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Proton

p+
1.672 621 925 95 × 10⁻²⁷ kg  ·  938.272 MeV
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The proton (p) is a baryon of spin ½, electric charge +e, and rest mass mp = 1.672 621 925 95(52) × 10⁻²⁷ kg, equivalent to a rest energy of 938.272 089 43(29) MeV (Mohr et al. 2025). It is a colour-singlet bound state of the quark and gluon fields of quantum chromodynamics with valence content uud; the quark masses contribute of order 1% of the proton mass, and the remainder is the kinetic and field energy of the confined system (Dürr et al. 2008). It is a quantum object, localised in detection and described in bound states and diffraction by a wavefunction of de Broglie wavelength λ = h/p; as the nucleus of hydrogen it sets the reduced mass of the electron–proton system through the ratio mp/me = 1836.152 673 426(32) (Mohr et al. 2025). Its internal structure is resolved in electron scattering: the elastic form factors give an rms charge radius rp = 0.840 75(64) fm (Mohr et al. 2025; Antognini et al. 2013; Xiong et al. 2019), and deep inelastic scattering at large momentum transfer exhibits Bjorken scaling, the signature of point-like constituents (Bloom et al. 1969; Breidenbach et al. 1969). The proton is a source of the electromagnetic field, carrying a Coulomb field and a magnetic moment of 2.792 847 344 63(82) nuclear magnetons; the departure of its g-factor, gp = 5.585 694 6893(16), from the Dirac value g = 2 is a consequence of its composite structure (Mohr et al. 2025; Schneider et al. 2017). It participates in the strong, electromagnetic and weak interactions. No decay has been observed; the partial lifetime for p → e⁺π⁰ exceeds 2.4 × 10³⁴ yr at 90% confidence (Takenaka et al. 2020). It is the dominant baryonic constituent of stars and of the interstellar medium, and proton–proton fusion is the first reaction of the pp chain in main-sequence stars (Adelberger et al. 2011). The proton was identified as a constituent of nuclei by Rutherford (1919).

02 · Equations

Governing equations

The proton is a composite state of quantum chromodynamics (QCD). Its mass, size and magnetic moment follow from that theory; its charge and its motion in external fields follow from electrodynamics. Numerical values are from the CODATA 2022 adjustment (Mohr et al. 2025) unless otherwise cited.

Master · QCD Lagrangian

QCD Lagrangian density

$$\mathcal L_{\rm QCD}=\sum_{q}\bar q\left(i\hbar c\,\gamma^\mu D_\mu-m_qc^2\right)q-\tfrac14\,G^a_{\mu\nu}G^{a\,\mu\nu},\qquad D_\mu=\partial_\mu-ig_s\,t^aA^a_\mu$$

The gluon field carries colour charge, so the coupling grows at large separation and the quarks are confined. The proton is the lowest-mass colour-singlet uud state and the lowest-mass baryon (Navas et al. 2024).

01 · Where the mass comes from

Mass–energy decomposition

$$\begin{gathered}m_pc^2=938.27\ \mathrm{MeV}\\ \approx\underbrace{2m_u+m_d}_{\sim 9\ \mathrm{MeV}}+\underbrace{E_{\rm kin}+E_{\rm gluon}+E_{\rm anom}}_{\sim 929\ \mathrm{MeV}}\end{gathered}$$

The three valence-quark masses account for approximately 1 % of the proton mass; the remainder is the kinetic and field energy of the confined colour field. Lattice QCD reproduces the measured mass to about 1 % (Dürr et al. 2008).

02 · Size

Charge radius and form factors

$$\begin{gathered}\frac{d\sigma}{d\Omega}=\left(\frac{d\sigma}{d\Omega}\right)_{\!\rm Mott}\frac{G_E^2+\tau G_M^2}{1+\tau}+2\tau G_M^2\tan^2\!\tfrac\theta2,\\ r_p^2=-6\hbar^2\frac{dG_E}{dQ^2}\Big|_{0}\end{gathered}$$

Elastic electron scattering (Hofstadter 1956) maps the proton’s charge; the slope at zero momentum transfer defines its radius, rp = 0.8407 fm, confirmed by muonic-hydrogen spectroscopy.

03 · Magnetism

Anomalous magnetic moment

$$\begin{gathered}\mu_p=g_p\,\frac{e\hbar}{4m_p}=2.792\,847\ \mu_N,\\ g_p=5.585\,694\,689\,3(16)\end{gathered}$$

A point-like Dirac particle has g = 2. The measured value g_p = 5.585 694 6893(16) implies an anomalous moment κ = 1.792 847, indicating internal structure (Frisch & Stern 1933; Schneider et al. 2017).

04 · Bound to an electron

Hydrogen with a finite-mass nucleus

$$\begin{gathered}E_n=-\frac{hcR_\infty}{n^2}\,\frac{1}{1+m_e/m_p},\\ \Delta E_{\rm hfs}=\tfrac43\,g_p\,\alpha^4\,\frac{m_e}{m_p}\,m_ec^2=5.87\ \mu\mathrm{eV}\ (\lambda=21\ \mathrm{cm})\end{gathered}$$

The finite nuclear mass reduces the magnitude of every level energy by the reduced-mass factor mₑ/m_p ≈ 1/1836. Coupling of the proton and electron magnetic moments splits the ground state into F = 0 and F = 1, separated by 1420.405 751 768(2) MHz (Essen et al. 1971).

05 · Fusion in stars

Coulomb barrier and tunnelling

$$\begin{gathered}p+p\to d+e^++\nu_e\ (+0.42\ \mathrm{MeV}),\\ P\propto\exp\!\left(-\sqrt{\frac{E_G}{E}}\right),\quad E_G=2\pi^2\alpha^2\mu c^2=493\ \mathrm{keV}\end{gathered}$$

At solar-core temperatures the mean thermal energy is of order 1 keV against a Coulomb barrier of order 500 keV, so the reaction proceeds by quantum tunnelling (Adelberger et al. 2011). The rate is further suppressed by the weak-interaction conversion required in p + p → d + e⁺ + νₑ.

06 · Inside the proton

Deep inelastic scattering

$$\begin{gathered}x=\frac{Q^2}{2P\!\cdot\!q},\\ F_2(x)=x\sum_q e_q^2\,f_q(x),\\ \int_0^1 x\sum_q f_q(x)\,dx\approx0.5\end{gathered}$$

At large Q² the structure function F₂(x, Q²) becomes approximately independent of Q², the scaling behaviour expected of scattering from free point-like constituents (Bloom et al. 1969; Breidenbach et al. 1969). Quarks carry approximately half the proton momentum; the remainder is carried by gluons.

07 · In a magnetic field

Cyclotron motion and Larmor precession

$$\begin{gathered}r=\frac{p}{eB},\\ \omega_c=\frac{eB}{\gamma m_p},\\ f_{\rm Larmor}=\frac{\gamma_p}{2\pi}B=42.577\ \mathrm{MHz\,T^{-1}}\times B\end{gathered}$$

The first relation gives the cyclotron radius of a proton in a transverse field; the last gives the Larmor precession frequency of the proton magnetic moment, γp/2π = 42.577 478 461(18) MHz T⁻¹ (Mohr et al. 2025).

08 · Stability

Baryon number conservation

$$\begin{gathered}\tau\!\left(p\to e^+\pi^0\right)>2.4\times10^{34}\ \mathrm{yr},\\ \tau_{\rm universe}\approx1.4\times10^{10}\ \mathrm{yr}\end{gathered}$$

Baryon number is conserved by the Standard Model gauge interactions, and no proton decay has been observed. Super-Kamiokande sets τ/B > 2.4 × 10³⁴ yr for p → e⁺π⁰ at 90 % confidence (Takenaka et al. 2020).

03 · Numbers

Measured properties

Recommended values from the CODATA 2022 adjustment (Mohr et al. 2025), the exact 2019 SI definition of the elementary charge, and Particle Data Group limits (Navas et al. 2024). Parenthesised digits give the standard uncertainty in the final digits shown.

+1.602 176 634 × 10⁻¹⁹ C
Charge · exact · SI 2019
938.272 089 43 MeV
Rest energy mpc² · Mohr et al. 2025
0.840 75 fm
rms charge radius · Mohr et al. 2025
τ > 2.4 × 10³⁴ yr
Partial lifetime p → e⁺π⁰ · Takenaka et al. 2020
QuantityValueStatusMeaning & convention
Electric charge+1.602 176 634 × 10⁻¹⁹ Cexact · SI 2019Exact by the 2019 SI definition; equal in magnitude to the electron charge to 1 part in 10²¹ (Bressi et al. 2011).
Mass1.672 621 925 95(52) × 10⁻²⁷ kgmeasured · 3.1 × 10⁻¹⁰1.007 276 466 578(83) u (Mohr et al. 2025); the quark masses contribute of order 1% (Dürr et al. 2008).
Rest energy938.272 089 43(29) MeVderived · mpEnergy of one proton at rest (Mohr et al. 2025).
Proton–electron mass ratio1836.152 673 426(32)measured · 1.7 × 10⁻¹¹Reduced-mass factor entering the hydrogen level energies (Mohr et al. 2025).
Spin½ ℏexact · representationQuark spins account for approximately 30% of the proton spin; the remainder is attributed to gluon spin and orbital angular momentum (Aidala et al. 2013).
Charge radius rp0.840 75(64) fmmeasured · 7.6 × 10⁻⁴rms charge radius (Mohr et al. 2025); the muonic-hydrogen (Antognini et al. 2013) and electron-scattering (Xiong et al. 2019) determinations are consistent.
Magnetic moment1.410 606 795 45(60) × 10⁻²⁶ J T⁻¹measured · 4.3 × 10⁻¹⁰2.792 847 344 63(82) nuclear magnetons, parallel to the spin (Mohr et al. 2025; Schneider et al. 2017).
g-factor5.585 694 689 3(16)measured · 2.9 × 10⁻¹⁰Departure from the Dirac value g = 2, a consequence of composite structure (Frisch & Stern 1933; Mohr et al. 2025).
Gyromagnetic ratio2.675 221 870 8(11) × 10⁸ s⁻¹ T⁻¹derived · 42.577 MHz T⁻¹Larmor precession frequency per unit field, γp/2π = 42.577 478 461(18) MHz T⁻¹ (Mohr et al. 2025).
Compton wavelength1.321 409 853 60(41) fmderived · h/mpcLength scale h/mpc (Mohr et al. 2025); not a spatial extent.
Quark contentu u dexact · quantum numbersValence quark charges +⅔, +⅔, −⅓; sea quarks and gluons carry the remaining momentum (Navas et al. 2024).
Lifetime (p → e⁺π⁰)> 2.4 × 10³⁴ yrlimit · 90 % CLTakenaka et al. (2020), Super-Kamiokande.
Sources · NIST CODATA 2022 (physics.nist.gov/constants) · PDG 2024 proton listing S016 · Takenaka et al. PRD 102, 112011 (2020) · Xiong et al. Nature 575, 147 (2019) · Antognini et al. Science 339, 417 (2013)
04 · Situations

Representative configurations

Six configurations computed from CODATA 2022 constants (Mohr et al. 2025): the hydrogen nucleus, deep inelastic electron scattering, proton–proton fusion in the solar core, the LHC beam, Larmor precession in a static field, and an extensive air shower. Each scene states any scale factor applied to the rendering.

05 · Latest research

Recent literature

Preprints and papers retrieved at page load, ordered by submission date. arXiv: categories hep-ex, hep-ph, nucl-ex, nucl-th and hep-lat, abstracts matching the proton charge radius, electromagnetic form factors, spin structure, parton distributions, deep inelastic scattering, the magnetic moment or proton decay. INSPIRE-HEP: the same terms in titles of high-energy-physics records. Dates are arXiv submission dates and INSPIRE record dates; no publisher issue dates are used.

Querying arXiv · INSPIRE-HEP …
DateSourceTitle and authorsReference
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06 · References

References

Declared representations
Valence quarks are illustrativeThe three points inside the hero are valence quarks moving within the confinement radius; their positions are illustrative, their colours cycle to show colour exchange.
The edge is an rms radiusThe 0.84 fm shell marks the rms charge radius (Mohr et al. 2025); the proton has no surface, and its density falls off smoothly.
The spin axis is a quantum numberNothing rotates. The axis shows the spin direction and its precession in a field.
Declared scalingWhere a physical ratio is below display resolution (the LHC bunch size, the fusion rate, atmospheric shower geometry), the applied scale factor is stated in the scene.
Sources
E. Rutherford (1919). Collision of α particles with light atoms IV. An anomalous effect in nitrogen. Phil. Mag. 37, 581.
L. Essen, R. W. Donaldson, M. J. Bangham & E. G. Hope (1971). Frequency of the hydrogen maser. Nature 229, 110.
E. G. Adelberger et al. (2011). Solar fusion cross sections. II. The pp chain and CNO cycles. Rev. Mod. Phys. 83, 195.
L. Evans & P. Bryant, eds. (2008). LHC Machine. J. Instrum. 3, S08001.
CERN (2022). LHC Run 3 begins: collisions at 13.6 TeV. CERN press release, 5 July 2022.
J. Matthews (2005). A Heitler model of extensive air showers. Astropart. Phys. 22, 387.
D. J. Bird et al. (1995). Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due to cosmic microwave radiation. Astrophys. J. 441, 144.
Telescope Array Collaboration (2023). An extremely energetic cosmic ray observed by a surface detector array. Science 382, 903.
G. Bressi et al. (2011). Testing the neutrality of matter by acoustic means in a spherical resonator. Phys. Rev. A 83, 052101.
C. A. Aidala, S. D. Bass, D. Hasch & G. K. Mallot (2013). The spin structure of the nucleon. Rev. Mod. Phys. 85, 655.
S. Navas et al., Particle Data Group (2024). Review of particle physics. Phys. Rev. D 110, 030001.
R. Frisch & O. Stern (1933). Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. Z. Phys. 85, 4.
R. Hofstadter & R. W. McAllister (1955). Electron scattering from the proton. Phys. Rev. 98, 217.
R. Hofstadter (1956). Electron scattering and nuclear structure. Rev. Mod. Phys. 28, 214.
E. D. Bloom et al. (1969); M. Breidenbach et al. (1969). High-energy inelastic e–p scattering at 6° and 10°. Phys. Rev. Lett. 23, 930; 935.
S. Dürr et al., BMW (2008). Ab initio determination of light hadron masses. Science 322, 1224.
A. Antognini et al. (2013). Proton structure from the measurement of 2S–2P transition frequencies of muonic hydrogen. Science 339, 417.
W. Xiong et al., PRad (2019). A small proton charge radius from an electron–proton scattering experiment. Nature 575, 147.
A. Takenaka et al., Super-Kamiokande (2020). Search for proton decay via p → e⁺π⁰ and p → μ⁺π⁰. Phys. Rev. D 102, 112011.
G. Schneider et al., BASE (2017). Double-trap measurement of the proton magnetic moment at 0.3 parts per billion precision. Science 358, 1081.
P. J. Mohr et al. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97, 025002 · physics.nist.gov/constants · PDG proton listing.