Corrections to the SM effective action and stability of EW vacuum

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1 Corrections to the SM effective action and stability of EW vacuum Zygmunt Lalak ITP Warsaw Corfu Summer Institute 2015 with M. Lewicki and P. Olszewski arxiv: (JHEP), arxiv: , and to appear soon with T. Krajewski arxiv: with O. Czerwińska and Ł. Nakonieczny arxiv:

2 Outline: SM effective potential SM phase diagram BSM physics via higher-order operators Gauge fixing in-dependence of tunneling rate Modifications of the vacuum properties due to expanding background

3 SM E ective potential V SM (µ) = Standard Model E ective potential m X i n i 64 2 M4 i " ln M 2 i µ 2! C i # For large field values m 2 << 2 and µ = the potential is very well approximated by ( V SM ( ) apple g 2! ln 4 y 2! 2 t y 2!! t ln g 2 2 4!! ln 2 g1 2 + g 2 2! ln 2 +3 ln 2 2 g1 2 + g 2 2!! ) 2 V SM ( ) e ( ) 4 4 classically quantum corrected 3

4 SM Metastability e < 0=) Metastability D. Buttazzo, et al. [arxiv: ]. G. Degrassi, et al. [arxiv: ]. See lectures by G. Degrassi Corfu

5 Tunneling Standard semiclassical formalism S. R. Coleman, Phys. Rev. D 15 (1977) C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16 (1977) with O(4) symmetric solution to euclidean equation of motion + 3 s ( ) q s = ~x 2 + x4 (s = 0) = 0 near at the the true vacuum (s = 1) = min at the false vacuum = v EW 5

6 Tunneling Action of the bounce solution Z ( S E = d 4 1 (x) 2 x + V ( (x))) =1 Z 1 = 2 2 dss 3 2 (s)+v( (s)), 2 allows us to calculate decay probability dp of a volume d 3 x dp = dtd 3 x S 2 E 4 2 det V 00 ( )] 2 + V 00 ( 0 )] 1/2 e S E. Simplifying normalisation factor replaced with width of the barrier / 0 size of the universe is T U = yr we can calculate the lifetime of the false vacuum (p( ) = 1) = 1 T 4 U 0 T U 4 e S E. 6

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8 New extrema created by quantum corrections (Coleman-Weinberg mechanism) = ~ condition for cancellation of corrections to the derivative of SM apple g g 2 1g g h 4 t 3(g g 2 2) 2 log g2 1 + g g2 4 log g y4 t log y2 t 2 running λ (2loop) 0.10 λ 0.05 λ RHS μ Hence sensitivity to New Physics

9 E ective potential with nonrenormalisable interactions We add new nonrenormalisable couplings (similar to V. Branchina and E. Messina, [arxiv: ].) V e ( ) ! 6 M 2 p + 8 8! 8 M 4 p. New Physics at Planck scale That modify the potential around the Planck scale: log = T U Figure: e ective potential with 6 = 1and 8 = 1. 9

10 Numerical vs Analytical again Figure: Decimal logatihm of lifetime of the universe in units of T U as a function of the nonrenormalisable 6(M p )and 8 (M p ) couplings, calculated numerically (left panel) and analytically (right panel). 10

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12 Magnitude of the suppression scale Approximate lifetime: New Physics at the scale M 1 = T U µ 4 ( min )TU 4 e min. Positive 6 and 8! stabilizing the potential e Figure: Scale dependence of 4 = V 4 with 6 = 8 = 1 for di erent values of suppression scale M. The lifetimes corresponding to suppression scales M =10 8, 10 12, are, respectively, log 10 ( T U )=1, 1302, 581 while for the Standard Model log 10 ( T U )=

13 Magnitude of the suppression scale Positive 8 and negative 6! New Minimum e Figure: Scale dependence of 4 = V 4 with 6 = 1and 8 =1for di erent values of suppression scale M. The lifetimes corresponding to suppression scales M = 10 8, 10 12, 10 16,are,respectively, log 10 ( T U )= 45, 90, 110 while for the Standard Model log 10 ( T U ) =

14 Gauge dependence of the tunneling rate It is well known that the e ective potential, and in general the e ective action, are gauge-dependent objects However, the statement about the spontaneous breaking of gauge symmetry is gauge invariant (N. K. Nielsen 1975) The gauge invariant observables are the values of the e ective potential at the extrema, and the tunneling rate between di erent minima When one computes the SM e ective potential in a straightforward manner (say naively), nothing looks gauge independent - neither the value of the e ective potential at the extrema (see L. Di Luzio and L. Mihaila 2014) nor the tunneling rate (ML,PO,ZL) 14

15 4 SM modification in running of Z due to gauge dependence Luca Di Luzio and Luminita Mihaila: arxiv: v1 The leading gauge dependence comes from the gauge-dependent anomalous rescaling of the field Contributes to: 1-loop potential γ function of the scalar field More important. One needs to remember that kinetic contribution to the action is muliplied by Z. L. Di Luzio, L. Mihaila /13

16 At one loop e ective potential contains gauge-dependent terms V 1, 1 = 256 h h 4 2 B g1 2 log h4 ( B g W g 2 2 ) i + W g2 2 log 3 h 12 2 W g4 2 ( 2 B g2 1 + W g 2 2 ) 64µ µ 4 3 (1) As pointed out by A. Andreassen, W. Frost and M. Schwartz 2014, who followed E. Weinberg and D. Metaxas 1996 and S. Coleman and E. Weinberg 1973, the key to save in the calculations the gauge independence of the potantial at the extrema is to realize, that to create extrema radiatively, loop corrections have to cancel between themselves or the tree-level contributions In CW model ~e

17 In the SM the equivalent condition is = ~ apple g g 2 1g g h 4 t 3(g g 2 2) 2 log g2 1 + g which holds at the extrema h = µ 6g2 4 log g y4 t log y2 t 2 Hence is of the order ~ g 4 and gives a higher order contribution It has been shown that that taking this relation into account in counting radiative contributions in the SM makes the value of the potential at the extrema gauge independent at LO (~ g 4 ) and NLO (~ g 6 ) 17

18 In general, the tunneling rate has the form =Ae B Weinberg and Mataxas argued that if the reordering of the radiative corrections used above holds everywhere, not only at the extrema, then the exponent B shall be gauge independent at the NLO. 18

19 Leading Order Gauge-fixing dependent terms in the e ective potential are order g 6 and corrections to the kinetic term are g 2. Observation, which allows one to ease the problem, is that once one includes in the euclidean action which is used to compute the bounce the renormalization factor in the 2-derivative term, and treats it consistently as a field dependent quantity, then one can go over to the new field variable h! p Z(h)h in terms of which the whole action becomes gauge independent at the modified leading order (that is assuming ~), and only mildly gauge dependent in the more standard expression, through small logarithmic terms. The LO procedure leading to gauge independent estimate the tunneling rate can easily be extended to the analysis of the role of the e ective nonrenormalisable operators, and the results shown correspond to such a case. 19

20 Gauge fixing in-dependence order g 6 20

21 Gauge fixing independence in abelian Higgs model in t Hooft gauge

22 Running and scaling

23 Explicit running

24 Renormalized effective action Action is explicitly µ-independent

25 Gauge fixing independence desired property: Nielsen [ = R C [ ] [ ]

26 Nielsen functions bounce is derived from the lowest nontrivial order, o(g 4 ), Lagrangian with these ingredients S B gauge fixing independent to the order g 6

27 Back to higher-order operators but Nielsen identity: Action supplemented this way is explicitly scale invariant and gauge fixing invariant

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29 Gauge dependence of the potential

30 Gravity Corrections in Curved Space

31 Effective action in curved background: gauge-less Higgs model

32 where:

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36 In Robertson-Walker background one may express curvature invariant through energy density and preassure RD

37 Quadratic part of the potential in RD in ds or MD

38 Critical Temperature in RD Critical Temperature in ds vs

39 Large field region Stability in RD Stability in ds

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42 Quantum gravity e ects: In Loop Quantum Cosmology holonomy corrections can be summarized as! 1 cr Hence, for given the correction becomes smaller.

43 Summary SM vacuum can be stabilized by higher order operators if they appear at suffciently low energy scale GeV SM vacuum lifetime can be dramatically shortened by higher order operators for any suppression scale Beyond the leading order one needs to define proper expansion of the action to demonstrate perturbatively the cancellation of gaugedependent contributions to the lifetime of the EW vacuum In the abelian Higgs model such a procedure can be carried out at the level of the renormalized effective action Peoperties of the electroweak vacuum - critical temperature and lifetime - can be modified by a fast expansion of the gravitational background