The Lagrangian

Elementary Strong Interaction Fenomenology

Diego Bettoni Academic Year 2011-12 Dirac Formalism

  i    m  0

 j     Conserved Current

 0  0 1 1 0 i  i  0   5                i 0  1 0 0 1        R     L  i    m  L 

D. Bettoni Fenomenologia Interazioni Forti 2 Gauge Invariance

    U   i 1  1     igA A    U U UA U Abelian D g D   U D  

      1    W      W  D    ig2 W i i ijk j k Non 2 g 2 Abelian

Gauge invariance requires the introduction of vector , which act as quanta of new interactions. In gauge theories the symmetries prescribe the interactions. D. Bettoni Fenomenologia Interazioni Forti 3 The Symmetries of the Standard Model

• U(1) invariance. All particles appear to have this kind of invariance, related to electromagnetism. It requires a vector , B, whose connection with the photon will be determined later. • SU(2) invariance. Non abelian gauge invariance (electroweak  isospin). It requires three vector bosons, Wi , one for each generator of SU(2). The physical W particles have definite electromagnetic charges.

W   W1  iW2  2 W   W1  iW2  2 W 0  W 3

 • SU(3) invariance. It requires eight vector bosons, Ga , the gluons, the quanta of the strong interaction, described by (QCD).

D. Bettoni Fenomenologia Interazioni Forti 4 The Lagrangian

• In order to obtain the Standard Model Lagrangian we start from the free particle Lagrangian and replace the ordinary derivative by the convariant derivative. It will contain two parts: Lgauge kinetic energies of the gauge fields Lferm covariant derivative  kinetic energies • Next we must specify the particles and their transformation properties under the three internal symmetries. • Notation

D. Bettoni Fenomenologia Interazioni Forti 5 Leptons 

e   P e   P  R R e L L e Left-handed and right-handed particles behave differently under electroweak SU(2) transformations: the R are SU(2) singlets, whereas the electrons L are put in doublets together with the L .

  e  eR SU(2) singlet L     SU(2) doublet e L • Rotations in electroweak SU(2) turn L electrons in L neutrinos and vv. • Ordinary spin: raising and lowering operators (vectors). • Strong isospin: pions (vectors). • : the W bosons connect the members of an electroweak doublet

• eR is not connected to any other state by electroweak transitions. - • p,q,r=1,2 e.g.: Lp L1=eL, L2=e L. D. Bettoni Fenomenologia Interazioni Forti 6 Quarks

 u  QL    d R ,uR d L • the index  describes how the quark transforms under color SU(3). • The basic representation is a triplet: ,, = 1,2,3 or r,g,b. • Color (e.g. r) and anticolor (e.g. r ). Singlet (rr+ gg+ bb) • All leptons are color singlets. • All quarks are color triplets. • The gluons generate the transitions from one color to another: they are the quanta of the strong interaction, but unlike photons they carry color charge. • There are eight “bi-colored” gluons (e.g. bg): octet representation of color SU(3).

D. Bettoni Fenomenologia Interazioni Forti 7 • In the Standard Model there are no R neutrinos:

– Experimentally ony L are observed. – masses are very small (but non-zero, oscillations).

– If right-handed neutrinos R exist either they are very heavy or they interact very weakly. • R and L were put in different electroweak SU(2) multiplets: that implies a violation of parity, since clearly the theory is not invariant under the reversal of the component of spin in the direction of motion. This is the way in which parity violation is described in the standard model, but it is not explained in a fundamental sense. • The same theory can be applied to the two other fermion families: (,, c,s) and ( ,,t,b). – The universe consists of fermions from the first generation. – The other families are produced in high-energy cosmic ray collisions and in particle accelerators. – No reason has been found for the existence of three families of particles with identical quantum numbers and interactions.

D. Bettoni Fenomenologia Interazioni Forti 8 The Quark and Lepton Lagrangian            D   i a Y i  a D   ig B  ig W ig G 1 2 2 2 3 2 

• B is the spin-one field needed to maintain the U(1) gauge invariance. g1 is the coupling strength (to be measured experimentally. Y is the generator of U(1), transformations, a constant, but in principle different for the different

fermions.  • Analogous remarks describe the SU(2) and SU(3) terms. We introduce 3 and, respectively, 8 vector bosons which  are needed to maintain the local i i 1 1 2 2 3 3 gauge invariance.  W  W  W  W

•D gives a zero result when it acts on a term of different matrix form. For i i example  W is a 22 matrix in SU(2) and it gives zero acting on eR, uR ,dR.  Lferm   f D f f  L,e R,Q L,u R ,dR f

D. Bettoni Fenomenologia Interazioni Forti 9 Gauging the Global Symmetries   Dirac kinetic energy Lagrangian for the first generation:      L  eR  eR  eL   eL  L  L

We put eL and L in a doublet, eR in a singlet.   L  ei  2 L Global SU(2) symmetry eR  eR L  ei L Global U(1) symmetry i  eR  e eR

 . We make these symmetries locals by introducing potentials Wi and B and by replacing  with the covariant derivative D . Thus we obtain the same result. Some attempts to extend the Standard Model proceed along these lines, by adding particles and symmetries and then gauging the symmetries.

D. Bettoni Fenomenologia Interazioni Forti 10 The Electroweak Lagrangian

• Since the term  is always present we will not write it. • All the calculations in SU(2) will be done only for the leptons. • Since the color labels of the quarks do not operate in the U(1) or SU(2) space, quarks will behave the same way as leptons for U(1) and SU(2) interactions.

D. Bettoni Fenomenologia Interazioni Forti 11 The U(1) Terms

  

 YL   YR   Lferm U 1,leptoni Li ig1 B L eRi ig1 B eR   2   2     L L     e   e L  L L  L

  g L U1 ,leptoni  1 Y  e e Y e  e B ferm 2 L L L L L R R R 

D. Bettoni Fenomenologia Interazioni Forti 12   

 The SU(2) Terms     i      i   SU2 ,leptoni  Li ig W L   Lferm  2     2  3 1 2     g2 W W  iW  L        L eL 1 2   3      2 W  iW W  eL      0      g2 W   2W  L     L eL        0   2    2W W  eL     g  W 0  2W e    2 e  L  L  L L   0   2  2W L W eL     g    2  W 0  2 e W   2e W   e  e W 0  2 L L L L L L L L 

D. Bettoni Fenomenologia Interazioni Forti 13 The

Electromagnetic interaction of particles of charge Q:   LEM  QA eL eL  eR eR   g1 g2 0    There are terms involving neutrinos  YL B  W  L  L  2 2  

  We assume the the electromagnetic field A is the orthogonal combination:   0 0 A  g2 B  g1YLW Z  g1YLB  g2W     g B  g Y W 0 g Y B  g W 0 A  2 1 L  Z  1 L 2  g 2  g 2Y 2 g 2  g 2Y 2 2 1 L 2 1 L

D. Bettoni Fenomenologia Interazioni Forti 14 

 

  g g   g  e e  1 Y B  2 W 0 e  e  1 Y B Terms involving electrons: L L  L   R R  R     2 2   2    g2 A  g1YL Z  0  g1YL A  g2Z  B  W    2 2 2 g 2  g 2Y 2 g  g Y 2 1 L 2 1 L

       g1g2YL  g1g2YR   A eL eL   eR eR     g 2  g 2Y 2  2 g 2  g 2Y 2    2 1 L   2 1 L 

  2 2 2   2   g1 YL  g2  g1 YRYL   Z eL eL    eR eR   2 g 2  g 2Y 2  2 g 2  g 2Y 2    2 1 L   2 1 L 

The term in A must be the usual electromagnetic current. The term in Z can be an additional interaction, to be checked experimentally.

D. Bettoni Fenomenologia Interazioni Forti 15 g g Y g g Y  e  1 2 L  e  1 2 R g 2  g 2Y 2 2 g 2  g 2Y 2 2 1 L 2 1 L

YR  2YL We can choose YL=-1, since any g 2  g 2Y 2 2 1 L change in YL can be absorbed by YL  e a redefinition of g . g1g2 1 g g Y  1  e  1 2 L 2 2 g2  g1 The theory we have been writing can be interpreted to contain the usual electromagnetic interaction, plus an additional neutral current interaction with Z for both electrons and neutrinos.

D. Bettoni Fenomenologia Interazioni Forti 16  g1 Define: sin W  g 2  g 2 2 1 W weak mixing angle g (Weinberg angle) cos  2 W 2 2 g2  g1

e g1   cos W e g2  sinW

2 g1 and g2 are written in terms of the known e (e /41/137) and the electroweak mixing angle, which needs to be measured or calculated some other way.

2 sin W  0.23

D. Bettoni Fenomenologia Interazioni Forti 17  -Z Coupling   

2 2 g 2  g1 g 2     Z L L   Z L  L 2 2cos W

g2 quantity to be associated to each L-Z vertex. 2cosW “electroweak charge” of the left-handed neutrino.

2 2 1 2 2 2  e e  g2  g1   2  2  cos W sin W  1 2  e2     2 2  cos W sin W  e   cos W sinW

D. Bettoni Fenomenologia Interazioni Forti 18 e-Z Coupling   

  g2  g2   g2   Z e e 1 2  e   e 1  L L  2 2  R R  2 2   2 g2  g1   g2  g1  g 2  g 2 e2  1 1  1 2      2 2 2 2  2 2  2 g2  g1 2 g2  g1  cos W sin W   e  1 2     sin W  eL Coupling cos W sin W  2   g 2 e2 cos sin 1   W W 2 2 2 g2  g1 cos W e  e  2 eR Coupling  sin W cos W sin W

D. Bettoni Fenomenologia Interazioni Forti 19   e f 2 T3  Q f sin W  cos W sin W This expression gives the electroweak charge of any fermion, i.e. the strength of the coupling to the Z.

f T3 is the eigenvalue of T3 for any fermion f. f For a singlet (f=eR,uR,dR ecc) T3 = 0. f For the upper member of a doublet (f=L,uL ecc) T3 = +1/2. f For the lower member of a doublet (f=eL,dL ecc) T3 = -1/2. Qf is the of the fermion in units of e: Qe=-1. Q=0, Qu=2/3, Qd=-1/3)

The electroweak theory contains both the electromagnetic interaction, mediated by the photon, and the weak neutral current, mediated by the Z0, which couples to any fermion with electric charge or electroweak isospin.

The strength of the Z0 interaction is not intrinsically small, but it gets reduced by the high value of its mass.

D. Bettoni Fenomenologia Interazioni Forti 20 The process e+e- +- (or e+e- +-) is not purely electromagnetic, but it has a weak component, due to the exchange of a Z0.

    e    e G G 

   0 e  e Z   d d d d  QED  (weak)  interference d d d d

2  G 2s G s The asymmetry comes from the interference term, the effect is of the order of 10 % for s = 1000 GeV2. D. Bettoni Fenomenologia Interazioni Forti 21 The Charged Current

The U(1) part of the Lagrangian contains only terms diagonal in the fermions, whereas the SU(2)part has also non diagonal terms.   g L  2  e W  e   W   charged current ferm  L L L L  2   1   e   1 5e V-A interaction L L 2 We thus expect W bosons and the associated charged current transitions. The observed charged currents occur with a strength much smaller than one would expect:

 2 g 2 e2 4  2 2   2 4 2sin W 137

D. Bettoni Fenomenologia Interazioni Forti 22 Example of a Charge Current:  decay

 n  p  e  e   885.7  0.8 s  d  u  e  e 

ue- ue- M = 80.425  0.038 GeV/c2 W G gg W- d e q2 << M2 d e W The interaction is practically pointlike, described by a 4-fermion couplgin g 2 G  2 MW As is the case with neutral currents, also for charged currents the charged current strenghts are reduced by the high value of the W mass. D. Bettoni Fenomenologia Interazioni Forti 23 The Quark Electroweak Terms

The SU(2) and spin structure of quarks and leptons are the same, consequently all the previous conclusions hold without modifications for the quarks: • They couple to the same gauge bosons W, Z0, . • Normal electromagnetic coupling to the photon.

• Charged current coupling generating transitions uL  dL, but no charged current transitions for uR e dR. • Neutral current transitions with a f fQT3 universal strength:  uL +2/3 +1/2 e f 2 d -1/3 -1/2 T  Q sin   L cos sin 3 f W W W uR +2/3 0

dR -1/3 0

D. Bettoni Fenomenologia Interazioni Forti 24 The Quark QCD Lagrangian

  g  ,   1,2,3 3  a a q  G q  2  a 1,...,8 • It contains only quarks, since leptons have no color charge. • In the electroweak case the Wi are related to states of electromagnetic charge because of the interaction with the photon. The gluons are electrically neutral, i.e. they have no interaction with the electromagnetic field. • Since the generators a are not all diagonal, the interaction with a gluon can change the color charge of a quark. • Gluons and quarks are confined within hadrons.

D. Bettoni Fenomenologia Interazioni Forti 25 The Second and Third Families   e             e           u  c  t         d    s b • All known phenomenology is consistent with the above replacements • It is unknown whether more families or additional quarks and leptons exist • All fermions of the three families have been observed experimentally. • The same set of gauge bosons (, W, Z0,g) interacts with all the families of fermions: – lepton universality – u- and d- universality

D. Bettoni Fenomenologia Interazioni Forti 26 The Fermion- Lagrangian

    L  eQ f f f A f  ,e,u,d   g   2  f f T 3 Q sin 2  f f Q sin 2 Z   L L f f W R R f W cos W f  ,e,u,d    g 2    uL d L  L eL W  h.c. 2     g  3  a  a  q  q G   2 qu,d

D. Bettoni Fenomenologia Interazioni Forti 27 Masses

• For fermions a mass term would be of the form m.

m  m R L  L R  Since L fermions are members of an SU(2) doublet, whereas the R

fermions are singlets, the termsRL and LR are not SU(2) singlets and would not give and SU(2) invariant Lagrangian, • For gauge bosons mass terms are of the form 1  m2 B B 2 B  which is clearly not invariant under gauge transformations. The solution of this problem lies in the .

D. Bettoni Fenomenologia Interazioni Forti 28