Dijet Invariant Mass Distribution in Top Quark Hadronic Decay with QCD Corrections
aa r X i v : . [ h e p - ph ] N ov Dijet Invariant Mass Distribution in Top Quark Hadronic Decaywith QCD Corrections
Hua-Sheng Shao ( a ) , Yu-Jie Zhang ( b ) and Kuang-Ta Chao ( a, c ) (a) Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China(b) Key Laboratory of Micro-nano Measurement-Manipulationand Physics (Ministry of Education) and School of Physics,Beihang University, Beijing 100191, China(c) Center for High Energy Physics,Peking University, Beijing 100871, China Abstract
The dijet invariant mass distributions from the hadronic decay of unpolarized top quark ( t → bW + followed by W + → u ¯ d ) are calculated, including the next-to-leading order QCD radiativecorrections. We treat the top decay in the complex mass scheme due to the existence of theintermediate state W boson. Our analytical expressions are also available in different dimensionalregularization schemes and γ strategies. Finally, in order to construct the jets, we use different jetalgorithms to compare their influences on our results. The obtained dijet mass distributions fromthe top quark decay are useful to distinguish these dijets from those produced via other sourcesand to clarify the issue about the recent CDF Collaborations’ W jj anomaly.
PACS numbers: 12.38.Bx,12.38.-t,14.65.Ha . INTRODUCTION Since the discovery of the top quark at the Tevatron[1, 2], the top quark has played aspecial role in searching for the electroweak symmetry breaking mechanism and new physicsbeyond the standard model. This can be attributed to the large mass of the top quark (about173 GeV ), which is almost 40 times larger than the next heaviest quark. As the Cabibbo-Kabayashi-Maskawa matrix element V tb approaches to 1, the top quark decays almost toa bottom quark and a W boson. Its decay width [3–6] is O ( GeV ), much larger than thetypical QCD scale Λ
QCD ∼ M eV , indicating that the top quark decay takes place beforehadronization. Therefore, nonperturbative effects are not important in the properties of thetop quark, and one can perturbatively calculate its physical quantities precisely, such astop quark’s spin correlation. At the Large Hadron Collider (LHC) at CERN, thousands oftop quarks are expected to be produced per year at 14 TeV. Hence a new era in top quarkresearch has arrived.On the other hand, very recently a dijet bump around 150 GeV in the
W jj channelhas been observed by the Collider Detector at Fermilab (CDF) at the Tevatron[7], and ithas attracted a lot of attention. There are some explanations within the standard modelfor this anomaly[8–10]. Some studies may indicate that single top production may play animportant role in the CDF dijet excess. Moreover, the D0 Collaboration reported that theirresults were consistent with the standard model’s prediction in the same channel[11]. Hencea careful investigation regarding the dijet in the single top production and decay is helpful.Even without this CDF anomaly, it is still useful to study the dijet distribution in the topquark decay, as a part of investigations for the top quark properties. Inspired by this, in thepresent study we will investigate the dijet mass distribution in the top quark decay. Thiswork also aims at understanding the properties of top quarks.There are a lot of works already about top quark decays[4, 12–28]. Generally, the QCDnext-to-leading order (NLO) radiative corrections to the top quark’s width amount to about − . − .
05% and 1 .
54% respectively. The nonvanishing m b effects [23–26] and finite width corrections[27] reduce the Born level width by about 0 .
27% and 1 . γ schemes. Section III tackles our scheme-independent analyti-cal expressions. Jet algorithms are recalled in Sec. IV and Sec. V discusses the results. Thefinal section contains the conclusion. II. DIMENSIONAL REGULARIZATION SCHEMES AND γ SCHEMES
Dimensional regularization has many advantages in dealing with ultraviolet, infrared andmass divergences encountered in high-order calculations in a unified manner. However,there are still some freedoms to handle these divergences in dimensional regularization. Inthis section we recall four modern versions of frequently used schemes and adopt the firstthree in the rest of this paper. The four schemes include the conventional dimensionalregularization (CDR), the ’t-Hooft-Veltman scheme (HV)[29], the four dimensional helicityscheme (FDH)[30–34], and the dimensional reduction scheme (DR)[35].In CDR, only the d = 4 − ǫ dimensional metric tensor is introduced, i.e. g µµ = d .The loop momentum and the spins of vectors, regardless of whether they are ”observed”or ”unobserved” are in d dimensions, whereas the spins of the spinor are in d s dimensionswith d s ≥ d . In this section, the observed states refer to the external states appearing in thehard part of the process without any subsequent hadronization. We treat d s of fermions asfour because it is distinct from d and always appears as a global factor in computations.HV and FDH have many advantages in helicity amplitude calculations, while FDH andDR are two supersymmmetric preserving schemes[31, 33–35]. We describe the schemes in aunified way as explained below: • To maintain the gauge invariance, all momentum integrals are integrated in d dimen-sions. • The dimensions of all observed particles (hard and noncollinear external particles) areleft in four dimensions. We call the hard and non-collinear external particles observed states and internal, soft, or collinear externalparticles unobserved states in this context. The dimensions of all unobserved particles (internal states and soft or collinear externalstates ) are treated in d s dimensions . Any explicit factors of dimension arising fromthese state should be labeled as d s temporary; these must be kept distinct from d atthe beginning.We treat internal states with d > d < d dimensions can be divided into a four-dimensional part and d − d and 4 − d -dimensionalquantities in DR. The expressions are analytic functions of d and they are continued to anydesired regions. Setting d s = d denoted in the above items, we obtain the HV scheme, whilesetting d s = 4 results in the FDH and DR schemes. As mentioned above, the d s arising fromdimensions of spinor space is just a global factor. Therefore, we can set this part of d s to beequal to 4. All of the above is summarized in Table I. TABLE I: Summary of dimensions in different regularization schemes.
Regularization schemes CDR HV FDH DR
Dimensions of momenta of observed particles d Dimensions of momenta of unobserved particles d d d d
Number of polarizations of observed massless vector bosons d − Number of polarizations of unobserved massless vector bosons d − d − Number of polarizations of observed massive vector bosons d − Number of polarizations of unobserved massive vector bosons d − d − Number of polarizations of fermions γ . γ = i ε µνρσ γ µ γ ν γ ρ γ σ , which is well defined in four dimensions. However, there are some problemswith this definition because antisymmetric tensor ε µνρσ lives in four dimensions only. In thenaive definition of γ , some obviously inconsistent equalities appear. If we keep all thefour-dimensional rules and cyclicity of the trace, the analytic continuation is forbidden[36].Therefore, one should at least change one of the properties to obtain a consistent result.To the best of our knowledge, there are two kinds of well-known γ strategies that have4een introduced; one is proposed by ’t-Hooft and Veltman and proved by Breitenlohnerand Maison [29, 37–39] (we call it the BMHV scheme), and the other one is introduced byKorner, Kreimer and Schilcher [36, 40, 41](we call it the KKS scheme).As a compromise, in the BMHV scheme the anticommutation relationship between γ and γ µ is violated, i.e. { γ , γ µ } 6 = 0. In fact, every d -dimensional quantity can be dividedinto a four-dimensional part and a − ε part, which implies that in this scheme d > γ anticommutes with a four-dimensional γ -matrix, while it commutes with a − ε -dimensional γ -matrix. This definition results in some ambiguousness of chiral vector current treatment,e.g. γ µ ± γ = ∓ γ γ µ in tree-level Feynman rules. For the current work, we take thesymmetric version as presented in [42, 43], i.e. γ µ − γ → γ γ µ − γ ,γ µ γ → − γ γ µ − γ γ µ . (1)The violation of anticommutation is also a violation of the Ward identity in axial-vectorcurrents. To prevent such a violation, additional renormalization is needed[3] (Readers whoare interested in dimensional renormalization issues can also refer to Refs.[44–49]). This willbe used in the next section. Although it is the first rigorously proven consistent scheme, theprocess of isolating four-dimensional and − ε parts in the Lorentz space often suffers fromcomplex practical calculations.On all accounts, the strategy of covariance violation in γ has some disadvantages incomplicated situations. On the other hand, the KKS scheme keeps the covariant anticom-mutations but forbids the cyclicity in the trace. In γ -matrix algebra (Clifford algebra), thereis a unique generator, which anticommutes with all other generators in infinite dimensions.This generator can be defined as the γ . To avoid the cyclicity in the trace, the ”readingpoint” must be chosen first, and all γ are moved to this point before a trace is taken. Thiscompromise recovers a correct anomaly as well.Finally, we also introduce the renormalization constants and the splitting functions inthe CDR, HV, and FDH dimensional regularization schemes used in this paper. In order toavoid calculating external self energy diagrams, we choose the on-shell scheme for external5egs. These constants are δZ OSt = − α s C F π (cid:18) ǫ UV + 2 ǫ IR − γ E + 3 ln (cid:18) πµ m t (cid:19) + 4 + 1 F DH (cid:19) ,δZ
OSq = − α s C F π (cid:18) ǫ UV − ǫ IR (cid:19) , (2)where δZ OSt , δZ OSq are on-shell(OS) wave function renormalization constants for top quarkand light quarks, respectively, γ E is the Euler constant, and 1 F DH is only nonvanishing inFDH scheme. The unpolarized Altarelli-Parisi splitting functions [50, 51] to O ( ǫ ) in HVand CDR schemes are all listed in the following: P qq ( z ) = C F z − z − ǫ C F (1 − z ) ,P gq ( z ) = C F − z ) z − ǫ C F z,P gg ( z ) = 2 N c (cid:18) z − z + 1 − zz + z (1 − z ) (cid:19) ,P qg ( z ) = z + (1 − z ) − ǫ z (1 − z ) , (3)while in FDH and DR these terms should be P qq ( z ) = C F z − z ,P gq ( z ) = C F − z ) z ,P gg ( z ) = 2 N c (cid:18) z − z + 1 − zz + z (1 − z ) (cid:19) + ǫ N c z (1 − z ) ,P qg ( z ) = z + (1 − z ) − ǫ z (1 − z ) . (4) III. SCHEME INDEPENDENCE AND ANALYTICAL EXPRESSIONS
As emphasized in Sec.II, there are some degrees of freedom to regularize possible di-vergences. Because of unitarity in QCD cross sections[51], we should expect the schemeindependence of the well-defined physical results. In this section, analytical results areprovided for top quark decay and subsequent hadronic decay, thus affirming the simplicityof these processes. Moreover, we also demonstrate that the off-shell effect in the top quark Our equations are the same as those in ref.[51]. The discrepancies in the O ( ǫ ) parts and refs.[31, 52, 53]were carefully discussed in ref.[51]. A. Corrections To t → bW + We first reproduce the well-known QCD corrections to the top quark decay[3–6](Feynmandiagrams generated by FEYNARTS [54] are shown in Fig.1). Because of the Cabibbo-Kabayashi-Maskawa (CKM) matrix elements 1 ≈ | V tb | ≫ | V ts | , | V td | , the branching ratioof t → bW + is almost 100%. For simplification, we set the CKM matrix to be diagonaland the mass of the b-quark equal to zero. As presented in previous works, the effect ofnonvanishing mass of the bottom quark is negligible. Following the notations of Ref.[5], thematrix element of the tree-level process t ( p t ) → b ( p b ) W + ( p W ) with averaging over the topquark’s spin and color is given by |M | = e m t s w m W (cid:0) − r (cid:1) (cid:0) r (cid:1) , (5)where r = m W m t and s w is the sine of Weinberg angle. We can get the leading-order widtheasily Γ = αm t s w m w (cid:0) − r (cid:1) (cid:0) r (cid:1) , (6)where we have used the electromagnetic fine-structure coupling constant α = e π .To check the regularization scheme independence of these results, we first derive theaveraged squared matrix element in the FDH regularization scheme within the naive orKKS γ scheme. The virtual terms and counter-terms for renormalization are given by (cid:0) |M v | + |M ct | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) − ǫ − − − r ) ǫ − − π − r ) − ln(1 − r ) r + 1 − r r (1 + 2 r ) ln(1 − r ) − (1 − r ) − ( r ) (cid:3) . (7)In order to see the scheme-dependent terms, we subtract the expressions in other schemesby the expressions in FDH with KKS γ treatment and use δ |M v/ct/real | . = |M v/ct/real | − |M v/ct/real | (cid:1) KKSF DH . These scheme-dependent terms are (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV = −|M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) KKSCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ
11 + 2 r (cid:20) r ǫ + 8 r − − r ln(1 − r ) (cid:21) , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVF DH = |M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVHV = |M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ
11 + 2 r (cid:20) r ǫ + 3 + 16 r − r ln(1 − r ) (cid:21) . (8)These scheme-dependent terms should be canceled exactly with real corrections originatedfrom soft and collinear regions. In process t ( p t ) → b ( p b ) W + ( p W ) g ( p g ), the real correctionexpressions in different schemes after integrating over the momentum of the radiative gluonare given by (cid:0) |M real | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) ǫ + − − r ) ǫ − π − r − r − r ) (1 − r )) − − r ) + 2 ln (1 − r ) − r (1 + r )(1 − r )(1 + r ) (1 + 2 r ) ln( r ) + 2 Li (1 − r ) (cid:21) , (cid:0) δ |M real | (cid:1) BMHVF DH = 0 , (cid:0) δ |M real | (cid:1) KKS/BMHVHV = |M | α s C F π = − (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV , (cid:0) δ |M real | (cid:1) KKS/BMHVCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ
11 + 2 r (cid:20) − r ǫ − r + 1 + 8 r ln(1 − r ) (cid:21) = − (cid:0) δ |M v | + δ |M ct | (cid:1) KKSCDR . (9)Combining all the results above, we find that the results in the three-dimensional regular-ization schemes in the KKS γ strategy are the same; however these are not consistent withthe BMHV γ scheme at present. In the BMHV γ scheme, the violation of anticommuta-tion also violates the Ward identities, which is also pointed out in Ref.[3]. Furthermore, to8 b W + Born t b W + g Virtual t b W + CT t b W + g Real 1 t b W + g Real 2
FIG. 1: Feynman diagrams in t → bW + . maintain the Ward identities, finite renormalization is made for axial-vector currents, (cid:0) Γ renµ (cid:1) F DH = (cid:18) − α s C F π (cid:19) Γ bareµ , (cid:0) Γ renµ (cid:1) HV/CDR = (cid:18) − α s C F π (cid:19) Γ bareµ , (10)where Γ µ represents the axial-vector current.Thus far, we get the unique result Γ = Γ (cid:26) − α s C F π (cid:20) π − − − r ) + 13(1 + 2 r ) − r − r )+2 ln( r ) ln(1 − r ) + 22 − r − r ) ln( r ) + 3 ln(1 − r )1 + 2 r − r )9(1 + 2 r ) + 4 Li ( r ) (cid:21)(cid:27) . (11)If we set r ≈ .
46, we get the well-known K factor (1 − . α s ). In general, we should include finite renormalization terms of coupling constants in FDH related to con-ventional MS scheme [31] to obtain the unique physical result. However, all of our processes underconsideration are only O ( α s ) at the QCD one-loop level. This finite renormalization is absent in ourcalculations. . Corrections To W + → u ¯ d With the same procedure described in the previous subsection, we obtain the analyticalresults for the subsequent decay of W boson[55, 56]. We labeled the momenta of the W boson,up (charm) quark, and down (strange) quark as p W , p u , p d respectively. The diagonalizationof CKM matrix and vanishing mass of light quarks guarantee a factor of 2 to the W boson’shadronic decay channel via the process W + → u ¯ d . The diagrams of QCD correction to thisprocess are all shown in Fig.2.The lowest-order squared matrix element and decay width are |M | = e m W s w , Γ = α m W s w . (12)The contributions of virtual terms and counter-terms are (cid:0) |M v | + |M ct | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m W (cid:19) ǫ (cid:18) − ǫ − ǫ + π − (cid:19) , (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV = −|M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) KKSCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m W (cid:19) ǫ (cid:18) ǫ + 1 (cid:19) , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVF DH/HV = −|M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m W (cid:19) ǫ (cid:18) ǫ + 2 (cid:19) . (13)For real corrections after the phase space integration over radiative gluon momentum, wearrive at (cid:0) |M real | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m W (cid:19) ǫ (cid:18) ǫ + 3 ǫ + 172 − π (cid:19) , (cid:0) δ |M real | (cid:1) BMHVF DH = 0 , (cid:0) δ |M real | (cid:1) KKS/BMHVHV = |M | α s C F π = − (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV , (cid:0) δ |M real | (cid:1) KKS/BMHVCDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m W (cid:19) ǫ (cid:18) − ǫ − (cid:19) = − (cid:0) δ |M v | + δ |M ct | (cid:1) KKSCDR . (14)After including the renormalization of the axial-vector current in the BMHV γ scheme,we obtain the scheme-independent answer for the decay width of process W + → u ¯ d ,Γ = Γ (cid:16) α s π (cid:17) . (15)10 + u ¯ d Born W + u ¯ dg Virtual W + u ¯ d CT W + u ¯ dg Real 1 W + u ¯ dg Real 2
FIG. 2: The Feynman diagrams in W + → u ¯ d . C. Corrections To t → bu ¯ d In this subsection, we present the analytical expressions of the top quark hadronic decay.The corresponding graphs are shown in Fig.3. As the mass of the top quark is 30 timeslarger than that of the bottom quark, we set the masses of all final states to be zero. Theeffect of the nonzero mass of the bottom quark is negligible in our results. Because of theintermediate-state W boson in t ( p t ) → bW + → b ( p b ) u ( p u ) ¯ d ( p d ), we treat this process inthe complex mass scheme[57, 58]. The Born amplitude squared with averaging over theinitial-state spin and color is given by |M | = 3 e | s w | (1 − y ) y (1 − y − z − r ) + ( r w ) , (16)where we have defined ( p t − p d ) = m t y, ( p t − p u ) = m t z, r = m W m t , w = Γ W m W . Here, y, z, r, w are all dimensionless variables. We keep the width of the W boson nonvanishing. The Born11evel decay width of this channel isΓ = α m t π | s w | (cid:2) r − r (cid:0) w r − r + 3 (cid:1) ln( r )+ r (cid:0) w r − r + 3 (cid:1) ln( 1 + w (1 − r ) + ( wr ) )6 w r − r − w r + 3 r − wr (cid:18) tan − ( w ) + tan − ( wr − r ) − π (cid:19)(cid:21) . (17)By expanding it in terms of w , to O ( w ) the above result can be expressed asΓ = α m t | s w | r (cid:2)(cid:0) − r (cid:1) (cid:0) r (cid:1) w − + (cid:16) r (1 − r ) ln( r − r ) + 6 r − r − (cid:17) π w + O ( w ) . (18)To leading order in w the result is consistent with the narrow-width-approximation (NWA),Γ t → bu ¯ d = Γ t → bW + × Γ W + → u ¯ d Γ W , with the Born width formulas of the top quark and W bosonexhibited in two previous subsections. The second term is an off-shell correction, which isabout − . w relative to the first term with r ≈ . γ strategy within FDH regularization scheme, at QCD one loop level the squared matrix elements after renormalization with the initial-state averaged is given by (cid:0) |M v | + |M ct | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) − ǫ + 4 ln(( y + z )(1 − y − z )) − ǫ + 6 ln(( y + z )(1 − y − z )) − (1 − y − z ) + 4 ln(1 − y − z ) ln( y + z ) − ( y + z ) − y + z )1 − y + 4 Li ( y + z ) + π − (cid:21) . (19)With the same rules as those stated in the previous subsections, the differences betweenother γ strategies/regularization schemes and the FDH scheme in KKS γ scheme are given Because of color flow,the W boson propagator is not involved in loops. The scalar one loop integrals withreal masses encountered in this process were already illustrated in Ref.[59]. However, some analyticalcontinuations should be made in calculating scalar one-loop integrals with complex arguments contrastto the ones with real arguments. (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV/CDR = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) − y − z )( y + z ) y (1 − y ) 1 ǫ − y (1 − y ) (cid:0) y + 22 yz + 11 z − y − z +4(1 − y − z )( y + z ) ln[(1 − y − z )( y + z )])] , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVF DH = |M | α s C F π , (cid:0) δ |M v | + δ |M ct | (cid:1) BMHVHV/CDR = |M | α s C F π . (20)To check our results, we also treat the numerators of loop amplitudes in four-dimensionsby adding the R terms at last. All of the results discussed above are recovered using thismethod. Because of the right-handed currents[60] of the R in the BMHV γ scheme, theunrenormalized virtual contributions are the same within the same γ treatment, and onlythe renormalization constants are different.The remaining regularization scheme-dependent terms should be canceled by the realradiation part. The scheme-dependent terms in real corrections resulted from the soft andcollinear region of phase space. The two cutoff phase space slicing method given by B.Harrisand J.Owens is used here[61]. The analytical result within the FDH and KKS regularizationscheme is given by (cid:0) |M sc | (cid:1) KKSF DH = |M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) ǫ − − y − z )( y + z )] − ǫ + 2 ln [ y z − y − z ] − (1 − y ) − (1 − z ) − ( y + z )+4 ln[(1 − y )(1 − z )( y + z )] ln( δ s δ c ) − δ c ) − δ s ) −
12 ln( δ s ) ln( δ c ) + 8 ln[(1 − y − z )( y + z )] ln( δ s )+6 ln ( δ s ) + 4 Li [ − − y − zy z ] + 22 − π (cid:21) , (21)where δ s and δ c are two parameters to isolate the soft and collinear regions, respectively.13he differences between other regularization schemes and the above scheme are (cid:0) δ |M sc | (cid:1) KKSHV/CDR = −|M | α s C F π Γ(1 − ǫ ) (cid:18) πµ m t (cid:19) ǫ (cid:20) − y − z )( y + z ) y (1 − y ) 1 ǫ − y (1 − y ) (cid:0) y + 22 yz + 11 z − y − z +4(1 − y − z )( y + z ) ln[(1 − y − z )( y + z )])]= − (cid:0) δ |M v | + δ |M ct | (cid:1) KKSHV/CDR , (cid:0) δ |M sc | (cid:1) BMHVF DH = 0 , (cid:0) δ |M sc | (cid:1) BMHVHV/CDR = |M | α s C F π . (22)In the BMHV γ scheme, we also obtain the scheme-independent results after includingthe finite renormalization to the axial-vector currents. This was done in order to maintainthe Ward identities as already shown in the last two subsections.In the hard noncollinear phase space region, we treat the squared matrix element of t ( p t ) → b ( p b ) u ( p u ) d ( p d ) g ( p g ) in four dimensions. Dimensionless variables are redefined asfollows: ( p t − p g ) = m t x, ( p t − p u ) = m t y, ( p t − p d ) = m t z, ( p u + p d ) = m t k, ( p u + p g ) = m t l, r = m W m t , w = Γ W m W . (23)The averaged squared amplitude is |M real | = 3 e g s C F | s w | m t (cid:26) k − r ) + ( wr ) − x ) (1 − y − z − k ) (cid:2) ( x −
3) ( k + l ) k + 2 ( xz − x − y − z + 4) k + (cid:0) (2 xz − x − y − z + 11) l − x − y − z + 4 y − yz + 15 z + (cid:0) y − y + z − z + 6 (cid:1) x − (cid:1) k + (1 − z ) (2 − x − y − z ) (1 − x − z ) − ( x + 2 y + 2 z − l − ( x + 2 z −
2) ( x + 2 y + 2 z − l (cid:3) + 1(2 − x − y − z − r ) + ( wr ) l ( x + y + z + k + l − (cid:2) (2 − x − y − z ) (1 − z − k − l ) − ( k + l ) (1 − z − k − l )+ (1 − y ) ( l − y (2 − x − y − z ))] } (24)There are two kinds of Breit-Wigner distributions of the W boson in Eq.(24). The firstterm originated from the first two real diagrams, while the second is contributed by the last14 bu ¯ d W + Born t bu ¯ d W + g Virtual 1 t bu ¯ d W + g Virtual 2 t bu ¯ d W + CT 1 t bu ¯ d W + CT 2 t bu ¯ d W + g Real 1 t bu ¯ d W + g Real 2 t bu ¯ d W + g Real 3 t bu ¯ d W + g Real 4
FIG. 3: Feynman diagrams in t → bu ¯ d . two final state radiative diagrams. Because of color flow, there is no interference observedbetween the first two and the last two diagrams. IV. JET ALGORITHMS AND PHASE SPACE
At high energy colliders, it was pointed out that the observed jets provided a view ofparton (e.g. gluon and quark) interactions occurring at short distances[62]. At leading-order (LO) level, partons can be naively treated as jets, while at NLO level this coarsetreatment often suffers from soft and collinear divergences. Therefore, an infrared-collinearsafe jet definition is necessary in investigating strong interaction physics. Nowadays, these15et definitions play important roles in collider physics. Following the jet definition descriptionin Refs.[63, 64], the requirements implemented in a jet algorithms are as follows: • simple to use in experiments and theoretical calculations, • infrared and collinear safe, • small hadronization corrections.At hadron colliders, a well-defined jet algorithm must be able to factorize initial-statecollinear singularities; they should also be isolated from the contamination of hadron rem-nants and underlying soft events.Since the advent of jet production in electron-positron and hadron colliders, it has be-come one main tool in QCD research. Many kinds of algorithms have been proposedand developed. Essentially, the two classes of jet algorithms present mainly the clus-tering algorithms[65, 66] and the cone-type algorithms[62, 67–69]. In the present study,we focused on the three popular inclusive clustering algorithms, namely the k ⊥ -clusteringalgorithm[63, 64, 70],the Cambridge / Aachen clustering algorithm(CA)[71, 72] and the anti- k ⊥ clustering algorithm[73] respectively. These three inclusive clustering algorithms can bedescribed uniformly: • Define a distance d ij = min ( p rT i , p rT j ) ∆ R ij R between each pair of protojets i and j, aswell as a distance d iB = p rT i between each protojet i and the beam,with r = +1 , , − k ⊥ ,CA, and anti- k ⊥ respectively. • Find the smallest of all the d ij and d iB and label it as d min . • If d min is a d ij , then cluster protojets i and j as a new protojet with a selected com-bination procedure. If the distance between protojet i and the beam is the shortest,set the protojet i aside and leave it without any further clustering as a possible jetcandidate. • Repeat the items above until there is no protojet left. • Perform some cuts (as in the experiment) to select jet(s) of interest.Here ∆ R ij = q ( η i − η j ) + ( φ i − φ j ) ( η and φ are rapidity and azimuthal angle respec-tively). As E can be measured at e + e − colliders rather than only p T at hadronic colliders,16ne should use E instead of p T and ∆ S ij = r ∆ θ ij + (cid:16) sin θ i + θ j ∆ φ ij (cid:17) instead of ∆ R ij at e + e − colliders.It was also emphasized in Ref.[70] that traditional cone-type jet algorithms were relatedto clustering algorithms by the approximation R cluster = 1 . × R cone .In the present study, we only used the three clustering algorithms with the E-schemerecombination to reconstruct our leading two jets from top quark hadronic decay in thenext section (one can also use other recombination procedures as suggested in Ref.[63] andreferences therein). In addition, we used hadron collider clustering algorithms and electron-positron collider clustering algorithms but without any cut in our calculation.The last topic of this section is about a phase-space integration treatment. Given that weshould reconstruct the four momenta of all final states in order to reconstruct two leadingjets, we built up the n-particle phase space iteratively by nested integration over invariantmasses and solid angles of outgoing particles, similar to the strategy in Ref.[74]. V. RESULTS
The dijet invariant mass distributions with different clustering jet algorithms are pre-sented in this section.As discussed in the previous section, we used two variations of clustering jet algorithmsin our top decay process in the c.m. frame of the top quark. In these two variations,we chose the distances defined at hadron colliders ( i.e. use p T ) and e + e − colliders( i.e.use E ), respectively, to reconstruct the final jets. Afterward, two leading jets in energy E were chosen to construct their invariant mass m jj . Here,we call the first types KT1, CA1,anti-KT1, while the second types are denoted as KT2, CA2, anti-KT2.The following input parameters are used: α − = 129 , α s ( m Z ) = 0 . ,m W = 80 . GeV, Γ W = 2 . GeV, m Z = 91 . GeV, Γ Z = 2 . GeV,s w = 1 − m W − i m W Γ W m Z − i m Z Γ Z = 0 . − . × − i, (25)with two groups of top quark mass and renormalization scale µ choices, i.e. m t = 175 GeV ,17 = 80 . GeV and m t = 172 . GeV , µ = m t = 172 . GeV .We varied the parameter R in the CA1 jet algorithm and compared its influence on ourresults in Fig.4. Only when R ≥ . R = 1 . R = 1 . R slightly changed our domainregion (110-150 GeV) both in LO and NLO level. The larger R reconstructs a smallernumber of final jets; it makes the number of events in the last bin (170-175 GeV) larger withlarger R . At LO, the distributions dropped sharply below 110 GeV and vanish below 100GeV, as shown on the upper panel of Fig.4. In contrast, a NLO QCD correction resulted inthe smooth descent of the low energy tail. The peak in Fig.4 (lower panel) between 80 GeVto 85 GeV is the W boson’s resonance.Histograms in Figs.5 and 6 establish the influences of clustering jet algorithms to thedijet invariant mass distribution. The LO distributions reconstructed by various algorithmsare almost indistinguishable. In comparison,there are some differences in the substructuresof NLO histograms. The combination sequence of protojets is responsible for these tinydistinctions . Soft protojets may be clustered before the hard ones in k ⊥ , while the situationmay be totally different in anti- k ⊥ . For comparison,we also plot the histograms with m t =172 . GeV and µ = 172 . GeV in Fig.7. As shown in Sec.III, the only scale µ dependence in O ( α α s ) is α s ( µ ), which is just a global factor and doesnot change our dijet invariant mass distribution significantly. However, the top quark mass dependencein our results is much more complicated. Therefore, we choose two top quark mass benchmark points toinvestigate its influence on our curves’ shape, and do not plot the scale dependence in this paper. Statistical uncertainties are also responsible for these differences in the histograms. They change ourresults by about 4 percent. M jj H GeV L d G t ® j + X d M jj H G e V G e V L LO R
CA1 = CA1 = CA1 = CA1 = CA1 =
60 80 100 120 140 1600.0000.0050.0100.0150.0200.025 M jj H GeV L d G t ® j + X d M jj H G e V G e V L NLO R
CA1 = CA1 = CA1 = CA1 = CA1 = FIG. 4: The LO (upper panel) and NLO (lower panel) dijet invariant mass distribution from topdecay with different R using the CA1 clustering jet algorithm ( m t = 175 GeV, µ = 80 . GeV ).Plotted are R=1.0 (solid line), 1.2 (short-dashed line), 1.3 (dotted line), 1.4 (long-dashed line), and1.5 (dot-dashed line), respectively. M jj H GeV L d G t ® j + X d M jj H G e V G e V L LO R
KT1 = CA1 = AntiKT1 =
60 80 100 120 140 1600.0000.0050.0100.0150.020 M jj H GeV L d G t ® j + X d M jj H G e V G e V L NLO R
KT1 = CA1 = AntiKT1 = FIG. 5: The influence on distribution with different clustering jet algorithms of the first type( m t = 175 GeV, µ = 80 . GeV ). LO is in the upper panel, while the lower panel is for the NLOresults. Plots are KT1 (solid line), CA1 (dashed line), and anti-KT1 (dotted line).
VI. CONCLUSIONS
We have performed QCD radiative corrections to the dijet production in the unpolarizedtop quark hadronic decay in the complex mass scheme. We carefully checked the indepen-20ence of dimensional regularization schemes and γ strategies in our analytical formalism.Applying different clustering jet definitions, we obtained our final dijet invariant mass dis-tributions. The obtained dijet mass distributions from the top quark decay are useful tounderstand the top quark properties and also to distinguish these dijets from those producedvia other sources. Therefore, these results are useful in investigating the recent CDF W jj anomaly and clarifying this interesting issue. Furthermore, a more careful investigation fortop and W boson associated production at hadron colliders will be definitely needed.
Acknowledgments
We are grateful to K. Wang for the help in some program techniques. We also thankJ. Gao, C. Meng, and Y.Q. Ma for useful discussions. This work was supported by theNational Natural Science Foundation of China (No.10805002, No.11021092, No.11075002,No.11075011), the Foundation for the Author of National Excellent Doctoral Disserta-tion of China (Grant No. 201020), and the Ministry of Science and Technology of China(2009CB825200). [1]
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KT2 = CA2 = AntiKT2 =
60 80 100 120 140 1600.0000.0050.0100.0150.0200.025 M jj H GeV L d G t ® j + X d M jj H G e V G e V L NLO R
KT2 = CA2 = AntiKT2 = FIG. 6: The influence on distribution with different clustering jet algorithms of the second type( m t = 175 GeV, µ = 80 . GeV ). LO is in the upper panel, while the lower panel is for the NLOresults. Plots are KT2 (solid line), CA2 (dashed line), and anti-KT2 (dotted line). M jj H GeV L d G t ® j + X d M jj H G e V G e V L LO R
KT2 = CA2 = AntiKT2 =
60 80 100 120 140 1600.0000.0050.0100.0150.020 M jj H GeV L d G t ® j + X d M jj H G e V G e V L NLO R
KT2 = CA2 = AntiKT2 = FIG. 7: The influence on distribution with different clustering jet algorithms of the second type( m t = 172 . GeV, µ = 172 . GeV ). LO is in the upper panel, while the lower panel is for the NLOresults. Plots are similar to Fig.6.). LO is in the upper panel, while the lower panel is for the NLOresults. Plots are similar to Fig.6.