We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model
Lagrangian density. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer. Therefore, the
global Poincaré symmetry, consisting of
translational symmetry,
rotational symmetry and the inertial reference frame invariance central to the theory of
special relativity must apply. The
local gauge symmetry is the
internal symmetry. The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see.
Kinetic terms A free particle can be represented by a mass term, and a
kinetic term that relates to the "motion" of the fields.
Fermion fields The kinetic term for a Dirac fermion is i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi where the notations are carried from earlier in the article. can represent any, or all, Dirac fermions in the standard model. Generally, as below, this term is included within the couplings (creating an overall "dynamical" term).
Gauge fields For the spin-1 fields, first define the field strength
tensor F^a_{\mu\nu}=\partial_{\mu}A^{a}_{ \nu} - \partial_{\nu}A^{a}_{ \mu} + g f^{abc} A^{b}_{\mu} A^{c}_{\nu} for a given gauge field (here we use ), with gauge
coupling constant . The quantity is the
structure constant of the particular gauge group, defined by the commutator [t_a, t_b] = if^{abc} t_c, where are the
generators of the group. In an
abelian (commutative) group (such as the we use here) the structure constants vanish, since the generators all commute with each other. Of course, this is not the case in general – the standard model includes the non-Abelian and groups (such groups lead to what is called a
Yang–Mills gauge theory). We need to introduce three gauge fields corresponding to each of the subgroups . • The gluon field tensor will be denoted by G^{a}_{\mu\nu}, where the index labels elements of the representation of color
SU(3). The strong coupling constant is conventionally labelled (or simply where there is no ambiguity).
The observations leading to the discovery of this part of the Standard Model are discussed in the article in quantum chromodynamics. • The notation W^a_{\mu\nu} will be used for the gauge field tensor of where runs over the generators of this group. The coupling can be denoted or again simply . The gauge field will be denoted by W^a_{\mu}. • The gauge field tensor for the of weak hypercharge will be denoted by , the coupling by , and the gauge field by . The kinetic term can now be written as \mathcal{L}_{\rm{kin}} = - {1\over 4} B_{\mu\nu} B^{\mu\nu} - {1\over 2} \mathrm{tr} W_{\mu\nu} W^{\mu\nu} - {1\over 2} \mathrm{tr} G_{\mu\nu} G^{\mu\nu} where the traces are over the and indices hidden in and respectively. The two-index objects are the field strengths derived from and the vector fields. There are also two extra hidden parameters: the theta angles for and .
Coupling terms The next step is to "couple" the gauge fields to the fermions, allowing for interactions.
Electroweak sector The electroweak sector interacts with the symmetry group , where the subscript L indicates coupling only to left-handed fermions. \mathcal{L}_\mathrm{EW} = \sum_\psi\bar\psi\gamma^\mu \left(i\partial_\mu-g^\prime{1\over2}Y_\mathrm{W}B_\mu-g{1\over2}\boldsymbol{\tau}\mathbf{W}_\mu\right)\psi where is the gauge field; is the
weak hypercharge (the generator of the group); is the three-component gauge field; and the components of are the
Pauli matrices (infinitesimal generators of the group) whose eigenvalues give the weak isospin. Note that we have to redefine a new symmetry of
weak hypercharge, different from QED, in order to achieve the unification with the weak force. The
electric charge , third component of
weak isospin (also called or ) and weak hypercharge are related by Q = T_3 + \tfrac{1}{2} Y_{\rm W}, (or by the
alternative convention ). The first convention, used in this article, is equivalent to the earlier
Gell-Mann–Nishijima formula. It makes the hypercharge be twice the average charge of a given isomultiplet. One may then define the
conserved current for weak isospin as \mathbf{j}_\mu = {1\over 2}\bar{\psi}_{\rm L} \gamma_\mu\boldsymbol{\tau}\psi_{\rm L} and for weak hypercharge as j_{\mu}^{Y}=2(j_{\mu}^{\rm em} - j_{\mu}^3)~, where j_{\mu}^{\rm em} is the electric current and j_{\mu}^3 the third weak isospin current. As explained
above,
these currents mix to create the physically observed bosons, which also leads to testable relations between the coupling constants. To explain this in a simpler way, we can see the effect of the electroweak interaction by picking out terms from the Lagrangian. We see that the SU(2) symmetry acts on each (left-handed) fermion doublet contained in , for example -{g\over 2}(\bar{\nu}_e \;\bar{e})\tau^+ \gamma_{\mu}(W^+)^{\mu} \begin{pmatrix} {\nu_e} \\ e \end{pmatrix} = -{g\over 2}\bar{\nu}_e\gamma_{\mu}(W^+)^{\mu}e where the particles are understood to be left-handed, and where \tau^{+}\equiv {1 \over 2}(\tau^1{+}i\tau^2)= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between and via emission of a boson. The symmetry, on the other hand, is similar to electromagnetism, but acts on all "
weak hypercharged" fermions (both left- and right-handed) via the neutral , as well as the
charged fermions via the photon.
Quantum chromodynamics sector The quantum chromodynamics (QCD) sector defines the interactions between
quarks and
gluons, with symmetry, generated by . Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by \mathcal{L}_{\mathrm{QCD}} = i\overline U \left(\partial_\mu-ig_sG_\mu^a T^a \right )\gamma^\mu U + i\overline D \left(\partial_\mu-i g_s G_\mu^a T^a \right )\gamma^\mu D. where and are the Dirac spinors associated with up and down-type quarks, and other notations are continued from the previous section.
Mass terms and the Higgs mechanism Mass terms The mass term arising from the Dirac Lagrangian (for any fermion ) is -m\bar{\psi}\psi, which is
not invariant under the electroweak symmetry. This can be seen by writing in terms of left and right-handed components (skipping the actual calculation): -m\bar{\psi}\psi=-m(\bar{\psi}_{\rm L}\psi_{\rm R}+\bar{\psi}_{\rm R}\psi_{\rm L}) i.e. contribution from \bar{\psi}_{\rm L}\psi_{\rm L} and \bar{\psi}_{\rm R}\psi_{\rm R} terms do not appear. We see that the mass-generating interaction is achieved by constant flipping of particle chirality. The spin-half particles have no right/left chirality pair with the same representations and equal and opposite weak hypercharges, so assuming these gauge charges are conserved in the vacuum, none of the spin-half particles could ever swap chirality, and must remain massless. Additionally, we know experimentally that the W and Z bosons are massive, but a boson mass term contains the combination e.g. , which clearly depends on the choice of gauge. Therefore, none of the standard model fermions
or bosons can "begin" with mass, but must acquire it by some other mechanism.
Higgs mechanism The solution to both these problems comes from the
Higgs mechanism, which involves scalar fields (the number of which depend on the exact form of Higgs mechanism) which (to give the briefest possible description) are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms. In the Standard Model, the
Higgs field is a complex scalar field of the group : \phi= \frac{1}{\sqrt{2}} \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, where the superscripts and indicate the electric charge () of the components. The weak hypercharge () of both components is . The Higgs part of the Lagrangian is \mathcal{L}_{\rm H} = \left [\left (\partial_\mu -ig W_\mu^a t^a -ig'Y_{\phi} B_\mu \right )\phi \right ]^2 + \mu^2 \phi^\dagger\phi-\lambda (\phi^\dagger\phi)^2, where and , so that the mechanism of
spontaneous symmetry breaking can be used. There is a parameter here, at first hidden within the shape of the potential, that is very important. In a
unitarity gauge one can set \phi^+=0 and make \phi^0 real. Then \langle\phi^0\rangle=v is the non-vanishing
vacuum expectation value of the Higgs field. v has units of mass, and it is the only parameter in the Standard Model that is not dimensionless. It is also much smaller than the Planck scale and about twice the Higgs mass, setting the scale for the mass of all other particles in the Standard Model. This is the only real fine-tuning to a small nonzero value in the Standard Model. Quadratic terms in and arise, which give masses to the W and Z bosons: \begin{align} M_{\rm W} &= \tfrac{1}{2}vg \\ M_{\rm Z} &= \tfrac{1}{2} v\sqrt{g^2+{g'}^2} \end{align} The mass of the Higgs boson itself is given by M_{\rm H}= \sqrt{2 \mu^2 } \equiv \sqrt{ 2 \lambda v^2 }.
Yukawa interaction The
Yukawa interaction terms are \mathcal{L}_\text{Yukawa} = (Y_\text{u})_{mn}(\bar{q}_\text{L})_m \tilde{\varphi}(u_\text{R})_n + (Y_\text{d})_{mn}(\bar{q}_\text{L})_m \varphi(d_\text{R})_n + (Y_\text{e})_{mn}(\bar{L}_\text{L})_m \tilde{\varphi}(e_\text{R})_n + \mathrm{h.c.} where Y_\text{u}, Y_\text{d}, and Y_\text{e} are matrices of Yukawa couplings, with the term giving the coupling of the generations and , and h.c. means Hermitian conjugate of preceding terms. The fields q_\text{L} and L_\text{L} are left-handed quark and lepton doublets. Likewise, u_\text{R}, d_\text{R} and e_\text{R} are right-handed up-type quark, down-type quark, and lepton singlets. Finally \varphi is the Higgs doublet and \tilde{\varphi} = i\tau_2\varphi^{*}
Neutrino masses As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the right-handed neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. An obvious solution is to simply
add a right-handed neutrino , which requires the addition of a new
Dirac mass term in the Yukawa sector: \mathcal{L}^\text{Dir}_{\nu} = (Y_\nu)_{mn}(\bar{L}_L)_m \varphi (\nu_R)_n + \mathrm{h.c.} This field however must be a
sterile neutrino, since being right-handed it experimentally belongs to an isospin singlet () and also has charge , implying (see
above) i.e. it does not even participate in the weak interaction. The experimental evidence for sterile neutrinos is currently inconclusive. Another possibility to consider is that the neutrino satisfies the
Majorana equation, which at first seems possible due to its zero electric charge. In this case a new
Majorana mass term is added to the Yukawa sector: \mathcal{L}^\text{Maj}_{\nu} = -\frac{1}{2} m \left ( \overline{\nu}^C\nu + \overline{\nu}\nu^C \right ) where denotes a charge conjugated (i.e. anti-) particle, and the \nu terms are consistently all left (or all right) chirality (note that a left-chirality projection of an antiparticle is a right-handed field; care must be taken here due to different notations sometimes used). Here we are essentially flipping between left-handed neutrinos and right-handed anti-neutrinos (it is furthermore possible but
not necessary that neutrinos are their own antiparticle, so these particles are the same). However, for left-chirality neutrinos, this term changes weak hypercharge by 2 units – not possible with the standard Higgs interaction, requiring the Higgs field to be extended to include an extra triplet with weak hypercharge = 2 (see
seesaw mechanism). Since in any case new fields must be postulated to explain the experimental results, neutrinos are an obvious gateway to searching physics
beyond the Standard Model. == Detailed information ==