The Higgs mechanism: spontaneous breaking of a local symmetry Consider a U(1) gauge theory with a complex scalar field (scalar QED) − 1 4 F µ ν F µ ν + ( D µ φ ) ∗ D µ φ − m 2 | φ | 2 − λ | φ | 4 ( D µ = ∂ µ − i e A µ ) L = φ = ρ e i θ Parameterize the complex scalar as modulus and phase: − 1 4 F µ ν F µ ν + ρ 2 ( ∂ µ θ − e A µ ) 2 + ∂ µ ρ ∂ µ ρ − m 2 ρ 2 − λ ρ 4 L = B µ ≡ A µ − 1 e ∂ µ θ is gauge invariant, and F µ ν = ∂ µ A ν − ∂ ν A µ = ∂ µ B ν − ∂ ν B µ 1 λ Again, for m 2 < 0 and > 0 the symmetry ρ = 2 ( v + H ) √ is broken and the scalar gets a vev − 1 4 F µ ν F µ ν + 1 2 e 2 v 2 B µ B µ + e 2 v HB µ B µ + 1 2 e 2 H 2 B µ B µ + 1 L = 2 ∂ µ H ∂ µ H − V ( H ) In this “unitary gauge”, the massless field A µ “eats” the phase and becomes the massive field B µ The remaining scalar H is also massive, and interacts with the gauge field
Spontaneous breaking of the SU(2)xU(1) gauge symmetry � ⇥ ϕ + (1 , 2 , +1 / 2) Introduce a SU(2) doublet of complex scalars: Φ ≡ ∼ ϕ 0 � µ τ i − i g � � D µ = ∂ µ − i g L S = ( D µ Φ ) † ( D µ Φ ) − m 2 Φ † Φ − λ ( Φ † Φ ) 2 2 W i 2 B µ The kinetic term determines the interactions between scalars and gauge bosons: Φ Φ V V g 2 , g ′ 2 g , g ′ Φ Φ V λ If m 2 < 0 and > 0 the mexican-hat potential induces a vev v for the doublet
1 � � 0 2 e i τ i θ i ( x ) Φ = We can parameterize the complex doublet as: v + H ( x ) √ θ i Gauge symmetry allows us to rotate away the via a SU(2) transformation ( unitary gauge ) = 1 2(2 λ v 2 ) H 2 + λ vH 3 + 1 4 λ H 4 V The kinetic term for the doublet contains mass and interaction terms for the weak gauge bosons � 1 ⇥ ( D µ Φ ) † ( D µ Φ ) = 1 µ + 1 8( g 2 + g ⇥ 2 ) Z µ Z µ 4 g 2 W µ + W � ( v + H ) 2 2 ∂ µ H ∂ µ H + (the photon remains massless) v W ± , Z W = 1 Z = 1 4( g 2 + g � 2 ) v 2 m 2 4 g 2 v 2 m 2 g 2 , g ′ 2 m 2 W ± , Z v W = 1 Note: Z cos 2 θ W m 2
The value of can be related to the constant in the low-energy effective Lagrangian G F v µ − − → e − ν µ ν e (four-fermion interaction) that describes the muon decay process e ν µ − G F 2 ν µ γ α (1 − γ 5 ) µ e γ α (1 − γ 5 ) ν e L e ff ⊃ √ G F Γ = G 2 F m 5 − 4 G F µ + O ( m 2 e /m 2 A = , µ ) √ 192 π 3 2 µ ν e In the Standard Model the muon decay is mediated by the exchange of a W boson e ν µ g 2 A ⇥ g g � 2 m 2 W W − µ ν e ⇥ m W = g v 2 G F ) − 1 / 2 � 246 GeV v = ( Equating the amplitudes and inserting we get: 2 This also allows us to derive another π α m 2 W (1 − m 2 W /m 2 Z ) = √ 2 G F relation among measurable quantities:
M ∝ m 2 The inclusion of diagrams with exchange of a scalar H H restores the unitarity of V V scattering at high energy: v 2 + . . . + + H The scalar mass cuts off the divergence. But unitarity is again at risk if m H is too large ∞ X M = 16 ⇡ (2 ` + 1) P ` (cos ✓ ) a ` ` =0 Unitarity conditions on the partial-wave decomposition of the amplitude: | Re( a ` ) | < 1 2 − m 2 m 2 W ⌧ m 2 H a 0 ( W L W L − → W L W L ) ≈ For H ⌧ s 8 π v 2 Thus, m H < 870 GeV (even stronger bounds by considering several processes at once)
Counting the bosonic degrees of freedom in the unbroken and broken phases: ( Φ ) A complex doublet 4+(4x2) = 12 unbroken symmetry: d.o.f. ( B , W i ) Four massless vector bosons ( H ) One real scalar : the Higgs boson 1+(3x3)+2 = 12 ( Z , W + , W − ) broken symmetry: Three massive vector bosons d.o.f. ( γ ) One massless vector boson The degrees of freedom corresponding to the three would-be-Goldstone bosons have been absorbed in the longitudinal components of the massive vector fields The renormalizability of the theory is still hidden in this unitary gauge, but it becomes manifest with different gauge choices (‘t Hooft, 1971)
The propagator of the massive vector boson depends on the choice of gauge: − g µ ν + k µ k ν Unitary gauge: � � i ∆ µ ν = k 2 − m 2 m 2 (no would-be-Goldstone boson) V V k µ k ν � � i − g µ ν + (1 − ξ ) ∆ µ ν = k 2 − m 2 k 2 − ξ m 2 V V Renormalizable gauge: i ∆ G = k 2 − ξ m 2 V The contributions of the unphysical would-be-Goldstone boson combine with those of the gauge boson, and we find the same results as in the unitary gauge (also, predictions for physical observables must not depend on the arbitrary parameter ) ξ
Fermion masses and flavor mixing We can generate the quark masses by building gauge-invariant interactions with the Higgs u i L q i u i d i ∼ (3 , 2 , +1 / 6) , R ∼ (3 , 1 , +2 / 3) , R ∼ (3 , 1 , − 1 / 3) L ≡ d i L � ⇥ ⇥ 0 ∗ Φ ≡ � Φ ∗ = ⇤ Φ ∼ (1 , 2 , +1 / 2) , ∼ (1 , 2 , − 1 / 2) − ⇥ − L Φ d j L � Φ u j − ( Y D ) ij q i R − ( Y U ) ij q i = + h . c . L Y R u L u L d L d L ϕ + ϕ 0 ϕ 0 ϕ − Y d Y u Y d Y u u R u R d R d R
Fermion masses and flavor mixing We can generate the quark masses by building gauge-invariant interactions with the Higgs u i L q i u i d i ∼ (3 , 2 , +1 / 6) , R ∼ (3 , 1 , +2 / 3) , R ∼ (3 , 1 , − 1 / 3) L ≡ d i L � ⇥ ⇥ 0 ∗ Φ ≡ � Φ ∗ = ⇤ Φ ∼ (1 , 2 , +1 / 2) , ∼ (1 , 2 , − 1 / 2) − ⇥ − L Φ d j L � Φ u j − ( Y D ) ij q i R − ( Y U ) ij q i = + h . c . L Y R u L u L d L d L ϕ + v v ϕ − Y d Y u Y d Y u u R u R d R d R
The matrices of Yukawa couplings can be diagonalized by bi-unitary transformations diag( Y d , Y s , Y b ) = V † diag( Y u , Y c , Y t ) = V † u Y U U u , d Y D U d u L → V u u L , u R → U u u R , Applying the same rotations to the quark fields: d L → V d d L , d R → U d d R the Yukawa interaction Lagrangian becomes (in the unitary gauge): − 1 tt + Y d ¯ ss + Y b ¯ cc + Y t ¯ � ⇥ = 2 ( v + H ) Y u ¯ uu + Y c ¯ dd + Y s ¯ L Y bb √ m q = Y q v Therefore the masses of the quarks are: √ 2
The neutral current couplings of the quarks to photon and Z are not affected by the rotation � L γ µ q i R γ µ q i ⇥ � L γ µ q i R γ µ q i ⇥ ⇤ ⇤ g i L + g i q i L + q i L q i R q i L ⊃ e i A µ + Z µ R R q i q i u L γ µ u L − u γ µ V u u L = u L γ µ u L → u L V † e.g. (and so on) On the other hand, the charged current couplings of the quarks to the W boson are affected: g g L γ µ d i L γ µ V � � d j 2 u i L W + 2 u i L W + CKM L ⊃ + h . c . − + h . c . √ √ → µ ij µ i i,j Therefore, charged interactions mix quarks of different flavor (neutral interactions don’t) ≡ V † CKM V u V d is the so-called Cabibbo-Kobayashi-Maskawa matrix ij The CKM matrix can be represented in terms of four independent parameters (e.g., three independent rotation angles and one complex phase)
An alternative representation of the CKM matrix is the so-called “Wolfenstein parametrization”: 1 − λ 2 A λ 3 ( ρ − i η ) λ 2 CKM = + O ( λ 4 ) 1 − λ 2 V A λ 2 − λ 2 A λ 3 (1 − ρ − i η ) − A λ 2 1 � � m � d 1 Summer14 A large number of flavor-violating m � s SM fit m � d processes allow for the determination � of the Wolfenstein parameters ¯ ρ , ¯ η 0.5 � V K ub V cb The good agreement between many 0 different measurements provides a � consistency check of the CKM picture -0.5 BR(B ) � � � sin(2 + ) � � (plot from UTfit collaboration) -1 -1 -0.5 0 0.5 1 �
ν i L l i ∼ (1 , 2 , − 1 / 2) , L ≡ e i ν i Among the SM leptons, there are no : L R e i R ∼ (1 , 1 , − 1) The only gauge-invariant Yukawa L Φ e j − ( Y E ) ij l i = + h . c . interaction that we can build gives L Y R a mass term for charged leptons: Again, we can diagonalize the Yukawa matrix with a bi-unitary transformation m l = Y l v diag( Y e , Y µ , Y τ ) = V † e Y E U e , √ 2 ν L → V e ν L , but now we are free to rotate the parallel to the : ν L e L e L → V e e L , e R → U e e R Therefore, the charged interaction does not mix leptons of different flavors: g L γ µ e i � 2 ν i L W + L ⊃ + h . c . − itself √ → µ i
Flavor oscillations in solar, atmospheric, and accelerator-produced neutrinos provide evidence of flavor mixing and (tiny) masses (the first clear sign of Beyond-the-SM physics!!!) This can be fixed by introducing N i R ∼ (1 , 1 , 0) “sterile” right-handed neutrinos: Then, gauge symmetry allows for both a Yukawa interaction and a “Majorana” mass term: h i 1 L Φ e j L e Φ N j R N j ( Y E ) ij l i R + ( Y N ) ij l i 2 M ij N i = + h . c . L Y − − R R After EWSB, the mass matrix for the neutrinos becomes (schematically): ◆ ✓ ν L � ✓ ◆ m D = Y N v 0 m D � L ν L N R ⊃ − with m D M N R √ 2 For M >> m D , this gives both light, almost-left neutrinos and heavy, almost-right neutrinos: m ν ≈ m 2 D (seesaw mechanism) M , m N ≈ M Introducing heavy sterile neutrinos does not affect SM phenomenology at the weak scale
Flavor oscillations in solar, atmospheric, and accelerator-produced neutrinos provide evidence of flavor mixing and (tiny) masses (the first clear sign of Beyond-the-SM physics!!!) This can be fixed by introducing N i R ∼ (1 , 1 , 0) “sterile” right-handed neutrinos: Then, gauge symmetry allows for both a Yukawa interaction and a “Majorana” mass term: h i 1 L Φ e j L e Φ N j R N j ( Y E ) ij l i R + ( Y N ) ij l i 2 M ij N i = + h . c . L Y − − R R After EWSB, the mass matrix for the neutrinos becomes (schematically): ◆ ✓ ν L � ✓ ◆ m D = Y N v 0 m D � L ν L N R ⊃ − with m D M N R √ 2 For M >> m D , this gives both light, almost-left neutrinos and heavy, almost-right neutrinos: Tape-cul m ν ≈ m 2 D (seesaw mechanism) M , m N ≈ M Introducing heavy sterile neutrinos does not affect SM phenomenology at the weak scale
Constraints on non-minimal Higgs sectors A single SU(2) doublet is the minimal option. Several scalars could contribute to EWSB m 2 However, constraints from W ρ ≡ ≈ 1 Z cos 2 θ W m 2 precision observables, e.g.: The contribution to the rho parameter from a given Higgs field depends on its SU(2) properties: M 2 � ✓ W 3 µ L ⊃ 1 + 1 M 0 2 ◆ 3 B µ ) 2 ( W µ 2 m 2 � W 1 µ W 1 µ + W 2 µ W 2 � M 0 2 M 00 2 W µ B µ ρ = m 2 M 00 Z = M 2 + M 00 2 m 2 W m γ = 0 tan θ W = , − → − → M 2 M i v 2 ⇥ I i ( I i + 1) − ( I 3 i ) 2 ⇤ P i ρ = For a set of Higgs fields : Φ i i v 2 2 P i ( I 3 i ) 2 Doublets are OK. Other SU(2) representations would change rho ( then v i must be small! )
The simplest non-minimal case: two-Higgs-doublet model h i 11 Φ † 22 Φ † 12 Φ † m 2 1 Φ 1 + m 2 m 2 = 1 Φ 2 + h . c . V 2 Φ 2 − + λ 1 1 Φ 1 ) 2 + λ 2 2 Φ 2 ) 2 + λ 3 ( Φ † 2 ( Φ † 2 ( Φ † 1 Φ 1 )( Φ † 2 Φ 2 ) + λ 4 ( Φ † 1 Φ 2 )( Φ † 2 Φ 1 ) ⇢ λ 5 � 1 Φ 2 ) 2 + h i 2 ( Φ † λ 6 ( Φ † 1 Φ 1 ) + λ 7 ( Φ † ( Φ † + 2 Φ 2 ) 1 Φ 2 ) + h . c . ϕ + i Two complex SU(2) doublets Φ i = => 8 degrees of freedom: 1 2 ( v i + ϕ R i + i ϕ I i ) √ H ± 5 physical states (3 neutral, 2 charged ) After EWSB: G 0 , G ± and 3 would-be-Goldstone bosons ( ) If the potential does not break CP, the neutral states � � � � � � H ϕ R cos α sin α 1 = are one pseudoscalar and two scalars ( ) A h , H h − sin α cos α ϕ R 2
Generating particle masses in many-Higgs-doublet models The gauge-boson masses receive a contribution from each Higgs vev Z = g 2 + g � 2 W = g 2 � � m 2 v 2 m 2 v 2 i , i 4 4 i i Also, each Higgs doublet has its own set of matrices for the couplings to the fermions: � � � q L � M U,D = y U,D Φ i y U q L Φ i y D − L Y i u R + i d R , v i = ¯ ¯ i i i i Φ SM Rotating the fields to a basis where one Higgs ( ) gets the vev and the others ( ) don’t Φ i � � q L � q L � Φ SM Y U u R + ¯ q L Φ SM Y D d R + Φ i y U q L Φ i y D − L Y i u R + i d R = ¯ ¯ ¯ i i y U,D Y U,D In general, the matrices are not diagonal in the basis where are diagonal i The non-SM doublets mediate Flavor-Changing Neutral Currents!!!
FCNC in Higgs-quark interactions are absent when Natural Flavor Conservation: only one doublet couples to each species of quarks q L � Φ 1 Y U u R + ¯ q L Φ 1 Y D d R (Type I) − L Y = ¯ e.g., in THDMs: q L � Φ 2 Y U u R + ¯ q L Φ 1 Y D d R (Type II) − L Y = ¯ FCNC can be suppressed if the matrices of non-SM Higgs Minimal Flavor Violation: couplings are made up of combinations of Y U and Y D Y U , 1 + � u Y U Y U † + . . . 1 + � d Y U Y U † + . . . y U i = A i y D i = A i Y D � � � � u d Only two sets of SU(3)xSU(2)xU(1) quantum numbers are allowed for an additional scalar whose Yukawa couplings transform like Y U and Y D under rotations in flavor space ( 1 , 2 ) 1/2 The usual THDMs ( 8 , 2 ) 1/2 The additional scalar is a color octet (Manohar & Wise, hep-ph/0606172) So far, no additional Higgs bosons did show up at colliders (nor did they manifest through contributions to flavor or EW observables)
FCNC in Higgs-quark interactions are absent when Natural Flavor Conservation: only one doublet couples to each species of quarks q L � Φ 1 Y U u R + ¯ q L Φ 1 Y D d R (Type I) − L Y = ¯ e.g., in THDMs: q L � Φ 2 Y U u R + ¯ q L Φ 1 Y D d R (Type II) − L Y = ¯ FCNC can be suppressed if the matrices of non-SM Higgs Minimal Flavor Violation: couplings are made up of combinations of Y U and Y D Y U , 1 + � u Y U Y U † + . . . 1 + � d Y U Y U † + . . . y U i = A i y D i = A i Y D � � � � u d Only two sets of SU(3)xSU(2)xU(1) quantum numbers are allowed for an additional scalar b whose Yukawa couplings transform like Y U and Y D under rotations in flavor space t t γ H + Z ( 1 , 2 ) 1/2 The usual THDMs t s b b ( 8 , 2 ) 1/2 H + The additional scalar is a color octet (Manohar & Wise, hep-ph/0606172) So far, no additional Higgs bosons did show up at colliders (nor did they manifest through contributions to flavor or EW observables)
Interlude: who ordered this particle? Three PRL papers in 1964 described the mechanism that gives mass to gauge bosons: (does not mention a physical scalar) (cites BE , mentions a massive scalar as an essential feature of the mechanism) (cites BE and H, mentions a scalar which is massless and decoupled ) Then Weinberg (1967) and Salam (1968) incorporated the mechanism in the EW theory and ‘t Hooft (1971) proved that spontaneously broken gauge theories are renormalizable
“Nobelitis” Symmetry breaking and the Scalar boson - evolving perspectives 1 Fran¸ cois Englert Service de Physique Th´ eorique Universit´ e Libre de Bruxelles, Campus Plaine, C.P.225 Five authors alive, only three Nobel slots...
The ending was unexciting... ...but some people just wouldn’t let go: [arXiv:1401.6924] “(...) the Nobel Committee [5] stated ‘The Goldstone theorem holds in the sense that that Nambu-Goldstone mode is there but it gets absorbed into the third component of a massive vector field.’ (...) It is shown in what follows that that is not a valid view and that a massless gauge particle necessarily remains in the theory.”
II) The hunt for the Higgs boson
The main contenders: • Large Electron-Positron Collider (LEP) at CERN (1989-2000): circular e + e - collider, center-of-mass energy up to 209 GeV; • Tevatron at Fermilab (1983-2012): _ circular pp collider, c.o.m. energy up to 2 TeV; • Large Hadron Collider (LHC) at CERN (2011-2012, 2015-? ): circular pp collider, c.o.m. energy up to 8 TeV (designed for 14 TeV).
Higgs boson couplings to the other SM particles The interaction Lagrangian contains (v + H) , thus HPP couplings are controlled by m P / v : 2 i m 2 2 i m 2 ff : i m f g µ ν , g µ ν , H ¯ W Z H W + v , µ W − H Z µ Z ν : ν v v Feynman rules: : 2 i m 2 2 i m 2 v 2 g µ ν , HH W + W v 2 g µ ν Z µ W − HH Z µ Z ν : ν (among fermions, only top, bottom and tau have sizable couplings to the Higgs) Loops of charged particles also induce Higgs-boson couplings to gluons and photons: g γ γ + q q, ℓ H H H W g γ ( Z ) γ ( Z ) α s α α 8 π v H G a µ ν G a 8 π v H A µ ν A µ ν 8 π v H A µ ν Z µ ν L ⊃ − C g − C γ − C γ Z µ ν (in practice, only the top, bottom and W contributions to the loops are relevant)
The decay rates of the Higgs boson depend only on its mass (the couplings are all fixed) 1 LHC HIGGS XS WG 2011 Higgs BR + Total Uncert b b WW ZZ gg t t -1 � � 10 c c -2 10 Z � � � -3 10 100 200 300 400 500 1000 M [GeV] H Decays to bottom quarks dominate at low mass, then WW (and ZZ ) for m H > 140 GeV Decays to two photons are loop-suppressed but easy to detect
LEP & Tevatron corner it
Higgs boson production at e + e - colliders The dominant processes are Higgs-strahlung and WW fusion: ν e H e − e − W − H Z e + W + e + Z ¯ ν e ✓ Well-defined energy and momentum in the initial state ✓ “Clean” experimental environment (no QCD background) ✓ Allows for precision studies of the Higgs boson properties (couplings, spin, parity...) - The cross section is small and it decreases with energy, high luminosity required - Synchrotron radiation makes circular machines unpractical above LEP2 energy ✦ The International Linear Collider (~500 GeV) could be the next Higgs factory
At LEP, the dominant channel was Higgs-strahlung followed by decay in bottom or tau pairs 1 CL s 95% CL limit on � 2 1 LEP LEP -1 (a) 10 � s = 91-210 GeV -2 m H > 114.4 GeV 10 Observed Expected for background -3 Observed -1 10 10 Expected for background -4 10 114.4 -5 10 115.3 -2 -6 10 10 20 40 60 80 100 120 100 102 104 106 108 110 112 114 116 118 120 m H (GeV/c 2 ) m H (GeV/c 2 ) ξ 2 = ( g HZZ /g SM HZZ ) 2 (CL s = CL s+b / CL b )
LEP’s parting shot: ~ 1.7 excess for m H ≈ 115 GeV σ Events / 3 GeV/c 2 LEP Tight – = 200-209 GeV � s 7 Data 6 Background Signal (115 GeV/c 2 ) 5 > 109 GeV/c 2 all 4 Data 18 4 Backgd 14 1.2 3 Signal 2.9 2.2 2 1 0 0 20 40 60 80 100 120 m Hrec (GeV/c 2 ) Was it the real thing? People kept arguing about it until the start of the LHC...
Higgs boson production at hadron colliders Higgs-strahlung VBF q q H q V H V q V q ¯ q V gluon fusion associated prod. with top/bottom g g t, b t, b H H g g t, b ✓ Synchrotron radiation negligible: high energies viable with circular machines ✓ Colored particles in initial state: large cross section due to the strong interaction - Energy and momentum of the initial-state partons not known event-by-event (PDFs) - Large QCD backgrounds, “messy” experimental environment
2 10 H+X) [pb] LHC HIGGS XS WG 2012 s = 8 TeV LHC p p → H ( N N L O + N N L L Q 10 C D + N L O E W ) → (pp p p → q q H ( N N L O Q σ C 1 pp D pp + N → L O → WH (NNLO QCD + NLO EW) E W ZH (NNLO QCD +NLO EW) ) pp → ttH (NLO QCD) -1 10 -2 10 80 100 200 300 400 1000 M [GeV] H • Gluon fusion is the dominant production mechanism both at the Tevatron and the LHC • VBF is the second-largest mechanism and can be easily separated from the background • Higgs-strahlung is the main channel for light Higgs at the Tevatron • Associated Higgs production with a top pair is rare and has difficult backgrounds
Tevatron experiments did their best, but it wasn’t enough Tevatron Run II Preliminary, L � 10.0 fb -1 95% CL Limit/SM Observed Tevatron Exclusion Tevatron Exclusion Expected w/o Higgs 10 10 ± 1 s.d. Expected ± 2 s.d. Expected 1 1 SM=1 June 2012 100 110 120 130 140 150 160 170 180 190 200 m H (GeV/c 2 ) Excluded at 95% CL: 147 GeV < m H < 180 GeV Broad excess (mostly from bb) for 115 GeV < m H < 140 GeV
Constraints on the Higgs mass from EW precision observables An exercise: let’s start from a set of well-measured electroweak (pseudo)- observables - fine-structure constant α = 1 / 137 . 03599911(46) (from Thomson scattering) - Fermi coupling constant G F = 1 . 16637(1) × 10 − 5 GeV − 2 (from muon decay) - Z-boson mass m Z = 91 . 1876(21) GeV (from LEP data) - leptonic width of the Z Γ � + � − = 83 . 984(86) MeV (from LEP data) - W-boson mass m W = 80 . 385(15) GeV (from LEP+Tevatron data) - effective leptonic Weinberg s 2 e ff = 0 . 23153(16) angle (from LEP+SLC data) ≡ (1 / 2 − s 2 e ff ) 2 − s 4 A LR ≡ σ L − σ R e ff e ff ) 2 + s 4 (1 / 2 − s 2 σ L + σ R e ff
At tree level, all of the observables can be expressed in terms of three parameters v, g, g � v, e, s ≡ sin θ W c ≡ cos θ W of the SM Lagrangian: or, equivalently, (also ) α = e 2 1 e v m W = e v s 2 e ff = s 2 , 4 π , G F = 2 v 2 , m Z = 2 sc, 2 s, √ √ √ 2 ⇤� ⌅ ⇥ 2 e 3 − 1 + 1 v 2 + 2 s 2 Γ � + � − = √ s 3 c 3 4 48 2 π Is this consistent with the experimental data? To check, we compute the three Lagrangian parameters in terms of the best-measured observables α , G F , m Z � √ 1 s 2 = 1 2 − 1 1 − 2 2 πα e 2 = 4 πα , v 2 = , √ G F m 2 2 2 G F 2 Z m W , s 2 e ff , Γ ⇥ + ⇥ − and we plug the resulting values of in the expressions for v, e, s tree-level predictions experimental values = 80 . 939 GeV 80 . 385 0 . 015 GeV ± m W s 2 = 0 . 21215 0 . 23153 0 . 00016 ± e ff Γ ⇥ + ⇥ − = 80 . 842 MeV 83 . 984 0 . 086 MeV ± Off by many standard deviations!!!
What happened? We tried to use the SM relations at tree level to predict some observables in terms of other observables, and we failed badly Obviously the tree level is not good enough! Radiative corrections to the relations between physical observables and Lagrangian params: e 2 v 2 m 2 2 s 2 c 2 + Π ZZ ( m 2 = Z ) Z + V V V V e 2 v 2 m 2 + Π W W ( m 2 Π V V ( q 2 ) = W ) W 2 s 2 1 � 1 − Π W W (0) � G F = + δ VB √ m 2 2 v 2 2 W ν µ ν µ µ µ + ... + e e Π WW W ¯ ¯ ν e ν e
α = e 2 � Π γγ ( q 2 ) ⇥ e − e − e − e − 1 + lim + q 2 4 π q 2 − > 0 e + e + e + e + γ Π γγ Π � γγ (0) this one is tricky: the hadronic contribution to cannot be computed perturbatively R had ( q 2 ) = σ had ( q 2 ) We can however trade it for another experimental observable: σ � + � − ( q 2 ) α ( m Z ) = e 2 � ⇥ 1 + Π γγ ( m Z ) α = 4 π 1 − ∆ α ( m Z ) m Z ∆ α ( m Z ) = ∆ α ⇥ ( m Z ) + ∆ α top ( m Z ) + ∆ α (5) had ( m Z ) calculable � ∞ had ( m Z ) = − m 2 R had ( q 2 )d q 2 ∆ α (5) Z Z ) = 0 . 02758 ± 0 . 00035 q 2 ( q 2 − m 2 3 π 4 m 2 π (This hadronic contribution is one of the biggest sources of uncertainty in EW studies)
All these corrections can be combined into relations among physical observables, e.g.: � � � √ � 1 2 + 1 1 − 2 2 πα m 2 W = m 2 (1 + ∆ r ) Z � G F m 2 2 Z ∆ r = ∆ α ( m Z ) − c 2 can be parameterized in terms of two ∆ r s 2 ∆ ρ + ∆ r rem universal corrections and a remainder: The leading corrections depend quadratically on but only logarithmically on : m t m H � m 2 + log m 2 ∆ ρ = Π ZZ (0) − Π W W (0) 3 α � t H + . . . ≈ m 2 m 2 s 2 m 2 m 2 16 π c 2 Z W Z W δ m 2 c 2 c 2 s 2 δ sin 2 θ e ff ≈ W c 2 − s 2 ∆ ρ , c 2 − s 2 ∆ ρ ≈ − m 2 W sin 2 θ e ff In the SM the predictions for and have been fully computed at m W the two-loop order, plus some leading (top/strong) corrections at three and four loops
The radiative corrections bring along a dependence of the experimental observables on all the parameters of the SM Lagrangian It is no longer possible to invert analytically the relations between observables and Lagrangian parameters. But we can still perform a statistical analysis: |O meas � O fit |/ � meas Measurement Fit 0 1 2 3 �� (5) �� had (m Z ) 0.02750 ± 0.00033 0.02759 m Z [ GeV ] m Z [ GeV ] 91.1875 ± 0.0021 91.1874 - compute radiative corrections � Z [ GeV ] � Z [ GeV ] 2.4952 ± 0.0023 2.4959 to all of the SM observables � 0 � had [ nb ] 41.540 ± 0.037 41.478 R l R l 20.767 ± 0.025 20.742 - fit the experimental data and A 0,l A fb 0.01714 ± 0.00095 0.01645 determine the most likely set A l (P � ) A l (P � ) 0.1465 ± 0.0032 0.1481 of Lagrangian parameters R b R b 0.21629 ± 0.00066 0.21579 R c R c 0.1721 ± 0.0030 0.1723 - compute predictions for all the A 0,b A fb 0.0992 ± 0.0016 0.1038 observables in terms of the A 0,c A fb 0.0707 ± 0.0035 0.0742 “best fit” Lagrangian A b A b 0.923 ± 0.020 0.935 A c A c 0.670 ± 0.027 0.668 - compare the predictions with A l (SLD) A l (SLD) 0.1513 ± 0.0021 0.1481 the experimental data and sin 2 � eff sin 2 � lept (Q fb ) 0.2324 ± 0.0012 0.2314 see if they are all consistent m W [ GeV ] m W [ GeV ] 80.385 ± 0.015 80.377 � W [ GeV ] � W [ GeV ] 2.085 ± 0.042 2.092 m t [ GeV ] m t [ GeV ] 173.20 ± 0.90 173.26 (LEP/TEV EWWG, 2012) 0 1 2 3 March 2012
Comparing predictions and experiment (LEP/TEV EWWG 2012) March 2012 March 2012 80.5 0.233 m t = 173.2 ± 0.9 GeV LHC excluded m H = 114...1000 GeV LEP2 and Tevatron LEP1 and SLD 68 % CL m H m W [ GeV ] lept 0.232 eff 80.4 sin 2 � m t 0.231 �� 80.3 m H [ GeV ] �� 68 % CL 114 300 600 1000 155 175 195 83.6 83.8 84 84.2 m t [ GeV ] � ll [ MeV ] (the LEP/Tevatron results favor a light Higgs boson)
Constraining the SM Higgs mass (LEP/TEV EWWG 2012) March 2012 m Limit = 152 GeV March 2012 6 Theory uncertainty � Z � Z � 0 �� (5) � had �� had = R 0 5 R l 0.02750 ± 0.00033 A 0,l A fb 0.02749 ± 0.00010 A l (P � ) A l (P � ) R 0 R b incl. low Q 2 data 4 R 0 R c A 0,b A fb A 0,c �� 2 A fb 3 A b A b A c A c A l (SLD) A l (SLD) 2 sin 2 � eff sin 2 � lept (Q fb ) m W m W � W � W 1 Q W (Cs) Q W (Cs) LEP LHC sin 2 ��� (e � e � ) sin 2 � MS excluded excluded sin 2 � W ( � N) sin 2 � W ( � N) 0 g 2 g L ( � N) 40 100 200 g 2 g R ( � N) m H [ GeV ] 2 3 10 10 10 M H [ GeV ] In March 2012, consistency of the SM required m H < 152 GeV at 95% C.L.
The LHC nails it
Sensitivity to individual search channels in the 2011 LHC data SM Expected limits Expected limits Expected limits CMS Preliminary, s = 7 TeV 2 10 Combined Combined Combined � -1 -1 -1 -1 Combined, L = 4.6-4.7 fb H H H bb (4.7 fb bb (4.7 fb bb (4.7 fb ) ) ) � � � / � -1 -1 -1 int H H H (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 95% CL limit on -1 -1 -1 H H H (4.7 fb (4.7 fb (4.7 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H WW (4.6 fb WW (4.6 fb WW (4.6 fb ) ) ) � � � -1 -1 -1 H H H ZZ ZZ ZZ 4l (4.7 fb 4l (4.7 fb 4l (4.7 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2q (4.6 fb 2l 2q (4.6 fb 2l 2q (4.6 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 10 95% CL expected 1 100 200 300 400 500 600 2 Higgs boson mass (GeV/c )
Sensitivity to individual search channels in the 2011 LHC data SM Expected limits Expected limits Expected limits CMS Preliminary, s = 7 TeV 2 10 Combined Combined Combined � -1 -1 -1 -1 Combined, L = 4.6-4.7 fb H H H bb (4.7 fb bb (4.7 fb bb (4.7 fb ) ) ) � � � / � -1 -1 -1 int H H H (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 95% CL limit on -1 -1 -1 H H H (4.7 fb (4.7 fb (4.7 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H WW (4.6 fb WW (4.6 fb WW (4.6 fb ) ) ) � � � -1 -1 -1 H H H ZZ ZZ ZZ 4l (4.7 fb 4l (4.7 fb 4l (4.7 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2q (4.6 fb 2l 2q (4.6 fb 2l 2q (4.6 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 10 95% CL expected 1 543 115 100 200 300 400 500 600 2 Higgs boson mass (GeV/c )
Sensitivity to individual search channels in the 2011 LHC data prevailing WW ZZ γ γ channel SM Expected limits Expected limits Expected limits CMS Preliminary, s = 7 TeV 2 10 Combined Combined Combined � -1 -1 -1 -1 Combined, L = 4.6-4.7 fb H H H bb (4.7 fb bb (4.7 fb bb (4.7 fb ) ) ) � � � / � -1 -1 -1 int H H H (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 95% CL limit on -1 -1 -1 H H H (4.7 fb (4.7 fb (4.7 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H WW (4.6 fb WW (4.6 fb WW (4.6 fb ) ) ) � � � -1 -1 -1 H H H ZZ ZZ ZZ 4l (4.7 fb 4l (4.7 fb 4l (4.7 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2q (4.6 fb 2l 2q (4.6 fb 2l 2q (4.6 fb ) ) ) � � � � � � -1 -1 -1 H H H ZZ ZZ ZZ 2l 2 2l 2 2l 2 (4.6 fb (4.6 fb (4.6 fb ) ) ) � � � � � � � � � 10 95% CL expected 1 543 115 100 200 300 400 500 600 2 Higgs boson mass (GeV/c )
Note how large rates for production and/or decay are not the end of the story: → b ¯ gg − → H − b dominant for light Higgs, but swamped by QCD background → ` ` 0 b ¯ q 0 − q ¯ → V H − b Needs leptons in the final state: SM Expected limits Expected limits CMS Preliminary, s = 7 TeV 2 10 Combined Combined � -1 -1 -1 H H bb (4.7 fb bb (4.7 fb ) ) Combined, L = 4.6-4.7 fb � � / -1 -1 � H H � � � � � � (4.6 fb (4.6 fb ) ) int -1 -1 95% CL limit on H H (4.7 fb (4.7 fb ) ) � � � � � � -1 -1 H H WW (4.6 fb WW (4.6 fb ) ) � � -1 -1 H H ZZ ZZ 4l (4.7 fb 4l (4.7 fb ) ) � � � � -1 -1 H H ZZ ZZ 2l 2q (4.6 fb 2l 2q (4.6 fb ) ) � � � � 10 95% CL expected (VH) (VBF) 1 110 115 120 125 130 135 140 145 150 155 160 2 Higgs boson mass (GeV/c )
The high-resolution channels: two photons and four leptons ℓ + γ ℓ − Z ( ∗ ) H H Z γ ℓ − ℓ + Both suppressed!!! (respectively by a loop factor and, for m H < 180 GeV, by the virtuality of the Z ) However, the precise reconstruction of the momenta of the particles in the final state produces a narrow peak around m H in the invariant-mass distribution
“I think we have it” [Rolf Heuer at CERN, 04/07/2012] H → γγ H → ZZ → 4 � -1 -1 s = 7 TeV, L = 5.1 fb ; s = 8 TeV, L = 19.7 fb CMS weights / GeV Events / 3 GeV 180 -1 L dt = 4.5 fb , s = 7 TeV ATLAS ∫ 35 Data Events / 3 GeV -1 16 ∫ L dt = 20.3 fb , s = 8 TeV 160 Data m =126 GeV kin D > 0.5 H S/B weighted sum 14 bkg Signal+background Z *,ZZ 30 γ 140 Signal strength categories 12 ∑ Background Z+X 10 120 Signal 8 25 m = 125.4 GeV H 100 6 4 80 20 2 60 0 110 120 130 140 150 m (GeV) 40 15 4 l 20 10 0 weights - fitted bkg 10 5 5 0 -5 ∑ 0 110 120 130 140 150 160 80 100 200 300 400 600 800 m [GeV] γ γ m (GeV) 4 l
“I think we have it” [Rolf Heuer at CERN, 04/07/2012] H → γγ H → ZZ → 4 � -1 -1 s = 7 TeV, L = 5.1 fb ; s = 8 TeV, L = 19.7 fb CMS weights / GeV Events / 3 GeV 180 -1 L dt = 4.5 fb , s = 7 TeV ATLAS ∫ 35 Data Events / 3 GeV -1 16 ∫ L dt = 20.3 fb , s = 8 TeV 160 Data m =126 GeV kin D > 0.5 H S/B weighted sum 14 bkg Signal+background Z *,ZZ 30 γ 140 Signal strength categories 12 ∑ Background Z+X 10 120 Signal 8 25 m = 125.4 GeV H 100 6 4 80 20 2 60 0 110 120 130 140 150 m (GeV) 40 15 4 l 20 10 0 weights - fitted bkg 10 5 5 0 -5 ∑ 0 110 120 130 140 150 160 80 100 200 300 400 600 800 m [GeV] γ γ m (GeV) ℓ 4 l ℓ γ ∗ Z ℓ ℓ
“I think we have it” [Rolf Heuer at CERN, 04/07/2012] H → γγ H → ZZ → 4 � -1 -1 s = 7 TeV, L = 5.1 fb ; s = 8 TeV, L = 19.7 fb CMS weights / GeV Events / 3 GeV 180 -1 L dt = 4.5 fb , s = 7 TeV ATLAS ∫ 35 Data Events / 3 GeV -1 16 ∫ L dt = 20.3 fb , s = 8 TeV 160 Data m =126 GeV kin D > 0.5 H S/B weighted sum 14 bkg Signal+background Z *,ZZ 30 γ 140 Signal strength categories 12 ∑ Background Z+X 10 120 Signal 8 25 m = 125.4 GeV H 100 6 4 80 20 2 60 0 110 120 130 140 150 m (GeV) 40 15 4 l 20 10 0 weights - fitted bkg 10 5 5 0 -5 ∑ 0 110 120 130 140 150 160 80 100 200 300 400 600 800 m [GeV] γ γ m (GeV) 4 l
Determination of the Higgs mass by ATLAS and CMS -1 -1 19.7 fb (8 TeV) + 5.1 fb (7 TeV) 10 7 Λ ln L H � � � tagged -2ln ATLAS Combined +4 CMS γ γ l H ZZ tagged 9 � -1 s = 7 TeV Ldt = 4.5 fb ∫ H → γ γ 6 H + H ZZ � � � � Combined: � -1 s = 8 TeV Ldt = 20.3 fb ∫ - 2 H ZZ* 4 l → → 8 stat. + syst. , (ggH,ttH), µ µ ZZ � � stat. only without systematics 5 (VBF,VH) µ 7 � � +0.26 +0.14 m = 125.02 (stat) (syst) H - 0.27 - 0.15 6 4 2 σ 5 3 4 3 2 2 1 1 σ 1 0 0 123 123.5 124 124.5 125 125.5 126 126.5 127 127.5 123 124 125 126 127 m (GeV) m [GeV] H H m H = 125.4 ± 0.4 ± 0.2 GeV m H = 125.0 ± 0.27 ± 0.15 GeV [ATLAS, 1406.3827] [CMS, 1412.8662]
Profile of a 125-GeV Higgs boson at the LHC with 8 TeV Theory predictions from the LHC Higgs cross-section Working Group, arXiv:1307.1347 σ ( pp → H ) = 19 . 3 +7%+8% σ ( pp → jjH ) = 1 . 6 +0 . 2%+2 . 6% − 8% − 7% pb , − 0 . 2% − 2 . 4% pb σ ( pp → WH ) = 0 . 70 +1%+2 . 3% σ ( pp → ZH ) = 0 . 42 +3 . 1%+2 . 5% − 1% − 2 . 3% pb , − 3 . 1% − 2 . 5% pb σ ( pp → ttH ) = 0 . 13 +3 . 8%+8 . 1% − 9 . 3% − 8 . 1% pb BR( H → b ¯ b ) = 57 . 7% , BR( H → WW ∗ ) = 21 . 5% , BR( H → ZZ ∗ ) = 2 . 6% , BR( H → τ + τ − ) = 6 . 3% , BR( H → gg ) = 8 . 6% , BR( H → γγ ) = 0 . 23% ( relative errors on the BRs range from 3% for bb to 10% for gg )
125 GeV is a lucky mass, several decays accessible ATLAS Prelim. (stat.) Total uncertainty σ -1 -1 m = 125.36 GeV sys inc. 1 on ( ) ± σ µ 19.7 fb (8 TeV) + 5.1 fb (7 TeV) σ H theory Phys. Rev. D 90, 112015 (2014) m = 125 GeV CMS H → γ γ + 0.23 Combined H - 0.23 = 1.00 0.14 µ ± p = 0.96 + 0.27 + 0.16 = 1.17 µ - 0.11 SM 0.27 - H tagged � � � arXiv:1408.5191 H ZZ* 4l + 0.34 → → = 1.12 0.24 µ ± - 0.31 + 0.40 + 0.21 = 1.44 µ - 0.11 - 0.33 H ZZ tagged � arXiv:1412.2641 = 1.00 0.29 µ ± H WW* l l + 0.16 → → ν ν - 0.15 + 0.23 0.17 + = 1.09 µ H WW tagged � 0.14 - - 0.21 = 0.83 0.21 µ ± arXiv:1409.6212 + 0.3 W,Z H b b → - 0.3 H tagged � � � + 0.4 + 0.2 = 0.5 µ - 0.2 - 0.4 = 0.91 0.28 µ ± ATLAS-CONF-2014-061 H + 0.3 → τ τ - 0.3 H bb tagged � + 0.4 + 0.3 = 0.84 0.44 = 1.4 µ ± µ - 0.3 0.4 - 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Best fit / � � -1 ∫ s = 7 TeV Ldt = 4.5-4.7 fb SM Signal strength ( ) µ -1 ∫ s = 8 TeV Ldt = 20.3 fb released 12.01.2015
The Higgs couplings to the other SM particles are proportional to their masses: -1 -1 19.7 fb (8 TeV) + 5.1 fb (7 TeV) 1/2 CMS /2v) t 1 WZ V or (g 68% CL 68% CL -1 10 95% CL 95% CL f λ SM Higgs SM Higgs b τ -2 10 µ (M, ) fit ε -3 10 68% CL 95% CL -4 10 0.1 1 10 100 Particle mass (GeV)
The angular distribution of the decay products allows to test spin and parity: CMS Preliminary -1 -1 19.7 fb (8 TeV) + 5.1 fb (7 TeV) Pseudoexperiments + gg X(2 ) ZZ + WW + → → γ γ 0.09 m Observed 0.08 + 0 0.07 + 2 m 0.06 0.05 0.04 0.03 0.02 0.01 0 -30 -20 -10 0 10 20 30 40 -2 ln(L / L ) × + 0 P J (spin 2 disfavored) (pseudoscalar disfavored)
The ultimate test of the Higgs mechanism: self-couplings = 1 2(2 λ v 2 ) H 2 + λ vH 3 + 1 The Higgs potential includes 4 λ H 4 V trilinear and quartic self-couplings: The three-Higgs coupling can be extracted from Higgs pair production. However, suppressed by phase space and diluted by other topologies. E.g., H g H g t t H ∗ g g H H The coupling could be measured with ~50% accuracy in a high-luminosity LHC run and with 10%-20% accuracy at the ILC with 1 TeV No hope to measure directly the four-Higgs coupling via three-Higgs production
Status of the EW fit after the Higgs discovery (Gfitter collaboration, 2014) 80.5 [GeV] 2 f sin ( � ) LEP+SLC ± 1 � 68% and 95% CL contours eff 80.48 2 f direct M and sin ( ) measurements � W eff W 2 f fit w/o M , sin ( ) and Z widths measurements � M 80.46 W eff 2 f fit w/o M , sin ( ) and M measurements � W eff H 2 f fit w/o M , sin ( ), M and Z widths measurements � 80.44 W eff H 80.42 80.4 80.38 M world comb. ± 1 � 80.36 W 80.34 80.32 G fitter SM Jul ’14 0.2308 0.231 0.2312 0.2314 0.2316 0.2318 0.232 0.2322 2 l sin ( ) � eff
Status of the EW fit after the Higgs discovery (Gfitter collaboration, 2014) 80.46 [GeV] 68% and 95% CL fit contour 2 f sin ( ) 1 � ± � eff 80.44 2 f w/o M and sin ( ) measurements � W eff W Present SM fit M 80.42 (300 fb -1 ) Prospect for LHC Prospect for ILC/GigaZ 80.4 Present measurement ILC precision 80.38 LHC precision M 1 ± � W 80.36 80.34 80.32 G fitter SM Jul ’14 0.231 0.2311 0.2312 0.2313 0.2314 0.2315 0.2316 0.2317 0.2318 0.2319 2 l sin ( ) � eff
The fate of the SM: stability of the electroweak vacuum
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ φ v
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ Including quantum corrections: V ( φ ) = m 2 ( µ ) | φ ( µ ) | 2 + λ ( µ ) | φ ( µ ) | 4 + ∆ V loop φ v
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ Including quantum corrections: V ( φ ) = m 2 ( µ ) | φ ( µ ) | 2 + λ ( µ ) | φ ( µ ) | 4 + ∆ V loop φ v φ At large , the potential is dominated by the quartic term: V ( φ ⇥ v ) � λ ( µ � φ ) | φ | 4
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ Including quantum corrections: V ( φ ) = m 2 ( µ ) | φ ( µ ) | 2 + λ ( µ ) | φ ( µ ) | 4 + ∆ V loop φ v φ At large , the potential is dominated by the quartic term: V ( φ ⇥ v ) � λ ( µ � φ ) | φ | 4 If the quartic coupling turns negative at some large scale, the potential is unstable
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ Including quantum corrections: V ( φ ) = m 2 ( µ ) | φ ( µ ) | 2 + λ ( µ ) | φ ( µ ) | 4 + ∆ V loop φ v φ At large , the potential is dominated by the quartic term: V ( φ ⇥ v ) � λ ( µ � φ ) | φ | 4 If the quartic coupling turns negative at some large scale, the potential is unstable The Higgs field can tunnel to a much larger value, destroying the EW vacuum
The fate of the SM: stability of the electroweak vacuum Tree-level scalar potential: V ( φ ) r − m 2 V 0 ( φ ) = m 2 | φ | 2 + λ | φ | 4 , v = λ Including quantum corrections: V ( φ ) = m 2 ( µ ) | φ ( µ ) | 2 + λ ( µ ) | φ ( µ ) | 4 + ∆ V loop φ v φ At large , the potential is dominated by the quartic term: V ( φ ⇥ v ) � λ ( µ � φ ) | φ | 4 If the quartic coupling turns negative at some large scale, the potential is unstable The Higgs field can tunnel to a much larger value, destroying the EW vacuum The lifetime of the EW vacuum must be longer than the age of the Universe (metastability)
m 2 H = 2 λ v 2 λ We can extract the weak-scale value of from . + higher orders λ Loops of SM particles determine the dependence of on the renormalization scale µ λ g 2 g 4 λ 2 λ Y 2 Y 4 λ ⇢ d λ 1 24 λ 2 + λ 12 Y 2 t + 12 Y 2 b + 4 Y 2 τ − 9 g 2 − 3 g 0 2 ⇤ ⇥ d log µ = 16 π 2 � +9 8 g 4 + 3 8 g 0 4 + 3 4 g 2 g 0 2 − 6 Y 4 t − 6 Y 4 b − 2 Y 4 + higher orders τ λ 2 Λ Large m H : prevails, grows with µ until it blows up at some scale (Landau pole) λ − Y 4 Λ Small m H : prevails, decreases with µ until it turns negative at (vacuum instability) λ t Λ is the scale at which new physics must rescue the SM (anyway, ) Λ ≤ Λ Planck
m H ≈ 125 GeV is right at the edge between the stability and metastability regions 0.10 180 10 7 10 8 10 9 Instability 3 s bands in 0.08 178 M t = 173.3 ± 0.8 GeV H gray L 10 10 a 3 H M Z L = 0.1184 ± 0.0007 H red L 0.06 10 11 Plots from 1307.3536v4 M h = 125.1 ± 0.2 GeV H blue L Higgs quartic coupling l Top pole mass M t in GeV 176 10 12 10 13 0.04 10 16 174 1,2,3 s 0.02 Meta - stability M t = 171.1 GeV 10 19 172 0.00 a s H M Z L = 0.1205 170 - 0.02 a s H M Z L = 0.1163 10 18 10 14 Stability M t = 175.6 GeV - 0.04 168 10 10 10 12 10 14 10 16 10 18 10 20 120 122 124 126 128 130 132 10 2 10 4 10 6 10 8 10 17 Higgs pole mass M h in GeV RGE scale m in GeV IF the SM is valid up to the Planck scale, the vacuum is most likely metastable
m H ≈ 125 GeV is right at the edge between the stability and metastability regions 0.10 180 10 7 10 8 10 9 Instability 3 s bands in 0.08 178 M t = 173.3 ± 0.8 GeV H gray L 10 10 a 3 H M Z L = 0.1184 ± 0.0007 H red L 0.06 10 11 Plots from 1307.3536v4 M h = 125.1 ± 0.2 GeV H blue L Higgs quartic coupling l Top pole mass M t in GeV 176 10 12 10 13 0.04 10 16 174 1,2,3 s 0.02 Meta - stability M t = 171.1 GeV 10 19 172 0.00 a s H M Z L = 0.1205 170 - 0.02 a s H M Z L = 0.1163 10 18 10 14 Stability M t = 175.6 GeV - 0.04 168 10 10 10 12 10 14 10 16 10 18 10 20 120 122 124 126 128 130 132 10 2 10 4 10 6 10 8 10 17 Higgs pole mass M h in GeV RGE scale m in GeV IF the SM is valid up to the Planck scale, the vacuum is most likely metastable But should we really buy that “IF”?
III) Beyond the Standard Model
The Standard Model does an excellent job in describing physics at the weak scale. Still, it is unlikely that it is valid all the way up to the scale of quantum gravity Observational arguments for BSM physics • The SM does not account for neutrino oscillations (this, however, can easily be fixed by adding heavy and sterile right-handed neutrinos to the theory) • The SM does not include a suitable candidate for Dark Matter, and cannot justify the matter-antimatter asymmetry in the Universe Theoretical arguments for BSM physics • The SM has many (>20) arbitrary parameters, and a rather complicated structure (“odd” gauge group, generation mixing, large mass hierarchies among fermions). It would be nice to embed it in a simpler and more predictive theory (e.g., a GUT). • Quantum corrections destabilize the Higgs mass inducing a quadratic dependence on the cutoff scale that regularizes the loop integrals (the hierarchy problem)
The hierarchy problem of the Standard Model The SM fermion masses are m f + + × × protected by chiral symmetry: · · · f L f R f L f R δ m f ∝ m f , thus if is small it stays so even after including quantum corrections m f There is no analogous mechanism to protect the scalar mass term: f L S m 2 H λ f λ f + + + × · · · · · · H H H H H f R λ S H H H The radiative corrections depend quadratically on the cutoff scale where New Physics kicks in: H ⊃ 3 G F Λ 2 ∆ m 2 2 m 2 W + m 2 Z + m 2 H − 4 m 2 � � √ t 2 π 2 4 If the validity of the SM extends up to the Planck scale (or the GUT scale) we need an extremely fine-tuned cancellation between the tree-level mass and the radiative corrections
Different approaches are possible: • New physics intervenes at the TeV scale (supersymmetry, composite Higgs models, ...) • The scale of quantum gravity is itself at the TeV (models with large extra dimensions) • Tough luck, live with fine tuning (SM up to high scales: “nightmare” scenario for LHC?)
Supersymmetry and the MSSM Fermions and bosons enter the quantum corrections to the Higgs mass with opposite sign f L S m 2 H λ f λ f + + × · · · H H H H H f R λ S H H H In a supersymmetric theory, each fermion has a bosonic partner with the same mass λ S = λ 2 and internal quantum numbers (their couplings to the Higgs are related, ). f Their quadratically divergent contributions to the Higgs mass cancel each other In the Minimal Supersymmetric Standard Model (MSSM) every SM particle is promoted to a supermultiplet (however, two Higgs supermultiplets are required) The superpartners must be heavier SUSY must be broken by explicit than the ordinary SM particles mass terms for the new particles λ 2 These SUSY-breaking masses M S are soft , i.e. ∆ m 2 16 π 2 M 2 H ∝ S they do not reintroduce quadratic divergences:
Supersymmetry and the MSSM Fermions and bosons enter the quantum corrections to the Higgs mass with opposite sign f L S m 2 H λ f λ f -1 + + × · · · H H H H H f R λ S H H H In a supersymmetric theory, each fermion has a bosonic partner with the same mass λ S = λ 2 and internal quantum numbers (their couplings to the Higgs are related, ). f Their quadratically divergent contributions to the Higgs mass cancel each other In the Minimal Supersymmetric Standard Model (MSSM) every SM particle is promoted to a supermultiplet (however, two Higgs supermultiplets are required) The superpartners must be heavier SUSY must be broken by explicit than the ordinary SM particles mass terms for the new particles λ 2 These SUSY-breaking masses M S are soft , i.e. ∆ m 2 16 π 2 M 2 H ∝ S they do not reintroduce quadratic divergences:
Composite Higgs models The hierarchy problem originates from the fact that the SM Higgs is an elementary scalar (therefore its mass cannot be protected by chiral or gauge symmetries) The classical alternative to the SM Higgs mechanism, i.e. dynamical symmetry breaking such as in Technicolor models, is disfavoured by flavour and electroweak precision tests An intermediate approach is possible: There is a light Higgs scalar (to satisfy the electroweak precision observables) but it is composite , the light remnant of a new strong dynamics responsible for EWSB To preserve EW observables, the particles of the strong sector should be above the TeV scale The composite Higgs can be lighter than the rest if it is a pseudo-Goldstone boson of a global symmetry of the strong sector (e.g. Little Higgs, Holographic Higgs, ...) Even if the new states are heavy, the composite nature of the Higgs should appear at the LHC: - high-energy growth of the V V V V cross sections - modified couplings of the Higgs to SM particles
Models with Large Extra Dimensions Supersymmetry helps the Higgs boson cross the “desert” between M EW and M Planck An alternative paradigm: there is no desert, and M Planck ~ M EW !!! The simplest scenario: The SM fields live on a 4-d “brane” Arkani-Hamed et al., hep-ph/9803315 but gravity propagates in the “bulk” n dimensions The “true” scale of quantum gravity can be compactified y i R over a radius R lower than the apparent 4-dim Planck scale: extra compactified y j dimensions M 2 Pl = M n +2 (2 π R ) n ∗ → R ≈ 10 10 m n = 1 , M ∗ = 10 TeV − n = 2 , M ∗ = 10 TeV − → R ≈ 0 . 1 mm SM fields Gravity is untested below 0.1 mm. For n = 2 the scale of quantum gravity 4-d spacetime the usual 4-d could be as low as 10 TeV spacetime (and even lower for larger n !!!)
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