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The bispectrum is an important statistics helpful for measuring theprimordial non-Gaussianity parameter $f_{\mathrm{NL}}$ to less than order unityin error, which would allow us to distinguish between single and multi-fieldinflation models. The Spectro-Photometer for the History of the Universe, Epochof Reionization and Ices Explorer (SPHEREx) mission is particularly well-suitedfor making this measurement with its $\sim$100-band all-sky observations in thenear-infrared. Consequently, the SPHEREx data will contain galaxies withspectroscopic-like redshift measurements as well as those with much largererrors. In this paper, we evaluate the impact of photometric redshift errors on$f_{\mathrm{NL}}$ constraints in the context of an updated multi-tracerforecast for SPHEREx, finding that the azimuthal averages of the first threeeven bispectrum multipoles are no longer sufficient for capturing most of theinformation (as opposed to the case of spectroscopic surveys shown in theliterature). The final SPHEREx result with all five galaxy samples and sixredshift bins is however not severely impacted because the total result isdominated by the samples with the best redshift errors, while the worse samplesserve to reduce cosmic variance. Our fiducial result of$\sigma_{f_{\mathrm{NL}}} = 0.7$ from bispectrum alone is increased by $18\%$and $3\%$ when using $l_{\mathrm{max}}=0$ and 2 respectively. We also explorethe impact on parameter constraints when varying the fiducial redshift errors,as well as using subsets of multi-tracer combinations or triangles withdifferent squeezing factors. Note that the fiducial result here is not thefinal SPHEREx capability, which is still on target for being$\sigma_{f_{\mathrm{NL}}} = 0.5$ once the power spectrum will be included.

Measuring $f_{\mathrm{NL}}$ with the SPHEREx Multi-tracer Redshift Space Bispectrum