NMR and μ-SR study on competing Heisenberg chain Cs₂Cu₂Mo₃O₁₂

S = 1/2 quasi one dimensional Heisenberg chain Cs₂Cu₂Mo₃O₁₂ has two dominant exchange interactions of the nearest-neig hbouring ferromagnetic J₁ = -93 K and the second nearest-neighbouring antiferromagnetic J₂ = +33 K. These competing interactions on the spin chain are expected to bring rich physics, such as frustration or nematic Tomonaga-Luttinger liquid (TL) phase. In order to investigate its ground state at zero and finite fields, we have performed ¹³³Cs-NMR experiments under the wide range of magnetic field up to 14 T, and μ-SR at zero field. In low-field region, the existence of long range magnetic order was demonstrated below 2 K. In the higher field region between 7 T and the saturation field H_s = 9.15 T, no evidence for magnetic order was confirmed at finite temperatures.