<p>We present analytical expressions for the transverse momentum dependent (unintegrated) gluon and quark densities in a proton derived at leading order of QCD running coupling and valid at both small and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. The calculations are performed using the Kimber-Martin-Ryskin/Watt-Martin-Ryskin prescription (in both differential and integral formulations) with different treatment of kinematical constraint, which reflects the angular and strong ordering conditions for parton emissions. As an input, the analytical solution of QCD evolution equations for conventional (collinear) parton distributions is applied, where the valence and nonsinglet quark parts obey the Gross-Llewellyn-Smith and Gottfried sum rules, and momentum conservation for the singlet quark and gluon densities is taken into account. Several phenomenological parameters are extracted from the combined fit to precision data on the proton structure function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> collected by the BCDMS, H1, and ZEUS Collaborations, comprising a total of 933 points from 5 datasets. Comparison with the numerical results obtained by other groups is presented, and the phenomenological application to the inclusive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>-jet production at the LHC is given.</p>

JINR

A.V. Kotikov

Russia

A.V. Lipatov

Updating TMD parton densities in a proton within the Kimber-Martin-Ryskin approach

In this work, we explore the discovery potential of the Inert Doublet Model (IDM) via the vector boson fusion (VBF) channel at a muon collider with centre-of-mass energy of 10 TeV. The Inert Doublet Model is a two-Higgs-doublet model variant with an unbroken discrete $\textit{ℤ}$ symmetry, featuring new stable scalar particles that can serve as dark matter candidates. Current dark matter data constrain the phenomenologically viable parameter space of the IDM and render certain collider signatures elusive due to tiny couplings. However, VBF-type processes can still exhibit significant enhancements compared to the Standard Model, presenting a promising avenue to probe the IDM at a high-energy muon collider. We consider as our specific target process $$ {\mu}^{+}{\mu}^{-}\to {\nu}_{\mu }{\overline{\nu}}_{\mu } AA\to {\nu}_{\mu }{\overline{\nu}}_{\mu } jj\mathcal{ll} HH $$ , where $\textit{H}$ and $\textit{A}$ are the lightest and second-lightest new scalars and $\textit{ℓ}$ can be electrons or muons. We perform both cut-based and machine-learning improved sensitivity analyses for such a signal, finding a population of promising benchmark scenarios. We additionally investigate the impact of the collider energy by comparing sensitivities to the target process at 3 TeV and 10 TeV. Our results provide a clear motivation for a muon collider design capable of reaching a 10 TeV centre-of-mass energy. We furthermore discuss constraints stemming from new-physics corrections to the Higgs to di-photon decay rate as well as the trilinear Higgs coupling in detail, using state-of-the-art higher-order calculations.

Germany

Johannes Braathen

Martin Gabelmann

Croatia

Tania Robens

Panagiotis Stylianou

Probing the Inert Doublet Model via vector-boson fusion at a muon collider

Romania

Victor E. Ambrus

France

Maxim N. Chernodub

Corrigendum to “Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism” [Physics Letters B 855 (2024) 138757/PLB 138795]

In a supersymmetric theory, large mass hierarchies can lead to large uncertainties in fixed-order calculations of the Standard Model (SM)-like Higgs mass. A reliable prediction is then obtained by performing the calculation in an effective field theory (EFT) framework, involving a matching to the full supersymmetric theory at the high scale to include contributions from the heavy particles, and a subsequent renormalisation-group running down to the low scale. We report on the prediction of the SM-like Higgs mass within the CP-violating next-to-minimal supersymmetric extension of the SM (NMSSM) in a scenario where all non-SM particles feature TeV-scale masses. The matching conditions are calculated at full one-loop order using two approaches. These are the matching of the quartic Higgs couplings as well as of the SM-like Higgs pole masses of the low- and high-scale theory. A comparison between the two methods allows us to estimate the size of terms suppressed by the heavy mass scale that are neglected in a pure EFT calculation as given by the quartic-coupling matching. Furthermore, we study the different sources of uncertainty which enter our calculation as well as the effect of CP-violating phases on the Higgs mass prediction. The matching calculation is implemented in a new version of the public program package $$\texttt {NMSSMCALC}$$ .

Christoph Borschensky

Vietnam

Thi Dao

Margarete Mühlleitner

Heidi Rzehak

Higgs mass predictions in the CP-violating high-scale NMSSM

In this paper, we address the reduced beauty cross section [<math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>] and the beauty structure function [<math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>], to study the beauty content of a proton. We calculate <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism by using the integral form of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution function (KMR and MRW-UPDF) with the angular ordering constraint (AOC) and the <math><mi>M</mi><mi>M</mi><mi>H</mi><mi>T</mi><mn>2014</mn></math> <math><mi>P</mi><mi>D</mi><mi>F</mi></math> set as the input. Recently Guiot and van Hameren demonstrated that the upper limit, <math><msub><mi>k</mi><mi>max</mi></msub></math>, of the transverse-momentum integration performed in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism should be almost equal to <math><mi>Q</mi></math>, where <math><mi>Q</mi></math> is the hard scale, otherwise it leads to an overestimation of the proton structure function [<math><msub><mi>F</mi><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>]. In the present work, we show that <math><msub><mi>k</mi><mi>max</mi></msub></math> cannot be equal to <math><mi>Q</mi></math> at low and moderate energy region, and also by considering the gluon and quark contributions to the same perturbative order and a physical gauge for the gluon, i.e., <math><msup><mi>A</mi><mi>μ</mi></msup><msubsup><mi>q</mi><mi>μ</mi><mo>′</mo></msubsup><mo>=</mo><mn>0</mn></math> in the calculation of <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism, we do not encounter any overestimation of the theoretical predictions due to different choices of <math><msub><mi>k</mi><mi>max</mi></msub><mo>&gt;</mo><mi>Q</mi></math>. Finally, the resulted <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> are compared to the experimental data and the theoretical predictions. In general, the extracted <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> based on the KMR and MRW approaches are in perfect agreement with the experimental data and theoretical predictions at high energies, but at low and moderate energies, the one developed from the KMR approach has better consistency than that of the MRW approach.

Iran

N. Olanj

Phenomenological investigation of the beauty content of a proton in the framework of <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization using Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distributions

We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-Martin-Schwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, <math><mi>α</mi><mo>=</mo><mi>a</mi><mo>/</mo><mi>T</mi></math>. In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.

Victor E. Ambruş

Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism

United States

Henning Bahl

Georg Weiglein

Erratum to: anyH3: precise predictions for the trilinear Higgs coupling in the Standard Model and beyond

The trilinear Higgs coupling  $$\lambda _{hhh}$$  <math> <msub> <mi>λ</mi> <mrow> <mi>hhh</mi> </mrow> </msub> </math>   of the detected Higgs boson is an important probe for physics beyond the Standard Model. Correspondingly, improving the precision of the theoretical predictions for this coupling as well as the experimental constraints on it are among the main goals of particle physics in the near future. In this article, we present the public Python code anyH3, which provides precise theoretical predictions for  $$\lambda _{hhh}.$$  <math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>hhh</mi> </mrow> </msub> <mo>.</mo> </mrow> </math>   The program can easily be used for any renormalisable model, where for the input the UFO format is adopted. It allows including corrections up to the full one-loop level with arbitrary values of the external squared momenta and features a semi-automatic and highly flexible renormalisation procedure. The code is validated against known results in the literature. Moreover, we present new results for  $$\lambda _{hhh}$$  <math> <msub> <mi>λ</mi> <mrow> <mi>hhh</mi> </mrow> </msub> </math>   in models which so far have not been investigated in the literature.

anyH3: precise predictions for the trilinear Higgs coupling in the Standard Model and beyond

We present the prediction of the electroweak  $$\rho $$  <math> <mi>ρ</mi> </math>   parameter and the W boson mass in the CP-violating Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) at the two-loop order. The  $$\rho $$  <math> <mi>ρ</mi> </math>   parameter is calculated at the full one-loop and leading and sub-leading two-loop order  $${\mathcal {O}}(\alpha + \alpha _t\alpha _s + \left( \alpha _t+\alpha _\lambda +\alpha _\kappa \right) ^2)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <mi>α</mi> <mo>+</mo> <msub> <mi>α</mi> <mi>t</mi> </msub> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>+</mo> <msup> <mfenced> <msub> <mi>α</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>λ</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>κ</mi> </msub> </mfenced> <mn>2</mn> </msup> <mo>)</mo> </mrow> </math>  . The new  $$\Delta \rho $$  <math> <mrow> <mi>Δ</mi> <mi>ρ</mi> </mrow> </math>   prediction is incorporated into a prediction of  $$M_W$$  <math> <msub> <mi>M</mi> <mi>W</mi> </msub> </math>   via a full supersymmetric (SUSY) one-loop calculation of  $$\Delta r$$  <math> <mrow> <mi>Δ</mi> <mi>r</mi> </mrow> </math>  . Furthermore, we include all known state-of-the-art SM higher-order corrections to  $$\Delta r$$  <math> <mrow> <mi>Δ</mi> <mi>r</mi> </mrow> </math>  . By comparing results for  $$\Delta \rho $$  <math> <mrow> <mi>Δ</mi> <mi>ρ</mi> </mrow> </math>   obtained using on-shell (OS) and  $$\overline{\text {DR}}$$  <math> <mover> <mtext>DR</mtext> <mo>¯</mo> </mover> </math>   renormalization conditions in the top/stop sector, we find that the scheme uncertainty is reduced at one-loop order by 55%, at two-loop  $${\mathcal {O}}(\alpha _s\alpha _t)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msub> <mi>α</mi> <mi>s</mi> </msub> <msub> <mi>α</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </math>   by 22%, and at two-loop  $${\mathcal {O}}(\alpha _t+\alpha _\kappa +\alpha _\lambda )^2$$  <math> <mrow> <mi>O</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>κ</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>λ</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math>   by 16%, respectively. The influence of the two-loop results on the  $$M_W$$  <math> <msub> <mi>M</mi> <mi>W</mi> </msub> </math>   mass prediction is found to be sub-leading. The new calculation is made public in the computer program NMSSMCALC. We perform an extensive comparison in the W-mass, Higgs boson mass and the muon anomalous magnetic moment prediction between our calculation and three other publicly available tools and find very good agreement provided that the input parameters and renormalization scales are treated in the same way. Finally, we study the impact of the CP-violating phases on the W-mass prediction which is found to be smaller than the overall size of the SUSY corrections.

The  $$\mathcal{O}(\alpha _t+\alpha _\lambda +\alpha _\kappa )^2$$  <math> <mrow> <mi>O</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>λ</mi> </msub> <mo>+</mo> <msub> <mi>α</mi> <mi>κ</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math>   correction to the  $$\rho $$  <math> <mi>ρ</mi> </math>   parameter and its effect on the W boson mass calculation in the complex NMSSM

In supersymmetric theories the Higgs boson masses are derived quantities where higher-order corrections have to be included in order to match the measured Higgs mass value at the precision of current experiments. Closely related through the Higgs potential are the Higgs self-interactions. In addition, the measurement of the trilinear Higgs self-coupling provides the first step towards the reconstruction of the Higgs potential and the experimental verification of the Higgs mechanism sui generis. In this paper, we advance our prediction of the trilinear Higgs self-couplings in the CP-violating Next-to-Minimal Supersymmetric extension of the SM (NMSSM). We provide the     $$\mathcal{O}(\alpha _t^2)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msubsup> <mi>α</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </math>   corrections in the gaugeless limit at vanishing external momenta. The higher-order corrections turn out to be larger than the corresponding mass corrections but show the expected perturbative convergence. The inclusion of the loop-corrected effective trilinear Higgs self-coupling in gluon fusion into Higgs pairs and the estimate of the theoretical uncertainty due to missing higher-order corrections indicate that the missing electroweak higher-order corrections may be significant.

The trilinear Higgs self-couplings at     $$\mathcal {O}(\alpha _t^2)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msubsup> <mi>α</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </math>   in the CP-violating NMSSM