D^{0}-D^{0} mass difference from a dispersion relation

SLAC-PUB-10342 LBNL–54319 WIS/05/04-Feb-DPP WSU–HEP–0401 hep-ph/0402204 The D 0 − D 0 mass difference from a dispersion relation arXiv:hep-ph/0402204v1 19 Feb 2004 Adam F. Falk,1 Yuval Grossman,2, 3, 4 Zoltan Ligeti,5 Yosef Nir,6 and Alexey A. Petrov7 1 Department of Physics and Astronomy, The Johns Hopkins University 3400 North Charles Street, Baltimore, MD 21218 2 Department of Physics, Technion–Israel Institute of Technology Technion City, 32000 Haifa, Israel 3 Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 4 Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064 5 Ernest Orlando Lawrence Berkeley National Laboratory University of California, Berkeley, CA 94720 6 Department of Particle Physics Weizmann Institute of Science, Rehovot 76100, Israel 7 Department of Physics and Astronomy Wayne State University, Detroit, MI 48201 Abstract We study the Standard Model prediction for the mass difference between the two neutral D meson mass eigenstates, ∆m. We derive a dispersion relation based on heavy quark effective theory that relates ∆m to an integral of the width difference of heavy mesons, ∆Γ, over varying values of the heavy meson mass. Modeling the mD -dependence of certain D decay partial widths, we investigate the effects of SU (3) breaking from phase space on the mass difference. We find that ∆m may be comparable in magnitude to ∆Γ in the Standard Model. I. INTRODUCTION The mixing and decay of K, B, and D mesons are sensitive probes of physics beyond the Standard Model. Among the many processes that one might study, flavor-changing neutral current D decays and D 0 − D 0 mixing provide unique information, because in the Standard Model (SM) they occur via loop diagrams involving intermediate down-type quarks. In particular, because of severe CKM and GIM suppressions, the mixing of D mesons is expected to be quite slow, and thus the D system is one of the most intriguing probes of new physics in low energy experiments [1]. We begin by recalling the formalism for heavy meson mixing. Using standard notation, the expansion of the off-diagonal terms in the neutral D mass matrix to second order in perturbation theory is given by  M− i Γ 2  12 = 1 X hD 0 |Hw∆C=1 |ni hn|Hw∆C=1|D0 i 1 hD 0|Hw∆C=2 |D 0 i + . 2mD 2mD n mD − En + iǫ (1) The first term represents the ∆C = 2 contributions that are local at the scale µ ∼ mD . It contributes only to M12 , and is expected to be very small unless it receives large enhancement from new physics. The second term in Eq. (1) comes form double insertion of ∆C = 1 operators in the SM Lagrangian and it contributes to both M12 and Γ12 . It is dominated by the SM contributions even in the presence of new physics. Two physical parameters that characterize the mixing are x= ∆m , Γ y= ∆Γ , 2Γ (2) where ∆m and ∆Γ are the mass and width differences of the two neutral D meson mass eigenstates and Γ is their average width. Because of the GIM mechanism the mixing amplitude is proportional to differences of terms suppressed by m2d,s,b /m2W , and so D 0 − D 0 mixing is very slow in the SM [2]. The contribution of the b quark is further suppressed by the small CKM elements |VubVcb∗ |2 /|Vus Vcs∗ |2 = O(10−6 ), and can be neglected. Thus, the D system essentially involves only the first two generations, and therefore CP violation is absent both in the mixing amplitude and in the dominant tree-level decay amplitudes, and will be neglected hereafter. Once the contribution of b quarks is neglected, the mixing vanishes in the flavor SU(3) limit, and it only arises at second order in SU(3) breaking if SU(3) breaking can be treated analytically [3] x , y ∼ sin2 θC × [SU(3) breaking]2 , 2 (3) where θC is the Cabibbo angle. Precise calculations of x and y in the SM are not possible at present, because the charm mass is neither heavy enough to justify inclusive calculations, nor is it light enough to allow a few exclusive channels to give a reliable estimate. According to Eq. (3), computing x and y in the SM requires a calculation of SU(3) violation in the decay rates. There are many sources of SU(3) violation, most of them involving nonperturbative physics in an essential way. In Ref. [3], SU(3) breaking arising from phase space differences was studied; computing them in two-, three-, and four-body D decays, it was found that y could naturally be at the level of one percent. This result can be traced to the fact that the SU(3) cancellation between the contributions of members of the same multiplet can be badly broken when decays to the heaviest members of a multiplet have small or vanishing phase space. This effect is manifestly not included in the OPE-based calculations of D 0 − D0 mixing, which cannot address threshold effects. The purpose of the present paper is to address the following question: if the dominant SU(3) breaking mechanism is indeed the one studied in Ref. [3], and it gives rise to y at the percent level, then can x naturally be comparably large? This is particularly relevant because the present experimental upper bounds on x and y are at the few times 10−2 level [4, 5] and are expected to significantly improve (for a review of the experimental situation, see Ref. [6]). To interpret the results from future measurements of x and y, and possibly establish the presence of new physics, we need to know the allowed range in the SM. In particular, since new physics can only contribute to x, an experimental observation of x ≫ y would imply a large new physics contribution to D 0 − D 0 mixing. Although y is determined by SM processes, its value still affects the sensitivity to new physics [7]. In this paper we study the SM predictions for x/y due to SU(3) breaking from final state phase space differences. In Sec. II we derive a dispersion relation using Heavy Quark Effective Theory (HQET) that relates ∆m to ∆Γ. To compute ∆m, we need a calculation of ∆Γ for varying heavy meson mass, so we review its calculation from Ref. [3] in Sec. III. In Sec. IV, we calculate ∆m and present numerical results. We find that despite the fact that SU(3) breaking in phase space affects x in a different way than it affects y, the final estimates of x and y are comparable. We present our conclusions in Sec. V and discuss the implications of our findings for experimental searches for new physics in D 0 − D0 mixing. 3 q pD D(pD ) ( ) ( ) ( ? q pD 6 ) ( ) ( ) -q D(pD ) FIG. 1: The correlator in Eq. (6). The black boxes denote the weak Hamiltonian, the wavy lines show external momenta inserted, and the gray area represents hadronic intermediate states. II. DERIVATION OF THE DISPERSION RELATION We start by reviewing the relevant formalism for D 0 − D 0 mixing. Equation (1) implies that the mass eigenstates are linear combinations of the weak interaction eigenstates, |D1,2 i = p |D 0i ± q |D 0 i. Since we neglect the effects of intermediate states containing a b quark, |D1,2 i are also CP eigenstates, CP |D± i = ±|D± i. Their mass and width differences are ∆m ≡ mD+ − mD− = 2M12 , ∆Γ ≡ ΓD+ − ΓD− = 2Γ12 . (4) Neglecting the small contribution from the local ∆C = 2 operators, Eq. (1) gives X hD 0 |Hw |nihn|Hw |D 0 i + hD 0 |Hw |nihn|Hw |D 0 i 1 P , 2mD mD − En n i 1 Xh 0 hD |Hw |nihn|Hw |D 0 i + hD0 |Hw |nihn|Hw |D 0 i (2π)δ(mD − En ) , (5) ∆Γ = 2mD n ∆m = where P denotes the principal value prescription, the sum is over all intermediate states, n, and it implicitly includes (2π)3 δ 3 (~pD − p~n ). To derive a dispersion relation between ∆m and ∆Γ, consider the following correlator ΣpD (q) = i Z d4 z hD(pD )| T [Hw (z) Hw (0)] |D(pD )i ei(q−pD )·z . (6) Here pD is a label given by the momentum of the on-shell D meson state (satisfying p2D = m2D ) and q−pD is an auxiliary four-vector that inserts external momentum to the weak interaction (see Fig. 1). There is no simple physical interpretation of Σ except at q = pD , where ΣpD (pD ) is related to physical properties of D mesons. Inserting a complete set of states in Eq. (6) and comparing with Eq. (5), we find −   i 1 ΣpD (pD ) = ∆m − ∆Γ . 2mD 2 (7) The correlator ΣpD (q) is an analytic function of q (but not of pD ) with a cut in the complex q 0 plane for q 0 > q |~q |2 + 4m2π for a fixed ~q. 4 To write the dispersion relation in terms of physical quantities, i.e., to give ΣpD (q) for q 6= pD a physical interpretation, we need to eliminate the heavy quark mass dependence from Eq. (6).1 The momentum of a heavy meson H containing a heavy quark Q can be written as pµH = mH v µ , with v 2 = 1. We can decompose Q as e (Q) (z) + . . . , Q(z) = e−imQ v·z hv(Q) (z) + e+imQ v·z h v (8) e (Q) respectively annihilate a heavy Q quark and create where the HQET fields hv(Q) and h v a heavy Q̄ antiquark with four-velocity v. Here and in the rest of this section the ellipses denote terms suppressed by a relative factor of ΛQCD /mc . The ∆C = 1 weak Hamiltonian contributing to neutral D meson mixing is h i X 4GF −imc v·z (c) imc v·z e (c) ∗ Hw = √ Vcq1 Vuq C O = Ĥ e h + e h + ... , i i w v v 2 2 i (9) where q1,2 = d or s, and the four-quark operators, suppressing their Dirac structures, are of the form e (c) + . . . . Oi ∼ q̄1 q2 ūc = e−imc v·z q̄1 q2 ūhv(c) + eimc v·z q̄1 q2 ūh v (10) In Eq. (9) Ĥw contains the light quark fields, the Wilson coefficients, and summation over operators. We also replace the QCD states |Di by HQET states |H(v)i, |D(p = mD v)i = √ mD |H(v)i + . . . . (11) The new states have a normalization that is independent of the heavy quark mass [9]. Then Eq. (6) yields ΣpD (q) = i mD Z d4 z hH(v)| T h nh e (c) (z) e−imc v·z Ĥw hv(c) (z) + eimc v·z Ĥw h v io e (c) (0) × Ĥw hv(c) (0) + Ĥw h v ei(q−pD )·z |H(v)i + . . . . i (12) e field each, The only nonzero contributions to this correlator involve a single h and h ΣpD (q) = i mD Z n h i e (c) (0) d4 z hH(v)| e−imc v·z T Ĥw hv(c) (z), Ĥw h v h io e (c) (z), Ĥ h(c) (0) + eimc v·z T Ĥw h w v v ei(q−pD )·z |H(v)i + . . . . (13) The two terms in Eq. (13) behave differently in the HQET limit mc → ∞ with q fixed. The term proportional to exp[i(q −pD −mc v)·z] oscillates infinitely rapidly and is integrated 1 The method of using HQET to derive a dispersion relation in the heavy quark mass was developed first in Ref. [8], where it was used to study the inclusive nonleptonic heavy meson decay rate. 5 out at the heavy scale. It should be removed from the effective theory and replaced by a local Hw∆C=2 contribution that can be included as a matrix element of ∆C = 2 operators. Such contributions are estimated to give rise to x and y at or below the 10−3 level [10–12],2 and since we are interested in the question whether x could be near the percent level, we can neglect them. By contrast, the term proportional to exp[i(q − pD + mc v) · z] becomes independent of mc as mc → ∞. Recalling that pD = mD v, we have ΣpD (q) = i mD Z h i e (c) (z), Ĥ h(c) (0) ei(q−Λ̄v)·z |H(v)i + . . . , d4 z hH(v)| T Ĥw h w v v (14) where Λ̄ = mD − mc + O(Λ2QCD /mc ). It is convenient to define Σv (q) = i Z h i e (Q) (z), Ĥ h(Q) (0) ei(q−Λ̄v)·z |H(v)i , d4 z hH(v)| T Ĥw h w v v (15) which is manifestly independent of the heavy quark mass. It follows that ΣpD (q) = mD Σv (q) + . . . , (16) and Eq. (7) becomes to leading order in ΛQCD /mc Σv (q) = −2 ∆m(E) + i ∆Γ(E) , where E ≡ √ (17) q 2 , and ∆m(E) and ∆Γ(E) can be interpreted as the mass and the width differences of neutral heavy mesons with mass E in HQET. Equation (17) shows that Σv (q) only depends on q 2 . Choosing a frame in which ~q = 0, we can use the analyticity of Σv (q) to write a dispersion relation, 1 Z∞ Im Σv (E, ~0) Σv (mD , ~0) = dE . π 2mπ E − mD + iǫ (18) Using Eq. (17), we obtain Z 1 ∆m = − P 2π "  ΛQCD ∆Γ(E) +O dE E − mD E 2mπ ∞ # . (19) Eq. (19) is the main result of this section. It expresses ∆mD in terms of a weighted integral of the width difference of heavy mesons, ∆Γ(E), over varying heavy meson masses, 2 In the OPE-based calculations, because mc /ΛQCD is not very large and subleading terms in the ΛQCD /mc expansion are enhanced by ΛχSB /ms [10], such terms dominate the short distance contribution [10–12]. 6 E. The heavy quark limit was essential in deriving this relation, since Σv (q) has a physical interpretation for arbitrary q, while for q 6= pD , ΣpD (q) does not. The O(ΛQCD /E) error in the integrand is a consequence of our reliance on this limit, and the resulting correction is O(1) in the small E region. Dispersion relations for ∆mD were considered previously in Ref. [13], where Im Σ(s) (with a different definition of Σ) was modeled, but it does not have a physical interpretation for s 6= m2D . To calculate x/y using the dispersion relation, we need to know ∆Γ as a function of the heavy meson mass. Examining Eq. (19), we expect that values of E close to mD give the largest contribution to x. In the next section we recall the calculation of ∆Γ(E) performed in Ref. [3]. If ∆Γ(E) is a decreasing function of E at least as a positive power, 1/E a with a > 0, then the dispersion relation does not require subtraction in order to converge. In the model we consider, ∆Γ(E) actually falls off as ∼ 1/E 2 , and we will argue that some kind of decreasing behavior is likely to hold model independently. III. CALCULATION OF THE LIFETIME DIFFERENCE The computation of x using Eq. (19), requires us to know ∆Γ for a heavy meson of varying mass. The calculation of ∆Γ cannot at present be done from first principles. In Ref. [3] ∆Γ was computed using a simple model in which SU(3) breaking was taken into account in calculable phase space differences, but neglected in the incalculable hadronic matrix elements. This approach was motivated by the fact that phase space differences alone can explain the experimental data in several cases; for example the ratio Γ(D2∗ → Dπ)/Γ(D2∗ → D ∗ π) [14], the large SU(3) breaking in Γ(D → K ∗ ℓν̄)/Γ(D → ρℓν̄) [15], and the lifetime ratio τDs /τD0 [16]. It certainly cannot explain all SU(3) violation, for example, Γ(D → ππ)/Γ(D → KK). The generic conclusion of Ref. [3] was that if multi-body final states close to the D threshold have significant branching ratios, then they can give rise to sizable contributions to ∆Γ that are absent in the OPE-based calculations. Our purpose in the next section will be to see whether the same mechanism can also give rise to x at or near the percent level. Here we review the analysis of Ref. [3]. We denote a set of final states F belonging to a certain representation R of SU(3) by FR . For example, for two pseudoscalar mesons in the octet, the possible representations for F = P P are R = 8 and 27. In Ref. [3] it was shown that yFR , the value which y would take 7 if elements of FR were the only channels open for D 0 decay, can be expressed as y FR P |Hw |nihn|Hw |D 0i =P = 0 0 n∈FR hD |Hw |nihn|Hw |D i n∈FR hD 0 P |Hw |nihn|Hw |D 0 i . 0 n∈FR Γ(D → n) n∈FR hD P 0 (20) The derivation of this relation assumes the absence of CP violation, so that hD0 |Hw |ni is related to hD 0 |Hw |n̄i, and uses the fact that both |ni and |n̄i belong to the same SU(3) multiplet. When the SU(3) breaking in the matrix elements is neglected, Eq. (20) gives a calculable contribution to yFR without any hadronic parameters. The numerator contains a combination of Clebsch-Gordan and CKM coefficients that ensures that yFR is proportional to m2s sin2 θC when the sum over all members of any given multiplet FR is performed, as required by Eq. (3). As an example, the contribution of the multiplet containing two pseudoscalar mesons in an SU(3) octet is given by 2  1 1 Φ(η, η) + Φ(π 0 , π 0 ) + Φ(π + , π − ) + Φ(K + , K − ) 2 2  1 1 + Φ(η, π 0 ) − Φ(η, K 0 ) − 2Φ(K + , π − ) − Φ(K 0 , π 0 ) 3 3  −1 1 1 0 − + 0 0 Φ(η, K ) + Φ(K , π ) + Φ(K , π ) × + O(sin4 θC ) , 6 2 y(P P )8 = sin θC (21) where Φ(n) is the phase space factor for D → n decay. Then y can be computed as the sum of the yFR ’s weighted with the D 0 decay rate to each representation, y= 1X yF Γ FR R  X n∈FR  Γ(D 0 → n) . (22) The yFR were computed for all P P , P V , and V V representations, and for the fully symmetric 3P and 4P final states [3]. The contribution of poles corresponding to nearby K resonances was shown to be small [3, 17]. Assuming that the values of y(4P )R for R = 8, 27, 27′ are typical for all R, it was found that the 4P final states give a contribution to ∆Γ at the percent level. The result is large because many of the decays in question are close to or above threshold, so the SU(3) cancellation in these multiplets is largely ineffective, yielding y(4P )R = O(0.1) [3]. Moreover, the D 0 branching ratio to four pseudoscalars is approximately 10%. We shall now use this model of SU(3) breaking, together with some assumptions about the energy dependences of the relevant decay rates, to compute x/y. 8 IV. CALCULATION OF THE MASS DIFFERENCE The crucial difference between the calculation of x and y is that once we assume that the only source of SU(3) breaking is from the final state phase space differences, the hadronic matrix elements cancel in y, but not in x. As determined by Eq. (19), x depends on ∆Γ(E), and so the E-dependence of the hadronic matrix elements does affect x. Using Eq. (19), we find for x/y, rF R Z xF 1 ≡ R =− P y FR π ∞ 2mπ yFR (E) ΓFR (E) dE . E − mD yFR (mD ) ΓFR (mD ) (23) We will quote our results in terms of rFR . To proceed further we need to understand or make some assumptions about the E-dependence of the decay rate to the final state F , ΓF (E). We define the dimensionless function gF (E) ∝ ΓF (E) , ΓF (mD ) (24) and we will study the E-dependence of this quantity. Note that the constant of proportionality in Eq. (24) cancels in the ratio rFR . Moreover, gF is expected to depend only on the final state F , and not on the SU(3) representation R. One can reconstruct x from xFR using a relation analogous to Eq. (22). Below we calculate rFR for several final states and then estimate the total x. First we will study F = P P , because it is a simple case that is interesting to understand in detail. Then we will turn to F = 4P , because it is the final state that can give y ∼ 1%. A. Two-body D → P P decays For decays to two pseudoscalar mesons, it is possible to develop a reasonable model of gP P (E). When mH ≫ ΛQCD , we may approximate the H → ππ amplitude with its factorized form. Here A(H → ππ) ∼ GF VCKM m2H fπ FH→π , where fπ is the pion decay constant and FH→π is the H → π form factor at q 2 = m2π . It has been shown that, as mH → ∞, FH→π ∝ (Λ/mH )3/2+X [18], where X arises from summing Sudakov logarithms of the form exp[Cαs (mH ) ln2 (mH /Λ)] ∼ (Λ/mH )X with X = −2πC/β0 . Since Γ ∝ |A|2 /mH , we conclude that gP P (E ≫ ΛQCD ) ∝ E −2X . The existing calculations suggest that |X| ≪ 1 [19], so we set X = 0 hereafter. 9 (25) 20 15 rPP 10 5 0 0.2 0.4 0.6 0.8 1 m1 (GeV) FIG. 2: Predictions for r(P P )8 (solid curve) and r(P P )27 (dashed curve) as functions of m1 . In the E → 0 limit our calculation is necessarily unreliable, as the derivation of Eq. (19) relied on HQET. Nevertheless, as a model we will take the behavior of the K → ππ amplitude in chiral perturbation theory. At leading order, this transition is mediated by an operator of the form Tr(∂µ Σ† O ∂ µ Σ), where Σ = exp[2iM/f ] and M is the meson octet. Since this term has two derivatives, it implies that the decay amplitude is proportional to m2K . Since this is the only dependence on mK in the amplitude, the E-dependence of the rate is gP P (E → 0) ∝ E 3 . (26) Based on these considerations, we employ the following simple model for gP P (E)   E 3 /(m21 m2 )    gP P (E) =  E/m2    1 for E < m1 , for m1 < E < m2 , (27) for E > m2 , where m1,2 are free parameters. The overall normalization cancels in the results. This model allows for a “chiral” region, E < m1 , an “intermediate” region, m1 < E < m2 , and a “high energy” region, E > m2 . In our calculations m1 is allowed to vary in the range 0.2−1.0 GeV, and m2 in the range 1.5 − 10 GeV. As we emphasized above, our derivation relies on HQET, so any strong dependence on scales below ∼ 1 GeV would signal an irreducible lack of reliability. In Fig. 2 we plot r(P P )8 (solid) and r(P P )27 (dashed) as a function of m1 , for m2 = 2 GeV. In this case all members of the final state representations are kinematically allowed and have large phase space, so we find that the result is dominated by cancellations below the scale mD . Therefore rP P is sensitive to the shape of gP P (E) at low energies, i.e., the value of m1 , 10 but changing m2 to 3 or 4 GeV has little effect on rP P . Because of the strong dependence on m1 , we should not trust this result. However, since y for these representations is very small, y(P P )8 = −0.018% and y(P P )27 = −0.0034% [3], these final states do not give sizable contributions to x in any case. When we consider decays to the lightest pseudoscalar octet, the dependence of these pseudo-Goldstone boson masses on ms is given by (for mu,d = 0) m2π = 0 , m2K = µms , m2η = 4 µms , 3 (28) where µ is a hadronic scale. We can then expand ∆Γ(E) for large E as  ∆ΓP P (E) = ΓP P (E) ms →0    c1 c2 × c0 + 2 + 4 + . . . . E E (29) Because SU(3) breaking in our approach comes from phase space differences, the coefficients ci depend quadratically on the masses of the final state particles. Since in Eq. (28) ms is always accompanied by µ and ∆Γ must be suppressed by m2s , we conclude that c0 = c1 = 0. The coefficient c2 can be proportional to µ2 m2s and is the leading nonvanishing term, implying a 1/E 4 suppression of ∆ΓP P (E) compared to ΓP P (E). However, the actual π, K, and η masses do not exactly satisfy Eq. (28) in the mu,d = 0 limit, nor the Gell-Mann-Okubo (GMO) relation, 3m2η = 4m2K − m2π . Violating the GMO relation is equivalent to adding a small term to m2K or to m2η of the form ε m2s . This changes the asymptotic behavior of ∆Γ(E), because now we can have c1 ∼ ε m2s . Since the D → P P decay is far from threshold, the SU(3) cancellation in this channel is very sensitive to the pseudoscalar meson masses. This can be verified analytically by expanding Eq. (21). As shown in Fig. 3 (again for m2 = 2 GeV), imposing the GMO relation on the π, K, and η masses decreases rP P significantly, in such a manner that yP P increases by roughly the same factor, while |xP P | is approximately stable at the (5 − 8) × 10−4 level. As discussed in Ref. [3], our results have little sensitivity to including or neglecting π − η − η ′ mixing. By contrast, for final states including vector mesons or heavier pseudoscalar representations, the masses of the mesons depend linearly on ms . Thus, for these final states, ∆ΓF (E)/ΓF (E) is simply proportional to m2s /E 2 for large E, and there is no strong dependence on the precise values of the hadron masses. This is the minimal suppression of ∆ΓF (E)/ΓF (E) consistent with group theory, i.e., Eq. (3), and our phase space model for SU(3) violation indeed gives such an effect. These results imply that the dispersion relation 11 2.5 2 rPP 1.5 1 0.5 0.2 0.4 0.6 0.8 1 m1 (GeV) FIG. 3: Predictions for r(P P )8 (solid curve) and r(P P )27 (dashed curve) as functions of m1 , imposing the GMO relation on the π, K, and η masses. in Eq. (19) converges for any final state F , for which ΓF (E) does not increase as E 2 or faster. This is very likely to be true for all final states (recall that ΓP P (E) ∼ constant for large E). B. Four-body D → 4P decays Now we turn to the 4P final state in the fully symmetric 8, 27, and 27′ representations of SU(3). We know even less about g4P (E) than about gP P (E), so we use two models to attempt to bracket roughly the uncertainties, g4P (E) = gP P (E) and ′ g4P (E) =    E/m1       for E < m1 , 1 for m1 < E < m2 , m2 /E for E > m2 . (30) ′ The choice of g4P (E) allows for the possibility that Γ(H → 4P ) may start to fall for large mH instead of remain constant. This alternative is motivated by the argument that because the quasi-two-body picture holds only in a small part of phase space, in most of the phase space the opening of many decay channels will reduce the rate. The left plot in Fig. 4 shows r(4P )8 (solid curve), r(4P )27 (long dashed curve), and r(4P )27′ (short dashed curve), as functions of m2 , using g4P (E) with m1 = 0.8 GeV. For m1 < 1 GeV > there is no dependence on m1 . The dependence of the curves on m2 is negligible for m2 ∼ ′ (E) instead, shown in the right plot in Fig. 4, then r(4P )R changes 3 GeV. If we use g4P roughly by a factor of two. We have explored other forms of g4P (E) as well, and we find 12 0 0.1 -0.2 0 -0.4 -0.1 r4P -0.6 r4P -0.2 -0.8 -0.3 -1 -0.4 1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 4.5 5 m2 (GeV) m2 (GeV) FIG. 4: Predictions for r(4P )8 (solid curve), r(4P )27 (long dashed curve), and r(4P )27′ (short dashed ′ (E) (right figure) in Eq. (30). curve), as functions of m2 for the models g4P (E) (left figure) and g4P that these two cases cover a reasonable range of predictions. In contrast to D → P P decays, for the 4P final state there is no strong dependence on the π, K and η masses. Because the decay is close to threshold, the dispersion integral is dominated by E near mD , where some of the 4P final states are kinematically forbidden, and so the sensitivity to the pseudoscalar meson masses is reduced. Imposing the GMO relation makes only a small difference; for example, for the (4P )8 representation the value r(4P )8 = −0.98 obtained with the g4P (E) model, m1 = 0.8 GeV, m2 = 3 GeV, and the physical meson masses (corresponding to the solid curve in the left plot in Fig. 4), would change to r(4P )8 = −0.87 if the GMO relation were imposed. V. DISCUSSION AND CONCLUSIONS It is likely that the dominant contributions to the mass and width differences in the D system have a long distance origin in the SM. Therefore, naively one would expect x and y to be of the same order of magnitude. We have derived a new dispersion relation (19) and used it to study this question. Our dispersion relation has the useful property that it relates the mass difference in the heavy neutral meson system at fixed heavy meson mass to the physical width difference of heavy mesons with varying mass. The advantage of using a dispersion relation that relates x to y is that we can use existing models for y to calculate x. Our dispersion relation is likely to converge without any subtraction, because the SU(3) breaking required to yield nonzero mixing introduces 13 an m2s /E 2 suppression in y(E). We have used a model in which SU(3) breaking arises from phase space differences, which may give a reasonable approximation to y(E) only when E is not very large. Since the derivation of the dispersion relation employed the heavy quark limit, it is essential not to interpret our analysis as a precise calculation for x. Instead, we used this model only to get a rough and qualitative prediction about the likely relation of x to y. To make numerical predictions we needed the heavy mass dependence of heavy meson partial widths to certain final states, which introduces some additional model dependence in our results. (For decays to two pseudoscalars, there are limits in which one can draw firmer conclusions about the mass dependence, which we have incorporated into the model.) We calculated the ratio x/y for P P and 4P final states. Our conclusion is that it is indeed likely that in the Standard Model, x is not much smaller than y in the D system. In our numerical study, we found that for the 4P final state, x/y varies roughly between −0.1 and −1. We conclude that if y is in the ballpark of +1% as expected if the 4P final states dominate y [3], then we should expect |x| between 10−3 and 10−2, and that x and y are of opposite sign. This estimate has a large uncertainty, and we can trust it only at the order of magnitude level. We have explored the sensitivity of this qualitative result to a number of the assumptions we have made, and have found that changing the details of the model does not significantly alter our conclusions. Furthermore, including some SU(3) breaking in the matrix elements cancels to some extent in x/y and does not induce dramatic changes. The significance of our result is clear only in the context of the experimental situation. The current bounds on x and y are at the level of a few percent, and the central question is whether their actual observation at or just below this level could be interpreted as a clear signal of physics beyond the Standard Model. We would argue that our analysis has taught us that, without further refinement, the answer is no. We have identified a real effect that could plausibly give x and y at the percent level, albeit with very large uncertainties. In general, an observation of x ≫ y would be an indication for new physics, but this could only be established if y were very small, at the 10−3 level. Such a situation could arise if new physics enhanced x but not y. Yet since one cannot exclude the possibility of cancellation between different SM contributions to y, even this outcome would not admit an unambiguous interpretation. However, if x were indeed enhanced by new physics, such new physics may also introduce 14 a sizable new CP violating phase which may be observable. Thus, we would argue that in D 0 − D 0 mixing, the only single measurement that could establish by itself the presence of new physics would be the observation of CP violation, which is very small in the Standard Model independent of hadronic effects. Acknowledgments It is a pleasure to thank Mark Wise, as usual, for helpful conversations. We are grateful to the Aspen Center for Physics for its hospitality while portions of this work were completed. A.F. was supported in part by the U.S. National Science Foundation under Grant PHY–9970781. Y.G. was supported in part by the Department of Energy, contract DEAC03-76SF00515 and by the Department of Energy under grant no. DE-FG03-92ER40689. Z.L. was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and by a DOE Outstanding Junior Investigator award. The work of Y.G. and Z.L. was also supported in part by the United States–Israel Binational Science Foundation (BSF) through Grant No. 2000133. Y.N. is supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, by EEC RTN contract HPRN-CT-00292-2002, by a Grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development, by the Minerva Foundation (München), and by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. A.P. was supported in part by the U.S. National Science Foundation under Grant PHY– 0244853, and by the U.S. Department of Energy under Contract DE-FG02-96ER41005. [1] G. Burdman and I. 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