1. Introduction
The interest in discovering new physics (NP) beyond the Standard Model (SM) at the Large Hadron Collider (LHC) is beyond doubt. Within the last decades, many distinct strategies have been put forward, most of them based on signatures in the transverse plane with respect to the beam’s axis, such mono-jets, missing transverse energy, displaced vertices, and so on. On the other hand, other kinds of rather “diffuse” signals have been examined in the literature (e.g., [
1,
2]), featuring the whole event (multiplicity distribution and moments, event shape variables, underlying event, etc.) as a key signature of NP. For instance, the so-called “soft bomb” scenario [
3] is characterized by high multiplicity events with nearly spherically distributed soft SM particles and a large amount of missing transverse energy. In particular, a strongly coupled hidden/dark sector could lead to a large angle emission of partons carrying a non-negligible amount of momentum and yielding a rather isotropic distribution of final-state particles all sharing a similar amount of energy. Note, however, that a likely complicated hidden sector (HS) beyond the SM may have limited observable effects at colliders, making detection from an SM background, and especially the discrimination between different models, difficult [
4,
5]. Therefore, alternative signatures, as proposed in this work, should be considered as complementary to other search strategies, as discussed in [
3].
Indeed, as is well known, (pseudo)rapidity and azimuthal particle correlations provide a crucial insight into the underlying mechanism of particle production (see [
6] for a review). Moreover, from general arguments based on causality, long-range correlations should have the origin at very early times after the collision. Therefore, if the parton shower were to be altered by the presence of a non-conventional state of matter, final-state particle correlations should be sensitive to it [
7,
8].
The two-particle correlation function is often defined in pseudorapidity and azimuthal space as [
9,
10,
11]
Here, S denotes the signal distribution built with particle pairs from the same event, while B denotes the background distribution constructed by particle pairs taken from different events. and denote, respectively, the pseudorapidity and azimuthal differences of particles 1 and 2—the indexes labeling the trigger and associate particles, respectively.
Typically, a complex structure of the correlation function is observed. In particular, an enhancement of the two-particle correlations is found at
in heavy-ion collisions [
12]. Because of its extended longitudinal (pseudorapidity) shape, as seen in the
–
plot, it is referred to as the (near-side) ridge. One-dimensional correlation functions
are obtained from Equation (
1) by integration over pseudorapidity along the range
to focus on long-range correlations.
The observed azimuthal anisotropy in heavy-ion collisions is commonly analyzed by means of a Fourier decomposition:
where the coefficients
are supposed to factorize as the product of the coefficients of the equivalent Fourier expansion of two single-particle densities. When applied to heavy-ion collisions, the different terms in the series of Equation (
2) find a “natural” interpretation according to a hydrodynamical model describing the very hot and dense matter resulting from the collision. In practice, up to five or six Fourier terms are taken into account in the analysis of the experimental data.
Remarkably, similar long-range ridge structures show up in proton–nucleus [
9,
10,
11,
13] and even proton–proton [
13,
14] collisions, under several conditions on events such as high multiplicity and a given transverse momentum range of charged particles. The interpretation of a positive
in these small systems is currently highly debated, and different observables have been proposed to probe new dynamical effects related to large hadronic densities [
15].
In this paper, we furthermore consider that hidden particles and states, stemming from Hidden Valley (HV) models [
16], can be formed at primary interactions in very high energy
collisions. Generically, an HV model consists of three sectors: (i) a HS containing v-particles charged under a valley group
but blind to the SM interactions, (ii) a visible sector including SM particles charged under the SM group
but neutral under
, and (iii) mediators connecting both visible and hidden sectors. Usually, the masses of the hidden sector particles are assumed to lie below the electroweak scale, while the mediators may have TeV-scale masses. The simplest possibility for
is a QCD-like scenario, with a strong (running) coupling constant
and confinement scale
. The SM sector could feebly couple to the HS (and the equivalent hadronic v-particles and states) via a neutral
or via heavy particles bearing both
and
charges. Here, we consider the latter possibility along the lines of the Monte Carlo (MC) study using PYTHIA [
17], where the hidden shower is basically controlled by two parameters: the coupling strength
, assumed to be a constant (i.e., no running is considered), and the lower cut-off scale set equal to 0.4 GeV as by default in QCD showers, consistent with a low hidden confinement scale
. Such a simplified picture is compatible with the expected walking behavior requiring a strong coupling over a large energy window along the showering before reaching
, thereby yielding a large number of hidden partons and final-state particles. At the end, the energy from the primary interaction is democratically shared by soft final-state SM particles, while no classical jet structure is expected, thereby adapting quite well to a soft-bomb scenario. As previously mentioned, the ultimate goal in this paper is to show that long-range azimuthal correlations among final-state particles should emerge as a consequence of such a scenario.
2. Hidden Valley Scenario
Focusing only on the particle content relevant to the study presented here, we collectively denote by
the (spin 1/2) hidden partners of the SM quarks, charged under both
and
, while
and
stand for the v-gluon and (spin 0) v-quark only charged under the
hidden group, respectively (the notation used follows that of [
17] for the HS in the PYTHIA 8 MC generator).
The unparticle scenario, which can be viewed as a special case of HV models, deserves special mention. Let us recall that, from a phenomenological point of view, an unparticle [
18] does not have a fixed invariant mass squared, but instead a continuous mass spectrum. As pointed out in the literature (see e.g., [
19]), direct detection of unparticle stuff at colliders should rely on peculiar missing energy distributions. The influence of unparticle production on particle correlations would become another useful tool to study such a scenario, as shown in this work.
For some parameter values of HV models, hidden particles could promptly decay back into SM particles, altering the subsequent conventional parton shower [
20] and yielding (among others [
4]) observable consequences, e.g., extremely long-range correlations, especially in azimuthal space [
21]. In this paper, we do not enter into details about specific models but limit ourselves to general features associated with the production of very massive objects on top of the parton shower and their observable consequences, mainly from kinematic constraints.
Our analysis focuses on
pair production via
or
fusion (see
Figure 1) subsequently decaying into a v-quark and an SM quark:
, where
X stands for an ensemble of radiated gluons and v-gluons which, in turn, will originate visible and hidden parton cascades. Note that a very massive
would be produced at a rather low velocity during the primary parton–parton interaction in
collisions at the LHC. In fact, assuming that the center-of-mass subenergy of the parton–parton interaction is of the order of or higher than twice the magnitude of the mediator mass (
), then
states can be on-shell pair-produced. Moreover, all (either SM or hidden) particles stemming from its decay should have access to a limited energy due to v-gluon radiation. In sum, final particles would “democratically” share the center-of-mass energy released in the primary collision, and rather soft and diffuse signatures are expected.
Below, we consider very heavy hidden sources moving non-relativistically for kinematic estimates involving angular distributions. Then, velocity may become a well-defined physical and meaningful quantity when dealing with heavy particles [
22]. Moreover, we assume an isotropic parton emission in the hidden particle rest frame, coming out from the primary interaction and slightly boosted in the laboratory reference frame due to a non-relativistic velocity of the above-mentioned very massive hidden source. Both assumptions provide the essential framework for our estimates and conclusions.
In the hidden source
rest frame, the product of the velocity
and the Lorentz factor
of the fragmenting v-quark is roughly given by
where the bare
q-mass is set equal to zero. The effective v-quark invariant mass, denoted as
, is defined in a similar way to conventional QCD jets, i.e.,
where
and
stand for the energy and three-momentum of the v-gluons emitted by the fragmenting v-quark, respectively, and the sum on
j runs over all emitted v-gluons. Even though the bare
-mass could be as light as 10 GeV,
can reach values close to
because of radiation, as happens in QCD jets [
17]. This would especially be the case for a strongly interacting hidden/dark sector, i.e., at large
. We look upon expression (
3) as providing an order of magnitude estimate of the v-quark velocity. Of course, large variations of the
factor will occur event by event because of the wide spread of
.
In their turn, bound v-states can be formed as v-gluons create new v-quark-antiquark pairs, as happens with gluons in a conventional QCD shower. In HV models with v-hadrons promptly decaying back into SM partons, a new SM parton cascade would be originated (coexisting with invisible particles), eventually leading to final-state SM particles as well. Furthermore, as the radiates more and more v-gluons and the mean value of its effective mass distribution shifts from towards , more and more energy is subtracted from the visible quark and its associated system of emitted gluons. In sum, a strong coupling should lead to small velocities of both SM and hidden particles.
Indeed, under a Lorentz boost of velocity
, the angular distribution of the final-state particles in the laboratory reference frame (LRF), with the latter almost coinciding with the fragmenting
reference frame, is given by [
23]
Here, with , and with the final-state particle velocity v in the rest frame. For , one can roughly set . A massive hidden object of spin zero, as assumed for the fragmenting v-quark in this work (leading to a nearly spherical distribution in the -quark reference frame), with being non-relativistic, plainly justifies such an approximation.
For practical purposes, the azimuthal distribution
can then be approximated by a Gaussian distribution for small
angles, namely
where
was interpreted as an azimuthal cluster “width” in [
24,
25]. As we are focusing on azimuthal angles, the particle trajectories are projected onto the transverse plane, hence the velocities
and
v and the Lorentz factor
actually correspond to transverse velocities. Large hidden source velocities lead to small
and thereby a more pronounced peak at
, in accordance with Equation (
5). Conversely, small velocities of the hidden source lead to flatter azimuthal distributions.
3. Results on Two-Particle Azimuthal Correlations
Substituting Equation (
3) into Equation (
6) for
, one gets
where
stands for the effective mass resulting from v-gluon radiation, as mentioned above.
Next, by Taylor expanding the exponential, we can identify the above expression with a cosine function such that
determines the leading Fourier component of the NP contribution from a given range of the effective v-quark mass. As reference values, we set
GeV and
GeV [
17], yielding the closest fractional number
This estimate can be extended to the mass interval of the v-quark invariant mass
GeV, leading to the NP contribution
from this
mass “slice”.
By integration of the product of the two single particle azimuthal distributions, one gets
In
Figure 2 we show the expected angular dependence of the corresponding Fourier term in the correlation function
for two reference benchmarks:
(a) GeV and
GeV, and
(b) GeV and
GeV. All hidden initial sources from the primary collision originating the subsequent visible/invisible shower are assumed to be non-relativistic (
g is taken of the order of 0.1 in Equation (
5)). A comparison with the
modulation, shown in the same plot, indicates that the HS contribution should yield a Fourier component dominated by this term (not yet considered so far in any analysis to our knowledge).
Actually, more fractional harmonic terms should be considered as the whole mass range of the effective mass
(up to
) is taken into account in the expected continuous spectrum obtained from radiation [
17]. Hence, the Fourier series should be more generally written as
where the extra
harmonic terms encode the angular anisotropies associated with massive hidden states modifying the parton shower and thereby correlations among final-state particles. The
term should be the leading component in these fractional harmonic terms. Note that the
term now has two contributions: a negative one (
) from the conventional series of Equation (
2) and another positive one (
) expected from a HS.
4. Discussion and Conclusions
Of course, the Fourier analysis using Equation (
11) will contain contributions from both the conventional partonic cascade and from the hidden sector. In order to enhance such a hypothetical NP contribution, extra selection cuts beyond high multiplicity and usual
ranges of charged hadrons should be applied on events, in particular high
leptons and/or missing transverse energy/momentum.
Indeed, multi-lepton signatures have already been used in the search of NP at the LHC (see e.g., [
26,
27]) as they are predicted by many models beyond the SM. For example, a cascade of partons/particles initiated by the decay of a heavy hidden particle can proceed though intermediate states, yielding electrons, muons, or tau leptons in the final state. Realistic requirements would imply an electron or muon with
GeV and
, a second electron or muon with slightly looser requirements, and a third electron, muon, or hadronically decaying tau. Moreover, lepton combinations of the same electric charge can be used to enhance the NP signature. Additional cuts can be large missing
since the hidden/conventional cascade can result in invisible particles at the end of the decay chain. Lastly, as the decay of hidden particles to bottom quarks can be significantly enhanced in many hidden models,
b-tagging would be another technique to enrich the sample with NP events.
Note that such proposed cuts (aside from a common high-multiplicity cut) could hardly be attributed to the formation of QGP or glass condensates, but are associated with the presence of NP.
Thereby, the sample would be enriched with NP events enhancing the ridge effect due to this non-standard mechanism. Then, the non-vanishing values of , and so on, resulting in a better fit than the conventional Fourier analysis, provide a hint of NP, complementary to other kinds of searches.
Let us finally remark that, based on the observation of a near-side ridge effect in hadronic collisions, an integrated luminosity on the order of tens of pb
would be needed to observe any possible NP emergent effect, provided that the HS production cross-section turns out to be large enough, which crucially depend on the
mass [
17]. In order to avoid pile-up effects, a dedicated low-luminosity run would be desirable at the LHC.
Summarizing, hidden/dark sector production on top of the parton shower in collisions can considerably alter final-state particle correlations, becoming a signature of NP. This conclusion bears a resemblance to the finding of the ridge phenomenon in heavy ion collisions. Specifically, more fractional harmonic terms should be included in the Fourier series when carrying out the analysis of the azimuthal correlation function , once appropriate NP selection cuts are applied to events. These extra fractional harmonic terms would capture the existence of very long range correlations.