# High Energy Behavior in Maximally Supersymmetric Gauge Theories in Various Dimensions

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## Abstract

**:**

## 1. Introduction

## 2. Spinor-Helicity Formalism in Various Dimensions and Amplitudes in D = 6, 8, 10 SYM Theories

#### 2.1. Spinor-Helicity Formalism

#### 2.2. On-Shell Momentum Superspace

## 3. The Structure of UV Divergences in the Leading, Subleading, etc. Orders of PT in SYM Theories

## 4. Properties of the Solutions and Numerical Analysis

#### 4.1. The Ladder Case

#### 4.1.1. D = 6

#### 4.1.2. D = 8

#### 4.1.3. D = 10

#### 4.2. The General Case

#### 4.2.1. D = 6

#### 4.2.2. D = 8

#### 4.2.3. D = 10

## 5. The Renormalization Procedure

#### 5.1. The Scheme Dependence

#### 5.2. Kinematically Dependent Renormalization

## 6. High Energy Behavior

## 7. Discussion

- (1)
- The on-shell scattering amplitudes contain the UV divergences that start from one loop (three loops) and do not cancel (except for the all-loop cancellation of the bubbles and triangles).
- (2)
- These divergences possess increasing powers of momenta (derivatives) with increasing order of PT. For the four-point scattering amplitude, this manifests itself as increasing power of the Mandelstam variables s or t. This means that the theory is not renormalizable by power counting.
- (3)
- Nevertheless, all the higher loop divergences are related to the lower ones via explicit pole equations which are the generalization of the RG equations to the case of non-renormalizable theories. The leading divergences are governed by the one-loop counter term, the subleading ones by the two-loop counter term, etc. This happens exactly as in the well known case of renormalizable interactions.
- (4)
- The summation of the leading and subleading divergences can be performed by solving the generalized RG equations. The solution to these equations depends on dimension and has a different form in different dimensions. For particular sets of diagrams, one can get an analytical solution, while in the general case it is only numerical.
- (5)
- In D = 6, the solution is characterized by the exponential function that decreases for some partial amplitudes and increases for the other as a function of $z={g}^{2}/\u03f5$. In D = 8 and D = 10, the solutions possess an infinite number of poles. This means that they do not have a finite limit when $z\to \infty $ ($\u03f5\to 0$) which would correspond to the finite answer when removing the regularization.
- (6)
- We reformulate the multiplicative renormalization procedure with replacement of the renormalization constant by an operator that depends on kinematics. As a result, one can construct a higher derivative theory that gives finite scattering amplitudes with a single arbitrary coupling g defined in PT within a given renormalization scheme. Transition to another scheme is performed by the action on the amplitude of the finite renormalization operator.
- (7)
- The high energy behavior of the amplitudes is governed by the generalized RG equations just as in renormalizable theories. In the three examples that we considered, this behavior is different but in all cases the amplitudes either increase with energy or hit the pole at finite energy, as in QED.
- (8)
- Thus, the maximal supersymmetric gauge theories at higher dimensions despite many attractive features still happened to be inconsistent at high energies. We hope that the methods of analysis developed here can be used in other non-renormalizable theories including gravity.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The universal expansion for the four-point scattering amplitude in SYM theories in terms of master integrals. The connected strokes on the lines mean the square of the flowing momentum.

**Figure 5.**Comparison of various approaches to solve Equation (67): PT, Pade, Ladder and Numerics. The PT and Pade curves coincide in a given interval.

**Figure 7.**Comparison of various approaches to solve Equation (68). The red and black lines are the numerical solutions described in the previous section between the first pole and between the first and the second ones. The green one is the PT. The blue one is the Pade approximation. The yellow one represents the Ladder analytical solution.

**Figure 8.**Comparison of PT (

**a**) and the ladder approximation (

**b**) in the region up to the first pole and (

**c**) the ladder approximation beyond the first pole. One can clearly see the pole structure of the function $\mathsf{\Sigma}$.

**Figure 9.**Comparison of various approaches to solve Equation (70). The red and black lines are the numerical solutions described in the previous section before the first pole and between the first and the second ones. The green one is the PT. The blue one is the Pade approximation. The yellow one represents the Ladder analytical solution.

**Figure 13.**Action of the Z-operator at the three loop level. The first, second and the last two diagrams in the right-hand side correspond to the three loop box counter terms and the third and fourth ones to the tennis-court counter terms.

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**MDPI and ACS Style**

Kazakov, D.; Bork, L.; Borlakov, A.; Tolkachev, D.; Vlasenko, D.
High Energy Behavior in Maximally Supersymmetric Gauge Theories in Various Dimensions. *Symmetry* **2019**, *11*, 104.
https://doi.org/10.3390/sym11010104

**AMA Style**

Kazakov D, Bork L, Borlakov A, Tolkachev D, Vlasenko D.
High Energy Behavior in Maximally Supersymmetric Gauge Theories in Various Dimensions. *Symmetry*. 2019; 11(1):104.
https://doi.org/10.3390/sym11010104

**Chicago/Turabian Style**

Kazakov, Dmitry, Leonid Bork, Arthur Borlakov, Denis Tolkachev, and Dmitry Vlasenko.
2019. "High Energy Behavior in Maximally Supersymmetric Gauge Theories in Various Dimensions" *Symmetry* 11, no. 1: 104.
https://doi.org/10.3390/sym11010104