/ CERN-TH.716e/94 February 1994 >¢~¤°5Lr/1/I
B Decays — Introduction and Overview
A. Ali*) Theory Division, CERN CH—1211 Geneva 23, Switzerland
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To be published in B Decays (Q"'! Edition}, World Scientific Publishers, Singapore (editor: S. Stone} (1.994}.
CERN-TH.7168/94
On leave of absence from DESY, Hamburg, FRG. OCR Output
OCR OutputOCR OutputB Decays — lntroduction and Overview
Ahmed Ali
Theory Division, CERN CH-1211 Geneva 23, Switzerland
Abstract This article gives an introduction and overview of B physics. It discusses the Kobayashi-Maskawa generalization of the Cabibbo—GIM matrix for flavour mixing to incorporate CP violation in the standard model (SM) of electroweak interactions, thereby postulating the existence of the b and t quarks. The dis covery of the T family of resonances by the Columbia-Fermilab-Stony Brook collaboration, in 1977, and its subsequent confirmation at DORIS II and later at CESR are reviewed. The bulk of the article is, however, devoted to the physics of B decays, which provide an almost ideal ground for testing QCD, involving both its perturbative and non—perturbative aspects. The rapport be tween current experiments and theoretical estimates obtained from frameworks embedded in QCD with varying degree of rigour is quantified. The main thrust of this review is in estimating the CKM matrix elements from B decays, which determine five of the nine matrix elements. This is then combined with the constraints from the CP-violation parameter \c|, in order to provide a profile of the CKM unitarity triangle. The interest in pursuing B physics at the ongoing and future experimental facilities is emphasized, in particular in rare B decays and CP violation, which will provide precision tests of the SM in the quark sector or perhaps indicate the existence of non-SM physics.
1. Weak-coupling Lagrangian for the b Quark and the CKM Matrix
The Lagrangian density of the standard model for electroweak interactions (hence forth called SM) can be symbolically written asl
C=£(f»W»B)+£(f,@)+£(WB»@)·V(‘I’) (I) OCR Output
"On leave of absence from DESY, Hamburg, FRG. where the symbols f, W, B and represent fermions (leptons and quarks), S U (2) L gauge bosons, WL, U (1)y gauge boson, Bu and the Higgs doublet field, respectively. The SM particle content together with the (weak) isospin properties of the basic quanta are given below, where we have assumed that there are three families of lep tons and quarks.
Fermions
L€pt¤¤S=(I/°) , (V") · , L(VT) L6 ifL, T <¤R,uR,m
uarks:t <;)L(E); , b; uR,dR,cR,. <)L
Gauge bosons
WuZW,} z W,} BN
Scalars
.. + q»:,,q>+=,(j;) (j,) (2) The various terms in the SM Lagrangian can be written by demanding S U (2) L ®U(1) gauge invariance, lepton-quark universality, and family—independence of the elec troweak interactions. Since the SM Lagrangian is given in any standard textbook on electroweak interactions;) we shall not reproduce it here. The main interest in B physics lies in the study of the first two terms in (1) involving fermions, gauge bosons and the Higgs fields, in particular flavour-changing transitions. In order to appreciate how flavour-changing transitions emerge in the SM, it is worth while to write the first two terms in (1) explicitly. The interactions between the fermions and gauge bosons has the form:
cw, W, B> = Z{?£10li+i§i1Z>’1§r+ ·z21pq2}+ Z @2210 nz, <3>OCR Output j=1 {:1 where j is the family index,
li:( I;) llL ’*:€R’ qi: ¤‘U*¢:“R= q?¥=“R¤· L 10 E DMZ 1D' 5 DNS . . Du = 8), —·— Zg '° Wu ·—Y Zg1Bu,(5
DL Z Gb · i91frBu» (4) where gl and gg are, respectively, the U (1)y and SU(2)L coupling constants, 0** (cz = 1,2,3) are the (weak) isospin Pauli matrices, and the Gell—Mann — Nishijima formula Q = I3 + Y defines the (weak) hypercharge of the quarks and leptons. The interaction term L( f, gb) involving the fermions and the Higgs fields has the Yukawa form
£(f7‘I’) = (UM), [ZW}; ]=1 + Z (*4,)<2‘i<1>¤’z+ (,, j,k=1
where QC _ ' * _ " ZU2¢¢0* _¢" - s with both and <1>° transforming as a (weak) isospin doublet. The hypercharge of the Higgs fields can be written by inspection. The Yukawa coupling constants (h;)j, (hq) jk and (h'q),k are arbitrary complex numbers and each term in (5) is independently SU(2)L ® U(1)y invariant due to the fact that the SU(2)L acts only on ZL and QL and on the Higgs doublet and °, whose products are S U (2) scalars. After spontaneous symmetry breaking S U (2) L ® U (1)y —> U (1)EM, which pre serves the symmetry under U (1) electromagnetism, the gauge bosons, fermions and the neutral scalar field gb acquire non-zero masses through the Higgs mechanism
V(¢) = M2I¢|2 + »\|¢|4; #2 < 0 ig) > 0, (6) with V(cb)m,,, at |
- 5(f,SSB ¢)= E:(mj)zZi]iq1 1 (y)+ i J=1 · -· Z {(mjk)U ituiiz + (mjk)1;1 ) (g)didfz1+ OCR Output b+ h- ¢·» (7) j,k=1 (mj); (hi): +1, (m1k)U (h'1)jk #3, (m1k)p (h’q)jk (8)
Since in the SM there are no right—handed fields ui? = 1,2,3), the neutrinos ui remain massless and the charged lepton mass matrix is diagonal. Hence, in the SM there are no family—changing leptonic interactions — an aspect that will soon come under experimental scrutiny from the ongoing and planned experiments dedicated to the neutrino mass and oscillation measurements. In (8), (mjk)U and (mjk)D are the (3 >< 3) quark mass matrices for the up- and down—type quarks, respectively. In order to write the Lagrangian in terms of the quark mass eigenstates one has to diagonalize the mass matrices (mjk)U and (m,k)D. This can be done with the help of two unitary matrices. It is customary to denote them by Vfp and VI? l (likewise for the down type quarks):
Vf°'°mUV§p` E (md,,,g_)U E Diag. (mu ,mc ,mt), (9) _ _um ·down VgmDVR ° li : (md,ag_)D : Dzag. (md ,m, ,mb), with Vfpl Vfp = 1, etc. Concentrating on the up—type quarks in (7) one can do the following manipulation : T "’”pl i2LmUuR = GLVITPV£mUV,QV,§‘puR (10) = ULVE" (md¢¤g.)U (V£"~R)» which shows that the physical quark fields (mass eigenstates) are:
tim : vg~¤uL : vg? CLUL it qu) Ph dL U5 : VId0wndL = Vrdcwn dLSL br
Likewise, ugysh = Vffpug, d};”’}h = V}§°"’"dR. Onecan now rewrite the term £( f, (D) hin the SM Lagrangian in terms of (u},)Phy’ and (di)Phy’, obtaining SSB - · Elf, Q) Z _ Q51+ 6 ZE {:1mqaqiqi + ZJ j=1 OCRmlaljlj Output v (12) where it is now understood that q1 = g (uliy+hs ulgys)h = u ctc., and we have dropped the superscript on the quark fields. The identification of the parameters mh, mm with the lepton and quark masses is now evident. In addition, the SM Lagrangian has speciiied the Higgs-fermion (Q5 f f ) Yukawa couplings and their Lorentz structure, as well as their C, P, and CP properties. All these symmetries are separately conserved in .C( f, SSBq$)and the Higgs-fermion Yukawa couplings are manifestly diagonal in flavour space. It should be emphasized that this is a consequence of the SM choice of a single doublet Higgs field since, otherwise, flavour-changing neutral—current (FCNC) transitions in the Higgs sector would be allowed. In general, generating fermion masses with their known values and keeping the FCNC transitions at an acceptable phenomenological level is a highly non—trivial problem. Finally, one can carry through the transformation (10) in the part of the SM Lagrangian describing the fermion-gauge boson couplings, L( f, W,B). Written in terms of the physical boson (W}, ZE, Au) and fermion fields, it is easy to show that the neutral current (NC) part of £( f, W, B) is manifestly flavour-diagonal. Thus, all flavour—changing transitions in the SM are confined to the the charged current (CC) sector. Denoting the quarks and leptons by f,(i : 1...6), the neutral current in the SM is given by: J:/YC ([3L —QS1H20w)i+€AuQg]fg.
The electroweak mixing angle in Jivc, denoted by OW, has its origin in the diag onalization of the gauge boson mass matrix, and it has the the usual definition cos HW = g2/\/gf + gg, with the electric charge defined as e E gz sin OW. It is easy to check that the neutral current interaction induced by the Z exchange violates P and C but conserves CP. The electromagnetic interaction conserves, of course, all three C, P, and CP separately.
We now proceed to discuss the texture of the charged current JEC, which is to be derived from the .C( f, W,B) part of the SM Lagrangian using the mass eigen states. The CC couplings in the SM involve only the left-handed fermions qi and I},. Concentrating on the hadronic (quark) part of LCC, one can now do the following manipulations:
LCC : —-——°·* ~w+d* ii. . 2\/§SlD26W ULSY ut L + C uv‘piv‘wv°“"lv¤~md+gggw,j§gg h.c. y M .._.._...C 3 -PhiW+ Vu? I _-’ 2x/Zsinww (UL l il down?” i L )VL Ph (dLij +11j us>C (14) OCR Output Thus, the charged current JS,C which couples to the W*, is
JS=C (&»E»{)L'7uVCKMd ( ) b8» L (15) where we have again dropped the superscript and VCKM E V£LpV1;isl0wnl a (3 >< 3) unitary matrix in the flavour space, first written down by Kobayashi and Maskawa in 19733. It is a generalization of the Cabibbo rotation4 for the three—quark—flavour (u, d, s) case, invented to keep the universality of weak interactions, which took the form of a (2 >< 2) matrix by the inclusion of c—quark with the GIM constructions, and is called the Cabibbo—Kobayashi—Maskawa (CKM) matrix. The charged current Lagrangian has the property that it has a (V — A) structure, hence it violates P and C maximally, conserves the electric charge and the lepton- and baryon—number sepa rately, but otherwise there are no restrictions on it except that VEKMVCKM = 1. ln general, LviolatesCC CP due to the possibility of a non-trivial phase in VCKM.
From this discussion it is clear that the flavour—changing transitions in the SM emerge from the Higgs-fermion Yukawa couplings and the diagonalization of the quark mass matrices after spontaneous symmetry breaking. As stated already, the Yukawa couplings in the standard model are arbitrary complex numbers and therefore there is no hope of understanding their origin within the standard model. The connection between the observed pattern of the fermion masses and mixing angles is one of the outstanding problems in particle physics. Perhaps, in a more fundamental framework these couplings may be derived from some deeper dynamics and/ or symmetry. Since the interpretation of the experimental results in the SM framework does not require a prior knowledge of this connection, we shall assume the validity of the CKM matrix as a consistent framework and study the consequences of this assumption in weak decays. The matrix elements of VCKM are determined by the charged current coupling to the we bosons. Symbolically this matrix can be written as:
VCKM 5- Wdl/Ld WsWs Wit WdVet Ws We (16)
The nine quantities V}, have to be determined experimentally via the couplings W*qq’ and this is the principal task of experimental flavour physics. Of the nine elements of VCKM, four involving the u,d,c,s quarks were already known before the advent of b physics. Their current values can be seen in the Particle Data Group (PDG) compilations? Two of the matrix elements involving the b quark, Vub and Kb, have been measured in direct decays of the B hadrons in experiments at the e+e" storage rings CESR and DORIS. This will be discussed quantitatively here and in the rest of this book. The remaining three elements Wd, V}, and VQI, are also accessible in OCR Output B decays through virtual transitions involving the couplings Wtb, l/Vt.§ and Wtd. Examples of these transitions are the |AB| = 2 processes, B0 ——B° rnixings, and AB| = 1 FCNC processes, such as rare B decays b —> (s,d) + *7 and b —> ($,:1) + l+l‘. Precision experiments involving B decays will completely determine the matrix VCKM, including the CP-violating phase. The original parametrization due to Kobayashi and Maskawa;3 was constructed from the rotation matrices in the iiavour space involving the angles 9, = 1,2,3) and a phase 6, where 0 § 9, f vr/2, O f 6 f 2vr, and R1j(0,¢) denotes a unitary rotation in the (i,j) plane by the angle 0 and the phase gb. The resulting representation, which we record for historical reasons, is:
C1 —51€3 —51$3 Z I SIC; C1C2C3 — S2S3€i6 CICZS3 + S2C3€i6 I , $152 CISQC3 + C2S3€w C1S2S3 — C2C3€1 with c, = cos 9,,.s, : sin 0,. This reduces to the usual Cabibbo form for 92 : 93 = O, with the angle 01, identified (up to a sign) with the Cabibbo angle. The Particle Data Group6, however, advocates a "standard” parametrization, which differs from VKM in assigning the complex phases (dominantly) to the (1,3) and (3,1) matrix elements of VCKM. We shall not write the exact PDG version of the CKM matrix since it is a cumbersome construct, but give instead the simpler approximate form, which follows from the a postcriori realization that cm 2 1 to a very high accuracy, due to the measurement of [Mb] = $13 2 U.003—0.006 (see below).
612613 $12613 5136-M13 VPDG 2 I --912623 612623 $23613 (19) b '6‘° $12523 · 612623$136-612523 623613 The three angles called 0,,, i yé j, can be chosen to lie in the range 0 §_ OH f 1r/ 2, making all the sines and cosines c,j and si, real and positive. The KM phase, called for obvious reasons 613, lies in the range 0 g 613 $ 21r. In the limit 013 = 023 = 0, the third generation decouples; identifying the Cabibbo angle with 012, one recovers the Cabibbo form. For the PDG parametrization,
ig.! = $12613 2* 5121
Mal Z $13s
lébl = $23613 2 $23
Another approximate but very popular form of the matrix VCKM is due to Wolfenstein.], who made the observation that (empirically) the following pattern for the VCKM matrix elements is suggested by data: lé1| 2 1, i= 1...3, OCR Output WI 1* \%1| ~ »\, Wal 2 Mel ~ A YG3| and |%3| ~ Ad', (21) with /\ 2 sin GC 2 0.221. Thus, it is useful to write a pcrturbativc form for VCKM. Dcnotmg th1s by Vwolfenstein, 1- E/\2 AA3(p — in) Wolfenstein = \ "" 3 1 " lp " iA2)‘4'7 A’\2 (22) _ A/\(1— p — uy) —A/\2 As for the previous iepresentations, Vwqlfenstein nas three real] parameters A? A and p, and a. phase ry. Since we shall be making extensive use of th1s parametrizauon, we write some relations involving the matrix elements of interest in this representation: Vub MEF, Kb W1 M/(1 — n)2 + "2’ l/Lb Ws (23) Kb The phase convention for VPDG is the same as that for Vwolfcnstcin.
1.1 The CKM unitarity triangles
The CKM matrix elements obey unitarity constraints, which state that any pair of rows, or any pair of columns, of the CKM matrix are orthogonal. This leads to six orthogonality conditions. The three involving the orthogonality of columns are listed below, with the quark pair in the parenthesis ( j k) representing the product of the j’th and k’th columns: (ds) ¥ Z Wdlé:
(sb) ¤ Z wc; (24) (db) = E WMS
Similarly, there are three more such orthogonality conditions on the rows: (¤¢)= Z WM} = 0, OCR Output (ci) = E WM} = 0. (25)
(ut) : Z Vujl/Q; = O.
The six orthogonality conditions can be depicted as six triangles in the complex plane. This is shown in fig. 1, in which each triangle is marked by the pair of quarks, representing the pair of columns, or of rows, whose orthogonality is represented by this triangle. Another way to view them is through the FCNC transitions in which the pair of quarks depicted participate. The triangle labelled (ds) represents the unitargy constraints on the transition s —+ d, which, for example, one encounters in the K °—K 0 transition. The two others in this group, namely (sb) and (db), are encountered in the FCNC transitions in B decays, such as particle—antiparticle mixings and rare B decays B —> K * + 7 and B ——> w + 7. These unitarity relations will, of course, be tested in CP violation experiments through the asymmetries that measure the angles in these triangles. The (uc), (ct) and (ut) triangles can be measured in FCNC transitions involving the Q = +2/3 quarks, i.e. c —> u, t —> c and t —> u. The analogous FCNC processes in this case are D°—D° and T°—T0 mixings, and decays such as D —> p+7, T —> D" +7 and T —-+ (D, D", p) + Z, etc. However, due to the fast decays of the charmed hadrons, which do not have any CKM suppression, unlike the K and B decays, and the even faster decays expected for the top quark, such FCNC transitions will be very difficult to measure. Of course, the sides of all of these triangles can be measured in CC decays of the quarks, as indicated in fig. 1. However, for the (ct) and (ut) triangles this would require the CKM matrix elements Wd and l/Q5, once again a dismal proposition in direct top quark decays due to the anticipated tiny magnitudes of these matrix elements (see below). The (uc) triangle involves CKM matrix elements that have all been measured, but getting a sharp profile of this triangle imposes enormous demands on the accuracy of the matrix elements involved. It should be noted that not all the sides and angles in these triangles are independent. Aleksan et al.8 have shown that only four of the eighteen angles in the six CKM unitarity triangles are independent and all CP asymmtries can be expressed in terms of these four angles. Unitarity constraints on the matrix elements resulting from VéKMl/CKM = 1 provide very powerful and interesting relations. In our opinion, the most interesting is the one stemming from the norm condition in the first and third row of VCKM: VMMQ + VLJCQ + Vutléi = 0 (26)
Since
Vud 2 wb 2 ls WL 2 —l/Lb, (27) the unitarity relation (26) simplifies: Wt + WZ = Wsléb, (28) OCR Output
10 ds uc ,_ Vudvus Vudvcg * Vcdvcs VMVG, ub cs
Sb ct . Vr=V¤¤ Vc¤V¤i§ ycdvtcll Vcsvcg'$/V V °° us ub Vcsvcg
db ut VMV Vudvu V¤uV1Z
Vcdvch
Figure 1: The six CKM unitarity triangles following from the orthogonality of the indicated pairs of columns, or of rows. The sides are drawn to the presently estimated lengths.
11 OCR Output B->(vn,lvp, B—>(vD,£vO', B —> D 1: B —> rz rz, p rt A (P. nl Bu ····> Ba B; .-»¤° 1:`,p‘° 1I:, • V ub lb V¢¤l
Via l )· Vcb l
C (0,0) B (0,1)
8;-) K,p°,D: K° Ba __>J/vK··DD’£ 7
Figure 2: The (db) unitarity triangle. The sides can be measured in CC and F CNC B transitions. The angles oz, B and 7 can be measured via CP violation in the B system. which can be conveniently depicted as a triangle relation in the complex plane, as shown in fig. 2. Thus, knowing the sides of the CKM-unitarity triangle, the three angles of the triangle a,B and 7 are determined. These angles are all related to the Kobayashi—Maskawa phase 6 (equivalently the phase 613 in VPDG orwthe phase ry in Vwolfenstein), and they can, in principle, be independently measured in various CP-violating B decays. As we shall discuss below, the matrix elements Vai, and V,,;, are already known from the CC B decays. With more data from B decays and an improved theory one would be able to determine them rather precisely. The matrix element Wd can, in principle, be determined from the rare decays b —+ d+ 7, d+ 1+1*, and B3—Bg mixing, the last of which already provides some constraint, which we shall discuss later. This set of experiments then provides another way of determining the triangle. The unitarity triangle represents common testing grounds for the SM elec troweak flavour physics, and simply stated the goal of future experiments in B physics is to overconstrain this triangle. We shall present a profile of this triangle in section 6, making use of our present theoretical and experimental knowledge in the quark sector.
There are yet more representations of VCKM. It has been pointed out by J arlskogg, that there are as many as 36 different parametrizations of the CKM matrix! Clearly, since the phases of the quark fields are unphysical quantities, the different parametriza tions, emerging from specific choices of these phases, must all be equivalent. The pa.rametrizati0n independent quantities are the absolute values of the matrix ele ments |V§j| and the area of the unitarity triangles in fig. 1, which is an invariant
12 OCR Output measure of CP violation. The area of the triangle can be expressed as
r€a_(A) : %.S]2S23S]3S1I16]3
LA/\6 [Wolfenstein], (29) for the PDG and Wolfenstein parametrizations of VCKM.
2. History: Discoveries at Fermilab, DORIS-II and CESR
In this section, we give a brief account of the discoveries concerning the T res onances and B hadrons. The resonance having the quantum numbers P JC : 1"', subsequently termed as T(1S) : T(9460) in the PDG tables, was discovered in 1977 at Fermilab by the Columbia—Fermilab—Stony Brook collaboration (CFS)1° in the pro cess p + nucleus ——> ;4+,u` +X. The background Drell-Yan process in hadron-hadron collisions follows a monotonously falling invariant dimuon mass distribution and is well calibrated. With a mass resolution of JM/M ~ 2%, the first data1° yielded the evidence for a narrow resonance, called T, at ~ 9.5 GeV. More data and an improved mass resolution led to a second narrow resonance, called T', at ~ 10.0 GeV and pos sibly a third one,T", around ~ 10.4 GeV 11*12. The results of the CFS collaboration are shown in fig. 3. The fact that T and T' had a significant branching ratio into the ,u+;1' final state to be observed at Fermilab indicated that the inclusive decay widths, 1"(T) and I`(T'), are probably substantially smaller than the experimental resolution, 0M ~ 200 MeV. This was subsequently confirmed by experiments in e"'e' annihilation at the energy upgraded DORIS13`15 and later at CESR16·17, which also established the quantum numbers J P C = l" and the (bb) bound state nature of the T family of resonances. Electron-positron annihilation experiments are particularly suitable to study the properties of heavy quark systems, open and bound. The background in this case arises from the QCD processes
66——>+ qé. qéy. (30) The hadronic cross sections can be expressed in terms of the Drell ratio: 0(e+e" -—> hadrons) R E ¤(¢+¤‘——> MWF) 2 = 3E:e§,1+—{—+...,(?§@>vr (31) where o4,(Q2) is the QCD effective coupling constant evaluated at Q2 = Egm, and eq, are the quark charges. Ignoring the QCD corrections, for the sake of simplicity,
13 OCR Output dig r1mdy|y=O ,6[_` ’°
Uh ••°, |• J I 0.2 1I ' •,•
·1v 10 I' OO
9 10 11 la `bl moss (GeV)
6 '/ 8 U IO H I2 \3 mass (GeV)
Figure 3: Invariant dimuon mass distributions in p + Be -+ ;1'*'y'X measured by the Columbia-Fermilab-Stony Brook Collaboration, providing evidence for the T and T' resonances. the background hadronic processes contribute Q units of R, which in absolute units corresponds to approximately the same cross section in nanobarns at Em, = 9.46 GeV. To this one should add about g unit of R due to e+e' —-—>·r"'·r' —> hadrons. The signal process, namely e*e' ——+ T —> hadrons, takes place via a single·photon annihilation of the e+e‘ pair, the coupling of the photon to the T and the subsequent decay of T into hadrons. This cross section can be expressed as
+ l.)7|’ 1`,,I`;,,d
+ _ _ a(e e ——> T ——-> hadron)M - —-TE_ M2)? + ¥ , (32) where M is the mass of the resonance, I` its total width, and 1`,, and PM; its partial widths into e+e' pair and hadrons, respectively.
Subsequent to the discovery of the T resonances at Fermilab, the e*e' storage ring DORIS at DESY was upgraded in energy to reach the T region. In the spring of 1978, the PLUTOI3 and DASP 11** collaborations at DESY confirmed the existence of the T resonance. They determined its mass to be 9.46 t 0.01 GeV, and its width in the range 25 keV < 1`(T) < 8 MeV, where the upper limit was set by the energy resolution of DORIS. Somewhat later, with additional rf cavities at DORIS, the T' was also confirmed with a mass 10.016 i 0.02 GeV by the DASP II and the Nal Pb-Glass detector (DESY-Heidelberg-Hamburg-Miinchen col1aboration)l5. Figure 4 shows the status of the T and T' resonances at the Tokyo conference in 1978. These
14 OCR Output + C¤hm•lu.•·FERMlLAU·${¤;.y/Troon, Colmborahon (l0kyc_Au9 76) NAI-F’b·G|,\s 0•·le:|orIOESV(0OfN5} T UCSV·I•e¤JHbr¤q·•v.m•uu1q-•·lungue»·. Cnnsuoulncn ( Auq 7B)
; I
·_l··-l—·\—-1l ;_A—- a_.-a-...I..--n...,.¤... •..-..¤...... L.-@ J‘ ---1` ": ·‘ ` ` ` ` "‘··— __
-¤.... 850 - -slcc ..L... asa __, _. . _ . _, igoo " »bls0"' total energy (GeV)
Figure 4: The T and 'I" resonances as seen in the hadrouic collisions (CFS Collabo ration) and e+e‘ annihilation (DESY-Ha.mburg~Heidelberg-Mi`1nchen Collaboration). measurements gave in additionl
I"(T —> e+e') (1.30 t 0.3) keV, B(T ~> #*1F) (2.3 ;l: 1.4)%, (33) I`(T’—» e+e“) (0.33 :l: 0.1) keV.
There was a third resonance in the Fermilab data. In particular, taking the mass difference m(T’) — m(T) = 558 zh 10 MeV measured at DORIS, a reanalysis of the data revealed a T" resonance, with AM (T" - T) = 950t50 MeV *2. This resonance was also confirmed in early 1980 by the CLEO‘“ and CUSBU collaborations at the e"'e‘ storage ring CESR, which had been commissioned for operation in late 1979. The three narrow T resonances measured by CLEO are shown in fig. 5. In addi tion to providing a more precise determination of the mass splitting AM (T — T'), the experiments at CESR provided the level spacing and the leptonic widths of the T" :
AM('I`" - T)(MeV) = 891.1 :1: 0.7 zi: 0.3 [CLEO], = 889.0 :l: 1.0 :l: 5.0 [CUSB], (34)
and
15 OCR Output 20}· i ' F
9.42 9.44 9.46 9.48 9.50
9.97 9.99 l0.0| |0.03 10.05
fi Li 1 * iii} 1
l0.3Z IOJ4 l0.36 IO.38 IO.40 W · Genie: ol man enuqy, GN
Figure 5: The T(1S), T(2S), and T(3S) resonances as measured by the CLEO de tector.
I`,.(T")/1`,,(T) = 0.35 :l: 0.04 zh 0.03 [CLEO], = 0.32 :b 0.04 [CUSB]. (35)
The three narrow resonances seen experimentally, with their measured electronic widths and the quantum numbers J =P C1" , readily lend themselves to interpreta tion in terms of 3 S; states of a bound system of heavy quark-antiquark. The general features of the number of S -states below the threshold of heavy-meson pair produc tion, their level spacings, and the leptonic decay widths (wave function at the origin) had been predicted in a number of models19'°3 based on their successful application in the J /¢• system°"·°°. In addition, the decay width 1",,(T) = 1.3t 0.3 keV favoured the assignment of charge for the new quark. Since the third charged lepton, ·r, had been discovered in 1975 26, with very probably its own attendant neutrino, lepton·quark universality had made the advent of a third quark family mandatory. Quite independently of this, and prior to the experimental discoveries of the r lepton and T resonances, Kobayashi and Maskawa in 1973 had postulated the existence of a third quark family in order to incorporate CP violation in the electroweak SM frame work, as we discussed at some length in the previous section. Thus, the stage was set for identifying the T resonances with the quark-antiquark bound states of the b quark.
Clearly, such an identification could only be tentative. What is needed for the
16 OCR Output OZ50I07•0IZ
" 'I`(|S Z L • ) ‘°’ 4 :.5}- L ¢<4S>‘b’
as
3‘° r<2s> T‘5,$’r
9.45 9.50 |0.00l0.05 \O.4O IO.5O lO.6O 10.5 |O.B II.! w, CENTER 0F MASS ENERGY (Gcv)
Figure 6: The e*'e’ total cross section measured by the CLEO Collaboration a). The energy region T(1S) through T(4S) b). The energy region T(4S) through T(6S
Kobayashi-Maskawa mechanism of CP violation to work is a third quark family, with SM assignmeng of charge, weak isospin, hypercharge, and canonical couplings of the (b,t)L doublet to the wi bosons. Moreover, all the CKM angles must be non zero. The bound (bb) resonances, which we shall henceforth label according to the PDG convention as 'I`(1S), 'I`(2S), and 'I`(3S), provided a determination of the elec tric charge of the b quark. The assignment of other quantum numbers required the production of hadrons (B = bei) with b—f1a.vour and their subsequent decays. In a number of potential models, it had been estimated that T(4S) should lie above the BB production threshold (see, for example, Quigg a.nd Rosner°°). In that case, the OZI suppression", which is at work for the T(1S) — T(3S) decays — hence their small widths — would become inoperative, giving rise to typical hadronic widths of T(4S), 'I`(5S), etc. They should show up as broad resonances with final-state topolo gies markedly different from the other three T resonances. The 'I`(4S) and higher resonances were found at CESR by the CLEOZS and CUSBZQ collaborations in 1980. Since then, both DORIS and CESR have accumulated high statistics at the T(4S), which is the principal source of B decays at present, though experiments at LEP and the Tevatron have lately provided very impressive results, in particular, on the lifetime measurements and properties of the B, and Ab hadrons. Scans of the e+e' total cross section showing the resonances T(1S) through 'I`(4S), and 'I`(4S) through 'I`(6S) are shown in figs. 6a and 6b, respectively. The (updated) parameters of these resonances are shown in table 1. In fig. 6b the structure in the cross section at 10.68 GeV is not completely understood.
17 OCR Output Table 1: Properties of the T family of resonances (from the PDG tables6)
T state | Mass (MeV) Pm, FCE (keV) IS I 9460.32 i 0.22 I 52.11 2.1 keV I 1.34 i 0.04 2S I 10023.30 ;k 0.31 I 43 i 8 keV I 0.586 j; 0.029 3S I 10355.3 i 0.5 I 24.3 :1: 2.9 keV I 0.44 j; 0.03 4S I 10580.0 i 3.5 I 23.8 i 2.2 MeV I 0.24 1 0.05 55 I 10865.0 5; 8.0 I 110 113 MeV I 0.311 0.07 6S I 11019.0 :1; 8.0 I 79 116 MeV I 0.13 i 0.030
Table 2: Measurements of Xystates. The Xb masses are calculated from the photon energies and M(T2S) = 10.02330 j; 0.00031 GeV and M('I`3$) = 10.3553 ;I: 0.0005 GeV (from the PDG tables6)
Xb(1P) T(2$) —» vxb(1P> xb(1P) -1r vT(1$) E., (MeV) MX (MeV) B(%) B(%) Xz>,(3P2) 109.6 -1; 0.6 9913.2 :1; 0.6 6.6 :1; 0.9 22 1 4 Xb1(3P1) 130.6 i 0.7 9891.9 :1; 0.7 6.7 i 0.9 35 1 8 Xb.,(3P0) 162.3 :1; 1.3 9859.8 11.3 4.3 11.0 < 6 (90%CL)
Xb(2P) T(3$) ——> vxz»(2P) > In addition to the triplet S-states seen directly in the e`*'e‘ annihilation channel, a number of photonic and hadronic transitions have also been seen in the decays of T(3S) and T(2S), providing evidence for other (bb) states. The discovery of the transition T(2S) —> T -1- mr 3°‘33 and also the transitions T(3S) —+ T(2S) + rrvr and 'I`(3S) —> T -1- 7r1r 333*, confirmed the bb bound state hadronic nature of the T resonances. Moreover, the triplet P-states Xb(1P) and X1,(2P) have been ob served, as shown in the level diagram (Hg. 7) taken from PDG3. The ARGUS 35, CLEO33, Crystal Ball37, and CUSB33 have each measured the X;,(1P) in the decays T(2S) —> Xb,7 ———> T(2S)#y7. The Xb(2P) states are measured mostly by the CUSB collaboration39. The present average of the Xb(1P) and X1,(2P) mass measurements, the photon energies and respective branching ratios are shown in table 2. The tran sition X1,0(1P) ——·> T(1S) -1- 7 has not yet been seen. While the triplet P-states are on a firm experimental footing, there is no experi mental evidence yet for the singlet-P states, called hb(1P) and hb(2P) in PDG tables. 18 OCR Output OCR OutputTable 3: Measurements of mr transitions in the decays of T(2S) and T(3S) (from the PDG tables';) Decay B(%) r(2S) -» r(1s)¤+¤· | (is.5 i 0.s)% —> T(1S)vr°rr° I (8.8 i1.1)% r(3$) -» r(2s)¤+¤· | (2.2 i 0.5)% --» T(lS)¢r"'rr` ) (3.63 al: 0.31)% —» T(2S) + X [ (10.1 $1.7)% The h),(1P) and hb(2P) with JPG = ll" cannot be produced directly in e+e' anni hilation or radiative decays of T's. The reaction T(3S) —> vrrrh), has been suggested in the literature as a probable process to observe the singlet P-states. However, the issue whether such a transition has been seen at a statistically significant level remains open. Likewise, the singlet S-state m,(1S), n),(2S) and m,(3S) have not been observed. The direct decays T(35') —> T(2S)rr7r and T(2S) ——> T(lS)7r¢r , involving hadrons, have been measured and their branching ratios are summarized in table 3. For a crit ical review of T spectroscopy and tests of QCD we refer to the comprehensive review by Buchmiiller and Cooper‘*° (see also Field‘“). Hadronic decays of the T resonances and their theoretical interpretation have been discussed by Wegener4 In this article we shall be mainly interested in the properties of B and B hadrons, which decay weakly. Their copious sources, apart from pp collisions at the CERN and Fermilab colliders and e"'e' annihilation at PEP, PETRA, KEK, LEP and the SLC, are the production and decays of the broad resonances T(4S) and T(5S That the T(4S) decays into a pair of BB mesons was confirmed in 1983 at CESR, and later at DORIS, where the B mesons were explicitly constructed. In addition, a number of properties of the T(4S)-decays has since been measured, all of which confirm the expected final states. A detailed account of the experimental data in B decays is given in a number of articles in this book"3"*5. We shall selectively take up some of these measurements and discuss their theoretical implications in the subsequent sections. 3. Evidence for B Decays Obeying the Standard Model In this section we review the present status of experimental measurements con cerning the properties of the b quark and their theoretical interpretation in the context provided by the standard model and QCD. The effective lowest-order weak interac tion Hamiltonian can be expressed in terms of J iv C and J E C, introduced earlier. 20 OCR Output HW ZTT; Jg¤m¤¤ [( + hx.+) (gy)]Jw~~¤ , (36) where Gp is the Fermi coupling constant. Perturbative QCD improvements in 'Hw have been worked out using the renormalization group techniques46"49, and in most numerical estimates presented below these improvements will be used. The quan titative description of the physical decay processes, with hadrons in the initial and (in most cases of interest) final states, requires the knowledge of the wave functions and/or hadronic form factors. The primary task of the theory therefore is to eval uate these non—perturbative functions. While there exists an outstanding exception, namely the decay B —> D’°‘ + fw, where the symmetries in the effective heavy-quark framework have been persuasively used to make (almost) model—independent pre diction of the relevant form factor, by and large estimating B hadron transitions quantitatively remains a model—dependent enterprise. There are several theoretical approaches, which have been used in quantitative studies of B decays and related phenomena, with varying degrees of rigour and suc cess. Examples are QCD sum rules 5°`6°, the & approach (Nc being the number of col0urs)61, spectator models with and without perturbative—QCD improvements and including explicit assumptions about the hadronic wave functions62”69, phenomeno logical models based on hadronic form factors with or without the assumption of factorization7°‘75, methods employing the inverse heavy—quark mass (1 / mq) expan sion in QCD with estimates of leading power corrections in some inclusive decays76*77, applications of the heavy—quark effective theory to exclusive pr0cesses78`91, which elaborated and sharpened earlier theoretical work on this subject 92, and last, but not least, lattice formulation of QCD93‘1°° . Clearly, a critical review of this liter ature is beyond the scope of this paper. Luckily, excellent recent reviews exist on many of these approaches77*9°*91·95, to which we refer for details. We shall discuss representative results on B decays obtained using some of these theoretical methods. Concentrating on 'Hw, testing the standard model in B decays requires establish ing the following properties: (U The V — A character of J E (ii) The weak-isospin doublet character of the b quark and determination of its weak charge (ISL —- Qsinz OW) and checking its consistency with other sin? GW measurements. (iii) A precise determination of the CKM matrix elements YQ, and Kd, from the CC B decays. (iv} Absence of FCNC transitions at O(GF) and their measurements at the GIM prescribed levels5. 21 OCR Output (za} Precise measurements ofthe CKM matrix elements |l/Q6], |V§,|, and |VQd| from FCNC processes, enabling precision tests of the CKM unitarity triangles. [vi} A consistent determination of the KM phase 6 in B decays and other CP vio lation phenomena (e, §, rare K decays, etc.). In the rest of this article we shall quantitatively discuss these points, using present data and suggesting future measurements. 3.1 (V — A) character 0f JS C We start with the assumption (justified later) that the inclusive decays of B hadrons, in particular their semileptonic decays, can be modelled on the QCD improved quark model decay Q —> qlvg. For b quark semileptonic decays involving CC interactions, one has two partonic transitions: b —-—> c€'17g, (37) 1* UE- D;. There exists a close analogy between the b quark decays and ;4 decay, p' —> 8-17cl/M, with the identification: lb? (C7 u)7Dl7 E-] (_} [#-76-71767 I//·’·]° This analogy holds in general; in particular, at the one loop level O(a) QED correc tions to n' decay and O(¢1,) QCD corrections to b semileptonic decays are related by simply replacing62‘63 1 8 4 CY ·—·* —O_,TT`Z/MA; = -0,, 3 {:1 3 where 12 a. : ——7£ (40) (33 — 2nf) ln(X§—) The semileptonic decay rates can then be read off the expression for the O(¢1) ra diatively corrected p-decay ratewl. The rates for b —+ (u,c)l?u; decays, setting ml = my, = O, are given by the expression: 2a m2 i raw —-»<~,c>4»r> -(§§,) 1`?Z1—¢f<~>, vm with Fg;) being the 1owest—order rate €i F?} = —;— I Wt I2 m5y(¤) 1927rS '>’ (42) OCR Output 22 where Tg = = u, b) and g(*r) = 1 —— 81*2 ·l- SW6 — 1*8 — 241*4 ln r. (43) The function f(1‘,) has the normalization : H-? and f(0.4) 2 2.3, relevant for the b —> u and b —> c transitions, respectively62*63*66. This gives ~ (15)% corrections reducing FSL (compared to Fg), which is relevant for estimating the semileptonic branching ratio B (B ——> X FW), to which we shall return later. It has been known for quite some time that the shape of the electron energy spectrum in ,u’ decay is sensitive to the space—time structure of the charged weak current J E C. ln particular, ignoring the electron mass, the electron energy spectrum in ;4—decay has the simple form: dp (p ——> eDe1x,,)~ 40:2 |3(1— x) + 2pMx -1, (44) dx 4 (§)] where :1: = 2Ec/mu, and pM is the Michel parameter with pM = (0,%) for the (V-l-A, V——A) CC interactions. The O(¢;x) QED corrections modify this resultlm. Ex perimentally, after correcting for me and QED effects, one gets6 pM = O.7518;l:0.0026, which is the present best evidence for the leptonic part of the current Jgc being V-A. Returning to the discussion of lepton energy spectrum in the semileptonic b decays, the corresponding expression for the V - A interaction is6 2 2 2 g ~ [(1-x)(i-x-1~2)+(2-x)(i+21~2-x)| , (45) where az = 2E;/mb. Again, there are O(o4,) QCD corrections to dl` /071;, which were calculated quite some time ago66·67. lt follows from these studies that the shape of the lepton energy spectrum is stable against O(cx,) corrections, except near the end-points Eg —> EW", which in perturbative treatment requires a Sudakov—type exponentiation. The QCD-corrected lepton energy spectra are process-dependent and the corresponding expressions for the transitions b —> cfu; and b —> Mw are known. These corrections can be explicitly taken into account to correct the (bare) Michel spectra in semileptonic b decays due to the perturbative-QCD effects. Unfortunately the similarity between the decays if ——+ C-Del/M and b —+ (u, c)l`1`}; ends here! While for the former the Sudakov-improved O(¢x) perturbative calculation is essentially the final result, for the latter one has to calculate the non perturbative effects stemming from the fact that both the initial and final quarks are bound into hadrons. Estimating the non—perturbative corrections to the lepton energy shape functions in B decays, B —> XC + ful, B —> Xufw (and corresponding distributions in FCNC radiative and semileptonic rare B decays, B —> (X,,Xd) + 7 and B ——> (X,,Xd) + €`*'€') is a non-trivial matter. This problem is formally equiva lent to predicting the structure function for the nucleon in deep inelastic scatterings, Ep —> (Z, 1/)X. For B decays, since at least the decaying quark is sufficiently heavy compared with the typical hadronic scale, it should be possible to use the heavy quark effective theory to relate some of the leading contributions in the processes 23 OCR Output listed above in a restricted kinematic domain. We shall discuss some suggestions en tertained recently in this direction later, but discuss first a simple ansatz to estimate the B—meson wave function effects on the lepton energy spectra. Apart from the quark mass in the final state, there are two effects that matter for the final—state distributions in inclusive semileptonic B decays, namely the (Fermi) motion of the b quark inside the B hadron (Doppler effect) and its off—shellness (mass defect). The energy-momentum constraint gives the mass defect (mf; — W2): W2 = M; + mi · 2MBx/I F I2 +m3» (46) where W is the b quark mass, gi is its three-momentum and mq the spectator quark mass in (B = bq). The momentum distribution of the b quark inside the B hadron is a model—dependent enterprise. If the decay quark mass is very large, the model dependence (due to effects of typical hadronic scale) should be weak, although the detailed distributions would depend on the precise form of the non—perturbative function, which is unknown. In the early papers on this subject66*67, the b quark was given a non-zero three-momentum having a Gaussian distribution determined by a phenomenological parameter Pp, to model the non—perturbative effects: 4>(1>) =4 ——-exp— I P I2 (t-" »/?1>i¤ Pi? (47) The lepton-energy spectrum is obtained by the convolution (p El 5 dl` dl` Pmaz ]; — = where pmq, is the maximum allowed value of p (with the energy-momentum con straint) and gh is the lepton—energy distribution from the decay of the b quark in motion, having an effective mass W and momentum p. In this model there are three 2 2 parameters that determine the lepton-energy shape: W, Pp and r = for b —> c (b —-> u) transitions. The energy-momentum constraint could be used to reduce them to two, fixing mq (whose variation is not essential for the present discus sion). This QCD-improved spectator model, often referred to as the Altarelli et al. model, has received quite a bit of experimental scrutiny‘3·2°2'2°5 and lately some theoretical criticism"26 due to its simplistic description of non—perturbative aspects in B decays. In an analysis by the CLEO collaboration, both the electron and muon spectra from the decays b ——> c€“ uq(Z = e,;i) have been fitted to determine the parameters mq (equivalently r) and Pp, with the result Pp = 0.30 :1: 0.09 GeV and mc = 1.680:}:0.086 GeV, with X2/(dof) = 17.4/31202. The result of an analysis by the ARGUS collaboration is very similar2°3. From the successful fit of the data using this QCD-improved spectator model, a number of quantitative statements will be made. 24 OCR Output While we defer the discussion of the CKM matrix elements to a subsequent section, it should be noted that, despite model dependence, the charged current JSC-induced transitions b —-> dw measured in B decays are consistent with the SM V — A form. As an alternative to using a specific artifact in modelling the transition amplitudes, and in the absence of any derivation of the non-perturbative shape functions from first principles, one could attempt to correct the inclusive lepton energy spectrum for the non-perturbative effects — in much the same way as is routinely done in comparing QCD-partonic predictions with inclusive jet distributions measured at LEP and elsewhere. This would allow to extract the effective Michel parameter for the b quark in a model-independent way. The interpretation of such measurements would require a modified calculation in which the lepton energy spectrum is to be calculated for a configuration in which the recoiling hadronic system in the decay B ——> X,/17 has to be interpreted with the attributes of a jet in terms of infra-red safe variables. The difference in the lepton energy spectra induced by the V — A and V + A currents in the transitions b —> (c, u)E1/[ is most marked near the end-point Eg -> E["‘"’. For the b —> u transition, where the u-quark mass corrections can be entirely neglected in the expression for the lepton energy spectra, one has the simple forms: dl" (49) _ _ 2 (V — A): 1;E—(b -—> ui ug) ~ 2x (3 — 22:), and d1_ (V + A): %(b -> u€`17g)~ l2;r2(1— :1:), (50) where the constants have been chosen to give the normalization fg dwg = 1. With higher statistics data, the end-point lepton energy spectrum in the transition b —-> uE‘17g should become reasonably well measured so as to both establish the Michel spectrum and determine the Michel parameter. While still on the point of establishing the chirality of the CC b —> c transitions, it should be remarked that the exclusive semileptonic decays, in particular B —+ D’°‘£1/l, have also been analysed. In this transition both the vector and axial—vector currents contribute and the shape of the lepton-energy spectrum is likewise sensitive to the chirality of J E C, assuming the universality of the leptonic charged current. A particular variation on this theme has been proposed by Korner and Schulerlm, who have argued that the forward—backward asymmetry of the charged lepton in the decay B —> D* + E' ug is sensitive to the chirality of J E C. This asymmetry is defined as: d1`(6)— dI`(1r — 9) “"" dI`(0) +dI`(1r—0)’ "/ZSHST (51) Here, 0 is the polar angle of the E' in the (Z` 17;) rest frame. Writing in terms of the helicity amplitudes H+, H. and H0, the asymmetry ay, can be expressed as: 3/4v€ fK<12(|H+|2 — IH-l2)d<12dP¢ (52) OCR Output af, Z f K<12(|H+|’· IH-I2 + |Ho|’)dq2d1¤z’ 25 where q2 = (pg — p,,)2 and K is the momentum of the D"‘ in the B rest frame. The variable r] is sensitive to the helicity of the b —> c current, with n = +1 (-1) for b —> obi; decay with V — A (V + A) b —> c transition, and § describes the helicity of the leptonic current, with § = +1(—1) for the lepton V — A(V+ A) currentws. In the SM, nf = +1, with both the bcW“ and Z` z7bW‘ mediated by the WE boson. However, if for some reasons both these couplings are mediated by a right-handed boson WR, then one also has nf = +1, and just from this measurement alone one would not be able to distinguish a. V — A current from a V + A. So, the universality of the leptonic current has to be assumed. We recall that there is ample and convincing evidence6 that the leptonic currents in both ,a and T decays are indeed the ones prescribed by the SM, namely V - A. The experimental situation concerning this test is being discussed in this volume by Stone‘*3. The result of the experimental analysis can be summarized as follows: afb(pg > 1 GeV) = 0.14 ;l; 0.06 :1; 0.03 [CLEO II], (53) = 0.13 — 0.15 [SM], where the range shown in the SM—based theoretical predictions covers a number of models for the helicity amplitudes Ht and H0. From this comparison, it is fair to conclude that with the assumption of a universal leptonic charged current, the V — A structure of the b —> c transition has been established. 3.2 Weak charge ofthe b quark The electromagnetic coupling of the b quark has been determined from the cross section e+e‘ -—->·y —» 'I`(nS) —> FE`. We now discuss the experimental determi nation of its weak—charge and -isospin. The SM vector and axial-vector NC couplings of the b quark can be read from (13): vf = 2(I;_{ — 2Qj sin2 0W), (54) af : v with Qb == —% and Ig = —%. In particular, ab measures the b quark weak-isospin and vb its weak charge. The couplings ab andvb can be determined by the production cross wsections a(e+e‘—> bb) (or B (Z°——> bb) at the 20 pole) and the forward-backward asymmetry, AQB, defined as , jg ggidcosa - [E, ggdcoso FB = 1 dab () dgb ‘ fo md cos 9 + f_1 md cos 0 In the quark—parton model, the expression for Rb E a(e+e°—-> bb)/a(e+e' —> ,u"'p') and AQB have simple forms. For s < mb, Rb = QE + 2Qbvcvtxz + O(x§), 26 OCR Output 3 C 2B = %> aaai, : +0.97 i 0.22 , (57) to be compared with the SM prediction acm, == +1.0. With the SM value ae = -1, this gives I§ = -0.48 zi: 0.11 , (58) in agreement with the SM value Ig = -0.5. This measurement constitutes a good determination of the b quark weak-isospin and of the doublet nature of the b quark. By implication, the other member of the isospin—doublet, namely the top quark, must exist! The number from a global analysis of the ratio R and the jet charge asymmetry ag, = -0.93 ;l: 0.13 is similarwg to the ones quotedabove. _At the Z0 pole, the decay width l`(Z°—> bb) and AIQTB also have a simple form (ignoring the fermion mass, electroweak and QCD corrections) _ 3 WZ —·* bb) = (bZ +¤§)» (59) b Z : ______3FB( 2vca€ ) 4(b§+¤3)vE+b%’ 2vba;, where AQB has to be corrected due to the phenomenon of B—B mixinglw. Defining Af}; as the measured quantity and X the B°—B° mixing probability: Ag=rn .,, $(1- 2x)· (60) The present LEP results are u1‘H3 A%B(Z) 0.099 dz 0.006, 0.117 t 0.010, (61) Fw 383 :l: 5 MeV. We shall discuss the quantity X later. The above measurement of the partial width can be compared with the SM expectation Pb; = 378 MeV, confirming the SM assignment of the b quark couplings. Also, since vb = 2(I§ + § sin2 0W), one could use A*};B(Z) and 1`(Z -———> bb) to determine sinz Bw. This gives, after correcting for the electroweak radiative effects which introduce (renormalization) scheme dependence of sing Ow, the following value for the quantity called sinafgjr;z “‘· sin0;;.72 : 0.2322 i 0.0011 (62) 27 OCR Output This number compares well with other independent measurements of the same quantityl sinOi?}2 = 0.2321 ;t 0.0006 . (63) The electroweak data from LEP and lower energies were combined in 1991 by Schaile and Zerwas, and a more general global fit was done to determine the weak isospin quantum numbers. This fit has since been updated by taking into account the more precise present data, givingu ‘*· IQL = -0.494 + 0.013 - 0.010 , I§R = -0.018 + 0.049 - 0.068 . (64) The above value of IQL is in very good agreement with the SM value IQL = -0.5. Also, as there are no right—handed WR bosons in the SM, IQR = 0, and the above analysis is consistent with it. We conclude that the SM values ab = -1,vb 2 -0.68 have been experimentally established in the b quark production experiments at LEP, PETRA, PEP and KEK. 3.3 Determination 0f]l{,b| from the mean B lifetime (Tb) We take up the present determination of the CKM matrix elements Vg, from a number of measurements in B decays. These include the mean B lifetime, the inclusive semileptonic decays B —> Xctz/[ , the exclusive decays B —> (D, D*)€1/l, analysed using models for the form factors as well as the heavy-quark effective theory approach. The smaller matrix element Md, is so far quantified only through the measurement of the end—point lepton energy spectrum coming from the decays B ——> Xugvg . Amusingly, the averaged B hadron lifetime is increasing as a function of time! The world average in 1990 115, (Tg) = (1.18 ;b 0.11) >< 10'12 s, moved up to (TB) (1.29i0.05) >< 10"12 s in 1992 6. The present world average, which is greatly influenced by the more precise LEP measurements, is"5·“6 (rg) = (1.49 ;k 0.04) >< 10`us. (65) The error on this quantity is now very small and it is natural to ask what we could learn from such a precise measurement. An honest answer is: were it not for the intrinsically imprecise value of the quark masses and non-perturbative effects, the B lifetime measurements would have determined the CKM matrix element |VQb| more precisely than any other method available on the market. An analysis of the lifetime measurements may be undertaken using the QCD—improved effective Hamiltonian for | AB | = 1 processes. Lumping together the QCD and phase-space corrections in the total decay width of the b quark in terms of two parameters, Kc and Ku, which multiply the CKM factors ]lQ;,|2 and |V§,b|2, respectively, one can express the QCD-improved spectator model decay rate as follows: Ggpms J•’ r E-¢vK. K., b 1921r3 Cb K + V 22). Kb I 66 ( ) OCR Output 28 Numerically, Kb = 2.8 :|: 0.8 and Ku 2 6.5, where the uncertainty in Kb is essentially due to mb and mb68*u7. We have neglected here the QCD and QED penguin contri butions as they are expected to be entirely negligible in the total decay rate. This expression can then be used to determine Kb from (TB), assuming that the QCD improved spectator model is an adequate description of the inclusive B decay rate. The biggest uncertainty in this approach lies in the value of mb. There now exist sev eral estimates of mb. Taking a value mb : 4.8 GeV H8, and using (rg) = 1.49 >< 10` s, one gets: v,,|’ : 7.59 X 10·3T—T4——- C Ac`l'Au(lVubl/ll/Lbl)2 (67) which, with the values Kb = 2.8 :}:0.8, Kb 2 6.5 and the present determination of the CKM matrix element ratio |l/bb)/|lQb| : 0.08 :t 0.02, yields: ml = 0.05118555 (68) This estimate, in view of the neglect of non-perturbative effects in the B decay width and the inherent sensitivity on mb, is subject to significant corrections at present. Of course, the basic assumption is that there are no significant non—spectator contribu tions splitting the various individual lifetimes of B hadrons, which are not included in the above estimates of average lifetimes. This assumption will be tested as the measurements of the individual B—hadron lifetimes become precise. With improved perturbative QCD estimates and theoretical insight in the definition of mb and mb, and with non-perturbative corrections becoming gradually known, mean B mean lifetime is potentially a quantitative method to determine |VQb|. At any rate, the estimates that one gets for |\Qb| from this method are in the right ball park, as we shall discuss below. 3.4 QCD—impr0ved spectator model and determination of Vbb from inclusive semileptonic decays In the following, we will quantify the accuracy of the spectator model by esti mating the semileptonic branching ratio in B decays, BSL. The role of non-spectator contributions due to the so-called weak annihilation and Pauli interference diagrams in meson decays have been analysed in a number of theoretical investigations69*1°°*H9, confirming that they are all small. We follow here the discussion by Altarelli and Petrarca69. In their analysis one gets somewhat different numbers for BSL, depending on the assumption of quark masses. With their choice of "light masses”, which also correspond to the current quark masses, and ("heavy masses”), which can be called the constituent quark masses, expressed in GeV as: mm = 0(0.16) , m, = 0.15(0.30), mb = 1.2(1.7) , mb = 4.6(5.0), (69) Altarelli and Petrarca obtain'; BSL = [12.2:1; 1.25]% (light masses), [14.4 zt 1.25)% (heavy masses). (70) OCR Output 29 Table 4: The B decay semileptonic branching ratios measured at the T(4S) 116*120 BSL[T](%) I Source 9.70 ;lz 0.6 i 0.4 I ARGUS MI 10.2 zi: 0.4 zlz 0.2 I ARGUS 10.65 zlz 0.05 zl; 0.4 I CLEO Il 10.0 zh 0.4 dz 0.3 I CUSB 12.0 zlz 0.5 zlz 0.7 I CB The experimental semileptonic branching ratios depend on the model used for the lepton energy spectra to correct for the low-energy leptons, which are not measured. For comparison with the Altrelli-Petrarca estimates, we use the lepton spectrum calculated using the so—called Altarelli et al. model67. The branching ratio measured at the 'I`(4S)U6*12° is given in table 4 and its model dependence is discussed by Stone in this volume43. The LEP average for the inclusive semileptonic branching ratio has been quoted as BSL = (11.0 i 0.5)%U6. One could reduce the theoretical dispersion on BSL by using the shape of the lepton energy spectrum and eliminating the models that fail to fit the data. Using this criterion, a model dependence of zlz0.4% has been estimatedlm. The CLEO analysis gives BSL = (10.8 i 0.2 zlz 0.4 i 0.4)%, which is in very good agreement with the one from LEP, despite the fact that the LEP data have a different mix of the various species of B hadrons. The corresponding ARGUS number is, however, somewhat smaller. The semileptonic B decay branching ratio in the QCD-improved parton model, represented here by the Altarelli—Petrarca estimate, is definitely higher than the present experimental measurements. We must conclude that there could be as much as 20% discrepancy here between the spectator model rate estimates and data. This is neither an outright victory nor a total defeat! J udged from this perspective, strong interaction effects in B decays are certainly weaker than in D decays, but not negli— gible. To be able to do better, one needs an understanding of the QCD dynamics in B decays. In this context, power corrections to the parton-model decay rate have been calculated recently."6, determining that the leading power corrections O(1/mg,) are not present in total decay rates. In this framework, the semileptonic decay width involving the b —> c transition can be expressed as: i F=FbI1+£f1<%)n§ · C2(%»)) (71) where the function has already been defined in the discussion of the spectator decay rates, where it was called g(x); the function fg appearing above is defined as follows76: f2(x) =1-Exz — 8:1:6 + isms + 81:4 lnzr. (72) OCR Output 30 The non—perturbative parameters /\1 and A2 may be related to the kinetic energy of the b quark inside the B meson and the mass splitting between the B and B"‘ mesons85, respectively. This determines A2, giving A2 = — m%)/4 2 0.12 GeV2 but a direct relationship of the parameter A1 to a physical quantity is not obvious. Estimates of /\1, using the QCD sum rules and potential models, vary in the range 0.1 to 1 GeV2 8*87. Likewise, the power corrections to the non-leptonic decay width is characterized by the same non-perturbative parameters. With these corrections, one gets a marginally smaller semileptonic branching ratio. For example, Bigi et al. estimate": 6BgL(B) 2 —0.02BRgL(B) 2 -0.0003 (73) One must conclude that, despite gallant efforts, the effect of the power corrections on inclusive b decay rates is far too feeble to be able to bridge the gap between QCD improved parton model and experiments — assuming that the experimental situation has consolidated. A matter related to the semileptonic branching ratios is that of the individual B hadron lifetimes. The QCD-improved spectator model gives almost equal lifetimes; the mentioned power corrections will split the B—baryon lifetime from those of B0, B and B,. However, estimates of these differencesm remain at the few per cent level. Perhaps the first hints of the inadequacy of this theoretical framework are visible in the data? The present situation is well depicted through the ratio of the Bi to the B0 lifetimes45: (74) Bi Qg =1-092;}; , which is slightly larger than unity, although consistent with it within the present errors. However, for the b baryons, the present measurements can be expressed as4 r(B*) — 7·(A;,) = 0.59 :1; 0.19 ps, (75) which is a 30 difference. The individual lifetime errors are still too large, i7% on T(B*) and T(B°) and $15% on T(B,) and r(Ab), to draw any drastic conclusions. Here, improved data from LEP and Tevatron, which are expected to reduce the present errors on lifetimes significantly, would clarify this issue. On the theoretical side, there exists a suggestion that the role of the weak annihilation diagrams in B baryon decays may be considerably larger than in B-meson decays due to the absence of the helicity suppression in the latterl°°. A quantitative analysis would, however, require estimates of the diquark overlap functions in the baryons, which in the op erator product language tantamounts to the baryon-matrix element of an effective four—fermion operator. This must be large if the present difference in the lifetimes r(B*) —·r(A;,) is to be dominantly ascribed to this operator, and deserves further theoretical work. The increased width of the B baryons should also reflect itself in their reduced semileptonic branching ratios, the measurements of which though challenging are worth the effort. Returning to the discussion of the matrix element |lQ;,|, we recall that the inclusive B-semileptonic decay width has also been used to determine |lQ;,|. One needs a 31 OCR Output non-perturbative model to incorporate the wave function/hadronic matrix element effects. The results in the Altarelli et al.67 and the ISGW model"'4 using the ARGUS, CLEO 1.5 and CLEO II data have been recently updated43*12°. While there is some dependence on both the data sets used in the analysis and the non—perturbative models, the resulting value of |lQb| is consistent with1 2°· I/Qbl = 0.039 i 0.001 ;I: 0.004 , where the larger error is attributed to theoretical dispersion. 3.5 Determination 0f |lQb| from l`(B ——> D*€ug) using form factor models One could also resort to specific exclusive semileptonic decays to determine IQ,. The simplest of the exclusive B decays are B —> DEW (involving Kb) and B —> rtw (involving IQ,). These involve only vector weak current and the matrix element depends on two form factors, in each case. For the decay B —> Dtug the form factors are defined as follows: DVB=( I AI } 1;—F —[(DB)) 2 + +F. 2 , 4EE1,,+(<1 )(pv PB)u (q MAI 77 ( I with qu = (pg ~— pD),,. Neglecting the lepton masses, only F+(q2) contributes. The semileptonic decays involving O" —> 1` transitions at the hadronic vertex, like B —> D*€1/[ and B —+ ptug, require, in general, four form factors. Again, in the approximation of neglecting mg, one of the axial form factors can be neglected and one has the Lorentz-invariant decomposition (for the decay B —> D"“Evg): 1 D"VB=——-—————-————————-2i ,,,,"’\.°V2, 78 ( I tl ) (4EBED_),/2(MB+MB)€tipsppe (q) ( ) 1 12* A B Z -—; M . A ’< I uI ) + D ) ](q )6,U 2A2(<12)/(MB + Mz>—)(€· PB)PB»I» (79) where 6,, is the polarization vector of the D"‘. A recent update of the ARGUS and CLEO data for the exclusive semileptonic branching ratios is shown in fig. 8. The decay mode B0 —> D*+€'u,, which is measured more accurately, has been used to determine the matrix element |lQ;,|. One also needs the lifetime. The numbers quoted below are based on using (T) = 1.495:0.04 X10`12 s, the present world-average. The error on |lQ;,| is somewhat larger if instead the presently measured lifetime of B0 45 T0 = 1.51 :I:0.10 >< 10"12 s is used. The values for HQ,] extracted from available data show some dispersion depending both on the data set used and theoretical models for the form factors. Typical values are1 2°· VQ;,| = 0.035 :I: 0.002 i 0.002 [ARGUS], (80) OCR Output VQ,] = 0.038 :l: 0.002 :l: 0.002 [CLEO ll]. 32 The semileptonic decays B —> Dfw and B —-> D*fw from ARGUS and CLEO and the determination of the matrix element HQ,] are also being discussed by Stoneq The numbers quoted here for the CLEO data are identical with his determination, lQ;,| = 0.038 i 0.002 i 0.002 from the exclusive decays B —+ D"‘fw. 3.6 HQET and its application in semileptonic B decays lt has been noted by a number of authors that, in the limit mq —> oo, there are new symmetries in the Hamiltonian of HQET, not present in the SM Lagrangiangmg These symmetries manifest themselves in many ways, in particular in the matrix el ements of the operators describing heavy hadron -> heavy hadron transitions. They can be shown to satisfy an S U (N j) symmetry, with N f being the number of quarks that can be treated as heavy. In addition, there are new symmetries in the spectrum of the states containing these heavy quarks, predicting, for example, that the mem bers of a (spin) multiplet are degenerate in mass. ln the present context, perhaps only the transitions b —> c + X qualify for this limit. It has been argued by Georgi92 that in the mQ —-> oo limit velocity becomes a good quantum number. The transitions B(v) —» (D,D")(v'), where the states are characterized by their four-velocities v and o', are related by an S U (2) _; rotation at the symmetry point v - v' = 1 (corresponding to zero recoil of the D, D*). Also the spin symmetry holds, relating the vector and axial—vector form factors of the B —> Dfw and B —> D*fw transitions. These can be compactified into the form of a Wigner—Eckart relation for the matrix element of the weak currents obeying an SU(2)Hav0m. >< SU(2)Spin symmetry: (H’(v’)|hL·I`hUIH(v)) = §(v · v')C(F)· (81) Here C(1`) is a Clebsch-Gordon coefficient (calculable for each I` = 1, 7,,, 7,,75, and {(2; ·v') is a non—perturbative function (reduced matrix element), called the Isgur-Wise function". Current conservation for I` = 7,, and h : h' implies § (v ·v') = 1, providing the normalization of the Isgur-Wise function at the symmetry point v · v' = 1. So a host of form factors in the weak decays of heavy hadrons into other (less) heavy hadrons in the heavy-quark limit is reduced to just one function, whose normalization at the symmetry point is known while its extrapolation away from this point could be measured experimentally or estimated in theory. We briefly review the application of the heavy—quark effective theory approach to the semileptonic decays B ——+ Dfw and B —+ D*fw81'83, with the idea of determining the matrix element Kb, which probably is the best use of this approach at present. To that end, one rewrites the Lorentz-invariant form factors introduced earlier in the four-velocity basis: (DIVAB) = \/(MBMD) I§+(v · v')(v + v’)t + é-(v · v’)(v - v')»I» (82) 33 OCR Output B(B —> D("*"’(’z/) B1‘2mcl1i11g Fraction 1%] OCR Output BO —> D+€‘D ARGUS .5 zh 0.5 zh 0.4 CLEO 1.5 .7 zh 0.6 iz 0.3 BU —> D*+€‘D ARGUS .5 t 0.7 t 0.7 ARGUS .8 zh 0.5 t 0.6 CLEO 1.5 4.2 zh0.5zh0.6 CLEO II .5 t 0.4 iz 0.4 BU -> D"+(”‘D ARGUS .3 zh 0.6 zh 0.5 B' -> D°L"D ARGUS .7 dz 0.6 t 0.4 CLEO 1.5 .7 zh 0.6 t 0.:3 B` -> D*°L"D ARGUS .6 t 0.7 zh 0.7 ARGUS .9 zh1.7 dz1.5 CLEO 1.5 4.4 zh 0.9 zh 0.9 CLEO II 5.4 dz 1.0 zh 1.5 0 1 2 3 4 5 6 7 B(B -> D<·><··>6u) [%] Figure 8: Measurements of the semileptonic branching ratios B(B°—>(')("") D£17) from ARGUS CLEO 1.5 and CLEO II data. (from Casse11°°). 34 ; \/MBMD·€V(v_,UI)€uVA0€*Uv}AvU’ (D*|A,,[B) : JMBMD* |g,.,,(t . t*)(t - by + 1)c; §A2(v · v’)(¢“' - v’)vt —€A.,(v · v’)(¢" · v’)vL (84) In the limit mb, mb —+ oo, these decays are governed by a single Isgur-Wise function §(v - v'), and the following relations hold: €+(v · v') = €v(v · v') = §A.(v · v') = §A.(v - v') = 6(U · U'), (85) €—(v · U') = {MU ·v') = U (86) Before applying these relations to the analysis of data, it is worth while to discuss the leading corrections to the Isgur-Wise relations just given. They are of two types: first, there are the perturbative QCD corrections to the bcJ,, vertex, for the estimates of which the renormalization group machinery can be used. Integrating out successively the heavy degrees of freedom, Mw, mb, mb, one could show in the leading logarithmic approximation (LLA) that the Isgur-Wise function gets renormalized according to8 — GL , ¤a(m¤»)/25 6 ¤8(mc) I [ - = so ·»> ————— (Mme,) (MA) · , so U) 87 <> where ' my - U') : é-fil-L-27 ` /(.U , Ul)2 _1 1¤(v. U'-)- ,/(U - t»)¢ 2-1)- L (88) 7 and aL(v - v' = 1) = 0. The power -6/25 is the so—called hybrid anomalous dirnensionml and emerges as one scales down in the quark masses mb —> mb; the second anomalous dimension is picked up when also the c quark is treated as heavy (compared with the QCD scale parameter A). The LLA results can be improved by doing a complete perturbative QCD calcu lation up to the desired order, good parts of which have already been done 83. The second, and in general more important corrections, involve estimates of O(1/mq) terms, which require a systematic expansion of all the kinematic terms giving rise to additional operators present in the HQET approach. These corrections break both symmetries alluded to above in the Isgur-Wise limit and introduce additional form factors. There is, however, an important exception. For currents, which correspond to a symmetry of the Lagrangian, it can be shown that at the symmetry point v · v' = 1, the O(1/mq) corrections to the function {A, (1; - v') vanish. In the LLA approximation one gets: - /25 6 . M A Amb) · I = l = S€A1(U; (G-,(Tnc) U ) CQCD) —- me 89 ( ) OCR Output 35 This result, due to Lukegl, is reminiscent of the good old Ademollo-Gatto theorem of the flavour-S U (3) vintagelm, where, however, the non—renorma1izability of the vector form factors in the leading order of S U (3)-symmetry breaking, such as is the case for F+(q2) in K;3 decays, holds at zero momentum transfer, q2 = 0. In contrast to this, Luke’s result holds at v · v' = 1, corresponding to the maximum momentum transfer q2 = q3,mI_. lt has been argued persuasively in the literature81‘83 that taking into account mass corrections in all the form factors, the function {A, in the decay B —+ D*€1/[ at the zero recoil point remains protected against O(1/mq) corrections. The decay B —-> Dfw in the same limit, on the other hand, receives contributions from both §+(v · v' = 1) and §-(v · v' = 1), the latter of which is unprotected against O(1/mq) corrections. This analysis suggests that the decay B —> D"€1/g is well suited for a model-independent estimate of Kb. It should, however, be emphasized that the great simplicity of the HQET approach as well as the non-renormalization theorem hold at zero recoil, at which point there are no data! Extrapolating the data to the symmetry point v · v' = 1, or more ambitiously extending the theory to explain the measured differential spectrum in the variable v · v', necessarily involves a parametrization of the Isgur-Wise function, which is a mode1—dependent enterprise. In this context we note that first calculations of the Isgur-Wise function {(22 · v') using lattice-QCD techniques, have been reported recently96*98 and have been applied to the data to extract a value of the CKM angle |lQb|. For a discussion of some of the problems concerning the lattice renormalization of the effective heavy quark theory and the treatment of unequal heavy quark masses in these transitions, see the work of Aglietti et al. 99 Since, O(1/mq) corrections to the Isgur-Wise function §A, (v · v' = 1) determining the rate for the decay B —> DTV; at the symmetry point are absent, deviation from the relation §A1(v · v' = 1) = 1 is dominantly of perturbative QCD origin. These give the following differential decay rates 1 df` B —> D*€ug GFZ ,93,1, 31 Z §Mg·(MB · MD—)Il/LbI2§i,, with the QCD corrected {A, given by: EAI : $6/25 [1 _,_ z(§ _ $$-9/25 + %x-12/2s _,_ %)ln(x) oz,(m;,) — cx,(m,,) 5 i_8oz@ _lHz]’ 2cv_,(r7L) (90) i37r 1: 1 — z where sc = cx,(mc)/a,(m;,), z = mc/mb, or,(mc) and cz_,(mb) are two-loop running coupling constants, and TTL is a scale parameter, which is not determined in this order, though the numerical results are quite insensitive to a particular choice. The constant Z has been shown to be scheme—independent, and it has a value: ~ 7 2 9403 N Z - 1r - 1.5608 (91) OCR Output 2257500 36 With the choice of parameters EQQ E 0.107, ?i@ »; 0.005. EQ z 0.001 (92) the QCD correction factor in the decay rate B -+ D"‘£ug turns out to be essentially 1, namely UQCD 2 0.99, leading to the very interesting result that the rate for this decay at the symmetry point remains, to a very high accuracy, unrenormalized from its Isgur-Wise value! ln this context, it should also be mentioned that there also exist theoretical estimates of the O((1/mQ)2) power correctionsgs. Although these corrections are model—dependent and the present estimates of their contribution at the level of O(&3%) may or may not be entirely realistic, it would be surprising if they are much larger, so as to significantly change the quantitative result. 3.7 Determination of llébl from HQET To extract |VQ;,| in a model-independent way, one has to extrapolate the data to the point y = v · v' : 1, for which one needs an ansatz for the lsgur-Wise function { The following parametrizations have been employed for this function to extract 14,;, from data123*124: Ar 1- p2(y -1), (93) 2 — ,+y(— 2 2 -1 —-€><1>(py — 1 > y+,). 2 (y -1- 1) 2¤° D = ¢><1>[—z>2(y -1)] A fit of the data on B —> D*Z1/[ was then attempted in terms of the parameters p and |lQ,|. The ARGUS collaboration have analysed their data for both the neutral BO —+ D*+€‘17( and charged B meson decay B‘—> D*°£‘17g. The present CLEO analysis is based on the neutral B—meson decay only. Apart from the semileptonic branching ratios for the modes B —+ D*E1/l, which have been given earlier, the mea sured qz (or equivalently v · v') dependence of the differential rate and the lifetime of the B mesons are needed to determine VQ),. With the branching ratios given already and TB = 1.49 :1: 0.04 ps, the results of a recent update of |lQb| based on HQET are shown in fig. 9 lm. Typical values from the ARGUS data are centred around Kbl = 0.045, and those of the CLEO data around HQ,] = 0.037. The errors are still significant, so that the two determinations are consistent with each other. A value of VQ,] in the range 0.050 2 |1Qb| 2 0.035 from the HQET and extrapolation to v -v’ = 1 represents the present state-of-the-art measurement of this quantity. Further theoretical and experimental work is needed to get a more precise value of |VQ;,|. As already noted, the first estimates of the Isgur-Wise function using the lattice·QCD methods have been presented 98. As the theoretical uncertainties of 37 OCR Output Experiment — Fit Wtbt B0 —> D‘+€’D 0.041 zh 0.005 h 0.00:3 ARGUS 0.048 t 0.007 zh 0.003 0.046 t 0.007 zh 0.003 0.045 ct 0.007 t 0.002 0.036 dz 0.004 t 0.004 CLEO II 0.038 zh 0.006 zh 0.004 0.038 zh 0.005 1 0.004 0.037 iz 0.005 zh 0.004 Average 0.038 zh 0.004 ARGUS 0.043 zh 0.005 and 0.041 zh 0.005 CLEO II 0.040 zh 0.005 B' —> D‘°F‘D 0.044 zh 0.007 ARGUS B 0.047 zh 0.009 0.048 zh 0.010 0.025 0.030 0.035 0.040 0.045 0.050 Wo Figure 9: Measurements of the CKM angle |Vcb| from data on the semileptonic decay B0 —> D*l"17 from ARGUS 123 and CLEO 124 and B' —> D*l'z7 from ARGUS 123 using the Isgur-Wise function {A1 . The parametrizations used are also indicated (from Cassel1°° 38 OCR Output this method get under control, one would have both a test of this method against data and a reliable function to extrapolate the data to the symmetry point. Future measurements at CLEO and LEP with significantly more statistics and, in a somewhat distant future, at the B factories being proposed in e+e` and hadron collisions, will greatly reduce the extrapolation ambiguities, so that eventually the HQET based technique will provide a truly quantitative measurement of HQ, Based on the various estimates presented here, and taking into account theoretical uncertainties as well as the differences in the present data, it is fair to conclude that the CKM matrix element lQ;,| has been determined by the present ARGUS and CLEO data to an accuracy of $12%. The present status of the |lQ;,| can be summarized as: VQ;,| = 0.042 ;}; 0.005, (94) giving: A = 0.86 ;l; 0.10 (95) Stone43 gets a value |lQ;,| = 0.038 zl; 0.003 as the present average, which is, within the quoted errors, compatible with the one given here. 3.8 Determination of Vub The dominance of the Vg, coupling has also been established by the so-called direct charm-counting method and the corresponding numbers are discussed by Browder et al. 44. This method can at best be used to put an upper bound on the matrix element Mb, since non—charm B decays involve both the CC decays b —> u +X and the F CN C penguins b —-—> (s, d) + X, the two being of comparable order of magnitude in the SM. The present evidence for the transition b —> u is based on semileptonic decays. In the B—meson rest frame, the maximum lepton energy from the b —+ c + ZW transitions is limited to E{"‘” = (MBZ — MD2)/2MB. Successful searches for the b —> u transition, leading to non-zero values for the matrix element Vub, which is a crucial element if the CKM model is to describe CP violation, were reported in 1989 by the CLEOIOZ and ARGUSIOS collaborations through precise measurements of the end-point lepton energy spectrum. In the mean while, the statistical significance of these transitions has become impeccable. For details of the experimental data and analysis, see Stone4 We show the past and present determinations of the ratio |V],;,|/ll/Qbl from the inclusive lepton energy spectrum in fig. 10, using the four indicated models for the spectrum. This figure shows that the present value of |l{,b|/|l{,;,| is considerably smaller than the earlier ones reported by ARGUS and CLEO. There is some model dependence in the extracted value. However, by consensus43*u6*12°, the present world average is: vuil/lm = 0.08 T 0.02 (96) Since, in the Wolfenstein parametrization, one has the relation |l£,;,|/ HQ,] = /\(p2+ 39 OCR Output Model - Experiment |V,,,/14,1 ARGUS 0.11 1:0.012 ACCMM CLEO 1.5 0.11 1:0.016 CLEO II 0.076 1:0.008 ARGUS —0-—— | 0.20 1:0.023 1SGw CLEO 1.5 0.18 1:0.026 CLEO II 0.101 t0.010 ARGUS 0.11 10.012 KS CLEO 1.5 0.10 1:0.014 CLEO II 0.056 t 0.006 ARGUS 0.13 1:0.015 WSB CLEO 1.5 0.13 t0.018 CLEO II 0.073 t0.007 0.00 0.05 0.10 0.15 0.20 0.25 lV».•/Val Figure 10: Determinations of |V,,;,|/|lQ;,| from the inclusive lepton energy spectrum in B decays. The data are from ARGUS *25, CLEO 1.5 126 and CLEO II 127. The extracted values are shown on the r.h.s., and the models used ACCMM", ISGW", KS" and WSB" are indicated on the l.h.s. (from Cassel1°°). 1;2)1/2, this translates into the following range: p2 + rf 2 0.36 ;t 0.09 (97) To get bounds on p and rp separately one has to use additional input, which we discuss in the subsequent sections. 4. FCNC '1‘ransitions—The SM Predictions and Experimental Status FCNC transitions have played a very important role in shaping the standard model in its formative stage. We recall here the role of the GIM mechanism in taming the AS = 1 and AS = 2 transitionss. This mechanism, which governs the s —+ d as well as the sd —+ sd transitions, is also operative in the FCNC decays of B hadrons. Testing the GIM predictions in B decays and mixing is crucial and equivalent in spirit to the quantum (loop) tests of the electroweak theory that have been undertaken at LEP and elsewhere. We take up the |AB| = 2 transitions in this section and relegate the discussion of the |AB| = 1 FCN C transitions to the next. 40 OCR Output The |AB| = 2 transitions induce particle-antiparticle mixing involving the neutral B mesons, B3—B3 and B3—B§’. Such mixings split the (otherwise) degenerate mass eigenstates, for example B3 and B3, into two non—degenerate states B3 and B3, which have, in general, also different lifetimes. We shall characterize the mass- and width differences by AM(B3 - B3) and AI`(B3 — B3), respectively (likewise for the Bg—B§ complex). There is general consensus among theoreticians that the quantity Al`/l" for the neutral B-meson complex B3—B3 is negligible5°*128. However, it has been argued that the lifetimes of the two mass eigenstates of the B2—BQ complex may differ by as much as 20%. In both cases AF/AM < 1 and hence we shall neglect the lifetime differences in what follows and concentrate on the observed mixing effects, interpreting them in terms of the mass difference AM/1`. 4.1 SM predictions for the mass difference The SM expressions for the mass differences are straightforward to calculate al though the QCD corrections are rather formidable. Fortunately they have been cal culated, including the effects of the large top-quark massug. Present measurements allow us to extract the values of only the mass difference AM(B3 — B3) and the quantity xd E AM(B3 — B3)/Pd for the B3 —> B3 complex, as discussed below. The mass difference to lifetime ratios in the SM can be expressed as: AM 2 2 1)" md = —,r(Bi- B?)%(%) = mwGiF—Maw. mw <98> 2 2 A M I.* T xs Vta " - Z -|’ 1 + 6 I() 103 ( ) OCR Output xd m 41 where 6 is an S U (3)—breaking parameter defined as, 6 = ( fE ELA — 1) > O 104 ( ) mBd fgd Measurement of xs/xd is then essentially equivalent to the measurement of the VCKM ratio $3 *2 The coupling constant products Bdf§d and Bsfgs are model-dependent. Until recently the scaling law fB(mB),/mg = constant, (105) was thought to be valid for the B system. This led to rather small values for f§d Bgd in the range 100 MeV § fBd\/Bgd $ 170 MeV. (106) However, recent lattice calculations and improved QCD sum rules have indicated that there are significant scaling violations in the region between m D and m B. Lattice QCD data are now usually interpreted as fm/MP ~f¤(1+f1/MP) (107) Abada et al.94 have estimated fl 2: -0.8 GeV, with probably also a (smaller) quadratic correction. These have led to larger estimates for f§dBBd, although the final number is still in a state of flux. Lattice calculations indicate that the ratio f B d / fg, could be calculated much more accurately than either fgd or fB,, due to the cancellation of some systematic uncer tainties. For example, Alexandrou et al. quoteg fgd = 188-246 MeV, fg, = 204-241 MeV. (108) Along the same lines, Abada et al.94 estimate fg,/fgd : 1.06 ;b 0.04; a recent av erage over a number of lattice calculations gives fB’/fgd = 1.08 ;l; 0.06 95. The corresponding numbers from recent QCD sum—rule updates typically predict5 fgd = (1.60 zi: 0.26)f,,, fg, = (1.86 ;b 0.31)f,,, (109) giving fg,/fgd = 1.16 i 0.05. Likewise, the bag factors for the B mesons are found to be given by the vacuum saturation value Bb 1* 1, within 15%. The implications of the measurement of xd in the SM is that it constrains the CKM matrix element product lQ§lQ;, as a function of mt, assuming some values for the hadronic matrix elements. We will use the following values in our analysis, presented in the next section, fB4 180 dz 50 MeV, BBd 1.0 :1; 0.2 , fen/BB. 210 dz 50 MeV. (110) 42 OCR Output The errors are large enough to cover most present lattice—based and QCD sum-rule based numbers. We now discuss the SM predictions for :1:,. On using eq. (22) to substitute for Wgléb, the SM-expression for x, can be written as: Ei2 (111) _ q 2 2 4 :1:, - TBSGZMWMBS s(fBBB,) UBSA /\ xtF(;rt). W The lifetime of the B, meson has now been measured at LEP and Tevatron. The present average for the BQ lifetimes is ‘*5· TB.; : (1.54 4 0.24) X 10·" S . (112) Within the measurement errors, this value is consistent with the averaged value of TB used in the previous section. The mass of the B;) meson has also now been measured and its present world average is (MB,) = 5373.2 2 4.2 MeV H6, thus removing one of the unknowns. We expect the QCD correction ng, to be equal to its Bd counterpart, i.e. ng, = 0.55. Since the CKM parameters A and 2\ have been determined, one is able to predict xs more confidentally than xd. The main uncertainties in x, are the t-quark mass and f§’BBa. Using the central values A = 0.84 and T(BE) = 1.54 ps, we obtain fz B 113 ( ) , Z 179 LL F . x (€) ) GV2xt (xt) With f§’BB5, given in eq. (110) this gives :1:, = (7.9 i 4.0) :ctF(xt). (114) The function a:tF(x,) is numerically equal to 2.03 for mt = 150 GeV, and for this value of mt, we get: x, = 16.0 i 8.1, (115) giving 2, 2 8-24. The standard model therefore predicts very large values for 2,. This is to be expected, since in the SM one has :1;,/xd 2 ll/Q,/Wdlz, which has been estimated to lie between 11 and 30 131, apart from SU (3) f;,,,,,,,,,—breaking effects. Using the central value for xd = 0.70, this gives x, 2 8-21. Due to these large values, time-dependent measurements are necessary to obtain x,. 4.2 Present status of Bg—Bg mixings Historically, the UA1 collaboration was the first to observe a signal of the B B mixing in the same-sign dilepton channeluz. However, the interpretation of the UA1 data was masked by the circumstance that their measurements were not able to distinguish the mixings in the BQLBQ and BQLBQ sectors. _The overwhelming theoretical prejudice at that time was that mixing in the Bg—Bg sector should be negligible. This prejudice was not entirely without experimental support! More pointedly, the top—quark mass was proclaimed to be in the range 30 — 50 GeV, based 43 OCR Output on an earlier analysis of the UA1 collaboration anno 1984. We recall that the time integrated mixing probability depends quartically on mt. The presumed low value of mt, taken together with the ITEP sum-rule based estimate of the coupling constant, which was the theoretical currency in that epoch, led to the erroneous conclusion5 12118.1] {Ed S 10` The ratio of same—sign to opposite-sign dileptons R((l+l+ + l"l")/l+l"), measured by the UA1 collaboration, was far too high to be understood only in terms of a large mixing in the Bg—BQ sector, although this hypothesis certainly helped to bring theoretical estimates and the UA1 measurements closer133. In that background, the announcement of a large mixing in the .82-52 sector by the ARGUS collaboration1 34 came as a stunning surprise to most members of the concerned community, includ ing the present author. Of course, retrospectively, with such a large mixing in the Bg—B3 sector, which has since been confirmed by CLEO135 and by experiments at LEP136`138, there is no problem in accommodating the large same—sign to opposite sign dilepton ratio of the UA1 collaboration. In the mean while, the UA1 data on mixing have been consolidatedwg and confirmed by the CDF collaborationlm and by a number of experiments in e+e' annihilation1‘“”“3. With the lower bound on the top-quark mass in direct searches at the Tevatron moved to 113 GeV 144 and the present theoretical estimates of the pseudoscalar coupling constant fg, likewise, gone up by approximately a factor 2, the large value of xd : 0.71 :}:0.07 is very comfortably accommodated in the standard model. This will be quantified below. Turning to the discussion of the B°—B° mixings in e+e' annihilation, we note that the following final states are produced at T(4S) r(4S) .. BgBg,BgBg, B§B§,’. (116) The ratio of the cross sections yd E ¤(BSB3) J; ¤(B§’BS) (H7) U(BdB3) can be expressed as (on the 'I`(4S) alone) wi + yi Iii N ”·"d·r;r1$1;g-2+1;, (H8) where Tg = is the ratio of the time-integrated probabilities. The variables Xd defined as Xd = xg/2(1 + IE5) and X, (defined analogously) are also very popular in the analysis of data. The mixing ratio Td (equivalently Xd) has been measured using two methods, namely through the same—sign to opposite-sign dilepton ratio, which we call Method I, and by reconstructing one B meson and tagging the other by the F from its semileptonic decay, which we call Method ll. With Method I, the mixing ratio rd at the T(4S) is obtained by using the relation: (N(l l )+ N(l+l+))(1+ A) rd Z (H9) OCR Output (N(l+l‘) -— N(l+l+))}\ 44 where the quantity A is defined as: ici (E) f 0 T 0 (120) and present measurements are consistent with A 2 1. For Method II, the correspond ing relation is: N B0 + °£‘ N(-BW`) + N(B3€+) The discussions of the ARGUS and CLEO results and their technical details can be seen elsewhere in this volume143. The present average for Xd from the ARGUS and CLEO measurements using the two methods is u2·“6· xd = 0.69 i 0.09 (122) Recently, thanks to the high-precision vertex detectors and LEP, the time evolu tion of the Bg—Bg oscillations has been measured by the ALEPH136 and DELPHI1 37 collaborations. First measurements of AM,) have also recently been reported by the OPAL collaborationws. As opposed to the time-integrated methods, which measure X,) at the T(4S) and a weighted average of Xd and X, in the continuum, the time dependent experiments measure AM directly. This method has a certain advantage as it does not depend on the lifetime and hence it removes a potential source of error due to the lifetime measurement uncertainties. In fact, this is the only method which would eventually allow a measurement of AM and hence 1, for the B2—B2 oscillations due to the large value ac, = O(10) expected in the standard model. ALEPH has used the time—dependent D*£ and dilepton final states to determine AM,) for the B3-B2 oscillation. Using the D*lZ method ALEPH gets1 36· AM,) = (3.44i§;$gi3;§3) X 10·4 ev, (123) which, on using Tgd = 1.44 j: 0.15 ps, yields: xd = 0-75l`3ilif3i3€ (124) The second method (using dileptons) gives1 36· AMd = (0-53f3lZZf31lt) PS`1» (125) which corresponds to M = 0-76f3ilif3il2 (126) The DELPHI collaboration analysed the time dependence of high-pq~ (py > 1 GeV) like-sign dileptons, estimating the lifetime by using the lepton impact parameter, obtaininglw: xd = 0.65fgjgg ;}; 0.11 (127) OCR Output 45 The recent result from OPAL, using time-dependent measurements, is l38· AM,) = (3.73 zh 0.60 :1: 0.13) >< 10`4 eV, (128) The results from the four experiments ARGUS, CLEO, ALEPH and DELPHI have been combined by Danilov to giveu 6· xd = 0.71 zi: 0.07 , (129) which corresponds to a 10% measurement of xd. Unfortunately, this precision is not matched by the theory, which is uncertain due to the imprecise knowledge of mt, fgd and Bd. The interest in xd lies in its potential role in determining the CKM matrix element (Vid). Later, we shall impose the constraints on the CKM parameters from xd. For the time-being, assuming 110 GeV $ mt f 200 GeV and the range of the coupling constants given above, one obtains: 0.005 $ |lQd| §_ 0.016 . (130) This is a first determination of VQ,) and has an uncertainty of a factor 3, assuming that the parameter ranges used are realistic. However, since this value lies within the unitarity constraints 6 |l4d| §_ 0.018, it is fair to conclude that the measaurement of wd is in line with the SM expectations. 4.3 Present status 0f B;’—B§’ mixings Finally, we discuss the present status of the mixing parameter, r,, characterizing the mass mixings in the B2—BQ complex. We recall that the data in the continuum above the BB production threshold is analysed in terms of a parameter X defined as: X :XdPd+XsPs (131) with Pd (P,) being the probability of finding a Bd (B,) meson in the fragmentation products of a b quark. The quantities X, = d,s) are related to r, = d,s), introduced earlier, by X, = For the continuum production, the assumption that the B and B mesons are produced incoherently holds and one has: l+l+ 1-1- 2 1 ¤(I+l‘) x +(1·X) The present measurements of the parameter X from the pp collider experiments (UA1, CDF and D0), the lower energy e+e' experiments (MAC and MARK II) and LEP average (ALEPH, DELPHI, L3, OPAL) have been recently summarized by Moser H2 and Forty“3 and are shown in table 5. The measurements of X is dominated by the LEP results. The measurements of Xd can be combined with the present world average of the weighted mixing ratio X : 0.121 :1: 0.010 , (133) OCR Output 46 Table 5: Present values of the mixing parameter X from the UA1, CDF, D0, MAC, MARK II and LEP measurements (from Moseruz). Source 0.148 ;I: 0.029 i 0.017 I UA1 0.176 d: 0.031 zi; 0.032 I CDF 0.140 zi; 0.03 nh 0.06 I D0 0.21+ 0.29 — 0.18 I MAC 0.17 + 0.15 — 0.08 I MARK II 0.117 :l: 0.010 I LEP average to determine X,. This requires a knowledge of the probabilities Pd and P,. With the assumed values Pd = 0.39, P, = 0.12, and PA = 0.12, used commonly, the measurements of X and Xd give X, = 0.45 t 0.11, or X, > 0.22 at 90% C.L. H2. lf on the other hand one uses Pd = 0.375 and P, = 0.15, one obtains X, = 0.38 zh 0.08, with the corresponding lower limit of 0.20. These measurements are consistent with the SM expectations X, 2 0.5. Unfortunately, the present experimental bound on z, is theoretically uninteresting, with sc, > 0.8. Time-dependent measurements on z, at LEP will be exploring the lower fringe of the expected domain of :1:,. Luckily, several proposals for doing dedicated B physics in hadronic collisions, such as will be carried out at the LHC“5 and HERA-B“6 experiments, are capable of reaching the entire 2, range, worked out in the preceding section. 5. Rare B Decays and the CKM Matrix Elements We give an overview of the FCNC B—decays mostly in the context of the stan— dard model. The Glashow-Iliopoulos-Maiani (GIM) construction, which is an inte gral part of the quark flavour mixing, forbids all FCNC transitions in SM in the Born approximations. Such transitions are governed by one-loop graphs (penguins and boxes), by virtue of which their rates get suppressed as compared with the usual (Born) charged current (CC)-induced transitions. FCNC B transitions, like their counterparts in K physics, provide information on the properties of the quarks, in par ticular their masses and the CKM matrix elements (IQ; i, j = l, 2, 3). In particular, one expects to effectively constrain mt and the CKM matrix elements, IQ,,i = d, s, b from such decays. We shall concentrate here on the so-called IAB I = 1, AQ = O decays, leading to transitions such as b -—> (s,d) + X, where X = g, 7, £+£‘, and l/17, as well as the purely leptonic decays B, —» E`*'£` and Bd —>£+€’. Inclusive rates and final state distributions in FCNC B decays and their CC counterparts can be calculated in perturbative QCD, which is the theoretical framework relied upon heavily in the literature1‘"*“8'15°. The role of QCD is here rather central. For example, it is known that the decay rates for B -> X, + 7 and B —> Xd + 7 get enhanced by factors of 47 OCR Output order 3 as a result of QCD corrections (here and in what follows the subscript q in B —> X, denotes the quark transition b —+ q). In a way this enhancement has been verified by the recent measurement of the exclusive decay mode B —+ K * + 7 by the CLEO collaboration151*152, having a branching ratio B(B —> K* + 7) = (4.5 j: 1.0 :1; 0.9) >< 10‘5. Though one needs a model to interpret this measurement, it is in line with the expectations that O(10—20)% of the inclusive decay rate B(B —> X, + 7) = (3.0 ;|; 1.2) >< 10'4, estimated in the SM155, will show itself in the exclusive mode B —> K * + 7. This measurement, together with the upper bound on the inclusive decay rate B —-> X + 7 < 5.4 >< 10"1, also reported by CLEO151, provides an upper and lower bound on the CKM matrix element ratio |V§,|/ |VQb|, which we discuss later. `While the decay mode B —> K * + 7 provides a calibration of the electromagnetic penguins in B decays, and we expect that the measurement of the inclusive mode B —> X, + 7 is around the corner, there exists an overriding theoretical interest in measuring the CKM-suppressed electroweak penguin B decays, B —> Xd + 7 and B ——+ Xd +€+€'. Representatives of such decays are the radiative decays Bd —> w + 7, (3,,;,13,,) —» p+7, and B, ——> K"‘+7 and the FCNC semileptonic decays Bu —+ (1r,p)+ E+€`, Bd —> (1r,w,p)+ €+E" and B, —> (K, K*) + £+E'. Their measurements can be combined with the corresponding CKM-allowed exclusive decay modes (B,;,Bu) —> K" +7, B, —->¢+7, (Bd, Bu) —~> (K, K"‘) +€+€' and B, —+ ¢+E+l", emerging from the decays B —-> X, + 7 and B —> X, + FE", whose measurements will precede those of the suppressed ones. We argue that the ratio of the relevant decay rates can be used to determine the CKM matrix-element ratio |VQd|/ |I/Q,} and we quantify this by explicit calculations carried out recently15°*5°. We also review the expected branching ratios for various B decay modes of interest in this paper. While our focus is on getting a trustworthy SM profile of FCNC B decays, we remark that the measurements of the electromagnetic penguin mode B ——> K * + 7 and the present upper bounds on the inclusive radiative and FCNC semileptonic branching ratios B(B ——> X, + 7) < (5.4) >< 10"4 152 and B —> X + ,a+;i` < 5.0 X 10“5 154 by the CLEO and UA1 collaboration, respectively, have been used to put constraints155*155 on non-SM scenarios157*158. In fact, the constraints from the FCNC semileptonic decay B —> X a+ p` are already quite strong, and they provide upper bounds on the effective neutral-current couplings bsZ° and the four-fermion coupling (bl";s)(E+I`,£`) (with I`; being the 7—matrices involving 7,, and 7,,75). In some non SM scenarios these couplings could be very large, enhancing the branching ratio in question. Similar constraints on the non-SM FCNC transitions in K decays were worked out by Dimopoulos and Ellis some time ago1 59 5.1 Inclusive radiative B decays The inclusive radiative decays B —> X +7 receive contributions from both the CC radiative transitions B —> (XC, Xu) + 7 and the FCNC transitions B —> (X,, X4) + 7. The CC decays depend on the CKM matrix elements |VQ;,| and |Vu;,|. The FCNC 48 OCR Output radiative decays B —> (X,,Xd) + 7, induced by the electromagnetic penguins, are essentially determined by the magnetic moment operators whose Wilson coeilicients are dominated by the top—quark contribution in the processes b —> (s,d) + 7 + The rates for these processes therefore depend on the top-quark mass mt and the CKM matrix elements H4,] (for B —+ X, + 7) and |lQd| (for B —> Xd + 7) 148'15° It follows that a precise measurement of the prompt photon energy spectra in B decays has, in principle, the potential of determining four CKM matrix elements: Kb), lléibl, |VQs|, and llédl. While it should be possible to carry out good measurements of the transitions B —> XC + 7 and B —-> X, + 7, by using the lower and higher frequency part of the inclusive photon spectra, respectively, the smaller components B —> Xu + 7 and B —> Xd + 7 will very probably require measurements of specific exclusive decay modes. We will concentrate here on the FCNC transitions and show how combining various inclusive and exclusive B decays the CKM matrix elements %d| and |l/Q,] may be determined. The calculations for the FCNC B decays have been done within the framework of an effective Hamiltonianlu. The operator basis for the effective Hamiltonian, responsible for the decay under consideration, consists of four-quark operators and the magnetic moment operators of dimension 6. Operators of higher dimension are suppressed by powers of the masses of the heavy particles (W-boson and top quark) which have been integrated out, and hence are not considered here. The complete set of operators relevant for the processes b -> d7, d7g can be symbolically written as: Hm ~ d> = —i-e.‘Ci Z cm vm. We note that all the CKM—angle-dependent quantities fi = t,c,u) in eq. (135) are of the same order of magnitude, which is most easily seen using the Wolfenstein parametrization? As discussed in the literature1"7‘”'°, the dominant contributions arise from the operators O1, O2, Oy and Og, whereas the operators 03,...,06 get coefhcients through operator mixing only, which numerically are small. The dominant operators read as follows: $(éLp7“bLC,)(dL.,7,,cL,;) —¥(z2Lp7“bL¤)(dL¤7wL0). £(ELC,'7*LbLC,)(¢fLp'y,,CLp) ··¥(7]L¤'YubL¤)(dLB'YuuLB)v (e/16 vrz) dc, a’"’ (TMR + msL)b¤ FW, (g,/16 W2) JG O"'W (mbR + TML) Tfp bi? (136) Here e and g, denote the QED and QCD coupling constants, respectively, and L = (1 —— 75)/2; R = (1 + 75)/2. The effects of QCD corrections, contained in 49 OCR Output the Wilson c0e$cicnts Cj(p), have been evaluated to leading logarithmic accuracy. This topic has received quite a bit of attention. Though the complete one-loop anoma lous dimension matrix is now available due to the work of Ciuchini et al.147 and by Misiaklu, we shall use the anomalous dimensions calculated in the truncated approx imation, using the work of Grinstein et al.“7, since the photon energy spectra and the phenomeological analysis based on this theoretical work has been done in this approximation. The numerical difference in the full and truncated approximations is small. At the renormalization scale p, which is expected to be of O(mb), the dominant coefficients read as follows1 47· Gif/L) [T]-6/23 _ 7,12/22.] C2(mW)’ C20,) Z g [,7-6/23 _|_ H12/23] C2(mW)’ CNN) Tl-16/23 {C7(mW) _ gg {n23w/ _ 1\ C2(mW) _ Egg 28/23[T]_1| @2(mw) l, CSW) UTiclsimwlM/23 ` g/23$2 _1C2(mW)inl + 5%% in/`m23 li Czimw) i’ (137) with 1; = At the scale p = mw, where the matching conditions between the full theory and effective (five—quark) theory are imposed 160, we have: C1(TTlW) Os = lv C-{(mw) l6x(3:v - 2) logx - (sv -1)(8x2 + 5x — 7)], C8(mw) [Gxlogx + (x —1)(x2 — 5x — 2) with 2 = We now discuss the changes in the operator basis for the B —> X,+7 CH·S€- FOI thé decays b —> sry and b —> sryg, the CKM factors fi are replaced by A, = V§;,l/Q'; i = u, c, i. The unitarity condition for the b —> s transition reads as: Z A, = 0. (139) As Au < /\,, or Ac, we can neglect Au. In this case the CKM matrix dependence in the effective Hamiltonian factorizes due to unitarity: 4G ~ Herdb ·* S) = _-TF M Z CAM) Oj(#)» (140) with A A O1 = (ELB'Y#bL¤)(·§L¤7uCLB)> O2 = (6Lw“bL¤)(5L¤m¢L¤)· (141) The remaining operators Oj are obtained from the corresponding Oj operators by simply replacing the d field by an s field. Furthermore, the Wilson coefficients in 50 OCR Output H,ff(b —> s) and H,fj(b ——> d) are identical. It follows from the above discussion that measuring the branching ratio involving the b —> .s transition, like the inclusive decay B ——> X, + 7, amounts to determining the ratio |l4,|/|V[,b|. In the b —> d transitions, however, the branching ratios will be functions of the Wolfenstein parameters A, p and ry, although in some cases of interest the top—quark—induced transitions dominate, whence the CKM matrix—element element dependence of the branching ratios also factorize in the ratio llédl/|VQ;,|. 5.2 Estimates 0fB —> X, + 7 in the SM In the estimates of B(B —-> X,+7), which we use here, the contribution of the QCD bremsstrahlung process b —> s + 7 + g and the virtual corrections to b —> s + 7are calculated in O(cm,) 1‘*8*1‘*9·153. The decay width I`(B —> X, + 7) is still domi nantly contributed by the magnetic moment term C7(p)O·,(;r). However, including the O(cm,) radiative corrections brings to the fore other operators in H,ff(b —> s) with their specific Wilson coefficients. The effect of these additional terms can be expressed in terms of a function K (sct, ir), which is a K -factor in the sense of QCD corrections. The branching ratio B (B ——> X, + 7) can be expressed in terms of the (well-measured) inclusive semileptonic branching ratio B (B —> X Eu,) 153 B(B —> X, + 7) cv IM2 |C7(=vt»#)|"K(¤¤».u) 62>< (0.11), (142) W |%b|g(mc/mb)(1_ 2/3%rLf(mc/mb)) where C·,-(1‘t,,u) E C7(,u), defined in H,ff earlier. The functions g(r) and have been introduced previously. In the above expression for B (B —> X, + 7), we have neglected the contribution from the decays b —> u -1- E ug in the denominator, since it is numerically inessential (more so since the recent CLEO and ARGUS data give V,,g,|/ [Val = 0.06-0.08). The above factor (0.11) represents the measured semileptonic branching ratio B(B ——> XE:/l). The functions K(:c,, p) and C7(a:,, p,) are worked out numerically for mt in the range 100 GeV $ mt $ 200 GeV and the QCD renormalization scale (= p) varied in the range mb /2 $ mb $ 2mb 153. The reason for including the p-dependence lies in the fact that at present only the first term in the perturbative expansion of the anomalous dimension matrix has been calculatedfu. The next-to-leading order terms in this matrix, together with the consistent calculation of the matrix elements of the operators entering in H,ff(b —-> s) (often called matching condition), are expected to reduce the pdependence of the branching ratio. Not having this information at hand, one has to ascribe a theoretical systematic error. A plausible way of doing this is to vary p in the range mb/2 f p g 2m,. It should be emphasized that this procedure is approximate, since in scaling below mb one should use an effective four quark theory to calculate the anomalous dimensions. This has not been done so far for the magnetic moment operators in question and hence the numbers presented are subject to some change. The strong coupling constant has been calculated from the 51 OCR Output 5.0 4.0 3.0 2.0 2.5 GeV 5 GeV 1.0 10 GeV 0.0 100 120 140 160 180 200 m, [GeV] Figure 11: Inclusive branching ratio B (B —> X, + 7) in the standard model assuming lQ,|/|lQ;,| = 1 as a function of the top-quark mass for three indicated values of the scale parameter p (from ref. (153)). two-loop 5-function, with N f = 5 and ls)A= 225 MeV, consistent with its present determination. The result for the branching ratio B (B —> X, + 7) obtained by setting the CKM matrix-element ratio |»\,|°/ lV,,|° = 1, expected in the SM, is shown in fig. 11, giving‘ 53· B(B -> X, + 7) = (3.0 ;l; 1.2) X 10'°, (143) for mt and p in the range 100 GeV $ m, §_ 200 GeV and 2.5 GeV $ p $ 10.0 GeV. Restricting the top-quark mass in the range predicted by the electroweak precision data, 164 ;t 27 GeV I", reduces the uncertainty in B (B -» X, + 7) only marginally, yielding B(B —-+ X, + 7) = (3.0 ;l: 1.0) x 10". Thus, the SM estimates at present are quantitative to only about approximately a factor 2 due to the indicated parametric dependence, with the scale ambiguity the dominant source of theoretical dispersion on B(B —+ X, + 7) 153*161. We now discuss the implications of an eventual measurement of the branching ratio B(B —> X, + 7) for the SM parameters. A much desired goal at present is to determine m, either directly or indirectly, through the loop corrections in which the top quark contributes. Unfortunately, the measurement of B (B -> X, + 7) would not be a big help here as the m,-dependence of B(B —> X, + 7) is rather mild. N umerically, one findsl (B(B -·> X, + 7); rn, = 200 GeV) 21.4 • (144) (B(B —> X, + 7),m, = 100 GeV) In contrast, the p-dependence of B(B —> X ,+7), in the present theoretical framework, is significantly more pronounced. For example, for m, = 140 GeV, it has been 52 OCR Output 100 GeV S m, § 200 GeV 2.5 GeV S p S 10 GeV ·¤ 10 Q2 5 f. .1;AA}, =.i._V 0.0 0.5 1.0 1.5 _ 2.0 2.5 :1.0 (IV`.!/lV.»|) Figure 12: Inclusive branching ratio B(B -> X, + 7) in the standard model as a function of the CKM matrix-element ratio (|1Q_,|/|l{,;,|)°, with the top-quark mass in the ranges 100 GeV $ m, $ 200 GeV and the QCD scale parameter p in the range 2.5 GeV $ p $ 10 GeV. The derived upper and lower (95% C.L.) bounds from the CLEO measurements are also shown (from ref. (153)). estimated to be”3· (B(B —> X, + 7); p = 2.5 GeV) 2 1.7 145 ( ) (B(B —• X, + 7),;; :10.0 GeV) Hence, an improvement on the bound on m, from a measurement of B(B —> X, + 7) with respect to the present LEP bounds is not anticipated, unless all the Wilson coefficients and matrix elements of the relevant operators have been systematically worked out to the desired level of theoreatical accuracy. In the SM context, the eventual measurement of B (B -> X, + 7) can be used to determine the CKM matrix element ratio /\,/ |Vd,|°. The unitarity constraint states that (»\,/ |V,,|)° 2 (X,/ |lQ,|)2 = HQ, |°. Since the CKM matrix element |lQ_,| is known from independent measurements, such as charmed hadron decays, the electromagnetic penguin decays B —» X, + 7 provide a quantitative test of CKM unitarity. The dependence of the branching ratio B(B —> X, + 7) on the CKM matrix element ratio |14,|’/|1Q,|’ is shown in fig. 12’f The two curves starting from the origin delimit the SM expectations with m, and p varied in the range 100 GeV $ m, $ 200 GeV and 2.5 GeV $ p $ 10.0 GeV. An upper bound on the CKM matrix·element ratio from the experimental upper limit on the inclusive branching ratio B(B —• X, + 7) can be readily obtained from fig. 12. For the CLEO upper limit B(B --» X, + 7) < 5.4 X 10" (95% C.L.), the corresponding upper bound is: MI/|V¤»I<1·77 (146) OCR Output 'Since |\4y| 2 1 to a very good approximation, we have set A, = |l4,|. 53 lt has been argued153 that this bound can be improved theoretically, if the QCD improved photon energy spectrum in the inclusive decay B —> X, + 7 is used in the experimental analysis. The detailed calculations of the photon energy (equivalently hadron mass) spectrum in the decays B —> X, + 7 have been worked out in the literature148"15°, where a model for the (non-perturbative) smearing due to the B meson wave function effects is also described. As discussed earlier in this report, this non-perturbative model66·67, which assumes a Fermi motion of the b quark with a Gaussian distribution, describes well the lepton energy spectrum in the semileptonic B decays and this spectrum has been used to determine the parameters of this modell 53 With the help of the photon energy spectrum it is now an easy matter to calculate the fraction of events in the photon energy interval 2.2 GeV f E, f 2.7 GeV (used in the experimental analysis"2) from the decays B ——> X, + 7. The details can be seen elsewherelsg. Using the QCD-corrected spectrum, the upper limit B(B ——> X, + 7) < 5.4>< l0'° (95% C.L.) goes down to 4.5 (4.8) x l0"*, assuming a fraction 0.85 (0.78) for the photons in the indicated energy range. ln the first theoretical analysis undertaken for the determination of the CKM parameter /\, from the electromagnetic penguins.153, an upper bound B(B —> X, + 7) < 4.8 X 10"‘ (95% C.L.) was used, giving (see fig. 12): WI/I\éb| < 1.67 (147) 5.3 Estimates 0fB(B —> K* + 7) and \l4,|/HQ, The CLEO measurements of the exclusive branching ratio B(B —> K* + 7) = (4.5 :1: 1.5 dz 0.9) >< l0‘5 yield, at 95% C.L., the following upper and lower limits1 52· 1.6 ><10"5 $ B(B -+ K* +7) S 7.4 x 10' (148) To convert these experimental bounds into the ones on the CKM matrix-element ratio, one has to calculate the exclusive branching ratio B (B —> K * + 7). For that purpose we introduce the following exclusive-to-inclusive B decay width ratio: NB —* K' + 7) R K " X, E —————————-—. ( / l r(B..x.+7> 149 ( ) The model dependence for the exclusive decay can now be discussed in terms of R(K"'/X,). For the sake of later discussion involving CKM-allowed and CKM suppressed exclusive radiative decays B —> V + 7, we define a transition form factor F1‘q2l“‘ - . (Vw Alfguvql/blB> = Zeuvpvcl. B PgPi/2F1 V . (q)r ) ¤ (150) where V is a vector meson (V = p,w,K* or rp) with the polarization vector c m and B is the generic B meson B,,,Bd or B,. The vectors pg, pv and q = pg — pv denote the four-momenta of the initial B—meson and the outgoing vector meson and 54 OCR Output photon, respectively. With the above definition, the exclusive decay widths are given by (B = B., or Bd): 1`(B —> K* + v) = §;qG}1)~¢llFi(0)|2Cv(<¤¢»mb)2(mi+23_`K1 2 2mZ) 3 1» (151) and by an analogous expression for I`(B, —> gb + 7). The branching ratios of these exclusive decays can again be written in terms of the inclusive semileptonic branching ratio: Bw _, [{+7).31*112 = 6 |Cv(¢¢»mb)|’1F{3”"`(0)12 (1 -mw?/mt)3 X(0_H)_ ¤|1éb|2y(mc/m¤»)(1-2/3%:f(mc/mw) (1 —m§/mi)3 ( 5 ) 1 2 The ratio of the exclusive-to-inclusive radiative-decay widths introduced earlier can now be expressed as (B = Bu or Bd): .. (1 —mK·2/m%)3m%1Ff(0)3‘“K`" R(K /X8) (153) (1 ‘ mi? K($¢»mb) The function K (2:,,mb) is almost independent of mt and has been estimated as K(:z:t, mb) 2 0.83, for mt in the range 100 GeV $ mt Q 200 GeV 133. In an analogous way, one defines the exclusive-to-inclusive ratio involving B, decays. The determination of the ratio R(K*/X,) has been the subject of many theoretical investigations, with many new calculations reported recently motivated by the CLEO measurements. We recall that several earlier estimates of this ratio based on quark models are in the ball park (5 to 15)% 132. However, there are some others, mostly based on the early-vintage QCD sum rules, which gave significantly higher values in the range (25 to 40)% for R(K*/X,) 133. Recently, the estimates of the branching ratio B (B —> K * + 7) in the QCD sum rule approach have been revised3°*134*163. The new calculations rest on a better footing, as they use three-point current correlators. The QCD sum rule updates yield: R(K*/X,) = (20:1:6)% 134, and (17:1:5)% 135, and the `significant decrease in the value of R(K*/X,) is to be traced to a higher value for the pseudoscalar coupling constant fg 2 180 MeV, now in fashion. The question of the stability of the traditional QCD sum rules applied in the decay B ——+ V + 7 has been studied at some length3°, in which the various contributions in this method and a new variant of it, the so-called light-cone QCD sum rules, are analysed in the mb —> oo limit. The light—cone QCD sum—rule method leads to a significant reduction in the decay width I`(B ——> K * + 7) and similarly for other decay widths for the generic B —> V + 7 decays, which we shall discuss later. The value obtained is Ff3"’K° = 0.32 :1: 0.05, taking rather liberal ranges for the sum—rule parameters. This gives R(K*/X,) = (16 1 5)% 33. In the QCD sum rule approach the effect of higher—order QCD corrections has still to be investigated. Very recently, a hybrid sum—rule method, called the hybrid moment—Laplace sum rule, has been used by Narison133 to calculate the form factor Ff3`“’K`, getting3"K° Ff= 0.313:007, where the quoted errors have been combined by the present author. This number is in 55 OCR Output very good agreement with the one obtained using the light-cone sum-rule method6 There also now exist two lattice—QCD-based estimates of the form factor in the decay B —+ K* + 7, obtained in the quenched approximation96*97. In lattice-QCD, one has to make an ansatz about extrapolating the result from the region mq 2 mc, where the present calculations are actually done, to the point mq = mb, where they are actually needed. At this stage, the lattice-QCD approach applied to the decay in question, has more of an exploratory character. The result of one of the lattice-analyses is quoted as96 RK· E I`(B —> K* + 7)/I`(b ——> s + 7) = (6.0 ;+; 1.2 i 3.4)%. The result presented by the UKQCD collaborationgf is also compatible, within theoretical and experimental errors, with the CLEO measurements. In the analysis which we adopt here for carrying out a quantitative discussion, the ratio R(K * / X,) has been determined by using the inclusive hadronic invariant mass distribution in B —> X, + 7 as an input, and estimating the K" fraction of the spectrum in the invariant mass region mg + m,, $ mx, § 0.97 GeV 153. (The upper value m§;" = 0.97 GeV conforms to the definition of K " used in the CLEO analysislsl.) The ratio of the resonating p-wave (i.e. K ’) and a non-resonating (i.e. non-K") contribution in the indicated interval of m X, has been estimated by comparison with the semileptonic decays D —> (K + vr) + Z + ul; these have been measured in three independent experiments, E691, MARK III and WA82‘67. This procedure yields1 53· R(K`”/X,) = (13 i 3)%, (154) where the error reflects the uncertainty in the B-meson wave—function model. So, a certain convergence in theoretical estimates of B(B —> K* + 7) is emerging, with R(K*/X,) =(10 to 15)%, covering most calculations. Using R(K*/X,) = (13 :l; 3)%, the branching ratio for the exclusive decay B(B —+ K * + 7) itself can be expressed 85153: B(B—> K*+7)=(4.0;t1.6:l;1.0)><10“5, (155) where the first error reflects both the uncertainty in R(K "‘ / X ,), given above, and that in mt in the range 100 GeV § mt $ 200 GeV; the second error is due to the variation in ,u in the range 2.5 GeV $ p $ 10.0 GeV. This estimate is in comfortable agreement with the CLEO measurementslsl, B(B —> K* + 7) = (4.5 1 1.0 ;l; 0.9) >< 10'5, which can be regarded as one more successful prediction of the standard model in the FCNC sector. It is easy to see that a more stringent upper bound on the ratio |I/Q, | / \IQb| follows at present from the 95% C .L. upper limit on the inclusive branching ratio B(B —+ X,+7). The measured branching ratio B ( B —> K *+7), however, provides at present the lower bound on the matrix element ratio, since the corresponding experimental lower bound on B(B -> X, + 7) is still being awaited. Using R(K*/X,)"‘“ = 16% from the Ali Greub analysis153 and the 95% C.L. experimental lower bound from CLEO given in (148), one gets a 95% C .L. lower bound on the inclusive branching ratio: B(B —-> X, + 7)} 1.0 ><10' (156) OCR Output 56 With this lower bound on B(B —> X, + 7), the following lower bounds on the CKM matrix element ratio follows: W.!/Im! > 0.50 , (157) which yields1 53· 0-50 S llisl/Il/i;5I $1.67. (158) The lower bound depends on R(K*/X,)m“”, which is model-dependent but, as dis cussed at length here, the number assumed is in the right ball—park. While these estimates are not yet completely quantitative and there is much room for both ex perimental and theoretical improvements, the point has been demonstrated that the electromagnetic penguins in B decays provide a determination of the matrix element V},\/ llébl, using the CKM unitarity constraints. Once the inclusive branching ratio B (B —-> X, + 7) has been measured, this model dependence can be largely removed and the route involving the exclusive decay B (B —> K * + 7) that we have undertaken here will no longer be necessary. Of course, the inclusive rate will provide a direct measurement of the ratio R(K* /X,), which can be used to discriminate among com peting theoretical models. The estimate of the CKM matrix element ratio from the electromagnetic penguins in B decays obtained above can be compared with the indi rect bounds due to the CKM unitarity that can be worked out from the fits reported by the Particle Data Group6 0.55 5 MSI/Mbl $1.68 (159) The two bounds on the ratio |VQ,| / HQ,] are quite comparable, although they have been derived by using very different considerations, which can be taken as a consistency check for the SM normalization of the electromagnetic penguins in B decays. With improved data and theory, these tests of the CKM unitarity will become much more precise. We discuss briefly the potential role of radiative rare B decays in determining the matrix element VQ,. The idea of relating the semileptonic decay B —> pfw (a decay mode yet to be measured) and the rare decay mode B -> K * + 7 (as well as other FCNC semileptonic B decays) has been discussed by Isgur and Wisewg and by Burdman and Donoghuewg. The point of their observation is that HQET, in the limit mg, —> oo, can be used to relate the form factor F1 appearing in the decay B ——-> K * +7 and the semileptonic decay form factors defined by: - Ks: IA- 1_ Z VU< (zn )|~m( 75) I (z¤)>2l/(Q2) mB+mK_61 I ¤•¤(}`)1/pd p 6 PP i(mB + mK·)A1(q2)e;+(*) . .. (160) where the ellipses denote terms proportional to (p + p’),, or qu. According to the HQET-based argument168*169: 2 2 2 2 -. +m —m». V(q) mg-}-mK· B-+R 2 q B Ix 2 ‘FiZ 1 (q) 2m A 7 B mB+mK_ + 2mB 1<<1) ( ) 57 OCR Output as far as the value of q2 is sufficiently close to the point qgmx = (mg — mK·)2 2 19 GeV2. Similar relations can of course be derived for other radiative decays. Using S U (3) symmetry, one could relate the semileptonic form factors in B —> K 'Zw, which are not experimentally accessible, to the ones which will be measured in the decays B —> ph,. This would then allow a determination of the CKM ratio \V,,;,|/|\&,|, which in the SM is approximately equal to the ratio }l€,;,|/|lQb|. The central problem in using predictions based on the HQET appraoch to the rare B-decays is whether the above relation is satisfied for small values of q2, as suggested in 168*69, and what is the size of the power 1/ mb corrections. In a recent paper, the HQET-based relation has been studied for general values of q2 available in the decays B —> p+£1/[ using the QCD sum rules on the light cone6°. The main result of this investigation is that the relation in (161) is fulfilled in the QCD sum rule approach to a good accuracy for finite b-quark masses and for all momentum transfers. The HQET-based prediction can then be used as and when data on CKM-suppressed semileptonic decay become available. It should, however, be stressed that there are important perturbative QCD corrections in the Wilson coefricients governing the radiative rare B decay B ·+ K ' + 7, which at present are known imprecisely due to the scale and mydependence, as discussed above. While still on this subject, we mention a recent proposallm, relating the integrated spectra in the inclusive decay B —+ Xu + Eu; and the radiative rare decay B —+ X, + 7, which has also been put forward as a means to determine the CKM matrix element Vub precisely. This relationship rests on the universal behaviour of the leading Sudakov spectra near the end-points Eg —> E}”"" and E., —> E,Q”“', which holds if one neglects the final-state quark masses in the radiative and semileptonic transitions. To give a quantitative content to this idea, however, further theoretical work is needed. In particular, the importance of the (non-universal) subleading corrections to these spectra has to be quantitatively understood. Again, a theoretically reliable calculation of the Wilson coefficients in the FCN C effective Hamiltonian governing the decays B —> X, + 7 will be essential to get reliable estimates of Vub from this method. 6. Importance and reasons for studying B decays In the preceding sections, we have discussed at length the implications of the measurements done so far in B-physics. These measurements have left little doubt about the weak isodoublet nature of the b quark and the chiral structure of its charged current being (V -— A), assuming the universal (V — A) nature of the leptonic charged current. In addition, the CKM matrix elements HQ], |I{,;,|, |I4,;| and |W,| have been measured (to varying degrees of precision), all obeying the expected pattern and unitarity constraints. Also, all present measurements in B decays are consistent with the statement that the FCNC transitions in [AB] = 1 and |AB| = 2 sectors are governed by the GIM mechanism. In many cases, however, the present precision on the CKM parameters is not very impressive. Likewise, the present measurements and bounds on the strength of the induced (loop) interactions involving B decays are such 58 OCR Output that non-SM interactions, with induced effective couplings only moderately smaller than the corresponding SM couplings, cannot be ruled out. It is therefore important to make the present measurements as precise as possible. The interest in future B-physics studies is manifold. We stress the following three broad areas of research: B decays provide a good laboratory to apply QCD in a quantitative fashion. ln this context, the methods of the heavy—quark effective theory need special emphasis, as they allow a systematic study of the hadronic matrix elements and structure functions that one encounters in B decays. Hopefully, the methods of the effective heavy quark theory can be combined with lattice—QCD techniques and other non—perturbative approaches, such as the QCD sum rules, to cover most transitions of interest in B decays. We have discussed some selected topics; they are discussed in more detail in other articles in this v0lume76·9° Precision studies of FCNC transitions in B hadron decays allow a search for quantum (loop) effects in the standard model. Their practical use in the SM lies in determining the CKM matrix elements involving the top quark, which would otherwise be difficult to measure, as well as the top-quark mass. This was illustrated in the previous section through the extraction of |W,| from the measurement of B(B —> K* + 7) and the present limit on B(B —> X, + 7). The CKM·suppressed F CNC decays have the potential of determining the CKM parameters p and n, and hence they will contribute in testing the unitarity of the CKM matrix. By the same token, rare B decays provide a window on new physics. B decays provide a multitude of final states, where CP violation effects could be measured. In fact, all the three angles of the unitarity triangle can be multiply constrained both by measuring the sides of the triangles, in CP—conserving pro cesses, and through CP-violating asymmetries in a number of B decays. A quantitative study of the resulting CP asymmetries would either lead to the conclusion that the KM phase is the sole source of CP violation or else provide experimental basis for new mechanism(s) for CP violation. In the rest of this section, we concentrate on the various facets of the CKM triangles, following from unitarity. In particular, we address the following questions: • What is the present profile of the CKM matrix? • What can FCNC B transitions, in particular B2 -B§ mixing and CKM-suppressed rare B decays, B —+ (p, tu) + 7, contribute in sharpening this profile? • What are the prospects of measuring CP violation in B decays? 59 OCR Output Table 6: Present status of the CKM matrix elements CKM matrix element I Present value Vud 0.9734 5: 0.0007 Vu, 0.2205 1: 0.0018 Kd 0.215 -1; 0.016 KQ, 0.98 ;+; 0.02 KQ, 0.042 i 0.005 VtbI/ Mb 0.08 4 0.02 Wd 0.005 — 0.016 Mal/1% 0.50 -1.67 Wh 0.998 — 0.9995 6.1 Present projile ofthe CKM unitarity triangle The six unitarity triangles that one encounters in the quark sector have been given earlier on in this article. It is obvious that measuring all these triangles independently will keep the experimental flavour physics community busy for some time to come! The unitarity triangle that has, in our opinion, the best chance to be measured precisely is the triangle (db) following from the constraint: Kali], + VQ.1lC,i,+ Vi¤1lCZ= O (162) This is so because the three terms here are expected to be of comparable magnitude, readily seen in the Wolfenstein parametrization of the CKM matrix. Using this representation of the CKM matrix, the (db) unitarity relation can be written as a triangle relation: V2 Vid u —-—· = + 1 163 ( ) mb M4. It is the profile of this triangle that we will concentrate on. We have discussed in the preceding sections the present knowledge of the CKM matrix elements. The result of this analysis is summarized in table 6. In terms of the Wolfenstein parameters, the information given there can be transcribed as follows: 0.86 zi; 0.10 , 0.2205 t 0.0018 , (164) pz + 02 0.36 t 0.09 , 0.005 A/\3\/(1* Pl2 + U2 S 0-016 , where the last inequality follows from the value of |V}d| obtained from xd. Further constraints on these parameters are obtained from The experimental value of lc 60 OCR Output is6: el = (2.26 i 0.02) >< 10` (165) Theoretglly, |e| is essentially proportional to the imaginary part of the box diagram for K°—K° mixing and is given byl n G2 fz M 6\/§7rAMKKM 2 ’l C1 C6 Ci _( CC ,xn)(y{n¢f:·.(y yi) U} + TIM?/tf2(?/t)A2/\4(1" P) (166) Here, the ry, are QCD correction factors, nec 2 0.82, my 2 0.62, nc, 2 0.35 for AQCD = 200 MeV 172, y, 5. and the functions f2 and fg are given by 1 9 1 3 1 3 mz ln xc MI) ‘i 1(1—Z+ z) Z_`§(1 - xy` 2(1— xp 3y y = l 2 ————i l —-—-— . f3(¤v.y) ¤ x 40 _ y)( + 1 _ y hw) (167) (The above form for f3(z, y) is an approximation, obtained in the limit sc < y. The exact expression is given by lnami and Lim16°.) One of the unknowns in eq. (166) is the top-quark mass. The present CDF bound from direct top searches gives mt > 113 GeV M4. While the exact range of mt from the electroweak analysis depends on a number of details, in particular on the value of 0;,111, a recent analysis based on the combined LEP data gives1 7‘· mt = 164 d; 27 GeV. The final parameter in the expression for |e| is BK, which involves the matrix element (K °|(E1"(1 —·y5)s)‘|T{_5). The evaluation of this matrix element has been the subject of much work, reviewed recentlylm. Although the entire range of BK is 1 /3 $ BK $ 1, the 1/ N and lattice approaches are generally considered more reliable. They give estimates, which can be stated as BK = 0.8 i 0.2 . In order to find the allowed unitarity triangle, a fitting programme was used to fit the CKM parameters A, p and 7; to the experimental values of |lQb|, |M,b/Kb], |c| and :1:,;. These fits"3 have since then been updated, and the results are being included here. The parameters used in this analysis have been discussed in the previous section and their input values from experimental measurements or theoretical estimates are listed in table 7. The results of the fit themselves are shown in fig. 13. In all these graphs, the solid line has X2 = X3,,,n+1, which corresponds to the 39% confidence level region';. For comparison, we include the dashed line, which is the 90% C.L. region (X2 = X3,,,,, + 4.6). (Note that, strictly speaking, these fits do not represent the true 61 OCR Output Table 7: Parameters used in the CKM fits Quantity I Input value EI, 0.042 :1: 0.005 VMI/IlQI,| I 0.08 :1; 0.02 r(B) 1.49 i 0.04 (ps) 6K (2.26 :1; 0.02) >< 10" 0.68 t 0.09 164 :1; 27 GeV BK 0.8 :1: 0.2 1.0 :1: 0.2 f(Bd) 180 i 50 MeV Table 8: “Probable values” for the Wolfenstein parameters p and r] following the analysis based on fig. 13 mt (GeV) I fa. (MeV) I Ba, (xw) x:..... 140 180 zh 50 I 1.0 :1: 0.2 I (-0.19, 0.29) I 0.04 165 180 :1; 50 I 1.0 :t 0.2 I (0.14, 0.31) I 0.19 190 180:1:50 I 1.0:1:0.2 I (0.21, 0.27) I 0.13 sense of confidence levels since some of the errors shown in table 7 are theoretical, and theoretical "errors” are not Gaussian! However, this method is not unreasonable, being quite similar to the practice of adding experimental statistical and systematic errors in quadrature.) It is clear that, as we pass from fig. 13a to 13c, the “most likely” unitarity triangle becomes more and more acute. However, it is also clear that there is a substantial overlap between the 90% C.L. regions. The “probable values” of the parameters (p, r;), corresponding to the triangles drawn in the figures, together with their X2, are given in table 8. We remark that the range of 1; is well constrained by the very accurate measure ment of Iel, and it depends only modestly on xd (and hence fg,) or, as we are going to argue in the next section, on the ratio 1,/xd. It should also be remarked that if the constraint from IcI is removed from this analysis, then the present data are not enough to prove that indeed one has a triangle in the p—n plane. The value of p is strongly affected by the xd measurement, which in turn depends strongly on both mt and fg,. It is therefore qualitatively clear that a good measurement of 2:, (and hence z,/xd) will be very effective in narrowing down the allowed range for p. Likewise, rare B decays such as B —> p + 7, B —> w + 7, B —+ p + FZ', etc., would also be very helpful. 62 OCR Output OCR OutputWe discuss here the impact of the CKM—suppressed rare B-decays on the CKM matrix elements. Whereas in the branching ratio for B —+ X, +7 discussed earlier, we had argued that the CKM matrix dependence can be factorized in an overall factor /\,|2, the case of inclusive B —> Xd + 7 decays is somewhat more complicated with respect to the factorization of the CKM parameters. These contributions bring to the fore a non-trivial dependence on the CKM matrix elements through terms propor tional to fu and fc. Explicit expressions for this dependence on the p and ry parameter in the Wolfenstein representation of the CKM matrix have been evaluatedlm. The dependence of the branching ratio can be written explicitly as: Bw —» xm) = Dilérlz 1- —¥——1>. ——-—L-—-D.i’ + -1-- l (1-p)2+n2 (1-0)*+*7* (1-p)’+n” (170) Here we have expressed ft and fc in terms of the Wolfenstein parameters A, /\, p and C`: = A%"(1—p+in)» 5, = -A A3. (171) The mt-dependence of the coefficients D, is rather weak, as shown in table 9. To get the inclusive branching ratio as a function of m,, we vary the CKM parameters p and n over the presently allowed range of the variables discussed previously. We have described this procedure in detail earlier. The resulting branching ratio B (B --> Xd + 7) is shown in fig. 14 as a function of mt, from which one obtains1 76· B(B —-> Xd + 7) = (0.8-3.2) >< 10' (172) for the top-quark mass in the range 100 GeV $ mt $ 200 GeV. We remark that the p-dependence of the branching ratio B(B —> X, + 7) is very significant, which can again be traced to the very significant QCD renormalization of the Wilson coefficients C,(mw) —> C,(p), in particular C7. A determination of the Wolfenstein parameters p and r; will follow if one is able to measure the inclusive radiative branching ratios B (B —> X, + 7) and B (B —> X.; +7). That a number of unknowns, such as the top-quark mass and the QCD scale param eter, drop out from this ratio is obvious from the expressions given earlier. However, measuring these inclusive rates is a formidable proposition, from the experimental point of view. It is perhaps easier to measure the exclusive radiative decays, as CLEO has demonstrated on the example of the decay B -> K * + 7. In the following we shall focus our attention on exclusive decays, such as BW; -> K" + 7, B, —>¢> + 7. For the generic radiative decay B —+ V + 7 the transition form factor F1(q2) has been defined in eq. (150).The expressions for the decay width and branching ratios for 64 OCR Output Table 9: Values of the coefficients Di entering in eq. (170) as a. function of mt (from ref. (149)) mt (GeV) | D1 100 0.16 0.21 0.04 0.14 120 0.17 0.20 0.04 0.13 140 0.19 0.18 0.04 0.12 160 0.20 0.17 0.04 0.11 180 0.21 0.17 0.03 0.10 200 0.22 0.16 0.03 0.10 the mode B —> K * + 7 in terms of F1(q2) have been given earlier. The corresponding expressions for the CKM-suppressed decay l"(B, —>¢ + 7) are defined analogously. Since the same short—distance~corrected coefficient function C7(m;,) enters in the Hamiltonian for CKM-allowed and CKM-suppressed modes, the QCD scaling for the two·body decays b -+ s + 7 and b —-> d + 7 is identical and does not affect the ratio of the decay widths I`(b —-> d + 7)/l`(b —> s + 7). The same applies for the exclusive decays such as B —> p + 7 and B —-> w + 7, which factorize in the CKM factors. From this observation a number of relations between the exclusive decay rates follow in the SM15°. This is exemplified by the decay rates for BM —+ p + 7 and Bud —> K"‘ + 7: l`(B~.¢··> P+ V) Mil? |FF(0)"”'2 _ W '?'—""_" - °"“' u,da (Bm -* K' + *7) lW»|2 |F1B"K (UW where QM is a phase-space factor: (mf + mf) (mb,. · mi? _ @*44 ‘ (174) 2 z 2 2 3 (mb + m,) (mam,. - mx-) The ratio (173) depends only on the CKM matrix elements and the ratio of form factors, while it is independent of the top—quark mass (and of the renormalization scale p). The transition form factors have been evaluated in the light-cone QCD sum-rule approach6°, yielding: B-•K'7 0.32 i 0.05, B-·(¤.w)7 0.24 zi: 0.04, F1Bs-*¢'Y 0.29 zh 0.05, FB;-•K°7 0.20 :1; 0.04. (175) For the ratios of the form factors, which are needed to determine the ratios of the CKM matrix elements, the following estimates have been reported B—+(p,w)·7 = 0.76 zi: 0.06 , (176) 65 OCR Output 5.0 4.0 BR(B ·• X, 7) 3.0 2.0 1.0 0.0 100 120 140m. [cw]160 180 200 Figure 14: Branching ratio for B —> Xd + 7 as a function of mt by varying over the allowed range in the (p,r;)-plane (from ref. (149)). F11?.-·K*+7 iw; = 0.66 ;i; 0.09 , (177) and Pwf,-•K°+'Y —-Eg-;?;-77 = 0.60 :1: 0.12. (178) The eventual measurements of the CKM-suppressed radiative decays (Bu, Bd) -» p + 7, Bd -+ w + 7 and B, —-> K* + 7 could then be used to determine the CKM parameters. This is illustrated in fig. 15, where the ratio of the branching ratios B(Bd -> w+')’)/B(B —+ K*+7) and B(B, —> K' +7)/B(B -+ K*+7) are plotted as a function of the CKM matrix element ratio |\4,|/ |l4,|. As a check of this method, we note that it predicts°° B(B -> K " +7) = (4.8;h1.5) >< 10*5, which agrees well with the experimental measurement by CLEO. We expect that the estimates of the ratio of the exclusive branching ratio should, in any case, be more stable against variation of the parameters in the QCD sum-rule method. So, the dependence of the rates on the CKM matrix elements shown in fig. 15 should be useful to determine the indicated ratio of the CKM matrix elements. We also note a recent calculation of the form factor ratio on the l.h.s. in eq. (176) *66, obtaining 0.88 ;l; 0.02, which is somewhat larger than the one given above. Apart from the radiative B decays, there is an entire class of FCNC semileptonic B decays, B —> X; +l+£’, B —> X; + l/17, where f = d,s, the purely leptonic decays (Bd, B,) —+"£+£`, and the 77 decays (Bd, B,) -1 77, which in the SM provide infor mation on the CKM matrix elements and top-quark mass 175. Once again, these tran 66 OCR Output .15 9 .1 .05 1, 05 O .1 .2 .3 .4 .5 O .1 .2 .3 .4 .5 IVo| / |V¤.| IVMI / |V¤| Figure 15: The ratios of the branching ratios B(Bd —» w + 7)/B(Bd -+ K' + 1) and B(B, ··* K` + 7)/B(B4 -> K` + 7) as a function of the CKM matrix—element ratio Wdl / |l&,|. The two curves represent theoretical model uncertainties (from ref. (60)). sitions can be classified as CKM-allowed and CKM-suppressed. Also, the ratios of the exclusive decay branching ratios, such as those involving the decays B —> (p, w) +€+€' and B -—> K' + WE', should have a dependence on the CKM matrix elements very similar to what we have shown for the corresponding decays in fig. 15. All of these remarks apply to the short-distance contribution. As is well known, for the F CN C semileptonic decays involving charged leptons, the long—distance contribution is quite important. So, an analysis where the long- and short-distance contributions have been treated coherently is required. The present best limit on the FCNC semileptonic de cay is from the UA1 collaboration, giving B(B -» X + p+p") < 5.0 >< 10'5 at 90 % C .L. 15*, which lies about an order of magnitude away from the SM estimates. The ex pected rates for a number of FCNC B decays that have been estimated inm, together with the present experimental bounds and the measured rate for B (B -» K " + 7), are given in table 10. We expect that some of these SM predictions will soon be tested in present experiments at CLEO and the Tevatron, but many more will be scrutinized in dedicated B facilities expected to operate in the later part of this decade. With these measurements, the structure of the FCNC processes will be completely pinned down and with that also the elements of the CKM matrix. 6.3 CP violation in the B system Mixing and CP violation in the B system can be described in much the same way 67 OCR Output OCR Outputas in the kaon system. In the B system, the states [B0) and {F1) are eigenstates of the strong and electromagnetic interactions, but not of the weak interactions, which are responsible for their decay. Taking into account the weak interactions, one writes the 2 >< 2 Hamiltonian (in the B°—B° basis) H = M - gr , (179) where the mass matrix [VI and the decay matrix l` are Hermitian. (Since neutral B’s do decay, H itself is not Hermitian.) CPT invariance implies that the diagonal components of H are equal, and if CP is conserved, M and P are real. Allowing for the possibility of CP violation, diagonalizing the Hamiltonian m M12]_i[ 7 1112] 1Mii2 m 2 Fitz '7 180 ( ) yields the eigenstates BL and BH (L and H denote the light and heavy states, re spectively): BL) = 1>|B°)+ <1|E6), BH) = PlB°) —<1|B°)» where [7-1+63- IMt2—·%F1g 9 1_€B \lM12_EF12 ln eq. (182), the symbol cg has been introduced to indicate the relation between this notation and that used normally in the kaon system. Angogously to the kaon system, a non-zero value of eg indicates CP violation in B°-B° mixing. However, there is a big difference in the B system — here it is found that P1; < M12 (recall that F12 ~ 2Mlg for kaons). This implies that eg is much smaller than cg, so tlg prospects for the observation of CP violation in the AB = 2 transitions of B°—B° mixing are virtually hopeless. Luckily for us, CP violation in B decays (AB = 1) can be large1"·155. (This is another difference between the K and B systems; in the kaon system, CP violation in K decays (e'/e) is small.) The most promising signal of CP violation in B decays comes by considering a CP eigenstate final state f to which both B° and B0 can decay. CP violation is manifested in a non-zero value of the time—dependent asymmetry ml ` AN): P B0 is — 1* EM ( ()—*f) ( ()··*f) (183) 001`(B (1) —> f)+1`(B (i) ··* f) The expression for this asymmetry is Af(t) = —lrnozf sin xqf , (184) OCR Output 69 in which zcq, q = d,s is the Bg—B? mixing parameter (eqs. (98), (99)), TB is the B lifetime, and 01 =‘— ;P;» @85) where pf is the ratio of the amplitudes for BQ and BQ to decay to f, ,/PE2 - P -M12 ,.-MM 188 ( ) Therefore _ = e'2'(°’M+‘(’°l => Ima) = - sin 2(4$M + op). (189) Note that, although ¢M and (bp each depend on the form chosen for the CKM matrix, the sum 45M + dm is convention—independent. In the Wolfenstein parametrization (eq. (22)), only the matrix elements Vid, and Wd have large phases. In the following we will ignore all small CKM phases. It is not difficult to relate ¢M and gbl; to theCKM matrix elements. Like xq, A/In is _calculable from the box diagram for Bf,°—BQ mixing (in fact, AM = 2\M12|). Thus, M12 ~ (I4;,lQQ)2, and therefore the phase information from the mixing is (2) = Y; , (2) P B, Wd P B_ :1. (190) The phase information in the decay depends on whether the b quark decays into a c or a u quark: (Pike,. = 7 (P;)b..c (191) ub There are therefore three classes of B decays into CP eigenstates that can have sizeable CP-violating asymmetries. These are related to the three angles cv, B, 7 of the unitary triangle as follows. • Bd decays with b -> cz for this class of decays, we have V oaf:-ig, Imorf=—sin2B. (192) me The classic example here is Bd —> J / \I!K S. For this particular final state, since J / \I/K8 is a CP-odd state, there is an additional minus sign in the asymmetry. 70 OCR Output Table 11: The allowed ranges for the CP asymmetries sin 204, sin 25 and sin 27, corresponding to the coriaints on p and ry shown in figs. 16. Values of mt and the coupling constant fgd \/BB', are stated. The _range for sin 25 includes an additional minus sign due to the CP of the final state J/WK; (updated from ref. (173)). mt (GeV) I fBd\/Bgd (MeV) I sin 204 I sin 25 140 180;I;50 1-1 I 0.15-0.97 I -1-1 165 180:t50 -1- 1 I 0.16-0.98 I -1-1 190 180 :I;50 -1 -1 I 0.18 - 0.97 I -1 -1 • Bd decays with b —> uz here we have Vub wd . cx =f, Ima =S1I12Q. ’’ 193 I I wc. One example of such a decay is Bd —> 1r+1r‘ • B, decays with b —> uz in this case IG . aj = %, Ima; = —s1n27 . V (194) ub This angle is the most difficult one to measure. One possible decay mode is B, -> pK_q. Like the final state J/\I1K_g, pKg is CP-odd, which changes the sign of the asymmetry. Some suggestions on measuring the angle 7 have been discussed recentlyg. The asymmetries in eqs. (192), (193) and (194) can be expressed straightforwardly in terms of the CKM parameters p and ry. The 90% C.L. constraints on p and ry found previously can be used to predict the ranges of sin 20, sin 25 and sin 27 allowed in the SM. The allowed ranges corresponding to each of the figures in figs. 16a — c are found in table 11. In this table we have assumed that the angle 5 is measured in Bd —> J / WK S, and have therefore included the extra minus sign due to the CP of the final state. Since the CP asymmetries all depend on p and rp, the ranges for sin 204, sin 25 and sin 27 shown in table 11 are correlated. That is, not all values in the ranges are allowed simultaneously. We illustrate this in figs. 16a —c by showing the region in sin 2a—sin 25 space allowed by the data, for various values of mt and _f§dBBd. From these figures one sees that the smallest values of sin 25 occur only in a small region of parameter space around sin 2a 2 0.2-0.6. Excluding this small tail, one expects the CP asymmetry in Bd -—> J / WK; to be at least 30%, which is good news for the B-physics community interested in CP violation 1‘*5·1‘6·18°*181. 71 OCR Output OCR OutputWith the SM being tested in the electroweak sector at the quantum level, one of the most important remaining tests of the standard model involves the precise deter mination of the CKM angles. B decays have contributed very significantly in shaping much of our knowledge of the CKM angles. An update has been presented in this report. While this information is at present not quite quantitative, future improve ments in experiment and theory will allow to determine precisely these fundamental constants of nature. The knowledge of the values of these parameters might give us some insight into the mechanism of spontaneous symmetry breaking, particularly as regards the fermion mass matrices. For example, it is known that SM and some un orthodox models, such as the extended technicolor, fulfil their charge in this respect very differently. New and crucial experiments are needed to test the SM and the competing models critically. Furthermore, this information would allow us to ascer tain whether the CKM matrix is the sole source of CP violation, or if new physics is needed 182. The CKM triangles provide a convenient framework to interpret present and planned experiments in simple terms. Presence of new physics, contributing to the various rates and asymmetries discussed in this report, will show itself through deviations from the CKM unitarity triangles 155*83. The interpretation of experi mental results, however, requires serious efforts in improving theory. We discussed the-state-of-the-art applications of the present theoretical framework in B decays. While some theoretical progress has been made, there is no room for being compla cent. Two examples, namely the B lifetime measurement (T(B)) = 1.49 :l: 0.04 ps and the mixing ratio xd = 0.71 zh 0.09, suffice to illustrate that the precision reached by the present experiments is not matched by theory. As the experimental presicion improves, this embarassment is likely to become even more acute. Theory is facing a real challenge in B physics. Let us hope that the experimental opportunities do not remain theoretically insurmountable! 8. Acknowledgements Helpful discussions with Ikaros Bigi, Volodya Braun, David Cassel, Michael Danilov, David London, Thomas Mannel, Stephan Narison, Matthias Neubert, Vivek Sharma, Hubert Simma, Sheldon Stone and Nicolia Uraltsev are gratefully acknowledged. David Cassel and David London have provided valuable input for this report. 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