Ultrahigh-Energy Cosmic Rays Particle Spectra from Crypton Decays

PHYSICAL REVIEW D 74, 115003 (2006) Ultrahigh-energy cosmic rays particle spectra from crypton decays

John Ellis,1 V.E. Mayes,2 and D. V. Nanopoulos2,3 1Theory Division, Physics Department, CERN, CH 1211 Geneva 23, Switzerland 2George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, Texas 77843, USA 3Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, Texas 77381, USA and Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece (Received 13 February 2006; published 7 December 2006) We calculate the spectra of ultrahigh-energy cosmic rays (UHECRs) in an explicit top-down model based on the decays of metastable neutral ‘‘crypton’’ states in a ﬂipped SU(5) string model. For each of the eight speciﬁc 10th-order superpotential operators that might dominate crypton decays, we calculate the spectra of both protons and photons, using a code incorporating supersymmetric evolution of the injected spectra. For all the decay operators, the total UHECR spectra are compatible with the available data. Also, the fractions of photons are compatible with all the published upper limits, but may be detectable in future experiments.

DOI: 10.1103/PhysRevD.74.115003 PACS numbers: 12.60.Jv, 04.60.m, 11.25.Wx, 96.50.S

I. INTRODUCTION heterotic string-derived flipped SU(5) model may have exactly the right properties to play this role [11,12]. The existence of cosmic rays with energies above the These ‘‘flipped cryptons’’ are bound by SU(4) hidden- Greisen-Zatsepin-Kuzmin (GZK) cutoff [1,2] is one of the sector interactions, and include four-constituent metastable most important open problems in high-energy astrophysics bound states called tetrons that are analogues of the three- [3,4]. These ultrahigh-energy cosmic rays (UHECRs) may constituent baryons of QCD, as well as two-constituent be a tantalizing hint of novel and very powerful astrophys- mesonlike states. Indeed, it was within this flipped SU(5) ical accelerators, or they may be harbingers of new micro- model that the confinement solution to avoiding the strin- physics via the decays of metastable supermassive gent experimental limits placed on fractional charges was particles. It is remarkable that we still do not know whether first pointed out, and this model remains the only example the UHECRs originate from macrophysics or microphys- to have been worked through in any detail [11,12]. ics. If there is no GZK cutoff, as suggested by the AGASA In general, tetrons may decay through Nth-order non- data [5], the sources of the UHECRs would need to be renormalizable operators in the superpotential, which local. In this case, since local magnetic fields are unlikely would yield lifetimes that are expected to be of the order of to have deflected significantly their directions of propaga- tion, the UHECRs would ‘‘remember’’ the directions of 2 N 2N3 their sources. Thus, one would expect some anisotropy in string Ms ; (1) the arrival directions of the UHECRs, associated either mX mX with discrete energetic astrophysical sources nearby, such 18 as BL Lac objects [4], or the distribution of (mainly where mX is the tetron mass and MS 10 GeV is the galactic) superheavy dark matter. No significant anisotropy string scale. The -dependent factor reflects the expected of the UHECRs has yet been seen, but the existing experi- dependence of high-order superpotential terms on the ef- ments have insufficient statistics to exclude one at the fective gauge coupling g. The mass scale associated with expected level [6]. On the other hand, the GZK cutoff these states is estimated using the renormalization group may be present in the HiRes data [7], in which case no for the SU(4) interactions to be 1012–1013 GeV, just exotic microphysics may be needed, and any astrophysical in the right range for their decays to produce the UHECRs. sources would be less restricted and more difficult to trace. The lifetimes of neutral tetrons without electric charge 11 17 The first batch of Auger [8] data are inconclusive on the has been estimated to lie in the range 0 10 –10 years, possible existence of the GZK cutoff. so that they may still be present in the Universe today, and Superheavy particles of the type required could have might produce the necessary flux of UHECRs if they are been produced gravitationally around the end of inflation sufficiently abundant. We have shown in our previous work [9]. Particularly interesting candidates for such superheavy [12] that the mesons and charged tetrons—whose present- particles are ‘‘cryptons,’’ bound states of the fractionally day abundances are subject to very stringent experimental charged constituents that arise generically [10] in the hid- limits—would have decayed with short lifetimes early in den sectors of models of particle physics derived from the the Universe. In the course of studying the possible tetron 1 heterotic string. Cryptons arising in the hidden sector of a lifetimes, we identified various 10th-order superpotential operators that might govern neutral tetron decays. Thus, in 1Such states may also be a generic feature of models con- this specific model of flipped cryptons we are able to go structed from intersecting D-branes. beyond generic statements regarding the injected UHECR

1550-7998=2006=74(11)=115003(11) 115003-1 © 2006 The American Physical Society JOHN ELLIS, V.E. MAYES, AND D. V. NANOPOULOS PHYSICAL REVIEW D 74, 115003 (2006) particle spectra that may result from their decays, and right range to produce the UHECR. However, as pointed make a number of specific predictions. out in the introduction, flipped SU(5) cryptons satisfy both This enables us to address an important experimental of these criteria, which makes them very attractive as a top- constraint on such crypton models of the UHECRs. down explanation of the UHECR. Indeed the flipped cryp- Although ‘‘top-down’’ models such as crypton decays ton is probably the most natural and most physically mo- may appear to be natural explanations for the UHECRs tivated of any top-down candidate. (if they exist), they generically share a potential drawback. There are three generic statements that can be presently The spectra of UHECRs that they produce might be ex- made about a decaying superheavy X particle explanation pected to have large photon fractions, in possible conflict for the UHECR: with the observation that most of the UHECR primaries (1) Since the superheavy relics may accumulate locally appear to be protons or nuclei. The Auger collaboration has in the galactic halo with an overdensity 104–5 over recently set an upper limit of 26% on the photon fraction the cosmological average, they may avoid the GZK above 1019 eV [13], and the limit may even be as low as cutoff. 7%–14% [14]), while an upper limit of 50% at energies (2) Because of the displacement of our solar system above 1020 GeV has been set by the AGASA collaboration with respect to the galactic plane, there should be [15]. some anisotropy in the arrival directions of the In this paper, we first give a review of relevant aspects of UHECR with respect to the galactic center. the flipped SU(5) heterotic string model. We then analyze (3) Photons tend to be the dominant component of the the specific primary multibody decay modes governed by UHECR flux produced by the superheavy X decay. the various different 10th-order superpotential operators However, the photons may scatter off the galactic found in our previous paper. We then calculate the radio background, which is poorly measured, and UHECR particle spectra that would be injected by these thus may be somewhat attenuated. decays using one of the most detailed and complete codes The injection spectrum produced by such a decaying currently available [16], paying particular attention to the superheavy relic X particles with number density nX and photon fraction. The total UHECR spectra obtained from lifetime X is proportional to the inclusive decay width: the various superpotential terms do not differ greatly. On halo nX 1 dX ! g1 the other hand, we find that different decay operators may E : (2) give rather different photon fractions, particularly at the X X dE highest energies. However, in every case, we find that the If a spherical halo of radius Rhalo and uniform number calculated spectra after supersymmetric evolution and density nX is assumed, then the galactic halo contribution fragmentation are compatible with the published upper to the UHECR will be given by limits on the photon fractions in various energy ranges, when we include the UHECR background resulting from a halo 1 halo J Rhalo E: (3) homogenous extragalactic distribution of sources and in- 4 corporate the pileup expected from the GZK effect. There In general, the X particles will decay into one or more is no need to appeal to the cosmic radio background to partons which hadronize into other particles g of the mini- absorb a significant fraction of the photons in order to bring mal supersymmetric standard model (MSSM), which carry them below the AGASA and other limits. a fraction x of the maximum available momentum MX=2 and a fraction z of the parton momentum. For such a decay, II. GENERIC SUPERHEAVY RELIC DECAY the inclusive decay width can be factored as Z ! X x 2 2 The basic idea in generic top-down explanations (see 1 d X g1 1 da y; ;MX [17,18]) is that the UHECR are produced via the decay of X dE a 0 a dy yx=z some relic particles or topological defects left over from g 2 Daz; ; (4) the inflationary epoch and which are locally clustered in g 2 the galactic halo as cold, dark matter with an overdensity where Daz; is the fragmentation function (FF) for cos 4–5 nX=nX 10 . The lifetime of such relics must exceed particles of type g into particles of type a, and is the the present age of the Universe in order for them to exist energy scale, most appropriately taken to be equal to the X today in sufficient abundance, however the lifetime must particle mass, MX. The evolution of the fragmenta- not be too large so that the decay rates produced are too tion function is governed by the Dokshitzer-Gribov- small to produce the UHECR. Furthermore, the relic mass Lipatov-Altarelli-Parisi (DGLAP) equations, which may 12 must be at least MX > 10 GeV in order to produce the be extended to include the MSSM. Thus, the determination UHECR energies observed. Typically, the lifetime of a of the expected UHECR flux from the superheavy X decay particle is expected to be inversely proportional to its essentially becomes the problem of starting with a set of mass, 1=M. Clearly it is not easy to have a particle initial decay partons and evolving the decay cascades via with both a large enough mass and a decay lifetime in the the fragmentation functions to find the end decay products

115003-2 ULTRAHIGH-ENERGY COSMIC RAYS PARTICLE SPECTRA ... PHYSICAL REVIEW D 74, 115003 (2006) and energy distribution. To evolve the fragmentation func- derived flipped SU(5) remains among the most fully de- tions up to the energy of the superheavy X decay, the veloped and realistic models yet derived from string. DGLAP equations must be solved numerically. Several Flipped SU(5) derives its name from the flipping of the groups have done such calculations for generic initial quark and lepton assignments relative to those in the c c c decay partons (usually into a quark-antiquark pair) [19– minimal SU(5) GUT [24]: uL;uL $ dL;dL and L;L $ c 22]. Perhaps the best such code is SHdecay [16]. This eL;eL for all three generations. This requires the gauge code calculates the fragmentation into the seven stable group to become SU5U1, so as to accommodate the MSSM particles (p, , e, neutralino LSP , e, , ) charges of the quarks and leptons, which fill up the 5, 10, and 1 representations of SU(5): for any given initial decay parton. In the case of flipped 0 1 c SU(5) cryptons, we have a specific model where the initial u1 decay partons in the cascade are known. B uc C B 2 C B c C u c c In addition to the UHECR flux produced by the super- f u ; F d ; 5 B 3 C 10 d L L heavy decay from X particles clustered in our galactic halo, @ e A L (5) there may be a background flux from sources outside of our e L galaxy, perhaps superheavy X decay in other galactic halos c c or in intergalactic regions. Generally, this flux is assumed l1 eL: to be due to a homogenous distribution of sources and The presence of a neutral component in the 10 allows exhibits a characteristic GZK pileup due to the fact that spontaneous GUT symmetry breaking to be achieved using they are produced nonlocally [23]. Since the GZK attenu- a 10 and a 10 of superheavy Higgs fields with the same and ation is much more severe for photons than for nucleons, opposite U(1) hypercharges. Their neutral components this background should be comprised primarily of nucle- develop large vacuum expectation values (vevs) breaking ons. Thus, in this scenario the UHECR flux observed on SU5U1!SU3SU2U1, while the elec- Earth will be the sum of this extragalactic background and troweak spontaneous breaking occurs through the Higgs the local galactic flux from decaying relics clustered in our doublets H2 and H2 : galactic halo. Because of the extragalactic component,

h fH ; H g; h fH ; H g: (6) superheavy X particle decay may only unambiguously 5 2 3 5 2 3 explain the UHECR ﬂux for energies E>4 1019 eV. The resulting economical doublet-triplet splitting mecha- However, we note that the extragalactic component may nism gives a large mass to the Higgs triplets H3; H3 by also be due partially to nonlocal superheavy X decay, as coupling them to states in 10 and 10 Higgs representations, well from astrophysical sources (‘‘bottom-up’’ produc- through trilinear superpotential couplings of the form tion). Thus, it is more accurate to say that a distinct signal c c FFh ! d h iH ; of superheavy X decay within our galactic halo would be H H 3 (7) the existence of an excess of events above this energy. The c c

lack of any events above this energy would not rule out the F F h ! dHhHiH3 ; (8) presence of a top-down component, but it would not pro- while keeping light the Higgs doublets H2; H2 light. vide an unambiguous reason for the introduction of such a The absence of any mixing between the Higgs triplets mechanism. H3; H3 in this dynamic doublet-triplet splitting mechanism provides a natural suppression of the d 5 operators III. STRING-DERIVED FLIPPED SU(5) that might otherwise mediate rapid proton decay, so that flipped SU(5) is probably the simplest unified model that Even before the first string models, there was strong can satisfy the experimental limits placed on the proton interest in flipped SU(5) as a possible grand unified theory lifetime [28]. (GUT), primarily because it did not require large Higgs String-derived flipped SU(5) belongs to a class of mod- representations and avoided the constraints of minimal els constructed using free fermions on the world sheet, SU(5) [24] without the extra gauge interactions required corresponding to compactification on the Z2 Z2 orbifold in larger groups such as SO(10) [25,26]. Interest in flipped at the maximally symmetric point in the Narain moduli SU(5) increased within the context of string theory, since space [29]. Although this model was originally constructed simple string constructions could not provide the adjoint in the weak-coupling limit, it is possible that it may be and larger Higgs representations required by other grand elevated in the strong-coupling limit to an authentic M unified theories, whereas they could provide the Higgs theory model and may at some point make contact with representations required for flipped SU(5). It was, more- models based on D-brane constructions. over, observed that flipped SU(5) provided a natural The full gauge group of the model is SU5U1 ‘‘missing-partner’’ mechanism for splitting the U14 SO10SO6’SU4, where the latter two electroweak-doublet and color-triplet fields in its five- factors are confining hidden-sector gauge interactions. The dimensional Higgs representations [27]. Heterotic string- matter spectrum comprises the following fields:

115003-3 JOHN ELLIS, V.E. MAYES, AND D. V. NANOPOULOS PHYSICAL REVIEW D 74, 115003 (2006) (i) Observable sector: The conventional matter fields TABLE I. The spectrum of hidden matter fields that are mass- may be regarded as three 16 representations of less at the string scale in the revamped flipped SU(5) model. We SO(10) that SU5U1 chiral multiplets display the quantum numbers under the hidden gauge group 1 3 c 5 SO10SO6U14, and subscripts indicate the electric Fi10; 2, fi5; 2, li 1; 2i 1; 2; 3; extra matter 1 3 c 5 1 charges. fields F410; 2, f45; 2, l41; 2 and F510; 2, 3 c 5 0 1 1 0 1 1 f55; 2, l51; 2; and four Higgs-like fields in the 10; 1; 6; 0; 2 ; 2 ; 0 20; 1; 6; 2 ; 0; 2 ; 0 0 1 1 1 0 1 1 10 representation of SO(10), that hi5; 1, 0; 1; 6; ; ; 0; 0; 1; 6; 0; ; ; 0 3 2 2 2 4 2 2 hi5; 1, i 1, 2, 3, 45. In our realization of the 00; 1; 6; 1 ; 0; 1 ; 0 model, we make the following flavor identifications 5 2 2 010; 1; 0; 1 ; 1 ; 0 010; 1; 1 ; 0; 1 ; 0 of the standard model states with the various string 1 2 2 2 2 2 010; 1; 1 ; 1 ; 0; 1 010; 1; 0; 1 ; 1 ; 0 representations: 3 2 2 2 4 2 2 0 1 1

10; 1; ; 0; ; 0 c c c c c c 5 2 2 tb: Q4d4u5L1l1; cs: Q2d2u2L2l2; c c c F~1=2 ; ; 1 ; 1 ; 1 ; 1 F~1=2 ; ; 1 ; 1 ; 1 ; 1 udee: Qdu1L5l5; (9) 1 1 4 4 4 4 2 2 1 4 4 4 4 2 F~1=21; 4; 1 ; 1 ; 1 ; 1 F~1=21; 4; 1 ; 1 ; 1 ; 6 1 where indicates a mixture of ﬁelds with the indices 3 4 4 4 2 4 4 4 4 2 ~1=2 1 3 1 ~1=2 1 1 1 1 F5 1; 4; 4 ; 4 ; 4 ; 0 F6 1; 4; 4 ; 4 ; 4 ; 2 1 and 3. The light Higgs doublets hd and hu (where F~1=21; 4; 1 ; 1 ; 1 ; 1 F~1=21; 4; 1 ; 1 ; 1 ; 1 hd is the Higgs doublet responsible for giving 1 4 4 4 2 2 4 4 4 2 ~1=2 1 1 1 1 ~1=2 1 1 1 1 mass to the down-type quarks and leptons and hu F3 1; 4; 4 ; 4 ; 4 ; 2 F3 1; 4; 4 ; 4 ; 4 ; 2 ~1=2 3 1 1 ~1=2 1 1 1 1 gives mass to the up-type quarks) are contained in F5 1; 4; ; ; ; 0 F6 1; 4; ; ; ; 4 4 4 4 4 4 2 the h1 and h45 string pentaplet representations, respectively. (ii) Singlets: There are ten gauge-singlet ﬁelds 45, , and singly charged tetrons , i 1; 2; 3; 4, , , , their ten i 12 23 31

F~ F~ F~ F~ ; F~ F~ F~ F~ ; (12) ‘‘barred’’ counterparts, and five extra fields II 3 5 5 5 5 3 3 3 1 5. (iii) Hidden sector: This contains 22 matter fields in the ~ ~ ~ ~ ~ ~ ~ ~ following representations of SO10h SU4h: F3F5F5F5; F5F3F3F3; (13) T 10; 1, 1; 6i 1 5; F~ 1; 4, F~ 1; 4 i i i i as well as neutral tetrons i 1 6. Flat potential directions along which the anomalous combination of hypercharges U1 0 0 ~ ~ ~ ~ A F~3F~3F~5F~5; F3F3F5F5: (14) are cancelled induce masses that are generally near the string scale for some, but not all, of these states. We have shown previously that the charged tetrons could have decayed with rather short lifetimes early in the history Depending upon the number of Ti and i states remaining massless, the SO(10) condensate scale is of the Universe [12], and so avoid any problems with 101415 GeV and the SU(4) condensate scale is charged dark matter particles [31]. 1113 ~ ~ However, the neutral tetrons can decay only via higher- 10 GeV [30]. The F3;5 and F3;5 states always remain massless down to the condensate scale. The order operators in the superpotential, which we have identified in [12], and may have lifetimes exceeding the age of U1i charges and hypercharge assignments are shown in the Table I below. the Universe. This may make them good candidates for The hidden-sctor matter fields are confined into crypton cold dark matter, and their slow decays are a possible bound states. These occur in ‘‘cryptospin’’ multiplets with source of the UHECRs. different permutations of confined constituents, analogous In the next section, we turn our attention to these inter- to the flavor SU(2), SU(3), and SU(4) multiplets of bound actions and examine the UHECR energy spectra that may states in QCD. The cryptospin multiplets contain doubly result from these specific tetron decays. charged tetrons IV. UHECR INJECTION SPECTRA

F~ F~ F~ F~ ; F~ F~ F~ F~ ; (10) 3 3 3 3 5 5 5 5 We have previously found [12] the following 10th-order superpotential operators through which the neutral tetrons ~ ~ ~ ~ ~ ~ ~ ~

F3F3F3F3; F5F5F5F5; (11) may decay:

0 F2F23145 h1 F2F22345 h1 F2F3F3445f2 F42345 h45f5 3145 c c c 2345 h1f2l2 f5l52345 h1f1l1; (15)

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0 F2F23145 h1 F2F22345 h1 F2F2 h1h1h45 F4F43145 h1 F4F4 h1h1h45 c c F43145 h45f5 F4 h1h45h45f5 F4 h1h1h1l5 3145 h1 h1h1h45f1l1 c c 3145 h1 2345 h1 h45h45h45 h1h1h45f2l2 f5l5: (16)

Using the flavor identifications we outlined above, these SHdecay [16]. This code calculates the fragmentation operators would give rise to the following neutral tetron into the seven stable MSSM particles (p, , e, neutralino decay modes: LSP , e, , ) for any given initial decay parton. The 0 c 3 0 c c 3 many-body decays distribute the total decay energy MX ! h ; ! e=e = h ; d d among the different particles. We include Higgs decays, 0 c 3 0 c ! bb hd ; ! bb hdhdhu; but we ignore the decays of the singlet fields, except to take (17) 0 c 3 0 c into account their kinematical effects on the primary quark ! tt hu ; ! tt huhuhd; and lepton spectra. We follow [32] in estimating the proba- 0 c c 2 0 c 3 ! cc dd ; ! ss hd : bility density nz that one decay parton carries off a fraction z of the total available decay energy MX: We note that there are several different possible decay n3 modes, any of which may be dominant, depending on nzn 1n 2z1 z (18) unknown features of the model dynamics that determine the relative values of their coefficients. In particular, the for n 3 decay partons. The resulting flux from the emis- sion of a given decay parton is then most important tetron decays could be into either leptons or quarks, and there are many different possibilities for the Z 1 3 i 3 dz x i 2 dominant flavors. E J EBx n D z; Mx; (19) x z z We plot in Figs. 1–8 below the expected UHECR energy spectra of photons and nucleons due to each of these where i p; ; e; ; e;;. possible tetron decay modes, as well as the maximum To obtain the total UHECR spectrum, we add to this the photon fractions expected. The energy spectra were calcu- background flux of nucleons that would be expected to 13 lated for a mass MX 2 10 GeV, using the fragmen- result from a homogenous distribution of extragalactic i 2 tation functions D x; MX generated by the code sources that exhibits the distinctive pileup due to the

FIG. 1 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c 3 ! ss hd .

115003-5 JOHN ELLIS, V.E. MAYES, AND D. V. NANOPOULOS PHYSICAL REVIEW D 74, 115003 (2006)

FIG. 2 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 ! cccddc2.

FIG. 3 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c 3 ! bb hd .

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FIG. 4 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c ! bb hdhdhu.

FIG. 5 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c 3 ! tt hu .

115003-7 JOHN ELLIS, V.E. MAYES, AND D. V. NANOPOULOS PHYSICAL REVIEW D 74, 115003 (2006)

FIG. 6 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c ! tt huhuhd.

FIG. 7 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c c 3 ! e=e = hd .

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FIG. 8 (color). The top panel shows the total UHECR spectrum and the bottom panel the photon fraction from the decay mode 0 c 3 ! hd .

GZK effect [23].2 The constant B in (19) is a normalization primary t quarks. We are thus led to hope that including the coefficent determined by the tetron number density and (model-dependent) decays of the singlets would not lifetime, viz. have large effects. The plots for lepton primaries shown in Figs. 7 and 8 are more distinctive, in that the photon halo nX 1 B R : (20) fractions rise to much larger values at energies above X MX 1020 eV.3 This dimensional coefficient B is not determined a priori, In Fig. 9 we compare the spectrum for one of the and must be fitted to the experimental data. In each of operators with primary b quarks, calculated for a crypton Figs. 1–8, we show the total spectrum obtained by sum- mass of 1013 GeV, with experimental data from the Fly’s ming the background and the fluxes of nucleons and pho- Eye, HiRes, AGASA, and Auger experiments [3,5,7,8]. tons resulting from tetron decay, and in a second panel we The AGASA flux has been scaled by a factor of 0.55 for display the gamma fractions: = p. We have as- consistency with the other data, and the normalizations for sumed no photon attenuation in the calculated spectra, the crypton decay contributions to these spectra has been although a strong attenuation cannot be excluded [34], adjusted for the different crypton masses. The limited because the galactic radio background has never been statistics for UHECRs with energies 1019 eV available accurately measured and its intensity is largely unknown in the present data sets do not offer any clear discrimination [35]. between crypton masses in the range 2 1013 GeV 12 Figures 1–6 are for quark primaries, and are ordered MX 10 GeV. In the case of a crypton mass according to the masses of the quarks involved. In the case 1012 GeV there is no clear signal of a crypton contribu- of the b quark, we show in Figs. 3 and 4 plots for two tion to the UHECR since the flux from such a decay is superpotential operators with different numbers of accom- essentially buried within the background from homoge- panying Higgs fields: the two plots are rather similar, and nous extragalatic sources. A clear signal of crypton decay, the same is true of the two plots shown in Figs. 5 and 6 for at least in this model, would require a lower limit on the 12 crypton mass MX 5 10 GeV in order to provide an 2However, we note that this model is likely to come under pressure from upper limits on high-energy cosmic-ray neutrinos [33]—private communication from S. Sarkar, see 3The photon fractions for second-generation quark primaries also http://www-thphys.physics.ox.ac.uk/users/SubirSarkar/ look somewhat flatter than those for b and t quarks, but this talks/munich05.pdf. difference is probably within the modelling uncertainties.

115003-9 JOHN ELLIS, V.E. MAYES, AND D. V. NANOPOULOS PHYSICAL REVIEW D 74, 115003 (2006) UHECRs, including all the possible 10th-order superpotential operators. The experimental data presently available are consistent with all the decay modes possible in this crypton framework. The total UHECR spectra are consistent with a contribution from cryptons weighing between 2 1013 GeV and 1012 GeV, although only a crypton 12 mass MX 5 10 GeV would provide an unambiguous signal over conventional explanations. The available upper limits on the possible photon fraction do not exclude any of the crypton models we have studied. In the future, the larger data set expected from Auger may be able to discriminate between crypton decays and other models of UHECRs, and also among different crypton models themselves. Greater statistics will enable the UHECR anisotropy to be measured with sufﬁcient accu- racy to discriminate crypton decay from a uniform distribution of astrophysical sources, and more accurate FIG. 9 (color). A comparison with the available data on the measurements of the photon fraction at higher energies UHECRs from the Fly’s Eye, HiRes, AGASA, and Auger experi- might offer some discrimination between models with 0 c 3 ments with the crypton decay model ! bb hd for MX 13 lepton and quark primaries, as seen by comparing 10 GeV. Figs. 1–6 with Figs. 7 and 8 above. Thus there is hope that, in the near future, we may ﬁnally excess of events above 4 1019 eV that could not be learn whether UHECRs have a macrophysical origin or a attributable to extragalactic astrophysical acceleration microphysical origin and, in the latter case, may start to mechanisms. discriminate between different microphysical models.

V. CONCLUSIONS ACKNOWLEDGMENTS We have carried as far as is possible at present the The work of D. V.N. is supported by DOE Grant modelling of ﬂipped crypton decay contributions to No. DE-FG03-95-ER-40917.

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