Table Of Content1
A Heavy Higgs Particle in the TeV Mass Range ?
a(cid:3) a a
Chuan Liu with Karl Jansen and Julius Kuti
a
Department of Physics 0319,University of California at San Diego, 9500 Gilman Drive, La Jolla, CA
92093-0319,USA
The (cid:12)rstsimulation results arepresented on Higgs mass calculations in the spontaneously broken phase ofthe
Higgs sector in the minimal Standard Model with higher derivative regulator. A heavy Higgs particle is found in
the TeVmass range in the presence of a complex conjugate ghost pair at higher energies. The ghost pair evades
easy experimental detection and a nonperturbative reinterpretation of the triviality Higgs mass bound becomes
necessary.
p
1. Introduction originalO(4)particleisgivenby!p = p2+m20
whereas the complex energy of the ghost state is
p
Following the systematic method that was de- (cid:10)p = p2+M2,withM=Mei(cid:2);thecomplex
veloped earlier [1] we report our (cid:12)rst simulation
phase(cid:2)isafunctionofm0 andM. Theoperator
results on a heavy Higgs particle in the presence (+)
a(cid:11);p creates the physical particle with momen-
of a higher derivative regulator in the Higgs sec-
(+)
tor of the minimal Standard Model. The quan- tum p and energy (cid:22)h!p. The operator b(cid:11);p cre-
tization procedure in Minkowski space-time with atesaghoststatewithmomentumpandcomplex
(+)
inde(cid:12)nite metric for the higher derivative Higgs energy h(cid:22)(cid:10)p; c(cid:11);p creates the complex conjugate
Lagrangian, ghost state. The interaction part of the Hamilto-
nian is proportional to the coupling constant and
1 1 3
L = (cid:0) 2 (cid:8)(cid:11)2(cid:8)(cid:11)(cid:0) 2M4 (cid:8)(cid:11)2 (cid:8)(cid:11) expressed in terms of creation and annihilation
operators.
1 2 2
(cid:0) m0 (cid:8)(cid:11)(cid:8)(cid:11)(cid:0)(cid:21)0 ((cid:8)(cid:11)(cid:8)(cid:11)) ; (1) Inthebrokenphase thespectrum is morecom-
2
plex. ThereisaHiggsparticlewithmassmH,and
with O(4) symmetry leads to the euclidean par-
three massless Goldstone excitations with resid-
tition function [1]
ual O(3) symmetry. In addition, there is a com-
Z (cid:18) Z (cid:19)
4 plexconjugatepairofheavyghoststateswiththe
Z = D(cid:8) exp (cid:0) d xLE (2)
internal quantum numbers of the Higgs particle,
andathree-component complexghostpair which
where LE is obtained from Eq. (1) by replac-
transformsas O(3)triplet according tothe quan-
ing the Minkowski operator 2 with the euclidean
tum numbers of the Goldstone particles. The
Laplacian.
heavy masses of the Higgs ghost and the Gold-
The general particle content of the model is
stone ghost are split by the symmetry breaking
exhibited in the symmetric phase by the Hamil-
mechanism.
tonian in terms of creation and annihilation op-
erators, H =H0+Hint,
2. Lattice Action and Phase Diagram
X
(+) ((cid:0)) (+) ((cid:0))
H0 = (cid:22)h[!p a(cid:11);p a(cid:11);p+ (cid:10)p b(cid:11);pb(cid:11);p
For non-perturbative computer simulations of
+ (cid:10)(cid:3)p c((cid:11)+;p) c(cid:11)((cid:0);p) ] (3) the higher derivative Lagrangian we introduce a
hypercubic lattice structure. The lattice spacing
wherethesummationisoverinternalO(4)indices
ade(cid:12)nes anewshortdistancescale withtheasso-
and momentum modes. The dispersion of the
ciated lattice momentum cut-o(cid:11) at (cid:3)=(cid:25)=a. We
(cid:3)SpeakerattheLattice93Conference,UCSD/PTH93-39 will have to worktowards the large (cid:3)=M limit in
2
ordertoeliminate(cid:12)nitelatticee(cid:11)ectsfromtheal- is shown in Fig. 1 in the (cid:21) = 1 limit. Tuning
ready regulated and (cid:12)nite theory. After rescaling the hopping parameter (cid:20) to the critical line for
p
the continuum (cid:12)eld components by(cid:8)(cid:11) = 2(cid:20)(cid:30)(cid:11) (cid:12)xed M corresponds to the limit of trivial scalar
the euclidean lattice action has the form (cid:12)eld theory in the continuum. In this limit, the
(cid:0)4 3
X 23 dimension eight operator M (cid:30)(cid:11)2 (cid:30)(cid:11) becomes
SE = [(cid:0)(cid:20)(cid:30)(cid:11)(2+ M4)(cid:30)(cid:11)+(1(cid:0)8(cid:20))(cid:30)(cid:11)(cid:30)(cid:11) irrelavant and while mH ! 0 in lattice units,
x MR=mH ! 1, so that ghost e(cid:11)ects from the
+(cid:21)((cid:30)(cid:11)(cid:30)(cid:11)(cid:0)1)2] (4) higher derivative termdisappear. Inournotation
mH isthephysicalHiggsresonancemassandMR
where 2 is the lattice Laplace operator, with the designates the real mass parameter of the exact
lattice spacing a set to one for convenience. The ghost pole location in the complex plane. The
massparameterandcouplingconstantarerelated continuum limit of the higher derivative theory
to the hopping parameter (cid:20) and lattice coupling without the underlying lattice structure is equiv-
2 2
(cid:21) by m0 =(1(cid:0)8(cid:20)(cid:0)2(cid:21))=(cid:20), (cid:21)0 =(cid:21)=(4(cid:20) ). alent to the tuning of (cid:20) towards zero along aline
Stochastic algorithms exhibit severe critical of (cid:12)xed MR=mH ratio. In this limit, which is
slowing down for small values of the regulator depicted in Fig. 1 as the solid line, the higher
mass M in the range M < 1. In this regime derivative term M(cid:0)4(cid:30)(cid:11)23(cid:30)(cid:11) cannot be treated
(cid:0)4 3
the M 2 term leads to a dramatic broadening as an irrelevant operator in the Lagrangian.
of the frequency spectrum of the Fourier modes.
We developed a Hybrid Monte Carlo algorithm
3. Results and Discussion
with fast Fourier acceleration which solved the
problem of critical slowing down very e(cid:11)ectively. Most of our computer simulations were per-
formed on lattices which have cylinder geome-
3 3
try in the size range 16 (cid:2) 40 to 20 (cid:2)40. On
each lattice with agiven parameter set 20;000to
60;000Hybrid Monte Carlo trajectories were ac-
cumulated in the Fourier transformed variables.
On each time slice (cid:28) the radial Higgs (cid:12)eld (cid:26)((cid:28))
and the O(4) unit vector n(cid:11)((cid:28)) at zero three-
momentum were de(cid:12)ned as
1 X
(cid:30)(cid:11)((cid:28))(cid:17) L3 (cid:30)a((cid:28);x)(cid:17)(cid:26)((cid:28))n(cid:11)((cid:28)) : (5)
x
The renormalized vacuum expectation value
v and the Higgs mass mH were determined
from the correlation functions hn(cid:11)(0)n(cid:11)((cid:28))i and
Figure 1. Thephase diagramofthe lattice model h(cid:26)(0)(cid:26)((cid:28))i.
at in(cid:12)nite bare coupling. The dotted line is cal- The (cid:12)tting procedure requires a theoretical
culated in the large-N expansion. The solid line understanding of the energy spectrum which is
displays the (cid:12)xedMR=mH ratiotowardsthecon- organized in the large volume limit according
tinuum limit of the higher derivative theory. to inverse powers of the linear spatial size L.
The tower of O(4) rotator states forms the most
The lattice model exhibits two phases, as ex- densely spaced energy levels with level spacing of
2 3 (cid:0)1
pected. ThesymmetricphasewithfullO(4)sym- the order (v L ) . The rotator states dominate
metry is separated by a second order phase tran- the correlation function hn(cid:11)(0)n(cid:11)((cid:28))i [7]. The
sition line from the broken phase which has a Goldstone excitations have a level spacing which
(cid:0)1
residual O(3)Goldstonesymmetryforevery(cid:12)xed isoftheorderL . Inthecorrelationfunctionsat
valueof(cid:21)inthe((cid:20);M)plane. Thephasediagram zero three-momentum only Goldstone pairs can
3
contribute whichhaveratherlargeenergies inthe To compare our new results with earlier Higgs
volumes we are considering. The largest level mass values, the ratio mH=v is plotted as afunc-
spacings correspond to radial Higgs excitations tion of the bare coupling constant in Fig. 2. If
with a (cid:12)nite energy gap mH above the ground our preliminary results will not change after a
state. more developed analysis, mH (cid:21) 1 TeV with a
Asimultaneous (cid:12)ttomH andv wasperformed ghostlocationinthemulti-TeVregionimpliesthe
on the data. The results are shown in Table 1 existence of a strongly interacting Higgs sector,
where all points correspond to a cylinder size of a scenario which was excluded in previous lat-
tice studies. Our heavy Higgs mass values in the
Table 1
TeV range do not represent conventional trivial-
Simulation results at M =0:8
itybounds [2{6]at(cid:12)xed bare coupling constants.
(cid:20) (cid:21)0=(10+(cid:21)0) mH v mH=v Since theghostpairevadeseasyexperimental de-
0:056 0:761 0:40(2) 0:058(2) 6:8(4) tection without violating unitarity, Lorentz in-
0:056 0:761 0:32(2) 0:045(1) 7:1(6) variance,oranyothersacredprinciples, anonper-
0:105 0:185 0:30(2) 0:066(2) 4:6(4) turbative reinterpretation of the triviality Higgs
0:115 0:086 0:21(3) 0:084(2) 2:5(4) mass bound becomes necessary.
To test the strongly interacting Higgs sector,
3
16 (cid:2) 40 except for the second entry with size
we also determined the renormalized coupling
3
20 (cid:2)40. The radial Higgs excitation in our sim-
constant in the symmetric phase. The prelimi-
ulations corresponds to a stationary state with
nary results show stronger couplings in compari-
real energy. Avoided level crossing with the low-
son with the lattice regulated O(4) model.
estGoldstonepairofzerototalmomentumwould
only occur in larger spatial volumes. In our 4. Acknowledgements
preliminary analysis the (cid:12)nite volume pro(cid:12)le of
the Higgs resonance was not developed. This This work was supported by the DOE under
rather crude approximation, where we identi(cid:12)ed Grant DE-FG03-90ER40546 and by the Texas
themeasured energy level ofthe radial excitation NationalResearchLaboratoryCommissionunder
with mH, is being subjected to systematic (cid:12)nite Grant RGFY93-206. The simulations were done
volume resonance analysis. atLivermoreNationalLaboratorywithDOEsup-
port for supercomputer resources.
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