CP-Violation in B_{q} Decays and Final State Strong Phases
aa r X i v : . [ h e p - ph ] N ov C P –Violation in B q Decays and Final State Strong Phases
Fayyazuddin ∗ National Centre for Physics &Department of Physics, Quaid-i-Azam University, Islamabad.
PACS: 12.15.Ji, 13.25.Hw, 14.40.Nd
Abstract
Using the unitarity, SU (2) and C -invariance of hadronic interactions, the bounds on finalstate phases are derived. It is shown that values obtained for the final state phases relevantfor the direct CP -asymmetries A CP ( B → K + π − , K π ) are compatiable with experimentalvalues for these asymmetries. For the decays B → D ( ∗ ) − π + ( D ( ∗ )+ π − ) described by twoindependent single amplitudes A f and A ′ ¯ f with differnt weak phases (0 and γ ) it is arguedthat the C -invariance of hadronic interactions implies the equality of the final state phase δ f and δ ′ ¯ f . This in turn implies, the CP -asymmetry S + + S − is determined by weak phase(2 β + γ ) only whereas S + − S − = 0 . Assuming factorization for tree graphs, it is shown thatthe B → D ( ∗ ) form factors are in excellent agreement with heavy quark effective theory.From the experimental value for (cid:16) S + + S − (cid:17) D ∗ π , the bound sin(2 β + γ ) ≥ .
69 is obtainedand (cid:16) S + + S − (cid:17) D ∗− S K + ≈ − (0 . ± .
08) sin γ is predicted. For the decays described by theamplitudes A f = A ¯ f such as B −→ ρ + π − : A ¯ f and B −→ ρ − π + : A f where theseamplitudes are given by tree and penguin diagrams with differnt weak phases, it is shownthat in the limit δ Tf, ¯ f → , r f, ¯ f cos δ f, ¯ f = cos α and S ¯ f S f = S +∆ SS − ∆ S = − q − C f q − C f . The CP asymmetries in the hadronic decays of B and K mesons involve strong final state phases.Thus strong interactions in these decays play a crucial role. The short distance strong interactionseffects at quark level are taken care of by perturbative QCD in terms of Wilson coefficients. TheCKM matrix, which connects the weak eigenstates with mass eigenstates, is another aspect ofstrong interactions at quark level. In the case of semi leptonic decays, the long distance stronginteraction effects manifest themselves in the form factors of final states after hadronization. Like-wise the strong interaction final state phases are long distance effects. These phase shifts essentiallyarise in terms of S-matrix which changes an ’in’ state into an ’out’ state viz. | f i in = S | f i out = e iδ f | f i out (1) ∗ [email protected]
1n fact, the CPT invariance of weak interaction Lagrangian gives for the weak decay B ( ¯ B ) → f ( ¯ f ) ¯ A ¯ f ≡ out h ¯ f |L w | ¯ B i = η f e iδ f A f ∗ (2)Taking out the weak phase φ , the amplitude A f can be written as A f = e iφ F f = e iφ e iδ f | F f | (3)Then Eq. (2) implies ¯ A ¯ f = e − iφ e iδ f F ∗ f = e − iφ F f It is difficult to reliably estimate the final state strong phase shifts. It involves the hadronicdynamics. However, using isospin, C-invariance of S-matrix and unitarity, we can relate thesephases. In this regard, following cases are of interest:Case (i): The decays B → f, ¯ f described by two independent single amplitudes A f and A ′ ¯ f with different weak phases: A f = h f |L W | B i = e iφ F f = e iφ e iδ f (cid:12)(cid:12) F f (cid:12)(cid:12) A ′ ¯ f = h ¯ f |L ′W | B i = e iφ ′ F ′ ¯ f = e iφ ′ e iδ ′ ¯ f (cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12) where the states | ¯ f i and | f i are C conjugate of each other such as states D ( ∗ ) − π + ( D ( ∗ )+ π − ), D ( ∗ ) − s K + ( D ( ∗ )+ s K − ), D − ρ + ( D + ρ − ).For case (i), there is an added advantage that the decay amplitudes A f and A ¯ f are given by treegraphs. Assuming factorization for tree amplitudes, it is shown that the form factors f B − D ( m π ), A B − D ∗ ( m π ), f B − D + ( m ρ ) obtained from the experimental branching ratios are in excellent agreementwith Heavy Quark Effective Theory (HQET). Hence factorization assumption is experimentallyon sound footing for these decays.Case (ii): The weak amplitudes A f = A ¯ f , A f = h f |L W | B i = (cid:2) e iφ F f + e iφ F f (cid:3) A ¯ f = h ¯ f |L W | B i = (cid:2) e iφ F f + e iφ F f (cid:3) as is the case for the following decays, B → ρ − π + ( f ) : A f , B → ρ + π − ( ¯ f ) : A ¯ f B s → K ∗− K + , B s → K ∗ + K − B → D ∗− D + , B → D ∗ + D − B s → D ∗− s D + s , B s → D ∗ + s D − s The C − invariance of S-matrix gives S ¯ f = S f which implies δ f = δ ′ ¯ f , δ f = δ f , δ f = δ f The time reversal invariance gives F f = out h f |L W | B i = in h f |L W | B i ∗ (4)2here L W is the weak interaction Lagrangian without the CKM factor such as V ∗ ud V ub . FromEq. (4), we have F ∗ f = out h f | S † L W | B i = X n S ∗ nf F n (5)It is understood that the unitarity equation which follows from time reversal invariance holdsfor each amplitude with the same weak phase. Above equation can be written in two equivalentforms:1. Exclusive version of Unitarity [1, 2]Writing S nf = δ nf + iM nf (6)we get from Eq (5) , Im F f = 12 X n M ∗ nf F n (7)where M nf is the scattering amplitude for f → n and F n is the decay amplitude for B → n .In this version, the sum is over all allowed exclusive channels. This version is more suitablein a situation where a single exclusive channel is dominant one. To get the final result,one uses the dispersion relation. In dispersion relation two particle unitarity gives dominantcontribution. From Eq.(7), using two particle unitarity, we get [1], Disc F ( B → f ′ ) ≈ πs Z −∞ M ∗ f ′ f F ( B → f ) dt (8)where t = − ~p (1 − cos θ ), | ~p | ≈ √ s. Eq.(8) is especially suitable to calculate rescatteringcorrections to color suppressed T -amplitude in terms of color favored T -amplitude as for ex-ample rescattering correction to color suppressed decay B → π ¯ D ( f ) in terms of dominantdecay mode B → π + D − ( f ). Before using two particle unitarity in this form, it is essentialto consider two particle scattering processes. SU (3) or SU (2) and C -invariance of S -matrix can be used to express scattering amplitudesin terms of two amplitudes M + and M − which in terms of Regge trajectories are given by[3, 4, 5] M (+) = P + f + A = − C P e − iπα p ( t ) / sin πα p ( t ) / s/s ) α ( t ) − C ρ e − iπα ( t ) sin πα ( t ) ( s/s ) α ( t ) (9) M ( − ) = ρ + ω = 2 C ρ − e − iπα ( t ) sin πα ( t ) ( s/s ) α ( t ) (10)For linear Regge trajectories, using exchange degeneracy, we have α ρ ( t ) = α A ( t ) = α ω ( t ) = α f ( t ) = α (0) + α ′ t,α p ( t ) = α p (0) + α ′ p ( t ) ,C f = C ω ; C A = C ρ ; C ω = C ρ (11)3e take α ≈ / α ′ ≈ − , α p (0) ≈ , α ′ p ≈ . − . Using SU (3) and taking γ ρD + D − = γ ρK + K − , we get C ρ = γ ρπ + π − γ ρK + K − = γ ρπ + π − γ ρD + D − = γ , γ = γ ρπ + π − ; γ ≈ . Hence for π + D − or π − K + scattering we get M = M (+) + M ( − ) = iC P e bt ( s/s )+2 γ ie α ′ (ln( s/s ) − iπ ) t ( s/s ) / (12)where b = α ′ P ln( s/s )For π ¯ D → π + D − , π K → π − K + M = ±√ M ( − ) = ± i √ C ρ e − iπα ( t ) / cos α ( t ) / s/s ) α ( t ) (13)From Eq.(8) and (13) with the use of dispersion relation, we obtain A ( B → π ¯ D ) F SI = √ γ (1 − i )16 π A ( B → π + D − )ln (cid:16) m B s (cid:17) + iπ/ π Z ∞ ( m B + m D ) dss − m B ( s/s ) α ( t ) = −√ ǫA ( B → π + D − ) e iθ (14)We get ǫ ≈ . , θ ≈ ◦ by putting s ≈ m B in ln( s/s ). Now A ( B → π + D − ) = T. Hencewith rescattering correction [6] A ( B → π ¯ D ) = − √ C − √ ǫT e iθ = − C √ h ǫb e iθ i (15)where 2 b = C/T.
Hence the final state phase shift δ C for the color suppressed amplitudeinduced by the final state interaction is given bytan δ C = ǫ/b sin θ ǫ/b cos θ → δ C ≈ ◦ (16)with b ≈ . , which we get fromΓ( B → π + D − )Γ( B + → π + ¯ D ) = 1(1 + 2 b ) ≈ . ± .
03 (17)For B → π K , the color suppressed T -amplitude with rescattering correction is given by − √ C + √ ǫT e iθ = − √ C h − ǫb e iθ i (18)where 2 b = C/T ≈ .
37 [7]. Hence δ C generated by the final state interaction is given bytan δ C = − ǫ/b sin θ − ǫ/b cos θ → δ C ≈ − ◦ (19)To conclude: The scattering amplitude M ( s, t ) for the two particle final state obtained ineq.(13) is used in the unitarity equation to generate the final state strong phase by rescatteringfor the color suppressed tree amplitude. 4. Inclusive version of Unitarity [2]This version is more suitable for our analysis. For this case, we write Eq. (5) in the form F ∗ f − S ∗ ff F f = X n = f S ∗ nf F n (20)Parametrizing S-matrix as S ff ≡ S = ηe i ∆ [5], 0 ≤ η ≤ , we get after taking the absolutesquare of both sides of Eq.(20) | F | (cid:2) (1 + η ) − η cos 2( δ f − ∆) (cid:3) = X n,n ′ F n S ∗ nf F ∗ n ′ S n ′ f (21)The above equation is an exact equation. In the random phase approximation [2], we can put X n ′ ,n = f F n S ∗ nf F n ′ S n ′ f = X n = f | F n | | S nf | = ¯ | F n | (1 − η ) (22)We note that in a single channel description [5, 8]:( F lux ) in − ( F lux ) out = 1 − | ηe i ∆ | = 1 − η = AbsorptionThe absorption takes care of all the inelastic channels.Similarly for the amplitude F ¯ f , we have F ∗ ¯ f − S ∗ ¯ f ¯ f F ¯ f = X ¯ n = ¯ f S ∗ ¯ n ¯ f F ¯ n (23)The C-invariance of S-matrix gives: S fn = h f | S | n i = h f | C − CSC − C | n i = h ¯ f | S | ¯ n i = S ¯ f ¯ n (24)Thus in particular C-invariance of S-matrix gives S ¯ f ¯ f = S ff = ηe i ∆ (25)Hence from Eq. (21), using Eqs. (22 − − η [(1 + η ) − η cos 2( δ f, ¯ f − ∆)] = ρ , ¯ ρ (26)where ρ = (cid:12)(cid:12) F n (cid:12)(cid:12) (cid:12)(cid:12) F f (cid:12)(cid:12) , ¯ ρ = (cid:12)(cid:12) F ¯ n (cid:12)(cid:12) (cid:12)(cid:12) F ¯ f (cid:12)(cid:12) (27)From Eq.(26), we get sin( δ f, ¯ f − ∆) = ± s − η η (cid:20) ρ , ¯ ρ − − η η (cid:21) / (28)5he maximum value for ρ , ¯ ρ is 1 and the minimum value for them is − η η . Hence we get thefollowing bounds: 1 − η η ≤ ρ , ¯ ρ ≤ ≤ δ f, ¯ f − ∆ ≤ θ − θ ≤ δ f − ∆ ≤ θ = sin − r − η η in the above bounds can be obtained from thescattering amplitude M ( s, t ) given in Eq.(12) obtain from Regge pole analysis. The s − wavescattering amplitude f is given by f ≈ πs Z − s M ( s, t ) (31)For the scattering amplitude M = M + + M − relevant for π + D − , π − K + and π + π − , we obtain fromEq.(31) using Eq.(12) f = f P + f ρ = 116 πs iC P b (cid:18) ss (cid:19) + 2 γ π s/s ) − iπ ( s/s ) − / (32)= i + (-0.08+0.08 i )0.17 i +(-0.08+0.08 i )0.16 i +(-0.16 ± i ) (33)where we have used s ≈ m B ≈ (5 . GeV . For C P we have used the values of reference [2]whereas for C ρ = γ ρπ + π − γ ρK + K − = γ ρπ + π − γ ρD + D − = γ and C ρ = γ ρπ + π − γ ρπ + π − = γ ≈
72 for πD, πK and ππ respectively.Using the relation S = ηe i ∆ = 1 + 2 if, where f is given by Eq. (33), the phase shift ∆ , theparameter η and the phase angle θ can be determined. One gets π + D − ( π − D + ) : ∆ ≈ − ◦ , η ≈ . , θ ≈ ◦ π − K + or π K : ∆ ≈ − ◦ , η ≈ . , θ ≈ ◦ π + π − : ∆ ≈ − ◦ , η ≈ . , θ ≈ ◦ (34)Hence we get the following bounds π + D − ( π − D + ) : 0 ≤ δ f, ¯ f − ∆ ≤ ◦ π − K + or π K : 0 ≤ δ f − ∆ ≤ ◦ (35) π + π − : 0 ≤ δ f − ∆ ≤ ◦ Further we note that for these decays, b -quark is converted into c or u quark : b → c ( u )+ ¯ u + d ( s ).In particular for the tree graph, the configuration is such that ¯ u and d ( s ) essentially go togetherinto a color singlet state with the third quark c ( u ) recoiling; there is a significant probability that6he system will hadronize as a two body final state [9]. This physical picture has been put on thestrong theoretical basis [10, 11], where in these references the QCD factorization have been proved.For the tree amplitude, factorization implies δ Tf = 0 . We, therefore take the point of view thateffective final state phase shift is given by δ f − ∆ . We take the lower bound for the tree amplitudeso that final state effective phase shift δ Tf = 0 . Thus for π + D − ( π − D + ) , δ Tf = δ ′ Tf = 0 . The decay B → π − K + is described by two amplitudes [7] A ( B → π − K + ) = − (cid:2) P + e iγ T (cid:3) = | P | (cid:2) − re i ( γ + δ + − ) (cid:3) (36)where P = − | P | e − iδ P , T = | T | e iδ T , δ + − = δ P , r = | T || P | The decay B → π K is described by the two amplitudes [7] A ( B → π K ) = − √ | P | (cid:2) r e i ( γ + δ ) (cid:3) (37)where C = | C | e iδ C , δ = δ C + δ P , r = | C || P | For these decays, we use the lower bounds in Eq.(35) for the tree amplitude so that the effectivefinal state phase δ T = 0 . The phase δ C is generated by rescattering correction and its value is -8 ◦ . For the direct CP asymmetries, the relevant phases are δ + − and δ . For the penguin amplitude,we assume that the effective final state phase δ P has the value near the upper bound. Thus wehave δ + − ≈ ◦ , δ ≈ ◦ . Now [7] A CP ( B → π − K + ) = − r sin γ sin δ + − RR = 1 − r cos γ cos δ + − + r − (38)Neglecting the terms of order r , we havetan γ tan δ + − = − A CP ( B → π − K + )1 − R (39)For B → π K A CP ( B → π K ) = ( R −
1) tan γ tan δ (40) R = 1 + 2 r cos γ cos δ + r Now the experimental values of A CP , R and R are [12] A CP ( B → π − K + ) = − . ± .
015 ( − . ± . A CP ( B → π K ) = − . ± .
11 ( − . ± ± . R = 0 . ± . R = 0 . ± .
068 7here the numerical values in the bracket are the latest experimental values as given in ref [7].With δ + − ≈ ◦ , we get from Eq.(39), γ = (60 ± ◦ . However for δ + − ≈ ◦ which one gets fromEq.(28) for ρ = 0 . , γ = (69 ± ◦ . We obtain the following values for A CP ( B → π K ) fromEqs.(39) and (40) A CP ( B → π K ) = (1 − R ) tan δ (1 − R ) tan δ + − A CP ( B → π − K + )= − . ± . , δ + − = 29 ◦ δ = 21 ◦ − . ± . , δ + − = 20 ◦ δ = 12 ◦ We conclude: The phase shift δ + − ≈ (20 − ◦ for π − K + is compatible with experimentalvalue of the direct CP − asymmetry for π − K + decay mode. For π + π − , δ + − ∼ ◦ is compatiblewith the value (33 ± +8 − ) ◦ obtained by the authors of ref.[7]. Finally we note that the actual valueof the effective phase shift ( δ f − ∆) depends on one free parameter ρ , factorization implies δ Tf = 0i.e. δ f − ∆ = 0 for the tree amplitude; for the penguin amplitude, δ Pf depends on ρ. However, fromthe experimental values of the direct CP -violation for π − K + , π − π + , it is near the upper bound.Finally we note that π + D − ( π − D + ) , π − K + , π − π + decays are s -wave decay whereas B → ρ + π − ( ρ − π + ) decays are p − wave decays. For p − wave, the decay amplitude f = 116 πs Z − s M ( s, t )(1 + 2 ts ) dt = 116 πs iC P (cid:20) b + 2 b s (cid:21) ( s/s )+ 2 γ π i (cid:20) s/s ) − iπ − s s/s ) − iπ ] ( s/s ) − / (cid:21) ≈ πs iC P b ( s/s ) + 2 γ π i s/s ) − iπ ( s/s ) − / + O (cid:18) s (cid:19) to be compared with Eq.(32). Now for the B → ρπ decay, only longitudinal polarization of ρ is effectively involved. Since the longitudinal ρ -meson emulates a pseudoscalar meson and if weassume same couplings as for pions, we conclude that the final state phase for ρπ should be of theorder 30 ◦ ; in any case it should not be greater than 30 ◦ . The upper bound δ f ≤ can be usedto select the several possible solutions in Table-2 [Section-4] obtained from the analysis of weakdecays B → ρ + π − ( ρ − π + ). In this section, we discuss the experimental tests to verify the equality (implied by C-invarianceof S-matrix) of phase shifts δ f and δ ¯ f for the weak decays of B mesons mentioned in section 1.8t is convenient to write the time-dependent decay rates in the form [13, 6] (cid:2) Γ f ( t ) − ¯Γ ¯ f ( t ) (cid:3) + (cid:2) Γ ¯ f − ¯Γ f ( t ) (cid:3) = e − Γ t n cos ∆ mt h(cid:16) | A f | − (cid:12)(cid:12) ¯ A ¯ f (cid:12)(cid:12) (cid:17) + (cid:16)(cid:12)(cid:12) A ¯ f (cid:12)(cid:12) − (cid:12)(cid:12) ¯ A f (cid:12)(cid:12) (cid:17)i +2 sin ∆ mt h Im (cid:0) e iφ M A ∗ f ¯ A f (cid:1) + Im (cid:16) e iφ M A ∗ ¯ f ¯ A ¯ f (cid:17)io (41) (cid:2) Γ f ( t ) + ¯Γ ¯ f ( t ) (cid:3) − (cid:2) Γ ¯ f ( t ) + ¯Γ f ( t ) (cid:3) = e − Γ t n cos ∆ mt h(cid:16) | A f | + (cid:12)(cid:12) ¯ A ¯ f (cid:12)(cid:12) (cid:17) − (cid:16)(cid:12)(cid:12) A ¯ f (cid:12)(cid:12) + (cid:12)(cid:12) ¯ A f (cid:12)(cid:12) (cid:17)i +2 sin ∆ mt h Im (cid:0) e iφ M A ∗ f ¯ A f (cid:1) − Im (cid:16) e iφ M A ∗ ¯ f ¯ A ¯ f (cid:17)io (42) Case (i):
Eqs. (41) and (42) give A ( t ) ≡ [Γ f ( t ) − ¯Γ ¯ f ( t )] + [Γ ¯ f ( t ) − ¯Γ f ( t )][Γ f ( t ) + ¯Γ ¯ f ( t )] + [Γ ¯ f ( t ) + ¯Γ f ]= 2 (cid:12)(cid:12) F f (cid:12)(cid:12)(cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12)(cid:12)(cid:12) F f (cid:12)(cid:12) + (cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12) sin ∆ mt sin (cid:0) φ M − φ − φ ′ (cid:1) cos (cid:0) δ f − δ ′ ¯ f (cid:1) (43) F ( t ) ≡ (cid:2) Γ f ( t ) + ¯Γ ¯ f (cid:3) − (cid:2) Γ ¯ f ( t ) + ¯Γ f (cid:3)(cid:2) Γ f ( t ) + ¯Γ ¯ f (cid:3) + (cid:2) Γ ¯ f ( t ) + ¯Γ f (cid:3) = (cid:12)(cid:12) F f (cid:12)(cid:12) − (cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12) (cid:12)(cid:12) F f (cid:12)(cid:12) + (cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12) cos ∆ mt − (cid:12)(cid:12) F f (cid:12)(cid:12)(cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12)(cid:12)(cid:12) F f (cid:12)(cid:12) + (cid:12)(cid:12) F ′ ¯ f (cid:12)(cid:12) sin ∆ mt cos (cid:16) φ M − φ − φ ′ (cid:17) sin (cid:0) δ f − δ ′ ¯ f (cid:1) (44)The effective Lagrangians L W and L ′ W are given by ( q = d, s ) L W = V cb V ∗ uq [¯ qγ µ (1 − γ ) u ][¯ cγ µ (1 − γ ) b ] L ′ W = V ub V ∗ cq [¯ qγ µ (1 − γ ) c ][¯ uγ µ (1 − γ ) b ] (45)(46)Hence for these decays φ = 0 , φ ′ = γ and φ M = ( − β, for B − β s , for B s (47) A f = h D − π + |L W | B i = F f ′ A ¯ f = h D + π − |L W ′ | B i = e iγ ′ F ¯ f A f s = h K + D − s |L W | B s i = F f s ′ A ¯ f s = h K − D + s |L W ′ | B s i = e iγ ′ F ¯ f s (48)9hus, we get from Eqs. (43) − (48) for B decays, A ( t ) = − r D r D sin ∆ m B t sin (2 β + γ ) cos (cid:16) δ f − δ ′ ¯ f (cid:17) F ( t ) = 1 − r D r D cos ∆ m B t − r D r D sin ∆ m B t cos (2 β + γ ) sin (cid:16) δ f − δ ′ ¯ f (cid:17) (49) A = − r D r D sin(2 β + γ ) (∆ m B / Γ)1 + (∆ m B / Γ) cos( δ f − δ ′ ¯ f ) (50)where r D = λ R b | F ′ ¯ f || F f | (51)For the decays, ¯ B s (cid:0) B s (cid:1) → D + s K − (cid:0) D − s K + (cid:1) ¯ B s (cid:0) B s (cid:1) → D − s K + (cid:0) D + s K − (cid:1) we get, A s ( t ) = − r D s r D s sin ∆ m B s t sin (2 β s + γ ) cos (cid:16) δ f s − δ ′ ¯ f s (cid:17) F s ( t ) = 1 − r D s r D s cos ∆ m B s t − r D s r D s sin ∆ m B s t cos (2 β s + γ ) sin (cid:16) δ f s − δ ′ ¯ f s (cid:17) (52)where r D s = R b | F ′ ¯ f s || F f s | (53)We note that for time integrated CP -asymmetry, A s ≡ R ∞ (cid:2) Γ fs ( t ) − ¯Γ fs ( t ) (cid:3) dt R ∞ (cid:2) Γ fs ( t ) + ¯Γ fs ( t ) (cid:3) dt = − r D s r r D s sin (2 β s + γ ) ∆ m B s / Γ s m B s / Γ s ) cos( δ f s − δ ′ ¯ f s ) (54)The experimental results for the B decays are as follows [12] D − π + D ∗− π + D − ρ + S − + S + : − . ± . − . ± . − . ± . ± . S − − S + : − . ± . − . ± . − . ± . ± .
018 (55)where S − + S + ≡ − r D r D sin(2 β + γ ) cos( δ f − δ ′ ¯ f ) S − − S + ≡ − r D r D cos(2 β + γ ) sin( δ f − δ ′ ¯ f ) (56)10or B s → D ∗− s K + , D − s K + , D − s K ∗ + , replace r D → r s , β → β s , δ f → δ f s , δ ′ ¯ f → δ ′ ¯ f s in Eq. (56).Since for B s , in the standard model, with three generations, gives β s = 0, so we have for theCP-asymmetries sin γ or cos γ instead of sin(2 β + γ ), cos(2 β + γ ). Hence B s -decays are moresuitable for testing the equality of phase shifts δ f s and δ ′ ¯ f s as for this case neither r s nor cos γ issuppressed as compared to the corresponding quantities for B . To conclude, for B q decays, theequality of phases δ f and δ ′ ¯ f for B d gives − S − + S + r D sin(2 β + γ ) − S − − S + B s decays, we get − S − + S + r D s r D s sin(2 β s + γ ) − S − − S + B s → D − s K + , D + s K − described by the tree diagrams, we have thecolor suppressed decays B → ¯ D K , D K . For these decays, − S − + S + r DK r DK sin(2 β + γ ) cos( δ ¯ D K s − δ ′ D ¯ K s ) − S − − S + r DK r DK cos(2 β + γ ) sin( δ ¯ D K s − δ ′ D ¯ K s ) r DK = R b (cid:12)(cid:12) C ′ D K s (cid:12)(cid:12)(cid:12)(cid:12) C ¯ D K s (cid:12)(cid:12) and the corresponding expression for B s → ¯ D φ, D φ . For the color suppressed decays B → ¯ D π , D π , we get similar expression as for B → D − π + , D + π − , with r D ≡ r D − π − , δ D − π + , δ ′ D − π + replaced by r D π , δ ¯ D π , δ ′ D π To determine the parameter r D or r D s , we assume factorization for the tree amplitude [7].Factorization gives for the decays ¯ B → D + π − , D ∗ + π − , D + ρ − , D + a − : | ¯ F ¯ f | = | ¯ T ¯ f | = G [ f π ( m B − m D ) f B − D ( m π ) , f π m B | ~p | A B − D ∗ ( m π ) , f ρ m B | ~p | f B − D + ( m ρ ) , f a m B | ~p | f B − D + ( a )] (59) | ¯ F ′ f | = | ¯ T ′ f | = G ′ [ f D ( m B − m π ) f B − π ( m D ) , f D ∗ m B | ~p | f B − π ( m D ∗ ) , f D m B | ~p | A B − ρ ( m D ) , f D m B | ~p | A B − a ( m B )] (60) G = G F √ | V ud || V cb | a , G ′ = G F √ | V cd || V ub | (61)The decay widths for the above channels are given in the table 1where we have used a | V ud | ≈ , f π = 131 M eV, f ρ = 209 M eV, f a = 229 M eV − MeV ×| V cb | ) Form Factor Form Factors h ( w ( ∗ ) )¯ B → D + π − (2 . | f B − D ( m π ) | . ± .
05 0 . ± . B → D ∗ + π − (2 . | A B − D ∗ ( m π ) | . ± .
04 0 . ± . B → D + ρ − (5 . | f B − D + ( m ρ ) | . ± .
11 0 . ± . B → D + a − (5 . | f B − D + ( m a ) | . ± .
31 0 . ± . | V cb | = (38 . ± . × − (62)we obtain the corresponding form factors given in Table 1.In terms of variables [14, 15]: ω = v · v ′ , v = v ′ = 1 , t = q = m B + m D ( ∗ ) − m B m D ( ∗ ) ω (63)the form factors can be put in the following form f B − D + ( t ) = m B + m D √ m B m D h + ( ω ) , f B − D ( t ) = √ m B m D m B + m D (1 + ω ) h ( ω ) A B − D ∗ ( t ) = m B + m D ∗ √ m B m D ∗ (1 + ω ) h A ( ω ) , A B − D ∗ ( t ) = m B + m D ∗ √ m B m D ∗ h A ( ω ) A B − D ∗ ( t ) = √ m B m D ∗ m B + m D ∗ (1 + ω ) h A ( ω ) (64)Heavy Quark Effective Theory (HQET) gives [14, 15]: h + ( ω ) = h ( ω ) = h A ( ω ) = h A ( ω ) = h A ( ω ) = ζ ( ω )where ζ ( ω ) is the form factor, with normalization ζ (1) = 1. For t = m π , m ρ , m a ω ( ∗ ) = 1 . . , . , .
508 (65)In reference [16], the value quoted for h A ( ω ∗ max ) is | h A ( ω ∗ max ) | = 0 . ± .
03 (66)Since ω ∗ max = 1 . | h A ( ω ∗ max ) | obtained in Table 1 is in remarkable agreementwith the value given in Eq. (66) showing that factorization assumption for B → πD ( ∗ ) decays isexperimentally on solid footing and is in agreement with HQET.From Eqs. (59) and (60), we obtain r D = λ R b | ¯ T ′ f || ¯ T ¯ f | = λ R b " f D ( m B − m π ) f B − π ( m D ) f π ( m B − m D ) f B − D ( m π ) , f D ∗ f B − π + ( m D ∗ ) f π A B − D ( m π ) , f D A B − ρ ( m D ) f ρ f B − D + ( m ρ ) (67)12here | V ub || V cd || V cb || V ud | = λ R b ≈ (0 . (0 . ≈ .
021 (68)To determine r D , we need information for the form factors f B − π ( m D ) , f B − π + ( m D ) , A B − ρ ( m D ).For these form factors, we use the following values [17, 18]: A B − ρ (0) = 0 . ± . , A B − ρ ( m D ) = 0 . ± . f B − π + (0) = f B − π (0) = 0 . ± . , f B − π + ( m D ∗ ) = 0 . ± . , f B − D ( m D ) = 0 . ± . r D ( ∗ ) = [0 . ± . , . ± . , . ± . r ∗ D gives − (cid:18) S + + S − (cid:19) D ∗ π = 2(0 . ± . β + γ ) (70)The experimental value of the CP asymmetry for B → D ∗ π decay has the least error. Hencewe obtain the following bounds sin(2 β + γ ) > .
69 (71)44 ◦ ≤ (2 β + γ ) ≤ ◦ (72)or 90 ◦ ≤ (2 β + γ ) ≤ ◦ (73)Selecting the second solution, and using 2 β ≈ ◦ , we get γ = (70 ± ◦ (74)Further, we note that the factorization for the decay ¯ B → D ∗− s π + gives¯ T = | V ub || V cs | f D ∗ s m B | ~p | f B − π + ( m D ∗ s ) (75)Using the experimental branching ratio for this decay, we get (cid:18) f D ∗ s f π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f B − π + ( m D ∗ s ) f B − π + (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 7 . ± . f B − π + (0) f B − π + ( m D ∗ s ) = 0 . ± .
09 (77)we get f D ∗ s = 279 ± M eV (78)Similar analysis for ¯ B → D − s π + gives (cid:18) f D s f π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f B − π ( m D s ) f B − π (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 . ± .
64 (79)13n using f B − π (0) f B − π ( m D s ) = 0 . ± .
05 (80)we get f D s = 201 ± M eV (81)Finally from the experimental branching ratio for the decay ¯ B s → D + s π − , we obtain f B s − D s (0) = 0 . ± .
18 (82) h (1 . . ± .
16 (83)To end this section, we discuss the decays ¯ B s → D + s K − , D ∗ + s K − for which no experimentaldata are available. However, using factorization, we getΓ( ¯ B s → D + s K − ) = (1 . × − ) | V cb f B s − D s ( m K ) | M eV (84)Γ( ¯ B s → D ∗ + s K − ) = (1 . × − ) | V cb A B s − D ∗ s ( m K ) | M eV (85)SU(3) gives | V cb f B s − D s ( m K ) | ≈ | V cb || f B − D ( m π ) | = (0 . ± . × − | V cb A B s − D ∗ s ( m K ) | ≈ | V cb || A B − D ∗ ( m π ) | = (0 . ± . × − (86)From the above equations, we get the following branching ratiosΓ( ¯ B s → D ( ∗ )+ s K − )Γ ¯ B s = (1 . ± . × − [(1 . ± . × − ] (87)For ¯ B s → D ∗ + s K − r D s = R b " f D ∗ s f B s − K + ( m D ∗ s ) f K A B s − D ∗ s ( m K ) (88)Hence we get − ( S + + S − D ∗ s K = (0 . ± .
08) sin(2 β s + γ )= (0 . ± .
08) sin γ (89)where we have used R b = 0 . , f D s f K = f D ∗ s f K = 1 . ± . , f B s − K + ( m D ∗ s ) = 0 . ± . A B s − D ∗ s ( m K ) = A B s − D ∗ s (0) = m B s + m D ∗ s √ m B s m D ∗ s [ h ( ω ∗ s = 1 . . ± . . ± .
03 (90)14
CP Asymmetries for A f = A ¯ f We now discuss the decays listed in case (ii) where A f = A ¯ f . Subtracting and adding Eqs. (42)and (41), we get, Γ f ( t ) − ¯Γ f ( t )Γ f ( t ) + ¯Γ f ( t ) = C f cos ∆ mt + S f sin ∆ mt =( C − ∆ C ) cos ∆ mt + ( S − ∆ S ) sin ∆ mt (91)Γ ¯ f ( t ) − ¯Γ ¯ f ( t )Γ ¯ f ( t ) + ¯Γ ¯ f ( t ) = C ¯ f cos ∆ mt + S ¯ f sin ∆ mt =( C + ∆ C ) cos ∆ mt + ( S + ∆ S ) sin ∆ mt (92)where C ¯ f,f = ( C ± ∆ C )= (cid:12)(cid:12) A ¯ f,f (cid:12)(cid:12) − (cid:12)(cid:12) ¯ A ¯ f,f (cid:12)(cid:12) (cid:12)(cid:12) A ¯ f,f (cid:12)(cid:12) + (cid:12)(cid:12) ¯ A ¯ f,f (cid:12)(cid:12) = Γ ¯ f,f − ¯Γ ¯ f,f Γ ¯ f,f + ¯Γ ¯ f,f = R ¯ f,f (1 − A ¯ f,fCP ) − R ¯ f,f (1 + A ¯ f,fCP )Γ(1 ± A CP ) (93) S ¯ f,f = ( S ± ∆ S ) (94)= 2Im[ e iφ M A ∗ ¯ f,f ¯ A ¯ f,f ]Γ ¯ f,f + ¯Γ ¯ f,f (95) A ¯ fCP = ¯Γ f − Γ ¯ f Γ ¯ f + ¯Γ f A fCP = ¯Γ ¯ f − Γ f Γ f + ¯Γ ¯ f (96) A CP = (Γ ¯ f + ¯Γ ¯ f ) − ( ¯Γ f + Γ f )(Γ ¯ f − ¯Γ ¯ f ) − ( ¯Γ f + Γ f ) (97)= R f A fCP − R ¯ f A ¯ fCP Γ (98)where R f = 12 (Γ f + ¯Γ ¯ f ) , R ¯ f = 12 (Γ ¯ f + ¯Γ f )Γ = R f + R ¯ f (99)15he following relations are also useful which can be easily derived from above equations R ¯ f,f R f + R ¯ f = 12 [(1 ± ∆ C ) ± A CP C ] (100) R ¯ f − R f R f + R ¯ f = [∆ C + A CP C ] (101) R ¯ f A ¯ fCP + R f A fCP R f + R ¯ f = [ C + A CP ∆ C ] (102)For these decays, the decay amplitudes can be written in terms of tree amplitude e iφ T T f andthe penguin amplitude e iφ P P f : A f = e iφ T e iδ Tf (cid:12)(cid:12) T f (cid:12)(cid:12) [1 + r f e i ( φ P − φ T ) e iδ f ] A ¯ f = e iφ T e iδ T ¯ f (cid:12)(cid:12) T ¯ f (cid:12)(cid:12) [1 + r ¯ f e i ( φ P − φ T ) e iδ ¯ f ] (103)where r f, ¯ f = (cid:12)(cid:12) P f, ¯ f (cid:12)(cid:12)(cid:12)(cid:12) T f, ¯ f (cid:12)(cid:12) , δ f, ¯ f = δ Pf, ¯ f − δ Tf, ¯ f .¯ A ¯ f = e − iφ T e iδ Tf (cid:12)(cid:12) T f (cid:12)(cid:12) [1 + r f e − i ( φ P − φ T ) e iδ f ]¯ A f = e − iφ T e iδ T ¯ f (cid:12)(cid:12) T ¯ f (cid:12)(cid:12) [1 + r ¯ f e − i ( φ P − φ T ) e iδ ¯ f ] (104)For B → ρ − π + : A f ; B → ρ + π − : A ¯ f ; φ T = γ, φ P = − β (105)For B → D ∗− D + : A Df ; B → D ∗ + D − : A D ¯ f ; φ T = 0 , φ P = − β (106)Hence for B → ρ − π + , B → ρ + π − , we have A f = (cid:12)(cid:12) T f (cid:12)(cid:12) e + iγ e iδ Tf [1 − r f e i ( α + δ f ) ] A ¯ f = (cid:12)(cid:12) T ¯ f (cid:12)(cid:12) e + iγ e iδ T ¯ f [1 − r ¯ f e i ( α + δ ¯ f ) ] (107)where r f, ¯ f = | V tb || V td || V ub || V ud | (cid:12)(cid:12) P f, ¯ f (cid:12)(cid:12)(cid:12)(cid:12) T f, ¯ f (cid:12)(cid:12) = R t R b (cid:12)(cid:12) P f, ¯ f (cid:12)(cid:12)(cid:12)(cid:12) T f, ¯ f (cid:12)(cid:12) (108)and for B → D ∗− D + , B → D ∗ + D − , we have A Df = (cid:12)(cid:12) T Df (cid:12)(cid:12) e iδ TDf [1 − r Df e i ( − β + δ Df ) ] A D ¯ f = (cid:12)(cid:12) T D ¯ f (cid:12)(cid:12) e iδ TD ¯ f [1 − r D ¯ f e i ( − β + δ D ¯ f ) ] (109)where r f, ¯ f = R t (cid:12)(cid:12) P Df, ¯ f (cid:12)(cid:12)(cid:12)(cid:12) T Df, ¯ f (cid:12)(cid:12) We now confine ourselves to B ( ¯ B ) → ρ − π + , ρ + π − ( ρ + π − , ρ − , π + ) decays only [19, 20]. Theexperimental results for these decays are [12] asΓ = R f + R ¯ f = (22 . ± . × − (110) A fCP = − . ± . , A ¯ fCP = 0 . ± .
12 (111) C = 0 . ± . , ∆ C = 0 . ± .
08 (112) S = 0 . ± . , ∆ S = − . ± .
10 (113)16ith the above values, it is hard to draw any reliable conclusion. Neglecting the term A CP C in Eqs. (100) and (101), we get R ¯ f,f = 12 Γ(1 ± ∆ C ) (114) R ¯ f − R f = ∆ C Using the above value for ∆ C , we obtain R ¯ f = (15 . ± . × − R f = (7 . ± . × − (115)We analyze these decays by assuming factorization for the tree graphs [10, 11]. This assumptiongives T ¯ f = ¯ T f ∼ m B f ρ | ~p | f + ( m ρ ) (116) T f = ¯ T ¯ f ∼ m B f π | ~p | A ( m π ) (117)Using f + ( m ρ ) ≈ . ± .
04 and A ( m π ) ≈ A (0) = 0 . ± .
03 and | V ub | = (3 . ± . × − ,we get the following values for the tree amplitude contribution to the branching ratiosΓ tree¯ f = (15 . ± . × − ≡ | T ¯ f | (118)Γ tree f = (7 . ± . × − ≡ | T f | (119) t = T f T ¯ f = f π A ( m π ) f ρ f + ( m ρ ) = 0 . ± .
12 (120)Now B ¯ f = R ¯ f | T ¯ f | = 1 − r ¯ f cos α cos δ ¯ f + r f (121) B f = R f | T f | = 1 − r f cos α cos δ f + r f (122)Hence from Eqs. (115) and (119), we get B ¯ f = 1 . ± . B f = 0 . ± .
11 (123)In order to take into account the contribution of penguin diagram, we introduce the angles α f, ¯ feff [21], defined as follows e iβ A f, ¯ f = | A f, ¯ f | e − iα f, ¯ feff e − iβ ¯ A ¯ f,f = | ¯ A ¯ f,f | e iα f, ¯ feff (124)With this definition, we separate out tree and penguin contributions: e iβ A f, ¯ f − e − iβ ¯ A ¯ f,f = | A f, ¯ f | e − iα f, ¯ f − | ¯ A ¯ f,f | e iα f, ¯ f = 2 iT f, ¯ f sin α (125) e i ( α + β ) A f, ¯ f − e − i ( α + β ) ¯ A ¯ f,f = | A f, ¯ f | e − i ( α f, ¯ feff − α ) − | ¯ A ¯ f,f | e i ( α f, ¯ feff − α ) = (2 iT f, ¯ f sin α ) r f, ¯ f e iδ f, ¯ f = 2 iP f, ¯ f sin α (126)17rom Eq. (125), we get2 | T f, ¯ f | R f, ¯ f sin α ≡ αB f, ¯ f = 1 − q − A f, ¯ f CP cos 2 α f, ¯ feff (127)sin 2 δ Tf, ¯ f = − A f, ¯ fCP sin 2 α f, ¯ feff − q − A f, ¯ f CP cos 2 α f, ¯ feff (128)cos 2 δ Tf, ¯ f = q − A f, ¯ f CP − cos 2 α f, ¯ feff − q − A f, ¯ f CP cos 2 α f, ¯ feff (129)From Eqs. (125) and (126), we get r f, ¯ f = 1 − q − A f, ¯ f CP cos(2 α f, ¯ feff − α )1 − q − A f, ¯ f CP cos 2 α f, ¯ feff (130) r f, ¯ f cos δ f, ¯ f = cos α − q − A f, ¯ f CP cos(2 α f, ¯ feff − α )1 − q − A f, ¯ f CP cos 2 α f, ¯ feff (131) r f, ¯ f sin δ f, ¯ f = − A f, ¯ fCP sin α − q − A f, ¯ f CP cos 2 α f, ¯ feff (132)Now factorization implies [22] δ Tf = 0 = δ T ¯ f (133)Thus in the limit δ Tf →
0, we get for Eq. (129)cos 2 α f, ¯ feff = − , α f, ¯ feff = 90 ◦ (134) r f, ¯ f cos δ f, ¯ f = cos α (135) r f, ¯ f sin δ f, ¯ f = − A f, ¯ fCP / sin α q − A f, ¯ f CP (136) r f, ¯ f = 1 + q − A f, ¯ f CP cos 2 α q − A f, ¯ f CP (137) ≈ cos α + 14 A f, ¯ f CP sin α (138)The solution of Eq. (135) is graphically shown in Fig. 1 for α in the range 80 ◦ ≤ α < ◦ for r f, ¯ f = 0 . , , . , . , .
30. From the figure, the final state phases δ f, ¯ f for various valuesof r f, ¯ f can be read for each value of α in the above range. Few examples are given in Table 2For α > ◦ , change α → π − α , δ f → π − δ f . For example, for α = 103 ◦ r f = 0 . , δ f = 154 ◦ , A fCP ≈ − . r f = 0 . , δ f = 138 ◦ , A fCP ≈ − . r f δ f A fCP ≈ − r f sin δ f sin α ◦ ◦ -0.190.25 46 ◦ -0.3682 ◦ ◦ -0.110.20 46 ◦ -0.2885 ◦ ◦ -0.100.15 54 ◦ -0.2486 ◦ ◦ -0.140.15 62 ◦ -0.2688 ◦ ◦ -0.19Table 2:These examples have been selected keeping in view that final state phases δ f, ¯ f are not toolarge. For A f, ¯ fCP , we have used Eq. (136) neglecting the second order term. An attractive optionis A fCP = A ¯ fCP for each value of α ; although A fCP = A ¯ fCP is also a possibility. A fCP = A ¯ fCP implies r f = r ¯ f , δ f = δ ¯ f .Neglecting terms of order r f, ¯ f , we have A CP ≈ α ( r ¯ f sin δ ¯ f − t r f sin δ f )1 + t = − A ¯ fCP − t A fCP t (139) C ≈ − t (1 + t ) ( A ¯ fCP + A fCP ) (140)∆ C ≈ − t t − t cos α (1 + t ) ( r ¯ f cos δ ¯ f − r f cos δ f ) (141)Now the second term in Eq. (141) vanishes and using the value of t given in Eq. (120), we get∆ C ≈ . ± .
06 (142)Assuming A ¯ fCP = A fCP , we obtain A CP = − − t t A ¯ fCP = (0 . ± . − A ¯ fCP ) (143) C ≈ − t (1 + t ) A ¯ fCP ≈ − (0 . ± . A ¯ fCP (144)Finally the CP asymmetries in the limit δ Tf, ¯ f → S ¯ f = S + ∆ S = 2Im[ e iφ M A ¯ f ∗ ¯ A ¯ f ]Γ(1 + A CP )= q − C f sin(2 α ¯ feff + δ )= − q − C f cos δ (145)19 f = S − ∆ S = 2Im[ e iφ M A ∗ f ¯ A f ]Γ(1 − A CP )= q − C f sin(2 α feff − δ )= q − C f cos δ (146)The phase δ is defined as ¯ A ¯ f = | ¯ A ¯ f | | ¯ A f | ¯ A f e iδ (147)To conclude:The final state strong phases essentially arise in terms of S -matrix, which converts an “ in ” stateinto an “ out ” state. The isospin, C -invariance of hadronic dynamics and the unitarity togetherwith two particle scattering amplitudes in terms of Regge trajectories are used to get informationabout these phases. In particular two body unitarity is used to calculate the final state phase δ C generated by rescattering for the color suppressed decays in terms of the color favored decays.In the inclusive version of unitarity, the information obtained for s -wave scattering from Reggetrajectories is used to derive the bounds on the final state phases. In particular, the value obtainedfor the final state phases δ + − = δ P ≈ ◦ − ◦ and δ = δ C + δ P ≈ ◦ , ◦ is found to becompatible with the experimental values for direct CP asymmetries A CP ( B → π − K + , π K ).For B → D ( ∗ ) − π + ( D ( ∗ )+ π − ), B s → D ( ∗ ) − s K + ( D ( ∗ )+ s K − ) decays described by two independentsingle amplitudes A f , A ′ ¯ f and A f s , A ′ ¯ f s with different weak phases viz. 0 and γ , equality of phases δ f = δ ′ ¯ f implies, the time dependent CP asymmetries − (cid:18) S + + S − (cid:19) = 2 r D ( ∗ )( s ) r D ( ∗ )( s ) sin(2 β ( s ) + γ ) (148) S + − S − A f can be determined in term of the form factors f B − D ( m π ) and A B − D ∗ ( m π ). The parameter r D ( ∗ ) can be expressed in terms of the ratios of the form factors f D f B − π ( m D )/ f π f B − D ( m π ) and f D ∗ f B − π + ( m D ∗ )/ f π A B − D ∗ ( m π ). From the experimental branchingratios, we have obtained the form factors f B − D ( m π ) and A B − D ∗ ( m π ) which are in excellent agree-ment with the prediction of HQET. We have also determined r D ∗ . For r D ∗ we get the value r D ∗ = 0 . ± . S + + S − for B → D ∗− π + decay: sin(2 β + γ ) > . B s → D ∗− s K + ( D ∗ + s K − ) decays, we predict − (cid:18) S + + S − (cid:19) = (0 . ± .
08) sin(2 β + γ )= (0 . ± .
08) sin γ in the standard model.In section-4, the decays B → ρ + π − ( ρ − π + ) for which decay amplitudes A ¯ f and A f are givenin terms of tree and penguin diagrams are discussed. We have analyzed these decays assuming20actorization for the tree graph. Factorization implies δ Tf = δ T ¯ f . In the limit δ Tf, ¯ f →
0, we haveshown that r f, ¯ f cos δ f, ¯ f = cos αr f, ¯ f ≈ cos α + A f, ¯ f CP sin α The first equation has been solved graphically, from which the final state phases δ f, ¯ f corre-sponding to various values of r f, ¯ f can be found for a particular value of α . The upper bound δ f, ¯ f ≤ obtained in Section-2, using unitarity and strong interaction dynamics based on Reggepole phenomonalogy can be used to select the solutions given in Table-2. Neglecting the terms oforder r f, ¯ f , we get using factorization ∆ C = 0 . ± . δ Tf, ¯ f →
0, we get S ¯ f S f = S + ∆ SS − ∆ S = − q − C f q − C f With the present experimental data, it is hard to draw any definite conclusion.
Acknowledgement
The author acknowledges a research grant provided by the Higher Edu-cation Commission of Pakistan as a Distinguished National Professor.
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Figure Caption:
Plot of equation r f cos δ ( f ) = cos α for different values of r. For 80 o ≤ α ≤ o . Where solidcurve, dashed curve, dashed doted curve, dashed bouble doted and double dashed doted curve arecorresponding to r = 0 . , r = 0 . , r = 0 . , r = 0 .
25 and r = 0 .3 respectively.22