GSI Oscillations as Laboratory for Testing of New Physics
aa r X i v : . [ h e p - ph ] J u l GSI Oscillations as Laboratory for Testing of New Physics
A. N. Ivanov ∗ and P. Kienle
2, 3 Atominstitut, Technische Universit¨at Wien, Stadionalle 2, A-1020 Wien , Austria Stefan Meyer Institut f¨ur subatomare Physik ¨Osterreichische Akademieder Wissenschaften, Boltzmanngasse 3, A-1090, Wien, Austria Excellence Cluster Universe Technische Universit¨at M¨unchen, D-85748 Garching, Germany (Dated: March 22, 2018)We analyse recent experimental data on the GSI oscillations of the hydrogen–like heavy Pm ions that is the time modulation of the K–shell electron capture (EC) decay rate. We follow the mech-anism of the GSI oscillations, caused by the interference of the neutrino flavour mass–eigenstates inthe content of the electron neutrino. We give arguments that these experimental data show i) an ex-istence of sterile neutrinos that is necessary for an explanation of a phase–shift, ii) an observation ofCP violation, related to a phase–shift, and iii) an influence of the Quantum Zeno Effect, explainingdifferent values of the amplitude and phase–shift for two runs of measurements with different timeresolutions and different numbers of consecutive measurements. For new runs of experiments onthe GSI oscillations we propose to measure the EC and bound–state β − –decay rates of the H–likeheavy ions Ag . These measurements should verify the A –scaling of the periods of the timemodulation, where A is the mass number of the parent ion, and give an important information onmasses of neutrino (antineutrino) mass–eigenstates. PACS numbers: 12.15.-y, 13.15.+g, 23.40.Bw, 03.65.Xp
I. INTRODUCTION
The measurements of the K–shell electron capture (EC) decays of the hydrogen–like (H–like) heavy ions p → d + ν e ,where p and d are the parent and daughter ions in their ground states and ν e is the electron neutrino, at GSI [1–4]in Darmstadt showed the time modulation of the rates of the number N d ( t ) of daughter ions. The experimental dataon the time modulation [1–4] have been fitted by the equation dN d ( t ) dt = λ EC ( t ) N p ( t ) , (1)where dN d ( t ) /dt is the rate of the number of daughter ions d , N p ( t ) is the number of the parent H–like heavy ions p in the ground hyperfine state (1 s ) F = ,M F = ± and λ EC ( t ) is the time–dependent EC decay rate in the laboratoryframe, given by λ EC ( t ) = λ EC (1 + a cos( ωt + φ )) . (2)Here t = 0 corresponds to the moment of the injection of parent ions into the Experimental Storage Ring (ESR), λ EC is the EC decay constant and a , T = 2 π/ω and φ are the amplitude, the period and the phase–shift of thetime modulation. The time modulation of the EC decay rates of the H–like heavy ions has been dubbed the “GSIoscillations” [5].As theoretical explanation of the GSI oscillations we have proposed the time modulation mechanism, caused by theinterference of the neutrino flavour mass–eigenstates in the final state of the EC decays [5–8]. It is well–known fromtheoretical and experimental investigations of the neutrino lepton flavour oscillations [9] that the electron neutrino | ν e i is a superposition | ν e i = P N ν j =1 U ∗ ej | ν j i of the neutrino flavour mass–eigenstates | ν j i with masses m j , where U ej arethe matrix elements of a N ν × N ν unitary mixing matrix U of the N ν neutrino flavour mass–eigenstates [9], which aretreated as Dirac particles. As has been shown in [5–8] the time modulation frequency of the EC decay rates of the H–like heavy ions, calculated in the rest frame of parent ions, is related to the masses of neutrino flavour mass–eigenstatesand the mass of parent ions M p by ω ij = ∆ m ij / M p , where ∆ m ij = m i − m j and i > j . Since the time modulationperiods, T ≈ (7 −
6) s, measured for the EC decay rates of the H–like heavy ions Pr , Pm and I [1–4],impose the constraint ∆ m ij ∼ − eV , the interferences between the neutrino flavour mass–eigenstates with i > j ∗ Electronic address: [email protected] and i ≥
3, having ∆ m ij larger compared with 10 − eV , should lead to a time modulation with periods by orders ofmagnitude smaller compared with the experimental values T ≈ (7 −
6) s. This implies that the time modulation ofthe EC decays of the H–like heavy ions Pr , Pm and I [1–4] may be induced by the interference ofthe neutrino flavour mass–eigenstates | ν i and | ν i with ∆ m , which we call as (∆ m ) GSI [5]. The combined value(∆ m ) GSI = 2 . × − eV , obtained from the experimental data on the periods of the time modulation of theEC decays of the H–like heavy ions Pr , Pm and I [5], by a factor of 2.9 exceeds the experimentalvalue (∆ m ) KL = 7 . × − eV , determined by the KamLAND Collaboration from the electron antineutrinooscillations ¯ ν e ←→ ¯ ν e [10]. However, as has been shown in [8], the neutrino flavour mass–eigenstates can acquiremass–corrections δm j in the strong Coulomb fields of the daughter ions. These mass–corrections change the massesof the neutrino flavour mass–eigenstates in the strong Coulomb fields of the daughter ions, i.e. m j → ˜ m j = m j + δm j .This gives (∆ m ) GSI = ˜ m i − ˜ m j and allows to explain an increase of (∆ m ) GSI with respect to (∆ m ) KL [5].As has been also discussed in [11], the corresponding mass–corrections to the antineutrino flavour mass–eigenstatesshould be taken into account for a correct elaboration of the experimental data by the KamLAND Collaboration.The time modulation mechanism of the EC decays of the H–like heavy ions, proposed in [5–8], explains i) thesuppression of the time modulation of the rates of the β + decays, i.e. p → d ′ + e + + = e , of the H–like heavy ions [12]and ii) the proportionality of the time modulation period of the EC decay rates to the mass number A of the parentions T = 4 πM p / (∆ m ) GSI ∼ A [1–4], the so–called A –scaling [5]. The suppression of the time modulation of the β + decay rates of the H–like heavy ions has been observed experimentally in [3, 4] and reported as a preliminary result.As has been found in [3, 4] the amplitude of the time modulation of the β + –decays a = 0 . β + decays of the H–likeheavy ions Pm have been reported in [13]. The suppression of the time modulation of the β + decay rates of theH–like heavy ions has been confirmed in high resolution measurements of the EC and β + decays of the H–like heavyions Pm with the amplitude of the time modulation a = 0 . a = 0 . et al. [15], is based on theassumption of the interference of two closely spaced energy levels of the daughter ions in the ground state. The mainproblem of this mechanism is the prediction of the time modulation for the β + decay rates of the H–like heavy ionswith the same modulation period as the EC decay rates [5, 16]. The explanation of the GSI oscillations by means ofa neutrino magnetic moment, proposed by Gal [17], suffers from the same problem as that by Giunti [14] and Kienert et al. [15]. In addition these two mechanisms as well as the mechanisms, proposed by Pavlichenkov [18, 19], Krainov[20], Lambiase et al. [21, 22] and Giacosa and Pagliara [23], do not predict the A –scaling of the periods of the timemodulation T ∼ A , observed in [1, 3, 4] and confirmed in the time modulation mechanism, caused by the interferenceof the neutrino flavour mass–eigenstates [5].Finally we would like to notice that a one–dimensional model of the GSI oscillations, proposed by Lipkin [24–26],being extended to 3–dimensions, leads to the EC decay rate in the rest frame of parent ions, given by [7] λ EC ( t ) = λ EC ( r.f. ) (cid:16) θ ( P sup − P sub ) sin(Ω L t )Ω L t (cid:17) (cid:16) X i>j U ∗ ei U ej ] cos( ω ij t ) (cid:17) , (3)where λ EC ( r.f. ) is the EC decay constant in the rest frame (r.f.) of parent ions, ω ij = ∆ m ij / M p , Ω L = 2 Q EC | δ~p | /M p and M p is a mass of parent ions. According to Lipkin [24–26], the GSI oscillations are caused by the incoherentcontributions of the EC decays from a superradiant | p i sup = cos θ | p ( ~p + δ~p ) i + sin θ | p ( ~p − δ~p ) i and subradiant | p i sub =sin θ | p ( ~p + δ~p ) i − cos θ | p ( ~p − δ~p ) i states of a parent ion, where θ is a mixing angle. The superradiant and subradiant states are formed with probabilities P sup and P sub , respectively, obeying the condition P sup + P sub = 1. If P sup = P sub the EC decay rate Eq.(3) reduces to ours [5]. For P sup = P sub Lipkin’s model predicts unobservable oscillations witha frequency Ω L = 2 Q EC | δ~p | /M p . One can show [7] that the term, oscillating with the frequency Ω L , appears also inthe β + decay rates of the H–like heavy ions [8]. This contradicts to both the experimental data [3, 4, 13] and ourtheoretical analysis [12].In spite of a certain success of the description of the time modulation in the EC and β + decays of the H–likeheavy ions by the interference of the neutrino flavour mass–eigenstates, such a mechanism of the GSI oscillationshas been criticised in publications [27–29], where i) the parent and daughter ions have been treated as the nuclei,unaffected by the measurements, and ii) energy–momentum conservation has been accepted for all decay channels p → d + ν j ( j = 1 , , . . . , N ν ) in the EC decay p → d + ν e . However, as has been pointed out in [5–7] and shownrecently in [30], energy and 3–momentum in the GSI experiments on the EC decays of the H–like heavy ions arenot conserved in the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) due to interactions of parent and daughterions with the resonant and capacitive pickups in the ESR [30]. Qualitatively the result of such interactions can bedescribed by the smearing of energy and momentum of daughter ions over the regions δE d ∼ π/δt d ∼ − eVand | δ~q d | ∼ ( M d /Q EC ) δE d ∼ − eV, calculated in the rest frame of parent ions (see section V). Since in the restframe of parent ions the differences of energies and 3–momenta of particles in the decay channels p → d + ν i and p → d + ν j are of order ω ij ∼ π/T ∼ − eV and | ~k i − ~k j | = | ~q i − ~q j | ∼ ( M d /Q EC ) ω ij ∼ − eV, such a violationof the energy–momentum conservation provides i) an overlap of the wave functions of the daughter ions in the decaychannels p → d + ν i and p → d + ν j , causing an overlap of these decay channels, and, as a result, ii) indistinguishabilityof the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) in the EC decay p → d + ν e , which is necessary for the interferenceof the neutrino flavour mass–eigenstates [5–7].In this paper we propose a theoretical analysis of recent experimental data on a time modulation of the EC decayrates of the H–like heavy ions Pm . In section II we present shortly the experimental data, given in Table 1 ofRef. [13]. In section III we show that in the mechanism of the interference of the neutrino flavour mass–eigenstatesa phase–shift of the GSI oscillations can be explained by extending the number of neutrino flavours from N ν = 3to N ν ≥
4. For an illustration we set N ν = 4 and show that the neutrino flavour mass–eigenstates | ν i and | ν i ,the interference of which is responsible for the observed time modulation, acquire a relative phase δ , violating CPinvariance. In section IV we show that different values of the amplitude and phase–shift of the time modulation,measured with time resolutions δt d = 32 ms and δt d = 64 ms and the number of consecutive measurements N = 1688and N = 844, can be explained by means of an influence of the Quantum Zeno Effect (QZE). We find the Zeno time asa function of the number of consecutive measurements N . This allows to obtain the amplitude a and the phase–shift∆ φ = π − φ as functions of the number of consecutive measurements N . In section V we discuss in detail the basicstatements of the quantum field theoretic approach to the GSI oscillations as the interference of the neutrino flavourmass–eigenstates. In the section VI we discuss the obtained results and analyse the periods of the time modulation ofthe EC and bound–state β − –decay rates of the H–like Ag heavy ions. These results can be used for new runs ofexperiments on the GSI oscillations for the verification of the A –scaling of the periods of the time modulation. Theyshould be also of great deal of importance for the estimate of the masses of neutrino (antineutrino) mass–eigenstates. II. RECENT EXPERIMENTAL DATA ON GSI OSCILLATIONS [13]
Recent experimental data on the GSI oscillations, reported in [13] (see Table 1 of Ref.[13]), show the time modulationof the EC decay rates of the H–like heavy ions Pm with i) the period T = 7 . a = 0 . φ = 2 . T = 7 . a = 0 . φ = 1 . δt = 32 ms and δt = 64 ms and the number of consecutivemeasurements N = 1688 and N = 844, respectively. III. PHASE–SHIFT OF GSI OSCILLATIONS AS A SIGNAL FOR STERILE NEUTRINOS AND CPVIOLATION
In the mechanism of the GSI oscillations, caused by the interference of the neutrino flavour mass–eigenstates [5–8, 30], a phase–shift can appear because of an extension of the 3–flavour structure of neutrinos with a certain leptoniccharge | ν α i = P j =1 U ∗ αj | ν j i , where U αj are the matrix elements of the 3 × U [9] and α = e, µ, τ ,to N ν –flavour structure | ν α i = P N ν j =1 U ∗ αj | ν j i with N ν ≥ j = 1 , , , . . . , N ν and α = e, µ, τ, . . . . New neutrinos | ν s i = P N ν j =1 U ∗ sj | ν j i with s = e, µ, τ are named sterile neutrinos. In order to illustrate an appearance of a phase–shiftin the time modulated term of the EC decay rates we consider below some extensions with N ν = 4.According to [31, 32], a 4 × sterile neutrino lepton flavours, which are 3 + 1 and 2 + 2 four–family neutrino flavour mixing,respectively. The 4 × U = U ( θ , U ( θ , U ( θ , U ( θ , δ ) U ( θ , δ ) U ( θ , δ ) ,U = U ( θ , U ( θ , U ( θ , U ( θ , δ ) U ( θ , δ ) U ( θ , δ ) , (4)for the 3 + 1 and 2 + 2 four–family neutrino flavour mixing, respectively, and [32] U = U ( θ , U ( θ , U ( θ , δ ) U ( θ , U ( θ , δ ) U ( θ , δ ) (5)for the 3 + 1 four–family neutrino flavour mixing. The 4 × U ( θ ij , δ k ) are constructed in analogy with 3 × δ ij are responsible for CP violation.Indeed, it is well–known that for neutrino flavour mass–eigenstates, treated as Dirac particles, a N ν × N ν neutrinoflavour mixing matrix is defined by N ν ( N ν − / θ ij and ( N ν − N ν − / | ν i and | ν i , neglecting the contributions of the mixing angles θ and θ ij for i = 1 , , j = 4 [34] thematrix elements U ej of the mixing matrices for the schemes 3+1 and 2+2 are equal to U ej = (cos θ , sin θ e − iδ , , θ is a mixing angle between the neutrino flavour mass–eigenstates | ν i and | ν i and δ is a CP violatingphase of the wave function of the neutrino flavour mass–eigenstate | ν i . It is defined relative to the phases of thewave functions of the neutrino flavour mass–eigenstates | ν j i ( j = 1 , , λ EC ( t ) = λ EC (1 + a cos( ωt − δ )) , (6)where λ EC and ω are the EC decay constant and frequency of the time modulation in the laboratory frame, relatedto the EC decay constant λ EC ( r.f. ) and frequency ω of the time modulation in the rest frame of parent ions as λ EC = λ EC ( r.f. ) /γ and ω = ω /γ , and γ = 1 .
43 is the Lorentz factor [13]. From the comparison of Eq.(6) withEq.(2) we define the phase–shift of a time modulation in terms of δ . This gives φ = − δ .It may be interesting to notice that one more possible signal for an existence of sterile neutrinos is a deficit of reactorantineutrinos at distances smaller than 100 m, observed in [35]. Recently such a deficit of reactor antineutrinos hasbeen confirmed in [36] for the new world average value τ n = 880 . .
1) s of the neutron lifetime [9]. The theoreticallifetime of the neutron τ n = 879 . .
1) s, agreeing well with the world average one τ n = 880 . .
1) s, has been recentlycalculated in [37].For further analysis of the experimental data on the GSI oscillations we propose to transcribe Eq.(2), defined inthe laboratory frame, into the form λ EC ( τ ) = λ EC (1 + a cos( ωτ − ∆ φ )) , (7)where τ = t − π/ω = t − T / φ = π − φ = π + δ . A change of a time dependence from t to τ = t − T /
T / . v/v ≈ × − during first 3 . φ , i.e. ∆ φ = 0 . φ = 1 . φ = 2 . φ = 1 . N = 1688 and N = 844, respectively. Since a phase δ is not measurable in the neutrino (antineutrino) lepton flavouroscillation experiments [31, 32], the GSI oscillations give an unprecedented possibility for an observation of such aphase in the EC decays of the H–like heavy ions. IV. QUANTUM ZENO EFFECT AND DEPENDENCE OF PARAMETERS OF GSI OSCILLATIONS ONTHE NUMBER OF CONSECUTIVE MEASUREMENTS
The experimental frequencies of the time modulation ω = 0 . ω = 0 . δt d = 32 ms and δt d = 64 ms and N = 1688 and N = 844consecutive measurements, respectively, do not show a dependence on the number of consecutive measurements N .In turn, the experimental data on the amplitude and phase–shift of the time modulation show, in principle, such adependence, which we explain below as an influence of the Quantum Zeno Effect (QZE) [38–48].For a quantum field theoretic analysis of the QZE one has to investigate a behaviour of a survival probability ofan unstable quantum state during short time intervals, which are much smaller than a total observation time. Sincea short time evolution of unstable quantum systems differs from the exponential decay law [38, 39], according tothe QZE, frequent consecutive measurements with short time intervals can prevent from an evolution of an unstablequantum state. Indeed, let ∆ τ be a time interval between two consecutive measurements such as τ = N ∆ τ , where τ is a total observation time and N is the number of consecutive measurements. Suppose that a time interval betweentwo consecutive measurements ∆ τ is small enough. In this case a survival probability of an evolution of an unstablequantum state during a time interval ∆ τ may be given by the form [38–48] P (∆ τ ) = 1 − (∆ τ ) /τ Z , (8)where τ Z is the Zeno time [38–48]. If a time interval ∆ τ is smaller compared with the Zeno time, i.e. τ Z ≫ ∆ τ , anevolution of an unstable quantum state can be significantly slowed down [47, 48]. In terms of a survival probabilitythis can be illustrated as follows. After N consecutive measurements a survival probability of an unstable quantumstate can be defined by [38–48] P N (∆ τ ) = (1 − (∆ τ ) /τ Z ) N = e − τ /Nτ Z . (9)In the limit N → ∞ a survival probability tends to unity, i.e. an unstable quantum state is left stable after continuousconsecutive measurements, carried out during an observation time τ . This is the QZE [38–48].In the case of the GSI experiments on the EC decays of the H–like heavy ions the QZE can, in principle, i) lead toa delay of the EC decays, i.e. to a decrease of the EC decay constants, and ii) prevent from a time modulation of theEC decay rates, leading to a decrease of the amplitudes and a change of the phase–shifts.Following [38–48] and using the results, obtained in [49], we analyse, first, an influence of the QZE on the lifetimeof the H–like heavy ions unstable under the EC decays. For this aim we calculate the Zeno time τ Z or a time scaleof an influence of frequent consecutive measurements on the lifetime of the H–like heavy ions, caused by the ECdecays. Skipping intermediate calculations we give the result. For example, for the H–like heavy ions Pm the Zeno time is equal to τ Z = p π/λ EC Q EC = p πγ/λ EC ( r.f. ) Q EC = 6 . × − s, where Q EC = 4 .
817 MeVand λ EC ( r.f. ) = 0 . − are the Q –value and the EC constant of the H–like heavy ions Pm [50]. Thus,in order to observe a delay of the EC decays of the H–like heavy ions Pm , i.e. a decrease of the EC decayconstant λ EC as a consequence of frequent consecutive measurements, a time interval ∆ τ between two consecutivemeasurements should be smaller compared to the Zeno time, i.e. τ Z ≫ ∆ τ . Since measurements of the EC decays ofthe H–like heavy ions with time intervals ∆ τ ≪ τ Z = 6 . × − s are unreal, an influence of frequent consecutivemeasurements on the value of the EC decay constant of the H–like heavy ions Pm can be neglected.For time intervals ∆ τ , which are commensurable with time resolutions of the GSI experiments, we define a survivalprobability of parent ions during a time interval ∆ τ = τ /N as follows [42–46] P (∆ τ ) = 1 − λ (∆ τ )∆ τ. (10)After N consecutive measurements a survival probability is given by P N (∆ τ ) = (1 − λ (∆ τ )∆ τ ) N = e − λ ( τ/N ) τ . (11)In the limit N → ∞ we arrive at the exponential decay law P ( τ ) = lim N →∞ P N (∆ τ ) = lim N →∞ e − λ ( τ/N ) τ = e − λ (0) τ , (12)where λ (0) = lim N →∞ λ ( τ /N ). In order to confirm our assertion that frequent consecutive measurements do notdelay the EC decays of the H–like heavy ions we have to show that λ (0) = λ EC . However, it is important to noticethat a linear dependence of the exponent of a survival probability Eq.(12) on the observation time τ , obtained in thelimit of infinite number of consecutive measurements, implies a suppression of a time modulation.In order to obtain a relation between λ (0) and the EC decay constant λ EC we use Eq.(7). This gives λ (∆ τ ) = 1∆ τ Z ∆ τ λ EC ( τ ′ ) dτ ′ = λ EC (1 + a cos(∆ φ )) + ∆ τ a λ EC ω sin(∆ φ ) == λ EC (1 + a cos(∆ φ )) + τN a λ EC ω sin(∆ φ ) . (13)Taking the limit N → ∞ we arrive at the relation λ (0) = λ EC (1 + a cos(∆ φ )) . (14)It is seen that the constant λ (0) coincides with the EC decay constant λ EC if the amplitude of a time modulation a vanishes in the limit N → ∞ .In order to find the amplitude and phase–shift of a time modulation as functions of a number of consecutive mea-surements N we have to calculate the Zeno time or a time scale of an influence of frequent consecutive measurementson the amplitude and phase–shift of the GSI oscillations. For this aim we follow [46] and define the required Zenotime as follows τ ( a ) Z = p /λ ′ (∆ τ ) | ∆ τ =0 = p /a λ EC ω sin(∆ φ ) = p γ/a λ EC ( r.f. ) ω sin(∆ φ ) , (15)where λ ′ (∆ τ ) is a derivative of λ (∆ τ ) with respect to ∆ τ . Using the experimental data λ EC ( r.f. ) = 0 . − [50], ω = 0 . − (or ω = 0 . − ) and γ = 1 .
43 we calculate the Zeno time τ ( a ) Z = 25 . / p a sin(∆ φ ) s . (16)This is a time scale of an influence of frequent consecutive measurements on the amplitude and phase–shift of theGSI oscillations. For the experimental values of the amplitudes a = 0 . a = 0 . φ = 0 . φ = 1 . δt d = 32 ms and δt d = 64 ms, respectively, we calculate the Zeno times τ ( a ) Z = 82(10) s and τ ( a ) Z = 78(10) s,respectively. One may see that the Zeno times are much larger than the time resolutions τ ( a ) Z ≫ δt d and commensurablewith the observation time 54 s [13]. This implies a strong influence of the QZE on the values of the amplitude andphase–shift of a time modulation.As a first step to the definition of the Zeno time, the amplitude and the phase–shift of the GSI oscillations asfunctions of the number of consecutive measurements N we set τ ( a ) Z = 13 . . N / s . (17)For N = 1688 and N = 844 we obtain τ ( a ) Z = 87(7) s and τ ( a ) Z = 73(6) s, which agree well with the values τ ( a ) Z =82(10) s and τ ( a ) Z = 78(10) s.Substituting Eq.(17) into Eq.(16) we define the product a sin(∆ φ ) as a function of the number of consecutivemeasurements a sin(∆ φ ) = 3 . / √ N. (18)For N = 1688 and N = 844 we get a sin(∆ φ ) = 0 . a sin(∆ φ ) = 0 . a sin(∆ φ ) = 0 . a sin(∆ φ ) = 0 . N a = 4 . / √ N . (19)The amplitudes of a time modulation a = 0 . a = 0 . N = 1688 and N = 844, agreewell with the experimental values a = 0 . a = 0 . φ we find thefollowing dependence on the number of consecutive measurements∆ φ = 2 π × . . /N / . (20)For N = 1688 and N = 844 the function Eq.(20) gives ∆ φ = 0 . φ = 1 . N → ∞ the amplitude a and phase–shift ∆ φ vanish, giving φ → π . This suppresses a time modulationof the EC decay rates and corroborates the equality λ (0) = λ EC , implying that the QZE does not affect the lifetimeof the H–like heavy ions.Using the functions Eq.(19) and Eq.(20) we may correct the dependence of the Zeno time t ( a ) Z on the number ofconsecutive measurements. For sufficiently large N in comparison to N = 1688 and N = 844 we obtain τ ( a ) Z = 0 . N . s . (21)In turn, for N = 1688 and N = 844 we get τ ( a ) Z = 90(10) s and τ ( a ) Z = 60(8) s, which do not contradict to the values τ ( a ) Z = 82(10) s and τ ( a ) Z = 78(10) s, respectively.Finally we would like to notice that the amplitude and phase–shift of a time modulation, obtained with the numberof consecutive measurements N = 1688 and N = 844, agree within one standard deviation. In spite of this factwe assume that they are affected by the QZE and define them as some functions of the number of consecutivemeasurements N , vanishing in the limit of the infinite number of consecutive measurements and suppressing a timemodulation as it is required by the QZE. V. QUANTUM FIELD THEORETIC ANALYSIS OF GSI OSCILLATIONS
A description of the electron neutrino | ν e i as a superposition of the neutrino flavour mass–eigenstates | ν j i withmasses m j , i.e. | ν e i = P N ν j =1 U ∗ ej | ν j i , where N ν is the number of neutrino flavours, implies an existence of N ν decaychannels p → d + ν j in the EC decay p → d + ν e . An indistinguishability of the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) in the EC decay p → d + ν e , which is the necessary condition of an interference of the neutrinoflavour mass–eigenstates | ν j i , requires an overlap of them. An overlap between the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) may occur only if energies and 3–momenta of the daughter ions are smeared with energy δE d and3–momentum | δ~q d | uncertainties, which are larger compared with the differences of energies and 3–momenta of theneutrino flavour mass–eigenstates and daughter ions, produced in two decay channels p → d + ν i and p → d + ν j .Since energy δE d and 3–momentum | δ~q d | uncertainties lead to a violation of energy–momentum conservation in thedecay channels p → d + ν j ( j = 1 , , . . . , N ν ) with accuracies δE d and | δ~q d | , a non–conservation of energy and 3–momentum in the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) is the sufficient condition of the appearance of aninterference between the neutrino flavour mass–eigenstates | ν j i in the rate of the EC decay p → d + ν e . In other words,a violation of energy–momentum conservation should guarantee that an overlap of the decay channels p → d + ν j ( j = 1 , , . . . , N ν ), as the necessary condition of an interference of the neutrino flavour mass–eigenstates, may befulfilled. Of course, energy δE d and 3–momentum | δ~q d | uncertainties should not violate the Fermi Golden Rule . Sucha requirement is fulfilled if δE d and | δ~q d | obey the constraints δE d ≪ T d = Q / M d and | δ~q d | ≪ Q EC , where Q EC isthe Q –value of the EC decay and T d = Q / M d is a kinetic energy of the daughter ions [5–7]. As has been shown in[30] a required violation of energy and momentum in the EC decays of the GSI experiments occurs due to interactionsof ions with the measuring apparatus, i.e. the resonant and capacitive pickups in the ESR.Since the origin of energy δE d and 3–momentum | δ~q d | uncertainties in the GSI experiments is the detection of thedaughter ions with a time resolution δt d , we have to check that the uncertainties δE d and | δ~q d | satisfy the constraintsnecessary for the appearance of the interference of the neutrino flavour mass–eigenstates. For quantitative analysisof the required overlap of the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) and violation of energy–momentumconservation we use i) energy and 3–momentum differences of the neutrino flavour mass–eigenstates and daughterions from two decay channels p → d + ν i and p → d + ν j , defined by ω ij = E i ( ~k i ) − E j ( ~k j ) = E d ( ~q j ) − E d ( ~q i )and | ~k i − ~k j | = | ~q i − ~q j | , where ( E j ( ~k j ) , ~k j ) and ( E d ( ~q j ) , ~q j ) are the energies and 3–momenta of the neutrino flavourmass–eigenstate | ν j i and the daughter ion d in the decay channel p → d + ν j , and ii) energy δE d and 3–momentum | δ~q d | uncertainties, induced by the detection of the daughter ions with a time resolution δt d . If δE d ≫ ω ij = E i ( ~k i ) − E j ( ~k j ) = E d ( ~q j ) − E d ( ~q i ) and | δ~q d | ≫ | ~k i − ~k j | = | ~q i − ~q j | the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) inthe EC decay p → d + ν e should be indistinguishable. Such an indistinguishability leads to the interference betweenthe decay channels without violation of the Fermi Golden Rule if δE d ≪ T d = Q / M d and | δ~q d | ≪ Q EC . This isthe basis of the mechanism of the GSI oscillations, caused by an interference of the neutrino flavour mass–eigenstates[5–8, 30]In the rest frame of parent ions the kinematics of particles in the EC decay channels p → d + ν j ( j = 1 , , . . . , N ν )is given by the relations: k p ( M p ,~ k d ( E d ( ~q j ) , ~q j ) + k j ( E j ( ~k j ) , ~k j ). The energies and 3–momenta of neutrinomass–eigenstates and the daughter ions are defined by E j ( ~k j ) = ( M p − M d + m j ) / M p , E d ( ~q j ) = ( M p + M d − m j ) / M p (22)and ~q j = − ~k j . The differences ω ij between energies of the neutrino flavour mass–eigenstates and daughter ions of twodecay channels p → d + ν i and p → d + ν j are equal to ω ij = E i ( ~k i ) − E j ( ~k j ) = E d ( ~q j ) − E d ( ~q i ) = ∆ m ij / M p , (23)where ∆ m ij = m i − m j . Since the neutrino flavour mass–eigenstates | ν j i are not detected, they move away fromdaughter ions with energies E j ( ~k j ) and 3–momenta ~k j . In principle, because of energy and 3–momentum conservationthe daughter ions, produced in the decay channel p → d + ν j , should go away with energies E d ( ~q j ) and 3-momenta ~q j = − ~k j . If it is so, one is able to distinguish a decay channel p → d + ν i from a decay channel p → d + ν j with i = j .In this case there are no interferences between decay channels p → d + ν i and p → d + ν j and, correspondingly, a timemodulation of the EC decay rate or of the rate of the number of daughter ions, caused by the EC decays. However,as has been pointed out in [5–7] and shown in [30] such a kinematics is not the case for the GSI experiments.Indeed, because of the detection with a time resolution δt d energies and 3–momenta of daughter ions, producedin the decay channels p → d + ν j ( j = 1 , , . . . , N ν ), are smeared with δE d ∼ π/δt d and | δ~q d | ∼ ( M d /Q EC ) δE d [5–7]. For example, for the EC decays of the H–like heavy ions Pm we get δE d ∼ π/δt d ∼ − eV and | δ~q d | ∼ ( M d /Q EC ) δE d ∼ − eV, where Q EC = 4 .
817 MeV is the Q –value of the EC decay Pm → Nd + ν e , M d is the daughter ion mass and δt d = 32 ms and δt d = 64 ms are the time resolutions of the GSI experiments [13].For the experimental value of the time modulation period T ≃ p → d + ν i and p → d + ν j are of order ω ij ∼ π/T ∼ − eV.In turn, the differences of 3–momenta of the neutrino flavour mass–eigenstates and of the daughter ions of the decaychannels p → d + ν i and p → d + ν j are of order | ~k i − ~k j | = | ~q i − ~q j | ≈ ( M d /Q EC ) (2 π/T ) ∼ − eV. Since δE d ≫ ω ij and | δ~q d | ≫ | ~k i − ~k j | = | ~q i − ~q j | , in the GSI experiments on the EC decays of the H–like heavy ions i) energy–momentumconservation is violated with accuracies δE d ∼ − eV and | δ~q d | ∼ − eV, respectively, and ii) the decay channels p → d + ν i and p → d + ν j are indistinguishable. This means that the daughter ions, produced in the decay channels p → d + ν j ( j = 1 , , . . . , N ν ), are not detected with 3–momenta ~q j and energies E d ( ~q j ) but they are detected with a3–momentum ~q and an energy E d ( ~q ), obeying the constraints | δ~q d | ≫ | ~q j − ~q | and δE d ≫ | E d ( ~q j ) − E d ( ~q ) | . Thus,in the GSI experiments on the EC decays of the H–like heavy ions the necessary and sufficient conditions for theinterference of the neutrino flavour mass–eigenstates are fulfilled and one may expect the appearance of the timemodulation of the EC decay rates [30].As has been shown in [5], the amplitude of the EC decay p → d + ν e , calculated with the ε –regularisation in therest frame of parent ions within a time dependent perturbation theory [51], takes the form A ( p → d ν e )( t ) = − δ M F , − √ q M p E d ( ~q ) M GT h ψ ( Z )1 s i X j U ej q E j ( ~k j ) e i (∆ E j − iε ) t ∆ E j − iε Φ d ( ~k j + ~q ) , (24)where ∆ E j = E d ( ~q ) + E j ( ~k j ) − M p is the difference of energies of the final and initial state in the decay channel p → d + ν j ( j = 1 , , . . . , N ν ) with a daughter ion, detected with a time resolution δt d and described by the wavefunction Φ d ( ~k j + ~q ), taken in the form of the wave packet and localised around ~k j + ~q ≈ | δ~q d | ∼ − eV. Then, M GT is a nuclear matrix element of the Gamow–Teller transition Pm → Nd and h ψ ( Z )1 s i is an average value of the Dirac wave function of the electron in the ground state of the H–like parent ion Pm [49, 52]. The Kronecker symbol δ M F , − implies that in the rest frame and with the spin quantisation axisanti–parallel to the neutrino 3–momentum parent ions are unstable under the EC decay in the hyperfine state 1 s F,M F with F = 1 / M F = − / P ( p → d ν e )( t ) = ddt | A ( p → d ν e )( t ) | , (25)where we have denoted [5] ddt | A ( p → d ν e )( t ) | = lim ε → ddt X M F | A ( p → d ν e )( t ) | = 6 M p E d ( ~q ) |M GT | |h ψ ( Z )1 s i| n N ν X j =1 | U ej | E j ( ~k j ) 2 π δ (∆ E j ) × | Φ d ( ~k j + ~q ) | + X ℓ>j | U eℓ || U ej | q E ℓ ( ~k ℓ ) E j ( ~k j ) | Φ d ( ~k ℓ + ~q ) || Φ d ( ~k j + ~q ) | (cid:16) πδ (∆ E ℓ ) + 2 πδ (∆ E j ) (cid:17) cos( ω ℓj t + φ ℓj ) o (26)with ω ℓj = ∆ m ℓj / M p and φ ℓj = arg U eℓ − arg U ej . We would like to note that before we take the limit ε → E ℓ − ∆ E j ) t + φ ℓj ) = cos(( E ℓ ( ~k ℓ ) − E j ( ~k j )) t + φ ℓj ). After theuse of Eq.(23) the time modulated terms become proportional to cos( ω ℓj t + φ ℓj ). Then, taking the limit ε → U eℓ → e − iβ e U eℓ e + iα ℓ [53]. It is obvious that the observables should be invariant under such transformations [53]. Sincethe decay rate Eq.(26) is an observable quantity, it should be invariant under phase transformations of the matrixelements of the mixing matrix. One may see that the time independent term of Eq.(26) is invariant under the phasetransformations U eℓ → e − iβ e U eℓ e + iα ℓ . In order to show that the time modulated term is also invariant quantity wepropose to rewrite it as follows X ℓ>j | U eℓ || U ej | q E ℓ ( ~k ℓ ) E j ( ~k j ) | Φ d ( ~k ℓ + ~q ) || Φ d ( ~k j + ~q ) | (cid:16) πδ (∆ E ℓ ) + 2 πδ (∆ E j ) (cid:17) cos( ω ℓj t + φ ℓj ) == X ℓ>j Re (cid:16) U eℓ U ∗ ej Φ d ( ~k ℓ + ~q )Φ ∗ d ( ~k j + ~q ) e iω ℓj t (cid:17)(cid:16) πδ (∆ E ℓ ) + 2 πδ (∆ E j ) (cid:17) . (27)Making the phase transformations of the matrix elements of the mixing matrix U eℓ → e − iβ e U eℓ e + iα ℓ and U ∗ ej → e + iβ e U ∗ ej e − iα j and the phase transformations of the wave functions of the daughter ions Φ d ( ~k ℓ + ~q ) → Φ d ( ~k ℓ + ~q ) e − iα ℓ and Φ ∗ d ( ~k j + ~q ) → Φ d ( ~k j + ~q ) e + iα j we leave the time modulated term unchanged.For the calculation of the EC decay rate we have to integrate P ( p → d ν e ) over the phase–volume of the final statesof the decays p → d + ν j ( j = 1 , , . . . , N ν ). For this aim we set zero masses of the neutrino flavour mass–eigenstateseverywhere in comparison to the Q –value of the EC decay, i.e. E i ( ~k i ) ≃ | ~k i | and E j ( ~k j ) ≃ | ~k j | . Then, due to therelations | ~k i − ~k j | ∼ − eV and | δ~q d | ≫ | ~k i − ~k j | , we may set ~k i ≈ ~k j ≈ ~k as | δ~q d | ≫ | ~k j − ~k | ( j = 1 , , . . . , N ν )[5–7]. This gives [5–7] λ EC ( t ) = 12 M p V Z P ( p → d ν e )( t ) d q (2 π ) E d d k (2 π ) E ν = λ EC ( r.f. ) (cid:16) X ℓ>j | U eℓ || U ej | cos( ω ℓj t + φ ℓj ) (cid:17) , (28)where V is a normalisation volume [5]. For the estimate of ∆ m ℓj with ℓ = j we have to take into account that thefrequencies ω = 0 . − and ω = 0 . − have been measured in the laboratory frame, where parent ionsmove with a velocity v = 0 .
71 and the Lorentz factor γ = 1 .
43 [1, 13]. This gives ∆ m ij = 4 πγM p /T ≃ . × − eV [5]. Since ∆ m j ∼ − eV for j = 1 , m j ∼ for j = 1 , , T ≈ | ν i and | ν i only.The value ∆ m ≈ . × − eV , which has been named in [5] as (∆ m ) GSI ≈ . × − eV , is 2.9 timeslarger than that reported by the KamLAND (∆ m ) KL = 7 . × − eV [10]. The solution of this problem interms of mass–corrections δm j to masses of the neutrino flavour mass–eigenstates | ν j i , i.e. m j → ˜ m j = m j + δm j ,has been proposed in [8] and discussed in [5] (see section I). VI. CONCLUSIVE DISCUSSION
We have proposed a theoretical analysis of recent experimental data on the GSI oscillations [13]. The new runs ofmeasurements of the EC and β + decays of the H–like heavy Pm ions have fully confirmed the results, reportedearlier in [1–4]. The new experimental data have shown the time modulation with the period T ≈ β + decays. A suppression of the time modulation of therate of the number of daughter ions from the β + decays is a strong argument [12] in favour of the mechanism of thetime modulation of the rates of the number of daughter ions from the EC decays as the interference of the neutrinoflavour mass–eigenstates | ν j i in the content of the electron neutrino | ν e i = P N ν j U ∗ ej | ν j i , proposed in [5–8].The new experimental data, obtained with the better time resolutions δt d = 32 ms and δt d = 64 ms, have shown adependence of the amplitude and phase–shift of the time modulation on the number of consecutive measurements N .Such a dependence we have explained assuming an influence of the Quantum Zeno Effect (QZE). We have shown thatthe influence of the QZE on the delay of the EC decays (or on the EC decay constant λ EC ) of the H–like heavy ionscan be neglected. In turn, the amplitude and phase–shift of the time modulation are strongly affected by the QZE. Wehave found that the amplitude a and the phase–shift ∆ φπ − φ depend on the number of consecutive measurements as a = 4 . / √ N and ∆ φ = 2 π × . . /N / , which fit well the experimental data (see section IV) and vanish inthe limit N → ∞ , suppressing a time modulation of the EC decay rates without influence on the EC decay constant λ EC .Of course, one may object against the influence of the QZE on the amplitude and phase–shift of the GSI oscilla-tions, since the experimental data on the amplitude and phase–shift of the GSI oscillations, measured with the timeresolutions δt d = 32 ms and δt d = 64 ms and the number of consecutive measurements N = 1688 and N = 844, arecommensurable within experimental uncertainties. Hence, for a confirmation of our analysis of the dependence of theamplitude and phase–shift of the GSI oscillations on the number of consecutive measurements, i.e. the influence of theQZE, it should be important to perform new runs of measurements with time resolutions δt d = 16 ms, δt d = 32 ms, δt d = 64 ms and δt d = 128 ms during the same observation time.The mechanism of GSI oscillations, caused by the interference of the neutrino flavour mass–eigenstates, allows toexplain the phase–shift of the time modulation by means of an extension of the number of neutrino flavours from N ν = 3 to N ν ≥
4, assuming an existence of so–called sterile neutrinos. For an illustration we have considered thecase with N ν = 4. We have shown that the known schemes of an inclusion of sterile neutrinos, i.e. (3 + 1) and(2 + 2) schemes, lead to an appearance of a phase–shift φ = − δ of a time modulation of the EC decays of the H–likeheavy ions, where δ is a phase of the wave function of the neutrino flavour mass–eigenstate | ν i , defined relative tothe phases of the wave functions of other neutrino flavour mass–eigenstates | ν j i ( j = 1 , , . . . , N ν ). The importantproperty of δ is to violate CP invariance. It is remarkable that in the considered schemes of an inclusion of sterile neutrinos a phase δ does not appear in the probabilities of the neutrino lepton flavour oscillations ν α ←→ ν β . Thismeans that the experiments on the GSI oscillations give unprecedented possibilities for an observation of such a phase,implying also an extension of the number of neutrino flavour mass–eigenstates from N ν = 3 to N ν ≥
4. We wouldlike also to note that a so–called reactor antineutrino flux anomaly [35] (see also [36]) may serve as one more hint ona low–energy confirmation of an existence of sterile neutrinos . For recent analysis of sterile neutrinos and estimatesof ∆ m j we refer to the paper by Martini et al. [54].0We have given a quantum field theoretic derivation of the EC decay rate of the H–like heavy ions with the timemodulation, published in [5–7]. We have discussed in more detail the necessary and sufficient conditions for theappearance of the interference of the neutrino flavour mass–eigenstates. We have accentuated that the necessary and sufficient conditions of this effect are related to i) an overlap of the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) inthe EC decay p → d + ν e and ii) a violation of energy–momentum conservation in the decay channels p → d + ν j ( j = 1 , , . . . , N ν ), caused by a detection of the daughter ions with a time resolution δt d . Such a detection introducesenergy δE d ∼ π/δt d ∼ − eV and 3–momentum | δ~q d | ∼ ( M d /Q EC ) δE d ∼ − eV uncertainties, which are largercompared with the differences of energies δE d ≫ ω ij ∼ π/T ∼ − eV and of 3–momenta | δ~q d | ≫ | ~k i − ~k j | = | ~q i − ~q j | ∼ − eV of the neutrino flavour mass–eigenstates and of the daughter ions, produced in the decay channels p → d + ν i and p → d + ν j . As result, the daughter ions are detected with an average 3–momentum ~q and an energy E d ( ~q ). This makes the decay channels p → d + ν i and p → d + ν j experimentally indistinguishable in the EC decay p → d + ν e .According to quantum mechanical principle of superposition [51], indistinguishability of the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) allows to calculate the amplitude of the EC decay p → d + ν e as a superposition of theamplitudes of the decays p → d + ν j ( j = 1 , , . . . , N ν ), multiplied by the matrix elements U ej of the N ν × N ν mixingmatrix U , where | U ej | defines the weight of the neutrino flavour mass–eigenstate | ν j i in the content of the electronneutrino | ν e i = P N ν j =1 U ∗ ej | ν j i . The time modulated EC decay rates can be calculated setting E j ( ~k j ) ≈ E ν = | ~k | as dynamical masses ˜ m j of the neutrino mass–eigenstates are much smaller than the Q –values of the EC decays,i.e. Q EC ≫ ˜ m j . Because of fulfilment of inequalities | δ~q d | ≫ | ~k j − ~k | ∼ | ~k + ~q | , δE d ≫ | E j ( ~k j ) − E ν | and δE d ≫ | M p − E d ( ~q ) − E ν | any deviations from the Fermi Golden Rule cannot be practically observable experimentally.We would like to emphasise that the authors [27–29], criticising the mechanism of the time modulation of the ECdecay rates, caused by the interference of the neutrino flavour mass–eigenstates, did not take into account that fact thatparent and daughter ions interact with the measuring apparatus, i.e. the resonant and capacitive pickups in the ESR,and such interactions lead to violation of energy and momentum in the EC decays in the GSI experiments. Assumingenergy–momentum conservation in the decay channels p → d + ν j ( j = 1 , , . . . , N ν ) they came to the conclusion thatthe mechanism of the interference of the neutrino flavour mass–eigenstates cannot be used for the explanation of theGSI oscillations. This is not a surprise, since dealing with kinematics ~q j = − ~k j and M p = E d ( ~q j ) + E j ( ~k j ) for thedecay channels p → d + ν j ( j = 1 , , . . . , N ν ), one is doomed to show the absence of the time modulation, caused bythe interference of the neutrino flavour mass–eigenstates | ν j i ( j = 1 , , . . . , N ν ) of the EC decay p → d + ν e .We would like also to mention our assertion that the orthogonality of the wave functions of the final states h dν i | dν j i ∼ δ ij in the decay channels p → d + ν i and p → d + ν j has no influence on the suppression of the time modulation asthe interference of neutrino flavour mass–eigenstates [5–7], has been recently confirmed by Murray Peshkin within hissimple two–channel model [55], invented for the analysis of our mechanism of the GSI oscillations as the interferenceof the neutrino flavour mass–eigenstates (see also [30]).For the further verification of the A –scaling of the periods of the time modulation of the H–like heavy ions in the GSIexperiments and the estimate of masses of neutrino (antineutrino) mass–eigenstates we propose to measure the weakdecays of the H–like Ag . In the ground state the odd–odd nucleus Ag with quantum numbers I π = 1 + has the unique feature of decaying as neutral atom with a halflife time of T / = 2 . .
15 % and 2 .
85 % [56], respectively. The H–like ions Ag are unstable underi) the EC decay Ag → Pd + ν e , ii) the bound–state β − –decay Ag → Cd + ˜ ν e and iii) the β − –decay Ag → Cd + e − + ˜ ν e . Storing H-like Ag ions in the ESR of heavy ions one may studythen a time modulation of the EC and bound-state β − –decays from the same parent H–like ion and thus compare theproperties of mixed massive neutrino and antineutrino flavour mass–eigenstates, emitted in these decays, respectively,directly in a CPT type of test.Following [12] we predict that the rates of the β − –decay Ag → Cd + e − + ˜ ν e should not show a timemodulation, whereas the EC and bound–state β − –decay rates should have a periodic time dependence with periods T EC and T β b , respectively. Following [5]–[11] we predict that for H–like ions Ag , moving with the Lorentz factor γ = 1 .
43, the period of the time modulation of the EC decay rate Ag → Pd + ν e should be equal to T EC = 4 πγM p (∆ m ) GSI ≃ A
20 = 5 . , (29)where (∆ m ) GSI = ( m + δm ) − ( m + δm ) . Here m and m are bare neutrino masses and δm and δm are the mass–corrections, caused by polarisation ν j → P ℓ ℓ − W + → ν j of the neutrino flavour mass–eigenstates ν j in the strong Coulomb field of the daughter ion Pd (see Table I) [5, 8, 11]. For the bound–state β − –decay Ag → Cd + ˜ ν e we predict the period of the decay rate modulation equal to T β b = 4 πγM p (∆ ¯ m ) GSI = (∆ m ) GSI (∆ ¯ m ) GSI T EC , (30)1where (∆ ¯ m ) GSI = ( ¯ m + δ ¯ m ) − ( ¯ m + δ ¯ m ) is the difference of the squared dynamical masses of the antineutrinoflavour mass–eigenstates, emitted in the bound–state β − –decay Ag → Cd + e − + ˜ ν e as constituents ofthe electron antineutrino | ¯ ν e i = P N ν j =1 U ej | ¯ ν j i . Then, ¯ m j for j = 1 , bare masses of the antineutrino flavourmass–eigenstates | ¯ ν j i . In case of CPT invariance we may set ¯ m j = m j for j = 1 ,
2. The mass–corrections δ ¯ m j for j = 1 ,
2, caused by polarisations of the antineutrino flavour mass–eigenstates ¯ ν j → P ¯ ℓ ¯ ℓW − → ¯ ν j in the strongCoulomb field of the daughter ions Cd , are given in Table. I. A X Z + δ ¯ m /δm (eV) 10 δ ¯ m /δm (eV) Cd / Pd + 8 . / − .
253 + 4 . / − . Cd and Pd , calculated for R = 1 . × A / [8, 11]. Using the masses m = 0 . m = 0 . Pm , Pr and I heavy ions, for(∆ m ) GSI in the period of the EC decay rate of Ag we obtain (∆ m ) GSI = 2 . × − eV . This agrees wellwith the values (∆ m ) GS , extracted from the periods of the EC decays of Pm , Pr and I heavy ions[5]. In turn, for (∆ ¯ m ) GSI we get (∆ ¯ m ) GSI = 2 . × − eV . This gives the modulation period of the bound–state β − –decay rate equal to T β b ≃ . T β b ≃ . β − –decay rates are ω EC = 2 π/T EC = 1 .
164 rad / s and ω β b = 2 π/T β b = 1 .
122 rad / s, respectively. VII. ACKNOWLEDGEMENT