CP Violation in D0-D0bar Mixing and Electric Dipole Moments in SUSY Alignment Models
aa r X i v : . [ h e p - ph ] M a y CP Violation in D − ¯ D Mixing and Electric Dipole Momentsin SUSY Alignment Models a W. ALTMANNSHOFER
Theoretical Physics Department,Fermilab, P.O. Box 500, Batavia, IL 60510, USA
We report on a study of CP Violation in D − ¯ D mixing and Electric Dipole Moments in theframework of supersymmetric alignment models. Both classes of observables are strongly sup-pressed in the Standard Model and highly sensitive to new sources of flavor and CP violationthat can be present in models of New Physics. Supersymmetric alignment models genericallypredict large non-standard effects in D − ¯ D mixing and we show that visible CP violationin D − ¯ D mixing implies lower bounds for the EDMs of hadronic systems, like the neutronEDM and the mercury EDM, in the reach of future experimental sensitivities. We also giveupdated constraints on the mass insertions of the Minimal Supersymmetric Standard Modelusing the current data on D − ¯ D mixing. Models of New Physics (NP) often contain new sources of flavor violation and are thereforestrongly constrained by experimental data on Flavor Changing Neutral Current (FCNC) pro-cesses. This is in particular the case for Supersymmetric (SUSY) extensions of the StandardModel (SM) like the Minimal Supersymmetric Standard Model (MSSM), as long as the SUSYdegrees of freedom are not far above the TeV scale 1 ,
2. This so-called SUSY flavor problem is forexample addressed in SUSY alignment models 3 , D − ¯ D mixing 3.On general grounds D − ¯ D mixing observables are highly sensitive probes of the flavor sectorof NP models 5. Especially CP violation in D − ¯ D mixing is strongly suppressed in the SM by a talk given at Rencontres de Moriond EW 2011, March 13 - 20, 2011 ( V ub V cb /V us V cs ) ∼ − and experimental evidence for it considerably above the per mill levelwould clearly point towards the presence of NP (see however 6).In the following we give updated bounds on the mass insertions of the MSSM using the latestexperimental data on D − ¯ D mixing, we analyze the predictions of SUSY alignment models forCP violation in D − ¯ D mixing and show that sizable CP violating effects in D − ¯ D mixingimply lower bounds on the Electric Dipole Moments (EDMs) of hadronic systems within thisclass of SUSY models. The presentation is mainly based on 7. D − ¯ D Mixing
The neutral D meson mass eigenstates D and D are linear combinations of the strong inter-action eigenstates, D and ¯ D | D , i = p | D i ± q | ¯ D i , qp = s M ∗ − i Γ ∗ M − i Γ , (1)where M is the dispersive part and Γ the absorptive part of the D − ¯ D mixing amplitude h D |H eff | ¯ D i = M − i , h ¯ D |H eff | D i = M ∗ − i ∗ . (2)The normalized mass and width differences in the D − ¯ D system, x and y , are given by x = ∆ M D Γ D = 2 τ D Re (cid:20) qp (cid:18) M − i (cid:19)(cid:21) , y = ∆Γ2Γ D = − τ D Im (cid:20) qp (cid:18) M − i (cid:19)(cid:21) , (3)with the lifetime of the neutral D mesons τ D = 1 / Γ D = 0 . D − ¯ D mixing is firmly established with the non-mixing hypothesis x = y = 0 excluded at 10 . σ
8. Still, at the current level of sensitivity, there is no evidence forCP violation in D − ¯ D mixing. The experimental data on both | q/p | and φ = Arg( q/p ) iscompatible with CP conservation, i.e. | q/p | = 1 and φ = 0. The most recent world averages asobtained by HFAG read 8 x = (0 . +0 . − . )% , y = (0 . ± . , | q/p | = 0 . +0 . − . , φ = ( − . +9 . − . ) ◦ . (4)These experimental results on D − ¯ D mixing lead to strong constraints on possible new sourcesof flavor violation in extensions of the Standard Model 9 , , δ that can be defined as the deviations of the up and downsquark mass matrices from universality in the super-CKM basis M q = ˜ m Q (11 + δ q ) , δ q = (cid:18) δ LLq δ LRq δ RLq δ RRq (cid:19) , q = u, d . (5)Complex flavor off-diagonal mass insertions lead to flavor and CP violating gluino – squark –quark interactions that typically lead to huge contributions to FCNC processes. Taking intoaccount only gluino box contributions in the so-called mass insertion approximation, neglectingfor simplicity renormalization group effects as well as setting the B -parameter to 1 in the eval-uation of the hadronic matrix elements, one finds for the MSSM contributions to the D − ¯ D mixing amplitude the following approximate expression b M NP12 ≃ α s ˜ m Q m D f D (cid:20)(cid:16) ( δ LLu ) + ( δ RRu ) (cid:17) g ( x g )3 + ( δ LLu ) ( δ RRu ) m D m c (cid:18) g ( x g )4 + g ( x g )12 (cid:19)(cid:21) , (6) b In our numerical analysis we implement the full set of 1 loop MSSM contributions that can be found e.g.in 11, we include 2 loop renormalization group running 12 and use the hadronic matrix elements given in 9.igure 1: Updated constraints on the mass insertions ( δ LLu ) and ( δ RRu ) from D − ¯ D mixing for a commonsquark and gluino mass of ˜ m Q = M ˜ g = 1 TeV. where m D is the mass and f D the decay constant of the neutral D mesons. The loop functions g , g and g depend on the ratio x g = M g / ˜ m Q of the gluino and squark masses and theirexplicit expression can be found e.g. in 1. In the limiting case of degenerate masses one has g (1) = − , g (1) = and g (1) = − . In (6) we neglected contributions from δ LRu and δ RLu mass insertions. They are given e.g. in 1.In Fig. 1 we show the allowed regions for the mass insertions ( δ LLu ) and ( δ RRu ) . Asthe SM contributions to M and Γ cannot be predicted in a reliable way, we allow themto vary in the range − . − < M SM12 < . − and − . − < Γ SM12 < . − andimpose the constraints (4) at the 2 σ level. The bounds on the mass insertions are obtained fora SUSY spectrum with a common squark and gluino mass of ˜ m Q = M ˜ g = M SUSY = 1TeV andswitching on one mass insertion at a time. They scale as δ u /M SUSY and hold barring accidentalcancellations among the different contributions in (6).The case where both LL and RR mass insertions are present simultaneously is particularlystrong constrained (see right plot of Fig. 1). Even for the rather heavy SUSY spectrum that weconsider, the mass insertions have to be smaller then about 5 · − . For maximal phases of themass insertions, the bounds are stronger by approximately a factor of 3. D − ¯ D Mixing in SUSY Alignment Models
A popular class of SUSY models that generically predict large NP effects in D − ¯ D mixing areSUSY alignment models 3. The quark-squark alignment mechanism occurs naturally in modelswith abelian horizontal symmetries that reproduce the observed hierarchy in the SM Yukawacouplings. Interestingly, in the framework of alignment it is possible to predict for a broad classof abelian flavor models both lower and upper bounds for the mass insertions 13.The most characteristic prediction of alignment models is the appearance of a large ( δ LLu ) mass insertion that leads to large effects in D − ¯ D mixing. Indeed, SU (2) invariance impliesa relation between the left-left mass insertions in the up and down sector( δ LLu ) = ( V δ
LLd V † ) ≃ ( δ LLd ) + λ m c L − m u L ˜ m Q + O ( λ ) , (7)where V is the CKM matrix, λ ≃ . m ˜ u L and m ˜ c L are the left handedup and charm squark masses, respectively. As abelian flavor symmetries do not impose anyrestriction on the mass splittings between squarks of different generations, they are expected tobe non-degenerate with natural order one mass splittings. Correspondingly, even for ( δ LLd ) = 0,which is approximately satisfied in alignment models to avoid the strong constraints from Kaonmixing, there is an irreducible flavor violating term of order λ leading to c − u transitions. Notethat this ( δ LLu ) is real to a good approximation.s shown in 13, the right-right mass insertion leading to c − u transitions is predicted tobe λ < | ( δ RRu ) | < λ in abelian flavor models with alignment. This mass insertion isnaturally expected to be complex. Therefore, all CP violating phenomena in D − ¯ D mixingare dominantly generated by the following combination of mass insertionsIm M NP12 ∝ Im[( δ LLu ) ( δ RRu ) ] . (8)In the following we focus on two observables sensitive to CP violation in D − ¯ D mixing:(i) the semileptonic asymmetry a SL and (ii) the time dependent CP asymmetry in decays to CPeigenstates S f .The semileptonic asymmetry in the decay to “wrong sign” leptons is defined as a SL = Γ( D → K + ℓ − ν ) − Γ( ¯ D → K − ℓ + ν )Γ( D → K + ℓ − ν ) + Γ( ¯ D → K − ℓ + ν ) = | q | − | p | | q | + | p | (9)and is a direct measure of CP violation in the mixing. However, as the decay rates to the“wrong sign” leptons are strongly suppressed by x + y , measurements of this asymmetry areexperimentally challenging.Also the time dependent CP asymmetry S f in decays to a common CP eigenstate f , akalifetime CP asymmetry ∆ Y f , is a sensitive probe of CP violation in D − ¯ D mixing 14 , S f = 2∆ Y f = 1Γ D (cid:16) ˆΓ ¯ D → f − ˆΓ D → f (cid:17) , (10) η CP f S f = η CP f Y f = x (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) sin φ − y (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) cos φ . (11)Here η CP f is the CP parity of the final state f . While singly Cabibbo suppressed decay modes canin principle be affected by new weak phases in the decay 16, possible effects in the lifetime CPasymmetry are strongly constrained by existing data on time integrated CP asymmetries 17 , η CP f S f is universal for all final statesand practically independent of direct CP violation in the decays. In fact, time dependent CPasymmetries are currently determined from the singly Cabibbo suppressed D → K + K − and D → π + π − modes and one has 8 η CP f S f = ( − . ± . . (12)Concerning Cabibbo favored decay modes, the most promising channel seems to be D → K S φ a SL and S f m < M / < | A | < m and 5 < tan β <
55. At the GUTscale we fix | ( δ RRu ) | = λ with an O(1) phase and set a mass splitting between the 1 st and 2 nd generation of squarks such that m ˜ u L = 2 m ˜ c L = 2 m . We find that in this setup the full range ofvalues for a SL and S f that is compatible with the experimental constraints (4) can be reached. Electric Dipole Moments represent very clean probes of CP violation in extensions of the SM 20.While the SM predicts EDMs far below the present experimental bounds 21 d Tl ≤ . × − e cm @ 90% C.L. , (13) d Hg ≤ . × − e cm @ 95% C.L. , (14) d n ≤ . × − e cm @ 90% C.L. , (15) igure 2: The semileptonic asymmetry a SL (left) and the neutron EDM d n (right) as a function of S f in SUSYalignment models. The gray region is excluded by the present data on S f . New Physics models that introduce new sources of CP violation are often strongly constrained bythese bounds. In particular in the MSSM with SUSY particles at the TeV scale, flavor diagonalCP violating phases of e.g. the gaugino masses, the higgsino mass or the trilinear couplings arestrongly constrained 22 at the level of 10 − .In the MSSM with flavor violating soft terms, large NP effects for the hadronic EDMs can benaturally generated (see e.g. 23). In particular, within SUSY alignment models, we find that thedominant SUSY contributions to the hadronic EDMs arise from “flavored” gluino – up squarkcontributions to the up quark (C)EDM. At the SUSY scale one has (cid:26) d u e , d cu (cid:27) ≃ − α s π m c M ˜ g A c ˜ m Q n f ( x g ) , f c ( x g ) o Im (cid:2) ( δ LLu ) ∗ ( δ RRu ) (cid:3) , (16)with the loop functions f and f c given e.g. in 23. Even though this contribution is suppressed bya double flavor flip, the corresponding up quark (C)EDM is sizable due to the chiral enhancementby the charm quark mass. As in alignment models ( δ LLu ) is real to an excellent approximation,the up quark (C)EDM (16) and CP violation in D − ¯ D mixing (8) is induced by the samecombination of mass insertions. As also the charm trilinear coupling A c that enters (16) isnaturally of the order of the gluino and squark masses, CP violating contributions to D − ¯ D mixing automatically also imply a non-zero up quark (C)EDM that in turn will induce EDMs ofhadronic systems like the neutron EDM d n or the mercury EDM d Hg , but not of the ThalliumEDM d Tl .In the right plot of Fig. 2 we show the correlation between the time dependent CP asymmetry S f and the neutron EDM d n in SUSY alignment models (we use the same setup as described atthe end of Sec. 3). We observe that visible CP violating effects in D − ¯ D mixing imply a lowerbound on the neutron EDM. For | S f | > .
1% we find d n > few · − e cm and simultaneouslyfor the mercury EDM d Hg > few · − e cm which is an interesting level in view of futureexperimental sensitivities. Electric Dipole Moments and CP violation in D − ¯ D mixing are examples of low energyobservables that are highly suppressed in the SM. Experimental evidence for them significantlyabove the tiny SM predictions would unambiguously signal the presence of NP.Supersymmetric alignment models generically predict large non-standard effects in D − ¯ D mixing 3. In addition, as we demonstrated in 7, large CP violating effects in D − ¯ D mixing inUSY alignment models, generically also imply lower bounds for the EDMs of hadronic systems,like the neutron EDM and the mercury EDM, within the future experimental sensitivities.Correspondingly, the simultaneous evidence of CP violation in the neutral D meson systemtogether with non-vanishing hadronic EDMs would strongly support the idea of SUSY alignmentmodels. Acknowledgments
I would like to thank the organizers for the invitation to Moriond EW 2011, Andrzej Burasand Paride Paradisi for the interesting collaboration and Stefania Gori for a reading of themanuscript. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.
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