The Standard Model Lagrangian
Elementary Particle PhysicsStrong Interaction Fenomenology
Diego BettoniAcademic Year 2011-12
D. Bettoni Fenomenologia Interazioni Forti 2
Dirac Formalism
0 mi
j Conserved Current
0110
00 0
i
ii
10
015
L
R
mi L
D. Bettoni Fenomenologia Interazioni Forti 3
Gauge Invariance
DD
D
U
igA
U 11
UUAUU
giA
Wig 22
D
kjijkii Wg
W 2
1
Abelian
Non Abelian
Gauge invariance requires the introduction of vector bosons, which act asquanta of new interactions.
In gauge theories the symmetries prescribe the interactions.
D. Bettoni Fenomenologia Interazioni Forti 4
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 boson, 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.
• SU(3) invariance. It requires eight vector bosons, Ga, the gluons, the
quanta of the strong interaction, described by Quantum ChromoDynamics (QCD).
302121 22 WWiWWWiWWW
D. Bettoni Fenomenologia Interazioni Forti 5
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:
kinetic energies of the gauge fieldscovariant derivative fermion kinetic energies
• Next we must specify the particles and their transformation properties under the three internal symmetries.
• Notation
gaugeL
fermL
D. Bettoni Fenomenologia Interazioni Forti 6
eLLeRR PePe
L
e
eL
Re
Leptons
• Rotations in electroweak SU(2) turn L electrons in L neutrinos and vv. • Ordinary spin: raising and lowering operators (vectors).• Strong isospin: pions (vectors).• Weak isospin: 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.
Left-handed and right-handed particles behave differently under electroweak SU(2) transformations: the electrons R are SU(2) singlets, whereas the electrons L are put in doublets together with the L neutrinos.
SU(2) singlet SU(2) doublet
D. Bettoni Fenomenologia Interazioni Forti 7
LL d
uQ
RR ud ,
Quarks
• 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 8
• In the Standard Model there are no R neutrinos:– Experimentally ony L are observed.– Neutrino 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 fermions 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.
• 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 gauge invariance.
• D gives a zero result when it acts on a term of different matrix form. For example is a 22 matrix in SU(2) and it gives zero acting on eR, uR ,dR.
D. Bettoni Fenomenologia Interazioni Forti 9
D
ai GigWigBYigai
222 321 D
332211 WWWW ii
RRLR duQeLf ,,,,f
ff DLferm
The Quark and Lepton Lagrangian
iiW
D. Bettoni Fenomenologia Interazioni Forti 10
LLLLRR eeee
L
RR
i
eeLeL
2
Ri
R
i
eee
LeL
Gauging the Global Symmetries
Dirac kinetic energy Lagrangian for the first generation:
Global SU(2) symmetry
Global U(1) symmetry
We put eL and L in a doublet, eR in a singlet.
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 theselines, by adding particles and symmetries and then gauging the symmetries.
D. Bettoni Fenomenologia Interazioni Forti 11
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 12
RR
RL eBYigieLBYigiLU
22
leptoni,1 11fermL
LLLL eeLL
BeeYeeYgU RRRLLLLL
2leptoni,1 1
fermL
The U(1) Terms
D. Bettoni Fenomenologia Interazioni Forti 13
002
0
02
0
02
321
2132
2ferm
222
22
2
22
2
2
2leptoni,2
WeeWeWeWg
eWWeWW
eg
eWWWW
eg
eWiWWiWWW
eg
LWigiLSU
LLLLLLLL
LL
LLLL
L
LLL
L
LLL
ii
L
The SU(2) Terms
D. Bettoni Fenomenologia Interazioni Forti 14
RRLL eeeeQA EML
LLL WgBYg
021
22
012 WYgBgA L 0
21 WgBYgZ L
221
22
012
LYgg
WYgBgA L
221
22
021
LYgg
WgBYgZ L
The Neutral Current
Electromagnetic interaction of particles of charge Q:
There are terms involving neutrinos
We assume the the electromagnetic field A is the orthogonal combination:
D. Bettoni Fenomenologia Interazioni Forti 15
BYgeeWgBYgee RRRLLL 2221021
221
22
12
LYgg
ZYgAgB L
221
22
210
LYgg
ZgAYgW L
221
22
21
221
22
22
221
221
22
2122
122
21
22
2
LL
LL
YggYYgee
YgggYgeeZ
YggYggee
YggYggeeA
LRRR
LLL
RRR
LLL
Terms involving electrons:
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 16
221
22
21
LYgg
Ygge L
221
22
21
2L
YggYgge R
21
221
22
2
ggYgg
eY
YY
LL
LR
21
22
211gg
ggeYL
We can choose YL=-1, since anychange in YL can be absorbed bya redefinition of g1.
The theory we have been writing can be interpreted to contain the usualelectromagnetic interaction, plus an additional neutral current interaction with Z for both electrons and neutrinos.
D. Bettoni Fenomenologia Interazioni Forti 17
21
22
2
21
22
1
cos
sin
ggg
ggg
W
W
W
W
eg
eg
sin
cos
2
1
23.0sin 2 W
Define:W weak mixing angle(Weinberg angle)
g1 and g2 are written in terms of the known e (e2/41/137)and the electroweak mixing angle, which needs to be measuredor calculated some other way.
D. Bettoni Fenomenologia Interazioni Forti 18
LLW
LL ZgZgg
cos22
221
22
WW
WW
WW
e
e
eegg
sincos
sincos
sincos21
22
2
21
2
2
2
221
22
-Z Coupling
W
gcos2
2 quantity to be associated to each L-Z vertex. “electroweak charge” of the left-handed neutrino.
D. Bettoni Fenomenologia Interazioni Forti 19
e-Z Coupling
21
22
21
21
22
22
21
2 gggee
ggggeeZ RRLL
WWW
WW
egg
egg
gg
2
2221
22
2
21
22
22
21
sin21
sincos
sin1
cos1
22
WWW
WW
W
e
ee
ggg
2
2
2
21
22
21
sinsincos
sincoscos
eL Coupling
eR Coupling
D. Bettoni Fenomenologia Interazioni Forti 20
Wff
WWQTe
2
3 sinsincos
T3f is the eigenvalue of T3 for any fermion f.
For a singlet (f=eR,uR,dR ecc) T3f = 0.
For the upper member of a doublet (f=L,uL ecc) T3f = +1/2.
For the lower member of a doublet (f=eL,dL ecc) T3f = -1/2.
Qf is the electric charge 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 reducedby the high value of its mass.
This expression gives the electroweak charge of any fermion, i.e. the strength of the coupling to the Z.
D. Bettoni Fenomenologia Interazioni Forti 21
The process e+e- +- (ore+e- +-) is not purelyelectromagnetic, but it has a weak component, due to theexchange of a Z0.
e
e
e
e
0Z
G G
ceinterferenddweak
ddQED
dd
dd
)(
s
2 sG2 G
The asymmetry comes from the interference term, the effect is of the order of10 % for s = 1000 GeV2.
D. Bettoni Fenomenologia Interazioni Forti 22
The Charged Current
WeWeg
LLLL22
fermL
eeLL51
21
137
2sin2
44
22
222
W
eg
The U(1) part of the Lagrangian contains only terms diagonal in the fermions, whereas the SU(2) part has also non diagonal terms.
charged current
V-A interaction
We thus expect W bosons and the associated charged currenttransitions. The observed charged currents occur with a strength muchsmaller than one would expect:
D. Bettoni Fenomenologia Interazioni Forti 23
Example of a Charge Current: decay
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.
eepn
eeud
s8.07.885
d
u e-
e
u e-
d e
g gG
W-
MW = 80.425 0.038 GeV/c2
q2 << M2W
The interaction is practically pointlike, described by a 4-fermion couplgin
2
2
WMgG
D. Bettoni Fenomenologia Interazioni Forti 24
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
universal strength:
Wff
WWQTe
2
3 sinsincos
f Q T3f
uL +2/3 +1/2dL -1/3 -1/2uR +2/3 0dR -1/3 0
D. Bettoni Fenomenologia Interazioni Forti 25
The Quark QCD Lagrangian
• 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.
8,...,13,2,1,
23
aqGqg aa
D. Bettoni Fenomenologia Interazioni Forti 26
The Second and Third Families
• 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
bt
sc
duee
D. Bettoni Fenomenologia Interazioni Forti 27
The Fermion-Gauge Boson Lagrangian
duq
aa
LLLL
duefWfRRWffLL
W
dueff
Gqqg
Wedug
ZQffQTffg
AffeQ
,
3
2
,,,
2232
,,,
2
h.c.2
sinsincos
L
D. Bettoni Fenomenologia Interazioni Forti 28
Masses
• For fermions a mass term would be of the form m.
Since L fermions are members of an SU(2) doublet, whereas the R fermions are singlets, the termsRL and LR are not SU(2)singlets and would not give and SU(2) invariant Lagrangian,
• For gauge bosons mass terms are of the form
which is clearly not invariant under gauge transformations.The solution of this problem lies in the Higgs mechanism.
RLLRmm
BBmB
2
21