Strong interactions in air showers

We study the role new gauge interactions in extensions of the standard model play in air showers initiated by ultrahigh-energy cosmic rays. Hadron-hadron events remain dominated by quantum chromodynamics, while projectiles and/or targets from beyond the standard model permit us to see qualitative differences arising due to the new interactions.


I. INTRODUCTION
The present study is concerned with the quest after the origin of mass in our universe and the phenomenology of ultra-high-energy cosmic rays: On one hand, the huge energies available in cosmic rays might give evidence for dynamical electroweak symmetry breaking. On the other hand, some observed cosmic-ray characteristics might be natural in the framework of dynamical electroweak symmetry breaking.
The standard model answers the question after the origin of mass of the weak gauge bosons and the standard model fermions by the elementary scalar Higgs mechanism. We have, however, never seen any elementary scalar particle in nature and the elementary Higgs remains only a prediction of the standard model. Furthermore, all scalar degrees of freedom that have been encountered so far turned out to be composites. In particular this is also true for the scalar of Ginzburg and Landau [1], which describes the generation of the photon mass (the inverse London screening length [2]) in superconductivity in a gauge invariant manner and thus the Meissner-Ochsenfeld [3] effect. It is the Abelian prototype for the standard model Higgs, but was identified by Bardeen, Cooper, and Schrieffer (BCS) [4] to be a composite of two electrons, the Cooper pairs. For more guidance consider the standard model without the Higgs sector. There the chiral symmetry breaking in quantum chromodynamics (QCD) would break the electroweak symmetry dynamically. (In order to avoid the subtleties arising from the alignment of the vacuum, for this gedanken experiment, we constrain ourselves to one generation of fermions, especially to only two quarks.) With the scale of quantum chromodynamics kept fixed this would result in massive weak gauge bosons, which are roughly 1000 times to light. The pions of quantum chromodynamics would become the longitudinal degrees of freedom of the weak gauge bosons, and would, thus, not appear in the spectrum. This is suggestive of employing an additional sector of electroweakly charged fermions that interact under a new force that grows strong at low energy scales, leads to chiral symmetry breaking among the fermions, and, simultaneously, breaks the electroweak symmetry. This approach is known as technicolour [5], which is to the elementary Higgs mechanism what the BCS theory is to the description by Ginzburg and Landau. As will be explained in detail in Sect. II, the difference between theories of dynamical symmetry breaking and those with some kind of elementary Higgs mechanism that will be crucial vis-à-vis cosmic ray physics, is the presence of additional gauge interactions. In that section, we are also going to discuss several different implementations of the dynamical elecroweak symmetry breaking paradigm in the context of cosmic-ray physics.
The dynamic electroweak symmetry breaking mechanism can manifest itself directly only at LHC energies or above. Hence, if we would like to see its traces we have to look at features at the high energy end of the cosmic ray spectrum, i.e., at cosmic-ray energies of at least 10 18 eV. The most prominent feature below this energy range is the "knee", an abrupt steepening of the spectrum between 10 15 and 10 16 eV. (Here and in the following, see, for example, Fig. 24.8 in [6].) Under the assumption that all the cosmic rays up to 10 18 eV have their origin inside the Milky Way, the knee could be the manifestation of the fact that all possible acceleration mechanisms within our galaxy have reached their maximal attainable energy.
From 10 18 eV onward cosmic rays are expected to be of extragalactic origin. In an interval of one order of magnitude around 10 19 eV the spectrum is flatter than after the knee. The place where this plateau begins is known as the "ankle." In case one merely extrapolates cosmicray-air cross sections straightforwardly to higher energies, the ankle structure must be explained by a changing composition of the cosmic rays. Under this assumption, the direction of change (to lighter or to heavier particles), however, inferred by the HiRes [7] and Auger [8] experiments, respectively, from the depth at which the shower maximum occurs in the atmosphere, do not agree. Conclusions about the composition change considerably once the assumption of the validity of a straightforward extrapolation of the scattering-cross sections is dropped [6]. Another explanation attempt for the ankle feature is a higher energy, e.g., cosmological flux that overtakes one of lower energy, e.g., a galactic flux [6], which, however, also would not overcome the composition issue.
At energies beyond 5 × 10 19 eV the spectrum should decay rapidly if the cosmic rays were of extragalactic origin because, according to Greisen,Zatsepin,and Kuzmin 2 (GZK) [9] at these energies the electrically charged nuclei should lose a considerable fraction of their energy by pion production after scattering with photons from the cosmic microwave background (CMB) when propagating over extragalactic distances. (The analogous production of electron-positron pairs sets in at lower energies around 10 17 eV, but is much less efficient in dispersing the cosmic ray energy than the heavier pions.) Trans-GZK events have been observed [10][11][12]. The hypothesis of an isotropic distribution of the highest-energy cosmic rays is consistent with observations, and a correlation with the position of known active galactic nuclei, which would give an indication for their origin, is not confirmed [13]. The AGASA experiment does not confirm the GZK suppression [11], whereas other experiments see some suppression [12][13][14]. Barring instrument errors, the explanation of trans-GKZ events requires either so far unknown galactic sources or particles that are not affected by the GKZ cutoff. These would have to be sufficiently weakly interacting particles from cosmological sources, which still have a cross section that allows them to be accelerated to the required energies and to interact with the atmosphere in order to be detected.
Taking stock, the three kinds of mechanisms brought forward to explain the ankle feature are changes in the cosmic-ray-air cross-section, the composition of the cosmic ray, and/or changes in their flux. In the following, we are going to investigate, which of these mechanisms are expected naturally in the presence of dynamical electroweak symmetry breaking. Moreover, we will study, whether the GZK cutoff would still have any bearing in this context. The paper is organised as follows. In section II we recall those details about dynamical electroweak symmetry breaking we need in order to proceed with our analysis. Thereafter, in section III we describe in detail the effect of dynamical electroweak symmetry breaking expected for the interaction of hadronic cosmic-ray particles with air. In subsection III B, we also allow for a component of the cosmic ray flux that is not made from standard model particles, but from particles that are associated with dynamical electroweak symmetry breaking. In the second part of that subsection, we extend our considerations also to cases, where the air shower is initiated by such a particle existing in a halo around the Earth and which is struck by a cosmic-ray particle. Finally, in Sect. IV we summarise our findings.

II. DYNAMICAL ELECTROWEAK SYMMETRY BREAKING
Let us start with technicolour. Fortunately, available data allows us to narrow down the set of viable technicolour models, which, otherwise, would contain a priori any theory exhibiting fermions with spontaneously broken chiral symmetry. For one, the space for electroweakly charged matter beyond what is already contained in the standard model is limited. (Still, sixth standard model generations need not be less favoured by data than three [15].) Technicolour alone provides the masses of the weak gauge bosons. It does not explain where the masses of the standard model fermions ψ come from. In the standard model this is accomplished by the Yukawa couplings to the Higgs H, ∼ ⟨H⟩ψψ. In the technicolour context the chiral condensate ⟨QQ⟩ of the techniquarks Q replaces the non-zero Higgs vacuum, and we could consider the operator ∼ ⟨QQ⟩ψψ. It is not renormalisable and in order to have a renormalisable theory it must be induced at energies above the technicolour chiral symmetry breaking scale. From Fermi's theory [16] and its UV completion in form of the , we know that these operators arise from the exchange of massive gauge bosons that acquire their mass at a higher energy scale (the electroweak scale) than the process under consideration (e.g., β decay). Applying this approach in the technicolour context is known as extended technicolour [18]. In those models, in general, the required four-fermion operator ∼QQψψ Λ 2 ETC , where Λ ETC denotes the scale at which the gauge bosons acquire their mass, are accompanied by two other types, ∼QQQ ′ Q ′ Λ 2 ETC and ∼ψψψ ′ ψ ′ Λ 2 ETC . The former, for example, contribute to the masses of potentially present pseudo-Nambu-Goldstone modes. The latter induce flavour-changing neutral currents, which are known to be very small in nature. They cannot be reduced enough by simply choosing the scale Λ ETC sufficiently big, as otherwise the top quark mass cannot be reached. (In view of the flavour-changing neutral currents, which have to be avoided, in the technicolour context the masses of all the standard model fermions apart from the top quark are natural, while in the context of the standard model order one Yukawa coupling would appear natural and, as a consequence, only the top mass.) In the standard model the flavour-changing neutral currents are suppressed by the Glashow-Iliopoulos-Maiani mechanism [19]. In connection to extended technicolour there are several ways of suppressing the flavour-changing neutral currents, some of which will be mentioned below and which allow for selecting viable models for dynamical electroweak symmetry breaking.
The details of the construction depend crucially on whether the techniquarks carry ordinary (QCD) colour or not. In case they do, we can unify the generation index of the standard model fermions with the technicolour index, G ETC ⊃ G TC × G gen. For three generations and the minimal number of two technicolours this could be realised as SU (5) ⊃ SU (3) gen. × SU (2) TC [20]. The SU (5) is broken down sequentially to SU (3) gen. × SU (2) TC at three different scales Λ ETC,1 > Λ ETC,2 > Λ ETC,3 to explain the generational mass hierarchy of the standard model fermions. Here the mass operator of the j-th generation contains the j-th scale, ∼QQψ j ψ j Λ 2 ETC,j . Flavour-changing-neutral-current transitions always oc-3 cur between generations, and a detailed analysis of the operators by which they are induced shows that they are suppressed relative to the above naïve discussion by powers of Λ ETC< Λ ETC> [21], where Λ ETC< (Λ ETC> ) is the smaller (bigger) of the two scales.
Technicolour models in which the techniquarks do not carry ordinary colour are more frugal when it comes to adding matter: There are (up to) a factor of three less additional fermion species from the viewpoint of the electroweak interactions. When it comes to the generation of the quark masses, some of the gauge bosons of the extended-technicolour sector must carry quantum chromodynamics colour. Thus, in this case SU (3) QCD must be embedded in the extended technicolour gauge group [22], G ETC ⊃ SU (3) QCD × G TC × G gen. (One can even consider models in which the extended technicolour sector does not commute with the weak interactions, which is known as noncommuting extended technicolour [23].) In both of the above types of models an enhancement of the techniquark condensate alleviates the tension between the small flavour-changing neutral currents and the comparatively large top mass. This can be accomplished in so-called walking theories. Because of a sufficient content of screening fermions, they almost evolve into an infrared fixed point [24]. It is only missed because the critical coupling for chiral symmetry breaking to occur is reached just before the would-be Caswell-Banks-Zaks fixed point [25]. As a consequence, the β function for the gauge coupling at relatively large coupling is close to zero, and thus, the gauge coupling can be strong already for a large range of scales before the chiral condensate finally forms. This entails a large anomalous dimension γ for the quark mass operator, which leads to a sizeable renormalisation of the quark condensate: Naïvely, the condensate ⟨QQ⟩ is expected to be of the order of Λ 3 TC , where Λ TC stands for the fundamental scale of the technicolour sector, i.e., the analogue of Λ QCD in quantum chromodynamics. The mass operator QQψψ Λ ETC , however, is generated at the scale Λ ETC , whereas the condensate is formed at the smaller scale Λ TC . As a consequence, the renormalisation of the condensate between these two scales must be taken into account, ⟨QQ⟩ ETC = ⟨QQ⟩ TC exp ∫ Λ ETC Λ TC dµ µ γ(µ). In a walking theory the anomalous dimension γ is approximately constant between these scales and the condensate is enhanced by a factor of (Λ ETC Λ TC ) γ .
From what was said above, walking theories require a sufficient amount of matter, while electroweak precision data prefers technicolour sectors with a small amount of matter. Walking with a relatively small amount of matter can be achieved by choosing the techniquarks in a different than the fundamental representation of the technicolour gauge group [27,28]. (Gauging only some, i.e., at least two, of the techniquarks under the electroweak interactions permits to further reduce the contribution to the oblique parameters [29], while allowing for walking dynamics. This is known as partially gauged technicolour [28].) Next to the additional gauge interactions, FIG. 1: β function of the gauge coupling α = g 2 (4π) as a function of the gauge coupling and gauge coupling as a function of the scale µ, for different amounts of matter (here, the number of fermion flavours) in the theory. A) Little matter, like in quantum chromodynamics; B) appearance of a perturbative Caswell-Banks-Zaks fixed point [25]; C) shape due to freezing out of fermions due to chiral symmetry breaking, as the would-be fixed point is situated at too large couplings; D) actual fixed point at small enough coupling in the presence of sufficient matter (inside the conformal window); E) quasiconformal case; F) asymptotic freedom is lost for too many flavours. The dashed line in the plot on the left-hand side represents the critical values of the gauge coupling for chiral symmetry breaking. (Taken from [26].) the additional bound states of the technicolour sector can become relevant for the physics of ultra-high-energy cosmic rays. The generically lightest bound states are the technipions. In models with minimal flavour symmetry there are only three pions, which become the longitudinal degrees of freedom of the weak gauge bosons. For larger chiral symmetries there are more pions, which from electroweak corrections obtain masses of the order of the Z mass [30]. In the context of the present study, models based on matter in higher-dimensional representations posses additional relevant features. (See,e.g.,Sect. III B 1.) Taking stock, for the present project, a relevant aspect is that in technicolour models with techniquarks that carry ordinary colour the technicolour sector couples to standard matter already through quantum chromodynamics, whereas in technicolour models with all fermions singlets of ordinary colour it couples to standard matter only through the extended-technicolour sector.
There are also other realisations of the dynamical electroweak symmetry breaking paradigm. Quite similar to technicolour is topcolour [31], where the condensation of the top quark due to a strong gauge interaction is responsible for the breaking of the electroweak symmetry. (The idea that the condensation of the top could be responsible for electroweak symmetry breaking [32] predates its gauge completion.) In order to implement the topcolour mechanism the SU(3) of quantum chromodynamics is embedded (at least) in SU (3)×SU (3), where the first two quark generations are charged under one of the SU(3)-s and the third generation under the second. {One could also think of using an additional U(1) [33].} At a scale M , this group is then broken down to the diagonal subgroup, which represents the SU(3) of quantum chromodynamics. This breaking induces a four-fermion interaction for the top quark. Once the coupling of this interaction reaches its critical value, the condensation of the top is triggered. For small M ≈ 1TeV the top quark would have to have a mass of about 600GeV. Only for M ≈ 10 15 GeV it may be as light as 160GeV [34]. Lowscale topcolour (M ⋘ 10 15 TeV) with the physical top mass can be achieved by combining it with a technicolour sector into topcolour assisted technicolour [31,35]. (This is a straightforward combination in the sense that in technicolour the extended technicolour sector generically induces all kinds of four-fermion interactions. Among them is also a four-top interaction. If it has the appropriate coupling strength both a technicolour and a top condensate can form.) Alternatively, by adding more matter a top seesaw [36] can be implemented to achieve a light enough top quark.
Furthermore, there is a class of models in which chiral and electroweak symmetry breaking do not occur simultaneously, but where the emergent Nambu-Goldstone fields supply the Higgs sector that is required for breaking the electroweak symmetry. This pseudo-Nambu-Goldstone Higgs has a compositeness scale that is decoupled from the electroweak symmetry breaking scale by virtue of the Goldstone theorem [37]. By arguments of naturality it may, however, not be too large. Subsequently, in these approaches an appropriate Higgs potential must be generated for the Nambu-Goldstone modes, such that electroweak symmetry breaking can occur. One type of these approaches is known as composite Higgs models [38,39]. (Nowadays, they are frequently analysed on AdS 5 [39], as are sometimes technicolour models [40].) These models are characterised by a vacuum misalignment parameter ξ = v 2 f 2 , where v is the analogue of the Higgs condensate and f represents the pion decay constant. ξ → 1 is commonly called the technicolour limit, while ξ → 0 is referred to a standard model limit, where the compositness scale is much larger than the electroweak scale. Another set of these models is known as Little Higgs [41], in which particular care is taken to eliminate radiative corrections to the Higgs mass appearing at one loop, by mimicking the enlarged Higgs sector of the minimal supersymmetric standard model. They feature yet more gauge bosons needed for generating the Higgs potential, which, in the lightest case, are heavier than 2TeV [42].

III. COSMIC-RAY-AIR INTERACTION
In the energy range of importance for dynamical electroweak symmetry breaking the cosmic rays are detected by observing the air showers they initiate. We must, hence, identify the principal contributions to the cross section of cosmic-ray with atmospheric particles. Let us start with hadronic cosmic-ray particles. (We will drop this assumption in Sect. III B.) Within the standard model their interaction with the atmosphere is dominated by quantum chromodynamics. Hence, the dominant contribution to the scattering by quantum chromodynamics sets the benchmark for all process arising from extensions of the standard model.
Since forward particles carry the largest fraction of the energy they dictate the evolution of the air shower [43]. At these large values of Feynman x F (the longitudinal momentum fraction) the partons in the incident cosmicray particle (the "projectile") are probed in the dilute region whereas the atmospheric particle (the "target") is practically at rest and appears dense due to the vast open phase space for gluons. Its parton distribution is probed at a small momentum fraction x. The dominant scattering process for the production of forward particles that drive the evolution of the shower is the liberation of quarks from the cosmic-ray particle by scattering off the strong gluon fields in the atmospheric particle. The quark essentially punches through the atmospheric particle and receives some transverse momentum that is still negligible compared to its longitudinal momentum, which is why one treats the process in the eikonal approximation. The gluon fields arise because at small momentum fraction x the emission of additional gluons is enhanced by large logarithms ln(1 x), the measure for the large available phase space, such that α QCD ln(1 x) is no longer perturbatively small-despite of the gauge coupling α QCD being small at large virtualities due to asymptotic freedom-and the corresponding series must be resummed to all orders in α QCD ln(1 x). (See Fig. 2.) This gives rise to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equations [44]. They predict a rapid growth of the gluon density-accompanied by a corresponding growth of the cross section-at small momentum fractions x or, equivalently, large energies at fixed virtuality. This rise has been observed at HERA to be ∝ x −0.3 at virtualities of more than (2GeV) 2 [45]. The power-law behaviour can be reproduced by the leading-order BFKL equation. There, however, the exponent comes out too large. Incorporating the next-to-leading order [46], which resums terms of order α QCD × [α QCD ln(1 x)] n leads to a growth rate that is more consistent with experiment [47]. In this context, an important aspect of orders beyond the leading is the running of the gauge coupling. Including a running coupling at the level of the leading order BFKL kernel also reduces the predicted growth rate. Be it leading order, next-to-leading order, fixed or running coupling, in the theoretical description laid out so far, the gluon density would keep growing ad infinitum for In hadron-hadron scattering in ultra-high-energy cosmic rays the leading parton from the projectile (top box, red, valence quarks shown, solid lines) probes the target (bottom box, blue) at very small momentum fraction. Therefore, the cross section is increased by the emission of additional gluons (curly lines, middle box, green), which is enhanced due to the large available phase space for this process.
decreasing gluon density. At variance to the Dokshitzer-Gribov-Altarelli-Parisi (DGLAP) evolution [48], which resums large logarithms ln Q 2 of virtuality, and where the transverse size of the gluons, i.e., ∼ Q −2 , becomes smaller faster than their number density increases, in the case of decreasing rapidity y = ln(1 x) the transverse size is constant. As a consequence, they must start overlapping at some point and non-linear effects must become important. In other words, in magnitude the loss term catches up with the gain term. (See Fig. 3.) Furthermore, a ceaselessly growing gluon density would lead to a violation of the Froissart bound [51] for the cross section and hence of unitarity. As a consequence, the increase of the gluon density should level off when it is of the order α −1 QCD [52]. Coming back to the picture of overlapping gluons, for a given rapidity y there must be a virtuality Q 2 QCD (y) below which the hadron/nucleus at a given impact parameter appears essentially black; the gluon density for smaller virtualities is saturated and cannot grow any further. The quantity Q 2 QCD (y) is known as saturation momentum, and can be interpreted as the density of gluons per unit of rapidity and per transverse area at saturation. Staying within the present overlap picture, the most parsimonious way of estimating it is to fix it at the point where the linear evolution equations would predict the scattering of a colour dipole of the size Q −2 QCD to be unity. This bound can be studied in much greater detail, e.g., in the framework of the Balitsky-Kovchegov (BK) equation [53], which also leads to the identification of geometric scaling at HERA [54] with travelling waves [55] in the BK approach. In fact, geometric scaling means that because of saturation the cross section for deep inelastic scattering, which, in general, can be a function independently of the virtuality Q 2 and the rapidity y becomes a function only of the ratio of the virtuality and the saturation momentum, Q 2 Q 2 QCD (y), the latter of which is a function of the rapidity y. As explained above, the fast quark punches through the atmospheric nucleus at a given impact parameter. Hence, the encountered colour charge density depends on the longitudinally integrated nuclear density profile ρ(r), i.e., the nuclear thickness function T (b) = ∫ dzρ(r = √ b 2 + z 2 ). If we adopt the normalisation ∫ d 2 bT (b) = 1 an additional factor of the total number of nucleons, i.e., the atomic mass number A must appear. From there the expected number of participating nucleons P is obtained by multiplying by the inelastic projectile-nucleon cross section σ N for the given centre-of-mass energy √ s, Finally, the initial condition for the evolution of the saturation momentum is given by [43] The evolution in rapidity of this initial condition using the leading order BFKL kernel with a constant coupling constant leads to [52,56] whereᾱ QCD = α QCD N QCD π, c ≈ 5, N QCD = 3 stands for the number of ordinary colours, and subleading orders 6 in y have been neglected. As mentioned before effects beyond the leading order like the running of the coupling constant play a role. Incorporating, hence, a running couplingᾱ the evolution slows down and one gets to leading order in large rapidities [57] stands for the number of quarks, and the momentum scale is set by the overall saturation momentum or the saturation momentum in an individual dipole, depending on which of the two is greater.
As next ingredient we need the cross section of a fast parton from the cosmic-ray particle on the dense gluons of the atmospheric nucleus. Here, the McLerran-Venugopalan model [58], which is formulated based on stochastic gluon fields, allows to determine an analytically closed expression [59] dσ QCD tot (The computation involves solving the equation of motion for the probe in a classical background field, a widely used approach in this subject area [60], and a subsequent averaging over configurations.) The nuclear density function is given by the Woods-Saxon distribution [61] ρ with the normalisation constant where Li n stands for the polylogarithm, a ≈ 0.5 fm is the nuclear surface thickness, and R = (1. [62] is the nuclear radius. When encountering the atmosphere, the cosmic ray particle is most likely to hit a nitrogen (A = 14) or oxygen (A = 16) nucleus and hence R ≈ 2 fm. The corresponding nuclear thickness function at large impact parameter-where it will be most important below-is given by Integration of Eq. (6) over the impact parameter b yields for large total rapidity y, which saturates the Froissart bound σ tot ∝ Y 2 . Herẽ We have used Eq. (2) to leading order in small P , Eq. (3) and Eq. (9) with b in the square root approximated by R as well as the approximation Moreover, we have taken π Λ 2 QCD as lower estimate for σ N , which actually grows larger than that at high energies. Accounting for the running of the coupling leads to instead.

A. Gauge interactions beyond the standard model
Extensions of the standard model that break the electroweak symmetry dynamically include additional gauge interactions. Here we are going to investigate, whether these could lead to comparable contributions to the interaction of hadronic cosmic rays with the atmosphere. In particular, we study whether a BFKL-like evolution could achieve this. In hadron-hadron scattering quantum chromodynamics is already in the saturation regime and leads to a sizeable cross section. Hence, in this system the onset of the BFKL evolution in additional gauge sectors is unlikely to show up as a prominent feature in the cosmic-ray spectrum. Thus, the question must be, whether the new gauge bosons also manage to form a dense system and whether this system contributes significantly to the cross section. The obstacles to be overcome by the evolution are the suppression factors that arise from the coupling to the valence quarks in the projectile and the target, respectively. The coupling in the target will lead to a reduction in initial density, while the coupling in the projectile leads to an overall suppression of the result even for asymptotically large energies.

With ordinary colour
Let us start with a technicolour model with some techniquarks that also carry ordinary colour. (See Fig. 4.) (This is not the case in all technicolour models, but their specific contribution offers a suitable starting point, and we can draw on the corresponding results in what follows.) These techniquarks can arise from the splitting of a gluon. [Thus, this process does not involve the extended technicolour sector. (Additional) contributions from that sector are discussed below.] Therefore, at large rapidity y and virtuality Q 2 , their density xQ(x, Q 2 ) inside hadrons and nuclei should be like that of heavy quarks. It is governed by the DGLAP evolution equation [48] for the emission of a fermion-antifermion pair from a gluon, where we summed over N f techniflavours, with trace normalisation factor T R and representation R under ordinary colour. Taking as initial condition that at the fundamental scale Λ TC the density of this fermion species is zero, the solution of this equation is approximated well by [49] Q for small enough momentum fraction x. Here, neglecting the scale dependence of the product of the coupling and the gluon distribution, gluon bremsstrahlung and taking the z integral to be dominated by small x z because of the rapid growth of the gluon distribution for small momentum fraction conspire to yield this result. Taking into account the integration volume by multiplication by (1 − x) improves the approximation for cases where x ≪1.
In the dilute regime, i.e., for Q 2 ≈ Λ 2 TC ≫ Q 2 QCD the gluon density is given by the Weizsäcker-Williams radiation [50] off A × N c separate quarks in the target, Consequently, where we have made use of Eq. (4). Therefore, the number of participants, the analogue of Eq. (1), is given by For a proton projectile the dipole density is obtained by setting A → 1 in Eq. (16), and thus the analogue of the cross section (6) at fixed x p and x t reads Carrying out the integration over the impact parameter, one finds to leading order in large rapidities where andR For the typical values N f = 2 = N TC , Λ TC = 10 3 Λ QCD , and ln(Q 2 Λ 2 TC ) = 1 the expression (radius) in Eq. (19) exceeds R for cᾱ TC y ≳ 14, which marks the onset of saturation in the technigluon sector. The total rapidity at the GZK cutoff is y = 26, which together with c ≈ 5 allows for perturbatively small values ofᾱ TC . In the case of a running coupling, the result reads which leads to the criterion 2cβ TC y ≳ 170. Withβ TC ⪆ 1 this requires y ≳ 17 in order to reach saturation.

Walking dynamics
As touched upon in the introduction, a walking coupling is a desired feature for a technicolour theory. A walking coupling is approximately fixed over a certain range of scales and only starts decreasing sizeably once the system is analysed at yet higher scales. This could be taken into account by implementing the corresponding two-loop β function in the vicinity of its fixed point when determining the saturation momentum. Approximating it by using the above results, the growth factor of the saturation momentum would take the form where y walk denotes the rapidity interval for which the system stays in the walking regime,ᾱ the coupling in the vicinity of the fixed point, andβ the effective value of 8 the β function once the coupling starts decreasing again. y walk is chosen such, that it takes the initial value Q 2 TC (0) of the saturation momentum to the scale Λ 2 ETC where the running sets in. Hence, we could write equivalently

So far
Taking stock, for asymptotically large values of the total rapidity y, where the logarithmic terms in the total cross sections after integration over the impact parameter can be neglected, every sector of gauge bosons-here gluons and technigluons-would contribute a structurally similar term, i.e., σ tot ∝ y 2 (σ tot ∝ y) for fixed (running) coupling. In reality, the logarithmic term is not negligible, but for quantum chromodynamics and technicolour with techniquarks that carry ordinary colour the saturation regime can be reached in ultra-high-energy cosmic rays. As the technigluon ladder cannot connect directly to the elementary constituents of the hadron projectile and target, but has to proceed through gluon and subsequent techniquark emissions on both sides, the contribution to the cross section from technicolour is typically only a thousands of that from quantum chromodynamics, where approximately one half of the orders of magnitude arises from the overall prefactor # and the other half from the smaller size of the dense system. Looking at the expected value for the transferred transverse momentum, however, at these high energies the two contributions are of the same order of magnitude because the technicolour scale is about a thousand times that of quantum chromodynamics. (See Fig. 5.) Because of the largely different fundamental scales and different final states there is no interference between the two sectors, and the expected values for the transverse momentum simply add up. For cosmic-ray phenomenology this means that the shower maximum for a given incident species and energy is actually shallower than expected from quantum chromodynamics alone. In exchange, the shower spreads out to larger lateral distances from its axis. Furthermore, the technicolour contribution is from relatively rare events as compared to quantum chromodynamics. This is compensated by being correspondingly more energetic. On the other hand, this should also lead to larger event-byevent fluctuations in shower depth and lateral spread for a given projectile species and incident energy. In order to carry out a fluctuation analysis, however, there are still not enough recorded events available in the high-energy tail of the cosmic-ray spectrum.

Within the standard model
We should also discuss whether the SU(2) L of the electroweak interactions, an integral part of the standard model, could achieve something similar completely on its y ! TC " TC ! QCD " QCD own. For this purpose the hadrons can be interpreted as dipoles right away albeit as rather dilute ones, as their scale is set by Λ QCD whereas the SU(2) L can only be considered in this context for scales much larger than the Z mass. Because of the smallness of the coupling constant α L ≈ 1 46, the evolution parameter cᾱ L Y is not sufficient for the hadrons to become densely populated by these gauge bosons even at the GZK cutoff.
To the contrary, if we use the weak gauge bosons as portal from the valence quarks to technidipoles instead of the gluon portal, relative to the latter contribution we pick up an additional factor of (times some colour factors), which enters in the overall prefactor of the cross section and the initial condition for the technigluon density and makes this contribution strongly subleading. The first fraction on the lefthand side accounts for the different couplings, the second for the difference in phase space. For the massless photon there would still be a logarithmic enhancement ln(Q 2 Λ 2 EW ) → ln(Q 2 m 2 u ) ≈ 20 ln(Q 2 Λ 2 EW ), which still cannot compensate for the powers of the weak interaction coupling. , which for this contribution may or may not carry ordinary colour. They are linked to the hadrons through the extended-technicolour sector, the analogue of the top Yukawa coupling. For techniquarks that carry ordinary colour all combinations of gluon portals (See Fig.4.) with top-quark-extended-technicolour portals at the target and projectile sides, respectively, contribute.

Without ordinary colour
After studying technicolour with techniquarks that are charged under ordinary colour, we continue for theories in which they are not. As pointed out in the introduction, these theories require less additional matter from the viewpoint of the electroweak interactions and are thus easier to accommodate within the bounds set by electroweak precision data. In these theories an ordinary gluon cannot emit a techniquark-antitechniquark pair. Thus this gateway is barred. The gauge bosons of the extended technicolour sector, however, must also carry ordinary colour in order to be able to give the ordinary quarks their mass. Hence, a valence quark in the hadrons can emit such an extended technigluon. (The entire consideration also goes through for a scalar extended-technicolour mechanism.) This would convert the quark to a techniquark, and we would have a technicolour dipole between the techniquark and the extended technigluon. The extended technigluon, however, is very heavy (for coupling to the first generation of the order 10 3 TeV) and we cannot put it on or even close to the mass shell with √ s about 10 2 TeV in the vicinity of the GZK cutoff. Therefore, we must consider a highly virtual extended technigluon that splits immediately into a quarkantitechniquark pair. (See Fig. 6.) This amounts to considering a four-fermion interaction involving two ordinary quarks and two techniquarks with the coupling strength m u,d ⟨T T ⟩, where ⟨T T ⟩ is the techniquark condensate, which is of the order of Λ 3 TC ≈ 0.1TeV 3 for running theories and larger for walking ones, while m u,d ≈ 10 −5 TeV. Consequently, after squaring the amplitude we must overcome a factor of (2π 3 ) −1 10 −8 Q 4 TeV 4 , where we have used the virtuality Q 2 to represent the momentum scale, and multiplying by (2π 3 ) −1 accounts for the phase space for a 1 → 3 process with two unresolved particles. To bring this factor anywhere close to unity requires again much more energy than is even available at the GZK cutoff.
The suppression is so overwhelming because of the smallness of the masses of the valence quarks compared to the technicolour scale (the equivalent of a small Yukawa coupling). Hence, we should take a look at contributions involving the top quark. (See Fig. 6.) Its distribution in the hadron is given by Eq. (16) with the replacement Λ TC → m t , If we couple the previous four-fermion interaction to the top or antitop, we only get an additional suppression of the order (2π 3 ) −1 m 2 t Λ 6 TC . The extended-technicolour scale for the extended technigluons coupling to the third generation of the standard model fermions is a few TeV. Hence, in detail depending on the value of the extended technicolour coupling, the mass of the extended technigluon is somewhat smaller than that value. Below that mass scale, we find the analogue of Eq. (19), but with whereR On this occasion we ignore additional combinatorial factors on the extended-technicolour side, involving, e.g., the rank of the gauge group etc. For the same benchmark values as above, but for Q 2 ≈ (3.14Λ TC ) 2 instead of Q 2 ≈ 2.72Λ 2 TC , # and # t are equal and so are the two total cross sections. For smaller values of Q 2 we have # > # t , while for larger values # t > #. Hence, in the typical range of parameters the two contributions are comparable. (See Fig. 7.) If, however, the virtuality Q 2 should reach the mass of the lightest extended technigluons, which are involved in linking the top quark to the technigluons, # t could become much bigger through resonance effects.
Contributions involving the extended-technicolour portal arise in all technicolour models that rely on this mechanism (including the scalar versions), as all of them need the four-fermion interaction to generate the standard-model fermion masses. Hence, in models in which the techniquarks carry ordinary colour both contributions appear. In that case they cannot simply be added, as the emission occurs at the same virtualities, and a mutual influence between the emitted technigluons should be considered.

More gauge bosons
We conclude this subsection with a comment on the potential participation also of the lightest extended technigluons, i.e., the ones that provide the masses for the heaviest generation of standard model fermions. They are expected to have a mass of a few TeV. One has to bear in mind that every emission in the gauge-boson ladder has to occur at a (much) larger scale than that. Hence, even in ultra-high-energy cosmic-ray hadron-hadron collisions the available energy does not suffice for a large number of emissions. In other words, there is not enough energy available to form a dense system of the lightest extended technigluons even at GZK-cutoff energies. Thus, they do not contribute significantly to the hadronhadron scattering cross section at the GZK cutoff. (This could have been a rather direct contribution, as in models where the techniquarks do not carry ordinary colour the extended technigluon sector unifies directly with quantum chromodynamics.)

B. Cosmic-ray particles beyond the standard model
So far, we have only considered effects of the presence of dynamical electroweak symmetry breaking in modifications of the hadron-hadron cross section. Such extensions of the standard model do, however, not only come with additional gauge interactions, but, in general, also with additional matter. A part of the cosmic-ray particles we observe could be made up from this matter. This deserves an investigation for at least two reasons. As already mentioned at the beginning, so far it has not been possible to find a satisfactory fit to cosmic-ray data by assuming that it is composed of hadrons and nuclei. This might be due to the commonly used straightforward extrapolation of the nuclear cross sections. The cosmic rays might, however, also contain non standard model particles. On the other hand, there is no consensus as to whether a sharp GZK cutoff is observed or not. If there were trans-GZK events that cannot come from local sources, we must explain, how these high-energy particles make it to us from their far-away sources without being affected by the GZK cutoff. At the same time they must be sufficiently strongly interacting with the atmosphere to be detected and also in order to be accelerated to high energies in the first place. (Nuclei are affected due to their electric charge. Standard model neutrinos do not interact strongly enough with the atmosphere, which in turn would require huge fluxes.)

Bound states
First of all, technicolour theories feature technihadrons. If they are sufficiently long lived to make it from the accelerating source to our atmosphere they can contribute to the cosmic-ray spectrum. The lightest technihadrons are typically the technipions because of their pseudo Nambu-Goldstone nature. In theories with techniquarks that transform under real representations of the technicolour gauge group there are also technibaryons among the technipions. If the technibaryon number is conserved by the extended technicolour sector the lightest technipion is stable and hence sufficiently long lived to make it from the source to the atmosphere. There can be plenty of other technihadrons, and they all have in common that they have valence techniquarks. There can also be bound states that do not have a direct analogue in the spectrum of quantum chromodynamics. In models where the techniquarks are in the adjoint representation, for example, there can also be bound states of techniquarks with technigluons, which at low energies resemble heavy leptons.
The sensitivity to a GZK-like cutoff is already altered for electrically charged technicolour bound states: The most potent mechanism for the energy loss of protons is the formation of a baryon resonance and the subsequent radiation of pions. The excitation spectrum in a technicolour bound state is (at least naïvely) three orders of magnitude harder than in a bounds state of quantum chromodynamics. Hence, such a channel must have a much higher threshold. Furthermore, if the cosmicray particle is already the lightest (and sufficiently stable) technicolour bound state, no lighter state can be radiated in the way necessary to cause a standard GZK bound. With this channel at least delayed to higher energies for the charged cosmic-ray particle there remains the much less potent channel for electron-positron production. Moreover, for an electrically neutral technicolour bound state not even this channel is present until the electric dipole moment of possible electrically charged constituents becomes relevant. Independent of the details a delay of the corresponding GZK cutoff to higher energies makes technicolour bound states good candidates for messengers for trans-GZK events. On top of that, due to the enhancement of their cross section at high energies, they can be accelerated effectively at the source.
The principal difference when a high energy technihadron strikes a nucleus in the atmosphere is that the suppression factor for the connection to the projectile is absent, as it is already a technidipole in the valence configuration, while the technigluon ladder connects to the ordinary target hadrons as before. (See Fig. 8.) As a consequence, the total cross section for the liberation of a techniquark from the incident technihadron by scattering on an atmospheric nuclei is given by Eqs. (19) and (22), respectively, but without the overall factor of #, the dependence on # remaining merely in the logarithm, whereR is defined in Eq. (21) and # in Eq. (20). Therefore, for the benchmark values of the parameters the cross section is increased by a factor of almost 36. That would also increase the expected value for the produced transverse momentum by the same factor. The average event would be very shallow and broad for a given energy and mass of the incident particle. For a technihadronic tech-  Fig. 6.) The ladder is linked to the hadrons through the extended-technicolour sector, i.e., the analogue of the top Yukawa coupling. For techniquarks that carry ordinary colour all combinations of top quark portals (See Fig.4.) with extended-technicolour portals at the target and projectile sides, respectively, contribute.
nipion we would at least expect the mass of the Z boson or above, which brings us in the mass range of a heavy nucleus, e.g., iron or beyond. One more difference is that, while the results for the hadronic projectiles still had to be convoluted with the parton distribution function of the hadron, the present cross sections still have to be convoluted with the parton distribution function of the technicolour bound state. Nevertheless, they are similar in so far as they are both in the dilute regime and thus dominated by the valence quarks.
For one putatively large contribution this does make a difference. Consider a theory with techniquarks that carry ordinary colour. There a gluon ladder could connect directly to a techniquark in a technicolour bound state. The scale in this bound state is, however, set by the technicolour scale. Hence, the corresponding ordinary colour dipole is extremely small and needs very high momenta to be resolved. Even for the large available rapidity the saturation momentum of quantum chromodynamics does not reach this scale. As a consequence, the hard momentum scale is set by Λ TC . There the coupling of quantum chromodynamics has already evolved to very small values,β QCD ln(Λ 2 TC Λ 2 QCD ), compared to the one of technicolour such that here log 10 (Q QCD Λ QCD ) ≈ 1.5, while log 10 (Q TC Λ TC ) ≈ 3. Another contribution that also would proceed through a gluon ladder and which would also be present irrespective of whether the techniquarks carry ordinary colour or not uses the topquark extendend-technicolour portal on the projectile side. The scale for the coupling of quantum chromodynamics would, however, still be set by the fundamental scale Λ TC of technicolour and the enhancement factor would still be comparatively small.
This leads to an early dominance of the technigluon contribution over the contribution from ordinary gluons. Hence, for non-hadronic targets the onset of BFKL evolution is not obliterated by the standard-model contribution. As a consequence, we should be able to see not only a contribution from an already dense, i.e., saturated technigluonic system, but from when the density starts growing rapidly, in analogy to what has been observed for the gluon density at HERA. For a hadronic projectile the contribution from the dense system of ordinary gluons hides the rapidly increasing but still comparatively small contribution from the technigluons.

Heavy leptons
Technicolour models with an odd number of fermion generations charged under SU (2) L require a family of additional leptons in order to cancel the Witten anomaly [64]. Data tells us that these leptons must be heavy (at least about 100GeV [6]) to have escaped observation so far. Therefore, like the top quark, they couple strongly through the extended technicolour sector, assuming of course that there is not yet another mass generation mechanism. (See Fig. 9.) Below the mass of the extended technigluon, in analogy to the top-quark portal for an incident ordinary hadron, we get a factor of Q 4 m 2 ν Λ 6 TC . At variance with the top-quark portal we do not have the suppression from the emission of the intermediary gluon and the top. Thus, for a neutrino with the mass of a top quark from the difference in prefactors we get an enhancement by relative to the technicolour contribution to the corresponding hadron-hadron cross section for our benchmark parameters. This would make also heavy neutrinos in the technicolour context to candidates for messengers for trans-GZK events. Among others, one could imagine a scenario where there is a flux of these neutrions (similarly of technihadrons), smaller than the overall effective particle flux below GZK energies and as such hidden. Then if the GZK cutoff is effective for all other species apart from  Fig. 6).
the heavy neutrino (and/or an appropriate technicolour bound state) its small flux would show up in trans-GZK events. Moreover, the enhancement of their cross section at high energies would also allow for an efficient acceleration at the source. (The above is different from the Z-burst scenario [63], where a high energy neutrino annihilates with a relic anti-neutrino in the halo of the earth, sending out a high-energy Z boson that interacts with the atmosphere.)

Non standard model target
It is conceivable that the particle struck in the atmosphere is not a standard model particle either, but one arising in an extension. If there is supposed to be a sizeable number of such events, the number of these targets in the vicinity of the earth must be large enough. In that case, they must interact sufficiently weakly at low energies in order to have eluded direct detection so far, and they must be sufficiently stable to have survived in sufficient numbers. Hence, electrically neutral particles would be likely candidates here. In the context of non-standard-model projectiles we have just mentioned neutral technihadrons, especially technipions, and heavy neutrinos in the technicolour context. For a techni- hadronic target (See Fig. 10.) the technigluon ladder can connect to both the projectile and the target without any suppression. For technicolour theories in which the techniquarks do not carry ordinary colour the contribution from quantum chromodynamics would be even more suppressed, as the ordinary gluons can only connect to the valence techniquarks through the extended-technicolour sector. Hence, already the onset of the BFKL evolution in the technigluon sector should be visible and not only the saturated regime. Moreover, even for theories in which the techniquarks do carry colour, the parton distribution functions in the bound states are governed by the technicolour scale and render the coupling of quantum chromodynamics small compared to that of technicolour. For this reason also there the contribution from technicolour dominates the contribution from quantum chromodynamics, and the beginning of the BFKL evolution in the technigluon sector is again not hidden by the contribution from quantum chromodynamics. The dominant cross section, which is induced by the exchange of the technigluon ladder, is given by As parameters for the thickness function T = T (b), which enters the initial conditions for the evolution of Q TCx , we choose the R=0.5fm for an ordinary proton but rescaled by the ratio Λ QCD Λ TC , i.e.,Ř = R×(Λ QCD Λ TC ). We do the same for all other corresponding length parameters.
(This is consistent with the scale for the tail of the nuclear density being set non-perturbatively by the mass of the lightest hadron, i.e., the pion [65], which in technicolour is raised to the mass of the weak gauge bosons.) For fixed gauge coupling the integration over the impact parameter while for running coupling we find The typical bound state radius makes these cross sections a priori smaller than the hadron-hadron cross sections by a factor of (Λ QCD Λ TC ) 2 . This does not impact the question whether the formed system is dense in the same way, as this depends on whether (here at fixed coupling) cᾱ TC y > ln from which the overall power of the ratio of the scales drops out. There remain additional powers of said scale inside the logarithm that lead to the condition cᾱ TC y +ln 2πaRn , which amounts to a strong enhancement of the density. In fact, it is parametrically the same as for proton-proton scattering through the exchange of ordinary glue with σ N = π Λ 2 QCD . (The analogous condition for a running coupling reads 2cβ TC y + ln 2πaRn .) Furthermore, these bound states are much heavier than a proton; they naturally have the mass of a Z or a top and can be even heavier. For a given laboratory-frame energy of the incident cosmic-ray particle this fact alone leads to an enhancement of the centre-of-mass energy, as it is approximately the geometric mean of the mass of the (approximately fixed) target and the laboratoryframe energy of the projectile. Hence, for a target mass of 100GeV we would gain one order of magnitude in centreof-mass energy relative to a proton of about 1GeV. This enhancement of energy would allow a further growth of the cross section. There arises also a qualitative difference. In Sec. III A 6 we had mentioned the contribution from the lightest extended technigluons, i.e., those responsible for the mass of the heaviest standard-model fermions. While in hadron-hadron scattering even at the GZK cutoff a rapid growth of the contribution should be hampered by the limited available energy, which is needed to emit enough extended technigluons with sufficiently high virtuality, the additional order of magnitude in centre-of-mass energy gained when a target is struck that is a bound state of the technicolour sector counteracts this obstacle. At the scales, where this becomes important, technicolour unifies with the aforesaid extended technigluons. At the corresponding scales, this increases the couplingᾱ TC = α TC N TC π which enters the growth of the gauge boson density through an increase of the number of technicolours N TC to the number of colours when it unifies into the lowest-scale extended technicolour sector. As for the expected value for the produced transverse momentum, the multiplication by the scale Λ TC compensates only for one of the two negative powers from the overall scale. Any combination of technihadrons and heavy neutrinos, respectively, as projectile and target, respectively, can occur. In the technicolour context, the dominant contribution to the scattering of two heavy neutrinos would come from the exchange of a technigluon ladder that connects to the leptons through the extendedtechnicolour sector, which leads only to a mild suppression due to the analogue of a large Yukawa coupling. (See Fig. 11.) The density of technidipoles in the neutrinos goes like ♯ν = m 2 ν Q 4 Λ 6 TC with a typical size of the order π Q 2 . The decay of their tail is again set non-perturbatively by the inverse mass of the technipions. Hence, as in the previous case we consideř R ≈ 0.5fm × Λ QCD Λ TC ≈ǎ. This leads to Compared to the previous case, saturation is slightly delayed owing to the elementary nature of the lepton as opposed to the previously regarded technicolour bound state; the scale for the transverse size is not set by the fundamental scale Λ TC itself, but we have to look at which size (∼ Q 2 ) of dipoles are contained in the wave function of the lepton and Q 2 is necessarily above Λ 2 TC if the exchange of a technigluon ladder is considered. The running coupling case is again obtained by the replacement cᾱ TC y → 2β TC y. The top-quark portals do not only lead to technicolour dipoles but also to dipoles of ordinary colour to which an ordinary gluon ladder can connect. The contribution is once more reduced relative to the technicolour contribution because the large momentum scale Q 2 leads to a very small coupling in this sector, The heavy target can also be struck by a hadronic cosmic-ray projectile. (See Fig. 12.) This leads to a suppression of the contribution to the cross section from the exchange of technigluons relative to the previous case, as they must connect to the projectile either through the top-quark portal or, if some of the techniquarks carry ordinary colour, through the emission of a techniquarkantitechniquark pair. The cross section for the latter case at fixed coupling for example reads The contribution from the top-quark (extended technicolour) portal is obtained by replacing # by # t from Eq. (27). The running case is obtained by replacing cᾱ TC y by 2cβ TC y.
In technicolour theories where the techniquarks do not carry ordinary colour the scattering of a hadronic instead of a non-hadronic projectile on a non-hadronic target leads to an enhancement of the contribution from the exchange of ordinary gluons relative to the case with a non-hadronic projectile since the gluons can connect directly to the projectile, while they connect to the target through the extend-technicolour sector. (The contribution to σ TC x x tot that is only present for techniquarks that carry ordinary colour, i.e., where the ordinary gluons can connect directly to the techniquarks, does not change by more than colour factors if a hadronic target is considered instead of the technicolour bound state.) This yields where the scale for the process is again set by the technicolour scale Λ TC and ♯t = m 2 t Q 2 Λ 6 TC .

C. Other realisations of dynamical electroweak symmetry breaking
As explained in the introduction, there are various other realisations of dynamical electroweak symmetry breaking. Topcolour, for example, contains an additional gauge sector made up of the gauge bosons that generate the required four-fermion interaction. They couple directly to standard-model fermions. In pure topcolour, their mass scale must, however, be extremely high to allow for a light enough top quark. In that case, they could not be excited even in ultra-high-energy cosmic ray events. To the contrary, if topcolour is combined with technicolour or a seesaw mechanism they can have a lower scale. A minimal setup for such a model is based on identifying the SU (3)  At virtualities below the corresponding unification scale there is still the contribution from the diagonal subgroup, i.e., quantum chromodynamics, studied at the beginning of Sect. III and depicted in Fig. 2. At virtualities above the mass m tg of the heavy gauge bosons, gauge bosons from both SU(3)-s can participate. The density of sufficiently small dipoles in the hadronic target at the unification scale is approximately given by [See also Eq. (15).] The density in a proton projectile is obtained by setting A = 1. Above the unification scale these dipoles can be seen as being made up to on half of one SU (3) and to one half of the other. Thus, we get Here, the evolution of the coupling with the β function of quantum chromodynamics is implemented consistently up to the unification scale where it matches the couplings of the two different SU(3)-s. Onward from there the two latter couplings evolve slightly faster than before and different from each other due to the quarks being split to the two sectors, but here we neglect this correction. Accordingly, a dense system is An √ 2πaR ≈ 13.5 for our benchmark values together with m tg = 700GeV [6]. For ln(Q 2 m 2 tg ) = 1 and Λ QCD = 250MeV this necessitates y ⪆ 46, which is predominantly due to the small coupling. Hence, for this particular implementation a sizeable contribution from the additional gauge bosons to the cross section appears unlikely. There remain plenty of variations of the topcolour principle to study, but as long as a unification with the coupling of quantum chromodynamics at scales of this order is required, it is not obvious, how they could lead to a significant part of the hadronhadron cross section. On a heavier target, e.g., a top pion, the centre-of-mass energy would be increased, but the issue with the small coupling remains.
In composite Higgs models the limit ξ → 1 is referred to as the technicolour limit. There the compositness scale is as low as in technicolour. Assuming that the binding of the constituents arises from a gauge interaction, it should be possible to resolve it in ultra-high-energy cosmic ray events in close analogy to the technicolour case. For smaller values of ξ this becomes more and more unlikely. A more concrete analysis can, however, only be carried out after a (partial) ultraviolet completion in the form of a gauge sector has been specified.
For little Higgs models one could also assume that the compositness is realised by new gauge interactions. A concrete analysis would again require specifying a concrete ultraviolet completion. Even before that, i.e., at lower energies, there are additional gauge fields that are required to generate the Higgs potential, which is needed to break the electroweak symmetry. The experimentally preferred value for their coupling is at least five times larger than the coupling of the SU(2) L [42]. The lower bound on their mass, however, is 2TeV [42], which makes it very energetically expensive to form a dense system in a hadron-hadron cosmic-ray event. A heavier target would increase the possibility of seeing these gauge bosons.
In all of these realisations of dynamical electroweak symmetry breaking there are also candidates for nonstandard-model cosmic-ray messengers. These are as diverse as the models they arise in. On top of that, in ! IR [eV] E cm /! IR FIG. 13: Ratio of the centre-of-mass energy Ecm to the infrared scale Λ IR for different mass scales M for fixed lab-frame energy E lab = 10 19 eV. The steeper lines from bottom to top are for the scattering by the exchange of a gauge-boson ladder on a proton (red), a technihadron (e.g., a technipion) with a mass of 100 GeV (green), a technihadron with a mass of 1 TeV (blue). Our benchmark values for technigluon ladders are where the lines end on the ordinate. The lines are relevant, for example, for contributions from extended technicolour. The shallower black line is for the scattering by the exchange of a gauge-boson ladder on a target that is a bound state of the same gauge interaction. many cases ultraviolet completions have to be specified in order to be able to carry out the corresponding analyses. Thus, this task must be left for later work. Nevertheless, if the target is much heavier than a proton, an enhanced centre-of-mass energy due to a heavy target may permit the resolution of some of the dynamical electroweak symmetry breaking mechanisms, which otherwise could not contribute significantly, as not enough emissions are possible for a given centre-of-mass energy and given infrared scale Λ IR . Here, we must distinguish two principal classes of cases. In one class the mass scale for the bound state (and thus the centre-of-mass energy √ s = √ E lab M ) and the infrared scale for the emissions are independent. For the other class they are coupled. Examples for the first case are: the technigluon ladder (Λ IR ↔ Λ TC ) for a hadronic target (M ↔ Λ QCD ) or the extended-technigluon ladder (Λ IR ↔ Λ ETC ) on a technihadronic (M ↔ Λ TC ) or a hadronic (M ↔ Λ QCD ) target. Examples for the second case arise for all gauge sectors that scatter on a target that is a state bound by the corresponding gauge interaction itself. Here the mass and infrared scales change in unison. As a consequence, for fixed lab-frame energy, the centre-of-mass energy is pro-portional to the square root of the infrared scale and the ratio falls only with 1 √ Λ IR instead of 1 Λ IR as in the other class of cases. That makes a significant contribution of gauge-boson ladder exchange in such processes possible for much higher scales than one could expect. (See Fig. 13.) IV. CONCLUSION We have studied dynamical electroweak symmetry breaking in the setting of ultra-high-energy cosmic rays. We have considered the possibility that ultra-highenergy cosmic rays offer the required energy to find direct signs for dynamic symmetry breaking and that some of the inconclusively explained features of the ultra-highenergy cosmic ray spectrum might have a natural explanation in such an extension of the standard model.
First of all this necessitates evaluating the contribution to ultra-high-energy cosmic ray events from within the standard model, which is dominated by quantum chromodynamics. The decisive question is whether quantum chromodynamics already explains the entire cross section deduced from the observation of ultra-high-energy cosmic rays, e.g., in air showers, or not. There are still considerable uncertainties in the extrapolation of the cross section due to quantum chromodynamics, where we were able to measure it reliably, i.e., in collider experiments. In this context, it would be of utmost importance to obtain data from proton-nucleus collisions at the Large Hadron Collider (LHC). This system would come closest to the situation expected in air showers and would reduce the uncertainties from having to extrapolate the part of the cross section stemming from quantum chromodynamics considerably.
Furthermore, extensions of the standard model that break the electroweak symmetry dynamically introduce new strongly interacting sectors into our description of the world, which-if they involve additional gauge interactions, and these are the ones, we have concentrated on in this paper-at increasing centre-of-mass energies should share the feature of a rapidly growing cross section with quantum chromodynamics. As a consequence, better knowledge from better data about the cross section of quantum chromodynamics will allow us to better judge on the extensions of the standard model.
In fact, there are two scenarios. Either, based on further studies and measurements, it becomes clear that quantum chromodynamics does explain the entire cosmic-ray-air cross section in which case we can use this information to rule out extensions that would have further modified said cross section, or it turns out that the cross section cannot be explained by quantum chromodynamics alone which implies that extra contributions are needed, which could be supplied by additional gauge interactions like the ones present in dynamical electroweak symmetry breaking.
It should also be emphasised that this is the point where the ultraviolet sector of dynamical electroweak symmetry breaking can be probed and where it can be told apart from an effective description in terms of composite fields in four dimensional or extra-dimensional setups.
In our investigation, we have distinguished different systems. We have started by studying the contribution of extensions to the case where the cosmic-ray projectile as well as the target in the atmosphere are ordinary hadrons. There the contribution from the extension is suppressed alone by the fact that, unlike quantum chromodynamics, it cannot connect directly to the constituents of the hadrons. Therefore, the onset of a HERAlike rapid growth of the cross section in the extension is probably hidden by the large contribution from quantum chromodynamics. Nevertheless, at energies around the ankle of the cosmic-ray spectrum a low-scale extension like technicolour can still form a dense system of gauge bosons (here technigluons) which, as they mediate much harder interactions than the ordinary gluons, can contribute a comparable amount of produced transverse momentum. For a given projectile species and incident energy this would lead to a shallower shower maximum. As the same average amount of transverse momentum is coming from relatively rare events the event-by-event fluctuations should become bigger. Moreover, the range of incident energies, where technicolour may impact the hadron-hadron cross section is in the vicinity of the ankle structure of the cosmic ray spectrum.
Sectors that break the electroweak symmetry dynamically also come with other candidates for the projectiles. These can be bound states of the new strongly interacting sector but also new heavy leptons needed to cancel anomalies. First of all, they do not see the GZK cutoff predicted for ordinary hadrons. (On photons from the cosmic microwave background nuclei photo-disintegrate, while protons are exited and radiate off pions.) The bound states of a new sector are much more deeply bound. Hence, the scale of the cutoff is correspondingly higher. If, on top, they are electrically neutral any such process is delayed even further. (A similar statement holds for neutrinos.) What the cross section is concerned, their constituents can couple directly to the technigluons and the corresponding suppression of the cross section drops out. To the contrary, quantum chromodynamics receives a two-fold penalty; apart from models where the techniquarks carry also ordinary colour it cannot couple directly to the constituents anymore, and, moreover, the momentum scale set by the bound-state wave function is much larger than the fundamental scale of quantum chromodynamics, which makes its gauge coupling very small. If one could isolate such events, even the onset of the HERA-like growth of the cross section could be visible. If the GZK cutoff were effective for ordinary hadrons, all trans-GZK events could be of the nature described here. Taken the other way round, given that bound states of a new strongly interacting sector do not see the GZK cutoff predicted for ordinary hadrons and in the absence of a fundamental reason why particles can practically not be accelerated to trans-GZK energies, the small number of observed trans-GZK events seems to indicate that these bound states are not stable enough to make it to our atmosphere in great numbers. The heavy leptons couple without big suppression to the technigluon sector through the analogue of a large Yukawa coupling. The direct coupling of all these non-hadronic projectiles to the strongly interacting sector also allows for their efficient acceleration at the source. Light neutrinos offer a different glance at dynamical electroweak symmetry breaking extensions of the standard model. In the absence of (the analogue) of a large Yukawa coupling they couple democratically to quarks and techniquarks through the exchange of weak gauge bosons. Despite the suppression by the weak coupling this offers a relatively clean look at standard-model and non-standard-model components of the cross section [66] also at neutrino observatories such as IceCube [67].
In technicolour the connection between the gauge sector and electroweak symmetry breaking is most direct and consequently the involved energy scales are the smallest possible. In all other realisations of dynamical electroweak symmetry breaking the connection is less direct and the energy scale for probing the additional gauge sector is delayed. That makes finding an increase of the corresponding contribution to the cross section energetically more and more difficult. Concretely, in topcolour the gauge interactions generate four-fermion interactions, which later break the electroweak symmetry. This is very similar to the extended technicolour sector; the part that gives the masses to the heaviest standard-model fermion generation comes in at the lowest scales. In composite and little Higgs theories a priori no gauge interaction is specified, but under the assumption that the Goldstone sector of these theories is generated in this way, it is detached from the electroweak scale by virtue of the Goldstone theorem. In the so-called technicolour limit of composite Higgs theories where the pion decay constant and the electroweak scale coincide a similar cosmic-ray phenomenology could be expected. A concrete analysis necessitates, however, specifying an ultraviolet completion for these models. This also holds for little Higgs theories only that there are additional gauge fields above at least 2TeV that generate the Higgs potential.
Additional space for discovering dynamical electroweak symmetry breaking through ultra-high-energy cosmic rays becomes available once we take into account the possibility that the target is not a hadron but a bound state of the extension or a heavy lepton, in the halo of the earth. To begin with, for a given lab-frame energy the centre-of-mass energy grows like the square root of the rest mass of the target. For the increase from the proton to the Z mass this already provides an additional order of magnitude. Thus, the prospect on discovering new gauge interactions by scattering on bound states they hold together is improved, as for these targets reaching the energy frontier is postponed. (See Fig. 13.) There are several paths that could be taken to further elaborate on the above results. It would be very beneficial to obtain data on the growth of the cross section of quantum chromodynamics with the energy at larger energies in a clean environment, which can be achieved in proton-nucleus collisions at the Large Hadron Collider. This is needed as better starting point for extrapolating the part of the cross section coming from quantum chromodynamics and as a milestone for understanding the growth of the cross section in an additional gauge sector. Before this data becomes available the above computations can already be developed further, i.e., e.g., taken to higher orders and/or beyond the mean-field approximation in the initial conditions as well as corrected for quantum fluctuations in the evolution of the parton distribution functions. That will also serve to get a better understanding of how to translate the quantum chromodynamics measurements to new gauge sectors. Moreover, the findings can then be implemented in a full shower simulation in order to quantify the statements about the shower profile and lateral spread. Furthermore, here, we mostly compare overall numbers. More pieces of information from ultra-high-energy cosmic-ray observations could be compared to our predictions, like, e.g., the secondary composition of the shower or the shower direction. An investigation of event-by-event fluctuations was already mentioned above, but requires a large number of recorded cosmic-ray events. Combining these investigations with information obtained from considering the acceleration mechanism at the source and/or the transport through space to the earth could also give additional insight [68].
In conclusion, studying dynamical electroweak symmetry breaking in the setting of ultra-high-energy cosmic rays can allow for the direct detection or exclusion of such a mechanism and may explain naturally structures in the cosmic-ray spectrum.