Dispersion relations for γ*γ* → ππ: helicity amplitudes, subtractions, and anomalous thresholds

Hoferichter, Martin; Stoffer, Peter

doi:10.1007/jhep07(2019)073

Dispersion relations for γγ → ππ: helicity amplitudes, subtractions, and anomalous thresholds

Journal Article · Mon Jul 15 00:00:00 EDT 2019 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/jhep07(2019)073· OSTI ID:1609329

Hoferichter, Martin ^[1]; Stoffer, Peter ^[2]

Univ. of Washington, Seattle, WA (United States); DOE/OSTI
Univ. of California, San Diego, CA (United States)

We present a comprehensive analysis of the dispersion relations for the doubly-virtual process γ^*γ^* → ππ. Starting from the Bardeen-Tung-Tarrach amplitudes, we first derive the kernel functions that define the system of Roy-Steiner equations for the partial-wave helicity amplitudes. We then formulate the solution of these partial-wave dispersion relations in terms of Omnès functions, with special attention paid to the role of subtraction constants as critical for the application to hadronic light-by-light scattering. In particular, we explain for the first time why for some amplitudes the standard Muskhelishvili-Omnès solution applies, while for others a modified approach based on their left-hand cut is required unless subtractions are introduced. In the doubly-virtual case, the analytic structure of the vector-resonance partial waves then gives rise to anomalous thresholds, even for space-like virtualities. We develop a strategy to account for these effects in the numerical solution, illustrated in terms of the D-waves in γ^*γ^* → ππ, which allows us to predict the doubly-virtual responses of the f₂(1270) resonance. In general, our results form the basis for the incorporation of two-meson intermediate states into hadronic light-by-light scattering beyond the S-wave contribution.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of California, San Diego, CA (United States); Univ. of Washington, Seattle, WA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: FG02-00ER41132; SC0009919

OSTI ID:: 1609329

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 7 Vol. 2019; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

References (83)

On the General Theory of Collisions for Particles with Spin Jacob, M.; Wick, G. C. Annals of Physics, Vol. 281, Issue 1-2 https://doi.org/10.1006/aphy.2000.6022	journal	April 2000
What we can learn from γγ→ππ,K $$\bar K$$ in the resonance region Morgan, D.; Pennington, M. R. Zeitschrift für Physik C Particles and Fields, Vol. 37, Issue 3 https://doi.org/10.1007/BF01578139	journal	September 1988
New dispersion relations and their application to partial-wave amplitudes Hite, G. E.; Steiner, F. Il Nuovo Cimento A, Vol. 18, Issue 2 https://doi.org/10.1007/BF02722827	journal	November 1973
Pion photon scattering Gourdin, M.; Martin, A. Il Nuovo Cimento, Vol. 17, Issue 2 https://doi.org/10.1007/BF02732027	journal	July 1960
On the solution of certain singular integral equations of quantum field theory Omnès, E. Il Nuovo Cimento, Vol. 8, Issue 2 https://doi.org/10.1007/BF02747746	journal	April 1958
Invariant amplitudes for virtual compton scattering off polarized nucleons free from kinematical singularities, zeros and constraints Tarrach, R. Il Nuovo Cimento A, Vol. 28, Issue 4 https://doi.org/10.1007/BF02894857	journal	August 1975
Dispersion relation for hadronic light-by-light scattering: two-pion contributions Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano Journal of High Energy Physics, Vol. 2017, Issue 4 https://doi.org/10.1007/JHEP04(2017)161	journal	April 2017
Roy-Steiner equations for pion-nucleon scattering Ditsche, C.; Hoferichter, M.; Kubis, B. Journal of High Energy Physics, Vol. 2012, Issue 6 https://doi.org/10.1007/JHEP06(2012)043	journal	June 2012
Dispersive approach to hadronic light-by-light scattering Colangelo, G.; Hoferichter, M.; Procura, M. Journal of High Energy Physics, Vol. 2014, Issue 9 https://doi.org/10.1007/JHEP09(2014)091	journal	September 2014
Dispersion relation for hadronic light-by-light scattering: theoretical foundations Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano Journal of High Energy Physics, Vol. 2015, Issue 9 https://doi.org/10.1007/JHEP09(2015)074	journal	September 2015
Dispersion relation for hadronic light-by-light scattering: pion pole Hoferichter, Martin; Hoid, Bai-Long; Kubis, Bastian Journal of High Energy Physics, Vol. 2018, Issue 10 https://doi.org/10.1007/JHEP10(2018)141	journal	October 2018
On the general theory of collisions for particles with spin Jacob, M.; Wick, G. C. Annals of Physics, Vol. 7, Issue 4 https://doi.org/10.1016/0003-4916(59)90051-X	journal	August 1959
Chiral perturbation theory to one loop Gasser, J.; Leutwyler, H. Annals of Physics, Vol. 158, Issue 1 https://doi.org/10.1016/0003-4916(84)90242-2	journal	November 1984
The two-photon particle production mechanism. Physical problems. Applications. Equivalent photon approximation Budnev, V. M.; Ginzburg, I. F.; Meledin, G. V. Physics Reports, Vol. 15, Issue 4 https://doi.org/10.1016/0370-1573(75)90009-5	journal	January 1975
Measurement of π−-meson polarizability in pion compton effect Antipov, Yu. M.; Batarin, V. A.; Bessubov, V. A. Physics Letters B, Vol. 121, Issue 6 https://doi.org/10.1016/0370-2693(83)91195-4	journal	February 1983
Chiral lagrangians for massive spin-1 fields Ecker, G.; Gasser, J.; Leutwyler, H. Physics Letters B, Vol. 223, Issue 3-4 https://doi.org/10.1016/0370-2693(89)91627-4	journal	June 1989
Charged pion polarizabilities to two loops Bürgi, Urs Physics Letters B, Vol. 377, Issue 1-3 https://doi.org/10.1016/0370-2693(96)00304-8	journal	June 1996
Phenomenological Lagrangians Weinberg, Steven Physica A: Statistical Mechanics and its Applications, Vol. 96, Issue 1-2 https://doi.org/10.1016/0378-4371(79)90223-1	journal	April 1979
Inelastic lepton (anti-) lepton scattering and the two-photon exchange approximation Bonneau, G.; Gourdin, M.; Martin, F. Nuclear Physics B, Vol. 54, Issue 2 https://doi.org/10.1016/0550-3213(73)90440-9	journal	April 1973
Meson pair production in two-photon processes Babelon, O.; Basdevant, J. L.; Caillerie, D. Nuclear Physics B, Vol. 114, Issue 2 https://doi.org/10.1016/0550-3213(76)90588-5	journal	November 1976
Chiral perturbation theory: Expansions in the mass of the strange quark Gasser, J.; Leutwyler, H. Nuclear Physics B, Vol. 250, Issue 1-4 https://doi.org/10.1016/0550-3213(85)90492-4	journal	January 1985
Two-pion production in photon-photon collisions Bijnens, Johan; Cornet, Fernando Nuclear Physics B, Vol. 296, Issue 3 https://doi.org/10.1016/0550-3213(88)90032-6	journal	January 1988
The role of resonances in chiral perturbation theory Ecker, G.; Gasser, J.; Pich, A. Nuclear Physics B, Vol. 321, Issue 2 https://doi.org/10.1016/0550-3213(89)90346-5	journal	July 1989
Low-energy photon-photon collisions to two-loop order Bellucci, S.; Gasser, J.; Sainio, M. E. Nuclear Physics B, Vol. 423, Issue 1 https://doi.org/10.1016/0550-3213(94)90566-5	journal	July 1994
Pion polarizabilities and charged pion-pair production to two loops Bürgi, U. Nuclear Physics B, Vol. 479, Issue 1-2 https://doi.org/10.1016/0550-3213(96)00454-3	journal	November 1996
ππ scattering Colangelo, G.; Gasser, J.; Leutwyler, H. Nuclear Physics B, Vol. 603, Issue 1-2 https://doi.org/10.1016/S0550-3213(01)00147-X	journal	June 2001
Low-energy photon–photon collisions to two loops revisited Gasser, J.; Ivanov, M. A.; Sainio, M. E. Nuclear Physics B, Vol. 728, Issue 1-3 https://doi.org/10.1016/j.nuclphysb.2005.09.010	journal	November 2005
Revisiting at low energies Gasser, J.; Ivanov, M. A.; Sainio, M. E. Nuclear Physics B, Vol. 745, Issue 1-2 https://doi.org/10.1016/j.nuclphysb.2006.03.022	journal	June 2006
Towards a data-driven analysis of hadronic light-by-light scattering Colangelo, Gilberto; Hoferichter, Martin; Kubis, Bastian Physics Letters B, Vol. 738 https://doi.org/10.1016/j.physletb.2014.09.021	journal	November 2014
Dispersive analysis of the γγ⁎ → ππ process Danilkin, Igor; Vanderhaeghen, Marc Physics Letters B, Vol. 789 https://doi.org/10.1016/j.physletb.2018.12.047	journal	February 2019
Roy–Steiner-equation analysis of pion–nucleon scattering Hoferichter, Martin; Ruiz de Elvira, Jacobo; Kubis, Bastian Physics Reports, Vol. 625 https://doi.org/10.1016/j.physrep.2016.02.002	journal	April 2016
Roy–Steiner-equation analysis of pion–nucleon scattering Meißner, U. -G.; Ruiz de Elvira, J.; Hoferichter, M. EPJ Web of Conferences, Vol. 137 https://doi.org/10.1051/epjconf/201713701014	journal	January 2017
The γγ Physics Program at BESIII Redmer, C. F. EPJ Web of Conferences, Vol. 166 https://doi.org/10.1051/epjconf/201816600017	journal	January 2018
COMPASS: meson spectroscopy and low-energy meson dynamics Friedrich, Jan M. EPJ Web of Conferences, Vol. 199 https://doi.org/10.1051/epjconf/201919901016	journal	January 2019
Bremsstrahlung of Very Low-Energy Quanta in Elementary Particle Collisions Low, F. E. Physical Review, Vol. 110, Issue 4 https://doi.org/10.1103/PhysRev.110.974	journal	May 1958
Kinematic Singularities and Threshold Relations for Helicity Amplitudes Jackson, J. D.; Hite, G. E. Physical Review, Vol. 169, Issue 5 https://doi.org/10.1103/PhysRev.169.1248	journal	May 1968
Invariant Amplitudes for Photon Processes Bardeen, W. A.; Tung, Wu-Ki Physical Review, Vol. 173, Issue 5 https://doi.org/10.1103/PhysRev.173.1423	journal	September 1968
Photo-Production of Neutral Mesons in Nuclear Electric Fields and the Mean Life of the Neutral Meson Primakoff, H. Physical Review, Vol. 81, Issue 5 https://doi.org/10.1103/PhysRev.81.899	journal	March 1951
Some General Relations between the Photoproduction and Scattering of π Mesons Watson, Kenneth M. Physical Review, Vol. 95, Issue 1 https://doi.org/10.1103/PhysRev.95.228	journal	July 1954
Structure analysis of the virtual Compton scattering amplitude at low energies Drechsel, D.; Knöchlein, G.; Korchin, A. Yu. Physical Review C, Vol. 57, Issue 2 https://doi.org/10.1103/PhysRevC.57.941	journal	February 1998
Fixed- t subtracted dispersion relations for Compton scattering off the nucleon Drechsel, D.; Gorchtein, M.; Pasquini, B. Physical Review C, Vol. 61, Issue 1 https://doi.org/10.1103/PhysRevC.61.015204	journal	December 1999
Determination of π ± meson polarizabilities from the γ γ → π + π − process Fil'kov, L. V.; Kashevarov, V. L. Physical Review C, Vol. 73, Issue 3 https://doi.org/10.1103/PhysRevC.73.035210	journal	March 2006
The reaction γγ→ π 0 π 0 and chiral loops Donoghue, John F.; Holstein, B. R.; Lin, Y. C. Physical Review D, Vol. 37, Issue 9 https://doi.org/10.1103/PhysRevD.37.2423	journal	May 1988
Photon-photon scattering, pion polarizability, and chiral symmetry Donoghue, John F.; Holstein, Barry R. Physical Review D, Vol. 48, Issue 1 https://doi.org/10.1103/PhysRevD.48.137	journal	July 1993
Dispersion representations and anomalous singularities of the triangle diagram Lucha, Wolfgang; Melikhov, Dmitri; Simula, Silvano Physical Review D, Vol. 75, Issue 1 https://doi.org/10.1103/PhysRevD.75.016001	journal	January 2007
Dispersive analysis on the f 0 ( 600 ) and f 0 ( 980 ) resonances in γ γ → π + π − , π 0 π 0 processes Mao, Yu; Wang, Xuan-Gong; Zhang, Ou Physical Review D, Vol. 79, Issue 11 https://doi.org/10.1103/PhysRevD.79.116008	journal	June 2009
Chiral extrapolation of light resonances from one and two-loop unitarized chiral perturbation theory versus lattice results Peláez, J. R.; Ríos, G. Physical Review D, Vol. 82, Issue 11 https://doi.org/10.1103/PhysRevD.82.114002	journal	December 2010
Pion-pion scattering amplitude. IV. Improved analysis with once subtracted Roy-like equations up to 1100 MeV García-Martín, R.; Kamiński, R.; Peláez, J. R. Physical Review D, Vol. 83, Issue 7 https://doi.org/10.1103/PhysRevD.83.074004	journal	April 2011
ω → π 0 γ * and ϕ → π 0 γ * transition form factors in dispersion theory Schneider, Sebastian P.; Kubis, Bastian; Niecknig, Franz Physical Review D, Vol. 86, Issue 5 https://doi.org/10.1103/PhysRevD.86.054013	journal	September 2012
Extracting the chiral anomaly from γ π → π π Hoferichter, Martin; Kubis, Bastian; Sakkas, Dimitrios Physical Review D, Vol. 86, Issue 11 https://doi.org/10.1103/PhysRevD.86.116009	journal	December 2012
Comprehensive amplitude analysis of γ γ → π + π − , π 0 π 0 and K ¯ K below 1.5 GeV Dai, Ling-Yun; Pennington, M. R. Physical Review D, Vol. 90, Issue 3 https://doi.org/10.1103/PhysRevD.90.036004	journal	August 2014
Dispersive analysis of ω / ϕ → 3 π , π γ * Danilkin, I. V.; Fernández-Ramírez, C.; Guo, P. Physical Review D, Vol. 91, Issue 9 https://doi.org/10.1103/PhysRevD.91.094029	journal	May 2015
Light-by-light scattering sum rules in light of new data Danilkin, Igor; Vanderhaeghen, Marc Physical Review D, Vol. 95, Issue 1 https://doi.org/10.1103/PhysRevD.95.014019	journal	January 2017
Radiative resonance couplings in γ π → π π Hoferichter, Martin; Kubis, Bastian; Zanke, Marvin Physical Review D, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevD.96.114016	journal	December 2017
Review of Particle Physics Tanabashi, M.; Hagiwara, K.; Hikasa, K. Physical Review D, Vol. 98, Issue 3 https://doi.org/10.1103/PhysRevD.98.030001	journal	August 2018
Rescattering Effects in the Hadronic-Light-by-Light Contribution to the Anomalous Magnetic Moment of the Muon Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano Physical Review Letters, Vol. 118, Issue 23 https://doi.org/10.1103/PhysRevLett.118.232001	journal	June 2017
Pion-Pole Contribution to Hadronic Light-By-Light Scattering in the Anomalous Magnetic Moment of the Muon Hoferichter, Martin; Hoid, Bai-Long; Kubis, Bastian Physical Review Letters, Vol. 121, Issue 11 https://doi.org/10.1103/PhysRevLett.121.112002	journal	September 2018
Pion Scattering Lengths Weinberg, Steven Physical Review Letters, Vol. 17, Issue 11 https://doi.org/10.1103/PhysRevLett.17.616	journal	September 1966
Unitarity Condition Below Physical Thresholds in the Normal and Anomalous Cases Mandelstam, S. Physical Review Letters, Vol. 4, Issue 2 https://doi.org/10.1103/PhysRevLett.4.84	journal	January 1960
Pion Scattering Lengths Capps, Richard H. Physical Review, Vol. 180, Issue 5 https://doi.org/10.1103/physrev.180.1449	journal	April 1969
Measurement of the πmeson polarizabilities via the γp → γπ+n reaction Ahrens, J.; Alexeev, V. M.; Annand, J. R. M. The European Physical Journal A, Vol. 23, Issue 1 https://doi.org/10.1140/epja/i2004-10056-2	journal	December 2004
Two photons into π0π0 Oller, J. A.; Roca, L. The European Physical Journal A, Vol. 37, Issue 1 https://doi.org/10.1140/epja/i2008-10600-0	journal	June 2008
Amplitude analysis of high statistics results on γγ→π+π- and the two photon width of isoscalar states Pennington, M. R.; Mori, T.; Uehara, S. The European Physical Journal C, Vol. 56, Issue 1 https://doi.org/10.1140/epjc/s10052-008-0648-9	journal	June 2008
MO analysis of the high statistics Belle results on γ γ→π + π −, π 0 π 0 with chiral constraints García-Martín, R.; Moussallam, B. The European Physical Journal C, Vol. 70, Issue 1-2 https://doi.org/10.1140/epjc/s10052-010-1471-7	journal	October 2010
Roy–Steiner equations for γγ→ππ Hoferichter, Martin; Phillips, Daniel R.; Schat, Carlos The European Physical Journal C, Vol. 71, Issue 9 https://doi.org/10.1140/epjc/s10052-011-1743-x	journal	September 2011
Couplings of light I=0 scalar mesons to simple operators in the complex plane Moussallam, B. The European Physical Journal C, Vol. 71, Issue 12 https://doi.org/10.1140/epjc/s10052-011-1814-z	journal	December 2011
Regge analysis of the ππ scattering amplitude Caprini, I.; Colangelo, G.; Leutwyler, H. The European Physical Journal C, Vol. 72, Issue 2 https://doi.org/10.1140/epjc/s10052-012-1860-1	journal	February 2012
Dispersive analysis of ω→3π and ϕ→3π decays Niecknig, Franz; Kubis, Bastian; Schneider, Sebastian P. The European Physical Journal C, Vol. 72, Issue 5 https://doi.org/10.1140/epjc/s10052-012-2014-1	journal	May 2012
Unified dispersive approach to real and virtual photon-photon scattering at low energy Moussallam, B. The European Physical Journal C, Vol. 73, Issue 9 https://doi.org/10.1140/epjc/s10052-013-2539-y	journal	September 2013
Single meson contributions to the muon’s anomalous magnetic moment Pauk, Vladyslav; Vanderhaeghen, Marc The European Physical Journal C, Vol. 74, Issue 8 https://doi.org/10.1140/epjc/s10052-014-3008-y	journal	August 2014
Dispersive analysis of the pion transition form factor Hoferichter, M.; Kubis, B.; Leupold, S. The European Physical Journal C, Vol. 74, Issue 11 https://doi.org/10.1140/epjc/s10052-014-3180-0	journal	November 2014
A new analysis of $\pi K$ scattering from Roy and Steiner type equations Büttiker, P.; Descotes-Genon, S.; Moussallam, B. The European Physical Journal C, Vol. 33, Issue 3 https://doi.org/10.1140/epjc/s2004-01591-1	journal	March 2004
Virtual Photon–Photon Scattering Hoferichter, Martin; Colangelo, Gilberto; Procura, Massimiliano International Journal of Modern Physics: Conference Series, Vol. 35 https://doi.org/10.1142/S2010194514604001	journal	January 2014
Hadron Polarizabilities Holstein, Barry R.; Scherer, Stefan Annual Review of Nuclear and Particle Science, Vol. 64, Issue 1 https://doi.org/10.1146/annurev-nucl-102313-025555	journal	October 2014
Roy-Steiner equations for pion-nucleon scattering Ditsche, Christoph; Hoferichter, Martin; Kubis, Bastian Proceedings of The 7th International Workshop on Chiral Dynamics — PoS(CD12) https://doi.org/10.22323/1.172.0064	conference	July 2013
Review of Particle Physics Beringer, J.; Arguin, J.; Barnett, R. Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/phppubdb-24149	text	January 2012
Review of Particle Physics Tanabashi, M.; Hagiwara, K.; Hikasa, K. Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2018-03100	text	January 2018
Virtual photon-photon scattering Hoferichter, Martin; Colangelo, Gilberto; Procura, Massimiliano arXiv https://doi.org/10.48550/arxiv.1309.6877	text	January 2013
Roy-Steiner-equation analysis of pion-nucleon scattering Hoferichter, Martin; de Elvira, Jacobo Ruiz; Kubis, Bastian arXiv https://doi.org/10.48550/arxiv.1510.06039	text	January 2015
Dispersion relation for hadronic light-by-light scattering: two-pion contributions Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano arXiv https://doi.org/10.48550/arxiv.1702.07347	text	January 2017
Review of Particle Physics Group, Particle Data; Amsler, C.; Doser, M. Elsevier https://doi.org/10.5167/uzh-11249	text	January 2008
Dispersion relation for hadronic light-by-light scattering: two-pion contributions Procura, Massimiliano; Stoffer, Peter; Hoferichter, Martin Springer https://doi.org/10.7892/boris.102040	text	January 2017
Dispersive approach to hadronic light-by-light scattering Stoffer, Peter; Procura, Massimiliano; Colangelo, Gilberto Springer https://doi.org/10.7892/boris.60234	text	January 2014

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