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Low energy theorems from QCD versus unitarity corrections to current algebra
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
LOW ENERGY THEOREMS FROM QCD
VERSUS UNITARITY CORRECTIONS TO CURRENT ALGEBRA J. Sa BORGES Uni~ersidade Federal do Rio de Janeiro, Instituto de Flsica, Odade Universitdria, llha do FuncMo, Rio de Janeiro, CEP. 21.944, Brazil Receded 13 June 1984
We compare the low energytheorem derivedfor the QCD Green's function with the unitarity corrected current algebra pion-pion scatteringamplitude. Second-orderunitarity corrections are presented as an alternativefor two-loop calculations
from the non-linearsigmamodel
1. IntroductiorL The success of the low-energy theorems of current algebra is a manifestation of massless chiral symmetric QCD. Among these theorems is the Weinberg prediction for pion-pion S-wave scattering length [ 1]. Since that prediction follows from a real scattering amplitude, people tried to supply this exact result with model dependent corrections, by several methods. To go beyond the soft-pion prediction for ~r~rscattering, the hard-meson method based on Ward identities of chiral SU(2) X SU(2) have been used by Schnitzer [2] ;he obtained a crossing symmetric amplitude in terms of scalars and electromagnetic pion form factors. From that representation, by estimating the soft-pion limit of the structure functions and propagators involved, we have introduced unitarity corrections to the Weinberg amplitude [3]. The three parameters of our crossing symmetric amplitude can be related to the scalar and electromagnetic pion radius and were used to fit experimental S- and P-wave phase-shifts. In a recent letter [4] Gasser and Leutwyler have obtained corrections to order m 2 to the low-energy theorem for pion-pion scattering. They used a method which allows one to determine these corrections by a simultaneous expansion of the Green's functions in QCD to first non-leading order in powers of the momenta and of the light quark masses [5]. The corresponding expansion of the rtlr scattering amplitude was obtained from the one-loop contribution to the pseudo-scalar four-point function. They showed that the resulting correction to the Weinberg S-wave scattering-length agrees with data within the errors when the four renormalization group invariant scales are estimated from other data. In a more recent paper Leutwyler concludes [6] that nonanalytic contributions to pion-pion scattering, required by chiral syrmuetry, which they had obtained, can not be all determined by unitarity corrections to the low energy theorem. Comparing our first-order corrected pion-pion amplitude obtained by a unitarization procedure with that obtained by Gasser and Leutwyler by evaluating the one-loop contributions to the pseudo-scalar four-point function we conclude that they coincide. Up to order p4 that coincidence indicates the equivalence between the loop expansion of a chiral symmetric effective lagrangian and the procedure used to compute unitarity corrections to soft pion theorems. That indication motivates me to present in this letter the tools for constructing second-order corrections to the soft-pion amplitude. 2. First-order unitarity corrections versus one-loop contribution to p i o n - p i o n scattering. Formula (4) of ref.
[4] can be directly compared with the first-order corrected amplitude of ref. [3] : A(1)(s, t, u) = ~h(01)(s) - ~h(21)(s) + ½~l(S - 2m2) 2 + [12h(1)(t) + (s - u) h~l)(t) - ~/J2(t - 2m2) 2 + (t ~ u)], 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2.1) 21
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
where
f4h(01)(s) = (2s - m2) 2 [g(s) + or0],
f4h(1)(s) = (s - 2m2) 2 [g(s) + or2] ,
f4htl)(s) = -~(s - 4m 2) g(s) + -~m2g(0) - 2Sal.
(2.2a, b) (2.2c)
In this formula 2 [g(0) - g(s)] is f(s) used in ref. [4], the subtraction constants a and seagull parameters ~ can be related to the constraints of ref. [4] ff we re-introduce the scale invariants/-3 and l-4 by eliminating F andM in favor of the pion decay constant/and mass m as in formula (3) of the same reference. We remark that the authors of ref. [4] used the non-linear sigma model to determine this low-energy expansion by evaluating the one-loop contribution to the pseudo-scalar four-point function and our method to solve the SU(2) X SU(2) Ward identities was to impose elastic urtitarity constraints to estimate form-factor and propagator corrections up to first order of the approximation. We claim that this comparison allows one to parallel the one-loop contribution to pion-pion scattering in the non-linear sigma model and the first-order unitarity corrections presented in ref. [3].
3. Second-order corrections. We present here the tools for constructing second-order unitarity corrections to the pion-pion amplitude as an alternative for two-loop calculations with the non-linear sigma model for we have in mind the proof due to Weinberg [7], that in a model with chiral symmetry the energy dependence of the amplitudes is determined by the number of derivatives of the pion field and by the number of loops in the diagrams. We are conjecturing the equivalence between loop expansion and our unitarization procedure. As in ref. [3] we will consider total isospin I = 0, 2 and I = 1 separately. (a) 1-- 0 and 2. The first-order corrections to the scalar form-factors and propagators have been evaluated from Im~l)(s) = (1/32~r)p(s) T~/A(s),
Im ~/(1)(s) = (1/327r)p(s),
(3.1a, b)
where p(s) = [(s - 4m)/s] 1/20(s - 4m 2) and the T~/A(s) are the current algebra amplitudes f2T~0A(s ) = 2S - m 2 ,
f2T~2A(s ) = 2m 2 - s.
The relation ([F/(s) - m2]/Al(S)) (0) = T~/g(s) leads to h(1)(s) = ~IA(s)[g(s)+ al] that appears in (2.2a, b) of ref. [3]. The next-order approximation of the relations Im Fl(s ) = (1/321r) p (s) TT(s ) Fl(S ) ,
Im A/(s) = (1/32rr) p (s)IF/(s) 12
(3.2a, b)
are
Im~2)(s) = (1/327r)p(s)(Re T(/1)+ f f A Re f}/1)),
Im 8(2)(s) = (1/3Err) 2p(s) Rej~/1)(s),
(3.3a, b)
so
Im h/(2)(s)= (1/32p)p(s) ETa/A(s) Re T~/1)(s), (3.4) (2) where h I is the second-order approximation for hi defined in (2.14) of ref. [3] and Re T~/1)(s) is the real part of the first-order corrected partial-wave amplitude computed from A(1)(s, t, u). Defining
h 2)(s) =
A(s)
we can write a three-fold subtracted dispersion relation from its imaginary parts, for example: Im ~0(s) = (2/32~r) p(s) Re T(01)(s), so
22
(3.5)
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
~0(s) = ~0(0) + s~b(0 ) + ½s2~(0) 2p(x) + 3 ~ ns32 ~ 4f~ 2 dx x3(x - s) ([(2x - m2) 2 p(x) + (x 2 - ~ m2x + 4-~-~m4)/p(x)
+ 4m 2 [1 + 2m2[3(s - 4m2)] In{[1 - p(x)]/[1 + p(x)])] ln[(1 - p)/(1 + p)] + bl x2 + b2 + b3}. This follows from (2.16) and the expressions for nA (s) defined in the appendix of ref. [3] where the combinations of the parameters b are also given. For constructing the Rrst-order amplitudes, one needs the integrals oo p(x) 32g2g(s) = (s - 4m 2) 4m J 2 (ix (x - 4-m-~-(x - s) = p(s) ln{[p(s) - l]/[p(s) + 11},
1 +1
f
ha(s) =
tng(t),
2t=(s - 4m2)(x - 1),
(3.6)
-1 for instance, 32zr20A(s) = - [ 1 / ( s - 4m2)] In 2 [(1 -p)[(1 + p)] + [1/m2p(s)] ln[(1 -p)[(1 + p)] + l[m 2 .
(3.7)
For the second-order calculations, as indicated above, we need the integrals 321r2G(s) = (s- 4m2)f ax" (xP(X)Reg(x)4m2)(x _ s) =(1]641r2)p2(s)ln2[(P - 1)/(p + 1)1 +-~r2fp 2 - 1) 4m2
(321r2)2Z(s)=(s'- 4m2) 4f~2 d x (x - 4m2)(xP(X)_ s) ln2 [(1 - p)/(1 +p)]
= ~p(s) ln[(p - 1)/(p + 1)] {In2 [(p - 1)/(p + 1)] + lr2).
In this way we obtain f4~0(s ) = 2{(2s - m2) 2 G(s) + 4m 4 [1 + 2m2[3(s - 4m2)] Z(s) + ~(3s 2 - 25m2s + 40m 4) G(s) +
2
- i9sg m 2 s + ~ m
4 + ~P0(s)]
g(s) + Qo(s)),
(3.8a)
f4~2(s ) = 2{(s - 2m2) 2 G(s) - 2m 4 [1 + 13m2/3(s - 4m2)] Z(s) + [(1 ls 2 - 32m2s + 6m 4) G(s) + [5~sr2S2 -1--~ m169 _2.a_ 2O61s3,,,,,4+ IP2(s)] g(s) + Q2(s)} ,
(3.8b)
where PO and P2 are given in the appendix of ref. [3] and the Q1 are polynomials whose coefficients are subtraction constants. (b) I = 1. The first-order corrections to the electromagnetic pion form-factor and to the vector propagator have been evaluated from Im ~l)(s) = (1/32n) p(s) T~IA(s),
Im 8(1)(s) = -~(s - 4m2)(1/3210 p ( s ) ,
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Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
where f2T~l A(s) : ~(s -- 4m 2) is the current algebra amplitude. Up to first order, the normalization conditions gave f2h~l)(s) =/~lA(s) g(s) + 1[121r2 -- ~ s a 1 ,
as in (2.2c) or in (2.15) of ref. [3]. The next-order approximation of the relations Im El(s) = (1[32~r) p(s) T~ (s) E l (s) ,
Im Av(S) = -~(s - 4m2)(1/B2~t) p ( s ) l F l ( s ) l 2
are"
I m ~ 2) = ( 1 / 3 2 0 #(s) [Re T(ll)(s) + T~IA Re~11)(s)],
Im 6(2)(s) = -~(1]327r) p(s)(s - 4m 2) Re fl(1)(s).(3.9a, b)
Using hi(s) = C v l 2 f - (2[Av)[F1 -- 1 -- ( l [ 2 f 2) Cv] 2 , as in (2.4) of reL [3], we obtain Im el(S) = (1[321r) 2p(s) Re T(ll)(s),
h~2)(s) = ( l / f 2) ~l(S),
so
.f4~l(S ) = 2(-~(s -- 4m2) 2 G(s) + 4m 2 [1 + t~ m2](s _ 4m 2) _ lOm4[3(s _ 4m2)2] Z(s) -
~ [s 2 -- 12m2s + 24m 4 + 60m6[(s -- 4m2)] O(s)
41 2-~ - ~ m 4 + ~-m6/(s + r..l_ 127oo2.4. - ~--?m
4 m 2) + ~Pl(s)] g(s)
+
Q1(s)),
where P1 is given in the appendix of ref. [3] and Q1 is a polynomial whose coefficientsare subtraction constants. Now, the second-order unitarity corrected pion-pion scatteringamplitude can be written as f2A(2)(s, t, u) = [(2s - m 2) qS0(s) - ~(2m 2 - s) ¢2(s) + [~(2m 2 - t) ¢2(0 + (s - u) ¢1(0 + (t "-~u)] .
(3.10)
The low-energy representation of pion-pion scattering is now composed of the current algebra amplitude f2A(O) = s - m 2, plus the first-order corrected amplitude A (1) given by (2.1) plus the second-order corrected am-
plitude A (2) shown in (3.10). To emphasise the content of this information we mention that, up to first-order correction, we have fixed the three parameters of the model in order to fit the experimental phase-shifts [3]. We shall shortly fit higher angular momentum phase-shifts with the actual improved representation in order to fix the new free parameters available. 4. Conclusions, We have stressed the possibility to compare the low energy structure off-shell amplitudes in QCD and the solution of the four-point function current algebra Ward identity satisfying unitarity constraints. After showing that our first-order corrected current algebra amplitude describing mr scattering coincides with that obtained by Gasser and Leutwyler, we have presented the tools for constructing second-order unitarity corrections. We conjecture the equivalence between loop expansion of a chiral symmetric model and the unitarization procedure developed here. We have also used a similar method for obtaining corrections to pion-kaon [8] and for pion-nucleon [9] current algebra amplitudes and we are looking for a possible connection of these representations and one-loop calculations from chiral effective lagrangians.
24
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
S. Weinberg, Phys. Rev. LetL 17 (1966) 616. H.J. Schnitzer, Phys. Rev. D2 (1970) 1621. J. S~ Borges, NucL Phys. B51 (1973) 189. J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 325. J. Gasser and H. Leutwyler, Phys. LetL 125B (1983) 321. H. Leutwyler, TH-3738 CERN (October 1983). S. Weinberg, Physica 96A (1979) 327. J. S~lBorges, Nucl. Phys. B109 (1976) 357. J. S~ Borges, Rev. Bras. F~sica, Special Edition (1980) 147.
25
PHYSICS LETTERS
13 December 1984
LOW ENERGY THEOREMS FROM QCD
VERSUS UNITARITY CORRECTIONS TO CURRENT ALGEBRA J. Sa BORGES Uni~ersidade Federal do Rio de Janeiro, Instituto de Flsica, Odade Universitdria, llha do FuncMo, Rio de Janeiro, CEP. 21.944, Brazil Receded 13 June 1984
We compare the low energytheorem derivedfor the QCD Green's function with the unitarity corrected current algebra pion-pion scatteringamplitude. Second-orderunitarity corrections are presented as an alternativefor two-loop calculations
from the non-linearsigmamodel
1. IntroductiorL The success of the low-energy theorems of current algebra is a manifestation of massless chiral symmetric QCD. Among these theorems is the Weinberg prediction for pion-pion S-wave scattering length [ 1]. Since that prediction follows from a real scattering amplitude, people tried to supply this exact result with model dependent corrections, by several methods. To go beyond the soft-pion prediction for ~r~rscattering, the hard-meson method based on Ward identities of chiral SU(2) X SU(2) have been used by Schnitzer [2] ;he obtained a crossing symmetric amplitude in terms of scalars and electromagnetic pion form factors. From that representation, by estimating the soft-pion limit of the structure functions and propagators involved, we have introduced unitarity corrections to the Weinberg amplitude [3]. The three parameters of our crossing symmetric amplitude can be related to the scalar and electromagnetic pion radius and were used to fit experimental S- and P-wave phase-shifts. In a recent letter [4] Gasser and Leutwyler have obtained corrections to order m 2 to the low-energy theorem for pion-pion scattering. They used a method which allows one to determine these corrections by a simultaneous expansion of the Green's functions in QCD to first non-leading order in powers of the momenta and of the light quark masses [5]. The corresponding expansion of the rtlr scattering amplitude was obtained from the one-loop contribution to the pseudo-scalar four-point function. They showed that the resulting correction to the Weinberg S-wave scattering-length agrees with data within the errors when the four renormalization group invariant scales are estimated from other data. In a more recent paper Leutwyler concludes [6] that nonanalytic contributions to pion-pion scattering, required by chiral syrmuetry, which they had obtained, can not be all determined by unitarity corrections to the low energy theorem. Comparing our first-order corrected pion-pion amplitude obtained by a unitarization procedure with that obtained by Gasser and Leutwyler by evaluating the one-loop contributions to the pseudo-scalar four-point function we conclude that they coincide. Up to order p4 that coincidence indicates the equivalence between the loop expansion of a chiral symmetric effective lagrangian and the procedure used to compute unitarity corrections to soft pion theorems. That indication motivates me to present in this letter the tools for constructing second-order corrections to the soft-pion amplitude. 2. First-order unitarity corrections versus one-loop contribution to p i o n - p i o n scattering. Formula (4) of ref.
[4] can be directly compared with the first-order corrected amplitude of ref. [3] : A(1)(s, t, u) = ~h(01)(s) - ~h(21)(s) + ½~l(S - 2m2) 2 + [12h(1)(t) + (s - u) h~l)(t) - ~/J2(t - 2m2) 2 + (t ~ u)], 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2.1) 21
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
where
f4h(01)(s) = (2s - m2) 2 [g(s) + or0],
f4h(1)(s) = (s - 2m2) 2 [g(s) + or2] ,
f4htl)(s) = -~(s - 4m 2) g(s) + -~m2g(0) - 2Sal.
(2.2a, b) (2.2c)
In this formula 2 [g(0) - g(s)] is f(s) used in ref. [4], the subtraction constants a and seagull parameters ~ can be related to the constraints of ref. [4] ff we re-introduce the scale invariants/-3 and l-4 by eliminating F andM in favor of the pion decay constant/and mass m as in formula (3) of the same reference. We remark that the authors of ref. [4] used the non-linear sigma model to determine this low-energy expansion by evaluating the one-loop contribution to the pseudo-scalar four-point function and our method to solve the SU(2) X SU(2) Ward identities was to impose elastic urtitarity constraints to estimate form-factor and propagator corrections up to first order of the approximation. We claim that this comparison allows one to parallel the one-loop contribution to pion-pion scattering in the non-linear sigma model and the first-order unitarity corrections presented in ref. [3].
3. Second-order corrections. We present here the tools for constructing second-order unitarity corrections to the pion-pion amplitude as an alternative for two-loop calculations with the non-linear sigma model for we have in mind the proof due to Weinberg [7], that in a model with chiral symmetry the energy dependence of the amplitudes is determined by the number of derivatives of the pion field and by the number of loops in the diagrams. We are conjecturing the equivalence between loop expansion and our unitarization procedure. As in ref. [3] we will consider total isospin I = 0, 2 and I = 1 separately. (a) 1-- 0 and 2. The first-order corrections to the scalar form-factors and propagators have been evaluated from Im~l)(s) = (1/32~r)p(s) T~/A(s),
Im ~/(1)(s) = (1/327r)p(s),
(3.1a, b)
where p(s) = [(s - 4m)/s] 1/20(s - 4m 2) and the T~/A(s) are the current algebra amplitudes f2T~0A(s ) = 2S - m 2 ,
f2T~2A(s ) = 2m 2 - s.
The relation ([F/(s) - m2]/Al(S)) (0) = T~/g(s) leads to h(1)(s) = ~IA(s)[g(s)+ al] that appears in (2.2a, b) of ref. [3]. The next-order approximation of the relations Im Fl(s ) = (1/321r) p (s) TT(s ) Fl(S ) ,
Im A/(s) = (1/32rr) p (s)IF/(s) 12
(3.2a, b)
are
Im~2)(s) = (1/327r)p(s)(Re T(/1)+ f f A Re f}/1)),
Im 8(2)(s) = (1/3Err) 2p(s) Rej~/1)(s),
(3.3a, b)
so
Im h/(2)(s)= (1/32p)p(s) ETa/A(s) Re T~/1)(s), (3.4) (2) where h I is the second-order approximation for hi defined in (2.14) of ref. [3] and Re T~/1)(s) is the real part of the first-order corrected partial-wave amplitude computed from A(1)(s, t, u). Defining
h 2)(s) =
A(s)
we can write a three-fold subtracted dispersion relation from its imaginary parts, for example: Im ~0(s) = (2/32~r) p(s) Re T(01)(s), so
22
(3.5)
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
~0(s) = ~0(0) + s~b(0 ) + ½s2~(0) 2p(x) + 3 ~ ns32 ~ 4f~ 2 dx x3(x - s) ([(2x - m2) 2 p(x) + (x 2 - ~ m2x + 4-~-~m4)/p(x)
+ 4m 2 [1 + 2m2[3(s - 4m2)] In{[1 - p(x)]/[1 + p(x)])] ln[(1 - p)/(1 + p)] + bl x2 + b2 + b3}. This follows from (2.16) and the expressions for nA (s) defined in the appendix of ref. [3] where the combinations of the parameters b are also given. For constructing the Rrst-order amplitudes, one needs the integrals oo p(x) 32g2g(s) = (s - 4m 2) 4m J 2 (ix (x - 4-m-~-(x - s) = p(s) ln{[p(s) - l]/[p(s) + 11},
1 +1
f
ha(s) =
tng(t),
2t=(s - 4m2)(x - 1),
(3.6)
-1 for instance, 32zr20A(s) = - [ 1 / ( s - 4m2)] In 2 [(1 -p)[(1 + p)] + [1/m2p(s)] ln[(1 -p)[(1 + p)] + l[m 2 .
(3.7)
For the second-order calculations, as indicated above, we need the integrals 321r2G(s) = (s- 4m2)f ax" (xP(X)Reg(x)4m2)(x _ s) =(1]641r2)p2(s)ln2[(P - 1)/(p + 1)1 +-~r2fp 2 - 1) 4m2
(321r2)2Z(s)=(s'- 4m2) 4f~2 d x (x - 4m2)(xP(X)_ s) ln2 [(1 - p)/(1 +p)]
= ~p(s) ln[(p - 1)/(p + 1)] {In2 [(p - 1)/(p + 1)] + lr2).
In this way we obtain f4~0(s ) = 2{(2s - m2) 2 G(s) + 4m 4 [1 + 2m2[3(s - 4m2)] Z(s) + ~(3s 2 - 25m2s + 40m 4) G(s) +
2
- i9sg m 2 s + ~ m
4 + ~P0(s)]
g(s) + Qo(s)),
(3.8a)
f4~2(s ) = 2{(s - 2m2) 2 G(s) - 2m 4 [1 + 13m2/3(s - 4m2)] Z(s) + [(1 ls 2 - 32m2s + 6m 4) G(s) + [5~sr2S2 -1--~ m169 _2.a_ 2O61s3,,,,,4+ IP2(s)] g(s) + Q2(s)} ,
(3.8b)
where PO and P2 are given in the appendix of ref. [3] and the Q1 are polynomials whose coefficients are subtraction constants. (b) I = 1. The first-order corrections to the electromagnetic pion form-factor and to the vector propagator have been evaluated from Im ~l)(s) = (1/32n) p(s) T~IA(s),
Im 8(1)(s) = -~(s - 4m2)(1/3210 p ( s ) ,
23
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
where f2T~l A(s) : ~(s -- 4m 2) is the current algebra amplitude. Up to first order, the normalization conditions gave f2h~l)(s) =/~lA(s) g(s) + 1[121r2 -- ~ s a 1 ,
as in (2.2c) or in (2.15) of ref. [3]. The next-order approximation of the relations Im El(s) = (1[32~r) p(s) T~ (s) E l (s) ,
Im Av(S) = -~(s - 4m2)(1/B2~t) p ( s ) l F l ( s ) l 2
are"
I m ~ 2) = ( 1 / 3 2 0 #(s) [Re T(ll)(s) + T~IA Re~11)(s)],
Im 6(2)(s) = -~(1]327r) p(s)(s - 4m 2) Re fl(1)(s).(3.9a, b)
Using hi(s) = C v l 2 f - (2[Av)[F1 -- 1 -- ( l [ 2 f 2) Cv] 2 , as in (2.4) of reL [3], we obtain Im el(S) = (1[321r) 2p(s) Re T(ll)(s),
h~2)(s) = ( l / f 2) ~l(S),
so
.f4~l(S ) = 2(-~(s -- 4m2) 2 G(s) + 4m 2 [1 + t~ m2](s _ 4m 2) _ lOm4[3(s _ 4m2)2] Z(s) -
~ [s 2 -- 12m2s + 24m 4 + 60m6[(s -- 4m2)] O(s)
41 2-~ - ~ m 4 + ~-m6/(s + r..l_ 127oo2.4. - ~--?m
4 m 2) + ~Pl(s)] g(s)
+
Q1(s)),
where P1 is given in the appendix of ref. [3] and Q1 is a polynomial whose coefficientsare subtraction constants. Now, the second-order unitarity corrected pion-pion scatteringamplitude can be written as f2A(2)(s, t, u) = [(2s - m 2) qS0(s) - ~(2m 2 - s) ¢2(s) + [~(2m 2 - t) ¢2(0 + (s - u) ¢1(0 + (t "-~u)] .
(3.10)
The low-energy representation of pion-pion scattering is now composed of the current algebra amplitude f2A(O) = s - m 2, plus the first-order corrected amplitude A (1) given by (2.1) plus the second-order corrected am-
plitude A (2) shown in (3.10). To emphasise the content of this information we mention that, up to first-order correction, we have fixed the three parameters of the model in order to fit the experimental phase-shifts [3]. We shall shortly fit higher angular momentum phase-shifts with the actual improved representation in order to fix the new free parameters available. 4. Conclusions, We have stressed the possibility to compare the low energy structure off-shell amplitudes in QCD and the solution of the four-point function current algebra Ward identity satisfying unitarity constraints. After showing that our first-order corrected current algebra amplitude describing mr scattering coincides with that obtained by Gasser and Leutwyler, we have presented the tools for constructing second-order unitarity corrections. We conjecture the equivalence between loop expansion of a chiral symmetric model and the unitarization procedure developed here. We have also used a similar method for obtaining corrections to pion-kaon [8] and for pion-nucleon [9] current algebra amplitudes and we are looking for a possible connection of these representations and one-loop calculations from chiral effective lagrangians.
24
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
S. Weinberg, Phys. Rev. LetL 17 (1966) 616. H.J. Schnitzer, Phys. Rev. D2 (1970) 1621. J. S~ Borges, NucL Phys. B51 (1973) 189. J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 325. J. Gasser and H. Leutwyler, Phys. LetL 125B (1983) 321. H. Leutwyler, TH-3738 CERN (October 1983). S. Weinberg, Physica 96A (1979) 327. J. S~lBorges, Nucl. Phys. B109 (1976) 357. J. S~ Borges, Rev. Bras. F~sica, Special Edition (1980) 147.
25