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Ian Lovejoy

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Investigating a dyon within a SU(2) Yang-Mills theory coupled to a Higgs field, the following differential equations arise (Julia and Zee, Phys. Review. D, 11:2227, 1975):

[tex]r^2J'' = 2JK^2[/tex]

[tex]r^2H'' = 2HK^2 - \mu^2r^2H(1 - \frac{\lambda}{e^2\mu^2r^2}H^2)[/tex]

[tex]r^2K'' = K(K^2 - J^2 + H^2 - 1)[/tex]

The BPS limit corresponds to taking [tex]\mu=0[/tex], leading to the simpler equations:

[tex]r^2J'' = 2JK^2[/tex]

[tex]r^2H'' = 2HK^2[/tex]

[tex]r^2K'' = K(K^2 - J^2 + H^2 - 1)[/tex]

Subject to the boundary conditions that J and H approach 0 and K approaches 1 as r approaches zero, and K approaches zero and H/r approaches ev as r approaches infinity.

One solution is J=0, H = evr(coth evr) - 1, K = evr/(sinh evr). This is the magnetic monopole. A solution with [tex]J\neq 0[/tex] would correspond to a monopole with an electric charge (a dyon).

However, I am completely stumped trying to find the explicit solution, which apparently exists according to A. Zee, Quantum Field Theory, page. 288. I have tried every straightforward generalization of the monopole I can think of, without success. Actually I'm unsure how to even derive the monopole solution, although it is easy to verify that the above solution works.

Can anyone give me a clue? Is there a general procedure for solving coupled differential equations of this form? Any hints or references would be much appreciated. Especially if there is a book outlining how to solve differential equations like this, I would very much like to purchase a copy for reference.

Thanks in advance for any help.

[tex]r^2J'' = 2JK^2[/tex]

[tex]r^2H'' = 2HK^2 - \mu^2r^2H(1 - \frac{\lambda}{e^2\mu^2r^2}H^2)[/tex]

[tex]r^2K'' = K(K^2 - J^2 + H^2 - 1)[/tex]

The BPS limit corresponds to taking [tex]\mu=0[/tex], leading to the simpler equations:

[tex]r^2J'' = 2JK^2[/tex]

[tex]r^2H'' = 2HK^2[/tex]

[tex]r^2K'' = K(K^2 - J^2 + H^2 - 1)[/tex]

Subject to the boundary conditions that J and H approach 0 and K approaches 1 as r approaches zero, and K approaches zero and H/r approaches ev as r approaches infinity.

One solution is J=0, H = evr(coth evr) - 1, K = evr/(sinh evr). This is the magnetic monopole. A solution with [tex]J\neq 0[/tex] would correspond to a monopole with an electric charge (a dyon).

However, I am completely stumped trying to find the explicit solution, which apparently exists according to A. Zee, Quantum Field Theory, page. 288. I have tried every straightforward generalization of the monopole I can think of, without success. Actually I'm unsure how to even derive the monopole solution, although it is easy to verify that the above solution works.

Can anyone give me a clue? Is there a general procedure for solving coupled differential equations of this form? Any hints or references would be much appreciated. Especially if there is a book outlining how to solve differential equations like this, I would very much like to purchase a copy for reference.

Thanks in advance for any help.

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