Coleman–Weinberg Potential in Practice
Radiative symmetry breaking can appear when the classical potential is scale free. The Coleman–Weinberg approach adds the one-loop vacuum energy, turning running couplings into a measurable shape for V_eff(φ). This calculator evaluates that shape at a chosen field point, helping compare parameter sets quickly and consistently across studies in research.
Model Inputs: φ, μ, m², and λ
The tree-level part uses V0 + 1/2 m^2 φ^2 + 1/4 λ φ^4. Enter μ as the renormalization scale that controls the logarithms. A practical rule is to keep μ near the dominant mass in the spectrum, which reduces large logs and improves numerical stability for perturbation theory in scans.
Loop Spectrum Table: m²(φ)=aφ²+b
Each one-loop entry defines a field-dependent mass with m²(φ)=aφ²+b. The coefficient a can represent a coupling-squared factor, while b captures masses or symmetry-breaking terms. At your chosen φ, the calculator computes m², m⁴, and ln(|m²|/μ²), then reports each contribution term so you can audit which particles dominate the loop sum.
Degrees of Freedom and Fermion Sign
Degeneracy n counts physical degrees of freedom, such as color, spin, and particle/antiparticle states. For fermions, the loop sign is negative, implemented here automatically through n_eff. This prevents accidental sign mistakes when exploring radiative corrections. Typical examples are n=1 for a real scalar and n=4 for a Dirac fermion field.
Scheme Constants and Scale Sensitivity
In the MS-bar convention, scalars and fermions use c=3/2, while gauge bosons commonly use c=5/6. You may override c to match alternative schemes or thresholds. Because the correction scales as m⁴/(64π²), even modest mass changes can matter. Shifting μ rescales the log and redistributes terms between tree and loop parts.
Interpreting V_tree, V_1loop, and V_eff
The summary separates V_tree, V_1loop, and V_eff so you can see whether radiative effects are small corrections or the driver of the potential. If V_1loop is comparable to V_tree, perturbation theory may be strained. Use the per-entry table to identify the largest term and reconsider its parameters or μ choice.
Using dV/dφ to Locate Extrema
Enable the numerical derivative to estimate dV/dφ using a central difference with step h. When the slope crosses zero, you may be near an extremum, and the sign change indicates stability trends. Choose h small enough for accuracy but large enough to avoid floating-point noise, especially when V_eff is flat.
Exportable Outputs and Reproducible Reports
After a computation, export results as CSV for spreadsheets or as PDF for archiving. The CSV captures inputs, the summary, and each one-loop row, supporting parameter sweeps and version tracking. The PDF prints the same tables in a format, making it easy to attach to notes, lab books, or reviews.