aa r X i v : . [ h e p - ph ] J un ∆ m D and ∆Γ D revisted Chuan-Hung Chen , ∗ , C. S. Kim † Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan National Center for Theoretical Sciences, Hsinchu 300, Taiwan Department of Physics & IPAP, Yonsei University, Seoul 120-479, Korea (Dated: October 23, 2018)
Abstract
The lifetime difference ( y D = ∆Γ D / D ) and mass difference ( x D = ∆ m D / Γ D ) of neutral D mesonhave been measured with y D = (0 . ± . x D = (0 . ± . K and B q systems, the current data indicate that y D /x D ∼ y D favors to be larger than x D . For explaining the experimental indication, we here study the D − ¯ D oscillation in the framework of unparticle physics. We demonstrate that the peculiar phaseappearing in off-shell unparticle propagator could play an important role on x D and y D . ∗ E-mail: [email protected] † E-mail: [email protected],
1n the Standard Model (SM), the most impressive features of flavor physics are theGlashow- Iliopoulos-Maiani (GIM) mechanism [1] and the large top quark mass. The formerresults in the cancelation between the first two generations so that the mass difference ∆ m K in the neutral K system could be suppressed, while the latter makes ∆ m B q (q = d, s) in the B q systems dominated by the short-distance (SD) top-quark effects [2]. Due to the precisionmeasurements and the sensitivity to the new physics, within the past decades enormousstudies have been done in K and B q mesons, which are composed of down type quarks.By the production of large number of D mesons at Tevatron and B factories worldwide,now the neutral charmed meson which is made up of up-type quarks also plays an importantrole on the test of the SM. By the world average, the current measurements with allowingCP violation (CPV) for D − ¯ D mixing are given by [3] x D = m H − m L Γ D = ∆ m D Γ D = (0 . ± . ,y D = Γ H − Γ L D = ∆Γ D D = (0 . ± . . (1)Combining the errors in quadrature, the ratio of y D to x D is estimated by y D x D = 1 . ± . . (2)Intriguingly, the current data not only show x D ∼ y D but also indicate that the former isslightly smaller than the latter.Due to the effective GIM mechanism and the absence of heavy quark enhancement, theSD SM predictions are several orders smaller than the data [4]. It is expected that the GIMsuppression factor might be lifted by long-distance (LD) effects [5–8]. With the exclusivetechnique [7, 8], the results in the SM are estimated to be [8] x SM D ≈ (0 . ± . , y SM D ≈ (0 . ± . , (3)where we have averaged the possible theoretical scenarios. Since the exclusive techniqueis based on the measurements of nonleptonic D decays, due to the limited accuracy ofexperimental data, the SM prediction on y D is still quite uncertain. Although the results inEq. (3) display the same tendency as the data, the values of x SM D and y SM D are quite smallerthan the experimental data. Thus, the ratio in the SM is estimated as y SM D x SM D = 2 . ± . . (4)2e see clearly that the central value by LD contributions is twice larger than that in Eq. (2).If we take the central values of data in Eq. (1) seriously, the SM results in Eqs. (3) and(4) obviously cannot match with the data consistently. For explaining the large x D ( y D )and y D /x D ∼
1, the incompatibility could be ascribed to new physics. In most extensionsof the SM, owing to the suppression of ( m c /m W ) [9] and the constraints of low-energymeasurements [10, 11], the SD contributions to y D with O (10 − ) is not favorable. Therefore,we are going to explore a peculiar new effect on the D − ¯ D mixing, especially on the y D ,where the associated new stuff is dictated by the scale or conformal invariance and named asunparticle [12, 13]. Some interesting applications of unparticle to various systems could bereferred to Refs. [13–18]. The unique character of unparticle is its peculiar phase appearing inthe off-shell propagator with positive squared transfer momentum [12]. Due to CP invariance,the imaginary (real) part of the phase factor leads to the absorptive (dispersive) effect of aprocess [19, 20]. In this Letter, we investigate how x D and y D are influenced by the phasefactor. Furthermore, in order to make the production of scale invariant stuff be efficient atLarge Hadron Collider (LHC), we will concentrate on the unparticle that carries the colorcharges of SU (3) c symmetry [17].Since there is no well established approach to give a full theory for unparticle interactions,we study the topic from the phenomenological viewpoint. In order to avoid fine-tuning theparameters for flavor changing neutral currents (FCNCs) at tree level, we assume thatthe unparticle only couples to the third generation of quarks before electroweak symmetrybreaking. Hence, the interactions obeying the SM gauge symmetry are expressed by1Λ d U U (cid:2) l R ¯ q ′ R γ µ T a q ′ R ∂ µ O aU + l L ¯ Q L γ µ T a Q L ∂ µ O aU (cid:3) , (5)where l R,L are dimensionless free parameters, q ′ R = t R , b R , Q TL = ( t, b ) L , { T a } = { λ a / } arethe SU (3) c generators (where λ a are the Gell-Mann matrices) normalized by tr ( T a T b ) = δ ab /
2. Λ U is the scale below which the unparticle is formed, and the power d U is determinedfrom the effective interaction of Eq. (5) in four-dimensional space-time when the dimensionof the colored unparticle O aU is taken as d U . Since we only concentrate on the phenomenaof up type quarks, the associated interactions are formulated by¯ U γ µ ( X R P R + X L P L ) T a U ∂ µ O aU , (6)where U T = ( u, c, t ), X R ( L ) is a 3 × X R ( L ) )=(0, 0, l R ( L ) / Λ d U U ).After spontaneous symmetry breaking of electroweak symmetry, we need to introduce two3nitary matrices V R,LU to diagonalize the mass matrix of up type quarks. In terms of physicaleigenstates and using the equations of motion, the interactions for c − u −O aU could be writtenas L cu O aU = m c Λ d U U ¯ u (cid:0) g Ruc P L + g Luc P R (cid:1) T a c O aU + h.c. , (7)where the mass of light quark has been neglected. And g χuc = λ χ ( V χU ) ( V χ ∗ U ) with χ = R, L ,in which the index of Arabic numeral (1, 2, 3) stands for ( u, c, t ) quark, respectively.By following the scheme shown in Ref. [18], the propagator of the colored scalar unparticleis written as Z d xe − ik · x h | T O a ( x ) O b (0) | i = i C S δ ab ( − k − iǫ ) − d U (8)with C S = A d U d U π ,A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) . (9)Combining Eqs. (7) and (8), the four fermion interaction for D -meson oscillation is given by H = C S m c (cid:18) m c Λ U (cid:19) d U e − id U π × (cid:2) ¯ u (cid:0) g Ruc P L + g Luc P R (cid:1) T a c (cid:3) . (10)For estimating the transition matrix elements, we use h ¯ D | ¯ uP R ( L ) c ¯ uP R ( L ) c | D i ≈ − ξ D m D f D , h ¯ D | ¯ uP R c ¯ uP L c | D i ≈ (cid:18)
124 + 14 ξ D (cid:19) m D f D , h ¯ D | ¯ u α P R c β ¯ u β P L c α | D i ≈ (cid:18)
18 + 112 ξ D (cid:19) m D f D , h ¯ D | ¯ u α P R ( L ) c β ¯ u β P R ( L ) c α | D i ≈ ξ D m D f D , (11)where ξ D = m D / ( m c + m u ) and f D is the decay constant of D meson. As a consequence,the dispersive and absorptive parts of D − ¯ D oscillation in the unparticle physics are foundby H U = M U − i U , M U = cos( d U π ) h U and Γ U = 2 sin( d U π ) h U with h U = C S m c (cid:18) m c Λ U (cid:19) d U m D f D × h(cid:16) g R uc + g L uc (cid:17) ξ D + 2 g Ruc g Luc i . (12)In order to study the x D and y D , we have to know their relations to M and Γ . Followingthe notation in Particle Data Group (PDG) [21], the mass and rate differences of heavy andlight D mesons could be formulated by∆ m D = Re (∆ ω HL ) , ∆Γ D = − Im (∆ ω HL ) (13)with ∆ ω HL = 2 s(cid:18) M − i (cid:19) (cid:18) M ∗ − i ∗ (cid:19) , (14)where M = M SM12 + M U and Γ = Γ SM12 + Γ U . If we define the relative phase between M and Γ to be φ D = arg ( M / Γ ), the ratio of rate difference to mass difference is obtainedby ∆Γ D ∆ m D = 2 r D − r D / R D cos φ D (15)with r D = | Γ || M | ,R D = q (1 − r D / + r D cos φ D . (16)We note that unlike the case in B q system where the sign of ∆Γ B q in the SM is certainand experimental data are consistent with SM prediction, the sign of ∆Γ D in the SM isuncertain; thus we use φ D = arg ( M / Γ ) for D -meson, instead of φ B = arg ( − M q / Γ q )for B q -meson. Hence, the ratio of y D to x D can be expressed by y D x D = ∆Γ D m D = r D cos φ D − r D / R D . (17)In order to illustrate the phase effect of unparticle and simplify the numerical estimates,we set Λ U = 1 TeV and g Ruc = g Luc = | g uc | e iθ , i.e. the couplings are vector-like. Since the SM5redictions are still quite uncertain, for numerical analysis we adopt the recent SM resultsto be [8] M SM12 = 0 .
13% ps − , Γ SM12 = 0 .
73% ps − , (18)where we adopt M SM12 = x SM D Γ D / SM12 = y SM D Γ D and we take only the central value of x SM D ( y SM D ) as input. Other relevant values used for numerical estimates are listed in Table I. TABLE I: Values used for numerical estimates [21]. m D [GeV] m c [GeV] f D [MeV] τ D [ps]1.864 1.3 206.7 0.41 With the chosen scenario for the free parameters and the taken numerical values, nowwe have to deal with three free parameters, i.e. the scale dimension d U , the magnitude ofcoupling g uc and its phase θ . Since the SM results are smaller than the current data, wefind that the influence of θ is insignificant when the constraints of measured x D and y D areincluded. In Fig. 1, we present the unparticle contributions to x D and y D as a function | g uc | (in units of 10 − ) and d U , where figure (a)-(d) stands for θ = (0 , π/ , π/ , π/
4) and solidand dotted line denotes x D and y D , respectively. It is clear that the allowed | g uc | is slightlychanged when θ is varied. For further understanding the θ -dependence, we plot x D and y D as a function of θ and d U with | g uc | = 1 . × − in Fig. 2, where the solid and dotted linecorresponds to x D and y D , respectively.We have studied the mixing parameter and lifetime difference of D − ¯ D oscillation inthe framework of unparticle physics, where the new stuff is dictated by scale or conformalinvariance. Unlike other models, due to the peculiar phase of unparticle , not only the mix-ing parameter x D but also the lifetime difference y D can be enhanced to fit the currentexperimental data, especially the experimental result of y D /x D ∼
1. We speculate that theunparticle or unparticle-like effects could be strongly verified, when x D ∼ y D ∼ few × − and y D /x D ∼ d U | g u c | d U | g u c | | d U | g u c | . . . . d U | g u c | (c)(a) (b)(d) FIG. 1: (a)-(d) the contours for ∆ m D (blue solid) and ∆Γ D (red dotted) as a function | g uc | and d U with θ = 0 , π/ , π/ , π/
4, respectively. The numbers on the curves are the data with 1 σ errors. d U θ (r a d ) . . . . FIG. 2: x D (blue solid) and y D (red dotted) as a function of θ and d U with | g uc | = 1 . × − . Acknowledgments
C.H.C would like to think Prof. Hai-Yang Cheng for useful discussions on the long-distanceeffects. C.H.C was supported in part by the National Science Council of R.O.C. under GrantNo. NSC-97-2112-M-006-001-MY3. C.S.K. was supported in part by the NRF grant fundedby the Korea government (MEST) (No. 2010-0028060) (No. 2011-0017430).7
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