Parton content of the real photon: astrophysical implications

0.12 MB·English

by  I. Alikhanov

Parton content of the real photon: astrophysical implications
∗
I. Alikhanov
Institute for Nuclear Research of the Russian Academy of Sciences,
60-th October Anniversary pr. 7a, Moscow 117312, Russia
We possess convincing experimental evidence for the fact that the real photon has non-trivial
parton structure. On the other hand, interactions of the cosmic microwave background photons
withhighenergyparticlespropagatingthroughtheUniverseplayanimportantroleinastrophysics.
Inthiscontext,toinvokethepartoncontentcouldbeconvenientforcalculationsoftheprobabilities
ofdifferentprocessesinvolvingthesephotons. Asanexample,thecrosssectionofinclusiveresonant
W+ boson production in the reaction νγ → W+X is calculated by using the parton language.
Neutrino–photon deep inelastic scattering is considered.
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PACSnumbers: 95.30.Cq,95.85.Ry,13.15.+g,13.60.Hb
0
0
2
I. INTRODUCTION On the other hand, the cosmic microwavebackground

n (CMB)photonsmayplayanimportantroleintheforma-
a
tionofcosmicrays(CR).Oneofthebrightestrepresenta-
J The parton model, according to which hadrons con-
tivesistheGreisen–Zatsepin–Kuzmin(GZK)limitonthe
6 sist of quarks antiquarks and gluons (partons), bound
energyofCR[15,16]. Forexample,protonsofenergiesof
2 together in different ways, has been very successful
over about 1020 eV would be decelerated by interaction
] in reproducing experiment. This provides a relatively withtheCMBphotons,mostlyduetoresonantpionpro-
explicit and transparent technique for the description
h duction, pγ ∆+ pπ0(nπ+). Other interesting pro-
p of high energy particle interactions. The distribu- → →
cesses, the νγ reactions and their possible astrophysical
- tions of partons inside hadrons are characterized by the
p implications, were extensively discussed in the literature
structure functions satisfying the Dokshitzer–Gribov–
e (see, e.g., Refs. [17, 18, 19, 20, 21, 22, 23, 24] and the
h Lipatov–Altarelli–Parisi(DGLAP) equations [1, 2, 3] or references cited therein). In this context, to invoke the
[ ones that are basically similar. Such a function is the
partoncontentoftherealphotoncouldbeconvenientfor

probability density of finding a parton in a hadron car-
4 calculations of the probabilities of such processes. Here,
rying a fraction of the total hadron’s momentum. Nu-
v we attempt to show the example of W+ boson produc-
merical solutions of the equations are in a remarkable
7 tionintheνγ scatteringwhichmayhaveimportantcon-
0 agreement with experimental measurements, especially sequences for astrophysics [17]. Studying this reaction
7 for the nucleon [4].
couldalsoprovideatestoftheuniversalityoftheparton
3
Photonsbeing involvedinhighenergyinteractionsare distribution functions of the photon.
.
3 also able to manifest hadronic structure. One can intu-
0 itivelycomprehendthissincethephotondirectlycouples
8
to quarks and therefore may split into quark–antiquark II. NEUTRINO–PHOTON REACTIONS
0
pairs. The parton contributions in two-photonprocesses WITHIN PARTON MODEL
:
v and some crucial peculiarities of the kinematic behav-
Xi ior of the photon structure function have been described Let us first consider inclusive on-shell W+ boson pro-
by Walsh and Zerwas [5]. The first work in studying duction in the reaction ν γ W+X at the resonance
r e →
a quantum-chromodynamicscorrectionstothenaivepoint- region using the parton language. We will view it from
like structure of the photon belongs to Witten [6]. This thecenter-of-mass(CMS)frameoftheν γ system. Here,
e
problem was also studied in Refs. [7, 8, 9]. Introducing for example, a substantial fraction of the CMB photons
the evolution equations, similar to the DGLAP ones, for will be of energies of about (εElab)1/2, where Elab is the
ν ν
photons as well as the properties of the corresponding neutrino energy in the laboratory frame defined as the
solutions were under scrutiny, for instance, in a series of frame in which the CMB is isotropic, ε is the CMB pho-
papers by Glu¨k, Reya, Grassie and Vogt [10, 11, 12]. A tonenergy(typicalvalueε 10−3eV[24]). Thisreaction
∼
formulationofhighenergyγpinteractionstakingintoac- is standardly factorizedinto two subprocesses: the emis-
countthehadronicpropertiesofthephotonwasproposed sion of a positron by the photon and annihilation of the
in Ref. [13]. neutrino with the positron into W+ (see Fig. 1a). Then
Today, we possess convincing experimental evidence the corresponding cross section may be written as
for the fact that the real photon has non-trivial parton
structure [14]. 1
σ(s)= σˆ(xs)fe(x,s)dx; (1)
Z γ
0
here s is the total CMS energy squared (s 4εElab),
∗E-mail:[email protected] fe(x,s) is the probability density function t≃o findνthe
γ
2
ν dfe(x,Q2) α
e e γ = k0(x), (4)
dlnQ2 2π
W + where α is the fine structure constant,
k0(x)=2[x2+(1 x)2] [27, 28]. Replacing in Eq. (4)
Q2bys,forthereaso−nexplainedabove,andchoosingthe
γ d, –u u, d – b) electron mass squared m2e as the lower integration limit,
one obtains
ν
e
α s
W + fe(x,s)= [x2+(1 x)2]ln . (5)
γ π − m2
e
e Note that the latter result is similar, for example, to
γ
the one from Ref. [29].
a)
Substituting Eqs. (2) and (5) into Eq. (1) and per-
forming the integration, one finally arrives at the cross
section
FIG. 1: Diagrams illustrating a) the inclusive reaction
νeγ → W+X. Neutrino annihilates with positron emitted 8αΓ2 s2 2m2 (s+Γ2 m2 )
bythephotonintoon-shellW+;b)chargedcurrentneutrino σ(s)= 2s+ − W − W
3 s3 (cid:20) Γm
scattering off quarks (antiquarks) coming from the photon. W
In this paper we take into account only the u and d quarks s m2 m
arctan − W +arctan W
(antiquarks) and neglect Cabibbo–Kobayashi–Maskawa mix-
×(cid:18) Γm Γ (cid:19)
W
ing.
Γ2m2 +m4 s
+(s 2m2 )ln W W ln . (6)
− W Γ2m2 +(s m2 )2(cid:21) m2
W − W e
positroninthephotoncarryingthefractionxofthetotal
The dependence of the cross section on s is displayed
photon’smomentum,andσˆ(xs)isthecrosssectionofthe
in Fig. 2a in comparison with calculations of the closely
annihilation subprocess. Note that we explicitly write
related process ν γ W+e− carried out by Seckel [17].
the s dependence of the function instead of the more Here m 80.4eGe→V, G 1.16 10−5 GeV−2,
traditionalQ2 one (4-momentumtransfer squared)since α(m2 )W ≃1/128 [30]. OnFe ≃can see×that the values given
we deal with an s-channel subprocess. W ≃
by Eq. (6) are about two times higher than those of
In the resonance region σˆ(xs) is given by the Breit–
Ref. [17].
Wigner formula [25]
Let us turn now to the charged current interaction of
the neutrino with the quark content of the photon (see
ΓΓ Fig. 1b). The corresponding cross section can be ob-
i
σˆ(xs)=24π , (2)
(xs m2 )2+m2 Γ2 tained in the same way as it is done for neutrino–proton
− W W scattering [25]:
where m is the mass of the W+ boson, Γ is the
W i
partialwidth of the initialchannel (the partialwidth for G2 s m2 2 1
the decay W+ ν e+), and Γ is the total decay width σν(s)= F W fq(Q2)+ fq¯(Q2) ,
of W+. In the l→eadieng order one can find that [26] π (cid:18)m2W +Q2(cid:19) (cid:18) γ 3 γ (cid:19)
(7)
with
G m3
Γ = F W, Γ=9Γ, (3)
i i
6π√2 1
fq(q¯)(Q2)= xfˆq(q¯)(x,Q2)dx, (8)
γ Z γ
where G is Fermi’s constant. 0
F
maTliosmdegteivrmeninine tRheef.fu[1n2c]t.ioInt ifsγef(axir,st)o, ewxepaecdtopfte(txh,es)fotro- where fˆγq(q¯)(x,Q2) is the probability density to find a
γ quarkq (antiquarkq¯)inthephotoncarryingthefraction
satisfy,uptofactorsassociatedwiththequarkcolorsand
x of the total photon’s momentum. Taking into account
fractional electric charges, the same evolution equation
only the densities of the lightest quarks u and d from
asthequarkdistributionsinthephotondo,providedthe
Ref. [11], we found that
gluons are excluded and one takes into account only the
electromagnetic interaction. Then, in the leading order
wewritethefollowingequationforthe positrondistribu- fq(Q2)=e2 α ln Q2 3 ; (9)
tion [12]: γ q2π (cid:18) m2 − 4(cid:19)
q
3
hadrons. The latter can be highly boosted and on de-
5 10 15 20 E lνa b (1022 eV)
caying (if unstable) may produce particles with energies
b)
m
νσ (10-11 νeγ e- X eoxfcteheediEngartthhetirheGdZeKcaylimpirto.duIfctistmocacyurrseaicnhtuhse wviictihnoiutyt
Q2=15 GeV2 significant energy loss, provided the incident quark mo-
Q2=10 GeV2
-12 Q2=5 GeV2 mentum pointed in the direction of the Earth. A similar
10
idea has been proposed, for example, in Ref. [31], when
b)
-13 photons would appear beyond the GZK limit from de-
10
100 200 300 400 500 600 700 800 900 1000 caysofhighlyboostedπ0,which,inturn,werethedecay
s (GeV2) productsofrealZ0bosonsexcitedinνν¯annihilation(the
162.5 175 187.5 200 212.5 E lνa b (1022 eV) so-called”Z-burst”mechanism). Butthereareproblems
b) here, mainly associated with the origin of such high en-
σ (m10-5 ννeeγγ WW ++e X- eorugrycnaseeu,trtihneosm, iEnνilamba≃l nmeu2Zt/ri4nεo(esneee,rgey.gr.,eqRueirf.ed[2t1o]).prIon-
duce hadrons is smaller than the latter one by about a
-6
10
factor of 400,and the corresponding cross section is also
suppressed by a factor αG . Anyway, one may expect
10-7 a) thatthese processeswereimFportantforhighenergyneu-
6000 6500 7000 7500 8000 8500 9000 trino absorption in the early Universe.
s (GeV2)
Throughout this paper we implicitly used the as-
FIG. 2: a) Dependence of the cross section of the inclusive sumption that the parton distributions are process–
reaction νeγ → W+X on s in the resonance region (solid independent, whichhas been experimentally justified for
curve). The same calculated for the closely related reaction the nucleon. For example, the functions phenomeno-
νeγ → W+e− [17] is shown by the dashed curve. b) De- logically derived from electron–nucleon and neutrino–
pendence of the cross section of the reaction νeγ → e−X nucleon deep inelastic scattering data are close to each
on s at some fixedvaluesof Q2 (Q2= 5 GeV2 – solid curve, other. Usingthemonecancorrectlypredicttheprobabil-
Q2=10GeV2–dashedcurve,Q2=15GeV2–dottedcurve). itiesofinclusiveproductionofµ+µ− pairsinpp¯collisions
Nisoctaelctuhlaattetdheatlaεb=ora1t0o−r3yeeVne.rgy of the neutrino Eνlab ≃s/4ε (Drell–Yan process) [32].
Analogously,theneutrino–photonreactionscouldpro-
vide an instrument for studying the universality of the
here e andm arethe electric chargeandmass ofthe
q q parton distributions in the photon.
quarkqrespectively(forantiquarkstheequationisanalo-
gous). NotethatEq.(9)isvalidinthelimitm2/Q2 1. We have discussed only the ν γ interactions. Mean-
q ≪ e
We set m = m = 0.2 GeV and α = 1/137. The while, all the things we said above may be straightfor-
u d
dependence of the cross section thus determined on s in wardly applied to the reactions involving the antineu-
the range 20 GeV2 s 1000 GeV2 at some values of trino. Likewise, heavier charged leptons can be consid-
Q2 is shown in Fig.≤2b. ≤ ered. One may also include neutral current interactions
This reaction may have interesting astrophysical im- intheneutrino–quarkscattering. Otherprocessesinvolv-
plications because the struck quark may fragment into ing the CMB photons can be treated in similar way.
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