Neutrinoless Double Beta Decay Half-Life Calculator

Introduction to 0νββ Half-Life Estimates

This 0νββ half-life calculator turns the standard light-Majorana-neutrino expression into a quick estimate you can explore with three inputs: the effective Majorana mass, the isotope-dependent phase-space factor, and the nuclear matrix element. It is meant to make the scale of neutrinoless double beta decay easier to inspect without digging through a long derivation every time you want to compare assumptions.

In neutrinoless double beta decay, a nucleus would emit two electrons without the neutrinos that accompany ordinary double beta decay. That difference is what makes the process so important: if the decay exists, it would imply lepton-number violation and strongly support the idea that neutrinos are Majorana particles rather than distinct from their antiparticles.

The reason people keep returning to 0νββ is that it sits at the intersection of several big questions. A confirmed signal would tie together neutrino mass generation, nuclear-structure calculations, and the broader puzzle of why matter dominates over antimatter in the observable universe. Since the expected half-lives are enormous, a calculator that shows how the estimate reacts to the inputs is useful even when it is only being used for a rough comparison.

Use this page as a compact consistency check, not as a substitute for a full theory paper or an experimental sensitivity study. The main value of a 0νββ half-life calculator is that it shows which assumptions move the result the most, and how quickly the estimate changes when you adjust the neutrino-mass scale or the nuclear inputs.

0νββ Background and Parameter Meaning

The physics behind neutrinoless double beta decay is usually organized into particle-physics, nuclear-structure, and phase-space pieces, and that same separation is what the calculator follows. The phase-space factor, usually written as $G_{0\nu }$, depends on the isotope and the available decay energy, while the nuclear matrix element $\left|M_{0\nu }\right|$ condenses the nuclear transition strength into a single number.

The particle-physics side is carried by the effective Majorana mass $m_{\beta \beta }$, which combines the neutrino masses, mixing angles, and Majorana phases into one parameter. Different authors may present those ingredients in slightly different conventions, so the same 0νββ half-life estimate can look a little different from paper to paper even when the underlying physics is the same.

This calculator follows the common light-neutrino exchange form and assumes that the numbers you enter already use compatible units: eV for $m_{\beta \beta }$, yr⁻¹ for $G_{0\nu }$, and a dimensionless magnitude for $\left|M_{0\nu }\right|$. That unit discipline matters because the page is evaluating a specific published-style expression rather than trying to infer or convert between conventions on the fly.

The scaling is the part most people want to understand first. Smaller $m_{\beta \beta }$ means a much longer half-life, a larger matrix element shortens the lifetime, and a larger phase-space factor also makes decay faster. In practice, that means isotope choice, nuclear-model choice, and assumptions about neutrino masses can all shift the predicted reach by a large amount.

How to Use This 0νββ Calculator

To use this 0νββ calculator, enter a value in each field and press Compute Half-Life. The first field is the effective Majorana mass $m_{\beta \beta }$ in eV, which is the most direct neutrino-physics input in the estimate.

The second field is the phase-space factor $G_{0\nu }$ in yr⁻¹. That value is usually taken from tabulated results for the isotope you care about. The third field is the magnitude of the nuclear matrix element $\left|M_{0\nu }\right|$, which comes from a nuclear-structure calculation or a paper summarizing one.

After submission, the page returns a predicted half-life $T_{1/2}$ in years and assigns one of its broad labels. Those labels are only a quick guide for comparing the estimate with the ranges built into the page; they are not tied to a specific detector, isotope, exposure, or background model.

Keep the units consistent. If you are copying parameters from a paper, check whether the authors use a different axial-coupling convention or a different normalization for the matrix element. The calculator does not try to convert between conventions; it simply evaluates the formula using the values you supply.

For intuition, hold $G_{0\nu }$ and $\left|M_{0\nu }\right|$ fixed and vary $m_{\beta \beta }$. That shows the inverse-square dependence on the neutrino-mass scale very clearly. You can also do the reverse and keep $m_{\beta \beta }$ fixed while changing the nuclear matrix element, which is a useful way to see how much nuclear-theory uncertainty matters at the same mass scale.

Formula for 0νββ Half-Life

For this neutrinoless double beta decay calculator, the half-life comes from the standard light-Majorana exchange relation:

Formula: T1_/2^− = G 0_ν | M 0_ν | 2(m_ββ / m_e) 2

${T{1}_{/2}}^{-}=G{0}_{\nu }\left|M{0}_{\nu }\right|{}^2\left(\frac{m_{\mathrm{\beta \beta }}}{m_e}\right){}^2$

Here, $G_{0\nu }$ is the phase-space factor with units of inverse time, computed from kinematics and Coulomb effects; $\left|M_{0\nu }\right|$ is the nuclear matrix element; $m_{\beta \beta }$ is the effective Majorana mass; and $m_e$ is the electron mass. The displayed form makes the quadratic dependence on both the matrix element and the effective mass ratio explicit.

The calculator evaluates the rearranged form for the half-life itself:

Formula: T 1_/2 = 1 / (G 0_ν |^M (m_ββ/m_e)^2)

$T{1}_{/2}=\frac{1}{G{0}_{\nu }{\left|}^M{\left(\frac{m_{\mathrm{\beta \beta }}}{m_e}\right)}^2}$

Using $m\_e=0.511\text{MeV}=5.11\times {10}^5\text{eV}$ keeps the units consistent with $m_{\beta \beta }$ in electronvolts. In the page script, that conversion is stored as 5.11e5 eV, and the result is computed as me² divided by G × M² × mββ².

The scaling is worth keeping in mind when you interpret the output. Doubling $m_{\beta \beta }$ makes the half-life four times shorter, while halving $\left|M_{0\nu }\right|$ makes it four times longer. That quadratic behavior is exactly why the neutrino-mass scale and the nuclear model both matter so much in 0νββ studies.

Worked 0νββ Example

Here is a worked 0νββ example using the default values loaded in the form: effective Majorana mass 0.05 eV, phase-space factor 1×10⁻¹⁴ yr⁻¹, and nuclear matrix element magnitude 5. With those inputs, the calculator returns a predicted half-life of 4.178×10²⁶ years, which the page classifies as Within next-generation sensitivity.

If you hold the isotope-dependent parameters fixed and lower the effective mass to 0.02 eV, the estimate grows to 2.611×10²⁷ years. Raising the effective mass to 0.15 eV brings the half-life down to 4.642×10²⁵ years, which the page tags as Currently testable. Those changes are exactly what the inverse-square dependence predicts.

A worked 0νββ example is most useful as a scaling check. It shows that, for the defaults on this page, the effective mass is the parameter that moves the answer the fastest, while the isotope-dependent phase-space factor and matrix element set the overall scale of the prediction.

Interpreting the 0νββ Result

The half-life number returned by this 0νββ calculator is a theoretical estimate, not a prediction that a detector will see a decay on a short timescale. A value around 10²⁶ years means the process is extraordinarily rare for a single nucleus, which is why real searches rely on large source masses, long run times, and aggressive background suppression.

On this page, the result labels are intentionally broad. Currently testable means the predicted half-life is below about 10²⁶ years, Within next-generation sensitivity covers the region below about 10²⁷ years, and Beyond next-generation reach means anything longer than that. Those thresholds are educational markers built into the page’s own classification logic rather than discovery or exclusion criteria.

Remember that the same measured half-life can imply different effective masses if a different nuclear matrix element is assumed. That is why papers often quote a band of inferred $m_{\beta \beta }$ values rather than a single number. This calculator makes that dependence easy to see by rerunning the estimate with several plausible $\left|M_{0\nu }\right|$ choices.

Limitations and Assumptions for 0νββ Half-Life Estimates

This neutrinoless double beta decay calculator is intentionally simple, so it is important to be clear about what it leaves out. It assumes the standard light-Majorana-neutrino exchange mechanism. Other 0νββ mechanisms discussed in the literature, such as heavy-particle exchange or right-handed currents, can change the relationship between half-life and the parameters people usually quote here, so the result should be treated as a comparison tool rather than a universal model.

The page also assumes that the $G_{0\nu }$ and $\left|M_{0\nu }\right|$ values you enter are already quoted in the convention you want to use. Published numbers can differ because of axial-coupling choices, short-range correlations, nuclear-model space, or operator treatment. The calculator does not convert between conventions or attempt to reconcile papers that use different normalizations.

It also omits detector-specific issues. Real sensitivity depends on isotope mass, enrichment, energy resolution, background level, exposure, and systematic uncertainties. A half-life that looks testable on the screen may still be difficult in a real experiment, and a longer half-life can sometimes become reachable only with a much larger source mass and lower background.

Finally, $m_{\beta \beta }$ is not a directly measured input in most contexts. It depends on the absolute neutrino masses, mixing angles, and unknown Majorana phases, so cancellations can make the effective mass small even when individual neutrino masses are not. This calculator is best used as a transparent way to see how those assumptions propagate into a half-life estimate.

Despite those limits, the tool is handy for lectures, quick cross-checks, and rough comparisons between isotopes. It condenses the core 0νββ relation into a form that is easy to experiment with and makes the scale of the search feel less abstract.