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Effective Field Theory Treatment of Monopole Production by Drell–Yan and Photon Fusion for Various Spins^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Calculations for Monopole Production Processes

#### 2.1. The Spin 0 Monopole

#### Pair Production by Photon Fusion

#### Pair Production by Drell–Yan

#### 2.2. The Spin $\frac{1}{2}$ Monopole

#### Pair Production by Photon Fusion

#### Pair Production by Drell–Yan

#### 2.3. The Spin 1 Monopole

#### Pair Production by Photon Fusion

#### Pair Production by Drell–Yan

## 3. A Comparison of the Total Cross Sections and Small Coupling Limits

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PF | Photon fusion |

DY | Drell–Yan |

SM | Standard Model |

## References

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**Figure 1.**Feynman-like tree-level graphs of (

**a**): a Standard Model Drell–Yan (DY) process for lepton production from quark annihilation, with appropriate electric charges ${q}_{\mathrm{e}}$; (

**b**) DY monopole–anti-monopole pair production from quark annihilation where g is the monopole’s magnetic charge; (

**c**) monopole–anti-monopole pair production via photon-fusion (PF) (for monopole spins 0,$\frac{1}{2}$, and 1); (

**d**) additional (contact) diagrams for monopole–anti-monopole pair production via PF (for monopole spins 0 and 1). The blob denotes the effective coupling. Wavy lines denote photons ($\gamma $), while continuous lines denote either fermions (quarks (q), antiquarks ($\overline{\mathrm{q}}$), and charged leptons (${l}^{\pm})$) or monopole (anti-monopole) fields M ($\overline{\mathrm{M}}$) [18].

**Figure 2.**Spin 0 monopole production by PF: (

**Left**) These plots show distributions for pair production in the centre of mass frame as functions of scattering angle $\theta $ and pseudo-rapidity $\eta $, which are focused in the central region. The monopoles have mass $M=1.5$ TeV and $\sqrt{{s}_{\gamma \gamma}}=2{E}_{\gamma}$, where ${E}_{\gamma}=6M$. (

**Right**) The total cross section varies slowly with monopole mass M at $\sqrt{{s}_{\gamma \gamma}}=4$ TeV until it drops off sharply in the kinematically forbidden region $M>\sqrt{{s}_{\gamma \gamma}}/2$.

**Figure 3.**Spin 0 monopole production by Drell–Yan (DY): (

**Left**) The figure shows that the production from massless quarks, with $M=1.5$ TeV and $\sqrt{{s}_{q\overline{q}}}=2{E}_{q}$ for ${E}_{q}=6M$, is predominantly concentrated in the central region. (

**Right**) The total cross section for pair production in dualised SQED is finite, as shown for $\sqrt{{s}_{q\overline{q}}}=4$ TeV, in the same way as FP production was.

**Figure 4.**Spin $\frac{1}{2}$ monopole production by PF: (Left) For $M=1.5$ TeV and ${E}_{q}=6M$, as $\tilde{\kappa}$ changes, the distributions change only by a scaling factor, and production is concentrated away from the central axis. (This contrast with the s=0 case is expected.) The $\tilde{\kappa}=0$, representing dualised QED, is unique as the only renormalisable, unitary case. (Right) For various values of $\tilde{\kappa}\ne 0$, the total cross section at $\sqrt{{s}_{q\overline{q}}}=4$ TeV diverges as $M\to 0$, where the monopole becomes non-relativistic.

**Figure 5.**Spin $\frac{1}{2}$ monopole production by DY: (Left) The angular and rapidity distributions for various values of the parameter $\tilde{\kappa}$ demonstrate rather more contrasting behaviours between the $\tilde{\kappa}$ cases, unlike the PF distributions, and also show a much more central production. Here, the monopole mass is $M=1.5$ TeV, and the quark energy is ${E}_{q}=6M$. (Right) For various values of $\tilde{\kappa}\ne 0$, the total cross section at $\sqrt{{s}_{q\overline{q}}}=4$ TeV diverges as $M\to 0$, where the monopole enters a non-relativistic regime.

**Figure 6.**Spin 1 monopole production by PF: (

**Left**) For different values of $\kappa $, at $M=1.5$ TeV, $\sqrt{{s}_{\gamma \gamma}}=2{E}_{\gamma}$, and ${E}_{\gamma}=6M$, the $\kappa =1$ distributions are uniquely unitary, showing a depression of the cross section in the central region. $\theta $ is the scattering angle, and $\eta $ is pseudo-rapidity. (

**Right**) The cross section for all $\kappa $, at $\sqrt{{s}_{\gamma \gamma}}=4$ TeV, diverges as $M\to 0$, where the monopole naturally becomes non-relativistic.

**Figure 7.**Spin 1 monopole production by DY: (

**Left**) In the massless quarks limit, with $M=1.5$ TeV and $\sqrt{{s}_{q\overline{q}}}=2{E}_{q}$ at ${E}_{q}=6M$, the value of $\kappa $ influences the behaviour of the distribution but does not on its own reflect the unitarity of the model in the $\kappa =1$ case. (

**Right**) The cross section for all $\kappa $, at $\sqrt{{s}_{q\overline{q}}}=4$ TeV, diverges as $M\to 0$, where the monopole naturally becomes non-relativistic.

**Figure 8.**The production cross sections for PF decidedly dwarf those for DY at $\sqrt{{s}_{qq/\gamma \gamma}}=4$ TeV. Shown here are (

**a**) $s=1$ monopole production in the $\kappa =1$ (SM-like) case; (

**b**) the $s=\frac{1}{2}$ case for $\tilde{\kappa}=0$; (

**c**) the only $s=0$ case, which has no magnetic moment; (

**d**) the $s=1$ monopole cross section with $\kappa =2$; and (

**e**) the $s=\frac{1}{2}$ monopole cross section with $\tilde{\kappa}=2$.

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**MDPI and ACS Style**

Baines, S.
Effective Field Theory Treatment of Monopole Production by Drell–Yan and Photon Fusion for Various Spins. *Proceedings* **2019**, *13*, 1.
https://doi.org/10.3390/proceedings2019013001

**AMA Style**

Baines S.
Effective Field Theory Treatment of Monopole Production by Drell–Yan and Photon Fusion for Various Spins. *Proceedings*. 2019; 13(1):1.
https://doi.org/10.3390/proceedings2019013001

**Chicago/Turabian Style**

Baines, Stephanie.
2019. "Effective Field Theory Treatment of Monopole Production by Drell–Yan and Photon Fusion for Various Spins" *Proceedings* 13, no. 1: 1.
https://doi.org/10.3390/proceedings2019013001