Results for Quantum Diagrams

Chapter VIII: One-loop Modification (Massive)

8.5 Results for Quantum Diagrams

As we attempted the calculation of the box and crossed box diagrams, two issues arose. Firstly, we were unable to calculate the𝐶_{𝜇 𝜈𝛼 𝛽𝛾} integral that we needed for this amplitude. The calculation, which boiled down to solving an arbitrary system

of 12 equations for 12 variables, used over 12GB of RAM from the computer that we were using and still did not finish the calculation.

Additionally, we realized that our method of solving these integrals was for integrals in a specific form that held for all of the integrals except for those used in the box and crossed box diagrams. Specifically, the triangle integrals needed to be in the form

∫ 𝑑⁴ℓ (2𝜋)⁴

𝐷(ℓ, 𝑝₁, ..., 𝑚₁, ...) (ℓ²−𝑚²

1) ( (ℓ±𝑝₁) −𝑚²

2) ( (ℓ±𝑝₂) −𝑚²

3) (8.16)

However, since the diagrams have a fourth propagator, and thereby a third momentum 𝑝₃, the Feyncalc function ScalarProductCancel put the triangle integrals into a form more like

∫ 𝑑⁴ℓ (2𝜋)⁴

𝐷(ℓ, 𝑝₁, ..., 𝑚₁, ...) (ℓ²−𝑚²

1) ( (ℓ± 𝑝₁± 𝑝₃) −𝑚²

2) ( (ℓ± 𝑝₂±𝑝₃) −𝑚²

3) (8.17) Thus, in order to continue to use the Passerino-Veltman decomposition, we would need to include all the permutations of including 𝑝₃in the solution for the integral.

While not physically impossible, this dramatically increases the number of coef- ficients needing to be solved for. In fact, in this method, when we tried to solve for 𝐶_{𝜇 𝜈𝛼}, it required a system of 13 equations with 13 variables. Since we had already not been able to solve the system of 12 equations and variables, we knew that these calculations were simply infeasible given our limited resources. Once we realized this, we knew we would be unable to finish the quantum amplitude calcu- lations. However, even though we were unable to calculate the full modification to the quantum potential due to the difficulties with the box and crossed box diagrams, we were still able to make excellent progress on this calculation. We here list the uncontracted amplitudes for the box and crossed box diagrams as well as the full amplitudes and potentials for all the other quantum diagrams.

The Box and Crossed Box Diagrams

Unfortunately, due to the extra complication of the various integrals needed by these two diagrams, we were unable to find the full contracted and simplified amplitudes.

We here show the uncontracted versions 𝑖 𝑀_2𝑎 =

∫ 𝑑⁴𝑙 (2𝜋)⁴𝜏

𝜇 𝜈

1 (𝑘₁, 𝑘₁+𝑙 , 𝑚₁)𝜏

𝜌 𝜎

1 (𝑘₁+𝑙 , 𝑘₂, 𝑚₁)𝜏

𝛼 𝛽

1 (𝑘₃, 𝑘₃−𝑙 , 𝑚₂)𝜏

𝛾 𝛿

1 (𝑘₃−𝑙 , 𝑘₄, 𝑚₂)

×

"

𝑖 (𝑘₁+𝑙)²−𝑚²

1

# "

𝑖 (𝑘₃−𝑙)²−𝑚²

2

#

𝑖 𝑃𝑚_{𝜇 𝜈𝛼 𝛽} 𝑙²−𝑚²

𝑖 𝑃𝑚_{𝜌 𝜎 𝛾 𝛿} (𝑙+𝑞)²−𝑚²

(8.18) 𝑖 𝑀_2𝑏 =

∫ 𝑑⁴𝑙 (2𝜋)⁴𝜏

𝜇 𝜈

1 (𝑘₁, 𝑘₁+𝑙 , 𝑚₁)𝜏

𝜌 𝜎

1 (𝑘₁+𝑙 , 𝑘₂, 𝑚₁)𝜏

𝛾 𝛿

1 (𝑘₃, 𝑘₄+𝑙 , 𝑚₂)𝜏

𝛼 𝛽

1 (𝑘₄+𝑙 , 𝑘₄, 𝑚₂)

×

"

𝑖 (𝑘₁+𝑙)²−𝑚²

1

# "

𝑖 (𝑘₃−𝑙)²−𝑚²

2

#

𝑖 𝑃𝑚_{𝜇 𝜈𝛼 𝛽} 𝑙²−𝑚²

𝑖 𝑃𝑚_{𝜌 𝜎 𝛾 𝛿} (𝑙+𝑞)²−𝑚²

(8.19) The Double-Seagull Diagram

Given the change to a massive graviton, the new uncontracted amplitude for the double-seagull diagram (see 6.3) takes the form

𝑀_4𝑎(𝑞) = 1 2!

∫ 𝑑⁴𝑙 (2𝜋)⁴𝜏

𝛼 𝛽𝛾 𝛿

2 (𝑘₁, 𝑘₂, 𝑚₁)𝜏

𝜇 𝜈 𝜎 𝜌

2 (𝑘₃, 𝑘₄, 𝑚₂)

𝑖 𝑃𝑚_{𝛾 𝛿𝜎 𝜌} 𝑙²−𝑚²

𝑖 𝑃𝑚_{𝛼 𝛽 𝜇 𝜈} (𝑙+𝑞)²−𝑚²

(8.20) We note the inclusion of the symmetry factor 1/2! due to the symmetry of the diagram. We were able to simplify the amplitude via Mathematica and remove all analytic and higher order nonanalytic terms. Under the nonrelativistic approxima- tion, the amplitude takes the form

𝑖 𝑀_4𝑎 =−32𝑖𝐺² 45𝑞²

271𝑚²𝑚₁𝑚₂log

−𝑚² 𝑞²

−168𝑚²𝑚₁𝑚₂

+ 2𝑖𝐺² 135𝑚₁𝑚₂

11595𝑚²

1𝑚²

2−1296𝑚²𝑚²

1−4926𝑚²𝑚₁𝑚₂+1948𝑚²𝑚₁𝑚₂

−1296𝑚²𝑚²

2

log

−𝑚² 𝑞²

(8.21) With a Fourier transform, we derive the contribution to the potential for the double seagull diagram

𝑉_4𝑎 = 8672𝐺²𝑚²𝑚₁𝑚₂ 45

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞² 𝑞²

− 1344 45𝑞²

𝐺²𝑚²𝑚₁𝑚₂ 𝑟

+ 𝐺²

135𝑚₁𝑚₂𝜋𝑟³

11595𝑚²

1𝑚²

2−1296𝑚²𝑚²

1−4926𝑚²𝑚₁𝑚₂ +1948𝑚²𝑚₁𝑚₂−1296𝑚²𝑚²

2

(8.22)

The Fully Quantum Vertex Correction Diagrams

We next examine the two vertex correction diagrams (see 5.cd in 6.4) which only contribute to the quantum portion of the amplitude and potential. Again, the changes to the uncontracted amplitude are minor and we still include the symmetry factor 1/2!.

𝑀_5𝑐(𝑞) = 1 2!

∫ 𝑑⁴𝑙 (2𝜋)⁴𝜏

𝜆 𝜅 𝜙𝜖

2 (𝑘₃, 𝑘₄, 𝑚₂)𝜏

𝛼 𝛽

1 (𝑘₁, 𝑘₂, 𝑚₁)𝜏^{(𝛾 𝛿)}

𝜇 𝜈 𝜌 𝜎 3 (𝑙 ,−𝑞)

×

𝑖 𝑃𝑚_{𝜇 𝜈𝜆 𝜅}

𝑙²−𝑚²

𝑖 𝑃𝑚_{𝜎 𝜌 𝜙𝜖} (𝑙+𝑞)²−𝑚²

𝑖 𝑃𝑚_{𝛼 𝛽𝛾 𝛿} 𝑞²−𝑚²

(8.23) 𝑀_5𝑑(𝑞) = 1

2!

∫ 𝑑⁴𝑙 (2𝜋)⁴𝜏

𝜌 𝜎 𝜇 𝜈

2 (𝑘₁, 𝑘₂, 𝑚₁)𝜏^{𝜆 𝜅}

1 (𝑘₃, 𝑘₄, 𝑚₂)𝜏⁽

𝜙𝜖)𝛼 𝛽𝛾 𝛿 3 (−𝑙 , 𝑞)

×

𝑖 𝑃𝑚_{𝜇 𝜈 𝛾 𝛿}

𝑙²−𝑚²

𝑖 𝑃𝑚_{𝜎 𝜌𝛼 𝛽} (𝑙+𝑞)²−𝑚²

𝑖 𝑃𝑚_{𝜆 𝜅 𝜙𝜖} 𝑞²−𝑚²

(8.24) These two diagrams simplify dramatically once run through the Mathematica pipeline. The nonrelativistic, Taylor expanded amplitudes become

𝑀_5𝑐 = 8𝑖𝐺² 405 𝑞²−𝑚²

4496𝑚²𝑚₁𝑚₂log

−𝑚² 𝑞²

+2145𝑚²𝑚₁𝑚₂

(8.25) 𝑀_5𝑑 = 8𝑖𝐺²

405 𝑞²−𝑚²

5922𝑚²𝑚₁𝑚₂log

−𝑚² 𝑞²

+85𝑚²𝑚₁𝑚₂

(8.26) We can then Fourier transform the expression to get the quantum contribution to the potential

𝑉_5𝑐 =−35968𝐺²𝑚²𝑚₁𝑚₂ 405

∫ 𝑑³𝑞 (2𝜋)³

𝑒^{𝑖𝒒·𝒓}

log −𝑞²

𝑞²−𝑚² + 286 27𝜋

𝐺²𝑚²𝑚₁𝑚₂𝑒^−𝑚𝑟 𝑟

(8.27) 𝑉_5𝑑 =−47376𝐺²𝑚²𝑚₁𝑚₂

405

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞² 𝑞²−𝑚² + 34

81𝜋

𝐺²𝑚²𝑚₁𝑚₂𝑒^−𝑚𝑟 𝑟

(8.28) Adding the two potentials together, we get

𝑉_5𝑐+5𝑑 =−83344𝐺²𝑚²𝑚₁𝑚₂ 405

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞²

𝑞²−𝑚² + 892 81𝜋

𝐺²𝑚²𝑚₁𝑚₂𝑒^−𝑚𝑟 𝑟

(8.29) The Vacuum Polarization Diagrams

Now for the vacuum polarization diagrams, we are no longer able to use the vacuum polarization tensor as we did for the massless case. Instead, we separately calculate

the two vacuum polarization diagrams with a graviton and ghost loop, respectively (see 6.5). For the ghost loop diagram, we use the massless ghost propagator and the 2ghost-graviton vertex factor gotten from Holstein as seen in Appendix A [21]. The uncontracted amplitudes take the form seen below

𝑀_6𝑎(𝑞) = 1 2!

∫ 𝑑⁴ (2𝜋)⁴𝜏

𝜌 𝜎

1 (𝑘₁, 𝑘₂, 𝑚₁)𝜏

𝜆𝜂𝛼 𝛽

3 𝜙𝜖(−ℓ,−𝑞)𝜏

𝜇 𝜈𝜓 𝜉

3 𝜄𝜃(𝑙 , 𝑞)𝜏

𝛾 𝛿

1 (𝑘₃, 𝑘₄, 𝑚₂)

×

𝑖 𝑃𝑚_{𝜌 𝜎𝜆𝜂} 𝑞²−𝑚²

𝑖 𝑃𝑚_{𝛼 𝛽𝜓 𝜉} (ℓ+𝑞)²−𝑚²

𝑖 𝑃𝑚_{𝜙𝜖 𝜄𝜃} (−ℓ)²−𝑚²

𝑖 𝑃𝑚_{𝜇 𝜈 𝛾 𝛿} 𝑞²−𝑚²

(8.30) 𝑀_6𝑏(𝑞) = 1

2!

∫ 𝑑⁴ (2𝜋)⁴𝜏

𝜌 𝜎

1 (𝑘₁, 𝑘₂, 𝑚₁)𝜏

𝜆𝜂𝛼 𝜙

𝑔 𝜙𝜖(−ℓ,−𝑞)𝜏

𝜇 𝜈 𝛽𝜖

𝑔 𝜄𝜃(𝑙 , 𝑞)𝜏

𝛾 𝛿

1 (𝑘₃, 𝑘₄, 𝑚₂)

×

𝑖 𝑃𝑚_{𝜌 𝜎𝜆𝜂}

𝑞²−𝑚²

𝑖𝜂_{𝛼 𝛽} (ℓ+𝑞)²−𝑚²

𝑖𝜂_𝜙𝜖 (−ℓ)²−𝑚²

𝑖 𝑃𝑚_{𝜇 𝜈 𝛾 𝛿} 𝑞²−𝑚²

(8.31) After undergoing contraction and simplification, we integrate and Taylor expand.

We note that the ghost is massless, so we use the integrals from [5] instead of our newly calculated ones. The simplified version of the amplitudes take the form

𝑖 𝑀_6𝑎 = 𝑖𝐺² 405 𝑞²−𝑚²

54469𝑚²𝑚₁𝑚₂log

−𝑚² 𝑞²

+4124𝑚²𝑚₁𝑚₂

(8.32) 𝑖 𝑀_6𝑏 =−56𝑖𝐺²𝑚²𝑚₁𝑚₂

45 𝑞²−𝑚² log

−𝑞²

(8.33) With a Fourier transform, these expressions become the expected potential terms

𝑉_6𝑎 =−54469𝐺²𝑚²𝑚₁𝑚₂ 405

∫ 𝑑³𝑞 (2𝜋)³

𝑒^{𝑖𝒒·𝒓}

log −𝑞²

𝑞²−𝑚² + 1031 405𝜋

𝐺²𝑚²𝑚₁𝑚₂𝑒^−𝑚𝑟 𝑟

(8.34) 𝑉_6𝑏 =−56𝐺²𝑚²𝑚₁𝑚₂

45

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞²

𝑞²−𝑚² (8.35)

C h a p t e r 9

FUTURE WORK

While we have found a satisfying end to this project, there are many more avenues down which it can be taken in the future. While we do not expect the quantum terms to contribute majorly to calculations of black hole inspirals, it is possible that they could be larger than expected. It is thus important to finish running all of the quantum diagrams through our Mathematica pipeline. In order to finish the last two diagrams, the box and cross box, we would need to calculate all the integrals which we could not find before. This would probably require a computer with more processing power and memory than the one we were using. By finishing this calculation, we would thus be able to compare both the classical and quantum terms to those in the massless graviton case and see how their dynamics differ.

Once this calculation is finished, the next step would be to create inspiral models from these potential terms and apply them to LIGO data. By comparing their predictions, it is possible to test the theory of massive gravity against general relativity to see which one fits the physical data better.

One way to confirm our results would be to rederive these amplitudes and thereby potentials via color-kinematic duality between gluons and gravitons. This provides a method to simplify the calculation, while also independently confirm our results.

We discussed this option further in Appendix E.

Another possible path forward would be to include extra interactions that would be suggested by the massive graviton theory. Since a massive graviton is predicted in multiple theories beyond the Standard Model, it could be important to add the other particle interactions and see how that might change the resultant potential. It would thereby produce a good means of testing some of these theories in addition to the basic massive gravity theory.

BIBLIOGRAPHY

[1] Yashar Akrami, S.F. Hassan, Frank Könnig, Angnis Schmidt-May, and Adam R. Solomon. Bimetric gravity is cosmologically viable. Phys. Lett.

B, 748:37–44, 2015. doi: 10.1016/j.physletb.2015.06.062.

[2] Zvi Bern, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng. Black Hole Binary Dynamics from the Double Copy and Effective Theory. JHEP, 10:206, 2019. doi: 10.1007/JHEP10(2019)206.

[3] L. Bernus, O. Minazzoli, A. Fienga, M. Gastineau, J. Laskar, and P. De- ram. Constraining the mass of the graviton with the planetary ephemeris INPOP. Phys. Rev. Lett., 123(16):161103, 2019. doi: 10.1103/PhysRevLett.

123.161103.

[4] N. Bjerrum-Bohr, John Donoghue, and B. Holstein. Quantum corrections to the schwarzschild and kerr metrics. Physical Review D, 68, 11 2002. doi:

10.1103/PhysRevD.68.084005.

[5] N. E. J Bjerrum-Bohr, John F. Donoghue, and Barry R. Holstein. Quan- tum gravitational corrections to the nonrelativistic scattering potential of two masses. Phys. Rev., D67:084033, 2003. doi: 10.1103/PhysRevD.71.069903, 10.1103/PhysRevD.67.084033. [Erratum: Phys. Rev.D71,069903(2005)].

[6] D.G. Boulware and Stanley Deser. Can gravitation have a finite range? Phys.

Rev. D, 6:3368–3382, 1972. doi: 10.1103/PhysRevD.6.3368.

[7] Clifford Cheung. TASI Lectures on Scattering Amplitudes. In Proceed- ings, Theoretical Advanced Study Institute in Elementary Particle Physics : Anticipating the Next Discoveries in Particle Physics (TASI 2016): Boul- der, CO, USA, June 6-July 1, 2016, pages 571–623, 2018. doi: 10.1142/

9789813233348_0008.

[8] S.R. Choudhury, Girish C. Joshi, S. Mahajan, and Bruce H.J. McKellar. Prob- ing large distance higher dimensional gravity from lensing data. Astropart.

Phys., 21:559–563, 2004. doi: 10.1016/j.astropartphys.2004.04.001.

[9] Diego A.R. Dalvit and Francisco D. Mazzitelli. Running coupling constants, Newtonian potential and nonlocalities in the effective action.Phys. Rev. D, 50:

1001–1009, 1994. doi: 10.1103/PhysRevD.50.1001.

[10] Ganesh Devaraj and Robin G. Stuart. Reduction of one loop tensor form- factors to scalar integrals: A General scheme. Nucl. Phys., B519:483–513, 1998. doi: 10.1016/S0550-3213(98)00035-2.

[11] John F. Donoghue. Introduction to the effective field theory description of gravity. In Advanced School on Effective Theories Almunecar, Spain, June 25-July 1, 1995, 1995.

[12] John F. Donoghue. The effective field theory treatment of quantum gravity.

AIP Conf. Proc., 1483(1):73–94, 2012. doi: 10.1063/1.4756964.

[13] R.Keith Ellis and Giulia Zanderighi. Scalar one-loop integrals for QCD.JHEP, 02:002, 2008. doi: 10.1088/1126-6708/2008/02/002.

[14] R.Keith Ellis, Zoltan Kunszt, Kirill Melnikov, and Giulia Zanderighi. One- loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts. Phys. Rept., 518:141–250, 2012. doi: 10.1016/j.physrep.2012.01.008.

[15] Marc H. Goroff and Augusto Sagnotti. The Ultraviolet Behavior of Einstein Gravity. Nucl. Phys., B266:709–736, 1986. doi: 10.1016/0550-3213(86) 90193-8.

[16] David J Griffiths. Introduction to elementary particles; 2nd rev. version.

Physics textbook. Wiley, New York, NY, 2008. URLhttps://cds.cern.

ch/record/111880.

[17] S.F. Hassan and Rachel A. Rosen. On Non-Linear Actions for Massive Gravity.

JHEP, 07:009, 2011. doi: 10.1007/JHEP07(2011)009.

[18] S.F. Hassan and Rachel A. Rosen. Bimetric Gravity from Ghost-free Massive Gravity. JHEP, 02:126, 2012. doi: 10.1007/JHEP02(2012)126.

[19] Kurt Hinterbichler. Theoretical Aspects of Massive Gravity. Rev. Mod. Phys., 84:671–710, 2012. doi: 10.1103/RevModPhys.84.671.

[20] Barry R. Holstein and Andreas Ross. Spin Effects in Long Range Electromag- netic Scattering. 2008.

[21] Barry R. Holstein and Andreas Ross. Spin Effects in Long Range Gravitational Scattering. 2008.

[22] Tristan Hübsch.Advanced Concepts in Particle and Field Theory. Cambridge University Press, 2015. doi: 10.1017/CBO9781316160725.

[23] Y. Iwasaki. Quantum theory of gravitation vs. classical theory. - fourth-order potential. Prog. Theor. Phys., 46:1587–1609, 1971. doi: 10.1143/PTP.46.

1587.

[24] Henrik Johansson and Alexander Ochirov. Color-Kinematics Duality for QCD Amplitudes. JHEP, 01:170, 2016. doi: 10.1007/JHEP01(2016)170.

[25] Henrik Johansson and Alexander Ochirov. Double copy for massive quantum particles with spin. JHEP, 09:040, 2019. doi: 10.1007/JHEP09(2019)040.

[26] I.B. Khriplovich and G.G. Kirilin. Quantum power correction to the Newton law. J. Exp. Theor. Phys., 95(6):981–986, 2002. doi: 10.1134/1.1537290.

[27] Michele Levi. Binary dynamics from spin1-spin2 coupling at fourth post- Newtonian order. Phys. Rev. D, 85:064043, 2012. doi: 10.1103/PhysRevD.

85.064043.

[28] R. Mertig, M. Bohm, and Ansgar Denner. FEYN CALC: Computer algebraic calculation of Feynman amplitudes. Comput. Phys. Commun., 64:345–359, 1991. doi: 10.1016/0010-4655(91)90130-D.

[29] Ivan J. Muzinich and Stamatis Vokos. Long range forces in quantum gravity.

Phys. Rev. D, 52:3472–3483, 1995. doi: 10.1103/PhysRevD.52.3472.

[30] J. C. Romao. Advanced Quantum Field Theory. Instituto Superior Tecnico, 2019. URL http://porthos.tecnico.ulisboa.pt/Public/textos/

tca.pdf.

[31] Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. New Develop- ments in FeynCalc 9.0. Comput. Phys. Commun., 207:432–444, 2016. doi:

10.1016/j.cpc.2016.06.008.

[32] Mark Srednicki. Quantum Field Theory. Cambridge Univ. Press, Cambridge, 2007. URLhttps://cds.cern.ch/record/1019751.

[33] K. S. Stelle. Classical Gravity with Higher Derivatives. Gen. Rel. Grav., 9:

353–371, 1978. doi: 10.1007/BF00760427.

[34] Gerard ’t Hooft and M.J.G. Veltman. One loop divergencies in the theory of gravitation. Ann. Inst. H. Poincare Phys. Theor. A, 20:69–94, 1974.

[35] M.J.G. Veltman. Quantum Theory of Gravitation. Conf. Proc. C, 7507281:

265–327, 1975.

A p p e n d i x A

VERTICES AND PROPAGATORS

Below we list the Feynman rules that were used in our calculation. For a derivation, see [4, 11, 12].

A.1 Massive Scalar Propogator

= 𝑖

𝑞²−𝑚²+𝑖𝜖

(A.1) A.2 Massless Graviton Propogator

= 𝑖 𝑃^{𝛼 𝛽𝛾 𝛿} 𝑞²+𝑖𝜖

(A.2) where

𝑃^{𝛼 𝛽𝛾 𝛿} = 1

2[𝜂^𝛼𝛾𝜂^𝛽𝛿+𝜂^𝛼𝛿𝜂^𝛽𝛾− 2

𝐷−2𝜂^{𝛼 𝛽}𝜂^{𝛾 𝛿}] (A.3) A.3 Massive Graviton Propagator

= 𝑖 𝑃

𝛼 𝛽𝛾 𝛿 𝑚 (𝑞, 𝑚)

𝑞²+𝑖𝜖

(A.4) where

𝑃

𝛼 𝛽𝛾 𝛿

𝑚 (𝑞, 𝑚) = 1 2

𝐹

𝛼𝛾

𝑚 (𝑞, 𝑚)𝐹

𝛽𝛿

𝑚 (𝑞, 𝑚) +𝐹^𝛼𝛿

𝑚 (𝑞, 𝑚)𝐹

𝛽𝛾

𝑚 (𝑞, 𝑚) − 2 𝐷−1𝐹

𝛼 𝛽

𝑚 (𝑞, 𝑚)𝐹

𝛾 𝛿 𝑚 (𝑞, 𝑚)

(A.5) and

𝐹

𝛼 𝛽

𝑚 (𝑞, 𝑚) =𝜂^{𝛼 𝛽}+ 𝑞^𝛼𝑞^𝛽 𝑚²

(A.6) A.4 Massless Ghost Propagator

= 𝑖𝜂^{𝛼 𝛽} 𝑝²+𝑖𝜖

(A.7)

A.5 2 Scalar, 1 Graviton Vertex Factor

=𝜏

𝜇 𝜈

1 (𝑝, 𝑝⁰, 𝑚) where

𝜏

𝜇 𝜈

1 (𝑝, 𝑝⁰, 𝑚)=−𝑖 𝜅 2

𝑝^𝜇𝑝^0𝜈+ 𝑝^0𝜇𝑝^𝜈 −𝜂^{𝜇 𝜈}( (𝑝· 𝑝⁰) −𝑚²)

(A.8) A.6 2 Scalar, 2 Graviton Vertex Factor

=𝜏

𝜇 𝜈 𝜌 𝜎

2 (𝑝, 𝑝⁰, 𝑚) where

𝜏

𝜇 𝜈 𝜌 𝜎

2 (𝑝, 𝑝⁰, 𝑚) =𝑖 𝜅²

𝐼^{𝜇 𝜈𝛼𝛿}𝐼

𝜌 𝜎 𝛽 𝛿 − 1

4

𝜂^{𝜇 𝜈}𝐼^{𝜌 𝜎𝛼 𝛽}+𝜂^{𝜌 𝜎}𝐼^{𝜇 𝜈𝛼 𝛽} (𝑝_𝛼𝑝⁰

𝛽+ 𝑝⁰

𝛼𝑝_𝛽)

− 1 2

𝐼^{𝜇 𝜈 𝜌 𝜎} − 1

2𝜂^{𝜇 𝜈}𝜂^{𝜌 𝜎}

[(𝑝· 𝑝⁰) −𝑚²]

(A.9) and

𝐼^{𝛼 𝛽𝛾 𝛿} = 1

2(𝜂^𝛼𝛾𝜂^𝛽𝛿+𝜂^𝛼𝛿𝜂^𝛽𝛾) (A.10)

A.7 3 Graviton Vertex Factor

=𝜏₃

𝜇 𝜈

𝛼 𝛽𝛾 𝛿(𝑘 , 𝑞) where

𝜏₃

𝜇 𝜈

𝛼 𝛽𝛾 𝛿(𝑘 , 𝑞) =−𝑖 𝜅

2 𝑃^{𝛼 𝛽𝛾 𝛿}

𝑘^𝜇𝑘^𝜈+ (𝑘 +𝑞)^𝜇(𝑘+𝑞)^𝜈+𝑞^𝜇𝑞^𝜈 − 3 2𝜂^{𝜇 𝜈}𝑞²

+2𝑞_𝜆𝑞_𝜎

𝐼 ^𝜎𝜆

𝛼 𝛽 𝐼 ^{𝜇 𝜈}

𝛾 𝛿 +𝐼 ^𝜎𝜆

𝛾 𝛿 𝐼 ^{𝜇 𝜈}

𝛼 𝛽 −𝐼 ^𝜇𝜎

𝛼 𝛽

𝐼 ^𝜈𝜆

𝛾 𝛿 −𝐼 ^𝜇𝜎

𝛾 𝛿

𝐼 ^𝜈𝜆

𝛼 𝛽

+

𝑞_𝜆𝑞^𝜇

𝜂_{𝛼 𝛽}𝐼 ^𝜈𝜆

𝛾 𝛿 +𝜂𝛾 𝛿 𝐼 ^𝜈𝜆

𝛼 𝛽

+𝑞_𝜆𝑞^𝜈

𝜂_{𝛼 𝛽}𝐼

𝜇𝜆

𝛾 𝛿 +𝜂𝛾 𝛿 𝐼

𝜇𝜆 𝛼 𝛽

−𝑞²

𝜂_{𝛼 𝛽}𝐼

𝜇 𝜈

𝛾 𝛿 +𝜂𝛾 𝛿 𝐼

𝜇 𝜈 𝛼 𝛽

−𝜂^{𝜇 𝜈}𝑞_𝜆𝑞^𝜎

𝜂_{𝛼 𝛽}𝐼 ^𝜎𝜆

𝛾 𝛿 +𝜂𝛾 𝛿 𝐼 ^𝜎𝜆

𝛼 𝛽

+

2𝑞_𝜆

𝐼 ^𝜆𝜎

𝛼 𝛽 𝐼 ^𝜈

𝛾 𝛿𝜎 (𝑘 −𝑞)^𝜇 +𝐼 ^𝜆𝜎

𝛼 𝛽 𝐼

𝜇

𝛾 𝛿𝜎 (𝑘−𝑞)^𝜈−𝐼 ^𝜆𝜎

𝛾 𝛿 𝐼 ^𝜈

𝛼 𝛽𝜎 𝑘^𝜇−𝐼 ^𝜆𝜎

𝛾 𝛿 𝐼

𝜇 𝛼 𝛽𝜎 𝑘^𝜈

+𝑞²

𝐼

𝜇 𝛼 𝛽𝜎 𝐼 ^{𝜈 𝜎}

𝛾 𝛿 +𝐼

𝜇 𝛾 𝛿𝜎 𝐼 ^{𝜈 𝜎}

𝛼 𝛽

+𝜂^{𝜇 𝜈}𝑞_𝜎𝑞_𝜆

𝐼

𝜆 𝜌

𝛼 𝛽 𝐼 ^𝜎

𝛾 𝛿 𝜌 +𝐼

𝜆 𝜌

𝛾 𝛿 𝐼 ^𝜎

𝛼 𝛽 𝜌

+

(𝑘²+(𝑘−𝑞)²)

𝐼

𝜇𝜎

𝛼 𝛽 𝐼 ^𝜈

𝛾 𝛿𝜎 +𝐼

𝜇𝜎

𝛾 𝛿 𝐼 ^𝜈

𝛼 𝛽𝜎 − 1

2𝜂^{𝜇 𝜈}𝑃_{𝛼 𝛽𝛾 𝛿}

− 𝐼

𝜇 𝜈

𝛾 𝛿 𝜂_{𝛼 𝛽}𝑘²+𝐼

𝜇 𝜈

𝛼 𝛽 𝜂_{𝛾 𝛿}(𝑘−𝑞)²! (A.11) A.8 2 Ghost, 1 Graviton Vertex Factor

=𝜏

𝜇 𝜈

𝑔 𝛼 𝛽(𝑘 , 𝑞) where

𝜏_𝑔

𝜇 𝜈

𝛼 𝛽(𝑘 , 𝑞)= 𝑖 𝜅 2

(𝑘²+ (𝑘+𝑞)²+𝑞²)𝐼

𝜇 𝜈

𝛼 𝛽 +2𝜂_{𝛼 𝛽}𝑘^𝜆(𝑘 +𝑞)^𝜎𝑃

𝜇 𝜈 𝜆𝜎

+2𝑞_𝛼𝑘^𝜆𝐼 ^{𝜇 𝜈}

𝛽𝜆 −2𝑞_𝛽(𝑘 +𝑞)^𝜆𝐼 ^{𝜇 𝜈}

𝛼𝜆 +𝑞_𝛼𝑞_𝛽𝜂^{𝜇 𝜈}

(A.12)

A p p e n d i x B

INTEGRALS USED IN THE MASSLESS CASE

We recreate the integrals used in the massless case here due to some copy errors noted in the integrals given by [5].

B.1 Bubble Integrals

𝐼 =

∫ 𝑑⁴ℓ (2𝜋)⁴

1

ℓ²(ℓ+𝑞)² = 𝑖 16𝜋²

[−𝐿] +... (B.1)

𝐼_𝜇 =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇

ℓ²(ℓ+𝑞)² = 𝑖 16𝜋²

1 2𝑞_𝜇𝐿

+... (B.2)

𝐼_{𝜇 𝜈} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈

ℓ²(ℓ+𝑞)² = 𝑖 16𝜋²

1

12𝐿 𝑞²𝜂_{𝜇 𝜈}− 1 3𝐿 𝑞_𝜇𝑞_𝜈

+... (B.3)

𝐼_{𝜇 𝜈𝛼} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼

ℓ²(ℓ+𝑞)² = 𝑖 16𝜋²

− 1

24𝐿 𝑞²3𝜂(𝜇 𝜈𝑞_𝛼) + 1

4𝐿 𝑞_𝜇𝑞_𝜈𝑞_𝛼

+... (B.4) 𝐼_{𝜇 𝜈𝛼 𝛽} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼ℓ_𝛽

ℓ²(ℓ+𝑞)² = 𝑖 16𝜋²

− 1

240𝐿 𝑞⁴3𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}) + 1

40𝐿 𝑞²6𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽) − 1

5𝐿 𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽+... (B.5)

where 𝐿 = 𝑙 𝑜𝑔(−𝑞²) and 𝑆 = √^𝜋2

−𝑞², and analytic and higher order nonanalytic terms are represented by the elipses.

B.2 Triangle Integrals

𝐽 =

∫ 𝑑⁴ℓ (2𝜋)⁴

1

ℓ²(ℓ+𝑞)²( (ℓ+𝑘)²−𝑚²

1)

= 𝑖 32𝜋²𝑚²

1

[−𝐿−𝑆] +... (B.6)

𝐽_𝜇 =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇

ℓ²(ℓ+𝑞)²( (ℓ+𝑘)²−𝑚²

1)

= 𝑖 32𝜋²𝑚²

1

𝑘_𝜇

−1− 1 2

𝑞² 𝑚²

1

𝐿− 1 4

𝑞² 𝑚²

1

𝑆

+𝑞_𝜇𝑞_𝜈

𝐿+ 1 2𝑆

+... (B.7)

𝐽_{𝜇 𝜈} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈

ℓ²(ℓ+𝑞)²( (ℓ+𝑘)²−𝑚²

1)

= 𝑖 32𝜋²𝑚²

1

𝑘_𝜇𝑘_𝜈

− 1 2

𝑞² 𝑚²

1

𝐿− 1 8

𝑞² 𝑚²

1

𝑆

+𝑞_𝜇

−𝐿− 3 8𝑆

+2𝑞(𝜇𝑘_𝜈) 1 2 + 1

2 𝑞² 𝑚²

1

𝐿+ 3 16

𝑞² 𝑚²

1

𝑆

+𝑞²𝜂_{𝜇 𝜈} 1 4𝐿+ 1

8𝑆

+... (B.8)

𝐽_{𝜇 𝜈𝛼} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼

ℓ²(ℓ+𝑞)²( (ℓ+𝑘)²−𝑚²

1)

= 𝑖 32𝜋²𝑚²

1

𝑘_𝜇𝑘_𝜈𝑘_𝛼

− 1 6

𝑞² 𝑚²

1

𝐿

+𝑞_𝜇𝑞_𝜈𝑞_𝛼

𝐿+ 5 16𝑆

+3𝑞(𝜇𝑘_𝜈𝑘_𝛼)1 3

𝑞² 𝑚²

1

𝐿+ 1 16

𝑞² 𝑚²

1

𝑆

+3𝑞(𝜇𝑞_𝜈𝑘_𝛼)

− 1 3− 1

2 𝑞² 𝑚²

1

𝐿− 5 32

𝑞² 𝑚²

1

𝑆

+3𝜂(𝜇 𝜈𝑘_𝛼) 𝑞² 1 12𝑞²𝐿

+3𝜂(𝜇 𝜈𝑞_𝛼)

− 1

6𝑞²𝐿− 1 16𝑞²𝑆

+... (B.9)

where 𝐿 = 𝑙 𝑜𝑔(−𝑞²) and 𝑆 = √^𝜋2

−𝑞²

, and analytic and higher order non-analytic terms are represented by the ellipses. We utilize the symmetrization convention used by [20] where

𝐴(𝜇₁𝜇₂𝜇₃... 𝜇_𝑛= 1

𝑛! 𝐴(𝜇₁𝜇₂𝜇₃... 𝜇_𝑛+𝐴(𝜇₂𝜇₁𝜇₃... 𝜇_𝑛+...

(B.10) For example, this means that 3𝜂(𝜇 𝜈𝑞_𝛼) =𝜂_{𝜇 𝜈}𝑞_𝛼+𝜂_𝜇𝛼𝑞_𝜈+𝜂_𝛼𝜈𝑞_𝜇.

B.3 Box and Cross Box Integrals

𝐾 =

∫ 𝑑⁴ℓ (2𝜋)⁴

1 ℓ²(ℓ+𝑞)²( (ℓ+𝑘₁)²−𝑚²

1) ( (ℓ−𝑘₃)²−𝑚²

2)

= 𝑖

16𝜋²𝑚₁𝑚₂ 𝐿

𝑞²

1− 𝑠−𝑠₀ 6𝑚₁𝑚₂

−𝑖2𝜋 𝐿 𝑞²

1 2√

𝑚₁𝑚₂

√ 𝑠−𝑠₀

1− 𝑠−𝑠₀ 8𝑚₁𝑚₂

(B.11) 𝐾⁰=

∫ 𝑑⁴ℓ (2𝜋)⁴

1 ℓ²(ℓ+𝑞)²( (ℓ+𝑘₁)²−𝑚²

1) ( (ℓ+𝑘₄)²−𝑚²

2)

= 𝑖

16𝜋²𝑚₁𝑚₂

− 𝐿 𝑞²

1− 𝑠−𝑠₀ 6𝑚₁𝑚₂

(B.12)

A p p e n d i x C

INTEGRALS USED IN THE MASSIVE CASE

C.1 Tadpole and Bubble Integrals

𝐴₀(𝑚) =

∫ 𝑑⁴ℓ (2𝜋)⁴

1

(ℓ²−𝑚²) =0 (C.1)

𝐵₀=

∫ 𝑑⁴ℓ (2𝜋)⁴

1 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 16𝜋²

1 𝑞²

− 𝑚²

1−𝑚²

2

2 log(𝑚1² 𝑚²

2

) +𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂)

(C.2) 𝐵_𝜇 =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 32𝜋²

𝑞_𝜇

− 1 𝑞²

− 1 𝑞⁴

(𝑚²

1−𝑚²

2)

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂)

+ 𝑖

64𝜋²𝑞⁴ 𝑞_𝜇

(𝑚²

1−𝑚²

2)² log

𝑚²

1

𝑚²

2

(C.3) 𝐵_{𝜇 𝜈} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 16𝜋²

𝑚²

1+𝑚²

2

6 1 𝑞²

− 1 12

𝜂_{𝜇 𝜈}+𝑚²

1−2𝑚²

2

3 1 𝑞⁴

+ 1 3

1 𝑞²

𝑞_𝜇𝑞_𝜈

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)³ 1

24 1 𝑞⁴

𝜂_{𝜇 𝜈}+

− 1 6

1 𝑞⁶

𝑞_𝜇𝑞_𝜈

log𝑚²

1

𝑚²

2

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)²

− 1 12

1 𝑞⁴

𝜂_{𝜇 𝜈}+1 3

1 𝑞⁶

𝑞_𝜇𝑞_𝜈

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) (C.4) 𝐵_{𝜇 𝜈𝛼} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 16𝜋²

− 𝑚²

1+3𝑚²

2

24 1 𝑞²

+ 1 24

3𝜂(𝜇 𝜈𝑞_𝛼) −𝑚²

1−3𝑚²

2

4 1 𝑞⁴ + 1

4 1 𝑞²

𝑞_𝜇𝑞_𝜈𝑞_𝛼

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)⁴

− 1 48

1 𝑞⁶

3𝜂(𝜇 𝜈𝑞_𝛼) +1 8

1 𝑞⁸

𝑞_𝜇𝑞_𝜈𝑞_𝛼

log 𝑚²

1

𝑚²

2

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)³ 1

24 1 𝑞⁶

𝜂(𝜇 𝜈𝑞_𝛼) +

− 1 4

1 𝑞⁸

𝑞_𝜇𝑞_𝜈𝑞_𝛼

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)

− 𝑚²

1+3𝑚²

2

24 1 𝑞⁴

𝜂(𝜇 𝜈𝑞_𝛼) +

− 𝑚²

1−3𝑚²

2

4 1 𝑞⁶

𝑞_𝜇𝑞_𝜈𝑞_𝛼

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) (C.5)

𝐵_{𝜇 𝜈𝛼 𝛽} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼ℓ_𝛽 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2) =

= 𝑖 16𝜋²

"

3𝑚⁴

1+2𝑚²

1𝑚²

2+3𝑚⁴

2

120

1 𝑞²

− 𝑚²

1+𝑚²

2

60 + 𝑞² 240

6𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}) +

𝑚⁴

1+4𝑚²

1𝑚²

2−9𝑚⁴

2

60

1 𝑞⁴

+ 𝑚²

1+6𝑚²

2

60 − 1

40

6𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽) + 𝑚⁴

1−6𝑚²

1𝑚²

2+6𝑚⁴

2

5

1 𝑞⁶

+ 𝑚²

1−4𝑚²

2

5 1 𝑞²

+ 1 5

1 𝑞²

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)⁵

− 1 480

1 𝑞⁶

6𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}) + 1

80 1 𝑞⁸

6𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽)+

− 1 10

1 𝑞¹⁰

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}

log 𝑚²

1

𝑚²

2

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)⁴ 1

240 1 𝑞⁶

6𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}) +

− 1 40

1 𝑞⁸

6𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽) + 1

5 1 𝑞¹⁰

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)²

"

− 𝑚²

1+𝑚²

2

60 1 𝑞⁴

6𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}) + 𝑚²

1+6𝑚²

2

60 1 𝑞⁶

6𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽) + 𝑚²

1−4𝑚²

2

5 1 𝑞⁸

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) (C.6)

𝐵_{𝜇 𝜈𝛼 𝛽𝛾} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼ℓ_𝛽ℓ_𝛾 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 16𝜋²

"

− 𝑚⁴

1+2𝑚²

1𝑚²

2+5𝑚⁴

2

240

1 𝑞²

+ 3𝑚²

1+5𝑚²

2

480 − 𝑞² 480

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾) +

− 𝑚⁴

1+7𝑚²

1𝑚²

2−20𝑚⁴

2

120

1 𝑞⁴

− 𝑚²

1+10𝑚⁴

2

120 + 1

60

10𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾) +

− 𝑚⁴

1−8𝑚²

1𝑚²

2+10𝑚⁴

2

6

1 𝑞⁶

− 𝑚²

1−5𝑚²

2

6 1 𝑞⁴

− 1 6

1 𝑞²

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂)

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)⁶ 1

960 1 𝑞⁸

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾) +

− 1 120

1 𝑞¹⁰

10𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾) +

1 12

1 𝑞¹²

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}𝑞_𝛾

log 𝑚²

1

𝑚²

2

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)⁵

− 1 480

1 𝑞⁸

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾) + 1

60 1 𝑞⁸

10𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾) +

− 1 6

1 𝑞¹²

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)³

"

3𝑚²

1+5𝑚²

2

480 1 𝑞⁶

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾) +

− 𝑚²

1+10𝑚²

2

120 1 𝑞⁸

10𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾) +

− 𝑚²

1−5𝑚²

2

6 1 𝑞¹⁰

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂)

− 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)

"

𝑚⁴

1+2𝑚²

1𝑚²

2+5𝑚⁴

2

240

1 𝑞⁴

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾) + 𝑚⁴

1+7𝑚²

1𝑚²

2−20𝑚⁴

2

120

1 𝑞⁶

10𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾) + 𝑚⁴

1−8𝑚²

1𝑚²

2+10𝑚⁴

2

6

1 𝑞⁸

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) (C.7) 𝐵_{𝜇 𝜈𝛼 𝛽𝛾 𝛿} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼ℓ_𝛽ℓ_𝛾ℓ_𝛿 (ℓ²−𝑚²

1) ( (ℓ+𝑞)²−𝑚²

2)

= 𝑖 16𝜋²

"

(𝑚²

1+𝑚²

2) (5𝑚⁴

1−2𝑚²

1𝑚²

2+5𝑚⁴

2) 1680

1 𝑞²

− 5𝑚⁴

1+6𝑚²

1𝑚²

2+5𝑚⁴

2

2240 +𝑞²

𝑚²

1+𝑚²

2

1120 − 𝑞⁴ 6720

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝜂_{𝛾 𝛿}) +𝑚⁶

1+2𝑚⁴

1𝑚²

2+9𝑚²

1𝑚⁴

2−20𝑚⁶

2) 210

1 𝑞⁴

− 𝑚⁴

1+4𝑚²

1𝑚²

2+15𝑚⁴

2

840

1 𝑞²

− 5𝑚²

1+12𝑚²

2

1680 − 𝑞² 840

45𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿) +𝑚⁶

1+9𝑚⁴

1𝑚²

2−54𝑚²

1𝑚⁴

2+50𝑚⁶

2) 210

1 𝑞⁶

+ 2𝑚⁴

1+22𝑚²

1𝑚²

2−75𝑚⁴

2

420

1 𝑞⁴

+ 𝑚²

1+15𝑚²

2

210 − 1

84

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿) +(𝑚²

1−2𝑚²

2) (𝑚⁴

1−10𝑚²

1𝑚²

2+10𝑚⁴

2) 7

1 𝑞⁸

+ (𝑚⁴

1−10𝑚²

1𝑚²

2+15𝑚⁴

2) 7

1 𝑞⁶

+ 𝑚²

1−6𝑚²

2

7 1 𝑞⁴

+1 7

1 𝑞²

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)⁷ 1

13440 1 𝑞⁸

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝜂_{𝛾 𝛿}) +

− 1 1680

1 𝑞¹⁰

45𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿) +

1 168

1 𝑞¹²

15𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿) +

− 1 14

1 𝑞¹⁴

𝑞_𝜇𝑞_𝜈𝑞_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿

log 𝑚²

1

𝑚²

2

+ 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)⁶

− 1

6720 1 𝑞⁸

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝜂_{𝛾 𝛿}) + 1 840

1 𝑞¹⁰

45𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿) +

− 1 84

1 𝑞¹²

15𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿) +1 7

1 𝑞¹⁴

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) + 𝑖

16𝜋² (𝑚²

1−𝑚²

2)⁴

"

𝑚²

1+𝑚²

2

1120 1 𝑞⁶

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝜂_{𝛾 𝛿}) +

− 5𝑚1²+12𝑚2² 1680

1 𝑞⁸

45𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞_𝛾𝑞_𝛿) +𝑚²

1+15𝑚²

2

210 1 𝑞¹⁰

15𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿) + 𝑚²

1−6𝑚²

2

7

1 𝑞¹²

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂)

− 𝑖 16𝜋²

(𝑚²

1−𝑚²

2)²

"

5𝑚⁴

1+6𝑚²

1𝑚²

2+5𝑚⁴

2

2240

1 𝑞⁴

15𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝜂_{𝛾 𝛿}) + 𝑚⁴

1+4𝑚²

1𝑚²

2+15𝑚⁴

2

840

1 𝑞⁶

45𝜂(𝜇 𝜈𝜂_{𝛼 𝛽}𝑞 𝛾 𝑞_𝛿) + 2𝑚⁴

1+22𝑚²

1𝑚²

2−75𝑚⁴

2

420

1 𝑞⁸

15𝜂(𝜇 𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿) + 𝑚⁴

1−10𝑚²

1𝑚²

2+15𝑚⁴

2

7

1 𝑞¹⁰

𝑞_𝜇𝑞_𝜈𝑞_𝛼𝑞_𝛽𝑞_𝛾𝑞_𝛿

#

𝑆 𝐿(𝑞, 𝑚₁, 𝑚₂) (C.8) C.2 Triangle Integrals

We used the𝐶₀integral as defined by [10]

𝐶₀=

∫ 𝑑⁴ℓ (2𝜋)⁴

1 (ℓ²−𝑚²

1) ( (ℓ+𝑘₁)²−𝑚²

2) ( (ℓ+𝑘₂)²−𝑚²

3)

= 𝑘²

1𝐵₀(𝑘₁, 𝑚₁, 𝑚₂) + ( (𝑘₁· 𝑘₂) −𝑘²

1)𝐵₀(𝑘₁+𝑘₂, 𝑚₁, 𝑚₃) − (𝑘₁·𝑘₂)𝐵₀(𝑘₁, 𝑚₂, 𝑚₃)

−(𝑘₁·𝑘₂)𝑓₁−𝑘²

1𝑓₂

(C.9) where

𝑓₁=𝑚²

1−𝑚²

2−𝑘²

1 (C.10)

𝑓₂=𝑚²

2−𝑚²

3+𝑘²

1− (𝑘₁− 𝑘₂)² (C.11)

All other non-scalar triangle integrals used were decomposed in terms of the 𝐶₀, 𝐵₀, and 𝐴₀ integrals. Each integral has the inputs 𝐶(𝑞, 𝑘₁, 𝑚, 𝑚, 𝑚1). We have Taylor expanded the expressions with respect to the graviton mass𝑚 up to order 2 and the exchanged momentum𝑞 up to order 4. We also only record the nonanalytic terms of these expressions.

𝐶_𝜇 =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇 (ℓ²−𝑚²

1) ( (ℓ+𝑘₁)²−𝑚²

2) ( (ℓ+𝑘₂)²−𝑚²

3)

=− 𝑖 64𝜋²𝑚²

1𝑞²

𝑚² 8𝑘

𝜇 1 log

−𝑚² 𝑞²

−𝑞^𝜇log 𝑚² 𝑚²

1

!

−2𝑞^𝜇log

−𝑚² 𝑞²

!

− 𝑖

128𝜋²𝑚⁴

1

−8𝑚²

1𝑘

𝜇 1log

−𝑚² 𝑞²

+6𝑚²𝑘

𝜇 1log

−𝑚² 𝑞²

+4𝑚²

1𝑞^𝜇log

−𝑚² 𝑞²

−2𝑚²𝑞^𝜇log

−𝑚² 𝑞²

+ 𝑖

1024𝜋²𝑚⁶

1

𝑞²

16𝑚²

1𝑘

𝜇 1log

−𝑚² 𝑞²

−16𝑚²𝑘

𝜇 1log

−𝑚² 𝑞²

−8𝑚²

1𝑞^𝜇log

−𝑚² 𝑞²

+6𝑚²𝑞^𝜇log

−𝑚² 𝑞²

− 𝑖

2048𝜋²𝑚⁸

1

𝑞⁴

−8𝑚²

1𝑘

𝜇 1log

−𝑚² 𝑞²

+10𝑚²𝑘

𝜇 1log

−𝑚² 𝑞²

+4𝑚²

1𝑞^𝜇log

−𝑚² 𝑞²

−4𝑚²𝑞^𝜇log

−𝑚² 𝑞²

+𝑂

𝑚³, 𝑞⁵

(C.12) 𝐶_{𝜇 𝜈} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈 (ℓ²−𝑚²

1) ( (ℓ+𝑘₁)²−𝑚²

2) ( (ℓ+𝑘₂)²−𝑚²

3)

=− 𝑖 128𝜋²𝑚²

1𝑞²

−8𝑚²𝑘^𝜈

1𝑞^𝜇log

−𝑚² 𝑞²

−8𝑚²𝑘

𝜇 1𝑞^𝜈log

−𝑚² 𝑞²

−𝑚²𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

! +6𝑚²

1𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

!

+16𝑚²𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

!

+ 𝑖

256𝜋²𝑚⁴

1

16𝑚²𝑚²

1log

−𝑚² 𝑞²

𝑔^{𝜇 𝜈}−8𝑚²

1𝑘^𝜈

1𝑞^𝜇log

−𝑚² 𝑞²

−8𝑚²

1𝑘

𝜇 1𝑞^𝜈log

−𝑚² 𝑞²

−40𝑚²𝑘

𝜇 1𝑘^𝜈

1log

−𝑚² 𝑞²

+24𝑚²𝑘^𝜈

1𝑞^𝜇log

−𝑚² 𝑞²

+24𝑚²𝑘

𝜇 1𝑞^𝜈log

−𝑚² 𝑞²

+10𝑚²

1𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

−18𝑚²𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

− 𝑖

2048𝜋²𝑚⁶

1

𝑞²

16𝑚⁴

1log

−𝑚² 𝑞²

𝑔^{𝜇 𝜈}−32𝑚²𝑚²

1log

−𝑚² 𝑞²

𝑔^{𝜇 𝜈}

−48𝑚²

1𝑘

𝜇 1𝑘^𝜈

1log

−𝑚² 𝑞²

+40𝑚²

1𝑘^𝜈

1𝑞^𝜇log

−𝑚² 𝑞²

+40𝑚²

1𝑘

𝜇 1𝑞^𝜈log

−𝑚² 𝑞²

+160𝑚²𝑘

𝜇 1𝑘^𝜈

1log

−𝑚² 𝑞²

−88𝑚²𝑘^𝜈

1𝑞^𝜇log

−𝑚² 𝑞²

−88𝑚²𝑘

𝜇 1𝑞^𝜈log

−𝑚² 𝑞²

−32𝑚²

1𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

+56𝑚²𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

+𝑂

𝑚³, 𝑞³

(C.13)

𝐶_{𝜇 𝜈𝛼} =

∫ 𝑑⁴ℓ (2𝜋)⁴

ℓ_𝜇ℓ_𝜈ℓ_𝛼 (ℓ²−𝑚²

1) ( (ℓ+𝑘₁)²−𝑚²

2) ( (ℓ+𝑘₂)²−𝑚²

3)

=

11𝑖 𝑚²𝑞^𝛼𝑞^𝜇𝑞^𝜈log

𝑚² 𝑚²

1

128𝜋²𝑞⁴

− 𝑖

256𝜋²𝑚²

1𝑞²

6𝑚²𝑚²

1𝑞^𝛼log 𝑚² 𝑚²

1

! 𝑔^{𝜇 𝜈}

+6𝑚²𝑚²

1𝑞^𝜇log 𝑚² 𝑚²

1

!

𝑔^𝛼𝜈 +6𝑚²𝑚²

1𝑞^𝜈log 𝑚² 𝑚²

1

!

𝑔^{𝛼 𝜇}−6𝑚²𝑘

𝜇 1𝑘^𝜈

1𝑞^𝛼log 𝑚² 𝑚²

1

!

−6𝑚²𝑘^𝛼

1𝑘^𝜈

1𝑞^𝜇log 𝑚² 𝑚²

1

!

−19𝑚²𝑘^𝜈

1𝑞^𝛼𝑞^𝜇log 𝑚² 𝑚²

1

!

−6𝑚²𝑘^𝛼

1𝑘

𝜇

1𝑞^𝜈log 𝑚² 𝑚²

1

!

−19𝑚²𝑘

𝜇

1𝑞^𝛼𝑞^𝜈log 𝑚² 𝑚²

1

!

−19𝑚²𝑘^𝛼

1𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

! +5𝑚²

1𝑘^𝜈

1𝑞^𝛼𝑞^𝜇log 𝑚² 𝑚²

1

!

+5𝑚²

1𝑘

𝜇

1𝑞^𝛼𝑞^𝜈log 𝑚² 𝑚²

1

! +5𝑚²

1𝑘^𝛼

1𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

!

−8𝑚²𝑘^𝜈

1𝑞^𝛼𝑞^𝜇log

−𝑚² 𝑞²

−8𝑚²𝑘

𝜇

1𝑞^𝛼𝑞^𝜈log

−𝑚² 𝑞²

−8𝑚²𝑘^𝛼

1𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

+51𝑚²𝑞^𝛼𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

!

+14𝑚²

1𝑞^𝛼𝑞^𝜇𝑞^𝜈log 𝑚² 𝑚²

1

!

+18𝑚²𝑞^𝛼𝑞^𝜇𝑞^𝜈log

−𝑚² 𝑞²

+ 𝑖

6144𝑚⁴

1𝜋²

48𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑔^{𝜇 𝜈}𝑚²

1+48𝑚²log

−𝑚² 𝑞²

𝑔^𝛼𝜈𝑞^𝜇𝑚²

1

−16 log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑘^𝜈

1𝑚²

1+48𝑚²log

−𝑚² 𝑞²

𝑔^{𝛼 𝜇}𝑞^𝜈𝑚²

1

−16 log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇 1𝑞^𝜈𝑚²

1−16 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑞^𝜈𝑚²

1

+96 log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑞^𝜈𝑚²

1+192𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑘^𝜈

1

−240𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇 1𝑘^𝜈

1−240𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑘^𝜈

1

+264𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑘^𝜈

1−240𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑞^𝜈

+264𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇

1𝑞^𝜈 +264𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑞^𝜈

−276𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑞^𝜈 − 𝑖 12288𝑚⁶

1𝜋²

−32 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑔^{𝜇 𝜈}𝑚⁴

1

+64 log

−𝑚² 𝑞²

𝑞^𝛼𝑔^{𝜇 𝜈}𝑚⁴

1−32 log

−𝑚² 𝑞²

𝑔^𝛼𝜈𝑘

𝜇 1𝑚⁴

1

+64 log

−𝑚² 𝑞²

𝑔^𝛼𝜈𝑞^𝜇𝑚⁴

1−32 log

−𝑚² 𝑞²

𝑔^{𝛼 𝜇}𝑘^𝜈

1𝑚⁴

1

+64 log

−𝑚² 𝑞²

𝑔^{𝛼 𝜇}𝑞^𝜈𝑚⁴

1+48𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑔^{𝜇 𝜈}𝑚²

1

−48𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑔^{𝜇 𝜈}𝑚²

1+48𝑚²log

−𝑚² 𝑞²

𝑔^𝛼𝜈𝑘

𝜇 1𝑚²

1

−48𝑚²log

−𝑚² 𝑞²

𝑔^𝛼𝜈𝑞^𝜇𝑚²

1+48𝑚²log

−𝑚² 𝑞²

𝑔^{𝛼 𝜇}𝑘^𝜈

1𝑚²

1

+160 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑘^𝜈

1𝑚²

1−224 log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇 1𝑘^𝜈

1𝑚²

1

−224 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑘^𝜈

1𝑚²

1+240 log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑘^𝜈

1𝑚²

1

−48𝑚²log

−𝑚² 𝑞²

𝑔^{𝛼 𝜇}𝑞^𝜈𝑚²

1−224 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑞^𝜈𝑚²

1

+240 log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇 1𝑞^𝜈𝑚²

1+240 log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑞^𝜈𝑚²

1

−248 log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑞^𝜈𝑚²

1−432𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑘^𝜈

1

+360𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇 1𝑘^𝜈

1+360𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑘^𝜈

1

−300𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑘^𝜈

1+360𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑘

𝜇 1𝑞^𝜈

−300𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑘

𝜇

1𝑞^𝜈−300𝑚²log

−𝑚² 𝑞²

𝑘^𝛼

1𝑞^𝜇𝑞^𝜈

+252𝑚²log

−𝑚² 𝑞²

𝑞^𝛼𝑞^𝜇𝑞^𝜈𝑞²+𝑂

𝑚³, 𝑞³

(C.14)

C.3 Box and Cross Box Integrals We used the 𝐷₀integral as defined by [10]

𝐷₀ =

∫ 𝑑⁴𝑙 (2𝜋)⁴

1 (𝑙²−𝑚²

1) ( (𝑙+𝑘₁)²−𝑚²

2) ( (𝑙+𝑘₂)²−𝑚²

3) ( (𝑙−𝑘₃)²−𝑚²

4)

= 1 𝑌₃

−𝑥₃₃𝐶₀ 𝑘₁, 𝑘₂, 𝑚²

2, 𝑚²

1, 𝑚²

3

+ (𝑥₃₃−𝑥₃₂)𝐶₀ 𝑘₁,−𝑘₃, 𝑚²

2, 𝑚²

1, 𝑚²

4

+ (𝑥₃₂−𝑥₃₁)𝐶₀ 𝑘₁−𝑘₂, 𝑘₂+𝑘₃, 𝑚²

2, 𝑚²

3, 𝑚²

4

+𝑥₃₁𝐶₀ 𝑘₂, 𝑘₂+𝑘₃, 𝑚²

1, 𝑚²

3, 𝑚²

4

(C.15) 𝐷⁰

0 =

∫ 𝑑⁴𝑙 (2𝜋)⁴

1

(𝑙²−𝑚1²) ( (𝑙+𝑘1)²−𝑚2²) ( (𝑙+𝑘2)²−𝑚3²) ( (𝑙+𝑘4)²−𝑚4²)

= 1 𝑌⁰

3

−𝑥⁰

33𝐶₀ 𝑘₁, 𝑘₂, 𝑚²

2, 𝑚²

1, 𝑚²

3

+ (𝑥⁰

33−𝑥⁰

32)𝐶₀ 𝑘₁, 𝑘₄, 𝑚²

2, 𝑚²

1, 𝑚²

4

+ (𝑥⁰

32−𝑥⁰

31)𝐶₀ 𝑘₁−𝑘₂, 𝑘₂−𝑘₄, 𝑚²

2, 𝑚²

3, 𝑚²

4

+𝑥⁰

31𝐶₀ 𝑘₂, 𝑘₂−𝑘₄, 𝑚²

1, 𝑚²

3, 𝑚²

4

(C.16) where

𝑓₁= 𝑓⁰

1=𝑚²

1−𝑚²

2−𝑘²

1 (C.17)

𝑓₂= 𝑓⁰

2=𝑚²

2−𝑚²

3+𝑘²

1− (𝑘₁−𝑘₂)² (C.18)

𝑓₃=𝑚²

3−𝑚²

4+ (𝑘₁−𝑘₂)²− (𝑘₁−𝑘₃)² (C.19) 𝑥₃₁ =−(𝑘₁·𝑘₂) (𝑘₂· 𝑘₃) − (𝑘₁· 𝑘₃)𝑘²

2 (C.20)

𝑥₃₂ =−(𝑘₁·𝑘₂) (𝑘₁· 𝑘₃) − (𝑘₂· 𝑘₃)𝑘²

1 (C.21)

𝑥₃₃ =𝑥⁰

33 =𝑘²

2𝑘²

1− (𝑘₁·𝑘₂)² (C.22)

𝑌₃=𝑥₃₁𝑓₁+𝑥₃₂𝑓₂+𝑥₃₃𝑓₃ (C.23) 𝑓⁰

3 =𝑚²

3−𝑚²

4+ (𝑘₁−𝑘₂)²− (𝑘₁+𝑘₄)² (C.24) 𝑥⁰

31 =(𝑘₁· 𝑘₂) (𝑘₂·𝑘₄) + (𝑘₁·𝑘₄)𝑘²

2 (C.25)

𝑥⁰

32 =(𝑘₁· 𝑘₂) (𝑘₁·𝑘₄) + (𝑘₂·𝑘₄)𝑘²

1 (C.26)

𝑌⁰

3=𝑥⁰

31𝑓⁰

1+𝑥⁰

32𝑓⁰

2+𝑥⁰

33𝑓⁰

3 (C.27)

(C.28) All other non-scalar triangle integrals used were decomposed in terms of the 𝐷₀, 𝐶₀, 𝐵₀, and

𝐴₀integrals

A p p e n d i x D

FOURIER TRANSFORMS

In this thesis, we make use of many 3D fourier transforms. We have accumulated these from several resources. The three that we use for the massless graviton calculations are listed below [5]:

∫ 𝑑³𝑞 (2𝜋)³𝑒^𝑖q·r

1

|q|² = 1

4𝜋𝑟 (D.1)

∫ 𝑑³𝑞

(2𝜋)³𝑒^𝑖q·r 1

|q| = 1 2𝜋²𝑟²

(D.2)

∫ 𝑑³𝑞

(2𝜋)³𝑒^𝑖q·rln(q²) =− 1 2𝜋𝑟³

(D.3)

∫ 𝑑³𝑞

(2𝜋)³𝑒^𝑖q·r=𝛿³(𝑟) (D.4)

∫ 𝑑³𝑞

(2𝜋)³𝑒^𝑖q·r𝑞²=𝛿³⁰⁰(𝑟) (D.5) For the massive graviton calculations, we reference [27] and use their transforms as shown below

𝐼 =

∫ 𝑑^𝑑𝑞 (2𝜋)^𝑑𝑒^𝑖q·r

1

(q²)^𝛼 = 1 (4𝜋)^𝑑/2

Γ(𝑑/2−𝛼) Γ(𝛼)

𝑟² 4

^𝛼−𝑑/2

(D.6)

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞² 𝑞²

= (D.7)

∫ 𝑑³𝑞 (2𝜋)³𝑒^{𝑖𝒒·𝒓}

log −𝑞²

(𝑞²−𝑚²) = (D.8)

A p p e n d i x E

COLOR-KINEMATICS DUALITY

Figure E.1: A tree level gluon inter- action

At one point in this thesis, we were consid- ering exploring the usefulness of utilizing color-kinematic duality as a way to check our answers. This duality interchanges the color and kinematic structures of the S- matrix. In practice, this means that the amplitudes of various particle interactions can be "squared" to produce an equivalent amplitude for interactions in other theories [24]. We were specifically interested in one such relation, that of "graviton = gluon²"

[7]. This means that if we took the expres-

sion for the scattering amplitude for fig. E.1 before integration and squared it, we would get the desired amplitude for the diagram in fig. E.2 Since great progress has already been made with the strong force, we originally thought we could rederive the amplitudes for our project through this independent method. We hoped that enough research would had been done in the field of massive gluons that this could be rea- sonably simple method to check our conclusions through an independent process.

However, after searching into the work done with massive gluons, we realized that while this procedure should work [25], it would require more calculation and effort than expected, since we would have had to calculate the majority of the massive gluon amplitudes that we needed. We expect that this process could be the source of an entirely new project or thesis and suggest this as a possibility to any interested readers.