Martin B. Einhorn

The weak mixing angle and unification mass in supersymmetric SU(5)

We consider the predictions for the weak mixing angle 0w and the scale M of unification in a supersymmetric extension of SU(5). with particular emphasis on the sensitivity to the number of Higgs multiplets. In the one-loop approximation, we also calculate the ratio nrJrn,. We discuss generally the effects of an intermediate threshold between the weak interaction scale and M and estimate the sensitivity of 6’w and M to the scale of supersymmetry breaking The evolution of the coupling constants of the supersymmetric SU(3) 8 SU(2) @ U(1) effective gauge theory is described and the two-loop corrections to 6’, and M are calculated. Supersymmetric unified models (SUMS) offer a potential solution to the unnaturalness or gauge hierarchy puzzle of standard grand unified models (GUMS) of electroweak and strong interactions [ 1,2]. Attempts to build realistic SUMS face theoretical and phenomenological difficulties [2,3], but it has been emphasized [4] that the most naive extension of SU(5) will not substantially alter the successful prediction of the weak mixing angle Bw, whereas the unification mass scale M tends to increase by several orders of magnitude, thereby suppressing the proton decay rate beyond the range of feasible observation. Were this a universal feature of SUMS, the observation of proton decay would discourage further attempts to utilize supersymmetry to render GUMS natural*. In this paper, we reconsider the predictions for 8, and M in SUMS, emphasizing the sensitivity to the number of light Higgs doublets. We also extend the previous lowest order results to two loops, our motivation being twofold: (i) The accuracy of the one-loop approximation depends upon the size of higher order corrections, and, since SUMS generally will contain many more particles than GUMS, one might a priori expect higher order corrections to be relatively more important. (ii) Even a relatively modest change in the unification mass M produces a large change in the proton lifetime r a M4. Another topic treated in the course of this investigation is the change in the prediction of the quark-lepton ratio m,/m, in going from the minimal SU(5) GUM to the minimal SU(5) SUM. Remarkably enough, we find only a small change in this seemingly successful consequence of SU(5) unification. *See note added in proof.

The effective potential and the renormalisation group

We discuss renormalisation-group improvement of the effective potential both in general and in the context of 0(N) scalar ~ and the Standard Model. In the latter case we find that absolute stability of the electroweak vacuum implies that m11~ l.95m~ —189GeV, for a3(Mz) = 0.11. We point out that the lower bound on m11decreases if a3(Mz) is increased.

Patterns of deviation from the standard model

Abstract Effective lagrangians can be used to parametrize the effects of physics beyond the Standard Model. Assuming the complete theory is a gauge theory, we determine which effective operators may be generated at tree level, and which are only generated at loop level. The latter are suppressed by factors of 1 16 π 2 and will therefore be quite difficult to detect. In particular, all operators changing the Standard-Model structure of the triplegauge-vector couplings fall into this category. We also point out that in certain cases, dimension-eight operators may be more important than dimension-six operators. We discuss both the linear and non-linear representation of the Higgs sector.

Heavy particles and the ϱ parameter in the standard model

Abstract A new formalism is presented for the calculation of the contribution δϱ to ϱ = Mw2/Mz2cos2θw from heavy particles transforming according to arbitrary representations of SU(2) × U(1). A conjecture for the necessary and sufficient conditions that δϱ⩾0 for all values of masses and mixing angles within a particular multiplet is formulated. A number of examples are given (all consistent with the conjecture) and the significance of improved knowledge of ϱ vis a vis the possible existence of undiscovered heavy particles discussed.

Interacting quantum field theory in de Sitter vacua

We discuss interacting quantum field theory in de Sitter space and argue that the Mottola-Allen (MA) vacuum ambiguity is an artifact of free field theory. The nature of the nonthermality of the MA vacua is also clarified. We propose analyticity of correlation functions as a fundamental requirement of quantum field theory in curved spacetimes. In de Sitter space, this principle determines the vacuum unambiguously and facilitates the systematic development of perturbation theory.

Higher mass scales and mass hierarchies

Abstract The impact of new interactions at very high energies on the spontaneous breaking of chiral symmetry in gauge field theories is studied. Described at low energies by higher dimension operators, these interactions can enhance the chiral condensate and even the Goldstone boson decay constant relative to the confinement scale. This can lead to important consequences in technicolor theories, including realistically large quark and lepton masses and the possibility of new physics at energies below the weak scale. In addition, small differences in strength of the four-fermion interactions, due say to isospin or flavor breaking, can be greatly magnified in the resulting quark and lepton mass spectrum.

Speculations on a strongly interacting Higgs sector

Abstract Using the 1/N expansion, we argue that the O2N Higgs-Goldstone model may be a good indicator of the behavior of the standard SU2 ⊗ U1 electroweak model in the non-perturbative limit of a strongly interacting Higgs sector. We emphasize that there remains a physical scalar particle or resonance σ (Higgs remnant), whose mass (and width) will be set by the weak scale. However, its coupling to vector bosons is expected to be much stronger than the standard model Higgs of comparable mass. This provides evidence that there is an upper limit to the Higgs mass in the hundreds of GeV, regardless of whether naturalness constraints are imposed on the parameters of the effective lagrangian. We conclude with some comments about the possible relevance of this particle to the radiative events observed at the CERN p p collider.

Entanglement interpretation of black hole entropy in string theory

We show that the entropy resulting from the counting of microstates of non extremal black holes using field theory duals of string theories can be interpreted as arising from entanglement. The conditions for making such an interpretation consistent are discussed. First, we interpret the entropy (and thermodynamics) of spacetimes with non degenerate, bifurcating Killing horizons as arising from entanglement. We use a path integral method to define the Hartle-Hawking vacuum state in such spacetimes and discuss explicitly its entangled nature and its relation to the geometry. If string theory on such spacetimes has a field theory dual, then, in the low-energy, weak coupling limit, the field theory state that is dual to the Hartle-Hawking state is a thermofield double state. This allows the comparison of the entanglement entropy with the entropy of the field theory dual, and thus, with the Bekenstein-Hawking entropy of the black hole. As an example, we discuss in detail the case of the five dimensional anti-de Sitter, black hole spacetime.

Squeezed states in the de Sitter vacuum

We discuss the treatment of squeezed states as excitations in the Euclidean vacuum of de Sitter space. A comparison with the treatment of these states as candidate no-particle states, or alpha vacua, shows important differences already in the free theory. At the interacting level alpha vacua are inconsistent, but squeezed state excitations seem perfectly acceptable. Indeed, matrix elements can be renormalized in the excited states using precisely the standard local counterterms of the Euclidean vacuum. Implications for inflationary scenarios in cosmology are discussed.

The bases of effective field theories

Abstract With reference to the equivalence theorem, we discuss the selection of basis operators for effective field theories in general. The equivalence relation can be used to partition operators into equivalence classes, from which inequivalent basis operators are selected. These classes can also be identified as containing Potential-Tree-Generated (PTG) operators, Loop-Generated (LG) operators, or both, independently of the specific dynamics of the underlying extended models, so long as it is perturbatively decoupling. For an equivalence class containing both, we argue that the basis operator should be chosen from among the PTG operators, because they may have the largest coefficients. We apply this classification scheme to dimension-six operators in an illustrative Yukawa model as well in the Standard Model (SM). We show that the basis chosen by Grzadkowski et al. [5] for the SM satisfies this criterion. In this light, we also revisit and verify our earlier result [6] that the dimension-six corrections to the triple-gauge-boson couplings only arise from LG operators, so the magnitude of the coefficients should only be a few parts per thousand of the SM gauge coupling if BSM dynamics respects decoupling. The same is true of the quartic-gauge-boson couplings.