Improved Lower Bounds on Partial Lifetimes for Nucleon Decay Modes

In the framework of a baryon-number-violating effective Lagrangian, we calculate improved lower bounds on partial lifetimes for proton and bound neutron decays, including $p \to \ell^+ \ell'^+ \ell'^-$, $n \to \bar\nu \ell^+ \ell'^-$, $p \to \ell^+ \nu\bar\nu$, and $n \to \bar\nu \bar\nu \nu$, where $\ell$ and $\ell'$ denote $e$ or $\mu$, with both $\ell = \ell'$ and $\ell \ne \ell'$ cases. Our lower bounds are substantially stronger than the corresponding lower bounds from direct experimental searches. We also present lower bounds on $(\tau/B)_{p \to \ell^+\gamma}$, $(\tau/B)_{n \to \bar\nu \gamma}$, $(\tau/B)_{p \to \ell^+ \gamma\gamma}$, and $(\tau/B)_{n \to \bar\nu \gamma\gamma}$. Our method relies on relating the rates for these decay modes to the rates for decay modes of the form $p \to \ell^+ M$ and $n \to \bar\nu M$, where $M$ is a pseudoscalar or vector meson, and then using the experimental lower bounds on the partial lifetimes for these latter decays.


I. INTRODUCTION
Although the Standard Model (SM) conserves baryon number 1 B, this is violated in many of its ultraviolet extensions. This violation is natural in grand unified theories (GUTs) [2]- [5], since these theories place quarks and (anti)leptons in the same representation(s) of the GUT gauge group. More generally, baryon number violation (BNV) is expected to occur in nature, because this is one of the necessary conditions for explaining the observed baryon asymmetry of the universe [6]. A number of dedicated experiments have been carried out since the early 1980s to search for proton decay and the decay of neutrons bound in nuclei. These experiments have obtained null results and have set resultant stringent upper limits for the rates of nucleon decays 2 .
In this paper, within the framework of a baryonnumber-violating effective Lagrangian, L ef f , we shall calculate improved lower bounds on partial lifetimes for a number of nucleon decays. Let us denote the rate for the decay of a nucleon N (where N = p or n) to a final state f.s. as Γ N →f.s. , which is the inverse of the partial lifetime, (τ /B) N →f.s. = [Γ N →f.s. ] −1 , where B denotes the branching ratio for this decay mode. Our method is to derive an approximate relation between the rate Γ N →f.s. for the decay of a nucleon to a final state f.s., N → f.s. (where N = p or n) and the rate for the decay Γ N →f.s. ′ to a different final state, denoted f.s. ′ . Combining this relation with the experimental lower bound on (τ /B) N →f.s. , we derive an approximate lower bound on (τ /B) N →f.s. ′ for each final state f.s. ′ . Our theoretical framework is minimal, in the sense that the only physics beyond the SM (BSM) is that is assumed is that which produces the basic set of local four-fermion operators in L ef f . Although the lower bounds that we derive are only approximate, they are useful because for 1 Recall that the violation of B by SU(2) L instantons in the SM is negligibly small at temperatures low compared with the electroweak scale [1]. 2 We shall use the term "nucleon decay" to mean the decay of a proton or the B-violating decay of a neutron bound in a nucleus.
many final states f.s. ′ they are more stringent than the lower bounds on (τ /B) N →f.s. ′ from direct experimental searches.
This paper is organized as follows. In Sect. II we discuss the four-fermion operators in L ef f . We present our lower bounds on (τ /B) for p → ℓ + γ and n →νγ in Sect. III. Sects. IV and V contain our lower bounds on (τ /B) for p → ℓ + ℓ ′+ ℓ ′− and n →νℓ + ℓ − , where ℓ and ℓ ′ denote e or µ, including both ℓ = ℓ ′ and ℓ = ℓ ′ cases. In Sects. VI and VII we derive lower bound bounds on (τ /B) for p → ℓ + νν and n →ννν. Here and below we use a symbolic notation in which ν may refer to an electroweak-doublet neutrino of some generation, or to an electroweak-singlet neutrino; the context will make clear the meaning. In Sect. VIII we remark on the application of our method to other decays, including p → ℓ + γγ and n →νγγ. Our conclusions are presented in Sect. IX and some relevant phase-space formulas are given in Appendix A.

II. EFFECTIVE LAGRANGIAN
Given the established experimental upper bounds on the rates for nucleon decays, it follows that the mass scale(s) characterizing the baryon-number-violating physics responsible for these days must be larger than the electroweak-symmetry-breaking (EWSB) scale, v ≃ 250 GeV. Hence, one can analyze these decays using an effective Lagrangian, L ef f , that is invariant not only with respect to color SU(3) c , but also with respect to gauge transformations of the electroweak gauge group, G EW = SU(2) L ⊗ U Y . With the original SM fermions, before the addition of any electroweak-singlet neutrinos, the fourfermion operators O i in L ef f that contribute to nucleon decays are as follows. We denote Q α a,L = Q 1α , where α, β, γ are SU(3) c indices, and a is a generation index, with d α and where C is the Dirac charge conjugation matrix satisfying Cγ µ C −1 = −(γ µ ) T , C = −C T and i, j, k, m are SU(2) L indices. As noted in [7], four-fermion operators with bilinears involving Dirac vector and tensor operators γ µ and σ µν = (i/2)[γ µ , γ ν ] can be transformed to the operators listed above via Fierz identities. These operators have ∆B = −1 and ∆L = −1, where L denotes total lepton number. After the discovery of nonzero neutrino masses and lepton flavor mixing, a natural generalization of the Standard Model has involved the introduction of a set of electroweak-singlet neutrinos ν s,R , s = 1, ..., n s , which are necessary to form Dirac neutrino mass terms via Yukawa couplings 3 a=1 ns s=1 y as [L a,L ν s,Rφ ] + h.c., whereφ ≡ iσ 2 φ * and φ = φ + φ 0 is the SM Higgs doublet. These ν s,R neutrinos also generically form Majorana bare mass terms ns s, s,s ′ ν T s,R Cν s ′ ,R + h.c., thereby explicitly breaking total lepton number L by 2 units. With the inclusion of these ν s,R , there are two additional types of operators for nucleon decay, namely and The generation indices (a 1 , a 2 , a 3 , a 4 ) in the O r with 1 ≤ r ≤ 4 and the indices (a 1 , a 2 , a 3 , s) in O r with r = 7, 8 will be left implicit in the notation. In terms of these fields, a minimal low-energy effective Lagrangian giving rise to nucleon decay can be written as where the second sum is over all of the generation indices (a 1 , a 2 , a 3 , a 4 ) in the operators O r , 1 ≤ r ≤ 4 and the 3 Two other operators vanish identically in the case a 1 = a 2 = a 3 = 1 relevant for nucleon decay, namely indices (a 1 , a 2 , a 3 , s) in O 7 and O 8 . Since these operators O r have Maxwellian dimension 6 (in mass units), the coefficients c r have dimension −2, and we write c r =c r /(M BN V ) 2 , where M BN V denotes an effective mass scale characterizing the baryon-number violation. In general,c r depends on the generational indices of fermion fields in O r ; this is again left implicit in the notation.
III. THE DECAYS p → ℓ + γ AND n →νγ We begin by deriving approximate lower limits, within this theoretical framework, on the partial lifetimes for the decays p → ℓ + γ and n →νγ, where ℓ + = e + or µ + and ν may be an electroweak-nonsinglet antineutrino,ν e ,ν µ , orν τ , or an electroweak-singlet antineutrino,ν s where 1 ≤ s ≤ n s . 4 In view of these possibilities, we omit a subscript onν here and in similar cases below. With our L ef f , the leading contributions to the decay p → ℓ + γ arise from the diagrams in Fig. 1. Fig. 1(a) shows a process in which r c r O r in L ef f (represented by the blob at the four-fermion vertex) with 1 ≤ r ≤ 4 and a 1 = a 2 = a 3 = 1, a 4 = 1 or 2 for ℓ + = e + or µ + , transforms an initial uu pair in the proton to ℓ + d c , and the d c annihilates with the d in the proton to produce the outgoing photon. This figure is also understood to include a process in which the d quark in the proton emits the photon, transitioning to a virtual d that undergoes the BNV process depicted by the blob, resulting in the outgoing ℓ + . Fig. 1(b) shows the analogous processes involving the BNV transformation du → ℓ + u c . In Fig.  1(c), the initial uud quarks in the proton are transformed via the BNV L ef f to a virtual s-channel ℓ + that then radiates the photon.
To proceed, we relate the amplitude for the p → ℓ + γ decay to an amplitude for p → ℓ + M decay, where ℓ + = e + or µ + and M denotes a neutral meson containing light quarks, such as π 0 , η, ρ 0 , or ω. Feynman diagrams for the decay p → ℓ + M are shown in Fig. 2. Both the isoscalar mesons η and ω and the isovector mesons π 0 and ρ 0 are relevant for this relation, since the electromagnetic current has both isoscalar and isovector parts, as embodied in the relation Q em = I 3 +(Y /2), where here I and Y denote isospin and hypercharge. The final states involving the vector mesons ω and ρ 0 share with the final state involving the photon the property that they are all vector (J = 1) particles. On the other hand, the final states involving the π 0 and η have (mass) 2 values 0.0182 GeV 2 and 0.301 GeV 2 that are smaller than the (mass) 2 values of 0.602 GeV 2 and 0.613 GeV 2 for the ω and ρ 0 bnd. , for various proton decays of the form p → ℓ + M , where ℓ + = e + or µ + and M = π 0 , η, ρ 0 , ω, with references, given in the first to third columns; (b) in the fourth column, our resultant estimated lower bounds on (τ /B) for p → ℓ + γ obtained from Eq. (9), denoted (τ /B) p→ℓ + γ;est. l.bnd. . All limits on (τ /B) are given in units of 10 33 yr. See text for further discussion.
and hence closer to the zero mass of the photon. In view of the complementary similarities (in spin and (mass) 2 ) of the ω and ρ 0 hadronic final states, on the one hand, and the π 0 and η hadronic final states, on the other, to the photon, we shall use all of these decay modes for our comparison. We list the experimental lower bounds on the partial lifetimes (τ /B) for relevant proton and bound neutron decay modes in Tables I and II. These and other limits listed here are at the 90 % confidence level.  The phase space factor for a decay of a nucleon N to a two-body final state with particles of masses m 1 and m 2 is given by Eq. (A5) in Appendix A. We list the values of (8π)R Table III for nucleon decays to various final states f.s. of relevance here.
Because the hadronic matrix elements M |L ef f |p and 0|L ef f |p that enter in the respective p → ℓ + M and p → ℓ + γ decays are different and the coefficients c r that enter into L ef f depend on the UV completion of the Standard Model that is responsible for the baryon-number violation, we will restrict our analysis to a rough esti- mate of the relation between the corresponding decays 5 A similar comment applies to n →νγ decays. We have where ℓ + = e + or µ + and e 2 = 4πα em is the squared electromagnetic coupling. The ratio of phase-space factors is included to take account of the difference in phase space for the p → ℓ + γ and p → ℓ + M decay modes. Equivalently, Note that although the branching ratios for the various decay modes in Eq. (9) depend on the UV completion of the SM, the basic relation (8) is between the absolute rates themselves, which does not depend on these branching ratios. We next make use of the experimental lower bounds on the partial lifetimes for various relevant proton decay modes, as displayed in Table I. These lower bounds on (τ /B) are all from the SuperKamiokande (SK) experiment; the bounds for p → ℓ + π 0 and p → ℓ + η are from Ref. [10], while the bounds for p → ℓ + ω and p → ℓ + ρ are from Ref. [11]. In the right-most column of Table I, we list our estimates for the lower bounds on p → ℓ + γ 5 For lattice calculations of hadronic matrix elements, see Ref. [9].
obtained by combining the relation (9) with the experimental lower bounds on (τ /B) p→ℓ + M given in the middle column, for ℓ + = e + and ℓ + = µ + . In view of the approximate nature of our estimated lower bounds on (τ /B) p→ℓ + γ in Table I, we list these only to one significant figure, and we follow this format with our estimates for other nucleon decay modes below.
We proceed to carry out the corresponding analysis for the decay n →νγ. The leading Feynman diagrams contributing to this decay are shown in Fig. 3. As noted above, since the antineutrino is not observed, it could either be an electroweak-nonsinglet,ν ℓ (i.e.,ν e , ν µ , orν τ ) or an EW-singlet,ν s . The leading contributions to the n →ν ℓ γ decay arise from the rele- Our procedure is again to obtain an approximate relation between the rates for n →νγ and for n →νM . In Figs. 4(a) and 4(b), we show diagrams contributing to the decay n →νM . By the same method as we used above, we obtain the approximate estimate We make use of the experimental lower bounds on (τ /B) for relevant decays n →νM , displayed in Table II. 6 In the right-most column of Table II, we list our estimates for the lower bounds on n →νγ obtained by combining the relation (10) with the experimental lower bounds given in the middle column. As is again evident from this table, our approximate lower bounds on (τ /B) n→νγ using the experimental limits on n →νπ 0 and n →νη are stronger than the bound from a direct experimental search, which is (τ /B) n→νγ > 0.550 × 10 33 yr [15].
In this section we calculate estimated lower bounds on the partial lifetimes of several proton decays of the form p → ℓ + ℓ ′+ ℓ ′− , where ℓ and ℓ ′ denote e or µ, including both of the cases ℓ = ℓ ′ and ℓ = ℓ ′ . Graphs for 6 Since the experiments do not observe theν, these experimental bounds are more general. For example, the lower bound on the partial lifetime for n →νπ 0 actually applies to any decay of the form n → x 0 π 0 , where x 0 is a neutral, weakly interacting particle or antiparticle that does not decay in the detector, and similarly for the other decay modes n → x 0 M . These subsume the case where x 0 =ν or x 0 = ν.
the above-mentioned decays are shown in Fig. 5. As discussed above, our theoretical framework for this and our other estimates is a minimal one in which we assume only the baryon-number-violating physics beyond the Standard Model that gives rise to L ef f . (If one were to assume other BSM physics involving new particles with masses much smaller than the GUT scale, then other graphs would become relevant (e.g., [17]) We denote the four-momenta of the p, ℓ + , ℓ ′− , and ℓ ′+ as p, p 3 , p 2 , and p 1 , respectively, and set q = p 1 + p 2 = p − p 3 . Let us first consider the qq annihilation producing a virtual photon, as indicated in Figs. 5(a) and 5(b). If the total angular momentum of the qq system is J = 0, then the matrix element for this sub-process is proportional to q λ [ū(p 2 )γ λ v(p 1 )] (whereū(p 2 ) and v(p 1 ) are Dirac spinors), which vanishes. In the corresponding terms involving Z exchange, if the angular momentum J of the qq subsystem is zero, then the matrix element for the sub-process is proportional to q λ [ū(p 2 )γ λ {−(1/2)P L + sin 2 θ W }v(p 1 )], where P L = (1 − γ 5 )/2. In this case, the vector part of the neutral current gives zero contribution and the axial-vector part gives a small term ∝ 2m ℓ ′ [ū(p 2 )γ 5 v(p 1 )], Thus, the dominant contributions from these graphs are expected to arise from qq configurations with J = 1. For these J = 1 terms, the contributions to the amplitude from the graphs with a virtual photon are expected to dominate over the contributions from the graphs with a virtual Z by a factor ∼ e 2 /(G F m 2 p ) ∼ 10 3 . The contribution from the virtual photon in the graph of Fig. 5(c) is similarly dominant over that from the virtual Z. It is thus natural to use, for comparison, proton decays to vector mesons, p → e + ρ 0 and p → e + ω. In contrast to the case with the p → ℓ + γ decays, this comparison connects a decay to a two-body final state to a decay with a three-body final state. Because the integral (A1) for the decay into a three-body final state involves details of L ef f in a nontrivial integration, while the corresponding integral for the decay into a two-body final state only involves a trivial angular integration (since the magnitudes of the 3-momenta of the two final-state particles are fixed), it is difficult to make a precise comparison between the rates for these two decays. For a rough approximation, we will simply take into account the differences in phase space for these decays, via the ratios of the two-body and three-body phase-space factors. Since the three-body phase space factor has dimensions of (mass) 2 and the mass scale is set by the initial nucleon mass, we will introduce a dimensionless three-body phase space factor for a decay to a given final state f.s., denotedR We are thus led to the estimate (11) or equivalently, Substituting the experimental lower bounds (τ /B) p→e + ρ 0 > 0.720 × 10 33 yr. and (τ /B) p→e + ω > 1.60 × 10 33 yr. [11] in Eq. (12) and taking account of the ratios of phase-space factors, we obtain estimated lower bounds on (τ /B) p→e + e + e − of 0.9 × 10 37 yr. and 2 × 10 37 yr, respectively. These are much stronger than the lower bound on the partial lifetime for this decay from a direct experimental search, which is [13] (τ /B) p→e + e + e − > 0.793 × 10 33 . Substituting the experimental lower bounds (τ /B) p→µ + ρ 0 > 0.570 × 10 33 yr. and (τ /B) p→µ + ω > 2.80 × 10 33 yr. [11] in Eq. (12) and computing the ratios of phase-space factors, we obtain estimated lower bounds on (τ /B) p→µ + e + e − of 0.6 × 10 37 yr and 3 × 10 37 yr, respectively. We conservatively list these as (τ /B) p→µ + e + e − > ∼ 10 37 yr in Table IV. Again, these are much stronger than the experimental lower bounds from direct searches, namely (τ /B) p→µ + e + e − > 0.529 × 10 33 and (τ /B) p→µ + µ + µ − > 0.675 × 10 33 [13]. These estimated lower bounds on partial lifetimes are summarized in Table IV, in comparison with the current lower bounds from direct experimental searches.
With regard to these and other nucleon decay modes for which our estimates yield lower bounds on the partial lifetimes that are much greater than existing bounds from direct experimental searches, we stress that this does not mean that these decay modes are not worth searching for in further experiments. If, for example, in the future, the decays p → e + π 0 and p → e + e + e − are both observed and the value of (τ /B) p→e + e + e − is significantly lower than a range estimated from Eq. (12), this would be doubly interesting, as evidence not only of baryon number violation incorporated in L ef f , but also of additional relevant physics beyond the Standard Model. This comment also applies for the other nucleon decay channels to be discussed below, for which our lower bounds on partial lifetimes are much higher than the lower bounds from direct experimental searches.

V.
n →νℓ + ℓ − AND n →νℓ + ℓ ′− DECAYS In this section we analyze the neutron decays n → νℓ + ℓ − and n →νℓ + ℓ ′− , where ℓ, ℓ ′ = e, µ. Feynman diagrams for n →νℓ + ℓ − decays are shown in Fig. 6. In the graphs in Figs. 6(a) and 6(b), theν could be an EW-doublet or an EW-singlet antineutrino, while in the graphs of Figs. 6(c) and 6(d), theν =ν ℓ is an EWdoublet antineutrino. Since an experiment would not observe theν, it would not distinguish between these possibilities. In the graphs of Figs. 6(a)-6(c), the charged (anti)leptons are of the same generation, while in Fig.  6(d), ℓ ′− may be of a different generation than ℓ + . As discussed before, the contributions of the diagrams with a virtual Z in Figs. 6(a)-6(c) are very small compared with the contributions of the diagrams of Figs. 6(a) and 6(b) with a virtual photon. This is also true of the graph with a virtual W in Fig. 6(d). Therefore, by arguments similar to those used for the analysis of p → ℓ + ℓ ′+ ℓ ′− decays, we estimate For the same reasons discussed in connection with the p → ℓ + ℓ ′+ ℓ ′− decays, when applying the relation (13), we will use n →νM decays with M being a vector meson, ρ 0 or ω. As an illustrative example, we consider the decay n →νe + e − . Since m 2 e /m 2 n = 3.0 × 10 −7 is negligibly small, it follows that, to very good accuracy,R 3 =R 3,0 (see Eq. (A10). Therefore, Eq. (13) takes the explicit form Of the two n →νM decay channels, the experimental lower bound on the channel with M = ω is the stronger one, so we focus on it. Substituting the lower bound (τ /B) n→νω > 1.08 × 10 32 from Ref. [13] in Eq. (13) and evaluating Eq. (14), we obtain the estimated lower bound (τ /B) n→νe + e − > ∼ 1.2 × 10 36 yr. This is much stronger than the lower bound from a direct search, namely (τ /B) n→νe + e − > 2.57 × 10 32 yr.
Next, again using the same experimental lower bound on (τ /B) n→νω and computing the ratio of phase-space factors using Eqs. (A5) and (A13) in Appendix A, we obtain the estimated lower bound (τ /B) n→νµ + µ − > ∼ 1.6 × 10 36 yr. This is a much more stringent lower bound than the one from a direct experimental search, which is (τ /B) n→νµ + µ − > 0.79 × 10 32 yr.
Finally, we discuss the decay n →νe ± µ ∓ . Only the Feynman diagram of Fig. 6(d) contributes to this decay, so we obtain the estimate Using the experimental lower bound on (τ /B) n→νω again, we obtain the estimate (τ /B) n→νe ± µ ∓ > ∼ 1 × 10 44 yr. This is much stronger than the bound from the direct search for a decay of this type, namely (τ /B) n→νµ + e − > 0.83 × 10 32 yr. [13].
In this section we derive an estimated bound for several different types of proton decays which are experimentally indistinguishable, namely (i) p → ℓ + ν ℓ ′ν ℓ ′ ; (ii) p → ℓ + ν ℓνℓ ′ ; and (iii) p → ℓ + ν ℓνs . We will refer collectively to these as p → ℓ + νν. Experimentally, these are all of the form p → ℓ + + missing, where "missing" denotes two neutral weakly interacting particles, antiparticles, or a particle-antiparticle pair, which do not decay in the detector. Experimental papers often use the symbolic notation p → ℓ + νν for all of these decays. Graphs that contribute to the decays (i)-(iii) are shown in Fig.  7. The graphs in Figs. 7(a)-7(c) contain an internal Z line and contribute to decays of type (i) and the subset of the decays of type (ii) in which ν ℓ = ν ℓ ′ . The graphs in Figs. 7(d) and 7(e) contain an internal W line and contribute to decays of the form (ii) and (iii), depending on whether theν emitted from the effective four-fermion BNV vertex is an EW-doublet antineutrino or an EWsinglet antineutrino.

VII. n →ννν
Finally, we consider the decays of the generic form n →ννν. Graphs that contribute to these decays are shown in Fig. 8. In the processes depicted in Figs. 8(a) and 8(b), theν that emanates from the BNV fourfermion vertex can be either an EW-doublet antineutrino (of any flavor) or an EW-singlet antineutrino, while the νν pair produced by the virtual Z are EW-doublet (anti)neutrinos. In Fig. 8(c), the two antineutrinos and the neutrino are all of EW-doublet type. We estimate FIG. 6: Feynman diagrams for n →νℓ + ℓ − and n →νℓ + ℓ ′− with ℓ, ℓ ′ = e, µ.
Using the experimental lower limit (τ /B) n→νω > 1.08 × 10 32 yr [13], as listed in Table I, we obtain the estimate (τ /B) n→ννν > ∼ 10 44 yr. This is much stronger than the lower bound from a direct experimental search, namely (τ /B) n→inv. > 0.58 × 10 30 yr [19]. This experimental lower bound was set by the KamLAND experiment from a search for gamma rays from the de-excitation of the 11 C nucleus that would result from the n → inv. decay of FIG. 8: Feynman diagrams for n →ννν decay. a neutron in a 12 C atom in its liquid scintillator detector.

VIII. OTHER DECAY MODES
Our method can also be applied to other nucleon decay modes, such as p → ℓ + γγ and n →νγγ, where here the γγ part of the final state are "continuum" photons, i.e., not photons that arise from a cascade decay such as p → ℓ + π 0 followed by π 0 → γγ or p → ℓ + η followed by η → γγ. Graphs for these decays with continuum diphotons are obtained from those for p → ℓ + γ in Fig.  1 and for n →νγ in Fig. 3 and hence are not shown separately. Using the same methods as before, we obtain and Using the experimental lower bounds on p → e + M and p → µ + M listed for M = ρ 0 , ω in Table I, the more stringent of which are for p → ℓ + ω, we obtain the estimates (τ /B) p→e + γγ > ∼ 2×10 37 yr. and (τ /B) p→µ + γγ > ∼ 3×10 37 yr. An experimental lower bound from a direct search is p → e + γγ decay mode, namely (τ /B) p→e + γγ > 1.0×10 32 yr. [20]. Our estimated lower bound is much stronger than this direct limit. We are not aware of any published experimental lower bound on (τ /B) p→µ + γγ .
By similar methods, we obtain the estimated lower bound (τ /B) n→νγγ > ∼ 10 36 yr. An experimental lower bound is (τ /B) n→νγγ > 2.39 × 10 32 yr from the IMB3 experiment [13]. This was an inclusive search for any events of this type, which also allowed for the possibility that the invariant diphoton mass was equal to m π 0 or m η to within the detector resolution [21]. Our estimated lower bound is again much stronger than this direct limit. These results are summarized in Table IV. One can also apply these techniques to relate other baryon-violating processes to each other. We have done this to derive improved bounds on certain ∆B = −2 dinucleon decays. These results are reported elsewhere.

IX. CONCLUSIONS
In this paper we have calculated estimated lower bounds on the partial lifetimes for several nucleon decays, including p → ℓ + ℓ ′+ ℓ ′− , n →νℓ + ℓ ′− , p → ℓ + νν, and n →ννν, where ℓ and ℓ ′ denote e or µ. We assume a minimal theoretical framework in which the only physics beyond the SM is that which produces the fourfermion operators in the baryon-number-violating effective Lagrangian responsible for these nucleon decays. Our method relies on relating the rates for these decay modes to the rates for decay modes of the form p → ℓ + M and n →νM , where M is a pseudoscalar or vector meson, and then using the experimental lower bounds on these latter decays. Although our estimates are rough, our lower bounds are substantially stronger than lower bounds on the partial lifetimes for these decays from direct experimental searches. We also present corresponding estimated lower bounds on partial lifetimes for the radiative decays p → ℓ + γ, n →νγ, p → ℓ + γγ, and n →νγγ. There are strong motivations for pushing the search for nucleon decay to greater sensitivity in many channels. It is hoped that this search will be carried out with current data and with future nucleon decay experiments.

Appendix A: Phase Space Formulas
In general, the decay rate of a parent particle N with four-momentum p satisfying p 2 = m 2 N to a final state (f.s.) consisting of n particles with four-momenta p i , 1 ≤ i ≤ n, is given by where a sum over polarizations of final-state particles and an average over the polarizations of the parent particle are understood; A N →f.s. denotes the amplitude for the decay; S is a symmetry factor to take account of possible identical particles in the final state; and the integration over the n-body final-state phase space is given by (A2) For the nucleon decays of interest here we have m N with N = p or N = n, and we denote p 2 i = m 2 i . It is useful to consider the phase space integration by itself, defining an n-body phase space factor R n as R n = dR n (A3) For our applications, we will sometimes want to explicitly indicate the final-state particles, and for this purpose, we will use the notation R (f.s.) n . For example, for p → e + π 0 , this phase space factor is written as R (e + π 0 ) 2 , and so forth for other decays.
Numerically, the right-hand side of Eq. (A12) has the value 0.889 for m = m µ with parent particle p or n.