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Particle unification: established theory and open proposals

Derive the gauge-theory framework, electroweak mixing and running couplings, then examine what grand unification and quantum gravity would need to explain.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Gauge symmetry and the Standard Model

Definitions & inputs. g coupling,T generators,Aμ gauge fields,ψ matter field,Fμν field strength.

  1. A gauge connection compensates local changes of the matter-field basis.

    Dμ=∂μ−igAμaTaD_\mu=\partial_\mu-igA_\mu^aT^a

  2. The commutator defines field strength including non-Abelian self-interactions.

    [Dμ,Dν]=−igFμνaTa[D_\mu,D_\nu]=-igF_{\mu\nu}^aT^a

  3. Gauge, matter, Higgs and Yukawa terms form the schematic structure.

    L=−14FμνaFaμν+ψˉiγμDμψ+LH+LY\mathcal L=-\tfrac14F_{\mu\nu}^aF^{a\mu\nu}+\bar\psi i\gamma^\mu D_\mu\psi+\mathcal L_H+\mathcal L_Y

Interpretation. The Standard Model uses SU(3)c×SU(2)L×U(1)Y, a product group rather than a single simple grand-unified gauge group. Gravity is not included.

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2. Electroweak mixing and masses

Definitions & inputs. g weakSU2 coupling,gprime hypercharge coupling,v Higgs expectation value,θW weak angle.

  1. Insert the Higgs expectation value into its kinetic term to obtain the neutral gauge mass matrix.

    ⟨H⟩=(0,v/2)T,∣DμH∣2⊃v28(gWμ3−g′Bμ)2\langle H\rangle=(0,v/\sqrt2)^{\mathsf T},\quad |D_\mu H|^2\supset\frac{v^2}{8}(gW_\mu^3-g'B_\mu)^2

  2. Rotate into the massless photon and massive neutral boson.

    Aμ=sin⁡θWWμ3+cos⁡θWBμ,Zμ=cos⁡θWWμ3−sin⁡θWBμA_\mu=\sin\theta_WW_\mu^3+\cos\theta_WB_\mu,\quad Z_\mu=\cos\theta_WW_\mu^3-\sin\theta_WB_\mu

  3. Diagonalization fixes the tree-level mixing and masses; the photon remains massless.

    tan⁡θW=g′/g,e=gsin⁡θW,mW=gv/2,mZ=vg2+g′2/2\tan\theta_W=g'/g,\quad e=g\sin\theta_W,\quad m_W=gv/2,\quad m_Z=v\sqrt{g^2+g'^2}/2

Interpretation. Electroweak unification does not imply electromagnetism and the weak force have equal low-energy behavior.

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3. Running couplings and grand unification

Definitions & inputs. αi=gi²/(4π),bi one-loop coefficients,μ renormalization scale,Λ candidate unification scale.

  1. Quantum corrections change effective couplings with scale under this sign convention.

    dgidln⁡μ=bigi316π2\frac{dg_i}{d\ln\mu}=\frac{b_i g_i^3}{16\pi^2}

  2. Differentiate1/α and integrate whilebi stays fixed.

    αi−1(μ)=αi−1(μ0)−bi2πln⁡(μ/μ0)\alpha_i^{-1}(\mu)=\alpha_i^{-1}(\mu_0)-\frac{b_i}{2\pi}\ln(\mu/\mu_0)

  3. A pairwise crossing can be solved algebraically; all relevant couplings and thresholds must agree for a unification claim.

    ln⁡(Λ/μ0)=2π[αi−1(μ0)−αj−1(μ0)]bi−bj\ln(\Lambda/\mu_0)=\frac{2\pi[\alpha_i^{-1}(\mu_0)-\alpha_j^{-1}(\mu_0)]}{b_i-b_j}

Interpretation. SU5, SO10 and related proposals embed gauge and matter representations. A crossing alone does not establish a GUT; proton decay, neutrinos, symmetry breaking and threshold corrections are independent tests.

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4. Effective operators, neutrinos and gravity

Definitions & inputs. Λ heavy scale,Od operator of mass dimension d,cd dimensionless coefficient,GFermi gravitational Newton constant G in Planck definitions.

  1. Symmetries organize possible effects of unresolved high-energy degrees of freedom.

    LEFT=LSM+∑d>4cdΛd−4Od\mathcal L_{EFT}=\mathcal L_{SM}+\sum_{d>4}\frac{c_d}{\Lambda^{d-4}}\mathcal O_d

  2. A simple heavy-neutrino seesaw and dimension-six decay scaling illustrate distinct potential probes, not universal predictions.

    mν∼y2v2/(2M),Γp∼mp5/Λ4m_\nu\sim y^2v^2/(2M),\quad\Gamma_p\sim m_p^5/\Lambda^4

  3. Dimensional combinations identify a scale at which quantum gravitational effects may be important; they do not derive a theory of everything.

    ℓP=ℏG/c3,EP=ℏc5/G\ell_P=\sqrt{\hbar G/c^3},\quad E_P=\sqrt{\hbar c^5/G}

Interpretation. String theory seeks quantum gravity with additional consistency structure; loop quantum gravity quantizes geometric degrees of freedom and is not by itself a particle GUT; asymptotic safety studies ultraviolet fixed points. None should be labeled experimentally confirmed unification.

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Graphical worked example

Illustrative one-loop running with b=−7 and α⁻¹(μ0)=10. This is not a demonstrated unification crossing. X axis: Scale ratio μ/μ0 (dimensionless). Y axis: Inverse coupling α⁻¹ (dimensionless).
Illustrative one-loop running with b=−7 and α⁻¹(μ0)=10. This is not a demonstrated unification crossing. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. SU2 generator count

Definitions & inputs. n2.

  1. Choose the governing model and isolate the requested quantity.

    dim⁡SU(n)=n2−1\dim SU(n)=n^2-1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    22−12^2-1

  3. Evaluate the expression; the result uses the units shown.

    Result=3 \mathrm{Result}=3\ {}

Interpretation. Gauge algebra count.

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Example 02. SU3 generator count

Definitions & inputs. n3.

  1. Choose the governing model and isolate the requested quantity.

    n2−1n^2-1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    32−13^2-1

  3. Evaluate the expression; the result uses the units shown.

    Result=8 \mathrm{Result}=8\ {}

Interpretation. Eight gluon gauge fields.

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Example 03. Standard Model gauge count

Definitions & inputs. SU3,SU2,U1.

  1. Choose the governing model and isolate the requested quantity.

    8+3+18+3+1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1212

  3. Evaluate the expression; the result uses the units shown.

    Result=12 \mathrm{Result}=12\ {}

Interpretation. Before symmetry-breaking field rotation.

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Example 04. SU5 generator count

Definitions & inputs. n5.

  1. Choose the governing model and isolate the requested quantity.

    52−15^2-1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2424

  3. Evaluate the expression; the result uses the units shown.

    Result=24 \mathrm{Result}=24\ {}

Interpretation. Hypothetical gauge embedding.

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Example 05. ExtraSU5 generators

Definitions & inputs. 24minus12SM.

  1. Choose the governing model and isolate the requested quantity.

    24−1224-12

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1212

  3. Evaluate the expression; the result uses the units shown.

    Result=12 \mathrm{Result}=12\ {}

Interpretation. Counting alone does not specify their masses.

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Example 06. SO10 generator count

Definitions & inputs. n10.

  1. Choose the governing model and isolate the requested quantity.

    n(n−1)/2n(n-1)/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10(9)/210(9)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=45 \mathrm{Result}=45\ {}

Interpretation. Lie algebra dimension.

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Example 07. Electron charge

Definitions & inputs. Left lepton T3−.5,Y−.5.

  1. Choose the governing model and isolate the requested quantity.

    Q=T3+YQ=T_3+Y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −.5−.5-.5-.5

  3. Evaluate the expression; the result uses the units shown.

    Result=−1 \mathrm{Result}=-1\ {}

Interpretation. Charge in units of positive e.

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Example 08. Neutrino charge

Definitions & inputs. T3+.5,Y−.5.

  1. Choose the governing model and isolate the requested quantity.

    Q=T3+YQ=T_3+Y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5−.5.5-.5

  3. Evaluate the expression; the result uses the units shown.

    Result=0 \mathrm{Result}=0\ {}

Interpretation. Same hypercharge doublet.

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Example 09. Up-quark charge

Definitions & inputs. T3+.5,Y1/6.

  1. Choose the governing model and isolate the requested quantity.

    Q=T3+YQ=T_3+Y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5+1/6.5+1/6

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6666667 \mathrm{Result}=0.6666667\ {}

Interpretation. Left quark doublet convention.

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Example 10. Down-quark charge

Definitions & inputs. T3−.5,Y1/6.

  1. Choose the governing model and isolate the requested quantity.

    Q=T3+YQ=T_3+Y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −.5+1/6-.5+1/6

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.3333333 \mathrm{Result}=-0.3333333\ {}

Interpretation. Same convention.

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Example 11. Weak angle

Definitions & inputs. Illustrative g.65,gprime.36.

  1. Choose the governing model and isolate the requested quantity.

    θW=arctan⁡(g′/g)\theta_W=\arctan(g'/g)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    arctan⁡(.36/.65)\arctan(.36/.65)

  3. Evaluate the expression; the result uses the units shown.

    Result=28.97971 deg\mathrm{Result}=28.97971\ \mathrm{deg}

Interpretation. Tree-level toy inputs.

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Example 12. Electromagnetic coupling

Definitions & inputs. Same couplings.

  1. Choose the governing model and isolate the requested quantity.

    e=gg′/g2+g′2e=gg'/\sqrt{g^2+g'^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .65(.36)/.652+.362.65(.36)/\sqrt{.65^2+.36^2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3149249 \mathrm{Result}=0.3149249\ {}

Interpretation. Natural-unit coupling.

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Example 13. W mass

Definitions & inputs. g.65,v246GeV.

  1. Choose the governing model and isolate the requested quantity.

    mW=gv/2m_W=gv/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .65(246)/2.65(246)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=79.95 GeV\mathrm{Result}=79.95\ \mathrm{GeV}

Interpretation. Tree-level exercise, not a precision mass prediction.

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Example 14. Z mass

Definitions & inputs. g.65,gprime.36,v246GeV.

  1. Choose the governing model and isolate the requested quantity.

    mZ=vg2+g′2/2m_Z=v\sqrt{g^2+g'^2}/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    123.652+.362123\sqrt{.65^2+.36^2}

  3. Evaluate the expression; the result uses the units shown.

    Result=91.39322 GeV\mathrm{Result}=91.39322\ \mathrm{GeV}

Interpretation. Same illustrative inputs.

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Example 15. Tree mass ratio

Definitions & inputs. Same couplings.

  1. Choose the governing model and isolate the requested quantity.

    mW/mZ=g/g2+g′2m_W/m_Z=g/\sqrt{g^2+g'^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .65/.652+.362.65/\sqrt{.65^2+.36^2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8747914 \mathrm{Result}=0.8747914\ {}

Interpretation. Equals cosθW at tree level.

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Example 16. Hypercharge normalization

Definitions & inputs. αY.01.

  1. Choose the governing model and isolate the requested quantity.

    α1=(5/3)αY\alpha_1=(5/3)\alpha_Y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (5/3)(.01)(5/3)(.01)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.01666667 \mathrm{Result}=0.01666667\ {}

Interpretation. Conventional SU5 normalization.

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Example 17. One-loop inverse coupling

Definitions & inputs. αinv0=10,b=−7,μ/μ0=10.

  1. Choose the governing model and isolate the requested quantity.

    α−1=10−bln⁡10/(2π)\alpha^{-1}=10-b\ln10/(2\pi)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10+7ln⁡10/(2π)10+7\ln10/(2\pi)

  3. Evaluate the expression; the result uses the units shown.

    Result=12.56527 \mathrm{Result}=12.56527\ {}

Interpretation. Fixed field content and sign convention.

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Example 18. Dimension-six amplitude suppression

Definitions & inputs. E100GeV,Λ10000GeV,c1.

  1. Choose the governing model and isolate the requested quantity.

    (E/Λ)2(E/\Lambda)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (100/10000)2(100/10000)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0001 \mathrm{Result}=0.0001\ {}

Interpretation. Scaling only; matrix elements can change observable effects.

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Example 19. Seesaw mass scale

Definitions & inputs. y.1,v246GeV,M10¹⁴GeV.

  1. Choose the governing model and isolate the requested quantity.

    mν=y2v2/(2M)m_\nu=y^2v^2/(2M)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .12(246)2/(2×1014)×109.1^2(246)^2/(2\times10^{14})\times10^9

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0030258 eV\mathrm{Result}=0.0030258\ \mathrm{eV}

Interpretation. GeV converted toeV; one-generation illustrative heavy-neutrino model.

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Example 20. Decay lifetime scaling

Definitions & inputs. Increase heavyΛby factor2,all other parameters fixed.

  1. Choose the governing model and isolate the requested quantity.

    τ′/τ=(Λ′/Λ)4\tau'/\tau=(\Lambda'/\Lambda)^4

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    242^4

  3. Evaluate the expression; the result uses the units shown.

    Result=16 \mathrm{Result}=16\ {}

Interpretation. Dimension-six scaling; no numerical proton lifetime is inferred.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.