Comment on ``Exact three-dimensional wave function and the on-shell t matrix for the sharply cut-off Coulomb potential: Failure of the standard renormalization factor''
aa r X i v : . [ nu c l - t h ] A ug Comment on “Exact three-dimensional wave function and theon-shell t matrix for the sharply cut-off Coulomb potential:Failure of the standard renormalization factor” Yuri V. Popov
Nuclear Physics Institute, Moscow State University, Moscow 119991, Russia ∗ Konstantin A. Kouzakov
Department of Nuclear Physics and Quantum Theory of Collisions,Faculty of Physics, Moscow State University, Moscow 119991, Russia † Vladimir L. Shablov
Obninsk State Technical University for Nuclear Power Engineering,Obninsk, Kaluga Region 249040, Russia (Dated: October 26, 2018)
Abstract
The solutions analytically derived by Gl¨ockle et al. [Phys. Rev. C , 044003 (2009)] for thethree-dimensional wave function and on-shell t matrix in the case of scattering on a sharply cut-offCoulomb potential appear to be fallacious. And their renormalization factor lacks mathematicalgrounds. PACS numbers: 21.45.Bc, 03.65.Nk ∗ Electronic address: [email protected] † Electronic address: [email protected]
1n a recent paper by Gl¨ockle et al. [1], nonrelativistic scattering of two equally chargedparticles with mass m interacting via potential V ( r ) = e r Θ( R − r ) was considered. Theauthors argued that they analytically derived the exact wave function and scattering ampli-tude for arbitrary values of a cut-off radius R . On this basis they obtained a renormalizationfactor which relates the scattering amplitude in the limit R → ∞ with the physical Coulombscattering amplitude. The purpose of this Comment is (i) to show that the analytical resultsof Gl¨ockle et al. [1] are erroneous for finite values of R and are mathematically ungroundedin the limit R → ∞ and (ii) to point out a different renormalization approach which is freefrom uncertainties associated with the cut-off renormalization.The authors of [1] made an unjustified premise that the solution of the Lippmann-Schwinger equation for r < R obeys the formΨ (+) R ( ~r ) = Ae i~p · ~r F ( − iη, , i ( pr − ~p · ~r )) , (1)where η = me p is a Sommerfeld parameter. The constant [11] A = 1 F ( − iη, , ipR ) (2)was determined in Ref. [1] by inserting (1) into the Lippmann-Schwinger equationΨ (+) R ( ~r ) = e i~p~r − µe π Z d r ′ r ′ e ip | ~r − ~r ′ | | ~r − ~r ′ | Θ( R − r ′ )Ψ (+) R ( ~r ′ ) (3)and solving the latter at r = 0. In fact, the correct form of the solution in the interior region r < R is Ψ (+) R ( ~r ) = 14 π Z d ˆ k A (ˆ k ) e ip ˆ k · ~r F ( − iη, , i ( pr − p ˆ k · ~r )) , (4)where the function A (ˆ k ) is defined on a unit sphere. To determine A (ˆ k ) one may employthe usual partial wave formalism (see, for instance, the textbook [2]). Consider the followingexpansion in Legendre polynomials: A (ˆ k ) = X l (2 l + 1) A l P l (ˆ p · ˆ k ) . (5)Matching the interior Lippmann-Schwinger solution and its derivative to the exterior onesat r = R yields A l = i ( pR ) − W ( ψ l , h (1) l )( pR ) , (6)2here h (1) l is a spherical Hankel function of the first kind and ψ l ( pr ) = e iσ l | Γ( l + 1 + iη ) | Γ(1 + iη ) (2 pr ) l (2 l + 1)! e − ipr F ( l + 1 − iη, l + 2 , ipr ) , with the Coulomb phase shift σ l = argΓ( l + 1 + iη ). It can be checked that A = A but A l ≥ = A , i.e. (1) is clearly invalid.The expression for the scattering amplitude (the on-shell t matrix) in Ref. [1] is invalid aswell, since it derives from the wave function (1). The limit of vanishing screening ( R → ∞ )has been considered previously in the literature (see, for instance, [2, 4, 5] and referencestherein). Using asymptotic forms of ψ l and h (1) l [3] one readily arrives at A l ≃ e − πη Γ(1 + iη ) e − iη ln(2 pR ) + O (cid:18) pR (cid:19) , (7)provided l ≪ pR . When l ≫ pR , the phase shifts behave as δ l → l ∼ pR is very hard to handle [4]. Thus, theconvergence A l → e − πη Γ(1 + iη ) e − iη ln(2 pR ) is not uniform, i.e. it depends on l , and thereforetaking the limit R → ∞ in (5) presents quite a challenge. Nevertheless, it can be shownthat the asymptotic form for the scattering amplitude is [2, 5] f R = e − iη ln(2 pR ) f c + f osc , (8)where f c is the physical Coulomb scattering amplitude. The first term in the right-handside of (8) appears because of (7). The term f osc oscillates rapidly like cos( qR ), where q isthe momentum transfer. It integrates out to zero with the incident wave packet and, hence,makes no contribution to the cross section as measured in typical experiments (see [2] fordetails).The amplitude derived in Ref. [1] in the limit R → ∞ resembles the form (8), howeverits derivation lacks mathematical grounds because it is carried out using (1) instead of theexact wave function (4). The wave function (1) can be presented as a product C R Ψ (+) c ,where Ψ (+) c is a Coulomb wave and C R is a constant ( C R →∞ → e − iη ln(2 pR ) ). The Coulombwave satisfies a homogeneous Lippmann-Schwinger equation [6]Ψ (+) c ( ~r ) = − µe π Z d r ′ r ′ e ip | ~r − ~r ′ | | ~r − ~r ′ | Ψ (+) c ( ~r ′ ) . (9)Let us introduce an auxiliary function which is a difference between the exact wave func-tion (4) and the wave function (1) in the limit R → ∞ : ψ R ( ~r ) = Ψ (+) R ( ~r ) − e − iη ln(2 pR ) Ψ (+) c ( ~r ) . (10)3ccording to (3) and (9), this function satisfies the following equation ( r < R ): ψ R ( ~r ) = ψ (0) R ( ~r ) − µe π Z d r ′ r ′ e ip | ~r − ~r ′ | | ~r − ~r ′ | Θ( R − r ′ ) ψ R ( ~r ′ ) , (11)with the inhomogeneous term ψ (0) R ( ~r ) = e i~p~r + µe π e − iη ln(2 pR ) Z d r ′ r ′ e ip | ~r − ~r ′ | | ~r − ~r ′ | Θ( r ′ − R )Ψ (+) c ( ~r ′ ) . (12)For r ≪ R one has approximately e ip | ~r − ~r ′ | | ~r − ~r ′ | ≈ e ipr ′ r ′ e − i~p ′ ~r , ~p ′ = p~r ′ r ′ , and it can be shown that ψ (0) R ( ~r ) ≈
0. However this does not imply that ψ (0) R ( ~r ) ≈ r < R . In fact, this is definitely not the case when r . R (within the partial wave formalismthis situation corresponds to the l . pR terms).Using (10), the scattering amplitude can be presented as f R = − µe π e − iη ln(2 pR ) Z d r ′ r ′ e − i~p ′ ~r ′ Θ( R − r ′ )Ψ (+) c ( ~r ′ ) − µe π Z d r ′ r ′ e − i~p ′ ~r ′ Θ( R − r ′ ) ψ R ( ~r ′ ) , (13)where ~p ′ = p ˆ r . Gl¨ockle et al. [1] have studied asymptotic behavior of the first term only.They unjustifiably have neglected the second term which, due to nontrivial properties of ψ R in the region r . R , potentially can provide a nonvanishing contribution to the scatteringamplitude in the limit R → ∞ . Thus, their analysis is incomplete and the validity of theirrenormalization factor is questionable. In this connection, the results of the numerical calcu-lations presented in Ref. [1] can not be a decisive argument in favour of the renormalizationfactor, for in this particular case one deals with divergent and rapidly oscillating quantities.Finally, it is useful to note that the renormalization treatments involving cut-off Coulombpotentials are of doubtful value from a practical viewpoint, especially in the case of many-body Coulomb scattering. In this respect, the methods based on regularization and renor-malization of the Lippmann-Schwinger equations in the on-shell limit are more efficient. Thetwo-particle case is fully explored: (i) the Green’s function is derived analytically both incoordinate and in momentum representations [7], (ii) an off-shell amplitude is known [8],and (iii) the rules for taking the on-shell limit are formulated [9]. This allows to generalizethe two-particle results to the many-particle case (see, for example, [10]).4 cknowledgments We wish to thank Sergue I. Vinitsky for drawing our attention to the paper by Gl¨ockle et al. [1] and to Akram Mukhamedzhanov for useful discussions. [1] W. Gl¨ockle, J. Golak, R. Skibi´nski, and H. Wita la, Phys. Rev. C , 044003 (2009).[2] J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions (Wiley,New York, 1972).[3]
Handbook of Mathematical Functions , edited by Milton Abramowitz and Irene A. Stegun(Dover Publishers, New York, 1972).[4] W. F. Ford, Phys. Rev. , B1616 (1964).[5] J. R. Taylor, Nuovo Cimento B , 313 (1974).[6] G. B. West, J. Math. Phys. , 942 (1967).[7] J. Schwinger, J. Math. Phys. , 1606 (1964).[8] C. S. Shastri, L. Kumar, and J. Callaway, Phys.Rev. A , 1137 (1970).[9] Yu. Popov, J. Phys. B: At. Mol. Phys. , 2449 (1981).[10] V. L. Shablov, V. A. Bilyk, and Yu. V. Popov, Phys. Rev. A , 042719 (2002).[11] The normalization factor π ) / is suppressed throughout.is suppressed throughout.