The mass and the width of the Higgs in the strongly interacting minimal standard model

Volume 237, number 3,4

PHYSICS LETTERS B

22 March 1990

THE MASS AND THE WIDTH OF THE HIGGS IN T H E S T R O N G L Y I N T E R A C T I N G M I N I M A L S T A N D A R D M O D E L A. D O B A D O l CER,V, CH-1211 Geneva 23, Switzerland

Received 30 October 1989

By the use of the [ 1, 1 ] Pad6 approximant to the one-loop amplitude for the WLWL--,WLWLscattering in the strongly interacting regime of the minimal standard model we build up a new amplitude for this process which has all the required physical properties such as analytieity, unitarity, absence of ghosts, etc. We also show that the l = J = 0 partial wave amplitude has a simple pole in the unphysical sheet which can be interpreted as the physical Higgs particle. The mass of this Higgsparticle is shown to be finite and in particular under 1.4 TeV in the limit 2(mH) going to infinity.

Probably the worst known aspect o f the standard model (SM) of the electroweak interactions is the one concerning the symmetry breaking o f the gauge symmetry S U ( 2 ) L × U ( 1 ) r t o U( 1 )cm, i.e. the symmetry breaking sector (SBS). Amongst the long list of models proposed in the literature to describe how this SBS could be (see ref. [ 1 ] for a recent review), the simplest one is the so-called minimal standard model (MSM). In this model, the symmetry breaking is driven by a doublet of complex scalars taking a vacuum expectation value v---250 GeV. The model has only one free parameter, namely the self-coupling of the scalars ~. and typically predicts the existence o f a new particle called the Higgs with a mass which is given at tree level by m2H=2)I.V'2. Nevertheless, it was pointed out some time ago [2 ] that if the value of 2 is large enough or equivalently if m , , as defined above, is in the TeV scale, the self-interactions o f the scalars become strong. In this case perturbation theory cannot be applied and one could expect the appearance of new states or resonances in the spectrum of the theory. This is referred to in the literature as the strongly interacting MSM (SIMSM). One interesting result obtained for the SIMSM is the so-called equivalence theorem [ 3 ]. This theorem states that, On leave of absence from Dcpartamento de Fisica Teorica de la Universidad Complutense de Madrid, E-28040 Madrid, Spain.

for energies well above the mass of the weak bosons (mw), the scattering amplitudes for the longitudinal components of these weak bosons (WL) are the same as the scattering amplitudes corresponding to the Goldstone bosons associated to the symmetI3' breaking. This result is important because it greatly simplifies the computation of the WL--WL elastic scattering amplitudes in the high energy regime. At the present moment, a lot of work is being done in order to search for the Higgs particle or for setting bounds on its mass (see ref. [4 ] for a recent review), but up to now the value of mu has remained essentially unconstrained. In this work, we will deal with the problem of the determination of the WLWL-" WLWL scattcring amplitudes in the region mH >> 1 TeV as a way of extracting information about the nature of the physical Higgs in this case. The plan of the paper will be the following: starting with the result for the WL--WL elastic scattering amplitudc to one loop recently obtained by Dawson and Willenbrok [5 ] and independently by Veltman and Yndurain [6 ] in the limit rnlel >>s>> rn~v ( , v / s = E is the centre-ofmass energy), we will study carefully the analytical and unitary behaviour o f this amplitude in the framework of the phenomenologlcal lagrangian approach followed in ref. [7]. Although the one-loop partial wave amplitudes have the fight cut structure in the complex plane, they are unitary only at low energies and hence they cannot be compared with the

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results of future experiments. On the other hand, the one-loop partial wave amplitudes are polynomials in s except for logarithmic factors, and cannot develop a resonant behaviour i.e., they have no poles in the unphysical sheet. Nevertheless, the appearance o f resonances is one of the main features expected in any strongly interacting dynamics. In order to solve these problems we will consider here not the one-loop result directly but the [ 1, 1 ] Pad6 approximant to it as the realistic result to be compared with experiments in the future. This technique has been applied very recently to the 7t-n elastic scattering amplitudes and the agreement obtained with the experimental data was quite impressive [ 8 ]. On the other hand the Pad6 approximants have been largely applied in potential theory and they have proved to be a good tool in the computation of the energies of discrete spectra (see ref. [9] for instance). This last use o f the Pad6 approximants will be useful for us here since we are interested in computing the physical Higgs mass. Wc will show that the [ 1, 1 ] Pad6 approximant to the one-loop result for the partial wave amplitudes is unitary and has a pole in the unphysical sheet o f the l = J = 0 channel which we claim has to be understood as the physical Higgs particle. The mass of this pole is always under 1.4 TeV and goes to zero (but very slowly) when rnH goes to infinity. In addition, the width of the physical Higgs will be also computed and compared with the naive result obtained at tree level. Finally, we will conclude with some comments about the relation between the results presented here and the ones obtained in other approaches to the problem. As was mentioned before, we will work in the framework of the model introduced in ref. [ 7 ]. This model is a phenomenological lagrangian approach to the dynamics of the SBS of the SM in the same spirit described by Weinberg in ref. [ 10 ] for the case ofpion dynamics. In fact this approach is much more general than MSM because it can be applied to any theory with a global symmetry breaking pattern S U ( 2 ) L × S U (2)R-*SU (2)L+R, as in the case o f the MSM. This pattern is important if we are interested in keeping the p parameter close to unity because o f the custodial symmetry SU(2)L+R [11]. The oneloop partial wave amplitudes for the WLWL--'WLWt. scattering were computed also in ref. [ 7 ] for the low458

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est J values ( J = 0 , 1 ). The general form of these partial wave amplitudes is

a1j(s)=a~°)(s)+a~))(s),

a}°~(s)=Kijs

a~J )(s) =s2[AH(1)2) +iBIj-CIj log(s/t ,2) ] . ( 1 ) Here v is an arbitrary, renormalization scale but

A,j(v 2) depends on it in such a way that a,j(s) is vindependent. For very low energies the approximation a~j(s)~_a~°)(s) applies and it is known sometime in the literature as the low energy theorems. On the other hand it is found that B~j=K~j and this means Im ars(s) = [a}°~ (s) [2. This last equation can be thought of as some kind of elastic unitarity at low energy. In ref. [ 7 ] it was argued that the amplitudes in eq. ( 1 ) have an acceptable unitarity behaviour for energies under 1.5 TeV. Now, if we want to reproduce the results o f ref. [5] or ref. [6] (both references agree), the parameters in eq. (1) have to be chosen to take the values in the l = J = 0 case given by

Koo-

1

16ztv 2'

Coo= 5 0 K~o 36~

'

Aoo(v 2) - 432(4g)3v4 - Coo log ~ A = 2 9 7 g , , / ~ - 1673.

, (2)

The values of these parameters corresponding to other channels can be read off from ref. [7 ]. We would like to stress here that in the strongly interacting case we do not expect mH to be the mass o f any physical particle or resonance but just a free parameter which defines the model and has to be set by the experiments. More precisely, the exact definition of mH as it appears in eq. (2) is m2~ =22(mn)v z, where 2 ( v ) is the well known renormalized coupling constant to oneloop:

~.0')

2(m.) =

1 - [32(rnt~)/2rt z ] log(t,/mn) 2n 2

_

3 log(ran~v) "

(3)

The last approximation corresponds to the limit rnH going to infinity where eq. (2) applies. It is interesting to realize that in this limit the renormalized coupling constant goes to zero at any other fixed energy scale v. Equivalently, we are studying here the case

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when 2 ( m n ) becomes large which means that 2 ( v ) becomes small for v << mn. It is important to stress too that the one-loop result in eq. (2) is not perturbative in 2 (mH) since the dependence on this constant is logarithmic. In fact, the expansion parameter in the loop expansion, in the large mH limit, is s/16nv 2 and not 2(mH) as already noticed in ref. [ 5 ]. Let us study now the analytical structure of the oneloop partial wave amplitudes in eq. ( 1 ). In this formula the variable s is a real number equal to the square of the centre-of-mass energy E. If we want to obtain the proper analytical amplitude promoting s to a complex variable we have to remember that in the computation o f the imaginary part o f e q . (1) in ref. [7] the prescription l o g ( - s ) = l o g ( s ) - i z c was used. Therefore, the appropriate extension ofeq. ( 1 ) to the whole complex plane is given by

a~°)(s)=K, js, a~J )(s) = s 2 [AIj( v 2) + ( B w / r c - Cw) log (sly 2) - (B~s/n) l o g ( - s / v z) ] .

(4)

In this formula s is a complex number and log(s) represents the standard sheet o f the analytical logarithm defined as log(s)=log(Isl)+iarg(s) with - n < arg(s) < n . The function in eq, (4) has two cuts (see fig. 1 ) in the real axis, one from - ( ~ to zero and another from zero to oo. O f course this is the expected analytical structure because we are dealing with partial wave amplitudes corresponding to the elastic scattering of massless particles (we are neglecting mw all the time). The partial wave amplitudes for the

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physical process in eq. ( 1 ) may be obtained by maki n g s = E ~ + i E in eq. (4), as is usually done. Nevertheless, the amplitudes of the form in eq. (4) have two serious drawbacks. First of all, as pointed out above, they are unitary only at low s. On the other hand, they have no poles in any sheet so they cannot really simulate a strongly interacting dynamics because of the absence of resonant behaviour. These two problems may in fact be solved in a simple and elegant way by the use of the Pad6 approximant of the loop series as was done in ref. [ 12]. If one considers the [ l, 1 ] Pad6 approximant o f the one-loop result as a reliable approximation to the true WLWL~ WLWL partial wave scattering amplitudes instead of the oneloop result itself, then a great improvement in the physical properties o f thcse partial wave amplitudes is obtained. This can be understood if one realizes that the [ 1, 1 ] Pad6 approximant as applied to the one-loop result is nothing but the sum of the infinite terms of the geometrical series generated by the oneloop result. The precise definition of the [ 1, 1 ] Pad6 approximant in our case is given by

a)°)(s) a~J'll(s) = l_a~))(s)/a~O)(s)"

(5)

This partial wave amplitude reproduces the same results as the one in eq. (4) up to terms of order s 3 or highcr [a)J.ll (s) =alj(S ) --t-O(s 3) ], i.e. where alj(S) is reliable. On the othcr hand, the Pad6 improved amplitude in eq. (5) has the same cuts as the one in eq. (4) but is exactly elastic unitary. This last property can be expressed in terms of the discontinuity along the cut on the positive real axis as lira [atlJ'lJ(EZ+i~)-al~'lJ(E2-i~) ] ~0

= 2ia~J.11 (E 2 + i~ )a~)'l J(E 2 - i~ ) . PItYSICAL

SIIEET

E~+ ~e LEFT CUT

RIGHT CUT

E7 - zf ~

~

H

E

E

T

Fig. I. Schematic picture of the cuts, shcets and some points of the a ~J'tl(s) partial wave amplitude considered in the tcxt.

(6 )

This cquation can be directly checked from eq. (4) and eq. (5). In fact, eq. (6) is a consequence o f -11'11(s)12 which applies for any Im a~)'ll(s) = I~ls value ofs. In addition, the Pad6 improved amplitude has other interesting physical properties that will be analyzed in the following. In order to clarify this last point let us study first under what conditions the partial wave amplitude ,~ ~ sI1,~1(s) has poles in the different regions of the complex plane where it is defincd. As is well known from the S-matrix theory, poles in the physical sheet of the partial wave amplitudes are 459

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physically unacceptable (ghosts). It is not difficult to show that the condition for avoiding such poles in the region close above the positive real axis is given by K22J >0, 7= A I j ( m 2 ) _ C # j

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intcracting case, m,i has to be considered as a free parameter of the model and not as the mass of any particle or resonance. Solving eq. (9) for mH we get [ { 24nv'~ 2 rnH(m):mexPLtl-~m )

(7)

- ~---~]

2

where m 2 is the solution of the equation m2 =

Ktj AH(m2) .

(8)

In the case where this last equation has no solutions with m 2 > 0 or the corresponding 7 is positive, no ghosts appear above the positive real axis. Let us consider now the conditions for having poles in the unphysical sheet of the Pad6 partial wave amplitudes aJ)'~l(s). The unphysical sheet has to be defined as usual as the analytical prolongation of the physical amplitude over the positive real axis into the region under the positive real axis (see fig. 1 ). The possibility o f having poles in this unphysical sheet is interesting because these poles can be interpreted as resonances with the same quantum numbers as the corresponding partial wave amplitudes. In fact the conditions for having these poles are closely related to the no-ghost conditions shown above. More specifically, if we have a solution of cq. (8) with m 2 > 0 and 7> 0 then we have a pole in the unphysical sheet ofaJj~''1 (s) at the point So= m2( 1 - i 7 ) . This pole can naturally be interpreted as a physical resonance in the corresponding channel with mass m and width F = roT. Let us apply the discussion above to the minimal standard model. In particular we will consider the l = J = 0 channel because this is the one with the same quantum numbers of the Higgs particle. By the use of eq. (2) one can immediately put eq. (8) in thc following form: m2=

18(4n)2V2

In fig. 2 a plot o f the values m in terms of rnn obtained by inverting this equation is shown. In this figure we see that no solution exists for m i f m H is under a critical value rnHc which is given by rnH~ = mH (me) -~ 5318 G e V , 24 mc = x / 2- r6nv~ 2666 G e V .

( 11 )

For values ofmH larger than mn~ two solutions for m appear starting from the critical value me. The upper one grows asymptotically as a linear function of rnn but the second one is a very slowly decreasing function o f mH. This last branch of the solutions is the physically relevant one. To see that, we can compute the y parameter by the use o f eq. (3) and eq. (7): 7(m)=7o(m)

1

l-(50/36n)yo(m)

'

m2 y o ( m ) - 16nv 2 •

(12)

In eq. ( 7 ) it was implicitly assumed that 171 << 1. For 10 2

I -

I01 /

/

...... /.1.

E

,./

................

..............

t

..........

I0 0

-3

(9)

d / 6 - 50 l o g ( m / m H ) " The meaning of this equation is the following. Provided we can solve it with m 2 > 0, this equation gives us the mass of the physical Higgs ( m ) (defined as the lowest mass resonance in the l = J = 0 channel of the WLWL--'WLWL scattering amplitude) as a function of rn H. As stressed in the introduction, in the strongly 460

1 0 - ltO 0

. . . .

., .~l 101 m

L ..... H

I 10 2

10 3

(TeV)

Fig. 2. The solid line corresponds to the solutions found for the mass of the physical Higgsrn versus the mjj parameter, the dashed line is the rn=mn line and the dotted lines separate different regions of the plot in terms of the value ofT.

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m > m c we have 7 ( m ) < 0 , 1 7 ( m ) l > 3 6 n / 5 0 and therefore eq. (12) cannot be applied i.e. there is no pole close to the real axis corresponding to the upper branch o f m in fig. 2. For m < m c , 7(m) >0, but only in the case

m2<

16ztv2 1 + 50/36zt

(13)

or equivalently rn<5.9v~_ 1475 GeV, the width hecomes smaller than the mass [y(m) < 1 ] and eq. (12) may start to be applied. By increasing mu in this branch the value of m decreases very slowly and the same happens with the corresponding width being asymptotically 7-~Yo. As far as m 2 and y are positive along this branch it can be said, from the discussion above, that it corresponds to a true pole in the unphysical sheet of the I = J = 0 partial wave amplitude and hence to a physical resonance. By identifying this resonance with the Higgs particle we arrive at the conclusion that for values ofmH greater than 10 TeV we have a Higgs particle with a mass around 1 TeV or less. Moreover, the mass of the physical Higgs decreases when mH goes to infinity but very slowly (for instance to get m = 0.3 TeV a value of mH of 5 × 10'9 GeV is required). Consequently, one result of our analysis of the Pad6 approximant to the one-loop computation of the WL--WL scattering amplitude is that in the limit mH--,• we always have a physical Higgs with a mass under 1.4 TeV. It is interesting to compare this result with the naive assumption of considering the mH parameter as the mass of the Higgs particle itself. In fig. 2 we have plotted the curve m = m H to compare it with the result of our analysis. It is amazing to realize that this curve is very close to the upper branch of our solution i.e., the one without physical meaning. Concerning the widths of the Higgs in the strongly interacting case, one can read off directly from eq. (12): m3 I m3 F ( m ) = 16ztv2 l_50m2/(24~zv) 2 _ 167rv2 ,

(14)

where the last approximation applies when F<< m and was already obtained in ref. [ 12 ]. This result reduces the lree-level one by a factor 2/3 when the relation m = m H is assumed. In the following, we sill try to make clear the meaning of the large mH limit considered here and its re-

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lation with other approaches to the problem. First of all we will deal with the issue of triviality (see ref. [ 13 ] and references quoted therein ). It is easy to understand how triviality may appear from eq. (3). If eq. (3) were the one describing the exact evolution of~. with the energy scale v, then the only consistent possibility would be to take ,;t(m~) = 0 because of the pole in this equation, i.e. the theory would be trivial. Of course, eq. (3) is only a one-loop result but at the present moment all the available evidence strongly support the idea that the SBS of the MSM is indeed trivial and therefore it cannot drive the expected breaking of symmetry of the model. From the practical point of view, this problem may be overcome by assuming that in fact the MSM is only a effective theory valid for energies under a certain cutoffA which probably is related with the onset of new physics. In order to understand what is the meaning of the large mH limit in that case, let us to back again to eq. (3). From this equation we see that the maximum possible cutoffAm~x is given by

/ 2n 2 Am~x=m,, exp~ 32~-7H ) / "--mH

(15)

where the last approximation applies in the large mn case, i.e. in the large 2 (rnH) case. Therefore, from the point of view of the effective approach to the MSM in this case, the right interpretation of mH is as the maximum value of the cutoff of the theory (Amax) and not as the mass of the physical Higgs boson mass. Of course, this result applies provided one define mu as m21=22(rnn)v 2 as it is the case in refs. [5,6]. Other definitions of rnn will require a different analysis. One interesting point which has been studied in the literature is the possibility of obtaining bounds on the Higgs mass based on the assumed triviality of the SBS of the MSM by requiring the Higgs mass being under the cutoff of the theory (A) (see ref. [ 14] and references quoted therein). Particularly interesting, are the attempt of some groups (see the references quoted in refs. [ 13,14]) of getting bounds on the ratio Higgs mass over A from lattice computations. One common result of this kind of approach is that the mass of the Higgs increases when the cutoff A decreases being the crossing around the 1 TeV scale which sets the absolute bound on the Higgs mass. On the other 461

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hand, o f one increases A again, the mass o f the Higgs decreases very slowly as it is expected to do from very general considerations. F o r instance, in ref. [ 14] it is quoted a Higgs mass under 170 GeV in the case A - - 1 0 t9 GeV. These results (including the value o f the absolutc b o u n d ) are very close to the one obtained by us in this p a p e r having into account the many orders o f magnitude involved ( r e m a i n we get m = 3 0 0 GeV for m . = 5 × l019 G e V ) . O f course, the only point is to identify the mass o f the Higgs as the physical one i.e. the one related with the pole in the Green functions ( m in our n o t a t i o n ) and the m , parameter as the c u t o f f A (in fact Amax) as explained above. Therefore, we arrive to the conclusion that the simple approach followed here is consistent with the almost proved triviality o f the SBS o f the SM and further results in the field. A n o t h e r very appealing approach to the p r o b l e m o f the determination o f the mass o f the physical Higgs was carried out some time ago by Einborn [ 15 ] and independently by Casalbuoni, D o m i n i c i and G a t t o [ 16 ] by the use o f a large N expansion o f the O ( N ) model. Their results at leading o r d e r for the WL--WL scattering a m p l i t u d e are very interesting since they obtain an unitary a m p l i t u d e that shows a pole in the I = J = 0 channel which is interpreted as the physical Higgs. This pole has the feature o f saturation i.e. it cannot be pushed beyond some point (in fact a line in the complex plane) corresponding to a mass o f the physical Higgs close to 900 GeV. The Pad6 approxim a n t o f the one-loop a m p l i t u d e s considered in this paper behave very much in the same way as the ones obtained in the large N expansion. Therefore, we can say that we have reobtained some o f the results o f rcfs. [ 15,16 ] in a quite different approach. Finally, we would like to stress that if the general discussion presented in this letter is a p p r o p r i a t e for the description o f the WL--WL elastic scattering, the

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existence o f a Higgs particle with a mass under 1.4 TeV becomes a generic prediction of the M S M for any value o f m . (or equivalently ofAma x in the triviality language) greater than 10 TeV. This p a p e r relied much on previous work carried out in collaboration with M.J. Herrero [7,12] and T.N. Truong [ 12 ]. The author would like to thank both o f t h c m for this collaboration and for useful discussions. This work has been supported by the Dep a r t a m e n t o de Postgrado y Especializacion, CSIC (Spain).

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