Abstract
Using the Resonance Chiral Theory Lagrangian, we perform a calculation of the vector form factor of the pion at the next-to-leading order (NLO) in the 1/N C expansion. Imposing the correct QCD short-distance constraints, one fixes the amplitude in terms of the pion decay constant F and resonance masses. Its low momentum expansion determines then the corresponding \( \mathcal{O}\left( {{p^4}} \right) \) and \( \mathcal{O}\left( {{p^6}} \right) \) low-energy chiral couplings at NLO, keeping control of their renormalization scale dependence. At μ 0 = 0.77 GeV, we obtain L 9(μ 0) = (7.9 ± 0.4) ∙ 10−3 and C 88(μ 0) − C 90(μ 0) = (−4.6 ± 0.4) ∙ 10−5 .
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References
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [SPIRES].
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. 158 (1984) 142 [SPIRES].
J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [SPIRES].
J. Gasser and H. Leutwyler, Low-energy expansion of meson form-factors, Nucl. Phys. B 250 (1985) 517 [SPIRES].
J. Bijnens, G. Colangelo and G. Ecker, The mesonic chiral Lagrangian of order p 6, JHEP 02 (1999) 020 [hep-ph/9902437] [SPIRES].
J. Bijnens, G. Colangelo and G. Ecker, Renormalization of chiral perturbation theory to order p 6, Annals Phys. 280 (2000) 100 [hep-ph/9907333] [SPIRES].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [SPIRES].
G. ’t Hooft, A two-dimensional model for mesons, Nucl. Phys. B 75 (1974) 461 [SPIRES].
E. Witten, Baryons in the 1/N expansion, Nucl. Phys. B 160 (1979) 57 [SPIRES].
M. Knecht and E. de Rafael, Patterns of spontaneous chiral symmetry breaking in the large-N c limit of QCD-like theories, Phys. Lett. B 424 (1998) 335 [hep-ph/9712457] [SPIRES].
S. Peris, M. Perrottet and E. de Rafael, Matching long and short distances in large-N c QCD, JHEP 05 (1998) 011 [hep-ph/9805442] [SPIRES].
M. Golterman and S. Peris, Large-N c QCD meets Regge theory: the example of spin-one two-point functions, JHEP 01 (2001) 028 [hep-ph/0101098] [SPIRES].
A. Pich, Low-energy constants from resonance chiral theory, PoS(CONFINEMENT8)026 [arXiv:0812.2631] [SPIRES].
G. Ecker, J. Gasser, A. Pich and E. de Rafael, The role of resonances in chiral perturbation theory, Nucl. Phys. B 321 (1989) 311 [SPIRES].
G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Chiral Lagrangians for massive spin 1 fields, Phys. Lett. B 223 (1989) 425 [SPIRES].
V. Cirigliano et al., Towards a consistent estimate of the chiral low-energy constants, Nucl. Phys. B 753 (2006) 139 [hep-ph/0603205] [SPIRES].
O. Catà and S. Peris, An example of resonance saturation at one loop, Phys. Rev. D 65 (2002) 056014 [hep-ph/0107062] [SPIRES].
I. Rosell, J.J. Sanz-Cillero and A. Pich, Quantum loops in the resonance chiral theory: the vector form factor, JHEP 08 (2004) 042 [hep-ph/0407240] [SPIRES].
I. Rosell, P. Ruiz-Femenia and J. Portoles, One-loop renormalization of resonance chiral theory: scalar and pseudoscalar resonances, JHEP 12 (2005) 020 [hep-ph/0510041] [SPIRES].
J.J. Sanz-Cillero, On the structure of two-point Green functions at next-to-leading order in 1/N C , Phys. Lett. B 649 (2007) 180 [hep-ph/0702217] [SPIRES].
L.Y. Xiao and J.J. Sanz-Cillero, Renormalizable sectors in resonance chiral theory: S → ππ decay amplitude, Phys. Lett. B 659 (2008) 452 [arXiv:0705.3899] [SPIRES].
J. Portoles, I. Rosell and P. Ruiz-Femenia, Vanishing chiral couplings in the large-N c resonance theory, Phys. Rev. D 75 (2007) 114011 [hep-ph/0611375] [SPIRES].
I. Rosell, P. Ruiz-Femenia and J.J. Sanz-Cillero, Resonance saturation of the chiral couplings at NLO in 1/N C , Phys. Rev. D 79 (2009) 076009 [arXiv:0903.2440] [SPIRES].
I. Rosell, Quantum corrections in the resonance chiral theory, hep-ph/0701248 [SPIRES].
K. Kampf, J. Novotny and J. Trnka, Renormalization and additional degrees of freedom within the chiral effective theory for spin-1 resonances, Phys. Rev. D 81 (2010) 116004 [arXiv:0912.5289] [SPIRES].
I. Rosell, J.J. Sanz-Cillero and A. Pich, Towards a determination of the chiral couplings at NLO in 1/N C :L 8 r(μ) and C 38 r(μ), JHEP 01 (2007) 039 [hep-ph/0610290] [SPIRES].
A. Pich, I. Rosell and J.J. Sanz-Cillero, Form-factors and current correlators: chiral couplings L 10(μ) and C 87(μ) at NLO in 1/N C , JHEP 07 (2008) 014 [arXiv:0803.1567] [SPIRES].
J.J. Sanz-Cillero and J. Trnka, High energy constraints in the octet SS-PP correlator and resonance saturation at NLO in 1/N C , Phys. Rev. D 81 (2010) 056005 [arXiv:0912.0495] [SPIRES].
J. Bijnens, E. Gamiz, E. Lipartia and J. Prades, QCD short-distance constraints and hadronic approximations, JHEP 04 (2003) 055 [hep-ph/0304222] [SPIRES].
S. Peris, Large-N C QCD and Pade approximant theory, Phys. Rev. D 74 (2006) 054013 [hep-ph/0603190] [SPIRES].
P. Masjuan and S. Peris, A rational approach to resonance saturation in large-N C QCD, JHEP 05 (2007) 040 [arXiv:0704.1247] [SPIRES].
M. Golterman, S. Peris, B. Phily and E. De Rafael, Testing an approximation to large-N C QCD with a toy model, JHEP 01 (2002) 024 [hep-ph/0112042] [SPIRES].
J.J. Sanz-Cillero, Spin-1 correlators at large-N C : matching OPE and resonance theory up to \( \mathcal{O}\left( {{\alpha_s}} \right) \), Nucl. Phys. B 732 (2006) 136 [hep-ph/0507186] [SPIRES].
M. Golterman and S. Peris, On the relation between low-energy constants and resonance saturation, Phys. Rev. D 74 (2006) 096002 [hep-ph/0607152] [SPIRES].
S. Weinberg, Precise relations between the spectra of vector and axial vector mesons, Phys. Rev. Lett. 18 (1967) 507 [SPIRES].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and resonance physics. Sum rules, Nucl. Phys. B 147 (1979) 385 [SPIRES].
J. Bijnens, G. Colangelo and P. Talavera, The vector and scalar form factors of the pion to two loops, JHEP 05 (1998) 014 [hep-ph/9805389] [SPIRES].
J. Bijnens and P. Talavera, Pion and kaon electromagnetic form factors, JHEP 03 (2002) 046 [hep-ph/0203049] [SPIRES].
F. Guerrero and A. Pich, Effective field theory description of the pion form factor, Phys. Lett. B 412 (1997) 382 [hep-ph/9707347] [SPIRES].
D. Gomez Dumm, A. Pich and J. Portoles, The hadronic off-shell width of meson resonances, Phys. Rev. D 62 (2000) 054014 [hep-ph/0003320] [SPIRES].
D. Gomez Dumm, A. Pich and J. Portoles, τ → (3π)ν τ decays in the resonance effective theory, Phys. Rev. D 69 (2004) 073002 [hep-ph/0312183] [SPIRES].
J.A. Oller, E. Oset and J.E. Palomar, Pion and kaon vector form factors, Phys. Rev. D 63 (2001) 114009 [hep-ph/0011096] [SPIRES].
A. Pich and J. Portoles, The vector form factor of the pion from unitarity and analyticity: a model-independent approach, Phys. Rev. D 63 (2001) 093005 [hep-ph/0101194] [SPIRES].
J.J. Sanz-Cillero, Renormalization group equations in resonance chiral theory, Phys. Lett. B 681 (2009) 100 [arXiv:0905.3676] [SPIRES].
J.J. Sanz-Cillero and A. Pich, Rho meson properties in the chiral theory framework, Eur. Phys. J. C 27 (2003) 587 [hep-ph/0208199] [SPIRES].
G.P. Lepage and S.J. Brodsky, Exclusive processes in quantum chromodynamics: evolution equations for hadronic wave functions and the form-factors of mesons, Phys. Lett. B 87 (1979) 359 [SPIRES].
G.P. Lepage and S.J. Brodsky, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev. D 22 (1980) 2157 [SPIRES].
Z.H. Guo, J.J. Sanz Cillero and H.Q. Zheng, Partial waves and large-N C resonance sum rules, JHEP 06 (2007) 030 [hep-ph/0701232] [SPIRES].
Z.-H. Guo and P. Roig, One meson radiative tau decays, Phys. Rev. D 82 (2010) 113016 [arXiv:1009.2542] [SPIRES].
Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [SPIRES].
V. Mateu and J. Portoles, Form factors in radiative pion decay, Eur. Phys. J. C 52 (2007) 325 [arXiv:0706.1039] [SPIRES].
J.J. Sanz-Cillero, One loop predictions for the pion VFF in resonance chiral theory, AIP Conf. Proc. 1317 (2011) 116 [arXiv:1009.6012] [SPIRES].
M. Gonzalez-Alonso, A. Pich and J. Prades, Determination of the chiral couplings L 10 and C 87 from semileptonic tau decays, Phys. Rev. D 78 (2008) 116012 [arXiv:0810.0760] [SPIRES].
P. Masjuan, S. Peris and J.J. Sanz-Cillero, Vector meson dominance as a first stepin a systematic approximation: the pion vector form factor, Phys. Rev. D 78 (2008) 074028 [arXiv:0807.4893] [SPIRES].
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Pich, A., Rosell, I. & José Sanz-Cillero, J. The vector form factor at the next-to-leading order in 1/N C : chiral couplings L9(μ) and C88(μ) − C90(μ). J. High Energ. Phys. 2011, 109 (2011). https://doi.org/10.1007/JHEP02(2011)109
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DOI: https://doi.org/10.1007/JHEP02(2011)109