Deep Minimum and a Vortex for Positronium Formation in Low-Energy Positron-Helium Collisions

: We ﬁnd a zero in the positronium formation scattering amplitude and a deep minimum in the logarithm of the corresponding differential cross section for positron–helium collisions for an energy just above the positronium formation threshold. Corresponding to the zero, there is a vortex in the extended velocity ﬁeld that is associated with this amplitude when one treats both the magnitude of the momentum of the incident positron and the angle of the scattered positronium as independent variables. Using the complex Kohn variational method, we determine accurately two-channel K-matrices for positron–helium collisions in the Ore gap. We ﬁt these K-matrices using both polynomials and the Watanabe and Greene’s multichannel effective range theory taking into account explicitly the polarization potential in the Ps-He + channel. Using the ﬁtted K-matrices we determine the extended velocity ﬁeld and show that it rotates anticlockwise around the zero in the positronium formation scattering amplitude. We ﬁnd that there is a valley in the logarithm of the positronium formation differential cross section that includes the deep minimum and also a minimum in the forward direction.


Introduction
Positronium (Ps) formation, a rearrangement process for positron collisions, is of interest in both theoretical and experimental studies [1][2][3][4][5][6]. Recently, we obtained two zeros in the Ps-formation scattering amplitude, f Ps , for positron-hydrogen collisions in the Ore gap [7]. We used the Kohn and inverse Kohn variational methods to compute the two-channel K-matrices, and from these we computed f Ps , the Ps-formation differential cross section (DCS) and the extended velocity field v ext that is associated with f Ps . (The extended velocity field is defined in reference [7].) Corresponding to the zeros in f Ps , there are deep minima in the logarithm of the Ps-formation DCS and vortices in v ext that are associated with f Ps [7].
Positron-helium collisions are more accessible experimentally than positron-hydrogen collisions. The absolute near-forward direction Ps-formation DCS has recently been measured for positron collisions from helium for an incident positron energy E i of 24.7 eV to 138 eV [4,6]. In this paper, we report a zero in f Ps for positron-helium collisions just above the Ps-formation threshold, a corresponding deep minimum in the logarithm of the Ps-formation DCS, a vortex in v ext that is associated with f Ps , and also a minimum in the forward direction Ps-formation DCS [8,9]. We use the complex Kohn variational method to compute two-channel K-matrices for s-, p-, dand f -wave positron-helium collisions in the Ore gap for a fine grid in k, where k is the magnitude of the momentum of the incoming positron k. We optimize the non-linear parameters in the short-range part of the trial wave function in the vicinity of the zero in f Ps to obtain accurate K-matrices expression for the extended velocity field v ext that is associated with f Ps . We present our results in Section 3. We show in Section 3.1 the deep minimum in the logarithm of the Ps-formation DCS and give the position of the zero in f Ps . In Section 3.2 we show the minimum in the forward-direction Ps-formation DCS. In both Sections 3.1 and 3.2, we discuss the convergence of results with respect to the maximum partial wave, max , which we use in the sum in the equation of f Ps . We show in Section 3.3 that the unit vector of the extended velocity fieldv ext = v ext /|v ext | rotates around the zero in f Ps and we present conclusions in Section 4. In Appendix A, we describe the complex Kohn th partial-wave trial functions that we use to compute the K-matrices. In Appendix B, we discuss some numerical investigations that we performed and also the WG MERT and polynomial fits.

Scattering Theory for Ps-Formation in Positron-Helium Collisions
Using the complex Kohn variational method, we compute partial-wave two channel K-matrices for positron-helium collisions in the Ore gap [1,13,19,[36][37][38][39][40]. In Appendix A we describe the complex Kohn trial wave function for the th partial wave. From the partialwave K-matrices, K , we determine the Ps-formation scattering amplitude f Ps , using where I is the 2 × 2 identity matrix, θ is the angle of the outgoing Ps, and P are Legendre polynomials [41]. The Ps-formation scattering amplitude, f Ps , has azimuthal symmetry. In Equation (1), the number 1 in the subscript refers to the initial channel i which is the positron-helium scattering channel and the number 2 in the subscript refers to the final channel, which is the Ps-He + channel [1]. Thus, the subscript 12 in Equation (1) refers to Ps formation so that f 12 is f Ps . The magnitude of the momentum of the incident positron k and of the outgoing Ps atom κ are related through energy conservation, which is given by where E is the total energy of the positron-helium system, E He + = −2 is the energy of the helium ion, E Ps = −0.25 is the energy of the Ps atom, whose mass M Ps is 2. We take the ground-state energy of the helium atom to be the variational determined energy E He = −2.903701104390 that corresponds to the variational wave function Φ He (r 2 , r 3 ) given in Refs. [19,[36][37][38] (see Appendix A). For this value of E He , the lowest limit of the Ore gap is 17.79 eV.
In our calculations of f Ps (k, θ), we sum to a finite , which we call max . The highest value of max that we consider is three, but we vary max from one to three to consider the convergence of the position of the zero in f Ps and of the corresponding Ps-formation DCS with respect to the number of partial waves included in Equation (1). We determine the Ps-formation DCS using [41]: We fit the complex Kohn K-matrices using polynomials and also the WG MERT [15] for which we explicitly take into account the polarization potential in the Ps-He + channel [15][16][17][18]. Using both the complex Kohn K-matrices and the fitted K-matrices, we determine the position of the zero in f Ps and thus in the corresponding Ps-formation DCS.

Extended Velocity Field Associated with the Ps-Formation Scattering Amplitude
Macek et al. related experimental measurements of a deep minimum in the triply differential cross section of the electron-impact ionization of helium to a vortex in a velocity field associated with the ionization amplitude [26]. A deep minimum in a differential cross section is a quantity that, in principle, is accessible to experimental measurements. While the extended velocity field associated with the Ps-formation scattering amplitude is not experimentally measurable, a zero in the Ps-formation differential cross section means that there is a vortex in this field. A discussion of the standard velocity field that is associated with the wave function in coordinate space is given in reference [42].
In our preceding paper [7] on the deep minima and vortices for Ps formation in positron-hydrogen collisions, we gave an expression for the velocity field associated with the Ps-formation scattering amplitude f Ps , namely, We referred to the velocity field of Equation (4) as the extended velocity field v ext , where we treated k and θ, and thus κ and θ, as independent variables [7]. Here, we determine the extended velocity field v ext that corresponds to f Ps for positron-helium collisions.
In the vicinity of a first-order isolated zero in the complex function f Ps , this function can be expressed in linear form where Im[b] = 0 [7,26,27,43,44]. In Equation (5), κ z and κ x are the zand x-components of the momentum of the outgoing Ps in the z-x plane, where the z-axis is taken to be parallel to the momentum of the incident positron k. The position of the zero in f Ps is at κ z = κ z0 and κ x = κ x0 . We treat the two components κ z and κ x as independent variables, which is consistent with treating k and θ as independent variables. Using the linear form of f Ps , which is a good approximation in the vicinity of the zero f Ps , one can determine the dominant term of v ext in this vicinity. Using this term, one can determine the circulation Γ as [7,27,42,44]: for a closed contour c, of anticlockwise orientation, enclosing only the isolated first-order zero in f Ps . We determine f Ps , the Ps-formation DCS and v ext using the sand p-wave K-matrices from the WG MERT and the dand f -wave from the polynomial fits. We refer to this specific set of K-matrices as the hybrid fit. In the vicinity of the zero of f Ps we fit f Ps , which we obtain using the hybrid fit to the power expansion: where we take max = 3. We substitute Equation (7) into Equation (4) to determine v ext in the vicinity of the zero in f Ps .  The three-dimensional plots of the common logarithm of the Ps-formation DCS, Figure 2a,b, show a deep minimum that corresponds to the zero in f Ps . The deep minimum lies in a valley ( Figure 2a). The valley extends to the forward direction so that there is a minimum in the forward direction (Figure 2b), which we discuss in Section 3.2. Table 1 presents the position of the zero in f Ps , (k 0 , θ 0 ) &(κ z0 , κ x0 ), which we determine using the complex Kohn K-matrices. We show the variation of the position of the zero with max , where max is the maximum value of that we use in Equation (1). We obtain a zero in f Ps with max = 1; thus, only the first two partial waves, = 0 and 1, are needed to obtain a zero. This is also the case for the first zero in f Ps for positron-hydrogen collisions [7]. For positron-helium collisions, we note that the θ 0 value of the position of the zero changes significantly with the inclusion of the d-wave while the k 0 value changes only slightly. However, the θ 0 value of the position of the zero changes only slightly by further adding the f -wave. There is no change in the k 0 value up to the fourth decimal figure with the inclusion of the f -wave. We can, thus, conclude that the value of the position of the zero has essentially converged with respect to partial waves by max = 3. This is reasonable since the k value of the position of the zero is just slightly higher than the k value for the Ps(1s)-formation threshold. Interestingly, for positron-helium collisions in the Ore gap, we obtain a single zero in f Ps ; whereas, for positron-hydrogen collisions in the Ore gap, we obtained two zeros in f Ps [7].
We also give in Table 1 the position of the zero in f Ps with max = 3, which we obtain using the hybrid, WG MERT and polynomial fits. The result using the WG MERT fits is in better agreement with the complex Kohn result than the result using the polynomial fits. Thus, it is important to explicitly take into account the polarization potential in the Ps-He + channel in the fitting of the K-matrices [16,18]. This is reasonable since the effective polarizability of Ps in the Ps-He + channel is significant [16].  Table 1. Position of the zero in f Ps , (k 0 , θ 0 ) & (κ z0 , κ x0 ), which we compute using the complex Kohn K-matrices. The table shows the convergence of the position of the zero with respect to the maximum partial wave max . It also gives the position of the zero in f Ps , which we compute using various fits, and taking max = 3. In Figure 3a, we show the logarithm of the Ps-formation DCS as a function of the angle θ at k 0 = 1.1487, the k value of the zero in f Ps with max = 3. In Figure 3b, we show the logarithm of the Ps-formation DCS as a function of k at θ 0 = 35.9 • , the θ value of the zero in f Ps with max = 3. We use the complex Kohn K-matrices to compute the Ps-formation DCS, and we compare the Ps-formation DCS for different values of max . The minimum in the logarithm of the Ps-formation DCS for max = 1 in these figures is not deep since the zero in f Ps for max = 1 is at a noticeably different position than for max = 3 (see Table 1). For the kinematics of the figures, we find that the Ps-formation DCS for max = 2 and max = 3 are almost identical. Thus, for these figures, we conclude that the Ps-formation DCS has converged sufficiently with respect to partial waves by = 3. We discuss in Appendix B other numerical investigations that we perform.

Minimum in the Ps-Formation DCS in the Forward Direction
We show in Figure 4 the logarithm of Ps-formation DCS in the forward direction as a function of the energy of the incident positron E i . There is a minimum in the Ps-formation DCS (θ = 0 • ), whose position for the calculation, where we use the complex Kohn Kmatrices and take max = 3, is at E i = 17.9 eV (k = 1.1472). The minimum is, therefore, at an energy lower than the lowest energy of 24.7 eV considered in the experimental measurements of the near-forward direction Ps-formation DCS for positron-helium collisions [4,6].   Figure 4a shows the convergence with respect to max of the logarithm of the Psformation DCS in the forward direction, which we compute using the complex Kohn Kmatrices. One can see that the cross sections for max = 2 and max = 3 are similar, especially in the region of the minimum. The minimum in the logarithm of the Ps-formation DCS is deepest for max = 1, which is not surprising since the θ value for the zero in f Ps is significantly closer to the forward direction for max = 1 than for the other two values of max (see Table 1). Figure 4b compares the Ps-formation DCS (plotted using a logarithmic scale) in the forward direction, which we compute using the complex Kohn K-matrices with the cross sections that we compute using the WG MERT, hybrid and polynomial fits. The cross sections, which we compute with the WG MERT and hybrid fits, are in excellent agreement with the cross section that we compute directly with the complex Kohn Kmatrices. However, the position of the minimum in the Ps-formation DCS (θ = 0 • ), which we compute with the polynomial fits, is noticeably different to the position that we compute using the complex Kohn K-matrices or the other fits. These results show that, for energies in the vicinity of the minimum in the Ps-formation DCS (θ = 0 • ), it is important in the fittings of the K-matrices to explicitly include the polarization potential in the Ps-He + channel.

Extended Velocity Field Associated with the Ps-Formation Scattering Amplitude
We consider f Ps to depend on two independent variables, k and θ, or alternatively κ z and κ x . We refer to the velocity field Equation (4), associated with this f Ps , as the extended velocity field v ext [7]. Using the hybrid fit of the complex Kohn K-matrices, we compute f Ps , the Ps-formation DCS, v ext andv ext = v ext /|v ext | for a (κ z , κ x ) grid. Figure 5a shows a density plot of the common logarithm of the Ps-formation DCS, the nodal lines of Re[ f Ps ] = 0 and Im[ f Ps ] = 0, andv ext . For a small (κ z , κ x ) grid that is in the vicinity of the zero in f Ps , we fit f Ps , which we determine from the hybrid fit of the complex Kohn K-matrices, to Equation (7). We use this fitted f Ps to determine the Ps-formation DCS, v ext andv ext . Figure 5b shows the logarithm of the Ps-formation DCS, the nodal lines of Re[ f Ps ] = 0 and Im[ f Ps ] = 0, andv ext that we obtain using the fitted f Ps . A small (κ z , κ x ) grid that is in the vicinity of the zero in f Ps . We use Equation (7) to fit f Ps that we compute using the hybrid fit, and we use the fitted f Ps to compute the Ps-formation DCS andv ext .
The rotation ofv ext is anticlockwise around the zero in f Ps , which is the same direction of rotation ofv ext for the first zero in f Ps for positron-hydrogen collisions in the Ore gap [7]. For Ps-formation in positron-hydrogen collisions, we previously found two zeros in f Ps in the Ore gap [7]. While the direction of rotation of v ext around the first zero in f Ps is anticlockwise, the direction of rotation of v ext is clockwise around the second zero [7]. Recently, for the electron-impact ionization of helium and for a number of different incident energies, it was found that the velocity field associated with the transition matrix element rotates anticlockwise around a zero in the transition matrix element [30]. In contrast, for the positron-impact ionization of helium, it was found that the direction of rotation of the velocity field is clockwise and, thus, in the opposite direction of rotation for electronimpact ionization of the same target [30]. However, one cannot assume that the direction of rotation of the velocity field is projectile dependent for the ionization of a given target, since it was found that, for both electron-and positron-impact ionization of atomic hydrogen, the velocity field associated with the transition matrix element rotates in the same direction (anticlockwise) around a zero [31].
For Ps-formation in positron-helium collisions, we use Equation (6) together with Equations (4) and (7) to compute the circulation Γ, and we obtain the value of 2π/M Ps to four significant figures.

Conclusions
Using complex Kohn K-matrices, we have computed the Ps-formation scattering amplitude, f Ps , and the corresponding DCS for positron-helium collisions in the energy range of the Ore gap. We found a deep minimum in the logarithm of the Ps-formation DCS that corresponds to a zero in f Ps [8,9]. Corresponding to the zero in f Ps , there is a vortex in the extended velocity field v ext that is associated with f Ps . We showed thatv ext rotates anticlockwise around the zero in f Ps [8,9], which is the same direction of rotation ofv ext around the first zero in f Ps for positron-hydrogen collisions in the Ore gap [7].
We saw that, to obtain a position of the zero in f Ps that compares well with the complex Kohn result, it is important to explicitly include the polarization potential in the Ps+He + channel in the fits of the K-matrices. For the minimum in the forward direction Ps-formation DCS, it is important to explicitly include the polarization potential in the fittings of the K-matrices for the sand p-waves. These findings are consistent with earlier findings that the WG MERT describes well the behaviour of the Ps-formation partial-wave cross sections near the Ps-formation threshold for positron collisions from hydrogen [16][17][18] and from helium [16,18].
The Ps-formation DCS near the forward direction has been recently measured for positron collisions for a number of targets including helium [4][5][6]. While a dip was found in the measured near-forward direction Ps-formation DCS for positron collisions from Ne, Ar, Kr and Xe [5], it was not observed in the cross section for positron-helium collisions for the energy range of the experiment [4,6]. However, we theoretically obtained a minimum in the Ps-formation DCS (θ = 0 • ) for positron-helium collisions at 17.9 eV, which is at a lower energy than the energies considered in the experiment for helium [4,6]. Interestingly, both the deep minimum and the minimum in the forward direction lie in a valley in the logarithm of the Ps-formation DCS, which is a function of k and θ.
We can conclude that, for the particular process of Ps formation in positron-helium collisions in the Ore gap, there is a deep minimum in the logarithm of the Ps-formation DCS, which is connected via the valley to the minimum in the logarithm of the Ps-formation DCS in the forward direction, and that there corresponds to the zero in f Ps a vortex in v ext that is associated with f Ps .  Data Availability Statement: Data is available upon request. Please email S.J.W. to request data. Additional material to that given in this paper may be found in reference [9].

Abbreviations
The following abbreviations are used in this manuscript:

DCS
Differential Cross Section WG MERT Watanabe and Greene's Multichannel Effective Range Theory

Appendix A. Complex Kohn th Partial Wave Trial Functions
We use the complex Kohn variational method to compute K-matrices for positronhelium collisions in the Ore gap. We describe briefly here the complex Kohn trial wave function for the th partial wave. The application of the Kohn-type variational method to positron-helium collisions may be found in Refs. [1,13,19,[36][37][38][39][40].
In the Ore gap, if one ignores positron annihilation, the only two open channels are elastic scattering and Ps(1s) formation. The total wave function Ψ(r 1 , r 2 , r 3 ) has two components, Ψ 1 and Ψ 2 , where Ψ 1 represents elastic positron-helium scattering and Ps(1s) formation, whilst Ψ 2 represents elastic Ps-He + scattering and helium formation [1]. In Figure A1, we show the position vectors for the positron-helium system. In this figure, N denotes the nucleus of the helium atom, He 2+ , which we take to be infinitely massive. The complex Kohn asymptotic form of the two components, Ψ 1 and Ψ 2 , for the th partial wave may be written in the form [13,19,47,48] and where Φ He + and Φ Ps are the ground-state wave functions of the helium ion and of the Ps atom, respectively. We take for the ground-state wave function of helium, Φ He (r 2 , r 3 ), the variational wave function given in Refs. [19,37]. This wave function is of Hylleraas form and is symmetric with respect to the two electrons in the helium atom. In Equations (A2)-(A5), Y ,0 is the spherical harmonic function, h (1) and h (2) are the two spherical Hankel functions that are related to the spherical Bessel function j and the spherical Neumann function n according to h (1) = j + in and h (2) = j − in , respectively [49][50][51]. The operator P 23 is the space exchange operator for the two electrons. The S ij in Equation (A1) are the partial wave S-matrix elements, where the S-matrix S is related to the K-matrix K by [19,41] In constructing the th partial-wave trial function Ψ t to use for all distances, we multiply the spherical Neumann functions in the asymptotic form of the two components (Ψ 1,asym and Ψ 2,asym ) by shielding functions that remove the singularities at the origin and which asymptotically tends to unity. We refer to these two resulting components, (Ψ 1,long and Ψ 2,long ), as the long-range components of the th partial-wave trial functions. For the s-wave, we add to the two long-range components highly correlated Hylleraas-type short-range terms of the form [13,19] which include all six interparticle distances (see Figure A1). In Equation (A7), α and β are non-linear parameters, the c i s are linear parameters, k i , l i , m i , n i , p i and q i are non-negative integer powers and q i is restricted to even powers only. The sum of these non-linear parameters, k i + l i + m i + n i + p i + q i , is less than or equal to ω, where ω is a non-negative integer [13,19,38]. To avoid the same Hylleraas-type short-range terms being repeated with the P 23 operation we require that l i ≥ n i , but if l i = n i then m i ≥ p i [13,19,36,38]. For the higher partial waves, we add to the two long-range components two sets of Hylleraas-type short-range terms of different symmetries, where the angular momentum is either on the positron (Equation (A8)) or on one of the electrons (Equation (A9)). In Equations (A8) and (A9), the linear parameters are c i (i = 1 → N 1 ) and d i (i = 1 → N 2 ), respectively, and N 1 (N 2 ) is the number of terms for the first (second) symmetry. We take N 1 = N 2 = N for all partial waves ( = 0 → 3) and ω values except the f -wave and ω = 4, where we take N 1 = N = 84 but N 2 = 36 to avoid linear dependence. For the s-, p-, dand f -waves, the highest value of ω we use is ω = 8, 6, 5 and 4, respectively. There are + 1 rotational harmonics [13,52] terms to consider that include the angular momentum on the incoming positron, angular momentum on one of the electrons and also mixed symmetry terms where the angular momentum is shared. However, following Van Reeth and Humberston [19], we do not explicitly include the mixed symmetry terms. The neglect of the mixed symmetry terms has been discussed in Refs. [19,39,48,53,54].
Following Van Reeth and Humberston [19], as well as adding the Hylleraas-type short-range terms to the two resulting components of the trial wave function, we add for the s-, pand d-waves polarized-orbital terms of the form [19] to each component, where θ a is the angle between the vectors ρ 2 and r 12 (see Figure A1). For the s-wave we consider four polarized-orbital terms, (g (κρ 2 ) = j (κρ 2 ) and n (κρ 2 ), n = 2 and 3), whereas for the pand d-waves we only consider two polarized-orbital terms, (g (κρ 2 ) = j (κρ 2 ) and n (κρ 2 ), n = 2). The shielding function f (ρ 2 ) removes the singularity at the origin arising from g (κρ 2 )/ρ n 2 . The polarized-orbital terms represent the configuration of the Ps atom distorted by the He + ion in the intermediate region. Adding these terms to the trial wave functions significantly improved the convergence of the K 22 matrix elements [19].

Appendix B. Numerical Investigations
We examine the convergence of the complex Kohn K-matrix elements at the k position of the zero in f Ps (k 0 = 1.1487) by computing the ratio: where K ij (ω) is a K-matrix element for a particular partial wave that we evaluate at a certain value of ω. The largest value of ω that we consider for the s-, p-, dand f -wave is 8, 6, 5 and 4, respectively. We find that for the s-wave R 11 (8) < 0.3%, R 12 (8) < 0.9%, R 21 (8) < 0.9% and R 22 (8) < 4%. For the p-wave, R ij (6) < 6% for all K-matrix elements and for the d-wave, R ij (5) < 4%. For the f -wave, the ratio R ij (4) is larger than the above percentages for K 11 and is significantly larger for K 22 . However, as one can see from Table 1, the f -wave has only a small effect on the position of the zero in f Ps .
In Table A1, we show the convergence in the position of the zero in f Ps with respect to ω for the s-wave only. We fix ω for the p-, d-, and f -wave at 6, 5 and 4, respectively. Both the k and θ values of the position of the zero, k 0 and θ 0 , in f Ps increase with increasing ω for the s-wave. The zero in f Ps is at a k value just above the Ps-formation threshold. We found that the position of the zero f Ps , especially the angle θ 0 , is extremely sensitive to the s-wave ω value. Thus, it is important to determine the s-wave K-matrix accurately at the k value of the zero in f Ps . In Table A2 we show the convergence in the position of the zero in f Ps with respect to the p-wave ω value in which we fix the ω values for the other partial waves (ω = 8 for the s-wave, ω = 5 for the d-wave and ω = 4 for the f -wave). Similar to the s-wave case, the angle θ 0 increases with increasing ω for the p-wave. However, in contrast, the k value decreases with increasing ω for the p-wave. The position of the zero is less sensitive to increasing ω for the p-wave than the s-wave. For the d-wave, we find that increasing ω from three to five does not change, to five significant figures, the k value of the position of the zero in f Ps , but it does change θ by less than 0.7 • . Furthermore, for the f -wave, increasing ω from zero to four does not change the k value to at least five significant figures, although θ changes by less than 0.1 • . Table A2. Convergence of the position (k 0 , θ 0 ) of the zero in f Ps with respect to variations in ω for the p-wave only. For the s-, dand f -wave, the ω value is 8, 5 and 4, respectively. ω k 0 θ 0 (deg.) 6 1.1487 35.9 5 1.1489 34.8 4 1.1491 33.7 In Table A3, we show the variation in the position of the zero (k 0 , θ 0 ) in f Ps with respect to the ω set , where ω = ω − i and i = 0 to 3. The ω set is the set that we use for our most elaborate calculations in which ω = 8, 6, 5 and 4 for the s-, p-, dand f -wave, respectively. The ω − i set, for i = 1 to 3, is where the ω value for each partial wave is decreased by i. We find that, while there is no change in the k 0 value to five significant figures when going from the ω set to the ω − 1 set, the θ 0 value changes in the second significant figure. Table A3. Convergence of the position (k 0 , θ 0 ) of the zero in f Ps with respect to the set ω = ω − i, where i = 0 to 3, and where for the ω set ω = 8, 6, 5, 4 for the s-, p-, dand f -wave, respectively. We compare in Figure A2 the logarithm of the Ps-formation DCS in the forward direction, which we compute with different ω = ω − i sets of complex Kohn K-matrices, where i = 0 to 3. One can see in the main figure of Figure A2 that the Ps-formation DCS (θ = 0 • ) is very stable with respect to ω over most of the energy range that we show. For a fine energy mesh in the vicinity of the minimum in the logarithm of the Ps-formation DCS (θ = 0 • ), one can clearly see a variation in the cross section for the different ω sets (see inset of Figure A2). We apply the WG MERT [15] in which the polarization potential is explicitly included in the Ps+He + channel [15][16][17][18] to fit the complex Kohn K-matrices. The WG MERT expansion of a K-matrix is expressed in terms of the quantum defect parameters that vary rapidly with energy and depend on the polarizability of Ps, and on the K P0 matrix elements that, in general, vary slowly with the energy and are related to short-range interactions [16]. Using the s-, p-, dand f -wave complex Kohn variational K-matrices and the polarizability of positronium, we extract the K P0 matrix elements [15][16][17][18] at different k values. We fit these elements to polynomials of powers of κ 2 , which enables us to determine K-matrices at any value of k. For the K P0 11 ( = 0), K P0 12 ( = 2) and K P0 11 ( = 3) matrix elements we chose cubic polynomials, whereas for the K P0 12 ( = 3) and K P0 22 ( = 3) matrix elements we chose quintic polynomials. We use quartic polynomials for the other K P0 ij matrix elements for the various partial waves. (For the p-wave we just use five values of k [1.16, 1.18, 1.20, 1.22 and 1.24] in obtaining fits for the K P0 ij matrix elements.) For the polynomial fits of the complex Kohn matrix elements K ij , we use for each partial wave the same order of polynomials as we use in the corresponding K P0 ij matrix elements.