On the "Field-Theoretical Approach" to the Neutron-Antineutron Oscillations in Nuclei
aa r X i v : . [ h e p - ph ] F e b ON THE ”FIELD-THEORETICAL APPROACH” TOTHE NEUTRON-ANTINEUTRON OSCILLATIONS INNUCLEI
Vladimir Kopeliovich ∗ , Institute for Nuclear Research of RAS60-th October Anniversary Prospect 7a, Moscow 117312
September 15, 2018
Abstract
It is argued that within the correct treatment of analytical properties of the transitionamplitudes, in particular, the second order pole structure, characteristic for the n − ¯ n tran-sition in nuclei, the ”infrared divergences” discussed in some papers, do not appear. Ex-plicit calculation with the help of diagram technique shows that the neutron-antineutronoscillations are strongly suppressed within a deuteron, as well as within arbitrary nucleus,in comparison with the oscillations in vacuum. The neutron-antineutron transition induced by the baryon numberviolating interaction (∆ B = 2) predicted within some variants of grand unifiedtheories (GUT) has been discussed in many papers since 1970 [1], see [2, 3,4, 5, 6].. Experimental results of searches for such transition are available, invacuum (reactor experiments [7], and references therein), in nucleus O [8] andin F e nucleus [9], see also the PDG tables.During the later time there have been many speculations that the neutron-antineutron oscillations in nuclei are not suppressed in comparison with the n − ¯ n transition in vacuum [10, 11]. The arguments were based on the ”true field-theoretical approach” to this problem. The result of [10] has been criticized ina number of papers [12, 13, 14, 15] which used somewhat different approaches(potential, S-matrix, diagram), and general physics arguments.However, in view of continuing publications [11] containing same state-ment as in [10], it seems to be necessary to analyze this problem just within the ∗ e-mail : [email protected] uantum field theory based approach used in [10, 11]. My consideration is closeto the approach of paper [14] where the diagram technique has been appliedto study neutron-antineutron transition in nuclei, although differs from [14]in some details. More recent realistic calculations of the neutron-antineutrontansition in nuclei can be found in [16, 17].Here we give first some general arguments based on analytical propertiesof amplitudes in favour of suppression of n − ¯ n transition in nuclei (section 3).The simplest example of the deuteron where the final result can be obtained inclosed form, is considered in details in section 4, and the result of [14] for thecase of the deuteron is reproduced. The analogy with the nucleus formfactorat zero momentum transfer is noted in section 5. To introduce notations, let us consider first the n ¯ n transition in vacuumwhich is described by the baryon number violating interaction (see, e.g. [2, 13,14]) V = µ n ¯ n σ / σ being the Pauli matrix. µ n ¯ n is the parameter which hasthe dimension of mass, to be predicted by grand unified theories and to bedefined experimentally . The n − ¯ n state is described by 2-component spinorΨ, lower component being the starting neutron, the upper one - the appearingantineutron. The evolution equation is i d Ψ dt = ( V + V )Ψ (1)with V = m N − iγ n / m N is the nucleonmass, γ n - the (anti)neutron normal weak interaction decay width, and we take γ ¯ n = γ n , as it follows from C P -invariance of weak interactions). Eq. (1) hassolutionΨ( t ) = exp [ − i ( µ n ¯ n t σ / V t )] Ψ = " cos µ n ¯ n t − iσ sin µ n ¯ n t exp ( − iV t )Ψ , (2)Here Ψ is the starting wave function, e.g. Ψ = (0 , T . In this case we havefor an arbitrary timeΨ(¯ n, t ) = − i sin µ n ¯ n t exp ( − iV t ) , Ψ( n, t ) = cos µ n ¯ n t exp ( − iV t ) , (3)which describes oscillation n − ¯ n . Evidently, for large enough observation times, t obs ≫ /µ n ¯ n , the average probabilities to observe neutron and antineutron are There is relation µ n ¯ n = 2 δm with the parameter δm introduced in [2]. The neutron-antineutron oscillationtime in vacuum is τ n ¯ n = 1 /δm = 2 /µ n ¯ n , see also [17] and references in this paper. qual if we neglect the natural decay of the neutron (antineutron): W (¯ n ) = | Ψ(¯ n ) | = | Ψ( n ) | = W ( n ) = 1 / . (4)This case is, however, of academic interest, only, since γ n ≫ µ n ¯ n It should bestressed that in the vacuum neutron goes over into antineutron, also discretelocalized in space state, which can go over again to the neutron, so the oscillationneutron to antineutron and back takes place.Since the parameter µ n ¯ n is small, expansion of sin and cos can be madein Eq. (3) at not too large times. In this case the average (over the time t obs ≪ /µ n ¯ n ) change of the probability of appearance of antineutron in vacuumis (for the sake of brevity we do not take into account the (anti)neutron naturalinstability which has obvious consequences) W (¯ n ; t obs ) /t obs = | Ψ(¯ n, t obs ) | /t obs ≃ µ n ¯ n t obs n → ¯ n is suppressed if the observation time is small, t obs ≪ /µ n ¯ n . Fromexisting data obtained with free neutrons from reactor the oscillation time isgreater than 0 . . sec ≃ . µ n ¯ n < . . − eV, (6)very small quantity.Recalculation of the quantity µ n ¯ n or τ n ¯ n from existing data on nucleistability [8, 9] is somewhat model dependent, and different authors obtainedsomewhat different results, within about 1 order of magnitude, see e.g. discus-sion in [14, 16, 17]. Most recent results for µ n ¯ n obtained from nuclear stabilitydata are close to (6) [16, 17], see also next section. In the case of nuclei the n − ¯ n line with the transition amplitude µ n ¯ n is the element of any amplitude describing the nucleus decay A → ( A −
2) + mesons , where ( A −
2) denotes a nucleus or some system of baryons It is a matter of simple algebra to calculate the integrals over time of the probabilities | Ψ( n, t ) | and | Ψ(¯ n, t ) | :. Z ∞ | Ψ( n, t ) | dt = 2 γ n + µ n ¯ n γ n ( γ n + µ n ¯ n ) , Z ∞ | Ψ(¯ n, t ) | dt = µ n ¯ n γ n ( γ n + µ n ¯ n )for neutron as initial state and for arbitrary, but different from zero γ n . The difference between both quantitiesis obvious, and disappears when γ n → ith baryonic number A −
2, see Fig. 1. The decay probability is thereforeproportional to µ n ¯ n , and we can write by dimension argumentsΓ( A → ( A −
2) + mesons ) ∼ µ n ¯ n m , (7)where m is some energy (mass) scale. For the result of [10, 11] to be correctthe mass m should be very small, m ∼ µ n ¯ n ∼ − eV , but we shall arguethat m is of the order of normal hadronic or nuclear scale, m ∼ m hadr ∼ (10 − M eV . We can obtain the same result from the above vacuum formula(5), if we take the time t obs ∼ /m hadr . ❥❦❦❥❥ ③ ✟✟✟✟✟(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✥✥✥✥✥✑✑✑✑✑✏✏✏✏✏ t µ n ¯ n / A A − mesonsA − n ¯ n (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅ Figure 1: The diagram describing n − ¯ n oscillation in nucleus A with subsequent annihilationof antineutron to mesons. The final state has the baryon number A − Indeed, the matrix element of any diagram containing such transition T ( A → ( A −
2) + mesons ) ∼∼ µ n ¯ n ( A − Z ) Z V ( A ; n, ( A − T (¯ n + ( A − → ( A −
2) + mesons )( E n − E n + iδ ) dE n ≃≃ − πi ( A − Z ) d ( V ˜ T ) dE n ( E n = E n ) , (8)according to the Cauchy theorem known from the theory of functions of complexvariable. E n is the neutron (antineutron) energy - integration variable, E n is the(anti)neutron on-mass-shell energy E n ≃ m N + ~p / m N . The energy-momentumconservation should be taken into account for the vertex V ( A → n + ( A − A −
1) system, and for the annihilation mplitude ˜ T . The case of the deuteron considered below is quite transparentand illustrative.The amplitude ˜ T which describes the annihilation of the antineutron, andthe vertex function V are of normal hadronic or nuclear scale and cannot, inprinciple, contain a very small factors in denominator (or very large factors, ofthe order of 10 , in the numerator). By this reason we come to the above Eq.(7), and the resulting decay width of the nucleus is very small,Γ( A → ( A −
2) + mesons ) < − µ n ¯ n , (9)at least 30 orders of magnitude smaller than the inverse time of neutron-antineutron oscillation in vacuum µ n ¯ n . From Eq. (7) or (9) we obtain µ n ¯ n ∼ q Γ( A → ( A −
2) + mesons ) m , (10)and when one tries to get restriction on µ n ¯ n from data on nuclei stability [8,9] the result is close to that from vacuum experiment [7], somewhat smaller,within one order of magnitude [5, 6, 14]. The result of [16] differs from thatof [14] for heavier nuclei, and the authors [16] come to the conclusion, thatexperiments with free neutrons from reactor could provide stronger restrictionon the neutron-antineutron transition parameter than experiments on stabilityof nuclear matter .According to [10, 11] the probability of the nucleus decay is proportionalto W ( t obs ) ∼ µ n ¯ n (cid:16) t obs (cid:17) (the process proceeds similar to the vacuum case),where t obs is the large observation time, of the order of ∼ µ n ¯ n is smaller than that given by Eq. (10),by about 15 orders of magnitude. Technical reason for strange result obtainedin [10, 11] is the wrong interpretation of the second order pole structure ofany amplitude containing the n − ¯ n transition. Instead of using the well de-veloped Feynman diagram technique, the author [10, 11] tries to construct thespace-time picture of the process by analogy with the vacuum case, which ismisleading, see also discussion in conclusions. There is, in fact some kind of competition between both methods, and final result will depend on theprogress to be reached in both branches of experiments — with free neutrons and with neutrons bound innuclei. Friedman and Gal [17] obtained the restriction τ n ¯ n > . . sec from the latest datum on O stabilityand using the potential approach. Experiments with ultracold neutrons in a trap have been proposed anddiscussed in [3, 18], but not performed till now. . We continue our consideration with the case of the deuteron whichis quite simple and instructive, and can be treated using standard diagramtechnique . The point is that in this case there is no final state containingantineutron — it could be only the p ¯ n state, by charge conservation. But thisstate is forbidden by energy conservation, since the deuteron mass is smallerthan the sum of masses of the proton and antineutron. Therefore, if the n − ¯ n transition took place within the deuteron, the final state could be only someamount of mesons. The amplitude of the process is described by the diagrams ❥ ③ PPPPP❵❵❵❵❵✥✥✥✥✥✏✏✏✏✏ t µ n ¯ n / d p mesonsn ¯ n (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅ Figure 2: The diagram describing n − ¯ n oscillation in the deuteron with subsequent annihilationof antineutron and proton to mesons. of the type shown in Fig. 2 and is equal to T ( d → mesons ) = ig dnp m N µ n ¯ n Z T (¯ np → mesons )( p − m N )[( d − p ) − m N ] d p (2 π ) . (11)The constant g dnp is normalized by the condition [20, 21, 22] g dnp π = κm N = vuut ǫ d m N , (12)which follows, e.g. from the deuteron charge formfactor normalization F d ( t =0) = 1, see next section. κ = √ m N ǫ d , ǫ d ≃ . M eV being the binding energyof the deuteron. For the vertex d → np we are writing 2 m N g dnp to ensure thecorrect dimension of the whole amplitude.The integration over internal 4-momentum d p in (11) can be made easilytaking into account the nearest singularities in the energy p = E , in the non-relativistic approximation for nucleons. As we shall see right now, the integral It has been considered in fact in [19] within reasonable framework of diagram technique, however, the authorhas drawn later wrong conclusions from this consideration. ver d p converges at small p ∼ κ which corresponds to large distances, r ∼ /κ .By this reason the annihilation amplitude can be taken out of the integrationin some average point, and we obtain the approximate equality T ( d → mesons ) = g dnp m N µ n ¯ n I dN N T (¯ np → mesons ) (11 a )with I dN N = i (2 π ) Z d p ( p − m N )[( d − p ) − m N ] ≃≃ i (2 π ) (2 m ) Z d p ( p − m N − ~p / (2 m N ) + iδ )( m d − m N − p − ~p / (2 m N ) − iδ ) == Z d p (2 π ) m N [ κ + ~p ] = 164 πm N κ , (13)This integral converges at small | ~p | ∼ κ , more details can be found in the nextsection. The decay width (probability) is, by standard technique,Γ( d → mesons ) ≃ µ n ¯ n g dnp I dN N m N Z | T (¯ np → mesons ) | d Φ( mesons ) , (14)Φ( mesons ) is the final states phase space. Our final result for the width of thedeuteron decay into mesons isΓ d → mesons ≃ µ n ¯ n πκ m N [ v σ ann (¯ np )] v → ≃ µ n ¯ n πκ m N h p c.m. σ ann ¯ np i p c.m. → , (15)where p c.m. is the (anti)nucleon momentum in the center of mass system. Thisresult is very close to that obtained by L.Kondratyuk (Eq. (17) in [14]) insomewhat different way, using the induced ¯ np wave function .The annihilation cross section of antineutron with velocity v on the pro-ton at rest equals σ (¯ np → mesons ) = 14 M N v Z | T (¯ np → mesons ) | d Φ( mesons ) . (16)According to PDG at small v , roughly, h v σ ann ¯ np i v → ≃ (50 − mb ≃ (130 − GeV − . So, we obtain µ n ¯ n ≤ . − eV , or τ n ¯ n > . sec if we take The result Eq. (17) in [14] can be rewritten in our notations asΓ d → mesons ≃ . µ n ¯ n m N κ [ v σ ann ¯ np ] v → , (17 ′ )which differs from our result by some numerical factor, close to 1 and not essential for our conclusions. ptimistically same restriction for the deuteron stability as it was obtainedfor the F e nucleus, T d ≃ T F e > . . yr [9]. Our result (15) is valid upto numerical factor of the order ∼
1, since we did not consider explicitly thespin dependence of the annihilation cross section and the spin structure of theincident nucleus. Same holds in fact for the results obtained in preceedingpapers, see [2, 14] e.g.Additional suppression factor in comparison with the case of a free neu-tron is of the order of µ n ¯ n /κ ∼ − in agreement with our former rough estimate (9), and disappears, indeed, whenthe binding energy becomes zero . The binding energy of the deuteron shouldbe very small, to give κ ∼ µ n ¯ n , to avoid such suppression. At such vanishingbinding energy the nucleons inside the deuteron are mostly outside of the rangeof nuclear forces, similar to the vacuum case.Results similar to (15) can be obtained for heavier nuclei, see [5, 6, 14, 16,17]. The physical reason of such suppression is quite transparent and has beendiscussed in the literature long ago (see [2, 13, 15] e.g.): it is the localizationof the neutron inside the nucleus, whereas no localization takes place in thevacuum case. In the case of the deuteron or heavier nucleus the annihilationof antineutron takes place, and final state is some continuum state containingmesons. By this reason the transition of the final state back to the incidentnucleus is not possible in principle, and there cannot be oscillation of the type,e.g. d → mesons → d . This is important difference from the case of the freeneutron. As we noted previously, the presence of the second order pole inintermediate energy variable is characteristic for the processes with the neutron- antineutron transition, but it is in fact not a new peculiarity, it takes place alsofor the case of the nucleus formfactor with zero momentum transfer, F A ( q = 0).Let us consider as an example the deuteron charge formfactor. In the zero rangeapproximation it can be written as F d ( q ) = i (2 mg dnp ) (2 π ) Z d p ( p − m N )[( d − p ) − m N ][( d − p + q ) − m N ] . (17) There is no final formula. for Γ d → mesons in [19] to be compared with our result (14) , (15). Numerically,however, the result of [19] is in rough agreement with our and [14] estimate. ehind the zero range approximation g dnp should be considered as a function ofthe relative n − p momentum, not as a constant. For q = 0 second order poleappears, and we come to the expression for F ( q = 0) containing the integral I dN N introduced above in Eq. (13): F d (0) = (2 m N g dnp ) I dN N . (18) ❥ ❥❢ γd n dp p (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅❅ Figure 3: The diagram describing the deuteron charge formfactor.
In the nonrelativistic approximation, when only the nearest in energy E = p singularities are taken into account, the integral over the energy has thestructure I dN N ∼ Z dE ( E − a + iδ )( E − b − iδ ) = − πi ( a − b ) , (19) a = m N + ~p / m N . b = m d − m N − ~p / m N , a − b = ǫ d + ~p /m N , and canbe calculated using the lower contour, or the upper contour, with the help offormulas known from the theory of functions of complex variables. After thiswe obtain F d ( q = 0) = g dnp m N π Z d p ( κ + ~p ) . (20)Since F d (0) = 1, this leads to the above mentioned normalization condition g dnp / (16 π ) = q ǫ d /m N .This relation between the constant g dnp and the binding energy of theweakly bound system (deuteron in our case) is known for a long time [20, 21, 22]. As it is known from nonrelativistic diagram technique, the wave function of the deuteron in momentumrepresentation is Ψ d ( ~p ) = g dnp / [4 π / ( κ + ~p )], therefore, the normalization of the charge formfactor F d (0) = 1follows from the normalization of the deuteron wave function, which is also well known from quantum mechanics. t was obtained in [20, 21, 22] using different methods, dispersion relation, forexample.If the infrared divergence discussed in [10, 11] takes place for the processof n − ¯ n transition in nucleus, it should take place also for the nucleus form-factor at zero momentum transfer. But it is well known not to be the case.There is no ”new limit on neutron - antineutron transition” [10]; instead, oneshould treat correctly singularities of the transition amplitudes in the complexenergy plane, in particular, the second order pole contribution to the transitionamplitudes. The author of [10, 11] tries to reconstruct the space-time pictureof the process, but the correspondence of this picture to the well justified am-plitude, as it appears from Feynman diagrams, is questionable. The infrareddivergence discussed in [10, 11] is an artefact of this inadequate space-timepicture of the whole process of n − ¯ n transition with subsequent antineutronannihilation. Another quite unrealistic consequence of this space-time pictureis the nonexponential law of the nucleus decay.Field-theoretical description of nuclear reactions and processes is poten-tially useful, it allows to study some effects which is not possible in principleto study in other way, e.g. relativistic corrections to different observables. Oneshould be, however, very careful to treat adequately analytical properties ofcontributing amplitudes. E.g., in the case of the parity violating amplitudeof np → dγ capture it was necessary to take into account contributions of allsingularities (poles) of the amplitude in the complex energy plane, not onlycontributions of the nearest poles in energy variable, as it is made usually innonrelativistic calculations. The nonrelativistic diagram technique developedup to that time turned out to be misleading for the case of physics problemconsidered in [23]. Besides, and it is the spesifics of the processes with photonemission, the contact terms should be reconstructed to ensure the gauge invari-ance of the whole amplitude of photon radiation [23]. Acknowledgements.
Present investigation, performed partly with ped-agogical purposes, has been initiated by V.M.Lobashev and V.A.Matveev whohave drawn my attention to the longstanding discrepance between papers [10,11] and the results accepted by scientific community ([1]-[5],[13, 14, 15] andreferences therein).I am thankful to A.E.Kudryavtsev for careful reading the manuscriptand many valuable remarks. I’m indebted also to A.Gal, B.Z.Kopeliovich, .I.Krivoruchenko, I.K.Potashnikova and to participants of the seminars ofthe INR of RAS for useful discussions and comments. Numerous discussionswith V.I.Nazaruk did not lead to any change of his way of thinking .This work was supported in part by Fondecyt (Chile), grant number1090236. References
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