al mass and momentum distributions are smeared out in a “tail" due to the correlated motion of a nearby nucleon. For both nucleons and pairs of nucleons, we assume that 10$\%$ of such decays are affected by the correlated motion of an additional nucleon [@1999corr]. Lepton rescattering within the nucleus is negligible.

The atmospheric $\nu$ MC sample corresponds to an exposure of 500 years for each of the four SK periods, 2000 years in total. Events in this sample are weighted assuming two-flavor mixing as is done in recent dinucleon and nucleon analyses [@2017mine; @2016miura; @2015jeff]. Details of the cross-sections and flux modeling used in this sample are discussed in recent SK nucleon decay analyses [@2016miura; @2017mine]. Event rates obtained from this sample are normalized to the relevant SK detector livetime.

Details of the neutron simulation and neutron tagging algorithm used for both the signal and atmospheric $\nu$ MC samples can be found in Ref. [@2016miura]. Neutron tagging can only be done for the SK-IV period; it reduces the expected number of background events by about 50% for our searches, and impacts signal efficiency by only a few percent.

![image](Fig2.png)

![(color online) Total mass ($M_{tot}$) and total momentum ($P_{tot}$) projections for $p \rightarrow \mu^+ \gamma$ after cut (A4). The red histogram shows atmospheric $\nu$ MC corresponding to 2000 years of SK exposure normalized to SK-I through SK-IV data. The selection criteria are indicated by the vertical blue lines.[]{data-label="fig:projection"}](Fig3.png)

Although the selection criteria for all ten modes are similar, the two single-nucleon decay modes have more background due to their lower total mass. We adapt our strategy, as is done in Ref. [@2016miura], to perform a two-box analysis which allows us to study free and bound protons separately.

The following selection criteria are applied to signal MC, atmospheric $\nu$ MC, and data:

1.  [Events must be fully contained in the inner detector with the event vertex wit