Electron Positron Annihilation to Z and Higgs Calculator

Explore collider kinematics with masses and luminosity controls. Estimate cross section, recoil and selected events. Plot energy scans for clearer threshold and peak behavior.

Calculated Results

Production is kinematically allowed for e⁺e⁻ → ZH.

Result Summary

Center-of-mass energy
250.000 GeV
Threshold energy (mZ + mH)
216.4376 GeV
Energy above threshold
33.5624 GeV
Lambda phase-space factor
0.245826
Z boson energy
110.255232 GeV
Higgs boson energy
139.744768 GeV
Common final-state momentum
61.976106 GeV
Z boson beta
0.562115 c
Higgs boson beta
0.443495 c
Z boson gamma
1.209103
Higgs boson gamma
1.115727
Unpolarized tree-level cross section
237.474256 fb
Polarization scaling factor
1.000000
Polarized effective cross section
237.474256 fb
Raw event yield
118737.128 events
Selected event yield
118737.128 events
Recoil mass
125.250000 GeV
Stationary annihilation rest energy
1.021998 MeV

Calculator Inputs

Use the responsive input grid below. It shows three columns on large screens, two on smaller screens, and one on mobile.

Leave blank to use the ideal calculated Z energy.
Leave blank to use the ideal Z mass value.

Plotly Energy Scan

The graph tracks polarized effective cross section and selected event yield versus center-of-mass energy.

Example Data Table

Example √s (GeV) mZ (GeV) mH (GeV) σpol (fb) Recoil Mass (GeV) Selected Events
Inclusive default setup 250.0 91.1876 125.25 237.4743 125.2500 118737.128
Z→ℓℓ, H→bb, 70% efficiency 250.0 91.1876 125.25 237.4743 125.2500 3258.743
Near threshold test 220.0 91.1876 125.25 105.9007 125.2500 52950.340

The example values are tree-level style estimates using the formulas implemented in this page.

Formula Used

Threshold condition:
√s ≥ mZ + mH
Two-body phase-space factor:
λ = [1 - (mH + mZ)² / s] × [1 - (mH - mZ)² / s]
Common final-state momentum:
p = √{[s - (mH + mZ)²][s - (mH - mZ)²]} / (2√s)
Final-state energies:
EZ = [s + mZ² - mH²] / (2√s)
EH = [s + mH² - mZ²] / (2√s)
Tree-level Higgsstrahlung cross section:
σ(e⁺e⁻ → ZH) = [GF² mZ⁴ / (96πs)] × (ve² + ae²) × √λ × [λ + 12mZ²/s] / [1 - mZ²/s]²
Electron couplings:
ve = -1 + 4sin²θW,   ae = -1
Approximate polarization scaling:
σpol = σunpol × polarization factor
Recoil mass:
mrecoil² = s + mvis² - 2√s Evis
Event yield:
Selected Events = σpol × Luminosity × BR(Z) × BR(H) × Efficiency

How to Use This Calculator

  1. Enter the collider center-of-mass energy √s in GeV.
  2. Set the Z and Higgs masses you want to study.
  3. Keep GF and sin²θW at their default values unless needed.
  4. Enter luminosity, chosen branching ratios, and detector efficiency.
  5. Optionally add beam polarizations for electron and positron beams.
  6. Optionally enter measured visible Z energy and mass for recoil studies.
  7. Set the graph scan range to visualize threshold and peak behavior.
  8. Press the calculate button to show the results above the form.
  9. Use the CSV and PDF buttons to save the output.

FAQs

1) How much energy is released when an electron and a positron annihilate each other?

For a stationary pair, the minimum released rest energy is 1.022 MeV, usually appearing as two 511 keV photons. In collider conditions, the total available center-of-mass energy can be much larger and may produce heavy states such as ZH once the threshold is exceeded.

2) What is the threshold energy for e⁺e⁻ → ZH?

The basic threshold is the sum of the final-state masses: mZ + mH. Using the default values here, that is about 216.4376 GeV. Below this energy, on-shell ZH production is not kinematically allowed.

3) Why does the cross section often look strongest near 240–250 GeV?

Near threshold, the phase space opens quickly, so the rate rises. At higher energies, the s-channel behavior falls. Their combination creates a broad maximum around the low few-hundred-GeV region for Standard Model style Higgsstrahlung studies.

4) What does recoil mass mean in this calculator?

Recoil mass is the invariant mass of everything opposite the reconstructed Z boson. If the measured Z energy and visible mass are accurate, the recoil peak can identify the Higgs without requiring direct reconstruction of its decay products.

5) Why do beam polarizations matter?

Electron and positron polarizations change the effective initial-state chiral weights. That can enhance or suppress the production rate. This page applies a compact scaling factor based on the left- and right-handed electron couplings to the Z boson.

6) Does this calculator include loop corrections or initial-state radiation?

No. It is a tree-level style calculator designed for fast studies and educational estimates. Precision collider analyses usually include radiative corrections, beam effects, acceptance modeling, and detector smearing.

7) How are event yields estimated?

The page multiplies the polarized effective cross section by integrated luminosity, then scales by the chosen Z branching ratio, Higgs branching ratio, and efficiency. This gives a selected-event estimate for your chosen channel and analysis setup.

8) Which units are used here?

Masses, energies, and momentum are reported in GeV. Cross section is shown in femtobarns. Luminosity is entered in inverse femtobarns. Event yields are simple counts, while beta and gamma are dimensionless relativistic quantities.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.