Table Of ContentGSI anomaly and spin-rotation coupling
Gaetano Lambiasea,b,c, Giorgio Papinic,d,e, Gaetano Scarpettaa,b,c
aDipartimento di Fisica ”E.R. Caianiello” Universita´ di Salerno, 84081 Baronissi (Sa), Italy.
bINFN, Sezione di Napoli Italy.
cInternational Institute for Advanced Scientific Studies, 89019 Vietri sul Mare (SA), Italy.
dDepartment of Physics, University of Regina, Regina, SK, S4S 0A2, Canada. and
ePrairie Particle Physics Institute, Regina, SK, S4S 0A2, Canada
3
1 We propose a model in which a recently reported modulation in the decay of the hydrogenlike
0 ions140Pr58+,142Pm60+ and122I52+ arisesfromthecouplingofrotationtothespinofelectronand
2 nucleus. The model shows that the spin-spin coupling of electron and nucleus does not contribute
to the modulation and predicts that the anomaly cannot be observed if the motion of the ions is
n
rectilinear, or if the ions are stopped in a target. It also supports the notion that the modulation
a
frequencyis proportional totheinverseof theatomic mass and that nomodulation is expected for
J
theβ+-decay.
3
1
PACSnumbers: 23.40.-s,27.60.+j
]
h
t 1. INTRODUCTION
-
l
c
u Experiments carried out at the storage ring ESR of GSI in Darmstadt [1–3] reveal an oscillation in the orbital
n electron capture and subsequent decay of hydrogenlike 140Pr58+, 142Pm60+ and 122I52+. The modulation has a
[
period of 7.069(8)s, 7.10(22)s and 6.1s respectively in the laboratory frame and is superimposed on the expected
2 exponential decay. The ”zero hypothesis” of a pure experimental decay has been excluded at the 99% C.L. and
v periodicinstabilities inthe storageringanddetectionapparatusalsoseemimprobablecausesofthe modulation. The
4 effect has been extensively studied in literature [4–7].
8
We show, in the model proposed below, that a modulation arises in the probability that the system, initially in
6
a superposition of hyperfine states (F = 3/2 and F = 1/2), finds itself again in such a superposition of hyperfine
0
. states after injection into the storage ring. The modulation has its origin in the spin-dependent part of the Thomas
5
precession,and is compatible with the observedESR modulation. The EC decay occurs for states with spin F =1/2
0
2 because decay from the spin 3/2 state is forbidden by the conservationof the F quantum number [2]. We stress that
1 the present paper differs in essential ways from [8] because it takes into account all the relevant features of the GSI
: experiment, such as bound states kinematics, dragging effects, Thomas precessions of nucleus and electron and QED
v
i and derives the probability of the observed modulation from the time evolution of nucleus plus electron once this
X
system is injected in the storage ring.
r The full Hamiltonian that describes the behavior of nucleus and bound electron in the external field B of the ring
a
is H =H +H , where H contains all the usual standard terms (Coulomb potential, spin-orbit coupling, etc.), and
0 1 0
H is (in units ~=c=1)
1
H = s I s Ω I Ω , (1)
1 e n
−A · − · − ·
4α4g
where Z3 n N 1014Hz,N (1) is the strength of spin-spin coupling, while
A≃ 3 ∼ × ∼O
Ω ω +ω(e) ω(e), (2)
e ≡ ge Th− c
Ω ω +ω(n) ω(n), (3)
n ≡ gn Th − c
representtheprecessionoftheelectronspinandtheusualspinprecessionofthenucleusinitsmotioninastoragering
that is assumed circular for simplicity. In (2) and (3), ω are the electron and nucleus spin precession frequencies
ge,n
duetotherespectivemagneticmomentsg andω(e,n) aretheangularcyclotronfrequencies. Theexplicitexpressions
e,n c
of all these quantities are given below. We refer (1) to a frame rotating about the x -axis in the clockwise direction
3
2
of the ions, with the x -axis tangent to the ion orbit in the direction of its momentum and write B = Buˆ , where
2 3
B =1.197T is the GSI value (we are assuming that this is the average value over the circumference).
TheThomasprecessionω(e,n) isrelatedtothestandardspin-rotationcouplingthatcanbederived,fortheelectron,
Th
from the spin connection coefficients of the Dirac equation in a rotating frame [9, 10]. In our derivation, we neglect
anystrayelectricfieldsandelectricfieldsneededtostabilizethenucleusorbits,aswellasallothereffectswhichcould
affect the Thomas precession [11].
We indicate by β and β the velocities of electron and nucleus relative to the lab frame. Using the composition
n
of velocities, the Lorentz factor γ =1/ 1 β2 of the electron can be written in the form γ =γ γ (1+Π), where
n e|n
−
Π = β β = β β cosθ, β is the velocity of the electron relative to the nucleus, γ = 1/ 1 β2, and
n · e|n n e|n e|n p n − n
γe|n =1/ 1−βe2|n. The explicit expression of β is also useful p
q
1 γ2Π
β = β + n β +γ β . (4)
γ (1+Π) e|n γ +1 n n n
n (cid:20) n (cid:21)
γ2 dβ
The Thomas precession of the electron in the lab frame is given by ω = β. The field B in the lab
Th
−γ+1 dt ∧
frame (where E=0) is transformed to the nucleus rest frame and gives E′ =γ β B and B′ =γ B on account of
n n n
dβ f β β f ∧
β B=0. Theequationsofmotionare e|n = e|n e|n e|n· e|n fortheelectronwithrespecttothenucleusand
n
· dt γ m−γ m
n e|n e|n
dβ Q
n = β Bforthenucleuswithrespecttothelabframe. Heref = e(E′+β B′)= eγ (β +β ) B.
dt Mγ n∧ e|n − e|n∧ − n e|n n ∧
n
Using dt=γ (1+Π)dt , taking β B=0, E β =0 and E β =0 (averagedoverthe decay time of the
n n e|n e|n e|n e|n n
· ∧ ∧
dβ e 1
ion in the storage ring), we find e|n = [β +β ] B. Neglecting spin-orbit coupling1, ω can
dt −mγ (1+Π) e|n n ∧ Th
e|n
be written as
eB 1 QB 1
ω = I I , (5)
Th e Q
m γ γ − M γ
e|n n n
where
β2 γ Π2
(γ γ )2 β2 + e|n +2Π+ n Y
e|n n n γn γn+1 − !
I ,
e
≡ (1+Π)[γ γ (1+Π)+1]
e|n n
γ Π 2
(γ γ )2 β2 1+ n X
e|n n " n(cid:18) γn+1(cid:19) − #
I ,
Q
≡ γ γ (1+Π)+1
e|n n
β2 [γ (2 cos2θ) sin2θ] β2 β2sin2θ
Y e|n n − − ,X e|n n .
≡ 3γ2 ≡ 3(γ +1)
n n
g e
The coupling of the electron magnetic moment µ = e s with the magnetic field is described by
e
− 2 m
e
1 γ
H = µ B′′ =µ B β(β B) .
ge γ e· e· − γ+1 ·
(cid:20) (cid:21)
1 The Coulomb interaction also contributes to the Thomas precession. It generates the spin-orbit coupling term in the Hamiltonian of
theelectronHC ∼ 2gme−e1r(dV/dr)s·L. Howevers·L= 21[j(j+1)−l(l+1)−s(s+1)]vanisheswhenj=sforl=0andj=l±1/2for
l6=0,thereforeHC =0inthegroundstate. Moreover,theeffectofAinπ= m1(p−eA)isnegligibleinthepresentcontext.
3
(β uˆ )2
Keeping only the quadratic term in β and using β(β B)= e|n· 3 B, we obtain
e|n · γ2(1+Π)2
n
γ2 (β uˆ )2
H =Υµ B, Υ 1 e|n e|n· 3 , (6)
ge e· ≡ − γ(γ+1)
and from it ds/dt=i[H ,s]=ω s=Υµ B which yields
ge ge ∧ e∧
g e
ω = e ΥB. (7)
ge −2m
e
Inordertoreferthespinprecessiontotheparticleorbit,theeffectivecyclotronfrequencyω(e)mustnowbesubtracted.
c
Its value is obtained by computing the instantaneous acceleration dβ/dt = ω(c)βuˆ . Omitting terms like a =
e 1 e|n
q E , β B and [B (β β )]β that vanish when averaged,as already pointed out, we find
me e|n e|n∧ · e|n∧ n n
ω(e) = eB βn1−(βe|n·uˆ1)2 + QB βn Ξ u , (8)
c −m β γ γ (1+Π)2 M γ β n 3
(cid:20) e e|n n n (cid:21)
1 γ Π
n
where Ξ 1+ and
n
≡ γ +1 γ +1
n (cid:18) n (cid:19)
β2
β2 + e|n +2Π+Π2
n γ2
β =v n .
u (1+Π)2
u
t
From (2),(5) and (8) we obtain
eB g I QBI +V
Ω = eΥ e U Q , (9)
e
−m 2 − γ γ − − M γ
e (cid:18) e|n n (cid:19) n
where Υ is defined in (6) and
U 1−(βe|n·uˆ1)2βn ,V βn 1+ γnΠ . (10)
≡ γ γ (1+Π)2 β ≡ β(1+Π) γ +1
e|n n (cid:18) n (cid:19)
Notice that the standard result Ω = eBa /m , where a =(g 2)/2 is the electron magnetic moment anomaly,
e e e e e
− | |−
is recovered in the limit Q=0.
The calculation of g -factors, based on bound state (BS) QED, can be carried out with accuracy even though, in
e
our case, the expansion parameter is Zα 0.4. The BS-QED calculation gives [12, 13]
≃
1+2 1 (αZ)2 α
gb =2 − + C(2)(αZ) , (11)
e 3 π
" p #
where C(2)(αZ) 1 + 1 (αZ)2 + 7(αZ)4. From (11) we obtain the values a 0.065122, a 0.0682112 and
≃ 2 12 2 e ≃ − e ≃ −
a =0.0505352for140Pr58+, 142Pm60+ and122I52+ respectively. Theadditionofmoreexpansionterms[14]doesnot
e
change these results appreciably.
e γ 1QB
ConsidernowthenucleuswithspinI. Thetermsof(3)areω =g µ B,whereµ = | | ,ω(n) = n− ,
gn n N N 2m Th − γ M
p n
QB
ω(n) = and give
c
Mγ
n
QB g A
Ω = n 1 . (12)
n
M 2 Z −
(cid:18) (cid:19)
4
2. PROBABILITY AND MODULATION
Let I,m and s,m be the eigenstates of the operators ˆI and ˆs. The total angular momentum operator is
I I s s
Fˆ =ˆs+| ˆI. Tihe angu|lar miomentum F assumes the values F =3/2,1/2, m = 3/2, 1/2,and m = 1/2
F=3/2 F=1/2
± ± ±
because I = 1, m = 1,0 and s = 1/2, m = 1/2. By making use of the raising and lowering operators
I s
Fˆ =ˆI +ˆs , we constr±uct the normalized and ortho±gonal states
± ± ±
3 3 1 1
φ , = 1,1 ,
1 F I s
≡ 2 2i | i 2 2i
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3 1 2 1 1 1 1 1
φ , = 1,0 , + 1,1 ,
2 F I s I s
≡ 2 2i 3| i 2 2i 3| i 2 −2i
r r
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
3 1
φ , =
3 F
≡ 2 −2i
(cid:12)
(cid:12) 2 1 1 1 1 1
= (cid:12) 1,0 , + 1, 1 ,
I s I s
3| i 2 −2i 3| − i 2 2i
r r
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3 3 1 1
φ , = 1, 1 ,
4 F I s
≡ 2 −2i | − i 2 −2i
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 1 1 1 1 2 1 1
φ , = 1,0 , 1,1 ,
5 F I s I s
≡ 2 2i 3| i 2 2i − 3| i 2 −2i
r r
(cid:12)1 1 (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
φ6 (cid:12) , F = (cid:12) (cid:12)
≡ 2 −2i
(cid:12)
(cid:12) 2 1 1 1 1 1
= (cid:12) 1, 1 , + 1,0 , .
I s I s
− 3| − i 2 2i 3| i 2 −2i
r r
(cid:12) (cid:12)
(cid:12) (cid:12)
The (6 6) matrix with elements φ Hˆ φ has th(cid:12)e eigenvalues (cid:12)
i 1 j
× h | | i
Ω
e
λ = A +Ω ,
1,4 n
−2 ± 2
(cid:18) (cid:19)
Ω ∆ Ω ∆
n ± n ±
λ = A , λ = A + ,
2,3 5,6
4 ∓ 2 − 4 4 ∓ 2 4
p p
where
∆ =9 2 4 Ω 4 Ω +4(Ω Ω )2,
± e n e n
A ± A ∓ A −
and the corresponding eigenstates i (i=1,...,6)
| i
B 1
±
1,4 = φ , 2,5 = φ + φ ,
1,4 2 5
| i | i 1+B2 1+B2
± ±
A q 1 q
±
3,6 = φ + φ ,
3 6
| i − 1+A2 1+A2
± ±
q q
where
9 2Ω +2Ω 3 ∆
e n −
A = A− ± ,
±
4√2(Ω Ω )
e n p
−
and
9 +2Ω 2Ω 3 ∆
e n +
B = A − ± .
±
4√2(Ω Ω )
e n p
−
5
In the limit Ω we obtain
e,n
A≫
i φ , (13)
i
| i≃| i
and
λ λ Ω +2Ω
1 3 e n
λ λ =λ λ = − = , λ λ = Ω +2Ω , (14)
1 2 2 3 1 4 e n
− − 2 − 3 − −
Ω 4Ω
e n
λ λ = − . (15)
5 6
− 3
Notice that in these expressions the -terms coming from the spin-spin coupling cancel out.
A
2.1. Modulation induced by quantum beats
These results must be now applied to the GSI experiment. Since the heavy nucleus decays via EC, only the states
withF =1/2arerelevant. ForsimplicityweconfineourselvestotheHilbertsubspacespannedbythestates 5 , 6 .
Here we follow [5] (see also [6, 7]). The decay processes involved in the GSI experiment 140Pr58+ 140Ce{5|8+i+| νi},
e
142Pm60+ 140Nd58++ν , and 122I52+ 122Te52++ν , can be schematically represented as →
e e
→ →
I F+ν , (16)
e
→
with obvious meaning ofthe symbols. At the initial instantt=0 (before injection into the ESR)the system nucleus-
electron is produced in a superposition of the states 5 , 6 ,
{| i | i}
6
I(0) = c a =c 5 +c 6 .
a 5 6
| i | i | i | i
a=5
X
with c 2+ c 2 =1. If one assumes, for simplicity, that the two states with energies λ and λ decay with the same
5 6 5 6
| | | |
rate Γ, at the time t the system evolves to the state
I(t) =e−Γt/2 c e−iλ5t 5 +c e−iλ6t 6 .
5 6
| i | i | i
(cid:0) (cid:1)
The probability of EC at time t reads
P (t)=e−Γt ν ,FS I(t) 2 =e−ΓtP¯ [1+a cos(ω t+ς)] , (17)
EC e EC 56 56
|h | | i|
where (see Eq. (15))
Ω 4Ω
e n
ω = λ λ = − , (18)
56 5 6
| − | 3
P¯ = ν ,FS 5 2 = ν ,FS 6 2, a = 2c c , and finally S is the interaction operator2. The phase ς comes
EC e e 56 5 6
formpos|hsiblep|h|asie|diffe|rhences|o|ftih|eamplitud|ec|| a|ndc andof ν ,FS 5 and ν ,FS 6 . As(17)and(18)show,
1 2 e e
|h | | i| |h | | i|
the modulation of the decay probability does not depend on .
A
2 If we consider the Hilbert space spanned by the states (13), {|1i,...,|6i}, then the probability (17) assumes the form PEC(t) ∼
[1+Pi<jaijcosωijt],withωij =|λj−λj|andi,j=1,5. Itcontains5termsofwhichonlyone(thatdueto(14)and(15))contributes
to the probability, while the others vanish because of EW selection rules. Assuming, therefore, that the states are equiprobable, the
magnitudes take the value a˜ij = 15 ≃0.2, with (i,j)={(1,2),(1,3),(1,4),(2,3),(5,6)}. The values obtained in the GSI experiments
are: a(Pr)=0.18(3), a(Pm)=0.23(4), a(I)=0.22(2)[18].
6
2.2. Estimate of γ
e|n
Wenowcomparethefrequenciesω /2π,givenby(18),withtheexperimentalvalues 0.14 Hzfoundfor140Pr58+
56
∼
and 142Pm60+ and 0.16Hz for 122I52+ and consider first the case ω =λ λ . We find
56 5 6
∼ −
eB m I¯ I +V¯ Z Z g A
| | p Υ¯(a +1) e U¯ + Q +4 n 1 =2π0.14Hz, (19)
e
3m m − γ γ − γ A A 2 Z −
p (cid:20) e (cid:18) e|n n (cid:19) n (cid:18) (cid:19)(cid:21)
where a bar on top means average values. These are computed by first expanding the quantities I , Υ, U and V
e,Q
in terms of Π < 1 and then averaging over the angle by means of cosnθ = 1+(−1)n. Using (β uˆ )2 = 1β2 ,
h i 2(n+1) e|n · i 3 e|n
i = 1,2,3, γ(Pr,Pm,I) = 1.43, g(Pr,Pm) = 2.5, g(I) = 0.94, up to (Π6) we obtain from (19) the numerical solutions
n n n
O
(see Fig. 1)
γ(Pr) 1.07904, γ(Pm) 1.08435, γ(I) 1.05902, (20)
e|n ∼ e|n ∼ e|n ∼
which must be compared with the Lorentz factors of the bound electron in the Bohr model γ(Pr) 1.0970, γ(Pm)
e|n ∼ e|n ∼
1.1040, and γ(I) 1.0776. The values (20) imply that the binding energies E = T +E = m[1 γ +(αZ)2],
e|n ∼ p − − e|n
where T ad E are kinetic and potential energies of the bound electron, are given by
p
E(Pr) 54.4keV, E(Pm) 58.2keV, E(I) 46.3keV,
∼− ∼− ∼−
in agreement with the values
E(Pr) = 49.5keV, E(Pm) = 53.1keV, E(I) = 39.6keV, (21)
R − R − R −
derived from the relativistic equation [20]
RZ2 (αZ)2 3
E = 1+ 1 , (22)
R − n2 n − 4n
(cid:20) (cid:18) (cid:19)(cid:21)
where R=13.6057eVand n=1.
3. CONCLUSIONS
InthispaperweexplaintheGSIanomalybymeansofasemiclassicalmodelbasedontheThomasprecessionofspins.
Themodelhasthefollowingconsequences: 1)Itavoidsallcriticismsraisedin[16]andin[17]becausetheHamiltonians
areessentiallydifferent. 2)Thereisnomodulationintheβ+-decaybranch[18]. ThisbecausetheThomasprecession,
when computed only for a decaying charged nucleus gives rise to a frequency Ω eB/m 107 Hz and the
n p
∼ ∼
probability e−Γt[1+acos(Ω t)]. The high frequency modulation term averages out to zero, and the probability
n
∼
obeys the standard exponential decay. 3) The GSI oscillations disappear when B = 0. 4) The model is consistent
with experiments on EC decays of neutral atoms in solid environments that have shown no oscillations/modulations
[15]. 5) The modelpredictsthatω A−1 ifthe three termsonthe l.h.s. of(19)areofthe sameorderofmagnitude
56
∼
(noticehoweverthattheA-termsonlyaffectthethirddigitofγ andarenot,therefore,arealdiscriminatingfeature
e|n
of our model). 6) The model is consistent with the absence of a periodic transfer from active (F = 1/2) to sterile
(F =3/2) states.
[1] Yu.A.Litvinov et al.,Phys.Rev. Lett.99, 262501 (2007).
[2] Yu.A.Litvinov et al.,Phys.Lett. B 664, 162 (2008).
[3] Yu.A.Litvinov et al.,nucl-ex/0509019.
[4] M. Faber, nucl-th/0801.3262. A.N. Ivanov et al., Phys. Rev. Lett. 101, 182501 (2008). A.N. Ivanov and P. Kienle, Phys.
Rev. Lett. 103, 062502 (2009). A. Gal, Nucl. Phys. A 842, 102 (2010). A.G. Cohen, S.L. Glashow, Z. Ligeti, Phys. Lett.
B678,191(2009). I.M.Pavlichenkov,Phys.Rev.C81,051602(R) (2010).H.J.Lipkins,hep-ph/0805.0435. H.Burkhardt
et al., hep-ph/0804.1099. V.V. Flambaum, Phys. Rev.Lett. 104, 159201 (2010). I. Borzov et al., Phys. At. Nucl. 71. 469
(2008). N. Winckleret al.,Phys. Rev.C 84, 014301 (2011); Phys. Lett.B 679, 36 (2009). J. Wuet al.,Phys. Rev.D 82,
045027 (2010). A.Merle, Prog. Part. Nucl.Phys. 64, 445 (2010). F. Giacosa and G. Pagliara, nucl-ph/1110.1669.
7
P
1.´108
5.´107
ΓeÈn
1.077 1.078 1.079 1.080 1.081 1.082
-5.´107
-1.´108
FIG. 1: P is defined by P≡ 3|em|Bphmmpe(cid:16)Υ˜(ae+1)− γe|Ineγn −U˜(cid:17)+ IQγ+nV˜ ZA +4ZA(cid:16)g2nZA −1(cid:17)i−2π0.14 Hz(19), with Υ˜, U˜ and
V˜ given by (10). The value of the unknown γ is obtained from P = 0 by using the experimental data and corresponds to
e|n
thecentral value of the range 6.98s.T .7.06s The plot refers to Pr. Similar plots can beobtained for Pm and I.
[5] C. Giunti, Phys. Lett.B 665, 92 (2008).
[6] H.Kienert, J. Kopp, M. Lindner, A.Merle, J. Phys. Conf. Ser. 136, 022049 (2008) [arXiv:hep-ph/0808.2389].
[7] A.Merle, Phys. Rev.C 80, 054616 (2009).
[8] G. Lambiase, G. Papini. and G. Scarpetta, arXiv:0811.2302[nucl-th].
[9] J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., 1999.
[10] G. Papini, G. Lambiase, Phys. Lett. A 294, 175 (2002). G. Papini, Phys. Rev. D 45, 077901 (2002). G. Lambiase, G.
Papini,Phys.Rev.D70,097901(2004).G.Lambiase,G.Papini.RPunzi,G.Scarpetta,Phys.Rev.D71,073011(2005).
[11] A.Silenko, Phys. Rev.Special Topics - Acc. and Beams 9, 034003 (2006).
[12] M. Vogel, et al.,Nucl.Instrum. Meth. B 235, 7 (2005). D. L. Moskovkin et al.,Phys.Rev. A 70, 032105 (2004).
[13] S.A. Blundellet al.,Phys.Rev.A 55, 1857 (1997).
[14] P.J. Mohr et al.,CODATA(2006) - Recommended Values of the FundamentalPhysics Constants: 2006.
[15] P.A.Vetteret al., Phys.Lett. B 670, 196 (2008).
[16] M. Faber, et al., arXiv:0906.3617 [nucl-th].
[17] T. Faestermann, arXiv:0907.1557.
[18] P.Kienle, Progr. In Part. And Nucl.Phys. 64, 439 (2010).
[19] Y.Litvinov F. Bosch, Rep.Prog. Phys. 74, 016301 (2011).Iwaniuk et al., Phys.Rev.C 84, 014301 (2011).
[20] C. Itzykson,J.-B. Zuber,Quantum Field Theory, McGrow-Hill International Book Company,1980.
