Why a splitting in the final state cannot explain the GSI-Oscillations
aa r X i v : . [ h e p - ph ] D ec Why a splitting in the final state cannot explain the GSI-Oscillations
Alexander Merle a Max–Planck–Institut f¨ur KernphysikPostfach 10 39 80, D–69029 Heidelberg, Germany
In this paper, I give a pedagogical discussion of the GSI anomaly. Using two different formula-tions, namely the intuitive Quantum Field Theory language of the second quantized picture as wellas the language of amplitudes, I clear up the analogies and differences between the GSI anomalyand other processes (the Double Slit experiment using photons, e + e − → µ + µ − scattering, andcharged pion decay). In both formulations, the conclusion is reached that the decay rate measuredat GSI cannot oscillate if only Standard Model physics is involved and the initial hydrogen-likeion is no coherent superposition of more than one state (in case there is no new, yet unknown,mechanism at work). Furthermore, a discussion of the Quantum Beat phenomenon will be given,which is often assumed to be able to cause the observed oscillations. This is, however, not possiblefor a splitting in the final state only.Keywords: GSI anomaly, neutrino oscillations, quantum theoryPACS: 14.60.Pq, 23.40.-s
1. INTRODUCTION
In the last months, a measurement of the lifetime ofseveral highly charged ions with respect to electron cap-ture (EC) decays at GSI Darmstadt [1] has caused a lot ofdiscussion: Instead of seeing only the exponential decaylaw, a superimposed oscillation has been observed. Thecause of this phenomenon, often referred to as
Darmstadtoscillations or GSI anomaly , is not yet clear and a hugedebate arose whether it could be related to neutrino mix-ing [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], ornot [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Alter-native attempts for an explanation involve spin-rotationcoupling [29, 30, 31, 32], the interference of the finalstates [33], or hyperfine excitation [34]. From the experi-mental side, two test-experiments (with, however, differ-ent systematics [35]) have been performed [36, 37].Several times in this discussion, the analogy of the GSI-experiment to the famous historical Double Slit experi-ment using, e.g., photons [38] has been drawn [6, 19, 26],which also led to lively debates at several meetings [39].In this article, I show that the intuitive Quantum FieldTheory (QFT) formulation of the problem always leadsto the correct result. As there is still a lot of discus-sion in part of the community, it might be useful to giveone more detailed explanation of the Quantum Mechanics(QM) involved. This can be done best by presenting easyand familiar examples that are not necessarily directlyrelated to the GSI anomaly but do involve the same log-ical steps and are not under dispute. To do this, I startwith the superposition principle and discuss the DoubleSlit experiment with photons, e + e − → µ + µ − scatter-ing, and the experiment performed at GSI. Afterwards,the language of amplitudes is used to further justify the a Email: [email protected]
QFT-treatment by carefully considering several cases,where π + decay serves as additional example before theconsiderations are applied to the GSI-experiment, too.Furthermore, the so-called Quantum Beats [40] will bediscussed, a well-known phenomenon that could indeedcause oscillatory decay rates. This has been argued tocause the observation at GSI in several places (see, e.g.,Refs. [2, 12, 13]). It will, however, be shown that thiscannot cause the observed behavior if a splitting is onlypresent in the final state.In the course of the paper, we will see that all three lan-guages naturally lead to the same result, namely thata splitting in the final state cannot explain the GSIanomaly. Depending on the field of the reader, one orthe other part may be clearer, but in the end it turnsout that the intuitive QFT picture is correct and in per-fect agreement with the results obtained using probabil-ity amplitudes or the language of Quantum Beats, whichare just different formulations of the same basic princi-ples.
2. THE QUANTUM FIELD THEORYFORMULATION OF THE PROBLEM
The starting point for the discussion is the superposi-tion principle in QM. One common formulation is [41]:“When a process can happen in alternative ways, we add the amplitudes for each possible way.” The problem inthe interpretation arises in the term “alternative ways”,because it is not a priori clear what the word “way” ac-tually means, as well as in the word “process”, whichexhibits similar ambiguities.Let us use the following terminology:
Process means a re-action with a well-defined initial and final state, whereas way is a particular intermediate state of a process. E.g.,the scattering reaction e + e − → µ + µ − is one single pro-cess, no matter by which way ( γ -, Z -, or H -exchange Category Double Slit e + e − → µ + µ − GSI experiment1 No slit-monitoring at all e + e − -collider N/A2A Monitoring & read out N/A GSI-like experiment with more kinematical accuracy2B Monitoring without read out N/A Actual GSI-experimentTable I: The classification of the three examples. at tree-level in the Standard Model (SM) of elementaryparticles) it is mediated. Z → ν e ν e and Z → ν µ ν µ are,however, two distinct processes.Using this terminology, the superposition principle canbe formulated in the following way:1. If different ways lead from the same initial to thesame final state in one particular process, then onehas to add the respective partial amplitudes to ob-tain the total amplitude. The absolute square ofthis total amplitude is then proportional to theprobability of the process to happen ( coherent sum-mation ).2. If a reaction leads to physically distinct final states,then one has to add the probabilities for the differ-ent processes ( incoherent summation ).If a certain situation belongs to category 1, an interfer-ence pattern will be visible (or oscillations, in case theinterfering terms have different phases as functions oftime), while if it belongs to category 2, there will be nointerference. The remaining question is at which pointthe measurement comes in. This can be trivially said forpoint 2: Either the experimental apparatus is sufficientlygood to distinguish between the different final states (2A)– then no summation whatsoever is necessary simply be-cause one can divide the data set into two (or more), onefor each of the different final states. If this is not the case(2B), the experiment will be able to lead to either of thefinal states, but one would not know which one had beenthe actual result – then one would simply have to addthe probabilities for the different final states to occur inorder to obtain the total probability.What if we do such a measurement for category 1? If wecan indeed distinguish several ways that a process canhappen, then this has to be done by some measurement.Since this measurement then has selected one particularway, we have actually transformed a situation belongingto category 1 into a situation of category 2. However,then there would be no terms to interfere with – the in-terference would have been “killed”.Let us now turn to Table I, that illustrates how our threeexamples fit into the categories 1, 2A, and 2B. Thesethree cases will be discussed one by one in the following. This is the “classical” situation of an experiment thatreveals the nature of QM. It has first been performed by Thomas Young [38] and has later been the major exam-ple to illustrate the laws of QM. Its basic procedure is thefollowing: Light emitted coherently by some source (e.g.a laser) hits a wall with two slits, both with widths com-parable to the wavelength of the light. If it hits a screenbehind the wall, one will observe an interference pattern,as characteristic for wave-like objects (category 1). Thereis, however, the interpretation of light as photons, i.e.,quanta of a well-defined energy. Naturally, one could askwhich path such a photon has taken, meaning throughwhich of the two slits it has travelled. The amazingobservation is that, as soon as one can resolve this bymonitoring the slits accurately enough, the interferencepattern will vanish, no matter if one actually reads outthe information of the monitoring (2A), or not (2B). Thereason is that, regardless of using the information or not,the measurement itself has disturbed the QM process ina way that the interference pattern is destroyed [42].The key point is that one cannot even say that the photontakes only one way: In the QM-formulation, amplitudesare added (and not probabilities), and hence the photondoes not take one way or the other (and we simply sumover the results), but it rather has a total amplitude thatincludes a partial amplitude to take way 1 as well as an-other partial amplitude way 2. By taking the absolutesquare of this sum of amplitudes, interference terms ap-pear.A QFT-formulation involving elementary fields onlywould be much more complicated: One would sum overthe amplitudes for the photon to interact with each elec-tron and each quark in the matter the slits are madeof, after having propagated to this particular particleand before further propagating to a certain point on thescreen. Of course, by using an effective formulation of thetheory, one can find a much more economical descriptionand the easiest one is to simply comprise all possible in-teractions into two amplitudes, one for going through thefirst and one for going through the second slit.Let us go back to this effective formulation: If there ismonitoring, one actually “kills” one of these two ampli-tudes, the other one remains, and the interference is de-stroyed. Whenever there has been such a measurement,the interference will vanishe. As we will see, the ques-tion is if in a certain situation a measurement has beenperformed (or is implicitly included in the process con-sidered), no matter if the corresponding information isread out, or not. e + e − → µ + µ − scattering at a collider Let us now consider the scattering of e + e − to form apair of muons. This is, differently from the Double Slitexperiment, a fundamental process where only a smallnumber of elementary particles is involved. If one wantsto calculate the scattering probability, the amplitude forthe process is again decisive. In the SM, there are onlythree possibilities for this process to happen at tree-leveland in all three of them the e + e − pair annihilates tosome intermediate (virtual) boson which in the end de-cays again, but this time into a µ + µ − pair. The inter-mediate particle can either be a photon, a Z -boson, ora Higgs scalar, see Fig. 1.Here, we have three different ways to form the process.The difference to the Double Slit experiment, however,is that these three ways cannot be separated easily. In areal collider-experiment we are not able to say that thereaction e + e − → µ + µ − has taken place by the exchangeof, e.g., a photon only, but it will always be the sum ofthe three diagrams (and a lot more, in case we includehigher orders). Hence, this process will always fall intocategory 1 and interference terms will appear.This is an easy and familiar example for the appearanceof interference terms in a real experiment. In the nextparagraph it will be shown exactly what is different inthe case of the GSI-experiment. The remaining question is what the situation lookslike for the GSI-experiment. Even though the QFT-calculation of what happens is pretty straightforward,fitting everything in the language used above might be abit more subtle. We will, however, see in Sec. 3.2 that theformulation in terms of amplitudes additionally justifiesthe result obtained here.Let us at first consider the Feynman diagram of the pro-cess involved in the GSI-experiment in Fig. 2 [17]: Here,in the absence of extreme kinematics (meaning that the Q -value of the reaction is large enough so that all neu-trino mass eigenstates can be emitted), the neutrino isproduced as electron neutrino. What happens to thisneutrino? Since it is not detected, it escapes to infinity inthe view of QFT (in the picture of second quantization).Physically, it loses its coherence after some propagationdistance and travels as a unique mass eigenstate.The key point is the following: Since the neutrino will notinteract before it loses its coherence, it must be asymp-totically a mass eigenstate. This can be shown easily:The coherence length of a (relativistic) neutrino is givenby [43] L coh = 2 √ σ x · p (∆ m ) ⊙ , (1)where σ x is the size of the neutrino wave packet, p isthe momentum of the mean value neutrino momentum in the limit m ν = 0, and (∆ m ) ⊙ = 7 . · − eV [44]is the solar neutrino mass square difference as knownfrom neutrino oscillation experiments. The question ishow to obtain an estimate for σ x : If the nucleus wasinside a lattice, one could estimate a width like the typ-ical interatomic distance, σ x ∼ L coh ∼ · m. Of course, this precision cannotbe reached in the GSI experiment. However, at leastduring the electron cooling [45], the nucleus will be lo-calized to some precision. Since the velocity of the nu-cleus is known, this information could in principle beextrapolated for each run. A fair estimate would then bethe average distance between two electrons in the cool-ing process, which is roughly given by 1 / √ n ∼ . n is the electron density [46]. This leads to a morerealistic coherence length of L coh ∼ · m. The pes-simistic case, where σ x is taken to be the approximatediameter 108 .
36 m /π [47] of the Experimental StorageRing (ESR) produces L coh ∼ · m. The mean freepath of a neutrino in our galaxy, however, is roughly1 · m (for an assumed matter density in the MilkyWay of 1 · − g / cm ), so the assumption that the neu-trino does not interact before losing its coherence is com-pletely safe.Even if we do not know in which of the three mass eigen-states the neutrino actually is, we know that it has tobe in one of them. This knowledge is somehow obtained“a posteriori”, since the mass eigenstate only reveals itsidentity after some propagation. But, by conservation ofenergy and momentum, one could treat the process as ifthe kinematical selection had already been present at theproduction point of the neutrino. This “measurement”is enforced by the physical conservation laws.An analogous reasoning is given by Feynman andHibbs [42], using the example of neutron scattering: Neu-trons prepared to have all spin up scatter on a crystal.If one of the scattered neutrons turns out to have spindown, one knows by angular momentum conservationthat it must have been scattered by a certain nucleus.In principle, by noting down the spin state of every nu-cleus in the crystal before and after the measurement,one could find the corresponding scattering partner of theneutron without disturbing it. No matter if this wouldbe difficult practically, by a physical conservation lawone knows that a particular scattering must have beenpresent, even if the corresponding nucleus is not “readout”. Accordingly, the corresponding interference van-ishes and the neutrons that have spin down after thescattering come out diffusely in all directions.This can also be formulated in the language of wavepackets: We have complete 4-momentum conservationfor each single component (which is a plane wave!) ofthe wave packets, but if we consider the whole wavepacket, its central momentum does not have to be con-served [48, 49]. However, all the different components canproduce both possible neutrino mass eigenstates, but fora certain kinematical configuration of parent and daugh-ter components only one of the mass eigenstates will ac- e + e - Μ + Μ - e + e - Μ + Μ - e + e - Μ + Μ - Γ Z H Figure 1: The diagrams contributing to e + e − → µ + µ − in the SM. PSfrag replacements undetected neutrinomass eigenstateparent iondaughter ion ν i Figure 2: The Feynman diagram for the GSI experiment. tually be produced.The rest is easy: If the GSI experiment had infinite kine-matical precision, one could read out which of the masseigenstates has been produced and it would clearly fallinto category 2A. Since, however, this information is notread out but could in principle have been obtained (e.g.by detecting the escaping neutrino), the GSI experimentfalls into category 2B and one has to sum over proba-bilities. This logic works because we know that the neu-trino is, after some propagation, no superposition of masseigenstates anymore, but just one particular eigenstatewith a completely fixed mass.A viewpoint closer to the amplitude formulation wouldbe: If the neutrino finally interacts, it has to “decide”which mass eigenstate it has, even if it was a superpo-sition of several mass eigenstates before. This is thenequivalent to the image of having produced one particu-lar mass eigenstate from the beginning on.
3. AMPLITUDES - PROBABLY THE EASIESTLANGUAGE TO USE
In this section, I use time-dependent amplitudes forthe different basis states to describe another example,namely charged pion decay, which I compare then to neu-trino oscillations (with referring to the actual situationin the GSI-experiment). The logical steps needed to un-derstand the familiar example of pion decay are exactlythe same as the ones needed to understand what is goingon at GSI. This description is clear enough to accountfor very different situations and allows for an easy andnearly intuitive understanding of the various cases. Fur-thermore, it yields an a posteriori justification of the viewused in the preceding section.
It is well-known that a charged pion (e.g. π + ) can de-cay into either a positron in combination with an electronneutrino, or into the corresponding pair of µ -like parti-cles. Let us consider the case of a pure (and normal-ized) initial state pion | π + i . As this state evolves withtime (and is not monitored), it will become a coherentsuperposition of the parent-state, as well as all possibledaughter states: | π + ( t ) i = A π ( t ) | π + i + A µ ( t ) | µ + ν µ i + A e ( t ) | e + ν e i , (2)where all time-dependence is inside the partial ampli-tudes A i . Of course, this state has to be normalizedcorrectly: |A π ( t ) | + |A µ ( t ) | + |A e ( t ) | = 1 , (3)with A π (0) = 1 and A µ (0) = A e (0) = 0. One canunderstand Eq. (2) in the following way: The state attime t is a coherent superposition of the basis states {| π + i , | µ + ν µ i , | e + ν e i} with time-dependent coefficients.Note that the basis states are orthogonal. The outcomeof a certain measurement is some state | Ψ i : All that adetector does is projecting on just this state | Ψ i . Ofcourse, different detectors will in general be describedby projections on different | Ψ i ’s, which is a reflection ofthe influence of the process of measurement on the mea-surement itself. If one wants to know the probability formeasuring that particular state, one has to calculate itaccording to the standard formula, P (Ψ) = |h Ψ | π + ( t ) i| . (4)The question is what | Ψ i looks like. To make that clear,let us discuss several cases: • The (trivial) case is that there has been no detec-tion at all: Then we have gained no information.This means that the projected state is just the time-evolved state itself (we do not know anything ex-cept for the time passed since the experiment hasstarted), and we get |h Ψ | π + ( t ) i| = |h π + ( t ) | π + ( t ) i| = 1 . (5)This result is trivial, since the probability for any-thing to happen must be equal to 1. • The next situation is when our experimental appa-ratus can give us the information that the pion hasdecayed, but we do not know the final state. Then,it can be either | µ + ν µ i or | e + ν e i and we remainwith a superposition of these two states. The onlyinformation that we have gained is that the ampli-tude for the initial pion to be still there is now zero, A π = 0 in Eq. (2). Then, the properly normalizedstate | Ψ i is | Ψ i = A µ ( t ) | µ + ν µ i + A e ( t ) | e + ν e i p |A µ ( t ) | + |A e ( t ) | . (6)The absolute value square of the corresponding pro-jection is |h Ψ | π + ( t ) i| = |A µ ( t ) | + |A e ( t ) | , (7)and if there is any oscillatory phase in the ampli-tudes, A k ( t ) = ˜ A k ( t ) e iω k t , it will have no effect dueto the absolute values. • What if we know that the initial pion is stillpresent? This sets A µ ( t ) = A e ( t ) = 0, and | Ψ i is just A π ( t ) | π + i / p |A π ( t ) | . The projection gives |h Ψ | π + ( t ) i| = |A π ( t ) | , (8)which again does not oscillate. • If one particular final state, let us say | e + ν e i , is de-tected, then A π ( t ) = A µ ( t ) = 0 and we get anotherterm free of oscillations: |h Ψ | π + ( t ) i| = |A e ( t ) | . (9)The question remains when we do get oscillations at all.The answer is: It depends on what our detector mea-sures. If, e.g., the detector measures not exactly the state | µ + ν µ i or | e + ν e i , but instead some (hypothetical) super-position (e.g., some quantum number which is not yetknown, under which neither µ + nor e + is an eigenstate,but some superposition of them), then one could measurethe following (correctly normalized!) state: | Ψ i = 1 √ (cid:0) | µ + ν µ i + | e + ν e i (cid:1) . (10)The squared overlap is |h Ψ | π + ( t ) i| = 12 (cid:2) |A µ ( t ) | + |A e ( t ) | + 2 ℜ (cid:0) A ∗ µ ( t ) A e ( t ) (cid:1)(cid:3) , where the 2 ℜ (cid:0) A ∗ µ ( t ) A e ( t ) (cid:1) -piece will, in general, lead tooscillatory terms. What has been done differently thanbefore? This time, we have done more than simply killingone or more amplitudes in Eq. (2), and this is the causeof oscillations: Whenever we are in a situation, in whichthe state playing the role of | Ψ i in Eq. (10) is physical,the corresponding projection will yield oscillatory terms.As we will see in a moment, this is exactly what happensin neutrino oscillations. Let us now turn to neutrino oscillations. Here, as wewill see, a state like | Ψ i in Eq. (10) can indeed be phys-ical in some situations. To draw a clean analogy to theexperiment done at GSI, we consider a hydrogen-like ionas initial state | M i that can decay to the state | Dν e i via electron capture. Since there was an electron in theinitial state, we know that the amplitude for producingthe mass eigenstate | ν i i is just U ei . If there is no relativephase between the two mass eigenstates, the neutrinoproduced in the decay is exactly the particle that wecall electron neutrino . In any case, due to different kine-matics, the two mass eigenstates will in general developdifferent phases in the time-evolution. This means that,in spite of the mixing matrix elements U ei being time-independent, there will be a phase between the two neu-trino mass eigenstates. Completely analogous to Eq. (2),the time-evolution of the initial state will be given by: | M ( t ) i = A M ( t ) | M i + U e A ( t ) | Dν i + U e A ( t ) | Dν i , (11)with |A M ( t ) | + | U e A ( t ) | + | U e A ( t ) | = 1 and A M (0) = 1. We can immediately look at different cases: • The parent ion is seen in the experiment: Thiskills all daughter amplitudes, A , ( t ) = 0. Theonly remaining amplitude is A M ( t ), very similar toEq. (8). With the proper normalization for | Ψ i onegets no oscillation again: |h Ψ | M ( t ) i| = |A M ( t ) | (12) • The next case corresponds to the GSI-experiment:One sees only the decay, but cannot tell which ofthe two neutrino mass eigenstates has been pro-duced. This leads to A M ( t ) = 0 and one has toperform a projection on the state | Ψ i = U e A ( t ) | Dν i + U e A ( t ) | Dν i p | U e A ( t ) | + | U e A ( t ) | . (13)Doing this with | M ( t ) i from Eq. (11) yields |h Ψ | M ( t ) i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | U e A ( t ) | · | U e A ( t ) | · p | U e A ( t ) | + | U e A ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) == | U e A ( t ) | + | U e A ( t ) | , (14)which exhibits no oscillations, but is rather an inco-herent sum over probabilities. This result is the jus-tification of the intuitive treatment in Sec. 2.3: Theelementary QM-discussion using probability ampli-tudes gives us just the correct prescription for howto sum up the amplitudes for the final states. • The GSI-experiment with infinite kinematical pre-cision: In this case, one could actually distinguish the states | Dν i and | Dν i . If one knows that | Dν i is produced (e.g., by having very precise informa-tion about the kinematics), one will again have nooscillation, |h Ψ | M ( t ) i| = |A ( t ) U e | , (15)just as in Eq. (9).These are in principle all cases that can appear. Onecan, however, have a closer look at the realistic situationin the GSI-experiment. Let us re-consider Eq. (11): Inreality, the parent ion will be described by a wave packetwith a finite size or, equivalently, a finite spreading inmomentum space, due to the Heisenberg uncertainty re-lation. If this wave-packet is broad enough that eachcomponent can equivalently decay into | Dν i or | Dν i ,then both of the corresponding amplitudes will actuallyhave the same phase ( A ( t ) = A ( t )), since they havethe same energy, and one can write Eq. (11) as | M ( t ) i = A M ( t ) | M i + A ( t ) [ U e | Dν i + U e | Dν i ] | {z } = | Dν e i . (16)Since the knowledge of the momentum of the parent ionis not accurate enough at the GSI-experiment to make adistinction between both final states | Dν k i , this is a real-istic situation. Of course, this does not at all change theabove argumentation, since the final state | Ψ i will expe-rience the same modification. The neutrino produced isan electron-neutrino, as to be expected.The question remains, why some authors come to theconclusion that there should be oscillations? The answeris simple: If the correspondence between time-evolvedinitial state and detected state is wrong, then oscilla-tions may appear. As example, we will consider the sit-uation that the kinematics of the parent and daughterare fixed so tightly, that indeed the production ampli-tudes for | Dν i and | Dν i are not equal. This wouldcorrespond to an extremely narrow wave packet in mo-mentum space. Let us, e.g, have in mind the extreme casewhen by kinematics only the production of ν is possible.This is no problem in principle and we would be used toit if neutrinos had higher masses, so that the Q -value ofthe capture was only sufficient to produce the lightestneutrino mass eigenstates. If only the disappearance ofthe parent is seen, the corresponding state | Ψ i , which isdetected, is given by Eq. (13) (with A ( t ) = 0 in the ex-treme case, but anyway with A ( t ) = A ( t )). The corre-sponding neutrino is, however, no electron-neutrino any-more (which would be U e | ν i + U e | ν i , with the samephase for both states)! Indeed this is no surprise at all,since the kinematics in the situation considered is so tightthat it changes the neutrino state which is emitted. Thisis a clear consequence of quantum mechanics, since forobtaining the necessary pre-knowledge (namely the veryaccurate information about the kinematics), one has todo a measurement that is precise enough to have an im-pact on the QM state. If one consideres the state from Eq. (13) as being theone emitted but then projects onto an electron neutrinostate, oscillations will appear: |h D, ν e | M ( t ) i| = | ( U ∗ e h Dν | + U ∗ e h Dν | ) ·· ( A M ( t ) | M i + U e A ( t ) | Dν i + U e A ( t ) | Dν i ) | == |A ( t ) | + |A ( t ) | + 2 ℜ ( A ( t ) A ∗ ( t )) . (17)This is, however, wrong: One has not used all the infor-mation that could in principle have been obtained! ButNature does not care about if one uses information or not,so this treatment does simply not correspond to whathas happened in the actual experiment. The oscillations,however, only arise due to the incorrect projection, andhave no physical meaning.The remaining question to obtain a complete understand-ing of the situation is if the neutrino that is emitted in theGSI-experiment oscillates. The answer is yes, of course.But to see that, we will have to modify our formalism abit. Knowing that an electron neutrino has been emittedcorresponds to A M ( t ) = 0 in Eq. (16), and the remaining(normalized) state is: | Ψ i = A ( t ) |A ( t ) | [ U e | Dν i + U e | Dν i ] . (18)Re-phasing this state and measuring the time from t ongives as initial state: | Ψ i = U e | Dν i + U e | Dν i . (19)This is the state which will undergo some evolution intime according to | Ψ( t ′ ) i = A ′ ( t ′ ) U e | Dν i + A ′ ( t ′ ) U e | Dν i , (20)with |A ′ ( t ′ ) U e | + |A ′ ( t ′ ) U e | = 1 and A ′ (0) = A ′ (0) = 1. If we ask what happens to this neutrinoif it is detected after some macroscopic distance, it isnecessary to take into account what has happend to thedaughter nucleus that has been produced together withthe neutrino, due to entanglement. The daughter nu-cleus, which is accurately described by a wave packet,is detected, but not with sufficient kinematical accuracyto distinguish the different components | D i of the wavepacket. The effect of such a non-measurement is studiedmost easily in the density matrix formalism. The densitymatrix ρ ′ corresponding to Eq. (20) is given by | Ψ( t ′ ) ih Ψ( t ′ ) | = |B ( t ′ ) | | D i| ν ih ν |h D | ++ |B ( t ′ ) | | D i| ν ih ν |h D | ++[ B ( t ′ ) B ∗ ( t ′ ) | D i| ν ih ν |h D | + h.c. ] , (21)where B k ( t ′ ) = A ′ k ( t ′ ) U ek . If the exact kinematics of thedaughter is not measured, then one has to calculate thetrace over the corresponding states. It gives ρ ≡ Z dD h D | ρ ′ | D i = |B ( t ′ ) | | ν ih ν | + (22)+ |B ( t ′ ) | | ν ih ν | + ( B ( t ′ ) B ∗ ( t ′ ) | ν ih ν | + h.c. ) . È a \È b \È c \ Ω ac Ω bc È a \È b \È c \ Ω ab Ω ac Figure 3: Type I (left) and type II (right) of the QuantumBeats settings.
If we want to know the probability to detect, e.g., a muonneutrino, | ν µ i = U µ | ν i + U µ | ν i , the correspondingprojection operator is given by P µ = | ν µ ih ν µ | , (23)and the probability to detect this state is P µ = Tr( P µ ρ ) = h ν |P µ ρ | ν i + h ν |P µ ρ | ν i . (24)Note, however, that the neutrino states | ν , i will alwaysbe orthogonal, since they correspond to eigenstates of dif-ferent masses (like an electron is in that sense orthogonalto a muon). The result is P µ = | U µ | |B ( t ′ ) | + | U µ | |B ( t ′ ) | ++[ U µ U ∗ µ B ∗ ( t ′ ) B ( t ′ ) + c.c. ] , (25)whose second line contains oscillatory contributions.These oscillation are indeed physical: Eq. (23) is a de-scription of a detector that is sensitive to ν µ ’s only. Ifit could not distinguish different neutrino flavours, theoscillation would vanish again.
4. QUANTUM BEATS
The last point to discuss are the so-called QuantumBeats (QBs) [40]. This phenomenon is known from Quan-tum Optics and has often been mentioned as possibleexplanation for the GSI anomaly. As we will see, thecorresponding language can be equally used to describethe GSI-experiment and (of course) yields the same re-sult as already obtained. Still, it is also useful to considerthe experiment from this point of view in order not to bemisled by claims that erroneously make QBs arising froma splitting in the final state responsible for the observa-tion at GSI.Normally, one considers atomic levels for this discussion,and we will stick to that here for illustrative purposes andgive the relation to the GSI-experiment at the end of eachsection. This way also easily clarifies the analogies to theQuantum Optics formulation.
Let us start with the classic example of QBs, namely anatom in a coherent superposition of three states | a i , | b i ,and | c i , where the first two states are above and closelyspaced compared to | c i . This setting is drawn on the leftpanel of Fig. 3 and is referred to as “type I”. First notethat the three levels correspond to different (but fixed)eigenvalues of the energy and are hence orthogonal vec-tors in Hilbert space. This is not at all changed by anenergy uncertainty which, however, makes it possible tohave a coherent superposition of the three states. Ini-tially, we assume the atom to be in such a superpositionof these states, but having emitted no photon yet. Ac-cordingly, the photon state can only be the vacuum | i γ .Then, the initial state of this system can be written as | Ψ(0) i = A | a i| i γ + B | b i| i γ + C | c i| i γ , (26)where |A | + |B | + |C | = 1. If this system undergoesa time-evolution, the lower state might be populated byde-excitation of the upper ones, which is done by photonemission. If the state | x i γ = a † x | i γ is assumed to de-scribe a state with one photon of frequency ω x , then thestate at time t can be written as | Ψ( t ) i = A ( t ) | a i| i γ + B ( t ) | b i| i γ + C ( t ) | c i| i γ ++ C ( t ) | c i| ac i γ + C ( t ) | c i| bc i γ , (27)where A (0) = A , B (0) = B , C (0) = C , C , (0) = 0,and |A ( t ) | + |B ( t ) | + |C ( t ) | + |C ( t ) | + |C ( t ) | = 1.Under the assumption that all levels are equally pop-ulated, the radiated intensity will be proportional to h Ψ( t ) | E ( , t ) | Ψ( t ) i , where E ( x , t ) = X k ,λ ǫ k ,λ (cid:16) a k ,λ e − ikx + a † k ,λ e + ikx (cid:17) (28)is the electric field operator and ǫ k ,λ is the electric fieldper photon of momentum k and polarization λ . Note thatthe creation and annihilation operators have only onenon-trivial commutation relation, namely [ a k ,λ , a † k ′ ,λ ′ ] = δ k , k ′ δ λ,λ ′ . In our case we obtain effectively: E ( , t ) = ǫ ac (1 + 2 a † ac a ac ) + (29)+ ǫ bc (1 + 2 a † bc a bc ) + 2 ǫ ac ǫ bc ( a † ac a bc e i ∆ t + a † bc a ac e − i ∆ t ) , where ∆ = ω ac − ω bc . Here, we have already used thatterms like, e.g., a ac give no contribution with | Ψ i fromEq. (27). Remember now, that the atomic states are or-thonormal. This means that one can, e.g., combine aterm proportional to h b | in h Ψ( t ) | only with the corre-sponding term | b i in | Ψ( t ) i . The corresponding combi-nation of amplitudes |B ( t ) | does, however, not oscillate,since any phase will be killed by the absolute value. Thisis also true for every term involving one of the constantparts of Eq. (29): E.g. the term proportional to C ∗ ( t ) C ( t )can involve a factor γ h | a † ac a ac a † ac | i γ = 0 , (30)because of a † ac acting on the left. There are, however,remaining oscillatory terms such as C ∗ ( t ) C ( t ) e i ∆ t , whichis proportional to γ h | a ac a † ac a bc a † bc | i γ = γ h | (1+ a † ac a ac )(1+ a † bc a bc ) | i γ = 1 . These terms cause the Quantum Beats for a type I atom.Actually, one could have expected this result intuitively:Both of the coherently excited upper levels can decay into the same state | c i via the emission of a photon. Hence,one cannot in any way determine the photon energy with-out measuring it directly. Without such a measurement,interference terms will appear.How is the situation for the GSI-experiment? In thiscase one simply has to replace the photon by the neu-trino. As explained in Ref. [17] for instance, a split-ting in the initial state could lead to an oscillatory be-havior. This splitting, however, would have to be tiny, ∼ − eV, a value which can hardly be explained. Fur-thermore, there exists preliminary data on the lifetimesof Pm with respect to β + -decay that shows no os-cillatory behavior [12]. An initial splitting in the nucleuswould lead to an oscillatory rate in this case, too. Ac-cordingly, if such a splitting is present in the initial state,it could be in the levels of the single bound electron, sincethis would then affect EC-decays while leaving β + -decaysuntouched. We can study a similar setting, namely an atom oftype II, shown on the right panel of Fig. 3. The cor-responding initial state would again be described byEq. (26), but its time-evolution would now look like | Ψ( t ) i = A ( t ) | a i| i γ + B ( t ) | b i| i γ + C ( t ) | c i| i γ ++ B ′ ( t ) | b i| ab i γ + C ′ ( t ) | c i| ac i γ , (31)where A (0) = A , B (0) = B , C (0) = C , B ′ (0) = 0, C ′ (0) = 0, and |A ( t ) | + |B ( t ) | + |C ( t ) | + |B ′ t ) | + |C ′ ( t ) | = 1. The square of the electric field has againthe form of Eq. (29), just with bc → ab . Due to theorthogonality of the atomic states, there are not too manycombinations which are possible: • |A ( t ) | , it does not oscil-late anyway. Hence, only the time-dependent partsin Eq. (29) (with bc → ab ) could lead to oscillations.But they are proportional to γ h | a † ac a ab | i γ = γ h | a † ab a ac | i γ = 0 . • |B ′ ( t ) | does not oscillate, too, and the time-dependent terms from the electric field yield γ h ab | a † ac a ab | ab i γ = γ h | a ab a † ac a ab a † ab | i γ = 0 and γ h ab | a † ab a ac | ab i γ = γ h | a ab a † ab a ac a † ab | i γ = 0 , which follows immediately from the action of a † ac to the left and of a ac to the right, respectively. • B ∗ ( t ) B ′ ( t ), this will oscillateanyway, so we will also have to check the constantterms in Eq. (29). The ones proportional to 1 arenaturally zero, γ h | ab i γ = γ h | a † ab | i γ = 0 . Theother terms are γ h | a † ac |{z} ← a ac | ab i γ = 0 , γ h | a † ab |{z} ← a ab | ab i γ = 0 , (32) γ h | a † ac |{z} ← a ab | ab i γ = 0 , and γ h | a † ab |{z} ← a ac | ab i γ = 0 , where the action of the operators to give zero isalways indicated by the arrow. The argumentationis analogous for the complex conjugated term.Hence, there can be no Quantum Beats for a single atomof type II! The intuitive reason is that, by waiting longenough, one could reach an accuracy in energy that isgood enough to distinguish the possible final states | b i and | c i . This would then be a way to determine theenergy of the emitted photon without disturbing it.To give an analogous argument for the GSI-experiment, one has to turn the comparison given inSec. 4.1 round and replace the atom by the neutrino andthe photon by the ion . The reason is that what is claimedto interfere in this situation is the neutrino states them-selves (see, e.g., Ref. [2]). This neutrino is not expectedto interact, before losing its coherence (cf. Sec. 2.3). How-ever, once it interacts, it has to decide for a certain masseigenstate. By monitoring this interaction, it would inprinciple be no problem to determine the neutrino’s mass(e.g., by exploiting the spatial separation of the masseigenstates far away from the source) and from this onecould easily reconstruct the kinematics of the daughterion in the GSI-experiment. Accordingly, no QBs are tobe expected in this situation. On the other hand, there is a situation in which wecan expect QBs even for atoms of type II, namely if wehave two of them. If these two atoms are separated bya distance which is smaller than the wavelength of theemitted photons, there is no way to resolve their sepa-ration in space and we have to write down a combinedinitial state for both atoms, 1 and 2: | Ψ(0) i = A | a i | a i | i γ + B | b i | b i | i γ + C | c i | c i | i γ + D , | a i | b i | i γ + D , | b i | a i | i γ + E , | a i | c i | i γ ++ E , | c i | a i | i γ + F , | b i | c i | i γ + F , | c i | b i | i γ . The corresponding time-evolution | Ψ( t ) i looks a bit com-plicated: A ( t ) | a i | a i | i γ + B ( t ) | b i | b i | i γ + C ( t ) | c i | c i | i γ ++ D ( t ) | a i | b i | i γ + D ( t ) | b i | a i | i γ ++ E ( t ) | a i | c i | i γ + E ( t ) | c i | a i | i γ ++ F ( t ) | b i | c i | i γ + F ( t ) | c i | b i | i γ ++ G ( t ) | b i | a i | ab i γ + G ( t ) | a i | b i | ab i γ ++ H ( t ) | c i | a i | ac i γ + H ( t ) | a i | c i | ac i γ ++ I ( t ) | b i | b i | ab i γ + I ( t ) | c i | c i | ac i γ ++ J ( t ) | b i | c i | ab i γ + J ( t ) | c i | b i | ab i γ ++ K ( t ) | b i | c i | ac i γ + K ( t ) | c i | b i | ac i γ . (33)One oscillatory term would then be, e.g., J ∗ K e − i ∆ t ,which is proportional to γ h ab | a † ab a ac | ac i γ = γ h | a ab a † ab a ac a † ac | i γ = (34)= γ h | (1 + a † ab |{z} ← a ab )(1 + a † ac a ac |{z} → ) | i γ = γ h | i γ = 1 . If the spatial separation is less than the photon wave-length, one cannot determine the photon energy, becauseone does not know which atom has emitted the radiation.Accordingly, we expect QBs.For the GSI-case, this possibility has to be taken intoaccount, because even for runs with one single EC only,there can have been more ions in the ring that were lostor decayed via β + . In this case (comparing the neutrinoagain with the photon), one has to replace the wave-length of the photon by the de Broglie wavelength of theneutrino. The neutrino energy should be of the same or-der as the Q -value of the EC-reaction, which is roughly1 MeV [1]. The corresponding wavelength is, however, λ = π ~ cEc ∼ − m, while the average distance betweentwo ions should be of the order of the storage ring [50],which is roughly 100 m [47]. Hence, this possibility isexcluded for the GSI-experiment.
5. CONCLUSIONS
A comparison of the GSI-experiment with several otherprocesses (the Double Slit experiment with photons, e + e − → µ + µ − scattering, and charged pion decay) hasbeen given. By using the language of QFT as well asthe intuitive formulation with probability amplitudes, Ihave shown that the situation at GSI cannot lead to anyoscillation of the decay rate, if the correct treatment ischosen and no additional assumptions (as, e.g., a split-ting in the initial state) are taken into account. Alsothe frequently mentioned possibility of Quantum Beatsof the final state cannot explain the observed oscillations,at least not in the standard picture. Hopefully this ar-ticle will contribute to the clarification of the physicalsituation in the experiment that has been performed atGSI. Acknowledgements
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