We apply the harmonic superspace approach for calculating the divergent part of the one-loop effective action of $6D$, ${\cal N}=(1,0)$ supersymmetric higher-derivative gauge theory with a dimensionless coupling constant. Our consideration uses the background superfield method allowing to carry out the analysis of the effective action in a manifestly gauge covariant and ${\cal N}=(1,0)$ supersymmetric way. We exploit the regularization by dimensional reduction in which the divergences are absorbed into a renormalization of the coupling constant. Having the expression for the one-loop divergences, we calculate the relevant $β$-function. Its sign is specified by the overall sign of the classical action which in higher-derivative theories is not fixed {\it a priori}. The result agrees with the earlier calculations in the component approach. The superfield calculation is simpler and provides possibilities for various generalizations.
We consider $6D$, ${\cal N}=(1,1)$ supersymmetric Yang-Mills theory formulated in ${\cal N}=(1,0)$ harmonic superspace and analyze the structure of the two-loop divergences in the hypermultiplet sector. Using the ${\cal N}=(1,0)$ superfield background field method we study the two-point supergraphs with the hypermultiplet legs and prove that their total contribution to the divergent part of effective action vanishes off shell.
We study the one-loop effective action for $6D,$ ${\cal N}=(1,0)$ supersymmetric Yang--Mills (SYM) theory with hypermultiplets and $6D,$ ${\cal N}=(1,1)$ SYM theory as a subclass of the former, using the off-shell formulation of these theories in $6D,$ ${\cal N}=(1,0)$ harmonic superspace. We develop the corresponding supergraph technique and apply it to compute the one-loop divergences in the background field method ensuring the manifest gauge invariance. We calculate the two-point Green functions of the gauge superfield and the hypermultiplet, as well as the three-point gauge-hypermultipet G…
We give a relatively simple explanation of the light-cone supergraph prediction for the UV properties of the maximally supersymmetric theories. It is based on the existence of a dynamical supersymmetry which is not manifest in the light-cone supergraphs. It suggests that N=4 supersymmetric Yang-Mills theory is UV finite and N=8 supergravity is UV finite at least until 7 loops whereas the $n$-point amplitudes have no UV divergences at least until $L=n+3$. Here we show that this prediction can be deduced from the properties of light-cone supergraphs analogous to the light-by-light scattering eff…
Using the covariant N=2 harmonic supergraph techniques we calculate the one-loop low-energy effective action of N=4 super-Yang-Mills theory in Coulomb branch with gauge group SU(2) spontaneously broken down to U(1). The full dependence of the low-energy effective action on both the hypermultiplet and gauge fields is determined. The direct quantum calculation confirms the correctness of the exact N=4 SYM low-energy effective action derived in hep-th/0111062 on the purely algebraic ground by invoking a hidden N=2 supersymmetry which completes the manifest N=2 one to N=4. Our results provide an e…
We enlarge the usual D=3 N=1 supergraph techniques to include the case of (explicitly or spontaneously) broken supersymmetric gauge theories. To illustrate the utility of these techniques, we calculate the two-loop effective potential of the SQED3 by using the tadpole and the vacuum bubble methods. In these methods, to investigate the possibility of supersymmetry breaking, the superfields must be shifted by $θ$ dependent classical superfields (vacuum expectation values), what implies in the explicit breakdown of supersymmetry in the intermediate steps of the calculation. Nevertheless, after st…
We continue studying $6D, {\cal N}=(1,1)$ supersymmetric Yang-Mills (SYM) theory in the ${\cal N}=(1,0)$ harmonic superspace formulation. Using the superfield background field method we explore the two-loop divergencies of the effective action in the gauge multiplet sector. It is explicitly demonstrated that among four two-loop background-field dependent supergraphs contributing to the effective action, only one diverges off shell. It is also shown that the divergences are proportional to the superfield classical equations of motion and hence vanish on shell. Besides, we have analyzed a possib…
We study systematically, through two loops, the divergence structure of the supersymmetric WZ model defined on the N=1/2 nonanticommutative superspace. By introducing a spurion field to represent the supersymmetry breaking term F^3 we are able to perform our calculations using conventional supergraph techniques. Divergent terms proportional to F, F^2 and F^3 are produced (the first two are to be expected on general grounds) but no higher-point divergences are found. By adding ab initio F and F^2 terms to the original lagrangian we render the model renormalizable. We determine the renormalizati…
In \cN = 2, 4 superconformal field theories in four space-time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in \cN = 1 superspace, we test the Dine-Seiberg proposal by comparing the two-loop F^4 quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) \cN = 4 SYM (i.e. \cN = 2 SYM …
We explicitly compute the complete three-loop (O(g^4)) contribution to the four-point function of chiral primary current-like operators <(q)^2 q^2 (q)^2 q^2> in any finite N=2 SYM theory. The computation uses N=2 harmonic supergraphs in coordinate space. Dramatic simplifications are achieved by a double insertion of the N=2 SYM linearized action, and application of superconformal covariance arguments to the resulting nilpotent six-point amplitude. The result involves polylogarithms up to fourth order of the conformal cross ratios. It becomes particularly simple in the N=4 special case.
We review the recent progress in studying the quantum structure of $6D$, ${\cal N}=(1,0)$ and ${\cal N}=(1,1)$ supersymmetric gauge theories formulated through unconstrained harmonic superfields. The harmonic superfield approach allows one to carry out the quantization and calculations of the quantum corrections in a manifestly ${\cal N}=(1,0)$ supersymmetric way. The quantum effective action is constructed with the help of the background field method that secures the manifest gauge invariance of the results. Although the theories under consideration are not renormalizable, the extended supers…
We develop a systematic approach to construct the one-loop ${\cal N}=4$ SYM effective action depending on both ${\cal N}=2$ vector multiplet and hypermultiplet background fields. Beginning with the formulation of ${\cal N}=4$ SYM theory in terms of ${\cal N}=2$ harmonic superfields, we construct the one-loop effective action using the covariant ${\cal N}=2$ harmonic supergraphs and calculate it in ${\cal N}=2$ harmonic superfield form for constant Abelian strength $F_{mn}$ and corresponding constant hypermultiplet fields. The hypermultiplet-dependent effective action is derived and given by in…
So far, quantum properties of N=1/2 nonanticommutative (NAC) super Yang--Mills theories have been investigated in the WZ gauge. The gauge independence of the results requires assuming that at the quantum level supergauge invariance is not broken by nonanticommutative geometry. In this paper we use an alternative approach which allows studying these theories in a manifestly gauge independent superspace setup. This is accomplished by generalizing the background field method to the NAC case, by moving to a momentum superspace where star products are treated as exponential factors and by developin…
In this article actions for N=4 SYM and N=8 supergravity are formulated in terms of a chiral superfield, which contains only the physical degrees of freedom of either theory. In these new actions, which originate from the lightcone superspace, the supergravity cubic vertex is the square of the gauge theory one (omitting the color structures). Amplitude calculations using the corresponding Feynman supergraph rules are tedious, but can be simplified by choosing a preferred superframe. Recursive calculations of all MHV amplitudes in N=4 SYM and the four-point N=8 supergravity amplitude are shown …
One-loop finiteness in higher-derivative $6D$, ${\cal N}=(1,0)$ super Yang-Mills -- hypermultiplet system
Joint Institute for Nuclear Research（俄罗斯联合核研究所）；Bogoliubov Laboratory of Theoretical Physics（博戈留博夫理论物理实验室）；Center of Theoretical Physics, Tomsk State Pedagogical University（托木斯克国立师范学院理论物理中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Moscow State University（莫斯科国立大学）
We consider supersymmetric gauge theories coupled to hyper multiplets on five and six dimensional orbifolds and determine the bulk and local fixed point renormalizations of the gauge couplings. We infer from a component analysis that the hyper multiplet does not induce renormalization of the brane gauge couplings on the five dimensional orbifold S^1/Z_2. This is not due to supersymmetry, since the bosonic and fermionic contributions cancel separately. We extend this investigation to T^2/Z_N orbifolds using supergraph techniques in six dimensions. On general Z_N orbifolds the gauge couplings do…
Using Schrödinger functional methods, we show that in the ${\cal N}=4$ SYM/Type IIB Supergravity correspondence the renormalisation of the boundary Newton and gravitational constants arising from bulk fields cancels when we sum over all the Kaluza-Klein modes of Supergravity. This accords with the expected finiteness of ${\cal N}=4$ SYM, and it is expected that other renormalisations cancel in a similar way.
We shall develop a general technique to obtain the super heat kernel coefficients of an arbitrary second order operator in $N=1$ superspace. We focus on the space of conformal supergravity here but the method presented is equally applicable for other types of superspace. The first three coefficients which determine the one-loop divergence of the corresponding quantum theory will be calculated. As an application we shall present the one-loop logarithmic divergence of super Yang-Mills theory coupled to a string dilaton $S$. This is the first superfield calculation for SYM with a non-trivial gaug…
We study the gauge dependence of one-loop divergences in a general matter-coupled $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory in the harmonic superspace formulation. Our analysis is based on the effective action constructed by the background superfield method, with the gauge-fixing term involving one real parameter $ξ_0$. A manifestly gauge invariant and ${\cal N}=(1,0)$ supersymmetric procedure for calculating the one-loop effective action is developed. It yields the one-loop divergences in an explicit form and allows one to investigate their gauge dependence. As compared to the minima…
We consider, in the harmonic superspace approach, the six-dimensional N=(1,0) supersymmetric Yang-Mills gauge multiplet minimally coupled to a hypermultiplet in an arbitrary representation of the gauge group. Using the superfield proper-time and background-field techniques, we compute the divergent part of the one-loop effective action depending on both the gauge multiplet and the hypermultiplet. We demonstrate that in the particular case of N=(1,1) SYM theory, which corresponds to the hypermultiplet in the adjoint representation, all one-loop divergencies vanish, so that N=(1,1) SYM theory is…
The two-loop (Euler-Heisenberg-type) effective action for N = 2 supersymmetric QED is computed using the N = 1 superspace formulation. The effective action is expressed as a series in supersymmetric extensions of F^{2n}, where n=2,3,..., with F the field strength. The corresponding coefficients are given by triple proper-time integrals which are evaluated exactly. As a by-product, we demonstrate the appearance of a non-vanishing F^4 quantum correction at the two-loop order. The latter result is in conflict with the conclusion of hep-th/9710142 that no such quantum corrections are generated at …
We consider the leading and subleading UV divergences for the four-point on-shell scattering amplitudes in D=8 N=1 sypersymmetric Yang-Mills theory within the spinor-helicity and superfield formalism. This theory belongs to the class of maximally supersymmetric gauge theories and presumably possesses distinguished properties beyond perturbation theory. We obtain the recursive relations that allow one to get the leading and subleading divergences in all loops in a pure algebraic way staring from the one loop (for the leading poles) and two loop (for the subleading ones) diagrams. As a particula…
In this paper we continue the study of perturbative renormalizations in an $\mathcal{N}=(0,2)$ supersymmetric model. Previously we analyzed one-loop graphs in the heterotically deformed CP$(N-1)$ models. Now we extend the analysis of the $β$ function and appropriate $Z$ factors to two, and, in some instances, all loops in the limiting case $g^2\to 0$. The field contents of the model, as well as the heterotic coupling, remain the same, but the target space becomes flat. In this toy $\mathcal{N}=(0,2)$ model we construct supergraph formalism. We show, by explicit calculations up to two-loop orde…
We take our first step toward getting finite results from the alpha'-corrected D=4 N=4 SYM theory with emphasis on the field theory techniques. Starting with the classical action of N=4 SYM with the leading alpha'-corrections, we examine new divergence at one loop due to the presence of the alpha'-terms. The new vertices do not introduce additional divergence to the propagators or to the three-point correlators. However they do introduce new divergence, e.g., to the scalar four-point function which should be canceled by extra counter-terms. We expect that the counter-terms will appear in the 1…
We show in two simple examples that for one-dimensional quantum chains with quantum group symmetries, the correlation functions of local operators are, in general, infrared divergent. If one considers, however, correlation functions invariant under the quantum group, the divergences cancel out.
We derive a new non-singular tree-level KLT relation for the $n=5$-point amplitudes, with manifest $2(n-2)!$ symmetry, using information from one-loop amplitudes and IR divergences, and speculate how one might extend it to higher n-point functions. We show that the subleading-color ${\cal N}=4$ SYM 5-point amplitude has leading IR divergence of $1/ε$, which is essential for the applications of this paper. We also propose a relation between the subleading-color ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity 1-loop 5-point amplitudes, valid for the IR divergences and possibly for the whole ampli…
For massless $ϕ^4$ theory, we explicitly compute the lowest order non-local contributions to the one-loop effective action required for the determination of the trace anomaly. Imposing exact conformal invariance of the local part of the effective action, we argue that the issue of quadratic divergences does not arise in a theory where exact conformal symmetry is only broken by quantum effects. Conformal symmetry can thus replace low energy supersymmetry as a possible guide towards stabilizing the weak scale and solving the hierarchy problem, if (i) there are no intermediate scales between the …
We compute the coefficient of the potential three-loop divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops.
We argue that N=4 supergravity is 3-loop UV finite because the relevant supersymmetric candidate counterterm is known to be SL(2, R)x SO(6) invariant, which violates the Noether-Gaillard-Zumino current conservation. Analogous arguments, based on the universality properties of groups of type E7, also apply to N=5,6,8 in 4,5,7 loops, respectively, since the 1/N BPS invariants break duality symmetry between Bianchi identities and quantum corrected vector field equations.
The light-cone gauge allows to single out a set of ``transverse'' fields (TF), whose Green functions are free from UV divergences in SYM N=4. Green functions with external lines involving the remaining fields do instead exhibit divergences: indeed those fields can be expressed, by solving their equations of motion, as composite operators in terms of ``transverse'' fields. A set of exact identities (bridge identities) automatically realize their insertions in a path-integral formulation.
Resummations of infinite sets of higher-order perturbative contributions are often needed both in thermal field theory and at zero temperature. For instance, the behaviour of the Standard Model (SM) effective potential extrapolated to very high energies is known to be extremely sensitive to higher-order effects. The 2PI effective action provides a systematic approach to consistently perform such resummations. However, one of its major limitations was that its loopwise expansion introduces residual violations of possible global symmetries, thus giving rise to massive Goldstone bosons in the spo…
We derive a compact all-loop-order expression for the IR-divergent part of the N=4 SYM four-gluon amplitude, which includes both planar and all subleading-color contributions, based on the assumption that the higher-loop soft anomalous dimension matrices are proportional to the one-loop soft anomalous dimension matrix, as has been recently conjectured. We also consider the Regge limit of the four-gluon amplitude, and we present evidence that the leading logarithmic growth of the subleading-color amplitudes is less severe than that of the planar amplitudes. We examine possible 1/N^2 corrections…
We study the stability of the gauge hierarchy in hidden-sector supergravity theories. We show that a destabilizing tadpole can appear if a theory has a gauge- and global-symmetry singlet with renormalizable couplings to the visible fields. We also find a quadratically divergent two-loop contribution to the supersymmetric effective potential. This term illustrates the difficulty of using the "LHC mechanism" to control quadratic divergences in theories with Planck-scale vevs.
Stueckelberg mechanism introduces a scalar field, known as Stueckelberg field, so that gauge symmetry is preserved in the massive abelian gauge theory. In this work, we show that the role of the Stueckelberg field is similar to the Kulish and Faddeev coherent state approach to handle infrared (IR) divergences. We expect that the light-front quantum electrodynamics (LFQED) with Stueckelberg field must be IR finite in the massless limit of the gauge boson. We have explicitly shown the cancellation of IR divergences in the relevant diagrams contributing to self-energy and vertex correction at lea…
Inspiring ourselves by the assumption that the notion of symmetry itself is insufficient to construct the consistent physics of the Desert (the so-called region of energies beyond the Standard Model) and some additional insights are needed, we suggests that high-energy theories must take into account the higher-order variations of fields. With this in mind we propose the generalization of the concepts of kinetic energy and free particle. It is shown that the theory founded on such principles reveals major features of the genuinely high-energy one, first of all, it appears to be free from ultra…
We compute the one-loop 1PI contributions to all the propagators of the noncommutative N=1, 2, 4 super Yang-Mills (SYM) U(1) theories defined by the means of the theta-exact Seiberg-Witten (SW) map in the Wess-Zumino gauge. Then we extract the UV divergent contributions and the noncommutative IR divergences. We show that all the quadratic noncommutative IR divergences add up to zero in each propagator.
