Gatland, Ian Robert

English

Scholarly Materials And Research @ Georgia Tech

CDD Poles in Dispersion Relations for Strong Interactions by Ian R. Gatland Georgia Institute of Technology, Atlanta June 14, 1966 Annual Report on Work Carried Out Under NSF Grant No. GP 4500 Technical Report on Research Carried out under NSF Grant No. GP 4500 (Georgia Tech Research Grant No. B 2015) entitled "CDD Poles in Dispersion Relations for Strong Interactions" by Ian R. Gatland Background In a previous analysis(1) of pion-pion scattering numerical solutions for the scattering amplitude have been found by iterating partial wave inverse amplitude dispersion relations coupled by unitarity and crossing equations. Only S and P waves are included so that there are three amplitudes to consider Ao (v), Al(v) and A2(v) corresponding to isospin 0, 1, and 2 respectively. An  and A2 are S wave amplitudes and Ai is P wave; v = (k/0 2 where k is the magnitude of the center of mass 3 momentum and 4 is the pion mass. Dispersion relations with one subtraction are used and first derivative crossing conditions applied so that only the pion-pion coupling constant, X, remains as a free parameter. Some of the solutions included a P wave resonance and these can be characterized as follows. For the S waves the inverse amplitude right cut is only slightly (less than 10%) dependent on the left cut in the dispersion relations. Thus we have for v)) 0, I = 0, 2 1 , Im A 	(v) = - 1 v/(v. + 1) 1 1/2 La (1 ) -2- by unitarity and from the dispersion relations .''' Im A-1 (v ' ) 1, ‘ 	-1 	v-vo 	I dv, 	Re A kv) = aI + — ID: 	  I 	 Tr , . (v -v) ( v -vo ) ,., 	, L+R where all  are the subtraction constants, vo is the subtraction point (vo = -2/3) and the integral runs over both the left cut, (- 	, -1), and the right cut, (0, 	If we neglect the left cut and use eqn. (1) for - 	, Im A1  kI v) for v > 0 we find 1 	-1 Re All	aI + h(v) where / V h \ v ) = 1 /V 7in i 11 +,V+1 	0.554 ( 	—  Tt 	v+1 1 - v+1 Equation (3) will be used whenever an expressed form of Re All (v) is needed. For the P wave inverse amplitude one finds an effective range behavior until well above the resonance energy. Thus we may write for and , v Re Al (v) = C(vR-v) Alv Im  (v) = v3/(v+1)] 1/2 1 with C and vR dependent on X. The amplitudes for v-1 may be found by first using the crossing equations for the imaginary parts and then the dispersion relations for the real parts. For present purposes it is more practical to use dispersion relations for the direct amplitude Re AI(v) 	 P dv = a + L+R 	(v'-v)(v'-vo) , Dm AI (v') 	(7) (2) (3) (4) -3- where again the integral runs over both the left and right cuts. The crossing equations are, for v $: -1, I = 0 or 2 -v-1 	. Im A (v) = dv,  3 Im Ao (v') + bI  ImA2(v ' ) 0 .1 + dI 3(1+2 v+1) Im Al(v') 0 	1 where bo = 13 , b2 = 3, do = 2 and d2 = -1 and -v-1  / 	Im Al(v) = — 	 (s i 2 v ) 7 Im 0( ) 	 A2 ( ') 1 dv' " .1__ v +1 	2 _m A  ,v, - 5 _., 7 Im  	v ] + 3(1+2 "1) Im Ai(v') v The parameters ao , al and a 2 as well as C and vR are determined for any particular X, by the consistency requirements of the crossing conditions. For the partial wave analyses these become  = 	A2\ V=V0 	V V=V0  3 a _ 1aAo vo 	- av (10)    In particular one finds ao = -51 and a2 = -2X. al, C and vR are not so simply determined but for X = -0.15 one finds al = -0.005, C = 19 and v 2 = 4.1. Problem The present problem centers around a modification of the solutions dis-cussed above. The dispersion relation analysis is arbitrary to the extent of poles in the inverse amplitudes (CDD poles) and these cannot be found by v' (8) (9) numerical iteration on a computer since they introduce infinities in the physical region. These correspond to zeros of the amplitude itself. However, in the presence of any inelastic scattering the elastic scattering has a lower bound which is non zero. The CDD poles still may exist but are moved off the real axis into the negative half v-plane. Such poles can be treated by the computer and when a program including inelastic scattering was run solutions with CDD poles appeared but with one peculiar property. The CDD pole always occurred close to the original resonance position 2) The result of this was to split the resonance into two, one each side of the CDD pole, but still close together. (3) It appears then that in any P wave resonance type solution a CDD pole may be placed close to the resonance position without disturbing the consistency requirements of the analysis, i.e. the crossing equations. Our purpose here is to justify this statement. Analysis  In the original discussion of CDD poles() their position and strength were arbitrary. On the other hand for that analysis a very simple form of crossing, (actually direct reflection) was used. Here we have partial wave coupling in the crossing equations so that more restrictions on the CDD pole may be expected. To start with we modify eqns. (5) and (6) to include a CDD pole at v = vp  with Re vp  = vR. Thus T(v-vR) Re All (v) = C(v -v)/v + k T2 4.( v _vR )2 1/2 -1 	 14: T2  Im A (v) = - 1 T2 1.( v _vR)2 (12) -5- where T = -Im v and A T is the strength of the CDD pole. The parameters N and T may be related to the real and imaginary parts of the P wave whose shift. For k large Im i(v) 	-1 , d  Re ul(v) 	' 	= ( Re ,S- 1 = 	77 	Re S = (13) Now using eqns. (1), (3), (4), (11) and (12) A/(v) (I = 0, 1, 2) may be found by inverting and then the Ai(v) for v—.; -I are determined by using first the crossing equations, eqns. (8) and (9), and then the dispersion relations, eqns. (7), since Im Ai(v) has then been determined for all v. The resulting 1 curves for Re A I , Im AI , Re AT and Im All  are shown in Figs. 1, 2 and 3 both with and without the CDD pole. We are basically only interested in the difference between the results with and without the pole and this corresponds to making further subtractions in the dispersion relations. This point will be clarified by a careful consideration of the crossing conditions. The salient point shown in the curves in Fig. 1 is the double resonance behavior in Re Al and Im Al . The two resonances are symmetrically positioned about v = vR and are both lower and narrower than the original resonance. As shown in an earlier paper (3) there combined effect in any dispersion integral is similar to that of the simple resonance except close to v = v R . On the other hand in the crossing equations the variable upper limit produces a change of slope of Im A(v) for v < -1 at the reflected resonance positions. When the CDD pole is present this occurs in two places and gives a somewhat different behavior as shown in Im A o (v) and Im A2(v) for v -I. These changes on the left cut are not important in the dispersion integrals in the low energy physical region due to the subtractions: their effect, if any, will be felt in the subtraction constants. Thus the consistency of the scheme with the CDD pole depends on satisfying the crossing c,nditions and, in particular leaving -6- the P wave subtraction constant unchanged. Using the dispersion relations (7) the crossing conditions (10) may be rewritten 3 al 	1 	Im Ao (v) 1 jiIm 	A2(v)  T( dv = 	 dv vo T (v-v0) 2 	 (v-vo) 2 L+R 	 L+R The curves in Fig. 1 were obtained by fairly crude numerical calculations, mainly for purposes of illustration, and are not accurate enough for use in eqn. (l1.). Instead we set up a precise analytic expression and then evaluate only the part which is effected by the CDD pole. For I = 0, 2 consider then -1 Im AI(v) 	 Im AI 	(v) Im AI 	(v) dv = 	 dv + 	 dv L+R (v-v0)2 	0 -v1-1 (v-v0 ) 2 (v-v0 ) 2 (14) -v -1 dv  7 	dv. 2  [3- Im Ao (v') + bI  Im A2 (v')1 (v-v0)2 	1  0 vi 7 lf dv' dI 	v+1,)  2(1+2 	Im A/(v') v 0 -v1-1 dv  ( V -Vo ) 2 _ where + dI R+ 	(15) -v-1 R±  = I 	dv  j 	dv' 3(1+2 v+1 ) Im Al(v') (16) (v-v0 ) 2 _ ,0 	 vt -7- and by a suitable choice of vi (vi = vR - 4T say) this is the only term effected by the CDD pole. The + designates the presence or absence of the CDD pole in Im Ai(v'). Writing PI for the CDD pole - independent terms we have Im AI(v) dv = PI + dI R+ (v-v0 ) 2 L+R Thus because the CDD pole is only effective in a limited range of v we may treat it as a perturbation which takes us from R to R. But in the unperturbed case we have 3al 	 \ 	-1 , , ) = O 2 + d2 R _ ) pole we require + d2 R4.) - ) if the first is not, (18)  (19)  (20)  (21)  ( Po 	,+ do vo = 2n and to satisfy the crossing conditions with the CDD 5a1 _ 1 	 -1 (p° 	do R(P2 	vo 	 +) = i.e., dn  0 = 	- R ) = - 	(R+ 	R 2n , 	+ - 	n The second condition is identically satisfied, even since by definition do _ 1 	d2- 2n - 7 = For the first condition to be satisfied we must have R + - R = 0 or, more in keeping with the spirit of the analysis, we require I 	!<.; I L'al I R+ - R - 	vo For the case X . -0.15, al = -0.005 and this reduces to R+ - R_ 	0.07 ( 17) (22)  (23)  -8- We now turn to the evaluation of R + and R - . Since the CDD pole effect is concentrated about v = v we shall only keep lowest order terms in v-vR. These approximations will be reviewed later. Thus g,(v-vR)/T vR Re A;1 .  CT 1.1rR -v  7 - T 	j 	1 4. [(V-VR )/TJ 2 	-0x +k x/(1 +x2 ) 	 (24) -1 	VI3 	11/2 Im A1  = -1 ---- +1 vR j 	1+x2 = - U - 101+X2 ) 	 (25) where 1/1 0 = CT/v.13, u =117R/(vR+ ld and x = (v-vR)/T. Inverting we find Im A - 	UX2 4- (A +u)  1  2  pJ x _(21( 0_024.u2)x2 	( 	u)2 (26) Making similar approximations in R + we write -v1 -1 and = 2v 	 v41 3(142 (v - 3 vo) 3 	vR Im Al(v') dv'   vl Using eqn. (26) we find -v -1 dv' Im Al(v') = f(-v -1 -vR) - f(v+1+vR ) - f(vi - vR) + f(vR - vi) v1 with f( v) =- 2 arctan( 2±I) + (Ø-11)T2 In I 1 + ( 27-7) 2 ,1 8 	p where and Further define -9- p 	TL 4 k0 	(0 -11 ) 2 3 	d q = T(QC+u)/2 = (3 	+ 6 + 4 vo)/12 q w= (vi + 1 + vo )/q x1= (vR + 1 +2 vo _ p)/ci 2 X 2 = (v + 1 + 7 vo + p)/q a = 6/1,-R  (0+u) and = -P2 = T(0-u)/2p 2 Then = 2 a i 1+x. i=1 1+( xj -ko) 2  In 	 X 2 x. 	1 	 2 	) x 	 f3i (xi 	+ 	) + 1+ x. 2 	 1+x. 2 1 1 4- 1 f 4- ar c tan ( xi 	(1) x + 21  ) 1+x.2 	i 1 (1 x. 1   1 +x. 2 1 - 	(1- 	xi)/a, - 10- This is to be compared with R or rather -V1 -1 	 -v -1 dv 	311+2 v+1) dv 	1/u  (v - 2 ) 3 3 v0 	 Vi 1+1e(v'-v ) 2 /u2 T2 where we have made the same approximations as in 	Values of R .: and R AR=R' -R' are given in table 1 for various values of 1<, T, and vi assuming C = 19 and vR = 4.1. The calculations indeed show that t R is very small for a wide range of values of T and 1< and is not sensitive to the value of vi provided it is somewhat less than vR' In fact we may use the freedom in vi to test the efficacy of the dispersion relations. By making vi. v2 we may evaluate the infinite tail of the integral. As shown in table 1 its effect is negligible both in R .I. ' and A R. Finally let us consider the approximations made in the analysis: Firstly the computational approximations. [v' 31/2. v 	71/2  . 	R v +1 _.), ! 	vR  +1 c(vR -v')/v' 	c(vR -v')/v R v+1 	v+1 (1+2 	) ---) v 	vR V • > vl V v 	vi (v -v0 ) 2 v 	- 3 vo )3 	 v The first two just constitute variations on the analytic approximations to the physical amplitudes and are not related to the OD pole. The third is the most suspect since it distinguishes between the unperturbed case with Im Al  large on each side of vR. This approximation will be refined in future work 2 Vo2 	 Vo (vR+T+l+vo ) 2 Vo 2 (vR+14v0 ) 2 E = R ) 1 	 + 12 , — ( ,AvR-T+1+v0 ) 2 by the addition of higher order terms in (v-vR). The fourth approximation is proportional to 2 	, (-v -1 - 3 .70) 3 E - 	  ( -v -1 -vo ) 2 (-vi -1) Vo 2 (v+l+v0 ) 2  and since the main contribution to R+ and R occurs near v = vR 	T and v = vR respectively the correction to A R will be of the order 1 3 V02 T2 R -5 vR which is negligible. Secondly there are the assumptions that the S wave right cuts are not effected by the modifications to the left cuts. However, if the changes near v = - vR -I are not noticeable at v = vo in the subtracting constant calculation they should also be negligible for 	0 in the dispersion integrals. Thirdly, there is the possibility of changes in the P wave right cut altering the effective range behavior independent of al. But arguments similar to these for the S wave right cut suggest that these may also be neglected. However, a more detailed analysis will be undertaken. One situation which has not been considered in detail here is that of a very weak CDD pole. In that case the resonance is not split (the variable P becomes imaginary) but only broadened and decreased in height. -12- Summary  The effect of adding a CDD pole at the resonance position in pion-pion scattering has been treated analytically and it has been shown that, provided the CDD pole is not too strong, i.e., the inelastic scattering is not too weak, the crossing conditions are still satisfied. It is interesting to note that for the,- -meson which may be a split resonance there is a strong inelastic contribution from the cu-meson. -13- References 1. B. H. Bransden and J. W. Moffat, N. Cimento 21 505 (1961). 2. B. H. Bransden, I. R. Gatland, and J. W. Moffat, Phys. Rev. 128 859 (1962). 3. I. R. Gatland and J. W. Moffat, Phys. Rev. 129 937 (1963). 4. L. Castillejo, R. H. Dalitz and F. J. Dyson, Phys. Rev. 101 453 (1956). -14- Table 1. Values of R4-' ' 	R and vi. T 1< ' and t R for C = 19, vR = 4.1 and various values of X , T vi 	 R+ 	 R ' 	 A R _ 0.01 0.2 2.5 0.145534 0.145535 -1 x 10 -6 0.1 0.2 2.5 0.145528 0.145535 -7 x 10-6  1 0.2 2.5 0.145460 0.145535 -8 x 10 -5 10 0.2 2.5 0.144540 0.145535 -1 x 10 -3 10 0.01 2.5 0.145520 0.145535 -1.5 x 10 -5 10 0.05 2.5 0.145429 0.145535 -1 x 10-4 10 0.1 2.5 0.145234 0.145535 -3 x 10-4 10 0.3 2.0 0.146730 0.147246 -5 x 10-4 10 0.5 1.5 0.147749 0.148290 -5 x 10-4 10 0.2 1.5 0.148420 0.148290 1.3 x 10-4 10 0.2 2 0.147088 0.147246 -1.6 x 10 -4 10 0.2 2.5 0.144540 0.145535 -1 x 10 -3 10 0.2 6 0.003348 0.002865 5 x 10-4 10 0.2 8 0.001058 0.001018 4 x 10-5 4 ReA -1 0 -1 5- 1.0 - 0.5 I Re A O -4 	-3 	-2 -1  2 	 4 	 6 	V 	8 - 0,5 1,0 -   Im A O 0.5 - -8 	-1 	-6 -5 	 -3  2 	 4 	 6 	 8 - 0.5 - 1 4 - I 2 - Im A -1 0 	I 	 r -8 	-7 	-6 -4 	-3 	-2 - 1 ♦ 	 -2 I 4 2 2 	4 	 6 	V 	8 	 NO C.D.O. POLE - - 	 WITH C.D.D. POLE / -3 	-2 	- 1 I / 	 I -2 -4 - 6 Figure 1. The S Wave, Tsospin 0, filiplitude without a CDD Pole (Continuous Line) and the Modifications Introduced Hy the ODD Pole (Draken Line). 6 	8 Re Ail 6 	11 	8 -2 	1 -10 - Re A l 2 Im A l 6 	V 8 - -2 - 20 - 10 - ImArl -a 	-6 2 4 1/ 10 - 1-20 -   NO C.O.O. POLE — — -- WITH C.O.D. POLE 1 -30 - -16- Figure 2. The P Wave Amplitude without a CDD Pole (Continuous Line) and the Modifications Introduced by the CDD Pole (Broken Line). 

CDD poles in dispersion relations for strong interactions

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CDD poles in dispersion relations for strong interactions

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