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Register a new userThree-loop effective potential of general scalar theory via differential equations
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We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using differential equations transformed into the favorable $epsilon$ form of dimensional regularization. The master integrals of the canonical basis thus obtained are expressed in terms of cyclotomic polylogarithms up to weight four. We also introduce an algorithm for the numerical evaluation of cyclotomic polylogarithms with multiple-precision arithmetic, which is implemented in the Mathematica package cyclogpl.m supplied here.
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For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the $overline{text{MS}}$ scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and derive $beta$ functions for scalar masses and cubic interactions from it. As an example, the results are applied to compute all renormalisation group equations in $U(n) times U(n)$ scalar theories.
Spontaneous breaking of quantum scale invariance may provide a solution to the hierarchy and cosmological constant problems. In a scale-invariant regularization, we compute the two-loop potential of a higgs-like scalar $phi$ in theories in which scale symmetry is broken only spontaneously by the dilaton ($sigma$). Its vev $langlesigmarangle$ generates the DR subtraction scale ($musimlanglesigmarangle$), which avoids the explicit scale symmetry breaking by traditional regularizations (where $mu$=fixed scale). The two-loop potential contains effective operators of non-polynomial nature as well as new corrections, beyond those obtained with explicit breaking ($mu$=fixed scale). These operators have the form: $phi^6/sigma^2$, $phi^8/sigma^4$, etc, which generate an infinite series of higher dimensional polynomial operators upon expansion about $langlesigmaranglegg langlephirangle$, where such hierarchy is arranged by {it one} initial, classical tuning. These operators emerge at the quantum level from evanescent interactions ($proptoepsilon$) between $sigma$ and $phi$ that vanish in $d=4$ but are demanded by classical scale invariance in $d=4-2epsilon$. The Callan-Symanzik equation of the two-loop potential is respected and the two-loop beta functions of the couplings differ from those of the same theory regularized with $mu=$fixed scale. Therefore the running of the couplings enables one to distinguish between spontaneous and explicit scale symmetry breaking.
For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the $overline{text{MS}}$ scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.
We study the decoupling effects in N=1 (global) supersymmetric theories with chiral superfields at the one-loop level. The examples of gauge neutral chiral superfields with the minimal (renormalizable) as well as non-minimal (non- renormalizable) couplings are considered, and decoupling in gauge theories with U(1) gauge superfields that couple to heavy chiral matter is studied. We calculate the one-loop corrected effective Lagrangians that involve light fields and heavy fields with mass of order M. The elimination of heavy fields by equations of motion leads to decoupling effects with terms that grow logarithmically with M. These corrections renormalize light fields and couplings in the theory (in accordance with the decoupling theorem). When the field theory is an effective theory of the underlying fundamental theory, like superstring theory, where the couplings are calculable, such decoupling effects modify the low-energy predictions for the effective couplings of light fields. In particular, for the class of string vacua with an anomalous U(1) the vacuum restabilization triggers the decoupling effects, which can significantly modify the low energy predictions for the couplings of the surviving light fields. We also demonstrate that quantum corrections to the chiral potential depending on massive background superfields and corresponding to supergraphs with internal massless lines and external massive lines can also arise at the two-loop level.
We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.
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