Quartic Divergences Research Articles

Massive gravitino scattering amplitudes and the unitarity cutoff of the new Fayet-Iliopoulos terms

We compute the 2 → 2 gravitino scattering amplitudes at tree level in supergravity theories where supersymmetry is spontaneously broken. In the unitary gauge, the gravitino becomes massive (of mass m3/2) by absorbing the Goldstino, and the scattering amplitudes of its longitudinal polarisations grow with energy as {kappa}^2{E}^4/{m}_{3/2}^2 , signaling a potential breakdown of unitarity at a scale {Lambda}^2sim {m}_{3/2}/kappa sim {M}_{textrm{SUSY}}^2 . As we show explicitly in the Polonyi model, this leading term is cancelled by the contributions coming from the scalar partner of the Goldstino (sgoldstino), restoring perturbative unitarity up to the Planck scale. This is expected since supersymmetry is spontaneously broken, in analogy with the situation occuring in the Standard Model, where massive gauge bosons scattering preserves unitarity at high energy once we consider the contributions from the Higgs boson. However, when supersymmetry is broken by the new Fayet-Iliopoulos D-term, with ungauged R-symmetry, the above cancellation does not occur. In this case, the unbroken phase is singular and there is no contribution able to cancel the quartic divergences of the amplitudes, leading to a cutoff Λ ~ MSUSY where the effective theory breaks down. The same behaviour is obtained when supersymmetry is non-linearly realised.

Counting of states in Higgs theories

We enumerate the micro-states in Higgs theories, addressing (i) the number of vacuum states and (ii) the appropriate measure in the quantum path integral. To address (i) we explicitly construct the set of ground state wave-functionals in the field basis focussing on scalar modes $\theta(x)$. Firstly, we show that in the limit in which the gauge coupling is zero, we obtain an infinite set of degenerate ground states at large volume distinguished by $\theta(x)\to\theta(x)+\theta_0$, spontaneously breaking the global symmetry, as is well known. We then show that at finite gauge coupling there is a unique ground state at large volume since the wave-functional only depends on $\nabla\theta$ in the IR, and we explain this at the level of the Lagrangian. Since gauge fields fall off exponentially from sources there are no conserved charges or symmetries in the Higgs phase; so the Higgs mechanism is the removal of symmetry from the theory. We show how physical features of defects, such as cosmic strings in the abelian Higgs model, are best understood in this context. Since there is a unique ground state, we address (ii) whether the path integral is a volume measure for the radial Higgs field $\mathcal{D}\rho\,\rho^{N-1}$ from the $N$ components of the Higgs multiplet, or a line measure $\mathcal{D}\rho$ as the $N-1$ would-be Goldstones can be removed in unitary gauge. We prove that the need to avoid quartic divergences demands a tower of counter terms that resum to exactly give the volume measure. So the size of the Hilbert space in the zero gauge coupling case and finite gauge coupling case are in one-to-one correspondence, despite the degeneracy of the ground state being lifted in the latter. As a cosmological application, we point out that the volume measure can make it exponentially more unlikely in $N(=4)$ for the Standard Model Higgs to relax to the electroweak vacuum in the early universe.

The Maxwell–Einstein system, Ward identities and the Vilkovisky construction

The gauge fixing dependence of the one-loop effective action of quantum gravity in the proper-time representation is investigated for a space of arbitrary curvature, and the investigation is extended to Maxwell–Einstein theory. The construction of Vilkovisky and DeWitt for removal of this dependence is then considered in general gauges, and it is shown that nontrivial criteria arising from a Ward identity of the theory must be obeyed by the regularization scheme, if the construction is to remove the gauge dependence of quadratic and quartic divergences. The results apply also to non-Abelian gauge theories; they are used to address the question of gauge dependence of asymptotic freedom arising through internal graviton lines at one-loop order as suggested by Robinson and Wilczek.

Considerations concerning the contributionsof fundamental particles to the vacuum energy density

The covariant regularization of the contributions of fundamental particles to the vacuum energy density is implemented in the Pauli-Villars, dimensional regularization, and Feynman regulator frameworks. Rules of correspondence between dimensional regularization and cutoff calculations are discussed. Invoking the scale invariance of free field theories in the massless limit, as well as consistency with the rules of correspondence, it is argued that quartic divergences are absent in the case of free fields, while it is shown that they arise when interactions are present.

Consistency of the pulsating universe with Einstein’s field equations

It is shown that the pulsating model of the universe can be derived from Einstein’s field equations with the addition of a traceless, negative-energy source tensor. Bounds on the negative-energy density are given. Earlier works on negative-energy sources as well as prevention of singularities are discussed. The results support earlier findings that one can obtain a bounce without modifying Einstein’s field equations, but instead, by violating the Hawking-Penrose strong energy condition. The negative-energy tensor is interpreted as the partially uncompensated zero-point energy of a weakly interacting, neutral massless fermion (which may or may not be a massless neutrino). It is shown explicitly that the zero-point quadratic divergences, reported earlier by Ford for the photon and the massless fermion, do not cancel, in contrast to the quartic divergences. To eliminate both the quadratic as well as the quartic divergences, three auxiliary massless fermions and six conformai scalars are introduced. The resulting multiplet is analogous to that of a massless multiplet forN = 4 supersymmetry with comparable helicity states. The corresponding small Casimir energy is evaluated. Since the model predicts a cosmic background of weakly interacting negative-energy radiation, greater in intensity than the CMB radiation, it can be experimentally falsified. A subordinate idea of the model, that there was an astrophysical- rather than a Planckian-size bounce, can be falsified by suitable computer simulations.

Quartic divergences and vector boson self-couplings

We study the radiative corrections to the ρ parameter induced by non-standard three and four vector boson self-couplings. The demand of the absence of quartically divergent terms to a limitation of the possible allowed terms, corresponding to hidden directions in coupling constant space.

Quadratic divergences and dimensional regularisation

We present a detailed analysis of quadratic and quartic divergences in dimensionally regulated renormalisable theories. We perform explicit three-loop calculations for a general theory of scalars and fermions. We find that the higher-order quadratic and quartic divergences are related to the lower-order ones by the renormalisation group β-functions.

A comment on induced gravity

Because of the non-perturbative ambiguity of the stress-energy tensor in theories with dynamical breaking of conformal invariance, there is no unique self-consistent way to compute the Newton's and cosmological constants of the corresponding induced gravitation theories, when leading quadratic or quartic divergences are present.

Quadratic and quartic one-loop divergences in gauge theories and their cancellation in supergravity

The quadratic and quartic divergences cancel in N = 2 supergravity in photon-photon scattering. They are defined by modified dimensional regularization as the residues of poles at d = 2 and d = 0 dimensions, respectively. Other definitions are shown to violate general coordinate invariance and Yang-Mills invariance. 't Hooft's algorithm for the one-loop divergences is extended to the case of higher-order divergences.

An effective lagrangian for multiphoton processes and a non-linear Born-Infeld lagrangian

Abstract An effective lagrangian for multiphoton processes (of Euler-Heisenberg type) is computed for an electromagnetic field interacting with massive scalar, fermion and vector particles. It is found that if the charges gI, masses mI and numbers NI of massive particles satisfy the “supersymmetric” conditions, NSmS4+3NVmV4−2NFmF4=0 and gI2/mI4 is I-independent, where I=S, F, V stands for scalars, fermions and vectors respectively, then not only quartic divergences (poles at n=0 in dimensional regularization) proportional to constant terms cancel out but also maximally helicity changing interaction terms vanish in the effective lagrangian. A possibility to relate a non-linear Born-Infeld lagrangian to the effective lagrangian is examined.

Regularization and renormalization of nonlinear pion-pion couplings

The fourth-order $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering resulting from nonlinear pion-pion couplings is investigated to show how such couplings can be regularized by the method of auxiliary fields. Special care is required in the choice of coupling terms involving the auxiliary fields to ensure that all quartic divergences are canceled by the $\ensuremath{\delta}(0)$ terms in the regularized effective interaction. It is found that four counterterms are necessary for the elimination of divergences in the process under consideration. In the limiting case of massless pions, the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering result is free from infrared divergence, satisfies Adler's condition, and requires only two counterterms for renormalization.

Origin shifts in divergent Feynman integrals

The surface terms arising from a shift of origin in divergent Feynman integrals are considered. Sum rules and recursion relations between these terms are derived for an arbitrary degree of divergence and tensor rank. These relations are explicitly solved for linear, quadratic, cubic, and quartic divergences.