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Dispersion sum rules for the Baryon spectrum in an O(3,1) dynamical group model
Volume
28B, number
PHYSICS
10
DISPERSION IN
SUM AN
O(3,l)
RULES
LETTERS
FOR
P. H. FRAMPTON The Enrico
THE
DYNAMICAL
Fermi
Received
BARYON
GROUP
1969
SPECTRUM
MODEL
*
and B. HAMPRECHT
Institute. Chicago,
17 March
The University Illinois,
28 January
of Chicago,
USA 1969
We study the saturation of superconvergence relations by the known N (isospin l/2) and A(isospin 3/2 ) baryon resonances. Two convenient sum rules occur in elastic n-A(1236) scattering. One sum rule has contributions only from N states, the other only from A states. For the pseudoscalar form factors we use an O(3,l) dynamical group model proposed by Barut and Kleinert [I]. The well known A spectrum is consistent with the A sum rule. The known N resonances (for which the phase shift analyses are
more ambiguous) are inadequate to saturate the N sum rule, and the nature of the missing contribution is briefly discussed,
The structure of the non-strange baryonic resonance spectrum in the mass region below 2 GeV is slowly becoming clarified through the extensive work by different groups on the pionnucleon phase shift analysis [2]. The A (isospin 3/2) spectrum, in particular, now seems to be fairly well known up to at least 1.9 GeV and possibly higher through the most recent CERN analysis. The N (isospin l/2) analyses are more ambiguous and there is disagreement amongst them about whether certain resonances exist. It therefore seems an opportune time to study whether the N and A resonant states so far proposed are adequate to saturate dispersion sum rules [3] and whether it is clear that resonant contributions must be added to achieve saturation. This then might provide at least an indication to the phase-shift analysts of where extra resonant states are to be expected from the theoretical considerations of Regge asymptotic behaviour and superconvergence. In this letter we derive two sum rules which are well suited for the present purpose and which occur in elastic or-~(1236) scattering. Assuming the absence of isospin 5/2 contributions, one of the sum rules has contributions only from the N intermediate states, the other only from A intermediate states. This feature then provides the opportunity of investigating separately the N and the A resonance spectra. The inte* This work was supported Energy Commission.
664
in part by the US Atomic
grands of the two sum rules we consider converge with greater rapidity than is needed for superconvergence and it is very reasonable to expect saturation from states below 2 GeV in mass. This expectation is supported a posteriori by the fact that any higher mass intermediate states we introduce always give very small contributions. Throughout we use the zero-width approximation and neglect background. It is possible, but we would regard it as exceedingly unlikely, that the errors introduced by these approximations are sufficient to obviate our main conclusion that not all of the low-mass N resonances have so far been isolated. To study saturation of the sum rules we require knowledge of the baryonic pseudoscalar form factors. Fortunately there is now available the O(3,l) dynamical group model of Barut and Kleinert which provides a fairly good approximation to the physical form factors in the momentum transfer region which is of interest here. This model has been used to correlate the decays of the baryon resonances and quite good agreement with experiment has been found [4]. In the following we first sketch the method for calculating the pseudoscalar form factors in the O(3,l) model. We then derive the superconvergence relations in r-~(1236) scattering and show how the O(3,l) form factors are used to find the contributions to the sum rules. Finally we give the results for the N and A resonance spectra. The simplest possible dynamical group that can be used to calculate the baryonic pseudo-
PHYSICS
Volume 28B. number 10
scalar form factors is the group 0(3, l), extended by parity [l]. The baryon states are assigned into unitary irreducible representations of 0(3, l), for which the representations are characterized by two Casimir operators j. and jl (we use the notation of Naimark’s book [5]). The value j, = f is fixed by the lowest spin in the representation and jl, which can have any pure imaginary value for the principal series, is fixed at jl = 5i by fitting the partial decay widths for the decay of the excited baryons into the ground state plus a pion [4]. A state at rest may be characterized by 1J J3 j,jI t) where J, J3 are the spin and its third component and I distinguishes the particular representation (or tower) to which the state is assigned. Eigenstates of parity are
fllJJ~j~.iltN)
= IJJ3jojlt)
= N(-1)J-$IJJ3jojl
tN>
$(5i)tN)
=gtBi
‘J
’
N
(5) (3)
where 71is the pion field operator and gt is the coupling constant associated with the tower t coupling to the ~(1236)~ system. These form factors are related to the matrix element of a finite Lorentz transformation between states at rest by [4] $J+
(I) = Re Vp(t;)
BP-
(5) = Im
&J(t)
(44
(4b)
where V?(5)
M@
= (AlP+A2)PPP
= (qx~(5i)Iexp(-i1M3)1JX~(5i)j
(6)
+ (A3B+A4)
Q”QN
+
(5)
is the quantity for which a closed form in terms of hypergeometric functions is given by Strom [6]. For the present work it is sufficient to use the exponential approximation to the Strom form, as described in the appendix of ref. 3,
(7)
whereP=p+p’, Q=q+q’. The amplitude A1 corresponds to triple helicity flip in the t-channel and its Regge asymptotic behaviour is given by Al(v,
(2)
We shall need to consider only baryon state in motion along the 3-axis and we write such a state as ] Jx 5 j,jl t N) where A is the helicity (A =J3) and { is the rapidity (5 =arsinh v/c). This state is obtained from the rest state by the operation of a boost operator exp(-iM3 5) in the usual way. The form factors that we shall need are of the form ~(1236)(n(JX0
T(v, t) = Ep (4’) Mpcu ucy (q) and
+ (A5~+A6kB”l
where N =f is the parity of the J=$ ground state, and in general is the ‘normality’ of the excited states
($5+(5i)
since these follow the exact value of expression (5) closely in the region of small momentum transfer. This means that the vertex function in eq. (3) will be represented in the superconvergence sum rules by a threshold factor times a function of the form exp( -o(2). Two superconvergence sum rules [3], which are ideally suited for our present investigation, exist in r-~(1236) elastic scattering [7]. Taking the s-channel process as n(P) + A(q) - n(p) + A(q’) and defining s = (p+ q)2, t = (p-p’)2, u = (p - q’)2, v = $(u -s) we may develop the T-matrix as follows:
+NIJJ3jo-jlt) (1)
PIJJ3jojltN)
17 March 1969
LETTERS
Consequently
0)“”
at fixed t = 0
.a(o)-3
(6)
the sum rule l”OImAl(v,O)dv=O
(2)
holds for both-i:ospin Tt=O and Tt=2 in the t-channel (the sum rule for Tt= 1 is trivial by crossing). Notice that from unitarity alone A1 must converge at least as fast as l/v2 for u -t cc), so that from this general consideration we expect the sum rule for A1 to converge rapidly. Using the isospin crossing matrix, and assuming absence of isospin 5/2 contributions, we find that the N and A contributions (in the s - and u channels) should cancel separately in eq. (9). The sum rules we consider are therefore 0
(N states only)
(10)
dv = 0
(A states only)
(11)
./"mImAl(v,O) dv =
-Co
1”
Im Al(v,O)
Wcnext write A1 (u, t) in terms of S-channel helicity amplitudes f “kf xi (u, t) because a zero width resonance approximation to the latter may be immediately related to the dynamical form factors BX+J~N(<) by Imf&hi
(:,t)
’ B$ ={J, N)R
(12)
=
N(‘tR) dJ,Q(es)BJ;’
N(
6(U-UR)
665
Volume 28B, number 10
PHYSICS
Table 1 Contributions to eq. (10) from the proposed isospin l/2 baryon resonances. Coefficient ofg2
t
Upper limit for g,2
N(939)
+0.47
g2 = 7.3
Pll(1470)
+0.16
< 2.6
D13(1518)
-0.71
< 0.37
S1I (1550)
+0.51
< 5.0
D15(1680)
+0.026
F15(1688)
+0.28
< 0.8
S11(1710)
10.29
< 6.0
%3(1730)
-0.23
< 2.0
Pll(1750)
+0.053
P13(1860)
-0.070
< 20
P17(1980)
co.005
< 140
Term
< 70
- 22~
(M; + M;)
- M;)2 (13)
4M2 M2 R A
The required relation pendent f ‘kf hi is
between A1 and the inde-
9
A1 =
whereEA=qo=qbandk=IqI=Iq/ inthe centre of mass frame and we have defined -s
f Xff hi = (15) = (cos fes)
-(Xi+Xfl
. 1 -/Xi-U (sin 2 63)
f
’ hf,Xi
The values of theg; are constrained by the requirement that the partial decay width for R- A(1236)a should be not greater than the total inelastic width of the resonance R seen in the TN phase shift analysis. The coupling constant for the nucleon intermediate state is fixed by the experimental decay width (A(1236) --t Na) = = 0.120 GeV. 666
Upper limit for gf
P33(1236)
-1.23
S31(1640)
+0.38
<
20
P33(1690)
-0.18
<
30
D33(1690)
-0.29
<
2
F35(1910)
+0.08
<
4
P31(1930)
+0.024
<
1.5
F37(1950) -0.005 D35(1960) and higher mass states negligible.
< 2.0
+ (M;
Coefficient of r$
+0.005
where the sum is over all intermediate states R, Qs is the c.o.m. scattering angle measured between the initial and final pion momenta, and UR is the value of v for s = Mi. The rapidity & is found from 4
17 March 1969
Table 2 Contributions to eq. (11) from the proposed isospin 3/2 baryon resonances. Term
and higher mass states negligible.
tanh2 IR =
LETTERS
50 < g2 < 100 <
40
The contributions to the sum rules, eqs. (10) and (ll), from all of the N and A resonances proposed in the analysis of the pion-nucleon phase shifts are given in tables 1 and 2 respectively, together with the corresponding upper limits on the coupling constants g:, taken from the known inelastic width in the nN system. We can draw the following conclusions from the results in tables 1 and 2. The approximations made in the calculation should first be summarized; they are: (i) zero-width resonances (ii) neglect of background and (iii) form factors from the O(3,l) model. The conclusions are (A) The N sum rule, eq. (10) and table 1 In this sum rule the nucleon N(939) term is large and positive. It cannot be cancelled by all the other proposed N resonances taken together, independent of how the resonances are assigned to particular O(3, 1) towers. Further, we find that spin $ and 2 contributions are positive in this energy region so that only a low-mass spin 3/2 state can lead to saturation if saturation is to be achieved by a single resonance with reasonable spin, mass and coupling constant. We note that if the debated Fl7 (1980) state is firmed as a resonance, and is the recurrence of a P13 state then the latter leads readily to saturation. This P13 state might couple only weakly to, and hence be difficult to detect in, the pionnucleon channel. Finally concerning this sum rule we observe that the states of mass greater than 2 GeV give completely negligible contributions and therefore that the sum appears to converge rapidly as expected from the general considerations described earlier in the letter. (B) The A sum rule, eq. (11) and table 2 Here the known A states can saturate the sum rule. We find many possible such fits and since, without consideration of other superconvergence relations, there is no reason to prefer one fit to
Volume 28B. number 10
PHYSIC S LETTERS
another, we cannot predict any of the coupling constants. This sum rule does give, however, an indication of how one should assign coupling constants in the O(3,l) model of pseudoscalar form factors [l]. The simplest possibility would be to assign a universal coupling constant g(tlt2) depending only on the two O(3,l) baryon towers coupled to the pion, and independent of the particular members of the towers. This works well for the decays of the baryon resonances (mass Ml) into the ground state (mass M2) of the same or a different tower [4]. Here of course the masses must satisfy the inequality Ml > M2+M, and we may consider the question: is it appropriate to use the same universal coupling constant g(tl, 12) when this mass inequality does not hold? Now for saturation of the superconvergence relation, eq. (ll), it is required that the couplings of h(1236) and F37(1940) (which fall naturally into the same O(3,l) representation) to ~(1236)n be quite different. Another counterexample is that the experimental widths for A(1236) + Nn and F15(l688) 4 A(1236)~ require the corresponding coupling constants to differ markedly, while the F15(l688) and nucleon are fitted into the same O(3,l) tower [4]. Thus it ap-
17 March 1969
pears that the answer to our question is negative and that, in the O(3,l) model, we can assume a universal coupling constant g(fl, t2) only where the inequality Ml > (M2 + M,) is satisfied by all of the vertices to be compared, as is the case, for example, in ref. 3. Further details of this work, together with a discussion of the extent to which the results for the N spectrum is independent of the assumption of the 0(3, 1) model and a treatment of other superconvergence relations which hold for elastic V- A(1236) scattering and in the inelastic process nN+ x~(1236), will be given elsewhere.
References
1. A. 0. Barut and H. Kleinert , Phys. Rev. Letters 18 (1967) 754. 2. A. Donnachie, Proc. 14th Intern. Conf. on Highenergy physics, p. 139. 3. V. De Alfaro, S. Fubini, G. Rossetti and G. Furlan, Phys. Letters 21 (1966) 576. 4. B. Hamprecht and H. Kleinert, to be published. 5. M. A. Naimark, Linear representations of the Lorentz group (MacMillan, New York) 1964. 6. S. StrBm, Arkiv far Fysik 33 (1966) 465. 7. P. H. Frampton, Nucl. Phys. B2 (1967) 518 and Oxford D. Phil. Thesis 1968 (unpublished).
*****
667
28B, number
PHYSICS
10
DISPERSION IN
SUM AN
O(3,l)
RULES
LETTERS
FOR
P. H. FRAMPTON The Enrico
THE
DYNAMICAL
Fermi
Received
BARYON
GROUP
1969
SPECTRUM
MODEL
*
and B. HAMPRECHT
Institute. Chicago,
17 March
The University Illinois,
28 January
of Chicago,
USA 1969
We study the saturation of superconvergence relations by the known N (isospin l/2) and A(isospin 3/2 ) baryon resonances. Two convenient sum rules occur in elastic n-A(1236) scattering. One sum rule has contributions only from N states, the other only from A states. For the pseudoscalar form factors we use an O(3,l) dynamical group model proposed by Barut and Kleinert [I]. The well known A spectrum is consistent with the A sum rule. The known N resonances (for which the phase shift analyses are
more ambiguous) are inadequate to saturate the N sum rule, and the nature of the missing contribution is briefly discussed,
The structure of the non-strange baryonic resonance spectrum in the mass region below 2 GeV is slowly becoming clarified through the extensive work by different groups on the pionnucleon phase shift analysis [2]. The A (isospin 3/2) spectrum, in particular, now seems to be fairly well known up to at least 1.9 GeV and possibly higher through the most recent CERN analysis. The N (isospin l/2) analyses are more ambiguous and there is disagreement amongst them about whether certain resonances exist. It therefore seems an opportune time to study whether the N and A resonant states so far proposed are adequate to saturate dispersion sum rules [3] and whether it is clear that resonant contributions must be added to achieve saturation. This then might provide at least an indication to the phase-shift analysts of where extra resonant states are to be expected from the theoretical considerations of Regge asymptotic behaviour and superconvergence. In this letter we derive two sum rules which are well suited for the present purpose and which occur in elastic or-~(1236) scattering. Assuming the absence of isospin 5/2 contributions, one of the sum rules has contributions only from the N intermediate states, the other only from A intermediate states. This feature then provides the opportunity of investigating separately the N and the A resonance spectra. The inte* This work was supported Energy Commission.
664
in part by the US Atomic
grands of the two sum rules we consider converge with greater rapidity than is needed for superconvergence and it is very reasonable to expect saturation from states below 2 GeV in mass. This expectation is supported a posteriori by the fact that any higher mass intermediate states we introduce always give very small contributions. Throughout we use the zero-width approximation and neglect background. It is possible, but we would regard it as exceedingly unlikely, that the errors introduced by these approximations are sufficient to obviate our main conclusion that not all of the low-mass N resonances have so far been isolated. To study saturation of the sum rules we require knowledge of the baryonic pseudoscalar form factors. Fortunately there is now available the O(3,l) dynamical group model of Barut and Kleinert which provides a fairly good approximation to the physical form factors in the momentum transfer region which is of interest here. This model has been used to correlate the decays of the baryon resonances and quite good agreement with experiment has been found [4]. In the following we first sketch the method for calculating the pseudoscalar form factors in the O(3,l) model. We then derive the superconvergence relations in r-~(1236) scattering and show how the O(3,l) form factors are used to find the contributions to the sum rules. Finally we give the results for the N and A resonance spectra. The simplest possible dynamical group that can be used to calculate the baryonic pseudo-
PHYSICS
Volume 28B. number 10
scalar form factors is the group 0(3, l), extended by parity [l]. The baryon states are assigned into unitary irreducible representations of 0(3, l), for which the representations are characterized by two Casimir operators j. and jl (we use the notation of Naimark’s book [5]). The value j, = f is fixed by the lowest spin in the representation and jl, which can have any pure imaginary value for the principal series, is fixed at jl = 5i by fitting the partial decay widths for the decay of the excited baryons into the ground state plus a pion [4]. A state at rest may be characterized by 1J J3 j,jI t) where J, J3 are the spin and its third component and I distinguishes the particular representation (or tower) to which the state is assigned. Eigenstates of parity are
fllJJ~j~.iltN)
= IJJ3jojlt)
= N(-1)J-$IJJ3jojl
tN>
$(5i)tN)
=gtBi
‘J
’
N
(5) (3)
where 71is the pion field operator and gt is the coupling constant associated with the tower t coupling to the ~(1236)~ system. These form factors are related to the matrix element of a finite Lorentz transformation between states at rest by [4] $J+
(I) = Re Vp(t;)
BP-
(5) = Im
&J(t)
(44
(4b)
where V?(5)
M@
= (AlP+A2)PPP
= (qx~(5i)Iexp(-i1M3)1JX~(5i)j
(6)
+ (A3B+A4)
Q”QN
+
(5)
is the quantity for which a closed form in terms of hypergeometric functions is given by Strom [6]. For the present work it is sufficient to use the exponential approximation to the Strom form, as described in the appendix of ref. 3,
(7)
whereP=p+p’, Q=q+q’. The amplitude A1 corresponds to triple helicity flip in the t-channel and its Regge asymptotic behaviour is given by Al(v,
(2)
We shall need to consider only baryon state in motion along the 3-axis and we write such a state as ] Jx 5 j,jl t N) where A is the helicity (A =J3) and { is the rapidity (5 =arsinh v/c). This state is obtained from the rest state by the operation of a boost operator exp(-iM3 5) in the usual way. The form factors that we shall need are of the form ~(1236)(n(JX0
T(v, t) = Ep (4’) Mpcu ucy (q) and
+ (A5~+A6kB”l
where N =f is the parity of the J=$ ground state, and in general is the ‘normality’ of the excited states
($5+(5i)
since these follow the exact value of expression (5) closely in the region of small momentum transfer. This means that the vertex function in eq. (3) will be represented in the superconvergence sum rules by a threshold factor times a function of the form exp( -o(2). Two superconvergence sum rules [3], which are ideally suited for our present investigation, exist in r-~(1236) elastic scattering [7]. Taking the s-channel process as n(P) + A(q) - n(p) + A(q’) and defining s = (p+ q)2, t = (p-p’)2, u = (p - q’)2, v = $(u -s) we may develop the T-matrix as follows:
+NIJJ3jo-jlt) (1)
PIJJ3jojltN)
17 March 1969
LETTERS
Consequently
0)“”
at fixed t = 0
.a(o)-3
(6)
the sum rule l”OImAl(v,O)dv=O
(2)
holds for both-i:ospin Tt=O and Tt=2 in the t-channel (the sum rule for Tt= 1 is trivial by crossing). Notice that from unitarity alone A1 must converge at least as fast as l/v2 for u -t cc), so that from this general consideration we expect the sum rule for A1 to converge rapidly. Using the isospin crossing matrix, and assuming absence of isospin 5/2 contributions, we find that the N and A contributions (in the s - and u channels) should cancel separately in eq. (9). The sum rules we consider are therefore 0
(N states only)
(10)
dv = 0
(A states only)
(11)
./"mImAl(v,O) dv =
-Co
1”
Im Al(v,O)
Wcnext write A1 (u, t) in terms of S-channel helicity amplitudes f “kf xi (u, t) because a zero width resonance approximation to the latter may be immediately related to the dynamical form factors BX+J~N(<) by Imf&hi
(:,t)
’ B$ ={J, N)R
(12)
=
N(‘tR) dJ,Q(es)BJ;’
N(
6(U-UR)
665
Volume 28B, number 10
PHYSICS
Table 1 Contributions to eq. (10) from the proposed isospin l/2 baryon resonances. Coefficient ofg2
t
Upper limit for g,2
N(939)
+0.47
g2 = 7.3
Pll(1470)
+0.16
< 2.6
D13(1518)
-0.71
< 0.37
S1I (1550)
+0.51
< 5.0
D15(1680)
+0.026
F15(1688)
+0.28
< 0.8
S11(1710)
10.29
< 6.0
%3(1730)
-0.23
< 2.0
Pll(1750)
+0.053
P13(1860)
-0.070
< 20
P17(1980)
co.005
< 140
Term
< 70
- 22~
(M; + M;)
- M;)2 (13)
4M2 M2 R A
The required relation pendent f ‘kf hi is
between A1 and the inde-
9
A1 =
whereEA=qo=qbandk=IqI=Iq/ inthe centre of mass frame and we have defined -s
f Xff hi = (15) = (cos fes)
-(Xi+Xfl
. 1 -/Xi-U (sin 2 63)
f
’ hf,Xi
The values of theg; are constrained by the requirement that the partial decay width for R- A(1236)a should be not greater than the total inelastic width of the resonance R seen in the TN phase shift analysis. The coupling constant for the nucleon intermediate state is fixed by the experimental decay width (A(1236) --t Na) = = 0.120 GeV. 666
Upper limit for gf
P33(1236)
-1.23
S31(1640)
+0.38
<
20
P33(1690)
-0.18
<
30
D33(1690)
-0.29
<
2
F35(1910)
+0.08
<
4
P31(1930)
+0.024
<
1.5
F37(1950) -0.005 D35(1960) and higher mass states negligible.
< 2.0
+ (M;
Coefficient of r$
+0.005
where the sum is over all intermediate states R, Qs is the c.o.m. scattering angle measured between the initial and final pion momenta, and UR is the value of v for s = Mi. The rapidity & is found from 4
17 March 1969
Table 2 Contributions to eq. (11) from the proposed isospin 3/2 baryon resonances. Term
and higher mass states negligible.
tanh2 IR =
LETTERS
50 < g2 < 100 <
40
The contributions to the sum rules, eqs. (10) and (ll), from all of the N and A resonances proposed in the analysis of the pion-nucleon phase shifts are given in tables 1 and 2 respectively, together with the corresponding upper limits on the coupling constants g:, taken from the known inelastic width in the nN system. We can draw the following conclusions from the results in tables 1 and 2. The approximations made in the calculation should first be summarized; they are: (i) zero-width resonances (ii) neglect of background and (iii) form factors from the O(3,l) model. The conclusions are (A) The N sum rule, eq. (10) and table 1 In this sum rule the nucleon N(939) term is large and positive. It cannot be cancelled by all the other proposed N resonances taken together, independent of how the resonances are assigned to particular O(3, 1) towers. Further, we find that spin $ and 2 contributions are positive in this energy region so that only a low-mass spin 3/2 state can lead to saturation if saturation is to be achieved by a single resonance with reasonable spin, mass and coupling constant. We note that if the debated Fl7 (1980) state is firmed as a resonance, and is the recurrence of a P13 state then the latter leads readily to saturation. This P13 state might couple only weakly to, and hence be difficult to detect in, the pionnucleon channel. Finally concerning this sum rule we observe that the states of mass greater than 2 GeV give completely negligible contributions and therefore that the sum appears to converge rapidly as expected from the general considerations described earlier in the letter. (B) The A sum rule, eq. (11) and table 2 Here the known A states can saturate the sum rule. We find many possible such fits and since, without consideration of other superconvergence relations, there is no reason to prefer one fit to
Volume 28B. number 10
PHYSIC S LETTERS
another, we cannot predict any of the coupling constants. This sum rule does give, however, an indication of how one should assign coupling constants in the O(3,l) model of pseudoscalar form factors [l]. The simplest possibility would be to assign a universal coupling constant g(tlt2) depending only on the two O(3,l) baryon towers coupled to the pion, and independent of the particular members of the towers. This works well for the decays of the baryon resonances (mass Ml) into the ground state (mass M2) of the same or a different tower [4]. Here of course the masses must satisfy the inequality Ml > M2+M, and we may consider the question: is it appropriate to use the same universal coupling constant g(tl, 12) when this mass inequality does not hold? Now for saturation of the superconvergence relation, eq. (ll), it is required that the couplings of h(1236) and F37(1940) (which fall naturally into the same O(3,l) representation) to ~(1236)n be quite different. Another counterexample is that the experimental widths for A(1236) + Nn and F15(l688) 4 A(1236)~ require the corresponding coupling constants to differ markedly, while the F15(l688) and nucleon are fitted into the same O(3,l) tower [4]. Thus it ap-
17 March 1969
pears that the answer to our question is negative and that, in the O(3,l) model, we can assume a universal coupling constant g(fl, t2) only where the inequality Ml > (M2 + M,) is satisfied by all of the vertices to be compared, as is the case, for example, in ref. 3. Further details of this work, together with a discussion of the extent to which the results for the N spectrum is independent of the assumption of the 0(3, 1) model and a treatment of other superconvergence relations which hold for elastic V- A(1236) scattering and in the inelastic process nN+ x~(1236), will be given elsewhere.
References
1. A. 0. Barut and H. Kleinert , Phys. Rev. Letters 18 (1967) 754. 2. A. Donnachie, Proc. 14th Intern. Conf. on Highenergy physics, p. 139. 3. V. De Alfaro, S. Fubini, G. Rossetti and G. Furlan, Phys. Letters 21 (1966) 576. 4. B. Hamprecht and H. Kleinert, to be published. 5. M. A. Naimark, Linear representations of the Lorentz group (MacMillan, New York) 1964. 6. S. StrBm, Arkiv far Fysik 33 (1966) 465. 7. P. H. Frampton, Nucl. Phys. B2 (1967) 518 and Oxford D. Phil. Thesis 1968 (unpublished).
*****
667