Evaluation of Entanglement Entropy in High Energy Elastic Scattering

Entanglement of the two scattered particles is expected to occur in elastic collisions, even at high energy where they are in competition with inelastic ones. We study how to evaluate quantitatively the corresponding entanglement entropy $S_{\rm EE}.$ For this sake, we regularize the divergences occurring in the formal derivation of $S_{\rm EE}$ using a regularization procedure acting on the two-particle Hilbert space of final states. A quantitative application is performed in proton-proton collisions at collider energies, comparing the results of $S_{\rm EE}$ with two different cut-offs and with a volume-regularization obtained by a prescription fixing the finite two-body Hilbert space volume. A significant entanglement is found which persists even at the highest available energies.


Introduction
Entanglement is a significant phenomenon in quantum theories and has been attracting many interests of scientists in various research areas. In this paper we are interested in the entanglement of scattering particles. How much are the particles entangled due to the scattering interaction? This is a simple and fundamental question. A way to answer it is to evaluate the entanglement entropy of the final state of particles. For this sake, two-body elastic scattering appears to be a case study for entanglement in the final state.
In Ref. [1] the entanglement in momentum Hilbert space in the scattering process has been studied, and the entanglement entropy of the final state of two particles has been calculated in weak coupling perturbation by applying the method developed by Ref. [2] for momentum space entanglement. Ref. [3] also has considered the entanglement in momentum Hilbert space for the elastically scattering particles, but has formulated non-perturbatively the entanglement entropy by the use of S-matrix theory [4,5]. Ref. [3], as a result, has derived an adequate formalism for the entanglement entropy and has suggested an entropy formula of the two-particle final state after the elastic scattering. Additionally the entanglement entropy in this formula includes the influential effect of inelastic processes which are present in the overall set of the possible final states at a given high energy.
However there is a problem of divergence in the entanglement entropy, which is caused by the infinite volume of the momentum Hilbert space in Refs, [1,3]. Indeed the formula in Ref. [3] is written in terms of not only physical observables, i.e., the elastic and total cross sections, but also the cut-off parameter for the infinite volume. One of the subjects in this paper is, starting from Ref. [3], to formulate a finite entanglement entropy formula by identifying the physical origin of the divergence in the entropy formula and using it to appropriately regularize this divergence.
As mentioned above, the entanglement in scattering process is a fundamental issue. For the sake of completion, we quote some works [6][7][8][9][10] related to this issue, while being of a different focus than ours. Ref. [6] has computed the variation of entanglement entropy in an elastic scattering of two interacting scalar particles at one-loop perturbative level. Ref. [7] has studied the entanglement entropy and mutual information in a fermion-fermion scattering. Ref. [8] are concerned with quantum measurement theory and relativistic scattering theory, and has studied the entanglement entropy of an apparatus particle scattered off and a set of system particles. Ref. [9] has suggested another derivation of the momentum space entanglement entropy in the scattering at weak coupling. Ref. [10] has discussed the entanglement entropy in a deep inelastic scattering.
In our study, having performed the regularization and using the obtained formula, it is interesting to evaluate the entanglement entropy for concrete particle scattering. We thus apply our formalism in high energy proton-proton scattering at Tevatron and LHC energies in order to evaluate the entanglement entropy of two-body elastic final states at the highest available energies.
The plan of our study is as follows: In Section 2 we reformulate the entanglement entropy of scattering particles, starting from Ref. [3], in order to determine the physical origin of the divergences one encounters and to properly regularize them. In Section 3, by using the entanglement entropy formulas obtained for different regularization procedures, we evaluate the regularized entanglement entropy in proton-proton scattering. We compare two different cut-off methods with the case of a volume-regularization given by an adequate prescription for the regularized Hilbert space volume without explicit cut-off procedure. Section 4 is devoted to a discussion of the results, an outlook on further directions and a conclusion.

Formulation of entanglement entropy
In this section, we start by recalling the formal derivation (see Ref. [3]) of the entanglement entropy. Then we reformulate the derivation in order to focus on the divergences one encounters. Our goal is to find the physical origin of these divergences, identify the divergent factor and propose the way to obtain a finite formula for the entanglement entropy of the two outgoing particles.

Density matrix
Let us consider elastic scattering of two particles A and B which have initial 3-momentum k and l respectively. Note that in the high energy regime inelastic scattering together with the elastic one have a large contribution. In fact, both types of scattering are related through the unitarity relations. Using the generic entanglement formalism, the statistical entanglement between the particles, A and B, with final 3-momentum respective p and q is expressed in terms of the entanglement entropy S EE as follows: One starts with the overall density matrix ρ in the Hilbert space spanned by two-body final states | p , q ≡ | p A ⊗ | q B . 1 One 1 Although the complete relativistic quantum numbers of a particle state are denoted by momentum and spin (or helicity) as | p , s , we focus only on the momentum Hilbert space in this paper. We will give some comments on the helicity in Section 4. defines a reduced density matrix as ρ A = tr B ρ, where one sums over the states of particle B. Then the entanglement entropy is given by S EE = − tr A ρ A ln ρ A , or equivalently by The overall density matrix reads where S is the S-matrix operator projecting the two-body initial state | k, l k, l | onto twobody final states. In Eq. (2.1), the integration measure is the Lorentz invariant one d 3 p 2E p for on-shell particles and N is a normalization ensuring the condition tr A tr B ρ = 1. Tracing out ρ with respect to the Hilbert space of particle B, we obtain the reduced density matrix, Taking into account energy-momentum conservation and the kinematics of elastic scattering | k, l → | p , q with | k| = | l | = | p | = | q |, one obtains where p , q |S| k, l ≡ δ (4) (P (4) p+q − P (4) k+l ) p , q |s| k, l with the notation P (4) for the center-ofmass energy-momentum vector. The density matrix (2.3) is normalized by its unit trace; where Eqs. (2.4) and (2.5) are expressed using the center-of-mass frame. Note that the δ(0) coming from the energy conservation in Eq. (2.3) cancels the similar one in Eqs. (2.5), leaving an overall δ (3) (0) due to the normalization. We shall discuss later the potential divergence related to this 3-dimensional δ-function.
One finally gets where for further purpose we quote and θ is the center-of-mass scattering angle.

Entanglement entropy
By performing the product of the n density operators of the form (2.6), one obtains the formal expression for the entanglement entropy through the calculation of tr A (ρ A ) n as The overall δ (3) (0) in the integration comes from taking the trace over the A particle's 3-momentum.
Let us now introduce the partial wave expansion of the reduced S-matrix element [4,5], where one used the known summation formula of Legendre polynomials P ℓ , together with the partial wave expansion of the scattering amplitude, and is the two-body S-matrix ℓ th partial wave. It becomes clear from Eq. (2.9) that the powers of δ-functions in Eqs. (2.5), (2.6) and (2.8) give rise to divergences. In order to exhibit these divergences for further regularization, we introduce the divergent full phase-space "volume", which we now prove that it is the key factor determining all divergences we encounter in the derivation of the entanglement entropy.
Inserting the S-matrix element (2.9) into the expression for N ′ in Eqs. (2.5), one obtains (2.14) With this expression one can reexpress Eq. (2.8) as where we reduce the momentum integration to the scattering angle and factorize out a constant prefactor in the last line of (2.15) between parentheses. This prefactor can be expressed in terms of the (infinite) phase-space volume (2.13), using the mathematical identity of δ-functions in spherical coordinates with azimuthal symmetry, In the cos θ → 1 limit we formally obtain for the inverse prefactor in (2.15), All in all we can rewrite Eq. (2.8) as where, using the orthogonality property of Legendre polynomials, P(θ) is of norm where the f ℓ are the partial wave components of the inelastic cross section related to the elastic ones τ ℓ through the unitarity relation, s ℓ s * ℓ = 1 − 2f ℓ , or equivalently Indeed, the standard expressions for physical scattering observables in terms of partial wave components τ ℓ and f ℓ read where the Mandelstam variable t = 2k 2 (cos θ − 1). We finally find the following expression for P(θ); Using Eq. (2.18), we write formally the entanglement entropy as not. Therefore we have to restore a projection of the two-body Hilbert space onto the set of interacting ones. We are thus led to interpret the divergence due to δ-functions and the "volume" V , as due to the infinite number of non-interacting two-body states. Hence an appropriate regularization is required.

Volume-regularization
As we pointed out in the previous subsection, the first term in Eq. (2.25) comes from the part of the two-body Hilbert space of the final states which does not correspond to the interacting states at the given energy. In an ideal cut-off independent way to avoid such non-interacting modes, we are led to note that the volume V could be regularized toṼ so that the first term vanishes, i.e., We call it the volume-regularization assumption. (2.28) However currently we do not know yet which could be an effective regularization of the partial wave components leading to the volume-regularization without modifying the observables.
The volume-regularization can thus be called ideal, since it only depends on measurable observables, and not on any cut-off. In the following sections, we try some concrete regularization methods, in order to obtain an approximation of the ideal determination (2.28) of the entanglement entropy and compare it with the one obtained from Eq. (2.28).
3 Evaluation of the regularized entanglement entropy

Cut-off regularization
We shall make use of the impact parameter b and the corresponding representation of observables, which correspond to a description of high-energy scattering observables, appropriate to our goal. The scattering amplitude (2.11) by the partial wave expansion is rewritten in the impact-parameter representation as where J n is the well-known Bessel function of order n. In other words, τ (b) is defined by this equation.
In actual physics experiments, τ ℓ for large ℓ, i.e., τ (b) for large b (because of bk ∼ ℓ), does not contribute to the scattering amplitude. Therefore we are led to a regularization truncating the large b modes by introducing a cut-off function c(b) satisfying lim b→∞ c(b) = 0, so that the amplitude becomeŝ This prescription gives an approximation of physical Hilbert space. Following this scattering amplitude, the differential elastic cross section becomes and the total, elastic and inelastic cross sections becomê Since the relationσ tot =σ el +σ inel is preserved by the regularization, f (b) is written in . This expression in the impact parameter space corresponds to Eq. (2.22).
Under the cut-off approximation, the volume of the regularized Hilbert spaceṼ is and the entanglement entropy (2.28) iŝ It is important to note thatP(θ) in Eq. (2.27) keeps to be a finite probability distribution verifying positivity and unit norm even under the cut-off approximation, i.e., since Eq. (3.5) leads to (3.10)

Step-function cut-off
In order for a concrete evaluation of the entanglement entropy, we, for instance, employ a step-function as the simplest cut-off function: The scattering amplitude (3.2) becomeŝ This cut-off truncates the modes whose impact parameter is larger than the maximal impact parameter 2Λ. Since the impact parameter b is related with angular momentum ℓ by b = ℓ/k, ℓ has an upper bound L defined by 2Λk ≡ L. Therefore one can also recognize the scattering amplitude asÂ = 16π L ℓ=0 (2ℓ + 1)τ ℓ P ℓ (cos θ). Simultaneously the cut-off regularizes the infinite volume V of the full Hilbert space aŝ (3.13) Then the condition (3.7) determines Λ such that Under the cut-off (3.11) we write the differential cross section (3.3), the total cross section (3.4) and the elastic cross section (3.5) as (3.17)

Gaussian cut-off
By concrete comparison with the step-function cut-off, let us consider a Gaussian cut-off function; corresponding to an impact-parameter width 2Λ. Then the differential cross section (3.3), the total cross section (3.4) and the elastic cross section (3.5) become This condition has the same expression as the one (3.14) in the step function cut-off.

Application: the diffraction peak approximation in proton-proton scattering at high energy
We concentrate on the proton-proton scattering, because we can use the experimental data given by the Tevatron (at √ s = 1800 GeV) and the LHC (at √ s = 7000, 8000, 13000 GeV), of which data are listed in Table 1. Note that the difference betweenp-p and p-p scattering at the Tevatron and LHC energies is not expected to be relevant in our study and thus has been neglected.
Since we must know the differential cross section dσ el dt as a function of t in order to evaluate the entanglement entropy (3.8), here we assume the diffraction peak model, which is described by the following scattering amplitude: where B is the slope parameter. We assume sufficiently high energy, so that s ≈ 4k 2 . The differential elastic cross section is and the elastic cross section is . (3.28)

Step-function cut-off
In terms of Eq. (3.28) we write down the truncated differential cross section (3.15), and compute the truncated cross sections (3.16) and (3.17), Then the condition (3.14) determining Λ becomes By using the data in Table 1, we numerically calculate the cut-off parameter Λ, the truncated cross sections (3.30) and the entanglement entropy (3.8), and the results are shown in Table 2.  Table 2: The cut-off (Λ), the cross sections (σ tot ,σ el ) and the entanglement entropy (Ŝ EE ) in the step-function regularization. The slope B is calculated by Eq.(3.27) from the experimental data of σ tot and σ el .

Gaussian cut-off
The differential cross section (3.19) truncated by the Gaussian cut-off with Eq. (3.28) is written down as (3.32) In the same way we calculate the truncated cross sections (3.20) and (3.21), so that (3.33) The condition (3.14) fixes Λ as Furthermore one can write down the entanglement entropy (3.8) aŝ The numerical evaluation of the cut-off, the total and elastic cross sections and the entanglement entropy are shown in Table 3.

Comparison with volume-regularization
In order to compare the cut-off regularizations with the volume-regularization, let us try to   Evaluating this in terms of the data in Table 1, we show the results in Table 4. The entanglement entropy monotonically increases according as the center-of-mass energy becomes higher.
The truncated cross sections in Table 2 by the step-function cut-off give a closer approximation to the experimental data in Table 1 better than those in Table 3 by the Gaussian cut-off. As shown in Fig. 1, actually the entanglement entropy obtained from the volumeregularization appears to be framed by the step-function one (above) and the Gaussian one (below).

Discussion, conclusion and outlook
In our study, we have evaluated the entanglement entropy S EE for the two particles elastically produced in a high-energy collision. For this sake, we have used a regularization procedure, in order to get rid of the divergences appearing in the formal derivation of S EE . These divergences happen to be related to the infinite "volume" of the full two-particle Hilbert space, be there coming from the interaction or not. It can be regularized by considering the Fig. 1: The entanglement entropy in three different regularizations with respect to the center-of-mass energy.
finite two-particle Hilbert space actually spanned by elastic collisions at a given energy. For the discussion we have first introduced a formulation of a finite entanglement entropyS EE using the formal definition supplemented with a regularized Hilbert space volume, which is defined by projecting out the volume of phase space spanned by the non-interacting final states responsible of the divergence. We then considered two explicit cut-off definitions, one using a step-function and the other with a Gaussian.
Summarizing our results, we found the following: i) The volume-regularized formulation provides an expression of the entanglement entropy in terms of physical observables (2.28); ii) In search of an adequate quantitative cut-off procedure defining the finite physical Hilbert space, we considered the case of proton-proton elastic scattering at the Tevatron and LHC energies. In a diffraction peak approximation as a simple example, we have compared the numerical results for the regularized entanglement entropyŜ EE in two different cut-offs, and we also compared them with the result for the entanglement entropyS EE (see Eq. (3.36)) from the volume-regularization.
iii) Since a cut-off dependence appears for the observables in the formula (2.28) and modifies their contribution to the entanglement entropy, the effect of the cut-off is to replace the observables in Eq. (2.28) with their expressions with the cut-off asŜ EE in Eq. (3.8).
The step-function cut-off appears to give a better approximation of the real observables than the Gaussian one. However, the result for the entanglement entropyS EE boils down to a framing of the volume-regularized entropy by the step-function one (above) and the Gaussian one (below).
iv) The trend of the overall results forS EE clearly demonstrates a non-zero entanglement entropy showing that a non-negligible entanglement is generated in a high-energy elastic collision, even in the presence of a large sector of inelastic reactions. Indeed, the entanglement entropy is different from zero and stays around unity, while increasing slightly with the center-of-mass energy. For instance, in the diffraction peak approximation, the volume-regularization gives Eq. (3.36); S EE = 1 + 2 ln 2 + ln σ el σ tot , which allows one to relate the entanglement entropy simply to the ratio σ el σtot . Higher is the ratio, larger is the entanglement entropy, which seems physically sound. Moreover, it is known that this ratio stays experimentally around 1/4, and thusS EE ∼ 1.
As an outlook, it would be useful to find a better cut-off procedure, which would leave the observables unchanged or only slightly changed by the regularization procedure. For example, an optimization calculation could be introduced to define the cut-off less arbitrarily as those we chose in our present study. Then the full set of experimental observables could be safely introduced in the calculation ofS EE , without cut-off dependence and diffraction peak approximation.
We have considered the entanglement entropy in the momentum Hilbert space. However the relativistic state of a particle also have a quantum number of spin (or helicity). Therefore one can consider the entanglement entropy in the momentum and spin Hilbert space, i.e., In a similar way as what we studied in this paper, such entropy will be formulated by the use of the S-matrix with respect to the helicity, which was studied in some literature [17,18]. Especially in high energy scattering of hadrons, where Pomeron exchange is dominant, the s-channel helicities of the particles are mostly conserved, and thus adding the spin boils down to extend our analysis to elastic scattering of quantum states with given momentum and given s-channel helicity.