Spin Pairs From the Vacuum
STAR’s spin-aligned Λ Λ̄ pairs read as the vacuum’s ³P₀ pair liberated by one seam snap: the Bell channel generalised from singlet to triplet, where baryon spin lives, and what a full correlation-tensor measurement would decide
The Measurement
In p+p collisions at \sqrt s = 200 GeV, the STAR experiment at RHIC reconstructed \Lambda\bar\Lambda, \Lambda\Lambda and \bar\Lambda\bar\Lambda pairs from about 600 million minimum-bias events and measured how their spins are correlated ([R217]). The \Lambda is a convenient spin meter. Its weak decay \Lambda\to p\pi^- throws the proton preferentially along the \Lambda’s spin, so the angle \theta^\star between the two decay (anti)protons, each boosted to its parent’s rest frame, records the spin–spin correlation:
\frac{1}{N}\frac{dN}{d\cos\theta^\star} = \tfrac12\bigl[1 + \alpha_1\alpha_2\,P_{\Lambda_1\Lambda_2}\cos\theta^\star\bigr].
The relative polarization P_{\Lambda_1\Lambda_2} is +\tfrac13 for a pair whose spins are fully parallel in a triplet, -1 for a singlet, and 0 for no correlation (Törnqvist [R218]). STAR found:
| Pair class | P_{\Lambda_1\Lambda_2} |
|---|---|
| \Lambda\bar\Lambda, short range (\lvert\Delta y\rvert<0.5, \lvert\Delta\phi\rvert<\pi/3) | \mathbf{0.181\pm0.035_\text{stat}\pm0.022_\text{sys}} (4.4\sigma) |
| \Lambda\bar\Lambda, long range | consistent with 0 |
| \Lambda\bar\Lambda, close in \phi but far in y | -0.012\pm0.073\pm0.022 |
| \Lambda\Lambda, \bar\Lambda\bar\Lambda, any range | consistent with 0 |
| K^0_SK^0_S (spin-0 control), PYTHIA 8.3 (no spin physics) | consistent with 0 |
The positive sign means parallel spins: the pairs are in a spin triplet. Only strange–antistrange pairs that are close in angle carry the correlation. Pairs from different parts of the event do not, and like-baryon pairs, which cannot come from one s\bar s pair, do not either.
STAR’s interpretation is the standard one. The QCD vacuum holds a condensate of quark–antiquark pairs with the vacuum’s quantum numbers J^{PC}=0^{++}. For a fermion–antifermion pair the parity is (-1)^{L+1}, so 0^{++} forces L=1 and S=1, locked to J=0: the ³P₀ state (Micu; Le Yaouanc et al. [R183]). The collision frees such a pair, each quark dresses itself into a hyperon, and in the \Lambda the strange quark carries all of the spin, so the hyperons inherit the quarks’ alignment. Two numbers then set the expected size. About 89% of measured pairs contain at least one \Lambda fed down from a heavier hyperon (mostly \Sigma^0\to\Lambda\gamma), which dilutes the correlation. With that dilution, the SU(6) quark model gives a ceiling of P = 0.096\pm0.004, and the Burkardt–Jaffe model, in which the \Lambda’s light quarks carry part of its spin, gives 0.015\pm0.002 ([R220]). The data sit above both.
What the Framework Already Said
This measurement lands on a claim the framework made before reading it, though not about this experiment. Superfluid Helium § ³He-B identifies the vacuum seen from outside a cell with the B phase of helium-3. That is the spin-triplet, p-wave condensate whose pairs are exactly ³P₀: orbital L=1 and spin S=1 locked opposite so that J=0. The same section reads the ³P₀ quark-pair model as the place “where we can watch the vacuum make fermion pairs, [and] it makes them in the B-phase channel.” Until now that was an argument from quantum numbers. STAR has measured its spin half directly: the pairs the vacuum gives up have parallel spins.
Two cautions from that section carry over unchanged. Any Lorentz-invariant vacuum has J=0, so the specific content is the internal split, S=1 against L=1, rather than J=0 itself. STAR now confirms the S=1 half. Its paper lists measuring the L=1 half, through the momentum distribution in the pair rest frame, as future work. Second, helium-3 pairs are fermion–fermion while the vacuum’s are fermion–antifermion, so the match is in angular-momentum content, not in the identity of the pair.
Where the pair comes from. In the framework a flux tube is a counter-rotating seam between quark orbital systems, and when a stretched seam stores about twice a quark mass it snaps and nucleates a quark–antiquark pair from the substrate (Proton Core § The pair creation threshold). The newborn pair is a closed B-phase unit. Each quark is an open seam whose residual spin is \hbar/2 (Spin-½ as a net angular momentum). The two residuals add to S=1, and the snapped seam between them carries the counter-rotation that cancels it: the counter-rotating layer with remainder zero. Reading the pair’s orbital L as that seam is the same identification the ³He-B section already flags, and it is flagged here too.
Why only close pairs. One snap makes one pair at one place, and its two members leave that place close in rapidity and azimuth. A \Lambda and a \bar\Lambda far apart in angle almost always come from different snaps, which share no seam and so have no reason to be aligned. The long-range null follows.
What does not discriminate. Both readings above have standard counterparts. Perturbative gluon splitting g\to q\bar q also gives triplet pairs, because the gluon is spin-1; PYTHIA puts that channel at a negligible share in STAR’s momentum range. The Lund string model also breaks a string locally, so it also predicts short-range-only correlations. The framework reproduces this measurement; the measurement does not single it out. The rest of this chapter asks where the framework has to say more than that, and where it can.
From Singlet to Triplet: the Channel Generalises
The Bell chapter builds its mechanism on singlet pairs. A half-quantum vortex channel joins the two particles, and its \pi winding encodes the constraint J_A+J_B=0 by pointing B’s end of the order parameter opposite A’s: \hat{\mathbf d}_0(B) = -\hat{\mathbf d}_0(A). A critic can fairly ask what channel a triplet pair carries. The answer is that very little changes, because what the channel encodes is the pair’s total angular momentum, and that is still zero.
The ³P₀ pair is a singlet with one end turned over. For a pair flying apart along the unit vector \hat{\mathbf k}, the ³P₀ spin state is the triplet with zero spin projection on \hat{\mathbf k}, which is ³He-B’s statement that a pair’s \hat{\mathbf d}-vector is locked to its orbital axis. That state is exactly the singlet with one partner rotated by \pi about \hat{\mathbf k}:
\lvert{}^3P_0;\hat{\mathbf k}\rangle_\text{spin} \;=\; \bigl(R_\pi(\hat{\mathbf k})\otimes\mathbb 1\bigr)\,\lvert\text{singlet}\rangle .
Since the particles separate along \hat{\mathbf k}, the channel lies along \hat{\mathbf k}, and the rotation is a half-turn of the order parameter about the channel’s own length. B’s end is no longer the point reflection of A’s but its mirror image in the plane perpendicular to the channel:
\hat{\mathbf d}_0(B) \;=\; \hat{\mathbf d}_0(A) - 2\bigl(\hat{\mathbf d}_0(A)\cdot\hat{\mathbf k}\bigr)\hat{\mathbf k}.
Spin components across the channel come out parallel and the component along it comes out opposed. That accounts for both halves of ³P₀: the transverse alignment is the S=1 STAR sees, and the frozen half-twist along the channel is where the L=1 is stored. The Part 4 derivation then goes through line by line. The twist wave still carries A’s snap rotation to B, and B’s updated axis is the mirror image of A’s outcome instead of its negative. The correlation becomes
\boxed{\;E(\hat{\mathbf a},\hat{\mathbf b}) \;=\; \hat{\mathbf a}\cdot\hat{\mathbf b} \;-\; 2\,(\hat{\mathbf a}\cdot\hat{\mathbf k})(\hat{\mathbf b}\cdot\hat{\mathbf k})\;}
which is the quantum-mechanical ³P₀ result, C_{ij} = \delta_{ij} - 2\hat k_i\hat k_j, with no net polarization on either particle (both checked numerically against the explicit Clebsch–Gordan state, scripts/lambda_3p0_check.py). Being a singlet with one end rotated, the state is still maximally entangled and reaches CHSH =2\sqrt2 when the detectors are rotated to match. The singlet mechanism does generalise. The channel is the same half-quantum vortex, and the triplet adds one frozen half-twist along it.
The channel does not drop shots here. The \Lambda’s own weak decay is the measurement, so the two “detectors” fire about 10^{-10} s and a few centimetres apart, essentially at random. A finite channel speed loses only shots simultaneous to within L/v_\text{ch}\lesssim10^{-16} s (Bell § Part 7), about one pair in 10^6. STAR’s pairs are always connected.
What STAR’s Number Can and Cannot Decide
STAR measures one number per pair class, the relative polarization, and that number is the trace of the correlation tensor: P = \tfrac13\operatorname{tr}C. The trace does not depend on how the pair’s axis \hat{\mathbf k} is oriented, so it cannot see the anisotropy that carries the entanglement. Averaged over all pair axes, the ³P₀ spin state becomes the uniform mixture of the three triplet states. That mixture is separable: it is exactly the ensemble of two parallel spins pointing along a common, random direction. STAR’s own paper notes that the entanglement question needs a full correlation-tensor analysis (Peres–Horodecki), and this is why.
Comparing local models with the channel makes it concrete. Write the tensor in the pair frame, with C_\perp the two components across \hat{\mathbf k} and C_\parallel the one along it. Every entry is multiplied by the same dilution factor D from hadronisation and feed-down. STAR’s PYTHIA-based estimate gives D=0.288 for SU(6), and D\le1 always:
| Picture | C_\perp | C_\parallel | P = \tfrac13\operatorname{tr}C | At D=0.288 |
|---|---|---|---|---|
| Channel on (= quantum ³P₀) | +1 | -1 | 1/3 | 0.096 |
| Channel off, hidden axis isotropic (Bell Part 4 baseline, mirrored) | +\tfrac13 | -\tfrac13 | 1/9 | 0.032 |
| Channel off, hidden axis born across \hat{\mathbf k} (a classical B-phase \hat{\mathbf d}-vector) | +\tfrac12 | 0 | 1/3 | 0.096 |
| No correlation | 0 | 0 | 0 | 0 |
Three things follow.
- The Bell chapter’s own channel-off baseline is disfavoured. With an isotropic hidden axis, the mirror correlation is diluted by the same \tfrac13 that turns -\cos\theta into -\cos\theta/3 for the singlet, so P\le\tfrac19 even with no feed-down at all. The data, 0.181\pm0.041, sit 1.7\sigma above that hard ceiling and 3.6\sigma above its feed-down-diluted value. Within the framework’s picture, STAR’s pairs need the channel on, or else a hidden axis that is born already perpendicular to the pair axis.
- STAR cannot tell the channel from that second local model. Both give the same trace. They differ in the component along the pair axis, -D for the channel and 0 for the local model. That is a clean target for the full tensor analysis STAR proposes, and Gong, Parida, Tu and Venugopalan have laid out how to extract it from hyperon decays [R219].
- The size of the signal decides whether entanglement can be seen at all. A diluted ³P₀ state, C = D\,\mathrm{diag}(1,1,-1), is entangled only if D>\tfrac13 (the sum of |C_i| must exceed 1; checked here by partial transpose). The SU(6)-plus-PYTHIA dilution, D=0.288, sits below that threshold: if it is right, STAR’s sample is separable however the tensor comes out. The measured central value implies D = 3\times0.181 = 0.54, above the threshold. The roughly 2\sigma excess therefore matters more than its significance suggests. It decides whether a Bell or Peres–Horodecki test on these pairs can succeed. One caveat: the tensor test also needs the pair axis to survive into the hyperons. Hadronisation, and especially feed-down, smear \hat{\mathbf k}, which lowers the effective D for the anisotropic part even though the trace is unaffected.
Where the Λ’s Spin Lives
STAR’s result says something about hadron structure as well as about the vacuum. The data favour the plain SU(6) picture, in which the strange quark carries essentially all of the \Lambda’s spin, and they disfavour Burkardt–Jaffe, which gives the light quarks a negative share. That is surprising given the proton spin puzzle: the proton’s quarks carry only about 35% of its spin, and STAR’s own polarized-jet program attributes about half of the remainder to gluons. Why would the \Lambda be simple when the proton is not?
The framework reads it through boundary parity. In the \Lambda (uds), the u and d are in isospin 0, which forces their pair into spin 0. In substrate terms their two odd-parity boundaries merge into an even-parity, closed whole, as a Cooper pair’s do (Conductors). A closed seam has remainder zero and carries no net angular momentum to the outside. That leaves the strange quark’s boundary as the \Lambda’s only open seam, and so the only place its \hbar/2 can live. The proton (uud) has no such option. Its two u quarks have identical flavour, so any pair containing both must be spin-1. In the SU(6) wavefunction the proton spends only half its weight in a closed spin-0 pair and the other half in open spin-1 pairs. An open pair’s seams join the junction’s counter-rotating sheath, and the proton is 99% sheath by energy (Mass as Rotational Energy). So its \hbar/2 is a small residual of a very large boundary angular momentum, shared between quark cores and the seams. The seams are the gluons (Standard Model § Bosons), and angular momentum held in the seam geometry shows up as gluon spin plus orbital angular momentum. The full development is in Spin-Statistics § Where Baryon Spin Lives.
This turns the gap into an ordering the framework can be held to. The quark-spin fraction of a baryon should track whether its non-spin-carrying pair can close.
- \Lambda, \Lambda_c, \Lambda_b (closed ud scalar pair): spin on the odd quark, SU(6)-like.
- \Sigma, \Xi, nucleon (pairs forced into spin 1, open): spin spread into seams, so proton-like reduced quark fractions.
For the heavy \Lambda_Q this matches heavy-quark symmetry and does not discriminate. The light \Lambda is where the framework departs from flavour-SU(3) rotations of the proton’s spin puzzle (the Burkardt–Jaffe logic), and STAR’s data side with the framework there. That is a postdiction, not a prediction. The framework also does not yet supply the proton’s 35%. That calculation is open as WIP-37.
The Excess Above the Quark-Model Ceiling
After feed-down the SU(6) ceiling is 0.096\pm0.004, and the data are 0.181\pm0.041, about 2\sigma higher. STAR calls this compatible, which is fair at 2\sigma. If it survives more data, though, something has to give.
The framework cannot get it from the quark level. A ³P₀ pair with the channel on has P=\tfrac13 before dilution, the same as quantum mechanics, and nothing in the substrate picture makes a pair more than fully aligned. What can move is the dilution D. Getting the measured value at the SU(6) per-quark transfers needs D\approx0.54 instead of 0.29. That means either a larger primary-pair fraction than PYTHIA’s 11%, or fed-down \Lambdas keeping more of their parent’s alignment than the SU(6) table gives them. STAR notes that PYTHIA’s intrinsic feed-down uncertainty is not included. The framework currently has no number for either. A substrate account of hadronisation, which would say how often a single snap yields a primary \Lambda\bar\Lambda rather than a \Sigma^0\bar\Lambda, is not built. This gap is open, not closed. The framework’s stake in it is the threshold above: whether D is above or below \tfrac13 decides whether these pairs can show their entanglement at all.
A Forward Prediction: Melting the Pairing
STAR suggests that in a quark–gluon plasma, where chiral symmetry is restored and the condensate is gone, \Lambda\bar\Lambda pairs might come out in spin singlets instead. The framework makes a more specific claim about what happens first.
In the framework, the ³P₀ lock is a property of the B-like ordered vacuum: a pair snapped from a closed seam inherits the B-phase split. A region dense and hot enough that seams no longer close is a small return to the state the boil passed through before the lock, when quarks had not yet knotted into baryons. There, no ordered B-like background exists to impose that split. The prediction is in two parts.
- The triplet correlation should fade to zero, not flip. As collision centrality and energy density rise, short-range P_{\Lambda\bar\Lambda} should fall from its p+p value toward zero, and it should do so in step with other signs of deconfinement. It should not pass smoothly through zero into negative values.
- A singlet signal needs anti-phase pairing, and that belongs elsewhere on the phase diagram. The framework’s singlet is the Cooper-like anti-phase pair, two same-chirality partners breathing \pi out of phase (Conductors). Its QCD analogue is pairing in cold, dense quark matter, not in the hot, nearly baryon-free plasma at top RHIC energy. So a negative P_{\Lambda\bar\Lambda} is predicted, if anywhere, at the high-baryon-density, low-\sqrt{s} end of the beam-energy scans (RHIC BES-II, and FAIR and NICA when they run), not in central Au+Au at 200 GeV.
A clean sign flip at top RHIC energy would count against this reading. The energy density at which the triplet signal disappears is not yet computed. The natural candidate is the seam’s own energy density, about 2 GeV/fm³ from the string tension (Proton Core § The constant string tension), but that is a scale, not a derivation.
Honest Accounting
Reproduced, not predicted. Parallel spins, short range only, null for like pairs. Each follows from the framework’s existing readings (B-phase vacuum pairs, one snap per pair), and each has a standard counterpart (³P₀ model, gluon splitting, Lund strings) that predicts the same thing.
Gap closed. The Bell mechanism is not singlet-only. The ³P₀ pair is a singlet with a frozen half-twist along its channel, the same half-quantum vortex carries it, and the derivation returns the exact quantum correlation C_{ij}=\delta_{ij}-2\hat k_i\hat k_j.
Sharpened. STAR’s trace observable rules against the framework’s own isotropic channel-off baseline (P\le\tfrac19) but cannot separate the channel from a local model with the hidden axis across the pair. The decisive measurement is the component along the pair axis, and it can succeed only if the dilution exceeds \tfrac13, which is the regime the measured central value points to.
Gap turned into an ordering. The \Lambda’s spin sits on the strange quark because its ud pair closes; the proton’s cannot fully close. The proton’s 35% itself is owed (WIP-37).
Open. The 2\sigma excess has no framework number. The melt threshold has a candidate scale, not a value. STAR’s 2017 discovery of global \Lambda polarization in Au+Au, read as a fluid vorticity of about 10^{22} s⁻¹ ([R221]), is the most vortical fluid ever measured. A vortex-substrate framework ought to have something to say about it, and does not yet.
Place in the Framework
This chapter connects four existing threads. Superfluid Helium § ³He-B supplies the vacuum’s pair state, Bell’s Theorem the channel that holds a pair together, Spin-Statistics the boundary-parity rule for where spin lives, and Proton Core the seam snap that makes the pair. STAR’s measurement is the first place all four act on one observable. None of them had to bend to fit it. Read against the popular picture of the vacuum as a froth of virtual pairs, it is also the first direct look at pairs the vacuum gives up, and they come out ordered (Quantum Foam § Popping the foam).