Cosmological analogies in the search for new physics in high-energy collisions
Miguel-Angel Sanchis-Lozano, Edward K. Sarkisyan-Grinbaum, Juan-Luis Domenech-Garret, Nicolas Sanchis-Gual
IIFIC/20-28, FTUV-20-06-11
Cosmological analogies in the search for new physicsin high-energy collisions
Miguel-Angel Sanchis-Lozano a , ∗ , Edward K. Sarkisyan-Grinbaum b , c , ∗∗ , Juan-LuisDomenech-Garret d , † and Nicolas Sanchis-Gual e , ‡ a Instituto de F´ısica Corpuscular (IFIC) and Departamento de F´ısica Te´oricaCentro Mixto Universitat de Val`encia-CSIC, Dr. Moliner 50, E-46100 Burjassot, Spain b Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland c Department of Physics, The University of Texas at Arlington, TX 76019, USA d Departamento de F´ısica A.I.A.N., Universidad Polit´ecnica de Madrid, E-28040 Madrid, Spain e Centro de Astrof´ısica e Gravita¸c˜ao - CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico -IST, Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal
Abstract
In this paper, analogies between multiparticle production in high-energy collisions and the timeevolution of the early universe are discussed. A common explanation is put forward under the as-sumption of an unconventional early state: a rapidly expanding universe before recombination (lastscattering surface), followed by the CMB, later evolving up to present days, versus the formationof hidden/dark states in hadronic collisions followed by a conventional QCD parton shower yieldingfinal-state particles. In particular, long-range angular correlations are considered pointing out deepconnections between the two physical cases potentially useful for the discovery of new physics.
August 26, 2020 ∗ E-mail address: Miguel.Angel.Sanchis@ific.uv.es ∗∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] a r X i v : . [ h e p - ph ] A ug Introduction
The study of correlations has decisively contributed to the advancement of scientific knowledge in allbranches of physics, from condensed matter and quantum information to particle physics and cosmology.In the latter case, the systematic study of correlations was considered in the context of the large structureof the universe [1]. In this sense, the homogeneity, thus long-range angular correlations of the cosmicmicrowave background (CMB) across the sky seen by the WMAP and Planck missions [2, 3], stronglysupports an inflationary era of the early universe [4, 5]. This proposal gave birth to a new paradigm inastrophysics and cosmology ultimately leading to the Standard Cosmological Model (ΛCDM).Similarly, the study of angular correlations in high-energy collisions has traditionally been a commontool to understand multiparticle production in particle collisions beginning with early studies of cosmicrays through to current investigations at the LHC. In earlier papers we show that one consequence of theproduction of a new still unknown stage of matter in high-energy hadronic collisions is to enhance long-range angular correlations among final-state particles [6–8]. This conclusion bears a certain resemblancewith the observed small temperature fluctuations of the CMB requiring an inflationary period right afterthe Big Bang.As is well known, analogies between different fields of knowledge have traditionally played an im-portant role in the advance of science. A paradigmatic example in physics is provided by the analogybetween superconductivity in condensed matter physics, and the vacuum screening currents leading tothe Higgs mechanism in elementary particle physics [9]. Although the physical origin may be totallydistinct (Cooper electron pairs versus a Higgs quantum field current), a mapping can be establishedbetween the equations governing both processes, as well as some specific relations between the theoryparameters. Actually, such an analogy proved to be a useful guide for getting a deeper understandingof the origin of mass and further developments of the electroweak theory.The main goal of this paper is showing an analogy between the cosmic evolution of the universe andmultiparticle production in high-energy collisions, as well as its consequences as a new way of huntinghidden/dark matter at the LHC and other future facilities. A caveat is in order however: in the formercase there is only one universe (ours) to be observed, while in the latter a large number of independentcollisions are statistically considered altogether. This difference should not alter the main consequencesof our analogy.
According to the firstly postulated Big Bang Theory, the angular scale of the horizon on the last scatteringsurface (when the CMB was emitted) should be θ (cid:39) ◦ [10]. This implies that strong temperatureinhomogeneities should show up above this scale in contrast to real measurements which reveal anextremely isotropic and homogeneous microwave background. To solve this problem, an inflationaryera in the very early universe was proposed, flattening all fluctuations up to very large opening anglescovering the entire celestial sphere.On the other hand, the emergence of large-scale features in the CMB are attributed to densityfluctuations in the early universe evolving into the large-scale structure as we see today. In fact, it iscommon wisdom that such small temperature fluctuations (of the order of 10 − K) are the seeds of thecurrent observed galaxy distributions, galaxy clusters and higher macro-structures of our universe.Two categories of temperature fluctuations observed in the CMB can be distinguished according tothe universal time evolution: (a) primary anisotropies, prior to decoupling, and (b) secondary anisotropiesdeveloping as the CMB propagates from the surface of the last scattering to the observer. The formerinclude temperature inhomogeneities due to photon propagation under metric fluctuations, the so-calledSachs-Wolfe (SW) effect. This effect shows up at rather large angles, i.e. for θ (cid:29) ◦ , where θ standsfor the angular separation of different directions in the present sky. Moreover, the primitive plasma also1nderwent acoustic oscillations prior to decoupling associated to a typical angular scale θ (cid:46) ◦ .On the other hand, once photons decoupled from baryons after recombination, the CMB propagatedthrough a large structure where the gravitational and inter-cluster gas which are not be necessarilyisotropic nor homogeneous on small spatial scales. Examples of such secondary anisotropies of the CMBinclude the Sunyaev-Zeldovich effect due to thermal electrons, and the integrated SW effect, induced bythe time variation of gravitational potentials. These effects are mainly expected to produce temperaturefluctuations on arc-minute scales. In this work, we shall consider them altogether under a commonparametrization of very-short-range correlations.Let us emphasize that what matters in our analogy on angular correlations is the existence of twowell differentiated steps in the evolution of the universe, before and after recombination. Therefore,rather than modeling an inflationary epoch in the primitive universe, we sill assume a linearly expandinguniverse whose scale factor reads: a ( t ) = t/t f , where t stands for the universal time and t f for the timeelapsed since the Big Bang to present. In Refs. [11, 12] a model of this kind was proposed and developedto explain the observed correlations of the CMB. Then, the maximum fluctuation size at any given time t can be estimated as λ max = 2 πR ( t ) with R as the cosmic horizon radius.Following the reasoning of [11], the comoving distance to the last scattering surface (at recombinationtime t rec ) reads r rec = ct f (cid:90) t f t rec dtt = ct f ln (cid:20) t f t rec (cid:21) . (1)Thus the maximum angular size θ of fluctuations associated to the CMB emitted at t rec is given by θ = λ max R ( t rec ) , (2)where R ( t rec ) = a ( t rec ) r rec = ct rec ln (cid:20) t f t rec (cid:21) . (3)Finally, one gets θ ∼ π ln [ t f /t rec ] (cid:39) π , (4)where the numerical estimate corresponds to t f = 13 . t rec = 3 . × yr. This value roughlyagrees with the curve determined from Planck data ( (cid:39) π/
3) as shown in [11]. Let us remark that Eq.(4) is considered here as a simple indicator of long-range angular correlations in the CMB, to be later“translated” to high-energy hadronic collisions.
Long-range angular correlations (both in pseudorapidity and azimuth) also show up in multiparticleproduction in both pp and heavy-ion collisions [13]. From general arguments based on causality, suchlong-range correlations can be traced back to the very early times after the primary parton-partonhadronic interactions. As stressed in Ref. [6], if the parton shower were to be altered by the presence ofa non-conventional state of matter, final-state particle correlations should be sensitive to it.We focus on strongly interacting dark sectors arising in a wide variety of new physics scenarios like e.g.the Hidden Valley [14–16]. Hidden Valley models predict the existence of a hidden/dark sector connectedto the Standard Model (SM) of particle physics through heavy mediators via different mechanisms (tree-level, higher loop diagrams). One of the most interesting situations from a phenomenological viewpointcorresponds to a QCD-inspired scenario with a hidden (running) coupling constant and a confinementscale Λ h . Then the hidden/dark quarks, which could be much lighter than the energy scale set by theheavy mediator M h , can form bound states at Λ h as hadrons in QCD. Usually the masses of the hidden2ector particles are assumed to lie below the electroweak scale while the mediators may have TeV-scalemasses. Therefore, it seems quite natural to expect large hierarchies between Λ h and the hidden quarkmasses m h [17]: m h ∼ Λ h (cid:28) M h . (5)where the condition on the relative value of Λ h ∼ m h can be taken quite loosely without changing themain conclusions of this paper.In our numerical estimates we assume that the strongly coupled hidden sector includes some familiesof hidden quarks that bind into hidden hadrons at energies below Λ h ∼ O (10) GeV, playing a similarrole as Λ QCD in the conventional strong interaction. Such a simplified picture is compatible with theexpected walking behaviour requiring a strong coupling over a large energy window before reaching Λ h ,thereby yielding a large number of hidden partons and ultimately high multiplicity events where theprimary energy is democratically shared by final-state particles.On the other hand, the SM sector could feebly couple to the hidden sector (and the equivalenthadronic hidden particles and states) with a substantial freedom in the form of the portal interaction:via a tree-level neutral Z (cid:48) or higher-order loops involving particles with charges of both SM and hiddensectors. In fact, for some (reasonable) values of the parameter space in hidden valley models, hiddenparticles can promptly decay back into SM particles, altering the subsequent conventional parton shower[16] and yielding (among others [17]) observable consequences, e.g. extremely long-range correlationsespecially in azimuthal space [7].The maximum length, i.e. time , of a parton shower initiated by a parton, down to a low virtualityscale Q , can be estimated as [18] L max (cid:39) EQ , (6)where E stands for the typical energy of the parton cascade; Q is expected to be of the order of Λ QCD for a conventional QCD cascade and of the order of Λ h for a dark cascade.In Fig. 1, we show pictorially the foreseen evolution after a primary hard parton-parton interactionproducing a hidden shower as a first stage of the cascade ultimately yielding final-state particles via aQCD parton cascade. Three steps can be distinguished: • Production of heavy mediators of mass O (10 ) GeV in the primary partonic collision. This isassumed to occur at a tiny fraction of a second ( (cid:39) c ), fixed by the energy scale M h . • Hidden shower and formation of hidden bound states (equivalent to recombination in cosmology),at typical time t h = M h /Q , where Q stands for the virtuality of the hidden shower. Assumingthat Q is of the order of Λ (cid:39)
100 GeV , one gets a time scale of the order of 1 fm/ c . • Once the hidden bound states (or a part of them) decay back to QCD partons (quarks and gluons),a “conventional” cascade takes place with typical time t QCD = Λ h / Λ ∼ several fm /c , wherewe have assumed that the typical energy of the now conventional parton shower is provided byΛ h ∼ m h . Since the successful running of the heavy-ion program at the LHC, it has become popular to compare theevolution of the universe, some seconds after the Big Bang, with the formation of very dense matter athigh temperature (presumably forming a soup of quarks and gluons) in hadronic collisions. It has evenbecome customary talking somewhat loosely about a “little Big Bang” at the LHC. Moreover, such aparallelism between the space-time developments of heavy-ion collisions and the early universe has been Hereafter natural units, c = (cid:126) = 1, will be used unless otherwise stated. am1(1 h M )fm1( Mtc hadro ni zation )fmseveral( tc d e t e c t o r s m1 Equivalent to recombination in cosmology
Inflation + early universe expansion
QCD cascade
Not at scale
Final-state particles
BoundHiddenstates
Hidden/dark cascade virtualityfinalparton, initial theofenergy,
QEQEtc pp collisions Figure 1: Time evolution of the parton shower in high-energy collisions with formation of an initialhidden stage of matter evolving into bound states (equivalent to recombination in cosmology) after atime t , followed by a decay back to SM quarks and gluon partons lasting about t QCD . There could alsobe hidden particles not decaying back to SM and therefore not detected, represented as dashed arrows.considered beyond purely outreach purposes as a source of physical inspiration, see e.g. Ref. [19]. In arecent paper [20], the authors established a correspondence between high-energy collions at future e + e − colliders (ILC and CLIC) and the CMB map.It should be mentioned that physics underlying angular correlations is completely different in thetwo cases: the cosmological evolution is fundamentally described by General Relativity whereas theparton cascade evolution in high-energy collisions is essentially governed by conventional or hiddenstrong interaction dynamics. However, on the one hand they share a common fact put forward toexplain long-range correlations: a rapidly growing initial-state. On the other hand, the typical values ofthe angular scales are (by coincidence) quite similar as we shall see. Indeed, primary long-range angularcorrelations are of the order of one radian, while secondary scales lie one order of magnitude or morebelow. Such a numerical concordance of scales, together with the fact that the time evolution in bothcases is not continuous but rather involves different well-defined steps, makes a connection between thetwo cases.Now, turning to high-energy collisions and naively applying the same expression (3) used for aparticular cosmological model, setting t h = M h / Λ , t QCD = Λ h / Λ and t f (cid:39) t h + t QCD (see Fig. 1),we get for the maximal azimuthal correlation angle φ (cid:39) π ln [ t f /t h ] (cid:39) π (7)for Λ h (cid:39)
10 GeV and M h (cid:39) .Such an order-of-magnitude estimate is in agreement with our earlier estimates [8] about the expected Of course, the actual situation in high-energy particle collisions at colliders is not the same as in an expanding universe,where space itself is being created as the expansion goes on. Nevertheless, one can still keep in mind the picture of agrowing particle horizon to be identified somehow with the radius of a growing sphere containing the developing partoncascade inside. (cid:126)n and (cid:126)n of the sky, the 2-point correlation function can be written as the ensemble-average product: C (cos ∆ θ ) = (cid:104) T ( (cid:126)n ) T ( (cid:126)n ) (cid:105) , (8)where isotropy and homogeneity of space have been assumed, and cos ∆ θ = (cid:126)n · (cid:126)n . It measures theconditional probability of having two CMB temperatures in the sky plane differing by ∆ θ = θ − θ .Furthermore, the 3-point angular correlation function is defined as [21] C (cos ∆ θ , cos ∆ θ ) = (cid:104) T ( (cid:126)n ) T ( (cid:126)n ) T ( (cid:126)n ) (cid:105) , (9)where now three different directions in the sky are labelled by three vectors (cid:126)n i , i = 1 , ,
3, and ∆ θ = θ − θ , ∆ θ = θ − θ . Note that actually only two angular differences are independent, here chosen∆ θ and ∆ θ , so that ∆ θ = θ − θ = ∆ θ − ∆ θ .Notice that already since some time ago the study of 3-point correlations has been recognized as apowerful probe of the origin and evolution of structures of the universe, see e.g. [22–24]. Specifically,non-Gaussian contributions to cosmological correlations should play a leading role in understanding thephysics of the early universe, when primordial seeds for large-scale structures were created, and theirsubsequent growth at later times. Interestingly, the correlation function method was recently proposedin Ref. [25] to distinguish between quantum and classical primordial fluctuations in a sense close to ourconsideration.In high-energy collisions, the 2-particle correlation function C ( φ , φ ) is similarly defined where φ i stands for the azimuthal emission angle of particle i measured on the transverse plane of a referenceframe whose z axis corresponds to the beams direction. Under rotation symmetry, the 2-point correlationfunction actually depends only on the azimuthal difference ∆ φ = φ − φ , i.e. C ( φ , φ ) = C (∆ φ ).Again, higher-order correlations are useful as well to get a deeper insight into multiparticle dynamicsin hadronic collisions [26]. A dependence of the correlations on angular differences is expected too, e.g.the 3-particle azimuthal correlation function C (∆ φ , φ ), where ∆ φ = φ − φ , ∆ φ = φ − φ . Letus point out that 3-point/particle correlations constitute the lowest-order statistical tool to check the non-Gaussianity of distributions. Furthermore, they can place strong constraints on underlying clusteringstructures, thereby becoming specially suited to uncover new physics in multiparticle production inparticle collisions as stressed in [27]. Generally speaking, 3-particle angular correlations may suggest theformation of primary clusters [28]; larger cluster sizes imply stronger 3-particle correlations.In our approach to multiparticle production, correlations are modeled by using Gaussian distributionsfor either cluster and final state particle production in high-energy collisions [29]. Thereby we make use ofGaussian widths to parametrize the typical correlation lengths in the different steps of hadron production.As shown in [7], the 3-particle correlation function can be written as: C (∆ φ , ∆ φ ) = 1 (cid:104) N h (cid:105) h (1) (∆ φ , ∆ φ ) + 1 (cid:104) N h (cid:105) h (2) (∆ φ , ∆ φ ) + h (3) (∆ φ , ∆ φ ) , (10)where each term on the r.h.s. represents the correlations due to one, two and three initial sources ofhidden particles, indicated by the upper index of the h -functions, produced in the same initial partonicinteraction; (cid:104) N h (cid:105) denotes the mean number of hidden sources per collision. Note that the h -functionsinclude the angular dependence due to all possible correlations, namely, particle correlations in clusters,cluster correlations and hidden source correlations. 5 - - - - - Δϕ Δ ϕ - - - Δϕ C on - diagonal Figure 2:
Left : Contour plot of the 3-particle correlation function C (∆ φ , ∆ φ ) of a 3-step cas-cade with an initial long-range contribution from a hidden sector; Right : Diagonal projection of C (∆ φ , ∆ φ ), where the peak at ∆ φ = ∆ φ = 0 is normalized to unity. The dotted (red), dashed(magenta) and thin solid (blue) curves show the contributions from one, two and three hidden parti-cles, respectively. The weighted sum is shown by the thick (turquoise) curve. Plots are taken from ourwork [7].The bigger long-range correlations for the 3-particle correlation function C (∆ φ , ∆ φ ) in a 3-stepcascade are given by the h (3) (∆ φ , ∆ φ ) term, associated to three initial hidden/dark particles: h (3) (∆ φ , ∆ φ ) ∼ exp (cid:34) − (∆ φ ) + (∆ φ ) − ∆ φ ∆ φ δ φ + δ φ (cid:35) (11)+ exp (cid:34) − (∆ φ ) δ φ + δ φ ) (cid:35) + exp (cid:34) − (∆ φ ) δ φ + δ φ ) (cid:35) + exp (cid:34) − (∆ φ ) + (∆ φ ) − φ ∆ φ δ φ + δ φ ) (cid:35) . Here, δ h φ and δ c φ represent the expected correlation length due to the first and second steps in theevolution of the parton cascade using a simplified model. In turn, correlations of particles from clustersare parametrized by δ φ , which can be referred to as the cluster decay width in the transverse plane(see [7, 29]). The full set of expressions for the 3-particle correlation function C (∆ φ , ∆ φ ) in a 3-stepcascade process can be found in [7].The plots of the 3-particle correlation function obtained from numerical estimates of a 3-step cascadewith an initial long-range contribution from a hidden sector are shown in Fig. 2. The left panel showsa typical (spiderweb) structure of the 3-particle correlations in a 2-dimensional (∆ φ , ∆ φ ) plot. Theright panel shows its projection along the diagonal ∆ φ = ∆ φ . The latter plot stands for 3-particlecorrelations disentangling the diverse sources of short-range and long-range angular correlations.Turning to the cosmological analogy, we apply the same analysis of angular correlations in the CMBas employed in high-energy collisions with the following correspondence in our Gaussian parametrization: δ h φ → δ h θ for long-range correlations, δ c φ → δ c θ for short-range correlations, and δ φ → δ θ for very-short-range (within-cluster) correlations, respectively.As commented above, two kinds of anisotropies can be distinguished in the CMB:(1) Primary anisotropies produced prior to recombination/decoupling, yielding rather long-range cor-relations mainly due to the SW effect. In our parametrization it corresponds to δ h θ (cid:39) δ c θ (cid:46) ◦ . 6 - - - - - Δθ Δ θ - - - Δθ C on - diagonal Figure 3:
Left : Contour plot of the 3-point correlation function C (∆ θ , ∆ θ ) in the cosmological caseusing a toy model to take into account different sources of short-range and long-range correlations; Right :Diagonal projection of C (∆ θ , ∆ θ ) showing the different contributions in analogy to Fig.2.(2) Secondary anisotropies developing as the CMB propagates from the last scattering surface to thepresent observer leading to very-short-range angular correlations. The set of such effects yields inour parametrization to δ θ (cid:28) ◦ .Notice that the angular scales showing up as anisotropies in the CBM are not too much differentfrom the expected azimuthal scales stemming from multiparticle production via a hidden sector on topof the partonic shower in high-energy collisions [7].Figure 3 shows the 3-point correlation function C (∆ θ , ∆ θ ) plots for cosmological estimates usinga simple model which takes into account the overall sources of short-range and long-range correlations.The left panel shows the 2-dimensional plot of C (∆ θ , ∆ θ ) as a function of ∆ θ and ∆ θ while theright panel shows the on-diagonal projection of the function. This figure is analogous to Fig.2 for high-energy collisions. Again, the 3-point correlation function is arbitrarily normalized to unity at ∆ θ =∆ θ = 0 since we are here interested rather in disentangling the different sources to angular correlations.Of course, a more realistic study should incorporate the absolute normalisation and relative weights usinga more detailed model. Note that the h (∆ θ , ∆ θ ) functions, i.e. the equivalent cosmological terms inEq.(10) for high-energy collisions, are similarly sensitive to very-short-range, short-range and long-rangecorrelations, respectively. This is an important result of our work.By comparing Figs. 2 and 3, an equivalent structure can be appreciated in both panels as expectedfrom the common existence of angular short-range and long-range correlations, no matter their physicalorigin. Again, as in the case of high-energy hadronic collisions, several distinct correlation scales clearlyshow up: short-range correlation lengths (secondary angular correlations), and a long-range correlationlength (primary angular correlations) associated to the early epoch of the universe. The diagonal pro-jection suggests a pattern which might be useful to disentangle possible sources of angular correlationspresent in the CMB. Short-range and very-short-range correlations are behind the peak structure whilelonger correlations determine the smooth falling off. Further detailed structure can vary depending onthe different underlying effects, but the overall behaviour is expected to be quite similar. In this paper, we discuss an intriguing similarity between long-range angular correlations observed in theCMB and those obtained from multiparticle production in high-energy collisions. Although the physical7rigin of such long-range angular correlations is completely different in the two physical situations,the analogy is supported by the following facts: the time evolution in both cases (yielding complexstructures from a primitive state of matter, either galaxies or final-state particles) is not continuousbut rather involves different well-defined steps, with similar angular scales. Based on this observationa common explanation has been proposed upon the assumption of the existence of an unconventionalearly state: an expanding universe before recombination/decoupling (last scattering surface), wherethe CMB was released, evolving up to present days, versus the formation of hidden/dark states inhadronic collisions followed by a conventional QCD cascade resulting in final-state particles. Usingsimple modeling, we show that 3-point/3-particle correlations should be a useful tool to disentanglethe different contributions to short-range and long-range correlations in the universe evolution or inmultiparticle production, highlighting deep connections between both fields in the search for new physicsand phenomena either at the LHC or future accelerators.
Acknowledgments
This work has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades,under grant FPA2017-84543-P and by Generalitat Valenciana under grant PROMETEO/2019/113 (EX-PEDITE). N.S-G is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) projects PTDC/FIS-OUT/28407/2017 and UID/FIS/00099/2020 (CENTRA), CERN/FIS-PAR/0027/2019, and by the Euro-pean Union’s Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 GrantNo. FunFiCO-777740.
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