The Higgs Boson
1. The Higgs Mechanism – extremely simplified 2. Significance of the recent discovery 3. The detectors (John Leung) 4. Thdhe data ( Martin Kwok)
Chu Ming-chung, Department of Physics, CUHK [email protected] 1 The “GGpod particle” http://press.web.cern.ch/press/PressReleases/Releases2012/PR17.12E.html 蘋果日報:「它是所有物質的質量之源，是促成宇 宙形成的重要粒子,…」http://www.youtube.com/watch?v=av_hWBQ7C_8 都市日報: 「有一種說法認為，找到「上帝粒子」，就找到 萬物之源。」 http://www.metrohk.com.hk/pda/pda_detail.php?id=189963&selectedDate=2012-07- 05&categoryID= all LA Times: “…the so-called God particle that theorists believe gives all other particles mass.”
Nownews:「『上帝粒子』被認為是宇宙中所有基本粒子 的質量之源，使得物質得以形成、凝聚、演化。」 http://www.nownews.com/2012/07/05/91-2831211.htm
2 Higgs Field (popu lar sci ence level)
Particle mass: inertia against acceleration
Higgs mechanism: medium → inertia (mass) Electron has Eg.: thffihe effective mass of fl electron i n a crystal + larger effffiective + + mass vacuum = fffggfilled of Higgs field = medium e- + Elementary particles acquire masses Lattice provides a potential ~ Higgs fldfield Different interactions with the medium → different masses
Excited states of Higgs field = Higgs particles Vacuum = lowest energy state, could be full of particles/energy 3 Animation from CERN Elementaryyp particles Gauge bosons
Why are the Carrier of masses so EM force different? CCfarrier of Why are and strong force g massless whlhile W and Z massive?
Carriers of weak force Figure from 4 Wikimedia Higgs Boson
weak EM strong
New force!
From Wikimedia 5 History Gauge fields must be massless! Can’t get weak and nuclear forces.
1954: Yang-Mills Gauge theory SU(2) nuclear force 1926: GGgauge theor y WlfWolfgang Pauli symmetries forces U(1) EM
C. N. Yang Robert Mills
6 Hermann Weyl Vladimir Fock 1963/4: ‘Higgs History mechanism’ massive gauge bosons
1960: Spontaneous 1961,62: SSB Symmetry Breaking in for relativistic superconductor gap field theory (massive photons), Robert Brout Peter Higgs Meissner effect, etc. ClCarl Hagen François Englert Nobel Prize 2008
Tom Kibble Gerald Guralnik
Jeffrey Goldstone Phil Anderson 南部 陽一郎 Yoichiro Nambu7 http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen Kibble_mechanism_%28history%29 http://nobelprize.org/nobel_prizes/physics/laureates/2008/phyadv08.pdf History
1973: asymptotic freedom, onlyyf for g aug e theories! Nobel Prize 2004
1967: electroweak unification SU(2)xU(1) gggauge theory + SSB Nobel Prize 1979 David Gross David Politzer Frank Wilzek
Steven Weinberg
Sheldon Glashow Abdus Salam8 Basic concepts
- Gauge symmetry and gauge interaction -Whyyggf must gauge fields be massless? - Spontaneous Symmetry Breaking - Higgs Mechanism: preserving gauge symmetry, yet massive gauge fields
9 Gauge Symmetry • Global symmetry: a transformation independent of (x, t) that leaves the system unchanged. Eg. rotating the angle coordinates of all particles y iio 1 constant, same foforr all particlesparticles anywhereanywhere 2
ij i - j independent of o x • Can we make the symmetry local?
o(,)xt or oi(t)? local gauge symmetry
• Yes, as long as each particle also carries oi(t) and lets others know: imagine putting in a string between each pair of particles carrying the information oi(t) and oj(t). ~ a gradient i ij i - j +oi - oj oi oj j Need a vector field to carry Gauge theory: add a vector field (gauge the info. gauge field 10 field) to restore the local gauge symmetry EM vector potential A A + Free particle Schrödinger Equation: 2 2 ixt (,) (,) xt tm2 = ||ei (x, t) Clearly have global gauge symmetry U(1): + Local U(1) Gauge Symmetry? ix () (()x,txtx)( (), )() ()' e Schrödinger Equation becomes: ix() ix () ix () ' eeie iħ ’ = -(ħ2 2/2m)[ - i]2 ’ t 2 1 iieAA '' ;where; where . tm2 Extra term e~ oi - oj
Free Schdhrödinger Equation d oes not ob ey l ocal U(1) symmetry! 11 RRgescuing Local G Ggyyauge Symmetry What if the particle is not free, but coupled to EM field? p p + qA (Gr iffit hs Ch. 7) EM Vector potential Q.M.: p -iħ + (i/ħ)qA A A + (ħ/q) gauge freedom: choose any (x) without affecting physics! The extra term in the free Schrödinger Equation can be cancelled by a gauge transformation in A! 2 2 s.t. iħt’ = -(ħ /2m)[ + (i/ħ)qA] ’ i.e., lllocal U(1) iritinvariant! A = gauge fifildeld EM coupling makes Schrödinger Equation local gauge invariant! Requiring local gauge invariant gives rise to interaction!
symmetry dynamics 12 Gauge theories U(1) → EM is the simplest gauge theory. SU(2) → Yang-Mills ( becomes a 2x2 unitary matrix) SU(3) → QCD ( becomes a 3x3 unitary matrix) All interactions are believed to be generated by gauge theories. Specialll relativity: gllblobal Lorentz covariance General relativity: local Lorentz covariance General covariance: physics is invariant w.r .t . coordinate choice Asymptotic freedom: needed for consistency of field theory, only true for gauge theories! But gauge fields must be massless! Weak interaction: short-ranged massive gauge fields! 13 Why must gauge fields be massless?
14 Euler-Lagrange Equation - Eq. of motion can be 'derived' from a Lagrangian
LTV Lqqt(,;)viaEuler(,;)ii via Euler- Lagrange Equation T = KE, V = PE dL L (1) dt qqii - Generalization to field theory:
qxqxii (), i i () LLL L (1) , (2) () i iii where LL Lagrangian density functional of i . symmetries of Lagrangian symmetries of equation of motion (eg. Lorentz and gauge invariant) 15 Mass of gauge fields Euler-Lagrange Eq. equation of motion (F = ma) from Lagrangian Can construct Lagrangian from known equations of motion
Can show that the mass of a particle (field) is given by the coefficient 2 2 of the quadratic term in L. Eg. L = ½ [ - m ] for a scalar field Think of 2 as number density of particles, each contributing m2 to energy density
2 Mass term for photons (if exists): -m A A But gauge transformation: A A + Mass of photon (gauge fields) would violate local gauge symmetry! Local U(1) symmetry photon has exactly zero mass! Good for EM, strong interactions, bad for weak (W, Z)! 16 Giving mass to gauge fields: StSpontaneous Symmetr y BrBrkieaking
17 Spontaneous Symmetry Breaki ng (SSB) Syyfggmmetries of the Lagrangian/Hamiltonian may not be realized in all states. Eg. H atom: rotational symmetry is observed in 1s, but not in some of the p states. Not SSB: 1s (g.s.) obeys rotational symmetry Eg. some 2p orbitals that ‘break’ rotational symmetry SSB = ground state does not obey a symmetry of the Lagrangian. The symmetry is not broken for the dynamics, just hidden for the ground state. No external field is needed for this to occur (spontaneous). Usuallyyppf happen for sy stems with many yp possible degenerate ground states, each ‘breaking’hides the symmetry, but all together ‘restore’reveals it. 18 Examples of SSB SSB: ground state of a system does not exhibit a symmetry of the Lagrangian
Axial s ymmetric p encil in vertical p osition The system is symmetric w.r.t. left/right for is not the ground state. each seat. The first person picking his/her glass The g. s. chooses a direction randomly: SSB. induces SSB. Note that SSB no symmetry. The Lagrangian (dynamics) has the symmetry. It’s not exhibited by one g.s., but restored if all degenerate g.s. are taken together. 19 More examples of SSB
Eg. ferromagnet: U = - si si 1 lowest energy state: all spins aligned
symmetric w.r.t. rotation: no preferred direction Zero external fifildeld T = 0 These are all degenerat e gr ound stat es, but ea ch ‘break s’ the symmetr y of the Hamiltonian. All possible g.s. together ‘restores’ the symmetry.
0 for any one possible g.s. = 0 0 indication of SSB = 0 iffg averaged over all p ossible g.s. order parameter condensate
External fields can also break the symmetry, but not SSB: Explicit
Symmetry Breaking. U = - si si 1 - B si 20 Vacuum
-In qqpyuantum physics, the vacuum is not emp pyffty. It’s full of quantum fluctuations (energy E, particle no. n, etc.). E - vacuum = lowest energy state
t
-usually, the average of fluctuations is zero: eg. = 0, … if the vacuum is symmetric V - SSB: possible to have symmetry broken spontaneously in the ground state eg. <> = 0 (cf . magneti zed medi um) o -o V o because of self interaction energy .
ground states21 Higgs Mechanism
-there exists a field (bosonic), which condenses in the vacuum (BEC), s.t. <> = o 0, because of Excitations self interaction V (P.E.) around o - Higgs fi eld i nteracts wi th WZW, Z: iiinteraction Higgs particles energy effective mass mA * 2 o oA A → ½ mA A A o -But W, Z are intrinsically massless gggauge symmetry ok degenerate BEC of ground states Higgs W
Illustration taken from: http://www.scholarpedia.org/article/Englert-Brout-22 Higgs-Guralnik-Hagen-Kibble_mechanism_(history) Higgs and fermi on mass
Fermion mass can also be generated by interacting with Higgs field : L = Lo + g YkYukawa coupling
SSB: <> 0 mass term of mi ~ gi <> Fermion masses generated from interacting with Higgs field!
Neutrinos, photons, gluons are assumed to be massless in the Standard Model they do not interact with Higgs field; remain massless.
23 Siggfnificance of the recent discovery - Found a new heavy scalar particle (m ~ 125 GeV), Higgs-like - New elementary particle! - If Higgs boson: Con firm Higgs Mec han ism, Stand ard Mod el (SM) Confirm gauge theory approach to interaction Confirm electroweak unification, boost confidence on GUT Discover a new force different from EM, Weak, Strong, gravity Constrain details of Higgs mechanism and physics beyond SM - If not Higgs: even more exciting! - giifves mass of e sifizes of atoms But not origin of all masses! Not even masses of elementary particles! 24 Mass offp a proton/ neutron
m = 938. 3 MeV _ quarks q q u glluons u proton _q d q
Most offy your mass is in p rotons/ neutrons proton = u, u, d + gluons + qq pairs u, d quark rest mass (few MeVs) negligible, gluons have zero rest mass Where is the mass of the proton? Major contributions: KE of quarks and gluons ~ 300 MeV PE stored in the gluons, particularly instantons 25 proven by mc2! The “God particle” 蘋果日報:「它是所有物質的質量之源，是促成宇 宙形成的重要粒子,…」http://www.youtube.com/watch?v=av_hWBQ7C_8 都市日報: 「有一種說法認為，找到「上帝粒子」，就找到 萬物之源。」 http://www.metrohk.com.hk/pda/pda_detail.php?id=189963&selectedDate=2012-07- 05&categoryID= all LA Times: “…the so-called God particle that theorists believe gives all other particles mass.”
Nownews:「『上帝粒子』被認為是宇宙中所有基本粒子 的質量之源，使得物質得以形成、凝聚、演化。」 http://www.nownews.com/2012/07/05/91-2831211.htm
26 The Higgs Boson
1. The Higgs Mechanism – extremely simplified 2. Significance of the recent discovery 3. The detectors (John Leung) 4. Thdhe data ( Martin Kwok)
Chu Ming-chung, Department of Physics, CUHK [email protected] 27 Classical field theory
Generalized coordinates: qi, q˙i ; i = 1, …, N
qi could even b e a f uncti on of ( x, t). L(xi,˙˙xi) L(qi,qi) Eg. normal modes of a string. • Field theory: use fifildelds as generali zed coordi nates
• Lagrangian Lagrangian density L(qi, qi˙) L(i, i)
i(x) is a field (i lab el s different field s) space-time coordinates
i is treated as an independent field ˙ Conjjgugate momentum densit y: i L/i ˙ Hamiltonian density: H(i ,i ) i i - L(i, i)
Ref.: Goldstein, ‘Classical Mechanics’. Landau, ‘Classical Field Theory’.28 Lagggrangian Mechanics Noether’s Theorem: symmetry → conservation
invariance under some operations Eg. translation invariance (x x + a) momentum conservation
i Global U(1) symmetry: e j = 0; j = E.M. current Gauge symmetry Charge conservation!
Local U(1) symmetry: ei (x) Maxwell equations!
Gauge fields dynamics (eg. electrodynamics)
29 Emmy Noether Spontaneous Symmetry Breaking 1. The mass term MlMassless, free KlKliein-GdGordon fildfield: L = ½ Vacuum: = 0 2 2 L = ½ [ - m ](1) mass = (-a)1/2 , a = coefficient of the quadratic term in (mass term) Mass is the energy needed for an excited state above the vacuum.
With interaction: addition of a potential can hide the mass term. Eg. 2 2 2 4 L = ½ + ½ -¼ (2) The mass i s not -i , (ldbh(would be tachyon, b ut not ph hil)ysical)
because the vacuum (ground state) is not = 0.
30 U = -½ 2 2 + ¼ 2 4 (3) U
Minima at = o; o /
2 degenerate vacuua: = o -o o Unstable (false) vacuum: = 0 Expand around = o: - (o) 2 2 2 4 g.s.: = 0 U = -½ ( o) + ¼ ( o) 2 2 2 2 2 2 4 2 2 2 = -½ - o - ½ o +¼+ ¼ + (3/2) o 2 3 2 3 2 4 o o + ¼ o = 22 – ¼ 4/2 3 +¼+ ¼24 (4)
2 2 4 2 3 2 4 L = ½ - + ¼ / -¼ (5)
KE for mass for = 21/2 potential for Zero mass around = 0, bhlbut unphysical. Physical excitations around = 0 have mass = 21/2 . 31 Note that (()2) and (()5) are identical, exceppft for a chang e of variable. Symmetry: L() = L(-) Spontaneous symmetry breaki ng (SSB): L() L(-) ( 3 term in (5)) Interaction with the medium (described by the potential) mass
32 2. SSB of continuous symmetry: Goldstone Theorem Generalize (2) to two (real) fields: 2 2 2 2 2 2 2 L = ½ 1 1 + ½ 2 2 + ½ (1 +2 ) - ¼ (1 +2 ) (6) 2 2 2 2 2 U Minima of U: 1 + 2 = o /
SSB: choose 1 = o, 2 = 0 as the vacuum
Expand around this vacuum: 1-o Mass term for : ½ 2 - ½ 2 2 = 0 2 o massless 2 2 L = ½ + ½ 2 2 - + (3+ 2) -¼2(2 + 2)2 + 4/(42)()) (7) 2 2 2 2 2 1 + 2 = o
1/2 SSB: pick this as acquires a mass 2 , but 2 becomes massless! the ground state Goldstone Theorem: Spontaneous breaking of a continuous symmetry gives rise to a massll(bless scalar (Nambu-Goldstone Boson) . 33 But there’s no massless scalar particle observed! 4. What i s Hi ggs M ech ani sm?
34 Higgs mechanism = SSB of gauge symmetry
Pictorially, it’ s clear that the symmetry in (6) is rotation in (1, 2) plane, can be represented in ‘polar form’ as symmetry w.r.t. .
1+ i2 * 2 * 2 * 2 L = ½ () + ½ ( ) - ¼ ( ) (8) L is clearly invariant w.r.t. → exp(i) (U(1) transformation).
Make local (x) → U(1) gauge theory:
* 2 * 2 * 2 L = ½ (D) D + ½ ( ) -¼ ( ) – (1/16)F F (()9)
D + iqA Must couple to A to make L gauge invariant!
Now SSB! 35 ) A o 36 / A gy 2 2
A i m p( F , but also ½ 2 … A ex ield ener
) )p( f m →
F e )
g + excitations A o au g ggf gy ( A = * – (1/16
/ A )
2 q ) 2
1/2 , 0 o Goldstone boson (massless)
2 )( ( )
= 2 / = 10 o A
( ()
m ½ q /
( ~ i +½ + . ) 2 ) is related to massive boson mass -2 o ( 2 A
- A 2
)
(real); / o massive scalar o - q
and SSB around 1
) and around the vacuum ½ p >= >= D + ½( + interaction terms = < term for gauge field! mass
→ L
The mass of the gauge field comes from gauging ( Note that Higgs mass (~ depends on the potential But there is still the unseen massless scalar (Goldstone boson) SSB: Ex Higgs Mechanism
Make use of the remaining gauge freedom in A to cancel 2!
A A + (i/qo) 2
* 2 2 2 L = ½ () - + ½(q /) A A – (1/16)F F + iiinteraction terms (11)
Higgs mechanism: SSB of gauge symmetry (SU(2))→ massive vector boson A. Goldstone boson got ‘eaten’ by the third polarization of A.
ElkhHihiklectroweak theory: Higgs mechanism to make W, Z heavy
A Higgs particle: (scalar particle(s), composite, …), mass related to those of W, Z, but depends on details of potential.
Higgs condensate fills the vacuum: <>= o 37 Higgs mechanism for superconductivity: Ginzb urg-LdLandau MMdlodel (x) = macroscopic complex wavefunction of Cooper pairs (Landau: superfluid of electrons) = | (x)|exp[i (x)] Density of pairs: | (x)|2 Cooper pairs are charged coupled to external EM fields Hamiltonian density: H = (1/4m)[D* D] + V() D + i2eA minimal coupling; q = 2e for a Cooper pair V() = a| (x)|2 + b| (x)|4
Taylor expansion around = 0; a, b depend on T
Local U(1) symmetry (x) 38 GGginzburg-Landau Model V Normal state, a > 0 for T > Tc: <> = 0 T > Tc massless A, massive 0 a < 0 for T < Tc: minima of V shifted to 2 |o| = -a/b,arbitrary phase T < Tc SSB: ground state of system picks a Goldstone mode: massless particular phase and breaks U(1): <> = (Cooper pair condensate) Higgs mode: o massive 2 2 2 (1/4m)[D* D] = (1/4m)[4e o A ] 1/2 A acquires an effective mass eo/m Meissner effect degenerate massive A (short range), ground states Illustration taken from: massive ()(gap) http://www.scholarpedia.org/article/Englert-Brout-Higgs- Guralnik-Hagen-Kibble_mechanism_(history) 39 SSB and mass - Ground state is the vacuum of a system - must identify the correct vacuum before finding the excitations and their masses! Eg. effective mass of e- + depends + + on the lattice arrangement. e- May even have didiirectional + dependence! Eg. this is g. s. (vacuum) not this Zero external field T = 0
0 = 0
~ o SSB of gauge fields: effective mass for gauge fields 2 →D , ~ o (1/8)mA A A Make use of gauge freedom of A to get rid of Goldstone bosons. Only massive boson (Higgs) and massive gauge fields remain.
41 Summary • Spontaneous Symmetry Breaking - Ground state of a system may not show the symmetries of the Lagrangian (eg. < > 0) - Need to expand around the true vacuum to identify the mass term - SSB of conti nuous symmetry → massless scalar (G(Gldoldstone bb)oson) • Higgs mechanism
- Assume existence of a scalar field with some continuous gauge syyymmetry → couppgling to g ggfauge field - SSB of gauge symmetry → massive gauge field - Use gauge freed om to eli mi nat e Gold st one boson - Electroweak: massive W, Z
42