Factorization and resummation of t-channel single top quark production
aa r X i v : . [ h e p - ph ] O c t Factorization and resummation of t-channel single top quarkproduction
Jian Wang, Chong Sheng Li ∗ ,
1, 2
Hua Xing Zhu, and Jia Jun Zhang
1, 3 Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing, 100871, China Center for High Energy Physics, Peking University, Beijing, 100871, China Nuclear Science Division, MS 70R0319,Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract
We investigate the factorization and resummation of t-channel single top (antitop) quark produc-tion in the SM at both the Tevatron and the LHC in SCET. We find that the resummation effectsdecrease the NLO cross sections by about 3% at the Tevatron and about 2% at the LHC. Andthe resummation effects significantly reduce the factorization scale dependence of the total crosssection. The transfer momentum cut dependence and other matching scale dependencies are alsodiscussed. We also show that when our numerical results for s- and t-channel single top productionat the Tevatron are combined, it is closer to the experimental result than the one reported in theprevious literatures. ∗ Electronic address: [email protected] . INTRODUCTION Top quark is an interesting particle in the standard model (SM) because of its large mass.It may play a special role in the electroweak symmetry breaking mechanism. Its propertiessuch as mass [1], lifetime [2], spin, couplings to other particles and production mechanismdeserve to be studied precisely.The hadronic production of the single top production provides an important opportunityto study the charged weak current interactions of the top quark, e.g., the structure of thethe
W tb vertex [3]. Besides, it is a background for many new physics processes. However,due to the difficulties to discriminate its signatures from the large background, it is onlyrecently that D0 [4] and CDF [5] collaborations have observed the single top production atthe Tevatron.In the three main production modes of the top quark, the t-channel is specially importantbecause of its largest cross section, which has been extremely studied, including the next-to-leading order (NLO) corrections in [6–12]. Their results show that the NLO correctionsare about 5% and 9% at the LHC and Tevatron, respectively. Also, parton shower MonteCarlo simulation was considered in [13, 14]. Threshold resummation for this process wascalculated in [15–17], where only large soft and collinear gluon corrections were resummedand the resummed cross section was expanded up to O ( α s ).In this paper, we use the framework of soft-collinear effective theory (SCET) [18–22]to give a resummed cross section of t-channel single top production, which contains allcontributions from large logarithms in hard, jet and soft functions to all orders. There is astrong motivation for performing this calculation because the hard functions of this process,compared with soft functions, is not small, but refs. [15–17] only resum the soft and jeteffects to next-to-next-to-leading logarithmic (NNLL) order, using the traditional method,and leave the hard effects unresummed. In the SCET approach, the different scales in aprocess can be separated because the soft and collinear degrees can be decoupled by theredefinition of the fields. At each scale, we only need to deal with the relevant degrees offreedom. Their dependencies on the scale are controlled by the renormalization group (RG)equations, and the hard, jet and soft effects can be resummed conveniently.This paper is organized as follows. In section II, we give the factorization and resum-mation formalism of this process in momentum space. In section III, each factor in the2esummed cross section is calculated. In section IV, we present the numerical results forthis process at the Tevatron and LHC, respectively. We conclude in section V. II. FACTORIZATION AND RESUMMATION FORMALISM
Following the same factorization formalism as in our previous paper [23], the partonicdifferential cross section of t-channel can be written as d ˆ σ thres d ˆ td ˆ u = X ij λ ,ij πN c ˆ s Z s / (2 E )0 dk + H IJ ( µ ) S JI ( k + , µ ) J ( s − E k + , µ ) , (1)with λ ,ij = e sin θ W | V ij | | V tb | (ˆ s − m t )ˆ s (ˆ t − M W ) . (2)We use the same definitions of all above notations as [23] and also choose the independentsinglet-octet basis in color space used in [23]. Because of the special color structure ofthis process at leading order (LO), the hard function matrix elements do not contribute tothe cross section except for H at the NLO accuracy. In SCET, there is a RG evolutionfactor connecting the hard scale µ h and the final common scale µ , which would containcontributions from non-diagonal elements in its anomalous dimension matrix. However,these contributions are so small, as we saw in ref. [23], that we can neglect them. Therefore,we can consider this channel as a double deep-inelastic-scattering (DDIS) process [8] andassume that the hard function H can be factorized into two parts, i.e. H up and H dn ,which represent corrections from the up fermion line and down fermion line in the Feynmandiagram 1, respectively. And eq. (1) can be simplified to d ˆ σ thres d ˆ td ˆ u = X ij λ ,ij πN c ˆ s Z s / (2 E )0 dk + H up ( µ ) H dn ( µ ) J ( s − E k + , µ ) S ( k + , µ ) . (3) III. THE HARD, JET AND SOFT FUNCTIONS AT NLO
At the NNLL accuracy, we need the explicit expressions of hard, jet and soft functionsup to NLO in perturbation theory. In this section, we show the calculations of them.3
IG. 1: The LO Feynman diagram of the t-channel single top production.
A. Hard functions
The hard functions are the absolute square of the Wilson coefficients of the operators,which can be obtained by matching the full theory onto SCET. In practice, we need tocalculate the one-loop on-shell Feynman diagrams of this process in both the full theoryand SCET. In dimensional regularization(DR), the facts that the IR structure of the fulltheory and the effective theory is identical and the on-shell integrals are scaleless and vanishin SCET imply that the IR divergence of the full theory is just the negative of the UVdivergence of SCET. After calculating the one-loop on-shell Feynman diagrams, we get thehard functions at NLO are as follows: H up ( µ h,up ) = 1 + C F α s ( µ h,up )4 π (cid:18) − − ˆ tµ h,up + 6ln − ˆ tµ h,up + c H,up (cid:19) , (4) H dn ( µ h,dn ) = 1 + C F α s ( µ h,dn )4 π (cid:18) − − t + m t µ h,dn m t + 10ln − t + m t µ h,dn m t + c H,dn (cid:19) , (5)with c H,up = −
16 + π , (6) c H,dn = − λ ln(1 − λ ) + 2ln (1 − λ ) + 6ln(1 − λ ) + 4Li ( λ ) − − π
6+ 2 m t ˆ u ˆ t (ˆ s − m t ) ln m t m t − ˆ t , (7)where λ = ˆ t/ (ˆ t − m t ). The hard functions have a well behaved expansion in powers of thecoupling constant, if µ h,up and µ h,dn are taken to be of order the natural scales, µ h,up ∼ p − ˆ t and µ h,dn ∼ ( − ˆ t + m t ) /m t , respectively. From eqs.(4, 5), we can write the RG equations of4ard functions as dd ln µ h,up H up ( µ h,up ) = (cid:18) cusp ln − ˆ tµ h,up + 2 γ Vup (cid:19) H up ( µ h,up ) , (8) dd ln µ h,dn H dn ( µ h,dn ) = (cid:18) cusp ln − ˆ t + m t µ h,dn m t + 2 γ Vdn (cid:19) H dn ( µ h,dn ) , (9)where Γ cusp is related to the cusp anomalous dimension of Wilson loops with light-likesegments [24], while γ Vup and γ Vdn accounts for single-logarithmic evolution. Their expressionsare shown in appendix A.After solving the RG equations, we get the hard functions at an arbitrary scale µ : H up ( µ ) = exp (cid:2) S ( µ h,up , µ ) − a Vup ( µ h,up , µ ) (cid:3)(cid:18) − ˆ tµ h,up (cid:19) − a Γ ( µ h,up ,µ ) H up ( µ h,up ) , (10) H dn ( µ ) = exp (cid:2) S ( µ h,dn , µ ) − a Vdn ( µ h,dn , µ ) (cid:3)(cid:18) − ˆ t + m t µ h,dn m t (cid:19) − a Γ ( µ h,dn ,µ ) H dn ( µ h,dn ) , (11)where S ( µ h,up , µ ) and a Vup are defined as [25] S ( µ h,up , µ ) = − Z α s ( µ ) α s ( µ h,up ) dα Γ cusp ( α ) β ( α ) Z αα s ( µ h,up ) dα ′ β ( α ′ ) , (12) a Vup ( µ h,up , µ ) = − Z α s ( µ ) α s ( µ h,up ) dα γ Vdn ( α ) β ( α ) , (13)and similarly for S ( µ h,dn , µ ), a Γ and a Vdn . B. Jet function
The jet function J ( p , µ ) is defined as [26] θ ( p ) p − J ( p , µ ) = 18 πN c Z d p ′ (2 π ) Tr h | ¯ χ ( − p ′ )¯ n/ χ ( − p ) | i . (14)The RG evolution of the jet function is given by [25] dJ ( p , µ ) d ln µ = (cid:18) − cusp ln p µ − γ J (cid:19) J ( p , µ ) + 2Γ cusp Z p dq J ( p , µ ) − J ( q , µ ) p − q . (15)To solve this integro-differential evolution equation, we use the Laplace transformed jetfunction: e j (ln Q µ , µ ) = Z ∞ dp exp( − p Q e γ E ) J ( p , µ ) , (16)5hich satisfies the the RG equation dd ln µ e j (ln Q µ , µ ) = (cid:18) − cusp ln Q µ − γ J (cid:19) e j (ln Q µ , µ ) . (17)Then the jet function at an arbitrary scale µ is given by J ( p , µ ) = exp (cid:2) − S ( µ j , µ ) + 2 a J ( µ j , µ ) (cid:3)e j ( ∂ η j , µ j ) 1 p (cid:18) p µ j (cid:19) η j e − γ E η j Γ( η j ) , (18)where η j = 2 a Γ ( µ j , µ ). The Laplace transformed jet function e j ( L, µ ) at NLO is e j ( L, µ ) = 1 + α s π (cid:26) Γ L + γ J L + c J (cid:27) , (19)where c J = (cid:0) − π (cid:1) C F . C. Soft function
The soft function S ( k + , µ ), which describe soft interactions between all colored particles,is defined as [26] S ( k + , µ ) = 1 N c Z dk + d k ′ s (2 π ) d k s (2 π ) h |O † ,fedcS ( k ′ s ) δ [ k + − n · k s ] O cdefS ( k s ) | i , (20)where O cdefS ( k s ) = Z d xe − ik s · x T (cid:20)(cid:0) Y † n b ( x ) Y n a ( x ) (cid:1) cd (cid:16) ( ˜ Y † v ( x ) ˜ Y n ( x ) (cid:17) ef (cid:21) . (21)Here T is the time-ordering operator required to ensure the proper ordering of soft gluonfields in the soft Wilson line [27], which is defined as Y n ( x ) = P exp (cid:18) ig s Z −∞ ds n · A as ( x + sn ) t a (cid:19) (22)for incoming Wilson lines, and˜ Y n ( x ) = P exp (cid:18) − ig s Z ∞ ds n · A as ( x + sn ) t a (cid:19) (23)for out going Wilson lines, respectively.The soft function can be calculated in SCET or in the full theory in the Eikonal approx-imation [28]. In DR, actually, we only need to calculate the non-vanishing real emissiondiagrams, as shown in figure 2, which give 6 a n b n n a n a n n b n b n a n n n a n b FIG. 2: Non-vanishing diagrams contributing to the soft function at NLO. The contribution fromthe left and right diagrams are denoted as S bt and S tt , respectively. S (1) bt ( k, µ ) = 2 g s C F µ ǫ (2 π ) d − Z ∞ d q + Z ∞ d q − Z d d − q ⊥ δ ( q + q − − q ⊥ ) δ ( k − n · q ) n b · v ( q · n b )( q · v ) , (24)and S (1) tt ( k, µ ) = − g s C F µ ǫ (2 π ) d − Z ∞ d q + Z ∞ d q − Z d d − q ⊥ δ ( q + q − − q ⊥ ) δ ( k − n · q ) 1( q · v ) , (25)respectively. Evaluating these integrals, we get S bt ( k, µ ) = δ ( k ) + 2 C F α s π (cid:26) (cid:20) ln k ˜ µ k (cid:21) [ k, ˜ µ ] ⋆ + δ ( k ) c Sbt (cid:27) , (26)and S tt ( k, µ ) = δ ( k ) − C F α s π (cid:26) (cid:20) k (cid:21) [ k, ˜ µ ] ⋆ + δ ( k ) c Stt (cid:27) , (27)respectively, where ˜ µ = µ/ p (2 n b ¯ b ) /n +21 = ( µ ( − ˆ u ) m t ) / (2( − ˆ t + m t ) E ). The explicitexpressions of c Sbt and c Stt are given in appendix B. And the soft function S ( k, µ ) = S bt ( k, µ ) + S bt ( k, µ ), similar to the jet function, satisfies the RG equation dd ln µ S ( k, µ ) = (cid:20) − cusp ln k ˜ µ + 2 γ S (cid:21) S ( k, µ ) + 2Γ cusp Z k dk ′ S ( k, µ ) − S ( k ′ , µ ) k − k ′ . (28)The solution to this equation is S ( k, µ ) = exp (cid:2) − S ( µ s , µ ) − a S ( µ s , µ ) (cid:3)e s ( ∂ η s , µ s ) 1 k (cid:18) k ˜ µ s (cid:19) η s e − γ E η s Γ( η s ) , (29)where η s = 2 a Γ ( µ s , µ ). The Laplace transformed soft function e s ( L, µ ) at NLO is given by e s ( L, µ ) = 1 + α s π (cid:26) Γ L − γ S L + c S (cid:27) , (30)where c S = (2 c Sbt − c Stt + π ) C F . 7 . Final resummed differential cross section After combining the hard, jet and soft function together, according to eq. (3), we obtainthe resummed differential cross section for t-channel single top production d ˆ σ thres d ˆ td ˆ u = X ij λ ,ij πN c ˆ s exp (cid:2) S ( µ h,up , µ F,up ) − a Vup ( µ h,up , µ F,up ) (cid:3)(cid:18) − ˆ tµ h,up (cid:19) − a Γ ( µ h,up ,µ F,up ) H up ( µ h,up )exp (cid:2) S ( µ h,dn , µ F,dn ) − a Vdn ( µ h,dn , µ F,dn ) (cid:3)(cid:18) − ˆ t + m t µ h,dn m t (cid:19) − a Γ ( µ h,dn ,µ F,dn ) H dn ( µ h,dn )exp (cid:2) − S ( µ j , µ F,up ) + 2 a J ( µ j , µ F,up ) (cid:3) (cid:18) m t µ j (cid:19) η j exp (cid:2) − S ( µ s , µ F,dn ) − a S ( µ s , µ F,dn ) (cid:3)(cid:18) m t ( − ˆ t + m t ) µ s ( − ˆ u ) (cid:19) η s e j ( ∂ η + L j , µ j ) e s ( ∂ η + L s , µ s ) 1 s (cid:18) s m t (cid:19) η e − γ E η Γ( η ) , (31)where η = η j + η s and L j = ln( m t /µ j ) , L s = ln( m t ( − ˆ t + m t )) / ( µ s ( − ˆ u )). In the aboveexpression, we have chosen µ = µ F,up or µ = µ F,dn to avoid the evolution of the partondistribution functions.If we set scales µ h,up , µ h,dn , µ j , µ s equal to the common scale µ , which is convenientlychosen as the factorization scale µ F,up = µ F,dn = µ F , then we recover the threshold singularplus distributions, which should appear in the fixed-order calculation. Up to order α s , we8ave λ ,ij πN c ˆ s d ˆ σ thres ij d ˆ td ˆ u = δ ( s ) + α s π (cid:26) (cid:20) ln( s /m t ) s (cid:21) + + (cid:2) γ J − γ S + ( L j + 2 L s )Γ (cid:3)(cid:20) s (cid:21) + (cid:27) + (cid:18) α s π (cid:19) (cid:26) (cid:20) ln ( s /m t ) s (cid:21) + + (cid:2)
92 ( L j + 2 L s )Γ −
12 (5 β − γ J − γ S )Γ (cid:3)(cid:20) ln ( s /m t ) s (cid:21) + + (cid:2) L j + (4 L s Γ + (5 γ J − γ S − β )Γ ) L j + 7Γ L s + (4 γ J − γ S − β ) L s − π +3( c H + c J + c S )Γ + ( γ J − γ S ) − β ( γ J − γ S ) + 3Γ (cid:3)(cid:20) ln( s /m t ) s (cid:21) + + (cid:2) Γ L j + { L s Γ + 12 (3 γ J − γ S − β )Γ } L j + { Γ L s + 2( γ J − γ S )Γ L s + ( c J + c S )Γ + ( γ J − γ S − β ) γ J − π + Γ } L j + 2Γ L s + ( γ J − γ S − β )Γ L s + {− π + 2( c J + c S )Γ +2(2 γ S − γ J + 2 β ) γ S + 2Γ } L s + 9 ζ Γ − ( 3 γ J − γ S − β
12 ) π Γ + ( γ J − γ S − β ) c J +( γ J − γ S − β ) c S + γ J − γ S + (cid:8) γ J − γ S + ( L j + 2 L s )Γ (cid:9) c H (cid:3)(cid:20) s (cid:21) + (cid:27) , (32)where c H = c H,up + c H,dn . We find that all O ( α s ) and two front O ( α s ) singular plus distri-bution terms coincide with Kidonakis’ [15].Including the remaining terms in the NLO result which we do not resum, we obtain thefinal resummed differential cross section d ˆ σ RES d ˆ td ˆ u = d ˆ σ thres d ˆ td ˆ u (cid:12)(cid:12)(cid:12) µ F,up ,µ F,dn ,µ h,up ,µ h,dn ,µ j ,µ s + d ˆ σ NLO d ˆ td ˆ u (cid:12)(cid:12)(cid:12) µ F,up ,µ F,dn − d ˆ σ thres d ˆ td ˆ u (cid:12)(cid:12)(cid:12) µ F,up = µ F,dn = µ h,up = µ h,dn = µ j = µ s . (33) IV. NUMERICAL DISCUSSION
In this section, we discuss the numerical results of threshold resummation in t-channelsingle top production via SCET. In our calculation, there are four scales, except the twofactorization scales, µ h,up , µ h,dn , µ j , µ s explicitly, which are all arbitrary in principle andour final results should not depend on them. However, because the Wilson coefficients ineach matching, expressed as hard, jet and soft functions, respectively, and the anomalousdimensions are evaluated in fixed-order perturbation theory, there are residual dependenceon these scales. To illustrate the reliability of our evaluation, first, we investigate these scaleuncertainties. In the discussion below, we focus on the scenario at the Tevatron and give a9 .50 0.75 1.00 1.25 1.50 1.75 2.000.700.720.740.760.780.800.820.840.860.880.90 up ( h , up ) h,up /sqrt(-t) fixed order RG improved dn ( h , dn ) h,dn /((-t+m t2 )/m t ) fixed order RG improved FIG. 3: The variations of H up ( µ h,up ) and H dn ( µ h,dn ) with µ h,up and µ h,dn , respectively. cut to p − ˆ t , which is the transfer momentum of this process, because, based on the viewpoint of effective theory, only processes of large transfer momentum are considered as hardprocesses with which we are concerned. A. Scale choices and uncertainties
First, we discuss the dependence of H up ( µ h,up ) on µ h,up . From eq.(4), in order to avoidlarge logarithms, we choose p − ˆ t as our natural hard(up) scale. The left curves in figure 3 il-lustrate the RG effects reduce the dependence of H up ( µ h,up ) on µ h,up . And the correctioninduced by the RG-improved H up ( µ h,up ) to the LO cross section is about − H dn ( µ h,dn ) on µ h,dn , where we choose ( − ˆ t + m t ) /m t as our natural hard(down) scale. Fromthese curves, we can see that the scale dependence is reduced and its RG-improved correctionto LO cross section is about − µ j . Unlike the case of hard functions,because we perform the integration convoluting the jet and soft functions analytically, wecan only choose the natural jet scale through the numerical results. In figure 4, we show thatthe natural jet scale is about 60 GeV around which the contribution of the fixed order jetfunction is minimal. Besides, the RG-improved jet function vary slowly, which indicates that For s-channel processes, the invariant mass of final state particles provides a natural cut to the transfermomentum √ ˆ s . .2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.91.01.11.2 J ( j ) j / 100GeV fixed order RG improved S ( s ) s / 100GeV fixed order RG improved FIG. 4: The variations of J ( µ j ) and S ( µ s ) with µ j and µ s , respectively. the scale dependence is significantly reduced. The correction induced by the RG-improvedjet function to the LO cross section is about +12%.The same case happens in the soft function. In principle, we may consider 2 E k ( − ˆ t + m t ) / ( − ˆ u ) /m t as the natural soft scale. But, in practice, from the numerical results infigure 4, we set the natural soft scale at 50 GeV, and find that the correction induced bythe RG-improved soft function to the LO cross section is about +14%. B. Resummed cross sections
We have chosen all the natural scales needed in this process. Now we give the numericalresults of the resummed cross section. When discussing each scale dependence, we fixed theother scales at the natural scales chosen in the last subsection. µ F,up µ F,dn µ F,up & µ F,dn σ LO (pb) 0 . +0 . − . . − . . . − . . σ NLO (pb) 0 . +0 . − . . − . . . − . . σ RES (pb) 0 . +0 . − . . − . . . − . . TABLE I: The variations of the resummed cross section and fixed order cross section with factor-ization scales at the Tevatron with √ S =1.96 TeV, choosing m t = 175 GeV and transfer momentumcut p − ˆ t >
80 GeV.
In table I, we vary the factorization scales over the ranges 100 GeV < µ
F,up , µ
F,dn < p − ˆ t/ < µ h,up < p − ˆ t , ( − ˆ t + m t ) /m t / < µ h,dn < − ˆ t + m t ) /m t ,30 GeV < µ j <
120 GeV and 25 GeV < µ s <
100 GeV. And we can see that theiruncertainties are all very small. µ h,up µ h,dn µ j µ s σ RES (pb) 0 . +0 . − . . − . . . − . . . − . . TABLE II: The µ h,up , µ h,dn , µ j and µ s scale dependencies of resummed cross section at the Tevatronwith √ S =1.96 TeV, taking m t = 175 GeV and transfer momentum cut p − ˆ t >
80 GeV.
In table III, we show how the value of m t affects our results. When the value of m t variesfrom 171 GeV to 175 GeV, the resummed cross sections vary by about 6%. m t (GeV) 171 172 173 174 175 σ LO (pb) 1.032 1.013 0.995 0.977 0.959 σ NLO (pb) 1.037 1.026 1.010 0.987 0.977 σ RES (pb) 1.008 0.997 0.982 0.959 0.948TABLE III: The m t dependence of resummed cross section at the Tevatron with √ S =1.96 TeV,taking transfer momentum cut p − ˆ t >
80 GeV.
Table IV gives the transfer momentum cut dependence of the cross sections. It shows thatthe resummed cross section gets smaller when the transfer momentum cut is decreased sincechoosing a smaller transfer momentum cut means that more hard effects are resummed.12ut as mentioned above, the transfer momentum cut can not be chosen too small for ahard process. Therefore, we choose 80 GeV as the natural transfer momentum cut in thenumerical calculations. p − ˆ t (GeV) > > > > > σ RES (pb) 0.920 0.936 0.948 0.957 0.964TABLE IV: The transfer momentum cut dependence of resummed cross section at the Tevatronwith √ S =1.96 TeV, taking m t = 175 GeV. In tables V, VI, VII, VIII, IX, and X, we present the results for the single top (anti-top) production at the LHC for different top quark mass with √ S = 7 , , and 14 TeV,respectively. We can see that the resummation effects decrease the NLO cross sections byabout 2% when the transfer momentum cut is chosen as 80 GeV. And the factorization scaledependencies of the cross section are reduced also. C. Combined s and t channel cross sections
Table XI shows the combined numerical results for s- [23] and t-channel single top pro-duction at the Tevatron. From table XI, we see that our result is closer to the experimentalresult [29] than the one reported in the previous literatures[15–17].
V. CONCLUSION
We have studied the factorization and resummation of t-channel single top (antitop)quark production in the Standard Model at both the Tevatron and the LHC with SCET.Our results show that the resummation effects decrease the NLO cross sections by about3% at the Tevatron and about 2% at the LHC, respectively. And the resummation effectssignificantly reduce the factorization scale dependence of the total cross section when the twofactorization scales vary simultaneously, compared with the NLO results. We also show thatwhen our numerical results for s- [23] and t-channel single top production at the Tevatron arecombined, it is closer to the experimental result [29] than the one reported in the previousliteratures[15–17] . 13 t (GeV) 171 172 173 174 175 σ LO (pb) 44 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 42 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 41 . − . . . − . . . − . . . − . . . − . . TABLE V: The cross sections for t-channel single top production at LHC with √ S =7 TeV, choosingtransfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are also shown. m t (GeV) 171 172 173 174 175 σ LO (pb) 90 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 86 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 84 . − . . . − . . . − . . . − . . . − . . TABLE VI: The cross sections for t-channel single top production at LHC with √ S =10 TeV,choosing transfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are alsoshown. m t (GeV) 171 172 173 174 175 σ LO (pb) 167 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 157 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 154 . − . . . − . . . − . . . − . . . − . . TABLE VII: The cross sections for t-channel single top production at LHC with √ S =14 TeV,choosing transfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are alsoshown.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China,under Grants No. 10721063, No. 10975004 and No. 10635030.14 t (GeV) 171 172 173 174 175 σ LO (pb) 23 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 22 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 22 . − . . . − . . . − . . . − . . . − . . TABLE VIII: The cross sections for t-channel single antitop production at LHC with √ S =7 TeV,choosing transfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are alsoshown. m t (GeV) 171 172 173 174 175 σ LO (pb) 52 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 49 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 48 . − . . . − . . . − . . . − . . . − . . TABLE IX: The cross sections for t-channel single antitop production at LHC with √ S =10 TeV,choosing transfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are alsoshown. m t (GeV) 171 172 173 174 175 σ LO (pb) 103 . − . . . − . . . − . . . − . . . − . . σ NLO (pb) 96 . − . . . − . . . − . . . − . . . − . . σ RES (pb) 95 . − . . . − . . . − . . . − . . . − . . TABLE X: The cross sections for t-channel single antitop production at LHC with √ S =14 TeV,choosing transfer momentum cut p − ˆ t >
80 GeV. The factorization scale uncertainties are alsoshown.
Appendix A: Relevant anomalous dimensions and matching coefficients
The various anomalous dimensions needed in our calculations can be found, e.g., in [25,28, 30]. We list them below for convenience of the reader. The QCD β function is β ( α s ) = − α s (cid:20) β α s π + β (cid:16) α s π (cid:17) + · · · (cid:21) , (A1)15 LO[8–10] Res.[15–17] Res.in SCET Experiments.[29]s-channel 0.99 pb 1.12 pb 1.04 pb —t-channel 2.15 pb 2.34 pb 2.04 pb —combined s- and t-channel 3.14 pb 3.46 pb 3.08 pb 2.76 pbTABLE XI: Combination of s- and t-channel single top production at the Tevatron with √ S =1.96 TeV. with expansion coefficients β = 113 C A − T F n f ,β = 343 C A − C A T F n f − C F T F n f ,β = 285754 C A + (cid:18) C F − C F C A − C A (cid:19) T F n f + (cid:18) C F + 15827 C A (cid:19) T F n f , (A2)where C A = 3, T F = 1 / n f is the number of active quark flavor.The cusp anomalous dimension isΓ cusp ( α s ) = Γ α s π + Γ (cid:16) α s π (cid:17) + · · · , (A3)withΓ = 4 C F , Γ = 4 C F (cid:20)(cid:18) − π (cid:19) C A − T F n f (cid:21) , Γ = 4 C F (cid:20) C A (cid:18) − π + 1145 π + 223 ζ (cid:19) + C A T F n f (cid:18) − π − ζ (cid:19) + C F T F n f (cid:18) −
553 + 16 ζ (cid:19) − T F n f (cid:21) . (A4)The other anomalous dimensions are expanded as eq. (A3), and their expansion coeffi-16ients are γ q = − C F ,γ q = C F (cid:18) −
32 + 2 π − ζ (cid:19) + C F C A (cid:18) − − π + 26 ζ (cid:19) + C F T F n f (cid:18) π (cid:19) ,γ Q = − C F ,γ Q = C F C A (cid:18) π − − ζ (cid:19) + 409 C F T F n f ,γ φ = 3 C F ,γ φ = C F (cid:18) − π + 24 ζ (cid:19) + C F C A (cid:18)
176 + 229 π − ζ (cid:19) − C F T F n f (cid:18)
23 + 89 π (cid:19) ,γ j = − C F ,γ j = C F (cid:18) −
32 + 2 π − ζ (cid:19) + C F C A (cid:18) − − π + 40 ζ (cid:19) + C F T F n f (cid:18) π (cid:19) . (A5) γ Vup , γ Vdn and γ S can be obtained from the anomalous dimensions above through the followingequations: γ Vup = 2 γ q ,γ Vdn = γ q + γ Q ,γ S = − γ φ − γ h + γ j . (A6) Appendix B: Calculation of the soft functions
In this appendix, we present the details of the calculation of the two O ( α s ) soft functions S (1) bt ( k, µ ) and S (1) tt ( k, µ ). We choose to do the calculation in the rest frame of top quark, inwhich the four-velocity of the top quark is v µ = (1 , , , n µb = (1 , , , q µ = q + ¯ n µb n b ¯ b + q − n µb n b ¯ b + q µ ⊥ , n µ = n +1 ¯ n µb n b ¯ b + n − n µb n b ¯ b + n µ ⊥ , (B1)17nd q · n = q + n − + q − n +1 n b ¯ b − | q ⊥ || n ⊥ | cos θ, q · v = q · ( n b + n ¯ b )2 = ( q + + q − )2 . (B2)Substituting these expressions into eq. (24), we get S (1) bt ( k, µ ) = g s C F µ ǫ (2 π ) d − Z ∞ d q + Z ∞ d q − Z dΩ d − (cid:18) q + q − n b ¯ b (cid:19) − ǫ δ ( k − q + n − + q − n +1 n b ¯ b + | q ⊥ || n ⊥ | cos θ ) n b · vq + ( q + + q − ) . (B3)Now redefine the integration variables q + and q − and define a = n +1 n − , then S (1) bt ( k, µ ) = g s C F µ ǫ (2 π ) d − Z ∞ d q + Z ∞ d q − Z dΩ d − (cid:18) n b ¯ b n +1 n − (cid:19) − ǫ δ ( k − q + − q − + 2 p q + q − cos θ ) n b · vq + ( aq + + q − ) . (B4)Introducing two variables x and y such that q + = kyx and q − = ky (1 − x ) = ky ¯ x , S (1) bt ( k, µ ) = g s C F µ ǫ (2 π ) d − (cid:18) n b ¯ b n +1 n − (cid:19) − ǫ k − − ǫ Z dΩ d − Z d xx − − ǫ (1 − √ x ¯ x cos θ ) ǫ ¯ x − ǫ ax + ¯ x . (B5)The singularity in the integrand can be isolated by x − − ǫ = − ǫ δ ( x ) + (cid:18) x (cid:19) + − ǫ (cid:18) ln xx (cid:19) + + O ( ǫ ) . (B6)Completing the above three parts of the integration separately and expanding1 k + (cid:18) ˜ µk + (cid:19) ǫ = − ǫ δ ( k + ) + (cid:20) k + (cid:21) [ k + , ˜ µ ] ⋆ − ǫ (cid:20) k + ln k + ˜ µ (cid:21) [ k + , ˜ µ ] ⋆ + O ( ǫ ) , (B7)we get S (1) bt ( k, µ ) = C F α s π (cid:26) (cid:20) ln k ˜ µ k (cid:21) [ k, ˜ µ ] ⋆ + δ ( k ) c Sbt (cid:27) , (B8)where c Sbt = − ln (1 + a ) − ( a ) + π .In a similar but simpler way, we can get c Stt = − a ). [1] Tevatron Electroweak Working Group and CDF and D0, [arXiv:0903.2503[hep-ex]]
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