B Machet

UNLOCKING THE STANDARD MODEL II . 1 GENERATION OF QUARKS . MASSES AND COUPLINGS

Keywords: 11, 15, Ex 11, 30 Qs

Unlocking the Standard Model. II. 1 generation of quarks. Masses and couplings.

Introduction

The extension of the Standard Model [START_REF] Glashow | Partial-symmetries of weak interactions[END_REF] for one generation of fermions advocated for in [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF] is endowed with two Higgs doublets, a "chiral" doublet

K =   p 1 -ip 2 -(s 0 + p 3 )   , < s 0 >= v √ 2 , (1)
and a "weak" doublet

H =   s 1 -is 2 -(p 0 + s 3 )   , < s 3 >= σ √ 2 , (2)
both isomorphic to the Higgs doublet of the Standard Model [START_REF] Glashow | Partial-symmetries of weak interactions[END_REF]. It constitutes the "smallest maximal extension" of the Glashow-Salam-Weinberg model. It is maximal in the sense that it incorporates all possible J = 0 scalars 1 LPTHE tour 13-14, 4 ème étage, UPMC Univ Paris 06, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France), Unité Mixte de Recherche UMR 7589 (CNRS / UPMC Univ Paris 06) 2 machet@lpthe.jussieu.fr (and pseudoscalars) that are expected for a given number of generations, and it is the smallest extension because it does not invoke a priori any physics "beyond the Standard Model" now any new type of particle. s 0 and s 3 have non-vanishing vacuum expectation values (VEV's) as written in [START_REF] Glashow | Partial-symmetries of weak interactions[END_REF]. In there, the symbols "s" and "p" stand respectively for "scalar" and "pseudoscalar", such that H and K are parity transformed of each other. Their components that we call generically h 0 , h 1 , h 2 , h 3 transform respectively by SU (2) L and SU (2) R according to

T i L . h j = -1 2 i ǫ ijk h k + δ ij h 0 , T i L . h 0 = -1 2 h i , (3)
and

T i R . h j = -1 2 i ǫ ijk h k -δ ij h 0 , T i R . h 0 = + 1 2 h i . (4)
The main steps of this works are the following. In section 2 we give the general formula for the mass of the W gauge bosons in terms of the two VEV's < s 0 > and < s 3 >. In section 3 we introduce Yukawa couplings of quarks to both Higgs doublets K and H. It could have looked more natural to first introduce the scalar potential, but it turns out that the latter gets strongly constrained by the former. After giving their general expression, from which we extract the u and d quark masses in terms of < s 0 > and < s 3 >, we investigate in section 4 their low energy limit by using the one-to-one correspondence demonstrated in [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF] between K, H and 4-sets of bilinear quark operators. At this limit, renormalizability is not a concern and Yukawa couplings can be rewritten in a very simple form in which, in particular, symmetries clearly show up. Using the Partially Conserved Axial Current hypothesis (PCAC) [START_REF] Adler | Current Algebras and Applications to Particle Physics[END_REF] [4] [START_REF] Alfaro | Currents in Hadron Physics[END_REF] and the Gell-Mann-Oakes-Renner (GMOR) [START_REF] Gell-Mann | Behavior of Current Divergences under SU(3) x SU(3)[END_REF] [5] relation enables to account for the mass of the pions and to determine the values of all but one Yukawa parameters. The last one is obtained by identifying the Goldstones of the spontaneously broken weak SU (2) L symmetry. A last constraint results from considering the π 0 -η system and requesting that it be devoid of any tachyonic state. This determines the quantity (m u -m d ) < ūudd > ((m u + m d ) < ūu + dd >) is determined by the GMOR relation).

We comment at length on fermion masses, and the important role of both Higgs doublets in their generation.

After gathering the values of the parameters in section 5, section 6 is devoted to the scalar potential. V (K, H) is chosen to be invariant by the chiral group U (2) L × U (2) R , which clearly identifies the Goldstone of chiral symmetry breaking. It only depends on two parameters, one quadratic and one quartic coupling. At low energy, it receives corrections from the bosonised (low energy) form of the Yukawa couplings, which yields an effective potential V ef f (K, H). A last constraint comes from minimizing V ef f at the known VEV's of the two Higgs bosons, which reproduce the pion and W masses. It determines the value of the quartic coupling and the masses of the two Higgs bosons. In section 7, we determine their couplings to quarks, gauge bosons and leptons.

Section 8 provides some additional considerations concerning symmetries, Goldstone and pseudo-Goldstone bosons. Several symmetries are at work and some fields play dual roles. We focus in particular on the custodial SU (2) symmetry and on the respective roles of < ūu + dd > and < ūudd >. Section 9 gives some remarks concerning more generations. Section 10 is a brief conclusion.

Kinetic terms for the Higgs doublets and gauge boson masses

The masses of gauge bosons arise from the kinetic terms

(D µ K) † D µ K + (D µ H) † D µ H (5)
for the two Higgs doublets K and H. D µ is the covariant derivative with respect to the group SU (2) L of weak interactions. Owing to the laws of transformations (3), the VEV's of s 0 and s 3 generate a mass m W for the W gauge bosons

m 2 W = g 2 2 < s 0 > 2 + < s 3 > 2 = g 2 v 2 + σ 2 4 , (6)
in which g is the SU (2) L coupling constant g ≈ .61 .

Yukawa couplings

We choose to first introduce Yukawa couplings because their low-energy limit (see section 4) will in particular constrain the effective scalar potential.

General expression

Quarks must be coupled to the two Higgs doublets K and H. Introducing the couplings ρ u and ρ d to K and λ u and λ d to H, the Yukawa Lagrangian writes 1

L Y ukawa = + ρ d u L d L K d R -ρ u u L d L (iτ 2 K * ) u R + λ d u L d L H d R + λ u u L d L (iτ 2 H * ) u R + h.c., (8)
which gives, explicitly,

L Y ukawa = - δ 1 v √ 2µ 3 (ūu + dd) + κ 12 σ √ 2ν 3 (ūu -dd) s 0 -δ 12 v √ 2µ 3 (ūu + dd) + δ 2 σ √ 2ν 3 (ūu -dd) s 3 + δ 1 v √ 2µ 3 ūγ 5 d p -+ dγ 5 u p + + (ūγ 5 u -dγ 5 d) p 3 + κ 12 σ √ 2ν 3 du p + -ūd p -+ (ūγ 5 u + dγ 5 d) p 3 - δ 12 v √ 2µ 3 dγ 5 u s + -ūγ 5 d s --(ūγ 5 u -dγ 5 d) p 0 + δ 2 σ √ 2ν 3 du s + + ūd s --(ūγ 5 u + dγ 5 d) p 0 . (9)
In ( 8) and (9) the signs have been set such that for positive < s 0 > and < s 3 >, the fermion masses are positive for positive ρ u,d and λ u,d (given that a fermion mass term is of the form -m ψψ). We introduced in (9) the parameters with dimension [mass] 2

δ 1 = ρ u + ρ d 2 √ 2µ 3 v , κ 12 = ρ u -ρ d 2 √ 2ν 3 σ , δ 12 = λ u + λ d 2 √ 2µ 3 v , δ 2 = λ u -λ d 2 √ 2ν 3 σ . ( 10
)
1 τ 2 is the Pauli matrix   0 -i i 0   .
The doublets K ≡ iτ 2 K * and H ≡ iτ 2 H * are isomorphic to K and H.

Fermion masses

We define the two quantum Higgs fields ς and ξ by shifting the scalar fields s 0 and s 3 occurring respectively in the Higgs doublets K and H (see (1),( 2)) according to

s 0 =< s 0 > +ς, s 3 =< s 3 > +ξ. (11)
The two VEV's (given in (1) and ( 2)) contribute to the fermion masses according to

m u = ρ u < s 0 > +λ u < s 3 >= vρ u + σλ u √ 2 , m d = ρ d < s 0 > +λ d < s 3 >= vρ d + σλ d √ 2 . ( 12
)
Additional remarks concerning fermion masses are written in subsection 4.4.

The low energy limit

At low energy we use the one-to-one correspondence between K, H and

K = 1 √ 2 v µ 3   φ 1 -iφ 2 -(φ 0 + φ 3 )   = v √ 2 µ 3   dγ 5 u -1 2 (ūu + dd) -1 2 (ūγ 5 u -dγ 5 d)   ≡   k 1 -ik 2 -(k 0 + k 3 )   , < ūu + dd > = µ 3 , H = 1 √ 2 σ ν 3   ξ 1 -iξ 2 -(ξ 0 + ξ 3 )   = σ √ 2 ν 3   du -1 2 (ūγ 5 u + dγ 5 d) -1 2 (ūu -dd)   ≡   h 1 -ih 2 -(h 0 + h 3 )   , < ūu -dd > = ν 3 , (13)
that has been established in [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF] and identify accordingly

(s 0 , p 1 , p 2 , p 3 ) ≃ (k 0 , k 1 , k 2 , k 3 ), (p 0 , s 1 , s 2 , s 3 ) ≃ (h 0 , h 1 , h 2 , h 3 ). ( 14
)

Rewriting Yukawa couplings

The first consequence of this correspondence is that, defining

m 2 12 = κ 12 + δ 12 , (15)
and expressing the bilinear quark operators in [START_REF] Machet | Unlocking the Standard Model. III. The case of N > 1 generations[END_REF] in terms of the components (s 0 , p), (p 0 , s) of K and H, the Yukawa couplings [START_REF] Machet | Unlocking the Standard Model. III. The case of N > 1 generations[END_REF] rewrite

L ef f Y ukawa = -δ 1 K † K - 1 2 m 2 12 (K † H + H † K) -δ 2 H † H, (16)
or, indifferently, since renormalizability is not an issue at low energy, as a sum of 4-fermion interactions

L ef f Y ukawa = -δ 1 K † K - 1 2 m 2 12 (K † H + H † K) -δ 2 H † H. ( 17
)
This bosonised form of the Yukawa couplings, only valid at low energy, will be later added to the scalar potential V (K, H) to define the low energy effective potential V ef f (K, H) (see subsection 6.2).

PCAC and the Gell-Mann-Oakes-Renner relation

Kinetic terms together with Yukawa couplings include in particular

(∂ µ K) † ∂ µ K -δ 1 K † K - 1 2 m 2 12 (K † H + H † K) + (∂ µ H) † ∂ µ H -δ 2 H † H + . . . (18)
and we now raise the issue whether, at low energy, the charged components of K can be identified with the charged pions. As we shall see in subsection 4.3 below, both δ 2 and m 2 12 have to vanish: the first to ensure that the breaking of the weak SU (2) L is accompanied by three true Goldstone bosons, and the second to ensure that the p 0 -p 3 system does not exhibit a tachyonic state. Eq. (18) reduces then to standard kinetic terms for unmixed doublets. Furthermore, the scalar potential will be chosen in such a way that the three pseudoscalar bosons inside K are Goldstone bosons in the absence of Yukawa couplings. So, due to the mass term proportional to δ 1 , the three "pions" inside K get a mass m at the simple condition that δ 1 = 1 2 m 2 . Owing to the Partially Conserved Axial Current (PCAC) hypothesis [START_REF] Adler | Current Algebras and Applications to Particle Physics[END_REF]

[4] i(m u + m d ) ūγ 5 d = √ 2f π m 2 π π + , (19)
which identifies the interpolating pion field with a bilinear quark operator, and to the corresponding Gell-Mann-Oakes-Renner relation [START_REF] Gell-Mann | Behavior of Current Divergences under SU(3) x SU(3)[END_REF] (

m u + m d ) < ūu + dd >= 2f 2 π m 2 π , (20)
p + ≡ p 1 + ip 2 = v √ 2
µ 3 dγ 5 u as it is defined in ( 16) and ( 14) can be identified at low energy with

p ± ≃ - iv f π π ± . (21)
So, the kinetic terms

(∂ µ K) † ∂ µ K, which contain in particular ∂ µ p + ∂ µ p -≡ ∂ µ p 1 ∂ µ p 1 + ∂ µ p 2 ∂ µ p 2 , will be normalized in the standard way if v = f π , (22)
such that

p ± ≃ -iπ ± . (23)
Then, the term proportional to δ 1 in (18) is a suitable pion mass terms if

δ 1 = m 2 π . (24)
Going back to the definition of δ 1 in (10) and using ( 22) and ( 20), (24) corresponds to

ρ u + ρ d = m u + m d √ 2f π . ( 25
) Since f π ≪ m W , (22) plugged into (6) entails σ ≈ 2m W g , (26)
which shows that the W 's get their mass essentially from the VEV of s 3 . The ratio of the VEV's of the two Higgs doublets comes out accordingly as

tan β = < s 3 > < s 0 > = σ/ √ 2 v/ √ 2 ≈ 2m W gf π ≈ 2856. (27)
They correspond respectively to the weak (m W ) and chiral (f π ) scale. Both scales can now coexist, unlike in the genuine Glashow-Salam-Weinberg model where the parity-transformed H of the Higgs doublet K is missing.

Eqs. ( 12) and ( 26) then determine λ u and λ d

(λ u + λ d ) = g 2 √ 2m W (m u + m d ), (λ u -λ d ) = g √ 2m W (m u -m d ) - f π √ 2 (ρ u -ρ d ) , (28)
that is

λ u = g 3m u -m d -2 √ 2f π (ρ u -ρ d ) 4 √ 2 m W , λ d = g 3m d -m u + 2 √ 2f π (ρ u -ρ d ) 4 √ 2 m W , (29)
in terms of ρ u -ρ d which is, at this point, still undetermined. However, the SU (2) L invariant Yukawa couplings that need to be introduced to provide fermions with "soft" masses also give, at low energy, among other couplings, a "soft" mass to s + and s -through the term proportional to δ 2 . The situation for s + and s -is different from that of the pions which can become pseudo-Goldstone bosons and stay as physical particles. The spontaneously broken SU (2) L symmetry requires true Goldstones, which can only go along with

Goldstones and pseudo-Goldstones

δ 2 = 0, (30)
which is accordingly a side-effect of weak symmetry breaking. Looking at [START_REF] Machet | Unlocking the Standard Model. IV. A natural candidate for dark matter[END_REF], one could think that ν 3 ≡< ūudd >= 0 could be a solution to δ 2 = 0. However, we shall see later in subsection 8.1 that < ūu > must be different from < dd > as a trigger of both weak and custodial symmetry breaking. So, (30) entails

λ u = λ d = g 4 √ 2 m W (m u + m d ). (31)
By ( 28), (31) determines

ρ u -ρ d = √ 2(m u -m d ) f π , (32)
and, combined with (25),

ρ u = 3m u -m d 2 √ 2f π , ρ d = 3m d -m u 2 √ 2f π . (33) 4.3.2 The p 3 -p 0 system
The (p 3 , p 0 ) or (k 3 , h 0 ) or, equivalently (π 0 , η) system gets endowed by the Yukawa couplings with a mass

matrix 1 2   2δ 1 m 2 12 m 2 12 2δ 2   . (34)
However, since δ 2 has been fixed to zero in subsection 4.3.1, this system now exhibits a tachyonic state unless

m 2 12 = 0 ⇔ (ρ u -ρ d ) ν 3 σ = -(λ u + λ d ) µ 3 v ⇔ m u -m d m u + m d = - 1 2 µ 3 ν 3 ≡ - 1 2 < ūu + dd > < ūu -dd > , (35)
in which we have used (15).( 10), (31), (33) and the definitions of µ 3 and ν 3 that were introduced in (13).

Eq. ( 35) is equivalent to2

< dd > < ūu > = 3m u -m d m u -3m d . ( 36
)
When this is realized, p 0 is a true Goldstone and p 3 keeps its mass m 2 π . They do not mix. This fits the picture of p 0 being the third Goldstone boson of the broken SU (2) L symmetry, and p 3 being the neutral member of the triplet of pseudo-Goldstone bosons of the broken chiral symmetry SU (2) L × SU (2) R down to the diagonal SU (2). Other considerations concerning symmetries will be given in section 8.

No scalar-pseudoscalar coupling

Yukawa couplings are seen on [START_REF] Machet | Unlocking the Standard Model. III. The case of N > 1 generations[END_REF] to potentially generate couplings between charged scalars, for example s -= σ √ 2ν 3 du and pseudoscalars, for example p + . It is the second important effect of the condition m 2 12 ≡ δ 12 +κ 12 = 0 obtained in subsection 4.3.2 to cancel these transitions.

The unitary gauge. Leptonic decays of pions

In the unitary gauge the crossed couplings between the W gauge bosons and the (derivative of the) SU (2) L Goldstone bosons p 0 , s + , s -are canceled, which leaves untouched the similar couplings between W and the three pions. Their proportionality to v = f π yields in particular leptonic decays of pions in agreement with the standard PCAC calculation.

Fermion masses versus the low energy effective Lagrangian

Fermions receive their masses from the VEV's of the two Higgs doublets K and H. From ( 12) and the values of the parameters that have been determined (see also section 5 below), it appears that < s The second point is the inadequacy to calculate quark masses from the low energy effective expression (16) of the Yukawa couplings and its set of parameters determined by low energy considerations. When plugged into (16) the conditions δ 2 = 0 and m 2 12 ≡ δ 12 + κ 12 = 0 demonstrated respectively in (30) and in (35) entail that quark masses come from the sole Higgs doublet K, by -δ 1 K † K. Going back to quark fields and writing it for example as the product -δ 1 K † K of scalar fields K times their equivalents in terms of bilinear quark operators K, which respects renormalizability, L ef f Y ukawa does, through quark-antiquark condensation, generate quark masses. They however come out as -δ 1

v 2 2µ 3 (ūu + dd) = -mu+m d 4
(ūu + dd), which is different from the masses obtained from the original Lagrangian ( 9)

-δ 1 v √ 2µ 3 (ūu + dd) < s 0 > +δ 12 σ √ 2ν 3 (ūu -dd) < s 0 > - v √ 2µ 3 (ūu + dd) < s 3 > ; (37)
using the expression for δ 12 deduced from ( 10) and (28), the genuine Lagrangian (37) yields the mass terms

-δ 1 v 2 2µ 3 (ūu + dd) + δ 12 vσ 2 ūu -dd ν 3 - ūu + dd µ 3 (39) = - f 2 π m 2 π 2 ūu + dd µ 3 + f 2 π m 2 π 2 ūu -dd ν 3 - ūu + dd µ 3 (39) = - 1 4 (m u + m d )(ūu + dd) from δ1 - 1 2 (m u -m d )(ūu -dd) from κ12 - 1 4 (m u + m d )(ūu + dd) from δ12 . (38)
In (38), unlike in L ef f Y ukawa , the terms proportional to δ 12 do not vanish because the bilinear fermion operators do not reduce to their low energy VEV's < ūudd >= ν 3 , < ūu + dd >= µ 3 . Furthermore, even if m u is set equal to m d , the part proportional to δ 12 , which describes H -K interplay, contributes to quark masses as much as the one proportional to δ 1 which comes from K alone. Therefore, neither the effective Lagrangian L ef f Y ukawa nor the "low energy truncation" of the model, that includes only one Higgs doublet, K, can correctly account for fermion masses (nor, of course, for the masses of the gauge bosons, problem which led to "technicolor" models [START_REF] Susskind | Dynamics of spontaneous symmetry breaking in the Weinberg-Salam theory[END_REF]). L ef f Y ukawa we shall accordingly only use to deal with low energy physics of scalars and pseudoscalars, in particular to build the effective scalar potential V ef f in subsection 6.2.

Summary of the parameters

By low energy considerations, we have determined the following parameters, introduced in particular in ( 8) and [START_REF] Machet | Unlocking the Standard Model. IV. A natural candidate for dark matter[END_REF]:

ρ u = 3m u -m d 2 √ 2f π , ρ d = 3m d -m u 2 √ 2f π , λ u = λ d = g(m u + m d ) 4 √ 2m W , δ 1 = m 2 π , δ 12 = -κ 12 = gf π m 2 π 2m W , δ 2 = 0, (m u + m d ) < ūu + dd > (20) = 2f 2 π m 2 π , (m u -m d ) < ūu -dd > (35) = -f 2 π m 2 π , v ≡ √ 2 < s 0 >= f π , σ ≡ √ 2 < s 3 >= 2m W g . (39)
These should be plugged into the renormalizable form (9) of the Yukawa Lagrangian. Note that, unlike its low energy avatar (16), it depends on κ 12 and δ 12 = -κ 12 , and not on m 2 12 = 0.

6 The scalar potential

6.1 A U (2) L × U (2) R invariant potential We shall consider a quartic U (2) L × U (2) R invariant potential V (K, H) = - m 2 H 2 (K † K + H † H) + λ H 4 (K † K) 2 + (H † H) 2 , (40)
which thus decomposes into two independent potentials, one for K and one for H. This is possible because (see [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF]) K and H are stable by both SU (2) L and SU (2) R and transform into each other by U (1) L and U (1) R (with the appropriate signs). This last symmetry dictates in particular the equality of the couplings (quadratic and quartic) for the two doublets.

SU (2)

L breaking by v = 0 and σ = 0 generates three Goldstone bosons in each Higgs multiplet: p ∈ K, the pseudoscalar singlet p 0 and the two charged scalars s ± ∈ H. This also fits the scheme according to which v = 0 and σ = 0 spontaneously break the chiral U (2) L × U (2) R down to U (1) × U (1) em (see [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF]); there, too, six Goldstones are generated. The pseudoscalar triplet p ∈ K gets a small mass from the SU (2) L invariant Yukawa couplings while the pseudoscalar singlet p 0 and the two charged scalars s ± ∈ H must be protected from this since they are also the three Goldstones to be eaten by the W gauge bosons (see section 4). p 0 plays a double role in that is also the Goldstone of the breaking of U (1) L × U (1) R down to the diagonal U (1), which, at the level of the algebra, is related to parity breaking. v = 0 is associated with < ūu + dd > = 0, responsible for the breaking of SU (2) L × SU (2) R down to SU (2)

with the three pions as (pseudo)-Goldstone bosons, while σ = 0 is associated with < ūudd > = 0 which is also responsible for the breaking of the custodial SU (2) into U (1) em and of the W mass.

Our choice for the potential amounts to requesting that, in the absence of Yukawa couplings, all fields are Goldstones but for the two Higgs bosons.

In the most general potential for two Higgs doublets the following terms have accordingly been discarded:

• (m 2 K † H + h.c
), with m ∈ C would mediate in particular transitions between scalars and pseudoscalars that should not occur classically;

• λ 4 (K † K)(K † H) + h.c., λ 5 (H † H)(K † H 2 ) + h.c
. with λ 4 , λ 5 ∈ C would also mediate unwanted classical transitions between scalars and pseudoscalars;

• λ 3 (K † H) 2 + h.c. with λ 3 ∈ C would in particular contribute to the mass of the neutral pion and not to that of the charged pions. Such a classical π + -π 0 mass difference which is not electromagnetic nor due to

m u = m d is unwelcome; • λ 1 (K † K)(H † H), λ 2 (K † H)(H † K), with λ 1 , λ 2 ∈
R would also spoil the Goldstone nature of the pions and η, the first because of terms proportional to < s 3 > 2 π 2 and < s 0 > 2 η 2 , the second because of terms proportional to < s 0 > 2 η 2 , < s 3 > 2 π 0 2 and < s 0 >< s 3 > π 0 η.

The low energy effective potential

At low energy, the renormalizable V (K, H) is supplemented by (-1)× the bosonised form of the Yukawa Lagrangian ( 16). This yields the effective potential

V ef f (K, H) = V (K, H) + δ 1 K † K + 1 2 m 2 12 (K † H + H † K) + δ 2 H † H = - m 2 H 2 (K † K + H † H) + λ H 4 (K † K) 2 + (H † H) 2 + δ 1 K † K + 1 2 m 2 12 (K † H + H † K) + δ 2 H † H. (41)
It is further simplified since we have shown that δ 2 = 0 and m 2 12 = 0 (see (30) and (31) in section 4) and V ef f accordingly reduces to

V ef f (K, H) = - m 2 H -2m 2 π 2 K † K - m 2 H 2 H † H + λ H 4 (K † K) 2 + (H † H) 2 . (42)
Last, to suitably reproduce the π and W masses, we know that it should have a minimum at values of v and σ

given by ( 22) and ( 26). The two equations

∂V ef f ∂s 0 <s 0 >= fπ √ 2
= 0 and

∂V ef f ∂s 3 <s 3 >= √ 2m W g = 0 yield respectively m 2 H = λ H < s 0 > 2 +2m 2 π and m 2 H = λ H < s 3 > 2 such that λ H = 2m 2 π < s 3 > 2 -< s 0 > 2 ≈ 2m 2 π < s 3 > 2 1 + < s 0 > 2 < s 3 > 2 = g 2 m 2 π m 2 W 1 + g 2 f 2 π 4m 2 W , (43)
which puts it definitely in the perturbative regime. It is because of the presence of m 2 π that λ H is different from zero. m π = 0 keeps accordingly the theory away from instability.

The masses of the two Higgs bosons ς and ξ

Since the effective scalar potential is now fully determined, one can calculate the masses of the two Higgs bosons ς and ξ defined in [START_REF] Branco | Theory and phenomenology of two-Higgs-doublet models[END_REF], which do not mix. One gets

m ξ = < s 3 > λ H ≈ √ 2 m π , m ς = < s 0 > λ H = m ξ < s 0 > < s 3 > ≈ m π gf π √ 2 m W ≈ 68 KeV, (44)
In particular, their ratio is that of the two VEV's

m ξ m ς = < s 3 > < s 0 > = 2m W /g f π (45)
which is also the ratio of the two scales involved in this 1-generation standard model, the weak scale ≃ m W and the chiral scale ≃ f π . The masses are small and justify a posteriori our low energy treatment of the scalar effective potential.

The composition of the two Higgs doublets is accordingly as follows. Inside the "chiral" doublet K one finds while m ξ will grow like the mass of the bound state. So, in the case of more generations, the presence of very light Higgs boson(s) with a mass lower than 100 M eV looks a robust feature as a damping effect of the weak scale m W but larger masses can be expected for some others. It would not be a surprise that, for 3 generations and up to some coefficient, the mass of one of the Higgs bosons be set by that of a bound state involving the top quark.

In the present case, the masses of the two Higgs bosons vanish at the limit m π → 0, that is, by the GMOR relation (20), either when < ūu + dd >→ 0 or when (m u + m d ) → 0. Since we have also determined (see (39)) that (m u -m d ) < ūudd > vanishes with m π , this limit corresponds either to < ūu >= 0 =< dd > or to m u = 0 = m d .

7 Couplings of the Higgs bosons ≡ δ 12 + κ 12 = 0 and δ 2 = 0 that we have obtained for the crossed couplings (see (39)) from low energy considerations would erroneously leave as the only couplings of quarks to Higgs bosons the ones present in -δ 1 K † K, in which, in particular, no coupling exists between the "quasi-standard" Higgs boson ξ, which belongs to H, and quarks. In order to properly determine these parameters, the original form (9) of the Yukawa couplings must instead be used.

Plugging therefore the definition ( 11) into (9) yields the following couplings of the Higgs bosons ς and ξ to quarks

-ς (ρ u ūu + ρ d dd) -ξ (λ u ūu + λ d dd) = -ς δ 1 v √ 2µ 3 (ūu + dd) + κ 12 σ √ 2ν 3 (ūu -dd) -ξ δ 12 v √ 2µ 3 (ūu + dd) + δ 2 σ √ 2ν 3 (ūu -dd) (46)
which exhibits, of course, the same structure as in (37) and which, using the values (39) of the parameters, δ 12 = -κ 12 and δ 2 = 0, yields

L Higgs-quarks = -ς 3m u -m d 2 √ 2f π ūu + 3m d -m u 2 √ 2f π dd -ξ g(m u + m d ) 4 √ 2m W (ūu + dd). (47)
The ς Higgs boson is more strongly coupled to quarks than ξ. Its coupling is still "perturbatively" since m u , m d ≪ f π . It however suggests that, for heavier quarks, some Higgs boson(s) could strongly couple to hadronic matter. As far as ξ is concerned, it looks at first sight "quasi-standard" because it is proportional to gm quark /m W . It is however not quite so because in the standard case we would have obtained

-g √ 2mW (m u ūu + m d dd) ξ.
The difference is that, though u and d have different masses, they now get coupled to ξ with equal strength: unlike in the genuine Glashow-Salam-Weinberg model, the heavier quark is no more strongly coupled than the lighter. Taking m d = γm u , γ > 1, the coupling -g(1+γ)

4 √ 2mW m u of ξ to u quarks can be very close to the standard one (it becomes identical for γ = 3, value at which < dd > vanishes, see footnote in subsection 4.3.2), while the one -g(1/γ+1) 4 √ 2mW m d of ξ to the heavier d is smaller than standard by the factor (1+γ) 4γ .

Couplings of Higgs bosons to gauge bosons

They arise from the kinetic terms [START_REF] Alfaro | Currents in Hadron Physics[END_REF]. Using ( 22) and ( 26)), one gets

L HiggsW W = gm W 2 W 2 µ ξ + g 2 f π 4 √ 2 W 2 µ ς. (48)
ξ couples accordingly in a "standard" way ≃ gm W to two W 's while the coupling of ς, O(g 2 f π ) is much smaller by a factor O(10 -3 ).

Couplings of Higgs bosons to leptons

Yukawa couplings to leptons need introducing four parameters, ρ e and ρ ν for s 0 and the quantum Higgs ς, λ e and λ ν for s 3 and the quantum Higgs ξ

L Y uk-lept = (ρ ν νν + ρ e ēe) s 0 -(λ ν νν + λ e ēe) s 3 + ρ ν + ρ e 2 νγ 5 e p -+ ēγ 5 ν p + + (νγ 5 ν -ēγ 5 e) p 3 + ρ ν -ρ e 2 ēν p + -νe p -+ (νγ 5 ν + ēγ 5 e) p 3 - λ ν + λ e 2 ēγ 5 ν s + -νγ 5 e s --(νγ 5 ν -ēγ 5 e) p 0 + λ ν -λ e 2 ēν s + + νe s --(νγ 5 ν + ēγ 5 e) p 0 . (49)
Using again ( 22) and (26 provides the lepton masses

m e = ρ e f π √ 2 + λ e √ 2m W g , m ν = ρ ν f π √ 2 + λ ν √ 2m W g . (50) 7.3

.1 The low energy limit

Let us use again the one-to-one correspondence between the components of the Higgs multiplets and bilinear quark operators [START_REF] Diaz | Phenomenological analysis of the two Higgs doublet model[END_REF]. Using PCAC (19) and the Gell-Mann-Oakes-Renner relation (20), we could relate the charged pion fields π ± and the charged pseudoscalar components p ± of the Higgs doublet K by (23). Yukawa couplings (49) are then seen to trigger, among others, leptonic decays of charged pions. These come in addition to the "standard ones" obtained from the W µ ∂ µ π crossed couplings that originate from the kinetic terms (5) at low energy (see subsection 4.3.4) and which agree with PCAC usual calculations.

This means that, in a first approximation (and it is not the goal of this work to go beyond), we should take

ρ ν ≈ 0 ≈ ρ e . (51)
In case observed leptonic pion decay turn out to differ from PCAC estimates, the issue could be raised whether (51) should be revisited.

In relation with (50) the choice (51) leads to a standard coupling of the Higgs boson ξ to leptons, proportional to gm lepton /m W , while the ones of ς vanish (or are extremely close to).

8 Symmetries again down to U (1) em . Let us show that < ūu > cannot be equal to < dd >. Indeed, for ν 3 = 0 one gets from ( 10)

δ 2 = 0 = κ 12 .
Then

m 2 12 = δ 12 = gf π m 2 π 2m W , (52)
in which we used the definition of δ 12 in [START_REF] Machet | Unlocking the Standard Model. IV. A natural candidate for dark matter[END_REF], the GMOR relation ( 20) and (28).

Performing the minimization of the effective potential V ef f (K, H) while still supposing that V (K, H) is

U (2) L × U (2) R invariant gives the two equations m 2 H = λ H < s 0 > 2 +2δ 1 + δ 12 < s 3 > < s 0 > , m 2 H = λ H < s 3 > 2 +δ 12 < s 0 > < s 3 > , (53)
which yield, since < s 3 >≫< s 0 > (see ( 22) and ( 26))

λ H ≈ 2δ 1 < s 3 > 2 + δ 12 < s 0 >< s 3 > = 3 2
g 2 m 2 π m 2 W . ( 54
)
The mass matrix of the s 0s 3 system becomes then (we use (53))

  ∂ 2 V ef f (∂s 0 ) 2 ≡ 2λ H < s 0 > 2 -δ 12 <s 3 > <s 0 > 1 2 ∂ 2 V ef f ∂s 0 ∂s 3 = 0 1 2 ∂ 2 V ef f ∂s 0 ∂s 3 = 0 ∂ 2 V ef f (∂s 3 ) 2 ≡ 2λ H < s 3 > 2 -δ 12 <s 0 > <s 3 >   ≈   -m 2 π 0 0 6m 2 π   . (55)
It exhibits, because of the term -δ 12

<s 3 > <s 0 > in ∂ 2 V ef f
(∂s 0 ) 2 , which comes from the low energy expression of Yukawa couplings, a tachyonic s Higgs boson m 2 ς ≈ -m 2 π . The theory with < ūu >=< dd > is thus unstable. Since we have everywhere supposed that the minimum of the effective potential fits the W and π masses, we conclude that chiral and weak symmetry breakings as they are observed are only possible for < ūu > =< dd >.

Unlike for the pions the masses of which are related to < ūu + dd > by the GMOR relation (20), there is no such relation between m W and < ūudd > (see the last line of (39)). Moreover, even when < ūu >=< dd > (that is, < ν 3 >= 0) < s 3 > can be equal to σ/ √ 2 because, in its expression [START_REF] Diaz | Phenomenological analysis of the two Higgs doublet model[END_REF], ν 3 cancels between the numerator and the denominator. This is why it looks opportune to rather speak of < ūu > =< dd > as the catalyst of weak (and custodial) symmetry breaking.

The custodial SU (2)

While (ūu + dd) gets annihilated by all generators of SU (2), (ūudd) does not, it only gets annihilated by [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF]). So, < ūu > =< dd > spontaneously breaks the custodial SU (2) down to U (1) em . In this breaking one expects two Goldstones. They are the excitations by T + and T -of the s 3 vacuum , that is the two scalars s + and s -eaten by W ± , and which coincide with the two charged Goldstones of the spontaneously broken weak SU (2) L .

T 3 = Q (see
The electroweak Lagrangian is invariant by the custodial SU (2) as soon as the W 's form an SU (2) vector. But, in the broken phase, the W 3 can only eat s 0 which is a SU (2) singlet. This is how the generation of the W mass breaks the custodial symmetry.

Goldstones and pseudo-Goldstones

Three true Goldstones are eaten by the W 's to get massive: they are p 0 , s + and s -, belonging to the doublet H.

p 0 is also the Goldstone of the U (1) L × U (1) R spontaneous breaking down to the diagonal U (1). The three p (the three pions) are the pseudo-Goldstones of the broken SU (2) L × SU (2) R down to SU (2).

The only non-Goldstones are the two Higgs bosons ξ and ς in the sense that, though their masses also vanish with m π , they do not seem connected with the breaking of any continuous symmetry. The first could only be excited by acting either on p 0 with T 3 L or T 3 R , or on p 3 with I L or I R . However, in a first approximation, neither p 0 nor p 3 , being a pseudoscalar, has a non-vanishing VEV. Likewise, H could only be excited either by acting on p 3 with T 3 L or T 3 R , or on p 0 with I L or I R . The same argumentation rejects thus both as Goldstone bosons, unless some additional spontaneously broken continuous symmetry is at work, which is to be uncovered.

A few hints for more generations

Before concluding, it is worth pointing at a few features concerning the case of a larger number N of generations (some information can also be found in [START_REF] Machet | Chiral scalar fields, custodial symmetry in electroweak SU (2) L × U (1) and the quantization of the electric charge[END_REF]). A more detailed study is postponed to [START_REF] Machet | Unlocking the Standard Model. III. The case of N > 1 generations[END_REF].

There are features of this work which only belong to the case of one generation. For example the fact that the η pseudoscalar meson (pseudoscalar singlet) becomes the longitudinal neutral W 3 . In the case of more generations, it may happen that this role is still held by the singlet ∝ ūγ 5 u + dγ 5 d + cγ 5 c + dγ 5 s + . . ., but it is no longer the η, or by another neutral combination. Though this can only be known by a precise study, it is likely that the η can then live again its life as a physical pseudoscalar meson.

Other features are certainly, at the opposite, robust, like the fact that there is a very light Higgs boson with mass ≤ f π ≈ 93 M eV . Likewise, from the expression (43) for the quartic Higgs coupling λ H , it seems reasonable to believe that, even if the mass of the pion gets replaced by the mass of a much heavier bound state, λ H will stay smaller than 1 and thus "perturbative". It can only get equal to 1 if m π is replaced by √ 2m W /g ≈ 168 GeV , such that one should only be careful when the "top" generation is concerned, for which "non-perturbative" phenomena could appear.

The logic of the present work and of [START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF] is that all (pseudo)scalar doublets isomorphic to the one of the Standard Model of Glashow, Salam and Weinberg [START_REF] Glashow | Partial-symmetries of weak interactions[END_REF] should be incorporated. This would stay an empty or meaningless statement without noticing that the standard Higgs doublet has transformations by the chiral group (3) (4) that are identical to those of bilinear quark operators. For one generation, this doubled the number of possible doublets, with parity distinguishing the two of them. In the case of N generations, it was shown in [START_REF] Machet | Chiral scalar fields, custodial symmetry in electroweak SU (2) L × U (1) and the quantization of the electric charge[END_REF] that there exists 2N 2 such doublets, divided, by parity again, in two sets. Their 8N 2 real components can be put in one-to-one relationship with the same number of scalar and pseudoscalar J = 0 mesons that occur for 2N flavors of quarks. The same logic as the one followed here requires accordingly that the Standard Model be then endowed with 2N 2 complex Higgs doublets. Among these, one expects in particular as many Higgs fields as there exist quark-antiquark < qi q i > condensates, that is, 2N . Owing to the large number of parameters involved, it looks of course too optimistic to think that one can easily calculate all masses and couplings as we did here. This path stays nevertheless in our opinion the most natural to follow, the underlying guess being that the mystery of Higgs boson(s) simply lies inside the one of scalar (and eventually pseudoscalar) J = 0 mesons.

Conclusion and prospects

As we re-built it, the Standard Model for one generation of fermions is complete in the sense that all masses and couplings of all fields present in the Lagrangian and of all J = 0 pseudoscalar mesons are determined.

Pions are accounted for with the correct decays and, of the four expected scalar mesons, two (the charged ones) become the longitudinal charged W ± while the last two are the Higgs bosons ς and ξ. Both have small masses and are perturbatively coupled and self-coupled. While ξ is expected to be close to standard, ς is extremely light and has peculiar properties that deserve a specific investigation concerning the role that it can hold in nature At this stage, no physics "beyond the Standard Model" looks needed 3 but, since the one generation case can only be considered as a "toy Standard Model", this is one among the features that should be carefully scrutinized for more generations of fermions [START_REF] Machet | Unlocking the Standard Model. III. The case of N > 1 generations[END_REF].

4. 3 . 1

 31 The charged Goldstones of the broken SU (2) L Since < s 3 > provides most of the mass of the W 's, the charged Goldstone bosons of the broken SU (2) L weak symmetry are, to a very good approximation, the excitations of s 3 by the generators T + L and T - L , that is s + and s -∈ H.

√ 2 = mu+m d 4 .√ 2 =

 242 3 >∈ H contribute to the u and d masses by the same amount σλu √ 2 = σλ d Then, < s 0 >∈ K contributes to the u mass by vρu √ 2 = 3mu-m d 4 and to the d mass by vρ d 3m d -mu 4 .

3

  pions and the very light scalar Higgs boson ς. As was shown in[START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF], they correspond respectively to a triplet and a singlet of the custodial SU (2) symmetry. Inside the "weak" doublet H, one finds the three Goldstones of the broken SU (2) L weak symmetry, the neutral pseudoscalar SU (2) singlet and two charged scalars inside the SU (2) triplet. The third component of this triplet is the second scalar Higgs boson ξ with mass ≈ m π . Note that the four particles ( π, ξ) with mass m π do not lie together inside the same SU (2) L doublet, nor do the three SU (2) L Goldstones and the very light Higgs boson ς.6.3.1 The roles of m W and m πIn our rebuilding of Standard Model with only one generation, we find that the masses of the two Higgs bosons are both proportional to m π and small. But they are not small in the same way. If m π is replaced by the mass of some heavier bound state m ≤ √ 2m W /g ≡< s 3 > ≈ 168 GeV , m ς will stay very small m ς ≤ f π ≈ 93 M eV

8. 1

 1 The roles of < ūu + dd > and < ūudd > < ūu + dd > = 0 is the signal for what is commonly called "chiral symmetry breaking", the breaking of SU (2) L × SU (2) R down to the diagonal SU (2). < ūudd > = 0 breaks SU (2) L , and the custodial SU[START_REF] Glashow | Partial-symmetries of weak interactions[END_REF]

[ 10 ]2N 2

 102 . As far as we can see, this minimal extension of the Standard Model is different from what other authors have been considering; it is different as a 2-Higgs doublet model[START_REF] Branco | Theory and phenomenology of two-Higgs-doublet models[END_REF] [12][START_REF] Diaz | Phenomenological analysis of the two Higgs doublet model[END_REF], and it is different in that, for a larger number of generations N > 1, it cannot stay as a 2-Higgs doublet model and should be endowed with Higgs doublets. A key ingredient to account simultaneously for the different scales in presence, weak and chiral, is parity doubling. It could only be uncovered through the one-to-one correspondence demonstrated in[START_REF] Machet | Unlocking the Standard Model. I. 1 generation of quarks. Symmetries[END_REF] between the Higgs fields and bilinear quark operators and detailed symmetry considerations. The breaking of parity has reflected here in the mass splitting of the two Higgs bosons, their ratio being precisely that of the two scales in presence.

< dd > vanishes for m d =

3mu. We shall see in subsection 7.1 that this is also the condition for the u quark to couple to the "standard" Higgs boson ξ like in the Glashow-Salam-Weinberg model.

The only hint in favor of it may be the vanishing of the masses of the two Higgs bosons at the chiral limit, which makes them appear "like pseudo-Goldstone bosons" (see subsection 8.3).

Acknowledgments: it is a great pleasure to thank O. Babelon, M. Capdequi-Peyranère, S. Davidson, M. Knecht, J. Lavalle, G. Moultaka, P. Slavich and M.I. Vysotsky for conversations, advice, and helping me to correct mistakes.


Bogdan Ludusan
email: bogdan.ludusan@irisa.fr

Stefan Ziegler
email: stefan.ziegler@irisa.fr

Guillaume Gravier
email: guillaume.gravier@irisa.fr

Integrating Stress Information in Large Vocabulary Continuous Speech Recognition

Keywords: speech recognition, stress, rhythm

In this paper we propose a novel method for integrating stress information in the decoding step of a speech recognizer. A multiscale rhythm model was used to determine the stress scores for each syllable, which are further used to reinforce paths during search. Two strategies for integrating the stress were employed: the first one reinforces paths through all the syllables with a value proportional to the their stress score, while the second one enhances paths passing only through stressed syllables, but with a constant value. The former strategy slightly outperforms the later, bringing a relative improvement of more than 2% over the baseline. Furthermore, the stress information proved to be a robust feature, by performing well even for foreign-accented speech.

Introduction

A popular approach explored nowadays to improve the performance of speech recognizers is the integration of prosodic information in the recognition process. Among the prosodic components and features investigated, stress seems suitable for speech recognition tasks. This is due to its intrinsic characteristics, stressed syllables exhibiting physical salience, phonemic stability and perceptual distinctiveness [START_REF] Mattys | The use of time during lexical processing and segmentation: A review[END_REF]. In other terms, their acoustic attributes are more distinguishable, they are less likely to suffer phonological modification and they are recognized better both in clear and noisy conditions. Besides the evidence coming from psycholinguistics studies, support for the use of stress information for recognition can be found also in speech processing works (e.g. [START_REF] Jenkin | Development and comparison of three syllable stress classifiers[END_REF]).

Despite the general agreement that stressed syllables are more intelligible than their unstressed counterparts, there are only a few recognition systems exploiting this kind of information (e.g. [START_REF] Wang | Lexical stress modeling for improved speech recognition of spontaneous telephone speech in the JUPITER domain[END_REF][START_REF] Van Den Heuvel | Modelling lexical stress in continuous speech recognition[END_REF][START_REF] Van Dalen | Lexical stress in continuous speech recognition[END_REF]). The systems proposed in [START_REF] Wang | Lexical stress modeling for improved speech recognition of spontaneous telephone speech in the JUPITER domain[END_REF] and [START_REF] Van Den Heuvel | Modelling lexical stress in continuous speech recognition[END_REF] used stress models only for the vowel of the stressed syllable, while the one presented in [START_REF] Van Dalen | Lexical stress in continuous speech recognition[END_REF] took into consideration also the consonants of the syllables. The results of these studies were mixed, some of them showing improvements when using stress information [START_REF] Wang | Lexical stress modeling for improved speech recognition of spontaneous telephone speech in the JUPITER domain[END_REF][START_REF] Van Dalen | Lexical stress in continuous speech recognition[END_REF], while others reporting no performance gain [START_REF] Van Den Heuvel | Modelling lexical stress in continuous speech recognition[END_REF].

We propose here a novel approach of combining stress knowledge in the search process by reinforcing paths passing through the stressed syllables. The proposed approach uses a quantitative representation of syllable stress obtained by means of a model of rhythm perception [START_REF] Todd | The auditory primal sketch: a multi-scale model of rhythm grouping[END_REF]. The model is based on a multiscale filtering mechanism and its corresponding representation, the rhythmogram, indicates the rhythmic grouping of auditory events. The rhythmogram was already applied to speech for the analysis of prosodic structure [START_REF] Todd | Visualization of rhythm, time and metre[END_REF] and of cross-linguistic differences in speech rhythm [START_REF] Lee | Towards an auditory account of speech rhythm: application of a model of the auditory primal sketch to two multi-language corpora[END_REF].

This paper is organized as follows: section 2 presents the stress detection procedure and the model on which it is based, the auditory primal sketch. The recognition system employed is described in section 3, while the corpus used in the experiments is presented in section 4. The results obtained are presented next, followed by a discussion on the behaviour and the performances of the proposed system. The paper concludes with some remarks and ideas for future work.

Stress Detection

The stress detection procedure is based on a multiscale model of rhythm perception, called the Auditory Primal Sketch [START_REF] Todd | The auditory primal sketch: a multi-scale model of rhythm grouping[END_REF]. The model aims at identifying the strong acoustic events in the speech signal.

The original model takes into account the effect of the peripheral auditory system on the speech signal by implementing all the steps necessary to model the human ear: the outer and middle ear's transfer functions, the frequency response of the basilar membrane and the response of the inner hair-cells. Next, the response of the auditory nerve is summed across all channels and lowpass filtered with a bank of Gaussian filters having different frequency widths. The peaks of the filters output are combined in a hierarchical representation, with the more prominent events being represented by peaks over more filter widths. This representation is called the rhythmogram.

The auditory nerve response proved to be appropriate for word-level analysis, but the authors argued that for larger time scale events, like phrases, it was more suitable to use the signal energy rather than the nerve response [START_REF] Todd | Visualization of rhythm, time and metre[END_REF]. Thus, we decided to use the energy of the speech signal for computing the rhythmogram in this study. Also, in order to be able to use the information coming from the rhythm model in any automatic process, it has to be quantized. First used in [START_REF] Lee | Towards an auditory account of speech rhythm: application of a model of the auditory primal sketch to two multi-language corpora[END_REF], this more compact representation of the rhythmogram consists of a sequence of events and their prominence value (P-value). The P-values were obtained by summing all peak values of a given event across all filter widths. Figure 1 illustrates the waveform of a sentence from the corpus used in the experiments (upper panel) along with the obtained rhythmogram (lower panel). The value above each event symbolizes its corresponding P-value.

After computing the rhythmogram of each phrase, the stress value for each syllable was determined in a similar manner to the one used for computing the stress accent score in [START_REF] Ludusan | On the use of the rhythmogram for automatic syllabic prominence detection[END_REF]:

• search inside the boundaries of each syllable and determine its corresponding rhythmogram events

• in case of multiple events corresponding to the same syllable, take only the event with the highest P-value

• the stress score is equal to the P-value of the event

• if no event was found within the boundaries of a syllable, set its stress score to 0

Being a preliminary study, the syllable boundaries were obtained by combining phone-level forced aligned data with the French syllabification rules proposed by Dell [START_REF] Dell | Consonant clusters and phonological syllables in French[END_REF].

Recognition System

The recognizer used in the experiments, Irene, is a twopass recognition system developed at IRISA. The first pass generates a word graph, while the second pass performs the rescoring of this graph using more complex models.

The first pass is based on the algorithm proposed by Ortmanns and Ney [START_REF] Ortmanns | A word graph algorithm for large vocabulary continuous speech recognition[END_REF] and it implements a beam-search strategy for decoding. The acoustic models employed are word-internal triphones with 4,019 distinct states and 32 Gaussians per state and word trigrams are used as language model. The rescoring pass uses 4-grams as language model and cross-word triphone models with 6,000 states and 32 Gaussians each. No speaker adaptation or morpho-syntactic information were used in the second pass.

The search strategy used by the recognizer aims at maximizing the following equation:

Q(j, t) = max i Q(i, t -1) + log(aij) + log(bj(yt)) (1)
where Q(j, t) is the score of the path in state j at time t, log(a ij ) the transition probability between states i and j and log(b j (y t )) the observation probability of y t when in state j.

In order to accommodate for the stress information, the previous equation was changed by adding a reinforcement factor:

Q(j, t) = max i Q(i, t -1) + log(aij) + log(bj(yt)) + str(t) • R (2)
where str(t) represents the stress score of the syllable at time t and R is the maximum reinforcement factor. The stress score varies between 0, meaning least stressed, and 1, representing most stressed. The parameter R is used for weighting the stress score and its value was obtained by optimizing the recognition performance on a development set.

The reasoning for making this implementation choice seems logical considering the results of studies showing that stressed syllables are better recognized than nonstressed syllables [START_REF] Mattys | The use of time during lexical processing and segmentation: A review[END_REF][START_REF] Jenkin | Development and comparison of three syllable stress classifiers[END_REF]. Thus, by reinforcing the paths passing through the stressed syllables, we effectively give those syllables a higher weight in the search process.

Materials

The data employed for the experiments was taken from the French ESTER2 evaluation campaign corpus [START_REF] Galliano | The ESTER 2 evaluation campaign for the rich transcription of French broadcasts[END_REF]. The corpus consists of mainly radio broadcast news, with a small part of it containing more spontaneous material, like debates. While all its recordings are in French, the corpus contains also news from African radio stations, which exhibit strong accents. Approximately 3.5 hours from the ESTER2 development set were used for tuning the parameters, while 4.5 hours from the ESTER2 test set were used for the experiments.

Results and Discussion

Two factors were considered when choosing the experimental settings. The first one concerned how the stress information should be integrated in the decoding process: either by adding to the path score a value proportional to the stress score of each syllable, or by reinforcing the path score with a constant value, but only for the stressed syllables. Here, a syllable was considered to be stressed if its stress score was greater than 0.5. The second factor addressed the issue of where, inside the recognition chain, the stress information should be added. A straightforward choice would be to add it to the first pass, because a better word graph before the rescoring pass would translate into increased performance. In case such an improvement was found after a stress-enhanced first pass, we were interested in knowing whether the stress would bring new information in the second pass, even if more complex acoustic models were being used.

A total of four tests were performed, by using either a stress-proportional or a constant value for reinforcing paths, and by integrating the stress information either only in the first pass or in both passes. Table 1 presents the Word Error Rate (WER) obtained on the development set with the baseline, the Irene recognition system, and the recognizer integrating stress information in the two previously explained configurations. The proportional reinforcing strategy improved recognition both when applied only in the first pass, as well as in both passes, while the strategy using a constant reinforcement posted a low gain. The results obtained on the test set are presented in Table 2. When applying the proportional stressenhancing scheme in the first step the performance was improved by 1.3% relative value, while its use in both passes brought a relative WER reduction of 2.1%. A Wilcoxon signed rank test was applied to the results obtained and it found that the differences are significant at the 1% and 0.1% level, respectively. Although the constant reinforcing strategy had a poor performance on the development set, it actually brought an improvement on the test set. The same statistical test showed a significant difference at the 0.1% level when integrating stress in both passes, with respect to the baseline. The better strategy among the two, the proportional one, was further used in experiments to provide a more detailed discussion of the behaviour and of the performances obtained by integrating stress information in the decoding step. First, we took a look at the WER on the development set for different values of R. Figure 2 shows the WER of the baseline (solid line), the results by varying R in the first pass (dashed line) and by varying it in the second pass (dotted line). One can see that the performance does not worsen even for high values of R, especially in the first pass, meaning that stress offers a robust source of information. A measure which can quantify the gain obtained by integrating stress information in the first pass is the Word Graph Error Rate (WGER). It is computed by determining the best sentence through the graph that matches the reference transcription and it offers a lower bound of the WER. As seen in Table 3, an absolute improvement of 0.6% in WGER was obtained by using stress information in the first step. Because this improvement is higher than the 0.3% decrease in WER after the same step, a new approach in the second pass was needed in order to fully take advantage of the gain in WGER. It can be seen from the results presented in Table 2 that the integration of stress information also in the rescoring pass does indeed achieve this goal.

Reinforcement

Reinforcement

Baseline Proposed WGER 12.7 12.1

Table 3: WGER (in %) after the first pass.

As mentioned in Section 4, the corpus contains also recordings exhibiting strong accents. Because the acoustic models were mostly trained on standard French broadcast news, we decided to examine how the stressenhanced recognizer behaves on this type of speech. It can be observed in Table 4 that the baseline performs worse on the strong-accented data, with a relative WER more than 50% higher than for non-accented speech. The stress-based decoding performs well even in this difficult conditions, bringing a higher relative improvement when used only in the first pass with respect to the non-accented speech (1.9% vs. 1.5%) and the same level of improvement when used in both passes (2.5%).

System

Conclusions

We investigated in this paper the usefulness of incorporating stress information in a large vocabulary speech recognition task. Differently from previous speech recognition studies using stress information, no prior stress knowledge is stored in the lexicon, nor was it used during the training stage. The stress information was obtained by means of a model of rhythm perception, which aims at finding strong acoustic events inside the speech signal.

A small but statistically significant improvement was obtained (0.5% absolute value). The improvements were found to be due to differences in the behaviour of the systems during hypothesis propagation, the addition of stress information helping to prune away some of the unreliable hypotheses. Also, it appears that the stress score used in this paper is a robust feature, bringing improvements when both simpler and more complex acoustic models were used and both in the presence of normal and strongaccented speech. Its robustness was further proved by the large interval of values that the parameter R can take without decreasing the recognition performance. As a future step, we plan to see how the recognition process behaves when automatically segmented syllables are used, or when only the neighbourhood around a rhythmogram event, and not the entire syllable, is reinforced. While in this study the role of the stress was to enhance paths equally inside a syllable, it would be interesting to use different reinforcement values inside the same syllable by weighting them based on each frame's acoustic likelihood.

Figure 1 :

 1 Figure 1: Waveform and rhythmogram of the sentence "Tous les groupes engagés dans les discussions ont apporté des amendements au texte."

Figure 2 :

 2 Figure 2: WER (in %) on the development set, for different values of the R parameter.

Table 1 :

 1 WER (in %) obtained on the development set.

	strategy	Baseline	Stress 1st only 1st & 2nd
	proportional constant	35.6	35.3 35.4	35.1 35.4

Table 2 :

 2 WER (in %) obtained on the test set.

	strategy	Baseline	Stress 1st only 1st & 2nd
	proportional constant	23.5	23.2 23.4	23.0 23.1

Table 4 :

 4 WER (in %) on the test set based on the type of accent present.

		Stress	Accent Normal Strong	Overall
	Baseline	-	20.0	31.5	23.5
	Proposed	1st only 1st & 2nd	19.7 19.5	30.9 30.7	23.2 23.0

Acknowledgements

This work was partially supported by the Agence Nationale de la Recherche through the ASH project.