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Application of a modified K-matrix formalism to ππ and KK scattering
Nuclear Physics B93 (1975) 429-445 © North-Holland Publishing Company
APPLICATION OF A MODIFIED K-MATRIX FORMALISM TO ~tn AND KI~ SCATTERING C.B. LANG Instirut far Theoretisehe Kernphysik, Universiti~t Karlsruhe, Germany Received 19 February 1975 (Revised 17 April 1975)
With the help of the recently proposed modified K- matrix formalism [1] we perform a coupled-channel calculation for nn and KK,.Crossed-channel influences are taken into account for zr~r~ n~r by iteration and for ~rrr--, KK and Kg~~ Kg. on a phenomenological basis. Direct-channel resonances are treated as coupled single-particle states. In the zrn channel we find good agreement with experiments below 1.2 GeV although we disagree with the results from linear current algebra models. The KK's wave phase shift above threshold agrees with the down solution of experiments. Our predictions for the s-wave amplitudes ~r~r~ KK below KK threshold are somewhat different to other authors; the higher partial waves are qualitatively similar to the prediction of other authors.
1. Introduction In ref. [1] we presented a method to include influences from coupled channels as well as crossing with the help of a modified K- matrix. The difference with the ordinary K- matrix approaches lies in the simple analytic off-shell behaviour of the intermediate one- and two-particle states. Direct channel resonances are introduced as coupled single-particle channels. The new K- matrix is then a real analytic function in the unitarity region, it does not have poles like normally required (e.g. Lovelace [2]). In order to learn how this scheme works in practice we want to deal with mesonmeson-scattering in a formalism with at most two coupled channels and only oneand two-particle intermediate states. We mean the coupled system of ~rTrand KI( scattering; we neglect the effects of other coupled channels (like the 4rr channel, which seems to be negligibly small below the Kg, threshold). For a fully self-consistent treatment we would have to include irK, KK and the crossed processes of 7rzr ~ 7rTrscattering. At that stage of our investigations however we restrict ourselves, insofar that we use experimental estimates for the nK s- and p-waves and no KK input at all. This seems to be acceptable and improves the transparency and feasibility a lot. We do iterate of course the nTr ~ 7r~r exchange processes to ob-
430
CB. Lang / K-matrix formalism
tain correct crossing behaviour. In rrzr scattering below 1 GeV there seems to be a good understanding now since a lot of experimental and theoretical work has been devoted to this subject. The up-down ambiguity in the s-wave phase shift around 900 MeV was solved in favour of the down solution by the analysis of the Berkeley experiment by Protopopescu et al. [3]. Different analysis of the CERN-Munich data (Estabrooks et al. [4], Grayer et al. [5], Hyams et al. [6]) led to better knowledge of the phases above 1 GeV although there are still ambiguities in that region. Froggatt and Petersen [7] tried to diminish the number of these ambiguities by means of a fixed t analysis. Finally Manner [8] presented data points in the very low energy region as well as above 1 GeV; although his p-wave scattering length is unexpectedly large the s-wave results agree with other data (for a review see ref. [9]). A good understanding of how much crossing restricts the number of theoretically possible unitary solutions was obtained in the investigations by Basdevant, Froggatt.and Petersen [ 10]. (To be referred as BFP). They came to the result that the s-wave scattering lengths and the value of that energy where 68 passes ½1r are still free parameters even if the p parameters are fixed. Different choices of 60 in the p region led to the observation that the s-wave / = 0 scattering length cannot be restricted better than within the lower limit - 0.05/a -1 and upper limit 0.7 g - 1 . Up to now there have not been so many investigations of the rrrt ~ KK and Kg, ~ KK amplitudes. Lovelace [2] unitarized the dual Veneziano formulas by using them as elements of the standard K- matrix; crossing was violated and held only asymptotically. Current algebra models as well as other approaches are reviewed by Petersen [ 11 ]. Fixed t dispersion relations for 7rK scattering are used by Martin [12] and Nielsen and Oades [13] (to be referred as NO) to extrapolate into the rrlr ~ KK channel. The phase in this channel is known below KK threshold because of extended unitarity (Mandelstam [14]), if one neglects other coupled channels like 4rr. The phase is then given through the rrTrphase and Omnes function techniques can be applied in order to get the 7rTr~ KK amplitudes out of LHC information. To avoid the extrapolation into unphysical regions Johannesson and Petersen [15] (referred to as JP) and Hedegaard-Jensen [16] (referred to as H-J) introduce partial~wave relations derived from hyperbolic dispersion relations. JP [15] use a parameterization of the Omnes function based on the formalism of BFP [10] and find stable results for the zrTr--* Kg, s-wave. H - J [16] calculates the s,p,d and f-waves below the KK threshold and derives also sum rules for zrK threshold parameters. Although the experimental situation for 7rK ~ 7rK s-waves is far from being clear all those authors agree that the scattering lengths are similar to the prediction from the linear current algebra model and that the general behaviour is like that indicated by NO [13]. All the higher partial waves in rr~r~ KK, seem to be dominated by resonance states. Only the s-wave is still controversial; JP [15] and H - J [16] agree on the sign of the imaginary part of the amplitude below KK threshold but diagree on the sign above and therefore on the sign of the KK scattering length. Beusch [17] obtained values for 60(KK ~ Kg,) that are negative
C B. Lang / K-matrix formalism
431
with an ambiguity above 1.2 GeV; these results would support JP [15]. The modulus above KK. threshold is known through the inelasticity ~70 from different experiments [3, 5, 6]. In sect. 2 we review briefly the relevant formulas for the modified K- matrix approach. For the details of the kinematics we refer to ref. [11]. We then develop our method to solve the equations. In sect. 3 we discuss our input, i.e. the number of the bare quantities and their derivation from the experimental informations. The resulting amplitudes for nTr -+ 7rlr, zrn ~ KK, KK -+ KK as well as the pion and kaon formfactors are presented in sect. 4. We end with some remarks on the general features of the formalism. 2. The modified K- matrix formalism
2.1. The basic equations The explicit derivation of the relevant equations can be found in ref. [ 1]. Here we give only a brief review on the main formulas and discuss their practical treatment. In each channel the K- matrix equation for the partial waves is just a matrixequation and can be solved explicitly:
~=~+kIdlaI.
(2.1)
The connection to the invariant amplitude T as it was defined in ref. [1] is
TI
I
l 1
fi, l = 16nail,/qf q ~ eN/~fei '
(2.2)
where the Bose factor e/= 1 for two different and ej = } for two identical particles in state j. We choose the t channel (s,t,u the Mandelstam variables) as that where nzr -+ KK and thus nrr ~ nTr, KK ~ KI(; in the u and t channels the corresponding crossed processes are 7rK ~ zrK, 7r~r~ nTr. For the detaiis of the kinematics we refer to Petersen [11 ]. The partial-wave expansion is
T~/16~ ~ _
1
~t
1li
(2l + 1) qfqiafi, tPt(z),
(2.3)
with isospin indices I. Let us give here the relation of our quantities alto r//and 6/ and the standard denotion o f the amplitudes. 7rTr~ 7rTr,
4/a 2 < t: q2l~(t) = "v/-f ~ 71 ,t 'lr t e 2iJ - 1)
q t2_ - ~(t-- 4U2), ~-+
K~,
4M2 < t:
qt Pt a[(t) = ~ qtp~ p2 = ~(t - 4M2),
2i ~ ( t ) = -g[(t)/16rtX/2 .
(2.4)
432
CB. Lang / K-matrix formalism
Here g ( t ) is the amplitude used in refs. [ 13, 15, 16]; # and M are the masses of the pion and the kaon. _ ~
KK, ~ KK, 4M 2 < t: p2tl a{(t) - ~ t
1,
I 2i6/
~trll e
-- 1).
(2.6)
The threshold behaviour of the partial waves has been accounted for explicitly and thus a l l behaves proportional to the scattering lengths at threshold. a = a(tthr)2/
tX~r = lim ~/q2l+l . q~O
(2.7)
The elements of the diagonal matrix d in eq. (2.1) are the propagators of the intermediate states multiplied with functions that assure correct threshold behaviour. dil I = q~l h ( t )
for 2-particle states,
d~, l = h (1, t ) ,
for single-particles states.
(2.8)
The functions h are in the two-particle case the Chew-Mandelstam functions and in the single particle case proportional to poles. The functions are adjusted to vanish at t = 0 and therefore aI(O) = kl(O). The general form of the functions h is given in appendix B of ref. [ 1 ]. We should note here that the requirements of extended unirarity are automatically fulfilled in this approach. The crossing postulate connects K with the crossed channel quantities K t = z t + Ots(T s - Ks) + ~0tu(T u - K u ) .
(2.9)
The convergence domain of ( T - K ) in the s- or the u-channel is that of Im T, i.e. the large Lehmann ellipse [ 1 ]. This relation can be projected into partial waves to give k t. K t is the K- matrix looked at in the t-channel, Ks, Ts, Ku, T u the corresponding K- and T-matrices looked at in the s- and u-channel quantum numbers. Ots and Otu are the crossing operators that relate the different channels, in our case isospin crossing matrices. The matrix z is crossing symmetric (z t = Otsr s = Oturu) and regular at those normal thresholds that are dealed with in the coupled channels equations. It has singularities corresponding to higher thresholds as well as pinch singularities at t = 0 ariving from the normal thresholds of the t- and u-channel and correspondingly for s = 0 and u=0. The first type of singularities could be removed in principle but the second type not at all. Because of the regularity of r in the interesting physical regions however we can assume that r can be approximated by a polynomial or even a constant there [19]. We identify r as the matrix for the description of the coupling parameters of the "bare" world [1 ] and discuss the choices of r in sect. 3. The above eq. (2.9) expresses the singularity structure of K in terms of crossed-channel quantities. In ref. [1 ] we suggested an iterative scheme to fred unitarity and crossing symmetric amplitudes. With a given r and d one sets K = r and solves eq. (2.1) to f'md
433
C.B. Lang / K-matrix formalism
unitary amplitudes; out of those with help of eq. (2.9) an iterated K- matrix, and out of this an unitary amplitudes again and so on until the iteration converges. In each step of the iteration we know the current version of the amplitude and therefore we have to find a way to calculate the real part of K for t > 0 out of its values in the crossed channels. 2.2. Dispersion relations f o r k
To find k we choose a two-step continuation of fixed t and partial-wave dispersion relations, both in the framework of the so-called discrepancy method. The first serves to get the values of the real part of k on a portion of its left-hand cut, the second to extrapolate it to positive values. The method we apply here is up, to small differences similar to that of NO [ 13], they however wanted to obtain directly partial wave amplitudes and therefore had a different cut structure than we have it for K t. For fixed negative t values we cannot write one common dispersion relation which contains both the u- and the s-channel cut contributions, like it is done for the amplitudes. For the amplitude one knows the real part both in the u- and in the s-channel physical regions. For k we know only the real part of ( T s - Ks) in the s-channel and the real part of ( T u - K u ) in the u-channel. Both quantities therefore have to be treated separately. As an example we discuss the contribution of ( T s - Ks); this function contains the unitarity cut of the physical s-channel processes. The crossed cuts of T s are removed because they occur also in Ks. This is true only up to a contribution of the order of r, but we neglect this part because we assume that r can be approximated by regular functions [ 1, 19]. The imaginary part o f the function is known for t > - 3 2 , the limit of convergence o f the partial-wave sum of the s-channel. The real part is known only in the physical region of the s-channel (and in small bands outside) because of lack o f convergence of the partial-wave summation. For each fixed value o f t a discrepancy function is defined through 1 f
sR
A(s) = Ts(s ) - Ks(s ) - 7r , ,
Im [Ts(s' ) - K s ( s ' ) ] s' - s
ds'
so
1 / -
SR Im [Ts(SR) -- Ks(SR) ] s'(s' - s)
ds'.
(2.1o)
sR s O is the value o f the s-channel threshold and s R the upper limit of our knowledge of the imaginary part (we took s R ~ 1300 MeV). The rightmost term removes the logarithmic singularity at s R and can be evaluated analytically. The discrepancy A(s) is an analytic function which has a cut for s R < s; it is therefore expected to behave well in the interesting region. Because of the knowledge of the real part of ( T s - K s ) for - 1 > z t we also know A(s) in that region; we then can extrapola-
434
C.B. Lang / K.matrix formalism
te it down into the region z t > - 1 which gives Re (T s - Ks) for zt E ( - 1 + 1). It turned out in our calculations that the discrepancy function can be well approximated through 2 nd order terms in s. The extrapolation of the discrepancy function gives the desired values of Re (Kt) in the unphysical region. Partial-wave projection leads to the values of the imaginary part o f k for t E ( - 3 2 / a 2, 0) and the real part of k for t E (tc, 0); here -32/a 2 < t c < 0 is a confidence value up to that one believes that the fixed t dispersion relation has given the correct real part (for lrzr ~ 7rTr,t c = -24bt 2 and for 7rTr~ KK, t c = -15/12 turned out to be an adequate value). Again we write a dispersion relation for the partial waves and define a discrepancy function 1 0 A(t)=k(t)-~f l~'?(t')dt'-l tL
tL t L i m k ( t L ) t~'~O "dt' , _o.
(2.11)
where the last term is added to remove the possible logarithmic singularity at t = t L ; its evaluation gives 1 tL -- ~ ~-
t L -- t
Im k(tL) In
tL
(2.12)
The discrepancy function A(t) has only a LHC for t < t L and because of our knowledge of Re k ( t ) for t E (tc, 0) we know A(t) there, i.e. in a region where it is holomorphic. We now follow refs. [18] to get an analytic interior -~ interior continuation of A(s) to the physical values 0 ~< t. We perform a conformal transformation to a new variable z so that the LHC _oo ~< t ~< t L is mapped unto the boundary o f an unifocal ellipse in the z plane and t = tc, 0 are mapped to the fociz = - I , + 1. Then-A(s) can be represented inside the ellipse and therefore throughout the whole cut s plane by a Legendre series in z (practically a parabola in z). It is clear that the parabola is only an "educated guess" - the correct extrapolation curve belongs to a manifold of curves with a boundary which is determined by our explicite and implicite assumptions on the behaviour of A(t) on the unknown part of the LHC. Because of the mapping however we expect that the extrapolating curve is at least an optimal one according to our knowledge of the values on the nearby part of the LHC of k. Through this extrapolation we find approximate values of A(t) for 0 ~< t and therefore also for k(0 ~< t). Like already proposed [1] we solve the system of eqs. (2.1) and (2.9) by iteration. We calculate a (n + 1)th iteration o f K t out of the previous nth iteration. We therefore have an operator equation of the form K = ~(K).
(2.13)
The fixpoint of this equation fulf'tlls the requirements of crossing (compare refs. [19, 1 ]) and unitarity; the latter is explicitly fulfilled at each iteration step. A minor complication is that the operator in eq. (2.13) is in general not a contracting one. The modification
CB. Lang / K-matrix formalism
K5n*l) -±rL'(n)+2 t~-t rt + Ots(T(n) - K(sn)) + ¢tu (T(n) - K(n))]
435
(2.14)
turns out to remove this problem and gives the equivalent contracting operator K = ½(K + ~ ( K ) ) .
(2.15)
One has to keep in mind that eqs. (2.9, 2.14) are equations for the single elements o f K t which are always functions of the corresponding crossed processes. The elements of k t therefore carry the available information of the exchange processes; they are rather flat functions for physical t values. All strong structure of the direct channel is given through coupled channels.
3. Input assumptions We have to distinguish between three types of input. First we need the masses of the bare particles that enter into the equations as intermediate states. Secondly we need values of the r matrix elements i.e. values of the bare couplings. In principle, namely for a full iteration of all possible involved channels, this would be enough. We however do not iterate the KK and the 7rK channels and therefore we need additional information on the K-matrix elements for KK -~ KK and nn -~ KK; this is the third type of input we use.
3.1. Masses The pion and kaon masses are given from experiments, this allows to compute the intermediate state two-propagator functions (appendix B of ref. [1 ]). We then have to decide upon the coupled one particle states. We assume one possible state in each of the non-exotic channels with the quantum numbers (1 = 0, I = 0), (l = 1, I = 1), (l = 2, I = 0). The masses of the bare p and f can be estimated through the following consideration. The solution of the inverse amplitude for p-wave elastic rrn scattering with only the p and the two-pion state as intermediate states is (t22) -1 = (k22 + k21 dp) -1 + d ~ r ,
(3.1)
where dp and d,~= are the intermediate state propagators of eq. (2.8) and k21 = r(~Ttp ). The zero of the real part of (t22) -1 must lie where the experimental phaseshift passes ~l n. This leads to the mass value of the bare p and similarly o f f as 27.5 #2 and 77 #2; this shifting in the p- and d-waves does not depend too much on the left-hand cut or on coupled channel effects. Once these values are fixed the resulting masses of the physical p and f(30/z 2 and 82/~ 2) remain stable in the iteration process. If a shift would occur one would have to readjust the bare masses and start again. In the s-wave the difference of the bare mass to the physically visible mass (which we define to be the energy value where 8 = ½7r) is much larger. The requirement
CB. Lang / K-matrixformalism
436
s(6 (s) = ½7r) E [35p 2, 40U 2 ]
(3.2)
covers the experimental results quite well and is consistent with a value of 27.5 #2 for the bare e mass too. This would support the idea that the bare e is a daughter of the bare p. It was not necessary to include a bare S* particle, we discuss that later in more detail.
3.2. Coupling We assume that the r matrix elements for the 2 ~ 2 channels vanish. This means that we neglect the effects of other coupled channels as well as the pinch singularities at t = 0; analyticity therefore is violated to some amount. We can expect however that the violation is not felt in the physical region. In addition it means that in the bare world status there are no 4-vertices (four point functions). The nonvanishing 4-vertex coupling behaviour that we can measure in practice comes from unitarization and the inclusion of crossing; one could call this a breaking of original symmetries by unitarity. This assumption implies that we need only the bare 3-vertex couplings (three-point functions). We assume that they are energy-independent constants and postulate in accordance with general symmetry ideas like SU(3): l"(Trrte) = T(KKe),
r(rrTrp) = 7"(KKp),
r(TrTrf) = 7"(KK,f) .
(3.3)
The elements r(pp), r(ee) and r(ff) vanish for reasons discussed in ref. [1]. The experimental evidence for the full widths of the physical p and f again determines via the relationship (3.1) the values z(rrrrp) = 0.224,
r(Trrrf) = 0.046.
(The first value corresponds a g2~rp/41r = 2.5). The value of r(rrrre) cannot be determined that way and is therefore choosen to act as a free parameter. We assign it two different values and have therefore two different solutions for the whole set of amplitudes named A (z(Trzte)= 1) and B(r(nne) = 0.8). This parameter is related to different so wave scattering lengths (and because of crossing to s2 and pl low-energy behaviour too) as well as to the slope of the phase shift where it passes through ½rr and therefore to the width of the physical e.
3.3. Information on 7rK and inelasticity ~70 To find values for the K- matrix element nTr ~ KK, we use experimental information on rrK -* irK. We parametrize the s~, s], pl ( K * ) , d~ (K**) in terms of our modified K- matrix; we neglect the exotic p~- and d] waves because they seem to be negligibly small. Similar to NO [13] we fix the scattering relationship between dicted from the linear Weinberg model. The corresponding relationship between l 1 a~, a 0 and a] is fulfilled; for details we refer to Petersen [11]. It turns out to be
C.B. Lang / K-matrix formalism
437
possible to parametrize the K- matrix elements of this process 7rK --> 7rK by linear functions. The resulting s-wave phase shifts lie within the boundary that was used by NO [13]. To find out how much the final results depend on the nK input we use two different versions for the s~ phase shift. In one version we take 6 (1000 MeV) ~ 60 ° and in the other 6 (1000 MeV) ~- 70 °. The s] phase goes almost like a straight line and becomes - 1 5 ° at 1000 MeV. The K* and K** resonances were incorporated as possible intermediate particles and their bare masses and couplings were choosen to give the physical masses and widths in the 7rK -+ 7rK amplitudes. Again we find coupling values of 0.224 for K* and 0.046 for K**! Out o f these phenomenological informations on rrK --, nK it is possible to calculate the K- matrix elements for nn ~ KK with the procedure described in sect. 2. The remaining problem is the determination of the K- matrix element for KI~ ~ KK; the amplitude has a LHC from two-pion exchange that overlaps the extended unitary cut. In ref. [1 ] we demonstrated that the corresponding element of the modified K- matrix is real. For the p-wave and the d-wave we assume that the K- matrix element for this process just vanishes. From the strong effect at KK threshold in the so wave however we know that it certainly must not be neglected there. We therefore assume that the K- matrix element for the KK -~ KK. so wave can be approximated by a linear function in t, vanishing at t = 0 as required in the linear current algebra models. The only free parameter is the slope which can be determined by requiring that the output inelasticity r~0 exhibits a behaviour as expected from the experiment; it should have a minimum between 1000 MeV and 1050 MeV and should be near to 0.4 at 1075 MeV. These requirements can be easily fulfilled by adjusting the free slope parameter and (although restricted only by one parameter) the resulting r/0 shows fine behaviour in all solutions.
4. Results
In the calculations we cover a range between 0 and 100/.t 2 in all relevant channels. The iteration for the K- matrix elements nTr ~ n n converges relatively fast, the results become almost stable after the 5 - 6 t h iteration. This comes mainly from the fact that our K- matrix represents only the crossed channels effects and therefore behaves very flat in the direct channel. The main effects of the direct channel come from the coupling to other states. Similar to refs. [13, 15, 16] we determine a "theoretical" error corresponding to the uncertainty o f the irK s-wave input. We always calculate two version o f a solution according to the two different values of 6~ at 1000 MeV. The resulting "error bars" o f the amplitudes however are very small, almost everywhere below KI~ threshold less than 5% and only near KI~ threshold approximately 10% of the values. These differences are smaller than those of H - J [16] and NO [13] for two reasons. Firstly we have only a 10% difference between the two s~ input phase shifts at 1000 MeV compared to a variation of 2 0 - 3 0 % in ref. [13 ]. Secondly, and
438
CB. Lang / K-matrix formalism
250"
ZOO'
150"
6o° (~Tt--,qC~)
100'
50"
I
°~0
I
4o0
I
I
6o0
I
I
800
I
I
1000
1200
for rrlr Fig. 1. Solutions A (solid line) and B (dashed line) for the I = 0 s-wave phase shift 80 u rrlr. Also shown are the phases of Estabrooks et al. [4] +, Hyams et al. [6] ~, Manner [81 * and Protopopescu et al. [31 #. 200
400
600
.........
800
1000
1200
';+1
F~g. 2. Solutions A (solid line) and B (dashed line) for the exotic nlr ~ lr~rs-wavephase shift 80. The experimental phases are from ref. [22], namely Cohen et al. ~k, Colton et al.~, Hoogland et al.-~and Losty et al.--I-. this is the main reason, the uncertainty enters in the K- matrix element for rtrt KI(. The amplitude however is a function of all involved K- matrix elements and of the possible intermediate single particle states which play the dominant role. In this mixture the effects of the uncertainty of the K- matrix element rtrt ~ KK, are suppressed. It is this dynamical feature which is mainly responsible for the small "theoretical error bars". 4.1. S waves In the s0 wave we assume that one "bare" particle is coupled to both ~Tr and KI( with equal coupling constant. We give here the results according to the values
C.B. Lang / K-matrix formalism
439
Table 1 Scattering lengths of solutions A and B (in parentheses) Process
a0
a2
nlr --, ~rTr
+ 0.45 (+ 0.26) - 0.30 + 1.04i (- 0.34 + 1.11i)
-0.010 + 0.039 (-0.015) (+ 0.026) - 0.018 ÷ 0.013i ( - 0.017 ÷ 0.012i)
KK,-~ KK
a]
a0Xl04
a~× 104
+ 23.2 (+ 18.2) 1.34 + 0.14i (1.34 + 0.13i)
+ 5.5 (+ 7.7)
1.0 and 0.8 for this constant. It is hardly believable that the physical value could be below 0.8, if our simplicity assumption of vanishing 7(2 ~ 2) should still hold. Solution A (1 "0) seems to be nearest to the experimental results (figs. 1-2). In the process zrn ~ zrTrwe find a good agreemerit with experiments; solution A prefers a behaviour according to the results of Estabrooks et al. [4] and Manner [8] whereas solution B (0.8) is closer to the Berkely points of Protopopescu et al. [3]. Both solutions lie between solutions 2 and 3 of BFP [10]. The values of the combination of the s-wave scattering lengths (2a00 - 5a02) lie on the boundary of the "universal band " [10]. The values of the scattering lengths can be found in table 1 ; the value for a 0 of solution A coincides with Manners [8] value. The slope of the phase shift 50 where it goes through ½zr is relatively small; this corresponds to a rather broad e(P, > 5/a). We however do not try to localize the exact position of the e in the unphysical sheet. The sudden rise of the phase shift near 1000 MeV comes from the opening of the KK channel at 50 #2. No additional bare particle is necessary to give this expected increase of the phase shift ("S*") and the expected behaviour of 70 (fig. 3). Above 1000 MeV the phase shift seems to stay below 270 ° in contrast to most curves from experiments [ 6 - 8 ] ; they increase above 270 ° near 1200 MeV and run to much higher values. This structure however might be due to higher resonances or thresholds and should then be accounted for by the introduction of additional bare particles or coupled channels. The nn -~ KK amplitudes (fig. 4) show in the region of extended unitarity a very strong onset of the e. The slope at t = 0 differs by a factor of two from the linear current algebra predictions [11]. This comes from the very broad e which is in our approach the most important symmetry breaking term. The main contributions to the s- and p-wave scattering lengths come from the effect of this resonance in the crossed and direct channels. A comparison with the prediction of refs. [15, 16] exhibits a clear difference in the behaviour of the scattering amplitude below the e region. To test our assumptions we actually calculated the amplitudes also for nonvanishing r(2 ~ 2) elements. We took the r elements to be half the values which the linear current algebra models [21 ] predict for the amplitudes. The results were more similar to those of ref. [16] in the e region. The price for it were however several bad features; the inelasticity 770 became small soon above 1000 MeV but remained small over a to long region; the exotic phase was increasing too fast (--45 ° at 1000 MeV).
440
C.B. Lang / K-matrix formalism
,
i
I
I
i
i
I
I
1,0
"
1.0
Fig. 3. The rrn inelastieities r/0, ~ and r/0 of solution A (solution B is indistinguishable). The data points are from ref. [3] # and ref. [6] ~.
lo
zo
3b ~~
( 'I
x'
"'
"
2)='-
-1.0
Fig. 4. The s-wave (I = 0) amplitudes a~0 for Irlr ~ KK, solutions A (solid line) and B (dashed line). In fig. 5 we compare the phase shift of the sO wave of KI~ -+ KI~with recent experimental data from Beusch [17]. Here we agree with JP [15] and get a large negative phase above KI~ threshold and a negative scattering length. The I = 2 rtlr -+ lrlr process is elastic without resonances. The scattering length and the phase shift (fig. 2) are completely determined by the exchange processes; solution A is in good agreement with the experiments, solution B seems to be too small.
CB. Lang / K-matrix formalism
1.0
1.1
t.2 I
441
1.3 i
I MK[
i
(GeV)
-50"
-1013"
Fig. 5. The s-wave ( / = 0) phase shift for KK "-~ KK, solutions A (solid line) and B (dashed line). The data points are from Beusch et al. [17]; above 1.2 GeV we took only data points of the down-type solution.
4.2. P waves The bare p meson in this process is coupled to nTr and KI( with equal bare coupling constants. The resulting phase shift in the zr~r~ ~rrr channel has the required resonating behaviour at the physical p mass 775 MeV with the width 150 MeV. The obtained value for the scattering length (table 1) lies near to the value 0.033, which is predicted by current algebra. We cannot reproduce M/inner's [8] value near 0.1. The K- matrix element for Kg~ ~ Ki~ is set to zero; it is therefore surprising that the inelasticity seems to be close to the experimental values (fig. 3). Our approach is no analysis of the data and is predictive only on the basis of the input assumptions; we therefore cannot say anything about a possible p (1600 MeV) resonance. The general behaviour o f the lrTr ~ Kg~ amplitude (fig. 6) is very similar to that of H - J [16] although our numerical values are somewhat larger. The only qualitative difference is that the solution of H - J chooses the current algebra value at t = 0 whereas our solution has a zero at t ~/a 2 and is small and negative at t = 0. This may be due to different methods of analytic extrapolation from the LHC. In fig. 6 we also show the KI( -~ KI~ amplitudes which are almost the same like those for zrn -~ KK. The formalism leads to "unitarized coupling constants" i.e. T- matrix elements for rrzr -~ bare p and KK -~ bare p. Because of extended unitarity the phase of these functions is the same as that for the corresponding 2 -~ 2 amplitude. The functions can be interpreted as form factors which are defined like those of Gellmann and Zachariasen [20] having a zero at the position of the bare pO. Removing this zero by multiplication with 1/(m20 - t) and normalizing to unity at t = 0 leads to expressions for the isovector c~ntribution to the electromagnetic form factors of the pion and the kaon. In fig. 7 we compare them with each other; because of the normalization to unity at t = 0 there is no factor of ½ in the kaon formfactor. They are equal up to a 10% difference in the p region.
C.B. Lang / K-matrix formalism
442
0.1
10 20 30 z,O 50 60 70 _.8_0~-t~ -0"11 "
0.1
-~R i . , -" lb " 2'0 3b 4'0 " 50 60 " 7'0 eb t(p.z)
-0.1 Fig. 6. Solutions A (solid line) and B (dashed line) for the I = 1 p-wave amplitudes for rr~r --* KK and KK --* KK,. I
,
I
"
I
I
I
'
I
I
30
fi IF~'"I'
20
10
10
20
30
40
50 60
70
t (~=)
Fig. 7. The squared absolute value o f the pion (full line) and the kaon (dashed line) isovector contribution to the form factors in the timelike region. Both are from solution A and normalized to unity at t = 0.
C.B. Lang / K.matrix formalism
443
4.3.D waves
A bare f meson is coupled to the 7rn and KK, channels with equal bare coupling constants again. The position and width of the f meson is bound to have the experimental values m 2 = 82/~2 and Ff = 170 MeV. The 7rTr~ nrr scattering lengths and phase shifts below 1000 MeV for I = 0 and 1 = 2 are similar to solution 3 of BFP [ 10]. The phase shifts (fig. 8) agree with the values of table 6 of [ 10] within the errors given there. Close above threshold the 1= 2 phase shift is positive but very small and very soon it becomes negative and reaches - 1 ° already before 1000 MeV energy. An unexpected feature is its turning point and the decrease towards zero and positive values. The absolute value of the phase shift however is very small and this behaviour might be due to the large extrapolation distance from the LHC. The extrapolated K- matrix element is extremely small and the value might become incorrect above 1 0 0 0 - 1 2 0 0 MeV. This is a reason to trust our results only up to 1200 MeV, Like in the p-wave we put the K- matrix element for KK ~ KK equal to zero and again we get a remarkable coincidence o f our resulting inelasticity r/0 (fig. 3) with experiment.
20'
10'
:
'
sbo
'
-I' \ -2*
\
\
\
\
N~.~js
Fi~. 8. Solutions A (solid line) and B (dashed line) for the I = 0 and I = 2 d-wave phase shifts
~ a n d 8 2 for ~v-+ ~t.
444
CB. Lang / K-matrix formalism
5. Conclusion The relevance of the involved processes and their relationship to other scattering processes has been discussed already by many authors. Let us therefore not go into much details of the results but let us point out only the main features. On basis of the proposed modified K- matrix formalism [ 1] it is possible to find solutions for a coupled channel system of scattering processes. Our way to proceed is first to specify the assumptions on the structure of the bare world, a world where the effects of unitarity and crossing have been removed, and then to solve the combined matrix equations. It is by no means obvious that 4-vertices have to be included, on the contrary we find that in the treated meson meson system it is sufficient to include only 3-vertices (i.e. three point functions) in the bare coupling behaviour. In terms of a Lagrangian description this would mean that no ~4 contact term is necessary. Qualitatively we therefore agree with BFP [10] who discuss 7rTrscattering with the help of Roy's equations. They conclude that a saturation of the requirements of unitarity and crossing still leaves the freedom of 5 parameters which they identify "~s ,,0 ,,2 rnp, l~p and the energy where 60 goes through ½n. Once these are satu~0' "0' rated the low energy behaviour is fully determined. They compare this situation with that of a Born term of a Lagrangian which contains also five parameters namely one from the contact term, two from e and two from p intermediate states. This is analogous to the freedom in our approach; two free parameters specify the bare e state and its coupling to ~rzr, two more specify the p. The fifth open parameter just happens to vanish. The intermediate l = 0 state leads to the characteristic s-wave phase shift and has the main responsibility for s- and p-wave low-energy parameters e.g. scattering lengths. The results are compatible with current algebra although they disagree with the more spcialized Weinberg model. We therefore conclude that the type of breaking an original symmetry through the inclusion of crossing and unitarity leads to different results than that proposed by Weinberg. The modified K- matrix seems to be an advantageous possibility to describe scattering data. The concept of single particle channels is easy to handle and prevents a pole parametrization of the K- matrix. Coupled channel effects like the S* phenomenon can be simply described by flat K- matrix elements. In addition the multichannel formalism leads to expressions for the form factors. The computational apparatus is rather large but the coincidence with experiments is remarkable. Further investigations might make it possible to find an easier way to determine the K- matrix elements out of the crossed channel processes. The formalism enables one to remove at least to a certain amount unitarity and crossing effects in order to find a simple structure behind the complicated scattering data. I would like to thank Professor G. H6hler for his constant interest in this work. Much profit I had from the fruitful discussions with Drs. J.L. Basdevant, J.L. Petersen, E. Pietarinen and I. Sabba-Stefanescu. I am thankful to Dr. E. Pietarinen for reading the manuscript.
CB. Lang / K-matrix formalism
445
References [1] [2] [3] [4] [5]
[6] [7] [8]
[91 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
[20] [21] [22]
C.B. Lang, Nucl. Phys. B93 (1975) 4 2 9 - 4 4 5 . C. Lovelace, Proc. Conf. on rr~rand K~r interactions (Argonne National Laboratory, 1969) 562. S.D. Protopopescu et al., Phys. Rev. D7 (1973) 1279. P. Estabrooks et al., Nucl. Phys. B79 (1974) 301. G. Grayer et al., Contribution to the 16th Int, Conf. on high-energy physics, Batavia, Sept. 1972; G. Grayer et al., Experimental meson spectroscopy, 1972, Proc. of the 3rd Int. Conf. Philadelphia, 1972, eds. F.W. Lai and A.H. Rosenfeld. B. Hyams et al., Nucl. Phys. B64 (1973) 134. C.D. Froggatt and J.L. Petersen, Contribution to the 17th Int. Conf. on high-energy physics, London, July 1974. W. M~inner, Contribution to the 17th Int. Conf. on high-energy Physics, London, July 1974; F. Wagner, Proc. of the 17th Int. Conf. on high-energy physics, London July 1974; D. Morgan, Proc. of the 17th Int. Conf. on high-energy physics, London, July 1974. J.L. Basdevant, C.D. Froggatt and J.L. Petersen, Nucl. Phys. B72 (1974) 413. J.L. Petersen, Phys. Reports 2 (1971) 155. B.R. Martin, Nucl. Phys. B45 (1972) 102. H. Nielsen and G.C. Oades, Nucl. Phys. B55 (1973) 301. S. Mandelstam, Phys. Rev. Letters 4 (t960) 84. N.O. Johannesson and J.L. Petersen, Nucl. Phys. B68 (1974) 397. N. Hedegaard-Jensen, Nucl. Phys. B77 (1974) 173. W. Beusch et al., Contribution to the 17th Int. Conf. on high-energy physics, London, July 1974. R. Cutkosky and B.B. Deo, Phys. Rev. 174 (1968) 1859; S. Ciulli, Nuovo Cimento 61A (1969) 787; 62A (1969) 301. W. Zimmermann, Nuovo Cimento 21 (1961) 249; S. Humble, Phys. Rev. 183 (1969) 1335; P.V. Landshoff and D.I. Olive, J. Math. Phys. 7 (1966) 1464." M. Gell-Mann and F. Zachariasen, Phys. Rev. 124 (1961) 953; N.M. Kroll, T.D. Lee and B. Zumino, Phys. Rev. 157 (1967) 1376. S. Weinberg, Phys. Rev. Letters 17 (1966) 616. W. Hoogland et al., Nucl. Phys. B69 (1974) 266; M.J. Losty et al., Nucl. Phys. B69 (1974) 185; D. Cohen et al., Phys. Rev. D7 (1973) 661; E. Colton et al., Phys. Rev. D3 (1971) 2028.
APPLICATION OF A MODIFIED K-MATRIX FORMALISM TO ~tn AND KI~ SCATTERING C.B. LANG Instirut far Theoretisehe Kernphysik, Universiti~t Karlsruhe, Germany Received 19 February 1975 (Revised 17 April 1975)
With the help of the recently proposed modified K- matrix formalism [1] we perform a coupled-channel calculation for nn and KK,.Crossed-channel influences are taken into account for zr~r~ n~r by iteration and for ~rrr--, KK and Kg~~ Kg. on a phenomenological basis. Direct-channel resonances are treated as coupled single-particle states. In the zrn channel we find good agreement with experiments below 1.2 GeV although we disagree with the results from linear current algebra models. The KK's wave phase shift above threshold agrees with the down solution of experiments. Our predictions for the s-wave amplitudes ~r~r~ KK below KK threshold are somewhat different to other authors; the higher partial waves are qualitatively similar to the prediction of other authors.
1. Introduction In ref. [1] we presented a method to include influences from coupled channels as well as crossing with the help of a modified K- matrix. The difference with the ordinary K- matrix approaches lies in the simple analytic off-shell behaviour of the intermediate one- and two-particle states. Direct channel resonances are introduced as coupled single-particle channels. The new K- matrix is then a real analytic function in the unitarity region, it does not have poles like normally required (e.g. Lovelace [2]). In order to learn how this scheme works in practice we want to deal with mesonmeson-scattering in a formalism with at most two coupled channels and only oneand two-particle intermediate states. We mean the coupled system of ~rTrand KI( scattering; we neglect the effects of other coupled channels (like the 4rr channel, which seems to be negligibly small below the Kg, threshold). For a fully self-consistent treatment we would have to include irK, KK and the crossed processes of 7rzr ~ 7rTrscattering. At that stage of our investigations however we restrict ourselves, insofar that we use experimental estimates for the nK s- and p-waves and no KK input at all. This seems to be acceptable and improves the transparency and feasibility a lot. We do iterate of course the nTr ~ 7r~r exchange processes to ob-
430
CB. Lang / K-matrix formalism
tain correct crossing behaviour. In rrzr scattering below 1 GeV there seems to be a good understanding now since a lot of experimental and theoretical work has been devoted to this subject. The up-down ambiguity in the s-wave phase shift around 900 MeV was solved in favour of the down solution by the analysis of the Berkeley experiment by Protopopescu et al. [3]. Different analysis of the CERN-Munich data (Estabrooks et al. [4], Grayer et al. [5], Hyams et al. [6]) led to better knowledge of the phases above 1 GeV although there are still ambiguities in that region. Froggatt and Petersen [7] tried to diminish the number of these ambiguities by means of a fixed t analysis. Finally Manner [8] presented data points in the very low energy region as well as above 1 GeV; although his p-wave scattering length is unexpectedly large the s-wave results agree with other data (for a review see ref. [9]). A good understanding of how much crossing restricts the number of theoretically possible unitary solutions was obtained in the investigations by Basdevant, Froggatt.and Petersen [ 10]. (To be referred as BFP). They came to the result that the s-wave scattering lengths and the value of that energy where 68 passes ½1r are still free parameters even if the p parameters are fixed. Different choices of 60 in the p region led to the observation that the s-wave / = 0 scattering length cannot be restricted better than within the lower limit - 0.05/a -1 and upper limit 0.7 g - 1 . Up to now there have not been so many investigations of the rrrt ~ KK and Kg, ~ KK amplitudes. Lovelace [2] unitarized the dual Veneziano formulas by using them as elements of the standard K- matrix; crossing was violated and held only asymptotically. Current algebra models as well as other approaches are reviewed by Petersen [ 11 ]. Fixed t dispersion relations for 7rK scattering are used by Martin [12] and Nielsen and Oades [13] (to be referred as NO) to extrapolate into the rrlr ~ KK channel. The phase in this channel is known below KK threshold because of extended unitarity (Mandelstam [14]), if one neglects other coupled channels like 4rr. The phase is then given through the rrTrphase and Omnes function techniques can be applied in order to get the 7rTr~ KK amplitudes out of LHC information. To avoid the extrapolation into unphysical regions Johannesson and Petersen [15] (referred to as JP) and Hedegaard-Jensen [16] (referred to as H-J) introduce partial~wave relations derived from hyperbolic dispersion relations. JP [15] use a parameterization of the Omnes function based on the formalism of BFP [10] and find stable results for the zrTr--* Kg, s-wave. H - J [16] calculates the s,p,d and f-waves below the KK threshold and derives also sum rules for zrK threshold parameters. Although the experimental situation for 7rK ~ 7rK s-waves is far from being clear all those authors agree that the scattering lengths are similar to the prediction from the linear current algebra model and that the general behaviour is like that indicated by NO [13]. All the higher partial waves in rr~r~ KK, seem to be dominated by resonance states. Only the s-wave is still controversial; JP [15] and H - J [16] agree on the sign of the imaginary part of the amplitude below KK threshold but diagree on the sign above and therefore on the sign of the KK scattering length. Beusch [17] obtained values for 60(KK ~ Kg,) that are negative
C B. Lang / K-matrix formalism
431
with an ambiguity above 1.2 GeV; these results would support JP [15]. The modulus above KK. threshold is known through the inelasticity ~70 from different experiments [3, 5, 6]. In sect. 2 we review briefly the relevant formulas for the modified K- matrix approach. For the details of the kinematics we refer to ref. [11]. We then develop our method to solve the equations. In sect. 3 we discuss our input, i.e. the number of the bare quantities and their derivation from the experimental informations. The resulting amplitudes for nTr -+ 7rlr, zrn ~ KK, KK -+ KK as well as the pion and kaon formfactors are presented in sect. 4. We end with some remarks on the general features of the formalism. 2. The modified K- matrix formalism
2.1. The basic equations The explicit derivation of the relevant equations can be found in ref. [ 1]. Here we give only a brief review on the main formulas and discuss their practical treatment. In each channel the K- matrix equation for the partial waves is just a matrixequation and can be solved explicitly:
~=~+kIdlaI.
(2.1)
The connection to the invariant amplitude T as it was defined in ref. [1] is
TI
I
l 1
fi, l = 16nail,/qf q ~ eN/~fei '
(2.2)
where the Bose factor e/= 1 for two different and ej = } for two identical particles in state j. We choose the t channel (s,t,u the Mandelstam variables) as that where nzr -+ KK and thus nrr ~ nTr, KK ~ KI(; in the u and t channels the corresponding crossed processes are 7rK ~ zrK, 7r~r~ nTr. For the detaiis of the kinematics we refer to Petersen [11 ]. The partial-wave expansion is
T~/16~ ~ _
1
~t
1li
(2l + 1) qfqiafi, tPt(z),
(2.3)
with isospin indices I. Let us give here the relation of our quantities alto r//and 6/ and the standard denotion o f the amplitudes. 7rTr~ 7rTr,
4/a 2 < t: q2l~(t) = "v/-f ~ 71 ,t 'lr t e 2iJ - 1)
q t2_ - ~(t-- 4U2), ~-+
K~,
4M2 < t:
qt Pt a[(t) = ~ qtp~ p2 = ~(t - 4M2),
2i ~ ( t ) = -g[(t)/16rtX/2 .
(2.4)
432
CB. Lang / K-matrix formalism
Here g ( t ) is the amplitude used in refs. [ 13, 15, 16]; # and M are the masses of the pion and the kaon. _ ~
KK, ~ KK, 4M 2 < t: p2tl a{(t) - ~ t
1,
I 2i6/
~trll e
-- 1).
(2.6)
The threshold behaviour of the partial waves has been accounted for explicitly and thus a l l behaves proportional to the scattering lengths at threshold. a = a(tthr)2/
tX~r = lim ~/q2l+l . q~O
(2.7)
The elements of the diagonal matrix d in eq. (2.1) are the propagators of the intermediate states multiplied with functions that assure correct threshold behaviour. dil I = q~l h ( t )
for 2-particle states,
d~, l = h (1, t ) ,
for single-particles states.
(2.8)
The functions h are in the two-particle case the Chew-Mandelstam functions and in the single particle case proportional to poles. The functions are adjusted to vanish at t = 0 and therefore aI(O) = kl(O). The general form of the functions h is given in appendix B of ref. [ 1 ]. We should note here that the requirements of extended unirarity are automatically fulfilled in this approach. The crossing postulate connects K with the crossed channel quantities K t = z t + Ots(T s - Ks) + ~0tu(T u - K u ) .
(2.9)
The convergence domain of ( T - K ) in the s- or the u-channel is that of Im T, i.e. the large Lehmann ellipse [ 1 ]. This relation can be projected into partial waves to give k t. K t is the K- matrix looked at in the t-channel, Ks, Ts, Ku, T u the corresponding K- and T-matrices looked at in the s- and u-channel quantum numbers. Ots and Otu are the crossing operators that relate the different channels, in our case isospin crossing matrices. The matrix z is crossing symmetric (z t = Otsr s = Oturu) and regular at those normal thresholds that are dealed with in the coupled channels equations. It has singularities corresponding to higher thresholds as well as pinch singularities at t = 0 ariving from the normal thresholds of the t- and u-channel and correspondingly for s = 0 and u=0. The first type of singularities could be removed in principle but the second type not at all. Because of the regularity of r in the interesting physical regions however we can assume that r can be approximated by a polynomial or even a constant there [19]. We identify r as the matrix for the description of the coupling parameters of the "bare" world [1 ] and discuss the choices of r in sect. 3. The above eq. (2.9) expresses the singularity structure of K in terms of crossed-channel quantities. In ref. [1 ] we suggested an iterative scheme to fred unitarity and crossing symmetric amplitudes. With a given r and d one sets K = r and solves eq. (2.1) to f'md
433
C.B. Lang / K-matrix formalism
unitary amplitudes; out of those with help of eq. (2.9) an iterated K- matrix, and out of this an unitary amplitudes again and so on until the iteration converges. In each step of the iteration we know the current version of the amplitude and therefore we have to find a way to calculate the real part of K for t > 0 out of its values in the crossed channels. 2.2. Dispersion relations f o r k
To find k we choose a two-step continuation of fixed t and partial-wave dispersion relations, both in the framework of the so-called discrepancy method. The first serves to get the values of the real part of k on a portion of its left-hand cut, the second to extrapolate it to positive values. The method we apply here is up, to small differences similar to that of NO [ 13], they however wanted to obtain directly partial wave amplitudes and therefore had a different cut structure than we have it for K t. For fixed negative t values we cannot write one common dispersion relation which contains both the u- and the s-channel cut contributions, like it is done for the amplitudes. For the amplitude one knows the real part both in the u- and in the s-channel physical regions. For k we know only the real part of ( T s - Ks) in the s-channel and the real part of ( T u - K u ) in the u-channel. Both quantities therefore have to be treated separately. As an example we discuss the contribution of ( T s - Ks); this function contains the unitarity cut of the physical s-channel processes. The crossed cuts of T s are removed because they occur also in Ks. This is true only up to a contribution of the order of r, but we neglect this part because we assume that r can be approximated by regular functions [ 1, 19]. The imaginary part o f the function is known for t > - 3 2 , the limit of convergence o f the partial-wave sum of the s-channel. The real part is known only in the physical region of the s-channel (and in small bands outside) because of lack o f convergence of the partial-wave summation. For each fixed value o f t a discrepancy function is defined through 1 f
sR
A(s) = Ts(s ) - Ks(s ) - 7r , ,
Im [Ts(s' ) - K s ( s ' ) ] s' - s
ds'
so
1 / -
SR Im [Ts(SR) -- Ks(SR) ] s'(s' - s)
ds'.
(2.1o)
sR s O is the value o f the s-channel threshold and s R the upper limit of our knowledge of the imaginary part (we took s R ~ 1300 MeV). The rightmost term removes the logarithmic singularity at s R and can be evaluated analytically. The discrepancy A(s) is an analytic function which has a cut for s R < s; it is therefore expected to behave well in the interesting region. Because of the knowledge of the real part of ( T s - K s ) for - 1 > z t we also know A(s) in that region; we then can extrapola-
434
C.B. Lang / K.matrix formalism
te it down into the region z t > - 1 which gives Re (T s - Ks) for zt E ( - 1 + 1). It turned out in our calculations that the discrepancy function can be well approximated through 2 nd order terms in s. The extrapolation of the discrepancy function gives the desired values of Re (Kt) in the unphysical region. Partial-wave projection leads to the values of the imaginary part o f k for t E ( - 3 2 / a 2, 0) and the real part of k for t E (tc, 0); here -32/a 2 < t c < 0 is a confidence value up to that one believes that the fixed t dispersion relation has given the correct real part (for lrzr ~ 7rTr,t c = -24bt 2 and for 7rTr~ KK, t c = -15/12 turned out to be an adequate value). Again we write a dispersion relation for the partial waves and define a discrepancy function 1 0 A(t)=k(t)-~f l~'?(t')dt'-l tL
tL t L i m k ( t L ) t~'~O "dt' , _o.
(2.11)
where the last term is added to remove the possible logarithmic singularity at t = t L ; its evaluation gives 1 tL -- ~ ~-
t L -- t
Im k(tL) In
tL
(2.12)
The discrepancy function A(t) has only a LHC for t < t L and because of our knowledge of Re k ( t ) for t E (tc, 0) we know A(t) there, i.e. in a region where it is holomorphic. We now follow refs. [18] to get an analytic interior -~ interior continuation of A(s) to the physical values 0 ~< t. We perform a conformal transformation to a new variable z so that the LHC _oo ~< t ~< t L is mapped unto the boundary o f an unifocal ellipse in the z plane and t = tc, 0 are mapped to the fociz = - I , + 1. Then-A(s) can be represented inside the ellipse and therefore throughout the whole cut s plane by a Legendre series in z (practically a parabola in z). It is clear that the parabola is only an "educated guess" - the correct extrapolation curve belongs to a manifold of curves with a boundary which is determined by our explicite and implicite assumptions on the behaviour of A(t) on the unknown part of the LHC. Because of the mapping however we expect that the extrapolating curve is at least an optimal one according to our knowledge of the values on the nearby part of the LHC of k. Through this extrapolation we find approximate values of A(t) for 0 ~< t and therefore also for k(0 ~< t). Like already proposed [1] we solve the system of eqs. (2.1) and (2.9) by iteration. We calculate a (n + 1)th iteration o f K t out of the previous nth iteration. We therefore have an operator equation of the form K = ~(K).
(2.13)
The fixpoint of this equation fulf'tlls the requirements of crossing (compare refs. [19, 1 ]) and unitarity; the latter is explicitly fulfilled at each iteration step. A minor complication is that the operator in eq. (2.13) is in general not a contracting one. The modification
CB. Lang / K-matrix formalism
K5n*l) -±rL'(n)+2 t~-t rt + Ots(T(n) - K(sn)) + ¢tu (T(n) - K(n))]
435
(2.14)
turns out to remove this problem and gives the equivalent contracting operator K = ½(K + ~ ( K ) ) .
(2.15)
One has to keep in mind that eqs. (2.9, 2.14) are equations for the single elements o f K t which are always functions of the corresponding crossed processes. The elements of k t therefore carry the available information of the exchange processes; they are rather flat functions for physical t values. All strong structure of the direct channel is given through coupled channels.
3. Input assumptions We have to distinguish between three types of input. First we need the masses of the bare particles that enter into the equations as intermediate states. Secondly we need values of the r matrix elements i.e. values of the bare couplings. In principle, namely for a full iteration of all possible involved channels, this would be enough. We however do not iterate the KK and the 7rK channels and therefore we need additional information on the K-matrix elements for KK -~ KK and nn -~ KK; this is the third type of input we use.
3.1. Masses The pion and kaon masses are given from experiments, this allows to compute the intermediate state two-propagator functions (appendix B of ref. [1 ]). We then have to decide upon the coupled one particle states. We assume one possible state in each of the non-exotic channels with the quantum numbers (1 = 0, I = 0), (l = 1, I = 1), (l = 2, I = 0). The masses of the bare p and f can be estimated through the following consideration. The solution of the inverse amplitude for p-wave elastic rrn scattering with only the p and the two-pion state as intermediate states is (t22) -1 = (k22 + k21 dp) -1 + d ~ r ,
(3.1)
where dp and d,~= are the intermediate state propagators of eq. (2.8) and k21 = r(~Ttp ). The zero of the real part of (t22) -1 must lie where the experimental phaseshift passes ~l n. This leads to the mass value of the bare p and similarly o f f as 27.5 #2 and 77 #2; this shifting in the p- and d-waves does not depend too much on the left-hand cut or on coupled channel effects. Once these values are fixed the resulting masses of the physical p and f(30/z 2 and 82/~ 2) remain stable in the iteration process. If a shift would occur one would have to readjust the bare masses and start again. In the s-wave the difference of the bare mass to the physically visible mass (which we define to be the energy value where 8 = ½7r) is much larger. The requirement
CB. Lang / K-matrixformalism
436
s(6 (s) = ½7r) E [35p 2, 40U 2 ]
(3.2)
covers the experimental results quite well and is consistent with a value of 27.5 #2 for the bare e mass too. This would support the idea that the bare e is a daughter of the bare p. It was not necessary to include a bare S* particle, we discuss that later in more detail.
3.2. Coupling We assume that the r matrix elements for the 2 ~ 2 channels vanish. This means that we neglect the effects of other coupled channels as well as the pinch singularities at t = 0; analyticity therefore is violated to some amount. We can expect however that the violation is not felt in the physical region. In addition it means that in the bare world status there are no 4-vertices (four point functions). The nonvanishing 4-vertex coupling behaviour that we can measure in practice comes from unitarization and the inclusion of crossing; one could call this a breaking of original symmetries by unitarity. This assumption implies that we need only the bare 3-vertex couplings (three-point functions). We assume that they are energy-independent constants and postulate in accordance with general symmetry ideas like SU(3): l"(Trrte) = T(KKe),
r(rrTrp) = 7"(KKp),
r(TrTrf) = 7"(KK,f) .
(3.3)
The elements r(pp), r(ee) and r(ff) vanish for reasons discussed in ref. [1]. The experimental evidence for the full widths of the physical p and f again determines via the relationship (3.1) the values z(rrrrp) = 0.224,
r(Trrrf) = 0.046.
(The first value corresponds a g2~rp/41r = 2.5). The value of r(rrrre) cannot be determined that way and is therefore choosen to act as a free parameter. We assign it two different values and have therefore two different solutions for the whole set of amplitudes named A (z(Trzte)= 1) and B(r(nne) = 0.8). This parameter is related to different so wave scattering lengths (and because of crossing to s2 and pl low-energy behaviour too) as well as to the slope of the phase shift where it passes through ½rr and therefore to the width of the physical e.
3.3. Information on 7rK and inelasticity ~70 To find values for the K- matrix element nTr ~ KK, we use experimental information on rrK -* irK. We parametrize the s~, s], pl ( K * ) , d~ (K**) in terms of our modified K- matrix; we neglect the exotic p~- and d] waves because they seem to be negligibly small. Similar to NO [13] we fix the scattering relationship between dicted from the linear Weinberg model. The corresponding relationship between l 1 a~, a 0 and a] is fulfilled; for details we refer to Petersen [11]. It turns out to be
C.B. Lang / K-matrix formalism
437
possible to parametrize the K- matrix elements of this process 7rK --> 7rK by linear functions. The resulting s-wave phase shifts lie within the boundary that was used by NO [13]. To find out how much the final results depend on the nK input we use two different versions for the s~ phase shift. In one version we take 6 (1000 MeV) ~ 60 ° and in the other 6 (1000 MeV) ~- 70 °. The s] phase goes almost like a straight line and becomes - 1 5 ° at 1000 MeV. The K* and K** resonances were incorporated as possible intermediate particles and their bare masses and couplings were choosen to give the physical masses and widths in the 7rK -+ 7rK amplitudes. Again we find coupling values of 0.224 for K* and 0.046 for K**! Out o f these phenomenological informations on rrK --, nK it is possible to calculate the K- matrix elements for nn ~ KK with the procedure described in sect. 2. The remaining problem is the determination of the K- matrix element for KI~ ~ KK; the amplitude has a LHC from two-pion exchange that overlaps the extended unitary cut. In ref. [1 ] we demonstrated that the corresponding element of the modified K- matrix is real. For the p-wave and the d-wave we assume that the K- matrix element for this process just vanishes. From the strong effect at KK threshold in the so wave however we know that it certainly must not be neglected there. We therefore assume that the K- matrix element for the KK -~ KK. so wave can be approximated by a linear function in t, vanishing at t = 0 as required in the linear current algebra models. The only free parameter is the slope which can be determined by requiring that the output inelasticity r~0 exhibits a behaviour as expected from the experiment; it should have a minimum between 1000 MeV and 1050 MeV and should be near to 0.4 at 1075 MeV. These requirements can be easily fulfilled by adjusting the free slope parameter and (although restricted only by one parameter) the resulting r/0 shows fine behaviour in all solutions.
4. Results
In the calculations we cover a range between 0 and 100/.t 2 in all relevant channels. The iteration for the K- matrix elements nTr ~ n n converges relatively fast, the results become almost stable after the 5 - 6 t h iteration. This comes mainly from the fact that our K- matrix represents only the crossed channels effects and therefore behaves very flat in the direct channel. The main effects of the direct channel come from the coupling to other states. Similar to refs. [13, 15, 16] we determine a "theoretical" error corresponding to the uncertainty o f the irK s-wave input. We always calculate two version o f a solution according to the two different values of 6~ at 1000 MeV. The resulting "error bars" o f the amplitudes however are very small, almost everywhere below KI~ threshold less than 5% and only near KI~ threshold approximately 10% of the values. These differences are smaller than those of H - J [16] and NO [13] for two reasons. Firstly we have only a 10% difference between the two s~ input phase shifts at 1000 MeV compared to a variation of 2 0 - 3 0 % in ref. [13 ]. Secondly, and
438
CB. Lang / K-matrix formalism
250"
ZOO'
150"
6o° (~Tt--,qC~)
100'
50"
I
°~0
I
4o0
I
I
6o0
I
I
800
I
I
1000
1200
for rrlr Fig. 1. Solutions A (solid line) and B (dashed line) for the I = 0 s-wave phase shift 80 u rrlr. Also shown are the phases of Estabrooks et al. [4] +, Hyams et al. [6] ~, Manner [81 * and Protopopescu et al. [31 #. 200
400
600
.........
800
1000
1200
';+1
F~g. 2. Solutions A (solid line) and B (dashed line) for the exotic nlr ~ lr~rs-wavephase shift 80. The experimental phases are from ref. [22], namely Cohen et al. ~k, Colton et al.~, Hoogland et al.-~and Losty et al.--I-. this is the main reason, the uncertainty enters in the K- matrix element for rtrt KI(. The amplitude however is a function of all involved K- matrix elements and of the possible intermediate single particle states which play the dominant role. In this mixture the effects of the uncertainty of the K- matrix element rtrt ~ KK, are suppressed. It is this dynamical feature which is mainly responsible for the small "theoretical error bars". 4.1. S waves In the s0 wave we assume that one "bare" particle is coupled to both ~Tr and KI( with equal coupling constant. We give here the results according to the values
C.B. Lang / K-matrix formalism
439
Table 1 Scattering lengths of solutions A and B (in parentheses) Process
a0
a2
nlr --, ~rTr
+ 0.45 (+ 0.26) - 0.30 + 1.04i (- 0.34 + 1.11i)
-0.010 + 0.039 (-0.015) (+ 0.026) - 0.018 ÷ 0.013i ( - 0.017 ÷ 0.012i)
KK,-~ KK
a]
a0Xl04
a~× 104
+ 23.2 (+ 18.2) 1.34 + 0.14i (1.34 + 0.13i)
+ 5.5 (+ 7.7)
1.0 and 0.8 for this constant. It is hardly believable that the physical value could be below 0.8, if our simplicity assumption of vanishing 7(2 ~ 2) should still hold. Solution A (1 "0) seems to be nearest to the experimental results (figs. 1-2). In the process zrn ~ zrTrwe find a good agreemerit with experiments; solution A prefers a behaviour according to the results of Estabrooks et al. [4] and Manner [8] whereas solution B (0.8) is closer to the Berkely points of Protopopescu et al. [3]. Both solutions lie between solutions 2 and 3 of BFP [10]. The values of the combination of the s-wave scattering lengths (2a00 - 5a02) lie on the boundary of the "universal band " [10]. The values of the scattering lengths can be found in table 1 ; the value for a 0 of solution A coincides with Manners [8] value. The slope of the phase shift 50 where it goes through ½zr is relatively small; this corresponds to a rather broad e(P, > 5/a). We however do not try to localize the exact position of the e in the unphysical sheet. The sudden rise of the phase shift near 1000 MeV comes from the opening of the KK channel at 50 #2. No additional bare particle is necessary to give this expected increase of the phase shift ("S*") and the expected behaviour of 70 (fig. 3). Above 1000 MeV the phase shift seems to stay below 270 ° in contrast to most curves from experiments [ 6 - 8 ] ; they increase above 270 ° near 1200 MeV and run to much higher values. This structure however might be due to higher resonances or thresholds and should then be accounted for by the introduction of additional bare particles or coupled channels. The nn -~ KK amplitudes (fig. 4) show in the region of extended unitarity a very strong onset of the e. The slope at t = 0 differs by a factor of two from the linear current algebra predictions [11]. This comes from the very broad e which is in our approach the most important symmetry breaking term. The main contributions to the s- and p-wave scattering lengths come from the effect of this resonance in the crossed and direct channels. A comparison with the prediction of refs. [15, 16] exhibits a clear difference in the behaviour of the scattering amplitude below the e region. To test our assumptions we actually calculated the amplitudes also for nonvanishing r(2 ~ 2) elements. We took the r elements to be half the values which the linear current algebra models [21 ] predict for the amplitudes. The results were more similar to those of ref. [16] in the e region. The price for it were however several bad features; the inelasticity 770 became small soon above 1000 MeV but remained small over a to long region; the exotic phase was increasing too fast (--45 ° at 1000 MeV).
440
C.B. Lang / K-matrix formalism
,
i
I
I
i
i
I
I
1,0
"
1.0
Fig. 3. The rrn inelastieities r/0, ~ and r/0 of solution A (solution B is indistinguishable). The data points are from ref. [3] # and ref. [6] ~.
lo
zo
3b ~~
( 'I
x'
"'
"
2)='-
-1.0
Fig. 4. The s-wave (I = 0) amplitudes a~0 for Irlr ~ KK, solutions A (solid line) and B (dashed line). In fig. 5 we compare the phase shift of the sO wave of KI~ -+ KI~with recent experimental data from Beusch [17]. Here we agree with JP [15] and get a large negative phase above KI~ threshold and a negative scattering length. The I = 2 rtlr -+ lrlr process is elastic without resonances. The scattering length and the phase shift (fig. 2) are completely determined by the exchange processes; solution A is in good agreement with the experiments, solution B seems to be too small.
CB. Lang / K-matrix formalism
1.0
1.1
t.2 I
441
1.3 i
I MK[
i
(GeV)
-50"
-1013"
Fig. 5. The s-wave ( / = 0) phase shift for KK "-~ KK, solutions A (solid line) and B (dashed line). The data points are from Beusch et al. [17]; above 1.2 GeV we took only data points of the down-type solution.
4.2. P waves The bare p meson in this process is coupled to nTr and KI( with equal bare coupling constants. The resulting phase shift in the zr~r~ ~rrr channel has the required resonating behaviour at the physical p mass 775 MeV with the width 150 MeV. The obtained value for the scattering length (table 1) lies near to the value 0.033, which is predicted by current algebra. We cannot reproduce M/inner's [8] value near 0.1. The K- matrix element for Kg~ ~ Ki~ is set to zero; it is therefore surprising that the inelasticity seems to be close to the experimental values (fig. 3). Our approach is no analysis of the data and is predictive only on the basis of the input assumptions; we therefore cannot say anything about a possible p (1600 MeV) resonance. The general behaviour o f the lrTr ~ Kg~ amplitude (fig. 6) is very similar to that of H - J [16] although our numerical values are somewhat larger. The only qualitative difference is that the solution of H - J chooses the current algebra value at t = 0 whereas our solution has a zero at t ~/a 2 and is small and negative at t = 0. This may be due to different methods of analytic extrapolation from the LHC. In fig. 6 we also show the KI( -~ KI~ amplitudes which are almost the same like those for zrn -~ KK. The formalism leads to "unitarized coupling constants" i.e. T- matrix elements for rrzr -~ bare p and KK -~ bare p. Because of extended unitarity the phase of these functions is the same as that for the corresponding 2 -~ 2 amplitude. The functions can be interpreted as form factors which are defined like those of Gellmann and Zachariasen [20] having a zero at the position of the bare pO. Removing this zero by multiplication with 1/(m20 - t) and normalizing to unity at t = 0 leads to expressions for the isovector c~ntribution to the electromagnetic form factors of the pion and the kaon. In fig. 7 we compare them with each other; because of the normalization to unity at t = 0 there is no factor of ½ in the kaon formfactor. They are equal up to a 10% difference in the p region.
C.B. Lang / K-matrix formalism
442
0.1
10 20 30 z,O 50 60 70 _.8_0~-t~ -0"11 "
0.1
-~R i . , -" lb " 2'0 3b 4'0 " 50 60 " 7'0 eb t(p.z)
-0.1 Fig. 6. Solutions A (solid line) and B (dashed line) for the I = 1 p-wave amplitudes for rr~r --* KK and KK --* KK,. I
,
I
"
I
I
I
'
I
I
30
fi IF~'"I'
20
10
10
20
30
40
50 60
70
t (~=)
Fig. 7. The squared absolute value o f the pion (full line) and the kaon (dashed line) isovector contribution to the form factors in the timelike region. Both are from solution A and normalized to unity at t = 0.
C.B. Lang / K.matrix formalism
443
4.3.D waves
A bare f meson is coupled to the 7rn and KK, channels with equal bare coupling constants again. The position and width of the f meson is bound to have the experimental values m 2 = 82/~2 and Ff = 170 MeV. The 7rTr~ nrr scattering lengths and phase shifts below 1000 MeV for I = 0 and 1 = 2 are similar to solution 3 of BFP [ 10]. The phase shifts (fig. 8) agree with the values of table 6 of [ 10] within the errors given there. Close above threshold the 1= 2 phase shift is positive but very small and very soon it becomes negative and reaches - 1 ° already before 1000 MeV energy. An unexpected feature is its turning point and the decrease towards zero and positive values. The absolute value of the phase shift however is very small and this behaviour might be due to the large extrapolation distance from the LHC. The extrapolated K- matrix element is extremely small and the value might become incorrect above 1 0 0 0 - 1 2 0 0 MeV. This is a reason to trust our results only up to 1200 MeV, Like in the p-wave we put the K- matrix element for KK ~ KK equal to zero and again we get a remarkable coincidence o f our resulting inelasticity r/0 (fig. 3) with experiment.
20'
10'
:
'
sbo
'
-I' \ -2*
\
\
\
\
N~.~js
Fi~. 8. Solutions A (solid line) and B (dashed line) for the I = 0 and I = 2 d-wave phase shifts
~ a n d 8 2 for ~v-+ ~t.
444
CB. Lang / K-matrix formalism
5. Conclusion The relevance of the involved processes and their relationship to other scattering processes has been discussed already by many authors. Let us therefore not go into much details of the results but let us point out only the main features. On basis of the proposed modified K- matrix formalism [ 1] it is possible to find solutions for a coupled channel system of scattering processes. Our way to proceed is first to specify the assumptions on the structure of the bare world, a world where the effects of unitarity and crossing have been removed, and then to solve the combined matrix equations. It is by no means obvious that 4-vertices have to be included, on the contrary we find that in the treated meson meson system it is sufficient to include only 3-vertices (i.e. three point functions) in the bare coupling behaviour. In terms of a Lagrangian description this would mean that no ~4 contact term is necessary. Qualitatively we therefore agree with BFP [10] who discuss 7rTrscattering with the help of Roy's equations. They conclude that a saturation of the requirements of unitarity and crossing still leaves the freedom of 5 parameters which they identify "~s ,,0 ,,2 rnp, l~p and the energy where 60 goes through ½n. Once these are satu~0' "0' rated the low energy behaviour is fully determined. They compare this situation with that of a Born term of a Lagrangian which contains also five parameters namely one from the contact term, two from e and two from p intermediate states. This is analogous to the freedom in our approach; two free parameters specify the bare e state and its coupling to ~rzr, two more specify the p. The fifth open parameter just happens to vanish. The intermediate l = 0 state leads to the characteristic s-wave phase shift and has the main responsibility for s- and p-wave low-energy parameters e.g. scattering lengths. The results are compatible with current algebra although they disagree with the more spcialized Weinberg model. We therefore conclude that the type of breaking an original symmetry through the inclusion of crossing and unitarity leads to different results than that proposed by Weinberg. The modified K- matrix seems to be an advantageous possibility to describe scattering data. The concept of single particle channels is easy to handle and prevents a pole parametrization of the K- matrix. Coupled channel effects like the S* phenomenon can be simply described by flat K- matrix elements. In addition the multichannel formalism leads to expressions for the form factors. The computational apparatus is rather large but the coincidence with experiments is remarkable. Further investigations might make it possible to find an easier way to determine the K- matrix elements out of the crossed channel processes. The formalism enables one to remove at least to a certain amount unitarity and crossing effects in order to find a simple structure behind the complicated scattering data. I would like to thank Professor G. H6hler for his constant interest in this work. Much profit I had from the fruitful discussions with Drs. J.L. Basdevant, J.L. Petersen, E. Pietarinen and I. Sabba-Stefanescu. I am thankful to Dr. E. Pietarinen for reading the manuscript.
CB. Lang / K-matrix formalism
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[6] [7] [8]
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