hole charge” or “connection hair”. It is important to notice that the
correction terms could lead to deviations from the Schwarzschild solution even
at large values of $r$. The logarithmic correction term will dominate over the
renormalized Schwarzschild term for radii satisfying

$\left|\ln(r/r^{*})\right|>\frac{2M_{\textsf{ren}}}{\alpha^{2}|\mu|}.$ (5.22)

This equation implies that perturbation theory breaks down at such large $r$.
Only at smaller $r$ are the metric perturbations small compared to the
background Schwarzschild spacetime.
This form of $g_{tt}$ is also particularly useful when we want to inquire the
location of the Killing horizon. To that end, we just need to determine where
the norm of the time-translation Killing vector field
$\mathcal{T}:=t^{\alpha}\partial_{\alpha}$ vanishes. One finds

$\displaystyle g_{\mu\nu}t^{\mu}t^{\nu}$ $\displaystyle=g_{tt}\overset{!}{=}0$
$\displaystyle\Longrightarrow$ $\displaystyle
2M_{\textsf{ren}}+\alpha^{2}\mu\ln(r/r^{*})-r\overset{!}{=}0.$ (5.23)

Numerical considerations show that this equation has generically two
solutions. The first one can even be found analytically in terms of Lambert’s
$W$-function and we call it the internal Killing horizon. Its location is
given by

$r_{\textsf{Internal Killing
horizon}}=-\alpha^{2}\mu\,W\left(-\frac{r^{*}}{\alpha^{2}\mu}\operatorname{e}^{-\frac{2M_{\textsf{ren}}}{\alpha^{2}\mu}}\right)\approx
r^{*}\operatorname{e}^{-\frac{2M_{\textsf{ren}}}{\alpha^{2}\,\mu}},$ (5.24)

where we have expanded Lambert’s $W$-function to first order in its argument;
$W(x)=x+\mathcal{O}(x^{2})$. We call this the internal Killing horizon because
the exponential factor is essentially zero and because a second solution,
which we call the outer Killing horizon, can be found numerically, which is
close to the standard Schwarzschild radius $r=2M$. There is no closed
analytical expression for the outer horizon, but we can solve equation (5.24)
to first order in $\alpha$ and we find the radius

$r_{\textsf{Outer Killing horizon}}=2M+\alpha\,c_{2}+\mathcal{O}(\alpha^{2}).$
(5.25)

Hence, the outer Killing horizon can lie below or above the standard
Schwarzschild horizon. The considerations of this subsection can also be
generalized to the case of an electro-vacuum and to a non-zero cosmological
constant. This generalization will be the subject of the next subsection.

###  V.2 Approximate electro-vacuum solution beyond GR for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}-2\Lambda$

To generalize the approximation scheme of the previous subsection to the case
of an electro-vacuum, we need to introduce the energy-momentum tensor

$\displaystyle
T^{\mu}{}_{\nu}=\textsf{diag}\left(\Lambda+\frac{Q^{2}}{r^{4}},\Lambda+\frac{Q^{2}}{r^{4}},\Lambda-\frac{Q^{2}}{r^{4}},\Lambda-\frac{Q^{2}}{r^{4}}\right),$
(5.26)

where $Q$ denotes the charge of the source. We choose again the ansatz

$\displaystyle g_{tt}$
$\displaystyle=g_{tt}^{(0)}+\alpha\,g_{tt}^{(1)}+\alpha^{2}\,g_{tt}^{(2)}$
$\displaystyle g_{rr}$
$\displaystyle=g_{rr}^{(0)}+\alpha\,g_{rr}^{(1)}+\alpha^{2}\,g_{rr}^{(2)}$
$\displaystyle\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=-r+\alpha\,\gamma^{(1)}+\alpha^{2}\,\gamma^{(2)},$ (5.27)

but now with

$\displaystyle g_{tt}^{(0)}$
$\displaystyle=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}\right)$
and $\displaystyle g_{rr}^{(0)}$ $\displaystyle=-\frac{1}{g_{tt}^{(0)}}.$
(5.28)

Just as in the previous subsection, we can solve the field equations order by
order. We do not give all details here as they are completely analogously
solved as for the vacuum case. Unfortunately, the solution for $\gamma^{(1)}$
is rather implicit:

$\displaystyle\gamma^{(1)}$
$\displaystyle=r\,c_{3}\left(1+\int^{r}\operatorname{e}^{-\frac{1}{2}\sum_{i=1}^{4}\frac{(6M+7\Lambda\,\rho^{3})\ln(\rho-
x_{i})}{3M+2\Lambda\rho^{3}}}\rho^{3}\,\ln(\rho)\,\mathrm{d}\rho\right)+r\,c_{4}\int^{r}\operatorname{e}^{-\frac{1}{2}\sum_{i=1}^{4}\frac{(6M+7\Lambda\,\rho^{3})\ln(\rho-
x_{i})}{3M+2\Lambda\rho^{3}}}\rho^{3}\,\mathrm{d}\rho,$ (5.29)

where $x_{i}$ are the four solutions to the quartic equation

$3Q^{2}-6M\,x_{i}-\Lambda\,x_{i}^{4}=0.$ (5.30)

However, we can discuss the cases $Q\neq 0$, $\Lambda=0$ and $Q=0$,
$\Lambda\neq 0$ separately. We begin by setting $\Lambda=0$. Then the metric
components at second order in perturbation theory are given by

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}-\frac{Q_{\text{ren}}^{2}}{r^{2}}+\alpha^{2}\mu\left(\frac{2M_{\text{ren}}}{r}-\frac{Q_{\text{ren}}^{2}}{r^{2}}\right)\ln\left(\frac{r^{2}}{r^{*2}}\left(\frac{2M_{\text{ren}}}{r}-\frac{Q_{\text{ren}}^{2}}{r^{2}}\right)\right)$
$\displaystyle g_{rr}$ $\displaystyle=-\frac{1}{g_{tt}},$ (5.31)

where we have defined

$\displaystyle\mu$ $\displaystyle:=-\frac{6c_{8}}{M}$ $\displaystyle
2M_{\text{ren}}$ $\displaystyle:=2M+\alpha
c_{2}+\alpha^{2}(c_{4}-2M(8c_{6}-4c_{7})-4c_{8})$ $\displaystyle
Q_{\text{ren}}^{2}$
$\displaystyle:=Q^{2}\left(1-\alpha^{2}(8c_{6}-4c_{7})-4\alpha^{2}\frac{c_{8}}{2M}\right).$
(5.32)

The $c_{i}$ are integration constants, where $c_{2}$ and $c_{4}$ comes from
solving the $\mathcal{M}_{tt}$ equations for $g_{rr}$ at first and second
order in $\alpha$, respectively, and $c_{6},c_{7},c_{8}$ come from solving the
connection equation $\mathcal{C}_{r}$ at second order in $\alpha$. The
integration constants in $g_{tt}$ were set to zero in order to have again
$\lim_{r\to\infty}g_{tt}=-1$. The scale $r^{*}$ can again be introduced via a
redefinition of $c_{6}$ and it ensures proper units.
Next, let us set $Q=0$ but keep $\Lambda$ different from zero. The solution
for the metric is now given by

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\frac{\Lambda_{\text{ren}}}{3}r^{2}+\alpha^{2}\mu\left(\frac{2M_{\text{ren}}}{r}+\frac{\Lambda_{\text{ren}}}{3}r^{2}\right)\ln\left(\frac{r^{*2}}{r^{2}}\left(\frac{2M_{\text{ren}}}{r}+\frac{\Lambda_{\text{ren}}}{3}r^{2}\right)\right)$
$\displaystyle g_{rr}$ $\displaystyle=-\frac{1}{g_{tt}},$ (5.33)

where we have defined

$\displaystyle\mu$ $\displaystyle:=\frac{4c_{8}}{18M}$ $\displaystyle
2M_{\text{ren}}$ $\displaystyle:=2M+\frac{\alpha(c_{2}+\alpha
c_{4})+24\alpha^{2}M(c_{6}-c_{7})}{9}$
$\displaystyle\frac{\Lambda_{\text{ren}}}{3}$
$\displaystyle:=\frac{\Lambda}{3}\left(1+\frac{12\alpha^{2}(c_{6}-c_{7})}{9}\right).$
(5.34)

The integration constants and scale $r^{*}$ arise in a similar fashion as for
the charged case above.

We note that using the functional equation of the logarithm in the limits
$Q\rightarrow 0$ and $\Lambda\rightarrow 0$, respectively, we obtain the
vacuum solutions from the previous subsection by absorbing the powers of $r$
in the logarithm in the prefactor $\mu$. Moreover, we see that the new
correction is simply the “background” GR potential times the logarithm of the
potential times or divided by $r^{2}$ for the charge and cosmological constant
cases, respectively. One can draw similar conclusions for the Killing horizons
as in the previous subsection, but we will not go into details here.

###  V.3 Approximate solutions beyond GR from constraints on the connection

In subsection IV.4 we have seen that the field equations can also be solved by
imposing one of the constraints (4.30) on $\Gamma^{r}{}_{\theta\theta}$. We
choose again to work with option 2 and we select the constraint

$\Gamma^{r}{}_{\theta\theta}=\pm\frac{r}{\sqrt{g_{rr}}}.$ (5.35)

We call these two cases (I±) and we first discuss (I+). As explained in
subsection IV.4, the constraint ensures that the connection field equation is
satisfied and we are hence left with the two metric field equations
$\mathcal{M}_{tt}$ and $\mathcal{M}_{rr}$. In order to find an approximate
solution to these equations, we choose again the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$ together with

$\displaystyle g_{tt}$ $\displaystyle=g_{tt}^{(0)}+\alpha\,g_{tt}^{(1)}$
$\displaystyle g_{rr}$ $\displaystyle=g_{rr}^{(0)}+\alpha\,g_{rr}^{(1)},$
(5.36)

where $g_{tt}^{(0)}$ and $g_{rr}^{(0)}$ are given by the standard
Schwarzschild solution (5.10). First of all, we notice that the field
equations $\mathcal{M}_{tt}$ and $\mathcal{M}_{rr}$ can be solved for
$\partial_{r}g_{tt}$ and $\partial_{r}g_{rr}$ in full generality:

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=\frac{g_{tt}\,\left(f(\mathbb{Q})\,g_{rr}\,r^{2}-4\left(1+\sqrt{g_{rr}}\right)f^{\prime}(\mathbb{Q})\right)}{2\left(2+\sqrt{g_{rr}}\right)\,r\,f^{\prime}(\mathbb{Q})}$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=-\frac{g_{rr}\left(f(\mathbb{Q})\,g_{rr}\,r^{2}-4\left(f^{\prime}(\mathbb{Q})+\sqrt{g_{rr}}\right)\left(f^{\prime}(\mathbb{Q})+\left(2+\sqrt{g_{rr}}\right)\,r\,\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})\right)\right)}{2\left(2+\sqrt{g_{rr}}\right)\,r\,f^{\prime}(\mathbb{Q})}.$
(5.37)

Moreover, it can be shown that the non-metricity scalar can be written as

$\mathbb{Q}=\frac{2\left(1+\sqrt{g_{rr}}\right)\left(g_{tt}\,\left(1+\sqrt{g_{rr}}\right)+r\,\partial_{r}g_{tt}\right)}{g_{tt}\,g_{rr}\,r^{2}}.$
(5.38)

After inserting the first line of (V.3) into the above expression of
$\mathbb{Q}$ and using the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, we can get rid of
$\partial_{r}\mathbb{Q}$ in the second line of (V.3) and any $\mathbb{Q}$
which appears from using the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$. Hence, we end up with
equations which only depend on the metric components and nothing else. After
inserting the ansatz (V.3) into these equations, we can expand them order by
order. At zeroth order, we find trivially satisfied equations, as had to be
expected. At first order, we find non-trivial equations which can be
integrated:

$\displaystyle g_{tt}^{(1)}$
$\displaystyle=\left(1-\frac{2M}{r}\right)c_{2}+\frac{32}{3M^{2}}\frac{\left(r-2M\right)^{\frac{3}{2}}}{r^{\frac{3}{2}}}+\frac{1}{M^{2}\,r^{3}}\left(\ln\left(1-\frac{2M}{r}\right)r^{2}\,\left(r-3M\right)+M\left(2M^{2}+2r^{2}+Mr\,\left(12+c_{1}\,r\right)\right)\right)$
$\displaystyle g_{rr}^{(1)}$
$\displaystyle=\frac{r}{(r-2M)^{2}}\left(c_{1}-\frac{1}{M}\ln\left(1-\frac{2M}{r}\right)-\frac{50M}{r^{2}}+\frac{46}{r}-\frac{16\sqrt{r-2M}(M-2r)(3M-r)}{3M\,r^{\frac{5}{2}}}\right)$
(5.39)

For the asymptotic limit we find

$\displaystyle\lim_{r\to\infty}g_{tt}$
$\displaystyle=-1+\alpha\left(\frac{32}{3M^{2}}+c_{2}\right)$ and
$\displaystyle\lim_{r\to\infty}g_{rr}$ $\displaystyle=1,$ (5.40)

which implies that we have to set $c_{2}=-\frac{32}{3M^{2}}$ in order to
obtain a standard asymptotically Minkowski solution. With this choice of
integration constant, we find that the first order corrections for large $r$
can be written as

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{32}{r^{2}}$
$\displaystyle-\frac{1}{g_{rr}}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{96}{r^{2}}$ (5.41)

where we have introduced the renormalized mass

$2M_{\text{ren}}:=2M-\alpha\left(\frac{32}{3M}+c_{1}\right).$ (5.42)

Hence, the first order beyond-GR corrections scale as $\frac{1}{r^{2}}$. As we
will see shortly, for the case (I-) we obtain a different scaling of the first
order corrections.

We now deal with (I-) analogously to (I+). The field equations are now given
by

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=\frac{g_{tt}\left(f(\mathbb{Q})\,r^{2}g_{rr}-4f^{\prime}(\mathbb{Q})\left(\sqrt{g_{rr}}-1\right)\right)}{2f^{\prime}(\mathbb{Q})\,r\left(\sqrt{g_{rr}}-2\right)}$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=-\frac{g_{rr}\left(4\left(\sqrt{g_{rr}}-1\right)\left(f^{\prime\prime}(\mathbb{Q})\,r\left(\sqrt{g_{rr}}-2\right)\partial_{r}\mathbb{Q}-f^{\prime}(\mathbb{Q})\right)+f(\mathbb{Q})\,r^{2}g_{rr}\right)}{2f^{\prime}(\mathbb{Q})r\left(\sqrt{g_{rr}}-2\right)},$
(5.43)

and the non-metricity scalar for (I-) takes the form

$\mathbb{Q}=\frac{2\left(\sqrt{g_{rr}}-1\right)\left(r\partial_{r}g_{tt}-\sqrt{g_{rr}}g_{tt}+g_{tt}\right)}{r^{2}g_{rr}g_{tt}}\
.$ (5.44)

After using $f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$ and applying the
ansatz (V.3), one finds the first order solutions

$\displaystyle g_{tt}^{(1)}$
$\displaystyle=\frac{(r-2M)}{3r}\left(\frac{\frac{6M}{r^{2}}+\frac{64}{M\sqrt{\frac{r}{r-2M}}}+\frac{6\ln\left(1-\frac{2M}{r}\right)}{r-2M}-\frac{6\ln\left(1-\frac{2M}{r}\right)}{M}+\frac{51-6c_{1}M}{2M-r}+\frac{39}{r}}{2M}+3c_{2}\right)$
$\displaystyle g_{rr}^{(1)}$
$\displaystyle=\frac{r}{(r-2M)^{2}}\left(-\frac{4\sqrt{\frac{r}{r-2M}}\left(68M^{2}r-24M^{3}-44Mr^{2}+8r^{3}\right)}{3Mr^{3}}+\frac{50M}{r^{2}}+\frac{\ln\left(1-\frac{2M}{r}\right)}{M}-\frac{46}{r}+c_{1}\right).$
(5.45)

The asymptotic limit is again easy to determine and we find

$\displaystyle\lim_{r\to\infty}g_{tt}$
$\displaystyle=-1+\alpha\left(\frac{32}{3M^{2}}+c_{2}\right)$ and
$\displaystyle\lim_{r\to\infty}g_{rr}$ $\displaystyle=1.$ (5.46)

Upon fixing $c_{2}=-\frac{32}{3M^{2}}$, we obtain
$\lim_{r\to\infty}g_{tt}=-1$, i.e., we have again an asymptotically Minkowski
solution. If we use this value for the integration constant $c_{2}$, we find
that the first order corrections at large $r$ are given by

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{8M^{3}}{5r^{5}}$
$\displaystyle-\frac{1}{g_{rr}}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{8M^{3}}{r^{5}},$
(5.47)

where the renormalized mass now reads

$2M_{\text{ren}}:=2M+\alpha\left(\frac{32}{3M}-c_{1}\right).$ (5.48)

Hence, we find that the corrections for the (I-) case scale like
$\frac{1}{r^{5}}$ and the solution is therefore virtually indistinguishable
from Schwarzschild at large radii.

###  V.4 A comment on isotropic coordinates

We want to mention here a different ansatz for the metric, and its connection
to the solutions above, namely isotropic coordinates. In these coordinates one
chooses not $g_{\theta\theta}=r^{2}$, as we did above in III.6, but
$g_{\theta\theta}=g_{RR}R^{2}$. We denote the isotropic radius by $R$ to
distinguish it from the Schwarzschild radial coordinate $r$. The line element
can then be written as

$\mathrm{d}s^{2}=-g_{tt}\mathrm{d}t^{2}+g_{RR}\left(\mathrm{d}R^{2}+R^{2}\mathrm{d}\Omega^{2}\right),$
(5.49)

were $g_{tt}$ and $g_{RR}$ are functions of $R$ only. The coordinate
transformation that links isotropic and Schwarzschild coordinates is given by

$\ln\left(\frac{R}{R_{0}}\right)=\int_{r_{0}}^{r}\mathrm{d}r\,\frac{\sqrt{g_{rr}}}{r}.$
(5.50)

Of course, since the difference between the isotropic and the Schwarzschild
ansatz is just a choice of coordinates, the results for the gravitational
potential are the same as for the ones discussed above; so this ansatz might
seem pointless. Nevertheless, we found an interesting link.
If we want to plug the isotropic ansatz into the equations of motion, we have
to choose an ansatz for the connection. But now we have an advantage in
isotropic coordinates; the spatial sections are flat. Isotropic coordinates
are thus very similar to cosmological spacetimes with flat spatial sections.
We can therefore try to use the coincident gauge for the connection, as it is
the natural connection of flat spacetimes in flat coordinates. In spherical
coordinates one then obtains the spherical connection, as used at zeroth order
perturbation theory in V.1, or in Zhao:2021 ; Lin:2021 , but just in isotropic
coordinates with $R$ instead of $r$.
Since the connection is fixed, we will not find any new connection hairs, but
curiously enough we do find completely consistent metric equations of motions;
moreover, the connection equations of motion are identically satisfied. Even
better, (5.28) is not a solution to the metric equations of motion, so it
might seem that this very simple ansatz gives again new black hole
corrections. But we have actually covered this already; if one performs the
coordinate transformation (5.50) and applies this to the spherical connection,
one obtains precisely the same connection as we had in case (I-). Hence (I-)
corresponds to an “isotropic ansatz” to the black hole problem in
$f(\mathbb{Q})$ gravity.

###  V.5 Exact vacuum solutions beyond GR for
$f(\mathbb{Q})=\mathbb{Q}^{\kappa}$

So far we have only discussed approximate solutions within the framework of
perturbation theory and we have seen that $f(\mathbb{Q})$ admits “connection
hair”. However, it is also possible to derive exact solutions of
$f(\mathbb{Q})$ gravity which go beyond GR and where the connection appears as
“hair”, with a new scale $r_{T}$ appearing in the metric components. The
starting point are again the field equations (IV.3) for option 2 since these
equations are simpler and only require the specification of initial
conditions, rather than the arbitrary choice of the connection components $c$
and $k$.
The main observation we need is that the metric field equations for option 2
imply that

$\displaystyle
g_{tt}g_{rr}=c_{1}\,\exp\left(2\int\mathrm{d}r\,\frac{\Gamma^{r}{}_{\theta\theta}+r}{\Gamma^{r}{}_{\theta\theta}}\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\right),$
(5.51)

as we had seen at the beginning of section V. Performing integrals is in
general a daunting task and it is often not possible to compute them
analytically. However, if we assume that $\Gamma^{r}{}_{\theta\theta}$ is of
the form

$\Gamma^{r}{}_{\theta\theta}=-\lambda\,r,$ (5.52)

where $\lambda\in\mathbb{R}\setminus\\{0\\}$ is an arbitrary constant, then
the integral becomes manageable. First of all, we notice that the choice
$\lambda=1$ would again give us the spherical connection of Zhao:2021 ;
Lin:2021 , which simply produces the GR solution for arbitrary $f$. Hence, we
have again a parametrization of $\Gamma^{r}{}_{\theta\theta}$ which allows us
to “deform” the spherical connection and “move away” from the GR solution.
Moreover, after inserting the ansatz (5.52) into the integral, we easily find

$g_{tt}\,g_{rr}=c_{1}\,\left(f^{\prime}(\mathbb{Q})\right)^{2\frac{\lambda-1}{\lambda}},$
(5.53)

where we have absorbed additional factors in the integration constant $c_{1}$.
We can use this equation to eliminate $g_{rr}$ and we are therefore left with
just $g_{tt}$ which needs to be determined. To that end, we use the metric
field equation $\mathcal{M}_{tt}$ for the pure vacuum case and we find

$\partial_{r}g_{tt}=\frac{c_{1}\,\lambda}{2(2\lambda-1)}\frac{f^{\prime}(\mathbb{Q})^{\frac{\lambda-2}{\lambda}}}{r}\left(f(\mathbb{Q})\,r^{2}+2\lambda\,f^{\prime}(\mathbb{Q})+2(2\lambda-1)\,r\,\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})\right)-g_{tt}\left(\frac{1}{r}+\frac{\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})}{\lambda\,f^{\prime}(\mathbb{Q})}\right)$
(5.54)

We have already made use of both metric field equations and we are thus only
left with the connection field equation $\mathcal{C}_{r}$. To analyze this
equation, it is convenient to first compute the non-metricity scalar
$\mathbb{Q}$ and then insert it into $\mathcal{C}_{r}$. For the non-metricity
scalar we find

$\mathbb{Q}=\frac{\lambda-1}{2\lambda-1}\frac{f(\mathbb{Q})\,r^{2}-2(\lambda-1)\,f^{\prime}(\mathbb{Q})}{r^{2}\,f^{\prime}(\mathbb{Q})}.$
(5.55)

At this point we need to make a choice for $f$ in order to continue. We
choose888Unfortunately we could not proceed with the more interesting choice
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{\kappa}$.
$f(\mathbb{Q})=\mathbb{Q}^{\kappa}$, where
$\kappa\in\mathbb{R}\setminus\\{0\\}$. After inserting this ansatz for $f$
into the above expression for $\mathbb{Q}$, we find that $\mathbb{Q}$ can be
written as

$\mathbb{Q}=-\frac{2\kappa\,(\lambda-1)^{2}}{1-\lambda+\kappa\,(2\lambda-1)}\frac{1}{r^{2}}.$
(5.56)

This is a very simple function of $r$ and when inserted into the connection
field equation one finds the polynomial equation

$(\kappa-1)\left(\kappa(8-14\lambda)+5(\lambda-1)+\kappa^{2}(8\lambda-4)\right)\left(\frac{\kappa(\lambda-1)^{2}}{(1-\lambda+\kappa(2\lambda-1))r^{2}}\right)^{\kappa}=0\
.$ (5.57)

This equation is solved by

$\displaystyle\lambda$ $\displaystyle=1,$ $\displaystyle\lambda$
$\displaystyle=\frac{5-8\kappa+4\kappa^{2}}{5-14\kappa+8\kappa^{2}}.$ (5.58)

The first solution simply produces the spherical connection. The second
solution is more interesting, but we also need to assume that
$\kappa\notin\\{\frac{1}{2},\frac{5}{4}\\}$ in order for this solution to be
well-defined. Moreover, it is now easy to check that all field equations,
$\mathcal{M}_{tt}$, $\mathcal{M}_{rr}$, $\mathcal{C}_{r}$, and
$\mathcal{M}_{\theta\theta}$, are satisfied. There is therefore no new
information we can gather and we can use all results obtained thus far to
integrate the equation (5.54). We find that $g_{tt}$ is given by

$\displaystyle g_{tt}=c_{1}\,r^{\beta}+c_{2}\,r^{\alpha},$ (5.59)

where $c_{2}$ is an integration constant and where we have defined

$\displaystyle\beta$
$\displaystyle:=\frac{8(2\kappa-3)(\kappa-1)\kappa}{5+4(\kappa-2)\kappa},$
$\displaystyle\alpha$
$\displaystyle:=\frac{(2\kappa-3)(5+4\kappa(2\kappa-3))}{5+4(\kappa-2)\kappa}.$
(5.60)

The solution for $g_{rr}$ then follows from (5.53) and we find

$\displaystyle
g_{rr}=\underbrace{\left(\frac{8\kappa^{2}-14\kappa+5}{4\kappa^{2}-8\kappa+5}\right)^{2}}_{=:C}\frac{c_{1}}{c_{1}+c_{2}\,r^{\gamma}},$
(5.61)

with $\gamma:=\alpha-\beta$. Observe that if we choose $\kappa=1$, which
corresponds to the choice $f(\mathbb{Q})=\mathbb{Q}$ and which implies
$\beta=0$, $\alpha=-1$, and $C=1$, we obtain

$\displaystyle g_{tt}$ $\displaystyle=c_{1}+\frac{c_{2}}{r}$ and
$\displaystyle g_{rr}$ $\displaystyle=\frac{c_{1}}{g_{tt}}.$ (5.62)

In other words: The GR solution is correctly reproduced by the equations
(5.59), (5.60), and (5.61). This is a reassuring consistency test and we can
now try to quantify how the solutions for $\kappa>1$ deviates from the
Schwarzschild solution by setting $c_{1}\equiv-r_{s}^{-\gamma}c_{2}$ and
$c_{2}\equiv-r_{T}^{-\gamma}$, where $r_{s}$ is the Schwarzschild radius and
$r_{T}$ is a new gravitational time dilation scale. The $g_{tt}$ and $g_{rr}$
components then read

$\displaystyle g_{tt}$
$\displaystyle=-\left(\frac{r}{r_{T}}\right)^{\beta}\left(1-\left(\frac{r_{s}}{r}\right)^{-\gamma}\right)$
and $\displaystyle g_{rr}$
$\displaystyle=\frac{C}{1-\left(\frac{r_{s}}{r}\right)^{-\gamma}}.$ (5.63)

In order to compare this to the Schwarzschild solution, we need to assume
$\gamma<0$. This condition is satisfied for all
$\kappa\in\mathbb{R}\setminus[\frac{5}{4},\frac{3}{2}]$; note that for
$\kappa=3/2$ we have the trivial flat solution $\alpha=\beta=0$. See also
Figure 1, which shows the exponents $\alpha$, $\beta$ and $\gamma$ as
functions of $\kappa$. This assumption is also reasonable since it leads to a
well-behaved limit of $g_{rr}$ as $r$ goes to infinity:

$\lim_{r\to\infty}g_{rr}=C.$ (5.64)

However, the situation for $g_{tt}$ looks quite different. Its limit for
$r\to\infty$ is given by

$\lim_{r\to\infty}g_{tt}=-\lim_{r\to\infty}\left(\frac{r}{r_{T}}\right)^{\beta},$
(5.65)

This limit does not behave well because it either diverges ($\beta>0$) or it
vanishes ($\beta<0$). It is only interesting for $\beta=0$, which is achieved
for $\kappa=0$, $\kappa=1$, and $\kappa=\frac{3}{2}$. But $\kappa=0$ is not
admissible, since then $f(\mathbb{Q})=1$, $\kappa=1$ is simply symmetric
teleparallelism, and $\kappa=\frac{3}{2}$ belongs to the range of excluded
$\kappa$ values. Hence, the interpretation of the metric at large radii is
difficult unless we are in standard symmetric teleparallelism, where the
metric reduces to the Schwarzschild solution.

We conclude this subsection by noting that this solution is only close to
Schwarzschild for $\kappa$ very close to unity. Even for $f=\mathbb{Q}^{2}$
one has $\alpha=2.6$ and $\beta=3.2$, which is far from Schwarzschild; Solar
system tests would have revealed such large deviations. These exact solutions
are thus physically not relevant. What is interesting, though, is that exact
solutions can be found which go beyond GR. Finding physically interesting
exact solutions is left as a challenge for future work.

Figure 1: Left panel: The exponents $\alpha$ (solid blue curve), $\beta$
(dashed red curve), and $\gamma$ (dotted orange curve) as functions of
$\kappa$. The shaded blue region indicates the range where both, $\alpha$ and
$\beta$ are negative. Right panel: The function $\gamma$ is negative except in
the shaded orange region which corresponds to the interval
$[\frac{5}{4},\frac{3}{2}]$.

##  VI Application to $f(\mathbb{T})$ gravity

There is yet another theory of gravity which is closely related to Symmetric
Teleparallelism and which together with GR forms the trinity gravitational
theories BeltranJimenez:2019  – Metric Teleparallelism (MT).

The underlying geometric framework of MT is closely related to the one of ST
studied in this paper and we can easily transfer some of our methods to MT in
order to discuss stationary and spherically symmetric solutions of
$f(\mathbb{T})$. Our discussion of $f(\mathbb{T})$ gravity will be brief in
the sense that we will only report the basic results, but we will not repeat
the lengthy derivations which are analogous to the ones for ST which have been
explained in great detail in this paper.
We begin by recalling basic definitions of MT and fixing our notation.
Subsequently, we will report the symmetry reduced form of the metric, the
connection, and the field equations of MT. Finally, we discuss some
approximate solutions to the field equations.

We give first a brief introduction to MT, and then note the stationary and
spherically symmetric Ansatz for the MT connection and the equations of
motion. Results are discussed last.

###  VI.1 Metric Teleparallelism and $f(\mathbb{T})$ gravity

Let $(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}_{\ \mu\nu})$ be a metric-affine
geometry, where $\mathcal{M}$ is a four-dimensional manifold, $g_{\mu\nu}$
denotes the components of the metric tensor of signature $(-,+,+,+)$, and
$\Gamma^{\alpha}{}_{\mu\nu}$ represents an affine connection. The latter is
now postulated to be flat and metric-compatible, but with non-trivial torsion.
That is, we postulate

$\displaystyle R^{\alpha}{}_{\beta\mu\nu}$ $\displaystyle\overset{!}{=}0$ and
$\displaystyle Q_{\alpha\mu\nu}$ $\displaystyle\overset{!}{=}0.$ (6.1)

As alluded to above, the only non-trivial object in this metric-affine
geometry is the torsion tensor, defined by

$\displaystyle T^{\alpha}{}_{\mu\nu}:=2\Gamma^{\alpha}{}_{[\mu\nu]}.$ (6.2)

Notice that the postulates of Symmetric Teleparallelism imply that the
connection is completely independent of the metric. This is no longer true in
Metric Teleparallelism because the postulate of metric compatibility obviously
involves the metric as well as the connection.

To define a MT theory which is equivalent to GR, one can proceed analogously
to II. To that end, we notice that due to the skew-symmetry of the torsion
tensor, there are three independent scalars which can be constructed from
$T^{\alpha}{}_{\mu\nu}$. One can then define the following linear combination
of these scalars:

$\mathbb{T}:=-\frac{1}{4}T_{\alpha\mu\nu}T^{\alpha\mu\nu}-\frac{1}{2}T_{\alpha\mu\nu}T^{\mu\alpha\nu}+T_{\alpha}T^{\alpha},$
(6.3)

where we have used the trace $T_{\alpha}:=T^{\mu}{}_{\alpha\mu}$ and we refer
to $\mathbb{T}$ as the torsion scalar. Analogously to II, one can easily
derive the following relation

$-\mathbb{T}+2\mathcal{D}_{\mu}T^{\mu}+\mathcal{R}=0,$ (6.4)

where $\mathcal{D}_{\mu}$ denotes the covariant derivative with respect to the
Levi-Civita connection and $\mathcal{R}$ is the Ricci scalar of the Levi-
Civita connection. This shows that if we define the action of MT as
$\mathcal{S}[g,\Gamma]:=\int\mathrm{d}^{4}x\,\sqrt{-g}\,\mathbb{T}$, we obtain
an action which is equivalent to Einstein-Hilbert, up to a boundary term.
Hence, in MT, just as in ST, only the Levi-Civita part of the connection
contributes and everything else drops out of the field equations. The
connection does not carry any physical degrees of freedom.
Let us now consider the non-linear extension of MT to $f(\mathbb{T})$ gravity.
The action defining this theory is simply given by

$\mathcal{S}[g,\Gamma;\lambda,\rho]:=\int_{\mathcal{M}}\mathrm{d}^{4}x\left(\frac{1}{2}\sqrt{-g}\,f(\mathbb{T})+\lambda_{\alpha}{}^{\beta\mu\nu}R^{\alpha}{}_{\beta\mu\nu}+\rho^{\alpha\mu\nu}Q_{\alpha\mu\nu}\right),$
(6.5)

where the tensor densities $\lambda_{\alpha}^{\ \beta\mu\nu}$ and
$\rho^{\alpha\mu\nu}$ act again as Lagrange multiplies which enforce the MT
postulates (6.1). The function $f$ is again arbitrary and only subjected to
the requirement that
$f^{\prime}(\mathbb{T}):=\frac{df(\mathbb{T})}{d\mathbb{T}}\neq 0$. As for
$f(\mathbb{Q})$ gravity, for generic $f$ we can no longer remove the
connection from the action by a boundary term. Hence, the connection equations
of motion are no longer trivial and the connection can propagate degrees of
freedom. By varying (6.5) with respect to the metric and the connection leads
to the equations of motion of $f(\mathbb{T})$ gravity Jimenez:2018 . These are
explicitly given by

$\displaystyle\mathcal{M}_{\mu\nu}:=(\nabla_{\alpha}+T_{\alpha})[S_{(\mu\nu)}{}^{\alpha}f^{\prime}(\mathbb{T})]+f^{\prime}(\mathbb{T})t_{\mu\nu}-\frac{1}{2}f(\mathbb{T})g_{\mu\nu}-T_{\mu\nu}$
$\displaystyle=0$
$\displaystyle\mathcal{C}_{\alpha\beta}:=-(\nabla_{\mu}+T_{\mu})\left[\frac{\sqrt{-g}}{2}f^{\prime}(\mathbb{T})S_{[\alpha}{}^{\mu}{}_{\beta]}\right]$
$\displaystyle=0,$ (6.6)

where $S_{\alpha}{}^{\mu\nu}$, the so-called torsion conjugate, and the
symmetric tensor $t_{\mu\nu}$ are defined by999Note that
$S_{\alpha}{}^{\mu\nu}$ is by definition antisymmetric in the last two
indices, while –by using the explicit form of $S_{\alpha}{}^{\mu\nu}$– one can
check that $t_{\mu\nu}$ is indeed symmetric, as required by the symmetry of
the metric equations of motion.

$\displaystyle S_{\alpha}{}^{\mu\nu}$
$\displaystyle:=\frac{\partial\mathbb{T}}{\partial
T^{\alpha}{}_{\mu\nu}}=-\frac{1}{2}T_{\alpha}{}^{\
\mu\nu}-{T^{[\mu}}_{\alpha}{}^{\nu]}-2\delta_{\alpha}{}^{[\mu}T^{\nu]}$
$\displaystyle t_{\mu\nu}$
$\displaystyle:=\frac{1}{2}S_{\mu}{}^{\alpha\beta}T_{\nu\alpha\beta}-T^{\alpha\beta}{}_{\mu}S_{\alpha\beta\nu}.$
(6.7)

Note the close similarity in the structure of these equations with the ST
counterparts, except for the form of the connection equations of motion. One
can then rewrite the metric equations of motion in the more useful form

$f^{\prime}(\mathbb{T})G_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(f(\mathbb{T})-f^{\prime}(\mathbb{T})\mathbb{T})+f^{\prime\prime}(\mathbb{T})S_{(\mu\nu)}{}^{\alpha}\partial_{\alpha}\mathbb{T}=T_{\mu\nu},$
(6.8)

with $G_{\mu\nu}$ being again the Einstein tensor with respect to the Levi-
Civita connection. For $f(\mathbb{T})=\mathbb{T}+2\Lambda$, this reduces to
the Einstein field equations with a cosmological constant.

In the next few subsections we will sketch the symmetry reduction of the
metric, the connection, and the field equations, and finally discuss some
perturbative solution. We will see that there are some similarities with
$f(\mathbb{Q})$.

###  VI.2 Symmetry reduction of the connection and metric

Unsurprisingly, the symmetry reduced metric has the same form as in
$f(\mathbb{Q})$, namely, it is given by (3.3). Moreover, we can apply the same
diffeomorphisms which we described in III.6 in order to bring the metric in
the even simpler form (3.50). This will not spoil the symmetry reduced form of
the connection for the same reasons given in III.6: The diffeomorphism
respects the symmetry of the metric-affine geometry and it can neither create
curvature nor non-metricity. Thus, we choose to work with the simple form
(3.50) of the metric from now on.

Finding a parametrization of the connection which is compatible with the
postulates of MT can be achieved analogously to what we did in the sections
III. We start with a general connection $\Gamma^{\alpha}{}_{\ \mu\nu}$, which
has $64$ components, and first apply the conditions for stationarity and
spherical symmetry. After that, we use the MT postulates (6.1) to further
eliminate connection components and knead the connection into the simplest
possible form. We only report the main results here.

After implementing the symmetry conditions (3.4), (3.6), (3.7), and (3.12), we
obtained the same connection as in Hohmann:2019 , but with the additional
property of being stationary, which simply means all connection components are
time-independent.

In order to implement the MT postulates, one can again split the arising
equations into algebraic101010Notice that the equations $Q_{\alpha\mu\nu}=0$
are purely algebraic and even linear in the connection, thus leading to unique
solutions. and differential equations for the connection. By plugging the
solutions of the algebraic equations into the differential equations, one can
ultimately find the following form of the connection:

$\displaystyle{\Gamma^{t}}_{rt}$
$\displaystyle=\frac{\partial_{r}g_{tt}}{2g_{tt}}\ ,\
{\Gamma^{t}}_{rr}=\pm\frac{r({\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\theta\phi}{}^{2})\partial_{r}g_{rr}-2g_{rr}\left({{\bar{\Gamma}}^{r}}_{\theta\phi}{}^{2}+{\Gamma^{r}}_{\theta\theta}({\Gamma^{r}}_{\theta\theta}-r\,\partial_{r}{{\Gamma^{r}}_{\theta\theta}})-r{{\bar{\Gamma}}^{r}}_{\theta\phi}\partial_{r}{{{\bar{\Gamma}}^{r}}_{\
\theta\phi}}\right)}{2r\sqrt{g_{tt}}\sqrt{{\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\
\theta\phi}{}^{2}}\sqrt{r^{2}-g_{rr}({\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\
\theta\phi}{}^{2})}}$ $\displaystyle{\Gamma^{t}}_{\theta\theta}$
$\displaystyle=\pm\frac{{\Gamma^{r}}_{\theta\theta}\sqrt{r^{2}-g_{rr}({\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\
\theta\phi}{}^{2})}}{\sqrt{g_{tt}}\sqrt{{\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\
\theta\phi}{}^{2}}}\ ,\
{\Gamma^{t}}_{\phi\phi}=\sin^{2}\theta\,{\Gamma^{t}}_{\theta\theta}$
$\displaystyle{\Gamma^{t}}_{\theta\phi}$ $\displaystyle=-{\Gamma^{t}}_{\
\phi\theta}=\frac{{\Gamma^{r}}_{\theta\phi}}{{\Gamma^{r}}_{\theta\theta}}\sin\theta\,{\Gamma^{t}}_{\theta\theta}$
$\displaystyle{\Gamma^{r}}_{rt}$
$\displaystyle=-\frac{g_{tt}}{g_{rr}}{\Gamma^{t}}_{rt}\ ,\
{\Gamma^{r}}_{rr}=\frac{\partial_{r}g_{rr}}{2g_{rr}}\ ,\
{\Gamma^{r}}_{\phi\phi}=\sin^{2}\theta\,{\Gamma^{r}}_{\theta\theta}\ ,\
{\Gamma^{r}}_{\theta\phi}=-{\Gamma^{r}}_{\phi\theta}=\sin\theta\,{\bar{\Gamma}^{r}}_{\
\theta\phi}$ $\displaystyle{\Gamma^{\theta}}_{\theta t}$
$\displaystyle=-\frac{g_{tt}}{r^{2}}{\Gamma^{t}}_{\theta\theta}\ ,\
{\Gamma^{\theta}}_{\phi t}=-\frac{g_{tt}}{r^{2}}{\Gamma^{t}}_{\phi\theta}\ ,\
{\Gamma^{\theta}}_{\theta r}=-\frac{g_{rr}{\Gamma^{r}}_{\theta\theta}}{r^{2}}\
,\ {\Gamma^{\theta}}_{\phi r}=\frac{g_{rr}\sin(\theta){{\bar{\Gamma}}^{r}}_{\
\theta\phi}}{r^{2}}$ $\displaystyle{\Gamma^{\theta}}_{r\theta}$
$\displaystyle=\frac{1}{r}\ ,\
{\Gamma^{\theta}}_{r\phi}=\sin\theta\,\frac{{\Gamma^{r}}_{\theta\theta}\partial_{r}{{\bar{\Gamma}}^{r}}_{\
\theta\phi}-\partial_{r}{\Gamma^{r}}_{\theta\theta}{{\bar{\Gamma}}^{r}}_{\theta\phi}}{{\Gamma^{r}}_{\theta\theta}{}^{2}+{{\bar{\Gamma}}^{r}}_{\theta\phi}{}^{2}}\
,\ {\Gamma^{\theta}}_{\phi\phi}=-\cos\theta\,\sin\theta$
$\displaystyle{\Gamma^{\phi}}_{\theta t}$
$\displaystyle=-\frac{g_{tt}{{\bar{\Gamma}}^{r}}_{\
\theta\phi}}{r^{2}\sin\theta\,{\Gamma^{r}}_{\theta\theta}}{\Gamma^{t}}_{\theta\theta}\
,\ {\Gamma^{\phi}}_{\phi t}=-\frac{g_{tt}}{r^{2}}\Gamma^{t}_{\ \theta\theta}\
,\
{\Gamma^{\phi}}_{r\theta}=-\frac{1}{\sin^{2}\theta}{\Gamma^{\theta}}_{r\phi}$
$\displaystyle{\Gamma^{\phi}}_{\theta r}$
$\displaystyle=-\frac{g_{rr}{{\bar{\Gamma}}^{r}}_{\theta\phi}}{r^{2}\sin\theta}\
,\ {\Gamma^{\phi}}_{\phi r}=-\frac{g_{rr}{\Gamma^{r}}_{\theta\theta}}{r^{2}}\
,\ {\Gamma^{\phi}}_{r\phi}=\frac{1}{r}\ ,\
{\Gamma^{\phi}}_{\theta\phi}={\Gamma^{\phi}}_{\phi\theta}=\cot\theta.$ (6.9)

with all other components vanishing. This result agrees with the one reported
in Hohmann:2019nat . As alluded to before, the connection and the metric are
not completely independent, because of $Q_{\alpha\mu\nu}=0$, and that is why
the metric appears in the above expressions for the connection components.
Moreover, the only free connection components are
$\Gamma^{r}{}_{\theta\theta}$ and ${\bar{\Gamma}}^{r}{}_{\theta\phi}$, where
the latter is defined in the fourth line above, both arbitrary functions of
$r$ only. In addition to being free in specifying these components, we can
also freely choose the sign $\pm$ which arises from taking the square roots of
the metric components. This sign has to be chosen such that it is the same for
all components, either always the upper or the lower one. Thus, we obtain two
distinct parametrizations for the connection which we denote by
$\Gamma^{\pm}$.

It is not surprising that the connection now has less free components than in
$f(\mathbb{Q})$ gravity: While the connections both have to fulfil the
symmetry and flatness conditions, in $f(\mathbb{Q})$ it has to fulfil
$T^{\alpha}_{\ \mu\nu}=0$, which are $24$ equations, while in $f(\mathbb{T})$
we must have $Q_{\alpha\mu\nu}=0$, which are $40$ equations. The connection in
$f(\mathbb{T})$ is thus more constraint. Also, note that the connection
components are not continuous at
$\Gamma^{r}{}_{\theta\theta}={\bar{\Gamma}}^{r}{}_{\theta\phi}=0$. For
instance, in the expression for $\Gamma^{t}{}_{\theta\theta}$, when taking
such a limit one has to take care with the order of the two limits
$\Gamma^{r}{}_{\theta\theta}\to 0$ and ${\bar{\Gamma}}^{r}{}_{\theta\phi}\to
0$.

###  VI.3 Symmetry reduced field equations for the metric and the connection

After having worked out the symmetry reduced form of the connection, and
having established that there are two distinct parametrizations corresponding
to a choice of sign, we can now consider the symmetry reduced field equations.
By plugging (3.50) and (VI.2) into the field equations (VI.1), one finds that
they have the following structure

Structure of metric field equations:
$\displaystyle\begin{pmatrix}\mathcal{M}_{tt}&\mathcal{M}_{tr}&0&0\\\
\mathcal{M}_{tr}&\mathcal{M}_{rr}&0&0\\\ 0&0&\mathcal{M}_{\theta\theta}&0\\\
0&0&0&\mathcal{M}_{\theta\theta}\,\sin^{2}\theta\end{pmatrix}$ Structure of
connection field equations:
$\displaystyle\partial_{r}\mathbb{T}\,f^{\prime\prime}(\mathbb{T})\begin{pmatrix}0&\mathcal{C}_{tr}&0&0\\\
-\mathcal{C}_{tr}&0&0&0\\\ 0&0&0&-\sin\theta\,\mathcal{C}_{\theta\phi}\\\
0&0&\sin\theta\,\mathcal{C}_{\theta\phi}&0\end{pmatrix},$ (6.10)

and we remark that the torsion scalar is explicitly given by

$\mathbb{T}=\frac{2}{r^{2}}\frac{(r+g_{rr}{\Gamma^{r}}_{\theta\theta})\partial_{r}g_{tt}+g_{tt}(1+g_{rr}+{\Gamma^{r}}_{\theta\theta}\partial_{r}g_{rr}+2g_{rr}{\partial_{r}\Gamma^{r}}_{\theta\theta})}{g_{rr}\,g_{tt}}$
(6.11)

for both choices of sign in $\Gamma^{\pm}$. Observe that the metric field
equations have the same structure as for $f(\mathbb{Q})$ gravity. Moreover,
one finds that the sign of $\Gamma^{\pm}$ not only has absolutely no effect on
the structure of the connection field equations; it does not enter these
equations at all! No matter which sign we choose, we obtain exactly the same
field equations. Hence, we can drop the distinction between the $+$ and $-$
choice.

The connection equations of motion have a very simple form, and are given by

$\displaystyle\partial_{r}\mathbb{T}\,f^{\prime\prime}(\mathbb{T})\,\mathcal{C}_{tr}$
$\displaystyle=\mathcal{M}_{tr}=0\ ,$ (6.12)
$\displaystyle\partial_{r}\mathbb{T}\,f^{\prime\prime}(\mathbb{T})\,\mathcal{C}_{\theta\phi}$
$\displaystyle=\partial_{r}\mathbb{T}\,f^{\prime\prime}(\mathbb{T})\,{\bar{\Gamma}}^{r}_{\
\theta\phi}=0.$ (6.13)

In particular, we note that no derivatives of the connection –apart from the
$\partial_{r}\mathbb{T}$ term– appear in the connection equations of motion.
The connection is thus not dynamical, and the connection equations of motion
are mere constraints.

Let us now look more closely at the off-diagonal metric field equation, which
reads

$\mathcal{M}_{tr}=-\frac{2\sqrt{g_{tt}}}{r^{2}}\Gamma^{r}{}_{\theta\theta}\sqrt{\frac{r^{2}-g_{rr}(\Gamma^{r}{}_{\theta\theta}{}^{2}+{\bar{\Gamma}}^{r}{}_{\theta\phi}{}^{2})}{\Gamma^{r}{}_{\theta\theta}{}^{2}+{\bar{\Gamma}}^{r}{}_{\theta\phi}{}^{2}}}\,\partial_{r}\mathbb{T}\,f^{\prime\prime}(\mathbb{T})=0\
.$ (6.14)

This equation is again structurally similar to the equation we obtained in
$f(\mathbb{Q})$ gravity, To solve it, we either have
$\mathbb{T}_{\text{sol}}=$ const., or
$f^{\prime\prime}(\mathbb{T}_{\text{sol}})=0$, or we end up with a constraint
equation for the connection.

Just as in $f(\mathbb{Q})$ gravity, the first two options will immediately
lead to trivially satisfied connection field equations. Using a similar
argument as in IV, one can show that
$f^{\prime\prime}(\mathbb{T}_{\text{sol}})=0$ immediately implies
$f(\mathbb{T})=a\,\mathbb{T}+b$ and hence one can only obtain the
Schwarzschild-deSitter-Nordström solution of GR. Also, if
$\mathbb{T}_{\text{sol}}$ is a constant, we find from (6.8) the GR field
equations

$G_{\mu\nu}+\Lambda_{\textsf{eff}}\,g_{\mu\nu}=\bar{T}_{\mu\nu},$ (6.15)

where $G_{\mu\nu}$ is the standard Einstein tensor with respect to the Levi-
Civita connection and where we have defined

$\displaystyle\Lambda_{\textsf{eff}}$
$\displaystyle:=\frac{1}{2}\frac{f(\mathbb{T}_{\textsf{sol}})-f^{\prime}(\mathbb{T}_{\textsf{sol}})\mathbb{T}_{\textsf{sol}}}{f^{\prime}(\mathbb{T}_{\textsf{sol}})}$
$\displaystyle\bar{T}_{\mu\nu}$
$\displaystyle:=\frac{1}{f^{\prime}(\mathbb{T}_{\textsf{sol}})}T_{\mu\nu}.$
(6.16)

This is exactly the same results obtained in IV for $f(\mathbb{Q})$ gravity!
Notice that in the case of $f(\mathbb{T})$ gravity, we can immediately
establish that solutions with $\mathbb{T}_{\text{sol}}=$ const. are not an
empty set. To that end, assume $\mathbb{T}_{\text{sol}}=$const. and solve
(6.11) for ${\Gamma^{r}}_{\theta\theta}$:

${\Gamma^{r}}_{\theta\theta}=-\frac{1}{\sqrt{g_{tt}\,g_{rr}}}\left(c+\int^{r}\mathrm{d}\rho\
\frac{g_{tt}(2+\rho(2+\mathbb{T}_{\textsf{sol}}\,\rho^{2}))+2\rho\,\partial_{\rho}g_{tt}}{4\sqrt{g_{tt}\,g_{rr}}}\right).$
(6.17)

Hence, the connection can, in principle, be fixed such that $\mathbb{T}$
becomes a constant.

Finally, the only choice left for solving the off-diagonal metric field
equation which does not force GR solutions on us is the constraint equation

$\Gamma^{r}{}_{\theta\theta}\sqrt{\frac{r^{2}-g_{rr}(\Gamma^{r}{}_{\theta\theta}{}^{2}+{\bar{\Gamma}}^{r}{}_{\theta\phi}{}^{2})}{\Gamma^{r}{}_{\theta\theta}{}^{2}+{\bar{\Gamma}}^{r}{}_{\theta\phi}{}^{2}}}=0.$
(6.18)

Once $\mathcal{M}_{tr}$ is fulfilled $\mathcal{C}_{tr}$ is also fulfilled.
Since the connection field equation $\mathcal{C}_{\theta\phi}=0$ uniquely
implies $\bar{\Gamma}^{r}{}_{\theta\phi}=0$, we find that the constraint
equation (6.18) has the two solutions

$\Gamma^{r}{}_{\theta\theta}=\pm\frac{r}{\sqrt{g_{rr}}}.$ (6.19)

With this, the off-diagonal metric field equation and all connection field
equations are satisfied and we find that the connection (VI.2) reduces to

$\displaystyle\Gamma^{t}_{\ rt}$
$\displaystyle=\frac{\partial_{r}g_{tt}}{2g_{tt}}\ ,\ \Gamma^{r}_{\
rr}=\frac{\partial_{r}g_{rr}}{2g_{rr}}\ ,\ \Gamma^{r}_{\
\theta\theta}=\pm\frac{r}{\sqrt{g_{rr}}}\ ,\ \Gamma^{r}_{\
\phi\phi}=\Gamma^{r}_{\ \theta\theta}\sin(\theta)^{2}\ ,$
$\displaystyle\Gamma^{\theta}_{\ r\theta}$ $\displaystyle=\frac{1}{r}\ ,\
\Gamma^{\theta}_{\ \theta r}=\mp\frac{\sqrt{g_{rr}}}{r}\ ,\ \Gamma^{\theta}_{\
\phi\phi}=-\cos(\theta)\sin(\theta)\ ,$ $\displaystyle\Gamma^{\phi}_{\ r\phi}$
$\displaystyle=\frac{1}{r}\ ,\ \Gamma^{\phi}_{\ \phi
r}=\mp\frac{\sqrt{g_{rr}}}{r}\ ,\ \Gamma^{\phi}_{\
\theta\phi}=\Gamma^{\phi}_{\ \phi\theta}=\cot(\theta)\ ,$ (6.20)

with all other components vanishing. We call these two solutions (III±). The
sign ambiguity here comes only from the choice of $\Gamma^{r}_{\
\theta\theta}$; the connection is the same for both $\Gamma^{\pm}$ for this
connection choice.

Note especially that the two choices (III±) are in complete analogy to the
choices of connection for $f(\mathbb{Q})$, namely (I±), in V.3. The important
difference –apart from the obvious differences in the forms of the whole
connections– is of course that in $f(\mathbb{T})$ the connection is not
dynamical. Its equations of motion completely fix the connection from the
start, and leave no room for a dynamical evolution. We can still have beyond
GR solutions for (III±), but they will not involve any connection hairs in
this sense.

We now have only three equations of motion left, namely $\mathcal{M}_{tt},\
\mathcal{M}_{rr}$, and $\mathcal{M}_{\theta\theta}$. One can check analogously
to IV.3 that $\mathcal{M}_{\theta\theta}$ follows from $\mathcal{M}_{tt}$ and
$\mathcal{M}_{rr}$ so we have only two equations left for the two metric
components $g_{tt}$ and $g_{rr}$.
The analogy between $f(\mathbb{T})$ and $f(\mathbb{Q})$ gravity for stationary
and spherically symmetric spacetimes comes to its conclusion when one notes
that for each case (III±) the exact equations of motion are identical for any
$f$ with those from $f(\mathbb{Q})$ gravity for the cases (I±); in particular
we have $\mathbb{T}=\mathbb{Q}$. These $f(\mathbb{T})$ black hole solutions
are thus merely a subset of $f(\mathbb{Q})$ solutions, where the connection is
fixed and given by (I±). This is the main result of this section. As we have
discussed the equations and their (approximate) solutions already in V.3, we
are done with these cases.

Note especially that the (approximate) beyond GR solutions found in
DeBenedictis:2016aze  are the same as we found for the case (III-). The
solution of Ruggiero:2015oka  corresponds to the case (III+). In these
references the $f(\mathbb{T})$ theory was discussed using tetrads instead of
the full connection, but one can check that the connections that were
constructed there are precisely the same as the ones we derived for (III±).

##  VII Conclusion

In this paper we have systematically derived and studied symmetry reduced
field equations for $f(\mathbb{Q})$ gravity and we have sketched how a similar
approach can be applied to $f(\mathbb{T})$ gravity. We began our analysis by
performing a systematic symmetry reduction of the metric affine geometry
described by $(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}{}_{\mu\nu})$. The main
results, which have been extensively discussed in section III, are the
following: (a) There are two classes of parametrizations for the connection
which guarantee that the connection is stationary, spherically symmetric,
torsionless, and flat, as required by the postulates of Symmetric
Teleparallelism (cf. subsections III.3 and III.4); (b) the connection in
coincident gauge fails to be spherically symmetric, which in part explains why
Zhao:2021 ; Lin:2021  were not able to find beyond-GR solutions in
$f(\mathbb{Q})$ which describe black holes; (c) the first parametrization
class of the connection (aka solution set 1) can be obtained from the second
class (aka solution set 2) by a double scaling limit (see subsection III.5);
(d) the metric can be brought into a simple diagonal form, which is
parametrized by only two arbitrary functions of $r$, without spoiling the
structure of the solution sets for the connection, as explained in III.6.
Hence, we have constructed the simplest, and yet most general metric-affine
geometry which is stationary, spherically symmetric, torsionless, and flat.

In section IV we have discussed the implications of the simple form of the
metric and the two parametrization classes of the connection for the field
equations of $f(\mathbb{Q})$ gravity. In subsection IV.1 we have formulated
precise conditions under which $f(\mathbb{Q})$ either reduces to Symmetric
Teleparallelism, gives rise to GR solutions for generic choices of $f$, or
produces beyond-GR solutions. In particular, this subsection fully explains
why Zhao:2021 ; Lin:2021  were not able to find any beyond-GR solutions.

Additionally, we have shown in subsection IV.2 that solution set 1, while
attractive because of its simplicity, is not viable when looking for beyond-GR
solutions. In the subsections V.3 and IV.4 we have extensively discussed the
self-consistency of the field equations, the number of degrees of freedom they
propagate, the initial data which needs to be specified, and additional
constraints on the connection which can appear. We have seen that the
connection becomes dynamical, in stark contrast with Symmetric
Teleparallelism, where the connection is unphysical, or that it can be
completely fixed by additional constraints. The latter option leads
nevertheless to beyond-GR solutions for the metric.

In section V we finally constructed explicit beyond-GR solutions. In the
subsections V.1, V.2, and V.3 we used a perturbative approach to construct
solutions for the pure vacuum as well as the electro-vacuum case and non-zero
cosmological constant case for the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, where $\alpha$ is assumed
to be small. We have done so for a dynamical connection as well as for one
which is fixed by the additional constraint described in IV.4.

In subsection V.5 we even succeeded in finding exact solutions which go beyond
GR. These solutions ultimately turned out to have undesirable properties,
which make them physically unattractive. But it is nevertheless interesting
that exact solutions can be found, given the complexity of the $f(\mathbb{Q})$
field equations.

Finally, in section VI we sketched how the same methods which have been
described in detail for $f(\mathbb{Q})$ can be applied to $f(\mathbb{T})$ in
order to perform a systematic symmetry reduction of the metric, the
connection, and the field equations. There are many (perhaps surprising)
structural similarities between $f(\mathbb{Q})$ and $f(\mathbb{T})$ which
facilitate the analysis of $f(\mathbb{T})$ gravity. Moreover, we have also
reported some beyond-GR solutions for $f(\mathbb{T})$.

In conclusion, we have succeeded in showing the consistency of the symmetry
reduced field equations of both, $f(\mathbb{Q})$ and $f(\mathbb{T})$ gravity,
we have formulated precise criteria under which beyond-GR solutions can exist,
and we have shown that the GR solutions can exist for arbitrary choices of
$f$. Moreover, we have discussed a few perturbative beyond-GR solutions to the
$f(\mathbb{Q})$ and $f(\mathbb{T})$ field equations. Whether these solutions
are stable or whether they lead to instabilities is beyond the scope of the
current analysis and will be left for future work. We have also not discussed
the question of formation processes in the context of $f(\mathbb{Q})$ or
$f(\mathbb{T})$ gravity. It would be interesting to understand whether a
realistic formation process could give rise to one of the solutions discussed
here or whether it leads to beyond-GR solutions at all. This question is also
left for future work.

Finally, we note that a similar symmetry reduction analysis to the one carried
out here can be performed for cosmological models. A detailed discussion will
be given elsewhere.

## Acknowledgements

LH is supported by funding from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme grant agreement
No 801781 and by the Swiss National Science Foundation grant 179740.

##  Appendix A Connection transformation when diagonalizing the metric

For the sake of eliminating doubt about whether the diffeomorphism described
in III preserves the structure of the solution sets of the connection, we
explicitly determine the transformation of the connection components. The
untransformed components are denoted by $\Gamma^{\alpha}{}_{\mu\nu}$, the
components after having applied $\phi_{2}$ are
$\bar{\Gamma}^{\alpha}{}_{\mu\nu}$, and after applying $\phi_{1}$ we call them
$\overline{\overline{\Gamma}}^{\alpha}{}_{\mu\nu}$. The same for the metric
components $g_{\mu\nu}$. The order in which we apply $\phi_{1}$ and $\phi_{2}$
is irrelevant. We then find that the components transform as

$\displaystyle\overline{c}$ $\displaystyle=c$ $\displaystyle\overline{k}$
$\displaystyle=k$ $\displaystyle{\overline{\Gamma}^{t}}_{rr}$
$\displaystyle={\Gamma^{t}}_{rr}-\frac{1}{g_{tt}{}^{3}\,{\Gamma^{r}}_{\theta\theta}}\left[c(k-2c)g_{tr}{}^{3}\,{\Gamma^{r}}_{\theta\theta}{}^{2}+g_{tr}{}^{2}\,g_{tt}\,{\Gamma^{r}}_{\theta\theta}\left(3c-k+3c(2c-k){\Gamma^{t}}_{\theta\theta}\right)-\right.$
$\displaystyle\left.-g_{tt}{}^{2}\,\partial_{r}g_{tr}\,{\Gamma^{r}}_{\theta\theta}+g_{tt}g_{tr}\left(g_{tt}{\Gamma^{r}}_{rr}{\Gamma^{r}}_{\theta\theta}-2(2c-k){\Gamma^{t}}_{\theta\theta}(1+c{\Gamma^{t}}_{\theta\theta})+\partial_{r}g_{tt}{\Gamma^{r}}_{\theta\theta}\right)\right]$
$\displaystyle{\overline{\Gamma}^{t}}_{\theta\theta}$
$\displaystyle={\Gamma^{t}}_{\theta\theta}-\frac{g_{tr}{\Gamma^{r}}_{\theta\theta}}{g_{tt}}$
$\displaystyle{\overline{\Gamma}^{r}}_{rr}$
$\displaystyle={\Gamma^{r}}_{rr}+\frac{c(k-2c)g_{tr}{}^{2}{\Gamma^{r}}_{\theta\theta}}{g_{tt}{}^{2}}+\frac{2cg_{tr}\left(1+(2c-k){\Gamma^{t}}_{\theta\theta})\right)}{g_{tt}}$
$\displaystyle{\overline{\Gamma}^{r}}_{\theta\theta}$
$\displaystyle={\Gamma^{r}}_{\theta\theta},$ (1.1)

and

$\displaystyle\overline{\overline{c}}$ $\displaystyle=\overline{c}$
$\displaystyle\overline{\overline{k}}$ $\displaystyle=\overline{k}$
$\displaystyle{\overline{\overline{\Gamma}}^{t}}_{rr}$
$\displaystyle=\frac{{\overline{\Gamma}^{t}}_{rr}}{\partial_{r}\overline{g}_{\theta\theta}{}^{2}}$
$\displaystyle{\overline{\overline{\Gamma}}^{t}}_{\theta\theta}$
$\displaystyle={\overline{\Gamma}^{t}}_{\theta\theta}$
$\displaystyle{\overline{\overline{\Gamma}}^{r}}_{rr}$
$\displaystyle=\frac{{\overline{\Gamma}^{r}}_{rr}\partial_{r}\overline{g}_{\theta\theta}-\partial^{2}_{r}\overline{g}_{\theta\theta}}{\partial_{r}\overline{g}_{\theta\theta}{}^{2}}$
$\displaystyle{\overline{\overline{\Gamma}}^{r}}_{\theta\theta}$
$\displaystyle={\overline{\Gamma}^{r}}_{\theta\theta}\partial_{r}\overline{g}_{\theta\theta}.$
(1.2)

All other components are zero. The derivative relations coming from the
flatness condition remain, as they come from the coordinate invariant
condition $R^{\alpha}{}_{\beta\mu\nu}=0$. The constants are thus unchanged,
and the diffeomorphisms can be absorbed in the arbitrary components
$\Gamma^{t}{}_{\theta\theta}$ and $\Gamma^{r}{}_{\theta\theta}$. The structure
is preserved, as expected.

##  Appendix B Approximate solutions for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{\kappa}$

The perturbative solutions for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$ derived in subsection V.1
can be generalize to the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{\kappa}$ for an integer $\kappa$
which satisfies $\kappa\geq 2$. The solutions are derived in complete analogy
to those for $f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, so we only
report the final results for the metric components here.

###  B.1 Connection hair solutions

This is the generalization of V.1, which we explicitly did for $\kappa=3,4$
for vacuum. It turns out one has to go to perturbation order $\kappa$ in
$\alpha$ in the metric, at which the first order correction of the connection
$\Gamma^{r}_{\ \theta\theta}=-r+\alpha\gamma^{(1)}$ enters in the form

$\displaystyle g_{tt}$
$\displaystyle=-\left(1-\frac{2M_{\text{ren}}}{r}\right)+\alpha^{\kappa}\frac{\mu}{r}\ln\left(\frac{r}{r^{*}}\right)\
,$ (2.3) $\displaystyle g_{rr}$ $\displaystyle=-\frac{1}{g_{tt}}\ .$ (2.4)

$\mu$ and $r^{*}$ are new scales coming from the connection integration
constants in $\gamma^{(1)}$ and $M_{\text{ren}}$ is the renormalized mass. We
suspect that this formula holds for all integer $\kappa\geq 2$.

###  B.2 Constraint solutions

This is the generalization of (I±) of V.3, which we explicitly did for
$\kappa=3,4,5,10$ for vacuum. It turns out that we only need to go to first
order metric perturbations. We then have found the formulas

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{2^{3\kappa-1}}{(2\kappa-3)r^{2\kappa-2}}+\mathcal{O}(r^{-2\kappa+1})\
,$ (2.5) $\displaystyle-\frac{1}{g_{rr}}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{(2\kappa-1)(\kappa-1)2^{3\kappa-1}}{(2\kappa-3)r^{2\kappa-2}}+\mathcal{O}(r^{-2\kappa+1})$
(2.6)

for (I+), and

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{(-1)^{\kappa}2\kappa(2M_{\text{ren}})^{2\kappa-}}{2^{\kappa}(4\kappa-3)r^{4\kappa-3}}+\mathcal{O}(r^{-4\kappa+2})\
,$ (2.7) $\displaystyle-\frac{1}{g_{rr}}$
$\displaystyle=-1+\frac{2M_{\text{ren}}}{r}+\alpha\frac{(-1)^{\kappa}2\kappa(2M_{\text{ren}})^{2\kappa-1}}{2^{\kappa}r^{4\kappa-3}}+\mathcal{O}(r^{-4\kappa+2})$
(2.8)

for (I-). We again suspect these formulas to hold for all integer $\kappa\geq
2$.

## References

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  * (8) D. Zhao, “Covariant formulation of f(Q) theory,” arXiv:2104.02483 [gr-qc].
  * (9) R.-H. Lin and X.-H. Zhai, “Spherically symmetric configurations in $f(Q)$ gravity,” arXiv:2105.01484 [gr-qc].
  * (10) M. Hohmann, “Metric-affine Geometries With Spherical Symmetry,” Symmetry 12 no. 3, (2020) 453, arXiv:1912.12906 [math-ph].
  * (11) S. Bahamonde and C. Pfeifer, “General Teleparallel Modifications of Schwarzschild Geometry,” arXiv:2010.02161 [gr-qc].
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&=\lambda^{2}\kappa_{f}^{2}I_{f}^{\pm}(y,y^{\prime})-\delta_{\phi}\chi^{\pm}_{0}(y,y^{\prime}).\end{split}$
(150)

Using the identities

$\displaystyle\begin{split}&\operatorname{tr}\Big{[}\langle\Psi(x)\bar{\Psi}(x^{\prime})\rangle_{0}\langle\Psi(x^{\prime})\bar{\Psi}(x)\rangle_{0}\Big{]}=-4\kappa_{f}^{2}\left(\frac{1}{|x-x^{\prime}|^{2(d-1)}}+\frac{1}{|\bar{x}-x^{\prime}|^{2(d-1)}}\right)\,,\\\
&\operatorname{tr}\Big{[}\langle\Psi(x)\bar{\Psi}(x^{\prime})\rangle_{0}\gamma_{5}\langle\Psi(x^{\prime})\bar{\Psi}(x)\rangle_{0}\gamma_{5}\Big{]}=4\kappa_{f}^{2}\left(\frac{1}{|x-x^{\prime}|^{2(d-1)}}-\frac{1}{|\bar{x}-x^{\prime}|^{2(d-1)}}\right)\,,\end{split}$
(151)

we see that the Feynman integral is

$\displaystyle I_{f}^{\pm}(y,y^{\prime})$
$\displaystyle=\int_{0}^{\infty}dz\int_{0}^{\infty}dz^{\prime}\chi^{\pm}(y,z)b^{\pm}(z,z^{\prime})\chi^{\pm}(z^{\prime},y^{\prime})\,,$
(152)

where we have defined

$\displaystyle\begin{split}b^{\pm}(z,z^{\prime})&=\int
d^{d-1}\mathbf{r}\left(\frac{1}{\big{(}\mathbf{r}^{2}+(z-z^{\prime})^{2}\big{)}^{d-1}}\pm\frac{1}{\big{(}\mathbf{r}^{2}+(z+z^{\prime})^{2}\big{)}^{d-1}}\right)\\\
&=\frac{2^{2-d}\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}\left(|z-z^{\prime}|^{-3+\epsilon}\pm|z+z^{\prime}|^{-3+\epsilon}\right)\,.\end{split}$
(153)

We will evaluate this integral with a trick that has been used in the
literature in similar situations PhysRevB.50.10009 ; Shpot:2019iwk ;
Dey:2020lwp . The idea is to split the integration region between
$z>z^{\prime}$ and $z<z^{\prime}$. By changing variables to
$Z=\frac{z}{z^{\prime}}$ and $Z=\frac{z^{\prime}}{z}$, one can carry out the
first integration in terms of $I_{b}^{\pm}$ defined in the previous section.
The result is:

$\displaystyle I_{f}^{\pm}(y,y^{\prime})$
$\displaystyle=y^{\epsilon+1}\Bigg{[}\int_{1}^{\infty}dZI^{\pm}_{b}(1,Z/\zeta)b^{\pm}(1,Z)Z^{-\epsilon}+\int_{1}^{1/\zeta}dZI^{\pm}_{b}(1,(Z\zeta)^{-1})b^{\pm}(1,Z)Z$
$\displaystyle\quad+\zeta^{-1-\epsilon}\int_{1/\zeta}^{\infty}dZI^{\pm}_{b}(1,\zeta
Z)b^{\pm}(1,Z)Z^{-\epsilon}\Bigg{]}\,.$ (154)

Remember that we are assuming $y<y^{\prime}$, such that $\zeta=y/y^{\prime}$.
Finally, all terms in (5.3.3) can be integrated using Mathematica181818The
only exception are integrals of the form $\int_{1}^{\infty}dZ(Z-1)^{a}$, but
they are zero in dimensional regularization.. The result for general
$\epsilon$ is not particularly illuminating and will not be needed later,
instead we focus on the result in the limit $\epsilon\to 0$. First, the
divergent piece is canceled in $\overline{\text{MS}}$ with the following
counterterm:

$\displaystyle\delta_{\phi}=-\frac{\lambda^{2}}{(4\pi)^{2}}\left(\frac{1}{\epsilon}+\frac{1}{2}(\gamma+\log\pi)+1\right)\,.$
(155)

The total diagram is now finite:

$\displaystyle\begin{split}\chi^{+}(y,y^{\prime})|_{\text{bubble}}&=\frac{\lambda^{2}}{64\pi^{2}}\frac{\sqrt{4yy^{\prime}}}{\sqrt{\zeta}}\Big{(}\log(4yy^{\prime})-\log\zeta-2\\\
&\qquad\qquad\qquad\quad+(1+\zeta)\log(1+\zeta)+(1-\zeta)\log(1-\zeta)\Big{)}+{\mathcal{O}}(\epsilon)\,,\\\
\chi^{-}(y,y^{\prime})|_{\text{bubble}}&=\frac{-\lambda^{2}}{64\pi^{2}}\sqrt{4yy^{\prime}\zeta}\Big{(}\log(4yy^{\prime})-\log\zeta-2\\\
&\qquad\qquad\qquad\quad+\frac{(1+\zeta)}{\zeta}\log(1+\zeta)-\frac{(1-\zeta)}{\zeta}\log(1-\zeta)\Big{)}+{\mathcal{O}}(\epsilon)\,.\end{split}$
(156)

####  5.3.4 Final result

We can obtain the full susceptibility at order $\epsilon$ by combining the
tree-level result (145), the tadpole diagram (149), and the fermion bubble
(156). We should evaluate the sum at the fixed point coupling
$\lambda_{*}^{2}=\frac{16\pi^{2}}{3}\epsilon$, and keep only terms up to order
$\epsilon$. The result is perfectly consistent with the bootstrap prediction
(143), and we identify

$\displaystyle\Delta_{\phi}^{(1)}=\frac{1}{6}\,,\qquad\hat{\Delta}_{0}^{(1)}=0\,,\qquad
s=+1\,.$ (157)

The anomalous dimension of $\phi$ in the Wess-Zumino model is well known in
the literature. One can obtain it by demanding that the superpotential has
$R$-charge $R(W)=3\mathit{r}_{\phi}=2$, so we find that
$\mathit{r}_{\phi}=2/3$. Using the relation between the $R$-charge and
conformal dimension we find $\Delta_{\phi}=\frac{d-1}{3}$, in perfect
agreement with (157). From this argument it is clear that $\Delta_{\phi}$ is
one-loop exact.

An interesting prediction of our calculation is the anomalous dimension of the
leading bulk operator in the OPE $\phi\times\phi\sim{\mathcal{O}}_{0}+\ldots$.
We calculated this for a general model in (129), and for the Wess-Zumino case
we get

$\displaystyle\Delta_{0}^{(1)}=\frac{2}{3}\qquad\Rightarrow\qquad\Delta_{0}=\frac{d+2}{3}=d-2\Delta_{\phi}\,.$
(158)

Recalling the selection rules of section 4.1.3, we see that the exchanged
operator is of the form ${\mathcal{O}}_{0}\sim(Q^{+})^{2}\bar{\Psi}$ where
$\bar{\Psi}$ is an antichiral primary operator. Indeed, the numerical
bootstrap applied to the Wess-Zumino model in Bobev:2015vsa ; Bobev:2015jxa
also provides strong evidence that the leading operator in the
$\phi\times\phi$ OPE is of this form. The agreement of our results with the
predictions from Bobev:2015vsa ; Bobev:2015jxa  provides a non-trivial sanity
check of our perturbative calculation. It would be interesting to consider
other particular models, for instance with extra boundary interactions or a
more complicated bulk, and see whether the anomalous dimension of the defect
operator changes. We hope to come back to this question in the future.

##  6 Conclusions

In this work we studied supersymmetric boundaries for $3d$ ${\mathcal{N}}=2$
superconformal theories. There are two possible choices characterized by $2d$
${\mathcal{N}}=(0,2)$ and ${\mathcal{N}}=(1,1)$ boundary algebras
respectively. After performing a careful superspace analysis of correlators
involving chiral fields, we observed in section 4 that the
${\mathcal{N}}=(1,1)$ choice can be analytically continued in the spacetime
dimension. This allowed us to compute superconformal blocks across dimensions
and opened the door for the $\epsilon$-expansion bootstrap in our
supersymmetric setup. We proved uniqueness of the first two orders in
$\epsilon$, and confirmed our general prediction for one specific model using
perturbation theory. We used standard perturbative and bootstrap techniques in
this analysis, but one could also try using alternative approaches, such as
Mellin space Kaviraj:2018tfd  or the equations of motion method of
Giombi:2020rmc .

An interesting follow up to our BCFT analysis is to consider higher
codimension defects. The algebraic approach to calculate superblocks of
section 4 is applicable to higher codimension, where the spacetime and defect
dimensions are allowed to change while the codimension is kept fixed. In
particular, the codimension-two blocks calculated in section 4.2 are
applicable to known examples, such as the Wess-Zumino model in the presence of
twist defects. In the same spirit of the present work, one can use the
$\epsilon$-expansion to study two-point functions of local operators. In
principle one can do explicit perturbative calculations as in Gaiotto:2013nva
, however it is perhaps simpler to set up a bootstrap problem and attempt to
solve it using the technology of inversion formulas Lemos:2017vnx ;
Liendo:2019jpu . One could also concentrate exclusively on three dimensions
and apply the numerical bootstrap on the line, analogous to what was done for
the twist defect in the $3d$ Ising model Gaiotto:2013nva  (the $1d$ bootstrap
for superconformal line defects has been studied in Liendo:2018ukf ; Gimenez-
Grau:2019hez ; Bianchi:2020hsz ).

A longer term goal is to include multi-point correlators in the analysis; this
is a program that has been underexplored even in the bosonic case, although
significant progress can be made using Calogero-Sutherland technology
Buric:2020zea . Finally, the study of free theories in the presence of
interacting defects has gotten some attention recently; in particular the
results of Behan:2020nsf  suggest the existence of a new conformal boundary
condition for the free scalar field. It would be interesting to repeat their
analysis in our supersymmetric setup, either for boundaries or higher
codimension defects.

###### Acknowledgements.

We are particularly grateful to E. Lauria for many discussions and
collaboration during several stages of this project. We thank C. Beem, A.
Bissi, I. Buric, Z. Liu, J. Rong, V. Schomerus, and A. Söderberg for useful
discussions and comments. We also thank the Simons Collaboration on the Non-
perturbative Bootstrap for many stimulating activities. This work is supported
by the DFG through the Emmy Noether research group “The Conformal Bootstrap
Program” project number 400570283.

##  Appendix A Details on three-dimensional boundaries

###  A.1 Conventions

In section 3 we work in Lorenzian signature with mostly plus metric
$\eta_{\mu\nu}=\text{diag}(-1,+1,+1)$. The gamma matrices are defined in terms
of the identity matrix $\mathbf{1}$ and Pauli matrices $\sigma^{i}$ as

$(\gamma^{\mu})_{\alpha\beta}\equiv\left(-\mathbf{1}_{\alpha\beta},(\sigma^{3})_{\alpha\beta},(\sigma^{1})_{\alpha\beta}\right)\,,\quad(\gamma^{\mu})_{\alpha}^{\phantom{{\alpha}}{\beta}}=(\gamma^{\mu})_{{\alpha}\gamma}\varepsilon^{\gamma{\beta}}\,,\quad(\gamma^{\mu})^{{\alpha}{\beta}}=\varepsilon^{{\alpha}\gamma}(\gamma^{\mu})_{\gamma}^{\phantom{\gamma}{\beta}}\,.$
(159)

With these conventions the gamma matrices are real and symmetric. Here and in
what follows we are raising and lowering spinor indices as
$\theta^{\alpha}=\varepsilon^{{\alpha}{\beta}}\theta_{\beta}$ and
$\theta_{\alpha}=\varepsilon_{{\alpha}{\beta}}\theta^{\beta}$, where

$\varepsilon^{12}=1,\quad\varepsilon_{12}=-1\,.$ (160)

The contraction of two spinors is defined as
$\theta^{2}=\varepsilon_{{\alpha}{\beta}}\theta^{\alpha}\theta^{\beta}$.
Finally, the spacetime Levi-Civita tensor is defined by:

$\varepsilon_{012}=-1,\quad\varepsilon^{012}=1\,.$ (161)

###  A.2 Superconformal algebra

The three dimensional Lorenz group $SO(2,1)$ is generated by
$\mathcal{M}_{\mu\nu}$. A generic element of the algebra ${\mathcal{J}}$
contains vector indices $\mu,\nu,\lambda,\ldots$ and spinor indices
${\alpha},{\beta},\ldots$, and each of them transforms under rotations as:

$\displaystyle\begin{split}&[\mathcal{M}_{\mu\nu},{\mathcal{J}}_{\lambda\ldots}]=i\left(\eta_{\mu\lambda}{\mathcal{J}}_{\nu\ldots}-\eta_{\nu\lambda}{\mathcal{J}}_{\mu\ldots}\right)\,,\\\
&[\mathcal{M}_{\mu\nu},{\mathcal{J}}_{\alpha\ldots}]=+\frac{i}{2}\varepsilon_{\mu\nu\lambda}(\gamma^{\lambda})_{\alpha}^{\phantom{{\alpha}}\beta}{\mathcal{J}}_{\beta\ldots},\\\
&[\mathcal{M}_{\mu\nu},{\mathcal{J}}^{\alpha\ldots}]=-\frac{i}{2}\varepsilon_{\mu\nu\lambda}(\gamma^{\lambda})_{{\beta}}^{\phantom{{\beta}}{\alpha}}{\mathcal{J}}^{\beta\ldots}\,.\end{split}$
(162)

With these identities it is easy to write down any commutator involving
${\mathcal{M}}_{\mu\nu}$. The rest of the $3d$ conformal algebra is:

$[\mathcal{D},\mathcal{P}_{\mu}]=i\mathcal{P}_{\mu},\quad[\mathcal{D},\mathcal{K}_{\mu}]=-i\mathcal{K}_{\mu},\quad[\mathcal{K}_{\mu},\mathcal{P}_{\nu}]=-2i\left(\mathcal{M}_{\mu\nu}+\eta_{\mu\nu}\mathcal{D}\right)\,.$
(163)

The $3d$ ${\mathcal{N}}=2$ superconformal algebra is given by $OSP(2|4)$.
Besides the conformal and $R$-symmetry generators, it contains four Poincaré
supercharges and four superconformal supercharges that anticommute as

$\displaystyle\begin{split}&\\{\mathcal{Q}_{\alpha},\bar{\mathcal{Q}}_{\beta}\\}=2(\gamma^{\mu})_{\alpha\beta}\mathcal{P}_{\mu},\quad\\{\mathcal{S}^{\alpha},\bar{\mathcal{S}}^{\beta}\\}=2(\gamma^{\mu})^{\alpha\beta}\mathcal{K}_{\mu},\\\
&\\{\mathcal{Q}_{\alpha},\bar{\mathcal{S}}^{\beta}\\}=-i\left(2\delta_{\alpha}^{\phantom{{\alpha}}\beta}(\mathcal{D}+i\mathcal{R})-\varepsilon^{\mu\nu\lambda}(\gamma_{\lambda})_{\alpha}^{\phantom{{\alpha}}\beta}\mathcal{M}_{\mu\nu}\right)\,,\\\
&\\{\bar{\mathcal{Q}}_{\alpha},\mathcal{S}^{\beta}\\}=-i\left(2\delta_{\alpha}^{\phantom{{\alpha}}\beta}(\mathcal{D}-i\mathcal{R})-\varepsilon^{\mu\nu\lambda}(\gamma_{\lambda})_{\alpha}^{\phantom{{\alpha}}\beta}\mathcal{M}_{\mu\nu}\right)\,.\end{split}$
(164)

The commutation relations between the conformal group and the supercharges are

$\displaystyle[\mathcal{D},\mathcal{Q}_{\alpha}]=\frac{1}{2}i\mathcal{Q}_{\alpha},\quad[\mathcal{D},\mathcal{S}^{\alpha}]=-\frac{1}{2}i\mathcal{S}^{\alpha},\quad[\mathcal{K}_{\mu},\mathcal{Q}_{\alpha}]=(\gamma_{\mu})_{\alpha\beta}\mathcal{S}^{\beta},\quad[\mathcal{P}_{\mu},\mathcal{S}^{\alpha}]=-(\gamma_{\mu})^{\alpha\beta}\mathcal{Q}_{\beta},$
$\displaystyle[\mathcal{D},\bar{\mathcal{Q}}_{\alpha}]=\frac{1}{2}i\bar{\mathcal{Q}}_{\alpha},\quad[\mathcal{D},\bar{\mathcal{S}}^{\alpha}]=-\frac{1}{2}i\bar{\mathcal{S}}^{\alpha},\quad[\mathcal{K}_{\mu},\bar{\mathcal{Q}}_{\alpha}]=(\gamma_{\mu})_{\alpha\beta}\bar{\mathcal{S}}^{\beta},\quad[\mathcal{P}_{\mu},\bar{\mathcal{S}}^{\alpha}]=-(\gamma_{\mu})^{\alpha\beta}\bar{\mathcal{Q}}_{\beta}\,.$
(165)

Lastly, all generators are neutral under $R$-symmetry, except the eight
supercharges:

$[\mathcal{R},\mathcal{Q}_{\alpha}]=-\mathcal{Q}_{\alpha}\,,\quad[\mathcal{R},\bar{\mathcal{Q}}_{\alpha}]=\bar{\mathcal{Q}}_{\alpha}\,,\quad[\mathcal{R},\mathcal{S}^{\alpha}]=-\mathcal{S}^{\alpha}\,,\quad[\mathcal{R},\bar{\mathcal{S}}^{\alpha}]=\bar{\mathcal{S}}^{\alpha}\,.$
(166)

###  A.3 Differential operators

In this appendix we present the action of our generators in terms of
differential operators in superspace. We consider an operator
${\mathcal{O}}(z)$ of dimension $\Delta$ and charge $r$ that transforms under
rotations in a representation dictated by matrices $s_{\mu\nu}$, which satisfy
the same commutation relations as ${\mathcal{M}}_{\mu\nu}$. Then, the
generators of the algebra act as:

$\displaystyle\left[\mathcal{D},\mathcal{O}(z)\right]$
$\displaystyle=i\left(\Delta+x^{\mu}\partial_{\mu}+\frac{1}{2}\theta^{\alpha}\partial_{\alpha}+\frac{1}{2}\bar{\theta}^{\alpha}\bar{\partial}_{\alpha}\right)\mathcal{O}(z)\,,$
(167) $\displaystyle\left[\mathcal{K}_{\mu},\mathcal{O}(z)\right]$
$\displaystyle=\Big{(}-2i\Delta
x_{\mu}-2ix_{\mu}x^{\nu}\partial_{\nu}+ix^{2}\partial_{\mu}$ $\displaystyle-
ix_{\mu}(\theta^{\alpha}\partial_{\alpha}+\bar{\theta}^{\alpha}\bar{\partial}_{\alpha})+i\epsilon_{\mu\nu\rho}(\gamma^{\nu})_{\alpha}^{\phantom{{\alpha}}\beta}x^{\rho}(\theta^{\alpha}\partial_{\beta}+\bar{\theta}^{\alpha}\bar{\partial}_{\beta})$
$\displaystyle+(\gamma_{\mu})_{\alpha\beta}\theta^{\alpha}\bar{\theta}^{\beta}(\theta^{\gamma}\partial_{\gamma}-\bar{\theta}^{\gamma}\bar{\partial}_{\gamma})-\frac{i}{2}\theta^{2}\bar{\theta}^{2}\partial_{\mu}$
$\displaystyle-2\mathit{r}(\gamma_{\mu})_{\alpha\beta}\theta^{\alpha}\bar{\theta}^{\beta}-2x^{\nu}\mathit{s}_{\mu\nu}-i\eta_{\mu\nu}\varepsilon^{\nu\rho\sigma}\mathit{s}_{\rho\sigma}\theta^{\alpha}\bar{\theta}_{\alpha}\Big{)}\mathcal{O}(z)\,,$
(168) $\displaystyle\left[\mathcal{M}_{\mu\nu},\mathcal{O}(z)\right]$
$\displaystyle=\left(\frac{i}{2}\varepsilon_{\mu\nu\rho}(\gamma^{\rho})_{\alpha}^{\phantom{{\alpha}}\beta}(\theta^{\alpha}\partial_{\beta}+\bar{\theta}^{\alpha}\bar{\partial}_{\beta})+ix_{\mu}\partial_{\nu}-ix_{\nu}\partial_{\mu}+\mathit{s}_{\mu\nu}\right)\mathcal{O}(z)\,,$
(169) $\displaystyle\left[\mathcal{R},\mathcal{O}(z)\right]$
$\displaystyle=\left(\bar{\theta}^{\alpha}\bar{\partial}_{\alpha}-\theta^{\alpha}\partial_{\alpha}+\mathit{r}\right)\mathcal{O}(z)\,,$
(170) $\displaystyle\left[\mathcal{S}^{\alpha},\mathcal{O}(z)\right]$
$\displaystyle=\Big{(}-i(\gamma_{\mu})^{\alpha\beta}x^{\mu}\partial_{\beta}+\bar{\theta}^{\alpha}x^{\mu}\partial_{\mu}+\frac{1}{2}(\gamma^{\mu})^{\alpha\beta}(\gamma^{\nu})_{\beta\gamma}\bar{\theta}^{\gamma}(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$
$\displaystyle-\theta^{\alpha}\bar{\theta}^{\beta}\partial_{\beta}+\bar{\theta}^{\alpha}\theta^{\beta}\partial_{\beta}-2\bar{\theta}^{\alpha}\bar{\theta}^{\beta}\bar{\partial}_{\beta}$
$\displaystyle-i(\gamma^{\mu})_{\beta\gamma}\theta^{\beta}\bar{\theta}^{\gamma}\bar{\theta}^{\alpha}\partial_{\mu}+2\Delta\bar{\theta}^{\alpha}-2\mathit{r}\bar{\theta}^{\alpha}+i\varepsilon^{\mu\nu\rho}(\gamma_{\rho})_{\beta}^{\phantom{{\beta}}{\alpha}}\bar{\theta}^{\beta}\mathit{s}_{\mu\nu}\Big{)}\mathcal{O}(z)\,,$
(171) $\displaystyle\left[\bar{\mathcal{S}}^{\alpha},\mathcal{O}(z)\right]$
$\displaystyle=\Big{(}i(\gamma^{\mu})^{\alpha\beta}x_{\mu}\bar{\partial}_{\beta}-\theta^{\alpha}x^{\mu}\partial_{\mu}-\frac{1}{2}(\gamma^{\mu})^{\alpha\beta}(\gamma^{\nu})_{\beta\gamma}\theta^{\gamma}(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$
$\displaystyle+\bar{\theta}^{\alpha}\theta^{\beta}\bar{\partial}_{\beta}-\theta^{\alpha}\bar{\theta}^{\beta}\bar{\partial}_{\beta}+2\theta^{\alpha}\theta^{\beta}\partial_{\beta}$
$\displaystyle+i(\gamma^{\mu})_{\beta\gamma}\bar{\theta}^{\beta}\theta^{\gamma}\theta^{\alpha}\partial_{\mu}-2\Delta\theta^{\alpha}-2\mathit{r}\theta^{\alpha}-i\varepsilon^{\mu\nu\rho}(\gamma_{\rho})_{\beta}^{\phantom{{\beta}}{\alpha}}\theta^{\beta}\mathit{s}_{\mu\nu}\Big{)}\mathcal{O}(z)\,.$
(172)

##  Appendix B Non-supersymmetric conformal blocks

In this appendix, we will derive the non-supersymmetric bulk and boundary
blocks for bulk two-point functions and bulk-boundary-boundary correlators.

###  B.1 Two-point function

The conformal blocks for a two-point function were first derived in
McAvity:1995zd  but we will follow Liendo:2012hy  in our approach.

##### Bulk channel:

The bulk-channel blocks can be found by acting on the two-point function (7)
with the bulk Casimir operator:

$\mathcal{C}_{\text{bulk,bos}}^{(12)}=-\mathcal{D}^{2}-\frac{1}{2}\left\\{\mathcal{K}^{\mu},\mathcal{P}_{\mu}\right\\}+\frac{1}{2}\mathcal{M}^{\mu\nu}\mathcal{M}_{\mu\nu}\,.$
(173)

The differential operators are well know, but they can also be obtained from
section A.3 by setting all Grassmann coordinates to zero. The Casimir
eigenvalues is $C_{\Delta,\ell}=\Delta(\Delta-d)+\ell(\ell+d-2)$, but only
operators with $\ell=0$ can appear in the bulk OPE. For the bulk channel, it
is convenient to define the blocks in terms of
$g^{\Delta_{12}}_{\Delta}(\xi)=\xi^{(\Delta_{1}+\Delta_{2})/2}f^{\Delta_{12}}_{\Delta}(\xi)$.
The resulting differential equation is

$\displaystyle\begin{split}\frac{\left({\mathcal{C}}_{\text{bulk,bos}}-C_{\Delta,\ell}\right)\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\rangle}{{(2x_{1}^{\bot})^{-\Delta_{1}}(2x_{2}^{\bot})^{-\Delta_{2}}\xi^{-(\Delta_{1}+\Delta_{2})/2}}}&=\bigg{[}4\xi^{2}(\xi+1)\partial_{\xi}^{2}+2\xi(2\xi-d+2)\partial_{\xi}\\\
&\qquad\quad-\big{(}\Delta(\Delta-d)+\Delta_{12}^{2}\xi\big{)}\bigg{]}g^{\Delta_{12}}_{\Delta}(\xi)=0\,,\end{split}$
(174)

which is solved by

$\displaystyle
g^{\Delta_{12}}_{\Delta}(\xi)=\xi^{\Delta/2}\,_{2}F_{1}\Big{(}\frac{\Delta+\Delta_{12}}{2},\frac{\Delta-\Delta_{12}}{2};\Delta+1-\frac{d}{2};-\xi\Big{)}.$
(175)

Whenever the superscript $\Delta_{12}$ is omitted, it is assumed that
$\Delta_{12}=0$.

##### Boundary channel:

In the boundary channel, the conformal blocks are eigenfunctions of the
boundary Casimir that acts on a single point

$\hat{\mathcal{C}}^{(1)}_{\text{non-
susy}}=-\mathcal{D}^{2}-\frac{1}{2}\left\\{\mathcal{K}^{a},\mathcal{P}_{a}\right\\}+\frac{1}{2}\mathcal{M}^{ab}\mathcal{M}_{ab}\,,$
(176)

where the index $a,b$ runs only on directions parallel to the boundary. The
eigenvalue of the boundary Casimir is
$\hat{C}_{\Delta,j}=\Delta(\Delta-d+1)+j(j+d-1)$, but once more we have to
take $j=0$ because only scalar operators appear in the BOE of a bulk scalar.
The resulting differential equation is

$\displaystyle\begin{split}\frac{\left(\hat{\mathcal{C}}_{\text{non-
susy}}-\hat{C}_{\Delta,0}\right)\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\rangle}{(2x_{1}^{\bot})^{-\Delta_{1}}(2x_{2}^{\bot})^{-\Delta_{2}}}&=\bigg{[}\xi(\xi+1)\partial_{\xi}^{2}+\frac{d}{2}(2\xi+1)\partial_{\xi}\\\
&\qquad\qquad\qquad\quad-\hat{\Delta}(\hat{\Delta}-d+1)\bigg{]}\hat{f}_{\hat{\Delta}}(\xi)=0\,,\end{split}$
(177)

which is solved by

$\displaystyle\hat{f}_{\hat{\Delta}}(\xi)=\xi^{-\hat{\Delta}}\,_{2}F_{1}\Big{(}\hat{\Delta},\hat{\Delta}-\frac{d}{2}+1;2\hat{\Delta}-d+2;-\frac{1}{\xi}\Big{)}\,.$
(178)

###  B.2 Three-point bosonic blocks

In this section we restrict to $d=3$. We start with considering the bosonic
correlator

$\langle\mathcal{O}_{1}(x)\hat{\mathcal{O}}_{2,j}(0)\hat{\mathcal{O}}_{3}(\infty)\rangle=\frac{\left(x^{a}\omega_{a}\right)^{j}}{(x^{\bot})^{\Delta_{1}+\hat{\Delta}_{23}}|x^{a}|^{j}}\mathcal{F}^{\text{3pt}}(\chi),\qquad\chi=\frac{|x^{a}|^{2}}{(x^{\bot})^{2}}\,,$
(179)

where the second operator has parallel spin $j$. We used index-free notation
to contract all vector indices, and $\omega_{a}$ is a null-vector. We need to
evaluate the eigenvalue equation

$\hat{\mathcal{C}}_{\text{bos}}^{(1)}\langle\mathcal{O}_{1}(x)\hat{\mathcal{O}}_{2,j}(0)\hat{\mathcal{O}}_{3}(\infty)\rangle=\hat{C}_{\hat{\Delta},0}\langle\mathcal{O}_{1}(x)\hat{\mathcal{O}}_{2,j}(0)\hat{\mathcal{O}}_{3}(\infty)\rangle\,,$
(180)

where $\hat{C}_{\hat{\Delta},0}$ is the boundary Casimir eigenvalue when the
parallel spin of the exchanged operator is zero. This gives us the
differential equation

$\left[4\chi(\chi+1)\partial^{2}_{\chi}+4\left((\hat{\Delta}_{23}+2)\chi+1\right)\partial_{\chi}-\hat{\Delta}(\hat{\Delta}-2)+\hat{\Delta}_{23}(\hat{\Delta}_{23}+2)-\frac{j^{2}}{\chi}\right]\hat{f}^{\text{3pt},\hat{\Delta}_{23}}_{\hat{\Delta},j}=0\,.$
(181)

The solution to equation (181) is once more given by a hypergeometric function

$\hat{f}^{\text{3pt},\hat{\Delta}_{23}}_{\hat{\Delta},j}(\chi)=\chi^{-\frac{1}{2}(\hat{\Delta}+\hat{\Delta}_{23})}\,_{2}F_{1}\Big{(}\frac{1}{2}\left(\hat{\Delta}+\hat{\Delta}_{23}-j\right),\frac{1}{2}\left(\hat{\Delta}+\hat{\Delta}_{23}+j\right);\hat{\Delta};-\frac{1}{\chi}\Big{)}\,.$
(182)

##  Appendix C More on blocks across dimensions

In this appendix, we provide more details on the derivation of the
superconformal blocks in any number of dimension. In section 4 we showed that
the supersymmetric part of the Casimir acting on a two-point function can be
written in terms of $Q$ supercharges acting on the two-point function. Our
current goal is to find equivalent expressions where the supercharges are
replaced by a differential operator, for example

$\displaystyle\begin{split}\big{\langle}0\big{|}\left[Q^{-}_{\dot{\alpha}},\phi_{1}(x_{1})\right]\left[Q^{+}_{\alpha},\bar{\phi}_{2}(x_{2})\right]\big{|}0\big{\rangle}\sim{\mathcal{D}}_{x}\big{\langle}0\big{|}\phi_{1}(x_{1})\bar{\phi}_{2}(x_{2})\big{|}0\big{\rangle}\,.\end{split}$
(183)

It was proposed in Bobev:2015jxa  that this can be achieved with
supersymmetric Ward identities. Here we give a quick summary of the strategy.
In our setup the supercharges ${\mathcal{Q}}_{A}$ and ${\mathcal{S}}_{A}$ are
preserved by the boundary, so the following Ward identities are satisfied:

$\displaystyle\begin{split}&\big{\langle}0\big{|}\big{\\{}Q^{\text{bdy}}_{1},\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}=0\,,\\\
&\big{\langle}0\big{|}\big{\\{}Q^{\text{bdy}}_{1},\left[Q^{-}_{2},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}=0\,,\\\
&\big{\langle}0\big{|}\big{\\{}Q^{\text{bdy}}_{2},\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}=0\,,\\\
\end{split}\begin{split}&\big{\langle}0\big{|}\big{\\{}S^{\text{bdy}}_{1},\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}=0\,,\\\
&\big{\langle}0\big{|}\big{\\{}S^{\text{bdy}}_{1},\left[Q^{-}_{2},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}=0\,.\\\
\phantom{1}\end{split}$ (184)

There are other Ward identities that can be considered, but these five are
sufficient for our purposes. At this point, it is hard to continue without an
explicit matrix representation for the Clifford algebra, so we focus on $d=3$
where $\Sigma_{\mu}=(\sigma_{3},\sigma_{1},\sigma_{2})$. Let us consider
explicitly the simplest Ward identity to show how to replace the supercharges
with differential operators in the general case. With elementary manipulations
we find:

$\displaystyle\begin{split}0&=\big{\langle}0\big{|}\big{\\{}Q^{\text{bdy}}_{2},\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\bar{\phi}_{2}(x_{2})\big{\\}}\big{|}0\big{\rangle}\\\
&=\big{\langle}0\big{|}\big{\\{}Q^{\text{bdy}}_{2},\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\big{\\}}\bar{\phi}_{2}(x_{2})\big{|}0\big{\rangle}-\big{\langle}0\big{|}\big{[}Q^{-}_{1},\phi_{1}(x_{1})\big{]}\big{[}Q^{\text{bdy}}_{2},\bar{\phi}_{2}(x_{2})\big{]}\big{|}0\big{\rangle}\\\
&=\big{\langle}0\big{|}\big{[}\big{\\{}Q_{2}^{+},Q^{-}_{1}\big{\\}},\phi_{1}(x_{1})\big{]}\bar{\phi}_{2}(x_{2})\big{|}0\big{\rangle}-\big{\langle}0\big{|}\big{[}Q^{-}_{1},\phi_{1}(x_{1})\big{]}\big{[}Q_{2}^{+},\bar{\phi}_{2}(x_{2})\big{]}\big{|}0\big{\rangle}\,.\end{split}$
(185)

In our conventions $\\{Q_{2}^{+},Q^{-}_{1}\\}=P_{2}+iP_{3}$ and also
$[P_{\mu},{\mathcal{O}}(x)]=-i\partial_{\mu}{\mathcal{O}}(x)$, so we conclude

$\displaystyle\left\langle
0\left|\left[Q^{-}_{1},\phi_{1}(x_{1})\right]\left[Q_{2}^{+},\bar{\phi}_{2}(x_{2})\right]\right|0\right\rangle=\left(-i\partial_{2}+\partial_{3}\right)\left\langle
0\left|\phi_{1}(x_{1})\bar{\phi}_{2}(x_{2})\right|0\right\rangle\,.$ (186)

The other Ward identities can be manipulated identically, but unlike the
example we showed they do not decouple, so one has to solve a simple linear
system of equations to obtain the terms we are interested in.

These steps can be automated in Mathematica and applied to all cases of
interest in $d=3,4$. The resulting differential operators ${\mathcal{D}}_{x}$
depend on $x_{i}^{\mu}$ and $\partial_{\mu,i}$ and take a complicated looking
form. However, we know that the Casimir operator
$C_{\text{bulk/bdy},\text{susy}}$ has to respect conformal invariance, so when
we combine all the contributing terms, the result has to be a differential
operator of the cross-ratio $\xi$. Indeed, in $d=3,4$ we find the following
results:

$\displaystyle\begin{split}ix_{12}^{\mu}\bar{\Sigma}_{\mu}^{{\dot{\alpha}}{\alpha}}\left\langle
0\left|\left[Q^{-}_{\dot{\alpha}},\phi_{1}(x_{1})\right]\left[Q^{+}_{\alpha},\bar{\phi}_{2}(x_{2})\right]\right|0\right\rangle&\to\big{(}4\xi\partial_{\xi}-2(\Delta_{1}+\Delta_{2})\big{)}{\mathcal{G}}(\xi)\,,\\\
ix_{1}^{\bot}\left\langle
0\left|\left\\{Q^{-}_{1},\left[Q^{-}_{2},\phi_{1}(x_{1})\right]\right\\}\bar{\phi}_{2}(x_{2})\right|0\right\rangle&\to(-\xi\partial_{\xi}-\Delta_{1}){\mathcal{F}}(\xi),\\\
ix_{1}^{\bot}\left\langle
0\left|\left\\{Q^{-}_{1},\left[Q^{-}_{2},\phi_{1}(x_{1})\right]\right\\}\phi_{2}(x_{2})\right|0\right\rangle&\to\big{(}{-}(\xi+1)\partial_{\xi}-\Delta_{1}\big{)}{\mathcal{F}}(\xi)\,.\end{split}$
(187)

From these results we can obtain (93), (96) and (101). Although the
intermediate differential operators were complicated, the final result takes a
remarkably simple form. Perhaps one could find a more direct method of
obtaining these results, and at the same time make it more manifest that the
result is indeed independent of $d$.

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