Bound states in monopoles: sources for UHECR ?

9 Pages·0.22 MB·English

by  E. Huguet

BOUND STATES IN MONOPOLES: SOURCES FOR UHECR ?
Eric HUGUET1 and Patrick PETER2
1D´epartement d’Astrophysique Stellaire et Galactique,
Observatoire de Paris-Meudon, 92195 Meudon, France,
and Universit´e Paris VII, place Jussieu, 75005 Paris, France,
2D´epartement d’Astrophysique Relativiste et de Cosmologie,
Observatoire de Paris-Meudon, UPR 176, CNRS, 92195 Meudon, France.
Email: [email protected],[email protected]
(February 1, 2008)
Bound states in monopoles are studied through a simplified, Witten-like model. As the overall
structure is determined in full details, it is shown in particular that only those states having a
vanishing angular momentum are allowed; for those, the energy spectrum is derived numerically
andanapproximationissetupthatallowsaneasydescriptionintermsofwavefunctions,usefullfor
9
9 further applications. The monopoles are then proposed as candidates of ultra high energy cosmic
9 rays, an hypothesisthat should soon betestable through the Pierre AugerObservatory.
1
PACS numbers: 98.80.Cq, 11.27+d
n
a
J

1
I. INTRODUCTION i.e., 137 in the case of electromagnetism (as ought to
2 ∼
be the case):

1 Among the various topological defects that may have
η 3 m
v formedduring phasetransitionsinthe earlyuniverse[1], Ω h2 1011 M , (1)
0 M ≃ 1014GeV 1016GeV
monopoles are absolutely unavoidable in Grand Unified
7 (cid:16) (cid:17) (cid:16) (cid:17)
Theories (GUT), according to which there was a time (here we note Ω the monopole density in units of the
3 M
1 when all symmetries were unified in a single semi-simple critical closure density, and h the Hubble constant in
0 gauge group, since electromagnetism is an exact (unbro- units of100km s−1 Mpc−1) sothat it suffices to lower
9 ken) symmetry still now. They are however usually con- η to < 109 Gev· in o·rder to cure the density problem.
9 sideredasasufficientnuisance[2](inthe sensethatthey This i∼sthe solutionwe shalladopthere,foritmightalso
h/ wouldveryrapidelycometodominatetheuniverse)that provide a useful explanation for the high energy cosmic
p inflationisgenerallyinvokedrightaftertheirtime offor- ray mystery [7]. (Note in this regard that one can also
- mation so that they are completely diluted [3] and thus simply assume the monopole overdensity problem to be
p
unobservable. Theseresultsarebasedontheby-nowvery cured somehow and investigate anyway the possibility
e
h standard view that they originated at the GUT phase that they give rise to high energy cosmic rays [8].) This
: transition, i.e., at an energy around 1015 1016 GeV. way, the possibility that monopoles are still present in
v −
Many ways out however have been proposed other than the universe is still reasonnable.
i
X inflationnary scenarios, among which the Langacker-Pi Once the mass scale is fixed, the relevant physics
r mechanism [4] which relies upon using cosmic strings to still needs be properly clarified in order to enable one
a connectthemonopolesandanti-monopolespairs,thereby to study the interactions between monopoles and other
effectively enhancing considerably the decay probability, particles. In this regard, models have been suggested
thus reducing the remnant monopole density. Although whereboundstatesofscalar[9],gaugevectorbosons[10]
themodel,initsoriginalpresentation,suffersfrommany or fermions [11] can form in SO(3) t’Hooft-Polyakov
drawbacks,ithasatleasttheadvantageofproposingan- monopoles [12]. Those modify the scattering solutions
other solution, not involving inflation and letting open and can enhance greatly the cross sections by means
other alternative possibilities. Domain walls have also of the Callan-Rubakov effect [13]. It is the same kind
been used to sweep them away [5]; in all cases, unde- of approach we wish to present here, although using
sirable monopoles are gotten rid of by means of higher a simplified Witten like model (generally used to de-
dimensional topological defects [6]. scribe current-carrying cosmic strings [14,15] or domain
Yet another alternative possible solution to the walls [16]) and a completely different method for calcu-
monopole excess problem is the simplest one, although lating the bound state energy levels. Our model, con-
completely overlooked until recently: it consists in not- traryto the other proposedmodels having bound states,
ing that the monopole density is in fact proportional to presents the advantage (apart from its simplicity) that
the fourthpower ofthe energyscaleη at whichthe sym- the bound states are present for dynamical reasons, i.e.,
metry breaking during which monopoles were generated theydonotexistforallvaluesoftheunderlyingparame-
took place, giventhe monopole mass m is essentially η ters. Thismeansthatwecanconsiderthatparticlesmay
M
timestheinverseofthecorrespondingcouplingconstant, be trapped in the monopole even though they might be-
1
long to representations that are not directly related to where the covariant derivative is
thatofthe monopolefields. Besides,weshallworkusing
theJulia–Zeefullynumericalsolution[17]andnottheex- DµΦa ≡∂µΦa−qεabcAbµΦc, (3)
plicit Bogomol’nyi–Prasad–Sommerfield (BPS) analytic
and the gauge field strength
limit [18].
In what follows, we shall first define our model and
Fa ∂ Aa ∂ Aa +qεa AbAc. (4)
show that bound states may be present in monopoles, µν ≡ µ µ− ν µ bc µ ν
provided their angular momentum vanish, exactly as in
A static monopole configuration then has the
the Callan-Rubakov mechanism [13]. This not-quite-
form [12,17], in spherical coordinates xi (r,θ,φ)
obvious result follows from the requirement that bound ≡
states do exist in the monopole, which in turn seriously xa
constrains the possible couplings between the fields in- Φa =ηh(r) , (5)
r
volved.
Having explored the full microscopic structure of a
monopole having bound states, we go on to investigate 1 K(r)
Aa = − εa xj, Aa =0. (6)
the possibility that, in a fashion similar to Witten su- i − qr2 ij 0
perconductingcosmicstringmodels[14],throughtheex-
istence of vortons [19] (rotating superconducting cosmic Thefieldequationsfortheconfiguration(5)and(6),with
string loops configurations), these monopoles could be the Σ field not taken into account, i.e. for the ordinary
thesourceofUltraHighEnergyCosmicRays(UHECR). ’t Hooft–Poyakovmonopole, read
Their acceleration,once they are coupled to electromag-
1 d dh 2
netism, poses no particular problem as long as an exter- (r2 )= hK2+λ η2h(h2 1) (7)
nal magnetic field exists in the accelerating region un- r2dr dr r2 φ −
der consideration. Similarly to the model based on vor-
tons [20], they can propagatealong huge (cosmologically d2K 1
speaking) distances, and thus have the ability to reach =q2η2Kh2+ K(K2 1), (8)
dr2 r2 −
us fairly easily. Their interaction cross-section with air
nuclei can be obtained through the Callan-Rubakov ef- with boundary conditions
fet [13] and is thus evaluated to be hadronic in nature.
The expected properties of the resulting UHECR distri- h(0)=K( )=0, h( )=K(0)=1. (9)
∞ ∞
butionareessentiallythosederivedinthevortoncase[20]
and can therefore reproduce the existing data. More
data, thanks for instance to the Pierre Auger Observa-
tory project [21], will give a definite answer concerning
this possibility. 1.0 h(ρ)
II. THE MONOPOLE STATE 0.8
ρ)
The model we shall use in what follows involves the K( 0.6
symmetrybreakingofanSO(3)invariancebymeansofa nd
a
HiggsfieldΦbelongingto the3representationofSO(3), ρ) K(ρ)
coupled to a complex scalar field Σ which we assume, h( 0.4
for the sake of simplicity, not to be coupled to the gauge
field Aµ of SO(3). This assumption should of course be
0.2
modifiedwhenonewantstoevaluatethelong-rangeelec-
tromagnetic interaction of the resulting monopole with
other particle, but we shall show latter on how this can 0.0
0.0 10.0 20.0
be achieved. With a metric convention having positive Distance to the center ρ
signature, we have the model
FIG.1. Amplitudeofthefieldfunctionsh(ρ)andK(ρ)vs
1 1 1
= (D Φa)†(DµΦ ) (∂ Σ)⋆(∂µΣ) Fa Fµν therescalled unitoflengthρ=rλ1/2η. Curvesareshownfor
L −2 µ a − 2 µ − 4 µν a q2/λ = 10−2 (solid line), 0.1 (daφshed line), 1 (long dashed
λ φ
φ(Φ Φa η2)2 f(Φ Φa η2)Σ2 line) and 10 (dot-dashed line).
a a
− 4 − − − | |
m2 λ
σ Σ2 σ Σ4, (2)
− 2 | | − 4 | |
2
Theseequationshavebeensolvednumericallyforvari- which will admit bound state eigensolutions (ω2 < m2)
σ
ousvaluesofthe onlyrelevantparameterq2/λ andpro- provided that the potential V satisfies either
φ
duce the characteristic curves on Figure 1, where h and
K are shown as functions of the rescalled unit of length V <0= ℓ=0 & m2 <2fη2, (14)
⇒ σ
ρ=r λ η.
φ
or
Thetotalenergyofthemonopoleisexpressiblesimply
in terpms of q2/λ and the Higgs field mass m = λ η
φ H φ dV
as R [0, [ ; =0 and V(R)<0. (15)
p ∃ ∈ ∞ dr r=R
(cid:12)
K′2 1 K2h2 (cid:12)
E =4πm ρdρ + h′2+ In the latter case, deno(cid:12)ting by a prime a derivative with
M HZ n(q2/λφ)ρ4 2 ρ2 respect to the rescalled distance ρ, one finds that the
1 (1 K2)2 minimum of the potential would be for ρ such that
+ (h2 1)2+ − , (10)
4 − 2(q2/λ )ρ4
φ λ
o φℓ(ℓ+1)=hh′ρ3. (16)
which is shown on Figure 2 as a function of q2/λ . 2f
φ
OnFigure3isplottedtherighthandsideofthisrelation
against ρ for three orders of magnitude of the relevant
103 underlyingparameterq2/λ ,anditisseenthatingeneral
φ
this function is of order unity throughout its range of
variation. Hence, in order for the condition (16) to be
102 satisfied, it is necessary that the left hand side be also
of order unity. Assuming λ and f to have comparable
φ
values (they are both quartic interaction term coupling
101
constants), this leads to the constraint
mH
E/
100 ℓ(ℓ+1)∼1, (17)
which shows that no bound state is expected for large
10−1 angular momentum. For this reason, we restricted our
analysis to vanishing angular momentum states ℓ=0.
10−2
10−3 10−2 10−1 100 101 102 103 104 105 106
q2/λ 2.0
φ
FIG.2. Variation of the total monopole energy as a func-
tion of q2/λ .
φ 1.5
III. THE SCALAR CONDENSATE 3ρρ)
h’( 1.0
ρ)
In this background, we now investigate the behaviour h(

ofthebosonicfieldΣbyfirstlookingatthefieldequation
in which the nonlinar term is omitted. Separating space
0.5
and time variables in the form
Σ(xµ)=σ(r)eiωtY (θ,φ), (11)
ℓm
0.0
0.0 5.0 10.0 15.0 20.0
thefieldequationforΣinthemonopolebackgroundgives Distance to the center ρ
the Schr¨odinger-like eigenvalue equation for the ampli-
tude of this field as
FIG. 3. Value of the function ρ3h(ρ)dh/dρ which de-
termines the possibility of bound states for nonzero val-
∆σ+V(r)σ =ω2σ, (12)
− ues of the angular momentum ℓ vs the rescalled unit of
length ρ = rλ1/2η. As on Figure 1, curves are shown for
where φ
q2/λ = 10−2 (solid line), 0.1 (dashed line), 1 (long dashed
φ
ℓ(ℓ+1) line) and 10 (dot-dashed line).
V(r)=[ +2fη2(h2 1)+m2] (13)
r2 − σ
3
Yet another way to convince oneself that the bound
states should be restricted to those with vanishing an- 1.0
gular momentum consists in investigating the full, non
perturbative, field equation for Σ, still within the sepa-
rated form (11), that is, assuming the resulting state to 0.8
be stillaneigenstateofthe angularmomentum. The ex-
istence of the non linear term would then imply |Yℓm|2 ρY() 0.6
to be a constant, i.e., independent of both angular vari- d
n
ables θ and φ. Thus, only the case ℓ= 0 can satisfy the ρ) a
non linear equation, so that the background σ field has K(
vanishing angular momentum. ρh(), 0.4
We now therefore consider again the ansatz (11) with
no spherical harmonics included and in the region of pa- 0.2
rameter space where a condensate might form, i.e. we
assume m2 < 2fη2. Setting ω = 0 yields the actual
σ
vacuum state as the solution of the field equations 0.0
0.0 5.0 10.0 15.0 20.0
ρ
1 d dh 2
(r2 )= hK2+λ η2h(h2 1)+2fη2hσ2, (18)
r2dr dr r2 φ −
FIG. 4. Fields function h(ρ) [full line], K(ρ) [dotted line]
d2K 1
=q2η2Kh2+ K(K2 1), (19) and Y(ρ) [dashed line] for the set of parameters α1 = 0.1,
dr2 r2 − α =0.6 and α =0.6.
2 3
1 d dσ
(r2 )=[m2 +2fη2(h2 1)]σ+λ σ3, (20)
r2dr dr σ − σ These fields represent the vacuum state out of which
the bound state solutions can be calculated.
with boundary conditions for σ as
dσ
(0)=0, lim σ =0. (21) IV. QUANTUM THEORY
dr r→∞
Rescalling the field σ through In order to calculate quantum effects related to the
monopole solution, we turn to the standardsolitonic ap-
σ
Y(ρ)= λ (22) proach [23]. We shall for now on consider the Higgs and
σ
m
σ gauge vector fields to represent a fixed background in
p
which the charged scalar field Σ evolves. This means we
and defining the dimensionless parameters as
describes its dynamics through an effective Lagrangian
m2 fm2 m4 density
α = σ , α = σ , α = σ , (23)
1 λση2 2 λσλφη2 3 λσλφη4 1
= ∂ Σ2 f(Φ Φa η2)Σ2
Σ µ a
L −2| | − − | |
allows a numerical calculation of the vacuum solution.
m2 λ
Such a solution, obtained by means of a Successive Over σ Σ2 σ Σ4, (24)
Relaxation method [22], is shown for a special set of pa- − 2 | | − 4 | |
rameters αi onFigure4(here andinwhatfollows,the andthelocalizedsolutionsarederivablefromtheeffective
backgroun{dp}arameter q2/λφ has been fixed to the arbi- potential V[Σ]
trary value 0.1).
1∂Σ
L = d3x = d3x V[Σ], (25)
Σ Σ
L 2 ∂t −
Z Z
1 λ
V[Σ]= d3x ( Σ)2+ σ Σ4
2 ∇ 4 | |
Z n
m2
+ f(Φ Φa η2)+ σ Σ2 . (26)
a
− 2 | |
(cid:20) (cid:21) o
The classicalsolutionΣ derivedearlieris then obtained
c
by minimizing V[Σ]:
δV
=0= ∆Σ =[2f(Φ Φa η2)+m2]Σ +λ Σ3,
δΣ ⇒ c a − σ c σ c
(27)
4
where, because of the U(1) symmetry in the scalar field = P,n(η),n(µ),q(η), ,q(η),q(µ), ,p(µ) ,
Σ, the later could have been chosen real. F | i j 1 ··· N 1 ··· M i
n
ExpandingV[Σ]aroundthe classicalsolutionandnot- n(η),n(µ) N,q(η),q(µ) R3 , (33)
i j ∈ α β ∈
ing respectively
o
and this space is assumed disconnected from the ordi-
η˜= 1 (ψ+ψ⋆) (28) naryfreeparticlespaceofthetheorywithoutamonopole.
√2 Cross sections will then be calculable by computing ma-
trix elements of the quantum field Σ4, which is seen to
and | |
involve essentially space integrals of the eigenfunctions.
1 Specificsuchcalculationsandtheirapplicationtoprimor-
µ˜= (ψ ψ⋆), (29)
dialcosmologywillbe the subjectofa further work,and
√2 −
for now on we shall concentrate on the actual structure
with ψ the quantum perturbation from the classical so- of both the self potential and the bound states.
lution Σ=Σ +ψ, one gets
c
V[Σ]=V[Σc]+Vint[η˜,µ˜] (30) V. THE SELF POTENTIALS; BOUND STATES
1 ∆ m 3
d3xη˜ f(Φ Φa η2) σ Σ η˜
a c
− 2 2 − − − 2 − 2
Z h i
1 ∆ m 1
d3xµ˜ f(Φ Φa η2) σ Σ µ˜,
a c
− 2 2 − − − 2 − 2
Z h i
where V [η˜,µ˜] comprises the interactionterms between
int 0.0
η˜ and µ˜, originating from the self coupling Σ4. These
| |
terms are not explicitely developed here since we are
only interested in the bound state solutions around the
monopole, i.e., the stationnary solutions on the basis of 2mσ −1.0
which the system can be quantized. We are therefore µ()V/
looking for the eigenmodes of the second derivatives of nd
a
V withrespectto the fieldsη˜andµ˜,whicharethenseen 2m σ
to satisfy the Schr¨odinger-like equations η()V/ −2.0
∆+2f(ΦaΦ η2)+m2 +3Σ2 η˜ =ω2η˜, (31)
− a− σ c i i i
h i
and −3.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
ρ
∆+2f(ΦaΦ η2)+m2 +Σ2 µ˜ =Ω2µ˜ . (32)
− a− σ c j j j
h i
The eigenmodes η˜ are of two different kinds: either ω FIG.5. Confiningpotentialofthefirst[V(µ)(r)]andsecond
i i
belongs to a discrete set, which is the case if ωi mσ, [V(η)(r)] kind in units of the Σ−field mass squared m2σ for
≤ the set of parameters α = 0.1, α = 0.2 and α = 0.1 as
so that η˜ represents a bound state localized around the 1 2 3
i
a function of the rescalled distance to the monopole core ρ.
monopole,orω >m canbeparametrizedbyacontinu-
i σ In this case, the minimum of V(µ)(r) is located at ρ = 0
ous index q that can be identified with a momentum. In
while that of V(η)(r) is for ρ = ρ 6= 0. The bound states
thelatercase,thecorrespondingmodesarediffusion,i.e. m
eigenvalues are indicated as the straight lines. The full lines
asymptotically free states, with momentum q. Similar
correspond to V(µ) and its associated eigenvalues, while the
considerations obviously apply also to the states µ˜ .
j dotted lines stand for V(η).
On the basis of this set of eigenmodes, and following
Goldstone and Jackiw [24], one can in principle build
The purpose of this section is to exhibit the various
the quantum theory of the monopoles by constructing
possible situations, depending on the underlying micro-
a Fock space as follows. The original monopole state,
scopic parameters, in which the field Σ can evolve. To
i.e., the classical solution with no bound state, can be
achieve this, we rewrite the eigenmode equations for η˜
boosted to aquire an arbitrary momentum P, thereby
and µ˜ in the form
generating the set P . The same can be done for
monopoles with any{|ocic}upation numbers in the bound [−∆+V(η)(r)]η˜=(ω2−m2σ)η˜,
states P,n(η),n(µ) . Finally, the Fock space is com- , (34)
{| i j i} ([ ∆+V(µ)(r)]µ˜ =(ω2 m2)µ˜
pleted by inclusion of the diffusion states labelled by the − − σ
momenta of the various ingoing and outgoing particles which define the self potentials V(η)(r) and V(µ)(r) by
(η) (η) (µ) (µ)
q , ,q ,q , ,p . Thus,anarbitraryquan- comparison with Eqs. (31) and (32). Some characteris-
{| 1 ··· N 1 ··· M i}
tum state belongs to the set tic shapes and amplitudes of these potentials are shown
5
on Figure 5, in rescalled units (the square of the carrier be scattered off the monopole, effectively ionizing it. As
mass),togetherwiththecorrespondingeigenvaluesinthe in the vorton case [20], three possible interactions can
same units. take place, depending on the proton energy (we keep
Thefigurerevealstheexistenceoftwodifferentkindsof working in the monopole center-of-mass frame): elastic
self potentials, respectively called of the first and second interaction, which was discussed in [7] and which will
kind, depending on the underlying microscopic parame- presumably yield an unobservable shower accompanied
ters. The first kind is characterized by an absolute min- with Cˇerenkov radiation, excitation interaction whereby
imum in the monopole core and corresponds to a weak a charge carrier will use the proton energy to move to a
coupling where f is small, while the second kind, hav- higher energy level, and finally ionization. The last two
ing a local maximum at r = 0 and a minimum for some casesarethemostimportantasoneexpectsthemtoyield
nonzero value of the distance to the monopole core, re- an observable signal; they will be compared to existing
flects the existence of a strong coupling. As a rule, and dataandusedformakingpredictionsinthePierreAuger
as could have been expected, the energy eigenstates are Observatory [21].
moreboundforconfiningpotentialsofthefirstkindthat Once a trapped particle has been moved up to an ex-
for second kind. Therefore, one can expect the lifetime cited state, it will decay into the minimum energy state,
ofthecorrespondingconfigurationstobestronglydepen- for instance in radiating a photon∗, assuming there does
dent on the parameters. exist such a channel. Calling ∆E = E E m the
n 0 σ
− ∼
The figure 6 illustrates some wavefunctions living in energydifferencebetweenthetwoeigenenergies(see,e.g.
self potentials of the first and second kinds. Fig. 5), such an observed primary photon would have
energy, seen on earth
E =γ∆E, (35)
obs
with γ = ε/m the monopole Lorentz factor. Here, ε is
M
0.30 the total monopole energy, while m η/e is its mass.
M ∼
It should be remarked that the monopole mass can be
0.20 much larger than the carrier mass, in practice as high
as a hundred times, so that the monopole itself must be
η2
d accelerated to much higher energies than the observed
η an1 0.10 1020 eV. We shall see however in the following section
η , 0 0.00 tishasutffithciisenits tnootioanivzeertyhesemrioounsopcoolnes,ttrhaeinot.bsIefrvthede eenneerrggyy
will be also of the order of that given on Eq. (35), with
−0.10 ∆E simply equal to E0, the minimum energy state. In
order that the reaction actually takes place, it is nec-
essary that the incident proton, which, as seen from the
−0.20
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 monopoleframe,ishavingthesameLorentzfactorγ,has
ρ
enough energy to excite the corresponding state. There-
fore, ∆E =γm , with m the proton mass, so that
p p
FIG.6. Anillustrativeexampleofsomewavefunctionsliv-
ingin a self potential ofthesecond kind. Theset of parame- ε
γ =( )1/2 106.5. (36)
ters has been chosen as on figure 5. m ∼
p
Thesewavefunctionsareallthatisneededtofullyclar- Suchanumberwouldseemtoimplystatesinamonopole
ify the internalstructure ofthe monopole. The expected at energies of the order of a few thousands TeV, and
spectrumwouldhaveline featuresasobtainedby revers- therefore far below the cosmological limit m < 109
M
ing the axes on Fig. 5. We therefore turn to these cross- GeV. This leaves a large range of possibilities f∼or the
sections before discussing the acceleration and propaga- ratio η/mσ.
tion mechanisms of monopoles. Various cases have to be taken into account if one
wantsto actuallyunderstandthedataintermsofbound
states in monopoles. First of all, the determination of
VI. MONOPOLE-PROTON INTERACTIONS
Contrary to the case of an ordinary (structureless)
∗
monopole,theconfigurationswehavebeenstudyinghave The term “photon” shall be used here in thegeneric sense
the ability to interact through deep-inelastic scattering of an unbound decay product of our Σ−field. In case Σ is
with air nucleus, a proton say to simplify matters. In- effectivelycoupledtoelectromagnetism, this“photon”would
deed, a particle trapped inside the monopole core can bean actual photon.
6
the energy levels, as discussed above, although specifi- Accelerating a magnetic monopole in the presence of
cally relevant to our case of a bosonic condensate, is es- a magnetic field is a very easy task: even the galactic
sentially the same in the case of fermionic bound states. magnetic field could do it [7] as the kinetic energy E
M
However,in the latter case, the Pauliexclusionprinciple of a monopole in a magnetic field B 10−6 G with a
∼
would apply and the overall distribution of populated coherence length D 300 pc would be of order
∼
states would be different: as all bosons would preferen-
tially occupy the same state (the lowest energy state), E =q BL 6 1019 eV B D ,
fermions on the other hand would fill all the states. The M M ∼ × 3 10−6G 300pc
(cid:18) × (cid:19)(cid:18) (cid:19)
expected spectrum would therefore be qualitatively dif-
(38)
ferent. The solid prediction here is the same as that for
the vortonsolutionto the UHECRproblem[20], namely
with q the magnetic charge (inversely proportional to
that of the existence of a line spectrum, the exact form M
the electric coupling constant). Using Eq. (38), one gets
of which needing a specific model to be determined. If
table I for the maximum energy acquired by a magnetic
such a detection was indeed achieved, the method pre-
monopole (adapted from [25]).
sented above would prove usefull to actually determine
the relevant particle physics model through a fit of the
data.
Magnetic fields, distance and energy
Identifying the primary as a monopole is also possible
through its interactions with air nuclei. In other words,
one needs to know the monopole atom cross section, or, object B (G) D (pc) E (eV)
M
asemphasizedabove,the monopoleprotoncrosssection.
Such a calculation is feasible in principle by making use Neutron Star 109.5−1013 10−12 1020−1024
of the quantum theory developed in Section IV and V, WhiteDwarf 104−108 10−9 1018−1022
andwillhavetobedoneincasearayspectrumisindeed AGN 103−104.5 10−3 1023−1024
observed in the Pierre Auger Observatory [21]. For the
SNR 10−5−10−4 10 1019−1020
time being, a rough evaluation will be enough.
RGlobes 10−5−10−4 105 1023−1024
A monopole such as described here interacts with a
fermionic field such as a proton through possible exi-
tations of the A modes, the so-called dyonic modes. TABLE I. Characteristic values of magnetic fields B and
0
Thisisaresonantexitationwhicheffectivelyconfinesthe associated coherencelengthsD forastrophysicalobjectscan-
fermioninthecoreofthemonopoleforalongtime,mak- didatesfor accelerating UHECR.Thethird column givesthe
ing the cross-section essentially hadronic (the proton is kinetic energy range that can be obtained for a monopole in
thenseenbythemonopoleasaquasi-boundstate). Once such a field.
trapped, the proton (or a quark therein) has time to in-
teract with the bound states. As a result [13], the inter-
action cross-sectionwe are seeking is given by:
σ q−2m−2, (37)
Mp ∼ p
numerically of the order of a few hundreds mb. Such
a large cross-section implies that most of the monopoles
arrivingintheatmospherewouldbedetectable,provided
they carry bound states.
VII. ACCELERATION AND PROPAGATION
Once the internal structure of a monopole is known,
it must be shown that they have the ability to be de-
tected as UHECR. It is the purpose of this section to
show that indeed many astrophysicalsites have the abil-
ity toacceleratethem. Moreover,there isno GZKcutoff
for them since they share the vorton features of being
effectively highly charged, a fact that is compensated
by their huge mass: the electromagnetic energy losses
are similar to that of a heavy nucleus having an electric
charge Z =1/2q=137/2 [7], but are reduced by inverse
powers of the mass [20].
7
Monopoles are however believed not to be bound to t canbeassumedtobe thesizeL ,sothatEqs. (42)
0 AGN
small objects such as stars (See Ref. [7] and references and (40) combine into
therein). Therefore, among the various candidates pre-
m 3 L 2
sentedonTableI,onlyActiveGalacticNuclei(AGN)and φ 1021ǫα M AGN s−1. (43)
Radio Galaxy (RG) lobes have a chance to be the likely AGN ≃ 1011Gev 10−3pc
(cid:16) (cid:17) (cid:16) (cid:17)
accelerators of monopoles. In both cases, the situation
Thespherewithz =1containsabaryonicmassM =
is such that the regions where the magnetic field is im- B
5.8 1054h−1g, with h the Hubble constant H in units
portant correspond also to regions where other particles of 7×5 kms−1Mpc−1. The total number of ga0laxies in
would be accelerated. These regions are believed to be such a ra·dius is therefore N = 3 1010h−1M−1, M
filled essentially with electromagnetic radiation together beingthemassofthegalaxyignunits×of1010solar10masse1s0,
withverylowhadronicdensity(inthesensethatprotons
withN approximatelyatenth ofthis value [27]. The
in such a medium would dominantly see the radiation AGN
expected flux Φ is now given by
andwouldotherwisehaveameanfreepath,withrespect M
to proton-proton interactions, larger than the accelera-
D −2
tion zone). In turn, this radiation is responsible for de- Φ 3 1019ǫα M m3 L2 h i cm−2 s−1,
AGN ≃ × 6 10 11 −3 1Gpc ·
grading the acceleratedproton energies, but is far below
(cid:16) (cid:17)
the threshold for interacting efficiently with monopoles. (44)
Therefore,the mean free path of the monopoles is larger
than the actual size of the acceleration region. As a re- with α6 = α/106, m11 = mM/1011GeV and L−3 =
sult, one does not expect any kind of cutoff in the injec- L /10−3pc. In Eq. (44), it should be remarked that
AGN
tion spectrum. the efficiency ǫ can exceed unity and in fact presumably
Finally, one needs to know the expected number of depends on the energy scale η.
such events. Let us assume for the sake of the argument
that AGN are the most likely accelerator candidates.
To estimate the flux Φ , we set φ the number VIII. CONCLUSIONS
AGN AGN
of monopoles emitted per unit time by a characteristic
AGN,N thenumberofAGNsuptoadistancewhich Monopoles represent the most generic prediction of
AGN
wearbitrarilyassumecorrespondstoaredshiftz =1and GUTmodelsandassucharealwayspresentedasapossi-
D the mean distance to the AGN under consideration. ble cosmologicalnuisance. This is because they are usu-
h i
One then obtains ally postulated to form at the GUT phase transition, in
whichcasetheirremnantdensityintheuniversewouldbe
Φ =φ N D −2. (39) proportionalto the fourth power of this energy scale [2],
AGN AGN AGNh i
adensityconsiderablylargerthanthecriticaldensityto-
The quantity φAGN is day. Going back to the original idea of Kephart and
Weiler [7], we consider instead their usefulness in ex-
φ =ǫt−1N , (40)
AGN 0 M plaining the mystery of the UHECRs by assuming the
simplestofallsolutiontothemonopoleproblem,namely
with ǫ the ratio between the power used to accelerate
thattheyareproducedataphasetransitiontakingplace
monopoles and the electromagnetic luminosity, i.e., a
atatemperaturescalenohigherthan109 GeV,implying
measure of the efficiency of our mechanism, while t is
0 a monopole mass m <1011 GeV. In this case, one can
the characteristic time for emitting a monopole. N is M
M safely consider that ma∼gnetic monopoles do exist in rea-
thenumberofmonopolespresentintheacceleratingzone
sonnablenumberintheuniverseandtheirstudybecomes
of the AGN. Typically N = αn L3 , where α is the
M M AGN less academic.
ratiobetweenthe hadronic(ρ )andmonopole(ρ )den-
b M Monopoles as such can hardly interact with air pro-
sities;n isthemeannumberdensityofmonopolesinthe
M tons to yield air showers as they must be topologically
universe and L the characteristic size of the acceler-
AGN stable: their expected signal would be an extremely dif-
ation region. Standard monopole formation mechanisms
ficult to observe Cˇerenkov shower[7]. However,they are
give [2]
obviouslyeasilyacceleratedtoenergiesmuchhighereven
m 3 than the world record observed until now of E
n 10−19 M cm−3, (41) 3 1020 eV [26]. One is therefore tempted to coWn.Rsi.de∼r
M ≃ 1011Gev ×
(cid:16) (cid:17) them as candidates for explaining those data. The point
which implies we want to make here is that a possibility is left opened
when one considers the nature of monopole.
m 3 L 3
N 1026α M AGN . (42) A monopole is a solitonic configuration of a winding
M ≃ 1011Gev 10−3pc localised Higgs field. In most reasonnable theories, such
(cid:16) (cid:17) (cid:16) (cid:17)
aHiggsfieldnotonlypermitthesymmetrybreaking,and
Considering the monopoles to be relativistics so that
thus the appearance of topological defects, but serves
their velocity is essentially that of light, the escape time
also as a means to provide masses to various particles
8
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We should like to thank S. Bonazzola, B. Carter,
sium, Paris (1998).
J. Dubau, R. Hakim, M. Lemoine, and H. D. Sivak for
many stimulating and enlightning discussions.
9