Inclusive Nucleon Decay Searches As a Frontier of Baryon Number Violation

UCI-TR-2019-24

Inclusive Nucleon Decay Searches as a Frontier of Baryon Number Violation

Julian Heeck1, ∗ and Volodymyr Takhistov2, † 1Department of Physics and Astronomy, University of California, Irvine Irvine, California, 92697-4575, USA 2Department of Physics and Astronomy, University of California, Los Angeles Los Angeles, California, 90095-1547, USA Proton decay, and the decay of nucleons in general, constitutes one of the most sensitive probes of high-scale physics beyond the Standard Model. Most of the existing nucleon decay searches have focused primarily on two-body decay channels, motivated by Grand Uniﬁed Theories and supersymmetry. However, many higher-dimensional operators violating baryon number by one unit, ∆B = 1, induce multi-body nucleon decay channels, which have been only weakly constrained thus far. While direct searches for all such possible channels are desirable, they are highly impractical. In light of this, we argue that inclusive nucleon decay searches, N → X+anything (where X is a light Standard Model particle with an unknown energy distribution), are particularly valuable, as are model-independent and invisible nucleon decay searches such as n → invisible. We comment on complementarity and opportunities for such searches in the current as well as upcoming large- scale experiments Super-Kamiokande, Hyper-Kamiokande, JUNO, and DUNE. Similar arguments apply to ∆B > 1 processes, which kinematically allow for even more involved ﬁnal states and are essentially unexplored experimentally.

CONTENTS I. INTRODUCTION

I. Introduction 1 Baryon number B and lepton number L are seemingly accidentally conserved in the Standard Model (SM), II. Nucleon decay operators 2 which makes searches for their violation extremely impor- A. Operator dimension d = 6 2 tant. So far we have not observed any B or L violating B. Operator dimension d > 6 3 processes despite decades of experimental investigation, C. UV-complete example 5 yet there are many reasons to expect that these symmetries could be broken. The linear combination B +L is in III. Exclusive nucleon decay searches 6 principle already violated by 3 + 3 units within the SM itself through non-perturbative instanton effects, albeit IV. Inclusive nucleon decay searches for ∆B = 1 highly suppressed [1]. From a more fundamental top- processes 8 down perspective, global symmetries such as U(1) and A. Model-independent and invisible mode B U(1) are expected to be violated at some level by quan- searches 8 L tum gravity effects [2, 3], which is however difficult to 1. Isotope fission products 9 quantify. Furthermore, baryon number violation is one 2. Nuclear de-excitation emission 10 of the key prerequisites for successful baryogenesis [4], B. N X+anything 11 which would address the observed baryon–antibaryon 1. →N asymmetry of our Universe. Explicit B-violation and as- (e±→, µ±, π±,K±, ρ±,K∗,±) + anything 11 sociated proton decay is a defining prediction of Grand 2. N (π0,K0, η, ρ, ω, K∗,0) + anything 12 Unified Theories (GUTs) [5, 6] that unify the three forces 3. N → γ + anything 12 of the SM into a single gauge group, offering an expla- 4. N → ν + anything 12 arXiv:1910.07647v2 [hep-ph] 8 Jan 2020 → nation for the observed charge quantization as well as V. Processes with ∆B > 1 12 gauge coupling unification. GUTs typically lead to effec- d B L A. ∆B = 2, ∆L = 0 13 tive dimension-six ( = 6) operators with ∆ = ∆ = 1 p e+π0 B. ∆B = 2, ∆L = 2 14 that induce two-body nucleon decays such as ± and n e+π−, mediated by heavy gauge bosons→ with C. ∆B > 2 15 → D. Inclusive searches for ∆B > 1 15 family-universal couplings [7–9]. It has to be stressed, however, that the significance of VI. Conclusions 15 nucleon decays stretches far beyond GUTs. ∆B = 0 processes generically appear in numerous theoretical6 exten- Acknowledgements 15 sions of the SM, such as supersymmetry (SUSY) [8], typically mediated at the renormalizable level by leptoquarks and diquarks [10–13]. A somewhat model-independent approach to study ∆B = 0 operators is through the ∗ [email protected] SM effective field theory6 (SMEFT), neglecting model- † [email protected] dependent interference effects [14–18]. This allows one to 2 identify higher-dimensional operators with a flavor struc- II. NUCLEON DECAY OPERATORS ture that can make rather unconventional nucleon decay channels dominant. Further, higher-dimensional d > 6 Several well-motivated theoretical models such as operators typically induce nucleon decays with multi- GUTs or R-parity-violating SUSY lead to nucleon de- body final states and even in simple UV-complete models cay, typically with specific two-body channels such as one can encounter more complicated nucleon decay chan- p e+π0 or p νK¯ + being dominant [8, 9]. In order nels such as n e+e−ν [19] or p π+π+e−νν [12, 16]. → → → → to discuss nucleon decay in its generality without being Hence, there is a vast landscape of possible motivated restricted to a certain model, we instead consider various nucleon decay modes of varying complexity. possible higher-dimensional operators that can mediate these processes, an approach1 that goes back to Refs. [14– 18]. We aim to determine which operators lead to two- Experimentally, an extensive nucleon decay search pro- body and which lead to multi-body nucleon decays. gram has been carried out over multiple decades, cover- ing more than 60 decay channels [20]. The most sensitive searches, coming from the Super-Kamiokande (SK) A. Operator dimension d = 6 experiment [21, 22] (see Ref. [23] for a review), have already pushed the nucleon lifetime limits for certain In the SMEFT, operators that exhibit ∆B = 1 start channels above 1034 yr [24], twenty-four orders of mag- to appear at operator mass dimension d = 6. Keeping nitude beyond the age of our Universe. The frontier of the flavor structure, these ∆B = ∆L = 1 operators can baryon number violation searches will be spearheaded be written as by next-generation large-scale underground neutrino experiments, namely the Jiangmen Underground Neutrino 1 αβγ C C d=6 = yabcd (da,αub,β)(Qi,c,γ ijLj,d) Observatory (JUNO) [25], the Deep Underground Neu- L 2 αβγ C C trino Experiment (DUNE) [26], and Hyper-Kamiokande + yabcd (Qi,a,αijQj,b,β)(uc,γ `d) (HK) [27]. It is of paramount importance to take full C C + y3 αβγ (Q Q )(Q L ) advantage of these considerable efforts and to identify abcd il jk i,a,α j,b,β k,c,γ l,d potential new signals in order to ensure that interesting 4 αβγ C C + yabcd (da,αub,β)(uc,γ `d) + h.c. , (1) channels are not overlooked due to theoretical biases. where α, β, γ denote the color, i, j, k, l the SU(2)L, and a, b, c, d the family indices [14–18]. u, d, and ` are the In this work we revisit nucleon decay channels arising right-handed up-quark, down-quark, and lepton fields, from higher-dimensional operators and discuss some of while Q and L are the left-handed quark and lepton dou- the possible resulting final states. While systematically blets, respectively. The yj couplings have mass dimen- searching through all of the kinematically allowed nu- sion 2 and the first-generation entries are constrained cleon decay channels with increased final-state complex- to be−< ( (1015) GeV)−2 due to the induced two-body ity would constitute the strongest probes, this approach nucleon decays.O Specifically, all of the above operators quickly becomes highly impractical beyond the simplest generate the well-constrained decay p e+π0 with a of the modes. In view of this, we highlight the impor- rate of order → inclusive nucleon decay searches tance of . Although these 2 searches are not as sensitive as exclusive ones looking at 1 yj Γ(p e+π0) 1111 . (2) a particular channel, they allow one to cover very broad 34 15 −2 → ' 2 10 yr (3 10 GeV) parameter space in a model-independent manner and are × × practically far more feasible. This approach is particu- A variety of other two-body nucleon decay channels are larly fruitful to revisit in view of the upcoming large-scale induced as well, including muon and kaon modes once experiments. we consider second-generation flavor indices. Three-body decay modes with similar rates are induced as well [29] but ultimately lead to weaker constraints. This paper is organized as follows: in Sec. II we dis- Operators in d=6 involving either charm, top, bot- cuss higher-dimensional operators that lead to nucleon tom or tau are seeminglyL unconstrained by nucleon de- decay and argue in particular that many of them lead cay since these particles are heavier than the proton; it to multibody final states that are not covered in cur- is however possible to go through heavy off-shell par- rent searches. In Sec. III we provide a brief overview of ticles and still induce nucleon decay, as emphasized in current and upcoming detectors as well as existing ex- Ref. [30] for operators involving a tau and more gener- clusive nucleon-decay searches. We discuss and propose ally in Ref. [31] (see also Ref. [32] for a UV-complete possible inclusive searches as well as model-independent signatures in Sec. IV. Sec. V is devoted to a short discussion of ∆B > 1 processes such as dinucleon decay, which would also profit from inclusive searches. Finally, 1 This approach does not cover the case of beyond-the-SM light + we conclude in Sec. VI. particles X that could lead for example to p → ` + X [28]. 3 example with scalar leptoquarks). As an extreme ex- quark couplings, we find that d > 14 (for k = 2) or d > 6 4 30 ample, Ref. [31] has considered the coupling y3333 and (for k = 15) in order to push the lifetime above 10 yr 0 shown that at two-loop level the simple decay n ν¯τ π (which corresponds to a reasonable lower bound on the is induced with an estimated rate → total nucleon lifetime, see discussion in Sec. IV). Clearly, nucleon decays can probe very high-dimensional opera- 4 2 0 1 y3333 tors and very high multiplicity, making it possible and Γ(n ν¯τ π ) . (3) → ' 1033 yr (5 108 GeV)−2 necessary to go beyond two-body decays mediated by × d = 6 operators. 4 11 −2 An even stronger limit y3333 . (10 GeV) has been As already pointed out by Weinberg [16], ∆B = 1 op- + estimated from p ν¯τ K in Ref. [33]. Despite the sup- d > lepton → erators with 6 can carry different total num- pression by loop factors, Cabibbo–Kobayashi–Maskawa ber ∆L, which can be used to make them dominant over mixing angles, and the Fermi constant GF , these limits the d = 6 operators. Interesting connections between are far stronger than any constraints from ∆B = 1 top ∆B, ∆L, and the mass dimension of the operator d were or tau decays on the same operator, making nucleon de- proven in Refs. [37, 38], cays clearly the best search channels. Notice that any operator involving tau leptons brings a missing-energy d is even ∆(B L) = 0, 4, 8, 12,..., (5) ↔ | − | tau neutrino in the final state, reducing the detection ef- d is odd ∆(B L) = 2, 6, 10, 14,..., (6) ficiency somewhat; turning this around, it is then crucial ↔ | − | to search for (two-body) nucleon decays involving one as well as the weak inequality neutrino, as these are the best channels of ∆B = 1 oper- 9 3 ators that involve tau leptons. As it is conceivable that dmin ∆B + ∆L (7) the UV completion that generates the ∆B = 1 operators ≥ 2| | 2| | also singles out tau leptons (or any other lepton flavor for for the minimum dimension dmin of an operator with ∆B that matter), one must not rely exclusively on searches and ∆L. For the cases of practical interest we give dmin in involving electrons and muons [34]. Fig. 1, adapted from Ref. [38]. Below we discuss ∆B = 1 The main conclusion of the above exercise is that any operators with d > 6 in order to establish the importance ∆B = 1 operator leads to nucleon decay, no matter the of multi-body nucleon decay searches. flavor structure. In the d = 6 case of Eq. (1) it is fur- At d = 7 one finds ∆B = ∆L = 1 operators that thermore always possible to close SM loops in order to induce e.g. n e−K+ or p −e−π+K+ [16]. In analogy generate a two-body nucleon decay. These might not al- to the d = 6 operators→ from→ above one can show that all ways be the dominant decay modes, but in light of the of these d = 7 operators induce two-body nucleon decays clean decay channels they are clearly the preferred way to at some loop level, irrespective of their flavor structure. search for the operators of Eq. (1). This picture changes While these might not necessarily be the dominant de- once we consider ∆B = 1 operators of mass dimension cay modes, they are clearly far easier to constrain exper- d > 6 as we discuss below. imentally. The currently best constrained channels are p νK+ and n `−π+, while many other two-body final→ states with ∆(→B L) = 2 have unfortunately not B. Operator dimension d > 6 been updated for| twenty− years| (listed below in Tab. I). At d = 8 one finds ∆B = ∆L = 1 operators of the form Assuming a ∆B = 1 operator of mass dimension d 6 uuQLφ¯φ¯, ddQLφφ, and dQQ`φφ in addition to simply 4−d ≥ with coefficient Λ we can estimate the amplitude for dressing the d = 6 operators of Eq. (1) with φ 2, where φ an k-body nucleon decay as md−k−1Λ4−d and the is the SM scalar doublet.3 Once again two-body| | nucleon M ∼ p decay width [35] decays are the best search channels. Starting at d = 9 the phenomenology becomes vastly 5−4k 3−2k 2k−4 2 2 π mp more interesting. Suppressing all indices and using a very Γ(N k particles) |M| , (4) → ∼ 2mp (k 1)!(k 2)! compact notation, the ∆B = 1 operators can be written − − in the form neglecting the final-state masses and possible symmetry factors (see Ref. [36] for loop-suppressed nucleon decays). 9 = ddd``¯`¯ , 9 = udd`L¯L,¯ O1 O2 The decay is suppressed for large d and large k, effectively 9 = ddd`L¯ L,¯ 9 = ddQ``¯L,¯ lowering the probed scales Λ2. Assuming conservatively O3 O4 9 = ddQLL¯L,¯ 9 = dQQ`L¯L,¯ that Λ should lie above TeV in order to evade LHC con- O5 O6 straints on the underlying colored mediator particles with 9 = uddL¯φ¯φφ¯ , 9 = uQQL¯φ¯φ¯φ¯ , O7 O8

2 3 We note that the ratio Γ(N → k1 particles)/Γ(N → k2 particles) Here and in the following we ignore operators that contain deriva- for k1 > k2 could also be larger than 1 and not follow the suppres- tives. A full list of these and other operators can be conveniently sion just from phase-factor considerations, as shown in Ref. [29]. obtained using the program Sym2Int [43, 44]. 4

∆( B 2 )/ L − L ∆B ) + /2 B ∆(

d 19 d 16 d 15 d 18 ≥ ≥ ≥ ≥ 3n 3ν nn nν¯ nn n¯ν¯ Instanton → → → d 15 d 12 d 9 d 12 d 15 ≥ ≥ ≥ ≥ ≥ nn 4ν nn νν nn ππ pp e+e+ nn 4¯ν → → → → → d 10 d 7 d 6 d 9 ≥ ≥ ≥ ≥ + + 0 + n 3ν n e−π p e π p e ν¯ν¯ → → → → d 10 d 5 d 5 d 10 ≥ ≥ ≥ ≥ 0ν4β 0ν2β 0ν2β 0ν4β ∆L

FIG. 1. Processes with baryon and lepton number violation by ∆B and ∆L units, respectively. We only show one example process, others are implied (e.g. nn → ππ also give n–¯n oscillation, pp → π+π+, and many more). “Instanton” refers to processes such as 3n → 3¯ν that break the same quantum numbers as non-perturbative electroweak instantons. 0ν2β (0ν4β) refers to neutrinoless double [39, 40] (quadruple [41]) beta decay. Final states with neutrinos make an experimental determination of ∆L impossible, but are shown here for the sake of brevity. Also shown is the minimal mass dimension d of the underlying eﬀective operator following Ref. [38]. In addition to total lepton number L, all operators and processes also carry lepton ﬂavor, which turns the above two-dimensional plot into a four-dimensional lattice [34, 42].

9 = dddL¯φφφ¯ , 9 = ddQ`¯φ¯φφ¯ , ments in the literature these operators do indeed gener- O9 O10 9 = dQQL¯φ¯φφ¯ , 9 = QQQ`¯φ¯φ¯φ¯ , (8) ate nucleon decays despite the fact that they necessar- O11 O12 ily contain charm or top quarks, in complete analogy to 9 =uuddd ¯ `¯ , 9 =uuddQ ¯ L,¯ O13 O14 the discussion in Ref. [31]. At loop level the operators 9 ¯ 9 ¯ 9 with arbitrary flavor structure induce the decays 15 =uddQQ ¯ ` , 16 =udQQQ ¯ L, 22,23 O O O + 9 ¯ ¯ 9 ¯ ¯ p ` ν¯ν¯ and n 3¯ν, which are experimentally con- 17 = udddQL, 18 = ddddd` , → → O O strained already. 9 = ddddQ¯ L,¯ 9 = dddQQ¯ `¯ , O19 O20 At d = 10 there are ∆B = ∆L = 1 as well as 9 ¯ ¯ 9 1 21 = ddQQQL, 22 = uuu`LL , ∆B = ∆L = 1 operators. The former are generically O O 3 9 expected− to be suppressed compared to d = 6 opera- 23 = uuQLLL . O tors, except for operators that involve three lepton fields with non-trivial flavor, in particular those operators that 9– 9 have recently been discussed in Ref. [34]; they 1 6 give rise to the very clean channels p e+µ−µ− and fulfilO O ∆B = ∆L = 1, but since they involve three lep- p µ+e−e−, discussed at length in Ref.→ [34]. The ∆B = ton fields they− can carry non-trivial lepton flavor num- →1 ∆L = 1 operators are of the form dddL¯L¯L¯φ¯ and lead ber, i.e. ∆L > 1 for one α e, µ, τ , which makes 3 α to− multi-body nucleon decays such as n `−ννK+, it possible| to| make them dominant∈ { over} d = 7 opera- p `−ννπ+π+ [12, 16, 46], and n 3ν, only→ the latter tors (which carry at most ∆L = 1) and then lead α being→ explicitly constrained so far. → to three-body [45] and four-body| | nucleon decays [34]. 9 9 7– 21 leads to ∆B = ∆L = 1 two-body nucleon For examples of ∆B = 1 operators with d > 10 we refer decaysO O reminiscent of the−d = 7 operators, although to Refs. [38, 46, 47]. If such operators are to dominate 9 9 13– 21 dominantly induces multi-meson final states, over lower-dimensional ones they should carry a differ- e.g.O nO `−K+π0, due to the large number of quark ent lepton (flavor) number, e.g. ∆L > 3. It is clear that → 9 9 1 | | fields. Finally, 22 and 23 violate ∆B = 3 ∆L = 1 [16] these lead to multi-lepton final states, typically accompa- and thus lead toO nucleonO decays with at least three an- nied by mesons. We note that the semi-inclusive invisible tileptons in the final state, typically accompanied by one neutron decay, n neutrinos, can carry away an arbi- or more mesons. We stress that contrary to many state- trary amount of lepton→ number and flavor via neutrinos 5 and is thus a particularly powerful decay to probe, not least because the experimental signature is the same no + matter the number or flavor of the final-state neutrinos. u K We come back to this channel in Sec. IV. The calculation and comparison of all possible nucleon s¯ final states induced by a given d 6 operator is of course impractical and is not attempted here. From the exam- n d ¯ ples discussed above we can however already glean our S1 e− simple main point: current nucleon decay searches cover ˜ S1∗ δ but a fraction of relevant modes and should by all means e− be extended in order not to miss new physics. From our d discussion it is clear that two-body nucleon decays are powerful probes of d = 6–8 operators, whereas d 9 op- µ+ ≥ erators, in particular those with ∆L > 1 or ∆Lα > 1, lead to multi-body final states that| mostly| lie| outside| of FIG. 2. Neutron decay n → K+µ+e−e− in the UV-complete current searches. Since nucleon decay is an extremely example. sensitive probe of new physics, even higher-dimensional d operators with 6 can give testable signals and should 9 be investigated. To constrain these operators one then the symmetries conserved by we enforce y3 = 0, y2 = O 9 (y2)jµ, y4 = (y4)ee. This leads to the desired operator has to either go through all kinematically allowed multi- O body final states (in general without knowing the angular with the effective suppression scale and spectral distributions) or focus on inclusive searches, ∗ ∗ 1 κ (y1)12(y2)1µ(y4)ee to be discussed in Sec. IV. Before, we must first discuss 5 2 2 2 (11) Λ ∼ m ¯ m ˜ mδ the status of exclusive searches in Sec. III. S1 S1 upon integrating out the new scalars, as illustrated by the Feynman diagram for n K+µ+e−e− in Fig. 2. The C. UV-complete example imposed symmetries ensure→ that all ∆B = 0 processes have to go through 9, as there is no lower-dimensional6 To illustrate the above discussion we give one sim- operator with the sameO quantum numbers. While it is ple UV-complete example for a potentially dominant possible to attach SM interactions to Fig. 2 to generate + − ∆B = 1 process that is not covered by existing exclu- three-body final states such as n µ e νe, these are → sive searches, closely following Ref. [34]. We consider the further suppressed by GF . From neutrino oscillations 9 d = 9 operator 1 from above, which includes a (dd) we already know that the lepton flavor symmetries we O pair that is antisymmetric in the flavor indices and thus imposed are actually broken, which calls into question contains a strange quark. We focus on one particularly our usage of them. However, if these symmetries are only interesting lepton flavor combination broken in the neutrino sector then the flavor-breaking effects in ∆B = 0 processes are suppressed by mν and 9 = (ds)(dµ)(¯ee¯)/Λ5 . (9) 6 O thus negligible [34]. Having motivated the existence of the operator 9, we Of the four globally conserved quantum numbers of the n K+µ+Oe−e− 9 can calculate the total decay rate for . SM, (B,Le,Lµ,Lτ ), the operator breaks the linear → O A simple analytic expression can only be obtained if we combination ∆(B 2Le + Lµ) = 6 but still conserves − neglect all final-state masses B + Le + Lµ, Le + 2Lµ, and Lτ . In fact, it is the lowest- dimensional operator with these properties, which is the m7 W 2 Γ(n K+µ+e−e−) n 0 , (12) reason it can be the dominant nucleon-decay operator. → ' 737280π5Λ10 As a UV completion we introduce the two scalar lep- where W (0.23 GeV)2 comes from the relevant nuclear toquarks S¯1 (3¯, 1, 2/3) and S˜1 (3¯, 1, 4/3) as well 0 ∼ − ∼ matrix element' K+ dsd n [49]. A numerical evaluation as the dilepton scalar δ (1, 1, 2), with the relevant h | | i interactions ∼ − including the kaon and muon masses yields instead 10 ¯∗ ˜ ˜∗ ∗ + + − − 1 100 TeV = y1ddS1 + y2d`S1 + y3uuS1 + y4``δ Γ(n K µ e e ) . (13) L 31 ˜ ¯∗ → ' 5 10 yr Λ + κS1S1 δ + h.c., (10) × The final-state electrons are typically above the with Yukawa coupling matrices yj and a scalar-potential Cherenkov threshold, while the kaon is below and thus in- coupling constant κ with units of mass [48]. While we visible in experiments such as SK. The muon momentum suppress the flavor indices, let us note that y1 and y3 are spectrum is shown in Fig. 3, which leads to roughly half antisymmetric and y4 symmetric. Assigning Le(δ) = 2, of the muons above and half below the Cherenkov thresh- B(S¯1) = 2/3, and Lµ(S˜1) = 3B(S˜1) = 1 and imposing old. No dedicated exclusive search for this (or similar) − 6

7 Other early searches looked for fragments of ﬁssion induced by nucleon decays within radioactive ore. As 6

1 we further discuss below, a particular advantage of such GeV 5 searches is that they are completely model indepen- in )

- dent. However, the amount of source material is limited e

- 4 and hence they are signiﬁcantly less sensitive than ded- e + μ

μ icated state-of-the-art exclusive searches. For exclusive + 3 dp searches, ﬁne-grained (typically iron-based) calorimeters, 2 such as in the Kolar Gold Field (KGF) [59], NUSEX [60],

K → ( n d Γ Soudan [61], and Fréjus[62] experiments, have been also Γ 1 1 utilized. In these experiments one can achieve high signal efficiency due to precise reconstruction and tracking. 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 However, these configurations do not scale well and hence

Muon momentump μ in GeV suffer from a limited fiducial mass (. 1 kton). The most sensitive searches come from large water- FIG. 3. Primary final-state muon spectrum of neutron decay Cherenkov (WC) experiments such as Homestake [63], n → K+µ+e−e−. Harvard–Purdue–Wisconsin (HPW) [64], Irvine– Michigan–Brookhaven (IMB) [65], and Kamiokande [66]. WC experiments can identify charged particles with high four-body decay has ever been performed, although it has efficiency and are readily scalable, enabling fiducial vol- been proposed long ago [19]. Nevertheless, constraints umes far greater than any other technique. The current on Γ(n K+µ+e−e−) can be obtained from inclusive most sensitive proton-decay limits, already exceeding searches,→ discussed extensively in Sec. IV. In particular, lifetimes of 1034 yr, come from the state-of-the-art WC a limit of τ(N µ+ + anything) > 12 1030 yr [20] ap- experiment SK [24], which boasts a fiducial volume of plies, neglecting→ any detection efficiency× penalties. As we 22.5 kton and has been collecting data for two decades. argue below, inclusive searches in SK should be able to WC detection relies on the presence of electromagneti- improve this bound by more than an order of magnitude, cally interacting particles in the final state of nucleon de- with even more progress expected with the future HK. cays. The observable signature from a particle traversing + + − − While a dedicated search for n K µ e e would of the detector is a Cherenkov radiation ring, classified ei- → course provide the best possible limit, it is clearly not ther as “showering” for e± and γ or “non-showering” for feasible to systematically study all possible multi-body µ± and π±. WC searches are particularly sensitive when nucleon decay channels. As we show in the next section, the final state is fully visible and one can efficiently re- even restricting the search to two- and three-body final construct the original parent nucleon, such as in the case states is a challenge. Inclusive searches on the other hand of the leading GUT modes p e+π0 and p µ+π0, provide a simpler handle on the final-state complexity where the pion is identified via→π0 γγ. To→ produce and deserve more attention. Cherenkov radiation, charged particles→ of mass m trav- eling in a medium of refraction index n must exceed the 2 Cherenkov momentum threshold of pth = m/√n 1. − III. EXCLUSIVE NUCLEON DECAY In water, n 1.33 and thus pth = 1.14m, which trans- SEARCHES lates to a required' minimum momentum of 0.58 MeV, 121 MeV, 159 MeV, and 563 MeV for e±, µ±, π±, and ± Several major directions have been pursued to ex- K , respectively. Hence, kaons from single nucleon decays, which have energy below mp/2 470 MeV where perimentally study nucleon decays. The main points ' of focus are the ability of an experiment to achieve mp is the proton mass, are always invisible in WC de- high signal-detection efficiency as well as scalability to tectors and one must rely on reconstructing products large fiducial masses/volumes, which is particularly crit- from their subsequent decays. Thus the typically leading SUSY GUT decay channel p νK+ suffers from a ical for such rare-event searches. Some of the earliest → searches were performed with scintillators, pioneered by low detection efficiency in WC detectors. Scintillation Reines et. al. [50–54]. These exclusive searches, focus- detectors have an advantage over WC in that there is no ing on some specific nucleon decay channels, assumed Cherenkov threshold and a higher light yield. However, that a final state of the decay contains an electromagnet- because emission of scintillation light is nearly isotropic ically interacting particle. Current large-volume scintil- the directional information is lost. Further, while scintillator neutrino experiments based on carbon 12C, such as lators are scalable, they are still behind in volume com- KamLAND [55] and Borexino [56], have fiducial masses pared to leading WC experiments. of . 1 kton and can be also utilized for nucleon-decay Several next-generation large-scale experiments will al- searches (e.g. Refs. [57, 58]). The main distinguishing low us to push nucleon decay searches even further. feature of these experiments is a very low sub-MeV en- The upcoming successor of SK, the HK WC experi- ergy threshold. ment, is expected to have a 187 kton fiducial volume 7

Γ−1 in its initial configuration [27]. This will allow HK to Channel |∆(B − L)| 1030 yr 35 probe nucleon lifetimes up to 10 yr. Since a large p → e+ + γ 0 41000 [72] fiducial volume benefits all the decay modes, improved p → e+ + π0 0 16000 [24] limits across the board are expected. On the scintilla- + tor front, the liquid scintillator experiment JUNO will p → e + η 0 10000 [73] have 20 kton fiducial volume [25] and will start taking p → e+ + ρ0 0 720 [73] data in a few years. The DUNE [26] experiment with p → e+ + ω 0 1600 [73] 40 kton fiducial volume based on liquid-argon time- p → e+ + K0 0 1000 [74] ∼ projection-chamber (LArTPC) technology will allow one p → e+ + K∗,0 0 84 [65] to track both light deposits as well as charge. Despite p → µ+ + γ 0 21000 [72] their smaller size compared to HK, the JUNO and DUNE + 0 detectors are particularly promising for decay modes in- p → µ + π 0 7700 [24] + volving kaons, such as p νK+, which are fully visible p → µ + η 0 4700 [73] and can be reconstructed→ with high efficiency. Addition- p → µ+ + ρ0 0 570 [73] ally, for the multi-body decay channels discussed in this p → µ+ + ω 0 2800 [73] article the efficiency in WC experiments could be de- p → µ+ + K0 0 1600 [75] creased, as it becomes difficult to reconstruct multiple p → ν + π+ 0,2 390 [76] overlapping Cherenkov rings, making alternative tech- p → ν + ρ+ 0,2 162 [65] nologies as those in JUNO and DUNE crucial. For ex- + ample, in ∆B = 2 n–n oscillations the resulting n is p → ν + K 0,2 5900 [77] ∗,+ subsequently captured on n or p and produces a slew of p → ν + K 0,2 130 [78] decaying pions [67, 68]. In DUNE, the multi-track nature n → e− + π+ 2 65 [79] (5300∗ [73]) of such decays could be utilized with high efficiency [69]. n → e− + ρ+ 2 62 [79] (217∗ [65]) We briefly note that planned next-generation dark- n → e− + K+ 2 32 [62] matter direct-detection experiments such as Argo [70] n → e− + K∗,+ 2 (based on liquid Argon) or Darwin [71] (Xenon) could n → e+ + π− 0 5300 [73] be able to achieve ultra-low sub-keV energy thresholds + − combined with a (10–100) ton fiducial mass and could n → e + ρ 0 217 [65] in principle also beO utilized for nucleon decay searches. n → e+ + K− 0 17 [65] However, their fiducial volume is still multiple orders of n → e+ + K∗,− 0 magnitude smaller than dedicated large-scale neutrino n → µ− + π+ 2 49 [79] (3500∗ [73]) experiments and would have a hard time competing with n → µ− + ρ+ 2 7 [79] (228∗ [65]) their exclusive searches, although their low threshold n → µ− + K+ 2 57 [62] could be beneficial for model-independent searches. n → µ+ + π− 0 3500 [73] Despite significant experimental efforts proton decay n → µ+ + ρ− 0 228 [65] or any other ∆B = 0 process has not been observed + − thus far. Almost all6 kinematically allowed two-body nu- n → µ + K 0 26 [65] cleon decay channels have been searched for in various n → ν + γ 0,2 550 [28] experiments, although several of the limits are outdated. n → ν + π0 0,2 1100 [76] We collect the strongest limits in Tab. I and also spec- n → ν + η 0,2 158 [65] ify the violation in (B L) symmetry that such decays n → ν + ρ0 0,2 19 [79] induce. We note that− a neutrino or an antineutrino of n → ν + ω 0,2 108 [65] any flavor in the final state will escape the detector be- 0 fore interacting and is hence invisible, which can result n → ν + K 0,2 130 [74] ∗,0 in a variation of ∆(B L) by 2 units. We do not in- n → ν + K 0,2 78 [65] clude ∆B = 1 multi-nucleon| − | decays such as pn e+n or pp e+∆+ here, some of which were searched→ for in → TABLE I. Kinematically allowed two-body nucleon decays Ref. [62]. We also omit showing processes that violate with 90% C.L. upper limit on the partial decay width Γ. Here, Lorentz or charge symmetry. ν can be a neutrino or antineutrino of any flavor, which does In two-body decay modes, kinematics and phase-space not change the observation signature. The column |∆(B −L)| considerations uniquely determine the resulting energy- indicates the violation of B − L in the decay, which depends momentum distribution of the resulting final-state parti- on whether ν is a neutrino or an antineutrino. An asterisk denotes a limit that has been translated using the properties cles, making these searches highly model independent. − + − + − + For multi-body decays, additional model dependence of WC detectors (e ∼ e ∼ γ, µ ∼ µ , π ∼ π ) but was not given explicitly in the references. Unconstrained channels from dynamics comes into play, as discussed for example are still subject to limits from inclusive searches, discussed in in Ref. [80]. Multi-body searches can furthermore suffer Sec. IV. See text for more details. from lowered detection efficiencies and enhanced system- atic uncertainties, e.g. from multiple hadronic/nuclear in- 8

Γ−1 teractions. Only a subset of all kinematically allowed Channel |∆(B − L)| 1030 yr three-body nucleon decay channels have been searched p → e− + e+ + e+ 0 793 [65] for so far, collected in Tabs. II and III along with the p → e− + e+ + µ+ 0 529 [65] ∆(B L) structure for the modes. To remain agnostic + + − ∗ regarding| − theoretical| models, a uniform phase-space en- p → e + e + µ 0 529 [65] ergy and momentum distribution for final-state particles p → e− + µ+ + µ+ 0 6 [64] (359∗ [65]) is typically assumed in such searches. p → e+ + µ− + µ+ 0 359 [65] We are not aware of any searches involving more than p → µ− + µ+ + µ+ 0 675 [65] three particles in the final state so we do not provide a p → e+ + 2ν 0,2 170 [81] table listing all kinematically allowed modes. Instead, p → µ+ + 2ν 0,2 220 [81] we discuss inclusive searches, which could in principle be p → e− + 2π+ 2 30 [62] (82∗ [65]) sensitive to arbitrary multi-body final states. p → e− + π+ + ρ+ 2 p → e− + K+ + π+ 2 75 [65] p → e+ + 2γ 0 100 [82] (793∗ [65]) IV. INCLUSIVE NUCLEON DECAY SEARCHES + − + FOR ∆B = 1 PROCESSES p → e + π + π 0 82 [65] p → e+ + ρ− + π+ 0 + − + ∗ As we have shown in Sec. II, without focusing on spe- p → e + K + π 0 75 [65] cific models many ∆B = 1 operators induce nucleon de- p → e+ + π− + ρ+ 0 cays with a multi-body final state. More so, these chan- p → e+ + π− + K+ 0 75∗ [65] nels could be dominant and often the simpler two-body p → e+ + 2π0 0 147 [65] decay modes are completely forbidden, e.g. by a lepton p → e+ + π0 + η 0 number or flavor symmetry. It is therefore necessary to p → e+ + π0 + ρ0 0 broaden the nucleon decay searches, which have primar- + 0 ily been focused on two-body channels, in order to also p → e + π + ω 0 cover multi-body channels. While this can be done by p → e+ + π0 + K0 0 performing exclusive searches looking at specific multi- p → µ− + 2π+ 2 17 [62] (133∗ [65]) body channels, an exhaustive search program quickly be- p → µ− + K+ + π+ 2 245 [65] comes impractical as the more complex modes are consid- p → µ+ + 2γ 0 529∗ [65] ered. Hence, fully model-independent as well as inclusive p → µ+ + π− + π+ 0 133 [65] N X + anything channel searches that are less depen- + − + ∗ dent→ on the number and details of final-state particles are p → µ + K + π 0 245 [65] + − + ∗ particularly important. We outline such searches below. p → µ + π + K 0 245 [65] p → µ+ + 2π0 0 101 [65] p → µ+ + π0 + η 0 + 0 0 A. Model-independent and invisible mode searches p → µ + π + K 0 p → ν + π+ + π0 0,2 + Model-independent searches allow us to probe all pos- p → ν + π + η 0,2 sible nucleon decay channels simultaneously and are not p → ν + π+ + ρ0 0,2 constrained to any specific final-state particle. The most p → ν + π+ + ω 0,2 stringent model-independent limits come from nuclear p → ν + π+ + K0 0,2 de-excitation emission searches. Such limits apply to all p → ν + ρ+ + π0 0,2 seemingly unconstrained channels described in Tabs. I, II, p → ν + K+ + π0 0,2 and III. Model-independent searches for neutron decays also constitute the main method to study invisible decays that TABLE II. Same as Tab. I but for three-body proton decays. leave no other trace in the detector, such as n invisible, We have not included the (anyways unconstrained) N → A + where the invisible final state could consist→ of an arbi- B + γ modes since they should be suppressed compared to trary combination of neutrinos and antineutrinos. Invis- the N → A+B channels. We have however kept N → A+2γ ible modes decaying to SM particles, such as n 3ν, because limits on this mode exist and one can imagine models could become significant in models based on extra→ di- with Γ(N → A + 2γ) > Γ(N → A + γ), e.g. via a decay chain mensions (e.g. [84, 85]) or partially unified Pati–Salam- N → A + (J → γγ) with an axion-like particle J. type constructions [86]. Proposals for invisible decays to more exotic states, such as dark fermions [87, 88] or un- particles [89], have been also suggested. Applying these searches to processes like p 3ν can test electric-charge conservation. → 9

Γ−1 Channel |∆(B − L)| 1030 yr for radioactive remnants due to nucleon decay within 58 n → ν + e− + e+ 0,2 257 [65] some favorable and abundant isotope, such as Ni 57Co [91] or 39K (38Ar, 38K) 37Ar [92]. These→ n → ν + e− + µ+ 0,2 83 [65] experiments must→ be placed deep→ underground to sup- + − ∗ n → ν + e + µ 0,2 83 [65] press background isotope production due to cosmic ray n → ν + µ− + µ+ 0,2 79 [65] interactions. Further, as radioactive decay happens dur- n → 3ν 0,2,4 0.58 [83] ing experimental observation large quantities of mate- n → e− + π+ + π0 2 29 [62] (52∗ [65]) rial are required. Typically, these experiments are not n → e− + π+ + η 2 easily scalable, with limits on nucleon decay lifetimes of 26–27 − + 0 τp 10 yr already requiring ton-scale fiducial mass. n → e + π + ρ 2 ∼ n → e− + π+ + ω 2 n → e− + π+ + K0 2 Due to their ultra-low keV-level thresholds and well- understood backgrounds, large DM experiments could n → e− + ρ+ + π0 2 − + 0 also constitute efficient detectors for radiogenic nucleon n → e + K + π 2 decay searches. Using 7 kg of fiducial mass and + − 0 ∼ 24 n → e + π + π 0 52 [65] 6 kg yr of exposure, a limit of τp 10 yr was ob- n → e+ + π− + η 0 tained∼ with· the DAMA liquid Xenon scintillator∼ [93]. A n → e+ + π− + ρ0 0 future large-scale Xenon experiment with fiducial mass + − of (10–100) tons, such as DARWIN [71], might thus n → e + π + ω 0 O n → e+ + π− + K0 0 18 [82] achieve an improved sensitivity reach by several orders of magnitude. n → e+ + ρ− + π0 0 n → e+ + K− + π0 0 A variation on the radiogenic approach is to search n → µ− + π+ + π0 2 34 [62] (74∗ [65]) − + for appearance of fragments from spontaneous fission of n → µ + π + η 2 a large nucleus into several sizable components. Here, − + 0 232 21 n → µ + π + K 2 using 7 kg hr exposure of Th a limit of τp 10 yr n → µ− + K+ + π0 2 was set in· Ref. [94]. However, using this method∼ effec- n → µ+ + π− + π0 0 74 [65] tively requires good background understanding and large n → µ+ + π− + η 0 quantities of very heavy elements. n → µ+ + π− + K0 0 n → µ+ + K− + π0 0 In the geochemical approach one searches for stable n → ν + 2γ 0,2 219 [65] nuclear residue associated with nucleon decays that have accumulated over billions of years within naturally abun- n → ν + π− + π+ 0,2 − + dant ore. This approach has been employed for double n → ν + ρ + π 0,2 beta-decay studies, especially when the resulting isotope − + n → ν + K + π 0,2 is a noble gas. When applied to nucleon decay by con- n → ν + π− + ρ+ 0,2 sidering processes such as 23Ne 22Ne, 39K 38Ar, 133 132 → → n → ν + π− + K+ 0,2 and Cs Xe, a nucleon decay lifetime sensitivity 23 → n → ν + 2π0 0,2 of 10 yr can be achieved [95], assuming spectroscopic ∼ 8 0 resolution of 1 part in 10 (10 ppb). An improved lifetime n → ν + π + η 0,2 25 0 0 limit of 10 yr was achieved in Ref. [96] by considering n → ν + π + ρ 0,2 ∼ 130 129 129 0 beta decays in Te Sb Xe, measured dou- n → ν + π + ω 0,2 ble beta-decay lifetime→ and known→ abundances as well 0 0 n → ν + π + K 0,2 as atmospheric contamination of various xenon isotopes in telluride ore. Due to the very long lifetime of the source material, these searches are not limited by fidu- TABLE III. Same as Tab. II, but for neutron decays. cial mass as radiochemical searches are. These searches require precise impurity identification. While purifica- tion levels of material, such as 130Te crystals produced 1. Isotope fission products for double beta-decay studies, in current experiments has greatly improved in recent years and approaches 1 part 14 A model-independent search for nucleon decays can in 10 [97], obtaining accurate understanding of accu- be performed by analyzing fission isotope products due mulated background contributions is highly non-trivial. to nucleon decay within an element [90]. If the fission fragment is stable (unstable), a geochemical (radio- While we envision that some improvement on such genic/radiochemical) search method can be used. searches is feasible, it is difficult for these searches to In the radiogenic/radiochemical approach one looks compete with nuclear de-excitation emission searches. 10

2. Nuclear de-excitation emission was obtained with the 3.2 kton KamLAND scintillator detector and 0.84 kton yr of data [83]. · Aside from analyzing fission fragments, another gen- In WC-based experiments, a de-excitation proton or an eral and model-independent signature of nucleon decay α would be below the Cherenkov threshold and thus in- is nuclear de-excitation emission. Every nucleon decay visible. A kicked-out neutron quickly thermalizes and dif- results in a hole within the host nucleus, e.g. in oxy- fuses, subsequently being captured on a hydrogen atom gen 16O (as relevant for WC-based detectors) or car- with an accompanying emission of a low-energy 2.2 MeV bon 12C (as relevant for scintillator-based detectors). γ-ray. A low-efficiency neutron tagging is currently being Such decays from an inner nuclear shell typically leave used in SK to gather this signal [105], which is below the the residual nucleus in an excited state, which subse- energy threshold of SK. The planned SK upgrade involv- quently de-excites by emission of secondary (n, p, α, γ) ing gadolinium doping [106] will allow us to detect neutrons with high efficiency due to 8 MeV γ-ray emission particles and also possibly further via β-decay of the ∼ residual radioactive nuclei. Emission steps can be iden- accompanying neutron capture on gadolinium. Since de- 4 tified with nuclear-shell models and energetic considera- excitations will generally result in neutrons, they could tions. The de-excitation particles accompanying nucleon be beneficial for WC searches. However, without addi- decay could be searched for as a model-independent sig- tional coincidence signatures the signal-background dis- nature [98–101]. Such signals, however, could be plagued crimination is non-trivial as neutrons also generically ap- by typical radioactive background within experiments as pear from neutrino interactions such as inverse β-decay + νe +p n+e . Previously, the importance of detecting well as neutrino interactions that result in similarly ex- → cited nuclear states [102]. neutrons associated with proton decay has been empha- For WC experiments, de-excitation γ’s are especially sized for heavy water (D2O) searches (e.g. in SNO) [107]. In particular, the proton decay search d n+? requires important for these searches. De-excitation γ emis- → sion with energies of a few MeV is expected with a less stringent implicit hypotheses regarding the stability large branching ratio (e.g. 44% for 6.18 MeV γ-ray for of the resulting daughter nuclei and is hence even less 15O [99]). However, this energy range is plagued by back- model dependent than some of the other searches dis- grounds associated with solar neutrinos and cosmogenics. cussed in this section. Placing the experiment very deep underground (e.g. 6000 Coincidence tagging of de-excitation emission together meter water equivalent for SNO+) and using a directional with some visible final state can provide additional dis- cut to suppress solar neutrino contributions, the SNO+ crimination of nucleon decay signal versus backgrounds, scintillator experiment was able to achieve a limit on in- as first suggested by Totsuka for SK [98]. However, this is visible proton decay of often difficult to utilize in practice. For example, in WC searches the nearly prompt emission of a low-energy de- τ inv. > 0.36 1030 yr (14) p × excitation γ could be hard to resolve when super-imposed with only 0.9 kton of water and 0.58 kton yr exposure ob- with an energetic Cherenkov ring e.g. from a final state tained during its water-filled WC phase· [103]. e± or µ±. A time separation between the energy depo- Large WC experiments that have significant expo- sitions, for example in p νK+ due to the 12 ns kaon → sure can take advantage of a relatively clean signature lifetime, is then beneficial and can be used to suppress from the higher energy γ-rays, which however have sup- background – as is done in SK analyses [77, 108–110]. pressed branching ratios. In particular, search for γ- Future detectors such as JUNO, DUNE, and HK are rays in the 19–50 MeV energy range has been used to expected to improve on these important channels. While set a lower partial-lifetime limit associated with n 3ν we are not aware of dedicated sensitivity studies, the of τ/Br 2 1031 yr by the Kamiokande experiment→ JUNO liquid-scintillator detector, which is analogous using ∼8.5 kton× yr exposure [104]. With the associ- to KamLAND but with a 20 times larger size (20 kt), ated cumulative∼ branching· ratio for such energetic γ’s could be expected to test invisible-nucleon decays be- of (1.4 0.7) 10−4 [100], the resulting lifetime limit is yond 1031 yr lifetime. Further, with low detection thresh- τ 5 ±1026 yr× [104]. With over 370 kton yr of exposure olds and sizable 40 kt mass, DUNE could also be very available∼ × for SK, a limit of order 1028 yr can· be achieved promising. To our knowledge no dedicated studies for de- using this method. excitation emission associated with nucleon decay from A coincidence multi-event signature associated with argon, as would be relevant for DUNE, have been per- several spatially and temporally correlated secondary formed thus far. particles from de-excitation can be employed to dif- Despite being derived as limits on invisible nucleon de- ferentiate signal from background (see discussion in cay, the discussed limits can be interpreted as the best Ref. [100]). Such low-energy MeV-level signals associated with de-excitation can be particularly advantageous for scintillator-based experiments, as considered by Borex- 4 ino [57] and KamLAND [83]. In particular, a limit on Taking into account both single-step and multi-step emission, neutrons are expected to appear with a combined branching ra- invisible nucleon decay of tio of ∼ 50% from de-excitation of 15O associated with neutron 16 τ inv. > 0.58 1030 yr (15) decays from an s1/2 state of O [100]. n × 11 current model-independent limits on the total nucleon 1979 analysis of Ref. [111]. We expect that SK and lifetime and are likely applicable to all otherwise uncon- later HK could significantly improve on this result, due strained modes in Tabs. I–III. We note that there is still to their large detector volume and exposure. An esti- some implicit dependence on the actual nucleon decay mate of such a limit can be obtained from the recent mode. For example, decay modes with highly energetic SK search that placed a limit of 170 1030 yr [81] on pions can potentially destroy the daughter nucleus rather the p e+νν channel, which bears close× similarity to than just leaving a hole, but we envision that such events the inclusive→ p e+ + anything mode. Lowering the would be captured with other searches. positron-momentum→ threshold employed in that analysis Finally, we comment that the invisible searches pre- and varying the energy distribution, or even adapting a sented here could also be interpreted as tests of forbidden pure counting analysis, is expected to yield an inclusive nuclear transitions, as was done in Ref. [104]. N e± + anything mode limit of similar order with available→ SK data, i.e. (100) 1030 yr. O × In the case of inclusive N µ± + anything channel B. N → X+anything with a muon, if the muon has enough→ momentum it leads to a µ-like ring in WC detectors. The current limit on Above we have discussed nucleon decay searches that this search is τ(N µ+ + anything) > 12 1030 yr [20], → × are nearly model independent and only make use of the converted from a 1981 Homestake experiment analysis of properties of the daughter nucleus that is created after Ref. [63] that used a 0.3 kton WC detector. We again nucleon decays within a nucleus. The current most sensi- expect SK and future HK to improve on this limit. Sim- tive searches in this category are from de-excitation emis- ilar to the inclusive search with an electron, the recent sion, with resulting lifetime limits approaching 1030 yr. SK search of p µ+νν that resulted in a lifetime limit 30 → This is more than four orders of magnitude below the of 220 10 yr [81] serves as a good indication of the × ± best exclusive search limits (Tab. I) and one or two or- potential limit on N µ + anything. → ders of magnitude below the predominant majority of Inclusive searches for charged mesons are feasible as other existing exclusive searches (e.g. Tabs. II and III). well, even though the mesons will eventually decay into In an effort to keep searches as model independent as pos- charged leptons and thus in principle will be covered by sible while utilizing the excellent particle identification of the searches outlined above. Nevertheless, designated exclusive searches, we now discuss inclusive nucleon de- charged-meson searches can provide additional useful in- cay searches, which correspond to N X + anything formation. In WC detectors the charged pion π± will with a light SM particle X of unknown→ energy. Without look similar to a µ±, albeit with a somewhat higher knowledge of the underlying energy distribution these are Cherenkov threshold, so the resulting limit can be ex- in general background-dominated searches. pected to be of similar order. An estimate of p π±+anything can be obtained from the SK search for→ the two-body decay p ν + π+ that yielded a limit of 1. N → (e±, µ±, π±,K±, ρ±,K∗,±) + anything 390 1030 yr [76]. Unlike→ this two-body decay, multi- body× decays will result in a smeared pion momentum – In general, from size and exposure considerations, WC causing an additional detection-efficiency penalty when experiments are expected to dominate inclusive searches the pion is below Cherenkov threshold. with a charged particle. However, the primary charged Charged mesons more massive than the pion particle of interest could be invisible in WC experiments (i.e. K±, ρ±,K∗,±) are always below Cherenkov thresh- due to its Cherenkov threshold. Hence, additional de- old when originating from decays of a single nucleon. tection efficiency can be expected in experiments with Thus, in WC experiments such mesons can only be iden- low-energy thresholds and where they are clearly visible, tified by their decay products. Due to their low momen- such as scintillators, especially for particles with a higher tum and short lifetime, they effectively decay at rest. mass and a larger Cherenkov threshold. There is thus not much dependence of the associated in- From electric-charge conservation it is clear that any clusive limits on momentum of the heavy meson and if proton decay will eventually result in at least one the decay is two body or multi-body. Hence, limits from positron, albeit potentially space-time-delayed if it orig- exclusive searches with such mesons and other states be- inated from the decay of a heavier charged final-state ing invisible, such as τ(p ν +K+) > 5900 1030 yr [77] ∗,+ → 30 × particle (e.g. muon or pion). Hence, p e± + anything and τ(p ν + K ) > 130 10 yr [78] from SK as → + × 30 is an important general inclusive search.→ Since the well as τ(p ν + ρ ) > 162 10 yr from IMB [65], → × positron will almost always have its momentum above can be readily re-interpreted as inclusive limits on p ± ± ∗,± → the Cherenkov threshold, unless the number of final-state (K , ρ ,K )+anything processes. particles is very high, a particularly promising sensitiv- We expect that SK can improve by more than an order ity for this general inclusive search is achievable from a over the existing IMB limits and even further improve- re-analysis of a one-ring e-like data sample in SK and ment can be achieved with future HK. Further, JUNO subsequently HK. The current best lifetime limit on the and DUNE are particularly promising for complemen- N e+ + anything search is 0.6 1030 yr, from a tary studies of these modes since they allow for an iden- → × 12 tification of the low-energy heavy charged mesons that Nevertheless, this search would provide some compli- are below Cherenkov threshold, with a particularly good mentary to the low-energy γ searches from de-excitation efficiency achievable for kaons. and is also close to the inclusive search on e±, as both result in an e-like ring in WC experiments.

2. N → (π0,K0, η, ρ, ω, K∗,0) + anything 4. N → ν + anything Inclusive searches involving neutral mesons M 0 rely on the electromagnetically interacting daughter particles of As neutrinos escape the detector before interacting, M 0 for detection. For example, in the rest frame π0 this search is effectively invisible if nucleon decays are → γγ decay results in photon lines of energy mπ/2, clearly considered to occur within the experimental fiducial vol- visible in WC detectors. For slow π0 these are back to ume. However, it has been suggested that by considering back, otherwise more collimated. Already in the two- such decays on macroscopic Earth-sized scales the addi- body nucleon decay the pion is not very energetic and tional cumulative neutrino flux from decays can be signif- carries energy below mN /2. Hence, the inclusive limit icant and observable. By attributing the experimentally 0 n π +anything is not particularly sensitive to pion observed νµ/νe fluxes to nucleon decays in Earth a limit momentum→ and one can estimate it to be of similar order of 1026 yr was estimated [111]. However, a further re- as the limit from existing SK search for n π0ν of analysis∼ by the Fréjusexperiment using 2 kton yr of 1100 1030 yr [76]. → data yielded a lower limit of 1025 yr [62] (see× their The× above considerations are even more applicable to Sec. 7 for flux estimation). Since∼ SK has already col- neutral mesons heavier than pion (i.e. K0, η, ρ0, ω, K∗,0), lected over 370 kton yr of data (e.g. [72]), we anticipate which can be approximated as decaying at rest. Hence, that an improved sensitivity× of up to 1028 yr could be the search limits of τ(n ν + K0) > 130 1030 yr [74] achieved by re-analysis of this channel∼ in SK using the from SK, τ(n ν + η) →> 158 1030 yr [65]× from IMB, full data set and even further improved upon in future τ(n ν + ρ0)→> 19 1030 yr [79]× from IMB, τ(n ν + HK. We note that exact search details depend on the ω) >→108 1030 yr [65]× from IMB and τ(n ν +K→∗,0) > assumptions about the flavor of emitted neutrinos. 78 1030×yr [65] from IMB can provide an→ estimate of magnitude× for sensitivity to such inclusive modes. As before, we expect that a re-analysis of the above V. PROCESSES WITH ∆B > 1 modes with the SK data-set can improve IMB’s limits by more than an order of magnitude and even better limits So far we have focused on processes violating baryon can be expected from HK. number by one unit, ∆B = 1, which are described by the lowest-dimension ∆B = 0 SMEFT operators. However, nucleon decays also constitute6 sensitive probes of dimen- 3. N → γ + anything sion d 6 operators beyond ∆B = 1. Such “multinucleon decays” with ∆B > 1 can be treated in complete Since we are envisioning multi-body nucleon decays, analogy to the processes discussed in previous sections. the emission of a photon is typically suppressed com- The main difference is that ∆B > 1 processes have more pared to the same channel without a photon. It is still available energy, which allows for the on-shell produc- useful to perform such an analysis for final states that are tion of tau leptons [114] as well as heavier mesons such otherwise difficult to detect, such as n γ + neutrinos. as D and φ, thus increasing the number of possible fi- As emphasized in Ref. [112], neutron decay→ into neutri- nal states. Appearance of visible heavier charged mesons nos corresponds to the sudden disappearance of the neu- such as K±, which are always below Cherenkov thresh- tron’s magnetic moment and should thus lead to elec- old when originating from single nucleon decays, also be- tromagnetic radiation. From a particle-physics perspec- comes possible. Only a few channels out of all possible tive this effect can be readily obtained from attaching a kinematically allowed two-body ∆B = 2 di-nucleon de- photon to the initial quarks in any diagram leading to cays have been experimentally studied thus far (17 out n neutrinos, in analogy to radiative τ decays [113]. of 118), as we summarize in Tabs. IV–VI. Even more Compared→ to low-energy de-excitation photons, these limited are searches for ∆B > 2 processes, as we briefly photons can reach energies of up to mN /2 and thus pro- comment on in Sec. V C. vide a far cleaner signature. However, with the probabil- Let us reiterate an important point: ∆B > 1 processes ity that a photon emitted from this process has energy do not need to be suppressed compared to ∆B = 1 nu- > 100 MeV being only 5 10−5 [112], it is unlikely cleon decays. In fact, it is possible to completely forbid that such a search will be∼ more× sensitive than the n all ∆B < n processes by a global symmetry, making invisible limit of Eq. (15). Further, the photon spectrum→ ∆B| = |n the dominant baryon number violating pro- here depends on the details of the final state and number cess [14]. This already occurs within the SM, where of emitted neutrinos, implying that obtained limits from baryon number is only broken by three units, so the such a search are less general than Eq. (15). proton remains stable. Models in which ∆B = 2 pro- 13

Γ−1 Γ−1 Channel |∆(B − L)| 1030 yr Channel |∆(B − L)| 1030 yr pp → e+ + e+ 0 4200 [72] nn → π0 + φ 2 pp → µ+ + µ+ 0 4400 [72] nn → 2η 2 pp → e+ + µ+ 0 4400 [72] nn → η + ρ0 2 pp → e+ + τ + 0 nn → η + ω 2 pp → π+ + π+ 2 72 [115] nn → η + η0 2 pp → π+ + ρ+ 2 nn → η + K0 2 pp → π+ + K+ 2 nn → η + K∗,0 2 pp → π+ + K∗,+ 2 nn → η + φ 2 pp → ρ+ + ρ+ 2 nn → 2ρ0 2 pp → ρ+ + K+ 2 nn → ρ0 + ω 2 pp → ρ+ + K∗,+ 2 nn → η0 + ρ0 2 pp → K+ + K+ 2 170 [116] nn → K0 + ρ0 2 pp → K+ + K∗,+ 2 nn → K∗,0 + ρ0 2 pp → K∗,+ + K∗,+ 2 nn → ρ0 + φ 2 − + nn → e+ + e− 2 4200 [72] nn → ρ + ρ 2 + − nn → e+ + µ− 2 4400 [72] nn → K + ρ 2 ∗,+ − nn → µ+ + e− 2 4400 [72] nn → K + ρ 2 − + nn → µ+ + µ− 2 4400 [72] nn → K + ρ 2 ∗,− + nn → e+ + τ − 2 nn → K + ρ 2 nn → τ + + e− 2 nn → 2ω 2 0 nn → 2ν 0,2,4 1.4 [83] nn → η + ω 2 0 nn → 2γ 2 4100 [72] nn → K + ω 2 ∗,0 nn → γ + π0 2 nn → K + ω 2 nn → γ + η 2 nn → ω + φ 2 0 0 nn → γ + ρ0 2 nn → η + K 2 0 ∗,0 nn → γ + ω 2 nn → η + K 2 − + ∗ nn → γ + η0 2 nn → K + K 2 170 [116] + ∗,− nn → γ + K0 2 nn → K + K 2 − ∗,+ nn → γ + K∗,0 2 nn → K + K 2 0 nn → γ + D0 2 nn → 2K 2 ∗,0 0 nn → γ + φ 2 nn → K + K 2 0 nn → π− + π+ 2 0.7 [62] (72∗ [115]) nn → K + φ 2 ∗,0 nn → π+ + ρ− 2 nn → 2K 2 ∗,− ∗,+ nn → K− + π+ 2 nn → K + K 2 nn → K∗,− + π+ 2 − + nn → π + ρ 2 TABLE V. Extension of Tab. IV. nn → K+ + π− 2 nn → K∗,+ + π− 2 nn → 2π0 2 404 [115] cesses dominate over ∆B = 1 have been discussed extensively in the literature (e.g. [12, 117–120]). In addi- nn → η + π0 2 0 0 tion to total baryon and lepton number as selection rules nn → π + ρ 2 for allowed processes [14] one can further consider lepton 0 nn → π + ω 2 ﬂavor [34, 42], which is the only other global quantum nn → η0 + π0 2 number not violated in SM interactions. nn → K0 + π0 2 nn → K∗,0 + π0 2 A. ∆B = 2, ∆L = 0

TABLE IV. Same as Tab. I, but listing all kinematically allowed two-body di-nucleon ∆B = 2 decays with 90% C.L. up- As illustrated in Fig. 1, ∆B = 2 processes start at per limit on the partial decay width Γ. d = 9 and do not involve leptons. The corresponding operators, such as uddudd, have been discussed at length in 14

Γ−1 Channel |∆(B − L)| 30 similarly np, pp mesons decays. While more compli- 10 yr → pn → e+ + ν 0,2 260 [28] cated multi-meson final states are also expected, calculations of their branching ratios have not been exten- pn → µ+ + ν 0,2 200 [28] sively performed. Experimentally, only a small subset of + pn → τ + ν 0,2 29 [28] such possible two-body final states has been searched for, pn → γ + π+ 2 namely nn 2γ, NN 0 ππ and NN 0 KK, with the pn → γ + ρ+ 2 latter being→ motivated by→ SUSY [125, 126].→ pn → γ + K+ 2 ∆B = 2 operators with ∆L = 0 involving leptons arise ∗,+ pn → γ + K 2 at d = 13 and should be suppressed compared to the pn → γ + D+ 2 d = 9 operators unless they carry lepton flavor [34, 42], pn → π+ + π0 2 170 [115] leading for example to nn µ+e− or nn µ+τ −. The → → pn → η + π+ 2 tau modes have not been tested thus far, but since these pn → π+ + ρ0 2 channels have a fully visible final state they should allow for better sensitivity than the already performed np pn → π+ + ω 2 τ +ν search in SK [28], which gave a limit of 29 1030 yr.→ pn → η0 + π+ 2 × 0 + The invisible di-nucleon decay nn neutrinos can pn → K + π 2 → pn → K∗,0 + π+ 2 be searched in a way analogous to the ∆B = 1 case by looking for nuclear de-excitation emission [127]. The pn → π+ + φ 2 0 + current best limits come from KamLAND [83], with a pn → π + ρ 2 30 limit τnn > 1.4 10 yr. Slightly lower limits around + 0 pn → K + π 2 a few 1028 yr have× been obtained by SNO+ [103] for pn → K∗,+ + π0 2 np ×invisible and pp invisible modes. We envi- → → pn → η + ρ+ 2 sion that JUNO will be able to test these modes with pn → η + K+ 2 lifetime sensitivity over an order of magnitude stronger 31 pn → η + K∗,+ 2 than KamLAND, approaching 10 yr. Dedicated studies of sensitivity to these modes in DUNE, JUNO, and HK pn → ρ+ + ρ0 2 would provide valuable complementary information. pn → K+ + ρ0 2 pn → K∗,+ + ρ0 2 pn → ρ+ + ω 2 pn → η0 + ρ+ 2 pn → K0 + ρ+ 2 B. ∆B = 2, ∆L = ±2 pn → K∗,0 + ρ+ 2 pn → ρ+ + φ 2 pn → K+ + ω 2 At dimension d = 12, we find ∆B = ∆L = 2 oper- pn → K∗,+ + ω 2 ators, an example being uuuuddee. For first-generation pn → η0 + K+ 2 quark-flavor indices, this simply induces the highly visi- pp `+`0+ pn → η0 + K∗,+ 2 ble , which has been discussed in the context of GUTs→ [128]. While the electron and muon modes have pn → K+ + K0 2 been searched for in SK [72], the one kinematically avail- + ∗,0 pn → K + K 2 able tau mode pp e+τ + has not. Similar to the case pn → K+ + φ 2 of nn µ+τ − we→ expect that SK can probe its lifetime pn → K∗,+ + K0 2 above→ 1031 yr and better than the np τ +ν search [28]. → pn → K∗,+ + K∗,0 2 HK will allow us to considerably improve on studies of these channels. We note that taking the quark indices of the d = 12 operators into account can lead to addi- TABLE VI. Extension of Tabs. IV and V. tional emission of kaons as well as other mesons in the final state, which we do not list here. Still at dimension d = 12, we find ∆B = ∆L = 2 the literature [117, 118, 121, 122]. Experimentally, they operators such as dddddde¯e¯. Here, the dominant− de- have been probed through neutron–antineutron (n–n) os- cay mode is nn `−`−π+π+, potentially with kaons cillations [123] as well as deuteron decays [124] and other instead of pions.→ This competes with the loop-induced di-nucleon decays such as nn ππ. Taking into account nn νν, for which experimental limits exist. Other the possible quark-flavor structure→ of these operators it than→ this invisible di-neutron decay, no two-body decay is clear that they can induce a variety of nn MM 0 fulfils ∆B = ∆L = 2, highlighting the importance of processes, with M and M 0 being mesons or photons,→ and multi-body final− states and inclusive searches. 15

C. ∆B > 2 landscape of possible baryon-number-violating processes exists beyond them. Baryon number violation by more than two units has Nucleon decay provides a unique opportunity to test not been well explored, with sparse theoretical [129] and baryon-number-violating operators of very high dimen- experimental [130–133] studies. As mentioned before, sionality and even beyond ∆B = 1, achieving sensitivity one particularly well-motivated case that falls into this not available with other techniques. These operators can category is ∆B = ∆L = 3, as mediated in the SM by dominate over the conventionally considered lower-order non-perturbative electroweak instantons. At low temper- dimension-six terms due to their lepton number and fla- atures and energies as in nuclear decays these processes vor quantum numbers. Going beyond the lower order, are highly suppressed, while rate calculations in high- a vast amount of additional possible operators can start energy collider setups are notoriously difficult [134, 135]. to contribute. Generically, such operators will result in While we do not discuss the various ∆B > 2 final states complicated nucleon decays with a multi-body final state. and operators (starting at d = 15) further, we stress that Most of the conducted nucleon decay searches have inclusive searches can easily cover these processes as well, been exclusive, focusing on decay processes with a par- as we describe below. ticular final state. We have identified a slew of single nucleon decay processes yet to be tested with exclusive studies. Further, exclusive searches for relatively clean D. Inclusive searches for ∆B > 1 channels like p `+`−`+ or n `−K+π0 could provide information indicative→ of d = 9→ and 10 operators. As discussed above, there is still significant untested Exclusive studies rapidly become unfeasible to test nu- parameter space for ∆B 2 processes, including about cleon decay beyond the simplest channels. As we argued a hundred ∆B = 2 two-body≥ final states with fixed in this work, a particularly important venue to further kinematics (see Tabs. IV, V, and VI). While exclusive explore baryon number violation in the future is through studies of inclusive nucleon decays N X+anything searches for these channels would clearly provide the → best sensitivity, inclusive searches again offer an inter- (where X is a light SM particle with unknown energy esting alternative. A particularly relevant aspect of the distribution), as well as model-independent and invisible decay searches such as n neutrinos (which can carry proposed inclusive ∆B = 1 searches from Sec. IV B to → stress is that they will automatically provide limits for away an arbitrary lepton number). Such searches allow ∆B > 1 channels. For example, an inclusive search for us to constrain multiple complicated nucleon decay chan- + nels simultaneously and broadly cover dimension d 9 p e +anything with positron energy between MeV ≥ → + ∼ baryon-number-violating operators. More so, inclusive and mp/2 would also constrain np e + anything or nnp e+ +anything. In comparison,→ dedicated ∆B > 1 ∆B = 1 searches will allow us to gain some insight into inclusive→ searches would allow us to widen the search win- ∆B > 1 processes as well. dow to positron energies above mp/2 or look for on-shell For the searches outlined above, we foresee that the tau leptons. WC detectors SK and HK will in general provide the fur- thest reach in sensitivity due to their unparalleled size. However, we stress that searches involving heavier final- VI. CONCLUSIONS state charged particles, e.g. kaons, can significantly ben- efit from studies in JUNO and DUNE. Further, the low- Baryon number violation is strongly motivated from energy sensitivity of JUNO and DUNE could be particu- many distinct theoretical considerations, including larly beneficial for nuclear de-excitation emission searches GUTs, supersymmetry, and baryogenesis. Thus, prob- (e.g. invisible decay channels). We are not aware of dedi- ing proton decay and other baryon-number-violating pro- cated sensitivity studies related to these aspects. Finally, cesses is of fundamental importance in order to learn let us stress that even though we envision nuclear decays as the best probe for ∆B = 0, other baryon-number- about physics beyond the SM. A slew of upcoming 6 large-scale experiments, in particular JUNO, DUNE, and violating searches as in meson or tau decays could pro- Hyper-Kamiokande, will be able to explore these provide important complementary information. cesses with unparalleled sensitivity. Hence, it is crucial to make the most of these efforts (as well as existing detectors, most notably SK) and probe as much parameter ACKNOWLEDGEMENTS space for baryon-number violation as possible. As we highlighted in this work, nucleon decay consti- We would like to thank Mu-Chun Chen, Huan Huang, tutes a premier probe of baryon number violation be- Ed Kearns, and Henry Sobel for discussions. The cause ∆B = 1 operators will result in nucleon decay re- work of J.H. is supported, in part, by the National gardless of their quark or lepton flavor structure. While Science Foundation under Grant No. PHY-1620638 most nucleon decay searches have focused on two-body and PHY-1915005 and by a Feodor Lynen Research decay channels (e.g. p e+π0), as motivated by minimal Fellowship of the Alexander von Humboldt Foundation. GUT models and simplified→ EFT considerations, a vast The work of V.T. is supported by the U.S. Department 16 of Energy (DOE) Grant No. DE-SC0009937. This Ref. [136] the authors estimate the branching ratios of work was performed in part at the Aspen Center for proton and neutron into two-body and some three-body Physics, which is supported by the National Science final states, induced by one of the d = 6 operators from Foundation grant PHY-1607611. J.H. further thanks Eq. (1), concluding that two-body decays currently pro- the CERN theory group for hospitality while this article vide the best limits, in particular p `+π0 and n νπ0. was finalized. A similar analysis is performed in Ref.→ [137] but for→ processes induced by d = 9 operators with ∆B = 2 and ∆L = 0. We agree with their conclusions, but stress once more that operators with d > 6 (d > 9) and differ- Note added: Recently, two articles came out that ent lepton numbers can significantly change the dominant bear relevance and are complimentary to our work. In nucleon (di-nucleon) decay modes.

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