- Open Access
Form Factors from String Amplitudes
Phys. Rev. Lett. 135, 021603 – Published 10 July, 2025
DOI: https://doi.org/10.1103/svb5-mhpc
Abstract
In this Letter, we propose a stringy model for -point tree-level form factor with the off-shell operator in the scalar and gluon theories from the bosonic string disk amplitude: open string states and one closed string state scatter on the disk. In the field-theory limit (), the “stringy form factor” reduces to the form factor, which helps us to investigate the hidden properties of the field-theory form factors, manifest the factorization and soft behaviors, and uncover more nontrivial relations between form factors and scattering amplitudes.
Physics Subject Headings (PhySH)
Article Text
References (108)
- H. Kawai, D. C. Lewellen, and S. H. H. Tye, Nucl. Phys. B269, 1 (1986).
- Z. Bern and D. A. Kosower, Phys. Rev. Lett. 66, 1669 (1991).
- Z. Bern and D. A. Kosower, Nucl. Phys. B362, 389 (1991).
- Z. Bern and D. A. Kosower, Nucl. Phys. B379, 451 (1992).
- Z. Bern, V. Del Duca, and C. R. Schmidt, Phys. Lett. B 445, 168 (1998).
- Z. Bern, L. J. Dixon, M. Perelstein, and J. S. Rozowsky, Nucl. Phys. B546, 423 (1999).
- E. Witten, Commun. Math. Phys. 252, 189 (2004).
- F. Cachazo, S. He, and E. Y. Yuan, Phys. Rev. D 90, 065001 (2014).
- F. Cachazo, S. He, and E. Y. Yuan, J. High Energy Phys. 07 (2014) 033.
- F. Cachazo, S. He, and E. Y. Yuan, Phys. Rev. Lett. 113, 171601 (2014).
- L. Mason and D. Skinner, J. High Energy Phys. 07 (2014) 048.
- S. He, F. Teng, and Y. Zhang, Phys. Rev. Lett. 122, 211603 (2019).
- S. He, F. Teng, and Y. Zhang, J. High Energy Phys. 09 (2019) 085.
- Z. Bern, in Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles (1992), pp. 471–536.
- H. Elvang and Y.-t. Huang, arXiv:1308.1697.
- J. M. Henn and J. C. Plefka, Scattering Amplitudes in Gauge Theories (Springer, Berlin, 2014), Vol. 883.
- G. Travaglini et al., J. Phys. A 55, 443001 (2022).
- C. R. Mafra and O. Schlotterer, Phys. Rep. 1020, 1 (2023).
- N. Berkovits, E. D’Hoker, M. B. Green, H. Johansson, and O. Schlotterer, in Snowmass 2021 (2022).
- M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory: Volume 1, Introduction, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1988).
- S. Stieberger, J. Phys. A 47, 155401 (2014).
- S. Stieberger and T. R. Taylor, Nucl. Phys. B881, 269 (2014).
- N. Arkani-Hamed, S. He, and T. Lam, J. High Energy Phys. 02 (2021) 069.
- N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, Phys. Rev. Lett. 103, 161602 (2009).
- S. Stieberger, arXiv:0907.2211.
- R. Kleiss and H. Kuijf, Nucl. Phys. B312, 616 (1989).
- Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. D 78, 085011 (2008).
- W. L. van Neerven, Z. Phys. C 30, 595 (1986).
- J. Maldacena and A. Zhiboedov, J. High Energy Phys. 11 (2010) 104.
- A. Brandhuber, B. Spence, G. Travaglini, and G. Yang, J. High Energy Phys. 01 (2011) 134.
- L. V. Bork, D. I. Kazakov, and G. S. Vartanov, J. High Energy Phys. 02 (2011) 063.
- A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini, and G. Yang, J. High Energy Phys. 10 (2011) 046.
- A. Brandhuber, G. Travaglini, and G. Yang, J. High Energy Phys. 05 (2012) 082.
- A. Brandhuber, B. Penante, G. Travaglini, and C. Wen, J. High Energy Phys. 08 (2014) 100.
- R. H. Boels, B. A. Kniehl, O. V. Tarasov, and G. Yang, J. High Energy Phys. 02 (2013) 063.
- L. V. Bork, J. High Energy Phys. 12 (2014) 111.
- R. Frassek, D. Meidinger, D. Nandan, and M. Wilhelm, J. High Energy Phys. 01 (2016) 182.
- L. V. Bork and A. I. Onishchenko, J. High Energy Phys. 12 (2016) 076.
- L. V. Bork and A. I. Onishchenko, J. High Energy Phys. 04 (2017) 019.
- G. Yang, Phys. Rev. Lett. 117, 271602 (2016).
- L. V. Bork and A. I. Onishchenko, Phys. Rev. D 97, 126013 (2018).
- L. Bianchi, A. Brandhuber, R. Panerai, and G. Travaglini, J. High Energy Phys. 02 (2019) 182.
- D. Nandan and G. Yang, Hidden structure in the form factors of SYM (2018).
- G. Yang, Sci. China Phys. Mech. Astron. 63, 270001 (2020).
- G. Lin and G. Yang, J. High Energy Phys. 02 (2024) 012.
- G. Lin and G. Yang, J. High Energy Phys. 02 (2024) 013.
To be precise, while we believe that a stringy model of form factors may exist on higher-genus Riemann surfaces, we restrict our focus here to the simplest case: the disk topology, which corresponds to the tree-level.
Some literature studies the one massive string amplitudes [105, 106, 107, 108].
- I. R. Klebanov and L. Thorlacius, Phys. Lett. B 371, 51 (1996).
- S. S. Gubser, A. Hashimoto, I. R. Klebanov, and J. M. Maldacena, Nucl. Phys. B472, 231 (1996).
- M. R. Garousi and R. C. Myers, Nucl. Phys. B475, 193 (1996).
- A. Hashimoto and I. R. Klebanov, Nucl. Phys. B, Proc. Suppl. 55, 118 (1997).
- A. Hashimoto and I. R. Klebanov, Phys. Lett. B 381, 437 (1996).
- A. Sen, Phys. Rev. D 68, 106003 (2003).
- A. Sen, Phys. Rev. Lett. 91, 181601 (2003).
- S. Stieberger and T. R. Taylor, Nucl. Phys. B903, 104 (2016).
If we actually choose the metric matrix (the same notation as in [56]) to be a specific value, we cannot realize the kinematic condition for the form factor (). Here, we actually generalize the kinematics part of disk string amplitude to the stringy form factor in (2) by mapping the kinematics , (, are defined in [56]). This map cannot be realized by any choice of the matrix .
- Z. Koba and H. B. Nielsen, Nucl. Phys. B10, 633 (1969).
- S. J. Parke and T. R. Taylor, Phys. Rev. Lett. 56, 2459 (1986).
- J. Broedel, O. Schlotterer, and S. Stieberger, Fortschr. Phys. 61, 812 (2013).
- J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, J. High Energy Phys. 06 (2017) 093.
- J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, J. High Energy Phys. 08 (2017) 135.
- J. Dong, S. He, and G. Lin, J. High Energy Phys. 02 (2023) 076.
- J. Dong, S. He, and L. Hou, Phys. Rev. D 105, 105007 (2022).
- Q. Cao, J. Dong, S. He, and F. Zhu, Phys. Rev. D 111, 065015 (2025).
- F. Cachazo, L. Mason, and D. Skinner, SIGMA 10, 051 (2014).
- C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, and P. H. Damgaard, J. High Energy Phys. 09 (2015) 129.
- Q. Cao, J. Dong, S. He, and C. Shi, Phys. Lett. B 856, 138934 (2024).
- Q. Cao, J. Dong, S. He, C. Shi, and F. Zhu, J. High Energy Phys. 09 (2024) 049.
- N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, J. High Energy Phys. 10 (2024) 231.
- L. Rodina, Phys. Rev. Lett. 134, 031601 (2025).
- K. Zhou, J. High Energy Phys. 03 (2025) 154.
- F. Cachazo, N. Early, and B. Giménez Umbert, J. High Energy Phys. 08 (2022) 252.
- N. Arkani-Hamed and C. Figueiredo, arXiv:2405.09608.
- Y. Zhang, Phys. Rev. D 111, 085004 (2025).
- Y. Zhang, J. High Energy Phys. 02 (2025) 074.
- B. Giménez Umbert and B. Sturmfels, J. High Energy Phys. 04 (2025) 076.
One of the subset or can be empty.
See Refs. [25, 56] for the detailed derivation and discussion.
- F. Cachazo, S. He, and E. Y. Yuan, J. High Energy Phys. 07 (2015) 149.
- C. Cheung, C.-H. Shen, and C. Wen, J. High Energy Phys. 02 (2018) 095.
- S. He, L. Hou, J. Tian, and Y. Zhang, J. High Energy Phys. 08 (2021) 118; 06 (2022) 37.
- P. Ramond, Phys. Rev. D 3, 2415 (1971).
- A. Neveu and J. H. Schwarz, Nucl. Phys. B31, 86 (1971).
- N. Berkovits, J. High Energy Phys. 04 (2000) 018.
- N. Berkovits, J. High Energy Phys. 09 (2004) 047.
- N. Berkovits, J. High Energy Phys. 10 (2005) 089.
- O. T. Engelund and R. Roiban, J. High Energy Phys. 03 (2013) 172.
- T. Ahmed and P. K. Dhani, J. High Energy Phys. 05 (2019) 066.
- T. Ahmed, P. Banerjee, A. Chakraborty, P. K. Dhani, and V. Ravindran, Phys. Rev. D 102, 061701(R) (2020).
- S. Stieberger, arXiv:2105.06888.
- N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon, and H. Thomas, arXiv:2309.15913.
- N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon, and H. Thomas, arXiv:2311.09284.
- N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, J. High Energy Phys. 04 (2025) 078.
- N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, Phys. Rev. D 110, 065018 (2024).
- N. Arkani-Hamed, C. Figueiredo, H. Frost, and G. Salvatori, J. High Energy Phys. 05 (2025) 051.
- N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, Phys. Rev. Lett. 134, 171601 (2025).
- N. Arkani-Hamed, C. Figueiredo, and G. N. Remmen, J. High Energy Phys. 04 (2025) 039.
- N. Arkani-Hamed, H. Frost, and G. Salvatori, arXiv:2412.21027.
- S. De, A. Pokraka, M. Skowronek, M. Spradlin, and A. Volovich, J. High Energy Phys. 09 (2024) 160.
- Q. Cao and F. Zhu, arXiv:2503.15860.
- N. Arkani-Hamed, S. He, T. Lam, and H. Thomas, Phys. Rev. D 107, 066015 (2023).
- N. Arkani-Hamed, S. He, G. Salvatori, and H. Thomas, J. High Energy Phys. 11 (2022) 049.
- G. Lin and G. Yang, Phys. Rev. Lett. 129, 251601 (2022).
- M. Guillen, H. Johansson, R. Lipinski Jusinskas, and O. Schlotterer, Phys. Rev. Lett. 127, 051601 (2021).
- S. P. Kashyap, C. R. Mafra, M. Verma, and L. A. Ypanaque, arXiv:2311.12100.
- S. P. Kashyap, C. R. Mafra, M. Verma, and L. Ypanaqué, J. High Energy Phys. 02 (2025) 215.
- C. R. Mafra, J. High Energy Phys. 11 (2024) 045.