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We consider a Yang-Mills-Higgs theory with spontaneous symmetry breaking ofthe gauge group G -> U(1)^r -> C(G), with C(G) being the center of G. We studytwo vacua solutions of the theory which produce this symmetry breaking. We showthat for one of these vacua, the theory in the Coulomb phase has the massspectrum of particles and monopoles which is exactly the same as the massspectrum of particles and solitons of two dimensional affine Toda field theory.That result holds also for N=4 Super Yang-Mills theories. On the other hand, inthe Higgs phase, we show that for each of the two vacua the ratio of thetensions of the BPS Z(N) strings satisfy either the Casimir scaling or the sinelaw scaling for G=SU(N). These results are extended to other gauge groups: forthe Casimir scaling, the ratios of the tensions are equal to the ratios of thequadratic Casimir constant of specific representations; for the sine lawscaling, the tensions are proportional to the components of the leftPerron-Frobenius eigenvector of Cartan matrix and the ratios of tensions areequal to the ratios of the soliton masses of affine Toda field theories.

BPS Z(N) String Tensions, Sine Law and Casimir Scaling and Integrable Field Theories