RRapidity gaps and ancestry ∗ Dung LE ANH, St´ephane MUNIER
Centre de physique th´eorique (CPHT), ´Ecole polytechnique, CNRS,Universit´e Paris-Saclay, Route de Saclay, 91128 Palaiseau, FranceThe recently discovered correspondence between the distribution of ra-pidity gaps in electron-nucleus diffractive processes and the statistics ofthe height of genealogical trees in branching random walks is reviewed.In addition, a new comparison of numerical solutions of exact equationsfor diffraction on the one hand, and for ancestry on the other hand, bothestablished in the framework of the color dipole model, is presented.
1. Rapidity gap distribution in deep-inelastic scattering
In the scattering of electrons off protons at very high energies, a par-ticularly striking – and a priori surprising – phenomenon was discoveredexperimentally at the DESY-HERA collider: hard diffraction . In a sig-nificant proportion of the events (about 10% overall), the proton left thecollision unaltered, while in the forward region of the scattered electron, ahadronic system was observed, as a result of the dissociation of the virtualphoton mediating the interaction (see Fig. 1). Diffractive events may belabeled by the size of the region void of particles surrounding the scatteredproton, which can be characterized by a Lorentz-invariant rapidity gap vari-able y . The latter fluctuates from event-to-event between (almost) zero andthe maximum available rapidity Y = ln ˆ s/Q , where ˆ s is the squared center-of-mass energy of the γ ∗ -proton/nucleus subreaction, and Q the virtualityof the photon. What has been observed in high-energy electron-proton scat-tering is also expected in electron-nucleus collisions at a future Electron-IonCollider (EIC).Some time ago, an equation for the distribution of rapidity gaps wasrigorously established by Kovchegov and Levin (KL) [2] in this context. ∗ Work presented at Diffraction and Low-x 2018, Reggio Calabria, August 2018, andsupported in part by the Agence Nationale de la Recherche under the project For background on all aspects of high-energy scattering, see the textbook of Ref. [1]. (1) a r X i v : . [ h e p - ph ] N ov diffraction printed on November 6, 2018 Fig. 1.
Diffractive dissociation event recorded in the H1 detector.
The highly-energetic proton (entering from the right) interacts elastically, picks a small trans-verse momentum compared to its longitudinal momentum and therefore does notleave any track in the detector. The virtual photon instead is converted into ahadronic system. The angular sector between the momentum of the scatteredproton and the produced hadronic system void of any activity is the rapidity gap.
But solving it analytically remains a formidable challenge. They did notaddress directly deep-inelastic scattering, but instead onium-nucleus scat-tering, which is straightforwardly related to the former when the interac-tion between the electron and the nucleus is mediated by a longitudinally-polarized virtual photon.Let us consider an onium of size r scattering off a big nucleus. In the KLformulation, the distribution of the rapidity gaps is the solution of a systemof two equations. The first one is the Balitsky-Kovchegov (BK) equation forthe rapidity evolution of the forward elastic S -matrix element. Introducingthe notation ¯ α ≡ α s N c /π , the BK equation reads ∂ y S ( r, y ) = ¯ α (cid:90) d r (cid:48) π r r (cid:48) ( r − r (cid:48) ) (cid:2) S ( r (cid:48) , y ) S ( r − r (cid:48) , y ) − S ( r, y ) (cid:3) . (1)The initial condition is given by e.g. the McLerran-Venugopalan (MV)model, S ( r, y = 0) = e − r Q ln( e +4 /r Λ ) , with Q MV the saturation mo-mentum of the nucleus. 1 /Q MV can be interpreted as the dipole size above The (dimensionless) total, elastic and inelastic cross sections per impact parameter may be derived from S , which is essentially real at high energy: σ tot = 2(1 − S ), σ el = (1 − S ) , σ in = σ tot − σ el = 1 − S . These formulas show, in particular, thatthe elastic cross section is maximum (and equal to the inelastic one) when S = 0. iffraction printed on November 6, 2018 which the scattering occurs with unit probability [i.e. S ( r (cid:29) /Q MV , (cid:28) S ( r, ˜ y ) which also obeys the BK equation ∂ ˜ y S ( r, ˜ y ) = ¯ α (cid:90) d r (cid:48) π r r (cid:48) ( r − r (cid:48) ) (cid:2) S ( r (cid:48) , ˜ y ) S ( r − r (cid:48) , ˜ y ) − S ( r, ˜ y ) (cid:3) , (2)with the initial condition S ( r, ˜ y = 0) = [ S ( r, y )] . In terms of S , the gapdistribution then reads dσ diff ( y | r, Y ) dy = ∂∂ ˜ y (cid:12)(cid:12)(cid:12)(cid:12) ˜ y = Y − y S ( r, ˜ y ) . (3)The work presented here may be viewed as an effort to find a solutionto the KL set of equations (1,2). But instead of trying to solve it bruteforce, which is technically extremely challenging, we develop a picture ofdiffractive scattering, from which what we believe should be the asymptoticsof the KL equation (almost) straightforwardly follow and which points to adeep link with ancestry problems in branching random walks. The presentwrite-up shortly summarizes the papers in Refs. [3, 4], before presentinga new numerical comparative study of exact equations for diffraction andancestry in the dipole model (see Sec. 3.2 below).
2. Picture of onium-nucleus scattering
In the onium restframe, the nucleus appears in a highly-evolved andoccupied state, while the dominant state of the onium is a bare quark-antiquark pair. Event-by-event fluctuations are negligible; S may be inter-preted as the “transparency” of the boosted nucleus.In the nucleus restframe instead, the whole evolution is in the onium,which appears typically as a set of many gluons (represented by dipoles inthe large number-of-color limit [5]), whose detailed content strongly fluctu-ates from event-to-event. For a scattering to occur, there should be at leastone gluon in this set which has a transverse momentum of the order of Q MV ,so that the whole state has a non-negligible probability to scatter with thenucleus. In this context, 1 − S can be interpreted as the probability thatthe Fock state of the onium contains at least one gluon with a transversemomentum of that magnitude. Equations (1,2) also follow quite straightforwardly from the Good-Walker picture,see Ref. [3]. diffraction printed on November 6, 2018 y -frame Let us now choose a frame in which the nucleus is boosted to rapidity y and the onium to rapidity ˜ y = Y − y . In order to have a diffractive eventexhibiting a gap y with unit probability, one needs at least one gluon in theFock state of the onium whose transverse momentum is smaller than thesaturation scale at rapidity y , Q s ( y ). Indeed, this condition makes surethat the elastic interaction cross section of the onium is significant. Thediffractive cross section dσ diff /dy is tantamount to this very probability. Astraightforward calculation leads to an elegant formula for the latter, oncenormalized to the total cross section:1 σ tot dσ diff dy = const × (cid:20) ¯ αY ¯ αy ( ¯ αY − ¯ αy ) (cid:21) / . (4)The overall numerical constant, of order unity, cannot be determined withinthe present approach. This formula is actually only valid in the so-called scaling region , defined by the following constraints on the parameters: 1 (cid:28) ln r Q s ( Y ) (cid:28) (cid:112) χ (cid:48)(cid:48) ( γ ) ¯ αY .
3. Ancestry
It has been known for some time that dipole evolution is a peculiarbranching random walk [6]. One of the main results of Refs. [3, 4] is thesurprising observation that the structure of the branches may be directlyrelated to an observable in high-energy physics.Boosting a bare onium of size r by one unit in the rapidity ˜ y opens thephase space for quantum fluctuations in the form of additional gluons popu-lating its Fock state. A one-gluon emission by the onium may be interpretedas the splitting of a color dipole into two dipoles, of different sizes. Upona further boost, each of these two dipoles may split independently throughthe same process. Thus one understands that QCD evolution is a branchingprocess of dipoles in rapidity, with a random walk in the sizes of the latter. Now boost to rapidity Y , take e.g. the two largest dipole in the Fockstate and track their first common ancestor. According to Ref. [7], the Denoting by χ ( γ ) the eigenvalue of the linearized BK equation about S ∼ − S = r γ and by γ the solution of the equation χ ( γ ) = γ χ (cid:48) ( γ ), one has Q s ( y ) = Q e ¯ αy χ (cid:48) ( γ ) / (¯ αy ) / γ . The relevant scale for the dipole sizes is logarithmic, and the relevant evolutionvariable is the scaled rapidity ¯ αY . Actually, the process is diffusive only if one looksat a fixed impact parameter, but this is what turns out to be relevant here. iffraction printed on November 6, 2018 rapidity y at which the ancestor branches is distributed as p ( y | r, Y ) = c p (cid:20) ¯ αY ¯ αy ( ¯ αY − ¯ αy ) (cid:21) / , with c p = 1¯ γ (cid:112) πχ (cid:48)(cid:48) ( γ ) . (5)The value of ¯ γ depends on which dipoles are picked: In the present case,¯ γ coincides with γ . The formula (5) was actually not established in thepeculiar context of dipole evolution of interest for particle physics, but wasargued to apply to a wide class of branching random walks. We also expectit to be correct (up to the overall numerical factor) for related quantities,such as the rapidity distribution of the common ancestor of all dipoles largerthan some given (large enough) size /Q MV . Equation (5) was found by assuming that the common ancestor was anunusually large object generated around the rapidity ˜ y = Y − y in the evo-lution of the onium [7]. This is exactly the same mechanism as in the caseof the diffraction problem (see Sec. 2.2). Hence, the two problems are inti-mately related: up to the overall normalization, which is determined in thecase of the genealogies but not in the case of diffraction, (1 /σ tot )( dσ diff /dy )corresponds to p ( y | r, Y ).In order to check quantitatively this correspondence between diffractionand ancestry, we have established exact equations for ancestry and havecompared their numerical solutions to those for diffraction. The distribution p > ( y | r, Y ) of the rapidity at which the first commonancestor of all dipoles larger than 1 /Q MV (or, alternatively, of a set of dipolesrandomly picked among the dipoles present in the Fock state at rapidity Y with some probability T ( r )) first split obeys the following equation: ∂ y p > ( y | r, y ) = ¯ α (cid:90) d r (cid:48) π r r (cid:48) ( r − r (cid:48) ) × (cid:2) p > ( y | r (cid:48) , y ) S ( r − r (cid:48) , y ) + S ( r (cid:48) , y ) p > ( y | r − r (cid:48) , y ) − p > ( y | r, y ) (cid:3) , (6)where S solves Eq. (1) with the initial condition S ( r,
0) = 1 − T ( r ). Theinitial condition for p > reads p > ( y | r, y ) = ¯ α (cid:90) d r (cid:48) π r r (cid:48) ( r − r (cid:48) ) [1 − S ( r (cid:48) , y )][1 − S ( r − r (cid:48) , y )] . (7) This holds true if r , ¯ αY and Q MV are such that one is in the scaling region definedat the end of Sec. 2.2. If one wanted to consider the common ancestor of all dipoles larger than 1 /Q MV ,then one would use as an initial condition T ( r,
0) = θ (ln r Q ). In our calculation, T = 1 − S where S is given by the MV model, which is a bit less sharp than the θ function. diffraction printed on November 6, 2018 ) (Y-y a - -
10 1
Ancestry Diffraction Theory
Fig. 2. Numerical solution of the equation for (1 /σ tot )( dσ diff /dy ) (labeled “Diffrac-tion”; from Ref. [4]) and for p > (“Ancestry”; from Ref. [8]) as a function of¯ α ( Y − y ), with the parameters set to ¯ αY = 20 and rQ MV = 4 × − . Thecontinuous line has been generated from the analytical formula (5) with ¯ γ = 1. The numerical solutions of the equations for p > and for (1 /σ tot )( dσ diff /dy )are shown in Fig. 2: Both are in very good agreement with Eq. (4), with anoverall constant of order 1. Trying to understand analytically this constantis one of our current goals. REFERENCES [1] Y.V. Kovchegov, E. Levin, Quantum chromodynamics at high energy (Cam-bridge University Press, 2012), ISBN 978-0521112574[2] Y.V. Kovchegov, E. Levin, Nucl. Phys.
B577 , 221 (2000), hep-ph/9911523 [3] A.H. Mueller, S. Munier, Phys. Rev.
D98 , 034021 (2018), [4] A.H. Mueller, S. Munier, Phys. Rev. Lett. , 082001 (2018), [5] A.H. Mueller, Nucl. Phys.
B415 , 373 (1994)[6] S. Munier, Sci. China Phys. Mech. Astron. , 81001 (2015), [7] B. Derrida, P. Mottishaw, EPL (Europhysics Letters)115