The spin-lattice relaxation rate of nuclear magnetic resonance in a clean superconductor without inversion center is calculated for arbitrary pairing symmetry and band structure, in the presence of strong spin-orbit coupling.PACS numbers: 74.25. Nf, 74.20.Rp, 74.70.Tx Most superconducting materials have an inversion center in their crystal lattices. Among a few exceptions are the heavy-fermion compounds CePt 3 Si [1] and UIr [2]. The peculiar feature of superconductivity without inversion center is that there are fewer pairing channels available than in a centrosymmetric crystal, because the spin degeneracy of the bands is lifted in the presence of strong spin-orbit coupling [3,4]. Other properties of non-centrosymmetric superconductors studied recently include paramagnetic limit and Knight shift [5,6,7,8,9,10,11], helical superconducting phases [12,13,14], and magnetoelectric phenomena [15,16,17]. The determination of the pairing symmetry however presents a considerable challenge. Although there are strong indications that the gap in CePt 3 Si has lines of nodes, the decisive proof, including the determination of the nodal locations, is still lacking.A powerful probe of the quasiparticle properties in superconductors is the spin-lattice relaxation rate T −1 1 of nuclear magnetic resonance (NMR). The presence of the coherence, or Hebel-Slichter, peak in T −1 1 below the critical temperature [18] provided an early strong support for the Bardeen-Cooper-Schrieffer theory of superconductivity [19]. More recently, the power-law behavior of the relaxation rate at low temperatures commonly observed in heavy-fermion compounds, see e.g. Ref.[20], has been used as an argument in favor of the existence of gapless excitations in those materials. Indeed, if there are line (point) nodes in the gap, thenThe measurements of the NMR relaxation rate in CePt 3 Si [22] indicate the likely presence of line nodes. On the other hand, a small Hebel-Slichter peak just below T c observed in Ref. [22] was not found in other experiments [23]. Theoretically, the standard analysis of the NMR relaxation rate in superconductors, see e.g. Ref.[19], is not directly applicable in the noncentrosymmetric case because of a complicated spin structure of the non-degenerate bands. The purpose of the present Rapid Communication is to calculate T −1 1 in a non-centrosymmetric superconductor with arbitrary pairing symmetry. Our analysis does not rely on any specific model for the band structure, the only assumption being that the bands are well split due to strong spinorbit coupling, which is the case for CePt 3 Si [3]. In these circumstances, it is convenient to use the exact band representation of the order parameter introduced in Ref.[3], see also Ref. [4], in which the possibility of superconducting states with lines of nodes at high-symmetry locations appears naturally. In the alternative approach developed in Ref.[8], the spin-orbit coupling is introduced using the Rashba model, and the order parameter becomes a mixture of spin-singlet and spin-t...