Nonrenormalization of the superstring tension

Atish Dabholkar, Jeffrey A. Harvey
1989 Physical Review Letters  
It is argued that the superstring tension is not renormalized in perturbation theory for vacua which preserve N= 1 spacetime supersymmetry. Some implications of this result for macroscopic superstrings are discussed, as well as some analogies between macroscopic superstrings and solitons in supersymmetric theories. PACS numbers: 11.17.+y, 11.30.Pb In superstring theory, the only really fundamental constant is the string tension p. All other constants of nature are, at least in principle,
more » ... to the string tension for a given vacuum configuration. It is of interest, therefore, to know how quantum corrections alter the classical value of the string tension because it is the renormalized string tension that would be measured at low energies. For example, if cosmic superstrings were to exist, the deficit angle measured from double images of quasars ould directly determine the renormalized superstring tension. Many authors have addressed the questions of the renormalization of low-energy couplings and mass renormalization for massive string states, but to our knowledge the renormalization of the string tension itself has not been addressed. As we will see, for spacetime supersymmetric vacua, the superstring tension receives no renormalization in perturbation theory. If the scale of supersymmetry breaking is much below the Planck scale, then we would expect the renormalized string tension to be very close to its classical value. To evaluate this renormalization we first of all have to define what we mean by the superstring tension. Consider a vacuum configuration of the form M x 5 ' & K. Here K can be any internal N=2 superconformal theory that results in N=1 spacetime symmetry. M is threedimensional Minkowski space and S' is a large circle of radius R, say in the z direction. A closed string state that wraps around this circle will look like a cosmic superstring in four dimensions. We can dePne the superstring tension as the energy per unit length of such a state with winding number one in the limit that R goes to infinity. To calculate the energy of this winding state we evaluate the mass shift in perturbation theory by evaluating the two-point function of the corresponding vertex operator on genus-g Riemann surfaces. Before proceeding with the calculation, let us discuss what we expect to find. At large distances a macroscopic superstring should look much like a cosmic axion string. ' Classically, an axion string receives an ultravioletdivergent contribution to its self-energy as a result of the surrounding static axion field. This is quite analogous to the quadratically divergent renormalization of the electron mass due to the static Coulomb field in the Dirac-Lorentz theory of classical electrons. Since there is an axion in the massless spectrum of the superstring, with a coupling to a macroscopic superstring which is like that of an axion string, we expect classically that this coupling should renormalize the string tension. Quantum corrections typically soften the classical ultraviolet divergence and superstring theory is in fact completely finite in the ultraviolet. We therefore expect an ultraviolet finite-but-nonzero contribution to the string tension from the interaction of the string with its various modes. Finiteness of string theory, of course, does not preclude infrared divergences of the amplitudes and such divergences, if present, signify important physics. In our case, there is a very good physical reason to expect the two-point function of a cosmic winding state to diverge in the infrared. A classical string coupled to an axion is much like a vortex line of a spontaneously broken global symmetry. The energy contained in the surrounding static axionic field of such a global string is we11 known to be infrared divergent and goes as pin(pR), where R is the infrared cutoff, the radius of the string in our case. This classical infrared divergence should show up in the two-point amplitude at the quantum level. Thus, if we take the radius R of the winding state to be sufficiently large, we should be able to extract the string-tension renormalization by looking at the coefficient in front of the leading-log divergence and comparing it with the classical result. We will show explicitly that the two-point function vanishes on the torus, and then argue that it vanishes to all orders in perturbation theory. We will then discuss how the apparent contradiction between this result and the above discussion can be resolved. Some connections between macroscopic superstrings and solitons in supersymmetric theories will also be presented. The mass shifts and decay rates of massive states are related to the real and the imaginary parts of the twopoint amplitude. For a stable winding state with vertex operator V the mass shift is directly given by the twopoint function. The two-point function 2 at order g is given by integrating the correlation function of vertex operators (VV) over the genus-g moduli space with the 478
doi:10.1103/physrevlett.63.478 pmid:10041085 fatcat:p2qombujt5exvbm6cpoccj6rcy