Quick viewing(Text Mode)

Standard Model, Higgs Boson and What Next?

Standard Model, Higgs Boson and What Next?

GENERAL ¨ ARTICLE , Higgs and What Next?

G Rajasekaran

One hundred years of Fundamental , start- ing with discoveries such asradioactivity and , have culminated in a theory which is called the Standard Model of High Energy Phy- sics. This theory is now known to be the basis of almost all of known physics except . We GRajasekaran is a give anelementary account of this theory in the theoretical physicist at the context of the recently announced discovery of Institute of Mathematical , Chennai and the .We conclude with brief re- Chennai Mathematical marks on possible future directions that this in- Institute, Siruseri (near ward bound journey maytake. Chennai). His of research is Field 1. Hundred Years of FundamentalPhysics Theory and HighEnergy Physics. The earlier partof the 20th century wasmarked by two revolutions that rocked the foundations of physics: and Relativity. Quantum mechanics becamethebasis for understanding , and then, coupled with specialrelativity, quan- tum mechanics provided the framework for understand- ing the andwhatliesinside. At the beginningof the 20th century, the quest for the understanding of the topped the agenda of fun- damentalphysics. This quest successively led to the unravelling of the atomic nucleus and then to the nu- cleon (the ortheneutron). Now we know that the itself is made of three . This is the level to which we have descended attheendof the 20th

Keywords century. The depth (or thedistance scale) probed thus 17 Standard Model, quantum field faris10¡ cm. theory, high energy physics, , , Higgs boson discovery, .


INWARDBOUND Standard Model is now known to be the Atoms Nuclei Quarks ? ¡! ¡! ¡! ¡! basis of almost ALL 8 12 13 17 of known physics 10¡ cm 10¡ cm 10¡ cm 10¡ cm except gravity. It is the dynamical theory This inward bound path of discovery unraveling the mys- of electromagnetism teries of and the holding it together { at and the strong and deeper and ever deeper levels { hasculminated, at the weak nuclear forces. end of the 20th century, in the theory of fundamental forces based on nonabelian gauge ¯elds, for which we have given a rather prosaic name: `The Standard Model of High Energy Physics'. In this theory, the strong forces operating within the nuclei and within the nucleons, aswellas the weak forces that were revealed through the discovery of radioactivity a hundred years ago are understood to be generalizations of the electrodynamics of Faraday and Maxwell. Electrodynamics was formulated around the year 1875 and its applications came in the 20th century. We owe a lot to the Faraday{Maxwell electrodynamics, for the applications of electrodynamic technologyhave become a part of modern life. People take out a small gadget from their pockets and speaktotheirfriends living hun- dreds or thousands of kilometers away; somebody in a spacelab turns a knob of an instrument and controls a spacecraftthat is hurtling across millionsof kilometers to a distant planet. All this hasbeenpossible only be- cause of electromagnetic waves. It turns out that the dynamics of strong and weak forces was formulated around 1975, almost 100 years after the Standard Model has formulation of electrodynamics. We may expect that been constructed by equally profound applications will follow, once the tech- generalizing the nologies of the strong and weak forces are mastered. century-old Thatmay be the technology of the . electrodynamicsof Faraday and Maxwell.


The four After this bird'seye view of one century of develop- fundamental ments, we now describe the four forces of and forces govern all then take up the Standard Model. of Nature. 2. FundamentalForcesof Nature The four fundamental forces are the strong, electromag- netic, weak and gravitational forces. Strong forces are responsible for binding nucleons into the nucleus (and for binding quarks into the nucleons). They are charac- terised by a strength parameter which is roughly one and 13 their range is 10¡ cm. Electromagnetic forces bind nu- clei and electronstoform atoms and and bind atoms or molecules to form solid matter. Their strength is measured by the ¯ne structure constant whose value is about 1/137 and their range is in¯nite. Weakin- teractions cause the beta decayof nuclei and also are responsible for the fusion reactions that power the 5 2 and stars. Their strength is 10¡ mp¡ and their range is 14 less than10¡ cm. Here mp is the of the proton. Gravity binds the planets into the solarsystem,stars into and so on. Although gravity is the weak- 40 2 est { its strength being 10¡ mp¡ { it becomes the dominant force for theUniverseatlarge, because of its in¯nite range and because of it being attractive only (unlike electromagnetism where attraction canbe cancelled by repulsion). In quantum theory the range of a force is inversely pro- portionaltothemass of the quantum thatisexchanged. Since the mass is zero, electromagnetic force me- diated by the exchange of is of in¯nite range. In quantum theory Since the strong between nucleons has ¯nite the range of a range, it hastobemediated by a quantum (or ) force is inversely of ¯nite mass. This is how Yukawa predicted the parti- cle thatwaslateridenti¯edas the , which we now proportional to the know to be a composite of a and an antiquark. mass of the We shall discusslater more about the ¯nite range of quantum that is the weak force and the quantum exchanged. Since grav- exchanged.

958 RESONANCE ¨October 2012 GENERAL ¨ ARTICLE ity has in¯nite range, quantum theory of gravity (if it Electrodynamics is constructed) will have its quantum, called , and weak forces with zero mass. have been unified The above textbook classi¯cation of the four fundamen- into electroweak tal forces has broken down. We now know thattheweak dynamics. Further force and electromagnetism are two facetsof one entity unification may be called electroweak force. Canonegofurther and unify achieved in future. the strong force with the electroweak force? It is possi- bletodosoand it is called `grand uni¯cation', but that is a speculative step which maybecon¯rmedonlyin the future. The grander uni¯cation will be uni¯cation with gravitation which we maycall `TotalUni¯cation' which was the dreamof Einstein. Perhaps that will be realized by theory and in the future. For the present we have the Standard Model which is a theory of the electroweak and strong andis based on a generalization of elecrodynamics. So let us start with electrodynamics. 3. Laws of Electrodynamics The laws of electrodynamics are expressed in termsof the following partialdi®erentialequations:

~ E~ =4¼½ ; (1) r¢ 1 @B~ ~ E~ + =0; (2) r£ c @t ~ B~ =0; (3) r¢ 1 @E~ 4¼ ~ B~ = ~j : (4) All laws of Nature are r£ ¡ c @t c written in the These laws were formulated by Maxwell on the basis language of of earlier experimentaldiscoveriesbyOersted, Ampµere, mathematics. But Faraday and many others. Actually, from his obser- they can be vations and extensive experimentalstudies of the elec- discovered and tromagnetic phenomena,Faradayhad built up an in- establishedtobetrue tuitive physical picture of the electromagnetic ¯eld and only by experiment.


Maxwell's laws have Maxwell made this pictureprecisebyhismathematical stood the test of formulation. Once Maxwell wrote down the complete for much more than a and consistent system of laws, very important conse- century. But the quences followed. He could show thathisequations ad- classical picture of mitted the existence of waves thattravelled with a ve- the electromagnetic locity thathecould calculate purely from electricalmea- surements to be 3 1010 cm per sec. Since the velocity field has to be £ replaced by quantum of light wasknown tobethis number, Maxwell proposed field theory. thatlightwas an electromagnetic wave. This was a great discovery since until thattimenobody knew whatlight was. Subsequently Hertz experimentally demonstrated the existence of the electromagnetic waves prediced by Maxwell. Maxwell's laws have stoodthetestof time for much more than a century. Even the two revolutions of rel- ativity and quantum mechanics havenotinvalidated them. In fact, Einstein resolved the confrontation be- tween Newton's laws of particle dynamics and Maxwell's laws of ¯eld dynamics in favour of thelatter. He had to modifyNewton's laws to be in conformity with the space{time picture of Maxwell's laws and this is how specialtheoryof relativity wasborn.Evenquantum me- chanics lefttheform of Maxwell's equations unchanged. However, there was a profound reinterpretation of the continuous Faraday{Maxwell¯eldwithitscontinuous energy distribution in space{time. It wasreplaced by Standard Model is discrete ¯eld-quanta or discrete packets of energy. This wascalled ¯eld and this was the birth of written in the . language of quantum field theory. All Let us brie°y compare the Faraday{Maxwell picture with forces or interactions quantum ¯eld theory. In the former, a charged particle, are mediated by say, a proton is surrounded by an electromagnetic ¯eld quanta, like the existing ateverypointinspace{time. If another charged photon which particle, anelectron is placed in this ¯eld, the ¯eld will mediates interact with the electron and that is how the electro- electromagnetic magnetic interaction between proton and electron is to interactions. be understood in the classical electromagnetic theory.


In quantum ¯eld theory, the proton emits an electro- magnetic quantum which is called the photon and the electron absorbs it and this is how the interaction be- tween the proton and electron is to be understood. Ex- change of the ¯eld-quanta is responsible for the interac- tion. This is depicted in the `' shown in Figure 1. This is a brief description of quantum ¯eld theory which is the basic language in which Standard Model of High Energy Physics is written. 4. Standard Model of High Energy Physics Figure 1. Standard Model consists of two parts, electroweak dy- namics that uni¯es electromagnetic and weakinterac- tions, and chromodynamics that governs strong interac- tions. In electrodynamics we have an electromagnetic ¯eld de- scribed by the pair of vector ¯elds (E~ ; B~ ) and the corre- sponding quantum is the photon. Analogously, in elec- troweakdynamics we have four types of generalized elec- ~ ~ tromagnetic ¯elds (Ei; Bi) with the index i going over 1to4,oneof them being the Faraday{Maxwell electro- magnetic ¯eld. Correspondingly there exist four elec- troweakquanta, also called electroweakgaugebosons. One of them is the photon °,mediating electromagnetic + interaction and the other three W , W ¡ and Z mediate weakinteraction. In Figure 2weillustrate anexample of weakinteraction, namely the decayof the into proton, electron and antineutrino. Neutron and proton are depicted as composites of three quarks udd and uud respectively. The d quark turns into a u quark by emitting the weak Electroweak quantum W ¡ which turns into a pair of (elec- interactions are tron and antineutrino). The electromagnetic and weak mediated by four interactions among the quarks and the leptons mediated quanta called by the electroweakquanta are pictured in the Feynman electroweak gauge diagrams of Figure 3. , photon being one of them.


The weak boson W mediates all nuclear beta decays including the decay of the neutron.

Figure 2.

Figure 3.

Exchange of W can generate a force between or can cause decay of a particle. Photon and Z boson can only generate forces among the particles.


The laws of electroweakdynamics (EWD) are given by Strong interactions the equations: are caused by that are ~ E~ + ::: =4¼½ ; (5) r¢ i i exchanged between quarks; there are 1 @B~i ~ E~ + + ::: =0; (6) eight gluons. r£ i c @t

~ B~i + ::: =0; (7) r¢ 1 @E~ 4¼ ~ B~ i + ::: = j~ : (8) r£ i ¡ c @t c i We shall explain the dots in the equationssoon. In (QCD) governing strong interactions we have eight types of generalized electro- ~ ~ magnetic ¯elds (E®; B®) with the index ® ranging over 1 to 8. The corresponding quanta are called gluons G® since it is their exchange between quarks thatbind or glue the quarks together to form the proton or neutron. This exchange of gluons between the quarks q (which maybeu or d)isshowninFigure 4. The laws of QCD are given by the same equations asthosefor EWD with the index i replaced by ®.Theanalogy of the laws of EWD and QCD to the originallaws of electrodynamics is obvious. If these generalizations of Maxwell's equations are as simple asmade out above, why did the Standard Model take another hundred years to be constructed? The an- swer lies in the dots in the equations expressing the laws Figure 4. of EWD and QCD. Let us go back to electrodynamics in which every electri- cally charged particle interacts with the electromagnetic ¯eld or (in the quantized version) emits or absorbs a pho- ton. But photon itself does not have and hence does not interact with itself.Inthegeneralization de- scribed above, there are twelve generalized charges, four in EWD and eight in QCD, correspondingto a similar


number of generalized electromagnetic ¯elds. In con- trast to which is just a number (positive, negative or zero), these generalized charges are matrices which do not commute witheach other and hence canbe called `nonabelian charges'1. Electrodynamics which is basedontheabelianchargeiscalled abeliangauge the- ory and the generalization based on nonabeliancharges is called nonabeliangauge theory. In contrast to photon, which is the abeliangauge quantum and does not carry the abelianelectric charge, the nonabeliangauge quanta themselves carry the nonabeliancharges and hence are self-interacting. These self-interactions are shown in Figure 5; both a cubic and a quartic exist. The nonlinearterms expressingthesecouplings are hidden Figure 5. behind our dots and it is these which make the theory of nonabelian gauge ¯elds much more complex thanthe simple Maxwell theory. Nonabeliangauge ¯elds were 1 In mathematics, algebras with introduced by Yang and Mills in 1954 and hence are commuting and noncommuting also called Yang{Mills (YM) ¯elds, but it took many objects are respectively called more importantsteps in the next twodecades before abelian and nonabelian alge- bras. this theory couldbe used to construct the correct Stan- dard Model. Aremark on gravity is appropriate atthispoint.What plays the role of `charge' in gravity? Obviously it is mass in Newton's theory, but is replaced by energy in Einstein's theory.Sincethegravitational¯elditself has energy, it hasto be self-interacting exactly asinthecase of the nonabelian gauge ¯eld carrying the nonabelian charge. However, unlike in thatcase where Yang and Mills showed that a cubic and a com- pletes the theory, in the gravitationalcase one hastoadd vertices of all orders (quintic, sextic,...). This is what Standard Model is makes Einstein's theory of gravitation much more in- the theory of tractable in the quantized version. In complexity, Yang{ nonabelian gauge Mills theory comes between the simple Maxwell theory quanta interacting and the complexEinsteintheory. among themselves and with quarks.


5. The Field and Particle Sectors Gaugesymmetry, The constituents of the accordingto the Stan- part of which will dard Model come in two categories which maybecalled be broken, makes the ¯eld sector and the particle sector. the twelve gauge bosons massless, In the ¯eld sector, we have the twelve gauge¯elds °, like the photon. + W , W ¡, Z, G1, G2,...,G8.Theirquanta are all particles with 1 (in units of ~), exactly like the ¯rst and most familiaroneamong them, the photon(°).Allsuch particles having spin equaltoanintegral multiple of ~ belong to the great family of `bosons' (particlesobeying Bose{Einstein statistics).

1 The particle sector consists of spin 2 particles belonging to the other great family of `' (particles that obey Fermi{Diracstatistics.) Among these,wehave already encountered the two quarks (u,d) and the two `leptons' (º,e). The quarks make up the nucleon and the nucleons make nuclei. Nuclei and electronsmake atoms, molecules and all known matter. The weak radioactive decays involve the º.Thusthequartet of particles con- sisting of a quark doublet and a doubletseemsto be su±cient to make up the whole Universe. However Nature has chocen to repeatthisquartet twicemore,so thatthereactually exist three `generations' of particle quartets each consisting of a quark doublet and a lep- ton doublet: (i) (u,d), (ºe,e); (ii) (c,s), (º¹,¹); (iii) (t,b), (º¿ ,¿). From a fundamental The existence of three generationsisrequired to explain point of view, all the 6 (the experimentally observed) matter{ asym- quarksandallthe6 metry which can solve the cosmological puzzle:howdid leptons are on an the Universe which started as a ¯reball with equalpro- equal footing. All of portion of matter and antimatter evolve into a state them were created in which hasonlymatter? But we shall not delve into equal numbers, but this question here except to mention thatKobayashi and the heavier particles Maskawa predicted, on this basis, the existenceof three decayed into the generations of quarks, even before the three generations lighter ones u,d,e and were experimentally discovered. the .


Due to the self- An important remark about quantum ¯eld theory is in interactions among order here. Although we divided the stu® of the Uni- the gluons of QCD, verse into a ¯eldsector and a particle sector, ¯elds have their quanta which are particles and in quantum ¯eld becomes weak at theory each particle in theparticle sector also hasits asymptoticallyhigh quantum ¯eld; electron, forinstance, is the quantum of energies (called the electron ¯eld. Thus quantum ¯eld theory uni¯es ‘’). ¯eld and particle concepts. But at low energies There is anincompleteness in our description of the the strength grows QCD sector. Both quarks and gluons are not seen di- enormously and rectly in any experiment. They are supposed to be per- confines the quarks manently con¯ned inside the proton and neutron. But and gluons within a this hypothesis of con¯nement which is supposed to be prison of the size of a property of QCDhas not been proved. This impor- the nucleon. tant theoreticalchallenge remains as a loophole. This problem is so intractable thatithas been announced as one of the millennium problems of mathematics. 6. Breaking and Higgs Remember the vast disparity between electromagnetism and the weak force asregards their ranges; one is of in¯- nite range and the other is short-ranged. How does elec- troweak uni¯cation cope with this breakdown of the elec- troweaksymmetry that is intrinsic to the uni¯cation? This is achievedbya spontaneous breakdown of sym- metry engineered by the celebrated which keeps photon massless while raising the of W and Z to ¯nite values.Thusweakinteraction gets a ¯nite range. The experimentaldiscoveryof W and Z with the masses predicted by the electroweaktheory was a great triumph for thetheory. The idea of spantaneous breakdown of symmetry (SBS) in high energy physics originates from Nambu although he applied it in a di®erentcontext. But the stumbling block was the Goldstone theorem. This predicted the existence of a massless spin zero boson as the conse- quence of SBS and prevented the application of SBS to

966 RESONANCE ¨October 2012 GENERAL ¨ ARTICLE construct any physically relevant theory, since such a Higgs showed that is not observed. (See Appendix I at spontaneous the end of this article for more on SBS.)Thusappar- breakdown of gauge ently one had to choose between the devil (massless W symmetry violates boson) and the deep sea (massless spin zero boson). Goldstone theorem It was Higgs who, in 1964, showed thatthisisnotcor- and Kibble extended rect. By using Goldstone's model (which is much sim- the idea to nonabelian pler than the originalNambu model), heshowedthat gauge symmetry. there is no Goldstone theorem if the symmetry thatis Electroweak dynamics broken is a gauge symmetry. The devil drinks up the was then constructed deep sea and comes out as a regularmassive spin one using Higgs–Kibble . No massless spin zero bosonis left. This mechanism. is called the Higgs mechanism. Many other authors also contributed to this idea. Earlier, Glashow had identi¯ed the correctversion of the Yang{Mills theory for the electroweak uni¯cation. By combining that with Higgs mechanism, Weinberg and Salam independently constructed the elecroweakpart of SM in 1967. There is a bonus. Higgs mechanism postulatestheexis- tence of a universal all-pervading ¯eld called the Higgs ¯eld and this ¯eld which gives masses to W and Z also gives masses to all the fermions of the particle sector, except to the neutrinos. Thus, in particular, the masses of the quarks and electron come from the Higgs ¯eld. But there is animportant byproduct of theHiggs mecha- nism: a massive spin zero boson, called theHiggsboson, must exist as a relic of the original Higgs ¯eld. High en- The ergy physicists searching for it in all the earlier particle were experimentally accelerators had failed to ¯nd it. So the announcement discovered in 1982 on 4 July 2012 that the Higgs boson hasbeen sighted exactly at their ¯nally at a mass of 125 GeV atthegigantic particle respective masses called Large Collider (LHC) atCERN, 80 and 91 GeV has been welcomed by everybody. More tests predicted by have to be performed to establish thattheparticle seen electroweak is indeed the Higgs boson. dynamics.


Box 1.

Year Winners Contribution to SM

1978 Glashow, Salam, Weinberg Construction of Electroweak Theory 1983 Rubbia, Van der Meer Discovery of W and Z 1990 Friedman, Kendal,Taylor ‘Observation’ of quarks inside proton 1999 ‘t Hooft, Veltman Proof of renormalizability of EW Theory 2004 Gross, Politzer, Wilczek Asymptotic freedom of YM Theory 2008 Nambu Spontaneous breaking of symmetry 2008 Kobayashi, Maskawa Matter–antimatter

Higgs boson In the last four decades, experimenters have succeeded remained as the only in con¯rming every component of the full SM with three missing component. generations of fermions. Higgs boson remained asthe With its discovery only missing piece. So with its discovery (assuming that announced on 4 July the discovery will be established by further tests), Stan- 2012 (to be dard Model hasemergedas the Standard Theory de- confirmed), Standard scribing Nature.Thisisa greatscienti¯c achievement. Model has emerged SM now deserves a better name! as the Standard We have now completed our descriptionof the SM. We Theory describing list in Box 1theNobel Prizes awarded so fartosome Nature. of the makers of the SM, the theorists who proposed it and the experimentalists who proved it to be right. 7. Beyond Standard Model Neutrinos: Neutrinos are massless in theStandard Mod- el. As already mentioned,Higgsmechanism does not give mass to neutrinos. About 15 years ago, experi- menters discovered that neutrinos do have tiny masses and this hasbeen hailed as a greatdiscoverysincethis mayshowushow togobeyondtheStandard Model. may betheportaltogobeyond SM and that is the importance of the India-based Neutrino Observa- tory (INO) which is about to come up in Tamil Nadu. : Astronomers have discovered thatmostof the matter in theUniverseisnotthekind we are famil- iarwith.Itiscalled dark matter since it does not emit

968 RESONANCE ¨October 2012 GENERAL ¨ ARTICLE or absorb light. Although this discovery has been made The elusive neutrinos already, nobody knows whatthisdark matter is and and the still more only physicists candiscoverthat. A dark matter exper- elusive dark matter iment also will be mounted in the INO cavern (suitably will be pursued in the extended). under-the-mountain In the last four decades after the Standard Model was laboratory of the India constructed, theoreticians have not been idle but have based Neutrino constructed many theories thatgobeyondtheSM.Of Observatory (INO). these we have already mentioned one, namely grand For more on INO, see uni¯cation. Another is thatpostulates www.ino.tifr.res.in the existence of a boson corresponding toevery known and vice versa.Thisisa very elegant symme- try thatleads to a better quantum ¯eld theory than the one on which SM has been built. But if it is right,we have to discover a whole new world of particles equalling our known world; remember we took a hundredyearsto discover the known particles starting withthe electron. There are many more theoreticalspeculations apart from grand uni¯cation and supersymmetry. Butnoneof them hasseenaniota of experimental support so far, even in the LHC. However LHC will have many moreyearsof operation; let us hope new things will be discovered. 8. Quantum Gravity The biggest loophole in SM is thatgravityhasbeen left out. The most successful attempt to construct quantum gravity is the . The role of quantum mechanics coupled with specialrel- ativity in providing the basis for the understandingof what lies inside the atomic nucleus (the microcosm) was mentioned at the beginning of this article.Ontheother hand, it is generalrelativity, which is also the theoryof gravitation that provides the framework for understand- ing the Universe atlarge (the macrocosm). It is a deep irony of Nature thatthetwinrevolutions of quantum and relativity thatpoweredtheconceptual


Quantum Gravity advances of the 20th century and that underlie all the remainsasthe subsequent scienti¯c developments, have a basic incom- most fundamental patibility between them. The marriage between quan- problem of tum mechanics and relativity has not been possible. By physics. relativity, here wemean generalrelativity since has already been combined with quantum me- chanics leading to quantum¯eldtheory. Gravity which gets subsumed into the very fabric of space and time in Einstein's generalrelativity hasre- sisted all attempts at being combined with the quan- tum world. Hence, quantum gravity hadbecomethe most fundamentalproblem of physics attheendof the twentieth century. This is in contrast to all theotherfundamental forces of Nature, namely electromagnetic, strong and weak forces, which have all been successfully incorporated into the quantum mechanical framework. The Standard Model of High Energy Physics thatwehave described is just that, and it leaves out gravity.Thisisthereason for the rise of StringTheory,for it promises to be a theory of quantum gravity. For the ¯rst time in history, we maybeglimpsing at a possible solution to the puzzle of quantum gravity. Actually, string theory o®ers much more than a quantum theory of gravity. It provides a quantum theoryof all theotherforces too. In other words, it can incorporate the standard model of HEP also, within a unifying framework that includes gravity. So, String Theory hasbeenhailed as the `Theory Of Ev- erything' and some theoretical physicists have even had `Dreams of a FinalTheory' { which is actually the title of an excellent book by Weinberg. But one is tempted String Theory is to say the best candidate There are more things in heaven andearth, Horatio, we have for a Than are dreamt of in yourphilosophy correct theory of quantum gravity. { Shakespeare(inHamlet, Act I, Scene V)


Idonotbelievethatstringtheoryorany theory will be The fundamental a or a ¯naltheory. But it is very scale of quantum likely that string theory is the fundamentaltheory for gravity is 10–33 cm, the 21st century. but we have reached only 10–17cm. We In string theory, a is replaced by a one- dimensional object called string asthefundamentalen- have a long way to 33 go and the Inward tity. Its length is about 10¡ cm which is the length scale of any theory of quantum gravity including string Bound Journey theory. The various vibrationalmodesof the string cor- continues. respond to all the elementary particles. String theory automatically contains quantum gravity and thatisits chief attraction. However thatisbought at a price. It works only if the number of space dimensionsis9and including time it is 10. Where are the extra six dimen- sions? They are curled up to form space bubbles existing 33 atdistance scales of the same 10¡ cm! Both the string and the extra curled-up dimensions will berevealed only when we can access such length scales. Remember we 17 have so farreached only 10¡ cm. We have a long way to go. Apart from string theory there are other approaches to quantum gravity. Only the future will tell us which is the right one. In any case, quantum gravity is animportant (albeit distant) frontier and the journey continues.

Suggested Reading

[1] Ashoke Sen, Search for a final theory of matter, Resonance, Vol.5, No.1, pp.4-13, 2000. [2] Rohini M Godbole and Sunil Mukhi, Nobel for a minus sign: The 2004 in Physics, Resonance, Vol.10, No.2, pp.33–51, 2005. [3] Avinash Khare, Subatomic physics – 100 not out and still going strong!, Resonance, Vol.2, No.12, pp.8–16, 1997. Address for Correspondence G Rajasekaran Institute of Mathematical Sciences Chennai 600 113, India. Email: graj@imsc.res.in


Appendix 1 ATutorialonSpontaneous Breakdown of Symmetry (SBS) Consider a simple mechanicalexample: a ball placedonthetopof a hill of circular cross section surrounded by a circularvalley (Figure A). This system has a circular symmetry, but theball is in an unstable equilibrium and will roll down into the bottom of the valley where it will reach a point of stable equilibrium. The ball could have cometoany point along the circular bottom of the valley, but once it has done it, the circular symmetry has been broken. Now replace the hill and valley problem with a problem of ¯eld theory. This is Goldstone's model of the ¯eldwhich hastwo components Á1 and Á2.Thequanta of the scalar¯eldhave spin zero and hence are bosons. The potentialenergyV of the¯eld system as a function of the ¯eld components is chosen to be exactly as in the mechanicalexample and hascircular symmetry in the ¯eld space(see Figure B). It has a maximum energy atpointAwhereÁ1 and Á2 are zero and a minimum all along a circle.

z z V (φ)

A β

x φ1 α y y φ2 Figure A Figure B

Itiswrongtochoosethemaximum of the potential(pointA)as the of the ¯eld system although the ¯eld haszerovalue atthatpointsince it is a state of unstable equilibrium. We canchooseany point along the circle of the minimum of V , astheground state of the system; however once we choose it, the circularsymmetry is broken.Thisisthemechanism of spontaneous breaking of symmetry.


An important consequence follows. Since it does not cost any energy to move around the circular trough of minimum potential, there exists a mass- less particle (the ® mode). As canbeseen from Figure B, movement along a direction normal to this circle (the ¯ mode)costspositivepotentialenergy and this corresponds to the massive particle. The massless mode is called the Nambu{ and this result is called the Goldstone Theo- rem (proved by Goldstone, Salam and Weinberg)whichstates thatSBSof any continuous symmetry results in the existence of the spin zero massless Nambu{Goldstone boson. By the addition of a massless spin one gaugebosontotheGoldstone model, Higgs showed thatthemassless spin zero boson is eaten up bythemassless spin one boson and as a result, the massless spin one boson becomes massive and the massless spin zero boson disappears.This is the Higgs mechanism. The massivespinzeroboson(thebeta mode) however exists and this is the Higgs boson which waseagerly searched for, and presumably found now. Note that in the ground state, the ¯eld is not zero, but is equaltotheradius of thecircleof minimum potential. This is the universalHiggs¯eldexisting everywhere, thatgivesmass to all the particles.

RESONANCE ¨ October 2012 973