# Bifurcation and stability in Yang-Mills theory with sources

• R. Jackiw
Invited Papers
Part of the Lecture Notes in Physics book series (LNP, volume 129)

## Keywords

Gauge Transformation Bifurcation Point Source Strength Temporal Gauge Weak Source
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## References

1. 1.
R. Jackiw, L. Jacobs and C. Rebbi, Phys. Rev. D 20, 474 (1979).Google Scholar
2. 2.
P. Sikivie and N. Weiss, Phys. Rev. D 20, 487 (1979).Google Scholar
3. 3.
J. Mandula, Phys. Rev. D 14, 3497 (1976).Google Scholar
4. 4.
An example of this solution, with a radially symmetric source, ϱa = δ q(r), is constructed by taking $$\vec A_a = - \vec E_a t + \delta _{a1} \vec \nabla (\alpha r^2 \frac{{d\phi }}{{dr}}), \vec E_a = \frac{{ - \hat r}}{{r^2 }}\frac{1}{\alpha }\{ \delta _{a3} \sin (\alpha r^2 \frac{{d\phi }}{{dr}}) + \delta _{a2} [1 - \cos (\alpha r\frac{{d\phi }}{{dr}})]\} ,\phi = - \frac{1}{{\nabla ^2 }}q$$. This is of the form (2.15) with $$U = exp(\frac{{i\sigma ^1 }}{2}\alpha r^2 \frac{{d\phi }}{{dr}})$$. The energy is $$\varepsilon = 8\pi \int\limits_0^\infty {\frac{{dr}}{{\alpha ^2 r^2 }}\sin ^2 } (\frac{\alpha }{2}r^2 \frac{{d\phi }}{{dr}})$$. Here a is an arbitrary parameter, which when set to zero gives the Abelian Coulomb solution. This configuration is essentially the “total screening solution” found by P. Sikivie and N. Weiss, Phys. Rev. Lett. 40, 1411 (1978), and Phys. Rev. D 18, 3809 (1978); except that they use a descrete parameter rather than our continuously varying α. The feasibility of generalizing the total screening solution was pointed out by P. Pirilä and P. Presnajder, Nucl. Phys. B 142, 229 (1978). The present formulation is given by Jackiw and Rossi, Ref. 5.Google Scholar
5. 5.
R. Jackiw and P. Rossi MIT preprint (to be published).Google Scholar
6. 6.
S. Coleman, in New Phenomena in Sub-Nuclear Physics, edited by A. Zichichi (Plenum, New York, 1977); S. Deser, Phys. Lett, 64B, 463 (1976).Google Scholar
7. 7.
The truth of this statement is manifest from (2.22c) for sufficiently small Q, so that the O(Q4)terms are negligible; and for charge densities q(r) which never change sign, so that q(r)q(r′)ř·ř′<q(r)q(r′). However, in Ref. 1 it is shown that even for charge densities with varying signs the inequality in (2.22c) is valid, and that numerical computation at large Q confirms the bound; see Fig. 1.Google Scholar
8. 8.
9. 9.
Another example of a solution which exists when the source is of sufficient magnitude is Sikivie and Weiss' “magnetic dipole solution”; see Ref. 4. The source which supports this solution is studied by Y. Leroyer and A. Raychaudhuri, Phys. Rev. D (in press).Google Scholar
10. 10.
For an introduction to current, mainly mathematical research on stability of motion see C. Siegel and J. Moser, Lectures on Celestial Mechanics, (Springer Verlag, Berlin, 1971). A simple discussion, referring to older physics research, is found in H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics, (Cambridge University Press, Cambridge, 1972).Google Scholar
11. 11.
This form for the fluctuation equations in a Coulomb background field was also given by M. Magg, Physics Letters 74B, 246 (1978).Google Scholar
12. 12.
J. Mandula, Physics Letters 67B, 175 (1978); M. Magg, Physics Letters 74B, 246 (1978).Google Scholar
13. 13.
L. Schiff, H. Snyder and J. Weinberg, Phys. Rev. 57, 315 (1940); K. Johnson, Harvard Ph.D theisis (1954) (unpublished); A. Migdal, Zh. Eksp, Teor, Fiz. 61, 2209 (1972) [English translation: Soviet Physics JETP 34, 1184 (1972)]; A. Klein and J. Rafelski, Phys. Rev. D 11, 300 (1975). For the delta-shell source (2.34) the instability sets in at Q=1.5. This value is not related in any transparent way to the magnitude of Q at the bifurcation. It does agree with the strength of a point source at the onset of instability, as determined in Ref. 12. However this coincidence is a consequence of the scaling properties of the delta-shell source, and is not expected for arbitrary extended sources.Google Scholar
14. 14.
The truth of this statement is manifest for charge densities which do not change sign, so that $$q(\vec r)q(\vec r') > 0$$. However, for spherically symmetric charge densities the proviso can be removed, see Ref. 15, below.Google Scholar
15. 15.
For spherically symmetric charge densities, we may take the solution described in Ref. 4. For small α it becomes an infinitesimal deformation of the Abelian Coulomb with energy $$\varepsilon = 2\pi \int\limits_0^\infty {drr^2 (\phi ')^2 - \frac{{\alpha ^2 \pi }}{6}} \int\limits_0^\infty {drr^6 (\phi ')^4 }$$, which is always less than the Coulomb energy, regardless of the sign of the source.Google Scholar
16. 16.
That the physics of the top is encountered in the Yang-Mills theory was previously remarked by J. Goldstone and R. Jackiw, Phys. Lett. 74B, 81 (1978). Indeed it was in the context of the formalism developed in this paper that some of the results sumarized here were first encountered.Google Scholar