Coherent $ρ^0$ photoproduction in bulk matter at high energies

0.13 MB·English

by  Elsa Couderc

Coherent ρ0 photoproduction in bulk matter at high energies
Elsa Couderc
Lawrence Berkeley National Laboratory, Berkeley CA 94720 USA
and D´epartement de physique, Ecole Normale Sup´erieure, Paris, France.
Spencer Klein
Lawrence Berkeley National Laboratory, Berkeley CA, 94720 USA
and Department of Physics, University of California, Berkeley, CA, 94720 USA
(Dated: January 9, 2009)
The momentum transfer ∆k required for a photon to scatter from a target and emerge as a ρ0
decreasesasthephotonenergykrises. Fork>3×1014 eV,∆kissmallenoughthattheinteraction
cannotbelocalizedtoasinglenucleus. Atstillhigherenergies,photonsmaycoherentlyscatterelas-
tically from bulk matter and emerge as a ρ0, in a manner akin to kaon regeneration. Constructive
9
interference from the different nuclei coherently raises the cross section and the interaction prob-
0
0 ability rises linearly with energy. At energies above 1023 eV, coherent conversion is the dominant
2 process; photons interact predominantly as ρ0. We compute the coherent scattering probabilities
in slabs of lead, water and rock, and discuss the implications of the increased hadronic interaction
n
probabilities for photons on ultra-high energy shower development.
a
J
PACSnumbers: PACSNumbers:13.60.-r,25.20.-x,95.55.Vj
9

] An understanding of the cross sections for photon in- constructive interference raises the cross section.
h
t teractions is important in many areas of physics. For Photon to ρ conversion occurs when a photon fluctu-
- example, several groups have searched for radio waves ates to a virtual qq pair which then scatters elastically
l
c [1] or acoustic pulses [2] produced by interactions of as- fromatargetnucleus,emergingasavectormeson[7]. As
u
trophysicalν withenergiesupto1025 eV.Theradioand thephotonenergykrises,therequiredmomentumtrans-
n e
acoustic frequency spectra and angular distribution de- fer ∆k = M2 /2k decreases, and the coherence length
[ ππ
pend on the distribution of moving electric charges (for l = ~/∆k rises (we take ~ = c = 1 throughout). Here
f
1 radio) and energy deposition (for acoustic). These dis- M is the final state mass. When k >3 1014 eV (for
ππ
v tributionsarecontrolledbytheparticleinteractionsthat M = M = 778 MeV/c2, the ρ pole m×ass), l > 0.2
1 ππ ρ f
governshowerdevelopment. Theexperimentalfluxlimits nm, the typical internuclear spacing, and coherence over
6
depend on the assumed character of these interactions. multiple nuclei becomes possible.
1
1 In this letter, we discuss hadronic interactions of pho- We consider a high-energy plane-wave photon travel-
. ling in the +z direction. The photon wave function is
1 tons [3], and introduce a new effect: coherent photon to
0 ρ0 conversion in bulk matter. These effects become im- mostly bare photon (|γb >), but with a qq component,
9 portant at energies above 1020 eV, but are not in many |qq > and heavier fluctuations (e.g. qqqq, and strange
0 quark pairs):
current calculations [4]. With these effects, photons
:
v produce hadronic showers, rather than electromagnetic ψ >=eikz 1 F2 γ >+F qq>+... (1)
i showers. Sincehadronicshowersaremuchlessaffectedby | − | b |
X (cid:2)p (cid:3)
the Landau-Pomeranchuk-Migdal(LPM) effect [5], they where F = 6.0210−2 is the amplitude for the photon to
r
a are much more compact; the presence of hadronic inter- fluctuate to a qq pair [7]. Heavier fluctuations are not
actions alters both the frequency spectrum and angular relevant here, and will not be further considered.
distributionoftheelectromagneticandacousticradiation The qq > scatters from nuclei at positions ~r , with
i
fromneutrinoinduced showers. Publishedlimits that do scatteri|ngamplitude f(θ), emergingwith momentum k~′.
not consider these effects may need to be revised.
Neglecting photon or ρ absorption, and photon scatter-
Besides it’s importance for ν searches, coherent con- ing (so qq do not scatter and then fluctuate back to a
version is very interesting in it’s own right, as one of a photon), the wavefunction at a depth L in the material
handful of examples where particles interact very differ- is
etinotnlys winitbhuilnkdmiviadtutearl ttahragnetwsiathreirseoplalatecdedabtoymdsi;stirnitbeurtaecd- ψ(L)>= 1 F2eikL γ >+F eikzif(θ)eik~i′(L~−r~i) ρ>.
interactions which are de-localized over multiple atoms | p − | b Xi |L~ −r~i| |
[6]. The otherprominentexamplesareLPMsuppression (2)
of bremsstrahlung and pair production [8, 9], kaon re- Thephotoncanbeabsorbedinthetarget,withabsorp-
generation[10],andcoherentneutrinoforwardscattering tionlengthα=1/n[σ (γA)+σ (γA)]. Hereσ (γA)
ee hadr ee
[11]. The LPM effect decreases the cross section for pair isthepairproductioncrosssection,σ (γA)isthepho-
hadr
production, due to destructive interference. In contrast, tonuclear cross section, and n is the target density, in
in the other examples, including coherent ρ conversion, atoms/volume.
2
The ρ can be lost by direct hadronic interaction, or Eq. (6) illustrates some of the features of the system.
by decay to π+π−, followed by π interactions. The de- When ∆kL 1 the cosine fluctuates rapidly, leading to
≫
cay itself does not affect the coherence [12], but two π incoherent ρ production. However, when the coherence
interact differently from one ρ. For simplicity, we take condition∆kL 1isfullfilled,thescatteringamplitudes
≪
β = 1/nσ (ρA) where σ (ρA) is the ρ nucleus cross add in phase, and production is coherent.
tot tot
section. The factor of 2 difference between two π and Aρinsideatargetisnotdirectlyobservable. However,
one ρ is small compared to the other uncertainties. This thewayitinteracts-electromagnetically,hadronically,or
alsoappliesfordirectπ+π− production,discussedbelow. throughρconversion-isindirectlyobservable,bystudy-
Including these absorption factors, and treating the ing the shower development. The probability that an
target as a homogenous medium with constant density incident photon interacts as a real ρ is the integrated
n, the wave function at depth L in the slab is probability for finding a ρ, multiplied by the probability
∞ 2π for a ρ to interact hadronically in length dz, or dz/β:
<ρ0 ψ(L)>=nF ηdη dψ
| Z Z L dz <ρ0 ψ(z)> 2
0 0
P(L)= | | | . (7)
× Z Ldze(ik−α2)zf(θ)e(ik′r−′β2)r′, (3) Z0 β
0 This equationis only properlynormalizedforP(L) 1.
≪
foranincidentplanewavemovingalongthe+zaxis. The The loss of photon intensity due to the coherent ρ reac-
integral over the target volume is in cylindrical coordi- tion is not included. For kaon regeneration, the analo-
nates where η is the radial distance from the z axis and gous problem has been solved recursively [16]. Here, we
ψ istheazimuthalangle. Weneglectscatteringeffectsat normalizetheprobabilitiestosumtooneinthicktargets.
large distances since absorption limits the effective size The numericalvalues of P(L)depend onσ (ρA) and
tot
of the target. the cross sectionfor a photon to interact hadonically (as
In an infinite medium only forward scattering, f(0), a ρ), σ (γA). They are related via
hadr
contributes to coherent scattering [10, 11, 13]. We as-
σ (γA)=F2σ (ρA). (8)
sumethattherealpartoff(0)issmall[7],corresponding hadr tot
to nearly complete absorption. Then, from the optical
We determine these using two different methods: a
theorem, f(0) =σ (ρA)k/4π.
| | tot ′ Glauber calculation based on HERA data on γp ρp,
After scattering, the system has the momentum k = →
using asoft Pomeronmodelto extrapolate the crosssec-
k ∆k. Since ∆k M2 , the ρ width affects the cal-
− ∝ ππ tions to higher energies, and a second calculation that
culation. The photon may also fluctuate directly into
an π+π− pair [14]; these two processes interfere and the includes generalized vector meson dominance (GVMD)
plus a component for direct photon interactions [17].
mass spectrum is a Breit-Wigner spectrum. Then,
We consider three materials: water, standard rock [9]
andlead. Forwater,weaddthe amplitudes forthe cross
k MππmρΓρ 2 sections for hydrogen and oxygen.
f(0,M )= A +B (4)
ππ 4π(cid:12) M2p m2+im Γ (cid:12) The Glauber calculation uses the optical theorem [18]
(cid:12)(cid:12) ππ− ρ ρ ρ (cid:12)(cid:12) and vector meson dominance model to link the total ρp
where A andB are t(cid:12)he energy-dependentamplit(cid:12)udes for cross section to the differential γp cross section:
ρ and direct ππ production respectively [15], and the
ρ0 width Γρ = 150 MeV. We assume that B/A is inde- 4√π dσ(γp ρp)
pendent of photon energy and target material. Integrat- σtot(ρp)= F r dt→ |t=0 (9)
ing dM f(0,M ) returns the traditional f(0). With
ππ ππ
this,Rσ(γp ρp)=A2. where t is the squared momentum transfer. dσ/dt(γp
We subs→titute ηdη = r′dr′ where r′ is the distance ρp comes from a fit to HERA data [19]. →
between the scattering point (η,z) and the observer at The crosssections fornuclear targetsarefound with a
(0,L), to evaluate the integrals [11]: quantum Glauber calculation [20]:
<ρ0 ψ(L)>=2πnF Mρ+5ΓρdMππf(0,Mππ) σtot(ρA)=2 d2~r 1 e−σtot(ρp2)TA(~r) (10)
| Z Z (cid:18) − (cid:19)
2mπ
e(ik′−β2)Le(i∆kL−α−2β)L 1 where T (~r) is the nuclear thickness function for a ma-
− (5) A
×(ik′ β) (i∆k α−β) terial with atomic number A, calculated from a Woods-
− 2 − 2
Saxon nuclear density, with a skin thickness of 0.53 fm
For simplicity, we give here the probability for a fixed and density ρ =1.16A1/3(1 1.16A−2/3).
0
M (although the full calculations include the wide ρ): −
ππ ThesecondapproachfollowsEngel,RanftandRoessler
P (L,M ) = 4π2 f(0)2F2n2 (ERR) [17]. It uses GVMD plus direct photon-quark
ρ0 ππ
| | interactions, to determine σ (γA). ERR predict a
e−αL+e−βL−2e−(α+2β)Lcos(∆kL)(6) steeper rise in the cross secthioadnrthan the Glauber cal-
× (k′2+ β2)(∆k2+ (α−β)2) culation. We parameterizetheir results. For W <1011
4 4 γp
3
Water-1e23 eV photon
Water y10-2
1 e+e- bilit Glauber
n in b 10-1 GERlaRuber Proba10-3 ERR
ectio10-2 10-4
s
ss 10-5
Cro10-3
10-6
10-4
13 16 19 22 25
10 10 10 10 10 10-7
Energy in eV
Standard rock
1 e+e- 101-80-3 10-2 10-1 1 10 102 103
Slab length in m
on in b 10-1 GERlaRuber FIG. 2: ρ probability as a function of depth in a slab for a
secti10-2 1023 eV photon incident on a water target. The two broad
ss peaksaround20and40 cmcorrespond to∆kL=2nπ atthe
o
Cr10-3 ρ pole mass.
Engel-Lead-1e24 eV photon
Le1a0d-4 1013 1016 1019 1022 Energy i1n0 2e5V ability 1 rho
102 ob ee
e+e- Pr10-1 hadron
n b 10 Glauber
n i 1 ERR
ectio 10-1 10-2
s
oss 10-2 10-3
Cr10-3
10-4 1013 1016 1019 1022 1025 101-40-3 10-2 10-1 1
Energy in eV Slab length in m
FIG. 1: The cross sections for photon interactions to e+e− FIG.3: Theprobabilityofa1024 eVphotoninteractingelec-
pairs, and for incoherent photonuclear interactions in the tromagnetically (dashedblueline),inanincoherenthadronic
Glauber and ERR models, for water (top), standard rock interaction(dottedline),orasacoherentlyproducedρ(solid
(middle) and lead (bottom). line), as a function of target depth in a lead target.
eV, the cross section is constant, while at higher ener- interact as a function of target thickness L. For L < 10
gies it rises as W0.2. The cross section scales as A0.887, cm, the probability of a photon interacting as a coher-
γp
normalized so σ(γPb) = 15 mb at low energies. Again, ently produced ρ is quite small. For L > 10 cm, the
σ (ρA) = σ (γA)/F2. The ERR cross sections are probability is higher than for photonuclear interactions
tot hadr
similartoanewerresultthatcomputedphotoncrosssec- and e+e−; it is the dominant interaction!
tions using a dipole model [21]. Figure 4 shows the probabilities for incident photons
Figure 1 comparesthe crosssections for e+e− produc- to interact electromagnetically (γ e+e− pair), via in-
→
tion and the two photonuclear models. σ (γA) is con- coherent hadronic interactions, and as coherently pro-
ee
stantatlowenergies,butfallsathigherenergieswhenthe ducedρ,asafunctionofenergy,ina100mthickslab;in
LPMeffectbecomesimportant[3,8]. Thehadroniccross the displayed energy range, 100 m is effectively infinite
sections rise slowly with energy. The hadronic curves thickness. At energies above 1023 eV (for ERR) to 1024
agree at low energies, but diverge as the energy rises. In eV (for the Glauber calculation), coherently produced ρ
the ERR (Glauber) model, above 5 1019 eV (5 1020 interactions dominate. This dominance will reduce the
× ×
eV),photonuclearinteractionspredominate,andphotons interaction length of photons. The reduction in shower
produce hadronic showers. The cross-over energy is al- length will have consequences for searches for ultra-high
most material-independent for solids. energy astrophysical ν [1, 4].
e
Figure2showstheprobabilityoffindingaρ0,Eq. (6), In conclusion, photons with energies above 1020 eV in
as a function of depth in a thick target. As with K solids or liquids are more likely to interact hadronically
s
from kaonregeneration[10], the amplitudes add and the ratherthanelectromagnetically. These hadronicshowers
probabilityinitiallyrisesasthesquareofthedepth. After arenotsubjecttotheLPMeffect,andsoareconsiderably
reachingamaximum, the probabilitydecreasesslowlyas more compact than purely electromagnetic calculations
ρabsorptioncompeteswithρproduction,andthendrops would predict. At energies above 1023 eV, photons are
rapidly as the bare photons are absorbed. most likely to interactby coherently convertinginto a ρ,
Figure3showstheprobabilityfora1024 eVphotonto andtheninteractinghadronicallyinthetarget. Thisisa
4
Water-Glauber Water-ERR
y y
bilit 1 bilit 1
a a
b b
o o
Pr Pr
10-1 10-1
10-2 rho 10-2 rho
ee ee
hadron hadron
10-3 1017 1019 1021 1023 1025 10-3 1017 1019 1021 1023 1025
Photon energy in eV Photon energy in eV
Standard rock-Glauber Standard rock-ERR
y y
bilit 1 bilit 1
a a
b b
o o
Pr Pr
10-1 10-1
10-2 rho 10-2 rho
ee ee
hadron hadron
10-3 1017 1019 1021 1023 1025 10-3 1017 1019 1021 1023 1025
Photon energy in eV Photon energy in eV
FIG. 4: The normalized probabilities for a photon to interact as an e+e− pair, via incoherent hadronic interaction (‘hadron’),
or as an elastically produced ρ (‘rho’) in a 100 m thick water or standard rock target. This is thick enough for almost total
absorption, so these results apply for an infinitely thick target.
new example of a coherent process in bulk matter, simi- included in the models used in searches for ultra-high
lar to kaon regenerationor coherent neutrino scattering. energy astrophysical ν .
e
Thiscoherentinteractionfurthershortenstheshowerde- WethankVolkerKochforusefulcomments. Thiswork
velopment. The reduction in photon interaction length wasfundedbytheU.S.NationalScienceFoundationand
will shorten ultra-high energy electromagnetic showers, the U.S. Department of Energy under contract number
altering the radio and acoustic emission. It should be DE-AC-76SF00098.
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