PoS(Bormio2012)028 http://pos.sissa.it/ ce. mposed oscillations on top of the ccur when going beyond the Breit- ion of oscillations in an analogous e with the measurement apparatus. ectron-capture. Moreover, we discuss n the case of the GSI anomaly are (pre- e concentrate on an oscillating behavior, gion. We propose that these oscillations ara, via Saragat 1, 44100 Ferrara, -von-Laue-Str. 1, 60438 Frankfurt am ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ can explain the socalled GSI anomaly, which measured superi Wigner distribution for an unstable state. Inremisiscent particular, of w the Rabi-oscillations, in the short-time re exponential law for hydrogen-like nuclides decaying via el We discuss deviations from the exponential decay law which o the possibility that the deviations from the Breit-Wigner i dominantely) caused by theThe interaction consequences of of the this unstable scenario, stat such as the non-observat experiment perfromed at Berkley, are investigated. Speaker. ∗ Copyright owned by the author(s) under the terms of the Creat c Francesco Giacosa (Oscillating) non-exponential decays of unstable states Institut für Theoretische Physik, Goethe Universität, Max Italy E-mail: [email protected] Main, Germany E-mail: [email protected] Giuseppe Pagliara Dipartimento di Fisica, Universitá di Ferrara and INFN Ferr 50th International Winter Meeting on Nuclear23-27 Physics January 2012 Bormio, Italy

PoS(Bormio2012)028 15 − tion is the time after t Francesco Giacosa goes to zero at times ) t ( , where p t Γ of Darmstadt has reported − e 1] and in Quantum Field The- ics: the electromagnetic decays ements on the system (inducing scuss the results obtained at the ti-Zeno effects theoretically pre- )= epending on the frequency of the [4] and even shorter for hadronic t s, which is still a quite active area explanation in terms of “standard” uantum theory, but also because it ( tions from the exponential law are e experiment which has proven the oson, are all described by the well ntial decay law, not only because it nuclei have been observed [15]. s allowed to clearly observe for the any different possible explanations short time scale, for instance 10 nances and of the Standard Model y difficult to observe experimentally p ble systems (via tunneling of hat the survival probability at small her consequences of our explanation 0 and at very late times, while at “in- ults is still lacking. By assuming that xponential decay with superimposed ical explanation of this phenomenon d approximation. In particular, at late y is already vanishingly small. On the tum systems, see for instance Ref. [11]. = e ions which decay via capture serve such deviations and to confirm the e performed during the non-exponential al success triggered a new interest of the e derivative of t is calculated from the microscopic interactions Γ 2 represents the survival probability. ) t ( p 0. In turn, this behavior allows for a quite peculiar modifica )= 0 ( ′ p On the other hand, it is a fact that both in Quantum Mechanics [ In 2008, an experiment at the Storage Ring of the GSI facility The decay law of unstable systems plays a crucial role in Phys s for the electromagnetic decaysdecays [2]. of It an is thus excited experimentally hydrogenpredictions very challenging of to the ob theory. Onlyfirst in time 1997 deviations cold from atom the experiment out exponential of decay a law trap) of [5].times unsta is In not exponential, particular, but this it experiment is rather has a shown Gaussian, t i.e. th close to the initial time, ory [2, 3] a purepresent exponential at decay times very law close is to not the obtained: initial preparation devia time the preparation of the unstable state and termediate” times the exponential lawtimes represents the a decay very law goo follows a power-law,because which is it however occurs ver at timesother for hand, which the the deviations survival at probabilit small times occur within a very regime, one can obtain a slower or faster decay of the system d physics community on the topic of deviations from the expone of the decay law induceda by collapse measurements: of the when state pulsed into measur the original undecayed state) ar measurements. Those two effects, calleddicted in Quantum Refs. Zeno and [6, An 7,existence 8, of 9], non-exponential decays have [10]. been then This experiment observed in the sam represents a new and deepopens confirmation the of possibility to the engineer predictions the decay ofAlso, of q the unstable general quan theory ofof measurement research, in benefits quantum from these mechanic experimental results [12]. the observation of non-exponential decays of[13]. hydrogen-lik Quite remarkably, theoscillations. survival probability These shows an datahave e stimulated been many proposed. discussions Presently, andand, there m more is important, an no experimental accepted confirmationthe theoret of phenomenon the observed res at GSI isquantum real, mechanical we effects present [14]. a possible Moreover,which we can present be furt proved orBerkeley disproved Lab where in no the oscillations near for future. the decays We of also the same di or measured in experiments, the decay law is simply given by (Oscillating) non-exponential decays of unstable states 1. Introduction of atoms, the decaysparticles of such radioactive as the nuclei, weakknown of exponential interaction decay hadronic law. bosons Once reso and the decay the rate Higgs b PoS(Bormio2012)028 , Γ . t ixt )= Γ − x − e (a.u.). ( e ) x Γ ( BW d d )= 200 t ( dx rame) and → = ∞ ) 2 BW ∞ − x p R Λ ( Francesco Giacosa d 1, )= = t ( 0 with unit probability, Γ a , . and thus = ) 1 x 2 t Λ ( / t d Γ to the experiments. However, therefore not depend from the ts only the contribution of the − tes is to assume that the decay e cutoff ral function) of the unstable state mediately leads to the exponential iMt ectrum: (i) It does not allow for the large energies is such that, while the .e. it corresponds to an Hamiltonian − omenta of the distribution, including e lows to correctly describe the spectral ed to cure these two problems in order appears a threshold which then, in the d propagator, i.e. the one obtained by epend on the time of “preparation” (in uantum Field Theory, once the interac- at large times, for which a power law is he decay products is known, the spectral . The “empirical” exponential decay law s. )= 2 t | ( ) t ( BW a | a t(a.u.) is a Breit-Wigner distribution 3 ) )= x t ( ( p d 0 is the Fourier transform of 0 Γ ) Γ t 0 ( 2 leading to Γ a / Γ = -0.1 1 10 100 i = - = -10 1 (the state is prepared at the instant 1 1 1 Λ Λ Λ − . t M )= Γ 0 − ( = e a is the mass of the unstable state (i.e. its energy in the rest f 1 pole )= M x t

( 1e-08 1e-12

0,0001 p p(t) , where 4 / 2 Γ [1], the survival amplitude Survival probability for different values of the low energy + ) 2 Γ 1). When calculating the Fourier transform, the integral ge x ) ( , i.e. the number of decays per unit time, is constant and does M d The exponential law works astonishingly well when compared The standard empirical approach to the decay of unstable sta Denoting the energy distribution function (alias the spect Γ − )= x ( 0 ( π 1 2 p simple pole located at there are two evident problems inexistence assuming of a a Breit-Wigner minimum sp of energyunbound (threshold from below. for (ii) the The decay), behavior i normalization of the can Breit-Wigner be at imposed (andalso thus the unitarity), average energy all of the theto m unstable state, build diverge. a We physically ne motivatedtion distribution Hamiltonian function. between In the Q unstable state/particlefunction and is t proportional to thethe imaginary resumming all part the of loops, the seefunction dresse Ref. in [16]. the This whole procedure energy al survival spectrum. probability, regulates In the particular, decay there law of the system Figure 1: (Oscillating) non-exponential decays of unstable states its decay width. Note that rate The exponential decay law turns into a power law at large time 2. A phenomenological approach to unstable states with is theoretically justified, if we assume that “history” of the unstable system,a in quantum particular mechanical it meaning) does of notsurvival the d probability unstable state. This im and the survival probability is just given by PoS(Bormio2012)028 in ch and o an 1 e want Λ robability Francesco Giacosa one or two orders of , (2.1) long time scale but deviations ixt e case of the GSI anomaly the − n be “natural” cutoffs determined e e energy, a low energy cut aw cannot be adjusted, one should interesting features emerge in the h the experimental apparatus that e, especially the oscillating behav- ients enter, for instance the Planck 4 olate between the pure exponential te falls to zero at threshold). When m and a pure oscillating probability / he larger the cutoff, the larger is the cated problem. All field theories are equence, for the survival probability: sts itself (in the Fourier transform) as 2 , we construct here the simplest phe- Γ w in Figs. 1 and 2 the effect of varying e of the oscillation as one can notice in there could be an “experimental” cutoff, e). , which controls the survival probability and its decay products (as the low-energy + 2 represents the probability that the unstable Γ ) and on the decay rate as a function of time. ) dt ). M t ) very close to the average energy of the unstable ( t dt − ( 2 p h x 4 has been made). The physical interpretation of this + Λ ( ( regulates the time after which the exponential law t 2 1 Λ dt dx Λ 2 / ) 1 = and Λ t Λ 1 t + ( − Λ M M dp Z − N , the survival probability remains basically very similar t 2 determines the bandwidth of the continuum of states into whi For such distribution, obviously all the momenta are finite ( have been already observed in experiments. In particular, w Λ 2 )= . t )= 1 Λ ( t ( h a ) = . The high energy cutoff regulates the behavior of the decay p might ) 0 t ( ), ( a ′ Γ p : 2 − defined as Λ ) t )= ( t h ( h shown in Fig. 2 emerging for “small” values of the cutoff(s) ( is such that ) t ( N h As expected, the exponential behavior dominates over a very The physical origin of the cutoff(s) can be twofold: there ca The interesting properties of the decay law explained befor valid until a cutoff ofenergy energy scale where some in new particle physicalnomenological physics. ingred model for Following the this spectral functionwe reasoning and, assume as a a Breit-Wigner cons spectrum corrected by two cuts in th (Oscillating) non-exponential decays of unstable states realized. The high energy behavior ofat the small spectral function times, represents unfortunately a much more compli where particular the average energy of thethese unstable two state). parameters We sho on the survival probability The quantity are clearly present. The low-energy cutoff a high energy cut state decays in the time interval between exponential law (see Ref.behavior of [14] and figures therein), but very at small times after the preparationan of oscillation the superimposed system to and the manife frequency exponential of decay the oscillation law. and T theFig. smaller 2 is (where, for the simplicity, amplitud the choice turns into a power law (inintroduce our another approach parameter the which index fixes ofvarying the how power fast the l the high-energy decay cutoff ra the unstable state can decay.decay The which occurs plots in shown presence in ofwhich Fig. a occurs in large 2 a bandwidth interp system continuu of two discrete levels ( phenomenon is quite natural: magnitude larger than system) where Rabi oscillations are obtained. by the microphysics of the interaction of the unstable state threshold and the high-energy cutoff mentioned above), but which is caused bymeasures the the interaction decay. of theexperimental As cutoff unstable we must system dominate will (i.e. wit discuss it is in the the smaller on following, in th 3. The peculiar case of the GSI anomaly ior of PoS(Bormio2012)028 1, ≃ dt ich / Γ = (3.1) Γ 32 dN . denotes 2 ≃ D Λ Λ Francesco Giacosa 13]). These results Pm) and 1 142 Pm the cutoff Pr or re fitted with superimposed 142 , (3.2) 140 esults based only on Quantum )) 0,8 , it has been found that avy ions storage ring at the GSI , seems very improbable in this t φ that the mass distribution of the + son. he cutoff originates from the inter- o be satisfactory, see Refs. [17, 18, ation of non-exponential decays of t Indeed, the experiment performed at ta by invoking oscillations, ut the origin of these oscillations is ω ( tions the survival probability amplitude cos 0,6 -like nuclide a e ν H + + 1 ( t t(a.u.) D 5 Nd, respectively). λ − → 142 60 e for different choices of the high energy cutoff ) ∝ M t ( Γ is very small. An intrinsic origin of this cutoff based on h dt Γ dN Λ Ce and is the decay width of the state, gives rise to oscillations wh − =10 =200 1 140 58 2 2 = − Exponential Λ Λ Pm in the electron-capture reactions of the form: has been used. We have shown that for dec 142 dt 2 dN 0224 s Λ . 0 = 1 = 0 0,2 0,4 Pr and 1 0

Λ

Γ 0,5 h(t) 140 = the number of unstable particles at the instant Λ represents the number of decay per time (see Fig. 3-5 of Ref. [ ) t ( dt N / eV, where Decay rate as a function of time dec denotes the ‘mother state’ (i.e. the unstable follow a simple exponential law. The experimental points we (a.u.). Also the exponential decay rate is shown for compari 15 2 − dN M Λ not The required value of the cutoff In Ref. [14] we have put forward an interpretation of the GSI r Calling 10 · = 5 1 . are qualitatively similar to those measured in Ref. [13]. 0 QED and QCD fundamental interactions,case. on A the more line promising ofaction direction Ref. of consists the [4] in unstable system assuming with that the t measuring apparatus. oscillations: Λ to point to what has beenfacility named of the Darmstadt. “GSI anomaly” seen InHydrogen-like at ions 2008 the Ref. he [13] reported the observ Mechanics: Following the discussionsmother of state is Sec. not a pure II, Breit-Wigner.of In we Eq. doing assumed (2.1) the calcula with stimulated the theoretical modellingnot of yet this clear: phenomenon, explanations b neutrino of spin the precession and observed quantum experimental19, beats da 20] seem and indeed refs. not therein. t where Figure 2: (Oscillating) non-exponential decays of unstable states where does the ‘daughter state’ (i.e. the nuclei PoS(Bormio2012)028 is re it Pm w 142 61 he quan- 2, has been nstable ions / Pr and w 140 59 entering into the ) of the unstable = Francesco Giacosa 2 Λ M Λ s to complete a turn µ = 5 1 . is introduced and named Λ 0 R , (3.4) = ∼ 2

Λ ale 4 stem Depend on the Measuring nto two particles is modified by dge, the first theoretical work on eed ntally proven by the observation / 2 As discussed in that paper, for the stable state from the experimental he decay of these unstable systems γ e) needed to describe the oscillation meter which measures the frequency ation between the unstable state and onnect the cutoff bility of an apparatus-induced cutoff ds on the charge to mass ratio of the wave function occurs instantaneously ratio, and thus the rotation frequency, n, see Refs. [11, 12] and refs. therein. + /overlaps with the measurement appa- le of minutes. Some important features the time-energy uncertainty relation: eV . (3.3) m has decayed or not. In this scheme, requency spectrum is identified within nstable Hydrogen-like 2 that the measurement itself can modify iEt ) etween the decay products beyond which − 15 R e − E − 10 cy of the daughter ion in the frequency spectrum. This me interval which lasts between the disappearance of the E ∼ ( 6 dE Pm respectively) is related to the cooling of the stream befo of the two decay products and the localization radius 2 / 2 1 v / w 142 w is the average energy (i.e. the mass resolution + t R − R ∆ R E in the case of Ref. [21] is quite clear: it is related to the E E Z ∼ Pr and

w ) if it is not disturbed by the measurement, and the cutoff Λ Γ N 140 , formally identical to Eq. (2.1) for )= ) t t ( . This measurement is clearly not an ‘ideal measurement’ in t ( 1 p p 200 ms. This means that the ‘measurement’ of the state of the u ≃ resolution . The interpretation of t R ∆ is a normalization constant, / v the width of the state (i.e. N γ = Another important experimental limitation concerns the ti 1 w , R proportional to the ratio of relative velocity its decay products and it is thus of the same type of Eq. (3.3). time needed by the measuring apparatus to destroy the correl expression (2.1) to the precision of the experiment through tum mechanical sense, according to which theas collapse soon of as the the wave functionratus of the (the unstable projection system postulate interacts of Quantum Mechanics). We c does not occur at every turn the following formula for the experimental apparatus can ascertain whether the syste “localization radius”, which corresponds to the distance b where This number is remarkably close toseen the in value the (mentioned abov GSI experiment.is It viable seems and therefore deserves that further thethe discussion. possi decay Indeed, law the of fact anof unstable the quantum system Zeno has and beenthis Anti-Zeno already problem effects experime [10]. is Ref. To ourApparatus” knowle and [21] recently “Does a new the interest Lifetime onIn this of Ref. issue an has grow [21] Unstable it Sy the is measuring analyzed apparatus how such the as decay a of bubble an chamber. unstable state A i length sc (Oscillating) non-exponential decays of unstable states the GSI storage ring is unique. After their “creation”, the u frequency of the mother iontime and interval the of 900 appearance ms of andcan the 1200 be frequen ms identified (for by the mass spectrometer. state, in the storage ring, theiraveragely frequency of rotation within the f ions are stored in a ringof equipped with rotation a of Shottky the Mass spectro ionsions. inside When the the ring. reaction (3.1) This takeschange. place, frequency the depen In charge this to mass wayfew the seconds experimentalists after their at preparation GSI and for canof a monitor period this of t a experiment coup are worth to be mentioned: While the ions n obtained by studying the dependenceapparatus: of the lifetime of the un PoS(Bormio2012)028 . + ) t → β ( ′ p M erved. is very oscilla- − w no )= t decay: ( h + d approximation β Francesco Giacosa eV: the deviations 15 − 0), the first peak is more ) evaluated starting from dt )= / 0 hold in the framework of our ( ′ dec p r than 10 The oscillations have a too small ent extremely fast. Thus, for the the Berkeley Lab [15] ], the non-occurrence of the Zeno r interpretation, the crucial fact is dN . (which represents the decay prob- s sense, the GSI experiment could inside a lattice, thus also phonons at ons and build a detailed theoretical easuring apparatus also in the case bsence of oscillations at the Berke- ) with the function rvable in this channel (see Fig. 2 to effort which is left for future work. ent) would assure that the quantity Λ ) part of our outlook. are analogous to the emergence of a is much shorter and, conversely, the to Ref. [24]. A detailed study of the cases sizably) via a t . to ( aster than the fitting curve in Eq. (3.2), t mental fitting curve of Eq. (3.2). Our ≫ ces from the GSI experiment w.r.t. the h -shell vacancy is formed and a photon w nstance the bubble chamber) r the determination of a time scale. For nsic cutoff emerging out of microscopic ∆ ect of the ‘collapse’ of the wave function y of Eq. (3.1) the emitted neutrino does also useful to clarify the following point: K Berkeley Λ 7 -like daughter state for this process. In the case of the H -like ions under study at the GSI do not decay only via the H : Our theoretical function ) t ( ′ Pm in relation to the ‘same’ process of Eq. (3.1) have been obs p 142 − refers to the ′ )= D and therefore the exponential decay is obtained to a very goo t ( 1 h − s 2 in our case. where 22 / decay channel: The , e w -10 ν + 17 = β + Λ + e As a next step we list the predictions and consequences(i) which The curve (iii) Berkeley-experiment: In the experiment performed at (ii) Indeed, one should go beyond these qualitative considerati vanishes for short times (a general feature due to the fact th -channel the corresponding cutoff turns out to be much large ) + t ′ + ( These issues go at therepresent very a heart wonderful of way Quantum to Mechanics: directly in investigate them. thi proposed interpretation. As already noticed in Ref.Berkeley [25], one: there in are peculiar theare differen latter, emitted the in atoms the are finalthat not state. very ionised soon However, and in after are the the framework electron of capture ou of Eq. (3.1), a is very soon emitted.cutoff Thus, is just much larger as in in the the Berkeley-experiment: previous case, from the exponential decay law are‘see’ very the small effect and of thus an unobse increasedat cutoff). variance This with discussion is the positron,not in interact the with electron-capture matter deca anda is mathematical therefore not description responsible of fo thetwo-channel two-channel problem case using we the refer formalism of Ref. [24] is also tions in the decay law for ability per unit time and unit ion and it is thus proportional Eq. (2.1) shows some peculiarh differences w.r.t. the experi pronounced than the others and the oscillationssee are Fig. damped 2 f and the detailed discussion and figures in Ref. [14] electron-capture mechanism of Eq. (3.1), but decay (in both β (Oscillating) non-exponential decays of unstable states typical measuring apparatuses in particle physics (as for i model for the interaction betweenof the the unstable GSI ions experiment, and butMoreover, the this it m will represents be a also important quite to demanding in study in the detail case the of eff theeffect GSI experiment. (and therefore As discussed the inmeasured clock Ref. is in [22, not the 23 reset experiment at coincides each (up measurem to a normalization D decay a positron is emitted which is absorbed by the environm large: 10 cutoff amplitude and period and cannot beley experiment observed. is Moreover, a the further a strongform argument against factor. an intri in most of the cases. The arguments in Ref. [21] leading PoS(Bormio2012)028 which are , Γ / -like ions may Λ H Francesco Giacosa nduced cutoff can explain -like ion: In the framework is quite remarkable that the suppressed oscillations in the H ng of the measurement proce- al decay law when the energy d rformed with an improved time hem unobservable: the standard ry of Research through the pro- , the period and the amplitude of d qualitative features are general, a sions with F. Bosch, Y. Litvinov, N. cutoff only, are also expected to be Ref. [13]. Notice that the same cutoff inty relation the physical origin of the e peak have been introduced. We have f Eq. (2.1). This will be possible once rease). ron-capture decay of r needed to obtain the time modulation tion of the mother state [14]. Inspired umerical example. ; a mild dependence of the period and s when the GSI experiment is repeated and thus exponential decay law) in the n of the system as a whole (unstable state ses, see Eq. (3.3), and thus the period and , we have studied a modified energy distri- 8 Pm corresponds to quite different ratios of 142 is almost uniquely related to the measurement process and is Λ Pr and 140 470 respectively. It is interesting that the measurement-i ∼ to the time uncertainty of the measuring apparatus at GSI. It Λ G.P. acknowledges financial support from the Italian Minist As an outlook for future works we mention the precise modelli (iv) Independence of the period and amplitude on the employe (v) Repetition of the experiment. If the GSI experiment is pe Finally, it should be stressed that, while the here describe In this work we have described deviations from the exponenti We have analyzed the consequences of our proposal: very much decay-channel because of a much larger cutoff, which makes t 32 and + originate from a similar modificationby of Ref. the [21], energy we distribu have linkedcutoff through the time-energy uncerta therefore independent of the employed mother nuclide. Thus the superimposed oscillations, whichcomparable: are this is controlled indeed by the case the of of Eq. the two (3.3) ions for studied in both ions β (Oscillating) non-exponential decays of unstable states of our interpretation, the cutoff gram Rita Levi Montalcini. F.Winckler, G. S. acknowledges Bishop, useful and discus D. Shubina. Acknowledgments dure and the detailed study of the two-channel problem. decay law holds here;electron-capture similarly, decay suppressed channel oscillations at ( theamplitude Berkeley on experiment the unstable ion;with more an suppressed increased oscillation time resolution (period and amplitude dec proposed that the oscillations seen in Ref. [13] in the elect bution in which cutoffs on the left and on the right sides of th distribution is not a Breit-Wigner function. In particular naturally these quite different ratios. resolution, we expect that the corresponding cutoffthe increa amplitude of the oscillations decrease, see Fig. 2 for aquantitative n analysis should go beyond thethat, simple as formula mentioned o above, a detailed studyplus of measurement) the will interactio be undertaken. 4. Conclusions ∼ cutoff obtained in this way, see Eq.of (3.3), 7 is s of measured the in same Ref. orde [13]. PoS(Bormio2012)028 , , 387 , Int. 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