NTUTH–95–04 May 1995
arXiv:hep-ph/9505300v1 13 May 1995
Magnetic Field Dependence of Muonium-antimuonium Conversion
Wei-Shu Hou and Gwo-Guang Wong
ment of Physics, National Taiwan University, Taipei, Taiwan 10764, R.O.C.
We study the magnetic ?eld dependence of muonium–antimuonium conversion induced by neutral (pseudo)scalar bosons. Only the SS operator contributes to the conversion of polarized muonium, but it gets quenched by a magnetic ?eld of strength 0.1 Gauss or stronger. Conversion induced by SS couplings for unpolarized muonium is independent of magnetic ?eld. Magnetic ?elds of 0.1 Tesla or stronger starts to suppress conversion induced by P P interactions in the lowest Breit-Rabi level, but gets partially compensated by a rise in conversion probability in the other unpolarized level. The e?ects of (S ? P )(S ? P ) and (S ? P )(S ± P ) operators behave in the same way as (V ? A)(V ? A) and (V ? A)(V ± A) operators, respectively.
The spontaneous conversion of muonium (hydrogen-like atom M = ?+ e? ) into antimuo? nium (M ) would violate the separate additive muon and electron numbers, but would remain consistent with multiplicative muon or electron number conservation. De?ning the e?ective coupling GM M via the interaction , ? GM ? HM M = √ M ?γλ (1 ? γ5 )e ?γ λ (1 ? γ5 )e + h.c., ? ? ? 2 the limit has just been improved  by an order of magnitude, GM M < 1.8 × 10?2 GF , ? (2) (1)
compared with the previous bound of 0.16 GF , where GF is the Fermi constant. The ultimate aim  is to reach the sensitivity level of 10?3 GF . The (V ?A)(V ?A) interaction of eq. (1) gained further theoretical footing when Halprin [5,6] pointed out that in left-right symmetric models with Higgs triplets, doubly charged ? scalars can mediate M–M transitions at tree level. The e?ective interaction is of (V ± A)(V ± A) form after Fierz rearrangement. The possibility of (V ? A)(V + A) interactions, induced by dilepton gauge bosons , was discussed by Fujii et al. . Interestingly, muonium conversion is more pronounced in the singlet channel, in contrast to the (V ? A)(V ? A) case where the conversion matrix element is of equal strength for both singlet and triplet muonium . Recently, Horikawa and Sasaki  pointed out further that (V ? A)(V + A) interactions are considerably less senstive to magnetic ?elds compared to the (V ? A)(V ? A) case. Since actual experiments are conducted in the presence of magnetic ?elds [2–4], this has important implications for the interpretation of the bound of eq. (2) for (V ? A)(V + A) interactions. In a recent paper  (see also [11,12]), we explored neutral (pseudo)scalar ? induced M–M oscillations. Since (S ? P )(S ± P ) and (V ? A)(V ± A) operators are related by Fierz transform, we wish to study the magnetic ?eld dependence of neutral scalar induced muonium conversion. Exotic neutral (pseudo)scalar bosons H and A could have couplings , 2
fH fA ? LY = √ ?e H + i √ ?γ5 e A + h.c., ? ? 2 2
while a discrete symmetry such as multiplicative  electron number Pe  forbids processes odd in number of electrons (plus positrons) like ? → eγ and ? → ee?. The resulting e?ective e ? Hamiltonian responsible for M–M conversion is, HM M = ?
2 f2 fH ?e ?e ? A2 ?γ5 e ?γ5 e ? ? ? ? 2m2 2mA H 2 1 fH f2 1 = + A (S 2 ? P 2 ) + 2 2 4 mH mA 4
2 f2 fH ? A (S 2 + P 2 ), m2 m2 H A
where S = ?e and P = ?γ5 e. Note that S 2 ? P 2 and S 2 + P 2 contain (S ? P )(S ± P ) and ? ? (S ? P )(S ? P ) interactions, respectively. In the “U(1) limit” of fH = fA and mH = mA the subleading (S ? P )(S ? P ) terms are completely absent. The matrix elements of eq. (4) and the accompanying phenomenology of eq. (3) have been discussed in ref. . To explore magnetic ?eld dependence of muonium conversion probabilities, consider the Hamiltonian for 1S muonium, H = H0 + a se · s? ? ?e · B ? ?? · B + HM M , ? ? ? (5)
where H0 gives the 1S energy E0 = ?α2 m/2, with reduced mass 1/m = 1/me + 1/m? , a ? 1.846 × 10?5 eV is the 1S muonium hyper?ne splitting, and ?e = ?ge ?B se , ?? = = ?
me +g? ?B m? s? , where ge ? g? ? 2 and ?B = e/(2me ) ? 5.788 × 10?9 eV/Gauss is the Bohr = = =
magneton. Introducing the dimensionless parameters, X, Y ≡ ?B B me ge ± g? , a m? (6)
we see that |Y | is just 1% smaller than |X|. Ignoring HM M for the moment, the four ? muonium Breit-Rabi energy levels  are, a 1 ±Y , 2 2 1 √ a ? + 1 + X2 , EM (1, 0) = E0 + 2 2 1 √ a ? ? 1 + X2 , EM (0, 0) = E0 + 2 2 EM (1, ±1) = E0 + 3
which correspond to the eigenstates |M; 1, ±1 = |M; 1, 0 = |M; ↑↑ , |M; ↓↓ , c |M; ↑↓ + s |M; ↓↑ , (8)
|M; 0, 0 = ?s |M; ↑↓ + c |M; ↓↑ , where the magnetic ?eld dependent “rotation” is 1 X s= √ 1? √ 2 1 + X2
1 X c= √ 1+ √ 2 1 + X2
We have labeled the Breit-Rabi energy levels with the weak ?eld basis |M; F, mF , i.e. √ the corresponding zero B ?eld (X, Y ?→ 0 and s, c ?→ 1/ 2) hyper?ne states. The “uncoupled” basis of |M; mse , ms? corresponds to the strong ?eld limit of X, Y ? 1 and ? s ?→ 0. In this limit, |M; 1, 0 ?→ |M; ↑↓ and |M; 0, 0 ?→ |M; ↓↑ , and the electron and muon hyper?ne spin-spin coupling is overwhelmed by the Zeeman e?ect. For antimuonium, again ignoring the e?ect of HM M , retaining the spin labels and with ? ? uncoupled basis |M ; mse , ms? , one simply ?ips X → ?X, Y → ?Y and hence interchange ? s ? c in eqs. (8) and (9). The notable changes are EM (1, ?1) = EM (1, ±1), ? since for given spin, the antiparticle magnetic moments have ?ipped sign, and, ? |M; 1, 0 = ? ? s |M; ↑↓ + c |M; ↓↑ , (11) (10)
? ? ? |M; 0, 0 = ?c |M ; ↑↓ + s |M; ↓↑ . ? We have the following energy di?erences between M and M eigenstates, EM (1, ±1) ? EM (1, ±1) = ±a Y, ? EM (1, 0) ? EM (1, 0) = ? EM (0, 0) ? EM (0, 0) = 0, ? √ EM (1, 0) ? EM (0, 0) = ?(EM (0, 0) ? EM (1, 0)) = a 1 + X 2 . ? ?
The e?ect of HM M can be treated as a perturbation. De?ne generically ? 4
? M|HM M |M = δ/2 ?
? (for simplicity, we take δ to be real) between any two Breit-Rabi M and M energy eigenstates, it was shown by Feinberg and Weinberg  that the time integrated probability for an initial muonium state to decay as antimuonium is ? P (M) = δ2 , 2(δ 2 + ?2 + λ2 ) (14)
where ? = EM ? EM is the energy di?erence, and λ ? 2.996 × 10?10 eV is the muon decay = ? rate. The physics is clear: muonium oscillation has to compete with muon decay and the damping from oscillations between two states that are very disparate in energy. The total transition probability is ? PT (M) =
? |cF,mF |2 P (F,mF ) (M)
? where |cF,mF |2 are the populations in muonium states of eq. (8), and P (F,mF ) (M ) are the probabilities for an initial (F, mF ) muonium state to decay as antimuonium. ? ? In principle P (1,0) (M ) and P (0,0) (M) also contain the probabilities of initial (1, 0) or (0, 0) muonium states to decay as (0, 0) or (1, 0) antimuonium, respectively, since (1, 0) and (0, 0) are mixtures of unpolarized states. To show that these are vanishingly small, it is useful to notice the hierarchy δ ? λ ? a, (16)
the ?rst of which follows from eq. (2) for any HM M model. Combining eqs. (12) and (14) we ? see that (1, 0) → (0, 0) and (0, 0) → (1, 0) transitions are extremely suppressed by hyper?ne splitting even for B = 0, and need not be considered. Similarly, although (1, ±1) → (1, ±1) transitions contribute in B = 0 limit, they become rapidly suppressed even for rather weak ? magnetic ?elds , because of the mismatch between Zeeman energy levels for M vs. M polarized states with same mF . ? In this work we will discuss only the relative magnetic ?eld dependence of M-M conversion probabilities. Di?erences in coupling strength for various e?ective operators at zero B 5
?eld can be found in refs. [5–7,10]. Hence, we normalize e?ective interactions to the conversion probability due to eq. (1) at zero magnetic ?eld, and in particular for equally populated (1/4 each) Breit-Rabi states (the latter condition would be removed at the end). Thus, the zero ?eld total transition probability is ? δ2 GM M ? ? PT (M; B = 0) ? 2 = 2.56 × 10?5 2λ GF ? which de?nes the parameter δ. For the (V ? A)(V ? A) case one basically multiplies the r.h.s. (right hand side) of eq. (13) by δmse mse δms? ms? , hence ? ? ? δ ? M ; 1, ±1|HM M |M; 1, ±1 = , ? 2 ? δ ? ? M; 1, 0|HM M |M; 1, 0 = M ; 0, 0|HM M |M; 0, 0 = √ , ? ? 2 1 + X2 and all other matrix elements vanish. One therefore ?nds the result  ? P (1,±1) (M) = P (1,
? (M ) = P (0,
? δ2 , ? 2(δ 2 + a2 Y 2 + λ2 ) ? δ2 0) ? (M) = ?2 . 2(δ + λ2 (1 + X 2 ))
? ? The magnetic ?eld dependence for PT (M ), as well as the separate probabilities P (1,±1) (M), ? ? P (1,0) (M) and P (0,0) (M ) of eq. (19), are plotted as “+” symbols in Figs. 1–4, respectively. The behavior is readily understood. Because of the aY energy splitting, the suppression of (1, ±1) modes sets in with B ?eld of just a few cG, and they become quenched for 0.1 G or higher. The mF = 0 “unpolarized” modes are oblivious to the magnetic ?eld until X becomes appreciable, i.e. for B ? a/2?B ? 1kG, and get quenched by ?elds of 1 Tesla or ? higher. The scale di?erence for Fig. 4 would be discussed shortly. The suppression in P (M) has been taken into account in the experimental limit of eq. (2) for the interaction of eq. (1). We have checked and con?rmed the result for the (V ? A)(V + A) case , P
? (M ) =
? δ2 , ? 6(δ 2 + a2 Y 2 + λ2 ) 6
? (M ), P
? (M ) = 6
√ 1 1+X 2 2 √ 1 1+X 2
? δ 2 + λ2
The results are also plotted in Figs. 1–4 as “×” symbols. Conversion in (0, 0) mode is the most prominent , but gets suppressed by up to a factor of 4/9 when the magnetic ?eld goes beyond ? 1kG. The (1, ±1) modes are quenched by magnetic ?elds of 0.1G or higher, ? just like in the (V ? A)(V ? A) case, but P (1,0) (M) actually grows with B ?eld around 1kG, and partially compensates for the drop in P (0,0) . We now state the results for the (pseudo)scalar induced interaction case. Details would be given elsewhere . For purely scalar interactions (fA = 0), we ?nd ? P (1,±1) (M ) = P (1,
? (M) = P (0,
? δ2 , ? 2(δ 2 + a2 Y 2 + λ2 ) ? δ2 0) ? (M ) = ?2 . 2(δ + λ2 )
Thus, aside from the familiar quenching of the (1, ±1) states, the (1, 0) and (0, 0) states are completely insensitive to magnetic ?elds . For purely pseudoscalar interactions (fH = 0), we ?nd ? P (1,±1) (M ) = 0, P
? (M), P (0,
? (M ) = 2
?1 + ?1 +
√ 1 1+X 2 2 √ 1 1+X 2
? δ 2 + λ2
In this case, muonium conversion occurs solely in the (0, 0) mode for zero magnetic ?eld . Conversion in the (1, 0) mode starts to grow from zero for ?eld strengths beyond ? 1kG, and partially compensates for the drop, by a factor of 4, in transition probability in the (0, 0) mode. We plot the results again in Figs. 1–4, with solid and dashed lines representing scalar and pseudoscalar case, respectively. The results for (S ? P )(S ? P ) and (S ? P )(S + P ) operators can be similarly obtained. With the same normalization conditions as described above, the results are also given in Figs. 1–4 for sake of comparison, with open circles representing the (S ? P )(S + P )case and open boxes representing the (S ? P )(S ? P ) case. Note that according to eq. (4), the 7
(S?P )(S?P ) operators should be subdominant compared to (S?P )(S+P ) operators. Pure (S ? P )(S ± P ) operators corresponds to complex neutral scalars , where the sneutrino ντ in SUSY models with R-parity breaking  as a special case. It is evident from Figs. ? 1–4 that the combinations (S ? P )(S ? P ) and (S ? P )(S + P ) behave in the same way as (V ? A)(V ? A) and (V ? A)(V + A), respectively. In the latter case, the two operators are related to each other by a Fierz transform. For the former case, although the operators can not be related to each other by a Fierz transform, the matrix elements are always in same proportion, which comes as a consequence of the nonrelativistic limit. Turning to discussions, we note that the assumption of equally populated Breit-Rabi levels is not a valid one, since this is determined by the muonium formation process and the magnetic ?eld strength. However, as a consequence of this assumption, the results in eqs. (19-22) and hence Figs. 2–4 all have an arti?cial factor of 4. To illustrate the e?ect of di?erently populated Breit-Rabi levels, normalizing again to the (V ? A)(V ? A) case at zero B ?eld, we take muonium states to be populated as  32%, 35%, 18% and 15%, ? respectively, for (F, mF ) = (0, 0), (1, +1), (1, 0), (1, ?1), and plot the results for PT (M) (i.e. analogous to Fig. 1) in Fig. 5. The (V ? A)(V ? A) and (S ? P )(S ? P ) results are unchanged, since the conversion matrix elements are the same for all modes. The purely scalar (SS) case is also unchanged, since |c10 |2 + |c00 |2 = |c1+1 |2 + |c1?1 |2 = 50% is the same as the equally populated case. The (V ? A)(V + A), (S ? P )(S + P ) and P P cases are somewhat modi?ed from the equally populated case, but the di?erence for 1kG ?eld is rather slight. ? Let us summarize our ?ndings. Purely scalar (SS) induced M-M transitions in polarized (i.e. (F, mF ) = (1, ±1)) modes are quenched by magnetic ?elds of 0.1 Gauss or higher, but conversion in unpolarized states (i.e. (1, 0) and (0, 0)) are independent of magnetic ?eld strength. Purely pseudoscalar (P P ) interactions do not induce conversion in (1, ±1) states, but exhibit compensating e?ects in the (1, 0) and (0, 0) channels, similar to the (V ?A)(V + A) case. Interactions of (S ?P )(S ±P ) and (V ?A)(V ±A) form have the same magnetic ?eld dependence since they are related by Fierz transform, while (S ? P )(S ? P ) 8
and (V ? A)(V ? A) interactions have the same ?eld dependence because of proportional conversion matrix elements.
We thank K. Jungmann for numerous communications and a copy of ref. . The work of WSH is supported by grant NSC 84-2112-M-002-011, and GGW by NSC 84-2811-M-002035 of the Republic of China.
 G. Feinberg and S. Weinberg, Phys. Rev. Lett. 6, 381 (1961); Phys. Rev. 123, 1439 (1961).  K. Jungmann et al., presented at 23rd INS “International Symposium on Nuclear and Particle Physics with Meson Beams in the 1-GeV/c Region”, Tokyo, Japan, March 15-18, 1995.  B. E. Matthias et al., Phys. Rev. Lett. 66, 2716 (1991).  K. Jungmann et al., PSI Experiment No. R-89-06.1.  A. Halprin, Phys. Rev. Lett. 48, 1313 (1982).  See also J. D. Vergados, Phys. Rep. 133, 1 (1986); D. Chang and W.-Y. Keung, Phys. Rev. Lett. 62, 2583 (1989); M. L. Swartz, Phys. Rev. D 40, 1521 (1989); P. Herczeg and R. N. Mohapatra, Phys. Rev. lett. 69, 2475 (1992).  H. Fujii, S. Nakamura and K. Sasaki, Phys. Lett. B 299, 342 (1993); H. Fujii et al., Phys. Rev. D 49, 559 (1994).  S. Adler, Phys. Lett. B 225, 143 (1989); P. H. Frampton and B.-H. Lee, Phys. Rev. Lett. 64, 619 (1990).  K. Horikawa and K. Sasaki, preprint YNU-HEPTh-107, March 1995.  W. S. Hou and G. G. Wong, preprint NTUTH-95-03, April 1995.  E. Derman, Phys. Rev. D 19, 317 (1979).  Neutral scalar exchange is also mentioned by B. E. Matthias, Ph.D. thesis (LA-12202-T, 1991), with some discussion of di?erence in magnetic ?eld dependence.  D. Chang and W.-Y. Keung, ref. .  See, for example, V.W. Hughes, Ann. Rev. Nucl. Sci. 16, 445 (1966); or, ref. . 10
 W. Sch¨fer, Ph.D. Thesis (1988), Universit¨t Heidelberg. a a  W. S. Hou, T. C. Luo and G. G. Wong, in prepartion.  A. Halprin and A. Masiero, Phys. Rev. D 48, R2987 (1993).
? FIG. 1. Magnetic ?eld dependence of total muonium conversion probability PT (M ) assuming |c1,1 |2 = |c1,0 |2 = |c1,?1 |2 = |c0,0 |2 = 1/4, and normalized to conversion strength of (V ? A)(V ? A) interaction at zero magnetic ?eld. Solid and dashed lines stand for SS and P P operators, respectively, while ?, 2,
stand for (S ? P )(S + P ), (S ? P )(S ? P ), (V ? A)(V + A) and
(V ? A)(V ? A) cases, respectively.
? FIG. 2. P (1,±1) (M ) vs. magnetic ?eld with same assumption as Fig. 1. ? FIG. 3. P (1,0) (M ) vs. magnetic ?eld with same assumption as Fig. 1. ? FIG. 4. P (0,0) (M ) vs. magnetic ?eld with same assumption as Fig. 1.2 The same as Fig. 1 except |c1,1 |2 = 0.35, |c1,0 |2 = 0.18, |c1,?1 | = 0.15 and
FIG. 5. |c00 |2 = 0.32.
This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9505300v1
This figure "fig2-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9505300v1
This figure "fig1-2.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9505300v1
This figure "fig2-2.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9505300v1
This figure "fig1-3.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9505300v1