Renormalization

Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.

The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum p^μ and ends up with 4-momentum r^μ. It emits a virtual photon carrying r^μ − p^μ to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum q^μ, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum q^μ uniquely, so all possibilities contribute equally and we must integrate.

This diagram's amplitude ends up with, among other things, a factor from the loop of

${\displaystyle -ie^{3}\int {\frac {d^{4}q}{(2\pi )^{4}}}\gamma ^{\mu }{\frac {i(\gamma ^{\alpha }(r-q)_{\alpha }+m)}{(r-q)^{2}-m^{2}+i\epsilon }}\gamma ^{\rho }{\frac {i(\gamma ^{\beta }(p-q)_{\beta }+m)}{(p-q)^{2}-m^{2}+i\epsilon }}\gamma ^{\nu }{\frac {-ig_{\mu \nu }}{q^{2}+i\epsilon }}.}$

The various γ^μ factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron. The factors of e are the electric coupling constant, while the ${\displaystyle i\epsilon }$ provide a heuristic definition of the contour of integration around the poles in the space of momenta. The important part for our purposes is the dependency on q^μ of the three big factors in the integrand, which are from the propagators of the two electron lines and the photon line in the loop.

This has a piece with two powers of q^μ on top that dominates at large values of q^μ (Pokorski 1987, p. 122):

${\displaystyle e^{3}\gamma ^{\mu }\gamma ^{\alpha }\gamma ^{\rho }\gamma ^{\beta }\gamma _{\mu }\int {\frac {d^{4}q}{(2\pi )^{4}}}{\frac {q_{\alpha }q_{\beta }}{(r-q)^{2}(p-q)^{2}q^{2}}}.}$

This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.

Similar loop divergences occur in other quantum field theories.

Figure 3. The vertex corresponding to the Z₁ counterterm cancels the divergence in Figure 2.

For example, in the Lagrangian of QED

${\displaystyle {\mathcal {L}}={\bar {\psi }}_{B}\left[i\gamma _{\mu }\left(\partial ^{\mu }+ie_{B}A_{B}^{\mu }\right)-m_{B}\right]\psi _{B}-{\frac {1}{4}}F_{B\mu \nu }F_{B}^{\mu \nu }}$

the fields and coupling constant are really bare quantities, hence the subscript B above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:

${\displaystyle \left({\bar {\psi }}m\psi \right)_{B}=Z_{0}{\bar {\psi }}m\psi }$

${\displaystyle \left({\bar {\psi }}\left(\partial ^{\mu }+ieA^{\mu }\right)\psi \right)_{B}=Z_{1}{\bar {\psi }}\left(\partial ^{\mu }+ieA^{\mu }\right)\psi }$

${\displaystyle \left(F_{\mu \nu }F^{\mu \nu }\right)_{B}=Z_{3}\,F_{\mu \nu }F^{\mu \nu }.}$

Gauge invariance, via a Ward–Takahashi identity, turns out to imply that we can renormalize the two terms of the covariant derivative piece

${\displaystyle {\bar {\psi }}(\partial +ieA)\psi }$

together (Pokorski 1987, p. 115), which is what happened to Z₂; it is the same as Z₁.

A term in this Lagrangian, for example, the electron–photon interaction pictured in Figure 1, can then be written

${\displaystyle {\mathcal {L}}_{I}=-e{\bar {\psi }}\gamma _{\mu }A^{\mu }\psi -(Z_{1}-1)e{\bar {\psi }}\gamma _{\mu }A^{\mu }\psi }$

The physical constant e, the electron's charge, can then be defined in terms of some specific experiment: we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the magnetic moment). The rest is the counterterm. If the theory is renormalizable (see below for more on this), as it is in QED, the divergent parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from Z₀ and Z₃).

The diagram with the Z₁ counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

Historically, the splitting of the "bare terms" into the original terms and counterterms came before the renormalization group insight due to Kenneth Wilson.[20] According to such renormalization group insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.

To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be independent of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of physical constants with changes in scale. This variation is encoded by beta-functions, and the general theory of this kind of scale-dependence is known as the renormalization group.

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity. This running does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction. For example, since the coupling in quantum chromodynamics becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon known as asymptotic freedom. Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

For example,

${\displaystyle I=\int _{0}^{a}{\frac {1}{z}}\,dz-\int _{0}^{b}{\frac {1}{z}}\,dz=\ln a-\ln b-\ln 0+\ln 0}$

is ill-defined.

To eliminate the divergence, simply change lower limit of integral into ε_a and ε_b:

${\displaystyle I=\ln a-\ln b-\ln {\varepsilon _{a}}+\ln {\varepsilon _{b}}=\ln {\tfrac {a}{b}}-\ln {\tfrac {\varepsilon _{a}}{\varepsilon _{b}}}}$

Making sure ε_b/ε_a → 1, then I = ln a/b.

Julian Schwinger discovered a relationship between zeta function regularization and renormalization, using the asymptotic relation:

${\displaystyle I(n,\Lambda )=\int _{0}^{\Lambda }dp\,p^{n}\sim 1+2^{n}+3^{n}+\cdots +\Lambda ^{n}\to \zeta (-n)}$

as the regulator Λ → ∞. Based on this, he considered using the values of ζ(−n) to get finite results. Although he reached inconsistent results, an improved formula studied by Hartle, J. Garcia, and based on the works by E. Elizalde includes the technique of the zeta regularization algorithm

${\displaystyle I(n,\Lambda )={\frac {n}{2}}I(n-1,\Lambda )+\zeta (-n)-\sum _{r=1}^{\infty }{\frac {B_{2r}}{(2r)!}}a_{n,r}(n-2r+1)I(n-2r,\Lambda ),}$

where the B's are the Bernoulli numbers and

${\displaystyle a_{n,r}={\frac {\Gamma (n+1)}{\Gamma (n-2r+2)}}.}$

So every I(m, Λ) can be written as a linear combination of ζ(−1), ζ(−3), ζ(−5), ..., ζ(−m).

Or simply using Abel–Plana formula we have for every divergent integral:

${\displaystyle \zeta (-m,\beta )-{\frac {\beta ^{m}}{2}}-i\int _{0}^{\infty }dt{\frac {(it+\beta )^{m}-(-it+\beta )^{m}}{e^{2\pi t}-1}}=\int _{0}^{\infty }dp\,(p+\beta )^{m}}$

valid when m > 0, Here the zeta function is Hurwitz zeta function and Beta is a positive real number.

The "geometric" analogy is given by, (if we use rectangle method) to evaluate the integral so:

${\displaystyle \int _{0}^{\infty }dx\,(\beta +x)^{m}\approx \sum _{n=0}^{\infty }h^{m+1}\zeta \left(\beta h^{-1},-m\right)}$

Using Hurwitz zeta regularization plus the rectangle method with step h (not to be confused with the Planck constant).

The logarithmic divergent integral has the regularization

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n+a}}=-\psi (a)+\log(a)}$

since for the Harmonic series ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{an+1}}}$ in the limit ${\displaystyle a\to 0}$ we must recover the series ${\displaystyle \sum _{n=0}^{\infty }1=1/2}$

For multi-loop integrals that will depend on several variables ${\displaystyle k_{1},\cdots ,k_{n}}$ we can make a change of variables to polar coordinates and then replace the integral over the angles ${\displaystyle \int d\Omega }$ by a sum so we have only a divergent integral, that will depend on the modulus ${\displaystyle r^{2}=k_{1}^{2}+\cdots +k_{n}^{2}}$ and then we can apply the zeta regularization algorithm, the main idea for multi-loop integrals is to replace the factor ${\displaystyle F(q_{1},\cdots ,q_{n})}$ after a change to hyperspherical coordinates F(r, Ω) so the UV overlapping divergences are encoded in variable r. In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as

${\displaystyle \left(1+{\sqrt {q}}_{i}q^{i}\right)^{-s}}$

so the multi-loop integral will converge for big enough s using the Zeta regularization we can analytic continue the variable s to the physical limit where s = 0 and then regularize any UV integral, by replacing a divergent integral by a linear combination of divergent series, which can be regularized in terms of the negative values of the Riemann zeta function ζ(−m).

A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[31] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

This approach covered the conceptual point and was given full computational substance[20] in the extensive important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

In more technical terms, let us assume that we have a theory described by a certain function ${\displaystyle Z}$ of the state variables ${\displaystyle \{s_{i}\}}$ and a certain set of coupling constants ${\displaystyle \{J_{k}\}}$. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the whole description of the physics of the system.

Now we consider a certain blocking transformation of the state variables ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$, the number of ${\displaystyle {\tilde {s}}_{i}}$ must be lower than the number of ${\displaystyle s_{i}}$. Now let us try to rewrite the ${\displaystyle Z}$ function only in terms of the ${\displaystyle {\tilde {s}}_{i}}$. If this is achievable by a certain change in the parameters, ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$, then the theory is said to be renormalizable.

The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.

The most important information in the RG flow is its fixed points. A fixed point is defined by the vanishing of the beta function associated to the flow. Then, fixed points of the renormalization group are by definition scale invariant. In many cases of physical interest scale invariance enlarges to conformal invariance. One then has a conformal field theory at the fixed point.

The ability of several theories to flow to the same fixed point leads to universality.

If these fixed points correspond to free field theory, the theory is said to exhibit quantum triviality. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.[32]

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