QCD correction to single top quark production at the ILC

F. Penunuri ,1 F. Larios ,2 and Antonio O. Bouzas 2

1Facultad de Ingenierıa, Universidad Autonoma de Yucatan, A.P. 150, Cordemex, Merida, Yucatan, Mexico2Departamento de Fısica Aplicada, CINVESTAV-Merida, A.P. 73, 97310 Merida, Yucatan, Mexico

(Received 9 February 2011; published 1 April 2011)

Single top quark production at the International Linear Collider (ILC) can be used to obtain high

precision measurements of the Vtb Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM) element as

well as the effective tbW coupling. We have calculated the QCD correction for the cross section in the

context of an effective vector boson approximation. Our results show a �10% increase due to the strong

interaction.

DOI: 10.1103/PhysRevD.83.077501 PACS numbers: 14.65.Ha, 12.15.�y, 12.38.Bx

I. INTRODUCTION

The top quark stands out as the heaviest known elemen-tary particle and its properties and interactions are one ofthe most important measurements for present and futurehigh energy colliders [1]. At the Tevatron and at the LHCthe process of single top quark production has been exten-sively studied [2].

The top quark is likely to provide us with the first cluesof physics beyond the standard model [3]. In fact, newphysics effects are probably already manifest in the recentforward-backward asymmetry observed at the Tevatron[4,5].

The planned International Linear Collider (ILC) willcollide electron and positron beams at an initial energyof 500 GeVand higher. It will provide a clean environmentfor the study of precision measurements.

The single top production processes at lepton and photon(eþe�, e�e�, �e and ��) colliders have been extensivelystudied at tree level in [6]. The reaction �e� ! �tb�e, isparticularly suitable for precision studies, as it does nothave the t�t background. Compared to the ILC eþe� !t �be� ��e process the �e� reaction can yield a larger pro-duction rate and is directly proportional to the Vtb term.Further studies have thus been done for this reaction. Inparticular, the QCD corrections have been studied in [7].Their conclusion is that the QCD correction is not verylarge (� 5%) so that this mode remains very well suited fora precise measurement of Vtb. The approach of [7] is to usethe effective vector boson approximation, or effective Wapproximation [8] (EWA) and to compute the QCD loopcorrections for the Wþ� ! t �b fusion process. Then, theconvolution with the fWþ=eþðxÞ distribution function is

applied to obtain the correction to the actual eþ� process.We would like to point out that the authors of [7] havemade a very clear and thorough presentation of the calcu-lation. In this work we use their analysis of theWþ� ! t �bprocess to estimate the QCD correction for the eþe� !t �be� ��e process of the ILC. Here, in addition to the con-volution with the Wþ boson distribution function we willuse the effective photon (as well as the effective Z boson)

approximation to obtain the QCD correction. We will usethe same input values as in [7] for masses and couplingconstants, except for the masses of top and bottom quarkswe take mt ¼ 173 GeV and mb ¼ 4:2 GeV.

II. VECTOR BOSON CONTRIBUTIONSAT TREE LEVEL

At tree level there are 20 diagrams for the processeþe� ! t �be� ��e [6]. We can list them in three differenttypes: (a) vector boson fusion, (b) vector boson exchangeand (c) eþe� annihilation (see Fig. 1). For the energy rangewe consider, one of the diagrams actually corresponds to t�tproduction, where one of the tops decays leptonically. Inorder to exclude t�t production from the single top processwe discard all events where the invariant mass of the decayproducts (e�, ��e, �b) falls inside an interval around the topmass mt ��M � Me�b � mt þ�M. We take the value�M ¼ 20 GeV as in [6].The effective W approximation relies on the fact that

the vector boson fusion diagrams become dominant whenheavy particles are produced in very high energy colli-sions [8]. In general, three conditions should be met forthe EWA to work well: (1) The mass of the vector boson(MW or MZ) should be much smaller than its energy, andthis can be met if we require MV � ffiffiffi

sp

=2, (2) For q �q

FIG. 1. The three types of diagrams for the eþe� ! t �be� ��e

process.

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production mq � MV , this is true for the top quark but

not for the bottom quark, and (3) One polarization modeshould be dominant so that interference effects can beneglected. Fortunately, in our case the mode W� ! t �bdominates for longitudinal W, and the modes with the Zboson WZ ! t �b give even lower contributions.

As expected, this method works very well for t�t produc-tion at high

ffiffiffis

pand to a lesser degree for single top, which

in our case can be seen as t �b production. In [7] the QCDcorrection to the process eþ� ! t �b ��e was calculated bydoing first the QCD correction to the Wþ� fusion into t �band then by taking the convolution with an effective Wþcoming from the initial positron (see Fig. 2). We follow thesame approach by doing the one loop QCD correction toWþ� ! t �b as well as WþZ ! t �b and then convolutingwith the effective distribution functions for Wþ, � and Z:

�ðeþe� ! t �b ��ee�Þ

¼ XWL;WT

Z 1

xminW

dxWfWþ=eþðxWÞ

�Z 1

0dx�f�=e�ðx�Þ�ðWþ� ! t �bÞðsÞ

þ XWL;T ;ZL;T

Z 1

xminW

dxWfWþ=eþðxWÞ

�Z 1

xminZ

dxZfZ=e�ðxZÞ�ðWþZ ! t �bÞðsÞ; (1)

where xminV ¼ 2MV=

ffiffiffis

p, s ¼ xWx�s or xWxZs, and the

structure functions can be found in [8]. The tree levelcross section for single top production at the ILC is shownin Fig. 3. The exact Born level calculation for the eþe� !t �be� ��e process is obtained with CalcHEP [9] and isshown by the solid line. We can see that the predictionof the EWA (dot-dashed curve) is in very good agreementwith the exact result for center of mass energies above1.5 TeV. However, for the energy range of the ILC theEWA values can be significantly lower. In particular, forffiffiffis

p ¼ 1000 GeV there is a 15% difference and forffiffiffis

p ¼500 GeV the EWA result is about one half of the exactvalue.

Some kinematical aspects of our calculation are worthdiscussing in more detail. Because in the dominant dia-grams for the complete process eþe� ! Wþ�Z� !t �be� ��e the virtual vector bosons get spacelike momentak2V � 0 (V ¼ W, Z) and are always far from their massshell, the EWA is known to work better when the vectorboson squared momenta are set to k2V ¼ 0 in the external

legs, as discussed in [10]. Nevertheless, when dealing witha process like t�t production one may set k2V ¼ M2

V , as thisintroduces only a small error of order MV=

ffiffiffis

p. For this

reason it is customary [7,8] to set the external massiveWþand Z on shell for convenience. We stress here, however,that the EWA requires [8,10] the limits of integration to betaken as defined in (1) for all polarization states of themassive vector bosons, regardless of whether one choosesto set k2V ¼ 0 or M2

V [11]. In our study we keep the EWAcondition k2Z ¼ 0 for the Z boson, but choose k2W ¼ M2

W forthe initial-stateWþ for calculational convenience. We havenumerically checked that indeed by setting k2W ¼ 0 we donot find a significant change in the result.Because of the kinematics of theWþZ ! t �b process, its

tree level scattering amplitude has a pole singularity withinits physical region [12], leading to divergent behavior aswe integrate it over the Mandelstam variable t (or the polarangle of the outgoing quark). This is due to the fact that, forenergy values from slightly above the threshold mt þmb

and up to�3 TeV, there exists a certain value of t such thatthe massive Z boson can actually decay into b �b, causingthe bottom quark propagator to hit its pole at that t. Bystrictly adhering to the EWA requirement k2Z ¼ 0, thesingularity is pushed outside the physical region and nodivergent integration appears in the computation. Noticethat a similar situation does not occur with the Wþ boson,as it cannot decay into t �b.Below, we will describe the QCD corrections to the

Wþ� and WþZ processes, including the DipoleSubtraction Method of infrared divergences. We have fol-lowed closely the analysis of theWþ�mode done by Kuhnet al. in [7].

III. QCD CORRECTION TOTHE Wþ�ðZÞ ! t �b PROCESS

The QCD loop correction to the Wþ�ðZÞ ! t �b processis given by 9 Feynman diagrams (see Fig. 2 of [7]). Therenormalization procedure involves only the quark’s wavefunction and mass parameter. Specific formulas can befound in [7]. Concerning the renormalization scale depen-dence we have also set �s at the scale � ¼ ffiffiffi

sp

for our

numerical calculation (it becomesffiffiffis

punder the convolu-

tion). The extraction of IR singularities is done with thesubtraction method of the dipole formalism [13]. Thismethod consists of adding and subtracting a so-calleddipole term:

�NLOðWþ�! t �bÞ¼Ztbg

½ðd�RÞ�¼0�ðd�B�dVdipoleÞ�¼0

þZtb½d�Vþd�B�I�¼0; (2)

where d�R comes from the real emission Wþ�ðZÞ ! t �bgprocess and d�B � dVdipole is the subtracting dipole term

that matches pointwise the singularities associated with theFIG. 2. The vector boson fusion diagrams for the Wþ�ðZÞ !t �b process.

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soft and/or collinear gluon. Both terms are calculated ind ¼ 4 dimensions. In the second integral the same dipoleterm has been partially integrated in the gluon phase spaceand then added to the virtual correction d�V . This sum isperformed in d ¼ 4� 2� dimensions (consistent with di-mensional regularization).

The general formula for the dipole term is found inEq. (5.16) of [13]. The specific expression in our case is

d�B � dVdipole ¼hVgt;bi2kg kt jM0ð~kgt; ~kbÞj2 þ ft $ bg; (3)

where

hVgt;bi¼8��sCF

�2

1�~ztð1�ygt;bÞ�~vgt;b

vgt;b

h1þ~ztþ m2

t

kg kti�;

~zt¼ kt kbðktþkgÞ kb ; ygt;b¼2

kg ktsxtb

;

~vgt;b¼�tb

xtb; vgt;b¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þagt;bÞ2�a2gt;b=zb

q;

agt;b¼ 2zbxtbð1�ygt;bÞ ;

~kb¼xb2Pþ�tb

�gt

ðkb�P kbs

PÞ;~kgt¼P� ~kb; P¼kWþk�;

and M0ð~kgt; ~kbÞ is the Born level Wþ� ! t �b amplitude

with one modification: the final state momenta kt and kbhave been replaced by ~kgt and ~kb, respectively. The other

variables are defined as in [7]: �q ¼ mq=ffiffiffis

p, zq ¼ �2

q,

xt ¼ 1þ zt � zb, xb ¼ 1þ zb � zt, xtb ¼ 1� zt � zb,�tb ¼ �ð1; zt; zbÞ, �gt ¼ �ð1; ðkg þ ktÞ2=s; zbÞ, and

�ðx; y; zÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2 � 2xy� 2xz� 2yz

p.

For the real emission correction we have prepared aFortran program that integrates the cross section for theWþ� ! t �bg process along with dipole subtraction. As itturns out, the subtraction term defined by the dipole for-malism in the first integral of Eq. (2) is actually a very goodapproximation to the real emission cross section in animportant part of the tbg phase space, so that the numericalresults we obtained were very small: about 2 orders ofmagnitude below the values obtained for the virtualcorrection.

The expression for the dipole term in the virtual correc-tion is

d�B � I ¼ jMdðWþ� ! t �bÞj2 �s

2�

� 1

�ð1� �Þ�4��2

s

��ðIgt;b þ Igb;tÞ; (4)

whereMdðWþ� ! t �bÞ is the Born level amplitude in d ¼4� 2� dimensions (the flux term of the t �b phase spaceintegration is understood). The dipole function is given byIgt;b ¼ CF½2Ieik þ Icollgt;b (also Igb;t ¼ Igt;bft $ bg), whereIeik and Icollgt;b are given by Eqs. (5.34) and (5.35) in [13]:

Ieik¼ xtb�tb

�ln

2�þ�2

6� ln ln½1�ð�tþ�bÞ2

�1

2ln2t�1

2ln2bþ2Li2ð�Þ�2Li2ð1�Þ

�1

2Li2ð1�2

t Þ�1

2Li2ð1�2

b�Icollgt;b

¼1

�þ3þ ln�tþ lnð1��bÞ�2ln½ð1��bÞ2�zt

� �b

1��b

� 2

xtb

��bð1�2�bÞþzt ln

�t

1��b

�; (5)

where 2 ¼ ðxtb � �tbÞ=ðxtb þ �tbÞ, t ¼ ðxtb � �tb þ2ztÞ=ðxtb þ �tb þ 2ztÞ, and b ¼ tft $ bg. These formu-las also appear in [7], except that in their Eq. (4.14) in Icollgt;b

the constant term should not be 5 but 3.Concerning the calculation of d�V , the details can be

found in [7]. We actually worked out this same computa-tion before doing the case for the Z boson. As expectedfrom the results shown in Fig. 3 the contribution fromWþZfusion is much smaller than the one fromWþ�. In fact, weonly considered the correction for the polarizations Wþlongitudinal and Z transversal as the other possibilitiesyield negligible contributions.Our results are shown in Fig. 4. The solid line in the left

panel is the same exact Born level result shown in Fig. 3.The dashed line is obtained by adding to the solid line theQCD correction computed within the EWA method. Theright panel in the figure shows the ratio of the QCDcorrection to the exact Born cross section (solid line). Asseen there, for

ffiffiffis

p> 1 TeV the QCD correction remains

roughly stable about 11%. Its slow decrease above

�2 TeV is due to the running of �sð� ¼ ffiffiffis

p Þ. On the otherhand, below 1 TeV the correction drops to 7% with

500 1000 1500 2000 2500

5

10

15

20

σe+

e-t

bν

()

e-fb

s GeV

WL Z

γ

γ

WL

TW

T

EWA total

Born exact

FIG. 3 (color online). The tree level contributions from Wþ�and WþZ fusion to the eþe� ! t �be� ��e process. The solid lineshows the exact calculation.

BRIEF REPORTS PHYSICAL REVIEW D 83, 077501 (2011)

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decreasing energy. This is an effect of the Born level EWApredicted cross section, that drops to �50% of the exactBorn level value as shown in Fig. 3. If instead of using theexact Born value as denominator we use the EWA Bornprediction (see dot-dashed line in Fig. 3) our results(dashed line) show a 17% increase in the Born level crosssection at

ffiffiffis

p ¼ 500 GeV.

It will be interesting to compare this result based on theeffective W approximation with a future more robust cal-culation based on the complete eþe� ! t �be� ��e process.

ACKNOWLEDGMENTS

The authors thank RedFAE, Conacyt and SNI for sup-port. F. L. thanks C.-P. Yuan for useful discussions.

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500 1000 1500 2000 2500

5

10

15

20

25

σe+

e-t

bν

()

e-fb

s GeV

Born + QCD correction

Born

500 1000 1500 2000 25006

8

10

12

14

16

18

s GeV

QC

D c

orre

ctio

n/B

orn

(%)

FIG. 4 (color online). The QCD correction fromWþ� andWþZ fusion to the eþe� ! t �be� ��e process. Left panel: exact Born levelcalculation (solid line) and Born plus QCD correction (dashed line). Right panel: ratio of QCD correction to exact Born cross section(solid line) and ratio of QCD correction to EWA Born cross section (dashed line).

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