Description of recent large-$q$ neutron inclusive scattering data from liquid $^4$He

We report dynamical calculations for large-$q$ structure functions of liquid $^4$He at $T$=1.6 and 2.3 K and compare those with recent MARI data. We extend those calculations far beyond the experimental range $q\le 29\Ain$ in order to study the approach of the response to its asymptotic limit for a system with interactions having a strong short-range repulsion. We find only small deviations from theoretical $1/q$ behavior, valid for smooth $V$. We repeat an extraction by Glyde et al of cumulant coefficients from data. We argue that fits determine the single atom momentum distribution, but express doubt as to the extraction of meaningful Final State Interaction parameters.


I. INTRODUCTION
In the following we discuss different aspects of the response of liquid 4 He. First we report predictions which are compared with most recent data. Next, we compute the response for q < ∼ 300Å −1 in order to study how Final State Interaction (FSI) effects vanish for those large q. In the end we present results of a new model-independent cumulant analysis of data in order to extract the single-atom momentum distribution and interaction parameters.
We address below recent precision data for temperatures below and above the transition temperature T c , which have been taken at the Rutherford ISIS facility by means of the MARI spectrometer. Those by Andersen et al span neutron momentum transfers 3 ≤ q(Å −1 ) ≤ 10 for T =1.42 K and 3 ≤ q(Å −1 ) ≤ 17 for T =2.5 K 1 , while Azuah's measurements covered 10 ≤ q(Å −1 ) ≤ 29 for T =1.6, 2.3 2 . The above expand in scope previous information taken a few years ago at the IPNS facility at Argonne for q ≤ 23.1Å −1 3 .
From inclusive cross sections one extracts the linear response. The latter is a function of two parameters q and ω, which in the scattering experiment are the momentum and energy, transferred from the projectile to the target. For medium and large q those responses contain information on the target, such as the momentum distribution of the constituents and prescribed manifestations of their interaction.
A dynamical calculation of the response requires as input the atom-atom interaction and groundstate information, which for the above q-regime are primarily the single-atom momentum distribution n(p) and the semi-diagonal two-particle density matrix.
Using variations of much the same theory, calculations have been made before for medium-q, as well as for the higher-q Argonne data [4][5][6][7][8] . To our knowledge no ab initio calculations of the MARI data have been performed until now.
The same MARI data have recently been approached in an entirely different fashion with the purpose to determine in a model-independent way the dominant coefficients in the cumulant expansions of the asymptotic and FSI parts of the response 9,10 . Good fits to the data were obtained, but those have little in common with dynamic calculations. The latter use as input n(p), additional groundstate information and V whereas, ideally, from cumulant fits n(p) and properties of V are extracted.
As a major result of the above analysis, Glyde et al report the reconstruction of the single-atom momentum distribution n(p) in good agreement with accurate theoretical predictions 12,13 . However, a less satisfactory feature is the extracted dominant FSI cumulant coefficient which, dependent on the analysis, is reported to be less than 0.65 times the calculated value. One then wonders whether the apparent partial fit may have consequences on the precision of the reconstructed n(p).
The following program emerges from the above observations. In Section II we outline an approach to high-q responses. In Section III we report computations of the high-q measurements using the MARI spectrometer and compare those predictions with the data. In addition we interpret responses computed out to very high q ≤ 300Å −1 . The results enable the study of the approach of the response to its asymptotic limit for systems with a strong short-range repulsion in the interaction between the constituents. In Section IV we present fits for cumulant parameters for T =2.3K and compare those with similar results by Glyde et al [9][10][11] . We discuss the discrepancy between the calculated and the extracted FSI parameters and attribute it to the truncation of the cumulant series. In the conclusion we estimate that both experimental and theoretical studies of the response of liquid 4 He at high q may have reached a degree of sophistication, beyond which there is little prospect to gain new information.

II. DESCRIPTIONS OF THE LINEAR RESPONSE FOR HIGH Q.
Consider for infinitely extended liquid 4 He the response per atom in the form with M the mass of a 4 He atom. ρ q (t) above is the density operator Strictly speaking, the symbol 0|...|0 should stand for a canonical average at given T , but we shall use instead the ground state in conjunction with T -dependent quantities.
In the last line of (1) we introduce the reduced response φ(q, y) with the energy loss ω, replaced by an alternative kinematic variable y = y(q, ω) 14,15 Upon substitution of (2) into (1) one generates two components of the response. In the incoherent part, one tracks the same particle when propagating in the medium, while in the coherent part one transfers momentum and energy to a particle distinct from the struck one.
For q > ∼ 8Å −1 the response is dominated by the incoherent part and the coherent part one can be safely disregarded.
For the description of the large-q response we shall exploit the theory of Gersch, Rodriguez and Smith (GRS) for smooth interactions V which leads to the following expansion for the reduced response in inverse powers of q or of the recoil velocity v q = q/M 14 (We use unitsh = c = 1 causing all quantities to have dimensions of powers ofÅ orÅ −1 ) ... etc.
The functionχ in (4d) resembles an eikonal phase. It differs from it because the integration limits on the line integral over the first component of V are not (−∞, ∞), as is appropriate for on-shell scattering. The finite limits correspond to off-shell scattering described in the coordinate representation. Moreover a second interaction is implicit in (4d). In the following we shall allude to the total expression (4d) as the generalized eikonal phase.
We recall the interpretation of the lowest order terms. For sufficiently large momentum transfer q, an atom with initial momentum p recoils with p ′ = |p + q| ≈ q ≫ p 2 1/2 , which is larger than the average momentum of an atom in the medium and is moreover in excess of any inverse length in the system. The recoiling atom moves therefore too fast to be affected by atom-atom collisions and the response is the asymptotic limit F 0 (y) for q, ω → ∞ at fixed y. Eq. (4b) shows its expression in terms of the single-atom momentum distribution, normalized as dp/(2π) 3 n(p) = 1.
Although the GRS theory is not a perturbation theory in the interaction V , the second term in the series (4a), linear in V , is entirely due to binary collisions (BC) between the hit and any other atom. It accounts for the dominant FSI collecting all contributions ∝ 1/q. This is achieved at the price of introducing the semi-diagonal two-particle density matrix ρ 2 in (4c).
In another publication Gersch and co-workers suggested an alternative representation for the reduced response 16 In Eq. (5a) the response is written as a convolution of its asymptotic limit and a FSI factor R(q, y). It is frequently convenient to use Fourier transformsF i (q, s),R(q, s).... In particular for the first two terms in (5b) one has (cf. (4b)-(4c)) with ρ 1 (s, 0) = ρ 1 (r − sq, r), the single-atom density matrix and ρ = ρ 1 (r, r), the number density.
We shall restrict ourselves below to various descriptions of FSI due to BC, starting from the corresponding cumulant form (5b) and using (6a) 16 whereR 2 (q, s) andΩ 2 (q, s) are the BC approximation to the corresponding quantities defined in (5a) and (5b). ζ 2 above is defined by with g 2 the pair-distribution function.
Eq. (7) is the most general cumulant form in the BC approximation for the FSI phasẽ Ω 2 (q, s) = ln[R 2 (q, s)], and distinguishes through ω 2 between different dynamical approaches.
For instance for smooth interactions V , which would allow for an expansion of the exponential in (7), comparison of (7), (5b) and (4d) shows ω 2 to be the generalized eikonal phase ω 2,V (q, r, s) =χ(q, r, s) For interactions with a strong short-range repulsion, the line integral over V in the (offshell) phase (4d) which enters the dominant BC FSI contributionF 1 (s), Eq. (6b), may produce large and even divergent integrals. The standard method to tackle those difficulties is by partial summation of selected higher order terms which amounts to replacing the ′ bare ′ V by a q-dependent effective interaction V → V ef f (q) =t(q), the latter being the off-shell t-matrix, in turn generated by V . Moreover the propagation in between collisions is described in the eikonal approximation 6,17 .
In an alternative regularization for an atom-atom interaction with a strong shortrange repulsion, one replaces the generalized eikonal phase (4d) by a semi-classical approximation 18,19 .
Whereas ω 2,V is strictly proportional to 1/q, this is no more the case for ω 2,t after the replacement V → V ef f (q). The above manifestly introduces q-dependence in coefficients of the GRS series (4a), (5b) and in particular in the BC approximation. Taking the latter in the cumulant form (7) adds to the blurring of the original 1/q dependence. This raises the question how the response approaches its asymptotic limit.
We start with a theoretical analysis of the first cause of additional q-dependence and focus on 4 He-4 He scattering for high lab momenta q. The latter is of distinct diffractive nature, typical for interactions with a strong, short-range repulsion. For those, the dominant imaginary part of the on-shell scattering amplitude t(q) ≈ Imf (q) ∝ iqσ tot q , where the total 4 He-4 He cross section σ q varies much slower than q itself 5 .
Without entering into details, we state that the off-shellt = V ef f in ω 2 , Eq. (7), can approximately be related to the on-shell scattering amplitude for elastic scattering. (See ref. 20 for a more extensive treatment of the parallel discussion for atomic nuclei). It can then be shown that the rigorous proportionality of the dominant BC FSI phaseΩ 2,V ∝ 1/q for a smooth, bare V still holds approximately forΩ 2,t .
Additional q-dependence is due to the use of the cumulant representation (7) but it will be small to the extent that FSI are. In conclusion, the reduced response described by (5) and (7) is expected to approximately preserve the 1/q signature of the dominant binary collision contribution. We shall return below to a numerical confirmation.

III. DYNAMICAL CALCULATIONS OF SELECTED MARI 4 HE DATA
We first mention and discuss the input elements which suffice for the BC approximation in any of the forms described in Section II.
a) The atom-atom interaction V Aziz 21 .
b) The single-atom momentum distribution n(p, T ) g wd (r) is an auxiliary function related to what in HNC formalism is called a form factor 25 .
The function A(r, s) formally adds all so-called Abe diagrams and is approximated in (13) by the 4-body Abe diagram A 4 (r, s), using in addition a scaling parameter ξ 8,26 .
Far less sophisticated and simpler is the GRS approximation 14 which interpolates ζ 2 between s = 0 and the Hartree limit for large s 14 .
Both options have drawbacks and fail for instance the extended unitarity test which can be written as Using a typical pair distribution function g(r), Ξ(s) above computed with ζ GRS  (16) is intrinsic in the GRS approximation (14), no matter what g(r) is used.
Another important constraint is the fact that the diagonal two-body density matrix should coincide with the pair distribution function: ζ 2 (r, 0) = g(r) (8b). While the GRS approximation fulfils that condition by construction, a full evaluation of the Abe terms is necessary in the HNC formalism. Demanding the boundary value condition to be fulfilled in the mean, one determines the so-called scaling ξ by minimizing the following quantity A particular choice of ζ 2 presumably matters for medium q, but for increasing q > ∼ 20Å −1 FSI contributions decrease in importance relative to the asymptotic response. A few-percent spread, due to uncertainty in the choice of ζ 2 , in anyhow small FSI terms will go unnoticed.
We thus opted for the expression (14) which is numerically much easier to handle than (13). d) Finally, for a comparison of actual data with predictions the latter have to be folded into the experimental resolution function (ER) E(q, y; T ) of the instrument. The E(q, y; T ) corresponding to the q ≥ 20Å −1 MARI data are given in Azuah's thesis 2 and have been fitted to the sum of two off-center gaussians. No ER, pertinent to lower q's were available to us, thus precluding an analysis for q ≤ 20Å −1 .
Until this point we did not specify the T -dependence of the theoretical responses. In fact one ought to employ quantities computed for given T . In fact there exist experimental data 28 and also PIMC studies 12,24 on the T dependence of the pair-distribution function g(r, T ). However, in view of the above arguments we shall use the one for T = 0. By the same token ζ 2 , Eq. (14), and consequently FSI effects will be independent of T . This leaves the single-particle density matrices, or equivalently the momentum distributions as the only T − dependent quantities in the present analysis. We took n 0 (T = 1.6K) = 0.087 and ρ N O 1 (0, s; T = 1.6K) = ρ 1 (0, s; T = 2.3K) from calculations for T = 1.54 and 2.5K 12,24 . The expression for the predicted response is therefore which in order to enable a comparison with data, has to be folded into ER For future reference we emphasize here that the FSI factor R is from (7) and (8) seen to be independent of the single particle density matrix ρ 1 (s, 0)/ρ. In particular for all but pure hard-core interactions lim q→∞R (q, s) = 1.
We thus computed the reduced response φ E (q, y; T ), for the q = 21, 23, 25, 29Å −1 sample out of the MARI data. In view of the steady decrease of FSI, this q-range and steps seems to be sufficient for our study. We emphasize in particular the case q = 23Å −1 , considered because it is the largest q in the older Argonne data sets 3 and for it we shall compare our results with others.
We start with a comparison of our predictions for T = 2.3K and the corresponding data 2 (Figs. 1 a-d). The overall agreement is very good. One notices that, whereas the central value for the theoretical response hardly changes for 21 ≤ q(Å −1 ) ≤ 29, the data for the same, folded in the ER, φ E (q, 0) show q-dependence present in E(q, y).
The agreement for T = 1.6K (see Figs. 2 a-d) is slightly worse. The slight staggering in the central region for q = 21Å −1 is probably of instrumental origin, but contrary to the T = 2.3K case, differences in E(q, y) for q = 21, 29Å −1 do not explain the small discrepancies in their central regions. We recall that exactly the same input is used as for T = 2.3K and that the only extra parameter is the condensate fraction n 0 (T = 1.6K).
We now reach our second topic. In spite of the fact that no data exist for q ≥ 29Å −1 , we have extended calculations up to q = 300Å −1 . The purpose of the exercise is to obtain theoretical information on the approach of the response to its asymptotic limit.
In Fig. 3a we present the part φ even (q, y, T = 2.3K) of the predicted response, computed with (7), (10) which is even in y. Even in the wings out to y ≈ 3.5Å −1 those coincide within 1% amongst themselves and with the asymptotic response F 0 (y), Eq. (4b). The above appears hardly changed, when predictions for 20 ≤ q(Å −1 ) ≤ 29 are included: only in the immediate neighborhood of y = 0, is there a < ∼ 2.5% difference.
In Fig. 3b we show qφ odd (q, y, T = 2.3K) which contains the part of the response which is odd in y. Having included the q signature of the dominant FS some residual q dependence is apparent in the extrema as well as in the wings, but the conclusion is clear: Neither the strong short-range repulsion in the atom-atom interaction which forces the use of V ef f (q) =t(q), nor the effect of the cumulant representation, much changes the 1/q signature of the dominant FSI term in the GRS series (4a) for smooth V . The above agrees with our arguments in Section II and with our previous results 5 .
All reported predictions are based on the use of the t-matrix, i.e. on (10). In Section II we also mentioned a semi-classical approximation (11) for FSI and found that, except for small s, there are considerable differences between the BC phases, calculated by means of (11) and (10).
Ultimately excellent agreement is obtained between the corresponding responses, computed with (5) and (7). Clearly both the t-matrix and the semi-classical method accurately describe the binary collision phase in the salient region just inside the classically forbidden region. Contributions from deeper penetration distances are strongly suppressed.
We conclude this Section, by a discussion and comparison of predictions for q = 23Å −1 by other authors. Since the various studies refer to different T and data have been taken at different instruments, the natural quantity to compare is the FSI factor R(q, y) assumed to be T -independent.  (7) with ω 2 → ω 2,t and ζ 2 → g 2 , the pair distribution function. In his Hard Core Perturbation Theory he disregarded the second part of the total phase χ, Eq. (4d), which is only permissible for a pure hard-core interaction. However, Silver actually constructed the off-shell t-matrix in (10), corresponding to the first part in (4d) from JWKB partial wave phase shifts for a realistic V , which in addition to strong short-range repulsion also included attractive components. Nevertheless, he neglected the second component in (4) which does not vanish when an attraction is present. We conclude with a path-integral method by Carraro and Koonin, who computed high-q FSI using a fixed scattering approximation for the entire system with a large, finite number of atoms 7 . The method requires the parallel calculation of the ground state wave function in order to construct the N-body density matrix, diagonal except for one particle, with N the number of atoms in the sample and which averages the response for fixed scatterers.
Results for R(q, y) for all cases discussed are assembled in Fig. 4 and show occasionally substantial differences. However, those are considerably smoothened by the single particle momentum distribution (Eq. (5)) or density matrix in (6), and ultimately produce quite similar responses 3,8 .

IV. CUMULANT EXPANSIONS OF THE RESPONSE.
We consider below a method which has extensively been applied in the past 29 before the rediscovery of the 1/q GRS expansion of the response (4a) 14 . Recently it has been brought to the fore again in an attempt to parameterize data without the intervention of a theory.
The method uses cumulant expansions of the Fourier transforms of the separate asymptotic and FSI parts of the response (5b), with coefficient functions related to moments of the response 9φ Using (5b) the various coefficient functions are related byμ n (q) =ᾱ n +β n (q) 30 . Data for the response φ exp E are then compared with the parameterization (20) where, as in (18) the right-hand side in (20a) includes ER. In principle, no a priori knowledge, or even meaning of the cumulant coefficient functions µ n (q) is necessary for a search.
However, it is natural to use motivated initial values such as calculated ones.
Consider the normal fluid, in which case the q-independent cumulant coefficientsᾱ n originating in the asymptotic part ofφ(q, s) can simply be expressed in terms of averages of even powers of the momentum of an atom, e.g.
The q-dependent coefficientsβ n (q) relate to the FSI factorsR(q, s),Ω(q, s) in (5b) and may be written as 9β For convenience we use q * , the momentum transfer parameter q in units of 10Å −1 . Eq. (23) displays q−dependence and defines coefficients z n,2m , which have the same dimensions and may be expressed in the same units asβ n . The above are operators for dynamical variables of the system, averaged over diagonal l-body density matrices and their derivatives of order l ≤ n. For the lowest order cumulant coefficient functions one has Contrary to the GRS series in 1/q, with coefficients depending on nearly-diagonal n-particle density matrices ρ n (r 1 − sq, r 2 , ...r n ; r 1 , ...r n ), the moment approach underlying the cumulant expansion does not produce a systematic q-dependence of the coefficient functions. For instance, allβ n (q) with odd n contain FSI contributions ∝ 1/q. Eqs. (24a) and (24d) illustrate this for the dominant FSI coefficient functionsβ n (q). Thusβ 3 (q) ∝ 1/q draws exclusively on the BC contribution (7), while the two components ofβ 5 are proportional to 1/q and (1/q) 3 due to binary, respectively higher order collision contributions, etc. It is the expansion of the semi-diagonal ρ 2 (r 1 − sq, r j ; r 1 , r j ) in s which produces an infinite number of contributionsβ 2n+1 (q), all of which have parts ∝ 1/q with coefficients depending on diagonal densities (cf. (24b)) and their derivatives.
We start with the threshold behavior of the computed BC FSI phaseΩ 2 (q, s), Eqs.
(7), (10) and in particular of its imaginary part which, according to (24a), produces the dominant FSI parameterβ 3 (q). We checked that, within a few percent q ImΩ 2 (q, s) is out to s ≈ 0.8Å independent of q. In particular we could extract the theoretical threshold value z 3 = q * βΩ2 3 (q) = 6q * lim s→0 [ImΩ 2 (q, s)/s 3 ] = 0.555Å −3 which over the entire range q ≤ 300Å −1 is to better than 1% independent of q. For the above reason one cannot determine the next order odd-n coefficients z 5 with reasonable precision.
Regarding parameters of even order in n one finds from ReΩ 2 (q, s), Eq. (24c) for the leading coefficient z 42 = q * 2β 4 (q) = −2.26Å −4 . It appears that for even n, the next-to-leading We return to Eq. (24b) which seems to provide an independent way to calculateβ 3 (q).
Using the same g(r) as in ζ GRS 2 (14) we compute q * β∇ 2 V 3 (q) = 0.56Å −3 . The agreement between q * β∇ 2 V 3 and q * βΩ 3 is very good, especially in view of the sensitivity of q * β∇ 2 V 3 on the precise shape of g(r) where the Laplacian of V is large. The extraordinary stability of the extracted q * βΩ 3 confirms the numerical consistency of the calculation.
As has been mentioned before, all FSI functions have been assumed to be T -independent and we have used values for T = 0. In order to estimate the influence of the temperature we have also calculated q * β∇ 2 V 3 using a g(r) obtained with PIMC at T = 2.8K 12 . The result q * β∇ 2 V 3 (T = 2.8K) = 0.47Å −3 confirms its sensitivity on the precise shape of g(r).
We now report several results for cumulant fits (21), obtained with the CERN MINUIT code and applied to the ten T = 2.3K data sets in the range 21 ≤ q(Å −1 ) ≤ 29 from Azuah's thesis 2 . We first note that the integrated strengths of the data there appear for all q to be approximately 1.4% in excess of the exact result 1. By construction that demand is exactly fulfilled by a cumulant expansion (20a) in, no matter what approximation. We also considered data cut at y ≈ 3.0 − 3.4Å −1 where statistical noise in anyhow very small responses may cause those to have negative values. The latter appear to hardly affect the extracted parameters.
Next, the above source of information does not contain numerical data and ER for the lower q data . As a consequence we had to limit ourselves to a small data base which is bound to influence the FSI parameter functions which increases with decreasing q.
Our results for T =2.3K are assembled in Table I. We entered in column 2 theoretical values for the parameters, calculated as indicated , or set to 0 when impossible to evaluate reliably. Notice that the negativez 6 , obtained from a limited s−range, will generate a non boundR(q, s), barring a convergent Fourier transform ofR(q, s). Actually we did not consider z 6 in the fits and restrict ourselves to a maximum of seven parameters as is discussed below.
Column 3 is the result of a 7-parameter fit for cumulant coefficients functions with theoretical q-dependence and encompasses therefore all ten data sets in our q-range. Columns 4 and 5 are 6-, respectively 5-parameter fits like in column 3, when first q * β 3 (q) alone and then also q * 2β 4 (q) are held at their theoretical values.
We now reach the extraction ofμ n (q) for each q and the determination, as opposed to the above assumption, of their q-dependence. This appeared to be impossible for the limited data set available to us. However, Glyde et al had access to far more data and we enter their results in the last column taken from Table I of Ref. 10 . As the reported z 4 is zero, a positive value of z 6 is required to give a convergent Fourier transform ofR(q, s).
We hinted in the Introduction, that the misfit of FSI parameters may reflect on the meaning of the parameters from which one builds the momentum distribution n(p). In fact, the above was the main reason why we repeated the fit of the cumulant expansion coefficients. Table I shows even smaller values for z 3 = q * β 3 (q) than reported in 9,10 . Although the fitted z 3 appears strongly correlated to other parameters, we remained worried by the misfit. The above observation led us to ponder about the very possibility to obtain reliable values for the cumulant coefficients for the response.
In fact, we found the results of column 5 in Table I most telling. Fixing the dominant FSI parameters z 3,4 at their theoretical values, one expectsᾱ 4 ,ᾱ 6 to settle close to their starting values. This appears not to be the case. In fact, the above happens to produce the worst agreement between the reconstructed and computed n(p) to be presented below.
To understand the above, we turn again to the calculatedΩ(q, s) which we recall, produces very good fits to the data. Although not called for in calculations using a dynamic theory, a comparison of (5b) and (20c) shows that the expansion ofΩ(q, s) in s produces in principle the cumulant coefficients and in particular the FSI parametersβ n (q). Examples are (24a) and (24c).
The complete cumulant series is of course equivalent to the exactΩ, but a truncation at some n obviously reproduces behavior up to some relatively low s. The crucial question is to what order one should go and the answer clearly depend on the effective s range of each of the component factors. Those are according to (5b), (18) and (21) the single-atom density distribution ρ 1 (0, s)/ρ, the FSI interaction factorR(q, s) (orΩ(q, s)) and the Fourier transformẼ(q, s) of the ER function.
In the tests to be reported we compare not data, but theoretical values of the input factors ρ 1 (0, s),R,Ω and their cumulant approximationsR cu,n ,Ω cu,n to order n. Fig. 5a shows over the relevant s-range reasonable adequacy of the cumulant expansion for ρ 1 (0, s)/ρ to order n = 6. Next, Figs. 5b,c give Re and Im parts ofΩ(q, s), respectivelyR(q, s) q = 23Å does not prevent rapid deterioration ofR(q, s),Ω(q, s) from 1.2Å −1 on. As mentioned before the estimated z 6 gives rise to a non-boundR(q, s). Since terms in the cumulant expansions (24) have alternating signs of powers in s, it requires knowledge of higher order β n in order to establish the proper order of the expansion for given s range.
We return to the interpretation of our results. The cited parameters for finite−n expansions indeed reproduce very good fits to data, but have no resemblance to the values of the parameters from actual (in case theoretical) FSI factorsR(q, s),Ω(q, s). The failure is due to the insufficiency of low order cumulant representationsR cu,n ,Ω cu,n and will only disappear if for sufficiently high n, FSI functions will coincide with their cumulant expansion out to the relevant medium s.
The above is to our opinion the source of the discrepancy between calculated and extracted FSI parameters. In spite of correlations between fitted parameters, there is sufficient meaning to the extracted FSI to justify the above conclusion. Were it not for the overwhelming role of the asymptotic part and its representation by a truncated cumulant series, one could not trust the extractedᾱ n .
The sensitivity of the fitting procedure can be judged from Fig. 6, where we reconstruct single-atom momentum distributions from the various α n . Dense dots, short dashes and spaced dots are results using columns 3-5 from Table I. Long dashes are from the last column, i.e. the fit of Glyde et al. All fits are roughly of comparable quality when judged as to their capacity to reproduce the computed distribution n(p, T = 2.5K) 12 , marked by circles. We recall that the theoretical n(p) has been used to compute α n in column 2 which produces the drawn curve in Fig. 6 . The moderate misfit for large s of the finite order cumulant expansion for ρ 1 (0, s) explains deviations from the underlying n(p).

V. SUMMARY AND CONCLUSIONS.
We addressed above three topics regarding the response of liquid 4 He, retrieved from inclusive scattering of neutrons. Using dynamics we first made predictions for the recent MARI data, taken at temperatures both below and above T c . We analysed the range of momentum transfers 21 ≤ q(Å −1 ) ≤ 29, for which we had available data and experimental resolution and obtained good agreement with experiment.
A second topic was the approach of the response to its asymptotic limit in q for fixed scaling variable y. For smooth interactions between constituents, that approach is rigorously ∝ 1/q, but the same is not guaranteed when a strong short-range repulsion is present in the atom-atom interaction V . We investigated theoretically the response for q ≤ 300Å −1 for the actual V with its short range repulsion and found re-confirmed our findings from a few years ago based on medium-q data: Final State Interaction contributions, over and above the asymptotic part still decrease approximately as 1/q.
Regarding the q-values for which the response has been measured we repeat what was already evident from the older data 3 : there is no additional information to be retrieved by increasing q by less than 20-30% and no new information at all at very high q.
Our last topic was a re-fit of the expansion coefficient functions, which parametrize data.
A previous analysis led to a single atom momentum distribution in good agreement with computed n(p, T ), but did not produce the main FSI coefficient function. Its influence in the large-q region is too marginal, to be extracted from the data with present accuracies.
This does not change our judgement that little can be added to understanding the data on the response of liquid 4 He at high momentum.
In spite of minor discrepancies, a summary of the treatment of the first two topics is definitely positive. Presumably for no atomic, molecular, nuclear or sub-nuclear system for which the response has been measured, has one reached asymptotia as clearly and as well understood as for liquid 4 He. Of course ′ asymptotia ′ is not simply the mathematical limit q → ∞ for fixed y. Increasing q requires increasing beam energy, ultimately beyond the ionization energy ≈ 39 eV. A description of the response then requires the inclusion of additional electronic degrees of freedom to translational ones.
Finally we venture an outlook for future explorations of inclusive scattering which almost certainly implies extension of experiments to larger momentum transfers q. Our judgement is based on Eqs. (5b) and (7). In that representation for the response, the FSI factorR(q, s) does not depend on the single particle density ρ 1 (0, s) and in particular not on the condensate fraction for T ≤ T c . Together with (19) this implies the universality of the asymptotic limit     Table I, including the one from theoretical starting values.
The curves correspond for increasing column number of the column in Table I   Cumulant coefficient functions from data at T =2.3K. The second column gives theoretical values. The third column gives 7-parameter fits with prescribed q * behavior from (renormalized) data. Column 4 give fits when q * β 3 (q) is fixed at its starting values. Column 5 are fits if in addition q * 2 β 4 (q) is fixed. In the last column are results by Glyde et al, who fitted µ n (q) from a large set of data for different q.