New Terms for the Compact Form of Electroweak Chiral Lagrangian
arXiv:0704.1075v2  [hep-ph]  3 May 2007
Preprint typeset in JHEP style - HYPER VERSION TUHEP-TH-07156
New Terms for the Compact Form of Electroweak
Chiral Lagrangian
Hong-Hao Zhang, Wen-Bin Yan, and J. K. Parry
Center for High Energy Physics & Department of Physics, Tsinghua University,
Beijing 100084, China
E-mail: zhanghonghao@tsinghua.org.cn, wenbin.yan@gmail.com,
jkparry@tsinghua.edu.cn
Xue-Song Li
Science College, Hunan Agricultural University, Changsha 410128, China
E-mail: lixuesong@tsinghua.org.cn
Abstract: The compact form of the electroweak chiral Lagrangian is a reformulation of
its original form and is expressed in terms of chiral rotated electroweak gauge fields, which
is crucial for relating the information of underlying theories to the coefficients of the low-
energy effective Lagrangian. However the compact form obtained in previous works is not
complete. In this letter we add several new chiral invariant terms to it and discuss the
contributions of these terms to the original electroweak chiral Lagrangian.
Keywords: Electroweak Chiral Lagrangian, Beyond Standard Model.
http://arxiv.org/abs/0704.1075v2
mailto:zhanghonghao@tsinghua.org.cn
mailto:wenbin.yan@gmail.com
mailto:jkparry@tsinghua.edu.cn
mailto:lixuesong@tsinghua.org.cn
http://jhep.sissa.it/stdsearch
So far the postulated Higgs particle of the standard model has not been observed in ex-
periments. We do not know what the electroweak symmetry breaking mechanism in nature
is. The electroweak chiral Lagrangian is a general low-energy description for electroweak
symmetry breaking patterns [1, 2], especially for those strong dynamical electroweak sym-
metry breaking mechanisms [3]. All the coefficients of the electroweak chiral Lagrangian,
in principle, can be fixed by experiments [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Due to the non-
perturbative property of the possible strong dynamics, it is difficult to relate the measured
coefficients of the electroweak chiral Lagrangian to underlying theories. Recently, the series
of work of Ref. [14, 15, 16, 17, 18, 19, 20, 21] successfully produced the predictions for the
coefficients of the chiral Lagrangian from the underlying theory of QCD, which lights up
the hope of building up the relationship between the coefficients of the electroweak chiral
Lagrangian and underlying strong dynamical models. As seen in Ref. [14], a crucial first
step of the derivation of the chiral Lagrangian from QCD is to reformulate the original
chiral Lagrangian in terms of chiral rotated external source fields. For the case of the elec-
troweak chiral Lagrangian, we need to similarly reformulate it in terms of chiral rotated
electroweak gauge fields in order to deduce the information of underlying theories. An
attempt of this line of thought is the work of Ref. [22], which tries to give a so-called com-
pact form of the electroweak chiral Lagrangian in terms of chiral rotated electroweak gauge
fields. However, there are still several relevant terms not included in the reformulation of
Ref. [22]. In this letter, we shall present 3 new terms for this reformulation and give their
contributions to the original electroweak chiral Lagrangian.
We refer the interested reader to Ref. [22] for all the details on the compact form of the
electroweak chiral Lagrangian and the relations between the compact form and the original
form. Besides the inner-product terms c2g
b′µV b
′ν , c8g
ν − dνV
a′µAν ,
and c9g
(dµV a
ν already included in the compact form of Ref. [22], there should be
3 new cross-product terms which are given by,
∆L = c′
2 + c′
a′b′(dµV
ν − dνV
ν + c′
a′b′(dµV a
where a′, b′ run from 1 to 2, and ǫ12 = −ǫ21 = 1. Here and henceforth V a
′µ and Aµ
are short for the chiral rotated gauge fields V
and A
in Ref. [22] respectively. There
are also 3 cross-product terms corresponding to the c15,17,25-terms in that paper, but they
vanish. Let us consider Eq. (1) term by term. From the definitions of Ref. [22], it is
straightforward to obtain the relation between the first term of Eq. (1) and the ordinary
terms of the electroweak chiral Lagrangian as follows,
2 = −4c′
[tr(XµXν)]
− [tr(XµX
µ)]2 − tr(XµXν)tr(τ
µ)tr(τ3Xν)
+tr(XµX
µ)[tr(τ3Xν)]
, (2)
where Xµ ≡ U
†(DµU). And the last two terms of Eq. (1) are respectively given by,
′b′(dµV
ν − dνV
b′µAν
– 1 –
tr(τ3Xν)
tr(τ3Xµ)tr(XµXν)− tr(τ
3Xν)tr(XµX
+ itr(W µνX
tr(τ3W µν)tr(τ
, (3)
with Wµν ≡ U
W aµνU , and
′b′(dµV a
(1− 4β1)c
tr(τ3Xµ)tr(τ3Xν)tr(XµXν)
−[tr(τ3Xµ)tr(τ
. (4)
From Eqs. (2), (3) and (4), we obtain the contributions of these 3 new terms to the original
electroweak chiral Lagrangian as follows,
∆α3 =
, ∆α4 = −4c
, ∆α5 = 4c
∆α6 = 4c
(1− 4β1)c
, ∆α7 = −4c
∆α9 = −
, ∆α10 = (1− 4β1)c
. (5)
The coefficients c′
2,8,9 and ci (i = 1, 2, . . . , 25) in this compact form of the electroweak chiral
Lagrangian are determined by the underlying ultraviolet theories. For example, if we take
the one-doublet technicolor model as the underlying theory, it can be shown that these 3
new coefficients c′
2,8,9 are all non-vanishing, and full details will be presented in forthcoming
publications [23].
In summary, we have provided 3 new terms to the compact form of the electroweak
chiral Lagrangian introduced in Ref. [22]. These additional terms were not considered in
the previous work. In this letter we have related these new terms to the original electroweak
chiral Lagrangian, which will be crucial in forthcoming studies of strong dynamical models.
Acknowledgments
We are indebted to Qing Wang for all our knowledge about the chiral Lagrangian and his
helps and supports for this work. This work is supported in part by the National Natural
Science Foundation of China.
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