尺度不變性的破壞對具有連續質量之粒子的影響

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(1)國立台灣師範大學物理研究所碩士論文指導教授：張嘉泓博士. 尺度不變性的破壞對具有連續質量之粒子的影響. 研究生：劉逸烽. 撰. 中華民國九十七年六月 1.

(2) NATIONAL TAIWAN NORMAL UNIVERSITY DEPARTMENT OF PHYSICS. The Effect of Broken Scale Invariance on Unparticle Physics. YI-FENG LIU JUNE 2008. Advising Professor: Dr. Chia-Hung Chang. 2.

(3) 物理. 96. 碩. 2. The Effect of Broken Scale Invariance on Unparticle Physics. 張嘉泓博士. 劉逸烽 695410323. 97. 3. 6. 24.

(4) 物理 695410323. 劉逸烽. The Effect of Broken Scale Invariance on Unparticle Physics. 4.

(5) Acknowledgments I acknowledge with gratitude the invaluable assistance, support and time given in the preparation of this thesis by my advisor Professor Chia-Hung Chang. Without him, this thesis would not have been possible. I thank him for his patience and encouragement that carried me on through difficult times, and for his insights and suggestions that helped to shape my research skills. His valuable feedback contributed greatly to this dissertation. He met with me for countless hours, providing editorial help. I also acknowledge with gratitude the assistance given by Dr. Li and Dr. Sze who read and commented of this thesis.. 5.

(6) Abstract. We study the muon pair production in electron-positron collision in the presence of an unparticle with a low energy scale invariance breaking. The breaking, possibly due to Higgs VEV, generates a mass gap μ in the unparticle continuous mass spectrum. This effect is incorporated into the unparticle propagator in a way that renders smooth transition to a massive particle propagator as unparticle dimension approach 1. We concentrate especially in the most likely case μ ≈ M Z and μ > M Z and discuss in muon pair production how to distinguish the original unparticle and the unparticle with mass gap μ.. 6.

(7) List of Figures Fig. 1: UV fixed point in QCD………………………………………………………...9 Fig. 2: Fixed point in QFT with scale invariance……………………………………...9 Fig. 3: Muon pair production...………………...………………………………..…...11 Fig. 4: The fractional change for axial-vector coupling..……………………..……...12 Fig. 5: The fractional change for vector coupling..…………….………………….....12 Fig. 6: Ru for axial-vector coupling with varying du and μ =115GeV..……….……..18 Fig. 7: Ru for axial-vector coupling with varying du and μ =500GeV……….……....19 Fig. 8: Ru for axial-vector coupling with varying μ and du=1.1………….......….......20 Fig. 9: Barger’s research with μ = 30GeV and du=1.1….....….……………………...21 Fig. 10: FBA for vector coupling with varying μ and du=1.1…………….......……...23 Fig. 11: Ru for vector coupling with varying μ and du=1.1..........…………………...24 Fig. 12: FBA for axial-vector coupling with varying μ and du=1.1..………………...25 Fig. 13: FBA for vector coupling with varying μ and du=1.7...….…………………..26 Fig. 14: Ru for vector coupling with varying μ and du=1.7...............………………..27 Fig. 15: FBA for axial-vector coupling with varying μ and du=1.7.......……………..28 Fig. 16: Ru for axial-vector coupling with varying μ and du=1.7.......……………….29 Fig. 17: FBA for vector coupling with varying du and μ =115GeV……………...…..30 Fig. 18: Ru for vector coupling with varying du and μ =115GeV...............………….31 Fig. 19: FBA for axial-vector coupling with varying du and μ =115GeV..…………..32 Fig. 20: FBA for vector coupling with varying du and μ =500GeV...………………..33 Fig. 21: Ru for vector coupling with varying du and μ =500GeV..…………………..34 Fig. 22: FBA for axial-vector coupling with varying du and μ =500GeV......………..35. 7.

(8) Contents. 1.Introduction…………………………………………………………….7 2. Pure unparticle…………………………………………………………8 3. Unparticle with a low energy scale invariance breaking……………..12 4. Vector unparticles phenomenology in muon pair production...............14 5. Conclusions..........................................................................................22 Appendix..................................................................................................23. 8.

(9) 1. Introduction Recently the physics of unparticle has attract a lot of attention. This interesting “stuff” is not like normal particle with fixed mass, but contains a continuous mass spectrum. Unparticle theory is essentially an extra conformal sector (called the unparticle sector) that couples with the standard model (SM) through heavy particle exchange. This theory will appear as the low energy effective theory of a hidden sector with a nontrivial infrared fixed point, and is hence dictated by the principle of scale invariance. Banks and Zaks[4] first brought out this idea. Scale invariance is the key to unparticle physics. A scale invariant theory by definition exhibits continuous mass spectrum stretching from zero to infinity. This scale invariant sector can have experimental signals radically different from those of normal particles and hence was called unparticle sector. Although Hrvoje Nikolic inferred a different conclusion, saying the “unparticle” word seems somewhat misleading and called it “uncanonical field” [1], we still use the original name because the unusual mass spectrum can become normal particle spectrum by shifting some parameters. The main signals of unparticle consist of real unparticle emissions and virtual unparticle exchange. It was noted by Georgi [2] that unparticle once generated does not decay and hence will appear like missing energy in collisions. The unusual aspect is that the phase space of an unparticle emission is equivalent to that of a “fractional” number of massless particles. This peculiar result will constitute clear signal if unparticle does exist. Virtual unparticle exchange also exhibits unexpected result. The unparticle propagator contains a cut in the time-like section of momentum space. Hence unparticle exchange contains both real and imaginary part for the whole cut from zero to infinity. Therefore it could interfere with any particle exchange with a pole at the particle mass. For example, in muon pair production from electron-positron collision, it could interfere with Z boson exchange even on the Z pole and produce large corrections, so there are different signals. However, astrophysics and cosmology constraints are sensitive to low mass channels and hence require that the interaction between unparticle and standard model particles to be very weak [3, 9, 10]. If these constraints are satisfied, low energy experiments will not be able to probe any aspects of unparticle physics. The way out of this is to observe that scale invariance will be broken somehow in the low energy area due to interaction of unparticle with Higgs. Breaking of conformal invariance makes things easier and unparticle interaction with SM does not need to be impossibly weak and we have a chance to find unparticle. But the scale invariance will emerge at higher energy, where unparticle physics will become relevant. That is, there is a conformal window. And the hope is that in high energy experiments like the LHC, we can indeed probe unparticle physics. In fact, from a recent calculation [3], 9.

(10) astrophysics and cosmology constraints can be escaped if scale invariance is broken at a scale μ 1 GeV and it makes unparticle detectable in colliders. In this paper, we study that the cross-section and forward-backward asymmetry of e + + e − → μ + + μ − when the energy scale μ are close to and larger than MZ. This is the most likely situation if the scale invariance breaking is introduced by the unparticle interaction with Higgs field. We start by reviewing the model of unparticle physics. Next we discuss the physics signal of unparticle. Then we started our discussion on how the scale invariance breaking effects can be introduced and modeled in unparticle physics. Finally we show the calculation using this “scale invariance broken propagator” of muon pair production physics. The result show the scale invariance broken unparticle will give rise to significantly different signals and the pattern is very different from the contribution of an ordinary unparticle and ordinary particle. 2. Pure Unparticle Unparticles are first discussed by Howard Georgi in 2007 [2]. The idea is based on scale invariance. In 1981, Banks and Zaks [4] observed that a QCD like theory with more than six flavors could have a nontrivial infrared fixed point. That means the theory will become scale invariant at a finite energy. Scale invariance is a powerful concept that has wide applications in many different disciplines of physics. Banks and Zaks combined information with strong coupling and large-NF massless fermions gauge theories. In string theory, scale invariance plays an even more fundamental role since it is part of the local diffeomorphism X Weyl reparametrization invariance group of the two-dimensional Riemann surfaces[4]. However, at the low energy world of particle physics, what we observe is a plethora of elementary and composite particles with a wide spectrum of masses [5]. In quantum field theory, the dilatation generator D for scale transformation does not commute with the spacetime translation generators Pμ, if we make a coordinate transformation x μ → λx μ , the 4-momentum will transform as P μ → λ−1 P μ , and the field will transform as φ → λ− dφ , where the dimension d depends on the field. However, even a massless theory, is more complicated due to the presence of a renormalization scale μ. And because the coupling constant g depends on μ, for dimensionless g, we need a dimensionful parameter Λ to eliminate the dimension of μ. Hence the scale invariance is broken because of the nonrenormalizable parameter, dg unless there exist a fixed point which makes = β = 0 . Figure 1 shows this d ln μ point, the UV fixed point in QCD, when μ → ∞ ⇒ β → 0 :. 10.

(11) Figure 1: UV fixed point in QCD Banks and Zakes concluded that a large class of continuum models with fermions in real representations of the gauge group do not have spontaneous chiral symmetry breaking. Hence the theory will be dictated by the scale invariance principle. Figure 2 shows the asymptotic free gauge theory with massless fermions has a non-trivial IR fixed point:. g*. UV. IR. Figure 2:Scale invariance, when μ ↓⇒ β → 0, g → g * Georgi, motivated by the Banks-Zaks theory, suggested that a scale invariant sector with a nontrivial infrared fixed-point behaves rather peculiar from the perspective of particle physics. Such a theory in low energy is dominated by a set of operators with anomalous (unconventional) dimensions. Most of the degree of freedom will be strong interacting above the scale μ = ΛU. If it becomes strong interacting near IR fixed point, they will disappear (and can be integrated out) in the low energy theory. Hence at low energy we can adopt an effective theory with these operators replaced by low energy effective field operators-denoted as unparticles, since they have continuous spectra, and all the other degrees of freedoms can be integrated out. By definition the unparticle exhibits continuous mass spectrum stretching from zero to infinity. This would be a problem if it interacts strongly with Standard model particles, we will discuss about this point later. Now, the two sets are hence designed to interact through exchange a hidden particle with large mass Mu. The nonrenormalizable couplings involving both SM and BZ fields are suppressed by Mu. These interactions 11.

(12) have the generic form. Cu OSM OBZ M uk. (2.1). where OSM are operators with mass dimension dSM built from SM fields and OBZ are operators with mass dimension dBZ built from BZ fields, i.e. hidden sector fields, and k = dBZ + dSM – 4. Cu is dimensionless coefficient function. Scale invariance in the hidden sector emerges at the energy scale Λu where the infrared fixed point appears. In the effective theory below Λu the interactions of Eq. (2.1) take the form. Cu ΛduBZ − d u OSM Ou M uk. (2.2). where du is the scaling dimension of the unparticle operator Ou. Since the low energy theory is scale invariant, the physics of the unparticle operators Ou will be dictated by scale invariance. In Ref. [2], Georgi showed by scale invariance the propagator of Ou can be totally determined up to a constant. He started with the vacuum matrix element: 0 Ou ( x)Ou+ (0) 0 = 0 Ou ( x)∑ n n Ou+ (0) 0 = ∫ e −ipx 0 Ou (0) P ρ ( P 2 ) 2. n. = ∫ e −ipx AdU (P 2 ) U. d −2. 4. d P = (2π ) 4 ∫. [∫ dM δ (M 2. 2. ]. − P 2 ) × eiPx AdU (P 2 ) U. d −2. d 4P (2π ) 4. d 4P (2π )4. (2.3). ∞. ⎡ 2 dU −2 d 4 P iPx ⎤ = AdU ∫ dM ⎢(M ) ∫ e δ ( M 2 − P 2 )⎥ 4 (2π ) ⎣ ⎦ 0 2. where the constant is irrelevant for physics and will be fixed as 1 Γ(d u + ) 16π 5 / 2 2 for the reason in phase space calculations. Ad u = (2π ) 2 d Γ(du − 1)Γ(2d u ) Then take time order and Fourier transformation, we can get. (. ). 4 iPx + ∫ d x e 0 T OU ( x)OU (0) 0 ==. AdU. (. 1 =i − P 2 − iε 2 sin (dU π ). AdU. ∞. ( ). ⎡ dM 2 ⎢ M 2 ∫ 2π 0 ⎣. dU − 2. i ⎤ 2 P − M + iε ⎥⎦ 2. ). (2.4). dU − 2. ∞. here we use. x p −1 π 2 ∫0 1 + x dx = sin pπ because it is well defined for negative P .. From the final formula, we can find that for non-integer dU there is a cut in the 12.

(13) space of P2, which has to be chosen at positive timelike P2 and could be seen as a combination of continuous poles. The unparticle, with a continuous mass spectrum, is very different from ordinary particle. Their effects might be detectable in missing energy distributions and interference with SM amplitudes [2, 6, 8, 14]. In missing energy observation, the unparticle phase space will look like fractional numbers of final massless particles. And due to the peculiar behavior of its propagator, the virtual exchange of an unparticle will interfere strongly with SM contributions. The beauty of the theory is that all of these effects can be deduced solely from the scale invariance of the unparticle sector without knowing the details of the theory. The muon pair production in electron-positron collision e + + e − → μ + + μ − is a beautiful example where the peculiar property of the unparticle propagator will show up [5]. The muon pair could be generated either through virtual photon, Z boson or unparticle production. As Georgi observed in his paper [2], the unparticle propagator contains both a real part and an imaginary part along the whole momentum range and hence it will interfere with Z boson and photon propagator even at Z pole, where the Z boson is dominated by its imaginary part. This point is brought home by calculating the fractional change in the total cross section.. Figure 3.. Assume that the interaction between leptons and a vector unparticle can be divided into a vector part and axial-vector part: CVU Λku+1−du C AU Λku+1−du μ + l γ l O l γ μ γ 5l Ouμ u μ M uk M uk. (2.5). For the convenience of discussion, we can define the couplings as: CVU =. CVU Λku+1−du M uk M Z1−du. C AU =. C AU Λku+1−du M uk M Z1−du. (2.6). The case of pure axial-vector coupling cVU = 0 is the most illustrative. In this case, interference only appears between virtual unparticle and virtual Z boson, since photon only couples vectorially. An unparticle propagator i. AdU. (. 1 − P 2 − iε 2 sin (dU π ). ). dU − 2. contains a phase φU = −(dU − 1)π along the whole momentum range. Hence it interferes with Z propagator everywhere and the leading contribution is of order 13.

(14) c AU. 2. instead of c AU 4 , we show it in Figure 4.. The effect is sensitive to the unparticle dimension. The real part of Eq. (2.4) is positive for 1 < d u < 1.5 and negative for 1.5 < d u < 2 and the real part of Z propagator is negative below Z pole and positive above Z pole. Hence the interference is destructive (constructive) below (above) the pole for 1 < d u < 1.5 and vice-versa for 1.5 < d u < 2 . This interesting behavior can be seen from the diagram above. In the case of pure vector coupling c AU = 0 , interference between virtual unparticle and virtual Z boson is small. The interference with photon dominates. The photon propagator is real and always positive. So we expect the interference is constructive for 1 < d u < 1.5 and destructive for 1.5 < d u < 2 , as show in Figure 5.. 3.. Unparticle with a Low Energy Scale Invariance Breaking Unparticles are very different from ordinary particles and hence are subjected to strict constraints from current experiments. Especially astrophysics and cosmology constraints are sensitive to low mass channels and hence require that the interaction between unparticle and standard model particles to be very weak. If these constraints are satisfied, low energy experiments will not be able to probe any aspects of 14.

(15) unparticle physics. However, this scale invariance has to be broken in the even lower energy area. The supernova experiment like BBN and SN 1987A constraints can be evaded provided scale invariance is broken at a scale μ sufficiently large compared to the relevant energy scales (≃ 1 MeV and ≃ TSN respectively). Although unparticle is fine when scale invariance is exact, it is not clear if they remain so when scale invariance is broken. If they are stable and μ is less than the top quark mass, it is not sufficient that they will decouple. Instead, they should remain out of equilibrium at all temperatures before BBN, at least up to the reheating temperature [12]. Even without these constraints, the scale invariance has to be broken anyway, because once Higgs boson has vacuum expectation value, it takes a most explicit form in operators involving both the unparticle sector and the SM Higgs boson[13]. The coupling of the following form is gauge invariant: 1 | H |2 OBZ (3.1) d BZ − 2 Mu with scale invariance, which will flow in the low energy to Cu ΛduBZ − d u M. d BZ − 2 u. | H |2 Ou. (3.2). Once the Higgs acquires a vacuum expectation value v, the operator (3.2) introduces a scale into conformal field theory, causing the unparticle sector to break the scale invariance at a scale μ, where Λ μ 4 − d u = ( u ) d BZ − d u M u2 − d u v 2 (3.3) Mu. Below it, the unparticle sector presumably becomes a traditional particle sector. This breaking scale invariance has the ironic consequence of saving the whole unparticle idea. The reason is that, if scale invariance is exact, unparticles are unlikely to be probed in colliders since there are strong constraints from astrophysics and cosmology [3, 9, 10]. These constraints are sensitive to a possible low mass channel and hence require that the interaction between unparticle and standard model particles to be very weak. In this case, the unparticles will not be detectible in colliders. The breaking of scale invariance at the scale μ immediately raises the mass spectrum to at least above μ. Hence the constraints will be loosened and the interaction between unparticle and SM particles need not be undetectably weak. It was shown that the breaking scale μ can be as small as 1 GeV. We expect that unparticle effects will be relevant for experiments energy s > μ and resonance-like behavior at μ is expected. Breaking scale invariance will modify the form of the unparticle propagator. Nobody knows how a scale invariance breaking term affects unparticle theory. J. Terning et al. in [7] suggested (and V. Barger et al used in Ref. [3]) that the density in 15.

(16) Eq. (2.3) can be written as 2. 0 Ou (0) P ρ ( P 2 ) = Ad u θ ( P 2 − μ 2 )( P 2 − μ 2 ) d u − 2. (3.4). instead of 0 Ou (0) P. 2. ρ ( P 2 ) = Ad ( P 2 ) d. u −2. u. .. With this prescription, the propagator for a spin 1 unparticles can then be rewritten as Adu PP (3.5) [ Δ F ( P 2 )]μν = [−( P 2 − μ 2 + iε )]du −2 (− g μν + a μ 2ν ) P 2 sin( d uπ ) Here [−( P 2 − μ 2 )]d. u −2. is defined as P 2 − μ 2. du −2. for P 2 < μ 2 and P 2 − μ 2. du −2. e −id uπ. for P 2 > μ 2 . Note that for P 2 < μ 2 , the propagator is real and hence the cut starts from P 2 = μ 2 and extends to infinity. The point of this hypothesis is that it is 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 ( P 2 − μ 2 ) 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. MZ. The calculation when μ < MZ is done by Barger et al [3]. We also case of μ Mz and s >> M Z . discuss the difference when s We plug the propagator (Eq. 3.5) into the amplitude M | M |2 = 2q 4 [(| ΔVV (q 2 ) |2 + | Δ AA (q 2 ) |2 + | ΔVA (q 2 ) |2 + | Δ AV (q 2 ) |2 ) (1 + cos 2 θ ) + (Re(Δ*VV (q 2 )Δ AA (q 2 )) + Re(Δ*VA (q 2 )Δ AV (q 2 ))) 4 cos θ ]. (4.1). For convenience, in these delta functions, we chose h = e = c = 1 , and θ = θw = 28.7° is weak mixing angle, doesn’t like the cosθ in Eq. 4.1 will be integrated as solid angle, 16.

(17) and e = g e = 4πα is coupling constant. For μ2 > q2 , these delta functions will as | ΔVV (q 2 ) |2 = [. Adu e 2 (sin 2 θ − 0.25) 2 e 2 q 2 − M z2 Cvu2 + ( μ 2 − q 2 ) d u − 2 ]2 + 2 du −2 2 2 2 2 2 2 2 q sin θ cos θ (q − M z ) + ( M z Γz ) 2 sin(d uπ ) MZ. M z Γz (sin 2 θ − 0.25) 2 e 2 ]2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ). +[. (4.2) e q −M C | Δ AA ( q 2 ) |2 = [ + 2 2 2 2 2 2 M 16 sin θ cos θ (q − M z ) + ( M z Γz ) 2. +[. 2. 2 Au 2 du − 2 Z. 2 z. Adu. 2 sin(d uπ ). ( μ 2 − q 2 ) d u − 2 ]2. M z Γz ]2 16 sin θ cos θ (q − M z2 ) 2 + ( M z Γz ) 2 e. 2. 2. 2. 2. (4.3) | ΔVA (q 2 ) |2 =| Δ AV (q 2 ) |2 =[. Adu e 2 (sin 2 θ − 0.25) q 2 − M z2 C C + AU2 du VU ( μ 2 − q 2 ) d u − 2 ]2 −2 2 2 2 2 2 2 4 sin θ cos θ (q − M z ) + ( M z Γz ) 2 sin( d uπ ) MZ. +[. e 2 (sin 2 θ − 0.25) M z Γz ]2 2 2 2 4 sin θ cos θ (q − M z2 ) 2 + ( M z Γz ) 2. (4.4) Re(Δ*VV (q 2 )Δ AA (q 2 )) =. e4 q 2 − M z2 16q 2 sin 2 θ cos 2 θ (q 2 − M z2 ) 2 + ( M z Γz ) 2. 1 e 4 (sin 2 θ − 0.25) 2 + 4 4 2 2 2 16 sin θ cos θ (q − M z ) + ( M z Γz ) 2 2 Adu q 2 − M z2 CVU ( μ 2 − q 2 ) du −2 + 2 du −2 2 2 2 2 2 2 16 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin( d uπ ). e2. +. 2 Adu e 2 C AU ( μ 2 − q 2 ) du −2 2 du −2 2 q MZ 2 sin( d uπ ). +. 2 Adu e 2 (sin 2 θ − 0.25) 2 q 2 − M z2 C AU ( μ 2 − q 2 ) du − 2 2 du −2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). 2 2 Ad2u C AU CVU + ( μ 2 − q 2 ) 2 du − 4 4 du − 4 2 MZ 4 sin (d uπ ). (4.5) Re(Δ*VA (q 2 )Δ AV (q 2 )) = +. e 4 (sin 2 θ − 0.25) 2 1 4 4 2 2 2 16 sin θ cos θ (q − M z ) + ( M z Γz ) 2. 2 2 Adu Ad2u CVU C AU C AU CVU e 2 (sin 2 θ − 0.25) q 2 − M z2 2 2 du −2 ( μ − q ) + ( μ 2 − q 2 ) 2 du −4 2 sin 2 θ cos 2 θ (q 2 − M z2 ) 2 + ( M z Γz ) 2 M Z2 du −2 2 sin( d uπ ) M Z4 du −4 4 sin 2 (d uπ ). (4.6) 2. 2. and for μ ＜q these delta functions will be 17.

(18) | ΔVV (q 2 ) |2 = [ +[. Adu C2 q 2 − M z2 e 2 (sin 2 θ − 0.25) 2 e 2 (q 2 − μ 2 ) du −2 cos(d u − 2)π ]2 + + 2vu du −2 2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) 2 sin( d uπ ) q MZ. Adu Cvu2 M z Γz (sin 2 θ − 0.25) 2 e 2 + (q 2 − μ 2 ) du −2 sin( d u − 2)π ]2 2 2 2 2 2 2 2 du − 2 2 sin( d u π ) sin θ cos θ (q − M z ) + ( M z Γz ) MZ. (4.7) e q −M C | Δ AA ( q 2 ) |2 = [ + 2 2 2 2 2 2 16 sin θ cos θ (q − M z ) + ( M z Γz ) M 2. +[. 2. 2 z. 2 Au 2 du −2 Z. M z Γz C + 2 2 2 16 sin θ cos θ (q − M z ) + ( M z Γz ) M e. 2. 2. 2. 2 AU 2 du − 2 Z. 2. Adu. 2 sin(d uπ ). Adu. 2 sin( d uπ ). ( q 2 − μ 2 ) du −2 cos(d u − 2)π ]2. (q 2 − μ 2 ) du −2 sin(d u − 2)π ]2. (4.8) | ΔVA (q 2 ) |2 =| Δ AV (q 2 ) |2 = [ +. e 2 (sin 2 θ − 0.25) q 2 − M z2 4 sin 2 θ cos 2 θ (q 2 − M z2 ) 2 + ( M z Γz ) 2. Adu C AU CVU (q 2 − μ 2 ) du −2 cos(d u − 2)π ]2 2 du −2 MZ 2 sin(d uπ ). +[. Adu e 2 (sin 2 θ − 0.25) M z Γz C C + AU2 du VU ( q 2 − μ 2 ) du −2 sin(d u − 2)π ]2 −2 2 2 2 2 2 2 4 sin θ cos θ (q − M z ) + ( M z Γz ) MZ 2 sin(d uπ ). (4.9) Re(Δ*VV (q 2 )Δ AA ( q 2 )) =. e q −M e (sin θ − 0.25) 1 + 2 2 2 2 2 2 4 4 2 2 2 16q sin θ cos θ (q − M z ) + ( M z Γz ) 16 sin θ cos θ (q − M z ) + ( M z Γz ) 2 4. 2. 2 z. 4. 2. 2. 2. +. 2 Adu CVU q 2 − M z2 (q 2 − μ 2 ) du −2 cos(d u − 2)π 2 du − 2 2 2 2 2 2 2 16 sin θ cos θ ( q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). +. 2 Adu M z Γz CVU (q 2 − μ 2 ) du −2 sin(d u − 2)π 2 du − 2 2 2 2 16 sin θ cos θ ( q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). +. 2 Adu e 2 C AU (q 2 − μ 2 ) du −2 cos(d u − 2)π 2 du − 2 2 q MZ 2 sin(d uπ ). +. 2 Adu e 2 (sin 2 θ − 0.25) 2 q 2 − M z2 C AU (q 2 − μ 2 ) du −2 cos(d u − 2)π 2 du − 2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). +. 2 Adu C AU e 2 (sin 2 θ − 0.25) 2 M z Γz (q 2 − μ 2 ) du −2 sin(d u − 2)π 2 du − 2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). e2 e2. 2. 2. 2. 2 2 Ad2u C AU CVU + ( q 2 − μ 2 ) 2 du − 4 4 du − 4 2 4 sin (d uπ ) MZ. (4.10) Re(Δ*VA (q 2 )Δ AV (q 2 )) =. e 4 (sin 2 θ − 0.25) 2 1 16 sin 4 θ cos 4 θ (q 2 − M z2 ) 2 + ( M z Γz ) 2. +. Adu e 2 (sin 2 θ − 0.25) q 2 − M z2 CVU C AU (q 2 − μ 2 ) du −2 cos(d u − 2)π 2 du − 2 2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). +. Adu CVU C AU e (sin θ − 0.25) M z Γz (q 2 − μ 2 ) du −2 sin(d u − 2)π 2 du − 2 2 2 2 2 2 2 2 sin θ cos θ (q − M z ) + ( M z Γz ) M Z 2 sin(d uπ ). +. 2 2 Ad2u C AU CVU ( q 2 − μ 2 ) 2 du − 4 M Z4 du −4 4 sin 2 ( d uπ ). 2. 2. 18. (4.11).

(19) So we can find the cross section variation between unparticle and photon or Z through computing dσ | M |2 | p f | 2 hc = ( )2 ⇒σ = | M |2 dΩ (4.12) 2 2 2 ∫ dΩ 8π (E1 + E2 ) | pi | (8π ) q 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 e 2 = 0.0917 ， M z = 91.19GeV ， hΓz = 2.5GeV . 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 M Z < μ and hence Z pole does not sit on unparticle propagator cut. The. s around Z pole is. smaller than μ. The propagator [Δ F ( P 2 )] =. Adu. 2 sin( d uπ ). [−( P 2 − μ 2 )]du −2. (4.13). is real but negative for s 2 = P 2 < μ 2 , 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 < M z and. destructive for s > M z , 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 s 2 = P 2 < μ 2 . And now the pattern in our case around the Z pole is the same as that of pure unparticle, i.e., constructive for. s < M z and. destructive for s > M z . 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 [−( P 2 − μ 2 )]du −2 is defined as 19. P2 − μ 2. du −2. for P 2 < μ 2 and.

(20) P2 − μ 2. du −2. e −iduπ for P 2 > μ 2 . Hence the propagator is real and negative for P 2 < μ 2 .. But for P 2 > μ 2 , 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 P 2 < μ 2 and constructive for P 2 > μ 2 when 1 < du < 1.5 . So the cutoff does not look like a particle pole but a discontinuous jump. When 1.5 < du < 2 , the pattern is both destructive for P 2 < μ 2 and P 2 > μ 2 . It looks like a strange negative pole, while particle poles usually enhance the cross section. axial-vector coupling withｄu=1.1，μ=115GeV. 1.02. axial-vector coupling withｄu=1.3，μ=115GeV. 1.02. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. 50. (a) du = 1.1. 150. 200. 250. (b) du = 1.3. axial-vector coupling withｄu=1.5，μ=115GeV. 1.002. 100. axial-vector coupling with ｄu=1.7，μ=115GeV. 1.002. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru 1. 1. 0.998. 0.998 0. 50. 100. 150. 200. 250. (c) du = 1.5 and enlarge 10 times. 0. 50. 100. 150. 200. 250. (d) du = 1.7 and enlarge 10 times. axial-vector coupling withｄu=1.9，μ=115GeV. 1.002 Cv=0,CA=0.026 mu=0 SM. Ru 1. 0.998 0. 50. 100. 150. 200. 250. (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. 20.

(21) 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. 1.04. axial-vector coupling withｄu=1.3，μ=500GeV. 1.02. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. 1.02. Ru. Ru. 1. 1. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. (a) du = 1.1. 300. 400. 500. 600. 700. 800. 900. 1000. (b) du = 1.3. axial-vector coupling withｄu=1.5，μ=500GeV. 1.002. 200. axial-vector coupling with ｄu=1.7，μ=500GeV. 1.01 Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.998. 0.99 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (c) du = 1.5 and enlarge 10 times. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (d) du = 1.7 and enlarge 2 times. axial-vector coupling withｄu=1.9，μ=500GeV. 1.01. Cv=0,CA=0.026 mu=0 SM. 1. Ru. 0.99. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (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. 21.

(22) axial-vector coupling withｄu=1.1，μ=105GeV. 1.04. axial-vector coupling withｄu=1.1，μ=115GeV. 1.04. Cv=0,CA=0.026 mu=0 SM. Cv=0,CA=0.026 mu=0 SM. 1.02. 1.02. Ru Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 0. 250. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. axial-vector coupling withｄu=1.1，μ=125GeV. 1.04. 50. axial-vector coupling withｄu=1.1，μ=150GeV. 1.04 Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. 1.02. 1.02 Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. axial-vector coupling withｄu=1.1，μ=200GeV. 1.04. 50. axial-vector coupling withｄu=1.1，μ=350GeV. 1.04. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. 1.02. 1.02. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. (f) μ = 250GeV. axial-vector coupling withｄu=1.1，μ=500GeV. 1.04. 50. axial-vector coupling withｄu=1.1，μ=900GeV. 1.04. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. 1.02. 1.02. Ru. Ru. 1. 1. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (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. 22.

(23) 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. vector coupling withｄu=1.1，μ=30. 1.04. 15. Cv=0.026,CA=0 mu=0. 10. SM 1.02. 5 FBA(%) 0. Ru. -5 1 Cv=0.026,CA=0. -10. mu=0. -15. SM. -20 0. 50. 100. 150. 200. 250. 0.98 0. (a) Vector coupling FBA. 100. 150. 200. 250. (b) Vector coupling RU. axial-vector coupling withｄu=1.1，μ=30. 20. 50. axial-vector coupling withｄu=1.1，μ=30. 1.04. 15 Cv=0,CA=0.026. 10. mu=0 1.02. 5. SM. FBA(%) 0. Ru. -5 1. -10. Cv=0,CA=0.026 mu=0. -15. SM. -20 0. 50. 100. 150. 200. 250. (c) Axial-vector coupling FBA. 0.98 0. 50. 100. 150. 200. 250. (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.. 23.

(24) 5.. Conclusions Just as the exchanges of ordinary massive particles whose effects may be detected. indirectly below threshold, unparticles will be detectable at energy scales slightly below μ. We study the muon pair production in electron-positron collision in the presence of an unparticle with a low energy scale invariance breaking. The breaking, possibly due to Higgs VEV, generates a mass gap μ in the unparticle continuous mass spectrum. This effect is incorporated into the unparticle propagator in a way that renders smooth transition to a massive particle propagator as unparticle dimension approach 1. We concentrate especially in the most likely case μ ≈ M Z and μ > M Z . The total cross section and forward-backward asymmetry as a function of center of mass energy are calculated and compared with the result of pure unparticle. Virtual unparticle contribution again interferes with those of virtual photon and Z boson and gives a deviation linear to virtual unparticle production. However the interfere pattern could be different from pure unparticle. The scale invariance breaking scale will give rise to an observable feature on the curve at the scale. The shape is sensitive to the unparticle dimension. It could be a discontinuous jump or a negative pole. Both are new and very different from the normal particle positive pole, as we would have naively expected for our case. After all, unparticle spectrum contains a cut, not a pole, even in the presence of a mass gap. The interference pattern near the Z pole could also be different from pure unparticle. For 1 < du < 1.5 , the destructive-constructive pattern is just the opposite of that with pure unparticle, and for 1.5 < du < 2 , the pattern will remain the same. We also show other pattern in Appendix, by changing μ and dU, the FBA or Ru display different pattern with vector and axial-vector coupling.. 24.

(25) 6.. Appendix vector coupling withｄu=1.1，μ=105GeV. 20. vector coupling withｄu=1.1，μ=115GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. mu=0. -15. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. vector coupling withｄu=1.1，μ=125GeV. 20. 50. vector coupling withｄu=1.1，μ=150GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. mu=0 -15. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 0. 250. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. vector coupling withｄu=1.1，μ=200GeV. 20. 50. vector coupling withｄu=1.1，μ=350GeV. 20. Cv=0.026,CA=0. 15. 15. mu=0 SM. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. Cv=0.026,CA=0 mu=0. 0. 50. 100. 150. 200. 250. SM. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. 900. 1000. (f) μ = 250GeV. vector coupling withｄu=1.1，μ=500GeV. 20. 50. vector coupling withｄu=1.1，μ=900GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. mu=0. -15. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. (g) μ = 500GeV (h) μ = 900GeV Figure 10: FBA for 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. 25.

(26) vector coupling withｄu=1.1，μ=105GeV. 1.04. vector coupling withｄu=1.1，μ=115GeV. 1.04. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. 1.02. 1.02. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. vector coupling withｄu=1.1，μ=125GeV. 1.04. 50. vector coupling withｄu=1.1，μ=150GeV. 1.04. Cv=0.026,CA=0 mu=0 SM 1.02. 1.02. Ru. Ru. 1. 1 Cv=0.026,CA=0 mu=0 SM. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. vector coupling withｄu=1.1，μ=200GeV. 1.04. 50. vector coupling withｄu=1.1，μ=350GeV. 1.04 Cv=0.026,CA=0 mu=0 SM. 1.02. 1.02. Ru. Ru. 1. 1 Cv=0.026,CA=0 mu=0 SM. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. 700. 800. 900. 1000. (f) μ = 250GeV. vector coupling withｄu=1.1，μ=500GeV. 1.04. 50. vector coupling withｄu=1.1，μ=900GeV. 1.04 Cv=0.026,CA=0 mu=0 SM. 1.02. 1.02. Ru. Ru. 1. 1 Cv=0.026,CA=0 mu=0 SM. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. (g) μ = 500GeV (h) μ = 900GeV Figure 11: Ru for 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. 26.

(27) axial-vector coupling withｄu=1.1，μ=105GeV. 20. axial-vector coupling withｄu=1.1，μ=115GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 0. 250. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. axial-vector coupling withｄu=1.1，μ=125GeV. 20. 50. axial-vector coupling withｄu=1.1，μ=150GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. -10. Cv=0,CA=0.026. Cv=0,CA=0.026. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. axial-vector coupling withｄu=1.1，μ=200GeV. 20. 50. axial-vector coupling withｄu=1.1，μ=350GeV. 20. Cv=0,CA=0.026. 15. 15. mu=0 SM. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. Cv=0,CA=0.026 mu=0 SM. 0. 50. 100. 150. 200. 0. 250. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. 900. 1000. (f) μ = 250GeV. axial-vector coupling withｄu=1.1，μ=500GeV. 20. 50. axial-vector coupling withｄu=1.1，μ=900GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5 -10. Cv=0,CA=0.026. -10. Cv=0,CA=0.026. mu=0. mu=0. -15. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. (g) μ = 500GeV (h) μ = 900GeV Figure 12: FBA 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. 27.

(28) vector coupling with ｄu=1.7，μ=105GeV. 20. vector coupling with ｄu=1.7，μ=115GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. vector coupling with ｄu=1.7，μ=125GeV. 20. 50. vector coupling with ｄu=1.7，μ=150GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. vector coupling with ｄu=1.7，μ=200GeV. 20. 50. vector coupling with ｄu=1.7，μ=350GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. (f) μ = 250GeV. vector coupling with ｄu=1.7，μ=500GeV. 20. 50. vector coupling with ｄu=1.7，μ=900GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (g) μ = 500GeV (h) μ = 900GeV Figure 13: FBA for vector coupling with varying μ and du=1.7. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different μ. 28.

(29) vector coupling with ｄu=1.7，μ=105GeV. 1.02. vector coupling with ｄu=1.7，μ=115GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. vector coupling with ｄu=1.7，μ=125GeV. 1.02. 50. vector coupling with ｄu=1.7，μ=150GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. vector coupling with ｄu=1.7，μ=200GeV. 1.02. 50. vector coupling with ｄu=1.7，μ=350GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. 700. 800. 900. 1000. (f) μ = 250GeV. vector coupling with ｄu=1.7，μ=500GeV. 1.02. 50. vector coupling with ｄu=1.7，μ=900GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. (g) μ = 500GeV (h) μ = 900GeV Figure 14: Ru for vector coupling with varying μ and du=1.7. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different μ. 29.

(30) axial-vector coupling with ｄu=1.7，μ=105GeV. 20. axial-vector coupling with ｄu=1.7，μ=115GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. axial-vector coupling with ｄu=1.7，μ=125GeV. 20. 50. axial-vector coupling with ｄu=1.7，μ=150GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%) 0. 0 -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. axial-vector coupling with ｄu=1.7，μ=200GeV. 20. 50. axial-vector coupling with ｄu=1.7，μ=350GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 250. 0. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. (f) μ = 250GeV. axial-vector coupling with ｄu=1.7，μ=500GeV. 20. 50. axial-vector coupling with ｄu=1.7，μ=900GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (g) μ = 500GeV (h) μ = 900GeV Figure 15: FBA for axial-vector coupling with varying μ and du=1.7. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different μ. 30.

(31) axial-vector coupling with ｄu=1.7，μ=105GeV. 1.02. axial-vector coupling with ｄu=1.7，μ=115GeV. 1.02 Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (a) μ = 105GeV. 100. 150. 200. 250. (b) μ = 115GeV. axial-vector coupling with ｄu=1.7，μ=125GeV. 1.02. 50. axial-vector coupling with ｄu=1.7，μ=150GeV. 1.02 Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (c) μ = 125GeV. 100. 150. 200. 250. (d) μ = 150GeV. axial-vector coupling with ｄu=1.7，μ=200GeV. 1.02. 50. axial-vector coupling with ｄu=1.7，μ=350GeV. 1.02. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 0. 250. (e) μ = 200GeV. 100. 150. 200. 250. 300. 350. 400. 450. 500. (f) μ = 250GeV. axial-vector coupling with ｄu=1.7，μ=500GeV. 1.02. 50. axial-vector coupling with ｄu=1.7，μ=900GeV. 1.02. Cv=0,CA=0.026. Cv=0,CA=0.026. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (g) μ = 500GeV (h) μ = 900GeV Figure 16: Ru for axial-vector coupling with varying μ and du=1.7. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different μ. 31.

(32) vector coupling withｄu=1.1，μ=115GeV. 20. vector coupling withｄu=1.3，μ=115GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. SM. -20. -20 0. 50. 100. 150. 200. 0. 250. (a) du = 1.1. 100. 150. 200. 250. (b) du = 1.3 vector coupling withｄu=1.5，μ=115GeV. 20. 50. vector coupling with ｄu=1.7，μ=115GeV. 20. 15. 15. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 50. 100. 150. 200. 250. (c) du = 1.5. 0. 50. 100. 150. 200. 250. (d) du = 1.7 vector coupling withｄu=1.9，μ=115GeV. 20 15 10 5 FBA(%) 0 -5. Cv=0.026,CA=0. -10. mu=0. -15. SM. -20 0. 50. 100. 150. 200. 250. (e) du = 1.9 Figure 17: FBA for vector coupling with varying du and μ =115GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.. 32.

(33) vector coupling withｄu=1.1，μ=115GeV. 1.02. vector coupling withｄu=1.3，μ=115GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. 0. (a) du = 1.1. 100. 150. 200. 250. 200. 250. (b) du = 1.3 vector coupling withｄu=1.5，μ=115GeV. 1.02. 50. vector coupling with ｄu=1.7，μ=115GeV. 1.02. Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. Ru. Ru. 1. 1. 0.98. 0.98 0. 50. 100. 150. 200. 250. (c) du = 1.5. 0. 50. 100. 150. (d) du = 1.7 vector coupling withｄu=1.9，μ=115GeV. 1.02 Cv=0.026,CA=0 mu=0 SM. Ru 1. 0.98 0. 50. 100. 150. 200. 250. (e) du = 1.9 Figure 18: Ru for vector coupling with varying du and μ =115GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.. 33.

(34) axial-vector coupling withｄu=1.1，μ=115GeV. 20. axial-vector coupling withｄu=1.3，μ=115GeV. 20. Cv=0,CA=0.026. 15. Cv=0,CA=0.026. 15. mu=0. mu=0. SM. SM. 10. 10. 5. 5. FBA(%). FBA(%). 0. 0. -5. -5. -10. -10. -15. -15. -20. -20 0. 50. 100. 150. 200. 250. 0. (a) du = 1.1. 150. 200. 250. 200. 250. axial-vector coupling with ｄu=1.7，μ=115GeV. 20. Cv=0,CA=0.026. 15. 100. (b) du = 1.3. axial-vector coupling withｄu=1.5，μ=115GeV. 20. 50. Cv=0,CA=0.026. 15. mu=0. mu=0. SM. 10. SM. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5. -10. -10. -15. -15. -20. -20 0. 50. 100. 150. 200. 250. (c) du = 1.5. 0. 50. 100. 150. (d) du = 1.7. axial-vector coupling withｄu=1.9，μ=115GeV. 20 Cv=0,CA=0.026. 15. mu=0 SM. 10 5 FBA(%) 0 -5 -10 -15 -20 0. 50. 100. 150. 200. 250. (e) du = 1.9 Figure 19: FBA 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.. 34.

(35) vector coupling withｄu=1.1，μ=500GeV. 20. vector coupling withｄu=1.3，μ=500GeV. 20. 15. 15. 10. 10 5. 5. FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. mu=0. -15. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 0. 1000. (a) du = 1.1. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (b) du = 1.3 vector coupling withｄu=1.5，μ=500GeV. 20. 100. vector coupling with ｄu=1.7，μ=500GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5 Cv=0.026,CA=0. -10. Cv=0.026,CA=0. -10. mu=0. -15. mu=0. -15. SM. -20. SM. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (c) du = 1.5. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (d) du = 1.7 vector coupling withｄu=1.9，μ=500GeV. 20 15 10 5 FBA(%) 0 -5. Cv=0.026,CA=0. -10. mu=0. -15. SM. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (e) du = 1.9 Figure 20: FBA for vector coupling with varying du and μ =500GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.. 35.

(36) vector coupling withｄu=1.1，μ=500GeV. 1.04. vector coupling withｄu=1.3，μ=500GeV. 1.02. 1.02. Ru. Ru. 1. 1 Cv=0.026,CA=0. Cv=0.026,CA=0. mu=0. mu=0. SM. SM. 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 0. (a) du = 1.1. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 700. 800. 900. 1000. (b) du = 1.3 vector coupling withｄu=1.5，μ=500GeV. 1.02. 100. vector coupling with ｄu=1.7，μ=500GeV. 1.02 Cv=0.026,CA=0 mu=0 SM. Ru. Ru. 1. 1. Cv=0.026,CA=0 mu=0 SM 0.98. 0.98 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (c) du = 1.5. 0. 100. 200. 300. 400. 500. 600. (d) du = 1.7 vector coupling withｄu=1.9，μ=500GeV. 1.02. Ru 1. Cv=0.026,CA=0 mu=0 SM 0.98 0 √s. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (e) du = 1.9 Figure 21: Ru for vector coupling with varying du and μ =500GeV. Light blue curves: unparticles with μ = 0, green curves: SM, blue curves: unparticles with different du.. 36.

(37) axial-vector coupling withｄu=1.1，μ=500GeV. 20. axial-vector coupling withｄu=1.3，μ=500GeV. 20. 15. 15. 10. 10 5. 5. FBA(%) 0. FBA(%) 0. -5. -5. Cv=0,CA=0.026. -10. Cv=0,CA=0.026. -10. mu=0. mu=0. -15. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 0. 1000. (a) du = 1.1. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (b) du = 1.3. axial-vector coupling withｄu=1.5，μ=500GeV. 20. 100. axial-vector coupling with ｄu=1.7，μ=500GeV. 20. 15. 15. 10. 10. 5. 5 FBA(%). FBA(%) 0. 0. -5. -5. -10. -10. Cv=0,CA=0.026 mu=0. -15. Cv=0,CA=0.026 mu=0. -15. SM. SM. -20. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (c) du = 1.5. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (d) du = 1.7. axial-vector coupling withｄu=1.9，μ=500GeV. 20 15 10 5 FBA(%) 0 -5 -10. Cv=0,CA=0.026 mu=0. -15 SM. -20 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. (e) du = 1.9 Figure 22: FBA 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.. 37.

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