Higher-order recoil corrections for triplet states of the helium atom
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Higher-order recoil corrections for triplet states of the heliumatom
Vojtˇech Patk´oˇs, V. A. Yerokhin, and Krzysztof Pachucki Faculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw, Poland Center for Advanced Studies, Peter the Great St. PetersburgPolytechnic University, 195251 St. Petersburg, Russia (Dated: July 11, 2018)
Abstract
Nuclear recoil corrections of order α m /M are calculated for the lowest-lying triplet states ofthe helium atom. It improves the theoretical prediction for the isotope shift of the 2 S − P transition energy and influences the determination of the He − He nuclear charge radii difference.This calculation is a step forward on the way towards the direct determination of the charge radiusof the helium nucleus from spectroscopic measurements.
PACS numbers: 31.30.Gs, 31.30.J- . INTRODUCTION The direct determination of the nuclear charge radius from the measured transition en-ergies has been so far carried out only for the hydrogen-like atoms [1]. In more complexsystems, the possibilities of such determination are limited by our insufficient knowledge ofthe QED effects. The main advantage of hydrogen-like atoms is that the relativistic electronwave function can be determined analytically in the limit of infinite nuclear mass. It is thenpossible to express all QED and nuclear recoil corrections within the Furry picture of QEDand calculate them either analytically in terms of the
Z α expansion or numerically to allorders in
Z α (where Z is the nuclear charge number and α is the fine structure constant).Calculations of QED effects in few-electron systems are much more difficult than in hy-drogen. Presently the best theoretical accuracy is achieved for the helium atom, whose (low-lying) energy levels are calculated rigorously within QED up to orders α m and α m /M [2, 3] (where m is the electron mass and M is the nuclear mass). The theoretical accuracyachieved in these calculations was not sufficient for determination of the charge radius of thehelium nucleus (i.e. α -particle). Significant progress, however, can be achieved by calculat-ing the next-order QED and nuclear recoil effects, namely α m /M and α m corrections.These calculations will bring the theoretical accuracy of the helium n = 2 transition ener-gies on a 10 kHz level, which will allow us to determine the α -particle charge radius with anaccuracy of a few parts of 10 − . Such a project is challenging but looks feasible, at least forthe triplet states.The most suitable transition for such a project is 2 S − P , which has already beenmeasured with sufficient accuracy [4, 5], E (2 S − P , He) centroid = 276 736 495 649 . .
1) kHz h . (1)The finite nuclear size contribution to this transition energy is E fs = 3 427 kHz h. Takinginto account that E fs is proportional to the nuclear charge radius squared, R , the expected10-kHz theoretical accuracy will determine the nuclear charge radius with 0 .
15% accuracy,∆ RR = 12 δE fs E fs ≈
12 103 427 ≈ . · − . (2)After the project is accomplished, we shall be able to compare the charge radius of the α -particle with the result from muonic helium, which is expected soon from the CREMA2ollaboration [6]. Such a comparison would be of particular interest in view of the dis-crepancy for the proton charge radius observed in the muonic hydrogen experiment [7, 8].Additional motivations for this project are the yet unexplained 4 σ discrepancy for the differ-ence in He and He nuclear charge radii [9] and plans to measure the charge radii differencefrom isotope shifts in helium-like ions [10]. In this work we make the first step towards theabsolute nuclear charge radius determination and calculate the nuclear recoil correction toorder α m /M for the 2 S and 2 P states of the helium atom.This paper is organized as follows. Section II introduces notations that will be usedthroughout the paper. Section III describes our approach to the calculation of the en-ergy levels by an expansion in the fine-structure constant α . Section IV reports the Foldy-Wouthuysen transformed NRQED Lagrangian, which is the starting point for our derivation.The derivation of the α m /M correction is presented in Section V. Section VI is devotedto the rearrangements of terms in such a way that all matrix elements become finite. Sec-tion VII presents the final formulas. Section VIII describes the numerical evaluation of allmatrix elements. Results and discussion are presented in Section IX. The principles of thedimensional regularization, details about the elimination of singularities, the simplificationof the formulas, and the reduction to the hydrogenic limit are presented in Appendices. II. NOTATIONS
We will use the following notations throughout the paper. The operators, energies, andwave functions for a nucleus with a finite mass M will be marked with indices “ M ”: X M , E M , φ M . The operators, energies, and wave functions in the infinite nuclear mass limit arewithout indices: X , E , φ . The recoil corrections to the operators and energies are denotedby δ M X and δ M E , X M ≡ X + mM δ M X + O (cid:16) mM (cid:17) , (3) E M = E + mM δ M E + O (cid:16) mM (cid:17) . (4)We also introduce the shorthand notations: h X i M ≡ h φ M | X | φ M i , (5)3nd δ M h X i ≡ (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) ~P I E − H ) ′ X (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29) + (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) X E − H ) ′ ~P I (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29) , (6)where ~P I is the momentum of the nucleus in the center of mass frame, and H , E , and φ arethe nonrelativistic Hamiltonian, energy, and the wave function in the infinite nuclear masslimit. III. NRQED APPROACH
According to QED theory, the expansion of energy levels in powers of α has the form E M ( α ) = E (2) M + E (4) M + E (5) M + E (6) M + E (7) M + O ( α ) , (7)where E M ( α ) ≡ E ( α, mM ), E ( n ) M is a contribution of order m α n and may include powers ofln α . E ( n ) M is in turn expanded in powers of the electron-to-nucleus mass ratio m/ME ( n ) M = E ( n ) + mM δ M E ( n ) + O (cid:16) mM (cid:17) . (8)Each term of the expansion E ( n ) M can be expressed as an expectation value of some effectiveoperator. Namely, E (2) M ≡ E M is the eigenenergy of the nonrelativistic Hamiltonian H (2) M ≡ H M with the eigenstate φ M H M = X a (cid:18) ~p a m − Zαr aI (cid:19) + X a>b X b αr ab + ~P I M . (9)Here ~P I is the momentum of the nucleus; in the center of mass system it is just ~P I = − P a ~p a . E (4) M is the expectation value of the Breit-Pauli Hamiltonian H (4) M [11], E (4) M = (cid:10) H (4) M (cid:11) M , (10)4 (4) M = X a (cid:20) − ~p a m + πZ α m δ ( r aI ) + Z α m ~σ a · ~r aI r aI × ~p a (cid:21) + X a
In order to derive the effective Hamiltonians H ( n ) M , and in particular H (6) M , we transform theQED Lagrangian to the NRQED form by using the Foldy-Wouthuysen (FW) transformation[16]. This transformation is the nonrelativistic expansion of the Dirac Hamiltonian in anexternal electromagnetic field, H = ~α · ~π + β m + e A , (13)where ~π = ~p − e ~A . The FW transformation SH F W = e i S ( H − i ∂ t ) e − i S = H + δH , (14)leads to a new Hamiltonian, which decouples the upper and lower components of the Diracwave function up to a specified order in the 1 /m expansion. In order to simplify the deriva-5ion of m /M α corrections, we start from FW Hamiltonian from Ref. [2], H F W = e A + π m − e m σ ij B ij − π m + e m (cid:8) σ ij B ij , p (cid:9) − e m (cid:16) ~ ∇ · ~E + σ ij (cid:8) E i , π j (cid:9)(cid:17) − e m (cid:8) ~p , ∂ t ~E (cid:9) + 3 e m (cid:8) σ ij E i p j , p (cid:9) + 1128 m [ p , [ p , e A ]] − m (cid:16) p ∇ ( e A ) + ∇ ( e A ) p (cid:17) + 116 m p , (15)where { x , y } and [ x, y ] stand for the anti-commutator and commutator, correspondingly, σ ij = 12 i [ σ i , σ j ] , (16) B ij = ∂ i A j − ∂ j A i , (17) E i = −∇ i A − ∂ t A i , (18)and apply further transformations. The first one S = − e m (cid:8) ~π , ~E (cid:9) (19)eliminates ∂ t ~E from H F W , δ H ≈ e m (cid:8) ~p , ∂ t ~E (cid:9) + e m ~E + 132 m [ p , [ p , e A ]] . (20)The second one S = e m σ ij { A i , π j } , (21)eliminates the transverse part ~E ⊥ = − ∂ t ~A , δ H ≈ e m σ ij { E i ⊥ , π j } − e m σ ij A i E j + i e m [ σ ij { A i , p j } , p ] . (22)The resulting new FW Hamiltonian is H F W = e A + π m − e m σ ij B ij − π m + e m (cid:8) σ ij B ij , p (cid:9) − e m (cid:16) ~ ∇ · ~E k + σ ij (cid:8) E i k , p j (cid:9)(cid:17) + e m σ ij E i k A j + i e m [ σ ij { A i , p j } , p ] + e m ~E k + 3 e m { p , σ ij E i k p j } + 5128 m [ p , [ p , e A ]] − m n p , ∇ ( e A ) o + 116 m p , (23)where ~E k = − ~ ∇ A . Since we are interested here in the leading O ( m/M ) term, the nucleuscan be treated nonrelativistically, so δ M H F W = 12 M (cid:0) ~P I + Z e ~A ) . (24)6 . THE HIGHER ORDER BREIT-PAULI HAMILTONIAN In this section we derive the effective operator H (6) M . The derivation is similar to thatin Ref. [2], including the use of the dimensional regularization. For the simplicity of thepresentation, all the derivations here will be performed in d = 3, but in such a way thatallows for a straightforward (and unique) generalization to the d = 3 − ǫ form. Thisgeneralization will be needed only for a few divergent terms, and details of the dimensionalregularization are presented in Appendix A.Using the nomenclature described in Appendix A, we denote by V the nonrelativisticinteraction potential V ≡ X a − Z αr aI + X a>b X b αr ab , (25)by E a the static electric field at the position of particle ae ~ E a ≡ −∇ a V = − Z α ~r aI r aI + X b = a α ~r ab r ab , (26)by ~ A a the vector potential at the position of particle a , which is produced by all otherparticles e A ia ≡ X b = a (cid:20) α r ab (cid:18) δ ij + r iab r jab r ab (cid:19) p jb m + α m σ kib r kab r ab (cid:21) − Zα r aI (cid:18) δ ij + r iaI r jaI r aI (cid:19) P jI M , (27)and by ~ A I the vector potential at the position of nucleus, which is produced by electrons e A iI ≡ X a α r aI (cid:18) δ ij + r iaI r jaI r aI (cid:19) p ja m . (28)Following Ref. [15], H (6) M is expressed as a sum of various contributions H (6) M = X i =1 , H Mi . (29) H M is the kinetic energy correction H M = X a p a m . (30) H M is a correction due to the static electric interaction, namely H M = X a (cid:18) e m ~ E a + 332 m { p a , e σ ija E ia p ja } + 5128 m [ p a , [ p a , V ]] − m n p a , ∇ a V o(cid:19) . (31)7 M is a correction to the Coulomb interaction between electrons, which comes from the 6 th term in H F W , namely − e m (cid:16) ~ ∇ · ~E k + σ ij (cid:8) E i k , p j (cid:9)(cid:17) . (32)If the interaction of both electrons is modified by this term, it can be obtained in thenon-retardation approximation, so H M = X a>b X b Z d k πk m (cid:18) k + 2 i σ ija k i p ja (cid:19) e i~k · ~r ab (cid:18) k + 2 i σ klb k k p lb (cid:19) = X a>b X b m (cid:26) − π ∇ δ ( r ab ) − π i σ ija p ia δ ( r ab ) p ja − π i σ ijb p ib δ ( r ab ) p jb +4 σ kia p ka (cid:20) δ ij π δ ( r ab ) + 1 r ab (cid:18) δ ij − r iab r jab r ab (cid:19)(cid:21) σ ljb p lb (cid:27) . (33) H M is the relativistic correction due to transverse photon exchange H M = X a − e m (cid:0) π a − e (cid:8) σ ija B ija , p a (cid:9)(cid:1) = X a e m (cid:0) { p a , ~p a · ~ A a } + { p a , σ ija ∇ ia A ja } (cid:1) . (34) H M comes from the remaining transverse photon exchange H M = X a (cid:18) e m σ ija E ia A ja + i e m [ σ ij {A ia , p ja } , p a ] (cid:19) . (35) H M comes from the double transverse photon exchange H M = X a e m A a + Z e M A I . (36) H M is a retardation correction in the nonrelativistic single transverse photon exchange E M = − e Z d k (2 π ) k (cid:18) δ ij − k i k j k (cid:19) (cid:20) X a = b X b (cid:28) φ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) p ia m + 12 m σ kia ∇ ka (cid:19) e i~k · ~r a ( H M − E M ) (cid:18) p jb m + 12 m σ ljb ∇ lb (cid:19) e − i~k · ~r b (cid:12)(cid:12)(cid:12)(cid:12) φ M (cid:29) − Z X b (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) P iI M e i~k · ~r I ( H − E ) (cid:18) p jb m + 12 m σ ljb ∇ lb (cid:19) e − i~k · ~r b (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29) − Z X a (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) p ia m + 12 m σ kia ∇ ka (cid:19) e i~k · ~r a ( H − E ) P jI M e − i~k · ~r I (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29)(cid:21) . (37)8his is the most complicated term in the evaluation, and we have to split it into four partswith no spin, single spin, and double spin terms, and the nuclear part E M = E M a + E M b + E M c + E M d . (38)The part with double spin operators is E M c = X a X a = b − e Z d k (2 π ) k ( σ kia k k ) ( σ lib k l )4 m D φ M (cid:12)(cid:12)(cid:12) e i~k · ~r a ( H M − E M ) e − i~k · ~r b (cid:12)(cid:12)(cid:12) φ M E . (39)One uses the commutation identity D e i~k · ~r a ( H M − E M ) e − i~k · ~r b E M + ( a ↔ b ) = Dh e i~k · ~r a , h ( H M − E M ) , e − i~k · ~r b iiE M = − m D(cid:2) p a , (cid:2) p b , e i~k · ~r ab (cid:3)(cid:3)E M (40)to express this correction in terms of the effective operator H M c , H M c = X a>b X b α m (cid:20) p a , (cid:20) p b , σ ija σ ijb r ab + σ ia σ jb r ab (cid:18) r iab r jab r ab − δ ij (cid:19)(cid:21)(cid:21) . (41)The part with no spin operator is E M a = X a = b X b − e Z d k (2 π ) k (cid:18) δ ij − k i k j k (cid:19)(cid:28) φ M (cid:12)(cid:12)(cid:12)(cid:12) p ia m n e i~k · ~r a ( H M − E M ) e − i~k · ~r b − ( H M − E M ) o p jb m (cid:12)(cid:12)(cid:12)(cid:12) φ M (cid:29) . (42)We subtracted here the term with k = 0. We ought to perform this in Eq. (37), but forsimplicity of writing we have not done it until now. We use another commutator identity e i~k · ~r a ( H M − E M ) e − i~k · ~r b − ( H M − E M ) =( H M − E M ) ( e i~k · ~r ab −
1) ( H M − E M ) + ( H M − E M ) (cid:20) p b m , e i~k · ~r ab − (cid:21) + (cid:20) e i~k · ~r ab − , p a m (cid:21) ( H M − E M ) + (cid:20) p b m , (cid:20) e i~k · ~r ab − , p a m (cid:21)(cid:21) (43)and the integration formula Z d k πk (cid:18) δ ij − k i k j k (cid:19) (cid:0) e i~k · ~r − (cid:1) = 18 r (cid:0) r i r j − δ ij r (cid:1) (44)9o obtain the effective operator H M a H M a = X a>b X b − α m (cid:26)(cid:2) p ia , V (cid:3) r iab r jab − δ ij r ab r ab (cid:2) V, p jb (cid:3) + (cid:2) p ia , V (cid:3) (cid:20) p b m , r iab r jab − δ ij r ab r ab (cid:21) p jb + p ia (cid:20) r iab r jab − δ ij r ab r ab , p a m (cid:21) (cid:2) V, p jb (cid:3) + p ia (cid:20) p b m , (cid:20) r iab r jab − δ ij r ab r ab , p a m (cid:21)(cid:21) p jb (cid:27) . (45)The part with the single spin operator is E M b = X a = b X b − i e m Z d k (2 π ) k (46) D φ M (cid:12)(cid:12)(cid:12)(cid:12)n e i~k · ~r a ( H M − E M ) e − i~k · ~r b σ kia k k p ib − σ ljb p ja k l e i~k · ~r a ( H M − E M ) e − i~k · ~r b o(cid:12)(cid:12)(cid:12)(cid:12) φ M E . With the help of the commutator in Eq. (43) and the integral Z d k π ~kk e i~k · ~r = i ~rr (47)one obtains H M b = X a>b X b α m (cid:26)(cid:20) σ ija r iab r ab , p a m (cid:21) (cid:2) V, p jb ] + (cid:20) p b m , (cid:20) σ ija r iab r ab , p a m (cid:21)(cid:21) p jb − (cid:2) p ja , V (cid:3) (cid:20) p b , σ ijb r iab r ab (cid:21) − p ja (cid:20) p a m , (cid:20) σ ija r iab r ab , p b m (cid:21)(cid:21)(cid:27) . (48)Finally, the nuclear part is E M d = − e Z d k (2 π ) k (cid:18) δ ij − k i k j k (cid:19) × ZM X a,b (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) p ia ( H − E ) (cid:18) p jb m + 12 m σ ljb ∇ lb (cid:19) e − i~k · ~r bI (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29) + h . c . = − Z αM X a,b (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) [ p ia , V ] (cid:20) H − E , ( r ibI r jbI − δ ij r bI )8 r bI p jb m − m σ ljb r lbI r bI (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) φ (cid:29) ≡ h φ | H M d | φ i . (49)We have checked that the non-recoil part agrees with that derived in [15] and that thespin-dependent recoil part agrees with that in [17]. Here, we are interested in the spin-independent part, which in the center-of-mass system ~P I = − P a ~p a is (from now on we use10tomic units m = 1) H M = X a p a ,H M = X a (cid:18) ( ∇ a V ) p a , [ p a , V ]] − n p a , ∇ a V o(cid:19) ,H M = X a>b X b (cid:26) − π ∇ δ ( r ab ) + 23 σ ija σ ijb (cid:20) ~p a π δ ( r ab ) ~p b − p ia r ab (cid:18) δ ij − r iab r jab r ab (cid:19) p jb (cid:21)(cid:27) ,H M = 18 X a (cid:20)X b = a (cid:26) p a , p ia (cid:18) δ ij r ab + r iab r jab r ab (cid:19) p jb (cid:27) − σ ija σ ijb (cid:8) p a , π δ ( r ab ) (cid:9) + ZM X b (cid:26) p a , p ia (cid:18) δ ij r aI + r iaI r jaI r aI (cid:19) p jb (cid:27)(cid:21) ,H M = X a = b,b σ ija σ ijb (cid:18) − ~r ab r ab · ∇ a V + 116 (cid:20)(cid:20) r ab , p a (cid:21) , p a (cid:21)(cid:19) ,H M = X a X b = a X c = a (cid:20) p ib (cid:18) δ ij r ab + r iab r jab r ab (cid:19) (cid:18) δ jk r ac + r jac r kac r ac (cid:19) p kc + σ ijb σ ijc ~r ab r ab ~r ac r ac + Z mM p ib (cid:18) δ ij r ab + r iab r jab r ab (cid:19) (cid:18) δ jk r aI + r jaI r kaI r aI (cid:19) p kc (cid:21) + X a X b Z mM × (cid:20) p ia (cid:18) δ ij r aI + r iaI r jaI r aI (cid:19) (cid:18) δ jk r bI + r jbI r kbI r bI (cid:19) p kb + σ ija σ ijb ~r aI r aI ~r bI r bI (cid:21) ,H M a = X a>b X b − (cid:26)(cid:2) p ia , V (cid:3) r iab r jab − δ ij r ab r ab (cid:2) V, p jb (cid:3) + (cid:2) p ia , V (cid:3) (cid:20) p b , r iab r jab − δ ij r ab r ab (cid:21) p jb + p ia (cid:20) r iab r jab − δ ij r ab r ab , p a (cid:21) (cid:2) V, p jb (cid:3) + p ia (cid:20) p b , (cid:20) r iab r jab − δ ij r ab r ab , p a (cid:21)(cid:21) p jb (cid:27) ,H M c = X a>b X b σ ija σ ijb (cid:20) p a , (cid:20) p b , r ab (cid:21)(cid:21) ,H M d = i Z mM X a,b ∇ ia V (cid:20) H − E , ( r ibI r jbI − δ ij r bI ) r bI p jb (cid:21) . (50)Further Hamiltonians H M . . . H M come from the high-energy contributions, so they areproportional to Dirac delta’s and we will account for them in the next paragraph. These H Mi form a general m α effective Hamiltonian for arbitrary atom and arbitrary state, neglectingthe spin-dependent operators.From now on we consider the specific case of the triplet states of the He atom, where the11xpectation value of δ ( r ab ) vanishes and almost all matrix elements become finite.The Breit-Pauli Hamiltonian of Eq. (11) is split into four parts (with r ≡ r , r aI ≡ r a and ~P ≡ ~p + ~p ) H (4) M = H MA + H MB + H MC + H MD , (51)where H MA = −
18 ( p + p ) + Z π δ ( r ) + δ ( r )] − p i (cid:18) δ ij r + r i r j r (cid:19) p j − Z mM (cid:20) p i (cid:18) δ ij r + r i r j r (cid:19) + p i (cid:18) δ ij r + r i r j r (cid:19)(cid:21) P j , (52) H MB = (cid:20) Z (cid:18) ~r r × ~p + ~r r × ~p (cid:19) − ~rr × ( ~p − ~p ) + Z mM (cid:18) ~r r + ~r r (cid:19) × ~P (cid:21) ~σ + ~σ , (53) H MC = (cid:20) Z (cid:18) ~r r × ~p − ~r r × ~p (cid:19) + 14 ~rr × ( ~p + ~p ) + Z mM (cid:18) ~r r − ~r r (cid:19) × ~P (cid:21) ~σ − ~σ , (54) H MD = 14 (cid:18) ~σ ~σ r − ~σ · ~r ~σ · ~rr (cid:19) . (55)The corresponding second-order correction is A M = X I = A,B,C,D h H MI E M − H M ) ′ H MI i M , (56)whereas the first-order contribution is given by B M = h H (6) M i M . (57) H (6) M consists of eleven parts according to Eq. (29) with H M . . . H M already defined and H M = Z mM (cid:18) − (cid:19) (cid:2) δ ( r ) + δ ( r ) (cid:3) , (58) H M = Z mM (cid:18) − π − ζ (3) π (cid:19) (cid:2) δ ( r ) + δ ( r ) (cid:3) , (59) H M = π Z (cid:18) − (cid:19) (cid:2) δ ( r ) + δ ( r ) (cid:3) , (60) H M = Zπ (cid:18) − − π + 32 π ln(2) − ζ (3) (cid:19) (cid:2) δ ( r ) + δ ( r ) (cid:3) . (61)Here H M is the high-energy pure recoil correction taken from hydrogenic results, H M standsfor the radiative recoil correction, and H M and H M stand for the one-loop and two-loopradiative corrections, correspondingly [18]. 12 I. ELIMINATION OF SINGULARITIES
The principal problem of the used approach is that both the first-order and the second-order contributions are divergent; the divergence cancels out only in the sum of these contri-butions. To achieve the cancellation of the divergences, we (i) regularize the divergent contri-butions by switching to d = 3 − ǫ dimensions, (ii) move singularities from the second-ordercontributions to the first-order ones, and (iii) cancel algebraically the 1 /ǫ terms. Moreover,we notice that the recoil corrections are of two types: (i) corrections due to the perturbationof the wave function φ , the energy of the reference state E , and the nonrelativistic Hamilto-nian H by the nuclear kinetic energy ~P / (2 M ), and (ii) corrections due to the extra recoiloperators in H (4) M and H (6) M . We will use this fact in the following derivations. A. Recoil correction from the second-order contribution
In this subsection we consider the recoil correction coming from the second-order matrixelements, i.e. the first term in Eq. (12), which is denoted by A M . The recoil correction fromthe second term in Eq. (12), denoted by B M , will be examined in the next subsection.The second-order contribution with H MA is divergent and has to be regularized. Regular-ization is performed by rewriting H MA in such a way that the singularities are moved fromthe second-order matrix element into the first-order ones, where they cancel each other. Todo this, we write H MA as H MA = H MR − (cid:26) H M − E M , Zr + Zr − r − mM (cid:18) Zr + Zr (cid:19)(cid:27) = H MR + (cid:8) H M − E M , Q M (cid:9) . (62)The operator Q M is the same as in [2] with the exception that it also includes a recoil part δ M Q . The regular part of operator H MA can be evaluated to yield H MR = H R + mM δ M H R , (63) H R | φ i = (cid:26) −
12 ( E − V ) − Z ~r · ~ ∇ r − Z ~r · ~ ∇ r + 14 ∇ ∇ − p i V ij ( r ) p j (cid:27) | φ i , (64) δ M H R | φ i = (cid:26) ( E − V ) (cid:18) ~P − (cid:28) ~P (cid:29)(cid:19) + 3 Z ~r · ~ ∇ r + 3 Z ~r · ~ ∇ r − Z p i V ij ( r ) P j − Z p i V ij ( r ) P j (cid:27) | φ i , (65)13here V = − Zr − Zr + 1 r , (66) V ij ( x ) = 12 x (cid:18) δ ij + x i x j x (cid:19) . (67)Moreover, the kinetic energy of the nucleus is h ~P / i = δ M E . After regularization, the firstterm in Eq. (12) takes the form A M = X a = R,B,C,D (cid:28) H Ma E M − H M ) ′ H Ma (cid:29) M + (cid:10) Q M ( H M − E M ) Q M (cid:11) M + 2 E (4) M (cid:10) Q M (cid:11) M − (cid:10) H (4) M Q M (cid:11) M = A M + A M . (68)where A M stands for the first term (i.e. the second-order contribution), and A M incorporatesthe remaining first-order matrix elements. Recoil corrections are obtained by perturbing thesecond-order matrix element by the kinetic energy of the nucleus and keeping the first-orderterms in the nuclear mass. So, δ M A is δ M A = X a = R,B,C,D (cid:28) H a E − H ) ′ (cid:20) ~P − δ M E (cid:21) E − H ) ′ H a (cid:29) + 2 (cid:28) H a E − H ) ′ [ H a − h H a i ] 1( E − H ) ′ ~P (cid:29) + 2 (cid:28) δ M H a E − H ) ′ H a (cid:29) , (69)while the first-order terms are A M = h Q ( H M − E M ) Q i M + 2 E (4) M h Q i M − h H (4) M Q i M + mM (cid:26) h Q ( H − E ) δ M Q i + 2 E (4) h δ M Q i − h H A δ M Q i (cid:27) . (70)Reduction of these terms will be left to the Appendix, and we present here the final result14or the recoil part δ M A = δ M (cid:28) − (cid:18) Z r + Z r (cid:19) − r + 14 (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 2 E (4) Q + Z ( Z − π (cid:20) δ ( r ) r + δ ( r ) r (cid:21) − p i (cid:18) Zr + Zr − r (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j + 12 (cid:20) p i , (cid:20) p j , r (cid:21)(cid:21) r (cid:18) δ ij + r i r j r (cid:19) + ( E − V ) Q + 18 p (cid:18) Zr + Zr (cid:19) p − p r p −
18 [ p , [ p , V ]] (cid:29) + δ M E (4) (cid:18) E + (cid:28) r (cid:29)(cid:19) + (cid:28) (cid:18) Z r + Z r (cid:19) − Z ~r · ~r r r + 32 E (4) r − EE (4) + 34 ( E − V ) (cid:18) Zr + Zr (cid:19) − p (cid:18) Zr + Zr (cid:19) p + 34 p i (cid:18) Zr + Zr (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j + 2 δ M E ( E − V ) Q + π Z δ ( r ) (cid:18) Z − r + 2 E + 2 Z (cid:19) + π Z δ ( r ) (cid:18) Z − r + 2 E + 2 Z (cid:19) + ~P (cid:20) E (cid:18) Zr + Zr (cid:19) − E r + 14 (cid:18) Zr + Zr (cid:19) − r (cid:18) Zr + Zr (cid:19) + 12 r (cid:21) ~P − X a Z P i (cid:18) δ ij r a + r ia r ja r a (cid:19) (cid:18) Zr + Zr − r (cid:19) p ja (cid:29) . (71) B. Recoil correction from the first-order terms
In this section we examine the recoil correction coming from the first-order matrix ele-ments, i.e. the second term in Eq. (12), which is denoted as B M . Using Eq. (29), B M canbe written as B M = h H (6) M i M = X i =1 ... h H Mi i M . (72)For each of the operators H Mi = H i + mM δ M H i , the recoil correction is the sum of two parts:(i) perturbation of the nonrelativistic wave function, E and H by the nuclear kinetic energyin the non-recoil part, and (ii) the expectation value of the recoil part δ M H i (if present).The derivation is straightforward but tedious, so we have moved the description of this15alculation to the Appendix and present only the final result for the recoil correction δ M B , δ M B = δ M (cid:28) (cid:18) Z r + Z r (cid:19) − (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 14 (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r + 4148 r + 1196 (cid:20) p , (cid:20) p , r (cid:21)(cid:21) + 12 ( E − V ) − p ( E − V ) p − πZ (cid:20) (cid:18) E + Z − r (cid:19) δ ( r ) + 2 (cid:18) E + Z − r (cid:19) δ ( r ) − p δ ( r ) − p δ ( r ) (cid:21) − π ∇ δ ( r ) + 12 p i (cid:0) E − V (cid:1) r (cid:18) δ ij + r i r j r (cid:19) p j − Z r i r j r r (cid:18) r i r j r − δ ij r (cid:19) − Z (cid:20) r i r p k (cid:18) δ jk r i r − δ ik r j r − δ ij r k r − r i r j r k r (cid:19) p j + (1 ↔ (cid:21) + 18 p k p l (cid:20) − δ il δ jk r + δ ik δ jl r − δ ij δ kl r − δ jl r i r k r − δ ik r j r l r + 3 r i r j r k r l r (cid:21) p i p j + 14 (cid:18) ~p r ~p + ~p r ~p (cid:19) + H + H (cid:29) + (cid:28) δ M E ( E − V ) − ~P ( E − V ) ~P − δ M E p p + 316 P p p − (cid:18) δ M E + 3 E + 3 ( Z − r − ~p · ~p (cid:19) πZ δ ( r ) + (1 ↔ δ M E p i r (cid:18) δ ij + r i r j r (cid:19) p j − ~P p i r (cid:18) δ ij + r i r j r (cid:19) p j + 1332 (cid:18) Z r + Z r (cid:19) + 1316 Z ~r · ~r r r (cid:29) + h δ M H (6) i , (73)where h δ M H (6) i = (cid:28) Z " p i ( E − V ) δ ij r + r i r j r ! + p i ( E − V ) δ ij r + r i r j r ! P j − Z (cid:20) p i p k δ ij r + r i r j r ! p k P j + p i p k δ ij r + r i r j r ! p k P j (cid:21) + Z ~r · ~r r r + Z (cid:20) p i (cid:18) δ ij r + r i r j r (cid:19) δ jk r + r j r k r ! + p i (cid:18) δ ij r + r i r j r (cid:19) δ jk r + r j r k r !(cid:21) P k + Z (cid:20) ~p r ~p + ~p r ~p + p i δ ij r + r i r j r ! δ jk r + r j r k r ! p k (cid:21) + Z ~r · ~r r r + Z ~r · ~r r r + Z (cid:18) r i r + r i r (cid:19) (cid:18) r i r j − δ ij r r − r i r j − δ ij r r (cid:19) r j r + Z (cid:20) p k r i r − δ ik r j r + δ jk r i r − δ ij r k r − r i r j r k r ! p j + (1 ↔ (cid:21) + Z r + Z r − Z r − Z r − Z π δ ( r ) + π δ ( r )] + δ M H + δ M H (cid:29) . (74)16t this point we have obtained all the terms contributing to the recoil correction. C. Cancellation of singularities
The first-order terms δ M A and δ M B could be further transformed using various identities,namely (cid:20) p , (cid:20) p , r (cid:21)(cid:21) = (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r + P i P j r i r j − δ ij r r , (75)1 r = 1 r + 12 (cid:18) ~p r ~p + ~p r ~p (cid:19) − (cid:18) E + Zr + Zr (cid:19) r − mM (cid:18) δ M E − ~P (cid:19) r , (76) Z r = ~p Z r ~p − (cid:18) E + Zr + Zr − r (cid:19) Z r + p Z r − mM (cid:18) δ M E − ~P (cid:19) Z r , (77) p i (cid:18) δ ij r + r i r j r (cid:19) p j = − H (4) M − ( E − V ) + 12 p p + Zπ (cid:2) δ ( r ) + δ ( r ) (cid:3) − mM (cid:20)(cid:0) E − V (cid:1) (cid:18) δ M E − ~P (cid:19) − δ M H (4) (cid:21) , (78) ∇ δ ( r ) = 2 ~p δ ( r ) ~p . (79)Using these identities we remove all the remaining singularities and transform the results intoa form suitable for numerical calculation. The final result for recoil correction is presentedin the next section. VII. FINAL FORMULA
The final results are split into seven parts: (i) the second-order and third-order matrixelements containing H R , (ii) the second-order and third-order matrix elements containing H B , (iii) the second-order and third-order matrix elements containing H C , (iv) the third-order matrix elements containing H D , (v) the first-order matrix elements with the referencestate and the perturbed wave function, and (vi) the remaining first-order terms with theexception of (vii) pure recoil, the radiative recoil and the recoil corrections to one-loop andtwo-loops radiative corrections. 17he final formula is then E recoil = E i + E ii + E iii + E iv + E v + E vi + E vii , (80) E i = * H R E − H ) ′ (cid:18) ~P − δ M E (cid:19) E − H ) ′ H R + (81)+ 2 * H R E − H ) ′ [ H R − h H R i ] 1( E − H ) ′ ~P + + 2 (cid:28) δ M H R E − H ) ′ H R (cid:29) ,E ii = * H B E − H ) (cid:18) ~P − δ M E (cid:19) E − H ) H B + + 2 * H B E − H ) H B E − H ) ~P + + 2 (cid:28) H B E − H ) δ M H B (cid:29) , (82) E iii = * H C E − H ) (cid:18) ~P − δ M E (cid:19) E − H ) H C + + 2 * H C E − H ) H C E − H ) ~P + + 2 (cid:28) H C E − H ) δ M H C (cid:29) , (83) E iv = * H D E − H ) (cid:18) ~P − δ M E (cid:19) E − H ) H D + + 2 * H D E − H ) H D E − H ) ~P + , (84)Here δ M E = (cid:28) ~P (cid:29) = − E + h ~p · ~p i . (85)For E v and E vi the results can be brought into a more suitable form by introducing set ofoperators Q i , see Tables I and II, E v = − E Z δ M h Q i + 18 Z (1 − Z ) δ M h Q i + 316 Z δ M h Q i − δ M h Q i + E + 2 E (4) δ M h Q i − E δ M h Q i + 78 δ M h Q i + E Z δ M h Q i + E Z δ M h Q i − E Z δ M h Q i − Z δ M h Q i + Z δ M h Q i − Z δ M h Q i− Z δ M h Q i − Z δ M h Q i + Z δ M h Q i − Z δ M h Q i + Z δ M h Q i + Z δ M h Q i + 138 δ M h Q i + Z δ M h Q i − δ M h Q i − Z δ M h Q i− E δ M h Q i − Z δ M h Q i + 14 δ M h Q i + 18 δ M h Q i (86)18nd E vi = (cid:28) − E − EE (4) − E δ M E − E + δ M E + 4 Z Z Q − Z (8 Z − Q + 3 E + 2 E δ M E + 6 E (4) + 2 δ M E (4) Q − δ M E Q + 2 E + δ M E Z Q + (3 E + δ M E ) ( Z Q − Z Q ) − Z Q + 52 Z Q − Z Q + 32 Z Q + Z Q + 32 Z Q + 32 Z Q − δ M E Q − Z Q + 38 Z Q + 1924 Z Q − E Z Q + 12 E Q − Z Q − Z Q + 32 Z Q + 516 Q + 316 Q − Q + Z Q + Z Q − Z Q + Z Q + Z Q + Z Q + Z Q − Z Q + Z Q (cid:29) . (87)Finally, E vii = h δ M H + δ M H i + δ M h H + H i . (88) VIII. NUMERICAL CALCULATIONS OF MATRIX ELEMENTS
The helium wave function for triplet states is expanded in a basis set of exponentialfunctions in the form of [19] φ ( S ) = N X i =1 v i (cid:2) e − α i r − β i r − γ i r − ( r ↔ r ) (cid:3) , (89) φ ( P ) = N X i =1 v i (cid:2) ~r e − α i r − β i r − γ i r − ( r ↔ r ) (cid:3) , (90)where α i , β i , and γ i are generated quasi-randomly with conditions: A < α i < A , β i + γ i > ε,B < β i < B , α i + γ i > ε,C < γ i < C , α i + β i > ε. (91)In order to obtain a highly accurate representation of the wave function, following Korobov[19], we use a double set of the nonlinear parameters of the form (89). The parameters A i , B i , C i , and ε are determined by the energy minimization, with the condition that ε > v i in Eq.(89) form a vector v , which is a solution of the generalized eigenvalue problem H v = E N v , (92)where H is the matrix of the Hamiltonian in this basis, N is the normalization (overlap)matrix, and E the eigenvalue, the energy of the state corresponding to v . For the solution ofthe eigenvalue problem with N = 100 , , , , , E (2 S ) = − .
175 229 378 236 791 306 , (93) E (2 P ) = − .
133 164 190 779 283 199 . (94)The calculation of matrix elements of the nonrelativistic Hamiltonian is based on the singlemaster integral, 116 π Z d r Z d r e − αr − βr − γr r r r = 1( α + β )( β + γ )( γ + α ) . (95)The integrals with any additional powers of r i in the numerator can be obtained by differ-entiation with respect to the corresponding parameter α , β or γ . The matrix elements ofrelativistic corrections involve inverse powers of r , r , r . These can be obtained by integra-tion with respect to a corresponding parameter, which leads to the following formulas116 π Z d r Z d r e − αr − βr − γr r r r = 1( β + α )( α + β ) ln (cid:18) β + γα + γ (cid:19) , (96)116 π Z d r Z d r e − αr − βr − γr r r r = 12 β (cid:20) π (cid:18) α + ββ + γ (cid:19) + Li (cid:18) − α + γα + β (cid:19) + Li (cid:18) − α + γβ + γ (cid:19)(cid:21) . (97)All matrix elements involved in the α m /M correction, see Tables I and II, can be expressedin terms of rational, logarithmic, and dilogarithmic functions, as above. The high qualityof the wave function allowed us to obtain accurate values of the matrix elements of Q i and δ M Q i operators. The corresponding numerical results are presented in Tables I and II.For the second-order matrix elements, the inversion of the operator E − H is performedin the basis of even or odd parity with l = 0 , , H A ; for 2 P , also H B and H D ), it is20ecessary to subtract the reference state from the implicit sum over states. This is obtainedby the orthogonalization with respect to the eigenstate with the closest-to-zero eigenvalueof H − E . This eigenvalue is not exactly equal to 0 because we use a basis set with differentparameters, which are obtained by minimization of that particular term. IX. RESULTS AND DISCUSSION
In this paper, we derived the complete recoil contribution of order α m /M to the energylevels of the triplet states of helium. The final result is given by Eqs. (81) - (88). It is acombination of various contributions of two types: (i) perturbations of the nonrelativisticwave function, energy, and Hamiltonian in the non-recoil matrix elements by the nuclearkinetic energy operator and (ii) expectation values of extra recoil operators. In Tables I andII the matrix elements of individual operators entering Eqs. (86) - (88) are presented.Results of our numerical calculation of E i . . . E vii for the 2 S and 2 P states are presentedin Table III. For the 2 S state, the total α m /M recoil correction is dominated by the Diracdelta-like term coming from the one-loop radiative correction, see Eq. (60); the result for theionization energy being − .
91 kHz. Contrary to that, for the 2 P state, the contributionsfrom E i . . . E vii are of similar size but of the opposite sign. So, the total correction to theionization energy is only − .
11 kHz in this case. Contributions of individual recoil terms tothe 2 S − P transition energy of helium are presented in Table IV.The obtained results can be used to improve the theoretical prediction of the He − Heisotope shift of the 2 S − P transition. In this case the total m /M α recoil correctioncalculated in this work is − . m /M α terms, we considered two typical contributions. One of them is the hydrogenicrecoil m /M α contribution (as evaluated in [20]) scaled by the expectation value of δ ( r )operator; whereas the second is the hydrogenic m α contribution with the δ ( r ) operatorperturbed by ~p · ~p . Since both contributions happen to be small and of opposite sign, wetook the largest one and multiplied it by a conservative coefficient of 2.The updated theoretical result for the He − He isotope shift allows us to improve theaccuracy of determination of the nuclear charge radii difference δR = R ( He) − R ( He),derived from the 2 S − P transition [9], namely δR [Cancio 2012] = 1 . and21 R [Shiner 95] = 1 . . This reduces slightly the discrepancy with the result fromthe 2 S − S transition [9], δR [Rooij 2011] = 1 . , but does not remove it entirely.In order to clarify this further one needs to calculate the complete α m /M recoil correctionalso for singlet states of helium. Acknowledgments
K.P. and V.P. acknowledge support by the National Science Center (Poland) Grant No.2012/04/A/ST2/00105, and V.A.Y. acknowledges support by the Ministry of Educationand Science of the Russian Federation (program for organizing and carrying out scientificinvestigations) and by RFBR (grant No. 16-02-00538). [1] P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod. Phys. , 035009 (2016).[2] K. Pachucki, Phys. Rev. A , 022512 (2006).[3] V. A. Yerokhin and K. Pachucki, Phys. Rev. A et al , Phys. Rev. Lett. , 023001 (2004), [(E) ibid , 139903 (2006)].[5] K. Pachucki and V. A. Yerokhin, J. Phys. Conf. Ser. , 012007 (2011).[6] R. Pohl, talk at the ECT workshop “The proton radius puzzle” (2016), unpublished.[7] R. Pohl et al., Nature (London) , 213 (2010).[8] A. Antognini et al., Science , 417 (2013).[9] K. Pachucki and V. A. Yerokhin, J. Phys. Chem. Ref. Data , 031206 (2015).[10] W. N¨ortersh¨auser, private communication .[11] H.A. Bethe and E.E. Salpeter, Quantum Mechanics Of One- And Two-Electron Atoms ,Plenum Publishing Corporation, New York (1977).[12] P.K. Kabir and E.E. Salpeter. Phys. Rev. , 1256 (1957); H. Araki Prog. Theor. Phys. ,619 (1957); J. Sucher, Phys. Rev. , 1010 (1957).[13] J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics , Editor T. Kinoshita, WorldScientific Singapore (1990).[14] K. Pachucki, J. Phys. B , 5123 (1998).[15] K. Pachucki, Phys. Rev. A , 012503 (2005).
16] C. Itzykson and J. B. Zuber,
Quantum Field Theory , McGraw–Hill, New York (1990).[17] K. Pachucki and J. Sapirstein, J. Phys. B , 803 (2003).[18] M.I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rep. , 63 (2001).[19] V. Korobov, Phys. Rev. A , 064503 (2000), Phys. Rev. A , 024501 (2002).[20] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. Lett. , 233002 (2015). Appendix A: Dimensional regularization
Since the triplet state wave function vanishes at r = 0, the electron-electron operatorsdo not lead to any singularities, and thus can be calculated directly in d = 3. Thereare, however, several terms arising from the electron-nucleus recoil operators, which needto be treated within the dimensional regularization in order to isolate the singular partof the operator. We essentially repeat the approach from [2], so only a brief introductionto dimensional regularization is presented here. The dimension of space is assumed to be d = 3 − ǫ . The surface area of the d -dimensional unit sphere isΩ d = 2 π d/ Γ( d/ , (A1)and the d -dimensional Laplacian is ∇ = r − d ∂ r r d − ∂ r . (A2)The photon propagator, and thus Coulomb interaction, preserves its form in the momentumrepresentation, while in the coordinate representation it is V ( r ) = Z d d k (2 π ) d πk e i~k · ~r = π ǫ − / Γ(1 / − ǫ ) r ǫ − ≡ C r − ǫ . (A3)The elimination of singularities will be performed in atomic units. In accordance with [2]this is achieved by transformation ~r → ( mα ) − / (1+2 ǫ ) ~r (A4)and pulling factors m (1 − ǫ ) / (1+2 ǫ ) α / (1+2 ǫ ) and m (1 − ǫ ) / (1+2 ǫ ) α / (1+2 ǫ ) from H and H (6) . Thenonrelativistic Hamiltonian of hydrogen-like systems is H = ~p − Z C r − ǫ , (A5)23nd that of helium-like systems is H = ~p ~p − Z C r − ǫ − Z C r − ǫ + C r − ǫ . (A6)The solution of the stationary Schr¨odinger equation H φ = E φ is denoted by φ ; we willnever need its explicit (and unknown) form in d -dimensions. Instead, we will use onlythe generalized cusp condition to eliminate various singularities from matrix elements withrelativistic operators. Namely, we expect that for small r ≡ r φ ( r ) ≈ φ (0) (1 − C r γ ) (A7)with some coefficient C and γ to be obtained from the two-electron Schr¨odinger equationaround r = 0, (cid:20) − ∇ − Z V ( r ) (cid:21) φ (0) (1 − C r γ ) ≈ E φ (0) (1 − C r γ ) . (A8)From cancellation of small r singularities on the left side of the above equation, one obtains γ = 1 + 2 ǫ, (A9) C = − Z π ǫ − / Γ( − / − ǫ ) . (A10)Therefore, the two-electron wave function around r = 0 behaves as φ ( ~r , ~r ) ≈ φ ( r = 0) (1 − C r ǫ ) . (A11)Apart from the Coulomb potential V ( r ) in the coordinate space, we need also other functions,which appear in the calculations of relativistic operators, namely V ( r ) = Z d d k (2 π ) d πk e i~k · ~r , (A12) V ( r ) = Z d d k (2 π ) d πk e i~k · ~r . (A13)They can be obtained from the differential equations − ∇ V ( r ) = V ( r ) , (A14) −∇ V ( r ) = V ( r ) , (A15)with the result V ( r ) = C r ǫ , (A16) V ( r ) = C r ǫ , (A17)24here C = 14 π ǫ − / Γ( − / − ǫ ) , (A18) C = 132 π ǫ − / Γ( − / − ǫ ) . (A19)Using V i , we calculate various integrals involving the photon propagator in the Coulombgauge, namely Z d d k (2 π ) d πk (cid:18) δ ij − k i k j k (cid:19) e i~k · ~r = δ ij V + ∂ i ∂ j V = π ǫ − / r − ǫ (cid:20) δ ij Γ( − / − ǫ ) r + 18 Γ(1 / − ǫ ) r i r j (cid:21) ≡ (cid:20) r (cid:0) r i r j − δ ij r (cid:1)(cid:21) ǫ = W ijǫ , (A20)and Z d d k (2 π ) d πk (cid:18) δ ij − k i k j k (cid:19) e i~k · ~r = δ ij V + ∂ i ∂ j V = π ǫ − / r − ǫ (cid:20) δ ij Γ(1 / − ǫ ) r + Γ(3 / − ǫ ) r i r j (cid:21) ≡ (cid:20) r (cid:0) δ ij r + r i r j (cid:1)(cid:21) ǫ . (A21)Now we are ready to remove the singularities from matrix elements of various operators. Byconvention we pull out a common factor (cid:2) (4 π ) ǫ Γ(1 + ǫ ) (cid:3) from all matrix elements. Then,for example, the matrix element h [ Z /r ] ǫ i with r = r is *(cid:20) Zr (cid:21) ǫ + = Z C Z d d r φ ( r ) r − ǫ = Z C φ (0) Z a d d r r − ǫ + Z Z a d r φ ( r ) r − = (cid:28) Z r (cid:29) + Z (cid:10) π δ d ( r ) (cid:11) (cid:18) ǫ + 2 (cid:19) , (A22)where (cid:28) r (cid:29) = lim a → Z d r φ ( r ) (cid:20) r Θ( r − a ) + 4 π δ ( r ) ( γ + ln a ) (cid:21) (A23)is the regularized form of 1 /r . The matrix element h [ Z /r ] ǫ i is (cid:28)(cid:20) Z r (cid:21) ǫ (cid:29) = Z C Z d d r φ ( r ) (cid:2) ∇ (cid:0) r − ǫ (cid:1)(cid:3) = Z C ( − ǫ ) φ (0) Z a d d r r − ǫ (cid:0) − C r ǫ (cid:1) + Z Z a d r φ ( r ) r − = (cid:28) Z r (cid:29) + Z (cid:10) π δ d ( r ) (cid:11) (cid:18) − ǫ + 8 (cid:19) , (A24)25 /r i in the above is again a regularized form of 1 /r , where 1 /a and ln a + γ are dropped,analogous to h /r i term. However, we do not need its explicit form because we can alwaysrewrite it in terms of h Z /r i using expectation value identities. Similarly, Z (cid:28)(cid:20) r (cid:18) δ ij + r i r j r (cid:19)(cid:21) ǫ ∇ i ∇ j (cid:20) r (cid:21) ǫ (cid:29) = (cid:28)(cid:20) Z r (cid:21) ǫ (cid:29) + 2 Z (cid:10) π δ d ( r ) (cid:11) , (A25) − Z (cid:28) W ijǫ ∇ i (cid:20) r (cid:21) ǫ ∇ j (cid:20) r (cid:21) ǫ (cid:29) = 14 (cid:28)(cid:20) Z r (cid:21) ǫ (cid:29) − Z (cid:10) π δ d ( r ) (cid:11) , (A26)and − i Z (cid:28) ∇ i (cid:20) r (cid:21) ǫ (cid:20) p , W ijǫ (cid:21) p j (cid:29) = 18 (cid:28) p i Z r (cid:0) δ ij r − r i r j (cid:1) p j (cid:29) + 18 (cid:28)(cid:20) Z r (cid:21) ǫ (cid:29) + 3 Z (cid:10) π δ d ( r ) (cid:11) . (A27)The last singular term appearing in these calculations is (cid:28) σ ij σ ij d (cid:20) Z r (cid:21) ǫ (cid:29) = (cid:28) d − (cid:20) Z r (cid:21) ǫ (cid:29) = 14 (cid:28)(cid:20) Z r (cid:21) ǫ (cid:29) + Z (cid:10) π δ ( r ) (cid:11) , (A28)where we used the identity σ ij σ ij = d ( d − . (A29)All the singular terms can now be expressed in terms h [ Z /r ] ǫ i and h [ Z /r ] ǫ i and usingthe expectation value identity (cid:20) Z r (cid:21) ǫ = ~p Z r ~p − (cid:18) E + Zr − r − p (cid:19) Z r − (cid:20) Z r (cid:21) ǫ (A30)they eventually cancel out. Appendix B: Derivation of δ M A Let us present here again the terms contributing to A M : A M = h Q ( H M − E M ) Q i M + 2 E (4) M h Q i M − h H (4) M Q i M + mM (cid:26) h Q ( H − E ) δ M Q i + 2 E (4) h δ M Q i − h H A δ M Q i (cid:27) = A M a + A M b + A M c + A M d + A M e + A M f . (B1)26he first three terms contain both recoil and non-recoil parts, while the latter three arerecoil only terms. Individual terms can be reduced by using expectation value identities: A M a = h Q ( H M − E M ) Q i M = 12 h [ Q, [ H M − E M , Q ]] i M = 12 h ( ∇ Q ) + ( ∇ Q ) i M + 14 mM h [ Q, [ ~P , Q ]] i = (cid:28) (cid:18) Z r + Z r (cid:19) + 14 r − (cid:18) ~r r − ~r r (cid:19) · ~rr (cid:29) M + mM (cid:28) (cid:18) Z r + Z r (cid:19) + 116 Z ~r · ~r r r (cid:29) , (B2) A M b = 2 E (4) h Q i M + 2 δ M E (4) (cid:18) E (cid:28) r (cid:29)(cid:19) , (B3) A M c = − h H (4) M Q i M = X + X + X + X , (B4)where X = − h δ M H (4) Q i = X a (cid:28) − Z P i (cid:18) δ ij r a + r ia r ja r a (cid:19) (cid:18) Zr + Zr − r (cid:19) p ja − Z (cid:18) δ ij r a + r ia r ja r a (cid:19)(cid:20) p ia , (cid:20) p ja , r a (cid:21)(cid:21)(cid:29) = X a (cid:28) − Z P i (cid:18) δ ij r a + r ia r ja r a (cid:19)(cid:18) Zr + Zr − r (cid:19) p ja + 14 Z r a + Z πδ ( r a ) (cid:29) . (B5)Here we used the identity (A25) from Appendix A to rewrite the singular term in the secondequality in (B5) as (cid:28) Z r a (cid:18) δ ij + r ia r ja r a (cid:19) ∇ ia ∇ ja r a (cid:29) = (cid:28) Z r a + Z π δ ( r a ) (cid:29) . (B6)Further (using h δ ( x ) /x i = 0 which is valid in dimensional regularization) X = − (cid:28)(cid:2) Z πδ ( r ) + Z πδ ( r ) (cid:3) Q (cid:29) M = (cid:28) Z ( Z − π (cid:18) δ ( r ) r + δ ( r ) r (cid:19)(cid:29) M , (B7) X = (cid:28) p i r (cid:18) δ ij + r i r j r (cid:19) p j Q (cid:29) M (B8)= (cid:28) − p i (cid:18) Zr + Zr − r (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j + 12 (cid:20) p i , (cid:20) p j , r (cid:21)(cid:21) r (cid:18) δ ij + r i r j r (cid:19)(cid:29) M ,X = 14 (cid:10)(cid:2) ( p + p ) − p p (cid:3) Q (cid:11) M = 14 (cid:10) ( p + p ) Q ( p + p ) + 12 [ p + p , [ Q, p + p ]] − p Q p − [ p , [ p , Q ]] (cid:11) M = X A + X B + X C + X D , (B9)27here X A = h ( E − V ) Q i M + 2 mM (cid:28) ( E − V ) Q (cid:18) δ M E − ~P (cid:19)(cid:29) (B10)= h ( E − V ) Q i M + mM (cid:28) δ M E ( E − V ) Q − ~P ( E − V ) Q ~P −
12 [ ~P , [ ~P , ( E − V ) Q ]] (cid:29) ,X B = − (cid:28)(cid:20) V + mM ~P , (cid:20) p + p , Q (cid:21)(cid:21)(cid:29) M (B11)= (cid:28) − (cid:18) Z r + Z r (cid:19) + 38 (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r (cid:29) M + mM (cid:28) (cid:18) Z r + Z r (cid:19) + 14 Z ~r · ~r r r (cid:29) ,X C = (cid:28) p (cid:18) Zr + Zr (cid:19) p − p r p (cid:29) M , (B12) X D = (cid:28) − (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:29) M . (B13)The remaining terms are A M d = mM (cid:10) [ Q, [ H − E, δ M Q ]] (cid:11) = mM (cid:10) ( ∇ Q )( ∇ δ M Q ) + ( ∇ Q )( ∇ δ M Q ) (cid:11) = mM (cid:28) − (cid:18) Z r + Z r (cid:19) + 38 (cid:18) Z~r r − Z~r r (cid:19) · ~rr (cid:29) , (B14) A M e = mM (cid:18) E (4) (cid:28) r (cid:29) − EE (4) (cid:19) , (B15) A M f = − mM h H A δ M Q i = F + F + F , (B16)where F = − mM (cid:28) Z π (cid:18) δ ( r ) r + δ ( r ) r (cid:19)(cid:29) , (B17) F = mM (cid:28) p i (cid:18) Zr + Zr (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j (cid:29) , (B18) F = 14 mM (cid:10)(cid:2) ( p + p ) − p p (cid:3) δQ (cid:11) = 14 mM (cid:10) ( p + p ) δQ ( p + p ) + 12 [ p + p , [ p + p , δQ ]] − p δQ p (cid:11) = F A + F B + F C , (B19)28nd where F A = mM (cid:28)
34 ( E − V ) (cid:18) Zr + Zr (cid:19)(cid:29) , (B20) F B = mM (cid:28) (cid:18) Z r + Z r (cid:19) − (cid:18) Z~r r − Z~r r (cid:19) · ~rr (cid:29) , (B21) F C = − mM (cid:28) p (cid:18) Zr + Zr (cid:19) p (cid:29) . (B22)Taking now only the recoil part of terms A M a . . . A M f we obtain the results: δ M A a = δ M (cid:28) (cid:18) Z r + Z r (cid:19) + 14 r − (cid:18) ~r r − ~r r (cid:19) · ~rr (cid:29) (B23)+ (cid:28) (cid:18) Z r + Z r (cid:19) + 116 Z ~r · ~r r r (cid:29) ,δ M A b = 2 E (4) δ M h Q i + 2 δ M E (4) (cid:18) E (cid:28) r (cid:29)(cid:19) , (B24) δ M A c = δ M (cid:28) Z ( Z − π (cid:18) δ ( r ) r + δ ( r ) r (cid:19) − p i (cid:18) Zr + Zr − r (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j + 12 (cid:20) p i , (cid:20) p j , r (cid:21)(cid:21) r (cid:18) δ ij + r i r j r (cid:19) + ( E − V ) Q − (cid:18) Z r + Z r (cid:19) + 38 (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r + 18 p (cid:18) Zr + Zr (cid:19) p − p r p − (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:29) + (cid:28) − Z X a P i (cid:18) δ ij r a + r ia r ja r a (cid:19) (cid:18) Zr + Zr − r (cid:19) p ja + 38 (cid:18) Z r + Z r (cid:19) + Z (cid:0) πδ ( r ) + πδ ( r ) (cid:1) + 2 δ M E ( E − V ) Q − ~P ( E − V ) Q ~P −
12 [ ~P , [ ~P , ( E − V ) Q ]] + 14 Z ~r · ~r r r (cid:29) ,δ M A d = (cid:28) − (cid:18) Z r + Z r (cid:19) + 38 (cid:18) Z~r r − Z~r r (cid:19) · ~rr (cid:29) , (B25) δ M A e = 32 E (4) (cid:28) r (cid:29) − EE (4) , (B26) δ M A f = (cid:28) − Z π (cid:18) δ ( r ) r + δ ( r ) r (cid:19) + 34 p i (cid:18) Zr + Zr (cid:19) r (cid:18) δ ij + r i r j r (cid:19) p j + 34 ( E − V ) (cid:18) Zr + Zr (cid:19) + 38 (cid:18) Z r + Z r (cid:19) − (cid:18) Z~r r − Z~r r (cid:19) · ~rr − p (cid:18) Zr + Zr (cid:19) p (cid:29) . (B27)29umming all of the recoil parts δ M A a . . . δ M A f and using the identity[ ~P , [ ~P , ( E − V ) Q ]] = 12 (cid:18) Z r + Z r (cid:19) + Z ~r · ~r r r − (cid:18) E + 2 Z − r (cid:19) π Z δ ( r ) − (cid:18) E + 2 Z − r (cid:19) π Z δ ( r ) (B28)we get the final result (71). Appendix C: Derivation of δ M B In the following we perform only derivation of terms B M . . . B M defined as B Mi = h H Mi i M (C1)and the evaluation of the remaining terms is trivial since they contain only Dirac delta-likecontributions. The expectation value of the kinetic term H M = 116 (cid:0) p + p (cid:1) (C2)is B M = 116 (cid:10) ( p + p ) − p p ( p + p ) (cid:11) M = (cid:28) (cid:20) V + mM ~P , (cid:20) p + p , V (cid:21)(cid:21) + 12 (cid:18) E − V + mM (cid:18) δ M E − ~P (cid:19)(cid:19) − p p (cid:18) E − V + mM (cid:18) δ M E − ~P (cid:19)(cid:19)(cid:29) M = (cid:28) (cid:2) ( ∇ V ) + ( ∇ V ) (cid:3) + 12 ( E − V ) − p ( E − V ) p + 316 [ p , [ p , V ]] (cid:29) M + mM (cid:28)
32 ( E − V ) (cid:18) δ M E − ~P (cid:19) − p p (cid:18) δ M E − ~P (cid:19) − (cid:18) Z r + Z r (cid:19) − Z ~r · ~r r r (cid:29) . (C3)The recoil correction δ M B is then δ M B = δ M (cid:28) Z r + Z r − (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 12 r + 12 ( E − V ) + 316 (cid:20) p , (cid:20) p , r (cid:21)(cid:21) − p ( E − V ) p (cid:29) + (cid:28) δ M E ( E − V ) − ~P ( E − V ) ~P + 14 (cid:18) Z r + Z r (cid:19) + 12 Z ~r · ~r r r − (cid:18) E + Z − r (cid:19) π Z δ ( r ) + (1 ↔ − p p (cid:18) δ M E − ~P (cid:19)(cid:29) . (C4)30ere we used [ ~P , [ ~P , ( E − V ) ]] = − (cid:18) Z r + Z r (cid:19) − Z ~r · ~r r r + 2 ( E − V ) (cid:2) π Zδ ( r ) + 4 π Zδ ( r ) (cid:3) . (C5)The operator H M is H M = X a ( ∇ a V ) (cid:2) p a , (cid:2) p a , V (cid:3)(cid:3) − (cid:8) p a , ∇ a V (cid:9) . (C6)For the sake of simplicity we split its expectation value into three parts, B M = (cid:28) (cid:2) ( ∇ V ) + ( ∇ V ) (cid:3) + 5128 (cid:0)(cid:2) p , (cid:2) p , V (cid:3)(cid:3) + (cid:2) p , (cid:2) p , V (cid:3)(cid:3)(cid:1) − (cid:0) p ∇ V + p ∇ V (cid:1)(cid:29) M = B M a + B M b + B M c . (C7)Term B M a = h ( ∇ V ) + ( ∇ V ) i M needs no further reduction. The remaining terms couldbe simplified to B M b = 5128 (cid:10)(cid:2) p + p , (cid:2) p , V (cid:3)(cid:3) + (cid:2) p + p , (cid:2) p , V (cid:3)(cid:3) − (cid:2) p , (cid:2) p , V (cid:3)(cid:3)(cid:11) M = − (cid:28)(cid:20) V + mM ~P , (cid:20) p + p , V (cid:21)(cid:21) + (cid:2) p , (cid:2) p , V (cid:3)(cid:3)(cid:29) M , (C8) B M c = − (cid:10)(cid:0) p + p (cid:1) ∇ V + (cid:0) p + p (cid:1) ∇ V − p ∇ V − p ∇ V (cid:11) M (C9)= − πZ (cid:28) (cid:20) E − V + mM (cid:18) δ M E − ~P (cid:19)(cid:21)(cid:0) δ ( r ) + δ ( r ) (cid:1) − p δ ( r ) − p δ ( r ) (cid:29) M . Taking now only the recoil parts of individual terms we get δ M B a = 18 δ M (cid:28) ( ∇ V ) + ( ∇ V ) (cid:29) , (C10) δ M B b = − δ M (cid:28) ( ∇ V ) + ( ∇ V ) + 12 (cid:2) p , (cid:2) p , V (cid:3)(cid:3)(cid:29) + 564 (cid:10)(cid:2) V, (cid:2) ~P , V (cid:3)(cid:3)(cid:11) , (C11) δ M B c = − πZ δ M (cid:28) (cid:18) E + Z − r (cid:19) δ ( r ) + 2 (cid:18) E + Z − r (cid:19) δ ( r ) − p δ ( r ) − p δ ( r ) (cid:29) − (cid:28)(cid:18) δ M E − ~P (cid:19) πZ (cid:0) δ ( r ) + δ ( r ) (cid:1)(cid:29) . (C12)31erm δ M B is then the sum of these three terms and takes the form δ M B = δ M (cid:28) − (cid:18) Z r + Z r (cid:19) + 116 (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r − (cid:20) p , (cid:20) p , r (cid:21)(cid:21) − πZ (cid:20) (cid:18) E + Z − r (cid:19) δ ( r ) + 2 (cid:18) E + Z − r (cid:19) δ ( r ) − p δ ( r ) − p δ ( r ) (cid:21)(cid:29) + (cid:28) (cid:18) Z r + Z r (cid:19) + 516 Z ~r · ~r r r − (cid:18) δ M E − E + 1 − Zr − ~p · ~p (cid:19) πZδ ( r ) + (1 ↔ (cid:29) . (C13)Operator H M is H M = 164 (cid:20) − π ∇ δ ( r ) + 43 p i (cid:18) δ ij π δ ( r ) + 1 r (cid:0) r i r j − δ ij r (cid:1)(cid:19) p j (cid:21) . (C14)Here we used the identity valid for triplets σ ij σ ij = 2 ~σ · ~σ = 2 (C15)to evaluate the spin product in H M . Since there is no singular term, we can use d = 3representation and the scalar product in the evaluation of the spin part. This will beassumed also in all the other terms where the spin product appears. In the case where theterm containing the spin product is singular and one has to use its d -dimensional form toevaluate such term, it will be explicitly stated. The expectation value of H M is B M = (cid:28) − π ∇ δ ( r ) + 148 p i (cid:18) δ ij π δ ( r ) + 1 r (cid:0) r i r j − δ ij r (cid:1)(cid:19) p j (cid:29) M = (cid:28) − p i r (cid:0) δ ij r − r i r j (cid:1) p j − π ∇ δ ( r ) (cid:29) M , (C16)where we used the expectation value identity h ~p δ ( r ) ~p i = − h∇ δ ( r ) i . (C17)Further, with the help of identity p i r (cid:0) δ ij r − r i r j (cid:1) p j = − (cid:20) p , (cid:20) p , r (cid:21)(cid:21) − π ∇ δ ( r ) (C18)we get the resulting recoil correction δ M B δ M B = δ M (cid:28) (cid:20) p , (cid:20) p , r (cid:21)(cid:21) − π ∇ δ ( r ) (cid:29) . (C19)32e split the correction due to operator H M = H + mM δ M H into two parts: the recoilcorrection to operator H , which we denote as B M a , and the expectation value of the recoilpart δ M H , which we denote as B M b . The non-recoil part of the operator H M is (omittingthe part with δ ( r ), which does not contribute for triplet states) H = 14 (cid:0) p + p (cid:1) p i r (cid:18) δ ij + r i r j r (cid:19) p j . (C20)The expectation value of this is B M a = 12 (cid:28)(cid:0) E − V (cid:1) p i r (cid:18) δ ij + r i r j r (cid:19) p j (cid:29) M + m M (cid:28)(cid:18) δ M E − ~P (cid:19) p i r (cid:18) δ ij + r i r j r (cid:19) p j (cid:29) = 12 (cid:28) p i (cid:0) E − V (cid:1) r (cid:18) δ ij + r i r j r (cid:19) p j − r (cid:18) δ ij + r i r j r (cid:19)(cid:20) p i , (cid:20) p j , r (cid:21)(cid:21)(cid:29) M + m M (cid:28)(cid:18) δ M E − ~P (cid:19) p i r (cid:18) δ ij + r i r j r (cid:19) p j (cid:29) . (C21)Recoil correction δ M B a is then δ M B a = δ M (cid:28) p i (cid:0) E − V (cid:1) r (cid:18) δ ij + r i r j r (cid:19) p j − r (cid:29) + (cid:28) (cid:18) δ M E − ~P (cid:19) p i r (cid:18) δ ij + r i r j r (cid:19) p j (cid:29) . (C22)The recoil part of H M is δ M H = Z (cid:18) p p i (cid:18) δ ij r + r i r j r (cid:19) P j + p p i (cid:18) δ ij r + r i r j r (cid:19) P j (cid:19) . (C23)The expectation value of this operator can then be reduced to δ M B b = Z (cid:28) (cid:0) E − V (cid:1)(cid:20) p i (cid:18) δ ij r + r i r j r (cid:19) P j + p i (cid:18) δ ij r + r i r j r (cid:19) P j (cid:21) − (cid:20) p p i (cid:18) δ ij r + r i r j r (cid:19) P j + p p i (cid:18) δ ij r + r i r j r (cid:19) P j (cid:21)(cid:29) = (cid:28) Z " p i (cid:0) E − V (cid:1) δ ij r + r i r j r ! P j + p i (cid:0) E − V (cid:1) δ ij r + r i r j r ! P j − (cid:18) Z r + Z r (cid:19) − Z (cid:2) πδ ( r ) + πδ ( r ) (cid:3) − Z (cid:20) p i p k δ ij r + r i r j r (cid:19) p k P j + p i p k (cid:18) δ ij r + r i r j r ! p k P j (cid:21)(cid:29) . (C24)When commuting E − V we used equation (A25) of Appendix A, in particular (cid:28) Z p ia , [ P j , E − V ]] (cid:18) δ ij r a + r ia r ja r a (cid:19)(cid:29) = (cid:28) − Z r a − Z π δ ( r a ) (cid:29) . (C25)33perator H M is H M = − (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 13 r + 148 (cid:18)(cid:20) p , (cid:20) p , r (cid:21)(cid:21) + (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:19) , (C26)where the spin product was again resolved using identity (C15). The expectation value is B M = (cid:28) − (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 13 r − (cid:18)(cid:20) V, (cid:20) p + p , r (cid:21)(cid:21) + (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:19)(cid:29) M . (C27)The recoil correction is then δ M B = δ M (cid:28) − (cid:18) Z~r r − Z~r r (cid:19) · ~rr + 16 r − (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:29) . (C28)The operator H M contains the recoil part δ M H , so we again split the calculation intotwo parts: the recoil correction due to H , denoted as δ M B a , and the expectation value of δ M H , which we denote as δ M B b . The non-recoil part of the operator H M is H = 18 p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + 18 p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + 12 r , (C29)where we used the identity from Eq. (A29). The recoil correction due to this operator issimply δ M B a = δ M (cid:28) p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + 18 p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + 12 r (cid:29) . (C30)The recoil part of operator H M contains a singular term with spin product, which we haveto evaluate using dimensional regularization. In particular, we use Eq. (A28) to get (cid:28) σ ija σ ija Z r a (cid:29) = (cid:28) Z r a + Z π δ ( r a ) (cid:29) . (C31)Using this and (C15) the expectation value of δ M H can be evaluated to δ M B b = (cid:28) Z (cid:20) p i (cid:18) δ ij r + r i r j r (cid:19)(cid:18) δ jk r + r j r k r (cid:19) + p i (cid:18) δ ij r + r i r j r (cid:19)(cid:18) δ jk r + r j r k r (cid:19) (cid:21) P k + 14 (cid:18) Z r + Z r + 23 Z ~r · ~r r r (cid:19) + Z (cid:20) πδ ( r ) + πδ ( r ) (cid:21) + Z (cid:20) p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + p i r (cid:18) δ ij + 3 r i r j r (cid:19) p j + 2 p i (cid:18) δ ij r + r i r j r (cid:19)(cid:18) δ jk r + r j r k r (cid:19) p k (cid:21)(cid:29) . (C32)34inally, we calculate correction due to the operator H M = H M a + H M c + H M d . We split itcorrespondingly into three parts, B M = B M a + B M c + B M d . The operator H M a reads H M a = − (cid:26)(cid:2) p i , V (cid:3)(cid:18) r i r j r − δ ij r (cid:19)(cid:2) V, p j (cid:3) + (cid:2) p i , V (cid:3)(cid:20) p , r i r j r − δ ij r (cid:21) p j + p i (cid:20) r i r j r − δ ij r, p (cid:21)(cid:2) V, p j (cid:3) + p i (cid:20) p , (cid:20) r i r j r − δ ij r, p (cid:21)(cid:21) p j (cid:27) . (C33)The recoil correction due to this operator is δ M B a = δ M (cid:28) − Zr i r Zr j r (cid:18) r i r j r − δ ij r (cid:19) + 14 (cid:18) Z~r r − Z~r r (cid:19) · ~rr − r (C34) − Z (cid:20) r i r p k (cid:18) δ jk r i r − δ ik r j r − δ ij r k r − r i r j r k r (cid:19) p j + (1 ↔ (cid:21) + 18 (cid:20) p j r (cid:0) δ jk r − r j r k (cid:1) p k + (1 ↔ (cid:21) + 14 r + 18 p k p l (cid:20) − δ il δ jk r + δ ik δ jl r − δ ij δ kl r − δ jl r i r k r − δ ik r j r l r + 3 r i r j r k r l r (cid:21) p i p j (cid:29) . The operator H M c is H M c = 124 (cid:20) p , (cid:20) p , r (cid:21)(cid:21) , (C35)where we used (C15) for the spin part. The corresponding recoil correction is simply δ M B c = δ M (cid:28) (cid:20) p , (cid:20) p , r (cid:21)(cid:21)(cid:29) . (C36)Finally, the operator H M d is H M d = i Z M X a,b r ia r a (cid:20) H − E, r ib r jb − δ ij r b r b p jb (cid:21) = i Z M X a,b r ia r a (cid:26)(cid:2) V, p jb (cid:3) r ib r jb − δ ij r b r b + (cid:20) p b , r ib r jb − δ ij r b r b (cid:21) p jb (cid:27) . (C37)The expectation value of this can then be written as δ M B d = W + W , (C38)where W = (cid:28) − Z X a,b,c = b r ia r a Zr jb r b − r jbc r bc ! r ib r jb − δ ij r b r b − Z πδ ( r b ) (cid:29) = (cid:28) Z r + Z r + Z ~r · ~r r r + Z ~r · ~r r r − Z πδ ( r ) + πδ ( r )]+ Z X b,c = b (cid:18) r i r + r i r (cid:19) r ib r jb − δ ij r b r b r jbc r bc (cid:29) . (C39)35ere we used the identity (A26) to rewrite the singular term as (cid:28) − Z r ib r jb − δ ij r b r b (cid:18) ∇ ib Zr b (cid:19) (cid:18) ∇ jb Zr b (cid:19)(cid:29) = (cid:28) Z r b − Z π δ ( r b ) (cid:29) . (C40)Further, W = (cid:28) i Z (cid:18) X a = b r ia r a (cid:20) p b , r ib r jb − δ ij r b r b (cid:21) p jb + X b r ib r b (cid:20) p b , r ib r jb − δ ij r b r b (cid:21) p jb (cid:19)(cid:29) (C41)= (cid:28) Z X a = b p kb r ia r a − δ ik r jb r b + δ jk r ib r b − δ ij r kb r b − r ib r jb r kb r b ! p jb + Z r + Z r + 3 Z πδ ( r ) + πδ ( r )] + Z X b p jb r b (cid:0) δ jk r b − r jb r kb (cid:1) p kb (cid:29) , where for the reduction of the singular term we used the identity (A27) from the Appendix,in particular i Z (cid:28) r ib r b (cid:20) p b , r ib r jb − δ ij r b r b (cid:21) p jb (cid:29) = (cid:28) Z r b + 3 Z πδ ( r b ) + Z p jb r b (cid:0) δ jk r b − r jb r kb (cid:1) p kb (cid:29) . (C42) Appendix D: Hydrogen limit
In this section we perform the reduction of our general formulas to the hydrogenic limitfor the S states, in order to demonstrate that the method reproduces the known resultsin agreement with the Dirac equation and with the hydrogenic recoil corrections. First wetreat the infinite nucleus mass limit and then the recoil correction.
1. Infinite nucleus mass limit
We obtain the hydrogenic limit by sending r → ∞ and consequently ~p → r → ∞ .The effective operator H (6) reduces in the hydrogenic limit to (writing r ≡ r ) H (6) = p
16 + 18 (cid:0) ∇ V (cid:1) + 5128 (cid:2) p , (cid:2) p , V (cid:3)(cid:3) − p ∇ V . (D1)The first-order contribution to energy B = h H (6) i is then B = (cid:28)
14 ( E − V ) p ( E − V ) + 18 Z r + 564 (cid:2) E − V, (cid:2) p , V (cid:3)(cid:3) −
316 ( E − V ) ∇ V (cid:29) = (cid:28) (cid:2) V, (cid:2) p , V (cid:3)(cid:3) + 12 ( E − V ) + 18 Z r − Z r − E
16 4 πZ δ ( r ) (cid:29) = (cid:28) Z r − E + 3 E Z r + 12 Z r − E
16 4 πZ δ ( r ) (cid:29) . (D2)36he operator H (4) reduces in the hydrogenic case to H (4) = − p Zπ δ ( r ) , (D3)which we again regularize as H (4) = H R + { H − E, Q } , (D4)with Q = − Z/ (4 r ) = V / H R = −
12 ( E − V ) − Z ~r · ~ ∇ r . (D5)The second-order contribution to energy is then A = (cid:28) H R E − H ) ′ H R (cid:29) + h Q ( H − E ) Q i + 2 (cid:10) H (4) (cid:11) h Q i − (cid:10) Q H (4) (cid:11) = (cid:28) H R E − H ) ′ H R (cid:29) + 132 (cid:10)(cid:2) V, (cid:2) H − E, V (cid:3)(cid:3)(cid:11) + E E (4) + 116 (cid:10)
V p (cid:11) = (cid:28) H R E − H ) ′ H R (cid:29) + (cid:28) − Z r + E E (4) + E − E Z r − Z r (cid:29) , (D6)where we have used h V i = 2 E . The sum of first- and second-order contributions is E (6) = (cid:28) H R E − H ) ′ H R (cid:29) (D7)+ (cid:28) Z r + 14 Z r − E + E E (4) + E Z r − E
16 4 πZ δ ( r ) (cid:29) = (cid:28) H R E − H ) ′ H R (cid:29) + (cid:28) ~p Z r ~p + 3 E Z r − E + E E (4) − E
16 4 πZ δ ( r ) (cid:29) , where we have used the indentity Z r = ~p Z r ~p − (cid:18) E + Zr (cid:19) Z r . (D8)The expectation values of operators appearing in the final result are for S states E = − Z n , (D9) E (4) = 3 Z n − Z n , (D10) (cid:28) Z r (cid:29) = 2 Z n , (D11) (cid:28) ~p Z r ~p (cid:29) = − Z n + 8 Z n , (D12) (cid:28) H R E − H ) ′ H R (cid:29) = Z (cid:18) − n + 2324 n − n − n (cid:19) , (D13) h πZ δ ( r ) i = 4 Z n . (D14)37ubstituting these values into energy we get the result E (6) = Z (cid:18) − n + 34 n − n − n (cid:19) , (D15)in agreement with the result from the Dirac equation obtained by expanding E D = Zα ) ( n − p − ( Zα ) ) ! − (D16)in the order α .
2. Recoil correction for hydrogenic limit
Here the perturbation of the nonrelativistic Hamiltonian reduces to ~P / M = ~p / M .This correction is then easily accounted for by making reduced mass rescaling r → rµ andexpanding the reduced mass as ( µ/m ) n ≈ − n mM . The total recoil correction will thenbe the sum of the correction due to reduced mass and the correction due to extra recoiloperators in H (6) M and H (4) M . First we examine the reduced mass correction.Rescaling the first-order operator H (6) and expanding up to the first order in nuclearmass we obtain the recoil correction (utilizing results from the infinite nucleus mass limit) δ M B = − B + (cid:28) − p
16 + 18 (cid:0) ∇ V (cid:1) (cid:29) = − B + (cid:28) − Z r + 5 E − E Z r − Z r (cid:29) . (D17)The second-order contribution due to reduced mass is δ M A = − A + (cid:28) p E − H ) ′ H (4) (cid:29) (D18)= − A + (cid:28) ( E − V ) E − H ) ′ H (4) (cid:29) + (cid:28) ( E − V ) ( H − E ) 1( E − H ) ′ H (4) (cid:29) = − A + (cid:28) ( E − V ) E − H ) ′ H R (cid:29) + (cid:28) − E E (4) − E + 2 E Z r + 34 Z r + 14 Z r (cid:29) . Summing now both terms δ M A and δ M B we get the total recoil correction due to thereduced mass rescaling E i , E i = δ M A + δ M B (D19)= − E (6) + (cid:28) ( E − V ) E − H ) ′ H R (cid:29) + (cid:28) − E − E E (4) + E Z r + 18 ~p Z r ~p (cid:29) , H (4) is δ M H (4) = − Z p i (cid:18) δ ij r + r i r j r (cid:19) P j , (D20)and the corresponding second-order correction to energy is δ M A = − Z (cid:28) p i (cid:18) δ ij r + r i r j r (cid:19) p j E − H ) ′ H (4) (cid:29) (D21)= − Z (cid:28) p i (cid:18) δ ij r + r i r j r (cid:19) p j E − H ) ′ H R (cid:29) + (cid:28) − EZ p i (cid:18) δ ij r + r i r j r (cid:19) p j − Z p i (cid:18) δ ij r + r i r j r (cid:19) p j + 14 Z r + Z π δ ( r ) (cid:29) . The correction due to the extra first-order recoil operators is δ M B = h δ M H (6) i = (cid:28) Z r + Z π δ ( r ) + Z p p i (cid:18) δ ij r + r i r j r (cid:19) p j + Z p i (cid:18) δ ij r + 3 r i r j r (cid:19) p j + i Z r i r (cid:20) H − E, r i r j − δ ij r r p j (cid:21)(cid:29) = δ M B a + δ M B b + δ M B c + δ M B d + δ M B e . (D22)The third and the fifth terms are δ M B c = (cid:28) Z E − V ) p i (cid:18) δ ij r + r i r j r (cid:19) p j (cid:29) = (cid:28) Z p i ( E − V ) (cid:18) δ ij r + r i r j r (cid:19) p j + Z (cid:20) p i , (cid:20) p j , r (cid:21)(cid:21) (cid:18) δ ij r + r i r j r (cid:19)(cid:29) = (cid:28) Z p i (cid:18) E + Zr (cid:19) (cid:18) δ ij r + r i r j r (cid:19) p j − Z r − Z π δ ( r ) (cid:29) , (D23)and δ M B e = (cid:28) i Z r i r (cid:26)(cid:2) V, p j (cid:3) r i r j − δ ij r r + (cid:20) p , r i r j − δ ij r r (cid:21) p j (cid:27)(cid:29) = (cid:28) Z r + 18 Z r − Z π δ ( r ) + Z p i r (cid:0) δ ij r − r i r j (cid:1) p j (cid:29) . (D24)The first-order contribution δ M B is the sum of all terms δ M B a . . . δ M B e and is δ M B = (cid:28) − Z r + 14 Z r − Z π δ ( r ) + Z p i (cid:18) E + Zr (cid:19) (cid:18) δ ij r + r i r j r (cid:19) p j + 14 ~p Z r ~p (cid:29) . (D25)39he correction E ii due to extra recoil operators is then the sum of δ M B and δ M A , E ii = δ M A + δ M B = − Z (cid:28) p i (cid:18) δ ij r + r i r j r (cid:19) p j E − H ) ′ H R (cid:29) + (cid:28) Z p i (cid:18) δ ij r + r i r j r (cid:19) p j − Z π δ ( r ) + 38 ~p r ~p − E Z r (cid:29) . (D26)Finally, the total recoil correction for the S state hydrogenic limit is the sum of thereduced mass scaling correction E i and the correction due to extra operators E ii and is δ M E (6) = E i + E ii (D27)= (cid:28) ( E − V ) E − H ) ′ H R (cid:29) − Z (cid:28) p i (cid:18) δ ij r + r i r j r (cid:19) p j E − H ) ′ H R (cid:29) − E (6) − E − E E (4) + (cid:28) ~p Z r ~p + Z p i (cid:18) δ ij r + r i r j r (cid:19) p j − Z π δ ( r ) (cid:29) . In addition to the operators already used, the expectation values are (cid:28) Z p i (cid:18) δ ij r + r i r j r (cid:19) p j (cid:29) = − Z n + 16 Z n , (D28) (cid:28) ( E − V ) E − H ) ′ H R (cid:29) = Z (cid:18) n − n + 32 n + 32 n (cid:19) , (D29) Z (cid:28) p i (cid:18) δ ij r + r i r j r (cid:19) p j E − H ) ′ H R (cid:29) = Z (cid:18) n − n + 3 n + 113 n (cid:19) . (D30)Using these expectation values the final result is δ M E (6) = Z (cid:18) n − n + 38 n + 18 n (cid:19) , (D31)in agreement with the result from the Dirac equation E MD = 1 − E D α . In particular, it vanishes for the hydrogenic ground state.40 ABLE I: Expectation values of operators Q i and the corresponding recoil corrections, with i =1 . . . S P h Q i i δ M h Q i i h Q i i δ M h Q i i Q = 4 πδ ( r ) 16 .
592 071 − .
748 907 15 .
819 309 − .
358 598 Q = 4 πδ ( r ) 0 0 0 0 Q = 4 πδ ( r ) /r .
648 724 − .
821 266 4 .
349 766 − .
576 147 Q = 4 πδ ( r ) p .
095 714 − .
638 077 4 .
792 830 − .
366 064 Q = 4 πδ ( r ) /r Q = 4 π ~p δ ( r ) ~p .
028 099 − .
163 026 0 .
077 524 − .
100 949 Q = 1 /r .
268 198 − .
272 645 0 .
266 641 − .
082 865 Q = 1 /r .
088 906 − .
182 363 0 .
094 057 − .
052 275 Q = 1 /r .
038 861 − .
121 355 0 .
047 927 − .
036 603 Q = 1 /r .
026 567 − .
113 712 0 .
043 348 − .
042 669 Q = 1 /r .
170 446 − .
338 455 4 .
014 865 − .
127 584 Q = 1 / ( r r ) 0 .
560 730 − .
147 101 0 .
550 342 − .
709 019 Q = 1 / ( r r ) 0 .
322 696 − .
657 458 0 .
317 639 − .
381 158 Q = 1 / ( r r r ) 0 .
186 586 − .
576 097 0 .
198 346 − .
295 115 Q = 1 / ( r r ) 1 .
242 704 − .
791 743 1 .
196 631 − .
687 288 Q = 1 / ( r r ) 1 .
164 599 − .
545 640 1 .
109 463 − .
554 378 Q = 1 / ( r r ) 0 .
112 360 − .
346 820 0 .
121 112 − .
166 459 Q = ( ~r · ~r ) / ( r r ) 0 .
011 331 − .
055 997 0 .
030 284 − .
030 290 Q = ( ~r · ~r ) / ( r r ) 0 .
054 635 − .
211 280 0 .
075 373 − .
104 553 Q = r i r j ( r i r j − δ ij r ) / ( r r r ) 0 .
027 082 − .
256 024 0 .
090 381 − .
166 239 Q = p /r .
751 913 − .
075 881 1 .
410 228 − .
635 740 Q = ~p /r ~p .
720 479 − .
901 955 15 .
925 672 − .
131 339 Q = ~p /r ~p .
243 754 − .
008 306 0 .
279 229 − .
572 398 Q = p i ( r i r j + δ ij r ) / ( r r ) p j .
002 750 − .
068 255 − .
097 364 − .
056 872 Q = P i (3 r i r j − δ ij r ) /r P j .
062 031 − .
336 586 − .
060 473 0 .
119 687 Q = p k r i /r ( δ jk r i /r − δ ik r j /r − δ ij r k /r − r i r j r k /r ) p j − .
009 102 0 .
035 209 0 .
071 600 − .
134 238 Q = p p .
488 198 − .
988 286 1 .
198 492 − .
171 122 Q = p /r p .
597 727 − .
106 766 3 .
883 405 − .
814 978 Q = ~p × ~p /r ~p × ~p .
070 535 − .
358 089 0 .
399 306 − .
076 373 Q = p k p l ( − δ jl r i r k /r − δ ik r j r l /r + 3 r i r j r k r l /r ) p i p j − .
034 780 0 .
177 968 − .
187 305 0 .
490 555 ABLE II: Expectation values of operators Q i with i = 31 . . .
50, the expectation value of the BreitHamiltonian E (4) and the first-order recoil corrections δ M E and δ M E (4) . S PQ = 4 πδ ( r ) ~p · ~p .
040 294 − .
457 224 Q = ( ~r · ~r ) / ( r r ) − .
005 797 − .
032 383 Q = ~p · ~p .
007 442 − .
064 572 Q = ~P /r ~P .
974 707 4 .
730 359 Q = ~P /r ~P .
232 372 1 .
127 146 Q = ~P /r ~P .
504 835 16 .
972 775 Q = ~P / ( r r ) ~P .
489 592 2 .
291 176 Q = ~P / ( r r ) ~P .
454 007 1 .
350 214 Q = ~P /r ~P .
438 804 0 .
413 144 Q = p p P .
324 509 24 .
527 699 Q = P p i ( r i r j + δ ij r ) /r p j .
151 748 0 .
067 201 Q = p i ( r i r j + δ ij r ) /r P j .
461 709 31 .
489 835 Q = p i ( r i r j + δ ij r ) / ( r r ) P j .
486 269 2 .
217 310 Q = p i p k ( r i r j + δ ij r ) /r p k P j .
100 915 2 .
527 505 Q = p i ( r i r j + δ ij r )( r j r k + δ jk r ) / ( r r ) P k .
540 877 0 .
467 623 Q = p i ( r i r j + δ ij r )( r j r k + δ jk r ) / ( r r ) p k .
006 782 − .
201 826 Q = ( ~r · ~r ) / ( r r ) − .
008 117 − .
028 621 Q = r i r j ( r i r j − δ ij r ) / ( r r ) − .
036 861 − .
057 404 Q = r i r j ( r i r j − δ ij r ) / ( r r r ) − .
089 086 − .
126 780 Q = p k r i /r ( δ jk r i /r − δ ik r j /r − δ ij r k /r − r i r j r k /r ) p j .
005 856 − .
092 036 E (4) − .
164 477 972 − .
967 358 377 δ M E .
182 671 509 2 .
068 591 766 δ M E (4) .
089 185 018 0 .
230 100 830 ABLE III: Individual α m /M recoil corrections to the ionization energies of the 2 S and 2 P states.Term 2 S PE i .
190 05 0 .
853 52(10) E ii .
044 46 0 .
032 33(5) E iii .
025 11 1 .
197 09(10) E iv .
018 48 0 .
013 00(6) E v − .
048 06 − .
977 21 E vi .
945 69 52 .
339 33Subtotal 0 .
175 72 1 .
458 05(16) E vii − .
867 15 − .
891 19Sum − .
691 45 − .
433 13(16) δ M E (6) (kHz) − . − . ABLE IV: Breakdown of theoretical contributions to the 2 S –2 P centroid transition energy for He, in MHz. The uncertainty due to approximate α contribution is assumed to be 1 MHz, i.e.four times less than in our previous work [3]. FNS is a finite nuclear size and NPOL the nuclearpolarizability corrections. ( m/M ) ( m/M ) ( m/M ) Sum α −
276 775 637 .
536 102 903 . − . −
276 672 738 . α −
69 066 . − . − . −
69 072 . α . − .
186 — 5 233 . α . − .
029 — 87 . α − . .
0) — — − . . .
427 — — 3 . − .
002 — — − . −
276 736 495 .
41 (1 . −
276 736 495 .
37 (4 . P – P [5] −
276 736 495 .
649 (2) ABLE V: Breakdown of theoretical contributions to the He − He isotope shift of the 2 S –2 P centroid transition energy, for the point nucleus, in kHz. EMIX is an additional correction in Hedue to the second-order hyperfine singlet-triplet mixing [9].( m/M ) ( m/M ) ( m/M ) Sum α
33 673 018 . − . . . α − . − . − . α − . − . α − . − . α . .
9) — — 0 . . − . − . . . . .9)