Abstract
In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized U(1) identity for even-point currents. The off-shell U(1) identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.
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References
G. Chen and Y.-J. Du, Amplitude relations in non-linear σ-model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
H. Tye and Y. Zhang, Remarks on the identities of gluon tree amplitudes, Phys. Rev. D 82 (2010) 087702 [arXiv:1007.0597] [INSPIRE].
Y.-X. Chen, Y.-J. Du and B. Feng, A proof of the explicit minimal-basis expansion of tree amplitudes in gauge field theory, JHEP 02 (2011) 112 [arXiv:1101.0009] [INSPIRE].
F. Cachazo, Fundamental BCJ relation in N = 4 SYM from the connected formulation, arXiv:1206.5970 [INSPIRE].
Z. Bern, T. Dennen, Y.-T. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, H. Johansson and T. Sondergaard, Monodromy-like relations for finite loop amplitudes, JHEP 05 (2011) 039 [arXiv:1103.6190] [INSPIRE].
J.J. Carrasco and H. Johansson, Five-point amplitudes in N = 4 super-Yang-Mills theory and N =8 supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
R.H. Boels and R.S. Isermann, New relations for scattering amplitudes in Yang-Mills theory at loop level, Phys. Rev. D 85 (2012) 021701 [arXiv:1109.5888] [INSPIRE].
R.H. Boels and R.S. Isermann, Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts, JHEP 03 (2012) 051 [arXiv:1110.4462] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≤ 4 supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].
Y.-J. Du and H. Lüo, On general BCJ relation at one-loop level in Yang-Mills theory, JHEP 01 (2013) 129 [arXiv:1207.4549] [INSPIRE].
S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].
R. Saotome and R. Akhoury, Relationship between gravity and gauge scattering in the high energy limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].
R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic duality for form factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
R.H. Boels, R.S. Isermann, R. Monteiro and D. O’Connell, Colour-kinematics duality for one-loop rational amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].
N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand oxidation and one-loop colour-dual numerators in N = 4 gauge theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, Y.-T. Huang and J. Nohle, Color-kinematics duality for pure Yang-Mills and gravity at one and two loops, arXiv:1303.6605 [INSPIRE].
J. Nohle, Color-kinematics duality in one-loop four-gluon amplitudes with matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
R. Monteiro and D. O’Connell, The kinematic algebra from the self-dual sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].
R. Monteiro and D. O’Connell, The kinematic algebras from the scattering equations, JHEP 03 (2014) 110 [arXiv:1311.1151] [INSPIRE].
J. Broedel and J.J.M. Carrasco, Virtuous trees at five and six points for Yang-Mills and gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on symmetric BCJ numerator, JHEP 08 (2014) 098 [arXiv:1403.6262] [INSPIRE].
S.G. Naculich, Scattering equations and virtuous kinematic numerators and dual-trace functions, JHEP 07 (2014) 143 [arXiv:1404.7141] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
Z. Bern and T. Dennen, A color dual form for gauge-theory amplitudes, Phys. Rev. Lett. 107 (2011) 081601 [arXiv:1103.0312] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, The construction of dual-trace factor in Yang-Mills theory, JHEP 07 (2013) 057 [arXiv:1304.2978] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on construction of dual-trace factor in Yang-Mills theory, JHEP 10 (2013) 069 [arXiv:1305.2996] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, Dual-color decompositions at one-loop level in Yang-Mills theory, JHEP 06 (2014) 157 [arXiv:1402.6805] [INSPIRE].
T. Bargheer, S. He and T. McLoughlin, New relations for three-dimensional supersymmetric scattering amplitudes, Phys. Rev. Lett. 108 (2012) 231601 [arXiv:1203.0562] [INSPIRE].
Q. Ma, Y.-J. Du and Y.-X. Chen, On primary relations at tree-level in string theory and field theory, JHEP 02 (2012) 061 [arXiv:1109.0685] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ relation of color scalar theory and KLT relation of gauge theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
K. Kampf, J. Novotny and J. Trnka, Recursion relations for tree-level amplitudes in the SU(N) nonlinear σ-model, Phys. Rev. D 87 (2013) 081701 [arXiv:1212.5224] [INSPIRE].
K. Kampf, J. Novotny and J. Trnka, Tree-level amplitudes in the nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].
F.A. Berends and W.T. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].
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ArXiv ePrint: 1412.3722
On leave from Center for Field Theory and Particle Physics, Department of Physics, Fudan University, China. (Yi-Jian Du)
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Chen, G., Du, YJ., Li, S. et al. Note on off-shell relations in nonlinear sigma model. J. High Energ. Phys. 2015, 156 (2015). https://doi.org/10.1007/JHEP03(2015)156
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DOI: https://doi.org/10.1007/JHEP03(2015)156