2$ but has the standard gauge invariance for $p=2$. The $p=2$ case of $\SS^{(1,0)}_{p}$ is thus $V^{(1,0)}$ in~\eqref{19}. Note also that the formal $p=1$ case of field content in table~\ref{T4} is the same as the scalar multiplet $S^{(1,0)}$ in~\eqref{18} but with the opposite chirality of the fermion, i.e.\ $\SS^{(1,0)}_{1}=S^{(0,1)}$. Thus the chiral anomalies of $\SS^{(1,0)}_{1}$ will be opposite in sign to the anomalies of $S^{(1,0)}$. In what follows we shall assume that $p >1$, as required by the actual construction of $\SS^{(1,0)}_{p}$ as a tensor product. From the canonical dimensions of the fields one can determine the power of derivatives in kinetic terms in the corresponding 6d Lagrangian \be \la{31} \begin{split} \mc L=& \varphi\,\Box^{2p-3}\, \varphi+\overline\psi^{+}\slashed{\partial}^{4p-5}\psi^{+} +V_{m}\,\Box^{2p-2}\,V_{m}+\overline\psi^{-}\slashed{\partial}^{4p-3}\psi^{-}+ \varphi'\,\Box^{2p-1}\,\varphi' \, , \end{split} \ee where each field is assumed to transform under $\SU(2)_{R}$ according to representations in table~\ref{T4}.\footnote{From the kinetic terms we can compute the corresponding numbers of dynamical d.o.f. for bosons and fermions ($p>2$) $\qquad \nu_{b} = (p+1)(2p-3)+(p-3)(2p-1)+6\,(p-1)(2p-2) = 4(2p-3)(2p-1), $ \noindent $ \nu_{f} = -2[p(4p-5)+(p-2)(4p-3)] = -4(2p-3)(2p-1), $ so that $\nu_{b}+\nu_{f}=0$. The above expressions are formally true also for $p=2$ when the vector is a gauge-invariant one but there is no second scalar $\vp'$ and this has the same effect as subtracting the gauge mode of $V_{m}$. } Let us first compute the corresponding chiral $\SU(2)_R$ and gravitational anomalies. The gravitational anomaly coefficients $\g,\d$ in~\eqref{11} here get contributions from the chiral fermions $\psi^+$ and $\psi^-$ and do not depend on powers of derivatives in the kinetic terms, i.e.\ are the same as for the usual $\slashed{\partial} $ fermions. As we have $p$ of $\psi^+$ fermions and $p-2$ of $\psi^-$ fermions the total contribution is as of $p - (p-2) =2$ positive-chirality fermions, i.e.\ it is the same as for the standard vector multiplet or for $V^{(1,0)}$ in~\eqref{19} (see table~\ref{T2}). Thus the $\g,\d$ coefficients are $p$-independent \be\la{32} \SS^{(1,0)}_{p}:\qquad \qquad \gamma \te = -\frac{7}{240},\qquad \qquad \delta = \frac{1}{60} \, . \ee The coefficients $\a$ and $\b$ in~\eqref{11} are proportional to the sums of second and fourth powers of the R-charges.\footnote{Here we consider the Cartan $\U(1)$ subgroup of the $\SU(2)_{R} $ as the 4th order anomaly $c_{2}^{2}$ is determined by the sum of fourth powers of the $J_{z}$ eigenvalue, see, e.g.,~\cite{Kumar-ml-2010am}.} Taking again into account that the chiral anomaly receives only the contributions of the two opposite chirality fermions in table~\ref{T4}, we get \be\la{33} \begin{split} \sum r^{2} &= \sum_{k=0}^{p-1}{\te \left(-\frac{p-1}{2}+k\right)^{2}}- \sum_{k=0}^{p-3}{\te \left(-\frac{p-3}{2}+k\right)^{2}} =\te \frac{1}{2}(p-1)^{2}, \\ \sum r^{4} &= \sum_{k=0}^{p-1}{\te \left(-\frac{p-1}{2}+k\right)^{4}}- \sum_{k=0}^{p-3}{\te \left(-\frac{p-3}{2}+k\right)^{4}} = \te \frac{1}{8}(p-1)^{4}. \end{split} \ee Including the normalization constants so that for $p=2$ we recover the vector multiplet values in table~\ref{T2}, we get \be \label{34} \SS^{(1,0)}_{p}: \qquad\qquad \alpha = -(p-1)^{4}\, ,\qquad \qquad \beta = -\frac{1}{2}(p-1)^{2} \, . \ee As expected, for $p=1$ the expressions for the chiral anomalies in~\eqref{32} and~\eqref{34} are the same as for $S^{(1,0)}$ in table~\ref{T2} up to an opposite overall sign. Next, let us turn to the conformal anomalies. To compute the conformal anomaly a-coefficient corresponding to $\SS^{(1,0)}_{p}$ we may use the general expression for $\aa$ of a field associated to a representation $(\Delta; h_1,h_2,h_3)$ of the conformal group $\SO(2,6)$ given in~\cite{Beccaria-ml-2014qea,Beccaria-ml-2015uta}:\foot{Here ${\bf{h}}=(h_1,h_2,h_3)$, \ $\bh= h_1 + h_2 + h_3$ and the dimension $\dd(\mathbf{h})$ of the $\SO(6)$ representation $\mathbf{h}$ is $ \dd(\mathbf{h}) =\frac{1}{12}(1+h_{1}-h_{2})(1+h_{2}-h_{3})(1+h_{2}+h_{3}) (2+h_{1}-h_{3}) (2+h_{1}+h_{3})(2+h_{1}+h_{2}) $. Note that $$ {\del \aa(\Delta;{\bf{h}}) \ov \del \Delta}= {\te \frac{(-1)^{2\bh} \dd(\bfh)}{96\times360}} (\Delta - h_1 + 5) (\Delta + h_1 -1) (\Delta - h_2 -4 ) (\Delta + h_2 -2) (\Delta - h_3 -3) (\Delta + h_3 -3)\, . $$ } \ba\no & \aa(\Delta;{\bf{h}}) = -\frac{(-1)^{2\bh} \dd(\bfh)}{96\times37800} \, (\Delta -3) \Big[15 (\Delta -3)^6 \\ & \qquad \qquad \qquad -21 (\Delta -3)^4 \left[h_3^2+h_1 \left(h_1+4\right)+h_2 \left(h_2+2\right)+5\right]\no \\ \no & \qquad \qquad \qquad + 35 (\Delta -3)^2 \big[\left(h_1+2\right)^2 \left(h_2+1\right)^2+\big(h_1 \left(h_1+4\right)+h_2 \left(h_2+2\right)+5\big) h_3^2\big] \\ & \qquad \qquad \qquad - 105 \left(h_1+2\right)^2 \left(h_2+1\right)^2 h_3^2\Big] \, . \la{35} \end{align} Summing up contributions of all fields with their multiplicities in table~\ref{T4}, we get\foot{Explicitly, the separate contributions $\aa(\Delta; h_1,h_2,h_3)$ are \be\notag \begin{split} &\aa(2p; 0,0,0) =\te -\frac{1}{1890}(p-1)^{7}+\frac{1}{540}\,(p-1)^{6}-\frac{1}{540}\,(p-1)^{5} \te +\frac{11}{12960}\,(p-1)^{3}-\frac{1}{2880}\,(p-1)^{2}+\frac{1}{72576}, \\ &\aa\bigg(2p+\frac{1}{2}; \frac{1}{2},\frac{1}{2},-\frac{1}{2}\bigg) \te = \frac{2}{945}\,(p-1)^{7}-\frac{1}{270}\, (p-1)^{6}-\frac{1}{270}\,(p-1)^{5}\te +\frac{1}{144}\,(p-1)^{4} +\frac{1}{810}\,(p-1)^{3}\\ &\qquad \qquad \qquad \qquad \te -\frac{1}{360}\,(p-1)^{2}+\frac{191}{1451520}, \\ &\aa(2p+1; 1,0,0) = \te -\frac{1}{315}\,(p-1)^{7}+\frac{1}{90}\,(p-1)^{5}-\frac{1}{240}\,(p-1)^{3}, \\ &\aa\bigg(2p+\frac{3}{2}; \frac{1}{2},\frac{1}{2},\frac{1}{2}\bigg) =\te \frac{2}{945}\,(p-1)^{7}+\frac{1}{270}\,(p-1)^{6}-\frac{1}{270}\,(p-1)^{5}-\frac{1}{144}\,(p-1)^{4} +\frac{1}{810}\,(p-1)^{3}\\ &\te \qquad \qquad \qquad \qquad+\frac{1}{360}\,(p-1)^{2}-\frac{191}{1451520}, \\ &\aa(2p+2; 0,0,0) =\te -\frac{1}{1890}\,(p\!-\!1)^{7}\!-\!\frac{1}{540}\,(p\!-\!1)^{6}\!-\!\frac{1}{540}\,(p\!-\!1)^{5} \!+\!\frac{11}{12960}\,(p\!-\!1)^{3}\!+\!\frac{1}{2880}\,(p\!-\!1)^{2}\!-\!\frac{1}{72576}. \end{split} \ee } \be \label{36} \SS^{(1,0)}_{p}: \qquad \qquad \aa =\te \frac{1}{72}\,(p-1)^{4}-\frac{1}{144}\,(p-1)^{2}+\frac{11}{34560} \, . \ee For $p=2$ this gives the value for $V^{(1,0)}$ in~\eqref{20}.\foot{The expression~\eqref{36} was derived assuming $p >1$; still, if we formally set $p=1$ we get the value opposite to the value of $S^{(1,0)}$ in table~\ref{T1} while the conformal anomaly of $\SS^{(1,0)}_{1}=S^{(0,1)}$ (cf.\ table~\ref{T4}) should be the same as of $S^{(1,0)}$.} It is straightforward also to directly compute the conformal anomaly on a Ricci flat background by assuming that Weyl-covariant kinetic operators appearing in the generalization of the Lagrangian~\eqref{31} to curved background factorize into a product of the standard 2-derivative operators (cf.\ appendix~\ref{A}). We then find \ba \SS^{(1,0)}_{p}: \ \ A_6\big|_{R_{mn}=0} &= \big[(p+1)\,(2p-3)+(p-3)\,(2p-1)\big]\, A_6(\varphi) +(p-1)\,(2p-2)\, A_6(V) \no \\ &\ \ \ \ + \big[p\,(4p-5)+(p-2)\,(4p-3)\big]\, A_6(\psi) =\te -\frac{1}{11520}\, E_{6} \, . \la{37} \end{align} Here $A_6(\varphi)$, $A_6(V) $ and $A_6(\psi) $ are conformal anomalies for standard Laplacians defined on scalars, vectors and spinors on a Ricci flat background. This expression was derived for $p >2$ but is also valid for $p=2$, i.e.\ for $V^{(1,0)}$ when the vector is a gauge-invariant one. Comparing~\eqref{37} to the general expression in~\eqref{6} and using the expression for $\aa$ in~\eqref{36} we conclude that in addition to the expected relation~\eqref{7}, i.e.\ $\cc_3= {1 \ov 6} (\cc_{1}-2\cc_{2})$, we get \be \SS^{(1,0)}_{p}: \qquad\qquad \cc_1 + 4 \cc_{2} =192 \aa - \te \frac{1}{60}= { 8 \ov 3} (p-1)^4 - { 4 \ov 3} (p-1)^4 + { 2 \ov 45} \, . \la{38} \ee Using the results for the chiral anomalies in~\eqref{32},~\eqref{34} we conclude that the value of the a-coefficient in~\eqref{36} is indeed consistent with the general expression~\eqref{12} of~\cite{Cordova-ml-2015fha}. Our suggested expressions~\eqref{14} for $\cc_1$ and $\cc_2$ give \be \la{39} \begin{split} \te \cc_{1} = \frac{4}{3}\,(p-1)^{4}-\frac{2}{3}\,(p-1)^{2}+\frac{1}{27}\, , \qquad \qquad \cc_{2} =\frac{1}{3}\,(p-1)^{4}-\frac{1}{6}\,(p-1)^{2}+\frac{1}{540} \, . \end{split} \ee These are indeed consistent with~\eqref{38}, thus providing an additional test of eq.~\eqref{14}. Let us note also that $\cc_{1}-4\,\cc_{2} = \frac{4}{135}$ is independent of $p$, i.e.\ is the same as for $V^{(1,0)}$.\foot{Once again, for $p=1$ the $\cc_i$ in~\eqref{39} are opposite to the values corresponding to $S^{(1,0)}$ in table~\ref{T1} which is due to the fact that we applied the relations between chiral and conformal anomalies valid for $(1,0)$ multiplets while the $p=1$ example is formally a $(0,1)$ scalar multiplet (cf.\ footnote above eq.~\eqref{12}).} ]]>

2$ level corresponds to higher derivative ``massive" $(2,0)$ superconformal multiplet in 6d (which we shall denote as CSG$_p$).\foot{Fields of ``massive" conformal group representations do not have gauge invariances that are present in the ``massless" case.} Equivalently, just like the higher derivative vector multiplets in table~\ref{T4} appear in the product of $p$ copies of $(1,0)$ hypermultiplet $S^{(1,0)}$, the \csg multiplets appear in the product of $p$ copies of $(2,0)$ tensor multiplet $T^{(2,0)}$~\cite{Gunaydin-ml-1984wc}. Below we shall compute the chiral (gravitational and R-symmetry) anomalies and the conformal anomaly $\cc$-coefficient in~\eqref{8} of these $(2,0)$ \csg multiplets complementing the result for their conformal anomaly $\aa$-coefficient found in~\cite{Beccaria-ml-2014qea,Beccaria-ml-2015uta}. The anomaly coefficients will all be proportional to the same factor $ 6 p (p-1) +1$ and will satisfy the expected relations~\eqref{16},~\eqref{17},~\eqref{77}. In particular, we will confirm the conjecture of~\cite{Beccaria-ml-2015uta} that the system of $(2,0)$ conformal supergravity coupled to 26 tensor multiplets is completely anomaly free. ]]>