A Lecture on the Standard Model Higgs

Home , Higgs boson, Hypercharge, Standard Model, Weak isospin

Eduardo Pontón

IFT-UNESP & ICTP-SAIFR The Higgs in the SM

Goal: To cover the theory of Electroweak Symmetry Breaking, understand the main properties of the Higgs boson resonance within the Standard Model (SM), and to review the experimental status pos-LHC Run I…

Outline: • The Standard Model: Highlights • SM Higgs Properties • Production • Decay Channels

• The Higgs Boson Discovery and Experimental Status

• Further Remarks on the SM Structure

• Further Reading (sample bibliography) The Standard Model of Particle Physics (Highlights) The Standard Model

The theory describing the properties of, and interactions between all known elementary particles

(… hence, in principle, of everything made of these particles!)

A theory based on a gauged SU(3) SU(2) U(1) ``symmetry" C ⇥ L ⇥ Y

Physics Translation:

Describes spin-1 (quantum) ﬁelds coupled to the other ﬁelds (i.e. spin-1/2 and spin-0) through

the Noether currents of a global SU(3) SU(2) U(1) internal symmetry: C ⇥ L ⇥ Y

= + J µaGa + J µiW i + J µB + LSM ··· 3 µ 2 µ 1 µ ··· 8 color currents hypercharge current 3 weak isospin currents

Until recently: all collider-experiment results could be understood just in terms of these interactions! The Standard Model

Furthermore:

The electroweak (EW) sector, i.e. SU(2) U(1) , is ``spontaneously broken” L ⇥ Y

Translation: the vacuum we live in is not invariant under the EW transformations, so that it selects a particular direction in the internal SU(2) U(1) space… L ⇥ Y … much as the ground state of a ferromagnet, at sufﬁciently low temperatures, selects one direction, even if the underlying physics is perfectly rotationally invariant.

Consequence: three of the EW spin-1 particles are massive (hence mediate short-range interactions) and one is massless (mediating long range interactions). + 0 These are the W ,W, the Z and the photon.

Physics: ``weak” lifetimes (e.g. beta-decay or muon decay) much longer than those of EM or strongly induced decays. The Symmetry Breaking Sector

0 A striking difference between the (massless) photon and the (massive) W ± and Z :

• The photon has two physical polarizations

• The massive vector bosons have three physical polarizations

Clearly, the three extra d.o.f. must arise from the EW symmetry breaking sector (and we have been studying their properties for decades)

These, together with the 125 GeV resonance discovered in 2012, nicely ﬁt into the simplest imaginable picture (at least as measured by counting d.o.f.) for the symm. breaking sector!

The Standard Model posits the existence of a scalar (spin-0) ﬁeld, transforming as a doublet of

SU(2)L and with hypercharge Y =1/2 . In the vacuum, this ``Higgs doublet” has a vacuum expectation value (VEV):

0 (The direction is conventional H = and arbitrary, given the h i v SU(2) x U(1) invariance) ✓ ◆ The Symmetry Breaking Sector

The Standard Model posits the existence of a scalar (spin-0) ﬁeld, transforming as a doublet of

SU(2)L and with hypercharge Y =1/2 . In the vacuum, this ``Higgs doublet” has a vacuum expectation value (VEV):

0 (The direction is conventional H = and arbitrary, given the h i v SU(2) x U(1) invariance) ✓ ◆

The observed d.o.f. can be parametrized as follows:

0 H+ H = ei ~ ~⌧ · v + 1 h 0 p2 ⌘ H ``eaten NGB's" ✓ ◆ ✓ ◆

The 125 resonance (the ``Higgs boson")

2 ``Higgs ﬁeld” potential: V (H)= H H v2 H = v 174 GeV † |h i| ⇡ (Most general renormalizable one) The Symmetry Breaking Sector

2 2 V (H)= H†H v H = v 174 GeV |h i| ⇡ 1 =2v2h2 + p2vh3 + h4 m2 =4v2 4 h

Thus, within the SM, a measurement of the Higgs mass ﬁxes the remaining free parameter:

mh 125 GeV 0.13 ⇡ ⇡ But so far we have no direct evidence for , i.e. from cubic and quartic Higgs self-interactions!

Measuring these interactions directly would constitute a non-trivial check of the SM h h hhhh h h Alternatively: any deviations would indicate a h hh more complicated symmetry breaking sector, h hh i.e. physics beyond the SM!

0-0

0-0 The Higgs Mechanism

(or how massive gauge bosons acquire a third polarization)

Gauge boson masses arise from the Higgs “kinetic term”:

µ (D H)†D H where DµH =(@µ igWµ ig0/2 Bµ)H LSM µ

Minimal coupling prescription: @ D µ ! µ The construction that leads to consistent interactions of spin-1 ﬁelds:

µ⌫ a ⌫ a @µW = JNoether conserved source

Can use gauge invariance to choose:

0 0 i ~ ~⌧ i ~ (x) ~⌧ H = e · 1 H0 = U(x)H = 1 with U(x) = e v + h ! v + h · ✓ p2 ◆ ✓ p2 ◆ In this (unitary) gauge:

µ 1 µ 1 2 2 0 (DµH0)†D H0 = @µh@ h +(v + h) 01(gW 0 + g0/2 B0 ) 2 p2 ⇥ µ µ 1 ✓ ◆ The Higgs Mechanism

In this (unitary) gauge:

µ 1 µ 1 2 2 0 (DµH0)†D H0 = @µh@ h +(v + h) 01(gW 0 + g0/2 B0 ) 2 p2 ⇥ µ µ 1 ✓ ◆ From the h-independent terms, we read the gauge boson masses:

2 2 MW = gv/p2 MZ = g + g0 v/p2 p where 1 1 2 0 3 Wµ± =Mass(W generationµ MassiWµ ) generation andZ Higgs= andcW couplingsW Higgs+ s couplingsW inB theµ SM intW the= SM tan ✓W = g0/g p2 ⌥ µ µ Gauge bosonsGauge (V = bosonsW ± or (VZ=)acquiremassviainteractionwiththeHiggsW or Z)acquiremassviainteractionwiththeHiggs This leads to one important prediction: MW± = cW MZ vacuum condensate.vacuum condensate.

The interactionsVVVVVV betweenVVVVVV the Higgs boson and the massive igMV ⌘µ⌫ 0 0 0 gauge bosons are also fixed: h 0 h h0 0 vvv vvv h h h Thus, Thus, 2 2 2 2 2 i(MV /v )⌘µ⌫ ghV V =2mV /v ,Knowsand2 aboutg hhVEWSB! V =2mV /v , 2 2 ghV V =2mV /v , and ghhV V =2mV /v , i.e., the Higgsi.e., the couplings Higgs| to couplings vector tobosons{z vector are bosons proportional} are proportional to the to the correspondingcorresponding boson squared-mass. boson squared-mass. 0 Likewise, byLikewise, replacing byV replacingwith theV Higgswith field theh Higgsin the field aboveh0 in diagrams, the above the diagrams, the Higgs self-couplingsHiggs self-couplings are also proportional are also proportional to the square to of the the square Higgs of mass: the Higgs mass: 2 2 3 3mh 2 3 3mh 2 g = λv = 3, and3mh g = λ = .3 3mh hhh 2 g = λv = , hhhhand 2 g 2= λ = . hhh v 2 v hhhhv 2 v2 The Rest of the SM

(with apologies to neutrinos)

The SM fermions come in the simplest (smallest) representations: singlets or fundamental

i i } u YQ =1/6 Qi = L uR

L i {z d i SU(3)C triplets Y =2/3 L dR u ✓ ◆ |

i } Yd = 1/3 i ⌫L i L = l {z SU(3)C singlets L li R YL =1/2 L | ✓ ◆ Y = 1 l ( i =1, 2, 3 label the three generations) SU(2)L doublets singlets | {z } | {z } The EW part is chiral, i.e. the quantum numbers for LH and RH fermions are different!

With the SM Higgs doublet, we can write the following gauge invariant terms:

0 ˜ 2 H ⇤ SM Q H˜ uuR Q HddR LLHelR +h.c. H i H⇤ = L L ⌘ H L ✓ ◆ Yukawa matrices The Rest of the SM

(with apologies to neutrinos)

With the SM Higgs doublet, we can write the following gauge invariant terms:

Q H˜ u Q H d L H l +h.c. LSM L u R L d R L e R 1 The generation of masses for quarks and leptons is especially elegant in = (v + h) u uthe+ SMd (in otherd + approachesl l to+h EWSB,.c. fermion(in mass unitary generation gauge) is often p L u R L d R L e R 2 achallenge).ThefermionscoupletotheHiggsﬁeldthroughthegauge One can furthermore diagonalize⇥ the Yukawainvariant matrices Yukawa couplings, (and choose e.g., real,⇤ positive eigenvalues) by rotating independently the LH and RH components (in generation space) 0 + 0 Yukawa = hu(¯uRuLΦ u¯RdLΦ ) hd(d¯RdLΦ ∗ + d¯RuLΦ−)+h.c. (the rotation matrices appear only in the interactionsL with the− charged W’s: the− CKM matrix)−

The quarks and charged leptons acquire mass when Φ0 acquires a vacuum One notices that: the eigenvalues above expectation value:

• The Higgs VEV leads to fermion masses: mf = f v ffff • There are Yukawa interactions proportional to the fermion mass: v h0 Thus, Hence the discovery of the Higgs boson amounts to the discovery of new, non-gauge interactions! ghff¯ = mf /v , … and with a veryi.e.,Higgscouplingstofermionsareproportionaltothecorresponding distinctive pattern of strengths! fermion mass. Particle content of the Standard The SM Field ContentModel Particle Something is content of missing… the Standard Model

Something is missing… Higgs Production h hh h

h hhh hh h Singlehhe−,u,d Higgs Production h h h h hh We have seenh that the Higgs hhcoupleshe+, u,¯ d¯ most stronglyhh to the heavier particles… − e−,u,d e−,u,d e ,u,d h h ∗ … but our beams (and, moduloe−,u,d DM, our universe) hare made mostly of the lightest particles! Z h h Z∗ hh h

+ ¯ − + ¯ ¯ e , u,¯ d e ,u,d e+, u,¯ d¯ e , u, d + ¯ ¯ Z e , u, d Z h Hopeless! e−,u,d h e− h q′ − ∗ h Z∗ e+, u,¯ d¯ e h Z Z∗,W∗ Z∗ e+Rather,, u,¯ d¯ look for processese−,u,d involving the heavy gauge bosons: e+ q¯ Z e+ h Z Z, W Z∗ Z

− ′ e h q h f1 f3 Z∗ + ¯ ¯ ∗ ∗ V e , u, d Z Z ,W h + − ¯ V e e Z q h Z, W f2 f4 Z∗ ′ q(At leptons colliders) h 0-0 ∗ ∗ (At hadron colliders) (Vector Boson Fusion or VBF) Z ,W + e Z

q¯ ′ q Z, W h Z∗,W∗ 0-0 0-0 ¯ q Z, W

0-0 h hh h

h hhh hh h hhe−,u,d h h h h hh h hhe+, u,¯ d¯ − e−,u,d e−,u,d e ,u,d h ∗ e−,u,d h Z h Z∗ h

+ ¯ + ¯ ¯ e , u,¯ d e+, u,¯ d¯ e , u, d + ¯ ¯ Z e , u, d Z e−,u,d h e− h q′ − ∗ h Z∗ Singleh eHiggshh Productionh Z Z∗,W∗ Z∗ h hh h h e+, u,¯ d¯ e+ q¯ Rather, look for processesZ involvingh ethe+ heavyhh gauge bosons: Z Z, W h Z hh − − − e ,u,d e ,u,d ′ −h e h q− ∗ h e ,u,d f1 f3 ∗ e ,u,d ∗ ∗ Z h Z Z ,W Z∗ V h h h + ¯ + ¯ ¯ e , u,¯ d e , u, d + ¯ V + +¯¯ ¯ eZ, u,¯ d e Z e q, u, d Z, W f2 Z f4 − ′ ′ e h − q h ′ ∗ e ∗ ∗ h0-0 q h q(At leptons colliders) hZ (At hadron∗ Zcolliders),W (Vector∗ ∗ Boson Fusion or VBF) Z∗,W∗ Z Z ,W

+ Or the heavier efermions: e+ q¯ q¯ ¯ Z Z Z, W Z, W q Z, W ¯ ¯ ¯ ¯ t t b b 0-0 g 0-0 g g g h h h h g g g g t t b b

• Directly sensitive to the top Yukawa coupling! • Pays price of small bottom Yukawa coupling • Pays phase space price • Pays phase space price

• Challenging, but doable at the LHC • Enhanced0-0 in some BSM scenarios! 0-0 Loop-induced Higgs Couplings

Loop induced Higgs boson couplings There is another way in which heavy particles can affect Higgs physics: while, being colorless, Higgs bosonthe coupling Higgs to boson gluons does not couple to gluons at tree-level, at loop level one has: At one-loop, the Higgs boson couples to gluons via a loop of quarks: 3 m2 q g N = A (⌧ ) ⌧ = h g 4 1/2 i i 4m2 e↵ g↵sNg a µ⌫ i qi h0 = hG G X Lhgg 24⇡M µ⌫ a 4/3 for ⌧ 1 W A (⌧) ⌧ q¯ g 1/2 ! 0 for ⌧ 1 ⇢ This diagram leads to an effective Lagrangian Ng counts roughly the number of quarks Loop induced Higgs boson couplings gα N heavier than h (the top in the SM case) eff = s g h0Ga Gµνa , Higgs boson coupling to gluonshgg µν Aside:L non-decoupling24πmW At one-loop, the Higgs boson couples to gluons via a loop of quarks:0 where Ng is roughly the number of quarks heavier than h .Moreprecisely, g q 2 m2 2 tgs qi 1 gs h0 Ng = F1/2(xi) ,xi 2 , ≡2 m 2 i ⇠ 16⇡ ⇥hm ⇠ 16⇡ v !q¯ g t This diagram leads to an effective Lagrangian where the loop function F1/2(x) 1 for x 1. • Fourth generationeff gαsN→g 0ofa quarksµνa≫ would induce a cross section hgg = h GµνG , about nineL times24πmW larger, hence highly disfavored!

0 where Ng is roughly the number of quarks heavier than h .Moreprecisely, • However, heavy fermionsm that2 do not owe most of their N = F (x ) ,xqi , g 1/2 i i ≡ m2 mass to iEWSB can be allowed.h ! where the loop function F (x) 1 for x 1. 1/2 → ≫ Figure 2.15: Real and imaginary parts of the W boson (left) and heavy fermion (right) amplitudes in the decay H γγ as a function of the mass ratios τ = M 2 /4M 2. → i H i

Figure 2.16: The partial width for the decay H γγ as a function of MH with the W and all third generation fermion contributions (solid)→ and with W and only the top quark contribution (dashed) and with the W and t quark contributions for m (dotted lines). t →∞ The NLO QCD corrections

The QCD corrections to the quark amplitude in the decay H γγ consist only of two–loop → virtual corrections and the corresponding counterterms; some generic diagrams are shown in Fig. 2.17. There are no real corrections since the decay H γγ + g does not occur due to → color conservation. The calculation can be done in the on–shell scheme, in which the quark

90 Higgs Production XS at the LHC

“Gluon fusion” g t 5 SM Higgs production 10 t H0 LHC ! [fb] g t gg ! h “Vector boson fusion” or qq qqh 10 4 ! qt qq ! qqh qg q W,t¯ Z q 0 W, Z H0 3 t 10 qg W, Z H0 q W,q¯ Z qq ! Wh t g q bb ! h “Associatedq productiont withW, Z W or Z” 2 gg,qq ! tth W, Z 10 q t W,H Z0 W, Z q¯ t 0 qb ! qth g H q¯ 0 qq ! Zh H TeV4LHC Higgs working group tth t 100 125 200 300 400 500 g t¯ mh [GeV] H0 g t

t¯ QCD Corrections

Characteristic diagrams of the QCD radiative corrections are shown in Fig. 3.19. They involve the virtual corrections to the gg H subprocess, which modify the LO fusion cross → It turns out that the previous ``leading order” contributionsection by a coe ﬃtocient ``gluon linear in αs ,andtheradiationofgluonsintheﬁnalstate.Inaddition,fusion” receives large radiative corrections from QCD (even EW corrections mustHiggs be bosons included). can be produced in gluon–quark collisions and quark–antiquark annihilation which contribute to the cross section at the same order of αs.

g Some representative examples of next-to- H g Q • • • leading order (NLO) diagrams are shown: g

KNLO NLO/LO 1.6 ! g q ⌘ ⇠ g q Higher-order corrections exhibit convergence Q • H q¯ g g • • Figure 3.19: Typical diagrams for the virtual and real QCD corrections to gg H. →

MSTW90 NNLO

(pb) The cross sections for the subprocesses ij H + X, i, j = g,q,q,canbewrittenas σ MSTW90 NLO → Inclusive productionαs cross section:αs σˆ = σ δ δ 1+CH (τ ) δ(1 τˆ)+DH(ˆτ, τ ) Θ(1 τˆ) (3.60) MSTW90 LO ij 0 ig jg Q π − ij Q π − 10 ! " # $ τ τ M 2 /s τ M 2 / m2 where• theQCD new scaling at NNLO: variable ˆ,supplementing H = H and Q = H 4 Q introduced 2 earlier, is defined at the parton level asτ ˆ = MH /sˆ; Θ is the step function. H H The coe• ffiBottomcients C (τ Q+)and topDij quark(ˆτ, τQ)havebeendeterminedinRefs.[180,286]forarbi- mass effects exact to NLO trary Higgs boson and quark masses and the lengthy analyticalexpressionshavebeengiven there [see• also NNLO2.3.3 for in some large details ontop the calculationmass limit and on the renormalization scheme]. § If all the corrections eq. (3.60) are added up, ultraviolet and infrared divergences cancel. However• NNLO collinear singularities partons are left over and are absorbed into the renormalization of the parton densities [84, 325] where the MS factorization scheme can be adopted. The• NLO final result EW for the corrections hadronic cross section (5%) at NLO can be cast into the form gg H H αs d H H H σ(pp H + X)=σ0 1+C τH L + σgg + σgq + σqq (3.61) 1 • Partial→ QCD N3LO partonsπ dτH △ △ △ 120 140 160 180 200 220 240 260 280 300 " # H mH (GeV) The coefficient C denotes the contributions from the virtual two–loop quark corrections regularized• Estimate by the infrared for singular mixed part ofEW-QCD the cross section corrections for real gluon emission. It splits Anastasiou et. al. arXiv:1107.0683 2 into the infrared term π ,atermdependingontherenormalizationscaleµR of the coupling

147 Energy dependence (LHC)

2 10 pp → H (NNLO+NNLL QCD + NLO EW) s= 14 TeV LHC HIGGS XS WG 2010 H+X) [pb] 10 → pp 8 TeV 14 TeV → qqH (NNLO QCD + NLO EW) (pp

σ pp ! pp → → WH (NNLO QCD + NLO EW) ZH (NNLO QCD +NLO EW) Factor of ~ 3 1 pp → ttH (NLO QCD)

10-1 102 100 200 300 400 500 1000 pp → H (NNLO+NNLL QCD + NLO EW) MH [GeV] H+X) [pb] LHC HIGGS XS WG 2014

→ 10 Leading-order convolution with PDF’s: (pp σ gg d pp → qqH (NNLO QCD + NLO EW) LO(pp h)=0 ⌧ L ! d⌧ 1 pp → WH (NNLO QCD + NLO EW) pp → ZH (NNLO QCD + NLO EW) G ↵2(µ2 ) N 2 µ s R g pp → bbH (NNLO and NLO QCD) 0 = | | 288p2⇡ -1 M = 125 GeV 10 pp → ttH (NLO QCD) H d gg 1 dx MSTW2008 L = g(x, µ2 )g(⌧/x, µ2 ) d⌧ x F F 7 8 9 10 11 12 13 14 Z⌧ s [TeV] Double-Higgs Production

The Higgs boson can also be pair produced. At the LHC, the main production channel is again gluon fusion, and again the higher-order corrections need to be included.

From a theoretical perspective, double-Higgs production gives a handle on the trilinear interaction:

g h g h g h

Aside comment:

g h g h The quartic self-interaction would enter in triple-Higgs production. g h g h

Unfortunately, this appears extremely Figure 2: Generic diagrams contributing to Higgs pair productionchallenging in gluon fusionat any at LO.foreseeable collider… The form factors F and F2 in F1 and F2 deﬁned as

2 2 2 F = c F + c and F = c F2 + c F c2 , (2.9) 1 t 3 2 t tt 3 contain the full quark mass dependence and can be found in [9]. In the heavy quark limit the form factors F, F2 and G2 approach 2 2 F ,F2 and G2 =0, (2.10) ! 3 !3

and F1 and F2 simplify to

lim 2 lim 2 2 F = (c + c ) ,F= ( c + c c2) . (2.11) 1 3 t 2 3 t tt We have furthermore introduced the abbreviations 2 MZ C ,c12c and c2 12c , (2.12) ⌘ hhh Q2 M 2 + iM ⌘ g ⌘ gg h h h with 2 3Mh c3 hhh = 2 . (2.13) MZ

2 The terms proportional to ct,respectivelyct in F1 and F2 and in front of the form factor G2 are the usual SM contributions including the modiﬁcations due to the rescaling of the top Yukawa coupling by ct. The contributions coming with c and c2 originate from the e↵ective two-gluon couplings to one and two Higgs bosons, while the term involving ctt is due to the novel two-Higgs two-top quark coupling. We use the e↵ective couplings to compute the NLO QCD corrections to Higgs pair production. They are composed of the virtual and the real corrections. Sample diagrams are shown

4 Higgs Decays masses are used. This is shown in Fig 2.3 where Γ(H b¯b)andΓ(H cc¯)aredisplayedas 2-Body Decays → → functions of the Higgs mass MH in the Born approximation, using only the running quark masses and with the full set of QCD corrections implemented. Note that the latter increase the partial widths by approximately+ 20%. Given that m 125 GeV, the two-body decays h tt¯ , h ZZ and h W W h 2 ⇡ !The additional correction at (αs)involveslogarithmsofthemassesofthelightquarks ! ! O are forbidden by energy conservation. and the heavy top quark and is given by [151]

2 2 2 + 2 α¯s 2 MH 1 2 mq Therefore, the 2-body decays are dominated by ¯ , followed by ⌧ ⌧ , cc¯ . The∆ decaysH = into1.57 log + log (2.12) bb π2 − 3 m2 9 M 2 ! t H " lighter fermions (s, µ, d, u, e, ⌫0s) are much further suppressed.Because of chiral symmetry, all this discussion holds true iftheHiggsparticlewereapseu- 2 doscalar boson; the only exception is that the correction ∆H would be diﬀerent, since it Using that the relevant fermions are much lighter than the Higgs:involves the quark masses which break the symmetry.

+ Gµmh 2 (h l l) m¯ (mh) ! ⇡ 4p2⇡ l

3Gµmh 2 ↵¯s(mh) 2 (h qq¯) m¯ (mh) 1+5.67 + (¯↵ ) ! ⇡ 4p2⇡ q ⇡ O s  Note that: pole m 4.9 GeV m¯ b(mh) 2.8 GeV b ⇡ ⇡ pole m 1.7 GeV m¯ c(mh) 0.6 GeV c ⇡ ⇡

pole m 1.7 GeV m¯ ⌧ (mh) 1.7 GeV ⌧ ⇡ ⇡ Figure 2.3: The partial widths for the decays H b¯b (left) and H cc¯ (right) as a function → → of MH .TheyareshownintheBornapproximation(dottedlines),including only the running quark masses (dashed lines) and with the full set of QCD corrections (solid lines). The input pole masses are mb =4.88 GeV and mc =1.64 GeV and the running strong coupling constant is taken at the scale of the Higgs mass and is normalized to αs(MZ )=0.1172.

2.1.3 The case of the top quark

For Higgs bosons decaying into top quarks, the QCD corrections do not lead to large loga-

rithms since mt is comparable to MH .However,thesecorrectionscanbesizable,inparticular near the threshold M 2m .Atnext–to–leading–order,theyaregivenby H ∼ t 3Gµ 2 3 4 αs t Γ(H tt¯)= MH m β 1+ ∆ (βt) (2.13) → 4√2π t t 3 π H # $ 76 Loop-induced Decays

We have seen that the loop-induced coupling to gluons ends up being the dominant one for Higgs production. Also that the dominant 2-body decay is into bottom quark pairs. Since

Loop induced Higgs boson couplings mb 1 Can expect the decay into b = 0.02 2 gluons to be sizable Higgs boson couplingv ⇡ to gluons ⇠ 16⇡

At one-loop, the Higgs boson couples to gluons via a loop of quarks: tree-level gluon fusion q g 8m3 (parton level) h0 h LO = 2 0 q¯ g ⇡ This diagram leads to an effective Lagrangian HiggsHiggsHiggs boson boson boson coupling coupling coupling to photons photons to photons Similarly, a coupling ofeff theg (neutral)αsNg 0 a Higgsµνa boson to two photons is induced at 1-loop order: hgg = h GµνG , AtAt one-loop, one-loop,At one-loop, the the Higgs theL Higgs boson boson24π bosonmW couples couples couples to to photons to photons photons via via via a a loop a loop loop of of ofcharged charged charged particles: particles: particles: + ++ + 0+ WWW where Ng is roughlyff f the numberγγ γ of quarks heavier thanWW+Wh .Moreprecisely,γγγ γγγ 2 mq 00 0 N = F (x ) ,x00 0 i , 0 00 hh h g 1/2 i hh h i 2 hhh ≡ mh ¯ !i γ γ γ ¯¯ f γγ W − γγ γ γ ff WW−− W where the loop function F (x) 1 for x 1. W −W−− 1/2 → ≫ IfIf charged chargedIf charged scalars scalars scalars exist, exist, they they they would would would contribute contribute contribute as as well. as well. well. These These These diagrams diagrams diagrams lead lead lead to to to an an an eff eectiveffffectiveective While suppressed,Lagrangian due to the great sensitivity to photons of our detectors, this loop-induced LagrangianLagrangian eff gαNγ 0 µν eff ggααNNγ 0 µν eff hγγ = γ 0h FµνFµν , coupling is extremely important h(discoveryγγL == channel)!hhFFµνµFν FThere, , is also a coupling of photon + Z. L hγγ 12π12mπmW where L 12πmWW wherewhere 2 2 m2 2i 2 m Nγ = Nci2ei Fj(xi) ,xi mi i2 . Nγ = Nciei Fj(xi) ,xi ≡ m. Nγ = Ni ciei Fj(xi) ,xi 2 2h . ≡≡mmh In the sum over loop particlesii !i of mass m , N =3for quarksh and 1 for color singlets, In the sum over loop particles!!i of mass m , Ni =3ci for quarks and 1 for color singlets, In the sum over loop particles i of mass mi i, Ncici =3for quarks and 1 for color singlets, ei is the electric charge in units of e and Fj(xi) is the loop function corresponding to ith eei isis the the electric electric charge in in units units of ofeeandandFFj((xxi))isis the the loop loop function function corresponding corresponding to toithith i particle (with spin j). In the limit of x j 1,i particleparticle (with (with spin spin j).). In In the the limit limit of ofxx ≫11, , ≫≫ 1/4 ,j=0, ⎧11//44,j,j=0=0, , Fj(x) 1 ,j=1/2 , −→⎧ ⎪ Fj(x) ⎪⎧⎪ 1 ,j=1/2 , Fj(x)−→ ⎪⎪⎨ 1 ,j=1/2 , −→ ⎨⎪⎪ 21/4 ,j=1. ⎨⎪ 21−/4 ,j=1. −⎪21/4 ,j=1. ⎪ ⎪− ⎪⎪⎩ ⎩⎪⎪ ⎩⎪ ¯ ¯ 3-Bodyt t Decays¯b ¯b g g g g h h h h Each additionalg particle ing the final state leadsg to a gsuppression of about one-loop factor. Hence, we should expect the following 3-body decays to be relevant. t t b b

Z Z W W

h ∗ ′ ′ h ∗ Z q, l h ∗ ′ ′ ∗ Z q, l h W q ,lW q ,l

¯ q,¯ ¯l ¯ ¯ q,¯ l q,¯ ¯l q, l

Recall that these vertices are a direct consequence of EWSB!

2 4 3GµMV 2 2 (h VV⇤)= m 0 R (M /m ) ! 16⇡3 h V T V h

7 10 2 40 4 0 =1, 0 = sin ✓ + sin ✓ W Z 12 9 W 9 W

3(1 8x + 20x2) 3x 1 1 x 3 R (x)= arccos (2 13x + 47x2) (1 6x +4x2) log x T (4x 1)1/2 2x3/2 2x 2 ✓ ◆

0-1 0-1 Higgs Branching Fractions

1 WW bb m 125 GeV h ⇡ ¯ gg ZZ LHC HIGGS XS WG 2013 BR(bb) 0.6 10-1 ⇡ BR(WW) 0.20 ⇡ BR(gg) 0.077 cc ⇡ BR(⌧⌧¯) 0.06 -2 ⇡ 10 BR(cc¯) 0.026 ⇡ Higgs BR + Total Uncert BR(ZZ) 0.025 Z ⇡ BR() 0.002 ⇡ -3 BR(Z) 0.001 10 ⇡ µµ 4.4MeV Tot ⇡

-4 125 10 80 100 120 140 160 180 200 MH [GeV] The Higgs Discovery The Higgs Discovery

Higgs Coupling Measurements A possible parameterization ( SM µi = i/i i ggH, V BF, V H, ttH for the observed rates 2 { } distinguishes between µ = B /BSM f ,WW,ZZ,bb,⌧⌧ production and decay f f f 2 { }

Assuming a single rescaling factor for production, one can report an overall signal strength per channel:

Signal strength(µ)= BR(f)/SM BR(f)SM ⇥ ⇥

ATLAS and CMS Preliminary ATLAS ATLAS and CMS Preliminary ATLAS LHC Run 1 CMS LHC Run 1 ATLAS+CMS CMS ± 1σ ATLAS+CMS ± 2σ µ ± 1σ ggF µγγ µ VBF µZZ µ WH µWW µ ZH

ττ µ µ ttH

µ µbb

0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Parameter value Parameter value Higgs Coupling Measurements

Another parameterization allows for Z , W , f , g,  a rescaling of the couplings

ATLAS and CMS Preliminary ATLAS ATLAS and CMS Preliminary LHC Run 1 CMS LHC Run 1 ATLAS+CMS ± 1σ κZ κ ≤ 1 κ V Z BRBSM=0 κ ± 1σ W ± 2σ κW κt

κt κτ

κb κτ

κg

κb κγ

κµ BRBSM 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Parameter value Parameter value Further Implications 2. Upper bound from precision tests of the Standard Model

Very precise tests of the Standard Model are possible given the large sample of electroweak data from LEP, SLC and the Tevatron. Although the Higgs bosonElectroweak mass (mh)isunknown,electroweakobservablesaresensitiveto Precision Tests mh through quantum corrections. For example, the W and Z masses are shifted slightly due to: The W and Z masses receive corrections that depend logarithmically on mh h0 h0

0 0 W ± W ± Z Z

These self-energy contributions affect several EW observables (oblique corrections) Measurement Fit |OmeasOfit|/meas The mh dependence0123 of the above radiative corrections is logarithmic. (5) Previous indirect determination of mh (m ) 0.02750 0.00033 0.02759 Nevertheless,had Z ± a global ﬁt of many electroweak observables can determine mZ [GeV] 91.1875 ± 0.0021 91.1874

Z [GeV] 2.4952 ± 0.0023 2.4959 0the preferred value of m (assuming that the Standard Model is the correct had [nb] 41.540 ± 0.037 41.478 h

Rl 20.767 ± 0.025 20.742 0,l Afbdescription0.01714 ± 0.00095 of the 0.01645 data).

Al(P) 0.1465 ± 0.0032 0.1481

Rb 0.21629 ± 0.00066 0.21579

Rc 0.1721 ± 0.0030 0.1723 0,b Afb 0.0992 ± 0.0016 0.1038 0,c Afb 0.0707 ± 0.0035 0.0742

Ab 0.923 ± 0.020 0.935

Ac 0.670 ± 0.027 0.668

Al(SLD) 0.1513 ± 0.0021 0.1481 2 lept sin eff (Qfb) 0.2324 ± 0.0012 0.2314

mW [GeV] 80.385 ± 0.015 80.377

W [GeV] 2.085 ± 0.042 2.092

mt [GeV] 173.20 ± 0.90 173.26

March 2012 0123 Custodial Symmetry

An important constraint arises from the “rho parameter”. At tree-level it reads

2 2 2 M [T (T + 1) Y ] VT,Y cT,Y ⇢ = W = T,Y | | M 2 cos2 ✓ 2Y 2 V 2 Z W P T,Y | T,Y | P which has been written for an arbitrary Higgs sector. Here we have

• T and Y : total SU(2) isospin and hypercharge of Higgs representation V = (T,Y ) , the vev of the corresponding Higgs with quantum numbers T and Y • T,Y h i 1(T,Y ) complex representation c = 2 • T,Y 1 ( (T,Y =0) real representation 2 2 • Normalization: Q = T 3 + Y Experimentally, rho is very close to one, as predicted in the SM (T = Y =1/2)

tR There are also small loop corrections, most tL tL Wµ W⌫ importantly from the top quark tL tL

tR Custodial Symmetry

It is extremely useful/important to understand why ⇢ 1 ! ⇡ In the limit of g0 =0 and vanishing Yukawa’s, the SM is invariant under

gauge global SU(2) SU(2) (modulo absence of ⌫R) L ⇥ R To make this more explicit, note that

0 + µ µ 1 H ⇤ H (DµH)†D H =Tr[(Dµ)†D ] where = 0 SU(2)L p2 H H ✓ ◆ DµH =(@µ igWµ)H SU(2) D =(@ igW ) R µ µ µ A bidoublet of SU(2) SU(2) L ⇥ R Since, after EWSB, v 12 2 , it follows that h i/ ⇥ ⇥ SU(2)gauge SU(2)global EWSB SU(2)global SU(2) L ⇥ R ! Diag ⌘ custodial

The charged and neutral W’s transform as a triplet of the custodial symmetry, hence they must be degenerate in this limit. The small mass difference between MW and MZ arises from the small hypercharge gauge coupling, and loop effects dominated by the top Yukawa The S-T Ellipse

The oblique corrections (dominant) are often parameterized in terms of the Peskin-Takeuchi S and T parameters. A ﬁt to the EW observables looks like this...

0.4 m This is another way of displaying the mt= 171.4 ± 2.1 GeV W m = 114...1000 GeV H prel. preference, within the SM, for a light Higgs

0.2 U≡0 However, indirect measurement were never considered deﬁnitive, since new physics Γ ll typically gives important contributions to S and T!

T 0 (that could allow reentering the ellipse)

m -0.2 t The fact that the Higgs turned out to be light gives further indication that nature does not 2 lept mH sin θeff 68 % CL like to play dirty tricks (conspiracies that lead -0.4 to cancellations)! -0.4 -0.2 0 0.2 0.4 S ⇢ =1+↵T Summary

• Higgs boson discovery: an essential step towards a full understanding of EWSB • A new type of force between fundamental particles has been established

• Higher-order calculations: theory/experiment interplay

• An extraordinary achievement! (especially on the experimental side) V v m

V ATLAS and CMS t

κ 1 LHC Run 1 Preliminary Z

or W F

v −1 Observed m

F SM Higgs boson κ

10−2 τ b 10−3 µ

10−4

10−1 1 10 102 Particle mass [GeV] Let us hope this is the point of departure into a new journey! Further Reading

• A classic reference: The Higgs Hunter’s Guide, Gunion, Haber, Kane and Dawson

• Another good reference: The Anatomy of Electroweak Symmetry Breaking Tome 1: The Higgs Boson in the Standard Model, Abdelhak Djouadi, arXiv: hep-ph/0503172

• A comprehensive, updated reference: Handbook of LHC Higgs Cross Sections 3: Higgs Properties, arXiv: 1307.1347

• See also the LHC Higgs Cross Section Working Group webpage:

https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CrossSections

as well as the ATLAS and CMS TWiki pages:

https://twiki.cern.ch/twiki/bin/view/AtlasPublic/HiggsPublicResults

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsHIG