Spontaneous Symmetry Breaking gauge symmetry
Consider a U(1) gauge invariant Lagrangian for a complex scalar field f(x): • 1 = F Fµn +(D f)†(Dµf) µ2f†f l(f†f)2 , D = ∂ + ieqA L 4 µn µ µ µ µ
iqq(x) 1 inv. under f(x) f (x)=e f(x) , Aµ(x) A (x)=Aµ(x)+ ∂µq(x) 7! 0 7! µ0 e If l > 0, µ2 < 0, the in terms of quantum fields h and c with null VEVs: L 1 f(x) [v + h(x)+ic(x)] , µ2 = lv2 Comments: ⌘ p2 1 1 1 p = F Fµn + (∂ h)(∂µh)+ (∂ c)(∂µc) (i) mh = 2l v L 4 µn 2 µ 2 µ mc = 0 2 2 2 2 l 2 2 2 1 4 lv h lvh(h + c ) (h + c ) + lv (ii) M = eqv (!) 4 4 A | | + eqvA ∂µc + eqA (h∂µc c∂µh) (iii) Term A ∂µc (?) µ µ µ 1 1 (iv) Add GF + (eqv)2 A Aµ + (eq)2 A Aµ(h2 + 2vh + c2) L 2 µ 2 µ
1. Gauge Theories 24 Spontaneous Symmetry Breaking gauge symmetry
Removing the cross term and the (new) gauge fixing Lagrangian: • 1 = (∂ Aµ xM c)2 LGF 2x µ A µ µ MA[Aµ∂ c + ∂µ A c] total deriv. 1 1 1 + = F Fµn + M2 A Aµ (∂ Aµ)2 + M ∂ (Aµc) )LLGF 4 µn 2 A µ 2x µ A µ 1 1 z }| { + (∂ c)(∂µc) xM2 c2 + ... 2 µ 2 A
and the propagators of Aµ and c are:
i kµkn Dµn(k)= gµn +(1 x) k2 M2 + ie k2 xM2 A " A # e i D(k)= k2 xM2 A c has a gauge-dependente mass: actually it is not a physical field! ) 1. Gauge Theories 25 Spontaneous Symmetry Breaking gauge symmetry
A more transparent parameterization of the quantum field f is • 1 f(x) eiqz(x)/v [v + h(x)] , 0 h 0 = 0 z 0 = 0 ⌘ p2 h | | i h | | i
iqz(x)/v 1 f(x) e f(x)= [v + h(x)] z gauged away! 7! p2 )
1 1 Comments: = F Fµn + (∂ h)(∂µh) L 4 µn 2 µ (i) m = p2l v l 1 h lv2h2 lvh3 h4 + lv4 4 4 (ii) MA = eqv | | 1 2 µ 1 2 µ 2 + (eqv) A A + (eq) A A (2vh + h ) (iii) No need for GF 2 µ 2 µ L
This is the unitary gauge (x •): just physical fields ) ! i kµkn Dµn(k) gµn + and D(k) 0 ! k2 M2 + ie M2 ! A " A # e e 1. Gauge Theories 26 Spontaneous Symmetry Breaking gauge symmetry
Brout-Englert-Higgs mechanism: [Anderson ’62] ) [Higgs ’64; Englert, Brout ’64; Guralnik, Hagen, Kibble ’64] The gauge bosons associated with the spontaneously broken generators become massive, the corresponding would-be Goldstone bosons are unphysical and can be absorbed, the remaining massive scalars (Higgs bosons) are physical (the smoking gun!)
– The would-be Goldstone bosons are ‘eaten up’ by the gauge bosons (‘get fat’) and disappear (gauge away) in the unitary gauge (x •) ! Degrees of freedom are preserved ) Before SSB: 2 (massless gauge boson) + 1 (Goldstone boson) After SSB: 3 (massive gauge boson) + 0 (absorbed would-be Goldstone)
– For loops calculations, ’t Hooft-Feynman gauge (x = 1) is more convenient: Gauge boson propagators are simpler, but ) Goldstone bosons must be included in internal lines )
1. Gauge Theories 27 Spontaneous Symmetry Breaking gauge symmetry
Comments: •
– After SSB the FP ghost fields (unphysical) acquire a gauge-dependent mass, due to interactions with the scalar field(s): id D (k)= ab ab k2 x M2 + ie a Wa e
– Gauge theories with SSB are renormalizable [’t Hooft, Veltman ’72]
UV divergences appearing at loop level can be removed by renormalization of parameters and fields of the classical Lagrangian predictive! )
1. Gauge Theories 28 2. The Standard Model
29 Gauge group and particle representations [Glashow ’61; Weinberg ’67; Salam ’68] [D. Gross, F. Wilczek; D. Politzer ’73]
The Standard Model is a gauge theory based on the local symmetry group: • SU(3) SU(2) U(1) SU(3) U(1) c ⌦ L ⌦ Y ! c ⌦ Q strong electroweak em
with the electroweak| {z symmetry} | spontaneously{z } broken to| the{z electromagnetic}
U(1)Q symmetry by the Brout-Englert-Higgs mechanism
The particle (field) content: (ingredients: 12 flavors + 12 gauge bosons + H) • Fermions I II III Q Bosons
1 2 spin 2 Quarks f uuu ccc ttt 3 spin 1 8 gluons strong interaction f ddd sss bbb 1 W ,Z weak interaction 0 3 ± Leptons f ne nµ nt 0 g em interaction f e µ t 1 spin 0 Higgs origin of mass 0
Q f = Q f 0 + 1
2. The Standard Model 30 Gauge group and particle representations
The fields lay in the following representations (color, weak isospin, hypercharge): • Multiplets SU(3) SU(2) U(1) I II III Q = T + Y c ⌦ L ⌦ Y 3 2 1 1 1 uL cL tL 3 = 2 + 6 Quarks (3, 2, 6 ) 0d 1 0s 1 0b 1 1 = 1 + 1 L L L 3 2 6 @ A @ A @ A ( , , 2 ) 2 2 3 1 3 uR cR tR 3 = 0 + 3 1 1 1 (3, 1, ) dR sR bR = 0 3 3 3 1 1 1 neL nµL ntL 0 = 2 2 Leptons (1, 2, 2 ) 0 e 1 0 µ 1 0 t 1 1 = 1 1 L L L 2 2 @ A @ A @ A (1, 1, 1) eR µR tR 1 = 0 1
(1, 1, 0) neR nµR ntR 0 = 0 + 0 1 Higgs (1, 2, 2 ) (3 families of quarks & leptons) From now on just the electroweak part (EWSM): SU(2) U(1) ) L⌦ Y 2. The Standard Model 31 The EWSM with one family (of quarks or leptons)
Consider two massless fermion fields f (x) and f (x) with electric charges • 0 Q = Q + 1 in three irreps of SU(2) U(1) : f f 0 L⌦ Y
0 1 1 = i f ∂/f + i f 0∂/f 0 f = (1 g ) f , f 0 = (1 g ) f 0 LF R,L 2 ± 5 R,L 2 ± 5
fL = iY ∂/Y + iy ∂y/ + iy ∂y/ ; Y = , y = f , y = f 0 1 1 2 2 3 3 1 0 1 2 R 3 R fL0 (1, y ) (1, y ) @ A 2 3 (2, y1) |{z} |{z} | {z } To get a Langrangian invariant under gauge transformations: • iy b(x) iT ai(x) si Y (x) U (x)e 1 Y (x), U (x)=e i , T = (weak isospin gen.) 1 7! L 1 L i 2 iy b(x) y (x) e 2 y (x) 2 7! 2 iy b(x) y (x) e 3 y (x) 3 7! 3
2. The Standard Model 32 The EWSM with one family gauge invariance
i Introduce gauge fields W (x) (i = 1, 2, 3) and B (x) through covariant derivatives: ) µ µ s D Y =(∂ igW + ig y B )Y , W i Wi µ 1 µ µ 0 1 µ 1 µ ⌘ 2 µ 9 Dµy2 =(∂µ + ig0y2Bµ)y2 F e e > ) L => Dµy3 =(∂µ + ig0y3Bµ)y3 > where two couplings g and g0 have been introduced and ;>
† i † Wµ(x) U (x)Wµ(x)U (x) (∂µU (x))U (x) 7! L L g L L 1 eBµ(x) Bµ(x)+e ∂µb(x) 7! g0 Add Yang-Mills: gauge invariant kinetic terms for the gauge fields ) 1 1 j = Wi Wi,µn B Bµn , Wi = ∂ Wi ∂ Wi + ge W Wk LYM 4 µn 4 µn µn µ n n µ ijk µ n (include self-interactions of the SU(2) gauge fields) and B = ∂ B ∂ B µn µ n n µ 2. The Standard Model 33 The EWSM with one family mass terms forbidden
Note that mass terms are not invariant under SU(2) U(1) , since LH and RH ) L⌦ Y components do not transform the same:
m ff = m( fL fR + fR fL)
Mass terms for the gauge bosons are not allowed either )
Next the different types of interactions are analyzed )
2. The Standard Model 34 The EWSM with one family charged current interactions
3 p † µ 1 Wµ 2Wµ F gY1g WµY1 , Wµ = • L 2 0p2W W3 1 µ µ e e @ A charged current interactions of LH fermions with complex vector boson field W : ) µ g µ † 1 1 2 CC = f g (1 g5) f 0Wµ + h.c. , Wµ (Wµ + iWµ) L 2p2 ⌘ p2
` d W W ⌫ u
⌫ u
W W
` d
2. The Standard Model 35 The EWSM with one family neutral current interactions
The diagonal part of • µ µ µ µ gY g W Y g0B (y Y g Y + y y g y + y y g y ) LF 1 µ 1 µ 1 1 1 2 2 2 3 3 3 neutral current interactions with neutral vector boson fields W3 and B ) e µ µ We would like to identify Bµ with the photon field Aµ but that requires:
y = y = y and g0y = eQ impossible! 1 2 3 j j ) Since they are both neutral, try a combination: ) 3 W cW sW Zµ s sin q , c cos q µ W ⌘ W W ⌘ W 0 1 0 1 0 1 Bµ ⌘ sW cW Aµ qW = weak mixing angle @ A @ A @ A 3 µ NC = Â yjg gT3sW + g0yjcW Aµ + gT3cW g0yjsW Zµ yj L j=1 ⇥ ⇤ ⇥ ⇤ s with T = 3 (0) the third weak isospin component of the doublet (singlet) 3 2 2. The Standard Model 36 The EWSM with one family neutral current interactions
To make A the photon field: • µ
(1) e = gsW = g0cW (2) Q = T3 + Y
Q f 0 where the electric charge operator is: Q = , Q = Q , Q = Q 1 0 1 2 f 3 f 0 0 Q f 0 @ A (1) Electroweak unification: g of SU(2) and g of U(1) are related ) 0 (2) The hyperchages are fixed in terms of electric charges and weak isospin: ) 1 1 y = Q = Q + , y = Q , y = Q 1 f 2 f 0 2 2 f 3 f 0
µ = eQ f g fA +(f f 0) LQED f µ !
RH neutrinos are sterile: y = Q = 0 ) 2 f 2. The Standard Model 37 The EWSM with one family neutral current interactions
The Z is the neutral weak boson field: • µ Z µ = e f g (v a g ) fZ +(f f 0) LNC f f 5 µ ! with fL 2 fL T3 2Q f sW T3 v f = , a f = 2sW cW 2sW cW
The complete neutral current Lagrangian reads: • = + Z LNC LQED LNC
f f