Is There Still a B -> pi K Puzzle?

0.13 MB·English

by  Seungwon Baek

Is There Still a B → πK Puzzle?
Seungwon Baek1 and David London2
1Institute of Physics and Applied Physics, Yonsei University, Seoul 120-479, Korea
2Physique des Particules, Universit´e de Montr´eal,
C.P. 6128, succ. centre-ville, Montr´eal, QC, Canada H3C 3J7
(Dated: February 2, 2008)
We perform a fit to the 2006 B → πK data and show that there is a disagreement with the
standard model. That is, the B → πK puzzle is still present. In fact, it has gotten worse than in
earlier years. Assuming that one new-physics (NP) operator dominates, we show that a good fit
is obtained only when the electroweak penguin amplitude is modified. The NP amplitude must be
sizeable, with a large weak phase.
PACSnumbers: 13.25.Hw,11.30.Er,12.15Ff,14.40.Nd
7
0
0
There are four B πK decays – B+ π+K0 (desig- πK puzzleisstillpresent,andisinfactmoreseverethan
n 2 nated as +0 below)→, B+ → π0K+ (0+→), Bd0 → π−K+ before. However, it now is present only in CP-violating
( +) and B0 π0K0 (00) – whose amplitudes are asymmetries. If one looks at only the measurements of
a − d →
J related by an isospin quadrilateral relation. There are thebranchingratios,asabove,onewillnotseeit. Wealso
5 nine measurements of these processes that can be made: examine the questionofwhichNP scenarioscanaccount
2 the four branching ratios, the four direct CP asymme- for the current B πK data. As such, this paper can
→
triesA ,andthemixing-inducedCPasymmetryS in be consideredanupdate ofRef. [7], in which the present
CP CP
2 B0 π0K0. Severalyearsago,thesemeasurementswere authors were involved.
v d →
in disagreement with the standard model (SM), leading We begin by writing the four B πK amplitudes us-
1 →
some authors to posit the existence of a “B πK puz- ing the diagrammatic approach. Within this approach
8
→
1 zle” [1]. Subsequently, there was an enormous amount [8], the amplitudes can be written in terms of six dia-
1 of work looking at B πK decays, both within the SM grams: the color-favored and color-suppressed tree am-
0 and with new physics→(NP). plitudes T′ and C′, the gluonic penguin amplitudes P′
7 ′ tc
Onesimple waytosee thisdiscrepancyistodefine the andP ,andthecolor-favoredandcolor-suppressedelec-
0 uc
ratios of branching ratios troweakpenguinamplitudesP′ andP′C . (Theprimes
/ EW EW
h on the amplitudes indicate ¯b s¯transitions.) The am-
p 1 BR[B0 π−K+]+BR[B¯0 π+K−] →
hep- RRn ≡ 22BBRR[B[Bdd+d0→→→ππ00KK+0]]++BBRR[[BB¯dd0d−→→→ππ00K¯K0−] ],. (1) plitudeAs+a0re=g−ivPent′cb+yPu′ceiγ − 13PE′CW ,
v: c ≡ BR[Bd+ →π+K0]+BR[Bd− →π−K¯0] √2A0+ =−T′eiγ −C′eiγ +Pt′c
Xi Within the SM, it is predicted that Rc is approximately P′ eiγ P′ 2P′C ,
equaltoR [2]. QCD-factorizationtechniques [3]canbe − uc − EW − 3 EW
r n
2
a used to take corrections into account. These yield [4] A−+ = T′eiγ +P′ P′ eiγ P′C ,
− tc− uc − 3 EW
R =1.16+0.22 , R =1.15+0.19 . (2) 1
n −0.19 c −0.17 √2A00 = C′eiγ P′ +P′ eiγ P′ P′C ,(5)
− − tc uc − EW − 3 EW
However, the pre-ICHEP04 data was [5]
where we have explicitly written the weak-phase depen-
Rn =0.76+−00..1100 , Rc =1.17+−00..1122 . (3) dence,(includingtheminussignfromVt∗bVts [Pt′c]),while
the diagrams contain strong phases. (The phase infor-
The values of Rn and Rc revealed a clear discrepancy mation in the Cabibbo-Kobayashi-Maskawa quark mix-
with the SM. Even the pre-ICHEP06 data was in dis- ingmatrixisconventionallyparametrizedintermsofthe
agreement with the SM. unitarity triangle, in which the interior (CP-violating)
However, the ICHEP06 data is angles are known as α, β and γ [9].) The amplitudes
forthe CP-conjugateprocessescanbeobtainedfromthe
R =1.00+0.07 , R =1.10+0.07 , (4)
n −0.07 c −0.07 above by changing the sign of the weak phase (γ).
WithintheSM,toagoodapproximation,thediagrams
which is now in agreement with the SM. This has led
P′ and P′C can be related to T′ and C′ using flavor
some authorsto claim thatthere is no longera B πK EW EW
→ SU(3) symmetry [10]:
puzzle [5, 6].
Inthepresentpaper,werevisitthequestionofwhether 3c +c 3c c
′ 9 10 ′ ′ 9 10 ′ ′
P = R(T +C )+ − R(T C ) ,
ornotthereisaB πK puzzle. Aswewillsee,theB EW 4 c +c 4 c c −
→ → 1 2 1− 2
2
3c +c 3c c
P′C = 9 10R(T′+C′) 9− 10R(T′ C′)(.6) The only way to account for the data within the SM
EW 4 c1+c2 −4 c1 c2 − is to include C′, since it appears in only one of the two
−
amplitudes √2A(B+ π0K+) and A(B0 π−K+).
Her∗e, the c∗i are Wilson coefficients [11] and R ≡ Thus, as a first step t→o remedy the probledm→, we follow
|(VtbVts)/(VubVus)|. In our fits we take R = 48.9±1.6 Refs. [15] and add only C′ [from Eq. (5)] to the ampli-
[12].
tudes of Eq. (8). From the approximate formulae
In Ref. [8], the relative sizes of the B πK diagrams
→
were roughly estimated as ′
T
(cid:12) (cid:12)
1:|Pt′c| , O(λ¯):|T′|, |PE′W| , ACP(0+)≈−2(cid:12)(cid:12)(cid:12)PTt′′c(cid:12)(cid:12)(cid:12)sinδT′sinγ C′ (10)
O(λ¯2):|C′|, |Pu′c|, |PE′CW| , (7) ACP(−+)≈−2(cid:12)(cid:12)(cid:12)Pt′c(cid:12)(cid:12)(cid:12)sinδT′sinγ−2(cid:12)(cid:12)(cid:12)Pt′c(cid:12)(cid:12)(cid:12)sinδC′sinγ ,
(cid:12) (cid:12) (cid:12) (cid:12)
where λ¯ 0.2. These estimates are expected to hold
′
approxim∼atelyin the SM. If one ignoresthe small (λ¯2) where δC′ is the strong-phase difference between C and
′ ′
O P , we see that a large value of C can give the cor-
diagrams, the four B πK amplitudes take the form tc | |
→ rect sign for ACP(−+) when sinδC′ has a different sign
A+0 = P′ , from sinδT′. This is confirmed numerically. A good
− tc fit is obtained: χ2 /d.o.f. = 1.0/3 (80%). We find
√2A0+ = T′eiγ +P′ P′ , ′ m′in ′
− tc− EW P =47 1eV, T =8.1 3.5eV, C =13.0 3.2eV,
√2AA−0+0 == −PT′′eiγ +P′Pt′c., (8) δ|CT′′/|=T′(1=54±1±.610)0◦.,|3δiCs|′r=equ(−ir±1ed54(±we7s)t◦r.|eHsso|twheavtecro,rn±roetlaettihonats
− tc− EW |have b|een ta±ken into account in obtaining this ratio).
Since this value is much larger than the naive estimates
Mode BR[10−6] ACP SCP ofEq.(7), this showsexplicitly that the′ B′→πK puzzle
is still present. In Ref. [7] (2004), C /T = 1.8 1.0
B+ →π+K0 23.1±1.0 0.009±0.025 | | ±
wasfound. We thus see that the puzzle has gottenmuch
B+ →π0K+ 12.8±0.6 0.047±0.026
worse in 2006. (See Ref. [16] for another approach.)
Bd0 →π−K+ 19.7±0.6 −0.093±0.015 We note in passing that the two fits give γ = (62.5
Bd0 →π0K0 10.0±0.6 −0.12±0.11 0.33±0.21 11.1)◦ and γ = (50.0 5.6)◦, respectively. Thus, bot±h
±
fits give values for γ which are consistent with the value
TABLEI:Branchingratios,directCPasymmetriesA ,and
CP obtained via a fit to independent measurements: γ =
fmouixrinBg-→indπuKceddeCcaPymasoydmesm.eTtrhyedSaCtPai(siftaakpepnlifcraobmleR) effosr.[t1h3e] 59.0+−64..49◦ [12].
and [14]. We can also perform a fit which incorporates all the
(λ¯2)termsinEq.(5)(i.e.P′ isincluded). Onceagain,
O uc
wefindagoodfit: χ2 /d.o.f.=0.7/1(79%)(thevalues
From the above, we see that the amplitudes min
′ ′
√2A(B+ → π0K′ +) and A(B′ d0 → π−′K+) are equal, up ionftAerCfePr(e0n+c)e)a.nHdoAwCePv(e−r,+a)samrealrleeprrvoadluuceeodfduCe′toisCfo–uPnudc,
to a factor of P . Since P < P , we therefore ex-
EW | EW| | tc| compensatedbyalargevalueof P′ : C′/T|′ =| 0.8 0.1,
pectthedirectasymmetryACP(0+)tobeapproximately ′ ′ | uc| | | ± ◦
P /T = 1.7 0.6. We also find that γ = (30 7) .
equal to ACP(−+): |Thuucs,th|e probl±emin C′/T′ hasbeenalleviated,b±utwe
T′ encounter new difficul|ties in| P′ /T′ and γ.
ACP(0+)≈−2(cid:12)(cid:12)(cid:12)Pt′c(cid:12)(cid:12)(cid:12)sinδT′sinγ ≈ACP(−+) , (9) Having established that th|eruecstill|is a B → πK puz-
(cid:12) (cid:12) zle, we now turn to the question of the type of new
′
where δT′ is the strong-phase difference between T and physics which can be responsible. There are a great
P′ . However,looking at the ICHEP06B πK data in many NP operators which can contribute to B πK
tc → →
Table 1, we see that these asymmetries are very differ- decays. However, this number can be reduced consid-
ent. Thus, the 2006 B πK data cannot be explained erably. All NP operators in ¯b s¯qq¯ transitions take
by the SM. It is only →by considering the CP-violating the form ij,q s¯Γ bq¯Γ q (q →= u,d,s,c), where the
ONP ∼ i j
asymmetries that one realizes this. Γ represent Lorentz structures, and color indices are
i,j
To investigate this numerically, we have performed suppressed. These operators contribute to the decay
a fit to the 2006 ICHEP06 B πK data using the B πK through the matrix elements πK ij,q B ,
→ → h |ONP | i
parametrization of Eq. (8). We find that χ2 /d.o.f. = whose magnitude is takento be roughlythe same size as
min
25.0/5 (1.4 10−4), indicating an extremely poor fit. the SM ¯b s¯ penguin operators. Each matrix element
× →
(The number in parentheses indicates the quality of the has its own NP weak and strong phase.
fit, anddepends onχ2 andd.o.f. individually. 50%or We begin with the model-independent analysis de-
min
more is a good fit; fits which are substantially less than scribedinRef.[17]andusedinRef.[7]. Hereitisargued
50% are poorer.) thatallNPstrongphasesarenegligible. Thisallowsfora
3
greatsimplification. IfoneneglectsallNPstrongphases, andA00); ′C,ueiΦ′uC and ′C,deiΦ′dC appearalsoinampli-
onecannowcombineallNPmatrixelementsintoasingle tudes withAno C′ (A+0 anAdA−+). Thus,ifone wishesto
NP amplitude, with a single weak phase: reproducealargeC′ (orP′ [20]),itisnecessarytoadd
EW
′,combeiΦ′; ′C,ueiΦ′uC and ′C,deiΦ′dC are not necessary.
XhπK|ONijP,q|Bi=AqeiΦq . (11) AIn light of thAis, we expect aAgood fit only for possibility
(i) above; (ii), (iii) and (iv) should have poor fits. (Note
Now, B πK decays involve only NP parameters re-
→ that fit (ii) might be marginally acceptable due to the
latedtothequarksuandd. Theseoperatorscomeintwo
classes, differing in their color structure: s¯αΓibαq¯βΓjqβ interference of PE′W and A′C,ueiΦ′uC.)
This is borne out numerically. We find (i)
and s¯ Γ b q¯ Γ q (q = u,d). The matrix elements of
α i β β j α
χ2 /d.o.f. = 0.6/3 (90%), (ii) χ2 /d.o.f. =
these operators can be combined into single NP ampli- min min
tudes, denoted ′,qeiΦ′q and ′C,qeiΦ′qC,respectively[18]. 4.0/3 (26%), (iii) χ2min/d.o.f. = 21.1/3 (0.01%), (iv)
Here, Φ′ and ΦA′C are the NAP weak phases; the strong χ2min/d.o.f.=12.8/4 (1.2%). (Regarding the d.o.f.’s: in
q q fits (i)-(iii), wefit to the three magnitudes P′ , T′ and
phases are zero. Each of these contributes differently to | tc| | |
thevariousB πK decays. Ingeneral, ′,q = ′C,q and |A| (NP), two relative strong phases, and the NP weak
Φ′q 6= Φ′qC. No→te that, despite the “colorA-supp6 reAssed” in- pInhafiste(Φiv,)s,oontheeodf.oth.fe. NfoPr tphaersaemtehtreeresfictasnisbe9a−b6so=rbe3d.
dexC,thematrixelements ′C,qeiΦ′qC arenotnecessarily into P′ , so the number of d.o.f.’s is increased by one.)
smaller than the ′,qeiΦ′q. A Thus|, atcs|expected, a good fit is found only if the NP is
A
The B πK amplitudes cannow be written in terms in the form of ′,combeiΦ′.
of the SM→amplitudes to O(λ¯) [P′ and T′ are related A
EW This is the same conclusion as that found in Ref. [7].
as in Eq. (6)], along with the NP matrix elements [18]:
Thus, not only is the B πK puzzle still present,
→
A+0 =−Pt′c+A′C,deiΦ′dC , (12) bu′,tcomitbeisiΦs′ti=ll p0oi(ntthinisgctoorwreastpdosndthsetosaNmPe tiynpethoefeNlePc-,
√2A0+ =P′ T′eiγ P′ Atroweakpeng6 uin amplitude). For this (good) fit, we find
tc− − EW
+ ′,combeiΦ′ ′C,ueiΦ′uC , |T′/P′|=0.09, |A′,comb/P′|=0.24, Φ′ =85◦. We there-
A −A forefindthat the NP amplitude mustbe sizeable,with a
A−+ =Pt′c−T′eiγ −A′C,ueiΦ′uC , large weak phase.
√2A00 = P′ P′ + ′,combeiΦ′ + ′C,deiΦ′dC , If one does not wish to neglect the NP strong phases,
− tc− EW A A
one can still reduce the number of NP operators. It can
where ′,combeiΦ′ ′,ueiΦ′u + ′,deiΦ′d. be shown that an arbitrary amplitude can be written in
A ≡−A A
Note that the above form of NP amplitudes still satis- terms of two others with known weak phases. This is
′
fies the isospinquadrilateralrelation. IfPtc is dominant, knownas reparametrizationinvariance(RI) [21]. Apply-
onewillalsofindthatRn Rc andthesumrulesforCP ing this to the NP amplitudes, and taking the two weak
≈
asymmetries given in Ref. [19] are satisfied. Therefore phases to be 0 and γ, we can write
the fact that the SM satisfies those sum rules does not
necessarilyruleoutthiskind′ ofNP.(Ofcourse,iftheNP XhπK|ONijP,q|Bi=Aq0+Aqγeiγ , (13)
is about the same size as P , one can find that R =R
and that the sum rules aretvciolatied.) n 6 c where the q contain only strong phases.
A(0,γ)
The value of γ is taken from independent measure- The B πK amplitudes can now be written as in
◦ →
ments (γ =59.0+6.4 [12]), leaving a total of 10 theoret- Eq. (12), where each of the NP amplitudes is expressed
ical parameters:−4P.9′ , T′ , ′,comb , ′C,u , ′C,d , 3 asinEq.(13)above. Takingγ =59.0+9.2◦ [12],thereare
| tc| | | |A | |A | |A | −3.7
NP weak phases and two relative strong phases. With atotalof15theoreticalparameters: theeightmagnitudes
only 9 experimental measurements, it is not possible to P′ , T′ , ′,comb , ′C,u , ′C,d , and seven relative
| tc| | | |A(0,γ) | |A(0,γ)| |A(0,γ)|
perform a fit. It is necessary to make some theoretical strongphases. Withonly9measurements,wemustagain
assumptions. make theoretical assumptions, and we consider the four
As in Ref. [7], we assume that a single NP amplitude possibilities (i)-(iv) described above.
dominates. We consider the following four possibilities: As before, we expect a good fit only for possibility (i).
(i)only ′,comb =0,(ii)only ′C,u =0,(iii)only ′C,d = Forthed.o.f.’s, inthefits thereareseventheoreticalpa-
0, (iv) AA′C,ueiΦ6′uC = A′C,deiAΦ′dC, A6 ′,comb = 0 (Aisospin6 - rameters: the four magnitudes |Pt′c|, |T′|, |Aq0| and |Aqγ|,
conserving NP). and three relative strong phases. In fit (ii), A (+0)
CP
However, before presenting the results of the fits, we and A (00) are identically zero, and S (00) = sin2β.
CP CP
candeducewhattheresultsshouldyield. Earlier. wesaw Thus, these measurements do not constrain the parame-
thatthe SMcanreproducethe ICHEP06B πK data, ters; there are only six constraining measurements. This
′ →
but only if C is anomalously large. Of the three NP is fewer than the number of parameters, so a fit cannot
matrix elements, only ′,combeiΦ′ appears only in ampli- bedoneinthiscase. Thefitcanbedoneincases(i),(iii),
tudesofdecayswhichrAeceivecontributionsfromC′ (A0+ andthed.o.f.is9 7=2. Infit(iv), q canbeabsorbed
− A0
4
into P′ , so the d.o.f. is 9 5=4. The results of the fits 133 (2004), PoS HEP2005, 193 (2006).
are: (it)c χ2 /d.o.f. = 0.5−/2 (79%), (iii) χ2 /d.o.f. = [2] H. J. Lipkin, Phys. Lett. B 445 (1999) 403; M. Gronau
18.5/2 (0.m01in%), (iv) χ2 /d.o.f. = 12.8/4m(i1n.2%). As andJ.L.Rosner,Phys.Rev.D59(1999)113002; J.Ma-
expected, onlyfit(i)[NmPin= ′,comb]givesa goodfit. We tias, Phys. Lett. B 520 (2001) 131.
find T′/P′ =0.09, q/P′ A=0.15, q/P′ =0.28. We [3] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachra-
| | |A0 | |A0 | jda,Phys.Rev.Lett.83,1914(1999),Nucl.Phys.B591,
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theNPeffects: negligibleNPstrongphases,andthegen- 333.
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ically identical, the conclusions are similar. [6] For example, see F. Schwab, talk given in
http://euroflavour06.ifae.es.
Ofthefournew-physicsmodelsexaminedinthispaper,
[7] S. Baek, P. Hamel, D. London, A. Datta and
only one produces a good fit to the 2006 B πK data:
case(i), ′,comb =0. Thequalityoffitisext→remelygood D. A. Suprun,Phys.Rev. D 71, 057502 (2005).
[8] M. Gronau, O. F. Hernandez,D. London and J. L. Ros-
A 6
here. The NP amplitude must be sizeable, with a large ner, Phys. Rev. D 50, 4529 (1994), Phys. Rev. D 52,
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the general case). Here A (0+) A ( +) is found (2006) 1.
d(ue′Ct,do=the0)inatnedrfe(rievn)c(eisoofspPCiEPn′W-coannsde≈rAvi′nCCg,uPeNi−ΦP′u)C.yiFelidtsv(eiiriy) [10] M(D1.9.9PN8i)er,ujobPlehraytnsad.nRdTe.vJ..MLL.e.tYRt.aon8s,n1e,Pr,5h0yP7sh6.yRs(.1e9vL9.e8t)Dt;. MB60.4,G401r3o,4n04a20u13,
A 6
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operator,and so the more general case might be used to
[12] CKMfitter Group, J. Charles et al., Eur. Phys. J. C 41,
explain the B πK data.
→ 1 (2005).
In conclusion, in this paper we have done a fit to the
[13] Heavy Flavor Averaging Group (HFAG),
2006 B πK data to see if new physics (NP) was still arXiv:hep-ex/0603003.
→
indicated. Note that a fit to the full data is necessary; [14] W. M. Yao et al. [Particle Data Group], Ref. [9];
if a subset of the measurements is used, erroneous con- B. Aubert et al. [BABAR Collaboration], Phys. Rev.
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′ tion, A. Bornheim et al., Phys. Rev. D 68, 052002
found. If the “small” diagram C is added, a good fit is
(2003); B. Aubert et al. [BABAR Collaboration],
′ ′
obtained. However, C /T = 1.6 0.3 is required. The
arXiv:hep-ex/0607106; B. Aubert et al. [BABAR Col-
| | ±
fact that this ratio is large indicates that the B πK laboration], rXiv:hep-ex/0608003; B. Aubert et al.
→
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We have also examined which type of NP can be re- 525 (2000); K. Abe et al. [Belle Collaboration],
arXiv:hep-ex/0609006.
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NP operator domina→tes, and consider (i) only ′,comb = [15] C. W. Chiang, M. Gronau, J. L. Rosner and
0, (ii) only ′C,u = 0, (iii) only ′C,d = 0, (ivA) isospin6 - JDH.EAP.S0u6p0r7u,n0,2P5hy(2s0.0R6e)v..D70,034020 (2004); S.Baek,
conserving NAP: ′6C,ueiΦ′uC = ′C,AdeiΦ′dC6, ′,comb =0. Of [16] Y.L.WuandY.F.Zhou,Phys.Rev.D72(2005)034037;
these, the only (Avery) good fiAt is found foAr model (i). It C. W. Chiang and Y. F. Zhou, JHEP 0612, 027 (2006);
correspondstoamodificationoftheelectroweakpenguin Y. L. Wu,Y.F. Zhou and C. Zhuang, Phys.Rev.D 74,
094007 (2006).
amplitude. Note thatthese results arethe sameas those
[17] A.DattaandD.London,Phys.Lett.B595,453(2004).
found in Ref. [7]. This shows that, although the puzzle
[18] A. Datta, M. Imbeault, D. London, V. Page, N. Sinha
is worse than in 2004, its resolution is the same.
and R.Sinha, hep-ph/0406192.
[19] M. Gronau and J. L. Rosner, Phys. Rev. D 74, 057503
Acknowledgments: This work was financially sup- (2006).
ported by Brain Korea 21 (BK21) project (SB) and by [20] M. Imbeault, D. London, C. Sharma, N. Sinha and
NSERC of Canada (DL). R. Sinha, arXiv:hep-ph/0608169.
[21] F. J. Botella and J. P. Silva, Phys. Rev. D 71, 094008
(2005);S.Baek,F.J.Botella,D.LondonandJ.P.Silva,
Phys.Rev.D72,036004(2005),Phys.Rev.D72,114007
(2005).
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