Open questions in particle physics

The Standard Model is the most precisely tested theory in the history of science, and also, by broad consensus, unfinished. The tension is not that any laboratory measurement contradicts it — none does, to the parts-per-billion of the electron's magnetic moment — but that the theory is silent on questions the universe plainly answers. It offers no particle for the cold dark matter that outweighs ordinary matter five to one across every galaxy and cluster. It cannot explain why the cosmos is made of matter and not equal parts matter and antimatter. And it contains numbers whose smallness looks less like accident than clue: a Higgs mass held light against enormous quantum corrections, three gauge couplings that nearly meet at a single high energy, fermion masses spread across twelve orders of magnitude with no organizing principle. These puzzles fall into two families. Some are empirical — dark matter and the baryon asymmetry are facts of observation with no Standard-Model account, and increasingly it is cosmology, not colliders, that forces the issue. Others are structural — the hierarchy problem and gauge unification are not contradictions but signs that the theory is a low-energy effective description of something deeper. This article surveys that frontier: it writes down the quantitative statement that makes each question sharp, and shows how a handful of proposed answers — a stable weakly interacting relic, supersymmetry, a grand unified group — recur across the list, so that several puzzles may share a single solution.

Key takeaways

The Standard Model passes every laboratory test yet cannot supply dark matter, generate the matter–antimatter asymmetry, or explain why the Higgs stays light — each demands physics beyond it.
Flat galactic rotation curves and the Planck cosmic budget $\Omega_b:\Omega_c:\Omega_\Lambda\simeq0.05:0.26:0.69$ show dark matter outweighs baryons about $5:1$ and must be cold (non-relativistic).
Quantum corrections to the Higgs mass grow as the cutoff squared, $\delta m_H^2\sim\Lambda^2/8\pi^2$ ; keeping $m_H=125$ GeV against $\Lambda=M_{\rm Pl}$ needs cancellation to one part in $10^{32}$ — the hierarchy problem.
Run the three gauge couplings up and they nearly meet; adding TeV-scale supersymmetry makes them meet cleanly at $M_{\rm GUT}\approx2\times10^{16}$ GeV with $\alpha_{\rm GUT}^{-1}\approx24$ — the quantitative case for grand unification.
Grand unified theories predict proton decay ( $p\to e^+\pi^0$ ); minimal SU(5) gives $\tau_p\sim10^{31}$ yr and is excluded, while SUSY-GUTs push $\tau_p\sim10^{34}$ – $10^{36}$ yr, at today's experimental edge.
The baryon-to-photon ratio $\eta\simeq6\times10^{-10}$ requires Sakharov's three conditions; the Standard Model has all three in principle but its CP violation is far too small.
Key idea: the Standard Model works superbly but leaves dark matter, the unnatural lightness of the Higgs, the near-convergence of the forces, and the matter–antimatter asymmetry unexplained — each demanding new physics, and several of these puzzles (dark matter as a stable supersymmetric relic tied to gauge unification) may share a single solution.

The core insight

The Standard Model works superbly but leaves dark matter, the unnatural lightness of the Higgs, the apparent convergence of the forces, and the matter–antimatter asymmetry unexplained — each demanding new physics, and several of these puzzles (for example dark matter as a stable supersymmetric neutralino, tied to gauge-coupling unification) may share a single solution.

Where this lives today

The open questions define the entire experimental frontier. Dark matter is hunted three ways: directly, in ultra-quiet underground detectors like LUX-ZEPLIN, whose 2024 results push the WIMP–nucleon cross section below $10^{-47}\ \text{cm}^2$ near $30$ GeV and are closing the classic weak-scale window; indirectly, via annihilation photons and antimatter with Fermi-LAT and AMS; and by production at the LHC, where missing-energy signatures would betray dark particles. The same LHC has searched exhaustively for supersymmetry and found none below roughly a TeV, tightening the naturalness tension the theory was invoked to relieve. Proton decay is watched by Super-Kamiokande, whose non-observation sets $\tau(p\to e^+\pi^0)>2.4\times10^{34}$ yr, excluding minimal SU(5); its successor Hyper-Kamiokande will extend the reach nearly tenfold. The baryon asymmetry drives a global CP-violation programme — LHCb, Belle II, and the long-baseline experiments DUNE and Hyper-K seeking leptonic CP violation that could source the excess through leptogenesis. And the muon's anomalous magnetic moment, measured at Fermilab to sub-ppm precision, remains a live discrepancy that may be the first crack. None has yet produced a discovery beyond the Standard Model — itself the central fact of contemporary particle physics.

The physical picture

The most direct evidence that the Standard Model is incomplete does not come from an accelerator at all — it comes from watching stars orbit. Newtonian gravity is unambiguous: if a galaxy's mass tracked its light, concentrated in the luminous bulge and disk, then a star far out in the sparse halo should orbit like a planet around the Sun, with speed falling as $v\propto1/\sqrt{r}$ . Vera Rubin's spectra of spiral galaxies in the 1970s showed the opposite: the orbital speed rises and then stays stubbornly flat thousands of parsecs beyond the last visible star. A flat curve is not a small anomaly to be explained away; read through Newton it says the enclosed mass keeps growing, $M(r)\propto r$ , in a region that is optically empty. The first figure is that measurement — the single cleanest picture of physics beyond the Standard Model.

Figure

Galactic rotation curve. If mass tracked the starlight, the orbital speed beyond the luminous disk would fall off as v ∝ 1/√r (Keplerian, amber dashed). Instead v stays flat near 210 km/s out to 30 kpc and beyond (blue), forcing the enclosed mass to keep growing as M(r) = v²r/G ∝ r — a dark halo outweighing the visible galaxy by roughly 5 to 10 times

The structural puzzles are harder to photograph but just as concrete. The one that has driven theory for forty years is the hierarchy problem: the Higgs is a scalar, and a scalar's mass receives quantum corrections that scale with the square of whatever cutoff the theory runs up to. If the Standard Model holds to the Planck scale, those corrections dwarf the observed $m_H=125$ GeV, and the bare parameter must be tuned against them to thirty-two decimal places. Supersymmetry offers an elegant escape — a boson loop for every fermion loop, opposite in sign, cancelling the dangerous term. The same superpartners, if they exist near the TeV scale, do something else remarkable: they bend the running gauge couplings so all three meet at one energy. The second figure is that convergence, the quantitative hint of grand unification a whole generation of experiments was built to test.

Figure

Running of the three inverse gauge couplings α₁⁻¹ (blue), α₂⁻¹ (amber), α₃⁻¹ (green) from M_Z upward. Starting from α₁⁻¹ = 59.0, α₂⁻¹ = 29.6, α₃⁻¹ = 8.5 at M_Z = 91 GeV, the Standard-Model slopes (grey dashed) miss one another; adding TeV-scale supersymmetry bends them so all three meet at M_GUT ≈ 2×10¹⁶ GeV with α_GUT⁻¹ ≈ 24 — the quantitative signature of grand unification

The core equations

Each open question becomes sharp only when it is written as a number the Standard Model cannot produce. We collect those statements here, one per puzzle, so the rest of the article can derive and test them. The first is the rotation-curve argument for dark matter — Newtonian, model-independent, and decisive.

v(r)=\sqrt{\frac{G\,M(r)}{r}}\ \xrightarrow{\ v=\text{const}\ }\ M(r)=\frac{v^2 r}{G}\propto r \qquad (1)

where: $v(r)$ — circular orbital speed at radius $r$ ; $M(r)$ — total mass enclosed within $r$ ; $G$ — Newton's constant. Reading: if light traced mass, $v$ would fall as $r^{-1/2}$ (Keplerian) beyond the luminous disk; a flat $v$ instead forces $M(r)\propto r$ and hence a density $\rho\propto1/r^2$ — a dark halo. The rotation curve fixes dark matter galaxy by galaxy; the cosmic microwave background fixes it for the universe as a whole. The precision fit of the $\Lambda$ CDM model to the CMB power spectrum partitions the total energy density into three pieces.

\Omega_b:\Omega_c:\Omega_\Lambda\simeq0.05:0.26:0.69,\qquad \Omega_{\rm tot}\simeq1 \qquad (2)

where: $\Omega_b$ — baryon density parameter; $\Omega_c$ — cold dark matter; $\Omega_\Lambda$ — dark energy; each measured in units of the critical density $\rho_c=3H_0^2/8\pi G$ . Reading: ordinary matter is one-twentieth of the budget, dark matter outweighs it roughly $5:1$ , and structure formation demands the dark component be cold. The Higgs mass is the sharpest structural puzzle. Because the Higgs is a fundamental scalar, its self-energy is quadratically sensitive to the highest scale $\Lambda$ at which the theory holds — unlike a fermion mass, which is only logarithmically sensitive.

\delta m_H^2\sim\frac{\lambda_t^2}{8\pi^2}\,\Lambda^2,\qquad \frac{m_H^2}{\Lambda^2}\sim\left(\frac{125\ \text{GeV}}{10^{19}\ \text{GeV}}\right)^2\sim10^{-34} \qquad (3)

where: $\delta m_H^2$ — one-loop correction to the Higgs mass-squared, dominated by the top ( $\lambda_t\approx1$ ); $\Lambda$ — ultraviolet cutoff. Reading: keeping $m_H$ at the electroweak scale forces the bare mass to cancel $\delta m_H^2$ to dozens of decimal places — the unnaturalness motivating new physics near a TeV. Whether the forces unify is a question about how the couplings run. At one loop each inverse coupling is linear in $\ln\mu$ , with a slope set entirely by the particle content through the beta coefficients $b_i$ .

\alpha_i^{-1}(\mu)=\alpha_i^{-1}(M_Z)-\frac{b_i}{2\pi}\ln\frac{\mu}{M_Z},\qquad i=1,2,3 \qquad (4)

where: $\alpha_1=\tfrac53\alpha_Y$ (GUT-normalized hypercharge), $\alpha_2$ (weak $SU(2)$ ), $\alpha_3$ (strong $SU(3)$ ); $b_i$ — one-loop beta coefficients; $M_Z=91.19$ GeV. Reading: three straight lines in $\ln\mu$ generically do not cross at one point, so a common intersection is a real, content-dependent prediction — and supersymmetric content makes it happen. Unification is not free: a group large enough to contain the Standard Model rotates quarks into leptons, allowing the proton to decay through exchange of a superheavy gauge boson $X$ .

\tau_p\sim\frac{M_X^4}{\alpha_{\rm GUT}^2\,m_p^5} \qquad (5)

where: $M_X\sim M_{\rm GUT}$ — leptoquark gauge-boson mass; $\alpha_{\rm GUT}$ — unified coupling; $m_p=0.938$ GeV — proton mass. Reading: the rate is suppressed by four powers of $M_X$ , so the lifetime is astronomically long but finite — a falsifiable prediction distinguishing unified models. Finally, the matter–antimatter asymmetry is a single small number extracted from both big-bang nucleosynthesis and the CMB.

\eta\equiv\frac{n_B-n_{\bar B}}{n_\gamma}\simeq6\times10^{-10} \qquad (6)

where: $n_B,\,n_{\bar B},\,n_\gamma$ — number densities of baryons, antibaryons, and photons today. Reading: for every $\sim1.6$ billion photons there is one net baryon and essentially no antimatter — a nonzero excess that a matter–antimatter-symmetric early universe plus the Standard Model cannot generate, requiring Sakharov's three conditions.

Derivation AFlat rotation curves demand a dark halo

The goal is to turn the qualitative observation "rotation curves are flat" into the quantitative statement that a galaxy must contain far more mass than its stars, distributed as an extended halo with density $\rho\propto1/r^2$ . We use only Newtonian gravity and circular-orbit balance — no assumption about what the dark matter is — and follow the logic from the Keplerian expectation, through the flat-curve data, to the enclosed-mass and density profiles that the data force.

Step 1. Balance gravity against the centripetal requirement for a star on a circular orbit: $GM(r)/r^2=v^2/r$ , so $v(r)=\sqrt{GM(r)/r}$ . If essentially all the mass sits inside the luminous disk, then for a star well outside it $M(r)$ is constant and $v\propto r^{-1/2}$ — the same Keplerian falloff that makes Neptune orbit slower than Mercury. Why: this is the null hypothesis "mass follows light", and it makes a specific, falsifiable prediction for the outer curve.

Step 2. Confront the data. Measured curves do not fall; beyond the disk they are flat, $v(r)\approx v_0\approx200$ – $250$ km/s, out to the largest radii traced by neutral-hydrogen 21-cm emission — often several times the optical radius. Why: a flat curve is qualitatively incompatible with the Keplerian prediction, so the premise "mass follows light" must be wrong; there is unseen mass.

Step 3. Invert for the enclosed mass. Setting $v=v_0$ constant in Step 1 gives $M(r)=v_0^2\,r/G\propto r$ — the mass keeps growing linearly with radius even where no stars are seen. Why: this is the model-independent core of the argument; it converts a velocity measurement directly into a statement about how much matter is present and where.

Step 4. Get the density profile. Since $M(r)=\int_0^r 4\pi r'^2\rho(r')\,dr'\propto r$ , differentiating gives $4\pi r^2\rho(r)\propto1$ , so $\rho(r)\propto1/r^2$ — an extended isothermal halo, unlike the sharply peaked starlight, with total mass $5$ – $10\times$ the luminous mass. Why: $\rho\propto1/r^2$ is the fingerprint of a diffuse dark component, and its mass is what the cosmic budget of equation (2) must account for.

Go deeper · Derivation B · 1 minThe hierarchy problem and its supersymmetric cancellation

Derivation BThe hierarchy problem and its supersymmetric cancellation

The aim is to show precisely why the Higgs mass is unnatural — its quantum correction grows as the square of the cutoff, $\delta m_H^2\sim\Lambda^2$ , not logarithmically — and then how pairing every fermion with two complex scalars of matched coupling cancels the quadratic term exactly, the structural reason supersymmetry solves the problem and why superpartners are expected near the TeV scale. We compute the leading fermion- and scalar-loop contributions and impose the supersymmetric relation between their couplings.

Step 1. Compute the fermion loop. A Dirac fermion of Yukawa coupling $\lambda_f$ coupling to the Higgs contributes to its self-energy a term $\delta m_H^2\big|_f=-\dfrac{\lambda_f^2}{8\pi^2}\,\Lambda^2+\dots$ , quadratically divergent and negative. For the top quark, $\lambda_t\approx1$ , this dominates. Why: a scalar mass has no symmetry protecting it — unlike a chiral fermion mass or a gauge-boson mass — so nothing forbids the quadratic dependence, and it is genuinely there.

Step 2. Compute the scalar loop. A complex scalar of quartic coupling $\lambda_S$ to the Higgs contributes $\delta m_H^2\big|_S=+\dfrac{\lambda_S}{16\pi^2}\,\Lambda^2+\dots$ , quadratically divergent but positive — opposite in sign to the fermion. Why: the relative minus sign between fermion and boson loops is the fixed fact of quantum field theory that any cancellation must exploit.

Step 3. Impose supersymmetry. Give each Dirac fermion two complex scalar partners with $\lambda_S=\lambda_f^2$ . Then the two scalar loops contribute $2\times\dfrac{\lambda_f^2}{16\pi^2}\Lambda^2=+\dfrac{\lambda_f^2}{8\pi^2}\Lambda^2$ , exactly cancelling the fermion loop of Step 1. The $\Lambda^2$ terms sum to zero. Why: supersymmetry is precisely the symmetry that ties boson and fermion couplings together in the ratio needed for this cancellation to be automatic rather than accidental.

Step 4. Read off the residual. If supersymmetry were exact the cancellation would be perfect, but partners are not degenerate; the leftover is $\delta m_H^2\sim\dfrac{\lambda_f^2}{16\pi^2}\,(m_S^2-m_f^2)\ln(\Lambda/m_S)$ — only logarithmically divergent and set by the superpartner splitting. Naturalness ( $\delta m_H^2\lesssim m_H^2$ ) then requires $m_S-m_f\lesssim\mathcal{O}(1\ \text{TeV})$ . Why: this is the prediction that made the LHC the natural place to look — if supersymmetry stabilizes the Higgs, its partners should sit within collider reach.

Go deeper · Derivation C · 1 minGauge-coupling unification from the one-loop RGE

Derivation CGauge-coupling unification from the one-loop RGE

The goal is to show that grand unification is a genuine quantitative claim, not a hope: given the couplings measured at $M_Z$ and a particle content fixing the coefficients $b_i$ , the requirement that all three inverse couplings meet at one scale is generically overdetermined, and that supersymmetric content satisfies it — with $M_{\rm GUT}\approx2\times10^{16}$ GeV and $\alpha_{\rm GUT}^{-1}\approx24$ — is the strongest circumstantial evidence for both unification and TeV supersymmetry. We write the running solution, count constraints against unknowns, and identify what makes the lines coincide.

Step 1. Write the running. Integrating the one-loop renormalization-group equation gives $\alpha_i^{-1}(\mu)=\alpha_i^{-1}(M_Z)-\dfrac{b_i}{2\pi}\ln(\mu/M_Z)$ : three straight lines when plotted against $\ln\mu$ , with intercepts fixed by experiment and slopes fixed by $b_i$ . Why: at one loop the running decouples line by line, so unification becomes a question of pure geometry — do three given lines share a point.

Step 2. Count the bookkeeping. Unification imposes two conditions, $\alpha_1^{-1}=\alpha_2^{-1}$ and $\alpha_2^{-1}=\alpha_3^{-1}$ , on two unknowns, the scale $M_{\rm GUT}$ and the common value $\alpha_{\rm GUT}^{-1}$ . Two of the three lines can always be made to cross somewhere; whether the third passes through the same point is a nontrivial prediction. Why: this is why unification is falsifiable — a generic $b_i$ produces a small triangle, not a point, and the data decide which.

Step 3. Try the Standard Model. With $b_i=(41/10,-19/6,-7)$ the lines miss: $\alpha_1$ and $\alpha_2$ cross near $10^{13}$ GeV, but $\alpha_3^{-1}$ passes several units away there — no common point. Why: the minimal Standard Model nearly unifies, tantalizingly, but the failure by a few percent is real and is exactly the gap new physics must close.

Step 4. Add supersymmetry. The superpartners change the running above the TeV scale to $b_i=(33/5,1,-3)$ , and now the three lines converge to a single point at $M_{\rm GUT}\approx2\times10^{16}$ GeV with $\alpha_{\rm GUT}^{-1}\approx24$ , the scheme also predicting $\sin^2\theta_W$ well. Why: that content motivated independently by the hierarchy problem also fixes unification is why supersymmetric grand unification became the leading beyond-Standard-Model paradigm.

Limiting cases

Case	Result	Meaning
Mass follows light	$v\propto r^{-1/2}$	Keplerian falloff — the null hypothesis, ruled out by flat curves
Flat rotation curve	$M(r)\propto r,\ \rho\propto1/r^2$	extended dark halo, $\sim5$ – $10\times$ the luminous mass
Cutoff $\Lambda\to M_Z$	$\delta m_H^2\lesssim m_H^2$	no hierarchy problem if the theory ends near the electroweak scale
Cutoff $\Lambda\to M_{\rm Pl}$	$\delta m_H^2/m_H^2\sim10^{32}$	maximal fine-tuning — the naturalness case for new TeV physics
Exact supersymmetry	$\delta m_H^2=0$	boson and fermion loops cancel identically; but no superpartners seen
Broken SUSY, $m_S\lesssim$ TeV	$\delta m_H^2\sim(m_S^2)\ln\Lambda$	log-divergent, natural — the target of LHC searches
Standard-Model running	couplings miss by a few percent	near-unification, but no single meeting point
MSSM running	meet at $2\times10^{16}$ GeV, $\alpha_{\rm GUT}^{-1}\approx24$	quantitative grand unification
Minimal SU(5), $M_X\sim10^{15}$ GeV	$\tau_p\sim10^{31}$ yr	excluded by Super-Kamiokande

Common trapNeutrinos are the dark matter

They are not, and this was the first candidate ruled out. Neutrinos are dark and present — a cosmic sea fills space — but they fail the decisive test: they are hot, not cold. Being nearly massless, relic neutrinos stayed relativistic through the epoch when galaxies began to seed, and relativistic particles free-stream out of small density perturbations, erasing them. A neutrino-dominated universe would build the largest structures first and fragment them into galaxies late — the opposite of the observed "bottom-up" growth confirmed by galaxy surveys and the CMB. Cosmology also caps the total: $\sum m_\nu\lesssim0.1$ eV gives $\Omega_\nu\lesssim0.005$ , an order of magnitude too little even if it were cold. So the dark matter must be cold, which is why the leading candidates — a WIMP or an axion — are heavy or born cold, never a Standard-Model neutrino. A separate trap: dark matter is not dark energy. Dark matter clumps and gravitates with $\Omega_c\simeq0.26$ ; dark energy, $\Omega_\Lambda\simeq0.69$ , is smooth, drives accelerated expansion, and is a distinct open question.

Worked examples

1 · Where do the forces unify? (fully worked)

Take the GUT-normalized inverse couplings at $M_Z=91.19$ GeV to be $\alpha_1^{-1}=59.0$ , $\alpha_2^{-1}=29.6$ , $\alpha_3^{-1}=8.5$ , and the supersymmetric beta coefficients $b_i=(33/5,\,1,\,-3)$ . Find the unification scale $M_{\rm GUT}$ and the unified coupling $\alpha_{\rm GUT}^{-1}$ , and confirm that the third line passes through the same point.

Step 1. Impose $\alpha_1^{-1}=\alpha_2^{-1}$ using equation (4). Writing $L=\ln(\mu/M_Z)$ , the condition is $59.0-\dfrac{6.6}{2\pi}L=29.6-\dfrac{1}{2\pi}L$ , i.e. $29.4=\dfrac{6.6-1}{2\pi}L=\dfrac{5.6}{2\pi}L$ . Solving, $L=\dfrac{29.4\times2\pi}{5.6}=33.0$ .

Step 2. Convert to an energy. $M_{\rm GUT}=M_Z\,e^{L}=91.19\times e^{33.0}=91.19\times2.1\times10^{14}=\mathbf{1.9\times10^{16}\ \text{GeV}}$ . This sits about two-and-a-half orders of magnitude below the Planck scale — comfortably in the regime where quantum gravity can be neglected.

Step 3. Evaluate the common coupling. $\alpha_{\rm GUT}^{-1}=\alpha_2^{-1}(\mu)=29.6-\dfrac{1}{2\pi}(33.0)=29.6-5.25=\mathbf{24.3}$ . A perturbative value ( $\alpha_{\rm GUT}\approx1/24$ ), so the one-loop treatment is self-consistent.

Step 4. Check the third line. The strong coupling at the same scale is $\alpha_3^{-1}=8.5-\dfrac{-3}{2\pi}(33.0)=8.5+15.75=24.25\approx24.3$ . All three inverse couplings meet at $M_{\rm GUT}\approx2\times10^{16}$ GeV with $\alpha_{\rm GUT}^{-1}\approx24$ — the agreement of the third line to better than one part in a hundred is the non-trivial success. Repeating with the Standard-Model $b_i$ instead makes $\alpha_3^{-1}$ land near $37$ at the $\alpha_1$ – $\alpha_2$ crossing, missing by a wide margin.

2 · The WIMP miracle (fully worked)

A stable particle in thermal equilibrium in the early universe freezes out with a relic abundance $\Omega_\chi h^2\approx\dfrac{3\times10^{-27}\ \text{cm}^3\,\text{s}^{-1}}{\langle\sigma v\rangle}$ . Show that a weak-scale annihilation cross section reproduces the observed cold-dark-matter density $\Omega_c h^2=0.12$ .

Step 1. Estimate the cross section of a weakly interacting particle of mass $m_\chi\sim100$ GeV. Dimensionally $\langle\sigma v\rangle\sim\dfrac{\pi\alpha_W^2}{m_\chi^2}$ ; with $\alpha_W\sim1/30$ this gives $\langle\sigma v\rangle\sim3\times10^{-26}\ \text{cm}^3\,\text{s}^{-1}$ — the generic size of a weak annihilation rate.

Step 2. Feed it into the freeze-out formula. $\Omega_\chi h^2\approx\dfrac{3\times10^{-27}}{3\times10^{-26}}=\mathbf{0.1}$ , within a factor of order unity of the measured $\Omega_c h^2=0.12$ .

Step 3. Turn it around. To hit $0.12$ exactly needs $\langle\sigma v\rangle=\dfrac{3\times10^{-27}}{0.12}=2.5\times10^{-26}\ \text{cm}^3\,\text{s}^{-1}$ — again the weak scale. A particle with a weak-scale mass and weak-strength couplings automatically freezes out with roughly the observed dark-matter density: the "WIMP miracle." That such a relic is also what supersymmetry predicts (the lightest neutralino) is why one candidate answers two open questions at once — the "shared solution" theme of this article.

3 · How fine-tuned is the Higgs?

Estimate the top-loop correction $\delta m_H^2$ if the Standard Model holds to the Planck scale $\Lambda=1.2\times10^{19}$ GeV, and compare it to the physical $m_H^2=(125\ \text{GeV})^2$ .

Step 1. Use equation (3) with $\lambda_t\approx0.94$ : $\delta m_H^2\sim\dfrac{\lambda_t^2}{8\pi^2}\Lambda^2=\dfrac{0.88}{79}\,(1.2\times10^{19})^2\approx1.7\times10^{36}\ \text{GeV}^2$ .

Step 2. Compare to the observed mass-squared, $m_H^2=(125)^2=1.56\times10^4\ \text{GeV}^2$ . The correction is larger by a factor $\dfrac{1.7\times10^{36}}{1.56\times10^4}\approx10^{32}$ .

Step 3. State the tuning. The bare mass-squared must cancel this correction so that only $1.56\times10^4\ \text{GeV}^2$ survives — a cancellation accurate to one part in $10^{32}$ . Nothing in the Standard Model requires it. This is the hierarchy problem quantified — and it is what supersymmetry, by cancelling the $\Lambda^2$ term, removes.

Physics of the portrait

The open questions have many parents, but one measurement more than any other forced the community to accept that most of the universe is invisible. The portrait is Vera Rubin, whose rotation curves turned a fringe suspicion into an inescapable fact.

Origins · 1 minVera Rubin and how this was discovered

Vera Rubin

1928–2016, Carnegie Institution of Washington; the rotation-curve evidence for dark matter · galactic dynamics and the missing-mass problem

Fritz Zwicky had inferred "dunkle Materie" — missing mass — in the Coma cluster as early as 1933, from galaxies moving too fast for the visible mass to bind them, but for four decades the result was treated as a curiosity. Vera Rubin, working with the instrument-builder Kent Ford in the 1970s, made the case undeniable. Using a sensitive image-tube spectrograph, she measured orbital speeds of stars and gas across dozens of spiral galaxies, beginning with Andromeda, and found the same thing every time: the curves went flat and stayed flat far beyond the luminous edge, exactly where Keplerian physics said they should fall. There was no way to blame instrumentation or a peculiar galaxy — the effect was universal. The implication was stark: either Newtonian gravity fails on galactic scales, or every spiral is embedded in a halo of unseen matter outweighing its stars several times over. Over the following decade, with corroborating evidence from gravitational lensing and the CMB, the community settled on the second conclusion. Rubin championed careful observation over fashionable theory and opened the field to the women who followed her. She never received the Nobel Prize her discovery arguably merited; the dark matter she made real remains, half a century on, the largest unsolved problem in physics.

History

1933

— Zwicky infers missing mass from galaxy velocities in the Coma cluster

1967

— Sakharov states the three conditions for generating a baryon asymmetry

1970

— Rubin and Ford measure flat rotation curves in the Andromeda galaxy

1974

— Georgi and Glashow propose the SU(5) grand unified theory

1981

— Dimopoulos, Georgi and others formulate the supersymmetric Standard Model

Going further

Texts. Thomson, Modern Particle Physics, §18.2 (open questions: dark matter, supersymmetry, unification), Fig. 18.2–18.4, Eq. 18.1, Table 18.1 · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 12 Afterword (§12.1 the Higgs, §12.2 grand unification, §12.3–12.5), Eqs. 12.1–12.4 · Halzen & Martin, Quarks and Leptons, Ch. 15 (grand unification and beyond) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.2 and Ch. 12 (the renormalization group) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 22–23. The Higgs vacuum value is $v=246$ GeV. Key numerical anchors used throughout: flat $v\Rightarrow M(r)\propto r\Rightarrow\rho\propto1/r^2$ ; the fermion/scalar $\Lambda^2$ cancellation holds when $\lambda_S=\lambda_f^2$ ; MSSM $b_i=(33/5,1,-3)$ give $M_{\rm GUT}=1.9\times10^{16}$ GeV and $\alpha_{\rm GUT}^{-1}=24.3$ from both $\alpha_2$ and $\alpha_3$ , while SM $b_i$ do not unify; the WIMP relic density $\Omega_\chi h^2\approx0.1$ for $\langle\sigma v\rangle=3\times10^{-26}\ \text{cm}^3/\text{s}$ ; $\delta m_H^2\sim10^{36}\ \text{GeV}^2$ for $\Lambda=M_{\rm Pl}$ , a tuning of $\sim10^{32}$ . Cosmological parameters ( $\Omega_b\simeq0.05$ , $\Omega_c\simeq0.26$ , $\Omega_\Lambda\simeq0.69$ , $\eta\simeq6\times10^{-10}$ ) are the standard Planck values; $\tau(p\to e^+\pi^0)>2.4\times10^{34}$ yr is the Super-Kamiokande limit.

Prerequisites. Standard Model Summary · The Higgs Mechanism · Running of the Strong Coupling. Leads to. Dark Matter · The Hierarchy Problem · Supersymmetry · Grand Unification · Proton Decay · The Matter–Antimatter Asymmetry.

Ask the tutor: Ask me to run the gauge couplings up by hand and show exactly why the Standard-Model lines miss while the supersymmetric ones meet, or to walk through why a single weak-scale particle can be both the dark matter and the cure for the hierarchy problem.

Is there any direct proof of dark matter, or just rotation curves?

Several independent lines, which is why the case is overwhelming rather than suggestive. Rotation curves show it in galaxies; cluster velocity dispersions (Zwicky's 1933 argument) show it on larger scales; gravitational lensing maps the mass directly through how it bends light, independent of dynamics; and the CMB acoustic peaks fix its cosmic abundance to a few percent. The most striking piece is the Bullet Cluster, a collision in which lensing shows the bulk of the mass has sailed straight through, spatially separated from the hot X-ray gas carrying most of the ordinary matter — hard to explain by modifying gravity, natural if most of the mass is collisionless dark particles. What is missing is only a laboratory detection of the particle itself.

If the Standard Model passes every test, why believe it is incomplete?

Because "passes every test" refers to laboratory measurements, and the theory's failures are elsewhere. It has no particle for the dark matter, no mechanism to make more matter than antimatter in the amounts observed, and no account of neutrino masses without new fields. Those are not aesthetic complaints — they are phenomena the universe exhibits and the theory cannot reproduce. On top of them sit the structural clues: the fine-tuning of the Higgs mass, the near-unification of the couplings, the strange pattern of fermion masses. A theory can be simultaneously the most accurate ever built and manifestly a piece of something larger, just as Newtonian gravity was extraordinarily accurate yet incomplete. The open questions are the seams where the larger structure shows through.

Has the failure to find superpartners at the LHC killed supersymmetry?

It has killed the simplest, most natural version and wounded the idea, but not buried it. Supersymmetry was motivated largely by naturalness — the promise that superpartners near a few hundred GeV would stabilize the Higgs mass. The LHC has excluded gluinos and squarks below roughly one to two TeV, reintroducing a "little hierarchy" tension: the partners are heavy enough that some residual tuning creeps back. What survives is the broader framework, which still delivers gauge-coupling unification and a dark-matter candidate with partners in the multi-TeV range, plus variants (split supersymmetry) that trade naturalness for those successes. The honest summary: low-scale natural supersymmetry is in serious trouble, while supersymmetry as a high-scale organizing principle remains viable but is no longer the odds-on favourite it was in 2000.

Does the Standard Model really have all three Sakharov conditions?

Yes, in principle — which is what makes its quantitative failure instructive. Baryon number is violated by non-perturbative electroweak "sphaleron" processes; C and CP are both violated, C maximally by the weak interaction and CP by the complex phase in the quark-mixing matrix; and departure from equilibrium could come from the electroweak phase transition. The problem is magnitude. The CP violation from the CKM phase, measured by the Jarlskog invariant $J\approx3\times10^{-5}$ , is far too small once the tiny quark-mass differences are folded in, and for a $125$ GeV Higgs the electroweak transition is a smooth crossover, not the first-order transition needed to fall out of equilibrium. The Standard Model gives $\eta\sim10^{-20}$ or less, against the observed $6\times10^{-10}$ — so new CP violation, often sought via leptogenesis, is required.

What is the difference between dark matter and dark energy?

They share a name and nothing else, and conflating them is a common error. Dark matter is matter: it clumps under gravity, forms halos, and makes up $\Omega_c\simeq0.26$ of the budget; its job is to hold galaxies together and seed structure, and the candidates are new particles. Dark energy is not matter at all: a smooth, unclumping component with $\Omega_\Lambda\simeq0.69$ and negative pressure, which makes the expansion accelerate rather than decelerate. Dark matter pulls things together; dark energy pushes the universe apart. They are distinct open questions with distinct explanations — a particle for dark matter, a cosmological constant or evolving field for dark energy — and solving one would say little about the other.

Why does proton decay matter if it has never been seen?

Because its non-observation is itself a powerful result that rules theories in and out. Any grand unified theory placing quarks and leptons in the same multiplet must allow transitions between them, so the proton becomes unstable, decaying through channels like $p\to e^+\pi^0$ . The lifetime scales as the fourth power of the unification mass, so bounding it directly probes physics at $10^{16}$ GeV — energies no accelerator will reach. Minimal SU(5) predicted $\tau_p\sim10^{31}$ yr; when IMB and Kamiokande saw no decays and pushed the bound past that, they cleanly falsified the simplest unified theory. Current limits above $10^{34}$ yr now constrain the supersymmetric versions. A single confirmed proton decay would be one of the most important discoveries in the history of physics — direct evidence that the forces truly unify.