2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsthe order of limits

Back to four dimensions — the direction that removes the extra dimensions is opposite to the direction in which geometry can compute

The question of this chapter is whether the models of 7-09 can be brought back to the four-dimensional theory of the prize problem. The knobs are three: the $N$ of $SU(N)$, the 't Hooft coupling $\lambda=g^2N$ at the scale $M_{KK}$ (7-04), and the radius of the circle of the extra dimension $R=1/M_{KK}$ (the spacing of the KK tower, 7-06). With $\Lambda$ the glueball mass, the ratio $\Lambda R$ measures how visible the extra dimensions are. The requirements of the problem statement (Jaffe–Witten §4, 0-03) are

and the third is that the correlation functions agree at short distances with the predictions of asymptotic freedom and perturbative renormalisation theory (§4; quotation in 0-03 §07) Known. The models, conversely, can compute where $N=\infty$, $\lambda\gg1$ and $\Lambda R$ is of order 1 (§03, §05).

Example: the Sakai–Sugimoto model fixes $M_{KK}\simeq949$ MeV and $\kappa=\lambda N_c/(216\pi^3)\simeq7.45\times10^{-3}$ from experimental values. For $N_c=3$, $\lambda\approx16.6$ Computed. This is the value in the coupling convention of Sakai–Sugimoto, and can differ from the $\lambda$ of (2) in the next section by a factor of about 2 because of the difference in convention. The numbers put into (2) below are rough guides. The 776 MeV of the $\rho$ meson is 0.82 times $M_{KK}$, at the same height as the tower.

Removing the extra dimensions means taking $R\to0$ with $\Lambda$ fixed. $\Lambda$ is born from the running of the coupling (7-01, 4-07). Rewriting the one-loop formula $\mu\,dg/d\mu=-b_0g^3$ ($b_0=11N/(48\pi^2)$) of (1) of 7-01 in terms of $\lambda=g^2N$ gives

Dividing both sides by $\lambda^2$ gives $d(1/\lambda)/d\ln\mu=11/(24\pi^2)$, so $1/\lambda$ is linear in $\ln\mu$. Below, $\lambda$ without an argument is the value at $\mu=M_{KK}$.

Lowering $\mu$ decreases the right-hand side, which becomes 0 at $\mu=M_{KK}\exp\big(-24\pi^2/(11\lambda)\big)$ (the one-loop coupling diverges here). This scale where the coupling becomes strong is a rough guide to $\Lambda$:

Physics. $\sim$ means "equal up to a constant multiple", and the constant changes with the definition of $\Lambda$ (the $\Lambda$ parameter, or the glueball mass) and with conventions. Solving the other way gives

(a constant multiple only adds a constant to the denominator, so it does not affect the direction of $R\to0$), and removing the extra dimensions is the same direction as $\lambda\to0$. Witten (1998 §4) writes 'Asymptotic freedom should say that as η→0, the particle masses become exponentially small' ($\eta$ is the $\lambda$ of this chapter), and Csáki–Ooguri–Oz–Terning, putting the masses as $M^2=f(\lambda)\Lambda_{UV}^2$, state 'To take the continuum limit Λ_UV → ∞, we have to simultaneously take λ → 0' Physics ((2) is (10) of 7-07). As a metaphor, $R$ plays the role of the lattice spacing; more precisely, it is the same logic as sending the bare coupling to 0 as the spacing $a\to0$ (5-05).

ComputedTwo loops: from $\mu\,dg/d\mu=-b_0g^3-b_1g^5$ ($b_1=\frac{34N^2}{3(16\pi^2)^2}$), the formula of 7-01 with the next term added, the right-hand side of (2) multiplied by $(11\lambda/48\pi^2)^{-51/121}$ ($51/121=b_1/(2b_0^2)$). The coefficients can change with convention, and the rows with large $\lambda$ are rough guides outside the range where (2) can be used (that is also why, at $\lambda=100$, the two-loop column is smaller than the one-loop one; §03). Example: $\Lambda R=10^{-3}$ needs $\lambda\le3.12$ (two loops 2.66), and $10^{-6}$ needs $\lambda\le1.56$ (1.41).

Gravitational geometry can compute when $\lambda\gg1$ (7-08). Csáki et al. evaluated the leading string correction to the $f$ in the $M^2=f(\lambda)\Lambda_{UV}^2$ of §02, and for the $0^{++}$ glueball of three-dimensional pure Yang–Mills (QCD₃) obtained

(the lowest $0^{++}$ state and the first five excited levels) Physics. $c_1<0$ means that with the ultraviolet cutoff $\Lambda_{UV}$ fixed, decreasing $\lambda$ lowers the mass, the direction consistent with the expectation of a continuum limit. The power in four dimensions is not in this paper. Below we borrow $p=3/2$ as a rough guide for four dimensions.

The same paper writes, for the three-dimensional case, of taking $g_4^2N\to0$ as $R\to0$ (to keep the QCD₃ coupling $g_3^2N=g_4^2N/R$ constant): 'This is the opposite of the limit that is required for the supergravity description to be valid'. The grounds for four dimensions are Witten's sentence in §02 and the sentence 'To take the continuum limit …' in the introduction of Csáki et al. Witten too writes of showing asymptotic freedom and computing masses at small $\eta$: 'This looks like a tall order, given our limited knowledge of worldsheet field theory with Ramond-Ramond fields in the Lagrangian' (world-sheet field theory is explained in §09).

We look at the tug of war in numbers, with $e_{KK}=\Lambda R$ (the constant multiple in (2) set to 1) for how visible the tower is, and $e_{\rm str}=\lambda^{-p}$ with coefficient 1 (borrowing (4), $p=3/2$) as the rough guide to the string correction. We call the value at the crossing point $\lambda^*$ of the two curves the floor.

$\log e_{KK}+p\log\lambda$ is increasing in $\lambda$ (at one loop and at two loops, if $p\ge1$), so there is exactly one crossing point. The one-loop $e_{KK}$ increases towards 1 and $e_{\rm str}$ decreases, so to the left of the crossing $e_{\rm str}$, and to the right $e_{KK}$, exceeds $F_p$. That is, in the one-loop estimate $F_p=\min_\lambda\max(e_{KK},e_{\rm str})$, and for any $\lambda$ the larger of the two guides does not fall below $F_p$ Computed.

With the two-loop formula things differ. $e_{KK}$ reaches its maximum 0.61 at $\lambda\approx51$, and beyond that decreases towards 0 as $\lambda\to\infty$. So if the upper limit on $\lambda$ is removed, the minimum of the max approaches 0 (for $p=3/2$: 0.067 up to $\lambda\le10^4$, 0.038 for $\lambda\le10^5$, 0.014 for $\lambda\le10^6$) Computed. This is the behaviour of the perturbative formula for the running of the coupling extended to the large-$\lambda$ side, outside the range where (2) can be used. The two-loop rows of Table 2 are crossing values, and also agree with the minimum of the max restricted to $\lambda\le316$ (the range of Figure 1).

ComputedThe numbers in this figure were computed on this machine (the curves are computed on the spot from the formulas; the floor agrees with Table 2).

There is a window where both are around 5–13%, which does not contradict ratios agreeing with the lattice to a few per cent (when the three-dimensional $0^{--}/0^{++}$ is compared with the dilaton as reference; changing the reference mode or the states gives deviations of 14–26%: 7-09). But what a theorem needs is a limit sending the errors to 0, and in the one-loop estimate the floor blocks it. Both guides are crude, with coefficient 1; the point is not the value of the floor itself but that there is no direction making both small at once. Example: putting the $\lambda\approx16.6$ of Sakai–Sugimoto straight into (2) gives $e_{KK}=0.27$ (two loops 0.41) and $e_{\rm str}=0.015$. Allowing for the difference of convention, halving $\lambda$ (8.3) or doubling it (33) gives $e_{KK}$ of 0.075–0.52 and $e_{\rm str}$ of 0.042–0.005, and their order does not swap Computed. This is a rough guide suggesting that the tower side remains large.

At $\lambda\gg1$ the KK tower comes at the same height as the glueballs. The squared masses of fluctuations on a fixed background are eigenvalues of Laplace-type operators, and changing the metric as $g\mapsto s^2g$ multiplies all eigenvalues by $s^{-2}$.

If the background has a single length $R$, glueballs and KK modes alike are some multiple of $1/R$ (the counting of powers is Known; the part saying that the background describes Yang–Mills is Physics). But the string tension involves $\lambda$, and Klebanov–Strassler write $T_s\sim g_sM\,m_{\rm glueball}^2$ (the same circumstance as 7-08 §07). That the tower does not separate is acknowledged by the authors in Table 3 (for Witten's background, by Csáki et al. and Klebanov–Strassler) Physics.

How supersymmetry remains

In Witten's type (1998 §4; fermions made antiperiodic on the circle), the fermions acquire masses of order $1/R$ at tree level, the scalars 'very plausibly get masses … at one-loop level', and it continues 'If this is so, the low energy theory will be the pure SU(N) theory without supersymmetry' (§2.2, on the three-dimensional case, has a sentence to the same effect: at weak coupling the long-distance behaviour is 'very plausibly' only a three-dimensional gauge field). Klebanov–Strassler and Maldacena–Núñez preserve $\mathcal N=1$ supersymmetry, so a further limit making the gluino (the fermion that is the supersymmetric partner of the gluon) heavy is needed. The five-dimensional model of Gürsoy–Kiritsis is of the kind (bottom-up) that puts a five-dimensional gravity model directly rather than deriving it from a string-theory background, and takes the β function as input ('using input … as well as QCD') Physics. Example: within the range we examined, among the models built from string-theory backgrounds (top-down), only Witten's type yields non-supersymmetric pure Yang–Mills, and that too at the same scale as the tower.

Gravitational geometry becomes classical as $N\to\infty$. The quantities of the gauge theory are arranged by 't Hooft's counting (7-04) into a sum over genus $h$.

What the gravity computation gives is the leading part of $F_0$ on the large-$\lambda$ side, and for $SU(3)$ ($1/N^2=1/9$) all the remaining terms matter. That the ratios move slowly with $N$ on the lattice (7-04) Physics does not give the existence of the theory at fixed $N$ either. There is a gap in the kinds of group too: D-branes give $U(N)$ and $SU(N)$, and orientifolds (backgrounds divided by an operation that reverses the orientation of strings) give $SO$ and $Sp$ (7-07). Within the range we searched we found no holographic dual of four-dimensional pure Yang–Mills for the exceptional groups $G_2,F_4,E_{6,7,8}$, and geometric engineering can make types ADE, and types B, C, F, G by folding with outer automorphisms (automorphisms of Dynkin diagrams), but within supersymmetric theories (7-07). Example: in the problem statement, $SU(2)$ and $G_2$ are each a single theory.

The gravity computation takes the supergravity limit $\lambda\to\infty$ after the $N\to\infty$ that makes the string classical, and uses $R$ as the unit.

The problem statement's side fixes $N$ and takes $R\to0$, that is, by (3), $\lambda\to0$. A standard theorem of analysis gives the condition under which two limits may be exchanged Known. A double sequence $a_{m,n}$ converges to $b_m$ as $n\to\infty$ uniformly in $m$ when

that is, when a single $n$ makes the errors for every $m$ small at once. Then the following holds.

Without uniformity the exchange is in general not allowed. Example: for $a_{m,n}=m/(m+n)$, $\lim_m\lim_n a_{m,n}=0$ while $\lim_n\lim_m a_{m,n}=1$ (since $\sup_m m/(m+n)=1$, the convergence to 0 as $n\to\infty$ is not uniform).

What is needed is that the $1/N$ corrections be uniformly small in $\lambda$, that the string corrections be uniformly small in $N$, and an unbroken road down to $\lambda\to0$. (4) is the leading term of the correction in three dimensions, neither uniformity nor a limit. Within the range we searched we found no result justifying this exchange. As a metaphor, the supergravity region ('fixed ultraviolet cutoff … in the strong ultraviolet coupling regime') is in the position of a coarse lattice at strong coupling. More precisely, on the strong-coupling lattice the existence of the infinite-volume limit is proved (Osterwalder–Seiler 1978, 5-04), but this correspondence stays a metaphor.

The problem statement is 'Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R⁴ and has a mass gap Δ > 0', and existence includes satisfying axiomatic properties (Jaffe–Witten §4) Known.

None of the rows of Table 5 is satisfied yet. Jaffe–Witten §6 write 'at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions', and the problem page of the Clay Mathematics Institute still writes 'no proof of this property is known' Known (the definition of the gap is in 4-11). Example: the ratios the models hit to a few per cent (7-09; only the three-dimensional ratio with the dilaton as reference) are also numbers obtained while satisfying none of the rows of Table 5.

The problem does not vanish; it moves. Witten, noting that a singularity appears in the geometry as $\eta\to0$, writes 'Apparently, in this approach, the mysteries of four-dimensional quantum gauge theory are encoded in the behavior of string theory near this singularity', and Csáki et al. write that under the conjecture the problem of QCD at large $N$ reduces to 'that of controlling the α′ corrections to the two-dimensional sigma-model with the Ramond-Ramond background' Physics. We give the meanings of the words briefly. $\alpha'$ is the square of the string length (the string tension is $1/(2\pi\alpha')$, (1) of 7-05). The two-dimensional sigma model is the field theory on the two-dimensional surface swept out by a string (the world sheet), whose fields are maps from the world sheet to spacetime. Ramond–Ramond fields are fields sourced by D-branes. The $\alpha'$ corrections are corrections coming from the curvature radius of spacetime being finite compared with the string length, and (4) evaluates the leading term of this expansion. The 'worldsheet field theory with Ramond-Ramond fields' that Witten named in §03 (a field theory on the world sheet whose Lagrangian contains Ramond–Ramond fields) is also this two-dimensional theory.

Example: neither arrow in (12) is a mathematical theorem (the left is unjustified, the right a conjecture), and the object at the right end has no non-perturbative definition either (7-13, 9-02).

What this chapter can say

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Next chapter to read: 7-11 Supersymmetry and Seiberg–Witten — an example where, under supersymmetry, a gap is visible from a duality.

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