Decay of Z Boson into Photon and Unparticle

0.26 MB·English

by  Kingman Cheung

Decay of Z Boson into Photon and Unparticle
Kingman Cheung1,2, Thomas W. Kephart3, Wai-Yee Keung4, and Tzu-Chiang Yuan2
1Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
2Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3Department of Physics and Astronomy,
Vanderbilt University, Nashville, Tennessee 37235, USA
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4Department of Physics, University of Illinois,
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Chicago, Illinois 60607-7059, USA
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J (Dated: February 2, 2008)

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1 Abstract

]
We study the decay of the standard model Z boson into unparticle plus a single photon through
h
p
- a one-loop process. As in the anomaly type decay, only the axial-vector part of the Z coupling
p
e
matching with the vector unparticle and/or the vector part of the Z coupling matching with the
h
[
axial-vector unparticle can give a nonzero contribution to the decay. We show that the photon

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v spectrum terminates at the end point in accord with Yang’s theorem. Existing data on single
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6 photon production at LEP I is used to constrain the unparticle sector.
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1
.
1 PACS numbers: 14.80.-j, 12.38.Qk,12.90.+b, 13.40.Em
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I. INTRODUCTION
It is well known that the Poincar´e symmetry group can be enlarged to the group of
conformal symmetry in any number of spacetime dimension D. From the structure of the
conformal algebra we learn that conformal symmetry implies scale invariance, while scale
invariance in general does not necessarily imply conformal symmetry. It is widely believed
that a local quantum field theory that is scale invariant will also be conformal invariant for
any D. Although a formal proof is still lacking, no counterexample has been found. For
D = 2 it was first conjectured in [1] and later shown in [2] that scale invariance does imply
conformal symmetry under broad conditions 1. One of the most important consequences of
scale invariance is that the single particle state must be either massless or have a continuous
mass spectrum. An obvious example for the former case is pure QED, where the photon is
exactly massless. Minimal coupling of the photon to charged particles in a gauge invariant
fashion guarantees the photon remains massless to all orders in the perturbation series.
However, particles with continuous mass distributions have been largely ignored in particle
physics due to lack of experimental evidence.
Recently, an interesting physical possibility for scale invariant stuff with continuous mass
distribution was pointed out by Georgi [4], who coined the term unparticle 2 to describe a
possible scale-invariant hidden sector sitting at an infrared fixed point at a high scale ΛU.
If the hidden sector carries standard model (SM) quantum numbers, it would be highly
constrained by existing experimental data. In Georgi’s scheme [4], the hidden sector com-
municates with the SM content via a messenger sector characterized by a high mass scale
M. At energy below M, one can integrate out the messenger sector and end up with the
effective operator suppressed by inverse powers of M of the following form
1
, (1)
MdSM+dUV−4OSMOUV
where and represent local operators of the SM and hidden sector with scaling
SM UV
O O
dimensions d and d respectively. As one scales down the theory from M, the hidden
SM UV
sector may flow to an infrared fixed point at the scale ΛU which, for example, can be
generated by quantum effects via dimensional transmutation. At the fixed point where the
1
However, a counterexample in the two-dimensional theory of elasticity was demonstrated in [3].
2
We use the term “unparticle” as an uncountable noun, so just like water, it has no distinct plural.
2
hidden sector becomes scale invariant, the above operator Eq. (1) has to be replaced by a
new set of operators of similar form
ΛdUV−dU
U
COUMdSM+dUV−4OSMOU , (2)
where U is the unparticle operator with a scaling dimension dU and CO is the unknown
O U
coefficient. Because the underlying theory is a scale invariant interacting theory, the scaling
dimension dU need not have a canonical value of integer or half-integer, unlike the free
boson or free fermion cases. The unparticle operator U can be characterized as scalar,
O
vector, tensor, spinor, etc., according to its Lorentz group representation. One prototype [4]
hidden sector that can give rise to unparticle is the weakly interacting Banks-Zaks [5] theory.
Another possibility is the strongly interacting magnetic phase of certain supersymmetric
QCD theories [6], as pointed out in [7]. A third possibility would be the hidden valleys
model which can be viewed as an unparticle sector with a large mass gap [8].
Even though the scale invariant sector remains unspecified, the 2-point function [4] and
the Feynman propagator [9, 10] of the unparticle field operator U can be determined
O
by scale invariance. We note if special conformal invariance is imposed it is shown in
a recent paper [11] that the form of vector and tensor unparticle propagators should be
modified. Consequently, the polarization sum of the vector and tensor unparticle have to
be modified as well. However, the new form of the polarization sum does not allow one
to impose transversality of the vector and tensor unparticle unless the scaling dimension
dU equals to 3. Many groups have pursued phenomenological studies of unparticle physics
[4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,
79, 80, 81, 82, 83], while more conceptual aspects of unparticle were explored by others
[84, 85, 86, 87, 88, 89, 90]. Unparticle does not have a fixed invariant mass, but instead has
a continuous mass spectrum. Thus, like a massless particle, the unparticle has no rest frame.
This implies that real unparticle is stable and cannot decay. Direct signals of unparticle can
nevertheless be detected in the missing energy and momentum distributions carried away
by the unparticle once it is produced in a process [4], while virtual unparticle effects can be
probed via interference with SM amplitudes [9, 10].
In a previous note [73], Li and two of us studied the decay of a SM Higgs (H) into vector
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unparticle plus a single photon. We showed that the photon energy spectrum for the process
is continuously smeared out near its end point and its branching ratio is comparable to that
of the discovery mode of H γγ for an intermediate mass Higgs. In this note, we study
→
the rare decay of the Z boson into unparticle plus a single photon, Z γ, via a triangular
→ U
loop of SM fermions. As in the Higgs decay, the energy spectrum of the photon would have
been monochromatic had the unparticle had a fixed mass. However, due to the nature of
the continuous mass spectrum of the unparticle, the resultant photon energy spectrum is
also continuous and the shape depends sensitively on the scaling dimension dU. But unlike
the case of H γ, and many other previously studied cases, the end point of the photon
→ U
energy spectrum in the decay Z γ goes to zero as governed by Yang’s theorem.
→ U
II. DECAY RATE OF Z γ
→ U
The interaction of spin-1 unparticle with a SM fermion f can be parameterized by a
U
term in the effective Lagrangian [4, 10]
1
f¯ λfγ λ′fγ γ fOµ (3)
Leff ∋ ΛdU−1 1 µ − 1 µ 5 U
U
(cid:16) (cid:17)
where λf and λ′f are the unknown vector and axial-vector couplings. Here we assume the
1 1
transversality of unparticle operator ∂ Oµ = 0 is satisfied and Oµ has both vector and axial-
µ U U
vector couplings to the SM fermions. The process Z γ is induced at one-loop level with
→ U
the standard model fermions circling in a triangle loop diagram. The photon always has
vector-type interactions with SM fermions. Possible types of interactions for the Z γ
−U −
vertex are either AVV or VAV, where V(A) denotes vector (axial-vector) interaction. The
other two possibilities of VVV and AAV vanish due to Furry’s theorem. Note that the
vector couplings of the Z boson are much smaller than the axial-vector couplings (at least
true for u, d and e), but we consider both types of couplings for the Z γ vertex.
−U −
TheamplitudesquareforZ γ canbeadaptedfromanearliercalculationinadifferent
→ U
context [91]
1
2 = z(1 z)2(1+z) 2m2 , (4)
|M| 2π4 − |A| Z
pol
X
where z = P2/m2. The loop amplitude is given by
U Z A
e2 1
= Nf Q gfλf +gfλ′f (z,η ) , (5)
A −sinθ cosθ ΛdU−1 C f A 1 V 1 I f
w w U Xf (cid:16) (cid:17)
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where the color factor Nf = 3(1) for f being a quark (lepton), gf = T /2 Q sin2θ and
C V 3f − f w
gf = T /2 are the vector and the axial-vector couplings of the Z boson to the fermion f
A 3f
respectively, Q is the electric charge of the fermion f and η = m2/m2. The loop function
f f f Z
(z,η) is given by
I
1 1 η η 1 η
(z,η) = + F (η) F G(η) G , (6)
I 1−z (2 1−z (cid:20) − (cid:18)z(cid:19)(cid:21)− 2(1−z) (cid:20) − (cid:18)z(cid:19)(cid:21))
where
2
2 sin−1 1 for x 1
F(x) = − 4x ≥ 4
 21 l(cid:16)n xx−+ 2q− π22(cid:17)−iπln xx−+ for x < 41
(cid:16) (cid:17)
and 
2√4x 1sin−1 1 for x 1
G(x) = − 4x ≥ 4
 √1−4x ln xx−+ q−iπ for x < 14
(cid:16) (cid:17)
with 
1 1
±
x = x .
2 ±s4 −
The amplitude square vanishes in the limit z 0 as governed by Yang’s theorem.
→
The differential decay width for Z γ is
→ U
1
dΓ = 2dΦ , (7)
2m |M|
Z
X
where 2 = 1 2 and the phase space factor dΦ is
|M| 3 pol|M|
P P
A
dΦ = dU (m2)dU−1zdU−2(1 z)dz (8)
16π2 Z −
with
16π5/2 Γ(dU + 1)
A = 2 . (9)
dU (2π)2dU Γ(dU 1)Γ(2dU)
−
Collecting all the pieces, we have
dΓ e4 m2 dU−1
= A m Z
dz 192π6sin2θwcos2θw dU Z Λ2U !
2
Nf Q gfλf +gfλ′f (z,η ) zdU−1(1 z)3(1+z) . (10)
×(cid:12) C f A 1 V 1 I f (cid:12) −
(cid:12)(cid:12)Xf (cid:16) (cid:17) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Integrating the abov(cid:12)e expression over z from 0 to 1, w(cid:12)e obtain the partial width of the
channel Z γ.
→ U
5
-3
10
λ = λ’ = 1
1 1
-4 Λ = 1 TeV
10 U
-5
10
)
V
e
G
( -6
10

z
d
/
Γ
d
-7
10
-8
10
-9
10
-5 -4 -3 -2 -1 0
10 10 10 10 10 10
z
FIG. 1: Spectrum dΓ(Z → Uγ)/dz, where z = PU2/m2Z, with λf1 = λ′1f = 1 and ΛU = 1 TeV for
dU = 1.1 2.0 (from top to bottom).
−
III. NUMERICAL RESULTS
If we ignore the fermion masses as we take z 0, then the relevant factor in Eq. (10)
→
scales as zdU−1(1+lnz), where the factor (1+lnz) comes from (z,0) [91]:
I
1 ln(z)
(z,0) = 1+ . (11)
I 2(1 z) 1 z!
− −
The photon energy E is given by m (1 z)/2. At z = 0, the photon energy reaches its
γ Z
−
end point and the unparticle behaves like a massless particle. Thus, as long as dU > 1
the photon energy spectrum vanishes at z = 0 in accord with Yang’s theorem. It is worth
mentioned that since λf and λ′f are viewed as effective couplings in Georgi’s scheme, the
1 1
following combination
Nf Q gfλf +gfλ′f (12)
C f A 1 V 1
Xf (cid:16) (cid:17)
needs not vanish when summed over SM fermions in contrast with the anomaly-induced
′
decay of Z Zγ in E models studied in [91]. In Fig. 1, we plot the spectrum dΓ/dz,
6
→
6
f ′f
TABLE I: Partial width of Γ(Z → Uγ) for λ1 = λ1 = 1 and ΛU = 1 TeV for dU = 1.1−2.0.
dU Γ(Z γ) (GeV)
→ U
1.1 1.4 10−5
×
1.2 8.5 10−6
×
1.3 3.8 10−6
×
1.4 1.5 10−6
×
1.5 5.5 10−7
×
1.6 1.9 10−7
×
1.7 6.5 10−8
×
1.8 2.2 10−8
×
1.9 7.0 10−9
×
2.0 2.3 10−9
×
where z = PU2/m2Z, for a range of dU = 1.1−2.0 (from top to bottom) with the democratic
assumption of λf1 = λ′1f = 1 for all SM fermions and ΛU is set to be 1 TeV. In Table I, the
partial width for Z γ is tabulated with the same input parameters. The partial width
→ U
is only sizable for dU 1.4.
≤
On the other hand, as the fermion mass inside the loop becomes infinitely heavy, one has
the following expansion for the loop function (z,η) valid for η ,
I → ∞
1 (1+2z)
(z,η) + . (13)
I ≈ 24η 360η2
Thus the loop amplitude in Eq.(5) is vanishingly small and the heavy fermion decouples
A
in this limit unlike the cases in the Higgs decay of H γγ,Zγ and γ.
→ U
Experimental searchs for e−e+ γX, where X represents a weakly interacting stable
→
particle, have been performed at the Z resonance [92]. No signal was found, and the 95%
C.L. upper limit on the branching ratio is [92]
B(Z γX) 1 3 10−6 , (14)
→ ≤ − ×
who’s range depends on E , the minimum energy cut of the photon. For E between 30
min min
GeV and m /2, the branching ratio upper limit is roughly a constant of about 1 10−6.
Z
×
It is clear from Fig. 1 that in Z γ, most contributions come from the region where
→ U
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f ′f
FIG. 2: Contour plot for the branching ratio B(Z γ) versus (dU,ΛU) with λ1 = λ1 = 1.
→ U
The shaded region to the left of the contour 10−6 is ruled out by the 95% C.L. upper limit on
B(Z γX) < 1 10−6.
→ ×
E 30 GeV. Therefore, we use the limit B(Z γ) < 1 10−6 to constrain the unparticle
γ
≥ → U ×
parameter space. In Fig. 2, we plot the contour of the branching ratio for Z γ as a
→ U
function of dU and ΛU assuming democratically as before that λf1 = λ′1f = 1 for all SM
fermions. One sees that existing limits from LEP I can already place useful constraints on
the hidden unparticle sector.
To recap, we have studied the rare decay of the Z boson into a single photon plus
unparticle that has both vector and axial-vector couplings to the SM fermions. Existing
limits from LEP I were used to constrain the parameters of the hidden unparticle sector
associated with vector and/or axial-vector unparticle. Despite having a peculiar photon
energy distribution in this 2-body decay, the branching ratio is rather minuscule and at best
of the order of 10−6 for small scaling dimension dU 1.4. For larger scaling dimension,
≤
the branching ratio is at least smaller by two orders of magnitude. Unless one can collect a
very large sample of Z boson, detection of the unparticle through this mode would be quite
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challenging.
Acknowledgments
This research was supported in parts by the NSC under grant number NSC 96-2628-M-
007-002-MY3, the NCTS of Taiwan (Hsinchu) as well as U. S. DOE under grant numbers
DE-FG02-84ER40173 and DE-FG05-85ER40226.
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