It is noted that spontaneous symmetry treaking implies
a finite cosmological term in the Einstein equation for
gravity. The present theories of weak and e.m. interactions
disagree violently with the experimental limit on such a
term unless an ad-hoc counter curvature is introduced

Veltman, M.J.G.

Utrecht University Repository

COSMOLOGY AND THE HIGGS MECHANISMM. VeltmanRockefeller University, New York, N.Y. 10021andUniversity of Utrecht, NetherlandsABSTRACTIt is noted that spontaneous symmetry treaking impliesa finite cosmological term in the Einstein equation forgravity. The present theories of weak and e.m. interactionsdisagree violently with the experimental limit on such aterm unless an ad-hoc counter curvature is introduced.- 2 -1. Spontaneous symmetry breaking is an essential ingredientin gauge theories of elementary particle interactions. Upto now, the breaking is achieved through the introduction ofscala'r particles, that subsequently develop a non-vanishingand finite vacuum expectation value. Many physicists feelnowadays the necessity for these Higgs bosons as a cumbersomeand not very appealing burden; in this note we wish to pointout another feature of this mechanism. The effect discussedhere occurs already in the tree approximation, and in thissense it is a classical effect, quite distinct from zeropoint energy shifts resulting from quantum corrections. Herewe want to state explicitly that there is no logical diffi-culty regarding the now circulating models. However, moreambitious theories of nature, that also include gravitationas part of the symmetry scheme may experience great difficul-t i e s .Briefly stated, the following happens. In case of aspontaneously broken symmetry the vacuum becomes a state inwhich a certain finite amount of energy-momentum density issmeared over all space. This may be sensed by gravitation,and we may expect a curvature of space-time due to this. Ifwe assume that, before symmetry breaking, space time isapproximately euclidean then after symmetry breaking to theamount needed to generate the appropriate masses for thevector mesons of weak interactions, a curvature of finite butoutrageous proportions result. The reason that no logicaldifficulty arises is that one can assume that space-time wasoutrageously "counter curved" before symmetry breakingoccurred. And by accident both effects compensate so pre-cisely as to give the (in this context) very euclidean uni-verse as observed in nature. Actually, in the context of anexploding universe, very entertaining speculations can bemade, but we will refrain from that.2. As a simple model we consider the Lagrangian of gravi-tation and a scalar complex isospinor field in inter-action with themselves and each other;The K-field is like the Higgs field in the Weinberg model.The gravitational field enters through the tensorFurther g denotes the determinant of ,and is the inverse of To first order in ,The constant is related to Newton'sconstant G as expressed by the equation If unitsare such that = c = 1 we find = 5.8 x fromG = (6.6T3 0.003) x Tofirst order in theequation determining the vacuum expectation value of thefield K is as usual:3- 4 -If we substitutewe obtain the Lagrangianwhere contains terms involving the fields Z and Theconstant C is determined by the parameters and ExpressingC in terms of and we findThe constant C has the dimension of a andwithin the context of weak interaction theories its magnitude,give or take a few orders of magnitude may be expected to beof the order for the purposes of weak interactions.In fact, in those cases one has M = vector boson mass, m =Higgs particle mass and f = weak interaction coupling constant.However, there are strong experimental limits on themagnitude of C. If the equations of motion corresponding tothe above Lagrangian are written down we find:where contains the and Z fields.This is the Einsteinequation with a cosmological term. Cosmological consider-ations yield the limitThis results in:which is clearly in violent disagreement with the numberquoted above ,The discrepancy can be removed by introducing a termin the Lagrangian. This ad hoc removal of a finiteeffect to suit experiment is in our opinion not very satis-factory.3» To provide for a little perspective we want to exhibitmore explicitly the way such symmetry breaking acts togenerate a vector meson mass. In the absence of symmetrybreaking, the equation of motion for a vector meson fieldis :- 5 -- 6 -The spontaneous symmetry breaking gives rise, in to aterm of the form and the equation becomes the equationof motion for a massive vector field:This is quite analogous to what happens to the Einsteinequations of gravity. However, in the vector-meson case thegauge symmetry of the theory forbids the introduction of anad hoc counter mass term. But in itself, if this mass gener-ation mechanism is taken seriously it is quite illogical notto take it serious with respect to gravitation.4. Some further consequences of spontaneous symmetrybreaking occur if the improved energy-momentum tensor is em-ployed. This amounts to the introduction of a termin the Lagrangian. After spontaneous breakdown aterm R and a term R Z appear. They are however toosmall to be of any direct consequence.5. Acknowledgments. The author is greatly indebted toProfessors G. Feinberg, H. Pagels and A. Pais for very stimu-lating discussions.- 7 -Re ferences1. For reviews and further references see:E.S. Abers and B.W. Lee, Phys.Reports 9C, No. 1 (1973).M. Veltman, Proceedings of the Bonn Conference, Aug. 1973.2. Notations and conventions as in:G. 't Hooft, Nucl. Phys. B35, 167 (1971) especiallyse ction 6;G. 't Hooft and M. Veltman, example of a gauge fieldtheory, Marseille conference, 19-23 June 1972, C.P. Korthals-Altes, ed.;G. 't Hooft and M. Veltman, CERN preprint TH 1723, Aug. 1973,to be published in Annales de l'Institut Henri Poincare.3. See for instance:C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation,W.H. Freeman and Co., 1973.

Cosmology and the Higgs mechanism

It is noted that spontaneous symmetry treaking implies

a finite cosmological term in the Einstein equation for

gravity. The present theories of weak and e.m. interactions

disagree violently with the experimental limit on such a

term unless an ad-hoc counter curvature is introduced

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http://dspace.library.uu.nl:8080/bitstream/handle/1874/7336/cosmology_and_the_higgs_mechanism.pdf?sequence=1

Cosmology and the Higgs mechanism

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