reduced to Georgi’s unparticle propagator as μ→0 and also to standard particle propagator with mass μ as dU →1. It’s interesting to see how the broken scale invariance will affect experiment observation.
4. Vector Unparticles Phenomenology in Muon Pair Production
It is generally assumed that for momentum smaller than the breaking scale μ, the unparticle effects will be reduced since the mass spectrum of unparticles will now start at μ. But sometimes it’s a bit confusing how that will happen. Rizzo in Ref. [11]
(and quoted by Berger [3]) actually cut off the propagator of an unparticle by a theta function )θ(P2−μ2 in his calculation of Drell-Yan process q+q→l++l− for μ Mz. In fact, this is dangerous since it toss away all the unparticle effects as s<μ. Indeed it’s the vacuum matrix element that should be cut off by a theta function as in Eq. (3.4). The difference may be important if the center of mass energy s is close to but a bit smaller than μ and the scale μ is close to Mz as is likely. To illustrate it, we study muon pair production in electron-positron collision e+ +e− →μ++μ− in the case of μ MZ. The calculation when μ < MZ is done by Barger et al [3]. We also discuss the difference when s Mz and s >>MZ.
We plug the propagator (Eq. 3.5) into the amplitude M
]
is weak mixing angle, doesn’t like the cosθ in Eq. 4.1 will be integrated as solid angle,
and e =ge = 4πα is coupling constant. For μ2 > q2 , these delta functions will as
2
So we can find the cross section variation between unparticle and photon or Z through computing
∫ Ω
= + ⇒
Ω= M d
q p
p E M c
d d
i
f 2
2 2 2
2 1
2 2 | |
) 8 (
2
|
|
|
| ) (E
| ) |
(8
σ π π
σ h
(4.12) where |pf| (or |pi|) is the magnitude of either outgoing (incoming) momentum.
We calculate the total cross section ratio RU ≡ σ (with unparticles) / σ (without unparticles) and the front-back asymmetry for vector and axial-vector unparticle couplings. The input constants are chosen as e2 =0.0917，Mz =91.19GeV ，
z =2.5GeV
hΓ . We find interesting difference with the original calculation with pure unparticle by Georgi [6].
We illustrate this difference by first concentrating on RU with axial-vector coupling.
As noted earlier, axial-vector coupling ensure interference effects are mainly with Z propagator. In following Figure 6, we list diagrams of RU versus s with a definite cutoff μ of 115GeV, but varying the dimension dU from 1.1 to 1.9. It’s interesting to observe for 1<du <1.5, these interfering pattern around the Z pole is the opposite to that of pure unparticle as depicted in Georgi’s [6]. This is because MZ <μ and hence Z pole does not sit on unparticle propagator cut. The s around Z pole is smaller than μ. The propagator
2 2 2
2 [ ( )]
) sin(
)] 2 (
[Δ = u − − du−
u d
F P
d
P A μ
π (4.13) is real but negative for s2 = P2 <μ2, just opposite to the real part of the complex propagator of a pure unparticle. And hence when the unparticle propagator interfere with the Z propagator, the pattern around Z pole is constructive for s <Mz and destructive for s >Mz, just opposite the case of pure unparticle. This could serve as the differentiating signal between a pure unparticle from an unparticle with low energy scale invariance breaking. However, for 1.5<du <2, the real part of the complex propagator of a pure unparticle change sign while the propagator Eq. 4.13 remains real and negative for s2 = P2 <μ2 . And now the pattern in our case around the Z pole is the same as that of pure unparticle, i.e., constructive for s <Mz and destructive for s >Mz. For this case, we’d need both the interference pattern around Z pole and that around the cutoff μ to distinguish a pure unparticle from an unparticle with low energy scale invariance breaking.
We can also observe the behavior of the propagator around the cutoff μ. Again the result is sensitive to the unparticle dimension. As noted earlier, in the propagator Eq.
3.5, the factor [−(P2 −μ2)]du−2 is defined as P2−μ2du−2 for P2 <μ2 and
μ du e iduπ
P2 − 2 −2 − for P2 >μ2. Hence the propagator is real and negative for P2 <μ2. But forP2 >μ2, it touches the cut and becomes complex. Its real part, which interferes with Z propagator, is positive for 1<du <1.5 and negative for 1.5<du <2. Hence the pattern is destructive for P2 <μ2 and constructive for P2 >μ2 when
5 . 1
1<du < . So the cutoff does not look like a particle pole but a discontinuous jump.
When 21.5<du < , the pattern is both destructive for P2 <μ2 and P2 >μ2. It looks like a strange negative pole, while particle poles usually enhance the cross section.
axial-vector coupling withｄu=1.1，μ=115GeV
0.98 1 1.02
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.3，μ=115GeV
0.98 1 1.02
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(a) du = 1.1 (b) du = 1.3
axial-vector coupling withｄu=1.5，μ=115GeV
0.998 1 1.002
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling with ｄu=1.7，μ=115GeV
0.998 1 1.002
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(c) du = 1.5 and enlarge 10 times (d) du = 1.7 and enlarge 10 times
axial-vector coupling withｄu=1.9，μ=115GeV
0.998 1 1.002
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(e) du = 1.9 and enlarge 10 times
Figure 6: Ru for axial-vector coupling with varying du and μ =115GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.
We also raise the cutoff μ to observe the effects, like Figure 7 and 8. If μ is much larger than Mz, the Z pole becomes far away from the unparticle cut and indeed the unparticle feels like a heavy ordinary particle as expected. And it can be observed that the interference pattern at Z pole disappears as decoupling sets in. The cross section will be close to what SM gives. The unparticle effects will be detected as we raise the energy up to the cutoff μ, either as a discontinuous jump or as a negative pole, depending on the unparticle dimension.
axial-vector coupling withｄu=1.1，μ=500GeV
0.98 1 1.02 1.04
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.3，μ=500GeV
0.98 1 1.02
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
(a) du = 1.1 (b) du = 1.3
axial-vector coupling withｄu=1.5，μ=500GeV
0.998 1 1.002
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling with ｄu=1.7，μ=500GeV
0.99 1 1.01
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
(c) du = 1.5 and enlarge 10 times (d) du = 1.7 and enlarge 2 times
axial-vector coupling withｄu=1.9，μ=500GeV
0.98 0.99 1 1.01
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
(e) du = 1.9 and enlarge 2 times
Figure 7: Ru for axial-vector coupling with varying du and μ =500GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.
axial-vector coupling withｄu=1.1，μ=105GeV
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.1，μ=115GeV
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(a) μ = 105GeV (b) μ = 115GeV
axial-vector coupling withｄu=1.1，μ=125GeV
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.1，μ=150GeV
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(c) μ = 125GeV (d) μ = 150GeV
axial-vector coupling withｄu=1.1，μ=200GeV
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.1，μ=350GeV
0.98 1 1.02 1.04
0 50 100 150 200 250 300 350 400 450 500
Ru
Cv=0,CA=0.026 mu=0 SM
(e) μ = 200GeV (f) μ = 250GeV
axial-vector coupling withｄu=1.1，μ=500GeV
0.98 1 1.02 1.04
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.1，μ=900GeV
0.98 1 1.02 1.04
0 100 200 300 400 500 600 700 800 900 1000
Ru
Cv=0,CA=0.026 mu=0 SM
(g) μ = 500GeV (h) μ = 900GeV
Figure 8: Ru for axial-vector coupling with varying μ and du=1.1. Light blue curves:
unparticles with μ = 0, green curves: SM, blue curves: unparticles with different μ.
As a consistency check, we also set μ = 30GeV and dU = 1.1, like Figure 9, and compare to Barger’s patterns in [3]. We get exactly the same curves. That means our programs should be correct. Another check: when we increase the experiment energy
s, these curves will come back close to Georgi’s theory.
vector coupling withｄu=1.1，μ=30
-20 -15 -10 -5 0 5 10 15 20
0 50 100 150 200 250
FBA(%)
Cv=0.026,CA=0 mu=0 SM
vector coupling withｄu=1.1，μ=30
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0.026,CA=0 mu=0 SM
(a) Vector coupling FBA (b) Vector coupling RU
axial-vector coupling withｄu=1.1，μ=30
-20 -15 -10 -5 0 5 10 15 20
0 50 100 150 200 250
FBA(%)
Cv=0,CA=0.026 mu=0 SM
axial-vector coupling withｄu=1.1，μ=30
0.98 1 1.02 1.04
0 50 100 150 200 250
Ru
Cv=0,CA=0.026 mu=0 SM
(c) Axial-vector coupling FBA (d) Axial-vector coupling RU
Figure 9: Barger’s research in [3]. We set μ = 30GeV and du=1.1 in our program then get the same curves as Barger’s. Light blue curves: unparticles with μ = 0, green curves:
SM, deep blue curves: unparticles with μ = 30GeV.