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We shall first discuss the technique known as “Barrelet zeros” [Barrelet, 1972] as it is applied toππ scattering, e.g. [Becker et al., 1979; Estabrooks and Martin, 1975; Hyams et al., 1975], because the situation there is much simpler and the technique becomes clear before applying it to the more complicated situation in decays.

Discrete Ambiguities in ππ Scattering Partial Waves In general, the partial-wave

sum for the scattering amplitude of two pions in the case where the highest angular momentum `occuring isL can be written

A(s, t) =

L

X

`=0

(2`+ 1)a`(s)P`(z= cosθ), (7.7) where the amplitude is expressed as function of the Mandelstam variabless, t and z= cosθcan be calculated as function of the two. This is a polynomial of degreeLinz and can thus be written as function of its complex-valuedBarrelet zeroszi(s), to wit

A(s, t) =f(s)

L

Y

i=1

(z−zi(s)). (7.8) The measured values are the cross-sections at real values of s, z which for fixed s are proportional to the modulus-squared of the amplitude, which for real scattering angles and therefore real values of zis

dσππ→ππ dt ∝ |A(s, t)| 2 _=|f(s)|2 L Y i=1 (z−zi(s))(z−zi∗(s)). (7.9) Besides being independent on the overall phase contained in f(s), the value of this expression is invariant under the substitution zi → zi∗ for any of the 2L combinations of the i ∈ {1, . . . L}. Therefore, from measurement alone the amplitude Eq. (7.8) can only be determined up to this 2L-fold ambiguity in the signs of the imaginary parts

of the Barrelet zeros. Since the Legendre polynomials form an orthogonal basis of the space of polynomials, each such ambiguous solution in the form of Eq. (7.8) can be converted back to a set of partial waves in the form of Eq. (7.7) and each such set of partial waves predicts the same angular distributions. The decision which such set of angular distributions catches the physics therefore needs additional input, such as behavior corresponding to known resonances, requirements from analyticity etc.

Discrete Ambiguities in Two-Pseudoscalar Final States in Scattering Experiments

Here, the situation is more complicated as one has to deal with two non-interfering contributions coming from natural and unnatural exchange waves, respectively. Addi- tionally, we have to take into account the magnetic quantum numberm which is usually

≤1. Following the methods of Refs. [Barrelet, 1972; Costa et al., 1980; Gersten, 1969; Martin et al., 1978; Sadovsky, 1991], one can write for the amplitudes after a judicious choice of linear combinations and variables [Chung, 1997]

g(u) =c 1 (1 +u2₎L 2L Y k=1 (u−uk), h+(u) =c0 u (1 +u2₎L L−1 Y k=1 (u2−rk). (7.10)

where u = tan(ϑGJ/2) and L is the maximum angular momentum considered. The

intensity is related to these in the following way: writing

h0(u) = 1 √ 2(g(u) +g(−u)), h−(u) = 1 2(g(u)−g(−u)), (7.11)

one has for the intensity function

I(Ω) = 1 4π h |h0(ϑGJ) + √ 2h−(ϑGJ) cosϕGJ|2+| √ 2h+(ϑGJ) sinϕGJ|2 i . (7.12) Relations between the coefficients of the polynomials in Eq. (7.10) and the partial waves are given in Ref. [Chung, 1997]. The ambiguous solutions are now obtained by sequen- tially trying all combinations of complex conjugates of the zeroes uk, rk in Eq. (7.10) and solving for the partial waves.

In the case under consideration, unnatural-exchange waves are absent from the data. In this case, ambiguities only arise when angular momenta of at least three units are present. Additionally, an incomplete set of waves, such as our main setP+,D+,G+, has

no ambiguities besides the trivial indeterminacy of the signs of the relative imaginary parts. We return briefly to the question of ambiguities when we include theF+wave in

Continuous Ambiguities In this section we set out to show how a fit with higher-rank density matrix is not possible in the two-pseudoscalar case when only the information from the angles ϑGJ, ϕGJ is taken into account. The intensity function for a rank-two

fit then has the general form

I(ϑGJ, ϕGJ) =I1(ϑGJ, ϕGJ) +I2(ϑGJ, ϕGJ), (7.13)

whereI1,I2are the incoherent contributions, which we assume both to be due to natural-

parity exchange. The goal of this section is to show that the intensity distributionI1+I2

can, by an appropriate choice of partial waves, be expressed with a single set of partial waves, i.e. any distribution where the physics requires rank two can be fitted with a rank-one density matrix. Since the case of rank one is a special case of the case of rank two (set all partial waves in the second component to zero, i.e. I2 = 0), it is then clear

that a fit with rank two or higher can’t be stable, one has to fit with rank one and then be careful in the physical interpretation to allow for incoherent contributions.

The assertion is easily proven by looking at the relations between the partial waves and the moments of the intensity distribution, which taken together are equivalent to the intensity distribution. The moments are defined in the usual way as

H(LM) =w dΩY_ML∗(ϑGJ, ϕGJ)I(ϑGJ, ϕGJ). (7.14)

Since this is linear one has H(LM) =H1(LM) +H2(LM), whereH(LM) refers to the

moments of I and H1(LM),H2(LM)to the moments ofI1,I2. For the case of only the

natural-parity wavesP+,D+being present, relevant to the low-mass range, the following

linearly independent moments obtain in terms of the production amplitudes,

H(00) =|P+|2+|D+|2, H(10) = √1 5Re(P ∗ +D+), H(40) =−4 21|D+| 2_. (7.15)

Writing P1, D1, P2, D2 for the production amplitudes of the partial waves the non-

interfering components, one finds from H(LM) = H1(LM) +H2(LM) that the same

angular distribution is described by the rank-one density matrix with waves P+, D+

given by

|D+|2 =|D1|2+|D2|2,

|P+|2 =|P1|2+|P2|2,

cos ∆φ= |P1||D1|cos ∆φ1+|P2||D2|cos ∆φ2

|P+||D+|

,

(7.16)

where the∆φdenote the phase differencesarg(D+/P+), and similar for∆φ1,∆φ2. This

result is unique up to a global phase and the sign of∆φ. Note that this solution always exists except in the uninteresting case where either |D+| or|P+| is zero and the phase

differences therefore becomes undefined. For the case with an additonal G+ wave, one

derives similar relations. Since the rank-one case can be reached by the limit D2 → 0, P2 →0, there is no unique best description of the data in a rank-two fit.1 In the special

case where cos ∆φ1= cos ∆φ2= cos ∆χ one can write

cos ∆φ=ξcos ∆χ, 0≤ξ ≤1, (7.17) where ξ is then a measure of coherence and ∆χ is interpreted as the physical phase difference which can be extracted if assumptions aboutξ are made [Martin et al., 1978]. This issue seems to have escaped the authors of analyses of the π−η and π−η0 final states with the exception of the authors of Ref. [Chung et al., 1999] who instead go on to show that for the case of a fit which accomodates Breit-Wigner resonances in both the D+ and P+ partial waves, one can find a rank-two density matrix which has these

same resonances in both non-interfering contributions. From this, they seem to conclude that a successful fit to the rank-one density matrix with both aP-wave resonance and a

D-wave resonance implies that even in the presence of non-interfering contributions, this interpretation should hold. How this exercise in algebra should have the stated physical conclusion remains unclear, as only for very specific relative strengths of the various partial waves one will observe the resonances present in the rank-two density matrix with the same functional form in the rank-one fit. Indeed, one can show that under fairly general circumstances, one can find coefficients such that P1 = αP, D1 = βD, P2 =γP, D2 =δD. For this it is not needed the P and D be Breit-Wigner functions,

contrary to what the authors of Ref. [Chung et al., 1999] seem to suggest. What is really shown by Eq. 7.16 is that any interpretation of the results of the mass-independent fit should allow for non-interfering contributions.