Uses and Abuses of the Coleman-Weinberg Effective Potential

Transcription

1 Uses and Abuses of the Coleman-Weinberg Effective Potential 27-Jan-2014 talk based loosely on: H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 MPI-K Particle and Astroparticle Sear

2 Outline - Define the Coleman-Weinberg Effective Potential - Demonstrate problems associated with extracting physical quantities. - Solve the problems - Time permitting, apply solution to, and discuss: - Coleman-Weinberg mechanism - RG-group improved vacuum stability analysis. Root problem: Effective potential is gauge dependent. 2

3 Coleman-Weinberg Effective Potential Phenomenological Definition Energy (expectation value of H) of state suitably localized at. Theoretical Definition Catalog of 1PI Green s functions at. subject to equivalent

4 Coleman-Weinberg Effective Potential This dual interpretation makes powerful theoretical tool. but is also quite fragile when applied to gauge theories. is gauge-dependent, and harbors serious gauge artifacts and great care must be taken in extracting physical quantities. a Applications found in literature: - Evaluate vacuum energy, condensates (VEVs), masses - Explore of vacuum structure - Study dimensional transmutation - Efficiently calculate (b.g. field method) counterterms correlation functions - Combine with kinetic terms to study metastable decay, solitons - finite T: phase transitions, thermodynamics 4

5 Literature G. Maiella (1979) M. Carrington (1992) S. Martin (2002) B, Grinstein, M. Trott (2008) 5

6 Method employed in literature Abelian-Higgs model Potential: Step 1: Fix a gauge Background renormalizable ( ). (Goldstone modes acquire gaugedependent mass: ) Step 2: Compute ective potential in gauge. Re Fixed order (# loops) in perturbation theory. oneloop Im But wait! There is an imaginary part! some 6

7 Method employed in literature Abelian-Higgs model Can t imize with an imaginary part Step 2 đ: Take real part (for now) Potential: Re Step 3: Minimize ective potential. (by brute force) Re Read off the physical quantities = correction to VEV (usually negative) = vacuum energy Im Im = Higgs mass-square 7

8 Problems 1. For phys. relevant models (incl. SM) are complex! suggests instability: Not the meta-stability kind, but of the Weinberg-Wu type. PRD 36, 2474 (1987) 2. is gauge-dependent, and physical quantities inherit the gauge-dependence. Can make the imaginary part vanish by moving to Feynman gauge (or bigger) Can make the real part larger (negative) by further raising. 8

9 Nielsen s Identities These statements on the ective potential are not unfounded. Nielsen s Identity for the Effective Potential: G. can be rearranged to: Can be traced back to original paper: variations with gauge-parameter is prop. to variations with field. Specifically, at critical points of V, V is gauge independent. Re 9

10 Three Questions: 1. Why does computed behave differently than as implied by Nielsen? Re Re 2. Exactly, which derived quantities are gauge-independent? vacuum energy -ind? correction to VEV Higgs mass -square exact definition: r 3. And how can we extract them in a gauge-independent manner? 10

11 Key observation: Expansion Method Brute force imization retains incomplete higher-order estimates. Must drop these incomplete estimates. H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 originally proposed in context of finite temperature. But works much better at zero-temperature. Calculation of the vacuum energy: 1. Insert here ( counts # of loops) 2. Effective potential will be a series in. as will its derivative: Minimization condition: 3. Solve by inversion of series : tree-level VEV quant. corr. Insert into imization condition. expand: Each power of h-bar must satisfy equality. establishes tree-level VEV GENUINE oneloop correction 11

12 Key observation: Expansion Method Brute force imization retains incomplete higher-order estimates. Must drop these incomplete estimates. H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 originally proposed in context of finite temperature. But works much better at zero-temperature. Calculation of the vacuum energy: Therefore, to order, the vacuum energy is obtained by evaluating the one-loop potential at the tree-level imum. 4. Insert into potential: expand once more 12

13 Expansion Method Theoretical Properties - Manifestly gauge-independent Re - No spurious imaginary part in broken phase - Valid for all extrema (incl. maxima and saddle) Practical advantage Difficult numerics associated with imization in a complicated, multiloop function is gone.! All that is needed is to find the extrema of the tree-level potential (easy), and the series generates the corrections. Im 13

14 Nielsen id. Vacuum expectation value VEV not gauge-independent! =) Re Reason: field-op. not invariant! My Interpretation: Ground state changes as is varied. M.E. gives mean of projection. Goldstone Higgs G H G H Need radial field-op. for -ind. VEV. 14

15 Nielsen id. Vacuum expectation value VEV not gauge-independent! =) Re Reason: field-op. not invariant! -expansion My Interpretation: method: Ground state changes as is varied. M.E. gives mean of projection. 1. agreement with Feynman diagrams. = Goldstone Higgs G H G H Need radial field-op. for -ind. VEV. 2. Always real-valued. 15

16 Higgs Masses Effective Potential Approximation small? Practical Advantage: much easier to compute than self-energy graphs. used often in SUSY models Theoretical Disadvantages: 1. Yields imaginary masses 2. is not gauge-invariant! R.J. Zhang Phys. Lett B 447, 89 (1999) Espinosa, Zhang, JHEP 03, 026 (2000) S. Martin, Phys Rev D 67, (2003) 3. ) size of error? not gauge-invariant; even if -expansion is used. 16

17 a little secret = 5 others Im Im Divergent in Landau gauge! Im Only revealed by -expansion. 17

18 Higgs Masses Effective Potential Approximation Practical Advantage: much easier to compute than self-energy graphs. Theoretical Disadvantages: Yields imaginary masses is not gauge-invariant! small? used often in SUSY models R.J. Zhang Phys. Lett B 447, 89 (1999) Espinosa, Zhang, JHEP 03, 026 (2000) S. Martin, Phys Rev D 67, (2003) 3. ) not gauge-invariant; even if Hidden divergence in gauge! Only revealed by -expansion. 18 -expansion is used.

19 vacuum energy correction to VEV Summary -ind? Gauge-independent only following -expansion. Matches Feynman pert. th. only following -expansion. matrix elements of non-gauge-invariant operators are not expected to be gauge-independent. Higgs mass -square exact definition: is approximating, not r Further, is gauge-dependent, and singular when Theoretical aboation. Never use. Application to more advanced settings? 1. Coleman-Weinberg Mechanism 2. RG-improved vacuum stability analysis 3. Metastability analysis 4. Phase transition analysis 19

20 Application #1: Coleman-Weinberg mech. Paradigm: Start with or (no mass scale) Insert small. quantum ects generate mass scale.,,, It is possible to analyze this case in a gauge-invariant fashion by using the -expansion method. here and here tree ( no longer counts # of loops) In this case, and, all other aspects of analysis goes through. both tree- and part of one-loop 20

21 Application #2: RG-improved vacua It is well known that in the Standard Model, there is a non-standard vacuum lying at extremely large field values: This is revealed only by an RGimprovement of the ective potential, Callan-Symanzik eqn: This improvement entails a resummation of logarithms across all orders in the loop expansion. EW phase non-standard phase A simple solution exists provided the R.G. coicient functions are gauge-independent. This is incompatible with the -expansion method needed to maintain gauge-independence. Nielsen identity: Unfortunately, anomalous dimensions are gauge-dependent. Still an open research problem. 21

22 22 Thank you!