where $u \equiv a/x_{12}$ and

$$c_n(\alpha, t) \equiv \frac{\Gamma(n + \frac{p}{2}) \Gamma(2t - n - \frac{p}{2}) \Gamma(2n - 2t + 2\alpha) \Gamma(3n + \frac{p}{2} - 2t + \alpha)}{(2\alpha + 4n + p - 4t) \Gamma(2n + \alpha) \Gamma(3n + \frac{p}{2} - 2t + 2\alpha)} \quad (C.5)$$

We can close the contour to the left, and pick up poles at $-n-p/2+2t = -k$, $k = 0, 1, 2, 3, \dots$. After summing over all these residues we find

$$R_n(\alpha) \equiv \int_{-i\infty}^{i\infty} \frac{dt}{2\pi i} c_n(\alpha, t) u^{4t-2\alpha-4n-p} \\ = \sum_{k=0}^{\infty} \frac{(-1)^k \Gamma(n + \frac{p}{2}) \Gamma(k + 2n + \alpha) \Gamma(k + n + 2\alpha - \frac{p}{2})}{4k!(\alpha + k + n)\Gamma(2n + \alpha)\Gamma(k + 2n + 2\alpha)} \left(\frac{1}{u^2}\right)^{\alpha+k+n} \quad (C.6)$$

Plugging this result into the expression for $I_p(\alpha)$ we get

$$I_p(\alpha) = \sum_{n=0}^{\infty} \frac{2^{2n+1} \pi^{p/2} \left(\frac{\alpha}{2}\right)_n \left(\frac{\alpha+1}{2}\right)_n}{n! \Gamma\left(\frac{p}{2}\right) \left(\frac{p}{2}\right)_n} R_n(\alpha) \\ = \frac{2^{-2\alpha} \pi^{\frac{p+1}{2}} \Gamma(2\alpha - p/2)}{\Gamma(1+\alpha)\Gamma(\alpha+1/2)} \left(\frac{1}{u^2}\right)^{\alpha} {}_3F_2\left(\alpha, \alpha, 2\alpha - \frac{p}{2}; \alpha + \frac{1}{2}, \alpha + 1; -\frac{1}{4u^2}\right) \quad (C.7)$$

For small $a$ and finite $x_{12}$ (namely small $u$) we find

$$I_p(\alpha) = \frac{\pi^{p/2} \Gamma(\alpha - \frac{p}{2})}{\Gamma(\alpha)} \left( \log(1/u^2) - \psi(\alpha) + \psi\left(\alpha - \frac{p}{2}\right) \right) \\ + u^{2\alpha - p} \left( \frac{\pi^{\frac{p}{2}+2} \csc^2\left(\frac{1}{2}\pi(p-2\alpha)\right) \Gamma(2\alpha - \frac{p}{2})}{\Gamma(\alpha)^2 \Gamma(p-2\alpha+1) \Gamma(-\frac{p}{2}+\alpha+1)^2} + O(u^2) \right), \quad (C.8)$$

where $\psi(x)$ is the digamma function. Similar tricks can be used in order to compute the second integral in (C.1), for which we find

$$\hat{I}_p(\alpha) = \sum_{n,k=0}^{\infty} \frac{(-1)^k \pi^{p/2} \Gamma(k+2n+\alpha) \Gamma(k+n+2\alpha-\frac{p}{2})}{\Gamma(\alpha) \Gamma(k+1) \Gamma(n+1) \Gamma(k+2(n+\alpha))} \left(\frac{1}{u^2}\right)^{\alpha+k+n} \quad (C.9)$$

Therefore after performing the infinite sum we obtain

$$\hat{I}_p(\alpha) = \frac{2^{1-2\alpha} \pi^{\frac{p+1}{2}} \Gamma(2\alpha - p/2)}{\Gamma(\alpha) \Gamma(\alpha + 1/2)} \left(\frac{1}{u^2}\right)^{\alpha} {}_2F_1\left(\alpha, 2\alpha - \frac{p}{2}; \alpha + \frac{1}{2}; -\frac{1}{4u^2}\right) \quad (C.10)$$

For small $u$ we have that

$$\hat{I}_p(\alpha) = \frac{2\pi^{p/2}\Gamma(\alpha - \frac{p}{2})}{\Gamma(\alpha)} + u^{2\alpha - p} \left( \frac{\pi^{\frac{p+1}{2}} 2^{2\alpha - p + 1} \Gamma(\frac{p}{2} - \alpha) \Gamma(2\alpha - \frac{p}{2})}{\Gamma(\alpha)^2 \Gamma(\frac{p}{2} - \alpha + \frac{1}{2})} + O(u^2) \right) \quad (C.11)$$

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
References

[1] A. Antunes et al., Towards bootstrapping RG flows: sine-Gordon in AdS, JHEP 12 (2021) 094
[arXiv:2109.13261] [INSPIRE].

[2] C.G. Callan Jr. and F. Wilczek, Infrared behavior at negative curvature, Nucl. Phys. B 340 (1990) 366 [INSPIRE].

[3] O. Aharony, M. Berkooz, D. Tong and S. Yankielowicz, Confinement in Anti-de Sitter Space, JHEP 02 (2013) 076 [arXiv:1210.5195] [INSPIRE].

[4] D. Carmi, L. Di Pietro and S. Komatsu, A study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200 [arXiv:1810.04185] [INSPIRE].

[5] Ankur, D. Carmi and L. Di Pietro, Scalar QED in AdS, JHEP 10 (2023) 089
[arXiv:2306.05551] [INSPIRE].

[6] A. Kakkar and S. Sarkar, On partition functions and phases of scalars in AdS, JHEP 07 (2022) 089 [arXiv:2201.09043] [INSPIRE].

[7] A. Kakkar and S. Sarkar, Phases of theories with fermions in AdS, JHEP 06 (2023) 009
[arXiv:2303.02711] [INSPIRE].

[8] M. Meineri, J. Penedones and T. Spirig, Renormalization group flows in AdS and the bootstrap program, arXiv:2305.11209 [INSPIRE].

[9] M. Kruczenski, J. Penedones and B.C. van Rees, Snowmass White Paper: S-matrix Bootstrap, arXiv:2203.02421 [INSPIRE].

[10] M.F. Paulos et al., The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133
[arXiv:1607.06109] [INSPIRE].

[11] S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, AdS bulk locality from sharp CFT bounds, JHEP 11 (2021) 164 [arXiv:2106.10274] [INSPIRE].

[12] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (1984)
[DOI:10.1017/cbo9780511622632].

[13] A. Antunes, E. Lauria and B.C. van Rees, A bootstrap study of minimal model deformations, arXiv:2401.06818 [INSPIRE].

[14] A.B. Zamolodchikov, Higher Order Integrals of Motion in Two-Dimensional Models of the Field Theory with a Broken Conformal Symmetry, JETP Lett. 46 (1987) 160 [INSPIRE].

[15] A.B. Zamolodchikov, Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [INSPIRE].

[16] G. Mussardo, Boundary Field Theory, in Statistical Field Theory, G. Mussardo ed., Oxford University Press Oxford (2020), p. 836–860 [DOI:10.1093/oso/9780198788102.003.0021].

[17] S. Fredenhagen, M.R. Gaberdiel and C. Schmidt-Colinet, Bulk flows in Virasoro minimal models with boundaries, J. Phys. A 42 (2009) 495403 [arXiv:0907.2560] [INSPIRE].

[18] A. Recknagel, D. Roggenkamp and V. Schomerus, On relevant boundary perturbations of unitary minimal models, Nucl. Phys. B 588 (2000) 552 [hep-th/0003110] [INSPIRE].

[19] K. Graham, On perturbations of unitary minimal models by boundary condition changing operators, JHEP 03 (2002) 028 [hep-th/0111205] [INSPIRE].
[20] P. Liendo, L. Rastelli and B.C. van Rees, *The Bootstrap Program for Boundary CFT$_d$, JHEP 07 (2013) 113* [arXiv:1210.4258] [INSPIRE].

[21] E. Gerchkovitz et al., *Correlation Functions of Coulomb Branch Operators*, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].

[22] E. Witten, *Multitrace operators, boundary conditions, and AdS / CFT correspondence*, hep-th/0112258 [INSPIRE].

[23] M. Hogervorst, M. Meineri, J. Penedones and K.S. Vaziri, *Hamiltonian truncation in Anti-de Sitter spacetime*, JHEP 08 (2021) 063 [arXiv:2104.10689] [INSPIRE].

[24] C. Copetti, L. Di Pietro, Z. Ji and S. Komatsu, *Taming Mass Gap with Anti-de-Sitter Space*, arXiv:2312.09277 [INSPIRE].

[25] V. Gorbenko, S. Rychkov and B. Zan, *Walking, Weak first-order transitions, and Complex CFTs*, JHEP 10 (2018) 108 [arXiv:1807.11512] [INSPIRE].

[26] N. Levine and M.F. Paulos, *Bootstrapping bulk locality. Part I: Sum rules for AdS form factors*, JHEP 01 (2024) 049 [arXiv:2305.07078] [INSPIRE].

[27] K. Skenderis, *Lecture notes on holographic renormalization*, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].

[28] S. Fichet, *On holography in general background and the boundary effective action from AdS to dS*, JHEP 07 (2022) 113 [arXiv:2112.00746] [INSPIRE].

[29] M. Bañados, E. Bianchi, I. Muñoz and K. Skenderis, *Bulk renormalization and the AdS/CFT correspondence*, Phys. Rev. D 107 (2023) L021901 [arXiv:2208.11539] [INSPIRE].

[30] D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, *Correlation functions in the CFT$_d$/AdS$_{d+1}$ correspondence*, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].

[31] I.R. Klebanov and E. Witten, *AdS / CFT correspondence and symmetry breaking*, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].

[32] M. Henningson and K. Skenderis, *The Holographic Weyl anomaly*, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].

[33] I. Papadimitriou and K. Skenderis, *Thermodynamics of asymptotically locally AdS spacetimes*, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].

[34] E. Lauria, P. Liendo, B.C. Van Rees and X. Zhao, *Line and surface defects for the free scalar field*, JHEP 01 (2021) 060 [arXiv:2005.02413] [INSPIRE].

[35] T. Nishioka, Y. Okuyama and S. Shimamori, *The epsilon expansion of the O(N) model with line defect from conformal field theory*, JHEP 03 (2023) 203 [arXiv:2212.04076] [INSPIRE].

[36] J.L. Cardy, *Scaling and renormalization in statistical physics*, Cambridge University Press (1996) [DOI:10.1017/cbo9781316036440] [INSPIRE].

[37] P.H. Ginsparg, *Applied conformal field theory*, in the proceedings of the *Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena*, Les Houches, France, June 28 – August 05 (1988) [hep-th/9108028] [INSPIRE].

[38] P. Di Francesco, P. Mathieu and D. Senechal, *Conformal Field Theory*, Graduate Texts in Contemporary Physics. Springer-Verlag, New York, U.S.A. (1997) [DOI:10.1007/978-1-4612-2256-9].
[39] G. Mussardo, Statistical field theory: an introduction to exactly solved models in statistical physics. Oxford University Press, New York, U.S.A. (2010)
[DOI: https://doi.org/10.1093/oso/9780198788102.001.0001].

[40] A. Recknagel and V. Schomerus, Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes, Cambridge University Press (2013) [DOI:10.1017/CBO9780511806476] [INSPIRE].

[41] F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].

[42] D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].

[43] D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].

[44] J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].

[45] J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].

[46] J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].

[47] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].

[48] T. Quella, I. Runkel and G.M.T. Watts, Reflection and transmission for conformal defects, JHEP 04 (2007) 095 [hep-th/0611296] [INSPIRE].

[49] M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].

[50] M. Meineri, J. Penedones and A. Rousset, Colliders and conformal interfaces, JHEP 02 (2020) 138 [arXiv:1904.10974] [INSPIRE].

[51] A. Homrich et al., The S-matrix Bootstrap IV: Multiple Amplitudes, JHEP 11 (2019) 076 [arXiv:1905.06905] [INSPIRE].

[52] J.L. Cardy, Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].

[53] A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].

[54] P. Ruelle and O. Verhoeven, Discrete symmetries of unitary minimal conformal theories, Nucl. Phys. B 535 (1998) 650 [hep-th/9803129] [INSPIRE].

[55] N. Ishibashi, The Boundary and Crosscap States in Conformal Field Theories, Mod. Phys. Lett. A 4 (1989) 251 [INSPIRE].

[56] D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].

[57] I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].

[58] I. Runkel, Structure constants for the D series Virasoro minimal models, Nucl. Phys. B 579 (2000) 561 [hep-th/9908046] [INSPIRE].

[59] V.S. Dotsenko and V.A. Fateev, Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].
[60] V.S. Dotsenko and V.A. Fateev, *Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge $c < 1$, Nucl. Phys. B 251 (1985) 691* [INSPIRE].

[61] V.S. Dotsenko and V.A. Fateev, *Operator Algebra of Two-Dimensional Conformal Theories with Central Charge $C \le 1$, Phys. Lett. B 154 (1985) 291* [INSPIRE].

[62] I. Esterlis, A.L. Fitzpatrick and D. Ramirez, *Closure of the Operator Product Expansion in the Non-Unitary Bootstrap*, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].

[63] J.L. Cardy and D.C. Lewellen, *Bulk and boundary operators in conformal field theory*, Phys. Lett. B 259 (1991) 274 [INSPIRE].

[64] C.P. Herzog and A. Shrestha, *Conformal surface defects in Maxwell theory are trivial*, JHEP 08 (2022) 282 [arXiv:2202.09180] [INSPIRE].

[65] C. Behan, L. Di Pietro, E. Lauria and B.C. Van Rees, *Bootstrapping boundary-localized interactions*, JHEP 12 (2020) 182 [arXiv:2009.03336] [INSPIRE].

[66] C. Behan, L. Di Pietro, E. Lauria and B.C. van Rees, *Bootstrapping boundary-localized interactions II. Minimal models at the boundary*, JHEP 03 (2022) 146 [arXiv:2111.04747] [INSPIRE].

[67] T. Nishioka, Y. Okuyama and S. Shimamori, *Comments on epsilon expansion of the $O(N)$ model with boundary*, JHEP 03 (2023) 051 [arXiv:2212.04078] [INSPIRE].