Background Field Method Research Articles

Landau background gauge fixing and the IR properties of Yang-Mills Green functions

We analyse the complete algebraic structure of the background field method for Yang--Mills theory in the Landau gauge and show several structural simplifications within this approach. In particular we present a new way to study the IR behavior of Green functions in the Landau gauge and show that there exists a unique Green function whose IR behaviour controls the IR properties of the gluon and the ghost propagators.

The background field method and the linearization problem for Poisson manifolds

The background field method (BFM) for the Poisson sigma model (PSM) is studied as an example of the application of the BFM technique to open gauge algebras. The relationship with Seiberg–Witten maps arising in non-commutative gauge theories is clarified. It is shown that the implementation of the BFM for the PSM in the Batalin–Vilkovisky formalism is equivalent to the solution of a generalized linearization problem (in the formal sense) for Poisson structures in the presence of gauge fields. Sufficient conditions for the existence of a solution and a constructive method to derive it are presented.

One-loop anomalous couplings and matching conditions in the nonlinear realization of the [formula omitted] Higgs model

We study the renormalization of the nonlinear realization of the SU ( 2 ) Higgs model in the modified minimal subtraction renormalization scheme. We propose that the effective field method with truncated operator series is trustworthy even when the Higgs boson is relatively light. Using the technique of background field method in the coordinate space, we derive the matching conditions at tree and one-loop order, both in the regions where the Higgs boson is heavy and where it is light. We obtain the complete one-loop anomalous couplings up to O ( p 4 ) in the SU ( 2 ) Higgs model. We observe that the contribution of the gauge bosons and the Goldstone bosons to the anomalous couplings at the one-loop level is significant. By establishing the correspondence between our coordinate space calculation and the momentum space calculation that exists in the literature, we find agreement in all the matching conditions in the heavy Higgs boson limit.

Background field method in stochastic quantization of [formula omitted] supersymmetric Yang–Mills theory

In the previous works, we proposed the stochastic quantization method (SQM) approach to N = 1 supersymmetric Yang–Mills theory (SYM). In four dimensions, in particular, we obtained the superfield Langevin equation and the corresponding Fokker–Planck equation which describe the underlying stochastic process manifestly preserving the global supersymmetry as well as the local gauge symmetry. The stochastic gauge fixing procedure was also applied to SYM 4 in the superfield formalism. In this note, we apply the background field method to SYM 4 in terms of the stochastic action principle in the SQM approach. The one-loop β-function for the gauge coupling agrees with that given by the path-integral approach, thereby confirming that the stochastic gauge fixing procedure with the background local gauge invariant Zwanziger's gauge fixing functions simulates the contributions from the Nielsen–Kallosh ghost as well as the Faddeev–Popov ghost at the one-loop level. We also show the equivalence of the effective stochastic action in the background field method to the standard one in SQM.

Relaxed super self-duality and [formula omitted] SYM at two loops

A closed-form expression is obtained for a holomorphic sector of the two-loop low-energy effective action for the N = 4 super Yang–Mills theory on its Coulomb branch where the gauge group SU ( N ) is spontaneously broken to SU ( N − 1 ) × U ( 1 ) and the dynamics is described by a single Abelian N = 2 vector multiplet. In the framework of the background-field method, this holomorphic sector is singled out by computing the effective action for a background N = 2 vector multiplet satisfying a relaxed super self-duality condition. At the two-loop level, the N = 4 SYM effective action is shown to possess no F 4 term (with F the U ( 1 ) field strength), in accordance with the Dine–Seiberg non-renormalization conjecture hep-th/9705057 and its generalized form given in hep-th/0310025. An unexpected outcome of our calculation is that no (manifestly supersymmetric invariant generating) F 6 quantum correction occurs at two loops. This is in conflict with previous results.

Electroweak pinch technique to all orders

The generalization of the pinch technique to all orders in the electroweak sector of the Standard Model within the class of the renormalizable 't Hooft gauges, is presented. In particular, both the all-order PT gauge-boson- and scalar-fermion vertices, as well as the diagonal and mixed gauge-boson and scalar self-energies are explicitly constructed. This is achieved through the generalization to the Standard Model of the procedure recently applied to the QCD case, which consists of two steps: (i) the identification of special Green's functions, which serve as a common kernel to all self-energy and vertex diagrams and (ii) the study of the (on-shell) Slavnov–Taylor identities they satisfy. It is then shown that the ghost, scalar and scalar-gauge-boson Green's functions appearing in these identities capture precisely the result of the pinching action at arbitrary order. It turns out that the aforementioned Green's functions play a crucial role, their net effect being the non-trivial modification of the ghost, scalar and scalar-gauge-boson diagrams of the gauge-boson- or scalar-fermion vertex we have started from, in such a way as to dynamically generate the characteristic ghost and scalar sector of the background field method. The pinch technique gauge-boson and scalar self-energies are also explicitly constructed by resorting to the method of the background-quantum identities.

Twist-3 distribution amplitude of the pion in QCD sum rules

We apply the background field method to calculate the moments of the pion two-particles twist-3 distribution amplitude (DA) $\phi_p(\xi)$ in QCD sum rules. In this paper,we do not use the equation of motion for the quarks inside the pion since they are not on shell and introduce a new parameter $m_0^p$ to be determined. We get the parameter $m_0^p\approx1.30GeV$ in this approach. If assuming the expansion of $\phi_p(\xi)$ in the series in Gegenbauer polynomials $C_n^{1/2}(\xi)$, one can obtain its approximate expression which can be determined by its first few moments.

Thermal heat kernel expansion and the one-loop effective action of QCD at finite temperature

The heat kernel expansion for field theory at finite temperature is constructed. It is based on the imaginary time formalism and applies to generic Klein-Gordon operators in flat space-time. Full gauge invariance is manifest at each order of the expansion and the Polyakov loop plays an important role at any temperature. The expansion is explicitly worked out up to operators of dimension 6 included. The method is then applied to compute the one-loop effective action of QCD at finite temperature with massless quarks. The calculation is carried out within the background field method in the $\overline{\mathrm{MS}}$ scheme up to dimension-6 operators. Further, the action of the dimensionally reduced effective theory at high temperature is also computed to the same order. Existing calculations are reproduced and new results are obtained in the quark sector for which only partial results existed up to dimension 6.

Relaxed super self-duality and effective action

A closed-form expression is obtained for a holomorphic sector of the two-loop Euler–Heisenberg type effective action for N=2 supersymmetric QED derived in hep-th/0308136. In the framework of the background-field method, this sector is singled out by computing the effective action for a background N=2 vector multiplet satisfying a relaxed super self-duality condition. The approach advocated in this Letter can be applied, in particular, to the study of the N=4 super-Yang–Mills theory on its Coulomb branch.

OPTIMIZED POST GAUSSIAN APPROXIMATION IN THE BACKGROUND FIELD METHOD

We have extended the variational perturbative theory based on the background field method to include the optimized expansion of Okopinska and the post Gaussian effective potential of Stansu and Stevenson. This new method provides a much simpler way to compute the correction terms to the Gaussian effective action (or potential). We have also renormalized the effective potential in 3+1 dimensions by introducing appropriate counterterms in the Lagrangian.

Two-loop renormalization of the electric charge in the standard model

We discuss the renormalization of the electric charge at the two-loop level in the Standard Model of the electroweak interactions. We explicitly calculate the expression of the complete on-shell two-loop counterterm using the Background Field Method and discuss the advantages of this computational approach. We consider the related quantity $\hat e(\mu)$, defined in the $\ms$ renormalization scheme and present numerical results for different values of the scale $\mu$. We find that the full 2-loop electroweak corrections contribute more than 10 parts in units $10^{-5}$ to the $\Delta \hat\alpha (\mz^2)$ parameter, obtaining $\hat\alpha^{-1}(\mz)= 128.12 \pm 0.05$ for $\Delta\alpha_{had}(\mz^2) =0.027572 \pm 0.000359$.

Brane–antibrane solution and SUSY effective potential in five-dimensional Mirabelli–Peskin model

A localized configuration is found in the 5D bulk-boundary theory on an S1/Z2 orbifold model of Mirabelli and Peskin. A bulk scalar and the extra (fifth) component of the bulk vector constitute the configuration. N=1 SUSY is preserved. The effective potential of the SUSY theory is obtained using the background field method. The vacuum is treated in a general way by allowing its dependence on the extra coordinate. Taking into account the supersymmetric boundary condition, the 1-loop full potential is obtained. The scalar-loop contribution to the Casimir energy is also obtained. Especially we find a new type which depends on the brane configuration parameters besides the S1 periodicity parameter.

Pinch technique self-energies and vertices to all orders in perturbation theory

The all-order construction of the pinch technique gluon self-energy and quark–gluon vertex is presented in detail within the class of linear covariant gauges. The main ingredients in our analysis are the identification of a special Green function, which serves as a common kernel to all self-energy and vertex diagrams, and the judicious use of the Slavnov–Taylor identity it satisfies. In particular, it is shown that the ghost-Green functions appearing in this identity capture precisely the result of the pinching action at arbitrary order. By virtue of this observation the construction of the quark–gluon vertex becomes particularly compact. It turns out that the aforementioned ghost-Green functions play a crucial role, their net effect being the non-trivial modification of the ghost diagrams of the quark–gluon vertex in such a way as to reproduce dynamically the characteristic ghost sector of the background field method. The gluon self-energy is also constructed following two different procedures. First, an indirect derivation is given, by resorting to the strong induction method and the assumption of the uniqueness of the S-matrix. Second, an explicit construction based on the intrinsic pinch technique is provided, using the Slavnov–Taylor identity satisfied by the all-order three-gluon vertex nested inside the self-energy diagrams. The process independence of the gluon self-energy is also demonstrated, by using gluons instead of quarks as external test particles, and identifying the corresponding kernel function, together with its Slavnov–Taylor identity. Finally, the general methodology for carrying out the renormalization of the resulting Green functions is outlined, and various open questions are briefly discussed.

Super background field method forN= 2 SYM

The implementation of the Background Field Method (BFM) for quantum field theories is analysed within the Batalin-Vilkovisky (BV) formalism. We provide a systematic way of constructing general splittings of the fields into classical and quantum parts, such that the background transformations of the quantum fields are linear in the quantum variables. This leads to linear Ward-Takahashi identities for the background invariance and to great simplifications in multiloop computations. In addition, the gauge fixing is obtained by means of (anti)canonical transformations generated by the gauge-fixing fermion. Within this framework we derive the BFM for the N=2 Super-Yang-Mills theory in the Wess-Zumino gauge viewed as the twisted version of Donaldson-Witten topological gauge theory. We obtain the background transformations for the full BRST differential of N=2 Super-Yang-Mills (including gauge transformations, SUSY transformations and translations). The BFM permits all observables of the supersymmetric theory to be identified easily by computing the equivariant cohomology of the topological theory. These results should be regarded as a step towards the construction of a super BFM for the Minimal Supersymmetric Standard Model.

On the background field method beyond one loop: a manifestly covariant derivative expansion in super Yang-Mills theories

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 superspace.

Three-loop Yang–Mills β-function via the covariant background field method

We demonstrate the effectivity of the covariant background field method by means of an explicit calculation of the 3-loop β-function for a pure Yang–Mills theory. To maintain manifest background invariance throughout our calculation, we stay in coordinate space and treat the background field non-perturbatively. In this way the presence of a background field does not increase the number of vertices and leads to a relatively small number of vacuum graphs in the effective action. Restricting to a covariantly constant background field in Fock–Schwinger gauge permits explicit expansion of all quantum field propagators in powers of the field strength only. Hence, Feynman graphs are at most logarithmically divergent. At 2-loop order only a single Feynman graph without subdivergences needs to be calculated. At 3-loop order 24 graphs remain. Insisting on manifest background gauge invariance at all stages of a calculation is thus shown to be a major labor saving device. All calculations were performed with Mathematica in view of its superior pattern matching capabilities. Finally, we describe briefly the extension of such covariant methods to the case of supergravity theories.

Pinch technique to all orders

The generalization of the pinch technique to all orders in perturbation theory is presented. The effective Green's functions constructed with this procedure are singled out in a unique way through the full exploitation of the underlying Becchi-Rouet-Stora-Tyutin symmetry. A simple all-order correspondence between the pinch technique and the background field method in the Feynman gauge is established.

The one-loop effective action for quantum electrodynamics on noncommutative space

In this paper we calculate the divergent part of the one loop effective action for QED on noncommutative space using the background field method. The effective action is obtained up to the second order in the noncommutative parameter theta and in the classical fields.

Giant graviton and quantum stability in the matrix model on app-wave background

We study classical solutions in Berenstein-Maldacena-Nastase (BMN) matrix model. A supersymmetric (1/2 BPS) fuzzy sphere is one of the classical solutions and corresponds to a giant graviton. We also consider other classical solutions, such as non-supersymmetric fuzzy sphere and harmonic oscillating gravitons. Some properties of oscillating gravitons are discussed. In particular, oscillating gravitons turn into usual supergravitons in the limit mu -> 0. Moreover, we calculate the one-loop effective action around the supersymmetric fuzzy sphere by the use of the background field method and show the quantum stability of the giant graviton. Also, the instability of the non-supersymmetric fuzzy sphere is proven.

FIELD DECOMPOSITION AND THE GROUND STATE STRUCTURE OF SU(2) YANG–MILLS THEORY

We compute the effective potential of SU(2) Yang–Mills theory using the background field method and the Faddeev–Niemi decomposition of the gauge fields. In particular, we find that the potential will depend on the values of two scalar fields in the decomposition and that its structure will give rise to a symmetry breaking.