In this Letter we compute the three-loop corrections to the beta functions of the three gauge couplings in the Standard Model of particle physics using the minimal subtraction scheme and taking into account Yukawa and Higgs self-couplings.PACS numbers: 11.10.Hi 11.15.BtRenormalization group functions are fundamental quantities of each quantum field theory and play an important role in various aspects. Besides controlling the energy dependence of parameters and fields they are also crucial for the resummation of large logarithms. Furthermore, renormalization group functions are important for the development of grand unified theories and the extrapolation of low-energy precision data to high energies, not accessible by collider experiments.As far as the strong interaction part of the Standard Model is concerned the corresponding gauge coupling beta function is known up to four-loop order [1][2][3][4][5][6][7][8][9][10] [21].) For a general theory based on a simple gauge group the three-loop corrections to the gauge coupling beta function have been calculated in Ref. [22]. In this Letter we provide results for the three-loop gauge coupling beta functions taking into account all sectors of the Standard Model, i.e., the gauge, Yukawa and Higgs boson self-couplings.Let us in a first step define the beta functions. We denote the three gauge couplings by α 1 , α 2 and α 3 and adopt a SU (5)-like normalization withwhere α is the fine structure constant, θ W the weak mixing angle and α s the strong coupling. In our calculation we consider in addition to the gauge couplings also the third-generation Yukawa couplings [34] α 4 = α t , α 5 = α b and α 6 = α τ , and the Higgs boson self-couplingwhere m x and M W are the fermion and W boson mass, respectively, and −λ(Φ † Φ) 2 is the part of the Lagrange density describing the quartic Higgs self interaction.The functions β i are obtained from the renormalization constants of the corresponding gauge couplings that are defined as g bare i = µ ǫ Z gi g i where α i = g 2 i /(4π). Exploiting the fact that the g bare i are µ-independent and taking into account that Z gi may depend on all seven couplings leads to the following formulawhere ǫ = (4 − d)/2 is the regulator of Dimensional Regularization with d being the space-time dimension used for the evaluation of the momentum integrals and the dependence of α i on the renormalization scale µ is suppressed. From Eq. (2) it is clear that the renormalization constants Z gi (i = 1, 2, 3) have to be computed up to three-loop order.In the modified minimal subtraction (MS) renormalization scheme the perturbative expansion of the gauge coupling beta functions can be written asIn this Letter we evaluate the three-loop terms (coefficients c ijk ) only for the gauge couplings (i.e. i = 1, 2, 3). For our calculation the beta functions for the Yukawa couplings are needed to the one-loop order and the treelevel expression [first term in Eq. (3)] is sufficient for β λ . In the MS scheme the beta functions are mass independent that allows us to use th...

We compute the beta functions for the three gauge couplings of the Standard Model in the minimal subtraction scheme to three loops. We take into account contributions from all sectors of the Standard Model. The calculation is performed using both Lorenz gauge in the unbroken phase of the Standard Model and background field gauge in the spontaneously broken phase. Furthermore, we describe in detail the treatment of γ 5 and present the automated setup which we use for the calculation of the Feynman diagrams. It starts with the generation of the Feynman rules and leads to the bare result for the Green's function of a given process.

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We consider the relations between the gauge couplings at the electroweak scale and the high scale where unification of the three gauge couplings is expected. Threshold corrections are incorporated both at the supersymmetric and at the grand unified scale and, where available, three-loop running and two-loop decoupling are employed. We study the impact of the current experimental uncertainties of the coupling constants and the supersymmetric mass spectrum on the prediction of the superheavy masses within the socalled minimal supersymmetric SU(5). As a main result of the three-loop analysis we confirm that minimal supersymmetric SU(5) cannot be ruled out by the current experimental data on proton decay rates.

The goal of this white paper is to give a comprehensive overview of the rich field of forward physics. We discuss the occurrences of BFKL resummation effects in special final states, such as Mueller-Navelet jets, jet gap jets, and heavy quarkonium production.It further addresses TMD factorization at low x and the manifestation of a semi-hard saturation scale in (generalized) TMD PDFs. More theoretical aspects of low x physics, probes of the quark gluon plasma, as well as the possibility to use photon-hadron collisions at the LHC to constrain hadronic structure at low x, and the resulting complementarity between LHC and the EIC are also presented. We also briefly discuss diffraction at colliders as well as the possibility to explore further the electroweak theory in central exclusive events using the LHC as a photon-photon collider. CONTENTS I. Introduction II. Manifestations of BFKL evolution A. Mueller-Navelet jets B. Toward precision studies of BFKL dynamics C. Unintegrated gluon distribution (UGD) D. BFKL resummation of NLO collinear factorization: Heavy quarkonium production E. Diffraction: Gaps between jets III. Hadronic Structure at low x and gluon saturation A. Color Glass Condensate and high gluon densities B. Transverse Momentum Dependent Parton Distribution Functions C. Complementarity between EIC and LHC D. Forward dijets: from LHC to EIC E. Color Glass Condensate beyond high energy factorization IV. Imprints of high gluon densities at low x at the LHC. A. Open QCD questions at a hadron-hadron collider: Parton fragmentation, mini-jets and their interplay with high parton densities B. Forward direct photon measurements 27 C. Top quark pair production as a tool to probe Quark-Gluon -Plasma formation 29 V. Ultra-peripheral collisions at hadronic colliders and exclusive reactions at the EIC 31 A. Existing measurements on diffractive vector meson photoproduction in UPCs 31 B. Future measurements on diffractive vector meson photoproduction in UPCs 33 C. The ratio of Ψ(2s) and J/Ψ photoproduction cross-sections as a tool to quantify non-linear QCD evolution 34 D. Planned measurements at the Electron Ion Collider 36 VI. Odderon discovery and diffractive jets 38 A. Soft diffraction and the Odderon discovery by the D0 and TOTEM experiments 38 B. Inclusive diffraction measurements at the LHC and sensitivity to the Pomeron structure 39 VII. Electroweak and Beyond the Standard Model Physics 41 A. Precision Proton Spectrometer (PPS) and ATLAS Forward Proton detector (AFP) at high luminosity 41 B. Non-elastic contribution in photon-photon physics 44 C. Exclusive production of Higgs boson 47 D. Anomalous quartic couplings with proton tagging 49 VIII. Conclusions 53 Acknowledgements 54 References 55