2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicseffective string theory

The string picture — treating the tube of confinement as a single string, a universal correction and a shape come out of its fluctuations

The flux tube of 7-01 §05, if its thickness is much smaller than the distance $R$ of the pair, is from afar a single line. Extended in the direction of Euclidean time, the line sweeps out a surface (the world sheet). The string picture is an effective theory that describes long distances by the fluctuations of this surface Physics. To a surface $S$ bounded by the rectangle $R\times T$ of the Wilson loop we give a weight fixed by its area $A(S)$.

With the minimal surface alone this is $e^{-\sigma RT}$, the area law. The question is what the fluctuations add. As a metaphor, it is the trembling of a stretched membrane. More precisely, (1) is not a formula derived from Yang–Mills but a starting point under the assumption that "a long tube exists, and its low excitations are only transverse fluctuations" (Athenodorou–Bringoltz–Teper 2011 §2; ABT below).

At strong coupling on the lattice (5-02) a formula close to (1) holds, and the weight $u$ of one plaquette plays the role of $e^{-\sigma_0}$ Known. But the area law also comes out for U(1), so it is no evidence of non-abelian confinement (Chatterjee 2018 §4, 5-06).

An action proportional to the area is called the Nambu–Goto action (the name is said to be after Nambu and Goto). We describe the surface by coordinates $t,x$ and a perpendicular displacement $h(t,x)\in\mathbb{R}^{d-2}$ (static gauge; $d$ is the dimension of spacetime). The area of a surface $X(t,x)$ is given by the surface-area formula learnt in multivariable calculus, $\int\!\!\int\sqrt{|X_t|^2|X_x|^2-(X_t\cdot X_x)^2}\,dt\,dx$ ($X_t=\partial_tX$, $X_x=\partial_xX$). Putting in $X=(t,x,h)$ gives the following.

If the fluctuations are gentle, expanding with $\sqrt{1+z}\approx 1+z/2$ gives

The second term is a massless free field with $d-2$ components, of the same form as the vibrating string of 2-02 (with $h=0$ at the ends). For a long string the higher terms are suppressed by powers of $1/(\sigma R^2)$ (ABT §2.1).

Example: the increase in length of a ripple $h=A_0\sin(\pi x/R)$ is $\pi^2A_0^2/(4R)$ in the quadratic approximation. With $R=1$ fm and $A_0=0.2$ fm this is 0.099 fm (0.092 fm exactly). Multiplying by $\sigma=1192$ MeV/fm ($\sqrt\sigma=485$ MeV, 7-01 §07) gives about 110 MeV Computed.

The second term of (3) is the action of a Euclideanised free field. When $T$ is long, $\langle W_{R,T}\rangle\approx e^{-V(R)T}$, so the ground-state energy of the Hamiltonian of the same field gives $V(R)$ (4-02). The modes of this field are $k_n=n\pi/R$ ($n=1,2,\dots$). Each mode is a harmonic oscillator of frequency $k_n$, and even in its lowest state it has the zero-point energy $k_n/2$ (2-06; $\hbar=c=1$). Summing over the $d-2$ components gives the following.

The sum diverges. Fluctuations finer than a length $a$ are outside the effective theory, so we cut off with $e^{-k_na}$.

Putting (5) into (4), the first term is $(d-2)R/(2\pi a^2)$, proportional to $R$, and is absorbed into the tension. What remains is the static potential.

$c_0$ is a non-universal constant. At $d=4$ this is $-\pi/(12R)$, the Lüscher term of 7-01 formula (6) (Lüscher–Symanzik–Weisz 1980, Lüscher 1981). It is a coefficient derived within the effective theory of long tubes, not a mathematical theorem Physics. The $1/R$ term is a finite part that does not diverge, so its coefficient does not depend on $a$. With $R=1$ and $a=0.2,\,0.1,\,0.05$, the remainder is $-0.2567,\,-0.2605,\,-0.2615$, approaching $-\pi/12$ Computed.

Closed strings and the Casimir energy of a cylinder

For a tube wound around a direction of period $L$ (flux going round a periodic boundary), $k_n=2\pi n/L$ occurs in both directions, positive and negative, and the same computation gives the following (ABT formula (9)).

This is the Casimir energy of a free field on a cylinder. Lucini–Teper 2001 extract the lattice $\sigma$ with formula (4) of their paper, $am_l=a^2\sigma L-\pi/(3L)$, and note: 'in using eqn(4) we are assuming that we have linear confinement' (5-08). A rough guide: at $R=1$ fm, $\pi/(12R)\approx52$ MeV is 4.3% of $\sigma R$, and at $L\sqrt\sigma=3$ (the lower limit of Lucini–Teper; about 1.22 fm converting with $\sqrt\sigma=485$ MeV, while Lucini–Teper themselves write ≃1.3 fm) the correction in (7) is 11.6% of the leading term Computed.

The modes $q_n$ of (3) are oscillators with mass $\sigma$ and frequency $k_n$, and the variance in the ground state is $1/(2\sigma k_n)$. At the midpoint of $h=\sum_n\sqrt{2/R}\,q_n\sin k_nx$ only odd $n$ survive, and $\langle h^2\rangle=\sum_{n\,\text{odd}}1/(\sigma\pi n)$. Summing up to $n\lesssim R/a$ gives a logarithm (Lüscher–Münster–Weisz 1981; Gliozzi–Pepe–Wiese 2010 formula (3)). It is a formula derived within the effective theory of long tubes Physics.

What is universal is the coefficient of $\ln R$; $R_0$ is not universal. The thickness grows without bound, but $w/R\to0$, so from afar it stays a thin line.

ComputedThe numbers in this figure were computed on this machine. Cutoff $a=0.25/\sqrt\sigma$, the band is $\pm w$, one transverse component. The oscillation is a classical picture, not the quantum fluctuation itself.

ComputedSummed exactly with the closed form $1/\sqrt{m^2(m^2+4)}$ for each mode. The increase agrees with (9) to 6 digits.

At $d=4$ the coefficient $1/(\pi\sigma)$ is 0.053 fm², and doubling $R$ increases $w^2$ by only 0.037 fm² Computed. The logarithmic growth has been confirmed on the lattice for SU(2) in (2+1) dimensions (Gliozzi–Pepe–Wiese 2010) Physics.

The levels of a closed string with no longitudinal momentum can be written in closed form in Nambu–Goto (with the sums of the left- and right-moving excitations both equal to $n$; ABT formula (21)). It is a formula that comes from naive light-cone quantisation (Aharony–Komargodski cite, as the source of this method, Arvis 1983, which treated a string with fixed ends).

Expanding gives $\sigma L+\frac{4\pi}{L}\big(n-\frac{d-2}{24}\big)+\cdots$, and $n=0$ returns to (7). The ground state becomes imaginary at $\sigma L^2\lt\pi(d-2)/3$, which in $d=4$ is $L_H\sqrt\sigma=\sqrt{2\pi/3}=1.447$. If the periodic direction is regarded as time, $1/L$ is the temperature (4-02). The number of string states grows exponentially with energy, so above some temperature the sum of the partition function stops converging. This point is called the Hagedorn transition (for the growth of the states, see the detour of 7-05) Physics. ABT write that $L_H$ can be regarded as 'the Hagedorn/deconfining transition in the Nambu-Goto model', and go on to say that pure gauge theories at large $N$ do not reach this length. As the length shrinks, a first-order deconfinement transition occurs before it, at $l_c\sqrt\sigma=\sqrt\sigma/T_c\simeq1.65$ (the $l_c$ below Table 3).

ComputedBy formula (10). The length $l_c\sqrt\sigma$ at which the tube disappears on the lattice is the product of $a\sqrt\sigma$ and $l_c/a$ in Table 3 of ABT: SU(3) 1.544, 1.559, SU(5) 1.636, SU(6) 1.676. All are slightly longer than $L_H$, and the two at larger $N$ lie on either side of ABT's 1.65.

How far does universality go? According to Aharony–Komargodski 2013 'all the terms up to (and including) order 1/L^5 are universal': the ground state agrees with (10) up to $1/L^5$, the excitations for $d\gt3$ deviate universally at $1/L^5$, and theory-specific terms enter from $1/L^7$. These are results derived within the effective theory of long tubes Physics. Of the quantisation in (10) they also write 'There is no rigorous derivation'. ABT measured about 20 levels of closed tubes on the lattice for four-dimensional SU(N), and reported that many are well described by (10) down to short lengths where the $1/L$ expansion does not converge. Some states with $J^P=0^-$ ($J$ is the angular momentum about the axis of the tube, $P$ the sign under reflection) deviate strongly Physics.

In the last column of Table 3, the first excitation $3.03\sqrt\sigma$ of the string at $L\sqrt\sigma=3$ is of the same order as the $0^{++}$ glueball of SU(3), $3.405\sqrt\sigma$ (7-01 Table 2) Computed (this does not imply a correspondence). Turning "there is a single scale" into a mathematical sentence is the subject of 7-12.

On the strong-coupling side of the lattice, shifting part of the surface by one lattice step adds plaquettes, each carrying a small weight $u$. Transverse fluctuations are suppressed exponentially, and the surface stays rigid (flat). In the world of (9) the variance diverges logarithmically, and the surface is rough.

For crystal surfaces there is a roughening transition between the two. Itzykson–Peskin–Zuber 1980 argued that in four-dimensional lattice gauge theory too, roughening occurs at a finite coupling without any phase transition of the vacuum, and that $\sigma(\beta)$ has an essential singularity there. An essential singularity here is a singularity of the kind where the function and all its derivatives stay finite, but the Taylor expansion no longer represents the function at that point. An example is the function set to $e^{-1/x^2}$ for $x\gt0$ and $0$ for $x\le0$; its Taylor series at $x=0$ has all coefficients $0$. However far the series is extended, this point cannot be seen. IPZ write 'The roughening transition is a natural barrier to the extrapolation of strong-coupling series for the string tension'. For SU(2), $\beta_r$ is about 2 (with a large error), and the authors note that their four-dimensional argument is 'for the most part, intuitive' Physics.

The three-dimensional Z(2) gauge theory is dual to the three-dimensional Ising model. Dual means that under a change of variables the partition functions of the two models agree (up to simple factors) (the detour of 2-09; IPZ cite Wegner 1971). Under this correspondence the surface of a Wilson loop becomes an interface of the Ising model (IPZ formula (8)) Known. Measuring the gauge coupling by $t=\tanh\beta$ (in terms of the temperature of the dual Ising model, $t=e^{-2J/T}$; IPZ formula (7)), the phase transition is at $t_c=0.6418$. The roughening point, by the series estimate IPZ cite, is $t_r\approx0.46$, before the phase transition Physics. The effective string theory applies only on the rough side (Gliozzi–Pepe–Wiese 2010) Physics.

On the rigid side of §07, the tension also depends on direction. The edge of a Wilson loop at angle $\theta$ to the lattice axes becomes a staircase, and the minimal surface has $RT(|\cos\theta|+|\sin\theta|)$ plaquettes. At leading order in strong coupling this gives the following.

At $45^\circ$ it is $\sqrt2$ times larger Computed. In the continuum limit rotation symmetry is expected to return and the tension to become constant Physics.

What shape does a region of fixed area (a two-dimensional droplet) take when it is enclosed by a line whose tension depends on direction? With the tension $\tau(\mathbf n)$ for each normal $\mathbf n$, we minimise $\oint\tau\,ds$ with the enclosed area fixed. The answer is given by the construction of Wulff 1901 (bibliographic data for the proof that the construction gives the minimum is in the table of sources) Known.

If the tension is constant the answer is a circle; for $\tau=|n_1|+|n_2|$ of (11) it is the square $\{\max(|x|,|y|)\le1\}$. The total at area 1 is 4 for the square and $8/\sqrt\pi=4.51$ for a circle Computed.

An example where the direction-dependent tension is known exactly is the Ising model on the square lattice (coupling $J$, 2-09). The $\beta=1/T$ of this section is the inverse temperature ($k_B=1$; not the $\beta$ of lattice gauge theory). The equilibrium shape of an island of the minority (a droplet) in the low-temperature phase is the following curve (the first exact analysis is attributed to Rottman–Wortis 1981) Known.

(13) is the boundary of $\mathcal W_\tau$ in (12) itself. Its size too is as in (12): the extent $\max\mathbf x\cdot\mathbf n$ in the direction $\mathbf n$ equals the tension $\tau(\mathbf n)$ (in units of $J$). So $x,y$ are not positions but coordinates measured in units of energy. The intercept with the $x$ axis (the tension in the axis direction) agrees to 5 digits with $2J+\beta^{-1}\ln\tanh\beta J$ from Onsager's exact solution Computed. As $T\to0$, (13) becomes a square, and the tension is the same staircase count as in (11). As $T\to T_c$ the right-hand side approaches 2, and for $u=\beta x$, $v=\beta y$ we have $\cosh u+\cosh v\approx2+(u^2+v^2)/2$, so it becomes a circle.

ComputedThe numbers in this figure and table were computed on this machine ((13) solved by bisection for each angle).

At low temperature the diagonal tension falls first. A stepped edge has a number of arrangements (entropy), and the corners round off from there. Approaching $T_c$ the correlation length becomes much longer than the lattice spacing, the orientation of the lattice is forgotten, and the shape becomes a circle. This is how the expectation that rotation symmetry returns in the continuum limit (5-05) looks in an exactly tractable model.

That the droplet takes the Wulff shape in the low-temperature two-dimensional Ising model has been shown from the microscopic model (Dobrushin–Kotecký–Shlosman 1992, 1993) Known.

The Wulff problem applies directly to droplets of a two-dimensional phase with fixed area like this one. A closed ring of flux in Yang–Mills has no constraint fixing its area, and left alone it shrinks. Still, a closed tube on the lattice must turn corners, so its energy depends on its shape through $\sigma(\theta)$ Physics. How this is used in the record kept on this machine is in 10-02.

The string picture is about massless transverse fluctuations, and the glueball masses do not come out of it directly. ABT examine the possibility that the $J^P=0^-$ levels that deviate from (10) represent 'the massive modes associated with the internal structure of the flux tube' Physics. Since the only length scale is $1/\sqrt\sigma$, $m\sim\sqrt\sigma$ looks natural, but it is not derived (7-01 §09, 7-12). The view as a sum over surfaces appears in the next chapter, 7-04, as the $1/N$ expansion.

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Next chapter to read: 7-04 Large N — a view that organises the sum over surfaces by $1/N$.

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