### Leading Order Coefficients (tilde_b1 and tilde_c1)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Provides the leading order terms for the coefficients \tilde{b_1} and \tilde{c_1} after considering the leading order in s and the factor |\Phi_1|. These are used for further simplification of the amplitude.

```latex
\tilde{b_1}= -\frac{35}{128}\frac{ i \alpha s^2}{ m^2}\left( \frac{\kappa}{2}\right)^6\left(m^2-t
ight)^4 \>
```

```latex
\tilde{c_1}=\frac{35}{64} i \alpha s^2 \left( \frac{\kappa}{2}\right)^6\left(m^2-t
ight)^4 \>
```

--------------------------------

### Reduction of \(\xi^{--}\) using LiteRed

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Shows the result of reducing the \(\xi^{--}\) amplitude using the LiteRed package, expressing it in terms of a 'triangle' integral and a coefficient \(c_2\).

```latex
\xi^{--}= c_2 \text{ triangle}
```

--------------------------------

### Integral Reduction Coefficients (b1 and c1)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Expresses the rational part of the integral \xi^{+-} as a linear combination of bubble and triangle integrals, with coefficients b1 and c1. Approximations from Gab and Andi's R^3 paper are applied.

```latex
\xi^{+-}= b_1 \text{ bubble} + c_1 \text{ triangle}
```

```latex
b_1=i \alpha \left( \frac{\kappa}{2}\right)^6 \frac{ s^2 \left(-420 m^6+818 m^4 s-434 m^2 s^2+333 s^3
ight) \left(m^4-2 m^2 t+t (s+t)
ight)^4}{6 \left(s-4 m^2
ight)^4}
```

```latex
c_1=4 i \alpha \left( \frac{\kappa}{2}\right)^6 \frac{ s^2 \left(35 m^8-10 m^6 s-6 m^4 s^2+8 m^2 s^3+4 s^4
ight) \left(m^4-2 m^2 t+t (s+t)
ight)^4}{\left(s-4 m^2
ight)^4}
```

--------------------------------

### Eikonal Phase Matrix (Leading Order)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Combines EH and FFR contributions into a single leading eikonal phase matrix for photon bending. This form is used before expanding around D=4.

```latex
\delta_0^{\gamma} = \left(rac{\kappa}{2}
ight)^2 m\,\omega \, f(-2,D-2) \, \mqty( 1 & \left(\dfrac{\alpha_\gamma}{8}\right) \dfrac{\xi''}{\bar{\mathscr{b}}^2} \\ \cr \left(\dfrac{\alpha_\gamma}{8}\right)  \dfrac{\xi''}{ \mathscr{b}^2} & 1) \> ,
```

--------------------------------

### R4 One-Loop Amplitude (Eikonal Approximation)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents the one-loop R4 amplitudes in the eikonal approximation for different particle helicity configurations. These expressions involve loop integrals I3 and I2.

```latex
\mathcal{A}^{(1)}_{R^4}  (1^\phi, 2^\phi, 3^{--}, 4^{++})  &\simeq  -  \cN_h\, \widetilde{\beta} \left( \dfrac{\kappa}{2}\right)^4 \, s^2 \left[ \dfrac{35}{4} \, ( m \omega)^4 \, I_3 (s;m) + \dfrac{93}{8}  (m \omega^2)^2 \, I_2 (s) \right] 
 \ ,
```

```latex
\mathcal{A}^{(1)}_{R^4}  (1^\phi, 2^\phi, 3^{++}, 4^{++})  &\simeq -  \beta\,\left( \dfrac{\kappa}{2}\right)^4 \,\sqr{3}{4}^4 \left[ 
     \dfrac{3}{4} \, (m \omega)^4 
     \, I_3 (s;m) 
 \right. 
\t+\left. \dfrac{55}{24}  \, (m\omega^2)^2 \, I_2 (s) \right] \ ,
```

--------------------------------

### Impact Parameter Amplitude Transformation

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

This snippet shows the transformation of amplitudes to impact parameter space, specifically for the EH case. It utilizes a general integral formula for calculating these transformations.

```latex
f(p, d):= \int\!{d^dq\over (2 \pi)^d} \, e^{i\vec{q} \cdot \vec{b}} \, |\vec{q} \, |^{p}\ = \ 
{2^p \pi^{-d/2} 
\Gamma\left( 
{{d+p}\over 2}\right)
\over 
\Gamma \left( - {p\over 2} 
right)
}
{1\over b^{\, d+p}}
.
```

--------------------------------

### EH Tree-Level Amplitude for Photon Deflection (Eikonal Approximation)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

This snippet shows the leading order EH tree-level amplitude for photon deflection, expressed in the eikonal approximation. It is used to test exponentiation properties.

```mathematica
    \cA^{(0)} (1^\phi, 2^\phi, 3^{-}, 4^{+}) \simeq i \left(\frac{\kappa}{2}\right)^2 \frac{(2m\,\omega)^2}{\left|\vec{q}\,\right|^{\, 2}} 

```

--------------------------------

### One-Loop Amplitudes in EH Gravity (Leading Terms)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Extracts the leading $\omega^3$ terms from one-loop amplitudes in Einstein-Hilbert gravity, relevant for checking exponentiation. These terms are then transformed to impact parameter space.

```latex
\left. \mathcal{A}_{\rm EH}^{(1)} (1^\phi, 2^\phi, 3^{--}, 4^{++}) \right|_{\omega^3} = \ \left(\frac{\kappa}{2}\right)^4 (2 m \omega)^4 \Big[ I_4(s, t; m) + I_4 (s, u; m)\Big] \ , \ \[.2em]
        \left. \mathcal{A}_{\rm EH}^{(1)} (1^\phi, 2^\phi, 3^{++}, 4^{++})\right|_{\omega^3} = 0 \ \ , 
```

--------------------------------

### Kinematic Parameterization of Four-Momenta

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Defines the four-momenta for incoming and outgoing scalar particles and gravitons in a center-of-mass frame. Note the convention where incoming particles have negative signs.

```latex
\begin{equation}
\begin{array}{lr}
\begin{tikzpicture}[scale=15,baseline={([yshift=-1mm]centro.base)}]
\def\x{0}
\def\y{0}

\node at (0+\x,0+\y) (centro) {};
\node at (-3pt+\x,-3pt+\y) (uno) {$p_1$};
\node at (-3pt+\x,3pt+\y) (due) {$p_2$};
\node at (3pt+\x,3pt+\y) (tre) {$p_3^{h_3}$};
\node at (3pt+\x,-3pt+\y) (quat) {$p_4^{h_4}$};

\draw [thick] (uno) -- (centro);
\draw [thick] (due) -- (centro);
\draw [vector,double] (tre) -- (centro);
\draw [vector,double] (quat) -- (centro);

\draw [->] (-2.8pt+\x,-2pt+\y) -- (-1.8pt+\x,-1pt+\y); 
\draw [->] (2.8pt+\x,-2pt+\y) -- (1.8pt+\x,-1pt+\y); 
\draw [->] (-1.8pt+\x,1pt+\y) -- (-2.8pt+\x,2pt+\y); 
\draw [->] (1.8pt+\x,1pt+\y) -- (2.8pt+\x,2pt+\y); 

%\node at (0+\x,0+\y) [draw, fill=gray!90!black, circle, inner sep=10pt] {};

\shade [shading=radial] (centro) circle (1.5pt);

\end{tikzpicture}
&
\begin{aligned}
p_4^\mu & = - (E_4,-\vec{p}+\vec{q}/2)\, ,  \ 
 p_1^\mu & =  -(E_1,\vec{p}-\vec{q}/2) \, , \ 
p_2^\mu & =  (E_2,\vec{p}+\vec{q}/2) \, , \ 
p_3^\mu & =  (E_3,-\vec{p}-\vec{q}/2)\ . 
\end{aligned}

\end{array}
\end{equation}
```

--------------------------------

### Amplitude \(\xi^{++}\) Decomposition

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Expresses the \(\xi^{++}\) amplitude as a sum of a 'bubble' integral multiplied by \(b_3\) and a 'triangle' integral multiplied by \(c_3\).

```latex
\xi^{++} = b_3 \text{ bubble} +c_3 	ext{triangle}
```

--------------------------------

### Eikonal Phase Matrix (Comparison with Literature)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

This matrix represents the eikonal phase, adapted for comparison with literature (Camanho et al.). It includes logarithmic terms and is presented in a specific format.

```mathematica
\delta_0 
=  
 \left({\kappa\over 2}\right)^2 {m \omega
 2 \pi }  
\begin{pmatrix} 
- \dfrac{1}{2 \epsilon} - \log b  & \left(\dfrac{\alpha^\prime}{4}\right)^2  \dfrac{3}{\bar{\mathscr{b}}^4}\
\cr \left(\dfrac{\alpha^\prime}{4}\right)^2  \dfrac{3}{{\mathscr{b}}^4} & - \dfrac{1}{2 \epsilon} - \log b  
\end{pmatrix}
```

--------------------------------

### Splitting Integral into Helicity and Rational Parts

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Splits the integral \mathcal{I}^{+-} into a helicity-dependent factor \Phi_1 and a rational function integral \xi^{+-}. This is achieved by dividing and multiplying by \langle 3 \; 2 \; 4 ]^4.

```latex
\mathcal{I}^{+-}=\dfrac{1}{\langle 3 \; 2 \; 4 ]^4} \left( \langle 3 \; 2 \; 4 ]^4 \; \mathcal{I}^{+-} \right) \equiv \Phi_1 \; \xi^{+-} \>
```

--------------------------------

### EH Tree-Level Amplitude in Impact-Parameter Space

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

This snippet presents the EH tree-level amplitude for photon deflection transformed into impact-parameter space. It is primarily for verifying exponentiation properties.

```mathematica
    \Tilde{\cA}^{(0)} (1^\phi, 2^\phi, 3^{-}, 4^{+}) \simeq i \left(\frac{\kappa}{2}\right)^2 m\,\omega\, f(-2,D-2) 

```

--------------------------------

### R^4 One-Loop Amplitudes in Impact-Parameter Space

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

These equations show the translation of the R^4 one-loop amplitudes into impact-parameter space, essential for analyzing scattering in that domain.

```latex
\begin{align}
\begin{split}
\left. \widetilde{\cA}^{(1)}_{ R^4} (1^\phi,2^\phi,3^{--},4^{++})\right|_{\omega} & = i \, \widetilde{\beta} \, \left(\frac{\kappa}{2}\right)^4 \, \frac{315}{512} \, \frac{m^2 \omega ^3}{2\pi b^5}\, \\ 
\left. \widetilde{\cA}^{(1)}_{ R^4} (1^\phi,2^\phi,3^{++},4^{++})\right|_{\omega} & = i \, \beta \, \left(\frac{\kappa}{2}\right)^4 \, \frac{315}{512}  \,  \frac{m^2 \omega ^3}{32 \pi b} \, \frac{1}{\bar{\mathscr{b}}^4} \\ 
\end{split}
\end{align}
```

--------------------------------

### Time Delay Formula

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents the formula for the time delay \(t^{(i)}\) experienced by a particle, obtained by differentiating the eikonal phase eigenvalues with respect to the energy \(\omega\).

```latex
\begin{equation}
\label{timedelay}
t^{(i)} = \frac{\partial \delta^{(i)}}{\partial \omega}
 \ .
\end{equation}
```

--------------------------------

### Impact Parameter Space Integrals

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Transformations from momentum space to impact parameter space using Fourier transforms for specific powers of $|\vec{q}|$.

```latex
\int\!{d^{D-2}q
over (2\pi)^{D-2}}\ e^{i\vec{q} \cdot \vec{b}}  \, |\vec{q} |^{3} & = {9
over 2 
pi} {1
over b^5} + 
\cO(D-4)\\[.2em]
\left({\partial 
over 
partial 
\bar{\mathscr{b}}}\right)^4
\int\!{d^{D-2}q
over (2
pi)^{D-2}}\ e^{i\vec{q} 
cdot 
\vec{b}}  \, |\vec{q} |^{-1} & = {105
over 32 
pi}{1
over b}{1
over 
\bar{\mathscr{b}}^4} + 
\cO(D-4)
 .
```

--------------------------------

### One-Loop Amplitude in EH + R^3 Theory (Helicity Flip)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Provides the one-loop helicity-flip amplitude for four-point scalar/graviton scattering in EH + R^3 theory, calculated in the eikonal approximation. This formula includes contributions from various integral functions and kinematic terms.

```latex
\mathcal{A}^{(1)}_{R^3}  (1^\phi, 2^\phi, 3^{++}, 4^{++}) \simeq \left({\kappa\over 2}\right)^4 \left({\alpha^\prime\over 4}\right)^2 \sqr{3}{4}^4 \, \bigg[ & (2 m \omega )^4  \big(I_4 (s, t;m) + I_4 (s, u;m) \big) -13 ( m^2 \omega)^2  I_3 (s;m) \ & +16 (m \omega)^2 s \, I_3(s) + \dfrac{153}{10} (m \omega)^2 I_2 (s) \bigg]
        \ . \ 
```

--------------------------------

### Leading Eikonal Phase Matrix Expansion around D=4

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

The leading eikonal phase matrix expanded around D=4, showing divergences and logarithmic terms relevant for scattering amplitudes.

```latex
\delta_0^{\gamma} = - \left(rac{\kappa}{2}
ight)^2 \frac{m\,\omega}{2\pi} \, \mqty( \dfrac{1}{4-D} + \log{b} & -\left(\dfrac{\alpha_\gamma}{8}\right) \dfrac{1}{2 \bar{\mathscr{b}}^2} \\ \cr -\left(\dfrac{\alpha_\gamma}{8}\right) \dfrac{1}{2 \mathscr{b}^2} & \dfrac{1}{4-D} + \log{b}) \> .
```

--------------------------------

### Subleading Eikonal Amplitudes (FFR Contributions)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Impact-parameter space representations of FFR contributions to the subleading eikonal phase at order \omega. One contribution is zero, the other is non-zero.

```latex
\def\arraystretch{2}
\begin{array}{rl}
\left. \widetilde{\cA}^{(1)}_{\rm FFR} (1^\phi,2^\phi,3^{-},4^{+})\right|_{\omega} &= \, 0 \> ,\\
\left. \widetilde{\cA}^{(1)}_{\rm FFR} (1^\phi,2^\phi,3^{+},4^{+})\right|_{\omega} &= \, i \left(\dfrac{\kappa}{2}\right)^4 \left(\dfrac{\alpha_\gamma}{8}\right) \dfrac{45}{1024\pi} \, \dfrac{m^2 \omega}{b}\, \dfrac{1}{\bar{\mathscr{b}}^2} \> ,
\end{array}
```

--------------------------------

### Deflection Angle Formula

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Provides the formula for the particle deflection angle \(\theta^{(i)}\) in the eikonal approximation, derived from the derivative of the eikonal phase eigenvalues with respect to the impact parameter.

```latex
\begin{equation}
\label{rebend}
\theta^{(i)}  \ = \ \frac{1}{\omega} \frac{\partial}{\partial b} {\delta^{(i)}} \ ,
\end{equation}
```

--------------------------------

### One-Loop Amplitude in EH Gravity (Helicity Flip)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents the one-loop amplitude for two-scalar two-graviton scattering in EH gravity with a helicity flip, computed in the eikonal approximation. It involves specific kinematic factors and integral functions.

```latex
\mathcal{A}_{\rm EH}^{(1)} (1^\phi, 2^\phi, 3^{++}, 4^{++}) & \simeq \ \left( \frac{\kappa}{2} \right)^4 \, \frac{\sqr{3}{4}^2}{\agl{3}{4}^2} (m^2 s)^2  \Big[ I_4(s, t;m) + I_4 (s, u;m)\Big] \ . \ 
```

--------------------------------

### One-Loop Amplitude in EH + R^3 Theory (No-Flip)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents the one-loop amplitude for four-point scalar/graviton scattering in EH + R^3 theory with no helicity flip, using the eikonal approximation. It includes the normalization factor \(\mathcal{N}_h\).

```latex
\mathcal{A}^{(1)}_{R^3}  (1^\phi, 2^\phi, 3^{--}, 4^{++}) \simeq \left({\kappa\over 2}\right)^4 \left({\alpha^\prime\over 4}\right)^2 \, \mathcal{N}_h \ \Big[& (m s)^4 \big(I_4 (s, t;m) + I_4 (s, u;m) \big)  + (m^2 s\, \omega )^2  I_3 (s;m) \ & + \frac{3}{2} (m s\, \omega )^2    I_2 (s) \Big]  \ , \ 
```

--------------------------------

### Leading Eikonal Phase in EH Gravity

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Calculates the leading eikonal phase in Einstein-Hilbert gravity. This is derived from the transformed amplitudes in impact parameter space.

```latex
\mathcal{\delta}_{0, \text{EH}} \ = \ \left({\kappa\over 2}\right)^2 ( m \omega) f (-2, D-2) \mathbf{\uno}_2
\ 
\simeq - \left({\kappa\over 2}\right)^2 \frac{m \omega}{2 \pi} \left[\frac{1}{4-D} + \log{b} \right]\mathbf{\uno}_2 + \cdots \ .
```

--------------------------------

### Modified Coefficients \(\tilde{b}_3\) and \(\tilde{c}_3\)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents modified versions of coefficients \(b_3\) and \(c_3\), denoted as \(\tilde{b}_3\) and \(\tilde{c}_3\), which are simplified and depend on \(m^2-t\).

```latex
\tilde{b_3}=\frac{55}{192 \; m^2} \; i \beta \; \delta \; \left(\dfrac{\kappa}{2}\right)^6 s^2 \left(m^2-t
ight)^4
```

```latex
\tilde{c_3}=\frac{3}{32} \;  i \beta \; \delta \; \left( \dfrac{\kappa}{2}\right)^6 s^2 \left(m^2- t
ight)^4
```

--------------------------------

### Transformed One-Loop Amplitude in Impact Parameter Space

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Shows the one-loop amplitude in impact parameter space for the EH case, after transformation and selection of leading terms. This is used to derive the subleading eikonal phase.

```latex
\left.\widetilde{\mathcal{A}}_{\rm EH}^{(1)} (1^\phi, 2^\phi, 3^{--}, 4^{++})\right|_{\omega^2}
= -\left({\kappa\over 2} \right)^4 {(m \omega)^2} \frac{2^{D-7} \Gamma ( D-4)}{\pi^{\frac{D}{2}} (D-4) \Gamma (3 - D/2)} \, {1\over b^{\, 2D-8}} \ .
```

--------------------------------

### One-Loop Amplitudes (Order \omega^2)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Impact-parameter space representations of one-loop amplitudes contributing at order \omega^2. These are used to test exponentiation properties.

```latex
\def\arraystretch{2.5}
\begin{array}{rl}
 \left. \widetilde{\cA}^{(1)} (1^\phi,2^\phi,3^{-},4^{+})\right|_{\omega^2} &= - \left(\dfrac{\kappa}{2}\right)^4 (m\,\omega)^2 \, \dfrac{f(D-6,D-2)}{2\pi (D-4)} \> , \\
 \left. \widetilde{\cA}^{(1)}_{\rm FFR} (1^\phi,2^\phi,3^{+},4^{+})\right|_{\omega^2} &= - \left(\dfrac{\kappa}{2}\right)^4 \dfrac{(m\,\omega)^2}{\bar{\mathscr{b}}^2} \, \left(D-3\right) \dfrac{f(D-6,D-2)}{2 \pi} \> ,
\end{array}
```

--------------------------------

### FFR One-Loop Box Terms (Eikonal Approximation)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

These snippets represent the one-loop box contributions in the FFR theory for graviton deflection. They are shown to vanish in the eikonal approximation.

```mathematica
    \left. \cA^{(1)}_{\rm FFR} (1^\phi,2^\phi,3^{--},4^{++})\right|_{\omega^2}  = - i e^2 \left( \frac{\kappa}{2}\right)^2 \left( \frac{\alpha_\gamma}{8} \right) \frac{m \omega^2}{32} \left| \vec{q} \, \right|  \simeq 0 \ ,
    \left. \cA^{(1)}_{\rm FFR} (1^\phi,2^\phi,3^{++},4^{++}) \right|_{\omega^2}  = 0

```

--------------------------------

### Eikonal Amplitude Consistency Condition

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Shows the consistency condition between the \(\omega^2\) term in the amplitude expansion and the square of the leading eikonal phase. This implies no new information from \(\widetilde{\cA}^{(1)}_{\omega^2}\).

```latex
\begin{align}
\label{exponentiation}
\widetilde{\cA}^{(1)}_{\omega^2} \ = \   {1\over 2} (\widetilde{\cA}^{(0)}_{\omega})^2
 \ . 
 \end{align}
```

--------------------------------

### R4 Tree-Level Amplitude (Zero)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Specifies that the R4 tree-level amplitude is zero for a configuration with two positive and two negative helicity particles where the negative helicity particles are adjacent.

```latex
\cA_{R^4}^{(0)} (1^{++},2^{++},3^{++},4^{--}) &= 0 \ \ .
```

--------------------------------

### Subleading Eikonal Matrix $\delta_{1,R^3}$

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

The subleading eikonal matrix $\delta_{1,R^3}$ derived from the $R^3$ interaction, expressed in terms of impact parameters $b$ and $\bar{\mathscr{b}}$.

```latex
\mathcal{\delta}_{1,R^3}
= 
 \left({\kappa
over 2}\right)^4 
\left({  \alpha^\prime 
over 4} \right)^2 
\dfrac{1}{256
pi}
, 
{ m^2 
omega
over   b}
\begin{pmatrix}
 - 
\dfrac{9}{b^4} & 
\dfrac{1365}{16} 
, 
\dfrac{1}{
\bar{\mathscr{b}}^4} 
\
\cr 
\dfrac{1365}{16} 
, 
\dfrac{1}{\mathscr{b}^4}  & - 
\dfrac{9}{b^4} 

\end{pmatrix}
 
```

--------------------------------

### Prefactor and \(c_2\) Recalculation

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Recalculates the coefficient \(c_2\) taking into account the prefactor \(|\Phi_2| = s^2\). This expression differs from the earlier one by a factor of \(s^2 m^4\).

```latex
|\Phi_2|=s^2\>
```

```latex
c_2=2 i \alpha  \left( \dfrac{\kappa}{2}\right)^6 m^4 s^4
```

--------------------------------

### Eikonal S-Matrix Expansion

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Represents the S-matrix in the eikonal approximation as an exponential of phases and as a series expansion in impact parameter space. Used to relate eikonal phases to amplitude corrections.

```latex
\begin{align}
\label{eq:Seikonal0}
S_{\rm eik} = 
\end{align}
```

```latex
\begin{align}
\label{eq:Seikonal}
S_{\rm eik} \ = \ 1 +  \widetilde{\cA}^{(0)}_{\omega} +  \widetilde{\cA}^{(1)}_{\omega^2} + \widetilde{\cA}^{(1)}_{\omega} + \ldots \ ,
\end{align}
```

--------------------------------

### Eikonal Phase Definitions

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Defines the leading eikonal phase \(\delta_0\) and the first subleading correction \(\delta_1\) in terms of the amplitude corrections in impact parameter space.

```latex
\begin{align} 
\label{delta0}
\delta_0 &= \  -i \, \widetilde{\cA}^{(0)}_{\omega}\,,
\end{align}
```

```latex
\label{delta1}
\delta_1 &= \  -i\,  \widetilde{\cA}^{(1)}_{\omega}\, 

```

--------------------------------

### Subleading Eikonal Amplitude (EH Contribution)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

The EH contribution to the subleading eikonal phase at order \omega, calculated in impact-parameter space and expanded around four-dimensional spacetime.

```latex
\left. \widetilde{\cA}^{(1)} (1^\phi,2^\phi,3^{-},4^{+})\right|_{\omega} = i \left(rac{\kappa}{2}
ight)^4\frac{15}{256\pi} \, \frac{m^2 \omega}{b} \> ,
```

--------------------------------

### Four-Momentum of Scattered Graviton in Spinor Notation

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Expresses the four-momentum of the scattered graviton (p3) in spinor notation, using components related to the momentum transfer squared and the magnitude of the momentum.

```latex
\begin{align}
\label{spainors}
\begin{split}
p_3 = \begin{pmatrix} 
\dfrac{\vec{q}^{\, 2}}{ 
 8 
|\vec{p}\, |}  & - 
\dfrac{\bar{q}}{2}
\vspace{0.3cm} \ 
- 
\dfrac{{q}}{2}  & 2  |\vec{p}\, |
\end{pmatrix} \ , 
\end{split}
\end{align}
```

--------------------------------

### R4 Amplitude (Helicity Preserving)

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Calculates the R4 amplitude in the helicity preserving case after integral reduction and eikonal approximation. Coefficients b1 and c1 are defined.

```latex
\mathcal{A}_{R^4}^{(1)}(1^{\phi},2^{\phi},3^{++},4^{--})= \ b_1 I_2(s) + c_1 I_3(s) \> ,
```

```latex
b_1= -\dfrac{35}{2} \, i \, \alpha \, \left( \dfrac{\kappa}{2}\right)^6 \, \dfrac{ s^2}{ m^2}\left(m^2-t\right)^4 \, \,
```

```latex
c_1=35 \ i \, \alpha  \,\left( \dfrac{\kappa}{2}\right)^6 \, s^2  \left(m^2-t\right)^4 \, \, .
```

--------------------------------

### Graviton Energy Approximation in Kinematic Limit

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Provides an approximation for the energy of the scattered gravitons (omega) in terms of the momentum magnitude |p| and the momentum transfer squared |q|^2, valid in the specified kinematic limit.

```latex
\begin{align}
\begin{split}
E_3=E_4 := \omega \simeq |\vec{p}\, |\left(1 + {\vec{q}^{\, 2} 
\over 8 \, \vec{p}^{\, 2}}
\right)
\ 
\end{split}
\end{align}
```

--------------------------------

### FFR Photon Tree Amplitude in Impact-Parameter Space

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Represents the non-vanishing helicity-flip amplitude transformed into impact-parameter space. Used in calculations of photon bending.

```latex
\widetilde{\cA}^{(0)}_{\rm FFR} (1^\phi, 2^\phi, 3^{+}, 4^{+}) \simeq i \left(\frac{\kappa}{2}\right)^2 \left(rac{\alpha_\gamma}{8}\right)\frac{m\,\omega}{\bar{\mathscr{b}}^2} \, \xi'' \, f(-2,D-2) \> ,
```

--------------------------------

### Tree-Level Helicity-Preserving Amplitude in R^3 Theory

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Shows that the tree-level helicity-preserving amplitude for four-point scalar/graviton scattering in EH + R^3 theory is zero.

```latex
\mathcal{A}_{R^3}^{(0)} (1^\phi, 2^\phi, 3^{--}, 4^{++}) = 0\ , 
```

--------------------------------

### Combined Box Integral in EH Gravity

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Presents the combined result for the $I_4(s, t) + I_4 (s, u)$ integral, which appears in the one-loop amplitudes of Einstein-Hilbert gravity.

```latex
I_4(s, t) + I_4 (s, u) \ = \ - {\frac{1}{8\pi}} {\frac{1}{m\omega}} {1\over D-4} {\abs{\vec{q} \,}^{D-6} }
.
```

--------------------------------

### Momentum Conservation Equations

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Defines the momentum conservation laws for external and loop momenta in a scattering process. Assumes all external momenta are outgoing and loop momenta flow from left to right.

```latex
p_1+p_2+l_1+l_2=0 \>, \hspace{0.5cm} p_3+p_4-l_1-l_2=0 \>
```

--------------------------------

### Coefficient \(c_2\) Calculation

Source: https://github.com/accettullihuber/spinorhelicity4d/blob/master/Package_functions/R4.txt

Provides the formula for the coefficient \(c_2\), which involves constants \(\alpha\), \(m\), \(s\), and \(\kappa\).

```latex
c_2=2 i \alpha m^4 s^2 \left( \frac{\kappa}{2}\right)^6
```