#
Production of Magnetic Monopoles via Photon Fusion: Implementation in MadGraph ^{†}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. MadGraph Universal FeynRules Output Model

#### 2.1. Monopole Couplings

#### 2.2. Generation and Validation of the MadGraph UFO Model

#### 2.3. LHC Phenomenology

#### 2.3.1. Kinematic Distributions

`NNPDF23`at lowest order (LO) [15]. For the PF method,

`LUXqed`[16] was used. This was done as

`LUXqed`provides a relatively small uncertainty in the photon distribution function of the proton [17]. For spin ½ plots, the value of the $\kappa $ parameter was taken to be zero, and for spin 1 plots, the value of the $\kappa $ parameter was taken to be one.

#### 2.3.2. Limiting Case of Large $\kappa $ and Small $\beta $ for Photon Fusion

**a. Spin ½ monopole scenario:**A magnetic moment-generating term $\kappa $ has been added to the Lagrangian of the spin ½ monopole case (Equation (7)). A dimensionless parameter $\tilde{\kappa}=\kappa M$ with M being the mass of the monopole has been varied from 0–10,000 for the photon fusion process with a photon-photon collision energy of 13 TeV. The cross-section for $\tilde{\kappa}=0$ (the Standard Model scenario) goes to zero at $\beta \to 0$ very fast, as can be seen from the third column of Table 7. However, for the non-zero $\tilde{\kappa}$, the cross-section values remain finite, even if the $\beta $ goes to zero. The same conclusion can be obtained from the left plot of Figure 6, where the cross-section has been plotted against the monopole mass for the proton-proton collision energy of 13 TeV.

**b. Spin 1 monopole scenario:**Similar to the spin ½ monopole scenario, the spin 1 monopole also has a magnetic moment term in the Lagrangian (Equation (8)). For the spin 1 monopole, $\kappa =1$ is the Standard Model case. Here, the cross-section for the $\kappa =1$ case goes to zero very fast as $\beta $ goes to zero (as seen from the third column of Table 8) at a photon-photon collision energy of 13 TeV. However, when $\kappa >1$, the cross-section remains finite as $\beta $ goes to zero. A similar conclusion can be drawn from the cross-section vs. monopole mass plot of Figure 7 (left), where the proton-proton collision energy of 13 TeV has been used.

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LHC | Large Hadron Collider |

ATLAS | A Toroidal LHC Apparatus |

MoEDAL | Monopole and Exotics Detector at the LHC |

PF | Photon Fusion |

DY | Drell–Yan |

UFO | Universal FeynRules Output |

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**Figure 1.**The Feynman Diagram produced by MadGraph for the spin 0 magnetic monopole scenario. Here, “mm+” suggests monopole, “mm-” shows the anti-monopole, and “a” describes the photon.

**Figure 2.**The Feynman Diagrams produced by MadGraph for spin ½ magnetic monopoles. Here, “mm+” suggests monopole, “mm−” shows the anti-monopole and “a” describes the photon.

**Figure 3.**The Feynman Diagrams produced by MadGraph for the spin 1 magnetic monopoles. Here “mm+” suggests monopole, “mm−” shows the anti-monopole, and “a” describes the photon.

**Figure 4.**Kinetic energy distribution comparison between the photon fusion and Drell–Yan processes for spin 0 (

**left**), spin ½ (

**middle**), and spin 1 (

**right**) monopoles. Here, $\beta $-dependent coupling has been used. For the PDFs,

`NNPDF23`was used for the Drell–Yan process and

`LUXqed`was used for the photon fusion process [10].

**Figure 5.**Pseudorapidity ($\eta $) distribution comparison between the photon fusion and Drell–Yan processes for spin 0 (

**left**), spin ½ (

**middle**), and spin 1 (

**right**) monopoles. Here, $\beta $-dependent coupling has been used. For the PDFs,

`NNPDF23`was used for the Drell–Yan process and

`LUXqed`was used for the photon fusion process [10].

**Figure 6.**The cross-section variation with the monopole mass at a proton-proton collision energy of 13 TeV is shown on the

**left**plot for the spin ½ monopoles. The transverse momentum (${p}_{T}$) distribution (

**middle**) and pseudorapidity distribution (

**right**) are shown for the spin ½ monopole at a proton-proton collision energy of 13 TeV [10].

**Figure 7.**The cross-section variation with the monopole mass at a proton-proton collision energy of 13 TeV is shown on the

**left**plot for the spin 1 monopoles. The transverse momentum (${p}_{T}$) distribution (

**middle**) and pseudorapidity distribution (

**right**) are shown for the spin 1 monopole at a proton-proton collision energy of 13 TeV [10].

**Table 1.**The cross-section for the spin 0 monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-independent coupling when no PDF is used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of the UFO model to that of the theoretical prediction. These ratios are very close to 1, suggesting excellent agreement between the theory and the MadGraph model [10].

Mass (GeV) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (UFO Model) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (Theory Values) | Ratio UFO Model/Theory |
---|---|---|---|

1000 | $1.518\times {10}^{4}$ | $1.5039\times {10}^{4}$ | $1.009$ |

2000 | $1.202\times {10}^{4}$ | $1.1945\times {10}^{4}$ | $1.006$ |

3000 | 9218 | $9108.09$ | $1.012$ |

4000 | 7366 | $7218.79$ | $1.020$ |

5000 | 6558 | $6519.68$ | $1.006$ |

6000 | 5378 | $5325.76$ | $1.010$ |

**Table 2.**The cross-section for the spin 0 monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-dependent coupling constant, when no PDF is used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of the UFO model to that of the theoretical prediction. These ratios are very close to 1, suggesting excellent agreement between the theory and the MadGraph model. Here, the sixth column shows the ratio of cross-sections as obtained by the UFO model for $\beta $-dependent coupling to $\beta $-independent coupling (shown in Table 1). This ratio varies as ${\beta}^{4}$, as expected from the theory [10].

Mass (GeV) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (UFO Model) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (Theory Values) | Ratio UFO Model/Theory | $\mathit{\beta}$ | Ratio $\mathit{\beta}$-dep/$\mathit{\beta}$-ind (UFO Model) |
---|---|---|---|---|---|

1000 | $1.4493\times {10}^{4}$ | $1.4336\times {10}^{4}$ | $0.99$ | 0.9881 | 0.9547 (~$0.{9881}^{4}$) |

2000 | $9.851\times {10}^{3}$ | $9.791\times {10}^{3}$ | $1.006$ | 0.9515 | 0.8196 (~$0.{9515}^{4}$) |

3000 | $5.685\times {10}^{3}$ | $5.640\times {10}^{3}$ | $1.007$ | 0.8871 | 0.6167 (~$0.{8871}^{4}$) |

4000 | 2847 | $2810.5$ | $1.013$ | 0.7882 | 0.3866 (~$0.{7882}^{4}$) |

5000 | 1094 | 1087 | $1.006$ | 0.639 | 0.1658 (~$0.{639}^{4}$) |

6000 | $117.8$ | $116.53$ | $1.011$ | 0.3846 | 0.022 (~$0.{3846}^{4}$) |

**Table 3.**The cross-section for the spin ½ monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-independent coupling when no PDF is used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of the UFO model to that of the theoretical prediction. As the ratios are very close to 1, this suggests excellent agreement between the theory and the MadGraph model [10].

Mass (GeV) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (UFO Model) | $\mathit{\sigma}$ (pb) $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-}$ (Theory Values) | Ratio UFO Model/Theory |
---|---|---|---|

1000 | $1.431\times {10}^{5}$ | $1.425\times {10}^{5}$ | $1.004$ |

2000 | $1.018\times {10}^{5}$ | $1.007\times {10}^{5}$ | $1.010$ |

3000 | $7.755\times {10}^{4}$ | $7.679\times {10}^{4}$ | $1.010$ |

4000 | $5.830\times {10}^{4}$ | $5.7404\times {10}^{4}$ | $1.016$ |

5000 | $3.817\times {10}^{4}$ | $3.797\times {10}^{4}$ | $1.005$ |

6000 | $1.691\times {10}^{4}$ | $1.6705\times {10}^{4}$ | $1.012$ |

**Table 4.**The cross-section for the spin ½ monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-dependent coupling constant, when no PDF is used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of the UFO model to that of the theoretical prediction. As these ratios are very close to 1, they suggest excellent agreement between the theory and the MadGraph model. Here, the sixth column shows the ratio of cross-sections as obtained by the UFO model for $\beta $-dependent coupling to $\beta $-independent coupling (shown in Table 3). This ratio varies as ${\beta}^{4}$, as expected from the theory [10].

Mass (GeV) | Ratio UFO Model/Theory | $\mathit{\beta}$ | Ratio $\mathit{\beta}$-dep/$\mathit{\beta}$-ind (UFO Model) | ||
---|---|---|---|---|---|

1000 | $1.364\times {10}^{5}$ | $1.358\times {10}^{5}$ | $1.004$ | 0.9881 | 0.9531 (~$0.{9881}^{4}$) |

2000 | $8.341\times {10}^{4}$ | $8.2551\times {10}^{4}$ | $1.010$ | 0.9515 | 0.8193 (~$0.{9515}^{4}$) |

3000 | $4.803\times {10}^{4}$ | $4.7554\times {10}^{4}$ | $1.010$ | 0.8871 | 0.6193 (~$0.{8871}^{4}$) |

4000 | $2.251\times {10}^{4}$ | $2.2156\times {10}^{4}$ | $1.012$ | 0.7882 | 0.3861 (~$0.{7882}^{4}$) |

5000 | 6362 | 6331 | $1.005$ | 0.639 | 0.1667 (~$0.{639}^{4}$) |

6000 | 370 | $365.5$ | $1.012$ | 0.3846 | 0.0219 (~$0.{3846}^{4}$) |

**Table 5.**The cross-section for the spin 1 monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-independent coupling when no PDF is used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of the UFO model to that of the theoretical prediction. The ratios, being very close to 1, suggest excellent agreement between the theory and the MadGraph model [10].

Mass (GeV) | Ratio UFO Model/Theory | ||
---|---|---|---|

1000 | $1.131\times {10}^{7}$ | $1.131\times {10}^{7}$ | $1.000$ |

2000 | $2.765\times {10}^{6}$ | $2.747\times {10}^{6}$ | $1.007$ |

3000 | $1.164\times {10}^{6}$ | $1.151\times {10}^{6}$ | $1.011$ |

4000 | $5.879\times {10}^{5}$ | $5.835\times {10}^{5}$ | $1.008$ |

5000 | $3.161\times {10}^{5}$ | $3.109\times {10}^{5}$ | $1.017$ |

6000 | $1.39\times {10}^{5}$ | $1.378\times {10}^{5}$ | $1.009$ |

**Table 6.**The cross-section for the spin 1 monopole as obtained by the theory and the MadGraph UFO model for the $\beta $-dependent coupling constant, when no PDF was used and the center-of-mass energy is 13 TeV. The fourth column shows the ratio of the cross-sections of UFO model to that of the theoretical prediction. The ratios, being very close to 1, suggest excellent agreement between the theory and the MadGraph model. Here, the sixth column shows the ratio of cross-sections as obtained by the UFO model for $\beta $-dependent coupling to $\beta $-independent coupling (shown in Table 5). This ratio varies as ${\beta}^{4}$, as expected from the theory [10].

Mass (GeV) | Ratio UFO Model/Theory | $\mathit{\beta}$ | Ratio $\mathit{\beta}$-dep/$\mathit{\beta}$-ind (UFO Model) | ||
---|---|---|---|---|---|

1000 | $1.078\times {10}^{7}$ | $1.0781\times {10}^{7}$ | $0.999$ | 0.9881 | 0.9531 (~$0.{9881}^{4}$) |

2000 | $2.277\times {10}^{6}$ | $2.2520\times {10}^{6}$ | $1.011$ | 0.9515 | 0.8235 (~$0.{9515}^{4}$) |

3000 | $7.214\times {10}^{5}$ | $7.1290\times {10}^{5}$ | $1.012$ | 0.8871 | 0.6198 (~$0.{8871}^{4}$) |

4000 | $2.275\times {10}^{5}$ | $2.2523\times {10}^{5}$ | $1.010$ | 0.7882 | 0.3870 (~$0.{7882}^{4}$) |

5000 | $5.256\times {10}^{4}$ | $5.1833\times {10}^{4}$ | $1.014$ | 0.639 | 0.1663 (~$0.{639}^{4}$) |

6000 | $3.034\times {10}^{3}$ | $3.014\times {10}^{3}$ | $1.007$ | 0.3846 | 0.0218 (~$0.{3846}^{4}$) |

**Table 7.**Photon fusion production cross-sections at a photon-photon collision energy of 13 TeV for the spin ½ monopole, $\beta $-dependent coupling, and various values of the $\tilde{\kappa}$ parameter [10].

Monopole | $\mathit{\beta}$ | $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\sigma}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{pb}\right)$ | |||
---|---|---|---|---|---|

Mass (GeV) | $\tilde{\mathit{\kappa}}=0$ | $\tilde{\mathit{\kappa}}=10$ | $\tilde{\mathit{\kappa}}=100$ | $\tilde{\mathit{\kappa}}=10,000$ | |

1000 | 0.9881 | $1.37\times {10}^{5}\pm 4.6\times {10}^{2}$ | $1.639\times {10}^{24}\pm 3.3\times {10}^{21}$ | $1.639\times {10}^{28}\pm 3.3\times {10}^{25}$ | $1.639\times {10}^{36}\pm 3.3\times {10}^{33}$ |

2000 | 0.9515 | $8.303\times {10}^{4}\pm 4.5\times {10}^{2}$ | $1.61\times {10}^{24}\pm 3.1\times {10}^{21}$ | $1.61\times {10}^{28}\pm 3.1\times {10}^{25}$ | $1.61\times {10}^{36}\pm 3.1\times {10}^{33}$ |

3000 | 0.8871 | $4.78\times {10}^{4}\pm 3.5\times {10}^{2}$ | $1.356\times {10}^{24}\pm 2.5\times {10}^{21}$ | $1.356\times {10}^{28}\pm 2.5\times {10}^{25}$ | $1.356\times {10}^{36}\pm 2.5\times {10}^{33}$ |

4000 | 0.7882 | $2.237\times {10}^{4}\pm 1.9\times {10}^{2}$ | $8.612\times {10}^{23}\pm 2.1\times {10}^{21}$ | $8.613\times {10}^{27}\pm 2.1\times {10}^{25}$ | $8.613\times {10}^{35}\pm 2.1\times {10}^{33}$ |

5000 | 0.639 | $6396\pm 61$ | $3.154\times {10}^{23}\pm 1.1\times {10}^{21}$ | $3.154\times {10}^{27}\pm 1.1\times {10}^{25}$ | $3.154\times {10}^{35}\pm 1.1\times {10}^{33}$ |

5500 | 0.5329 | $2256\pm 22$ | $1.247\times {10}^{23}\pm 4.5\times {10}^{20}$ | $1.247\times {10}^{27}\pm 4.5\times {10}^{24}$ | $1.247\times {10}^{35}\pm 4.5\times {10}^{32}$ |

5800 | 0.4514 | $886.5\pm 7.8$ | $5.28\times {10}^{22}\pm 2.5\times {10}^{20}$ | $5.28\times {10}^{26}\pm 2.5\times {10}^{24}$ | $5.28\times {10}^{34}\pm 2.5\times {10}^{32}$ |

6000 | 0.3846 | $367.2\pm 3$ | $2.294\times {10}^{22}\pm 7.6\times {10}^{19}$ | $2.294\times {10}^{26}\pm 7.6\times {10}^{23}$ | $2.294\times {10}^{34}\pm 7.6\times {10}^{31}$ |

6200 | 0.3003 | $97.19\pm 0.77$ | $6.43\times {10}^{21}\pm 3.3\times {10}^{19}$ | $6.43\times {10}^{25}\pm 3.3\times {10}^{23}$ | $6.43\times {10}^{33}\pm 3.3\times {10}^{31}$ |

6400 | 0.1747 | $5.846\pm 0.025$ | $4.065\times {10}^{20}\pm 1.5\times {10}^{18}$ | $4.065\times {10}^{24}\pm 1.5\times {10}^{22}$ | $4.065\times {10}^{32}\pm 1.5\times {10}^{30}$ |

6490 | $0.0554$ | $0.017\pm 2.27\times {10}^{-5}$ | $1.27\times {10}^{18}\pm 8.74\times {10}^{14}$ | $1.27\times {10}^{22}\pm 8.74\times {10}^{18}$ | $1.27\times {10}^{30}\pm 8.74\times {10}^{26}$ |

**Table 8.**Photon fusion production cross-sections at a photon-photon collision energy of 13 TeV for the spin 1 monopole, $\beta $-dependent coupling, and various values of the $\kappa $ parameter [10].

Monopole Mass (GeV) | $\mathit{\beta}$ | $\mathit{\gamma}\mathit{\gamma}\to {\mathbf{mm}}^{+}{\mathbf{mm}}^{-},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\sigma}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{pb}\right)$ | ||
---|---|---|---|---|

$\mathbf{\kappa}=\mathbf{1}$ | $\mathbf{\kappa}=\mathbf{100}$ | $\mathbf{\kappa}=\mathbf{10},\mathbf{000}$ | ||

1000 | 0.9881 | $1.086\times {10}^{7}\pm 1.4\times {10}^{5}$ | $4.939\times {10}^{15}\pm 1\times {10}^{13}$ | $5.033\times {10}^{23}\pm 2.1\times {10}^{21}$ |

2000 | 0.9515 | $2.275\times {10}^{6}\pm 1.6\times {10}^{4}$ | $2.844\times {10}^{14}\pm 4.9\times {10}^{11}$ | $2.879\times {10}^{22}\pm 9.8\times {10}^{19}$ |

3000 | 0.8871 | $7.198\times {10}^{5}\pm 6.6\times {10}^{3}$ | $4.518\times {10}^{13}\pm 1.5\times {10}^{11}$ | $4.536\times {10}^{21}\pm 1.2\times {10}^{19}$ |

4000 | 0.7882 | $2.273\times {10}^{5}\pm 2.2\times {10}^{3}$ | $9.079\times {10}^{12}\pm 2.7\times {10}^{10}$ | $9.002\times {10}^{20}\pm 3.2\times {10}^{18}$ |

5000 | 0.639 | $5.232\times {10}^{4}\pm 4.9\times {10}^{2}$ | $1.513\times {10}^{12}\pm 9.2\times {10}^{9}$ | $1.5\times {10}^{20}\pm 9.3\times {10}^{17}$ |

5500 | 0.5329 | $1.785\times {10}^{4}\pm 1.6\times {10}^{2}$ | $4.49\times {10}^{11}\pm 1.7\times {10}^{9}$ | $4.466\times {10}^{19}\pm 2.9\times {10}^{17}$ |

5800 | 0.4514 | $7118\pm 62$ | $1.658\times {10}^{11}\pm 1.1\times {10}^{9}$ | $1.624\times {10}^{19}\pm 8.4\times {10}^{16}$ |

6000 | 0.3846 | $3025\pm 24$ | $6.72\times {10}^{10}\pm 2.5\times {10}^{8}$ | $6.627\times {10}^{18}\pm 3.7\times {10}^{16}$ |

6200 | 0.3003 | $836.9\pm 6.3$ | $1.764\times {10}^{10}\pm 1\times {10}^{8}$ | $1.733\times {10}^{18}\pm 1\times {10}^{16}$ |

6400 | 0.1747 | $53.42\pm 0.23$ | $1.066\times {10}^{9}\pm 3.9\times {10}^{6}$ | $1.05\times {10}^{17}\pm 3.8\times {10}^{14}$ |

6490 | $0.0554$ | $0.1694\pm 0.00065$ | $3.293\times {10}^{6}\pm 5.6\times {10}^{3}$ | $3.244\times {10}^{14}\pm 5.6\times {10}^{11}$ |

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**MDPI and ACS Style**

Santra, A.
Production of Magnetic Monopoles via Photon Fusion: Implementation in MadGraph ^{†}

. *Proceedings* **2019**, *13*, 4.
https://doi.org/10.3390/proceedings2019013004

**AMA Style**

Santra A.
Production of Magnetic Monopoles via Photon Fusion: Implementation in MadGraph ^{†}

. *Proceedings*. 2019; 13(1):4.
https://doi.org/10.3390/proceedings2019013004

**Chicago/Turabian Style**

Santra, Arka.
2019. "Production of Magnetic Monopoles via Photon Fusion: Implementation in MadGraph ^{†}

" *Proceedings* 13, no. 1: 4.
https://doi.org/10.3390/proceedings2019013004