ecture III: Higgs Mechanism Spontaneous Symmetry Breaking The Higgs Mechanism Mass Generation for eptons Quark Masses & Mixing
III.1 Symmetry Breaking One example is the infinite ferromagnet the nearest neighbor interaction between spins or magnetic dipole moments is invariant under the group of spatial rotations SO(3) Above the Curie temperature, T>T C, in the absence of a magnetic field we have the paramagnetic phase, where we encounter an exact symmetry of the medium spontaneous magnetization is 0 as no preferred direction in space exists SO(3) symmetry Now we impose an external magnetic field that tends to align spins The SO(3) symmetry is broken down to SO(2) symmetry of rotations about the external field direction the full symmetry is restored when the field is turned off For temperatures T<T C, the system is in the ferromagnetic phase In the absence of an external magnetic field, the configuration of lowest energy have a non-zero spontaneous magnetization, because the nearest neighbor force favors alignment of all spins SO(3) symmetry is spontaneously broken down to SO(2), where the direction of spontaneous symmetry breaking is random
III.1 Symmetry Breaking The measurable properties of the infinite ferromagnet do not depend upon its orientation, which is due to its original SO(3) symmetry the ground state is infinitely degenerate In the presence of an external magnetic field that explicitly breaks the SO(3) symmetry, a particular direction for the the spontaneous magnetization is selected In contrast to the paramagnetic case, the spontaneous magnetization does not return to 0 when the magnetic field is turned off For rotational invariance to be broken spontaneously, it is crucial that the ferromagnet is infinite in extent Then rotations from one degenerate ground state another would require a rotation of an infinite number of dipoles (analog to statement in field theory that degenerate vacua lie in distinct Hilbert spaces)
III.2 The Higgs Mechanism In the SM 2 complex scalar fields are introduced Its self interactions provide a mechanism for spontaneous symmetry breaking (SSB) giving masses to gauge bosons & fermions It also gives rise to a new scalar particle: the H-boson We extend the agrangian by adding are and F = D μ 2 V 2 kinetic energy where 2 denotes and F fermions (see later) potential is Yukawa coupling of to (3.1) Most general renormalizable form of scalar potential V is: V = μ 2 2 + 4 (3.2)
III.2 The Higgs Mechanism Write 2 scalars as = + 0 (3.3) where + and 0 are each complex fields with QN s: + T T 3 1/2Y 1/2 1/2 1/2 1 0 1/2-1/2 1/2 0 Q V() V() In a classical theory with μ 2 <0, ground state of 2 occurs at Quantum analog is a non-vanishing expectation value of 2 in physical vacuum state Appearance of this non-vanishing vacuum expectation value selects a preferred direction in weak isospin-hypercharge space this spontaneously breaks the SU(2) U(1) Y symmetry 2 = 1 2 μ 2 μ2
III.2 The Higgs Mechanism It is convenient to introduce modulus of vacuum expectation value of v 2 = μ2 2 Conventional perturbation theory is formulated for fields with zero vaccum expectation value (VEV) So it is appropriate to separate out VEV & redefine scalar doublet (x) = exp i (x) 0 ( 2v v + H(x) ) 2 (3.4) where the real fields 1 (x), 2 (x), 3 (x) and H(x) have zero VEV By finite gauge transformations under SU(2) with eliminate explicit appearance of (x) in (x) exp i(x) (x) = 2 0 ( v + H(x) ) 2 (x) = (x) / v (3.5)
III.2 The Higgs Mechanism In this unitary gauge degrees of freedom vanish but essentially reappear as longitudinal components of W ± & Z, when these acquire masses they have been eaten by gauge fields The Goldstone theorem states that massless spin-0 particles in the theory, whenever a continuous symmetry is broken Physically, they embody zero-energy excitations that were previously described by symmetry transformations For global symmetry breaking such Goldstone bosons are inescapable but in gauge symmetry breaking by Higgs mechanism something special happens: There are 3 Goldstone bosons represented by fields They are gauged away from scalar sector to reappear in the gauge field sector as longitudinal components of W ± & Z 0 this procedure gives them masses
III.2 The Higgs Mechanism The covariant derivative on the isodoublet field in terms of A,W & Z is: D μ = μ + ieqa μ + 1 2 ig + W μ + + W μ ( ) + ig ( 1 x Q ) Z Z 2 3 W μ (3.6) So the couplings are e, g, g Z for EM, charged current & neutral current weak interactions, respectively Here we have defined the raising & lowering operators + = 2T + = 0 1 0 0 & = 2T = 0 0 1 0 (3.7) (3.8) So for example + e = 0 1 0 0 0 1 =, e + e = 0 e = 0, + e = e In the unitary gauge (x) has only one neutral component (x) = 1 2 0 v + H(x) (3.9)
III.2 The Higgs Mechanism The kinetic energy term of the scalar field becomes 1 D μ (x) = igw ( + v + H(x) ) 1 2 μ 2 μ H(x) 1 g Z v + H(x) 2 Z μ Now the scalar field agrangian becomes: ( ) (3.10) ( ) 2 + 1 4 g2 W + μ W μ ( v + H(x) ) 2 + 1 g 8 Z ( ) 2 = 1 2 μ H(x) V 1 v + H(x) 2 The v 2 terms provide W & Z boson mass terms with M W 2 W μ + W μ ( ) 2 2 Z μ Z μ v + H(x) (3.11) + 1 M 2 Z 2 Z μ Z μ (3.12) M W = 1 gv 2 (3.13) M Z = 1 g v = M cos 2 Z W W (3.14)
III.2 The Higgs Mechanism The photon remains massless The kinetic & potential energy terms in scalar field agrangian give ( 1 H(x) ) 2 1 ( ) 2 μ 2 2μ2 H 2 (x) + 1 4 μ2 v 2 1 + 4H3 (x) + H 4 (x) (3.15) v 3 v 4 This describes the physical Higgs scalar boson of mass M H = 2μ 2 H H H (3.16) H with cubic & quartic self interactions The Higgs field has no electromagnetic interactions! H H H Its interactions with other gauge fields are given by cubic & quartic terms 1 4 g2 W + μ W μ + 1 g 2 Z 8 Z μ Z μ (3.17) ( ) ( H 2 (x) + 2vH(x) ) (3.17) which are completely specified by the gauge couplings
III.2 The Higgs Mechanism We will consider physical implications of Higgs particle interactions later Note that SSB based on a VEV for + would have unacceptable physical consequences Final result of the Higgs mechanism: vacuum everywhere can emit or absorb neutral colorless quantum of Higgs field that carries weak isospin & hypercharge W ±, Z 0 & fermions that couple to H(x) acquire masses but & gluons that cannot couple to it remain massless
III.3 epton Mass Generation Spontaneous symmetry breaking will generate the -mass, if we add a Yukawa interaction between lepton & fields that is renormalizable & invariant under SU(2) U(1) Y gauge transformations Y = G ( R ) + ( ) (3.18) R where is a further coupling constant & G = and = + 0 (3.19) Substituting the unitary gauge form of (x), we get v v (3.20) Y = G G H = e,μ, 2 = e,μ, 2 For example, considering the 1 st family, the e acquires mass of v m e = G e 2 as well as an electron coupling to the Higgs boson (3.21)
III.3 epton Mass Generation Replacing G e by 2 m e /v and using the H-e coupling is = m e 1 e v H = 2 4 e e G F m e H e e (3.22) The coupling is proportional to m e i.e. very small: G e =2.910-6 In a similar way we can write the interactions of the higgs boson to μ field and the field The Higgs mechanism for generating masses introduces arbitrary coupling parameters for each fermion mass i.e. no fundamental understanding of mass values We have not included the neutrinos here, since we assumed m =0 thus no R state exists that would be required to produce a mass term and have an interaction with the Higgs field So, in the SM the lepton states are e μ e μ,, 1v= 2G F & e R,μ R, R (3.23)
III.4 Quark Masses & Mixing Quark masses are also generated by Yukawa couplings to the scalar fields in a similar way as for the leptons However, since u and & quarks both have -handed & R-handed states the Yukawa interaction is more complicated here We need to introduce 9 complex couplings for the Q=2/3 quarks and 9 complex couplings G ij for the Q=-1/3 quarks The Higgs couples to quark weak eigenstates (u 1,u 2,u 3 ) & (d 1,d 2,d 3 ) that are different from the mass eigenstates (u,c,t) & (d,s,b) they are related by unitary transformations U,R & D,R performed separately for -handed & R-handed states The interaction of the quarks with the vacuum expectation value of the scalar field yields quark mass terms u 1 ( u 1, u 2, u 3 ) M u u R 2 + hc & u ITRG lecture week Oslo 3 06-03-2006 G ij d 1,d 2,d 3 d 2 + hc d G. Eigen, 3 IFT Bergen ( ) R M d d 1 (3.24) (3.25)
where III.4 Quark Masses & Mixing M ij u = v 2 G ij & M d ij = v 2 G ij (3.26) (3.27) For the weak eigenstates the mass matrices are not diagonal each depends on 9 complex parameters With the help of the unitary transformations U,R & D,R the mass matrix is diagonalized to obtain the observed masses m u, m c, m t and m d, m s, m b In weak interactions mediated by the W bosons (charged current) the different unitary transformations introduce family mixing d 1 ( u 1, u 2, u 3 ) μ d 2 We introduce the mixing matrix d 3 d ( ) U D μ s = u, c, t b (3.28) V U D (3.29)
III.4 Quark Masses & Mixing Using the mixing matrix V, the charge current weak interactions for quarks are d V ij W ( u, c, t) μ V Then the quark weak eigenstates are s b (3.30) By convention, the mixing is ascribed completely to Q=-1/3 states by defining u c t d', s', b' weak isospin doublets q i d' s' b' d = V s b & u R,c R,t R,d R,s R,b R weak isospin singlets q j (3.31) (3.32) The important prediction of the gauge theory is that all fermion doublets & singlets, whether leptons or quarks, enter the weak interaction with the same electroweak coupling strengths g & g