Production of B c or B ¯ c meson and its excited states via t ¯ -quark or t -quark decays
aa r X i v : . [ h e p - ph ] N ov Production of B c or ¯ B c meson and its excited states via ¯ t -quark or t -quark decays Chao-Hsi Chang , , ∗ , Jian-Xiong Wang † and Xing-Gang Wu ‡ CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China. § Institute of Theoretical Physics, Chinese Academy of Sciences,P.O.Box 2735, Beijing 100080, P.R. China Institute of High Energy Physics, P.O.Box 918(4), Beijing 100049, China Department of Physics, Chongqing University, Chongqing 400044, P.R. China (Dated: October 28, 2018)
Abstract
The production of ( b ¯ c )-quarkonium ( ¯ B c meson and its excited states) or ( c ¯ b )-quarkonium ( B c meson and its excited states) via top quark t or top anti-quark ¯ t decays, t → ( b ¯ c ) + c + W + or¯ t → ( c ¯ b ) + ¯ c + W − , respectively is studied within the framework of NRQCD. In addition to theproduction of the two color-singlet S -wave states | ( b ¯ c )( S ) i or | ( c ¯ b )( S ) i and | ( b ¯ c )( S ) i or | ( c ¯ b )( S ) i , the production of the P -wave excited ( b ¯ c ) or ( c ¯ b ) states, i.e. the four color-singlet P -wave states | ( b ¯ c )( P ) i or | ( c ¯ b )( P ) i , and ( b ¯ c )( P J ) i or ( c ¯ b )( P J ) i (with J = (1 , , P -waveexcited states the contributions from the two color-octet components | ( b ¯ c )( S ) i or | ( c ¯ b )( S ) i and | ( b ¯ c )( S ) i or | ( c ¯ b )( S ) i are also taken into account. We quantitatively discuss thepossibility and the advantages in experimental studies of B c or ¯ B c meson and its excited states viathe indirect production at LHC in high luminosity runs and at LHC possible upgraded versionssuch as SLHC, DLHC, TLHC etc. in future. PACS numbers: ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] § Not correspondence address. . INTRODUCTION Since B c meson was observed by CDF in 1998 [1], the interests in studying of the meson B c are increasing and a new stage of the studies has been started. As pointed by the authorsof Refs.[2, 3], the product cross-section at Large Hadron Collider (LHC), CERN, is muchgreater than that at Tevatron, Fermilab, therefore, more precise experimental studies of B c meson are expected at the forthcoming of LHC. Indeed, some progress at LHC in theexperimental study of B c physics can be achieved, especially at the beginning stage of LHC(LHC just runs at ’low’ luminosity L = 10 ∼ cm − s − ). However, when LHC runsat higher luminosity, such as the design luminosity L = 10 cm − s − later on, due to therequests coming from the main purposes of LHC (such as searching for Higgs particle andSUSY partners etc) and the restricts from the abilities to record the events for the detector(e.g. the detector cannot record too frequently events etc), the condition for triggering theevents occurring in collisions has to be set such that too many B c events via the directhadronic production according to the theoretical estimate of ‘direct’ B c -production[2, 3, 4,5, 6] are cut off. As a result, the direct hadronic production of B c cannot be expected tomake much progress in B c -meson study in the high luminosity stage of LHC runs. Baringthe situation pointed out here and the possible upgrade for LHC (SLHC, DLHC and TLHCetc[7]) in mind, the possibility to study B c meson experimentally via indirect productionof B c meson, namely, via producing a huge amount of top antiquarks ¯ t and their decays,is worthwhile to think seriously about. It is because that at LHC no matter how high theluminosity, the produced top quark(s) shall never be cut off by the trigger condition setdown for any experimental purposes, and the frequency of the t -quark production can bestood up for the detector always. Furthermore, the mechanisms for producing the doublyheavy mesons, such as η c , η b , B c , · · · and J/ψ , Υ, B ∗ c , · · · , are interesting too. The indirectproduction of B c or ¯ B c (or B − c ), including its excited states, via ¯ t -decays or t -decays mayoffer some knowledge on the mechanisms, therefore, this paper is devoted to study theindirect production of B c or ¯ B c meson via ¯ t -decays or t -decays. Without confusing andfor simplifying the statements, later on we will not distinguish B c and ¯ B c unless necessary,and all results for B c and ¯ B c obtained in the paper are symmetric in the interchange fromparticle to anti-particle.The doubly heavy meson production via top quark decays is special interesting from the2oint view of precise testing perturbative quantum chromodynamics (pQCD)[8]. The meson¯ B c is the ground state of the heavy-flavored binding system ( b ¯ c ), and it is unique ‘doublyheavy-flavored’ anti-meson in Standard Model and is stable for strong and electromagneticinteractions. Although in literature the ‘direct’ hadornic production of ¯ B c meson has beenthoroughly studied, e.g. see Refs. [2, 3, 4, 5, 6] (references therein) and CDF discoveredthe meson which just come from the ‘direct’ production, as a compensation to understandthe production mechanisms, it is quite interesting that to study the production of ¯ B c meson‘indirectly’ through t -quark decays, especially, considering the fact that numerous t -quarksmay be produced at LHC. The theoretical studies of the direct production[2, 3, 4, 5, 6] isbased on NRQCD [9], so now we study the indirect production based on NRQCD too.In the framework of effective theory of NRQCD, a doubly heavy meson is considered asan expansion of a series Fock states. The relative importance among the infinite ingredientsis accounted by the velocity scaling rule. Namely the physical state of ¯ B c , ¯ B ∗ c , h ¯ B c and χ J ¯ B c can be decomposed into a series of Fock states as follows: | ¯ B c i = O ( v ) | ( b ¯ c ) ( S ) i + O ( v ) | ( b ¯ c ) ( P ) g i + · · ·| ¯ B ∗ c i = O ( v ) | ( b ¯ c ) ( S ) i + O ( v ) | ( b ¯ c ) ( P J ) g i + · · · (1)and | h ¯ B c i = O ( v ) | ( b ¯ c ) ( P ) i + O ( v ) | ( b ¯ c ) ( S ) g i + · · ·| χ J ¯ B c i = O ( v ) | ( b ¯ c ) ( P J ) i + O ( v ) | ( b ¯ c ) ( S ) g i + · · · , (2)here v is the relative velocity between the components. The thickened subscripts of the( b ¯ c ) denote for color indices, for color singlet and for color-octet; the relevant angularmomentum quantum numbers are shown in the parentheses accordingly. According to thevelocity scaling rule of NRQCD, the probability of each Fock state in the expansion isproportional to a definite power in v as indicated as that in Eqs.(1,2). Since the valueof v is around 0 . ∼ . P -wave production might be comparable with thosefrom the color-singlet components. So we shall consider the two color-octet components: | ( b ¯ c ) ( S ) g i and | ( b ¯ c ) ( S ) g i , in addition to those color-singlet components in the mesons¯ B c , ¯ B ∗ c , h ¯ B c and χ J ¯ B c . 3he calculations of the process are very complicated and lengthy by using the conventionaltrace techniques to calculate the amplitude square due to the two massive particles andbound state effects. To shorten the calculations and to make the results more compact, weadopt the method used in Ref.[10] to do the calculations. For convenience, we will call itas the ‘new trace amplitude approach’. Under this approach, we first arrange the wholeamplitude into several orthogonal sub-amplitudes M ss ′ according to the spins of the t -quark( s ′ ) and c -quark ( s ), and then do the trace of the Dirac γ matrix strings at the amplitudelevel, which result in explicit series over some independent Lorentz-structures, and finally,we obtain the square of the amplitude. During the calculating, some useful tricks have alsobeen introduced to make the expressions more compact. More detail of the techniques andall the necessary expressions for the amplitudes of t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ) with( b ¯ c )-quarkonium in | ( b ¯ c )( S ) i , | ( b ¯ c )( S ) i , | ( b ¯ c )( P ) i and ( b ¯ c )( P J ) i (with J = (1 , , | ( b ¯ c )( S ) i and | ( b ¯ c )( S ) i can be obtained from thosefrom the two color singlet S -wave components by changing the overall color-factor and thecorresponding matrix elements. This approach is different from that of the spinor techniquesor the so called ‘helicity amplitude approach’, which has been proposed in Ref.[11] andimproved in Ref.[12]. Under the ‘helicity amplitude approach’, one can also derive compactresults that can be further expressed by spinor-products at the amplitude level and thendo the numerical calculations . The above two methods are complement to each otherand both can derive simple and compact expressions at the amplitude level. The ‘helicityamplitude approach’ is more suitable for the numerical calculations and is more quickerfor calculations, since full components of the helicity amplitude can be evaluated at theamplitude level. While for the ‘new trace amplitude approach’, at the amplitude level onlythe coefficients of the basic Lorentz structures are numerical. However from the amplitudederived from the ’new trace amplitude approach’, one can sequentially result in the squaredamplitude, which is more easier to be compared with the results derived by the traditionaltrace techniques.According to Refs.[13], one may expect at LHC to produce ∼ t ¯ t -pairs per year Here, we refer to Ref.[5] for an example on the ‘helicity amplitude approach’, where full processes of theapproach from the formulae deduction to the numerical calculation can be seen explicitly. More over sometricks to simplify the massive amplitudes can be found there. L = 10 cm − s − . Considering the possible upgrade for LHC and theestimate by Refs.[13], we will assume that one may obtain ∼ − t ¯ t -pairs per year toexamine the possibility to observe the meson B c via decay of the produced t ¯ t -pairs preciselyand to examine the advantages in the indirect way to observe the meson B c and its excitedstates.The paper is organized as follows. In Sec.II, we show our calculation techniques for theprocess t ( p ) → ( b ¯ c )( p )+ c ( p )+ W + ( p ). Then we present numerical results and make somediscussions on the properties of the ( b ¯ c )-production through t -quark decays in Sec.III. Thefourth section is reserved for a summary. All necessary expressions are put in the appendicesfinally. II. CALCULATION TECHNIQUES
Under the NRQCD framework [9, 14], the total decay width for the production of ( b ¯ c )-quarkonium through the channel t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ) takes the form:Γ = X n H n ( t → ( b ¯ c ) + c + W + ) × hO n i N col , (3)where N col refers to the number of colors, n stands for the involved state of b ¯ c -quarkonium. N col = 1 for singlets and N col = N c − , h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i = | √ N c h | χ + c ψ b | b ¯ c ( S ) i| [1 + O ( v )] , (4) h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i = | √ N c h | χ + c σψ b | b ¯ c ( S ) i| [1 + O ( v )] , (5) h b ¯ c ( P ) |O ( P ) | b ¯ c ( P ) i = | √ N c h | χ + c ( − i ↔ D ) ψ b | b ¯ c ( P ) i| [1 + O ( v )] , (6) h b ¯ c ( P ) |O ( P ) | b ¯ c ( P ) i = | √ √ N c h | χ + c ( − i ↔ D · σ ) ψ b | b ¯ c ( P ) i| [1 + O ( v )] , (7) h b ¯ c ( P ) |O ( P ) | b ¯ c ( P ) i = | √ √ N c h | χ + c ( − i ↔ D × σ ) ψ b | b ¯ c ( P ) i| [1 + O ( v )] , (8) h b ¯ c ( P ) |O ( P ) | b ¯ c ( P ) i = | √ N c h | χ + c ( − i ↔ D ( i ) σ ( j ) ) ψ b | b ¯ c ( P ) i| [1 + O ( v )] , (9)where the subscript 1 or 8 indicates that the operator is a color singlet or a color octet, ψ + is the Pauli-spinor field that create a heavy quark, χ is the Pauli-spinor that create Here as suggested in Ref.[14], an overall factor 1 / N c is introduced into the color-singlet matrix elements. D µ = ∂ µ + igA µ is the gauge-covariant derivative, A is the SU (3)-matrix-valued gauge field. The operator ↔ D is the difference between the covariant derivativeacting on the spinor to the right and on the spinor to the left, which is defined by χ † ↔ D ψ = χ † ( D ψ ) − ( D χ ) † ψ . The notation T ( ij ) is for the symmetric traceless component of a tensor: T ( ij ) = ( T ij + T ji ) / − T kk δ ij . Furthermore, we have1 √ N c h | χ + c ψ b | b ¯ c ( S ) i = 1 √ π ¯ R S (Λ)[1 + O ( v )] (10)1 √ N c h | χ + c σψ b | b ¯ c ( S ) ( ǫ ) i = 1 √ π ¯ R S (Λ) ǫ [1 + O ( v )] (11)1 √ N c h | χ + c σ ( − i ↔ D ) ψ b | b ¯ c ( P ) ( ǫ ) i = s π ¯ R ′ P (Λ) ǫ [1 + O ( v )] (12)1 √ √ N c h | χ + c σ ( 12 ↔ D · σ ) ψ b | b ¯ c ( P ) i = s π ¯ R ′ P (Λ)[1 + O ( v )] (13)1 √ √ N c h | χ + c σ ( − i ↔ D × σ ) ψ b | b ¯ c ( P ) ( ǫ ) i = s π ¯ R ′ P (Λ) ǫ [1 + O ( v )] (14)1 √ N c h | χ + c σ ( 12 ↔ D ( i ) σ ( j ) ) ψ b | b ¯ c ( P ) ( ǫ ) i = s π ¯ R ′ P (Λ) ǫ ij [1 + O ( v )] , (15)where ¯ R S (Λ) is the average radial wavefunction for 1 S state averaged over a region of size1 / Λ centered at origin, ¯ R ′ P (Λ) is the average derivative of the radial wavefunction at originof size 1 / Λ. For convenience, we shall take ¯ R S (Λ) and ¯ R ′ P (Λ) to be the phenomenologicalvalues R S (0) and R ′ P (0), which may be derived from the QCD potential models and relateto certain observable such as the width for electromagnetic annihilation etc.Although we do not know the exact values of the two decay color-octet matrix elements, h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i and h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i , we know that they are one orderin v higher than the S -wave color-singlet matrix elements according to NRQCD scale rule.More specifically, based on the velocity scale rule, we have: h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i ≃ ∆ S ( v ) · h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) ih b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i ≃ ∆ S ( v ) · h b ¯ c ( S ) |O ( S ) | b ¯ c ( S ) i , (16)where the second equation comes from the vacuum-saturation approximation. ∆ S ( v ) is ofthe order v or so, and we take it to be within the region of 0.10 to 0.30, which is in consistentwith the identification: ∆ S ( v ) ∼ α s ( M v ) and has covered the possible variation due to thedifferent ways to obtain the wave functions at the origin ( S -wave) and the first derivative ofthe wave functions at the origin ( P -wave) etc.6 − c ( p ) W − ( p ) c ( p ) t ( p ) B − c ( p ) W − ( p ) c ( p ) t ( p ) FIG. 1: Feynman diagrams for t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ), where b ¯ c -quarkonium is in eightstates, i.e. the six color-singlet states ( S ) , ( S ) , ( P ) and ( P J ) (with J = (1 , , S ) and ( S ) respectively.. The left and the right diagram are for thehard scattering amplitudes A and A respectively. The Feynman diagrams for the process of t ( p ) → b ¯ c ( p ) + c ( p ) + W + ( p ) are shown inFig.(1), where b ¯ c -quarkonium is in eight states: six color-singlet components ( S ) , ( S ) ,( P ) and ( P J ) (with J = (1 , , S ) and ( S ) respectively. Based on the phase-space integration simplification as shown in the AppendixA, the decay width of the process can be written in the form: d Γ = 3 | ¯ M | π m t × hO n i N col ds ds , (17)where the extra factor 3 in the numerator comes from the sum of the c -quark color, | ¯ M | is the mean square of the hard scattering amplitude, i.e. | ¯ M | = × P |A + A | with A and A are two amplitudes of the process, s = ( p + p ) and s = ( p + p ) .We deal with either L = 0 or L = 1 state here. The two hard scattering amplitudes ofthe process, which correspond to the left and right diagram of Fig.(1), can be written as A S =0 ,L =01 = i C ¯ u i ( p , s ) " γ µ Π p ( q )( p + p ) γ µ p + p + m b ( p + p ) − m b ε ( p ) P L u j ( p , s ′ ) | q =0 (18) A S =0 ,L =02 = i C ¯ u i ( p , s ) " γ µ Π p ( q )( p + p ) ε ( p ) P L p + p + m t ( p + p ) − m t γ µ u j ( p , s ′ ) | q =0 (19)and A S =1 ,L =01 = i C ¯ u i ( p , s ) " γ µ ε αs ( p )Π αp ( q )( p + p ) γ µ p + p + m b ( p + p ) − m b ε ( p ) P L u j ( p , s ′ ) | q =0 (20) A S =1 ,L =02 = i C ¯ u i ( p , s ) " γ µ ε αs ( p )Π αp ( q )( p + p ) ε ( p ) P L p + p + m t ( p + p ) − m t γ µ u j ( p , s ′ ) | q =0 (21)and A S =0 ,L =11 = i C ε αl ( p )¯ u i ( p , s ) ddq α " γ µ Π p ( q )( p + p ) γ µ p + p + m b ( p + p ) − m b ε ( p ) P L u j ( p , s ′ ) | q =0 (22)7 S =0 ,L =12 = i C ε αl ( p )¯ u i ( p , s ) ddq α " γ µ Π p ( q )( p + p ) ε ( p ) P L p + p + m t ( p + p ) − m t γ µ u j ( p , s ′ ) | q =0 (23)and A S =1 ,L =11 = i C ε Jαβ ( p )¯ u i ( p , s ) ddq α " γ µ Π βp ( q )( p + p ) γ µ p + p + m b ( p + p ) − m b ε ( p ) P L u j ( p , s ′ ) | q =0 (24) A S =1 ,L =12 = i C ε Jαβ ( p )¯ u i ( p , s ) ddq α " γ µ Π βp ( q )( p + p ) ε ( p ) P L p + p + m t ( p + p ) − m t γ µ u j ( p , s ′ ) | q =0 (25)with the color factor C = C s or C o for color-singlet and color-octet respectively, C s = gg s √ δ ij and C o = gg s √ ( √ T a T b T a ) ij ( √ T b stands for the color of the color-octet b ¯ c state). ε ( p ) isthe polarization vector of W + , P L = − γ and P R = γ . q , p and p are the relativemomentum between the two constitute quarks of ( b ¯ c )-quarkonium and the momenta of thesetwo constitute quarks respectively. More explicitly, we have p = m c M p + q and p = m b M p − q, (26)where M ≃ m b + m c . ε αs ( p ) and ε αl ( p ) are the polarization vectors relating to the spinand the orbit angular momentum of ( b ¯ c )-quarkonium, ε Jαβ ( p ) is the polarization tensor forthe spin triplet P -wave states with J = 0, 1 and 2 respectively. The covariant form of theprojectors can be conveniently written asΠ p ( q ) = −√ M m b m c ( p − m c ) γ ( p + m b ) , (27)and Π αp ( q ) = −√ M m b m c ( p − m c ) γ α ( p + m b ) , (28)To do the simplification of the projector, the following simplification shall be useful:Π p (0) = 12 √ M γ ( p + M ) , Π αp (0) = 12 √ M γ α ( p + M ) , (29)and ddq α Π p ( q ) | q =0 = √ M m b m c γ γ α ( p + m b − m c ) , (30) ddq α Π βp ( q ) | q =0 = − √ M m b m c h γ α γ β ( p + m b − m c ) − g αβ ( p − m c ) i . (31)Here the properties: p α ε α = 0 and p α ε αβ = p β ε αβ = 0. p = m c M p and p = m b M p , areapplied. After substituting all these relations into the amplitudes and doing the possible8implifications, the amplitudes then be squared, summed over the freedoms in final state andaveraged over the ones in initial state. And the selection of the appropriate total angularmomentum quantum number is done by performing the proper polarization sum. If definingΠ αβ = − g αβ + p α p β M , (32)the sum over polarization for a spin triplet S-state or a spin singlet P-state is given by X J z ε α ε ∗ α ′ = Π αα ′ , (33)where J z = s z or l z respectively. In the case of P J states, as for the three multiplets ε Jαβ ( p )with J = 0, 1 and 2, the sum over the polarization is given by ε (0) αβ ε (0) ∗ α ′ β ′ = 13 Π αβ Π α ′ β ′ (34) X J z ε (1) αβ ε (1) ∗ α ′ β ′ = 12 (Π αα ′ Π ββ ′ − Π αβ ′ Π α ′ β ) (35) X J z ε (2) αβ ε (2) ∗ α ′ β ′ = 12 (Π αα ′ Π ββ ′ + Π αβ ′ Π α ′ β ) −
13 Π αβ Π α ′ β ′ . (36)All the terms for the squared hard scattering amplitudes | ¯ M | for S-wave and P-wavestates are put in Appendix B accordingly, where the detail of the calculation techniques arealso attached for convenience. To effectuate the calculations and to make the results morecompact, we adopt the method used in Ref.[10]. As stated in the Introduction, we call itthe ‘new trace amplitude approach’ for convenience. Under the approach, we arrange thewhole amplitude into several orthogonal sub-amplitudes M ss ′ according to the spins of the t -quark ( s ′ ) and c -quark ( s ) first, and then do the trace of the Dirac γ matrix strings at theamplitude level by properly dealing with the massive spinors, which result in explicit seriesover some independent Lorentz-structures. The expressions for these coefficients of all theconsidered channels are put in Appendix B. With the help, one can do the square of theamplitude easily. As a cross-check of our results, we adopt the traditional trace techniquesand also the FDC[15] package to derive the numerical results of the mentioned processes.Indeed we find a well agreement among these methods.As a comparison and for later usages, let us present the width for the two body decay t ( p ) → b ( p ) + W + ( p ), which is dominant for the t -quark decay:Γ = G F m t | ~ p | √ π h (1 − y ) + x (1 + y − x ) i , (37)where | ~ p | = m t q (1 − ( x − y ) )(1 − ( x + y ) ), m w = m t x and m b = m t y .9 II. NUMERICAL RESULTS OF DIFFERENTIAL CROSS-SECTIONS
In numerical calculations, we take the parameters as follows: m b = 4 . , m c = 1 . , m t = 176 GeV , m w = 80 .
22 GeV , α s (2 m c ) = 0 . , (38)and g = 2 √ m w q G F / √
2. Then, the decay width of t → W + + b isΓ( t → W + + b ) = 1 . . (39)And the decay widths of t → ( b ¯ c ) + W + + c are:Γ t → ( b ¯ c )[( S ) ] = 0 .
79 MeV (40)Γ t → ( b ¯ c )[( S ) ] = 0 .
57 MeV (41)Γ t → ( b ¯ c )[( P ) ] = 0 .
057 MeV (42)Γ t → ( b ¯ c )[( P ) ] = 0 .
034 MeV (43)Γ t → ( b ¯ c )[( P ) ] = 0 .
070 MeV (44)Γ t → ( b ¯ c )[( P ) ] = 0 .
075 MeV (45)Γ t → ( b ¯ c )[( S ) ] = 0 . × v MeV (46)Γ t → ( b ¯ c )[( S ) ] = 0 . × v MeV (47)where v ≃ (0 . ∼ . t → ( b ¯ c ) + W + + c and t → W + + b , since uncertanties from the electro-weak couplingcan be cancelled out: Γ t → ( b ¯ c )[( S ) ] Γ( t → W + + b ) = 4 . × − (48)Γ t → ( b ¯ c )[( S ) ] Γ( t → W + + b ) = 3 . × − (49)Γ t → ( b ¯ c )[( P ) ] Γ( t → W + + b ) = 0 . × − (50)Γ t → ( b ¯ c )[( P ) ] Γ( t → W + + b ) = 0 . × − (51)Γ t → ( b ¯ c )[( P ) ] Γ( t → W + + b ) = 0 . × − (52)Γ t → ( b ¯ c )[( P ) ] Γ( t → W + + b ) = 0 . × − (53)Γ t → ( b ¯ c )[( S ) ] Γ( t → W + + b ) = 0 . × − v (54)10 −15 −10 −5 s (GeV ) d Γ / d s ( G e V − ) −13 −12 −11 −10 −9 −8 −7 −6 s (GeV ) d Γ / d s ( G e V − ) FIG. 2: The invariant mass distributions for t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ). The left is for d Γ /ds and the right is for d Γ /ds . The diamond line, the cross line, the dotted line, the solidline, the dashed line and the dash-dot line are for ( b ¯ c )-quarkonium in Fock states: ( S ) , ( S ) ,( P ) , ( P ) , ( P ) and ( P ) respectively. −1 −0.5 0 0.5 110 −7 −6 −5 −4 −3 −2 −1 cos θ d Γ / d c o s θ ( M e V ) −1 −0.5 0 0.5 110 −5 −4 −3 −2 −1 cos θ d Γ / d c o s θ ( M e V ) FIG. 3: The differential distributions d Γ /d cos θ (Left) and d Γ /d cos θ (Right) for t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ). The solid line, the dotted line, the diamond line, the dashed line, thetriangle line and the dash-dot line are for ( b ¯ c )-quarkonium in Fock states: ( S ) , ( S ) , ( P ) ,( P ) , ( P ) and ( P ) respectively. Γ t → ( b ¯ c )[( S ) ] Γ( t → W + + b ) = 0 . × − v . (55)Let us show some more characteristics of the decay t → ( b ¯ c ) + W + + c . The differentialdistributions of the invariant masses s and s , i.e. d Γ /ds and d Γ /ds are shown in Fig.2.While the differential distributions of cos θ and cos θ , i.e. d Γ /d cos θ and d Γ /d cos θ is11hown in Fig.3, where θ is the angle between ~p and ~p , and θ is the angle between ~p and ~p respectively in the t -quark rest frame ( ~p = 0). It can be found that the largest differentialcross-section of d Γ /d cos θ is achieved when θ = 180 ◦ , i.e. the ( b ¯ c )-quarkonium and W + moving back to back in the rest frame of t -quark. And the largest differential cross-sectionof d Γ /d cos θ is achieved when θ = 0 ◦ , i.e. the ( b ¯ c )-quarkonium and c -quark moving inthe same direction. IV. DISCUSSIONS AND SUMMARY
In the present paper, we have studied the decay channel t ( p ) → b ¯ c ( p ) + c ( p ) + W + ( p )in the leading α s calculation but with the v -expansion up to v , where b ¯ c -quarkonium isin one of the eight states: the six color-singlet states ( S ) , ( S ) , ( P ) and ( P J ) (with J = (1 , , S ) and ( S ) respectively. In literature, only( S ) state has been studied [8], however it can be found that all the other considered statescan be sizable in addition to the ( S ) state.As mentioned in the Introduction, about 10 t ¯ t per year will be produced in the stage ofhigh luminosity run at LHC and t -quark events always trigger the detector, then accordingto the present estimate it is possible to accumulate about 10 B c events a year via ¯ t -quarkdecay at LHC. Moreover, the indirect production of the ( b ¯ c )-quarkonium may be traced backto t -quark decay and has the characteristics in θ , θ shown in Figs.(2,3) etc, that may beused to identify the ( b ¯ c )-quarkonium events. Thus there may be some advantages in ( b ¯ c )-quarkonium studies via the indirect production in comparison with the direct production.Especially when LHC is really upgraded to SLHC, DLHC and TLHC, so 10 ∼ t ¯ t peryear may be produced, one may expect much progress in ( b ¯ c )-quarkonium studies is achievedthen. Acknowledgments:
This work was supported in part by the Natural ScienceFoundation of China (NSFC) and by the Grant from Chongqing University.12
PPENDIX A: FORMULAE FOR THE PHASE SPACE INTEGRATION
In this section, we derive the phase space of t ( m t , p ) → ( b ¯ c )( m , p ) + c ( m , p ) + W + ( m , p ). The decay width is proportional to the phase space: d Γ ∝ p
00 3 Y i =1 d p i (2 π ) (2 p i ) (2 π ) δ ( p − p − p − p ) , (A1)where p i = ( p i , ~p i ) = ( p i , p i , p i , p i ). Furthermore, in the rest frame of top quark ( p = m t ),we have d Γ ds ds ∝ m t )(2 π ) d p d p d p δ ( p − m ) δ ( p − m ) δ ( p − m ) θ ( p ) θ ( p ) · θ ( p ) δ ( s − ( p + p ) ) δ ( s − ( p + p ) ) δ ( p − p − p − p ) ∝ m t )(2 π ) d p d p δ ( p − m ) δ (( p − p − p ) − m ) δ ( p − m ) θ ( p ) · θ ( p ) θ ( m t − p − p ) δ ( s − ( p − p ) ) δ ( s − ( p − p ) ) ∝ m t π d ~p d ~p δ ( p − ~p − m ) δ ( p − ~p − m ) θ ( p ) θ ( p ) · θ ( m t − p − p ) δ ( s + m − m − m t p + 2 p p − ~p · ~p ) ∝ | ~p || ~p | m t π d Ω sin θ dθ dφ θ ( p ) θ ( p ) θ ( m t − p − p ) · δ ( s + s − m t − m + 2 p p − | ~p | · | ~p | cos θ ) (A2) ∝ m t π θ ( p ) θ ( p ) θ ( m t − p − p ) θ ( X ) , (A3)where p = m t + m − s m t and p = m t + m − s m t , | ~p | = q p − m and | ~p | = q p − m . The stepfunction θ ( X ) is determined by ensuring | cos θ | ≤
1, wherecos θ = s + s − m t − m + 2 p p | ~p || ~p | . All these step functions leads to the integration ranges: s min1 = m + m − h ( s − m t + m )( s − m + m ) + q λ ( s , m t , m ) λ ( s , m , m ) i s (A4) s max1 = m + m − h ( s − m t + m )( s − m + m ) − q λ ( s , m t , m ) λ ( s , m w , m ) i s (A5) s min2 = ( m + m ) (A6) s max2 = ( m t − m ) (A7)where λ ( x, y, z ) = ( x − y − z ) − yz . 13urthermore, we can obtain the cos θ distribution from Eq.(A2): d Γ dcosθ ds ∝ J m t π θ ( p ) θ ( p ) θ ( m t − p − p ) θ ( Y ) , (A8)where the extra Jacoiban J = −| ~p || ~p | (cid:12)(cid:12)(cid:12)(cid:12) − p m t + | ~p | ( m + m t − s ) cos θ m t √ m +( m t − s ) − m ( m t + s ) (cid:12)(cid:12)(cid:12)(cid:12) (A9)and s = 1 | ~p | cos θ − ( m t − p ) ( ( m + m t ) | ~p | cos θ − ( m t − p )( m t ( m − m + m t p ) − m p ) − m t | ~p | cos θ h − m ( m + 2( m t − p ) − s − | ~p | cos θ )+ m + ( m − s ) i / ) (A10)The θ ( X ) function determines the boundary of s : s min ≤ s ≤ ( m t − m ) (A11)where s min = m m t + m ( m t cos θ + m (cos θ − m m t √ Ym t − m (1 − cos θ ) (A12)with Y = ( m − m + m − m t ) − (cid:16) m − m − m + m t ) m +(( m − m ) − m t )(( m + m ) − m t ) (cid:17) cos θ + 4 m m cos θ . (A13)It should be noted that the integration over cos θ should be from 1 to −
1. The distributionfor θ can be obtained in a similar way. APPENDIX B: AMPLITUDE OF THE PROCESS t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ) Now let us illustrate the method to calculate the amplitude squared. In general, theamplitude for the process in which two massive fermions with spin projection s and s ′ areinvolved can be written as M ss ′ = ¯ u s ( p ) Au s ′ ( p ) , (B1)14here p and p denote the momenta of the top quark and the charm quark with mass m t and m c , and A is an explicit string of Dirac γ matrices for the process, which can be readfrom Eqs.(18,19,20,21,22, 23,24,25).Let us introduce a massless spinor u − ( k ) with a light-like momentum k and negativehelicity first. Thus u − ( k ) is satisfied with the following projection relation: u − ( k )¯ u − ( k ) = ω − k , (B2)where ω − = (1 − γ ) /
2. By introducing another spacelike vector k which satisfies therelations: k · k = − , k · k = 0 , (B3)then the other, massless, independent and positive helicity spinor u + ( k ) may be constructed: u + ( k ) = k u − ( k ) . (B4)It is easy to check that u + ( k ) is satisfied with the projection u + ( k )¯ u + ( k ) = ω + k , (B5)where ω + = (1 + γ ) /
2. Using these massless spinors, one can construct the massive spinorsfor the fermion and antifermion as follows: u s ( q ) = ( q + m ) u − ( k ) / q k · q, (B6) u − s ( q ) = ( q + m ) u + ( k ) / q k · q, (B7)with the spin vector s µ : s µ = q µ m − mq · k k µ . Using the above identities, we can write down the amplitude M ± s ± s ′ with four possible spinprojections in the trace form for the γ -matrices: M ss ′ = N T r [( p + m t ) · ω − k · ( p + m c ) · A ] (B8) M − s − s ′ = N T r [( p + m t ) · ω + k · ( p + m c ) · A ] (B9) M − ss ′ = N T r [( p + m t ) · ω − k k · ( p + m c ) · A ] (B10) M s − s ′ = N T r [( p + m t ) · ω + k k · ( p + m c ) · A ] (B11)15ith the normalization constant N = 1 / q k · p )( k · p ). Thus the squared unpolarizedmatrix elements can be written as | M | = | M ss ′ | + | M − s − s ′ | + | M − ss ′ | + | M s − s ′ | . Next, we are to simplify the calculation. For such purpose, we recombine these M ± s ± s ′ into M n ( n = 1 , · · · ,
4) as follows: M = 1 √ M ss ′ + M − s − s ′ ) = N √ T r [( p + m t ) · 6 k · ( p + m c ) · A ] (B12) M = 1 √ M ss ′ − M − s − s ′ ) = N √ T r [( p + m t ) · 6 k γ · ( p + m c ) · A ] (B13) M = 1 √ M s − s ′ − M − ss ′ ) = N √ T r [( p + m t ) · 6 k k · ( p + m c ) · A ] (B14) M = 1 √ M s − s ′ + M − ss ′ ) = N √ T r [( p + m t ) · γ k k · ( p + m c ) · A ] (B15)and then we have | M | = | M | + | M | + | M | + | M | . In order to write down A asexplicitly and simply as possible, we set the vector k : k = p − αp , (B16)where the coefficient α is determined by the requirement that k be a light-like vector: α = p · p m t ± ∆ m t with ∆ = q ( p · p ) − m t m c . Furthermore, if choosing k : k · p = 0 and k · p = 0. Andthen the resultant M n can be simplified as M = L T r [( p + m t ) · ( p + m c ) · A ] (B17) M = − L T r [( p + m t ) · γ · ( p + m c ) · A ] (B18) M = L T r [( p + m t ) · 6 k · ( p + m c ) · A ] (B19) M = L T r [( p + m t ) · γ · 6 k · ( p + m c ) · A ] (B20)with 2 L = 1 √ p · p + m c m t and 2 L = 1 √ p · p − m c m t . The value of k is arbitrary, and we take its explicit form as k µ = iκǫ µνρσ p ν p ρ p σ , (B21)16here κ is a suitable normalization constant and k can be expressed as k = κγ [( p · p ) p + ( p · p ) p − ( p · p ) p − 6 p · 6 p · 6 p ] . Substituting k into Eqs.(B19,B20), we obtain M = M ′ + κ [( p · p ) m c − ( p · p ) m t ] M (B22) M = M ′ − κ [( p · p ) m c + ( p · p ) m t ] M (B23)where M ′ = κ L T r [( p + m t ) · γ · 6 p · ( p + m c ) · A ] (B24) M ′ = κ L T r [( p + m t ) · 6 p · ( p + m c ) · A ] (B25)The amplitudes for the process t ( p ) → ( b ¯ c )( p ) + c ( p ) + W + ( p ) can be expanded oversome basic Lorentz structures: M i ( n ) = m X j =1 A ij ( n ) B j ( n )( i = 1 − , M ′ i ( n ) = m X j =1 A i ′ j ( n ) B j ( n ) ( i = 3 ,
4) (B26)where m is the number of basic Lorentz structure B j ( n ), whose value dependents on the( b ¯ c )-quarkonium state n : e.g. m = 3 for n = ( b ¯ c )[ S ] , m = 11 for n = ( b ¯ c )[ S ] and( b ¯ c )[ P ] and m = 30 for n = ( b ¯ c )[ P J ] . As for A j ( n ) and A j ( n ), they can be expressed by A j ( n ) = A ′ j ( n ) + κ [( p · p ) m c − ( p · p ) m t ] A j ( n ) (B27) A j ( n ) = A ′ j ( n ) − κ [( p · p ) m c + ( p · p ) m t ] A j ( n ) (B28)The explicit expression for A , j ( n ) and A ′ , ′ j ( n ) of each state shall be listed in the followingsubsections.To short the notation, we define some dimensionless parameters r = m b m t , r = m c m t , r = m w m t , r = Mm t and u = p · p /m t = 12 m t ( s − m B c − m w ) , v = p · p /m t = 12 m t ( s − m c − m w ) ,w = p · p /m t = 12 m t ( m t + m w − s ) , x = p · p /m t = 12 m t ( s − m B c − m c ) ,y = p · p /m t = 12 m t ( m t + m B c − s ) , z = p · p /m t = 12 m t ( m t + m c − s ) , s = ( p + p ) , s = ( p + p ) and s = ( p + p ) , which satisfy the relation: s + s + s = m t + m c + m w + m B c . And the short notations for the denominators are d = 1( p + p ) p + p ) − m b , d = 1( p + p ) p + p ) − m t ,d = m t ( p + p ) p + p ) − m b , d = m t ( p + p ) p + p ) − m t ,d = 1( p + p ) m t (( p + p ) − m t ) . Furthermore, the following relations are useful to short the expressions: y + z + w = 1 , x + u + r = y, x + v + r = z, u + v + r = w.
1. Coefficients for spin-singlet S-wave state: ( b ¯ c )[ S ] There are three basic Lorentz structures B j for the case of ( b ¯ c )[ S ] , which are B = p · ǫ ( p ) m t , B = p · ǫ ( p ) m t , B = im t ε ( p , p , p , ǫ ( p )) , (B29)where the short notation ε ( p , p , p , ǫ ( p )) = ε µνρσ p µ p ν p ρ ǫ σ ( p ). The values of the coeffi-cients A j and A ′ j are A = 2 L m t r (cid:16) d ( r r + r (2( v + x ) + r + 4 r ) − r (3 z − v + r u − w + 2) − x )+ r ( − v + r ( − u + 3 v + 4 x + r ( − r ) + r ))) + d r (2 x ( r + 2 r ) +( v + r − r r + 4 r ) r ) (cid:17) (B30) A = − L m t √ r (cid:16) − xd (1 + r ) + d ( u + r + r + 3 r r − r (3 + 2 r )) r + d (1 + r )( u + (1 + r ) r ) (cid:17) (B31) A = − L m t √ r (cid:16) d (1 + r ) + d r (cid:17) (B32) A ′ = κm t L r (cid:16) d r ( − x ( v + r + r (2 + r )) + ( v ( r − r ) + r ( r + r + 3 r r − r (3 + 2 r ))) r ) + d (4 x r + 2 x ( r + r ( − v + r ) + r ( − r ) + r ( − u + v + 2 r )) + r ((1 + r ) r (2 u + 2 r + ( − r ) r + r ( − r )) + v (2 u + ( − r ) r ))) (cid:17) (B33) A ′ = − κm t L √ r (cid:16) ( d (2 u ( u + x ) + r − ( u + 2 x ) r − r + r ( − r ) + r ( u − r ) +18 (2( u + x ) + 3( − r ) r − r ) + r ( − u − x + r (3 u + 2 x + ( − r ) r − r ))) + d ( − x + u ( r − r ) r + ( − r − r ) r − x ( u + 4 r r ))) (cid:17) (B34) A ′ = − κm t L √ r (cid:16) ( d (2( u + x ) + 2 r + 3 r r + r + r ) + d ( − x + ( r − r ) r )) (cid:17) (B35)And the values of the coefficients A j and A ′ j are A = − A L L − L m t r (cid:16) xd r − d ( − v + ( r − r ) r ) r − d r r (cid:17) (B36) A = − A L L − L m t √ r (cid:16) d ( − x + r r − r r ) + d ( u + r ) (cid:17) (B37) A = − A L L − L d m t √ r (B38) A ′ = A ′ L L − κm t L √ r (cid:16) d ( r r + r (2 u − v + x + z ) + r ( z − v − x )) + d ( r − r ) r r − xd ( r + r ) (cid:17) (B39) A ′ = A ′ L L − κm t L √ r (cid:16) d r + d r ( x + y ) (cid:17) (B40) A ′ = A ′ L L + κm t d √ r L (B41)Here for convenience an overall factor C s has been contracted out from these coefficients.Then the square of the amplitude | M i | can be conveniently obtained with the help ofEqs.(B26,B27,B28).
2. Amplitude for spin-triplet S-wave state: [ S ] There are eleven basic Lorentz structures B j for the case of ( b ¯ c )[ S ], which are B = ǫ ( s z ) · ǫ ( p ) , B = im t ε ( p , p , ǫ ( s z ) , ǫ ( p )) , B = im t ε ( p , p , ǫ ( s z ) , ǫ ( p )) ,B = im t ε ( p , p , ǫ ( s z ) , ǫ ( p )) , B = p · ǫ ( p ) p · ǫ ( s z ) m t , B = p · ǫ ( p ) p · ǫ ( s z ) m t ,B = p · ǫ ( p ) p · ǫ ( s z ) m t , B = p · ǫ ( p ) p · ǫ ( s z ) m t , B = ip · ǫ ( p ) m t ε ( p , p , p , ǫ ( s z )) ,B = im t ε ( p , p , p , ǫ ( p )) p · ǫ ( s z ) , B = im t ε ( p , p , p , ǫ ( p )) p · ǫ ( s z ) , (B42)19here the polarization vector ǫ ( s z ) related to the spin angular momentum of the spin-tripletstate. The values of the coefficients A j and A ′ j are A = − L m t √ r (cid:16) − ( d r ( x − r ( z + r ) + r ( y + z + r ))) + d (2 u ( v + x ) + x ( r − x ( r − r + v (2 r − r + 2 r ( u + ( r − r ) + r ( u − x + ( u + x ) r +(2 r + w − r + ( − r ) r )) (cid:17) (B43) A = − L m t √ r (cid:16) d (1 + 2 u + r + r + r r + 2 r ) + d (1 + r ) r (cid:17) (B44) A = 2 L m t √ r (cid:16) − d r r + d (4( v + x ) + r (1 − r + 2 r + r )) (cid:17) (B45) A = − L m t √ r (cid:16) ud + d ( − r + r ) r + d r ( − − r + 2 r ) (cid:17) (B46) A = − L m t √ r (cid:16) d (1 + r )(1 + r ) + d (1 + r + 2 r ) r (cid:17) (B47) A = 2 L m t √ r (cid:16) − d r r + d (2( v + x ) + r (1 − r + 2 r + r )) (cid:17) (B48) A = − d L m t √ r (1 + r ) (B49) A = 2 L m t √ r ( d + d r − d r + d r ) (B50) A = − d L m t √ r (B51) A = 4 d L m t √ r (B52) A ′ = − κm t L √ r (cid:16) ( d r ( − ( x (2( u + x ) + r )) − ( u + x ) r + ( v − x ) r + r ( x + ( u + x ) r +(1 + r ) r )) + d (2 x ( x + ( u + x ) r ) + x (2( v − − r + r ) r − ( v − x )(1 + r ) r + vr − r r ( u + r + r r ) + r ( u + r + r r )(2( u + x ) + r ( − r )))) (cid:17) (B53) A ′ = κm t L √ r (cid:16) − ( d r (2 x + r + r − r (1 + r ) + r )) + d ( − xr + 2( u + x + ur ) + r ( r + r − r r )) (cid:17) (B54) A ′ = − κm t L √ r (cid:16) ( d (2 x − r r + r ) r + d (2 x (1 + r ) − ur + 2 vr + r (1 + r − r ) r )) (cid:17) (B55) A ′ = κm t √ r L (cid:16) d r + d (2( u + x ) + r (1 + r + 2 r + r )) (cid:17) (B56)20 ′ = κm t L √ r (cid:16) d r (2 v − x + r + r + r r + r − r ) + d (2( u + x ) − xr + r ( r + r − r r )) (cid:17) (B57) A ′ = − κm t L √ r (cid:16) ( d (2 x − r r + r ) r + d ( r r + 2( x + xr + vr ) + r ( − u + x ) + r − r ))) (cid:17) (B58) A ′ = − κd m t √ r L ( u + 2 x + r ) (B59) A ′ = κm t √ r L (cid:16) d r + d (2( u + x ) + r (1 + r + r )) (cid:17) (B60) A ′ = κd m t r L √ r (B61) A ′ = − κd m t √ r L (B62)and A = A ′ = 0. And the values of the coefficients A j and A ′ j are A = − A L L + 4 L m t √ r (cid:16) d ( x − r r + r ) r + d (( v + 2 x ) r + 2 r r − r ( x + 2 r r ) + r ( y − u )) (cid:17) (B63) A = − A L L − L m t √ r ( d r + d r ) (B64) A = − A L L + 4 d L m t r √ r (B65) A = − A L L + 4 d L m t √ r (B66) A = − A L L − L m t √ r ( d r + d r ) (B67) A = − A L L + 4 d L m t r √ r (B68) A = − A L L − d L m t √ r (B69) A = − A L L + 4 d L m t √ r (B70) A ′ = A ′ L L − κm t L √ r (cid:16) − x d − xr ( − ( d + 2 d )( r − r ) + d r ) + r ( − ud r + vd r + r r ( − d r − d r )) (cid:17) (B71) A ′ = A ′ L L − κm t L √ r ( d ( r − r ) r + d (2( u + x ) + r )) (B72)21 ′ = A ′ L L + κd m t L √ r (2 x + r r ) (B73) A ′ = A ′ L L − κd m t r L (B74) A ′ = A ′ L L − κm t L √ r ( d r + d ( u + x + y )) (B75) A ′ = A ′ L L + κd m t L √ r (2 x + r r ) (B76) A ′ = A ′ L L − κd m t r L (B77)and for j = 9 , ,
11 and m = 7 , , ,
11, we have A j = − A j L L and A ′ m = A ′ m L L . Here for convenience an overall factor C s has been contracted out from these coefficients.Then the square of the amplitude | M i | can be conveniently obtained with the help ofEqs.(B26,B27,B28) and Eq.(33).
3. Amplitude for spin-singlet P-wave state: [ P ] The basic structures are similar to the case of S , only one needs to replace the po-larization vector ǫ ( s z ) related to the spin angular momentum of the spin-triplet S-state tothe present ǫ ( l z ) that is related to the radial angular momentum of the spin-singlet P-wavestate. Secondly, we need to change coefficients there to the present case. The values of thecoefficients A j and A ′ j are A = L m t r r r (cid:16) − ( d ( r − r ) r ( x + r ( r + z ) + r ( r + y + z ))) − d r r (2 ur +( r + r − r )( x + 2( v + x ) r + 2 r r + r (2 r + y )) + d r ( r ( x + yr )+ r ( x − u ( v + x ) + 2 r − r ( − u + v + r ) + r ( v − u + 3 x + r )) − r ( r ( x − ( w − r + 2 r ) + r ( v + 2 vr + 2 x (1 + r ))+ r ( u + x + r (2 + 3 r )))) (cid:17) (B78) A = L m t r r r ( − d ( r − r − r ) r + 2 d r r (1 + r )(2 ur + ( r + r − r ) − d r ( − r (1 + r ) + r r (1 + r ) + r + ( r + 2( u + r )) r )) (B79)22 = L m t r r r ( d r ( r r + ( r + v + x + 3 z ) r − r r ( r + r )) + r ( d ( r − r ) r +2 d r r (2 ur + ( r + r − r ))) (B80) A = − L m t √ r r r ( d ( r − r ) + d ( r (1 + r ) − r (1 + r ) + 2( u + r ))) (B81) A = − L m t r r r ( d (1 + r )( r + r (1 − r + r )) r + d ( − r − r )( r − r ) r +2 r r ( d (1 + r )(2 ur + ( r + r − r ) + 2( d ( r r + r ( r − v + x − z )) − r ( r ( u − w + 2) − v − x + 3 z ) + r ( − v + r ( − u + 3 v + 4 x + r ( − r ) + r ))) + d r (2 x ( r + 2 r ) + ( v + r − r r + 4 r ) r )))) (B82) A = L m t r r r ( d r ( r r + ( r − v − x + 3 z ) r − r r ( r + r ))+ r ( d ( r − r ) r + 2 d r (2 v (2 r − r + 4( r − r r + 4 x ( r + ( r − r )+ r (2( u + 2 x ) r + ( − v − ( − r ) r + r ) r + 2( − r ) r )))) (B83) A = 2 L m t r r √ r ( d ( r − r )(1 + r ) r + 2 r r ( d (1 + r )( u + r + r r ) − d (2 x − ur +2 r r + r − r + r ( u − r ) r + r ) + r (2 r − u + 2 y )))) (B84) A = − L m t r r √ r ( d ( r − r )(1 + r − r ) r + 4 d r r (1 + r )( u + r + r r )) (B85) A = − d L m t √ r r r (B86) A = 2 L m t r r √ r ( − d r r (1 + r ) + d r ) (B87) A = 4 L m t √ r ( d (1 + r ) + d r ) (B88) A ′ = − κm t L r r r (cid:16) d ( r − r ) r (2 x ( u + x ) − xr + ( u + x ) r + ( x − v ) r + r ( x ( r − ur + ( r − r )) − d r r (2 ur + ( − r + r ) r )( − x + r ( − ur + vr + r (1 + r − r ) r ) − x (2 u + r ( − r + r ))) + d r (2 x ( r ( − r + r ) + r ) + x ( − r (2 u + r ) + r ( r (2( − u + v ) + 2 r + r ) + r ) + r (2 ur − ( − v +2 r + r ) r + ( − r ) r )) + r ( − vr ( − ( r − r )( r − r ) + r ) + r ( u (2 u + r − r ) + r (1 + r ) r + ( − u + r ) r + ur ( − r + 2 r ) − r r ( r + r − − r ) r )))) (cid:17) (B89) A ′ = κm t L r r r (cid:16) d r r (2 ur + ( − r + r ) r )(2 x + r ) + d ( r − r ) r (2 x + r +23 + r r + r + r ) + d r (2 x ( r − r r + r ) + r (2 u (1 + r ) + r r + r r + r − r ( r + r r ))) (cid:17) (B90) A ′ = κm t L r r r (cid:16) d ( r − r )(2 x + r ( r + r )) r + 4 xd r r (2 ur + ( r + r − r )+ d r (2 v ( r − r ) r − x ( r ( r − r ) + r ) + r r ( r (1 + r ) − r (1 + r ) +2( u + r ))) (cid:17) (B91) A ′ = − κm t √ r L r r (cid:16) d ( r − r ) r + 2 d r r (2 ur + ( − r + r ) r ) + d ( − u + x ) r + r r − (1 + 3 r ) r + r (2( u + x ) − r r + r )) (cid:17) (B92) A ′ = κm t L r r r (cid:16) d ( r − r )(2 x − v + r + r − r + 3 r r ) r + d r ((2 u + r + r + r ( r + r ) − r (( − r ) r + r )) r + 2 x ( r − r r + r )) − r r ( d (2 ur +( − r + r ) r )( u + x + y ) + 2( d r ( − x ( v + r + r (2 + r )) + ( v ( r − r ) + r ( r + r + 3 r r − r (3 + 2 r ))) r ) + d (4 x r + 2 x ( r + r ( − v + r ) + r (3 r −
1) + r (2 u − v + 2 r )) + r ((1 + r ) r (2 u + 2 r + ( r − r + r ( − r )) + v (2 u + ( − r ) r ))))) (cid:17) (B93) A ′ = κm t L r r r (cid:16) d ( r − r )(2 x + r ( r + r )) r + 4 d r r (4 x r − (1 + r ) r r ( − u +(1 + r − r ) r ) + vr (2 u − (1 + r ) r + r ) + 2 x (2 ur + ( v + (1 + r )( r − r + r r )) + d r (2 v ( r − r ) r + 2 x ( r − r r + ( − r ) r ) + r r ( r (1 + r ) − r (1 + r ) + 2( u + r ))) (cid:17) (B94) A ′ = κm t L r r √ r (cid:16) d ( r − r ) r ( x + y ) + 2 d r r (2 u ( u + x ) + ( u + 2 x )( r − r + ur − r r + ( r − r ) + 2 d r r ( − x + 2 x ( − u + 2 r r − r ( r + r )) + r ( r ( r + r − r r − r (2 + r ) + r (1 + r + 2 r )) − u (2 r + r − r ))) (cid:17) (B95) A ′ = − κm t L r r √ r (cid:16) d ( r − r ) r + 2 d r r (4 u ( u + x ) + (4 x ( r −
1) + u (4 r − r +(2 u − r − r ) r + 2( − r ) r ) + d r ( − u + x ) r + r r − (1 + r ) r + r (2( u + x ) − r r + r )) (cid:17) (B96) A ′ = − κm t L r r (cid:16) − d r + 2 d r (2 ur + ( − r + r ) r ) (cid:17) (B97) A ′ = − κd m t L √ r (2( u + x ) + r (1 + r + r )) (B98)24 ′ = κm t L r r √ r (cid:16) d ( r − r ) r + 2 r r ( − xd + d (2 r − r − r ) r + d (2( u + x )+ r (1 + r + r ))) (cid:17) (B99)And the values of the coefficients A j and A ′ j are A = − A L L + 2 L m t r r r ( − d ( r − r ) r ( x + r r ) − d r r ( x + 2 r r )(2 ur +( − r + r ) r ) + d r ( xr + r r ( y − u − v + 2 z ) − r ( x ( r + 2 r )+ r ( v + 2 r ( r + r ))))) (B100) A = − A L L + 2 L m t r r r (4 ud r r + r ( r ( d ( r − r ) + 2 d r ( − r + r )) +( d r − ( d + d ) r ) r )) (B101) A = − A L L + 2 d L m t √ r r (B102) A = − A L L + 2 d L m t √ r r r ( r − r ) (B103) A = − A L L − L m t r r r (8 ud r r + 8 d r r ( − xr + ( − v + ( r − r ) r ) r ) + r (4 d r r ( − r + r ) + 2( − ( d r ) + d r + 4 d r r ) r + d ( − r +3 r r + r + r + ( − r ) r ))) (B104) A = − A L L + 2 L m t r r ( d r + 4 d r (2 r x + r r ) − r ( r r − v + 2 z ))) (B105) A = − A L L + 4 L m t r r √ r ( d ( r − r ) r + 2 r r ( − xd + d (2 r − r − r ) r + d ( u + r ))) (B106) A = − A L L − L m t r r √ r ( d ( r − r ) r + 4 d r r ( u + r )) (B107) A = − A L L − d L m t √ r (B108) A = − A L L + 8 d L m t √ r (B109) A ′ = A ′ L L − κm t L r r √ r (2 d r r (2 x + r r )(2 ur + ( − r + r ) r ) + d ( r − r ) r ( xr + r ( y − u )) + d r ( − x − r r r + vr − xr +2 xr ( r + r ) − r ( y + x ) + r r ( u + r ( r + 2 r )))) (B110) A ′ = A ′ L L − κm t √ r L r r ( d ( r − r ) r + d ( y + u + x )) (B111)25 ′ = A ′ L L − κd m t √ r L r r ( − x + ( r − r ) r ) (B112) A ′ = A ′ L L − κd m t r L r r (B113) A ′ = A ′ L L + κm t L r r r (2 d r ( − xr ( r − r ) r + r r ( − vr + r (4 u + 3 r + r (2 r −
1) + r (5 r − − r ) + r ))) + r ( − x (8 d r r ( r + 2 r ) + 2 d r ) − r ( d ( r − r ) − d r ( r − r ) r + d (2 u + r )))) (B114) A ′ = A ′ L L − κm t L r r √ r ( d ( − x + ( r − r ) r ) r − d r r ( vr + r r + r (2 x + r r ) − yr )) (B115) A ′ = A ′ L L + 2 κm t √ r L ( d r + d ( x + y )) (B116) A ′ = A ′ L L − κm t √ r L r r ( d r + 4 d r r ( x + y )) (B117) A ′ = A ′ L L + 2 κd m t √ r L (B118) A ′ = A ′ L L − κd m t √ r L (B119)and A = − A L L and A ′ = A ′ L L . (B120)Here for convenience an overall factor C s has been contracted out from these coefficients.Then the square of the amplitude | M i | can be conveniently obtained with the help ofEqs.(B26,B27,B28) and Eq.(33).
4. Amplitude for spin-triplet P-wave state: [ P J ] There are totally thirty independent basic Lorentz structures B j for the case of( b ¯ c )[( P J ) ], which are B = 1 m t p α ǫ β ( p ) ε Jαβ , B = 1 m t p · ǫ ( p ) ε Jαα , B = 1 m t p · ǫ ( p ) ε Jαα ,B = 1 m t p α ǫ β ( p ) ε Jαβ , B = iε Jαβ m t ε ( p , p , p , α ) ǫ β ( p ) , B = iε Jαα m t ε ( p , p , p , ǫ ( p )) ,B = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β , B = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β ,B = iε Jαβ m t ε ( p , p , α, β ) p · ǫ ( p ) , B = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β , = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β , B = iε Jαβ m t ε ( p , p , α, β ) p · ǫ ( p ) ,B = iε Jαβ m t ε ( p , p , α, β ) p · ǫ ( p ) , B = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β ,B = iε Jαβ m t ε ( p , p , α, ǫ ( p )) p β , B = iε Jαβ m t ε ( p , p , α, β ) p · ǫ ( p ) ,B = iε Jαβ m t ε ( p , α, ǫ ( p ) , β ) , B = iε Jαβ m t ε ( p , α, ǫ ( p ) , β ) , B = iε Jαβ m t ε ( p , α, ǫ ( p ) , β ) B = iε Jαβ m t ε ( p , p , p , α ) p · ǫ ( p ) p β , B = iε Jαβ m t ε ( p , p , p , α ) p · ǫ ( p ) p β ,B = iε Jαβ m t ε ( p , p , p , · ǫ ( p )) p α p β , B = iε Jαβ m t ε ( p , p , p , · ǫ ( p )) p α p β ,B = iε Jαβ m t ε ( p , p , p , · ǫ ( p )) p α p β , B = ε Jαβ m t p · ǫ ( p ) p α p β ,B = ε Jαβ m t p · ǫ ( p ) p α p β , B = ε Jαβ m t p · ǫ ( p ) p α p β , B = ε Jαβ m t p · ǫ ( p ) p α p β ,B = ε Jαβ m t p · ǫ ( p ) p α p β , B = ε Jαβ m t p · ǫ ( p ) p α p β . (B121)Noting the fact that ε , αβ is the symmetric tensor and ε αβ is the anti-symmetric tensor,and the fact that ε αα = ε αα = 0, one may observe that the terms involving the followingcoefficients do not have any contributions to the square of the amplitude, so one can safelyset the following coefficients to zero: A ij ( P ) = 0 for i = (1 − , j = (9 , , , , , ,
19) (B122) A ij ( P ) = 0 for i = (1 − , j = (2 , , , , , , ,
30) (B123) A ij ( P ) = 0 for i = (1 − , j = (2 , , , , , , , , , . (B124)The coefficients A j , A J , A ′ j and A ′ j that are the same for all the three [ P J ] states (with J = 1 , , A = 8 d L m t √ r , A = − d L m t √ r , A = − d L m t √ r (B125) A = 8 d L m t √ r , A ′ = − κd m t r L √ r , A ′ = 2 κd m t r L √ r (B126) A ′ = A ′ = 0 (B127)and for j = 20 , , , A j = − A j L L and A ′ j = A ′ j L L .27he coefficients A j , A J , A ′ j and A ′ j that are the same for both [ P ] and [ P ] : A = 4 L m t √ r ( d (1 + r )(2 ur + ( r + r − r ) − d r ( x − r ( z + r ) + r ( r + y + z )) + d (2 u ( v + x ) + x ( r −
1) + 2 x ( r − r + v (2 r − r +2 r ( u + ( r − r ) + r ( u − x + ( u + x ) r + ( w − r ) r + ( r − r )))(B128) A = − d L m t √ r ( − x + 2 u ( v + x ) + ( u − v − x ) r + ( − u + v ) r − r + r ( r − v − x + 3 z ) + r ( − v − x + r ( − v + 3 x + r ( − r )))) (B129) A = 0 (B130) A = − d L m t √ r (1 + 2 u + r r + 2 r + r ) (B131) A = 4 L m t √ r ( d (1 + r ) r + d (1 + 2 u + r r + 2 r + r )) (B132) A = − L m t √ r ( d ( r + v + x + 3 z ) − d r r ) (B133) A = 4 d L m t √ r ( r + v + x + 3 z ) (B134) A = 4 L m t √ r (2 ud + ( d ( − r + r ) + d ( − r + r )) r ) (B135) A = − d L m t √ r (2 u + 2 r + ( − r ) r + r ( − r )) (B136) A = − L m t √ r ( d (1 + r )(1 + r ) + d ( r − v − x + 3 z ) − d r r ) (B137) A = − L m t √ r ( d + d r − d r + d r ) (B138) A ′ = κm t L √ r ( d (2 x + 2 u r + 2 r r + ( u − v + 3 x ) r + ( − u + v ) r + r + r ( x + r + 5 r ) + r ( − v − x + r (3 r + 4 u − v + 4 z − xr ( u + v + w −
2) + r (( u − v + 2 x ) r + ( − u + 2 v + 3 x ) r +3 r + r + x ( u + v + w + 2 x − d ( − x ( u + x ) + xr + r r +( v − u − x ) r + r − xr (1 + r ) + r ( v − x + r + 2 r ) + r r ( u + 2 v + r (2 + r ))) r + d (2 ur + ( − r + r ) r )( x + y )) (B139) A ′ = − κd m t L √ r (2 x + 2 u r + ( − u + 2 v ) xr + 2 r r + ( u − v + 3 x ) r +( − u + v ) r + r + r r (1 + 5 r ) + r ( − v − x + r ( − u + v +3 x + r (3 + 4 r ))) + r (2 x + x ( − v + r (2 + 3 r )) +28 ( u − v + r ( − u + 2 v + r (3 + r ))))) (B140) A ′ = κd m t L √ r (2 ur + ( − r + r ) r ) (B141) A ′ = − κd m t L √ r ( − u + x ) + r + r ( − r ) + r (2 x + ( − r ) r − r ) − r (2 u + r + r )) (B142) A ′ = κm t L √ r ( d ( − u + x ) + r + r ( − r ) + r (2 x + ( − r ) r − r ) − r (2 u + r + r )) + d (2 x + r + r ( − r ) + r + 2 r ) r ) (B143) A ′ = κm t L √ r ( d (2 xr + 2 xr − r r + r ) + d (2 x − r r + r (2( v + x ) + r − r )+ r ( − u + 2 v + r − r ))) (B144) A ′ = κd m t L √ r ( − x + 2 r r + r (2 u − v − r + r ) + r ( − v + x ) + r ( − r )))(B145) A ′ = − κm t √ r ( d r + d (2( u + x ) + 2 r + 5 r r + 3 r + r )) L (B146) A ′ = κd m t √ r L (2( u + x ) + (1 + 2 r + 3 r ) r ) (B147) A ′ = − κm t L √ r ( d ( − x + 2 r r + r (2( u − v + x ) + ( − r ) r ) + r ( − v + x ) + r ( − r ))) + d ( − u + x ) + r + r (2 r −
1) + r (2 x + ( r − r − r ) − r ( r + r )) + d ( − x + ( r − r ) r ) r ) (B148) A ′ = − κm t √ r L ( d r + d (2( u + x ) + 2 r + 3 r r + r + r )) (B149)and A = − A L L − L m t √ r ( d ( x − r r + r ) r − d (2 ur + ( r + r − r ) + d (( v + 2 x ) r + 2 r r − r ( x + 2 r r ) + r ( y − u ))) (B150) A = − A L L + 8 d L m t √ r (( v + 2 x ) r + 2 r r − r ( x + 2 r r ) + r ( y − u )) (B151) A = − A L L (B152) A = − A L L − d L m t √ r (B153) A = − A L L + 8 L m t √ r ( d r + d r ) (B154) A = − A L L − d L m t r √ r (B155)29 = − A L L + 8 d L m t r √ r (B156) A = − A L L − d L m t √ r (B157) A = − A L L + 8 d L m t √ r (B158) A = − A L L − L m t √ r ( d r + d r ) (B159) A = − A L L − d L m t √ r (B160) A ′ = A ′ L L − κm t L √ r (2 x d + xr ( − (( d + 2 d )( r − r )) + d r ) + r ( − ( vd r ) + d r r + d r ( u + r ))) (B161) A ′ = A ′ L L − κd m t L √ r ( x (2 r − r − r ) r − x + r ( vr − ur − r r )) (B162) A ′ = A ′ L L (B163) A ′ = A ′ L L − κd m t L √ r (2( u + x ) + r ) (B164) A ′ = A ′ L L + 2 κm t L √ r ( d ( r − r ) r + d (2( u + x ) + r )) (B165) A ′ = A ′ L L − κd m t L √ r (2 x + r r ) (B166) A ′ = A ′ L L + 2 κd m t L √ r (2 x + r r ) (B167) A ′ = A ′ L L + 2 κd m t r L (B168) A ′ = A ′ L L − κd m t r L (B169) A ′ = A ′ L L − κm t L √ r (2 ud + 2 x ( d + d ) + d r r + d r ) (B170) A ′ = A ′ L L + 2 κd m t r L . (B171)The remaining coefficients for the case of [ P ] : A = L m t r r √ r ( d (1 + r )( r − r + r ( u + r ) + r ( − u + r (2 + r ))) + d ( r +2 xr (1 + r ) + r (7 r −
1) + r r (6 u + ( r − r ) + r ( u + r (9 r − − r (2 x (1 + r ) + r ( r (3 + 2 r ) − u ))) + 2 d r r (1 + r )( − ur − r − r ) r )) (B172) A = − L m t r r r ( d ( − r r + r ( r + v + x + z ) + r r (2 x + r (2( u + v + 2 x ) + r (3 r − r ( v + r (3 r − u − v − w − x + 5 z + 2) − r ( v + 2 x +3 r − r ( − v + r ))) + d ( r r + r ( r − v + 4 z ) + r ( v + 2 x + r + 6 r ) + r r (6 r − v − x + 13 z )) r + 2 d r r (2( v + x ) + r (2 + r + 3 r ))(2 ur + ( − r + r ) r )) (B173) A = L m t r r √ r ( − d ( r − r )(1 + r ) + d ( r + 6 r r + r )) (B174) A = 0 (B175) A = 4 d L m t √ r ( r − v − x + 3 z ) (B176) A = 4 L m t √ r ( d (1 + r )(1 + r ) + d (1 + r + 2 r ) r ) (B177) A = 8 d L m t √ r (1 + r ) (B178) A = 4 d L m t √ r (1 + r ) (B179) A ′ = − κm t L r r √ r (2 d r r (2 ur + ( r − r ) r )( x + y ) + d (4 x ( r − r )+( r + r (1 + 6 r ) + r (3 u + r + 4 r ) + r ( u + r (6 + 13 r ))) r +2 x ( u ( r − r ) + 4 r r )) + d ( r ( − u ( u + x ) − ( u + 2 x )(2 r − r − ur +2 r r + (1 − r ) r ) + r ( − r r + r (1 + r )( x + y ) + u ( u + x + y )))) (B180) A ′ = κm t L r r r (2 xd ( r − r ( v + ( − r ) r ) + r ( v + r (2 + r ))) r + d ( v ( r + 3 r ) + r ( r − r − r r + r (3 + 2 r ))) r + 2 d r r (2 ur +( − r + r ) r )(2 x + vr + r r + r ( r − y )) + d (4 x r ( r − r ) +2 x ( r + r − r ( v − r ) + r ( − u + v − − r ) r ) + r r ( − u + r − r )) + v ( r (1 + r ) + 2 r ( u + r ) + r ( − u + r − r )) r + r r ( − u ( r + r ( − r )) + r ( r − r +2 r r − r r − r − r r + r )))) (B181) A ′ = − ( κm t L r r √ r ( − xd ( r − r ) + d (2 r + r ( r − − r ( − u − x + r + r ) − r ( r + 2 u − v + x + z )) + d ( r + 3 r ) r + 2 d r r (2 ur + ( r − r ) r )))(B182) A ′ = 2 κd m t √ r L (B183)31 ′ = − κd m t L √ r (2( x + ( v + x ) r + vr ) + r r + r ( − u + x ) + r − r )) (B184) A ′ = − κm t L √ r ( − ( d ( − v + 2 x − r + r ( − r )) r ) + d (2( u + x ) − xr + r ( r + r − r r ))) (B185) A ′ = 2 κd m t √ r L ( x + y ) (B186) A ′ = κd m t √ r L (2( u + x ) + (1 + 2 r + r ) r ) (B187)and A = − A L L − L m t r r √ r ( − xd ( − r + r ) + 2 d r r (2 ur + ( − r + r ) r ) + d r (2 r r + r ) + d ( − ur + ur − r r + r r ( − r + r ))) (B188) A = − A L L + 2 L m t r r r (2 xd ( r − r )( − r + r ) + d r ( v (2 r − r ) +( r − r ) r ( − r − r + r )) + r ( − ( d r (2( r − r ) r + ( r + 3 r ) r )) +4 d r r ( − ur − ( − r + r ) r ))) (B189) A = − A L L − d L m t r r √ r ( − r + r ) (B190) A = − A L L (B191) A = − A L L + 8 d L m t r √ r (B192) A = − A L L + 8 L m t √ r ( d r + d r ) (B193) A = − A L L + 16 d L m t √ r (B194) A = − A L L + 8 d L m t √ r (B195) A ′ = A ′ L L + κm t √ r L r r ( d r (2( r − r ) r + ( r + 3 r ) r ) + d r ( x + y )) (B196) A ′ = A ′ L L − κm t L r r r (2 d r r (2 x + r r )(2 ur + ( − r + r ) r ) + d r ( r r (2 r r + r ) + 2 x (2 r r + r (2 r + r ))) + d r (2 xr + 2 xr r + r r ( v + 3 r − r r ) + r ( r r − yr + r (2 u + v + 2 x + 2 r )))) (B197) A ′ = A ′ L L − κd m t r L r r (B198)32 ′ = A ′ L L (B199) A ′ = A ′ L L + 2 κd m t L √ r (2 x + r r ) (B200) A ′ = A ′ L L + 2 κm t L √ r ( d r + d ( u + x + y )) (B201) A ′ = A ′ L L − κd m t r L . (B202)The remaining coefficients for the case of [ P ] state: A = − L m t r r √ r ( d (1 + r )( u + r + r ( − r )) r − d r ( u + ( − r ) r ) +2 r r ( − ( d ( − x + 2 u ( v + x ) + ( u − v − x ) r + r r − r + r (2( v + x ) + r + 4 r ) + r (2 u + w −
2) + r ( − v − x + r ( u − r − v + w + 3 z −
2) + r )))) + d ( x + r r + r ( − v − x + ( r − r ) + r ( r + u − v + 2 z )) r ))(B203) A = 2 L m t r r √ r ( d ( − x + r ( r + v + x + z ) − r ( r − v + 3 z )) r + x (2 d r r (2 u +( r − r + 3 r )) + d r ) + r ( d r + 2 d r ( ur (1 + 2 r + 3 r ) + v (2 u + r (2 r − − r + 3 r r + r ) + r (( r − r + 3 r ) + r ) r )))(B204) A = 2 L m t √ r r r ( d − d r + ( d + d ) r ) (B205) A = 2 L m t r r r (4 ud r ( r − r ) r + r (2 d r (1 + r )( − r − r ) r + d (2 r − r ( − r ) + r (3 + r )) + d ( r − r ) r )) (B206) A = 4 L m t r r √ r ( r r ( d (1 + r ) r + d (1 + 2 u + r r + 2 r + r )) − d (1 + r ) r )(B207) A = − L m t r r r ( d ( r (2 + r ) + r ) r + 2 d r r (2 ur + ( − r + r ) r )) (B208) A = − L m t r r r ( − d ( r ( − r ) − r (1 + r )) r + d r + 2 r r (4 ud r +2 d ( − r + r ) r + r ( d ( r + v + x + 3 z ) − d r r ))) (B209) A = 4 d L m t √ r ( r + v + x + 3 z ) (B210) A = 2 d L m t r r √ r (B211) A = L m t r r r (8 ud r r + r ( d ( − ( r ( − r )) + 3 r (1 + r )) +33 d r r ( − r + r ) + d ( r − r ) r )) (B212) A = 4 L m t √ r (2 ud + ( d ( − r + r ) + d ( − r + r )) r ) (B213) A = 4 L m t r r √ r ( − ud r r − ( d ( r − r ) + d r r ( − r + r )) r ) (B214) A = L m t √ r r r ( d − d r + 2 d r + d r − d r ) (B215) A = − L m t r r r ( d ( − r r + r (4( v + x ) + r ) + r ( v + 4 x + 3 r − r ( − u + 3 v + r )) + r ( v + 4 x − r ( u − v + w + x + 3 z − r + d ( r r − r ( v + r ) + r ( r − v − x + 3 z )) r + 2 d r r (2( v + 2 x ) + r (2 + r + 3 r ))(2 ur + ( − r + r ) r )) (B216) A = L m t r r r ( − d r r ( − u − ( − r + r ) r )(2 ur + ( − r + r ) r ) + d r ( u ( r − r ) − x (1 + r ) + ( r − − r ) r ) + d r ( u (4 r − r ( r − − ( r − r ) + r (( r − r )(1 + r + 2 r ) + r (1 + 2 r − r ) r + 2 r ))) (B217) A = L m t r r r ( d ( xr ( − r ) − r r − r ( r + v + x + 3 z ) + r ( − x + r ( r + 3 v + z ))) r − d r ( x + r r ) − d r r (2 x + r r )(2 ur +( − r + r ) r )) (B218) A = − L m t r r √ r ( d (1 + r ) r − d r + 2 r r ( d (1 + r )(1 + r ) − d ( r − v − x + 3 z ) + d r r )) (B219) A = − L m t √ r ( d + d r − d r + d r ) (B220) A ′ = − κm t L r r √ r (2 d r r ( − x ( x + ( u + x ) r ) + x (2 − v + 2 r − r ) r +( v − x )(1 + r ) r − vr + r r ( u + r + r r ) + r ( u + r + r r ) · ( − u + x ) + r − r )) + r (2 xd ( − u + r ) + d r ( u + 2 v + r + r + r r + r + r ) + d (2 u + 2 ux + ur − r r − r ( x + y ) + r (1 + r )( x + y )) − d r r ( − ( x (2( u + x ) + r )) − ( u + x ) r +( v − x ) r + r ( x + ( u + x ) r + (1 + r ) r )))) (B221) A ′ = κm t L r r √ r (2 d r r (2 x ( x + ( u + x ) r ) + x ( − v − r + r ) r − ( v − x )(1 + r ) r + vr − r r ( u + r + r r ) + r ( u + r + r r ) · u + x ) + r ( − r ))) + r ( d (2 x + xr + ( r − r ) r r ) + d ( − ( x (2( u + x ) + r )) + xr r + (2 v − x ) r − r (2( u + x ) + r ) + r ( x + 2( u + x ) r + (1 + 2 r ) r )))) (B222) A ′ = κm t √ r L r r ( d (2( u + x ) + r (2 r −
1) + r + 3 r r + r ) + d (2 x + r )) (B223) A ′ = κm t L r r √ r ( − ( d r (2 x + r )) + d r (2( u + x ) + r + r ( − r ) + r ) − d r r (2 x ( r −
1) + 2 u ( r − − r ) + ( r − r + ( r − r )) (B224) A ′ = κm t L r r √ r (2 d r r (2 x + r + r ( r −
1) + r + 2 r ) r − d r (2 x + r ) − d r r ( − xr + 2( u + x + ur ) + r ( r + r − r r ))) (B225) A ′ = κm t L r r r (2 d r r (1 + r )(2 ur + ( r − r ) r ) − d r ( − ( r (1 + r )) +(1 + r − r ) r + r (2 x + r + r + r ))) (B226) A ′ = κm t L r r √ r (4 xd r r + 4 xd r r − ud r r − xd r + 4 xd r r r +4 vd r r r + 2 d r r r − d r r r + 2 d r r r + 2 d r r r − d r r r + d r + 2 d r r (2 ur + ( − r + r ) r ) − d r (2( u + x ) − r (1 + 2 r ) + r + r (1 + 2 r + r ))) (B227) A ′ = − κm t L r √ r ( d ( r − r ) r + d r (2 x (1 + r ) − ur + 2 vr + r (1 + r − r ) r )) (B228) A ′ = κm t L r r r ( d ( r r − r ( z + r ) − r (2 v + r (2 + r ))) r +2 d r r (2 ur + ( − r + r ) r )) (B229) A ′ = − ( κm t L r r √ r ( d ( − u + x ) − r (1 + 2 r ) + r − r r + r ) r +2 d r r (2 ur + ( − r + r ) r ) + d r (2 x + r ))) (B230) A ′ = − κm t √ r L r r ( r r ( d r + d (2( u + x ) + 2 r + 5 r r + 3 r + r )) − d r ) (B231) A ′ = κd m t √ r L (2( u + x ) + (1 + 2 r + 3 r ) r ) (B232) A ′ = κm t L r r √ r ( d (2( u + x ) + 2 r + r ( − r ) + r + 3 r ) r +2 d r r (2 ur + ( − r + r ) r ) + d r (2 x + r )) (B233) A ′ = κm t L r r r (2 xd ( v + r + r ) r + d ( v + r − r r + 2 r ) r +35 d r r (2 ur + ( − r + r ) r )(4 x + r ( v + r + r − r r )) + d r ( v ( − u + 3 r (1 + r ) − r (3 + r )) r − r (2 u + 2 r + r ( r −
1) + r − r (1 + r (4 + r ) + 2 r ) + r ( − r + 2( u + r ))) r +2 x (2 r ( r − r ) + ( − v + r ) r ))) (B234) A ′ = − κm t L r r √ r (2 d r r ( u + 2 x + r (2 + r ) + 2 r + 4 r r + 3 r )(2 ur +( − r + r ) r ) + d r ( − u − u (2 r + r ( − r ) + 3 r (1 + r )) − x ( u + r + r ( r − − (( r − r − r + 2 r + 4 r r + 3(1 + r ) r ) r + r r ) + d r (4 x + ( u + r + r − r + 3 r r ) r + 2 x ( u + 2 r ))) (B235) A ′ = − ( κm t L r r √ r ( d (2 x ( u + x ) + xr + 4 r r − (4 u + 2 v + x ) r − r − r + r ( − v + ( − r ) r ) − r ( x + r (6 r − u − v − x + 6 z ))) r + d (2 x + r − r ) r ( x + r r ) + 2 d r r ( x + 2 r r )(2 ur + ( r − r ) r ))) (B236) A ′ = − ( κm t L r r √ r ( − ud r r + 4 vd r r r + 2 d r r r − d r r r +2 d r r r + 2 d r r r − d r r r + d r + d r (2( u + x ) + r + r ( − r ) + r ) + 2 d ( − r ) r r (2( u + x ) + r (1 + r + r )) +2 x ( − d r + 2 r r ( d + d r − d r + d r )))) (B237) A ′ = − κm t √ r L r r ( r r ( d r + d (2( u + x ) + 2 r + 3 r r + r + r ) − d r )) (B238)and A = − A L L − L m t r r √ r (2 r r ( d ( xr + xr − r r + r ) + d ( − r r + r ( z + r ) + r (3 z − u − v ))) + d ( u + r + r ) r + d r ) (B239) A = − A L L + 4 L m t r r √ r ( d ( − x + ( r − r ) r ) r +2 d r r ( − ( v + 2 x ) r − r r + r ( x + 2 r r ) − r ( − u + x + r ))) (B240) A = − A L L + 4 d L m t √ r r r (B241) A = − A L L − L m t r r √ r ( − d r + 2 d r r r ) (B242) A = − A L L + 8 L m t r r √ r ( − ( d r ) + d r r r + d r r r ) (B243) A = − A L L − d L m t r r √ r ( r + r ) (B244)36 = − A L L − L m t r r √ r (2 d r r + 3 d r ) (B245) A = − A L L + 8 d L m t r √ r (B246) A = − A L L (B247) A = − A L L + 6 d L m t √ r r r (B248) A = − A L L − d L m t √ r (B249) A = − A L L + 8 d L m t √ r (B250) A = − A L L + 2 d L m t √ r r r (B251) A = − A L L − L m t r r r ( d r (2 r r + ( v + 4 x + 3 r ) r − r r (2 r + r )) + r ( d ( r − r ) r + 4 d r r (2 ur + ( − r + r ) r ))) (B252) A = − A L L + 2 L m t r r √ r ( − d r r (2 ur + ( − r + r ) r ) − d r (2 x + r )+ d r (3 u − r + 2 r + r ( r + r ))) (B253) A = − A L L − d L m t √ r r r (3 x + ( r − r ) r ) (B254) A = − A L L − L m t r r √ r ( d r + 2 r r ( − ( d r ) + d r )) (B255) A = − A L L − d L m t √ r (B256) A ′ = A ′ L L + κm t L r r √ r (2 d r r ( − x − xr ( − r + 2 r + r ) + r ( − ur + vr − r r )) + r ( d r + d ( r − r )( x + y ) + 2 d r r ( xr − r ( x + r )))) (B257) A ′ = A ′ L L + κm t L r r √ r ( − d r ( − xr + r ( x + r )) +2 d r r ( − x + x (2 r − r − r ) r + r ( − ur + vr − r r ))) (B258) A ′ = A ′ L L − κd m t √ r L r r ( − r + r ) (B259) A ′ = A ′ L L − κm t L r r √ r ( d ( r − r ) r + 2 d r r (2( u + x ) + r )) (B260) A ′ = A ′ L L + 2 κm t L √ r ( d ( r − r ) r + d ( u + x + y )) (B261)37 ′ = A ′ L L − κm t L r r r ( d r ( r − r r − r r ) +2 d r r (2 ur + ( − r + r ) r )) (B262) A ′ = A ′ L L − κm t ( d ( − r + r ) r + 2 d r r (2 x + r r )) L r r √ r (B263) A ′ = A ′ L L + 2 κd m t L √ r (2 x + r r ) (B264) A ′ = A ′ L L + 2 κd m t √ r L r (B265) A ′ = A ′ L L − κd m t √ r L r r ( r − r ) (B266) A ′ = A ′ L L + 2 κd m t r L (B267) A ′ = A ′ L L − κd m t r L (B268) A ′ = A ′ L L + κd m t √ r L r r ( r − r ) (B269) A ′ = A ′ L L − κm t L r r r (2 d r r (4 x + r r )(2 ur + ( − r + r ) r ) + d r (2 xr + r r ) + d r (4 xr ( r − r ) + r r (3 v + r − r r ) − r r (4 u + 2 v + x − y + z ))) (B270) A ′ = A ′ L L + κm t √ r L r r ( d ( r − r ) r + 4 d r r (2 ur + ( − r + r ) r ) + d ( − xr + 2 r r − r ( u + r ) + r (3 u + 2 x − r r + r ))) (B271) A ′ = A ′ L L − κd m t √ r L r r ( xr − xr + 3 r r ) (B272) A ′ = A ′ L L − κm t L r r √ r ( d ( r − r ) r + 2 r r ( d ( u + x + y ) − d (2 x + r r )))(B273) A ′ = A ′ L L + 2 κd m t r L . (B274)Here for convenience an overall factor C s has been contracted out from these coefficients.Then the square of the amplitude | M i | can be conveniently obtained with the help of38qs.(B26,B27,B28) and Eqs.(34, 35,36). [1] CDF Collaboraten, F. Abe, et al. , Phys. Rev. Lett. , 2432 (1998); Phys. Rev. D58 , 112004(1998).[2] Chao-Hsi Chang and Yu-Qi Chen, Phys.Rev. D , 4086(1993); Chao-Hsi Chang, Yu-Qi Chen,Guo-Ping Han and Hong-Tan Jiang, Phys.Lett. B , 78(1995); Chao-Hsi Chang and Xing-Gang Wu, Eur.Phys.J. C , 267(2004).[3] A.V. Berezhnoi, A.K. Likhoded and M.V. Shevlyagin, Phys. Atom. Nucl. , 672(1995); S.S.Gershtein, V.V. Kiselev, A.K. Likhoded, A.V. Tkabladze, Phys. Usp. , 1(1995).[4] Chao-Hsi Chang, Jian-Xiong Wang and Xing-Gang Wu, Phys.Rev. D , 114019(2004);Chao-Hsi Chang, Cong-Feng Qiao, Jian-Xiong Wang and Xing-Gang Wu, Phys.Rev. D ,074012(2005).[5] Chao-Hsi Chang, Chafik Driouich, Paula Eerola and Xing-Gang Wu, Comput. Phys. 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