b nce tion. One dent cross al impact le t showing invariant Physics Letters B 581 (2004) 161–166 www.elsevier.com/locate/physlet Ultra peripheral heavy ion collisions and the energy depende of the nuclear radius C.G. Roldão, A.A. Natale Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, SP, Brazil Received 21 October 2003; received in revised form 20 November 2003; accepted 26 November 2003 Editor: W. Haxton Abstract To estimate realistic cross sections in ultra peripheral heavy ion collisions we must remove effects of strong absorp method to eliminate these effects make use of a Glauber model calculation, where the nucleon–nucleon energy depen sections at small impact parameter are suppressed. In another method we impose a geometrical cut on the minim parameter of the nuclear collision (bmin> R1 +R2, whereRi is the radius of ion ‘i’). In this last case the effect of a possib nuclear radius dependence with the energy has not been considered in detail up to now. Here we introduce this effec that for final states with small invariant mass the effect is negligible. However when the final state has a relatively large mass, e.g., an intermediate mass Higgs boson, the cross section can decrease up to 50%.  2003 Published by Elsevier B.V. e n ng he lear of ay ng er rly ect he lved efs. n– ely in act nt- nt ut) oid ad- lei er al th s– en Collisions at relativistic heavy ion colliders lik the relativistic heavy ion collider RHIC/Brookhave and the large hadron collider LHC/CERN (operati in its heavy ion mode) are mainly devoted to t search of a quark–gluon plasma in central nuc reactions. In addition to this important feature heavy-ion colliders, ultra peripheral collisions m give rise to a huge luminosity of photons openi the possibilities of studying two-photon and oth interactions as reviewed in Refs. [1–3]. In the ea papers on peripheral heavy ion collisions the eff of strong absorption was not taken into account. T separation of the strong interactions effects was so by using impact parameter space methods in R E-mail addresses: roldao@ift.unesp.br (C.G. Roldão), natale@ift.unesp.br (A.A. Natale). 0370-2693/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physletb.2003.11.066 [4–6]. In order to obtain a truly peripheral photo photon interaction one has to remove complet the central collisions, i.e., we must enforce that the cross section calculation the minimum imp parameter,bmin, should be larger thanR1 +R2, where Ri is the nuclear radius of the ion ‘i ’ [4]. The photon distributions can be described using the equivale photon approximation (EPA) with the requireme of minimum impact parameter (or geometric c discussed above [3,6]. The above method is not the only manner to av events where hadronic particle production oversh ows theγ –γ interaction, i.e., events where the nuc physically collide. An alternative is to use the Glaub model for heavy ion collisions [7]. It is a semiclassic model picturing the nuclei moving in a straight pa along the collision direction, and gives the nucleu nucleus interaction in terms of the interaction betwe http://www.elsevier.com/locate/physletb 162 C.G. Roldão, A.A. Natale / Physics Letters B 581 (2004) 161–166 ibu- of - ill he t er e to tor ion is the ave and by tal eV ar tion the ions m very he m he oton nted vy ot t is rgy ions tric of ral ard th in ker– a- - ss ess ss oing listic 6], ed w- ial a- n for the constituent nucleons and nuclear density distr tions. If we write the cross section for the collision two nucleusA andB as a function of the impact para meter (b), the elastic (el) peripheral cross section w be given by (1)σel = ∫ d2b [ 1− exp (−ABσ0TAB(b)/2 )]2 , whereA andB are the nucleon numbers,σ0 is the total nucleon–nucleon cross section and (2)TAB(b)= ∫ dQ2 (2π2) FA ( Q2)FB( Q2)eıQb, whereFA(B) are nuclear form factors. Eq. (1) and t form (2) for TAB(b) are valid only if one can neglec the finite range of the nuclear interaction. If at high energies the total cross section increases both du strength and due to the range the equation forTAB(b) should take this into account. The exponential fac in Eq. (1) is the one responsible for the suppress of the inelastic collisions. Theσ0 total nucleon– nucleon cross section that appears in Eq. (1) known to be dependent on the energy. Actually increase of hadron–hadron total cross sections h been theoretically predicted many years ago [8] these predictions have been accurately verified experiment [9]. For instance, the proton–proton to cross section roughly double as we go from a few G up to the Tevatron energies. In ultra peripheral heavy ion collisions it is cle how this energy dependence of the cross sec enters in the Glauber approximation. However same is not true when we compute the cross sect with the EPA and the requirement of a minimu impact parameter. It seems that cross sections in peripheral heavy ion collisions calculated within t Glauber method turn out to be slightly different fro the ones computed with the geometric cut [10]. The nuclear radius certainly expands with t increase of the energy in the same way as the pr expands, and this expansion should be impleme in the geometrical cut calculation of peripheral hea ion collisions. As far as we know this effect has n been discussed in detail in the literature, and i the purpose of this Letter to introduce the ene dependence of the nuclear radius in the calculat of peripheral heavy ion collisions when the geome cut method is used. In order to introduce the energy dependence the nuclear radius in the calculations of periphe heavy ion collisions we start discussing a stand computation of the photon distribution in the ion wi the geometric cut method. The photon distribution the nucleus can be described using the Weizsäc Williams approximation (or EPA) in the impact par meter space. Denoting byF(x) dx the number of pho tons carrying a fraction betweenx andx + dx of the total momentum of a nucleus of chargeZe, we can de- fine the two-photon luminosity through (3) dL dτ = 1∫ τ dx x F(x)F (τ/x), whereτ = ŝ/s, ŝ is the square of the center of ma (c.m.s.) system energy of the two photons ands of the ion–ion system. The total cross section of the proc ZZ→ ZZX is (4)σ(s)= ∫ dτ dL dτ σ̂ (ŝ), where σ̂ (ŝ) is the cross section of the subproce γ γ → X. There remains only to determineF(x). In the literature there are several approaches for d so, and we choose the conservative and more rea photon distribution of Ref. [6]. Cahn and Jackson [ using a prescription proposed by Baur [4], obtain a photon distribution which is not factorizable. Ho ever, they were able to give a fit for the different luminosity which is quite useful in practical calcul tions: (5) dL dτ = ( Z2α π )2 16 3τ ξ(z), where z = 2MR √ τ , M is the nucleus mass,R its radius andξ(z) is given by (6)ξ(z)= 3∑ i=1 Aie −biz, which is a fit resulting from the numerical integratio of the photon distribution, accurate to 2% or better 0.05< z < 5.0, and whereA1 = 1.909,A2 = 12.35, A3 = 46.28,b1 = 2.566,b2 = 4.948, andb3 = 15.21. For z < 0.05 we use the expression (see Ref. [6]) (7) dL dτ = ( Z2α π )2 16 3τ ( ln ( 1.234 z ))3 . C.G. Roldão, A.A. Natale / Physics Letters B 581 (2004) 161–166 163 f of is rue me al rm side ce, ther n to for s - s an en- ibu- all the the we en- om- ius nd g ral he ius g to the n in ere ated ase lso we dius ose ven h ro- ns the he e to the tion that - ory on e ion ll ion l to sed en- lep- med Eq. (3) is written in a factorised form, which o course is valid only if one neglects the exclusion central collisions into account. Therefore Eq. (3) not the most general form [3,6], and the same is t for Eq. (5). The calculation assumes that the sa radiusR is used for both ionsbmin = 2R but also to have a cutoff for the individual impact parameterb1 and b2 (which either is necessary to eliminate fin state interaction, or which takes into account the fo factor effects, that is, the decrease of the charge in the nucleus). Especially when looking, for instan an intermediate mass Higgs boson production or o non-strongly interacting particles there is no reaso assume that the size of the individual cutoff radii b1 andb2 scales in the same way asbmin. Therefore the calculation overestimates the dependence onR a bit. The condition for realistic peripheral collision (bmin > R1 + R2) is present in the photon distribu tions showed above. To obtain the above equation elastic Gaussian form factor and an energy indep dent nuclear radius giving byRion = 1.2A1/3 fm have been used. A more accurate Woods–Saxon distr tion for symmetric nuclei would produce some sm deviations, but for our purposes the expressions for luminosity described above are enough. However expression for the nuclear radius is exactly the one believe that should be changed by its energy dep dent expression, and the problem is to have a phen enologically sensible expression for the nuclear rad increase with the energy. In the heavy ions colliders nucleus like Au a Pb will collide with a great amount of energy, goin from 200 GeV/nucleon (Au at RHIC) up to 5.5 TeV/nucleon (Pb at LHC), and the ultra periphe collisions can be computed with the help of t photon distribution described above. If the ion rad increase with the energy, the value correspondin bmin will also become greater, and consequently cross section must decrease. This is easily see the many examples calculated in the literature wh the cross section for a given process is concentr at some moderate impact parameter and decre whenb increases. Of course the Lorentz factor is a important to determine this behavior. Therefore, if introduce the energy dependence in the nuclear ra we expect lower rates for a given process than th obtained in the usual calculations, and this effect, e s if it is small, could be important if we have a hig precision measurement. The authors of Ref. [11] modelled the particle p duction process in ultrarelativistic heavy ion collisio in terms of an effective scalar field produced by colliding objects, in their work they showed that t nuclear cross sections increase with the energy du a logarithmic increase of the nuclear radius with energy. We shall use this reference to obtain a rela between the nuclear radius and the incident energy is the following: (8)R2 H (s)= 1+ 2 d R′ γE + (δ+ 1) d R′ ln ( A √ s ε0 ) . R′ =RP +RT � 2.4A1/3 fm (RP (RT )means projec tile (target) and we assumeRP = RT = Rion), √ s is the energy of the projectile nucleus in the laborat rest frame. The nuclear density for a nucleusA at dis- tancex from its center is modelled by a Woods–Sax distribution for symmetric nuclei, (9)ρWS(x)= ρ0 (1+ exp[ (x−Rion) d ]) , whered = 0.549 fm, andρ0 can be found when th Wood–Saxon density is normalized by the condit∫ d3x ρ(x)=A. And ε0 is equal to (10)ε0 =MZd [ 16 π2g2ρ4 0 R′ d3 (RT RP ) (δ−7)/4 Γ 2( δ+1 4 ) ]2/(δ+1) . MZ is the nuclear mass. The coupling constantg and the parameter for the mass spectrumδ were estimated in Ref. [11] and they are equal toδ = −0.56 and g = 3.62 fm(7+δ)/2. The radiusR2 H (s) appearing in Eq. (8) at sma energies gives a nuclear radius larger thanRion, for this reason we have assumed the following normalizat (11)R2(s)= R2 H(s)R 2 ion R2 H(s =M2 Z) , whereRion = 1.2A1/3 fm. With this normalization factor we assure that when the ion energy is equa its mass, the nuclear radius will be equal toRion. It is the radius given by Eq. (11) that should be u in Eqs. (5)–(7). Typical values forRion andR(s) are showed in Table 1. To show the effects of the nuclear radius dep dence on the energy we computed production of tons pairs (muons and taus) and resonances for 164 C.G. Roldão, A.A. Natale / Physics Letters B 581 (2004) 161–166 ase ss m- nu- the d t ven tary own ee ne a in of ith the ree we y the than nd . en ne oss e . The ns. ver re lear he th t so hat ider rger ant ces. ion ss the s and for ed Table 1 Values forRion andR(s), Eq. (11), in fm. The energies ( √ s ) are in TeV/nucleon Ion √ s Rion R(s) Au 0.2 6.98 7.29 Ca 7.2 4.10 4.75 Pb 5.5 7.11 7.61 by the photon–photon fusion. In the resonance c we considered theηc meson and an intermediate ma Higgs boson with a mass equal to 115 GeV. We co puted the cross sections for two cases: in one the clear radius is energy independent (and equal toRion). In the second case the radius obeys Eq. (11). For an invariant mass of the photon pair above threshold √ ŝ > 2ml , a lepton pair can be produce in two-photon collisions (γ γ → l+l−) and the lowes order QED cross section for this subprocess is gi by [3] σ ( γ γ → l+l− ) (12)= 4πα2 ŝ βl [ (3− β4 l ) 2βl ln ( 1+ βl 1− βl ) − 2+ β2 l ] , whereβl = √ 1− 4m2 l /ŝ is the velocity of the pair in the γ γ rest frame,ml is the lepton mass, √ ŝ is the c.m. system energy of the two photons andα is the fine-structure constant. Using this elemen cross section in Eq. (4) we obtained the rates sh in Table 2. The calculation was performed for thr different ions with different beam energies, the o of RHIC (Au) and the ones expected at LHC (C and Pb). The cross sections were integrated in a b energy equal to 1< √ ŝ < 10 GeV. The third column of Table 2 shows the cross section computed w a constant nuclear radius and the fourth column one with the energy dependent radius. For the th different ions the cross sections decrease when consider the energy dependent radius described b Eq. (11). In all the cases the decrease is smaller 10% and is negligible considering the theoretical a experimental uncertainties involved in the problem In Table 3 it is possible to see the results wh the subprocess analyzed isγ γ → τ+τ− with 2mτ <√ ŝ < 10 GeV. The general behavior of theτ pair production cross sections is very similar to the o observed previously in Table 2. Of course, the cr Table 2 Cross sections of the processZZγγ → ZZµ+µ−. The cross sectionsσRi (σR(s)) given in the third (fourth) column are th ones computed with the energy independent (dependent) radius last column shows the ratio between the third and fourth colum The cross sections are in mbarn and the energies ( √ s ) are in TeV/nucleon Ion √ s σRion σR(s) Ratio Au 0.2 2.127 1.947 1.09 Ca 7.2 0.643 0.588 1.09 Pb 5.5 106.4 101.3 1.05 Table 3 The same as in Table 2, but for the subprocessγ γ → τ+τ− Ion √ s σRion σR(s) Ratio Au 0.2 6.972× 10−4 5.727× 10−4 1.22 Ca 7.2 5.176× 10−3 4.604× 10−3 1.12 Pb 5.5 0.759 0.718 1.05 sections for producing tau pairs are smaller. Howe the collision of Au–Au and Ca–Ca are now mo sensitive to the energy dependence of the nuc radius, producing an effect larger than 10%. T rates for tau pairs production in Pb collision wi a c.m. energy equal to 5.5 TeV/nucleon, with and without energy dependence in the ion radius are no different. As we shall discuss later the larger cut t we perform in the impact parameter when we cons the energy dependent radius removes photons of la energy. Therefore for final states with larger invari masses we may expect a larger effect. Let us now consider the case of heavy resonan To estimate the production of one resonanceR formed by a photon–photon fusion in peripheral heavy collisions we use the following elementary cro section in Eq. (4), (13)σ(γ γ → R)= 8π2 MRs Γ (R→ γ γ )δ ( τ − M2 R s ) , whereMR is the resonance mass andΓ (R→ γ γ ) its decay width into two photons. In Table 4 we show results obtained for two-photon production ofηc in peripheral heavy ion collisions withMηc = 2.979 GeV andΓ (ηc → γ γ ) = 6.6 keV. The ratio of the cros sections considering the two scenarios are 1.16 1.11 for Au and Ca ions, respectively, and 1.06 the Pb ion. Finally, in Table 5 it can be observ the values corresponding to the subprocessγ γ → H C.G. Roldão, A.A. Natale / Physics Letters B 581 (2004) 161–166 165 on o t is he gly by rgy arly rgy the e ive e of ter as ber e per he act ith n a vy ys avy ar n– In on the not at The n the 1] lear ons al t of will of an lem ltra ion, of ust own ith ross o to this 3], a– er the his onal q) do Table 4 The same as in Table 2, but for the subprocessγ γ → ηc Ion √ s σRion σR(s) Ratio Au 0.2 2.147×10−3 1.846×10−3 1.16 Ca 7.2 2.897×10−3 2.614×10−3 1.11 Pb 5.5 0.437 0.413 1.06 Table 5 The same as in Table 2, but for the subprocessγ γ →H Ion √ s σRion σR(s) Ratio Ca 7.2 9.970× 10−10 6.789× 10−10 1.47 Pb 5.5 1.854× 10−8 1.387× 10−8 1.34 with MH = 115 GeV, where we used the Higgs bos two-photon decay width found in Ref. [12]. We d not show the result for RHIC energies because i too small. The values of Table 5 indicate that t production cross sections for both ions are stron affected by the inclusion of a radius described Eq. (11). In the case of Ca collision with a c.m. ene of 7.2 TeV/nucleon the cross sections decrease ne to half of the value obtained in the case of a ene independent radius. The situation is less drastic for Pb ion with √ s = 5.5 TeV/nucleon, but the ratio is still large (= 1.34). This is the only situation that w investigated where the Pb collision is clearly sensit to the use of Eq. (11) (or to the energy dependenc the nuclear radius). The fact that a sharp cutoff in impact parame space atbmin should be replaced by a smooth one w already discussed in Ref. [2]. Comparing the Glau model calculation with the one with a sharp cutoff w could expect significant deviations present at the up end of the invariant mass distribution. Looking at t photon luminosity we see that only the smallest imp parameter contribute significantly to the events w large invariant masses. Imposing the cut onbmin but now with the radius described by Eq. (11) we obtai more realistic calculation of the very peripheral hea ion collisions. In conclusion, we discussed the two different wa to compute cross sections for ultra peripheral he ion collisions. In the Glauber method it is quite cle how the increase with the energy of the nucleo nucleon cross section enters in the calculation. the calculation with the geometrical cut imposed the impact parameter, the nucleon, as well as nuclear, radius expansion with the energy was introduced up to now. It was noticed in Ref. [10] th there was a difference between the two methods. difference was small and had some dependence o invariant mass of the final states. The work of Ref. [1 prescribe a very precise way to introduce the nuc radius dependence with the energy. We believe that the estimative of the cross secti in ultra peripheral collisions with the geometric cut method just changing the radius independen the energy by the one dependent of the energy give realistic predictions for any invariant mass the final state. The effect is of order of 50% for intermediate mass Higgs boson. Turning the prob the other way around we may also say that if the u peripheral collisions are measured with high precis we may have a new way to study the increase the nuclear radius with the energy. To do so we j have to measure the cross sections for very kn final states with small and large invariant masses w high precision, there should be a decrease of the c sections as a function of the invariant mass as we g larger and larger energies. Note added Some comments on the effects discussed in Letter were also made by Klein and Nystrand in [1 where the Fig. 3 gives the reduction in gamm gamma luminosity (for gold at RHIC) for a Glaub calculation of hadronic interactions compared to one with geometrical cut. Acknowledgements We are grateful to Y. Hama for discussions. T research was supported by the Conselho Naci de Desenvolvimento Científico e Tecnológico (CNP (A.A.N.) and by Fundação de Amparo à Pesquisa Estado de São Paulo (FAPESP) (C.G.R.). References [1] C.A. Bertulani, G. Baur, Phys. Rep. 163 (1988) 299. 166 C.G. Roldão, A.A. Natale / Physics Letters B 581 (2004) 161–166 ov, al de . ter- 57; igh 96) r- 40. 7. [2] G. Baur, J. Phys. G 24 (1998) 1657. [3] G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Y. Kharl Phys. Rep. 364 (2002) 359. [4] G. Baur, in: T. Kodama, et al. (Eds.), CBPF Internation Workshop on Relativistic Aspects of Nuclear Physics, Rio Janeiro, 1989, World Scientific, Singapore, 1990, p. 127. [5] G. 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