Abstract
A method to unitarize the scattering amplitude produced by infinite-range forces is developed and applied to Born terms. In order to apply S-matrix techniques, based on unitarity and analyticity, we first derive an S-matrix free of infrared divergences. This is achieved by removing a divergent phase factor due to the interactions mediated by the massless particles in the crossed channels, a procedure that is related to previous formalisms to treat infrared divergences. We apply this method in detail by unitarizing the Born terms for graviton-graviton scattering in pure gravity and we find a scalar graviton-graviton resonance with vacuum quantum numbers (JPC = 0++) that we call the graviball. Remarkably, this resonance is located below the Planck mass but deep in the complex s-plane (with s the usual Mandelstam variable), so that its effects along the physical real s axis peak for values significantly lower than this scale. This implies that the corrections to the leading-order amplitude in the gravitational effective field theory are larger than expected from naive dimensional analysis for s around and above the peak position. We argue that the position and width of the graviball are reduced when including extra light fields in the theory. This could lead to phenomenological consequences in scenarios of quantum gravity with a large number of such fields or, in general, with a low-energy ultraviolet completion. We also apply this formalism to two non-relativistic potentials with exact known solutions for the scattering amplitudes: Coulomb scattering and an energy-dependent potential obtained from the Coulomb one with a zero at threshold. This latter case shares the same J = 0 partial-wave projected Born term as the graviton-graviton case, except for a global factor. We find that the relevant resonance structure of these examples is reproduced by our methods, which represents a strong indication of their robustness.
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References
Z. Bern, Perturbative quantum gravity and its relation to gauge theory, Living Rev. Rel. 5 (2002) 5 [gr-qc/0206071] [INSPIRE].
S. B. Giddings, The gravitational S-matrix: Erice lectures, Subnucl. Ser. 48 (2013) 93 [arXiv:1105.2036] [INSPIRE].
R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
C. Cheung, TASI Lectures on Scattering Amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics : Anticipating the Next Discoveries in Particle Physics (TASI 2016): Boulder U.S.A., June 6 — July 1 2016, R. Essig and I. Low eds., World Sceintific, Singapore (2018) pp. 571–623 [DOI] [arXiv:1708.03872] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
J. F. Donoghue, M. M. Ivanov and A. Shkerin, EPFL Lectures on General Relativity as a Quantum Field Theory, arXiv:1702.00319 [INSPIRE].
M. Ciafaloni, D. Colferai and G. Veneziano, Infrared features of gravitational scattering and radiation in the eikonal approach, Phys. Rev. D 99 (2019) 066008 [arXiv:1812.08137] [INSPIRE].
D. J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett. B 197 (1987) 81 [INSPIRE].
G. ‘t Hooft, The Scattering matrix approach for the quantum black hole: An Overview, Int. J. Mod. Phys. A 11 (1996) 4623 [gr-qc/9607022] [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Towards an S-matrix description of gravitational collapse, JHEP 02 (2008) 049 [arXiv:0712.1209] [INSPIRE].
S. B. Giddings and R. A. Porto, The Gravitational S-matrix, Phys. Rev. D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].
F. Bezrukov, D. Levkov and S. Sibiryakov, Semiclassical S-matrix for black holes, JHEP 12 (2015) 002 [arXiv:1503.07181] [INSPIRE].
G. Dvali, C. Gomez, R. S. Isermann, D. Lüst and S. Stieberger, Black hole formation and classicalization in ultra-Planckian 2 → N scattering, Nucl. Phys. B 893 (2015) 187 [arXiv:1409.7405] [INSPIRE].
R. Alonso and A. Urbano, Amplitudes, resonances, and the ultraviolet completion of gravity, Phys. Rev. D 100 (2019) 095013 [arXiv:1906.11687] [INSPIRE].
D. Blas, J. Martin Camalich and J. A. Oller, Scalar resonance in graviton-graviton scattering at high-energies: The graviball, Phys. Lett. B 827 (2022) 136991 [arXiv:2009.07817] [INSPIRE].
H. Fritzsch and P. Minkowski, Psi Resonances, Gluons and the Zweig Rule, Nuovo Cim. A 30 (1975) 393 [INSPIRE].
UKQCD collaboration, A Comprehensive lattice study of SU(3) glueballs, Phys. Lett. B 309 (1993) 378 [hep-lat/9304012] [INSPIRE].
C. J. Morningstar and M. J. Peardon, The Glueball spectrum from an anisotropic lattice study, Phys. Rev. D 60 (1999) 034509 [hep-lat/9901004] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: Calculations with improved operators, JHEP 06 (2004) 012 [hep-lat/0404008] [INSPIRE].
M. Albaladejo and J. A. Oller, Identification of a Scalar Glueball, Phys. Rev. Lett. 101 (2008) 252002 [arXiv:0801.4929] [INSPIRE].
M. Chanowitz, Chiral suppression of scalar glueball decay, Phys. Rev. Lett. 95 (2005) 172001 [hep-ph/0506125] [INSPIRE].
S. J. Brodsky and F. J. Llanes-Estrada, Using QCD Counting rules to Identify the Production of Gluonium, Phys. Lett. B 793 (2019) 405 [arXiv:1810.08772] [INSPIRE].
V. Mathieu, N. Kochelev and V. Vento, The Physics of Glueballs, Int. J. Mod. Phys. E 18 (2009) 1 [arXiv:0810.4453] [INSPIRE].
F. J. Llanes-Estrada, Glueballs as the Ithaca of meson spectroscopy: From simple theory to challenging detection, Eur. Phys. J. ST 230 (2021) 1575 [arXiv:2101.05366] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72 (1994) 2996 [gr-qc/9310024] [INSPIRE].
J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67 (2003) 084033 [Erratum ibid. 71 (2005) 069903] [hep-th/0211072] [INSPIRE].
C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7 (2004) 5 [gr-qc/0311082] [INSPIRE].
T. Han and S. Willenbrock, Scale of quantum gravity, Phys. Lett. B 616 (2005) 215 [hep-ph/0404182] [INSPIRE].
G. Veneziano, Large N bounds on, and compositeness limit of, gauge and gravitational interactions, JHEP 06 (2002) 051 [hep-th/0110129] [INSPIRE].
J. F. Donoghue and T. Torma, Infrared behavior of graviton-graviton scattering, Phys. Rev. D 60 (1999) 024003 [hep-th/9901156] [INSPIRE].
S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
A. Pich, Chiral perturbation theory, Rept. Prog. Phys. 58 (1995) 563 [hep-ph/9502366] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].
J. A. Oller and E. Oset, Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the σ, f0 (980), a0 (980) scalar mesons, Nucl. Phys. A 620 (1997) 438 [Erratum ibid. 652 (1999) 407] [hep-ph/9702314] [INSPIRE].
I. Caprini, G. Colangelo and H. Leutwyler, Mass and width of the lowest resonance in QCD, Phys. Rev. Lett. 96 (2006) 132001 [hep-ph/0512364] [INSPIRE].
J. R. Pelaez, From controversy to precision on the sigma meson: a review on the status of the non-ordinary f0 (500) resonance, Phys. Rept. 658 (2016) 1 [arXiv:1510.00653] [INSPIRE].
R. Garcia-Martin, R. Kaminski, J. R. Pelaez and J. Ruiz de Elvira, Precise determination of the f0 (600) and f0 (980) pole parameters from a dispersive data analysis, Phys. Rev. Lett. 107 (2011) 072001 [arXiv:1107.1635] [INSPIRE].
E791 collaboration, Experimental evidence for a light and broad scalar resonance in D+ → π− π+ π+ decay, Phys. Rev. Lett. 86 (2001) 770 [hep-ex/0007028] [INSPIRE].
FOCUS collaboration, Dalitz plot analysis of D+ and D+ decay to π+ π− π+ using the K matrix formalism, Phys. Lett. B 585 (2004) 200 [hep-ex/0312040] [INSPIRE].
CLEO collaboration, Dalitz plot analysis of the D+ → π− π+ π+ decay, Phys. Rev. D 76 (2007) 012001 [arXiv:0704.3954] [INSPIRE].
DM2 collaboration, Study of the J/ψ Decay Into Five Pions, Nucl. Phys. B 320 (1989) 1 [INSPIRE].
BES collaboration, The sigma pole in J/psi → ωπ+ π−, Phys. Lett. B 598 (2004) 149 [hep-ex/0406038] [INSPIRE].
J. A. Oller, Coupled-channel approach in hadron-hadron scattering, Prog. Part. Nucl. Phys. 110 (2020) 103728 [arXiv:1909.00370] [INSPIRE].
J. A. Oller, Unitarization Technics in Hadron Physics with Historical Remarks, Symmetry 12 (2020) 1114 [arXiv:2005.14417] [INSPIRE].
U. Aydemir, M. M. Anber and J. F. Donoghue, Self-healing of unitarity in effective field theories and the onset of new physics, Phys. Rev. D 86 (2012) 014025 [arXiv:1203.5153] [INSPIRE].
X. Calmet, The Lightest of Black Holes, Mod. Phys. Lett. A 29 (2014) 1450204 [arXiv:1410.2807] [INSPIRE].
F. Bloch and A. Nordsieck, Note on the Radiation Field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].
D. R. Yennie, S. C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
P. P. Kulish and L. D. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4 (1970) 745 [INSPIRE].
J. Ware, R. Saotome and R. Akhoury, Construction of an asymptotic S matrix for perturbative quantum gravity, JHEP 10 (2013) 159 [arXiv:1308.6285] [INSPIRE].
H. Hirai and S. Sugishita, IR finite S-matrix by gauge invariant dressed states, JHEP 02 (2021) 025 [arXiv:2009.11716] [INSPIRE].
H. Hannesdottir and M. D. Schwartz, A Finite S-Matrix, arXiv:1906.03271 [INSPIRE].
E. Himwich, S. A. Narayanan, M. Pate, N. Paul and A. Strominger, The Soft \( \mathcal{S} \)-Matrix in Gravity, JHEP 09 (2020) 129 [arXiv:2005.13433] [INSPIRE].
R. Dalitz, On higher Born approximations in potential scattering, Proc. Roy. Soc. Lond. A A 206 (1951) 509.
D. R. Brill and J. B. Hartle, Method of the Self-Consistent Field in General Relativity and its Application to the Gravitational Geon, Phys. Rev. 135 (1964) B271 [INSPIRE].
P. R. Anderson and D. R. Brill, Gravitational geons revisited, Phys. Rev. D 56 (1997) 4824 [gr-qc/9610074] [INSPIRE].
B. Guiot, A. Borquez, A. Deur and K. Werner, Graviballs and Dark Matter, JHEP 11 (2020) 159 [arXiv:2006.02534] [INSPIRE].
G. Dvali and C. Gomez, Self-Completeness of Einstein Gravity, arXiv:1005.3497 [INSPIRE].
G. Dvali and C. Gomez, Black Hole’s Quantum N-Portrait, Fortsch. Phys. 61 (2013) 742 [arXiv:1112.3359] [INSPIRE].
G. Dvali, Nature of Microscopic Black Holes and Gravity in Theories with Particle Species, Int. J. Mod. Phys. A 25 (2010) 602 [arXiv:0806.3801] [INSPIRE].
I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B 436 (1998) 257 [hep-ph/9804398] [INSPIRE].
M. T. Grisaru, P. van Nieuwenhuizen and C. C. Wu, Gravitational Born Amplitudes and Kinematical Constraints, Phys. Rev. D 12 (1975) 397 [INSPIRE].
H. Lehmann, Analytic properties of scattering amplitudes as functions of momentum transfer, Nuovo Cim. 10 (1958) 579 [INSPIRE].
A. D. Martin and T. D. Spearman, Elementary Particle Theory, North-Holland Publishing Company, Amsterdam, The Netherlands (1970).
R. Landau, Quantum mechanics. Vol. 2: A second course in quantum theory, WILEY-VCH, Weinheim, Germany (1995).
S. G. Naculich and H. J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [arXiv:1101.1524] [INSPIRE].
M. E. Rose, Elementary Theory of Angular Momentum, John Wiley & Sons, Inc. New York, U.S.A. (1957).
P. M. Stevenson, Optimized Perturbation Theory, Phys. Rev. D 23 (1981) 2916 [INSPIRE].
S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics, Phys. Rev. D 28 (1983) 228 [INSPIRE].
N. Su, A brief overview of hard-thermal-loop perturbation theory, Commun. Theor. Phys. 57 (2012) 409 [arXiv:1204.0260] [INSPIRE].
P. Baratella, C. Fernandez, B. von Harling and A. Pomarol, Anomalous Dimensions of Effective Theories from Partial Waves, JHEP 03 (2021) 287 [arXiv:2010.13809] [INSPIRE].
D. Kapec, M. Perry, A.-M. Raclariu and A. Strominger, Infrared Divergences in QED, Revisited, Phys. Rev. D 96 (2017) 085002 [arXiv:1705.04311] [INSPIRE].
S. Choi and R. Akhoury, BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes, JHEP 02 (2018) 171 [arXiv:1712.04551] [INSPIRE].
J. D. Dollard, Asymptotic convergence and the coulomb interaction, J. Math. Phys. 5 (1964) 729.
H. J. Schnitzer, Current algebra beyond the tree approximation, Phys. Rev. D 2 (1970) 1621 [INSPIRE].
L. S. Brown and R. L. Goble, Pion-Pion Scattering, Current Algebra, Unitarity, and the Width of the Rho Meson, Phys. Rev. Lett. 20 (1968) 346 [INSPIRE].
J. A. Oller and E. Oset, N/D description of two meson amplitudes and chiral symmetry, Phys. Rev. D 60 (1999) 074023 [hep-ph/9809337] [INSPIRE].
J. A. Oller, The Case of a WW dynamical scalar resonance within a chiral effective description of the strongly interacting Higgs sector, Phys. Lett. B 477 (2000) 187 [hep-ph/9908493] [INSPIRE].
J. A. Oller and U. G. Meissner, Chiral dynamics in the presence of bound states: Kaon nucleon interactions revisited, Phys. Lett. B 500 (2001) 263 [hep-ph/0011146] [INSPIRE].
T. Ericson and W. Weise, Pions and Nuclei, Clarendon Press, Oxford, U.K. (1988) [INSPIRE].
L. Castillejo, R. H. Dalitz and F. J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101 (1956) 453 [INSPIRE].
J. A. Oller, E. Oset and J. R. Pelaez, Meson meson interaction in a nonperturbative chiral approach, Phys. Rev. D 59 (1999) 074001 [Erratum ibid. 60 (1999) 099906] [Erratum ibid. 75 (2007) 099903] [hep-ph/9804209] [INSPIRE].
A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234 (1984) 189 [INSPIRE].
D. C. Dunbar and P. S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].
D. C. Dunbar and P. S. Norridge, Infinities within graviton scattering amplitudes, Class. Quant. Grav. 14 (1997) 351 [hep-th/9512084] [INSPIRE].
B. W. Lee, C. Quigg and H. B. Thacker, Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass, Phys. Rev. D 16 (1977) 1519 [INSPIRE].
N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, The Hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263 [hep-ph/9803315] [INSPIRE].
L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].
L. Randall and R. Sundrum, An Alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].
G. F. Giudice and M. McCullough, A Clockwork Theory, JHEP 02 (2017) 036 [arXiv:1610.07962] [INSPIRE].
S. W. Weinberg, Steven; Hawking and W. E. Israel, Ultraviolet divergences in quantum theories of gravitation in General Relativity: an Einstein Centenary Survey, Cambridge University Press, Cambridge, U.K. (1979) [INSPIRE].
M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5 [INSPIRE].
A. Codello, R. Percacci and C. Rahmede, Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation, Annals Phys. 324 (2009) 414 [arXiv:0805.2909] [INSPIRE].
K. Falls, D. F. Litim, K. Nikolakopoulos and C. Rahmede, Further evidence for asymptotic safety of quantum gravity, Phys. Rev. D 93 (2016) 104022 [arXiv:1410.4815] [INSPIRE].
D. Blas, O. Pujolàs and S. Sibiryakov, Consistent Extension of Hořava Gravity, Phys. Rev. Lett. 104 (2010) 181302 [arXiv:0909.3525] [INSPIRE].
C. F. Steinwachs, Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity, arXiv:2004.07842 [INSPIRE].
R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge, U.K. (1966).
J. A. Oller, New results from a number operator interpretation of the compositeness of bound and resonant states, Annals Phys. 396 (2018) 429 [arXiv:1710.00991] [INSPIRE].
J. A. Oller and E. Oset, Theoretical study of the γγ → meson-meson reaction, Nucl. Phys. A 629 (1998) 739 [hep-ph/9706487] [INSPIRE].
J. A. Oller, Scalar mesons and chiral symmetry, Soryushiron Kenkyu 102 (2000) 33 [hep-ph/0007349] [INSPIRE].
J. A. Oller, A Brief Introduction to Dispersion Relations, SpringerBriefs in Physics, Springer, Berlin, Germany (2019) [DOI] [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG Review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020) 113 [arXiv:1902.08191] [INSPIRE].
R. A. Briceno, J. J. Dudek, R. G. Edwards and D. J. Wilson, Isoscalar ππ scattering and the σ meson resonance from QCD, Phys. Rev. Lett. 118 (2017) 022002 [arXiv:1607.05900] [INSPIRE].
M. Albaladejo and J. A. Oller, On the size of the sigma meson and its nature, Phys. Rev. D 86 (2012) 034003 [arXiv:1205.6606] [INSPIRE].
C. Hanhart, J. R. Pelaez and G. Rios, Quark mass dependence of the rho and sigma from dispersion relations and Chiral Perturbation Theory, Phys. Rev. Lett. 100 (2008) 152001 [arXiv:0801.2871] [INSPIRE].
A. Salas-Bernárdez, F. J. Llanes-Estrada, J. Escudero-Pedrosa and J. A. Oller, Systematizing and addressing theory uncertainties of unitarization with the Inverse Amplitude Method, SciPost Phys. 11 (2021) 020 [arXiv:2010.13709] [INSPIRE].
G. ‘t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
E. Witten, Baryons in the 1/N Expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
J. R. Pelaez, On the Nature of light scalar mesons from their large Nc behavior, Phys. Rev. Lett. 92 (2004) 102001 [hep-ph/0309292] [INSPIRE].
N. Kaiser, P. B. Siegel and W. Weise, Chiral dynamics and the low-energy kaon-nucleon interaction, Nucl. Phys. A 594 (1995) 325 [nucl-th/9505043] [INSPIRE].
E. Oset and A. Ramos, Nonperturbative chiral approach to S-wave \( \overline{K}N \) interactions, Nucl. Phys. A 635 (1998) 99 [nucl-th/9711022] [INSPIRE].
R. H. Dalitz and S. F. Tuan, A possible resonant state in pion-hyperon scattering, Phys. Rev. Lett. 2 (1959) 425 [INSPIRE].
J. A. Oller, Final state interactions in hadronic D decays, Phys. Rev. D 71 (2005) 054030 [hep-ph/0411105] [INSPIRE].
J. A. Oller, On the resonant and nonresonant contributions to B → ρπ, eConf C 0304052 (2003) WG412 [hep-ph/0306294] [INSPIRE].
H. Leutwyler, Physics of the light quarks, Subnucl. Ser. 45 (2009) 185 [arXiv:0808.2825] [INSPIRE].
J. Gasser, H. Leutwyler and M. E. Sainio, Form-factor of the sigma term, Phys. Lett. B 253 (1991) 260 [INSPIRE].
G. Dvali and M. Redi, Black Hole Bound on the Number of Species and Quantum Gravity at LHC, Phys. Rev. D 77 (2008) 045027 [arXiv:0710.4344] [INSPIRE].
G. R. Dvali, G. Gabadadze, M. Kolanovic and F. Nitti, Scales of gravity, Phys. Rev. D 65 (2002) 024031 [hep-th/0106058] [INSPIRE].
G. Dvali and C. Gomez, Quantum Information and Gravity Cutoff in Theories with Species, Phys. Lett. B 674 (2009) 303 [arXiv:0812.1940] [INSPIRE].
G. Dvali, Black Holes and Large N Species Solution to the Hierarchy Problem, Fortsch. Phys. 58 (2010) 528 [arXiv:0706.2050] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity, Phys. Rev. D 102 (2020) 046014 [arXiv:2006.02375] [INSPIRE].
P. van Nieuwenhuizen and C. C. Wu, On Integral Relations for Invariants Constructed from Three Riemann Tensors and their Applications in Quantum Gravity, J. Math. Phys. 18 (1977) 182 [INSPIRE].
S. L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022 [INSPIRE].
S. L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II, Phys. Rev. 139 (1965) B1638 [INSPIRE].
J. A. Oller and D. R. Entem, The exact discontinuity of a partial wave along the left-hand cut and the exact N/D method in non-relativistic scattering, Annals Phys. 411 (2019) 167965 [arXiv:1810.12242] [INSPIRE].
J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications Inc., Amsterdam, The Netherlands (1983).
M. Sugawara and A. Kanazawa, Subtractions in Dispersion Relations, Phys. Rev. 123 (1961) 1895 [INSPIRE].
G. Somogyi, Angular integrals in d dimensions, J. Math. Phys. 52 (2011) 083501 [arXiv:1101.3557] [INSPIRE].
J. Zinn-Justin, Critical Phenomena: field theoretical approach, Scholarpedia 5 (2010) 8346.
Y. Nishida and D. T. Son, An ϵ-expansion for Fermi gas at infinite scattering length, Phys. Rev. Lett. 97 (2006) 050403 [cond-mat/0604500] [INSPIRE].
Y. Nishida, Ground-state energy of the unitary Fermi gas from the E-expansion, Phys. Rev. A 79 (2009) 013627 [arXiv:0808.3826] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, M. S. Ruf et al., Two-Loop Four-Graviton Scattering Amplitudes, Phys. Rev. Lett. 124 (2020) 211601 [arXiv:2002.12374] [INSPIRE].
M. H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
Z. Bern, H. Ita, J. Parra-Martinez and M. S. Ruf, Universality in the classical limit of massless gravitational scattering, Phys. Rev. Lett. 125 (2020) 031601 [arXiv:2002.02459] [INSPIRE].
J. H. Schwarz, Superstring Theory, Phys. Rept. 89 (1982) 223 [INSPIRE].
B. Bellazzini, M. Lewandowski and J. Serra, Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity, Phys. Rev. Lett. 123 (2019) 251103 [arXiv:1902.03250] [INSPIRE].
J. Tokuda, K. Aoki and S. Hirano, Gravitational positivity bounds, JHEP 11 (2020) 054 [arXiv:2007.15009] [INSPIRE].
L. Alberte, C. de Rham, S. Jaitly and A. J. Tolley, Positivity Bounds and the Massless Spin-2 Pole, Phys. Rev. D 102 (2020) 125023 [arXiv:2007.12667] [INSPIRE].
A. Guerrieri, J. Penedones and P. Vieira, Where Is String Theory in the Space of Scattering Amplitudes?, Phys. Rev. Lett. 127 (2021) 081601 [arXiv:2102.02847] [INSPIRE].
M. Kruczenski, J. Penedones and B. C. van Rees, Snowmass White Paper: S-matrix Bootstrap, arXiv:2203.02421 [INSPIRE].
Z.-H. Guo and J. A. Oller, Meson-baryon reactions with strangeness −1 within a chiral framework, Phys. Rev. C 87 (2013) 035202 [arXiv:1210.3485] [INSPIRE].
Z.-H. Guo and J. Oller, in preparation.
I. Antoniadis and S. P. Patil, The Effective Planck Mass and the Scale of Inflation, Eur. Phys. J. C 75 (2015) 182 [arXiv:1410.8845] [INSPIRE].
M. Kleban, M. Mirbabayi and M. Porrati, Effective Planck Mass and the Scale of Inflation, JCAP 01 (2016) 017 [arXiv:1508.01527] [INSPIRE].
N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, H. D. Kim and D. Pinner, Solving the Hierarchy Problem at Reheating with a Large Number of Degrees of Freedom, Phys. Rev. Lett. 117 (2016) 251801 [arXiv:1607.06821] [INSPIRE].
C. Csáki, TASI lectures on extra dimensions and branes, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2002): Particle Physics and Cosmology: The Quest for Physics Beyond the Standard Model(s), Boulder U.S.A, 2–28 June 2002, pp. 605–698 [hep-ph/0404096] [INSPIRE].
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Blas, D., Camalich, J.M. & Oller, J.A. Unitarization of infinite-range forces: graviton-graviton scattering. J. High Energ. Phys. 2022, 266 (2022). https://doi.org/10.1007/JHEP08(2022)266
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DOI: https://doi.org/10.1007/JHEP08(2022)266