Instantaneous Bethe Salpeter Equation and Its Exact Solution

Transcription

1 Commun. Theor. hys. Beijing, China pp c International Academic ublishers Vol. 43, No. 1, January 15, 2005 Instantaneous Bethe Salpeter Equation and Its Exact Solution CHANG Chao-Hsi, 1,2 CHEN Jiao-Kai, 2,3 LI Xue-Qian, 1,4 and WANG Guo-Li 4,5 1 CCAST World Laboratory,.O. Box 8730, Beijing , China 2 Institute of Theoretical hysics, the Chinese Academy of Sciences,.O. Box 2735, Beijing , China 3 Graduate School of the Chinese Academy of Sciences, Beijing , China 4 Department of hysics, Nankai University, Tianjin , China 5 Department of hysics, Fujian Normal University, Fuzhou , China Received July 19, 2004 Abstract We present an approach to solve Bethe Salpeter BS equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and welldetermined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium, the exact solutions precisely present corrections to those of the corresponding Schrödinger equation in order v 1 v is the relative velocity for eigenfunctions, in order v 2 for eigenvalues, and the mixing between S and D components in J C = 1 states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the Ov corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states. ACS numbers: St, Dr, Ds Key words: instantaneous BS equation, exact solutions, positronium, relativistic corrections Bethe Salpeter BS equation [1 is a very good tool to treat various relativistic bound state systems. For a fermion-antifermion binding system, the BS equation is written as p 1 m 1 χ q p 2 m 2 d 4 k = i 2π 4 V, q, kχ k, 1 where χ q is the BS wave function, is the total momentum, q is relative momentum, and V, q, k is the so-called BS kernel between the electron and positron in the bound state, p 1, p 2 are the momenta of the constituent electron 1 and positron 2, respectively. The total momentum and the relative momentum q are related to the momenta of the fermion-antifermion as follows: p 1 = α 1 q, α 1 = m 1 /m 1 m 2, p 2 = α 2 q, α 2 = m 2 /m 1 m 2. The BS wave function χ q satisfies the normalization condition: d 4 qd 4 q 2π 4 [ Tr χ q [S1 1 p p 1S2 1 p 2δ 4 q q 0 = 2ip 0. 2 V, q, q χ q If the kernel V, q, k of the BS equation has the behavior: V, q, k =0 = V q, k in = 0 frame C.M.S of the concerned bound state, the BS equation is called as instantaneous one, and may be derived to a Schrödinger equation accordingly it was firstly realized by E.E. Salpeter. [2,3 As an example, the BS equation for positronium in the lowest order has the Coulomb kernel as follows: V, q, k =0 = γ 0 V v γ 0 = γ 0 4πα q k 2 γ0. Since in Coulomb gauge, the transverse-photon exchange between electron and positron for the e e bound states is considered as higher order, so with V, q, k as above in the lowest order the BS equation is instantaneous exactly. Thus, when we present the approach to solving an instantaneous BS equation exactly, for convenience and applications, we will take positronium as an example and explore the common features of the solution. The key step for the approach is to derive the instantaneous BS equation to a set of coupled and self-consistent integration equations for the components scalar functions of the BS wave functions without any approximation. The set of equations may not have analytic solutions, but they may be solved numerically under controlled accuracy. Here we will present the interesting features of the exact solutions for the S-wave positronium briefly, while we will put the details and those for -wave ones in Refs. [4 and [5. The approach, which we present, in fact essentially is to follow E.E. Salpeter derivation, but without any approximation. Thus as done in Refs. [2 and [3, first of all we introduce the instantaneous BS wave function dq 0 ϕ q i 2π χ q 0, q, 3 The project supported by National Natural Science Foundation of China under Grant Nos and

2 114 CHANG Chao-Hsi, CHEN Jiao-Kai, LI Xue-Qian, and WANG Guo-Li Vol. 43 then the BS equation 1 is re-written as χ q 0, q = S 1 f pµ 1 η q S2 f pµ 2, 4 where S 1 f p 1 and S 2 f p 2 are the propagators of the fermion and anti-fermion respectively with the definition of the BS-nut d 3 k η q 2π 3 V q, k ϕ k. 5 For general applications, here we keep m 1 m 2 at this moment, although the final application in this paper is to positronium m 1 = m 2. The propagators can be decomposed as with ijs i f Jpµ i = Λ i q Jq 0 α i M ω i iɛ ω i = m 2 i q 2, Λ i q Jq 0 α i M ω i iɛ 6 Λ ± i q = 1 2ω i [γ 0 ω i ± Jm i γ q, where J = 1 for the fermion i = 1 and J = 1 for the anti-fermion i = 2. It is easy to check Λ ± i q Λ i q = γ0, Λ ± i q γ0 Λ i q = 0, Λ ± i q γ0 Λ ± i q = Λ± i q. 7 Based on Eq. 7, Λ ± can be considered as energy projection operators, and they are complete for the projection. For below discussions let us introduce the notations q as ϕ ±± ϕ ±± q Λ ± 1 q γ0 ϕ q γ 0 Λ ± 2 q. 8 Because of the completeness of the projection for Λ ±, we have: ϕ q = ϕ q ϕ q ϕ q ϕ q for the BS wave function ϕ q. As done by E.E. Salpeter to derive the instantaneous BS equation to a corresponding Schrödinger equation, we carry out a contour integration for the time-component q 0 on both sides of Eq. 4, and obtain ϕ q = Λ 1 q η q Λ 2 q Λ 1 q η q Λ 2 q, 9 M ω 1 ω 2 M ω 1 ω 2 M is the eigenvalue and applying the complete set of the projection operators Λ ± i q to Eq. 9 further, we obtain the four equations: M ω 1 ω 2 ϕ q = Λ 1 q η q Λ 2 q, 10 M ω 1 ω 2 ϕ q = Λ 1 q η q Λ 2 q, 11 ϕ q = 0, ϕ q = Now the normalization condition reads: q 2 T dq [ T 2π 2 tr ϕ ϕ ϕ ϕ = M M M M Note that equations 12 do not contain eigenvalue M, so essentially they are constraints on the BS wave functions ϕ q, and because of the completeness of the projectors Λ ±, only the whole set of Eqs are equivalent to Eq. 9. While E.E. Salpeter [2 and the authors of Ref. [3 would like to connect the above coupled equations to the Breit equation, [6,7 so they combined Eqs into one operator equation in C.M.S. of the bound state: Mϕ q γ 0 H 1 q γ 0 ϕ q ϕ q γ 0 H 2 q γ 0 = Λ 1 q η q Λ 2 q Λ 1 q η q Λ 2 q, 14 with the definitions H 1 q m 1 β q α, H 2 q m 2 β q α, β = γ 0, α = β γ. Namely with the definition H 1,2 they considered the equation Eq. 14 as an operator representation of the coupled equations Eqs In fact, exactly to say it is not correct. Equation 14 is not completely equivalent to the coupledequations Eqs Now let us show the un-equivalence as follows. When applying the project operator Λ 1 q γ0 γ 0 Λ 2 q to Eq. 14 we obtain Eq. 10, when applying the project operator Λ 1 q γ0 γ 0 Λ 2 q to Eq. 14 we obtain the Eq. 11, whereas when applying Λ ± 1 q γ0 γ 0 Λ 2 q to Eq. 14, then we obtain the homogeneous equations: [M ω 1 q ω 2 q ϕ q = 0, [M ω 1 q ω 2 q ϕ q = 0 15 with ω 1,2 = m 2 1,2 q 2. The two equations of Eq. 15 not only have the trivial solutions Eq. 12, but also have non-trivial solutions. Therefore, we conclude that equation 14 with the definitions of H 1,2 is not completely equivalent to the BS equation: Eqs Especially, when further approximations on Eq. 14 are made without checking Eq. 12 on the obtained solutions precisely, certain misleading may occur, i.e. the obtained solutions may not be that of the instantaneous BS equation Eq. 9 with η as Eq. 5. The roles of Eqs are important in the present approach. If one really starts with the equation Eq. 14 alone, then it means that one solves the equations Eqs. 10 and 11 and the homogeneous equation Eqs. 15 together. To see the differences, we will solve Eq. 14 with the definitions of H 1,2 and the BS equation: Eqs respectively, and present the results precisely in Ref. [8. The authors of Refs. [2 and [3 etc. thought that, of the coupled equations Eqs , only the so-called positive equation Eq. 10 is important for nonrelativistic binding systems, so they considered it only, and ignored the rests, and then they further assumed that the spin structure of the BS wave function, e.g., for S-wave positronium in c.m.s. = 0, the BS wave function has the formulation precisely ϕ 0 1 S 0 q = [γ 0 1γ 5 f q 16

3 No. 1 Instantaneous Bethe Salpeter Equation and Its Exact Solution 115 for J C = 0 1 S 0 ; and ϕ 1 3 S 1,λ q = [ γ 0 1 A λ f q, A λ ɛ λ γ ɛ λ γ ɛ λ γ, = 0, γ = γ 3 17 for J C = 1 3 S 1. Here 0 is introduced for the zeroth component of γ-matrices in spherical harmonics coordinate to avoid confusions in notation. With the spin structure, making the non-relativistic expansion for ω 1,2 m 1,2 1 q 2 /2m 1,2 and to change the eigenvalue from E to ε M m 1 m 2 as well, they finally turned the equation Eq. 10 for the binding systems exactly to a Schrödinger equation accordingly. To pursue the exact solutions of the instantaneous BS equation 4, we should return to Eqs as a whole without any approximation. Although it is the case for a fermion and anti-fermion system for positronium m 1 = m 2 as an example to illustrate the approach, the approach we present here is also applicable in the cases m 1 m 2, where there is no charge parity quantum number C. [9 In = 0 frame C.M.S., the most general formulation of the BS wave function for the bound states positronium with the quantum numbers J C = 0 1 S 0 may be written as where ϕ1 S 0 q = γ 0 γ 5 ϕ 1 q γ 5 ϕ 2 q 4π q γ5 3 2m Θϕ 3 q 4π 3 γ0 γ 5 q 2m Θϕ 4 q, 18 Θ [Y 1 1 γ Y 11 γ Y 10 γ, γ = γ 3, γ γ1 iγ 2 2, γ = γ1 iγ 2 2, and Y lm Y lm θ q, φ q are spherical harmonics. First of all, to apply Eq. 12 to Eq. 18, we obtain the constraints for the components of the wave functions precisely: ϕ 1 q = 1 2 ϕ 4 q, ϕ 3 q = utting the constraints Eq. 19 into Eq. 18 of the states 0, the components ϕ 3, ϕ 4 are replaced, then the wave function contains two independent components ϕ 1, ϕ 2 only, ϕ1 S 0 q = γ 0 γ 5 ϕ 1 q γ 5 ϕ 2 q 4π 3 γ0 γ 5 q m Θϕ 1 q. 20 By straightforward calculations including to integrate out the relative angles, equations 10 and 11 for positronium, finally, are turned into the coupled radius equations M 2mf 1 2mf 1 2mf 2 = α s π m k ω q d k Q 0 f 2, M 2mf 2 2 ω2 m f 1 2mf 2 = α s m k π ω q d k Q 0 q k m 2 Q 1 f 1 21 with the normalization for the BS wave function q 2 d q 4ω 2π 2 m f 1f 2 = M. 22 Here q 2 Q n Q k 2 n 2 q k n = 0, 1,... are the n-th the Legendre functions of the second kind. Here to shorten the notations we have introduced f 1,2 : ϕ 1 q f 1, ϕ 2 q f 2 and ω = ω 1,2 m 1 = m 2. Especially, we note here that equations 21 are coupled and self-consistent equations in respect to two independent scalar functions f 1,2 respectively and there is not any approximation. They are exact they may be solved numerically under controlled accuracy and may be applicable even to extremely relativistic case. To see the general features for positronium and to reach to the accuracy α 2, we further solve the coupled equations Eq. 21 numerically: to expand them and f 1,2 on the bases of the exact solution of the Schrödinger equation, then to diagonalize the obtained matrix equations on the bases. The corresponding eigenfunctions are expanded as f j i q = nl i,nl R nl q, 23 where j denotes the j-th eigenvalue and eigenfunction, R nl q n and l are the principal and angular quantum numbers: nl = 1S, 2S, 2, 3S, 3,... denotes the Schrödinger radius solution in momentum representation for the positronium their precise expression can be found in Ref. [10. The final results the values of the fine structure constant α and the electron mass are taken so accurate as given by DG, [11 the eigenvalues M j 2m E j, E 1 = [85 ; E 2 = [71 ; E 3 = [18 ; Note that the two digital numbers in the squared brackets in Eq. 24 mean that only the last two digital numbers should be replaced when the solution of the instantaneous BS equation change into that of the Schrödinger equation accordingly. The resultant eigenfunctions corresponding to the eigenvalues are listed in Table 1.

4 116 CHANG Chao-Hsi, CHEN Jiao-Kai, LI Xue-Qian, and WANG Guo-Li Vol. 43 Table 1 NL WF i,1s The expansion coefficients i,nl for the J C = 0 states. i,2s i,3s i,4s E 1 f S 0 f E 2 f S 0 f E 3 f S 0 f In fact, according to Table 1 and up to the order v accuracy, the eigenfunctions for J C = 0 1 S 0 states can be approximately simplified to write as { } 4π ϕ1 S 0 q γ 0 q 1 Θγ0 fγ 5 Ov 2, 25 3 m i.e. f 1 q f 2 q f = R nl=0 q. The solutions for the 1 positronium states are more interesting than those of 0. Similar to the case of 0 states for the general form of the wave functions from Eq. 18 four components to Eq. 20 two components, now having the constraints Eq. 12 applied, the general form of the BS wave functions for 1 states are reduced from a general form eight components to the following form four components, π π ϕ 1,λ q = 2 3 C λ A λ f D λγ 5 A λ γ 0 f 2 where f i ϕ i q i = 1, 2, 3, 4, A λ is the same as in Eq. 17 and C λ B λ f 3 B λ 6ɛ λ γ Y ɛ λ γ ɛ λ γ Y 2 1 ɛ λ γ ɛ λ γ 2ɛ λ γ Y 20 3 ɛ λ γ ɛ λ γ Y 21 6ɛ λ γ Y 22, 5 3 D λγ 5 B λ γ 0 f 4, 26 C λ q [ ɛ λ m Y 1 1 ɛ λ Y 11 ɛ λ Y 10, D λ q [ ɛ λ m γ ɛ λ γ Y 11 ɛ λ γ ɛ λ γ Y 1 1 ɛ λ γ ɛ λ γ Y 10. From Eq. 26, one may see clearly that f 3,4 correspond to D-wave components, while f 1,2 to S-wave ones and the formulism Eq. 26 is suitable to describe the S D component mixing in 1 states. From Eqs. 10 and 11 and to carry out the angular integration, finally we obtain the coupled radius equations: M 2mf 1 = q 2 π m f 4 2mf 1 2m2 4ω 2 f 2 α k [ 3m π q d 5 k 9π M 2mf 2 = q 2 π m f 3 2 2m 2 ω 2 f 1 2mf 2 α k 1 k [ 3 m π q d 3 M 2mf 3 = 2mf 3 2 2m 2 ω 2 f 4 4 π 3 m 3 q 2 5 m f 2 α k [ q π q d k k 3mω Q 1 m ω Q 2 M 2mf 4 = 1 2m 2 4ω 2 3 m f 3 2mf 4 4 π 3 q 2 5 m f 1 α π q k 2 mω Q q 1f 4 k 3 mω Q 1 m ω Q 0 f 2, 5 q k 1 π mω Q q 1f 3 k 3 mω Q 1 m ω Q 0 f 1, f 4 2 π 3 q k 5 mω Q 1f 2, k m k [ q d ω Q 2 2 q k 3 mω Q 1 f 3 2 π 3 q k 5 mω Q 1f The normalization condition of the BS wave functions now becomes q 2 d q 6ω 2π 2 mπ f 3f 4 f 3 f 2 f 1 f 4 3f 1 f 2 = 2M. 28 To be self-consistent, the four coupled equations for f 1,2,3,4 Eq. 27 are the request to describe the S D wave mixing. With the same method as the case for 0, we solve the set of coupled equations Eq. 27 numerically and obtain the results for eigenvalues M j 2m = E j as follows: E 1 3 S 1 = [85 ; E 2 3 S 1 = [12 ;

5 No. 1 Instantaneous Bethe Salpeter Equation and Its Exact Solution 117 E 3 3 S 1 = [18 ; E 4 3 D 1 = [18 ; The two digital numbers in the squared brackets in Eq. 29 have the same meaning as those in Eq. 24, i.e., they are the last two digital numbers which are different from those eigenvalues of the Schrödinger equation. The coefficients i,nl for the eigenfunctions are listed in Table 2. According to Table 2, for J C = 1 3 S 1 states with the approximation to α 2 v 2, we can simplify f 1 q f 2 q f = R nl=0 q and f 3 q f 4 q 0 Dwave components vanish, i.e. [ π q ϕ3 S 1,λ q 2 3 m C λ A λ 2 π 3 q m D λγ 5 A λ γ 0 f. 30 While for J C = 1 3 D 1 states with the approximation to α 2 v 2, we can simplify f 1 q f 2 q 0 S-wave components vanish and f 3 q f 4 q f = R nl=2 q, i.e. [ 5 ϕ3 D q q 1,λ 2 3 m C λ B λ 5 q 3 m D λγ 5 B λ γ 0 f. 31 We may see that the exact solution obtained here for the wave function contains the order v = q /m corrections to the approximate ones obtained by E.E. Salpeter approach as described above, namely, we obtain the correction terms explicitly proportional to v in Eqs. 25, 30 and 31. An interesting fact is that the S D wave mixing for 1 states due to the instantaneous interaction is fixed by the exact solution. To compare the eigenvalues Eq. 24 for 0 and Eq. 29 for 1 with those of the Schrödinger solution the degenerate ones without any splitting, we may also see that the hyperfine splitting purely from the instantaneous interaction for spin singlet and triplet states is in order of α 2 v α, and there is a remarkable splitting between n[ 3 S 1 and n[ 3 D 1 n 3 for 1 states E S 1 E3 Sch for Schrödinger equation E D 1 and E α 2 due to the fact that the BS equation Eq. 4 is relativistic. We should emphasize several points for the approach to solving instantaneous equations exactly: i The set of Eqs. 10, 11, and Eq. 12 are treated equally ; ii We had better to solve the derived equations, such as Eq. 21 for 0 and Eq. 27 for 1 etc. [5 directly, i.e. to keep the equations with linear eigenvalue originally, but not to solve them by changing them to the equations with quadratic even higher eigenvalues etc.; iii To obtain the clean S-D wave mixing in the states 1 etc., we had better to de-composite the BS wave function to the equations for scalar components in polar coordinate. In the literature, when solving an instantaneous BS equation, in many cases to Eq. 12 less attention was payed, the gauge invariance for radiative transitions was broken quite often. [4 Note that the exact precise equations derived from an instantaneous BS equation with a different kernel may be different from those for positronium as described above, but the present approach may be applicable and exact solutions can be obtained accordingly. NL WF 1S Table 2 The expansion coefficients i,nl for the J C = 1 states. 2S 3S f E 1 f S 1 f f f E 2 f S 1 f f f E 3 f S 1 f f f E 4 f D 1 f f D 4S 4D

6 118 CHANG Chao-Hsi, CHEN Jiao-Kai, LI Xue-Qian, and WANG Guo-Li Vol. 43 In summary, for the positronium example with Coulomb instantaneous interaction being considered, the values and the spin structure of the BS wave functions, the relativistic corrections to the eigenvalues energy level and the possible mixing, such as that between S and D components in J C = 1 states etc. are determined by solving the instantaneous BS equation exactly. The general feature on the quantities is that the corrections to the eigenvalues energy level of the Schrödinger equation are of the order α 2 v 2 v is the relative velocity between electron and positron and those to the wave functions spin structure are of the order Ov. Although based on the specific system of positronium, the general features should not be changed much for the other systems which may be described by an instantaneous BS equation. Since the wave functions contain relativistic corrections in order Ov, thus if the relevant calculations declare that the relativistic effects are taken into account, the relativistic corrections from the wave functions should be considered carefully. For the effective theories, such as NRQED [12 and NRQCD, [13 the relativistic effects from the wave functions may be involved either at matching the underlined theory to the effective one or into the non-perturbative matrix elements properly, but we should note here that in the literature many calculations under the framework of the effective theories when matching the underlined theory to the effective one, the spin projectors, where the relative motions between the two composite elements fermion and anti-fermion were ignored, i.e., equations 16 and 17 were adopted. If the corrections even in order v 1 from the wave functions are taken into account, the spin projectors for the matching should take those as Eqs. 25, 30 and 31. Since we have not involved the contributions from transverse photon exchange and the annihilation of electronpositron as high order ones yet, so at present stage for positronium we do not compare the exact solutions of the instantaneous BS equation with experimental data. We will present the comparisons soon elsewhere when the mentioned contributions are taken into account. Acknowledgments The author C.H. Chang would like to thank Stephen L. Adler for valuable discussions and to thank Xiang-Dong Ji for very useful discussions. The authors would like to thank Tso-Hsiu Ho and Cheng-Rui Ching for valuable suggestion and encouragement. References [1 E.E. Salpeter and H.A. Bethe, hys. Rev [2 E.E. Salpeter, hys. Rev [3 C. Itzykson and J. Zuber, Quantum Field Theory, McGraw-Hill Inc., rinted in the United State of America. [4 Chao-Hsi Chang, Jiao-Kai Chen, and Guo-Li Wang, hepth/ [5 Chao-Hsi Chang, Jiao-Kai Chen, Xue-Qian Li, and Guo- Li Wang, Exact Solutions of Instantaneous Bethe Salpeter Equation for S-Wave and -Wave ositronium in Coulomb Gauge, in preparation. [6 G. Breit, hys. Rev [7 G. Breit and G.E. Brown, hys. Rev ; G.E. Brown and D.G. Ravenhall, roc. Roy. Soc. London A [8 Chao-Hsi Chang and Jiao-Kai Chen The Instantaneous Bethe Salpeter Equation and Its Analog: the Breit Like Equation, nucl-th/ [9 Chao-Hsi Chang, Jiao-Kai Chen, Xue-Qian Li, and Guo- Li Wang, The Exact Solution of the Instantanious BS Equation for µ e or µ e System, in preparation. [10 H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, OHG, Berlin Göttingen Heidelberg, Gemerny [11 article Data Group, hys. Rev. D [12 W.E. Caswell and G.. Lepage, hys. Lett [13 Geoffrey T. Bodwin, Eric Braaten, and G. eter Lepage, hys. Rev. D ; Erratum, ibid. D