PoS(Corfu2012)131 http://pos.sissa.it/ [email protected] † ∗ Speaker. Invited talks presented at the 2001 Corfu Summer Institute on Elementary Particle Physics. The theory of symmetry breaking in presencetrack. of gauge Particular fields emphasis is is presented, placed following upon the historical the underlying concepts. ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c ⃝ Proceedings of the Corfu Summer Instituteand 2012 Gravity" "School and Workshops on ElementarySeptember Particle 8-27, Physics 2012 Corfu, Greece François Englert Service de Physique Théorique, Université LibreBoulevard de du Bruxelles, Triomphe, Campus B-1050 Plaine, Bruxelles, C.P.225, Belgium E-mail: A Brief Course in Spontaneous SymmetryII. Breaking Modern Times: The BEH Mechanism PoS(Corfu2012)131 François Englert mesons); theories with fundamental massive − W 2 and + W ]. The Brout-Englert-Higgs (BEH) theory is based on a mechanism, inspired 3 , 2 Spontaneous breaking of a Lie group symmetry was discussed by Robert Brout in “The Pale- Before turning to an exposé of the BEH mechanism, we shall in section II review, in the The discovery of subatomic structures and of the concomitant weak and strong interaction The solution of the latter problem came from the theory proposed in 1964 by Brout and Englert Gravitational and electromagnetic forces are long range and hence can be perceived directly It was known in the first half of the twentieth century that, at the atomic level and at larger ] and by Higgs [ 1 olitic Age”. I review here its essential features in the quantum field theory context. 2. Spontaneous Breaking of a Global Symmetry context of quantum field theory, the analysisa given continuous by symmetry. Robert Brout Section of III thetheory spontaneous explains approach breaking the of of BEH Brout mechanism. andabelian Englert We gauge present wherein groups the the is quantum breaking induced field dynamical mechanism by symmetry for scalar breaking both bosons. from abelian fermion and Weapproach condensate. non also of present Higgs. We then their Finally turn approach wethe to in explain well-known the the the applications equation renormalization of case the of issue. of BEH motion Into mechanism the section with quest particular IV, we for emphasis briefly unification. on concepts review Some relevant comments on this subject are made in section V. from the spontaneous symmetry breaking ofby a continuous Robert symmetry, Brout, discussed in adapted the tomechanism previous talk gauge unifies theories long range and and invector short particular mesons range masses to forces from non a mediated abelian fundamental bysolution theory gauge vector of containing theories. mesons, the only weak by massless The interaction vector deriving puzzle fields. the of and It nature. opened led the to way a to modern perspectives on unified laws short range forces raisedtheory. the question The of Fermi howcurrent-current theory to of interaction, cope weak was with interactions, predictive shortconfronted formulated in many range in experimental lowest forces data. terms order in However, ofof perturbation quantum it uncontrollable a was theory field quantum clearly four and divergences inconsistent Fermi at successfully in highquantum point-like higher energies. electrodynamics, order In because the order Fermi words,solved in theory by contradistinction is with smoothing not the renormalizable.vector point-like particle interaction This exchange by (the difficulty a so-called could massive, not and be therefore short range, charged [ without the mediation of highlyphysics, sophisticated initiated by technical the devices. Galilean inertialtook The about principle, development three is of centuries surely large to tributary achieve scale to a this successful circumstance. description of It long then range effects. charged vector mesons are notto renormalizable be either. insuperable obstacles In for thevectors. formulating early a nineteen theory sixties, with short there range seemed forces mediated by massive distance scales, all phenomena appear toof be governed quantum by electrodynamics. the laws of classical general relativity and A Brief Course in SSB II 1. Introduction PoS(Corfu2012)131 (2.1) -field ϕ and the 0 . Its mass , one may ⟩ 1 = 2 , ϕ ] and Nambu ⟩ ̸ √ 0 ⟨ positive. The 4 / François Englert ⟩ ) > ¯ 1 (Fig.1). ψψ 2 ⟨ ϕ ϕ λ 2 i ⟨ , + ]. Here, symmetry is 2 6 ) 1 ϕ ϕ ∗ ϕ = ( ( λ ϕ + is clearly illustrated in the Fig.1. ϕ 2 ∗ ϕ ϕ 2 µ . Writing − ) ]. is rotated into an equivalent vacuum by the ϕ symmetry is called global because the group 6 ∗ is always present in spontaneous breakdown ⟩ ) = ) 1 ϕ 1 ϕ 1 ( ϕ ϕ ∗ ( ⟨ V ϕ has negative mass and acquires a positive mass U 3 ( is massless. The latter is the NG boson of broken 1 V ϕ 2 ϕ is broken by the fermion condensate . The symmetry [ ) ϕ ) and we select, say, the vacuum with α 1 α 5 with i ( λ γ e i / U ( 2 ) → µ ϕ ∗ exp ϕ = ϕ ( 2 ⟩ V 1 ϕ group − ⟨ ) ϕ 1 µ ( ∂ ∗ U ] is the spontaneous breaking of chiral symmetry induced by a fermion con- ϕ 5 µ is depicted in Fig.1 . ∂ . Hence ) 0 ϕ = ∗ is constant in space-time. It is broken by a vacuum expectation value of the = ϕ ⟩ ( L α 2 V ϕ . The chiral phase group ⟨ 1 In the classical limit, the origin of the massless NG boson See the detailed discussion in Brout’sIt lecture, is section often VII. called the Higgs boson in the literature. Around the unbroken vacuum the field Let us first illustrate the occurrence of this massless Nambu-Goldstone (NG) boson in a simple The Lagrangian density, Spontaneous symmetry breaking was introduced in relativistic quantum field theory by Nambu Recall that spontaneous breakdown of a continuous symmetry in condensed matter physics 1 2 given, at the classical level, by the minimum of around the broken vacuum where the field of a symmetry. Inintroduced the by context Brout and of myself, the and BEH by Higgs. mechanism We analyzed shall label in itThe the the vacuum following BEH characterized section, boson by the it order was parameter parameter model of a complex scalar field with is invariant under the massless mode is identified with thesmall pion. explicit The breaking latter of gets the its symmetry, tiny justin mass as the (on a spin the small wave hadron external spectrum. scale) magnetic fromunderstanding field This of a imparts interpretation strong a interaction of small physics. the gap General pionin features mass of relativistic spontaneous constituted quantum symmetry a breakdown field breakthrough theory inbroken were our by further non formalized vanishing by vacuum Goldstoneexhibit expectation [ the values appearance of of scalar apend fields. massless on mode the The out significance method of of is thevariables the scalar of designed degenerate fields. more to vacuum fundamental The and fields, latter as does coulditeness would not be be affects really elementary the details or de- case of represent in collective the thebut original model not Nambu the considered, model. existence such Compos- of as the massless the excitations behavior encoded at in high the degeneracy momentum of transfer, the vacuum. U(1) symmetry. The massive scalar describes the fluctuations of the order parameter in analogy to the BCS theory of superconductivity. The problem studied by Nambu [ implies a degeneracy of thecollective ground modes state, appear whose and energies as go aexemplified to in consequence, zero particular in when by the absence spin wavelength of waves goesis in long to the a range infinity. rotation Heisenberg This forces, invariance. ferromagnet. was There, the broken symmetry densate A Brief Course in SSB II and Jona-Lasinio [ choose is the analog ofby the Robert inverse Brout longitudinal while susceptibilityinverse the of transverse vanishing susceptibility. the of The Heisenberg scalar the ferromagnet boson NG discussed boson mass corresponds to the vanishing of its potential PoS(Corfu2012)131 . ) x ( q α = → µ and µ ∂ ) (3.1) ) J x x is the e ( 3 2 / ϕ d at space 1 ϕ 2 generated 2 ϕ 0 ϕ G ∂ )+( ⟩ x 1 ( François Englert ϕ µ ⟨ A ∫ have a NG pole at invariance . The field → 0 ) = 0 1 ) ( Q x = = . As in the abelian case ( ⟩ ̸ U x µ rotates the vacuum by an 2 -invariant action and if the 3 A q A ) d ϕ G ⟨ Q , A is non zero in the chosen vac- α ϕ i ⟩ µν ( aBA 1 F T ϕ aBA ⟨ i exp µν T = F ⟩ B = 1 ] 1 4 cannot vanish at zero four-momentum ϕ . Expressing the propagator in terms of ⟩ B φ ] ⟩ µ ⟩ ] − ϕ 2 BEH massive boson ) (inverse) transverse susceptibility (inverse) transverse ∂ NG massless boson 2 , ′ (inverse) longitudinal susceptibility ) ϕ a ϕ x , ∫ ϕ , ( Q ∗ Q 2 [ [ Q = ⟨ ϕ ϕ [ ⟨ ( ⟨ a ) 4 x V Q ( . The operator − µ transforming according to x 3 ) can be extended to a local ϕ ) ,s , symmetry is broken and the vacuum is degenerate TJ x d A µ ⟨ ( 0 ϕ such that 2.1 D µ J µ ∗ ∂ A ∫ V B -propagator must have a pole at ϕ 2 ϕ µ . If the dynamics is governed by a ϕ = Fig. 1 D Q aAB = T L 2 on the equations of motion, and hence zero mass. φ be scalar fields spanning a representation of the Lie group 0 A = ϕ symmetry in Eq.( 0 ) q 1 ( gives a charge at zero momentum. In general, U 1 has 2 ϕ ϕ 0 µ ∂ = 2 by introducing a vector field q ϕ → ) at zero space momentum. Such rotation costs no energy and thus the field -rotations. The conserved charges are . In the classical limit, this charge is, around the chosen vacuum, x − ( 2 . G 2 α ϕ 0 ϕ ϕ The proof is immediately extended to the spontaneous breaking of a semi-simple Lie group The global This can be formalized and generalized by noting that the conserved Noether current ) x µ ( = ∂ α 1 2 i The corresponding Lagrangian density is global symmetry. Let Feynman diagrams we see thatmassless the NG boson. by the (antihermitian) matrices 3. The BEH Mechanism 3.1 From global to local symmetry under momenta ϕ uum. This implies that thebecause propagator its integral over space-time is precisely potential has minima for non vanishing above, the propagators of theq fields field A Brief Course in SSB II angle e involves only PoS(Corfu2012)131 ) ) 2.1 3.2 (3.4) (3.2) (3.3) breaks appar- rotated ) h ) or ( 1 ( is invariant 3.1 U V François Englert . any ν c 0 A µ b A → h abc e f . µ − A µ a ν and the potential ∂ A ν − ∂ in the Lagrangian Eq.( µ a aAB ν µ − A A T , hence on the long range character of A a ν µ µ , A ∂ A µ µν = ∂ can be realized by extending the Lagrangian a , -field simply picks up one of the degenerate always develops an expectation value. When = h F h µν G 1 ∗ F a a ϕ µν µν ϕ F F 5 , + 4 1 and generated by B ∗ h − ϕ ϕ G µ ϕ V . The tiny aAB 0 − ieA T A µ a = ) − ⟩ ̸ ϕ ; in this way, we can reach in the limit ϕ eA 1 h µ µ ϕ − ⟨ D ∂ ( A A = ϕ ∗ ) µ ϕ ∂ ϕ µ symmetry and the field µ D = ) D 1 A ( ) = ( ϕ U µ G , we still have D 0 L where ( = | ̸ are constant in space time. Let us take h real. The presence of the field ϕ | ∗ belongs to the representation of h at , . A ) to incorporate non-abelian Yang-Mills vector fields h ) ϕ G , the symmetry of the action is restored but, when the symmetry is broken by a minimum of ∗ relies on the massless character of the gauge fields 0 The local symmetry, or gauge invariance, of Yang-Mills theory, abelian or non abelian, The success of quantum electrodynamics based on local U(1) symmetry, and of classical gen- To understand the difference, let us break the symmetries explicitly. To the Lagrangian Eq.( When the symmetry is extended from global to local, one can still break the symmetry by an Local invariance under a semi-simple Lie group 3.1 ϕϕ → ( under eral relativity based on a localthe generalization relevance of of Poincare local invariance, symmetry providesmetry for ample has the a evidence fundamental for description significance of rootedlaws natural in at causality laws. a and fundamental in One level, the of expects existence whichof that of charge the exact conservation local conservation appears strength sym- as of the local prototype.symmetry As symmetry alone an are we example violated cite in the presence of fact black that holes. conservation laws resultingently from a global the forces they transmit, as the addition of a mass term for explicitly the global h where destroys gauge invariance. Butbe short as range fundamental as forces, the suchlaws. electromagnetic as ones To reach the despite a basic the weak description apparent interaction ofto absence such forces, of forces a one seem exact mass is conservation to tempted of to the linkthe gauge the problem violation fields of of which spontaneous conservation broken would symmetry arise is from different spontaneous for global symmetry and breaking. forwe local add However symmetry. the term Here, vacua in perfect analogy with theferromagnet. infinitesimal As magnetic in field statistical which mechanics, orients spontaneous thethe broken magnetization limit global of symmetry of a can vanishing be external recovered symmetryevidence in breaking. by changing The the degeneracy phase of of the vacuum can be put into A Brief Course in SSB II with covariant derivative external “magnetic” field. However in the limitany of gauge vanishing dependent magnetic local field operator the will expectation tendtry, value to of it zero because, cost in no contradistinction energy to in globalthen symme- the no limit ordered configuration to in change group the space relative which orientation can of be neighboring protected from “spins”; disordering there fluctuations. is Eq.( vacuum. V PoS(Corfu2012)131 . The is arbi- A η (b) (a) (c) , by non zero V François Englert Fig. 2 ). Let us choose a gauge 3.2 . The gauge parameter ν a . Fig.2a pictures the spontaneously symmetry. This can be achieved by A B ν ∂ G µ a A µ ∂ q 1 − ) 6 η . 5 2 ( . Recalling that the explicit presence of a gauge vector 4 ]. 8 of BEH fields. In Fig.2 we have represented fluctuations of this parame- symmetry. This choice has no observable consequences and only masks the degeneracy of ) ⟩ 1 A ( ϕ U ⟨ -direction and in an internal space direction orthogonal to the direction q B and the vacuum is gauge invariant A 3 The global symmetry can now be spontaneously broken, for suitable potential For a detailed proof, see reference [ Note that for global symmetry breaking, one can always choose a linear combination of degenerate vacua which is In perturbation theory, gauge invariant quantities are evaluated by choosing a particular gauge. To this end, it is often necessary, in particular for non abelian gauge theories, to include Fadeev-Popov ghosts terms Consider the Yang-Mills theory defined by the Lagrangian Eq.( 3 4 5 trary and has no observable consequences. expectation values invariant under, say, the One imposes the gauge conditionsubsets by of adding graphs to satisfying the the Ward action Identities a gauge fixing term and one sums over in the action. These contribute when closed gauge field loops are included in the computation. 3.2 Solving the dilemma the vacuum which is guaranteedanalyzed by by a Robert superselection Brout rule. (section Theall V Hilbert the of space finite “The excitations splits Paleolitic on indeed, Age”), each as into degenerate in an vacuum. the infinite ferromagnetic number case of orthogonal spaces formed by mass breaks gauge invariance, wemass are without thus breaking the faced local with symmetry? a dilemma. How can gauge fields acquire which preserves Lorentz invariance andadding a to residual the Lagrangian global a gauge fixing term As a consequence, the vacuumin is group generically space non as degenerate the andbroken external points field in goes no to particular zero. direction Local gauge symmetry cannot be spontaneously A Brief Course in SSB II orthogonal direction depicted in the figure has been labeled ter in the spatial PoS(Corfu2012)131 (3.5) are just local , leaving long ⟩ exemplified in François Englert A ϕ H ϕ / ⟨ G Gauge field Complex scalar field , 2 q is / 2 µ ν q q A µ q η + 2 q / 7 ν q ), is given by the one-loop diagrams of Fig.3. The µ 2 ] Fig. 3 q q 1 3.1 − . µν G g of = H µν 0 D these fluctuations can only induce global rotations in the internal space. ∞ → λ is the gauge parameter. It can be put equal to zero, as in the Landau gauge used in reference . ). λ η ) Breaking by BEH bosons In absence of symmetry breaking, the lowest order contribution to the self-energy, arising Let us first examine the abelian case as realized byIn the the complex covariant scalar gauges, the field free propagator of the field α We shall see in the next sections how these considerations are realized in quantum field the- The degrees of freedom of the NG fields were present in the original gauge invariant action Clearly as 3.1 ], but we leave it arbitrary here to illustrate explicitly the role of the NG-boson. 1 from the covariant derivative terms in Eq.( range forces only in a subgroup Eq.( where ory, giving rise to an apparentsymmetry breaking, breakdown gauge of invariant vector symmetry: masses despite will be the generated absence in of a spontaneous coset local 3.3 The quantum field theory approach [ and cannot disappear. But what makes localis internal precisely space rotations the unobservable in fact a that gaugefields. theory they The can be absorption absorbed ofthey through are the coupled, gauge long and transformations range transfers by NG topolarization. the them fields Yang-Mills the renders missing massive degrees those of freedom gauge which fields becomes to their which third rotations in the internal space and henceinduce are unobservable only gauge gauge fluctuations. transformations Hence and the its NG excitations bosons disappear from the physical spectrum. broken vacuum of the gauge fixedlength Lagrangian. Fig.2b and 2c represent fluctuations of finite wave- In absence of gauge fields,continuous symmetries, such to fluctuations massless NG would mode. give In rise, a gauge as theory, fluctuations in of spontaneously broken global A Brief Course in SSB II [ PoS(Corfu2012)131 . η (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) and, as we 6 François Englert NG propagator BEH tadpole , . 2 2 q q , . / / ) 2 2 0 ν ν 2 q q q q q , ( = µ µ ⟩ q q , ⟩ ̸ Π A 1 η η 2 , . ) ϕ ϕ , leading to the gauge field propagator ⟩ ⟨ ν ⟨ ) 2 0 1 + + q 2 2 ⟩ ϕ 1 q µ q 2 2 ) ⟨ = bCA ( ϕ q q q )] 2 2 ⟨ 2 2 e / / T ϕ ab 2 8 i q − q 2 ν ν e Fig. 4 2 ( Π 2 q q ) to the non abelian case described by the action µ + q ) aBC q µ µ Π = ) = ∗ ) acquires a pole ν 1 q q − 2 q 2 3.9 ϕ µν T − 2 q ( µ ⟩ − − g µ ( 3.6 q 1 . The additional diagrams are depicted in Fig.4. In this 2 q B [ 2 ∗ 1 Π 2 µν µν ϕ √ − q = ( ϕ g g ⟨ 2 2 ) and ( = is regular at q µν e = = in Eq.( ) ϕ ) is transverse and hence does not affect the gauge parameter µν Π 2 ) 3.6 g 2 q µν µν ( q ) = 3.6 D D ( 2 Π = ( q Π ( -field remains massless. µ µν ab ab -field gets a mass A Π Π µ A and the NG field 1 ϕ ) is straightforward. One gets from the graphs depicted in Fig.5, The transversality of polarisation tensors is a consequence of the Ward Identities alluded to in the preceding section. The generalization of Eqs.( Symmetry breaking adds tadpole diagrams to the previous ones. To see this write 3.2 6 Eq.( and, in lowest order perturbation theory, the gauge field propagator becomes which shows that the The BEH field is The transversality of the polarization tensorshall reflects see the below, gauge the invariance regularity of of theThis the theory guarantees polarization that scalar the signals the absence of symmetry breaking. The polarization tensor in Eq.( self-energy (suitably regularized) takes the form of a polarization tensor where the scalar polarisation case, the polarisation scalar A Brief Course in SSB II PoS(Corfu2012)131 (3.14) (3.15) (3.16) François Englert contributions to 2 q / propagator. We shall 1 2 q / 1 which leaves invariant the , one easily verifies that the 2 G a , µ 2 . Fig. 5 a of 2 = µ . q 2 / ⟩ H = 2 q ν A q ). The singular 2 does not enter the computation of the q ϕ q µ ⟨ V q , 3.12 b b η bCA ) λ T + ( ν e 2 . q ) aBC ) and ( ∗ / λ 2 µ ( 9 ν a T e q ⟩ 3.6 1 µ µ B 3 = ∗ q ∑ − λ ϕ C ), which preserve transversality while giving mass to the 2 ⟨ − = q of the mass matrix the propagator for the massive gauge 2 e ) a µν ν ) is: . 3.13 g 2 µ q ⟩ = q µ A 3.10 q = ab ϕ 3.15 ⟨ µ around its minimum. Hence the mass matrix decouples from the − µν a ) and ( a a V D µν 3.9 g ), which yields the second graph of Fig.5, can only couple the tadpoles to 3.2 ). 4.5 are the three polarization vectors which are orthonormal in the rest frame of the ) λ µ ( e ) Dynamical symmetry breaking β In this way, the NG bosons generate massive propagators for those gauge fields to which they Note that (as in the abelian case) the scalar potential The gauge invariance is expressed, as it was in absence of symmetry breaking, through the numerator in their propagator Eq.( are coupled. Long range forcesnon vanishing only expectation survive values in the subgroup particle. other scalar fields through groupare rotations the and eigenvectors hence with zero couple eigenvalue themthe of expansion only the of to scalar the the mass potential NG matrix given bosons. by These the quadratic term in where the tadpole at the tree level consideredLagrangian above. Eq.( An explicit example of this feature will be given for the transversality of the polarization tensors Eqs.( gauge field propagator. This isthe because Lagrangian the Eq.( trilinear term arising from the covariant derivatives in from which follows the mass matrix In terms of the non-zero eigenvalues gauge fields, stem from the long range NG boson fields encoded in their A Brief Course in SSB II vectors takes the same form as Eq.( the polarization scalars Eqs.( verify below that this polethe has gauge no field propagator observable transfers effect the aspolarization degrees such. of of the freedom On massive of vectors. the the NG Indeed, other bosons on hand, to the its the mass third shell absorption degree in of PoS(Corfu2012)131 2 5 µ q / Γ 1 term, (3.17) (3.18) (3.19) (3.20) (3.21) ) 2 p ( 1 Σ . ) François Englert A ( , µν ) the axial vertex 2 F , / ) q µν 1 ) which comes from NG F − 4 1 . p ) = ( symmetry. Chiral anomalies 2 − 1 µ 3.21 ) ) m − ( 1 , V S 2 ( ( 5 ). It is a consequence of the ) Σ γ 2 U ) or ( µν q . + ( × F axiovector propagator axiovector fermion propagator ) 3.15 0 5 acquires a non vanishing ) A γ ( 1 µν ) 3.12 ) ̸= ( 2 F Π 2 . 2 p ) 1 4 / U ), ( ( ν µ q 2 µ 1 ) and ( q − q q Σ 3.9 µ + depicted in Fig.6 5 µ q γ p ) = − A µ 2 3.10 ( ) m − 1 A q ψ 10 2 2 ( 2 , we get 5 − p ) q γ q S ( A a gauge invariant mass → Fig.6 ( µ 2 5 µν 7 Σ µ Π ¯ g (taking for simplicity ν5 2 µ ) = ψγ ( Γ γ q 2 p A 2 A m 0 / e e µ q γ → − 2 = lim q − ) = ) µ 2 p A V µν ( , m ψ 2 ( Π µ / 1 q Σ ¯ ψγ + at are abelian field strength for V p . In leading order in e of the axial vector field ) µ5 ( 0 m Γ ) 5 A − A ( µ = µν ( F Γ 0 2 µν Π µ q F L q = and acquires in this approximation L ) µ V A ( µν F This example illustrates the fact that the transversality of the polarization tensor used in the The validity of the approximation, and in fact of the dynamical approach, rests on the high momentum behavior of The pole in the vertex function induces a pole in the suitably regularized gauge invariant The symmetry breaking giving mass to gauge vector bosons may arise from the fermion con- The Ward identity for the chiral current 7 boson contribution. the fermion self energy, but this problem will not be discussed here. singularity in the vacuum polarisation scalars Eqs.( quantum field theoretic approach totrue mass whether vector generation masses is arise through a fundamental consequence fundamentalcondensate. BEH of bosons a The or through Ward generation fermion identity. of Thisapproximation” gauge used is invariant to masses get is the therefore propagators Eqs.( not contingent upon the “tree with The field polarization tensor shows that if the fermion self-energy thus a dynamical mass densate breaking chiral symmetry. This is illustrated by the following chiral invariant Lagrangian Here A Brief Course in SSB II are eventually canceled by adding in the required additional fermions. develops a pole at PoS(Corfu2012)131 2 in ], q 2 / 2 ν ϕ ], the q (3.22) (3.23) (3.24) (3.25) (3.26) µ 3 field in µ ∂ ] q ⟩ µ 1 − A ϕ ⟨ ] indicates its e µν 1 / g field defined in François Englert 1 [ µ B in accordance with , 2 . pole mentioned in the ⟩ ], he derived the BEH 1 µν ν 2 3 ϕ 2 q F q ⟨ q µ / 2 . = q 1 e µ . 2 µ B a 1 0 B . µ ν } = ∂ + µ µ − 2 A B a ⟩ ν 2 1 µ B ⟩ ϕ , / 1 2 µ ⟨ a ν ϕ ∂ in the transverse projector 0 e ⟨ q µ ], which states that, in relativistic quantum 2 0 µ 7 − = = e − , q 2 = } 2 6 ϕ + µν µ − 2 q µ ), the expectation value of the BEH boson to be q A G µν ∂ ⟩ µν 11 ) is the conventional massive vector propagator. It 1 3.8 G g ⟩{ ϕ ν 1 ⟨ ] ) differs from a conventional free massive propagator ∂ ϕ = e and 3 3.26 ⟨ , 2 e − 2 2 q 2 3.15 ϕ / = ϕ 2 , ν µ a µ q ∂ 0 µν ∂ µ to first order, one gets the classical equations of motion to that µ ⟩ { 1 F 2 q field into this massive vector field = − µ ν ϕ 1 ϕ 2 ⟨ ∂ ∂ − µ µ q e A B µν µ − ), taking as in Eq.( g ∂ µ A ≡ is a massive vector field with mass squared 3.1 . Second the NG pole at to zero) η µ = µν a ); this will be made explicit in the next section. B µ η D B 3.14 ) while the second term is) a ) pure which gauge converts propagator the due to the NG boson ( ) as (putting ) shows that ). The massive vector propagator Eq.( In this formulation, we see clearly how the Goldstone boson is absorbed into aThe redefined equation of motion approach is classical in character but, as pointed out by Higgs [ Shortly after the above analysis was presented, Higgs wrote two papers. In the first one [ From the action Eq.( , and expanding the NG field 3.24 3.24 3.15 3.11 3.25 ⟩ 1 ϕ Eq.( Eq.( formulation of the BEH mechanism in the quantum field theory terms of reference [ The first term in themay right be hand viewed side as of the Eq.( (non-abelian generalization of the) free propagator ofEq.( the 3.5 The renormalization issue in two respects. First theof presence the gauge of parameter the unobservable longitudinal term reflects the arbitrariness Eq.( massive vector field which has noquantum longer field explicit theory gauge approach invariance. is The related same to phenomenon the in unobservability the of the validity in the quantum regime.ability We of now the show BEH how theory. the latter formulation signals the renormaliz- is unconventional. Its significance is made clear by expressing the propagator of the Defining one gets order A Brief Course in SSB II 3.4 The equation of motion approach [ ⟨ discussion of Eq.( field theory, spontaneous symmetry breakingNG of a bosons, continuous fails global in symmetry the implies case zero of mass gauge field theory. In the second paper [ he showed that the proof of the Goldstone theorem [ theory in terms of the classical equations of motion, which he formulated for the abelian case. Eq.( PoS(Corfu2012)131 ) ]. Q 0 9 [ ϕ , 8 + charac- ϕ ( 2 is usually a µ e ), is the clue / ν propagator. In François Englert q 3.26 µ µ q B with corresponding − pole in the projector, ) 2 1 µν ( q g / U 1 × . The BEH bosons ) acts on left-handed fermions θ 2 ) ( 2 cos ( SU / e and the electric charge SU ′ = . Breaking occurs in such a way that in the numerator of the massive gauge ′ Y 2 g 2 / , + q 1 θ 3 . The / ′ ν = T propagator. It is the transversality of the self Y q ′ sin µ µ = µ Y / B q ′ e B Q 12 g − = g µν and g ]. as free propagators, which is only true in a gauge invariant a charge is T θ 10 µ ) µ a 1 B . ( ) v gA U , and 0 ] ( ) we get the mass matrix µ 11 ] was not complete because closed Yang-Mills loops, which would have required 2 A 9 ) which appeared in the field theoretic approach contains thus, in the √ 3.14 / 1 and their 3.15 ) = 2 ( ⟩ ) and ( ϕ ⟨ SU 3.11 ]. To prove this statement, one must verify that the theory is unitary, a fact which 9 propagator. This is in sharp contradistinction to the numerator ). As already mentioned, the transversality is a consequence of a Ward identity and there- µ a A The proof given in reference [ Using Eqs.( The most dramatic application of the BEH mechanism is the electroweak theory, amply con- In the electroweak theory, the gauge group is taken to be The propagator Eq.( On this basis we suggested that the BEH theory constitutes indeed a consistent renormalizable 3.15 8 are in a doublet of the introduction of Fadeev-Popov ghosts were not included. 4.1 The electroweak theory [ firmed by experiment.mulate Considerable Grand work Unified theories has ofwell been known non done, ideas gravitational and using interactions. then evoke thebosons, We the because BEH shall construction they of summarize mechanism, raise regular here potentially to monopolesthe these important and for- attempts conceptual flux to issues. lines using include We BEH shall gravityfocuses in also in this mention the context briefly unification on an quest, interesting in geometrical interpretation the of so the called BEH mechanism. M-theory approach, and of the consistency of thewas BEH achieved by theory. ’t Hooft A and full Veltman [ proof that the theory is renormalizable4. and Consequences unitary generators and coupling constants generates an unbroken subgroup, coupled tois which characterized by is the massless photon field. Thus the vacuum fore does not depend on theexample tree presented approximation. above This but fact is was already proven suggested in from more the general dynamical terms in a subsequent publication is not apparent in the “renormalizable” covariant gauges because of the covariant gauges, the transverse projector A Brief Course in SSB II but would be manifest inthe the unitary “unitary gauge however, gauge” renormalization defined from power inat counting the the is free not free manifest. theory level, The by between equivalence, theory the the where their difference is the unobservable NG propagator appearing in Eq.( field theory [ field The importance of this factmines is the power that counting the of transversality irreduciblequantum diagrams. of field It the theory is self-energy formulation then in is straightforward covariant renormalizable to verify by gauges that power deter- the counting. BEH expressed in terms of the mixing angle teristic of the conventional massive vector field only. The electromagnetic charge operator is energy in covariant gauges, whichEq.( led in the “tree approximation” to the transverse projector in PoS(Corfu2012)131 (4.2) (4.3) (4.4) (4.1) , be it a Z and François Englert  W . 0 = 2 A M . , 1 ) − led to tentative Grand Unifica- 2 ) g where the string has been put (see ) G . 3 ) 2 + ( π π ′ 2 ′ √ 4 , 2 g = SU gg ( g , one can take the line-singular potential = . ϕ = ( − θ 0 2 ∇ 4 ( ⃗ Z 2 v ′ eg ) v = 2 ∈ θ ′ gg 0 0 r = ⃗ g as , theory, the Dirac string in non abelian gauge 2 − Z -axis cos ) z G 2 M 1 0 0 0 2 b g 13 increase logarithmically with the energy scale, the − ( n n , r r ) 1 2 π 2 U 0 0 0 0 0 ( 1 g 2 g iab ( π 2 ε g 4 2 = v 4 U ]. Breaking occurs through vacuum expectation values 4 v π g 4 = = eg 12 = | [ 2 − A = ⃗ ) µ 2 W | 3 ai ( M A , SU 2 g × 2 ) 4 v 1 ( = to the the Fermi coupling U + v × 2 W ) M 2 ( SU In contradistinction to the string in the In electromagnetism, monopoles can be included at the expense of introducing a Dirac string An example is the SO(3) monopole, represented in Fig.8, arising from the potential The discovery that confinement could be explained by the strong coupling limit of quan- Although the electroweak theory has been amply verified by experiment, the existence of ]. The latter creates a singular potential along the string terminating at the monopole. For containing 13 Fig.7). The unobservability ofthe the Dirac string condition implies that its fictitious flux be quantized according to of BEH fields and unificationgroup can makes be the small realized gauge at coupling high of energies because while the renormalization [ instance to describe a point-like monopole located at This potential has a singularity along the negative 4.3 Monopoles, flux tubes and electromagnetic duality tum chromodynamics based on the “color” gauge group converse is true for the asymptotically free non abelian gauge groups. groups can be removed by aing gauge the singularity line for singularity well to chosen a quantized point magnetic like charges, singularity. reduc- 4.2 Grand unification schemes This permits to relate whose diagonalization yields the eigenvalues A Brief Course in SSB II the BEH boson has, asboson yet, is not more been sensitive confirmed.genuine to elementary It dynamical field should assumptions or be than aHence noted the manifestation observation that of of massive the its a vectors physics mass composite ofmechanism and due at the width work. to BEH is a of more particular elaborate interest mechanism. for further understanding of the tion schemes where electroweak andG strong interaction could be unified in a simple gauge group PoS(Corfu2012)131 2 one can ) . Namely, y 2 3 G ( François Englert SO is the subgroup of the Z ]. For a general Lie group . This implies that a bound 15 must be a curve continuously B y r where 3 G z Fig. 7 isospace Z ) means that the regular monopole ) . Gauging out only occurs for the 3 Z ( 4.4 / ˜ G SO × = of higher rank [ 1 x G G space ) A 3 14 ( . 2 . This is the origin for the unconventional factor of y 2 SO B Z Z r / Dirac string such that are characterized by θ ) Fig. 8 2 ϕ z ( G G of SU ˜ by a BEH field belonging to the adjoint group G ) = ) with the monopole is a space-time fermion. In this way, fermions 3 1 ]. ( ( 2 / 3 16 U gauging out 1 A SO x unobservability B r 3 ) as z 4.4 ) in Eq.( 1 π x 2 . 2 The construction of regular monopoles hasThe interesting mixing conceptual between implications. space and isospace indices in Eq.( One can define regular monopoles in a limit in which the BEH-potential vanishes. These This procedure can be extended to Lie groups = ]. π , the possibility of gauging out the Dirac string depends on the global properties of 14 4 state of a scalar ofcan isospin be made out of bosons [ is invariant under the diagonal subgroup of Breaking the symmetry to remove the point singularity to[ get the topologically stable ’t Hooft-Polyakov regular monopole G A Brief Course in SSB II the mapping of a smalldeformable circle to surrounding zero. the Dirac Closedcenter string curves of onto in the universalcurve covering corresponding to the unit element( of PoS(Corfu2012)131 ) N ( U zero length 2 N François Englert massless vectors 2 ]. This led, for the N 18 Fig. 9 coincident branes. There are 15 N symmetry is completely broken, is a relativistic analog of G Dp-branes invariance. At rest, BPS Dp-branes can separate from each other in the ) ]. N ( massless scalars for each dimension transverse to the branes. The open string 17 2 U N BPS Dp-branes coincide, they admit massless excitations from the when all the branes are at distinct location in the transverse space, because strings N N ) 1 ( U When The BEH mechanism operates within the context of gauge theories. Despite the fact that grand The BEH-mechanism, when oriented strings with both end attachedand on additional the sector has local transverse dimensions at no cost of energy. Clearly this can break the symmetry group from first time, to an interpretationquantum of states. the area Here entropy weBEH mechanism. of shall some explain black how holes Dp-branes in yield terms a of geometrical a interpretation counting of of the up to joining two different branes have finiteonly length remaining and hence massless now excitations describe are finitesame then mass brane. due excitations. to The the zero length strings with both ends on the unification schemes reach scales comparable tothat Yang-Mills the fields Planck offer any scale, insight there into quantum was,which gravity. a had The priori, some only approach no success, to indication in quantum particular gravity entropies, in are the the context of superstring a theoryapproaches quantum approaches interpretation (Type and of IIA, the the IIB, black possible Type holes mergingclassical I of limit and the would the five be two perturbative 11-dimensional heteroticdiscovery supergravity. strings) of into Of Dp-branes an particular along elusive interest M-theory which in whose the that ends context is of the open strings can move [ 4.4 A geometrical interpretation of the BEH mechanism are the BPS monopoles.that They electromagnetic admit duality a can supersymmetric be extensions realizedelectric at in and a magnetic which fundamental charge there level, could are namely be that indications realized the by interchange equivalent but of distinct actions. superconductivity. The latter may beis viewed then as channeled a into quantized condensation flux of tubes.into electric In quantized charges. confinement, tubes. it Magnetic is Therefore the flux electric-magnetic electric dualityconfinement flux suggests which is that, is at channeled a some condensation fundamental level, BEH of mechanism magnetic [ monopoles and constitutes the magnetic dual of the A Brief Course in SSB II PoS(Corfu2012)131 is is ∝ = a d ) mn (4.6) (4.7) (4.8) (4.9) (4.5) µν µ define l mn F A y ) ( ∗ , and ] n dimensions j nm y 1 > [ , François Englert + − m is p , . a gauge fields given ⟩ ) 2 (= ) ]. , ) m i mn j | N mn )] j ] mn 20 y y j l n ( y , − x ( fermions A i 2 appearing in the covariant . The gauge invariance is , 19 − i + } N + l ( m i m mn A 1 ) x [ x ) | µ 2 { )( n ] A j mn j k n y ⟩⟨ x . The A n | , − . In this way Dp-branes provide a ] i mn j = ( n . k m may be viewed in the string picture as µ A x A [ j A ( mn . The mass of the vector meson , j gauge fields [ N mn 1 4 i y ) , y ) m and − 1 A 2 by − ( [ N ) + | i kl ( m n µ in a defining representation. U δ x m A ⃗ mn U 2 A ⟨ ) µ ) δ where n the directions transverse to the branes. − , n N j D m ) i m ( i x m ⃗ ; x 2 j ∑ up to 16 x ⃗ , A , U i ( − ) = 1 µ are the field theoretic expression of the unobservable 1 4 m ) NG bosons. To identify the latter we consider the N ⟩ ∝ D ) x ( = ⃗ n n N | 1 2 NG bosons, as can be checked directly from the coupling [( j x a U ⃗ 2 mn ] = ( ) − j ab A ( m − 2 | a Dp-branes. This action is the reduction to N δ A ) m ). N m , Tr x − acquire a mass N ⃗ ⟨ i = ( i mn 2 n y + 4.5 A are the BEH expectation values and the b ∥ ( N ) , two zero eigenvalues corresponding to the eigenvectors ][ ( matrix elements of j } ) ̸= matrices, that is, up to an equivalence, by all diagonal matrices µν mn j n l mn m 2 A is a generator of F y m y x } ⃗ , N ( a < i { µν V i mn ∂ T A 2 m a F x brane coordinates and [ ), namely ( ) ∂ . { 4 1 i 1 Tr n k mn A , 4.5 = 1 4 y + Tr µ i ( m where p − D ∂ A = as coordinates transverse to the brane i a . Label the = } A T V } µ i m ai x L mn { A δ in the Lagrangian Eq.( i m TrD hermitian scalar fields = x µ ) , as is easily checked by computing the quadratic terms in i labels the { . These are the required n A ) N A x ⃗ l n µ = , x to − a ̸= } As mentioned above, the breaking of The states of zero energy are given classically, and hence in general because of supersymmetry, This symmetry breaking is induced by the expectation values This symmetry breaking mechanism can be understood as a BEH mechanism from the action The Lagrangian is i 2 T − m A i mn N l m x a ⃗ ( µν x x of ( by the non diagonal elements the number of transverse space dimensions. The mass matrix for the fields due to the stretched strings joining branesidentifies separated the in the dimensions transverse to thethen branes. the One mass shift dueThe to unobservable the NG bosons stretching of the otherwise massless open string vector excitations. and has for each pair We write Here the diagonal elements of 10-dimensional supersymmetric Yang-Mills with d longitudinal modes of thegeometrical strings interpretation joining of the the BEH branes mechanism. if where describing low energy excitations of A Brief Course in SSB II derivatives F ensured, as usual, byscalar unobservable potential ( in Eq.( { by all commuting PoS(Corfu2012)131 ]. The can be are the i 21 } A , t i mn x D 20 [ { François Englert 0 = -branes are largely 0 p 1961 246. 124 matrices commute at large dis- , of the BEH expectation values 0 } = i mn x p { = i 9 A go to zero. This suggests that the space-time i mn -brane coordinates (viewed as partons in the in- y 0 (1961) 345; Phys. Rev. 17 which enters the covariant derivative t (31 August 1964) 321. 122 A 13 ]) are the analog, for (1961) 154. 21 19 (19 October 1964) 508. (15 September 1964) 132. 13 (1960) 380. 12 4 . Hence they get locked in their ground state when the D have a positive potential energy proportional to the distance squared between n ) ) then describes a pure quantum mechanical system where the n which define in string theory D x ⃗ and } 4.5 − i m m x m case, although they label now classical collective position variables of the quantum { x ⃗ ( 0 ⊥ ̸= . The p -branes mn ) 0 Y. Nambu and G. Jona-Lasinio, Phys. Rev. J. Goldstone, Il Nuovo Cimento F. Englert and R. Brout, Phys. Rev. Lett. P.W. Higgs, Physics Letters P.W. Higgs, Phys. Rev. Lett. Y. Nambu, Phys.Rev.Lett. y ⃗ References of the original work on the BEH theory have been fully dated. The dates given here are the publication Originally the BEH mechanism was conceived to unify the theoretical description of long Physics, as we know it, is an attempt to interpret the apparent diversity of natural phenomena in It may be worth mentioning the interesting situation which occurs when N 9 ( [2] [3] [4] [5] [6] [1] dates; submission dates follow the same sequence. range and short range forces.didate for The further success unification. of the Grandclose electroweak unification to theory schemes, the made where scale the the of scale mechanism quantumto gravity of a include effects, unification can- raised gravity. is the pushed possibility Thisthe that trend unification developments might towards of also the string have theory questThe and for latter from unification was its received then connection a viewedbative with further string as eleven-dimensional theories a impulse supergravity. would classical from merge. limitsuggestive In of of that the a existence context, hypothetical of the an M-theory geometrization underlying into of non which the commutative geometry. BEH all mechanism pertur- is References Lagrangian Eq.( finite momentum frame in reference [ terms of general laws. By essencelaws. then, it incites one towards a quest for unifying diverse physical 5. Remarks geometry exhibits non commutativity atan small essential distances, element a of quantum feature gravity. which may well turn out to be A Brief Course in SSB II dynamical variable. The timeput equal component to zero, leavingSU a constraint which amounts to restrict the quantum statesin to the singlets of mechanical system and not vacuum expectationdom values. The nondiagonal quantum degrees of free- tance scale and define geometrical degreesshort of distances freedom. where However the these potential matrices energies do of not the commute at the D separated from each other. In this way, the D0-brane PoS(Corfu2012)131 20 , invited talk at François Englert , Interscience (1967) 1264; A. Salam, 19 (1974) 403, (JETP Lett. 20 (1974) 1974. (1989) 2073. 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Particle , Physics, 1997). June 24-27, 1992; (Printed in H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev.P.A.M. Lett. Dirac, Phys. Rev. G ’t Hooft, Nucl. Phys. F. Englert and P. Windey, Phys. Rev. B125 Lett. S.L. Glashow, Nucl. Phys. J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. S. Elitzur, Phys. Rev. F. Englert, R.Brout and M. Thiry, Il Nuovo Cimento For a detailed history on this subject, see M. Veltman in (1974) 199). 1997 Solvay Conference, Publishers J. Wiley ans Sons, p 18. [9] [7] [8] [20] [21] [16] [17] [18] [19] [14] [15] [11] [12] [13] [10] A Brief Course in SSB II