Theoretical and Mathematical Physics, 192(1): 1016–1027 (2017)
ULTRAVIOLET DIVERGENCES IN D=8 N =1 SUPERSYMMETRIC
D. I. Kazakov
We consider the leading and subleading UV divergences for the four-point on-shell scattering amplitudes
in the D=8 N=1 supersymmetric Yang–Mills theory in the planar limit for ladder-type diagrams. We
obtain recurrence relations that allow obtaining the leading and subleading divergences in all loops purely
algebraically starting from the one-loop diagrams (for the leading poles) and the two-loop diagrams (for
the subleading poles). We sum the leading and subleading divergences over all loops using diﬀerential
equations that are generalizations of the renormalization group equations to nonrenormalizable theories.
We discuss the properties of the obtained solutions and the dependence of the constructed counterterms
on the scheme.
Keywords: amplitude, maximal supersymmetry, UV divergence
Dedicated to D. V. Shirkov
In recent years, we have seen signiﬁcant interest in calculating scattering amplitudes in maximally
supersymmetric Yang–Mills (SYM) theories ,  and maximally supersymmetric gravity  (theories
with the maximum group of supersymmetries in a given spatial dimension are said to be maximally su-
persymmetric). We speak of scattering amplitudes on the mass shell with the calculations in the so-called
planar limit where we consider the theory with the number of colors N
→∞under the condition that
the coupling constant g satisﬁes the relations g
→ const. The biggest successes were
achieved in the case of the four-dimensional N=4 supersymmetric gauge theory and four-dimensional N=8
supergravity theory. In these theories, all or almost all ultraviolet (UV) divergences cancel, and we have a
factorization of higher radiative corrections. The last property is manifested in the so-called Bern–Dixon–
Smirnov (BDS) ansatz  in the four-dimensional N=4 SYM theory and stimulated similar attempts in
other maximally supersymmetric theories in higher dimensions. A review of various results of calculating
amplitudes (S-matrices) in gauge theories in diﬀerent dimensions can be found in .
This paper is devoted to calculating the UV divergences in the eight-dimensional N=1 SYM theory for
the four-point scattering amplitudes on the mass shell in the planar limit in all orders of the perturbation
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Oblast,
Russia; Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia,
Physics Faculty, South Federal State University, Rostov-on-Don, Russia.
This research was supported by the Russian Foundation for Basic Research (Grant Nos. 14-02-00494 and 17-
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoretich eskaya i
Matematicheskaya Fizika, Vol. 192, No. 1, pp. 89–102, July, 2017. Original article submitted September 27, 2016.
2017 Pleiades Publishing, Ltd.