Higgs mechanism

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In particle physics, the Brout–Englert–Higgs mechanism or Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism, is one of the possible mechanisms to produce spontaneous electroweak symmetry breaking in a gauge invariant theory. It allowed establishing the unification between the electromagnetic theory and the weak nuclear theory, which was called unified field theory, for which Steven Weinberg, Sheldon Lee Glashow and Abdus Salam would obtain the Nobel Prize in 1979.

The Higgs mechanism is the process that gives mass to elementary particles. The particles gain mass by interacting with the Higgs field that permeates all of space. More precisely, in a gauge theory, the Higgs mechanism endows gauge bosons with mass through the absorption of Nambu–Goldstone bosons derived from spontaneous symmetry breaking.

The simplest implementation of the mechanism adds an extra Higgs field to the gauge theory. The spontaneous breaking of the underlying local symmetry triggers the conversion of the components of this Higgs field to Goldstone bosons that interact (at least some of them) with the other fields of the theory, in order to produce mass terms for (at least some of) the gauge bosons. This mechanism can also leave behind elementary scalar (spin-0) particles, known as Higgs bosons.

In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the weak gauge, W±, and Z bosons via electroweak symmetry.

The mechanism was proposed in 1962 by Philip Warren Anderson. The relativistic model was developed in 1964 by three independent groups: Robert Brout and Francois Englert; Peter Higgs; and Gerald Guralnik, C.R. Hagen, and Tom Kibble. On July 4, 2012, the Large Hadron Collider (LHC) at CERN announced results consistent with the Higgs particle, but stressed that more tests are needed to confirm the full mechanism.

History and name

2010 APS J.J. Sakurai Prize - Kibble, Guralnik, Hagen, Englert, Brout

This mechanism is also known as the Brout–Englert–Higgs mechanism, Higgs–Brout–Englert–Guralnik–Hagen–Kibble mechanism, or mechanism from Anderson–Higgs. In 1964, it was initially proposed by Robert Brout and François Englert, and independently by Peter Higgs and by Gerald Guralnik, C. R. Hagen, and Tom Kibble. It was inspired by the BCS Theory of symmetry breaking superconductivity based on the Ginzburg-Theory. Landau, Yoichiro Nambu's work on the structure of a vacuum, and Philip Anderson's ideas that superconductivity could be relevant to relativity, electromagnetism, and other classical phenomena. The name Higgs mechanism was given by Gerardus 't Hooft in 1971. The three original articles by Guralnik, Hagen, Kibble, Higgs, Brout, and Englert proposing this mechanism were Recognized as Fundamentals at Physical Review Letters 50th Anniversary Celebration.

Fields and particles

The second half of the 20th century was a time of discovery of new elementary particles, new forces and, above all, new fields. Space can be filled with a wide variety of invisible influences that have all kinds of effects on ordinary matter. Of all the new fields that have been discovered, the one that has the most to teach us about the landscape is the Higgs field. There is a general relationship between particles and fields. For each type of particle in nature there is a field and for each type of field there is a particle. Thus fields and particles bear the same name. The electromagnetic field could be called a photon field. The electron has a field, so does the quark, the gluon, and every member of the cast of Standard Model characters, including the Higgs particle.

The Higgs field

In the conception of the Standard Model of particle physics, the Higgs boson as well as other bosons (already found experimentally) and linked in this theory, are interpreted from the Goldstone Boson where each part of the symmetry breaking generates a field, for which the elements that live in this field are their respective bosons. There are theories created from the fear of the non-existence of the Higgs boson where its appearance is not necessary. The Higgs field is the mathematical entity where it exists, its interpretation with the theory is the product of it with the other fields that comes out through the rupture mechanism, this product gives us the coupling and the interaction of it, with this interaction with the other fields we bequeath the mass generator feature.

(see Higgs field)

Mathematical formulation

We introduce an additional field Φ whose final effect will be to set a self-interaction potential and a spontaneous electroweak symmetry breaking (so the symmetry group will change SU(2)_L × U(1)_Y → U(1)_em). Due to the conditions that are required of the theory, it will be a doublet (of SU(2)_L) of complex scalar fields (Higgs doublet):

${displaystyle Phi (x)={left({begin{matrix}phi ^{+}\phi ^{0}end{matrix}}right)}={frac {1}{sqrt {2}}}{left({begin{matrix}phi _{1}+mathrm {i} phi _{2}\phi _{3}+mathrm {i} phi _{4}end{matrix}}right)}}$

Higgs doublets

Potential of double well in a field theory with spontaneous rupture of symmetry.

The total number of entries (dimensional number of the vector) of the Higgs is not determined by theory and could be anything. However, the minimum version of the SM has only one of these doublets.

The system will then be described by a Lagrangian of the form:

{displaystyle {mathcal {L}}_{SBS}=({mathcal {D}}_{mu }Phi)^{dagger }({mathcal {D}}^{mu }Phi)-V(Phi)}

such that:

{displaystyle V(Phi)=mu ^{2}Phi ^{dagger }Phi -lambda (Phi ^{dagger }Phi)^{2}}

where V(Φ) is the simplest renormalizable (and therefore gauge invariant) potential. For spontaneous symmetry breaking to occur, it is necessary that the expected value of the Higgs field in vacuum is not zero. For λ > 0, if μ² < 0, the potential has infinitely many non-zero solutions (see figure 1), in which only the norm of the Higgs field is defined:

{displaystyle |Phi |^{2}=Phi ^{dagger }Phi =-{frac {mu ^{2}}{2lambda }}={frac {upsilon ^{2}}{2}}}

Ground state

The ground state is therefore degenerate and is not invariant under any original symmetry group transformation SU(2)_L × U(1)_Y, however, the ground state will be invariant under a group of minor symmetry U(1)_em (which is in fact only a subgroup of the previous group). The fact that the symmetry group before the introduction of the Higgs boson or field was SU(2)_L × U(1)_Y and after its introduction is a minor group U(1)_em, is expressed by theoretical physicists as saying that the Higgs boson symmetry breaks SU(2)_L × U (1)_Y in U(1)_em" (which is equivalent to what has been expressed a little more formally before).

The value of υ indicates the energy scale at which electroweak symmetry breaking occurs. The break SU(2)_L × U(1)_Y --> U(1)_em occurs when a particular void state is selected. The usual choice is the one that makes φ₃ non-zero:

{displaystyle Phi (x)={left({begin{matrix}phi ^{+}\phi ^{0}end{matrix}}right)}longrightarrow {frac {1}{sqrt {2}}}{left({begin{matrix}0\upsilon end{matrix}}right)}}

Particle spectrum

The spectrum of resulting physical particles is built by performing small oscillations around a vacuum, which can be parameterized in the form:

{displaystyle Phi (x)={frac {1}{sqrt {2}}},e^{mathrm {i} {frac {{vec {xi }}(x)cdot {vec {tau }}}{upsilon }}}{left({begin{matrix}0\upsilon +mathrm {h} (x)end{matrix}}right)}}

where the vector ${displaystyle {vec {xi }}(x)}$ and the h(x) scale are small fields corresponding to the four degrees of real freedom in the field. The three fields ${displaystyle {vec {xi }}(x)}$ are the Goldstone Bones, of null mass, which appear when a continuous symmetry is broken by the fundamental state (Goldstone Theorem).

At this point we still have 4 balls gauge (W)ⁱ_μ(x) and B_μ(x)) and 4 scales ( ${displaystyle {vec {xi }}(x)}$ and h(x)), all of them without mass, which is equivalent to 12 degrees of freedom (it should be noted that a null mass vector boson has two degrees of freedom, while a massive vector boson acquires a new degree of freedom due to the possibility of having longitudinal polarization: 12 = 4 [massless vectorial buns] × 2 + 4 [massless scales]). P. W. Higgs was the first to realize that the Goldstone theorem is not applicable to gauge theories, or at least can be circumvented by a convenient selection of representation. So, just choose a transformation:

{displaystyle U(xi)=e^{-mathrm {i} {frac {{vec {xi }}(x)cdot {vec {tau }}}{upsilon }}}}

so that:

${displaystyle {begin{matrix}Phi ^{prime }&=&U(xi)Phi ={frac {1}{sqrt {2}}}{left({begin{matrix}0\upsilon +mathrm {h} (x)end{matrix}}right)}qquad \left({frac {{vec {tau }},{vec {mathrm {W} }}_{mu }^{prime }}{2}}right)&=&U(xi)left({frac {{vec {tau }},{vec {mathrm {W} }}_{mu }}{2}}right)U^{-1}(xi)-{frac {mathrm {i} }{g}}(partial _{mu }U(xi))U^{-1}(xi)\mathrm {B} _{mu }^{prime }&=&mathrm {B} _{mu }qquad end{matrix}}}$

with which the three fields of nonphysical Higgs disappear ${displaystyle {vec {xi }}(x)}$ . We must apply these transformations on the sum of the Lagrangianas for bosons and femions:

{displaystyle {mathcal {L}}={mathcal {L}}_{bos.}+{mathcal {L}}_{ferm.}+{mathcal {L}}_{SBS}}

At the end of the process, three of the four gauge bosons acquire mass by absorbing each of the three degrees of freedom removed from the Higgs field, thanks to the couplings between the gauge bosons and the Φ field present in the kinetic component of the Lagrangian SBS:

{displaystyle ({mathcal {D}}_{mu }Phi)^{dagger }({mathcal {D}}^{mu }Phi)={frac {upsilon ^{2}}{8}}[mathrm {g} ^{2}(W_{1mu }^{2}+W_{2mu }^{2})+(mathrm {g} W_{3mu }-mathrm {g} ^{prime }B_{mu })^{2}]}

On the other hand, the vacuum of the theory must be electrically neutral, which is why there is no link between the photon and the field of Higgs, h(x), so that it maintains a null mass. In the end, we get three massive gauge balls (WW^±_μ, Z_μ), a massless gauge ball (A_μ) and a scaler with mass (h), so we continue to have 12 degrees of freedom (like before: 12 = 3[mass vectorials] × 3 + 1[massless vectorial syndrome] × 2 + 1[scalar]). The physical states of the gauge balls are then expressed according to the original states and the electrodebil mixing angle ${displaystyle theta _{mathrm {W} }}$ :

{displaystyle {begin{matrix}mathrm {W} _{mu }^{pm }&=&{frac {1}{sqrt {2}}}(mathrm {W} _{mu }^{1}mp mathrm {W} _{mu }^{2})qquad \mathrm {Z} _{mu }&=&cos {theta _{mathrm {W} }}mathrm {W} _{mu }^{3}-sin {theta _{mathrm {W} }}mathrm {B} _{mu }\mathrm {A} _{mu }&=&sin {theta _{mathrm {W} }}mathrm {W} _{mu }^{3}+cos {theta _{mathrm {W} }}mathrm {B} _{mu }end{matrix}}}

Blending Angle

The mixing angle ${displaystyle theta _{mathrm {W} }}$ , is defined according to weak coupling constants, gand electromagnetic, g ́according to:

{displaystyle tan {theta _{mathrm {W} }}equiv {frac {mathrm {g} ^{prime }}{mathrm {g} }}}

The predictions of the boson masses at the tree level are:

{displaystyle {begin{matrix}mathrm {M_{W}} &=&{frac {1}{2}}mathrm {g} upsilon qquad \&&\mathrm {M_{Z}} &=&{frac {1}{2}}upsilon {sqrt {mathrm {g} ^{2}+{mathrm {g} ^{prime }}^{2}}}end{matrix}}}

where (e is the electric charge of the electron):

{displaystyle {begin{matrix}mathrm {g} &=&{frac {e}{sin {theta _{mathrm {W} }}}}\mathrm {g} ^{prime }&=&{frac {e}{cos {theta _{mathrm {W} }}}}end{matrix}}}

Higgs boson mass

The mass of the Higgs boson is expressed as a function of λ and the value of the symmetry breaking scale, υ, as:

{displaystyle mathrm {m_{H}^{2}} =2lambda upsilon ^{2}}

The measure of the partial width of the decay:

{displaystyle mu rightarrow nu _{mu }{bar {nu _{mathrm {e} }}}mathrm {e} }

at low energies in the SM allows to calculate the Fermi constant, G_F, with great precision. And since:

{displaystyle upsilon =({sqrt {2}}mathrm {G_{F}})^{-{frac {1}{2}}}}

a value of υ = 246 GeV is obtained. However, the value of λ is unknown and therefore the mass of the Higgs boson in the SM is a free parameter of the theory.

Gauge bosons and fermions

Similarly to the case of gauge bosons, fermions acquire mass through so-called Yukawa couplings, which are introduced through a series of new terms in the Lagrangian:

{displaystyle {mathcal {L}}_{YW}=lambda _{mathrm {e} }{bar {ell }}_{L}Phi mathrm {e} _{R}+lambda _{mathrm {u} }{bar {mathrm {q} }}_{L}{tilde {Phi }}mathrm {u} _{R}+lambda _{mathrm {d} }{bar {mathrm {q} }}_{L}Phi mathrm {d} _{R}+{mbox{h.c. + 2ª y 3ª familias}}}

where:

{displaystyle {begin{matrix}ell _{L}&=&{left({begin{matrix}mathrm {e} \nu _{mathrm {e} }end{matrix}}right)}_{L},{left({begin{matrix}mu \nu _{mu }end{matrix}}right)}_{L},{left({begin{matrix}tau \nu _{tau }end{matrix}}right)}_{L}\&&\mathrm {q} _{L}&=&{left({begin{matrix}mathrm {u} \mathrm {d} end{matrix}}right)}_{L},{left({begin{matrix}mathrm {c} \mathrm {s} end{matrix}}right)}_{L},{left({begin{matrix}mathrm {t} \mathrm {b} end{matrix}}right)}_{L}end{matrix}}}

In the same way as before, the transformation is applied to the left-handed part of the fermions, while the right-handed part is not transformed:

{displaystyle ell '_{L}=U(xi)ell _{L};qquad mathrm {e} '_{R}=mathrm {e} _{R}}

{displaystyle mathrm {q} '_{L}=U(xi)q_{L};qquad mathrm {u} '_{R}=mathrm {u} _{R};~mathrm {d} '_{R}=mathrm {d} }

And finally the masses of the fermions are obtained according to:

{displaystyle {begin{matrix}mathrm {m} _{mathrm {e} }&=&lambda _{mathrm {e} }{frac {upsilon }{sqrt {2}}}\mathrm {m} _{mathrm {u} }&=&lambda _{mathrm {u} }{frac {upsilon }{sqrt {2}}}\mathrm {m} _{mathrm {d} }&=&lambda _{mathrm {d} }{frac {upsilon }{sqrt {2}}}\...end{matrix}}}

It is convenient to note at this point that the determination of the Higgs boson mass does not directly explain the fermionic masses since they depend on the new constants λ_e, λ_u, λ_d,... On the other hand, the value of the couplings of the Higgs boson with the different fermions and bosons is also deduced, which are proportional to the coupling constants gauge and the mass of each particle.

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