REFERENCE IC/8O/32
A INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
INDUCED HIGGS COUPLINGS AND SPONTANEOUS SYMMETRY BREAKING
Abdus Sal am
and
INTERNATIONAL Victor Elias ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1980 MIRAMARE-TRIESTE y&& JL LJLEAL us IC/80/32
International Atomic Energy Agency and United nations Educational Scientific and Cultural Organization
IHTEHMATIONAL CENTRE FOR THEORETICAL PHYSICS
INDUCED HIGGS COUPLINGS AND SPONTANEOUS SYMMETRY BREAKING •
Abdus Sal am International Centre for Theoretical Phyaica, Trieste, Italy, and Imperial College, London, England,
and
Victor Elias Department of Physics, University of Toronto, Ontario, Canada.
MRAMABE - TRIESTE Karen I960
* To be submitted for publication.
Consider a qnuqe theory of scalar (^) and vector bosons. Perturbative renormalizability implies the existence of a term in the scalar field potential in order to cancel the logarithmic infinity induced by the gauge interaction (Fig.l). ABSTRACT However, it has been recently demonstrated that self-energy and vertex insertions in a given scalar-amplitude graph {such
as those in Fig. 1) are mimicked by replacing 9renormaiize
-1- -2- we find that Consider the case of a scalar field f. transforming as a triplet under SUI2) £ (j[ ~ ef +$, +
In fact, ^LIC is negative for any gauge model, as Tr{M ' ]
4; is always greater than zero. Had we not substituted g
for g, A^a would have corresponded to a negative infinity
to be absorbed in primitive A_ff terms. If A^a is rendered (1) finite by substitution of g for g, a primitive Ag term
(w and K are orthonormal to the uncoupled gauge field is no longer required to absorb infinities. However, in the \fyj~ ^(4ty'* $ (V *• ^» '*£//£ where Wj , _ correspond JT*' to7~-matrices.] In Fig. lc , w and W_j exchanges lead absence of such a term, the induced coefficient of g to equal contributions to the induced ff potential. The is negative and spontaneous symmetry breaking is impossible. 3) h) sum of these contributions is ' This situation can be remedied by introducing a sufficient number of fermions into the theory. The graphs
shown in Fig. 2 induce positive dF couplings that are opposite
in sign to those of Fig.lc because of the closed fermion
C2) loops. It is non-trivial, however, to change the sign of the
The bracketed integral is manifestly UV-finite. In Eef.r. induced potential by adding fermions to the theory. Suppose, arguments are presented invoking a spectral representation for example, that we almost saturate asymptotic freedom of of g to show that the bracketed integral is IR-finite as the SU(2) gauge coupling by incorporating 10 fermion doublets well. Upon Wick rotation to Euclidean^ j_ K' •* (fl^~ n f ~^ f in the theory. For simplicity, we assume that there is
only one free Yukawa parameter h and a Yukawa interaction
given by -U- ("0 where A,B and b are all greater than zero. The solution
This Yukawa interaction can be expressed as for h(t) may be written in the form
(5) t90
where Fx is a column vector containing all 2 0 fermions. where R= hfy/f ^ ( The induced £ coupling from graphs in Fig. 2 is given and o(£j~ EyY*)/f*(tJj i . The gauge by 3) ~n coupling constant is asymptotically free provided b ,> 0. In Appendix I, the Yukawa coupling is shown to be asymptot-
ically free provided fj> 0 and 0 £ R £ j . since vhere ;ft) — n 0 *j£/ where the trace is understood to be the fermion mass i\. obtained through spontaneous symmetry (Wer Dirac Indices as veil 3) breaking is related to the gauge boson mass by AVF will be finite providea ti is replaced by the running Yukawa coupling constant h(-k }: l 7= /°°A cU' JT V( (T) (10) The expression in brackets [ Eq. (10)] increases monoton- VJe now wish to compare the magnitudes of^lv and ically as R approaches its maximum allowed value of } y . In Appendix I the renormalization group equations [note that u(t> > 1 for all tl . Therefore of g" and H are shown to be of the form x ^Jh»J: fi/Tu 4 •2 J (ID Ji (8) -5- -6- Since //4 ^ 1 in our example (Appendix I), we see that and a "positive-eigenvalue" asymptotically free solution X(i)~ kf LhJ [k>o] is guaranteed to exist. Note that the coefficients y andtf~ arc proportional to the coefficients (12) of the divergent part of diagrams in Figs, lc and .2., re- in which case spectively. In our example, /> = 72, Since the factor {-J/z f- $~fZ/£j ) is negative in »ur example, we have is the same as the criterion guaranteeing a positive-eigen- failed to prove that Vln is necessarily positive, despite value asymptotically free solution for X • This is hardly the fact that the addition of even one more fermion doublet surprising, as the same combinatorics manifestly appear from Figs, lc and f . In terms of the notation of Ref.8, . destroys asymptotic freedom for the gauge coupling constant. in which scalar fields p. transform according to Under what circumstances will the induced potential be guaranteed to be positive? Suppose the SU(2) model P.. a sufficient condition for the existence of a positive in- duced coefficient of (P. w a1 i; is that [o( ,p,/\/>, ^are all positive] . If h =/ g , its maximum allowed value, and if {p -/ -7- initial values R for the Yukawa couplings. Suppose, for example, that Qp -f(T ) < 0, thereby guaranteeing that Eq. (15) has a positive eigenvalue solution for ^ when (19) the Yukawa coupling h is at its limiting eigenvalue solution htyrffW . If f-'V(o)/ffc) and where the matrix is chosen to be anything less than f , the eigenvalue solution disappears, as hftj now falls faster with t than $(ty$ ' The corresponding induced - -potential is positive provided Xjf^, MF^J ^ f "2 Lf However, we have already shown that this is true when is expressed in terms of Yukawa-coupling-constant eigen- 12) value solutions h Ct) = fc f ty- manifestly a continuous function of R[Eq. (10)]. There- The criterion of Eq. (18) is also sufficient for the fore, if the difference between ~TUUJS$mr] and / Zlf^^ existence of a positive-eigenvalue solution for the running is finite, there will exist some domain of R. f: f^ fj within any primitive potential which I [H~,KU] remains greater than y* TCfjmi^j • The present in the theory. We claim, however, that the induced potential remains positive over this domain of R. real significance of theories containing such positive-eigen- Thus the presence of positive-eigenvalue solutions to a value solutions, which can hardly be expected to be ra- given theory serves to indicate the possible existence of diatively stable, is that such theories are also likely an asymptotically free theory of a non-eigenvalue character to contain positive induced potentials after running coupling in which positive quartic scalar field couplings are en- constants have been incorporated to allow removal of all tirely induced. primitive £ couplings. There is an important qualitative difference between Acknowledgements the positive-eigenvalue and the induced potential approaches One of us (Victor Elias) wishes to thank N. Isgur, A. Namazie, to total asymptotic freedom; only the latter approach is and T. Sherry for useful discussions. successful for a continuous domain of input parameters - the -10- -0- Appendix I The solution to Eqs. (A.4,5) is found by setting 8) The Running Yukawa Coupling Constant in which case tA.6) The Yukawa interactioon betweeb n f fermion doublets 2-7* At 8) and one fermion triplet i mam y be expressed as Eq. (A.6) is inteyrated using Eq. (A.5): (A.I) In this equation there are f(f + l)/2 independent A - - 'At (A.7) The associated running coupling constants satisfy and the solution (A.8} is obtained inn which f/V = £^f°J/f Yt/J -i £A0f? (A. 2) , and /P - /T ^ - A Wf *fq In the text we consider the case of a single independent The gauge coupling is asymptotically free provided Yukawa parameter such that b } 0, in which case f £ 10. The Yukawa coupling is asymp- totically free provided £<'*) (A-3) O , A sufficient condition for this to be the case is that i The running value of h is then seen to satisfy the following Consider the RHS of Eg. (A.6): differential equation: (A.9) Eq. (A.9) has critical points at k=0 and k=y. The criterion for stability is that (dF/dk i ) < 0. Now As/^/^V _ )~~s The running gauge coupling constant equation is and {^etf-/an i^./J" f . Suppose 3 (A.5) Since n- h/J is positive, the k= f critical point is unphysical. Therefore, physical values of k have stability where = (2 I - 2 properties determined by the k=0 critical point, which is -11- -12- REFERENCES unstable. However, if f}>Q , the k=0 critical point is stable. k(t) will go to zero as t^«9 provided C£'k(t)<"f. 1) A. Salam and J. Strathdee, Phys. Rev. , 4713 (1978). Therefore k(t) is asymptotically free provided f > 0 and V. Elias, Phys. Rev. D2^, 262 (1979). The OSA <.f • case R=/* corresponds to k being con- 3) S. Coleman and E. Weinberg, Phys. Rev. D7^ 1888 (1973) . stant over all t. Note for the example of SU(2) that fc) Our combinatoric and sign conventions for diagrammatic r f - (8S -3C>Jt/(l2Si~3) • Therefore to have an asymp- contributions to the potential are consistent with totically free Yukawa coupling, f must be greater than or those of fief. 3. Since the primitive potential A equal to U, corresponds to a vertex of -i^, an "induced vertex" +/t corresponds to an induced potential cAf /'' i we determine the sign of the induced potential by multiplying the diagrammatic coefficient of^ by +t > The sign of the integral in Eq. (2) is opposite to the sign appearing in Eq. (11) of Jlef. 1, which is in error. 5) The form of g"(-k ) in the 1R region is the subject of some speculation — see R. Anishetty, M. Baker, S.K. Kim, J.S. Bell and F. Zachariesen, Univ. of Washington pre- print RL0-1388-789. g) The existence of a suitably defined renormalizationpoint such that t = /« is implicit throughout. Thus and 7) D-J. Gross and F. Wilczek, Phys. Rev. D8^ 3633 (1973). 8) T.P. Cheng, E. Eichten, and L.-F. Li, Phys. Rev. D9, 2259 (1974). 9) Such positive eigenvalue type solutions are discussed -111- -13- FIG I a in N.-P. Chang, Phys. Rev. Dl£, 2706 (1974); E.S. Fradkin and Q.K. Kalashnikov, Phys. Lett. 59B, 159 (1975); E. Ma, Phys. Rev. D.L7, 623 (1978). 10) Note that a negative diagrammatic f coefficient to the FIG.I divergent part of an integral corresponds to a positive term in the appropriate renormalization group equation ^-function. These signs are sorted out properly in D. Bailin, Weak Interactions, (Sussex University Press, London, 1977), pp. 345-350. 11) See Eq. (2.8) of Eef.S, as corrected in fnotngte 9 of Cliang\Op.Clt Bef.9l- FIG.IC 12) Tl f>j mfj is displayed as a function of R in Eq. (10) . is If Iif,*r7 finite, then function of R over this domain. FIG.2 Figure Captions figure 1. One loop contributions to the £ -potential induced by interaction with the gauge field r.iultiplet. Figure One loop contributions to the -potential induced by the Yukawa interaction. -15- -16- ic/lcj/l22 K. TAFITR fiilA!!: A note on the violation o'1 IC/7&/y8 V.£. fiC3'rfin and E. TOIiATTl: Local fielri corrections to tVo bsndinr INT,HE!1." conserviit ion. . INT.:
IC/79/l46 C, TOME: Change of elastic conatanta induced by point defects in INT.RSP.* hep crystals.
IC/79/1^7 P. STOVlttEK and J. TOIAR: Quantum mechanics in a discrete space-time.
IC/79/l48 M.T. TELIs Quaternionic form of unified Lorentz tranaformationa. IC/60/5 RIAZUDDIN: Two-body D-meson decays in non-relativistic quart model. 1ST.REP.* INT.REP.»
IC/79/l49 tf, KROLIKOWSKI: Primordial quantum chromodynaraica: of a Fjrmi- IC/80/6 G. ALBERI, M. BLESZYNSKI, T, JAPOSZEWICZ and S. SANTOS: Deuteron INT.REP.* Boae couple of coloured preons. INT.REP.* D-wave and the non-eikonal effects in tensor asymmetries in elastic proton-deuteron scattering. IC/79/154 RIAZIFDDIN and PATYAZUDDIN: A model for el«ctroweak interactions oased on INT .REP,* the left-right symmetric gauge «roup U,{2) ®UR{2). IC/80/7 A.M. Kurtatov and D.P. Sankovich: On the one variational principle in INT.HEP.* quantum statistical mechanics. IC/79/158 B, JULIA and J.F. LUCIANI: Son-linear realizations of compact and non- INT.REP.* compact IC/80/3 G. Strata.n: On the alpha decay "branching ratios of nuclei around A = 11 1ST.REP.* Ic/79/159 J. BOHACIK, P. LICHARD, A. NOGOVA and J. FXSUTt Monta Carlo quarlc- INT.REP.* parton model and deep inelastic electroproduction.
IC/79/162 P. BUDIKI: Reflections and internal symmetry.
IC/80/12 M.V. MIHAILOVIC and M.A. NAGARAJAN: A proposal for calculating the IC/79/16U P. BUDINI: On conformal covariance of spinor field equations. INT.HEP.* importance of exchange effects in rearrangement collisions.
IC/80/ll* W. KROLIKOWSKI: Lepton and quark families aa quantum-dynamical systems. • INT. REP.* ',
IC/79/167 T.D. PALEV: Para-Bose and para-Fermi operators as generators of INT.REP,* orthosymplectic Lie superalgebras. IC/80/17 NAMIK K. PAK: Introduction to instantons in Yang-Mills theory. (Part I 1ST.REP.•
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