Muonium Decay. Abstract

Transcription

1 Muonium Decay BNL-HET-99/20, hep-ph/ Andrzej Czarnecki and William J. Marciano Physics Department, Brookhaven National Laboratory Upton, NY Abstract Muonium (M = µ + e ) bound state effects are shown to decrease the muon decay rate by a factor of 1 5α 2 m e /m µ = That small 1.3 ppm shift is near the sensitivity of proposed muon lifetime measurements. The annihilation rate Γ(M ν e ν µ ) is also computed and found to be Γ(µ + e + ν e ν µ ). Other potential medium effects on stopped muon decay are briefly discussed. The muon lifetime, τ µ, is very well measured. Its current world average [1] τ µ = (40) 10 6 sec (1) exhibits an uncertainty of only 20 ppm. From that lifetime, the Fermi constant (denoted here by G µ ) is determined via the defining relationship [2 4] ( )( τµ 1 =Γ(µ all) = G2 µ m5 µ m 2 192π f e 1+ 3 m 2 ) µ (1 + R.C.), 3 m 2 µ 5m 2 W f(x) 1 8x +8x 3 x 4 12x 2 ln x, (2) where R.C. stands for QED radiative corrections to muon decay as calculated in an effective local V-A theory. Other standard model electroweak loop corrections as well as possible new physics effects are absorbed in G µ. The R.C. in (2) have been computed [2 5] through O(α 2 ) and higher order logs have been obtained using the renormalization group [6]. Altogether, one finds [7,8] R.C. = α [ ( 25 2π 4 π2 + m2 e 48 ln m ) ] µ 18 8π 2 m e [ 1+ α π m 2 µ ( 2 3 ln m µ m e 3.7 ) + α2 π 2 ( 4 m 9 ln2 µ 2.0ln m ) ] µ +C +... m e m e α 1 = (40) (3) where C corresponds to unknown non-logarithmic O(α 3 /π 3 ) corrections which are assumed to be insignificant. Employing (1,2,3) leads to G µ = (1) 10 5 GeV 2. (4) 1

2 The Fermi constant can be compared with other precise measurements such as α, m Z, sin 2 θ W, m W, etc., and used to test the consistency of the Standard Model at the quantum loop level. For example, G µ = πα/ 2m 2 W(1 m 2 W /m 2 Z)(1 r) where r represents calculable electroweak radiative correction [9]. In that role, G µ helped predict the top quark s mass before its discovery and currently constrains the Higgs mass to relatively low preferred values m H < 220 GeV. In addition, it provides a sensitive probe of new physics such as SUSY, Technicolor, Extra Dimensions etc. [7]. Recently, there have been several proposals [10 12] to further improve the measurement of τ µ (and thereby G µ ) by as much as a factor of 20, bringing its uncertainty down to an incredible ±1 ppm. Of course, such an improvement can only by fully utilized if the other electroweak parameters with which it is compared reach a similar level of precision and radiative corrections to their relationship are computed at least through 2 loops. Those advances appear unlikely in the foreseeable future. Nevertheless, given the fundamental nature and importance of G µ, such efforts should be strongly encouraged and pushed as far as possible. In that spirit, we examine in this paper several theoretical concerns that must be addressed in any ±1 ppm study of τ µ. Precision muon lifetime experiments involve stopping µ + in material. At some level, the medium in which it comes to rest will affect the muon decay rate. The most straightforward issue to consider is the formation of muonium (M = µ + e ) and the bound state effect on τ µ. The actual fraction of stopped µ + that form muonium and decay while in that bound state is very medium dependent. It can range from a small fraction in metals to nearly 100% in some materials. If a significant modification of τ µ occurred in muonium, a correction would have to be applied. For example, a published study [13] has claimed a reduction factor for the bound state decay rate. Such a large 484 ppm shift would be difficult to correct for at the ±1 ppm level unless the muonium formation fraction was very precisely known. It would also impact the interpretation of existing τ µ measurements [14] and might indicate the possibility of other large medium effects. Given its potential importance, we have reexamined the muonium bound state effect on the muon lifetime. As we shall show, the leading correction turns out to be small O(α 2 m e /m µ ) and remarkably simple to calculate. We begin by recalling some basic properties of muonium. It is a µ + e Coulombic bound state with hydrogenic features. The reduced mass m = m em µ m µ + m e 0.995m e (5) is slightly below the electron mass, but the difference is inconsequential for the considerations here and can be neglected. The energy levels are given by E n = α2 m 2n = eV/n2, n =1,2... (6) For the n = 1 ground state the virial theorem tells us that the average (electron) kinetic energy is T = E 1 =13.5eV (7) 2

3 while the average bound state potential energy is V = E 1 T =α 2 m= 27.0eV. (8) Also, the n = 1 (electron) momentum distribution is given by [15] 4πp 2 ψ(p) 2 = 32 ( ) 4 π αm αm p 2. (9) p 2 + α 2 m 2 It peaks at p 2 = α 2 m 2 /3 which is somewhat below the average p 2 = α 2 m 2. In that configuration the e and µ + have average velocities β e α =1/137, β µ αm e /m µ. Of course, (9) represents a non-relativistic approximation and should not be used indiscriminately for large p. In fig. 1, we display the (electron) kinetic energy probability distribution corresponding to (9). Muonium bound state corrections to the muon decay rate must vanish as α or m e 0. There are no O(αm e /m µ ) corrections, instead the leading effect is O(α 2 m e /m µ ). That correction arises because the available decay energy of the bound muon is reduced below m µ. Such an effect can be simply accounted for by the replacement [16] m µ m µ + V (10) where V is given by (8). (For n>1, V is reduced by 1/n 2.) That approximation ignores O(α 3 m e /m µ )ando(α 2 m 2 e/m 2 µ) corrections (e.g. relativistic dilation and e + e scattering effects) which are suppressed by extra powers of α or m e /m µ.itleads(forn= 1) to [17] or to a lifetime relation ( ) Γ(M e + ν e ν µ e )= 1 5α 2m e Γ(µ + e + ν e ν µ ) free m µ = Γ(µ + e + ν e ν µ ) free (11) τ M = τ µ. (12) Our 1.3 ppm decay rate reduction is nearly 400 times smaller than a previous claim [13] and just at the sensitivity limit of proposed future experiments. It can be reliably used to correct the measured muon lifetime in muonium. In the case of muonium, the bound state can also decay via annihilation (i.e. electron capture), M ν e ν µ. We have computed that rate for the n = 1 ground state and find [18,19] Γ(M ν e ν µ )=48π ( ) 3 αme Γ(µ + e + ν e ν µ ) m µ Γ(µ + all). (13) That rate is much too small to affect muon lifetime measurements, even at the 1 ppm level. Our result is in accord with the more general analysis of Ref. [20]. 3

4 Our prescription for computing the leading muonium bound state correction provides a simple procedure for estimating other stopping medium effects on the muon lifetime [21]. One replaces m µ by m µ + V where V is the average electromagnetic potential of the muon in its surroundings. The decay rate is then reduced by a factor 1+5 V /m µ for V m µ. So, for example, in metals where stopped µ + do not generally form muonium [22,23], the muon decay rate is still reduced. Conduction band electrons screen the µ + charge with local densities which may in fact not be so different than the orbital electron density of muonium. If that is the case, the screening potential will be similar to (8) and roughly a 1 ppm muon decay rate reduction will result. (Generally, the screening in metals results in a smaller potential (in magnitude) than in muonium.) Other materials may exhibit even smaller µ + potentials which give rise to much less than a 1 ppm reduction. For example, a stopped µ + in Helium would experience very little screening because the electron binding in Helium is stronger than muonium. In some materials V for a stopped µ + may be somewhat larger than muonium; however, we have not examined that possibility. Other long distance medium effects such as radiation damping [24], outer bremsstrahlung, Čerenkov radiation [25], etc., can modify the spectrum of final muon decay products but should not directly impact the muon lifetime. (Rearrangement of the spectrum would not influence lifetime measurements in counting experiments.) They may, however, indirectly affect τ µ through V, e.g. the properties of screening conduction electrons in metals will be dependent on qualities such as the dielectric constant of the medium. In most cases, we expect such effects to remain small; but each situation should be separately scrutinized. In summary, we have found that the lifetime of muons in muonium is about 1.3 ppm longer than than that of free muons. The main effect comes from replacing m µ by m µ + V in the decay rate formula, where V is the average electromagnetic potential of the µ + e system. A similar analysis can approximate other medium effects if V is known. In general, such corrections are likely to be near or below the ±1 ppm goal of proposed experiments. ACKNOWLEDGMENTS We thank D. Hertzog and A. Sirlin for stimulating discussions that rekindled our interest in this problem. This work was supported by the DOE contract DE-AC02-98CH

5 REFERENCES [1]C.Casoet al. (Particle Data Group), Eur. Phys. J. C3, 1 (1998). [2] S. Berman and A. Sirlin, Ann. Phys. 20, 20 (1962). [3] T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959). [4]S.Berman,Phys.Rev.112, 267 (1958). [5] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999), hep-ph/ [6] M. Roos and A. Sirlin, Nucl. Phys. B29, 296 (1971). [7] W. J. Marciano, Fermi constants and New Physics, to be published in Phys. Rev. D, hep-ph/ [8] Y. Nir, Phys. Lett. B221, 184 (1989). [9] A. Sirlin, Phys. Rev. D22, 971 (1980). [10] R. Carey and D. Hertzog et al., A precision measurement of the positive muon lifetime using a pulsed muon beam and µlan detector, proposaltopsi. [11] J. Kirkby et al., FAST Proposal to PSI (May 1999). [12] S. N. Nakamura et al., Precise measurement of the µ + lifetime and test of the exponential decay law, RIKEN-RAL Proposal. [13] L. Chatterjee, A. Chakrabarty, G. Das, and S. Mondal, Phys. Rev. D46, 5200 (1992), E: D49, 6247 (1994). [14] D. Hertzog, private communication. [15] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Plenum Publ. Corp., New York, 1977). [16] For an orbital electron in muonium with binding E and kinetic energy T,themuon decay phase-space scale is reduced from m µ to m µ + E T = m µ + V. The resulting muon decay rate is reduced by (1 + V/m µ ) 5 for that configuration. Integrating over the orbital electron momentum distribution (Eq. (9) or its relativistic generalization for large p) replaces V by V up to terms of O( V 2 /m 2 µ ) and leads to the overall suppression factor V /m µ. [17] After completing this analysis, we were informed that Richard Hill recently carried out an explicit calculation of muonium decay and obtained the same leading correction as our Eq. (11). We thank Professor T. Kinoshita for communicating that result to us. [18] Our result in Eq. (12) is about an order of magnitude smaller than a similar calculation in Ref. [13]. [19] Muonium annihilation was originally estimated in B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33, 549 (1958) [Sov. Phys. JETP 6, 429 (1958)]. [20] P.-J. Li, Z.-Q. Tan, and C.-E. Wu, J. Phys. G14, 525 (1988). [21] For a consideration of medium effects on muonium binding energies see G. Feinberg and S. Weinberg, Phys. Rev. 123, 1439 (1961). [22] J. Brewer, K. Crowe, F. Gygax, and A. Schenk, in C. S. Wu and V. W. Hughes, eds., Muon Physics (Academic Press, New York, 1977), vol. 3, p. 1. [23] A. Seeger, in G. Alefeld and J. Völkl, eds., Topics in Applied Physics (Springer Verlag, New York, 1978), vol. 28, p [24] For discussions of radiation damping in free muon decay, see W. Marciano, G. Marques, and N. Papanicolaou, Nucl. Phys. B96, 237 (1975); L. Matsson, Nucl. Phys. B12, 647 (1969); 5

6 D. A. Ross, Nuovo Cimento 10A, 475 (1972); D. Atwood and W. Marciano, Phys. Rev. D41, 1736 (1990). [25] In a recent preprint, A. Widom, Y. Srivastava, and J. Swain, hep-ph/ (1999), have claimed that Čerenkov radiation and other long distance impedances can significantly modify the muon lifetime in condensed matter mediums. In our view, the approach and results presented in that paper are not well founded. 6

7 FIGURES 1dΓ ΓdT T[eV] FIG. 1. Normalized electron kinetic energy distribution in muonium decay from the 1S level. 7