
Volume 132B, number 4,5,6
PHYSICS LETTERS
References [1] H.D. Politzer, Phys. Rep. 14C (1974) 129. [2] T. Appelquist and H. Georgi, Phys. Rev. D 8 (1973) $4000. [3] S.L. Adler, Phys. Rev. D 10 (1974) 132. [4] E. Poggio, H. Quinn and S. Weinberg, Phys. Rev. D13 (1978) 1958. [5 ] A.A. Logunov, L.D. Soloviev and A.N. Tavkhelidze, Phys. lett. 24 B (1967) 181. [6] K.G. Chetyrkin, N.V. Krasnikov and A.N. Tavkhelidze, Phys. Lett. 76B (1978) 83. [7] K.G. Chetyrkin and N.V. Krasnikov, Nucl. Phys. B 119 (1977) 174. [8] N.V. Krasnikov and A.A. Pivovarov, Phys. Lett. 112 B (1982) 397.
406
1 December 1983
[9] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448,519; for a review, see, for example: V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 191 (1981) 301; L.J. Reinders, Spectroscopy with QCD sum rules, RL81-078 (1981). [10] J. Schwinger, Phys. Rev. 128 (1962) 2425. [11] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172. [12] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Teor. y Mat. Fyz. 45 (1980) 313. [13] N.N. Tavkhelidze and V.F. Tokarev, e + e - ~ hadrons in Schwinger's model, preprint INR P-243 (1982).
PHYSICS LETTERS
1 December 1983
THE SCHWINGER MODEL AND SUM RULES A.A. PIVOVAROV, N.N. TAVKHELIDZE and V.F. TOKAREV Institute for Nuclear Research of the Academy o f Sciences of the USSR, Moscow 11 7312, USSR Received 24 May 1983 Revised manuscript received 23 August 1983
It is shown with the example of the two dimensional strictly solvable model that the knowledge of the asymptotic behaviour of the correlation function of two interpolating currents at the great euclidean momenta does not make it possNle to recover the parameters of the resonance on the basis of using the standard sum rules. This is due to the fact that the resonance lies out of the asymptotic freedom region. The use of the sum rules in x-space, however, allows us to fix resonance parameters reliably.
1. Introduction. As is well known the perturbation theory calculations within the QCD framework are reliable only in the deep euclidean region [ 1 - 4 ] . One o f the methods of getting information on the low energetic behaviour o f the theory is the use of one or another of the sum rules. The finite energy sum rules [ 5 - 7 ] , being the mathematical expression of the local q u a r k - h a d r o n duality idea, allow us to get interesting information about the radial excitations of many resonances [8] which is fairly reliable at the perturbation theory level. To study the lowest lying resonance we have to go behind the limits of the perturbation theory and to take into account the nonperturbative effects which determine the physics of great distances. In refs. [9] the authors suggested a method of taking into account the so-called power corrections to the Green function of the different currents on the basis'of using the Wilson expansion. The use of this technique allowed them to show in many cases that arising power corrections really define the parameters of the physical spectrum in the lowenergy region with reasonable assumptions about its qualitative character. However the motivation of the technique used is in general of an heuristic character and so it is attractive to check this method on the example of the strictly solvable model of field theory.
402
In the present paper we show that this method is a reliable way to determine the resonance parameters if the resonance lies in the asymptotic freedom region only. This means that the power series expansion of the corresponding correlator converges rather well in the neighbourhood of the resonance. Such a situation really takes place in the configuration space of the model and it is not so in the m o m e n t u m space. The absence o f the hard correlation between scales of the asymptotic freedom o f the Green functions in x- and p-spaces is quite unexpected. 2. The model. As a model for our research we choose the two-dimensional massless electrodynamics [10] which has many attractive features which make it resemble QCD. The model is asymptotically free and the confinement phase is realized in it explicitly [11,12]. The lagrangian of this model has the form .(2 = - ¼ ( F u r ) 2 + ~-(i~ + e~4)~b,
(I)
where Fur = 3 uA v - 8vAu is the electromagnetic field strength tensor, ff is the massless spinor field in two-dimensional s p a c e - t i m e . This model has been very well studied. There is the operator solution expressing the interacting fields q) and A u through the free (pseudo) scalar field ~ with the mass m = e/x/7, the massless scalar field r~ and the massless spinor field
~0
0.03 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 ] $ 03.00 © 1983 North-Holland
Volume 132B, number 4,5,6
PHYSICS LETTERS
p0. =
if(x) = :exp l i v e r s (Y.(x) + r/(x))] "4; ° (x),
A , ( x ) - (x/~/e)%~O v (2(x) + rl(x)), e O1 = - e 10 = 1.
(2)
In the unitary gauge the spectrum consists of the massive (pseudo) scalar field 2; and the fields r/and qj0 generate the complex structure of the ground state [11,12]. We investigate the physical spectrum in the channel with quantum numbers JP = 0 - . The interpolating current for the corresponding meson has the form J5(x) = i~75 ~(x) and the use of the operator solution (2) leads to the representation /'5 (x) = (m7/2rr): sin(Zx/~y-(x)) :,
(3)
where the normal ordering is taken with respect to the vacuum of the free field £, 7 = exp C, C is the Euler constant, C = 0.577 .... The exact expression for the correlator of the currents]5(x ) has the form
n(x) =
1 December 1983
= (m272]4rr2)sh[(4rc/i)A(x, m2)],
(4)
+ m2],/2.
(7 con'd)
The power series expansion of the function (4) in the short distance limit has the form (in euclidean space-time) ll(x) = (m 2/27r2x 2) { 1 - z(L x - 2)
+ (z2/2!)(L2x - ~ L x + ~ -- 274) -(z3/3!)(L3x -'£~xlS/2 +(s9_5_ + 674)Lx --~-317 _ 1274)
+ (z4/4!)[L 4 -llL3x __ \(635972
--
423,4)Lx
+ (~ -+
102116
- 123,4)L 2 --
303,4 ] + O(z5LSx) ),
(8) where z = (½rex) 2 ,
L x = ln(y2z).
It is easy to get the corresponding expansion in momentum space which is true in the great momentum limit (remind once more that we work in euclidean space-time) ~(p2) =
f dxil(x)eipX = ~1 {ln(uZ/pZ) + m2 /p2
where A(x, m 2) is the propagator of the free scalar field of the mass m,
+ (m2/p2)2(Lp - ¼)
A(x, m 2) = (i/2rOKo(m(-x 2 + i0) 1/2)
+ (m2/p2)3(2L 2 + 2Lp + 474 --T)35
and Ko(z ) is the McDonald function of the zero order. For the two point function (4) the K/~llen-Lehmann representation takes place
+ (mZ/pZ)4 [6L3 _ ~p33/2 _ (1~__~3 + 3674)Lp
n(x) = 1 ;
p(s) l A ( x , s ) d s ,
(5)
0 here O(s) is the spectral density, the explicit expression of which is [13]
O(s) = FS(s - m 2) + F ~
_ _ 2 2 k ~22k+1(s), k=l (2k + 1)!
(6)
where F = 2m272 = 6.34 ... m 2 , and ~2n is the n-particle phase volume
i=t 2p 0 5(pl + "" + Pn - q),
(7)
41s
6974 - 24f(3)] +
O(L4p/plO)},
(9)
where Lp = ln(m2/p2),/a is the renormalization mass, ~'(3) = 1.202 .... Note that having calculated the Fourier transformation of the expression (8) we omit the infinitly differentiable terms like X 2n which lead to the contact terms of the form ([]p)n6(p) in the momentum space and can be omitted in the great momentum limit. It is seen from (9) that the expansion parameter is the effective coupling constant ap = m2/p 2 which tends to the zero in the high energy limit. From (9) and (5) the asymptotic value for p(s) follows paS(s) = 1. Having used the representation (7) we found the explicit formula for P3(S) which gives p(9m 2) = 1.92 .... p(25m 2) = 0.89 .... 403
3. The method. Now we shall try to calculate the resonance parameters F and m in the formula (6) b y the sum rule method. We write the sum rules in the form, suggested b y ITEP group (9) d._[s= / } M f pexp(s)e-s/p2 112
2
[2n~(p2) ] _= II(M2) ' (10)
where II~M2) is the Borel transformation of the function 2n II(p 2) from (9) we get II(M 2) = 1 + M - 2 - - M - 4 ( L M
+ 0.673)
+ M - 6 ( L 2 + 0.846 L M + 17.7) - M-8(L3M + 1.02 L 2 - 68.5 L m + 36.9) +O(L4/MIO),
1 December 1983
PHYSICS LETTERS
Volume 132B, number 4,5,6
LM=lnM
2.
(11)
Here and in the following we suppose m 2 = e2/n =1. As peXP(s) in formula (10) we chose the fit in the form peXP(s) = f R 8(s - m2R) + O(s - So),
(12)
which takes into account the existence of the resonance and the beginning of the asymptotic continuum at some energy s o . We know that in the exact spectral density (6) the continuum starts from the three particle threshold (3m) 2 = 9 but in the formula (12) we can (and even must) use some "effective threshold" allowing to optimize the continuum contribution to the correlator (11) rather than the true value o f the threshold energy. This freedom of the choice o f s 0 arises due to rough fitting the form of the exact p(s) when s > (3m) 2 and can partially compensate for the roughness o f this approximation. The relations (10), (12) give
8 m R / m R = ~1- ( g 2 /m 2R - 1 ) ( S f l ' / f l ' + 6fl/fl),
(15)
where (l = dI'l(M2)/dM2 . It is seen that when M 2 ~ m 2 the accuracy of the resonance mass determination is proportional to one o f the calculation of the functions l'l and (I' but when M 2 >> m ~ the error increases considerably due to the presence of the great factor M 2/m2R . (When M 2 = m 2 the function H' must be equal to zero - in real cases because of error 61:I' it is only in the vicinity of zero and one has to calculate accurately the uncertainty %). So the reliable determination of the resonance mass is possible if we can work in the region of small values o f M 2 , namely, M 2 --~ m 2 , otherwise the error in the mass determination will be very great. The expansion (11) however does not work at this point I1(1) = 1 + 1 - 0.673 + 17.7 - 36.9 + ... In fig. 1 we plot m R versusM 2 for several values o f s 0 and in different orders of the expansion parameter O~M = m 2 / M 2 . It happens to be impossible to calculate the resonance parameters b y the standard sum rule technique taking into account the expansion o f the exact function H(M 2) to fourth order on a M only. Note that in the expansion (9) we neglect the contact terms which can be essential in the resonance region that can be easily seen from the expansion (8) if we take it at the point (mx) 2 = 1. It is worth noticing that the representation (5), written in the form
mR y ~
II(M2) - e x p ( - s o / M 2) = (I(M 2)
11/i/1"j = (FR/M2)exp(-m2/M
2)
~ ......
(13)
and the resonance mass is m 2 - M 2 (d/dM 2 )(M 2 I'I(M2))
F~
(14)
,
l](M 2 ) In the linear approximation we find the guaranteed accuracy of the determination o f this parameter from eq. (14) 404
,
,rt
.
J
,
[ .
,
4r~
.
~
30
M~
Fig. 1. m R versus M 2 . 2 ( 3 , 4 ) . So = 6, 2 n d ( 3 r d , 4 t h ) a p p r o x i m a t i o n o n c~M. - - (-. -) - so = 9(4,5), 4th approximation on aM"
Volume 132B, number 4,5,6
PHYSICS LETTERS
n(x) = ~ J p(s)Ko(xx/s)ds, 4rr2 ~)
(16)
is the set of sum rules depending on the parameter x. From this point x will define the length of the euclidean vector x u. Note that the continuum in p(s) is pressed due to the exponential decreasing of function Ko(XX/~) in the limit s -+ oo Having substituted the fit (12) in the sum rules (16) we obtain
FR½X2Ko(mR x) = 2n2x 2 H(x) - X/~OXgl(xX/So) =f(x),
(17)
where K t ( z ) is the McDonald function of the first order. After differentiation by x and excluding F R we get the equation for the m R determination
E - I ( m R x) = 2 - xf'/f, E(z) =Ko(z)/zKI(Z),
f ' ( x ) - df/dx.
The relation,between the accuracy of the calculation of the functions f(x) and f'(x) and one of the parameter m R analogous to (15) is
8 m R/m R = P(mRxXSf'/f + 8f/f), P(z) = [ 2 E ( z ) - 1]E(z)/[1 -K2(z)/K2(z)].
(18)
Eq. (18) means that a greater accuracy of the deter-
1 December 1983
mination of m R will be obtained if one works in the region of large x since IP(z)l-~ 1 when z-~ oo and [P(z)I -~ oo when z -+ 0. In fig. 2 we plot the value of m R as a function of 1/x 2 (that analogous to M 2 in p-space). When 1Ix 2 is small (~1) the continuum contribution is about 10% only, P(1) ~- 0.6, and the results depend on the accuracy of the expansion of the exact function If(x). We see that the successive approximations on a x = m2x 2 converge rather well (m R) = 1.84, m ~ ) = 0.82, m ~ ) = 1.00) for the optimal choice of continuum threshold sO = 6. The fourth approximation allows us to fix the resonance parameter with great accuracy m R = 1.00, F R = 6.18. In the region of large 1/x 2 the continuum contribution is not small but it is correct enough due to the optimal choice ofs 0 . So, one can think that the difference between the contribution of the truth continuum and that of effective one is much smaller than the contribution itself (for instance if the continuum amounts 50% of the magnitude of the function f ( x ) it doesn't mean that all this contribution is wrong). Really when 1/x 2 = 30 the continuum contribution is about 80% but the parameters of the resonance are m R = 1.13, F R = 6.78 and we find still reasonable agreement with the exact values. This fact reflects the successful chocie o f s 0. For s O = 9 we have m R -- 1.42, F R = 9.53 and for sO = 4.5 - m R = 0.9, F R = 5.36. Note that in this point there exists almost no difference between the exact function [I(x) and its approximations. Note the function P(z) belongs to the interval 0.6 < P < 5.7 when 1 < 1/x z < 30, while the maximum of analogous function in p-space in eq. (15) can reach the value 15. Thus we may say that inx-space does exist the large and well expressed region of stability of the resonance parameters relative to changing sum rules parameter x. In this region the successive approximations converge rapidly and the influence of the continuum is small. In p-space it is not so.
L 5
10
~
~
.
.
.
30 .
-
~/X a
Fig. 2. m R v e r s u s 1 / x 2. 2 ( 3 , 4 ) . So = 6, 2 n d ( 3 r d , 4 t h ) a p p r o x i m a t i o n o n c~x . - - (-. -) - so = 9(4,5) 4th approximation o n a x.
The authors are indebted to K.G. Chetyrkin, N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, M.E. Shaposhnikov, A.N. Tavkhelidze, F.V. Tkachov for useful discussions. We are grateful to the referee for careful reading of the manuscript and valuable comments. 405
Volume 132B, number 4,5,6
PHYSICS LETTERS
References [1] H.D. Politzer, Phys. Rep. 14C (1974) 129. [2] T. Appelquist and H. Georgi, Phys. Rev. D 8 (1973) $4000. [3] S.L. Adler, Phys. Rev. D 10 (1974) 132. [4] E. Poggio, H. Quinn and S. Weinberg, Phys. Rev. D13 (1978) 1958. [5 ] A.A. Logunov, L.D. Soloviev and A.N. Tavkhelidze, Phys. lett. 24 B (1967) 181. [6] K.G. Chetyrkin, N.V. Krasnikov and A.N. Tavkhelidze, Phys. Lett. 76B (1978) 83. [7] K.G. Chetyrkin and N.V. Krasnikov, Nucl. Phys. B 119 (1977) 174. [8] N.V. Krasnikov and A.A. Pivovarov, Phys. Lett. 112 B (1982) 397.
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[9] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448,519; for a review, see, for example: V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 191 (1981) 301; L.J. Reinders, Spectroscopy with QCD sum rules, RL81-078 (1981). [10] J. Schwinger, Phys. Rev. 128 (1962) 2425. [11] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172. [12] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Teor. y Mat. Fyz. 45 (1980) 313. [13] N.N. Tavkhelidze and V.F. Tokarev, e + e - ~ hadrons in Schwinger's model, preprint INR P-243 (1982).