The emergent black ring: a note on increasing ratio σ el (s)/ σ tot (s) at the LHC
TThe emergent black ring: a note on increasingratio σ el ( s ) /σ tot ( s ) at the LHC. S.M. Troshin, N.E. Tyurin
NRC “Kurchatov Institute”–IHEPProtvino, 142281, Russia
Abstract
We discuss the relations between the elastic and inelastic cross-sectionsvalid for the shadow and reflective modes of the elastic scattering. Consider-ations are based on the unitarity arguments. It is shown that the redistributionof the total interaction probability between the elastic and inelastic interac-tions can lead to increasing ratio of σ el ( s ) /σ tot ( s ) at the LHC energies inpresence of the reflective scattering mode. The form of the inelastic overlapfunction becomes peripheral due to the negative feedback. In the absorptivescattering mode, the mechanism of this increase is a different one since theimpact parameter dependence of the inelastic interactions probability is cen-tral in this case. A short notice is also given on the slope parameter and theleading contributions to its energy dependence in the both modes.Keywords: Impact parameter; Elastic scattering; Unitarity. a r X i v : . [ h e p - ph ] J u l bsorptive and reflective scattering modes It is known that the upper bounds for the inelastic cross–section [1,2] are differentcompared to the well known Froissart–Martin bound for the total cross-sections.This fact can be considered as a hint for a different asymptotic energy dependenceof the inelastic cross–section compared to the total and elastic ones. Such differ-ence indeed takes place in case of the reflective scattering mode [3–5] presence athigh values of the collision energy. The reflective scattering mode implies unitar-ity saturation at asymptotics. This mode is relevant for the energy region of veryhigh energies. There are strong corroborative indications of its existence [6–8]despite the data analyses at the LHC energy of √ s = 13 TeV [9] are still treatedas being nonconvergent ones [10].First, we briefly mention what the absorptive and reflective scattering modesare. Matrix element of the elastic scattering is related to the corresponding partialamplitude f l ( s ) by the relation S l ( s ) = 1 + 2 if l ( s ) , (1)and the partial amplitude f l ( s ) is constrained by the unitarity equation:Im f l ( s ) = | f l ( s ) | + h l,inel ( s ) (2)where h l,el ( s ) ≡ | f l ( s ) | . (3)Eq. (2) means that there are profound interrelations between elastic and inelasticinteractions, see for discussion e.g. [11]. At high energies, it is a common prac-tice to use an impact parameter representation making replacement l = b √ s/ as well as use an assumption on smallness of the real part of elastic scattering am-plitude. As usual, we adopt this approximation for a qualitative discussion. Thearguments [13, 14] are based on the theoretical results and numerical estimations.The assumption is in agreement with the small value of the ratio of the real toimaginary part of the forward scattering amplitude [15]. And corollary to that, theunitary upper bound for [ Re f ( s, b )] is only 1/4 of the corresponding bound for [ Im f ( s, b )] .Unitarity written in the impact parameter representation (the real part of thescattering amplitude is neglected) provides relations for the differential distribu-tions of the elastic and inelastic collisions over b : h el ( s, b ) = f ( s, b ) , (4) h inel ( s, b ) = f ( s, b )(1 − f ( s, b )) . (5) Note, that impact parameter is a conserved quantity at high energies [12]. f ( s, b ) variation is limited by the values from the interval ≤ f ≤ and f = 1 / means the complete absorption of the initial state, i.e. S = 0 , and h el ≤ , but h inel ≤ / .Absorptive scattering mode corresponds to the interval < f ≤ / , whilethe reflective scattering mode appears in the region of s and impact parameters b such that the amplitude f acquires values in the range of / < f ≤ . As itfollows from the analysis [7], one can safely assume that the elastic scattering hasan absorptive nature in the energy region √ s ≤ TeV, i.e. in this energy region: f ≤ / at all values of the impact parameter b and the following inequality takesplace since f ≤ / : h el ( s, b ) ≤ h inel ( s, b ) ≤ / , (6)where h el,inel are the respective overlap functions. Therefore, in the above energyregion, the elastic and inelastic cross–section should obey the relation σ inel ( s ) ≥ σ el ( s ) , (7)since σ el,inel ( s ) = 8 π (cid:90) ∞ bdbh el,inel ( s, b ) . (8)Eq. (7) is an important inequality for discussion of the absorptive versus reflectivemodes, but not quite new, e.g. the Pumplin bound [16] for inelastic diffractionpresupposes such a relation between elastic and inelastic cross-sections since ithas been obtained in a absorptive approach .Thus, the absorption is the reason for the inelastic interactions dominance. Butthe opposite claim is not valid, from dominance of the inelastic interactions onecannot conclude that the absorption only takes place in the whole region of theimpact parameter variation .When the energy becomes greater than the threshold value s r , the scatter-ing picture at small values of the impact parameter ( b ≤ r ( s ) , where S ( s, b = r ( s )) = 0 and S ( s, b ) becomes negative at b ≤ r ( s ) ), starts to acquire a reflectivecontribution and the Eq. (7) transforms into the following two inequalities: (cid:90) ∞ r ( s ) bdbh el ( s, b ) < (cid:90) ∞ r ( s ) bdbh inel ( s, b ) (9)and r ( s )2 ≥ (cid:90) r ( s )0 bdbh el ( s, b ) > r ( s )8 > (cid:90) r ( s )0 bdbh inel ( s, b ) . (10) Generalized upper bound when both – absorptive and reflective – modes are presented hasbeen obtained in [17]. The threshold value s r is determined by S ( s r , b = 0) = 0 . Note, that r ( s ) ∼ ln s at s → ∞ [4].
3t happens due to unitarity constraint for h el and appearance of the new relationbetween h el and h inel at b < r ( s ) , i.e. ≥ h el ( s, b ) > / > h inel ( s, b ) , (11)(see Fig. 1). Eq. 10 leads to the lower bound for integrated elastic cross–section, σ el > πr ( s ) and (since σ tot ( s ) = σ el ( s ) + σ inel ( s ) ) to the upper bound for the total cross–section of inelastic interactions, σ inel ( s ) < σ tot ( s ) − πr ( s ) . The above heuristic inequalities are valid at finite energies, s > s r , the rigorousasymptotic ones have been obtained in [1, 2]. bh elhinel1/4 r(s) Figure 1: Schematic representation of the overlap functions h el,inel impact param-eter dependencies at the energy s > s r .The illustrative schematic representation of Fig. 1 reflects qualitatively thepicture corresponding to the quantitative, model–independent impact parameteranalysis performed in [6–8].To confront above inequalities with the data, the explicit forms of the impactparameter dependencies of the functions h el,inel are needed. Currently, there areuncertainties under extraction of such information from the available data. Butit should be noted here that analysis performed in [6] gives the following values4 el ( s, b = 0) = 0 . and h el = 1 / at b (cid:39) . fm at the energy √ s = 8 TeV forthe central form of the impact parameter dependence of elastic overlap function(cf. Fig.19 of ref. [6]). h el and h inel Fig.19 of ref. [6] includes the second curve corresponding to a peripheral impactparameter dependence of h el ( s, b ) implying that elastic scattering amplitude ac-cording to unitarity (cf. Eqs.(2)-(5)) is determined by elastic overlap function inthe region of large impact parameters. But, the scattering amplitude (both the realand the imaginary parts of the amplitude) is exponentially decreasing with b inthe region of large impact parameters due to analyticity and becomes small. Suchdecreasing dependence leading to smallness of the amplitude combined with uni-tarity contradicts to a peripheral form of the impact parameter dependence of theelastic overlap function h el ( s, b ) since at large values of impact parameter the scat-tering amplitude is determined by the inelastic collisions according to unitarity: f (cid:39) h inel , (12)while h el ( s ) = o ( f ) at large values of b . The option of central inelastic overlapfunction and peripheral elastic overlap function is not considered therefore.Thus, we have f ( s, b = 0) = 0 . and h inel ( s, b = 0) = 0 . (i.e. shallowminimum at b = 0 ) and h inel ( s, b = 0) < h inel ( s, b (cid:39) .
35) = 1 / [6]. Hence,already at the LHC energy √ s = 8 TeV the black disk limit is exceeded at smallvalues of collision impact parameter. Due to monotonic energy dependence ofthe scattering amplitude the effect should be more pronounced at √ s = 13 TeV.Such black disk limit exceeding has been discussed more than 25 years ago [3].There are, of course, ambiguities in extraction of b –dependent amplitude of elasticscattering from experimental data related to the presence of the two signs of asquare root of differential cross–section (cf. [14]), but no one seems to be objectiveto the principal statement on the reaching a black disc limit at b = 0 at the LHCenergies. An incomplete list of the relevant references includes the followingones [7, 8, 14, 18, 19].We adopt the point that this limit has already been exceeded at the LHC ener-gies and represent then the scattering amplitude f ( s, b ) in the region ≤ b ≤ r ( s ) and a fixed the LHC energy in the form f ( s, b ) = 12 [1 + α ( s, b )] (13)with small positive function α ( s, b ) such that α ( s, b = r ( s )) = 0 . The value of α at b = 0 and √ s = 8 TeV is of order of 10 % [6]. The elastic overlap function can5e written at the LHC energies and b ≤ r ( s ) in the form h el ( s, b ) = 14 [1 + 2 α ( s, b ) + α ( s, b )] , (14)while inelastic overlap function has a negative deviation (hollowness) of the sec-ond order on α : h inel ( s, b ) = 14 [1 − α ( s, b )] . (15)Thus, it is evident that due to a smallness of deviation from the black disc limit at the LHC energies exceeding of this limit should be detected more easily underthe studies of the elastic overlap overlap function than under the studies of theinelastic one at the LHC energies. Due to the black disk limit exceeding elasticoverlap function gets a positive feedback and the inelastic one gets a negativefeedback.Eq. (10) can be used for a qualitative explanation of the ratio increase withenergy at the LHC [22] where the reflective scattering mode seems to appear.Indeed, we observe the mentioned redistribution of the total probability increasewith the energy growth in favor of the elastic interactions under transition to areflective mode (see Fig. 1, Eqs. (14) and (15)). In this regard, it is helpful to recallthat σ el ( s ) ∼ ln s , but σ inel ( s ) ∼ ln s and σ el ( s ) /σ tot ( s ) → at s → ∞ [4].It also is instrumental to keep in mind that in the reflective scattering mode S < . The following relations are relevant for clarification of the energy dependenciesof the functions h el,inel [5]: ∂h el ( s, b ) ∂s = (1 − S ( s, b )) ∂f ( s, b ) ∂s (16)and ∂h inel ( s, b ) ∂s = S ( s, b ) ∂f ( s, b ) ∂s . (17)Thus, the following limit takes place, namely, h inel ( s, b = 0) → when f ( s, b =0) approaches unitary limit, i.e. f ( s, b = 0) → at s → ∞ .It should also be noted that the above mechanism is not relevant for the ab-sorptive scattering mode where the inelastic overlap function has a central impactparameter dependence. At the energies where only absorptive scattering takesplace the ratio σ el ( s ) /σ tot ( s ) increase is due to a faster increase of elastic scat-tering probability at small and moderate values of impact parameter compared tothe probability of collision with the same values of the impact parameter leadingto the inelastic interactions. This limit corresponds to a complete absorption of the initial state The CERN ISR energies are in the absorptive scattering region [20]. The slope parameter
These results have some implications for the slope parameter B ( s ) of the forwardelastic peak, B ( s ) = ddt ln dσdt | t = o . (18)This quantity is determined by the average value of the impact parameter squared,i.e. (cid:104) b (cid:105) tot ≡ (cid:90) ∞ b f ( s, b ) bdb/ (cid:90) ∞ f ( s, b ) bdb. (19)In its turn, according to unitarity the energy dependence of (cid:104) b (cid:105) tot is determinedby the ones of cross–sections σ el,inel and average values (cid:104) b (cid:105) el,inel .In the absorptive scattering mode the major contribution to B ( s ) comes frominelastic processes: σ inel ( s ) (cid:104) b (cid:105) inel ≥ σ el ( s ) (cid:104) b (cid:105) el , (20)where (cid:104) b (cid:105) el,inel ≡ (cid:90) ∞ b h el,inel ( s, b ) bdb/ (cid:90) ∞ h el,inel ( s, b ) bdb. (21)But the elastic interactions give a subdominant contribution to the slope parameterin this mode. The relation Eq. (20) is a direct result of Eq. (6).Contrary, in the reflective scattering mode, due to unitarity saturation, the maincontribution to B ( s ) comes from the elastic scattering decoupled from the inelasticinteractions [21] and the following limiting dependence at s → ∞ should takeplace due to the elastic scattering asymptotic dominance: B ( s ) → (cid:104) b (cid:105) el / . (22)It should be noted again that elastic scattering dominance at s → ∞ is a naturalresult of a self–damping of the inelastic channels’ contributions in this limit [23],i.e. it is a consequence of the aforementioned redistribution between the proba-bilities of elastic and inelastic interactions due to unitarity. At the same time, theaverage values (cid:104) b (cid:105) el,inel ( s ) have similar dependence in the limit of s → ∞ [21]: (cid:104) b (cid:105) el,inel ( s ) ∼ r ( s ) . Redistribution of probabilities with corresponding decrease with energy of theinelastic interactions’ probability at small impact parameters is a result of the re-flective scattering mode appearance at the LHC energies. One could expect aspeed up of the ratio σ el ( s ) /σ tot ( s ) increase compared to the absorptive mode atlower energies due to starting decrease with energy of the inelastic interactionprobability at the point of b = 0 and in its vicinity ( S < ). The region where re-flective scattering mode is expected to give a noticeable contribution corresponds7o the LHC energy region. Contrary, one should expect a slow down of the ratio σ el ( s ) /σ tot ( s ) increase at the LHC energies in case of asymptotic saturation of theblack disk limit at s → ∞ (cf. [24]). The mode is called reflective since the phases of incoming and outgoing waves dif-fer by π . In optics, its appearance is associated with density increase of a reflectingmedium beyond some threshold value. Such medium has a higher refractive indexthan the one incoming wave arrived from. In QCD, one can assume that the colorconducting medium is formed in the transient state of hadron interactions and it isresponsible for reflective scattering mode [25]. There is a link between structureof proton responsible for reflective scattering and that of the chiral models. Thechiral models describe baryon as consisting from an inner core with a baryoniccharge and an outer cloud surrounding the core. Presence of the inner repulsivecore is in agreement with the DVCS data analysii [26] and with the indirect (pres-ence of the second exponential slope in dσ/dt ) LHC data at √ s = 13 TeV [9].These results are in favor of the two–scale structure of proton. The interpretationsof the results of the CLAS detector at Jefferson Laboratory based on the existenceof the extended substructures inside the proton was discussed also in [27].The reflective scattering mode appears when the value of elastic overlap func-tion exceeds the black disk limit, i.e. it becomes greater than 1/4 at b = 0 . Due tounitarity relation in the impact parameter representation the value of the inelasticoverlap function becomes less than 1/4 at b = 0 . The latter phenomenon wascalled hollowness and widely discussed in the recent papers. We mention onlysome of them here, i.e. [19, 28–30]. It is evident that hollowness and reflectivescattering are, in fact, represent the “two sides of the same medal”, those are re-lated by unitarity and appear when the black disc limit is exceeded. The value ofthe ratio r e ( s ) ≡ σ el ( s ) /σ tot ( s ) at s > s r is correlated with the degree of reflection(albedo in optics), while the value of the ratio r i ( s ) ≡ σ inel ( s ) /σ tot ( s ) is corre-lated with the degree of the hollowness. The sum of these two ratios r e + r i = 1 due to unitarity. Discussion of relation of the reflective scattering with a periph-eral form of the inelastic overlap function (hollowness) and the particular effectsof the reflective scattering mode in hadron production can be found e.g. in [33].The peripheral form of h inel ( s, b ) would lead, in particular, to slow down of themean multiplicity growth at the LHC [34].The appearance of these new twin phenomena (reflective scattering and hol-lowness) is a manifestation of the fact that the restricting assumption on the sole The ratio r i ( s ) can be related to the real to imaginary part ratio of forward scattering amplitude[31] due to the local dispersion relations [32] . both scattering modes — the absorptive and the reflective ones —- under descrip-tion of the energy evolution of the scattering amplitude. Constraining assumptionon the existence of the single scattering mode (absorption only) has no firm phys-ical ground. Reflective scattering and hollowness can be associated with an effectof the self–dumping intermediate inelastic channels contribution [23] at the LHCenergies. The self-dumping can arise as a result of randomization of the phases ofmultiparticle production amplitudes. This randomization in its turn can be consid-ered as a consequence of the color–conducting collective state of hadronic matterformation under the central over impact parameter hadron collisions (with highmultiplicities) and subsequent stochastic decay of this state and hadronization intothe multiparticle final states [25]. Important role of the phases for the inelasticoverlap function dependence has been discussed in [36]. Their mutual cancella-tion has been proposed as an explanation for the diffraction peak appearance inelastic scattering [37].It is also useful here to pay an attention again to the results of the seminalpaper [38] on the alarming tendencies in the analytical properties of the scatteringamplitude observed in the approach based on the exponential unitarization whena simple pole singularity in a phase function of energy turns into an essentialsingularity in the scattering amplitude after the unitarization procedure. Acnowledgements
We are grateful to T. Cs ¨org˝o, J. Kaˇspar and E. Martynov for the interesting anduseful correspondence on the amplitude impact parameter analysis.
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