THE SM VACUUM AND HIGGS INFLATION Javier Rubio based on arxiv:1412.3811 F. Bezrukov, J.R., M.Shaposhnikov 25th International Workshop on Weak Interactions and Neutrinos (WIN2015)
A fundamental scalar ATLAS+CMS, arxiv:1503.07589 m H = 125.09 ± 0.21(stat) ± 0.11 (syst) GeV low or high, depending on your taste...but certainly particular...
Higgs inflation at tree level L p = M P 2 + hh 2 R 1 g 2 2 (@h)2 4 (h2 vew) 2 2 Moving to the Einstein frame g µ = 2 (h)g µ 2 =1+ h h 2 L = M P 2 U( ) g 2 R 1 2 gµ µ M 2 P Scale invariance at h M P / p h All the non-linearities moved to the scalar sector U( )= 4 ( 2 v 2 EW) 2 U( )= M 4 P 4 2 h 1 e p 2/3 M P (1 + hv 2 EW M 2 P 2 ) for for h<m P / h h>m P / h F. L. Bezrukov and M. Shaposhnikov Phys. Lett. B659 (2008) 703 706
A sufficiently flat potential Scale invariance JF --> shift symmetry EF U 1.0 0.8 0.6 T T 10 5 U 0 0.4 0.2 0.0 U( ) ' M 4 P 4 2 h 1 e p 2/3 M P 2 0 1 2 3 4 5 6 /M P 2 h 10 11 Only / 2 h is important
The primordial spectra Scalar pert. Tensor pert. P S (k) =A s k k ns 1+ 1 2 s ln(k/k )+ 1 6 s (ln(k/k )) 2 +... P T (k) =A t k k nt + 1 2 t ln(k/k )+... s d2 n s d ln k 2 n s ' 1 2 N ' 0.97 r ' 12 N 2 ' 0.0033 HI
On the edge of stability SM remains perturbative all the way up till the inflat./planck scale Does the SM vacuum remains stable till those scales? µ d (µ) d log(µ) = +# 2 +... #y 4 t Non trivial interplay between Higgs self-coupling and top quark Yukawa coupling 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0-0.02-0.04 (no gravity) y t =0.9176, m t =170.0 y t =0.9235, m t =171.0 y t =0.9294, m t =172.0 y t =0.9359, m t =173.1 y t =0.9413, m t =174.0 y t =0.9472, m t =175.0 100000 1e+10 1e+15 1e+20 µ(gev) µ, GeV
Top quark & vac. instability y crit t = 0.9223 + 0.00118 s 0.1184 Mh 125.03 +0.00085 0.0007 0.3 Stable Metastable Stable Metastable Tevatron LHC/Tevatron M t =172.38±0.66 GeV 127 126.5 Stable Metastable 1e+18 1e+16 (µ 0 )=0 M h, GeV 126 125.5 125 µ 0, GeV 1e+14 1e+12 1e+10 124.5 124 0.91 0.92 0.93 0.94 0.95 0.96 y t (µ=173.2 GeV) CMS 1e+08 1e+06 0 0.01 0.02 0.03 0.04 0.05 y t -y t crit (µ=173.2 GeV) See F. Bezrukov, M. Shaposhnikov J.Exp.Theor.Phys. 120 (2015) 335-343 and references therein
Imagine that the top and Higgs masses are measured with a precision enough to conclude that our vacuum is not completely stable... Should we abandon HI?? Should/Must new physics appear below or µ 0 at the scale? How did we end in the right EW minimum?
HI is non-renormalizable Quantum corrections should be introduced by interpreting the theory as an EFT in which a particular set of higher dim. operators are included but... Which set of operators? The most general approach Add all kind of Planck scale suppressed operators (in the Einstein frame) This would automatically spoil the flatness of the potential. Indeed, it would generically kill all large-field models of inflation. The poor man s or self-consistent approach 1. Add only the higher dimensional operators generated by radiative corrections ( i.e, those needed to make the theory finite at every order in PT). 2. Require the procedure to maintain the symmetries of the original theory (in particular scale invariance at large field values). This provides a (partially) controllable link between the low and high energy parameters of the model and selects a particular set of UV completions. F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. Sibiryakov JHEP 1101 (2011) 016, See also C.P. Burguess, S.P. Patil, M.Trott JHEP 1406 (2014) 010
Top quark in Higgs background L F = y f p h! LA = y f 2 L t ( + )= y t p 2 F ( + ) t t = y t p 2 F ( ) t t + y t p 2 df ( ) d t t +... t At low energies At high energies F = F 0 (0) = 1 F h p 2 y f p F ( ) 2 F = M P p 1 e apple 1/2 h y t F 0 F 0 (1) =0 t Consider the propagation of the top quark Add counterterms to cancel divergencies at small
One-loop effective potential At low energies At high energies F = F 0 (0) = 1 F = const. F 0 ( 0 )=0 Add counterterms to cancel divergencies new tree level
Eff. behaviour of coupling constants yt 0.55 0.50 0.45 0.40 m h = 125.5 GeV m t = 173.1 GeV Neglecting the running of 1 and yt1 1 between µ M P / h and M P / p h y t (µ)! y t (µ)+ y t F 02 0.35 10-8 10-6 10-4 0.01 1 k m Dashed dy t = 0.025 Dotted dy t = -0.025 1 +... (µ)! (µ)+ 10 3 l " F 02 + 1 3 F 00 F The shape of the asymptotics is maintained 8 6 4 2 0-2 m h = 125.5 GeV m t = 173.1 GeV dl = -0.003 dl = 0 dl = 0.003 2 1# 10-8 10-6 10-4 10-2 10 0 k m Dashed dy t = 0.005 Dotted dy t = -0.005
Restoring Higgs inflation (M P / ) y t y t (M P / ) (M P / ) y t y t (M P / ) Higgs inflation requires absolute stability of the SM vacuum Higgs inflation can be possible even in the case of a metastable vacuum. 10 5 Non-critical Critical 10 3 l 0-5 h -10 m h = 125.5 GeV m t = 173.1 GeV 10-10 10-8 10-6 10-4 10-2 10 0 k m But how to avoid finishing in the wrong vacuum???
Sketch of effective potential (not to scale!)
Sketch of effective potential (not to scale!) Inflation takes place with the standard initial conditions
Sketch of effective potential (not to scale!) Higgs oscillates and particle creation takes place. J. Garcia-Bellido, D.G. Figueroa, J.R., Phys.Rev. D79 (2009) 063531 F. Bezrukov, D. Gorbunov and M. Shaposhnikov JCAP 0906 (2009) 029 J. Repond, J. R, M. Shaposhnkov, in preparation
Sketch of effective potential (not to scale!) V T = 1 X 6 2 P Z 1 0 k 4 dk 1 k (m) e k(m)t 1 I f t h e r e h e a t i n g temperature is large enough the wrong minimum disappears. arxiv:1412.3811 F. Bezrukov, J.R., M.Shaposhnikov
Sketch of effective potential (not to scale!) The field settles down at the true EW minimum and stays there till the present time.
Symmetry restoration r 10 7 10 6 10 5 10 4 10 3 10 2 r c r F r B r W r Z Higgs Fermions Bosons W bosons Z bosons 10 1 0 50 100 150 200 250 j T R ' 1.8 10 14 GeV 10 7 U U 0 8 6 4 2 0-2 T = 8 x 10 13 GeV T = 7 x 10 13 GeV T = 6 x 10 13 GeV T = 5 x 10 13 GeV T = 0-4 0.00 0.05 0.10 0.15 0.20 0.25 10 3 kf T + ' 7 10 13 GeV T R >T + Higgs inflation can be possible even if our vacuum is not completely stable
What about lifetime? 4 2 SM + x-coupling SM 10 7 U U 0 0-2 -4 0.00 0.05 0.10 0.15 0.20 0.25 10 3 kf Z. Lalak,M. Lewicki, P. Olszewki JHEP 1405 (2014) 119, V. Branchina, E. Messina Phys.Rev.Lett. 111 (2013) 241801 etc... For SM computation see : J. R. Espinosa and M. Quiros, Phys. Lett. B 353 (1995) 257 J. R. Espinosa, G. F. Giudice and A. Riotto, JCAP 0805 (2008) 002
CONCLUSIONS The Higgs field can inflate the Universe HI provides universal predictions if the UV completion respects SI n s ' 1 2 N ' 0.97 r ' 12 N 2 ' 0.0033 The relation of these predictions to LE observables contains an irreducible theoretical uncertainty. UV completion? Higgs inflation can be possible even if our vacuum is metastable The HI scenario is just a particular realization of a general idea. Vacuum instability is not necessarily a problem if: The potential is modified below the scale of inflation The reheating process is efficient enough as to make the wrong minimum disappear temporally.
BACKUP SLIDES
The new finite parts should be promoted to new independent coupling constants with their own RG equations... d d log µ = (, 1,...) d 1 d log µ = 1(, 1,...) dy t d log µ = y t (y t,y t1,...) dy t1 d log µ = y t1 (y t,y t1,...)............ but since we are dealing with a non-renormalizable theory, the set of RG equations is not closed...
Truncation The one-loop diagrams associated to the new counterterms are given by The two-loop contributions generated by the original Lagrangian 3. In order to have a good perturbative expansion, the finite parts must be of the same order (in power counting) than the loops producing them. O( 2,y 4 ) y O(y 3,y ) O(y 2 )
Example 1. Compute the quadratic lagrangian (Jordan F.) 2. Get rid of the mixings in the quadratic action 3. Read out the cutoff from higher order operat.
Strong Coupling p h h M P P M P h S h h 2 M P G Weak Coupling M P p h M P h h A consistent EFT : Cutoffs are parametrically larger than all the energy scales involved in the history of the Universe
Respect scale invariance -> Dimensional regularization The choice of µ In renormalizable theories, it is arbitrary and field-independent We modify this prescription in our non-renormalizable theory! y t 2 M P 2 2 x h P-II = µ 2 R + X n a n n P-I M P x h M P x h f e f 0 Jordan frame def. Einstein frame equiv. P-I P-II M 2 P + h h 2 M 2 P P-I P-II M 2 P M 4 P /(M 2 P + h h 2 )
Lets look at the asymptotics (µ( )) = 0 + b log 2 p 1 e apple q e! b ' 2.3 10 5 0 = 0 (m h,m t ) 1 0 q e = q e (m h,m t ) O(1) q e p 1 p 1 p 1 e apple e 2b( ) UNIVERSALITY CRITICALITY NO INFLATION 1.0 0.8 RGE TREE 1.05 1.00 0.95 N=56 N=59 4 3 U é U 0 0.6 0.4 0.2 0 & b/16 0.0 0 1 2 3 4 5 6 fêm P /M P U é U 0 0.90 0.85 0.80 0.75 0.70 0 ' b/16 10 3 e 0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 fêm P 12 10 8 6 4 2 /M P U é U 0 2 1 0. b/16 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 fêm P /M P