nLab non-perturbative effect




An effect in non-perturbative quantum field theory that cannot be seen in perturbative quantum field theory is called a non-perturbative effect.

More in detail, theories with instanton field configurations (such as in Yang-Mills theory, hence in QCD and QED) or branes (such as in string theory), etc., are expected to have observables which as functions of the coupling constant gg are transseries of the form

(1)Z(g)= na ng n+e A/g na n (1)g n+e B/g 2 nb n (1)g n, Z(g) = \sum_n a_n g^n + e^{-A/g} \sum_n a_n^{(1)} g^n + e^{-B/g^2} \sum_n b_n^{(1)} g^n \,,

where the first sum is the Feynman perturbation series itself and where the terms with a non-analytic dependence of the form exp(A/g)\exp(-A/g) or exp(A/g 2)\exp(-A/g^2) are the contributions of the instantons. Since all the derivatives of the functions ge 1/gg \mapsto e^{-1/g} or ge 1/g 2g \mapsto e^{-1/g^2} vanish at coupling constant g=0g = 0, the Taylor series of this part of the observable does not appear in perturbative QFT, even though it is present. Therefore this is called a non-perturbative effect.

Related is resurgence theory. See also at perturbation theory – Divergence/convergence for more.

Notice that, while non-perturbative effects are generally the least understood, they are not exotic but ubiquitous: Essentially the entire parameter space of a generic “physical theory” is non-perturbative. In fact, the non-perturbative region of parameter space is the complement of an infinitesimally thickened point (labeled “perturbative” in the following graphics):


Schwinger effect

Confinement and the mass gap

A central example of a non-perturbative effect is confinement (hence the “mass gap problem”) in Yang-Mills theory at low temperature. Perturbation theory is not suited to explain this (e.g Espiru 94, section 7).

Quark-gluon plasma

At the other extreme of high temperature QCD, also the quark-gluon plasma, while now deconfined is thought to be strongly coupled.

Hadronic physics in flavour and g2g-2 anomalies

Non-perturbative effects in hadron-physics affects the discussion of possible beyond-standard model physics as seen in

  1. flavour anomalies (Nierste 18, 26/59)

  2. muon anomalous magnetic moment (Jegerlehner 18b, section 2)

Worldsheet and brane instantons in string/M-theory

Consider the M-theory scales

and the string theory scales

Then under the duality between M-theory and type IIA string theory these scales are related as follows:

P=g st 1/3 s,AAAR 10=g st s \ell_P \;=\; g_{st}^{1/3} \ell_s \,, \phantom{AAA} R_{10} \;=\; g_{st} \ell_s


s=R 10/g st,AAA P=g st 2/3R 10 \ell_s \;=\; R_{10}/ g_{st} \,, \phantom{AAA} \ell_P \;=\; g_{st}^{-2/3} R_{10}


g st=(R 10/ P) 3/2,AAA s= P(R 10/ P) 1/2. g_{st} \;=\; (R_{10}/\ell_P)^{3/2} \,, \phantom{AAA} \ell_s \;=\; \ell_P (R_{10}/\ell_P)^{-1/2} \,.

Hence a membrane instanton, which on a 3-cycle C 3C_3 gives a contribution

exp(vol(C 3) P 3) \exp\left( - \frac{ vol(C_3) }{ \ell^3_P } \right)


  1. if the cycle wraps, C 3=C 2S 10 1C_3 = C_2 \cup S^1_{10}, a worldsheet instanton

    exp(vol(C 3) P 3)=exp(R 10vol(C 2)g st s 3)=exp(vol(C 2) s 2) \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 } \right) \;=\; \exp\left( - \frac{ R_{10} vol(C_2) }{ g_{st} \ell_s^3 } \right) \;=\; \exp\left( - \frac{ vol(C_2) }{ \ell_s^2 } \right)
  2. the cycle does not wrap, a spacetime instanton contribution, specifically a D2-brane instanton?

    exp(vol(C 3) P 3)=exp(vol(C 3)/ s 3g st) \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 } \right) \;=\; \exp\left( - \frac{ vol(C_3)/\ell_s^3 }{ g_{st} } \right)

(This unification of the two different non-perturbative effects in perturbative string theory (worldsheet instantons and spacetime instantons), to a single type of effect (membrane instanton) in M-theory was maybe first made explicit in Becker-Becker-Strominger 95. Brief review includes Marino 15, sections 1.2 and 1.3).


In field theory

Discussion of traditional algebraic quantum field theory as being about non-perturbative effects (and therefore lacking any interesting models in 3+1 dimensions, so far…):

Genral introduction and toy examples (e.g. phi^4-theory or anharmonic oscillator):

Discussion for phi^4 theory is in

  • Marco Serone, from 2:46 on in A look at ϕ 2 4\phi^4_2 using perturbation theory, January 2018 (recording)

Discussion for QCD includes

  • Marcos Mariño, Instantons and Large NN – An introduction to non-perturbative methods in QFT (pdf)

and in (super-)Yang-Mills theory and string theory is in

In QCD (hadrodynamics) via operator product expansion:

In phenomenology

In flavour anomalies:

In cosmology

In cosmology:

In string theory

The form of the contribution of non-perturbative effects in string theory was originally observed in

  • Stephen Shenker, The Strength of nonperturbative effects in string theory, presented at the Cargese Workshop on Random Surfaces, Quantum Gravity and Strings, Cargese, France, May 1990 (spire)

The interpretation via D-branes of non-perturbative effects in the string coupling constant is due to

The identification of non-perturbative contributions in string theory with brane contributions is due to

Review includes

Reviews specifically in type II string theory include

  • Hugo Looyestijn, Non-perturbative effects in type IIA string theory, Master Thesis 2006 (pdf)

  • Angel Uranga, Non-perturbative effects and D-brane instanton resummation in string theory (pdf, pdf)

In M-Theory

The relation of non-perturbative effects in string theory to M-theory goes back to

In Becker-Becker-Strominger 95 it was realized that the worldsheet instantons and D-brane instantons of string theory unify to membrane instantons (see Marino 15, section 1.3)

Last revised on December 4, 2023 at 19:57:37. See the history of this page for a list of all contributions to it.