Strongly interacting Higgs sector in the minimal Standard Model?

Physics Letters B 309 (1993) 127-132 North-Holland

PHYSICS LETTERS B

Strongly interacting Higgs sector in the minimal Standard Model ?* Karl Jansen 1, Juhus Kutl 2 and C h u a n Llu 3 Department of Physics 0319, Umverslty of Cahfornla at San Diego, 9500 Gdman Drive, La JoUa, CA 92093-0319, USA Received 9 December 1992, revised manuscript received 12 April 1993 Editor H Georgl

The triviality Hlggs mass bound IS studied with a higher derivative regulator in the spontaneously broken phase of the four dimensional 0(4) symmetric scalar field theory with quartlc self-interaction The phase diagram of the 0(4) model is determined in a Monte Carlo simulation which interpolates between the hypercublc lattice regulator and the higher derivative regulator in continuum space-time The same method can be used to calculate the Higgs mass bound in continuum space-time In a large-N analysis, when compared with a hypercublc lattice, we find a relative increase in the triviality bound of the higher derivative regulator suggesting a strongly interacting Hlggs sector In the TeV region with neghglble dependence on regulator parameters When the higher derivative regulator mass is brought close to the Hlggs mass the model requires a more elaborate analysis of complex ghost states in scattering amplitudes

1. Introduction

We develop a systematic method to investigate in c o n t i n u u m space-time the possibility of a strongly interacting Higgs sector in the m i m m a l Standard Model with a heavy Hlggs particle in the TeV region This question, which is directly relevant for the Hlggs physics program of the Superconducting Supercolhder Laboratory, has never been answered in a reliable fashion, although speculations and phenomenological approximations are well-known We report here some results on our non-perturbatlve investigation of the problem In earher lattice studies the trlviahty upper b o u n d for the Hlggs mass was found at 640 GeV under some well-defined set of conditions [ 1-3 ] with a lattice mom e n t u m cut-off at 4 TeV One can show from the equivalence theorem [4] that lattice artifacts with the 4 TeV cut-off remain hidden to a few percent accuracy below 1 3 TeV center of mass energy (the physics reach of the SSC) m the experimentally rele-

* 1

2 3

This work was supported by the US Department of Energy under Grant DE-FG03-91ER40546 E-mail address jansen@higgs ucsd edu E-mall address kutl@sdphjk ucsd edu E-mail address chuan@higgs ucsdedu

ElsevierScience Publishers B V

vant WLWL cross sections, if the Hlggs mass is kept below the upper b o u n d There has been great concern that this finding was an artifact of the lattice regulator which breaks euchdean invarlance In fact, the first significant increase of the Hlggs mass b o u n d was reported [5] within the Symanzlk improvement program on a hypercublc lattice structure [6] Similar results on different lattice structures, with higher dimensional lattice operators in the interaction term, were also reported [7,8] In this work we replace the lattice regulator with a higher derivative kinetic term in the Hlggs lagranglan which, accordmg to conventional thinking, acts as a Pauh-Vi11ars regulator preserving all the relevant symmetries of the theory We wdl investigate the regulator dependence of the theory without lattice artifacts The phase diagram of the 0 ( 4 ) model will be determined in a Monte Carlo simulation which interpolates between the hypercubxc lattice regulator and the higher derivative regulator m c o n t i n u u m space-time The same method can be used to calculate the triviality H~ggs mass b o u n d in c o n t i n u u m space-time In a large-N analysis *~1, when compared with a hypercubic lattice, we find a relative increase m the trivial#1 Variants of the large-N analysis in the study of the O(N) model were developed earlier [7-10] 127

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lty b o u n d o f the higher derivative regulator suggesting a strongly interacting Higgs sector m the continuum with negligible dependence on the regulator mass parameter Consider the euclidean action of the O (4) symmetric scalar field theory in four dimensions with a higher derivative term in the kinetic energy,

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smoothly interpolate between the lattice regulated theory and the finite Paull-Vlllars theory in the continuum In computer simulations the lattice spacing a is set to one for convenience, and m 2, M 2 are measured m lattice units After rescahng the continuum fields by 0 = ( o ~ the euclidean lattice action can be written as

1)

1

V]3

__ ~m00 I 2

0 + ~ 0 ( 0 " 0 ) 2] ,

(1)

where the scalar field 0 has four components and [] is the euclidean Laplace operator, the bare parameters m 2, 2o are the same as in the ordinary 0 ( 4 ) theory The higher derivative term M-41--q 3 acts as a P a u h Villars regulator with mass parameter M It represents a m i n i m a l modification o f the continuum model, if we want to render the field theory and its euclidean path integral, Z = ftd0l

exp{-SE[0]},

(2)

finite To understand the physics o f higher derivative regulator effects we had to derive the euclidean path integral from the hamlltonian formulation in the Hllbert space of q u a n t u m states We also had to understand the spectrum o f the hamlltonian, the role of ghost states in scattering processes, and the closely related issues o f unltarity and microscopic acausahty effects Results along these lines were reported in previous publications [ 11,12 ]

2. Phase diagram and Pauli-Villars limit F o r non-perturbative computer investigation of the higher derivative lagrangian we introduce a hypercublc lattice structure The lattice spacing a defines a new short distance scale in the theory with the associated lattice m o m e n t u m cut-off at A = ~z/a We will have to work towards the large A / M limit of the continuum theory in order to eliminate finite lattice effects from the already regulated and finite theory The presence of the lattice parameter a allows us to 128

+ 2(tp t,O-- 1)21 ,

~0 + (1 - 8x)~o • q~

(3)

where the continuum operator [] is replaced by the equivalent hypercubic lattice operator [] = ~ u ( J x , x + u + J~,x-u - 2Jx,x) The lattice points are labeled by the variable x, and/~ designates unit vectors along the four independent directions The mass parameter and couphng constant are related to the hopping parameter ~c and the lattice coupling 2 by m 2 = (1 - 8/¢ - 22)/K, 20 = ,~/4t¢ 2 In the limit o f infinite 2 the radial mode of the scalar field becomes frozen at [~ox] = 1 m our normahzatlon This is the limit o f the non-hnear e - m o d e l where we expect to find the maximal upper b o u n d for the Hlggs boson mass The lattice model defined by eq (3) can be studled non-perturbatlvely in computer simulations The standard and popular stochastic algorithms which are based on local updating schemes are ineffective when applied to this particular lattice model, exhibiting severe problems of critical slowing down The very long autocorrelatlon times became particularly prohibitive for small values o f the P a u h - V d l a r s mass parameter, in the range M < 1 In this regime the M - 4 p 6 term of the kinetic energy in the higher derivative lattice lagranglan leads to a dramatic broadening of the elgenfrequency spectrum of the Fourier modes which is the primary source o f very long autocorrelatlons We developed a version of the H y b r i d Monte Carlo Algorithm ( H M C ) where the elgenmodes were accelerated by Fast Fourier Transformation ( F F T ) techtuques This new algorithm solved the problem of critlcal slowing down very effectively The H M C algorithm with F F T techniques allowed us to study the lattice model of eq (3) at the physically important values o f the parameters

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O3

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Hlggs phase

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aM/(l+aM) Fig 1 The phase diagram of the interpolated Pauh-Vlllars lattme model at infinite bare coupling is shown The dotted hne xs the analytic form of the critical line in the large-N approximation The data points of critical x values are simulation results m the 0 ( 4 ) model The solid line displays the fixed M/rnH = 10 ratio towards the aM ~ 0 Pauh-Villars limit of the higher derivative continuum theory, it is stretched by a factor of 5 along the x-axis for better separation from the phase transmon lane The lattice model defined by the euchdean partition function has two phases, as expected The phase diagram is shown in fig 1, in the 2 = oc limit The phase transition line is plotted as the dotted line from a large-N calculation and the full circles represent our simulation results The large-N a p p r o x i m a t i o n also predicts that the phase transition as o f second order along the whole transition line, in agreement with the simulation results In the symmetric phase we find the original massive particle (with four components in the intrinsic 0 ( 4 ) space), and also a complex ghost pair with intrinsic 0 ( 4 ) symmetry whose mass scale is set by the P a u h - V l l l a r s mass p a r a m e t e r M In the broken phase a Higgs particle was found with mass ran, and three massless Goldstone excitations with residual 0 ( 3 ) symmetry In addition, the 4-component heavy ghost particle and its complex conjugate partner also appear in the spectrum o f the broken phase The continuum limit, which eliminates the underlying lattice structure, is equivalent to the tuning o f K to its critical value for fixed bare coupling, so that the

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physical Higgs mass mE --+ 0 in lattice units At the same time the physical (renormahzed) mass MR o f the regulator particle also decreases towards MR ~ 0 in lattice units, but the ratio roll~MR has to be kept fixed by the appropriate adjustment o f M In this limit the higher derivative continuum theory is obtained with finite mass scale for the ghost states in Hlggs mass units The solid hne in fig 1 displays the fixed M/mH ---- l0 ratio in a large-N calculation Moving on the line o f constant M/mH towards the a M ~ 0 limit is indicated by arrows as we are approaching the higher derivative continuum theory At fixed lattice spacing a, the a M -~ oo hmlt will produce the ordinary 0 ( 4 ) field theory on a hypercubic lattice Approaching the dotted critical line for any fixed M corresponds to the hmlt of trivial free field theory

3. The Higgs mass bound and large-N The Hlggs mass bound in the simple 0 ( 4 ) model on a hypercubic lattice structure was determined earher [ 1-3 ] The calculation had a few important ingredients To obtain non-vanishing mH/V values we had to move away from the critical line of the phase diagram where the ratio vanishes (v denotes the vacuum expectation value o f the Hlggs field) The ratio m n / v grows as a function of decreasing A/mE, when moving away from the critical line An approximate upper bound on mH/V exists in the region where A l m n 1s not large enough to keep lattice cut-off effects reasonably small in physical quantities For fixed A/mH the largest m n / v ratio is obtained at Infinite bare Hlggs coupling When A / m E = 2~z is chosen, one finds the ratio mH/V = 2 6 at 20 = oo which leads to the 640 GeV Hlggs mass b o u n d It corresponds to a rather small renormahzed quartlc coupling constant, excluding a strongly interacting Hlggs sector in the simple O (4) model with hypercubic lattice regulator The reason one should not push the lattice cut-off much lower than A/mH = 2~z IS as follows In physical cross sections lattice cut-off effects (breakdown o f euclidean invarlance) begin to appear when A/mH gets smaller A somewhat arbitrary but useful measure of lattice cut-off effects is the ratio R of the perturbative continuum cross section for elastic Goldstone particle scattering at 90 ° and the equivalent regulator dependent cross section at finite cut-off Deviations from 129

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R = 1 indicate the size ofcut-offdependence [2] For

A/ml~ >I 2n, and center-of mass energies W <~ A/n, the cut-off effects in R do not exceed the few percent level [2] In the region A / n <~ W <~ 2 A / n there is a rapid growth in R, and for W >t 2A/n the cut-off effects become large, as expected when one gets close to the lattice m o m e n t u m cut-off As a test of the above scenario on lattice regulator dependence, the Higgs mass b o u n d can be estimated in the large-N approximation as we interpolate between the simple lattice O (N) model and the continu u m O ( N ) model with Pauh-Villars regulator The mass of the Hlggs particle is given by

mHM -- C ( a M , 2 o ) e x p

16/r2V2 ~ . -~m2n ,] ,

5

-q

lltt 1

2

3

4

5

~, ~ i 05 pva.nt R

"-~

1

(b) 0 95

(4)

where the amplitude C(aM,2o) is calculable from a Goldstone loop diagram At fixed M / m n , Cvv(A0) = llmaM~oC(aM, Ao) is the higher derivative (Pauh-Vlllars) regulator limit of the amplitude in c o n t i n u u m space-time At fixed a, CLAT(,~0) = h m a M ~ [ a M C (aM, A0) ] corresponds to the original hypercubic lattice theory without higher derivative regulator term (the reason for the prefactor a M in front of the amplitude C(aM,20) is the replacement of m n / M by a m n on the left side of eq (4) in the M ~ c~ limit) Numerically both amplitudes are the largest in the 20 ~ ~ limit where we find Cpv = e 1/4 and fLAT = e 2 896, for N = 4 Using eq (4) and the numerical value of CLAT we plot m n / v as the dotted line of fig 2a against the lattice correlation length ~/. = (arnn) -~ of the simple 0 ( 4 ) model at 20 = oo The solid line is the plot of m n / v against ~Pv = M / m H in the higher derivative c o n t i n u u m theory, with Pauh-Villars regulator mass M and Cpv in eq (4) The two independent curves are plotted together by using a j o i n t numerical ~-axis for the convenience of comparison Since Cpv as significantly smaller than CLAT, the solid P V curve runs above the dotted lattice curve As was found earlier, in the a M ---, oo hmlt, mlYa = 0 5 corresponds to few percent lattice effects in the physical cross section of elastic WLWc scattering at center of mass energies W <~ 2mn [2] In fig 2a arrow points to the dotted curve at this Hlggs correlation length representing the upper b o u n d of the simple 0 ( 4 ) model on the hypercubic lattice in the large-N approximation At fixed M / m n = 4, an the a M ~ 0 130

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Fig 2 (a) The ratio mH/v Is plotted m the large-N approximation of the lattice model and the higher derivative continuum model at 20 = oo using a joint ~-axls for the convenience of comparison, (L = aml.l, and gPV = M/mH, respectively Arrows mark the respective H~ggs mass bounds (b) The ratio R of the perturbatlve continuum cross section for elastic Goldstone particle scattering at 90° and the equivalent regulator dependent cross section at finite cut-off is plotted The dotted line represents R m the large-N calculaUon with Pauh-Vdlars regulator m the continuum theory, and the sohd hne is the same calculation for the lattice model of the higher derivative theory Deviations from R = 1 indicate the size of regulator effects

Pauh-Vlllars limit, we find a comparable few percent deviation from R = 1 (ghost effects) in the WLWL cross section of the higher derivative c o n t i n u u m theory for W <~ 2m14, as shown in fig 2b In the figure we show the ratio R as a function of the center of mass energy W when evaluated in the Pauh-Vlllars regulated theory In the c o n t i n u u m (dotted line), and the lattice model at a M = 0 8 (solid line) In fig 2a arrow points to the solid curve at ~PV = 4 representing the upper b o u n d of the higher derivative c o n t i n u u m 0 ( 4 ) model in the large-N approximation In fact, the ratio M/ml4 = 4 is a rather conservative choice since R deviates from 1 on the few percent level even at the much lower value of M / m H ~ 3 which would imply a larger Hxggs mass b o u n d A change in the Hlggs mass b o u n d can be found by comparing the ratio mH/V m the two different schemes,

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(mn/v)Pv (mH/V)tAT =

,/In(amn)EAX + In CLAT(20)

(5)

Using the large-N numbers for CLAT and Cpv at 2o = ~ , we find from fig 2a and eq (5) that the Higgs mass bound increases from m H ~ 3V to m H ~ 5V as we interpolate from the hypercubic lattice action to the continuum higher derivative regulator In a related large-N analysis [7,8] the CLOT and Cpv coefficients were also calculated However, the large relative increase of the Higgs mass bound in the higher derivative continuum model with respect to the simple lattice model was not pointed out We have to emphasize at this point that it is the relatzve increase o f the Higgs mass b o u n d as described by eq (5) which is the i m p o r t a n t hint from the large-N a p p r o x i m a t i o n As was pointed out earlier [ 10 ], N = 4 is too far from N = oo for any rehable absolute estimate of the Hlggs mass b o u n d by using the large-N expansion The main problem is that the N + 8 factor in the fl-functlon and other perturbatlve formulae creates a mismatch between perturbation theory and large-N for N -- 4 This mismatch disappears only for N >> 8 The difficulty is clearly illustrated by fig 7 6 o f r e f [8] where the width o f the Hlggs resonance is plotted as a function o f the Hlggs mass mH The Hlggs self-coupling is perturbatlve in the mass region mn <~ 700 GeV and, therefore, the width o f the Hlggs resonance from the large-N analysis should agree with the perturbatlve calculation Instead, a large discrepancy is exhibited between perturbation theory and the large-N expansion which cannot produce reliable absolute results at N=4 It is Indicative, however, that the relative Increase o f the Hlggs mass b o u n d is remarkably stable eq (5) was derived from the large-N result o f eq (4) where the finite width o f the Hlggs resonance was neglected The finite width corrections, and N + 8 effects in Hartree approximation, lead to a more c o m p h c a t e d numerical analysis which lowers the absolute positions o f the curves in fig 2a, but the relative increase o f the upper b o u n d in the higher derivative continuum theory, when c o m p a r e d with the simple 0 ( 4 ) model on the lattice, remains approximately as large as in the simplest large-N expansion It is suggestive that the large relative increase o f the upper b o u n d is a

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robust feature o f the higher derivative theory In this case we expect the Higgs mass b o u n d to be driven Into the TeV mass range wtthout noticeable ghost effects in the accessible energy range of WLWL scattering at the Supercolhder Our first simulation results on the mass spectrum are supporting this scenario, with the possibility of a strongly interacting Hlggs sector in the minimal Standard Model

4. Conclusions We beheve that there are two complementary views on the higher derivative lagranglan o f the Hlggs model According to conventional thinking, the higher derivative kinetic term acts as an ordinary cut-off on a mass scale set by M In this letter, we studied the Hlggs mass bound under the condition that cut-off effects associated with the regulator remain hidden from experiments The possibility o f a strongly mteracting Higgs sector was raised in the minimal Standard Model, with two independent parameters mE and v Scattering anaphtudes were restricted to the energy range below the mass of the conjugate ghost pair where dependence on the mass scale M is almost neghglble In a more unconventional approach, the higher derivative lagrangian is analyzed as a logically conSlstent and finite field theory [12] with calculable scattering a m p h t u d e s at arbitrary energies In this scenario a heavy ghost particle with complex mass is a d d e d to the minimal Higgs sector with observable physical consequences A strongly interacting Higgs sector emerges naturally, but the physical scattering processes depend on the new mass parameter M in a non-trivial fashion The non-perturbative computer investigation of the higher derivative lagranglan required the introduction of an underlying hypercublc lattice structure It is important to work in the large A / M limit where lattice effects are negligible compared with ghost effects, so that the finite continuum theory is simulated We hope to return to further non-perturbatlve investigations o f both viewpoints in the higher derivative Higgs model

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Acknowledgement We thank Jeroen Vlnk and other members of the Particle Theory G r o u p at U C San Diego for useful conversations and comments We are also grateful to Howard Georgl and a referee who suggested improvements in the presentation This work was supported by the DOE under G r a n t DE-FG03-91ER40546 The s~mulations were done at Llvermore National Laboratory with DOE support for supercomputer resources

References [1]J Kutl, L Lm, Y Shen, Nucl Phys (Proc S u p p l ) B 4 (1988)397, Phys Rev Lett 61 (1988)678 [2] M Luscher and P Welsz, Phys Lett B 212 (1988) 472 [3] A Hasenfratz, K Jansen, C B Lang, T Neuhaus and H Yoneyama, Phys Lett B 199 (1987) 531, A Hasenfratz, K Jansen, J Jersak, C B Lang, T NeuhausandH Yoneyama, Nucl Phys B 317 (1989) 81

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[ 4 ] B W Lee, C Q m g g a n d H B Thacker, Phys Rev D 16 (1977) 1519, J M Cornwall, D Levln and G Tlktopoulos, Phys Rev D 10 (1974) 1145 [5] M Gockeler, H Kastrup, T Neuhaus and F Zxmmermann, Nucl Phys (Proc Suppl ) B 26 (1992) 516 [6] K Symanzlk, Nucl Phys B 226 (1983) 187 [7] U M Heller, H Neuberger and P Vranas, Phys Lett B 283 (1992) 335, [8] U M Heller, H Neuberger and P Vranas, preprlnt FSU-SCRI-92-99 (1992) [ 9 ] M B Elnhorn, Nucl Phys B 246 (1984)75, MB Emhornand D N Wdhams, Phys Lett B 211 (1988) 4570 ]10] J Kutx, L Lln and Y Shen, Lattice Hxggs Workshop, eds B Berg et al (World Sclentffic, Singapore, 1988) p 186 [11] K Jansen, J Kutl and C Llu, preprmt UCSD-PTH92-39 (1992), talk presented by J KutL in Proc Conf LATTICE 92 (Amsterdam), to be pubhshed [12] K Jansen, J Kutl and C Lau, Phys Lett B 309 (1993) 119