Impact of colored scalars on $D^0-\bar D^0$ mixing in diquark models

Inspired by the recent observation of the $D^0-\bar D^0$ mixing, we explore the effects of colored scalars on the $\Delta C=2$ process in diquark models. As an illustration, we investigate the diquarks with the quantum numbers of (6, 1, 1/3) and (6, 1, 4/3) under $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge symmetries, which contribute to the process at one-loop and tree levels, respectively. We show that $\Delta m_D$ gives the strongest constraint on the free parameters. In addition, we find that the small couplings can be naturally interpreted by the suppressed flavor mixings if the diquark of (6, 1, 4/3) only couples to the third generation.

where x D ( y D ) denotes the mass (lifetime) difference parameter and y DCP is the mixing parameter including the CP violating information. That is, if no CP violation is found in the D-meson oscillation, we have y DCP = y D . Due to these data, lots of studies on the physics beyond Standard Model (SM) have been done [14][15][16][17][18][19][20]. In particular, the possible extensions of the SM for the D 0 -D 0 mixing have been investigated by the authors in Ref. [21] in great detail.
In this note, we explore the issue in scalar diquark models which were not included in the previous discussions [21]. In the literature, the motivation to study the light colored scalar could be traced to the solution for the strong CP problem [22], where to avoid the domain-wall problem on the spontaneous CP violating mechanism, the models were constructed in the framework of grand unified theories (GUTs), e.g. SU(5) gauge symmetry. The associated new source of CP violation on K and B systems was also studied in Refs. [23,24]. In addition, since the scalar sector in the SM has not been tested experimentally, it is plausible to assume the existence of other possible scalars in the gauge symmetry of Accordingly, the general scalar representations could be (1, 2) 1/2 , (8, 2) 1/2 , (6, 3) 1/3 , (6, 1) 4/3,1/3,−2/3 , (3, 3) −1/3 , (3, 1) 2/3,−1/3,−4/3 , where the first (second) argument in the brackets denotes the representation in color (weak isospin) space and the number in the subscript corresponds to the hy- [25]. Besides the SM Higgs doublet, it has been shown in Ref. [25] that when the hypothesis of minimal flavor violation (MFV) is imposed, only the representation (8, 2) 1/2 could avoid FCNCs at tree level. As a result, due to the suppression of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the masses of light quarks, the loop induced D 0 -D 0 oscillation in the color octet model is also negligible. Therefore, in the following analysis, we will concentrate on the situation of color triplet and sextet. In terms of involved Feynman diagrams, we find that m D can be produced by the diquark models through both box and tree diagrams. The various possible scalar diquarks are presented in Table 1 Open access under CC BY license.
Open access under CC BY license.  Table 1 Various diquark models for the D 0 -D 0 mixing with = iτ 2 .

Model
H Interaction Flavor symmetry Diagram Due to the antisymmetric property in flavor indices, Model (6) cannot contribute to the C = 2 process at tree level. It is worth mentioning that the colored sextet scalar also exists in a class of partial unification theories based Since our purpose is to show the influence of diquarks on the D 0 -D 0 oscillation, we are not planning to calculate the contributions of each model shown in Table 1. For comparison, we use Models (4) and (7), which have the same color structure, to illustrate the diquark effects. It is expected that the contributions of other models should be similar in order of magnitude.

Diquark (6, 1, 1/3)
We first write the interaction of quarks and the diquark of (6, 1, 1/3) as where f ij denote the couplings of diquark and various flavors, is the chiral projection operator and H αβ 6 is a weak gauge-singlet and colored sextet scalar with α and β being the color indices.
After Fierz transformation, since the structure of four-fermion interactions becomesd i Γ d jcm Γ c n , obviously lifetime and mass differences in the D 0 -D 0 mixing cannot be induced at tree level. However, they can be produced at one-loop where the box diagrams are sketched in Fig. 1.
To formulate the C = 2 effective Hamiltonian, we set the flavor indices in Fig. 1 to be k = = c and j = n = u. Hence, by including Wick contractions and neglecting the external momenta and the internal masses of light quarks, the effective Hamiltonian for Fig. 1 is written as with C D = i f * ic f iu . Here, we have used the propagator for the sextet scalar as (4) Fig. 2. Diquark-mediated flavor diagram for D + → π + (π 0 , φ).
With the loop integral (3) can be expressed as Using the transition matrix element given by with m D = 2|M 12 | = 2| D|H C =2 |D |. Taking τ D = 4.10 × 10 −13 s, m D = 1.86 GeV, f D = 0.222 GeV [28] and B D = 0.82 [29], in order to fit the current experimental data, the unknown parameters should satisfy With m H ∼ 1 TeV and x D ∼ 8 × 10 −3 , the free parameter |C D | is about 0.06. Besides a serious constraint on the free parameters from m D , for comparison, we consider other possible limits from rare nonleptonic D decays, such as D + → π + π 0 and D + → π + φ decays, which are Cabibbo-suppressed and Cabibbo and color-suppressed processes in the SM, respectively. Their current measurements are [28] B D + → π + π 0 = (1.24 ± 0.07) × 10 −3 , According to the interactions in Eq. (2), flavor diagrams for D decays are given in Fig. 2 with q = d and s. Based on the decay constants and transition form factor, defined by 0 q γ μ γ 5 with P = p 1 + p 2 and k = p 1 − p 2 , the decay amplitudes by the naive factorization approach for D + → π + (π 0 , φ) are given by Here, we have taken the approximation of m 2 π ≈ 0 and set N c = 3.
As a result, the branching ratios (BRs) are known as For simplicity, we use the central values of the data to obtain the upper limits of the parameters. With f π = 0.13 GeV, f φ = 0.237 GeV and f + (0) = f 0 (0) = 0.624 [30], we get By comparing with Eq. (8), we see clearly that unless there exist strong cancelations among the free parameters, the constraints from the D 0 -D 0 mixing are much stronger than those from D decays. By examining Fig. 1, it is easy to find that down-type quarks involve in the internal loop. In other words, the K 0 −K 0 mixing, denoted by m K , might give a strict constraint on the parameters. To understand the influence of m K , by the similar calculations on m D , the formula is given by with C K = j=u,c,t f sj f * dj , where we have set m t m H . Clearly, although some parameters such as f sc f * dc and f su f * du appear in both m D and m K , in general the C K and C D are different parameters. Adopting this viewpoint, the constraint from m K might not have an influence on the constraint from m D . Nevertheless, in some special case, C K and C D are strongly correlated. Therefore, it is interesting to survey both constraints in a little bit of detail. Hence, according to Eqs. (7) and (15), we get With the data of m D = x D Γ D ∼ 0.008Γ D ≈ 1.3 × 10 −14 GeV and m K = 3.483 × 10 −15 GeV [28], the constrained relation from D and K systems is Clearly, the result implies that when |C K | and |C D | are not regarded as independent parameters, both m K and m D will give similar constraints on the free parameters.
However, one can find that the behavior of couplings f ij is symmetric in flavors. To illustrate the result, the derivation is given as follows: where we have used H H ūγ μ P R cūγ μ P R c +ū β γ μ P R c αūα γ μ P R c β . (20) Using the transition matrix element of Eq. (6), we find Consequently, the mixing parameter of the D 0 -D 0 mixing is From the values taken previously, the bound on the free parameters is found as If we adopt m H ∼ 1 TeV and x D ∼ 8 × 10 −3 , the constraint on the parameter is | Re( f cc f * uu )| ∼ 5.76 × 10 −7 . In particular, for f cc ∼ f uu one gets | f cc | ∼ | f uu | ∼ 7.6 × 10 −4 .
At the first sight, it seems that the small couplings are finetuned. However, the smallness in fact could be related to the suppressed flavor mixings. To demonstrate the conjecture, we propose that before the electroweak symmetry breaking, the scalar diquark only couples to one flavor, such as the top quark. Accordingly, the interaction is set to be