equation_database.arxiv_2506_23162

Functions

bibtex()	arXiv, Conversion of photons to dileptons in the Kroll-Wada and parton shower approaches
equation_2_1([rho, W_pair, W_gamma, alpha, ...])	\[\begin{split}\rho = \frac{W_{\text{pair}}}{W_{\gamma}} = \frac{2 \alpha}{3 \pi} \int_{2m_e}^E \,\mathrm{d}{M_{ee}} \left(\frac{k'}{k}\right) \frac{(E+M_R)^2 + M_R^2 -M_{ee}^2}{(E+M_R)^2 + M_R^2} \\\cdot \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \left[ \frac{R_T}{M_{ee}} + \frac{2(E+M_R)^2 M_{ee} }{(2EM_R +E^2 + M_{ee}^2)^2}R_L\right]\end{split}\]
equation_2_10([p_1, p_2])	\[p = p_1 + p_2\]
equation_2_11([k_T, p, z])	\[p_1 = z p + k_T\]
equation_2_12([k_T, p, z])	\[p_2 = (1-z) p - k_T\]
equation_2_13([p, M_ee, m_e, p_dot_p2, ...])	\[p^2 = M_{ee}^2 = 2m_e^2 + 2 p_1\cdot{}p_2 = 2m_e^2 + 2 ( z (1-z) p^2 - k_T^2 - (2z-1) p\cdot{}k_T)\]
equation_2_14([p_1, m_e, z, p, p_dot_kT, k_T])	\[p_1^2 = m_e^2 = z^2 p^2 + k_T^2 + 2 z p \cdot{} k_T\]
equation_2_15([M_ee, m_e, z, k_T])	\[M_{ee}^2 = 2m_e^2 + 2 ( z (1-z) M_{ee}^2 - k_T^2 ) - \frac{2z-1}{z} (m_e^2 - z^2 M_{ee}^2 - k_T^2 )\]
equation_2_16([z, m_e, M_ee, k_T])	\[z = \frac 1 2 \pm \frac 1 2 \sqrt{ 1 - 4 \frac{m_e^2}{M_{ee}^2} + 4k_T^2 }\]
equation_2_17([alpha, e_e, m_e, M_ee, ...])	\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + \frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}\]
equation_2_18([alpha, e_e, M_ee, m_e, ...])	\[\approx \frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \,\mathrm{d} N_{\gamma^*} \left( 1 - \frac{m_e^2}{M_{ee}^2} - 4 \frac{m_e^4}{M_{ee}^4} \right)\]
equation_2_19([P_gamma_to_ee, e_e, z, m_e, M_ee])	\[P_{\gamma\to ee}^m = e_e^2 \left( 1- 2z+ 2z^2 + 2\frac{m_e^2}{M_{ee}^2}\right)\]
equation_2_2([N_gamma_star, dN_ee, dM_ee, ...])	\[\frac{1}{N_{\gamma^*}} \frac{\,\mathrm{d} N_{ee}}{\,\mathrm{d} M_{ee}} = \frac{2 \alpha}{3 \pi} \frac{1}{M_{ee}} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right)S\approx \frac{1}{M_{ee}}\]
equation_2_20([alpha, M_ee, y_minus, ...])	\[\frac{\,\mathrm{d} \mathcal P^m_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P^m_{\gamma \to ee}(z) \,\mathrm{d} z\]
equation_2_21([alpha, e_e, M_ee, m_e])	\[=\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + 2\frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}\]
equation_2_22([M_ee, Q, p_T, z])	\[M_{ee}^2 = Q^2 = \frac{p_T^2}{z(1-z)}\]
equation_2_23([dPhi_FF_ant, f_FF_Kallen, ...])	\[\,\mathrm{d} \Phi^{\text{FF}}_\text{ant} = \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} s_{IK} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{ij} \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}\]
equation_2_24([a_bar_e_gamma, s_IK, y_ij, ...])	\[\bar{a}_{e/\gamma}^{\text{FF},\gamma} = \frac{1}{s_{IK}} \frac 1 2 \frac{1}{y_{ij} + 2 \mu_e^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 \mu_e^2}{y_{ij} + 2 \mu_e^2}\right]\]
equation_2_25([dP_gamma_to_ee, dM_ee_sq, ...])	\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}^m}{\,\mathrm{d} M_{ee}^2} \propto 4 \pi \alpha e_e^2 \frac 1 2 \frac{1}{M_{ee}^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 m_e^2}{M_{ee}^2}\right] \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}\]
equation_2_26([Gamma_ijk, y_ij, y_jk, y_ik, ...])	\[0< \Gamma_{ijk} = y_{ij} y_{jk} y_{ik} - y_{jk} \mu_i^2 - y_{ik} \mu_j^2\]
equation_2_27([z, m_e, M_ee])	\[z = \frac 1 2 \pm \frac 1 2 \sqrt{1-\frac{4m_e^2}{M_{ee}^2}}\]
equation_2_28([y_ij, mu_e, y_jk, y_ik])	\[1 = y_{ij} + 2 \mu_e^2 + y_{jk} + y_{ik}\]
equation_2_29([y_plus, y_minus, M_ee, m_e, s])	\[y_\pm = \frac{\pm\sqrt{\left(M_{ee}^{2} - 2 m_{e}^{2}\right)^{-1} \left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)} + \left(- M_{ee}^{2} + s\right) }{2 s }\]
equation_2_3([d2N_ee, dM_ee_sq, alpha, ...])	\[\frac{\,\mathrm{d}^2 N_{ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{3 \pi} \frac{1}{M_{ee}^2} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \,\mathrm{d} N_{\gamma^*}\]
equation_2_31([dP_gamma_to_ee, dM_ee_sq, ...])	\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - \frac{M_{ee}^2}{s}\right)^3\]
equation_2_4([alpha, M_ee, m_e, dN_gamma_star])	\[\approx \frac{\alpha}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - 6 \frac{m_e^4}{M_{ee}^4} - 8 \frac{m_e^6}{M_{ee}^6} \right) \,\mathrm{d} N_{\gamma^*}\]
equation_2_5([P_g_to_qq, T_F, z])	\[P_{g \to qq}(z) = T_F ( z^2 + (1-z)^2)\]
equation_2_6([P_gamma_to_ee, e_e, z])	\[P_{\gamma \to ee}(z) = e_e^2 ( z^2 + (1-z)^2)\]
equation_2_7([dP_gamma_to_ee, alpha, dQ_sq, ...])	\[\,\mathrm{d} \mathcal P_{\gamma \to ee} = \frac{\alpha}{2\pi} \frac{\,\mathrm{d} Q^2}{Q^2} P_{\gamma \to ee}(z) \,\mathrm{d} z\]
equation_2_8([dP_gamma_to_ee, dM_ee_sq, ...])	\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} P_{\gamma \to ee}(z) \,\mathrm{d} z\]
equation_2_9([dP_gamma_to_ee, dM_ee_sq, ...])	\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P_{\gamma \to ee}(z) \,\mathrm{d} z\]

equation_database.arxiv_2506_23162.bibtex()

arXiv, Conversion of photons to dileptons in the Kroll-Wada and parton shower approaches

@article{Jezo:2025jyu,
    author = "Je{\v{z}}o, Tom{\'a}{\v{s}} and Klasen, Michael and Puck Neuwirth, Alexander",
    title = "{Conversion of photons to dileptons in the Kroll-Wada and parton shower approaches}",
    eprint = "2506.23162",
    archivePrefix = "arXiv",
    primaryClass = "hep-ph",
    reportNumber = "MS-TP-25-18",
    month = "6",
    year = "2025"
}

equation_database.arxiv_2506_23162.equation_2_1(rho=rho, W_pair=W_pair, W_gamma=W_gamma, alpha=alpha, m_e=m_e, E=E, M_ee=M_ee, k_prime=k_prime, k=k, M_R=M_R, R_T=R_T, R_L=R_L)

\[\begin{split}\rho = \frac{W_{\text{pair}}}{W_{\gamma}} = \frac{2 \alpha}{3 \pi} \int_{2m_e}^E \,\mathrm{d}{M_{ee}} \left(\frac{k'}{k}\right) \frac{(E+M_R)^2 + M_R^2 -M_{ee}^2}{(E+M_R)^2 + M_R^2} \\\cdot \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \left[ \frac{R_T}{M_{ee}} + \frac{2(E+M_R)^2 M_{ee} }{(2EM_R +E^2 + M_{ee}^2)^2}R_L\right]\end{split}\]

Returns:

\(\rho = \frac{W_{pair}}{W_{\gamma}}\), \(\frac{W_{pair}}{W_{\gamma}} = \frac{2 \alpha \int\limits_{2 m_{e}}^{E} \frac{k_{prime} \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right) \left(\frac{2 M_{ee} R_{L} \left(E + M_{R}\right)^{2}}{\left(E^{2} + 2 E M_{R} + M_{ee}^{2}\right)^{2}} + \frac{R_{T}}{M_{ee}}\right) \left(M_{R}^{2} - M_{ee}^{2} + \left(E + M_{R}\right)^{2}\right)}{k \left(M_{R}^{2} + \left(E + M_{R}\right)^{2}\right)}\, dM_{ee}}{3 \pi}\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

\rho = \frac{W_{\text{pair}}}{W_{\gamma}} = \frac{2 \alpha}{3 \pi}     \int_{2m_e}^E \,\mathrm{d}{M_{ee}} \left(\frac{k'}{k}\right) \frac{(E+M_R)^2 + M_R^2 -M_{ee}^2}{(E+M_R)^2 + M_R^2}    \\\cdot    \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right)    \left[ \frac{R_T}{M_{ee}} + \frac{2(E+M_R)^2 M_{ee} }{(2EM_R +E^2 + M_{ee}^2)^2}R_L\right]

equation_database.arxiv_2506_23162.equation_2_2(N_gamma_star=N_gamma_star, dN_ee=dN_ee, dM_ee=dM_ee, alpha=alpha, M_ee=M_ee, m_e=m_e, S=S)

\[\frac{1}{N_{\gamma^*}} \frac{\,\mathrm{d} N_{ee}}{\,\mathrm{d} M_{ee}} = \frac{2 \alpha}{3 \pi} \frac{1}{M_{ee}} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right)S\approx \frac{1}{M_{ee}}\]

Returns:

\(\frac{dN_{ee}}{N_{\gamma star} dM_{ee}} = \frac{2 S \alpha \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right)}{3 \pi M_{ee}}\), \(\frac{2 S \alpha \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right)}{3 \pi M_{ee}} = \frac{1}{M_{ee}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\frac{1}{N_{\gamma^*}} \frac{\,\mathrm{d} N_{ee}}{\,\mathrm{d} M_{ee}} = \frac{2 \alpha}{3 \pi} \frac{1}{M_{ee}} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right)S\approx \frac{1}{M_{ee}}

equation_database.arxiv_2506_23162.equation_2_3(d2N_ee=d2N_ee, dM_ee_sq=dM_ee_sq, alpha=alpha, M_ee=M_ee, m_e=m_e, dN_gamma_star=dN_gamma_star)

\[\frac{\,\mathrm{d}^2 N_{ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{3 \pi} \frac{1}{M_{ee}^2} \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \,\mathrm{d} N_{\gamma^*}\]

Returns:

\(\frac{d2N_{ee}}{dM_{ee sq}} = \frac{\alpha dN_{\gamma star} \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right)}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\frac{\,\mathrm{d}^2 N_{ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{3 \pi} \frac{1}{M_{ee}^2}  \sqrt{ 1 - \frac{4 m_e^2}{M_{ee}^2}} \left( 1 + \frac{2 m_e^2}{M_{ee}^2}\right) \,\mathrm{d} N_{\gamma^*}

equation_database.arxiv_2506_23162.equation_2_31(dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq, alpha=alpha, e_e=e_e, M_ee=M_ee, s=s)

\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - \frac{M_{ee}^2}{s}\right)^3\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha e_{e}^{2} \left(- \frac{M_{ee}^{2}}{s} + 1\right)^{3}}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2}  =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2}  \left( 1 - \frac{M_{ee}^2}{s}\right)^3

equation_database.arxiv_2506_23162.equation_2_4(alpha=alpha, M_ee=M_ee, m_e=m_e, dN_gamma_star=dN_gamma_star)

\[\approx \frac{\alpha}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - 6 \frac{m_e^4}{M_{ee}^4} - 8 \frac{m_e^6}{M_{ee}^6} \right) \,\mathrm{d} N_{\gamma^*}\]

Returns:

\(\frac{\alpha dN_{\gamma star} \left(1 - \frac{6 m_{e}^{4}}{M_{ee}^{4}} - \frac{8 m_{e}^{6}}{M_{ee}^{6}}\right)}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\approx \frac{\alpha}{3\pi} \frac{1}{M_{ee}^2} \left( 1 - 6 \frac{m_e^4}{M_{ee}^4} - 8 \frac{m_e^6}{M_{ee}^6} \right)  \,\mathrm{d} N_{\gamma^*}

equation_database.arxiv_2506_23162.equation_2_5(P_g_to_qq=P_g_to_qq, T_F=T_F, z=z)

\[P_{g \to qq}(z) = T_F ( z^2 + (1-z)^2)\]

Returns:

\(P_{g to qq}{\left(z \right)} = T_{F} \left(z^{2} + \left(1 - z\right)^{2}\right)\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

P_{g \to qq}(z) =  T_F ( z^2 + (1-z)^2)

equation_database.arxiv_2506_23162.equation_2_6(P_gamma_to_ee=P_gamma_to_ee, e_e=e_e, z=z)

\[P_{\gamma \to ee}(z) = e_e^2 ( z^2 + (1-z)^2)\]

Returns:

\(P_{\gamma to ee}{\left(z \right)} = e_{e}^{2} \left(z^{2} + \left(1 - z\right)^{2}\right)\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

P_{\gamma \to ee}(z) =  e_e^2 ( z^2 + (1-z)^2)

equation_database.arxiv_2506_23162.equation_2_7(dP_gamma_to_ee=dP_gamma_to_ee, alpha=alpha, dQ_sq=dQ_sq, Q=Q, P_gamma_to_ee=P_gamma_to_ee, z=z, dz=dz)

\[\,\mathrm{d} \mathcal P_{\gamma \to ee} = \frac{\alpha}{2\pi} \frac{\,\mathrm{d} Q^2}{Q^2} P_{\gamma \to ee}(z) \,\mathrm{d} z\]

Returns:

\(dP_{\gamma to ee} = \frac{\alpha dQ_{sq} dz P_{\gamma to ee}{\left(z \right)}}{2 \pi Q^{2}}\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

\,\mathrm{d} \mathcal P_{\gamma \to ee} = \frac{\alpha}{2\pi} \frac{\,\mathrm{d} Q^2}{Q^2}  P_{\gamma \to ee}(z) \,\mathrm{d} z

equation_database.arxiv_2506_23162.equation_2_8(dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq, alpha=alpha, M_ee=M_ee, P_gamma_to_ee=P_gamma_to_ee, z=z, dz=dz)

\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} P_{\gamma \to ee}(z) \,\mathrm{d} z\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha dz P_{\gamma to ee}{\left(z \right)}}{2 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2}  P_{\gamma \to ee}(z) \,\mathrm{d} z

equation_database.arxiv_2506_23162.equation_2_9(dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq, alpha=alpha, M_ee=M_ee, y_minus=y_minus, y_plus=y_plus, P_gamma_to_ee=P_gamma_to_ee, z=z)

\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P_{\gamma \to ee}(z) \,\mathrm{d} z\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha \int\limits_{y_{minus}}^{y_{plus}} P_{\gamma to ee}{\left(z \right)}\, dz}{2 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2}  = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2}  \int_{y_{-}}^{y_+}P_{\gamma \to ee}(z) \,\mathrm{d} z

equation_database.arxiv_2506_23162.equation_2_10(p_1=p_1, p_2=p_2)

\[p = p_1 + p_2\]

Returns:

\(p_{1} + p_{2}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

p = p_1 + p_2

equation_database.arxiv_2506_23162.equation_2_11(k_T=k_T, p=p, z=z)

\[p_1 = z p + k_T\]

Returns:

\(k_{T} + p z\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

p_1 = z p + k_T

equation_database.arxiv_2506_23162.equation_2_12(k_T=k_T, p=p, z=z)

\[p_2 = (1-z) p - k_T\]

Returns:

\(- k_{T} + p \left(1 - z\right)\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

p_2 = (1-z) p - k_T

equation_database.arxiv_2506_23162.equation_2_13(p=p, M_ee=M_ee, m_e=m_e, p_dot_p2=p_dot_p2, p_dot_kT=p_dot_kT, z=z, k_T=k_T)

\[p^2 = M_{ee}^2 = 2m_e^2 + 2 p_1\cdot{}p_2 = 2m_e^2 + 2 ( z (1-z) p^2 - k_T^2 - (2z-1) p\cdot{}k_T)\]

Returns:

\(p^{2} = M_{ee}^{2}\), \(M_{ee}^{2} = 2 m_{e}^{2} + 2 p_{dot p2}\), \(2 m_{e}^{2} + 2 p_{dot p2} = - 2 k_{T}^{2} + 2 m_{e}^{2} + 2 p^{2} z \left(1 - z\right) - 2 p_{dot kT} \left(2 z - 1\right)\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

p^2 = M_{ee}^2 = 2m_e^2 + 2 p_1\cdot{}p_2 =  2m_e^2 + 2  ( z (1-z) p^2 - k_T^2 - (2z-1) p\cdot{}k_T)

equation_database.arxiv_2506_23162.equation_2_14(p_1=p_1, m_e=m_e, z=z, p=p, p_dot_kT=p_dot_kT, k_T=k_T)

\[p_1^2 = m_e^2 = z^2 p^2 + k_T^2 + 2 z p \cdot{} k_T\]

Returns:

\(p_{1}^{2} = m_{e}^{2}\), \(m_{e}^{2} = k_{T}^{2} + p^{2} z^{2} + 2 p_{dot kT} z\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

p_1^2 = m_e^2 = z^2 p^2 + k_T^2 + 2 z p \cdot{} k_T

equation_database.arxiv_2506_23162.equation_2_15(M_ee=M_ee, m_e=m_e, z=z, k_T=k_T)

\[M_{ee}^2 = 2m_e^2 + 2 ( z (1-z) M_{ee}^2 - k_T^2 ) - \frac{2z-1}{z} (m_e^2 - z^2 M_{ee}^2 - k_T^2 )\]

Returns:

\(M_{ee}^{2} = 2 M_{ee}^{2} z \left(1 - z\right) - 2 k_{T}^{2} + 2 m_{e}^{2} - \frac{\left(2 z - 1\right) \left(- M_{ee}^{2} z^{2} - k_{T}^{2} + m_{e}^{2}\right)}{z}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

M_{ee}^2 =  2m_e^2 + 2  ( z (1-z) M_{ee}^2 - k_T^2 ) - \frac{2z-1}{z}  (m_e^2 - z^2 M_{ee}^2  - k_T^2  )

equation_database.arxiv_2506_23162.equation_2_16(z=z, m_e=m_e, M_ee=M_ee, k_T=k_T)

\[z = \frac 1 2 \pm \frac 1 2 \sqrt{ 1 - 4 \frac{m_e^2}{M_{ee}^2} + 4k_T^2 }\]

Returns:

\(z = \frac{\sqrt{4 k_{T}^{2} + 1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}}{2} + \frac{1}{2}\), \(z = \frac{1}{2} - \frac{\sqrt{4 k_{T}^{2} + 1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}}{2}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

z = \frac 1 2  \pm \frac 1 2  \sqrt{ 1 - 4 \frac{m_e^2}{M_{ee}^2} + 4k_T^2 }

equation_database.arxiv_2506_23162.equation_2_17(alpha=alpha, e_e=e_e, m_e=m_e, M_ee=M_ee, dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq)

\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + \frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha e_{e}^{2} \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{m_{e}^{2}}{M_{ee}^{2}}\right)}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2}  =\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2}  \left(  1  + \frac{m_e^2}{M_{ee}^2}\right)  \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}

equation_database.arxiv_2506_23162.equation_2_18(alpha=alpha, e_e=e_e, M_ee=M_ee, m_e=m_e, dN_gamma_star=dN_gamma_star)

\[\approx \frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \,\mathrm{d} N_{\gamma^*} \left( 1 - \frac{m_e^2}{M_{ee}^2} - 4 \frac{m_e^4}{M_{ee}^4} \right)\]

Returns:

\(\frac{\alpha dN_{\gamma star} e_{e}^{2} \left(1 - \frac{m_{e}^{2}}{M_{ee}^{2}} - \frac{4 m_{e}^{4}}{M_{ee}^{4}}\right)}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\approx \frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2}  \,\mathrm{d} N_{\gamma^*} \left( 1 - \frac{m_e^2}{M_{ee}^2} - 4 \frac{m_e^4}{M_{ee}^4} \right)

equation_database.arxiv_2506_23162.equation_2_19(P_gamma_to_ee=P_gamma_to_ee, e_e=e_e, z=z, m_e=m_e, M_ee=M_ee)

\[P_{\gamma\to ee}^m = e_e^2 \left( 1- 2z+ 2z^2 + 2\frac{m_e^2}{M_{ee}^2}\right)\]

Returns:

\(P_{\gamma to ee} = e_{e}^{2} \left(2 z^{2} - 2 z + 1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right)\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

P_{\gamma\to ee}^m = e_e^2 \left( 1- 2z+ 2z^2 + 2\frac{m_e^2}{M_{ee}^2}\right)

equation_database.arxiv_2506_23162.equation_2_20(alpha=alpha, M_ee=M_ee, y_minus=y_minus, y_plus=y_plus, z=z, P_gamma_to_ee=P_gamma_to_ee, dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq)

\[\frac{\,\mathrm{d} \mathcal P^m_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2} = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2} \int_{y_{-}}^{y_+}P^m_{\gamma \to ee}(z) \,\mathrm{d} z\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha \int\limits_{y_{minus}}^{y_{plus}} P_{\gamma to ee}{\left(z \right)}\, dz}{2 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyMathematicaASCIIUnicode

\frac{\,\mathrm{d} \mathcal P^m_{\gamma \to ee}}{\,\mathrm{d} M_{ee}^2}  = \frac{\alpha}{2\pi} \frac{1}{M_{ee}^2}  \int_{y_{-}}^{y_+}P^m_{\gamma \to ee}(z) \,\mathrm{d} z

equation_database.arxiv_2506_23162.equation_2_21(alpha=alpha, e_e=e_e, M_ee=M_ee, m_e=m_e)

\[=\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2} \left( 1 + 2\frac{m_e^2}{M_{ee}^2}\right) \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}\]

Returns:

\(\frac{\alpha e_{e}^{2} \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}} \left(1 + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right)}{3 \pi M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

=\frac{\alpha e_e^2}{3\pi} \frac{1}{M_{ee}^2}  \left(  1  + 2\frac{m_e^2}{M_{ee}^2}\right)  \sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}

equation_database.arxiv_2506_23162.equation_2_22(M_ee=M_ee, Q=Q, p_T=p_T, z=z)

\[M_{ee}^2 = Q^2 = \frac{p_T^2}{z(1-z)}\]

Returns:

\(M_{ee}^{2} = Q^{2}\), \(Q^{2} = \frac{p_{T}^{2}}{z \left(1 - z\right)}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

M_{ee}^2 = Q^2 = \frac{p_T^2}{z(1-z)}

equation_database.arxiv_2506_23162.equation_2_23(dPhi_FF_ant=dPhi_FF_ant, f_FF_Kallen=f_FF_Kallen, s_IK=s_IK, Gamma_ijk=Gamma_ijk, dy_ij=dy_ij, dy_jk=dy_jk, dphi=dphi)

\[\,\mathrm{d} \Phi^{\text{FF}}_\text{ant} = \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} s_{IK} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{ij} \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}\]

Returns:

\(dPhi_{FF ant} = \frac{dphi dy_{ij} dy_{jk} f_{FF Kallen} s_{IK} \theta\left(\Gamma_{ijk}, 1\right)}{32 \pi^{3}}\),

Original LaTeXLaTeXMathMLSympyOctavePythonCC++ASCIIUnicode

\,\mathrm{d} \Phi^{\text{FF}}_\text{ant} = \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} s_{IK} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{ij} \,\mathrm{d} y_{jk}  \frac{\,\mathrm{d} \phi}{2 \pi}

equation_database.arxiv_2506_23162.equation_2_24(a_bar_e_gamma=a_bar_e_gamma, s_IK=s_IK, y_ij=y_ij, y_ik=y_ik, y_jk=y_jk, mu_e=mu_e)

\[\bar{a}_{e/\gamma}^{\text{FF},\gamma} = \frac{1}{s_{IK}} \frac 1 2 \frac{1}{y_{ij} + 2 \mu_e^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 \mu_e^2}{y_{ij} + 2 \mu_e^2}\right]\]

Returns:

\(a_{bar e \gamma} = \frac{\frac{2 \mu_{e}^{2}}{2 \mu_{e}^{2} + y_{ij}} + y_{ik}^{2} + y_{jk}^{2}}{2 s_{IK} \left(2 \mu_{e}^{2} + y_{ij}\right)}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

\bar{a}_{e/\gamma}^{\text{FF},\gamma} = \frac{1}{s_{IK}} \frac 1 2 \frac{1}{y_{ij} + 2 \mu_e^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 \mu_e^2}{y_{ij} + 2 \mu_e^2}\right]

equation_database.arxiv_2506_23162.equation_2_25(dP_gamma_to_ee=dP_gamma_to_ee, dM_ee_sq=dM_ee_sq, alpha=alpha, e_e=e_e, M_ee=M_ee, y_ik=y_ik, y_jk=y_jk, m_e=m_e, f_FF_Kallen=f_FF_Kallen, Gamma_ijk=Gamma_ijk, dy_jk=dy_jk, dphi=dphi)

\[\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}^m}{\,\mathrm{d} M_{ee}^2} \propto 4 \pi \alpha e_e^2 \frac 1 2 \frac{1}{M_{ee}^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 m_e^2}{M_{ee}^2}\right] \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{jk} \frac{\,\mathrm{d} \phi}{2 \pi}\]

Returns:

\(\frac{dP_{\gamma to ee}}{dM_{ee sq}} = \frac{\alpha dphi dy_{jk} e_{e}^{2} f_{FF Kallen} \left(y_{ik}^{2} + y_{jk}^{2} + \frac{2 m_{e}^{2}}{M_{ee}^{2}}\right) \theta\left(\Gamma_{ijk}, 1\right)}{16 \pi^{2} M_{ee}^{2}}\),

Original LaTeXLaTeXMathMLSympyOctavePythonCC++ASCIIUnicode

\frac{\,\mathrm{d} \mathcal P_{\gamma \to ee}^m}{\,\mathrm{d} M_{ee}^2} \propto 4 \pi \alpha e_e^2 \frac 1 2 \frac{1}{M_{ee}^2} \left[y_{ik}^2 + y_{jk}^2 + \frac{2 m_e^2}{M_{ee}^2}\right] \frac{1}{16 \pi^2} f^{\text{FF}}_\text{Källén} \Theta(\Gamma_{ijk}) \,\mathrm{d} y_{jk}  \frac{\,\mathrm{d} \phi}{2 \pi}

equation_database.arxiv_2506_23162.equation_2_26(Gamma_ijk=Gamma_ijk, y_ij=y_ij, y_jk=y_jk, y_ik=y_ik, mu_i=mu_i, mu_j=mu_j)

\[0< \Gamma_{ijk} = y_{ij} y_{jk} y_{ik} - y_{jk} \mu_i^2 - y_{ik} \mu_j^2\]

Returns:

\(0 > \Gamma_{ijk}\), \(\Gamma_{ijk} = - \mu_{i}^{2} y_{jk} - \mu_{j}^{2} y_{ik} + y_{ij} y_{ik} y_{jk}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

0< \Gamma_{ijk} = y_{ij} y_{jk} y_{ik} - y_{jk} \mu_i^2 - y_{ik} \mu_j^2

equation_database.arxiv_2506_23162.equation_2_27(z=z, m_e=m_e, M_ee=M_ee)

\[z = \frac 1 2 \pm \frac 1 2 \sqrt{1-\frac{4m_e^2}{M_{ee}^2}}\]

Returns:

\(z = \frac{\sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}}{2} + \frac{1}{2}\), \(z = \frac{1}{2} - \frac{\sqrt{1 - \frac{4 m_{e}^{2}}{M_{ee}^{2}}}}{2}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

z = \frac 1 2 \pm \frac 1 2 \sqrt{1-\frac{4m_e^2}{M_{ee}^2}}

equation_database.arxiv_2506_23162.equation_2_28(y_ij=y_ij, mu_e=mu_e, y_jk=y_jk, y_ik=y_ik)

\[1 = y_{ij} + 2 \mu_e^2 + y_{jk} + y_{ik}\]

Returns:

\(1 = 2 \mu_{e}^{2} + y_{ij} + y_{ik} + y_{jk}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

1 = y_{ij} + 2 \mu_e^2 + y_{jk} + y_{ik}

equation_database.arxiv_2506_23162.equation_2_29(y_plus=y_plus, y_minus=y_minus, M_ee=M_ee, m_e=m_e, s=s)

\[y_\pm = \frac{\pm\sqrt{\left(M_{ee}^{2} - 2 m_{e}^{2}\right)^{-1} \left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)} + \left(- M_{ee}^{2} + s\right) }{2 s }\]

Returns:

\(y_{plus} = \sqrt{\frac{\left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)}{M_{ee}^{2} - 2 m_{e}^{2}}} + \frac{- M_{ee}^{2} + s}{2 s}\), \(y_{minus} = - \sqrt{\frac{\left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)}{M_{ee}^{2} - 2 m_{e}^{2}}} + \frac{- M_{ee}^{2} + s}{2 s}\),

Original LaTeXLaTeXMathMLSympyOctaveMathematicaPythonCC++FortranRustASCIIUnicode

y_\pm =  \frac{\pm\sqrt{\left(M_{ee}^{2} - 2 m_{e}^{2}\right)^{-1} \left(M_{ee}^{2} - s\right) \left(M_{ee}^{4} - 2 M_{ee}^{2} m_{e}^{2} - M_{ee}^{2} s + 6 m_{e}^{2} s\right)} + \left(- M_{ee}^{2} + s\right) }{2 s }