We discuss effects of reflective scattering for hadron and heavy nuclei
collisions at the LHC and asymptotical energies. It is shown that the
reflective scattering might lead to decreasing matter density with energy
beyond the LHC energies. Limiting form of energy dependence of hadron density
is obtained. Unitarity upper bound for the absolute value of the real part of
elastic scattering amplitude and two-particle inelastic binary reactions
amplitudes in impact parameter representation is two times less than the
corresponding bound for the imaginary part of the elastic scattering amplitude.
The former limit restricts a possible odderon contribution.Comment: 7 pages, 1 figure, talk given at the EDS'09 Conferenc

Troshin, Sergey

Tyurin, Nikolay

English

arXiv

We discuss effects of reflective scattering for hadron and heavy nuclei collisions at the LHC and asymptotical energies. It is shown that the reflective scattering might lead to decreasing matter density with energy beyond the LHC energies. Limiting form of energy dependence of hadron density is obtained. Unitarity upper bound for the absolute value of the real part of elastic scattering amplitude and two-particle inelastic binary reactions amplitudes in impact parameter representation is two times less than the corresponding bound for the imaginary part of the elastic scattering amplitude. The former limit restricts a possible odderon contribution

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Reflective Elastic Scattering at LHCSergey Troshin, Nikolay TyurinIHEP, Protvino, 142281 RussiaWe discuss effects of reflective scattering for hadron and heavy nuclei collisions at the LHCand asymptotic energies. It is shown that the reflective scattering might lead to decreasingmatter density with energy beyond the LHC energies. A limiting form of energy dependenceof the hadron density is obtained. The unitarity upper bound for the absolute value ofthe real part of the elastic scattering amplitude and two-particle inelastic binary reactionamplitudes in impact parameter representation is two times less than the correspondingbound for the imaginary part of the elastic scattering amplitude. The former limit restrictsa possible odderon contribution.1 Reflective ScatteringA new physical interpretation of unitarity saturation in elastic scattering as a reflective scat-tering was proposed in [1] proceeding from optical analogy. This interpretation is related tothe non-perturbative aspects of strong interactions and follows from the specific property ofthe unitarity saturation when the elastic S-matrix becomes negative and S(s, b)|b=0 → −1 ats → ∞. It should be noted that S(s, b) = 1 + 2if(s, b), where f(s, b) is the elastic scatteringamplitude in the impact parameter representation.In particular, we would like to note that the possible values of the elastic S matrix can benegative (in the pure imaginary case). Transition to the reflective scattering mode is naturallyreproduced by the U -matrix form of elastic unitarisation. The elastic scattering S-matrix(2 → 2 scattering matrix element) in the impact parameter representation is written in thisunitarisation scheme in the form of a linear fractional transform:S(s, b) =1 + iU(s, b)1− iU(s, b) , (1)where U(s, b) is the generalised reaction matrix, which is considered to be an input dynamicalquantity. For simplicity we consider the case of a pure imaginary U -matrix and make thereplacement U → iU in (1). The reflective scattering mode (S(s, b) < 0) starts to appear at theenergy sR, which is determined as a solution of the equation U(sR, b = 0) = 1. At s > sR theelastic scattering acquires ability for reflection, while the inelastic overlap function hinel(s, b)gets a peripheral impact parameter dependence in the region s > sR. It should be noted thatunitarity condition for the elastic scattering amplitude F (s, t), which can be written in the formImF (s, t) = Hel(s, t) + Hinel(s, t), (2)where Hel,inel(s, t) are the corresponding elastic and inelastic overlap functions introduced byVan Hove [2]. The functions Hel,inel(s, t) are related to the functions hel,inel(s, b) via the75Fourier-Bessel transforms, i.e.Hel,inel(s, t) =spi2∫ ∞0b db hel,inel(s, b)J0(b√−t). (3)The elastic and inelastic cross-sections can be obtained as follows:σel,inel(s) ∼ 1sHel,inel(s, t = 0). (4)Saturation of unitarity leads to the peripheral dependence of hinel(s, b). It is a manifestation ofthe self-damping of the inelastic channels at small impact parameters. The function hinel(s, b)reaches its maximum value at b = R(s), note thatR(s) ∼ 1Mln s,while an elastic scattering (due to reflection) occurs effectively at smaller values of impactparameter, i.e. 〈b2〉el < 〈b2〉inel. At the values of energy s > sR the equation U(s, b) = 1has a solution in the physical region of impact parameter values, i.e. S(s, b) = 0 at b = R(s).Fig. 1 shows the regions where elastic S-matrix has positive and negative values. Of course,bS(s,b) >0S(s,b)=0ss=sR<02reflective scattering with probability |S(s,b)|S(s,b)region I, no reflective scatteringregion IIFigure 1: Regions of positive (absorptive scattering) and negative values (absorptive and reflectivescattering) of the function S(s, b) in the s and b plane.the reflective scattering exists only in the elastic channel. For example, all inelastic binaryreactions have amplitudes f˜(s, b) in the impact parameter representation, which satisfy theinequality |f˜(s, b)| ≤ 1/2, while the elastic scattering amplitude satisfies |f(s, b)| ≤ 1. It followsfrom unitarity equation in the impact parameter representation:Imf(s, b) = hel(s, b) + hinel(s, b);amplitudes f˜(s, b) contribute to hinel(s, b). The upper bound for the elastic scattering cross-section is four times higher than the upper bound for the inelastic cross-section as it was recentlyS TROSHIN, AND N TYURIN76demonstrated in [3]. Scattering dynamics in the elastic channel such as pp → pp is thereforestrikingly different, e.g. from the inelastic binary diffractive process, such as pp → pN , whereN is an isobar. The latter reactions should have a peripheral impact parameter profile, whichcan be related to the dominating contribution of helicity-flip amplitudes. The unitarity limitand black disk limit are the same for the inelastic overlap function, but those limits are differentfor the elastic overlap functionhel(s, b) ≡ |f(s, b)|2.The unitarity limit for the elastic overlap function is four times higher than the black disk limit.This is an important point for consideration of exclusive limit of inclusive reactions. Saturationof unitarity leads to suppression of the inelastic cross-section, i.e. at fixed impact parameter(b < R(s)) hinel(s, b) → 0 at s →∞ andσel(s) ∼ R2(s), σinel(s) ∼ R(s). (5)Thus, Hel(s, t) which has the following t-dependenceHel(s, t) ∼ RJ1(R√−t)√−t ,dominates over Hinel(s, t), which depends on t likeHinel(s, t) ∼ RJ0(R√−t),at −t = 0, but it is not the case for the scattering in the non-forward directions. In this regionthese two functions have similar energy dependencies proportional to R1/2(s) at rather largefixed values of −t. The mean impact parameter values for elastic and inelastic interactions havealso similar energy dependencies〈b2〉el(s) ∼ R2(s), 〈b2〉inel(s) ∼ R2(s), (6)but the value of impact parameter averaged over all interactions (cf. [4])〈b2〉tot(s) =σel(s)σtot(s)〈b2〉el(s) +σinel(s)σtot(s)〈b2〉inel(s)acquires the main contribution from elastic scattering according to Eq. (5). Therefore, theinelastic intermediate states will give subleading contribution to the slope of diffraction coneB(s),B(s) ≡ ddtln(dσdt)∣∣∣∣t=0,at asymptotic energies. Indeed, since B(s) ∼ 〈b2〉tot(s), it can be written in the formB(s) = Bel(s) + Binel(s),where Bel(s) ∼ R2(s), while Binel(s) ∼ R(s). It should be noted that both terms Bel(s) andBinel(s) are proportional to R2(s) in case of the absorptive scattering.Under reflective scattering, the behaviour of the function Hinel(s, t) is determined by aperipheral impact parameter profile and its −t dependence is different. Meanwhile, the elasticREFLECTIVE ELASTIC SCATTERING AT LHC77overlap function Hel(s, t) has similarities with that function in the case of approach whereabsorption is only presented. As a result, zeroes and maxima of the functions Hel(s, t) andHinel(s, t) will be located at different values of −t and zeroes and maxima of ImF (s, t) will alsobe located at different position in the cases of absorptive and the reflective scattering. In thecase of reflective scattering, dips and maxima will be located in the region of lower values of −t.We would like to note that the presence of reflective scattering enhances the large −t regionby factor√−t compared to absorptive scattering. Despite that these two mechanisms leadat the asymptotics to the significant differences in the total, elastic and inelastic cross-sectiondependencies, their predictions for the differential cross-section of elastic scattering are not somuch different at small and moderate values of −t.2 Intermittent Remark on Unitarity and Real Part ofScattering AmplitudeIt is evident that there are serious difficulties in accounting all known dynamical issues andlimitations into a particular phenomenological model. But it is equally difficult to expect thatthe model inconsistent with unitarity (i.e. the one violating probability conservation law) wouldadequately reflect the dynamics of hadron interaction and provide reliable predictions. To fulfillunitarity condition under a model construction of the elastic amplitudes, it is natural to useunitarisation approaches such as eikonal or U -matrix, which consider amplitudes in the impactparameter space. They automatically guarantee that elastic amplitude in the impact parameterrepresentation will obey unitarity condition.Despite that the full implementation of unitarity is not possible nowadays (cf. e.g. [5]), theamplitude in the impact parameter space should not exceed unity anyway. However, when theamplitude F (s, t) is constructed in the s and t representation, it is a priori not evident that theparticular form of this amplitude being transformed into the impact parameter space f(s, b)would satisfy unitarity. This remains to be true, even when the model under considerationleads to the predictions for observables which explicitly agree with axiomatic bounds, e.g. suchas well known Froissart-Martin bound for the total cross-sections. Getting agreement withexperimental data at finite energies and asymptotic bounds at s → ∞ is not enough since awide class of functional dependencies can successfully describe experimental data and providecorrect asymptotic behaviour. Additional steps to justify that the impact parameter amplitudeis at least less than unity in the whole region of kinematic variables are necessary.In the above remarks we supposed that imaginary part of scattering amplitude is a domi-nating one. Further unitarity restriction exists for models which do not suppose domination ofimaginary part of scattering amplitude, such as models with maximal odderon contribution [6].Indeed, unitarity condition in the impact parameter representation for the elastic scatteringamplitude can be rewritten in the form:Imf(s, b)[1− Imf(s, b)] = [Ref(s, b)]2 + hinel(s, b).Since 0 ≤ Imf(s, b) ≤ 1, we obtain that unitarity limits the real part of scattering amplitudein the following way[Ref(s, b)]2 ≤ 1/4,−12√1− 4hinel(s, b) ≤ Ref(s, b) ≤ 12√1− 4hinel(s, b).S TROSHIN, AND N TYURIN78The function Ref(s, b) can be sign changing one in contrast with Imf(s, b)). This limitation,as it was already mentioned, is essential for the models with odderon and is indirectly in favourof the standard procedure of neglecting the real part of scattering amplitude compared to itsimaginary part. It also is evident that absolute value of the real part and imaginary part ofelastic scattering amplitude f(s, b) cannot reach their maximal values simultaneously, moreoverwhen Imf(s, b) → 1, saturating unitarity limit at large values of s in the region b < R(s), thenRef(s, b)→ 0 in this kinematic region. It should be noted that this saturation does not supposethat Ref(s, b) vanish everywhere. It means that [Ref(s, b)]2 should have a peripheral impactparameter profile. The same conclusion is valid when Imf(s, b) → 1/2, saturating the black disklimit at large values of s in the region b < R(s), then Ref(s, b)→ 0 because hinel(s, b) → 1/4 inthis region. The above difference in the impact parameter profiles would result in the differentenergy dependencies of ImF (s, t = 0) and ReF (s, t = 0) bringing maximal odderon on the edgeof contradiction with unitarity (or black disk) limit saturation. Of course, unitarity or blackdisk limits saturation itself does not follow from axiomatic field theory, but we would like tonote, that it is much more natural to expect that it could be a manifestation of a maximalstrength of strong interaction instead of behaviour of the real part of the forward scatteringamplitude in the form ReF (s, t = 0) ∼ s ln2 s as it happens in the models incorporating themaximal odderon regime.3 Reflective Scattering and DeconfinementPossible existence of the reflective scattering at very high energies implies that confinementbecomes stronger and stronger as the collision energy increases and proton collisions resemblemore and more collisions of hard spheres. In this section we address one aspect of the broadproblem of transition to the deconfined state of matter, namely, we discuss the role of thereflective scattering on the energy dependence of density in the percolation mechanism of thetransition to the deconfined state of matter.The main idea of the percolation mechanism of deconfinement is a formation in the certainvolume of a connected hadron cluster due to increasing temperature and/or hadron density [7],i.e. when vacuum as a connected medium disappears, the deconfinement takes place. Thisprocess has typical critical dependence on particle density. Thus, it was proposed to use per-colation to define the states of matter and consider the disappearance of a large-scale vacuumas the end of hadronic matter existence [7, 8].The probability of reflective scattering at b < R(s) and s > sR is determined by themagnitude of |S(s, b)|2; this probability is equal to zero at s ≤ sR and b ≥ R(s) (region I inFig. 1). At the energies s > sR reflective scattering will mimic the presence of a repulsivecore in hadron and meson interactions. Presence of the reflective scattering can be accountedfor using van der Waals method (cf. [9]). This approach was used originally for description ofthe fluids behaviour starting from the gas approximation by means of taking into account thenonzero size of molecules. Consider central collision of two identical nuclei having N hadrons intotal with centre-of-mass energy√s per nucleon and calculate hadron density nR(T, µ) = N/Vin the initial state at given temperature T and baryochemical potential µ in the presence ofreflective scattering. The effect of the reflective scattering of hadrons is equivalent to decreaseof the volume of the available space which the hadrons are able to occupy in the case whenreflective scattering is absent. Thus following to van der Waals method, we must then replaceREFLECTIVE ELASTIC SCATTERING AT LHC79volume V byV − pR(s)VR(s)N2,i.e. we should writen(T, µ) =NV − pR(s)VR(s)N2,where n(T, µ) is hadron density without account for reflective scattering and pR(s) is the aver-aged over volume VR(s) probability of reflective scattering:pR(s) =1VR(s)∫VR(s)|S(s, r)|2d3x.The volume VR(s) is determined by the radius of the reflective scattering. Here we assumespherical symmetry of hadron interactions, i.e. we replace impact parameter b by r and ap-proximate the volume VR(s) by VR(s) ' (4pi/3)R3(s). Hence, the density nR(T, µ) is connectedwith corresponding density in the approach without reflective scattering n(T, µ) by the followingrelationnR(T, µ) =n(T, µ)1 + α(s)n(T, µ),where α(s) = pR(s)VR(s)/2. Let us now estimate change of the function nR(T, µ) due to thepresence of reflective scattering. We can approximate pR(s) by the value of |S(s, b = 0)|2 whichtends to unity at s → ∞. It should be noted that the value √sR ' 2TeV [10]. Below thisenergy there is no reflective scattering, α(s) = 0 at s ≤ sR, and therefore corrections to thehadron density are absent. Those corrections are small when the energy is not too much higherthan sR. At s ≥ sR the value of α(s) is positive, and presence of reflective scattering diminisheshadron density. We should expect that this effect would already be noticeable at the LHCenergy√s ' 5 TeV in Pb+Pb collisions. At very high energies (s →∞)nR(T, µ) ∼ 1/α(s) ∼ M3/ ln3 s.This limiting dependence for the hadron density appears due to the presence of the reflectivescattering which results in similarity of head-on hadron collisions with scattering of hard spheres.It can be associated with saturation of the Froissart-Martin bound for the total cross-section.It should be noted that this dependence has been obtained under assumption on sphericalsymmetry of hadron interaction region. Without this assumption, limiting dependence of thehadron density in transverse plane can only be obtained, i.e. transverse plane density of hadronswould have then the following behaviournR(T, µ) ∼M2/ ln2 s.To conclude this section, we would like to note that the lower densities of hadron matter areneeded for percolation (and transition to the deconfined state) in the presence of reflectivescattering. It might be useful to note that the rescattering processes are also affected by thereflective scattering. Reflective scattering would lead to noticeable effects at the LHC energiesand beyond and could help in searches of the deconfined state and studies of properties oftransition mechanism to this state of matter which might proceed by means of percolation.Thus, it will affect description of initial state dynamics in nuclear interactions at the LHCenergies by introducing notion of limiting density of strongly interacting matter at respectiveenergies.S TROSHIN, AND N TYURIN804 ConclusionThus, at very high energies there would be two different regions of impact parameter distancesin particles collisions, namely the outer region (peripheral collisions) where elastic scatteringhas exclusively a shadow origin and inner region (central collisions) where reflective and absorp-tive scattering give competing contributions, reflective scattering contribution increases whileabsorptive scattering contribution decreases at fixed impact parameter. It is not surprisingthat the model with reflective scattering contribution leads to significantly higher values fortotal and elastic cross-sections at the LHC energies1 while it renders the standard values forthe inelastic cross-section. In the geometric terms, the generic scattering picture at fixed en-ergy beyond the black disc limit can be described as a scattering off a partially reflective andpartially absorptive disk surrounded by the black ring (which becomes grey at larger values ofthe impact parameter). The evolution with energy is characterised by increasing albedo due tothe interrelated increase of reflection and decrease of absorption at small impact parameters.This picture predicts that the scattering amplitude at the LHC energies is beyond the blackdisk limit at small impact parameters and it provides explanation for the regularities observedin cosmic rays studies, e.g. the existence of the knee in the cosmic rays spectrum. It leadsalso appearance of limiting density dependent on energy which takes place only at very highenergies and has an origin related to unitarity saturation.AcknowledgementsWe are grateful to J.-R. Cudell, M. Islam, L. Jenkovszky, A. Krisch, U. Maor, A. Martin,E. Martynov, V. Petrov and D. Sivers for the interesting comments and discussions.References[1] S.M. Troshin, N.E. Tyurin, Int. J. Mod. Phys. A22 4437 (2007).[2] L. Van Hove, Nuovo Cimento, 28 798 (1963).[3] A. Martin, arXiv: 0904.3724, talk given at this Conference.[4] Z. Ajduk, Nuovo Cim. A15, 390 (1973).[5] P.V. Landshoff, arXiv: 0903.1523.[6] R. Avila, P. Gauron, B. Nicolescu, Eur. Phys. J. C49 581 (2007);E. Martynov, B. Nicolescu, Eur. Phys. J. C56 57 (2008).[7] P. Castorina, K. Redlich, H. Satz. Eur. Phys. J. C59 67 (2009).[8] H. Satz, arXiv:0803.1611.[9] J. Cleymans, K. Redlich, H. Satz, E. Suhonen, Z. Phys. C33 151 (1986).[10] P.M. Nadolsky, S.M. Troshin, N.E. Tyurin, Z. Phys. C69 131 (1995).1The value of total cross-section about 150 mb is predicted for pp-interactions at the LHC starting up energy√s = 7 TeV.REFLECTIVE ELASTIC SCATTERING AT LHC81

Reflective Elastic Scattering at LHC

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