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Understanding Radiative Corrections in the Standard Model Theory

Learn about radiative corrections to the Standard Model, including basics like gauge couplings and Higgs potential, top quark contributions to precision observables, decoupling theorem, running of coupling constants, and implications for precision measurements in particle physics.

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Understanding Radiative Corrections in the Standard Model Theory

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  1. Radiative Corrections to the Standard Model S. Dawson TASI06, Lecture 3

  2. Radiative Corrections • Good References: • Jim Wells, TASI05, hep-ph/0512342 • K. Matchev, TASI04, hep-ph/0402031 • M. Peskin and T. Takeuchi, Phys. Rev. D46, (1992) 381 • W. Hollik, TASI96, hep-ph/9602380

  3. Basics • SM is SU(2) x U(1) theory • Two gauge couplings: g and g’ • Higgs potential is V=-22+4 • Two free parameters • Four free parameters in gauge-Higgs sector • Conventionally chosen to be • =1/137.0359895(61) • GF =1.16637(1) x 10-5 GeV -2 • MZ=91.1875  0.0021 GeV • MH • Express everything else in terms of these parameters

  4. Example:  parameter • At tree level, • Many possible definitions of ; use At tree level GCC=GNC=GF

  5.  parameter (#2) Top quark contributes to W and Z 2-point functions qq piece of propagator connects to massless fermions, and doesn’t contribute here

  6. Top Corrections to  parameter (Example) • 2-point function of Z • Shift momentum, k’=k + px Nc=number of colors n=4-2 dimensions

  7. Top Corrections to  parameter (#2) • Shift momenta, keep symmetric pieces • Only need g pieces

  8. Top Corrections to  parameter (#3) • Only need answer at p2=0 • Dimensional regularization gives:

  9. Top Corrections to  parameter (#4) • 2-point function of W, Z Lt=1-4sW2/3 Rt=-4sW2/3

  10. Top quark doesn’t decouple • Longitudinal component of W couples to top mass-- eg, tbW coupling: • Decoupling theorem doesn’t apply to particles which couple to mass For longitudinal W’s:

  11. Why doesn’t the top quark decouple? • In QED, running of  at scale  not affected by heavy particles with M >>  • Decoupling theorem: diagrams with heavy virtual particles don’t contribute at scales  << Mif • Couplings don’t grow with M • Gauge theory with heavy quark removed is still renormalizable • Spontaneously broken SU(2) x U(1) theory violates both conditions • Longitudinal modes of gauge bosons grow with mass • Theory without top quark is not renormalizable • Effects from top quark grow with mt2 Expect mt to have large effect on precision observables

  12. Modification of tree level relations • r is a physical quantity which incorporates 1-loop corrections

  13. Corrections to GF • GF defined in terms of muon lifetime • Consider 4-point interaction,with QED corrections • Gives precise value: • Angular spectrum of decay electrons tests V-A properties

  14. GF at one loop Vertex and box corrections are small (but 1/ poles don’t cancel without them)

  15. Renormalizing Masses • Propagator corrections form geometric series

  16. Running of , 1 • (0)=1/137.0359895(61) • (p2)= (0)[1+]

  17. Running of , 2 k+p Calculate in n=4-2 dimensions • Photon is massless:(0)=0   p k

  18. Running of , 3 Contributions of heavy fermions decouple: Contributions of light fermions:

  19. Running of , 4 • From leptons: lept(MZ)=0.0315 • Light quarks require  at low p2 • Strong interactions not perturbative • Optical theorem plus dispersion relation:

  20. Have we seen  run? Langacker, Fermilab Academic Series, 2005

  21. Final Ingredient is sW2 • sW is not an independent parameter

  22. Predict MW • Use on-shell scheme: • r incorporates the 1-loop radiative corrections and is a function of , s, Mh, mt,…

  23. Data prefer light Higgs • Low Q2 data not included • Doesn’t include atomic parity violation in cesium, parity violation in Moller scattering, & neutrino-nucleon scattering (NuTeV) • Higgs fit not sensitive to low Q2 data • Mh< 207 GeV • 1-side 95% c.l. upper limit, including direct search limit Direct search limit from e+e-Zh

  24. Logarithmic sensitivity to Mh • Data prefer a light Higgs 2006

  25. Understanding Higgs Limit MW(experiment)=80.404  0.030

  26. Light Higgs and Supersymmetry? • Adding the extra particles of a SUSY model changes the fit, but a light Higgs is still preferred

  27. How to define sin2W • At tree level, equivalent expressions • Can use these as definitions for renormalized sin2W • They will all be different at the one-loop level On-shell Z-mass Effective sW(eff)

  28. sin2w depends on scale • Moller scattering, e-e-e-e- • -nucleon scattering • Atomic parity violation in Cesium

  29. S,T,U formalism • Suppose “new physics” contributes primarily to gauge boson 2 point functions • cf r where vertex and box corrections are small • Also assume “new physics” is at scale M>>MZ • Two point functions for  , WW, ZZ, Z

  30. S,T,U, (#2) • Taylor expand 2-point functions • Keep first two terms • Remember that QED Ward identity requires any amplitude involving EM current vanish at q2=0

  31. S,T,U (#3) • Use JZ=J3-sW2JQ Normalization of JZ differs by -1/2 from Lecture 2here

  32. S,T,U (#4) • To O(q2), there are 6 coefficients: • Three combinations of parameters absorbed in , GF, MZ • In general, 3 independent coefficients which can be extracted from data

  33. S,T,U, (#5) SM contributions in , GF, and MZ Advantages: Easy to calculate Valid for many models Experimentalists can give you model independent fits

  34. Limits on S & T • A model with a heavy Higgs requires a source of large (positive) T • Fit assumes Mh=150 GeV

  35. Higgs can be heavy in models with new physics • Specific examples of heavy Higgs bosons in Little Higgs and Triplet Models • Mh  450-500 GeV allowed with large isospin violation (T=) and higher dimension operators We don’t know what the model is which produces the operators which generate large T

  36. Compute S,T, U from Heavy Higgs • SM values of S, T, U are defined for a reference Mh0

  37. Using S,T, & U • Assume new physics only contributes to gauge boson 2-point functions (Oblique corrections) • Calculate observables in terms of SM contribution and S, T, and U

  38. Conclusion • Radiative corrections necessary to fit LEP and Tevatron data • Radiative corrections strongly limit new models

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