DIMINISHED UPPER BOUNDS ON THE UNIFICATION MASS SCALES FOR HEAVY HIGGS BOSON MASSES

DIMINISHED UPPER BOUNDS ON THE UNIFICATION MASS SCALES FOR HEAVY HIGGS BOSON MASSES
DIMINISHED UPPER BOUNDS ON THE UNIFICATION MASS SCALES FOR HEAVY HIGGS BOSON MASSES

Modern Physics Letters A

1Vol.21,No.4(2006)1–4

c Worl

d Scienti?c Publishing Company

3DIMINISHED UPPER BOUNDS ON THE UNIFICATION MASS SCALES FOR HEA VY HIGGS BOSON MASSES

5V.ELIAS ?,S.HOMAYOUNI and D.J.JEFFREY

Department of Applied Mathematics,The University of Western Ontario,

7London,Ontario N6A 5B7,Canada

?velias@uwo.ca

9Received 23December 2005

We consider dominant three-,four-and ?ve-loop contributions to λ,the quartic scalar

11coupling-constant’s β-function in the Standard Model.We ?nd that these terms acceler-

ate the evolution of λto nonperturbative values,thereby lowering the uni?cation bound

13for which scalar-couplings are still perburbative.We also ?nd that these higher order

contributions imply a substantial lowering of λitself before the anticipated onset of

15nonperturbative physics in the Higgs sector.

The dominant running coupling constants of the standard model evolve with μ,the renormalization scale,according to two-loop renormalization group equations

μdλ

16π2 4λ2+12λh 2?36h 4?9λg 22?9100g 41

+27

4g 4

2 +13λ3

?24λ2h 2?3λh 4

+180h 6+80λg 23h 2?192h 4g 2

3+··· ,(1)

μdh

16π2 94g 22h ?

17

(16π2)2 ?12h 5?2λh 3+1

dμ=

1

(16π2)2 ?26g 53?2h 2g 33+11

2g 22g 3

3+··· ,(3)

1

2V.Elias,S.Homayouni&D.J.Je?rey

μdg2

16π2 ?19

(16π2)2 3510g21g32?3 dμ

=

1

10

g31

+

1

50

g51+

27

5

g23g31?

17 dμ

Y=4Y2?

26

Diminished Upper Bounds on the Uni?cation Mass Scales for Heavy Higgs Boson Masses3

Fig.1.Top curve,upper bound on uni?cation mass scale M with no higher-than-2-loop input.

Middle curve,upper bound with three-?ve loop contributions to theβ-function forλ,but withλassumed perturbative up toλFP/2,as in top curve.Bottom curve,upper bound with three-?ve loop contributions and concomitant reduction in how largeλcan be before it is nonperturbative.

This expression has a bearing both on howλmax is obtained,as well as how 1

rapidlyλitself evolves toλmax.

Prior calculations of the upper bound of the uni?cation mass scale assumed λmax was equal(or related)to its“?xed-point”valueλFP,de?ned as where the two-loop and one-loop contribution to(8)are equal:

Y=6/13orλFP~=73.

In a two-loop world,this would be near a?xed point in the RG equation(1), 3

particularly asλis so dominant a coupling constant compared to the others in Eq.(1).In fact,people have advocated for various reasons thatλmax be5λFP/2 5

or even smaller.6The top curve of Fig.1shows,given a choice of M H,the corre-sponding value of the upper bound M for the uni?cation mass scale,given that 7

λmax=λFP/2=36.The intermediate curve in Fig.1shows forλmax=λFP/2how the upper-bound M on the uni?cation mass scale decreases if the three-,four-and 9

?ve-loop terms in Eq.(8)are incorporated into theλβ-function(1).

However,the additionalβ-function terms in(8)make any referencing to 11

λFP irrelevant.Theβ-function series(8)does not monotonically decrease unless

4V.Elias,S.Homayouni&D.J.Je?rey

Y<0.084(λ<13.3).Henceλmax=13.3is an upper bound on the value ofλfor 1

which perturbative Higgs sector physics may still be possible,in that four-and?ve-loop terms in(8)are equal.The evolution of the coupling constantλshould also be 3

inclusive of the three-,four-and?ve-loop terms of Eq.(8),as in the middle curve, since such terms are comparable whenλmax=13.3.When we augment Eq.(1)with 5

these three-?ve loop terms in Eq.(8),and impose the additional requirement that the upper bound onλfor perturbative physics is13.3,we obtain the lowest of the 7

three curves in Fig.1.

Figure1shows that a given value for M,the upper bound for the uni?cation 9

mass scale,now corresponds to substantially smaller values of the Higgs mass when Eq.(8)augments the Eq.(1)β-function,and whenλmax=13.3.This separation 11

becomes pronounced when M<105GeV.By incorporating Eq.(8),we?nd that

a Higgs mass of304GeV can occur in a theory only if uni?cation is prior to 13

100TeV;a Higgs mass of360GeV can occur only if uni?cation is prior to10TeV;

and that Higgs mass in excess of460GeV would involve nonperturbative physics 15

immediately.In the prior analysis(top curve)this same nonperturbative bound would be in excess of800GeV.

17

We reiterate that the reduction we?nd in the uni?cation mass-scale upper bound M for a given choice of M H is itself conservatively taken.The choice 19

λmax=13.3assumes perturbative physics even when three-,four-and?ve-loop contributions to theβ-function(8)are comparable in magnitude.One could argue 21

forλmax=6.65via whatever reasoning already employed in the past for choosing λmax=λFP/2instead ofλFP.We also note that the e?ect of higher-than-2loop 23

contributions becomes unimportant for Higgs masses in the vicinity of200GeV.

We?nd,for example of a200GeV Higgs boson mass,that the upper bound on 25

the uni?cation mass scale is1012GeV;for a190GeV Higgs boson mass,the upper bound on the uni?cation mass scale goes up to1015GeV.In the prior two-loop 27

analysis(λmax=λFP/2)the same values of the uni?cation mass scale are achieved by Higgs masses only10GeV or so larger than those quoted above.

29

Acknowledgments

We are grateful for the support from the Natural Sciences and Engineering Research 31

Council of Canada.

References

33

1. C.Ford,D.R.T.Jones,P.W.Stephenson and M.B.Einhorn,Nucl.Phys.B395,17 35

(1993)Appendices.

2.V.Elias,Phys.Rev.D20,262(1979).

37

3.K.Riesselmann,Acta.Phys.Polon.B27,3661(1996).

4.H.Kleinert et al.,Phys.Lett.B319,545(1993).

39

5.T.Hambye and K.Riesselmann,Phys.Rev.D55,7255(1997).

6.K.Riesselmann and S.Willenbrock,Phys.Rev.D55,311(1997).

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