Proton decay testing low energy SUSY

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 $\tau(p \to \pi^0 e^+)$ already excludes gluinos and winos heavier than $\sim 120$ and 40 TeV, respectively, if their mass ratio is $M_3/M_2 \sim 3$. Next generation nucleon decay experiments are expected to bring these upper bounds down to $\sim 10$ and 3 TeV.

Proton decay would be the key evidence for grand unified 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 operators 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 typically 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 unification of the gauge couplings is precise (or exact) within the minimal SUSY Standard Model (MSSM) without threshold 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 often 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 unification (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 scaleμ in the * Unlike other upper bounds on SUSY masses based on the arguments of the Higgs boson mass [5] or the neutralino relic abundance [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
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 where summation is understood for the repeated indices and ijk is the antisymmetric tensor and Plugging the concrete values of b i , δ i and b η i into these expressions, one gets with The SUSY mass parameters appearing in this Letter should be understood as the magnitude of the corresponding parameters because phase factors do not affect RG running. In most models, the sfermion contributions to T and Ω are negligible (i.e. X T ∼ X Ω ∼ 1). In particular, these contributions vanish if the masses are degenerate within the SU(5) multiplets,5 i = (d c R ,l) i , 10 i = (q,ũ c R ,ẽ c R ) i . One can explicitly check that for a degenerate spectrum, ln Ω = C = 0.
To see roles of T , Ω and C in gauge unification, we substitute Eq. (4) into Eq. (1) and obtain where Eq. (3) has also been used. It is clear that the exact unification for the general case is obtained when the righthand-side (RHS) becomes i-independent, that is at T = M * s [8] and the exact unification scale is given by The unified gauge coupling is related to that of the degenerate case as Away from the exact unification, we define a candidate unification scale M U and a semi-unified coupling This scale can be computed from a low energy spectrum as M U = , and at this scale the gauge couplings are given by Using this formula, a measure of gauge coupling unification, which we define where the dots represent higher order terms of It is interesting that 3 depends only on T at the leading order [8].
Our argument so far is based on the one-loop renormalization group equations (RGEs). It turns out that the relations Eqs. (15), (16) and (18) Fig. 1 the result of our numerical scan. All numerical scans presented in this Letter use a two-loop RGE code including the effect of the top Yukawa coupling, following [9]. We use tan β = 10 but a variation of tan β results in negligible effects. The SUSY breaking parameters are uniformly scanned in the logarithmic scale within [m min , 10 3 TeV]. We take m min = 1.5 TeV for M 3 and 200 GeV for M 2 , µ and m A . The sfermion masses are assumed to be universal (≡ mf ) for simplicity and m min = 1 TeV is used. We also vary α 3 (m Z ) = 0.1184 (7), according to the 1-σ uncertainty.
The top plot in Fig. 1 tests the predicted relation Eq. (18) (dashed line). We see that the exact unification occurs only when the SUSY masses are arranged such that T computed by Eq. (9) is within a certain range [1,4] TeV centred around ∼ 2 TeV. The width of T for exact unification comes mainly from the uncertainty on M * s due to the variation of α 3 (m Z ). † The middle plot shows the correlation between Ω and the exact unification scale, M G . Hereafter, we require a precise gauge unification, | 3 | < 0.1%.     . We see that high scale SUSY tends to predict a smaller unified coupling, α G , but the variation is small and only up to ∼ 10% between the TeV and PeV scale SUSY mass points.
An interesting observation follows from the last two plots of Fig. 1. High scale SUSY, where the unification scale is lower, in general leads to a rapid proton decay, p → e + π 0 . This is because the rate Γ(p → e + π 0 ) scales as α G /(M G ) 4 , where the X, Y boson mass is identified as where This implies that the smallest mass in the LHS is bounded from above by the RHS of Eq. (20). When this bound is saturated, M 2 = µ = m A . The upper limit on the individual parameters are obtained, for example, as τ p / y e a r s < 1 · 1 0 3 5 Precise Gauge Unification the lightest among the three, respectively. A tendency is observed that M 2 is close to the upper limit if M 2 is the lightest. This is due to the higher power for M 2 in Eq. (20) than for µ and m A . At each point we calculate τ p→π 0 e + based on [10,11] ‡ using α G and M G obtained by the twoloop RGE code. The horizontal black-dashed and redsolid lines represent the boundaries where all points below them have the lifetime shorter than the quoted values. In particular, the region below the red line is excluded by the current limit: τ p→π 0 e + > 1.7 · 10 34 years [13].
The upper bound on the gluino mass can be found by eliminating µ 4 m A in Eq. (10) by using Eq. (9) as with As previously, the RHS of Eq. (23) is bounded from above by the experimental lower limit on the wino mass. If the SUSY breaking mechanism is specified, the ratio of gluino and wino masses is usually predicted. Assuming the value of R ≡ M 3 /M 2 , the following upper bounds can be derived: where We show in Fig. 3 our scan in the (M 3 , M 2 ) plane with the colour-code indicating (µ 4 m A ) 1 5 . As previously, the black-dashed and red-solid lines represent the future and current bounds on τ p→π 0 e + . It is evident that M 3 and M 2 are highly sensitive to the proton lifetime and constrained by it from above. This is in direct contrast to collider searches, constraining these parameters from below. Unlike M 3 and M 2 , µ and m A are almost insensitive to the proton lifetime, which follows from the lower power of M P D in Eq. (26). On the other hand, they are highly sensitive to R. In particular, µ is typically a TeV for R = 1 whereas it is O(100) TeV for R = 7. The implication of this to naturalness and phenomenology are studied in detail in [2,7,14].
It is remarkable that the current proton lifetime limit already excludes the gluino and wino masses larger than 200 and 30 TeV for R ∼ 7 (e.g. AMSB) and 120 and 40 TeV for R ∼ 3 (e.g. CMSSM, GMSB), respectively. Next generation nucleon decay experiments are expected to improve the current τ p→π 0 e + limit by a factor of ten [10], which will result in tightening the upper bounds on gluino and wino masses further down to (M 3 , M 2 ) (10, 3) TeV for R ∼ 3 and (M 3 , M 2 ) (15, 2) TeV for R ∼ 7. These bounds are close to the lower mass limits (M 3 , M 2 ) (10, 2.7) TeV [15,16], which are expected to be obtained at future 100 TeV hadron-hadron colliders.
We have investigated the link between the proton lifetime τ p→π 0 e + and the supersymmetric spectrum under the assumption of vanishing GUT thresholds. It has been shown that most of the allowed mass range of gluinos and winos will be probed by future collider and proton lifetime experiments. It will also be interesting to extend this study to models with non-vanishing GUT threshold corrections (see e.g. [17]).