Gap Equations, Dimensional Transmutation, and Physical Mass

A gap equation is a stationarity condition for an effective action. Its nonzero solution shows that a regulated saddle has generated a scale, but the symbol appearing in that saddle is a physical mass only after it is matched to a declared observable. The matching can be direct, as for the leading large-$N$ $O(N)$ vector propagator; indirect, as for gauge-invariant channels in $\mathrm{CP}^{N-1}$; or different for distinct excitations, as in the Gross–Neveu fermion and auxiliary-field channels.

Required background. The O(N) model as a strong-coupling laboratory supplies the regulated leading gap equation. Helpful background. Vector models and auxiliary large-N saddles supplies the determinant expansion and the order in $1/N$.

Renormalizing the O(N) saddle

At leading large $N$, with $t_0=Ng_0^2$, a circular Euclidean momentum cutoff gives Mariño 2015, § 6.2, pp. 194–200

$$\frac{1}{t_0} =\frac{1}{4\pi} \ln\!\left(1+\frac{\Lambda^2}{m^2}\right).$$

For $\Lambda\gg m$, separate the logarithmic divergence by defining

$$\frac{1}{t_R(\mu)} \equiv \frac{1}{t_0} -\frac{1}{4\pi} \ln\!\frac{\Lambda^2}{\mu^2}.$$

The continuum saddle equation becomes

$$\frac{1}{t_R(\mu)} =\frac{1}{4\pi} \ln\!\frac{\mu^2}{m^2}, \qquad m =\mu\exp\!\left[-\frac{2\pi}{t_R(\mu)}\right].$$

Holding the bare theory fixed and differentiating with respect to $\mu$ gives

$$\left. \mu\frac{\mathrm dt_R}{\mathrm d\mu} \right|_{N=\infty} =-\frac{t_R^2}{2\pi}.$$

Finite-$N$ corrections carry powers of both $t_R$ and $1/N$; a bare $O(1/N)$ symbol would not specify their coupling order. The displayed right-hand side for $m$ is RG invariant at leading $N=\infty$. The calculation is dimensional transmutation: the dimensionless renormalized coupling at a reference scale is replaced by the dimensionful invariant $m$. A finite redefinition of $t_R$ rescales the associated $\Lambda$ parameter, so a bare exponential without a named scheme is not a universal numerical prediction. The integral and RG checks are reproduced in the chapter’s benchmark.

The exact cutoff equation contains a useful check:

$$m =\frac{\Lambda} {\sqrt{e^{4\pi/t_0}-1}}.$$

Expanding at small $t_0$ recovers the transmuted exponential; expanding outside that regime has no continuum weak-bare-coupling interpretation.

Five different quantities called a gap

The same letter $m$ is often used for logically distinct objects:

1. Auxiliary saddle: a constant stationary value such as $\lambda=m_{\mathrm{aux}}^2$ or $\sigma=m_{\mathrm{aux}}$. This is defined by the effective action and its regulator.
2. Inverse correlation length: for a specified Euclidean operator $\mathcal O$, $$\langle\mathcal O(x)\mathcal O(0)\rangle_c \sim e^{-\lvert x\rvert/\xi_{\mathcal O}}$$ at large separation, up to powers and multiparticle effects.
3. Pole mass: an isolated pole of a Lorentzian two-point function at $p^2=M_{\mathrm{pole}}^2$, for a stable state created by the declared operator.
4. Screening mass: the inverse range extracted from a static response or spatial correlator. At zero temperature in a Lorentz-invariant vacuum it can coincide with a particle mass in a suitable channel; at finite temperature or density it need not.
5. Finite-volume spectral gap: $E_1(L)-E_0(L)$. It approaches an infinite-volume mass only after the state and the $L/\xi\to\infty$ limit are controlled.

The pole and correlation-length definitions can also be channel-dependent. A theory may have a lightest scalar, vector, or topological excitation with different masses. A branch cut can determine large-distance behavior without an isolated one-particle pole.

Observable matching in three models

O(N): direct at leading order

For the large-$N$ $O(N)$ saddle,

$$\langle n^a(p)n^b(-p)\rangle \propto \frac{\delta^{ab}}{p^2+m_{\mathrm{aux}}^2} +O(1/N).$$

The field $n^a$ is a physical local operator, so

$$m_{\mathrm{aux}} =\xi_n^{-1} =M_{\mathrm{vector}} \qquad\text{at leading }N=\infty.$$

At subleading order, the self-energy shifts the pole and the relation to a chosen RG scale. The equality is an output of the propagator, not a definition applied to every multiplier.

CP(N−1): charged saddle, gauge-invariant spectrum

In the projective model the constraint saddle again gives $\lambda=m_{\mathrm{aux}}^2$, and a gauge-fixed $z$ propagator contains $p^2+m_{\mathrm{aux}}^2$. But $z$ carries the redundant $U(1)$ charge. A physical claim must instead use a gauge-invariant operator such as $z^\dagger T^A z$, a Wilson-line-dressed bilocal, or a finite-volume energy. Its spectral density can begin at a multiparticle threshold, and the induced gauge dynamics can reorganize the spectrum Coleman 1985, ch. 8, §§ 2.2–2.3, pp. 358–367. Therefore

$$m_{\mathrm{aux}}\ne \text{automatically a gauge-invariant pole mass}.$$

Gross–Neveu: fermion mass versus sigma channel

In the discrete-chiral Gross–Neveu model, a constant Hubbard–Stratonovich saddle $\sigma_0$ enters the fermion inverse propagator as

$$S_F^{-1}(p)=p\!\!\!/-\sigma_0.$$

At leading large $N$, the fermion pole mass is $M_F=\lvert\sigma_0\rvert$. The propagating fluctuation $\delta\sigma$ is governed by a fermion bubble, not by the number $\sigma_0^2$ inserted into a free scalar propagator. Its pole or threshold must be computed separately. Moreover, in a continuous-chiral variant the amplitude saddle does not license a finite-$N$ continuous order parameter Gross and Neveu 1974, §§ II–IV.

These distinctions are aligned with the chapter’s model-regime map and observable-and-control comparison.

Finite volume and order of limits

To study a zero-temperature spectrum, compactify only space to a circle of length $L$. It is useful to begin on a Euclidean torus with inverse temperature $\beta$ and spatial size $L$:

$$\frac{1}{t_0} =\frac{1}{\beta L} \sum_{n_0,n_1\in\mathbb Z} \frac{1}{(2\pi n_0/\beta)^2+(2\pi n_1/L)^2+m_{\beta,L}^2}.$$

Taking $\beta\to\infty$ first turns the temporal sum into a continuous frequency integral:

$$\frac{1}{t_0} =\frac1L\sum_{n\in\mathbb Z} \int\frac{\mathrm dp_0}{2\pi} \frac{1}{p_0^2+(2\pi n/L)^2+m_L^2} =\frac1{2L}\sum_{n\in\mathbb Z} \frac{1}{\sqrt{(2\pi n/L)^2+m_L^2}}.$$

With the same ultraviolet regulator in the sum and integral, Poisson resummation gives

$$I_L(m)-I_\infty(m) =\frac1\pi\sum_{\ell=1}^{\infty}K_0(\ell mL) \sim \frac{e^{-mL}}{\sqrt{2\pi mL}}$$

for $mL\gg1$, where the last expression is the leading $\ell=1$ term. When $mL\lesssim1$, zero modes and finite-size effects are not small. One must specify whether the limits are

$$\Lambda\to\infty,\qquad L\to\infty,\qquad N\to\infty$$

and in what order. A square $L\times L$ Euclidean torus has temperature $1/L$ and is not the zero-temperature Hamiltonian problem. Taking $N\to\infty$ first can also create a sharp saddle or apparent order that infrared fluctuations modify at every finite $N$.

A practical mass-identification test

Given a nonzero saddle parameter, ask:

• Which renormalized action and scheme define it?
• Which gauge-invariant operator or finite-volume state is being measured?
• Is the signal an isolated pole, a threshold, or exponential screening?
• At what order in $1/N$, coupling, lattice spacing, and volume is the relation controlled?
• What dimensionless ratio can be checked by an independent method?

The final question is particularly valuable. Ratios such as $M_2/M_1$, $M/\Lambda_{\overline{\mathrm{MS}}}$, or $ML$ remove one arbitrary unit, though they can still depend on the continuum scheme or state definition.

Common pitfalls

Dropping the subtraction scale. Writing $m=\Lambda e^{-c/t_0}$ is a regulator-level result. Continuum comparisons require a renormalized coupling or a named $\Lambda$ parameter.

Reading a pole from the effective potential. The effective potential fixes zero-momentum stationary points. A pole requires the momentum-dependent second variation and analytic continuation.

Ignoring the operator channel. “The correlation length” is unambiguous only when the lightest state couples to the operator being measured. Selection rules can make another channel decay with a shorter length.

Exercises

1. Verify directly that

$$m(\mu) =\mu e^{-2\pi/t_R(\mu)}$$

is invariant under the leading beta function.

Solution

Taking a logarithmic derivative gives

$$\mu\frac{\mathrm d}{\mathrm d\mu}\ln m =1+\frac{2\pi}{t_R^2} \mu\frac{\mathrm dt_R}{\mathrm d\mu}.$$

Using $\mu\,\mathrm dt_R/\mathrm d\mu=-t_R^2/(2\pi)$ makes the right-hand side vanish. The equality is accurate to the same order as the beta function and saddle.

2. Suppose a gauge-invariant scalar correlator has spectral density

$$\rho(s)=0\quad(s<4m^2), \qquad \rho(s)>0\quad(s\ge4m^2),$$

with no isolated pole. What scale controls its long-distance Euclidean decay?

Solution

The first spectral support occurs at the two-particle threshold $\sqrt{s}=2m$. The correlator therefore decays as $e^{-2m\lvert x\rvert}$ times a dimension-dependent power determined by the threshold behavior. Its inverse correlation length is $2m$, but there is no scalar particle of pole mass $2m$. This is why a threshold and a pole must be distinguished.

3. Starting from the zero-temperature spatial-circle sum, use Poisson resummation to derive the $K_0$ finite-size correction.

Solution

For

$$f(p)=\frac{1}{2\sqrt{p^2+m^2}},$$

Poisson resummation gives

$$\frac1L\sum_{n\in\mathbb Z}f(2\pi n/L) =\sum_{\ell\in\mathbb Z} \int\frac{\mathrm dp}{2\pi} e^{i\ell Lp}f(p).$$

The $\ell=0$ term is the infinite-volume integral. For $\ell\ne0$,

$$\int\frac{\mathrm dp}{2\pi} \frac{e^{i\ell Lp}}{2\sqrt{p^2+m^2}} =\frac{1}{2\pi}K_0(\lvert\ell\rvert mL).$$

Combining positive and negative $\ell$ gives

$$I_L-I_\infty =\frac1\pi\sum_{\ell=1}^{\infty}K_0(\ell mL).$$

Since $K_0(x)\sim\sqrt{\pi/(2x)}e^{-x}$, the leading correction is $e^{-mL}/\sqrt{2\pi mL}$.

4. A saddle gives $m_{\mathrm{aux}}>0$, but a gauge-invariant correlator has only a continuum beginning at $2m_{\mathrm{aux}}$. Classify the auxiliary scale, the inverse correlation length, and the pole content.

Solution

$m_{\mathrm{aux}}$ is the stationary scale defined by the regulated effective action. The correlator’s first spectral support lies at energy $2m_{\mathrm{aux}}$, so its inverse correlation length is $2m_{\mathrm{aux}}$ up to the threshold power law. Because there is no isolated delta-function contribution or propagator pole below the cut, the channel has no one-particle pole at either $m_{\mathrm{aux}}$ or $2m_{\mathrm{aux}}$. This is precisely the distinction between saddle, threshold, correlation length, and particle mass.

Continue

Apply the test to The Gross–Neveu Model and Dynamical Mass Generation and The CP(N−1) Model.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 8, §§ 2.2–2.3. DOI.
• Gross, David J., and André Neveu. “Dynamical Symmetry Breaking in Asymptotically Free Field Theories.” Physical Review D 10 (1974): 3235–3253. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 6. DOI.