CERN-TH-2017-154 Proton decay testing low energy SUSY Stefan Pokorski, 1, 2 Krzysztof Rolbiecki, 1 and Kazuki Sakurai 1 1 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093 Warsaw, Poland 2 Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland We show that gauge coupling unification in SUSY models can make a non-trivial interconnection between collider and proton decay experiments. Under the assumption of precise gauge coupling unification in the MSSM, the low energy SUSY spectrum and the unification scale are intertwined, and the lower bound on the proton lifetime can be translated into upper bounds on SUSY masses. We found that the current limit on τ (p → π 0 e + ) already excludes gluinos and winos heavier than ∼ 120 and 40 TeV, respectively, if their mass ratio is M3/M2 ∼ 3. Next generation nucleon decay experiments are expected to bring these upper bounds down to ∼ 10 and 3 TeV. Proton decay would be the key evidence for grand uni- fied theories (GUTs) [1]. Among possible decay channels, a special role is played by the p → π 0 e + mode for which the dominant contribution may come from the D = 6 op- erators depending almost exclusively on the X, Y boson mass and the unified gauge coupling. This is in contrast to the other channels induced by D = 5 operators, which depend on many more parameters, though the rate is typ- ically larger than the p → π 0 e + mode. The main point we want to emphasise and make very explicit in this Letter is that τ p→π 0 e + carries an important information about the low scale supersymmetric (SUSY) spectrum. To this end we assume here that the unifica- tion of the gauge couplings is precise (or exact) within the minimal SUSY Standard Model (MSSM) without thresh- old corrections of GUT scale particles [2]. In fact, there exists a class of models where these corrections are absent or highly suppressed (see e.g. [3]). On the other hand, GUT threshold corrections in conventional models are of- ten too large compared to the typical mismatch of gauge couplings at a high scale in the MSSM (see [4] for a recent discussion). This means that the well-known “success of gauge coupling unification in the MSSM”, if not a mere accident, may favour the aforementioned class of models as the correct theory of grand unification. Under the assumption of precise gauge coupling unifi- cation (GCU) in the MSSM, we show that the low energy SUSY spectrum and the unification scale are intertwined, and the lower bound on the proton lifetime τ p→π 0 e + can be translated into upper bounds on SUSY masses. * This leads to an interesting interconnection between the proton decay experiments and the collider searches, particularly in view of the future progress on both fronts, in cornering supersymmetric spectrum from above and from below. At the one-loop the gauge couplings at scale ˜ μ in the * Unlike other upper bounds on SUSY masses based on the argu- ments of the Higgs boson mass [5] or the neutralino relic abun- dance [6], these bounds dependent neither on the ratio of the Higgs vacuum expectation values, tan β ≡ vu/v d , nor the assumption of R-parity conservation and the thermal history of the universe. MSSM is given by 2π α i (˜ μ) = 2π α i (m Z ) - b i ln  ˜ μ m Z  + s i . (1) where α 1 ≡ 3 5 α Y , i =1, 2, 3 represents the gauge group, b i =( 33 5 , 1, -3) are the one-loop β-function coefficients for the MSSM and s i ≡ X η b η i ln  m η m Z  (2) are the threshold corrections of SUSY particles. For SUSY particle η, the mass and its contribution to b i are given by m η and b η i , respectively. In the special case where all SUSY particles are mass degenerate at M s , the threshold correction can be writ- ten as s i = δ i ln(M s /m Z ) with δ i ≡ (b i - b SM i ), where b SM i =( 41 10 , - 19 6 , -7) are the one-loop β-function coeffi- cients for the Standard Model (SM). In this case, exact gauge unification α 1 (˜ μ)= α 2 (˜ μ)= α 3 (˜ μ) ≡ α * G is achieved by the particular values of M s and ˜ μ: M * s , M deg* G , satis- fying 2π α * G - 2π α i (m Z ) + b i ln  M deg* G m Z  = δ i ln  M * s m Z  (3) for all i. It should be kept in mind that the quantities M * s , M deg* G and α * G are not variables but constants defined as the solution to the above three simultaneous equations. Coming back to the general case, let us decompose the vector s i into three independent vectors as [7] s i = δ i ln  T m Z  + b i ln Ω + C. (4) The solution to this set of equations is given by ln  T m Z  = v i s i /D (5) lnΩ = u i s i /D (6) C =  ijk δ j b i s k /D (7) arXiv:1707.06720v1 [hep-ph] 20 Jul 2017