charming new physics in rare b decays and...

Charming new physics in rare B decays and mixing?

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Jaeger, Sebastian, Leslie, Kirsten, Kirk, Matthew and Lenz, Alexander (2018) Charming new physics in rare B decays and mixing? Physical Review D (PRD), 97 (1). pp. 1-8. ISSN 2470-0029

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Charming new physics in rare B decays and mixing?

Sebastian Jäger* and Kirsten Leslie‡

University of Sussex, Department of Physics and Astronomy, Falmer, Brighton BN1 9QH, United Kingdom

Matthew Kirk† and Alexander Lenz§

IPPP, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom

(Received 3 February 2017; published 31 January 2018)

We conduct a systematic study of the impact of new physics in quark-level b → cc̄s transitions on Bphysics, in particular rare B decays and B-meson lifetime observables. We find viable scenarios where asizable effect in rare semileptonic B decays can be generated, compatible with experimental indications andwith a possible dependence on the dilepton invariant mass, while being consistent with constraints fromradiative B decay and the measured Bs width difference. We show how, if the effect is generated at the weakscale or beyond, strong renormalization-group effects can enhance the impact on semileptonic decays whileleaving radiative B decay largely unaffected. A good complementarity of the different B-physicsobservables implies that precise measurements of lifetime observables at LHCb may be able to confirm,refine, or rule out this scenario.

DOI: 10.1103/PhysRevD.97.015021

I. INTRODUCTION

Rare B decays are excellent probes of new physics atthe electroweak scale and beyond, due to their strongsuppression in the Standard Model (SM). Interestingly,experimental data on rare branching ratios [1,2] and angulardistributions for B → Kð�Þμþμ− decay [2,3] may hint at abeyond-SM (BSM) contact interaction of the formðs̄LγμbLÞðμ̄γμμÞ, which would destructively interfere withthe corresponding SM (effective) coupling C9 [4–6],although the significance of the effect is somewhat uncer-tain because of form-factor uncertainties as well asuncertain long-distance virtual charm contributions [7].However, if the BSM interpretation is correct, it requiresreducing C9 by Oð20%Þ in magnitude. Such an effectmight arise from new particles (see e.g. [8]), which might inturn be part of a more comprehensive new dynamics.Noting that in the SM, about half of C9 comes from (short-distance) virtual-charm contributions, in this article we askwhether new physics affecting the quark-level b → cc̄stransitions could cause the anomalies, affecting rare B

decays through a loop. The bulk of these effects would alsobe captured through an effective shift ΔC9ðq2Þ, with apossible dependence on the dilepton mass q2. At the sametime, such a scenario offers the exciting prospect ofconfirming the rare B-decay anomalies through correlatedeffects in hadronic B decays into charm, with “mixing”observables such as the Bs-meson width difference stand-ing out as precisely measured [9] and under reasonabletheoretical control. This is in contrast with the Z0 andleptoquark models usually considered, where correlatedeffects are typically restricted to other rare processes andare highly model dependent. Specific scenarios of hadronicnew physics in the B widths have been consideredpreviously [10], while the possibility of virtual charmBSM physics in rare semileptonic decay has been raisedin [11] (see also [12]). As we will show, viable scenariosexist, which can mimic a shift ΔC9 ¼ −Oð1Þ while beingconsistent with all other observables. In particular, verystrong renormalization-group effects can generate largeshifts in the (low-energy) effective C9 coupling from smallb → cc̄s couplings at a high scale without conflicting withthe measured B̄ → Xsγ decay rate [13].

II. CHARMING NEW PHYSICS SCENARIO

We consider a scenario where new physics affects theb → cc̄s transitions. This could be the case in modelscontaining new scalars or new gauge bosons, or stronglycoupled new physics. Such models will typically affectother observables, but in a model-dependent manner. Forthis paper, we restrict ourselves to studying the new effects

*[email protected]†[email protected]‡[email protected]§[email protected]

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 97, 015021 (2018)

2470-0010=2018=97(1)=015021(8) 015021-1 Published by the American Physical Society

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induced by modified b → cc̄s couplings, leaving construc-tion and phenomenology of concrete models for futurework. We refer to this as the “charming BSM” (CBSM)scenario. As long as the mass scale M of new physicssatisfies M ≫ mB, the modifications to the b → cc̄s tran-sitions can be accounted for through a local effectiveHamiltonian,

Hcc̄eff ¼4GFffiffiffi

2p V�csVcb

X10i¼1

ðCciQci þ Cc0i Qc0i Þ: ð1Þ

We choose our operator basis and renormalization schemeto agree with [14] upon the substitution d → b, s̄ → c̄,ū → s̄:

Qc1 ¼ ðc̄iLγμbjLÞðs̄jLγμciLÞ; Qc2 ¼ ðc̄iLγμbiLÞðs̄jLγμcjLÞ;Qc3 ¼ ðc̄iRbjLÞðs̄jLciRÞ; Qc4 ¼ ðc̄iRbiLÞðs̄jLcjRÞ;Qc5 ¼ ðc̄iRγμbjRÞðs̄jLγμciLÞ; Qc6 ¼ ðc̄iRγμbiRÞðs̄jLγμcjLÞ;Qc7 ¼ ðc̄iLbjRÞðs̄jLciRÞ; Qc8 ¼ ðc̄iLbiRÞðs̄jLcjRÞ;Qc9 ¼ ðc̄iLσμνbjRÞðs̄jLσμνciRÞ; Qc10 ¼ ðc̄iLσμνbiRÞðs̄jLσμνcjRÞ:

ð2Þ

The Qc0i are obtained by changing all the quark chiralities.We leave a discussion of such “right-handed current”effects for future work [15] and discard the Qc0i below.We split the Wilson coefficients into SM and BSM parts,

Cci ðμÞ ¼ Cc;SMi ðμÞ þ ΔCiðμÞ; ð3Þ

where Cc;SMi ¼ 0 except for i ¼ 1, 2 and μ is the renorm-alization scale.

III. RARE B DECAYS

The leading-order (LO), one-loop CBSM effects inradiative and rare semileptonic decays may be expressedthrough “effective” Wilson coefficient contributionsΔCeff9 ðq2Þ and ΔCeff7 ðq2Þ in an effective local Hamiltonian,

Hrsleff ¼ −4GFffiffiffi

2p V�tsVtbðCeff7 ðq2ÞQ7γ þ Ceff9 ðq2ÞQ9VÞ; ð4Þ

where q2 is the dilepton mass and

Q7γ ¼emb16π2

ðs̄LσμνbRÞFμν;

Q9V ¼α

4πðs̄LγμbLÞðl̄γμlÞ:

For q2 small (in particular, well below the charmresonances), ΔCeff9 ðq2Þ and ΔCeff7 ðq2Þ govern the theoreti-cal predictions for both exclusive (B → Kð�Þlþl−;

Bs → ϕlþl−, etc.) and inclusive B → Xslþl− decay, upto OðαsÞ QCD corrections and power corrections to theheavy-quark limit that we neglect in our leading-orderanalysis. Similarly, ΔCeff7 ð0Þ determines radiative B-decayrates. We will neglect the small CKM combinationV�usVub, implying V�csVcb ¼ −V�tsVtb, and focus on real(CP-conserving) values for the Cci . From the diagramshown in Fig. 1 (left), we then obtain

ΔCeff9 ðq2Þ ¼�Cc1;2 −

Cc3;42

�h −

2

9Cc3;4; ð5Þ

ΔCeff7 ðq2Þ ¼mcmb

�ð4Cc9;10 − Cc7;8Þyþ

4Cc5;6 − Cc7;8

6

�; ð6Þ

with Ccx;y ¼ 3ΔCx þ ΔCy and the loop functions

hðq2; mc; μÞ ¼ −4

9

�lnm2cμ2

−2

3þ ð2þ zÞaðzÞ − z

�; ð7Þ

yðq2; mc; μÞ ¼ −1

3

�lnm2cμ2

−3

2þ 2aðzÞ

�; ð8Þ

where aðzÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffijz − 1jp arctan 1ffiffiffiffiffiffiz−1

p and z ¼ 4m2c=q2. Ournumerical evaluation employs the charm pole mass.We note that only the four Wilson coefficients ΔC1…4

enter ΔCeff9 ðq2Þ. Conversely, ΔCeff7 ðq2Þ is given in terms ofthe other six Wilson coefficients ΔC5…10. The appearanceof a one-loop, q2-dependent contribution to Ceff7 is a novelfeature in the CBSM scenario. Numerically, the loopfunction aðzÞ equals one at q2 ¼ 0 and vanishes atq2 ¼ ð2mcÞ2. The constant terms and the logarithm accom-panying yðq2; mcÞ partially cancel the contribution fromaðzÞ and they introduce a sizable dependence on therenormalization scale μ and the charm quark mass. Sincea shift ofΔCeff7 ðq2Þ is strongly constrained by the measuredB → Xsγ decay rate, we do not consider the coefficientsΔC5…10 in the remainder and focus on the four coefficientsΔC1…4, which do not contribute to B → Xsγ at 1-looporder. Higher-order contributions can be important if newphysics generates ΔCi at the weak scale or beyond, as istypically expected. In this case, large logarithms lnM=mBoccur, requiring resummation. To leading-logarithmicaccuracy, we find

FIG. 1. Leading CBSM contributions to rare decays (left), andto width difference ΔΓs and lifetime ratio τðBsÞ=τðBdÞ (right).

JÄGER, LESLIE, KIRK, and LENZ PHYS. REV. D 97, 015021 (2018)

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ΔCeff7 ¼ 0.02ΔC1 − 0.19ΔC2− 0.01ΔC3− 0.13ΔC4; ð9Þ

ΔCeff9 ¼8.48ΔC1þ1.96ΔC2−4.24ΔC3−1.91ΔC4; ð10Þ

if ΔCi are understood to be renormalized at μ ¼ MW andΔCeff7;9 at μ ¼ 4.2 GeV. It is clear that ΔC1 and ΔC3contribute (strongly) to rare semileptonic decay but onlyweakly to B → Xsγ.

IV. MIXING AND LIFETIME OBSERVABLES

A distinctive feature of the CBSM scenario is thatnonzero ΔCi affect not only radiative and rare semileptonicdecays, but also tree-level hadronic b → cc̄s transitions.While the theoretical control over exclusive b → cc̄smodes is very limited at present, the decay width differenceΔΓs and the lifetime ratio τðBsÞ=τðBdÞ stand out as beingcalculable in a heavy-quark expansion [16]; see Fig. 1(right). For both observables, the heavy-quark expansiongives rise to an operator product expansion in terms of localΔB ¼ 2 (for the width difference) or ΔB ¼ 0 (for thelifetime ratio) operators. The formalism is reviewed in [17]and applies to both SM and CBSM contributions. Forthe Bs width difference, we have [18] ΔΓs ¼ 2jΓs;SM12 þΓcc̄12j cosϕs12, where the phase ϕs12 is small. Neglecting thestrange-quark mass, we find

Γcc12¼−G2FðV�csVcbÞ2m2bMBsf2Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−4x2c

p576π

×f½16ð1−x2cÞð4Cc;22 þCc;24 Þþ8ð1−4x2cÞ×ð12Cc;21 þ8Cc1Cc2þ2Cc3Cc4þ3Cc;23 Þ−192x2c×ð3Cc1Cc3þCc1Cc4þCc2Cc3þCc2Cc4Þ�Bþ2ð1þ2x2cÞ×ð4Cc;22 −8Cc1Cc2−12Cc;21 −3Cc;23 −2Cc3Cc4þCc;24 Þ ~B0Sg;

ð11Þ

with xc ¼ mc=mb. B, ~B0S are defined through

hBsjðs̄LγμbLÞðs̄LγμbLÞB̄si ¼2

3M2BSf

2BsB; ð12Þ

hBsjðs̄iLbjRÞðs̄jLbiRÞjB̄si ¼1

12M2Bsf

2Bs~B0S; ð13Þ

with values taken from [19]. For our numerical evaluationof Γcc12, we split the Wilson coefficients according to (3),subtract from the LO expression (11) the pure SM con-tribution and add the NLO SM expressions from [20].In general, a modification of Γcc12 also affects the semi-leptonic CP asymmetries. However, since we considerCP-conserving new physics in this paper and since thecorresponding experimental uncertainties are still large, thesemi-leptonic asymmetries will not lead to an additionalconstraint.

In a similar manner, for the lifetime ratio, we find

τBsτBd

¼�τBsτBd

�

SM

þ�τBsτBd

�

NP

; ð14Þ

where the SM contribution is taken from [21] and�τBsτBd

�

NP

¼ G2FjVcbVcsj2m2bMBsf2BsτBsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4x2c

p144π

× fð1 − x2cÞ½ð4Cc;21;2 þ Cc;23;4ÞB1 þ 6ð4Cc;22 þ Cc;24 Þϵ1�

− 12x2cðCc1;2Cc3;4B1 þ 6Cc2Cc4ϵ1Þ −M2Bsð1þ 2x2cÞðmb þmsÞ2

× ½ð4Cc;21;2 þ Cc;23;4ÞB2 þ 6ð4Cc;22 þ Cc;24 Þϵ2�g; ð15Þ

subtracting the SM part and defining B1, B2, ϵ1, ϵ2 as

hBsjðb̄LγμsLÞðs̄LγμbLÞjBsi ¼1

4f2BsM

2BsB1; ð16Þ

hBsjðb̄RsLÞðs̄LbRÞjBsi ¼1

4

�MBs

ðmb þmsÞ�2

f2BsM2BsB2; ð17Þ

hBsjðb̄LγμTAsLÞðs̄LγμTAbLÞjBsi ¼1

4f2BsM

2Bsϵ1; ð18Þ

hBsjðb̄RTAsLÞðs̄LTAbRÞjBsi¼1

4

�MBs

ðmbþmsÞ�2

f2BsM2Bsϵ2;

ð19Þ

with values taken from [22]. We interpret the quark massesas MS parameters at μ ¼ 4.2 GeV.

V. RARE DECAYS VERSUSLIFETIMES—LOW-SCALE SCENARIO

We are now in a position to confront the CBSM scenariowith rare decay and mixing observables, as long as weconsider renormalization scales μ ∼mB. Then the loga-rithms inside the h function entering (5) are small and ourleading-order calculation should be accurate. Such ascenario is directly applicable if the mass scale M of thephysics generating the ΔCi is not too far above mB, suchthat lnðM=mBÞ is small. Fig. 2 (left) shows the experi-mental 1σ allowed regions for the width difference andlifetime ratio (from the web update of [23]) in theðΔC1;ΔC2Þ plane. The central values are attained on thebrown (solid) and green (dashed) curves, respectively.The measured lifetime ratio and the width differencemeasurement can be simultaneously accommodated fordifferent values of the Wilson coefficients: in the ΔC1-ΔC2plane, we find the SM solution, as well as a solution aroundΔC1 ¼ −0.5 and ΔC2 ≈ 0. In the ΔC3-ΔC4 plane, we have

CHARMING NEW PHYSICS IN RARE B DECAYS AND … PHYS. REV. D 97, 015021 (2018)

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a relatively broad allowed range, roughly covering theinterval ½−0.9;þ0.7� forΔC3 and ½−0.6;þ1.1� forΔC4. Forfurther conclusions, a considerably higher precision inexperiment and theory is required for ΔΓs and τBs=τBd .Also shown in the plot are contour lines for the contributionto the effective semileptonic coefficient ΔCeff9 ðq2Þ, both forq2 ¼ 2 GeV2 and q2 ¼ 5 GeV2 We see that sizable neg-ative shifts are possible while respecting the measuredwidth difference and the lifetime ratio. For example, a shiftΔCeff9 ∼ −1 as data may suggest could be achieved throughΔC1 ∼ −0.5 alone. Such a value for ΔC1 may well beconsistent with CP-conserving exclusive b → cc̄s decaydata, where no accurate theoretical predictions exist. On theother hand, ΔCeff9 only exhibits a mild q2-dependence.Distinguishing this from possible long-distance contribu-tions would require substantial progress on the theoreticalunderstanding of the latter.We can also consider other Wilson coefficients, such as

the pair ðΔC3;ΔC4Þ (right panel in Fig. 2). A shift ΔCeff9 ∼−1 is equally possible and consistent with the widthdifference, requiring only ΔC3 ∼ 0.5.

VI. HIGH-SCALE SCENARIO AND RGE

A. RG enhancement of ΔCeff9If the CBSM operators are generated at a high scale then

large logarithms lnM=mB appear. Their resummation isachieved by evolving the initial (matching) conditionsCiðμ0 ∼MÞ to a scale μ ∼MB according to the coupledrenormalization-group equations (RGE),

μdCjdμ

ðμÞ ¼ γijðμÞCiðμÞ; ð20Þ

where γij is the anomalous-dimension matrix. As is wellknown, the operators Qci mix not only with Q7 and Q9, butalso with the 4 QCD penguin operators P3…6 and thechromodipole operator Q8g (defined as in [24]), which inturn mix into Q7. Hence the index j runs over 11 operatorswith ΔB ¼ −ΔS ¼ 1 flavor quantum numbers in orderto account for all contributions to C7ðμÞ that are propor-tional to ΔCiðμ0Þ. Most entries of γij are known at LO[14,24–30]; our novel results are (i ¼ 3, 4)

γð0ÞQci

~Q9¼

�4

3;4

9

�

i; γð0ÞQci P4 ¼

�0;−

2

3

�

i;

γeffð0ÞQci Q7 ¼�0;224

81

�

i;

where ~Q9 ¼ ð4π=αsÞQ9VðμÞ and γeffð0ÞQci Q7 requires a two-loopcalculation. (See appendix for further technical informa-tion.) Solving the RGE for μ0 ¼ MW, μ ¼ 4.2 GeV, andαsðMZÞ ¼ 0.1181, results in the CBSM contributions toΔCeff7 and ΔCeff9 in (9), (10) as well as

0BBB@

ΔC1ðμÞΔC2ðμÞΔC3ðμÞΔC4ðμÞ

1CCCA¼

0BBB@

1.12 −0.27 0 0−0.27 1.12 0 00 0 0.92 0

0 0 0.33 1.91

1CCCA

0BBB@

ΔC1ðμ0ÞΔC2ðμ0ÞΔC3ðμ0ÞΔC4ðμ0Þ

1CCCA:

ð21Þ

A striking feature are the large coefficients in the ΔCeff9case, which are Oð1=αsÞ in the logarithmic counting. Thelargest coefficients appear for ΔC1 and ΔC3, which at thesame time practically do not mix into Ceff7 . This means thatsmall values ΔC1 ∼ −0.1 or ΔC3 ∼ 0.2 can generateΔCeff9 ðμÞ ∼ −1 while having essentially no impact on theB → Xsγ decay rate. Conversely, values for ΔC2 or ΔC4that lead to ΔCeff9 ∼ −1 lead to large effects in Ceff7and B → Xsγ.

B. Phenomenology for high NP scale

The situation in various two-parameter planes is depictedin Fig. 3, where the 1σ constraint from B → Xsγ is shownas blue, straight bands. (We implement it by splittingBRðB → XsγÞ into SM and BSM parts and employ thenumerical result and theory error from [31] for the former.The experimental result is taken from the web update of[23].) The top row corresponds to Fig. 2, but contours ofgiven ΔC9 lie much closer to the origin. All six panelstestify to the fact that the SM is consistent with all datawhen leaving aside the question of rare semileptonic Bdecays—the largest pull stems from the fact that theexperimental value for τBs=τBd is just under 1.5 standarddeviations below the SM expectation, such that the black(SM) point is less than 0.5σ outside the green area. Our

FIG. 2. Mixing observables versus rare decays in the CBSMscenario. Left: ðΔC1;ΔC2Þ plane, Right: ðΔC3;ΔC4Þ plane. Ineach case, all Wilson coefficients are renormalized at μ ¼4.2 GeV and those not corresponding to either axis set to zero.The black dot corresponds to the SM, i.e.ΔCi ¼ 0. The measuredcentral value for the width difference is shown as brown (solid)line together with the 1σ allowed region. The lifetime ratiomeasurement is depicted as green (dashed) line and band.Overlaid are contours of ΔCeff9 ð5 GeV2Þ ¼ −1;−2 (black,dashed) and ΔCeff9 ð2 GeV2Þ ¼ −1;−2 (red, dotted), as computedfrom (5), and of ΔCeff9 ¼ 0 (black, solid).


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main question is now: can we have a new contributionΔCeff9 ∼ −1 to rare semileptonic decays, while beingconsistent with the bounds stemming from b → sγ, ΔΓsand τBs=τBd? This is clearly possible (indicated by theyellow star in the plots) if we have a new contributionΔC3 ≈ 0.2, see the three plots of the ΔCi − ΔC3 planes inFig. 3 (right on the top row, left on the middle row and lefton the lower row). In these cases, theΔCeff9 ∼ −1 solution iseven favored compared to the SM solution. A joint effect inΔC2 ≈ −0.1 and ΔC4 ≈ 0.3 can also accommodate ourdesired scenario, see the right plot on the lower row, whilenew BSM effects in the pairs ΔC1;ΔC2 and ΔC1;ΔC4alone are less favored. One could also consider three or allfour ΔCi simultaneously.

C. Implications for UV physics

Our model-independent results are well suited tostudy the rare B-decay and lifetime phenomenology ofultraviolet (UV) completions of the Standard Model. Anysuch completion may include extra UV contributionsto C7ðMÞ and C9ðMÞ, correlations with other flavor

observables, collider phenomenology, etc.; the details arehighly model-dependent and beyond the scope of ourmodel-independent analysis. Here we restrict ourselvesto some basic sanity checks.Taking the case of ΔC1ðMÞ ∼ −0.1 corresponds to a

naive ultraviolet scale

Λ ∼�4GFffiffiffi

2p jV�csVcbj × 0.1

�−1=2

∼ 3 TeV:

This effective scale could arise in a weakly-coupledscenario from tree-level exchange of new scalar or vectormediators, or at loop level in addition from fermions; or theeffective operator could arise from strongly-coupled newphysics. For a tree-level exchange, Λ ∼M=g� where g� ¼ffiffiffiffiffiffiffiffiffig1g2

pis the geometric mean of the relevant couplings. For

weak coupling g� ∼ 1, this then givesM ∼ 3 TeV. Particlesof such mass are certainly allowed by collider searches ifthey do not couple (or only sufficiently weakly) to leptonsand first-generation quarks. Multi-TeV weakly coupledparticles also generically are not in violation of electroweakprecision tests of the SM. Loop-level mediation wouldrequire mediators close to the weak scale which may beproblematic and would require a specific investigation; thisis of course unsurprising given that b → cc̄s transitions aremediated at tree level in the SM. The same would be true ina BSM scenario that mimics the flavor suppressions in theSM (such as MFV models). Conversely, in a strongly-coupled scenario we would haveM∼ g�Λ∼4πΛ∼ 30 TeV.This is again safe from generic collider and precisionconstraints, and a model-specific analysis would berequired to say more.Finally, as all CBSM effects are lepton-flavor-universal,

they cannot on their own account for departures of thelepton flavor universality parameters RKð�Þ [32] from theSM values as suggested by current experimental measure-ments [33]. However, even if those departures are real,they may still be caused by direct UV contributions to ΔC9.For example, as shown in [5], a scenario with a muon-specific contributionΔCμ9 ¼ −ΔCμ10 ∼ −0.6 and in additiona lepton-universal contribution ΔC9 ∼ −0.6, which mayhave a CBSM origin, is perfectly consistent with all rare-B-decay data, and in fact marginally preferred.

VII. PROSPECTS AND SUMMARY

The preceding discussion suggests that a precise knowl-edge of width difference and lifetime ratio, as well asBRðB → XsγÞ, can have the potential to identify anddiscriminate between different CBSM scenarios, or rulethem out altogether. This is illustrated in Fig. 4, showingcontour values for future precision both in mixing andlifetime observables. In each panel, the solid (brown andgreen) contours correspond to the SM central values ofthe width difference and lifetime ratio (respectively). Thespacing of the accompanying contours is such that the area

FIG. 3. Mixing observables versus rare decays, for ΔCirenormalized at μ0 ¼ MW . Color coding as in Fig. 2, B → Xsγconstraint shown in addition (straight blue bands).


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between any two neighboring contours corresponds to aprospective 1σ-region, assuming a combined (theoreticaland experimental) error on the lifetime ratio of 0.001 and acombined error on ΔΓs of 5%. The assumed future errorsare ambitious but seem feasible with expected experimentaland theoretical progress. Overlaid is the (current) B → Xsγconstraint (blue). The figure indicates that a discriminationbetween the SM and the scenario where ΔC9 ≈ −1, whileBRðB → XsγÞ is SM-like is clearly possible. A crucial roleis played by the lifetime ratio τBs=τBd : in e.g. the ΔC3 −ΔC4 case a 1σ deviation of the lifetime ratio almostcoincides with theΔC9 ¼ −1 contour line; a further precisedetermination of ΔΓs could then identify the point on thisline chosen by nature. Further progress on B → Xsγ in theBelle II era would provide complementary information.In summary, we have given a comprehensive, model-

independent analysis of BSM effects in partonic b → cc̄stransitions (CBSM scenario) in the CP conserving case,focusing on those observables that can be computed in aheavy-quark expansion. An effect in rare semileptonic Bdecays compatible with hints from current LHCb and B-factory data can be generated, while satisfying the B → Xsγconstraint. It can originate from different combinations ofb → cc̄s operators. The required Wilson coefficients are sosmall that constraints from B decays into charm are noteffective, particularly if new physics enters at a highscale; then large renormalization-group enhancements arepresent. Likewise, there are no obvious model-independentconflicts with collider searches or electroweak precisionobservables. A more precise measurement of mixingobservables and lifetime ratios, at a level achievable atLHCb, may be able to confirm (or rule out) the CBSMscenario, and to discriminate between different BSMcouplings. Finally, all CBSM effects are lepton-flavor-universal; the current RK and RK� anomalies would eitherhave to be mismeasurements or require additional lepton-flavor-specific UV contribution to C9; such a combinedscenario has been shown elsewhere to be consistent with allrare B-decay data and also presents the most generic wayfor UV physics to affect rare decays. With the statedcaveats, our conclusions are rather model independent. It

would be interesting to construct concrete UV realizationsof the CBSM scenario, which almost certainly will affectother observables in a correlated, but model-dependentmanner.

ACKNOWLEDGMENTS

We would like to thank C. Bobeth, P. Gambino, M.Gorbahn, and especially M. Misiak for discussions. Thiswork was supported by an Institute of Particle PhysicsPhenomenology (IPPP) Associateship. S. J. and K. L.acknowledge support by UK Science and TechnologyFacilities Council (STFC) Consolidated Grants No. ST/L00504/1 and No. ST/P000819/1, an STFC studentship,and a Weizmann Institute “Weizmann-UK MakingConnections” grant. A. L. and M. K. are supported by theSTFC IPPP grant.

APPENDIX: TECHNICAL ASPECTS OF THEANOMALOUS-DIMENSION CALCULATION

Here we provide additional technical informationregarding our results on anomalous dimensions enteringin the RGE (20).A set ofWilson coefficients that containsC7,C9, andCc1…4

and is closed under renormalization necessarily also containsfour QCD-penguin coefficientsCPi multiplying the operatorsP3…6 (we define them as in [24]) and the chromodipolecoefficientC8g, resulting in an 11 × 11 anomalous-dimensionmatrix γ. If the rescaled semileptonic operator ~Q9ðμÞ ¼ð4π=αsðμÞÞQ9VðμÞ is used then to leading order γijðμÞ ¼αsðμÞ=ð4πÞγð0Þij , with constant γð0Þij . As is well known, thismatrix is scheme-dependent already at LO [28]. A scheme-independent matrix γeffð0Þ can be achieved by replacing C7and C8 by the scheme-independent combinations

Ceff7 ¼ C7 þXi

yiCi; ðA1Þ

Ceff8 ¼ C8 þXi

ziCi; ðA2Þ

where

hsγjQijbi ¼ yihsγjQ7γjbi; ðA3ÞhsgjQijbi ¼ zihsγjQ8gjbi; ðA4Þ

to lowest order and the sums run over all four-quark operators.We find thatyi and zi vanish forQc1…4, leavingonly the knowncoefficients yPi ¼ ð−1=3;−4=9;−20=3;−80=9Þi and zPi ¼ð1;−1=6; 20;−10=3Þi (i ¼ 3…6) [24]. The BSM correctionΔCeff9 in (5), (10) coincides with the (BSM correction to the)coefficient C9 of Q9V to LL accuracy.Many of the elements of γeffð0Þ are known [14,25–28],

except for γeffð0ÞQci Q7γ , γeffð0ÞQci Q8g

, γeffð0ÞQci Pj , and γeffð0ÞQci

~Q9, for i ¼ 3, 4. The

latter can be read off from the logarithmic terms in (5), and

FIG. 4. Future prospects for mixing observables. Dashed:contours of constant width difference, dotted: contours ofconstant lifetime ratio. See text for discussion.


015021-6

the mixing into Pi follows from substituting gauge cou-pling and color factors in diagram Fig. 1 (left). This gives

γð0ÞQci

~Q9¼

�−8

3;−

8

9;4

3;4

9

�

i; γð0ÞQci P4 ¼

�0;4

3; 0;−

2

3

�

i;

for i ¼ 1, 2, 3, 4, with the mixing into CP3;5;6 vanishing.The leading mixing into Ceff7 arises at two loops [29] and

is the technically most challenging aspect of this work. Ourcalculation employs the 1PI (off-shell) formalism and themethod of [30] for computing UV divergences, whichinvolves an infrared-regulator mass and the appearance of aset of gauge-non-invariant counterterms. The result is

γeffð0ÞQci Q7 ¼�0;416

81; 0;

224

81

�

iði ¼ 1; 2; 3; 4Þ:

Our stated results for i ¼ 1, 2 agree with the resultsin [24,26], which constitutes a cross-check of ourcalculation.We have not obtained the 2-loop mixing of Cc3;4 into

C8g and set these anomalous dimension elements tozero. For the case of Cc1;2 where this mixing is known,

the impact of neglecting γeffð0Þi8 on ΔCeff7 ðμÞ is small [theonly change being −0.19ΔC2 → −0.18ΔC2 in (9)]. Weexpect a similarly small error in the case of ΔC3;4.

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