当前位置：文档库 › A simultaneous explanation of the large phase in B_s-bar B_s mixing and B -- pipi pi K puz

A simultaneous explanation of the large phase in B_s-bar B_s mixing and B -- pipi pi K puz

a r X i v :0809.3300v 2 [h e p -p h ] 25 S e p 2008

SINP/TNP/2008/19,RECAPP-HRI-2008-012,DFTT-26/2008

A simultaneous explanation of the large phase in

B s ?

A SM s e ?2iβs ,(1)

where βs ≡arg(?V ts V ?tb /V cs V ?cb

)has the value 0.018±0.001in the SM.UT?t has got two solutions [1]:φB s (deg)=?19.9±5.6,A NP s /A SM s

=0.73±0.35;φB s (deg)=?68.2±4.9,A NP s /A SM s =1.87±0.06.(2)

The SM expectation of φB s is zero.But the above numbers show that φB s deviates from zero by

more than 3.7σfor the ?rst solution,while the second solution is signi?cantly more distant from the

SM expectation1.It should be noted that here the theoretical uncertainty is small,so a statistically signi?cant non-zeroφB

would constitute an unambiguous NP https://www.wendangku.net/doc/2118316441.html,bining the two UT?t solutions,the allowed range of the mixing-induced CP-asymmetry in the B s system is given by Sψφ∈[0.35,0.89]at95%C.L.[2],where Sψφ≡sin2(|βs|?φB

(ii)TheπK puzzle:The observed direct CP-asymmetries in theπK channel[3],

a CP(B d→π?K±)=?0.097±0.012,a CP(B±→π0K±)=0.050±0.025,(3) imply that?a CP=a CP(B±→π0K±)?a CP(B d→π?K±)=0.14±0.029di?ers from the naive SM expectation of zero at4.7σlevel.In the QCD factorization approach,?a CP=0.025±0.015,which di?ers from the experimental value by3.5σ.This is quite reliable as most of the model-dependent uncertainties cancel in the di?erence[4].

On the other hand,the following CP-conserving observables,as ratios of branching ratios[3]

R n=1B0d→π+K?]

→π0

BR[B+→π+K0]+[B?→π?

BR(B0d→π±π?)

=0.51±0.10,(6)

is in con?ict with the expected relation BR(B0d→π±π?)>>BR(B0d→π0π0).More speci?cally, what is expected,based on di?erent theoretical models(naive factorization[5],PQCD[6],QCDF [7]),is BR(B0d→π0π0)?O(λ2)BR(B0d→π±π?),while what is observed is BR(B0d→π0π0)?O(λ)BR(B0d→π±π?).On the other hand,

R a=

BR(B0d→π?π+)

1The UT?t collaboration have presented an updated estimate at ICHEP2008(talk by M.Pierini):φB

s=(?19±

7)?∪(?69±7)?,which shows a2.6σdiscrepancy with the SM expectation.In any case,as long as this deviation from the SM value remains sizable,the numerical exercise leading to our conclusion holds.We thank D.Tonelli of the CDF Collaboration for bringing this to our notice.

jack up B0→π0π0branching ratio but then B0→ρ0ρ0branching ratio goes out of control.Thus,a collective explanation for all anomalies is hard to obtain.

To account for the large phase in b→s transition,several new physics models have already been proposed[11].In this short paper,we show that some selective R-parity(more speci?cally,baryon-number)violating couplings can not only provide a large phase encountered in B s-

λ′′ijk U c i D c j D c k,(8)

where the antisymmetry in the last two indices impliesλ′′ijk=?λ′′ikj.Our selection of B-RPV couplings is motivated through the following chain of arguments:

(i)First,we take only those product couplings which contribute to B s-B d mixings via one-loop box diagrams.These areλ′′i13λ′′?i12andλ′′i23λ′′?i21respectively,where i corresponds to all the three singlet up-type?avors.

(ii)λ′′i13λ′′?i12,for i=2,contributes at tree level to b→c

B d mixing through one-loop box graphs.But,nevertheless,we refrain from using λ′′213λ′′?212to avoid any overwhelming tree level new physics imposition on the‘sin2βgolden channel’. (iii)For a simultaneous solution of theπK puzzle,we expect to generate a numerically meaningful contribution to B±→K±π0.The corresponding quark level process b→su

u(B→ππ)at the tree level.

(iv)Thus we are left with two combinations:λ′′113λ′′?112andλ′′123λ′′?121.These consist of three inde-pendent couplings:λ′′113,λ′′112andλ′′123.The strongest constraint onλ′′113comes from n?

2with R=?λ

,the ratio between the

hadronic and supersymmetry breaking scale.For R～10?3,the constraint is very strong:λ′′112～10?7; while for R～10?6,it gets pretty relaxed:λ′′112～1.The upper bound onλ′′123is1.25arising from the requirement of perturbative uni?cation.

B-RPV contributions to observables:The product couplingλ′′113λ′′?112triggers b→s transition, whileλ′′123λ”?121leads to b→d transition.We de?ne:

h(b→s)≡λ′′?

113λ′′112,h(b→d)≡λ′′?

123

λ′′121.(9)

These combinations contribute to B q–

192π2M2?q

R M B

?ηB

f2B

B B

q ?S0(x u)+?S0(x d)

,(10)

where

0(x)=1+x

(1?x)3

.(11)

Above,we have assumed the relevant squarks,?u R and?q R,to be mass degenerate,and we have denoted the common squark mass by?m.

The product coupling h(b→s)also contributes at tree level to non-leptonic B decays like b→d us,like B+→K0π+,B+→K+π0,B d→K0π0,B d→K+π?,B s→φπ0,B s→π+π?,B s→K+K?and their CP conjugate decays3.Similarly,h(b→d)provides new tree level contribution to di?erent B→ππdecay modes4.Thus,di?erent decay rates receive di?erent amount of SM and B-RPV contributions,and the net amplitude in each case amounts to their coherent sum5. The SM amplitude is calculated in the naive factorization model[5].Considering the uncertainties in any such calculation,we rely on observables which are either the ratio of branching ratios or CP-asymmetries(in B→πK modes).For the direct CP-asymmetries to proceed we need a sizable strong phase di?erence between the SM and the B-RPV amplitudes,which may be generated from?nal state interaction and rescattering.Indeed,the weak phases of the B-RPV couplings are free parameters. For simplicity,we have not considered the mixing between the B-RPV operators and the SM operators between the scale M W and m b.The dominant e?ect,which is just a multiplicative renormalization of the B-RPV operator,can be taken into account by interpreting the B-RPV couplings to be valid at the m b scale and not at the M W scale(thus,one should be careful in using the constraints on the couplings and in comparing di?erent limits,though the numerical di?erences are not expected to be signi?cant).

Numerical inputs:Unless otherwise mentioned,all numbers are taken from[3].The measured values of the mass di?erences(?M q)are

?M d=(0.507±0.005)ps?1,?M s=(17.77±0.10(stat)±0.07(syst))ps?1.(12) We require sin2βto lie between0.75±0.04(the SM?t value with V ub as input)and0.681±0.025 (measured from the golden channel B d→J/ΨK S).

B d and B d→π+π?withλ′-type couplings was studied in[18].

5It should be noted that for simplicity of our analysis we have neglected the contributions arising from R-parity conserving sector in all these cases.The leading contributions from this sector to non-leptonic B decays would come at one-loop order,whereas the B-RPV contributions in those decays would proceed at tree level.

We also use the recent lattice values of the bag factors[19]

f B s B B

B B

=1.20±0.06,(13)

and the short distance factors

ηB

d =ηB

=0.55,S0(x t)=2.327±0.044.(14)

The relevant CKM elements are[20]

|V td|=8.54(28)×10?3,|V ts|=40.96(61)×10?3,γ=(75±25)?,(15) while the other elements are taken to be?xed at their central values.

Results:We proceed by making two assumptions or working conditions:

(i)The strong phase di?erence between the SM amplitude and the corresponding BSM amplitude is the same irrespective of whether it is b→s or b→d transition.This assumption relies on?avor SU(3)symmetry.

(ii)In order to calculate the amplitudes for di?erent non-leptonic decay modes we have followed naive factorization approach and considered10%uncertainty over the SM amplitudes to cover the di?erent(model-dependent)non-factorizable corrections.For B d→π0π0mode we have taken this uncertainty to be20%,since the SM branching ratio for this mode is N c sensitive[5].

There are?ve parameters which we like to constrain:the magnitude of two product couplings

(|λ′′?

123λ′′121|and|λ′′?

113

λ′′112|),their weak phases(ΦD≡Arg(λ′′?

123

λ′′121)andΦS≡Arg(λ′′?

113

λ′′112)),and

the common strong phase di?erence between the NP and the SM amplitude(δS).We vary all of them simultaneously,and constrain them by requiring consistency with the observables?a CP,R n,R c,Rππ, R a,sin2β,?M d,?M s andφB

.We also use R=BR(B0→π+π?)/BR(B0→π+K?)=0.259±0.023 [3]to constrain those parameters.Our results are plotted in Fig.1and Fig.2.Throughout our analysis we have taken?m=300GeV;a few percent variation of it will not qualitatively alter our conclusions. Although we varied all the parameters simultaneously,in Fig.1a we projected the allowed region in

a two-dimensional space of the magnitude(|λ′′?

113λ′′112|)and phase(ΦS)of h(b→s).The red(right-

side)patches are allowed solutions when all the?ve parameters pass through the?lters of?M d,

sin2β,?a CP,R,R n and R c;while the blue(left-side)patches are zones allowed by?M s andφB

s only.There are small overlaps between the allowed regions from the two sets.The overlaps signify

a common solution for all the three puzzles.With increasing statistics and with further reduction in

theoretical uncertainties,the overlap may increase or decrease,i.e.it may or may not be possible to

simultaneously address all the riddles with B-RPV interactions.In Fig.1b,we displayed the allowed

zone in the plane ofΦS andδS.We note at this stage thatΦS has four sets of solutions,one in each

quadrant,and for each such set there is an associated patch ofδS.

Note that Rππhas been deliberately kept out of the above list of constraints.If we include it,then to

accommodate large BR(B0d→π0π0),only two sets ofδS are allowed,one in the interval(100→165)?

and the other in(195→245)?.SinceδS has been assumed to be the common strong phase di?erence,

its limitations of the b→d sector in?ltrate into the b→s sector as well,thus eliminatingΦS solutions

in the second and the third quadrants.The?nally allowed values ofΦS lie in the range(10→60)?

and(275→340)?.Clearly,if we relax the assumption of equality of the strong phase di?erence(i.e.

a commonδS),ΦS solutions in all the four regions will be allowed.

Figure 1:(Left panel-1a):The allowed zone in the plane of the magnitude of h (b →s )and its weak phase (ΦS )is shown.The red patches (on the right side)are scatter plots of the allowed parameters obtained by using ?M d ,sin 2β,?a CP ,R ,R n ,R c and R a ;while the blue patches (on the left side)correspond to the space allowed by ?M s and φB s only.(Right panel-1b):The allowed patches in the plane of the strong phase di?erence (δS )and ΦS are displayed.

Fig.2a is a zoomed version of Fig.1a,except that in Fig.2a we have included all possible constraints at the same time.For illustration,out of the two allowed sets of ΦS ,the one within the range (10→60)?has been shown.Fig.2b is an equivalent description replacing the magnitude and weak phase of h (b →s )by those of h (b →d ).Note that the constraint on |h (b →d )|is one order of magnitude tighter than |h (b →s )|,primarily because the SM prediction of the B d mixing is relatively more Figure 2:(Left panel-2a):Zoomed version of Fig.1a,only that all constraints are now used,and focussed in the ?rst quadrant solution of ΦS .(Right panel-2b):Similar to Fig.2a,but in the space of the magnitude and phase of h (b →d ).

Conclusions:In this paper,we wanted to solve three puzzles in B physics,namely,the large phase in B s mixing,a more than 3.5σdiscrepancy between CP-asymmetries in charged and neutral B decays in πK modes,and a signi?cantly larger than expected neutral B decay in π0π0channel.Here we make two remarks:(i )the theoretical uncertainty in the estimation of the B s mixing phase is small and hence a large non-zero phase would constitute a clinching signal for new physics;(ii )but,on account of large hadronic uncertainties associated with the πK and ππmodes,the discrepancies observed in ?a CP and R ππ,though tantalizing,are not conclusive.In fact,to get rid of these theoretical uncertainties as much as possible,we considered the di?erence between CP-asymmetries and the

relative branching ratios.Yet,from a conservative point of view,instead of entering into a debate whether the discrepancies constitute‘puzzles’or‘non-puzzles’,all that we wanted to emphasize in this paper is that if one can?gure out a new dynamics beyond the SM that causes a simultaneous and systematic movement of all those theoretical estimates towards better consistency with experimental data,then that source of new physics calls for special attention.As an illustration,we advanced the case of explicit baryon-number violating part of supersymmetry,and we have used only two product couplings,constructed out of three individual ones,to explain all the data.One should keep track of it in the LHC data analysis,as such interactions would give lots of?nal state jets.

In fact,even within the B physics context,it may be possible to infer our choices of B-RPV couplings(or,similar type diquark couplings)from the following observations:the coupling h(b→s) will contaminate B s→K+K?(b→su

u at the quark level)which is also used to determine

γ[22].Any statistically di?erent measurement ofγbetween these two methods will strengthen our hypothesis.Moreover,either of the two methods would yieldγdi?erent from the value extracted from B→πK.We stress again that the falsi?ability of our hypothesis,under the assumptions spelt above, can be judged from Fig.1a by noting that the common solution zone in the parameter space arising from the‘B s-set’and the other data set may shrink or expand as more data accumulate.LHCb will de?nitely shed more light to these issues.

Acknowledgements:GB acknowledges a partial support through the project No.2007/37/9/BRNS of BRNS(DAE),India.KBC acknowledges hospitality at the Regional Centre for Accelerator-based Particle Physics of HRI,Allahabad,during part of his work.SN’s work is supported in part by MIUR under contact2004021808-009and by a European Community’s Marie-Curie Research Training Net-work under contract MRTN-CT-2006-035505“Tools and Precision Calculations for Physics Discoveries at Colliders”.We thank D.Tonelli and A.Kundu for their helpful comments on the manuscript. References

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