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Institut de Physique Nucl´eaire

arXiv:hep-ph/9308247v1 9 Aug 1993Positronium Spectroscopyina Magnetic FieldJan GovaertsInstitut de Physique Nucl´eaireUniversit´e Catholique de Louvain2, Chemin du CyclotronB-1348 Louvain-la-Neuve, BelgiumAbstractHyperfine spectroscopy of positronium formed in the presence of a static mag-netic field is considered. Generalising the situation hitherto developed in theliterature, the magnetic field is not assumed to be parallel to the momentumof incoming polarised positrons, while the possibility of electron polarisation isalso included in the analysis.

The results are of application to high sensitivitypositron polarimeters used in current β decay experiments.hep-ph/9308247

1IntroductionHyperfine structure in positronium is a basic physical fact with wide ranging impli-cations. The energy difference between singlet and triplet states, and their lifetimesdominated by disintegrations into two and three photons, respectively, are observ-ables providing crucial tests for quantum electrodynamics (QED)[1] and the StandardModel[2] of electroweak and strong interactions.

On a more practical level, positroniumphysics is also essential in a series of technological developments both in the design ofdetectors for particle physics experiments and in applications to problems of solid statephysics (for a detailed review with references to the original literature, see Refs. [1, 3]).In particular, hyperfine structure of positronium has been put to use since overthirty years[4] in measurements of positron polarisations.

One instance where suchexperiments come immediately to bear on the structure of fundamental interactionsis in the case of positrons emitted in the β decay of radioactive nuclei. Specifically,any deviation from purely right-handed polarisation of positrons emitted in the βdecay of polarised nuclei would point to physics beyond the Standard Model in itselectroweak sector.

Actually, experiments with this purpose in mind and using sucha positronium based positron polarisation measurement technique, have been pursuedat our Institute[5, 6, 7]. Results are promising[5, 6], and compete well with limitson physics beyond the Standard Model obtained from experiments at much higherenergies.Typically in such β decay experiments, one is interested in measuring the longitu-dinal polarisation of positrons emitted parallel or antiparallel to the direction in whichthe decaying nucleus is polarised.

The emitted positron is stopped in some specificmedium where positronium is formed. The medium being placed in a strong magneticfield1, the decay spectrum of positronium formed provides information[4] on the po-larisation of the incoming positron.

The analysis of actual experimental positroniumspectra usually assumes that, at the location where positronium is formed, the ap-plied magnetic field is exactly parallel to the direction in which the positron is emitted.Since, strictly speaking, such an assumption can never be correct in practice, the moregeneral situation has to be considered, namely when the direction of the magnetic fieldis arbitrary with respect to the positron momentum. However, the present author hasbeen unable to find in the literature[3, 4, 9, 10] a detailed discussion of this point,hence this note addressing the problem specifically.In sect.2, a brief review of positronium hyperfine structure in the absence of anyexternal electromagnetic field is presented, with also the purpose of specifying our no-tation.

Sect.3 develops the discussion of the positronium ground state and its hyperfinestructure in the presence of an arbitrary static magnetic field. Sects.

4 and 5 considerlifetimes and time evolution of hyperfine populations, respectively, while sect.6 endswith some conclusions.1Since positrons and electrons have opposite electric charges, an external electric field does notaffect positronium states nor their intrinsic decay rates (however, “pick-up” processes of externalelectrons or other “quenching” effects can affect these rates[3]). Nevertheless, an external electric fieldcan indeed improve[8] the positronium formation rate in a given medium.1

2Hyperfine Structure of PositroniumAs is well known, the 1S positronium ground state is in fact split into two hyperfinelevels of total spin S = 0 and S = 1.Their energy difference is due to spin-spininteractions between the positron and the electron, to relativistic corrections to theirkinetic energies and to virtual pair annihilation in the S = 1 channel2.A non relativistic representation of the associated wave functions is sufficient for ourpurposes, as well as being justified. Accordingly, wave functions separate into spaceand spin components, with the space component being identical for both hyperfinestates.

Namely, the S = 0 singlet or parapositronium state is given by a wave functionof the form3| 0, 0 >= ψ(r)1√2 [ |+ > |−> −|−> |+ > ] . (1)Here, ψ(r) is the space wave function of the 1S ground state, while the second factor inthe r.h.s.

of this expression is the spin wave function. The convention used throughoutis that in the product of two spin ket vectors, the first always stands for the electronspin component whereas the second stands for the positron spin component.

Thesecomponents are taken with respect, say to the z axis of some reference frame. Lateron, this axis will of course correspond to the direction in which the β decay positron isemitted.

It is also assumed that the space wave function ψ(r) is properly normalised,4πZ ∞0dr r2 | ψ(r) |2= 1 ,(2)and that the basis vectors |+ > and |−> in spin space both for the electron and forthe positron are normalised in the usual way, namely< +|+ > = 1 = < −|−> ,< +|−> = 0 = < −|+ > . (3)Consequently, the | 0, 0 > state in (1) is also of norm 1.Similarly, the three components m = ±1, 0 of the S = 1 triplet orthopositroniumstate are given by the normalised state vectors| 1, m = 1 >= ψ(r) |+ > |+ > ,| 1, m = −1 >= ψ(r) |−> |−> ,(4)| 1, m = 0 >= ψ(r)1√2 [ |+ > |−> + |−> |+ > ] .The hyperfine states in (1) and (4) are eigenstates of the total Hamiltonian H0 of thesystem in the absence of an external magnetic field.

In the non relativistic limit, thisHamiltonian is comprised of the ordinary Schr¨odinger type Hamiltonian H(C)—which,apart from the usual kinetic term, only includes the Coulomb interaction between the2The latter effect is absent in the S = 0 channel due to charge conjugation. A photon has C = −1,whereas the S = 0 and S = 1 states have C = +1 and C = −1, respectively.3Note that a compactified notation for state vectors is used throughout, whereby their isotropicradial dependence is not displayed explicitly.2

two oppositely charged particles—to which diverse relativistic corrections are added.The latter contributions correspond to spin-spin interactions between the electron andthe positron, to relativistic corrections to their kinetic energies, to virtual pair anni-hilation effects for orthopositronium states and to further QED radiative corrections.Restricting for a moment the discussion to the non relativistic purely Coulomb Hamil-tonian, the singlet and triplet states above are degenerate eigenstates of H(C) withenergy4E(C) = −14α2mc2 = −6.803 eV . (5)In fact, in this purely Coulomb limit, the space wave function is simplyψ(C)(r) =πa3−1/2 e−r/a ,(6)with the positronium radiusa =2¯hαmc = 1.0584 ˚A .

(7)Even though the complete wave function ψ(r) in (1) departs from the simple radialdependence in (6), the parameter a in (7) gives a measure of the spatial extension ofthe positronium ground state.Relativistic effects just mentioned lift the singlet-triplet degeneracy5 according tothe eigenvaluesH0 | 0, 0 >= E0 | 0, 0 > ,H0 | 1, m = ±1, 0 >= E1 | 1, m = ±1, 0 > ,(8)with energies[3]E0 =−14 + α2−14 −564+ O(α3, α2 ln α−1)α2mc2 ,(9)E1 =−14 + α2 112 −564 + 14+ O(α3, α2 ln α−1)α2mc2 ,(10)and the hyperfine difference[13, 14]∆E = E1 −E0=h73 −απ329 + 2 ln 2+ 56α2 ln α−1 + O(α2)i14α4mc2= 8.41 × 10−4 eV . (11)Except for their first term corresponding to the purely Coulomb contribution (5),the different contributions of order α4 in (9) and (10) are as follows.

Both in E0 andin E1, the first such contribution is that of the spin-spin interaction energy, while the4Throughout, α, m and c of course stand for the fine structure constant, the electron and positronmass and the speed of light, respectively. Numerical values for these parameters are from Ref.

[11].5In the purely Coulomb situation of (5), the singlet-triplet degeneracy is due[12] to a dynami-cal SO(4) symmetry explicitly broken by effects now considered. Nevertheless, the S = 1 tripletstates m = ±1, 0 remain degenerate since, in the absence of external electromagnetic fields, the totalHamiltonian H0 is invariant under rotations in space.3

second—common to both expressions—is that due to relativistic corrections to thetotal kinetic energy. Finally, the third term of order α4 in E1 follows from virtual pairannihilation in the orthopositronium triplet state.To conclude, let us consider positronium lifetimes.

Due to charge conjugation prop-erties, the singlet state can only decay into an even number of photons and the tripletstates into an odd number (beginning of course with three photons). Therefore in verygood approximation, parapositronium decays predominantly into two photons and or-thopositronium into three photons, since compared to each of these processes, the ratefor any further photon pair emission is suppressed each time by an additional power ofα2.

Hence, only 2γ and 3γ decay processes are considered in this note, and these twochannels are assumed to encompass all possible decay modes of positronium.The associated lifetimes, including radiative corrections, have been computed withinQED[15, 16]. For the singlet state, one has the 2γ decay rateλS= 12α5 mc2¯hh1 −απ5 −π24+ 23α2 ln α−1 + O(α2)i= (0.125209 × 10−9 s)−1 = 7.98665 × 109 s−1 .

(12)Similarly for triplet states, their 3γ decay rate isλT=29πα6 mc2¯h (π2 −9)h1 −(10.266 ± 0.008) απ −13α2 ln α−1 + O(α2)i= (142.074 × 10−9 s)−1 = 7.03859 × 106 s−1 . (13)Note the rather large ratioλSλT= 1134.695 .

(14)3Coupling to a Magnetic FieldLet us now consider positronium states formed in the presence of some external staticmagnetic field ⃗B = (Bx, By, Bz). For all practical purposes, certainly always realisedin actual experimental conditions, it will be assumed that whenever ⃗B(⃗x) might havea non vanishing gradient, this gradient is nevertheless negligible on the scale of thespatial extension of the positronium state, namelya | ⃗∇Bi || ⃗B |≪1 ,for all i = x, y, z .

(15)Here, a is the positronium radius of (7). Effectively, one may then consider the magneticfield ⃗B not only to be static but also to be constant, which is thus the situation to beassumed in the analysis developed in this note.

Furthermore, the field ⃗B is not taken tobe necessarily parallel to the z axis with respect to which spin eigenstates were definedin the previous section, since in practical applications, the latter axis is often definedby the positron momentum instead.4

The presence of the magnetic field ⃗B induces an additional interaction term in thetotal Hamiltonian for the positronium system. The total Hamiltonian H now includesthe previous Hamiltonian H0 with its lowest energy eigenstates in (1) and (4), to whichthe magnetic coupling to magnetic moments is added, namelyHB = −⃗µ−· ⃗B −⃗µ+ · ⃗B .

(16)Here, ⃗µ−and ⃗µ+ are the electron and positron magnetic moments, respectively. Interms of their spin operators ⃗σ−/2 and ⃗σ+/2, respectively, we have6⃗µ± = ∓µ⃗σ±2 .

(17)The magnetic dipole moment µ is given byµ = g e¯h2m ,(18)with the gyromagnetic factor[17, 18, 19, 20, 1]g = 2(1 + α2π +34ζ(3) −3ζ(2) ln2 + 12ζ(2) + 197144 απ2+ O απ3!). (19)Therefore, the magnetic energy contribution to the total Hamiltonian readsHB = 12µ(⃗σ−−⃗σ+) · ⃗B .

(20)To complete this list of notations, it turns out that the parameter setting thephysical scale of magnetic fields in the present system, is the combination2µ∆E =13.628575 Tesla ,(21)leading to the definition of the positive quantityx = 2µ∆E | ⃗B | =| ⃗B |3.63 Tesla . (22)In addition, it proves convenient7 to introduce the following combinations of the Bxand By components of the magnetic field ⃗B,B+ = Bx + iBy√2,B−= Bx −iBy√2.

(23)Given the total HamiltonianH = H0 + HB ,(24)6Here, ⃗σ are the usual Pauli matrices defining the spin 1/2 representation of the (double coveringSU(2)) of the rotation group SO(3) in three dimensions.7The actual reason why these definitions are convenient is the fact that it is the spin 1/2 represen-tation of the three dimensional rotation group which appears throughout.5

it is now a simple matter to proceed diagonalising it for its lowest energy states withspherical symmetry, namely in the sector of positronium 1S hyperfine states of theprevious section. First, one easily findsH | 0, 0 >= E0 | 0, 0 > +µBz | 1, 0 > −µB−| 1, 1 > +µB+ | 1, −1 > ,(25)H | 1, 0 >= E1 | 1, 0 > +µBz | 0, 0 > ,(26)H | 1, 1 >= E1 | 1, 1 > −µB+ | 0, 0 > ,(27)H | 1, −1 >= E1 | 1, −1 > +µB−| 0, 0 > .

(28)Given these results, eigenstates of H and their eigenvalues can be derived aftersome work.The corresponding four eigenstates are denoted | ψS′ >, | ψT ′ > and| ψ± >. In the limit of vanishing magnetic field ⃗B, these states reduce—possibly upto some phase—to the singlet | 0, 0 >, the triplet | 1, 0 > and the triplet | 1, ±1 >states, respectively, hence the notation.

In particular, the | ψS′ > and | ψT ′ > statesare referred to as the “pseudo-singlet” and “pseudo-triplet” states, respectively.The pseudo-singlet state is given by| ψS′ >=1√2 (1 + x2)−1/4(√1 + x2 + 1)−1/2 ×n(√1 + x2 + 1) | 0, 0 > −−2µ∆EBz | 1, 0 > + 2µ∆EB−| 1, 1 > −2µ∆EB+ | 1, −1 >o,(29)with the eigenvalueES′ = −12∆Eh√1 + x2 −1i+ E0 . (30)The pseudo-triplet state is| ψT ′ >=1√2 (1 + x2)−1/4(√1 + x2 −1)−1/2 ×n(√1 + x2 −1) | 0, 0 > ++ 2µ∆EBz | 1, 0 > −2µ∆EB−| 1, 1 > + 2µ∆EB+ | 1, −1 >o,(31)with the eigenvalueET ′ = +12∆Eh√1 + x2 + 1i+ E0 .

(32)Finally, the remaining two states are| ψ± >=1√2√B2x+B2y| ⃗B|| 1, 0 > ++ 1√2Bz±| ⃗B|| ⃗B|B−√B2x+B2y | 1, 1 > + 1√2−Bz±| ⃗B|| ⃗B|B+√B2x+B2y | 1, −1 > ,(33)with the degenerate eigenvalueE± = ∆E + E0 = E1 . (34)Here, upper (resp.

lower) signs in the r.h.s. of (33) correspond to the state | ψ+ >(resp.

| ψ−>).6

By construction, the four states just given not only diagonalise the total Hamilto-nian H, but are also orthonormalised. Namely, these states are orthogonal by pairs andare normalised to 1.

Of course, orthogonality is automatic for non degenerate statesbut not for degenerate ones; by construction, the states | ψ+ > and | ψ−> given abovedo indeed have a vanishing inner product.Before commenting on these expressions, it is useful to consider the limit in whichthe magnetic field ⃗B is parallel to the z axis, namely when Bx = 0 and By = 0.Of course, in this limit, the eigenvalues of the four states above remain unchanged,since their energies only depend on the variable x, i.e. the norm of the magnetic field.However, the states specified above, which diagonalise the total Hamiltonian H, thenreduce to| ψS′ >= cos θ | 0, 0 > −sin θ | 1, 0 > ,(35)| ψT ′ >= η { sin θ | 0, 0 > + cos θ | 1, 0 > } ,(36)| ψ± >= ±e±iηω | 1, ±η > .

(37)In these expressions, η = sign(Bz) is the sign of Bz, ω is some arbitrary phase whosevalue is dependent on the manner in which the limit Bx = 0, By = 0 is taken8, andthe mixing angle θ is defined bycos θ = 1√2s1 +1√1 + x2 ,sin θ = η1√2s1 −1√1 + x2 . (38)Of course, these expressions coincide with those usually found in the literature[4, 10, 3],in which case it is customary to take | ψ± >=| 1, ±1 > and the z axis along the magneticfield ⃗B, namely η = +1.

In particular, note that in the limit of an infinite magneticfield ⃗B, the pseudo-singlet and pseudo-triplet states in (35) and (36) further reduce to(in the notation of (1))| ψS′ >= −η ψ(r) | −η >| +η > ,| ψT ′ >= +η ψ(r) | +η >| −η > . (39)In order to comment on the physical significance of these results, let us first considerthe case where the magnetic field is parallel to the z axis.

Even though the vector ⃗Bthen explicitly breaks rotational invariance of the positronium system in vacuum, therestill remains the symmetry of arbitrary rotations around the z axis. Consequently, thespin projection m on that axis still defines a good quantum number for positroniumstates (of vanishing angular momentum.).

Therefore, the states | 1, 1 > and | 1, −1 >must remain eigenstates of the total Hamiltonian, whereas the other two states withm = 0, namely | 0, 0 > and | 1, 0 >, are now allowed to mix.Moreover, for theformer two states with m = ±1, since the electron and positron have their spins thenaligned and since their magnetic moments are equal in norm but opposite, the magneticcoupling energy HB vanishes identically, leading for these two states to the same energyeigenvalue E1 as in the absence of any field ⃗B. Finally, for the remaining two states8Indeed, the coefficients of the states | 1, 1 > and | 1, −1 > in (33) are non analytic functions ofBx and By.7

with m = 0 diagonalising the total Hamiltonian H, in the limit9 where the magneticcoupling HB is much larger than all other contributions to H, namely for magneticfields whose magnitude is much larger than (∆E/(2µ) = 3.63 Tesla), the state oflowest (resp. highest) energy, i.e.

| ψS′ > (resp. | ψT ′ >), is the one for which theelectron and positron magnetic moments are both aligned parallel (resp.

antiparallel)to the magnetic field ⃗B, or equivalently the one for which the electron spin is alignedantiparallel (resp. parallel) to ⃗B and the positron spin is parallel (resp.

antiparallel)to ⃗B.These properties—indeed, solely expected on physical grounds independently of anyexplicit calculation—are beautifully confirmed by the results above in the case whereBx = 0 and By = 0. At this stage, it thus appears that an explicit calculation servesthe purpose only of determining the mixing angle θ in (38) between the two m = 0states, and of deriving the energy eigenvalues ES′ and ET ′ in (30) and (32), as functionsof the magnetic field ⃗B = (0, 0, Bz).Consider now the general situation when the direction of the magnetic field ⃗B isarbitrary with respect to the z axis.

In fact, the results just discussed can be usedin order to understand the general case as well. Indeed, the only difference betweenthe two configurations is that the axis with respect to which the spin part of statevectors is expanded, is different.

Hence, by an appropriate change of basis in the spinsector, effected through a rotation in the spin 1/2 representation, the eigenstates ofthe total Hamiltonian H in the arbitrary case (Bx, By) ̸= (0, 0) can in principle beconstructed from the expressions of these states when (Bx, By) = (0, 0). Therefore,since under such a rotation in spin space representations of spin 0 and of spin 1 areinvariant, only the states | 1, m = 0, ±1 > can mix among themselves.

Consequently,in the general case, both the pseudo-singlet and pseudo-triplet states should be givenas some superposition of all four states | 0, 0 > and | 1, m = 0, ±1 >, with in particularthe coefficient of the | 0, 0 > component independent of the components of the magneticfield but only dependent on its norm | ⃗B |, whereas the remaining two states | ψ± >can only involve the three states | 1, m = 0, ±1 >. In addition, for all four eigenstates,the coefficients of the states | 1, m = 0, ±1 > must depend on all three components Bx,By and Bz of the magnetic field.Indeed, these are features of the results in (29), (31) and (33), which therefore findtheir origin in the fact that spin 0 and spin 1 representations are invariant under spacerotations.

However, only an explicit calculation—either along the lines just sketched orby direct diagonalisation of the Hamiltonian as done in this note—can determine thespecific mixing coefficients defining each of the eigenstates of the total Hamiltonian H.Incidentally, note that the same argument of invariance under space rotations explainswhy eigenvalues of H remain unchanged when the magnetic field ⃗B is no longer parallelto the z axis, i.e. why these eigenvalues only depend on the norm | ⃗B | of the magneticfield.9This situation was pointed out to the author by J. Deutsch.8

4Positronium LifetimesAs a first application of results so far, let us compute decay rates for all four eigenstatesof the total Hamiltonian H in the presence of a magnetic field ⃗B. Actually, such acalculation is rather straightforward, given the decay rates λS and λT of the singletand triplet states | 0, 0 > and | 1, m = 0, ±1 >, respectively.First, consider decay rates into two photons.

Due to charge conjugation, only the| 0, 0 > state has 2γ decays. Therefore, the 2γ decay rate of the pseudo-singlet state| ψS′ > isλ(2γ)S′= 12"1 +1√1 + x2#λS = λS cos2 θ .

(40)For the pseudo-triplet state | ψT ′ >, we haveλ(2γ)T ′= 12"1 −1√1 + x2#λS = λS sin2 θ . (41)Finally, the 2γ decay rate of the remaining two states | ψ± > isλ(2γ)±= 0 .

(42)Similarly, for the 3γ decay rates of these states in the same order, one findsλ(3γ)S′= 12"1 −1√1 + x2#λT = λT sin2 θ ,(43)λ(3γ)T ′= 12"1 +1√1 + x2#λT = λT cos2 θ ,(44)λ(3γ)±= λT . (45)In these expressions, the angle θ is defined in (38).Finally, total decay rates—ignoring the much suppressed rates into four or morephotons—are simplyλS′ = 12(λS + λT) + 121√1 + x2(λS −λT) = λS cos2 θ + λT sin2 θ ,(46)λT ′ = 12(λS + λT) −121√1 + x2(λS −λT) = λS sin2 θ + λT cos2 θ ,(47)λ± = λT .

(48)As ought to be expected, these expressions only depend on the norm of the magneticfield ⃗B, but not on its direction. Indeed, since the choice of axis with respect to whichspin states are expanded does not affect the calculation of decay rates10, that axis can10Indeed, positronium states can “remember” the direction and polarisation of the incoming positrononly through the populations of the four Hamiltonian eigenstates (this is the topic of the next section).Decay rates are intrinsic properties of each of these states, and as such, are thus independent of anyvariable possibly affecting positronium formation.9

always be taken along the magnetic field, in which case decay rates can only dependon | Bz |, namely the norm of the magnetic field.Incidentally, note that total 2γand 3γ decay rates—λS and λT, respectively—are independent of ⃗B—a consequenceof unitarity.It is also easy to check that the statistical decay rates into two and three photons,i.e.the average of each of these rates over the four Hamiltonian eigenstates, areindependent of the magnetic field. These statistical rates thus coincide with their valueswhen ⃗B = ⃗0, namely λS/4 and 3λT/4 for two and three photon decays, respectively.Finally, let us remark that the difference between the total decay rates for thepseudo-triplet and | ψ± > states,λT ′ −λT = 12(λS −λT)"1 −1√1 + x2#= (λS −λT) sin2 θ ,(49)is a quantity always positive for all values of the magnetic field.5Positronium PopulationsLet us now address the specific topic of this note; the formation of positronium ina medium placed in a magnetic field.

In view of applications, the incoming positronis assumed to have a polarisation P+ (−1 ≤P+ ≤+1). This polarisation P+ is theexpectation value of the positron spin projected onto its momentum, the latter vectorthus also defining the z axis for spin quantisation from now on.

Therefore, up to aphysically irrelevant overall phase, the spin component of the incoming positron wavefunction is given by1√2q1 + P+ | + > + eiφ+ 1√2q1 −P+ | −> . (50)Here, φ+ is an arbitrary phase difference—thus possibly leading to observable physicaleffects—between the two spin components defining a positron state of polarisation P+.Similarly, it will be assumed11 that the positron capturing electron has a polarisationP−(−1 ≤P−≤+1) along the same z axis.

Though in most practical applications, thepositronium formation medium is at temperatures such that electrons are effectivelynot polarised, some experiments at very low temperatures are being planned, for whichan investigation of possible effects due to electron polarisation in the applied magneticfield might therefore be found useful. Consequently, again up to a physically irrelevantoverall phase, the spin part of the electron wave function is also of the form1√2q1 + P−| + > + eiφ−1√2q1 −P−| −> ,(51)where φ−is another arbitrary phase shift.Hence, at the moment of positronium formation, it is assumed that the spin com-ponent of the positronium state vector | ψ, t = 0 > is simply given by the tensor11The interest of this possibility was pointed out to the author by F. Gimeno-Nogues.10

product12 of the electron and positron spin vectors in (50) and (51), while the spacepart of the state vector is of course the wave function ψ(r) in (1). In order to obtainthe time evolution of the associated state vector, and thus also the time dependence ofits decay products, the resulting wave function | ψ, t = 0 > of formed positronium hasto be expanded in the basis of eigenstates of the total Hamiltonian H in the presenceof the magnetic field ⃗B.

This change of basis thus defines coefficients CS′, CT ′ and C±such that| ψ, t = 0 >=Xa=S′,T ′,+,−Ca | ψa > . (52)Explicit expressions for these coefficients are given in Appendix A.

Time evolution ofthe positronium state formed is then given by| ψ, t >=Xa=S′,T ′,+,−Ca | ψa > exp−i¯hEat −12λat,(53)with the quantities Ea and λa (a = S′, T ′, +, −) defined in (30), (32), (34) and (46),(47) and (48).Given these expressions, time evolution of each of the populations associated toeach of the four states | ψa > (a = S′, T ′, +, −) is simply obtained as| Ca |2 e−λat ,a = S′, T ′, +, −. (54)Expressions for all observables of interest are then easily written down.

For example,2γ and 3γ production rates are13, respectively,R(2γ)(t) =Xa=S′,T ′,+,−λ(2γ)a| Ca |2 e−λat ,(55)R(3γ)(t) =Xa=S′,T ′,+,−λ(3γ)a| Ca |2 e−λat ,(56)while the total photon production rate—simply the sum of the latter two rates—isitselfR(t) =Xa=S′,T ′,+,−λa | Ca |2 e−λat . (57)The rather lengthy expressions for the populations at t = 0, namely the coefficients| Ca |2 (a = S′, T ′, +, −), are given in Appendix B.

Results probably more relevantat this point are the same coefficients averaged 14 over the phase shifts φ−and φ+.Indeed, one ought to expect that for most sources where the β decay process responsible12Note that it is always possible to “rotate away” one of the two phases φ−or φ+—but not both—by an appropriate rotation around the z axis. However, dependence on the cancelled phase thenreappears through the Bx and By components of the rotated magnetic field.13Note that when actual experimental data are considered, the 2γ production rate in (55) shouldalso include a “fast” component due to direct pair annihilation of incoming positrons with electronsof the positronium forming medium.14This average does not amount to setting eiφ± = 0 in the original expressions for the coefficientsCa, and can only be applied once the complete expressions of Appendix B for the coefficients | Ca |2have been obtained.11

for positron production is taking place, positron states with all possible phase shiftsφ+ are statistically populated, thereby justifying an average of the final positroniumpopulations over φ+. Similarly, in the positronium forming medium, one should alsoexpect that all electron states with different phase shifts φ−are statistically populated,again justifying an average over the phase φ−.

Under these assumptions15, the averagedpopulations are given by| CS′,T ′ |2 = 14 [1 −P−P+] ∓141√1 + x22µ∆E Bz [P−−P+] ++ 12"1 ∓1√1 + x2# B−B+| ⃗B |2 P−P+ ,(58)and| C± |2 = 14"1 ± Bz| ⃗B |P−# "1 ± Bz| ⃗B |P+#. (59)In the r.h.s.

of these two expressions, the upper (resp. lower) sign refers to | CS′ |2(resp.

| CT ′ |2) and | C+ |2 (resp. | C−|2), respectively.

Note that the sum of the fourphase averaged populations does indeed reduce to 1, as ought be the case since, bydefinition, the state | ψ, t = 0 > is normalised to 1.In turn, phase averaged photon production rates R(2γ)(t), R(3γ)(t) and R(t) aredefined as in (55), (56) and (57), of course involving now the phase averaged coefficients| Ca |2 (a = S′, T ′, +, −) just given. Clearly, these averaged photon rates are morereadily amenable to experimental measurement than are the non phase averaged ratesconsidered previously.6ConclusionsThis note reports on the calculation of hyperfine positronium populations formed in thepresence of an arbitrary magnetic field—whose gradient is assumed to be vanishinglysmall over the spatial extension of the positronium bound state—for arbitrary positronand electron polarisations.

The analysis generalises results available[4, 10, 3] in theliterature in two respects. On the one hand, the magnetic field is not assumed to benecessarily aligned along the momentum of the incoming positron.

On the other hand,allowing for possible electron polarisation effects enables the present results to be ofapplication to positron polarimeters operated at very low temperatures.Expressions derived in this note provide the basic information required in any ex-perimental analysis of results obtained using positron polarimeters based on eithertime or energy distributions of positronium decay photons. As a simple but explicitillustration of relevance to current β decay experiments[5, 6, 7], let us consider for15Note that taking such averages is even more justifiable in the instance—often realised in practice—that the magnetic fields present in an experimental set-up possess an axial symmetry along the axisof incoming positrons.

Indeed, as was noticed previously, either of the phases φ−or φ+ can always be“rotated away” by an appropriate rotation around the z axis, then also a symmetry transformationof the magnetic fields.12

example the phase averaged photon production time spectrum R(t) when electrons inthe positronium formation medium are not polarised (P−= 0). Expressions derived inthe previous section then lead toR(t) = 12λTe−λT t + 14λT ′"1 −1√1 + x22µ∆E BzP+#e−λT ′t++ 14λS′"1 +1√1 + x22µ∆E BzP+#e−λS′t .

(60)However, since the pseudo-singlet decay rate λS′ is much larger than the pseudo-tripletone λT ′, the pseudo-singlet contribution in the r.h.s. of (60) becomes effectively neg-ligible after a nanosecond or so, leaving only the first two terms.

Hence in effect, thephoton time spectrum R(t)R(t ≥1 ns) ≈12λTe−λT t + 14λT ′ [1 −ǫP+] e−λT ′t ,(61)with the analysing powerǫ =1√1 + x22µ∆E Bz ,(62)provides the means of measuring positron polarisations. Note that, when comparedto the usual result obtained for a magnetic field ⃗B assumed to be parallel to theincoming positron momentum, the sole effect of non vanishing components Bx and Byin this simple example is to decrease the effective analysing power ǫ multiplying thepositron polarisation P+.

Nevertheless, there certainly exist other instances where theeffects of non vanishing components Bx and By have to be properly accounted for whenanalysing actual experimental data. The results of this note provide the basis for suchinvestigations.AcknowledgementsUseful discussions with J. Deutsch and F. Gimeno-Nogues are gratefully acknowledged.13

Appendix AThis Appendix gives the coefficients of the linear combination of the total Hamiltonianeigenstates defining the state vector of positronium formed in the presence of a magneticfield ⃗B (see (53)).For the pseudo-singlet and pseudo-triplet states, one hasCS′,T ′ =1√2 (1 + x2)−1/4 (√1 + x2 ± 1)−1/2××(12√2q1 + P−q1 −P+ eiφ+ −q1 −P−q1 + P+ eiφ−(√1 + x2 ± 1) ∓∓12√2q1 + P−q1 −P+ eiφ+ +q1 −P−q1 + P+ eiφ− 2µ∆E Bz ±± 12q1 + P−q1 + P+2µ∆E B+ ∓(63)∓12q1 −P−q1 −P+ ei(φ−+φ+) 2µ∆E B−,where the upper (resp. lower) sign refers to the coefficient CS′ (resp.

CT ′).Similarly, the coefficients of the remaining two states | ψ± > areC± = 14q1 + P−q1 −P+ eiφ+ +q1 −P−q1 + P+ eiφ− qB2x + B2y| ⃗B |++12√2q1 + P−q1 + P+Bz± | ⃗B || ⃗B |B+qB2x + B2y+(64)+12√2q1 −P−q1 −P+ ei(φ−+φ+) −Bz± | ⃗B || ⃗B |B−qB2x + B2y.Here again, the upper (resp. lower) sign refers to the coefficient C+ (resp.

C−).14

Appendix BThis Appendix gives the populations of positronium states formed in the presence ofa magnetic field, for electrons and positrons of initial polarisation P−and P+, respec-tively. With the same conventions as to upper and lower signs as in Appendix A, theresults are as follows.The | ψS′,T ′ > populations are| CS′,T ′ |2= 12 (1 + x2)−1/2 (√1 + x2 ± 1)−1×× 12(1 + x2)1/2(√1 + x2 ± 1) [1 −P−P+] ∓∓12(√1 + x2 ± 1) 2µ∆E Bz [P−−P+] + 2µ∆E2(B−B+)P−P+ ∓∓12(√1 + x2 ± 1)q1 −P 2−q1 −P 2+ cos(φ−−φ+) −(65)−14 2µ∆E2 q1 −P 2−q1 −P 2+hB−eiφ+ + B+e−iφ+i hB−eiφ−+ B+e−iφ−i±±12√2 2µ∆E q1 −P 2+hB−eiφ+ + B+e−iφ+i √1 + x2 ± 1) ∓2µ∆E Bz P−∓∓12√2 2µ∆E q1 −P 2−hB−eiφ−+ B+e−iφ−i (√1 + x2 ± 1) ± 2µ∆E Bz P+).Similarly, the | ψ± > populations are| C± |2= 14B2z| ⃗B |2 [1 + P−P+] ± 14Bz| ⃗B |[P−+ P+] + 12B−B+| ⃗B |2 ±±14√2q1 −P 2+"1 ± Bz| ⃗B |P−# " B−| ⃗B |eiφ+ + B+| ⃗B |e−iφ+#±±14√2q1 −P 2−"1 ± Bz| ⃗B |P+# " B−| ⃗B |eiφ−+ B+| ⃗B |e−iφ−#+(66)+ 18q1 −P 2−q1 −P 2+" B−| ⃗B |eiφ+ + B+| ⃗B |e−iφ+# " B−| ⃗B |eiφ−+ B+| ⃗B |e−iφ−#.Note that in each of these two equations, the last three terms could be combinedfurther into the product of two terms.

However, results in the form given here aremore amenable to the phase average discussed in the main text. Incidentally, it isstraightforward to check that the sum of all four populations does indeed reduce to 1,since, by construction, the state | ψ, t = 0 > is normalised to 1.15

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