The Higgs Boson and Electroweak Symmetry Breaking

1. The Minimal Standard Model

M. E. Peskin SLAC Summer Institute, 2004 When I was a student, I was told that the goal of elementary particle physics was to learn the basic laws of the strong and weak interactions.

Today, this is a solved problem.

These laws are explained by the “Standard Model”:

The strong, weak, and electromagnetic interactions are described by a Yang-Mills gauge theory with the gauge group SU(3) x SU(2) x U(1) . But, in science, the solution to every problem leads to new questions at a deeper level.

To describe the weak interactions in a Yang-Mills theory, the gauge symmetry must be spontaneously broken.

We know that this happens, but we do not know why.

My personal view is that this is the greatest of the Great Puzzles.

It is the key to further progress in microscopic physics.

You are likely to solve this problem during your careers, from discoveries at the next generation of accelerators. In these lectures, I will give an introduction to the problem of Electroweak Symmetry Breaking (EWSB):

1. The Minimal Standard Model

I will discuss the properties of the Higgs boson in the simplest version of the Standard Model. I hope this will be a useful starting point for analyzing experiments on the Higgs boson and EWSB.

2. Models of Electroweak Symmetry Breaking

I will present three models that might explain EWSB. I hope this will help you to appreciate the variety of possible approaches. Eventually, experiment will decide which approach is correct. Elements of the SU(2) x U(1) electroweak theory (Glashow, Salam, Weinberg) add to the known quarks, leptons, bosons one scalar 1 1 field with I = Y = + 2 2 The Lagrangian for ϕ is 2 1 2 1 2 L = |D ϕ| − V (|ϕ|) − (F a ) − (G ) µ 4 µν 4 µν + (coupling to quarks and leptons)

Assume V ( | ϕ | ) is such that ! ϕ " is nonzero: 2 2 4 e.g., V ( | ϕ | ) = µ | ϕ | + λ | ϕ |

2 with µ < 0 The field ϕ has the general structure: π+ ϕ = ! (v + h + iπ0)/√2 " ± 0 π , π are Goldstone bosons −iαa x τ a 0 ϕ(π+, π0) = e ( ) ! v/√2 " In the theory with global symmetry, they are massless. In the theory with gauge symmetry, they are gauge degrees of freedom, and become part of W, Z h(x) is the Higgs boson field. It corresponds to v → v + h(x) 2 Dµϕ | | " 2 g g − − g g 0 = ( 0 v/√2 ) W +σ+ + W σ + W 0σ3 + B !√2 √2 2 2 ! " v/√2 # 2 ! ! v !− 1 " ! = [g2W +W + ( gW 0 + g B)2] (1) 4 2 − The boson mass eigenstates are 3 Z = cos θwW − sin θwB 3 A = sin θwW + cos θwB (1) where g g! cos θw = sin θw = !g2 + g!2 !g2 + g!2 Then g2 g2 + g!2 m2 = v2 m2 = v2 W 4 Z 4 Notice the nontrivial relation mW /mZ = cos θw With this orientation, it is straightforward to work out the couplings of the Higgs boson.

Since in Higgs appears from v → v + h , its W, Z vertices are:

2 2 W+ W m Z Z m = 2i W g = 2i Z g v µν v µν h h

2 The potential above gives mh = !2|µ | = !λ/2v and the vertex h m2 h = −3i h v h The Higgs couples to fermions through scalar and pseudoscalar operators: f LfR , f RfL These are the operators used to build mass terms. But since left- and right-handed fermions have different SU(2)xU(1) quantum numbers, it is not possible to build such terms without the Higgs field. Using ν L = = u e u d e− Q R R R ! "L ! d "L we can form † † L = −λeeRϕ · L − λddRϕ · Q − λeuRϕα#αβQβ 1 1 2 1 1 Y = +1 − − Y = − + + 2 2 3 2 6 0 λf put ϕ = , then mf = v ! " ! v/√2 " √2 In the Standard Model, fermion masses arise only through EWSB. Using v → v + h we find the Higgs coupling to fermions: _ m f f = −i f v h If the fermion mass matrix is diagonal, the Higgs coupling is also flavor-diagonal.

Here is a direct argument. Start from the most general Lagrangian with flavor-mixing: L − ij i † j · · · = λe eRϕ L + We can represent any complex matrix as a product of unitary and real diagonal matrices: † λe = VeRDeVeL

Now transform eR → VeReR , eL → VeLeL

This removes flavor violation in the Higgs couplings. Can these unitary transformations show up elsewhere in the theory ?

For leptons, if λ ν = 0 , we can rotate L → VeLL

This completely removes V eL from the theory. Then, e.g. ν e is defined to be the neutrino emitted in β-decay. If neutrinos have mass, ν e might not be a mass eigenstate.

For quarks, V d L , V u L can still appear in weak boson couplings: µ → µ † ZµuLγ uL ZµuLγ (VuLVuL)uL = 1 + µ → + µ † Wµ uLγ dL Wµ uLγ (VuLVdL)dL = VCKM So, the only flavor-violating interactions in the Minimal Standard Model are hadronic weak interactions with CKM mixing. What constraints do we have on the

Higgs mass and the coupling of 500 Q (GeV) the MSM ? 103

Renormalization group evolution 400 implies that the MSM can be correct 106 up to high energies only in a limited 300 (GeV) range of parameters. H 1010 m 1015 d 3 2 − 1 4 · · · 200 λ = λ λt + 19 d log Q 2π2 ! 32 " 10 100 So for small λ t, λ → ∞ at high Q; for large , λ → −∞ 0 m 0 100 200 300 So, only if h is between about 140 and 5–97 mt (GeV) 8303A01 180 GeV can the MSM be the whole story. Lindner Fits to precision electroweak data also constrain the Higgs boson mass in the MSM. Marciano will discuss this in his lectures.

LEP EWWG: m h < 2 4 0 G e V (95% conf.) From the couplings of the Higgs boson, we can determine the Higgs decay modes. The decay pattern of the Higgs has a quite intricate variation with energy.

First, h → ff N λ2 m2 → c f Ncα f Γ(h ff) = mh = 2 2 mh 16π 8sw mW The heaviest species dominate. But, be careful; masses should be evaluted at a common scale. At Q ∼ 2mt the M S masses are

τ: 1.7 c: 0.5 b: 2.7 t: 160 GeV

− Next, h → W +W , Z0Z0 α m3 m2 m2 m4 h → W W h − 4 W 1/2 − W W Γ( ) = s2 m2 (1 m2 ) 1 4 m2 + 12 m4 16 w W h ! h h " α m3 m2 m2 m4 h → ZZ h − 4 Z 1/2 − Z Z Γ( ) = s2 m2 (1 m2 ) 1 4 m2 + 12 m4 32 w W h ! h h " 3 2 It seems odd that this should go as m h/mW To understand this, note that the decay is dominated by helicity = 0 W, Z bosons, for which - W+ W m2 m2 2E E m2 2i W !∗ · !∗ ∼ 2i W + − ∼ i h = + − 2 h v v mW v The helicity 0 weak bosons arose as eaten Goldstone bosons; and indeed + - λv m2 i = i h = 2 v h + − For m h > 2 m W , h → W W is the dominant mode.

For m h < 2 m W , the mode h → b b is still suppressed, so the modes h → W W ∗ , h → ZZ∗ can compete with it.

W* W h 2-jet mass distributions in h→WW*, ZZ* decays m(h) = 120 GeV 0.16

0.14

0.12

0.1

0.08

0.06

0.04 Z*

0.02 W*

0 0 20 40 60 80 100 120 M(W), M(Z) h → gg , h → γγ , h → γZ0

These processes occur through loop diagrams involving heavy particles.

g g t W t h h h

2 3 ααs m Γ(h → gg) = h · 2 2 2 2 mh ! 2mW 144π sw mW 3 ! !2 → α !21 − · 2 2! Γ(h γγ) = 2 2 ! 3 ( ) ! 144π sw ! 4 3 !

Notice that these decays measure sum rules over the spectrum of heavy particles with QCD/electroweak interactions that obtain mass from the Higgs boson. Now put all of the pieces together: here are the branching fractions for a heavy, intermediate, and light Higgs boson in the MSM

Inverting the decay modes, we find many processes by which the Higgs boson can be created and studied. Three of the most important are: h gg → h “gg fusion”, the process with the largest g g cross section for Higgs production at the LHC − e+e → Zh , qq → W h , Zh h Z “Higgs-strahlung”; this process measures the Z* h-Z and h-W couplings, which are present only because h generates the mass of these particles h W +W − → h

“WW fusion”, with W radiated from W W initial state q or e These processes will be described in the afternoon lectures; for a detailed treatment, see The Higgs Hunter’s Guide, by Dawson, Gunion, Haber, and Kane. The Minimal Standard Model is a perfectly consistent theory of EWSB, but there is something troubling about it.

You might ask, why is electroweak symmetry broken in this model ?

2 The answer is, because µ < 0 . It is easy to make this problem appear worse.

2 For example, µ is not computable even in principle in the MSM, because most of the diagrams contributing to 2 µ are quadratically ultraviolet divergent

λ 2 ∼ Λ (4π)2

16 If the cutoff Λ is a very high scale, e.g. Λ ∼ 10 GeV it is very difficult to understand how m h could be as small as 100 GeV.

This is called the “gauge hierarchy problem”. But isn’t it enough that we do not know any particles or forces that could provide a physical mechanism for EWSB ? It is as if we wanted to explain the tides without knowing about the existence of the moon.

My cat knows that the moon is there; it may take a $5 B accelerator to find the Higgs boson.

But conceptually, there is no difference. We have to interrogate Nature and find the missing ingredients to solve the puzzle.