Editing Quantum field theory (section)

===Renormalization===
{{Main|Renormalization}}

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The [[renormalisation]] procedure is a systematic process for removing such infinities.

Parameters appearing in the Lagrangian, such as the mass {{math|''m''}} and the coupling constant {{math|''λ''}}, have no physical meaning — {{math|''m''}}, {{math|''λ''}}, and the field strength {{math|''ϕ''}} are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off {{math|Λ}}, obtain expressions for the physical quantities, and then take the limit {{math|Λ → ∞}}. This is an example of [[regularization (physics)|regularization]], a class of methods to treat divergences in QFT, with {{math|Λ}} being the regulator.

The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of {{math|''ϕ''<sup>4</sup>}} theory, the field strength is first redefined:
:<math>\phi = Z^{1/2}\phi_r,</math>
where {{math|''ϕ''}} is the bare field, {{math|''ϕ<sub>r</sub>''}} is the renormalized field, and {{math|''Z''}} is a constant to be determined. The Lagrangian density becomes:
:<math>\mathcal{L} = \frac 12 (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 m_r^2\phi_r^2 - \frac{\lambda_r}{4!}\phi_r^4 + \frac 12 \delta_Z (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 \delta_m\phi_r^2 - \frac{\delta_\lambda}{4!}\phi_r^4,</math>
where {{math|''m<sub>r</sub>''}} and {{math|''λ<sub>r</sub>''}} are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
:<math>\delta_Z = Z-1,\quad \delta_m = m^2Z - m_r^2,\quad \delta_\lambda = \lambda Z^2 - \lambda_r</math>
are constants to be determined. The first three terms are the {{math|''ϕ''<sup>4</sup>}} Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or [[dimensional regularization]]); call the regulator {{math|Λ}}. Compute Feynman diagrams, in which divergent terms will depend on {{math|Λ}}. Then, define {{math|''δ<sub>Z</sub>''}}, {{math|''δ<sub>m</sub>''}}, and {{math|''δ<sub>λ</sub>''}} such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit {{math|Λ → ∞}} is taken. In this way, meaningful finite quantities are obtained.{{r|peskin|page1=323–326}}

<!--"Is it true?" The renormalization procedure will lead to the same quantitative result and physical prediction irrespective of the regularization scheme chosen. -->It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The [[Standard Model]] of elementary particles is a renormalizable QFT,{{r|peskin|page1=719–727}} while [[quantum gravity]] is non-renormalizable.{{r|peskin|zee|page1=798|page2=421}}

====Renormalization group====
{{Main|Renormalization group}}

The [[renormalization group]], developed by [[Kenneth G. Wilson|Kenneth Wilson]], is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.{{r|peskin|page1=393}} The way in which each parameter changes with scale is described by its [[beta function (physics)|''β'' function]].{{r|peskin|page1=417}} Correlation functions, which underlie quantitative physical predictions, change with scale according to the [[Callan–Symanzik equation]].{{r|peskin|page1=410–411}}

As an example, the coupling constant in QED, namely the [[elementary charge]] {{math|''e''}}, has the following ''β'' function:
:<math>\beta(e) \equiv \frac{1}{\Lambda}\frac{de}{d\Lambda} = \frac{e^3}{12\pi^2} + O\mathord\left(e^5\right),</math>
where {{math|Λ}} is the energy scale under which the measurement of {{math|''e''}} is performed. This [[differential equation]] implies that the observed elementary charge increases as the scale increases.<ref>{{cite arXiv |last=Fujita |first=Takehisa |eprint=hep-th/0606101 |title=Physics of Renormalization Group Equation in QED |date=2008-02-01 }}</ref> The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.{{r|peskin|page1=420}}

The coupling constant {{math|''g''}} in [[quantum chromodynamics]], a non-Abelian gauge theory based on the symmetry group {{math|[[special unitary group|SU(3)]]}}, has the following ''β'' function:
:<math>\beta(g) \equiv \frac{1}{\Lambda}\frac{dg}{d\Lambda} = \frac{g^3}{16\pi^2}\left(-11 + \frac 23 N_f\right) + O\mathord\left(g^5\right),</math>
where {{math|''N<sub>f</sub>''}} is the number of [[quark]] [[flavour (particle physics)|flavours]]. In the case where {{math|''N<sub>f</sub>'' ≤ 16}} (the Standard Model has {{math|''N<sub>f</sub>'' {{=}} 6}}), the coupling constant {{math|''g''}} decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as [[asymptotic freedom]].{{r|peskin|page1=531}}

[[Conformal field theories]] (CFTs) are special QFTs that admit [[conformal symmetry]]. They are insensitive to changes in the scale, as all their coupling constants have vanishing ''β'' function. (The converse is not true, however — the vanishing of all ''β'' functions does not imply conformal symmetry of the theory.)<ref>{{Cite journal |last1=Aharony |first1=Ofer |last2=Gur-Ari |first2=Guy |last3=Klinghoffer |first3=Nizan |arxiv=1501.06664 |title=The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants |journal=Journal of High Energy Physics |volume=2015 |issue=5 |pages=31 |date=2015-05-19 |bibcode=2015JHEP...05..031A |doi=10.1007/JHEP05(2015)031 |s2cid=115167208 }}</ref> Examples include [[string theory]]<ref name="polchinski1">{{cite book |last=Polchinski |first=Joseph |date=2005 |title=String Theory |volume=1 |publisher=Cambridge University Press |isbn=978-0-521-67227-6 |author-link=Joseph Polchinski }}</ref> and [[N = 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]].<ref>{{cite arXiv |last=Kovacs |first=Stefano |eprint=hep-th/9908171 |title={{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory and the AdS/SCFT correspondence |date=1999-08-26 }}</ref>

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off {{math|Λ}}, ''i.e.'' that the theory is no longer valid at energies higher than {{math|Λ}}, and all degrees of freedom above the scale {{math|Λ}} are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable [[effective field theory]].{{r|peskin|page1=402–403}} The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.{{r|shifman|page1=2}} According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off {{math|Λ}} from calculations in such a theory merely indicates that new physical phenomena appear at scales above {{math|Λ}}, where a new theory is necessary.{{r|zee|page1=156}}