Abstract
We discuss the soft behaviour of open string amplitudes with gluons and massive states in any dimension. Notwithstanding non-minimal couplings of massive higher spin states to gluons, relying on OPE and factorization, we argue that the leading and subleading terms are universal and identical to the ones in Yang-Mills theories. In order to illustrate this, we compute some 4-point amplitudes on the disk involving gluons, massive states and, for the bosonic string, tachyons. For the superstring, we revisit the structure of the massive super-multiplets at the first massive level and rewrite the amplitudes in D = 4 in the spinor helicity formalism, that we adapt to accommodate massive higher spin states. We also check the validity of a recently obtained formula relating open superstring amplitudes for mass-less states at tree-level to SYM amplitudes, by factorisation on two-particle massive poles. Finally we analyse the holomorphic soft limit of superstring amplitudes with one massive insertion.
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References
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS (2010) 010 [arXiv:1102.4632] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
F.E. Low, Scattering of light of very low frequency by systems of spin 1/2, Phys. Rev. 96 (1954) 1428 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
D.J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].
S. He, Y.-t. Huang and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].
F. Cachazo and E.Y. Yuan, Are Soft Theorems Renormalized?, arXiv:1405.3413 [INSPIRE].
M. Bianchi, S. He, Y.-t. Huang and C. Wen, More on Soft Theorems: Trees, Loops and Strings, arXiv:1406.5155 [INSPIRE].
A.J. Larkoski, D. Neill and I.W. Stewart, Soft Theorems from Effective Field Theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].
B.U.W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [INSPIRE].
N. Afkhami-Jeddi, Soft Graviton Theorem in Arbitrary Dimensions, arXiv:1405.3533 [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Perturbative gravity at null infinity, Class. Quant. Grav. 31 (2014) 225008 [arXiv:1405.5122] [INSPIRE].
B.U.W. Schwab, Subleading Soft Factor for String Disk Amplitudes, JHEP 08 (2014) 062 [arXiv:1406.4172] [INSPIRE].
Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE].
M. Bianchi and L. Lopez, Pair Production of small Black Holes in Heterotic String Theories, JHEP 07 (2010) 065 [arXiv:1002.3058] [INSPIRE].
M. Bianchi, L. Lopez and R. Richter, On stable higher spin states in Heterotic String Theories, JHEP 03 (2011) 051 [arXiv:1010.1177] [INSPIRE].
M. Bianchi and P. Teresi, Scattering higher spins off D-branes, JHEP 01 (2012) 161 [arXiv:1108.1071] [INSPIRE].
W. Black and C. Monni, High energy string-brane scattering for massive states, Nucl. Phys. B 859 (2012) 299 [arXiv:1107.4321] [INSPIRE].
D.P. Skliros and M.B. Hindmarsh, Covariant Vertex Operators for Cosmic Strings, arXiv:0911.5354 [INSPIRE].
D. Skliros and M. Hindmarsh, String Vertex Operators and Cosmic Strings, Phys. Rev. D 84 (2011) 126001 [arXiv:1107.0730] [INSPIRE].
E. Dudas and J. Mourad, String theory predictions for future accelerators, Nucl. Phys. B 575 (2000) 3 [hep-th/9911019] [INSPIRE].
D. Chialva, R. Iengo and J.G. Russo, Cross sections for production of closed superstrings at high energy colliders in brane world models, Phys. Rev. D 71 (2005) 106009 [hep-ph/0503125] [INSPIRE].
M. Bianchi and A.V. Santini, String predictions for near future colliders from one-loop scattering amplitudes around D-brane worlds, JHEP 12 (2006) 010 [hep-th/0607224] [INSPIRE].
W.-Z. Feng, D. Lüst, O. Schlotterer, S. Stieberger and T.R. Taylor, Direct Production of Lightest Regge Resonances, Nucl. Phys. B 843 (2011) 570 [arXiv:1007.5254] [INSPIRE].
W.-Z. Feng and T.R. Taylor, Higher Level String Resonances in Four Dimensions, Nucl. Phys. B 856 (2012) 247 [arXiv:1110.1087] [INSPIRE].
L.A. Anchordoqui, H. Goldberg, D. Lüst, S. Stieberger and T.R. Taylor, String Phenomenology at the LHC, Mod. Phys. Lett. A 24 (2009) 2481 [arXiv:0909.2216] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
J.E. Paton and H.-M. Chan, Generalized veneziano model with isospin, Nucl. Phys. B 10 (1969) 516 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
I.G. Koh, W. Troost and A. Van Proeyen, Covariant Higher Spin Vertex Operators in the Ramond Sector, Nucl. Phys. B 292 (1987) 201 [INSPIRE].
S. Ferrara and B. Zumino, Transformation Properties of the Supercurrent, Nucl. Phys. B 87 (1975) 207 [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
L.A. Barreiro and R. Medina, RNS derivation of N-point disk amplitudes from the revisited S-matrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].
L.A. Barreiro and R. Medina, Revisiting the S-matrix approach to the open superstring low energy effective lagrangian, JHEP 10 (2012) 108 [arXiv:1208.6066] [INSPIRE].
S. Stieberger and T.R. Taylor, Multi-Gluon Scattering in Open Superstring Theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].
S. Stieberger and T.R. Taylor, Supersymmetry Relations and MHV Amplitudes in Superstring Theory, Nucl. Phys. B 793 (2008) 83 [arXiv:0708.0574] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string, JHEP 05 (2015) 137 [arXiv:1502.05258] [INSPIRE].
M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].
C. Kalousios and F. Rojas, Next to subleading soft-graviton theorem in arbitrary dimensions, JHEP 01 (2015) 107 [arXiv:1407.5982] [INSPIRE].
B.U.W. Schwab, A Note on Soft Factors for Closed String Scattering, JHEP 03 (2015) 140 [arXiv:1411.6661] [INSPIRE].
M. Ademollo et al., Soft Dilations and Scale Renormalization in Dual Theories, Nucl. Phys. B 94 (1975) 221 [INSPIRE].
W.-M. Chen, Y.-t. Huang and C. Wen, From U(1) to E8: soft theorems in supergravity amplitudes, JHEP 03 (2015) 150 [arXiv:1412.1811] [INSPIRE].
M. Bianchi and A.L. Guerrieri, On the soft limit of closed string tree level amplitudes with massive states, in preparation.
S. Stieberger and T.R. Taylor, Graviton as a Pair of Collinear Gauge Bosons, Phys. Lett. B 739 (2014) 457 [arXiv:1409.4771] [INSPIRE].
S. Stieberger and T.R. Taylor, Graviton Amplitudes from Collinear Limits of Gauge Amplitudes, Phys. Lett. B 744 (2015) 160 [arXiv:1502.00655] [INSPIRE].
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Bianchi, M., Guerrieri, A.L. On the soft limit of open string disk amplitudes with massive states. J. High Energ. Phys. 2015, 164 (2015). https://doi.org/10.1007/JHEP09(2015)164
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DOI: https://doi.org/10.1007/JHEP09(2015)164