= '''Effective Theories for a light Axion-Light-Particle (ALP)''' =

== Authors ==

I. Brivio^a,b)^, M.B. Gavela^a)^, L. Merlo^a)^, K. Mimasu^c,d)^, J.M. No^c,e)^, R. del Rey^a)^, V. Sanz^c)^

''
a) Departamento de Física Teórica and Instituto de Física Teórica, Universidad Autónoma de Madrid, Spain[[BR]]
b) NBIA and Discovery Center, Niels Bohr Institute, University of Copenhagen, Denmark[[BR]]
c) Department of Physics and Astronomy, University of Sussex, UK[[BR]]
d) Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Louvain, Belgium[[BR]]
e) Department of Physics, King’s College London, UK''

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== The model description ==

The Standard Model (SM) is extended with the inclusion of a generic CP-odd scalar field, that is a singlet under the SM gauge symmetries. 
This new field is assumed to be an Axion-Like-Particle, i.e. the (pseudo) Nambu-Goldstone boson of a spontaneously broken U(1) symmetry, associated with a heavy scale f,,a,,.
As such, the ALP field respects an approximate shift symmetry that requires its couplings to be purely derivative and weighted by inverse powers of f,,a,,. Moreover, its mass must be much lighter than f,,a,,.
The models supplied here assume the benchmark values f,,a,,= 1 TeV, m,,a,, = 1 MeV.


Two major frameworks for the electroweak sector are considered: a linear realization of electroweak symmetry breaking (EWSB), typical of weakly coupled new physics, and a non-linear one, such as those in constructions involving new strongly interacting physics. In the first scenario, the ALP is coupled to a "linear" EFT (also called SMEFT), in which the Higgs boson belongs to an exact SU(2) doublet and the interaction terms are organized in an expansion in mass dimension.
In the latter scenario, instead, two independent fields are introduced for the EW Goldstone bosons and for the physical Higgs, and one constructs a "chiral" EFT (sometimes also called non-linear or HEFT), in which the effective operators are ordered according to a more complex counting that, for the Goldstone bosons sector, reduces to a derivative one.

The models presented here implement complete sets of operators for ALP interactions with the bosonic sector of the SM, up to next-to-leading terms in the effective expansion.
This corresponds to 4 operators of dimension 6 for the linear EFT and to 18 invariants with up to 4 derivatives for the chiral EFT. The operators are defined with an NDA normalization, according to the notation used in  [https://arxiv.org/abs/1701.05379/ arXiv:1701.05379].




The Feynrules models supplied contain only the definition of the ALP particle and of its interactions, and should therefore be loaded alongside the SM Lagrangian.


Please report bugs or questions to:  ilaria.brivio@nbi.ku.dk

== References ==


* I. Brivio, M.B. Gavela, L. Merlo, K. Mimasu, J.M. No, R. del Rey, V. Sanz, ''ALPs Effective Field Theory and Collider Signatures'', [https://arxiv.org/abs/1701.05379/ arXiv:1701.05379] 


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== Files ==

'''FeynRules sources'''

* [attachment:"ALP_chiral_Feynrules.tar.gz"] The Feynrules model file for the chiral EFT.
* [attachment:"ALP_linear_Feynrules.tar.gz"] The Feynrules model file for the linear EFT.

'''Mathematica notebook'''

* [attachment:"ALPs_EFT.nb"] Demo notebook

'''Pre-exported UFO models'''

* [attachment:"ALP_chiral_UFO.tar.gz"] The UFO model file for the chiral EFT.
* [attachment:"ALP_linear_UFO.tar.gz"] The UFO model file for the linear EFT.