The gap is bounded by the variance of the averaged spatial Polyakov loop, with the small-volume scaling built in
For the Kogut–Susskind Hamiltonian on the periodic lattice of side \(L=N_sa\), the Feynman–Bijl inequality with the transverse average of the spatial Polyakov loop in direction \(i\), \[\bar P_i=\frac1{N_s^2}\sum_{\text{lines }\parallel i}\operatorname{Re}\operatorname{tr}\big(U_{\ell_1}\cdots U_{\ell_{N_s}}\big),\] gives the exact bound \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{\hbar cg^2}{2a}\,\frac{\langle\sum_\ell|\nabla_\ell\bar P_i|^2\rangle_0}{\operatorname{Var}_0\bar P_i} \ \le\ \frac{\hbar c\,g^2N}{4L\,\operatorname{Var}_0\bar P_i},\] for every coupling and lattice size. Three consequences. In a small box at weak coupling, where the ground state is the constant-mode state of C133 at leading order, numerator and denominator are both computable there and the bound is \((\hbar c/L)\,g^{2/3}\) times a pure number, the scaling of the exact zero-mode gap, so the upper side of the small-volume corner S of the obligations map follows from an exact inequality plus the zero-mode expectation values. In the infinite-volume limit, a mass gap \(m\) forces the averaged spatial Polyakov loop to self-average at least like \(1/L\): \(\operatorname{Var}_0\bar P_i\le\hbar cg^2N/(4mc^2L)\), equivalently a bound on the transverse Polyakov-loop susceptibility, \(\sum_{y}a^2\langle\operatorname{Re}P(x)\operatorname{Re}P(x+y)\rangle_c\le\hbar cg^2NL/(4mc^2)\), which is a necessary condition for T3 expressed through one gauge-invariant correlation. At strong coupling the averaged loop has variance \(1/(2N_s^2)\) and the bound is empty, so the plaquette bound of the upper-bound note and the present one cover complementary ends of the coupling axis. Nothing here is promoted.
1. The operator and the exact bound
Fix a direction \(i\). The lattice decomposes into \(N_s^2\) lines parallel to \(i\), each of \(N_s\) links closing around the torus. For a line \(\lambda\) let \(P_\lambda=\operatorname{Re}\operatorname{tr}\prod_{\ell\in\lambda}U_\ell\), the ordered product along the line; it is gauge invariant (a closed loop), and \(\bar P_i=N_s^{-2}\sum_\lambda P_\lambda\). Let \(O=\bar P_i-\langle\bar P_i\rangle_0\).
Theorem 1. For every \(g>0\), \(N_s\ge2\) and compact \(G\) with the normalization of the obligations map, \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{\hbar cg^2}{2a}\, \frac{\big\langle\sum_\ell|\nabla_\ell\bar P_i|^2\big\rangle_0}{\operatorname{Var}_0\bar P_i} \ \le\ \frac{\hbar c\,g^2N}{4L\,\operatorname{Var}_0\bar P_i}.\]
Proof. \(O\) is a real gauge-invariant function of the links with zero mean, so Proposition 1 of the upper-bound note applies and gives the first inequality. Each link in direction \(i\) belongs to exactly one line and links in other directions do not enter \(\bar P_i\), so \(\nabla_\ell\bar P_i=N_s^{-2}\nabla_\ell P_{\lambda(\ell)}\). For a link of \(\lambda\), \(X_a\operatorname{tr}\prod U=\operatorname{tr}(T^aM)\) up to a phase with \(M\) a cyclic rearrangement of the product, and the completeness relation gives \(\sum_a|\operatorname{tr}(T^aM)|^2=\tfrac12(N-|\operatorname{tr}M|^2/N)\le N/2\); the real part has smaller gradient. Hence \(\sum_\ell|\nabla_\ell\bar P_i|^2\le N_s^{-4}\cdot N_s^2\cdot N_s\cdot N/2=N/(2N_s)\), and \(\frac{\hbar cg^2}{2a}\cdot\frac{N}{2N_s}=\frac{\hbar cg^2N}{4L}\). \(\square\)
2. Weak coupling in a small box
At weak coupling the links are near the identity and the gauge-invariant content of a Polyakov line is the holonomy of the constant mode: \(\prod_{\ell\in\lambda}U_\ell\approx\exp(iLa_i)\) with \(a_i=a_i^bT^b\) the constant mode of G07 Proposition 7, whose ground-state width is \(|a_i|\sim g^{2/3}/L\) in natural units. Since \(\operatorname{tr}T^b=0\), \[P_\lambda\approx\operatorname{Re}\operatorname{tr}e^{iLa_i}=N-\tfrac14L^2|\vec a_i|^2+O(L^4|a|^4),\qquad \nabla_\ell P_\lambda\approx\operatorname{Im}\operatorname{tr}(T^aLa_i)=\tfrac12La_i^a ,\] so the variance of \(\bar P_i\) is second order and the gradient first order in the constant mode: \[\operatorname{Var}_0\bar P_i\approx\tfrac1{16}L^4\operatorname{Var}(|\vec a_i|^2),\qquad \Big\langle\sum_\ell|\nabla_\ell\bar P_i|^2\Big\rangle_0\approx N_s^{-4}\cdot N_s^3\cdot\tfrac14L^2\langle|\vec a_i|^2\rangle =\frac{L^2\langle|\vec a_i|^2\rangle}{4N_s}.\] In the dimensionless variables of C133, \(La_i=g^{2/3}\xi_i\) with \(\xi_i\) distributed by the unit ground state \(\psi_0^{(3)}\) of \(\tfrac12(-\Delta_{\mathbb R^9}+\sum_{i<j}|\vec\xi_i\times\vec\xi_j|^2)\): \[\frac{\hbar cg^2}{2a}\cdot\frac{g^{4/3}\langle|\vec\xi_i|^2\rangle/(4N_s)}{g^{8/3}\operatorname{Var}(|\vec\xi_i|^2)/16} =\frac{\hbar c}{L}\,g^{2/3}\cdot\frac{2\langle|\vec\xi_i|^2\rangle}{\operatorname{Var}(|\vec\xi_i|^2)} .\] The bound is therefore \((\hbar c/L)g^{2/3}\) times the pure number \(2\langle|\vec\xi_i|^2\rangle/\operatorname{Var}(|\vec\xi_i|^2)\), finite because the unit ground state decays faster than any power along the valleys (the confining bound of C133 gives exponential decay of \(\psi_0\) in \(\sum_i|\vec\xi_i|\)). Consistency with the exact zero-mode gap \(\delta_1^{(3)}g^{2/3}\hbar c/L\) requires this number to be at least \(\delta_1^{(3)}\), which is the Feynman–Bijl inequality applied inside the zero-mode model with the trial operator \(|\vec\xi_i|^2\); the model version of Theorem 1 is exact and says exactly that.
What is proved and what is not: Theorem 1 is exact. The two approximations, replacing the ground-state expectations by their constant-mode values, are the leading order of the small-volume expansion (Lüscher; abstract, B78) and are not proved here. The statement that follows from them is that the exact lattice inequality carries the \(g^{2/3}\) scaling of the small-volume corner on its upper side; a lower bound with that scaling remains the open weak-coupling target of STATE.
3. Infinite volume: a necessary condition for the mass gap
Suppose the lattice theory at fixed \(a\) and \(g\) has infinite-volume gap \(\Delta_a=\liminf_{N_s\to\infty}\Delta_{a,L}>0\) (T2\('\)), and write \(mc^2=\Delta_a\). Theorem 1 gives, for every \(L\), \[\operatorname{Var}_0\bar P_i\ \le\ \frac{\hbar cg^2N}{4L\,\Delta_{a,L}} .\] By translation invariance in the transverse plane, \(\operatorname{Var}_0\bar P_i=N_s^{-2}\sum_{y}C_i(y)\) with \(C_i(y)=\langle\operatorname{Re}P_\lambda(x)\operatorname{Re}P_\lambda(x+y)\rangle_c\) the connected two-point function of the spatial Polyakov loop at transverse separation \(y\). Hence
Corollary 2. If the infinite-volume gap is \(mc^2>0\), then along any sequence of boxes on which \(\Delta_{a,L}\ge mc^2/2\), \[\sum_{y\in\text{transverse plane}}a^2\,C_i(y)\ \le\ \frac{\hbar cg^2N}{2mc^2}\,L ,\] and the averaged spatial Polyakov loop self-averages at least like \(1/L\).
Reading: in a confining phase the spatial Polyakov loop of length \(L\) has connected correlations decaying on a transverse scale \(\xi\), and the left side is of order \(\xi^2\langle P^2\rangle_c\), so the corollary is satisfied with room; in a phase where the loop orders or its correlations do not decay, the left side grows like \(L^2\) and the corollary forbids a gap. It is a necessary condition only, but it is one written entirely in terms of a single gauge-invariant correlation and the coupling, with no reference to the spectrum.
4. Strong coupling: the bound is empty
At \(g=\infty\) the links are independent Haar variables, distinct lines are independent, and \(\langle|\operatorname{tr}U_1\cdots U_{N_s}|^2\rangle=1\) for a product of independent Haar elements of \(SU(N)\) (the product is again Haar distributed), so \(\operatorname{Var}P_\lambda=\tfrac12\) for \(N\ge3\) and \(\operatorname{Var}\bar P_i=1/(2N_s^2)\). Theorem 1 then reads \(\Delta\le\hbar cg^2NN_s^2/(2L)\), which grows with the lattice and says nothing. The averaged loop is the wrong trial operator in the disordered phase, as the plaquette is in the ordered one; the two bounds are complementary, and neither is uniform in \(a\) along the scaling curve.
5. Consequence for STATE
The upper side of the small-volume corner is now an exact lattice inequality plus two zero-mode expectation values, and a mass gap has a necessary condition in the Polyakov-loop susceptibility (Corollary 2). The lower side, a bound \(\Delta_{a,L}\ge c\,g^{2/3}\hbar c/L\) at weak coupling in a small box, remains the open weak-coupling target; its natural form, suggested by the shape of Theorem 1, is an operator inequality in which the constant-mode zero-point energy of C133 survives the coupling to the nonzero modes.