Anomalously large O 8 and long-distance chirality from A CP [ D 0 →( ρ 0 ,ω)γ](t)
EEdinburgh/12/16CP -Origins-2012-23DIAS-2012-24 Anomalously large O and long-distance chirality from A CP [ D → ( ρ , ω ) γ ]( t ) James Lyon a,b, & Roman Zwicky a,b, a School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ,Scotland b School of Physics & Astronomy, University of Southampton, Highfield, SouthamptonSO17 1BJ, UK
Abstract:
A large CP-asymmetry ∆ A CP has been reported in the D → π + π − /K + K − system. Atpresent it remains unclear whether this is due to incalculable strong interaction matrixelements or genuine new physics (NP). Amongst the latter a new weak phase in thechromomagnetic operator O has emerged as a promising candidate. Extending earlierideas we show that the interference of long-distance (LD) terms with the O matrixelement, which has a large strong phase, gives rise to direct CP-violation at the level ofa few percent in D → ( ρ , ω ) γ and D +( d,s ) → ( ρ + , K ∗ + ) γ for reference values Im[ C NP ] (cid:39) . · − . This is two orders of magnitude above a Standard Model (SM) estimate. Thecontribution of Im[ C NP ], which is dependent on the model of NP, is governed by theLD strong phase which vanishes in the chiral limit at leading order. The question ofwhether this is significantly changed by radiative corrections is an open and interestingquestion that we discuss. Furthermore we point out that the relative size of left- andright-handed (photon polarisation)-LD amplitudes can be measured, in principle, throughtime-dependent CP (TDCP) asymmetries in the case where they are both sizeable whichis supported by SM estimates. Thus determination of the latter provides interestinginformation on the LD-chirality independent of NP. We comment on the origin of theLD contribution, which we believe to be dominated by weak annihilation (WA), in theappendix. [email protected] [email protected] a r X i v : . [ h e p - ph ] O c t Introduction A CP in D → ππ/K K Recent LHCb [2] and CDF [3] data both report significant evidence for CP-violation inthe charm system. The combined value leads to a world average [3]∆ A CP = A K + K − CP − A π + π − CP = − . · − , (1)where A f CP ≡ Γ[ D → f ] − Γ[ ¯ D → f ]Γ[ D → f ] + Γ[ ¯ D → f ] , (2)is a shorthand for the time integrated CP-asymmetry, for a case where the final state f is a CP-eigenstate. ∆ A CP is a convenient quantity since systematic experimental errorscancel. It is worthwhile to add that if SU (3) F , or more precisely U -spin, were a goodsymmetry then A K + K − CP = − A π + π − CP . In the quantity the ∆ A CP TDCP-asymmetry partcancels. Effects can though remain through time-acceptance differences in the π - and K -system. The latter is estimated to be small, e.g. [2], and thus in summary direct ( i.e.time-independent CP-asymmetry) is expected to be responsible for the relatively largevalue of ∆ A CP .Sizeable direct CP-asymmetries, c.f. appendix C, necessitate large strong (CP-even)and weak (CP-odd) phase differences in two amplitudes of comparable size. The rea-son CP-violation is believed to small in the charm system is that the weak phasesare suppressed by four powers of the Cabibbo angle, leading to the naive expectation∆ A CP (cid:39) few · − . In the non-leptonic case the QCD matrix elements, which determinethe strong phase as well as the ratio of amplitudes, are difficult to compute from firstprinciples as the size of the charm mass is neither suited to chiral nor heavy quark the-ory. Thus the question of whether the large central value (1), should it remain, is dueto NP [4, 5, 6, 7] or somewhat unexpected strong dynamics [8, 9, 10, 11], such as in the∆ I = 1 / K → ππ system , is an open question at present. It is fortunate thatcontributions due to new ∆ I = 3 / It was pointed out quite some time ago [14] that an enhancement of the triplet transition, in theSU(3)-flavour classification, may lead to sizeable CP-violation. E.g. A P P CP (cid:39) . · − which would leadto | ∆ A CP | (cid:39) . · − which is not far off the central value in (1). | ∆ C | = 1 chromomagnetic operator O ≡ − gm c π ¯ uσ · G (1 + γ ) c , O (cid:48) ≡ − gm c π ¯ uσ · G (1 − γ ) c (3)( σ · G = σ µν G µνa λ a / D - ¯ D -mixing. Note, the O ( (cid:48) )8 -operators are of the ∆ I = 1 / I = 3 / O (cid:48) is the structure which is the less abundant helicity inthe SM due left-handedness of the weak interactions; [ C (cid:48) /C ] | SM ≈ m u /m c .To get an idea of the size of the NP contribution [16] one might resort to naivefactorisation (NF) e.g. [13]. Slightly extending the notation in [16] one gets,∆ A NP CP | NF (cid:39) − . (cid:16) Im[ C NP ] − Im[ C (cid:48) NP ] (cid:17) sin( δ ) , (4)where δ is the unknown strong phase difference between the KK and ππ states which isexpected to be sizeable. Note since the sign of sin δ is unknown the sign of the differenceof the two Wilson coefficients is currently not determined by the D → ππ/KK -system.Throughout this paper, if not otherwise stated, the Wilson coefficient are understood tobe evaluated at the charm scale. Since the decay of a J P ( D ) = 0 − particle into two J P ( π/K ) = 0 − particles necessitates parity violation only the γ -part in (3) contributesand therefore results in opposite signs of Im[ C NP ] and Im[ C (cid:48) NP ] in (4) respectively. It isnoted that a value of (cid:16) Im[ C NP ] − Im[ C (cid:48) NP ] (cid:17) sin( δ ) (cid:39) . · − naive factorisation (NF) (5)could account for the central number in (1). One has to bear in mind that (5) is dueto NF and could easily be out by a few factors. Following the literature we shall takeIm[ C ( (cid:48) ) NP ] = 0 . · − as a reference value, which is two to three orders of magnitudeabove the SM-value for C , c.f. appendix B.1, and suppressed by an additional factor m u /m c in the case of C (cid:48) . A CP in D → V γ
As lamented above the situation in D → ππ/KK remains unclear. Thus the question ofwhether a value like (5) leads to an effect that can be estimated theoretically and measuredexperimentally. It was pointed out in reference [17] that sizeable direct CP-violation in D → ( ρ , ω ) γ can be induced through Im[ C ] provided that the LD amplitude carries Note this is the sign convention of [15] but opposite to references [16, 17]. O does not have astrong phase. Our work improves on [17] in that it includes the O matrix element per sewhich carries a strong phase and thus does not rely on a size strong LD phase. The lattervanishes in the chiral limit at leading order for WA which we believe to be dominant. Byhow much corrections change the LD phase is an interesting open question for which werefer the reader at this stage to the conclusion and appendices A.1.1 and B.3.Other channels and effects that were proposed are the electric dipole moment of thenucleon [16, 18], CP-asymmetry in D → φ → K + K − [19] and D → V ( → P P ) → K + K − [20]. We argue that the strong phase of the LD contribution, which we believe tobe WA, is small due to chiral suppression at leading order in appendix A.Another question is what type of NP models could induce such values as in (5) withoutviolating existing constraints. Among the NP models inducing (5) are supersymmetricmodels [13, 16, 21], Randall-Sundrum flavour anarchy [22] and models of partial com-positeness [23], whereas in fourth family models it seems more difficult to accommodate[10].The paper is organised as follows: In section 2 we introduce notation and remindthe reader of why CP-violation in a charm system is supposed to be small in the SM.Section 3 is the main part of this paper and consists of detailing the two amplitudes3.1 and an estimate of direct and time dependent CP-violation in section 3.2 using thematrix elements of the operator O ( (cid:48) )8 [24] computed within light-cone sum rules (LCSR).Conclusion and discussion are presented in section 4. An important part of our work isthe discussion around the estimate of the LD contribution. In appendix A it is arguedthat the WA contribution is the dominant LD effect and we comment on the possible sizeof the strong phase in A.1.1. Appendix B includes a discussion of the SM CP-asymmetryin B.2 as well as the a discussion of the C -effect, noted in [17], in B.3. | ∆ C | = 1 Hamiltonitan
Following, closely, the notation of [15] we write the effective ∆ C = 1 SM Hamiltonian asfollows H eff = λ d H d + λ s H d + λ b H peng , λ D ≡ V ∗ cD V uD , D = d, s, b (6)3nd H q = G F √ (cid:88) i =1 C qi O qi + h . c . , q = d, s O q = (¯ uL µ q )(¯ qL µ c ) , O q = (¯ u α L µ q β )(¯ q β L µ c α ) λ b H peng = G F √ C O + C (cid:48) O (cid:48) + C O + C (cid:48) O (cid:48) + ... ) (7)with L µ ≡ γ µ (1 − γ ) and α, β being colour indices. The Hamiltonian H peng contains allthe SD transitions including electric (A.10) and chromomagnetic (3) operators as well asthe four quark operators with structure different from O , . As compared to [15] we haveabsorbed the λ b into the Wilson coefficient which is non-standard for the SM contribution,but convenient for our presentation as we can write: C ( (cid:48) )8 = C ( (cid:48) )SM8 + C ( (cid:48) )NP8 (8)with C NP8 as in the literature. Since λ d,s = O ( λ ) and λ b = O ( λ ), where λ (cid:39) .
226 [1] isthe Wolfenstein parameter [1], one gets using the unitarity relation λ d + λ s + λ b = 0 , ⇒ λ d (cid:39) − λ s , λ b (cid:39) , (9)where the symbol (cid:39) above is to be understood as up to corrections of O ( λ ). The factthat the third generation decouples up to O ( λ ) is the reason why in the SM the genericexpectation for CP-violation is A CP (cid:39) few · O ( λ ) as mentioned in the introduction. Hier-archies in amplitudes might lead to an enhancement or further suppression. Throughoutthis paper we shall implicitly employ the Wolfenstein parametrisation.4 .2 D → V γ transitions
In this subsection we shall introduce some notation which aims to ease the presentationin the following subsection. We write the amplitude as follows : A [ D → V γ ] ≡ (cid:104) V γ |H eff | D (cid:105) = A P A P , = A L (cid:18) P + P (cid:19) + A R (cid:18) P − P (cid:19) (10)with P = 2 (cid:15) ραβγ (cid:15) ∗ ρ η ∗ α p β q γ , P = 2 i { ( p · q )( η ∗ · (cid:15) ∗ ) − ( η ∗ · q )( p · (cid:15) ∗ ) } , (11)where η ( p ) and (cid:15) ( q ) stand for the vector meson and photon polarisation tensors respec-tively. The two structures A L ( R ) ≡ ( A ± A ) , (12)of negative mass dimension, correspond to left- and right-handed polarised photons. Therate [33], in our conventions, is given byΓ[ D → V γ ] = 132 π m D (cid:18) − m V m D (cid:19) (cid:0) |A | + |A | (cid:1) . (13) D → V γ
The operators (3) consist of c → u transitions of the FCNC type. In a heavy-to-lighttransition for which LCSR can make predictions [24] the c -quark can pair with a u , d or s -quark leading to the transitions D → ( ρ , ω ) γ , D + → ρ + γ and D + s → K ∗ + γ respectively.These transitions shall be investigated. Note it is only for the neutral D -system thatoscillations and thus TDCP asymmetries are feasible.As previously mentioned and outlined in appendix C direct CP-violation originates inits minimal form by two amplitudes with weak and strong phase difference. These twoamplitudes will be given by the LD and O contributions respectively, to be discussedbelow. The amplitudes A , up to phases are often denoted by A PC , PV in the literature e.g. [33, 17]. Theacronyms PC and PV stand for parity-conserving and -violating respectively. The sign convention for the epsilon tensor is given by tr[ γ γ a γ b γ c γ d ] = 4 i(cid:15) abcd and are the ones usedin the classic textbook of Bjorken & Drell. .1 Amplitudes with strong and weak phase difference LD: The LD contributions are generated by four quark operators amongst which arethe WA (Fig.1,left) and the quark loop (QL) (Fig.1,middle;right) topologies. Weshall summarise the essential points below and refer the reader to appendix A formore details. The only global assumptions we need for our argument are that theLD contribution is a) dominant, b) has a small weak phase and c) a small strongphase. Points a) and b) are a direct consequence of the CKM/Cabibbo-hierarchy:WA terms are proportional to λ d,s (cid:39) O ( λ ) which has a small weak phase O ( λ ) andis large compared to the SD part which is proportional to λ b (cid:39) O ( λ ) (7).Turning to the smallness of the strong phase let us summarise the essence ofappendix A. Branching ratios are compatible with WA-dominance from LCSR-predictions of WA diagrams [26]. QCD estimates suggests that WA dominatesQL by almost or about two orders of magnitude, as the latter are suppressed bytwo additional loops with respect to the former. We thus expect the lion’s shareof the strong phase to originate from radiative corrections. Further discussions aredeferred to appendix A.1.1.We extract the amplitudes for D → ( φ, ¯ K ∗ ) γ from experimental branching ratios.The latter differ from LCSR predictions [26] by about a factor of 2 which translatesinto a factor √ (cid:39) . √ A L | LD (cid:39) − . · − c V m D , A R | LD (cid:39) − . · − c V m D , { D → ( ρ , ω ) } , A L | LD (cid:39) . · − c V m D , A R | LD (cid:39) . · − c V m D , { D +( d,s ) → ( ρ + , K ∗ + ) } . (14) As previously emphasized we do not distinguish between the ρ and ω as the effect largely cancels inthe ratio. : The amplitude of the chromomagnetic operator (3) is parametrised by , A i | = (cid:104) V γ |H eff | | D (cid:105) = G F √ (cid:16) em c π (cid:17) c V (cid:40) ( C + C (cid:48) ) G (0) i = 1( C − C (cid:48) ) G (0) i = 2 , (15)where H eff | = G F √ ( C O + C (cid:48) O (cid:48) ). Therefore G , (0) corresponds to the matrixelements, with on-shell photon q = 0, (cid:104) V γ |O ( (cid:48) )8 | D (cid:105) = (cid:16) em c π (cid:17) c V ( G (0) P ± G (0) P ) , (16)which are analogous to the penguin matrix element for T and T Eq. (A.11). Thevariable e = √ πα > G (0) = G (0) which implies that O and O (cid:48) generate solely left- and right-handed ampli-tudes respectively. Moreover G D → ρ γ (0) (cid:39) G D → ωγ (0), G D + s → K ∗ + γ (0) (cid:39) G D + → ρ + γ (0)to an accuracy sufficient for our purposes. We shall therefore not distinguish be-tween them . The imaginary part, relevant for the CP-asymmetry, is found to beIm[ G D (0)] (cid:39) − . , Im[ G D + (0)] (cid:39) − . , (17)where numbers were rounded. The values in (17) are sizeable compared to typicalestimates T D (0) (cid:39) T D + (0) (cid:39) . O operator as compiled in [17]. Thedifference in the numerical value of neutral and charged matrix element in Eq. (17)originate from the charges of the valence quarks of the mesons. Using the referencevalue for Im[ C ( (cid:48) )8 ] the relevant ratios |A , | / A , | LD | are around 10 − and thus thescale for CP-violation is set at the percent level. Since the photon polarisation is not easy to measure in practice a slightly inclusive rateΓ[ D → V γ ] = Γ[ D → V γ L ]+Γ[ D → V γ R ] is measured. We parametrise the corresponding The factor c V is inserted to absorb trivial factors due to the ω ∼ (¯ uu + ¯ dd ) / √ , ρ ∼ (¯ uu − ¯ dd ) / √ c V = −√ ρ in c → d , c V = √ ω & ρ and c V = 1otherwise. Note in the overall CP-asymmetry this factor will drop out. In fact the ratio of the WA to the G (0) form factor is well approximated by R = r ρ /r ω where r X = ( f ⊥ X ) / ( m X f (cid:107) X ) is the ratio of the tensor to the vector decay constant. Information on this ratioexists only sparsely in the literature. Similar remarks apply to the D + s → K ∗ + and D + → ρ + -transitions. A L,R = A ± A = l L,R e iδ L,R + g L,R e i ∆ L,R e iφ L,R (18)with l L ( R ) = | l ± l | , l , ≡ A , | LD g L ( R ) e i ∆ L ( R ) = G F √ (cid:16) em c π (cid:17) c V | C ( (cid:48) )8 | (2 G (0)) G (0) = | G (0) | e iδ , C = | C | e iφ L C (cid:48) = | C (cid:48) | e iφ R , (19)where ∆ L,R , δ L,R and φ L,R are the strong and weak phase of (15) respectively leaving thequantities l L ( R ) , g L ( R ) real-valued. In the equation above we have made use of G (0) = G (0), found at leading twist [24], implying that O and O (cid:48) solely contribute to the left-and right-handed amplitude respectively and in addition leads to ∆ L = ∆ R . The latteris not true when the contribution due to Im[ C ( (cid:48) )7 ] is included, in which case the formulaefor g L,R have to be modified according to Eq. (A.26) in appendix D.In the case where the two photon polarisations are not distinguished the formula forCP-violation is slightly more complicated than the one given in Eq. (A.25). The generalformulae and a derivation, including TDCP-asymmetries, can be found in Ref. [25] forexample. We find A CP ( D → V γ ) = − n ( g L l L sin(∆ L − δ L ) sin( φ L ) + g R l R sin(∆ R − δ R ) sin( φ R )) (20) n ≡ l L + l R + 2 ( g L l L cos(∆ L − δ L ) cos( φ L ) + g R l R cos(∆ R − δ R ) cos( φ R )) + g L + g R ) . Assuming l L ( R ) (cid:29) g L ( R ) and imposing ∆ ≡ ∆ L = ∆ R one gets: A CP ( D → V γ ) (cid:39) − l L + l R ( g L l L sin(∆ − δ L ) sin( φ L ) + g R l R sin(∆ − δ R ) sin( φ R )) . (21)In the absence of a computation, and in view of the chiral suppression at leading order,we shall set the LD phases δ L,R (18) to zero in remaining formulae. This allows us toexpress A CP in terms of quantities discussed at the beginning of the paper: A CP ( D → V γ ) = − l L + l R G F √ (cid:16) em c π (cid:17) Im[ G (0)] c V ( l L Im[ C ] + l R Im[ C (cid:48) ]) . (22)8his formula, modulo notation, reduces to A CP (A.25) for l = l (i.e. l R = 0).With m c = 1 . A CP ( D → ( ρ , ω ) γ ) = (cid:16) − . C NP ] − . C (cid:48) NP ] (cid:17) c B = (cid:18) − . (cid:18) Im[ C NP ]0 . · − (cid:19) − . (cid:18) Im[ C (cid:48) NP ]0 . · − (cid:19)(cid:19) c B , (23)with an estimated of uncertainty of about 45%, to be discussed below, and c B ≡ (cid:18) . × − B ( D → ( ρ , ω ) γ ) (cid:19) / (24)being the correction factor for the yet to be measured branching ratios. In going from(21) to (23) we have used the fact that the imaginary part of C SM8 , which contains theCKM prefactors, is negligible with respect to the values (5). For the charged transitionswe get A CP ( D +( d,s ) → ( ρ + , K ∗ + ) γ ) = (cid:16) . C NP ] + 0 . C (cid:48) NP ] (cid:17) c B = (cid:18) . (cid:18) Im[ C NP ]0 . · − (cid:19) + 0 . (cid:18) Im[ C (cid:48) NP ]0 . · − (cid:19)(cid:19) c B , (25)with an estimated of uncertainty of about 45% to be discussed below. Note, the dominanceof the O -contribution, in both neutral and charged case, is due to the A L ( R ) hierarchy inEq.(14). In fact the different sensitivity of ∆ A CP , A CP ( D → ( ρ , ω ) γ ) and A CP ( D +( d,s ) → ( ρ + , K ∗ + ) γ with respect to Im[ C ] and Im[ C (cid:48) ] gives a handle to discriminate between theindividual contributions of the two chromomagnetic operators.Let us turn to the discussion of the uncertainty. The major uncertainty comes from theestimate of the O matrix elements which we estimate to be around 35% [24]. Then thereis the phase of the WA contribution, δ L,R , for which we assign an uncertainty | δ L,R | = 10 ◦ (c.f. appendix A.1.1) which leads to a 20% uncertainty. Amongst the LD contributionsthe combination l L + l R is taken from experiment but the ratio l L /l R which we took from[26] could have uncertainties, say, at the 20%-level. Adding the three sources discussedabove in quadrature, as they would seem uncorrelated, we get about 45% uncertainty. Afew additional remarks are in order. In appendix B.2 we estimate the SM contribution tobe of the order of 10 − which is negligible. Furthermore we refrain from including at thispoint the uncertainty due to the C -effect discussed in appendix B.3. We would like to9ention though that it cannot be excluded, depending on the model and the LD phase,that the C and C -effect conspire to cancel significantly in the CP-asymmetry. As a result of D - ¯ D oscillations CP-asymmetries are time dependent for the neutral me-son, which will lead to novel features. In particular TDCP asymmetries do not necessitatea strong phase difference in the two amplitudes. Thus in principle we have to adjust theamplitudes to include the C -effect, from g L,R
Eqs. (18,19) to ˜ g L,R (A.26) as detailed inappendix D. Indications are though that these effects are overshadowed by the dominanceof the LD amplitudes l L,R .Important mixing parameters of the D - ¯ D system are the mass and width difference,the mixing phase φ D as well as the ratio | p/q | of the parameters p and q translatingbetween the flavour and mass eigenstates. The latest HFAG values [27] are x D = ∆ m D Γ = 0 . · − , (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) D =1 . ,y D = ∆Γ D
2Γ = 0 . · − , φ D = − . . ◦ [ − . . ◦ ] , (26)where Γ = ( τ D ) − is the inverse lifetime of the D -mesons and ∆ m D and ∆Γ D are thedifference of the heavy and the light D -meson mass and width respectively. We quotethe values with no direct CP-violation except for φ D , where we also quote the resultfor allowed direct CP-violation as it is rather different in the central value, althoughcompatible within errors.In view of the closeness of | p/q | D to unity in (26) we shall assume | p/q | D = 1 in whichcase the TDCP-asymmetry assumes the following form, A CP ( D → V γ )[ t ] = S sin(∆ m D t ) − C cos(∆ m D t )cosh( ∆Γ D t ) − H sinh( ∆Γ D t ) , (27)where the convention A CP (0) = − C is somewhat awkward but standard. The formulaefor S and H are given in appendix D and C = − A CP from the previous section. Let usdefine the LD chirality-asymmetry (ratio) by χ LD ≡ l − l l + l = 2 l L l R l L + l R ∈ [ − , . (28)Note for the values in (14) we get χ LD (cid:39) . χ LD (cid:29) − ,10 L,R (cid:29) ˜ g L,R , which both seem true, and once more set δ L,R = 0 we get an interestingexpression for for H and S , H [ S ] (cid:39) l L l R l L + l R · ( − ξ cos[sin]( φ D ))= χ LD · ( − ξ cos[sin]( φ D )) , (29)which directly measures the ratio of the LD chirality structure times the cosine and sineof the mixing angle of the D -system. The variable ξ = ± V -meson whose values can be found in appendix D. With ξ ( ρ , ω ) = 1 we get H [ D → ( ρ , ω ) γ ] (cid:39) − . φ D ) , S [ D → ( ρ , ω ) γ ] (cid:39) − . φ D ) . (30)Let us emphasize once more that this relation is valid in the case where a left- and right-handed amplitude are of comparable size and dominate all the other contributions. Theseratios do not depend on the branching ratio and therefore do not have a correction factor c B (24).The experimental tractability of S and or H depends on the angle φ D . Should φ D (26),that is to say sin φ D , turn out to be sizeable then S could be measured as for B → K ∗ γ at the B-factories. If cos φ D is sizeable, which is what the value in (26) indicates, thenone would need to focus on H . The latter might be measured, in analogy to B s → φγ case [25], in the rates D → ( ρ , ω ) γ and the one for ¯ D without flavour-tagging, whichhas experimental advantages, though it has to be added that the relatively small widthdifference in the D system, y D /y B s (cid:39) .
1, means that roughly a hundred times more datahas to be accumulated to achieve the same precision on H in the D - as in the B s -system. Partly building up on ideas in [17] we have shown how Im[ C ( (cid:48) )8 ] become observable inCP-asymmetries in D → V γ . Setting the LD phases δ L,R = 0 (18), in the absence of acomputation, we got A CP ( D → ( ρ , ω ) γ ) (cid:39) (cid:18) − . (cid:18) Im[ C NP ]0 . · − (cid:19) − . (cid:18) Im[ C (cid:48) NP ]0 . · − (cid:19)(cid:19) c B ,A CP ( D +( d,s ) → ( ρ + , K ∗ + ) γ ) (cid:39) (cid:18) . (cid:18) Im[ C NP ]0 . · − (cid:19) + 0 . (cid:18) Im[ C (cid:48) NP ]0 . · − (cid:19)(cid:19) c B , (31)11ith c B (24) a correction factor for a yet to be measured branching fraction of D → ρ γ .Uncertainties are in the 45%-range, c.f. section 3.2, for which we have not included theuncertainty of the Im[ C ]-effect on which we comment in the the last paragraph of thissection. The SM contribution is negligible, down by two orders of magnitude c.f. appendixB.2, for reference values of Im[ C ( (cid:48) )NP8 ] used throughout this paper. An obviously interestingaspect is that the Wilson coefficients of the two chiralities of the chromomagnetic operatorenter with different sensitivity. Even the relative sign is different between the D → V γ and the D → KK, ππ case (4),∆ A NP CP (cid:39) − . (cid:16) Im[ C NP ] − Im[ C (cid:48) NP ] (cid:17) sin( δ ) c NF , (32)where c NF is a correction factor for using naive factorisation of order one. Note whereasthe sign of the entire contribution is predicted for D → V γ this is not the case for thenon-leptonic mode as the sign of sin( δ ) remains undetermined.The chirality of the photon is an interesting aspect and deserves some discussionin comparing it to the b -sector. In b → ( d, s ) γ transitions the left-handed amplitudedominates over the right-handed amplitude as a result of the large b -quark mass andthe V - A interactions. This pattern might be broken by physics beyond the SM and canbe measured in TDCP-asymmerties [28]. The situation in D → V γ is rather different.Whereas it is still true that the left-handed amplitude is larger than the right-handedamplitude, e.g. (14) it is not very significant since the c -quark mass is smaller. Thisneither-nor situation has two consequences.First since the amplitudes themselves are LD dominated the TDCP-asymmetries arenot sensitive to novel right-handed currents. On a positive note TDCP-asymmetries mea-sure the LD chirality asymmetry χ LD (28) and thus can provide interesting informationon LD dynamics and could serve as validation criteria for theoretical tools. Let us addthat the feasibility of the measurement depends on the definite value of the mixing phase φ D as commented on at the end section 3.3. Second the fact that the CP-asymmetries in D → V γ are more sensitive to C than C (cid:48) (31) follows from A L > A R , c.f. Eqs. (14,12).Thus the direct CP-asymmetry in D → V γ is not a good place either to look for right-handed currents as reflected in (31) in the low relative sensitivity of C (cid:48) versus C .On the speculative side it is of course possible that NP contributes to SM or non-SMoperators of the WA-type, O d , (7) possibly with new weak phases. Allowing for thelatter and parametrising a strong phase for the yet to be computed O ( α s )-corrections In addition, extensions of the SM which remotely follow the pattern of minimal flavour violation | C (cid:48) /C | (cid:39) m u /m c predict small effects in C (cid:48) per se. Note in [15] it is the GIM-combination (9), O d , − O s , , which is severely constrained through (cid:15) (cid:48) /(cid:15) innew weak phases but not the individual operators O d ( s )1 , of down and strange per se. L ( R ) → l L ( R ) e i Φ L ( R ) , one gets: H [ S ] = χ LD · ( − ξ cos[sin]( φ D − Φ L − Φ R ) cos( δ L − δ R )) (33)Needless to say that χ LD is then affected by the NP.At last let us give an outlook and hint how the current work could be improved. On thetheoretical side it would benefit from a thorough reassessment of the WA contributions,that is to say the full O ( α s )-correction and using updated values of the photon DA [29]and other parameters. In particular the radiative corrections would allow an estimate ofthe strong phase and the inclusion of the C -effect [17], c.f. appendix B.3, to the centralvalue rather than as an error estimate. Our discussion in appendix A.1.1 suggests thatthe convergence of the α s -expansion is at least good enough to get the sign and a roughidea of the value of the angle. This would allow exclusion of scenarios where the C -and C -effects conspire to cancel each other in the direct CP-asymmetry, c.f. appendixB.3. The prominence of WA in the isospin asymmetry in b → s processes provides yetanother motivation for their reassessment. Furthermore it might be interesting to extendthis work from D → V γ to D → V l + l − as the latter might be easier to deal with atthe LHCb where the photon final state remains challenging at present. The estimatedbranching ratios for D → ( ρ , ω ) γ are a factor of 30 below the current experimentallimit . Should the CP-violation in non-leptonic charm decays (1) remain one would hopethat A CP ( D → V γ ) is going to be measured at some future flavour factory or possibly atthe upgraded LHCb.
Acknowledgments
We are grateful to Ikaros Bigi, Franz Muheim, Steve Playfer, Stefan Recksiegel, NikolaiUraltsev, Yuehong Xie and especially Jernej Kamenik, Gino Isidori and Ayan Paul fordiscussions and or correspondence. RZ is grateful to the extended Southampton latticegroup meeting where the news of CP-violation in charm were discussed in an informalmanner. RZ gratefully acknowledges the support of an advanced STFC fellowship.
A Long-distance contributions
In section A.1 we extract the LD contributions in D → ( ρ , ω ) γ and D +( d,s ) → ( ρ + , K ∗ + ) γ from experiment with minimal theoretical input. In section A.2 we comment on the For the D +( d,s ) → ( ρ + , K ∗ + ) γ decays there are not even limits, possibly because they are experimen-tally more challenging. A.1 Estimate of LD contribution
Our basic assumption is that WA, as discussed in section A.2, dominates over QL. There-fore we feel justified to use LCSR result in [26] as a guideline for the ratio of left- toright-handed amplitude as well as the sign of each of them. More precisely we infer thatthe strong phases are relatively small, to be discussed in more detail in section A.1.1,and thus we assume the central value to be real valued. This information is sufficient toextract the amplitudes from experimental branching ratios.The measured rates of the D → V γ -type are [1], B ( D → ¯ K ∗ γ ) = 3 . · − , B ( D → φγ ) = 2 . · − , (A.1)of the Cabibbo allowed and singly Cabibbo suppressed type. From the rate (13), with τ D (cid:39) . · − s , we get x [ ¯ K ∗ ] ex = 1 . · − (10%) GeV − , x [ φ ] ex = 2 . · − (10%) GeV − (A.2)where we have introduced the following shorthand, x [ V ] ≡ ( |A | + |A | ) D → V γ , whichis the rate (13) modulo phase space factors. The relation between A , (10) and A PC(PV) of reference [26], for c ¯ u → d ¯ d (e.g. D → ( ρ , ω ) γ ) transition, is given by A | WA , O ( α ) = − (cid:18) G F √ λ d a (cid:19) (cid:18) f V m V c V (cid:19) A PC(PV) (A.3)where a = C + C / (cid:39) − . x [ ¯ K ∗ ] LCSR x [ ¯ K ∗ ] ex (cid:39) . √ With respect to reference [26] the labels 1 and 2 are chosen the other way around in H eff (7). → ( ρ , ω ) γ and D +( d,s ) → ( ρ + , K ∗ + ) γ and scale them by a factor of √ (cid:39) . Indoing so we implicitly rely on the sign and the ratio both of which would seem morereliable than the individual amplitudes. In following our recipe above we get, A | LD (cid:39) . × A | WA , O ( α ) , (A.5)using the notation l ≡ A | LD we get l (cid:39) − . · − c V m D , l (cid:39) − . · − c V m D , { D → ( ρ , ω ) } ,l (cid:39) . · − c V m D , l (cid:39) . · − c V m D , { D +( d,s ) → ( ρ + , K ∗ + ) } . (A.6)In turn this gives a result for the branching ratios of B ( D → ( ρ , ω ) γ ) | (A.5) (cid:39) . · − , B ( D + s → K ∗ + γ ) (cid:39) B ( D + → ρ + γ ) | (A.5) (cid:39) . · − (A.7)Comparing B ( D → ( ρ , ω ) γ ) above with B ( D → φγ ) PDG (A.1) we note that the formeris down by about a factor of four. A factor of two comes from c V = 2, i.e. the valencequark content of ρ and ω . Another factor 1 . m V f V -prefactors. The remaining discrepancy might partly be due to neglecting the strangequark mass. It might be questioned whether 2 m s , which results from the axial current, isreally that small a parameter. At last we note that the value for the ρ -branching ratiois well below the current limit of 2 . · − [1]. A.1.1 Strong phase in WA-process
Here we shall add comments on the the strong phase δ LR (18) in WA, that is to say,we discuss the suppression at leading order and reflect on the reliability of perturbationtheory for this computation.The photon can either be emitted from the initial state meson or the final state meson,c.f. Fig.1(left). At leading order O ( α s ) the former does not lead to an imaginary partand the latter is chirally suppressed. More precisely the vector part vanishes identically We should add that this itself does not settle the issue of the hierarchy between the WA and QL, tobe discussed in subsection A.2 ,as the decays above do not have QL or at least if the ¯ uu component ofthe φ is neglected.
15y vector current conservation and the axial part, which corresponds to A , is partiallyconserved; that is to say proportional to 2 m d and thus negligible as noted in [26, 30] .The reason we can trust perturbation theory, beyond questions of convergence of theseries, is that m γ = 0 and m D (cid:39) (1 .
86 GeV) are sufficiently far away from the narrowpeaks such as the ω ( m ω (cid:39) .
78 GeV). One can get an idea of this by looking at thefamous σ ( e + e − → e + e − ) cross section plot [1] in the region of √ s ≈ m D for example.Note the latter probes the electromagnetic current which is not the same but definitely hassimilarities to the ¯ dγ µ (1 − γ ) d current relevant for WA. The partonic over the hadroniccross section at m D indicates about a 25% correction and the 3-loop perturbative QCDresult is roughly in between the two. Thus it seems fair to say that in the region of m D perturbative QCD is neither great nor bad, somewhere in between. Let us now turn to thequestion of the size of the O ( α s )-corrections from another point of view. The O matrixelement might be regarded as a radiative correction to the O matrix element, at least thepart where the gluon does not connect to the spectator quark. The latter contribution isabout 1 / / . × / . × / / (cid:39) .
085 which has to be compared to the referencevalue T (0) = 0 . O ( α s )-correctionaround the 15%-level .In view of the discussion above we shall take 25% as a, probably conservative, referencevalue for radiative corrections. If we assume that real and imaginary part share out thisvalue in an equal way we end up with | δ L | (cid:39) ◦ and | δ R | (cid:39) ◦ . Needless to saythat a computation of the radiative correction would add more weight to these kind ofdiscussions. One might be sceptical about the convergence of the α s -expansion in the m D region yet it would seem that the first correction should at least give a reasonableestimate of the phases δ L,R and in particular of their directions which is relevant forexcluding cancellation effects between the C - and C -effect as discussed in appendix B.3. Note there are also four quark operators other than O , whose matrix elements are not chirallysuppressed but since they originate from SD physics their Wilson coefficients are tiny in the SM aspreviously discussed. There are two further aspects that one could bring in. Only a limited number of graphs generate animaginary part at O ( α s ). On the other hand the radiative correction are colour-enhanced as with respectto tree graphs. Possibly these two effects just balance each other off so we shall not give it any furtherthought here. The two phases ought to have the same sign as a remnant of the heavy quark chirality structure. .2 Weak annihilation versus quark loops - theory point of view One may distinguish two types of LD contributions according to the quark level transitionsis c ¯ u → d ¯ d or c → ud ¯ d ( s ¯ s ) generated by the weak operators O d,s , (7) for instance. Fromthe viewpoint of quarks and gluons the first type is known as weak annihilation (WA)(Fig. 1, left) and we shall name the second type quark loops (QL) (Fig. 1, middle;right).The WA contributions have been computed in 1995 for B → V γ and D → V γ in [31]and B → ( ρ , ω ) γ in [31, 30] at O ( α s ). We hasten to add, for further understanding ofour arguments below, that QL of the type shown in Fig. 1(middle,right) are evaluated inan 1 /m c (1 /m b )-expansion for c ( b ) → u ( d, s ) γ , although in principle one could computethem in the exclusive case with LCSR which does not adhere to a 1 /m c (1 /m b )-picture.We shall argue that we believe the WA to dominate over QL: • Generic observation:
QL and WA are generated by the same weak operator, O d,s , and O d , (7) respectively, yet the QL is down by two loops with respect to WA .This is the case because the single quark loop Fig. 1(middle) vanishes exactly byvirtue of gauge invariance. The photon polarisation Π µν ( q ) = ( q g µν − q µ q ν )Π( q )vanishes for q = 0 when contracted with the photon polarisation tensor. Thissuggests a natural hierarchy WA (cid:29) QL in the types of charm transitions discussedin this paper. • Test-case in B -physics: One would think that the perturbative picture is valid in B -physics. Taking numbers from [32] for the WA and QL one gets: |A QL / A W A | B − → ρ − γ (cid:39) · − . To be more precise, for A QL we have taken the charm loop contributionwhere the gluon is radiated into the final state vector-meson. The latter, as men-tioned above, does not depend on the 1 /m b -expansion. Note WA for B → ρ γ isaccidentally small because of cancellations between tree-level and penguin four quarkoperator contributions. We do not expect the same to take place for D → ( ρ , ω ) γ since those cancellation are between tree and penguin four quark operator contri-butions and the latter are tiny in D -physics. • Test-case in D -physics: Does this hierarchy remain intact for the D -physics. De-spite the obvious fact that the α s ( m c )-expansion and the 1 /m c -expansion are lesstrustworthy it seems hard to see how a hierarchy of two order of magnitude can beoverthrown. Taking the contribution Fig. 1(right) for the QL from [31], which does In principle there is a GIM suppression of the QL in addition which is though not very effective forthe matrix elements [31]. WA is Cabibbo suppressed with respect to QL in B -physics. In comparing the WA and QL pro-cesses/diagrams we, of course, do not take CKM hierarchies into account, especially because they are notpresent in the charm decays we are interested in. /m c -expansion, and the estimates of [26] one gets a number, |A QL / A W A | (cid:39) · − , which is surely accidentally close to the one for the B − → ρ − γ . An estimateof the size of 1 /m c -corrections, for WA, can be gained by looking at the chiralitystructure of the D ( B ) → V γ result in LCSR. In the heavy quark limit it is believedthat A = A (i.e. A R = 0). The results in [26] indicates that this is fulfilled forthe D ( B ) → ρ γ transition at 57%(70%)-level respectively. This does not suggesta dramatic breakdown, by which we mean one or two orders of magnitude, of the1 /m c -expansion.Thus our analysis suggests that WA dominates QL by roughly two orders of magnitude.We shall briefly comment on another approach. In the extensive work [33] the two transi-tions were modelled with hadronic data. It would seem that WA corresponds to the pole(P) terms and QL to the vector-meson dominance (VMD) part. Comparable numbersfor P and VMD were found which is not in line with the arguments above . One mightwonder how the vanishing of the single quark loop in Fig. 1(middle) is reflected in this for-malism. A problem is that the signs of the couplings of the VMD models are not known,that is to say their absolute values can sometimes be taken from experiment. Thus theformalism might overestimate the contributions as it cannot capture cancellations, whichgauge invariance almost suggests to be present. A similar point of view has been takenin [34] by one of the authors of [33] in chapter 3.1.3. c d ¯ u ¯ dγD ρ , ω c u ¯ u γ ρ , ωD d, s c u ¯ u γ ρ , ωD d, s Figure 1:
A selection of LD diagrams for D → ( ρ , ω ) γ . Note it is the fact that the ρ /ω carryboth ¯ dd and ¯ uu components that makes the same operator O d,s (7) contribute to both (WA &QL) topologies. (left) Weak annihilation (WA). (middle) Quark loop (QL). This contributionvanishes, exactly, for on-shell photon by virtue of gauge invariance as discussed in the text.(right) QL example of O ( α s )-correction. This diagram has a sizeable imaginary part which canbe inferred from the computation for c → uγ in reference [31]. We note that in [33] the P-part receives no contribution in A ( ↔ A PV ) which is a fact that is notreflected in the LCSR computation [26, 30]. A CP ( D → V γ ) other than through C NP8For our discussion it is convenient to write the amplitude as follows,
A (cid:39) λ d e iδ d A d + λ s e iδ s A s + λ b e iδ b A b , (A.8)which is similar to (A.24) with the exception that the unitarity relation (9) has not beenused and that the weak phases are contained within λ d,s,b (6). As argued in appendix Awe expect the lion’s share of A d to be covered by WA which has, presumably, a smallstrong phase which we shall neglect ( δ d → O ( λ ) which fulfils, e.g. [35],Im[ λ d ] = 0 , Im[ λ s ] = A λ η , Im[ λ b ] = − A λ η (A.9)where A , ρ and η are the other three Wolfenstein parameters and A λ η (cid:39) . · − .Eq.(A.9). The fact that | Im[ λ b,s ] | (cid:39) . · − indicates small CP-asymmetries , of thatorder.Thus it remains to identify contributions with sizeable strong phases δ s,b and ampli-tudes A s,b for which we see two major sources. First the matrix element of O e.g. (17)[24] and second the matrix element of O d,s [31] (c.f. Fig. 1(right) for a contribution) givingrise, effectively, to an O -operator. The latter as well as its matrix element analogous to(16) are defined and parameterised respectively as follows, O ( (cid:48) )7 ≡ − m c e π ¯ uσ µν F µν (1 ± γ ) c (A.10) (cid:104) V γ |O ( (cid:48) )7 | D (cid:105) = (cid:16) em c π (cid:17) c V ( T (0) P ± T (0) P ) . (A.11) B.1 Effective Wilson coefficients C eff7 , ( m c ) Let us state that we do not intend to give a critical review of the treatment of Wilsoncoefficients in the charm sector, e.g. of whether it makes sense to include light-quarksinto SD contributions evaluated in perturbation theory . We shall simply follow theliterature. It is fortunate that the SD contributions turn out to be subdominant in theSM. One might be tempted to say that if WA dominates by another two order of magnitudes then thisimplies that the CP asymmetry is automatically below 10 − . This is not correct as in this way of thinkingthe absolute value of λ b should be factored into A b and then Im[ λ b / | λ b | ] (cid:39) O (1) is not small any more. We are grateful to Ikaros Bigi and Ayan Paul to draw our attention to this point. C , ( m c ) and matrix elements which can be rewritten in terms of O , which we denoteby δC eff7 , ( m c ) C eff7 , ( m c ) = C , ( m c ) + δC eff7 , ( m c ) . (A.12)From a conceptual point of view the Wilson coefficient can be divided into two furthersub-parts, C , ( m c ) = C ( m W )7 , ( m c ) + C ( m b )7 , ( m c ) . (A.13)The notation above is non-standard but hopefully useful for clarity. For the reminder ofthis section we closely follow the notation of [36]. For C eff8 only C ( m W )8 ( m c ) = η c η b C ( m W ), η b = α s ( m W ) /α s ( m b ) and η c = α s ( m b ) /α s ( m c ), is known explicitly in the literature. For C eff7 all three parts are known which we shall quote, almost explicitly, below, C ( m W )7 ( m c ) = (cid:20) η c η b C ( m W ) − (cid:16) η c η b − η c η b (cid:17) C ( m W ) (cid:21) C ( m b )7 ( m c ) = − λ b (cid:88) i,j C j ( m b ) X ji η z i c , (A.14)where i = 1 .. j = 1 ..
6. Note C ( m W )7 ( m c ) describes the evolution directly from m W to m c and C ( m b )7 ( m c ) originates from integrating out the b -quark at the m b -scale and runningfrom m b to m c . We hasten to add that the above expressions are given in the leadinglogarithm approximation. The term from the four quark matrix element is given by [31] δC eff7 ( m c ) = α s ( m c )4 π C ( m c ) (cid:0) λ s f [( m s /m c ) ] + λ d f [( m d /m c ) ] (cid:1) . (A.15)The strong phase results from the charmed meson’s four momentum cutting the diagramthrough light quark lines. The contribution of C ( m c ) vanishes whereas the C , , , ( m c )have not been given but are small as they originate from SD contributions which them-selves are small. In fact the numerical hierarchy is as follows [31]: | C ( m W )7 ( m c ) | (cid:39) · − (cid:28) | C ( m b )7 ( m c ) | (cid:39) · − (cid:28) | δC eff7 ( m c ) | = 5 · − . (A.16)The hierarchy between the first two was noted in [33] and numerically improved in [31].The fact that matrix element dominates the Wilson coefficient was pointed out in [31].The expression of C ( m b )7 ( m c ) for operators other than O , was given recently in ref.[36]. As20entioned previously we are not aware of explicit results for C ( m b )8 ( m c ) and δC eff8 ( m c ) inthe literature, yet they can be expected to be close to their C -counterparts as they differonly by colour factors. Excluding cancellation effects we would expect them to equal upto O (1 /N c ) effects, say equal to about 30 − C ( m b )8 ( m c ) ≈ C ( m b )7 ( m c ) and δC eff8 ( m c ) ≈ δC eff7 ( m c ), are good for ourpurposes . Furthermore with C ( m c ) (cid:39) C ( m b )8 ( m c ) ≈ C ( m b )7 ( m c ) (cid:39) ( − . . i ) · − we see that the SM value is two to three order of magnitude below the reference valueIm[ C NP ] (cid:39) . · − . B.2 A CP ( D → V γ ) in the SM In the SM we identify three main sources contributing to the direct CP-asymmetry: a) C ( m c ) (cid:39) C ( m b )8 ( m c ) b) δC eff7 ( m c ) and c) δC eff8 ( m c ). Right-handed operators O ( (cid:48) )7 , arenegligible in the SM as Wilson coefficients as well as matrix elements are suppressed. Aspreviously mentioned we shall use C ( m c ) ≈ C ( m c ) for cases a) and c) which is good upto 1 /N c corrections. Note, as the leading LD amplitude is proportional to λ d , it is only λ s or λ b that can contribute to the direct CP-asymmetry.a) It is found that [31] C ( m b )7 ( m c ) (cid:39) . λ b (cid:39) (0 . − . i ) · − (A.17)and assuming, as discussed above, C ( m b )8 ( m c ) ≈ C ( m b )7 ( m c ), we get that this contri-bution compares with C NP8 in A CP as follows:0 .
06 Im[ λ b ]Im[ C NP8 ] ≈ − . · − . (A.18)b) It is found that δC eff7 ( m c ) = (0 . . i ) · − λ s + cλ d , (A.19)where the imaginary part, other than λ s , corresponds to a strong phase. The number c is of no importance for CP-violation as it can be absorbed into WA which isproportional to λ d and much larger. The contribution A CP compares with C NP8 asfollows: Im[ λ s ]Im[(0 . . i ) · − ] T (0)Im[ C NP8 ]Im[ G (0)] (cid:39) − · − , (A.20) Though the values C ( m W )7 , ( m c ) differ substantially for various reasons but this is of no concern asthey are small. T (0) = 0 . G (0)] = − . δC eff8 ( m c ) ≈ δC eff7 ( m c ) and this leads to a result forc) with Im[ G (0)] /T (0) (cid:39) / C NP8 ] (5)-contribution and with the value in (23) we get A CP | SM ( D → ( ρ , ω ) γ ) ≈ ( − . c B )( − · − ) ≈ · − , (A.21)with c B a correction factor for the branching ratio (24) which we set to unity in the laststep. We refrain from quoting a specific uncertainty. We would though think that thevalue catches the right order of magnitude. As possible criticisms one could advocate forexample the estimate C ( m b )8 ( m c ) ≈ C ( m b )7 ( m c ) and question the accuracy of local dualityin (A.19). We refer the reader to appendix A.1.1 for related discussions. The chargedcase is obtained by replacing Im[ G D ] → Im[ G D + ] in (A.20) and this would lead to A CP | D + SM ≈ . c B ( − · − ) ≈ − · − . B.3 A CP ( D → V γ ) through Im[ C NP7 ] and a strong LD-phase In reference [17] the idea was put forward that C ( m NP ) mixes into C ( m c ), e.g. Eq. (A.14)for the SM evolution. More precisely depending on the model and the scale of NP, M NP ,it was put forward [17] that this leads to comparable values . An important point isthat C ( m c ) hardly affects D → ππ/KK because of α -suppression and is therefore notconstrained by the latter. Following [17] we shall assume only SM degrees of freedombelow the scale M NP = 1 TeV and that the NP part of the Wilson coefficients is muchlarger than the SM part. Amending the notation of (A.14) to include the running of sixquarks above the top threshold one gets C (1 TeV)8 ( m c ) ≈ . C (1 TeV) , (A.22) C (1 TeV)7 ( m c ) ≈ . C (1 TeV) − . C (1 TeV) ≈ . C (1 TeV) − . C ( m c ) , and the analogous equations for the O (cid:48) , -operators. Eq. (A.22) exposes the dependenceof C ( m c ) on the scale M NP and C ( (cid:48) )7 ( M NP ). We shall somewhat arbitrarily choose thevalue Im[ C ( (cid:48) )NP7 ( m c )] ≈ − . C ( (cid:48) )NP8 ( m c )] as a reference values. This follows the modeldependent assumption | Im[ C ( (cid:48) )7 (1 TeV)] | (cid:28) | Im[ C ( (cid:48) )8 (1 TeV)] | in [17]. Note our normalisation of O differs from [17] by a factor of Q u which translates in to Q u C = C IK ,where IK stands for the authors of [17]. O matrix element itself, as opposed to δC eff7 , does not carry a strong phaseand the LD strong phase vanishes at leading order in the chiral limit, as discussed inappendix A.1.1, we did not include this effect in our results (23,25). In fact we estimatedthat the phases could be around | δ L,R | (cid:39) ◦ and we shall investigate how the CP-asymmetry changes. It is then useful to rewrite the g L amplitude as in (A.26) with thereplacement: [ C (2 G (0)) + C (2 T (0))] (A.22) → [2Im[ C ]( G (0) − . T (0) (cid:124) (cid:123)(cid:122) (cid:125) F )] (A.23)For T (0) = 0 . G D (0) (cid:39) − . − . i (cid:39) . e − i ◦ [24] one gets F (cid:39) − . − . i =0 . e − i ◦ . Thus a correction of the LD phase δ L,R = ± ◦ leads to a strong phasedifference between the two amplitudes in the range of 10 ◦ to 30 ◦ which corresponds to arescaling of the CP asymmetry by factors sin(10 ◦ ) / sin(20 ◦ ) (cid:39) . ◦ ) / sin(20 ◦ ) (cid:39) . O ( α s ) computation would presumably give an indication of the sign of the LD phase aswell as its size and would allow to make firmer statements. C Formulae for direct CP-violation
In this appendix we collect some formulae which are useful throughout the text. We shallparametrise an amplitude as follows, A ( D → f ) = A a e iδ a e iφ a + A b e iδ b e iφ b , (A.24)with weak (CP-odd) phases φ and strong (CP-even) phases δ separated to leave A a,b real.Note that in the SM the decomposition (A.24) is sufficient as one might use unitarity (9)to eliminate one amplitude to arrive at two amplitudes. Using the notation ∆ ≡ A a A b , δ ( φ ) ab = δ ( φ ) a − δ ( φ ) b the CP-asymmetry becomes: A CP [ D → f ] = − δ ab ) sin( φ ab )∆1 + 2∆ cos( δ ab ) cos( φ ab ) + ∆ (cid:28) (cid:39) − δ ab ) sin( φ ab )∆ . (A.25)23n the second line we have assumed a hierarchy between the amplitudes which is the casefor D → ( ρ , ω ) γ as studied in this paper. D Formulae for TDCP-violation
The replacement due to the relevance of O as described in subsection 3.3 is as follows: g L e iδ e iφ L → ˜ g L e i ∆ L e i Φ L = G F √ (cid:16) em c π (cid:17) c V [ C (2 G (0)) + C (2 T (0))] , (A.26)and for g R is given by the following replacements: L → R and C , C → C (cid:48) , C (cid:48) . Noteunlike before we cannot assume a common strong phase as the ratios C /C and C (cid:48) /C (cid:48) might not necessarily be the same. This is why the strong phase ∆ carries a chiralitylabel. The symbol Φ denotes the weak phase. 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