The effect of Topcolor Assisted Technicolor, and other models, on
  Neutrino Oscillation
arXiv:0704.0369v1  [hep-ph]  3 Apr 2007
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OCHA-PP-270, YITP-07-09, VPI-IPNAS-07-02
The effect of Topcolor Assisted Technicolor, and other models,
on Neutrino Oscillation
Minako Honda1, Yee Kao2, Naotoshi Okamura3, Alexey Pronin2, and Tatsu Takeuchi2∗
1Physics Department, Ochanomizu Women’s University, Tokyo 112-8610, Japan
2Physics Department, Virginia Tech, Blacksburg VA 24061, USA
3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
New physics beyond the Standard Model can lead to extra matter effects on
neutrino oscillation if the new interactions distinguish among the three flavors
of neutrino. In Ref. 1, we argued that a long-baseline neutrino oscillation ex-
periment in which the Fermilab-NUMI beam in its high-energy mode2 is aimed
at the planned Hyper-Kamiokande detector3 would be capable of constraining
the size of those extra matter effects, provided the vacuum value of sin2 2θ23
is not too close to one. In this talk, we discuss how such a constraint would
translate into limits on the coupling constants and masses of new particles in
models such as topcolor assisted technicolor.4
1. Introduction
When considering matter effects on neutrino oscillation, it is customary to
consider only the W -exchange interaction of the νe with the electrons in
matter. However, if new interactions beyond the Standard Model (SM) that
distinguish among the three generations of neutrinos exist, they can lead
to extra matter effects via radiative corrections to the Zνν vertex which
effectively violate neutral current universality, or via the direct exchange of
new particles between the neutrinos and matter particles.
For instance, topcolor assisted technicolor4 treats the third generation
differently from the first two and the Z ′ in this class of models couples
more strongly to the ντ than to the νe or νµ. In Extended Technicolor
(ETC) Models, such as that of Appelquist, Piai, and Shrock,5 the neutral
technimesons, which mix with the Z, couple to different generation fermions
differently, distinguishing among νe, νµ, and ντ . The diagonal ETC gauge
bosons also couple to the different generations differently, as well as the
∗Presenting Author
http://arxiv.org/abs/0704.0369v1
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large variety of leptoquark states in the model. Flavor distinguishing matter
effects from diagonal ETC and leptoquarks are induced by ETC gauge
boson mixing.
The effective Hamiltonian that governs neutrino oscillations in the pres-
ence of neutral-current lepton universality violation, or new physics that
couples to the different generations differently, is given by1
H = Ũ
λ1 0 0
0 λ2 0
0 0 λ3
 Ũ † = U
0 0 0
0 δm221 0
0 0 δm231
U † +
a 0 0
0 0 0
0 0 0
be 0 0
0 bµ 0
0 0 bτ
where U is the MNS matrix,
a = 2EVCC , VCC =
2GFNe = Ne
, (2)
is the usual matter effect due to W -exchange between νe and the electrons,
and be, bµ, bτ are the extra matter effects which we assume to be non-equal.
We define the parameter ξ as
bτ − bµ
= ξ . (3)
Then, the effective Hamiltonian can be rewritten as
H = Ũ
λ1 0 0
0 λ2 0
0 0 λ3
 Ũ † = U
0 0 0
0 δm221 0
0 0 δm231
U †+a
1 0 0
0 −ξ/2 0
0 0 +ξ/2
 , (4)
where we have absorbed the extra b-terms in the (1, 1) element into a.
The extra ξ-dependent contribution in Eq. (4) can manifest itself when
a > |δm231| (i.e. E & 10GeV for typical matter densities in the Earth) in
the νµ and ν̄µ survival probabilities as
P (νµ → νµ) ≈ 1− sin2
2θ23 −
δm231
P (ν̄µ → ν̄µ) ≈ 1− sin2
2θ23 +
δm231
, (5)
where
∆ ≈ ∆31c213 −∆21c212 , ∆ij =
δm2ij
L , cij = cos θij , (6)
and the CP violating phase δ has been set to zero. As is evident from
these expressions, the small shift due to ξ will be invisible if the value of
sin2 2θ23 is too close to one. However, if the value of sin
2 2θ23 is as low
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as sin2 2θ23 = 0.92 (the current 90% lower bound), and if ξ is as large
as ξ = 0.025 (the central value of from CHARM/CHARM II6), then the
shift in the survival probability at the first oscillation dip can be as large as
∼ 40%. If the Fermilab-NUMI beam in its high-energy mode2 were aimed at
a declination angle of 46◦ toward the planned Hyper-Kamiokande detector3
in Kamioka, Japan (baseline 9120 km), such a shift would be visible after
just one year of data taking, assuming a Mega-ton fiducial volume and
100% efficiency. The absence of any shift after 5 years of data taking would
constrain ξ to1
|ξ| ≤ ξ0 ≡ 0.005 , (7)
at the 99% confidence level.
In the following, we look at how this potential limit on ξ would translate
into constraints on the Z ′ in topcolor assisted technicolor, and various types
of leptoquarks. A more comprehensive analysis will be presented in Ref. 7.
2. Topcolor Assisted Technicolor
Though there are several different versions of topcolor assisted technicolor,4
we consider here the simplest in which the quarks and leptons transform
under the gauge group
SU(3)s × SU(3)w × U(1)s × U(1)w × SU(2)L (8)
with coupling constants g3s, g3w, g1s, g1w, and g, respectively. It is assumed
that g3s ≫ g3w and g1s ≫ g1w. SU(2)L is the usual weak-isospin gauge
group of the SM. The first and second generation fermions are assumed to
be charged only under SU(3)w×SU(2)L×U(1)w, while the third generation
fermions are assumed to be charged only under SU(3)s × SU(2)L ×U(1)s.
The U(1) charges for both cases are set equal to the SM hypercharge. At
scale Λ ∼ 1 TeV, technicolor, which is included in the model to generate the
W and Z masses, is assumed to become strong and generate a condensate
(of something which is left unspecified) which breaks the two SU(3)’s and
the two U(1)’s to their diagonal subgroups:
SU(3)s × SU(3)w → SU(3)c , U(1)s × U(1)w → U(1)Y , (9)
which we identify with the usual SM color and hypercharge groups. The
massless unbroken U(1) gauge boson Bµ and the massive broken U(1) gauge
boson Z ′µ are related to the original U(1)s × U(1)w gauge fields Ysµ and
Ywµ by
Z ′µ = Ysµ cos θ1 − Ywµ sin θ1
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Bµ = Ysµ sin θ1 + Ywµ cos θ1 (10)
where
tan θ1 =
. (11)
The currents to which the Bµ and Z
µ couple to are:
1sYsµ+g1wJ
1wYwµ = g
′ (cot θ1J
1s − tan θ1J
1s + J
1w)Bµ ,
where
. (13)
The current J
1s + J
1w is the SM hypercharge current, and g
′ is the
SM hypercharge coupling constant.
The exchange of the Z ′ leads to the current-current interaction
(cot θ1J1s − tan θ1J1w) (cot θ1J1s − tan θ1J1w) , (14)
the J1sJ1s part of which does not contribute to neutrino oscillations on the
Earth, while the J1wJ1w part is suppressed relative to the J1wJ1s part by
a factor of tan2 θ1 ≪ 1. Therefore, we only need to consider the J1sJ1w
interaction which only affects the propagation of ντ . The effective potential
felt by ντ due to this interaction is
Vντ =
, (15)
and the effective ξ is
ξTT =
Vντ − Vνµ
(g′/MZ′)
(g/MW )2
tan2 θW
sin2 θW
The limit |ξTT | ≤ ξ0 = 0.005 then translates into:
MZ′ ≥ MZ
sin2 θW
≈ 440GeV . (17)
Unfortunately, this potential limit from the measurement of ξ is weaker
than what is already available from precision electroweak data,8 and from
direct searches for pp̄ → Z ′ → τ+τ− at CDF.9,10
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3. Generation Non-diagonal Leptoquarks
The interactions of leptoquarks with ordinary matter can be described in
a model-independent fashion by an effective low-energy Lagrangian as dis-
cussed in Ref. 11. Assuming the fermionic content of the SM, the most gen-
eral dimensionless SU(3)C × SU(2)L ×U(1)Y invariant couplings of scalar
and vector leptoquarks satisfying baryon and lepton number conservation
are given by:
L = LF=2 + LF=0 , (18)
where
LF=2 =
ejL − dciLνjL) + g
γµejL) + g
γµejR)
V +2µ
γµνjL) + g
γµejR)
V −2µ
γµejL)Ṽ
2µ + (u
γµνjL)Ṽ
ejL)S
3 − (uciLejL + dciLνjL)S
νjL)S
, (19)
LF=0 =
(uiRejL) + h
(uiLejR)
(uiRνjL)− hij2R(diLejR)
(diRejL)S̃
2 + (diRνjL)S̃
(uiLγ
µνjL + diLγ
µejL) + h
(diRγ
µejR)
V 01µ
2(uiLγ
µejL)V
3µ + (uiLγ
µνjL − diLγµejL)V 03µ +
2(diLγ
µνjL)V
Here, the scalar and vector leptoquark fields are denoted by S and V , their
subscripts indicating the dimension of their SU(2)L representation, and the
superscripts indicating the sign of the weak-isospin of each component. We
allow for generation non-diagonal couplings with the indices i and j indicat-
ing the quark and lepton generation numbers, respectively. The subscript L
or R on the coupling constants indicate the chirality of the lepton involved
in the interaction. For simplicity, color indices have been suppressed. The
leptoquarks S1, ~S3, V2, Ṽ2 carry fermion number F = 3B + L = −2, while
the leptoquarks S2, S̃2, V1, ~V3 have F = 0. The interactions that affect neu-
trino oscillation are those with (ij) = (12) or (13).
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Table 1. Constraints on the leptoquark couplings with all the leptoquark
masses set to 300 GeV. To obtain the bounds for a different leptoquark mass
MLQ, simply rescale these numbers with the factor (MLQ/300 GeV)
LQ CLQ δλ
upper bound from |ξ| ≤ ξ0
S1 +3 |g
|2 − |g13
|2 0.01
~S3 +9 |g
|2 − |g13
|2 0.003
S2 −3 |h
|2 − |h13
|2 0.01
S̃2 −3 |h̃
|2 − |h̃13
|2 0.01
V2 +6 |g
|2 − |g13
|2 0.005
Ṽ2 +6 |g̃
|2 − |g̃13
|2 0.005
V1 −6 |h
|2 − |h13
|2 0.005
~V3 −18 |h
|2 − |h13
|2 0.002
It is straightforward to calculate the effective potentials due to the ex-
change of these leptoquarks, as well as the effective values of ξ.7 Assuming a
common mass for leptoquarks in the same SU(2)L weak-isospin multiplet,
the effective ξ due to the exchange of any particular type of leptoquark can
be written in the form
ξLQ = CLQ
δλ2LQ/M
g2/M2
δλ2LQ
. (21)
Here, CLQ is a constant prefactor, and δλ
LQ represents
δλ2LQ = |λ12LQ|2 − |λ13LQ|2 , (22)
where λ
is a generic coupling constant. The values of CLQ and δλ
for the different types of leptoquark are listed in Table 1. The constraint
|ξLQ| ≤ ξ0 translates into:
MLQ ≥
|CLQ||δλ2LQ|
2GF ξ0
|CLQ||δλ2LQ| × (1700GeV) . (23)
Alternatively, one can fix the leptoquark mass and obtain upper bounds on
the leptoquark couplings:
∣δλ2LQ
2GF ξ0
M2LQ ≈
300GeV
. (24)
The values when MLQ = 300GeV are listed in the rightmost column of
Table 1. Thought it is often stated that generation non-diagonal couplings
of leptoquarks are strongly constrained by the absence of flavor chang-
ing neutral currents, it is only the products of the (ij) = (12) and (13)
couplings with other couplings that are constrained.12 The limits on the
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individual couplings can be improved considerably. The current leptoquark
mass bounds from direct searches at the Tevatron, LEP, and HERA are in
the 200∼300 GeV range assuming generation diagonal couplings set equal
4πα. At the LHC, leptoquarks, if they exist, can be expected to be
pair-produced copiously through gluon-gluon fusion. The expected sensi-
tivity is up to about 1.5 TeV.13 Depending on the value assumed for δλ2LQ,
the bound from Eq. (23) can be competitive.
Acknowledgments
We would like to thank Drs. Andrew Akeroyd, Mayumi Aoki, Masafumi
Kurachi, Robert Shrock, and Hiroaki Sugiyama for helpful discussions. This
research was supported in part by the U.S. Department of Energy, grant
DE–FG05–92ER40709, Task A (Kao, Pronin, and Takeuchi).
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