Abstract
The Equivalence Theorem is commonly used to calculate perturbatively amplitudes involving gauge bosons at energy scales higher than gauge boson masses. However, when the scalar sector is strongly interacting the theory is nonperturbative. We show that the Equivalence Theorem holds in the large N limit at nexttoleading order by calculating the decay widths h > W+W and h <FONT FACE=Symbol>® p</FONT>+pi. We also show, in the same scheme of calculations, that unitarity is fulfilled for the process h <FONT FACE=Symbol>® p</FONT>+pi.
Higgs boson decay in the large N limit
John Morales^{I,II}; R. Martinez^{I}; Rafael Hurtado^{I,II}; Rodolfo A. Diaz^{I}
^{I}Universidad Nacional de Colombia, Departamento de Física, Bogotá, Colombia
^{II}Centro Internacional de Física, Bogotá, Colombia
ABSTRACT
The Equivalence Theorem is commonly used to calculate perturbatively amplitudes involving gauge bosons at energy scales higher than gauge boson masses. However, when the scalar sector is strongly interacting the theory is nonperturbative. We show that the Equivalence Theorem holds in the large N limit at nexttoleading order by calculating the decay widths h ® W^{+}W^{} and h ® p^{+}p^{}. We also show, in the same scheme of calculations, that unitarity is fulfilled for the process h ® p^{+}p^{}.
I. INTRODUCTION
The Standard Model (SM) of the electroweak interactions, based on the SU(2)_{L }Ä U(1)_{Y} gauge symmetry [1], is a successful theory and agrees with most experimental results [2]. However, the scalar sector responsible for the symmetry breaking of the SM is not well known and it has not been tested yet. This sector gives masses to the particles of the model, fermions and gauge fields, when the scalar field has a non vanishing Vacuum Expectation Value (VEV) after the symmetry breaking. In the scalar sector a Higgs particle appears with a mass given by = 2lv^{2}, where l is the coupling constant of the selfinteracting term and v is the VEV (v » 246 GeV). m_{h} is an unknown parameter so far.
Nevertheless, the precision tests of the SM impose strong bounds to the Higgs mass when the scalar sector is weaklycoupled. The results from LEP Electroweak Working Group analysis yield m_{h} = GeV (68% CL) [3], and an upper limit of m_{h} < 260 GeV with onesided 95% CL [3]. The direct search of the Higgs boson done at LEP gives a lower limit of m_{h} > 114.4 GeV [3]. On the other hand, it is possible to have different models beyond the SM with a heavy Higgs with a mass lying in the TeV scale for a strongly interacting scalar sector. However, for this scenario to be held, the new physics contributions must cancel those ones introduced by the heavy Higgs particle at low energies [4].
If the SM is an effective theory derived from a more fundamental one, then there is an associated L scale for the appearance of new physics. The use of theoretical arguments, like unitarity [5], triviality [6] and vacuum stability [7], may allow to get constraints for these two parameters (L, m_{h}) [8].
The upper limit for the Higgs mass can be obtained by triviality considerations in the Higgs sector [9]. When the quartic coupling constant l in the scalar sector of the Higgs potential is renormalized introducing a cutoff L, the coupling goes to zero when L goes to infinity, implying that m_{h} goes to zero. This is not the case for the SM, because it needs a massive scalar particle at low energies to explain experimental results, and then the SM can be considered as an effective theory below a given energy scale. If we knew this scale we could predict the Higgs mass. Further, if the SM had a Higgs with a mass around 1 TeV, then the scalar sector would be strongly interacting and the underlying theory would become nonperturbative[10].
The amplitude for a heavy Higgs decaying into two longitudinally polarized gauge bosons reads [11]
By considering that in the perturbation expansion the l^{2} term must be smaller than the l term, it is found that l(m_{h}) » 7 implying that m_{h} » 1 TeV. On the other hand, using the scattering process WW ® ZZ mediated by a Higgs particle, which might be important in future collider experiments like LHC and linear colliders, the cross section for energies >> m_{h} at two loops level is given by [12]
This cross section is negative for some values of l which means that the perturbative expansion breaks down. Considering that the l^{2} term must be smaller than the l term, a necessary condition to have a convergent series is l » 4, in this case m_{h} » 700 GeV [13]. The above scenarios correspond to the limit between weaklycoupled and stronglycoupled scalar sectors.
In the Marciano and Willenbrock paper [14] they calculated the decays of a heavy Higgs boson up to O(g^{2}/) in perturbation theory using the Equivalence Theorem (ET) [15], from which the amplitude with gauge bosons longitudinally polarized at energies O(q^{2} >> ) is equivalent to the same amplitude but changing the corresponding longitudinal components by the wouldbe Goldstone bosons. For Higgs masses of the order of m_{h} » 1 TeV and m_{h} » 1.3 TeV the radiative corrections for the decay h ® W^{+}W^{} are 7.3% and 12%, respectively. At this scale the scalar sector is stronglycoupled and the theory is nonperturbative. It is obvious that the amplitude at nexttoleading order breaks the perturbative expansion because all Feynman rules are proportional to the Higgs mass. For strongly interacting models is necessary to use a nonperturbative method to calculate radiative corrections and get bounded amplitudes. While it has been shown that the ET holds order by order in perturbation theory, it has not been confirmed that it does in nonperturbative calculations.
Due to the importance of studying the Higgs dynamics in nonperturbative regimes, a formalism was introduced in Ref. [16] which uses Chiral Perturbation Theory(cPT) [17]. Amplitudes are obtained as a power expansion in the energy, this implies that the conventional ET does not hold anymore [18]. Thus, a new formalism is necessary to have an effective theory [19].
The large N limit is an alternative approach that predicts bounded positive defined amplitudes, consistent with pion dispersion [20], and useful to study the symmetry breaking of the strongly interacting sector [21]. The scalar sector of the SM can be modelled by a Linear Sigma Model O(4) and then generalized to a model with O(N + 1) symmetry. This method has been applied to study the Higgs boson at TeV energy scales [10, 22]. We show that the large N limit can predict amplitudes that fulfill the ET and the unitarity condition at nexttoleading order for the SM, with a strongly interacting scalar sector, by using the h ® W^{+}W^{} and h ® p^{+}p^{} processes.
In section 2 we introduce the Gauged Linear Sigma Model O(N + 1). In section 3 we calculate the Higgs decay widths, h ® W^{+}W^{} and h ® p^{+}p^{}, in the large N limit and we show that the ET holds at nexttoleading order. In section 4 we show that the amplitude h ® p^{+}p^{} satisfies unitarity in the large N limit. In section 5 we give our conclusions.
II THE O(N + 1) MODEL
It is well known that the Linear Sigma Model represents the symmetry breaking O(N + 1) ® O(N) with N wouldbe Goldstone bosons which belong to the fundamental irreducible representation of the remaining symmetry O(N). For the purposes of this work the wouldbe Goldstone bosons will be named like pions p. For a gauge invariant model under SU(2)_{L }Ä U(1)_{Y} local symmetry the large N limit for the SM is defined as
L_{g} = L_{YM }+ (D_{m}F)^{}(D^{m}F)  V(F^{2})
with F^{} = (p_{1},p_{2},...,p_{N},s) and F^{2} = F^{}F. As usual L_{YM} is the YangMills Lagrangian of the SM and the covariant derivative is defined as
where = (i/2) are the generators of the SU(2)_{L} gauge group and = (i/2)M^{Y} is the generator of the U(1)_{Y} 4gauge group. The M matrices are given by [23]
which belong to an irreducible representation of the O(N + 1) Lie algebra with i,j = 1,2,3 and a,b = 1,2,...,N + 1. The matrices which belong to the adjoint representation of the SU(2)_{L} Lie algebra are given by
and, the corresponding matrix for the U(1)_{Y} Lie algebra reads
where dots represent zeros. In this form we have a global O(N + 1) symmetry with a local SU(2)_{L }Ä U(1)_{Y} symmetry.
The Higgs potential, invariant under O(N + 1), can be written as
Aligning the vacuum state as áfñ_{0}º (0,...,v), with F^{2} = v^{2} = 2µ^{2}/l, the global symmetry O(N + 1) is broken to O(N) and the local symmetry is broken as SU(2)_{L} Ä U(1)_{Y} ® U(1)_{Q}. By defining the Higgs field as h = s  v, we find the following expression
The gauge boson masses are obtained from the kinetic term,
where the mass eigenstates are given by
and q_{W} is the Weinberg angle with tan q_{W} = g'/g. The fields, with m_{W} = gv/2 » 80.6 GeV masses, are the charged gauge bosons, and the Z_{m} field, with m_{Z} = v(g^{2} + g'^{2})^{1/2}/2 » 91.2 GeV mass, is the weak neutral gauge boson. The A_{m} field is the massless photon.
The Lagrangian has terms of the form g^{2}v¶^{m}p_{a} /4, mixing gauge bosons with wouldbe Goldstone bosons, which can be cancelled by gauge fixing. We choose the Landau gauge (x = 0) because in this gauge a lot of Feynman diagrams cancel or suppress, the p_{a} fields do not couple to the ghost fields, and their propagators are massless. The final Lagrangian can be written as
Thus we have a gauge theory spontaneously broken with the = (p_{1},p_{2},p_{3}) fields as the wouldbe Goldstone bosons of the broken symmetry SU(2)_{L} Ä U(1)_{Y}/U(1)_{Q} and p_{a} fields as the wouldbe Goldstone bosons of the broken global symmetry O(N + 1)/O(N).
The theory for the large N limit makes sense when N ® ¥ and gives rise to finite amplitudes for different processes. To get finite amplitudes is necessary to choose appropriate parameters in the large N limit. We will take the following definition
in order to use perturbative expansion of the strongly interacting sector as a function of the l parameter. With this definition, physical masses must be finite and independent of N in the large N limit. From the masses
we obtain for the other parameters of the model in the large N limit the following values
Finally, we obtain the Feynman rules necessary to calculate the decay widths for h ® W^{+}W^{} and h ® p^{+}p^{} in the large N limit, see Fig. 1.
III. THE HIGGS BOSON DECAY AND THE EQUIVALENCE THEOREM
The SM in the large N limit is associated with the O(N + 1)/O(N) and SU(2)_{L }Ä U(1)_{Y}/U(1)_{Q} global and local symmetry breaking schemes respectively. We can calculate the amplitudes for h ® W^{+}W^{} and h ® p^{+}p^{} decays in order to show that the ET holds in the proposed scenario.
Feynman diagrams at tree level in this approximation are of the order of O(g) or O(g') and of the order of O(1/) in the large N limit . The decay widths at tree level for h ® W^{+}W^{} and h ® p^{+}p^{} processes are given by
To obtain the amplitudes at nexttoleading order is necessary to introduce the radiative corrections. First we calculate the selfenergy of the scalar particle h, whose Feynman diagrams at nexttoleading order are shown in Fig. 2. In this case, the selfenergy at one loop level with p_{a} fields into the loops is of the order of 1/N times N where N is the number of degrees of freedom running into the loop. Therefore, radiative corrections are of the order of one in the large N limit. The same analysis can be done for the selfenergy diagram with l loops. It has two vertices with hpp and l  1 vertices with four p_{a} and is of the order of (1/)^{2}1/N)^{l1} times N^{l} the number of pion fields running into the l loops. Consequently, the selfenergy diagram with l loops is of the order of one in the large N limit. However, the one irreducible particle function (1IP) for selfenergy diagram with , Z_{m} into the loop is of the order of 1/N, which is negligible in the large N limit.
After doing all calculations by using dimensional regularization, with d = 4  e integrals from the loops, we find (see ^{appendix}appendix)
with I_{q} given by
where D = 2/e + log4p  g_{e} and µ is the renormalization scale. The choice of the renormalization scale µ is arbitrary. Therefore, we shall adopt µ 1 TeV as a reasonable choice. We have taken into account that only the one irreducible particle functions are important in perturbation theory for the renormalization of parameters such as the mass and the wave function [24].
From the selfenergy calculation the wave function renormalization of the Higgs boson can be obtained as
The contributions to p^{±}, W^{±} and Z_{m} selfenergies are proportional to 1/N and in the large N limit we obtain that
To calculate the Higgs decays at this order, vertex corrections have to be included as well, as shown in Fig. 3.
The radiative corrections of hW^{+}W^{} vertex displayed in Fig. 3(a) are suppressed since they are of the order of 1/N^{2} becoming negligible in our approximation. The hW^{+}W^{} vertex can be written at this level as
For the hp^{+}p^{} vertex corrections shown in Fig. 3(b), the pions into the loops give the most important contributions and can be written as
where the W^{±} contributions into the loops are suppressed by a 1/N factor with respect to the p_{a} contributions. Similarly, the contribution with a higgs running into the loop is also suppressed by a 1/N factor.
The wave function renormalization of the Higgs particle Eq.(14) and the vertex radiative correction Eq. (3) diverge. To obtain finite amplitudes the Higgs mass has to be renormalized [10]
The real part of the Z_{h} function Eq.(14) is given by
In the same way, the real part of the vertex correction Eq.(3) can be expressed as
Hence, the vertex corrections Eq.(19) multiplied by the factors of the renormalized wave functions
Z_{p} give rise to the h ® p^{+}p^{} amplitude at nexttoleading order and can be written at O(g^{2}/ ) as
The same procedure is done for the h ® W^{+}W^{} amplitude where the vertex corrections are multiplied by the factors of the renormalized wave functions Z_{W}, and can be written at O(g^{2}/ ) as
In order to show that the ET holds for nonperturbative nexttoleading order, we calculate Higgs decays into gauge bosons and pions in the large N limit. We then compare the decay widths as obtained from the decay amplitudes for h ® p^{+}p^{} and h ® W^{+}W^{} in Eqs. (20) and (21) respectively. Such decay widths are given by
and
In Fig. 4 we display the ratio G(h ® p^{+}p^{}) /G(h ® W^{+}W^{}) as a function of the Higgs mass including nexttoleading order corrections. From this figure it can be seen that such quotient tends to one for large Higgs masses ( 4.5 TeV), showing the validity of the ET at high energies.
IV. UNITARY IN THE LARGE N LIMIT
As a consequence of unitarity of the Smatrix, i. e. S^{}S = 1, the Optical Theorem is obtained . By defining S = 1 + iT, where the T is called the transition matrix, we have
and since four momentum is conserved in the transition from initial state iñ to final state fñ, we can always write
and
Inserting a complete set of intermediate states qñ we find
and from the identity (24) we can obtain the Cutkosky's rule[25]
where the sum runs over all possible sets of intermediate states q_{i}.
Applying this identity to the decay G(h ® p^{+}p^{}) we find
In the lefthand side of the previous equation we have the imaginary part of the product of Eq. (3) times the wave function
Z_{p}, resulting
For the righthand side, we have to multiply the amplitudes calculated in the large N limit
i.e.,
by
i.e.,
Then the righthand side of Eq. (29) becomes
where 1/4 is the symmetry factor for identical bosons in the final state.
From Eq. (29) we define
and the integral over q_{b} can be written as
Integrating the fourdimensional delta function in Eq.(36) we obtain
In the centerofmass frame
the integral in Eq. (36) can be rewritten as
and by using p^{2} =
Comparing equations (30) and (39) we see that the Higgs decay G(h ® p^{+}p^{}) calculated in the large N limit at nexttoleading order fulfills the unitarity condition.
V. CONCLUSIONS
We have shown that nonperturbative calculations at nexttoleading order in the large N limit for the case of a Higgs decaying into W^{±} and p^{±} fulfill the ET. In particular, we found that the decay widths G(h ® W^{+}W^{}) and G(h ® p^{+}p^{}) get values that are basically identical for heavy Higgs bosons i.e. 4.5 TeV.
On the other hand, we have also shown that calculations in the same scheme for the Higgs decaying into pions respect unitarity. This results open the possibility to study strongly interacting systems as could be the case of the SM with a heavy Higgs boson.
Acknowledgments
We thank to Alexis Rodriguez and Marek Nowakowski for reading the manuscript and their comments. We also thank COLCIENCIAS, DIB, and DINAIN for their financial support.
REFERENCES
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[13] U. Nierste and K. Riesselmann, Phys. Rev. D 53, 6638 (1996).
[14] W.J. Marciano and S.S.D. Willenbrock Phys. Rev. D 37, 2509 (1988).
[15] B.W. Lee, C. Quigg, and H. Thacker Phys. Rev. D 16, 1519 (1977). J.M. Cornwall, D.N. Levin, and G. Tiktopoulus, Phys. Rev. D 10, 1145 (1974). M.S. Chanowitz and M.K. Gallaird Nucl. Phys. B 261, 379 (1985). G.K. Gounaris, R. Kogerler, and H. Neufeld, Phys. Rev. D 34, 3257 (1986). H.J. He and W. Kilgore, Phys. Rev. D 55, 1515 (1997).
[16] A. Dobado and M. Herrero, Phys. Lett. B 228, 495 (1989). J. Donoghue and C. Ramirez, Phys. Lett. B 234, 361 (1990).
[17] J. Gasser and H. Leutwyler, Ann. of Phys. 158, 142 (1984). S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); M.S. Chanowitz, M. Golden, and H. Georgi, Phys. Rev. D 36, 1490 (1987); A. Longhitano, Phys. Rev. D 22, 1166 (1980); Nucl. Phys. B 188, 118 (1981).
[18] H. Veltman, Phys. Rev. D 41, 264 (1990).
[19] A. Dobado, J.R. Pelaez, and M. T. Urdiales, Phys. Rev. D 56, 7133 (1997).
[20] A. Dobado, A. Lopez, and J. Morales, Il Nuovo Cimento A 108, 335 (1995); A. Dobado and J. Morales, Phys. Rev. D 52, (1995) 2878.
[21] S. Coleman, R. Jackiw, and H. D. Politzer, Phys. Rev. D 10 2491 (1974); Dolan and Jackiw, Phys. Rev. D 9, 3320 (1974); M. Kobayashi and T. Kugo, Prog. Theor. Phys. 54, 1537 (1975); W. Bardeen and M. Moshe, Phys. Rev. D 28, 1372 (1983).
[22] R.Casalbuoni, D. Dominici, and R. Gatto, Phys. Lett. B 147, 419 (1984); M.B. Einhorn, Nucl. Phys. B 246, 75 (1984); A. Dobado and J.R. Pelaez, Phys. Lett. B 286, 136 (1992).
[23] H. Georgi, Lie Algebras in Particle Physics, Addison Wesley, Frontiers in Physics. New York 1982.
[24] M. E. Peskin, D. V. Schroeder, An introduction to quantum field theory, Westview Press (1995), Chap. 7.
[25] R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).
Received on 13 October, 2004
In this appendix we show the explicit calculation of a Feynman diagram with l loops in the large N limit that contributes to the Higgs boson selfenergy.
where 1/2l is the symmetry factor of the diagram. The first factor corresponds to the initial and final loops times the vertices with three particles, the second factor represents the product of the l  1 internal vertices with four interacting fields and the last factor correspond to l  2 loops. Each loop contributes with an N factor, as they have N circulating pions.
 [1] S. Weinberg, Phys. Lett. 19, 1264 (1967);
 S.L. Glashow, Nucl. Phys. B 20, 579 (1961).
 [2] Particle Data Group, Phys. Rev. D 66, 1 (2002).
 [4] R. S. Chivukula and N. Evans, Phys. Lett. B 464, 244 (1999);
 R. S. Chivukula, N. Evans, and C. Hoelbling. Phys. Rev. Lett. 85, 511 (2000);
 J. Bagger, A. Falk, and M. Schwartz, Phys. Rev. Lett. 84, 1385 (2000);
 Michael E. Peskin, Phys. Rev. D 64, 093003 (2001).
 [5] B. Lee, C. Quigg, and H. Thacker, Phys. Rev. D 16, 1519 (1977);
 W. Marciano, G. Valencia, and S. Willenbrock, Phys. Rev. D 40, 1725 (1989).
 [6] N. Cabibbo, L. Maiani, G. Parisi, and R. Petronzio, Nucl. Phys. B 158, 295 (1979).
 [7] M. Sher, Phys. Rep. 179, 273 (1989);
 M. Quiros, Perspectives on Higgs Physics II, Ed. G.L. Kane, World Scientific, Singapore [arXiv: hepph/9703412].
 [8] Christopher Kolda, Hitoshi Murayama, JHEP 0007, 035 (2000).
 [9] D.J.E Callaway, Phys. Rep. 167, 241 (1988);
 T. Hambye and K. Riesselmann, Phys. Rev. D 55, 7255 (1997);
 J.S. Lee and J.K. Kim, Phys. Rev. D 53, 6689 (1996).
 [10] A. Dobado, J. Morales, J.R. Pelaez, and M.T. Urdiales, Phys. Lett. B 387, 563 (1996);
 A. Ghinculov, T. Binoth, and J.J. van der Bij, Phys. Rev. D 57, 1487 (1998).
 [11] A. Ghinculov, Nucl. Phys. B 455, 21 (1995);
 A. Frink et. al., Phys. Rev. D 54, 4548 (1996).
 [12] P. Maher, L. Durand, and K. Riesselmann, Phys. Rev. D 48, 1061 (1993); 52, 553 (1995);
 K. Riesselmann, Phys. Rev. D 53, 6626 (1996).
 [13] U. Nierste and K. Riesselmann, Phys. Rev. D 53, 6638 (1996).
 [14] W.J. Marciano and S.S.D. Willenbrock Phys. Rev. D 37, 2509 (1988).
 [15] B.W. Lee, C. Quigg, and H. Thacker Phys. Rev. D 16, 1519 (1977).
 J.M. Cornwall, D.N. Levin, and G. Tiktopoulus, Phys. Rev. D 10, 1145 (1974).
 M.S. Chanowitz and M.K. Gallaird Nucl. Phys. B 261, 379 (1985).
 G.K. Gounaris, R. Kogerler, and H. Neufeld, Phys. Rev. D 34, 3257 (1986).
 H.J. He and W. Kilgore, Phys. Rev. D 55, 1515 (1997).
 [16] A. Dobado and M. Herrero, Phys. Lett. B 228, 495 (1989).
 J. Donoghue and C. Ramirez, Phys. Lett. B 234, 361 (1990).
 [17] J. Gasser and H. Leutwyler, Ann. of Phys. 158, 142 (1984).
 S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967);
 M.S. Chanowitz, M. Golden, and H. Georgi, Phys. Rev. D 36, 1490 (1987);
 A. Longhitano, Phys. Rev. D 22, 1166 (1980); Nucl. Phys. B 188, 118 (1981).
 [18] H. Veltman, Phys. Rev. D 41, 264 (1990).
 [19] A. Dobado, J.R. Pelaez, and M. T. Urdiales, Phys. Rev. D 56, 7133 (1997).
 [20] A. Dobado, A. Lopez, and J. Morales, Il Nuovo Cimento A 108, 335 (1995);
 A. Dobado and J. Morales, Phys. Rev. D 52, (1995) 2878.
 [21] S. Coleman, R. Jackiw, and H. D. Politzer, Phys. Rev. D 10 2491 (1974);
 Dolan and Jackiw, Phys. Rev. D 9, 3320 (1974);
 M. Kobayashi and T. Kugo, Prog. Theor. Phys. 54, 1537 (1975);
 W. Bardeen and M. Moshe, Phys. Rev. D 28, 1372 (1983).
 [22] R.Casalbuoni, D. Dominici, and R. Gatto, Phys. Lett. B 147, 419 (1984);
 M.B. Einhorn, Nucl. Phys. B 246, 75 (1984);
 A. Dobado and J.R. Pelaez, Phys. Lett. B 286, 136 (1992).
 [23] H. Georgi, Lie Algebras in Particle Physics, Addison Wesley, Frontiers in Physics New York 1982.
 [24] M. E. Peskin, D. V. Schroeder, An introduction to quantum field theory, Westview Press (1995), Chap. 7.
 [25] R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).
appendix
Publication Dates

Publication in this collection
13 Oct 2005 
Date of issue
Sept 2005
History

Received
13 Oct 2004