$h_{[a_{1},b_{1}],[a_{2},b_{2}]}$ are the R-symmetry blocks, solving the
$SO(5)$ quadratic Casimir equation and $\beta_{i}$ are unknown constants to be
determined up to an overall factor. The Ward Identities constrain the
coefficients up to an overall constant which is normalised to
$\beta_{5}=\frac{-1}{5}$ for the lowest conformal weight $g_{2,2}$ to match
the normalisation conventions.

$\displaystyle\beta_{1}$ $\displaystyle=\frac{3}{70}\beta_{5}$
$\displaystyle\beta_{2}$ $\displaystyle=-\frac{3}{70}\beta_{5}$
$\displaystyle\beta_{3}$ $\displaystyle=-\frac{1}{4}\beta_{5}$
$\displaystyle\beta_{4}$ $\displaystyle=\frac{1}{4}\beta_{5}$
$\displaystyle\beta_{6}$ $\displaystyle=\frac{3}{40}\beta_{5}$ (4.31)

This gives a fully fixed superconformal block which depends on the two
spacetime and 5 R-symmetry cross-ratios. In turn, this block can be decomposed
like the correlator in terms of $F_{i}$ and $f_{i}$ functions. In terms of the
Ward identity channels, this gives

$\displaystyle f^{0}_{\mathcal{D}_{2},\mathcal{D}_{2}}$ $\displaystyle=1$
(4.32) $\displaystyle f^{1}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=\sum_{k,l}\frac{(k+1)(l+1)(2(k+1)l+3k+2)}{\nu_{2}^{2}\chi_{1}^{2}(k+l+1)}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.33) $\displaystyle f^{2}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=-\sum_{k,l}\frac{(k+1)(l+1)(2k(l+1)+3l+2)}{\nu_{2}^{2}\chi_{1}^{2}(k+l+1)}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.34) $\displaystyle f^{3}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=-\sum_{k,l}\frac{(k+1)(k+2)(l+1)(l+2)}{2\nu_{2}^{2}\chi_{1}^{2}}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.35)

One can define the bloc

$\displaystyle
G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}=\frac{3(2)_{k+l}}{(5)_{k}(5)_{l}}\chi_{1}^{k+2}\nu_{2}^{l+2}$
(4.36)
$\displaystyle\sum_{k,l}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}=3\chi_{1}^{2}\nu_{2}^{2}F_{2}\left(2,1,1;5,5\,|\,\chi_{1},\nu_{2}\right)$
(4.37)

and the function $f^{3}_{\mathcal{D}_{2},\mathcal{D}_{2}}$ is then the
derivative of this Appell function. What is remarkable in these explicit forms
of the blocks above (4.32) and listed in the Appendix C.2 is that the
superconformal blocks are simple functions of the cross-ratios (essentially of
the same form as the Appell functions of the scalar conformal blocks), even in
the case where the functions came from the sum of 50 terms.444This being said,
one must be careful of which variable to use in the OPE expansion as an
awkward choice can lead to quite cumbersome expressions.

#### Superconformal Blocks in the Asymmetric Channel

In the other channel, where we have the OPE expansion

$\displaystyle\mathcal{D}_{1}\times\mathcal{D}_{2}\sim\mathcal{D}_{1}+\mathcal{D}_{3}+\sum_{h}\mathcal{L}^{h}_{0,[0,1]}$
(4.38)

in addition to the one above. This can be seen in the diagram

$\mathcal{D}_{2}(X_{5})$$\mathcal{D}_{1}(X_{1})$$\mathcal{D}_{1}(X_{4})$$\mathcal{D}_{1}(X_{2})$$\mathcal{D}_{1}(X_{3})$$\mathcal{I}+\mathcal{D}_{2}+\mathcal{L}^{h_{1}}_{0,[0,0]}$$\mathcal{D}_{1}+\mathcal{D}_{3}+\mathcal{L}^{h_{2}}_{0,[0,1]}$
Figure 4.2: OPE diagram for the five-point function
$\langle\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}\rangle$
for the Asymmetric OPE channel, where the same operators are exchanged from
(12) and (45).

The OPE limits in this diagram can be seen at the vertices, and we can choose
variables that vanish in these limits; for example, it will be useful to write
the results in terms of

$\displaystyle\nu_{1}$ $\displaystyle=\frac{\chi_{1}-\chi_{2}}{1-\chi_{2}}$
$\displaystyle\nu_{2}$ $\displaystyle=1-\chi_{2}$ (4.39)

In this expansion, the non-vanishing blocks are given by

$\displaystyle\mathcal{A}=$ $\displaystyle
c_{110}^{2}c_{112}\mathcal{G}_{\mathcal{I},\mathcal{D}_{1}}+c_{112}^{3}\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{1}}+c_{112}c_{123}^{2}\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{3}}+\sum_{h_{1}}c_{112}c_{11h_{1}}^{2}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{D}_{1}}$
(4.40)
$\displaystyle+\sum_{h_{1}}c_{11h_{1}}c_{13h_{1}}c_{123}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{D}_{3}}+\sum_{h_{2}}c_{112}c_{12h_{2}}^{2}\mathcal{G}_{\mathcal{D}_{2},\mathcal{L}^{h_{2}}_{0,[0,1]}}+\sum_{h_{1},h_{2}}c_{11h_{1}}c_{12h_{2}}c_{1h_{1}h_{2}}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{L}^{h_{2}}_{0,[0,1]}}$

In particular, the topological limit is given by

$\displaystyle
g^{0}=c_{\mathcal{D}_{1},\mathcal{I}}+c_{\mathcal{D}_{2},\mathcal{D}_{1}}+c_{\mathcal{D}_{2},\mathcal{D}_{3}}$
(4.41)

###  4.4 Free Theory Data

We can find the free theory OPE coefficients in both channels from the block
expansion above. The Free result is obtained by Wick contracting the
elementary fields and gives555To do this, we compute the six-point GFF result
and pinch points 5 and 6.

$\displaystyle\mathcal{A}=\sqrt{2}\left(\frac{r_{1}}{\chi_{1}^{2}}+\frac{r_{2}}{\chi_{2}^{2}}+\frac{s_{1}}{(\chi_{1}-1)^{2}}+\frac{s_{2}}{(\chi_{2}-1)^{2}}+\frac{t}{(\chi_{1}-\chi_{2})^{2}}+1\right)$
(4.42)

which gives the $F_{i}=\sqrt{2}$ and

$\displaystyle f_{0}^{(0)}$ $\displaystyle=6\sqrt{2}$ (4.43) $\displaystyle
f_{1}^{(0)}$
$\displaystyle=\frac{\sqrt{2}(1-2\chi_{1})}{(\chi_{1}-1)\chi_{1}}$ (4.44)
$\displaystyle f_{2}^{(0)}$
$\displaystyle=\frac{\sqrt{2}(1-2\chi_{2})}{(\chi_{2}-1)\chi_{2}}$ (4.45)
$\displaystyle f_{3}^{(0)}$
$\displaystyle=-\frac{\sqrt{2}}{(\chi_{1}-\chi_{2})^{2}}$ (4.46)

For the symmetric channel, this gives us the following:

$\displaystyle
c_{\mathcal{I},\mathcal{D}_{2}}=c_{\mathcal{D}_{2},\mathcal{I}}=c_{112}=\sqrt{2}$
(4.47) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{2}}=c_{112}^{2}c_{222}=4\sqrt{2}$ (4.48)
$\displaystyle
c_{\mathcal{L}^{\Delta}_{0,[0,0]},\mathcal{D}_{2}}=c_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,0]}}=c_{112}c_{11\Delta}c_{22\Delta}=\left(\frac{1+(-1)^{\Delta}}{2}\right)\frac{\sqrt{\pi}2^{-2\Delta+\frac{1}{2}}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.49) $\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,0]}}=c_{11\Delta_{1}}c_{11\Delta_{2}}c_{2\Delta_{1}\Delta_{2}}=-\frac{\left(\pi\,2^{-2\Delta_{1}-2\Delta_{2}-\frac{3}{2}}(\Delta_{1})_{3}(\Delta_{2})_{3}\Gamma(\Delta_{1}+\Delta_{2})\right)}{\Gamma\left(\Delta_{1}+\frac{3}{2}\right)\Gamma\left(\Delta_{2}+\frac{3}{2}\right)}$
(4.50)

where the last coefficient is vanishing for $\Delta_{1}$ and $\Delta_{2}$ odd.
For the asymmetric channel, this gives:

$\displaystyle c_{\mathcal{I},\mathcal{D}_{1}}=c_{110}^{2}c_{112}=\sqrt{2}$
(4.51) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{1}}=c_{112}^{3}=2\sqrt{2}$ (4.52)
$\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{3}}=c_{112}c_{123}^{2}=3\sqrt{2}$ (4.53)
$\displaystyle
c_{\mathcal{L}^{\Delta}_{0,[0,0]},\mathcal{D}_{1}}=c_{112}c_{11\Delta}^{2}=\left(\frac{1+(-1)^{\Delta}}{2}\right)\frac{\sqrt{\pi}2^{-2\Delta-\frac{1}{2}}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.54) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,1]}}=c_{112}c_{12\Delta}^{2}=-\frac{\sqrt{\pi}2^{-2\Delta-\frac{7}{2}}(\Delta-2)\left((-1)^{\Delta}(\Delta+1)(\Delta+2)-12\right)\Gamma(\Delta+4)}{3(\Delta+1)\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.55) $\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,1]}}=c_{11\Delta_{1}}c_{12\Delta_{2}}c_{1\Delta_{1}\Delta_{2}}$
(4.56)
$\displaystyle=\frac{1+(-1)^{\Delta_{1}}}{2}\frac{2\sqrt{2}(\Delta_{1}-1)\Delta_{1}(\Delta_{1}+1)(\Delta_{2}-\Delta_{1})\Gamma(\Delta_{1}+3)\Gamma(\Delta_{2}-1)\Gamma(\Delta_{1}+\Delta_{2}+4)}{\Gamma(2\Delta_{1}+3)\Gamma(2\Delta_{2}+3)}$
$\displaystyle\text{for}\quad\Delta_{2}>\Delta_{1}\quad\text{otherwise}$
$\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,1]}}=0$
(4.57)

This gives the OPE coefficients for the protected operators

$\displaystyle c_{112}$ $\displaystyle=\sqrt{2}$ (4.58) $\displaystyle
c_{222}$ $\displaystyle=2\sqrt{2}$ (4.59) $\displaystyle c_{123}$
$\displaystyle=\sqrt{3}$ (4.60)

which agree with [37] Those of the uncharged long operators

$\displaystyle c_{11\Delta}$
$\displaystyle=\left(\frac{1+(-1)^{\Delta}}{2}\right)\sqrt{\frac{\sqrt{\pi}2^{-2\Delta-1}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}}$
(4.61) $\displaystyle c_{22\Delta}$ $\displaystyle=2c_{11\Delta}$ (4.62)
$\displaystyle c_{2\Delta_{1},\Delta_{2}}$
$\displaystyle=\sqrt{\frac{2\Delta_{1}^{2}(\Delta_{1}+1)(\Delta_{1}+2)\Delta_{2}^{2}(\Delta_{2}+1)(\Delta_{2}+2)\Gamma(\Delta_{1}+\Delta_{2})^{2}}{(\Delta_{1}-1)(\Delta_{2}-1)\Gamma(2\Delta_{1}+2)\Gamma(2\Delta_{2}+2)}}\qquad\\{\Delta_{1},\Delta_{2}\\}\,\text{even}$
(4.63)

where the last expression is only valid when both $\Delta_{1}$ and
$\Delta_{2}$ are even given the form of the OPE expansion.

For the charged channel, just as for the protected operators, the Ward
Identities only constrain the blocks up to a constant. Unlike the protected
operators, no topological sector fixes this constant. We thus impose a minimal
condition of reality and choose a normalisation factor of
$-\frac{(-1)^{\Delta+1}\left((-1)^{\Delta}(\Delta+1)(\Delta+2)-12\right)}{\Delta+1}$
for the $\mathcal{G}_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,1]}}$ block
which gives the OPE coefficient for the charged long and mixed
uncharged/charged long, respectively:

$\displaystyle c_{12\Delta}$
$\displaystyle=\sqrt{\frac{\sqrt{\pi}2^{-2\Delta-4}(\Delta-2)\Gamma(\Delta+4)}{3\Gamma\left(\Delta+\frac{3}{2}\right)}}$
(4.64) $\displaystyle c_{1\Delta_{1}\Delta_{2}}$
$\displaystyle=\sqrt{\frac{12\Delta_{1}\Delta_{2}(\Delta_{2}-1)(\Delta_{1}-\Delta_{2})^{2}\left((\Delta_{2}-1)_{4}\right){}^{2}\Gamma(\Delta_{1}+\Delta_{2}+4)^{2}}{\Gamma(2\Delta_{1}+2)\Gamma(2\Delta_{2}+2)\left((\Delta_{1}-1)_{3}\right){}^{2}(\Delta_{1}-1)_{4}(\Delta_{2}-2)_{6}}}\qquad\Delta_{1}\,\,\text{even
and }\,\Delta_{2}>\Delta_{1}$ (4.65)

One can already recognise patterns observed in four-point functions, such as
the absence of uncharged long operators of odd dimensions. This is, however,
not the case for the charged long operators
$\mathcal{L}^{\Delta_{2}}_{0,[0,1]}$. For the final OPE coefficient, we
observe a bound which seems to come from the OPE

$\displaystyle\mathcal{D}_{1}\times\mathcal{L}^{\Delta_{1}}_{0,[0,0]}\sim\sum_{\Delta_{2}>\Delta_{1}}\mathcal{L}^{\Delta_{2}}_{0,[0,1]}+...$
(4.66)

###  4.5 Conformal Bootstrap

We will focus on bootstrapping the function$f_{3}$ since it has both braiding
and crossing symmetry and deducing the rest of the functions from there. The
crossing is

$\displaystyle f_{3}(\chi_{1},\chi_{2})$
$\displaystyle=f_{3}(1-\chi_{2},1-\chi_{1})$ (4.67)

and the braiding is

$\displaystyle f_{3}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
f_{3}(\chi_{2},\chi_{1}).$ (4.68)

For the first order functions, we start with the Ansatz which satisfies this
crossing and braiding

$\displaystyle f_{3}^{(1)}(\chi_{1},\chi_{2})$
$\displaystyle=p(\chi_{1},\chi_{2})+r_{1}(\chi_{1},\chi_{2})\log(\chi_{1})+r_{1}(\chi_{2},\chi_{1})\log(\chi_{2})+r_{1}(1-\chi_{1},1-\chi_{2})\log(1-\chi_{1})$
$\displaystyle+r_{1}(1-\chi_{2},1-\chi_{1})\log(1-\chi_{2})+r_{5}(\chi_{1},\chi_{2})\log(\chi_{2}-\chi_{1})$
(4.69)

where

$\displaystyle p(\chi_{1},\chi_{2})$ $\displaystyle=p(\chi_{2},\chi_{1})$
(4.70) $\displaystyle p(\chi_{1},\chi_{2})$
$\displaystyle=p(1-\chi_{2},1-\chi_{1})$ (4.71) $\displaystyle
r_{5}(\chi_{1},\chi_{2})$ $\displaystyle=r_{5}(\chi_{2},\chi_{1})$ (4.72)
$\displaystyle r_{5}(\chi_{1},\chi_{2})$
$\displaystyle=r_{5}(1-\chi_{2},1-\chi_{1})$ (4.73)

The easiest way to see the effect of crossing and braiding is for the
individual R-symmetry channels where the relevant transformations are:

$\displaystyle F_{0}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}\left(\frac{\chi_{2}-1}{\chi_{2}-\chi_{1}},\frac{\chi_{2}}{\chi_{2}-\chi_{1}}\right)$
(4.74) $\displaystyle F_{1}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{2}-\chi_{1}}{\chi_{2}-1},\frac{\chi_{2}}{\chi_{2}-1})$
(4.75) $\displaystyle F_{2}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}})$ (4.76) $\displaystyle
F_{3}(\chi_{1},\chi_{3})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{1}}{\chi_{1}-1},\frac{\chi_{2}-\chi_{1}}{1-\chi_{1}})$
(4.77) $\displaystyle F_{4}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{1}{\chi_{1}},\frac{\chi_{2}}{\chi_{1}})$ (4.78)

So there are only three rational functions to fix. The rational functions can
only have poles when operators collide; that is

$\displaystyle\\{\chi_{1},\chi_{2},1-\chi_{1},1-\chi_{2},\chi_{2}-\chi_{1},\frac{1}{\chi_{1}},\frac{1}{\chi_{2}}\\}\rightarrow
0$ (4.79)

Additionally, the OPE expansion dictates the behaviour of the overall function
as well as the logarithms in the OPE limits; this gives the following boundary
conditions

$\displaystyle r_{1}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O(\chi_{1}^{1},(1-\chi_{2})^{0})$ (4.80) $\displaystyle
r_{5}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O((1-\chi_{2})^{-2},(\chi_{1}-\chi_{2})^{0})$ (4.81) $\displaystyle
f_{3}(\chi_{1},\chi_{2})$ $\displaystyle\sim
a_{\mathcal{D}_{1},\mathcal{I}}(\chi_{1}-\chi_{2})^{-2}+O(\chi_{1}^{0},(1-\chi_{2})^{0},(\chi_{1}-\chi_{2})^{0})$
(4.82)

Additionally, the anomalous dimension’s growth constrains the poles’ maximal
order in the other variables. For example, $r_{1}(\chi_{1},\chi_{2})$ cannot
have poles in $\\{\chi_{1},1-\chi_{2}\\}$ so one can parametrise it as

$\displaystyle r_{1}(\chi_{1})$
$\displaystyle=\frac{\chi_{1}\sum_{\\{k,l\\}=0}r_{1,kl}\chi_{1}^{k}\chi_{2}^{l}}{\chi_{2}^{N_{r_{1}}}(\chi_{2}-\chi_{1})^{M_{r_{1}}}}$
(4.83)

where the growth of the anomalous dimension fixes

$\displaystyle N_{r_{1}}$ $\displaystyle=l+2$ $\displaystyle M_{r_{1}}$
$\displaystyle=l$ (4.84)

Likewise, one can use the crossing symmetry to best parametrise

$\displaystyle r_{5}(\chi_{1},\chi_{2})$
$\displaystyle=\left(\frac{\tilde{r_{5}}(\chi_{1},\chi_{2})}{\chi_{1}^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(\chi_{2},\chi_{1})}{\chi_{2}^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(1-\chi_{1},1-\chi_{2})}{(1-\chi_{1})^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(1-\chi_{2},1-\chi_{1})}{(1-\chi_{2})^{N_{r_{5}}}}\right)$
(4.85)

where the growth of the anomalous dimension sets

$\displaystyle N_{r_{5}}=l+1$ (4.86)

Possible cancellations between the rational functions and the logarithms in
the OPE limits mean that the boundary conditions for $p_{1}$ are relaxed from
those of $f_{3}$ to

$\displaystyle p_{1}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O(\chi_{1}^{-1},(1-\chi_{2})^{-1},(\chi_{1}-\chi_{2})^{-2})$ (4.87)

so we can parameterise $p_{1}$ accordingly as

$\displaystyle p_{1}$
$\displaystyle=\frac{p_{1,1}}{(\chi_{2}-\chi_{1})^{2}}+\frac{\sum_{k,l}p_{k,l}\chi_{1}^{k},\chi_{2}^{l}}{\chi_{1}\chi_{2}(1-\chi_{1})(1-\chi_{2})}$
(4.88)

The crossing symmetry (which still relates some coefficients of $p_{1}$), the
boundary conditions of $f_{1},f_{2},f_{3}$ from the OPE expansion, as well as
the constraint that $f_{0}$ is a constant are all extremely constraining along
with the Regge behaviour fixes all the coefficients up to an overall constant
and $p_{1,1}$. At this point, two physical inputs are needed, known from
integrability: The OPE coefficient of the protected operator.

$\displaystyle
c_{\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}}=2-\frac{3}{2\sqrt{2\lambda}}-\frac{9}{16\sqrt{2}\lambda}+O(\lambda^{-3/2})$
(4.89)

and the value of the topological sector at large N

$\displaystyle
f_{0}=\frac{6\mathbb{I}_{2}^{2}}{\lambda\mathbb{I}_{1}^{2}}\,\frac{2(\mathbb{I}_{1}-2)(\mathbb{I}_{1}+28)+\lambda(2\mathbb{I}_{1}-19)}{\sqrt{3\lambda-(\mathbb{I}_{1}-2)(\mathbb{I}_{1}+10)}}$
(4.90)

where

$\displaystyle\mathbb{I}_{i}=\frac{\sqrt{\lambda}I_{0}(\sqrt{\lambda})}{I_{a}(\sqrt{\lambda})}$
(4.91)

and $I_{a}$ are modified Bessel functions of the first kind. The explicit
expansion at strong coupling is

$\displaystyle f_{0}$
$\displaystyle=6\sqrt{2}-\frac{33}{\sqrt{2\lambda}}+\frac{189}{8\sqrt{2}\lambda}+O(\lambda^{-3/2})$
(4.92)

This fixes the free rational functions to

$\displaystyle r_{1}(\chi_{1})$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\lambda}}\frac{\chi_{1}\left(\chi_{1}^{2}-3\chi_{1}\chi_{2}+4\chi_{2}^{2}\right)}{\chi_{2}^{2}(\chi_{2}-\chi_{1})^{3}}$
(4.93) $\displaystyle r_{5}(\chi_{1}-\chi_{2})$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\lambda}}\left(\frac{1}{\chi_{1}^{2}}+\frac{1}{(\chi_{1}-1)^{2}}+\frac{1}{\chi_{2}^{2}}+\frac{1}{(\chi_{2}-1)^{2}}\right)$
(4.94) $\displaystyle p(\chi_{1},\chi_{2})$
$\displaystyle=\frac{1}{2\sqrt{2}\sqrt{\lambda}}\left(\frac{4}{(\chi_{1}-1)(\chi_{2}-1)}+\frac{4}{\chi_{1}\chi_{2}}+\frac{19}{(\chi_{1}-\chi_{2})^{2}}\right)$
(4.95)

This gives a degenerate anomalous dimension in all the channels, independent
of multiplying OPE coefficients. This solves the mixing between all these
operators

$\displaystyle\gamma^{(1)}_{\Delta}=\frac{-1}{2\sqrt{\lambda}}\Delta(\Delta+3)$
(4.96)

###  4.6 Comments

The low dimensionality of this system, along with the high symmetry of the
insertions, allows for a simple analysis of a presumably complex quantity: the
five-point correlator of operator insertions on the Wilson line. The system is
set up to allow the recursive analytic conformal bootstrap to be implemented,
and the first-order bootstrap solution is presented where the higher-order
solutions will be found in [8]. The structure of the superconformal blocks is
a non-trivial result in itself, which enables the bootstrap process for the
defect correlators but could potentially also inform more general setups such
as the bulk-defect blocks presented in section 3.4 of [34]. The bootstrap of
the five-point quantity means that the CFT one can access is much greater and
includes the OPE coefficients with 2 long operators. The analysis of the six-
point function in [8] will also shed light on the full long OPE coefficient
$c_{\Delta_{1}\Delta_{2}\Delta_{3}}$.

##  Chapter 5 Effective Theories in AdS2

“Deep in the human unconscious is a pervasive need for a logical universe that
makes sense, But the real universe is always one step beyond logic.” -
Princess Irulan

– Frank Herbert, Dune

A common trait of the theories above, as well as other defect theories (such
as the 1/2-BPS minimal surface string solution in
$\text{AdS}_{3}\times\text{S}^{3}\times\text{S}^{3}\times\text{S}^{1}$ [61]
and in $\text{AdS}_{3}\times\text{S}^{3}\times T^{4}$), is that they have an
effective theory description in $\text{AdS}_{2}$. For this reason, it is
beneficial to understand features of QFT in $\text{AdS}_{2}$ in general; such
as generic diagrams, properties from different interaction vertices, and tools
to compute and interpret quantities within this framework effectively.

As introduced in subsection 2.3.1, boundary (and bulk) correlators in
$\text{AdS}_{2}$ are computed using Witten diagrams. These have the same
structure as Feynman diagrams. However, the difference in metric does bring
complications when it comes to propagators but simplifications when it comes
to the symmetries of the correlators. The conformal symmetry can help solve
some of these diagrams, particularly involving exchange diagrams. Another
method is to use the embedding space formalism to map the problem to flat-
space coordinates, though the bulk propagators remain difficult. Yet another
method uses the “dimension-independent representation of Conformal
Theories”[64] provided by the Mellin transform. However, this description is
redundant for one-dimensional defects and does not allow for the Mellin
transforming of analytically-obtained results. This Chapter will present two
methods specific to one-dimensional defects addressing these issues. The first
is a tool for Witten diagrams in $\text{AdS}_{2}$ that uses the one-
dimensional boundary to simplify direct spacetime computations of contact
diagrams. The second is a Mellin formalism tailored to one-dimensional systems
that provides a useful tool for studying effective theories in
$\text{AdS}_{2}$.

###  5.1 Explicit Computations in AdS2

Perturbative computations in Anti-de-Sitter space are done through Witten
diagrams, whose structure has been studied extensively [23, 118, 56, 63, 117,
102, 94, 95, 96, 109, 65]. Yet, the complexity relative to their flat-space
counterpart is still a roadblock to perturbative analysis. There is still a
search for the full equivalent of Feynman rules [108]. As such, the most
efficient methods for perturbative correlators are through the conformal
bootstrap [47, 53]. However, explicit computations remain a reliable way to
make progress in perturbation theory and can provide some insight into
assumptions that may simplify the bootstrap process. First-order four-point
correlators with quartic interactions in the strong coupling limit can be
written in terms of $D$-functions [56], which are four-point Witten contact
diagrams. At higher order, with loops and exchanges corresponding to
additional integrated bulk points, some diagrams can be related to contact
integrals through differential equations [63, 120]. As such, the $n$-point
$D$-functions are used beyond the first order and can be seen as a starting
point to build ‘master integrals’ for Witten diagrams. AdS2 is a perfect place
to look at these integrals as it provides a simple framework with relevance in
its own right (e.g. in defect theories) and corresponds to a diagonal limit
($\chi=\bar{\chi}$ for four points) of its higher-dimensional counterparts.

Boundary correlators in AdS2 enjoy a one-dimensional conformal symmetry. In
higher dimensions, the conformal symmetry simplifies perturbative
computations. However, the structure of AdS2 provides a framework in which
another elementary method can be used to compute perturbative quantities: the
residue theorem. Using contour integration for one of the AdS2 integrals, the
contact diagram in $\lambda_{n}\phi_{\Delta}^{n}$ theory for $n$ scalars of
low conformal weights is remarkably simple, leading to the results for the
integral 2.33

$\displaystyle I_{\Delta=1,n}(x_{i})$
$\displaystyle=\frac{\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\log\left(\frac{x_{ij}}{2i}\right),$ (5.1) $\displaystyle
I_{\Delta=2,n}(x_{i})=\sum_{i}\sum_{j\neq
i}\frac{-\pi}{2(2i)^{2n-4}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-5}}{\prod_{k\neq
j,k\neq i}x_{ik}^{2}x_{jk}^{2}}\log\frac{x_{ji}}{2i}\right)$
$\displaystyle+\sum_{i}\sum_{j\neq
i}\partial_{x_{i}}\frac{-\pi}{(2i)^{2n-2}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-4}}{\prod_{k\neq
j,k\neq i}x_{ik}^{2}x_{jk}^{2}}\log\frac{x_{ji}}{2i}\right),$ (5.2)

Above, $I_{\Delta,n}(x_{i})$ is the integral corresponding to the Witten
contact diagram of $n$ fields $\phi_{\Delta}$ of conformal dimension $\Delta$
inserted at positions $x_{i}$ defined in 2.33. The normalisation $C_{\Delta}$
in (2.31) and the vertex coupling have been factored out. These expressions
are in terms of the operators’ positions and combine naturally into the cross-
ratios obtained with the usual conformal transformations (see discussion in
section 1.1 around equation (1.17) and Appendix D.1).

####  5.1.1 Contour Integration for Witten Diagrammatics

In the case of massless scalar fields, the integral corresponding to the
contact Witten diagram with $n$ external points is

$\displaystyle
I_{\Delta=1}(x_{1},...,x_{n})=\int_{0}^{\infty}dzz^{n-2}\int_{-\infty}^{\infty}dx\frac{1}{\Pi_{i=1}^{n}(z^{2}+(x-x_{i})^{2})}.\vspace{.4cm}$
(5.3)

The advantage of working in AdS2 is that since the boundary only has one
dimension, the integrated boundary coordinate $x$ can be analytically
continued to the complex plane and the integral can be evaluated with the
residue theorem. The contribution from the contour around infinity
($\mathcal{C}_{\infty}$ in Figure 5.1) vanishes since the integrand is
appropriately bounded at large $|x|$.

$x_{i}+iz$$x_{i}-iz$$x\in\mathbb{C}$$\mathcal{C}_{\mathbb{R}}$$\mathcal{C}_{\infty}$
Figure 5.1: Contour used for the integral over the variable $x$ parametrising
the AdS2 boundary. The contour can be chosen to close in the upper or lower
complex half-plane since the integrand is appropriately bounded at large
$|x|$.

The integrand in (5.3) has $2n$ poles at

$\displaystyle x=x_{j}\pm iz$ (5.4)

where $1\leq j\leq n$, with residues

$\displaystyle\pm\frac{1}{2iz\Pi_{i\neq j}(x_{ij}^{2}+2izx_{ij})},$ (5.5)

which are depicted in Figure 5.1. Since, for $z>0$, the poles in the upper
half-plane (UHP) come with a positive sign and those in the lower half-plane
(LHP) come with a minus sign, the result is independent of the choice of
closing the contour. However, when $z$ is real, these poles will have an
additional factor $\textrm{sgn}(z)$. This is because the poles cross the
$x\in\mathbb{R}$ axis when $z$ crosses $0$.

We are thus left with the integral

$\displaystyle
I(x_{i})=\pi\int_{0}^{\infty}dzz^{n-3}\sum_{j=1}^{n}\frac{1}{\Pi_{1=1,i\neq
j}^{n}(x_{ij}^{2}+2izx_{ij})}.$ (5.6)

The integrand of (5.6) has a leading large $z$ behaviour

$\displaystyle\pi\sum_{j=1}^{n-1}\frac{z^{-1}}{(2i)^{n-2}}\frac{1}{\Pi_{i=1,i\neq
j}^{n}x_{ij}}+O(z^{-2}),$ (5.7)

which vanishes thanks to the identity

$\displaystyle\sum_{j\in J}\frac{1}{\Pi_{i\in J,i\neq j}x_{ij}}=0,$ (5.8)

so the integral is convergent for $n\geq 3$ as expected. Notice that the
integrand of (5.6) has the same parity as the number of external fields. This
leads to a simplification in computing the odd $n$-point functions since the
$z$-integral can be extended to the range $\\{-\infty,\infty\\}$ and evaluated
using contour integration once again (see Appendix D.1). The symmetry of the
integrand dictates that all contact diagrams where $\sum\Delta$ is odd will
lead to a rational function and not a logarithm.

For generic $n$, equation (5.6) can be evaluated explicitly with the pole-
matched, partial fraction decomposition of the integrand

$\displaystyle\sum_{j}\frac{z^{n-3}}{\Pi_{k\neq
j}\left(2ix_{kj}(z-i\frac{x_{kj}}{2})\right)}$
$\displaystyle=\frac{1}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq j}x_{ik}x_{kj}}\frac{-1}{(z+a_{ij})},$
(5.9)

where

$\displaystyle a_{ij}=\frac{x_{ij}}{2i}.$ (5.10)

Using this decomposition, we obtain logarithm functions whose branch cut is
chosen to be on the negative real axis. The choice of the branch of the
logarithm is arbitrary since we do not cross any branch cut in the definite
integration.111The author thanks Luke Corcoran for a discussion on this point.
The convergent commuting of the sum and the integral is ensured by only taking
the upper bound $\Lambda$ to infinity at the end of computations. This gives
the result

$\displaystyle I(x_{i})$
$\displaystyle=\lim_{\Lambda\rightarrow\infty}\frac{-\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\left(\ln(a_{ij}+\Lambda)-\ln(a_{ij})\right),$ (5.11)

which can be simplified by averaging over the permutation of the two indices
$i$ and $j$. The first consequence is that the divergent term cancels in both
cases. In the even-$n$ case

$\displaystyle\log(\Lambda)\sum_{i\neq j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}=0.$ (5.12)

In the odd-$n$ case, we have a vanishing leading term since

$\displaystyle\ln(\Lambda-i\frac{x_{ij}}{2})-\ln(\Lambda+i\frac{x_{ij}}{2})\xrightarrow{\Lambda\rightarrow\infty}0.$
(5.13)

Thus, we can write the result as

$\displaystyle I(x_{i})$ $\displaystyle=\frac{\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\ln\left(\frac{x_{ij}}{2i}\right),$ (5.14)

which is a real quantity for both the even case

$\displaystyle I_{even}(x_{i})$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\ln\left(x_{ij}^{2}\right),$ (5.15)

and the odd-$n$ case

$\displaystyle I_{odd}(x_{i})$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\left(\ln(a_{ij})-\ln(-a_{ij})\right)$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\left(i\pi\sum_{i>j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq
i\neq j}x_{ik}x_{jk}}-i\pi\sum_{i<j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\right)$
$\displaystyle=\frac{\pi^{2}}{2(2i)^{n-3}}\sum_{i>j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq
i\neq j}x_{ik}x_{jk}}.$ (5.16)

The correlator (2.38) follows from (5.1.1) and (5.15). This matches known
literature for the case of the four-point functions, for example;

$\displaystyle
I_{\Delta=1,n=4}=-\frac{\pi}{2}\left(\frac{\log\left(\chi_{1}\right)}{1-\chi_{1}}+\frac{\log\left(1-\chi_{1}\right)}{\chi_{1}}\right).$
(5.17)

More cases are listed in Appendix D.1.

####  5.1.2 Massive Scalar Fields

The method used in subsection 5.1.1 is compelling in the generic $n$ case but
quickly increases in complexity when $\Delta>1$. For $\Delta=2$, the result
can still be computed with this method relatively efficiently.222Another
method can be used to obtain the massive $n$-point functions from the massless
cases, as seen in 2.3.1 . The integral

$\displaystyle I_{\Delta=2}(x_{i})=\int dzz^{2n-2}\int
dx\frac{1}{\Pi_{i=1}^{n}(z^{2}+(x-x_{i})^{2})^{2}}.$ (5.18)

is evaluated by contour integration for the $x-$integral and partial fraction
decomposition for the $z-$integral. Double poles lead to the less compact
formula

$\displaystyle I_{\Delta=2,n}=\sum_{i}\sum_{j\neq
i}\frac{-\pi}{2(2i)^{2n-4}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-5}}{\prod_{k\neq
j,k\neq i}x_{kj}^{2}x_{ki}^{2}}\ln\frac{x_{ji}}{2i}\right)$
$\displaystyle+\sum_{i}\sum_{j\neq
i}\partial_{x_{i}}\frac{-\pi}{(2i)^{2n-2}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-4}}{\prod_{k\neq
j,k\neq i}x_{kj}^{2}x_{ik}^{2}}\ln\frac{x_{ji}}{2i}\right),$ (5.19)

which is derived in Appendix D.1. One expects a similar structure at higher
$\Delta$, with a double sum over the external coordinates $x_{i,j}$ and
$\partial^{2\Delta}$ derivatives and $\Delta$ terms. Some evidence of this is
the pinching presented in subsection 2.3.1 though subtleties in the order of
limits prevent a general analysis. As such, the residue method loses its
efficiency as we increase the dimension of the external operators.

####  5.1.3 An Application: Topological Correlators

We now consider non-Abelian gauge theories in AdS2 and an alternative
construction to the Witten diagram computation in Appendix A.2 of [212]. For
consistency with the notation in [212], we denote the boundary coordinate by
$t$ instead of $x$. We review the setting of [212] where the strong coupling
action is that of Yang-Mills theory in AdS2, completed with a regulating
boundary term

$\displaystyle S_{YM}$
$\displaystyle=\frac{1}{2g_{YM}^{2}}\int_{\text{AdS}_{2}}dx^{2}\sqrt{-g}\textrm{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right)$
(5.20) $\displaystyle S_{b^{y}}$
$\displaystyle=\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}-2A^{i}F_{\mu
i}n^{\mu}\right),$ (5.21)

where $\mu,\nu$ are the indices in the bulk coordinates of AdS2, $i$ those of
the boundary coordinates, and $n^{\mu}$ is a unit vector normal to the
boundary of AdS2.

In radial coordinates, the equation of motion is solved by

$\displaystyle F_{r\varphi}$ $\displaystyle=Q\sinh r$ $\displaystyle
A_{\varphi}$ $\displaystyle=Q(\cosh r-1)$ $\displaystyle A_{r}$
$\displaystyle=0,$ (5.22)

where $Q=Q_{a}T^{a}$ is an element of the Lie algebra of the theory, and in
the following, indices $a,b,a_{i}$ are those of the gauge algebra. This gives
the on-shell action

$\displaystyle\left(S_{tot}\right)_{\textrm{on-
shell}}=-2\pi\frac{\textrm{Tr}(Q^{2})}{g_{YM}^{2}}.$ (5.23)

To relate the boundary fields to the bulk fields, the variation of the bulk
action needs to be written in terms of the variation of the boundary field

$\displaystyle\delta S_{tot}$
$\displaystyle=\frac{2}{g_{YM}^{2}}\int_{\partial
B}dx\sqrt{-\gamma}\textrm{Tr}\left(A^{i}\delta a\right)$ $\displaystyle a$
$\displaystyle=\lim_{x^{\mu}\rightarrow\partial B}\left(A_{i}-F_{\mu
i}n^{\mu}\right),$ (5.24)

where $a$ is thus the corresponding boundary field and $i$ is the index
corresponding to the boundary coordinate ($t$). The on-shell action (5.23) can
be written in terms of the boundary fields $a$ through the equation

$\displaystyle a(\varphi)$
$\displaystyle=-uQu^{-1}+iu\partial_{\varphi}u^{-1}$ (5.25) $\displaystyle
u_{0}Qu_{0}^{-1}$
$\displaystyle=\frac{i}{2\pi}\log\left(P\exp\left(i\int_{0}^{2\pi}d\varphi
a(\varphi)\right)\right),$ (5.26)

where the $\varphi-$dependant large gauge transformations at the boundary are
parametrised by $u$ and $P\exp$ denotes the usual path ordered exponential.
The expression for the on-shell action is then proportional to the trace of
(5.26) squared,

$\displaystyle\textup{Tr}(Q^{2})=\textup{Tr}((u_{0}Qu_{0}^{-1})^{2})=-\frac{1}{4\pi^{2}}\Omega(a).$
(5.27)

The expression for $\Omega(a)$ is a standard result in quantum mechanics and
is solved by the Magnus expansion [213, 214]

$\displaystyle\exp\left(\Omega\right)=P\exp\left(i\int d\varphi
a(\varphi))\right).$ (5.28)

This can be used to find the dual correlators through the holographic
dictionary

$\displaystyle<j^{a}(\varphi_{1})j^{b}(\varphi_{2})>$
$\displaystyle=\frac{\delta^{ab}}{4\pi g_{YM}^{2}}$ (5.29)
$\displaystyle<j^{a}(\varphi_{1})j^{b}(\varphi_{2})j^{c}(\varphi_{3})>$
$\displaystyle=-\frac{f^{abc}\textrm{sgn}{\varphi_{12}\varphi_{23}\varphi_{31}}}{4\pi
g^{2}_{YM}}$ (5.30)
$\displaystyle<j^{a_{1}}(\varphi_{1})j^{a_{2}}(\varphi_{2})j^{a_{3}}(\varphi_{3})j^{a_{4}}(\varphi_{4})>$
$\displaystyle=-\frac{f^{aa_{1}a_{2}}f^{aa_{3}a_{4}}}{4\pi
g^{2}_{YM}}\left(\textrm{sgn}{\varphi_{12}\varphi_{24}\varphi_{43}\varphi_{31}}-\textrm{sgn}{\varphi_{21}\varphi_{14}\varphi_{43}\varphi_{32}}\right)$
$\displaystyle\quad+(2\leftrightarrow 3)+(2\leftrightarrow 4),$ (5.31)

where the indices $a,b,c,a_{i}$ are those of the gauge algebra. Through Witten
diagrams, these correlators of boundary terms can be computed explicitly using
the contour integral method detailed above. The bulk-to-boundary propagators
in Poincaré coordinates for the gauge field $A_{\mu}$ [118, 215] are

$\displaystyle G_{\mu}(z,t;t_{i})$
$\displaystyle=\frac{z^{2}+(t-t_{i})^{2}}{2\pi
z}\partial_{\mu}\left(\frac{t-t_{i}}{z^{2}+(t-t_{i})^{2}}\right),$ (5.32)

or explicitly

$\displaystyle G_{z}(z,t;t_{i})$
$\displaystyle=\frac{t_{i}-t}{\pi\left((t-t_{i})^{2}+z^{2}\right)}$
$\displaystyle G_{t}(z,t,t_{i})$
$\displaystyle=\frac{z^{2}-(t-t_{i})^{2}}{2\pi
z(t-t_{i})^{2}+z^{2}}.\vspace{3mm}$ (5.33)

The on-shell action is a pure boundary term

$\displaystyle S_{on-shell}$
$\displaystyle=\frac{1}{2g_{YM}^{2}}\int_{\text{AdS}_{2}}dx^{2}\sqrt{-g}\textrm{Tr}\left(D_{\mu}A_{\nu}F^{\mu\nu}\right)+\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}-2A^{i}F_{\mu
i}n^{\mu}\right)$
$\displaystyle=\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}+A^{i}F_{i\mu}n^{\mu}\right).\vspace{3mm}$
(5.34)

Explicitly, in the $(z,t)$ Poincaré coordinates, this gives333Note that the
vector pointing out of the boundary goes in the $-z$ direction.

$\displaystyle S_{on-shell}$ $\displaystyle=-\frac{1}{g_{YM}^{2}}\int
dtz\textrm{Tr}\left(A_{t}A_{t}-zA_{t}F_{tz}\right)|_{z=0}.$ (5.35)

The two-point correlators are given by Wick contractions acting on this term

$\displaystyle<a^{a}(t_{1})a^{b}(t_{2})>$ $\displaystyle=\lim_{z\rightarrow
0}-\frac{1}{2g_{YM}^{2}}\int
dt\delta^{ab}zG_{t}(z,t,t_{2})\left(G_{t}(z,t;t_{1})+z\partial_{[z}G_{t]}(z,t;t_{1})\right)$
(5.36) $\displaystyle=\lim_{z\rightarrow
0}\frac{(t_{1}-t_{2})^{2}\delta^{ab}}{4\pi
g_{YM}^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}$ (5.37)
$\displaystyle=\frac{\delta^{ab}}{4\pi g^{2}_{YM}}.$ (5.38)

The three-point vertex is

$\displaystyle S_{3}=-\frac{1}{g_{YM}^{2}}\int
dtdzz^{2}f_{abc}A^{a}_{z}A^{b}_{t}\partial_{[z}A^{c}_{t]},$ (5.39)

which gives a correlator

$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{2})a^{c}(t_{3})\rangle=\frac{1}{g_{YM}^{2}}\text{Perm}\left(f^{abc}I(t_{1},t_{2},t_{3})\right),$
(5.40)

where we define the single-Wick-contracted integral

$\displaystyle I(t_{1},t_{2},t_{3})=\int
dtdzz^{2}G_{z}(z,t;t_{1})G_{t}(z,t;t_{2})\partial_{[z}G_{t]}(z,t;t_{3}).$
(5.41)

The anti-symmetrised derivative removes the $t_{3}$ dependence, and the parity
of this integrand under $z\rightarrow-z$ is the same as that of the odd $n$
massless scalar case (see subsection 5.1.1), so we can evaluate both the $z$
and $t$ integrals with a complex contour.444There is a convergence issue in
$I({t_{1},t_{2},t_{3}})$, which is solved when considering the sum of the Wick
contractions. The leading term ($t^{-1}$) is always cancelled by the odd
permutation of the indices $(1,2,3)$ and the next-to-leading term is
convergent. Extending the $z$ variable to the entire real line, we have

$\displaystyle I({t_{1},t_{2},t_{3}})$
$\displaystyle=\int_{-\infty}^{\infty}dt\int_{0}^{\infty}dz\frac{(t-t_{1})\left((t-t_{2})^{2}-z^{2}\right)}{4\pi^{3}z\left((t-t_{1})^{2}+z^{2}\right)\left((t-t_{2})^{2}+z^{2}\right)}$
(5.42)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dz\,\,\textrm{sgn}(z)\frac{(t-t_{1})\left((t-t_{2})^{2}-z^{2}\right)}{4\pi^{3}z\left((t-t_{1})^{2}+z^{2}\right)\left((t-t_{2})^{2}+z^{2}\right)}.$
(5.43)

This integral has a $t_{i}$-independent contribution from the behaviour at
$t\rightarrow\infty$. However, this is cancelled by the permutation and the
antisymmetry of the structure constants. We will therefore ignore this
contribution and evaluate the integral by contour integration. The
$t-$integral evaluates to

$\displaystyle I({t_{1},t_{2},t_{3}})$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dz}{z}\frac{2z(t_{1}-t_{2})+i(t_{1}-t_{2})^{2}+4iz^{2}}{8\pi^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}\textrm{sgn}(z)^{2}$
(5.44)
$\displaystyle=\int_{-\infty}^{\infty}\frac{dz}{z}\frac{2z(t_{1}-t_{2})+i(t_{1}-t_{2})^{2}+4iz^{2}}{8\pi^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}.$
(5.45)

The factors of $\textrm{sgn}(z)$ cancel and leave an analytic function in $z$.
This integral also has a pole at 0 and at $\infty$, which can also be
cancelled using the antisymmetry of the structure constants of the algebra by
considering a combination of Wick contractions; for example,
$f^{a_{1},a_{2},a_{3}}\left(I_{t_{1},t_{2},t_{3}}-I_{t_{2},t_{1},t_{3}}\right)$.
With this in mind, the integral can be evaluated using contour integration.
The only remaining pole is at $z=\pm i\frac{t_{1}-t_{2}}{2}$ and therefore the
integral will have a factor of $\textrm{sgn}(t_{1}-t_{2})$, multiplying the
residue at that point (see Figure 5.2).

$i(t_{2}-t_{1})$$-i(t_{2}-t_{1})$$z\in\mathbb{C}$$\mathcal{C}_{\mathbb{R}}$$\mathcal{C}_{\infty}$
Figure 5.2: Contour used for the $z-$integral in equation (5.45). The origin
of the topological factor sgn$(t_{1}-t_{2})$ is clear in this setup. The
contour is closed in the UHP (the same analysis holds for the LHP closing).
The pole contained within this contour depends on the sign of $(t_{1}-t_{2})$
where, in this example, we have shown the case $t_{1}<t_{2}$.

The single-Wick-contracted integral is then

$\displaystyle I(t_{1},t_{2},t_{3})=\frac{1}{2\pi}\textrm{sgn}(t_{1}-t_{2}),$
(5.46)

and therefore

$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{2})a^{c}(t_{3})\rangle=\frac{1}{8\pi
g_{YM}^{2}}\left(\sum_{\sigma(\\{1,2,3\\})}\left(f^{a_{1}a_{2}a_{3}}\textrm{sgn}(t_{1}-t_{2})\right)\right)|_{\\{a_{1},a_{2},a_{3}\\}\rightarrow\\{a,b,c\\}}.$
(5.47)

Using the total antisymmetry of the structure constants, we have

$\displaystyle\sum_{\sigma(\\{1,2,3\\})}\left(f^{a_{1}a_{2}a_{3}}\textrm{sgn}(t_{1}-t_{2})\right)|_{\\{a_{1},a_{2},a_{3}\\}\rightarrow\\{a,b,c\\}}=-2f^{abc}\,\textrm{sgn}(t_{12}t_{23}t_{31}),$
(5.48)

which gives the final result

$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{1})a^{c}(t_{1})\rangle=-\frac{1}{4\pi
g_{YM}^{2}}f^{abc}\,\textrm{sgn}(t_{12}t_{23}t_{31}).$ (5.49)

This agrees with equation (5.30) through a simple change of coordinates.555For
higher-point functions, there is the subtlety that the boundary field $a$ is
not the boundary limit of the gauge field $A_{\mu}$, but rather has a
dependence on both $A_{\mu}$ and $F_{\mu\nu}$. This implies that the bulk-to-
boundary propagator receives corrections from multi-source terms. These
questions are addressed in [212].

####  5.1.4 A Basis of Interaction Terms

This subsection considers deformations from generalized free field theory
produced by effective interactions in a bulk AdS2 field theory. In this
holographic AdS2/CFT1 setup, the background AdS2 metric is not dynamical,
corresponding to the absence of a stress tensor in the boundary CFT1.
According to the usual dictionary, a massive free scalar field $\Phi$ in AdS2
is dual to a boundary 1d generalised free field $\phi$. We deform this theory
with quartic self-interactions with an arbitrary number $L$ of derivatives

$\displaystyle S=\int
dxdz\,\sqrt{g}\,\big{[}\,g^{\mu\nu}\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi+m^{2}_{\Delta_{\phi}}\Phi^{2}+g_{L}\,(\partial^{L}\Phi)^{4}\,\big{]}\,,\qquad
L=0,1,\dots$ (5.50)

where we use the AdS2 metric in Poincaré coordinates
$ds^{2}=\frac{1}{z^{2}}(dx^{2}+dz^{2})$. The mass
$m^{2}_{\Delta_{\phi}}=\Delta_{\phi}(\Delta_{\phi}-1)$ is fixed in units of
the AdS radius so that $\Delta_{\phi}$ is the dimension (independent of
$g_{L}$) of the field $\Phi$ evaluated at the boundary, $\phi(x)$.666When we
introduce an interaction, such as (5.50), there will be Witten diagrams
contributing to the mass renormalization of $\Phi$. We can always choose the
bare mass so that the dictionary is preserved and $\Delta_{\phi}$ is not
modified. We will limit our analysis to tree-level correlators and thus
consider only contact diagrams, whose building blocks are the $D$-functions
[194, 56, 97] reviewed in Appendix D.1. The writing $(\partial^{L}\Phi)^{4}$
above is symbolic, denoting a complete and independent set of quartic vertices
with four fields and up to $4L$ derivatives.777The fact that a complete and
independent basis of vertices is labelled by 1/4 the number of derivatives can
be seen using integration by parts and the equations of motion, or by noticing
that the counting of physically distinct four-point interactions is equivalent
to the counting of crossing-symmetric polynomial $S$-matrices in 2D Minkowski
space (see discussion in [55, 49]). In the following, we will present a
particularly convenient basis for these interactions, allowing us to derive a
closed-form expression for the tree-level correlator in Mellin space. Consider
the interaction Lagrangian

$\mathcal{L}_{L}=g_{L}\left[\prod_{k=0}^{L-1}\left(\tfrac{1}{2}\partial_{\mu}\partial^{\mu}-(\Delta_{\phi}+k)(2(\Delta_{\phi}+k)-1)\right)\Phi^{2}\right]^{2}\,.$
(5.51)

This looks like a very complicated term, but it contains four fields $\Phi$
and $4L$ derivatives, so by the argument above, it must be effectively a
linear combination of operators like $(\partial^{\ell}\Phi)^{4}$ for $\ell\leq
L$. The advantage of this interaction is that the corresponding correlator
computed via Witten diagrams reads

$\displaystyle
f^{(1)}_{L}(z)=\frac{4^{L-1}\pi^{-\frac{3}{2}}\Gamma(2\Delta_{\phi}-\frac{1}{2}+2L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})^{4}}z^{2\Delta_{\phi}}\,(1+z^{2L}+(1-z)^{2L})\bar{D}_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(z)$
(5.52)

where the $\bar{D}$-functions are listed in Appendix D.1. If one starts with
some specific $4L$-derivative interaction, such as $(\partial^{L}\Phi)^{4}$,
the explicit computation through Witten diagrams shows several other
combinations of $D$-functions with different weights. Nevertheless, by the
argument above, these results cannot be independent of those obtained using
$\mathcal{L}_{L}$ and therefore, the result must be expressible as a linear
combination $\sum_{\ell}a_{\ell}f^{(1)}_{\ell}(z)$. This requires a series of
non-trivial identities among $\bar{D}$ functions, some of which are derived in
[2].

Here we consider the interaction Lagrangian (5.51) and show that it leads to
the correlator (5.52) using Witten diagrams. The result of the Wick
contractions is

$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}=\\\
\sum_{\text{perms}}g_{L}\int\frac{dxdz}{z^{2}}\mathcal{D}\left(K_{\Delta_{\phi}}(x,z;x_{1})K_{\Delta_{\phi}}(x,z;x_{2})\right)\mathcal{D}\left(K_{\Delta_{\phi}}(x,z;x_{3})K_{\Delta_{\phi}}(x,z;x_{4})\right)\,,$
(5.53)

where we defined

$\displaystyle\mathcal{D}=\prod_{k=0}^{L-1}\left(\frac{1}{2}\partial_{\mu}\partial^{\mu}-(\Delta_{\phi}+k)(2(\Delta_{\phi}+k)-1)\right)$
(5.54)

acting on the bulk point, and we used the bulk-to-boundary propagator with the
conventions of (2.31). Using the identity recursively

$\displaystyle-2x_{ij}^{2}\Delta^{2}\tilde{K}_{\Delta+1}(x,z;x_{i})\tilde{K}_{\Delta+1}(x,z;x_{j})=(\tfrac{1}{2}\partial_{\mu}\partial^{\mu}-\Delta(2\Delta-1))(\tilde{K}_{\Delta}(x,z;x_{i})\tilde{K}_{\Delta}(x,z;x_{j}))\,,$
(5.55)

which can be derived from (D.34), we obtain

$\displaystyle\mathcal{D}\left(\tilde{K}_{\Delta_{\phi}}(x,z;x_{1})\tilde{K}_{\Delta_{\phi}}(x,z;x_{2})\right)=(-2x_{12}^{2})^{L}\left(\frac{\Gamma(\Delta_{\phi}+L)}{\Gamma(\Delta_{\phi})}\right)^{2}\tilde{K}_{\Delta_{\phi}+L}(x,z;x_{1})\tilde{K}_{\Delta_{\phi}+L}(x,z;x_{2})\,.$
(5.56)

Inserting this into equation (5.53), summing over the permutations and
remembering the definition of $\mathcal{C}_{\Delta}$ in (2.31), we get

$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}_{L}=g_{L}[(x^{2}_{13}x^{2}_{24})^{L}+(x^{2}_{12}x^{2}_{34})^{L}+(x^{2}_{14}x^{2}_{23})^{L}]\times\\\
2^{2L-1}\pi^{-2}\left(\frac{\Gamma(\Delta_{\phi}+L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})}\right)^{4}D_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(x_{1},x_{2},x_{3},x_{4})\,.$
(5.57)

Using (D.35) we immediately get

$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}_{L}=g_{L}(1+\chi^{2L}+(1-\chi)^{2L})\times\\\
\frac{4^{L-1}\pi^{-\frac{3}{2}}\,\Gamma(2\Delta_{\phi}-\frac{1}{2}+2L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})^{4}\,(x^{2}_{13}\,x^{2}_{24})^{\Delta_{\phi}}}\bar{D}_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(\chi)$
(5.58)

in perfect agreement with (5.52).

###  5.2 Mellin Amplitudes

In the higher-dimensional case, the Mellin representation of conformal
correlators [64, 109] has proven to be an excellent tool, especially for the
study of holographic CFTs [65, 108, 120]. The counting of independent cross-
ratios for an $n$-point correlation function of local operators in a
$d$-dimensional CFT is identical to that of independent variables scattering
in $d+1$ dimensions. The Mellin representation, or Mellin amplitude, makes
this correspondence manifest, expressing the correlators in a form that is the
natural AdS counterpart of flat-space scattering amplitudes. This construction
has several nice features. First, the Mellin amplitude has simple poles
located at the values of the twist of exchanged operators (there are, however,
infinitely many accumulation points of such poles). Secondly, the crossing
symmetry of the correlator maps to the amplitude crossing symmetry. Finally,
the language of Mellin amplitudes is particularly suitable for large $N$ gauge
theories, where perturbation theory is described in terms of Witten diagrams.
An extensive introduction to higher dimensional Mellin amplitudes is presented
in Appendix E.1. In the following, we will use these properties as guiding
principles for the definition of Mellin amplitudes for 1d CFTs.

####  5.2.1 Nonperturbative Mellin Amplitude

Let $f(t)$ be a function describing a four-point correlator in a 1d CFT where
the cross ratio $t$ is defined as

$\displaystyle t=\frac{\chi}{1-\chi}.$ (5.59)

We define the one-parameter Mellin transform $\mathcal{M}_{a}[f(t)]$ as

$\displaystyle\mathcal{M}_{a}[f(t)]==\int_{0}^{\infty}dt\,\left(\frac{t}{1+t}\right)^{a}f(t)\,t^{-1-s}\,,$
(5.60)

and Mellin amplitude $M_{a=0}(s)$ as

$\displaystyle
M(s)=\frac{1}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\int_{0}^{\infty}dt\,f(t)\,t^{-1-s}\,$
(5.61)

which we multiply by an overall factor for future convenience. In this case,
the crossing relation becomes

$\displaystyle f(t)$ $\displaystyle=t^{2\Delta_{\phi}}f(\frac{1}{t}),$
$\displaystyle M(s)$ $\displaystyle=M(2\Delta_{\phi}-s)\,.$ (5.62)

The goal of our discussion is to infer the analytic properties of the Mellin
amplitude $M(s)$ from the physical requirements on the correlator $f(t)$.
First, following [216], we recall a general result for the one-dimensional
Mellin transform (5.61).

##### A Theorem

Consider a function $F(t)$ in the vector space $\mathcal{F}_{H}^{\Theta}$ of
complex valued functions that are holomorphic for $\textrm{arg}(t)\in\Theta$
and obey

$|F(t)|\leq\frac{C(h)}{|t|^{h}}\qquad h\in H\,,$ (5.63)

where H is a subset of $\mathbb{R}$, typically of the form
$H=(h_{min},h_{max})$. Consider also the function $\hat{M}(s)$ in the vector
space $\mathcal{M}_{H}^{\Theta}$ of complex valued functions that are
holomorphic for $\text{Re}(s)\in H$ and exponentially suppressed in the limit
$|\text{Im}(s)|\rightarrow\infty$

$\displaystyle|\hat{M}(s)|\leq
K(\textrm{Re}(s))e^{-|\text{Im}(s)\sup_{\Theta}\textrm{arg}(t)|}\qquad|\text{Im}(s)|\to\infty\,.$
(5.64)

These two vector spaces exist independently but the following theorem holds

Theorem:

Given a function $F(t)\in\mathcal{F}_{H}^{\Theta}$, its Mellin transform
$\mathcal{M}[F](s)$ exists and $\mathcal{M}[F](s)\in\mathcal{M}_{H}^{\Theta}$.
Furthermore, $\mathcal{M}^{-1}\mathcal{M}[F](t)=F(t)$ for any
$\arg(t)\in\Theta$. Conversely, given $\hat{M}(s)\in\mathcal{M}_{H}^{\Theta}$
its inverse Mellin transform exists and
$\mathcal{M}^{-1}[\hat{M}](s)\in\mathcal{F}_{H}^{\Theta}$.
Furthermore, $\mathcal{M}\mathcal{M}^{-1}[\hat{M}](s)=\hat{M}(s)$ for any
$s\in H+i\mathbb{R}$.

This is a classical result for the one-dimensional Mellin transform, so we
will not prove it here. Instead, we will discuss how the physical 1d
correlator violates the hypothesis of the theorem and how we can overcome this
issue. The convergence of the $s$-channel OPE for $|\arg(t)|<\pi$ ensures that
the function $f(t)$ is indeed analytic in a sectorial domain $\Theta$.
Nevertheless, the condition (5.63) is violated in two ways :

  * •

When light operators ($\Delta<\Delta_{\phi}$) are exchanged in the OPE, the
region $H$ is not well defined and the Mellin transform does not exist. This
issue is analogous to the higher dimensional case of [216] and we will solve
it by implementing a finite number of subtractions in subsection 5.2.2.

  * •

The correlator $f(t)$ is not bounded for $t\to e^{i\pi}$ where it has a
singularity controlled by the Regge limit (5.86). This issue does not spoil
the existence of the Mellin transform, but it gives a result that is not
bounded by (5.64).

To understand this second point, let us present a simple example which will be
useful to explain the issue. Consider the function
$F(t)=(\frac{t}{1+t})^{2\Delta_{\phi}}$. This function is analytic for
$|\arg(t)|<\pi$ and gives a convergent integral (5.61) for
$0<\text{Re}(s)<2\Delta_{\phi}$. However, although the bound (5.63) holds
along the real axis, it is violated for $t\to e^{i\pi}$. The Mellin transform
of this function reads

$\int_{0}^{\infty}dt\left(\frac{t}{1+t}\right)^{2\Delta_{\phi}}t^{-1-s}=\frac{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}{\Gamma(2\Delta_{\phi})}\,.$
(5.65)

From this explicit expression we see immediately that for
$|\text{Im}(s)|\to\infty$ the r.h.s. is not bounded by
$e^{-\pi|\text{Im}(s)|}$. It is bounded by

$\frac{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}{\Gamma(2\Delta_{\phi})}\leq
K(\textrm{Re}(s))|\text{Im}(s)|^{2\Delta_{\phi}-1}e^{-\pi|\text{Im}(s)|}\qquad|\text{Im}(s)|\to\infty\,.$
(5.66)

The exponential decay is correctly predicted by the theorem, while the
additional polynomial divergence can be related to the behaviour of the
function $F(t)$ for $t\to e^{i\pi}$. In section 5.2.3, we will show that this
is a specific instance of a general relation between the large $s$ asymptotics
of $M(s)$ to the Regge limit of $f(t)$.

####  5.2.2 Convergence and Subtractions

Let us now discuss the convergence of the integral (5.61). Let $f(t)$ be well-
behaved for $t\in\mathbb{R}^{+}$; we do not want divergences in $t$ other than
at $t=0$ and $t\rightarrow\infty$. This behaviour coincides with the CFT1
correlators we are interested in. Consider the behaviour of $f(t)$ close to
$t=0$. Using the conformal block expansion (1.22), we find that the leading
power is $f(t)\sim t^{\Delta_{0}}$ where $\Delta_{0}$ is the dimension of the
lightest exchanged operator. Analogously, using the crossing symmetry relation
(5.62), we find that the large $t$ behaviour of $f(t)$ is $f(t)\sim
t^{2\Delta_{\phi}-\Delta_{0}}$. Therefore the integral converges in the strip

$\displaystyle 2\Delta_{\phi}-\Delta_{0}<\text{Re}(s)<\Delta_{0}\,,$ (5.67)

which is a well-defined interval _only for $\Delta_{0}>\Delta_{\phi}$_. To
give a nonperturbative definition of the Mellin transform, which allows for
lighter operators to be exchanged, we need to perform some subtractions along
the lines of [216].888See in particular Appendix B in [216] for the one-
dimensional case. One obvious example is GFF, where the identity operator is
exchanged. We will consider this case explicitly in Appendix E.2. For the
moment, we consider the Mellin transform of the connected part of the
correlator. Let us consider the following subtractions

$\displaystyle f_{0}(t)$
$\displaystyle=f_{\text{conn}}(t)-\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}t^{\Delta+k}\,,$
(5.68) $\displaystyle f_{\infty}(t)$
$\displaystyle=f_{\text{conn}}(t)-\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}t^{2\Delta_{\phi}-\Delta-k}\,,$
(5.69)

where, for convenience, we write
$c_{\Delta_{\phi}\Delta_{\phi}\Delta}^{2}\equiv c_{\Delta}$. For the function
$f_{0}(t)$ we subtracted the $s$-channel contribution of all the operators
(primaries and descendants) with scaling dimension below the threshold
$\Delta=\Delta_{\phi}$, making use of the series expansion of the
hypergeometric function in the conformal block. This improves the behaviour of
the function at $t=0$. On the other hand, for $f_{\infty}(t)$ we subtracted
all the $t$-channel operators below the threshold, thus improving the
behaviour at $t=\infty$. The idea is to split the integral (5.61) into two
parts, which are defined on (possibly non-overlapping) semi-infinite regions
of the complex $s$ plane

$\displaystyle\psi_{0}(s)$
$\displaystyle=\int_{0}^{1}dt\,f_{\text{conn}}(t)\,t^{-1-s}$
$\displaystyle\text{Re}(s)$ $\displaystyle<\Delta_{0}\,,$ (5.70)
$\displaystyle\psi_{\infty}(s)$
$\displaystyle=\int_{1}^{\infty}dt\,f_{\text{conn}}(t)\,t^{-1-s}$
$\displaystyle\text{Re}(s)$ $\displaystyle>2\Delta_{\phi}-\Delta_{0}.$ (5.71)

When the two regions do not overlap, we analytically continue $\psi_{0}(s)$
and $\psi_{\infty}(s)$ by considering the integrals of the functions (5.68)
and (5.69) and adding a finite number of poles

$\displaystyle\\!\\!\\!\\!\psi_{0}(s)$
$\displaystyle=\int_{0}^{1}\\!\\!\\!dt\,f_{0}(t)\,t^{-1-s}+\\!\\!\\!\\!\\!\\!\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\\!\\!\\!\\!\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\tfrac{(-1)^{k}}{k!}\tfrac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\tfrac{1}{s-\Delta-k}\,,\qquad\small{\text{Re}(s)<\tilde{\Delta}_{0}}\,,$
(5.72) $\displaystyle\\!\\!\\!\\!\psi_{\infty}(s)$
$\displaystyle=\int_{1}^{\infty}\\!\\!\\!\\!\\!\\!dt\,f_{\infty}(t)\,t^{-1-s}+\\!\\!\\!\\!\\!\\!\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\\!\\!\\!\\!\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\tfrac{(-1)^{k}}{k!}\tfrac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\tfrac{1}{s-2\Delta_{\phi}+\Delta+k}\,,\small{\text{Re}(s)>2\Delta_{\phi}-\tilde{\Delta}_{0}}\,,$
(5.73)

where $\tilde{\Delta}_{0}>\Delta_{\phi}$ is the lightest exchanged operator
above the threshold (notice that this operator could be either a primary or a
descendant). Both these functions are now well-defined on the non-vanishing
strip $2\Delta_{\phi}-\tilde{\Delta}_{0}<\text{Re(s)}<\tilde{\Delta}_{0}$ and
therefore their sum yields a well-defined Mellin transform

$\displaystyle M(s)$
$\displaystyle=\frac{\psi_{0}(s)+\psi_{\infty}(s)}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\,,$
$\displaystyle 2\Delta_{\phi}-\tilde{\Delta}_{0}$
$\displaystyle<\text{Re(s)}<\tilde{\Delta}_{0}\,.$ (5.74)

The price to pay is a deformation of the integration contour in the inverse
Mellin transform, which reads

$\displaystyle f(t)=\int_{\mathcal{C}}\frac{ds}{2\pi
i}\,\Gamma(s)\Gamma(2\Delta_{\phi}-s)\,M(s)\,t^{s}\,.$ (5.75)

To understand the form of the contour $\mathcal{C}$, we need to discuss the
analytic structure of $M(s)$. One can follow the above-mentioned strategy to
extend the definition (5.74) to the whole complex $s$ plane. To analytically
continue $\psi_{0}(s)$ from the region $\text{Re}(s)<\Delta_{0}$ to the region
$\text{Re}(s)<\tilde{\Delta}_{0}$ we subtracted a few exchanged operators in
$f(t)$ and added a finite number of poles in (5.72). By adding more and more
poles, we can further extend the area of analyticity. We then conclude that
the Mellin block expansion defined by

$M(s)=\frac{\psi_{0}(s)+\psi_{\infty}(s)}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}$
(5.76)

with

$\displaystyle\psi_{0}(s)$
$\displaystyle=\sum_{\Delta}\sum_{k=0}^{\infty}c_{\Delta}\frac{(-1)^{k+1}\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{k!\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\frac{1}{s-\Delta-k}\,,$
(5.77) $\displaystyle\psi_{\infty}(s)$
$\displaystyle=\sum_{\Delta}\sum_{k=0}^{\infty}c_{\Delta}\frac{(-1)^{k}\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{k!\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\frac{1}{s-2\Delta_{\phi}+\Delta+k}$
(5.78)

provides a representation of $M(s)$, which is valid on the whole complex $s$
plane (excluding the point at infinity, which will be discussed in detail in
subsection 5.2.3). In particular, the expression (5.76) immediately allows us
to read off the position of the poles of $M(s)$.999In principle, there could
be an additional singularity at $\infty$, but we postpone this discussion to
section 5.2.4. For any exchanged primary operator of dimension $\Delta$, two
infinite sequences of poles run to the right of $s=\Delta$ and to the left of
$s=2\Delta_{\phi}-\Delta$. Following the common nomenclature, we denote them
as

$\displaystyle\text{\emph{right} poles}:s_{R}$
$\displaystyle=\Delta+k\,,~{}~{}\qquad\,\,\qquad k=0,1,2,\dots$ (5.79)
$\displaystyle\text{\emph{left} poles}:s_{L}$
$\displaystyle=2\Delta_{\phi}-\Delta-k\,,\qquad k=0,1,2,\dots$ (5.80)
$\displaystyle\text{Res}[M(s)]|_{s_{L}}$
$\displaystyle\equiv-\text{Res}[M(s)]|_{s_{R}}=\frac{(-1)^{k}\Gamma(2\Delta)\Gamma(\Delta+k)}{k!\,\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)}\,.$
(5.81)

Notice that the precise identification of the sum over $k$ in (5.77) with the
sum over descendants in the block expansion is a consequence of the choice
$a=0$ in (5.60). The different choices of $a$ in (5.60) would lead to a less
transparent correspondence between poles and conformal descendants.

Given this structure of poles, we can now precisely define the contour
$\mathcal{C}$ in (5.75). The contour $\mathcal{C}$ is chosen in such a way as
to leave all the _right_ poles of $M(s)$ on its right and all the _left_ poles
on its left. Suppose the lightest exchanged operator has dimension
$\Delta_{0}>\Delta_{\phi}$. In that case, no analytic continuation is required
in (5.70) and (5.71) (in other words, the set of left and right poles do not
overlap) and any contour within the interval (5.67) will suffice, see the
straight one on the left in Figure 5.3.

When lighter operators are exchanged, the contour needs to be deformed because
the set of right poles intersects with the set of left poles. Figure 5.3 shows
an example with a single operator below the threshold. It is clear from the
picture that a more complicated situation arises when a left and a right pole
coincide. For instance, this happens in the GFF case, which we address in
Appendix E.2. More generally, this happens whenever there is an exchanged
operator with dimension $\Delta=\Delta_{\phi}+\frac{\mathbb{Z}}{2}$. We do not
expect this to be the case in a generic spectrum.

$s$$\mathcal{C}$$\Delta_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\Delta_{0}$

$s$$\mathcal{C}$$\Delta_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\Delta_{0}$$\tilde{\Delta}_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\tilde{\Delta}_{0}$

Figure 5.3: Left: The contour for the inverse Mellin transform when
$\Delta_{0}>\Delta_{\phi}$. Left poles are marked in red and right poles are
green. Right: When $\Delta_{0}<\Delta_{\phi}$ left and right poles intersect,
the contour must be deformed.

We conclude this section by noticing that we can perform the sum over $k$ in
(5.76), resumming all the conformal descendants in a crossing symmetric Mellin
block expansion

$\displaystyle M(s)$
$\displaystyle=\frac{1}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\sum_{\Delta}\,c_{\Delta}[\mathcal{F}_{\Delta}(s)+\mathcal{F}_{\Delta}(2\Delta_{\phi}-s)]\,,$
(5.82) $\displaystyle\mathcal{F}_{\Delta}(s)$
$\displaystyle=\frac{{}_{3}F_{2}({\Delta,\Delta,\Delta-s};{2\Delta,1+\Delta-s};-1)}{\Delta-s}\,.$
(5.83)

It is worth noting the importance of these finite subtractions in the context
of generalized free field theory. This simple yet subtle example is shown in
Appendix E.2.

####  5.2.3 Regge Limit and Mellin Boundedness

In this subsection, a bound on the large $s$ behaviour of the Mellin amplitude
$M(s)$ will be derived using the Regge behaviour of the function
$f(t)$.101010In terms of the cross-ratio $t$, this corresponds to the limit
$t\to e^{i\pi}$ described in (5.86). The Regge limit is 111111This limit can
be taken along any direction excluding the real line to avoid the branch cuts,
but for definiteness we take it along the imaginary axis.

$\displaystyle z\to\frac{1}{2}+i\infty.$ (5.84)

This limit can be understood in terms of the higher-dimensional correlator in
the diagonal limit, where it corresponds to the $u$-channel Regge
limit.121212In 1d there is no $u$-channel OPE expansion as it is impossible to
bring $x_{1}$ close to $x_{3}$ without $x_{2}$ in-between. However, one can
resort on the higher dimensional picture to understand that while the
$u$-channel OPE would correspond to $z\to i\infty$ and $\bar{z}\to-i\infty$,
the $u$-channel Regge limit is $z,\bar{z}\to i\infty$. In particular, four-
point functions of a unitary CFT are bounded in the Regge limit [217, 218],
and we have [55]

$\displaystyle\left|\tilde{g}\left(\textstyle{\frac{1}{2}}+iT\right)\right|\,\text{is
bounded as }T\rightarrow\infty.$ (5.85)

Translating into the $t$ cross-ratio (5.59), the line parametrized by
$z=\frac{1}{2}+i\,\xi$ is mapped into the unit circle $t=e^{i\theta}$ for
$\theta\in(-\pi,\pi)$ and the Regge limit occurs when $\theta\to\pi$. The
Regge boundedness condition (5.85) for the function $f(t)$ then reads

$f(e^{i\theta})=\mathcal{O}\left((\pi-\theta)^{-2\Delta_{\phi}}\right)\qquad\theta\to\pi.$
(5.86)

Looking at the direct definition of the Mellin transform (5.61), it may seem
surprising that the large $s$ behaviour is controlled by a region ($t\sim-1$)
which is far away from the integration contour. We must start by considering
the inverse Mellin transform (5.75) where the contour $\mathcal{C}$ is a
straight line parametrized by $s=c+i\eta$ for some constant
$2\Delta-\tilde{\Delta}_{0}<c<\tilde{\Delta}_{0}$ (the additional poles that
are included in (5.72) and (5.73) for the analytic continuation will not
affect this argument) and $\eta\in\mathbb{R}$. We take $t=e^{i\theta}$ and we
integrate over $\eta$

$f(e^{i\theta})=e^{ic\theta}\int_{-\infty}^{\infty}d\eta\,\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\,M(c+i\eta)\,e^{-\theta\eta}\,.$
(5.87)

We are interested in the behaviour of the integrand for $|\eta|\to\infty$. In
this limit

$\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\sim
e^{-\pi|\eta|}\,\eta^{2\Delta_{\phi}-1}\qquad|\eta|\to\infty\,.$ (5.88)

This means that the Gamma function prefactor accounts for the exponential
behaviour (5.64) of $\hat{M}(s)$ for $|\text{Im}(s)|\to\infty$ predicted by
the theorem in subsection 5.2.1. This essentially motivates our choice of
prefactor in (5.61). In particular, the exponential in (5.88) combined with
that in (5.87) shows that the regime $\theta\to\pm\pi$ is controlled by the
region $\eta\sim\mp\infty$. Let us make this more precise by defining

$\displaystyle H(\eta)$
$\displaystyle\equiv\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\,M(c+i\eta)\,e^{\pi|\eta|}$
(5.89)

so that the integral (5.87) can be rewritten as

$f(e^{i\theta})=e^{ic\theta}\int_{0}^{\infty}d\eta\,H(-\eta)\,e^{-\eta(\pi-\theta)}+e^{ic\theta}\int_{0}^{\infty}d\eta\,H(\eta)\,e^{-\eta(\pi+\theta)}\,,$
(5.90)

where we recognize two Laplace transforms of the functions $H(\pm\eta)$. A
singular behaviour for $\theta=\pi$ originates from the first term in the sum
(5.90), while the singularity at $\theta=-\pi$ arises from the second term.
More specifically, Tauberian theorems for the Laplace transform imply that for
a function $H(\eta)\sim k\,\eta^{\alpha}$ as $\eta\to\infty$ then

$\int_{0}^{\infty}d\eta\,H(\eta)\,e^{-\eta(\pi+\theta)}\sim
k\,\Gamma(\alpha+1)(\theta+\pi)^{-\alpha-1}\qquad\theta\to-\pi$ (5.91)

and similarly for the case $\theta\to\pi$. We are then led to the conclusion
that the Regge behaviour (5.86) is reproduced by asking that

$H(\eta)\sim|\eta|^{2\Delta_{\phi}-1}\qquad|\eta|\to\infty\,.$ (5.92)

Combining this with (5.89) and (5.88) we conclude that

$M(c+i\eta)=\mathcal{O}(|\eta|^{0})\qquad|\eta|\to\infty\,.$ (5.93)

Assuming that no Stokes phenomenon occurs for physical correlators, we can
extend this behaviour for any $\text{arg}(s)$ such that

$M(s)=\mathcal{O}(|s|^{0})\qquad|s|\to\infty\,.$ (5.94)

The absence of Stokes phenomenon is an assumption for which we do not have
proof. This assumption, however, is verified in all our examples and was also
made in the higher-dimensional case [216].

We conclude this subsection with an important remark about the perturbative
regime, which we will consider in subsection 5.2.5. The result (5.94) is valid
for the full non-perturbative Mellin amplitude. If the correlator contains a
small parameter, it is often the case that, order-by-order in the perturbative
expansion, the Regge behaviour is worse than in the full non-perturbative
correlator.131313A typical example of this phenomenon is the function
$\frac{1}{1-gz}$, which is regular for $z\to\infty$ but its expansion at small
$g$ is more and more divergent. In Appendix E.3, we illustrate this in detail
in the context of the analytic sum rules discussed in the next section. Given
this aspect, it is thereby useful to formulate our result in a more general
form. Let us consider a correlator $\tilde{f}(z)$ with a Regge behaviour

$\tilde{f}(z)=\mathcal{O}(z^{2\Delta_{\phi}+n})\qquad z\to\frac{1}{2}+i\infty$
(5.95)

for some positive integer $n$, then the associated Mellin amplitude will have
a large $s$ asymptotics

$M(s)=\mathcal{O}(|s|^{n})\qquad|s|\to\infty\,.$ (5.96)

####  5.2.4 Sum Rules

A common way to express the well-known fact that an arbitrary set of CFT data
does not necessarily lead to a consistent CFT is through a set of sum rules
for the CFT data. In the following, we will start with our Mellin amplitude
definition and derive an infinite set of sum rules. As we mentioned in the
Introduction, these sum rules are not dispersive, according to the definition
of [122]. This is related to the behaviour at infinity obtained using our one-
dimensional definition. In subsection 5.2.3, we described how the product of
Gamma functions in our definition (5.61) leads to a nice behaviour for the
Mellin amplitude $M(s)$ at $s=\infty$. However, introducing that prefactor
also leads to the appearance of spurious poles in the integral (5.75). In a
generic CFT, it is not expected that operators with the exact dimension
$s=2\Delta_{\phi}+n$ are present in the spectrum. Thus, the poles of the Gamma
functions must be compensated by zeros in the Mellin amplitude. This strategy
was used in [216, 219] to derive dispersive sum rules for the higher
dimensional case, where the Mellin amplitude needs to have double zeros. Here,
we will use the same idea to derive a new set of sum rules characterised by
single zeros of the Mellin amplitude. This makes these sum rules different and
less powerful than the dispersive ones. Still, we believe that their
derivation and the check of their validity on a set of known examples provide
an important consistency check of our results.

One may be concerned because the presence of single or double zeroes for the
Mellin amplitude seems to be related to the choice of the prefactor in (5.61).
This is not the case. The choice to factor out a prefactor in (5.61) is
related to having a polynomial behaviour for the function $M(s)$ as
$s\to\infty$. If we were to pick a different prefactor (for instance, using
Gamma function squared, leading to double poles for the Mellin amplitude), the
Mellin amplitude would contain an essential singularity at $s=\infty$, and
this divergence would have to be compensated by the function $F_{p}(s)$, which
we will use in (5.100) to derive our sum rules. We can safely conclude that
choosing a prefactor is a convenient trick, but it does not affect the
resulting sum rules.

Finally, let us emphasize some important differences compared to the higher-
dimensional strategy of the Mellin Polyakov bootstrap [54, 52, 53]. The
derivation of the non-perturbative Polyakov consistency conditions used in
[216, 219] is quite subtle due to accumulation points in the twist spectrum of
higher-dimensional CFTs. In our case, however, the situation is simpler. The
twist accumulation points are related to the presence of a spin or,
equivalently, to the need of introducing two Mandelstam variables. For us
there is no spin and the only quantum number is the scaling dimension of the
operators. Therefore, we do not expect any accumulation point in the spectrum
and we will be able to impose the conditions (5.98) without recurring to any
analytic continuations.

We start by summarizing the main properties of the Mellin amplitude $M(s)$ in
(5.61):

  * •

$M$ is crossing symmetric

$M(s)=M(2\Delta_{\phi}-s)\,.$ (5.97)

  * •

$M$ has poles at the location of the physical exchanged operators in the two
channels, i.e. $s=\Delta+k$ and $s=2\Delta_{\phi}-\Delta-k$ for
$k\in\mathbb{N}\,.$

  * •

Generically, $M$ has single zeros compensating the poles of the prefactor

$\displaystyle M(2\Delta_{\phi}+k)=0\quad\text{and}\quad
M(-k)=0\quad\text{for}\quad k\in\mathbb{N}\,.$ (5.98)

Some of these zeros might be absent if the spectrum contains protected
operators.

  * •

$M$ is bounded for $|s|\to\infty$, see (5.94) .

  * •

$M$ admits a crossing-symmetric Mellin block expansion

$\displaystyle M(s)=\sum_{\Delta}\,c_{\Delta}\,M_{\Delta}(s)$ (5.99)

with $M_{\Delta}(s)$ given by the comparison with (5.82).

The properties above will allow us to define a set of sum rules along the
lines of [216, 219]. Let $\omega_{p}$ be the functional

$\displaystyle\omega_{p_{i}}=\oint_{\mathbb{C}|_{\infty}}\frac{ds}{2\pi
i}M(s)F_{p_{i}}(s)\,,$ (5.100)

where the contour here is a very large circle around infinity. When
$F_{p_{i}}(s)$ is a sufficiently suppressed function at $s\to\infty$, we can
take the limit of infinite radius for the circle, and we get

$\displaystyle\omega_{p_{i}}[M]=0\,.$ (5.101)

For a nonperturbative Mellin amplitude characterised by the asymptotic
behaviour (5.94) it is sufficient to ask that $F_{p_{i}}(s)\sim
s^{-1-\epsilon}$ for $\epsilon>0$ as $|s|\to\infty$. As we mentioned at the
end of subsection 5.2.3, when considering a perturbative expansion around GFF,
the Regge behaviour may worsen and a sufficiently suppressed function $F$
would be required as detailed in Appendix E.3.

The strategy to derive the sum rules consists of deforming the integration
contour in (5.100) to include all the poles of the integrand such that

$\displaystyle\omega_{p_{i}}=\sum_{s^{*}}\text{Res}_{s=s^{*}}\left[M(s)\right]F_{p_{i}}(s^{*})+\sum_{s^{**}}M(s^{**})\text{Res}_{s=s^{**}}\left[F_{p_{i}}(s)\right]=0\,.$
(5.102)

This equation already looks like a sum rule, but it depends on the value
$M(s^{**})$ of the Mellin amplitude at the poles of $F_{p_{i}}(s)$. To avoid
this issue, one can choose $F_{p_{i}}(s)$ to have simple poles at the position
of the zeros of $M(s)$. Therefore, we need a function $F_{p_{i}}(s)$ with
poles at $s=-k$ or at $s=2\Delta_{\phi}+k$. Furthermore, the function
$F_{p_{i}}(s)$ must not be crossing symmetric. Indeed, using the position of
the poles in (5.79) and (5.80) and crossing symmetry for the residues (5.81)
we get

$\displaystyle\omega_{p_{i}}=\sum_{s_{R}}\text{Res}_{s=s_{R}}(M(s))(F_{p_{i}}(s_{R})-F_{p_{i}}(2\Delta_{\phi}-s_{R}))\,,$
(5.103)

so that any crossing symmetric function $F$ would lead to a trivial vanishing
of $\omega_{p_{i}}$. Using the explicit expression for the residues (5.81), we
find the set of sum rules

$\displaystyle\sum_{\Delta,k}c_{\Delta}\frac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)}(F_{p_{i}}(\Delta+k)-F_{p_{i}}(2\Delta_{\phi}-\Delta-k))=0\,.$
(5.104)

A natural choice for the function $F$ is

$\displaystyle F_{p_{1},p_{2}}(s)=\frac{1}{(s+p_{1})(s+p_{2})}\,,\qquad
p_{1},p_{2}\in\mathbb{N}\,.$ (5.105)

Notice that, despite the function $F_{p_{1},p_{2}}(s)\sim\frac{1}{s^{2}}$ for
$s\to\infty$, thanks to (5.103) only the crossing antisymmetric part of it
matters, i.e. $F_{p_{1},p_{2}}(s)-F_{p_{1},p_{2}}(2\Delta-s)$, and one can
easily check that this combination decays as $\frac{1}{s^{3}}$ for
$s\to\infty$. Using this function, we can derive the nonperturbative sum rules

$\displaystyle\sum_{\Delta,k}c_{\Delta}\tfrac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)}\tfrac{2(\Delta+k-\Delta_{\phi})(p_{1}+p_{2}+2\Delta_{\phi})}{(\Delta+k+p_{1})(\Delta+k+p_{2})(2\Delta_{\phi}-\Delta-k+p_{1})(2\Delta_{\phi}-\Delta-k+p_{2})}=0\,.$
(5.106)

Performing the sum over $k$, one obtains sum rules of the form

$\displaystyle\sum_{\Delta}c_{\Delta}\alpha_{\Delta}=0$ (5.107)

with

$\displaystyle\alpha_{\Delta}$
$\displaystyle=\frac{\Gamma(\Delta)}{\Gamma(2\Delta)\Gamma(2\Delta_{\phi}-\Delta)}\left(\mathcal{F}_{p_{1},p_{2}}(\Delta)+\mathcal{F}_{-2\Delta_{\phi}-p_{1},-2\Delta-
p_{2}}(\Delta)\right)\,,$ (5.108)
$\displaystyle\mathcal{F}_{p_{1},p_{2}}(\Delta)$
$\displaystyle=\tfrac{1}{p_{1}-p_{2}}\left(\tfrac{{}_{3}F_{2}(\Delta,p_{1}+\Delta,1+\Delta-2\Delta_{\phi};2\Delta,1+p_{1}+\Delta;1)}{(p_{1}+\Delta)}-\tfrac{{}_{3}F_{2}(\Delta,p_{2}+\Delta,1+\Delta-2\Delta_{\phi};2\Delta,1+p_{2}+\Delta;1)}{(p_{2}+\Delta)}\right)\,.$
(5.109)

As already mentioned in the Introduction, these sum rules differ from those
found in [55, 49, 216, 219], which are dispersive sum rules having double
zeros at the dimension of double twist operators (twist in higher-$d$). Our
functionals $\alpha_{\Delta}$ have single zeros at $\Delta=2\Delta_{\phi}+k$
for $k\in\mathbb{N}$ and $k\neq p_{1},p_{2}$, implying that the functional
changes sign at any of these zeros. The absence of any positivity property
makes these sum rules less powerful and harder to use with the standard method
of the modern conformal bootstrap. Recent developments in this direction have
these positivity conditions [220] and have improved upon this method,
confirming some of the results presented in [2].

####  5.2.5 Perturbative Results

In this subsection, we consider the Lagrangian presented in (5.51) and use the
Mellin transform to find the corresponding four-point correlators and
anomalous dimension for generic $L$. Using (5.52) as a basis for
$4L$-derivative results, we can take its Mellin transform. The first step is
to compute the Mellin transform of the function
$\bar{D}_{\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}}(t)$. In this
section we consider the reduced Mellin amplitude $\hat{M}(s)\equiv
M(s)\Gamma(s)\Gamma(2\Delta_{\phi}-s)$ and we need to compute

$\displaystyle\hat{M}_{\Delta_{\phi}}(s)=\int_{0}^{\infty}dt\,\bar{D}_{\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}}(t)\,\Big{(}\frac{t}{1+t}\Big{)}^{2\Delta_{\phi}}\,t^{-1-s}\,.$
(5.110)

A closed-form expression for the $\bar{D}$ functions is unavailable, and
dealing with integral representations is quite hard. Therefore, we considered
the case of integer $\Delta_{\phi}$, where simple explicit expressions for the
$\bar{D}$ functions are known (see (D.39)-(D.41)) and we inferred the general
form

$\displaystyle\hat{M}_{\Delta_{\phi}}(s)$ $\displaystyle=\pi\csc(\pi
s)\,\Big{(}\pi\cot(\pi
s)P_{\Delta_{\phi}}(s)-\sum_{k=1}^{2\Delta_{\phi}-1}\frac{P_{\Delta_{\phi}}(k)}{s-k}\Big{)}\,,$
(5.111) $\displaystyle P_{\Delta_{\phi}}(s)$
$\displaystyle=2\sum_{n=0}^{\Delta_{\phi}-1}(-1)^{n}\frac{\Gamma(2n+1)\Gamma^{4}(\Delta_{\phi})\Gamma(\Delta_{\phi}+n)}{\Gamma^{4}(n+1)\Gamma(\Delta_{\phi}-n)\Gamma(2(\Delta_{\phi}+n))}(2\Delta_{\phi}-s)_{n}(s)_{n}\,,$
(5.112)

The functions $P_{\Delta_{\phi}}(s)$ are effectively just polynomials of order
$2\Delta_{\phi}-2$. Defining

$\displaystyle Q_{\Delta_{\phi}}(s(s-2\Delta_{\phi}))\equiv
P_{\Delta_{\phi}}(s)\,,$ (5.113)

we have, for the first few cases

(5.120)

The functions $P_{\Delta_{\phi}}(s)$ can also be rewritten as

$\displaystyle P_{\Delta_{\phi}}(s)$
$\displaystyle=2\frac{\Gamma(\Delta_{\phi})^{4}}{\Gamma(2\Delta_{\phi})}{}_{4}{F}_{3}(\\{\textstyle\frac{1}{2},s,1-\Delta_{\phi},2\Delta_{\phi}-s\\};\\{1,1,\Delta_{\phi}+\frac{1}{2}\\};1)\,,$
(5.121)

Notice that in cross-ratio space a closed-form expression for the $\bar{D}$
functions is not known, while in Mellin space it looks reasonably simple, at
least for integer $\Delta_{\phi}$. This is similar to what happens in the
higher dimensional case, where this occurrence is even more striking as the
reduced Mellin transform of the $\bar{D}$ functions is simply a product of
Gamma functions.141414It is often said that the Mellin transform of contact
interactions is one, but this assumes that the correct product of Gamma
function has been factored out [64, 109]. In the one-dimensional case, we
could not find such a simple representation for the contact interactions, but
the fact we could write the result in a closed form is already a notable
improvement compared to cross-ratio space, and as we will see, it will allow
us to extract new CFT data successfully.

Knowing the Mellin transform for the $\bar{D}$ functions, it is simple to
compute the Mellin transform of (5.52)

$\displaystyle\hat{M}^{(1)}_{L}(s)$
$\displaystyle=\int_{0}^{\infty}dt\,f^{(1)}_{L}(t)\,\Big{(}\frac{t}{1+t}\Big{)}^{2\Delta_{\phi}}\,t^{-1-s}=\,\sum_{k=0}^{2L}c_{k,L}\,\hat{M}_{\Delta_{\phi}+L}(s+k)\,,$
(5.122) $\displaystyle 2c_{k,L}$
$\displaystyle=\frac{\Gamma(2L+1)}{\Gamma(k+1)\Gamma(2L-k+1)}+\delta_{k,0}+\delta_{k,2L}\,.$
(5.123)

Notice that the presence of double poles for integer values of $s$ in (6.1) is
not in contradiction with the general single-pole structure of the
nonperturbative Mellin amplitude (5.76)-(5.78), but just a consequence of the
perturbative expansion of those single poles at this first order of
perturbation theory as detailed in Appendix E.3. Moreover, the structure of
(6.1) is such that both single and double poles cancel (as they should) within
the region of convergence (5.67) of the integral (5.110), which in this case
($\Delta_{0}=2\Delta_{\phi}$) is $0<\text{Re}(s)<2\Delta_{\phi}$. The
cancellation of the double poles is evident, given the poles of $\cot(\pi s)$
and the explicit poles in the sum. The cancelling of the single poles stems
from a property of the polynomial $P_{\Delta_{\phi}}(s)$, which ensures the
cancellation of the finite part in the expression multiplying $\csc(\pi s)$
when expanded around integer values of $s$, $0<s<2\Delta_{\phi}$. As we will
see, this structure is consistent with the OPE expansion.

We stress that equation (5.122) is a closed-form expression for the first-
order perturbation around GFF generated by a quartic interaction with any
number of derivatives. Under the assumption that the deformation from GFF
described by these interactions only modifies two-particle data, see
(5.128)-(5.129) below, one can extract these data. In particular, in section
5.2.6, we will show that the anomalous dimension of two-particle operators
receives the following correction

$\Delta=2\Delta_{\phi}+2n+g_{L}\hat{\gamma}_{L,n}(\Delta_{\phi})$ (5.124)

with

$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\frac{\Gamma(L+\Delta_{\phi})^{4}}{\Gamma(2L+2\Delta_{\phi})}\sum_{p=0}^{2n}\sum_{k=2L-p}^{2L}\sum_{l=0}^{k+p-2L}(-1)^{k}c_{k,L}\times$
(5.125)
$\displaystyle\frac{(4\Delta_{\phi}+2n-1)_{p}(-2n)_{p}(2L-k-p)_{l}(1-\Delta_{\phi}-L)_{l}(2\Delta_{\phi}+k+p)_{l}(\tfrac{1}{2})_{l}}{(l!)^{3}(2\Delta_{\phi})_{p}(\tfrac{1}{2}+\Delta+L)_{l}}\,.$

To compare these results with those computed with bootstrap methods in [55,
49] for $L=0,1,2,3$, we have to change the basis in the space of couplings.
Since the bootstrap approach is blind to the specific values of the couplings
$g_{L}$ in (5.124), one needs to establish a criterium to organize the set of
independent data. The criterium that is used in [55, 49] consists of setting

$\gamma_{L,n}(\Delta_{\phi})=0\qquad n<L\,.$ (5.126)

In our approach, this is implemented by taking a linear combination

$\gamma_{L,n}=\sum_{\ell=0}^{L}a_{\ell}\hat{\gamma}_{\ell,n}$ (5.127)

and fixing the $L+1$ $a_{\ell}$ coefficients in (5.127), using the $L$
conditions (5.126) and the normalization $\gamma_{L,L}(\Delta_{\phi})=1$.
Following this strategy in section 5.2.6, we will reproduce the known results
for $L\leq 3$ and present new results for $L\leq 8$ at any $\Delta$ and $n$.
We stress, however, that equation (5.125) is valid for any $L$, so, up to the
algorithmic procedure of fixing the $a_{\ell}$ coefficients, one can easily
extract the result for any given $L$.

Below we will show how the interaction (5.51) leads to the correlator (5.52)
through explicit Witten diagrammatics. We will also see, for the cases
$L=0,1,2$, how other interaction terms lead to results that can be rearranged
as linear combinations of the eigenfunctions (5.52). We will then proceed to
extract CFT data in the bootstrap normalization.

####  5.2.6 CFT Data

Given the closed-form expression (5.122) for the perturbative Mellin
amplitude, we can use it to extract the CFT data

$\displaystyle\Delta$
$\displaystyle\equiv\Delta_{n,L}=2\Delta_{\phi}+2n+g_{L}\,\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})+\dots\,,$
(5.128) $\displaystyle c_{\Delta}$ $\displaystyle\equiv
c_{n,L}=c_{n}^{(0)}(\Delta_{\phi})+g_{L}\,c_{L,n}^{(1)}(\Delta_{\phi})+\dots\,,$
(5.129)

where $n\in\mathbb{N}$ and $c_{n}^{(0)}$ is given in (E.81). In
(5.128)-(5.129), we assume that the AdS interaction only modifies the CFT data
of the two-particle operators exchanged in GFF. If we insert (5.128)-(5.129)
in the general Mellin OPE expansion (5.76) at first order in $g_{L}$, one
obtains double and single poles at $s=2\Delta_{\phi}+p$, $p\in\mathbb{N}$. One
can then compare the corresponding residues with the ones in the tree-level
Mellin amplitude (5.122), which amounts to solving the equations formally
written as

$\displaystyle\sum_{n}c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\,{\textstyle{\frac{(-1)^{p}\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\lim_{s\rightarrow
2\Delta_{\phi}+p}(s-2\Delta_{\phi}-p)^{2}\hat{M}^{(1)}_{L}(s)\,,$ (5.130)
$\displaystyle\sum_{n}(c_{L,n}^{(1)}(\Delta_{\phi})\\!+c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\partial_{n})\,{\textstyle{\frac{(-1)^{p}\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\text{Res}_{s=2\Delta_{\phi}+p}\hat{M}^{(1)}_{L}(s)$
(5.131)

for the leading-order corrections $\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
and $c_{L,n}^{(1)}(\Delta_{\phi})$. Since the function $\hat{M}^{(1)}_{L}(s)$
is known explicitly, it is possible to write down a linear system for the
anomalous dimensions $\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$. To do this,
let us use the form (6.1) for $\hat{M}^{(1)}_{\Delta_{\phi}}(s)$ to rewrite
(5.130) as

$\displaystyle\sum_{n}c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\,{\textstyle{\frac{\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\sum_{k=0}^{2L}(-1)^{k}c_{k,L}P_{\Delta_{\phi}+L}(2\Delta_{\phi}+p+k)\,.$
(5.132)

Notice that the sum on the left hand side of (5.132) is truncated by the
$\Gamma(p-2n+1)$ in the denominator. This means that at a fixed value of $p$,
equation (5.132) provides an invertible linear system which can be solved for
$\gamma$. The system can also be inverted explicitly using the identity

$\displaystyle\sum_{p=0}^{2n}\frac{(4m+4\Delta_{\phi}-1)\,\Gamma(p+4\Delta_{\phi}+2m-1)(-1)^{p}}{\Gamma(2m-p+1)\,\Gamma(2\Delta+p)^{2}}\frac{\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}=\delta_{m,n}$
(5.133)

yielding (5.125):

$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\frac{\Gamma(L+\Delta_{\phi})^{4}}{\Gamma(2L+2\Delta_{\phi})}\sum_{p=0}^{2n}\sum_{k=2L-p}^{2L}\sum_{l=0}^{k+p-2L}(-1)^{k}c_{k,L}\times$
(5.134)
$\displaystyle\frac{(4\Delta_{\phi}+2n-1)_{p}(-2n)_{p}(2L-k-p)_{l}(1-\Delta_{\phi}-L)_{l}(2\Delta_{\phi}+k+p)_{l}(\tfrac{1}{2})_{l}}{(l!)^{3}(2\Delta_{\phi})_{p}(\tfrac{1}{2}+\Delta+L)_{l}}\,.$

Equation (5.134) has been derived under the assumption that $\Delta_{\phi}$
takes integer values. However, one can argue that it holds for any
$\Delta_{\phi}$ by noticing that the result for $L=0$ agrees with that of
[55], which has been obtained without assuming integer $\Delta_{\phi}$.
Furthermore, equation (5.122) implies that the CFT data at $L=0$ fully
determines that at higher $L$ or, in other words, equation (5.134) could be
rewritten as a combination of anomalous dimensions for $L=0$. This shows that
(5.134) holds for any $\Delta_{\phi}$.

Notice that all the sums in (5.134) are finite, so for a given value of $L$
and $n$, it is straightforward to extract the value of the anomalous dimension
$\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$. It turns out expression (5.134) can
be rewritten as

$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})\,,$
(5.135)

where

$\displaystyle\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})=\tfrac{\sqrt{\pi}4^{-2\Delta-L+1}\Gamma(2\Delta)^{2}\Gamma(L+\frac{1}{2})\Gamma(L+\Delta)^{4}\Gamma(L+2\Delta-\frac{1}{2})\Gamma(n+\Delta+\frac{1}{2})\Gamma(L-n+\Delta)}{\Gamma(L+1)\Gamma(L+\Delta+\frac{1}{2})^{2}\Gamma(L+2\Delta)\Gamma(n+\Delta)^{3}\Gamma(2n+2\Delta-\frac{1}{2})\Gamma(L+n+\Delta+\frac{1}{2})}\,,$
(5.136)

while $\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})$ is a polynomial in $n$ and in
$\Delta_{\phi}$ of degree $6L$. It is easy to extract these polynomials from
(5.125), a short Mathematica notebook was attached to the [2] arXiv submission
where the function FindPolynomial[L,$\Delta$,n] allows to extract
$\hat{\mathcal{P}}_{L,n}$ for many values of $L$ (the function gets slower and
slower at higher $L$, but in principle it works for any $L$).

To make contact with the bootstrap results, the coefficients $a_{\ell}$ in the
definition (5.122) of $M_{L}(s)$ should be chosen to have the bootstrap
normalization, namely $\gamma^{(1)}_{\ell,L}=0$ for $0\leq\ell<L$, and
$\gamma^{(1)}_{L,L}=1$. In this case, we have

$\displaystyle\gamma_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\mathcal{G}_{L,n}(\Delta_{\phi})\mathcal{P}_{L,n}(\Delta_{\phi})\,,$
(5.137)

where

$\displaystyle\mathcal{G}_{L,n}(\Delta_{\phi})=\frac{4^{-L}\left(L+\frac{1}{2}\right)_{\Delta_{\phi}}\left(L+\Delta_{\phi}\right)_{\Delta_{\phi}}(-L+n+1)_{\Delta_{\phi}-1}\left(L+n+\Delta_{\phi}+\frac{1}{2}\right)_{\Delta_{\phi}-1}}{\Gamma\left(\Delta_{\phi}\right)\left(\Delta_{\phi}\right)_{3L}\left(2L+\Delta_{\phi}+\frac{1}{2}\right)_{\Delta_{\phi}-1}\left(L+2\Delta_{\phi}-\frac{1}{2}\right)_{2L}\left(n+\frac{1}{2}\right)_{\Delta_{\phi}}\left(n+\Delta_{\phi}\right)_{\Delta_{\phi}}}\,,$
(5.138)

while $\mathcal{P}_{L,n}(\Delta_{\phi})$ is a polynomial of degree $4L$ in $n$
and $5L$ in $\Delta_{\phi}$. The explicit polynomials for the first few values
of $L$ are detailed in Appendix E.4 and, up to $L=3$, they perfectly agree
with the result of [49]. The attached Mathematica notebook has values of $L$
ranging from $L=0$ to $L=8$ as well as a function
FindBootstrapPolynomial[L,$\Delta$,n] to compute
$\mathcal{P}_{L,n}(\Delta_{\phi})$ for arbitrary $L$.

####  5.2.7 Bootstrapping the Anomalous Dimension

Since the interaction terms in (5.51) form a basis, we can use the results for
the anomalous dimensions to directly bootstrap the result and find the
resulting effective action in AdS2. Consider an effective theory of massless
scalar in $\text{AdS}_{2}$ with the Lagrangian based on the one in (5.50)

$\displaystyle S=\int
dxdz\,\sqrt{g}\,\big{[}\,g^{\mu\nu}\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi+m^{2}_{\Delta_{\phi}}\Phi^{2}+\sum_{L}g_{L}\,(\partial^{L}\Phi)^{4}\,\big{]}\,,\qquad
L=0,1,\dots$ (5.139)

From the Mellin space analysis, the resulting anomalous dimension will be

$\displaystyle\hat{\gamma}_{n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\sum_{L}g_{L}\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})\,.$
(5.140)

Explicitly for massless fields, we have

$\displaystyle\gamma^{(1)}_{h}=\frac{2g_{0}}{(h-1)h}+\frac{3g_{1}\left((h-1)^{2}h^{2}-4\right)}{35(h-1)h}+\sum_{L\geq
2}g_{L}\hat{\gamma}_{L}^{(1)}.$ (5.141)

For a given Regge bound $\gamma^{1}\sim_{\Delta>>1}\Delta^{l}$, this series
truncates. Additionally, the requirement that the anomalous dimension vanishes
for a certain protected operator fixes the ratio of the two coefficients, for
example:

$\displaystyle\gamma^{(1)}_{h=2}$ $\displaystyle=0$
$\displaystyle\qquad\Rightarrow g_{0}$ $\displaystyle=0$ (5.142)
$\displaystyle\gamma^{(1)}_{h=4}$ $\displaystyle=0$
$\displaystyle\qquad\Rightarrow g_{0}$ $\displaystyle=-6g_{1}$ (5.143)

A particular choice stands out, which is for $\gamma^{(1)}_{h=\Delta=1}=0$.
This gives a polynomial anomalous dimension proportional to the quadratic
Casimir:

$\displaystyle\gamma^{(1)}_{h,(\gamma_{h=1}=0)}$
$\displaystyle=\frac{g_{0}}{2}h(h-1)$ (5.144)

where $g_{1}=\frac{35}{6}g_{0}$.

This analysis also points to one of the limits of the analytic conformal
bootstrap. When the constraint from the growth of the anomalous dimension
depends on the perturbative order, each order has additional contact terms
which can potentially be added and the number of additional constraints to fix
the full correlator increases. Up to this point, in the bootstrap of 1d dCFTs
this has not yet been an issue (at third-order in ABJM and fourth-order in
$\mathcal{N}=4$ SYM). However, this may indicate an upper limit for the order
of perturbation one can bootstrap.151515The author thanks Miguel Paulos for
useful comments about this fact. The next and final Chapter will reflect on
this and other areas of the analytic bootstrap that can be improved.

##  Chapter 6 Optimisation of the Bootstrap Procedure

End? No, the journey doesn’t end here.

– J.R.R. Tolkien, The Lord of the Rings

As illustrated in the first Chapters of this thesis, the analytic conformal
bootstrap is an extremely powerful tool to compute correlators in the context
of conformal line defects in holographic theories. However, many improvements
can still be made in a number of the steps essential to the bootstrap process.
The first concerns the Ansatz. As this is the starting point of the analytic
bootstrap process, its accuracy and reliability are crucial. Additionally,
there are limits to the bootstrap procedure; these take the form of the
increased number of constants to fix at each order of the bootstrap and the
resolving the mixing problem. Finally, improvements can still be made for the
physical input. All-order results may not be available depending on the setup,
especially in non-supersymmetric cases. However, there is hope that QFT
methods for non-perturbative computations might pave the way for these all-
order results through lattice computations.

###  6.1 Ansatz and Mellin Bootstrap

The first obvious choice of improvement is through the Ansatz: a rigorous
treatment along the lines of the master integral’s formalism for flat-space
amplitudes is yet to be carried out. There has been work done in this
direction in Mellin space [108] and for contact integrals as seen above [4],
and the diagrams’ structure is relatively well understood. However, in
physical examples, the results are much simpler than the individual diagrams
(usually at least one order of transcendentality less). This has already been
observed for supergravity amplitudes in $\text{AdS}_{5}\times\text{S}^{5}$ and
could be linked to the exchange diagram being expressed as a finite truncation
of the originally infinite sum of $D-$functions[63]. There is some evidence
that this could be linked to Yangian symmetry[101] given both ABJM and
$\mathcal{N}=4$ are integrable theories. Such an explicit realisation of
Yangian symmetry would be a powerful additional tool for the bootstrap [221,
222].

The requirements of having well-defined functions under crossing and braiding
are already highly constraining. However, additional factors linked to the
reasons above seem to favour powers of logarithms to polylogarithms in the
results obtained in Chapters 3 and 4. Additionally, the Ansatz will change
drastically depending on the point around which one is expanding. This
difference is visible between the free theories at weak and strong coupling,
where the weak coupling sector seems to have a quicker increase in
transcendentality [79, 119]. Finally, there might be simpler frameworks for
the Ansatz and the results; for example in [102], the Mellin amplitude was
essential for an all-order result. In the perturbative results above, there is
a drastic simplification between the generic $D-$functions and the result for
correlators. For example, the $D_{1111}$ function, corresponding to a four-
point contact diagram with external massless scalars, has the Mellin transform
(as defined in section 5.2)

$\displaystyle\hat{M}_{\Delta_{\phi}=1}(s)$ $\displaystyle=\pi\csc(\pi
s)\,\Big{(}2\pi\cot(\pi s)-\frac{2}{s-1}\Big{)}$ (6.1)

Comparing this to the ABJM four-point functions, one immediately sees the
simpler forms. The first-order solution $f^{(1)}(\chi)$ in equation 3.166 has
the Mellin transform

$\displaystyle\hat{M}[f^{(1)}(\chi)](s)$
$\displaystyle=6\epsilon\Gamma(-s-2)\Gamma(s-1),$ (6.2)

and the second order is simple enough to be written as

$\displaystyle\hat{M}[f^{(2)}(\chi)](s)$
$\displaystyle=6(\epsilon^{2}p_{0}(s)+1)\Gamma(-s+\epsilon^{2}p_{1}(s)-2)\Gamma(s+\epsilon^{2}p_{2}(s)-1)|_{\epsilon^{2}}$
(6.3) $\displaystyle p_{0}(s)$
$\displaystyle=\frac{30\gamma(s-2)((s-1)s+2)-2s(2s(s((s-5)s+17)-21)+69)+298}{3(s-3)(s-2)^{2}}$
(6.4) $\displaystyle p_{1}(s)$
$\displaystyle=\frac{(s+2)(s(s(s(2s-9)+8)+34)+3)}{6(s-3)(s-2)}$ (6.5)
$\displaystyle p_{2}(s)$
$\displaystyle=-\frac{s\left(s\left(2s^{2}+s-7\right)-31\right)+38}{6(s-2)}.$
(6.6)

Additionally, the differential operators relating $f(\chi)$ to the other
correlators of elements of the displacement multiplet are linear in the cross-
ratio. Therefore, the Mellin transform will correspond to a linear combination
of shifted Mellin amplitudes and will therefore keep the ‘simple form’. For
example, the massless bosonic correlator is expressed in 3.114 in terms of two
functions $G_{1}(\chi)$ and $G_{2}(\chi)$. These can be expressed in terms of
a differential equation acting on $f(\chi)$ in equation (3.116) and can be
written in terms of the Mellin amplitude

$\displaystyle G_{1}(\chi)$ $\displaystyle=f(\chi)-\chi
f^{\prime}(\chi)+\chi^{2}f^{\prime\prime}(\chi)$ (6.7) $\displaystyle
G_{2}(\chi)$
$\displaystyle=-\chi^{2}f^{\prime}(\chi)-\chi^{3}f^{\prime\prime}(\chi)$ (6.8)
$\displaystyle M[G_{1}(\chi)](s)$ $\displaystyle=(s^{2}+1)M[f(\chi)](s)$ (6.9)
$\displaystyle 6(s^{2}+1)\Gamma(-s-2)\Gamma(s-1)$ (6.10) $\displaystyle
M[G_{2}(\chi)](s)$ $\displaystyle=-(s-3)(s-1)M[f(\chi)](s-1)$ (6.11)
$\displaystyle=-6(s-3)(s-1)\Gamma(-s-1)\Gamma(s-2).$ (6.12)

This can also be done for the correlator of the mixed fluctuations detailed in
B.49, for example:

$\displaystyle M[h(\chi)]$
$\displaystyle=(s-2)(s-1)((s-2)s+3)M[f(\chi)](s)-(-3+s)(-2+s)(-1+s)^{2}M[f(\chi)](s-1)$
(6.13) $\displaystyle=12(1-3s+s^{2})\Gamma(s+1)\Gamma(-s-2).$ (6.14)

The Mellin transform from [2] seems promising at tree-level given the
simplifications which occur when compared the generic $D-$functions. Even
beyond this, there seems to be a clear perturbative structure which might be
exploited (for example 6.3). Moreover, the different aspects of the bootstrap
up to the normalisation can be implemented directly in Mellin space.

###  6.2 Physical Non-Perturbative Input

One of the dangers of the ever-increasing bound on the large-$\Delta$
behaviour of the anomalous dimension is that one has an ever-increasing number
of terms to fix with the bootstrap.111This is discussed in subsection 5.2.7
This implies that there might be a hard upper limit beyond which one cannot
use the same algorithm to constrain the correlators via the analytic
bootstrap. Even in the lucky cases of the 1/2-BPS defect theories in ABJM and
$\mathcal{N}$=4 SYM, there is still a constant to fix at each order requiring
input from other computations. In the latter, the topological sector is
amenable to a localisation computation which gives an all-order result. In the
former, the integrated correlator conditions [195] relate to another quantity,
the Bremsstrahlung function, which can be computed non-perturbatively [180,
181]. In theory, another such integrated correlator condition should exist for
the four-point function [79], and one could also use the analogous quantities
for higher-point functions. This is one of the difficulties when extending
these methods to systems with fewer symmetries; they often lack non-
perturbative results.

One way to counter this is to use the non-perturbative framework developed for
the standard model: Lattice Field Theory. A well-defined discretised theory
would provide non-perturbative data for analytic results and give insight into
the whole range of the AdS/CFT correspondence. However, such a discretisation
which preserves the symmetries, matches with continuum data and is
renormalisable is hard to come by. This section presents the analysis carried
out in [3] of the cusped Wilson line in $\mathcal{N}=4$ SYM studied in [223,
208, 209, 205], uncovering a discretisation which preserves more symmetry.
Despite this, the lattice model requires fine-tuning to have finite
quantities.

####  6.2.1 Green-Schwarz AdS${}_{5}\times\text{S}^{5}$

The Green-Schwarz AdS${}_{5}\times\text{S}^{5}$ string is expected to be
defined at the non-perturbative level. A valid question is whether the non-
perturbative regime of the $\sigma$-model, which describes the
AdS${}_{5}\times~{}\text{S}^{5}$ string at tree-level in string perturbation
theory, is accessible through a lattice discretization of the worldsheet
(while target space remains continuous). This question is motivated by the
success of the lattice as a UV non-perturbative regulator of Quantum
Chromodynamics. This approach has been pioneered in [224, 223, 209, 225],
where a lattice-discretized version of $S_{\text{cusp}}$ has been introduced
and also used to perform Monte Carlo simulations.222Other lattice approaches
to AdS/CFT include [226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236,
237, 238, 239, 240, 241, 242], see also [243] and references therein. The goal
of this system is to have a benchmark for a case where analytic results are
still known to extend to systems where they are not.

####  6.2.2 $U(1)\times SU(4)$ Invariant Discretization

In the framework of the AdS/CFT correspondence [140, 141], the expectation
value of a light-like cusped Wilson loop in $\mathcal{N}=4$ super Yang-Mills
is equal to the partition function of an open string propagating in
AdS${}_{5}\times\text{S}^{5}$ space and ending on the loop at the AdS
boundary. In practice, one writes

$\left\langle\mathcal{W}_{\text{cusp
}}\right\rangle=\int\mathcal{D}Y\mathcal{D}\Psi\,e^{-S_{\text{cusp}}(X_{\text{cl}}+Y,\Psi)}\equiv
e^{-\frac{f(g)}{8}V_{2}}\,.$ (6.15)

This gauge-fixed configuration in AdS${}_{5}\times$S5 has a superstring action
describing quantum fluctuations around this null-cusp background [244]

$\displaystyle S^{\text{cont}}_{\rm cusp}=g\int dtds$
$\displaystyle\Bigg{\\{}\left|\partial_{t}x+\tfrac{m}{2}x\right|^{2}+\tfrac{1}{z^{4}}\left|\partial_{s}x-\tfrac{m}{2}x\right|^{2}+\left(\partial_{t}z^{M}+\tfrac{m}{2}z^{M}+\tfrac{i}{z^{2}}z_{N}\eta_{i}\left(\rho^{MN}\right)_{\phantom{i}j}^{i}\eta^{j}\right)^{2}$
(6.16)
$\displaystyle+\tfrac{1}{z^{4}}\left(\partial_{s}z^{M}-\tfrac{m}{2}z^{M}\right)^{2}+i\,\left(\theta^{i}\partial_{t}\theta_{i}+\eta^{i}\partial_{t}\eta_{i}+\theta_{i}\partial_{t}\theta^{i}+\eta_{i}\partial_{t}\eta^{i}\right)-\tfrac{1}{z^{2}}\left(\eta^{i}\eta_{i}\right)^{2}$
$\displaystyle+2i\,\Big{[}\tfrac{1}{z^{3}}z^{M}\eta^{i}\left(\rho^{M}\right)_{ij}\left(\partial_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}-\tfrac{i}{z}\eta^{j}\left(\partial_{s}x-\tfrac{m}{2}x\right)\right)$
$\displaystyle\qquad+\tfrac{1}{z^{3}}z^{M}\eta_{i}\big{(}{\rho^{M}}^{\dagger}\big{)}^{ij}\left(\partial_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}+\tfrac{i}{z}\eta_{j}\left(\partial_{s}x-\tfrac{m}{2}x\right)^{*}\right)\Big{]}\,\Bigg{\\}}\,,$

where

  * •

$x$ is a complex bosonic field whose real and imaginary parts parametrize the
fluctuations of the string (in light-cone gauge) at the boundary of AdS5

  * •

$z^{M}$ are six real bosonic fields333$z=\sqrt{z^{M}z^{M}}$ is the radial
coordinate of the AdS5 space, while $u^{M}=z^{M}/z$ identifies points on
$S_{5}$.

  * •

the Graßmann-odd fields
$\theta^{i}=(\theta_{i})^{\dagger},~{}\eta^{i}=(\eta_{i})^{\dagger},\,i=1,2,3,4$
are complex anticommuting variables (no Lorentz spinor indices appear);

  * •

the matrices $(\rho^{MN})_{i}^{\hphantom{i}j}=(\rho^{[M}\rho^{\dagger
N]})_{i}^{\hphantom{i}j}$ are the $SO(6)$ generators. $\rho^{M}_{ij}$444By
convention, we will write the indices of $\rho$ as down and those of
$\rho^{\dagger}$ as up. are the (traceless) off-diagonal blocks of $SO(6)$
Dirac matrices $\gamma^{M}$ in chiral representation.

The massive parameter $m$ keeps track of the dimensionful light-cone momentum
$P_{+}$, set to 1 in [245]. The action (6.16) is invariant under a $U(1)\times
SU(4)$ global symmetry defined by

$\displaystyle z^{M}\to\text{Ad}(U)^{MN}z^{N}\ ,\quad\theta^{i}\to
U^{i}_{\phantom{i}j}\theta^{j}\ ,\quad\eta^{i}\to
U^{i}_{\phantom{i}j}\eta^{j}\ ,$ (6.17) $\displaystyle x\to e^{i\alpha}x\
,\quad\theta^{i}\to e^{i\alpha/2}\theta^{i}\ ,\quad\eta^{i}\to
e^{-i\alpha/2}\eta^{j}\ ,$ (6.18)

where $U$ is an element of $SU(4)$ and its representative in the adjoint,
$\text{Ad}(U)$, is an element of $SO(6)$. While the original Green-Schwarz
AdS${}_{5}\times\text{S}^{5}$ string action is invariant under diffeomorphisms
and $\kappa$-symmetry, these local symmetries have been fixed by the choice of
light-cone gauge in equation (6.16). Notice that the action is not invariant
under worldsheet rotations, parity ($s\rightarrow-s$), or time reversal
($t\rightarrow-t$).

To define the lattice-discretised theory, we need to provide a discretised
action and an explicit expression for the measure. We choose to use a flat
measure for the fields but we keep in mind that this choice is quite arbitrary
as it is not invariant under reparametrisation of the target AdS${}_{5}\times
S_{5}$ target space. Given a generic observable $A$, expectation values in the
lattice discretised theory are defined by

$\langle A\rangle=\frac{1}{Z_{\text{cusp}}}\int dxdx^{*}d^{6}zd^{4}\theta
d^{4}\theta^{\dagger}d^{4}\eta d^{4}\eta^{\dagger}\,e^{-S_{\text{cusp}}}A\,,$
(6.19)

where $df\equiv\prod_{s,t}df(s,t)$ is the discretised measure. The partition
function $Z_{\text{cusp}}$ is fixed by the requirement $\langle 1\rangle=1$,
and $S_{\text{cusp}}$ refers now to the discretised action that we choose to
be

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\rm cusp}=g\sum_{s,t}a^{2}$
$\displaystyle\\!\\!\\!\\!\\!\Bigg{\\{}\\!\\!\left|b_{+}\hat{\partial}_{t}x+\tfrac{m}{2}x\right|^{2}\\!\\!+\tfrac{1}{z^{4}}\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}\\!\\!+\big{(}b_{+}\hat{\partial}_{t}z^{M}+\tfrac{m}{2}z^{M}+\tfrac{i}{z^{2}}z^{N}\eta_{i}(\rho^{MN})_{\phantom{i}j}^{i}\eta^{j}\big{)}^{2}$
(6.20)
$\displaystyle+\tfrac{1}{z^{4}}\big{(}\hat{\partial}_{s}z^{M}\hat{\partial}_{s}z^{M}+\tfrac{m^{2}}{4}z^{2}\big{)}+2i\,\big{(}\theta^{i}\hat{\partial}_{t}\theta_{i}+\eta^{i}\hat{\partial}_{t}\eta_{i}\big{)}-\tfrac{1}{z^{2}}\left(\eta^{i}\eta_{i}\right)^{2}$
$\displaystyle+2i\,\Big{[}\tfrac{1}{z^{3}}z^{M}\eta^{i}\left(\rho^{M}\right)_{ij}\big{(}b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}-\tfrac{i}{z}\eta^{j}\big{(}b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\big{)}\big{)}$
$\displaystyle\qquad+\tfrac{1}{z^{3}}z^{M}\eta_{i}\big{(}{\rho^{M}}^{\dagger}\big{)}^{ij}\big{(}b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}+\tfrac{i}{z}\eta_{j}\big{(}b_{-}\hat{\partial}_{s}x^{*}-\tfrac{m}{2}x^{*}\big{)}\\!\big{)}\\!\Big{]}\Bigg{\\}}\,.$

The action is written in terms of the forward and backward discrete
derivatives

$\hat{\partial}_{\mu}f(\sigma)\equiv\frac{f\left(\sigma+ae_{\mu}\right)-f\left(\sigma\right)}{a}\,,\qquad\bar{\partial}_{\mu}f(\sigma)\equiv\frac{f\left(\sigma\right)-f\left(\sigma-
ae_{\mu}\right)}{a}\,$ (6.21)

where $e_{\mu}$ is the unit vector in the direction $\mu=0,1$, and $\sigma$ is
a shorthand notation for $(s,t)$.

Notice that the proposed discretized action (6.20) depends on four parameters:
$g$, $m$, and the auxiliary parameters $b_{\pm}$. The discretized action
$S_{\rm cusp}$ reduces to the desired continuum action $S^{\text{cont}}_{\rm
cusp}$ in the naïve $a\to 0$ limit if $b_{\pm}\to 1$. However, as we will
discuss in detail, the naïve choice $b_{\pm}=1$ produces undesired UV
divergences at one loop. The values of $b_{\pm}$ need to be tuned so that
these UV divergences cancel. This is a sign that the lattice regulator does
not manage to reproduce the cancellation of UV divergences that occurs in
dimensional regularisation.

An important feature of the proposed discretised action and measure is that
they are invariant under the full $U(1)\times SU(4)$ internal symmetry group.
This is in contrast to the discretisation previously presented in [209]. The
key ingredient is the use of forward and backward discrete derivatives for
both the bosonic and fermionic parts of the action. This is normally avoided
for fields that satisfy first-order equations of motion (usually fermions)
since it breaks parity and time reversal. This is not an issue in our case
because these symmetries are already broken in the continuum action. In [209],
instead, the symmetric derivative was used and, as in lattice QCD, a Wilson-
like term was included to solve the resulting doubling problem while breaking
either the $U(1)$ or the $SU(4)$ symmetry. The perturbative series is obtained
on the lattice as in the continuum by expanding the action around its minima.
The $SU(4)$ symmetric point is a singularity for the action because of the
terms proportional to inverse powers of the radial coordinate $z$.555All
fields vanish in this point. Consequently, the minimum of the action
spontaneously breaks the internal symmetry. In the continuum, an absolute
minimum of the action is given by $x=x^{*}=0$ and $z^{M}=\delta^{M6}$, and any
other absolute minimum is obtained by acting with the $SU(4)$ symmetry. One
can easily check that these minima are relative minima for the discretised
action. We parametrise the fluctuations around the chosen minimum in the same
way as it is done in the continuum [244]

$z=e^{\phi}\,,\qquad z^{a}=e^{\phi}\frac{y^{a}}{1+\frac{1}{4}y^{2}}\,,\qquad
z^{6}=e^{\phi}\frac{1-\frac{1}{4}y^{2}}{1+\frac{1}{4}y^{2}}\,,\qquad
y^{2}=\sum_{a=1}^{5}(y^{a})^{2},\,$ (6.22)

where $a=1,\dots,5$. In terms of the new variables $\phi$ and $y^{a}$, the
path-integral measure over the $z^{M}$ fields reads

$\displaystyle\prod_{M=1}^{6}dz^{M}=e^{\sum_{s,t}\left\\{6\phi+5\log\left(1+\frac{y^{2}}{4}\right)\right\\}}\,d\phi\prod_{a=1}^{5}dy^{a}\,.$
(6.23)

The contribution of the Jacobian determinant above can be conveniently
included in the effective action

$\displaystyle
S_{\text{eff}}=S_{\text{cusp}}-\sum_{s,t}\left\\{6\phi+5\log\left(1+\frac{y^{2}}{4}\right)\right\\}\,,$
(6.24)

in terms of which the expectation values of observables are

$\langle A\rangle=\frac{1}{Z_{\text{eff}}}\int dxdx^{*}d\phi d^{5}yd^{4}\theta
d^{4}\theta^{\dagger}d^{4}\eta d^{4}\eta^{\dagger}\,e^{-S_{\text{eff}}}A\,.$
(6.25)

Notice that the sum in (6.23) does not come with the corresponding $a^{2}$
factor, which means that in the naïve continuum limit it diverges like
$a^{-2}$. This should not be surprising: in the continuum, this term would be
proportional to $\delta^{2}(0)$ which yields a quadratic divergence in a hard-
cutoff regularization (but it is set to zero in dimensional regularization).

The perturbative expansion, i.e. the expansion in powers of $g$, is obtained
by splitting the action $S_{\text{eff}}=S_{0}+S_{\text{int}}$, where $S_{0}$
contains all quadratic terms in the fields with a coefficient proportional to
$g^{-1}$, and $S_{\text{int}}$ contains all other terms. Notice that
$S_{\text{int}}$ also contains $g$-independent quadratic terms, which come
from expanding the Jacobian determinant. We focus here on the leading-order
quadratic action

$\displaystyle S_{0}=g\,a^{2}\sum_{s,t}\Bigg{\\{}$
$\displaystyle\left|b_{+}\hat{\partial}_{t}x+\tfrac{m}{2}x\right|^{2}+\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}$
(6.26)
$\displaystyle+b_{+}^{2}(\hat{\partial}_{t}y^{a})^{2}+mb_{+}y^{a}\hat{\partial}_{t}y^{a}+(\hat{\partial}_{s}y^{a})^{2}$
$\displaystyle+b_{+}^{2}(\hat{\partial}_{t}\phi)^{2}+mb_{+}\phi\hat{\partial}_{t}\phi+(\hat{\partial}_{s}\phi)^{2}+m^{2}\phi^{2}+2i\left(\theta^{i}\hat{\partial}_{t}\theta_{i}+\eta^{i}\hat{\partial}_{t}\eta_{i}\right)$
$\displaystyle+2i\eta^{i}(\rho^{6})_{ij}\left(b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}\right)+2i\eta_{i}({\rho^{6}}^{\dagger})^{ij}\left(b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}\right)\Bigg{\\}}\,.$

####  6.2.3 One-Point Functions

Let us turn to the one-point functions of the perturbative fields. Notice that
$\langle x\rangle=0$ because of the $U(1)$ symmetry, and $\langle
y^{a}\rangle=0$ because of the $SO(5)\subset SO(6)\simeq SU(4)$ which leaves
the perturbative vacuum invariant. $\phi$ is the only field with a non-
vanishing one-point function, which has been calculated in dimensional
regularisation [245, 244, 246]. This one-point function and any $n$-point
function of bare fields are not expected to be UV finite. It is known that
$\langle\phi\rangle$ is UV divergent in dimensional regularisation, and we
will see that it turns out to be UV divergent also in the lattice
regularisation. The interest in this one-point function lies in the fact that
it appears as a sub-diagram in any other $n$-point function, and ultimately
its UV divergence contributes to any physical observable. We will give an
example of this mechanism in the next subsection.

There are two classes of vertices contributing to the one-point function of
$\phi$: single-field vertices coming from the measure

$\displaystyle S_{\phi}=-6\sum_{s,t}\phi\ ,$ (6.27)

and three-field vertices coming from the action

$\displaystyle S_{\phi\bullet\bullet}$ $\displaystyle=$ $\displaystyle
g\sum_{s,t}a^{2}\bigg{\\{}-4\phi\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}+c_{+}\hat{\partial}_{t}\phi\hat{\partial}_{t}(\phi^{2})+\hat{\partial}_{s}\phi\hat{\partial}_{s}\phi^{2}-4\phi(\hat{\partial}_{s}\phi)^{2}$
(6.28) $\displaystyle\hskip
45.5244pt+2c_{+}\hat{\partial}_{t}y^{a}\hat{\partial}_{t}(\phi
y^{a})-c_{+}\hat{\partial}_{t}\phi\hat{\partial}_{t}(y^{2})+2\hat{\partial}_{s}y^{a}\hat{\partial}_{s}(\phi
y^{a})-\hat{\partial}_{s}\phi\hat{\partial}_{s}(y^{2})-4\phi(\hat{\partial}_{s}y^{a})^{2}$
$\displaystyle\hskip
45.5244pt-4i\phi\left[\eta^{i}(\rho^{6})_{ij}\left(b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}\right)+\eta_{i}({\rho^{6}}^{\dagger})^{ij}\left(b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}\right)\right]\bigg{\\}}\
.$

Notice that the insertion of $S_{\phi}$ produces a tree-level diagram, while
the insertion of $S_{\phi\bullet\bullet}$ produces a one-loop diagram.
However, because of the mismatch in the power of $g$ in $S_{\phi}$ and
$S_{\phi\bullet\bullet}$, all these diagrams contribute to the same order in
$g$, yielding

$\displaystyle\langle\phi\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{gm^{2}a^{2}}+\frac{2}{gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{-}|\hat{q}_{1}|^{2}+\tfrac{m^{2}}{4}}{c_{+}|\hat{q}_{0}|^{2}+c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{2}}$
(6.29)
$\displaystyle-\frac{1}{2gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{0}|^{2}-|\hat{q}_{1}|^{2}}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}+m^{2}}-\frac{5}{2gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{0}|^{2}-|\hat{q}_{1}|^{2}}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}}$
$\displaystyle-\frac{8}{gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{|\hat{q}_{0}|^{2}+c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}+O(g^{-2})\
.$

With the special choice $b_{\pm}=\bar{b}_{\pm}$, i.e. $c_{\pm}=1$, one can use
the symmetry of the integrals under $p_{0}\leftrightarrow p_{1}$ exchange to
simplify

$\displaystyle\langle\phi\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{g}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+\frac{m^{2}}{4}}+O(g^{-2})$
(6.30) $\displaystyle=$
$\displaystyle\frac{1}{g}\left\\{\frac{1}{4\pi}\log\frac{(am)^{2}}{4}+\frac{1}{4\pi}-I_{0}^{(0,0)}+O(a\log
a)\right\\}+O(g^{-2})\ ,$

which is logarithmically divergent. The definition of the numerical constant
$I_{0}^{(0,0)}\simeq 0.355$ is given in the appendix of [3]. Notice that the
measure, fermion-loop and $x$-loop contributions are separately quadratically
divergent, and the cancellation of these divergences is highly non-trivial.

In the general case $c_{\pm}=1+(am)\delta c_{\pm}$ where $\delta
c_{\pm}=O(a^{0})$ and, after a lengthy calculation, one gets

$\displaystyle\langle\phi\rangle=\frac{1}{g}\bigg{\\{}\frac{-8\delta
c_{+}+\delta c_{-}}{\pi
a}+\frac{1}{4\pi}\log\frac{(am)^{2}}{4}+\frac{1}{4\pi}-I_{0}^{(0,0)}+\frac{8\delta
c_{+}^{2}-\delta c_{-}^{2}}{2\pi}+O(a\log a)\bigg{\\}}+O(g^{-2})\ .$ (6.31)

Notice that the naïve choice $b_{\pm}=1$ corresponds to the choice $\delta
c_{\pm}=\mp 1/2$ which yields indeed a linear divergence for
$\langle\phi\rangle$:

$\displaystyle\langle\phi\rangle=\frac{1}{g}\left\\{\frac{9}{2\pi a}+O(\log
a)\right\\}+O(g^{-2})\ .$ (6.32)

####  6.2.4 Two-Point Function

We turn now to the two-point function of the field $x$, which we calculate at
one loop. We will use the two-point function to extract the dispersion
relation of the $x$ particle propagating on the worldsheet. In dimensional
regularisation and at one loop [244], both the two-point function and the
dispersion relation are UV finite without the need for renormalisation. We
will also see that this is true at one loop in lattice perturbation theory,
provided that one has chosen $c_{\pm}=1$. The naïve choice $b_{\pm}=1$
generates UV divergences in the dispersion relation. A valid question is
whether these divergences can be eliminated with a renormalisation procedure.

There are two classes of vertices contributing to the two-point function of
$x$ at one loop: three-field vertices

$\displaystyle S_{xx^{*}\bullet}$ $\displaystyle=$ $\displaystyle
g\sum_{s,t}a^{2}\bigg{\\{}-4\phi\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}$
(6.33) $\displaystyle\hskip
51.21495pt+2\eta^{i}\rho^{6}_{ij}\eta^{j}\left(b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right)-2\eta_{i}({\rho^{6}}^{\dagger})^{ij}\eta_{j}\left(b_{-}\hat{\partial}_{s}x^{*}-\tfrac{m}{2}x^{*}\right)\bigg{\\}}\
,$

and four-field vertices

$\displaystyle
S_{xx^{*}\bullet\bullet}=8g\sum_{s,t}a^{2}\phi^{2}\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}\
,$ (6.34)

combined to give Feynman diagrams with the three different topologies
illustrated in Figure 6.1. Notice that the tadpole contribution will be
proportional to $\langle\phi\rangle$.

Figure 6.1: Topologies of diagrams contributing to the two-point function
$\langle xx^{*}\rangle$ at one-loop in the discretized model in (6.20) .

On general grounds, one sees that the two-point function has the following
form

$\\!\\!\\!\\!\\!\langle\tilde{x}(p)x^{*}(0)\rangle=\frac{1}{g}\left\\{c_{+}|\hat{p}_{0}|^{2}+c_{-}|\hat{p}_{1}|^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(c_{-}|\hat{p}_{1}|^{2}+\frac{m^{2}}{4}\right)\Pi_{a}(p)+O(g^{-2})\right\\}^{-1}\
.$ (6.35)

The factor $\left(c_{-}|\hat{p}_{1}|^{2}+\tfrac{m^{2}}{4}\right)$ comes from
the fact that, in all interaction vertices, $x$ always appears in the
combination $\left(b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right)$ or its
complex conjugate. The function $\Pi_{a}(p)$ has a representation in terms of
amputated Feynman diagrams, and it is explicitly given by

$\displaystyle\Pi_{a}(p)$ $\displaystyle=$
$\displaystyle-4g\langle\phi\rangle+4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}+m^{2}}$
(6.36)
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{c_{+}|\hat{q}_{0}|^{2}+c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{2}}\frac{1}{c_{+}|\widehat{p+q}_{0}|^{2}+|\widehat{p+q}_{1}|^{2}+m^{2}}$
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{\hat{q}_{0}}{|\hat{q}_{0}|^{2}+c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}\frac{\widehat{p+q}_{0}^{*}}{|\widehat{p+q}_{0}|^{2}+c_{+}|\widehat{p+q}_{1}|^{2}+\frac{m^{2}}{4}}\
.$

All integrals in the above formula are logarithmically divergent, while the
term proportional to $\langle\phi\rangle$ generally contains a linear
divergence. Up to terms that vanish in the $a\to 0$ limit, one can replace
$c_{\pm}=1$ in the above integrals, obtaining the simpler expression

$\displaystyle\Pi_{a}(p)$ $\displaystyle=$
$\displaystyle-4g\langle\phi\rangle+4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+m^{2}}-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{|\hat{q}|^{2}+\frac{m^{2}}{2}}\frac{1}{|\widehat{p+q}|^{2}+m^{2}}$
(6.37)
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{\hat{q}_{0}}{|\hat{q}|^{2}+\frac{m^{2}}{4}}\frac{\widehat{p+q}_{0}^{*}}{|\widehat{p+q}|^{2}+\frac{m^{2}}{4}}+O(a\log
a)\ .$

As in the continuum, the leading divergence of the above integrals does not
depend on the external momentum, therefore the subtracted quantity
$\Delta\Pi_{a}(p)=\Pi_{a}(p)-\Pi_{a}(0)$ has a finite $a\to 0$ limit given by
the corresponding continuum integrals, i.e.

$\displaystyle\Delta\Pi_{0}(p)$ $\displaystyle=$
$\displaystyle-8\int_{-\infty}^{\infty}\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{1}^{2}+\frac{m^{2}}{4}}{q^{2}+\frac{m^{2}}{2}}\left\\{\frac{1}{(p+q)^{2}+m^{2}}-\frac{1}{q^{2}+m^{2}}\right\\}$
(6.38)
$\displaystyle-8\int_{-\infty}^{\infty}\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{0}}{|\hat{q}|^{2}+\frac{m^{2}}{4}}\left\\{\frac{p_{0}+q_{0}}{(p+q)^{2}+\frac{m^{2}}{4}}-\frac{q_{0}}{q^{2}+\frac{m^{2}}{4}}\right\\}+O(a\log
a)\ ,$

while all the divergences are contained in

$\displaystyle\Pi_{a}(0)=-4g\langle\phi\rangle-4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+\frac{m^{2}}{4}}+\frac{1}{\pi}+O(a\log
a)\ ,$ (6.39)

where we have used the symmetry of the integrals under $p_{0}\leftrightarrow
p_{1}$ exchange to simplify them.

With the choice $c_{\pm}=1$, using equation (6.30) one immediately sees that
all divergences cancel and $\Pi_{0}(0)=1/\pi$. The two-point function is
finite in the continuum limit and

$\displaystyle\lim_{a\to
0}\langle\tilde{x}(p)x^{*}(0)\rangle=\frac{1}{g}\left\\{p^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\Pi_{0}(p)+O(g^{-2})\right\\}^{-1}.$
(6.40)

The two-point function has poles at $p_{0}=\pm iE(p_{1})$ for every value of
$p_{1}$, where $E(p_{1})$ is the energy of a single excitation with the
quantum numbers of the field $x$, propagating on the worldsheet with momentum
$p_{1}$. In the continuum limit, this is found to be

$\displaystyle E(p_{1})^{2}$ $\displaystyle=$ $\displaystyle
p_{1}^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\Pi_{0}\left(\sqrt{p_{1}^{2}+\frac{m^{2}}{2}},p_{1}\right)+O(g^{-2})$
(6.41) $\displaystyle=$ $\displaystyle
p_{1}^{2}+\frac{m^{2}}{2}-\frac{1}{gm^{2}}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)^{2}+O(g^{-2})\
,$

where we have used the on-shell value of $\Pi_{0}$. The obtained dispersion
relation coincides with the result in [244].666When comparing to [244], notice
that one has to redefine the worldsheet coordinates, resulting in square
masses of the fluctuations rescaled by a factor of four.

However in the general case $c_{\pm}=1+(am)\delta c_{\pm}$ where $\delta
c_{\pm}=O(a^{0})$, $\Pi_{a}(0)$ and $E(p_{1})$ inherit the linear divergence
from $\langle\phi\rangle$. Using equation (6.31) one obtains

$\displaystyle\Pi_{a}(0)=\frac{32\delta c_{+}-4\delta c_{-}}{\pi
a}+\frac{1-16\delta c_{+}^{2}+2\delta c_{-}^{2}}{\pi}+O(a\log a)\ .$ (6.42)

For instance, for the naïve choice $b_{\pm}=1$, which corresponds to $\delta
c_{\pm}=\mp 1/2$, one obtains for the dispersion relation

$\displaystyle
E(p_{1})^{2}=p_{1}^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\left[-\frac{18}{\pi
a}+O(\log a)\right]+O(g^{-2})\ .$ (6.43)

It is interesting to notice that once we have set $b_{\pm}=1$, the divergence
in the dispersion relation cannot be eliminated by renormalising the remaining
available parameters, i.e. $g$ and $m$. In other words, the choice $b_{\pm}=1$
is not stable under renormalisation. On the other hand, if one allows the
coefficients $b_{\pm}$ to be renormalised along with $m$ and $g$, then the
divergences in the dispersion relation are eliminated, e.g. by choosing

$\displaystyle
b_{+}=1+\frac{1}{g_{R}}\frac{\frac{a\,m_{R}}{8}}{2+\frac{a\,m_{R}}{2}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\
,$ (6.44) $\displaystyle
b_{-}=1-\frac{1}{g_{R}}\frac{1+\frac{5a\,m_{R}}{8}}{2+\frac{a\,m_{R}}{2}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\
,$ (6.45) $\displaystyle
m^{2}=m_{R}^{2}\left[1+\frac{1}{2g_{R}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\right]\
,$ (6.46) $\displaystyle g=g_{R}\left[1+O(g^{-1})\right]\ .$ (6.47)

This choice yields a dispersion relation in the continuum limit of the same
form as equation (6.41), except that the mass $m$ needs to be replaced by its
renormalised counterpart $m_{R}$. One could also see that the one-loop
renormalisation of the coupling constant can be chosen so that the cusp
anomaly is finite. With this discussion, we do not want to imply that the
chosen lattice theory is renormalisable (we do not know this). However, we
conclude that if the lattice theory is renormalisable, then it is not
sufficient to renormalize $m$ and $g$; one also needs to introduce additional
coefficients in the action and either fine-tune their tree-level value or
renormalise them.

##  Chapter 7 Conclusions and Outlook

In the first Chapters (2, 3, 4), the analytic conformal bootstrap was
introduced and applied to two examples: the four-point correlator of operator
insertions on the 1/2-BPS Wilson line in ABJM and the five-point correlator of
operator insertions on the 1/2-BPS Wilson line in $\mathcal{N}=4$ SYM. In the
first example, the superspace structure was developed in order to relate all
the correlators of fields in the displacement multiplet to a single function
$f(z)$. This function was then bootstrapped entirely up to third order in a
strong coupling expansion and to all orders in the double-scaling limit. The
mixing problem for the CFT data was solved at first order to obtain these
results. The CFT data (generically mixed) was then extracted up to third
order.

In the second example, multipoint Ward identities and selection rules were
used to compute the superblock expansion of the five-point correlator of
1/2-BPS operators inserted on the 1/2-BPS Wilson line in $\mathcal{N}=4$ SYM.
This allows for the computation of the first-order correlator through the
analytic conformal bootstrap. These higher-point correlators access a wider
range of CFT data, which were then extracted. The analysis of the different
channels confirms that the anomalous dimension does not contribute to mixing
at this order.

Extensions to this work fall into two parallel categories: bootstrapping
correlators and improving the constraints from the bootstrap. One can easily
find more results to bootstrap; by increasing the order of perturbation, the
number of insertions, or by considering different setups. Higher-order
contributions increase in complexity due to the mixing of operators and
somewhat because of the type of functions appearing in the ansatz. The higher-
point correlators increase in complexity because of the wealth of possible
operators exchanged in the many OPEs. Systems with lower supersymmetry will
have fewer constraints from the corresponding selection rules. As such, these
three extensions will have upper limits as to what can be fixed by the
symmetry.

When looking at improving the bootstrap, many constraints have not yet been
explored. The first category are additional constraints from the symmetries of
the theory. For example, integrated correlator constraints relate the
$n$-point results to functions known through integrability and should exist
for most superconformal defects and for increasing number of insertions [196].
Integrability has also been extremely powerful in determining the spectrum of
the 1/2-BPS setup in $\mathcal{N}=4$ SYM[149]. This symmetry should also
constraint the perturbative correlators through Yangian symmetry. The second
category covers more general defects setups and can be explored in toy models.
Three examples are the constraints from single-valuedness, from the flat-space
limit and from non-perturbative sum rules.

Looking forwards, an immediate extension is the unmixing of the CFT data at
higher orders, which would enable the computation of the correlators at the
next orders in perturbation theory. One way to do this is to look at more
general correlators, such as was done to solve the first-order mixing in the
1/2-BPS Wilson line in ABJM. A particular family of interest in this case are
the 1/3-BPS operators, which are charged under R-symmetry. These form an
infinite tower of protected operators which may have a topological limit.
Additionally, these operators preserve the same symmetries as the 1/2-BPS
operators on the 1/3-BPS Wilson line, whose parallel computation would be
enlightening. A direct extension of the 5-point bootstrap which is underway is