New test of Lorentz symmetry using ultrahigh-energy cosmic rays
NNew test of Lorentz symmetry using ultrahigh-energy cosmic rays
Luis A. Anchordoqui
1, 2, 3 and Jorge F. Soriano
1, 2 Department of Physics & Astronomy, Lehman College, City University of New York, NY 10468, USA Department of Physics, Graduate Center, City University of New York, NY 10016, USA Department of Astrophysics, American Museum of Natural History, NY 10024, USA
We propose an innovative test of Lorentz symmetry by observing pairs of simultaneous parallel exten-sive air showers produced by the fragments of ultrahigh-energy cosmic ray nuclei which disintegratedin collisions with solar photons. We show that the search for a cross-correlation of showers in arrivaltime and direction becomes background free for an angular scale (cid:46) ◦ and a time window O (10 s).We also show that if the solar photo-disintegration probability of helium is O (10 − . ) then the hunt forspatiotemporal coincident showers could be within range of existing cosmic ray facilities, such as thePierre Auger Observatory. We demonstrate that the actual observation of a few events can be used toconstrain Lorentz violating dispersion relations of the nucleon. Ever since Greisen, Zatsepin, and Kuzmin (GZK)pointed out that the pervasive radiation fields make theuniverse opaque to the propagation of ultrahigh-energy( E (cid:38) GeV) cosmic rays (UHECRs) [1, 2], it became ev-ident that the actual observation of the GZK e ff ect wouldprovide strong constraints on Lorentz invariant break-ing e ff ects. This is because if Lorentz invariance is bro-ken in the form of non-standard dispersion relations forvarious particles, then absorption and energy loss pro-cesses for UHECR interactions would be modified; seee.g. [3–11]. In particular, the GZK interactions (photo-pion production and nucleus photo-disintegration) arecharacterized by well defined energy thresholds (nearthe excitation of the ∆ + (1232) and the giant dipole reso-nance, respectively), which can be predicted on the basisof Lorentz invariance. Therefore, the experimental con-firmation that UHECR processes occur at the expectedenergy thresholds can be considered as an indirect pieceof evidence supporting Lorentz symmetry under colos-sal boost transformations.A suppression in the UHECR flux at E (cid:38) . GeVhas been established beyond no doubt by the HiRes [12],Auger [13], and Telescope Array (TA) [14] experiments.By now (in Auger data) the suppression has reached astatistical significance of more than 20 σ [15]. This sup-pression is consistent with the GZK prediction that inter-actions with universal photon fields will rapidly degradethe energy of UHECRs. Intriguingly, however, there arealso indications that the source of the suppression maybe more complex than originally anticipated.Observations of the rate of change with energy of themean depth-of-shower-maximum X max seem to indicatethat the cosmic ray composition becomes lighter as en-ergy increases toward E ∼ . GeV from below [16],fueling a widespread supposition that extragalactic cos-mic rays are primarily protons. However, Auger high-quality, high-statistics data, when interpreted with theleading LHC-tuned shower models, exhibit a strong like-lihood for a composition that becomes gradually heav-ier with increasing energy; namely, 1 . (cid:46) (cid:104) ln A (cid:105) (cid:46) . (cid:46) E (cid:46) . [17–20]. Within uncertainties, thedata from TA are consistent with these findings [21, 22]. For E (cid:38) . GeV, the indication of an anisotropy atan intermediate angular scale of 13 ◦ (significant at the4 . σ level [23]) [24] points to a similar nuclear composi-tion. Note that for E / Z = GeV, typical deflections ofUHECRs crossing the Galaxy are about 10 ◦ , where Ze isthe nucleus charge [25].For a uniform source distribution, the simultaneousfit to the UHECR spectrum and composition ( X max andits fluctuations) imposes severe constraints on modelparameters: (i) hard source spectra and (ii) a maxi-mum acceleration energy E max (cid:46) . Z GeV [26–28].Hence, under the assumption of a uniform source distri-bution, the data seem to favor the so-called “ disappoint-ing model” [29] wherein it is postulated that the “end-ofsteam” for cosmic accelerators is coincidentally near theputative GZK cuto ff , with the exact energy cuto ff deter-mined by data. This interpretation encompasses a radi-cally di ff erent viewpoint in which the maximum energyof the most powerful cosmic ray accelerators would beobserved for the first time, and therefore could call intoquestion limits on the violation of Lorentz invariancededuced using the observed suppression in the UHECRspectrum [30–32].Very recently, one of us put forward a multi-dimensional reconstruction of the individual emissionspectra (in energy, arrival direction, and nuclear com-position) to study the hypothesis that primaries areheavy nuclei subject to GZK photo-disintegration, andto determine the nature of the extragalactic sources [33].In this paper we introduce an alternative approach toprobe Lorentz invariance using UHECRs. We propose tosearch for a cross-correlation in arrival time and directionof the secondary nucleon (of energy E / A ) produced viaphoto-disintegration of an UHECR nucleus (of energy E and baryon number A ) and the associated survivingfragment (of baryon number A − (i) the Lorentz factor (which isequivalent to energy per nucleon) is conserved for photo-disintegration and (ii) the trajectory of cosmic rays withina magnetic field is only rigidity-dependent; the relevantquantity for the separation among fragments (hereafteridentified with subindices 1 and 2) is | Z / A − Z / A | . a r X i v : . [ h e p - ph ] F e b A simple dimensional argument constrains the dis-tance to the photo-disintegration site. Assuming theenergy di ff erence between nucleons inside the nucleusis given by the binding energy E ∼ MeV, the di ff erencein velocity of the secondary products is δ v = (cid:112) E / M ∼ (cid:112) − / A , (1)where M (cid:39) A GeV is the mass of the parent nucleus. Thedi ff erence in the time of flight of the secondary productsis then δ t ∼ δ L = ( L / Mpc) γ δ v × cm , (2)where L is the distance to the photo-disintegration siteand γ ( = E / M at Earth) contracts this length. For a si-multaneous observation of the two secondaries at Earth,we demand δ L (cid:46) R ⊕ ( ∼ cm), which yields γ ∼ √ A ( L / Mpc) . (3)For the particular range 10 (cid:46) γ (cid:46) , whichspans the UHECR spectrum, (3) constrains the photo-disintegration site to a distance (cid:46) kpc. It has long beenknown that UHECR nuclei scattering o ff the universalradiation fields have a mean free path (cid:29) kpc [34]. More-over, we know the devil is in the detail and so the numberof GZK interactions which would lead to a simultane-ous observation of their secondary products on Earth isessentially negligible.Of particular interest here, UHECR nuclei en route to Earth also interact with the solar radiation fieldand photo-disintegrate [35, 36]. The nuclear photo-disintegration process has two characteristic regimes.There is the domain of the giant dipole resonance (GDR),where a collective nuclear mode is excited with thesubsequent emission of one (or possible two nucleons),and the high energy plateau, where the excited nucleusdecays dominantly by two nucleon and multi-nucleonemission. The energy range of the GDR in the nucleusrest frame spans 10 (cid:46) ε (cid:48) / MeV (cid:46)
30, and the plateauextends up to the photo-pion production threshold (i.e.,photon energy ε (cid:48) ∼
150 MeV).The background radiation field can be described bya Planckian spectrum, with a temperature of the solarsurface T s (cid:39) . L (cid:12) = π r c (cid:82) d ε ε dn / d ε , yielding dnd ε = . × ε exp( ε/ T s ) − (cid:18) r AU (cid:19) − (eV cm) − , (4)where r is the spherical radial coordinate centered at theSun. In the rest frame of the nucleus, the energy ε ofthe solar photons (in the rest frame of the Sun) is highlyblue-shifted to ε (cid:48) = εγ (1 + β cos α ) ∼ γ ε c α/ , (5) where β = (cid:112) − /γ ∼ c α/ = cos( α ( (cid:96) ) / α ( (cid:96) ) is the angle between the momenta of photonand nucleus in the Sun’s reference frame, with (cid:96) thecoordinate along the path of the nucleus; i.e., cos α = ˆ (cid:96) · ˆ r .The GDR cross section in the narrow width approxi-mation is σ ( ε ) = π σ Γ δ (2 γ ε c α/ − ε ) , (6)where Γ and σ are the GDR width and cross section atmaximum; the factor of 1 / σ = . A mb, Γ = ε = . A − . MeV for A > ε = . A . MeV for A ≤ σ ( ε ) ≈ A / η A = − exp (cid:34) − (cid:90) ∞ d (cid:96) λ ( (cid:96) ) (cid:35) , (7)where 1 λ ( (cid:96) ) = (cid:90) ∞ σ ( ε ) dnd ε c α/ d ε (8)is the inverse photo-disintegration mean-free-path [39].Integration of (7) yields: 10 − (cid:46) η A (cid:46) − for iron, and10 − (cid:46) η A (cid:46) − for helium and oxygen. These valuesof η A are in agreement with the estimates in [39–41].Since the secondary fragments have slightly di ff erentrigidities the deflection in the interplanetary magneticfield will result in two separate extensive air showers, ar-riving essentially at the same time and from the same di-rection in the sky [39, 40]. More specifically, the averageseparation of the shower on Earth can be parametrizedby [41] (cid:104) δ L (cid:105) A = A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z A − Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) E GeV (cid:19) − km , (9)where E is the energy of the parent nucleus. The averageseparation of showers as estimated in [40] is somewhatsmaller. For a given experiment, each nuclear species hasa critical energy above which (cid:104) δ L (cid:105) A would be comparableto the size of the instrumented area. As benchmark weconsider a 3 ,
000 km array of detectors, with interspac-ing of about 1.5 km. For He, (9) yields (cid:104) δ L (cid:105) He ∼
50 km at E ∼ . GeV. However, for Fe, at the same energy (9)leads to (cid:104) δ L (cid:105) Fe ∼
260 km, and so the separation distancebetween the showers would be out of detection range.Because the intensity of cosmic rays is steeply fallingwith energy, contributions from the counting rate at thecritical energy dominate the integrated event rate. Exist-ing estimates of the event rate at UHECR facilities [39–42]are subject to large uncertainties, mainly because η A and (cid:104) δ L (cid:105) A depend strongly on A and the nuclear compositionof UHECRs is poorly known.Herein, we assume a nuclear composition dominatedby helium at E (cid:38) . GeV that becomes gradually heav-ier with increasing energy; see e.g. Fig. 4 of [27]. Wefurther assume that the photo-disintegration probabilityof helium on the solar photons is η He ∼ − . . These twoassumptions together lead to an expected integrated fluxof dFdt d Ω dA ( E > . GeV) ∼ × − km − sr − yr − , (10)where E denotes the energy of the parent nucleus. Thisflux is in agreement with the one shown in Fig. 3 of [40].Moreover, as exhibited in Fig. 2 of [40], for η He ∼ − . and E (cid:38) GeV, we have 20 (cid:46) (cid:104) δ L (cid:105) He / km (cid:46)
50. Theflux derived herein, using a helium saturated spectrumabove 10 . GeV, is larger than the intensity derivedin [41] using the spectrum of [43]. Whichever flux cal-culation one may find more convincing, it seems mostconservative at this point to depend on experiment (ifpossible) to resolve the issue.The 3 ,
000 km surface detector array of the PierreAuger Observatory is fully e ffi cient at E (cid:38) . GeV [44].From January 2004 until December 2016 this facility hasaccumulated an exposure [24] E ( E > . GeV) = . × km sr yr . (11)At lower energies, the trigger e ffi ciency of the surfacedetector array decreases smoothly and becomes roughly30% at 10 . GeV [44]. To get a rough estimate of theexposure available to probe spatiotemporal correlationsof air showers in an experiment like Auger we scaledown E ( E > . GeV) by a factor of 0.3. This leads to E (10 . < E / GeV < . ) (cid:38) × km sr yr . (12)For He, ∆ E = E − E ∼ γ GeV. For E > . GeV,(10) and (12) lead to an expected integrated rate whichis consistent with 1 event.It is clear that for a signal O (1) event we must learnhow to properly conduct background rejection to ascer-tain whether the observation of a few events is due to physics or statistics . Moreover, to calculate a meaning-ful statistical significance in the shower cross-correlationanalysis, it is important to define the search procedure a priori in order to ensure it is not inadvertently devisedespecially to suit the particular data set after having stud-ied it. With the aim of avoiding accidental bias on thenumber of trials performed in selecting the cuts, we nowconduct a phenomenological analysis of the potentialbackground to define the angular and temporal cuts.We start by selecting a reference direction on the sky d . We define θ as the angle between d and other direc-tion on the sky d . We define φ as the angular distancebetween a reference axis, placed on the normal plane tothe vector pointing in the direction d , and the projection on that plane of a vector pointing towards d . With thisconstruction, θ ∈ [0 , π ] and φ ∈ [0 , π ].The expected fraction of events that will be containedin a cap of the sphere of within an angle α to the direction d , for all φ , and in a time interval t is f ( α, t ) = tT (cid:90) π d φ (cid:90) α d θ π sin θ = tT (1 − cos α ) , (13)where T is the time span for the experiment. In a sampleof N events, we expect µ ( α, t ) = N f ( α, t ) events in theangle-time window defined above. The actual numberof events in that window will be distributed following aPoisson distribution of mean µ ( α, t ). The probability ofobserving k events in an angle-time window is then p k ( α, t ) = e − µ ( α, t ) µ ( α, t ) k k ! . (14)In Fig. 1 we show the function log p ( α, t ), for α ∈ [0 ◦ , ◦ ] and t ∈ [0 s ,
10 s], which gives the probabilityof measuring 2 events in an angle-time window specifiedby the pair ( α, t ). Since (cid:80) ∞ k = p k ( α, t ) p ( α, t ) (cid:46) − (15)in our ( α, t ) range of interest, p ( α, t ) practically accountsfor the probability of having not only two, but anyamount of events above one.The quantity p ( α, t ) is then the p -value for observinga coincidence of two detections in a background onlyhypothesis. To quantify this in a more comprehensibleway, one can use the usual relation between p -values and σ levels following a normal distribution p = (cid:34) − erf (cid:32) z √ (cid:33)(cid:35) , (16)being z the number of standard deviations from themean. The relation between p and z is shown in Fig. 2.One can check by inspection that for the whole range of α and t , the p -value for observing two or more eventstogether in a small angle-time window is a more than5-sigma e ff ect against the background. Hence, the ac-tual observation of a few pairs of cross-correlated eventswould become the smoking gun to set model indepen-dent constraints on Lorentz invariance violation.Strictly speaking, a nucleus with baryon number A and charge Ze would have a non-standard dispersionrelation of the form E A , Z = p A , Z + M A , Z + ζ A , Z p n + A , Z M n Pl , (17)where E A , Z is the nucleus energy, p A , Z is the absolutevalue of its 3-momentum, and M A , Z its mass. Here, - - - - - - events - - - - - - events - - - - - - events - - - - - -
10 yr10 events - - - - - -
10 yr10 events - - - - - -
10 yr10 events - - - - - -
20 yr10 events - - - - - -
20 yr10 events - - - - - -
20 yr10 events FIG. 1: log p ( α, t ) for di ff erent total number of events (from left ro right N = , , ), and di ff erent lifetimes of the experiment(from top to bottom T / yr = , , T =
10 yr and T =
20 yr are scales compatible with Auger, while T = M Pl ≈ GeV is the Planck mass and ζ A , Z are Lorentz-violating parameters of the nucleus. In the rest frameof the Sun we assume that only baryons have non-standard dispersion relations (note that the solar pho-ton fields have too low energy for Lorentz invariantbreaking e ff ects to be relevant in their dispersion rela-tions) and so one can easily obtain a threshold relation,which constrains the ζ A , Z coe ffi cients when confrontedwith data. Actually, since we expect nuclear physics to have negligible Lorentz breaking e ff ects we can write ζ A , Z = ζ/ A , where ζ regulates deviations from Lorentzsymmetry of the nucleon. The baryon number A ofthe original disintegrated nucleus can simply be deter-mined by estimating the energies of the primaries ofthe two air showers, A = + E / E , where E is theenergy of the less energetic shower. With the < A is obtained with a reso- - - - - - - - FIG. 2: Relation between the p -value and z , the number of stan-dard deviations away from the mean, for a normal distribution. lution σ ( A ) / A < . √ − / A ), which is around 20% fora Helium primary, or σ ( A = ∼ .
85, allowing its dif-ferentiation from other primaries with A around 4. Thisprovides an univocal (model independent) determina-tion of the nuclear composition and thereupon boundsthe threshold energy interval to be compatible with ex-perimental results on photo-nuclear interactions [47–56].For He, the photo-excitation cross section of the GDRhas a threshold ε (cid:48) th ≈
20 MeV [57]. The GDR decays bythe statistical emission of a single nucleon, leaving anexcited daughter nucleus ( A − ∗ . The probability foremission of two (or more) nucleons is smaller by an or-der of magnitude. The excited daughter nuclei typicallyde-excite by emitting one or more photons of energies1 (cid:46) (cid:15) (cid:48) / MeV (cid:46)
5, in the nuclear rest frame [37]. For sim- plicity, herein we neglect the de-excitation process andconsider the photo-disintegration reaction with two in-coming particles (nucleus + photon) and two outgoingparticles (nucleus + nucleon). Though we are primar-ily interested in helium photo-disintegration, the ensu-ing discussion will be framed in a general context. Theenergy-momentum 4-vectors for the four particles in therest frame of the Sun are: ( E , p ), for the incoming nu-cleus; ( ε, k ), for the photon; ( E , p ), for the nucleon; and( E , p ), for the outgoing nucleus. The relation describ-ing the conservation of energy and momentum is givenby ( E + ε ) − ( p + k ) = ( E + E ) − ( p + p ) . (18)We are interested in studying the energy thresholds forwhich the relation (18) holds.According to the threshold theorem, at an upper orlower threshold the incoming particle momenta are alwaysanti-parallel and the final particle momenta are parallel [58].This applies for dispersion relations E ( p ) depending on p ≡ | p | , and being a monotonically increasing functionof that variable, when energy and momentum are con-served additive quantities. Then, to obtain the thresh-old conditions, one can make use of p · k = − p k and p · p = p p . Since we neglect Lorentz invariant break-ing e ff ects on the solar photon fields we take ε = k .In threshold conditions the reaction is collinear and so p − k = p + p . Since k is much smaller than the othermomenta, we have p ≈ p + p . Following [10], we de-fine p = κ p and p = (1 − κ ) p , with 0 < κ <
1. Now,neglecting the mass di ff erence between the proton andthe neutron ( M A , Z = A m p , where m p is the proton mass),the energy conservation relation is found to be ξ A (1) + ε p (cid:104) + (cid:112) + ξ A (1) (cid:105) = κ ξ A − ( κ ) + (1 − κ ) ξ (1 − κ ) + κ (1 − κ ) (cid:104) (cid:112) + ξ A − ( κ ) (cid:112) + ξ (1 − κ ) − (cid:105) , (19)where ξ A ( κ ) = (cid:32) A m p κ p (cid:33) + ζ A (cid:18) κ pM Pl (cid:19) n . (20)Note that for M A , Z (cid:28) p A , Z ≡ p (cid:28) M Pl , ξ A ( κ ) (cid:28)
1. Ex-panding the square roots to first order in the ξ functions, (cid:32) + ε p (cid:33) ξ A (1) + ε p = κ ξ A − ( κ ) + (1 − κ ) ξ (1 − κ ) . (21)Since all the ξ -functions are of the same order and ε (cid:28) p ,the term εξ A (1) / p is negligible in comparison to the restof the terms, and so (21) becomes ξ A (1) + ε p = κ ξ A − ( κ ) + (1 − κ ) ξ (1 − κ ) . (22) After some algebra, (22) can be rewritten as ζ g ( κ ) (cid:32) pm p (cid:33) (cid:18) pM Pl (cid:19) n + ε pm p − [1 − (1 − κ ) A ] κ (1 − κ ) = , (23)where g ( κ ) = A − κ n + ( A − − (1 − κ ) n + . (24)We next consider the threshold configuration for a pho-ton with energy ε (cid:48) th ≈
20 MeV. In the rest frame ofthe Sun, the photon energy is ε th and the UHECR isboosted with speed β in the direction of Earth. Fora head on collision, k points in the opposite direction FIG. 3: Sensitivity to ζ as a function of κ for n = n = ε (cid:48) th =
20 MeV and E = . GeV. Theembedded box details the restricted interval of κ for which ζ >
0. The shaded band indicates the region for which ζ < ζ null . and so the photon energy in the nucleus rest frame is ε (cid:48) = γ ( ε + β k ) = γε (1 + β ). The threshold energy in therest frame of the Sun is then ε th = (cid:115) − β + β ε (cid:48) th . (25)Since p = β E , we can write ζ = [1 − (1 − κ ) A ] κ (1 − κ ) − β E ε (cid:48) th m p (cid:115) − β + β × (cid:16) β E / m p (cid:17) − (cid:0) β E / M Pl (cid:1) − n g ( κ ) . (26)We take E ≈ . GeV and so γ ∼ . With this in mind,we adopt the following expansion (cid:115) − β + β = γ + O (cid:32) γ (cid:33) , (27)and set β ≈ ζ as a functionof κ , ζ = (cid:32) [1 − (1 − κ ) A ] κ (1 − κ ) − A ε (cid:48) th m p (cid:33) ( m p / E ) ( M Pl / E ) n g ( κ ) , (28)where we have used E = γ Am p . As an illustration, inFig. 3 we show the sensitivity for probing ζ as a func-tion of κ , assuming observation of a few spatiotemporalcoincident showers near the critical energy.Despite the assumption of Lorentz invariance viola-tion, we want to preserve the time-like character of phys-ical trajectories. For a particle with four momentum p µ ,this means that p µ p µ > + , − , − , − ) metric signature.Using (17) this condition creates a lower bound ζ > ζ null ,with ζ null ≡ − A (cid:18) m p E (cid:19) (cid:18) M Pl E (cid:19) n , (29) assuming β ≈ ζ . In Fig. 3 we show the limiting value ζ null andthe (shaded) prohibited region. We conclude that witha detection of a few spatiotemporal coincident showerswe will be able to constrain ζ at the level of ζ ∼ × − for n =
0, and ζ ∼ − for n = g ( κ ) ≤ n = ,
1, using (28) and (29) we canrewrite the time-like condition as[1 − (1 − κ ) A ] A κ (1 − κ ) + A g ( κ ) < ε (cid:48) th m p . (30)Using (30) we study the dependence on A and ε (cid:48) th of thelimiting values κ min and κ max , such that the time-likecondition is satisfied for all κ ∈ [ κ min , κ max ]. Note thatnear the limits of the interval [ κ min , κ max ], d ζ/ d κ is largecompared to ε (cid:48) th / m p , for 10 (cid:46) ε (cid:48) th / MeV (cid:46)
20 [57]. Thus,the intervals of κ which satisfy (30) barely depend on ε (cid:48) th , which can be assumed to be zero. For a fixed n , the κ limits only depend on A . The values of κ min and κ max for n = n =
1, the values arewithin a distance of ∼ − of those for n =
0. As can beseen, the values are considerably close to 0 and 1, withintersections at [0 . , .
96] for A = O (10 − . ), then the unambiguousobservation of the extensive air showers that would beproduced almost simultaneously by the secondary frag-ments is within reach of UHECR experiments. This isbecause our analysis of spatiotemporal correlations in-dicates that for angular scales (cid:46) ◦ and a time windowof O (10 s) the signal is background free. Detection of afew events will be enough to constrain Lorentz invari-ant breaking e ff ects in the range 10 (cid:46) γ (cid:46) . Sucha detection will also provide valuable information onthe UHECR nuclear composition, which is independentof the hadronic interaction models used to describe thedevelopment of air showers, and therefore such infor-mation develops complementary to studies of the X max distribution and its fluctuations. FIG. 4: Allowed (shaded) κ region as a function of A , for n = κ min and κ max , respectively. Acknowledgments
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