On the Symmetry Foundation of Double Soft Theorems

Double-soft theorems, like its single-soft counterparts, arises from the underlying symmetry principles that constrain the interactions of massless particles. While single soft theorems can be derived in a non-perturbative fashion by employing current algebras, recent attempts of extending such an approach to known double soft theorems has been met with difficulties. In this work, we have traced the difficulty to two inequivalent expansion schemes, depending on whether the soft limit is taken asymmetrically or symmetrically, which we denote as type A and B respectively. We show that soft-behaviour for type A scheme can simply be derived from single soft theorems, and are thus non-preturbatively protected. For type B, the information of the four-point vertex is required to determine the corresponding soft theorems, and thus are in general not protected. This argument can be readily extended to general multi-soft theorems. We also ask whether unitarity can be emergent from locality together with the two kinds of soft theorems, which has not been fully investigated before.Comment: 45 pages, 7 figure

Assessing the detailed time course of perceptual sensitivity change in perceptual learning.

The learning curve in perceptual learning is typically sampled in blocks of trials, which could result in imprecise and possibly biased estimates, especially when learning is rapid. Recently, Zhao, Lesmes, and Lu (2017, 2019) developed a Bayesian adaptive quick Change Detection (qCD) method to accurately, precisely, and efficiently assess the time course of perceptual sensitivity change. In this study, we implemented and tested the qCD method in assessing the learning curve in a four-alternative forced-choice global motion direction identification task in both simulations and a psychophysical experiment. The stimulus intensity in each trial was determined by the qCD, staircase or random stimulus selection (RSS) methods. Simulations showed that the accuracy (bias) and precision (standard deviation or confidence bounds) of the estimated learning curves from the qCD were much better than those obtained by the staircase and RSS method; this is true for both trial-by-trial and post hoc segment-by-segment qCD analyses. In the psychophysical experiment, the average half widths of the 68.2% credible interval of the estimated thresholds from the trial-by-trial and post hoc segment-by-segment qCD analyses were both quite small. Additionally, the overall estimates from the qCD and staircase methods matched extremely well in this task where the behavioral rate of learning is relatively slow. Our results suggest that the qCD method can precisely and accurately assess the trial-by-trial time course of perceptual learning