arXiv:0704.0377v3  [hep-ph]  22 Dec 2008
The lifetime of unstable particles in electromagnetic fields
Daniele Binosi1 and Vladimir Pascalutsa1, 2
1ECT* Trento, Villa Tambosi, Villazzano, I-38050 TN, Italy
2Institut für Kernphysik, Johannes Gutenberg Universität, Mainz D-55099, Germany
(Dated: October 30, 2018)
Abstract
We show that the electromagnetic moments of unstable particles (resonances) have an absorptive
contribution which quantifies the change of the particle’s lifetime in an external electromagnetic
field. To give an example we compute here the imaginary part of the magnetic moment for the
cases of the muon and the neutron at leading order in the electroweak coupling. We also consider
an analogous effect for the strongly-decaying ∆(1232) resonance. The result for the muon is
Imµ = eG2Fm
3/768π3, with e the charge and m the mass of the muon, GF the Fermi constant,
which in an external magnetic field of B Tesla give rise to the relative change in the muon lifetime
of 3 × 10−15 B. For neutron the effect is of a similar magnitude. We speculate on the observable
implications of this effect.
PACS numbers: 13.40.Em, 13.35.-r, 12.15.Lk, 23.40.-s
http://arxiv.org/abs/0704.0377v3
I. INTRODUCTION
The electromagnetic (e.m.) moments of a particle are among the few fundamental quan-
tities which describe the particle properties and as such have thoroughly been studied. The
most renowned examples are the magnetic moments of the electron and the muon which
have been measured to unprecendented accuracy and yielded a number of physical insights,
see[1] for recent reviews. What is far lesser known is that the e.m. moments of unstable
particles are complex numbers in general [2, 3]. Their imaginary part reflects, of course, the
unstable nature of the particle, however, the precise interpretation has been missing. In this
paper we work out the relation, suggested first by Holstein [4], which should exist between
the imaginary part of the magnetic moment and the effect of an external magnetic field on
particle’s lifetime.
The argument for such a relation is very simple. The (self-)energy of the particle with a
lifetime τ has an absorptive part, which has an interpretation of the width Γ = 1/τ . The
particle’s magnetic moment ~µ in the presence of magnetic field ~B induces the change in
the energy: −~µ · ~B. The latter contribution can then also change the width, provided the
magnetic moment has an absorptive part (Imµ 6= 0).
The decay properties of unstable particles, such as muon or neutron are extremely
well studied and are widely used for the precise determination of the Standard Model
parameters[5, 6]. There are also a plethora of studies of how these particles behave in
e.m. fields. A well-known example is the search for the neutron’s electric dipole moment[7].
In view of these studies it is compelling to investigate how the decay properties of unstable
particles may be affected by e.m. fields.
The lifetime of unstable quantum-mechanical systems is known to be affected by an
e.m. field. Positronium provides a textbook example[8], where the effect arises due to the
admixture of para- (S = 0) and ortho- (S = 1) positronium states with orbital momentum
l = 0 by the magnetic field interacting with the magnetic moments of the constituents. As
the result, already in the field of B = 0.2 Tesla, the lifetime of ortho-positronium decreases
by almost a factor of 2.
It is far from obvious how the same kind of an effect can arise for an elementary unstable
particle, e.g., the muon. The above-mentioned relation between the imaginary part of the
magnetic moment and the lifetime change may, therefore, provide us with both an interpre-
µ νµ µ
FIG. 1: The muon self-energy contributing to its decay width.
tation for the imaginary part of the magnetic moment and the means to compute the effect
of the lifetime change.
In the following we examine in detail the case of the muon, compute the leading contri-
bution to Imµ and the corresponding effect on the lifetime. Then we will briefly discuss the
cases of the neutron and of the ∆-resonance.
II. MUON DECAY (µ → e νeνµ)
The leading contribution to the muon decay width arises at two-loop level, see Fig. 1. For
our purposes, the W propagators in this graph can safely be assumed to be static — Fermi
theory. We also neglect the mass of the electron in the loops, since it leads to an under-
percent correction of O(me/m); here and in what follows, m is the muon mass. The graphs
with other Standard Model fermions (e.g., quarks) in the loops need not to be considered
here, because they cannot give any contribution to the muon width.
Using dimensional regularization, we compute this graph in d = 4 − 2ǫ dimensions (in
the limit ǫ → 0+),[14]
Σ (p/) =
(2π)d
2γµ(1− γ5) (p/− k/) γν
(p− k)2 + iε
Πµν(k). (1)
where MW is the W -boson mass, g = |e|/ sin θW is the electroweak coupling related to the
Fermi constant by GF/
2 = g2/8M2W , e is the charge, θW is the Weinberg angle, and
Πµν(k) =
d (d− 2)
(4π)d/2(d− 1)
Γ (ǫ)Γ (1− ǫ)2
Γ (2− 2ǫ)
× (−k2)−ǫ
k2gµν − kµkν
is the one-loop correction to the polarization tensor of the W boson. The decay width can
then be found as Γ = −2 ImΣ (p/ = m). A brief calculation shows that the self-energy has
the following form:
Σ (p/) = v(s) p/ (1− γ5) , (3)
with s = p2 and the scalar function v given by:
v(s) = −
G2F s
3(4π)4
− 2γE − 2 ln
+O(ǫ)
, (4)
where γE = −Γ ′(1) is the Euler’s constant. The absorptive part of this function stems from
the logarithm [ln(−s− iε) = ln s− iπ, for s > 0]:
Im v(s) = −
G2F s
384π3
. (5)
Terefore, the width is Γ = −2m Im v(m2), and the muon lifetime:
τ = 192π3/(G2Fm
5) ≃ 2.187× 10−6 sec, (6)
This result is of course long-known due to the seminal work of Feynman and Gell-Mann on
Fermi theory[9]. It is in a percent agreement with the experimental value[5]:
τ (exp) = (2.19703± 0.00004) 10−6 sec, (7)
The discrepancy is due the neglect of the electron mass and some radiative corrections,
c.f.[10]. We now investigate the influence of the e.m. field on the leading contribution given
by Eq. (6).
Let us denote by Σ (x, y;Aµ) the self-energy in the presence of an external e.m. field Aµ.
It is obtained by minimal substitution (∂µ → ∂µ − ieAµ) of the derivatives of all charged
fields into the self-energy of Fig. 1. Expanding in the e.m. coupling, we obtain:
Σ [x, y;Aµ] = Σ (i∂/
x) δ4(x− y)
dz Λµ(x, y; z)Aµ(z) +O(e
2A2), (8)
where Σ (i∂/) is the already computed self-energy in the vacuum, while Λ is the e.m. vertex
correction of Fig. 2, with static W ’s.
Denoting p (p′) the 4-momentum of the initial (final) muon and assuming the on-shell
situation (p2 = p′
= p · p′ = m2), the vertex correction has in the momentum space the
following general form:
Λµ(p′, p) = e
F γµ +G
(p+ p′)µ
+ FA γ
, (9)
where F , G and FA are complex numbers. Note that eF/2m is the correction to the magnetic
moment, and eF + eG is the correction to the electric charge. The Ward-Takahashi (WT)
identity:
(p′ − p) · Λ(p′, p) = e [Σ (p/)− Σ (p/′)] (10)
µ µνµ
FIG. 2: Electromagnetic correction to the muon decay.
with the self-energy in Eq. (3) leads to the following conditions:
F +G = −v(m2)− 2m2v′(m2), FA = v(m2) . (11)
Therefore, the term FA is in fact necessary by the e.m. gauge invariance. The γ5 terms, in
both self-energy and the vertex, are shown to vanish when summing over all the fermions
in Standard Model[11]. However, this does not happen for the imaginary part because the
heavier fermions do not contribute.
The expression for the graph in Fig. 2 is (in Fermi theory) given by
Λµ(p′, p) = −
64M4W
(2π)d
(2π)d
γβ(1− γ5)
2γα(1− γ5) (p/′ − k/1) γµ (p/− k/1) γβ (k/1 − k/2)
(k1 − k2)2 (p− k1)2 (p′ − k1)2
After a lengthy calculation we obtain the following result:
ImF =
384π3
, ImG =
, ImFA = −
384π3
, (13)
hence satisfying the gauge-invariance conditions Eq. (11), for Im v given by Eq. (5).
We would like to emphasize here that, of course, not only the magnetic moment, but
also the charge operator receives an imaginary contribution, equal to e Im(F +G). However,
through the WT identity, this contribution is completely fixed by the momentum dependence
of the self-energy, and therefore is not independent. The same holds for FA. We thus
discuss only the effect of the absorptive part of the magnetic moment, here given by Imµ =
e ImF/2m = eG2Fm
3/768π3.
The energy of the magnetic moment interacting with the magnetic field is equal to −µBz,
with Bz being the projection of the field along the muon spin. Then the total energy, in the
muon rest-frame, is given by: m − (i/2)Γ− µBz. We thus deduce that the absorptive part
yields the following change in the muon width:
∆Γ = 2 ImµBz =
192π3
Bz , (14)
while the change in the lifetime is ∆τ = −(∆Γ/Γ) τ , for ∆Γ/Γ ≪ 1.
Given this result, we conclude that the positively-charged muons live shorter (longer) in
a uniform magnetic field if their spin is aligned along (against) the field. For the relative
change in the width we find:
|eBz|
<∼ 3× 10−15B T−1, (15)
where B is the strength of the field in Tesla. Therefore, in moderate magnetic fields the
change in the muon lifetime is tiny, well beyond the present experimental accuracy (which
is at the ppm level). We will dwell on this more in the concluding part of the paper, but for
now we turn to a more technical issue.
It is interesting to observe that the result of Eq. (13), can simply be obtained by the
minimal substitution into Eq. (3), rather than into the electron propagator in Eq. (1). To
show this we go to coordinate space and hence write the self-energy as Σ (x, y) = Σ (i∂/
) δ(x − y). The minimal substitution to the first order in e leads to the following vertex
correction:
Λ̃µ(x, y; z) = − δ/δAµ(z)Σ (i∂/ + eA/ ) δ(x− y) |A=0 . (16)
Note that in general this is different from the vertex function in Eq. (8), since in the latter
the minimal substitution is performed also in the internal lines. The general form of Eq. (9),
of course, applies here as well, but now the scalar functions are completely specified by the
self-energy:
F̃ = −v(m2), G̃ = −2m2 v′(m2), F̃A = v(m2) . (17)
Substituting the explicit form of Im v, we see that this method unambiguously leads to
exactly the same result [Eq. (13)] as the full calculation. We emphasize though, that this
method cannot always work (see, e.g., Ref.[12]), as will also be clear from the following
examples. Nevertheless, it is worthwhile to investigate this method further, since knowing
whether it is applicable a priori can enormously facilitate the calculations.
III. NEUTRON DECAY AND THE ∆-RESONANCE
We consider now the neutron β-decay. Assuming exact V − A interaction (gA = 1) and
neglecting the electron mass (but not the proton mass, mp), the corresponding two-loop
self-energy can still be written in the form of Eq. (3). We introduce δ = (s −m2p)/2s and
treat it as a small parameter, since in the physical case (where s = m2n), δ ≃ 1.293 × 10−3.
A simple calculation then yields:
Im v(s) = −G
F |Vud|2
s2 δ5, (18)
where Vud is the relevant quark-mixing (CKM) matrix element. We note in passing that
this result leads to the lifetime of τn ≈ 622 sec, to be compared with the experimental
value of 886 sec. This 30% disagreement is largely due to the fact that in reality the axial
coupling gA deviates from 1. However, for our order-of-magnitude estimate this discrepancy
is unimportant.
What is important is that the derivative of the self-energy is enhanced by one power of
Im v′(s) = −(GF |Vud|)
s δ4 . (19)
and this opens the possibility for the enhancement of the effect in the lifetime. Namely, the
relative change in the neutron width then goes as
|∆Γn|
∼ µN |Bz|
mn −mp
<∼ 3× 10−14B T−1, (20)
where µN ≃ 3.15× 10−14 MeV T−1 is the nuclear magneton. A more precise analysis of this
effect for the neutron is beyond the scope of this paper. We focus instead on the example
of the ∆-resonance, where such an enhancement will be shown to be even more dramatic,
at least qualitatively.
The ∆ resonance decays strongly into the pion and nucleon, ∆ → πN , and the cor-
responding self-energy, to leading order in chiral effective-field theory, yields the following
result for the absorptive part[3]:
ImΣ∆(p/) = −23πλ
3C2 (α p/+mN ) , (21)
where the isospin symmetry is assumed, e.g., mp = mn = mN . The constant C =
hAm∆/8πfπ ≃ 1.5, where hA represents the πN∆ coupling and is fitted to the empira-
cal width of the ∆, fπ ≃ 93 MeV is the pion-decay constant, and m∆ = 1232 MeV is the ∆
π(b)(a)
FIG. 3: The leading chiral-loop correction to the magnetic moment of the ∆.
mass. For simplicity we neglect the pion mass (i.e., take the chiral limit). Then, in Eq. (21),
λ = (s−m2N)/2s, α = 1− λ. For s = m∆, λ ≈ (m∆ −mN)/mN ∼ 1/3 is a small parameter
in the chiral effective-field theory with ∆’s (see Ref.[13] for a recent review), and will so be
treated here too.
The absorptive part of the magnetic dipole moment of the ∆ arises at this order from
graphs in Fig. 3. These graphs, computed in Ref.[3], in the chiral limit yield the following
result (upto λ4 terms):
ImF (a) = 4πC2(λ− 3λ2 + 43
λ3) ,
ImG(a) = 4πC2(−λ+ 4λ2 − 71
λ3) ,
ImF (b) = 4πC2(λ2 + 1
λ3) , (22)
ImG(b) = −32
πC2λ3 ,
where F and G correspond with the decomposition in Eq. (9), with the superscript referring
to the corresponding graphs in Fig. 3; FA is absent in this case, of course.
First of all, we observe that this result satisfies the WT conditions, Eq. (11), for each of
the four charge states of the ∆,
∆++ : Im [F (a) +G(a) + F (b) +G(b)] = −2 ImΣ ′∆ ,
∆+ : Im [1
(F (a) +G(a)) + 2
(F (b) +G(b))] = − ImΣ ′∆ ,
∆0 : Im [−1
(F (a) +G(a)) + 1
(F (b) +G(b))] = 0 , (23)
∆− : − Im [F (a) +G(a)] = ImΣ ′∆ ,
where Σ ′∆ = ∂/∂p/Σ∆(p/)|p/=m∆, and hence ImΣ ′∆ = 4πC2(−λ2 + 73λ
At the same time, the ‘naive’ minimal-substitution procedure [Eq. (16)], that happens
to work for the muon, fails here miserably. It would predict that the magnetic moment
contribution would go with the same power as the self-energy [Eq. (17)], which for the
absorptive part means Imµ ∼ ImΣ (m∆) ∼ λ3. In reality it goes as λ. E.g., for the ∆+:
Imµ∆+ = (e/2m∆) Im[
F (a) + 2
F (b)]
π µNC
2 λ+O(λ2). (24)
The fact that the self-energy goes as λ3, while Imµ as λ has as a consequence the enhance-
ment of the lifetime change in the magnetic field by two powers of λ.
Quantitatively such enhancements of the lifetime change over the lifetime by the phase-
space volume do not make much difference in the above examples. However, it shows that
it might be useful to look for manifestations of the lifetime change in the medium where the
phase-space volume can be varied.
IV. CONCLUSIONS AND OUTLOOK
We have examined her a concept of the ‘absorptive magnetic moment’ — an intrinsic
property of an unstable particle, together with the width or the lifetime. It manifests itself
in the change of the particle’s lefetime in an external magnetic field, see Eq. (25) below.
We have computed this quantity for the examples of muon, neutron and ∆-resonance to
leading order in couplings. In all the three cosidered cases the effect on the lifetime is tiny
for normal magnetic fields: in a uniform field of 1 Tesla the change in the lifetime is of order
of 10−13 percent, at most.
In the case of the muon we have computed this effect to the leading order in the elec-
troweak coupling; the change in the lifetime is
∆τ = −2 ImµBz τ 2 = −96π3eBz/(G2Fm7) , (25)
or, numerically, |∆τ | <∼ 6× 10−21 (B/T) sec. A direct measurement of this effect is therefore
beyond the present experimental precision. Nevertheless, it is worthwhile to investigate
the effect of the magnetic field on the differential decay rates, with the hope that some
asymmetries could show a significantly bigger sensitivity.
A notable feature of this effect is that the relative change of the lifetime is inversely
proportional to the phase space. It goes as (mn −mp)−1 in the neutron case, and as (m∆ −
−2 in the ∆-resonance case. (The difference in power is apparently because the neutron
decays solely into fermions while the ∆ has a boson in the decay product.) One can expect
that in the conditions where the phase-space is significantly reduced, e.g. for the neutron in
nuclear medium, the effect of the lifetime change may become measurable.
Especially interesting would be to evaluate the manifestations of this effect in neutron star
formations. Not only the phase-space of the neutron decay is shrinking, the protons decay
too, and all that occurs in magnetic fields as large as 1010 Tesla. Even larger fields can be
achieved in atomic or nuclear systems. Finally, it is worthwhile to point out that in lattice
QCD studies strong magnetic fields are standardly used to compute the electromagnetic
properties of hadrons. Combined with the lattice techniques of extracting the width, the
relation between the absorprive part and the lifetime change may allow to compute the
former on the lattice for unstable hadrons.
Acknowledgments
We thank Barry Holstein and Marc Vanderhaeghen for a number of insightful discus-
sions. The work of V.P. is partially supported by the European Community-Research In-
frastructure Activity under the FP6 ”Structuring the European Research Area” programme
(HadronPhysics, contract RII3-CT-2004-506078).
[1] M. Passera, J. Phys. G 31, R75 (2005); J. P. Miller, E. de Rafael and B. L. Roberts, Rept.
Prog. Phys. 70, 795 (2007).
[2] L. V. Avdeev and M. Y. Kalmykov, Phys. Lett. B 436, 132 (1998).
[3] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. Lett. 94, 102003 (2005); Phys. Rev. D 77,
014027 (2008).
[4] B. R. Holstein, unpublished.
[5] W. M. Yao et al. [Particle Data Group],“Review of particle physics,”J. Phys. G 33, 1 (2006).
[6] D. Tomono [RIKEN RAL R77 Collaboration], AIP Conf. Proc. 842, 906 (2006); K. R. Lynch,
AIP Conf. Proc. 870, 333 (2006); J. S. Nico, AIP Conf. Proc. 870, 132 (2006); A. P. Serebrov
et al., arXiv:nucl-ex/0702009.
[7] P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999); C. A. Baker et al., Phys. Rev. Lett. 97,
131801 (2006).
[8] J. L. Basdevant and J. Dalibard, “Quantum Mechanics Solver,” (Springer, Berlin, 2005).
[9] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958).
[10] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999).
[11] A. Czarnecki and B. Krause, Nucl. Phys. Proc. Suppl. 51C, 148 (1996).
[12] J. H. Koch, V. Pascalutsa and S. Scherer, Phys. Rev. C 65, 045202 (2002).
[13] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007).
[14] Our conventions are: metric (+,−,−,−), ε0123 = +1, γ5 = iγ0γ1γ2γ3, γ’s stand for Dirac
matrices and their totally-antisymmetric products: γµν = 1
[γµ, γν ], γµνα = 1
{γµν , γα},
γµναβ = 1
[γµνα, γβ].

  We show that the electromagnetic moments of unstable particles (resonances)
have an absorptive contribution which quantifies the change of the particle's
lifetime in an external electromagnetic field. To give an example we compute
here the imaginary part of the magnetic moment for the cases of the muon and
the neutron at leading order in the electroweak coupling. We also consider an
analogous effect for the strongly-decaying $\Delta$(1232) resonance. The result
for the muon is Im$ \mu = e G_F^2 m^3/768 \pi^3$, with $e$ the charge and $m$
the mass of the muon, $G_F$ the Fermi constant, which in an external magnetic
field of $B$ Tesla give rise to the relative change in the muon lifetime of
$3\times 10^{-15} B$. For neutron the effect is of a similar magnitude. We
speculate on the observable implications of this effect.

arXiv:0704.0377v3  [hep-ph]  22 Dec 2008
The lifetime of unstable particles in electromagnetic fields
Daniele Binosi1 and Vladimir Pascalutsa1, 2
1ECT* Trento, Villa Tambosi, Villazzano, I-38050 TN, Italy
2Institut für Kernphysik, Johannes Gutenberg Universität, Mainz D-55099, Germany
(Dated: October 30, 2018)
Abstract
We show that the electromagnetic moments of unstable particles (resonances) have an absorptive
contribution which quantifies the change of the particle’s lifetime in an external electromagnetic
field. To give an example we compute here the imaginary part of the magnetic moment for the
cases of the muon and the neutron at leading order in the electroweak coupling. We also consider
an analogous effect for the strongly-decaying ∆(1232) resonance. The result for the muon is
Imµ = eG2Fm
3/768π3, with e the charge and m the mass of the muon, GF the Fermi constant,
which in an external magnetic field of B Tesla give rise to the relative change in the muon lifetime
of 3 × 10−15 B. For neutron the effect is of a similar magnitude. We speculate on the observable
implications of this effect.
PACS numbers: 13.40.Em, 13.35.-r, 12.15.Lk, 23.40.-s
http://arxiv.org/abs/0704.0377v3
I. INTRODUCTION
The electromagnetic (e.m.) moments of a particle are among the few fundamental quan-
tities which describe the particle properties and as such have thoroughly been studied. The
most renowned examples are the magnetic moments of the electron and the muon which
have been measured to unprecendented accuracy and yielded a number of physical insights,
see[1] for recent reviews. What is far lesser known is that the e.m. moments of unstable
particles are complex numbers in general [2, 3]. Their imaginary part reflects, of course, the
unstable nature of the particle, however, the precise interpretation has been missing. In this
paper we work out the relation, suggested first by Holstein [4], which should exist between
the imaginary part of the magnetic moment and the effect of an external magnetic field on
particle’s lifetime.
The argument for such a relation is very simple. The (self-)energy of the particle with a
lifetime τ has an absorptive part, which has an interpretation of the width Γ = 1/τ . The
particle’s magnetic moment ~µ in the presence of magnetic field ~B induces the change in
the energy: −~µ · ~B. The latter contribution can then also change the width, provided the
magnetic moment has an absorptive part (Imµ 6= 0).
The decay properties of unstable particles, such as muon or neutron are extremely
well studied and are widely used for the precise determination of the Standard Model
parameters[5, 6]. There are also a plethora of studies of how these particles behave in
e.m. fields. A well-known example is the search for the neutron’s electric dipole moment[7].
In view of these studies it is compelling to investigate how the decay properties of unstable
particles may be affected by e.m. fields.
The lifetime of unstable quantum-mechanical systems is known to be affected by an
e.m. field. Positronium provides a textbook example[8], where the effect arises due to the
admixture of para- (S = 0) and ortho- (S = 1) positronium states with orbital momentum
l = 0 by the magnetic field interacting with the magnetic moments of the constituents. As
the result, already in the field of B = 0.2 Tesla, the lifetime of ortho-positronium decreases
by almost a factor of 2.
It is far from obvious how the same kind of an effect can arise for an elementary unstable
particle, e.g., the muon. The above-mentioned relation between the imaginary part of the
magnetic moment and the lifetime change may, therefore, provide us with both an interpre-
µ νµ µ
FIG. 1: The muon self-energy contributing to its decay width.
tation for the imaginary part of the magnetic moment and the means to compute the effect
of the lifetime change.
In the following we examine in detail the case of the muon, compute the leading contri-
bution to Imµ and the corresponding effect on the lifetime. Then we will briefly discuss the
cases of the neutron and of the ∆-resonance.
II. MUON DECAY (µ → e νeνµ)
The leading contribution to the muon decay width arises at two-loop level, see Fig. 1. For
our purposes, the W propagators in this graph can safely be assumed to be static — Fermi
theory. We also neglect the mass of the electron in the loops, since it leads to an under-
percent correction of O(me/m); here and in what follows, m is the muon mass. The graphs
with other Standard Model fermions (e.g., quarks) in the loops need not to be considered
here, because they cannot give any contribution to the muon width.
Using dimensional regularization, we compute this graph in d = 4 − 2ǫ dimensions (in
the limit ǫ → 0+),[14]
Σ (p/) =
(2π)d
2γµ(1− γ5) (p/− k/) γν
(p− k)2 + iε
Πµν(k). (1)
where MW is the W -boson mass, g = |e|/ sin θW is the electroweak coupling related to the
Fermi constant by GF/
2 = g2/8M2W , e is the charge, θW is the Weinberg angle, and
Πµν(k) =
d (d− 2)
(4π)d/2(d− 1)
Γ (ǫ)Γ (1− ǫ)2
Γ (2− 2ǫ)
× (−k2)−ǫ
k2gµν − kµkν
is the one-loop correction to the polarization tensor of the W boson. The decay width can
then be found as Γ = −2 ImΣ (p/ = m). A brief calculation shows that the self-energy has
the following form:
Σ (p/) = v(s) p/ (1− γ5) , (3)
with s = p2 and the scalar function v given by:
v(s) = −
G2F s
3(4π)4
− 2γE − 2 ln
+O(ǫ)
, (4)
where γE = −Γ ′(1) is the Euler’s constant. The absorptive part of this function stems from
the logarithm [ln(−s− iε) = ln s− iπ, for s > 0]:
Im v(s) = −
G2F s
384π3
. (5)
Terefore, the width is Γ = −2m Im v(m2), and the muon lifetime:
τ = 192π3/(G2Fm
5) ≃ 2.187× 10−6 sec, (6)
This result is of course long-known due to the seminal work of Feynman and Gell-Mann on
Fermi theory[9]. It is in a percent agreement with the experimental value[5]:
τ (exp) = (2.19703± 0.00004) 10−6 sec, (7)
The discrepancy is due the neglect of the electron mass and some radiative corrections,
c.f.[10]. We now investigate the influence of the e.m. field on the leading contribution given
by Eq. (6).
Let us denote by Σ (x, y;Aµ) the self-energy in the presence of an external e.m. field Aµ.
It is obtained by minimal substitution (∂µ → ∂µ − ieAµ) of the derivatives of all charged
fields into the self-energy of Fig. 1. Expanding in the e.m. coupling, we obtain:
Σ [x, y;Aµ] = Σ (i∂/
x) δ4(x− y)
dz Λµ(x, y; z)Aµ(z) +O(e
2A2), (8)
where Σ (i∂/) is the already computed self-energy in the vacuum, while Λ is the e.m. vertex
correction of Fig. 2, with static W ’s.
Denoting p (p′) the 4-momentum of the initial (final) muon and assuming the on-shell
situation (p2 = p′
= p · p′ = m2), the vertex correction has in the momentum space the
following general form:
Λµ(p′, p) = e
F γµ +G
(p+ p′)µ
+ FA γ
, (9)
where F , G and FA are complex numbers. Note that eF/2m is the correction to the magnetic
moment, and eF + eG is the correction to the electric charge. The Ward-Takahashi (WT)
identity:
(p′ − p) · Λ(p′, p) = e [Σ (p/)− Σ (p/′)] (10)
µ µνµ
FIG. 2: Electromagnetic correction to the muon decay.
with the self-energy in Eq. (3) leads to the following conditions:
F +G = −v(m2)− 2m2v′(m2), FA = v(m2) . (11)
Therefore, the term FA is in fact necessary by the e.m. gauge invariance. The γ5 terms, in
both self-energy and the vertex, are shown to vanish when summing over all the fermions
in Standard Model[11]. However, this does not happen for the imaginary part because the
heavier fermions do not contribute.
The expression for the graph in Fig. 2 is (in Fermi theory) given by
Λµ(p′, p) = −
64M4W
(2π)d
(2π)d
γβ(1− γ5)
2γα(1− γ5) (p/′ − k/1) γµ (p/− k/1) γβ (k/1 − k/2)
(k1 − k2)2 (p− k1)2 (p′ − k1)2
After a lengthy calculation we obtain the following result:
ImF =
384π3
, ImG =
, ImFA = −
384π3
, (13)
hence satisfying the gauge-invariance conditions Eq. (11), for Im v given by Eq. (5).
We would like to emphasize here that, of course, not only the magnetic moment, but
also the charge operator receives an imaginary contribution, equal to e Im(F +G). However,
through the WT identity, this contribution is completely fixed by the momentum dependence
of the self-energy, and therefore is not independent. The same holds for FA. We thus
discuss only the effect of the absorptive part of the magnetic moment, here given by Imµ =
e ImF/2m = eG2Fm
3/768π3.
The energy of the magnetic moment interacting with the magnetic field is equal to −µBz,
with Bz being the projection of the field along the muon spin. Then the total energy, in the
muon rest-frame, is given by: m − (i/2)Γ− µBz. We thus deduce that the absorptive part
yields the following change in the muon width:
∆Γ = 2 ImµBz =
192π3
Bz , (14)
while the change in the lifetime is ∆τ = −(∆Γ/Γ) τ , for ∆Γ/Γ ≪ 1.
Given this result, we conclude that the positively-charged muons live shorter (longer) in
a uniform magnetic field if their spin is aligned along (against) the field. For the relative
change in the width we find:
|eBz|
<∼ 3× 10−15B T−1, (15)
where B is the strength of the field in Tesla. Therefore, in moderate magnetic fields the
change in the muon lifetime is tiny, well beyond the present experimental accuracy (which
is at the ppm level). We will dwell on this more in the concluding part of the paper, but for
now we turn to a more technical issue.
It is interesting to observe that the result of Eq. (13), can simply be obtained by the
minimal substitution into Eq. (3), rather than into the electron propagator in Eq. (1). To
show this we go to coordinate space and hence write the self-energy as Σ (x, y) = Σ (i∂/
) δ(x − y). The minimal substitution to the first order in e leads to the following vertex
correction:
Λ̃µ(x, y; z) = − δ/δAµ(z)Σ (i∂/ + eA/ ) δ(x− y) |A=0 . (16)
Note that in general this is different from the vertex function in Eq. (8), since in the latter
the minimal substitution is performed also in the internal lines. The general form of Eq. (9),
of course, applies here as well, but now the scalar functions are completely specified by the
self-energy:
F̃ = −v(m2), G̃ = −2m2 v′(m2), F̃A = v(m2) . (17)
Substituting the explicit form of Im v, we see that this method unambiguously leads to
exactly the same result [Eq. (13)] as the full calculation. We emphasize though, that this
method cannot always work (see, e.g., Ref.[12]), as will also be clear from the following
examples. Nevertheless, it is worthwhile to investigate this method further, since knowing
whether it is applicable a priori can enormously facilitate the calculations.
III. NEUTRON DECAY AND THE ∆-RESONANCE
We consider now the neutron β-decay. Assuming exact V − A interaction (gA = 1) and
neglecting the electron mass (but not the proton mass, mp), the corresponding two-loop
self-energy can still be written in the form of Eq. (3). We introduce δ = (s −m2p)/2s and
treat it as a small parameter, since in the physical case (where s = m2n), δ ≃ 1.293 × 10−3.
A simple calculation then yields:
Im v(s) = −G
F |Vud|2
s2 δ5, (18)
where Vud is the relevant quark-mixing (CKM) matrix element. We note in passing that
this result leads to the lifetime of τn ≈ 622 sec, to be compared with the experimental
value of 886 sec. This 30% disagreement is largely due to the fact that in reality the axial
coupling gA deviates from 1. However, for our order-of-magnitude estimate this discrepancy
is unimportant.
What is important is that the derivative of the self-energy is enhanced by one power of
Im v′(s) = −(GF |Vud|)
s δ4 . (19)
and this opens the possibility for the enhancement of the effect in the lifetime. Namely, the
relative change in the neutron width then goes as
|∆Γn|
∼ µN |Bz|
mn −mp
<∼ 3× 10−14B T−1, (20)
where µN ≃ 3.15× 10−14 MeV T−1 is the nuclear magneton. A more precise analysis of this
effect for the neutron is beyond the scope of this paper. We focus instead on the example
of the ∆-resonance, where such an enhancement will be shown to be even more dramatic,
at least qualitatively.
The ∆ resonance decays strongly into the pion and nucleon, ∆ → πN , and the cor-
responding self-energy, to leading order in chiral effective-field theory, yields the following
result for the absorptive part[3]:
ImΣ∆(p/) = −23πλ
3C2 (α p/+mN ) , (21)
where the isospin symmetry is assumed, e.g., mp = mn = mN . The constant C =
hAm∆/8πfπ ≃ 1.5, where hA represents the πN∆ coupling and is fitted to the empira-
cal width of the ∆, fπ ≃ 93 MeV is the pion-decay constant, and m∆ = 1232 MeV is the ∆
π(b)(a)
FIG. 3: The leading chiral-loop correction to the magnetic moment of the ∆.
mass. For simplicity we neglect the pion mass (i.e., take the chiral limit). Then, in Eq. (21),
λ = (s−m2N)/2s, α = 1− λ. For s = m∆, λ ≈ (m∆ −mN)/mN ∼ 1/3 is a small parameter
in the chiral effective-field theory with ∆’s (see Ref.[13] for a recent review), and will so be
treated here too.
The absorptive part of the magnetic dipole moment of the ∆ arises at this order from
graphs in Fig. 3. These graphs, computed in Ref.[3], in the chiral limit yield the following
result (upto λ4 terms):
ImF (a) = 4πC2(λ− 3λ2 + 43
λ3) ,
ImG(a) = 4πC2(−λ+ 4λ2 − 71
λ3) ,
ImF (b) = 4πC2(λ2 + 1
λ3) , (22)
ImG(b) = −32
πC2λ3 ,
where F and G correspond with the decomposition in Eq. (9), with the superscript referring
to the corresponding graphs in Fig. 3; FA is absent in this case, of course.
First of all, we observe that this result satisfies the WT conditions, Eq. (11), for each of
the four charge states of the ∆,
∆++ : Im [F (a) +G(a) + F (b) +G(b)] = −2 ImΣ ′∆ ,
∆+ : Im [1
(F (a) +G(a)) + 2
(F (b) +G(b))] = − ImΣ ′∆ ,
∆0 : Im [−1
(F (a) +G(a)) + 1
(F (b) +G(b))] = 0 , (23)
∆− : − Im [F (a) +G(a)] = ImΣ ′∆ ,
where Σ ′∆ = ∂/∂p/Σ∆(p/)|p/=m∆, and hence ImΣ ′∆ = 4πC2(−λ2 + 73λ
At the same time, the ‘naive’ minimal-substitution procedure [Eq. (16)], that happens
to work for the muon, fails here miserably. It would predict that the magnetic moment
contribution would go with the same power as the self-energy [Eq. (17)], which for the
absorptive part means Imµ ∼ ImΣ (m∆) ∼ λ3. In reality it goes as λ. E.g., for the ∆+:
Imµ∆+ = (e/2m∆) Im[
F (a) + 2
F (b)]
π µNC
2 λ+O(λ2). (24)
The fact that the self-energy goes as λ3, while Imµ as λ has as a consequence the enhance-
ment of the lifetime change in the magnetic field by two powers of λ.
Quantitatively such enhancements of the lifetime change over the lifetime by the phase-
space volume do not make much difference in the above examples. However, it shows that
it might be useful to look for manifestations of the lifetime change in the medium where the
phase-space volume can be varied.
IV. CONCLUSIONS AND OUTLOOK
We have examined her a concept of the ‘absorptive magnetic moment’ — an intrinsic
property of an unstable particle, together with the width or the lifetime. It manifests itself
in the change of the particle’s lefetime in an external magnetic field, see Eq. (25) below.
We have computed this quantity for the examples of muon, neutron and ∆-resonance to
leading order in couplings. In all the three cosidered cases the effect on the lifetime is tiny
for normal magnetic fields: in a uniform field of 1 Tesla the change in the lifetime is of order
of 10−13 percent, at most.
In the case of the muon we have computed this effect to the leading order in the elec-
troweak coupling; the change in the lifetime is
∆τ = −2 ImµBz τ 2 = −96π3eBz/(G2Fm7) , (25)
or, numerically, |∆τ | <∼ 6× 10−21 (B/T) sec. A direct measurement of this effect is therefore
beyond the present experimental precision. Nevertheless, it is worthwhile to investigate
the effect of the magnetic field on the differential decay rates, with the hope that some
asymmetries could show a significantly bigger sensitivity.
A notable feature of this effect is that the relative change of the lifetime is inversely
proportional to the phase space. It goes as (mn −mp)−1 in the neutron case, and as (m∆ −
−2 in the ∆-resonance case. (The difference in power is apparently because the neutron
decays solely into fermions while the ∆ has a boson in the decay product.) One can expect
that in the conditions where the phase-space is significantly reduced, e.g. for the neutron in
nuclear medium, the effect of the lifetime change may become measurable.
Especially interesting would be to evaluate the manifestations of this effect in neutron star
formations. Not only the phase-space of the neutron decay is shrinking, the protons decay
too, and all that occurs in magnetic fields as large as 1010 Tesla. Even larger fields can be
achieved in atomic or nuclear systems. Finally, it is worthwhile to point out that in lattice
QCD studies strong magnetic fields are standardly used to compute the electromagnetic
properties of hadrons. Combined with the lattice techniques of extracting the width, the
relation between the absorprive part and the lifetime change may allow to compute the
former on the lattice for unstable hadrons.
Acknowledgments
We thank Barry Holstein and Marc Vanderhaeghen for a number of insightful discus-
sions. The work of V.P. is partially supported by the European Community-Research In-
frastructure Activity under the FP6 ”Structuring the European Research Area” programme
(HadronPhysics, contract RII3-CT-2004-506078).
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[14] Our conventions are: metric (+,−,−,−), ε0123 = +1, γ5 = iγ0γ1γ2γ3, γ’s stand for Dirac
matrices and their totally-antisymmetric products: γµν = 1
[γµ, γν ], γµνα = 1
{γµν , γα},
γµναβ = 1
[γµνα, γβ].
