The Lorentz-invariant boundary action of the confining string and its universal contribution to the inter-quark potential

We study the boundary contribution to the low energy effective action of the open string describing the confining flux tube in gauge theories. The form of the boundary terms is strongly constrained by the requirement of Lorentz symmetry, which is spontaneously broken by the formation of a long confining flux tube in the vacuum. Writing the boundary action as an expansion in the derivatives of the Nambu-Goldstone modes describing the transverse fluctuations of the string, we single out and put in a closed form the first few Lorentz invariant boundary terms. We also evaluate the leading deviation from the Nambu-Goto string produced by the boundary action on the vacuum expectation value of the Wilson loop and we test this prediction in the 3d Ising gauge model. Our simulation attains a level of precision which is sufficient to test the contribution of this term.