641 HIGGS BOSONS John ELLIS CERN - GENEVA Switzerland ABSTRACT The theoretical motivations for the existence of Higgs bosons are reviewed, and their properties in simple models are described. The Higgs boson mass is very uncertain, but there may be a prefe­ rence for it to be about IO GeV. Experiments to look for Higgs bosons are discussed. RESUME On passe en revue les motivations theoriques pour l'existence des bosons de Higgs, et on decrit leurs proprietes dans des modeles simples. La masse du boson de Higgs est tres indeterminee, mais pourrait bien etre de l'ordre de IO GeV. On discute des experiences pour chercher des bosons de Higgs. 642 I. INTRODUCTION Why should Higgs bosons exist ? What are th<!ir properties ? How can one detect them experimentally ? These are the three universal questions about Higgs bosons that will be addressed in this talk. The motivations for their existence are that theories with just mas­ sive vector bosons and fermions blow up at high energies,which means that higher order corrections to theit low energy predictions are infinite and uncalculable1). Sensible renormalizab le gauge theories which make finite predictions, such as the 2) Weinberg-Salam model ,must3) use spontaneous symmetry breaking to give particle masses, and as far as we know this entails the existence of Higgs Bosons. These considerations are discussed in more detail in section 2. The Higgs bosons are scalars with an almost totally indeterminate mass, probably between about 7 GeV and TeV in the simplest version of the Weinberg-Salam model, and perhaps with a slight preference for a mass around I 0 GeV4) , The Higgs couplings are very precisely fixed in the simplest model. Those to fermions are proportional to their masses, and therefore the Higgs likes to decay into pairs of the heaviest fermions kiner1atically accessible, e.g. - H -+ T+ T , bb , t't, . These properties are reviE�wed in section 3. Because of their uncertain mass, the best experiments to search for the Higgs boson are less well-defined than those designed to look for the w± and Present experiments only constrain m to be 15 MeV. Promising Z0 • H � ways to look for heavier Higgs bosons include heavy onium + H + QQ y (for mH 2 , 0 decays (for mH m 2 , e+e- + z" + H (for mH up to < mq) Z ) -S z / ) "' 100 GeV with LEP)S . If the Higgs weighs more than 1 00 GeV, which certainly seems theoretically to be a possibility, then only its i�ndirect effects may be visible for the foreseeable future. Experiments to seareh for Higgs bosons are discussed in section 4. 2. MOTIVATIONS Clearly we would like our physical theories to make finite predic­ tions . This means that higher order diagram corrections to the lowest order diagrams should be calculable, as is the case in QED . However, weak interaction theories have always been beset with infinities. For example, the old 4-fermi theory had many incalculable divergences which arose from the circumstance that its lowest order cross-section (cf. fig. I) rose (the centre-of-mass a energy) 2 at large s , which meant that higher order diagrams (see fig. 2) were infinite. It was known for a long time that one way of reducing these divergence 643 problems was to introduce intermediate vector bosons w± and into the current­ Z0 current lagrangian. This made the cross·-sections for fermion-fermion scattering more manageable, but now the cross-sections for ff + w+w- rose too fast at large s , and higher order diagrams like that in fig. 3 were still infinite. It was realized that one way to minimize these residual problems was to give the vector bosons couplings among themselves which were of the gauge theoretical form as in fig. 4. This step removed many infinities, but there were still some + - + - left. That from e e + w w was very small, being proportional to me , but 2 that in w+w- w+w- was very large, being proportional to rn The only w • 2) + known recipe to remove these remaining divergences which occured with massive vector bosons was to introduce scalar boson exchanges as well, as in fig. 5. These Higgs bosons would be responsible for the other particles ' masses, and would have couplings linked to mf or respectively. 3) m; It has in fact been shown that the only theories with cross-sec­ tions that are sufficiently well-behaved at high energies l/s) that their (0 a higher order diagrams are calculable are in fact gauge theories with mass gene­ rated by spontaneous symmetry breaking using Higgs fields. Furthermore, in these mode�s at least one - and possibly many more - of the Higgs fields actually appear as physical, detectable particles. So, if Higgs bosons exist and are so important, what are their properties ? 3. PROPERTIES To discuss the properties of Higgs bosons we will start by looking l)2)6) at the simplest gauge theory, the Weinberg-Salam , , model with a simple + complex weak isospin doublet of Higgs fields � = (� , �0 ). The Higgs self­ couplings take the form of a potential ( I) where must be 0 and µ2 is genErally taken to be 0 . > > As seen in fig. 6, the potential II) then has a minimum at a non-zero value of v2 (2) 2 The lagrangian must also contain gauge-covariant kinetic terms for the Higgs bosons µ ' i wi � (3) 1. 2 1 2 644 When the value of [¢[ goes to the value (2) minimizing the potential these (I), kinetic terms give terms that are masses for the vector bosons . When a this happens , Higgs fields Wtwtget "eaten" to become longitudinal polarization 3 states of the and Z0 w± + ¢ its complex conjugate + - ¢ and (4) , ¢ But 'the fourth field become$ a physieal Higgs boson H ¢0 + � H + v (5) We said before that the couplings of this Higgs boson were connected with parti­ cle masses . When ¢ takes the value v (2) a coupling of the type g ff ¢ + fH + gfH ff(H v) which means that (6) Correspondingly, the kinetic term contains a piece (3) which tells us immediately that 4m2 2 w v (7) and that 2m 2 w (8) v wh ich again exhibits the connectiJn with mass that we "xpected from section 2. The couplings (6) and are totally fixed by the theory, unfortu­ (8) nately the Higgs mass is totally arbitrary. From we find that (I) 2 2/ 2 (9) = � >w2 µ2 We saw above in equation that v 2 is. directly connected to the known (7) = );""' quantity GF , but the individual values of µ2 and are totally unknown at th is point . 4) An attractive possibility is that actually µ 0 . This is (a) = for economy (b) because µ is theoretically a divergent quantity if it is non-zero (c) because is in some sense already constrained to be small. The µ meaning of this last point is that modern grand unification theories contemplate mass scales of the order of 1015 to 1019 GeV . But in order for perturbation 7) theory to be valid for the weak interactions , mus t be 0 which < (I), gives via (2) a limi t on µ 0(103) GeV m TeV 0) µ < � H < I (I Since is in th is sense absolutely microscopic, it is nstural to suppose that for some as yet unknown reason it is in fact zero . But then how does spontaneous symmetry breaking work The answer is ? 8) provided by radiative corrections (fig. 7) to the Higgs potential which (I) have the form (I I) When added to V(¢) gives a minimum at 1¢1 # 0,which would mean that (I I) , m and m would be determined purely the ma sses of all the particles , mf w H dynamically, and none would be put in by hand . In fact 2 (12) � which can be seen as a radiative correction to the naive formula (9) for 0. µ = Taking sin2e 0.20 in equation (12) gives w = 10.4 GeV (13) 2 For various reasons such as the uncertainty in sin 8w and the possible exis­ tence of heavy fermions , the value (13) should be given a sizeable error (0(± 2) GeV but it at least indicates an interesting ball-park to search for ?) a Higgs boson. The value (12) is not actually a strict lower limit on . With � o ons it is possible that the potential have a minimum the radiative c rrecti (II) at /¢1 # 0 even if µ2 is slightly O. The requirement that the absolute < ) minimum of the potential be at 1¢/ # 0 in fact constrains9 7.3 GeV ( 14) > O) 0.20 . One could be even more radical l , and suppose that the world is not in fact sitting at an ab solute minimum of the potential , but merely at a local minimum. This will be unstable, but its lifetime will exceed the age 646 of the universe years) if (101 0 26 MeV (15) > 0 This is probably the absolute lower limit on , but more reasonable limits (JO) � would be those of equations and (14) , with perhaps a slight preference for the neighbourhood of JO GeV (12). couplings (6) and tell us that Higgs bosons will like to The (8) decay into pairs of the heaviest particles kinematically available - which may be a very distinctive signature Table indicates the dominant decay modes ! I and branching ratios for different values of m . Different branching ratios H 4) and production rates for a Higgs boson with mass around JO �eV are shown in more detail in fig. We see that in this range decays into T+T- and cc 8 are to be preferred .