Abstract
Using unitarity, analyticity and crossing symmetry, we derive universal sum rules for scattering amplitudes in theories invariant under an arbitrary symmetry group. The sum rules relate the coefficients of the energy expansion of the scattering amplitudes in the IR to total cross sections integrated all the way up to the UV. Exploiting the group structure of the symmetry, we systematically determine all the independent sum rules and positivity conditions on the expansion coefficients. For effective field theories the amplitudes in the IR are calculable and hence the sum rules set constraints on the parameters of the effective Lagrangian. We clarify the impact of gauging on the sum rules for Goldstone bosons in spontaneously broken gauge theories. We discuss explicit examples that are relevant for WW-scattering, composite Higgs models, and chiral perturbation theory. Certain sum rules based on custodial symmetry and its extensions provide constraints on the Higgs boson coupling to the electroweak gauge bosons.
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References
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
T.N. Pham and T.N. Truong, Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].
J. Distler, B. Grinstein, R.A. Porto and I.Z. Rothstein, Falsifying Models of New Physics via WW Scattering, Phys. Rev. Lett. 98 (2007) 041601 [hep-ph/0604255] [INSPIRE].
S.M. Roy, Exact integral equation for pion pion scattering involving only physical region partial waves, Phys. Lett. B 36 (1971) 353 [INSPIRE].
I. Caprini, G. Colangelo and H. Leutwyler, Regge analysis of the ππ scattering amplitude, Eur. Phys. J. C 72 (2012) 1860 [arXiv:1111.7160] [INSPIRE].
G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088] [INSPIRE].
M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].
A. Falkowski, S. Rychkov and A. Urbano, What if the Higgs couplings to W and Z bosons are larger than in the Standard Model?, JHEP 04 (2012) 073 [arXiv:1202.1532] [INSPIRE].
I. Low, R. Rattazzi and A. Vichi, Theoretical Constraints on the Higgs Effective Couplings, JHEP 04 (2010) 126 [arXiv:0907.5413] [INSPIRE].
M.G. Olsson, Low-energy p-wave π-π interaction, Phys. Rev. 162 (1967) 1338.
B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Roy equation analysis of ππ scattering, Phys. Rept. 353 (2001) 207 [hep-ph/0005297] [INSPIRE].
A. Urbano, Remarks on analyticity and unitarity in the presence of a Strongly Interacting Light Higgs, arXiv:1310.5733 [INSPIRE].
B. Grinstein, C.W. Murphy, D. Pirtskhalava and P. Uttayarat, Theoretical Constraints on Additional Higgs Bosons in Light of the 126 GeV Higgs, JHEP 05 (2014) 083 [arXiv:1401.0070] [INSPIRE].
D. Espriu and F. Mescia, Unitarity and causality constraints in composite Higgs models, Phys. Rev. D 90 (2014) 015035 [arXiv:1403.7386] [INSPIRE].
A.V. Manohar and V. Mateu, Dispersion Relation Bounds for ππ Scattering, Phys. Rev. D 77 (2008) 094019 [arXiv:0801.3222] [INSPIRE].
P.D.B. Collins, Regge theory and particle physics, Phys. Rept. 1 (1971) 103 [INSPIRE].
L.L. Foldy and R.F. Peierls, Isotopic Spin of Exchanged Systems, Phys. Rev. 130 (1963) 1585 [INSPIRE].
M.R. Pennington and J. Portoles, The Chiral Lagrangian parameters, \( {\overline{l}}_1,{\overline{l}}_2 \) , are determined by the ρ-resonance, Phys. Lett. B 344 (1995) 399 [hep-ph/9409426] [INSPIRE].
B. Ananthanarayan, D. Toublan and G. Wanders, Consistency of the chiral pion-pion scattering amplitudes with axiomatic constraints, Phys. Rev. D 51 (1995) 1093 [hep-ph/9410302] [INSPIRE].
B. Bellazzini, C. Csáki and J. Serra, Composite Higgses, Eur. Phys. J. C 74 (2014) 2766 [arXiv:1401.2457] [INSPIRE].
G.F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06 (2007) 045 [hep-ph/0703164] [INSPIRE].
R.S. Chivukula, M.J. Dugan and M. Golden, Analyticity, crossing symmetry and the limits of chiral perturbation theory, Phys. Rev. D 47 (1993) 2930 [hep-ph/9206222] [INSPIRE].
V. Mateu, Universal Bounds for SU(3) Low Energy Constants, Phys. Rev. D 77 (2008) 094020 [arXiv:0801.3627] [INSPIRE].
H.D.I. Abarbanel and C. Itzykson, Relativistic eikonal expansion, Phys. Rev. Lett. 23 (1969) 53 [INSPIRE].
M. Levy and J. Sucher, Eikonal approximation in quantum field theory, Phys. Rev. 186 (1969) 1656 [INSPIRE].
G.F. Giudice, R. Rattazzi and J.D. Wells, Transplanckian collisions at the LHC and beyond, Nucl. Phys. B 630 (2002) 293 [hep-ph/0112161] [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press, Cambridge U.K. (1996).
L.L. Foldy and H. Kottler, The internal symmetry crossing matrices for an arbitrary compact group, Annals Phys. 48 (1968) 541.
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Bellazzini, B., Martucci, L. & Torre, R. Symmetries, sum rules and constraints on effective field theories. J. High Energ. Phys. 2014, 100 (2014). https://doi.org/10.1007/JHEP09(2014)100
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DOI: https://doi.org/10.1007/JHEP09(2014)100