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Addressing the Higgs Naturalness Problem: Insights from φ⁴-Theory and Renormalization Group Evolution

This study delves into the Higgs naturalness problem, particularly with respect to quadratic divergence in φ⁴-theory. We analyze the success of the Standard Model (SM) and the fine-tuning issues associated with the Higgs mass. Our approach focuses on the renormalization group evolution of the Higgs mass, revealing that the mass remains stable at approximately 100 GeV due to self-corrections in the SM. Ultimately, we discuss the implications of these findings, challenging previous notions about the necessity of new symmetries or particles to resolve naturalness.

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Addressing the Higgs Naturalness Problem: Insights from φ⁴-Theory and Renormalization Group Evolution

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  1. Thoughts on Higgs naturalness problem Zheng-Tao Wei Nankai University 第十届粒子物理、核物理和宇宙学交叉学科 前沿问题研讨会，三亚，2011.12.17-21。

  2. Z. Wei, L. Bian, arXiv:1104.2735. • Introduction: • Higgs naturalness problem • Quadratic divergence in φ4-theory • Our approach • Summary

  3. Introduction • The SM is very successful. • Higgs mechanism provides mass for • everything. Higgs—God particle. • The crucial purpose of LHC is to search • and study Higgs.

  4. Measurements of SM Higgs mass from • ATLAS and CMS: • Exclude: 141—467 GeV; • Remain: 114—141 GeV. • 12.13 new results discover hint of Higgs. • ATLAS MH=126 GeV (3.6σ) • excludes: 112-115GeV, >131GeV • CMS MH=124 GeV (2.6σ) • excludes: 127-600GeV

  5. Higgs naturalness • problem Fine-tuning: bare and counter-term fine-tuning at (102/1019)2~10-34 Higgs is unnatural.

  6. History The Origins of lattice gauge theory,Kenneth G. Wilson, 2004

  7. Dimensional regularization is not physical. = c Λ2 • He did the first explicit calculation.

  8. Some scenarios of solution: • Veltman’s condition: • New symmetry: • SUSY, scale invariance, … • New particle, dimension: • composite Higgs, little Higgs, • extra dimension, .…

  9. A modern review on naturalness: arXiv: 0801.2562. Naturalness problems in physics: 1. Higgs mass, 2. fermion mass, 3. cosmological constant, …

  10. Is it really a problem, • or just an illusion? • SM is renormalizable, mH independent of Λ. • What can the equation tell us? • -----Chuan-Hung Chen’s question • One-loop result may be misleading. • Some examples: • Asymptotic freedom, g->0, large Log • Sudakov form factor, F(Q2)->exp{-c’ln2(Q2/m2)} • large double-Log 11

  11. Our idea • To study RG evolution of mH with energy • due to quadratic divergence. • What’s the asymptotic behavior of mH • in the short-distance? 12

  12. Quadratic divergence in φ4-theory • φ4-theory is simple and provides an ideal place to • study renormalization and RGE. The mass renormalization is additive, not multiplicative.

  13. Pauli-Villas regularization:

  14. Renormalization scheme Fujikawa’s idea: counter-term renormalized quantities Thus,

  15. Renormalization group equation for scalar mass The new mass anomalous dimension is negative.

  16. Solution for the case with μ2>>m2 m2 decreases as energy scale increase.

  17. Another way to look at RGE • A new concept The bare quantities are the renormalized parameters at the UV limit. m0=m(μ→∞)

  18. Our approach to Higgs naturalness • The counter-term changes to

  19. RGE for Higgs mass The bare mass is μ-independent, • The evolution is with respect to scale μ, • not lnμ. • The new mass anomalous dimension • is proportional to -mH2.

  20. Solution of the RGE Where mV is called by “Veltman mass”. • The Higgs mass is an exponential damping • function when energy scale increases. • The Higgs mass in the UV limit approaches • “Veltaman mass” mV. • The bare mass is not divergent, but finite.

  21. Peculiarity of the SM: • 1. The couplings are proportional to masses. • 2. The evolutions of coupling constants and • masses are correlated with each other.

  22. End of the naturalness problem? • The Higgs mass about 100 GeV order is stable. • The Higgs naturalness problem is solved by • radiative corrections themselves within SM. • New symmetry and new particles are • unnecessary.

  23. But, our start point is wrong: It is wrong in sign!!!

  24. Exponential damping becomes growing. • Naturalness problem becomes more serious.

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