Braz. J. Phys. vol.35 número3A

Higgs Boson Decay in the Large

N

Limit

John Morales1,2, R. Martinez1, Rafael Hurtado1,2, and Rodolfo A. Diaz1

Universidad Nacional de Colombia1_{, Departamento de F´ısica, Bogot´a, Colombia,} and Centro Internacional de F´ısica2, Bogot´a, Colombia

Received on 13 October, 2004

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 non-perturbative. We show that the Equivalence Theorem holds in the largeNlimit at next-to-leading order by calculating the decay widths h→W+W− andh→π+_π−_{. We also show, in the same scheme of}

calculations, that unitarity is fulfilled for the processh→π+π−.

I. INTRODUCTION

The Standard Model (SM) of the electroweak interactions, based on theSU(2)L⊗U(1)Y gauge symmetry [1], is a suc-cessful theory and agrees with most experimental results [2]. However, the scalar sector responsible for the symme-try 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 ap-pears with a mass given bym2_h=2λv2, whereλis the coupling constant of the self-interacting term andvis the VEV (v≈246 GeV).mhis an unknown parameter so far.

Nevertheless, the precision tests of the SM impose strong bounds to the Higgs mass when the scalar sector is weakly-coupled. The results from LEP Electroweak Working Group analysis yieldmh=114+₋69₄₅GeV (68% CL) [3], and an upper limit ofmh<260 GeV with one-sided 95% CL [3]. The direct search of the Higgs boson done at LEP gives a lower limit of

mh>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 funda-mental one, then there is an associatedΛscale for the appear-ance 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(Λ,mh)[8].

The upper limit for the Higgs mass can be obtained by triv-iality considerations in the Higgs sector [9]. When the quar-tic coupling constantλin the scalar sector of the Higgs po-tential is renormalized introducing a cut-offΛ, the coupling goes to zero whenΛgoes to infinity, implying thatmhgoes to zero. This is not the case for the SM, because it needs a massive scalar particle at low energies to explain experimen-tal 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 non-perturbative[10].

The amplitude for a heavy Higgs decaying into two

longi-tudinally polarized gauge bosons reads [11]

A

_{(h→ZZ,WW)≈λ(mh)}

1+2.8λ(mh) 16π2 +62.1

_λ(m

h) 16π2

2

.

(1) By considering that in the perturbation expansion theλ2_term

must be smaller than theλterm, it is found thatλ(mh)≈7 im-plying thatmh≈1 TeV. On the other hand, using the scattering processWW→ZZmediated by a Higgs particle, which might be important in future collider experiments like LHC and lin-ear colliders, the cross section for energies√s>>mhat two loops level is given by [12]

σ(s) = 1

8πsλ(s) 2

1−42.65λ(s)

16π2+2477.9

_λ

(s)

16π2

2

.

(2) This cross section is negative for some values of λ which means that the perturbative expansion breaks down. Consid-ering that theλ2_{term must be smaller than theλterm, a}

nec-essary condition to have a convergent series isλ_≈4, in this casemh≈700 GeV [13]. The above scenarios correspond to the limit between weakly-coupled and strongly-coupled scalar sectors.

In the Marciano and Willenbrock paper [14] they calculated the decays of a heavy Higgs boson up to

O

_(g2_m2

h/m2W)in per-turbation theory using the Equivalence Theorem (ET) [15], from which the amplitude with gauge bosons longitudinally polarized at energies

O

_(q2_>>m2

W)is equivalent to the same amplitude but changing the corresponding longitudinal com-ponents by the would-be Goldstone bosons. For Higgs masses of the order ofmh≈1 TeV and mh≈1.3 TeV the radiative corrections for the decayh_→W+W−are 7.3% and 12%, re-spectively. At this scale the scalar sector is strongly-coupled and the theory is non-perturbative. It is obvious that the am-plitude at next-to-leading order breaks the perturbative expan-sion because all Feynman rules are proportional to the Higgs mass. For strongly interacting models is necessary to use a non-perturbative 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 non-perturbative calculations.

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 dis-persion [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 ModelO(4)and then gen-eralized to a model withO(N+1)symmetry. This method has been applied to study the Higgs boson at TeV energy scales [10, 22]. We show that the largeN limit can predict ampli-tudes that fulfill the ET and the unitarity condition at next-to-leading order for the SM, with a strongly interacting scalar sector, by using theh→W+W−andh→π+_π−_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−andh→π+_π−_{, in the largeNlimit and we show} that the ET holds at next-to-leading order. In section 4 we show that the amplitudeh→π+_π−_{satisfies unitarity in the} largeNlimit. In section 5 we give our conclusions.

II. THEO(N+1)MODEL

It is well known that the Linear Sigma Model represents the symmetry breaking O(N+1)→O(N) with N would-be Goldstone bosons which would-belong to the fundamental irre-ducible representation of the remaining symmetryO(N). For the purposes of this work the would-be Goldstone bosons will be named like pions π. For a gauge invariant model under

SU(2)L⊗U(1)Y local symmetry the largeNlimit for the SM is defined as

L

_g=

L

_{Y M+ (DµΦ)}†_(Dµ_Φ)−V(Φ2₎

withΦ†_{= (π}

1,π2,···,πN,σ)andΦ2=Φ†Φ. As usual

L

Y Mis the Yang-Mills Lagrangian of the SM and the covariant deriv-ative is defined as

DµΦ=∂µΦ−igTL·WµΦ+ig′T3RBµΦ,

whereTL=−(i/2)MLare the generators of theSU(2)Lgauge group and T₃R=−(i/2)MY _{is the generator of the U(1)}

Y 4gauge group. TheMmatrices are given by [23]

Mi jab=−i(δaiδbj−δbiδaj)

which belong to an irreducible representation of theO(N+1)

Lie algebra withi,j=1,2,3 anda,b=1,2, . . . ,N+1. The matrices which belong to the adjoint representation of the

SU(2)LLie algebra are given by

ML₁ =



       

0 0 0 ··· −

0 0 − ··· 0 0 + 0 ··· 0 . . . . . . . . . . . .

+ 0 0 ··· 0



       

ML₂=



       

0 0 + ··· 0 0 0 0 ··· − − 0 0 ··· 0 . . . . . . . . . . . . 0 + 0 ··· 0



       

ML3 =



       

0 + 0 ··· 0

− 0 0 ··· 0 0 0 0 _··· +

. . . . . . . . . . . . 0 0 − ··· 0



       

and, the corresponding matrix for theU(1)YLie algebra reads

MY =



       

0 + 0 ··· 0

− 0 0 ··· 0 0 0 0 ··· −

. . . . . . . . . . . . 0 0 + ··· 0



       

where dots represent zeros. In this form we have a global

O(N+1)symmetry with a localSU(2)L⊗U(1)Y symmetry. The Higgs potential, invariant underO(N+1), can be writ-ten as

V(Φ2_{) =−µ}2_Φ2₊λ

4(Φ

2₎2_{. (3)}

Aligning the vacuum state asφ 0≡(0, . . . ,v), withΦ2=v2=

2µ2/λ, the global symmetryO(N+1)is broken toO(N)and the local symmetry is broken asSU(2)L⊗U(1)Y →U(1)Q. By defining the Higgs field ash=σ−v, we find the following expression

L

_{g =}

L

_{Y M+}1

2(Dµπa)

†_(Dµ_π a) +

1 2(Dµh)

†_(Dµ_h) (4)

− 1

2m

2 hh

2

−λ(π2+h2)2−4λvh(π2+h2).

The gauge boson masses are obtained from the kinetic term,

1 2

_gv

2

2

WµaWaµ+ 1 2

g′v

2

2 BµBµ−

gg′v2

4 W

3

µBµ (5)

where the mass eigenstates are given by

Wµ+ = (Wµ1−iWµ2)/

√

2

W_µ− = (W1

µ+iWµ2)/

√

2 (6)

Zµ = cosθWWµ3−sinθWBµ Aµ = sinθWWµ3+cosθWBµ

andθW is the Weinberg angle with tanθW =g′/g. TheWµ± fields, withmW =gv/2≈80.6 GeV masses, are the charged gauge bosons, and theZµfield, withmZ=v(g2+g′2)1/2/2≈ 91.2 GeV mass, is the weak neutral gauge boson. TheAµfield is the massless photon.

The Lagrangian has terms of the formg2v∂µ_π

L

_{[π, W,B,h] = −}1

2πa✷πa− 1

2h(✷+m

2

h)h−λ(π2a+h2)2

− 4λvh(π2 a+h2)−

g

2∂ µ_π1(W3

µπ2−Wµ2π3)

− g

2∂ µ_π2(W1

µπ3−Wµ3π1)− g

2∂ µ_π3(W2

µπ1−Wµ1π2) + g∂µ_h(W

µ·π)− g′

2(π1∂µπ2−π2∂µπ1)B µ

−g∂µhπ3Bµ + 1

2m

2

WWµ·W µ+ 1 2m

2

BBµBµ−mWmBWµ3Bµ + g

2

8 (Wµ·π)(W µ

·π) +g′ 2

v

4 hBµB µ

+ g′ 2

8 H

2_B µBµ−

gg′

4 h

2_W3 µBµ−

gg′v

2 hW

3 µBµ

+ g 2

8 h

2_W µ·W µ+

g2v

4 hWµ·W µ₊g′

2

8 BµB µ_π·π

+ gg′

4 W

3

µBµπ·π− gg′

2 π3Bµ(W µ 1π1+W

µ 2π2) + g′mWBµ(W1µπ2−W2µπ1) +

gg′

2 Bµ(W µ

1π2−W2µπ1)h

+

L

_{Y M. (7)}

Thus we have a gauge theory spontaneously broken with the π= (π1,π2,π3)fields as the would-be Goldstone bosons of the

broken symmetrySU(2)L⊗U(1)Y/U(1)Qandπafields as the would-be Goldstone bosons of the broken global symmetry

O(N+1)/O(N).

The theory for the largeNlimit makes sense whenN→∞ and gives rise to finite amplitudes for different processes. To get finite amplitudes is necessary to choose appropriate para-meters in the largeNlimit. We will take the following defini-tion

λ≈1/N (8)

in order to use perturbative expansion of the strongly interact-ing sector as a function of theλparameter. With this defini-tion, physical masses must be finite and independent ofN in the largeNlimit. From the masses

m2h = 2λv2≈const m2W =

g2v2

4 ≈const

m2Z =

(g2_+g′2 )v2

4 ≈const (9)

we obtain for the other parameters of the model in the largeN

limit the following values

v≈√N , g≈1/√N, g′≈1/√N. (10)

Finally, we obtain the Feynman rules necessary to calculate the decay widths forh→W+W−andh→π+_π−_{in the large}

Nlimit, see Fig. 1.

III. THE HIGGS BOSON DECAY AND THE

EQUIVALENCE THEOREM

The SM in the largeNlimit is associated with theO(N+

1)/O(N)andSU(2)L⊗U(1)Y/U(1)Q global and local sym-metry breaking schemes respectively. We can calculate the amplitudes forh→W+W−andh→π+_π−_{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/√_{N)in the}

largeNlimit . The decay widths at tree level forh→W+W−

andh→π+_π−_{processes are given by}

Γ(h→W+W−) = g 2_m3

h 64πm2

W

1−4m

2 W m2

h

1/2

1−4m

2 W m2

h

+12m 4 W m4

h

,

Γ(h→π+π−) = g 2_m3

h

w_µ

h

h

w−

µ

=

ig

µν

gm

√

W

N

πa

h

h

π

b

=

₋

i

gm

2

h

2

m

W

√

N

δ

ab

π+

h

h

π−

=

−

i

gm

2

h

2

m

W

√

N

πa

π+

πb

_π−

=

−

i

g

2

_m

2

h

4

m

_W2

N

δ

ab

πa

πc

π

b πd

=

−

i

g

2

_m

2

h

4

m

2_W

N

δ

ab

δ

cd

ha

p

h

h

=

i

p

2

₋

_m

2

h

fa

p π±,πa

f

=

i

p

2

Figure 1. Feynman rules in the Landau gauge for the SM in the largeNlimit

To obtain the amplitudes at next-to-leading order is neces-sary to introduce the radiative corrections. First we calculate the self-energy of the scalar particleh, whose Feynman dia-grams at next-to-leading order are shown in Fig. 2. In this case, the self-energy at one loop level withπafields into the loops is of the order of 1/NtimesNwhereNis the number of degrees of freedom running into the loop. Therefore, radiative corrections are of the order of one in the largeN limit. The

same analysis can be done for the self-energy diagram withl

loops. It has two vertices withhππandl−1 vertices with four πaand is of the order of(1/

√

N)2₍˙_1/N)l−1_timesNl_the num-ber of pion fields running into thelloops. Consequently, the self-energy diagram withlloops is of the order of one in the largeNlimit. However, the one irreducible particle function (1IP) for self-energy diagram withWµ±,Zµinto the loop is of the order of 1/N, which is negligible in the largeNlimit.

−i

∏

h

(q2) ₌ h

hπac πb

hh

+ hh

πa cπc

πb c

πd hh

+

h

hπacπc πb

cπe πd

c πf

hh

+· · · ·+ h_h

Wµ+

ow

Wµ−

hh +· · ·

O(1 N)

Figure 2. Next-to-leading order Feynman diagrams that contribute to the self-energy of the Higgs boson in the largeNlimit

After doing all calculations by using dimensional regular-ization, withd=4−εintegrals from the loops, we find (see appendix)

−iΠh(q2) = g2_m4

h 8m2

W Iq

1−ig2m2h 8m2_W Iq

(12)

withIqgiven by

Iq= i

16π2

∆+2−logq

2 µ2−iπ

(13)

where∆=2/ε+log4π₋γεandµis the renormalization scale.

have taken into account that only the one irreducible particle functions are important in perturbation theory for the renor-malization of parameters such as the mass and the wave func-tion [24].

From the self-energy calculation the wave function renor-malization of the Higgs boson can be obtained as

Zh=1+

g2

256m2

Wπ2q2

1+ g2m2h 128m2

Wπ2

∆+2−logq_µ22−iπ

2

q2_=m2

hR

.(14)

The contributions toπ±_,W±_andZ

µself-energies are

propor-tional to 1/Nand in the largeNlimit we obtain that

Zπ±=ZWµ=ZZµ=1. (15)

To calculate the Higgs decays at this order, vertex correc-tions have to be included as well, as shown in Fig. 3.

(a)h

h pu

Wµ+

v Wν−

=

h

h u

Wµ+

v Wν−

+

h h

Wµ+

yz

W_ν−

u Wµ+

v Wν−

+···

(b) h

h pe

π+

d

π−

=

h h

πa

πb

e

π+

d

π−

+

h h

πa

πc

πb

πd

e

π+

d

π−

+···

+

h h

Wµ+

yz

W_ν−

e

π+

d

π−

+···

+ h h

π+

π+

π−

π−

Figure 3. Feynman diagrams in the largeNlimit which contribute to vertex interactions.(a)h_→W+_W−_,(b)h→π+_π−_.

The radiative corrections ofhW+W−vertex displayed in Fig. 3(a) are suppressed since they are of the order of 1/N2_becoming

negligible in our approximation. ThehW+W−vertex can be written at this level as

h

h pu

Wµ+

v Wν−

=igµν

gmW

√

N . (16)

For thehπ+_π−_{vertex corrections shown in Fig. 3(b), the pions into the loops give the most important contributions and can be} written as

h

h pe

π+

d

π−

= −ig

2mW√N



 

1

1 m2

h +₁₂₈g_m22

wπ2

∆+2₋logq_µ22−iπ



 

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]

1

m2_h R

≡ 1

m2_h+

g2(∆+2)

128π2_m2 W

. (17)

The real part of theZhfunction Eq.(14) is given by

Z_h1/2=1+ g 2_m2

hR

162_m2 Wπ2

×

1− 2g

2_m2

hR 128m2_Wπ2log

m2_hR µ2 +

g4m4_hR 82_m4

W162π4

log

m2_hR µ2

2

−π2

1−

g2_m2

hR 128m2_Wπ2

logm 2

hR µ2 +iπ

4 .

In the same way, the real part of the vertex correction Eq.(3) can be expressed as

h Kp e π+ d π−

= −igm 2 hR

2mW

√ N×       1+ g2_m2

hR 128m2_Wπ2log

m2

hR µ2 −

g4_m4

hR 82_m4

W162π4

log m2 hR µ2 2

−π2

1−

g2_m2

hR 128m2

Wπ2

logm 2

hR µ2 +iπ

2      

Hence, the vertex corrections Eq.(19) multiplied by the factors of the renormalized wave functions Z_h1/2Zπ give rise to the

h_→π+_π−_{amplitude at next-to-leading order and can be written at}

O

_(g2_m2

h/mW2)as

A

_(h→π+_π−_{) =} gm

2 hR

2mW√N

     1+

g2m2_hR 128m2_Wπ2log

m2_hR µ2

1−

g2_m2

hR 128m2_Wπ2

logm 2

hR µ2 +iπ

2 +

g2_m2

hR 162_m2

Wπ2

1−

g2_m2

hR 128m2

Wπ2

logm 2

hR µ2 +iπ

4      . (20)

The same procedure is done for theh→W+W−amplitude where the vertex corrections are multiplied by the factors of the renormalized wave functionsZ_h1/2ZW, and can be written at

O

(g2m2h/m2w)as

A

_(h→W+_W−_{) =} gm

2 hR

2mW

√ N      1+

g2_m2

hR 162_m2

Wπ2

1−

g2_m2

hR 128m2_Wπ2

logm 2

hR µ2 +iπ

4      . (21)

In order to show that the ET holds for non-perturbative next-to-leading order, we calculate Higgs decays into gauge bosons and pions in the largeN limit. We then compare the decay widths as obtained from the decay amplitudes forh→π+_π−_and

h→W+W−in Eqs. (20) and (21) respectively. Such decay widths are given by

Γ(h→π+π−) = g 2_m3

hR

64πm2 WN      1+

g2_m2

hR 64π2_m2

W log m2 hR µ2 1−

g2_m2

hR 128π2_m2

W

logm 2

hR µ2 +iπ

4 +

g2_m2

hR 128π2_m2

W

1−

g2_m2

hR 128π2_m2

W

logm 2

hR µ2 +iπ

and

Γ(h→W+W−) = g 2_m3

hR

64πm2 WN

1−4m

2 W m2

hR

1−4m

2 W m2

hR +12m

4 W m4

hR



   

1+

g2m2_hR 128π2_m2

W

1−

g2_m2

hR 64π2_m2

W

logm 2

hR µ2 +iπ

4



   

. (23)

In Fig. 4 we display the ratioΓ(h→π+_π−_)/Γ(h→W+_W−_{)as a function of the Higgs mass including next-to-leading order} corrections. From this figure it can be seen that such quotient tends to one for large Higgs masses (mhR4.5 TeV), showing the

validity of the ET at high energies.

m

hR

(

GeV

)

Γ(

h

→

π

+

π

−

)

Γ(

h

→

W

+

W

−

)

0 2000 4000 6000 8000 10000 1

1.2 1.4 1.6 1.8 2 2.2 2.4

Figure 4. The quotientΓ(h→π+_π−_)/Γ(h→W+_W−_{)versus the}

renormalized Higgs mass, in the largeNlimit at next-to-leading order. This Fig. shows that both decay widths tend to be equal formhR4.5TeV;

showing the validity of the ET at high energies.

IV. UNITARITY IN THE LARGENLIMIT

As a consequence of unitarity of theS-matrix, i. e.S†S=1, the Optical Theorem is obtained . By defining S=1+iT,

where theT is called the transition matrix, we have

−i(T−T†) =T†T (24) and since four momentum is conserved in the transition from initial state|i to final state|f , we can always write

f|T|i = (2π)4δ4_(p

f−pi)

T

f i (25)

and

f|T†|i =i|T|f ∗(2π)4_δ4_(p

f−pi)

T

i f∗. (26)

Inserting a complete set of intermediate states|q we find

f|T†T|i =

∑

n

_n

∏

i=1 Z

d3qi (2π)3_2E

i

f|T†|qi qi|T|i (27)

and from the identity (24) we can obtain the Cutkosky’s rule[25]

2Im(

T

_{i f) =}

∑

n

n

∏

i=1 Z

d3qi (2π)3_2E

i

T

∗

f qi

T

iqi(2π) 4_δ4_(i→

∑

i

qi) (28)

where the sum runs over all possible sets of intermediate statesqi. Applying this identity to the decayΓ(h→π+_π−_{)we find}

2Im



p

Kpe

p+

d p−



 =

Z

d3qa (2π)3_2E

a d3qb (2π)3_2E

b

(2π)4δ4_(p

−qa−qb)

×



p

Kp

e qb

d qa





∗



qa

d

qb

e pe

p+

d p−



. (29)

Z_h Zπ, resulting

2Im(

A

_(h→π+_π−_{)) =}

g3_m4

hR 8m3_W16π√N

1−

g2_m2

hR 8m2

W16π2

logm 2

hR µ2 +iπ

2. (30)

For the right-hand side, we have to multiply the amplitudes calculated in the largeNlimit

h

h pe

qb

d qa

=Z

1/2 h Zπ



h

h e

qb

d qa

+

h

h e

qb

d qa

+···





i.e.,

A

_{(h→πaπb) =} gm

2 hR

2mW

√

Nδab



 

1

1− g

2_m2

hR 8m2

W16π2

log(qa+qb)2 µ2 +iπ

+

g2_m2

hR 162_m2

Wπ2

1− g

2_m2

hR 8m2

W16π2

logm 2

hR µ2 +iπ

2



   

(31)

by

qa

d

qb

e pe

p+

d p−

=Z

2

π





qa

d

qb

e e

p+

d p−

+

qa

d

qb

e

e p+

d p−

+···





i.e.,

A

_(πaπb→π+_π−_{) =} g

2_m2 hR

4m2 WN

δab



 

1

1− g

2_m2

hR 8m2

W16π2

log(qa+qb)2 µ2 +iπ





. (32)

Then the right-hand side of Eq. (29) becomes

[

A

_{(h→πaπb)]}∗

A

_(πaπb→π+_π−_{) =} 1

(2π)2×

1 4×

g3m4_hR 8m3

W

√ N

1−

g2_m2

hR 8m2

W16π2

log(qa+qb)2 µ2 +iπ

2

= 1

4f(qa,qb) (33)

where 1/4 is the symmetry factor for identical bosons in the final state. From Eq. (29) we define

M= 1

4π2 Z

d3qa 2Ea

d3qb 2Eb

δ4_(p−q

a−qb)×[

A

(h→πaπb)]∗

A

_(πaπb→π+_π−_{) (34)}

and the integral overqbcan be written as

Z d3qb

2Eb =

Z ∞

−∞d

4_q

Integrating the four-dimensional delta function in Eq.(36) we obtain

M = 1

16π2 Z

|qa|2d|qa|dΩ 2Ea

δ[(p−qa)2]Θ(p0−qa0)f(qa,p−qa)

= 1

16π2 Z

E

0

|qa|dEadΩ 2 δ[p

2_−2p·q

a+q2a]f(qa,p−qa). (36)

In the center-of-mass frame

p= (E,p), qa= (Ea,qa) = (E′,q), qb= (Eb,qb) = (E′,q) (37)

the integral in Eq. (36) can be rewritten as

M = 1

16π2 Z

E

0

|qa|dE′dΩ 2 δ[E

2_−2EE′_−p2_+2p·q]f(q

a,p−qa)

= g

3_m4 hR

8m_W316π2√_N

|qa|

1−

g2_m2

hR 8m2

W16π2

logp2 µ2 +iπ

2 dΩ

2| −2E| (38)

and by usingp2=m2_h R

M=

g3m4_hR 8m3

W16π

√ N

1−

g2_m2

hR 8m2

W16π2

logm 2

hR µ2 +iπ

2. (39)

Comparing equations (30) and (39) we see that the Higgs de-cayΓ(h→π+_π−_{)calculated in the largeNlimit at} next-to-leading order fulfills the unitarity condition.

V. CONCLUSIONS

We have shown that non-perturbative calculations at next-to-leading order in the largeNlimit for the case of a Higgs de-caying intoW±andπ±_{fulfill the ET. In particular, we found} that the decay widthsΓ(h→W+W−)andΓ(h→π+_π−_)get values that are basically identical for heavy Higgs bosons i.e.

mhR4.5 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 in-teracting 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 sup-port.

VI. APPENDIX

In this appendix we show the explicit calculation of a Feyn-man diagram withlloops in the largeNlimit that contributes to the Higgs boson self-energy.

h K

···

l−loop´s K

h = 1

2l

−igm2_h

2mW

√

N

(−NIq)

×

−ig2m2_h

4m2 WN

l−1

× (−NIq)l−2 = 1

2l(−im 2 h)l+1

−g2_I q

4m2_W

l

(40)

where 1/2lis the symmetry factor of the diagram. The first factor corresponds to the initial and final loops times the ver-tices with three particles, the second factor represents the

contributes with anNfactor, as they haveNcirculating pions.

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