Implications of the LHCb discovery of CP violation in charm decays

The recent measurement of $\Delta A_{CP}$ by the LHCb collaboration requires an ${\cal O}(10)$ enhancement coming from hadronic physics in order to be explained within the SM. We examine to what extent can NP models explain $\Delta A_{CP}$ without such enhancements. We discuss the implications in terms of a low energy effective theory as well as in the context of several explicit NP models.


Introduction to ∆A CP
The LHCb experiment has announced discovery of direct CP violation in singly Cabibbo suppressed D decays [1], Here In ∆A CP effects of indirect CP violation approximately cancel out [2]. (Due to different decay time acceptances between the K + K − and π + π − modes, a small residual effect of indirect CP violation remains.) Thus, ∆A CP is a manifestation of CP violation in decay. The updated world average for the direct and indirect CP violating contributions to this asymmetry are [3] ∆A dir CP = (−1.64 ± 0.28) × 10 −3 , A ind CP = (+0.28 ± 0.26) × 10 −3 .
The singly Cabibbo suppressed D 0 (D 0 ) decay amplitudes A f (A f ) to a final CP eigenstate f can be written as [2] where η CP = ±1 is the CP eigenvalue of f , the dominant singly Cabibbo suppressed "tree" amplitude is denoted by A T f e ±iφ T f , and r f parameterizes the relative magnitude of all subleading amplitudes (often called "penguin" amplitudes), which carry different strong (δ f ) and weak (φ f ) phases. Then The Standard Model (SM) contribution to the individual asymmetries is CKM suppressed by a factor of Naively, there is a further loop suppression by a factor of order α s /π ∼ 0.1. One cannot exclude an enhancement factor of order 10 from hadronic physics [4][5][6][7], in which case (3) is accounted for by SM physics. Yet, it is not implausible that new physics (NP) dominates ∆A CP [8,9] (indeed, QCD-based LCSR calculations [10] support the latter option.) In the following we assume that hadronic factors do not significantly alter the magnitude of the relevant effects; Thus, NP is required to explain the measured ∆A CP . We analyze the implications of Eq. (3) on candidate models. We phrase our findings in terms of which NP models can or cannot account for the measurement, assuming that the SM contribution is negligible. Relaxing this assumption, the same implications can be conservatively read as upper bounds on the NP model parameters.
In 2011, experimental evidence for ∆A CP [11] prompted several related studies [4,5,8,[12][13][14]. We provide an update to some of the relevant results, taking into account the recent discovery with a central value smaller by a factor of ∼ 4 as well as all applicable existing bounds.
We begin with an effective field theory (EFT) analysis in Section 2. We follow with specific examples of models in which the measured ∆A CP is explained: 2HDM in Section 3, the MSSM in Section 4 and models with vector-like up-quarks in Section 5. We conclude in Section 6.

Non-renormalizable operators
The relevant effects of new physics at a scale much higher than the electroweak breaking scale can be represented by the following effective Hamiltonian [8]: where q = {d, s, b, u, c}, the list of operators includes and the primed operators are related to the non-primed ones via A ↔ −A and γ 5 ↔ −γ 5 . The SM and NP contributions to ∆A CP can be parameterized as where ∆R SM,NP = R SM,NP K + R SM,NP π , and R SM,NP K are the ratios of subleading amplitudes to the leading SM amplitude, after factorizing out the CKM dependence and the Wilson coefficient (the loop factor for R SM K ). Thus the SM alone can explain the measured value of ∆A CP for Im(∆R SM ) ≈ 13. In the following we conversely adopt the naive expectation, Im(∆R SM ) ∼ Im(∆R NP ) ∼ 2 (the factor of 2 is inspired by the U-spin limit, in which A SM K + K − ≈ −A SM π + π − .) With this assumption, the measurement requires the existence of NP with a Wilson coefficient satisfying and the scale of NP can naively be estimated as ∼ < 37 TeV.

Constraints from
The contributions of H eff |∆c|=2 to D 0 − D 0 mixing are computed using the following formula: where all relevant parameters and hadronic matrix elements are defined in Ref. [15].
Using the up-to-date 95% C.L regions for the mixing parameters [3], we obtain the following bounds: Following Ref. [8], we can relate the two sets of Wilson coefficients via We then change basis to Q s−d i , and take the counter-terms to zero to arrive at the bounds on the ∆c = 1 operators, presented in Table 1. We conclude that the operators Q

Constraints from /
Following Ref. [8], we use the master formula for / , evaluating the matrix elements induced by the |∆s| = 1 operators at the large N c limit. The NP contribution is then given by NP ≈ 10 2 Im 3.5C i , and ρ = m K /m s . Taking the conservative bound | / | NP < | / | exp ≈ 1.7 × 10 −3 , the imaginary parts of the |∆s| = 1 Wilson coefficients are constrained. These are related to the |∆c| = 1 coefficients of interest via The resulting bounds on the |∆c| = 1 Wilson coefficients are presented in Table 2. Comparing these bounds to Eq. (11), we conclude that the operators Q Im(C We note that the set of operators, }, are relevant to neither D 0 − D 0 mixing nor | / |, and therefore are unconstrained. Table 3 summarizes which ∆c = 1 operators can contribute to ∆A CP at a level comparable to the current measured value. Table 3: Classification of new physics operators Q i according to whether upper bounds on Im(C NP i ) from D 0 −D 0 mixing and / are (i) much weaker than 9×10 −5 ("allowed"), (ii) of order 9×10 −5 ("marginal"), or (iii) much stronger than 9 × 10 −5 ("disfavored").

Allowed
Marginal Disfavored

2HDM
As a first example of an explicit NP model that can account for the measurement of ∆A CP , we consider a two-Higgs-doublet model (2HDM), where a second scalar doublet, is added to the SM. A contribution to ∆A CP arises if φ 0 couples to uū and cū, generating both D 0 → K + K − and D 0 → π + π − . Since all couplings besides uū and cū are irrelevant to this analysis, we take a conservative approach, considering minimal examples where Φ couples to u R and is aligned with a single down-type LH mass eigenstate. This allows us to evade tree-level scalar mediated FCNC in the down sector. Assuming alignment with the quark doublet that has b L as its down-type quark, we have [12] where U L1,2,3 = u L , c L , t L . Thus, the neutral scalar φ 0 couples u R to u L and c L : λV cb φ 0c L u R + λV ub φ 0 uūL u R . Integrating out the φ 0 field, these couplings lead to the effective four-quark coupling.
The contribution to ∆A CP , using Eq. (10), can be written as and I CKM is defined in Eq. (7). What is needed then to account for (3) is Thus, for Im(∆R φ ) ∈ {0.2 − 2}, we need G The scalar exchange contributes to D 0 − D 0 mixing via box diagrams. Requiring that this contribution is not larger than the experimental constraints from ∆m D gives [12] |λ| 4 32π 2

GeV
or, equivalently so, taking into account (25), the new contribution is negligible, allowing for the required G 0 /G F to explain ∆A CP .

Constraints from /
The same Yukawa couplings of φ 0 that contribute to direct CP violation in D decays, contribute unavoidably also to direct CP violation in K decays. The former effect comes at tree level and modifies ∆A CP . The latter effect comes via box diagrams, involving φ 0 and a W -boson, and modifies / . Upon integration out of φ 0 and W , we obtain the following effective four-quark coupling: where x φ ≡ m 2 φ 0 /m 2 W , and the loop function is given by , we read off the corresponding Wilson coefficient, Following Ref. [16], we use and where ∆C i = C u i − C d i . At the matching scale, our model generates ∆C 6 (m φ 0 ) = C u 6 (m φ 0 ), and ∆C 5 (m φ 0 ) = 0. Taking the conservative bound Re( / ) φ < Re( / ) Exp ≈ 1.66 × 10 −3 , we reach the constraint 'C u(ds) 6 (m φ 0 ) < 2.23 × 10 −7 .
(33) Figure 1 presents the various constraints together with curves for which Eq. (25) is satisfied with three representative values taken for Im(∆R φ ). We conclude that ∆A CP can be explained within this model, depending on the value of Im(∆R φ ). For Im(∆R φ ) ≈ 1, the mass of the neutral scalar is bounded to be m φ ∼ < 235 GeV, while for Im(∆R φ ) ≈ 0.2 it is bounded to be very light, and subject to further constraints. For Im(∆R φ ) 1.5, the mass is unconstrained.
We note the following points: • Two alternative choices for the Yukawa matrices such that only one down-type mass eigenstate is involved exist, with Φ aligned with the doublet containing either d L or s L . These suffer from large contributions to D 0 − D 0 mixing, and therefore cannot account for ∆A CP .
• It may seem surprising that this model can account ∆A CP even though it contributes via the operator Q c−u 6 , disfavored by the EFT analysis. This is explained by the existence of additional contributions within this model to / , which interfere destructively. These are not taken into account in the EFT approach. Therefore this model evades the EFT conclusions regardless of the mass scale of the new scalars.
• We note that mid-range masses for the charged scalar (450 GeV ∼ < m φ − ∼ < a few TeV) are constrained by LHC dijet searches [17][18][19]. These would result in a further constraint in the (|λ|, m φ 0 ) plane, depending on the mass splitting between the neutral and charged scalars. Charged scalar masses below 450 GeV or above a few TeV are not constrained by these bounds.

MSSM
As a second example for candidate NP models to explain the measurement of ∆A CP , we consider the MSSM. The dominant supersymmetric contribution to ∆A CP is likely to come from loops involving gluinos and up-squarks. These contribute to the chromomagnetic operators Q 8 and Q 8 , which are very weakly constrained by D 0 − D 0 mixing and / . The dominant source of CP violation is likely to be the chirality-changing and flavor-changing mass-squared insertion [13], wherem 2 is the average up-squark mass, andM 2u LR is the left-right block in the 6 × 6 up-squark masssquared matrix. In the approximation that only two squark generations are involved, we can express this parameter in terms of the supersymmetric mixing angles, (K u L,R ) ij and the mass-squared splitting between the squarks, ∆m 2 ij : One can estimate the supersymmetric contribution as [13] ∆A CP = 1.5 × 10 −3 Im(δ LR ) 2.5 × 10 −4 1 TeṼ m × Im(∆R SUSY ).
Thus in order to explain ∆A CP we require In MFV models [14], and the contribution is negligible. In Froggatt-Nielsen (FN) models [14,20], whereã is the typical scale of the trilinear scalar coupling. When comparing Eq. (39) to Eq. (37), it seems that FN-SUSY models are plausible candidates to account for ∆A CP . One has to take into account, however, the FN relations with other entries of the squark mass-squared matrices, and, in particular, Assuming phases of order one (which we do to explain ∆A CP ), the flavor-diagonal parameters are bounded by the EDM constraints. The resulting bounds are [14] ( Comparing to Eq. (36), we see that within FN, Im(∆R SUSY ) 3 is required in order to explain ∆A CP . In more elaborate flavor schemes (as in, for example, Ref. [21]) it is possible that Eq. (37) is satisfied for Im(∆R SUSY ) ≈ 2.

Vector-like quarks
The relevant coupling for ∆A CP is U u cu , which also contributes at tree level to ∆m D , and at loop level to / .

Constraint from D 0 − D 0 mixing
The constraint from ∆m D can be calculated using the effective operators of Ref. [8]. The relevant ∆c = 2 operator is (ū L γ µ c L ) 2 = 1 4 Q cu 1 . Using Eq. (16) for the current bound on Re(C cu 1 ), we arrive at

Constraints from /
A contribution to / arises via a W -loop, inducing the operators Q u(ds) 1,5 We calculate the relevant Wilson coefficients and arrive at Using Eq. (18), the constraint on these coefficients is given by The constraint on C (c−u)(ds) 5 is more stringent, implying ∆A CP arises in this model through the tree level annihilation diagramcu →ūu, which contributes to the ∆c = 1 four quark operators, The coefficients of these operators in this model are given by Using Eq. (10), the contribution to ∆A CP can be written as when we have taken Im(∆R Z 1 ) ≈ Im(∆R Z 5 ) ≡ Im(∆R Z ). Thus in order to explain the measurement we require which (under the assumption of Im(∆R Z ) ≈ 2) is allowed by Eqs. (43,46). We note that this model is viable despite the fact that it induces the EFT-disfavored operator Q (c−u) 5 (see Table 3), as its contibution to ∆A CP is subleading to that of the operator Q (c−u) 1 .

Discussion
We have addressed the question of how easily can the new measurement of ∆A CP be explained using benchmark NP models. We have followed the assumption that no significant hadronic enhancements are present, and derived the constraints coming mainly from measurements of D 0 − D 0 mixing and / . We find that non-generic though still simple NP models can account for the measured asymmetry. Three candidate NP models were discussed -2HDM, MSSM and vector-like up-quarks. Our assumption of no significant hadronic enhancements is implemented by allowing at most Im(∆R SM,NP ) ≈ 2, in our Eq. (10). We find that: • Both a 2HDM where scalar (cū), (uū) couplings are present and models with vector-like up-quarks inducing (cū) Z couplings can account for the measured asymmetry.
• The MSSM combined with flavor frameworks (MFV, FN) is unable to produce the desired contribution (FN requires Im(∆R FN ) 3). The MSSM with a generic flavor structure is unconstrained.
Ref. [6] studied the scenario where the SM accounts for ∆A CP with mild SU (3) breaking effects but a strong enhancement of ∆U = 0 transitions. They obtain two predictions: U -spin invariant strong phases should be large, and A CP (K + K − ) ≈ −A CP (π + π − ). Interestingly, in all three models that we analyzed the new physics operators that account for ∆A CP do not introduce new sources of U -spin breaking, and thus the latter prediction does not favor the SM over these models.
In all three specific new physics models, the flavor structure is not in the minimal flavor violation class, and in fact it is non-generic. Thus, it is difficult to make definite predictions for the modification of other flavor changing and/or CP violating processes. Yet, it is unlikely that the only significant modification would be to singly Cabibbo suppressed charm decays. This situation motivates a broad flavor precision program, such as in the LHCb and BELLE-II experiments.
Of course, a direct search for the new degrees of freedom required by the various models is also well motivated. The upper bound on the scale of new physics is model dependent, and varies from few tens of TeV in the low energy EFT, to hundreds of GeV in the 2HDM.