UV complete me: positivity bounds for particles with spin
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Springer
Received: August 16, Revised: February 1, Accepted: February 22, Published: March 5,
2017 2018 2018 2018
Claudia de Rham,a,b Scott Melville,a Andrew J. Tolleya,b and Shuang-Yong Zhoua,c a
Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K. b CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, U.S.A. c Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone. Keywords: Effective Field Theories, Scattering Amplitudes ArXiv ePrint: 1706.02712
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)011
JHEP03(2018)011
UV complete me: positivity bounds for particles with spin
Contents 1 Introduction
2
2 From helicity to transversity 2.1 Helicity formalism 2.2 Transversity formalism
4 4 8 11 12 14 17 19 21 22
4 Extensions 4.1 Two mass eigenstates 4.2 Multiple mass eigenstates
24 24 29
5 Discussion
29
A Analyticity and causality
30
B Crossing relations from multispinors B.1 Transversity states B.2 Multispinors
34 34 35
C Explicit examples C.1 Scalar-scalar C.2 Scalar-spinor C.3 Spinor-spinor C.4 Scalar-vector
39 39 40 41 43
D Crossing relations from Lorentz rotations D.1 Transversity crossing relations D.2 Helicity crossing relations
44 44 46
E Discrete symmetries of helicity and transversity amplitudes
49
F Properties of Wigner’s matrices
52
–1–
JHEP03(2018)011
3 Positivity bounds 3.1 Unitarity and the right hand cut 3.2 Crossing and the left hand cut 3.3 Dispersion relation 3.4 Positivity bounds for particles with spin 3.5 General higher order positivity bounds 3.6 Simple example
1
Introduction
(s-channel)
A+B →C +D
and
(u-channel)
¯ → C + B. ¯ A+D
(1.1)
For spins S > 0, this requirement forces one to consider only real polarizations in the forward limit. A discussion of the bounds for spinning particles in the forward scattering limit, which applies to more general polarizations, addressing the nontrivial issues with analyt
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