Exclusive muon-pair productions in ultrarelativistic heavy-ion collisions: Realistic nucleus charge form factor and differential distributions
aa r X i v : . [ nu c l - t h ] A p r Exclusive muon-pair productions in ultrarelativistic heavy-ioncollisions: Realistic nucleus charge form factor and differentialdistributions
M. K lusek-Gawenda ∗ and A. Szczurek
1, 2, † Institute of Nuclear Physics PAN, PL-31-342 Cracow, Poland University of Rzesz´ow, PL-35-959 Rzesz´ow, Poland (Dated: October 16, 2018)
Abstract
The cross sections for exclusive muon pair production in nucleus - nucleus collisions are calculatedand several differential distributions are shown. Realistic (Fourier transform of charge density)charge form factors of nuclei are used and the corresponding results are compared with the crosssections calculated with monopole form factor often used in the literature and discussed recentlyin the context of higher-order QED corrections. Absorption effects are discussed and quantified.The cross sections obtained with realistic form factors are significantly smaller than those obtainedwith the monopole form factor. The effect is bigger for large muon rapidities and/or large muontransverse momenta. The predictions for the STAR and PHENIX collaboration measurements atRHIC as well as the ALICE and CMS collaborations at LHC are presented.
PACS numbers: 25.75.-q,25.75.Dw,25.20Lj ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION In Fig.1 we show the basic QED mechanism of the exclusive production of muon pairs.The shaded circles represent the coupling of photons to large-size objects – nuclei. In themomentum space this is done in terms of electromagnetic form factors of nuclei. In the caseof scalar nuclei there is only one form factor – the charge form factor of the nucleus. A A A A µ + µ − γγ FIG. 1: (Color online) The Born diagram for the exclusive dimuon production.
It was recognized long ago that the production rate of leptons in ultrarelativistic heavyion collisions is enhanced considerably by the coherent effects and large charge of collidingions [1]. Many results have been presented in the literature since then (for reviews of thefield see e.g. [2, 3]). Recently, there was a growing theoretical interest in estimating higher-order QED corrections [4–7]. In most of the practical calculations of exclusive dileptonproduction a simple monopole charge form factor of the nucleus was used. While it may besufficient for estimating the total cross section, it may be not sufficient for calculations ofthe differential cross sections. The importance of including realistic charge form factors wasdiscussed recently for exclusive production of pairs of ρ mesons [8].Most of the existing calculations concentrated on total cross section, interesting theoreti-cal quantity, which cannot be, however, measured in practice, neither at RHIC nor at LHC.The experiments running at RHIC and those planned at LHC demand severe cuts on leptontransverse momenta or on their rapidities.It is the aim of the present analysis to make realistic estimates of the cross sectionsincluding the experimental cuts. We shall compare the results obtained with monopole formfactor used in the literature and the results obtained with realistic form factor being Fouriertransform of the charge density of the nucleus. We shall perform the calculation in theequivalent photon approximation (EPA) in the impact parameter space as well as in themomentum space. While the impact parameter EPA allows to include easily absorptioneffects due to the size of colliding nuclei, the momentum space approach allows to studyeasily several differential distributions. In our calculation we shall include experimentallimitations of the STAR and PHENIX detectors at RHIC and those of the ALICE and CMSdetectors at LHC. 2 I. FORMALISMA. Charge form factor of nuclei
The charge distribution in nuclei is usually obtained from elastic scattering of elec-trons from nuclei [9]. The charge distribution obtained from these experiments is oftenparametrized with the help of two–parameter Fermi model [10]: ρ ( r ) = ρ (cid:18) exp (cid:18) r − ca (cid:19)(cid:19) − , (2.1)where c is the radius of the nucleus and a is the so-called diffiusness parameter of the chargedensity. FIG. 2: (Color online) The ratio of the charge distibution ( ρ ) to the density in the center of nucleus( ρ ). Fig. 2 shows the charge density normalizationed to unity at r = 0. The correct normal-ization is: ρ Au (0) = . A f m − for Au nucleus and ρ P b (0) = . A f m − for Pb nucleus.The form factor ( F ) is the Fourier transform of the charge distribution [9]. If ρ ( r ) isspherically symmetric then the form factor is a function of photon virtuality ( q ) only: F ( q ) = Z πq ρ ( r ) sin ( qr ) rdr = 1 − q h r i
3! + q h r i . . . . (2.2)Fig. 3 shows the moduli of the form factor as a function of momentum transfer. Theresults are depicted for the gold (solid line) and lead (dashed line) nuclei for realisticcharge distribution. The realistic form factor is obtained as a Fourier transform of therealistic charge density which we take from the literature [9]. Here one can see manyoscillations characteristic for relatively sharp edge of the nucleus. For comparison we showthe monopole form factor often used in the literature. The two form factors coincide onlyin a very limited range of q and with larger value of q the difference between them becomes3 IG. 3: (Color online) The moduli of thecharge form factor F em ( q ) of the Au and P b nuclei for realistic charge distributions.For comparison we show the monopole formfactor for the same nuclei. FIG. 4: (Color online) The monopole formfactor for the values of Λ reproducing chargeradius of Au and P b nuclei and for com-parison for Λ = 0 .
08 GeV often used in theliterature. larger and larger.The monopole form factor [11] given by the simple formula: F ( q ) = Λ Λ + q (2.3)leads to a simplification of many formulae for photon-photon collisions. In our calculationΛ is adjusted to reproduce the root mean square radius of a nucleus (Λ = q
091 GeV, • for P b : < r > / = 5 . ⇒ Λ = 0 .
088 GeV.Different values of Λ are used in the literature, ranging from 80 to 90 MeV. Fig. 4 shows themonopole form factor with Λ adjusted to reproduce the rms radius of the charge distribution.
B. Equivalent Photon Approximation
The equivalent photon approximation is the standard semi–classical alternative to theFeynman rules for calculating cross sections of electromagnetic interactions [12]. This isillustrated in Fig. 5 where we can see a fast moving nucleus with the charge Ze . Due tothe coherent action of all the protons in the nucleus, the electromagnetic field surrounding(the dashed lines are lines of electric force for a particles in motion) the ions is very strong.4 FIG. 5: (Color online) Equiva-lent photon approximation. b b b FIG. 6: (Color online) The quantitiesused in the impact parameter calcula-tion.
This field can be viewed as a cloud of virtual photons. These photons are often consideredas real. They are called ”equivalent” or ”quasireal photons”. In the collision of two ions,these quasireal photons can collide with each other or with the other nucleus. So the strongelectromagnetic field is used as a source of photons to induce electromagnetic reactions onthe second ion. We consider very peripheral collisions. It means that the distance betweennuclei is bigger than the sum of the radii of the two nuclei ( b > R + R ∼ = 14 f m ). Fig. 6explains the quantities used in the impact parameter calculation. We can see a view in theplane perpendicular to the direction of motion of the two ions. In order to calculate thecross section of a process it is convenient to introduce a new kinematic variable: x = ωE A ,where ω is the energy of the photon and the energy of the nucleus E A = γAm proton = γM A ,where M A is the mass of the nucleus and γ is the Lorentz factor.The total cross section can be calculated by the convolution: σ (cid:16) AA → µ + µ − AA ; s AA (cid:17) = Z ˆ σ (cid:16) γγ → µ + µ − ; W γγ = √ x x s AA (cid:17) d n γγ ( x , x , b ) . (2.4)The effective photon fluxes can be expressed through the electric fields generated by thenuclei: d n γγ ( x , x , b ) = 1 π d b | E ( x , b ) | π d b | E ( x , b ) | × S abs ( b ) δ (2) ( b − b + b ) d x x d x x . (2.5)The presence of the absorption factor S abs ( b ) assures that we consider only peripheralcollisions, when the nuclei do not undergo nuclear breakup. In the first approximation thiscan be expressed as: S abs ( b ) = θ ( b − R A ) = θ ( | b − b | − R A ) . (2.6)Thus in the present case, we concentrate on processes with final nuclei in the ground state.The electric field strength can be expressed through the charge form factor of the nucleus: E ( x, b ) = Z √ πα em Z d q (2 π ) e − i bq qq + x M A F em (cid:16) q + x M A (cid:17) . (2.7)5ext we can benefit from the following formal substitution:1 π Z d b | E ( x, b ) | = Z d b N ( ω, b ) ≡ n ( ω ) (2.8)by introducting effective photon fluxes which depend on energy of the quasireal photon ω and the distance from the nucleus in the plane perpendicular to the nucleus motion −→ b .Then, the luminosity function can be expressed in term of the photon flux factors attributedto each of the nucleid n γγ ( ω , ω , b ) = Z θ ( | b − b | − R A ) N ( ω , b ) N ( ω , b ) d b d b d ω d ω . (2.9)The total cross section for the AA → µ + µ − AA process can be factorized into the equivalentphotons spectra ( n ( ω ) ) and the γγ → µ + µ − subprocess cross section as (see e.g.[13]): σ (cid:16) AA → µ + µ − AA ; s AA (cid:17) = Z ˆ σ (cid:16) γγ → µ + µ − ; W γγ (cid:17) θ ( | b − b | − R A ) × N ( ω , b ) N ( ω , b ) d b d b d ω d ω , (2.10)where W γγ = √ ω ω is energy in the γγ subsystem. Eq. (2.10) is a generalization of thesimple parton model formula (see e.g.[2]): σ (cid:16) AA → µ + µ − AA (cid:17) = Z ˆ σ (cid:16) γγ → µ + µ − ; √ ω ω (cid:17) n ( ω ) n ( ω ) d ω d ω . (2.11)Additionally, we define Y = ( y µ + + y µ − ), rapidity of the outgoing dimuon system which isproduced in the photon–photon collision. Performing the following transformations: ω = W γγ e Y , ω = W γγ e − Y , (2.12)d ω d ω = W γγ W γγ d Y , (2.13)d ω d ω → d W γγ d Y where (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ( ω , ω ) ∂ ( W γγ , Y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = W γγ , (2.14)formula (2.10) can be rewritten as: σ (cid:16) AA → µ + µ − AA ; s AA (cid:17) = Z ˆ σ (cid:16) γγ → µ + µ − ; W γγ (cid:17) θ ( | b − b | − R A ) × N ( ω , b ) N ( ω , b ) W γγ b d b d W γγ d Y . (2.15)Finally, the cross section can be expressed as the five-fold integral: σ (cid:16) AA → µ + µ − AA ; s AA (cid:17) = Z ˆ σ (cid:16) γγ → µ + µ − ; W γγ (cid:17) θ ( | b − b | − R A ) × N ( ω , b ) N ( ω , b ) 2 πb m d b m d b x d b y W γγ W γγ d Y , (2.16)where b x ≡ ( b x + b x ) / b y ≡ ( b y + b y ) / ~b m = ~b − ~b have been introduced. Thisformula is used to calculate the total cross section for the AA → AAµ + µ − reaction as wellas the distributions in b = b m , W γγ = M µ + µ − and Y ( µ + µ − ).6ifferent forms of form factors are used in the literature. We compare the equivalentphoton spectra for an extended charge distribution (realistic case) to the monopole case.The dependence of the photon flux on the charge form factors can be found in [2]: N ( ω, b ) = Z α em π b ω Z u J ( u ) F vuut (cid:16) bωγ (cid:17) + u b (cid:16) bωγ (cid:17) + u du , (2.17)where J is the Bessel function of the first kind and q is the four-momentum of the quasirealphoton. The calculations with the help of realistic form factor are rather laborious, so oftena simpler monopole form factor is used [11]. Introducing monopole form factor to (2.17) onegets: N ( ω, b ) = Z α em π ω ωγ K bωγ ! − s ω γ + Λ K b s ω γ + Λ !! , (2.18)where K is the modified Bessel function of the second kind. FIG. 7: (Color online) The equivalent photon number as a function of impact parameter (integratedover ω ), see Eq. (2.19). Fig. 7 shows the distribution of the equivalent photon number as a function of the impactparameter N ( b ) = Z N ( ω, b ) dω. (2.19)We present the results for gold and lead nuclei, for realistic and monopole form factors. Herewe do not impose any sharp cutoff on the impact parameter. One can see that for small b the flux factor with monopole form factor is bigger. For large b the results obtained withthe help of realistic and monopole form factors are almost the same.In addition in Fig. 8 we show the ratio of equivalent photon fluxes obtained with the helpof realistic form factor to that for the monopole form factor. The oscillations in b are due to7 IG. 8: (Color online) The ratio of the flux factor obtained with realistic charge distribution tothat with the monopole form factor as a function of impact parameter.FIG. 9: (Color online) The equivalent photon number n ( ω ), see Eq. (2.20). Left panel: b ∈ (0 , b ∈ (14 , step-like distribution of the charge in the nucleus. The results for lower ( √ s NN = 200 GeV)and higher ( √ s NN = 5 . N ( ω ) = Z πbN ( ω, b ) db. (2.20)Here we consider the integral over full range of the impact parameter (left panel) and for b > R A (right panel). One can see that the difference between monopole and realistic formfactor for both gold and lead nuclei is not significant. The quantity shown depends rather8eakly on the photon energy. FIG. 10: (Color online) The elementary cross section for the γγ → µ + µ − reaction as a function ofthe photon-photon energy. In Fig. 10 we show the energy dependence of the elementary γγ → µ + µ − cross sectionused in our EPA calculations [1]: σ (cid:16) γγ → µ + µ − (cid:17) = 4 πα em W γγ (2.21) × ( " W γγ m µ (1 + v ) m µ W γγ − m µ W γγ ! − m µ W γγ W γγ ! v ) , where v = vuut − m µ W γγ . (2.22)This formula is often called the Breit-Wheeler formula. C. Momentum space calculation
We consider a genuine 2 → p a + p b → p + p + p + p . In the momentum space approach the cross section for the production ofa pair of particles can be written as: σ = Z s |M| (2 π ) δ ( p a + p b − p − p − p − p ) × d p (2 π ) E d p (2 π ) E d p (2 π ) E d p (2 π ) E . (2.23)9 a p b p p q q p p p p q q β µναt u FIG. 11: (Color online) Amplitude of the considered process. On the left one can see the t -channelamplitude and on the right - the u -channel amplitude. Using d p i E i = dy i d p it = dy i p it dp it dφ i (2.24)Eq. (2.23) can be rewritten as: σ = Z s |M| δ ( p a + p b − p − p − p − p ) 1(2 π ) × ( dy p t dp t dφ ) ( dy p t dp t dφ ) (cid:16) dy d p t (cid:17) (cid:16) dy d p t (cid:17) . (2.25)In the above formula p it are transverse momenta of outgoing nuclei and leptons, φ , φ areazimuthal angles of outgoing nuclei. Additionally, we introduce a new auxiliary quantity p m = p t − p t (2.26)and benefitting from 4-dimensional Dirac delta function properties, Eq. (2.25) can be writtenas: σ = Z s |M| δ ( E a + E b − E − E − E − E ) δ ( p z + p z + p z + p z ) 1(2 π ) × ( dy p t dp t dφ ) ( dy p t dp t dφ ) dy dy d p m . (2.27)The energy-momentum conservation gives the following system of equations that has to besolved for discrete solutions ( √ s − E − E = q m t + p z + q m t + p z , − p z − p z = p z + p z , (2.28)where m t , m t are the so-called transverse masses of outgoing nuclei which are defined as: m it = p it + m i . (2.29)We wish to make the transformation from ( y , y ) to ( p z , p z ). The transformationJacobian takes the form: J k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z ( k ) q m t + p z ( k ) − p z ( k ) q m t + p z ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.30)10here k numerates discrete solutions of Eq. (2.28). Thus the cross section for the 2 → σ = Z X k J − k ( p t , φ , p t , φ , y , y , p m , φ m ) 12 q s ( s − m ) |M| π ) × ( p t dp t dφ ) ( p t dp t dφ ) 14 dy dy d p m . (2.31)For photon-exchanges, considered here, it is convenient to change the variables p t → ξ =log ( p t ), p t → ξ = log ( p t ). The lepton helicity dependent amplitudes of the processshown in Fig. 11 can be written as: M λ ,λ ( t -channel) = e F ch ( q ) ( p a + p ) α − i g αµ q + iε ¯ u ( p , λ ) i γ µ i [( p − 6 q ) + m µ ]( q − p ) − m µ × i γ ν v ( p , λ ) − i g νβ q + iε ( p b + p ) β e F ch ( q ) (2.32)and M λ ,λ ( u -channel) = e F ch ( q ) ( p a + p ) α − i g αµ q + iε ¯ u ( p , λ ) i γ ν i [( p − 6 q ) + m µ ]( q − p ) − m µ × i γ µ v ( p , λ ) − i g νβ q + iε ( p b + p ) β e F ch ( q ) . (2.33)These amplitudes are calculated numerically. Finally, to calulate the total cross sectionone has to calculate the 8-dimensional integral inserting M λ ,λ = M λ ,λ ( t -channel) + M λ ,λ ( u -channel) into Eq. (2.31). We shall compare the impact parameter EPA resultswith the exact Quantum Electrodynamics results.
III. RESULTS
Let us start from the presentation of the results obtained in the impact parameter EPA.In Fig.12 we show the distribution in the impact parameter b for typical RHIC energy √ s NN = 200 GeV. The contributions from distances smaller than b = 2 R A are cut off andconsistently with θ -function in Eq. (2.16). We clearly see a huge contribution from distanceslarge compared to the nuclear size. The distribution with realistic charge falls off somewhatquicker than that for the monopole charge form factor. This is better visualized in the rightpanel where the ratio of the corresponding cross sections is shown.The difference of the cross sections for the monopole and exact charge form factors atlarge impact parameter b shown in the figure is especially intriguing in the light of theequality of the photon flux factors at large b or b (see Fig.7). How to understand thisquite nonintuitive result? In Fig.13 we show the distribution of dσ/db db in ( b , b ) withthe severe restriction for the impact parameter b ∈ (480,520) fm. We see two pronouncedpeaks at ( b ≈ b, b ≈
0) and ( b ≈ , b ≈ b ). This demonstrates a strong preference of By exact we mean the correct inclusion of the 2 → IG. 12: (Color online) The cross section as a function of the impact parameter for the
AuAu → µ + µ − AuAu reaction calculated in the equivalent photon approximation. In the left panel we showthe results for realistic charge distribution (solid line) and for monopole form factor (dashed line).On the right side we depict the ratio :
RAT IO = d σ (cid:16) F REALIST ICem (cid:17) / d σ (cid:16) F MONOP OLEem (cid:17) . ( f m ) b (f m ) b ) ( nb / f m db / db σ d -6 -4 -2
10 1 (a) (fm) b (f m ) b (b) FIG. 13: d σ d b d b as a function of b and b in lego (left) and contour (right) representation for b ∈ (480,520) fm. asymmetric production of the pair: close to the trajectory of one or the other nucleus, wherethe form factor details are important (see Fig.7). This point was never discussed so far inthe literature.The distributions shown in Fig.12 are purely theoretical, that is cannot be easily mea-sured. Let us come now to the distributions which could, at least in principle, be mea-sured. Fig.14 shows the distribution in the dimuon subsystem energy. The distributions in12 γγ = M µ + µ − falls steeply off. In the right panel we show the ratio of the cross sections forrealistic charge distribution to that for the monopole charge form factor. At W γγ = 10 GeVthe two distributions differ already by a factor of about 5 which clearly shows limitations ofthe calculations with analytic charge form factors. FIG. 14: (Color online) The cross section for Au − Au scattering as a function of photon–photoncenter–of–mass energy W γγ = M µ + µ − in EPA. In the right panel we show the ratio of ”realistic”to ”monopole” form factor. Finally, in analogy to the AA → AAρ ρ reaction studied in Ref.[8], in Fig.15 we show thedistribution in the dimuon pair rapidity. As for the ρ ρ production we see a huge differencebetween the results of the two calculations for large dilepton rapidities. Measurements ofdileptons in forward directions would be therefore very useful to understand the role ofrealistic charge distribution. The relative effect is shown in the right panel of the figure.The preliminary calculation in the impact parameter space clearly shows how importantcan be studying of differential distributions to pin down the effects of realistic charge density.Not all of the distributions can be easily addressed in the impact parameter approach. TheFeynman diagram approach in the momentum space seems to be a better alternative tostudy the differential distributions.Now we come to the presentation of results obtained in the momentum space approachwith details outlined in Section II. Fig.16 shows distributions in muon rapidities (identicalfor µ + and µ − ). No other limitations or kinematical cuts have been included here. As in theprevious cases we show distributions obtained with the monopole and realistic charge formfactor. The effect of the oscillatory character of F ch ( q ) and in particular its first minimumis reflected by a smaller cross section at larger rapidities compared to the results obtainedwith monopole form factor. This is due to the fact that on average at large rapidities largerfour-momentum squared transfers ( t or t ) are involved. In reality, one effectively integratesover a certain range of t and t . The relative effect is shown in the right panel.Fig.17 shows the situation (the ratio of the two calculations) in the two dimensional space:( y , y ). Clearly at mid rapidities, where on average rather small t and t are involved, theuse of the approximate monopole form factor is justified. This is not the case at the edges13 IG. 15: (Color online) The cross section as a function of Y = (cid:0) y µ + + y µ − (cid:1) (left panel) forrealistic and monopole form factors (left) calculated in EPA and their ratio (right).FIG. 16: (Color online) The cross section as a function of y µ + , y µ − for realistic and monopole formfactor calculated in the momentum space (left panel). Their ratio is shown in the right panel. of the ( y , y ) plane where due to kinematics | t | or/and | t | are larger.Up to now we have discussed ”a theoretical situation” when all the muons are accepted.In practice one can measure only muons with transverse momenta larger than a certainvalue, characteristic for a given detector. We shall consider now cases relevant for concreteexperimental situations.The calculations in the literature concentrated mostly on the total cross section. InFig.18 we present the dependence of the total cross section on the lower cut-off in theimpact parameter. We present EPA results for realistic (lower solid line) and monopole14 y -4 -2 0 2 4 y -4-2024 ] dy / dy σ R A T I O [ d FIG. 17: The ratio of two-dimensial distributions d σ (cid:16) F REALIST ICem (cid:17) / d σ (cid:16) F MONOP OLEem (cid:17) in y and y .FIG. 18: (Color online) The compilation of the results obtained in different approaches for thetotal cross section for AuAu → AuAuµ + µ − at √ s NN = 200 GeV. (upper solid line) form factors. The cross section without the cut-off is by 15% larger thanthat for b cut = 14 fm. This result is smaller than the corresponding results obtained withinmomentum space calculations, shown as the horizontal dashed lines. Different methods hasbeen used in the literature to calculate the total cross section for the AuAu → AuAuµ + µ − process. For comparison we show also results obtained recently by Jentschura and Serbo(JS) [14] in the momentum space EPA and by Baltz et al. [15] in the b-space EPA. The JSresult should be compared to our momentum space calculation with monopole form factor.Our exact calculation is in this case larger than their EPA calculation by about 24%. Thisshows the precision of the momentum space EPA. The Baltz et al. result is significantly15ower than our b-space EPA result. In their calculations the cuts were imposed rather on b and b , instead on b in our case. If we impose additional cuts on b and b in Eq. (2.16) weget the point in the lower-right corner. If the cut on b is not imposed we get the point inthe lower-left corner. The result of Baltz et al. differs from both these values, the solutionbeing most probably a different form factor used in their case.Now we will continue reviewing our predictions for the differential distributions. Let usstart with the ALICE detector. The ALICE collaboration can measure only forward muonswith psudorapidity 4 < η < p t > FIG. 19: (Color online) Invariant mass distribution d σ d M µ + µ − (left) and muon transverse momentumdistribution d σ d p t = d σ d p t (right) for ALICE conditions: y , y ∈ (3 , p t , p t ≥ W NN = 5 . The distribution in rapidity is shown in Fig.20. We present the cross sections for both(realistic, monopole) form factors and their ratio.Double differential distribution of the muon rapidity and transverse momentum is shownin Fig.21. These are our predictions which could be studied experimentally in the future.The small irregularities seen in the two-dimensional spectra for realistic form factor are theconsequence of the oscillatory character of the nucleus charge form factor. The distributionfor the monopole form factor is more smooth.In Fig.22 we show the ratio of the cross sections shown in the previous figure. Hugedeviations from the unity can be seen. The reminiscence of the oscillating form factor canbe seen also in the ratio. Experimental confirmation of this behaviour would be very useful.Moreover it would demonstrate whether our understanding of the nuclear effects is correct.Large deviations from the predictions presented here would be surprising.Let us come now to the predictions for the CMS detector. In contrast to the ALICE16
IG. 20: (Color online) d σ d y = d σ d y (left) and their ratio (right) for the ALICE conditions: y , y ∈ (3 , p t , p t ≥ W NN = 5 . ( G e V ) t p y ( nb / G e V ) t dp / dy σ d -10 -7 -4 -1 (a) ( G e V ) t p y ( nb / G e V ) t dp / dy σ d -10 -7 -4 -1 (b) FIG. 21: Double differential cross section d σ d y d p t for realistic (left) and monopole (right) formfactors for ALICE conditions y , y ∈ (3 , p t , p t ≥ W NN = 5 . detector, CMS can measure midrapidity values with -2.5 < y , y < t and t therefore the efects of the realistic form factorsare expected to be smaller. Fig.23 confirms the expectations. Even for muon transversemomenta of 50 GeV one obtains damping with respect to the result obtained with themonopole form factor by a factor of about two only.The cross section dependence on the muon rapidity is shown in Fig.24. Rather largecross section of the order of 0.1 mb is expected within the CMS acceptance. The averagedeviation with respect to the monopole form factor is about 20% (see the left panel).The two-dimensional distributions within the main CMS detector are shown in Fig.25.17 t p y ] t dp / dy σ R A T I O [ d -6 -5 -4 -3 -2 -1
10 1
FIG. 22: Ratio of the cross sections d σ d y d p t for the ALICE conditions: y , y ∈ (3 , p t , p t ≥ W NN = 5 . d σ d p t (left) and the ratio(right) for the CMS conditions: y , y ∈ ( − . , . p t , p t ≥ W NN = 5 . Big modifications with respect to the monopole case can be seen for large p t and | y µ + , y µ − | ∼ y , y ) plane. Herethe distributions obtained with the monopole and realistic form factors are rather similar,but one should realize that these distributions are dominated by muons with small transversemomenta that are only slightly bigger than the experimental acceptance p t > . t and t values.The same processes can be also studied at the being presently in the operation RHIC.Here STAR and PHENIX detectors can be used. The distribution of the muon transverse18 IG. 24: (Color online) The muon rapidity distribution d σ d y (left) and the ratio (right) for the CMSconditions: y , y ∈ ( − . , . p t , p t ≥ W NN = 5 . ( G e V ) t p y -2 -1 0 1 2 ( nb / G e V ) t dp / dy σ d -4 -2
10 1 (a) ( G e V ) t p y -2 -1 0 1 2 ( nb / G e V ) t dp / dy σ d -4 -2
10 1 (b) FIG. 25: d σ d y d p t for realistic (left) and monopole (right) form factors for the CMS conditions: y , y ∈ ( − , . , p t , p t ≥ W NN = 5 . momentum is shown in Fig.28. The STAR rapidity cuts -1 < y , y < p t = 10 GeVthe damping factor is as big as 100! Experiments at RHIC have a potential to confirm thisprediction.In general, one could also inspect the rapidity distributions. Our predictions are shown19 t p y -2 -1 0 1 2 ] t dp / dy σ R A T I O [ d FIG. 26: The ratio of the realistic and monopole cross sections d σ d y d p t for the CMS conditions: y , y ∈ ( − . , . p t , p t ≥ W NN = 5 . ( G e V ) y -2 -1 0 1 2 y -2-1012 ( nb ) dy / dy σ d (a) ( G e V ) y -2 -1 0 1 2 y -2-1012 ( nb ) dy / dy σ d (b) FIG. 27: d σ d y d y for realistic (left) and monopole (right) form factors for the CMS conditions: y , y ∈ ( − . , . p t , p t ≥ W NN = 5 . in Fig.29. We predict the 30-40 % cross section damping with respect to the referencecalculation (monopole charge form factor).The two-dimensional distributions in muon rapidity and muon transverse momenta areshown in Fig.30 for the realistic and monopole form factors. Their ratio is presented inFig.31. Again as for the transverse momentum distribution (see Fig.28) a huge dampingcan be observed. The irregular structure of the ratio reflects the strong nonmonotonicdependence of the charge form factors of Au nuclei on t and t . For completeness inFig.32 we show the distribution of the dimuon invariant mass. The effect of the form factoroscillations shows up at large dimuon invariant masses where the ”realistic” cross section is20 IG. 28: (Color online) d σ d p t (left) and the ratio (right) for the STAR conditions: y , y ∈ ( − , p t , p t ≥ W NN = 200 GeV.FIG. 29: (Color online) d σ d y (left) and the ratio (right) for the STAR conditions: y , y ∈ ( − , p t , p t ≥ W NN = 200 GeV. rather small.The PHENIX collaboration can measure muons in a rather limited range of rapiditiesshown in Fig.33. We have given names to the four possible regions (squares) in the figure.In spite of these limitations, still interesting measurements can be done. As an examplein Fig.34 and Fig.35 we show our predictions for SQUARE1 and SQUARE2, respectively(the results for SQUARE3 and SQUARE4 are not shown as can be obtained by symmetry).Again large deviations from the monopole form factor results are predicted.21 G e V ) t p y -1-0.5 0 0.5 1 ( nb / G e V ) t dp / dy σ d -9 -6 -3
10 1 (a) ( G e V ) t p y -1-0.5 0 0.5 1 ( nb / G e V ) t dp / dy σ d -9 -6 -3
10 1 (b) FIG. 30: d σ d y d p t for realistic (left) and monopole (right) form factor for the STAR conditions y , y ∈ ( − , p t , p t ≥ W NN = 200 GeV. t p y -1-0.5 0 0.5 1 ] t dp / dy σ R A T I O [ d -5 -4 -3 -2 -1
10 1
FIG. 31: The ratio of the two-dimensional distributions from the previous figure for the STARconditions: y , y ∈ ( − , p t , p t ≥ W NN = 200 GeV. IV. CONCLUSIONS
The production of charge leptons in heavy ion collisions was proposed recently as a ”lab-oratory” for studying Quantum Electrodynamics effects, in particular the multiple photonexchanges. While very interesting theoretically it is still nonrealistic because of other ap-proximations made in the calculations.In this paper we have presented a study of the role of charge density for the differentialdistributions of muons produced in exclusive ultra-peripheral production in ultrarelativisticheavy ion collisions. Most of the calculations in the literature use so-called monopole charge22
IG. 32: (Color online) Invariant mass distribution d σ d M µ + µ − for the STAR conditions: y , y ∈ ( − , p t , p t ≥ W NN = 200 GeV. y y . . − . − . . . − . − .
123 4
FIG. 33: (Color online) The muon rapidity regions available by the PHENIX detector. form factor, which allows to write several formulae analytically. While it may be reason-able for the total rate of the dimuon production it is certainly too crude for differentialdistributions and for the cross sections with extra cuts imposed on transverse momenta ofmuons.We have performed calculations in the Equivalent Photon Approximation in the impactparameter space and in the momentum space using Feynman diagrammatic approach. Thefirst method is very convenient to include absorption effects, while the second one allows tostudy differential distributions.Our calculations show that the results obtained with the realistic and the approximate23
IG. 34: (Color online) SQUARE 1: d σ d y (left) and d σ d y (right) for the PHENIX conditions: 1.2 < | y , y | < p t , p t ≥ W NN = 200 GeV.FIG. 35: (Color online) SQUARE 2: d σ d y (left) and d σ d y (right) for the PHENIX conditions: 1.2 < | y , y | < p t , p t ≥ W NN = 200 GeV. form factors can differ considerably, in some parts of the phase space even by orders ofmagnitude. The effects related to the charge distribution in nuclei are particularly importantat large rapidities of muons and at large transverse momenta of muons.We have also discussed the role of absorption effects which can be easily estimated in theimpact parameter space. This allows to estimate the absorption effects for the total rateor for the rapidity distribution of the dimuon pairs. Estimating this effect in the case ofdifferential distributions is not simple, but could be studied in the future.We have presented predictions for the STAR and PHENIX detectors at RHIC as well as24or the ALICE and CMS detectors at LHC. In all cases we have found significant deviationsfrom the reference calculation for the monopole form factor. It would be interesting to pindown the effects discussed here and verify the present predictions in future studies at LHC.Both ALICE and CMS detectors could be used in such studies.In practice such studies may not be simple as an efficient trigger for the peripheralcollisions is required. The multiphoton exchanges leading to additional excitation of nucleiand subsequent emission of neutrons could be useful in this context (see e.g. [15]). Theneutrons could be then measured by the Zero Degree Calorimeters. First measurements ofthis type for e + e − pair emission have been already performed by the STAR and PHENIXcollaborations [16, 17].In the present calculation we have restricted to lowest-order QED calculations paying aspecial attention to realistic form factors and absorption effects and totally ignored higher-order corrections. How important are the QED higher-order correction was demonstratedrecently in Refs.[7, 14]. While Jentschura and Serbo [14] argue that the higher-order cor-rections are rather small, Baltz [7] finds a huge reduction of the integrated cross section ofthe order of 20%. Cleary the discrepancy should be clarified in the future. It would be alsovery interesting to calculate the higher-order corrections for differential distributions whichwill be measured at LHC . The latter calculations seem to us rather difficult technically.In the moment it seems precocious to answer the question whether the processes dis-cussed here could be used as a luminosity monitor for heavy ion collisions at LHC. In ouropinion, first these processes should be measured and compared to theoretical calculations.In addition, the influence of the absorption effects and multiphoton processes on differentialdistributions should be studied in more detail. Acknowledgments
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