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On the low-energy πN scattering amplitudes
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
O N T H E L O W - E N E R G Y rrN S C A T T E R I N G A M P L I T U D E S L.M. NATH and A.Q. SARKER International Centre for Theoretical Physics, Trieste, Italy and Department of Physics, Dacca University, Dacca-2, Bangladesh Received 25 July 1974 Revised manuscript received 20 August 1974 The pion-nucleon scattering amplitudes with current algebra constraints involving "remainder terms" are calculated by saturating them with the N*(1236) contribution having off-~ass shell effects and compared with experiments. Since the estimate of the so-called o term by Cheng and Dashen [1 ] (CD) from the low-energy rrN scattering amplitudes, there has been considerable controversy over the subject. The value of the o term ~110 MeV, as deterrnined by CD, is quite large compared with many other estimates*. It is, moreover, hard to understand such large value of the o term from the point of view of the SU(3) ® SU(3)'symmetry breaking. Further criticism of the work of CD was made on two different considerations. The broad area subtraction technique used by CD seems to over-emphasise the low-energy data points considerably, as first pointed out by H6hler, Jakob and Strauss [3] (HJS), and the methods of analytic continuation of the amplitudes from the physical region to unphysical points are ambiguous and doubtful unless one can estimate with reasonable certainty the contributions of the N*(1236) and other higher resonances to the rrN scattering amplitudes at the relevant unphysical points. Brown, Pardee and Peccei [4] (BPP) referred to these latter contributions as the "remainder terms" of the amplitudes. Since the work of CD, the experimental position on the low-energy cross section data and the phase shift analy sis of the rrN scattering have considerably improved, due to the recent works of Bussey et al. [5 ] and the parameters, such as the coupling constant f, the scattering lengths a 1 , a 3 as well as the dependence of the amplitudes on the energy ~, momentum transfer t (say, within the Lehmann ellipse) are now known with far better accuracy from the analysis of Bugg, Carter and Carter [6] (BCC), HJS and Nielsen and Oades [7] (NO). A more recent and independent determination of the sigma term by Langbein [ 13] from the same data gives ONN(t = 292 ) = 46 -+ 14 MeV. The main difference between the work of Langbein and that of NO is that the former finds it impossible to make a reliable estimate of the coefficient of t 2 in the expansion of A(+)(0, t, ~2). In a private communication Dr. E. Reya suggested to the authors that the experimental term quadratic in t may not be used, as there are several sources of errors which are of the order/~4. From the theoretical point of view, the significant contribution to remove the o term controversy was made by BPP, and Osypowski [8], who have written the current algebra constraints on the amplitude as an identity involving a "remainder term" and kept the plans on the mass-shell. BPP further pointed out that this remainder term for the amplitude A(+)0,, t,/a 2) at the unphysical point ~ = 0, t = 2# 2 (sometimes referred to as the ChengDashen point) receives its largest contributions from the N*(1236) and is of the order of ( m 2 / M 3 , ) . Otherwise the determination of the o term from A(+)(0, t = 2/~2,/~2) would not make sense as the terms o-1 and A (+)(0, t = 2g 2,/a 2) are of the order of m~-1. We used the letter/~ to denote the mass of the charged plan. We point out that the BPP and Osypowski calculations for the N*(1236) contributions were done with N*(1236) on the mass shell. While the latter author has taken the usual rrNN* interaction Lagrangian, BPP used a modified interaction and neglected the contact terms.
* We refer to ref. [2] for the collection of up-to-date references on the different determinations of the a term. 213
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
We have re-examined the calculations of the low-energy lrN scattering amplitudes in the spirit of BPP and Osypowski and evaluated the N*(1236) contributions; however, by taking into consideration its off-mass shell effects in a Lorentz covariant way following the earlier work of Nath, Etemadi and Kirnel [9] (NEK). The results obtained go a considerable way (further than those obtained by BPP) in understanding the low-energy behaviour of the IrN scattering amplitudes. They remove some of the doubts that were expressed in the works of Cheng and Dashen and others, and sort out the points of controversy in a general way. It is therefore considered worthwhile to report some of these results here. One has the four independent amplitudes A (+)(~, t,/z2), A(-)(v, t, #2), B (+)(v, t,/a2) and B (-)(v, t, #2) for the 7rN scattering (we follow the convention and notations of BPP). The current algebra constraints on these amplitudes or any suitable linear combinations of them can be expressed as, e.g., A(+)(0, t,/a 2) =g2/m + ONN(t)/F2.. + A (R+)(0,t,/a2),
(1) ~V/.x
G(0, t,#2) - lim u-l[A(-)(~,t,#2)-vB(-)(v,t,#2)] ~-+0
-
g2 +t,l~.r)+ lim GR(V,t,#2). 2m 2
2F 2 It
(2)
v~0
Here ONN(t) is the usual o term which is a function of the momentum transfer variable t, m is the nucleon mass, g2/4rr(14.28 +--0.18) is the rrN coupling constant, F~r = 92.56 MeV is the pion decay constant and FVl(t) is the nucleon isovector electromagnetic form factor (F~(0) = 1). The "remainder terms", denoted by the suffix R, are those terms that will get contributions from N*(1236) and higher resonances. To take into account the off-mass-shell effect of the N*(1236) propagator in a general way, the 7rNN* interaction Lagrangian depends on an arbitrary parameter Z, first introduced by NEK. From purely theoretical considerations, these authors were able to fix the value of Z to be 4- The pure on-shell calculation corresponds to Z = - {-, while our results agree with those of BPP for Z = - -~. ' However, we have kept Z as a free parameter and its determination from rrN scattering experiments will be discussed in detail elsewhere. Here we present those results that are independent of and weakly dependent on the parameter Z and compare them with the experimental results. We fred
A(+)(O t,/a 2) = ~2g'2 N*''
9
F 2M+m ( . 2 ~] + ( t _ 2 # 2 ) g * 2 [ 2 Y(M,m,#) L~\M2]] 3 "l_"M--m 3M2(M2_m 2)
(2M+m)# 4 3~)2"_1
]
+ (t-202)g*2 {(1 - ~ ) m - 2•M}, 9M 2
(3)
where ~ = 2Z(2Z+l),
V~= 2Z +1,
(4)
and g*2 = 4.526 #-2, the aNN* coupling constant obtained from the 116 MeV width of N*(1236) and Y(M, m,/a) = 3.885 M 3 . From (3) we obtain the important result (+) 2 - 2_g.2 m +2M l / a 4 ] AN*(0'2# 'is2) 9 M2_m2~M2] '
(5)
i.e. the N*(1236) contribution to the amplitude A (+)(O,2 la2, la2) at the Cheng-Dashen point is independent of the N*(1236) off-mass-shell parameter Z. \, • 1± It is easy to see that the spm~resonances do not contribute toA (+) at u = 0, t = 2# 2 and the contributions of the higher spin . ga- resonances to A (+) are at least two orders of magnitude smaller than of that of N*(1236). Hence, the largest contribution to the remainder of A (+) at v = 0, t = 2/*2 comes from N*(1236) as given by (5), and is very small ("0.0093/a-l). Hence, one can safely calculate the ONN term from (1). Using the experimentally [7] determined quantity A(+)(0, 2/a2,/a 2) -g2/m = +1.05 + 0.14/a -1 in (1) one finds after using (5)
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Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
aNN(2//2) = 63.5 -+8.6 MeV
(6)
a similar result already reported by NO [7]. (See, however, the smaller value of ref. [13] ). We note that even though Z is given, we do not have any prediction about the t-dependence of A(+)(0, t,//2) from the relation (3), as we do not know the t-dependence of the o term. To get an idea as to what kind of slope aNN(t) might have, we write ONN(t) = ONN(2#2 ) [1 + o'(t - 2//2)].
(7)
Now using the experimental information about A(÷)(0, t,//2)exp given by NO [7] in the form A(+)(0, t,//2)exp --g2/m = b(t -2gt 2) + 1.05 -+ 0.14/~-1
(8)
with bexp = 1.32 • 0.05//-3, we obtain from (1) and (3): o'
f0.79 = /0"42 ± 0"10
Z=½, Z1
(9)
which are quite large. We have calculated the contributions of N*(1236) to the amplitudes/~(+)(v, t,/fi) and A (-)(v, t,//2) and found that the t-slope parameters for the choice limv_.,o v-1B(+)(v, t,//2) and limv+ 0 v-lA(-)(v, t, #2) are also independent of Z, 13=~-~ l i ~
/~(+)(v,t,V 2)
,=2• 2
-
3(M2- m2) 2
+
(10,11)
3(-M-2~ m-~] '
where X(M, m,//) is a long expression involving the masses M, m and//and is equal to 4.469 M 2. From (11) we have the prediction 3th = --0.167//-5,
(12)
which should be compared with the number obtained by HJS and NO from the analysis of the experimental data (HJS and NO have an extra minus factor for their B (-+) amplitudes): 3~xp =
-0.19 -+0//-5 (HJS)
-0.14 + 0.03//-5 (NO)" Although HJS did not quote any experimental errors, the agreement between (12) and (13) is quite good. We also have
- ~ lim v-lA(-)(v,t,//2) o=~ v~O = -0.076//-4.
2rag*2 t=2~2' 9M2(M2-m2) 2
M 2 ( M + m ) - Y(M,m,//)
(M2-m2)
(13 )
J (16)
The experimental values of,v, after taking account of the contributions due to the isovector form factor, ( --0.20 + 0//-4 (HJS) aexp = {[_0.12 + 0.03//-4 (NO) So the prediction (15)does not seem to agree well with the experimental result (17). We also obtain
(17)
215
Volume 52B, number 2 G(0, 2 V2, U2) -
g2
PHYSICS LETTERS +
1 +4.16(/22/m 2)
2m 2
=/-0885 [ - 0 . 8 7 6 -+ 0.041/2-2,
2F 2 Z-
30 September 1974
2g .2 ~M+2m/22 -i M2 +4Mm+m2/x4 - ½(T/- V~-- ~)/22] 9M 2 [ . ~ 2(MZ_m2) 2
(18)
(19)
4,
We note that the Z-dependent term in (18) is only the last term on the right-hand side. The factor ( r / - x / ~ - ~) 1 This being multiplied by the factor/22 then gives a small negative contribuvanishes at Z = - ~ and is - 4 at Z = ~-. tion (2.7%-1.7% only) compared to the sum of the rest of the terms on the right-hand side of (18). In estimating the error in (19) we assumed that the slope o f F , ( t ) has an error of 18% [10]. The prediction (19) is to be compared with the experimental values obtained by Adler [11], HJS and NO [ -0.834 + 0/2-2 G(0, 2/d2,/22)exp = l - 0 . 7 0 + 0.4 # - 2 { - 0 . 7 8 + 0.03/2-2
(Adler) (HJS) (NO)
The agreement between (19) and (20) is good, although we note that Adler used quite old lrN scattering data in his analysis. We have estimated the contributions of N*(1520) spm~ " a, resonance to (15) and (18) and found them to be of • 1 the order of 14% and 1.4%. The contribution to (11) is of the order of 4%. The contributions of the other spin ~resonances to these relations are also expected to be small. From our work and those of BPP, Osypowski, I-IJS, and others, one has now a reasonably clear understanding, from the theoretical point of view, of the low-energy behaviour of the 7rN scattering amplitudes. It should be pointed out that one should in principle be able to calculate the off-shell contributions of the resonances to the amplitude also using the dispersion relation technique, where the real and imaginary parts of the amplitude are related to each other through the dispersion integral. The advantage of the field theoretical method used here is that it gives the off-mass-shell effects in an explicit manner through the parameter Z. However, it would be interesting to be able to derive equivalent results from the dispersion relation method [12]. We are grateful to Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. We would also like to thank Professor G. Furlan for helpful discussions and comments.
[1] T.P. Cheng and R. Dashen, Phys. Rev. Lett. 26 (1974) 594. [2] M.D. Scadron and L.R. Thebaud, Phys. Lett. B46 (1973) 257; E. Reya, Phys. Lett. B43 (1973) 213 and Chiral symmetry breaking and meson-nucleon-sigmacommutators. A review, to be published in Rev. Mod. Phys., July 1974, [3] G. HiShler, H,P. Jakob and R. Strauss, Phys. Lett. B35 (1974) 445; Nucl. Plays. B39 (1972) 237. [4] L.S. Brown, W.J. Pardee and R.D. Peceei, Phys. Rev. D4 (1971) 2801. [5] P.J. Bussey et al., NucL Phys. B58 (1973) 333. [6] D.V.Bugg, A.A. Carter and J.R. Carter, Phys. Lett. B44 (1973) 278; Lettere al Nuovo Cimento 8 (1973) 639; 9 (1974) 336. [7] H. Nielsen and G.C. Oades, Nucl. Phys. B72 (1974) 310. [8] E.T. Osypowski, Nucl. Phys. B21 (1970) 615. [9] L.M. Nath, B. Etemadi and J.D. Kimel, Phys. Rev. D3 (1971) 2153. [10] L.H. Chan et al., Phys. Rev. 141 (1966) 1298. [11] S.L. Adler, Phys. Rev. 140 (1965) B736. [12] G. HiShlerand P. Stichel, Z. Physik 245 (1971) 387. [13] W. Langbein, preprint. University of Trier-Kaiserslautern,submitted to the XVII Intern. Conf. on High energy physics (London 1974).
216
PHYSICS LETTERS
30 September 1974
O N T H E L O W - E N E R G Y rrN S C A T T E R I N G A M P L I T U D E S L.M. NATH and A.Q. SARKER International Centre for Theoretical Physics, Trieste, Italy and Department of Physics, Dacca University, Dacca-2, Bangladesh Received 25 July 1974 Revised manuscript received 20 August 1974 The pion-nucleon scattering amplitudes with current algebra constraints involving "remainder terms" are calculated by saturating them with the N*(1236) contribution having off-~ass shell effects and compared with experiments. Since the estimate of the so-called o term by Cheng and Dashen [1 ] (CD) from the low-energy rrN scattering amplitudes, there has been considerable controversy over the subject. The value of the o term ~110 MeV, as deterrnined by CD, is quite large compared with many other estimates*. It is, moreover, hard to understand such large value of the o term from the point of view of the SU(3) ® SU(3)'symmetry breaking. Further criticism of the work of CD was made on two different considerations. The broad area subtraction technique used by CD seems to over-emphasise the low-energy data points considerably, as first pointed out by H6hler, Jakob and Strauss [3] (HJS), and the methods of analytic continuation of the amplitudes from the physical region to unphysical points are ambiguous and doubtful unless one can estimate with reasonable certainty the contributions of the N*(1236) and other higher resonances to the rrN scattering amplitudes at the relevant unphysical points. Brown, Pardee and Peccei [4] (BPP) referred to these latter contributions as the "remainder terms" of the amplitudes. Since the work of CD, the experimental position on the low-energy cross section data and the phase shift analy sis of the rrN scattering have considerably improved, due to the recent works of Bussey et al. [5 ] and the parameters, such as the coupling constant f, the scattering lengths a 1 , a 3 as well as the dependence of the amplitudes on the energy ~, momentum transfer t (say, within the Lehmann ellipse) are now known with far better accuracy from the analysis of Bugg, Carter and Carter [6] (BCC), HJS and Nielsen and Oades [7] (NO). A more recent and independent determination of the sigma term by Langbein [ 13] from the same data gives ONN(t = 292 ) = 46 -+ 14 MeV. The main difference between the work of Langbein and that of NO is that the former finds it impossible to make a reliable estimate of the coefficient of t 2 in the expansion of A(+)(0, t, ~2). In a private communication Dr. E. Reya suggested to the authors that the experimental term quadratic in t may not be used, as there are several sources of errors which are of the order/~4. From the theoretical point of view, the significant contribution to remove the o term controversy was made by BPP, and Osypowski [8], who have written the current algebra constraints on the amplitude as an identity involving a "remainder term" and kept the plans on the mass-shell. BPP further pointed out that this remainder term for the amplitude A(+)0,, t,/a 2) at the unphysical point ~ = 0, t = 2# 2 (sometimes referred to as the ChengDashen point) receives its largest contributions from the N*(1236) and is of the order of ( m 2 / M 3 , ) . Otherwise the determination of the o term from A(+)(0, t = 2/~2,/~2) would not make sense as the terms o-1 and A (+)(0, t = 2g 2,/a 2) are of the order of m~-1. We used the letter/~ to denote the mass of the charged plan. We point out that the BPP and Osypowski calculations for the N*(1236) contributions were done with N*(1236) on the mass shell. While the latter author has taken the usual rrNN* interaction Lagrangian, BPP used a modified interaction and neglected the contact terms.
* We refer to ref. [2] for the collection of up-to-date references on the different determinations of the a term. 213
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
We have re-examined the calculations of the low-energy lrN scattering amplitudes in the spirit of BPP and Osypowski and evaluated the N*(1236) contributions; however, by taking into consideration its off-mass shell effects in a Lorentz covariant way following the earlier work of Nath, Etemadi and Kirnel [9] (NEK). The results obtained go a considerable way (further than those obtained by BPP) in understanding the low-energy behaviour of the IrN scattering amplitudes. They remove some of the doubts that were expressed in the works of Cheng and Dashen and others, and sort out the points of controversy in a general way. It is therefore considered worthwhile to report some of these results here. One has the four independent amplitudes A (+)(~, t,/z2), A(-)(v, t, #2), B (+)(v, t,/a2) and B (-)(v, t, #2) for the 7rN scattering (we follow the convention and notations of BPP). The current algebra constraints on these amplitudes or any suitable linear combinations of them can be expressed as, e.g., A(+)(0, t,/a 2) =g2/m + ONN(t)/F2.. + A (R+)(0,t,/a2),
(1) ~V/.x
G(0, t,#2) - lim u-l[A(-)(~,t,#2)-vB(-)(v,t,#2)] ~-+0
-
g2 +t,l~.r)+ lim GR(V,t,#2). 2m 2
2F 2 It
(2)
v~0
Here ONN(t) is the usual o term which is a function of the momentum transfer variable t, m is the nucleon mass, g2/4rr(14.28 +--0.18) is the rrN coupling constant, F~r = 92.56 MeV is the pion decay constant and FVl(t) is the nucleon isovector electromagnetic form factor (F~(0) = 1). The "remainder terms", denoted by the suffix R, are those terms that will get contributions from N*(1236) and higher resonances. To take into account the off-mass-shell effect of the N*(1236) propagator in a general way, the 7rNN* interaction Lagrangian depends on an arbitrary parameter Z, first introduced by NEK. From purely theoretical considerations, these authors were able to fix the value of Z to be 4- The pure on-shell calculation corresponds to Z = - {-, while our results agree with those of BPP for Z = - -~. ' However, we have kept Z as a free parameter and its determination from rrN scattering experiments will be discussed in detail elsewhere. Here we present those results that are independent of and weakly dependent on the parameter Z and compare them with the experimental results. We fred
A(+)(O t,/a 2) = ~2g'2 N*''
9
F 2M+m ( . 2 ~] + ( t _ 2 # 2 ) g * 2 [ 2 Y(M,m,#) L~\M2]] 3 "l_"M--m 3M2(M2_m 2)
(2M+m)# 4 3~)2"_1
]
+ (t-202)g*2 {(1 - ~ ) m - 2•M}, 9M 2
(3)
where ~ = 2Z(2Z+l),
V~= 2Z +1,
(4)
and g*2 = 4.526 #-2, the aNN* coupling constant obtained from the 116 MeV width of N*(1236) and Y(M, m,/a) = 3.885 M 3 . From (3) we obtain the important result (+) 2 - 2_g.2 m +2M l / a 4 ] AN*(0'2# 'is2) 9 M2_m2~M2] '
(5)
i.e. the N*(1236) contribution to the amplitude A (+)(O,2 la2, la2) at the Cheng-Dashen point is independent of the N*(1236) off-mass-shell parameter Z. \, • 1± It is easy to see that the spm~resonances do not contribute toA (+) at u = 0, t = 2# 2 and the contributions of the higher spin . ga- resonances to A (+) are at least two orders of magnitude smaller than of that of N*(1236). Hence, the largest contribution to the remainder of A (+) at v = 0, t = 2/*2 comes from N*(1236) as given by (5), and is very small ("0.0093/a-l). Hence, one can safely calculate the ONN term from (1). Using the experimentally [7] determined quantity A(+)(0, 2/a2,/a 2) -g2/m = +1.05 + 0.14/a -1 in (1) one finds after using (5)
214
Volume 52B, number 2
PHYSICS LETTERS
30 September 1974
aNN(2//2) = 63.5 -+8.6 MeV
(6)
a similar result already reported by NO [7]. (See, however, the smaller value of ref. [13] ). We note that even though Z is given, we do not have any prediction about the t-dependence of A(+)(0, t,//2) from the relation (3), as we do not know the t-dependence of the o term. To get an idea as to what kind of slope aNN(t) might have, we write ONN(t) = ONN(2#2 ) [1 + o'(t - 2//2)].
(7)
Now using the experimental information about A(÷)(0, t,//2)exp given by NO [7] in the form A(+)(0, t,//2)exp --g2/m = b(t -2gt 2) + 1.05 -+ 0.14/~-1
(8)
with bexp = 1.32 • 0.05//-3, we obtain from (1) and (3): o'
f0.79 = /0"42 ± 0"10
Z=½, Z1
(9)
which are quite large. We have calculated the contributions of N*(1236) to the amplitudes/~(+)(v, t,/fi) and A (-)(v, t,//2) and found that the t-slope parameters for the choice limv_.,o v-1B(+)(v, t,//2) and limv+ 0 v-lA(-)(v, t, #2) are also independent of Z, 13=~-~ l i ~
/~(+)(v,t,V 2)
,=2• 2
-
3(M2- m2) 2
+
(10,11)
3(-M-2~ m-~] '
where X(M, m,//) is a long expression involving the masses M, m and//and is equal to 4.469 M 2. From (11) we have the prediction 3th = --0.167//-5,
(12)
which should be compared with the number obtained by HJS and NO from the analysis of the experimental data (HJS and NO have an extra minus factor for their B (-+) amplitudes): 3~xp =
-0.19 -+0//-5 (HJS)
-0.14 + 0.03//-5 (NO)" Although HJS did not quote any experimental errors, the agreement between (12) and (13) is quite good. We also have
- ~ lim v-lA(-)(v,t,//2) o=~ v~O = -0.076//-4.
2rag*2 t=2~2' 9M2(M2-m2) 2
M 2 ( M + m ) - Y(M,m,//)
(M2-m2)
(13 )
J (16)
The experimental values of,v, after taking account of the contributions due to the isovector form factor, ( --0.20 + 0//-4 (HJS) aexp = {[_0.12 + 0.03//-4 (NO) So the prediction (15)does not seem to agree well with the experimental result (17). We also obtain
(17)
215
Volume 52B, number 2 G(0, 2 V2, U2) -
g2
PHYSICS LETTERS +
1 +4.16(/22/m 2)
2m 2
=/-0885 [ - 0 . 8 7 6 -+ 0.041/2-2,
2F 2 Z-
30 September 1974
2g .2 ~M+2m/22 -i M2 +4Mm+m2/x4 - ½(T/- V~-- ~)/22] 9M 2 [ . ~ 2(MZ_m2) 2
(18)
(19)
4,
We note that the Z-dependent term in (18) is only the last term on the right-hand side. The factor ( r / - x / ~ - ~) 1 This being multiplied by the factor/22 then gives a small negative contribuvanishes at Z = - ~ and is - 4 at Z = ~-. tion (2.7%-1.7% only) compared to the sum of the rest of the terms on the right-hand side of (18). In estimating the error in (19) we assumed that the slope o f F , ( t ) has an error of 18% [10]. The prediction (19) is to be compared with the experimental values obtained by Adler [11], HJS and NO [ -0.834 + 0/2-2 G(0, 2/d2,/22)exp = l - 0 . 7 0 + 0.4 # - 2 { - 0 . 7 8 + 0.03/2-2
(Adler) (HJS) (NO)
The agreement between (19) and (20) is good, although we note that Adler used quite old lrN scattering data in his analysis. We have estimated the contributions of N*(1520) spm~ " a, resonance to (15) and (18) and found them to be of • 1 the order of 14% and 1.4%. The contribution to (11) is of the order of 4%. The contributions of the other spin ~resonances to these relations are also expected to be small. From our work and those of BPP, Osypowski, I-IJS, and others, one has now a reasonably clear understanding, from the theoretical point of view, of the low-energy behaviour of the 7rN scattering amplitudes. It should be pointed out that one should in principle be able to calculate the off-shell contributions of the resonances to the amplitude also using the dispersion relation technique, where the real and imaginary parts of the amplitude are related to each other through the dispersion integral. The advantage of the field theoretical method used here is that it gives the off-mass-shell effects in an explicit manner through the parameter Z. However, it would be interesting to be able to derive equivalent results from the dispersion relation method [12]. We are grateful to Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. We would also like to thank Professor G. Furlan for helpful discussions and comments.
[1] T.P. Cheng and R. Dashen, Phys. Rev. Lett. 26 (1974) 594. [2] M.D. Scadron and L.R. Thebaud, Phys. Lett. B46 (1973) 257; E. Reya, Phys. Lett. B43 (1973) 213 and Chiral symmetry breaking and meson-nucleon-sigmacommutators. A review, to be published in Rev. Mod. Phys., July 1974, [3] G. HiShler, H,P. Jakob and R. Strauss, Phys. Lett. B35 (1974) 445; Nucl. Plays. B39 (1972) 237. [4] L.S. Brown, W.J. Pardee and R.D. Peceei, Phys. Rev. D4 (1971) 2801. [5] P.J. Bussey et al., NucL Phys. B58 (1973) 333. [6] D.V.Bugg, A.A. Carter and J.R. Carter, Phys. Lett. B44 (1973) 278; Lettere al Nuovo Cimento 8 (1973) 639; 9 (1974) 336. [7] H. Nielsen and G.C. Oades, Nucl. Phys. B72 (1974) 310. [8] E.T. Osypowski, Nucl. Phys. B21 (1970) 615. [9] L.M. Nath, B. Etemadi and J.D. Kimel, Phys. Rev. D3 (1971) 2153. [10] L.H. Chan et al., Phys. Rev. 141 (1966) 1298. [11] S.L. Adler, Phys. Rev. 140 (1965) B736. [12] G. HiShlerand P. Stichel, Z. Physik 245 (1971) 387. [13] W. Langbein, preprint. University of Trier-Kaiserslautern,submitted to the XVII Intern. Conf. on High energy physics (London 1974).
216