Entanglement in Reggeized Scattering using AdS/CFT

The eikonalized parton-parton scattering amplitude at large $\sqrt{s}$ and large impact parameter, is dominated by the exchange of a hyperbolic surface in walled AdS$_5$. We construct the entangled density matrix following from the exchanged surface, and show that the ensuing entanglement entropy is fixed by the intercept of the Reggeized trajectory. The largest eigenvalues of the entangled density matrix obey a cascade equation in rapidity, reminiscent of non-linear QCD evolution. We suggest that they describe the probability distributions of wee-quanta at low-x that maybe measurable at the future eIC.


I. INTRODUCTION
Entanglement in quantum mechanics is still one of the most subtle concept that permeates our description of the quantum world.The canonical example is the entangled Einstein-Podolsky-Rosen pair whereby the measurement of one of the state in the pair forces the state of the partner.This conundrum has recently been revisited in many areas of physics, ranging from low-dimensional quantum critical systems [1,2] to wormholes in gravity [3,4].
Recently the holographic principle was used to derive the entanglement of boundary conformal field theories in terms of pertinent area of finite dimensional surfaces in bulk [5], reviving the idea that the entanglement entropy bears similarities with the Bekenstein entropy for black holes [6].These relationships are important in our understanding of the concept of information storage or loss whether in quantum mechanics or around a black hole.
Current high multiplicity pp collisions at collider energies display rapid collectivization [7], an indication of early entropy deposition and thermalization.This leads us to ask about the origin of this fast scrambling of information in the prompt phase of the process.One of the purpose of this letter is to show that parton-parton scattering at large √ s is highly entangled, with an entanglement entropy matching the thermodynamical entropy initially discussed in [8].Entanglement entropies in the context of perturbative QCD evolution were recently discussed in [9,10].
Below we briefly review the Reggeization of the partonparton scattering at large √ s, through the exchange of a minimal surface using the AdS/CFT correspondence.The transverse fluctuations on the surface are shown to be entangled with an entropy that equals that of critical conformal field theories in lower dimension.The largest eigenvalues of the entangled density matrix describe the probability distributions of wee-quanta at low-x.
In the eikonal limit, the probe and target partons support Wilson lines running along the light cone and sourcing gluon fields [19].Specifically, the parton-parton scattering amplitude in the large number of colors N c limit, is dominated by the octet contribution (t = −q 2 ) with the connected Wilson loop correlator traced over colors, and subject to the normalization W ii = 1.The Wilson lines W ij are evaluated along C 1,2 on the light cone at fixed separation b = |b| as illustrated in Fig. 1 following [15,16,19].The averaging in (2) is over the Yang-Mills gauge fields.
In the double limit of large N c and large gauge coupling λ = g 2 N c , this correlator can be calculated using holography.In leading order, the correlator involves a closed surface exchange in a slice of AdS 5 with a metric for 0 ≤ z ≤ z 0 with D ⊥ = 3.The AdS/CFT correspondence allows for the evaluation of (2) through the minimal surface attached to C 1,2 [20] WW ∼ e − 1 2πα Amin ≡ e −Smin The minimal area will be sought by analytically continuing the extremal surface from Minkowski to Euclidean signature as we now show.
For large impact parameter b, the extremal surface is composed of two straight strips joined by a surface at z = z 0 as shown in Fig. 1.The two straight strips contribute about 1 with the normalization W ii = 1.To assess the joining surface at z = z 0 , we use the Polyakov action in the conformal gauge with mostly positive Minkowski signature with ) as the metric is nearly flat.The boundary conditions at z = 0 transfer almost unchanged to z = z 0 .We will assume that these boundaries are straight lines with rapidity angles χ/2 and −χ/2 for σ = 0, 1 respectively, with χ = lns.These boundary conditions are effectively similar to the boundary conditions on a D0 brane scattering set up with at σ = 0 (and similarly at σ = 1) For large b, the extremal solution to the equation of motion stemming from the Polyakov action, i.e. ∂ 2 a x = 0, that satisfies (6) at z = z 0 , is the hyperbolic surface The induced world-sheet metric associated to ( 7) is conformal, ds 2 W = b 2 cosh 2 (χτ )(−dτ 2 + dσ 2 ) in conformity with the gauge choice in (5).It is free of the wormhole discussed in [4].Using the analytical continuation τ → iτ , we have ds 2 W → b 2 cos 2 (χτ )(dτ 2 +dσ 2 ) which describes the conformal world-sheet of an instanton with period T P = 2π/χ and finite action A more thorough characterization of this instanton and its relation to the Schwinger mechanism on the worldsheet can be found in [17].Following the AdS/CFT correspondence, we insert (8) into ( 5) and define β = bT P , to obtain (4) as We have included the 1-loop quantum correction restricted to the instanton strip and ignored the warping in the holographic direction.
The instanton period or better tunneling time β plays the role of an inverse temperature for D ⊥ massless bosonic modes confined to a box of length b as shown in (10).As a result, the exchanged surface carries also a finite thermodynamical entropy in agreement with the observation in [8].Inserting ( 9) in (1) and carrying the transverse Fourier transform yields the Reggeized scattering amplitude T ∼ is α P (t) with the trajectory (R = z 0 ) also in agreement with [17].The thermal entropy (11) per unit rapidity χ is fixed by the Reggeized intercept α P (0).We now show that ( 11) is also a measure of quantum entanglement or entanglement entropy.

III. ENTANGLEMENT ENTROPY
The transverse fluctuations follow from the Polyakov action (5) with x → x ⊥ with fixed end-points.The transverse string can be thought as a collection of N string bits connected by identical springs and discretized as follows [21,22] 1 where the summation over i = 1, ..., D ⊥ is subsumed and √ α = 1 2 for simplicity.( 13) describes N coupled harmonic oscillators in D ⊥ dimensions, with a transverse Hamiltonian where K is a banded matrix with positive eigenvalues.
Ignoring warping (large b), the ground state wave function of ( 14) is where Ω is the square root of K (see below).The transverse string density matrix is Ψ To quantify the entanglement of the string bits in transverse space, we follow Srednicki [6] and define the entanglement density where we used the notation [x] → [x; x] with dim x = n and dim x = N − n.The positive eigenvalues of (17) follow by diagonalization The entanglement entropy is the Von-Neumann entropy for the transverse string IV. NUMERICAL ANALYSIS For general n the p l can only be obtained numerically.For that, we fix the end points through the boundary condition x N +1 = x 1 = 0. Without loss of generality, we set N = 2p.Since K is real symmetric, it diagonalizes by ortogonal rotation with K = U † K D U and Ω = U † √ K D U .The eigenvalues and eigen-vectors of K are respectively with k labeling the eigenvalues and n labeling the entries, k, n = 1, 2, ..2p.The matrices U, Ω can be found in explicit form, with U kn = α n k and with C an overall unimportant constant.Given Ω mn , the derivation of the entanglement entropy is essentially an exercise in the diagonalization of nested Gaussians as in [6] to which we refer for details.With the above in mind, the entanglement entropy between the subsystem with size [n] and size [N − n] can be calculated by splitting the matrix Ω as and defining the squared matrix β through with the eigenvalue spectrum βN,n v N,n,i = χ N,n,i v N,n,i .
For each transverse dimension 1, ..., D ⊥ , the eigenvalues of the entangled density matrix (18) are [6] p with The entanglement entropy ( 19) is then FIG.3: Largest eigenvalue ξ300,n,1 versus n in the full range 0 < n < 300, with the mid-chain periodicity manifest: dots are the numerical results and the solid line is a best fit (29).The insert shows ξ400,n,1 versus ln(n) in the range 3 ≤ n ≤ 10.In Fig. 2 we show our results for S E (n, N = 2n) versus ln(n) per D ⊥ , in the range 50 ≤ n ≤ 250, Because of the mid-point symmetry of the chain, the last equation follows.We have checked that for the string with periodic boundary conditions, i.e. x N +1 = x 1 , ( 27) is also recovered with 1/6 → 1/3 (2 boundary points).( 27) is identical to the thermodynamical entropy (11) for 1 n N with the identification of the rapidity χ = ln(n).It is consistent with results from conformal field theories and spin chains with central charge D ⊥ [1] (1 boundary point).
In terms of ( 16), the squared transverse size of the string grows logarithmically as R 2 ⊥ (N ) = D ⊥ α ln N (units restored) [17,21].From (27) it follows that the entropy growth is proportional to R 2 ⊥ (n) (n/N 1) which is reminiscent of the Bekenstein entropy.
V. WEE-QUANTA The entanglement entropy is dominated by the largest two eigenvalues ξ N,n,i=1,2 with ξ N,n,1 = 0.155 ln(n) as shown in Fig. 3, which reproduces the entropy (27) for small ln(n).The eigenvalue distribution decreases exponentially, i.e. ξ 2n,n,i>2 ≈ e −a|i| , as shown in Fig. 4 for 2n = 300 and a = 3.65.The dependence of ξ 300,n,1 on n is shown by the dots on the semi-circle-like in Fig. 3, with the best fit for N = 300 and ∆ = 0.067.Using (29) in (24) gives the dominant eigenvalues or probabilities (n N = 300) with 0.963 → 1 for D ⊥ = 1.For large ln(n), the numerical analysis is more intensive, but we expect ∆ = 0.067 → 1 6 as required by the entropy constraint (27).In general, we have D ⊥ independent copies of string chains, each with l 1,...,D ⊥ , n 1,...,D ⊥ = n.For fixed l, n, (30) is replaced by which satisfies a cascade equation in rapidity If we identify x = 1 n as the fraction of longitudinal momentum carried by each of the transverse n-string bits, then l ≈ D ⊥ /x ∆ is the mean number of wee-quanta at low-x.In terms of l , (31) is a negative binomial distribution.For D ⊥ = 1 and modulo ∆, (31) is similar to the probability to find l-wee-dipoles inside a hadron at rapidity χ = ln(n) following from a model of non-linear QCD evolution [10].

VI. CONCLUSIONS
In walled AdS 5 , parton-parton scattering at large √ s is dominated by the exchange of a hyperbolic surface that Reggeizes [15][16][17].The surface is spatially entangled with an entropy that is fixed by the Reggeized intercept.The entanglement entropy coincides with the one in critical 2-dimensional conformal field theories and spin chains with a central charge D ⊥ [1,2].It maybe at the origin of the fast scrambling of information and collectivization in pp collisions as recently reported by the CMS collaboration [7].The largest eigenvalues of the entangled density matrix obey a cascade equation in rapidity.They describe the stringy probability distributions of wee-quanta at low-x, that maybe measurable in deep-inelastic scattering at the future eIC.