Higgs theory Resummation and theoretical uncertainties Andrea Banfi The HiggsSingl eboson-logari tinhm theic r Standardesummatio Modeln Terascale Monte Carlo School 2014 - DESY Hamburg 12/03/2014 Outline The Brout-Englert-Higgs mechanism The Higgs boson in the Standard Model Stability of the Higgs potential Custodial symmetry A historical prelude: Goldstone theorem To understand the importance of the Brout-Englert-Higgs mechanism we need to go back to the Goldstone theorem Our main characters: µ A conserved current @µJ =0 A set of scalar fields φ a , on which the current can act [Jµ, φa]=itabφb Example: U (1) current and a complex scalar field φ : [ Jµ, φ]=iφ If 0 φ 0 =0 (spontaneous symmetry breaking) the theory contains one h | | i6 massless “Goldstone” boson Example of a Lagrangian leading to Goldstone bosons µ 2 2 = @ φ⇤@ φ + µ φ⇤φ λ(φ⇤φ) L µ − Note: the scalar field φ does not need to be an elementary field Proof of the Goldstone theorem The basic object entering the Goldstone theorem is the commutator of the current with the scalar field 4 ip x 4 0 [Jµ(x), φ(0)] 0 = d p e · δ (pN p) 0 Jµ(0) N N φ(0) 0 h | | i " − h | | ih | | i Z XN ip x 4 e− · δ (pN p) 0 φ(0) N N Jµ(0) 0 − − h | | ih | | i# XN Exercise. Derive the above expression Crucial point of Goldstone theorem is the Lorentz decomposition of the expectation value on the complete set of states N | i δ4(p p) 0 J (0) N N φ(0) 0 = ip ⇥(p0)⇢(p2) N − h | µ | ih | | i µ XN δ4(p p) 0 φ(0) N N J (0) 0 = ip ⇥(p0)⇢(p2) N − h | | ih | µ | i − µ XN Proof of the Goldstone theorem Crucial point of Goldstone theorem is the Lorentz decomposition of the expectation value on the complete set of states N | i δ4(p p) 0 J (0) N N φ(0) 0 = ip ⇥(p0)⇢(p2) N − h | µ | ih | | i µ XN δ4(p p) 0 φ(0) N N J (0) 0 = ip ⇥(p0)˜⇢(p2) N − h | | ih | µ | i µ XN Exercise. Show that, because of causality, ⇢ ˜(p2)= ⇢(p2) − Proof of the Goldstone theorem The basic object entering the Goldstone theorem is the commutator of the current with the scalar field 1 0 [J (x), φ(0)] 0 = @ dm2∆(x, m2)⇢(m2) h | µ | i µ Z0 2 4 0 2 2 ipx ipx 2 2 ∆(x, m ) d p ⇥(p )δ(p m ) e e− ( + m )∆(x, m )=0 ⌘ − − ) ⇤ Z µ ⇥ ⇤ 2 2 Exercise. Show that @ µ J =0 and Q 0 =0 ⇢ ( m )= N δ ( m ) with N =0 , | i6 ) 6 i.e. there exists at least one massless state, a Goldstone boson Problems with Goldstone theorem There are as many massless Goldstone boson as the number of broken generators. However, no one has ever seen any. What is their fate? Longitudinal waves in plasmas Maxwell’s equations in Lorenz gauge kµA˜µ(k)=0 k2A˜µ(k)=(!2 c2~k2)A˜µ(k)= J˜µ(k) − − If J ˜ µ ( k ) A˜ µ ( k ) we can induce a mass term for the electromagnetic field / In spite of a local conservation law, the corresponding gauge bosons are not massless [Anderson ‘64] Example. In the propagation of a plasma with velocity of sound c s 2 2~ 2 0 ~ ! c k 2 0 ~ ~ 2 ~ J (!, k)= − !p A (!, k) J (!,k)= !pA (!,k) −!2 c2~k2 ? − ? − s ne2 ! p = is the plasma frequency of the medium r m What have longitudinal waves in plasma have to do with Goldstone theorem? Longitudinal waves in plasma In a plasma, the ions are still, giving a uniform charge density ⇢0 = nee The fluctuations of the electron density give a longitudinal wave, that propagates with the speed of sound of the plasma Relation to Goldstone theorem In a non-relativistic theory, there exists a special reference frame n µ (e.g. the rest frame of the ions in a plasma) p ⇢(p2) p ⇢ (p2,n p)+n ⇢ (p2,n p)+C n δ4(p) µ ! µ 1 · µ 2 · 3 µ µ Imposing current conservation @µJ =0 p ⇢ + n ⇢ p δ(p2)⇢ + p2n p (nk) ⇢ µ 1 µ 2 ! µ 4 µ − µ 5 In non-relativistic theories, ⇢ 4 can be zero, thus⇥ avoiding the problem⇤ of Goldstone bosons. Problem. In relativistic theories there seem to be no preferred reference frame, yet Goldstone bosons seem not to exist LETTERS 16 September 1964 Volume 12, number 2 PHYSICS to the interactions, Aß M, E is the en- µ Vµ strong ergy of 0-transition. According to experimental /T 0.988: 0.004. data Tµ µ° = 1 Substituting the numbers into (1) we obtain D' the disagreement between t T µ/ Tµ=1.003 and the theory and experiment will be in our case 1.5 * 0.4%. When discussing this result one should tr that in (1) the terms ýW take into consideration only e2 In e-2 were correctly taken into account but Fig. 1. the terms ^- e2 were discarded. It seems to us that the conclusion that in the µ vµ theory of weak interaction with intermediate W- meson 0- and µ-constants must be with good ac- curacy the same (taking into account the correc- G' tions due to the inter- D' electromagnetic and weak actions), is in favour of the weak interaction the- iW ory with W-meson unlike the four-fermion theory. More detailed paper will be published else- e ve where. Fig. 2. The author is indebted to B. V. Geshkenbein, 1. Yu. Kobsarev, L. B. Okun, A. M. Perelomov, well into account the radiation correction to the 1. Ya. Pomeranchuk, V. S. Popov, A. P. Rudik and ß-decay constant found by Berman 3) and Kino- M. V. Terentyev for valuable discussions. 4) shita and Sirlin we obtain for the muon life time Tµ=]- 3e2 A2 Aß Mµ2 i e2 in '1) +327T 2E _35 ' References To02 µ2 1) M. V. Terentyev (in is the life time by B. L. Ioffe, print). where T µo muon calculated 2) T. D. Lee, Phys. Rev. 128 (1962) 899. means of universal theory of four fermion inter- 3) S. M. Berman, Phys. Rev. 112 (1958) 267. action with a constant taken from ß-decay without 4) T. Kinochtta, A. Sirlin, Phys. Rev. 113 (1959) any corrections, Aß is the cut off momentum due 1652. Higgs’ solution to Goldstone***** problem BROKEN SYMMETRIES, MASSLESS PARTICLES AND GAUGE FIELDS P. W. HIGGS Tait Institute of Mathematical Physics, University of Edinburgh, Scotland Received 27 July 1964 Recently a number ofpeople have discussed ever, gave a proof that the failure of the Goldstone 1, the Goldstone theorem -2): that any solution of a theorem in the nonrelativistic case is of a type Lorentz-invariant theory which violates an inter- which cannot exist when Lorentz invariance is im- nal symmetry operation of that theory must con- posed on a theory. The purpose of this note is to 3) tain a massless scalar particle. Klein and Lee show that Gilbert's argument fails for an impor- showed that this theorem does not necessarily ap- tant class of field theories, that in which the con- ply in non-relativistic theories and implied that served currents are coupled to gauge fields. 4), their considerations would apply equally wgll to Following the procedure used by Gilbert let 4), Lorentz-invariant field theories. Gilbert how- us consider a theory of two hermitian scalar fields 132 Higgs’ solution to Goldstone problem In order to quantise gauge theories, one needs to introduce a gauge-fixing µ condition, e.g. Coulomb gauge nµA =0 i 0 [A (x), φ (0)] 0 = p ⇢ + p2n p (np) ⇢ + C n δ4(p) h | µ 1 | i|F.T. µ 1 µ − µ 2 3 µ Combining with Maxwell’s equations ⇥ ⇤ @ F µ⌫ = J ⌫ F = @ A @ A µ µ⌫ µ ⌫ − ⌫ µ + i 0 [J (x), φ (0)] 0 = p2n p (np) ⇢(p2, (np)) h | µ 1 | i|F.T. µ − µ ⇥ ⇤ + In a broken gauge symmetry there are no massless Goldstone bosons The Higgs model VoLUME 1$, NUMBER 16 PHYSICAL REVIEW LETTERS 19 OcTQBER 1964 BROKEN SYMMETRIES AND THE MASSES OF GAUGE BOSONS Peter W. Higgs Tait Institute of Mathematical Physics, University of Edinburgh, Edinburgh, Scotland (Received 31 August 1964) In a recent note' it was shown that the Gold- about the "vacuum" solution y, (x) =0, y, (x) = y, : stone theorem, ' that Lorentz-covaria. nt field in which spontaneous breakdown of s "(s (np )-ep A =0, (2a) theories 1 0 ) symmetry under an internal Lie group occurs contain zero-mass particles, fails if and only if (&'-4e,'V" ) = (2b) the conserved currents associated with the in- (y,')f(&y, 0, ternal group are coupled to gauge fields. The purpose of the present note is to report that, s r"'=eq (s"(c, ) ep A -t. (2c) V 0 p,1 0 p, as a consequence of this coupling, the spin-one quanta of some of the gauge fields acquire mass; Equation (2b) describes waves whose quanta have the longitudinal degrees of freedom of these par- (bare) mass 2po(V"(yo'))'"; Eqs. (2a) and (2c) ticles (which would be absent if their mass were may be transformed, by the introduction of new zero) go over into the Goldstone bosons when the var iables coupling tends to zero. This phenomenon is just fl =A -(ey ) '8 (n, (p ), the relativistic analog of the plasmon phenome- p.