CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server UW/SEA 97-01
The Higgs Particle as a Self Consistent Bound State
H. J. Lubatti∗
Department of Physics
University of Washington, Seattle, WA 98195
e-mail [email protected]
I. J. Muzinich
Institute for Nuclear Theory
University of Washington, Seattle, WA 98195
email [email protected]
(January 1997)
Abstract
We consider Higgs-Higgs scattering in a minimal Higgs model with strong cou-
pling which leads to the Higgs as a bound state. The partial wave amplitude is
unitarized using general principles of analyticity. We find λ(mH)/16π ∼ 1/2.
λ A Higgs mass of approximately 1 TeV follows from mH = 2 v (spontaneous q symmetry breaking). PACS: 11.55.-m; 14.80.B; 14.80.Bn
Typeset using REVTEX
∗Corresponding author: University of Washington, Department of Physics, Box 351560, Seattle,
WA 98195-1560 U.S.A. phone: (206) 543-8964; fax: (206) 685-9242
1 In this paper we consider a framework where the Higgs is a bound state and calculate its mass and coupling. A minimal Higgs model with spontaneous symmetry breaking is assumed. We study elastic Higgs-Higgs scattering and employ unitarity and analyticity of the scattering amplitude and constrain the mass of the bound state to be the same as the range of the force that produces the bound state (fig. 1). Historically, this is referred to as the bootstrap procedure. Fundamental to this approach is strong coupling which is necessary to produce a bound state. Applying the bootstrap procedure we calculate the coupling constant. The Higgs mass is obtained from the spontaneous symmetry breaking relation between the coupling constant, mass, and vacuum expectation value. We use the N/D representation of the partial wave amplitude which contains the con- straints of analyticity and unitarity. The approach unitarizes the lowest order Born graphs. This leads to the well know Cauchy integral representations (dispersion relations) of the N- and D-functions which are used for our calculations. The standard model unitary gauge Higgs Lagrangian is used [1] to define vertices and couplings:
1 2 1 2 2 λ 3 λ 4 L = (∂µh) − m h + mh − h , (1) 2 2 s 2 4 where m is the Higgs mass and h the Higgs field. From spontaneous symmetry breaking the
λ masses are related to coupling constants by m = 2 v, mZ = mW / cos θW , m = ev/2sinθW, q v = 246 GeV, and θW is the Weinberg angle. The cubic term in Eq. (1) gives rise to an attractive force. We note that not all con- tributions to the interactions are attractive; zero range polynomial terms and finite parts after renormalization could be repulsive. The quartic term is typical of such interactions. Other intermediate states in the s-channel which include tt, W W, ..., and multiparticle states are neglected. We consider only the single, symmetrized Higgs exchange in the elastic Higgs-Higgs channel (fig. 1). The unitary L = 0 partial wave elastic scattering amplitude expressed in terms of the
2 phase shift is
√ M =(8π s/p)eiδ sin δ. (2)
In the center of mass frame, one has the standard kinematics
s =4p2+4m2 t=−2p2(1 − cos θ) u = −2p2(1 + cos θ),
where p, E,andθare the center of mass momentum, energy, and scattering angle. We now have all of the necessary ingredients to proceed with a bootstrap calculation. The partial wave amplitude, Eq. (2), can be decomposed into an N/D representation,
M = N/D, where the N-function has a left hand cut, and the D-function contains the usual right hand cut required by elastic unitarity. This decomposition can be established quite generally and rigorously [2,3]. The motivation comes from potential theory or Fredholm theory for two body scattering. The standard application of unitarity and analyticity leads to
1 ∞ D(p2)=1− dx ρ(x) N(x)/(x − p2 − iε)(3) πZ0 1 ∞ N(p2)=B(p2)− dx ρ(x) N(x) B(p2) − B(x) /x − p2 − iε, (4) π 0 Z where ρ(x) is the two body phase space factor p/16πE and B is the (t, u) symmetrized Born term including the quartic term (fig. 1). B is given by
2 1 2 9λm 1 1 B(p )=−λ+ d cos θ 2 + 2 (5) 2 Z−1 m − t m − u 9λm2 4p2 = −λ + ln 1+ . 4p2 m2!
Eq. (5) has a left hand cut at p2 = −m2/4, corresponding to the rigorously established domains of analyticity. The normalization of the N and D functions is chosen to agree with
3 the renormalized quartic theory at high p2.1 We calculate D to O(λ)andNto O(λ2)using Eqs. (3 and 4). The integral over B(x) in Eq. (3) yields a divergent term which renormalizes the coupling constant λ and a convergent, attractive potential term from the logrithmic term. We have used the MS renormalization scheme [1].
We set D(λ, p2) = 0 at the physical Higgs mass s = m2, p2 = −3m2/4 (the bootstrap condition). As expected for a physical bound state the derivative ∂D/∂p2 is negative in the neighborhood of p2 = −3m2/4. The Higgs mass scales out of Eq. (3) and we determine the coupling constant, λ(m). The Higgs mass is obtained from the spontaneous symmetry
λ breaking relation m = 2 v. q The numerical integration of Eq. (3) (relativistic kinematics) yields for the coupling constant and Higgs mass:
λ ≈ 0.5,m=0.87 TeV. 16π
Having obtained a self-consistent value of the Higgs mass, we next examine the consis- tency of the residue of the output pole with the input Born terms of Eq. (5). The amplitude
in the neighborhood of the bound state is M = r/m2 − s.2 The residue of the pole is −1 3 2 ∂D 3 2 r ≡ 4N − 4 m ∂p2 −4m . Evaluating this yields a residue which is approximately two times larger than the input pole. We consider this reasonable qualitative agreement in view of the approximations made. The agreement between the input and output residue can be significantly improved by replacing the quartic term by a phenomenological contact interaction. The dominant decay modes of a high mass Higgs are pairs of longitudinally polarized W ’s and Z’s with a characteristic signature of leptons in the final state. LHC experiments taking data at the full design luminosity (105 pb−1) for one year will be sensitive to Higgs of mass 800 GeV.
1We do not consider CDD ambiguities [3] or other arbitrary parameters in Eqs. (3 and 4).
2The N-function is real and positive at the bound state pole.
4 One can view the approach adopted in this paper as an example of a specific dynamical realization of a strongly coupled heavy Higgs. If the Higgs mass is large, there are non-trivial strong interactions and it is likely to be composite. Related work on strong interactions of the Higgs sector and the related W and Z pair channels has also been communicated by a number of authors [4]. As is well known, the renormalization group for the λh4/4theory indicates a sharp increase of the coupling as the mass scale increases above the Higgs3.This has been interpreted as indicative of new physics. From the minimal assumptions of analyticity and unitarity of the Higgs-Higgs elastic scattering amplitude where the dominant force is taken to be Higgs exchange in the crossed
channel we find that λ(m)/16π ≈ O(1) (strong coupling). By further assuming spontaneous symmetry breaking we obtain a Higgs mass of approximately 1 TeV. Our result is consistent with the bound obtained by Lee, Quigg and Thacker from an application of partial wave unitarity to tree graphs for Higgs-Higgs and longitudinal vector boson scattering [5]. IJM thanks the Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support in the initial stages of this work. HJL acknowledges NSF contract PHY-9515490 for support of this research. IJM thanks Bill Marciano for repeated conversations on the question of the Higgs boson and other issues associated with the electro-weak sector of the standard model. He also thanks
λ 4 Peter Arnold for a discussion of the renormalization group for the 4 h theory. We also thank Steven Frautschi and Geoffrey Chew for discussions concerning general principles of S-matrix theory, and Marshall Baker for several very helpful conversations on S-matrix theory and applications of the N/D formalism.
3 4 3 2 The beta function for λh /4 theory is given at one loop by β(λ)=16π2 λ . The coupling constant
3λ(Mo) M evolution corresponding to this beta function is λ(M)=λ(M )/(1 − 2 ln ). 0 16π M0
5 REFERENCES
[1] M. Peskin and D. Schroeder, An introduction to quantum field theory (Addison Wesley, Reading MA, 1994).
[2] A. deAlfaro and T. Regge, Potential scattering (North Holland Publishing, Amsterdam Netherlands, 1963).
[3] S.C. Frautschi, Regge poles and s-matrix theory (W.A. Benjamin Inc., 1963).
[4] D. Sivers and J.L. Uretsky, Phys. Rev. Lett. 68 (1992) 1649; L.A.P.Bala’zs Phys. Rev. D 48 (1993) 1310.
[5] B.W. Lee, C. Quigg, and H. Thacker, Phys. Rev. Lett. 38 (1977) 883 and Phys. Rev. D 16 (1977) 1519.
6 FIGURES
h h h @@ @@ @ √ @ iλ @ −i √3 λm@ − @ 2 h @ @@ h h h
(a)
bb "" A b " h b " h h A h b " "b A " b A " b h A h A A A h A h h A h A A
(b)
FIG. 1. a) Quartic and cubic vertices; b) t- and u-channel Born terms for Higgs-Higgs scattering.
7