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arxiv: 2606.18364 · v1 · pith:HALNKOTEnew · submitted 2026-06-16 · ✦ hep-ph · hep-ex· hep-th

The Landscape of Composite Higgs Models

Daniel Murnane This is my paper

Pith reviewed 2026-06-26 23:34 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-th
keywords composite HiggsnaturalnessBayesian tuning4DCHMcollider phenomenologymodel extensionsdark matter
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The pith

A Bayesian tuning measure applied to Composite Higgs models finds limited naturalness improvements from common extensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a new way to measure how finely tuned the parameters of the Higgs must be, grounding it in Bayesian probability. This measure is then used to evaluate the minimal four-dimensional composite Higgs model and versions that include composite leptons or a dark matter particle. The analysis includes global fits to collider data that exclude large parts of the model's parameter space. Understanding any naturalness gains from these additions helps decide which directions in new physics are most promising for explaining the lightness of the Higgs without extreme fine-tuning.

Core claim

Through global fits and the application of the new tuning measure, the work concludes that extensions to the minimal 4D composite Higgs model, such as embedding a composite tau or adding a dark matter candidate, provide no substantial benefit to naturalness in the composite Higgs sector.

What carries the argument

The N-site four-dimensional composite Higgs model, an effective theory where the Higgs emerges as a pseudo-Goldstone boson from a strong dynamics sector at higher scales, combined with the proposed Bayesian tuning measure that quantifies the improbability of the observed Higgs mass parameter.

If this is right

  • Collider searches impose strong exclusion limits on the minimal model's parameters.
  • Embedding composite leptons in different representations yields no significant reduction in tuning.
  • The next-to-minimal model with a dark matter candidate shows comparable or worse naturalness metrics.
  • The Bayesian tuning measure allows quantitative comparison of naturalness across different composite Higgs setups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The results imply that achieving better naturalness may require going beyond these simple extensions to more intricate model structures.
  • The measure could be tested for consistency by applying it to other beyond-Standard-Model scenarios with known naturalness properties.
  • Collider experiments might prioritize searches motivated by other considerations if naturalness gains remain minimal.

Load-bearing premise

The new tuning measure correctly captures the concept of naturalness in a way that matches physical intuition and that the four-dimensional models sufficiently describe the relevant physics at collider energies.

What would settle it

A future collider observation of a light composite resonance whose parameters require extreme tuning according to the measure but appear in data would challenge the exclusion bounds and naturalness conclusions.

Figures

Figures reproduced from arXiv: 2606.18364 by Daniel Murnane.

Figure 1.1
Figure 1.1. Figure 1.1: A pair of balancing rocks (also called precarious boulders) in the Karlu Karlu (Devil’s Marbles) Conservation Park ical fine-tuning to account for the gravitational hierarchy problem. In this case, we would have no excuse that the theory is “undoubtedly effective”. Instead, we might continue searching for another theory that describes the universe equally well, mo￾tivated by nothing more than naturalness… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Visualisation of some low-dimensional symmetries in the classical vacuum and then shifted to a physical vacuum. If the fields are redefined as having zero vev, i.e. ϕ → ϕ ′ = ϕ + ⃗f , which leaves the redefined Lagrangian invariant under some smaller group h ∈ H, then we say this set {⃗f }H is invariant under the subgroup H < G. Casually, we say that the vev has spontaneously broken the global symmetry G… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The Higgs mechanism sketched using the convention established by S. Weinberg in [304] A proof of this theorem will be outlined here (based on the strategies found in [62, 29, 235]), although the full quantum-theoretic argument contains many subtleties. At the semi-classical level, a much more elegant proof is provided by the CCWZ construction, described later in section 2.5. Consider a set of scalar fiel… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Diagrammatic expansion of the quantum Coleman-Weinberg effective potential. Each order of the n-leg 1PI summation is itself a series of m-loop diagrams. [92] argues that the first layer of the summation V (1) eff is a sufficient approximation. Proving the adequacy of this truncation is outside the scope of this section, but is perfectly well-explained in [64]. For now, we will be content with seeing clea… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The manifold of matter ψ and symmetry π fields, around the origin, with a linear subgroup leaving the vacuum expectation invariant by the broken degrees of freedom π aˆ and the vacuum direction ⃗f . The matrix U is the Goldstone matrix - a spacetime-dependent transformation of the vacuum in the direction of the coset G/H. Now we assemble the physics and group theory: we denote the space of physical field… view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: The notation of "hidden local symmetry" situations such as those described in section 2.8: dynamical electroweak symmetry breaking. Combining an arbitrary non-linear symmetry with a gauge symmetry can be described by fig. 2.5. Consider a NGB matrix Ω(x) transforming linearly as the fundamental of a global group g ∈ Gglobal ≡ GL and the antifundamental of a local group h ∈ Hlocal ≡ HR ⊂ GL 19 GL × HR : Ω(… view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: An overview on the bounds of the Higgs mass, in order to avoid triviality, vac￾uum instability, and EW precision exclusion. Reproduced from [275] The vector boson mass terms are therefore proportional to the gauge coupling, and taking gH → ∞ allows them to be decoupled. Applying the equations of motions to the vector bosons leads to the solution i(AH)µ = eµ, as defined in the CCWZ section. In this limit … view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: A typical contribution to the mass of the W boson the theory, many of which are charged and massive. There is a residual (SU(3)u × SU(3)d )L + R flavour symmetry which leaves all up-type quark masses degenerate, and the same for down-type quarks. It does not give leptons mass. One might also add the fact that we’ve discovered a Higgs boson to the list of shortcomings, but actu￾ally the sigma field around… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A general Goldstone boson breaking pattern [PITH_FULL_IMAGE:figures/full_fig_p048_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: A typical FCNC allowed in CHMs without custodial symmetry ken group, which we denote as Q = H ∩ H. In this case, dim(Q)=dim(U(1)em) = 1, thus dim(H0 ∪ H1) = dim(H0) + dim(H1) − dim(H0 ∩ H1) (131) = 3 + 4 − 1 = 6 = dim(G) (132) We can see that for the SM Higgs pattern, there are no remaining Goldstone bosons. In other words, all the degrees of freedom in the coset G/H are "eaten" by gauging H. For the hig… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The prototypical symmetry-breaking diagram, illustrating the invariant vacuum space SO(3)/SO(2) ∼ S2 is given by ⃗f ′ , with an invariant plane given by the red disc. Inspecting the geom￾etry of the situation, there are no remaining symmetries - the intersection between the local S1 (red) and global S1 (blue) is only two points unable to be transformed between, due the “special" nature of the groups. The… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Visualisation of SO(4)/SO(3) symmetry breaking, including a misalignment of the vacuum that leaves a SO(2) symmetry remaining [PITH_FULL_IMAGE:figures/full_fig_p052_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: A sketch of "vacuum misalignment" (VM) - i.e. the mixing of two groups’ genera￾tors, which may change due to one-loop corrections, leading to a vev ⟨h⟩. Before VM, the vacuum is invariant under SO(4)H ∼ SU(2)L × SU(2)R. Afterwards, SO(4)H′ mixes TL, TR with previously broken generators X. pairs of SM fermions amounts to an "elementary pNGB Higgs". We will be for￾bidding this form of interaction for reaso… view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: The types of vacuum (mis)alignment in Composite Higgs models 3.2.5 Calculating the Higgs Potential and Vacuum Misalignment our goal To use the Coleman-Weinberg 1-loop procedure to find the Higgs potential in terms of the gauge form factors. From our experience with the Coleman-Weinberg formalism of section 2.3, we know that we can approximate the interaction of a weakly-interacting quantised field with a… view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Series of gauge boson interactions with the classical background Higgs field. These are contributions to the 1-loop effective potential The propagator is the inverse of the interaction of W± µ with only itself, in eq. (166) 10 . We propose that it is iGµν = i Π(0) (p 2) (PT)µν − ζ ig2 p 2 (PL)µν (182) We can see that this is the propagator, by the definitions of the projection operators (PT)µν = ηµν − pµ… view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: The Composite Higgs paradigm of vacuum misalignment which is strictly negative. Inserting this behaviour into the potential equation eq. (198) leads to a local minimum of the potential at ⟨h⟩ = 0. There are also solutions at ⟨h⟩ = f nπ 2 . The energy scale between the two sets of solution is too great to tunnel in finite time. Regardless, assuming f > 1 TeV, these solutions do not correctly reproduce EWS… view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Series of fermion interactions with the classical background Higgs field. These are contributions to the 1-loop effective potential as we did in the gauge boson case, we are interested in applying the Feynman rules to diagrams of the sort in fig. 3.9, where the top diagrams have propagator form factors i /pΠ q,t 0 and vertex form factors i/pΠ q,t 1 . The bottom diagrams have total propagators  i /p(Π q … view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Roads to Composite Higgs 3.3.1 Roads to a pNGB Composite Higgs A Higgs that emerges as a pNGB from hidden global symmetries is a convincing way to ameliorate the Hierarchy Problem. But what does the hidden global symme￾try describe - what degrees of freedom does it represent and how are some of these "broken" to a smaller set? And how would composite matter also be described by this group, especially wh… view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: An overview of electroweak-symmetry-breaking scenarios with Higgs dynam￾ics, organised by the vacuum misalignment angle θ (ξ = v 2/ f 2 = sin2 θ). The composite pseudo-Goldstone Higgs interpolates continuously between the ele￾mentary Standard-Model Higgs (ξ → 0, f → ∞) and technicolor (ξ → 1, v = f); little-Higgs models realise the same small-ξ pNGB Higgs but protect its mass through collective symmetry… view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: A generic Moose diagram 1. A set of k sites of fermions multiplets, Lmatter = ψ¯ α 1 (/∂ − m1)ψ α 1 + ψ¯ α 2 (/∂ − m2)ψ α 2 + ... + ψ¯ α N(/∂ − mN)ψ α N (227) each in a fundamental17 of Gi , where G1 ∼ G2 ∼ ... ∼ Gk for convenience. Denote each fermion ψ α i , where (G flavour index) α = 1, ..., M and (site index) i = 1, ..., N. 2. A set of gauge bosons Ai,µ = A a i,µ S a that promote each G invariance … view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: A two-site model [PITH_FULL_IMAGE:figures/full_fig_p074_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: A Moose diagram describing the Hidden Local Symmetry limit of a gauged Moose model We remind that we discriminate between those gauge generators that are shared between sites and those that are not. This is summarised in the diagram from section 2.5.3 with some additional structure we had not included at that stage, in fig. 3.15. It can also be written as the Moose diagram in fig. 3.17. G global 1+2 , T… view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: The generators of a 2-site Moose model, with K a , Tˆ a ∈ T a and Pˆa ∈ X a to refer to the sigma model Global: Sites: Local: ψ1 G1 E × × Ω1 ψ2 G2 H [PITH_FULL_IMAGE:figures/full_fig_p077_3_15.png] view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: A two-site model [PITH_FULL_IMAGE:figures/full_fig_p077_3_16.png] view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: A Moose diagram describing scenario (1), (2) & (3), L = Lmatter + Lgauge + Lgoldstone. E ⊕ H is the diagonal subgroup of those two gauged groups. This chain of link fields and gaugings can be extended to an N-site Moose model, as in fig. 3.18. However, recall that we only want the physical Goldstone bosons corresponding to an overall G/H breaking. Therefore, we would like to gauge away all but those cor… view at source ↗
Figure 3.18
Figure 3.18. Figure 3.18: A Moose diagram describing scenario (1)+(2)+(3) with gH → ∞. 3.5 general composite matter our goal To describe an effective Lagrangian for any matter in the Minimal Com￾posite Higgs model. Just as the transformation property of the NGBs determined their interactions and repre￾sentation, so we consider how bosonic and fermionic resonances can transform under either a non-linear g ∈ G, or a linear h ∈ H. … view at source ↗
Figure 3.19
Figure 3.19. Figure 3.19: Fermion couplings in the N-site Moose (with HLS on site N) g i ρL,R , the composite gauge couplings, are free parameters. We see from eq. (104) that Eµ is the correct term to include in trace of ρ fields, to cancel the transformed field’s h∂µh † term. Dealing with composite fermions is somewhat easier, using the CCWZ ideas developed previously. Again, we ask how the resonances transform under our Moose … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Posterior distribution plot, as a uniform prior is “spent" on the region ∆x, with the purchased area being the Occam Factor. Adapted from [246]. Let us make our intuitions rigorous. For a distribution of observables, we assume the Laplace approximation of the likelihood function. This is a multivariate Taylor series to first order around a peak in the likelihood around parameter point xmax logLO(x) ≃ log… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Exponential scaling of higher order tuning when increasing the number of ob￾servables, given a model of np = 25. normalise each order - to average the tunings as we do with the first order. For example, a second order tuning may be normalised to be ∆ 2 norm. =  no 2 −1  ∆ ab + ∆ ac + ... + ∆ yz (380) thereby removing the scaling. However, this does not limit correctly to our criterion 3, of a single … view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The configurations available for one source of double-tuning amongst four ob￾servables Configuration 1 algebraically satisfies criterion 3 in a simple extension of eq. (367) to unordered pairs over no observables ∆2 = 1 no − 1 ( no 2 ) ∑ { a,b }|b<a ∆ ab 2 b→c→d −−−−→ ∆ ab 2 . (381) However, calculating eq. (381) for configuration 2 gives more unordered pairs, and thus a factor of 4/3 above configuration… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Comparisons of the higher order tuning approximation to the orthogonalisation procedure. The approximation generally overestimates tuning, but not signifi￾cantly [PITH_FULL_IMAGE:figures/full_fig_p102_4_4.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Posterior distribution plot. Numerical integration of Ldx = LdX can be achieved with ordered sampling of (a), assuming a monotonically increasing likelihood function, as in (b). Points v2 and v5 are thrown away as they are not more likely than v1 and v4 respectively. computationally easier, but it now matches our usual goal of exploring likely param￾eter space. Thus, we receive a posterior distribution a… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Ellipsoidal nesting in MultiNest for a population of 4, replacing 2 points per gen￾eration. For each generation a, b, ..., each vector is ordered from highest likelihood to lowest: {a1 , a2, a3, a4}. Thus a3 and a4 are replaced by b vectors, which are then re-ordered. Elliptical contours are drawn with average likelihoods L a ,L b , .... These are integrated over, as in fig. 5.1 (b). 5.2.2 Diver Evolutio… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Prototypical genetic algorithm search vs. differential evolution. Compare with fig. 5.4 A very simple example of this procedure is given in fig. 5.3a, for two generations. It will be useful to compare this choice of simple genetic operations with the more complex one used in this work: differential evolution. Diver Implementation In order to produce a well-sampled analysis of the model’s fine tuning, we … view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Breeding process for a generic GA (a), and differential evolution (b) [PITH_FULL_IMAGE:figures/full_fig_p108_5_4.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: The Minimal 4D Composite Higgs Model For reference, we note that this is an extension of the 2-site "Discrete CHM (DCHM)", as described by [266] that allows for calculability of the potential. On the other hand, it is a simplification of the 3-site DCHM, where we have enforced cer￾tain couplings taken to zero. Indeed, in the M4DCHM, one can interpolate between these two cases by taking g2 → ∞, performing… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: The two- and three-site models, to compare with the M4DCHM considered in this work The typical "vanilla" M4DCHM considers only a partially composite top quark in the fundamental representation, as its contribution to the Higgs potential is domi￾nant. In this thesis we will variously consider the following extensions: • A partially composite bottom quark (3rd-generation M4DCHM) • Partially composite lepto… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Contributions to S and T deviations from the SM. Resonance searches We include Drell-Yan search channels from LEP, Tevatron, and run-1 and -2 LHC, for BSM heavy resonances. Bosonic resonances will appear in decays such as ρ ± → W±h, and ρ 0 → t ¯t. These are listed in table 6.3, table 6.4 and table 6.5. Masses in the full model can be calculated with the bosonic mass matrices constructed from the couplin… view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Tuning of the breaking scale and fermionic composite partner masses [PITH_FULL_IMAGE:figures/full_fig_p125_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: The tuning behaviour of the lightest composite vector resonances Finally, it should be noted that although the differential evolution scanner pro￾duced a convergent global fit, the posterior results are sparsely sampled, and fur￾ther work should be done to obtain a better understanding of the global behaviour. However, we observed a best likelihood fit of mU ∼ 2050 GeV, mD ∼ 4000 GeV in the composite fer… view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: The tuning behaviour of SM Higgs and top masses [PITH_FULL_IMAGE:figures/full_fig_p127_6_6.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Allowed interactions in the minimal model (excluding the 5-5-5 and 14-14-14 embedding). between representations, as will be detailed shortly. The link terms would typi￾cally depend on representation, but we are enforcing that the elementary fields fill full multiplets in matching representations, in order to couple with their respective partner. Tracing over this term handles the case that the terms are … view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Allowed interactions in the minimal model, for the 5-5-5 and 14-14-14 embedding. The kinetic coupling is included here explicitly to remind that there is indeed coupling between the up-type fermions and down-type. where Ψu , Ψ˜ u ∼ 5Xu and Ψd , Ψ˜ d ∼ 5Xd . In the case of a generation of quarks, X u = 2/3 and X d = −1/3. The composite Yukawa couplings for a 5-5-5 generation are Ly = Yu(Ψ¯ u LΦ)(Φ †Ψ˜ u R… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: A comparison of non-normalised (a) and normalised (b) fine tunings in the mass of top partners A comparison of our new tuning with less sophisticated tuning measures can be seen in the bottom right panel of fig. 7.4, which shows the fine tuning for the LM4DCHM5-5-5 5-5-5 model as a function of the vacuum misalignment ξ. Our measure gives higher values for fine tuning relative to the single tuning ∆1 or t… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Tuning in the LM4DCHM5-5-5 5-5-5 model as a function of Higgs coupling ratios, light￾est scalar resonance mass, top partner masses, and vacuum misalignment [PITH_FULL_IMAGE:figures/full_fig_p141_7_4.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: The tuning of Higgs-tau coupling modifications magnitude argument, the tuning at these low masses does not prefer this to the pre￾vious models. However, where a natural symmetric representation shows a sharp rise in the fine tuning with better top partner mass exclusion limits and more pre￾cise Higgs coupling measurements, the present model remains relatively untuned even at top partner masses of m27/6 =… view at source ↗
Figure 7.6
Figure 7.6. Figure 7.6: Tuning in the LM4DCHM5-5-5 14-14-10 model as a function of Higgs coupling ratios, lightest scalar resonance mass, top partner masses, and vacuum misalignment [PITH_FULL_IMAGE:figures/full_fig_p143_7_6.png] view at source ↗
Figure 7.7
Figure 7.7. Figure 7.7: Tuning in the LM4DCHM5-5-5 14-1-10 model as a function of Higgs coupling ratios, lightest scalar resonance mass, top partner masses, and vacuum misalignment [PITH_FULL_IMAGE:figures/full_fig_p144_7_7.png] view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: The Next-to-Minimal 4D Composite Higgs Model moose diagram 135 [PITH_FULL_IMAGE:figures/full_fig_p145_8_1.png] view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: Two examples of the GB potential. On the left, c1 = 1, c2 = 1, c3 = −0.1,sθ = 0.7 with ξ = 15.7, corresponding to no EWSB. On the right, c1 = 0.1, c2 = −0.2, c3 = 0.1,sθ = 0.7, with ξ = 0.48. Satisfying the condition ξ < 1 allows for the possibil￾ity of EWSB. Note that we have changed the basis from φ, η to h,s, but that the masses are the same due to ⟨η⟩ = 0 being a stationary point. This can be laborio… view at source ↗
Figure 8.3
Figure 8.3. Figure 8.3: Comparison of each model’s lightest top partner vs. naive tuning can be found in reference [59]. This agrees with a first-order expectation, since NM4DCHM observables are generically proportional to M4DCHM observables ac￾cording to mNMCHM ∝ mMCHM sin θ, and θ is a free parameter. To understand the different contributions to the higher-order tuning, we show in fig. 8.5 to fig. 8.7 various first-order tuni… view at source ↗
Figure 8.4
Figure 8.4. Figure 8.4: Comparison of higher-order tuning (defined in eq. (357)) in the Higgs-gluon, -top and -bottom coupling deviation (as defined in eq. (415) and eq. (489)) between the minimal and next-to-minimal models. Precision bounds (denoted by coloured lines) are defined in [PITH_FULL_IMAGE:figures/full_fig_p154_8_4.png] view at source ↗
Figure 8.5
Figure 8.5. Figure 8.5: Comparison of the first-order tuning (as defined in eq. (357)) contribution from the Higgs mass, in the Higgs-gluon, -top and -bottom coupling deviation. The red line shows the expected precision of a 250 GeV ILC, green a 500 GeV ILC, and blue a high-luminosity 1 TeV ILC. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p155_8_5.png] view at source ↗
Figure 8.6
Figure 8.6. Figure 8.6: Comparison of the first-order tuning contribution from the top mass, in the Higgs-gluon, -top and -bottom coupling deviation. The red line shows the expected precision of a 250 GeV ILC, green a 500 GeV ILC, and blue a high￾luminosity 1 TeV ILC [PITH_FULL_IMAGE:figures/full_fig_p155_8_6.png] view at source ↗
Figure 8.7
Figure 8.7. Figure 8.7: Comparison of the first-order tuning contribution from the vacuum misalignment ξ, in the Higgs-gluon, -top and -bottom coupling deviation. The red line shows the expected precision of a 250 GeV ILC, green a 500 GeV ILC, and blue a high￾luminosity 1 TeV ILC. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p156_8_7.png] view at source ↗
Figure 8.8
Figure 8.8. Figure 8.8: Comparison of higher-order tuning in the Higgs-vector boson coupling devia￾tion, between the minimal and next-to-minimal models. The red line shows the expected precision of a 250 GeV ILC, green a 500 GeV ILC, and blue a high￾luminosity 1 TeV ILC [PITH_FULL_IMAGE:figures/full_fig_p156_8_8.png] view at source ↗
Figure 8.9
Figure 8.9. Figure 8.9: Comparison of higher-order tuning in the Higgs-photon (loop) coupling devia￾tion, between the minimal and next-to-minimal models. Future ILC bounds are below the cut-off f > 800 GeV. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p157_8_9.png] view at source ↗
Figure 8.10
Figure 8.10. Figure 8.10: Comparison of the lightest vector resonance mass vs higher-order tuning, be￾tween models. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p157_8_10.png] view at source ↗
Figure 8.11
Figure 8.11. Figure 8.11: Comparison of the lightest fermionic resonance mass vs higher-order tuning, between models. Note that the lightest resonance may be either the singlet (yellow) or doublet (maroon) [PITH_FULL_IMAGE:figures/full_fig_p157_8_11.png] view at source ↗
Figure 8.12
Figure 8.12. Figure 8.12: Comparison of vacuum misalignment vs higher-order tuning, between models. The sin θ = 1 limiting case is more interesting. Here, the elementary top quark does not couple with the singlet eigenstate, and eq. (166) and eq. (486) become (considering only the subset of terms containing the η field) Lη θ→π/2 −−−−→ (∂η) 2 + g 2 f 2 4 sin2 φ f cos2 η f W2 η→−η = Lη (503) This Z2(η) symmetry is explored in refe… view at source ↗
Figure 8.13
Figure 8.13. Figure 8.13: The top quark mixing parameter sin θ vs (top row left) higher order tuning and (top row right) naive tuning. (a) [PITH_FULL_IMAGE:figures/full_fig_p159_8_13.png] view at source ↗
Figure 8.14
Figure 8.14. Figure 8.14: Mass of the singlet in GeV, with singlet-quark coupling deviation (as defined in eq. (415)) as the third dimension (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p159_8_14.png] view at source ↗
Figure 8.15
Figure 8.15. Figure 8.15: The singlet mass (in GeV) vs (a) higher order tuning and (b) naive tuning with sin θ as the third dimension, for points with θ ∈ {π/4, π/2} [PITH_FULL_IMAGE:figures/full_fig_p159_8_15.png] view at source ↗
Figure 8.16
Figure 8.16. Figure 8.16: The singlet mass (in GeV) vs (a) higher order tuning and (b) naive tuning with sin θ as the third dimension, for points with θ ∈ {0, π/4}. In fig. 8.15, we show our higher-order tuning measure, and the naive tuning mea￾sure, vs mS for points with θ ∈ {π/4, π/2}, indicating that the points of lowest tuning have sinθ values close to 1. This implies that the Z2 symmetry exists to stabilise a dark matter ca… view at source ↗
Figure 8.17
Figure 8.17. Figure 8.17: Top partner masses vs. full tuning, broken into region 1 (left) and region 2 (right), as defined by eq. (505) [PITH_FULL_IMAGE:figures/full_fig_p161_8_17.png] view at source ↗
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While the Standard Model (SM) of particle physics contains the most precise set of predictions ever devised by humanity, that precision comes at a cost. The strange nature of the Higgs particle requires its parameters to be tuned so precisely that if the SM is indeed the true description of reality, one is forced to wonder how such a miracle as galactic structure and life could occur. Instead, we search in this work for a natural explanation. The concept of naturalness is comprehensively explored, and a new tuning measure proposed, with an aim to place it on well-defined Bayesian footing. We then turn this measure on to the analysis of a class of intriguing new physics - Composite Higgs models. These effective models are the result of a plethora of underlying theories, and they allow the production of a naturally light Higgs particle, appearing as the SM Higgs at low energy. We establish the background required to appreciate the N-site 4D Composite Higgs model, and subsequently focus on the simplest incarnations of this class. A global fit is performed on the Minimal 4D Composite Higgs model (M4DCHM), with strong exclusion bounds placed on collider search channels. We analyse any improvement in tuning that could be gained from several extensions to this model. The Leptonic M4DCHM is explored, with a composite tau lepton embedded in various representations. The possibility of a dark matter candidate existing in the Next-to-Minimal 4DCHM is considered. Ultimately, we are able to define what, if any, benefit to naturalness can come to the Composite Higgs sector by introducing these extensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a new tuning measure intended to place naturalness on Bayesian footing, applies it to the Minimal 4D Composite Higgs Model (M4DCHM) through a global fit that yields strong exclusion bounds on collider channels, and examines whether leptonic extensions or the Next-to-Minimal 4DCHM improve the tuning relative to the baseline, ultimately assessing any naturalness benefit from these extensions.

Significance. If the tuning measure is rigorously derived from well-motivated priors and the global fits are reproducible, the work would supply a quantitative framework for comparing naturalness across composite Higgs scenarios and their extensions, clarifying whether leptonic or dark-matter extensions yield measurable improvement.

major comments (2)
  1. [Tuning measure section] The section introducing the tuning measure asserts that it places naturalness on well-defined Bayesian footing, yet provides no explicit derivation of the prior on the Higgs-mass parameter or composite scale, no marginal-likelihood expression, and no demonstration that the measure alters posterior odds rather than reproducing a conventional fine-tuning metric.
  2. [Global fit section] The global-fit section reports strong exclusion bounds on collider search channels for the M4DCHM but supplies no description of the data sets employed, the treatment of experimental and theoretical errors, or the statistical procedure used to obtain the posterior, rendering the bounds impossible to assess or reproduce.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments identify key areas where additional detail is needed to support the claims regarding the Bayesian tuning measure and the reproducibility of the global fit. We address each point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Tuning measure section] The section introducing the tuning measure asserts that it places naturalness on well-defined Bayesian footing, yet provides no explicit derivation of the prior on the Higgs-mass parameter or composite scale, no marginal-likelihood expression, and no demonstration that the measure alters posterior odds rather than reproducing a conventional fine-tuning metric.

    Authors: We agree that the derivation of the priors on the Higgs-mass parameter and composite scale, along with the marginal-likelihood expression and explicit comparison to posterior odds, requires more detail to fully substantiate the Bayesian foundation. In the revised manuscript we will add a dedicated subsection that derives the prior choices from first principles, presents the marginal likelihood explicitly, and demonstrates how the measure modifies posterior odds relative to conventional fine-tuning metrics, including the relevant equations and prior justification. revision: yes

  2. Referee: [Global fit section] The global-fit section reports strong exclusion bounds on collider search channels for the M4DCHM but supplies no description of the data sets employed, the treatment of experimental and theoretical errors, or the statistical procedure used to obtain the posterior, rendering the bounds impossible to assess or reproduce.

    Authors: We acknowledge that the absence of explicit information on the datasets, uncertainty treatment, and statistical procedure limits reproducibility. The revised version will include a new subsection detailing the specific collider datasets employed, the incorporation of experimental and theoretical uncertainties into the likelihood function, and the full Bayesian inference procedure used to obtain the posterior, including sampling methods, to enable assessment and reproduction of the reported bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The abstract introduces a new tuning measure aimed at Bayesian footing and applies it to global fits of the M4DCHM and extensions, but the provided text contains no equations, self-citations, or derivations that reduce the measure or naturalness benefit to a fit or prior result by construction. No load-bearing self-citation chain or self-definitional step is exhibited. The central claim remains independent of its own inputs per the available content, consistent with the most common honest finding of no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are specified in sufficient detail to populate the ledger.

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Works this paper leans on

295 extracted references · 85 canonical work pages · 54 internal anchors

  1. [1]

    (606) for a volume spanned by three vectors

    Now,du 1 1 is just a scalar, and it can be factored out of the square root Vol(f(dP)) = vuuuut (J1)o 1(J1)o 1 (J1)p 1 (J2)p 2 (J1)q 1(J3)q 3 (J2)o 2(J1)o 1 (J2)p 2 (J2)p 2 (J2)q 2(J3)q 3 (J3)o 3(J1)o 1 (J3)p 3 (J2)p 2 (J3)q 3(J3)q 3 du1 1du2 2du3 3du1 1du2 2du3 3 = q |J JT|(Vol(du))2 (609) This formula is the analogue of the above eq. (606) for a volume s...

  2. [2]

    Then the fundamental embedding is given by Ψ5 =   Ψ4 Ψ1   , where,Ψ 4 = 1√ 2   iΨ−− −iΨ ++ Ψ−− +Ψ ++ iΨ−+ −iΨ +− Ψ−+ +Ψ +−   ,Ψ 1 =Ψ 00

    This reflects the decomposition underSO(4)of the fundamental5∼4⊕1. Then the fundamental embedding is given by Ψ5 =   Ψ4 Ψ1   , where,Ψ 4 = 1√ 2   iΨ−− −iΨ ++ Ψ−− +Ψ ++ iΨ−+ −iΨ +− Ψ−+ +Ψ +−   ,Ψ 1 =Ψ 00 . (615) e.1.3Antisymmetric Embedding We explored in section A.2.5how one can always build higher representations by taking the ten...

  3. [3]

    Ψ4/ √ 2 ΨT 4 / √ 2 (2/ √ 5)Ψ1 ! , where, (629) Ψ9 = 1 2   ˆΨ0,0 2,+ −Ψ 0,0 4 i ˆΨ0,0 2,− ˆΨ1,1 + + ˆΨ−1,−1 + i( ˆΨ1,1 − − ˆΨ−1,−1 − ) − ˆΨ0,0 2,+ −Ψ 0,0 4 i( ˆΨ1,1 + − ˆΨ−1,−1 + ) ˆΨ1,1 − − ˆΨ−1,−1 − Ψ0,0 4 − ˆΨ0,0 1,− i ˆΨ0,0 1,+ Ψ0,0 4 + ˆΨ0,0 1,−   and theΨ 4 andΨ 1 are as given previously. As in the10case, we have defined some convenient f...

  4. [4]

    +p 4 AR(m1,m 2,m 3,m 4,∆) =∆ 2 m2 1m2 2 +m 2 2m2 3 −p 2(m2 1 +m 2 2 +m 2 3 +m 2

  5. [5]

    The expressions for theSO(4)decomposed form factors are to be found originally in [76]

    +p 4 AM(m1,m 2,m 3,m 4,∆ 1,∆ 2) =∆ 1∆2m1m2m4(m2 3 −p 2) B(m1,m 2,m 3,m 4,m 5) =m 2 1m2 2m2 3 −p 2 m2 1m2 2 +m 2 1m2 3 +m 2 2m2 3 +m 2 2m2 5 +m 2 3m2 4 +p 4 m2 1 +m 2 2 +m 2 3 +m 2 4 +m 2 5 −p 6 (634) The precise expressions for the source terms in this study are slightly different from both [58,76], so we present them in full for each representation. The ...

  6. [6]

    +m 2 1m2 2 , (638) ˆM[m1,m 2,m 3] = m1m2m3∆2 p4 −p 2(m2 1 +m 2 2 +m 2

  7. [7]

    (642) Full correlators: Πq 0 = 1 y2 tL + ˆΠqL 0 ,Π q1 1 = ˆΠqL 1 , (643) Πu 0 = 1 y2 tR + ˆΠuR 0 +s 2 θ ˆΠuR 1 ,Π u 1 =−2 ˆΠuR 1 , (644) Mu 1 = ˆMu 1

    +m 2 1m2 2 (639) Broken correlators: ˆΠqL 0 = ˆΠ[mT,m ˜T,m YT ], ˆΠqL 1 = ˆΠ[mT,m ˜T,m YT +Y T]− ˆΠ[mT,m ˜T,m YT ], (640) ˆΠuR 0 = ˆΠ[m ˜T,m T,m YT ], ˆΠuR 1 = ˆΠ[m ˜T,m T,m YT +Y T]− ˆΠ[m ˜T,m T,m YT ], (641) ˆMu 0 = ˆM[mT,m ˜T,m YT ], ˆMu 1 = ˆΠ[mT,m ˜T,m YT +Y T]− ˆΠ[mT,m ˜T,m YT ]. (642) Full correlators: Πq 0 = 1 y2 tL + ˆΠqL 0 ,Π q1 1 = ˆΠqL 1 , (64...

  8. [8]

    URL:https://math.stackexchange.com/q/40141 (version:2017-07-13)

    Jack Schmidt (https://math.stackexchange.com/users/583/jack-schmidt).How can there be multiple irreducible representations of a group each having distinct di- mension?Mathematics Stack Exchange. URL:https://math.stackexchange.com/q/40141 (version:2017-07-13). eprint:https : / / math . stackexchange . com / q / 40141. url:https://math.stackexchange.com/q/40141

    2017

  9. [9]

    Physics Stack Exchange

    Jonathan Gleason (https://physics.stackexchange.com/users/3397/jonathan- gleason).Vacuum Expectation Value and the Minima of the Potential. Physics Stack Exchange. URL:https://physics.stackexchange.com/q/75845(version: 2017-07-22). eprint:https : / / physics . stackexchange . com / q / 75845.url: https://physics.stackexchange.com/q/75845

    2017

  10. [10]

    Search for resonant $WZ$ production in the fully leptonic final state in proton-proton collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

    M. Aaboud et al. “Search for resonantWZproduction in the fully leptonic final state in proton-proton collisions at √s=13 TeV with the ATLAS detec- tor.” In:Phys. Lett.B787(2018), pp.68–88.doi:10.1016/j.physletb.2018. 10.021. arXiv:1806.01532 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2018 2018

  11. [11]

    Search for heavy resonances decaying to a $Z$ boson and a photon in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector

    Morad Aaboud et al. “Search for heavy resonances decaying to aZboson and a photon inppcollisions at √s=13 TeV with the ATLAS detector.” In: Phys. Lett.B764(2017), pp.11–30.doi:10.1016/j.physletb.2016.11.005. arXiv:1607.06363 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2016.11.005 2017

  12. [12]

    Search for low-mass resonances decaying into two jets and produced in association with a photon using $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

    Morad Aaboud et al. “Search for low-mass resonances decaying into two jets and produced in association with a photon usingppcollisions at √s=13 TeV with the ATLAS detector.” In:Phys. Lett.B795(2019), pp.56–75.doi: 10.1016/j.physletb.2019.03.067. arXiv:1901.10917 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2019.03.067 2019

  13. [13]

    Search for new high-mass phenomena in the dilepton final state using36fb −1 of proton-proton collision data at √s=13 TeV with the ATLAS detector

    Morad Aaboud et al. “Search for new high-mass phenomena in the dilepton final state using36fb −1 of proton-proton collision data at √s=13 TeV with the ATLAS detector.” In:JHEP10(2017), p.182.doi:10.1007/JHEP10(2017)

    work page doi:10.1007/jhep10(2017 2017

  14. [14]

    arXiv:1707.02424 [hep-ex]

    Pith/arXiv arXiv

  15. [15]

    Search for new resonances decaying to a $W$ or $Z$ boson and a Higgs boson in the $\ell^+ \ell^- b\bar b$, $\ell \nu b\bar b$, and $\nu\bar{\nu} b\bar b$ channels with $pp$ collisions at $\sqrt s = 13$ TeV with the ATLAS detector

    Morad Aaboud et al. “Search for new resonances decaying to aWorZboson and a Higgs boson in theℓ +ℓ−b¯b,ℓνb ¯b, andν ¯νb¯bchannels withppcollisions at √s=13 TeV with the ATLAS detector.” In:Phys. Lett.B765(2017), pp.32– 52.doi:10.1016/j.physletb.2016.11.045. arXiv:1607.05621 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2016.11.045 2017

  16. [16]

    Search for pair production of Higgs bosons in the $b\bar{b}b\bar{b}$ final state using proton--proton collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

    Morad Aaboud et al. “Search for pair production of Higgs bosons in theb ¯bb¯b final state using proton–proton collisions at √s=13 TeV with the ATLAS detector.” In:Phys. Rev.D94.5(2016), p.052002.doi:10.1103/PhysRevD.94. 052002. arXiv:1606.04782 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94 2016

  17. [17]

    Searches for heavy $ZZ$ and $ZW$ resonances in the $\ell\ell qq$ and $\nu\nu qq$ final states in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector

    Morad Aaboud et al. “Searches for heavyZZandZWresonances in the ℓℓqqandννqqfinal states inppcollisions at √s=13 TeV with the ATLAS detector.” In:JHEP03(2018), p.009.doi:10.1007/JHEP03(2018)009. arXiv: 1708.09638 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2018)009 2018

  18. [18]

    A search for high-mass resonances decaying to $\tau^{+}\tau^{-}$ in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector

    Georges Aad et al. “A search for high-mass resonances decaying toτ +τ− in ppcollisions at √s=8 TeV with the ATLAS detector.” In:JHEP07(2015), p.157.doi:10.1007/JHEP07(2015)157. arXiv:1502.07177 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2015)157 2015

  19. [19]

    Analysis of events with $b$-jets and a pair of leptons of the same charge in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector

    Georges Aad et al. “Analysis of events withb-jets and a pair of leptons of the same charge inppcollisions at √s=8 TeV with the ATLAS detector.” In: JHEP10(2015), p.150.doi:10.1007/JHEP10(2015)150. arXiv:1504.04605 [hep-ex]. 193 194 bibliography

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2015)150 2015

  20. [20]

    Search for a Charged Higgs Boson Produced in the Vector-boson Fusion Mode with Decay $H^\pm \to W^\pm Z$ using $pp$ Collisions at $\sqrt{s}=8$ TeV with the ATLAS Experiment

    Georges Aad et al. “Search for a Charged Higgs Boson Produced in the Vector-Boson Fusion Mode with DecayH ± →W ±ZusingppCollisions at √s=8 TeV with the ATLAS Experiment.” In:Phys. Rev. Lett.114.23 (2015), p.231801.doi:10.1103/PhysRevLett.114.231801. arXiv:1503.04233 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.114.231801 2015

  21. [22]

    Search for high-mass diboson resonances with boson- tagged jets in proton-proton collisions at √s=8 TeV with the ATLAS de- tector

    Georges Aad et al. “Search for high-mass diboson resonances with boson- tagged jets in proton-proton collisions at √s=8 TeV with the ATLAS de- tector.” In:JHEP12(2015), p.055.doi:10 . 1007 / JHEP12(2015 ) 055. arXiv: 1506.00962 [hep-ex]

    Pith/arXiv arXiv 2015

  22. [23]

    Search for high-mass dilepton resonances in pp collisions at sqrt(s) = 8 TeV with the ATLAS detector

    Georges Aad et al. “Search for high-mass dilepton resonances in pp colli- sions at √s=8 TeV with the ATLAS detector.” In:Phys. Rev.D90.5(2014), p.052005.doi:10.1103/PhysRevD.90.052005. arXiv:1405.4123 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.052005 2014

  23. [24]

    Search for New Phenomena in Dijet Mass and Angular Distributions from $pp$ Collisions at $\sqrt{s}$ = 13 TeV with the ATLAS Detector

    Georges Aad et al. “Search for new phenomena in dijet mass and angular distributions fromppcollisions at √s=13TeV with the ATLAS detector.” In: Phys. Lett.B754(2016), pp.302–322.doi:10.1016/j.physletb.2016.01.032. arXiv:1512.01530 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2016.01.032 2016

  24. [25]

    Search for pair production of a new heavy quark that decays into aWboson and a light quark inppcollisions at √s=8 TeV with the ATLAS detector

    Georges Aad et al. “Search for pair production of a new heavy quark that decays into aWboson and a light quark inppcollisions at √s=8 TeV with the ATLAS detector.” In:Phys. Rev.D92.11(2015), p.112007.doi:10.1103/ PhysRevD.92.112007. arXiv:1509.04261 [hep-ex]

    Pith/arXiv arXiv 2015

  25. [26]

    Search for pair-produced heavy quarks decaying to Wq in the two-lepton channel at sqrt(s) = 7 TeV with the ATLAS detector

    Georges Aad et al. “Search for pair-produced heavy quarks decaying to Wq in the two-lepton channel at √s=7 TeV with the ATLAS detector.” In: Phys. Rev.D86(2012), p.012007.doi:10.1103/PhysRevD.86.012007. arXiv: 1202.3389 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.012007 2012

  26. [27]

    Search for production of $WW/WZ$ resonances decaying to a lepton, neutrino and jets in $pp$ collisions at $\sqrt{s}$ = 8 TeV with the ATLAS detector

    Georges Aad et al. “Search for production ofWW/WZresonances decaying to a lepton, neutrino and jets inppcollisions at √s=8 TeV with the ATLAS detector.” In:Eur. Phys. J.C75.5(2015). [Erratum: Eur. Phys. J.C75,370(2015)], p.209.doi:10.1140/epjc/s10052-015-3593-4,10.1140/epjc/s10052-015- 3425-6. arXiv:1503.04677 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-015-3593-4 2015

  27. [28]

    Search for vector-likeBquarks in events with one iso- lated lepton, missing transverse momentum and jets at √s=8TeV with the ATLAS detector

    Georges Aad et al. “Search for vector-likeBquarks in events with one iso- lated lepton, missing transverse momentum and jets at √s=8TeV with the ATLAS detector.” In:Phys. Rev.D91.11(2015), p.112011.doi:10.1103/ PhysRevD.91.112011. arXiv:1503.05425 [hep-ex]

    Pith/arXiv arXiv 2015

  28. [29]

    Search for WZ resonances in the fully leptonic channel using pp collisions at sqrt(s) = 8 TeV with the ATLAS detector

    Georges Aad et al. “Search for WZ resonances in the fully leptonic channel using pp collisions at sqrt(s) =8TeV with the ATLAS detector.” In:Phys. Lett.B737(2014), pp.223–243.doi:10.1016/j.physletb.2014.08.039. arXiv: 1406.4456 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2014.08.039 2014

  29. [30]

    Search for Fermion-Pair Decays $Q\Qbar \to (\t\Wmp)(\tbar\Wpm)$ in Same-Charge Dilepton Events

    T. Aaltonen et al. “Search for New Bottomlike Quark Pair DecaysQ ¯Q→ (tW ∓)(¯tW±)in Same-Charge Dilepton Events.” In:Phys. Rev. Lett.104(2010), p.091801.doi:10.1103/PhysRevLett.104.091801. arXiv:0912.1057 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.104.091801 2010

  30. [31]

    Search for New Particles Leading to Z+jets Final States in $p\bar{p}$ Collisions at $\sqrt{s}=1.96$ TeV

    T. Aaltonen et al. “Search for New Particles Leading toZ+jets Final States inp ¯pCollisions at √s=1.96-TeV.” In:Phys. Rev.D76(2007), p.072006.doi: 10.1103/PhysRevD.76.072006. arXiv:0706.3264 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.76.072006 2007

  31. [32]

    The minimal com- posite Higgs model

    Kaustubh Agashe, Roberto Contino, and Alex Pomarol. “The minimal com- posite Higgs model.” In:Nuclear Physics B719.1(2005), pp.165–187

    2005

  32. [33]

    A Custodial symmetry forZb ¯b

    Kaustubh Agashe et al. “A Custodial symmetry forZb ¯b.” In:Phys. Lett.B641 (2006), pp.62–66.doi:10 . 1016 / j . physletb . 2006 . 08 . 005. arXiv:hep - ph/0605341 [hep-ph]. bibliography 195

    arXiv 2006

  33. [34]

    A custodial symmetry for Zbb

    Kaustubh Agashe et al. “A custodial symmetry for Zbb.” In:Physics Letters B641.1(2006), pp.62–66

    2006

  34. [35]

    RS1, custodial isospin and precision tests

    Kaustubh Agashe et al. “RS1, custodial isospin and precision tests.” In:JHEP 08(2003), p.050.doi:10 . 1088 / 1126 - 6708 / 2003 / 08 / 050. arXiv:hep - ph / 0308036 [hep-ph]

    2003

  35. [36]

    Clockwork Goldstone Bosons

    Aqeel Ahmed and Barry M. Dillon. “Clockwork Goldstone Bosons.” In:Phys. Rev.D96.11(2017), p.115031.doi:10 . 1103 / PhysRevD . 96 . 115031. arXiv: 1612.04011 [hep-ph]

    Pith/arXiv arXiv 2017

  36. [37]

    CRC Press,2003

    Ian JR Aitchison and Anthony JG Hey.Gauge theories in particle physics, Vol- ume II: QCD and the Electroweak Theory. CRC Press,2003

    2003

  37. [38]

    A geometric formulation of Higgs effective field theory: measuring the curvature of scalar field space

    Rodrigo Alonso, Elizabeth E Jenkins, and Aneesh V Manohar. “A geometric formulation of Higgs effective field theory: measuring the curvature of scalar field space.” In:Physics Letters B754(2016), pp.335–342

    2016

  38. [39]

    Geometry of the scalar sector

    Rodrigo Alonso, Elizabeth E Jenkins, and Aneesh V Manohar. “Geometry of the scalar sector.” In:Journal of High Energy Physics2016.8(2016), p.101

    2016

  39. [41]

    Realistic Composite Higgs Models

    Charalampos Anastasiou, Elisabetta Furlan, and Jose Santiago. “Realistic Composite Higgs Models.” In:Phys. Rev.D79(2009), p.075003.doi:10.1103/ PhysRevD.79.075003. arXiv:0901.2117 [hep-ph]

    Pith/arXiv arXiv 2009

  40. [42]

    Challenging weak scale supersym- metry at colliders

    Greg W. Anderson and Diego J. Castano. “Challenging weak scale supersym- metry at colliders.” In:Phys. Rev.D53(1996), pp.2403–2410.doi:10.1103/ PhysRevD.53.2403. arXiv:hep-ph/9509212 [hep-ph]

    Pith/arXiv arXiv 1996

  41. [43]

    Measures of fine tuning

    Greg W. Anderson and Diego J. Castano. “Measures of fine tuning.” In:Phys. Lett.B347(1995), pp.300–308.doi:10.1016/0370- 2693(95)00051- L. arXiv: hep-ph/9409419 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0370- 1995

  42. [44]

    Naturalness and superpartner masses or when to give up on weak scale supersymmetry

    Greg W. Anderson and Diego J. Castano. “Naturalness and superpartner masses or when to give up on weak scale supersymmetry.” In:Phys. Rev. D52(1995), pp.1693–1700.doi:10 . 1103 / PhysRevD . 52 . 1693. arXiv:hep - ph/9412322 [hep-ph]

    arXiv 1995

  43. [45]

    Naturalness Lowers the Upper Bound on the Lightest Higgs Boson Mass in Supersymmetry

    Greg W. Anderson, Diego J. Castano, and Antonio Riotto. “Naturalness low- ers the upper bound on the lightest Higgs boson mass in supersymmetry.” In:Phys. Rev.D55(1997), pp.2950–2954.doi:10.1103/PhysRevD.55.2950. arXiv:hep-ph/9609463 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.55.2950 1997

  44. [46]

    Vector-like top/bottom quark partners and Higgs physics at the LHC

    Andrei Angelescu, Abdelhak Djouadi, and Grégory Moreau. “Vector-like top/bottom quark partners and Higgs physics at the LHC.” In:Eur. Phys. J.C76.2(2016), p.99.doi:10 . 1140 / epjc / s10052 - 016 - 3950 - y. arXiv: 1510.07527 [hep-ph]

    Pith/arXiv arXiv 2016

  45. [47]

    The Littlest Higgs

    N. Arkani-Hamed et al. “The Littlest Higgs.” In:JHEP07(2002), p.034.doi: 10.1088/1126-6708/2002/07/034. arXiv:hep-ph/0206021 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2002/07/034 2002

  46. [48]

    Electroweak symmetry breaking from dimensional deconstruction

    Nima Arkani-Hamed, Andrew G Cohen, and Howard Georgi. “Electroweak symmetry breaking from dimensional deconstruction.” In:Physics Letters B 513.1-2(2001), pp.232–240

    2001

  47. [49]

    (De)constructing dimensions

    Nima Arkani-Hamed, Andrew G. Cohen, and Howard Georgi. “(De)constructing dimensions.” In:Phys. Rev. Lett.86(2001), pp.4757–4761.doi:10 . 1103 / PhysRevLett.86.4757. arXiv:hep-th/0104005 [hep-th]

    Pith/arXiv arXiv 2001

  48. [50]

    Twisted su- persymmetry and the topology of theory space

    Nima Arkani-Hamed, Howard Georgi, and Andrew G Cohen. “Twisted su- persymmetry and the topology of theory space.” In:Journal of High Energy Physics2002.07(2002), p.020. 196 bibliography

    2002

  49. [51]

    Effective field theory for massive gravitons and gravity in theory space

    Nima Arkani-Hamed, Howard Georgi, and Matthew D Schwartz. “Effective field theory for massive gravitons and gravity in theory space.” In:Annals of Physics305.2(2003), pp.96–118

    2003

  50. [52]

    The minimal moose for a little Higgs

    Nima Arkani-Hamed et al. “The minimal moose for a little Higgs.” In:Jour- nal of High Energy Physics2002.08(2002), p.021

    2002

  51. [53]

    Andreas Arvanitogeorgos and Andreas Arvanitoge ¯orgos.An introduction to Lie groups and the geometry of homogeneous spaces. Vol.22. American Mathe- matical Soc.,2003

    2003

  52. [54]

    A global fit of the MSSM with GAMBIT

    Peter Athron et al. “A global fit of the MSSM with GAMBIT.” In:Eur. Phys. J.C77.12(2017), p.879.doi:10 . 1140 / epjc / s10052 - 017 - 5196 - 8. arXiv: 1705.07917 [hep-ph]

    arXiv 2017

  53. [55]

    GAMBIT: The Global and Modular Beyond-the-Standard- Model Inference Tool

    Peter Athron et al. “GAMBIT: The Global and Modular Beyond-the-Standard- Model Inference Tool.” In:Eur. Phys. J.C77.11(2017). [Addendum: Eur. Phys. J.C78,no.2,98(2018)], p.784.doi:10 . 1140 / epjc / s10052 - 017 - 5513 - 2 , 10 . 1140/epjc/s10052-017-5321-8. arXiv:1705.07908 [hep-ph]

    arXiv 2017

  54. [56]

    Global fits of GUT-scale SUSY models with GAMBIT

    Peter Athron et al. “Global fits of GUT-scale SUSY models with GAMBIT.” In:Eur. Phys. J.C77.12(2017), p.824.doi:10.1140/epjc/s10052-017-5167-0. arXiv:1705.07935 [hep-ph]

    work page doi:10.1140/epjc/s10052-017-5167-0 2017

  55. [57]

    New measure of fine tuning

    Peter Athron and David J Miller. “New measure of fine tuning.” In:Physical Review D76.7(2007), p.075010

    2007

  56. [59]

    Exceptional composite dark matter

    Guillermo Ballesteros, Adrián Carmona, and Mikael Chala. “Exceptional composite dark matter.” In:The European Physical Journal C77.7(2017), p.468

    2017

  57. [60]

    Nonlinear realization and hidden local symmetries

    Masako Bando, Taichiro Kugo, and Koichi Yamawaki. “Nonlinear realization and hidden local symmetries.” In:Physics Reports164.4-5(1988), pp.217–314

    1988

  58. [61]

    Improving Fine-tuning in Composite Higgs Models

    Avik Banerjee, Gautam Bhattacharyya, and Tirtha Sankar Ray. “Improving Fine-tuning in Composite Higgs Models.” In:Phys. Rev.D96.3(2017), p.035040. doi:10.1103/PhysRevD.96.035040. arXiv:1703.08011 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.96.035040 2017

  59. [62]

    Upper bounds on supersym- metric particle masses

    Riccardo Barbieri and Gian Francesco Giudice. “Upper bounds on supersym- metric particle masses.” In:Nuclear Physics B306.1(1988), pp.63–76

    1988

  60. [63]

    A125GeV composite Higgs boson versus flavour and electroweak precision tests

    Riccardo Barbieri et al. “A125GeV composite Higgs boson versus flavour and electroweak precision tests.” In:JHEP05(2013), p.069.doi:10.1007/ JHEP05(2013)069. arXiv:1211.5085 [hep-ph]

    Pith/arXiv arXiv 2013

  61. [64]

    Chiral dynamics and heavy quark symmetry in a solvable toy field-theoretic model

    William A Bardeen and Christopher T Hill. “Chiral dynamics and heavy quark symmetry in a solvable toy field-theoretic model.” In:Physical Review D49.1(1994), p.409

    1994

  62. [65]

    UV descriptions of composite Higgs models without elementary scalars

    James Barnard, Tony Gherghetta, and Tirtha Sankar Ray. “UV descriptions of composite Higgs models without elementary scalars.” In:JHEP02(2014), p.002.doi:10.1007/JHEP02(2014)002. arXiv:1311.6562 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2014)002 2014

  63. [66]

    Collider constraints on tuning in composite Higgs models

    James Barnard and Martin White. “Collider constraints on tuning in compos- ite Higgs models.” In:JHEP10(2015), p.072.doi:10.1007/JHEP10(2015)072. arXiv:1507.02332 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2015)072 2015

  64. [67]

    Constraining fine tuning in Composite Higgs Models with partially composite leptons

    James Barnard et al. “Constraining fine tuning in Composite Higgs Models with partially composite leptons.” In:JHEP09(2017), p.049.doi:10.1007/ JHEP09(2017)049. arXiv:1703.07653 [hep-ph]

    Pith/arXiv arXiv 2017

  65. [68]

    Radiative corrections to the composite Higgs mass from a gluon partner

    James Barnard et al. “Radiative corrections to the composite Higgs mass from a gluon partner.” In:JHEP10(2013), p.055.doi:10.1007/JHEP10(2013)

    work page doi:10.1007/jhep10(2013 2013

  66. [69]

    bibliography 197

    arXiv:1307.4778 [hep-ph]. bibliography 197

    Pith/arXiv arXiv

  67. [70]

    Composite Higgses

    Brando Bellazzini, Csaba Csáki, and Javi Serra. “Composite Higgses.” In: Eur. Phys. J.C74.5(2014), p.2766.doi:10.1140/epjc/s10052- 014- 2766- x. arXiv:1401.2457 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052- 2014

  68. [71]

    Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that

    Jeremy Bernstein. “Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that.” In:Rev. Mod. Phys.46(1Jan.1974), pp.7–48. doi:10.1103/RevModPhys.46.7.url:https://link.aps.org/doi/10.1103/ RevModPhys.46.7

    work page doi:10.1103/revmodphys.46.7.url:https://link.aps.org/doi/10.1103/ 1974

  69. [72]

    On composite two Higgs doublet models

    Enrico Bertuzzo et al. “On composite two Higgs doublet models.” In:Journal of High Energy Physics2013.5(May2013), p.153.issn:1029-8479.doi:10 . 1007/JHEP05(2013)153.url:https://doi.org/10.1007/JHEP05(2013)153

    work page doi:10.1007/jhep05(2013)153 2013

  70. [73]

    Quantum infrared instabilities of gauge and gravity coupled higgs fields

    Srijit Bhattacharjee. “Quantum infrared instabilities of gauge and gravity coupled higgs fields.” In: (2013).url:http://hdl.handle.net/10603/37546

    2013

  71. [74]

    Marco Billo.Introduzione alla Teoria dei Gruppi.2005.url:http://personalp ages.to.infn.it/~billo/didatt/gruppi/liegroups.pdf

    2005

  72. [75]

    Spontaneous symmetry breaking in strong and electroweak interactions

    Tomas Brauner. “Spontaneous symmetry breaking in strong and electroweak interactions.” In:arXiv preprint hep-ph/0606300(2006)

    Pith/arXiv arXiv 2006

  73. [76]

    Precision Correc- tions to Fine Tuning in SUSY

    Matthew R. Buckley, Angelo Monteux, and David Shih. “Precision Correc- tions to Fine Tuning in SUSY.” In:JHEP06(2017), p.103.doi:10 . 1007 / JHEP06(2017)103. arXiv:1611.05873 [hep-ph]

    Pith/arXiv arXiv 2017

  74. [77]

    Goldstone and Pseudo-Goldstone Bosons in Nuclear, Particle and Condensed-Matter Physics

    C. P . Burgess. “Goldstone and pseudoGoldstone bosons in nuclear, particle and condensed matter physics.” In:Phys. Rept.330(2000), pp.193–261.doi: 10.1016/S0370-1573(99)00111-8. arXiv:hep-th/9808176 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-1573(99)00111-8 2000

  75. [78]

    The dynamics of Composite Higgses

    Giacomo Cacciapaglia. “The dynamics of Composite Higgses.” In:J. Phys. Conf. Ser.623.1(2015), p.012006.doi:10.1088/1742-6596/623/1/012006

    work page doi:10.1088/1742-6596/623/1/012006 2015

  76. [79]

    Fundamental Composite (Goldstone) Higgs Dynamics

    Giacomo Cacciapaglia and Francesco Sannino. “Fundamental Composite (Gold- stone) Higgs Dynamics.” In:JHEP04(2014), p.111.doi:10.1007/JHEP04(2 014)111. arXiv:1402.0233 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep04(2 2014

  77. [80]

    Fundamental composite (Gold- stone) Higgs dynamics

    Giacomo Cacciapaglia and Francesco Sannino. “Fundamental composite (Gold- stone) Higgs dynamics.” In:Journal of High Energy Physics2014.4(2014), pp.1–26

    2014

  78. [81]

    Structure of Phenomenological Lagrangians. II

    Curtis G. Callan et al. “Structure of Phenomenological Lagrangians. II.” In: Phys. Rev.177(5Jan.1969), pp.2247–2250.doi:10.1103/PhysRev.177.2247. url:https://link.aps.org/doi/10.1103/PhysRev.177.2247

    work page doi:10.1103/physrev.177.2247 1969

  79. [82]

    Triviality pursuit: Can elementary scalar particles ex- ist?

    David J.E. Callaway. “Triviality pursuit: Can elementary scalar particles ex- ist?” In:Physics Reports167.5(1988), pp.241–320.issn:0370-1573.doi:https: //doi.org/10.1016/0370-1573(88)90008-7.url:http://www.sciencedire ct.com/science/article/pii/0370157388900087

    work page doi:10.1016/0370-1573(88)90008-7.url:http://www.sciencedire 1988

  80. [83]

    UV Completions of Composite Higgs Models with Partial Compositeness

    Francesco Caracciolo, Alberto Parolini, and Marco Serone. “UV Completions of Composite Higgs Models with Partial Compositeness.” In:JHEP02(2013), p.066.doi:10.1007/JHEP02(2013)066. arXiv:1211.7290 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2013)066 2013

Showing first 80 references.