# $C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories

## TL;DR

This paper confirms the correct value of the Weyl anomaly coefficient c_3 for higher derivative (1,0) superconformal 6d vector multiplets, linking it to the C_T coefficient and refining anomaly relations.

## Contribution

It provides an independent verification of the Weyl anomaly coefficient c_3 for (1,0) superconformal 6d vector multiplets, clarifying the value of the free parameter in anomaly relations.

## Key findings

- Confirmed the value of c_3 coefficient for the vector multiplet.
- Derived relations between c_3, C_T, and entropy derivatives.
- Computed conformal anomalies using Seeley-DeWitt coefficients.

## Abstract

In arXiv:1510.02685 we proposed linear relations between the Weyl anomaly $c_1, c_2, c_3$ coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories. These relations were determined up to one free parameter $\xi$ and its value was then conjectured using some additional assumptions. A different value for $\xi$ was recently suggested in arXiv:1702.03518 using a different method. Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet. This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar. We find the corresponding value of $c_3$ which is proportional to the coefficient $C_T$ in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space. We present some general results of computation of the Renyi entropy and $C_T$ from the partition function on $S^1 \times \mathbb H^{d-1}$ for higher derivative conformal scalars, spinors and vectors in even dimensions. We also give an independent derivation of the conformal anomaly coefficients of the higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.