Physics Letters B 788 (2019) 436–441
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Derivation of spontaneously broken gauge symmetry from
the consistency of effective field theory II: Scalar field self-interactions
and the electromagnetic interaction
D. Djukanovic
a
, J. Gegelia
b,c
, Ulf-G. Meißner
d,b,c,∗
a
Helmholtz Institute Mainz, University of Mainz, D-55099 Mainz, Germany
b
Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany
c
Tbilisi State University, 0186 Tbilisi, Georgia
d
Helmholtz Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received
19 November 2018
Accepted
26 November 2018
Available
online 28 November 2018
Editor:
B. Grinstein
Keywords:
Effective
field theory
Renormalization
Gauge
symmetry
We extend our study of deriving the local gauge invariance with spontaneous symmetry breaking in
the context of an effective field theory by considering self-interactions of the scalar field and inclusion
of the electromagnetic interaction. By analyzing renormalizability and the scale separation conditions of
three-, four- and five-point vertex functions of the scalar field, we fix the two couplings of the scalar field
self-interactions of the leading order Lagrangian. Next we add the electromagnetic interaction and derive
conditions relating the magnetic moment of the charged vector boson to its charge and the masses of
the charged and neutral massive vector bosons to each other and the two independent couplings of the
theory. We obtain the bosonic part of the Lagrangian of the electroweak Standard Model as a unique
solution to the conditions imposed by the self-consistency conditions of the considered effective field
theory.
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Local gauge invariance is taken as an input in the construction
of the Standard Model [1]. On the other hand, a gauge-invariant
theory with the spontaneous symmetry breaking can be derived by
demanding tree-order unitarity of the S-matrix [2–5]. The modern
point of view considers the Standard Model as an effective field
theory (EFT) [1], in which tree-order unitarity is in any case vi-
olated
at sufficiently high energies. This motivated us to address
the issue of deriving the Lagrangian of the electroweak interac-
tion
from the conditions of self-consistency of EFT. In Ref. [6]we
started with constructing the most general Lorentz-invariant EFT
Lagrangian of three interacting massive vector bosons and a scalar.
Non-trivial relations between the coupling constants of the inter-
action
terms of the most general Lorentz-invariant Lagrangian of
a scalar and vector bosons are imposed by the conditions of con-
sistency
with the second class constraints which must be satisfied
by the systems with spin-one particles [7]. Further restrictions on
the interaction terms are imposed by the condition of perturbative
*
Corresponding author.
E-mail
address: meissner@hiskp.uni-bonn.de (U.-G. Meißner).
renormalizability in the sense of an EFT and scale separation. The
last condition requires that contributions of higher order terms of
the effective Lagrangian in physical quantities are suppressed by
some large scale(s) (for more details see Ref. [6]). To achieve this
scale separation we have to demand that the divergences of loop
diagrams contributing in physical scattering amplitudes generated
by the leading order Lagrangian should be removable by renormal-
izing
the parameters of the leading order Lagrangian alone.
In
Ref. [6]we considered three- and four-point vertex functions
and required perturbative renormalizability and scale separation.
This led to conditions imposed on the interaction terms such that
we obtained the Lagrangian of spontaneously broken gauge sym-
metry
in the unitary gauge except that the coupling constants of
the self-interactions of the scalar field remained unfixed. In the
current work we analyse one-loop diagrams contributing to three-,
four- and five-point functions of the scalar field and constrain the
two free couplings of the self-interactions.
Next,
in close analogy to Ref. [8]we “switch on” the electro-
magnetic
interaction. By demanding perturbative renormalizability
of the obtained effective Lagrangian, we relate the magnetic mo-
ment
of the charged vector boson and the mixing of the neutral
vector bosons to other parameters of the effective Lagrangian. This
https://doi.org/10.1016/j.physletb.2018.11.044
0370-2693/
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.