We consider realizations of a gauged B-L symmetry in the context of the Dark Dimension scenario, where the SM lives on a codimension one brane in 5d spacetime. The B-L can naturally be a bulk gauge symmetery leading to a global symmetry on the SM brane, and have its gauge anomaly canceled by charged bulk modes. This naturally leads to the existence of 3 right-handed neutrinos propagating in the dark dimension. Allowing for Higgsing of B-L by a bulk scalar at the Higgs scale, results in a massive gauge field with $\textit{m}$ ~ 100 GeV and weak coupling $\textit{g}$ ~ 10 which is allowed by current bounds. The model also predicts a natural matching $\textit{m}$ ~ $\textit{m}$ ~ Λ , thereby providing a theoretical explanation for the observed coincidence between neutrino masses and the Dark Energy scale. It also predicts a tower of sterile right-handed neutrinos in the $\textit{keV}$ mass range.

Spain

Miguel Montero

United States

Cumrun Vafa

Switzerland

Irene Valenzuela

Neutrinos, B-L symmetry and the dark dimension

Inspired by the investigation of Kalb–Ramond strings and hairy black holes by Duncan et al., in this paper we investigate two approaches to determine the existence of asymptotic flatness and how the presence of cosmic string axionic torsion hair affects wormholes and black holes. In the case of Einstein–Cartan–Kalb–Ramond (ECKR) gravity, it is shown that by assuming asymptotic flatness in ECKR equations, the KR vanishes at both sides far away from the wormhole throat, indicating the absence of torsion hair. On the other hand, in the case of teleparallel gravity, we show that in a Schwarzschild black hole with a cosmic string, by considering the cosmic string tension as constant, the torsion invariant vanishes, showing no signatures of torsion hair. When a dynamical cosmic string with variable string tension is introduced in torsional wormholes, it is shown that no asymptotic flatness is obtained, and torsion does not vanish asymptotically. This indicates that dynamical cosmic strings must generate an imprint of torsion hair at wormholes away from the throat. We argue that in the future, the ideas here may be used to detect torsion from an astrophysical point of view.

Croatia

L. Andrade

On dynamical stringy asymptotic flat Einstein–Cartan–Kalb–Ramond and teleparallel wormholes

A modified gravitational model whose action is given by an arbitrary function of the Ricci scalar, the matter Lagrangian density, a scalar field, and its kinetic term is investigated as an extension of the gravitational sector including an additional dynamical degree of freedom. Within this framework, the causal structure of rotating cosmological solutions is analyzed by considering Gödel and Gödel-type spacetimes as background geometries used as theoretical probes of the model consistency. Different matter sources are examined, including a perfect fluid and scalar-field configurations. It is found that the standard Gödel metric is not compatible with the scalar sector of the theory unless the model reduces to the General Relativity limit. In contrast, Gödel-type geometries admit a wider class of solutions whose causal properties depend on the model parameters and on the matter content. In particular, perfect-fluid sources may lead to either causal or noncausal configurations, whereas scalar-field configurations constrain the geometry to the causal limit, preventing the formation of closed timelike curves, highlighting that the scalar field plays a nontrivial dynamical role distinct from that of a cosmological constant.

Brazil

L. Evangelista

M. Silva

J. Moretti

A. Santos

Causality and its violation in $$f(R,\mathcal {L}_m,\phi ,g^{\mu \nu }\nabla _\mu \phi \nabla _\nu \phi )$$ gravity

We consider type IIB string theory on AdS × $\textit{S}$ /ℤ orbifold spaces with generic $\textit{L}$. Recent localisation results in the dual 4d $$ \mathcal{N}=2 $$ circular quiver gauge theories provide us with strong coupling expansions of certain correlators involving twisted half-BPS operators. To leading order, these results have been matched to an effective theory for massless twisted string states, which can be constructed by resolving the orbifold singularity and considering localised supergravity modes on the resolution cycles. Applying this reasoning to subleading order in strong coupling, we observe that for $\textit{L}$ ≠ 2, 3, 4, 6, a naive reduction of the 10d ($\textit{α′}$) -correction does not result in the correct coefficients to match the localisation result. We explain this mismatch by the appearance of twisted sector resonances in string amplitudes involving external twisted sector states. We perform the low-energy expansion of a “twisted” Virasoro-Shapiro amplitude and recover the expected coefficients, suggesting that the orbifold resolution and the low-energy expansion can not be interchanged directly. Finally, we comment on the long-quiver limit, $\textit{L}$ → ∞, in the context of the low-energy effective action.

United Kingdom

Carlos Barredo Martínez

Germany

Torben Skrzypek

Exploring the twisted sector of ℤ $\textit{L}$ orbifolds: matching $\textit{α′}$-corrections to localisation

We report a search for an invisible particle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math> in the decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">→</mo><mi>γ</mi><mi>X</mi></mrow></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">→</mo><mtext>invisible</mtext></math>), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math> can be interpreted as a massless or massive dark photon. No evidence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math> was found, based on 13 candidate events consistent with a predicted background of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>12.66</mn><mo>±</mo><mn>4.4</mn><msub><mrow><mn>2</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>2.1</mn><msub><mrow><mn>3</mn></mrow><mrow><mi>syst</mi></mrow></msub></mrow></math> events. Upper limits on the branching ratio of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>K</mi><mi>L</mi><mn>0</mn></msubsup><mo stretchy="false">→</mo><mi>γ</mi><mi>X</mi></math> were set for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math> mass range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>≤</mo><mn>425</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. For massless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math>, the upper limit was <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3.4</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math> at the 90% confidence level, improving the previous indirect bound by over three orders of magnitude. This result yielded a new lower bound of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4.1</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math> on the probed effective mass scale governing flavor-changing dark-photon couplings, representing approximately two orders of magnitude improvement on existing constraints. For massive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math>, the upper limits in the searched mass region ranged from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup><mo stretchy="false">)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>.

China

T. Wu

Taiwan

Y.C. Tung

Y.B. Hsiung

South Korea

J.K. Ahn

Japan

M. Gonzalez

E.J. Kim

T.K. Komatsubara

K. Kotera

S.K. Lee

G.Y. Lim

C. Lin

T. Matsumura

H. Nanjo

Y. Noichi

T. Nomura

T. Nunes

K. Ono

J. Redeker

N. Shimizu

S. Shinohara

K. Shiomi

R. Shiraishi

Y. Tajima

Y.W. Wah

H. Watanabe

T. Yamanaka

H.Y. Yoshida

Search for Dark Particles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">→</mo><mi>γ</mi><mi>X</mi></mrow></math> at the KOTO Experiment

The decay of light bosonic dark matter particles can produce a bright electron/positron (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>) flux that can be strongly constrained by local Voyager observations of the direct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> flux, as well as 511 keV line and x-ray continuum observations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> emission. We carefully analyze the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> yield and resulting cosmic ray and x-ray spectra from theoretically well-motivated light dark matter models, including (a) electrophilic axion-like particles, (b) dark photons, (c) scalars, and (d) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>i</mi></msub><mo>−</mo><msub><mi>L</mi><mi>j</mi></msub></math> vector bosons. We use the morphology and spectrum of the INTEGRAL 511 keV line data, the eROSITA x-ray continuum spectrum and the Voyager <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> spectrum to constrain the decay lifetime and coupling of each dark matter model. We find that 511 keV observations typically set world-leading limits on bosonic dark matter decay below masses of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, while eROSITA observations provide the strongest constraints in the range from 1–10 GeV. Finally, we forecast future limits from 21 cm line searches with next-generation HERA data.

Sweden

Thong T.Q. Nguyen

Pedro De la Torre Luque

Italy

Isabelle John

Shyam Balaji

Pierluca Carenza

Tim Linden

INTEGRAL, eROSITA and Voyager constraints on light bosonic dark matter: ALPs, dark photons, scalars, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>i</mi></msub><mo>−</mo><msub><mi>L</mi><mi>j</mi></msub></math> vectors

We compute the complex, long-distance two-photon-exchange amplitude which contributes to the rare <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mi mathvariant="normal">L</mi></msub><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> decay from lattice QCD. We use a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>24</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> physical-pion-mass gauge field ensemble at an inverse lattice spacing of 1.023 GeV and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mi>∞</mi></mrow></msub></mrow></math>-based formalism. Our implementation strategies for all five non-SU(3)-flavor-suppressed diagram topologies are given in detail. We achieve a 25% statistical precision on the dispersive part of this long-distance amplitude. This calculation is carried out with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> quark flavors and therefore requires the addition of counterterms to compensate for the absence of the Glashow-Iliopoulos-Maiani mechanism. These counterterms are not included in the current calculation and will be the subject of a second paper. Although a direct comparison to experiment cannot yet be made because of those omitted counterterms, the present exploratory calculation allows one to identify principal sources of statistical uncertainty in this calculation. The precision of our results is limited by the reconstruction of the physical contribution of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> intermediate state, for which various strategies are tested and compared.

Peter Boyle

En-Hung Chao

Norman Christ

Ceran Hu

Luchang Jin

Yidi Zhao

Exploratory calculation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mi mathvariant="normal">L</mi></msub><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> decay from lattice QCD at physical pion mass

A search for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>K</mi><mrow><mi mathvariant="normal">S</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo stretchy="false">)</mo></mrow><mn>0</mn></msubsup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> decays is performed using proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5.4</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>. No <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>K</mi><mrow><mi mathvariant="normal">S</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo stretchy="false">)</mo></mrow><mn>0</mn></msubsup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> signals are found and upper limits are set for the first time on the branching fractions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></mrow><mo>&lt;</mo><mn>1.4</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></mrow><mo>&lt;</mo><mn>6.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow></math>, at the 90% confidence level.

Netherlands

R. Aaij

A.S.W. Abdelmotteleb

C. Abellan Beteta

F. Abudinén

T. Ackernley

A.A. Adefisoye

B. Adeva

M. Adinolfi

P. Adlarson

France

C. Agapopoulou

C.A. Aidala

Z. Ajaltouni

S. Akar

K. Akiba

P. Albicocco

J. Albrecht

R. Aleksiejunas

CERN

F. Alessio

P. Alvarez Cartelle

R. Amalric

S. Amato

J.L. Amey

Y. Amhis

L. An

L. Anderlini

M. Andersson

P. Andreola

M. Andreotti

S. Andres Estrada

A. Anelli

D. Ao

C. Arata

F. Archilli

Z. Areg

M. Argenton

Costa Rica

S. Arguedas Cuendis

L. Arnone

A. Artamonov

M. Artuso

E. Aslanides

R. Ataíde Da Silva

M. Atzeni

B. Audurier

J.A. Authier

D. Bacher

I. Bachiller Perea

S. Bachmann

M. Bachmayer

J.J. Back

P. Baladron Rodriguez

V. Balagura

A. Balboni

W. Baldini

Z. Baldwin

L. Balzani

H. Bao

J. Baptista de Souza Leite

C. Barbero Pretel

M. Barbetti

I.R. Barbosa

R.J. Barlow

M. Barnyakov

S. Barsuk

W. Barter

J. Bartz

Poland

S. Bashir

B. Batsukh

P.B. Battista

A. Bay

A. Beck

M. Becker

F. Bedeschi

I.B. Bediaga

N.A. Behling

S. Belin

A. Bellavista

K. Belous

I. Belov

I. Belyaev

G. Benane

G. Bencivenni

E. Ben-Haim

A. Berezhnoy

R. Bernet

S. Bernet Andres

A. Bertolin

F. Betti

J. Bex

O. Bezshyyko

S. Bhattacharya

J. Bhom

M.S. Bieker

N.V. Biesuz

A. Biolchini

M. Birch

F.C.R. Bishop

A. Bitadze

A. Bizzeti

T. Blake

F. Blanc

Search for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mrow><mi mathvariant="normal">S</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo stretchy="false">)</mo></mrow></mrow><mn>0</mn></msubsup><mo stretchy="false">→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> decays at LHCb

We present the nonrelativistic QCD (NRQCD) calculations at the next-to-leading order (NLO) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> vector, axial-vector, tensor, and axial-tensor form factors, and obtain complete analytical expressions for the form factors, along with their asymptotic forms in the hierarchical heavy quark limit. Our results show that the NLO corrections are both sizable and well behaved in the low squared transfer momentum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> region. Using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>NRQCD</mi><mo>+</mo><mtext>lattice</mtext></mrow><mo>+</mo><mi>z</mi></math>-series method, we further provide theoretical predictions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> form factors across the full physical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> range. Based on these predicted form factors, we finally compute the decay widths and branching fractions for the semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub><mo>+</mo><mi>l</mi><msub><mi>ν</mi><mi>l</mi></msub></math>.

Wei Tao

Ya-Hui Zhao

Li-Ting Wang

Qin Chang

Zhen-Jun Xiao

Form factors for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub><mo>+</mo><mi>l</mi><msub><mi>ν</mi><mi>l</mi></msub></math> at NLO in QCD

It is well known that torsion waves in teleparallel Einstein gravity may induce gravitational waves (GWs). In this paper we show that a modification to Holst gravity induces low-frequency torsion waves of torsional frequency $$\sim {10^{12}Hz}= 1THz$$ as GWs sourced from astrophysical black holes. Of course, since here we use the almost Riemann-flat manifold approximation, there are no Riemannian GWs as in teleparallel spacetime but only torsionful gravitational waves. This result in Holst gravity allows us to determine an upper bound for the Barbero–Immirzi (BI) parameter as $$\gamma<<<10^{-58}$$ , which agrees with the BI bound obtained by Aliberti and Lambiase by considering matter–anti-matter asymmetry in Holst gravity. Unfortunately, this very interesting result suffers from the usual torsion ambiguity. To remedy this situation, we proceed with the BI parameter promoted to a field ore BI scalarization. Again we find a BI parameter with a bound of $$10^{-58}$$ from a terahertz-frequency torsion wave. The BI parameter is promoted to a field, and with specific approximation one obtains a BI wave equation with a plane wave solution, similar to those obtained by Taveras and Yunes. From this wave equation constant the BI parameter implies zero torsion, and consequently general relativity (GR), but the converse is not true. Higgs field variation is regarded as responsible for massive torsion. This comes from an analogy between the Higgs field and BI scalar.

Zhi-fu Gao

Torsion mass generation induced by Einstein–Cartan gravity with a Barbero–Immirzi Higgs field analogue