Lorentz invariance violation in top-down scenarios of ultrahigh energy cosmic ray creation

0.16 MB·English

by  James R. Chisholm

FERMILAB-Pub-03/183-A
Lorentz invariance violation in top-down scenarios of ultrahigh energy cosmic ray
creation
James R. Chisholm and Edward W. Kolb
Fermilab Astrophysics, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, and
Enrico Fermi Institute, University of Chicago, Illinois 60637
(Dated: February 1, 2008)
The violation of Lorentz invariance (LI) has been invoked in a number of ways to explain issues
dealingwithultrahighenergycosmicray(UHECR)productionandpropagation. Thesetreatments,
however, have mostly been limited to examples in the proton-neutron system and photon-electron
system. InthispaperweshowhowabroaderviolationofLorentzinvariancewouldallowforaseries
ofpreviouslyforbiddendecaystooccur,andhowthatcouldleadtoUHECRprimariesbeingheavy
baryonicstates or Higgs bosons.
4
PACSnumbers: 11.30.Cp,96.40.De,98.70.Sa
0
0
2
I. INTRODUCTION produce superheavy UHECR primaries.

n
a
J UHECRs are an enduring mystery. Without the in-
8 troduction of new particles or interactions, evading the II. BREAKING THE LAW
2 Greisen–Zatsepin–Kuz’min (GZK) cutoff [1, 2] requires
unidentified nearbysources. Evenwithoutthe GZKcut- A. Modified Dispersion Relations

2 off,the“bottom-up”approachfacesthechallengeoffind-
v
ing in nature anacceleratorcapable of energiesin excess There are two approaches to breaking LI commonly
8 of 1020eV. used in the literature. The first [16, 20, 21, 25, 26] is
8
2 “Top-down”scenariosassumethattheUHECRsresult genericallyto modify the dispersionrelationsfor aparti-
6 from the fragmentation of a ultra-high energy hadronic cle of mass m and 3-momentum p= p.
| |
0 jet produced by cosmic strings [3] or by the decay of The second method, which we will not utilize here, is
3 a supermassive particle [4]. In the supermassive par- instead to write down a particle Lagrangian [27] which
0 ticle (wimpzilla) scenario, the UHECRs are of galactic includes Lorentz and CPT violating terms. A mapping
/
h origin, resulting from the decay of relic supermassive of the Lagrangian terms to a dispersion relation is non-
p (M > 1013GeV) particles. Wimpzillas can be produced trivial(see [23]for a simple case); however,to first order
p- copio∼uslyintheearlyuniverse[5],thussolvingtheenergy the changes to the photon and electron propagators in-
e problem. Since they would cluster in the dark matter duce shifts to cγ and ce [17]. Similarly, it was shown by
h halo of our galaxy [6, 7, 8], they also solve the distance [28] that loop quantum gravity effects produce modified
v: problem. Detailed analysis of these decays, however, dispersionrelationssimilartothoseconsideredelsewhere
i show that at high energy top-down scenarios produce and here.
X more photons than protons [9, 10, 11] in the UHECR The simplest way to break LI, as shown in [17], is to
r spectrum seen at Earth. write down a dispersion relation for a particle species i
a
as
The top-down prediction of photon preponderance in
UHECRsistheonemajorprobleminanotherwisesimple E2 =p2c2+m2c4. (1)
explanation. Results from the Fly’s Eye [12], Haverah i i i i i
Park [13], and AGASA [14] cosmic ray experiments all
This is changing the “speed of light” or Maximum At-
indicate that at energies above around 1019 to 1020eV,
tainable Velocity (MAV) for each particle to something
protons are more abundant than photons in UHECRs. slightly different than c.1 The MAVs for different par-
While photons are disfavored, it is not possible to be
ticle species are assumed a priori to be different in that
sure that the primary is indeed a proton.
and similar treatments [26].
The idea of violating LI (see Ref. [15] for a broad Insuchacase,itispossible forpreviouslyunallowable
overview) has recently been studied in the context of reactionstooccur,suchasp ne+ν [16,17],γ e+e−
e
UHECRs for proton decay [16, 17], atmospheric shower [25, 26] and γ νν¯ [25].→A conclusion reach→ed, us-
development[18],andmodificationstothecutoffsofpro- ing the first reac→tion, is that neutrons may make up the
ton and photon spectra due to cosmic background fields dominantbaryoniccomponentofthe UHECRprimaries.
[19, 20, 21, 22, 23, 24]. As shown in Ref. [25], LI vi-
olation can also lead to vacuum photon decay, and in
general the decay of particles to more massive species.
We exploit this fact to show how particles produced in 1 Althoughfactorsofcareexplicitlyincludedinequations,westill
top-down decays can undergo an “inverse cascade” to takeittobedimensionlesswithcγ =c=1.
2
This, strictly, would be true if one was only tuning the B. Example: Photon Decay
proton and neutron MAVs and leaving all others equiv-
alent (= c). On the other hand, keeping every MAV as In this section we consider the tree-levelphoton decay
individually tunable gives an overwhelming number of process γ e+e−, which is kinematically forbidden for
free parameters. c = c. Fo→llowing Ref. [17], we define the parameter δ
e γe
In order to examine the consequences of varying all as3
particle MAVs using the fewest number of free parame-
ters, we extend the method used in Ref. [21] and write δ =c2 c2. (9)
γe − e
the dispersion relation as
Using astrophysical constraints [29, 30], δ can be lim-
γe
E2 = p2c2+m2c4+p2c2f(p/Mc) ited to the range 10−16 > δγe > 10−17. For δγe > 0,
−
+m2c4g(p/Mc)+mc2pch(p/Mc). (2) photon decay occurs above a photon energy threshold
given by
Here, f(x),g(x), and h(x) are dimensionless universal 2m c2
functions having the property f(0) = g(0) = h(0) = 0, E = e ec. (10)
th
δ
so that as p/M 0 the normal dispersion relations are γe
→
recovered. Here M is the mass scale that determines the
ThisdecayrateofwascompputedinColeman&Glashow
relative degree of Lorentz violation; for our purposes we
(1997) as
set it at the Planck mass M 1019 GeV. Expanding
≈
these functions in Taylor series about x=0 and keeping 1 δ 3/2
terms of O(p2) and smaller gives Γ= 2α cγ2e E 1−(Eth/E)2 . (11)
(cid:18) e (cid:19) h i
E2 = m2c4+mc2pc 1g′(0)m Fordecayproductenergyaboveabout2 1020eV,pho-
2 M tons will outnumber protons, so we set E×to this value
(cid:18) (cid:19) th
1 m 1 m2 and suggestively rewrite the decay rate above threshold
+p2c2 1+ 2h′(0)M + 6g′′(0)M2 . (3) as:
(cid:18) (cid:19)
Γ E δ
The g′(0) term can be neglected for two reasons: at (1 km)−1 γe . (12)
¯hc ≈ 2 1020 eV 3 10−29
p mc it would be experimentally detected as devia- (cid:18) × (cid:19)(cid:18) × (cid:19)
≪
tions in the non-relativistic kinetic energy; for p mc ThiswouldruleoutallbutaterrestrialsourceofUHE
≫
it is negligible to the quadratic p term. Note that to photons. The value δ = 3 10−29 is used is a lower
γe
this (lowest) order, f(x) has no effect.2 For this section, limit; if δ would be any lo×wer the energy threshold
γe
we examine the cases of massless (photon) and massive wouldbecometoohigh. Thismeansthatonlyδ (3
γe
(electron) particles: 10−29,10−16) will correct for the photon overabun∈danc×e
in top-down scenarios.
E2 = m2c4+p2c2 1+ 1h′(0)m + 1g′′(0)m2 (4) Now we see what is required to have δγe in the neces-
e e 2 M 6 M2 saryrangeto allowforthis photondecay. UsingEqs.(6)
(cid:18) (cid:19)
E2 = p2c2. (5) and (9), we can write
γ γ
1 m 1 m2
We can now define δ = c2h′(0) c2g′′(0) . (13)
γe −2 M − 6 M2
1 m 1 m2
c2 =c2 1+ h′(0) + g′′(0) (6) Since the functions g(x) and h(x) are a priori ar-
e 2 M 6 M2
(cid:18) (cid:19) bitrary, so are the values and signs of their deriva-
tives. It is argued in Ref. [21] that too strongly vary-
ing functions would be unphysical, and that the deriva-
m2 =m2 1+ 1h′(0)m + 1g′′(0)m2 −2 (7) tives should be of order unity.4 Adopting this, we can
e 2 M 6 M2 ignore the g′′(0) term in the above equation. In this
(cid:18) (cid:19)
case, in order to have photon decay we need to require
to then write the dispersion relationin the familiar form
of Eqn. 1 as
Ee2 =m2ec4e+p2ec2e. (8) 3 Thisisidentical toξ−η forn=2inthenotationofRef.[29].
4 Notethatifthisholdsalsoforf(x),weerredinnotincludingthe
f′(0)terminEq.(3),asthatdominatesovertheh′(0)termwhen
p>∼mc. Wewillnotapologizemuchforthis,aswearelookingat
the“lowestorder”changeinphenomenology (thechange inthe
2 Issuesofphotondispersioninthevacuum, asdiscussedinRefs. “speedsoflight”)andnot“nextorder”changes(suchasvacuum
[20,29],willnotbeaddressedhere. photon dispersion).
3
h′(0) negative. Then δ mc2/2M 3 10−23 and exactly zero momenta [17]. Since the c are the max-
γe X
∼ ∼ × { }
E = 2m c2c 2M/mc2 = 2√2m Mc2 2 1017eV. imum attainable velocies for the decay products, that
aTlhtlhoiwsvaabllueereaonfegδeγp.e is(logarithmically)einthe≈mid×dle ofour imIptliiesstthhiastlaλst<pmoiinnt{cwXe}f.ocus on: min cX in our case
{ }
Returning to our original impetus of addressing the corresponds to max mX . In the ultrarelativistic limit,
{ }
feasibilityof“TopDown”scenarios,weseethatrequiring EA pAcA and so on, and the particle with the lowest
≈
h′(0) 1 is sufficient to eliminate any photons among speed of light will carry most of the momentum (in the
the U∼HE−CR primaries. This model, however, eliminates minimum energy case). Further, in this threshold case,
any protons as well, which we will now address. λ is also the velocity of the incident particle [17]. Since
λ c intheregimeweareconsidering,thatmeansthat
A
≈
the decaycanonlyoccurif c >min c . FromEq.14,
A X
{ }
C. Making the impossible possible that implies that mA <max mX .
{ }
Inotherwords,aboveacertainthreshold,adecaymay
Unlike previousinvestigations[17,29] wherethe MAV only be allowed if at least one decay product is more
for each particle could be chosen independently, this is massivethanthedecayingparticle. Thiscanhaveserious
not so in our case. Since Eq. 2 holds for all types of consequences for the particle content of UHECRs, as we
particles, it is straightforwardto see other effects of this will now address.
levelofLorentzinvarianceviolation. UsingEq.6,wecan
in general write for two particles A and X
III. THE INVERSE CASCADE
1 m m
δ = c2 c2 = h′(0)c2 A− X
AX A− X 2 M In this model, for each individual reaction there is a
(mX −mA)c2, (14) thresholdenergyET abovewhichparticlescanonlydecay
≈ 2M ifatleastoneofthedecayproductsismoremassive. For
the decay A B + ..., where B is the most massive
where we have used h′(0) 1. →
∼− decay product, mB >mA, and m... mB the threshold
The consequencesof this for particle decaysare as fol- ≪
energy goes like
lows,as firstnoted by [17], and we reproducetheir argu-
ment here.
m2 m2
Consider the decay A X , where X is a col- E c3 B − A. (19)
lection of massive particle→s. {We}want to {find}the min- T ∼ s c2A−c2B
imum possible energy E where this can occur. We
min
can always lower final state energy by removing trans- Using equation 14, this reduces to
verse components of momenta, so we’re working in one
dimension only. We want to minimize E c2 M(m +m ) h′(0) −1/2. (20)
T A B
∼ 1
(cid:18) (cid:19)
E( p )= m2 c4 +p2 c2 (15) p
{ X} X X X X Due tothe largevalueofthePlanckmass,this thresh-
XX q old is around the same order of magnitude for a wide
subject to the constraint range of reactions involving particles of mass much less
than M.
G( p )= p p =0. (16) ConsideranUHECRofmassm0producedwithenergy
{ X} XX X!− A 0E0ca≫n bEeTa. dIefcnaoyrmpraoldlyuc(tatofenaermgoierseEma≪ssivEeTp),arptaicrlteic1le,
sTthaent{s.pXU}sinagrethvearmiaebtlheos,daonfdLapnAg,rEanAge>mEulmtiipnliearrse,cEonis- tlihgehntehreprero(dEuc≫tsEthTa)t, 0wicllanredsuecltayasinwtoell1;.thWeesiugbnsotraenttihael
minimized when fractionoftheincidentenergywillbecarriedbythemost
massive decay product.
∂E ∂G
=λ , (17) Now, we have a particle of energy E1 < E0 and mass
∂pX ∂pX m1 > m0. If it can, it will also decay into a particle
which becomes 2 with m2 > m1 and energy E2 < E1. As the decays
continue into more and more massive final states, the
p c2 p c2 final energy continues to decrease, until at some point
λ= X X = X X . (18)
EX m2Xc4X +p2Xc2X wmeNre>achma0.paSritniccelewNe awriethnoenwerbgeyloEwNth<reEshToladn,dthmeaNss
Thusallproductsmovewpiththesamevelocityλ. Note particle can decay as usual into N-1 and so on into less
that λ < c for all X. For the example we consider, in massive particles.
X
the event of massless particles among the decay prod- So, to summarize, this inverse cascade decay process
ucts, they would minimize final state energy by carrying occurs in two stages.
4
1. Decay up: Decays into more and more massive, thanthetopquark. Ifitislighterthanthetop,theHiggs
yet less energetic particles, when E >E , and is unstable to H tt¯.
T
→
2. Decay down: Regular decay schemes once E <
E .
T
C. Tree-level Strong
IV. PANDORA’S BOX (NEWLY ALLOWED Since the bare QCD vertex preserves quark flavorand
REACTIONS) charge,notmuchhappensinthebarequarksector,apart
fromthereversingsofstrongdecayssuchasN ∆πand
→
π ρπ. Justasforthemasslessphoton,freegluondecay
We saw in Section II that at sufficiently high energies
→
is allowed via g qq¯.
particles can only decay if one of the decay products is
→
moremassive. InSectionIIIwesawhowthiscanproduce
an “inverse cascade” to more and more massive particle
D. One-loop processes
species. In this section we address the specifics of how
this cascade occurs, by examining which previously for-
One could formulate many more decays at the 1-loop
bidden decays can now occur. All of the following are
assumedto be occuringwellabovethreshold,andwelist level, but the rates for these for them are suppressed
them in several classes. compared to the tree level decays. For this reason, we
do not consider them further here, except to note two
interesting decays of the photon:
γ ν ν¯ (QED and weak process). Note that since
A. Tree-level QED → l l
neutrinos are so much lighter than their heavy partners
and the quarks,the threshold for photon to neutrino de-
Here the bare QED vertex γ Q+Q− where Q is
cay could be very low depending on how light the neu-
→
any charged particle is allowed as a decay. This also
trinos are.
includesfinalstateswithextraphotonsandboundstates
γ Zγ (weak process). This photon Z-Cerenkov de-
of Q+Q−, like γ π0γ. →
cay occurs due to a W-loop.
→
B. Tree-level weak E. Spin up
It is in the weak sector that most of the interesting For bound states (and now restricting ourselves to
decays in the inverse cascade occur, due to the flavor- bound states of three valence quarks) there exist higher
changing W vertex. mass resonances that only differ in the mass and total
In this Lorentz-violating scheme, weak decays (such angular momentum (e.g., the ∆ and higher resonances
as that of the neutron) can happen in reverse, allowing for the protonand neutron). Since the state with higher
proton decay via p ne+ν and flavor changing decays angularmomentum is more energetic,it is more massive
such as n Λπ0. → and thus processes like N N∗π, N N∗∗π and so
→ → ∗ →
Not only that, decays to weak bosons become permis- on can act to “spin up” the particle to more and more
sible, such as ν l±W∓ and l± ν W±. Included massive states.
l l
in this are decays→of leptons to hea→vier leptons due to Thus,eventhealltopquarkbaryontttisn’tcompletely
virtual W’s. In the quark sector,this allows decays such stable, but can decay via spin up.
as p nW+ and so on. Generally, the flavor changing
weak→decay q q′W will convert all quarks to t quarks.
→ V. ESTIMATES OF DECAY RATES
In the absence of a fourth generation, a bare t quark is
stable.
Since the Z is massive, decays such as q,l,ν Forthe photondecayγ e+e−, the decayrateΓwell
Z q,l,ν (“Z-Cerenkov”)become permissib{le as wl}ell.→ above threshold was →
l
{For th}e bosons; W+ t¯b is allowed, and Z-Cerenkov 1
→ Γ = αδ E. (21)
is possible for the W as well (this is a bare weak vertex, γ→ee 2 γe
as is W WZZ and W WZγ). The Z is unstable
to the de→cay Z tt¯. → Weadaptthisformofthedecayratetootherprocesses
→ to make estimates, and argue for it as follows. At ultra-
NowconsidertheHiggs: indeedwemightbeabovethe
relativistic energies,the only energy scale is the incident
energy of weak symmetry breaking. Including a single
energy E, so Γ E. It is proportional also to the ef-
massive Higgs H would allow; f,W,Z f,W,Z H ∝
{ } → { } fective coupling constant involved in the decay, and to
(“Higgs-Cerenkov” for fermions and weak bosons) and
kinematic factors. We schematically write this as
W,Z W,Z HH. In the absence of new physics
{ } → { }
beyondtheStandardModel,theH isstableifitisheavier Γ (coupling) (kinematics) E. (22)
∼ × ×
5
Forthephotondecay,thecouplingisαandthekinematic Second, and more interesting, if the initial UHE pri-
factor is δ . We generalize these factors as follows. mary is of high enough energy, the decay up sequence
γe
We (first) consider decays of the form A B + C may occur and leave particles that can no longer decay
→
where m <m , m m . At threshold and beyond, into moremassiveparticles,but arestillabovethreshold
A B B C
≫
most of the momentum of the final state is carried by for decay into less massive particles. These Most Mas-
the particle with the lowest speed of light, thus highest sive Particles (MMPs) would then be the most energetic
mass (particle B). The relevant kinematical factor will componentoftheUHECRpopulation. Whatexactlythe
then be δ . This will continue to be true when final MMPsaredependsonthemodelofparticlephysicsused.
AB
states with more than 2 particles are allowed, as long as Of particles knownto existknow, the MMPs would con-
B continues to be the most massive particle. sist of top quarks and excited bound states of top (tt¯
The coupling factor is determined by looking at the mesons and ttt baryons) quarks. If the Higgs is more
tree level Feynman diagrams and counting vertices in- massive than the top, then the MMPs would consist of
volved in the reaction. single Higgs bosons. Including Supersymmetry and its
Consider then the decay of the proton; in addition to expected spectrumof more massivesuperpartners,if the
the inverse of the neutron decay p ne+ν (“leptonic” decay rate is still rapid enough the MMP would be the
e
decay), the decay p nW+ (“bo→sonic” decay) is also most massive superparticle (MMSP).
→
allowed. Of course, in the first outcome, our proposed solution
For the bosonic decay; E √m M 2.8 1019 of using LI violation to solve the photon abundance is-
T W
∼ ∼ ×
eV, and as most of the energy goes into the W, δ sues in top-down models is not obviously successful. For
pW
m /2M 4 10−18. Thevertexisasingleweakvertex∼; bothoutcomes,infact,onewouldhavetore-examinethe
W
with coup∼ling×α =α/sin2θ 0.0316. Thus, at E = decayprocessstartingwiththeinitialdecayofthesuper-
W W
∼
E ,Γ 4 eV. massive relic (which is assumed to be sub-threshold, so
T
For∼the leptonic decay; E M(2m ) 4.5 1018 it decays “normally”), and follow the decay products to
T p
∼ ∼ ×
eV, and as most of the energy goes into the n, δ see what results.
pn
(m m )/2M 5 10−23. pThis reaction has tw∼o Simulations of MMP-induced shower development
n p
weak−vertices wit∼h co×upling α2 10−3. Thus, at would have to be performed to see if they are consis-
E =E ,Γ 2.2 10−7 eV. W ∼ tent with actual events. It was shown in Ref. [32] that
T
∼ ×
Notethatthedistancetravelledbeforedecayis 1/Γ, stableUHEprimariesofmassgreaterthanabout50GeV
and givenh¯c=6.57 10−24 eV pc, the 1/Γ for th∼ese re- shouldbe distinguishable fromprotonprimariesby their
actions is incredibly×small: 10−24 pc for the bosonic and atmospheric shower profiles. As the lightest conceivable
10−17 pc for the leptonic. In the context of a UHECR, MMP (a tt¯ meson) would be much heavier than this
assuming comparable rates (within a few orders of mag- bound, the result of [32] might be used to exclude the
nitude)tothe above,this inversecascadehappensessen- MMP as the primary for UHE showers. However, a key
tially immediately, so that by the time any decay prod- assumptionisthattheprimaryisstable. ForaMMPjust
uctsreachtheEarth,theyhavegonethroughbothstages above the threshold energy, it may take only a few colli-
of the cascade and are now sub-threshold particles. sionsbeforetheMMPenergyisloweredbelowthreshold,
where it will decay “normally” (e.g., tt¯ W+W−) into
→
a shower of particles, perhaps mimicking the shower of
VI. CONSEQUENCES FOR UHECR a nucleus UHECR. A MMP with incident energyfarther
COMPOSITION above threshold would then penetrate deeper. In any
event, further investigation into MMP UHECR shower
Given the rapid rate of the inverse cascade process, profiles is warranted.
there can be two outcomes for the UHECR population Also of concern is that, while energy and momentum
at earth. are conserved in a certain preferred frame, they are nec-
First,ifboththeDecayUpandDecayDownsequences essarily not conserved in all other frames. As such, due
occur, then the consituent particles could, depending on to the motion of the earth and the solar system, small
theexactdecaychainsandnatureofinitialUHEprimary, departures from energy conservation might be observed
consist of protons, electrons, photons, and their antipar- as well in the UHECR atmospheric showers.
ticles. Each particle spectrum would exhibit a cutoff at ThisworkwassupportedinpartbytheDepartmentof
theirprospectivethresholdenergies,whichareallrelated Energy and by NASA (NAG5-10842) and by NSF grant
by the h′(0) parameter. PHY 00-79251.
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