When one thinks of quantum field theory, one’s mind is undoubtedly drawn to Feynman diagrams. The näıve view these diagrams as merely a concise and standardized way of describing collisions and processes. Feynman diagrams, however, encode a great deal of information, for each one corresponds exactly to the probability amplitude of the transition it describes. In other words, it is easily possib...

In this paper we show how to define the UV completion of a scalar field theory such that it is both UV-finite and perturbatively unitary. In the UV completed theory, the propagator is an infinite sum of ordinary propagators. To eliminate the UV divergences, we choose the coefficients and masses in the propagator to satisfy certain algebraic relations, and define the infinite sums involved in Fe...

Feynman diagrams for gluon tree amplitudes are studied in the gauge and any number of spacetime dimensions. The color-kinematics combinations $\Delta=n_s-n_t-n_u$ numerators explicitly calculated $N=4,5,6$ gluons to see whether relation $\Delta=0$ is satisfied. This a tedious task because presence four-gluon vertices, large diagrams, numerators, $\Delta$ involved, especially when $N=6$. For on-...

with the arguments xj being, in general, rational functions (typically, simple ratios) of kinematic invariants of the original Feynman diagram, and the parameters {γj} and {σk} being linear combinations of the exponents of propagators and the dimension of space-time. The γj and σk are called upper and lower parameters, respectively. These statements follow from the multiple Mellin-Barnes repres...

The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.