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Magnetic moment of the deuteron as probe of relativistic corrections S.G. Bondarenko, V.V.Burov Laboratory of Theoretical Physics, JINR, 141980 Dubna,Russia M.Beyer Max Planck AG, Rostock University, 18055 Rostock, Germany S.M.Dorkin Far East State University, 690000 Vladivostok, Russia Abstract: The calculation of the magnetic moment of the deuteron in the framework 6 of the Bethe–Salpeter approach is performed. The relativistic corrections are calculated 9 analytically andestimated numerically. Itisshownthatthemaincontributions aredueto 9 partial waves with positive energies and P–waves. A comparison with the non-relativistic 1 schemes of calculations including mesonic exchange currents is made. n a J 1 Introduction 5 2 1 The relativistic description of the deuteron has a long history. This problem still remains v present although non-relativistic schemes of calculations are widely used to analyze reactions 0 with the deuteron. There are different approaches which take into account relativistic effects. 4 0 They can be separated into three groups: (i) the Bethe–Salpeter approach [1], (ii) a reduction 1 of the Bethe–Salpeter equation (quasi-potential, light front dynamics), and (iii) prescriptional 0 6 treatments. 9 The Bethe–Salpeter approach is the most consistent one. However, it contains some dif- / h ficulties that do not allow to establish a direct link to the non-relativistic calculations – e.g. t - the absence of a non-relativistic reduction of arbitrary kernels or the problem to interpret the l c abnormal parity states. u n Non-relativistic calculations with mesonic exchange currents give a reasonable description : v of the electromagnetic processes of the deuteron (e.g. elastic scattering and electro disintegra- Xi tion). It is important to note that those mesonic exchange currents treated in a non–relativistic r framework are partially included in the above mentioned relativistic calculations (i) and (ii) a without additional assumptions. This has been demonstrated e.g. in ref. [2] for the electro disintegration of the deuteron in the formalism of light front dynamics. Here, we present the investigation of a simple electromagnetic property of the deuteron, the magnetic moment, in the framework of the Bethe–Salpeter approach, and show the connection to the results of a non-relativistic calculation. 2 Basic definitions The main object of the Bethe–Salpeter approach is the vertex function Γ(P,p) which obeys the Bethe–Salpeter equation, here written within the matrix formalism: d4k P P [Γ(P,p)C] = i v(p,k)Γ(1)S(1)( +k)[Γ(P,k)C]S˜(2)( k) Γ(2), (1) (2π)4 2 2 − Z 1 where C is charge conjugation matrix, v(p,k)Γ(1) Γ(2) is the interaction kernel, and S(i), S˜(i) ⊗ are Fermi propagators. The vertex function for the deuteron can be decomposed into basis states of definite values of orbital momentum L, spin S and ρ–spin ρ [3], [4]. The general form in the rest frame is given by Γ (P,p) = g (p , p ) Γα (p), (2) M α 0 | | M α X where g are radial functions, and Γα (p) are spin – angular momentum parts. α M In order to exhibit the ρ-spin dependence, the spin – angular momentum functions Γα (p) M may be replaced via Γα (p) Γα˜,ρ1ρ2(p), where M ≡ M pˆ +m 1+γ pˆ m Γα˜,++(p) = 2 0 Γ˜α˜ (p) 1 − , M 2E(m+E) 2 M 2E(m+E) q pˆ m 1+γ q pˆ +m ΓαM˜,−−(p) = 2E1(−m+E) − 2 0 Γ˜αM˜ (p) 2E2(m+E), (3) q pˆ +m 1+γ qpˆ +m ΓαM˜,+−(p) = 2E2(m+E) 2 0 Γ˜αM˜ (p) 2E2(m+E), q pˆ m 1 γ q pˆ m ΓαM˜,−+(p) = 2E1(−m+E) −2 0 Γ˜αM˜ (p) 2E1(−m+E), q q with α˜ L,S,J , m the nucleon mass, pˆ= γ pµ, and Γ˜α˜ given by ∈ { } µ M α˜ √8π Γ˜α˜ M 3S ξˆ 1 M 3D 1 ξˆ + 3(pˆ pˆ )(pξ ) p 2 1 −√2 M 2 1 − 2 M | |− 3P 3 h1ξˆ (pˆ pˆ ) (pξ ) p 1i 1 2 2 M 1 − 2 − M | |− 1P q h √3(pξ ) p 1 i 1 M − | | There are eight states in the deuteron channel (instead of two in the non-relativistic case), viz. 3S++, 3D++, 3S , 3D , 3Pe, 3Po, 1Pe, 1Po (notation: 2S+1Lρ1ρ2). The normalization 1 1 1−− 1−− 1 1 1 1 J condition for this functions can be written as: P +P = 1, P = P +P , + + 3S++ 3D++ − 1 1 P = P3S−− +P3D−− +P3Pe +P1Po +P3Pe +P1Pe, (4) − 1 1 1 1 1 1 introducing pseudo-probabilities P that are negative for the states 3S , 3D , 3Pe, 3Po, 1Pe, α 1−− 1−− 1 1 1 1Po, and positive for 3S++, 3D++ [3]. The calculation with realistic vertex functions give the 1 1 1 following values: 3D++ 3D 3Pe +3 Po 1Pe +1 Po 1 1−− 1 1 1 1 [%] 4.8 6 10 4 0.88 10 2 2.5 10 2 [3] − − − − · − · − · [%] 5.1 3.4 10 4 9 10 2 2.4 10 2 [5] − − − − · − · − · It is obvious that the main contribution to the normalization is due to the states with positive energies, and the contribution of the P–states is larger than that of the negative energies states by at least one order of magnitude. 2 3 The calculation of the magnetic moment The matrix element of the electromagnetic current used to evaluate the magnetic moment of the deuteron is given by ie P M J PM = d4p ′ ′ x h | | i 2π2M Z P γ qˆ qˆγ Tr Γ¯M′(P′,p′)S(1)( 2′ +p′)[γ1F1(s)(q2)− 1 4−m 1F2(s)(q2)]ΨM(P,p) , (5) (cid:26) (cid:27) where Ψ (P,p) = S(1)(P + p)Γ (P,p)S˜(2)(P p), the isoscalar form factors of the nucleons M 2 M 2 − are given by Fs (q2), and p = p+q/2, P = P +q. The magnetic moment then is evaluated 1,2 ′ ′ via 1 m M = +1 J M = 0 µ = √2lim h ′ | x| i, (6) D eM η 0 √η√1+η → with η = q2/4M2. Since the integral (5) vanishes at q2 = 0, we expand the integrand of (5) − in powers of √η. This is done in the Breit system. To this end the vertex functions need to be transformed from the rest system to the Breit system using the following formulae. Γ (P(B),p(B)) = Λ(P(B)) Γ (P(0),p(0)) Λ 1(P(B)), M M − Γ (P (B),p (B)) = Λ 1(P(B)) Γ (P(0),p (0)) Λ(P(B)), M ′ ′ − M ′ where Λ(P(B)) = (M +Pˆ(B)γ )/ 2M(E(B) +M), and the vectors P(B), P (B), p(B), p (B) are 0 ′ ′ connected with the respective veqctors in the rest system by P(B) = P(0), p(B) = p(0), P (B) = 1P(0), p (B) = 1p (0). (7) ′ − ′ − ′ L L L L The Lorentz transformation matrix is of the form L √1+η 0 0 √η − 0 1 0 0 =  . (8) L 0 0 1 0    √η 0 0 √1+η   −    From (7) it follows that p (0) = p (B) = (p(B) + 1qB) = 2p(0) + 1 q(B). To simplify ′ L ′ L 2 L 2L notation the vector p(0) will be denoted as p(0) p = (p ,p ,p ,p ). The components of the 0 x y z ≡ vector p (0) p are then given by ′ ′ ≡ p = (1+2η)p 2√η 1+ηp Mη, ′0 0 − z − p = p , p = p , q (9) ′x x ′y y p = (1+2η)p 2√η 1+ηp +M√η 1+η. ′z z − 0 q q With the help of the transformations (7), the integral (5) may then be written as ie P(0) P′M′ Jx PM = d4p Tr Γ¯M′(P(0),p′)S(1)( +p′) h | | i 2π2M ( 2 × Z γ qˆ qˆγ Λ(P(B))[γ F(s)(q2) 1 − 1F(s)(q2)]Λ(P(B))Ψ (P(0),p)[Λ 1(P(B))]2 , (10) × 1 1 − 4m 2 M − ) 3 where the wave function ΨM(P(0),p) and the vertex function Γ¯M′(P(0),p′) are taken in the deuteron rest frame. It is obvious from (10) that the sources of √η terms may appear in (i) the matrix Λ(P(B)), (ii) the propagator S(1)(12P(0) + p′), and (iii) the vertex function Γ¯M′(P(0),p′). In detail this reads for the matrix (i) γ qˆ qˆγ 1 κ Λ(P(B))[γ F(s)(q2) 1 − 1F(s)(q2)]Λ(P(B)) = (γ +√ηγ γ γ (γ qˆ qˆγ )), 1 1 − 4m 2 2 1 1 3 0 − 4m 1 − 1 [Λ 1(P(B))]2 = 1+√ηγ γ , (11) − 0 3 and for the propagator (ii) P(0) P(0) 4Mp 2p γ +(M 2p )γ S(1)( +p) = S(1)( +p) 1+√η z √η z 0 − 0 3. (12) ′ 2 2 " (P(0) +p)2 m2#− (P(0) +p)2 m2 2 − 2 − The corrections due to the vertex function (iii) may be separated into (α) corrections due to the radial wave functions, and (β) due to the spin – angular momentum function. This will be demonstrated in the following using only one component of the vertex function (2), i.e. Γ¯M′(P(0),p′) = g(p′0,|p′|) Γ¯M′(p′). (13) For the radial part we get the following expansion ∂ M 2p ∂ g(p , p ) = g(p , p )+√ηp 2 + − 0 g(p , p ). (14) ′0 | ′| 0 | | z − ∂p p ∂ p 0 | | 0 n | | | |o The spin – angular momentum part can be written (e.g. for the 3D++ state) as 1 Γ¯M′(p) = N(E′)(m−pˆ′1)Γ˜(p′,ξ)(m+pˆ′2), (15) where p z pˆ = pˆ +√η(M 2p )[ γ γ ], ′1 1 − 0 E 0 − 3 p z pˆ = pˆ +√η(M 2p )[ γ +γ ], (16) ′2 2 − 0 E 0 3 1 1 M 2p 0 N(E ) = = 1 √η − (m+2E)p , (17) ′ z 2E (m+E ) 2E(m+E) − E2(m+E) ′ ′ (cid:16) (cid:17) and corrections to Γ˜(p,ξ) can be calculated using (16)–(17). ′ 4 Results of calculations The general formula for magnetic moment can be written as µ = µ +µ +µ + 1 2 − − 3 1 µ = (µ +µ )(P +P ) (µ +µ )P +R , + p n 3S1++ 3D1++ − 2 p n − 2 3D1++ + 1 5 µ2− = −(µp +µn)P3S1−− +P3S1−− + 2(µp +µn)P3D1−− − 4P3D1−− +R2−, 1 1 1 µ1 = (µp +µn)(P3Pe +P3Po) (P3Pe +P3Po) (P1Pe +P1Po)+R1 , − −2 1 1 − 4 1 1 − 2 1 1 − 4 where R are the relativistic correction terms, viz. a 1 3S++ 1 3S++ m 3S++ 2m 3S++ R = (µ +µ )H 1 H 1 H 1 1 G 1 + −3 p n 1 − 2 2 − M 3 − − M 1 − (cid:18) (cid:19) 1 3D++ 1 3D++ m 3D++ 2m 3D++ (µ +µ 3)H 1 H 1 H 1 + 1 G 1 + −6 p n − 1 − 2 2 − M 3 − M 2 (cid:18) (cid:19) √2 m 3S++,3D++ + µ +µ (1+ ) H 1 1 , 3 p n − M 1 (cid:20) (cid:21) 1 2m 3S−− m 3S−− m 3S−− R = µ +µ (1 ) H 1 H 1 + H 1 2− −3 p n − − M 1 − M 2 M 3 − (cid:20) (cid:21) 1 4m 3D−− m 3D−− m 3D++ 3 2m −6 µp +µn −1+ M H1 1 − MH2 1 + MH3 1 + 4(1− M )P3D1−− + (cid:20) (cid:21) √2 m 3S−−,3D−− + µ +µ (1 ) H 1 1 , 3 p n − − M 1 (cid:20) (cid:21) 1 2m 1 3Pe,3Po R1− = 2 1− M µp +µn + 2 P3P1e +P3P1o −[µp +µn +1]H4 1 1 + (cid:18) (cid:19)(cid:20) (cid:21)(cid:16) (cid:17) 1 2m 1Pe,1Po +2 1− M P1P1e +P1P1o −2H4 1 1 + (cid:18) (cid:19)(cid:16) (cid:17) 3Pe,1Pe 3Po,1Po 3Pe,1Po 3Po,1Pe +G 1 1 +G 1 1 +G 1 1 +G 1 1, 4 4 5 6 and Hα,α′(Hα,α Hα) and Gα are integrals of the form i dp p 2d p with the integrands i i ≡ i i 4π−2M 0| | | | given by (compare with [6]) R H1α,α′ = 1− mE φα(p0,|p|)φα′(p0,|p|) ×( ((EE −+MM//22)),, ffoorr αα((αα′′)) ==33 SS11−+−+,,33DD11−+−+, (cid:16) (cid:17)h i 1 M 2 (E M/2), for α =3 S++,3D++, Hα = (E ) φ (p , p ) − 1 1 2 E − 2 α 0 | | ×( (E +M/2), for α =3 S1−−,3D1−− (cid:16) (cid:17)h i H3α = pE20 φα(p0,|p|) 2, H4α,α′ = mEp0 φα(p0,|p|)φα′(p0,|p|) (cid:16) (cid:17)h i (cid:16) (cid:17)h i 3S++ 2E +4m+3M M 2 G 1 = E φ (p , p ) 1 12E − 2 3S1++ 0 | | (cid:16) (cid:17)h i 3D++ E 4m+3M M 2 G 1 = − E φ (p , p ) 2 12E − 2 3D1++ 0 | | (cid:16) (cid:17)h i Gα4,α′ = −2√M2Em2p0 φα(p0,|p|)φα′(p0,|p|) (cid:16) (cid:17)h i 3Pe,1Po √2M √2M 4m2 G5 1 1 = (µp +µn) 2 − 2 (1− M2 ) φ3P1e(p0,|p|)φ1P1o(p0,|p|) (cid:16) (cid:17)h i G63P1o,1P1e = (µp +µn)√22M − √22M(1− mE22(1+ M4p202)) φ3P1o(p0,|p|)φ1P1e(p0,|p|) (cid:16) (cid:17)h i The relativistic corrections R can be estimated numerically for a separable model [7]. They a are smaller than the dominant terms by at least one order of magnitude. Taking into account different orders of pseudo-probabilities one gets µ = µ +∆µ (18) d NR 5 3 1 µ = (µ +µ )+ (µ +µ )P , NR p n 2 p n − 2 3D1++ ∆µ = R +∆µ + 1 − 3 ∆µ1 = (µp +µn)(P3Pe +P3Po) (µp +µn)(P1Pe +P1Po) − −2 1 1 − 1 1 1 1 (P3Pe +P3Po) (P1Pe +P1Po), −4 1 1 − 2 1 1 The contribution R is negative and ∆µ is positive. An estimation gives the following + 1 − results R /µ = (5.8 11) 10 2 %, ∆ /µ = 0.2 % (19) + NR − µ1 NR − − × − 5 Conclusion We have shown that the expression for the magnetic moment in the Bethe–Salpeter approach canbewritten inaformcloser tonon-relativistic calculations. The additionaltermsinequation (18) can be considered as relativistic corrections to the non-relativistic formula. The experimental value of the magnetic moment is known with a high accuracy: µ = exp 0.857406(1). The non-relativistic value reflects only the D-state probability. Whereas in the relativistic corrections P-states play the dominant role. The magnitude of the corrections can be compared with the contributions of mesonic exchange currents to the magnetic moment as extracted from ref. [8]. The main contribution is due to the pair term, which leads to ∆µ/µ = 0.21 0.22% for different forms of the NR − Bonn potential. The same size of this correction as compared to (19) may be considered as an indication that both corrections are of the same physical origin. This work has been partially supported by the Deutsche Akademische Austauschdienst. References [1] E.E.Salpeter and H.A.Bethe, Phys. Rev. C84, 1232 (1951). [2] B. Desplanques et.al. Preprint IPNO/TH 94-78 [3] M.J.Zuilhof, J.A.Tjon Phys.Rev. C6,2369 (1980). [4] J.J.Kubis, Phys. Rev. D6, 547 (1972). [5] K.Kazakov, private communication. We are grateful to K. Kazakov for providing us with this numbers prior to publication [6] N. Honzava, S. Ishida, Phys. Rev. C45, 47 (1992). [7] G.Rupp, J.A.Tjon. Phys. Rev. C 37,1729 (1988). [8] V.V. Burov, V.N. Dostovalov, S.E. Suskov. Elementary Particles and Atomic Nuclei 23(3),721 (1992). 6