Proton Decay: Improving the sensitivity through nuclear dynamics?

The kinematics of the decay of a bound proton is governed by the proton spectral function. We evaluate this quantity in 16O using the information from nuclear physics experiments. It also includes a correlated part. The reliability of this evaluation is sufficient to open the possibility of correlated cuts in the missing mass and momentum variables in order to identify the decay events from the bound protons with a possible increase of the signal to noise ratio.


I. INTRODUCTION
Proton decay is an important window for theories beyond the standard model. Several decay channels have been experimentally explored, leading to lower limits for the corresponding proton lifetimes. One of them is the pionic decay mode into a neutral pion and a positron. The signature of a decay process is that the sum of the four momenta of the decay products should reconstitute the proton four-momentum. For a free proton at rest this means a vanishing total three-momentum and a missing mass equal to the free proton mass. The pionic decay experiment at Kamiokande [1,2] is performed with water Cerenkov detectors in such a way that 8 out of 10 of the decaying protons are embedded in an oxygen nucleus, where their spectrum is modified. In a pure shell model description corrections are applied to incorporate the shell model momentum distribution and corresponding binding of the shell model orbits of the oxygen nucleus. However, Yamazaki and Akaishi [3] pointed out that this procedure does not take into account the correlations of the decaying proton with the neighboring nucleons. Using a correlation function deduced from the Reid soft core potential, they evaluated the effect on the invariant mass spectrum which acquires a broad low energy tail representing ≃ 10% of the total decay. It is customary in the decay problem to introduce the missing three momentum P miss = Σ i P i , sum of the momenta of the decay particles, and the missing mass, These quantities also refer to the decaying proton. They are related to the momentum and energy of the residual nucleus which can be in an excited state defined by: P miss = P A−1 and Σ i E i = E * A−1 . Since the state of the residual nucleus for each decay event is not known, there is no strict constraint to identify a proton decay. In the pure shell model case where the smearing is already present, the effect is rather mild and controllable. But as shown in Ref. [3] this is not the complete story. It is the aim of this work to evaluate these distributions. The issue at stake is if the broadening is so big that a substantial fraction of genuine decay event are lost in the background. In other words the lower limit on the proton lifetime deduced from the absence of events within a certain domain in the missing momentum and missing energy variables must take into account the portion of decay events outside this domain. It is therefore useful to have the best possible probability distributions. Our result applies to any decay channel but for illustrative purposes we will often refer to the pionic channel. It is valid as well for neutron decay with disappearance of the hadron.
In order to give a feeling for the importance of the modification introduced by correlations, in terms of particlehole (ph) excitations, correlations translate into the existence of 2p2h excited states mixed into the nuclear ground state. The decay of a correlated particle leaves the nucleus into an excited state with one hole and one particle-one hole, for which we want to evaluate the excitation energy. Beyond the energy associated with one hole creation as evaluated in the shell model, there is the energy of the particle-hole which can be approximated by E ph = P 2 ph /2M where P ph is the momentum exchanged between the correlated pair. Neglecting the momentum of the hole which has a relatively narrow distribution, P ph is also the opposite of the missing momentum.
The missing mass square is then given by where M * is the nucleon mass reduced by the energy necessary for the hole creation. The missing mass square evolves approximately parabolically with P miss . To il-lustrate the expected effect let us take an approximate value M * ≃ 900 MeV. For a typical exchanged momentum P ph = 300 MeV/c, which is also the value of the missing momentum, the missing mass value turns out to be 800 MeV. These two missing values happen to be on the border line of the domain in which Refs. [1,2] interpret an event as a proton decay one (no event in fact fell into this domain). We therefore expect that in the analysis a substantial fraction of the correlated decays escapes detection. Moreover, the future experiments aiming to improve the current limits on the proton decay will have to introduce even tighter cuts to avoid the background due to the atmospheric neutrino interactions making the effects discussed even more important. The above qualitative argument is made quantitative in the next section.

II. PROTON SPECTRAL FUNCTION
Since the decay of the bound proton occurs instantaneously on the scale of nuclear interactions one can express the quantities relevant for the bound proton decay in terms of the nuclear spectral function which describes the probability to find a nucleon in the nucleus with momentum k and produce a residual A − 1 system with excitation energy E after an instantaneous removal of this nucleon. The spectral function is related to the single nucleon momentum distribution as and it is normalized as In order to resolve the spectral function at the high resolution relevant for the proton decay one needs to use probes which transfer large energies and momenta, above 1 GeV, to the nucleons in the nuclei. Such studies were performed in the last few years using proton and electron beams of high energies. It was observed [4] that the ratios of (e, e ′ ) cross sections off nuclei and the deuteron ( 3 He) are independent of x, Q 2 for 1.3 < x < 2 and Q 2 ≥ 1.5 GeV 2 corresponding to the kinematics where the electron can scatter only off the correlated nucleon -nucleon pair with internal momenta ≥ 300 MeV/c. Moreover in (e, e ′ p) or (p, 2p) reactions on nuclei at large Q 2 , a strong correlation was observed between the emission of a fast proton and that of a nucleon (predominantly neutron) in the opposite direction [5,6]. These studies confirmed theoretical expectations of the presence of significant short-range correlations (SRC) in nuclei -for instance in 12 C the probability P12 C to find a nucleon with momentum ≥ 300 MeV/c is a factor of ∼ 5 ± 0.5 larger than in the deuteron. The current models of the deuteron give P D in the range 3 ÷ 4%, and this corresponds to P16 O = .15 ÷ .2. The data also support the expectation that most of this probability is due to the pn -tensor correlations (see e.g. [7,8]), which are a specific case of 2p-2h excitations. For a review and detailed references see [9].
In the many-body models of nuclei with realistic NN potential the high momentum component with momenta between 300 and 600 MeV/c originates from the interplay of attraction and repulsion at distances ≤ 1.2 fm. Hence we have used two spectral functions [10,11] calculated in such models to analyze the effect of the nuclear structure on the detection of the bound proton decay.
For the purposes of the analysis of the proton decay events it is convenient to choose as variables the threemomentum of the decaying proton and the square of the bound proton mass V = M 2 miss which fixes its offshellness. The two spectral functions of Ref. [10,11] have an uncorrelated part, S 0 (P miss , E), and a correlated one, In the first model we use S 0 as calculated with the Skyrme force and renormalized by a factor 0.8. The correlated part, S 1 represents 20% of the total spectral function and it is given by the model of [10]. In this model where k 0 = 300 MeV/c; (the second model we considered provides the same result as far as this quantity is concerned). The ratio of 16 O and deuteron high momentum components in these models varies in the range of 3 ÷ 6 for 300 < k < 600M eV /c which is rather close to the value of the ratio ∼ 5 obtained from the analysis of the hard phenomena and in particular x > 1 data (see the review in [9]). Smaller value of the total probability than in a phenomenological estimate is mainly due to a later onset of the dominance of the short-range correlation regime. The model of Ref. [10] for the correlated part of the spectral function is based on the notion of the factorization of the two-body momentum distribution for high values of the relative and small values of the center of mass momenta of the pair and it is valid in this regions; this factorization was justified within a many-body approach in Ref. [12] and shown to hold for 16 O within the many-body calculation of Ref. [7]. It also gives a correct dependence for the center of mass of the correlated pair, as measured in Refs. [5] and [6]. We choose for the relative motion of the pair in the two-body momentum distribution a parametrization which reproduces well the high momentum tail of the deuteron in the region of interest and leads to a good description of the high momentum tail of n16 O (k) [13,14].
The spectral function of Ref. [11] has been obtained within the Local Density Approximation [15], in which the (e, e ′ p) data on single nucleon knock-out at low missing energy [16] is combined with the results of accurate theoretical calculations of the nuclear matter spectral function at different densities [17]. A direct measurement of the correlation component of the spectral func- tion of 12 C, obtained measuring the (e, e ′ p) cross section at missing momentum and energy up to ∼ 800 MeV and ∼ 200 MeV, respectively, has been recently carried out at Jefferson Lab by the E97-006 Collaboration [18]. The data resulting from the preliminary analysis appear to be consistent with the theoretical predictions based on the spectral function of Ref. [11]. The quantity P 2 miss S 0 (P miss , V ) is shown in Figs. 1, 2 in three dimensional plots, and P 2 miss S 0 (P miss , M miss ) is shown in Figs. 3, 4 in contour plots, for the two considered models. The strength is concentrated over three or four stripes in the P miss , M miss plane in the two cases; they correspond to the occupied shells of 16 O: the P 1/2 , P 3/2 and S 1/2 states in the case of calculation with the Skyrme force and to an additional P 3/2 state in the case of Ref. [11].
The energies and widths of the occupied states in the two models we considered are the following. In the first model, as calculated from the Hartree-Fock Skyrme model with shell model parameters which describe the (p, 2p) and (p, pn) data of Refs. [19,20], they are 12.06 MeV with a width ≃ 5 MeV for the P 1/2 state, the P 3/2 state has 18.63 MeV with a width of 5 MeV; the S state is quite broad with a width of ≃ 40 MeV for an energy The behavior in the momentum P miss can be inferred from the expression of V which in the uncorrelated case is: Expanding the square in Eq. (5) and neglecting the P 4 miss term we obtain: where ǫ α are the values of the proton shells energies. The coefficients C α are 1.07 and 1.06 for the P and S proton shells, respectively. As for the correlated spectral function, P 2 miss S 1 (p miss , V ) is represented in Figs. 1, 2 in three dimensional plots, and P 2 miss S 1 (p miss , M miss ) is  [10], P 2 miss S1(Pmiss, Mmiss). We show with black solid lines the cut Pmiss < 250 MeV and V < 640 MeV, quoted in Refs. [1] and [2] in both panels.
shown in Figs. 3, 4 in contour plots; it has a similar behavior than the corresponding uncorrelated quantity but it is broader. One can see from Figs. 3, 4 that the correlated spectral functions in the two models exhibit differences; a detailed comparison of the two spectral functions is out of the scope of the present paper. Nevertheless, these differences do not affect our conclusions, as appears in the following, since most of the strength is concentrated along the stripe of maximum strength, even if the second model is more peaked at low momenta and it is more narrow around the center of the stripe. The center of the corresponding stripe obeys the following equation in the V, P miss plane for both models: where V is expressed in GeV 2 and P miss in GeV/c, which is close to the approximate expression of Eq. (1).
The concentration of the strength of the spectral func-  Fig. 3, but with the model spectral function of Ref. [11].
tion in limited regions of space which project in some bands in the M miss , P miss plane suggests a complementary analysis of the data specifically aimed at the decay of the 16 O protons. It consists in the following: to look for events which, in this plane, fall in one or several, depending of the accuracy of the data, regions of this plane selected to cover the lines of the maximum of the (uncorrelated or correlated) strengths defined in Eqs. (6,7) so as to maximize the number of significant events while minimizing the background, i.e., the area. Correlated events can also be included in this way. The calculated proton spectral function in 16 O is sufficiently reliable, as it is established in connection with various nuclear physics experiments, to allow for this possibility.
This kind of analysis, if feasible, precludes the subsequent distortions of the pion kinematics after emission by the proton. We will comment later on that. A technical remark: for the correlated part, when transforming coordinates from (P miss , E) to (P miss , V ) as in Eq. (5), we impose that V stays positive, so we forbid a certain region of E, P miss space to be accessible; this in turn means the normalization of the correlated spectral function is not The total spectral function, S0 + S1, within the two considered models. Top: the mixed model of S0 from the Skyrme force and S1 from Ref. [10]; bottom: the model of Ref. [11].
We have checked that integrating over negative values of V gives the missing normalization, 0 −∞ dV dP miss S 1 (P miss , V ) = 0.02. In Fig. 5 we show the total spectral function S 0 + S 1 , in the two considered models, as a function of P miss and M miss , while in Fig. 6 we present the normalization integral of the spectral function It represents the number of events lost by applying a cut on the missing mass such that only the events which correspond to a missing mass larger than this particular value M miss are kept, irrespective of the momentum.  [10] and the red curves correspond to the model of Ref. [11]; separate contributions are shown for the uncorrelated/correlated part within both models. 4 and 5, the number of nuclear events is reduced by a factor of N = 0.83 using the model of Ref. [10] and by a factor of N = 0.80 using the model of Ref. [11]. However, future experiments are likely to have to introduce tighter cuts in order to reduce the background from the interactions of atmospheric neutrinos. If, for example, the cut √ V > 900 MeV is imposed, ≃ 44% of the events are removed, namely 26% (25%) from uncorrelated events and 18% (19%), the near totality of correlated events in the first (second) considered model. With a tight constraint on M miss the fraction lost is quite appreciable if no other precaution is taken. The correlated analysis that we discussed may allow a better efficiency.
There are other effects which reduce the contribution of the bound nucleon decays. This includes a reduced phase volume which is ∝ √ V for decays with production of light particles. Furthermore the very mechanism of the proton decay may be sensitive to the nuclear correlations. For example, if the decay amplitude is proportional to the three quark wave function at the origin, see e.g. [21], the effect of suppression of the point-like configurations in bound nucleons [22,23] would contribute, reducing the rate of the decay by about 14% for the M p − √ V = 100 MeV cut.
All the effects that we have discussed are genuine medium effects on the decay amplitude. They are not the whole story. The subsequent history of the pion, rescatterings / absorption in oxygen further reduces the number of "observable" pions. The inelastic scattering of pions clouds the message on the kinematics since the inelastically scattered pion ejects a nucleon. The corresponding point in the M miss , P miss plane would be likely to fall outside the interesting regions delimited from the proton spectral function that we discussed in this work. Therefore for an analysis of the type suggested in this work, inelastically scattered pions may be considered as lost events. It may represent a reduction factor of about 0.6. This is usually taken care of through a Monte Carlo evaluation which is beyond the scope of the present work.
In conclusion we have introduced, in the problem of the identification of the decay events of the protons bound in the oxygen nucleus, the use of proton spectral function. It allows the prediction of the location in the M miss , P miss plane of the decay events. Our spectral function has an uncorrelated and a correlated part. It is has been tested against a number of nuclear physics experiments and the reliability of our prediction is sufficient to be exploitable. We considered two models for the spectral function and the conclusions on the correlated cuts holds even if the two models exhibit some difference for the spectral functions. It appears that for the future nucleon decay experiments with tight cuts on the mass of the products of the proton decay it may be interesting to consider, as a complementary information for the decay events of the oxygen protons, correlated cuts on the mass and missing momentum obtained from the spectral function in order to decrease the background to signal ratio. The price to pay for this type of analysis is the loss of decay events where a pion produced in the decay is inelastically scattered, which clouds the reconstitution of the proton spectral function in oxygen. The loss in intensity however is moderate and it may be compensated by the advantage of a decrease of the background.