Neutrino emission from a GRB afterglow shock during an inner supernova shock breakout
aa r X i v : . [ a s t r o - ph ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 21 November 2018 (MN L A TEX style file v2.2)
Neutrino emission from a GRB afterglow shock during aninner supernova shock breakout
Y.W. Yu , ⋆ , Z.G. Dai † , and X.P. Zheng ‡ Department of Astronomy, Nanjing University, Nanjing 210093, China Institute of Astrophysics, Huazhong Normal University, Wuhan 430079, China
21 November 2018
ABSTRACT
The observations of a nearby low-luminosity gamma-ray burst (GRB) 060218 associ-ated with supernova SN 2006aj may imply an interesting astronomical picture wherea supernova shock breakout locates behind a relativistic GRB jet. Based on this pic-ture, we study neutrino emission for early afterglows of GRB 060218-like GRBs, whereneutrinos are expected to be produced from photopion interactions in a GRB blastwave that propagates into a dense wind. Relativistic protons for the interactions areaccelerated by an external shock, while target photons are basically provided by theincoming thermal emission from the shock breakout and its inverse-Compton scat-tered component. Because of a high estimated event rate of low-luminosity GRBs,we would have more opportunities to detect afterglow neutrinos from a single nearbyGRB event of this type by IceCube. Such a possible detection could provide evidencefor the picture described above.
Key words: gamma rays: bursts — elementary particles
Gamma-ray burst (GRB) 060218 associated with SN 2006ajdiscovered by Swift (Campana et al. 2006) provides a newexample of low-luminosity GRBs (LL-GRBs), as its isotropicequivalent energy ( ∼ × erg) is 100 to 1000 times lessbut its duration ( T = 2100 ±
100 s) is much longer thanthose of conventional high-luminosity GRBs. More interest-ingly, besides an usual non-thermal component in its earlyX-ray spectrum, a surprising thermal component was ob-served by the Swift XRT during both burst and afterglowphases. Fitting with a blackbody spectrum, the temperatureof this thermal component was inferred to be kT ∼ .
17 keVduring the first 3 ks. When t >
10 ks, however, the peak en-ergy of the blackbody decreased and then passed throughthe Swift UVOT energy range at ∼
100 ks (Campana et al.2006; Blustin 2007).To explain the prompt emission, in principle, a modelbased on the internal dissipation of relativistic ejecta maybe valid in the case of GRB 060218. The relativistic GRBejecta, which interacts with a dense wind surrounding theprogenitor, has also been required to understand a power-law decaying afterglow in X-ray and radio bands about ∼ ⋆ [email protected] (YWY) † [email protected] (ZGD) ‡ [email protected] (XPZ) standard afterglow model are involved (Soderberg et al.2006; Fan et al. 2006; Waxman et al. 2007). Furthermore, assuggested by Wang & M´esz´aros (2006), the soft X-ray ther-mal emission could arise from a shock breakout, namely, ahot cocoon that breaks out from the supernova ejecta andthe stellar wind. In detail, a more rapid part of a jet mov-ing in the envelope and the dense wind is accelerated toa highly-relativistic velocity to produce the GRB, while aslower part of the jet together with the outermost parts ofthe envelope becomes a mildly relativistic cocoon (M´esz´aros& Rees 2001; Ramirez-Ruiz et al. 2002; Zhang et al. 2003),which locates behind the GRB blast wave.Although the GRB blast wave runs in front of the shockbreakout, the thermal emission from the latter outshinesthe former persistently until the emission of the breakoutis switched off. Thus, the emission properties of the GRBblast wave (consisting of external shock-accelerated elec-trons and protons) should be influenced by the incomingthermal photons significantly during both burst and earlyafterglow phases. On one hand, the cooling of the relativis-tic electrons, which upscatter the thermal photons, could bedominated by inverse-Compton (IC) radiation rather thansynchrotron radiation (Wang & M´esz´aros 2006). On theother hand, inferred from the observations, the intensity ofthe thermal emission could be comparable in the same bandto the one of the prompt emission due to internal dissipa-tions and much larger than the one of the afterglow emissiondue to an external shock. Therefore, the thermal photons as c (cid:13) Y.W. Yu, Z.G. Dai and X.P. Zheng target photons for photopion interactions could also play animportant role in the energy loss of the relativistic protonsand thus influence or even dominate neutrino emission ofthe GRB blast wave.It has been widely studied that conventional GRBs inthe standard internal-external shock model emit high energyneutrinos during the burst, early afterglow, and X-ray flarephases (Waxman & Bahcall 1997, 2000; Dai & Lu 2001; Der-mer 2002; Dermer & Atoyan 2003; Asano 2005; Murase &Nagataki 2006a, 2006b; Murase 2007; Gupta & Zhang 2007).In contrast to the conventional GRBs, LL-GRBs may have amuch higher event rate (several hundred to thousand eventsGpc − yr − ), which is inferred from the fact that two typ-ical nearby LL-GRBs, i.e., GRB 060218 and GRB 980425,have been observed within a relatively short period of time(Cobb et al. 2006; Soderberg 2006; Liang et al. 2007). Thehigh event rate implies that the contribution from LL-GRBsto the diffuse neutrino background may be important andthat we have more opportunities to detect neutrinos froma very nearby single LL-GRB event. Therefore, Murase etal. (2006) and Gupta & Zhang (2006) recently studied theneutrino emission properties of LL-GRBs during their burstphase using the internal shock model, in which the targetphotons for photopion interactions are mainly provided byinternal shock-driven non-thermal emission. In this paper,however, we will focus on the early afterglow neutrino emis-sion. During this phase, relativistic protons are acceleratedby an external shock (rather than internal shocks) and tar-get photons are dominated by the incoming thermal emis-sion and even its IC scattered component.This paper is organized as follows. In section 2 we brieflydescribe the dynamics of a GRB blast wave propagating intoa surrounding dense wind. In section 3, we give the photondistribution in the blast wave by considering the incomingthermal emission and its IC scattered component, but theweak synchrotron radiation of the electrons is ignored. Insection 4, neutrino spectra are derived formally with an en-ergy loss timescale of protons due to photopion interactions.In section 5, we calculate the timescale and then the neu-trino spectra using the target photon spectra obtained insection 3 and an experiential fitting formula of the crosssection of photopion interactions. In addition, the peak ofa neutrino spectrum is also estimated analytically by using∆ − approximation. Finally, a summary is given in section 6. We consider a GRB jet with isotropic equivalent energy E = 10 E erg (hereafter Q x = Q/ x ) expandinginto a dense wind medium with density profile ρ ( r ) = Ar − . Here, the coefficient A is determined by the massloss rate and velocity of the wind of the progenitor, i.e., A = ˙ M/ πv w = 5 . × g cm − A ∗ , where A ∗ ≡ [ ˙ M / (10 − M ⊙ yr − )][ v w / (10 km s − )] − . From Dai & Lu(1998) and Chevalier & Li (2000), we get the Lorentz factorand radius of the GRB blast wave (i.e., the external-shockedwind gas) respectively asΓ = (cid:16) E πAc t (cid:17) / = 3 . E / A − / ∗ t − / , (1) r = (cid:16) Et πAc (cid:17) / = 3 . × cm E / A − / ∗ t / . (2)They satisfy r = 8Γ ct , which gives rise to a relationship, t ′ = (16 / t , between the dynamic time t ′ measured in therest frame of the blast wave and the observed time t (Dai &Lu 1998).As the circum-burst wind materials are swept up andshocked, most of the heated electrons before cooling con-centrate at the minimum Lorentz factor γ ′ e,m ∼ ¯ ǫ e m p m e Γ =659 ¯ ǫ e, − E / A − / ∗ t − / . The symbol ¯ ǫ e ≡ ǫ e ( p − / ( p − ǫ e is the usual equipartition factor of the hot electronsand p is the electron’s energy distribution index (where p > ǫ B of the inter-nal energy is assumed to be occupied by a magnetic field,and then the strengthen of the magnetic field is calculatedby B ′ ∼ (32 πǫ B Γ ρc ) / = 78G ǫ / B, − E − / A / ∗ t − / . Fi-nally, the other energy (a fraction of ǫ p = 1 − ǫ e − ǫ B ) is car-ried by the accelerated protons. For these protons, we canestimate their maximum energy by E p, max = 2 eB ′ r/ . × GeV ǫ / B, − E / A / ∗ t − / by equating the accel-eration time to the shorter of the dynamic time and thesynchrotron cooling time (Razzaque el al. 2006). However,the minimum energy of the protons is unknown, but thecorresponding Lorentz factor γ ′ p, min is thought to be closeto ∼ Γ. The photons in the GRB blast wave have two origins, i.e., theblast wave self and the inner supernova shock breakout. Theelectrons in the blast wave emit photons via synchrotron andIC scattering processes. Moreover, as analyzed by Wang &M´esz´aros (2006), the synchrotron radiation (peaking withinX-ray band) of the blast wave electrons is inferred from theobservations to be much weaker than the incoming thermalemission, and thus the cooling of the electrons should bedominated by their IC scattering off the thermal photons.Therefore, in following calculations, we consider the thermalemission and its subsequent IC scattered component only.The properties of the supernova shock breakout havebeen unclear to date. We suppose that it has a constantblackbody temperature of kT = 0 . kT ) − and a con-stant radius of the emission region of R = 10 cm R .The lifetime t SB of this high-temperature emission is aboutthousands of seconds, which is considered to be severalto several ten times longer than the duration of theGRB. Then, the isotropic equivalent luminosity and en-ergy of the shock breakout can be estimated by L SB =4 πR σT = 1 . × erg s − ( kT ) − R and E SB = 1 . × erg ( kT ) − R t SB , , respectively. Meanwhile, it is easyto write the monochromatic number density of these thermalphotons at the breakout as n ( E γ ) = 8 πh c E γ exp( E γ /kT ) − πk T h c φ (cid:16) E γ kT (cid:17) , (3)where the function φ ( x ) = x / ( e x − r , we can calculate the density of the thermalphotons in the blast wave by multiplying a factor ( R/r ) to Eq. (3). Subsequently, after Lorentz transformation, we c (cid:13) , 000–000 fterglow Neutrino Emission from LL-GRBs obtain the density of the incoming photons in the blast wavemeasured in its rest frame by n ′ in ( E ′ γ, in ) = R r n (Γ E ′ γ, in ) = R r πk T h c φ (cid:18) E ′ γ, in E ′ γ, pk1 (cid:19) , (4)where E ′ γ, pk1 ≡ kT / Γ is the peak energy of the black bodyspectrum. When these photons cross the blast wave, a partof them should be upscattered by the relativistic electrons.The energy of the IC scattered photons can be estimated by E ′ γ, IC = 2 γ ′ e,m E ′ γ, in and the corresponding density by n ′ IC ( E ′ γ, IC ) = τ γ ′ e,m n ′ in (cid:18) E ′ γ, IC γ ′ e,m (cid:19) = τ γ ′ e,m R r πk T h c φ (cid:18) E ′ γ, IC E ′ γ, pk2 (cid:19) , (5)where E ′ γ, pk2 ≡ γ ′ e,m kT / Γ. The probability of the scatter-ing is represented by the photon optical depth of the blastwave, τ = σ T ( A/m p r ) = 6 . × − E − / A / ∗ t − / , where σ T is the Thomson cross section. According to this esti-mation, Wang & M´esz´aros (2006) predicted that the earlyafterglow spectra of GRB 060218-like GRBs may have a bi-modal profile peaking at E γ, pk1 = 0 . kT ) − (6)and E γ, pk2 = 0 .
26 GeV ¯ ǫ , − ( kT ) − E / A − / ∗ t − / . (7)Thus, a significant sub-GeV or GeV emission componentaccompanying the thermal emission would be detectablewith the upcoming Gamma-ray Large Area Space Telescope ,which could provide evidence for the GRB jet.
Since relativistic protons in the GRB blast wave are im-mersed in the photon field described above, the protonswould lose their energy to produce mesons such as π and π ± etc, and subsequently generate neutrinos by the decay of π ± , i.e., π ± → µ ± + ν µ (¯ ν µ ) → e ± + ν e (¯ ν e ) + ¯ ν µ + ν µ . Duringthese processes, the energy loss rate of a proton with energy E ′ p = γ ′ p m p c can be calculated by (Waxman & Bahcall1997) t ′− π ≡ − E ′ p dE ′ p dt ′ = c γ ′ p Z ∞ ˜ E th σ π ( ˜ E ) ξ ( ˜ E ) ˜ E × "Z ∞ ˜ E/ γ ′ p n ′ ( E ′ γ ) E ′− γ dE ′ γ d ˜ E, (8)where σ π ( ˜ E ) is the cross section of photopion interactionsfor a target photon with energy ˜ E in the proton’s rest frame, To obtain this expression, an isotropic target photon field isrequired. However, in our model, the radially incoming photonfield is seen by the protons in the blast wave anisotropically. Thisgives an extra complication for a more realistic consideration. Forsimplicity, we ignore the anisotropic effect in our calculations. ξ is the inelasticity defined as the fraction of energy loss ofa proton to the resultant pions, and ˜ E th = 0 . h − R t ′ ( dt ′ /t ′ π ) i .In our scenario, if the shock-breakout emission could last fora period of t SB , the fraction of the energy loss of the protonsto pions could be calculated by f π = 1 − exp − Z t ′ SB dt ′ t ′ π ! , (9)where t ′ SB = (16 / t SB . In order to calculate t ′ π , the crucialinput in the model is the target photon spectrum n ′ ( E ′ γ ).From Eqs. (4) and (5), we know that n ′ ( E ′ γ ) depends onboth r and Γ and thus the value of t ′ π could evolve withtime. However, if t ′ π is independent of time or varies withtime slowly, Eq. (9) can be also approximated by f π ≈ − exp( − t ′ SB /t ′ π ) ≈ min[ t ′ SB /t ′ π ,
1] (10)as usual, especially for analytical calculations.To be specific, the energy loss of the protons is shared by π ± and π with a certain ratio. Unfortunately, it is not easyto fix this ratio due to the complications arising from varioussingle-pion and multipion production processes. In followingcalculations, we simply take it to be a constant, π ± : π =2 : 1, as in Asano (2005). Furthermore, two resultant muon-neutrinos from the decay of a π ± could inherit half of thepion’s energy roughly evenly. Therefore, we can relate theneutrino energy E ν to the energy loss of the primary protonby E ν = 14 ξE p , (11)and give an observed time-integrated muon-neutrino spec-trum by E ν φ ν ≡ πD l E ν dN ν dE ν = 14 πD l f π E p dN p dE p , (12)where D l is the luminosity distance of the burst. As usual,we assume the energy distribution of the shock-acceleratedprotons to be ( dN p /dE p ) ∝ E − p , where the proportionalcoefficient can be calculated by ǫ p E/ ln( E p, max /E p, min ).In addition, because of the presence of the magneticfield, the ultrahigh energy pions and muons would lose theirenergy via synchrotron radiation before decay. This leads tobreaks in the neutrino spectrum at (Murase 2007) E sπν, b = 14 E π, b = 14 Γ (cid:18) πm π c σ T m B ′ τ π (cid:19) / = 1 . × GeV ǫ − / B, − E / A − ∗ t / , (13) E sµν, b = 13 E µ, b = 13 Γ (cid:18) πm µ c σ T m B ′ τ µ (cid:19) / = 8 . × GeV ǫ − / B, − E / A − ∗ t / , (14)where τ π = 2 . × − s and τ µ = 2 . × − s are themean lifetimes of pions and muons in their rest frames.Above E sν, b , the neutrino flux would be suppressed by afactor ( E ν /E sν, b ) − (Rachen & M´esz´aros 1998; Razzaque etal. 2006). However, as pointed out by Asano & Nagataki(2006), neutral kaons can survive in the magnetic field, while c (cid:13) , 000–000 Y.W. Yu, Z.G. Dai and X.P. Zheng the ultrahigh-energy charged pions and muons cool rapidly.Moreover, because kaons have a larger rest mass than pionsand muons, charged kaons can reach higher energy althoughthey also suffer from synchrotron cooling. Thus, decay ofkaons, which is not taken into account in our calculations,may dominate neutrino emission above ∼ − GeV.Now, by inserting Eqs. (4) and (5) into Eq. (8) andthen into Eq. (9) to get f π , we can easily obtain the ob-served neutrino spectra from Eq. (12) for our scenario. Theremaining task is only to express the cross section σ π ( ˜ E )and inelasticity for photopion interactions. Since the cross section of photopion interactions peaks at˜ E ∆ ≃ . E in Eq. (8) can be roughly approximated by t ′− π ≈ c γ ′ p σ π, ∆ ξ ∆ ˜ E ∆ δ ˜ E Z ∞ ˜ E ∆ / γ ′ p n ′ ( E ′ γ ) E ′− γ dE ′ γ , (15)where σ π, ∆ ≈ . ξ ∆ ≈ .
2, and the peak width isabout δ ˜ E ≈ . f π = min[ t ′ SB /t ′ π ,1] toobtain f π = min (cid:26) ς R r πk T h c E ′ γ, pk E ∆ σ π, ∆ ξ ∆ δ ˜ E Γ t SB × ε ∗ [ ε ∗ − ln( e ε ∗ − , (cid:9) . (16)where ς = 1 and τ / (2 γ ′ e,m ) for pre- and post-upscattered target photons, respectively. The dimension-less variable ε ∗ is defined by ε ∗ ≡ E ∆ / (2 γ ′ p E ′ γ, pk ) =3 ξ ∆ Γ ˜ E ∆ m p c / (8 E ν E γ, pk ). In the case of f π <
1, the peakvalue of f π reading f π, pk = 3 ς R r πk T h c E ′ γ, pk E ∆ σ π, ∆ ξ ∆ δ ˜ E Γ t SB (17)is at ε ∗ = 1 .
8, which gives rise to the relationship betweenthe peak energies of the neutrino and photon spectra as E ν, pk E γ, pk = 0 . GeV . Considering the bimodal distri-bution of the target photons peaking at E γ, pk1 and E γ, pk2 ,two peaks are also expected in the resultant neutrino spec-trum but only the one determined by E γ, pk1 could fall intothe high energy range ( E ν > TeV) of our interest at E ν, pk = 4 . × GeV ( kT ) − − E / A − / ∗ t − / , . (18)In other words, the target photons for photopion interactionsof interest are contributed by the incoming thermal emissionmainly. The value of the differential neutrino fluence at E ν, pk reads[ E ν φ ν ] pk = 2 . × − erg cm − ǫ p ( kT ) − R A ∗ D − l, . , (19)which is calculated by using the peak value of f π as f π, pk = 0 .
02 ( kT ) − R E − A ∗ . (20)On the other hand, when f π = 1, E ν φ ν would reach anupper limit as 1 . × − erg cm − ǫ p E D − l, . , which isdetermined by the total energy carried by the protons inthe GRB blast wave. Although it is convenient and effective to use the∆ − approximation to estimate the peak of a neutrino spec-trum, the ∆ − approximation would lead to an remarkableunderestimation of the neutrino flux above the peak energydue to the non-zero cross section of photopion interactionsin high energy regions. So, for more careful calculations, weprovide an experiential fitting formula for the cross sectionas shown in Eq. (A5), which is extrapolated from experimen-tal data taken from particle data group (Yao et al. 2006).However, since we can not find a simple expression for theinelasticity, we take ξ = 0 . f π in Eq. (10) is feasible to someextent for the thermal seed photon-dominated photopion in-teractions, but not for the IC scattered photon-dominatedinteractions. This difference of these two kinds of interactionarises from different temporal behaviors of t ′ π .Next let’s discuss the detectability of the afterglow neu-trinos, using the following fitting formula for the probabilityof detecting muon-neutrinos by IceCube (Ioka et al. 2005;Razzaque et al. 2004) P ν = 7 × − (cid:16) E ν . × GeV (cid:17) β , (21)where β = 1 .
35 for E ν < . × GeV, while β = 0 . E ν ≥ . × GeV. The number of muon events frommuon-neutrinos above TeV energy is given by N µ = A det Z TeV φ ν P ν dE ν , (22)where A det ∼ is the geometrical detector area. In-serting Eqs. (12) and (21) into the above integral, we ob-tain N µ ∼ . E = 1, A ∗ = 10,( kT ) − = 2, R = 1 and t SB , = 3) inferred from GRB060218 for a very nearby LL-GRB at 50 Mpc, where a LL-GRB event is expected to be observed within a many-yearsobservation. According to this estimation, we expect opti-mistically that IceCube may be able to detect afterglow neu-trinos from one LL-GRB event in the following decades. Ifsuch a detection comes true, the afterglow neutrino emissionaccompanying the soft X-ray thermal and sub-GeV or GeVemissions from a GRB 060218-like GRB event would pro-vide strong evidence for the picture that a supernova shockbreakout locates behind a relativistic GRB jet, and furtherwould be used to constrain the model parameters severely.Besides the possible detection of neutrinos from a singleLL-GRB event, the contribution to the neutrino backgroundfrom LL-GRBs is also expected to be important. We can es-timate the diffuse muon-neutrino flux arising from afterglowneutrino emission of LL-GRBs by (Waxman & Bahcall 1998;Murase et al. 2006) E ν Φ ν ∼ c πH f π f b ǫ p E p dN p dE p R LL (0) f z c (cid:13) , 000–000 fterglow Neutrino Emission from LL-GRBs -10-8-6-4 by IC scattered photonsby thermal seed photons l og ( E / e r g c m - ) log ( E / GeV ) Figure 1.
The time-integrated afterglow muon-neutrino ( ν µ +¯ ν µ )spectra for one GRB event. The solid lines are calculated by usingthe expressions for f π in Eq. (9) and for σ π ( ˜ E ) in Eq. (A5). Upper panel : The contributions to the total neutrino emission bythe two target photon components are represented by the dashedand dotted lines, respectively.
Lower panel : The dashed line isobtained by an approximation for the time integration as in Eq.(10), and the peak estimated by the ∆ − approximation is labeledby an open circle. In all cases, we take the model parameters E , A ∗ , ( kT ) − , R , D l, . and t SB , to be unity and theequipartition factors ǫ e = 0 . , ǫ B = 0 . ǫ p = 0 . = 2 . × − GeV cm − s − sr − ǫ p ( kT ) − R A ∗ × f b (cid:18) R LL (0)500Gpc − yr − (cid:19) (cid:16) f z (cid:17) , (23)where H = 71km s − Mpc − , f b is the beaming factor, and f z is the correction factor for the possible contribution fromhigh-redshift sources. In the above estimation, the approxi-mative value of f π in Eq. (20) is applied. By comparing Eq.(23) to Eq. (3) of Murase et al. (2006), we find that, for LL-GRBs, the contribution to the diffuse neutrino backgroundby the early afterglow neutrino emission may be relativelysmaller than or even comparable to (e.g., for model param-eters ǫ p = 0 .
6, ( kT ) − = 2, R = 1, and A ∗ = 10) that bythe burst neutrino emission.Finally, we would like to refer the reader to neutrinooscillation, which will change neutrino flavor ratio from ν e : ν µ : ν τ ≃ ∼ The surprising soft X-ray thermal emission during bothburst and afterglow phases of GRB 060218/SN 2006aj wasproposed to be due to the breakout from a strong stellarwind of a radiation-dominated shock. This shock breakoutwas further thought to locate behind a relativistic GRB jet, which is required by understanding the burst emissionand the power-law decaying afterglow emission. Wang &M´esz´aros (2006) suggested that a sub-GeV or GeV emissionproduced by IC scattering of the thermal photons by therelativistic electrons in the GRB blast wave could give evi-dence for this astronomical picture. In this paper, we stud-ied another possible implication, namely, afterglow neutrinoemission. The neutrinos are produced by photopion interac-tions of relativistic protons, which could be accelerated by arelativistic external shock. The target photons in the interac-tions are contributed by the incoming thermal emission andits upscattered component. By considering the high eventrate of LL-GRBs, we argue optimistically that the afterglowneutrinos from very nearby (several tens of Mpc) LL-GRBsmay be detected by IceCube in the following decades. Webelieve the detection of these expected afterglow neutrinosis helpful to uncover the nature of GRB 060218-like GRBs.
ACKNOWLEDGEMENTS
We would like to thank the referee for helpful commentsand suggestions. This work is supported by the NationalNatural Science Foundation of China (grants 10221001 and10640420144) and the National Basic Research Program ofChina (973 program) No. 2007CB815404. Y.W.Y. is alsosupported by the Visiting PhD Candidate Foundation ofNanjing University and partly by the National Natural Sci-ence Foundation of China (grants 10603002 and 10773004).
APPENDIX A: CROSS SECTION FITS
The fits to the total cross section for pγ interactions havebeen widely studied (e.g., Rachen 1996; M¨ucke et al. 2000).In physics, this cross section is contributed by resonant ex-citations and direct (non-resonant) single-pion productionprocesses in the resonant energy regions and by statisticalmultipion production processes mainly and diffractive scat-tering slightly in the high energy region ( ˜ E > σ R ( ˜ E ) = s ˜ E σ ¯Γ s ( s − m R c ) + ¯Γ s , (A1)where √ s = ( m p c + 2 m p c ˜ E ) / is the total energy of thecolliding photon and proton in the mass-center frame, and m R and ¯Γ are the nominal mass and width of the reso-nance, respectively. The coefficient σ is determined by theresonance angular momentum and the electromagnetic exci-tation strength. For nine important resonances in pγ inter-actions, we take the related parameters from M¨ucke et al.(2000) and list them in Table 1. Then, the total cross sectioncontributed by these resonances can be written as σ ( ˜ E ) = h − exp (cid:16) . − x . (cid:17)i X σ R ( ˜ E ) , (A2)where the suppression factor in the square bracket repre-sents the threshold ˜ E th = 0 . pγ interactions with x = ˜ E/ GeV. Moreover, Rachen (1996) found that the crosssection in the high energy region ( ˜ E > c (cid:13) , 000–000 Y.W. Yu, Z.G. Dai and X.P. Zheng
Table A1.
Parameters for resonances.Name m R c /GeV ¯Γ/GeV σ /µ barn∆(1232) 1.231 0.11 31.125 N (1440) 1.44 0.35 1.389 N (1520) 1.515 0.11 25.567 N (1535) 1.525 0.1 6.948 N (1650) 1.675 0.16 2.779 N (1680) 1.68 0.125 17.508∆(1700) 1.69 0.29 11.116∆(1905) 1.895 0.35 1.667∆(1950) 1.95 0.3 11.116 fitted by σ ( ˜ E ) = h − exp (cid:16) . − x . (cid:17)i × (cid:0) . y . + 0 . y − . (cid:1) mbarn . (A3)where y = s/ GeV . Subtracting the two components ex-pressed by Eqs. A2 and A3 from the experimental data takenfrom particle data group (Yao et al. 2006), we find the resid-uals to the total cross section exhibit a broken power-lowbehavior, which yields σ ( ˜ E ) = h − exp (cid:16) . − x . (cid:17)i × . x − . (cid:20)(cid:16) . x (cid:17) + 1 (cid:21) − . × (A4) (cid:20)(cid:16) . x (cid:17) + 1 (cid:21) . (cid:20)(cid:16) . x (cid:17) + 1 (cid:21) − . mbarn . Roughly speaking, the above formula could be related withthe direct single-pion production processes. Finally, com-bining the three components, we can express the total crosssection of pγ interactions by σ π ( ˜ E ) = (cid:26) σ + σ , for 0 . < ˜ E < . σ + σ + σ , for ˜ E > . . (A5)We confront this experiential formula with the experimentaldata in Fig. A1. It can be seen the fits is good, although theformula can not describe the detailed physics. REFERENCES
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