A $U$-Spin Anomaly in Charm CP Violation

Recent LHCb data shows that the direct CP asymmetries of the decay modes $D^0\rightarrow \pi^+\pi^-$ and $D^0\rightarrow K^+K^-$ have the same sign, violating an improved $U$-spin limit sum rule in an unexpected way at $2.1\sigma$. From the new data, we determine for the first time the imaginary part of the CKM-subleading, $U$-spin breaking $\Delta U=1$ correction to the $U$-spin limit $\Delta U=0$ amplitude. The imaginary part of the $\Delta U=0$ amplitude is determined by $\Delta a_{CP}^{\mathrm{dir}}$. The corresponding strong phases are yet unknown and could be extracted in the future from time-dependent measurements. Assuming $\mathcal{O}(1)$ strong phases due to non-perturbative rescattering, we find the ratio of $U$-spin breaking to $U$-spin limit contributions to the CKM-subleading amplitudes to be $(173^{+85}_{-74})\%$. This highly exceeds the Standard Model (SM) expectation of $\sim 30\%$ $U$-spin breaking, with a significance of $1.95\sigma$. If this puzzle is confirmed with more data in the future, in the SM it would imply the breakdown of the $U$-spin expansion in CKM-subleading amplitudes of charm decays. The other solution are new physics models that generate an additional $\Delta U=1$ operator, leaving the $U$-spin power expansion intact. Examples for the latter option are an extended scalar sector or flavorful $Z'$ models.

An improved version of the sum rule Eq. (1.5) is given as [35,42,50,53] . (1.6) The sum rules Eqs. (1.5, 1.6) belong to a category of U -spin sum rules which are based on the complete interchange of s and d quarks [56][57][58]. Inserting the experimental measurements listed in Table I below, we obtain Γ(D 0 → K + K − ) Γ(D 0 → π + π − ) = 2.81 ± 0.06 (1.7) and − a dir CP (D 0 → π + π − ) a dir CP (D 0 → K + K − ) = −3.01 +0.95 −5.95 , (1.8) i.e. altogether The improved U -spin limit sum rule Eq. (1.6) is broken at 2.1σ, because Eq. (1.9) has the "wrong" sign. While U -spin breaking is expected, because U -spin is only an approximate symmetry of QCD, the amount of breaking goes beyond the Standard Model expectations In this article, we analyze the implications of the new charm CP measurements in more detail, extracting the CKM-subleading ∆U = 1 contributions to the amplitudes of D 0 → K + K − and D 0 → π + π − decays. In the SM these are generated from the tensor product of the U -spin limit ∆U = 0 operator with the U -spin breaking triplet operator [25,59].
After briefly reviewing the application of SU(3) F methods in charm decays in Sec. II, we summarize our notation in Sec. III. In Sec. IV we recapitulate how to completely solve the system of two-body D 0 decays to kaons and pions. We also show explicitly how to extract in principle the strong phases of the CKM-subleading ∆U = 1 and ∆U = 0 hadronic matrix elements from time-dependent CP violation. In Sec. V we present our numerical results. Finally, in Sec. VI we give predictions and options for interpretations in terms of new physics models that can be tested with future and more precise data. We conclude in Sec. VII.

II. REVIEW OF SU(3) F -BREAKING IN CHARM DECAYS
The application of SU(3) F methods in particle physics have their roots in spectroscopy, namely the "eightfold way" for the description of the spectrum of the meson and baryon octets [60,61]. In spectroscopy, SU(3) F has proven to be an extremely useful ordering principle. For example, SU(3) F -limit predictions agree with the baryon octet mass splitting with an accuracy of 10% [62]. Furthermore, the Gell-Mann-Okubo mass formula [60,63] demonstrated that by including SU(3) F -breaking effects in a systematic way the precision of predictions can be significantly improved. We know therefore that SU(3) F is a very trustable technique for the particle spectrum. The question is if the same applies also to decay rates, in particular for charm decays.
The nominal size of SU(3) F -breaking for decay amplitudes can be estimated from the ratio of the decay constants [64] f In the strict SU(3) F limit, neglecting also differences from phase space effects, we have: Coming now to the second example, as B(D → K S K S ) vanishes in the SU(3) F limit, we can estimate the corresponding amplitude-level SU(3) F breaking roughly as and, due to Bose symmetry, see e.g. Ref. [37] A We note that if one would adopt additional theory assumptions in terms of a 1/N c power counting [28,67] on top of the SU(3) F expansion, in the 1/N c limit one can factorize the tree amplitude of non-leptonic charm decays, see e.g. Ref. [38]. However, in this case the factorizable U -spin breaking of the tree amplitudes alone does not suffice in order to explain the SU(3) F breaking in Eq. (2.2) [68]. In the topological diagram approach, besides the tree amplitude T , the branching ratios of D 0 → K + K − and D 0 → π + π − depend also on exchange diagrams E and SU(3) F -breaking combinations of penguin contractions of the tree operator P break , see the parametrizations in Refs. [32,38]. Therefore, under the assumption of a 1/N c power counting, in order to explain Eq. (2.2), additional contributions to the SU(3) F -breaking have to come from these contributions. At first glance this seems counterintuitive, as E and P break are formally 1/N c suppressed relative to T , which would also affect the possible amount of SU(3) F breaking. However, there are two contributions to these respective topological diagrams, which are both suppressed by 1/N c , and which stem from the Hamiltonian [38] where we use C 2 = 1.2 and C 1 = −0.4 [38]. The estimate Eq. (2.14) reproduces the measurement Eq. (2.7) up to 8%. At the same time, the fit in Ref. [38] finds that the large SU(3) F -breaking exchange diagrams together with the broken penguin can also explain Eq. (2.2).
Global fits in the pure group-theoretical approach [35] agree with the approach employing topological diagrams [38] in that the maximal needed linear SU(3) F breaking in the CKMleading amplitudes is given as ε ∼ 30%.
We can test the SU(3) F expansion also beyond linear order breaking effects. For the ratio The experimental branching ratio measurements give seek to test the validity of U -spin at every possible opportunity.
Below, from recent data, we identify a new puzzle that appears in the CKM-subleading amplitude contributions to charm decays, as opposed to the CKM-leading contributions discussed above.

III. NOTATION
We use the notation of Ref. [11] which we shortly summarize in this section. In the SM, the Hamiltonian of SCS charm decays has the U -spin structure Amplitudes of SCS charm decays can then be written as We use the following parametrization of U -spin related two-body D 0 decays to kaons and pions [11,32] The amplitudes are normalized such that B(D → P P ) = |A| 2 · P(D, P, P ) , (3.9) and [29,36,53] a dir CP = Im Furthermore, we write the amplitudes without CKM factors as A(f ) for CF and DCS decays and Following Ref. [11] we also use the observable combinations (3.14) The strong phase between CF and DCS D 0 decays is defined as For convenience, we define the strong phases δ KK and δ ππ slightly different from Ref. [11] as

IV. SOLVING FOR UNDERLYING THEORY PARAMETERS
In the convention of Ref. [11], the parametrization Eqs. (3.4)-(3.7) has the following eight real parameters, not counting the normalization t 0 : Re(t 1 ), Im(t 1 ),t 2 ,s 1 , Re(p 0 ), Im(p 0 ), Re(p 1 ), Im(p 1 ). (4.1) We can solve the complete system to order O(ε 2 ) as follows [11] Re(t 1 ) = −R Kπ , (4.2) Note that tan δ Kπ ≈ δ Kπ . Furthermore, we have to O(ε 2 ): The parameters Re(p 0 ) and Re(p 1 ) can be determined from time-dependent measurements. In the following, we write the equations for Re(p 0 ) and Re(p 1 ) in a more convenient form in terms of the phases cot δ KK and cot δ ππ . These are related to the parametrization Eqs. (3.4)-(3.7) as and can be obtained from the subleading, non-universal contributions to the time-dependent CP violation observable ∆Y f , where [6] A and to very good precision [6] Here, x, y, |q/p| and φ are the parameters of D 0 − D 0 mixing, see Refs. [6,21,70,71] for details. Rearranging Eq. (4.13), we extract cot δ f from ∆Y f as (4.14) In terms of cot δ KK , cot δ ππ , ∆a dir CP and Σa dir CP we obtain the following expressions for Re(p 0 ) and Re(p 1 ) to order O(ε 2 ):  Tables I, II and apply Eqs. From time-dependent CP violation we obtain for the strong phases  [9]. However, like ∆Y K + K − and ∆Y π + π − , they do not yet show a significant final-state dependence.
Our results for all parameters in Eq. (3.8) are given in Table III. As a result of Eq. (5.4, we have basically no information on the real parts Re(p 0 ) and Re(p 1 ). We will therefore not include them in the discussion any further.
With future data on time-dependent CP violation this assumption can be tested and improved. From Eq. (4.10), it follows then for the ratio of the magnitude of the U -spin breaking contribution to the CKM-subleading amplitude A b (ππ) (A b (KK)) to the corresponding Uspin limit contribution: Note that also Eq. (4.10) is formally valid at O(ε 2 ) and is still valid at O(ε) when Im(p 1 ) ∼ O(1). The above implies that for Im(p 1 ) ∼ O(1) the methodology of Sec. IV still enables a consistent parameter extraction with the exception of the parameter Re(p 0 ). However, with current data we have in any case no sensitivity to this parameter. As can be seen from Table III,  . For this illustration we fix ∆a dir CP and R KK,ππ to their central values and vary Σa dir CP away from its measured value (blue). In red we show the current experimental data for Σa dir CP and the resulting value for 1/2 Im(p 1 )/Im(p 0 ) including 1σ errors. For the estimate of the region of 30% U -spin breaking (yellow) we assume that the strong phases are O(1), such that |1/2 Im(p 1 )/Im(p 0 )| ≈ 1/2|p 1 |/|p 0 | ≤ 30%.

VI. PREDICTIONS AND NEW PHYSICS INTERPRETATIONS
The large U -spin breaking of (173 +85 −74 )% that we find in Eq. (5.5) indicates large contributions from ∆U = 1 operators in the CKM-subleading amplitude of SCS charm decays.
This leads to an O(1) breaking of the U -spin limit sum rule [35,42,50,53] see the discussion in Sec. I. As the decays D 0 → K + K − and D 0 → π + π − are also connected to a wider class of decays via SU(3) F symmetry, we expect that the U -spin limit sum rule [35,42,50,53] is also broken at O(1). In Ref. [39] improved versions of the sum rules Eqs. (6.1, 6.2) are formulated that account for the first order SU(3) F breaking effects from all topological diagrams except for the penguin contraction of tree operators (P and P A in the notation therein). Therefore, we predict that also the sum rules [39] S( = 0 (6.4) are broken at O(1). Here, δ(d) ≡ arg(A Σ (d)) and the function S(d) can be found in Ref. [39].
Further SU(3) F sum rules are given in Refs. [50,79]. For a general treatment of U -spin sum rules at any order see Ref. [25]. In light of the puzzle posed by the U -spin expansion of charm decays, also a further test of the respective isospin structure is very important [55].
Recently, in Ref. [20] it has been shown that Z models can induce large U -spin breaking between a dir CP (D 0 → K + K − ) and a dir CP (D 0 → π + π − ), depending on the charge assignments of the quarks under an additional U (1) group. With the new data, charm CP asymmetries can be used effectively to probe and explore the parameter space of such models further.
For example, the specific charge assignments of Z models considered in Ref. [20] lead to opposite signs for a dir CP (D 0 → K + K − ) and a dir CP (D 0 → π + π − ), see Fig. 3 therein, whereas the most recent data indicates a dir CP (D 0 → K + K − ) > 0 and a dir CP (D 0 → π + π − ) > 0. The exploration of the U -spin puzzle with future and more precise measurements including sum rule tests is important for a complete understanding of the CKM-subleading amplitudes of SCS charm decays and in order to further probe the parameter space of ∆U = 1 models.

VII. CONCLUSIONS
Assuming the Standard Model, from recent measurements of charm CP violation in the single decay channels D 0 → K + K − and D 0 → π + π − we extract for the first time the imaginary part Im(p 1 ) of the U -spin breaking ∆U = 1 contribution to the CKM-subleading amplitudes. We obtain 1/2 Im(p 1 ) Im(p 0 ) = (−173 +74 −85 )% , (7.1) where Im(p 0 ) is the U -spin limit ∆U = 0 contribution to the CKM-subleading amplitudes which is determined by ∆a dir CP . The strong phases ofp 0,1 are yet unknown. Assuming O(1) strong phases due to non-perturbative rescattering, the result implies very large U -spin breaking, which exceeds the SM expectation of ∼ 30% by almost a factor six, at 1.95σ.
It is crucial to probe this anomaly further with more data and test the U -spin expansion also in additional decays, using the sum rules listed in Sec. VI. Most importantly, we need improved time-dependent measurements, such that we can extract the strong phases ofp 0,1 from data. In order to test the pattern of the SU(3) F expansion, measurements of CP asymmetries of basically all singly-Cabibbo suppressed decays are necessary.
We encourage experimental collaborations to extract the underlying theory parameters using the methodology described in Sec. IV directly from the data, enabling the most comprehensive treatment of all correlations.
If the U -spin anomaly is confirmed with more data in the future, this would imply either a breakdown of the U -spin expansion in the Standard Model, or a sign for new physics with an additional ∆U = 1 operator, for example from additional scalar particles or a flavorful Z .