Solution

The hierarchy problem is visible in the one-loop top-quark contribution to the Higgs boson mass parameter,

δ m_{H}^{2} \sim - \frac{3∣ y _{t} ∣ ^{2}}{8 π ^{2}} Λ^{2} .

(1)

In a supersymmetric theory the corresponding stop loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughly

∣ δ m_{H}^{2} ∣ \sim \frac{3∣ y _{t} ∣ ^{2}}{8 π ^{2}} m_{t}^{2} lo g \frac{Λ}{m _{t}} .

(2)

The two relevant one-loop Feynman diagrams are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the naturalness expectation

m_{t}

, and broadly other sparticle masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.

R-parity is

R_{p} = (- 1)^{3 (B - L) + 2 s},

(3)

where

B

is baryon number,

L

is lepton number and

s

is spin angular momentum. Standard Model particles have

R_{p} = + 1

, while their superpartners have

R_{p} = - 1

. The renormalizable R-parity-violating superpotential of the Minimal supersymmetric Standard Model is

W_{RPV} = \frac{1}{2} λ_{ijk} L_{i} L_{j} E_{k}^{c} + λ_{ijk}^{'} L_{i} Q_{j} D_{k}^{c} + \frac{1}{2} λ_{ijk}^{''} U_{i}^{c} D_{j}^{c} D_{k}^{c} + κ_{i} L_{i} H_{u},

(4)

with

λ_{ijk} = - λ_{jik}

and

λ_{ijk}^{''} = - λ_{ikj}^{''}

Combining

λ_{i 1 k}^{'} L_{i} Q_{1} D_{k}^{c}

with

λ_{11 k}^{''} U_{1}^{c} D_{1}^{c} D_{k}^{c}

produces proton decay through exchange of a virtual right-handed down-type squark. The tree diagram has

u

and

d

entering the baryon-number-violating vertex, an internal

d_{R k}

line, and the lepton-number-violating vertex emitting

\overset{ν}{ˉ}_{i}

and a light quark. Hadronization gives

p ⟶ π^{+} + \overset{ν}{ˉ}_{i} .

(5)

The invisible particle is an antineutrino of any family

i = 1, 2, 3

, and electric-charge conservation requires a positively charged pion. The antisymmetry of

λ_{11 k}^{''}

requires

k = 2

3

Integrating out a squark of mass

m_{q}

gives a dimension-six interaction with coefficient

C \sim λ^{'} λ^{''} / m_{q}^{2}

. Ignoring order-one hadronic and phase-space factors,

Γ_{p} \sim \frac{∣ λ ^{'} λ ^{''} ∣ ^{2}}{8 π} \frac{m _{p}^{5}}{m _{q}^{4}} .

(6)

For

m_{p} \sim 1 GeV

and

m_{q} \sim 1 0^{4} GeV

, the factor before the couplings is about

1 0^{- 18}

1 0^{- 17} GeV

. A lifetime above

1 0^{34}

years is about

1 0^{65} GeV^{- 1}

, so

Γ_{p} ≲ 1 0^{- 65} GeV

and

∣ λ^{'} λ^{''} ∣ ≲ 1 0^{- 24}

(7)

at order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.

The matter-parity factor