We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives.
As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Notwithstanding the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the Lorentz invariance is recovered at the end. We develope this new setting for a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are outlined. This method …
As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Despite the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the explicit Lorentz invariance is recovered at the end. We develop this new setting on the grounds of a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are worked o…
In this paper, we consider an infinite derivative scalar field action with infinite derivative kinetic and interaction terms. We establish that the theory is unitary if the correlation functions are formulated in Euclidean space and then analytically continued in their external momenta to Minkowski space.
We carry out the extension of the covariant Ostrogradski method to fermionic field theories. Higher-derivative Lagrangians reduce to second order differential ones with one explicit independent field for each degree of freedom.
An alternative version of Hamiltonian formalism for higher-derivative theories is presented. It is related to the standard Ostrogradski approach by a canonical transformation. The advantage of the approach presented is that the Lagrangian is nonsingular and the Legendre transformation is performed in a straightforward way.
Higher Derivative (HD) Field Theories can be transformed into second order equivalent theories with a direct particle interpretation. In a simple model involving abelian gauge symmetries we examine the fate of the possible gauge fixings throughout this process. This example is a useful test bed for HD theories of gravity and provides a nice intuitive interpretation of the "third ghost" occurring there and in HD gauge theories when a HD gauge fixing is adopted.
We carry out the extension of the Ostrogradski method to relativistic field theories. Higher-derivative Lagrangians reduce to second differential-order with one explicit independent field for each degree of freedom. We consider a higher-derivative relativistic theory of a scalar field and validate a powerful order-reducing covariant procedure by a rigorous phase-space analysis. The physical and ghost fields appear explicitly. Our results strongly support the formal covariant methods used in higher-derivative gravity.
Higher derivative terms in the effective action of certain Yang-Mills theories can be severely constrained by supersymmetry. We show that requiring sixteen supersymmetries in quantum mechanical gauge theory determines the $v^6$ term in the effective action. Even the numerical coefficient of the $v^6$ term is fixed in terms of lower derivative terms in the effective action.
We study the problem of decoupling of heavy chiral superfields in four-dimensional $N=1$ supersymmetric field theories with Lorentz-invariant and Lorentz-violating higher-derivative terms. We demonstrate that the earlier found effect of large logarithmic quantum corrections, due to heavy chiral superfields, takes place not only if the theory possesses quantum divergences, but also for essentially finite theories involving higher derivative terms, both Lorentz-invariant and Lorentz-breaking ones.
We discuss the contribution of ADHM multi-instantons to the higher-derivative terms in the gradient expansion along the Coulomb branch of N=2 and N=4 supersymmetric SU(2) gauge theories. In particular, using simple scaling arguments, we confirm the Dine-Seiberg nonperturbative nonrenormalization theorems for the 4-derivative/8-fermion term in the two finite theories (N=4, and N=2 with N_F=4).
We show that the recently constructed higher-derivative 6D SYM theory involves an internal chiral anomaly breaking gauge invariance. The anomaly is cancelled when adding to the theory an adjoint matter hypermultiplet.
We review some results of applying the higher covariant derivative regularization to the investigation of quantum corrections structure in N=1 supersymmetric theories. In particular, we demonstrate that all integrals, defining the Gell-Mann--Low function in supersymmetric theories, are integrals of total derivatives. As a consequence, there is an identity for Green functions, which does not follow from any known symmetry of the theory, in N=1 supersymmetric theories. We also discuss how to derive the exact $β$-function by methods of the perturbation theory.
We consider warped AdS_3 black holes in generic higher derivatives gravity theories in 2+1 dimensions. The asymptotic symmetry group of the phase space containing these black holes is the semi-direct product of a centrally extended Virasoro algebra and an affine u(1) Kac-Moody algebra. Previous works have shown that in some specific theories, the entropy of these black holes agrees with a Cardy-like entropy formula derived for warped conformal field theories. In this paper, we show that this entropy matching continues to hold for the most general higher derivative theories of gravity. We also …
Interacting theories with higher derivatives involve ghosts. They correspond to instabilities that display themselves at the classical level. We notice that comparatively "benign" mechanical higher-derivative systems exist where the classical vacuum is stable with respect to small perturbations and the problems appear only at the nonperturbative level. We argue that benign higher-derivative field theories exist which are stable with respect to small fluctuations with nonzero momenta. A particular example is the 6D N=2 higher-derivative SYM theory, which is finite and unitary at the perturbativ…
We study the one-loop quantum corrections for higher-derivative superfield theories, generalizing the approach for calculating the superfield effective potential. In particular, we calculate the effective potential for two versions of higher-derivative chiral superfield models. We point out that the equivalence of the higher-derivative theory for the chiral superfield and the one without higher derivatives but with an extended number of chiral superfields occurs only when the mass term is contained in the general Lagrangian. The presence of divergences can be taken as an indication of this equ…
In this work, we employ renormalization group methods to study the general behavior of field theories possessing anisotropic scaling in the spacetime variables. The Lorentz breaking symmetry that accompanies these models are either soft, if no higher spatial derivative is present, or it may have a more complex structure if higher spatial derivatives are also included. Both situations are discussed in models with only scalar fields and also in models with fermions as a Yukawa like model.
In this work, we consider local and nonlocal higher-derivative generalizations of the super-Chern-Simons theory and four-dimensional SQED. In contrast to previous studies, the models studied here also have higher-derivative terms in the matter sector. For these models, we calculate the one-loop superfield effective potential.
The development of instability in the dynamics of theories with higher derivatives is traced in detail in the framework of the Pais-Uhlenbeck fourth oder oscillator. For this aim the external friction force is introduced in the model and the relevant solutions to equations of motion are investigated. As a result, the physical implication of the energy unboundness from below in theories under consideration is revealed.
We investigate the relation between the $S$-matrix unitarity ($SS^{\dagger}=1$) and the renormalizability, in theories with negative norm states. The relation has been confirmed in many theories, such as gauge theories, Einstein gravity and Lifshitz-type non-relativistic theories by analyzing the unitarity bound, which follows from the $S$-matrix unitarity and the norm positivity. On the other hand, renormalizable theories with a higher derivative kinetic term do not necessarily satisfy the unitarity bound essentially because the unitarity bound does not hold due to the negative norm states. I…
We use the novel Higgs mechanism of arXiv:0803.3218 to determine the leading higher-derivative corrections to the Euclidean N = 8 Bagger-Lambert-Gustavsson field theory. The result matches that previously found for Lorentzian 3-algebras, pointing to a universal answer for all maximally supersymmetric 3-algebra theories. We also comment on the extension to the lower-supersymmetric case of ABJM theory.
We consider BTZ black holes and flat space cosmologies in generic higher derivative gravity theories in 2+1 dimensions. Our goal is to prove the match between the bulk Iyer-Wald entropy and the field theory entropy for various symmetry algebras (CFT$_2$, Warped CFT$_2$, BMS$_3$). We also discuss phase transitions in higher curvature theories, and argue that, in the flat case, there is strictly speaking no phase transition in the grand canonical ensemble.
We propose and canonically quantize a generalization of the two-dimensional massive fermion theory described by a Lagrangian containing third-order derivatives. In our approach the mass term contains a derivative coupling. The quantum solution is expressed in terms of three usual fermions. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used to solve a theory including a current-current interaction.
We systematically analyze the effective action on the moduli space of (2,0) superconformal field theories in six dimensions, as well as their toroidal compactification to maximally supersymmetric Yang-Mills theories in five and four dimensions. We present a streamlined approach to non-renormalization theorems that constrain this effective action. The first several orders in its derivative expansion are determined by a one-loop calculation in five-dimensional Yang-Mills theory. This fixes the leading higher-derivative operators that describe the renormalization group flow into theories residing…
Energy momentum tensors of higher-derivative free scalar conformal field theories in flat spacetime are discussed. Two algorithms for the computation of energy momentum tensors are described, which accomplish different goals: the first is brute-force and highlights the complexity of the energy momentum tensors, while the second displays some features of their geometric origin as variations of Weyl invariant curved-space actions. New compact expressions for energy momentum tensors are given and specific obstructions to defining them as conformal primary operators in some spacetime dimensions ar…
The combined method of Higher Covariant Derivatives and Pauli-Villars regularization to regularize pure Yang-Mills theories is formulated in the framework of Batalin and Vilkovisky. However, BRS invariance is broken by this method and suitable counterterms should be added to restore it. The one loop counterterm is presented. Contrary to the scheme of Slavnov, this method is regularizing and leads to consistent renormalization group functions, which are the same as those found by other regularization schemes.
In this work we analyze systems described by Lagrangians with higher order derivatives in the context of the Hamilton-Jacobi formalism for first order actions. Two different approaches are studied here: the first one is analogous to the description of theories with higher derivatives in the hamiltonian formalism according to [Sov. Phys. Journ. 26 (1983) 730; the second treats the case where degenerate coordinate are present, in an analogy to reference [Nucl. Phys. B 630 (2002) 509]. Several examples are analyzed where a comparison between both approaches is made.
We speculate on Dyson series for the $S$-matrix when the interaction depends on derivatives of the fields. We stick on two particular examples: the scalar electrodynamics and the renormalised $ϕ^4$ theory. We eventually give evidence that Lorentz invariance is satisfied and that usual Feynman rules can be applied to the interaction Lagrangian.
We give simple arguments for new non-renormalization theorems on higher derivative couplings of gauge theories to supergravity, with sixteen supersymmetries, by considerations of brane-bulk superamplitudes. This leads to some exact results on the effective coupling of D3-branes in type IIB string theory. We also derive exact results on higher dimensional operators in the torus compactification of the six dimensional (0, 2) superconformal theory.
In this work, we systematically analyze higher derivative terms in the supersymmetric effective actions for three dimensional scalar field theories using $\mathcal{N} =1$ superspace formalism. In these effective actions, we show that auxiliary fields do not propagate and their effective actions can be expressed in terms of the physical fields. So, the theory does not change its field content upon addition of higher derivative terms. We use derivative expansion to generate four, five and six dimensional terms for an interacting scalar field theory with $\mathcal{N} =1$ supersymmetry. We show th…
We propose a type of non-anticommutative superspace, with the interesting property of relating to Lee-Wick type of higher derivatives theories, which are known for their interesting properties, and have lead to proposals of phenomenologicaly viable higher derivatives extensions of the Standard Model. The deformation of superspace we consider does not preserve supersymmetry or associativity in general; however, we show that a non-anticommutative version of the Wess-Zumino model can be properly defined. In fact, the definition of chiral and antichiral superfields turns out to be simpler in our c…
The problem of computing the effective nonrelativistic potential $U_{D}$ for the interaction of charged scalar bosons within the context of D-dimensional electromagnetism with a cutoff, is reduced to quadratures. It is shown that $U_3$ cannot bind a pair of identical charged scalar bosons; nevertheless, numerical calculations indicate that boson-boson bound states do exist in the framework of three-dimensional higher-derivative electromagnetism augmented by a topological Chern-Simons term.
We study the leading irrelevant operators along the flat directions of certain supersymmetric theories. In particular, we focus on finite N=2 (including N=4) supersymmetric field theories in four dimensions and show that these operators are completely determined by the symmetries of the problem. This shows that they are generated only at one loop and are not renormalized beyond this order. An instanton computation in similar three dimensional theories shows that these terms are renormalized. Hence, the four dimensional non-renormalization theorem of these terms is not valid in three dimensions…
We investigate some classical and quantum aspects of a general class of higher derivative theories of gravity. We propose a generalized version of the so-called Teyssandier gauge condition and we investigative its implications on the linearized field equations. An exhaustive investigation on the particle spectra is done, including a discussion on the appearance of ghost-like particles. We investigate the UV properties and we determine the power-counting renormalizability of the theory. Finally we probe a conjecture which relates renormalizability with the cancellation of Newtonian singularitie…
We consider a general non-Abelian renormalizable ${\cal N}=1$ supersymmetric gauge theory, regularized by higher covariant derivatives without breaking the BRST invariance, and calculate one-loop divergences for a general form of higher derivative regulator and of the gauge fixing term. It is demonstrated that the momentum integrals giving the one-loop $β$-function are integrals of double total derivatives independently of a particular choice of the higher derivative term. Evaluating them we reproduce the well-known result for the one-loop $β$-function. Also we find that the three-point ghost …
Structure of quantum corrections in ${\cal N}=1$ supersymmetric gauge theories is investigated in the case of using the regularization by higher covariant derivatives. It is demonstrated that this regularization allows revealing some interesting features which lead to the exact relations between the renormalization group functions. In particular, the NSVZ equation, which relates the $β$-function to the anomalous dimension of the matter superfields, naturally appears in this case. We briefly review the all-loop derivation of this equation and the construction of a simple renormalization prescri…
We first analyze the anti-BRST and double BRST structures of a certain higher derivative theory that has been known to possess BRST symmetry associated with its higher derivative structure. We discuss the invariance of this theory under shift symmetry in the Batalin Vilkovisky (BV) formalism. We show that the action for this theory can be written in a manifestly extended BRST invariant manner in superspace formalism using one Grassmann coordinate. It can also be written in a manifestly extended BRST invariant manner and on-shell manifestly extended anti-BRST invariant manner in superspace form…
For a general ${\cal N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the $β$-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the $β$-function containing the Yukawa couplings and compare it with the …
We discuss why the Slavnov higher covariant derivative regularization appeared to be an excellent instrument for investigating quantum corrections in supersymmetric gauge theories. For example, it allowed to demonstrate that the $β$-function in these theories is given by integrals of double total derivatives and to construct the NSVZ renormalization prescription in all loops. It was also used for deriving the non-renormalization theorem for the triple gauge-ghost vertices. With the help of this theorem the exact NSVZ $β$-function was rewritten in a new form, which revealed its perturbative ori…
Recently Seiberg and Witten have proposed that noncommutative gauge theories realized as effective theories on D-brane are equivalent to some ordinary gauge theories. This proposal has been proved, however, only for the Dirac-Born-Infeld action in the approximation of neglecting all derivative terms. In this paper we explicitly construct general forms of the 2n-derivative terms which satisfy this equivalence under their assumption in the approximation of neglecting (2 n+2)-derivative terms. We also prove that the D-brane action computed in the superstring theory is consistent with the equivale…
In this paper based on the basic principles of gauge/gravity duality we compute the hall viscosity to entropy ratio in the presence of various higher derivative corrections to the dual gravitational description embedded in an asymptotically $ AdS_{4} $ space time. As the first step of our analysis, considering the back reaction we impose higher derivative corrections to the abelian gauge sector of the theory where we notice that the ratio indeed gets corrected at the leading order in the coupling. Considering the probe limit as a special case we compute this leading order correction over the f…
We extend the perturbative approach developed in an earlier work to deal with Lagrangians which have arbitrary higher order time derivative terms for both bosons and fermions. This approach enables us to find an effective Lagrangian with only first time derivatives order by order in the coupling constant. As in the pure bosonic case, to the first order, the quantized Hamiltonian is bounded from below whenever the potential is. We show in the example of a single complex fermion that higher derivative interactions result in an effective mass and change of vacuum for the low energy modes. The sup…
Theories with higher derivatives involve linear instabilities in the Hamiltonian commonly known as Ostrogradski ghosts and can be viewed as a very serious problem during quantization. To cure {this} , we have considered the properties of antilinearty that can be found inherently in the non-Hermitian Hamiltonians. Owing to the existence of antilinearity, we can construct an operator, called the $V$-operator, which acts as an intertwining operator between the Hamiltonian and its hermitian conjugate. We have used this $V$-operator to remove the linear momenta term from the higher derivative Hamil…
We present for the first time a ghost-free higher-derivative chiral model with a propagating auxiliary F-term field (highest component of the chiral multiplet). We obtain this model by removing a ghost in a higher derivative chiral model, with Higgsing it in terms of an auxiliary vector superfield. Depending on the sign of the quadratic derivative term of the chiral superfield, the model contains two ghost free branches of the parameter regions. We find that supersymmetry is spontaneously broken in one branch while it is preserved in the other branch. As a consequence of dynamical F-term field…
Dyonic non-Abelian local/semi-global vortex strings are studied in detail in supersymmetric/non-supersymmetric Yang-Mills-Higgs theories. While the BPS tension formula is known to be the same as that for the BPS dyonic instanton, we find that the non-BPS tension formula is approximated very well by the well-known tension formula of the BPS dyon. We show that this mysterious tension formula for the dyonic non-BPS vortex stings can be understood from the perspective of a low energy effective field theory. Furthermore, we propose an efficient method to obtain an effective theory of a single vorte…
There are currently many string inspired conjectures about the structure of the low-energy effective action for super Yang-Mills theories which require explicit multi-loop calculations. In this paper, we develop a manifestly covariant derivative expansion of superspace heat kernels and present a scheme to evaluate multi-loop contributions to the effective action in the framework of the background field method. The crucial ingredient of the construction is a detailed analysis of the properties of the parallel displacement propagators associated with Yang-Mills supermultiples in N-extended super…
We perform the UV deformation of the Green's function in free scalar field theory using a discrete heat kernel method. It is found that the simplest UV deformation based on the discretized diffusion equation leads to the well-known Pauli-Villars effective Lagrangian. Furthermore, by extending assumptions on the discretized equation, we find that the general higher derivative theory is derived from the present UV deformation. In some specific cases, we also calculate the vacuum expectation values of the scalar field squared in nontrivial background spaces and examine their dependence on the UV …
We show that some higher derivative theories have a BRST symmetry. This symmetry is due to the higher derivative structure and is not associated to any gauge invariance. If physical states are defined as those in the BRST cohomology then the only physical state is the vacuum. All negative norm states, characteristic of higher derivative theories, are removed from the physical sector. As a consequence, unitarity is recovered but the S-matrix is trivial. We show that a class of higher derivative quantum gravity theories have this BRST symmetry so that they are consistent as quantum field theorie…
We present exact classical solutions of the higher-derivative theory that describes the dynamics of the position modulus of a probe brane within a five-dimensional bulk. The solutions can be interpreted as static or time-dependent throats connecting two parallel branes. In the nonrelativistic limit the brane action is reduced to that of the Galileon theory. We derive exact solutions for the Galileon, which reproduce correctly the shape of the throats at large distances, but fail to do so for their central part. We also determine the parameter range for which the Vainshtein mechanism is reprodu…
The contributions of the matter superfields and of the Faddeev--Popov ghosts to the $β$-function of ${\cal N}=1$ supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the $β$-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the $β$-function beyond the one-loop approximation is …
We consider the canonical quantization of a generalized two-dimensional massive fermion theory containing higher odd-order derivatives. The requirements of Lorentz invariance, hermiticity of the Hamiltonian and absence of tachyon excitations suffice to fix the mass term, which contains a derivative coupling. We show that the basic quantum excitations of a higher-derivative theory of order 2N+1 consist of a physical usual massive fermion, quantized with positive metric, plus 2N unphysical massless fermions, quantized with opposite metrics. The positive metric Hilbert subspace, which is isomorph…
The ultraviolet completion of gauge theories by higher derivative terms can dramatically change their behavior at high energies. The requirement of asymptotic freedom imposes very stringent constraints that are only satisfied by a small family of higher derivative theories. If the number of derivatives is large enough ($n > 4$) the theory is strongly interacting both at extreme infrared and ultraviolet regimes whereas it remains asymptotically free for a low number of extra derivatives ($n \leqslant 4$). In all cases the theory improves its ultraviolet behavior leading in some cases to ultravi…
The article is devoted to the investigation of the Noether currents and integrals of motion in the special subclass of theories with higher field derivatives -- theories under differential field transformations in action (DFTA). Under fairly general assertions, we derive a simple representation for the integrals as sums of "old" integrals of motion, terms, that vanishes on the "old" equations of motion and surface integrals. We show that for some cosmological theories of that kind with high order derivatives the last contribution can violate gauge invariance, specifically, diffeomorphisms. The…
We construct classical theories for scalar fields in arbitrary Carroll spacetimes that are invariant under Carrollian diffeomorphisms and Weyl transformations. When the local symmetries are gauge fixed these theories become Carrollian conformal field theories. We show that generically there are at least two types of such theories: one in which only time derivatives of the fields appear and the other in which both space and time derivatives appear. A classification of such scalar field theories in three (and higher) dimensions up to two derivative order is provided. We show that only a special …
For the general renormalizable ${\cal N}=1$ supersymmetric gauge theory we investigate renormalization of the Faddeev--Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general $ξ$-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev--Popov ghosts defined in terms of the renormalized coupling constant and examin…
