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  • Introduction to the Standard Model

    Origins of the Electroweak Theory

    Gauge Theories

    The Standard Model Lagrangian

    Spontaneous Symmetry Breaking

    The Gauge Interactions

    Problems With the Standard Model

    (Structure Of The Standard Model, hep-ph/0304186)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • The Weak Interactions

    Radioactivity (Becquerel, 1896)

    decay appeared to violate energy(Meitner, Hahn; 1911)

    Neutrino hypothesis (Pauli, 1930)

    e (Reines, Cowan; 1953)

    (Lederman, Schwartz, Steinberger;1962)

    (DONUT, 2000) ( , 1975)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Fermi theory (1933)

    Loosely like QED, but zero range (non-renormalizable) and non-diagonal (charged current)

    pe

    e

    n

    J J

    e e

    e e

    J J e

    e

    e e

    e e

    W

    pe

    e

    n

    g g W +

    e e

    e e

    g g

    Typeset by FoilTEX 1

    H GFJJ

    J pn+ee [np, ee]

    J np+ee [pn, ee ( ee)]

    GF ' 1.17105 GeV2 [Fermi constant]

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Fermi theory modified to include , decay

    strangeness (Cabibbo)

    quark model

    heavy quarks (CKM)

    mass and mixing

    parity violation (V A) (Lee, Yang; Wu; Feynman-Gell-Mann)

    Fermi theory correctly describes (at tree level) Nuclear/neutron decay (npee)

    , decays (ee; , , )

    , K decays (++, 0e+e; K++, 0e+e, +0)

    hyperon decays (p; n; +e+e)

    heavy quark decays (cse+e; bc, c)

    scattering (ee; np| {z }elastic

    ; NX| {z }deepinelastic

    )

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Fermi theory violates unitarity at high energy (non-renormalizable)

    pe

    e

    n

    J J

    e e

    e e

    J J e

    e

    e e

    e e

    W

    pe

    e

    n

    g g W +

    e e

    e e

    g g

    Typeset by FoilTEX 1

    (eeee)G2F s

    , s E2CM

    pure S-wave unitarity: < 16s

    fails for ECM2

    GF 500 GeV

    Born not unitary; often restored by H.O.T.

    Fermi theory: divergent integralsd4k6 k +mek2 m2e

    6 kk2

    pe

    e

    n

    J J

    e e

    e e

    J J e

    e

    e e

    e e

    W

    pe

    e

    n

    g g W +

    e e

    e e

    g g

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Intermediate vector boson theory (Yukawa, 1935; Schwinger, 1957)

    pe

    e

    n

    J J

    e e

    e e

    J J e

    e

    e e

    e e

    W

    pe

    e

    n

    g g W +

    e e

    e e

    g g

    Typeset by FoilTEX 1

    pe

    e

    n

    J J

    e e

    e e

    J J e

    e

    e e

    e e

    W

    pe

    e

    n

    g g W +

    e e

    e e

    g g

    Typeset by FoilTEX 1

    GF2

    g2

    8M2Wfor MW Q

    no longer pure S-wave

    eeee better behaved

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • e

    W W +

    e e+

    g g W 0

    W W +

    e e+

    g

    g

    Z

    d s

    d sK0

    K0

    Typeset by FoilTEX 2

    but, e+eW+W violatesunitarity for

    s > 500 GeV

    k/MW for longitudinalpolarization (non-renormalizable)

    introduce W 0 to cancel

    fixes W 0W+W and e+eW 0

    vertices

    requires[J, J

    ] J0

    (like SU(2)U(1))

    not realistic

    e

    W W +

    e e+

    g g W 0

    W W +

    e e+

    g

    g

    Z

    d s

    d sK0

    K0

    Typeset by FoilTEX 2

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Glashow model (1961) (W, Z, , but no mechanism for MW,Z)

    Weinberg-Salam (1967): Higgs mechanism MW,Z

    Renormalizable (1971) (t Hooft, )

    Flavor changing neutral currents (FCNC)

    very large K0 K0 mixing

    GIM mechanism (c quark) (1970)

    c discovered (1974)

    e

    W W +

    e e+

    g g W 0

    W W +

    e e+

    g

    g

    Z

    d s

    d sK0

    K0

    Typeset by FoilTEX 2

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Weak neutral current(1973)

    QCD (1970s)

    W,Z (1983)

    Precision tests (1989-2000)

    CKM unitarity ( 1995-)

    t quark (1995)

    mass (1998-2002)

    Measurement Fit |Omeas!Ofit|/"meas0 1 2 3

    0 1 2 3

    #$had(mZ)#$(5) 0.02758 0.00035 0.02767

    mZ [GeV]mZ [GeV] 91.1875 0.0021 91.1874%Z [GeV]%Z [GeV] 2.4952 0.0023 2.4959"had [nb]"

    0 41.540 0.037 41.478RlRl 20.767 0.025 20.742AfbA

    0,l 0.01714 0.00095 0.01643Al(P&)Al(P&) 0.1465 0.0032 0.1480RbRb 0.21629 0.00066 0.21579RcRc 0.1721 0.0030 0.1723AfbA

    0,b 0.0992 0.0016 0.1038AfbA

    0,c 0.0707 0.0035 0.0742AbAb 0.923 0.020 0.935AcAc 0.670 0.027 0.668Al(SLD)Al(SLD) 0.1513 0.0021 0.1480sin2'effsin

    2'lept(Qfb) 0.2324 0.0012 0.2314mW [GeV]mW [GeV] 80.410 0.032 80.377%W [GeV]%W [GeV] 2.123 0.067 2.092mt [GeV]mt [GeV] 172.7 2.9 173.3

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Gauge Theories

    Standard Model is remarkably successful gauge theory of themicroscopic interactions

    Gauge symmetry (apparently) massless spin-1 (vector, gauge) bosons

    Interactions group, representations, gauge coupling

    Like QED (U(1)), but gauge self interactions for non-abelian

    Application to strong (short range) confinement

    Application to weak (short range) spontaneous symmetry breaking(Higgs or dynamical)

    Unique renormalizable field theory for spin-1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • QED

    Free electron equation,(i

    xm

    ) = 0,

    is invariant under U(1) (phase) transformations,(i

    xm

    ) = 0, where ei

    Not invariant under local (gauge) transf.,

    ei(x), x (~x, t)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Introduce vector field A ( ~A, ):(i

    x+eA m

    ) = 0,

    (e > 0 is gauge coupling) is invariant under

    ei(x), AA 1

    e

    x

    Quantization of A massless gaugeboson

    Gauge invariance , long rangeforce, prescribed (up to e) amplitudefor emission/absorption

    e p

    e p

    e e

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Non-Abelian

    n non-interacting fermions of same mass m:(i

    xm

    )a = 0, a = 1 n,

    invariant under (global) SU(n) group, 1...n

    exp(i Ni=1

    iLi)

    1...n

    .Li are nn generator matrices (N = n21); i are real parameters

    [Li, Lj] = icijkLk

    (cijk are structure constants)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Gauge (local) transformation: ii(x)(i

    xabg

    Ni=1

    AiLiab mab

    )b = 0

    Invariant under

    1...n

    U~A ~L ~A ~L U ~A ~LU

    1 +i

    g(U)U1

    U ei~~L

    (1)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Gauge invariance implies:

    N (apparently) massless gaugebosons Ai

    Specified interactions (up to gaugecoupling g, group, representations),including self interactions

    Ai

    b

    a

    igLiab

    Typeset by FoilTEX 1

    g g2

    Typeset by FoilTEX 1

    Generalize to other groups, representations, chiral (L 6= R)

    Chiral Projections: L(R) 12(1 5)(Chirality = helicity up to O(m/E))

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • The Standard Model

    Gauge group SU(3)SU(2)U(1); gauge couplings gs, g, g(ud

    )L

    (ud

    )L

    (ud

    )L

    (ee

    )L

    uR uR uR eR(?)dR dR dR e

    R

    ( L = left-handed, R = right-handed)

    SU(3): u u u, d d d (8 gluons)

    SU(2): uL dL, eL eL (W); phases (W 0)

    U(1): phases (B)

    Heavy families (c, s, , ), (t, b, , )

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Quantum Chromodynamics (QCD)

    LSU(3) = 1

    4F iF

    i +r

    qri 6D qr

    F 2 term leads to three and four-point gluon self-interactions.

    F i = Gi G

    i gsfijk G

    j G

    k

    is field strength tensor for the gluon fields Gi, i = 1, , 8.gs = QCD gauge coupling constant. No gluon masses.

    Structure constants fijk (i, j, k = 1, , 8), defined by

    [i, j] = 2ifijkk

    where i are the Gell-Mann matrices.

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • i = i 00 0

    , i = 1, 2, 3

    4 =

    0@ 0 0 10 0 01 0 0

    1A 5 =0@ 0 0 i0 0 0i 0 0

    1A6 =

    0@ 0 0 00 0 10 1 0

    1A 7 =0@ 0 0 00 0 i

    0 i 0

    1A8 = 1

    3

    0@ 1 0 00 1 00 0 2

    1A

    The SU(3) (Gell-Mann) matrices.

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Quark interactions given by qri 6D qrqr = rth quark flavor; , = 1, 2, 3 are color indices

    Gauge covariant derivative

    D = (D) = + igs Gi L

    i,

    for triplet representation matrices Li = i/2.

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Quark color interactions:

    Diagonal in flavor

    Off diagonal in color

    Purely vector (parity conserving)

    Gi

    u

    u

    igs2 i

    Typeset by FoilTEX 1

    Bare quark mass allowed by QCD, but forbidden by chiral symmetryof LSU(2)U(1) (generated by spontaneous symmetry breaking)

    Additional ghost and gauge-fixing terms

    Can add (unwanted) CP-violating term

    L = g2s

    322FiF

    i, F i 12F i

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • QCD now very well established

    Short distance behavior (asymptotic freedom)

    Confinement, light hadron spectrum (lattice)

    gs = O(1) (s(MZ) = g2s/4 0.12) Strength + gluon self-interactions confinement Yukawa model dipole-dipole

    Approximate global SU(3)LSU(3)R symmetry and breaking(,K, are pseudo-goldstone bosons)

    Unique field theory of strong interactions

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Quasi-Chiral Exotics

    (J. Kang, PL, B. Nelson, in progress)

    Exotic fermions (anomaly-cancellation)

    Examples in 27-plet of E6 DL + DR (SU(2) singlets, chiral wrt U(1))

    (E0

    E

    )

    L

    +

    (E0

    E

    )

    R

    (SU(2) doublets, chiral wrt U(1))

    Pair produce D + D by QCD processes (smaller rate for exotic leptons)

    Lightest may decay by mixing; by diquark or leptoquark coupling;or be quasi-stable

    22nd Henry Primakoff Lecture Paul Langacker (3/1/2006)

    Quantum Chromodynamics (QCD)

    Modern theory of the strong interactions

    NYS APS (October 15, 2004) Paul Langacker (Penn)

    9. Quantum chromodynamics 7

    0.1 0.12 0.14

    Average

    Hadronic Jets

    Polarized DIS

    Deep Inelastic Scattering (DIS)

    ! decays

    Z width

    Fragmentation

    Spectroscopy (Lattice)

    ep event shapes

    Photo-production

    " decay

    e+e

    - rates

    #s(M

    Z)

    Figure 9.1: Summary of the value of s(MZ) from various processes. The valuesshown indicate the process and the measured value of s extrapolated to = MZ .The error shown is the total error including theoretical uncertainties. The averagequoted in this report which comes from these measurements is also shown. See textfor discussion of errors.

    theoretical estimates. If the nonperturbative terms are omitted from the fit, the extractedvalue of s(m ) decreases by 0.02.

    For s(m ) = 0.35 the perturbative series for R is R 3.058(1+0.112+0.064+0.036).The size (estimated error) of the nonperturbative term is 20% (7%) of the size of theorder 3s term. The perturbation series is not very well convergent; if the order 3s termis omitted, the extracted value of s(m ) increases by 0.05. The order 4s term has beenestimated [47] and attempts made to resum the entire series [48,49]. These estimates canbe used to obtain an estimate of the errors due to these unknown terms [50,51]. Anotherapproach to estimating this 4s term gives a contribution that is slightly larger than the3s term [52].

    R can be extracted from the semi-leptonic branching ratio from the relationR = 1/(B( e) 1.97256); where B( e) is measured directly or extractedfrom the lifetime, the muon mass, and the muon lifetime assuming universality of lepton

    December 20, 2005 11:23

    18 9. Quantum chromodynamics

    0

    0.1

    0.2

    0.3

    1 10 102

    GeV!

    s(

    )

    Figure 9.2: Summary of the values of s() at the values of where they aremeasured. The lines show the central values and the 1 limits of our average.The figure clearly shows the decrease in s() with increasing . The data are,in increasing order of , width, decays, deep inelastic scattering, e+e eventshapes at 22 GeV from the JADE data, shapes at TRISTAN at 58 GeV, Z width,and e+e event shapes at 135 and 189 GeV.

    The value of s at any scale corresponding to our average can be obtainedfrom http://www-theory.lbl.gov/ianh/alpha/alpha.html which uses Eq. (9.5) tointerpolate.

    References:1. R.K. Ellis et al., QCD and Collider Physics (Cambridge 1996).2. For reviews see, for example, A.S. Kronfeld and P.B. Mackenzie, Ann. Rev. Nucl.

    and Part. Sci. 43, 793 (1993);H. Wittig, Int. J. Mod. Phys. A12, 4477 (1997).

    3. For example see, P. Gambino, International Conference on Lepton PhotonInteractions, Fermilab, USA, (2003); J. Butterworth International Conference onLepton Photon Interactions, Upsala, Sweden, (2005).

    December 20, 2005 11:23

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • The Electroweak Sector

    LSU(2)U(1) = Lgauge + L + Lf + LYukawa

    Gauge part

    Lgauge = 1

    4F iF

    i 1

    4BB

    Field strength tensors

    B = B BF i = W

    i W

    i gijkW

    jW

    k , i = 1 3

    g(g) is SU(2) (U(1)) gauge coupling; ijk is totally antisymmetric symbol

    Three and four-point self-interactions for the Wi

    B and W3 will mix to form , Z

    g g2

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • U(1): j exp(igyj)j, yj = qj t3j = weak hypercharge

    Scalar part

    L = (D)D V ()

    where =+

    0

    is the (complex) Higgs doublet with y = 1/2.

    Gauge covariant derivative:

    D =

    ( + ig

    i

    2W i +

    ig

    2B

    )

    where i are the Pauli matrices

    Three and four-point interactionsbetween gauge and scalar fields g g2

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Higgs potential

    V () = +2+ ()2

    Allowed by renormalizability and gaugeinvariance

    Spontaneous symmetry breaking for 2 < 0

    Vacuum stability: > 0.

    Quartic self-interactions+

    0

    0

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Fermion part

    LF =Fm=1

    (q0mLi 6Dq

    0mL + l

    0mLi 6Dl

    0mL

    + u0mRi 6Du0mR + d

    0mRi 6Dd

    0mR + e

    0mRi 6De

    0mR+

    0mRi 6D

    0mR

    )L-doublets

    q0mL =(u0md0m

    )L

    l0mL =(0me0m

    )L

    R-singlets

    u0mR, d0mR, e

    0mR,

    0mR(?)

    (F 3 families; m = 1 F = family index;0 = weak eigenstates (definite SU(2) rep.), mixtures of mass eigenstates (flavors);quark color indices = r, g, b suppressed (e.g., u0mL). )

    Can add gauge singlet 0mR for Dirac neutrino mass term

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Different (chiral) L and R representations lead to parity and chargeconjugation violation (maximal for SU(2))

    Fermion mass terms forbidden by chiral symmetry

    Triangle anomalies absent for chosen hypercharges (quark-leptoncancellations)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Gauge covariant derivatives

    Dq0mL =

    ( +

    ig

    2 iW i + i

    g

    6B

    )q0mL

    Dl0mL =

    ( +

    ig

    2 iW i i

    g

    2B

    )l0mL

    Du0mR =

    ( + i

    2

    3gB

    )u0mR

    Dd0mR =

    ( i

    g

    3B

    )d0mR

    De0mR = ( ig

    B) e0mR

    Read off W and Bcouplings to fermions W i

    ig2i

    15

    2

    Bigy

    15

    2

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Yukawa couplings (couple L to R)

    LYukawa =F

    m,n=1

    [umnq

    0mLu

    0mR +

    dmnq

    0mLd

    0nR

    + emnl0mne

    0nR (+

    mnl

    0mL

    0mR)

    ]+ H.C.

    mn are completely arbitrary Yukawa matrices, which determinefermion masses and mixings

    d, e terms require doublet =(+

    0

    )with Y = 1/2

    u (and ) terms require doublet

    =(

    0

    )with Y = 1/2

    nR

    mL

    mn

    Typeset by FoilTEX 1

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • In SU(2) the 2 and 2 are similar i 2 =(

    0

    )transforms as a 2 with Y = 12 only one doublet needed.

    Does not generalize to SU(3), most extra U(1), supersymmetry,SO(10) etc need two doublets.(Does generalize to SU(2)LSU(2)RU(1) and SU(5))

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Spontaneous Symmetry Breaking

    Gauge invariance implies massless gauge bosons and fermions

    Weak interactions short ranged spontaneous symmetry breakingfor mass; also for fermions

    Color confinement for QCD gluons remain massless

    Allow classical (ground state) expectation value for Higgs field

    v = 0||0 = constant

    v 6= 0 increases energy, but important for monopoles, strings,domain walls, phase transitions (e.g., EWPT, baryogenesis)

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Minimize V (v) to find v and quantize = v

    SU(2)U(1): introduce Hermitian basis

    =(+

    0

    )=

    (12(1 i2)

    12(3 i4

    ),

    where i = i .

    V () =1

    22

    (4i=1

    2i

    )+

    1

    4

    (4i=1

    2i

    )2

    is O(4) invariant.

    w.l.o.g. choose 0|i|0 = 0, i = 1, 2, 4 and 0|3|0 =

    V ()V (v) =1

    222 +

    1

    44

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • For 2 < 0, minimum at

    V () = (2 + 2) = 0

    =(2/

    )1/2SSB for 2 = 0 also; must

    consider loop corrections

    12

    (0

    ) v the generators L1, L2, and L3 Y

    spontaneously broken, L1v 6= 0, etc (Li = i2 , Y = 12I)

    Qv = (L3 + Y )v =(

    1 00 0

    )v = 0 U(1)Q unbroken

    SU(2)U(1)YU(1)Q

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

  • Quantize around classical vacuum

    Kibble transformation: introduce new variables i for rollingmodes

    =1

    2eiPiLi

    (0

    +H

    ) H = H is the Higgs scalar No potential for i massless Goldstone bosons for global

    symmetry

    Disappear from spectrum for gauge theory (eaten) Display particle content in unitary gauge

    = eiPiLi =

    1

    2

    (0

    +H

    )+ corresponding transformation on gauge fields

    PiTP 2007 (July 16, 2007) Paul Langacker (IAS)