Perturbative calculations in quantum field theory are based on S-matrix expansion and calculating the Feynman diagrams. These Feynman diagrams are related to the scattering cross-sections and decay rates by appropriate formulae.

  1. Can anyone enlighten me about how the non-perturbative calculations are performed in any quantum field theory? For example, I have reasonable familiarity with instantons. But don't know how to calculate the effects of instantons and to make measurable predictions from it.

  2. What are the typical quantities that one can calculate (like the scattering cross-section, decay rates etc in perturbative approach) in the non-perturbative approach?

  3. Is there a general rule (such as Feynman diagram calculation in perturbative approach) to calculate non-perturbativily calculable effects?

The whole non-perturbative scheme of calculation is not quite clear to me.

Note: If this question is too broad to answer, it would suffice to know "how will an instanton calculation be mathematically associated to some measurable quantity (like Feynman amplitude calculation is related to cross-section.)".


closed as too broad by AccidentalFourierTransform, JamalS, heather, Martin, rob Jan 9 '17 at 9:17

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ In quantum mechanics nonperturbative calculations of GS properties ensue from combining first order perturbation theory with variational methods (see here: physics.stackexchange.com/q/240506) A similar approach also works in QFT for some problems (see here: physics.stackexchange.com/q/284368 ) Hope this helps. $\endgroup$ – Lewis Miller Jan 5 '17 at 19:26
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    $\begingroup$ There is no "non-perturbative scheme of calculation", you use whatever is appropriate to the issue at hand. Non-perturbative calculations encompass non-equilibrium field theory, lattice field theory, standard path integral computations, and many other subfields. This is too broad a question; and for how instantons can be associated to some measureable quantity this again has many answers, depending on which theory you are talking about, the answer in Peccei-Quinn theory is certainly different from that in QCD is different again from supergravity´, and so on. $\endgroup$ – ACuriousMind Jan 6 '17 at 14:25

1) The observables in field theory are (T-ordered) correlation functions. These correlation functions have perturbative (P) and non-perturbative (NP) contributions, but the relationship between the correlators and observables is obviously the same, independent of whether the correlator is dominated by P or NP effects. For example, the correlation fucntion of the QCD vector currents $$ \Pi_{\alpha\beta}(x) = \langle j_\alpha(x) j_\beta(0)\rangle $$ is related to the famous $R$ ratio of $e^+e^-\to {\it hadrons}$ over $e^+e^-\to\mu^+\mu^-$, $$ R(s)\sim \pi\, {\rm Im}\,\Pi(q^2=s+i\epsilon), $$ where $\Pi_{\alpha\beta}(q)=(g_{\alpha\beta}q^2-q_\alpha q_\beta)\Pi(q^2)$, and I have dropped some factors related to quark charges. The correlation function is known perturbatively to four or five loops (I lost track), and it has a calculable instanton contribution.

2) Ordinary perturbation theory proceeds by expanding around the trivial vacuum. Non-perturbative effects arise from expanding around non-trivial saddle points, $A_\mu=A_\mu^0+\delta A_\mu$, where $A_\mu^0$ is the field of a (multi) instanton, monopole, etc. At leading order this is a completely classical calculation, and at higher order it involves propagators in the background field of a (multi) instanton (etc.). You can view these background field propagators and vertices as a new set of Feynman rules.

3) There are many subtleties in the interplay of P and NP effects. For example, P-theory is in general divergent (not even Borel resummable), and any attempt to define the perturbative sum typically involves NP ambiguities of the form $\exp(-2/g)$, where $g$ is the coupling. These ambiguities have to cancel against higher order NP ambiguities, a phenomenon known as resurgence.

4) In practice the trick is to find correlation functions that vanish to all orders in perturbation theory, have calculable non-perturbative effects, and are related to an interesting physical observable. A possible example would be the $U(1)_A$ puzzle in QCD, because the mass difference (quark mass effects ignored) betwee the $\eta'$ and the pion vanishes to all orders in perturbation theory. This mass difference has an instanton contribution, but it is not reliably calculable (due to the IR problem of instanton physics in QCD).

5) There are some interesting calculations that have been performed that satisfy the criteria in 4). These include: i) The gluino condensate in ${\cal N}=1$ SUSY Yang-Mills [1], ii) The $\eta'$ mass in high density QCD [2], iii) Certain correlation functions in QCD [3], iv) The quark condensate and pion decay constant in deformed QCD [4].


As for Your narrow question, just an example.

Suppose the naïve PCAC equation for the sub-group $U_{A}(1)$ of the full global chiral symmetry group $U_{L}(3)\times U_{A}(3)$ of the QCD. It is given by $$ \tag 1 \partial_{\mu}J^{\mu}_{5} \simeq f\partial^{2}\eta{'} = -m_{\eta{'}}^{2}f\eta{'}, $$ where $\eta{'}$ would be the QCD's ninth pseudo-scalar pseudo-goldstone boson, $f \simeq f_{\pi}$ is the associated quantity determining its decay width, and $m_{\eta'}$ is the mass generated by non-zero $u,d,s$-quarks masses. This naive equation $(1)$, however, is modified by the axial anomaly: $$ \tag 2 f_{\pi}\partial^{2}\eta{'} \simeq -m_{\eta{'}}^{2}f_{\pi}\eta{'} + \frac{3g^{2}}{16\pi^{2}}G_{\mu\nu}^{a}\tilde{G}^{\mu\nu,a}, $$ where $a$ denotes the color indice and $G$ is the gluon field strength. Eq. $(2)$ says us that the anomaly generates an effective interaction term $$ L_{\text{int}} = \frac{3g^{2}}{16 \pi^{2}}\frac{\eta{'}}{f_{\pi}}G_{\mu\nu}^{a}\tilde{G}^{\mu\nu,a} $$ One can obtain the correction to the self-energy of $\eta'$ by calculating the following Green function: $$ \Pi(p^{2}) \equiv \frac{9g^{4}}{256 \pi^{4}f_{\pi}^{2}} \int d^{4}x e^{ipx}\langle \text{vac}|T\big(G(x)\tilde{G}(x)G(0)\tilde{G}(0)\big)|\text{vac}\rangle $$ In particular, its value for $p = 0$ generates the correction $\Delta m_{\eta'}^{2}$ to $\eta{'}$ mass, which in fact is much larger than the initial "bare" $m_{\eta{'}}^{2}$, giving rise to the solution of the so-called $U_{A}(1)$ problem in the QCD.

The integral is proportional to the so-called topological susceptibility $\kappa (p^{2})$, defined as $$ \kappa (p^{2}) \equiv \int d^{4}x e^{ipx}\langle \text{vac}|T\big(G(x)\tilde{G}(x)G(0)\tilde{G}(0)\big)|\text{vac}\rangle $$ It, as we know, is generated non-perturbatively by instantons, since is determined by fluctuations of the squared topological charges. It can be given in a form $$ \tag 3 \Pi(p^{2}) \simeq \frac{1}{2f_{\pi}^{2}}\int d^{4}x e^{ipx}\left(\frac{\delta^{2}E(\theta)}{\delta\theta(x)\delta\theta(0)}\right)_{\theta = 0}, $$ where $E(\theta)$ is the QCD's $\theta$-vacua effective potential defined by the Euclidean path integral $$ \tag 4 e^{-E(\theta)} \equiv \int DG_{\mu}D\bar{\psi}D\psi e^{-S_{\text{QCD}} - \int d^{4}x\theta G_{\mu\nu}\tilde{G}^{\mu\nu}} $$ The computations are given in the following article, Sec. 5. Some general ideas about how to compute this quantity are given in Weinberg's QFT Vol. 2, 23.7. The essential thing is that $(3)$ is generated by the fluctuations around the stationary points of the exponent $(4)$, which are instanton solutions.

  • $\begingroup$ Two minor comments: 1) It is hard to make equ.(1,2) precise, except, possibly, in the large N limit. 2) The $G\tilde{G}$ correlator has both perturbative and non-perturbative contributions. We expect the integral over $x$ (the $p\to 0$ limit) to be purely non-perturbative. $\endgroup$ – Thomas Jan 6 '17 at 15:39

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