Introduction to the Kaluza–Klein tower mass calculator
A Kaluza–Klein tower is the sequence of masses that appears when a field is allowed to propagate in a compact extra dimension. This calculator keeps that tower visible: enter the zero-mode mass m₀, the compactification radius R, and the highest mode you want to inspect, and it lists the masses from n = 0 up to that limit. Because the higher modes are generated from the same relation, the page lets you compare the baseline state with each excitation instead of inferring the spectrum from a single summary value.
The three inputs are enough to show the pattern, but they also make unit choices important. A radius entered in the wrong scale or a mass copied from a different convention will still produce a table, yet the table will no longer match the model you intended to study. For that reason, this calculator works best as a quick spectrum checker: it shows whether the tower rises slowly, rises sharply, or stays close to the zero mode over the range you care about.
The discussion below explains how to read the tower, how to choose inputs for a simple compactification setup, how the formula works mode by mode, and which assumptions matter when comparing the output with a more complete field-theory or model-building calculation.
How the Kaluza–Klein tower calculator turns m₀, R, and n into a spectrum
The Kaluza–Klein tower calculator applies one relation to every mode. The n = 0 entry is the baseline mass, while each higher entry includes quantized momentum from the compact dimension. Rather than giving a rough estimate for the tower, the result panel writes out individual masses so you can inspect the spacing and see how the sequence changes as n grows.
If you want to know how quickly levels separate, whether the first excitation sits near a threshold, or how strongly the tower compresses as R grows, the output makes the comparison direct. A smaller R increases the mass gap between adjacent modes. A larger R places excited states closer together. This distinction is useful when two compactification choices need to be compared without repeating the arithmetic by hand.
How to use the Kaluza–Klein tower mass calculator
To generate a Kaluza–Klein spectrum, enter the base mass and compactification scale first, then choose how many levels should appear in the table. The form uses GeV for mass and meters for the radius, while n is a dimensionless whole-number label.
- Enter the Zero-Mode Mass m₀ in GeV.
- Enter the Compactification Radius R in meters.
- Enter a whole number for the Maximum Mode n.
- Select Generate Spectrum and review the mode-by-mode table.
When comparing two radii, keep m₀ and the maximum mode fixed. Then the difference between the two output tables is specifically the effect of the compact dimension. As a useful direction check, reducing R must raise every excited mass, whereas changing m₀ raises or lowers the entire tower without changing the momentum scale ħc/R.
Inputs for a physically consistent Kaluza–Klein tower
The three Kaluza–Klein tower inputs are simple, but their meaning should remain consistent with the model being examined. The zero-mode mass m₀ is the mass of the n = 0 state. The compactification radius R is the size of the compact direction and controls the level spacing. The maximum mode is simply the final integer level shown; it is not itself a mass or a unit conversion.
Use the units printed in the form literally. In particular, R must be entered in meters because the page uses ħc in GeV·m. If a source presents the radius in inverse GeV, femtometers, or another natural-unit convention, convert it before entering it here. An apparently surprising tower is often a conversion issue rather than a feature of the spectrum.
If the radius is uncertain, calculate a smaller-radius and a larger-radius case. Shrinking R spreads the KK levels farther apart; enlarging it packs them toward the zero mode. This gives a useful range of outcomes and makes it easier to decide whether the first few excitations are meaningfully separated at the scale being studied.
Formulas for the Kaluza–Klein tower mass spectrum
For this simple Kaluza–Klein compactification, the mass of mode n combines the zero-mode mass in quadrature with the momentum contribution from the compact dimension. The n = 0 row equals m₀ because its extra-dimensional term vanishes. Each positive n adds a contribution proportional to n and inversely proportional to R.
The mode masses follow the relation used by the page:
Here, ħc is the conversion factor used by the calculator, approximately 1.973269804 × 10⁻¹⁶ GeV·m. The radius occurs in the denominator, which is why a smaller compact dimension makes the excited modes heavier. The calculator reports the resulting masses in GeV. Since n = 0 removes the second term, the zero-mode row is an immediate check that the entered m₀ has been carried through correctly.
Worked example: a 1 GeV zero mode at R = 1×10⁻¹⁵ m
A numerical Kaluza–Klein tower example makes the pattern concrete. Suppose m₀ = 1 GeV, R = 1×10⁻¹⁵ m, and the maximum mode is n = 3. The approximate compactification scale ħc/R is 0.197 GeV, so the extra-dimensional term becomes increasingly important as the mode number rises. The calculator reports the following masses:
- n = 0: 1.000 GeV
- n = 1: 1.019 GeV
- n = 2: 1.075 GeV
- n = 3: 1.163 GeV
The square root means the mass does not rise by a perfectly fixed number of GeV from row to row, especially near the zero mode. At much larger n, however, the behavior approaches a spacing governed by ħc/R. If the same example used a smaller radius, all of the positive modes would climb more rapidly; if it used a larger radius, they would move nearer to 1 GeV.
Comparison table: the third Kaluza–Klein mode at different radii
This comparison holds m₀ = 1 GeV and n = 3 fixed while changing only R. It isolates the effect of compactification scale and provides a quick reasonableness check for a generated spectrum.
| Scenario | Radius R (m) | Mode n | Result mₙ (GeV) | Interpretation |
|---|---|---|---|---|
| Smaller radius | 5×10⁻¹⁶ | 3 | 1.550 | A tighter compact dimension pushes the third excitation much farther from the zero mode. |
| Reference radius | 1×10⁻¹⁵ | 3 | 1.163 | This is the middle case used in the worked example. |
| Larger radius | 2×10⁻¹⁵ | 3 | 1.043 | A looser compact dimension keeps the same mode close to the baseline mass. |
The direction of the effect is the important lesson: when R shrinks, the KK shift grows; when R expands, it fades. Altering m₀ instead moves the baseline of every row, while the compactification scale remains what controls the excited-state separation.
How to interpret a Kaluza–Klein tower mass result
The Kaluza–Klein results panel lists masses one mode at a time. Read the n = 0 row as the baseline, then compare the positive modes with it and with one another. A sensible output has nondecreasing masses for nonnegative n, and reducing the entered radius should make the nonzero rows heavier. If either expectation fails, check the radius unit and the input values before drawing a physical conclusion.
The table is intended for immediate inspection rather than as a complete model record. If you save a run in notes or use it in a comparison study, save m₀, R, and the maximum mode with it. The same set of masses can only be interpreted correctly when those assumptions accompany the result.
Limitations and assumptions of this Kaluza–Klein mass spectrum
This Kaluza–Klein mass calculator uses the simple compact-dimension relation displayed above, so it is a spectrum estimator rather than a full extra-dimensional model. It assumes the relevant compactification behavior can be represented by the stated radius and a straightforward integer mode tower. Boundary conditions, warping, localized terms, mixing, radiative corrections, and model-specific degeneracies are not added automatically.
Displayed values are rounded to three decimal places, so small differences from a higher-precision hand calculation are normal. The table also limits the maximum mode to keep the browser output readable. For a detailed phenomenology, model-building, or numerical-analysis task, use this page to establish the simple tower and then apply the additional assumptions required by the specific theory.
Mini-game: phase-lock a Kaluza–Klein mode
This optional 75-second phase-lock challenge turns the tower idea into a fast visual exercise. A numbered KK pulse travels inward along the compact ring. Select the matching numbered gate when the pulse crosses its glowing resonance band. Each clean lock builds a streak; a wrong gate or missed pulse costs stability. The gates begin to rotate and pulses accelerate as the run develops.
Controls: tap or click a numbered gate at the gold ring, or press 0–5. The mini-game is separate from the calculator and never changes its result.