Check details of S-wave amplitude implementations

[Thomas Latham]

Check through the K-matrix implementation (particularly given Wenbin's findings regarding the phase-space definitions in different papers).

[John Back]

Below is a detailed summary of the questions posed by Wenbin to Tom, which is the reason for this ticket, with my (John) additional remarks:

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1. Wenbin: As far as I know, the K matrix should be Hermitian, while if looking at Appendix A.14 (John: reference unknown for this), if f_ij^scat is not symmetric (which is the case in the parameterisation), K matrix could not be Hermitian. Do I miss something in this part?

Tom: I think f_ij^scat is symmetric in actual fact. What makes you think it isn't?

Wenbin: The statement in the paper said that if i != 1, then f_ij = 0.

Tom: This statement refers only to the f_ij^prod in the production vector, not to the f_ij^scat in the K-matrix.

Wenbin: ​http://uk.arxiv.org/pdf/0804.2089v2.pdf see in page 10.

*John: The f_ij term here is indeed the scattering term. The K matrix is calculated by first finding the upper half of the matrix, so only considering the first row of f_ij (i=1) is OK. Then, the bottom half of the K matrix is set to be the symmetric counterpart of the upper half. In other words, Laura++ does produce a symmetric K matrix as expected.

2. Wenbin: How is the phase-space matrix rho calculated if the invariant mass is less than the threshold of other resonances besides pipi? For the rho of 4pi states, does it follow the one in the theory paper (​http://uk.arxiv.org/pdf/hep-ph/0204328) or it is just a simple implementation as others?

Tom: I just checked the FOCUS paper that we also refer to at the top of the code (hep-ex/0312040) and they have this other form for the phase space factor. So I guess we got it from there.

*John: The phase space matrix is continued analytically by becoming imaginary if the invariant mass of the system becomes less than the threshold of other resonances.

The multi-body phase space (4pi) matrix term uses a parameterisation that I think was meant to represent the theoretical integral quoted in hep-ph/0204328. I am not sure where this parameterisation was obtained(!), but it does produce a rapidly rising function that somewhat agrees with the curve shown in a thesis regarding a BaBar K-matrix analysis: see Fig 4.9 in

​http://www.slac.stanford.edu/pubs/slacreports/reports17/slac-r-872.pdf.

However, the parameterisation also rapidly rises at low values of the invariant mass squared of the system (s < 0.2), which is not correct. I've re-calculated the double integral quoted in hep-ph/0204328, which can be approximated very closely with a six-order polynomial in s. But the curvature of this new parameterisation does not exactly match that shown in Fig 4.9 of the above thesis. I am also not sure about the "gamma" term used in Eq 4 in ​​http://uk.arxiv.org/pdf/hep-ph/0204328 for the energy-dependent width Gamma(s) = gamma*rho³_1(s) and whether the mass of the rho M is just the pole mass of 770 MeV, or actually should be a "Breit-Wigner" expression; the theory paper does not explicitly make this clear.

Also, the analytic continuation at s = 1 is different between the Laura++ code and the theory paper. In Laura++, the phase space term is sqrt( (s - (4*m_pi)²)/s ), whereas the theory paper omits the square-root. Fig 4.9 in the above thesis supposedly uses the expression (s - (4*m_pi)²)/s at s = 1, advocated by hep-ph/0204328. However, Fig 4.9 is inconsistent with this; at s = 1, the phase-space factor should be ~0.69 according to the non-square-root formula, whereas the plot shows a value of ~0.83 which matches the square-root form!

3. Wenbin: For the eta eta prime phase space, you use sqrt( (s - sumetaeta')*(s-diffetaeta') )/s, while in the theory paper you refer to, it suggests to use sqrt( (s - sumetaeta')/s )?

Tom: I just checked the FOCUS paper that we also refer to at the top of the code (hep-ex/0312040) and they have this other form for the phase space factor. So I guess we got it from there.

Wenbin: For the phase space factor of eta, eta', I also take a look at Jordi Garra Tico's code, they use the one as in the theory paper and not as in the FOCUS. I will run the code to check how much this phase space factor changes the results (the main difference comes from the part m <= (m_eta' - m_eta) where the phase space of eta, eta' becomes real).

*John: There appears to be some inconsistency about this in the theory and experimental papers. From the PDG, the kinematic factor for the system should include the m_eta' - m_eta term; see Eq 46.16 in ​http://pdg.lbl.gov/2014/reviews/rpp2014-rev-kinematics.pdf. When the system has two particles of the same mass, the difference term gives the factor unity. Also, Eqs. 10 and 13 in the K pi S-wave theory paper ​http://uk.arxiv.org/abs/hep-ph/9705401 (from the same authors!) uses the difference term when the particles have different mass; see also Eq 40 in ​http://hadron.physics.fsu.edu/~e852/reviews/pwakmx_a.ps.

So I think the expression in hep-ph/0204328 is in fact wrong, or the authors simply forgot to include the case when the two particles have different mass. Laura++ uses the difference term shown in several other theory papers and the PDG kinematic reference.

I will try to contact Jordi to compare what the codes do.

[John Back]

Equations for the multibody (4pi) phase space factor in hep-ph/0204328

[John Back]

The 4pi phase space factor in the thesis SLAC-R-872

[John Back]

Various tests of the 4pi phase space factor in Laura++

[John Back]

Wenbin: image of the S-wave using the m_eta - m_eta' term for 4pi

[John Back]

Wenbin: image of the S-wave without using the m_eta - m_eta' term for 4pi

[John Back]

Jordi has replied to confirm that the 4pi phase space factor used in Laura++ is the same one used in the BaBar analysis. Also, the eta-eta' mass difference term was not included in the BaBar model, following the advice from the theory paper ​http://uk.arxiv.org/pdf/hep-ph/0204328 about "avoiding false singularities in the physical region". However, in Laura++ we continue analyticity for s below the threshold for a given resonance by setting sqrt(-number) to i*sqrt(number). So we should probably keep the mass difference term for the phase space factors, following the advice from the PDG kinematics section and other theory papers.

As shown in ​https://laura.hepforge.org/trac/attachment/ticket/7/Laura_rho4pi.png, the BaBar (current Laura++ code) 4pi phase-space term grows almost exponentially for s below ~0.1, owing to its 1/s and 1/s² terms. A new 6-order polynomial in s has been developed by trying to integrate the double integral in the hep-ph/0204328 theory paper. This does not exactly match the curvature of the BaBar model, but is reasonably close (see the Laura_rho4pi.png figure). Furthermore, the new parameterisation avoids the large increase for s < 0.1, and continues analytically from s = 1 with the usual kinematic square-root factor. So I think the code in LauKMatrixPropagator::calcFourPiRho() needs to be updated with this new function.

[John Back]

(In [190]) Corrected the eta-eta' and 4pi phase space factors in LauKMatrixPropagator,

which is used for the K-matrix amplitude:

• calcEtaEtaPRho() does not include the mass difference term m_eta - m_eta' following the recommendation in hep-ph/0204328 and from advice from M Pennington
• calcFourPiRho() incorporates a better parameterisation of the double integral of Eq 4 in hep-ph/0204328 which avoids the exponential increase for small values of s (~< 0.1)
• More detailed comments are provided in the above two functions to explain what is going on and the reason for the choices made

addresses #7

[John Back]

The plot ​https://laura.hepforge.org/trac/attachment/ticket/7/Laura_rho4pi.png showed a new parameterisation of the 4pi phase space double integral which did not include the "width" of the 4pi state gamma ~ 0.3 GeV. Including this actually gives perfect agreement with the BaBar model, also avoiding the exponential growth for s < 0.1; see ​https://laura.hepforge.org/trac/attachment/ticket/7/Corrected_Laura_rho4pi.png

[John Back]

Corrected version of the 4pi phase space double integral comparisons

[John Back]

ROOT macro used to calculate and parameterise the 4pi phase space integral

[John Back]

Plots showing the T amplitude for the pi pi S-wave K-matrix

[John Back]

The file ​https://laura.hepforge.org/trac/attachment/ticket/7/KMatrixTAmpPlots.png shows plots of the T amplitude for the pi pi S-wave K-matrix, using examples/PlotKMatrixTAmp.cc; these match those in SLAC-R-872.