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Sixty years after the experimental discovery of CP violation in the quark sector, the existence of a similar CP violation in the lepton sector is still to be established. Actually, the structure of such a violation depends crucially on the origin of the neutrino masses. In an attempt at categorizing the leptonic sources of CP violation, we studied the $\nu$SM, the Standard Model extended with three generations of sterile neutrinos, that can interpolate continuously between the Dirac and Majorana scenarios of neutrino masses. In particular, we perform a classification of the Jarlskog-like flavor invariants entering CP-violating observables and we study their suppression with the heavy Majorana mass in the seesaw limit of the model. To simplify the construction of the invariants, we introduce a graph-based method. With the guidance of the Hilbert series and plethystic logarithm of the theory, we construct the \emphgenerating and \emphprimary sets of invariants for the $\nu$SM for the first time. Unlike in the Standard Model and some other theories, we find that the numbers of generating invariants and the syzygies among them cannot immediately be read off from the plethystic logarithm, but require a more careful examination. Our analysis reveals that the \emphgenerating set contains 459 invariants, out of which 208 are CP-even and 251 are CP-odd. In the seesaw limit of the $\nu$SM, we show that all parameters of the UV theory can be captured in the effective theory with a certain suppression with the heavy Majorana mass, while these parameters can only appear in a \emphflavor-invariant way with a \emphhigher mass suppression. Furthermore, we discuss how the necessary and sufficient conditions for CP violation can be captured by utilizing these invariants. Along the way, we present useful algorithms to enumerate and build the flavor invariants.

We investigate the renormalization of the radiative decays of the Higgs to two gauge bosons in the Standard Model Effective Field Theory at mass dimension eight. Given that these are loop-level processes, their one-loop renormalization can be phenomenologically important when triggered by operators generated through the tree-level exchange of heavy particles (assuming a weakly coupled UV model). By computing the tree-level matching conditions of all relevant extensions of the Standard Model, we demonstrate that this effect is indeed present in the $h\to \gamma Z$ decay at dimension eight, even though it is absent at dimension six. In contrast, the $h\to gg$ and $h\to \gamma\gamma$ decays can only be renormalized by operators generated by one-loop processes. For UV models with heavy vectors, this conclusion hinges on the specific form of their interaction with massless gauge bosons which is required for perturbative unitarity. We study the quantitative impact of the possible logarithmic enhancement of $h\to \gamma Z$, and we propose an observable to boost the sensitivity to this effect. Given the expected increased precision of next-generation high-energy experiments, this dimension-eight contribution could be large enough to be probed and could therefore give valuable clues about new physics by revealing some of its structural features manifesting first at dimension eight.

Small instantons which increase the axion mass due to an appropriate modification of QCD at a UV scale $\Lambda_{\rm SI}$, can also enhance the effect of CP-violating operators to shift the axion potential minimum by an amount, $\theta_{\rm ind}$, proportional to the flavorful couplings in the SMEFT. Since physical observables must be flavor basis independent, we construct a basis of determinant-like flavor invariants that arise from instanton calculations containing the effects of dimension-six CP-odd operators at the scale $\require{cancel}\Lambda_{\cancel{\rm CP}}$. This new basis provides a more reliable estimate of the shift $\theta_{\rm ind}$, that is severely constrained by neutron electric dipole moment experiments. In particular, for the case of four-quark, semi-leptonic and gluon dipole operators, these invariants are then used to provide improved limits on the ratio of scales $\require{cancel}\Lambda_{\rm SI}/\Lambda_{\cancel{\rm CP}}$ for different flavor scenarios. The CP-odd flavor invariants also provide a classification of the leading effects from Wilson coefficients, and as an example, we show that a semi-leptonic four-fermion operator is subdominant compared to the four-quark operators. More generally, the flavor invariants, together with an instanton NDA, can be used to more accurately estimate small instanton effects in the axion potential that arise from any SMEFT operator.

We study how the coupling between axion-like particles (ALPs) and matter can be obtained at the level of on-shell scattering amplitudes. We identify three conditions that allow us to compute amplitudes that correspond to shift-symmetric Lagrangians, at the level of operators with dimension 5 or higher, and we discuss how they relate and extend the Adler's zero condition. These conditions are necessary to reduce the number of coefficients consistent with the little-group scaling to the one expected from the Lagrangian approach. We also show how our formalism easily explains that the dimension-5 interaction involving one ALP and two massless spin-1 bosons receive corrections from higher order operators only when the ALP has a non-vanishing mass. As a direct application of our results, we perform a phenomenological study of the inelastic scattering $\ell^+\ell^- \to \phi h$ (with $\ell^\pm$ two charged leptons, $\phi$ the ALP and $h$ the Higgs boson) for which, as a result of the structure of the 3-point and 4-point amplitudes, dimension-7 operators can dominate over the dimension-5 ones well before the energy reaches the cutoff of the theory.

Axions and axion-like particles (ALPs) are ubiquitous in popular attempts to solve supercalifragilisticexpialidocious puzzles of Nature. A widespread and vivid experimental programme spanning a vast range of mass scales and decades of couplings strives to find evidence for these elusive but theoretically well-motivated particles. In the absence of clear guiding principle, effective field theories (EFTs) prove to be an efficient tool in this experimental quest. Hilbert series technologies are a privileged instrument of the EFT toolbox to enumerate and classify operators. In this work, we compute explicitly the Hilbert series capturing the interactions of a generic ALP to the Standard Model particles above and below the electroweak symmetry scale, which allow us to build bases of operators up to dimension 8. In particular, we revealed a remarkable structure of the Hilbert series that isolates the shift-symmetry breaking and preserving interactions. In addition, with the Hilbert series method, we enumerate the sources of CP violation in terms of CP-even, CP-odd and CP-violating operators. Furthermore, we provide an ancillary file of the Hilbert series up to dimension 15 to supplement our findings, which can be used for further analysis and exploration.

A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10 TeV centre of mass energy are being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon colliders design, physics and detector studies. The aim is to provide a global perspective of the field and to outline directions for future work.

In the electroweak sector of the Standard Model, CP violation arises through a very particular interplay between the three quark generations, as described by the Cabibbo--Kobayashi--Maskawa (CKM) mechanism and the single Jarlskog invariant $J_4$. Once generalized to the Standard Model Effective Field Theory (SMEFT), this peculiar pattern gets modified by higher-dimensional operators, whose associated Wilson coefficients are usually split into CP-even and odd parts. However, CP violation at dimension four, i.e., at the lowest order in the EFT expansion, blurs this distinction: any Wilson coefficient can interfere with $J_4$ and mediate CP violation. In this paper, we study such interferences at first order in the SMEFT expansion, $\mathcal{O}(1/\Lambda^2)$, and we capture their associated parameter space via a set of 1551 linear CP-odd flavor invariants. This construction describes both new, genuinely CP-violating quantities as well as the interference between $J_4$ and CP-conserving ones. We call this latter possibility \textitopportunistic CP violation. Relying on an appropriate extension of the matrix rank to Taylor expansions, which we dub \emphTaylor rank, we define a procedure to organize the invariants in terms of their magnitude, so as to retain only the relevant ones at a given precision. We explore how this characterization changes when different assumptions are made on the flavor structure of the SMEFT coefficients. Interestingly, some of the CP-odd invariants turn out to be less suppressed than $J_4$, even when they capture opportunistic CPV, demonstrating that CP-violation in the SM, at dimension 4, is \textitaccidentally small.

This report summarizes the work of the Energy Frontier Topical Group on EW Physics: Heavy flavor and top quark physics (EF03) of the 2021 Community Summer Study (Snowmass). It aims to highlight the physics potential of top-quark studies and heavy-flavor production processes (bottom and charm) at the HL-LHC and possible future hadron and lepton colliders and running scenarios.

The precise measurement of physics observables and the test of their consistency within the standard model (SM) are an invaluable approach, complemented by direct searches for new particles, to determine the existence of physics beyond the standard model (BSM). Studies of massive electroweak gauge bosons (W and Z bosons) are a promising target for indirect BSM searches, since the interactions of photons and gluons are strongly constrained by the unbroken gauge symmetries. They can be divided into two categories: (a) Fermion scattering processes mediated by s- or t-channel W/Z bosons, also known as electroweak precision measurements; and (b) multi-boson processes, which include production of two or more vector bosons in fermion-antifermion annihilation, as well as vector boson scattering (VBS) processes. The latter categories can test modifications of gauge-boson self-interactions, and the sensitivity is typically improved with increased collision energy. This report evaluates the achievable precision of a range of future experiments, which depend on the statistics of the collected data sample, the experimental and theoretical systematic uncertainties, and their correlations. In addition it presents a combined interpretation of these results, together with similar studies in the Higgs and top sector, in the Standard Model effective field theory (SMEFT) framework. This framework provides a model-independent prescription to put generic constraints on new physics and to study and combine large sets of experimental observables, assuming that the new physics scales are significantly higher than the EW scale.

Diboson production processes provide good targets for precision measurements at present and future hadron colliders. We consider $Vh$ production, focusing on the $h \to b\bar b$ decay channel, whose sizeable cross section makes it accessible at the LHC. We perform an improved analysis by combining the 0-, 1- and 2-lepton channels with a scale-invariant $b$-tagging algorithm that allows us to exploit events with either a boosted Higgs via mass-drop tagging or resolved $b$-jets. This strategy gives sensitivity to 4 dimension-6 SMEFT operators that modify the $W$ and $Z$ couplings to quarks and is competitive with the bounds obtained from global fits. The benefit of the $h\to b\bar b$ decay channel is the fact that it is the only $Vh$ channel accessible at the LHC Run 3 and HL-LHC, while at FCC-hh it is competitive with the effectively background-free $h\to \gamma\gamma$ channel assuming $\lesssim 5$% systematic uncertainty. Combining the boosted and resolved categories yields a 17% improvement on the most strongly bounded Wilson coefficient at the LHC Run 3 with respect to the boosted category alone (and a 7% improvement at FCC-hh). We also show that, at FCC-hh, a binning in the rapidity of the $Vh$ system can significantly reduce correlations between some EFT operators. The bounds we obtain translate to a lower bound on the new physics scale of $5$, $8$, and $20$ TeV at the LHC Run 3, HL-LHC, and FCC-hh respectively, assuming new-physics couplings of order unity. Finally, we assess the impact of the $Vh$ production channel on anomalous triple gauge coupling measurements, comparing with their determination at lepton colliders.

Revealing the Higgs pair production process is the next big challenge in high energy physics. In this work, we explore the use of interpretable machine learning and cooperative game theory for extraction of the trilinear Higgs self-coupling in Higgs pair production. In particular, we show how a topological decomposition of the gluon-gluon fusion Higgs pair production process can be used to simplify the machine learning analysis flow. Furthermore, we extend the analysis to include $q\bar{q}\to hh$ production, which is strongly suppressed in the Standard Model, to extract the trilinear Higgs coupling and to bound large deviations of the light-quark Yukawa couplings from the Standard Model values. The constraints on the rescaling of the trilinear Higgs self-coupling, $\kappa_\lambda$, and the rescaling of light-quark Yukawa couplings, $\kappa_u$ and $\kappa_d$, at HL-LHC (FCC-hh) from single parameter fits are: \begineqnarray \kappa_\lambda&=&[0.53,1.7] \;\;([0.97, 1.03])\nonumber \newline \kappa_u&=&[-470,430] \;\;([-58,55])\nonumber \newline \kappa_d&=&[-360,360] \;\;([-26,28])\nonumber \endeqnarray We show that the simultaneous modification of the Yukawa couplings can dilute the constraints on the trilinear coupling significantly. We perform similar analyses for FCC-hh. We discuss some motivated flavourful new physics scenarios where such an analysis prevails.

Based on the framework of Standard Model Effective Field Theory, we performed a few global fits, each containing a subset of dimension-6 operators, for the measurements that are expected at future colliders. The fit for the Higgs and electroweak sector improves what has been done for the European Strategy Update in 2020 on both EFT treatments and experimental inputs. A new comprehensive fit is performed focusing on 4-fermion interactions at future colliders. Top-quark sector is studied in a dedicated fit which restricts the operators and measurements to be directly related to top-quark. A small subset of CP-violating operators involving bosonic fields alone are also investigated. Various running scenarios for future e+e- and Muon Colliders that are suggested in the Snowmass 2021 discussion are considered in the global fits. The outcomes from each fit are expressed in terms of either direct constraint on Wilson Coefficients or precision on Higgs and electroweak effective couplings.

It is generally believed that global symmetries, in particular axion shift symmetries, can only be approximate. This motivates us to quantify the breaking of the shift invariance that characterizes the couplings of an axion-like particle (ALP), and to identify proper order parameters associated to this breaking. Focusing on the flavorful effective Yukawa couplings to Standard Model fermions, we work out explicit conditions for them to maintain an exact axion shift symmetry. Those conditions are given in terms of Jarlskog-like flavor-invariants and can be directly evaluated from the values of the different Yukawa couplings. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate this construction by matching the axion EFT to UV models, and by showing that the renormalization group running closes on those shift-breaking flavor-invariants, as it should on any complete set of order parameters. Furthermore, the study of the invariants' CP-parities indicate that all but one are CP-odd, hence the assumption of CP conservation suffices to cancel all but one sources of shift-breaking in the theory. We also investigate similar conditions in the low-energy EFT below the electroweak scale, and comment on relations inherited from a UV completion which realizes the electroweak symmetry linearly. Finally, we discuss the order parameter associated to the non-perturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.

The International Linear Collider (ILC) is on the table now as a new global energy-frontier accelerator laboratory taking data in the 2030s. The ILC addresses key questions for our current understanding of particle physics. It is based on a proven accelerator technology. Its experiments will challenge the Standard Model of particle physics and will provide a new window to look beyond it. This document brings the story of the ILC up to date, emphasizing its strong physics motivation, its readiness for construction, and the opportunity it presents to the US and the global particle physics community.

Adding interpretability to multivariate methods creates a powerful synergy for exploring complex physical systems with higher order correlations while bringing about a degree of clarity in the underlying dynamics of the system.

The perspective of designing muon colliders with high energy and luminosity, which is being investigated by the International Muon Collider Collaboration, has triggered a growing interest in their physics reach. We present a concise summary of the muon colliders potential to explore new physics, leveraging on the unique possibility of combining high available energy with very precise measurements.

In the path towards a muon collider with center of mass energy of 10 TeV or more, a stage at 3 TeV emerges as an appealing option. Reviewing the physics potential of such muon collider is the main purpose of this document. In order to outline the progression of the physics performances across the stages, a few sensitivity projections for higher energy are also presented. There are many opportunities for probing new physics at a 3 TeV muon collider. Some of them are in common with the extensively documented physics case of the CLIC 3 TeV energy stage, and include measuring the Higgs trilinear coupling and testing the possible composite nature of the Higgs boson and of the top quark at the 20 TeV scale. Other opportunities are unique of a 3 TeV muon collider, and stem from the fact that muons are collided rather than electrons. This is exemplified by studying the potential to explore the microscopic origin of the current $g$-2 and $B$-physics anomalies, which are both related with muons.

This report presents the results of the Models and Effective Field Theories Subgroup of the Off-Shell Interpretations Task Force in the LHC Higgs Working Group. The main goal of the subgroup was to discuss and advance the potential impact of off-shell Higgs measurements on searches for BSM physics carried out in the EFT framework or as benchmark model studies. In the first contribution, the off-shell potential to resolve flat directions in parameter space for on-shell measurements is studied. Furthermore, the sensitivity of off-shell measurements to SMEFT dimension-6 operators for the gg $\to$ ZZ process is discussed, and studies of explicit models that are testable in off-shell production are reviewed. In the second contribution, the SMEFT effects in the off-shell gluon fusion and electroweak processes are discussed. Subsequently, the computation of integrated and differential effects using SMEFT@NLO and MG5_aMC@NLO, or JHUGen and MCFM, is demonstrated. On that basis, a study of the prospects of obtaining additional SMEFT constraints - beyond those from existing global fits - by utilising the off-shell process is presented. For clarification, a revised introduction, definition and discussion of the Higgs basis parametrisation of the SMEFT is given in the third contribution. In short notes on the SMEFT, the Higgs basis with an additional constraint is discussed and relations between the Higgs and Warsaw bases are presented. Lastly, an overview of EFT calculations and tools is given.

The truncation of the standard-model effective field theory, its validity and the associated uncertainties have been discussed in meetings of the LHC EFT WG. Proposals were made by participants to address these issues. No consensus was reached and no formal recommendation is therefore put forward at this time. None of the proposals has been approved or validated and further work is needed to establish a prescription. This note aims at summarizing the proposals and points of debate.

We explore the double copy of effective field theories (EFTs), in the recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator "seeds" with trace-like permutation properties. This construction has the advantage of being easily applicable to any multiplicity, which we exemplify up to 6-point. It employs the linear map between color factors formed by single traces of generators and by products of the structure constants, which also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

As SMEFT is a framework of growing importance to analyze high-energy data, understanding its parameter space is crucial. The latter is commonly split into CP-even and CP-odd parts, but this classification is obscured by the fact that CP violation is actually a collective effect that is best captured by considering flavor-invariant combinations of Lagrangian parameters. First we show that fermion rephasing invariance imposes that several coefficients associated to dimension-six operators can never interfere with operators of dimension $\leq4$ and thus cannot appear in any physical observable at $\order{1/\Lambda^2}$. For those that can, instead, we establish a one-to-one correspondence with CP-odd flavor invariants, all linear with respect to SMEFT coefficients. We explicitly present complete lists of such linear CP-odd invariants, and carefully examine their relationship to CP breaking throughout the parameter space of coefficients of dimension $\leq 4$. Requiring that these invariants all vanish, together with the Jarlskog invariant, the strong-CP phase, and the 6 CP-violating dimension-6 bosonic operators, provides $699(+1+1+6)$ conditions for CP conservation to hold in any observable at leading order, $\cO\(1/\Lambda^2\)$.

We study whether higher-dimensional operators in effective field theories, in particular in the Standard Model Effective Field Theory (SMEFT), can source gauge anomalies via the modification of the interactions involved in triangle diagrams. We find no evidence of such gauge anomalies at the level of dimension-6 operators that can therefore be chosen independently to each others without spoiling the consistency of SMEFT, at variance with recent claims. The underlying reason is that gauge-invariant combinations of Goldstone bosons and massive gauge fields are allowed to couple to matter currents which are not conserved. We show this in a toy model by computing the relevant triangle diagrams, as well as by working out Wess--Zumino terms in the bosonic EFT below all fermion masses. The same approach applies directly to the Standard Model both at the renormalisable level, providing a convenient and unusual way to check that the SM is anomaly free, as well as at the non-renormalisable level in SMEFT.

The associated production of a $b\bar{b}$ pair with a Higgs boson could provide an important probe to both the size and the phase of the bottom-quark Yukawa coupling, $y_b$. However, the signal is shrouded by several background processes including the irreducible $Zh, Z\to b\bar{b}$ background. We show that the analysis of kinematic shapes provides us with a concrete prescription for separating the $y_b$-sensitive production modes from both the irreducible and the QCD-QED backgrounds using the $b\bar{b}\gamma\gamma$ final state. We draw a page from game theory and use Shapley values to make Boosted Decision Trees interpretable in terms of kinematic measurables and provide physics insights into the variances in the kinematic shapes of the different channels that help us complete this feat. Adding interpretability to the machine learning algorithm opens up the black-box and allows us to cherry-pick only those kinematic variables that matter most in the analysis. We resurrect the hope of constraining the size and, possibly, the phase of $y_b$ using kinematic shape studies of $b\bar{b}h$ production with the full HL-LHC data and at FCC-hh.

The future 100 TeV FCC-hh hadron collider will give access to rare but clean final states which are out of reach of the HL-LHC. One such process is the $Zh$ production channel in the $(\nu\bar{\nu} / \ell^{+}\ell^{-})\gamma\gamma$ final states. We study the sensitivity of this channel to the $\mathcal{O}_{\varphi q}^{(1)}$, $\mathcal{O}_{\varphi q}^{(3)}$, $\mathcal{O}_{\varphi u}$, and $\mathcal{O}_{\varphi d}$ SMEFT operators, which parametrize deviations of the $W$ and $Z$ couplings to quarks, or, equivalently, anomalous trilinear gauge couplings (aTGC). While our analysis shows that good sensitivity is only achievable for $\mathcal{O}_{\varphi q}^{(3)}$, we demonstrate that binning in the $Zh$ rapidity has the potential to improve the reach on $\mathcal{O}_{\varphi q}^{(1)}$. Our estimated bounds are one order of magnitude better than projections at HL-LHC and is better than global fits at future lepton colliders. The sensitivity to $\mathcal{O}_{\varphi q}^{(3)}$ is competitive with other channels that could probe the same operator at FCC-hh. Therefore, combining the different diboson channels sizeably improves the bound on $\mathcal{O}_{\varphi q}^{(3)}$, reaching a precision of $|\delta g_{1z}| \lesssim 2 \times 10^{-4}$ on the deviations in the $ZWW$ interactions.

From general analyticity and unitarity requirements on the UV theory, positivity bounds on the Wilson coefficients of the dimension-8 operators composed of 4 fermions and two derivatives appearing in the Standard Model Effective Field Theory have been derived recently. We explore the fate of these bounds in the context of models endowed with a Minimal Flavor Violation (MFV) structure, models in which the flavor structure of higher dimensional operators is inherited from the one already contained in the Yukawa sector of the Standard Model Lagrangian. Our goal is to check whether the general positivity bounds translate onto bounds on the Yukawa coefficients and/or on elements of the CKM matrix. MFV fixes the coefficients of dimension-8 operators up to some multiplicative flavor-blind factors and we find that, in the most generic setup, the freedom left by those unspecified coefficients is enough as not to constrain the parameters of the renormalizable Yukawa sector. On the contrary, the latter shape the allowed region for the former. Requiring said overall coefficients to take natural $\mathcal{O}(1)$ values could give rise to bounds on the Yukawa couplings. Remarkably, at leading order in an expansion in powers of the Yukawa matrices, no bounds on the CKM entries can be retrieved.

We study axion effective field theories (EFTs), with a focus on axion couplings to massive chiral gauge fields. We investigate the EFT interactions that participate in processes with an axion and two gauge bosons, and we show that, when massive chiral gauge fields are present, such interactions do not entirely originate from the usual anomalous EFT terms. We illustrate this both at the EFT level and by matching to UV-complete theories. In order to assess the consistency of the Peccei--Quinn (PQ) anomaly matching, it is useful to introduce an auxiliary, non-dynamical gauge field associated to the PQ symmetry. When applied to the case of the Standard Model (SM) electroweak sector, our results imply that anomaly-based sum rules between EFT interactions are violated when chiral matter is integrated out, which constitutes a smoking gun of the latter. As an illustration, we study a UV-complete chiral extension of the SM, containing an axion arising from an extended Higgs sector and heavy fermionic matter that obtains most of its mass by coupling to the Higgs doublets. We assess the viability of such a SM extension through electroweak precision tests, bounds on Higgs rates and direct searches for heavy charged matter. At energies below the mass of the new chiral fermions, the model matches onto an EFT where the electroweak gauge symmetry is non-linearly realised.

To aid contributions to the Snowmass 2021 US Community Study on physics at the International Linear Collider and other proposed $e^+e^-$ colliders, we present a list of study questions that could be the basis of useful Snowmass projects. We accompany this with links to references and resources on $e^+e^-$ physics, and a description of a new software framework that we are preparing for $e^+e^-$ studies at Snowmass.

The increase in luminosity and center of mass energy at the FCC-hh will open up new clean channels where BSM contributions are enhanced at high energy. In this paper we study one such channel, $Wh \to \ell\nu\gamma\gamma$. We estimate the sensitivity to the $\mathcal{O}_{\varphi q}^{(3)}$, $\mathcal{O}_{\varphi {W}}$, and $\mathcal{O}_{\varphi \widetilde {W}}$ SMEFT operators. We find that this channel will be competitive with fully leptonic $WZ$ production in setting bounds on $\mathcal{O}_{\varphi q}^{(3)}$. We also find that the double differential distribution in the $p_T^h$ and the leptonic azimuthal angle can be exploited to enhance the sensitivity to $\mathcal{O}_{\varphi \widetilde {W}}$. However, the bounds on $\mathcal{O}_{\varphi {W}}$ and $\mathcal{O}_{\varphi \widetilde {W}}$ we obtain in our analysis, though complementary and more direct, are not competitive with those coming from other measurements such as EDMs and inclusive Higgs measurements.

This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments.

We present an overview of the capabilities that the International Linear Collider (ILC) offers for precision measurements that probe the Standard Model. First, we discuss the improvements that the ILC will make in precision electroweak observables, both from W boson production and radiative return to the Z at 250 GeV in the center of mass and from a dedicated GigaZ stage of running at the Z pole. We then present new results on precision measurements of fermion pair production, including the production of b and t quarks. We update the ILC projections for the determination of Higgs boson couplings through a Standard Model Effective Field Theory fit taking into account the new information on precision electroweak constraints. Finally, we review the capabilities of the ILC to measure the Higgs boson self-coupling.

LEP precision on electroweak measurements was sufficient not to hamper the extraction of Higgs couplings at the LHC. But the foreseen permille-level Higgs measurements at future lepton colliders might suffer from parametric electroweak uncertainties in the absence of a dedicated electroweak program. We perform a joint, complete and consistent effective-field-theory analysis of Higgs and electroweak processes. The full electroweak-sector dependence of the $e^+e^- \to WW$ production process is notably accounted for, using statistically optimal observables. Up-to-date HL-LHC projections are combined with CEPC, FCC-ee, ILC and CLIC ones. For circular colliders, our results demonstrate the importance of a new $Z$-pole program for the robust extraction of Higgs couplings. At linear colliders, we show how exploiting multiple polarizations and centre-of-mass energies is crucial to mitigate contaminations from electroweak parameter uncertainties on the Higgs physics program. We also investigate the potential of alternative electroweak measurements to compensate for the lack of direct $Z$-pole run, considering for instance radiative return to these energies. Conversely, we find that Higgs measurements at linear colliders could improve our knowledge of the $Z$ couplings to electrons.

This document aims to provide an assessment of the potential of future colliding beam facilities to perform Higgs boson studies. The analysis builds on the submissions made by the proponents of future colliders to the European Strategy Update process, and takes as its point of departure the results expected at the completion of the HL-LHC program. This report presents quantitative results on many aspects of Higgs physics for future collider projects of sufficient maturity using uniform methodologies. A first version of this report was prepared for the purposes of discussion at the Open Symposium in Granada (13-16/05/2019). Comments and feedback received led to the consideration of additional run scenarios as well as a refined analysis of the impact of electroweak measurements on the Higgs coupling extraction.

The discovery of the Higgs boson in 2012, by the ATLAS and CMS experiments, was a success achieved with only a percent of the entire dataset foreseen for the LHC. It opened a landscape of possibilities in the study of Higgs boson properties, Electroweak Symmetry breaking and the Standard Model in general, as well as new avenues in probing new physics beyond the Standard Model. Six years after the discovery, with a conspicuously larger dataset collected during LHC Run 2 at a 13 TeV centre-of-mass energy, the theory and experimental particle physics communities have started a meticulous exploration of the potential for precision measurements of its properties. This includes studies of Higgs boson production and decays processes, the search for rare decays and production modes, high energy observables, and searches for an extended electroweak symmetry breaking sector. This report summarises the potential reach and opportunities in Higgs physics during the High Luminosity phase of the LHC, with an expected dataset of pp collisions at 14 TeV, corresponding to an integrated luminosity of 3 ab$^{-1}$. These studies are performed in light of the most recent analyses from LHC collaborations and the latest theoretical developments. The potential of an LHC upgrade, colliding protons at a centre-of-mass energy of 27 TeV and producing a dataset corresponding to an integrated luminosity of 15 ab$^{-1}$, is also discussed.

We use the current CMS and ATLAS data for the leptonic $pp \to WW, WZ$ channels to show that diboson production is, for a broad class of flavour models, already competitive with LEP-1 measurements for setting bounds on the dimension six operators parametrising the anomalous couplings between the quarks and the electroweak gauge bosons, at least under the assumption that any new particle is heavier than a few TeV. We also make an estimate of the HL-LHC reach with $3$ ab$^{-1}$. We comment on possible BSM interpretations of the bounds, and show the interplay with other searches for a simplified model with vector triplets. We further study the effect of modified $Z$-quark-quark couplings on the anomalous triple gauge coupling bounds. We find that their impact is already significant and that it could modify the constraints on $\delta g_{1z}$ and $\delta \kappa_\gamma$ by as much as a factor two at the end of HL-LHC ($\lambda_\gamma$ is only marginally affected), requiring a global fit to extract robust bounds. We stress the role of flavour assumptions and study explicitly flavour universal and minimal flavour violation scenarios, illustrating the differences with results obtained for universal theories.

Future neutron-antineutron ($n$-$\bar n$) oscillation experiments, such as at the European Spallation Source and the Deep Underground Neutrino Experiment, aim to find first evidence of baryon number violation. We investigate implications of an improved $n$-$\bar n$ oscillation search for baryogenesis via interactions of $n$-$\bar n$ mediators, parameterized by an effective field theory (EFT). We find that even in a minimal EFT setup, there is overlap between the parameter space probed by $n$-$\bar n$ oscillation and the region that can realize the observed baryon asymmetry of the universe. The mass scales of exotic new particles are in the TeV-PeV regime, inaccessible at the LHC or its envisioned upgrades. Given the innumerable high energy theories that can match to, or resemble, the minimal EFT that we discuss, future $n$-$\bar n$ oscillation experiments could probe many viable theories of baryogenesis beyond the reach of other experiments.

This note proposes common standards and prescriptions for the effective-field-theory interpretation of top-quark measurements at the LHC.

The International Linear Collider is now proposed with a staged machine design, with the first stage at $\sqrt{s}=$~250 GeV and an integrated luminosity goal of 2~ab$^{-1}$. One of the questions for the machine design is the importance of positron polarization. In this report, we review the impact of positron polarization on the physics goals of the $250$ GeV stage of the ILC and demonstrate that positron polarization has distinct advantages.

We perform a global effective-field-theory analysis to assess the precision on the determination of the Higgs trilinear self-coupling at future lepton colliders. Two main scenarios are considered, depending on whether the center-of-mass energy of the colliders is sufficient or not to access Higgs pair production processes. Low-energy machines allow for ~40% precision on the extraction of the Higgs trilinear coupling through the exploitation of next-to-leading-order effects in single Higgs measurements, provided that runs at both 240/250 GeV and 350 GeV are available with luminosities in the few attobarns range. A global fit, including possible deviations in other SM couplings, is essential in this case to obtain a robust determination of the Higgs self-coupling. High-energy machines can easily achieve a ~20% precision through Higgs pair production processes. In this case, the impact of additional coupling modifications is milder, although not completely negligible.

The International Linear Collider is now proposed with a staged machine design, with the first stage at 250 GeV with a luminosity goal of 2 ab-1. In this paper, we review the physics expectations for this machine. These include precision measurements of Higgs boson couplings, searches for exotic Higgs decays, other searches for particles that decay with zero or small visible energy, and measurements of e+e- annihilation to W+W- and 2-fermion states with improved sensitivity. A summary table gives projections for the achievable levels of precision based on the latest full simulation studies.

We point out that the stringent lower bounds on the masses of additional electrically neutral and charged Higgs bosons crucially depend on the flavour structure of their Yukawa interactions. We show that these bounds can easily be evaded by the introduction of flavour-changing neutral currents in the Higgs sector. As an illustration, we study the phenomenology of a two Higgs doublet model with a Yukawa texture singling out the third family of quarks and leptons. We combine constraints from low-energy flavour physics measurements, LHC measurements of the 125 GeV Higgs boson rates, and LHC searches for new heavy Higgs bosons. We propose novel LHC searches that could be performed in the coming years to unravel the existence of these new Higgs bosons.

The cause of the screening of the weak interactions at long distances puzzled the high-energy community for more nearly half a century. With the discovery of the Higgs boson a new era started with direct experimental information on the physics behind the breaking of the electroweak symmetry. This breaking plays a fundamental role in our understanding of particle physics and sits at the high-energy frontier beyond which we expect new physics that supersedes the Standard Model. The Higgs boson (inclusive and differential) production and decay rates offer a new way to probe this frontier.

In this Letter we explore the potential of probing new light force-carriers, with spin-independent couplings to the electron and the neutron, using precision isotope shift spectroscopy. We develop a formalism to interpret linear King plots as bounds on new physics with minimal theory inputs. We focus only on bounding the new physics contributions that can be calculated independently of the Standard Model nuclear effects. We apply our method to existing Ca+ data and project its sensitivity to possibly existing new bosons using narrow transitions in other atoms and ions (specifically, Sr and Yb). Future measurements are expected to improve the relative precision by five orders of magnitude, and can potentially lead to an unprecedented sensitivity for bosons within the 10 keV to 10 MeV mass range.

Precision study of electroweak symmetry breaking strongly motivates the construction of a lepton collider with center-of-mass energy of at least 240 GeV. Besides Higgsstrahlung ($e^+e^- \to hZ$), such a collider would measure weak boson pair production ($e^+e^- \to WW$) with an astonishing precision. The weak-boson-fusion production process ($e^+e^- \to \nu \bar{\nu} h$) provides an increasingly powerful handle at higher center-of-mass energies. High energies also benefit the associated top-Higgs production ($e^+e^-\to t\bar th$) that is crucial to constrain directly the top Yukawa coupling. The impact and complementarity of differential measurements, at different center-of-mass energies and for several beam polarization configurations, are studied in a global effective-field-theory framework. We define a "global determinant parameter" (GDP) which characterizes the overall strengthening of constraints independently of the choice of operator basis. The reach of the CEPC, CLIC, FCC-ee, and ILC designs is assessed.

The Higgs self-coupling is notoriously intangible at the LHC. It was recently proposed to probe the trilinear Higgs interaction through its radiative corrections to single-Higgs processes. This approach however requires to disentangle these effects from those associated to deviations of other Higgs-couplings to fermions and gauge bosons. We show that a global fit exploiting only single-Higgs inclusive data suffers from degeneracies that prevent one from extracting robust bounds on each individual coupling. We show how the inclusion of double-Higgs production via gluon fusion, and the use of differential measurements in the associated single-Higgs production channels WH, ZH and ttH, can help to overcome the deficiencies of a global Higgs-couplings fit. In particular, we bound the variations of the Higgs trilinear self-coupling relative to its SM value to the interval [0.1, 2.3] at 68% confidence level at the high-luminosity LHC, and we discuss the robustness of our results against various assumptions on the experimental uncertainties and the underlying new physics dynamics. We also study how to obtain a parametrically enhanced deviation of the Higgs self-couplings and we estimate how large this deviation can be in a self-consistent effective field theory framework.

Higgs boson compositeness is a phenomenologically viable scenario addressing the hierarchy problem. In minimal models, the Higgs boson is the only degree of freedom of the strong sector below the strong interaction scale. We present here the simplest extension of such a framework with an additional composite spin-zero singlet. To this end, we adopt an effective field theory approach and develop a set of rules to estimate the size of the various operator coefficients, relating them to the parameters of the strong sector and its structural features. As a result, we obtain the patterns of new interactions affecting both the new singlet and the Higgs boson's physics. We identify the characteristics of the singlet field which cause its effects on Higgs physics to dominate over the ones inherited from the composite nature of the Higgs boson. Our effective field theory construction is supported by comparisons with explicit UV models.

This paper addresses the question of whether the International Linear Collider has the capability of discovering new particles that have not already been discovered at the CERN Large Hadron Collider. We summarize the various paths to discovery offered by the ILC, and discuss them in the context of three different scenarios: 1. LHC does not discover any new particles, 2. LHC discovers some new low mass states and 3. LHC discovers new heavy particles. We will show that in each case, ILC plays a critical role in discovery of new phenomena and in pushing forward the frontiers of high-energy physics as well as our understanding of the universe in a manner which is highly complementary to that of LHC. For the busy reader, a two-page executive summary is provided at the beginning of the document.

If the gamma-gamma resonance at 750 GeV suggested by 2015 LHC data turns out to be a real effect, what are the implications for the physics case and upgrade path of the International Linear Collider? Whether or not the resonance is confirmed, this question provides an interesting case study testing the robustness of the ILC physics case. In this note, we address this question with two points: (1) Almost all models proposed for the new 750 GeV particle require additional new particles with electroweak couplings. The key elements of the 500 GeV ILC physics program---precision measurements of the Higgs boson, the top quark, and 4-fermion interactions---will powerfully discriminate among these models. This information will be important in conjunction with new LHC data, or alone, if the new particles accompanying the 750 GeV resonance are beyond the mass reach of the LHC. (2) Over a longer term, the energy upgrade of the ILC to 1 TeV already discussed in the ILC TDR will enable experiments in gamma-gamma and e+e- collisions to directly produce and study the 750 GeV particle from these unique initial states.

We discuss the conditions for an effective field theory (EFT) to give an adequate low-energy description of an underlying physics beyond the Standard Model (SM). Starting from the EFT where the SM is extended by dimension-6 operators, experimental data can be used without further assumptions to measure (or set limits on) the EFT parameters. The interpretation of these results requires instead a set of broad assumptions (e.g. power counting rules) on the UV dynamics. This allows one to establish, in a bottom-up approach, the validity range of the EFT description, and to assess the error associated with the truncation of the EFT series. We give a practical prescription on how experimental results could be reported, so that they admit a maximally broad range of theoretical interpretations. Namely, the experimental constraints on dimension-6 operators should be reported as functions of the kinematic variables that set the relevant energy scale of the studied process. This is especially important for hadron collider experiments where collisions probe a wide range of energy scales.

The recently observed excess in diphoton events at around 750\u2009GeV can be satisfactorily described in terms of a new spin-0 real singlet with effective interactions to the gauge bosons. In this letter we first review the current constraints on this setup. We further explore the production in association with a gauge boson. We show the potential of this channel to unravel current flat directions in the allowed parameter space. We then study the potential of two different asymmetries for disentangling the CP nature of such a singlet in both gluon fusion and vector-boson fusion. For this matter, we perform an estimation of the efficiency for selecting signal and background events in eight different decay modes, namely $4\ell, 2j\, 2\ell, 2j\ell\,E_T^\text{miss}, 2\gamma\,2j, 4\ell\,2j, 2\ell\,\gamma\,2j, 4j\,2\ell$ and $4j\,\ell\,E_T^\text{miss}$. We emphasize that the very different couplings of this new singlet to the Standard Model particles as well as the larger mass provide a distinctive phenomenology with respect to Higgs searches. We finally show that a large region of the parameter space region could be tested within the current LHC run, the dominant channel being $2\gamma\, 2j$.

We review strongly coupled and extra dimensional models of electroweak symmetry breaking. Models examined include warped extra dimensions, bulk Higgs, "little" Higgs, dilaton Higgs, composite Higgs, twin Higgs, quantum critical Higgs, and "fat" SUSY Higgs. We also discuss current bounds and future LHC searches for this class of models.

Recently, a new mechanism to generate a naturally small electroweak scale has been proposed. It exploits the coupling of the Higgs to an axion-like field and a long era in the early universe where the axion unchains a dynamical screening of the Higgs mass. We present a new realization of this idea with the new feature that it leaves no signs of new physics up to a rather large scale, 10^9 GeV, except for two very light and weakly coupled axion-like states. One of the scalars can be a viable Dark Matter candidate. Such a cosmological Higgs-axion interplay could be tested with a number of experimental strategies.

We summarize the physics case for the International Linear Collider (ILC). We review the key motivations for the ILC presented in the literature, updating the projected measurement uncertainties for the ILC experiments in accord with the expected schedule of operation of the accelerator and the results of the most recent simulation studies.

The European School of High-Energy Physics is intended to give young physicists an introduction to the theoretical aspects of recent advances in elementary particle physics. These proceedings contain lecture notes on the Standard Model of electroweak interactions, quantum chromodynamics, flavour physics, physics beyond the Standard Model, neutrino physics, and cosmology.

This paper describes the physics case for a new fixed target facility at CERN SPS. The SHiP (Search for Hidden Particles) experiment is intended to hunt for new physics in the largely unexplored domain of very weakly interacting particles with masses below the Fermi scale, inaccessible to the LHC experiments, and to study tau neutrino physics. The same proton beam setup can be used later to look for decays of tau-leptons with lepton flavour number non-conservation, $\tau\to 3\mu$ and to search for weakly-interacting sub-GeV dark matter candidates. We discuss the evidence for physics beyond the Standard Model and describe interactions between new particles and four different portals - scalars, vectors, fermions or axion-like particles. We discuss motivations for different models, manifesting themselves via these interactions, and how they can be probed with the SHiP experiment and present several case studies. The prospects to search for relatively light SUSY and composite particles at SHiP are also discussed. We demonstrate that the SHiP experiment has a unique potential to discover new physics and can directly probe a number of solutions of beyond the Standard Model puzzles, such as neutrino masses, baryon asymmetry of the Universe, dark matter, and inflation

A comprehensive review of physics at an e+e- Linear Collider in the energy range of sqrts=92 GeV--3 TeV is presented in view of recent and expected LHC results, experiments from low energy as well as astroparticle physics.The report focuses in particular on Higgs boson, Top quark and electroweak precision physics, but also discusses several models of beyond the Standard Model physics such as Supersymmetry, little Higgs models and extra gauge bosons. The connection to cosmology has been analyzed as well.

We study the off-shell Higgs data in the process $pp\to h^{(*)} \to Z^{(\ast)}Z^{(\ast)}\to 4\ell$, to constrain deviations of the Higgs couplings. We point out that this channel can be used to resolve the long- and short-distance contributions to Higgs production by gluon fusion and can thus be complementary to $pp\to ht\bar t$ in measuring the top Yukawa coupling. Our analysis, performed in the context of Effective Field Theory, shows that current data do not allow one to draw any model-independent conclusions. We study the prospects at future hadron colliders, including the high-luminosity LHC and accelerators with higher-energy, up to 100 TeV. The available QCD calculations and the theoretical uncertainties affecting our analysis are also briefly discussed.

We present the activities of the "New Physics" working group for the "Physics at TeV Colliders" workshop (Les Houches, France, 3--21 June, 2013). Our report includes new computational tool developments, studies of the implications of the Higgs boson discovery on new physics, important signatures for searches for natural new physics at the LHC, new studies of flavour aspects of new physics, and assessments of the interplay between direct dark matter searches and the LHC.

We present eHDECAY, a modified version of the program HDECAY which includes the full list of leading bosonic operators of the Higgs effective Lagrangian with a linear or non-linear realization of the electroweak symmetry and implements two benchmark composite Higgs models.

The Higgs production and decay rates offer a new way to probe new physics beyond the Standard Model. While dynamics aiming at alleviating the hierarchy problem generically predict deviations in the Higgs rates, the current experimental analyses cannot resolve the long- and short-distance contributions to the gluon fusion process and thus cannot access directly the coupling between the Higgs and the top quark. We investigate the production of a boosted Higgs in association with a high-transverse momentum jet as an alternative to the $t\bar{t}h$ channel to pin down this crucial coupling. Presented first in the context of an effective field theory, our analysis is then applied to models of partial compositeness at the TeV scale and of natural supersymmetry.

We study deformations of the SM via higher dimensional operators. In particular, we explicitly calculate the one-loop anomalous dimension matrix for 13 bosonic dimension-6 operators relevant for electroweak and Higgs physics. These scaling equations allow us to derive RG-induced bounds, stronger than the direct constraints, on a universal shift of the Higgs couplings and some anomalous triple gauge couplings by assuming no tuning at the scale of new physics, i.e. by requiring that their individual contributions to the running of other severely constrained observables, like the electroweak oblique parameters or $\Gamma(h \rightarrow \gamma\gamma)$, do not exceed their experimental direct bounds. We also study operators involving the Higgs and gluon fields.

This report summarizes the work of the Energy Frontier Higgs Boson working group of the 2013 Community Summer Study (Snowmass). We identify the key elements of a precision Higgs physics program and document the physics potential of future experimental facilities as elucidated during the Snowmass study. We study Higgs couplings to gauge boson and fermion pairs, double Higgs production for the Higgs self-coupling, its quantum numbers and $CP$-mixing in Higgs couplings, the Higgs mass and total width, and prospects for direct searches for additional Higgs bosons in extensions of the Standard Model. Our report includes projections of measurement capabilities from detailed studies of the Compact Linear Collider (CLIC), a Gamma-Gamma Collider, the International Linear Collider (ILC), the Large Hadron Collider High-Luminosity Upgrade (HL-LHC), Very Large Hadron Colliders up to 100 TeV (VLHC), a Muon Collider, and a Triple-Large Electron Positron Collider (TLEP).

We study the impact of Higgs precision measurements at a high-energy and high-luminosity linear electron positron collider, such as CLIC or the ILC, on the parameter space of a strongly interacting Higgs boson. Some combination of anomalous couplings are already tightly constrained by current fits to electroweak observables. However, even small deviations in the cross sections of single and double Higgs production, or the mere detection of a triple Higgs final state, can help establish whether it is a composite state and whether or not it emerges as a pseudo-Nambu-Goldstone boson from an underlying broken symmetry. We obtain an estimate of the ILC and CLIC sensitivities on the anomalous Higgs couplings from a study of WW scattering and hh production which can be translated into a sensitivity on the compositeness scale 4\pi f, or equivalently on the degree of compositeness \xi=v^2/f^2. We summarize the current experimental constraints, from electroweak data and direct resonance searches, and the expected reach of the LHC and CLIC on \xi and on the scale of the new resonances.

The discovery by the ATLAS and CMS experiments of a new boson with mass around 125 GeV and with measured properties compatible with those of a Standard-Model Higgs boson, coupled with the absence of discoveries of phenomena beyond the Standard Model at the TeV scale, has triggered interest in ideas for future Higgs factories. A new circular e+e- collider hosted in a 80 to 100 km tunnel, TLEP, is among the most attractive solutions proposed so far. It has a clean experimental environment, produces high luminosity for top-quark, Higgs boson, W and Z studies, accommodates multiple detectors, and can reach energies up to the t-tbar threshold and beyond. It will enable measurements of the Higgs boson properties and of Electroweak Symmetry-Breaking (EWSB) parameters with unequalled precision, offering exploration of physics beyond the Standard Model in the multi-TeV range. Moreover, being the natural precursor of the VHE-LHC, a 100 TeV hadron machine in the same tunnel, it builds up a long-term vision for particle physics. Altogether, the combination of TLEP and the VHE-LHC offers, for a great cost effectiveness, the best precision and the best search reach of all options presently on the market. This paper presents a first appraisal of the salient features of the TLEP physics potential, to serve as a baseline for a more extensive design study.