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Register a new userTowards general scalar-Yukawa renormalisation group equations at three-loop order
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Tom Steudtner
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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.
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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.
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.
We present the complete 2-loop renormalisation group equations of the superpotential parameters for the supersymmetric standard model including the full set of R-parity violating couplings. We use these equations to do a study of (a) gauge coupling unification, (b) bottom-tau unification, (c) the fixed-point structure of the top quark Yukawa coupling, and (d) two-loop bounds from perturbative unification. For large values of the R-parity violating coupling, the value of alpha_S(M_Z) predicted from unification can be reduced by 5% with respect to the R-parity conserving case, bringing it to within 2sigma of the observed value. Bottom-tau Yukawa unification becomes potentially valid for any value of tanbetasim 2-50. The prediction of the top Yukawa coupling from the low tanbeta, infra-red quasi fixed point can be lowered by up to 10%, raising tanbeta up to a maximum of 5 and relaxing experimental constraints upon the quasi-fixed point scenario. For heavy scalar fermion masses of order 1 Tev the limits on the higher family Delta L ot=0 operators from perturbative unification are competitive with the indirect laboratory bounds. We calculate the dependence of these bounds upon tan beta.
The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $epsilon$ and also for the field anomalous dimension to order $epsilon^2$. An exact marginal operator for the full RG equations is also constructed.
We present analytical results for the $N_f^4$ and $N_f^3$ terms of the five-loop Beta function, for a general gauge group. While the former term agrees with results available from large-$N_f$ studies, the latter is new and extends the value known for SU(3) from an independent calculation.
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