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Particle Physics — Equation Sheet

The complete equation reference for particle physics — relativistic kinematics, conservation laws, QED, weak and electroweak interactions, CKM mixing, QCD, neutrino oscillations, and the Standard Model.

10 sections · 60 equations

Relativistic Kinematics

Four-Momentum
pμ=(E, p),p2=E2−∣p∣2=m2p^\mu = (E,\,\mathbf{p}), \qquad p^2 = E^2 - |\mathbf{p}|^2 = m^2
Natural units (c = ℏ = 1). The four-momentum magnitude is the Lorentz-invariant mass squared.
Invariant Mass of a System
M2=(∑ipi)2=(∑iEi)2−∣∑ipi∣2M^2 = \left(\sum_i p_i\right)^2 = \left(\sum_i E_i\right)^2 - \left|\sum_i \mathbf{p}_i\right|^2
Invariant mass of a multi-particle system. Equal to the CM frame energy squared.
Mandelstam Variables (2→2)
s=(p1+p2)2,t=(p1−p3)2,u=(p1−p4)2s=(p_1+p_2)^2,\quad t=(p_1-p_3)^2,\quad u=(p_1-p_4)^2
Lorentz-invariant combinations for 2→2 scattering (1,2 in; 3,4 out). s is the CM energy squared; t and u are momentum transfers.
Mandelstam Sum Rule
s+t+u=∑i=14mi2s + t + u = \sum_{i=1}^{4}m_i^2
Constraint relating the Mandelstam variables. In the massless limit s + t + u = 0.
Threshold CM Energy
sthr=∑fmf\sqrt{s_{\rm thr}} = \sum_f m_f
Minimum CM energy needed to produce final-state particles of masses m_f. Below this threshold the reaction is kinematically forbidden.
Rapidity
y=12ln⁡E+pzE−pz,Δy is Lorentz-boost invarianty = \frac{1}{2}\ln\frac{E + p_z}{E - p_z}, \qquad \Delta y \text{ is Lorentz-boost invariant}
Rapidity of a particle along the beam (z) axis. Differences in rapidity are invariant under longitudinal boosts.
Pseudorapidity
η=−ln⁡tan⁡θ2\eta = -\ln\tan\frac{\theta}{2}
Massless approximation to rapidity expressed through the polar angle θ from the beam axis. Widely used in collider detector geometry.
Lorentz-Invariant Flux Factor
F=4(p1⋅p2)2−m12m22=4∣pcm∣sF = 4\sqrt{(p_1\cdot p_2)^2 - m_1^2 m_2^2} = 4|\mathbf{p}_{\rm cm}|\sqrt{s}
Appears in the denominator of the cross-section formula. The second form holds in the CM frame.

Quantum Numbers & Conservation Laws

Gell-Mann–Nishijima Formula
Q=I3+Y2,Y=B+S+C+B′+TQ = I_3 + \frac{Y}{2}, \qquad Y = B + S + C + B^\prime + T
Electric charge Q from weak isospin I₃ and hypercharge Y. Y includes baryon number B, strangeness S, charm C, bottomness B′, and topness T.
Baryon Number
B=13(nq−nqˉ)B = \frac{1}{3}(n_q - n_{\bar{q}})
Each quark carries B = +1/3, each antiquark B = −1/3. B is conserved exactly in the Standard Model.
Parity of a Two-Particle State
P=P1P2(−1)LP = P_1 P_2 (-1)^L
Intrinsic parities P₁, P₂ times (−1)^L from the orbital angular momentum L. Fermion–antifermion pair has intrinsic parity −1.
Charge Conjugation (neutral mesons)
C=(−1)L+SC = (-1)^{L+S}
C eigenvalue for a neutral quark–antiquark state with orbital angular momentum L and total spin S. Applies only to C-eigenstates.
CP Transformation
(CP) ∣ψ(p,s)⟩=ηCP ∣ψˉ(−p,s)⟩(CP)\,|\psi(\mathbf{p},s)\rangle = \eta_{CP}\,|\bar{\psi}(-\mathbf{p},s)\rangle
Combined parity P and charge conjugation C. CP is almost — but not exactly — conserved (CP violation observed in K and B mesons).
CPT Theorem
L(x)→CPTL(x)⇒mparticle=mantiparticle\mathcal{L}(x) \xrightarrow{CPT} \mathcal{L}(x) \quad \Rightarrow \quad m_{\rm particle} = m_{\rm antiparticle}
Any local, Lorentz-invariant quantum field theory is invariant under the combined CPT transformation. Implies equal masses and lifetimes for particles and antiparticles.

Decay Rates & Resonances

Exponential Decay Law
N(t)=N0 e−Γt=N0 e−t/τ,τ=1ΓN(t) = N_0\,e^{-\Gamma t} = N_0\,e^{-t/\tau}, \qquad \tau = \frac{1}{\Gamma}
Number of undecayed particles at time t. τ is the mean lifetime; Γ is the total decay width (in natural units where ℏ = 1).
Branching Ratio
BR(i)=ΓiΓtotal,∑iBR(i)=1\text{BR}(i) = \frac{\Gamma_i}{\Gamma_{\rm total}}, \qquad \sum_i \text{BR}(i) = 1
Fraction of decays going through channel i. Partial width Γ_i and total width Γ_total are related by unitarity.
Two-Body Decay Rate
Γ=∣pcm∣8πm2 ∣M∣2\Gamma = \frac{|\mathbf{p}_{\rm cm}|}{8\pi m^2}\,|\mathcal{M}|^2
Decay rate for a particle of mass m into two bodies; |p_cm| is the CM frame momentum of either final-state particle.
Relativistic Breit-Wigner Cross Section
σ(s)=4πk2 (2J+1)(2s1+1)(2s2+1) ΓiΓf(s−mR)2+Γ2/4\sigma(s) = \frac{4\pi}{k^2}\,\frac{(2J+1)}{(2s_1+1)(2s_2+1)}\,\frac{\Gamma_i\Gamma_f}{(\sqrt{s}-m_R)^2 + \Gamma^2/4}
Cross section near a resonance of spin J, mass m_R, and total width Γ. Γ_i and Γ_f are partial widths to the initial and final states.
Natural Line Width
Γ⋅τ=ℏ≈6.58×10−25 GeV⋅s\Gamma \cdot \tau = \hbar \approx 6.58 \times 10^{-25}\text{ GeV·s}
Energy-time uncertainty relation for an unstable particle. A wide resonance has a short lifetime; a stable particle has Γ = 0.

QED Processes & Cross Sections

Fine Structure Constant
α=e24π≈1137.036(at q2=0)\alpha = \frac{e^2}{4\pi} \approx \frac{1}{137.036} \quad (\text{at } q^2 = 0)
Dimensionless QED coupling constant in natural units. Increases logarithmically with energy: α(m_Z) ≈ 1/128.
e⁺e⁻ → μ⁺μ⁻ Total Cross Section
σ(e+e−→μ+μ−)=4πα23s,s≫mμ2\sigma(e^+e^- \to \mu^+\mu^-) = \frac{4\pi\alpha^2}{3s}, \quad s \gg m_\mu^2
Leading-order QED result. Provides the standard "point cross section" reference unit at colliders.
R Ratio (e⁺e⁻ → hadrons)
R=σ(e+e−→hadrons)σ(e+e−→μ+μ−)=Nc∑qQq2R = \frac{\sigma(e^+e^- \to \text{hadrons})}{\sigma(e^+e^- \to \mu^+\mu^-)} = N_c\sum_q Q_q^2
Sum runs over quark flavors kinematically accessible at energy √s. N_c = 3 colors; experimentally confirms color. At high energy R ≈ 11/3.
Thomson Cross Section
σT=8π3re2=8πα23me2≈6.65×10−29 m2\sigma_T = \frac{8\pi}{3}r_e^2 = \frac{8\pi\alpha^2}{3m_e^2} \approx 6.65 \times 10^{-29}\text{ m}^2
Classical electron cross section for low-energy (ω ≪ m_e) Compton scattering. r_e = α/m_e ≈ 2.82 fm is the classical electron radius.
Compton Wavelength Shift
λ′−λ=hmec(1−cos⁡θ)=λC(1−cos⁡θ)\lambda^\prime - \lambda = \frac{h}{m_e c}(1-\cos\theta) = \lambda_C(1-\cos\theta)
Wavelength shift of a photon scattered by angle θ off a free electron. λ_C = h/(m_e c) ≈ 2.43 pm is the Compton wavelength.
Klein-Nishina Formula (Compton)
dσdΩ=re22 ⁣(ω′ω) ⁣2 ⁣(ωω′+ω′ω−sin⁡2θ)\frac{d\sigma}{d\Omega} = \frac{r_e^2}{2}\!\left(\frac{\omega^\prime}{\omega}\right)^{\!2}\!\left(\frac{\omega}{\omega^\prime} + \frac{\omega^\prime}{\omega} - \sin^2\theta\right)
Full relativistic differential cross section for Compton scattering. Reduces to Thomson formula in the low-energy limit ω → 0.
Rutherford Scattering Cross Section
dσdΩ=Z12Z22α24Ecm2sin⁡4(θ/2)\frac{d\sigma}{d\Omega} = \frac{Z_1^2 Z_2^2 \alpha^2}{4E_{\rm cm}^2 \sin^4(\theta/2)}
Differential cross section for Coulomb (electromagnetic) scattering. Diverges as θ → 0 due to the infinite range of the photon.

Weak Interactions

Charged-Current (V−A) Lagrangian
LCC=−GF2 fˉ1γμ(1−γ5)f2⋅fˉ3γμ(1−γ5)f4+h.c.\mathcal{L}_{\rm CC} = -\frac{G_F}{\sqrt{2}}\,\bar{f}_1\gamma^\mu(1-\gamma^5)f_2\cdot\bar{f}_3\gamma_\mu(1-\gamma^5)f_4 + \text{h.c.}
Fermi effective theory for charged-current weak interactions. The (1 − γ⁵) projectors select left-handed (V−A) currents.
Fermi Coupling Constant
GF2=g28mW2≈1.1664×10−5GeV2\frac{G_F}{\sqrt{2}} = \frac{g^2}{8m_W^2} \approx \frac{1.1664\times10^{-5}}{\text{GeV}^2}
Low-energy effective coupling of the weak interaction. Measured from muon decay; g is the SU(2) coupling and m_W ≈ 80.4 GeV.
Muon Decay Rate
Γ(μ−→e−νˉeνμ)=GF2mμ5192π3\Gamma(\mu^- \to e^-\bar{\nu}_e\nu_\mu) = \frac{G_F^2 m_\mu^5}{192\pi^3}
Dominant muon decay at tree level. The m⁵ dependence makes the muon (τ ≈ 2.2 μs) much longer-lived than heavier particles.
Pion Leptonic Decay Rate
Γ(π−→ℓ−νˉℓ)=GF2∣Vud∣2fπ28π mπmℓ2 ⁣(1−mℓ2mπ2) ⁣2\Gamma(\pi^- \to \ell^-\bar{\nu}_\ell) = \frac{G_F^2|V_{ud}|^2 f_\pi^2}{8\pi}\,m_\pi m_\ell^2\!\left(1-\frac{m_\ell^2}{m_\pi^2}\right)^{\!2}
Helicity suppression ∝ m_ℓ² explains why π → eν is strongly suppressed relative to π → μν despite more phase space.
Weak Neutral Current (Z coupling)
LNC=−gcos⁡θW fˉγμ ⁣(T3−Qsin⁡2θW−T3γ5) ⁣Zμf\mathcal{L}_{\rm NC} = -\frac{g}{\cos\theta_W}\,\bar{f}\gamma^\mu\!\left(T_3 - Q\sin^2\theta_W - T_3\gamma^5\right)\!Z_\mu f
Z⁰ couples to both vector (T₃ − Q sin²θ_W) and axial-vector (T₃) currents. The coupling depends on weak isospin T₃ and charge Q.

CKM Matrix & Flavor Mixing

CKM Matrix
(d′s′b′)=VCKM(dsb)=(VudVusVubVcdVcsVcbVtdVtsVtb)(dsb)\begin{pmatrix}d^\prime\\s^\prime\\b^\prime\end{pmatrix} = V_{\rm CKM}\begin{pmatrix}d\\s\\b\end{pmatrix} = \begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}
Cabibbo-Kobayashi-Maskawa matrix rotating mass-eigenstate quarks (d, s, b) to weak-eigenstate quarks (d′, s′, b′). Unitary: V†V = 1.
Wolfenstein Parametrization
VCKM≈(1−λ2/2λAλ3(ρ−iη)−λ1−λ2/2Aλ2Aλ3(1−ρ−iη)−Aλ21)V_{\rm CKM} \approx \begin{pmatrix}1-\lambda^2/2 & \lambda & A\lambda^3(\rho-i\eta)\\-\lambda & 1-\lambda^2/2 & A\lambda^2\\A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1\end{pmatrix}
Expansion in λ ≈ 0.225 (Cabibbo angle). A ≈ 0.814, ρ ≈ 0.117, η ≈ 0.353. The complex phase η is the sole source of CP violation in the quark sector.
Unitarity Triangle
VudVub∗+VcdVcb∗+VtdVtb∗=0V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0
One of six unitarity conditions. Represented as a triangle in the complex plane; its non-zero area quantifies CP violation.
Jarlskog Invariant
J=Im(VusVcbVub∗Vcs∗)≈3.2×10−5J = \mathrm{Im}(V_{us}V_{cb}V_{ub}^*V_{cs}^*) \approx 3.2\times10^{-5}
Rephasing-invariant measure of CP violation. Equals twice the area of any unitarity triangle. The same J appears in all CP-violating observables.
GIM Mechanism — FCNC Cancellation
∑q=u,c,tVqi∗Vqj=0(i≠j)\sum_{q=u,c,t} V_{qi}^*V_{qj} = 0 \quad (i\neq j)
CKM unitarity causes flavor-changing neutral currents (FCNC) to vanish at tree level and be suppressed at loop level. Explains the smallness of K⁰–K̄⁰ mixing.

Electroweak Theory

Electroweak Gauge Boson Masses
mW≈80.4 GeV,mZ≈91.2 GeV,mH≈125.1 GeVm_W \approx 80.4\text{ GeV}, \quad m_Z \approx 91.2\text{ GeV}, \quad m_H \approx 125.1\text{ GeV}
Measured masses of the W±, Z⁰, and Higgs bosons. All acquire mass through the Higgs mechanism after electroweak symmetry breaking.
Weinberg Angle (electroweak mixing)
sin⁡2θW≈0.2312,tan⁡θW=g′g\sin^2\theta_W \approx 0.2312, \qquad \tan\theta_W = \frac{g^\prime}{g}
Mixing angle between the SU(2) coupling g and the U(1)_Y coupling g′. Determines the relative coupling strengths of W and Z bosons.
Z Partial Width to Fermion Pair
Γ(Z→ffˉ)=GFmZ36π2 Nc(gVf 2+gAf 2)\Gamma(Z\to f\bar{f}) = \frac{G_F m_Z^3}{6\pi\sqrt{2}}\,N_c\bigl(g_V^{f\,2} + g_A^{f\,2}\bigr)
g_V^f = T₃ − 2Q sin²θ_W, g_A^f = T₃. N_c = 3 for quarks, 1 for leptons. Total Γ_Z ≈ 2.495 GeV determines the number of light neutrino families.
Number of Light Neutrino Families
Nν=ΓZinvΓ(Z→ννˉ)=2.984±0.008N_\nu = \frac{\Gamma_Z^{\rm inv}}{\Gamma(Z\to\nu\bar{\nu})} = 2.984 \pm 0.008
Inferred from the invisible Z width at LEP. Confirms exactly three generations of light (m_ν < m_Z/2) neutrinos.
W Partial Width
Γ(W→ℓν)=GFmW36π2≈226 MeV\Gamma(W\to\ell\nu) = \frac{G_F m_W^3}{6\pi\sqrt{2}} \approx 226\text{ MeV}
Leptonic partial width; each lepton family contributes equally. Hadronic partial width has an additional N_c|V_{qq′}|² factor.
Rho Parameter
ρ=mW2mZ2cos⁡2θW=1+O(α)\rho = \frac{m_W^2}{m_Z^2\cos^2\theta_W} = 1 + \mathcal{O}(\alpha)
Tree-level prediction of ρ = 1 from the Higgs doublet structure; radiative corrections provide sensitivity to top quark and Higgs masses.

QCD & Strong Interactions

QCD Color Factors (SU(3))
CF=Nc2−12Nc=43,CA=Nc=3,TF=12C_F = \frac{N_c^2-1}{2N_c} = \frac{4}{3}, \quad C_A = N_c = 3, \quad T_F = \frac{1}{2}
Casimir invariants for SU(3): C_F for quarks (fundamental representation), C_A for gluons (adjoint), T_F for fundamental generators.
Running QCD Coupling
αs(μ2)=2πβ0ln⁡(μ/ΛQCD),ΛQCD≈200 MeV\alpha_s(\mu^2) = \frac{2\pi}{\beta_0\ln(\mu/\Lambda_{\rm QCD})}, \qquad \Lambda_{\rm QCD}\approx 200\text{ MeV}
Leading-order running of α_s. α_s(m_Z) ≈ 0.118; the coupling grows at low energies, driving confinement.
Factorization Theorem (hadronic cross sections)
σ(AB→X)=∑a,b∫ ⁣dxa dxb fa(xa,μF2) fb(xb,μF2) σ^(ab→X;μF,μR)\sigma(AB\to X) = \sum_{a,b}\int\!dx_a\,dx_b\,f_a(x_a,\mu_F^2)\,f_b(x_b,\mu_F^2)\,\hat{\sigma}(ab\to X;\mu_F,\mu_R)
Hard process σ̂ is computed perturbatively; soft content of the hadrons A, B is absorbed into parton distribution functions f(x, μ_F²).
DGLAP Evolution Equation (quark)
μ2∂qi(x,μ2)∂μ2=αs2π∫x1 ⁣dzz[Pqq(z) qi(x/z)+Pqg(z) g(x/z)]\mu^2\frac{\partial q_i(x,\mu^2)}{\partial\mu^2} = \frac{\alpha_s}{2\pi}\int_x^1\!\frac{dz}{z}\Bigl[P_{qq}(z)\,q_i(x/z) + P_{qg}(z)\,g(x/z)\Bigr]
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations describe how PDFs evolve with the factorization scale μ_F².
Color Confinement Scale
ΛQCD≈200–300 MeV,rconfinement∼ΛQCD−1≈1 fm\Lambda_{\rm QCD} \approx 200\text{–}300\text{ MeV}, \qquad r_{\rm confinement} \sim \Lambda_{\rm QCD}^{-1} \approx 1\text{ fm}
Below this energy scale perturbative QCD breaks down; quarks and gluons are confined into hadrons. Not yet proven from first principles.
String Tension (linear confinement)
V(r)≈−4αs3r+κr,κ≈0.18 GeV2≈0.9 GeV/fmV(r) \approx -\frac{4\alpha_s}{3r} + \kappa r, \qquad \kappa \approx 0.18\text{ GeV}^2 \approx 0.9\text{ GeV/fm}
Cornell potential for a quark-antiquark pair: Coulomb-like at short range (gluon exchange) plus a linear "string" term at long range (confinement).

Neutrino Oscillations

PMNS Mixing Matrix
(νeνμντ)=UPMNS(ν1ν2ν3)\begin{pmatrix}\nu_e\\\nu_\mu\\\nu_\tau\end{pmatrix} = U_{\rm PMNS}\begin{pmatrix}\nu_1\\\nu_2\\\nu_3\end{pmatrix}
Pontecorvo-Maki-Nakagawa-Sakata matrix mixing weak-eigenstate neutrinos (e, μ, τ) with mass eigenstates (1, 2, 3). Parametrized by three angles θ₁₂, θ₁₃, θ₂₃ and a CP phase δ.
Neutrino Oscillation Probability
P(να→νβ)=∣∑iUαi∗ Uβi e−imi2L/2E∣2P(\nu_\alpha\to\nu_\beta) = \left|\sum_i U_{\alpha i}^*\,U_{\beta i}\,e^{-im_i^2 L/2E}\right|^2
Probability that a neutrino created as flavour α is detected as flavour β after traveling distance L with energy E.
Two-Flavor Oscillation
P(να→νβ)=sin⁡2(2θ)sin⁡2 ⁣(1.27 Δm2[eV2] L[km]E[GeV])P(\nu_\alpha\to\nu_\beta) = \sin^2(2\theta)\sin^2\!\left(\frac{1.27\,\Delta m^2[\text{eV}^2]\,L[\text{km}]}{E[\text{GeV}]}\right)
Simplified formula for two-flavor mixing. The oscillation length L_osc = 4πE/Δm² sets the scale of observable oscillations.
Solar Mass-Squared Splitting
Δm212=m22−m12≈7.53×10−5 eV2\Delta m_{21}^2 = m_2^2 - m_1^2 \approx 7.53\times10^{-5}\text{ eV}^2
Measured from solar neutrino flux (MSW effect) and reactor experiments (KamLAND). Establishes the "solar" mixing angle θ₁₂ ≈ 33°.
Atmospheric Mass-Squared Splitting
∣Δm312∣≈2.51×10−3 eV2|\Delta m_{31}^2| \approx 2.51\times10^{-3}\text{ eV}^2
Measured from atmospheric and accelerator neutrino experiments (Super-K, T2K, NOvA). Sets the "atmospheric" mixing angle θ₂₃ ≈ 45°.
MSW Resonance Condition (matter effects)
2 GFne=Δm22Ecos⁡2θ\sqrt{2}\,G_F n_e = \frac{\Delta m^2}{2E}\cos 2\theta
Mikheyev-Smirnov-Wolfenstein effect: neutrino oscillations are enhanced when the matter-induced potential √2 G_F n_e matches the vacuum oscillation frequency. Explains the solar neutrino problem.

Standard Model Summary

SM Gauge Group
GSM=SU(3)c×SU(2)L×U(1)YG_{\rm SM} = SU(3)_c \times SU(2)_L \times U(1)_Y
The gauge symmetry of the Standard Model: SU(3) for colour (QCD), SU(2)_L for weak isospin (left-handed), U(1)_Y for weak hypercharge.
SM Lagrangian (schematic)
LSM=−14Fμν2+iψˉD̸ψ+∣DμH∣2−V(H)+yfψˉHψ+h.c.\mathcal{L}_{\rm SM} = -\tfrac{1}{4}F_{\mu\nu}^2 + i\bar{\psi}\not{D}\psi + |D_\mu H|^2 - V(H) + y_f\bar{\psi}H\psi + \text{h.c.}
Gauge kinetic terms, fermion kinetic + minimal coupling, Higgs kinetic, Higgs potential (SSB), and Yukawa fermion-mass terms.
SM Quark Charges
Qu=+23,Qd=−13;generations: (u,d), (c,s), (t,b)Q_u = +\tfrac{2}{3}, \quad Q_d = -\tfrac{1}{3}; \qquad \text{generations: } (u,d),\,(c,s),\,(t,b)
Electric charges of up-type (u, c, t) and down-type (d, s, b) quarks in units of e. Top quark mass m_t ≈ 173 GeV is by far the largest Yukawa coupling.
Fundamental Coupling Constants (at m_Z)
αem(mZ)≈1128,αs(mZ)≈0.118,GF≈1.166×10−5 GeV−2\alpha_{\rm em}(m_Z) \approx \frac{1}{128}, \quad \alpha_s(m_Z) \approx 0.118, \quad G_F \approx 1.166\times10^{-5}\text{ GeV}^{-2}
The three measured input parameters of the SM gauge sector at the Z pole. All other precision electroweak observables can be predicted from these.
GUT Unification Scale
MGUT∼1016 GeV,α1(MGUT)=α2(MGUT)=α3(MGUT)M_{\rm GUT} \sim 10^{16}\text{ GeV}, \qquad \alpha_1(M_{\rm GUT}) = \alpha_2(M_{\rm GUT}) = \alpha_3(M_{\rm GUT})
The three SM gauge couplings almost unify (exactly in MSSM) at ~10¹⁶ GeV, suggesting a Grand Unified Theory above this scale.
Hierarchy Problem
mH2=mH,02+δmH2,δmH2∼ΛUV216π2m_H^2 = m_{H,0}^2 + \delta m_H^2, \qquad \delta m_H^2 \sim \frac{\Lambda_{\rm UV}^2}{16\pi^2}
Quadratic sensitivity of the Higgs mass to UV physics. Keeping m_H ≈ 125 GeV requires fine-tuning at the level of (m_H/M_Pl)² ∼ 10⁻³⁴ unless new physics (SUSY, composite Higgs) intervenes.