We formulate the equivalence between the longitudinal weak-boson and the
Goldstone boson as a criterion for sensitively probing the electroweak symmetry
breaking mechanism and develop a precise power counting rule for chiral
Lagrangian formulated electroweak theories. With these we semi-quatitatively
analyze the sensitivities to various effective operators related to
electrowaeak symmetry breaking via weak-boson scatterings at the CERN Large
Hadron Collider (LHC).Comment: 6 pages, LaTex, 1 postscript figure included using psfig.te

He, Hong-Jian

Kuang, Yu-Ping

Yuan, C. -P.

English

arXiv

We formulate the equivalence between the longitudinal weak-boson and the Goldstone boson as a criterion for sensitively probing the electroweak symmetry breaking mechanism and develop a precise power counting rule for chiral Lagrangian formulated electroweak theories. With these we semi-quatitatively analyze the sensitivities to various effective operators related to electrowaeak symmetry breaking via weak-boson scatterings at the CERN Large Hadron Collider (LHC)

He, H J

Kuang Yu Ping

Yuan, C P

CERN Document Server

July, 1995 VPI-IHEP-95-12
TUIMP-TH-95/67
MSUHEP-50712
Longitudinal/Goldstone Boson Equivalence and Phenomenology
of Probing the Electroweak Symmetry Breaking 
Hong-Jian He
Department of Physics and Institute of High Energy Physics
Verginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0435, USA
Yu-Ping Kuang
Institute of Modern Physics, Tsinghua University, Beijing 100084, China
C.{P. Yuan
Department of Physics and Astronomy, Michigan State University
East Lansing, Michigan 48824, USA
Abstract
We formulate the Equivalence between the longitudinal weak-boson and the
Goldstone boson as a criterion for sensitively probing the electroweak symmetry
breaking mechanism and develop a precise power counting rule for chiral La-
grangian formulated electroweak theories. With these we semi-quantitatively
analyze the sensitivities to various eective operators related to electroweak
symmetry breaking via weak-boson scatterings at the CERN Large Hadron
Collider (LHC).
Recent LEP/SLC experiments can test the electroweak (EW) theory to the accu-
racy of one-loop corrections, and support the spontaneously broken SU(2)  U(1)
gauge theory as the correct theory of the EW interactions. However, light Higgs boson
has not been found, and the current experiments are insensitive to the spontaneous
symmetry breaking (SSB) sector of the theory, compatible with a wide range of the
Higgs boson mass 60GeV mH  1TeV. So the SSB mechanism in the EW theory is
still a mystery, and it is thus important to probe all possible SSB mechanisms: either
weakly or strongly interacting.
We know that only the longitudinal component V aL of the weak-boson V
a (W,Z0)
(arising from \eating" the would-be Goldstone boson (GB)) is sensitive to the SSB
Talk presented by Yu-Ping Kuang at First International Conference on Frontiers of Physics,
Aug.5-9,1995,Shantou,China.
1
sector, while the transverse component V aT is not. The physical V
a
L scattering ampli-
tude is quantitatively related to the corresponding GB amplitude by the electroweak
Equivalence Theorem (ET)13 which comes from the following ET identity23
T [V a1L ;    ; V
an
L ; ] = C  T [−i
a1;    ;−ian; ] +B ; (1)
C  Ca1mod   C
an
mod ;
B 
Pn
l=1( C
al+1
mod   C
an
modT [v
a1;    ; val;−ial+1;    ;−ian; ]
+ permutations of v0s and 0s ) ;
va  vV a ; v
  L − k
=Ma = O(Ma=E) ; (Ma = MW ;MZ) ;
(2)
where a’s are GB elds, and  denotes other possible physical in/out states.
The renormalization scheme-dependent constant modication factor Camod has been
generally studied in Ref.2-3, which can be exactly simplied as Camod = 1 in certain
convenient renormalization schemes34.
For strongly interacting SSB models, the V aL -amplitude on the L.H.S. of (1) is
experimentally measurable, while the GB-amplitude on the R.H. S. of (1), though
not directly measurable, carries the information about the SSB mechanism. Similar
to V aT , the B-term in (1) is not sensitive to the SSB mechanism. If, under certain
conditions, the B-term can be neglected, (1) reveals the equivalence between the V aL -
amplitude and the GB-amplitude. In this case the V aL -scattering experiments can be
used to sensitively and unambiguously probe the SSB mechanism. When B is not
negligible, measurements of the V aL , V
a
T and B amplitudes with higher precision will
be required for probing the SSB mechanism, and those expreriments at LHC will be
harder.
The conditions for neglecting the B-term in (1), i.e. the condition for the validity
of the ET, is actually subtle. We rst note that the spin-0 GB’s are invariant under
the proper Lorentz transformations, while, on the contrary, VL, VT and B are Lorentz
non-invariant. Therefore the ratio of the B-magnitude relative to the GB-amplitude
in (1) is Lorentz frame dependent. So neglecting B makes sense only if the Lorentz
frame belongs to a group of frames whithin which Lorentz transformation does not
signicantly enhance B. We call such frames safe frames. The condition for a Lorentz
frame to be safe is given in Ref.3, which is
Ej  kj MW ; ( j = 1; 2;    ; n ) ;
where Ej is the energy of the j-th external V aL -line. For a given process, Ej can
be easily obtained from the kinematics. So this condition is a convenient criterion
for judging whether the experimental center of mass frame is safe or not for a given
process, i.e. it can discriminate processes which are not sensitive for probing the SSB
mechanisma. With this consideration, the ET can be precisely formulated as3
T [V a1L ;    ; V
an
L ; ] = C  T [−i
a1;    ;−ian; ] +O(MW =Ej−suppressed); (3a)
aSee the example given in Ref.3.
2
Ej  kj MW ; ( j = 1; 2;    ; n ) ; (3b)
B  C  T [−ia1;    ;−ian; ] : (3c)
(3b) and (3c) are the conditions for neglecting the B-term in (1) (for the validity
of the ET), or the conditions for sensitively probing the SSB mechanism via V aL -
scattering experiments. Here we see the profound physical content of the ET, i.e. ET
is not merely a tool for simplifying calculations.
The next thing is to realize the quantitative meaning of the condition (3c). To
a given order N in a perturbative expansion, the amplitude T can be written as
T =
PN
‘=0 T‘ with T0 > T1;    ; TN . Let Tmin = fT0;    ; TNgmin. Then, to the
precision of Tmin, condition (3c) precisely implies3
B  O(
M2W
E2j
)T0[−ia1;    ;−ian; ] +O(
MW
Ej
)T0[V
ar1
Tj
;−iar2 ;    ;−iarn ; ]
 Tmin[−ia1;    ;−ian; ] :
(4)
In the chiral Lagrangian formulated EW theory (CLEWT), the O(E2) leading
amplitude T0 is model-independent. Thus, for probing the SSB mechanism, we should
take into account the next-to-leading O(E4) amplitude T1, i.e. Tmin = T1 . By
means of our power counting rule (6), we can estimate that for leading contributions,
T1 = O(
E4
f2
2 f
4−n
 ) and B = O(g
2f4−n )
b. Thus condition (4) requires
M2W
E2
 1
4
E2
2
,
or (0:7TeV=E)4  1 . So the probe is generally sensitive when E  1 TeV which is
possible at the LHC.
In the CLEWT, the Lagrangian can be written in the following form6
Leff = LG + LF + L(2) + L(2)0 +
14X
n=1
Ln =
X
n
‘n
f
rn
an
On(W ; B ; DU; U; f; f);
(5)
where LG,LF are the kinetic terms of the gauge elds and fermions. The explicit
formula for Leff is given in Ref.56, in which L(2);L(2)0;L111 are CP conserving,
and L1214 are CP violating. Here, the dimensionless coecients ‘n’s can be naturally
regarded as of O(1) 7. In Ref.5, we developed the following power counting rule in
the CLEWT for the S-matrix element T
T = cTfDT
 
f

!NO  E
f
!DE0 E

DEL MW
E
ev
H(lnE=) ;
NO =
X
n
an ; DE0 = 2 +
X
n
Vn(dn − 2) + (iF +
1
2
eF) ; DEL = 2L ;
(6)
where the dimensionless coecient cT contains possible powers of gauge couplings
( g; g0 ) and Yukawa couplings ( yf ) from the vertices in T , which can be easily
bIn the CLEWT, f = 246GeV and the eective cut-o  ’ 4f ’ 3:1TeV .
3
Table 1. Contributions of the model-dependent operators to the WW!WW amplitudes
Operators T1[4] T1[3;WT ] T1[2; 2WT ] T1[; 3WT ] T1[4WT ]
L(2)0 ‘0
E2
2 ‘0 g
fE
2 ‘0 g
2 f
2

2 ‘0 g
3 f
3

E2 /
L1;13 / ‘1;13 e2g
fE
2 ‘1;13 e
4 f
2

2 ‘1;13 e
2g fE2 ‘1;13 e
2g2
f2
2
L2 ‘2 e2
E2
2 ‘2 e
2g fE2 ‘2 e
2 E2
2 ‘2 e
2g fE2 ‘2 e
2g2
f2
2
L3 ‘3 g2
E2
2 ‘3 g
E
f
E2
2 ‘3 g
2 E2
2 ‘3 g
3 fE
2 ‘3 g
4 f
2

2
L4;5 ‘4;5
E2
f2
E2
2
‘4;5 g
E
f
E2
2
‘4;5 g
2 E2
2
‘4;5 g
3 fE
2
‘4;5 g
4 f
2

2
L6;7;10 / / / / /
L8;14 / ‘8;14 g3
fE
2
‘8;14 g
2 E2
2
‘8;14 g
3 fE
2
‘8;14 g
4 f
2

2
L9 ‘9 g2
E2
2
‘9 g
E
f
E2
2
‘9 g
2 E2
2
‘9 g
3 fE
2
‘9 g
4 f
2

2
L11;12 / ‘11;12 g
E
f
E2
2 ‘11;12 g
2 E2
2 ‘11;12 g
3 fE
2 ‘11;12 g
4 f
2

2
determined from the vertices. H is a function of ln(E=) insensitive to E, where
 denotes the relevant renormalization scale. dn is the number of derivatives in the
type-n vertex, Vn is the number of type-n vertices in T , iF and eF are numbers of
internal and external fermion lines, respectively.
With this counting rule, we can estimate the sensitivities to probing specic oper-
ators in (5) via various W -W scattering amplitudes. In Table-1, we list the results in
the important WW channel as a typical example. We rst see that L6;7;10 do not
contribute to this channel. Table-1 then shows that the 4WL channel can probe L4;5
most sensitively, while the contributions of L(2)0; L2;3;9 to this channel lose E-power
dependence by a factor-2. This channel cannot probe L1;8;1114. L1;8;1114 can only
be probed via channels with WT (’s), among which L11;12 are most dominant though
they are still suppressed by a factor gf=E relative to the leading contributions to
the 4WL channel. L1;8;13;14 are generally suppressed by higher powers of the factor
gf=E and are thus less sensitive. For a more complete classication, see Table-3 in
Ref.5.
We have further calculated the number of events per [100fb−1 GeV] at the LHC
from our counting rule (6) combined with the eective-W approximation. We have
compared them with the corresponding available explicit calculations in Ref.8 for
a few typical examples. The comparison shows that the deviations are reasonably
within a factor of 2  3 which is of the same order as the uncertainty of the eective-W
approximation. Therefore our power counting rule does give correct semi-quantitative
results and is thus very useful and convenient for making a systematical analysis
for the sensitivities to probing the SSB mechanism at the LHC and future linear
colliders. In the typical case with ‘n  O(1), the number of the LHC events for the
W+W+ channel are shown in Fig.1. By comparing with the events from B, we see
4
that the probe of L4;5 are most sensitive, that of L3;9;11;12 are marginal, and that of
L(2)0;L1;2;8;13;14 are insensitive. More of the details are given in Ref.5.
5
Figure 1: Sensitivities of operators L(2)0;L114 whith ‘n  O(1), at the 14 TeV LHC
.
6
Acknowledgements H.J.H. is supported in part by the U.S. DOE under grant
DEFG0592ER40709; Y.P.K. is supported by the National NSF of China and the
FRF of Tsinghua University; C.P.Y. is supported in part by the NSF under grant
PHY-9309902.
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7


Longitudinal/Goldstone boson equivalence and phenomenology of probing the electroweak symmetry breaking

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