$\displaystyle\frac{3}{2}\delta(1-z)+2\mathcal{D}_{0}(z)-1-z\,,$
$\displaystyle p^{(0)}_{qg}(z)$ $\displaystyle=$ $\displaystyle
1-2z+2z^{2}\,,$ $\displaystyle p^{(0)}_{gq}(z)$ $\displaystyle=$
$\displaystyle\frac{2}{z}-2+z\,,$ $\displaystyle p^{(0)}_{gg}(z)$
$\displaystyle=$
$\displaystyle\frac{11}{6}\delta(1-z)+2\mathcal{D}_{0}(z)+\frac{2}{z}-4+2z-2z^{2}\,,$
$\displaystyle p^{(0)}_{gg,F}(z)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\delta(1-z)\,,$ $\displaystyle p^{(0)}_{\gamma
q}(z)$ $\displaystyle=$ $\displaystyle\frac{2}{z}-2+z\,,$ (133)

and the next-to-leading quark-to-photon and gluon-to-photon splitting
functions read

$\displaystyle p^{(1)}_{\gamma q}(z)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{9}{2}z+\left(-8+\frac{1}{2}z\right)\log
z+2z\log(1-z)+\left(1-\frac{1}{2}z\right)\log^{2}z$
$\displaystyle+\left[\log^{2}(1-z)+4\log
z\log(1-z)+8\text{Li}_{2}(1-z)-\frac{4}{3}z\right]p^{(0)}_{\gamma q}(z)\,,$
$\displaystyle p^{(1)}_{\gamma g}(z)$ $\displaystyle=$
$\displaystyle-2+6z-\frac{82}{9}z^{2}+\frac{46}{9z}+\left(5+7z+\frac{8}{3}z^{2}+\frac{8}{3z}\right)\log
z$ (134) $\displaystyle+(1+z)\log^{2}z\,.$

##  Appendix B Integrated $X^{0}_{3}$ Fragmentation Antenna Functions

We express the integrated fragmentation antenna functions in terms of
splitting functions (133) and colour-ordered infrared singularity operators,
which read

$\displaystyle\mathbf{I}^{(1)}_{q\bar{q}}(\epsilon,s_{q\bar{q}})$
$\displaystyle=$
$\displaystyle-\frac{e^{\epsilon\gamma_{E}}}{2\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}\right]{\mathcal{R}}(-s_{q\bar{q}})^{-\epsilon}\,,$
$\displaystyle\mathbf{I}^{(1)}_{qg}(\epsilon,s_{qg})$ $\displaystyle=$
$\displaystyle-\frac{e^{\epsilon\gamma_{E}}}{2\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{5}{3\epsilon}\right]{\mathcal{R}}(-s_{qg})^{-\epsilon}\,,$
$\displaystyle\mathbf{I}^{(1)}_{gg}(\epsilon,s_{gg})$ $\displaystyle=$
$\displaystyle-\frac{e^{\epsilon\gamma_{E}}}{2\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{11}{6\epsilon}\right]{\mathcal{R}}(-s_{gg})^{-\epsilon}\,,$
$\displaystyle\mathbf{I}^{(1)}_{q\bar{q},F}(\epsilon,s_{q\bar{q}})$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\mathbf{I}^{(1)}_{qg,F}(\epsilon,s_{qg})$ $\displaystyle=$
$\displaystyle\frac{e^{\epsilon\gamma_{E}}}{2\Gamma(1-\epsilon)}\frac{1}{6\epsilon}{\mathcal{R}}(-s_{qg})^{-\epsilon}\,,$
$\displaystyle\mathbf{I}^{(1)}_{gg,F}(\epsilon,s_{gg})$ $\displaystyle=$
$\displaystyle\frac{e^{\epsilon\gamma_{E}}}{2\Gamma(1-\epsilon)}\frac{1}{3\epsilon}{\mathcal{R}}(-s_{gg})^{-\epsilon}\,.$
(135)

The invariant masses that appear in these pole terms and in the normalisation
factors of the integrated antenna functions are always constructed from three-
parton invariants as $q^{2}=s_{12}+s_{13}+s_{23}$ and $Q^{2}=-q^{2}$.

###  B.1 Initial-Final Configuration

The unintegrated $X^{0}_{3}$ antenna functions in the initial-final
configuration were introduced in Daleo:2006xa . We recall their expressions
here and give the results for their integrated form differential in the final-
state momentum fraction.

The quark-initiated quark-quark antenna function in the initial-final
configuration reads

$A^{0}_{3}(\hat{1}_{q},3_{g},2_{q})=\frac{1}{s_{123}}\left(\frac{2s_{12}^{2}}{s_{13}s_{23}}+\frac{2s_{12}}{s_{13}}+\frac{2s_{12}}{s_{23}}+\frac{s_{23}}{s_{13}}+\frac{s_{13}}{s_{23}}\right)+\mathcal{O}(\epsilon)\,.$
(136)

For the integration of the fragmentation antenna function we need to specify
which parton in the final state is identified. In case the final-state gluon
is identified we find

$\displaystyle\mathcal{A}^{0,{\rm id.}g}_{3,\hat{q}}(x,z)$ $\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-x)p^{(0)}_{gq}(z)+\frac{1}{2}-\frac{x}{2}+\frac{z}{4}+\frac{xz}{4}+\frac{1}{2}z\delta(1-x)$
$\displaystyle+\left(-\frac{1}{4}-\frac{x}{4}+\frac{1}{2}\mathcal{D}_{0}(x)+\frac{1}{2}\delta(1-x)\left(\log(1-z)+\log(z)\right)\right)p^{(0)}_{gq}(z)\bigg{]}+\mathcal{O}(\epsilon)\,,$

and for the case of an identified final-state quark we have

$\displaystyle\mathcal{A}^{0,{\rm id.}q}_{3,\hat{q}}(x,z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{q\bar{q}}^{(1)}(\epsilon,-Q^{2})\delta(1-z)\delta(1-x)$
(138)
$\displaystyle+\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\left(\delta(1-z)p^{(0)}_{qq}(x)+\delta(1-x)p^{(0)}_{qq}(z)\right)+\frac{9}{16}\delta(1-z)\delta(1-x)$
$\displaystyle+\delta(1-z)\left(\frac{1}{2}-\frac{x}{2}+\mathcal{D}_{1}(x)-\frac{1}{2}(1+x)\log(1-x)-\frac{1+x^{2}}{2(1-x)}\log(x)\right)$
$\displaystyle+\delta(1-x)\left(\frac{1}{2}-\frac{z}{2}+\mathcal{D}_{1}(z)-\frac{1}{2}(1+z)\log(1-z)+\frac{1+z^{2}}{2(1-z)}\log(z)\right)$
$\displaystyle-\frac{3}{8}\left(\delta(1-z)p^{(0)}_{qq}(x)+\delta(1-x)p^{(0)}_{qq}(z)\right)+\frac{1}{4}p^{(0)}_{qq}(x)p^{(0)}_{qq}(z)$
$\displaystyle+\frac{3}{4}-\frac{x}{4}-\frac{z}{4}-\frac{xz}{4}\bigg{]}+\mathcal{O}(\epsilon)\,.$

In the subtraction of quark-photon collinear limits the antenna function
$A^{0}_{3}(\hat{1}_{q},3_{\gamma},2_{q})$ is used. Its unintegrated form
coincides with (136) and we have $\mathcal{A}^{0,{\rm
id.}\gamma}_{3,\hat{q}}=\mathcal{A}^{0,{\rm id.}g}_{3,\hat{q}}$.

The unintegrated $D$-type quark-initiated quark-gluon antenna function is

$\displaystyle D^{0}_{3}(\hat{1}_{q},2_{g},3_{g})$ $\displaystyle=$
$\displaystyle\frac{1}{s_{123}^{2}}\bigg{(}\frac{2s_{123}^{2}s_{13}}{s_{12}s_{23}}+\frac{2s_{12}s_{123}^{2}}{s_{13}s_{23}}+\frac{s_{123}s_{23}}{s_{12}}+\frac{2s_{12}s_{13}}{s_{23}}$
(139)
$\displaystyle+s_{12}+\frac{s_{123}s_{23}}{s_{13}}+4s_{123}+s_{13}\bigg{)}+\mathcal{O}(\epsilon)\,.$

It is symmetric under the exchange of gluons 2 and 3. Therefore, there is only
one corresponding integrated fragmentation antenna function, i.e.

$\displaystyle\mathcal{D}^{0,{\rm id.}g}_{3,\hat{q}}(x,z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{qg}^{(1)}(\epsilon,-Q^{2})\delta(1-x)\delta(1-z)$
(140)
$\displaystyle+\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\left(\delta(1-x)p^{(0)}_{gg}(z)+\delta(1-z)p^{(0)}_{qq}(x)\right)+\frac{11}{16}\delta(1-x)\delta(1-z)$
$\displaystyle+\delta(1-z)\left(\frac{1}{2}-\frac{x}{2}+\mathcal{D}_{1}(x)-\frac{1}{2}(1+x)\log(1-x)-\frac{1+x^{2}}{2(1-x)}\log(x)\right)$
$\displaystyle+\delta(1-x)\left(\mathcal{D}_{1}(z)+\left(-2+\frac{1}{z}+z-z^{2}\right)\log(1-z)+\frac{(1-z+z^{2})^{2}}{(1-z)z}\log(z)\right)$
$\displaystyle-\frac{3}{8}\delta(1-x)p^{(0)}_{gg}(z)-\frac{11}{24}p^{(0)}_{qq}(x)\delta(1-z)+\frac{1}{4}p^{(0)}_{gg}(z)p^{(0)}_{qq}(x)-1-\frac{1}{2x}-x+\frac{z}{2}$
$\displaystyle+\frac{z}{x}+\frac{xz}{2}-\frac{z^{2}}{2}-\frac{z^{2}}{x}-\frac{xz^{2}}{2}\bigg{]}+\mathcal{O}(\epsilon)\,.$

The three-quark quark-gluon antenna functions have the form

$\displaystyle E^{0}_{3}(\hat{1}_{q},2_{q^{\prime}},3_{\bar{q}^{\prime}})$
$\displaystyle=$
$\displaystyle\frac{1}{s_{123}^{2}}\left(\frac{(s_{12}+s_{13})^{2}}{s_{23}}-\frac{2s_{12}s_{13}}{s_{23}}+(s_{12}+s_{13})\right)+\mathcal{O}(\epsilon)\,,$
(141) $\displaystyle E^{0}_{3}(\hat{1}_{q^{\prime}},2_{q^{\prime}},3_{q})$
$\displaystyle=$
$\displaystyle-\frac{1}{s_{123}^{2}}\left(\frac{(s_{13}+s_{23})^{2}}{s_{12}}-\frac{2s_{13}s_{23}}{s_{12}}+(s_{13}+s_{23})\right)\,+\mathcal{O}(\epsilon)\,.$
(142)

The first antenna function is symmetric in the final-state quark pair. There
is one corresponding integrated fragmentation antenna function, i.e.

$\displaystyle\mathcal{E}^{0,{\rm id.}q^{\prime}}_{3,\hat{q}}(x,z)$
$\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-x)p^{(0)}_{qg}(z)-\frac{1}{2x}+\frac{p^{(0)}_{qg}(z)}{2x}+\frac{1}{2}\mathcal{D}_{0}(x)p^{(0)}_{qg}(z)$
$\displaystyle+\delta(1-x)\left(\frac{1}{2}-\frac{1}{2}p^{(0)}_{qg}(z)+\frac{1}{2}\left(\log(1-z)+\log(z)\right)p^{(0)}_{qg}(z)\right)\bigg{]}+\mathcal{O}(\epsilon)\,.$

The only unresolved limit of the antenna function in (142) is the flavour-
changing initial-final collinear limit. However, identifying the final-state
parton $q^{\prime}$ prevents it from becoming collinear to the initial-state
since any jet function will require a minimum transverse momentum of the
identified particle. Therefore, the only integrated fragmentation antenna
function corresponding to (142) identifies the final-state quark $q$. We find

$\displaystyle\mathcal{E}^{0,{\rm id.}q}_{3,\hat{q}^{\prime}}(x,z)$
$\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-z)p^{(0)}_{gq}(x)-\frac{1}{2}+\frac{x}{2}+\frac{1}{2}x\delta(1-z)$
$\displaystyle-\left(\frac{1}{2}-\frac{1}{2}\mathcal{D}_{0}(z)+\delta(1-z)\left(\frac{1}{2}-\frac{1}{2}\log(1-x)+\frac{\log(x)}{2}\right)\right)p^{(0)}_{gq}(x)\bigg{]}+\mathcal{O}(\epsilon)\,.$

The remaining quark-initiated antenna function is

$G^{0}_{3}(\hat{1}_{q^{\prime}},2_{q^{\prime}},3_{g})=-\frac{1}{s_{123}^{2}}\left(\frac{(s_{13}+s_{23})^{2}}{s_{12}}-\frac{2s_{13}s_{23}}{s_{12}}\right)+\mathcal{O}(\epsilon)\,.$
(145)

The $G^{0}_{3}$ antenna function at hand only contains the flavour-changing
initial-final limit. Using the same reasoning as for the $E^{0}_{3}$ antenna
function, we find only one integrated fragmentation antenna function:

$\displaystyle\mathcal{G}^{0,{\rm id.}g}_{3,\hat{q}^{\prime}}(x,z)$
$\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-z)p^{(0)}_{gq}(x)-\frac{1}{2}+\frac{x}{4}-\frac{z}{2}+\frac{xz}{4}+\frac{1}{2}x\delta(1-z)$
(146)
$\displaystyle-\left(\frac{1}{4}+\frac{z}{4}-\frac{1}{2}\mathcal{D}_{0}(z)+\delta(1-z)\left(\frac{1}{2}-\frac{1}{2}\log(1-x)+\frac{\log(x)}{2}\right)\right)p^{(0)}_{gq}(x)\bigg{]}$
$\displaystyle+\mathcal{O}(\epsilon)\,.$

The gluon-initiated quark-anti-quark antenna function is given by

$A^{0}_{3}(2_{\bar{q}},\hat{1}_{g},3_{q})=-\frac{1}{s_{123}}\left(\frac{2s_{23}^{2}}{s_{12}s_{13}}+\frac{s_{13}}{s_{12}}+\frac{s_{12}}{s_{13}}+\frac{2s_{23}}{s_{12}}+\frac{2s_{23}}{s_{13}}\right)+\mathcal{O}(\epsilon)$
(147)

and its integrated form with an identified quark reads

$\displaystyle\mathcal{A}^{0,{\rm id.}\,q}_{3,\hat{g}}(x,z)$ $\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-z)p^{(0)}_{qg}(x)-1+\frac{p^{(0)}_{qg}(x)}{2z}+\frac{1}{2}\mathcal{D}_{0}(z)p^{(0)}_{qg}(x)$
(148)
$\displaystyle-\frac{1}{2}\delta(1-z)\left(-1+p^{(0)}_{qg}(x)(\log(x)-\log(1-x))\right)\bigg{]}+\mathcal{O}(\epsilon)\,.$

As explained in Daleo:2006xa , the gluon-initiated $D^{0}_{3}$ antenna
function has to be decomposed into a flavour-preserving and flavour-changing
piece. The two resulting antenna functions are

$\displaystyle D^{0}_{3}(\hat{1}_{g},2_{g},3_{q})$ $\displaystyle=$
$\displaystyle\frac{1}{s_{123}^{2}}\left(\frac{s_{12}^{2}}{s_{23}}+\frac{2s_{13}^{3}}{s_{12}s_{23}}+\frac{4s_{13}^{2}}{s_{12}}+\frac{2s_{23}^{3}}{s_{12}(s_{12}+s_{13})}+\frac{6s_{13}s_{23}}{s_{12}}\right.$
(149)
$\displaystyle\left.+\frac{3s_{12}s_{13}}{s_{23}}+\frac{4s_{23}^{2}}{s_{12}}+6s_{12}+\frac{4s_{13}^{2}}{s_{23}}+9s_{13}+9s_{23}\right)+\mathcal{O}(\epsilon)\
,$ $\displaystyle D^{0}_{3,g\to q}(\hat{1}_{g},2_{q},3_{g})$ $\displaystyle=$
$\displaystyle-\frac{1}{s_{123}^{2}}\left(\frac{s_{13}^{2}}{s_{12}}+\frac{2s_{23}^{3}}{s_{12}(s_{12}+s_{13})}+\frac{3s_{13}s_{23}}{s_{12}}+\frac{4s_{23}^{2}}{s_{12}}\right)+\mathcal{O}(\epsilon)\,.$
(150)

For the flavour-preserving antenna function the quark or the gluon in the
final state can be identified. In the former case we find

$\displaystyle\mathcal{D}^{0,{\rm id.}q}_{3,\hat{g}}(x,z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{qg}^{(1)}(\epsilon,-Q^{2})\delta(1-x)\delta(1-z)$
(151)
$\displaystyle+\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\left(\delta(1-z)p^{(0)}_{gg}(x)+\delta(1-x)p^{(0)}_{qq}(z)\right)+\frac{11}{16}\delta(1-x)\delta(1-z)$
$\displaystyle+\delta(1-z)\left(\mathcal{D}_{1}(x)+\left(-2+\frac{1}{x}+x-x^{2}\right)\log(1-x)-\frac{(1-x+x^{2})^{2}}{(1-x)x}\log(x)\right)$
$\displaystyle+\delta(1-x)\left(\frac{1}{2}-\frac{z}{2}+\mathcal{D}_{1}(z)-\frac{1}{2}(1+z)\log(1-z)+\frac{1+z^{2}}{2(1-z)}\log(z)\right)$
$\displaystyle-\frac{3}{8}\delta(1-z)p^{(0)}_{gg}(x)-\frac{11}{24}p^{(0)}_{qq}(z)\delta(1-x)+\frac{1}{4}p^{(0)}_{gg}(x)p^{(0)}_{qq}(z)-\frac{5}{2}+\frac{1}{2x}+\frac{x}{2}$
$\displaystyle-\frac{x^{2}}{2}-z+\frac{z}{2x}+\frac{xz}{2}-\frac{x^{2}z}{2}\bigg{]}+\mathcal{O}(\epsilon)$

and in the latter case

$\displaystyle\mathcal{D}^{0,{\rm id.}g}_{3,\hat{g}}(x,z)$ $\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-x)p^{(0)}_{gq}(z)-\frac{7}{2}+\frac{1}{x}+x-x^{2}+z-\frac{z}{2x}-\frac{xz}{2}+\frac{x^{2}z}{2}$
(152)
$\displaystyle+\left(\frac{1-2x+x^{2}-x^{3}}{2x}+\frac{1}{2}\mathcal{D}_{0}(x)+\frac{1}{2}\delta(1-x)\left(\log(1-z)+\log(z)\right)\right)p^{(0)}_{gq}(z)$
$\displaystyle+\frac{1}{2}z\delta(1-x)\bigg{]}+\mathcal{O}(\epsilon)\,.$

The only integrated fragmentation antenna function of the flavour-changing
$D^{0}_{3}$ antenna identifies the final-state gluon. It reads

$\displaystyle\mathcal{D}^{0,{\rm id.}g}_{3,g\to q,\hat{g}}(x,z)$
$\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-z)p^{(0)}_{qg}(x)-\frac{3}{2}+\frac{1}{x}-\frac{z}{2x}+\frac{1}{2}\mathcal{D}_{0}(z)p^{(0)}_{qg}(x)$
$\displaystyle-\delta(1-z)\left(-\frac{1}{2}-\frac{1}{2}\log(1-x)p^{(0)}_{qg}(x)+\frac{1}{2}\log(x)p^{(0)}_{qg}(x)\right)\bigg{]}+\mathcal{O}(\epsilon)\,.$

There are two gluon-initiated gluon-gluon antenna functions:

$\displaystyle F^{0}_{3}(\hat{1}_{g},2_{g},3_{g})$ $\displaystyle=$
$\displaystyle\frac{1}{s_{123}^{2}}\left(\frac{2s_{123}^{2}s_{23}}{s_{12}s_{13}}+\frac{2s_{123}^{2}s_{13}}{s_{12}s_{23}}+\frac{2s_{12}s_{123}^{2}}{s_{13}s_{23}}\right.$
(154)
$\displaystyle\left.+\frac{2s_{13}s_{23}}{s_{12}}+\frac{2s_{12}s_{23}}{s_{13}}+\frac{2s_{12}s_{13}}{s_{23}}+8s_{123}\right)+\mathcal{O}(\epsilon)\,,$
$\displaystyle G^{0}_{3}(\hat{1}_{g},2_{q^{\prime}},3_{\bar{q}^{\prime}})$
$\displaystyle=$
$\displaystyle\frac{1}{s_{123}^{2}}\left(\frac{(s_{12}+s_{13})^{2}}{s_{23}}-\frac{2s_{12}s_{13}}{s_{23}}\right)+\mathcal{O}(\epsilon)\,.$
(155)

Both antenna functions are symmetric under the exchange of parton 2 and 3.
Therefore, for each antenna function there is only one integrated
fragmentation antenna function. We find

$\displaystyle\mathcal{F}^{0,{\rm id.}g}_{3,\hat{g}}(x,z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{gg}^{(1)}(\epsilon,-Q^{2})\delta(1-x)\delta(1-z)$
(156)
$\displaystyle+\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\left(\delta(1-z)p^{(0)}_{gg}(x)+\delta(1-x)p^{(0)}_{gg}(z)\right)+\frac{121}{144}\delta(1-z)\delta(1-x)$
$\displaystyle+\delta(1-z)\left(\mathcal{D}_{1}(x)+\left(-2+\frac{1}{x}+x-x^{2}\right)\log(1-x)-\frac{(1-x+x^{2})^{2}}{(1-x)x}\log(x)\right)$
$\displaystyle+\delta(1-x)\left(\mathcal{D}_{1}(z)+\left(-2+\frac{1}{z}+z-z^{2}\right)\log(1-z)+\frac{(1-z+z^{2})^{2}}{(1-z)z}\log(z)\right)$
$\displaystyle-\frac{11}{24}\left(\delta(1-z)p^{(0)}_{gg}(x)+\delta(1-x)p^{(0)}_{gg}(z)\right)+\frac{1}{4}p^{(0)}_{gg}(x)p^{(0)}_{gg}(z)$
$\displaystyle-(2-x+x^{2})(2-z+z^{2})\bigg{]}+\mathcal{O}(\epsilon)$

and

$\displaystyle\mathcal{G}^{0,{\rm id.}q^{\prime}}_{3,\hat{g}}(x,z)$
$\displaystyle=$
$\displaystyle\left(Q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}\delta(1-x)p^{(0)}_{qg}(z)+\frac{1}{2x}p^{(0)}_{qg}(z)+\frac{1}{2}\mathcal{D}_{0}(x)p^{(0)}_{qg}(z)$
$\displaystyle+\delta(1-x)\left(\frac{1}{2}-\frac{1}{2}p^{(0)}_{qg}(z)+\frac{1}{2}\left(\log(1-z)+\log(z)\right)p^{(0)}_{qg}(z)\right)\bigg{]}+\mathcal{O}(\epsilon)\,.$

###  B.2 Final-Final Configuration

The unintegrated $X^{0}_{3}$ antenna functions in the final-final
configuration can be found in GehrmannDeRidder:2005cm . We recall their
expressions here and give the results for their integrated form differential
in the final-state momentum fraction.

The tree-level three parton quark-anti-quark antenna function reads

$A^{0}_{3}(1_{\bar{q}},3_{g},2_{q})=\frac{1}{s_{123}}\left(\frac{2s_{12}^{2}}{s_{13}s_{23}}+\frac{2s_{12}}{s_{13}}+\frac{2s_{12}}{s_{23}}+\frac{s_{23}}{s_{13}}+\frac{s_{13}}{s_{23}}\right)+\mathcal{O}(\epsilon)\,.$
(158)

It is symmetric under the exchange of the quark pair. We find two integrated
fragmentation antenna functions. Identifying the gluon, we have

$\displaystyle\mathcal{A}^{0,{\rm id.}g}_{3,\bar{q}}(z)$ $\displaystyle=$
$\displaystyle\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{gq}(z)+\frac{1}{4}+\frac{z}{8}+\left(-\frac{3}{8}+\frac{1}{2}\log(1-z)+\frac{\log(z)}{2}\right)p^{(0)}_{gq}(z)\bigg{]}$
(159) $\displaystyle+\mathcal{O}(\epsilon)$

and in case the quark is identified, we find

$\displaystyle\mathcal{A}^{0,{\rm id.}q}_{3,\bar{q}}(z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{q\bar{q}}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{qq}(z)+\frac{3}{8}-\frac{z}{8}+\left(\frac{47}{16}+\frac{\pi^{2}}{6}\right)\delta(1-z)$
(160)
$\displaystyle+\mathcal{D}_{1}(z)-\frac{1}{2}(1+z)\log(1-z)+\frac{1+z^{2}}{2(1-z)}\log(z)-\frac{3}{8}p^{(0)}_{qq}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$

In the subtraction of quark-photon collinear limits the antenna function
$A^{0}_{3}(1_{\bar{q}},3_{\gamma},2_{q})$ is used. Its unintegrated form
coincides with (158) and we have $\mathcal{A}^{0,{\rm
id.}\gamma}_{3,\bar{q}}=\mathcal{A}^{0,{\rm id.}g}_{3,\bar{q}}$.

The tree-level quark-gluon antenna function can be expressed as

$D^{0}_{3}(1_{q},2_{g},3_{g})=d^{0}_{3}(1_{q},2_{g},3_{g})+d^{0}_{3}(1_{q},3_{g},2_{g})$
(161)

with the sub-antenna

$d^{0}_{3}(1_{q},2_{g},3_{g})=\frac{1}{s_{123}^{2}}\left(\frac{2s_{123}^{2}s_{13}}{s_{12}s_{23}}+\frac{s_{123}s_{23}}{s_{12}}+\frac{s_{12}s_{13}}{s_{23}}+\frac{s_{12}}{2}+2s_{123}+\frac{s_{13}}{2}\right)+\mathcal{O}(\epsilon)\,.$
(162)

In the sub-antenna at hand gluon 3 acts as a hard radiator while the full
antenna (161) also contains the soft limit of gluon 3. The reference particle
used in the definition of the momentum fraction has to be a hard radiator.
Therefore, if we want to use the quark-gluon antenna function with the gluon
as the reference particle we have to use the sub-antenna in which the
reference gluon is a hard radiator. Integrating (162) and remaining
differential in the gluon momentum fraction, we find

$\mathcal{D}^{0,{\rm
id.}g}_{3,g}(z)=\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{gq}(z)+\frac{5}{8}+\frac{z}{8}+\left(-\frac{11}{24}+\frac{1}{2}\log(1-z)+\frac{\log(z)}{2}\right)p^{(0)}_{gq}(z)\bigg{]}+\mathcal{O}(\epsilon)$
(163)

and for the case where the quark momentum is identified we have

$\displaystyle\mathcal{D}^{0,{\rm id.}q}_{3,g}(z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{qg}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{qq}(z)+\frac{3}{4}-\frac{z}{8}+\left(\frac{167}{48}+\frac{\pi^{2}}{6}\right)\delta(1-z)$
(164)
$\displaystyle+\mathcal{D}_{1}(z)-\frac{1}{2}(1+z)\log(1-z)+\frac{1+z^{2}}{2(1-z)}\log(z)-\frac{11}{24}p^{(0)}_{qq}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$

When the quark acts as a reference particle we can integrate the full antenna
function (161). In this case the fragmentation antenna function reads

$\displaystyle\mathcal{D}^{0,{\rm id.}g}_{3,q}(z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{qg}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{gg}(z)+\frac{5}{3}-\frac{13z}{12}+\frac{13z^{2}}{12}$
(165)
$\displaystyle+\left(\frac{49}{16}+\frac{\pi^{2}}{6}\right)\delta(1-z)+\mathcal{D}_{1}(z)+\left(-2+\frac{1}{z}+z-z^{2}\right)\log(1-z)$
$\displaystyle+\frac{(1-z+z^{2})^{2}}{(1-z)z}\log(z)-\frac{3}{8}p^{(0)}_{gg}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$

The last quark-gluon antenna function is

$E^{0}_{3}(1_{q},2_{q^{\prime}},3_{\bar{q}^{\prime}})=\frac{1}{s_{123}^{2}}\left(\frac{(s_{12}+s_{13})^{2}}{s_{23}}-\frac{2s_{12}s_{13}}{s_{23}}+(s_{12}+s_{13})\right)+\mathcal{O}(\epsilon)\,.$
(166)

It is symmetric under the exchange of particle 2 and 3. Phase space
integration with reference particle $q$ and identified particle $q^{\prime}$
yields

$\mathcal{E}^{0,{\rm
id.}q^{\prime}}_{3,q}(z)=\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{qg}(z)+\frac{2}{3}+\left(-\frac{17}{12}+\frac{1}{2}(\log(1-z)+\log(z))\right)p^{(0)}_{qg}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$
(167)

In case the primary quark $q$ is identified, we have

$\mathcal{E}^{0,{\rm
id.}q}_{3,\bar{q}^{\prime}}(z)=-4\mathbf{I}_{qg,F}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{12}-\frac{11}{12}\delta(1-z)+\frac{1}{3}\mathcal{D}_{0}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$
(168)

The first gluon-gluon antenna function is

$F^{0}_{3}(1_{g},2_{g},3_{g})=f^{0}_{3}(1_{g},2_{g},3_{g})+f^{0}_{3}(1_{g},3_{g},2_{g})$
(169)

with the sub-antenna

$f^{0}_{3}(1_{g},2_{g},3_{g})=\frac{1}{s_{123}^{2}}\left(\frac{2s_{123}^{2}s_{13}}{s_{12}s_{23}}+\frac{s_{13}s_{23}}{s_{12}}+\frac{s_{12}s_{13}}{s_{23}}+\frac{8s_{123}}{3}\right)+\mathcal{O}(\epsilon)\,.$
(170)

In (169) we have fixed $1_{g}$ to be the hard radiator which is used as a
reference particle in the definition of the momentum fraction. Consequently,
it does not contain the sub-antenna $f^{0}_{3}(2_{g},1_{g},3_{g})$.
Integration of (169) over the phase space while remaining differential in the
final-state momentum fraction yields

$\displaystyle\mathcal{F}^{0,{\rm id.}g}_{3,g}(z)$ $\displaystyle=$
$\displaystyle-2\mathbf{I}_{gg}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{gg}(z)+\frac{4}{3}-\frac{11z}{12}+\frac{11z^{2}}{12}$
(171)
$\displaystyle+\left(\frac{523}{144}+\frac{\pi^{2}}{6}\right)\delta(1-z)+\mathcal{D}_{1}(z)+\left(-2+\frac{1}{z}+z-z^{2}\right)\log(1-z)$
$\displaystyle+\frac{(1-z+z^{2})^{2}}{(1-z)z}\log(z)-\frac{11}{24}p^{(0)}_{gg}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$

The second gluon-gluon antenna function is

$G^{0}_{3}(1_{g},2_{q^{\prime}},3_{\bar{q}^{\prime}})=\frac{1}{s_{123}^{2}}\left(\frac{(s_{12}+s_{13})^{2}}{s_{23}}-\frac{2s_{12}s_{13}}{s_{23}}\right)+\mathcal{O}(\epsilon)\,.$
(172)

It is symmetric under the exchange of particle 2 and 3. Using the gluon as
reference particle and remaining differential in the momentum fraction of
$q^{\prime}$, the phase space integration gives

$\mathcal{G}^{0,{\rm
id.}q^{\prime}}_{3,g}(z)=\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{2\epsilon}p^{(0)}_{qg}(z)+\frac{1}{2}+\left(-\frac{17}{12}+\frac{1}{2}(\log(1-z)+\log(z))\right)p^{(0)}_{qg}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$
(173)

In case of an identified gluon we have

$\mathcal{G}^{0,{\rm
id.}g}_{3,\bar{q}^{\prime}}(z)=-2\mathbf{I}_{gg,F}^{(1)}(\epsilon,q^{2})\delta(1-z)+\left(q^{2}\right)^{-\epsilon}\bigg{[}-\frac{1}{6}-\frac{z}{6}-\frac{11}{12}\delta(1-z)+\frac{1}{3}\mathcal{D}_{0}(z)\bigg{]}+\mathcal{O}(\epsilon)\,.$
(174)

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# Discovery of a radio galaxy at z = 5.72

A. Saxena1, M. Marinello1,2, R. A. Overzier2, P. N. Best3, H. J. A.
Röttgering1, K. J. Duncan1, I. Prandoni4, L. Pentericci5, M. Magliocchetti6,
D. Paris5, F. Cusano7, F. Marchi5, H. T. Intema1 and G.K. Miley1

1Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands
2Observatório Nacional, Rua General José Cristino, 77, São Cristóvão, Rio de
Janeiro, RJ, CEP 20921-400, Brazil
3Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, EH9 3HJ Edinburgh, UK
4INAF-Instituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy
5INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00040 Monteporzio
(RM), Italy
6IAPS-INAF, Via Fosso del Cavaliere 100, I-00133 Rome, Italy
7INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via P.
Gobetti 93/3, I-40129 Bologna, Italy E-mail<EMAIL_ADDRESS>
(Accepted 2018 July 12. Received 2018 July 5; in original form 2018 June 4)

###### Abstract

We report the discovery of the most distant radio galaxy to date, TGSS
J1530+1049 at a redshift of $z=5.72$, close to the presumed end of the Epoch
of Reionisation. The radio galaxy was selected from the TGSS ADR1 survey at
150 MHz for having an ultra-steep spectral index, $\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{1.4 GHz}}}=-1.4$ and a compact morphology obtained using
VLA imaging at 1.4 GHz. No optical or infrared counterparts for the radio
source were found in publicly available sky surveys. Follow-up optical
spectroscopy at the radio position using GMOS on Gemini North revealed the
presence of a single emission line. We identify this line as Lyman alpha at
$z=5.72$, because of its asymmetric line profile, the absence of other
optical/UV lines in the spectrum and a high equivalent width. With a
Ly$\alpha$ luminosity of $5.7\times 10^{42}$ erg s-1 and a FWHM of $370$ km
s-1, TGSS J1530+1049 is comparable to ‘non-radio’ Lyman alpha emitters (LAEs)
at a similar redshift. However, with a radio luminosity of $\log
L_{\textrm{150 MHz}}=29.1$ W Hz-1 and a deconvolved physical size $3.5$ kpc,
its radio properties are similar to other known radio galaxies at $z>4$.
Subsequent $J$ and $K$ band imaging using LUCI on the Large Binocular
Telescope resulted in non-detection of the host galaxy down to $3\sigma$
limits of $J>24.4$ and $K>22.4$ (Vega). The $K$ band limit is consistent with
$z>5$ from the $K-z$ relation for radio galaxies and helps rule out low
redshifts. The stellar mass limit derived using simple stellar population
models is $M_{\textrm{stars}}<10^{10.5}$ $M_{\odot}$. Its relatively low
stellar mass and small radio and Ly$\alpha$ sizes suggest that TGSS J1530+1049
may be a radio galaxy in an early phase of its evolution.

###### keywords:

galaxies: individual: TGSS J1530+1049 – galaxies: high-redshift

††pubyear: 2018††pagerange: Discovery of a radio galaxy at z = 5.72–Discovery
of a radio galaxy at z = 5.72

##  1 Introduction

High-redshift radio galaxies (HzRGs) are thought to be the progenitors of
local massive elliptical galaxies and generally contain large amounts of dust
and gas (Best et al., 1998; Carilli et al., 2002a; Reuland et al., 2004; De
Breuck et al., 2010). They are also among the most massive galaxies at their
redshift (Overzier et al., 2009) and are often found to be located at the
centre of clusters and proto-clusters of galaxies (Pentericci et al., 2000;
Venemans et al., 2002; Röttgering et al., 2003; Miley et al., 2004; Hatch et
al., 2011; Orsi et al., 2016). Studies of their environment can give insights
into the assembly and evolution of the large scale structure in the Universe.
Miley & De Breuck (2008) provide an extensive review about the properties of
distant radio galaxies and their environments.

Radio galaxies at $z>6$, in the Epoch of Reionisation (EoR), are of particular
interest as they could be used as unique tools to study the process of
reionisation in detail. At these redshifts, the hyper-fine transition line
from neutral hydrogen atoms, with a rest-frame wavelength of 21 cm, falls in
the low-frequency radio regime and can be observed as absorption signals in
spectra of luminous background radio sources such as radio galaxies (Carilli
et al., 2002b; Furlanetto & Loeb, 2002; Xu et al., 2009; Mack & Wyithe, 2012;
Ewall-Wice et al., 2014; Ciardi et al., 2015). Such 21 cm absorption signals
from patches of neutral hydrogen clouds in the early Universe could in
principle be observed by current and next-generation radio telescopes such as
the Giant Metre-wave Radio Telescope (GMRT; Swarup et al. 1991), the Low
Frequency Array (LOFAR; van Haarlem et al. 2013), the Murchinson Widefield
Array (MWA; Tingay et al. 2013) and the Square Kilometer Array (SKA; Dewdney
et al. 2009). This unique application motivates searches for bright enough
radio galaxies at the highest redshifts from deep all-sky surveys at low radio
frequencies.

Finding powerful radio galaxies at increasingly large distances or redshifts,
however, is challenging. HzRGs are rare and flux-limited samples have shown
that their space densities fall off dramatically at $z>2-3$ (Dunlop & Peacock,
1990; Willott et al., 2001; Rigby et al., 2011, 2015). Although a number of
quasars at $z>5$ are known, with a few also being radio-loud (see Bañados et
al., 2015, for example), the same cannot be said for radio galaxies -
previously, the only known radio galaxy at $z>5$ was TN J0924$-$2201 at
$z=5.19$ (van Breugel et al., 1999). If the orientation-based unification of
radio galaxies and quasars is valid (see Morabito et al., 2017, for example),
the number of radio quasars and galaxies at any given epoch should be
comparable and it may be possible that many of the already unidentified radio
sources are at $z>5$. The scarcity of $z>5$ radio galaxies could therefore be
due to the relative difficulty in first identifying these sources amongst the
wider radio source population, and then obtaining spectroscopic redshifts for
these radio galaxies, which are optically much fainter than quasars. Dedicated
spectroscopic follow-up of radio sources such as the WEAVE-LOFAR survey (Smith
et al., 2016), and the upcoming major optical facilities such as the Extremely
Large Telescope (ELT), the Thirty Meter Telescope (TMT), the Giant Magellan
Telescope (GMT) and the James Webb Space Telescope (JWST) will help overcome
these difficulties and potentially help characterise a number of USS radio
sources.

The key requirement for gathering enough statistics for meaningful studies of
radio galaxies at high redshifts are deep low-frequency radio surveys covering
large areas on the sky. Surveys such as the TIFR GMRT Sky Survey Alternative
Data Release 1 (TGSS; Intema et al. 2017) and the currently ongoing surveys
using LOFAR (Shimwell et al., 2017) are opening up new parameter spaces for
searches for radio galaxies at $z\geq 6$. Using TGSS, which covers the entire
radio sky north of $-53$ declination at a frequency of 150 MHz and achieving a
median noise level of $3.5$ mJy beam-1, we launched a campaign to hunt for
fainter and potentially more distant HzRGs, with the ultimate aim of
discovering radio galaxies that could be suitable probes of the EoR (Saxena et
al., 2018). In this paper, we report the discovery of a radio galaxy at a
redshift of $z=5.72$, TGSS J1530+1049, which was pre-selected as part of our
sample of high-redshift radio galaxy candidates.

The layout of this paper is as follows. In Section 2 we present details about
the initial source selection criteria and follow-up radio observations at high
resolution for TGSS J1530+1049. In Section 3 we present the new optical
spectroscopy and infrared imaging obtained for TGSS J1530+1049 and expand upon
the data reduction methods. In Section 4 we describe how the redshift for this
source was determined. In Section 5 we study the emission line and radio
properties of this source and set constraints on its stellar mass. We also
compare the observed properties to galaxies at the same epoch from the
literature. Finally, in Section 6 we summarise the findings of this paper.
Throughout this paper we assume a flat $\Lambda$CDM cosmology with $H_{0}$ =
70 km s-1 Mpc-1 and $\Omega_{m}=0.3$. Using this cosmology, at a redshift of
5.72 the age of the Universe is 0.97 Gyr, and the angular scale per arcsecond
is 5.86 kpc.

##  2 Source selection

Our two stage selection process is based on first isolating compact radio
sources with an ultra-steep spectrum (USS; $\alpha<-1.3$, where
$S_{\nu}\propto\nu^{\alpha}$) at radio wavelengths, that has historically been
very successful at finding HzRGs from wide area radio surveys (Röttgering et
al., 1994; Blundell et al., 1999; De Breuck et al., 2000; Afonso et al.,
2011), and then combining it with optical and/or infrared faintness
requirements. The relation that exists between the apparent $K$-band magnitude
of radio galaxies and their redshift, known as the $K-z$ relation, (Lilly &
Longair, 1984; Jarvis et al., 2001; Willott et al., 2003; Rocca-Volmerange et
al., 2004) gives further strength to the argument of selecting USS sources
that are also faint at near-infrared wavelengths in a bid to isolate HzRGs
(Ker et al., 2012). Deep near-infrared imaging of promising USS candidates can
therefore serve as an independent way to set constraints on the redshifts of
radio sources. HzRGs are expected to be very young and therefore, have small
sizes at the highest redshifts (Saxena et al., 2017): implementing an
additional criterion that puts an upper limit on the angular sizes of radio
sources has the potential to increase the efficiency of pin pointing the
highest-redshift sources in a wide area radio survey.

Combining all of these selection methods, we first shortlisted 588 candidates
with an ultra-steep spectrum ($\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{1.4 GHz}}}<-1.3$) and compact morphologies, out of a
total of 65,996 sources that had spectral index information from TGSS and
FIRST/NVSS. This sample probes fainter flux densities than previous large area
searches and has flux limits that ensure that a new parameter space in flux
density and spectral index is probed, where a large number of undiscovered
HzRGs are expected to lie (Ishwara-Chandra et al., 2010). From this shortlist,
we then retained in our sample only those radio sources that are blank in all
available optical surveys such as the Sloan Digital Sky Survey DR12 (SDSS;
Alam et al. 2015) and the Pan-STARRS1 survey (PS1; Chambers et al. 2016), and
infrared surveys such as ALLWISE using the WISE satellite (Wright et al.,
2010) and the UKIDSS surveys (Lawrence et al., 2007) to maximise the chances
of finding radio galaxies at the highest redshifts. This led to a final sample
of 32 very promising HzRG candidates. Details of the sample selection can be
found in Saxena et al. (2018).

Figure 1: The location of TGSS J1530+1049 in the flux density$-$spectral index
parameter space. The large orange points show the parameter space probed by
the Saxena et al. (2018) sample and the smaller grey points show radio sources
from De Breuck et al. (2000), scaled to an observed frequency of 150 MHz using
the spectral indices provided for individual sources. Also shown for
comparison is TN J0924$-$2201 at $z=5.2$ (van Breugel et al., 1999). TGSS
J1530+1049 is fainter than the previously studied large area samples and
offers a new window into fainter radio galaxies at high redshifts.

High resolution imaging using the Karl G. Jansky Very Large Array (VLA) for
the 32 candidates, including for TGSS J1530+1049 (RA: 15:30:49.9, Dec:
$+$10:49:31.1) is presented in Saxena et al. (2018), which was used to obtain
morphologies and sub-arcsecond localisation of the expected positions of the
host galaxies, enabling blind spectroscopic follow-up. TGSS J1530+1049 in
particular showed a compact morphology, which was fitted with a single
Gaussian. With a flux density of $S_{\textrm{\tiny{150 MHz}}}=170\pm 34$ mJy,
TGSS J1530+1049 is one of the brightest sources in the sample. At 1.4 GHz, it
has a flux density $S_{\textrm{1.4 GHz}}=7.5\pm 0.1$ mJy, giving a spectral
index of $\alpha=-1.4\pm 0.1$. With a relatively small (deconvolved) angular
size of $0.6\pm 0.1$ arcsec, TGSS J1530+1049 was deemed to be a promising HzRG
candidate. We show the location of TGSS J1530+1049 in the flux
density$-$spectral index parameter space in Figure 1.

TGSS J1530+1049 is not detected in any of the PS1 bands ($g$, $r$, $i$, $z$
and $y$). This source also happens to lie in the sky area covered by the
UKIDSS Large Area Survey (LAS), and is not detected down to (Vega) magnitude
limits of $y>20.5$, $J>20.0$, $H>18.8$ and $K>18.4$. We show the image
obtained from stacking all of the LAS bands with radio contours overlaid in
Figure 2. Lastly, this source is also not detected in any of the ALLWISE
bands. These non-detections coupled with the ultra-steep radio spectral index
and compact radio morphology are in line with expectations of a high-redshift
host galaxy and made TGSS J1530+1049 a prime candidate for follow-up
spectroscopy.

Figure 2: Stacked $y$, $J$, $H$ and $K$ band image from the UKIDSS Large Area
Survey, with contours (starting from 0.5 mJy, in a geometric progression of
$\sqrt{2}$) from the 1.4 GHz VLA map (Saxena et al., 2018) overplotted for
TGSS J1530+1049. The radio source is compact and has an ultra-steep spectral
index. A non-detection in the UKIDSS LAS $K$ band down to a magnitude limit of
18.4 Vega ($\sim 20.3$ AB) made TGSS J1530+1049 a promising HzRG candidate and
a prime target for spectroscopic follow-up.

##  3 Observations

###  3.1 Gemini GMOS spectroscopy

A long-slit spectrum of TGSS J1530+1049TGSS J1530+1049 was taken using GMOS on
Gemini North on 28 April, 2017 (Program ID: GN-2017A-Q-8; PI: Overzier) using
the filter GG455_G0305 and the R400_G5305 grating giving a wavelength coverage
of $5400-10800$ Å and a resolution of roughly $R\sim 1500$. The central
wavelength was set to 7000 Å. The total length of the slit was 300 arcseconds
and the slit width was chosen to be 1.5 arcseconds so that it covers the
entire radio emission footprint detected in the VLA image. As the host galaxy
of the radio source was undetected in all available all-sky optical/IR
surveys, we performed blind offsetting from a bright star, which ensures
positional accuracy to within 0.1 arcseconds, to the centroid of the radio
emission. The VLA observations ensured sub-arcsecond localisation of the
expected position of the host galaxy and the relatively large slit-width
provided insurance against minor positional uncertainties. We took 3 exposures
of 800 seconds each, giving a total of 2400 seconds of on-source exposure
time. The standard star EG131 was observed for flux calibration.

We used the Gemini IRAF package for reducing the data, which includes the
standard steps for optical spectrum reduction. Briefly, the bias frames were
mean stacked in a master bias which was subtracted from all other images
acquired. Pixel-to-pixel sensitivity was corrected through the flat field
image taken during the day of the observations. The wavelength solution was
derived from the arc lamp frame taken immediately after the science
observations, and applied to the science frame and standard star. The 2D
images were then combined in a single frame, rejecting possible cosmic rays.
The sky lines were removed and flux calibration was achieved using the
standard star spectrum.

A single emission line with a peak at 8170 Å and a spatial extent of $\sim 1$
arcsecond was detected in the reduced 2D spectrum at the expected position of
the radio galaxy. No other line associated with this source was detected. No
continuum was detected either bluewards or redwards of this line either. To
ensure that the line detection is indeed real and not due to an artefact or
contamination by cosmic rays, we looked at the individual frames, both raw and
sky subtracted, to ensure that the detection (although marginal) was present
in each science frame. The three frames are shown in Figure 3. The top panels
show the raw frames and the bottom panels the sky subtracted frames. The
emission line is clearly present in all three frames, ensuring that the
detection is real. The extracted 1D spectrum with a 1 arcsecond aperture
showing the detected emission line is shown in Figure 4. We give details about
line identification in Section 4.

Figure 3: Raw (top panels) and sky subtracted (bottom panels) 2D frames shown
for the three individual exposures taken using GMOS on Gemini. Traces of the
emission line are visible in all three frames, ensuring that the detected line
is real and not a consequence of cosmic rays or artefacts. There is some
cosmic ray residual left over in the second frame but that does not
contaminate the emission line signal. Figure 4: Extracted 1D spectrum showing
the single emission line detection centred at 8170 Å from the GMOS 2D
spectrum. No other line or continuum is detected. Shaded regions mark the
presence of sky lines in the spectrum.

###  3.2 Large Binocular Telescope NIR imaging

Imaging in the $J$ and $Ks$ bands using LUCI (formerly known as LUCIFER;
Seifert et al. 2003) on the Large Binocular Telescope (LBT) was carried out in
two separate runs, with the first on 1 February 2018 and the second on 11 May
2018 (Program ID 2017_2018_43; PI: Prandoni). The average seeing throughout
the observations was $0.6-0.8$ arcseconds. In the first run, the on-source
exposure time was 720 ($12\times 60$s) seconds in $J$ (central wavelength of
1.247 microns) and 1200 ($20\times 60$s) seconds in $Ks$ (central wavelength
of 2.194 microns). In the second run we obtained additional 3600 ($30\times
120$s) seconds in $J$ and 3000 ($50\times 60$s) seconds in $Ks$, giving a
total on-source exposure time of 4320 seconds in $J$ band and 4200 seconds in
$Ks$ band.

The LUCI data reduction pipeline developed at INAF-OAR was used to perform the
basic reduction such as dark subtraction, bad pixel masking, cosmic ray
removal, flat fielding and sky subtraction. Astrometric solutions for
individual frames were obtained and the single frames were then resampled and
combined using a weighted co-addition to form a deeper image. The
$4^{\prime}\times 4^{\prime}$ field-of-view of LUCI contained many bright
objects detected in both 2MASS and the UKIDSS Large Area Survey, which were
used to calibrate the photometry of the images in both bands.

The median and standard deviation of the background in both images was
calculated by placing 5000 random apertures with a diameter of 1.5 arcseconds.
We measure $3\sigma$ depths of $J=24.4$ and $Ks=22.4$. Aperture photometry
performed on both $J$ and $Ks$ (from here on we denote $Ks$ as simply $K$)
images using photutils (Bradley et al., 2017) at the peak of the radio
emission using an aperture of diameter 1.5 arcseconds yield magnitudes that
are lower than the $3\sigma$ depths in both images. Smoothing the $K$ band
image with a $3\times 3$ pixel Gaussian kernel reveals a faint source very
close to the peak radio pixel, as shown in Figure 5, but it is not entirely
clear if this indeed the host galaxy and there is no faint detection even in
the smoothed $J$ band image. A summary of the observations is given in Table
1.

Figure 5: $K$-band image from the Large Binocular Telescope (LBT), which has been smoothed with a $3\times 3$ pixel Gaussian kernel, with radio contours (same as Figure 2) from the VLA at 1.4 GHz overlaid. The measured magnitude at the radio position with a 1.5 arcsecond aperture is fainter than the $3\sigma$ depth of the image, giving $K>22.4$. When the image is smoothed, however, faint emission is visible around the peak of the radio emission. The magnitude limit is consistent with $z>5$ following the $K-z$ relation for radio galaxies. For comparison, the $z=5.2$ radio galaxy TN J09224$-$2201 has a K-band magnitude of 21.3 (van Breugel et al., 1999). Table 1: Observation log. Telescope | Instrument | Date | Exp. time (sec)
---|---|---|---
Gemini N | GMOS long-slit | 28-04-2017 | 2400 ($3\times 800$s)
LBT | LUCI J-band | 01-02-2018 | 720 ($12\times 60$s)
|  | 09-05-2018 | 3600 ($30\times 120$s)
| Total |  | 4320
LBT | LUCI Ks-band | 01-02-2018 | 1200 ($20\times 60$s)
|  | 09-05-2018 | 3000 ($50\times 60$s)
| Total |  | 4200

##  4 Redshift determination

We identify the single emission line detected in the GMOS spectrum as
Ly$\alpha$ $\lambda 1216$, giving a redshift of $z=5.720\pm 0.001$, which is
shown in Figure 6. Other plausible identifications of this emission line could
be [O iii] $\mathrm{\lambda}5007$, giving a redshift of $z\approx 0.63$ or
H$\alpha$ $\mathrm{\lambda}6563$ at $z\approx 0.25$. These can be ruled out
given the non-detection of other bright lines expected in the wavelength range
covered. For example, if the observed line is [O iii] at $z=0.63$, then the [O
ii] $\lambda 3727$ line would be observed at $\sim 6075$ Å, unless the galaxy
has a particularly high [O iii]/[O ii] ratio as in the case of very low
metallicity objects. An unresolved [O ii] $\lambda\lambda 3726,3729$ doublet
at a redshift of $z\approx 1.2$ could be a possibility, but the absence of
other expected UV/optical lines common in AGN and radio galaxy spectra, such
as C ii] $\lambda 2326$ or Mg ii $\mathrm{\lambda\lambda}2797,2803$, which are
on average a factor of $2-4$ times fainter than [O ii] (De Breuck et al.,
2001), makes this possibility unlikely. Another possibility could be C iv
$\lambda 1549$ giving $z=4.27$, but this can be ruled out because in this case
Ly$\alpha$, which is often much brighter (for example, there are no radio
galaxies in De Breuck et al. (2001) with Ly$\alpha$ flux lower than C iv
$\lambda 1549$ flux), would be expected at $\sim 6410$ Å and we do not see any
signs of an emission line in that region, which is free from bright sky lines.

Figure 6: Rest-frame 1D spectrum showing the asymmetric Ly$\alpha$ line
profile at a redshift of $z=5.720$. Also shown is the best-fit Gaussian to the
emission line. The peak of the fitted Gaussian is slightly redder than the
peak of the line, suggesting asymmetry in the emission line. This is also
clear from the excess towards the redder parts of the Gaussian. Top: The 2D
GMOS spectrum showing the detected Ly$\alpha$ line. The spatial extent of the
emission is roughly 1 arcsecond, which is also the aperture size used to
extract the 1D spectrum.

We fit a Gaussian to the emission line (shown in Figure 6) to measure an
integrated line flux of $F_{Ly\alpha}=1.6\pm 0.2\times 10^{-17}$ erg s-1 cm-2.
The total measured Ly$\alpha$ luminosity is $L_{{Ly\alpha}}=5.7\pm 0.7\times
10^{42}$ erg s-1. The full width at half maximum (FWHM) after correcting for
the instrumental FWHM is $370\pm 30$ km s-1. Since no continuum is detected in
the spectrum (down to $1\sigma$ depth of $4.0\times 10^{-19}$ erg s-1 cm-2),
we can only put a lower limit on the rest-frame equivalent width (EW) of the
line, EW0 $>40$ Å. Table 2 presents a summary of emission line measurements
for TGSS J1530+1049.

###  4.1 Skewness and equivalent width

To further confirm our redshift determination, we quantify the asymmetry of
the emission line following the prescriptions laid out by Kashikawa et al.
(2006), by calculating the S-statistic and the weighted skewness parameter. A
measure of the skewness of the emission line is particularly useful when
dealing with spectra with a single emission line and can help differentiate
Ly$\alpha$ emission at high redshifts from [O ii], [O iii] or H$\alpha$
emission from lower redshift galaxies (Rhoads et al., 2003; Kurk et al., 2004;
Kashikawa et al., 2006). We measure the skewness $S=0.31\pm 0.14$ and the
weighted skewness $S_{w}=6.44\pm 2.97$ Å, which are consistent with what is
observed for confirmed Ly$\alpha$ emitters at high redshift (Kashikawa et al.,
2006, 2011; Matthee et al., 2017).

To check what possible values of skewness could be obtained from an unresolved
[O ii] doublet, we simulated the doublet with all possible ratios
($0.35<j_{\lambda 3729}/j_{\lambda 3726}<1.5$), convolved with the instrument
resolution. We find that the skewness measured for the emission line seen in
the spectrum ($S=0.31$) is only possible for $j_{\lambda 3729}/j_{\lambda
3726}<0.7$. These line ratios correspond to the high electron density regime
when the line would be collisionally de-excited, and hence unlikely to be as
strong as observed, with previous studies of the [O ii] doublet in high-z
galaxies (Steidel et al., 2014; Shimakawa et al., 2015; Sanders et al., 2016)
also finding much higher line ratios on average. This helps drive the
interpretation of the observed emission line more towards a Ly$\alpha$ at high
redshift.

Further, an EW $>40$ Å for an [O ii] line originating from a presumably
massive radio galaxy at $z\approx 1.2$ would be at the extreme end of the EW
distribution (Bridge et al., 2015), including for radio-loud quasars
(Kalfountzou et al., 2012). This EW value is also incompatible with the line
ratios that would give rise to the observed skewness, as in regions of very
high electron densities the [O ii] line is expected to be weaker due to
collisional de-excitation. Therefore, we can practically rule out the [O ii]
doublet as a possible identification of this emission line. An EW0 $>40$ Å,
however, is typical for Ly$\alpha$ emission seen in galaxies at $z\approx 5.7$
(see Kashikawa et al., 2011, for example) and generally consistent with the
$z\sim 6$ galaxy population (De Barros et al., 2017).

###  4.2 K-z relation for radio galaxies

Finally, a strong indicator of a high-redshift nature of the host galaxy is
the non-detection in $K$ band down to a $3\sigma$ limiting magnitude of 22.4
(Figure 5) using aperture photometry at the peak pixel of the radio emission.
For comparison, TN J0924$-$2201 at $z=5.2$ has a magnitude of $K=21.3\pm 0.3$
and our measurement of $K>22.4$ is consistent with $z>5$ and helps rule out
lower redshifts owing to the $K-z$ relation for radio galaxies (note that the
luminosity and the spectral index rule out that it is a star-forming galaxy).
We expand upon this point in Section 5.3. The additional non-detection in $J$
band down to a $3\sigma$ limit of 24.3 further favours a high redshift galaxy
and supports the argument that the line we see is indeed Ly$\alpha$ and not [O
ii].

##  5 Discussion

###  5.1 Emission line measurements

Table 2: Spectroscopic redshift and emission line measurements for TGSS J1530+1049 through GMOS spectroscopy. Property | Measurement
---|---
$z_{\textrm{spec}}$ | $5.720\pm 0.001$
$F_{\textrm{Ly}\alpha}$ |  $1.6\pm 0.2\times 10^{-17}$ erg s-1 cm-2
$L_{\textrm{Ly}\alpha}$ |  $5.7\pm 0.7\times 10^{42}$ erg s-1
FWHM |  $370\pm 30$ km s-1
EW${}_{\textrm{obs}}$ |  $>40$ Å

The Ly$\alpha$ luminosity and FWHM measured for TGSS J1530+1049 are lower than
what is seen for typical HzRGs at $z>4$ (see Spinrad et al., 1995; De Breuck
et al., 1999; van Breugel et al., 1999; Miley & De Breuck, 2008, for examples)
and more consistent with those measured for ‘non-radio’ Ly$\alpha$ emitting
galaxies (LAEs) at this redshift (Rhoads et al., 2003; Ouchi et al., 2008;
Kashikawa et al., 2011; Lidman et al., 2012; Matthee et al., 2017). However,
the FWHM for TGSS J1530+1049 is consistent with that of a very faint radio
galaxy VLA J123642$+$621331, with a 1.4 GHz flux density of $S_{\textrm{1.4
GHz}}=0.47$ mJy, discovered at $z=4.424$ (Waddington et al., 1999). This
galaxy has a FWHM of $\approx 420$ km s-1 and a Ly$\alpha$ luminosity $\approx
2\times 10^{42}$ erg s-1, which is weaker than TGSS J1530+1049. VLA
J123642$+$621331 is however, not detected in TGSS at 150 MHz down to a noise
level of 3.5 mJy beam-1, suggesting a relatively flat spectral index or a
spectral turnover at low radio frequencies. We present some comparisons of the
Ly$\alpha$ properties we measure for TGSS J1530+1049 with other HzRGs at $z>4$
and also non-radio LAEs at $z=5.7$ in Table 3.

Table 3: Comparison of Ly$\alpha$ emission line properties of TGSS J1530+1049 reported in this paper with some of the known radio galaxies at $z>4$, and the much fainter radio galaxy at $z=4.42$ (all marked as RG), in addition to several confirmed LAEs at $z\approx 5.7$ from the literature. |  | $F_{Ly\alpha}$ | $L_{Ly\alpha}$ | FWHMLyα |
---|---|---|---|---|---
Name | $z$ | ($\times 10^{-17}$ erg s-1 cm-2) | ($\times 10^{42}$ erg s-1) | (km s-1) | Reference
TGSS J1530+1049 | 5.72 | 1.6 | 5.7 | 370 | This work
TN J0924$-$2201 (RG) | 5.19 | 3.4 | 9.6 | 1500 | (van Breugel et al., 1999)
J163912.11$+$405236.5 (RG) | 4.88 | 18.5 | 47.0 | 1040 | (Jarvis et al., 2009)
VLA J123642$+$621331 (RG) | 4.42 | 0.6 | 2.0 | 420 | (Waddington et al., 1999)
LALA J142546.76$+$352036.3 | 5.75 | 1.6 | 6.7 | 360 | (Rhoads et al., 2003)
S11 5236 | 5.72 | 14.9 | 9.1 | 200 | (Lidman et al., 2012)
SDF J132344.8$+$272427 | 5.72 | 1.1 | 3.7 | 366 | (Kashikawa et al., 2011)
HSC J232558+002557 | 5.70 | 3.6 | 12.6 | 373 | (Shibuya et al., 2018)
SR6 | 5.67 | 7.6 | 25.0 | 236 | (Matthee et al., 2017)

A statistical sample of radio galaxies at $z\sim 6$ is needed to understand
whether they are more like LAEs at high redshift or whether a majority of them
continue being very different systems, surrounded by extremely overdense
regions and forming stars intensively. The relatively underluminous Ly$\alpha$
would be one signature of a significantly neutral intergalactic medium (IGM)
during the late stages of the EoR. Weaker Ly$\alpha$ emission may also be
caused by significant absorption in a cold and dusty medium surrounding the
radio galaxy. The presence of cold gas and dust has been reported in many
HzRGs, including TN J0924$-$2201 (see Klamer et al., 2005, for example) and
dedicated observations to look for molecular gas and dust in a statistically
significant sample of radio galaxies at $z>5$ are required to better
characterise their surrounding medium.

###  5.2 Radio properties

TGSS J1530+1049 has a flux density of 170 mJy at a frequency of 150 MHz and
7.5 mJy at 1.4 GHz (Saxena et al., 2018). Using the standard $K$-corrections
in radio astronomy and assuming a constant spectral index of $\alpha=-1.4$, we
calculate a rest-frame radio luminosity of $\log L_{\textrm{150 MHz}}=29.1$
and $\log L_{\textrm{1.4 GHz}}=28.2$ W Hz-1, which places this source at the
most luminous end of the radio luminosity function at this epoch (Saxena et
al., 2017). For comparison, TN J0924$-$2201 has a $K$-corrected radio
luminosity of $\log L_{\textrm{1.4 GHz}}=29.3$ W Hz-1 using a spectral index
of $\alpha=-1.6$ (van Breugel et al., 1999). TGSS J1530+1049 is close to an
order of magnitude fainter than TN J09224$-$2201 at 1.4 GHz, but remains by
far the brightest radio source observed this close to the end of the epoch of
reionisation.

The deconvolved angular size determined by Saxena et al. (2018) at 1.4 GHz for
TGSS J1530+1049, which remains unresolved, is 0.6 arcseconds, which translates
to a linear size of 3.5 kpc. This size is smaller than the size of TN
J0924$-$2201 (van Breugel et al., 1999) and in line with predictions at $z\sim
6$ from Saxena et al. (2017), as radio galaxies in the early Universe are
expected to be young and very compact (Blundell et al., 1999). In Table 4 we
compare the radio properties of TGSS J1530+1049 with all currently known radio
galaxies at $z>4$. This was done by querying the TGSS ADR1 catalog to
determine flux densities for all $z>4$ radio galaxies at 150 MHz, which were
then used to calculate radio powers using the standard $K$-corrections. We
find that TGSS J1530+1049 is comparable to many of the $z>4$ radio galaxies
when looking at radio properties alone.

Table 4: A comparison of the radio properties of TGSS J1530+1049 with other known radio galaxies at $z>4$. Flux densities at 150 MHz are measured from the TGSS catalog. |  | $S_{150}$ | $\log L_{\textrm{150}}$ | $S_{1400}$ |  | Size |
---|---|---|---|---|---|---|---
Name | $z$ | (mJy) | (W Hz-1) | (mJy) | $\alpha^{150}_{1400}$ | (kpc) | Reference
TGSS J1530+1049 | 5.72 | 170 | 29.1 | 7.5 | $-1.4$ | 3.5 | This work
TN J0924$-$2201 | 5.19 | 760 | 29.6 | 71.1 | $-1.1$ | 7.4 | (van Breugel et al., 1999)
J163912.11$+$405236.5 | 4.88 | 103 | 28.2 | 22.5 | $-0.7$ | $-$ | (Jarvis et al., 2009)
RC J0311$+$0507 | 4.51 | 5981 | 30.3 | 500.0 | $-1.1$ | 21.1 | (Parijskij et al., 2014)
VLA J123642$+$621331 | 4.42 | undetected | $-$ | 0.5 | $-$ | $-$ | (Waddington et al., 1999)
6C 0140$+$326 | 4.41 | 860 | 29.4 | 91.0 | $-1.0$ | 17.3 | (Rawlings et al., 1996)
8C 1435$+$635 | 4.25 | 8070 | 30.4 | 497.0 | $-1.2$ | 21.1 | (Lacy et al., 1994)
TN J1123$-$2154 | 4.11 | 512 | 29.1 | 49.3 | $-1.0$ | 5.5 | (Reuland et al., 2004)
TN J1338$-$1942 | 4.10 | 1213 | 29.5 | 120.8 | $-1.0$ | 37.8 | (Reuland et al., 2004)

TGSS J1530+1049 has a spectral index of $\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{1.4 GHz}}}=-1.4$, which is ultra-steep but flatter than
TN J0924$-$2201 at $z=5.19$, which was selected because of its spectral index
of $\alpha^{\textrm{\tiny{365 MHz}}}_{\textrm{\tiny{1.4 GHz}}}=-1.6$.
Interestingly, at lower radio frequencies the spectral index of TN
J0924$-$2201 appears to flatten dramatically. The 150 MHz flux density
measured in TGSS (Intema et al., 2017) for TN J0924$-$2201 is $760\pm 76$ mJy,
giving a low frequency spectral index $\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{365 MHz}}}=-0.16$. If the spectral index were to be
calculated only using the flux densities at frequencies of 150 MHz and 1.4
GHz, the inferred spectral index would be $\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{1.4 GHz}}}=-1.06$, making it not strictly ultra-steep
($\alpha<-1.3$). This implies that in a search for ultra-steep spectrum radio
sources using data at 150 MHz and 1.4 GHz, such as Saxena et al. (2018), TN
J0924$-$2201 would be missed entirely. Indeed, an indication of a generally
flatter spectral index at lower frequencies is visible in the spectral indices
calculated for known $z>4$ radio galaxies between frequencies of 150 MHz and
1.4 GHz in Table 4. A large majority of these radio sources were selected for
having an ultra-steep spectral index in a higher frequency range, but appear
to have a flatter spectral index when calculated between 150 MHz and 1.4 GHz.

Spectral flattening or even a turnover at low radio frequencies is expected in
radio galaxies at increasingly higher redshifts due to: a) Inverse Compton
(IC) losses due to the denser cosmic microwave background that affect the
higher frequencies and result in a steeper high frequency spectral index, and
b) free-free or synchrotron self absorption due to the compact sizes of radio
sources at high redshifts that can lead to a turnover in the low frequency
spectrum (see Callingham et al., 2017, and references therein). Saxena et al.
(2018) have reported evidence of flattening of the low-frequency spectral
index in candidate HzRGs and observations at intermediate radio wavelengths
for sources like TGSS J1530+1049 are essential to measure spectral flattening
and constrain various energy loss mechanisms that dominate the environments of
radio galaxies in the early Universe. Additionally, search techniques for
radio galaxies at even higher redshifts could be refined by possibly using
radio colours instead of a simple ultra-steep spectral index selection. The
LOFAR Two Metre Sky Survey (Shimwell et al. 2017, Shimwell et al. submitted)
will eventually provide in-band spectral indices at 150 MHz and could
potentially be used to identify HzRG candidates more efficiently.

We also draw attention towards the radio galaxy J163912.11$+$405236.5 at
$z=4.88$ (Jarvis et al., 2009), that has a spectral index of
$\alpha^{\textrm{\tiny{325 MHz}}}_{\textrm{\tiny{1.4 GHz}}}=-0.75$ and is not
an ultra-steep spectrum radio source. Interestingly, there is evidence of
spectral flattening at lower frequencies with a 150 MHz flux density of 103.5
mJy, giving a spectral index $\alpha^{\textrm{\tiny{150
MHz}}}_{\textrm{\tiny{325 MHz}}}=-0.56$, which is flatter than that at higher
frequencies. This source was targeted for spectroscopic follow-up owing to the
faintness of its host galaxy at $3.6$ $\mu m$. The very faint radio galaxy VLA
J123642$+$621331 at $z=4.42$ (Waddington et al., 1999) is also not strictly an
ultra-steep spectrum source at high radio frequencies
($\alpha^{\textrm{\tiny{1.4 GHz}}}_{\textrm{\tiny{4.8 GHz}}}=-0.94$) and is
too faint to be detected in TGSS. This source was also selected based on its
optical and infrared faintness, suggesting that a considerable fraction of
HzRGs may not be ultra-steep at all and therefore, be missed in samples
constructed using the ultra-steep spectrum selection technique.

Indeed Ker et al. (2012) have shown that selecting infrared-faint radio
sources (IFRS) could be more efficient at isolating HzRGs from large samples
when compared to radio selection alone. This has been confirmed
observationally through many previous studies (Norris et al., 2006; Collier et
al., 2014; Herzog et al., 2014; Maini et al., 2016; Singh et al., 2017).
However, the caveat is that deep infrared photometry over large sky areas is
required to effectively implement such a selection, which can be expensive.
The recently concluded UKIRT Hemisphere Survey (UHS; Dye et al. 2018) has the
potential to be extremely useful in the identification of promising HzRG
candidates in the Northern Hemisphere, particularly from the LOFAR surveys
(Shimwell et al. 2017, Shimwell et al. submitted), as a combination of a
relaxed spectral index steepness criterion and infrared-faintness could be
more effective at isolating HzRGs.

###  5.3 Stellar mass limits

The non-detection of the host galaxy down to a $3\sigma$ depth of $K=22.4$ can
be used to set limits on the stellar mass for TGSS J1530+1049 using simple
stellar population synthesis modelling. To do this, we make use of the python
package smpy111https://github.com/dunkenj/smpy, which is designed for building
composite stellar populations in an easy and flexible manner, allowing for
synthetic photometry to be produced for single or large suites of models (see
Duncan & Conselice, 2015). To build stellar populations, we use the Bruzual &
Charlot (2003) model with a Chabrier (2003) initial mass function (IMF) and
solar metallicity (Willott et al., 2003), a formation redshift $z_{f}=25$ and
assume a maximally old stellar population that has been forming stars at a
constant rate (Lacy et al., 2000). We follow the Calzetti et al. (2000) law
for dust attenuation and use values of $A_{v}=0.15$ (moderate extinction) and
$0.5$ mag (dusty), which are commonly seen in massive galaxies at $5<z<6$
(McLure et al., 2006). The synthetic photometry is produced for different
stellar masses, which we then convolve with the $K$ band filter to calculate
apparent $K$ magnitudes over a redshift range $0-7$.

Figure 7: The ‘$K-z$’ diagram for radio galaxies, showing stellar mass limits
derived from stellar population synthesis modelling for TGSS J1530+1049 (black
triangle). The $K$-band $3\sigma$ limit gives $M_{\textrm{stars}}<10^{10.25}$
$M_{\odot}$ for $A_{v}=0.15$ mag, and $M_{\textrm{stars}}<10^{10.5}$ for
$A_{v}=0.5$ mag. Also shown are $K$-band magnitudes and redshifts for known
radio galaxies in the literature (grey points; see text), with TN J0924$-$2201
at $z=5.19$ (orange circle). The $K$-band limits for TGSS J1530+1049 further
help exclude lower redshift measurements from incorrect line identification.

The $K$ magnitude limit for TGSS J1530+1049 fits well with a stellar mass
limit of $M_{\textrm{stars}}<\sim 10^{10.25}$ $M_{\odot}$ for $A_{v}=0.15$
mag, and $M_{\textrm{stars}}<\sim 10^{10.5}$ for $A_{v}=0.5$ mag. We note here
that thanks to the excellent seeing for $K$-band observations ($0.6-0.8$
arcseconds), and since at $z\sim 6$ the host galaxy is expected to be small,
any aperture correction is only expected to be at the level of a few tenths of
a magnitude at most, or $0.1-0.2$ in the logarithmic stellar mass, which is
smaller than the uncertainty from dust extinction corrections.

We find that the stellar mass limits we infer are in agreement with the $J$
band $3\sigma$ limit from LBT. The photometry predicted by the models in the
optical bands from PS1 ($g$, $r$, $i$, $z$, $y$) is also consistent with the
non-detections that we report. This stellar mass limit places TGSS J1530+1049
towards the $>M^{*}$ end of the galaxy stellar mass function at $z\sim 6$ (see
Duncan et al., 2014, for example). For comparison, we show the apparent $K$
band magnitudes of other radio galaxies in the literature, taken from the
3CRR, 6CE, 6C* and 7C$-$I/II/III samples (Willott et al., 2003), in Figure 7.
Also shown is the $K$ magnitude for TN J09224$-$2201 at $z=5.2$ (van Breugel
et al., 1999), which is best fit with a stellar mass of $10^{10.5}$
$M_{\odot}$ for $A_{v}=0.15$ mag and $10^{11}$ $M_{\odot}$ for $A_{v}=0.5$
mag.

We also show the $K$-band magnitude limit for the UKIDSS LAS as a dashed black
line in Figure 7. TGSS J1530+1049 was initially selected due to its non-
detection in LAS. However, these magnitude limits alone were not sufficient to
constrain the very high redshift nature of the host galaxy. With deeper LBT
observations in $K$ band, we show that TGSS J1530+1049 follows the trend in
the $K-z$ plot for radio galaxies. It is also clear that a low redshift
solution that would arise if the detected emission line in Section 4 is not
Lyman alpha (for example, $z\approx 1.2$ if the line is [O ii]) would be hard
to explain using galaxy evolution models with the inputs and assumptions
outlined above and those generally used to model radio galaxy spectra
(Overzier et al., 2009).

A stellar mass limit of $M_{\star}<10^{10.5}$ $M_{\odot}$ for TGSS J1530+1049
(based on the outlined assumptions) is almost an order of magnitude lower than
the sample of radio galaxies studied by Rocca-Volmerange et al. (2004), where
the highest stellar masses are seen for radio galaxies in the redshift range
$2-4$. This suggests that TGSS J1530+1049 is still evolving and is in the
process of building up its stellar mass. This interpretation is in line with
TGSS J1530+1049 being a relatively young radio galaxy owing to its small radio
size, and lower stellar masses may be expected from radio galaxies close to or
into the epoch of reionisation. Robust detections at optical and infrared
wavelengths are required to properly characterise the galaxy spectral energy
distribution in order to understand better the star formation history of TGSS
J1530+1049.

##  6 Conclusions

In this paper we have presented the discovery of the highest redshift radio
galaxy, TGSS J1530+1049, at $z=5.72$. The galaxy was initially selected at 150
MHz from TGSS (Saxena et al., 2018) and was assigned a high priority for
spectroscopic follow-up owing to its compact morphology and faintness at
optical and near-infrared wavelengths. The conclusions of this study are
listed below:

  1. 1.

Long-slit spectroscopy centered at the radio position of the source revealed
an emission line at 8170 Å, which we identify as Lyman alpha at $z=5.720$. We
rule out alternative line IDs owing to the absence of other optical/UV lines
in the spectrum, the asymmetrical nature of the emission line characteristic
of Lyman alpha at high redshifts that we quantify using the skewness parameter
and the high observed equivalent width of the emission line.

  2. 2.

Deep $J$ and $K$ band imaging using the Large Binocular Telescope led to no
significant detection of the host galaxy down to $3\sigma$ limits of $K>22.4$
and $J>24.4$. The limits in $K$ can be used as an additional constraint on the
redshift, owing to the relation that exists between $K$ band magnitude and
redshift of radio galaxies. The magnitude limit is consistent with $z>5$,
practically ruling out a redshift of $z\approx 1.2$ that would be expected if
the emission line were an unresolved [O ii] $\lambda\lambda 3726,3729$
doublet, which is the most likely alternative line identification.

  3. 3.

The emission line is best fitted with a skewed Gaussian, giving an integrated
line flux of $F_{Ly\alpha}=1.6\times 10^{-17}$ erg s-1 cm-2, a Ly$\alpha$
luminosity of $5.7\times 10^{42}$ erg s-1, an equivalent width of EW $>40$ Å
and a FWHM of $370$ km s-1. These values are more consistent with those
observed in non-radio Lyman alpha emitting galaxies at this redshift and much
lower than those corresponding to typical radio galaxies at $z>4$.

  4. 4.

The radio luminosity calculated at 150 MHz is $\log L_{\textrm{150}}=29.1$ W
Hz-1, which places it at the most luminous end of the radio luminosity
function at this epoch. The deconvolved angular size is 3.5 kpc, which is in
line with the compact morphologies expected at high redshifts. We find that
the radio properties of TGSS J1530+1049 are comparable to other known radio
galaxies at $z>4$ but the compact size suggests that it is a radio galaxy in
an early phase of its evolution. A joint study of the Ly$\alpha$ halo and the
radio size of this source may provide one of the earliest constraints on the
effects of radio-mode feedback.

  5. 5.

We use the $K$ band limit to put constraints on the stellar mass estimate
using simple stellar population synthesis models. Assuming a constant star
formation history and a maximally old stellar population, we derive a stellar
mass limit of $M_{\textrm{stars}}<\sim 10^{10.25}$ $M_{\odot}$ for
$A_{v}=0.15$ mag, and $M_{\textrm{stars}}<\sim 10^{10.5}$ for $A_{v}=0.5$ mag.
These limits are almost an order of magnitude lower than typical radio galaxy
masses in the redshift range $2-3$ and suggest that TGSS J1530+1049 may still
be in the process of assembling its stellar mass, which is in line with it
being a relatively young radio galaxy.

An effective application of deep radio surveys covering very large areas on
the sky has been demonstrated by this discovery of the first radio galaxy at a
record distance after almost 20 years. With the more sensitive, large area
surveys currently underway with LOFAR (LoTSS; Shimwell et al. 2017, Shimwell
et al. submitted), there is potential to push searches for radio galaxies to
even higher redshifts. Discovery of even a single bright radio galaxy at $z>6$
would open up new ways to study the epoch of reionisation in unparalleled
detail, through searches for the 21cm absorption features left behind by the
neutral hydrogen that pervaded the Universe at high redshifts.

## Acknowledgements

The authors thank the referee for useful comments and suggestions. AS would
like to thank Jorryt Matthee, David Sobral, Reinout van Weeren and Nobunari
Kashikawa for fruitful discussions. AS, HJR and KJD gratefully acknowledge
support from the European Research Council under the European Unions Seventh
Framework Programme (FP/2007-2013)/ERC Advanced Grant NEWCLUSTERS-321271. RAO
and MM received support from CNPq (400738/2014-7, 309456/2016-9) and FAPERJ
(202.876/2015). PNB is grateful for support from STFC via grant ST/M001229/1.
IP acknowledges funding from the INAF PRIN-SKA 2017 project 1.05.01.88.04
(FORECaST).

This paper is based on results from observations obtained at the Gemini
Observatory, which is operated by the Association of Universities for Research
in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of
the Gemini partnership: the National Science Foundation (United States), the
National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia,
Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência,
Tecnologia, Inovação e Comunicações (Brazil). This paper also contains data
from the Large Binocular Telescope (LBT), an international collaboration among
institutions in the United States, Italy and Germany. LBT Corporation partners
are: The University of Arizona on behalf of the Arizona university system;
Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft,
Germany, representing the Max-Planck Society, the Astrophysical Institute
Potsdam, and Heidelberg University; The Ohio State University, and The
Research Corporation, on behalf of The University of Notre Dame, University of
Minnesota, and University of Virginia.

This work has made extensive use of ipython (Pérez & Granger, 2007), astropy
(Astropy Collaboration et al., 2013), aplpy (Robitaille & Bressert, 2012),
matplotlib (Hunter, 2007) and topcat (Taylor, 2005). This work would not have
been possible without the countless hours put in by members of the open-source
developing community all around the world.

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