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 a , on which the current can act [Jµ, a]=itabb

Example: U (1) current and a complex scalar field : [ Jµ, ]=i

If 0 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 = @ ⇤@ + µ ⇤ (⇤) L µ Note: the scalar field 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), (0)] 0 = d p e · (pN p) 0 Jµ(0) N N (0) 0 h | | i " h | | ih | | i Z XN ip x 4 e · (pN p) 0 (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 4(p p) 0 J (0) N N (0) 0 = ip ⇥(p0)⇢(p2) N h | µ | ih | | i µ XN 4(p p) 0 (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

4(p p) 0 J (0) N N (0) 0 = ip ⇥(p0)⇢(p2) N h | µ | ih | | i µ XN 4(p p) 0 (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), (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 )(p m ) e e ( + m )(x, m )=0 ⌘ ) ⇤ Z µ ⇥ ⇤ 2 2 Exercise. Show that @ µ J =0 and Q 0 =0 ⇢ ( m )= N ( 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 4(p) µ ! µ 1 · µ 2 · 3 µ µ Imposing current conservation @µJ =0

p ⇢ + n ⇢ p (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 _3 5 ' 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), (0)] 0 = p ⇢ + p2n p (np) ⇢ + C n 4(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), (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. 0 p, 1' non to which Anderson' has drawn attention: G =8 B -BB =F that the scalar zero-mass excitations of a super- IL(. V p. V V p, LL(V conducting neutral Fermi gas become longitudi- into the form nal plasmon modes of finite mass when the gas is charged. Pv 2 2 8 B =0, 8 t" +e 8 =0. (4) The simplest theory which exhibits this be- v y 0 havior is a gauge-invariant version of a model used by Goldstone' himself: Two real' scalar Equation (4) describes vector waves whose quanta fields y„y, and a real vector field A interact have (bare) mass ey, . In the absence of the gauge through the Lagrangian density field coupling (e =0) the situation is quite differ- ent: Equations (2a) and (2c) describe zero-mass 2 2 and In L =-&(&v ) -@'7v ) scalar vector bosons, respectively. pass- 1 2 ing, we note that the right-hand side of (2c) is

2 2 JL(, V just the linear approximation to the conserved —V(rp + y ) -P'~ 1 2 P,v current: It is linear in the vector potential, where gauge invariance being maintained by the pres- ence of the gradient term. ' V p =~ p -eA When one considers theoretical models in 1 jL(, 1 p, 2' which spontaneous breakdown of symmetry under a semisimple group occurs, one encounters a p2 +eA {p1' variety of possible situations corresponding to the various distinct irreducible representations to which the scalar fields may belong; the gauge F =8 A -BA PV P, V V field always belongs to the adjoint representa- tion. ' The model of the most immediate inter- e is a dimensionless coupling constant, and the est is that in which the scalar fields form an metric is taken as -+++. I. is invariant under octet under SU(3): Here one finds the possibil- simultaneous gauge transformations of the first ity of two nonvanishing vacuum expectation val- kind on y, + iy, and of the second kind on A ues, which may be chosen to be the two Y=0, Let us suppose that V'(cpa') = 0, V"(&p,') ) 0; then I3=0 members of the octet. There are two spontaneous breakdown of U(1) symmetry occurs. massive scalar bosons with just these quantum Consider the equations [derived from (1) by numbers; the remaining six components of the A treating ~y„ay„and & as small quantities] scalar octet combine with the corresponding governing the propagation of small oscillations components of the gauge-field octet to describe

508 The Higgs model

Consider a pair of scalar field 1 , 2 coupled to an electromagnetic field Aµ 1 1 1 = (D )(Dµ )+ (D )(Dµ ) V 2 F F µ⌫ L 2 µ 1 1 2 µ 2 2 | | 4 µ⌫ 2 2 2 1 + 2 Dµ1 = @µ1 eAµ2 Dµ2 = @µ2 + eAµ1 = | | 2 The potential exhibits spontaneous symmetry breaking, e.g. 0 0 = v =0 , h | 1| i 6 i.e. it has a minimum for V ( 2) | | v = | | p2 Example. The potential V ( 2)= µ2 2 + ( 2)2 | | | | 2 | | has a minimum for µ2 1 = | | r 2 Higgs model: mass for the photon

Expand all fields around the minimum

2 V000 2 2 1 v + ⌘1 2 ⌘2 V ( ) V + v ⌘ ' ' | | ' 0 2 1 and linearise the equations of motions, assuming A ⌘ ⌘ v µ ⇠ 1 ⇠ 2 ⌧ µ µ @ (@µ⌘1 + evAµ)=@ Aµ0 =0 Aµ0 = Aµ + @µ(⌘2/ev)

µ 2 @ Fµ⌫ = J⌫ ev(@µ⌘2 + evAµ)= (ev) Aµ0 ' V ( 2) | |

This is the Anderson condition, giving

a mass m A = ev to the photon

Note. The fact that the photon gets a mass relies only on the fact that the scalar field gets a VEV 1

2 VOLUME 13, NUMBER 9 PHYSICAL REVIEW LETTERS 31 AUGUST 1964

*Work supported in part by the U. S. Atomic Energy (1962). They predict a branching ratio for decay mode Commission and in part by the Graduate School from (1) of -10~. funds supplied by the %isconsin Alumni Research 6N. P. Samios, Phys. Rev. 121, 275 (1961). Foundation. ~The best previously reported estimate comes from ~R. Feynman and M. Gell-Mann, Phys. Rev. 109, the limit on K2 —p + p . The 90% confidence level is — 13 (1958). )g&p)2&10 (g& „(t: M. Barton, K. Lande, L. M. Leder- 2T. D. Lee and C. N. Yang, Phys. Rev. 119, 1410 man, and %illiam Chinowsky, Ann. Phys. {N.Y. ) 5, {1960);S. B. Treiman, Nuovo Cimento 15, 916 (1960). 156 (1958). The absence of the decay mode p+ —e++e+ 3S. Okubo and R. E. Marshak, Nuovo Cimento 28, +e is not a good test for the existence of neutral cur- 56 {1963);Y. Ne'eman, Nuovo Cimento 27, 922 {1963). rents since this decay mode may be absolutely forbid- 4Estimates of the rate for X+-7l++e++e due to in- den by conservation of muon number: G. Feinberg duced neutral currents have been calculated by several and L. M. Lederman, Ann. Rev. Nucl. Sci. 13, 465 authors. For a list of previous references see Mirza A. {1963). Baqi B6g, Phys. Rev. 132, 426 {1963). S. N. Biswas and S. K. Bose, Phys. Rev. Letters The calculation5M. Baker and S. Glashow, NuovoofCimento Englert25, 857 12, and176 (1964). Brout

BROKEN SYMMETRY AND THE MASS OF GAUGE VECTOR MESONS* F. Englert and R. Brout Faculte des Sciences, Universite Libre de Bruxelles, Bruxelles, Belgium (Received 26 June 1964)

It is of interest to inquire whether gauge those vector mesons which are coupled to cur- vector mesons acquire mass through interac- rents that "rotate" the original vacuum are the tion'; by a gauge vector meson we mean a ones which acquire mass [see Eq. (6)]. Yang-Mills field' associated with the extension ~e shall then examine a particular model of a Lie group from global to local symmetry. based on chirality invariance which may have a The importance of this problem resides in the more fundamental significance. Here we begin possibility that strong-interaction physics orig- with a chirality-invariant Lagrangian and intro- inates from massive gauge fields related to a duce both vector and pseudovector gauge fields, system of conserved currents. ' In this note, thereby guaranteeing invariance under both local we shall show that in certain cases vector phase and local y, -phase transformations. In mesons do indeed acquire mass when the vac- this model the gauge fields themselves may break uum is degenerate with respect to a compact the y, invariance leading to a mass for the orig- Lie group. inal Fermi field. ~e shall show in this case Theories with degenerate vacuum (broken that the pseudovector field acquires mass. symmetry) have been the subject of intensive In the last paragraph we sketch a simple study since their inception by Nambu. ' ' A argument which renders these results reason- characteristic feature of such theories is the able. possible existence of zero-mass bosons which (1) Lest the simplicity of the argument be 'y' tend to restore the symmetry. We shall shrouded in a cloud of indices, we first con- show that it is precisely these singularities sider a one-parameter Abelian group, repre- which maintain the gauge invariance of the senting, for example, the phase transformation theory, despite the fact that the vector meson of a charged boson; we then present the general- acquires mass. ization to an arbitrary compact Lie group. ~e shall first treat the case where the orig- The interaction between the and the A y & inal fields are a set of bosons qA which trans- fields is form as a basis for a representation of a com- H. =ieA y~8 y-e'y*yA A ' pact Lie group. This example should be con- int p. p, p, p, sidered as a rather general phenomenological model. As such, we shall not study the par- where y =(y, +iy, )/v2. We shall break the ticular mechanism by which the symmetry is symmetry by fixing &y) e0 in the vacuum, with broken but simply assume that such a mech- the phase chosen for convenience such that anism exists. A calculation performed in low- &V) =&q ') =&q,)/~2. est order perturbation theory indicates that %'e shall assume that the application of the 321 Higgs model: massive scalar boson

Expand all fields around the minimum

2 V000 2 2 1 v + ⌘1 2 ⌘2 V ( ) V + v ⌘ ' ' | | ' 0 2 1 and linearise the equations of motions, assuming A ⌘ ⌘ v µ ⇠ 1 ⇠ 2 ⌧ 2 (⇤ + v V000)⌘1 =0

This is the equation for a scalar ``Higgs’' boson of mass mH = v V000 p

Note. The fact that a boson is found relies on the reliability of the linear approximation, i.e. quantum fluctuations

leave ⌘ 1 around the minimum The Higgs field in the Standard Model

The Standard Model for EW interactions has SU (2) U (1) gauge symmetry ⇥ i i i ijk j k Fµ⌫ = @µW⌫ @⌫ Wµ g2✏ WµW⌫ 1 µ⌫ 1 µ⌫ gauge = Tr(Fµ⌫ F ) Bµ⌫ B L 4 4 Bi = @ B @ B µ⌫ µ ⌫ ⌫ µ

To give a mass to the vector bosons we introduce a complex Higgs doublet

+ = 0 ✓ ◆

with a Lagrangian µ =(D )⇤(D ) V (⇤) LH µ g ~ D = I @ + i 1 B + ig W~ µ µ 2 µ 2 2 · µ ⇣ ⌘ The BEH mechanism in the Standard Model

Impose spontaneous symmetry breaking

2 1 3 + i4 v = 0 1 0 = v =0 0 † 0 = p2 1 + i2 h | | i 6 )h| | i 2 ✓ ◆

This is enough to give mass to the vector bosons through the coupling with the Higgs field, without any information on the potential

1 1 2 Wµ± = Wµ iWµ 2 p2 ⌥ µ vg2 + µ (D )⇤(Dµ) Wµ W ! 2 ⇣ 2 ⌘ 2 µ v 3 g2 g1g2 W3 + (Wµ Bµ) 2 µ 8 g1g2 g B ✓ 1 ◆✓ ◆ Vector bosonElectroweak mass eigenstates unification

Diagonalising the mass matrix one obtains the masses of the eigenstates, the The new boson B mixes with the weak boson W 3 W and Z bosons and the photon !

! B

! W3 ! ! Z !

!

! Z The results of the mixing are the photon and a new gauge boson , electrically v v 2 2 neutral like theM Wphoton= g 2 MZ = g + g M =0 2 2 1 2 q The VEV of the Higgs field

To measure the VEV of the Higgs field we need the coupling g 2 , which we obtain from weak decay, involving left particles only

⌫ g1 ~ L = ew = iLD/L¯ Dµ = I @µ + i BµYw + ig2 W~ µ L L 2 2 · ✓ ◆ ⇣ ⌘ νµ

νµ µ− W − µ− e− e− _ νe _ νe

2 GF g2 = 2 p2 8MW p 1/2 v v =( 2GF ) 246 GeV M = g ) ' W 2 2 The Higgs potential in the Standard Model

The Higgs potential in the Standard Model is the simplest compatible with spontaneous symmetry breaking and renormalisability 2 2 2 µ V ()= µ ⇤ + (⇤) v = r Expanding around the minimum of the potential, as follows

1 p2+ = p2 v + H + ia0 ✓ ◆

we find a single neutral Higgs boson H with a mass mH = vp2

Recently LHC has discovered a boson compatible with the Higgs of the Standard Model, with a mass m 125 GeV H ' This is enough to fix all the parameters of the Higgs potential

µ 88.8 GeV 0.13 ' ' Stability of the Higgs potential

In order to have spontaneous symmetry breaking, we need to have > 0

But the value of can change with the value of problem with the stability ) of the electroweak vacuum

Stable Metastable

To compute the running of we need to introduce the interaction of the Higgs boson with fermions, which are Yukawa interactions Fermion masses

To compute the running of we need to introduce the interaction of the Higgs boson with fermions, which are Yukawa interactions

= hˆijQ¯ d hˆijQ¯ ˜u hˆijL¯ e +h.c. LYukawa d L,i R,j u L,i R,j L i R,j

In the Standard Model ˜ = i 2 ⇤ (otherwise 2HDM) At the minimum of the Higgs potential 1 0 1 v = ˜ = p2 v p2 0 ✓ ◆ ✓ ◆

after diagonalisation of the Yukawa couplings each fermion acquires a mass v mf = hf p2 Higgs interactions

Expanding , and V ( ) around the minimum of the Higgs Lgauge LYukawa potential we find all interactions of SM particles with the Higgs g g = g ffH¯ + HHH H3 + HHHH H4 L Hff 6 24 g + V V µ g H + HHV V H2 V µ HV V 2 ⇣ ⌘

In the SM all these couplings are constrained and proportional to the masses of fermions and to the masses of boson squared 2 2 W =1 mf 2mV 2mV gHff = gHV V = gHHV V = 2 v v v Z =1/2

2 2 3mH 3mH g = gHHHH = HHH v v2 This is enough to compute the radiative corrections to the Higgs potential RG flow of the SM couplings

The running with energy of the SM couplings has been computed up to three loops 0.10 0.000 1.0 0.08 3s bands in g3 l yt Mt = 173.3 ± 0.8 GeV gray b 0.8 a3 MZ = 0.1184 ± 0.0007 red 0.06 l Mh = 125.1 ± 0.2 GeV blue -0.005 0.6 g2 H L quartic

g1 0.04 H L H L Higgs couplings 0.4 coupling H L the SM -0.010

0.02 of

0.2 quartic m in TeV l Mt = 171.1 GeV

0.0 yb Higgs 0.00 3s bands in function as MZ = 0.1205 -0.015 M = 173.3 ± 0.8 GeV gray 102 104 106 108 1010 1012 1014 1016 1018 1020 t

Beta a M = 0.1184 ± 0.0007 red -0.02 3 Z RGE scale m in GeV as MZ = 0.1163 Mh = 125.1 ± 0.2 GeV blue MtH = 175.6L GeV H L -0.04 H L H L Figure 1: Renormalisation of the SM gauge couplings g1 = 5/3gY ,g2,g3, of the top, bottom H L -0.020 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 and ⌧ couplings (y , y , y ), of the Higgs quartic coupling and of the Higgs mass parameter m10. 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10H 10L 10 t b ⌧ p All parameters are defined in the ms scheme. We include two-loop thresholds at the weak scale RGE scale m in GeV RGE scale m in GeV and three-loop RG equations. The thickness indicates the 1 uncertainties in M ,M , ↵ . All SM couplings stay small± until the Planckt h scale3 no problem with 1.5 0.8 Planck mass, we find the following values of the SM parameters: ) perturbativity 1s bands in M = 173.3 ± 0.8 GeV gray Mt MW 80.384 GeV m t h 0.6 g1(MPl)=0.6154 + 0.0003 173.34 0.0006 (61a)1.0 m a3 MZ = 0.1184 ± 0.0007 red GeV 0.014 GeV W 10 12 The Higgs potential✓ becomes◆ potentially unstable at aroundM h⇤=I125.1=± 10 0.2 GeV blue10 GeV g (M )=0.5055 (61b) 2 Pl ê H L L ↵ (M ) 0.1184 0.4 3 Z W H L H L g (M )=0.4873 + 0.0002 (61c) m 3 Pl m h ê 0.5 m H L 0.0007 h t Mt ↵3(MZ ) 0.1184 m

yt(MPl)=0.3825 + 0.0051 173.34 0.0021 (61d) contribution 0.2 GeV 0.0007 and ✓ ◆ t ê top m H ê

M 0.0 l t h b

(M )= 0.0143 0.0066 173.34 +(61e)m

Pl ê

GeV l 0.0 ✓ ◆ b 3s bands in ↵3(MZ ) 0.1184 Mh +0.0018 +0.0029 125.15 M = 173.3 ± 0.8 GeV gray 0.0007 GeV -0.5 t ✓ ◆ -0.2 as MZ = 0.1184 ± 0.0007 red Mh m(M )=129.4GeV+1.6GeV 125.15 +(61f) Mh = 125.1 ± 0.2 GeV blue Pl GeV ✓ ◆ H L Mt ↵3(MZ ) 0.1184 -1.0 -0.4 H L H L 0.25 GeV 173.34 +0.05 GeV 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 GeV 0.0007 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 H 10 L 10 10 10 10 ✓ ◆ RGE scale m in GeV RGE scale m in GeV All Yukawa couplings, other than the one of the top quark, are very small. This is the well- known flavour problem of the SM, which will not be investigated in this paper. The three gauge couplings and the top Yukawa coupling remain perturbative and are fairly Figure 2: Upper: RG evolution of (left) and of (right) varying Mt, ↵3(MZ ), Mh by weak at high energy, becoming roughly equal in the vicinity of the Planck mass. The near 3. Lower: Same as above, with more “physical” normalisations. The Higgs quartic coupling equality of the gauge couplings may be viewed as an indicator of an underlying grand± unification even within the simple SM, once we allow for threshold corrections of the order ofis 10% compared around with the top Yukawa and weak gauge coupling through the ratios sign() 4 /yt ascaleofabout1016 GeV (of course, in the spirit of this paper, we are disregarding the acute | | and sign() 8 /g2, which correspond to the ratios of running masses mh/mt and mh/mW , naturalness problem). It is amusing to note that the ordering of the coupling constants at | | p respectively (left). The Higgs quartic -function is shown in units of its top contribution, (top p 16 contribution) = 3y4/8⇡2 (right). The grey shadings cover values of the RG scale above the t Planck mass M 1.2 1019 GeV, and above the reduced Planck mass M¯ = M /p8⇡. Pl ⇡ ⇥ Pl Pl

18 Stability of the EW vacuum

To study the stability of the EW vacuum one looks at the Higgs effective potential for H v 2 1 2 2 yt 3 e↵ 3 (4 ln 6 + 3 ln 3) 3yt ln e↵ (H) 4 ' (4⇡)2 2 2 Ve↵ (H) H  ✓ ◆ ' 4 3 g2 5 3 g2 + g2 5 g2 ln 2 + (g2 + g2)2 ln 1 2 8 2 4 6 16 1 2 4 6 ✓ ◆ ✓ ◆

180 1000 6 10 1 10 8 107 108 9 1s bands in 200 Instability Instability 10 M =125.1±0.2 GeV 178 h 10 red dotted 19 10 10800 LCDM 16 a3=0.1184±0.0007 14 -500 11 10 10 gray dashed GeV 150 12 Non 176 12 10 GeV 10 decay in 9 13 H L

t stability - in 10

8 yr

- 600 t

M 7 10

Meta perturbativity M 6 1016 in H L 5 174 4 vacuum -1000 mass 100 10 time LI =10 GeV mass s of 1,2,3

s - Meta-stability 1 bands in M =125.1±0.2 GeV 400 pole pole h 10 19 Life 172 10 red dotted Top Top 50 Stability a3=0.1184±0.0007 Probability 10-1500 gray dashed 200 CDM 170 H L 10 1018 1014 Stability 0 173.3±0.8 H L 168 -2000 0 50 100 150 200 12010 122 124 126 128 130 132 1 1017 171 172 173 174 175 176 171 172 173 174 175 176 Higgs pole mass Mh in GeV Higgs pole mass Mh in GeV Pole top mass Mt in GeV Pole top mass Mt in GeV

Figure 3: Left: SM phase diagram in terms of Higgs and top pole masses. The plane is Figure 7: Left: The probability that electroweak vacuum decay happened in our past light-cone, divided into regions of absolute stability, meta-stability, instability of the SM vacuum, and non- taking into account the expansion of the universe. Right: The life-time of the electroweak perturbativity of the Higgs quartic coupling. The top Yukawa coupling becomes non-perturbative vacuum, with two di↵erent assumptions for future cosmology: universes dominated by the cos- for Mt > 230 GeV. The dotted contour-lines show the instability scale ⇤I in GeV assuming mological constant (⇤CDM) or by dark matter (CDM). ↵3(MZ )=0.1184. Right: Zoom in the region of the preferred experimental range of Mh and Mt (the grey areas denote the allowed region at 1, 2, and 3). The three boundary lines correspond to 1- variations of ↵3(MZ )=0.1184 0.0007, andNote the that grading ( of⇤ the)isnegativeintheSM. colours indicates the size ± I of the theoretical error. Figure 6 shows the SM phase diagram in terms of the parameters (MPl)andm(MPl). The sign of each one of these parameters corresponds to di↵erent phases of the theory, such that

The quantity e↵ can be extracted from the e↵ective(M potentialPl)=m at(M twoPl)=0isatri-criticalpoint. loops [112] and is explicitly given in appendix C. The region denoted by ‘ h M ’ corresponds to the case in which eq. (79)isnotsatisfied h i⇡ Pl and there is no SM-like vacuum, while the Higgs field slides to large values. In the region of 4.3 The SM phase diagram in termspractical of Higgs interest, and the top upper masses limit on m is rather far from its actual physical value m = Mh, although it is much stronger than M ,theultimateultravioletcuto↵ of the SM. A much more The two most important parameters that determine the various EW phases of the SM arePl the stringent bound on m can be derived from anthropic considerations [131] and the corresponding Higgs and top-quark masses. In fig. 3 we update the phase diagram given in ref. [4] with our band in parameter space is shown in fig. 6. We find it remarkable that the simple request of the improved calculation of the evolution of the Higgs quartic coupling. The regions of stability, existence of a non-trivial Higgs vacuum, without any reference to naturalness considerations, metastability, and instability of the EW vacuum are shown both for a broad range of Mh and gives a bound on the Higgs bilinear parameter m. Unfortunately, for the physical value of , Mt, and after zooming into the region corresponding to the measured values. The uncertainty the actual numerical value of the upper bound is not of great practical importance. from ↵3 and from theoretical errors are indicated by the dashed lines and the colour shading along the borders. Also shown are contour lines of the instability scale ⇤I .

As previously noticed in ref. [4], the measured6.3 values Lifetime of Mh and ofMt theappear SM to be vacuum rather special, in the sense that they place the SM vacuum in a near-critical condition, at the border The measured values of Mh and Mt indicate that the SM Higgs vacuum is not the true vacuum between stability and metastability. In the neighbourhood of the measured values of M and of the theory and that our universe ish potentially unstable. The rate of quantum tunnelling out M , the stability condition is well approximated by t of the EW vacuum is given by the probability d}/dV dt of nucleating a bubble of true vacuum within↵ a3 space(MZ ) volume0.1184dV and time interval dt [132–134] M > 129.6GeV+2.0(M 173.34 GeV) 0.5GeV 0.3GeV . (64) h t 0.0007 ± 4 S(⇤ ) d} dt dV e B . The quoted uncertainty comes only from higher order perturbative corrections. Other non-= ⇤B (80)

19

29 Custodial symmetry

Rearrange the Higgs field as a 2x2 complex matrix

1 0⇤ + 2 2 = V ()= µ Tr(†)+[Tr(†)] p2 0 ✓ ◆

The Higgs potential is invariant under SU(2) SU(2) L ⇥ R

SU(2) : U SU(2) : U † L ! L R ! R The kinetic term

µ g2 g1 Tr(D ) D Dµ = @µ + i ~ W~ µ i Bµ3 µ † 2 · 2 is invariant under SU (2) SU (2) only if g0 0 L ⇥ R ! µ =Tr(D )†D Linv µ |g1=0 Custodial symmetry

After spontaneous symmetry breaking SU (2) SU (2) is broken L ⇥ R 1 v 0 0 0 = UL 0 0 U † = 0 0 h | | i p2 0 v ) h | | i R 6 h | | i ✓ ◆

However U 0 0 U † = 0 0 , leaving a residual SU (2) symmetry, Lh | | i L h | | i L+R called ``custodial’’ symmetry

+ In the limit g 0 , W ,W ,Z form a triplet under SU(2) MW = MZ 1 ! L+R ) For g =0 , one defines the ⇢ parameter as 1 6 2 2 2 MW g2 2 MW 2 = 2 2 = cos ✓W ⇢ 2 2 =1(treelevel) MZ g1 + g2 ) ⌘ MZ cos ✓W

Experimentally, ⇢ =1 . 0050 0 . 0010 , suggesting that custodial symmetry is exp ± broken very mildly [ALEPH, DELPHI, L3, OPAL, SLD Phys. Rep 427 (2006) 257] Thus the Higgs vacuum expectation value breaks the global symmetry in the pattern

SU(2) SU(2) SU(2) + . (63) L × R → L R

This is called “custodial symmetry;” actually, some authorsrefertoSU(2)R by this name [6], while others reserve it for SU(2)L+R [7]. Since SU(2) is a three-dimensional group, the number of broken generators is 3+3 3=3. These give rise to three massless Goldstone bosons, which aretheneatenbytheHiggs− mechanism to provide the mass of the W +, W −,andZ bosons, 1 M 2 = g2v2 W 4 1 M 2 = (g2 + g′2)v2 . (64) Z 4

Thus 2 2 MW g 2 2 = 2 ′2 =cos θW (65) MZ g + g or 2 MW ρ = 2 2 =1 (66) MZ cos θW at tree level.

A Exercise 3.3 - Show that Wµ transforms as a triplet under global SU(2)L and a singlet under SU(2)R,andhenceasatripletundertheunbrokenSU(2)L+R. Thus, in the limit g′ 0, W +,W−,Z form a triplet of an unbroken global symmetry. This → ′ explains why MW = MZ in the g 0limit. Custodial symmetry also helps→ us understand properties of the theory beyond tree level. ′ Due to the unbroken SU(2)L+R in the g 0limit,radiativecorrectionstotheρ parameter → 2 in Eq. (66) due to gauge and Higgs bosons must be proportional to g′ .Forexample,the leading correction to the ρ parameter from loops of Higgs bosons (Fig. 2) in the MS scheme is 11G M 2 sin2 θ m2 ρˆ 1 F Z W ln h . (67) Breaking of custodial2 symmetry2 in the SM ≈ − 24√2π MZ This correction vanishes in the limit g′ 0(sin2 θ 0). The custodial symmetry protects → W → the tree-levelHiggs relation loopρ =1fromradiativecorrections,andhenceit’sname.Thislea corrections to W and Z masses ding

h h 2 2 2 11GF MZ sin ✓W mH ⇢ g =0 ln 06 p 2 m2 + | ' 24 2⇡ Z

Figure 2: Virtual Higgs-boson loops contribute to the W and Z masses. Yukawa interactions break custodial symmetry unless hu = hd 16 t t

WWZZ

b t

2 2 2 3GF 2 2 mt mb mt ⇢ hu=hd + mt + mb 2 2 2 ln 2 Figure 3: Virtual top-quark loops contribute to the W| and6 Z masses.' 8⇡2p2 m m m ✓ t b b ◆ correction, proportional to ln mh,allowsustoboundtheHiggs-bosonmassfromprecision electroweak measurements. Custodial symmetry also helps us understand radiative corrections due to massive fermions, as shown inGet Fig. 3.an The idea leading of correction SM m due H to from loops ofEW top aprecisionnd bottom quarks data is [8] 2 2 2 3GF 2 2 mt mb mt ρˆ 1+ mt + mb 2 2 2 ln 2 . (68) ≈ 8π2√2 ! − m m m " t − b b

Exercise 3.4 - Show that this correction vanishes in the limit mt = mb. Exercise 3.5 - Show that the t, b Yukawa couplings have a custodial symmetry in the limit mt = mb. This leading correction, proportional to the square of the fermion mass, allowed us to predict the top-quark mass from precision electroweak measurementsbeforeitwasdiscovered. Thus we see that custodial symmetry is vital to our understanding of the electroweak sector. However, the physical Higgs boson itself is not really necessary. As long as the electroweak-symmetry-breaking mechanism possesses custodial symmetry, the ρ parameter equals unity at tree level and is protected from large radiative corrections. Let’s develop an effective field theory of electroweak symmetry breaking that makes custodial symmetry manifest [9]. A simple way to do this is to replace the matrix field Φ with another matrix field, Σ,whichcontainstheGoldstonebosons, πi (which are eaten by the weak vector bosons), but does not contain a physical Higgsboson: v σ·π Φ ΣΣ= ei v . (69) → 2 The Lagrangian for electroweak symmetry breaking (EWSB) is the analogue of Eq. (51), v2 = Tr (D Σ)†DµΣ . (70) LEWSB 4 µ The Goldstone-boson matrix field Σ transforms under custodial symmetry as

SU(2)L : Σ LΣ → † SU(2)R : Σ ΣR → † SU(2) + : Σ LΣL . L R → 17 Breaking of custodial symmetry in the SM

m = 152 GeV 6 March 2012 Limit Theory uncertainty (5) ∆αhad = 5 0.02750±0.00033 0.02749±0.00010 4 incl. low Q2 data 2 3 ∆χ

2

1 LEP LHC excluded excluded 0 40100 200

mH [GeV] Learning outcomes

In this lecture we have learnt

The BEH Higgs gives a mass to vector bosons via spontaneous symmetry breaking

The form of the potential gives rise (or not) to an incomplete multiplet of scalar “Higgs” bosons

The Standard Model contains only one neutral Higgs boson

The observed mass of the SM Higgs boson is compatible with a metastable Higgs vacuum

The SM Higgs potential possesses a custodial symmetry which, from experimental data, is broken very mildly