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Recently we found compelling evidence for a gravitational wave backgroundwith Hellings and Downs (HD) correlations in our 15-year data set. Thesecorrelations describe gravitational waves as predicted by general relativity,which has two transverse polarization modes. However, more general metrictheories of gravity can have additional polarization modes which producedifferent interpulsar correlations. In this work we search the NANOGrav 15-yeardata set for evidence of a gravitational wave background with quadrupolarHellings and Downs (HD) and Scalar Transverse (ST) correlations. We find thatHD correlations are the best fit to the data, and no significant evidence infavor of ST correlations. While Bayes factors show strong evidence for acorrelated signal, the data does not strongly prefer either correlationsignature, with Bayes factors $\sim 2$ when comparing HD to ST correlations,and $\sim 1$ for HD plus ST correlations to HD correlations alone. However,when modeled alongside HD correlations, the amplitude and spectral indexposteriors for ST correlations are uninformative, with the HD processaccounting for the vast majority of the total signal. Using the optimalstatistic, a frequentist technique that focuses on the pulsar-paircross-correlations, we find median signal-to-noise-ratios of 5.0 for HD and 4.6for ST correlations when fit for separately, and median signal-to-noise-ratiosof 3.5 for HD and 3.0 for ST correlations when fit for simultaneously. Whilethe signal-to-noise-ratios for each of the correlations are comparable, theestimated amplitude and spectral index for HD are a significantly better fit tothe total signal, in agreement with our Bayesian analysis.

The NANOGrav 15-year data set: Search for Transverse Polarization Modes in the Gravitational-Wave Background