Bs Mixing and Electric Dipole Moments in MFV
aa r X i v : . [ h e p - ph ] J u l B s Mixing and Electric Dipole Moments in MFV
Brian Batell ( a ) and Maxim Pospelov ( a,b )( a ) Perimeter Institute for Theoretical Physics, Waterloo, ON, N2J 2W9, Canada ( b ) Department of Physics and Astronomy, University of Victoria,Victoria, BC, V8P 1A1 Canada
Abstract
We analyze the general structure of four-fermion operators capable of introducing CP -violation preferentially in B s mixing within the framework of Minimal Flavor Violation.The effect requires a minimum of O ( Y u Y d ) Yukawa insertions, and at this order we finda total of six operators with different Lorentz, color, and flavor contractions that lead toenhanced B s mixing. We then estimate the impact of these operators and of their closerelatives on the possible sizes of electric dipole moments (EDMs) of neutrons and heavyatoms. We identify two broad classes of such operators: those that give EDMs in the limitof vanishing CKM angles, and those that require quark mixing for the existence of non-zeroEDMs. The natural value for EDMs from the operators in the first category is up to anorder of magnitude above the experimental upper bounds, while the second group predictsEDMs well below the current sensitivity level. Finally, we discuss plausible UV-completionsfor each type of operator.June 2009 . Introduction Studies of the B d mesons in the last decade [1] have confirmed the economical Cabibbo-Kobayashi-Maskawa (CKM) mechanism of flavor and CP -violation implicit in the StandardModel (SM), described entirely by three mixing angles and a single phase. Despite a con-tinuing improvement in precision, a practical question arises as to the best strategy to lookfor new flavor/ CP -violating effects in B -systems. One formidable opportunity is presentedby the B s system, where the CP -violation due to the SM CKM phase is naturally small,so that a large amount of CP violation would necessarily be attributable to a New Physics(NP) source. In other words, CP violation in the B s system belongs to the same CKM-background-free category of tests as electric dipole moments (EDMs) of neutrons and heavyatoms.It is therefore intriguing that recent tests of CP violation in the decays of B mesonsat the Tevatron experiments show a deviation from the predictions of the SM. Specifically,both the CDF and D0 experiments have reported correlated measurements of the widthdifference ∆Γ s and CP -violating phase φ J/ψφs from an analysis of the angular distributionsof the flavor-tagged decays B s → J/ψφ . Early results [2, 3] showed a deviation with theSM at the 2 . σ level [4], while a more recent preliminary measurement [5] is consistent withthe SM, albeit still with relatively large error bars. More recently, the D0 collaborationannounced [6] the measurement of the same-sign dimuon asymmetry in B -decays: A b sl ≡ N ++ b − N −− b N ++ b + N −− b = − (9 . ± . ± . × − , (1)where N ++ b is the number of events in which two antimuons are produced from the decaysof a B and ¯ B meson, i.e. b ¯ b → µ + µ + X . This is to be compared to the SM prediction, A b sl (SM) = (2 . +0 . − . ) × [7, 8], and thus shows a 3 . σ deviation from the predicted value.Given the tighter constraints in the well-studied B d system, this measurement therefore alsohints at a NP source of CP violation in B s mixing. Indeed, interpreted at face value, theD0 result [6] implies the presence of a CP -violating part of the B s − ¯ B s ,Im(∆ M ) ∼ O ( | M SM12 | ) ∼ O (10 psec − ) . (2)We refer the reader to the detailed numerical fits of the data recently performed in Refs. [9,10]. We focus on NP contributions to ∆ M , but see [11] for a recent analysis of the possibleNP contributions to Γ . Perhaps the ultimate test of CP in B s systems will be deliveredby the LHCb experiment in the near future.If these recent measurements from the Tevatron experiments are to be ascribed to a NPsource of CP violation, a natural question is how such NP contributes to EDMs. On theexperimental front, significant progress has been achieved in the measurement of Hg EDM,where the upper bound has been improved by a factor of 7 [12], | d Hg | < . × − e cm . (3)This considerably tightens all bounds on flavor-conserving hadronic and semi-leptonic CP -violating operators, and in particular implies rather strong bounds on the color EDMs ofquarks. 1n this note we investigate a possible connection between CP violation in B s mixing andEDMs within the framework of Minimal Flavor Violation (MFV) [13, 14]. This frameworkassumes that NP, should it induce flavor change, preserves the re-parametrization indepen-dence of the SM flavor physics. In other words, the flavor transitions are governed by theCKM matrix V CKM and the eigenvalues of the Yukawa couplings, but new CP -violatingphases introduced in a flavor-universal way are allowed [15, 16]. We identify all ∆ F = 2four-fermion operators leading to the preferential introduction of CP violation in the B s system. To leading order in Yukawa insertions, the required operators arise at O ( Y u Y d ).These are specific realizations of the class of operators pointed out in Ref. [17], where thegeneral consequences of Yukawa insertions at any order were investigated. The origin of CP violation is flavor-blind, and the enhancement of its effect in B s relative to B d is governed bythe ratio of Yukawa couplings y s /y d [17]. Notice that despite the fact that the total number ofYukawa insertions is rather large, the effect is not necessarily hopelessly small simply becausethe scale of the Yukawa coupling in the down-type sector is very uncertain. In particular, theYukawa couplings of b -quark may not lead to any suppression at all, y b ∼ O (1), as happensin large tan β two-Higgs doublet models. An example of a model in this category would bethe large tan β supersymmetric model, where even in the limit of very heavy superpartnersthe Higgs exchange leads to important effects both in flavor physics [18] and EDMs [19, 20].For an MFV vs EDM discussion, see [21]. See also Ref. [22] for an early study of SUSYeffects on lepton asymmetries in B systems.Normalizing the size of these CP -violating four-fermion operators to a putative signal in B s decays, i.e. to the maximal size consistent with the mass splitting, we next address thequestion of EDMs. We point out that all these operators and their close relatives can befurther subdivided into two broad classes. The first class is contains the scalar-pseudoscalarLorentz structure S × P and survives in the limit of V CKM →
1. The neutron EDM ispredicted typically close to experimental bounds and the natural size of the mercury EDMis up to one order of magnitude above the experimental limits. The second class of operatorshas the structure of left- times right-handed vector current, ( V − A ) × ( V + A ). To have CP violation, this class of operators requires more than one generation, and as a result allEDMs acquire additional suppression by V ts or V td , bringing them well below the sensitivityof modern EDM experiments. We provide examples of possible UV completions for bothtypes of operators.In Section 2 we present a detailed classification of these operators, along with the estimateof their plausible size that makes them detectable in the B s mixing. Section 3 provides somebackground information on EDMs and estimates the contribution from each operator to d Hg and d n . Finally, Section 4 contains a discussion of possible UV completions and presents ourconclusions. C P -violating MFV operators for the B s mixing The rules of the game in MFV are defined by the requirement to retain the existing flavorre-parametrization freedom of the SM. Since the right-handed rotations remain unphysical,2e use this freedom to put all Yukawa couplings to the Hermitian form Y † u = Y u , Y † d = Y d , (4)which allows us for shorter expressions.A four-fermion operator relevant for B s mixing can be written in the following generalform, O = (¯ b Γ A s )(¯ b Γ ′ A s ) , (5)where Γ ( ′ ) A stands for all possible Lorentz and color contractions. Lifting such operators to theflavor space according to the MFV rules, one encounters two types of MFV operators: thosethat give identical CP -violating NP contributions in both B d and B s mixing in comparisonto their SM values, and those that are enhanced in B s mixing [17]. We are interested in thelatter, and therefore specify the following criteria for selecting the operators:1. O violates CP and contains ∆ F = 2 transitions.2. The contribution of O to the mixing in B s is enhanced over B d by y s /y d .3. O survives the limit of y c , y u →
0, and involves no more than one power of y s or y d .4. O is a local Lorentz scalar mediated by the exchange of particles heavier than m B .5. The total number of Yukawa insertions in O is minimized.Needless to say, we also require an overall electric and color neutrality of O , but do notimpose any SU (2) × U (1) requirements as those can be easily satisfied by an appropriatenumber of Higgs insertions. While the first four conditions on the list are crucial for ourdiscussion, the last one is for book-keeping purposes only, as any Y u insertions can be takenas ( Y u ) n , resulting in a proliferation of powers of top quark Yukawa, but not bringing anyadditional numerical smallness. The condition of locality explicitly forbids a situation wherethe B -mixing is mediated by a neutral particle with a mass close to m B s or m B d , whichcan enhance the mixing in either of these two systems via a resonance, with a possibility ofdistorting MFV relations.When generalizing (5) to the full flavor space, one has to remember that flavor indicescan be contracted outside of a quark pair that has its Lorentz/color indices contracted. Wechoose to eliminate such operators using completeness in the flavor sector, but then haveto account for all possible Lorentz and color structures. Calling the left- and right-handeddown-quarks as Q and D , with the above guiding conditions at hand, we arrive at thefollowing set { O s } of the effective CP -violating operators: O = i ( ¯ Q k Y u Y d D k ) ( ¯ D l Y d [ Y d , Y u ] Q l ) + ( h.c. ) , (6) O = i ( ¯ Q k Y u Y d D l ) ( ¯ D l Y d [ Y d , Y u ] Q k ) + ( h.c. ) ,O = i ( ¯ D k Y d [ Y d , Y u ] Y d γ µ D k ) ( ¯ Q l Y u γ µ Q l ) ,O = i ( ¯ D k Y d [ Y d , Y u ] Y d γ µ D l ) ( ¯ Q l Y u γ µ Q k ) ,O = i ( ¯ D k Y d Y u Y d γ µ D k ) ( ¯ Q k [ Y d , Y u ] γ µ Q l ) ,O = i ( ¯ D k Y d Y u Y d γ µ D l ) ( ¯ Q l [ Y d , Y u ] γ µ Q k ) .
3n these expressions, [ , ] denote commutators in the flavor space, superscripts k and l showthe contraction of color SU (3) indices while the Lorentz and flavor indices are contractedwithin each parentheses. From the point of view of ∆ F = 2 flavor transitions, the set { O s } is clearly over-complete. Indeed, e.g. to leading order in the strange quark Yukawa couplingseveral of them lead to the same (¯ b L s R )(¯ b R s L ) or (¯ b L γ µ s L )(¯ b R γ µ s R ) operators. However, weshould not count these operators as a priori redundant as they might contain differences in∆ F = 0 channels and thus have different manifestations in the EDMs. All of these operatorscontain commutators in the flavor space and therefore vanish in the limit of V CKM → O − O are clearly dictated by the properties of Hermiticity,the choice of flavor structure in O and O is not unique. One can take, for example, i ( ¯ Q k Y u Y d D k ) ( ¯ D l Y d Y u Q l ), which also leads to CP -violation in B s mixing. However, thiscan be reduced to O since a structure analogous to O with an anti-commutator instead ofcommutator gives a vanishing contribution to ∆ F = 2 operators.Choosing the normalization constant to be G F / √
2, we combine these operators into aneffective CP -odd Lagrangian weighted with dimensionless coefficients c i : L CP = G F √ X i =1 .. c i O i . (7)Next, we reduce Eq. (7) to the subset of operators leading to ∆ F = 2 transitions for B -mesons, finding two independent structures for each light flavor: L CP ∆ F =2 = i G F √ y t y b V ∗ tb X q = d,s y q V tq (cid:2) C SLR (¯ b L q R )(¯ b R q L ) + C V LR (¯ b L γ µ q L )(¯ b R γ µ q R ) (cid:3) + ( h.c. ) , (8)where the Wilson coefficients are related to the original classification as follows: C SLR = 2( c − c − c ); C V LR = c + c − c (9)These operators should be evolved using perturbative QCD from the scale where theyare generated to the B -meson energy scale. It is hard to do this in general since we do notknow the actual scale where these operators are generated. A reasonable assumption is thatthis scale is rather large, comparable to the EW scale, in which case we can directly use theresults of QCD evolution and the calculated matrix elements already present in the literature(see e.g. [23]). This produces the following estimate of the CP -odd mixing part of B s .Im( M ) ≃ G F √ m B s F B s y t y b y s | V ∗ tb V ts | × ( P C V LR + P C SLR ) (10) ≃ (10psec − ) × y b y s − [ c − c − c + 0 .
33 ( c − c − c )] , where P ≃ − .
62 and P ≃ .
46 are from Ref. [23]. The eigenvalues of Yukawa matrices arenormalized at a high-energy scale. We have disregarded the small complex phase of V ∗ tb V ts ,and took this product to be equal to 0.04, and y t ≃
1. The overall coefficient in (10) ischosen to be very close to half of the measured absolute value of ∆ M B s .4esides the presence of six unknown Wilson coefficients, an additional uncertainty inthe estimate (10) is the value of the combination of Yukawa couplings from the down-quarksector. In the SM such a combination is hopelessly small, but at large tan β this combinationcan be as large as 10 − , so that (10) will cause large effect in B s mixing. It should also be saidthat at very large tan β the relation between measured masses and eigenvalues of the Yukawacouplings, e.g. y s /y b ≃ m s /m b weakens considerably because of the possibility of very largecorrections to the mass operator [24]. The result (10) shows that there is some room for thegeneration of { O s } at the weak scale with nearly maximal tan β , a point emphasized recentlyin Ref. [25].Current experiments are capable of detecting only the maximal amount of CP -violating B s mixing [26–28]. Therefore, as a benchmark, we choose to equate the the imaginary part ofthe NP contribution to ∆ m B s / ≃
10 psec − . This benchmark fixes the combination of y b y s times the linear combination of Wilson coefficients. We shall now use this as an approximateinput for the estimates of the natural size of EDMs in this framework.
3. Natural size of EDMs from { O s } and its extensions Electric dipole moments of heavy atoms and neutrons (see Refs. [29, 30] for reviews) isa powerful probe of new CP -violating physics at and above the weak scale. EDMs do notrequire flavor transition and therefore may be induced by NP even in the limit of V CKM → CP -violatingphases, all operators in the set { O s } vanish at V CKM →
1. Moreover, it turns out thatoperators O and O do not contain CP -violating terms for ∆ F = 0 processes. Indeed, theflavor commutator requires the presence of quarks from two different generations, e.g. s and b , which makes such operators ∝ (¯ b L s R )(¯ s R b L ). These in turn can be Fierz-transformed tothe products of s - and d - vector and axial vector currents, (¯ s R γ µ s R )(¯ b L γ µ b L ), that alwaysconserve CP . Retaining only y b y s ( d ) -proportional contributions, we choose to eliminate all( V − A ) × ( V + A ) operators with Fierz transformations, arriving at the following ∆ F = 0component of the effective Lagrangian (7): L CP ∆ F =0 = i G F √ y t y b X q = d,s y q | V tb V tq | (cid:2) ( c − c )(¯ b kL b kR )(¯ q lR q lL ) + ( c − c )(¯ b kL b lR )(¯ q lR q kL ) (cid:3) + ( h.c. ) . (11)Thus, even before estimating the actual size of the EDMs, we can conclude that there existnatural choices of operators within MFV, such as O and O , that contribute to CP -violationin B s but do not lead to EDMs.Turning to the actual size of the EDMs induced by (11), we expect them to be very smallon account of the additional suppression by | V td | ∼ − . There are several pathways forthe Lagrangian (11) to contribute to EDM observables. Integrating out the b -quark inducesEDMs d q and color EDMs ˜ d q of light quarks that in turn lead to a neutron EDM, as well as CP -odd nuclear forces that manifest in d Hg . It is easy to see that L CP ∆ F =0 mixes with d q and˜ d q only at two-loop level.A good proxy for the strength of CP -odd nuclear forces is given by the CP -odd pion-nucleon and η -nucleon coupling constants. Besides being induced by the two-loop ˜ d q , these5ouplings also receive a more direct contribution from (11) [19, 31] via the “heavy quarkcontent” of a nucleon, h N | m b ¯ bb | N i ≃
65 MeV. Taking the ¯ bb ¯ diγ d part of L CP ∆ F =0 , we estimatethe strength of the pion-nucleon coupling constant to be g πNN = ( c − c ) G F √ | V tb V tq | y t y b y d × η QCD R QCD f π , (12)where R QCD is a factorized combination of QCD condensates and nucleon matrix elementsof quark bi-linears: R QCD = h N | ¯ bb | N ih | ¯ dd | i − h N | ¯ dd | N ih | ¯ bb | i ∼ × − (GeV) . (13)The numerical value of R QCD is obtained assuming the following values of the quark con-densates and matrix elements: h N | ¯ bb | N i ∼ . × − , h | ¯ bb | i = −h G a µν i α s / (12 πm b ) ∼− (55 MeV) , h N | ¯ dd | N i ∼
4, and h | ¯ dd | i ∼ − (250MeV) . There is also an enhancementcoefficient η QCD ∼ . b L b R ¯ d R d L operator from the UV scaledown to the scale m b . All of these steps produce the following estimate of the pion-nucleoncoupling constant: g πNN ∼ × − × y b y s − ( c − c ) , (14)where we also used y d /y s ∼ . m b ≪ m [32–34]:˜ d − loopd = α s m b π G F √ | V tb V tq | y t y b y d × ln Λ UV m b × (cid:18) c − c + 76 ( c − c ) (cid:19) (15) ≃ y b y s − (cid:18) c − c + 76 ( c − c ) (cid:19) × × − cm . (16)In the second relation we also used Λ UV ∼ G − / F and chose α s ∼ .
15. This result should becompared with the limit on the CP -odd pion-nucleon coupling in the isospin = 1 channelextracted from the mercury EDM [12] and the implied limit on color EDMs of quarks [12,29, 35]: | g πNN | < − ; | ˜ d d − ˜ d u | < × − cm . (17)It shows that the choice y b y s c i ∼ O (10 − ) motivated by maximal CP -violation in B s mixing(10) yields EDMs that are three orders of magnitude below the experimental bounds. Eventaking into account significant theoretical uncertainties involved in estimates (12) and (16)as well as nuclear/QCD uncertainties in extracting (17) from d Hg , one can conclude that theEDMs induced by operators from { O s } are well below current levels as well as anticipatedfuture experimental sensitivity benchmarks.One of the main reasons why the results (12) and (16) are so small is the strong sup-pression coming from the factor | V td | ≃ − , which is a consequence of the commutators inflavor space present in every member of { O s } . In this respect, it is reasonable to investigatewhether close “flavor relatives” of O i give EDMs in the limit V CKM →
1. We define a flavorrelative as a modification of an operator O i where some flavor-structure is added/removed6ithin each quark pair in a way consistent with MFV. For example, a “minimal” flavorrelative of O would be the operator ( ¯ Q k Y d D k )( ¯ D l Y d Q l ). It is easy to see that all operators O − O that involve a product of left- and right-handed currents do not have flavor relativesthat give large EDMs. Indeed, removing any of the Y u or Y d insertions leads to operatorsthat conserve CP . Therefore, flavor relatives of O − O always require V CKM = 1 to induceEDMs, and these EDMs are small according to (12) and (16). On the contrary, there existsflavor relatives of O and O that do give EDMs in the limit of V CKM → O , → i ( ¯ Q k Y d D k )( ¯ D l Y d Q l ) + ( h.c. ) → iy b y d (¯ b L b R )( ¯ d R d L ) + ( h.c. ) . (18)If UV physics generates these relatives of O and O with similar size Wilson coefficients,the resulting EDMs are four orders of magnitude above (12) and (16), on the order of˜ d d ∼ × − cm. This corresponds to EDMs right at the current level of experimentalsensitivity for the neutron [36] and about one order of magnitude above the current boundsfor mercury. The effective field theory approach does not allow one to make a more refinedstatement before the UV physics is specified.As a final comment in this section, it may still be possible that D0 same-sign dimuonasymmetry [6] has a non-negligible contribution from a NP source that does not differentiatebetween B s and B d by an extra factor of y s /y d , and for whatever reason the presence ofNP in B d has not been detected elsewhere. In this case, the spectrum of CP -violatingoperators broadens rather considerably. Some of these operators, such as pure left-handedtype i ( ¯ Q [ Y u , Y d ] γ µ Q )( ¯ QY u γ µ Q ) and its flavor relatives, cannot induce large EDMs. Othersinvolve the chirality flip, ( ¯ QY u Y d D )( ¯ QY u Y d D ), and their flavor/Lorentz relatives may inducesignificant EDMs even at one-loop level: i ( ¯ QY d σ µν D )( ¯ QY d σ µν D ) → iy b y d (¯ b L σ µν b R )( ¯ d L σ µν d R ) → d − loopd . (19)If the combination y b y d corresponds to a choice of maximal tan β , such one-loop EDMs willbe very large, being enhanced relative to (16) by O (100 × V − td ) ∼ , and indeed severalorders of magnitude above all EDM bounds. A detailed analysis of such operators fallsoutside the scope of the present paper.
4. Discussion
In this paper we have presented the explicit form of the CP -odd ∆ B = 2 MFV operatorsthat predominantly contribute to the CP -violation of B s mesons [17]. Since the CP -violatingproperties are governed by the flavor-universal phase in front of these operators, one mighthave expected large effects for EDMs. On the contrary, we have found that { O s } requiresnon-zero CKM mixing matrix elements for the EDMs to exist. This extra | V td | suppressionplaces EDMs directly induced by { O s } well below the experimental bounds. At the sametime “close flavor relatives” of scalar operators O and O give EDMs on the order of 10 − cm, comparable or even somewhat larger than the current best limits [12]. On the otherhand, operators of the type O − O that involve the product of left- and right-handedcurrents ( V − A ) × ( V + A ) do not have flavor relatives that generate EDMs in the limit V CKM →
1. Our general analysis has implications for the specific UV completion schemes7hat may be responsible for generating operators O i that lead to preferential CP -violationin the B s system. The exchange of MFV scalars [9,37] can generate O and/or O operators and their flavor relatives, and there is no good argument why EDMsshould be small. In the minimal supersymmetric model (MSSM), Higgs exchange at largetan β combined with SUSY radiative corrections to the mass sector of down-type quarks canbe a significant source of ∆ F = 2 operators. If CP -violation is introduced in an MFV-likefashion, it has to be sourced by the relative phase of the µ -parameter in the superpotentialand the gluino mass parameter. To have an effect on B s , this phase would have to be nearlymaximal. Besides the effects induced by O and O discussed in this paper, there will be ofcourse the one-loop tan β -enhanced EDMs that will be sensitive to the scale of the light scalarquark and lepton masses in excess of ∼ m A ( H ) are kept under a TeV, there are still non-vanishing contributions to atomic EDMson account of hadronic and semileptonic four-fermion operators, e.g. ¯ qq ¯ eiγ e [19]. Giventhe analysis of this paper and of Ref. [19], such EDMs will be at the current limits ofexperimental observability (or even slightly above current bounds). It would be interestingto investigate CP -violation in the B s system and EDMs in SUSY models extended by neutralchiral superfields (NMSSM) on account of possibly light mediators and new sources for the CP -violating phases. Vector and pseudoscalar exchange.
The exchange by neutral vector particles such as Z -bosons (or hypothetical Z ′ ) is the most relaxed possibility with respect to the EDM con-straints because it results only in O − O operators. The Z -boson, of course, does not haveany flavor-changing couplings, and those would have to be generated by integrating out NP.An explicit example of such couplings consistent with MFV was given recently in modelswith extra vector-like quarks [38]. These models may also have additional, and not neces-sarily small effects in B d and K -mesons from pure left-handed operators, ¯ QY u Q ¯ QY u Q , andthe associated CKM phase. The derivatively-coupled pseudoscalar particles, with couplings f − a ∂ µ a ¯ Dγ µ D and alike, are in principle capable of inducing similar effects if f a is under aTeV, but on account of the extra derivative would necessarily have to be relatively light, m a ∼ m B . Exchange by particles transforming nontrivially under flavor.
Lastly, the exchange byparticles that transform non-trivially under flavor-rotations (see e.g. Refs. [38, 39]) is alsocapable of inducing { O s } . In this case, however, the correspondence between spin of themediators and our operator classification may be different than in the case of the mediatorsthat transform trivially under flavor. For examples, exchange by scalars that transform as(3 , ¯3) under the SU (3) Q and SU (3) D would generate operators ¯ Q f D j ¯ D j Q f where j, f areflavor indices. After a Fierz transformation, this operator obtains the ( V − A ) × ( V + A )structure and therefore requires additional CKM suppression to induce EDMs.8 cknowledgements M.P. would like to thank the participants of the JHU workshop for a number of usefuldiscussions. The work of M.P. is supported in part by NSERC, Canada, and research at thePerimeter Institute is supported in part by the Government of Canada through NSERC andby the Province of Ontario through MEDT.
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