Superficial degree of divergence

The superficial degree of divergence , English superficial degree of divergence is, a size of the quantum field theory . The superficial degree of divergence of a 1-particle irreducible Feynman diagram is the upper limit with which strength this diagram diverges in the ultraviolet range : ${\displaystyle D}$

• for the diagram does not diverge ${\displaystyle D<0}$
• for it diverges logarithmically at most ${\displaystyle D=0}$
• for it diverges at most polynomially in order . ${\displaystyle D>0}$ ${\displaystyle D}$

The superficial degree of divergence depends on the structure of the theory, i.e. which fundamental interactions are permissible, the dimension of spacetime and the outer particles of the diagram.

A theory in which only a finite number of diagrams have a superficial degree of divergence can be renormalized or super-renormalized, otherwise it cannot be renormalized. The connection between renormalization and diverging diagrams is established by the BPHZ theorem . ${\displaystyle D\geq 0}$

background

Loops of internal, virtual particles can appear in Feynman diagrams , the momentum of which is not determined. Therefore this impulse has to be integrated. Each loop thus gives a factor

${\displaystyle \int {\frac {\mathrm {d} ^{d}k}{(2\pi )^{d}}}}$

in spacetime dimensions. ${\displaystyle d}$

On the other hand, every internal propagator through which the loop momentum flows leads to a factor for bosons and for fermions . ${\displaystyle k}$ ${\displaystyle k^{-2}}$ ${\displaystyle k\!\!\!/^{-1}}$

The following applies to the superficial degree of divergence:

${\displaystyle D=dL-2G_{B}-G_{F}}$

in which

• ${\displaystyle L}$ denotes the number of loops,
• ${\displaystyle G_{B}}$ the number of boson propagators
• ${\displaystyle G_{F}}$ the number of fermionic propagators,

Knowing the structure of the theory, this expression can be rewritten in terms of the number of outer particles and the number of vertices.

example

In quantum electrodynamics in the four-dimensional Minkowski space there is only one vertex. Here two fermions couple to a boson. This results in the superficial degree of divergence in quantum electrodynamics

${\displaystyle D=4-{\frac {3}{2}}N_{F}-N_{B}}$

With

• the number of outer fermions ${\displaystyle N_{F}}$
• the number of outer bosons . ${\displaystyle N_{B}}$

Therefore there are ten diagrams with . Six of them are identically zero due to various theorems ( fermion number conservation , furry theorem ). The four non-disappearing graphs are: ${\displaystyle D\geq 0}$

• the photon self-energy ( ) ${\displaystyle N_{B}=2,D=2}$
• the fermion self-energy ( ) ${\displaystyle N_{F}=2,D=1}$
• the vertex correction ( ) and ${\displaystyle N_{F}=2,N_{B}=1,D=0}$
• the diagram of light-light scattering ( ). ${\displaystyle N_{B}=4,D=0}$

Quantum electrodynamics is therefore a renormalizable theory.

Furthermore, it turns out that the Ward identity reduces the actual degree of divergence of the photon self-energy to , the chiral symmetry to that of the electron self-energy and the gauge invariance to that of the light-light scattering . ${\displaystyle {\tilde {D}}}$ ${\displaystyle {\tilde {D}}=0}$ ${\displaystyle {\tilde {D}}=0}$ ${\displaystyle {\tilde {D}}<0}$

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 381-393 (English).