Unparticle Physics and A_{FB}^b on the Z pole
aa r X i v : . [ h e p - ph ] J a n Unparticle Physics and A bF B on the Z pole Mingxing Luo , Wei Wu and Guohuai Zhu Zhejiang Institute of Modern Physics, Department of Physics,Zhejiang University, Hangzhou, Zhejiang 310027, P.R. China
Abstract
An attempt has been made to address the 3 σ anomaly of the forward-backwardasymmetry of b quark in LEP data via an unparticle sector. For most part of theparameter space except certain particular regions, the anomaly could not be explainedaway plausibly, when constraints from other LEP observables are taken into account. Email: [email protected] Email: [email protected] Email: [email protected]
Introduction
During the last 35 years or so, the Standard Model (SM) has been well tested by experiments.In particular, the LEP experiment is one of the most impressive, in which the statisticaluncertainty is reduced by the huge number of events and the systematic uncertainty reducedby the clean experiment environment. The physical analysis was well documented. Forexample, a recent report [1] summarized all of the precision electroweak measurements onthe Z resonance.Overall, the LEP data can be well interpreted by the SM, except for few small deviations.One such conspicuous case is the forward-backward asymmetry of the bottom quark on the Z resonance, A ,bF B , which differs from the SM prediction by approximately three standarddeviations. Deviations as such may well be statistical fluctuations. If it is not, there could betwo remedies. One possibility is high-order corrections within the SM, which is, however, notsupported by recent calculations [2]. Another explanation is, of course, due to new physicseffect [3]. In this paper, we will see to the possibility whether the so-called unparticle sectorwould provide such an explanation. Unfortunately, the answer turns out to be negative formost part of the parameter space.The notion of unparticle sector was suggested recently by Georgi [4]. It is supposedto be a hidden sector which has non-trivial conformal behavior at the low energy limit. Itsinteraction with the SM sector is through an intermediate sector which is of high energy scaleand its effects may well appear at the TeV scale. Non-trivial fixed point in the infrared arecommonly used in condensed matter physics to describe second order phase transitions, butrarely encountered in particle physics. However, the existence of such infrared fixed pointsin Yang-Mills theories was realized many years ago [5]. In a gauge theory with suitablenumber of massless fermions, one does have nontrivial infrared fixed points, which ensuresa non-trivial conformal sector in the infrared. Actually, it was argued by Seiberg [6] that aconformal window could exist in supersymmetric SU ( N C ) gauge theories with N f fermionsif 3 N c / < N f < N c . But phenomenological implications of these exotic possibilities havenot been addressed seriously.Admittedly, one has yet to iron out a consistent framework for the unparticle physics andthere are many theoretical issues to be examined carefully. For instance, it is well known that S -matrices cannot be defined in conformal field theories, as one cannot define asymptoticstates in these theories. On the other hand, the unparticles must interact with the SMparticles to be relevant, but such interactions definitely break down the scale invariance.Nevertheless, one may as well take such a novel framework as a working hypothesis andthen push forward to see how far it can take us. In the framework of effective field theory,one may discuss phenomenologies of the effective unparticle operators without concrete un-derstanding on their dynamics at high energy. One interesting point is that [4], thanks tothe property of scale invariance, the unparticle looks like a non-integral number of missingmassless particles in the detector. In this direction, there have been quite a bit activities towork out such implications [7].Interestingly, a queer phase [4] appears in the unparticle propagator in the time-like1egion, which leads to novel interferences between unparticles and the SM processes very dif-ferent from the familiar pattern. Potentially interesting phenomenologies could be observedat present or future colliders such as LEP and LHC.In most cases, leading order effects of new physics are their interferences with the SMones. However, Z pole is usually not a good place to see interference effects between theSM and new physics, as amplitudes from the SM and new physics are out of phase by 90degrees. The extra phase factor in the unparticle propagator changes things drastically [4].The unparticle amplitude interferes with the SM one fully which may give a considerablecontribution on the pole. This provides a new opportunity to address the deviation of theforward-backward asymmetry of b quark in LEP measurements. In this paper, we discussthis possibility systematically.The paper is organized as follows: In section 2, several physical observables on the Z resonance are discussed. Section 3 introduces basic notions of an unparticle sector. Adetailed numerical analysis is presented in section 4 on effects of the unparticles on physicalobservables measured in LEP, with a particular emphasis on A ,bF B . The results are finallysummarized in section 5. Z Resonance [8]
In Standard Model, the differential cross section for e + e − → f ¯ f through the s channel is dσd cos θ = βs π [( | G LL | + | G RR | )(1 + β cos θ ) + ( | G LR | + | G RL | )(1 − β cos θ ) + 2(1 − β ) ℜ e ( G LL G ∗ LR + G RR G ∗ RL )] , (1)and the total cross section is obtained by integrating out the θ angle. The cross sections forscatterings of left- and right-handed electrons on unpolarized positrons are σ X = βs π [( | G XL | + | G XR | )(1 + 1 / β ) + 2(1 − β ) ℜ e ( G XL G ∗ XR )] , (2)where s is the center of mass energy square, β = (1 − m /s ) / , m is the mass of the massivefermion, and G XY ’s are G XY ( s ) = X A g X ( A → e + e − ) g Y ( A → f ¯ f ) ∗ ∆ A ( s ) . (3)Here A is either γ or Z ; X, Y = L or R are the chiralities of fermions, the propagators are∆ A ( s ) { A = γ, Z } = (cid:26) s , s − M Z + iM Z Γ Z (cid:27) , (4) This effect was also noticed in Drell-Yan process by Cheung et al. in [7]. g X ( A → f ¯ f ) { A = γ, Z } = n − eQ f , − e sin θ W cos θ W ( I f − Q f sin θ W ) o , (5)where Q f and I f are the electric charge and weak isospin of f , respectively. Note that forleft-handed (right-handed) fermions, I f are taken to be ± / Z resonance is σ had = X q σ ( e + e − → q ¯ q ) ≃ σ b + 2 σ d + 2 σ u , (6)if u, d , s , c quarks are regarded as massless. Here and hereafter, the superscript denotesquantities on the Z resonance.The left-right polarization asymmetry is defined as A LR = σ L − σ R σ L + σ R , (7)here the luminosity-weighted e − beam polarization magnitude is supposed to be 1. The e + e − final state is excluded here because it contains t -channel subprocess of photon exchangewhich could dilute the result. µ + µ − and τ + τ − final states are considered in a complementaryanalysis [1], which therefore will not be included in the following discussions. With theseselection rules, A LR is measured by summing hadronic final states in SLD experiment. Thus, A LR = X q (cid:0) σ ,qL − σ ,qR (cid:1) /σ had . (8)To distinguish relatively heavy flavors from light ones, one defines R b = σ b σ had , R c = σ c σ had . (9)Finally, one defines the forward-backward asymmetry of f ¯ f production on the Z resonance: A ,fF B = σ ,fF − σ ,fB σ ,fF + σ ,fB , (10)where σ ,fF = Z π/ dσ f d cos θ d cos θ, σ ,fB = Z ππ/ dσ f d cos θ d cos θ . (11)Shown in table 1 are the latest experiment data and SM global fit [1] of these physicalobservables. All observables are well consistent with the SM except A ,bF B , which deviates byalmost three standard deviations. 3easurement SM fit σ had (nb) 41 . ± .
037 41 . ± . R b . ± . . ± . R c . ± . . ± . A LR (SLD) 0 . ± . . ± . A ,lF B . ± . . ± . A ,cF B . ± . . ± . A ,bF B . ± . . ± . Z resonance in the second column andSM global fits in the third column [1]. Interactions of vector-like unparticle with SM fermions can be approximated by an effectiveLagrangian L int = c fV U M ( d U − Z ¯ f γ µ f U µV + c fA U M ( d U − Z ¯ f γ µ γ f U µA . (12)Following conventions in ref. [4], the couplings c fV U and c fA U are normalized in terms of thethe Z boson mass. Scalar unparticles may also couple to the SM fermions and thus affect A ,bF B . The derivation are actually very similar to the case of vector unparticles, thoughnumerically they might be different. However as discussed recently by Fox et al. in [7], thescale invariance of the unparticles may break down if scalar unparticles are coupled to theHiggs. Even if such coupling does not exist at tree level, it could be regenerated throughloop diagrams. Therefore we choose not to discuss scalar unparticles in the following.One hopes that the unparticle sector could account for the roughly 3 σ deviation betweenthe SM prediction and the LEP measurement on A ,bF B . On the other hand, unparticlesshould not affect other observables too much so as not to invalidate agreements between theSM global fit results and their LEP measurements. Thus, c fV U and c fA U have to be flavor-dependent. For simplicity, we assume that the unparticle couplings with the SM fermions4re universal except those with the b quark, c fV U = ( c V U ( f = b ) ,λc V U ( f = b ) ; c fA U = ( c A U ( f = b ) ,λc A U ( f = b ) ; (13) | λ | > A ,bF B deviation.Following [4], the vector-like unparticle operators are assumed to be transverse and thepropagator is given by Z d xe iP x < | T U µV ( A ) ( x ) U µV ( A ) (0) | > = i A d U − g µν + P µ P ν /P sin( d U π ) (cid:16) − P − iǫ (cid:17) d U − , (14)with A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) . (15)It is then straightforward to calculate unparticle contributions to the process e + e − → f ¯ f .Following procedures in section 2 and define∆ U ( s ) = A d U − P − iǫ ) d U − sin( d U π ) , g L,R ( U → f ¯ f ) = c V U ∓ c A U M d U − Z ( f = b ) ,λ c V U ∓ c A U M d U − Z ( f = b ) . (16)The unparticle contributions are taken into account by letting A = γ , Z , and U in Eqs.(1-3).At Z pole, the SM amplitude is almost pure imaginary while normally the new physicscontribution is real. Therefore it is hard to observe interference effects at or near Z pole.However as first discussed by Georgi in the second paper of [4], the phase e − i ( d U − π in Eq.(14) causes the unparticle amplitude to be complex in the time-like region and thus providesa novel possibility for unparticles to interfere with the SM amplitude at the Z resonance.Note that the unparticle sector introduces four free parameters: c A U , c V U , λ , and d U . We now discuss unparticle contributions to physical observables on the Z resonance andcompare them with the LEP data. Since our main concern is about A ,bF B , we shall firstconsider the influence of the unparticle sector on this quantity.For convenience, one may express the forward-backward asymmetry in the massless limitin terms of vector and axial-vector couplings A F B = 32 (cid:16) ℜ e ( G ∗ V V G AA ) + ℜ e ( G ∗ V A G AV ) | G V V | + | G AA | + | G V A | + | G AV | (cid:17) . (17) The formula in massless limit is quoted solely for the purpose of the qualitative discussions. The b quarkmass effects are taken into account in the numerical analysis. u -0.100.10.20.30.4 D A FB (cid:144)H Λ È c a H v L U È L (a) u -0.100.10.20.30.4 D A FB (cid:144)HÈ c a H v L U ÈL (b) Figure 1: Changes of forward-backward asymmetries for (a) e + e − → b ¯ b in unit of λ | c A ( V ) U | and (b) e + e − → ℓ ¯ ℓ in unit of | c A ( V ) U | versus d U on the Z pole. The solid, dot, short-dashedand long-dashed lines represent cases of (i) c A U = c V U , (ii) c A U = − c V U , (iii) c V U = 0 , c A U = 0and (iv) c A U = 0 , c V U = 0, respectively.The definition of G xy with x, y = V or A is the same as those for the left and right ones inEq. (3), with g V,A = ( g R ± g L ) / b quark andleptons on the Z pole, due to the unparticle sector. For example, A ℓF B = A ℓ,SMF B + x ∗ | c A ( V ) U | + O ( | c A ( V ) U | ) , and the coefficient x denotes the change of A ℓF B in unit of | c A ( V ) U | , as shown in Figure 1,while the O ( | c A ( V ) U | ) term is neglected.At the resonance, the QED amplitude is very small compared with the weak one. Thisleads to the ordering pattern of the SM amplitude: G AA ≫ G V A , G AV ≫ G V V . Thus, theleading interference effect between the unparticle sector and the SM amplitude arises fromthe term ℜ e ( G SMAA G U ∗
V V ). Consequently, the contribution from the unparticle coupling c V U is the largest. In Fig.1, the long-dashed line with c V U = 0 has almost negligible effect on A F B . Note also that the unparticle contribution to A ,ℓF B is λ -independent, as seen fromFig. 1b. To be consistent with experimental observations, one hopes the change of A ,bF B to be relatively large to interpret the deviation. On the other hand, the change of A ,ℓF B should be small enough, say within the 1 σ experimental error. Similarly, constraints on theunparticle couplings could also be obtained from the forward-backward asymmetry of the c quark, which however are less restricted compared with those from the leptons and thereforenot shown here.The unparticle sector also affects other physical observables. We now discuss its impacton the hadronic cross section σ had , the ratios R b and R c , and the left-right polarization asym-metry A LR . As seen from Table 1, measurements on these quantities are well consistent withthe SM fits. Constraints on the unparticle couplings can be obtained by these observables.For the hadronic cross section, σ had ∝ | G V V | + | G AA | + | G V A | + | G AV | . Z pole from the unparticle sectorversus d U . For illustration, | λ | is chosen to be 4. The minus sign of λ in cases (b), (c), (d) ischosen so as to have negative contributions to A ,bF B , as required by the LEP data. In case (a),for any given sign of λ , the contribution to A ,bF B oscillates in sign with d U . The positive signas chosen is for illustration, but our conclusion does not depend on this particular choice.Solid lines from top down show different inputs for c A ( V ) U = 0 .
25, 0 .
2, 0 .
15, 0 .
1, 0 .
05, and0 .
01, respectively. Horizental dashed lines denote limits of 1 σ experimental errors.Again, the | G AA | term dominants on the Z resonance in the SM. The leading interferenceterm between the unparticles and the SM part is proportional to ℜ e ( G SMAA G U ∗ AA ). Therefore, σ had is quite sensitive to c A U , but much less so to c V U , as shown in Fig.2, where the ratio∆ σ had = ( σ had − σ SMhad ) /σ had represents the difference for the hadronic cross section with orwithout unparticle contributions. Here and hereafter, we choose four typical scenarios to beexamined, in which the unparticle couplings are vector, axial-vector, left-handed and right-handed, respectively. At the quark level, σ had contains the production of u , d , s , c and b quarks but only the b quark part depends on λ . Therefore the constraints obtained from σ had is not sensitive to the value of λ . Note also that in Fig.2, | ∆ σ had | is plotted on a logarithmicscale. The dips in Fig. (2b), (2d) means that the unparticle contributions vanish at thesespecific d U values, and ∆ σ had changes sign across these dips.. R b and R c are both measured precisely at LEP experiments, as shown in Table 1. The7igure 3: Unparticle contributions to R b on the Z pole versus d U . Parameters and conven-tions are the same as those in Fig. 2.experimental error of R b is about 5 times smaller than that of R c . Naturally, one anticipatesthat R b gives stricter constraint on the unparticle couplings than R c , which is verified by ournumerical analysis. Thus, only results from R b are plotted in Fig. 3. By similar reasoningas those for σ had , R b is quite sensitive to c A U but much less so to c V U . Note that λ appearsonly in the unparticle coupling with b quarks, so the λ dependence appears in the change of R b only.Finally we consider the left-right polarization asymmetry A LR . Contrary to the crosssection-related observables, such as σ had and R b , A LR could lead strong constraints on theparameter c V U instead of c A U , as shown in Fig.4. Since the quoted value of A LR is measuredby SLD experiment by summing over all hadronic final states, just like the case of σ had , thecurves in Fig.4 are not sensitive to the value of λ .Now we are ready to combine all constraints obtained from A ,ℓF B , σ had , R b and A LR , tosee whether there are areas in the parameter space to interpret the observed deviation of A ,bF B . Again four typical scenarios under consideration are chosen to be axial-vector, vector,left-handed and right-handed couplings between the unparticles and the SM part. Corre-sponding results are drawn in Fig. 5(a)-(d), respectively. It seems difficult for the unparticlesector to be able to account for the observed deviation on A ,bF B . Note that constraints from8igure 4: Unparticle contributions to Left-Right asymmetry on the Z pole versus d U . Pa-rameters and conventions are the same as those in Fig. 2. R b have similar λ -dependence as those from A ,bF B . Although only the cases with λ = 4 and6 are shown in Fig. 5, it is not difficult to go through detailed numerical investigationsto check that the scenarios with axial-vector, left-handed and right handed couplings (Fig.5(a),(c),(d),(e),(g),(h)) are almost completely excluded for any reasonable value of λ . How-ever, the scenario with pure vector coupling (Fig. 5(b),(f)) is more subtle: Here the moststringent constraint is from A ,ℓF B , which is independent on λ . Therefore for a larger λ , thecurves of A ,bF B may be lowered and the anomaly in A ,bF B can be explained away. For example,if λ ≥
6, a large pure vector unparticle coupling with b quarks c bV U > . A ,bF B to be consistent with the LEP data (Fig.5(f)). In general, inthe particular region where the unparticle coupling with b quarks is predominately vector-like and λ is substantially larger than 1, the anomaly in A ,bF B seems to be explained away.Whether this provides a plausible solution indeed, it heavily relies on one’s taste. Some mayargue that it is a little far stretched.Off-resonance data, as those discussed by Bander et al. and Cheung et al. in [7], providessimilar but weaker constraints in most cases, which are not included here as they are notparticularly illuminating. 9 Summary
In this paper, we have discussed the possibility whether the 3 σ deviation of the forward-backward asymmetry of b quark between the LEP measurements and the SM fit could beaccounted for by an unparticle sector. By considering constraints from other observables,namely the hadronic cross section σ had , the ratio R b , the left-right asymmetry A LR and theleptonic forward-backward asymmetry A ,ℓF B , which are all well consistent with the SM fits, itseems quite difficult to explain the A ,bF B anomaly by the notion of unparticles. Specifically, ifthe unparticle couplings with the SM fermions are axial-vector, left-handed or right-handed,it is almost impossible to interpret the 3 σ deviation of A ,bF B . In the particular region wherethe unparticle coupling with b quarks is predominately vector-like and λ is substantiallylarger than 1, the anomaly in A ,bF B seems to be explained away. Whether this provides aplausible solution indeed, it heavily relies on one’s taste. Acknowledgments
This work is supported in part by the National Science Foundation of China under grantNo.10425525 and No.10645001.
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