soft and collinear functions for the standard model

Soft and collinear functions for the standard model

Jui-yu Chiu, Andreas Fuhrer, Randall Kelley, and Aneesh V. Manohar

Department of Physics, University of California at San Diego, La Jolla, California 92093, USA(Received 5 September 2009; published 22 January 2010)

Radiative corrections to high-energy scattering processes were given previously in terms of universal

soft and collinear functions. This paper gives the collinear functions for all standard model particles, the

general form of the soft function, and explicit expressions for the soft functions for fermion-fermion

scattering, longitudinal and transverse gauge boson production, single W=Z production, and associated

Higgs production. An interesting subtlety in the use of the Goldstone boson equivalence theorem for

longitudinal Wþ production is discussed.

DOI: 10.1103/PhysRevD.81.014023 PACS numbers: 12.15.Lk, 12.38.Cy, 13.40.Ks, 13.85.�t

I. INTRODUCTION

Hard scattering processes can be described using soft-collinear effective theory (SCET) [1–4]. SCET was ex-tended to broken gauge theories [5–8] and used to computethe renormalization group improved amplitude for stan-dard model scattering processes at high energy. The effec-tive theory formalism sums the electroweak Sudakovcorrections using renormalization group evolution inSCET. The strong and electroweak radiative correctionsto hard scattering processes were formulated in terms ofcollinear and soft functions in Ref. [8]. The result gives anefficient way of computing the effective theory radiativecorrections in terms of a collinear function for each parti-cle, and universal soft functions. Electroweak radiativecorrections also have been computed previously usingfixed-order methods [9–31].

The soft and collinear functions were given in Ref. [8]for an SUð2Þ gauge theory. The complete standard modelexpressions are more involved because of custodial SUð2Þviolation, and because the right- and left-handed quarksand leptons have different quantum numbers. In this paper,we give the explicit collinear running and matching func-tions for each standard model particle, as well as the softfunctions for some important processes such as fermion-fermion scattering, gauge boson pair production, and asso-ciated Higgs boson production. We will use the notationsand conventions of Ref. [8], and assume that the reader isfamiliar with the results presented there. The split into softand collinear contributions is not unique, and we use thedefinition in Ref. [8]. The soft functions for QCD correc-tions have been obtained previously [32].

A collinear function F ðF!PÞ gives the amplitude F ! Pfor the field F to produce a particle P, analogous to theh0j�jpi factor in the Lehman-Symanzik-Zimmerman re-duction formula. Particularly interesting are the collinearfunctions for � ! WL and W ! WT , W

3 ! �, B ! �,W3 ! ZT , B ! ZT , and � ! ZL in the Higgs-gauge sec-tor. In most cases, there is a unique F, e.g. uL is onlyproduced by the quark doublet field Q, and so the Q ! uL

collinear function is also referred to as the uL collinearfunction. The subscript on a fermion field refers to chi-rality, and on a fermionic particle, to helicity. Thus, theuR ! uR collinear function is the amplitude for a right-handed u field, with projector ð1þ �5Þ=2, to produce aright-handed u quark, with spin parallel to momentum. Thedifference between helicity and chirality is orderm=E, andhigher order in the SCET power counting.We first present plots of the collinear functions in Sec. II

obtained using formulæ given later in Sec. III of this paper.There is an interesting subtlety in the Goldstone bosonequivalence theorem for Wþ

L arising from infrared diver-gences due to photon exchange, which is discussed in thissection. The general form of the soft functions, and somestandard soft matrices are given in Sec IV. These are thenused to compute the soft functions for fermion scattering inSec. V, longitudinal and transverse gauge boson productionin Sec. VI, single W, Z production in Sec. VII, and gluonscattering in Sec. VIII. Appendix A gives the analyticformula for integrating a SCET anomalous dimension in-cluding terms up to the three-loop cusp.The effective field theory (EFT) computation is given by

matching from the standard model onto SCET at a scale�h, running to �l at which the W, Z, H, t are integratedout, and then running usingQCDþ QED to a factorizationscale�f at which the hadronic scattering cross-sections are

computed by convolution with the parton distribution func-tions. The final answer is independent of the choice of�h;l;f, but in practice has some dependence on these quan-

tities due to neglected higher order terms. The �h;l depen-

dence was shown in Ref. [8] to be less than 1% forprocesses other than transverse WT production, for whichthe �h dependence was almost 10%.

II. PLOTS OF COLLINEAR FUNCTIONS

In this section, we give numerical plots for the collinearfunctions for the standard model, and discuss some inter-esting features of the collinear corrections. The collinear

PHYSICAL REVIEW D 81, 014023 (2010)

1550-7998=2010=81(1)=014023(23) 014023-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.014023

radiative corrections are process independent, and have thesame value in all scattering processes.

The collinear functions are given by running the col-linear anomalous dimension from �h to �l using theanomalous dimensions in Sec. III A, matching at �l usingexpDC in Sec. III B, and then running from �l to �f using

the anomalous dimensions in Sec. III C. In equations,

logF ðF!PÞð �n � p;�f;�hÞ¼ �

Z �l

�f

d�

��Pð �n � p;�Þ þDðF!PÞ

C ð �n � p;�lÞ

�Z �h

�l

d�

��Fð �n � p;�Þ: (1)

The collinear corrections are functions of �n � p ¼ 2E,where E is the particle energy,1 and depend linearly onlog �n � p to all orders in perturbation theory [35]. They aredefined after zero-bin subtraction to avoid double countingwith the soft contribution [36–43]. These subtractions arenecessary for soft-collinear factorization [44] to hold.

The collinear functions were used to compute 2 ! 2scattering processes in Ref. [8], where we used �h ¼ffiffiffiffiffis0

pfor the high-scale matching. In the partonic center-

of-mass frame, all four partons have energy 2E ¼ �n � p ¼ffiffiffiffiffis0

p. For this reason, we have used �h ¼ �n � p in the

collinear function plots, to reduce the number of variablesby one. The low scales are chosen to be �f ¼ �l ¼ MZ.

There is a tiny dependence on the Higgs mass—the rateschange by less than one part in 104 ifmH is varied between200 and 500 GeV. In the plots, mH ¼ 200 GeV. The col-linear anomalous dimension can be integrated analyticallyusing the results in Appendix A. If the factorization scale�f is below MZ, there is an additional contribution to the

collinear function from QCDþ QED running from MZ to�f, which is given separately. The total collinear function

is the product of the �h ! MZ and MZ ! �f collinear

functions.Figure 1 shows the collinear functions for quarks. The

collinear functions for the c and s are identical to those forthe u and d, respectively. The t and b quarks have slightlydifferent collinear functions because of Higgs corrections,and the mass of the t. In a 2 ! 2 scattering process such asuL �uL ! dL �dL, one has a collinear function in the ampli-tude for each external particle, so the rate depends on theproduct of the fourth powers of the uL and dL collinearfunctions. Thus, a 10% correction in Fig. 1 changes the rateby more than a factor of 2. The difference between theheavy- and light-quark collinear functions arises from

FIG. 1 (color online). Plot of the collinear functions against �n �p for (a) lower panel: uL (dotted green), uR (solid cyan), tL(dashed red), tR (short-dashed blue); (b) upper panel: dL (dottedgreen); dR (solid cyan); bL (dashed red); and bR (short-dashedblue).

FIG. 2 (color online). Plot of the collinear functions against �n �p for uL (dotted green), uR (solid cyan), dL (dashed red), and dR(short-dashed blue).

1The collinear functions depend on the Lorentz frame throughthe null vector n. The n dependence is cancelled by a corre-sponding n dependence in the soft functions, by reparametriza-tion invariance [33,34].

CHIU et al. PHYSICAL REVIEW D 81, 014023 (2010)

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Higgs contributions due to the tR Yukawa coupling to the

quark doublet QðtÞ, and due to the switch from SCET tobHQET fields for the t.

Figure 2 shows the collinear functions for u and d on thesame plot. The left- and right-handed quarks have differentcollinear functions because of the difference in SUð2Þquantum numbers. There is a small difference betweenuR, dR due to the different Uð1Þ quantum numbers, whichlead to different Uð1Þ anomalous dimensions. There is aneven smaller difference between uL, dL due to differencesin the low-scale matching from Z exchange due to thedifferent Z couplings. Figure 3 shows the collinear func-tions for the leptons. The corrections are smaller than forquarks because there are no QCD corrections.

If the factorization scale is chosen below MZ, there isadditional collinear running from QCD and QED. TheQCD collinear running is the same for all quarks, and thelog of the QED running is proportional to the electriccharge. Figure 4 show the collinear running below MZ

for �f ¼ 30, 50 GeV for quarks, gluons and electrons.

These multiply the collinear running from �h to MZ.The collinear functions for massless gauge bosons are

shown in Fig. 5. The corrections to the gluon are due toQCD, and are large because of the large value of CA. Thereare two collinear functions for photon production, depend-ing on the source of the photon. The W3 � B and Z� �fields are related by

Z ¼ cos�WW3 � sin�WB; A ¼ sin�WW

3 þ cos�WB:

(2)

At tree level theW3 ! � amplitudes is sin�W , and the B !� amplitude is cos�W . The photon can be emitted by whatstarted out as either a W3 or B field at high energy, and theradiative corrections shown by the solid red and dotted redcurves in Fig. 5 multiply the tree-level amplitudes. The

FIG. 3 (color online). Plot of the collinear functions against �n �p for �L (dashed blue), eL (dotted red), and eR (solid red).

FIG. 4 (color online). Plot of the collinear functions due torunning from MZ to �f against �n � p for electrons with �f ¼30 GeV (solid red) and �f ¼ 50 GeV (dashed blue) are shown

in the upper panel. The QCD correction for quarks with �f ¼30 GeV (solid red) and �f ¼ 50 GeV (dashed blue) and gluons

with �f ¼ 30 GeV (dotted green) and �f ¼ 50 GeV (dot-

dashed cyan) are shown in the lower panel.

FIG. 5 (color online). Plot of the collinear functions against �n �p forW ! � (solid red), B ! � (dotted red), and gluons (dashedblue).

SOFT AND COLLINEAR FUNCTIONS FOR THE STANDARD . . . PHYSICAL REVIEW D 81, 014023 (2010)

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correction forW ! � is much larger because of the SUð2Þcontribution.

Figure 6 gives the collinear functions for the massivegauge bosons and Higgs. The lower panel shows the col-linear functions for the transverse and longitudinal W, i.e.for W ! WT and � ! WL, since the transverse W canonly come from the W field and the longitudinal W fromthe� field. The radiative corrections are different, becauseat high energies,WT is part of theW� gauge field, whereas

WL is part of the scalar field �. The SUð2Þ corrections toW� depend on the adjoint Casimir CA ¼ 2, whereas the

corrections to� depend on the fundamental Casimir CF ¼3=4. The Uð1Þ corrections also differ. At high energies, theWL remembers that it originated from the scalar field viaspontaneous symmetry breaking. The upper panel gives theplots for the neutral boson sector. The transverse Z canarise from either W3 or B, as for the photon, and the twocases are shown in solid and dotted red. The B ! ZT

amplitude has smaller corrections (as for B ! �), so athigh energies, ZT is produced mainly via B ! ZT , eventhough at tree level, it is the W3 ! Z amplitude whichdominates. The � ! ZL and � ! H amplitudes have

similar shapes, since both are mainly given by the radiativecorrections to the scalar doublet �. There are two ampli-tudes for ZT , W ! Z and B ! Z, but only one for ZL,� ! Z.

III. COLLINEAR FUNCTIONS

The formulæ for the collinear functions are given in thissection. They were obtained using the procedure given inRef. [8]. The main complication arises from custodialSUð2Þ symmetry breaking in the standard model. In loopgraphs, one has to distinguish between W and Z exchangeas well as the mt-mb mass difference. The collinear func-tions, computed from one-loop graphs such as Fig. 7 aresummarized in Table I. The anomalous dimension �C gives

the running between the high-scale �h �ffiffiffis

pand the low-

scale �l �MZ, and the matching DC gives the collinearmatching at the low-scale �l. The �C column can also beused to obtain the anomalous dimension in SCET� be-

tween �l and the factorization scale �f. This table is a

generalization of Table II of Ref. [8], which gave thecollinear functions in the SUð2Þ theory. In the weak inter-actions, the two members of an SUð2Þ doublet can havedifferent masses. As a result, in computing Fig. 7, theinternal and external fermions can have different masses;e.g. the internal fermion can be a b quark, and the externalone, a t quark. This complication did not arise for theSUð2Þ theory with massless fermions considered inRef. [8]. The collinear functions in Table I include thepossibility of different internal and external masses. mint

is the mass of the internal particle in the loop, andffiffiffiffiffiffip2

pis

the mass of the external particle. The functions fF;S are

given in Appendix B of Ref. [7], and vanish for masslessparticles, fFð0; 0Þ ¼ fSð0; 0Þ ¼ 0. The c row is for fermi-ons, � for scalars, B? for an external transversely polar-ized gauge boson, H for the physical Higgs field, and ’a

for the Goldstone bosons, which are used to computelongitudinally polarized gauge bosons using the equiva-lence theorem.Table I gives the results in a compressed form, from

which the standard model results can be extracted. The T �T factor has to be taken apart into individual gauge bosoncontributions

�T � T ! �sT � Tþ �2t � tþ �1Y � Y; (3)

FIG. 6 (color online). Plot of the collinear functions against �n �p for (a) lower panel: WT (solid red), WL (dashed blue);(b) upper panel: W ! ZT (solid red), B ! ZT (dotted red), ZL

(dashed blue), and H (short-dash, dark green).

p2m2int

M2

FIG. 7. One-loop collinear graph, where the internal and ex-ternal particles can have different masses, e.g. mint ¼ mb andp2 ¼ m2

t . The wavyþ solid line is the collinear gauge boson,and the dashed line is a collinear fermion or scalar.


014023-4

summing over the SUð3Þ, SUð2Þ, and Uð1Þ contributions,where T are the QCD generators, and t are the SUð2Þgenerators. This form is convenient for computing thecollinear anomalous dimension �C, which is mass inde-pendent. The electroweak couplings constants are �2 ¼�em=sin

2�W and �1 ¼ �em=cos2�W .

The low-scale collinear matching DC depends on thegauge boson masses, so the SUð2Þ �Uð1Þ part has to berewritten in terms of the W, Z and � contributions,

�2t � tþ �1Y � Y ! 12�Wðtþt� þ t�tþÞ þ �ZtZ � tZþ �emQ �Q; (4)

where �W ¼ �2 and �Z ¼ �em=ðsin2�Wcos2�WÞ and tZ isthe Z charge, tZ ¼ t3 � sin2�WQ. A useful identity for theW contribution is

12 ðtþt� þ t�tþÞ ¼ t � t� t3 � t3: (5)

The matching DC depends on the gauge boson andfermion masses. The value of DC is given using Table Iand Eqs. (3) and (4) and using M ! MW in the W termsandM ! MZ in the Z terms. The photon and gluon do notcontribute to DC, since they are not integrated out at thelow-scale �l �MZ, and are dropped. Furthermore, in fFand hF, the internal fermion mass is equal to the externalfermion mass for the Z term, but is different for theW term.For example, for an external t quark, p ! mt, mint ! mt

for the t2Z term and p ! mt, mint ! mb for the tþt� andt�tþ terms. Explicit formulæ for the standard model par-ticles are given below using this procedure.

The wave function factors �R’;H;W can be found in

Refs. [45–48]. They are defined as the residue of thetwo-point Green’s function at the pole,

G� R

p2 �M2þ finite; (6)

with R ¼ 1þ �R. R is obtained using the two-point

function renormalized in the MS scheme, and is finite.We use the convention of Ref. [45] and denote the finitewave function correction by R, and reserve Z for theinfinite renormalization counterterms. There is one impor-

tant point to remember—the wave function graphs have tobe computed as an EFT matching condition. This meansthat the graphs are computed using dimensional regulari-zation to regulate the infrared divergences, setting all low-energy scales such as mb to zero, and retaining only thefinite part.2 R can be obtained from the expressions interms of Passarino-Veltman functions using

A0ðm2Þ ¼ �m2

�1

�UVþ 1� ln

m2

�2

�; (7)

and

B0ðp2; m1; m2Þ ¼ �ð�Þe��2�Z 1

0dx½m2

1xþm22ð1� xÞ

þ p2xð1� xÞ��;

B00ð�M2; m1; m2Þ ¼ @B0ðp2; m1; m2Þ

@p2

��p2¼�M2; (8)

where we follow the conventions of Ref. [46]. In particular,the infrared divergent functions needed are

B0ð0; 0; 0Þ ¼ 1

�UV� 1

�IR(9)

and

B00ð�M2; 0;MÞ ¼ 1

M2

�1

2�IRþ 1� 1

2log

M2

�2

�; (10)

which are replaced by 0 and ð1� 1=2 logM2=�2Þ=M2,respectively, in R.In Refs. [6,7], the radiative corrections for massive

particles were computed. In the region below the particle

TABLE I. The collinear anomalous dimension and low-scale matching. LM ¼ logðM2=�2Þ, Lp ¼ logð �n � pÞ=�, and � ¼ E=m. Therows are c : fermion, �: non-Higgs scalar multiplet, hv: HQET field, B?: transverse gauge boson, H: Higgs, ’a: Goldstone bosons(i.e. longitudinal gauge bosons using the equivalence theorem and mutiplying by E). The results are in R¼1 gauge. �W;h;’ andRW;h;’

are the wave function contributions. p2 is m2 for the external particle, and mint is the mass of the internal particle.

Field �C DC

c �4T � T½4Lp � 4� þ �c

�4T � T½2LMLp � 1

2LM2 � 2LM � 52

12 þ 2þ fFðp2=M2; m2int=M

2Þ� þ 12�Rc

� �4T � T½4Lp � 2� þ ��

�4T � T½2LMLp � 1

2LM2 � LM � 52

12 þ 1þ fSðp2=M2; m2int=M

2Þ� þ 12�R�

hv�4T � T½4 logð2�Þ� þ �h

�4T � T½2LM log2�� þ 1

2�Rhv

B? �4T � T½4Lp � 2� þ �W

�4T � T½2LMLp � 1

2LM2 � LM � 52

12 þ 1þ fSð1; 1Þ� þ 12�RW

H �4T � T½4Lp � 2� þ �H

�4T � T½2LMLp � 1

2 LM2 � LM � 52

12 þ 1þ fSðm2h=M

2; 1Þ� þ 12�RH

’a �4T � T½4Lp � 2� þ �’

�4T � T½2LMLp � 1

2LM2 � LM � 52

12 þ 1þ 23 fSð1; 1Þ þ 1

3 fSð1; m2h=M

2Þ� þ 12�R’

2See, for example, Refs. [35,49,50] for a more extensivediscussion and explicit examples. In Eqs. (7)–(10), the subscriptsUV and IR indicate whether the divergence is ultraviolet orinfrared. The integrals are done in 4� 2� dimensions, so �UV ¼�IR ¼ �.


014023-5

mass, the particle can be treated as a boosted heavy quarkeffective theory (bHQET) field [51,52]. The anomalousdimension and low-scale matching for bHQET fields isgiven in the row hv. For massive particles, the collinearanomalous dimension involves log2�, where � ¼ E=m isthe boost factor, rather than logð �n � pÞ=� ¼ logð2EÞ=�.The bHQET formula is needed for top-quark pair produc-tion, and for W and Z production.

In addition to gauge boson exchange, there are radiativecorrections due to scalar exchange graphs. In the standardmodel, these arise from Higgs exchange. As shown inRefs. [6,7], scalar exchange vertex graphs are 1=Q2 sup-pressed, and only the wave function graphs are leadingorder in the SCET power counting. Thus, we can includeHiggs corrections in the effective theory through theircontribution to R.

The anomalous dimension between �h and �l is inde-pendent of the low-energy scales, including the electro-weak symmetry breaking scale, and so can be computed inthe unbroken gauge theory. The collinear functions dependon �n � p ¼ 2E, where E is the energy of the particle. The �ndependence, or Lorentz frame dependence, is cancelled bya corresponding frame dependence in the soft functions.

The left-handed quark doublets will be denoted by QðiÞL ,

where i ¼ u, c, t is a flavor index, the right-handed charge

2=3 quarks by UðiÞR or uR, cR, tR, the right-handed charge

�1=3 quarks by DðiÞR , or dR, sR, bR, the left-handed lepton

doublets by LðiÞL , i ¼ e, �, �, and the right-handed lepton

singlets by EðiÞR or eR, �R, �R. Written in terms of SUð2Þ

components, QðiÞ is

QðiÞ ¼ UðiÞL

D0ðiÞL

" #¼ UðiÞ

L

VijDðjÞL

" #; (11)

where the primed down-type quarks are weak eigenstatefields, and the unprimed fields are mass eigenstates. All thelepton and down-type quark masses can be neglected in ourcalculation, so we can work in the weak eigenstate basis,the Cabibbo-Kobayashi-Maskawa quark-mixing matrix Vdoes not enter the SCET computation, and the generationnumber is conserved. Once the radiative corrections have

been computed, one can make the replacement D0ðiÞL !

VijDðjÞL to compute the amplitudes in terms of mass-

eigenstate fields.

A. Running from �h to �l �MZ

The collinear anomalous dimensions for the runningfrom �h �Q to �l �MZ are listed below. The gauge

coupling constants are the MS values in the theory with

six dynamical quark flavors, and yt ¼ffiffiffi2

pmt=v is the

t-quark Yukawa coupling. The top-quark multiplets havedifferent collinear running than the other quarks because ofthe large Yukawa coupling yt.

Qðu;cÞL :

1

4

�4

3�s þ 3

4�2 þ 1

36�1

��4 log

�n � p�

� 3

�; (12)

QðtÞL :

1

4

�4

3�s þ 3

4�2 þ 1

36�1

��4 log

�n � p�

� 3

�þ 1

2

y2t162

;

(13)

uR, cR:

1

4

�4

3�s þ 4

9�1

��4 log

�n � p�

� 3

�; (14)

tR:

1

4

�4

3�s þ 4

9�1

��4 log

�n � p�

� 3

�þ y2t

162; (15)

dR, sR, bR:

1

4

�4

3�s þ 1

9�1

��4 log

�n � p�

� 3

�; (16)

LðeÞL , Lð�Þ

L , Lð�ÞL :

1

4

�3

4�2 þ 1

4�1

��4 log

�n � p�

� 3

�; (17)

eR, �R, �R:

�1

4

�4 log

�n � p�

� 3

�: (18)

The gauge-field anomalous dimension at one-loop inR¼1 gauge is

� ¼ 2CA � b0; (19)

where b0 is the coefficient of the first term in the �function,

�dg

d�¼ �b0

g3

162þ . . . (20)

so that the collinear factor �C for transverse gauge bosonsis

�

4ð4CALp � b0Þ: (21)

It is more convenient to write the anomalous dimensionsfor W3 and B instead of Z and �, to avoid off-diagonalmixing terms in the renormalization group evolution due tothe running of sin2�W .


014023-6

g (transverse gluons):

�s

4

�12 log

�n � p�

� 7

�; (22)

WT (transverse W1;2;3):

�2

4

�8 log

�n � p�

� 19

6

�; (23)

BT (transverse B):

�1

4

�41

6

�: (24)

The scalar anomalous dimension for the unphysicalGoldstone bosons, needed for longitudinal gauge bosonproduction using the equivalence theorem, and for Higgsproduction is

’;H:

1

4

�3

4�2 þ 1

4�1

��4 log

�n � p�

� 4

�þ 3

y2t162

: (25)

The yt term in the� anomalous dimension affects the ratesfor H, WL and ZL production at the few percent level.

B. Matching at �l �MZ

The matching corrections at �l �MZ have to be com-puted in the broken electroweak theory, using Table I,Eq. (4) and the discussion following it. The matching canbe computed for each particle, and is shown schematically

in Fig. 8. The collinear gauge-invariant operator ½WyEWc �

in SCETEW matches onto ½Wy�c � in SCET�. The differ-

ence between the collinear Wilson lines is that WyEW con-

tains gluons,W andB gauge fields which are the dynamical

fields in SCETEW whereas Wy� contains gluons and pho-

tons, the dynamical gauge fields in SCET�. The matching

coefficients are given by integrating out theW and Z. Onceagain, the collinear matching is more complicated due tocustodial SUð2Þ violation. Thus, in the quark doubletQL ¼ðu; dÞL, there are separate matching functions for uL anddL, etc.

The low-scale matching for the quark doublet can bewritten as

½WyEWQL� ! expDðQL!ULÞ

C ½Wy�UL�

expDðQL!DLÞC ½Wy

�DL�

" #: (26)

The quantity ½WyEWQL�a is collinear gauge invariant, and

has an index a. Equation (26) implies that the a ¼ 1 term

matches to ½Wy�UL� and the a ¼ 2 term to ½Wy

�DL�, withamplitudes expDðQL!ULÞ

C and expDðQL!DLÞC , respectively.

The other cases listed below use a similar notational con-vention. The collinear functions DC are zero at tree level.The remaining fermionic collinear matching functions

are defined by

½WyEWUR� ! expDðUR!URÞ

C ½Wy�UR�;

½WyEWDR� ! expDðDR!DRÞ

C ½Wy�DR�;

½WyEWLL� !

expDðLL!�LÞC �L

expDðLL!ELÞC ½Wy

�EL�

24

35;

½WyEWER� ! expDðER!ERÞ

C ½Wy�ER�:

(27)

The collinear matching functions are

DðQL!ULÞC ð�Þ ¼ g2LUDZð�Þ þ 1

2DWð�Þ;DðQL!tLÞ

C ð�Þ ¼ g2LUDZð�Þ þ 12DWð�Þ þ FtLð�Þ;

DðQL!DLÞC ð�Þ ¼ g2LDDZð�Þ þ 1

2DWð�Þ;DðQL!b0Þ

C ð�Þ ¼ g2LDDZð�Þ þ 12DWð�Þ þ Fb0Lð�Þ;

DðUR!URÞC ð�Þ ¼ g2RUDZð�Þ;DðtR!tRÞ

C ð�Þ ¼ g2RUDZð�Þ þ FtRð�Þ;DðDR!DRÞ

C ð�Þ ¼ g2RDDZð�Þ;DðLL!�LÞ

C ð�Þ ¼ g2L�DZð�Þ þ 12DWð�Þ;

DðLL!ELÞC ð�Þ ¼ g2LeDZð�Þ þ 1

2DWð�Þ;DðER!ERÞ

C ð�Þ ¼ g2ReDZð�Þ;

(28)

where gLU ¼ 1=2� 2=3sin2�W , gRU ¼ �2=3sin2�W , etc.are the Z charges of the fermions, and

DZð�Þ ¼ �Z

4

�2 log

M2Z

�2log

�n � p�

� 1

2log2

M2Z

�2

� 3

2log

M2Z

�2� 52

12þ 9

4

�;

DWð�Þ ¼ �W

4

�2 log

M2W

�2log

�n � p�

� 1

2log2

M2W

�2

� 3

2log

M2W

�2� 52

12þ 9

4

�;

(29)

where �W ¼ �em=sin2�W , �Z ¼ �em=ðsin2�Wcos2�WÞ.

The additional contributions for the t, b quarks are givenby

FIG. 8. Collinear matching graphs for ½Wyc �. The � is the½Wyc � operator, the solid line is c , and the double line is Wy.


014023-7

FtLð�Þ ¼��s

4

4

3þ�em

4

4

9

��1

2log2

m2t

�2� 1

2log

m2t

�2þ2

12þ 2

�

þ�W

4

1

2fF

�m2

t

M2W

;0

�þ�Z

4g2LtfF

�m2

t

M2Z

;m2

t

M2Z

�þð�RtL ��RuLÞ;

FtRð�Þ ¼��s

4

4

3þ�em

4

4

9

��1

2log2

m2t

�2� 1

2log

m2t

�2þ2

12þ 2

�

þ�Z

4g2RtfF

�m2

t

M2Z

;m2

t

M2Z

�þð�RtR ��RuR Þ;

Fb0Lð�Þ ¼ �W

4

1

2fF

�0;

m2t

M2W

�þð�RbL ��RdLÞ: (30)

The �s and �em terms are from the QCD and QED cor-rections due to the transition from SCET to bHQET fields.The functions fF;S are given in Appendix B of Ref. [7].

(�RtL � �RuL) is the difference in wave function correc-

tions for the t and a massless quark. The massless wavefunction contribution has already been included in DW;Z.

The H, ’ and gauge boson matching has mixing effectsdue to graphs such as those in Fig. 9. The graphs are oforder h�i=ð �n � pÞ and are subleading in the SCET powercounting.

The matching function for the Higgs doublet has someinteresting features. The Higgs doublet is

� ¼ 1ffiffiffi2

p ’2 þ i’1

vþH � i’3

� �; (31)

with ’� ¼ ð’1 � i’2Þ= ffiffiffi2

p. There are two neutral gauge

bosons, the Z and �, but only one neutral unphysicalGoldstone boson, the’3. One could try a matching relationof the form

½WyEW�� ! expDð�!’þÞ

C ½Wy�’þ�

1ffiffi2

p expDð�!HÞC H � iffiffi

2p expDð�!’3Þ

C ’3

24

35;

(32)

analogous to the fermionic case discussed above. A match-ing of this kind, which was used in Ref. [8] for the SUð2Þtheory, is not possible for the standard model. The ’þpropagator in the full theory has photon corrections shownin Fig. 10. The graphs are infrared divergent, but theinfrared divergence cancels between the two diagrams so

that the ’þ propagator is not infrared divergent in theelectroweak theory. In the theory below �l, the W bosonshave been integrated out, and the second diagram is absent,so that the ’þ propagator is infrared divergent. Thus, theinfrared divergences do not match between the theoriesabove and below �l.The resolution of this paradox is that ’þ is not a

physical field and is gauge dependent. At the scale �l,the Higgs doublet matches, not to the Higgs H and un-physical Goldstone bosons ’þ and ’3, but to H andlongitudinal gauge bosons WL and ZL. WL is treated as abHQET field, and the WL propagator has an infrareddivergence from Fig. 11, so there is still an infrared diver-gence in the effective theory. However now, the amplitudethat must be matched is forWL, not ’

þ, and is given by theamplitude for ’þ multiplied by the equivalence theoremfactor E, which is the radiative correction factor in theequivalence theorem [45,53–59]. There is an infrared di-vergence in E that matches the infrared divergence in theeffective theory. The standard model one-loop values forEW;Z needed for longitudinalW and Z production are given

in Appendix C.The matching Eq. (32) should instead be written as

½WyEW�� ! expD

ð�!WþL Þ

C ½Wy�WL�

1ffiffi2

p expDð�!HÞC H� iffiffi

2p expDð�!ZLÞ

C ZL

24

35:

(33)

The collinear functions are

FIG. 9. One-loop collinear graphs that induce mixing betweenthe gauge and Higgs sectors.

γ

γ

W

FIG. 10. Photon corrections to the ’þ propagator.

γ

W+ W+

FIG. 11. Photon corrections to the bHQET Wþ propagator.


014023-8

Dð�!WLÞC ¼ �W

4

1

4

�FW þ fS

�1;M2

H

M2W

��þ �W

4

1

4

�FW þ fS

�1;

M2Z

M2W

��þ �Z

4g2’þ

�FZ þ fS

�M2

W

M2Z

;M2

W

M2Z

��

þ �W

4s2W

�1

2log2

M2W

�2� log

M2W

�2þ 2

12þ 2

�þ 1

2�R’þ þ logEW;

Dð�!ZLÞC ¼ �W

4

1

2

�FW þ fS

�M2

Z

M2W

; 1

��þ �Z

4

1

4

�FZ þ fS

�1;M2

H

M2Z

��þ 1

2�R’3 þ logEZ;

Dð�!HÞC ¼ �W

4

1

2

�FW þ fS

�M2

H

M2W

; 1

��þ �Z

4

1

4

�FZ þ fS

�M2

H

M2Z

; 1

��þ 1

2�RH;

(34)

where

FW;Z ¼ 2 logM2

W;Z

�2log

�n � p�

� 1

2log2

M2W;Z

�2� log

M2W;Z

�2

� 52

12þ 1; (35)

and EW;Z are the equivalence theorem factors for theW andZ. The expression for EW is the same as that for the SUð2Þtheory given in Ref. [8]. EZ is given by a similar expres-sion, see Ref. [45] for details. There are corrections to theequivalence theorem from �� Z mixing at two-loops, ifone does not use background field gauge [60].

The gauge-field collinear matching involves �� Zmix-ing. The collinear functions are defined by

½Wyg?� ! expDðg!gÞC g?;

½WyW�?� ! expDðW!WÞ

C W�? ;

½WyW3?� ! cW expDðW!ZÞ

C Z? þ sW expDðW!�ÞC A?;

½WyB?� ! �sW expDðB!ZÞC Z? þ cW expDðB!�Þ

C A?;

(36)

(sW ¼ sin�W , cW ¼ cos�W), so that all the collinear func-tions vanish at tree level. The complications of �� Zmixing only enter the effective theory at the low-scalematching at �l.The gluon matching is

Dðg!gÞC ¼ �s

4

1

3log

m2t

�2: (37)

There is a nontrivial gluon collinear matching from the top-quark vacuum polarization graph, since the top quark isintegrated out at the scale �l and is no longer a dynamicalfield. Processes involving external top quark can still becomputed using bHQET fields for the top.The other gauge-field collinear functions are

DðW!WÞC ¼ �W

4c2W

�FZ þ fS

�M2

W

M2Z

;M2

W

M2Z

��þ �W

4c2W

�FW þ fS

�1;

M2Z

M2W

��þ �W

4s2W½FW þ fSð1; 0Þ�

þ �W

4s2W

�1

2log2

M2W

�2� log

M2W

�2þ 2

12þ 2

�þ 1

2�RWþ ;

DðW!ZÞC ¼ �W

42

�FW þ fS

�M2

Z

M2W

; 1

��þ 1

2�RZ þ tan�WR�!Z;

DðB!ZÞC ¼ 1

2�RZ � cot�WR�!Z;

DðW3!�ÞC ¼ �W

42½FW þ fSð0; 1Þ� þ 1

2�R� þ cot�WRZ!�;

DðB!�ÞC ¼ 1

2�R� � tan�WRZ!�:

(38)

The definitions of R�!Z and RZ!�, which arise from �� Z mixing, are given in Appendix B.

C. Running below �l �MZ

The collinear anomalous dimensions for the running below�l �MZ are listed below. The gauge coupling constants are

the MS values in the theory with five dynamical quark flavors.


014023-9

ðu; cÞL;R, ð �u; �cÞL;R:1

4

�4

3�s þ 4

9�em

��4 log

�n � p�

� 3

�; (39)

ðd; s; bÞL;R, ð �d; �s; �bÞL;R:1

4

�4

3�s þ 1

9�em

��4 log

�n � p�

� 3

�; (40)

t treated as a bHQET field hv:

1

4

�4

3�s þ 4

9�em

�ð4 log2�� 2Þ; (41)

W�T;L treated as a bHQET field hv:

1

4ð�emÞð4 log2�� 2Þ; (42)

ZT;L treated as a bHQET field hv:

0; (43)

H treated as a bHQET field hv:

0; (44)

ðe;�; �ÞL;R, ð �e; ��; ��ÞL;R:�em

4

�4 log

�n � p�

� 3

�; (45)

ð�e; ��; ��ÞL;R, ð ��e; ��; ��ÞL;R:0; (46)

g:

�s

4

�12 log

�n � p�

� 23

3

�; (47)

�:

�em

4

�80

9

�: (48)

IV. SOFT FUNCTIONS

The high-energy scattering amplitude can be separatedinto soft and collinear contributions, the precise definitionsof which have been given in Ref. [8]. With the definitionsused there, the soft-anomalous dimension and low-scalematching take on a simple and universal form, which isvalid as long as all particles in the process are energetic.The expressions are valid even for energetic heavy parti-cles such as the top quark (with E mt). For nonrelativ-istic heavy particles, the soft function is more complicated,and depends both on the direction of the heavy particlen, and its velocity �. It reduces to our result in the limit� ! 1.

The universal soft functions is

USðni; njÞ ¼ log�ni � nj � i0þ

2(49)

in terms of which, the soft-anomalous dimension and low-scale matching are

�S ¼ �ð�ð�ÞÞ��X

hijiTi � TjUSðni; njÞ

�;

DS ¼ Jð�ð�Þ; LMÞ��X

hijiTi � TjUSðni; njÞ

�;

(50)

where, at one-loop,

�ð�ð�ÞÞ ¼ �ð�Þ4

4; Jð�ð�Þ; LMÞ ¼ �ð�Þ4

2 logM2

�2:

(51)

The soft-anomalous dimension is mass independent, butthe soft matching depends on the gauge boson mass M. Inthe computations, we have used the three-loop value for �[61], and the results of Refs. [62,63].The soft function has a simple form when written using

the color-operator notation [64]. For practical calculations,one needs to write the soft function as a matrix in the spaceof gauge-invariant operators. In this section, we give theexplicit matrices needed for some scattering processes.The QCD parts of these matrices have been obtainedpreviously [32]. The electroweak part is considerablymore involved, because the SUð2Þ �Uð1Þ symmetry isbroken, and this enters into the low-scale soft function DS.For the standard model, one has to use Eq. (4) and (5) to

obtain the soft-anomalous dimension and low-scale match-ing. For a given process, the SUð3Þ, SUð2Þ and Uð1Þmatrices are defined by

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ;

S1 ¼ �Xhiji

YiYjUSðni; njÞ;

(52)

in terms of which, the soft-anomalous dimension is

�S ¼ �s

S3 þ �2

S2 þ �1

S1: (53)

For the low-scale matching, one has to use Eq. (4) and(5) with M ! MW and M ! MZ for the W and Z terms.This gives


014023-10

DS ¼ �Wð�Þ4

2 logM2

W

�2

��X

hijiðti � tj � t3it3jÞUSðni; njÞ

�

þ �Zð�Þ4

2 logM2

Z

�2

��X

hijitZitZjUSðni; njÞ

�

¼ �Wð�Þ4

2 logM2

W

�2

�S2 þ

Xhiji

t3it3jUSðni; njÞ�

þ �Zð�Þ4

2 logM2

Z

�2

��X


�: (54)

The soft function has a universal form when written inthe operator form Eq. (50). For numerical computations, itis more convenient to choose a basis of gauge-invariantoperators, and write the soft-anomalous dimension andmatching as a matrix in the chosen basis. The soft factorP

hijiTi � TjUSðni; njÞwas computed for some simple cases

in Ref. [8] for an SUðNÞ gauge theory. Certain soft matri-ces occur in several different computations. These refer-ence matrices are for SUð3Þ:

Sð3Þ ¼ � 8

3i1þ

73T þ 2

3U 2ðT �UÞ49 ðT �UÞ 0

24

35;

Sð3Þ0 ¼ � 4

3i;

Sð3;gÞ ¼ � 13

3i1

þ0 0 U� T

0 32 ðT þUÞ 3

2 ðU� TÞ2ðU� TÞ 5

6 ðU� TÞ 32 ðT þUÞ

2664

3775 (55)

for SUð2Þ:

Sð2Þ ¼ � 3

2i1þ ðT þUÞ 2ðT �UÞ

38 ðT �UÞ 0

" #;

Sð2Þ0 ¼ � 3

4i;

Sð2;gÞ ¼ � 11

4i1þ 0 U� T

2ðU� TÞ ðT þUÞ

" #(56)

and for Uð1Þ:

Sð1Þðq1; q2; q3; q4Þ ¼ �i

2ðq21 þ q22 þ q23 þ q24Þ

þ ðq1q4 þ q2q3ÞT� ðq1q3 þ q2q4ÞU;

Sð1Þðqf; qiÞ ¼ �iðq2i þ q2fÞ þ 2qiqfðT �UÞ:(57)

For scattering kinematics, s > 0, t < 0, and u < 0, andthe variables T, U are defined by [32]

T ¼ log�t

sþ i; U ¼ log

�u

sþ i: (58)

The conventions for s, t, u are the same as those in Ref. [8].

V. SOFT FUNCTIONS FOR FERMIONSCATTERING

The soft-anomalous dimension and low-scale matchingmatrices will now be computed for some scattering pro-cesses. In these examples, the anomalous dimension and

matching depend on matrices S3;2;1, Rð0Þ, and DW;Z. The

equations for the anomalous dimension and matching havethe same form in each case; the matrices take on differentvalues depending on the process.

A. Two doublets

Consider first the case of fermion scattering, Q �Q !Q �Q, where all particles are electroweak doublet quarks.At the high-scale, one matches onto four-quark SCEToperators

C11�QðcÞ4 taTA��PLQ

ðcÞ3

�QðuÞ2 taTA��PLQ

ðuÞ1

þC21�QðcÞ4 ��TAPLQ

ðcÞ3

�QðuÞ2 ��TAPLQ

ðuÞ1

þC12�QðcÞ4 ta��PLQ

ðcÞ3

�QðuÞ2 ta��PLQ

ðuÞ1

þC22�QðcÞ4 ��PLQ

ðcÞ3

�QðuÞ2 ��PLQ

ðuÞ1 ; (59)

where the first index is 1 for ta � ta and 2 for 1 � 1 inSUð2Þ, and the second index is 1 for Ta � Ta and 2 for 1 �1 in SUð3Þ. Equation (59) is written in schematic form toemphasize the gauge structure of the operator. The actual

operator in SCET should be written withQ ! WyðQÞn;p, etc.

The subscripts 1–4 on the fields are a reminder that theSCET fields have momentum labels p1 � p4. We havechosen to label the two fields by u and c to make it easyto discuss related processes such as Drell-Yan by replacingsome quark fields by lepton fields. The one-loop values forCij at the high scale are given in Ref. [7].

The group theory sums needed for the soft-anomalousdimension matrix are

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ 1 �Sð3Þ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ Sð2Þ � 1;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðQðcÞÞ; YðQðuÞÞÞ1 � 1

in terms of the reference matrices Sð3;2;1Þ given in

Eqs. (55)–(57). For quark doublets, YðQðcÞÞ ¼ YðQðuÞÞ ¼1=6.The soft-anomalous dimension is given by Eq. (53)

using Eq. (60) for S3;2;1. At the low-scale �l �mZ, the

operators Eq. (59) match onto a linear combination of


014023-11

O11 ¼ ½ �cL4TA��cL3�½ �uL2TA��uL1�O21 ¼ ½ �cL4TA��cL3�½ �d0L2TA��d0L1�O31 ¼ ½�s0L4TA��s

0L3�½ �uL2TA��uL1�

O41 ¼ ½�s0L4TA��s0L3�½ �d0L2TA��d0L1�

O51 ¼ ½�s0L4TA��cL3�½ �uL2TA��d0L1�O61 ¼ ½ �cL4TA��s

0L3�½ �d0L2TA��uL1�

O12 ¼ ½ �cL4��cL3�½ �uL2��uL1�O22 ¼ ½ �cL4��cL3�½ �d0L2��d0L1�O32 ¼ ½�s0L4��s

0L3�½ �uL2��uL1�

O42 ¼ ½�s0L4��s0L3�½ �d0L2��d0L1�

O52 ¼ ½�s0L4��cL3�½ �uL2��d0L1�O62 ¼ ½ �cL4��s

0L3�½ �d0L2��uL1�;

(61)

with coefficients Cij. The matching matrix is

C ia ¼ RijCja; (62)

or equivalently,

C ¼ ðR � 1ÞC; (63)

since the electroweak matching does not change the colorstructure of the operators. At tree level R is

Rð0Þ ¼

14 1

� 14 1

� 14 1

14 1

12 0

12 0

266666666666664

377777777777775: (64)

Once again, we see the additional complication in thestandard model due to the Uð1Þ sector. In the pure SUð2Þtheory, the matching was SUð2Þ invariant; here the opera-tors have to be broken apart into individual fields ofdefinite charge.

The one-loop soft matching due toW and Z exchange iscomputed using Eq. (54),

Rð1ÞS;W ¼ �W

42 log

M2W

�2

�S2 þ

Xhiji

t3it3jUSðni; njÞ�;

Rð1ÞS;Z ¼ �Z

42 log

M2Z

�2

��X


�;

(65)

and the total soft-matching at one-loop is

R ¼ Rð0Þ þ Rð1ÞS;W þ Rð1Þ

S;Z: (66)

To evaluate Rð1ÞS;W , R

ð1ÞS;Z, we need to evaluate the group

theory factors in Eq. (65). The S2 term acting on Eq. (61)is a group-invariant Casimir operator, and can be thoughtof as S2 acting on the original basis Eq. (59) beforeSUð2Þ �Uð1Þ breaking, and so acting on the low-energy

basis Eq. (61) is equal to Rð0ÞS2, whereS2 is the matrix inEq. (60). The t3it3j and tZitZj terms are diagonal in the

basis Eq. (61), and we define them to be DW and DZ,respectively, so that the soft-matching matrices are

Rð1ÞS;W ¼ �W

42 log

M2W

�2½Rð0ÞS2 þDWR

ð0Þ�;

Rð1ÞS;Z ¼ �Z

42 log

M2Z

�2½DZR

ð0Þ�:(67)

This equation is valid for all the scattering processes wewill consider. TheW matching has aS2 term, which is thesame matrix that enters the soft-anomalous dimension, andtheW and Z matchings have extra diagonal matricesDW;Z

that depend on the process.For the doublet scattering case, Eq. (61), DW;Z are

DW ¼ diagðw1;�w1;�w1; w1; w2; w2Þ þ 1

2i1;

DZ ¼ diagðz1; z2; z3; z4; z5; z5Þ;w1 ¼ � 1

2ðT �UÞ;

w2 ¼ � 1

2ðT þUÞ;

z1 ¼ 2gLcgLuðT �UÞ � iðg2Lc þ g2LuÞ;z2 ¼ 2gLcgLdðT �UÞ � iðg2Lc þ g2LdÞ;z3 ¼ 2gLsgLuðT �UÞ � iðg2Ls þ g2LuÞ;z4 ¼ 2gLsgLdðT �UÞ � iðg2Ls þ g2LdÞ;z5 ¼ ðgLugLc þ gLdgLsÞT � ðgLugLs þ gLdgLcÞU

� i

2ðg2Lc þ g2Lu þ g2Ls þ g2LdÞ:

(68)

The results Eqs. (67) and (68) hold for all cases whereboth fermions are doublets. For example, if the final quarkdoublet is replaced by a lepton doublet, one gets four-fermion operators for the Drell-Yan process q �q !�þ��. The four-quark operators only have the tensorstructure 1 � 1 in color space and the anomalous dimen-

sion is Eq. (53) with Sð3Þ ! Sð3Þ0 and YðQðcÞÞ !YðLð�ÞÞ ¼ �1=2 in Eq. (60). The unit matrix in color spaceis now a 1� 1 matrix instead of a 2� 2 matrix. The low-scale matching is obtained from Eq. (68) with the obviousreplacement gLc ! gL�, gLs ! gLe. A similar result holdsif the initial doublet is a lepton doublet and the final is aquark doublet, or if both are lepton doublets (in which case,S3 ! 0).


014023-12

B. One doublet and one singlet

The second case is where one fermion is a doublet andthe other is a singlet. As an example, consider

C1�QðcÞ4 ��TAPLQ

ðcÞ3 �u2�

�TAPRu1

þC2�QðcÞ4 ��PLQ

ðcÞ3 �u2�

�PRu1: (69)


S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3Þ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ Sð2Þ0;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðQðcÞÞ; YðuRÞÞ:

(70)

At the low-scale �l �mZ, the operators Eq. (59) matchonto a linear combination of

O11 ¼ ½ �cL4TA��cL3�½ �uR2TA��uR1�O21 ¼ ½ �s0L4TA��s

0L3�½ �uR2TA��uR1�

O12 ¼ ½ �cL4��cL3�½ �uR2��uR1�O22 ¼ ½ �s0L4��s

0L3�½ �uR2��uR1�:

(71)

The matching matrix is

C ia ¼ RiCa ) C ¼ ðR � 1ÞC; (72)

since the matching leaves the color structure unchanged.At tree level R is

Rð0Þ ¼ 11

� �: (73)

At one-loop, the soft-matching matrices due to W and Zexchange are

DW ¼ diagðw1; w1Þ;DZ ¼ diagðz1; z2Þ;w1 ¼ 1

4i;

z1 ¼ 2gLcgRuðT �UÞ � iðg2Lc þ g2RuÞ;z2 ¼ 2gLsgRuðT �UÞ � iðg2Ls þ g2RuÞ;

(74)

and the soft matching is given by Eq. (67).Equations (72)–(74) apply to all cases where one fer-

mion is a weak doublet, and the other is a weak singlet,with the obvious replacement of the Z charges for leptondoublets.

C. Two singlets

The last case is if both fermions are weak singlets, forexample, the operators

C1 �c4��TAPRc3u2�

�TAPRu1 þ C2 �c4��PRc3u2�

�PRu1;

(75)

which match to

O1 ¼ ½ �cR4��TAcR3�½ �uR2��TAuR1�

O2 ¼ ½ �cR4��cR3�½ �uR2��uR1�:(76)


S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3Þ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ 0;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðcRÞ; YðuRÞÞ;

(77)

which are used in Eq. (53) to obtain the soft-anomalousdimension.The one-loop matching condition is

C a ¼ RCa ) C ¼ ðR � 1ÞC; (78)

with Rð0Þ ¼ 1 at tree level, and the soft-matching contri-bution is

Rð1ÞS;W ¼ 0;

Rð1ÞS;Z ¼ �Z

42 log

M2Z

�2½ðT �UÞ2gRcgRu � iðg2Rc þ g2RuÞ�:

(79)

One can similarly obtain the results for right-handedleptons by replacing the quark Z charges by the corre-sponding lepton charges.

VI. SOFT FUNCTIONS FOR ELECTROWEAKGAUGE BOSON PAIR PRODUCTION

A. Doublets

The kinematics for the electroweak gauge boson pairproduction is shown in Fig. 12.We first start with gauge boson production by left-

handed quarks, which are electroweak doublets, and inter-

p2

W+(p3)

p1

W-(p4)

FIG. 12. Pair production qðp1Þ þ �qðp2Þ ! Wþðp3Þ þW�ðp4Þ. Time runs vertically.


014023-13

act with the W and B gauge bosons of the SUð2Þ and Uð1Þinteractions. The operator basis is

O1 ¼ �QðuÞ2 QðuÞ

1 Wa4W

a3 O2 ¼ �QðuÞ

2 tcQðuÞ1 i�abcWa

4Wb3

O3 ¼ �QðuÞ2 taQðuÞ

1 B4Wa3 O4 ¼ �QðuÞ

2 taQðuÞ1 Wa

4B3


1 B4B3; (80)

where only the gauge structure has been shown. The op-erators at the high scale are best written in terms of theSUð2Þ and Uð1Þ gauge fields W and B, rather than themass-eigenstate fields Z and �. Note that �abcWa

3Wb4 � 0

since the two W fields have momentum labels p3 and p4,which are different. In this basis

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3Þ0 � 1;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ diagðS2a;S2b;S2b;S2cÞ;

S2a ¼��i

11

4

�1þ 0 U� T

2ðU� TÞ T þU

" #;

S2b ¼ � 7

4iþUþ T;

S2c ¼ � 3

4i;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðQðuÞÞÞ1; (81)

and the soft-anomalous dimension between the scales Q

and �l �MZ is given by Eq. (53), where the matricesSð3;2;1Þ are given by Eqs. (81).

At the low-scale �l �MZ, the operators Eq. (80) matchonto

O1 ¼ �uL2uL1Wþ4 W

�3 O2 ¼ �uL2uL1W

�4 W

þ3

O3 ¼ �uL2uL1Z4Z3 O4 ¼ �uL2uL1A4Z3

O5 ¼ �uL2uL1Z4A3 O6 ¼ �uL2uL1A4A3

O7 ¼ �dL2dL1Wþ4 W

�3 O8 ¼ �dL2dL1W

�4 W

þ3

O9 ¼ �dL2dL1Z4Z3 O10 ¼ �dL2dL1A4Z3

O11 ¼ �dL2dL1Z4A3 O12 ¼ �dL2dL1A4A3

O13 ¼ �uL2dL1Wþ4 Z3 O14 ¼ �uL2dL1W

þ4 A3

O15 ¼ �uL2dL1Z4Wþ3 O16 ¼ �uL2dL1A4W

þ3

O17 ¼ �dL2uL1Z4W�3 O18 ¼ �dL2uL1A4W

�3

O19 ¼ �dL2uL1W�4 Z3 O20 ¼ �dL2uL1W

�4 A3:

(82)

The subscripts 3, 4 represent outgoing label momenta p3

and p4, and the gauge indices are to be treated as those on aquantum field, i.e. they represent the charge on the anni-hilation operator. These operators are to be treated in thesame manner as terms in a Lagrangian. Thus, e �e !Wþðk1ÞW�ðk2Þ is given by C4 with p4 ¼ k1 and p3 ¼k2, plus C5 with p4 ¼ k1 and p3 ¼ k2.The tree-level matching is

C i ¼ ðRð0ÞÞijCj; Rð0Þ ¼

1 12 0 0 0

1 � 12 0 0 0

c2W 0 � 12 sWcW � 1

2 sWcW s2WsWcW 0 1

2 c2W � 1

2 s2W �sWcW

sWcW 0 � 12 s

2W

12 c

2W �sWcW

s2W 0 12 sWcW

12 sWcW c2W

1 � 12 0 0 0

1 12 0 0 0

c2W 0 12 sWcW

12 sWcW s2W

sWcW 0 � 12 c

2W

12 s

2W �sWcW

sWcW 0 12 s

2W � 1

2 c2W �sWcW

s2W 0 � 12 sWcW � 1

2 sWcW c2W0 � 1ffiffi

2p cW 0 � 1ffiffi

2p sW 0

0 � 1ffiffi2

p sW 0 1ffiffi2

p cW 0

0 1ffiffi2

p cW � 1ffiffi2

p sW 0 0

0 1ffiffi2

p sW1ffiffi2

p cW 0 0

0 � 1ffiffi2

p cW � 1ffiffi2

p sW 0 0

0 � 1ffiffi2

p sW1ffiffi2

p cW 0 0

0 1ffiffi2

p cW 0 � 1ffiffi2

p sW 0

0 1ffiffi2

p sW 0 1ffiffi2

p cW 0

266666666666666666666666666666666666666666666666664

377777777777777777777777777777777777777777777777775

: (83)


014023-14

The one-loop soft matching is given by Eq. (67), where

DW ¼ diagðw1;w2;w4;w4;w4;w4;w2;w1;w4;w4;w4;w4;w3;w3;w3;w3;w3;w3;w3;w3Þ;DZ¼ diagðz1;z2;z7;z7;z7;z7;z3;z4;z8;z8;z8;z8;z5;z5;z6;z6;z5;z5;z6;z6Þ;w1¼T�Uþ5

4i; w2¼�TþUþ5

4i; w3¼�1

2ðTþUÞþ3

4i; w4¼ 1

4i;

z1¼ 2gLugWðU�TÞ� iðg2Luþg2WÞ; z2¼ 2gLugWðT�UÞ� iðg2Luþg2WÞ; z3¼ 2gLdgWðU�TÞ� iðg2Ldþg2WÞ;z4¼ 2gLdgWðT�UÞ� iðg2Ldþg2WÞ; z5¼�gLdgWTþgLugWU� iðgLugLdþgLugW �gLdgWÞ;z6¼gLugWT�gLdgWU� iðgLugLdþgLugW �gLdgWÞ; z7¼�ig2Lu; z8¼�ig2Ld; (84)

and

gW ¼ 1� sin2�W ¼ cos2�W (85)

is the Z charge of the Wþ boson.The above equations can also be used to compute radia-

tive corrections to gauge boson pair production by thelepton electroweak doublet, with the obvious replacementof quark Z charges by the corresponding lepton ones, andS3 ! 0.

B. Singlets

Electroweak singlet (right-handed) quarks can produceelectroweak gauge bosons. For example gauge boson pro-dution by right-handed u quarks. The operator generated attree level is

O ¼ �uR2uR1B4B3; (86)

with

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3Þ0;

S2 ¼ �Xhiji


S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðuRÞÞ;

(87)

and the soft-anomalous dimension between the scales Qand �l �MZ is given by Eq. (53), where the matricesSð3;2;1Þ are given by Eqs. (87).

At the scale �l, the operator O matches to Oi,

O 1 ¼ �uR2uR1Z4Z3 O2 ¼ �uR2uR1A4Z3

O3 ¼ �uR2uR1Z4A3 O4 ¼ �uR2uR1A4A3:(88)

The tree-level matching is

Rð0Þs2W

�sWcW�sWcWc2W

26664

37775; (89)

and the one-loop soft-matching contribution is

Rð1ÞS;W ¼ 0; Rð1Þ

S;Z ¼ �Z

42 log

M2Z

�2ð�ig2RuÞRð0Þ: (90)

The above can also be used for right-handed d quarks withYðuRÞ ! YðdRÞ, and for right-handed leptons withYðuRÞ ! YðeRÞ and S3 ! 0.Right-handed u quarks can produce electroweak gauge

bosons via

O ¼ �uR2uR1Wa4W

a3 ; (91)

which is not present at tree level, sinceWa does not coupleto uR. For this operator,

S3 ¼ �Xhiji


S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ �2i;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðuRÞÞ:

(92)

At the low scale, the operator matches to

O1 ¼ �uR2uR1Wþ4 W

�3 O2 ¼ �uR2uR1W

�4 W

þ3

O3 ¼ �uR2uR1Z4Z3 O4 ¼ �uR2uR1A4Z3

O5 ¼ �uR2uR1Z4A3 O6 ¼ �uR2uR1A4A3;

(93)

and the matching condition is C ¼ RC with tree-levelvalue


014023-15

Rð0Þ ¼

11c2W

sWcWsWcWs2W

2666666664

3777777775: (94)

The one-loop matching due to W and Z exchange isgiven by Eq. (67) with

DW ¼ diagðw1; w1; 0; 0; 0; 0Þ;DZ ¼ diagðz1; z2; z3; z3; z3; z3Þ;w1 ¼ i;

z1 ¼ �2gRugWðT �UÞ � iðg2Ru þ g2WÞ;z2 ¼ 2gRugWðT �UÞ � iðg2Ru þ g2WÞ;z3 ¼ �ig2Ru:

(95)

The results for dR and eR are given by gRu ! gRd, gRe forthe Z charge in DZ.

C. Longitudinal bosons via QL�QL ! ’’

For longitudinal W production, we also need the resultsfor external unphysical Goldstone boson ’ fields, whichare contained in the Higgs multiplet �. The operators are

O1 ¼ �QðuÞtaQðuÞ�y4 t

a�3 O2 ¼ �QðuÞQðuÞ�y4�3: (96)

The gauge current ið�yTaD��D��yTa�Þ produces

operators of this form, weighted by a label momentumfactor P�

4 � P�3 , which is included in the operator coef-

ficients, and is antisymmetric in 3 $ 4.The group theory sums needed for the soft-anomalous

dimension matrix are

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3Þ0 � 1;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ Sð2Þ;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYð�Þ; YðQðuÞÞÞ1:

(97)

At the low scale, the operators match to 20 operators,which have the same structure as the gauge boson operatorsin Eq. (82), with the replacement W� ! ’�, W3 ! ’3,B ! H.

O1 ¼ �uL2uL1’�4 ’

þ3 O2 ¼ �uL2uL1’

34’

33

O3 ¼ �uL2uL1H4’33 O4 ¼ �uL2uL1’

34H3

O5 ¼ �uL2uL1H4H3 O6 ¼ �dL2dL1’�4 ’

þ3

O7 ¼ �dL2dL1’34’

33 O8 ¼ �dL2dL1H4’

33

O9 ¼ �dL2dL1’34H3 O10 ¼ �dL2dL1H4H3

O11 ¼ �uL2dL1’34’

þ3 O12 ¼ �uL2dL1H4’

þ3

O13 ¼ �dL2uL1’�4 ’

33 O14 ¼ �dL2uL1’

�4 H3:

(98)

The convention chosen for the scalar fields is

� ¼ 1ffiffiffi2

p ’2 þ i’1

vþH� i’3

� �; (99)

with ’� ¼ ð’1 � i’2Þ= ffiffiffi2

p, so that ’a / iTah�i. The ac-

tion of T3 on the neutral fields is

T3H ¼ i

2’3; T3’

3 ¼ � i

2H: (100)

This causes mixing between ’3 and H. Under custodialSUð2Þ symmetry, the H is a singlet, and ’3 belongs to atriplet, so ’3 �H mixing is forbidden by custodial SUð2Þ.In the standard model, custodial SUð2Þ is violated byhypercharge, and ’3 �H mixing is allowed. In the resultsderived below, there is ’3 �H mixing from W and Zexchange. In the limit �W ¼ �Z and MW ¼ MZ, whencustodial SUð2Þ is restored, the two mixing contributionscancel.The tree-level matching is

Rð0Þ ¼

14 1

� 18

12

i8 � i

2

� i8

i2

� 18

12

� 14 1

18

12

� i8 � i

2

i8

i2

18

12

� 12ffiffi2

p 0

i2ffiffi2

p 0

� 12ffiffi2

p 0

� i2ffiffi2

p 0

26666666666666666666666666666666666666666664

37777777777777777777777777777777777777777775

; (101)

and the one-loop matching is given by Eq. (67), with


014023-16

DW ¼ ðw2;DW1;w1;DW2;DW3;DW4Þ; DW1 ¼

w0 w3 �w3 w0

�w3 w0 �w0 �w3

w3 �w0 w0 w3

w0 w3 �w3 w0

2666664

3777775; DW2 ¼

w0 �w3 w3 w0

w3 w0 �w0 w3

�w3 �w0 w0 �w3

w0 �w3 w3 w0

2666664

3777775;

DW3 ¼w4 iw4

�iw4 w4

" #; DW4 ¼

w4 �iw4

iw4 w4

" #;

w0 ¼ 1

4i; w1 ¼ 1

2ðT�UÞþ1

2i; w2 ¼�1

2ðT�UÞþ1

2i; w3 ¼ 1

4iðT�UÞ; w4 ¼�1

4ðTþUÞþ1

4i;

DZ ¼ ðz1;DZ1; z2;DZ2;DZ3;DZ4Þ; DZ1 ¼

z3 �z4 z4 �z5

z4 z3 z5 z4

�z4 z5 z3 �z4

�z5 �z4 z4 z3

2666664

3777775; DZ2 ¼

z6 �z7 z7 �z5

z7 z6 z5 z7

�z7 z5 z6 �z7

�z5 �z7 z7 z6

2666664

3777775;

DZ3 ¼z8 �z9

z9 z8

" #; DZ4 ¼

z8 z9

�z9 z8

" #;

z1 ¼ 2g’þgLuðT�UÞ� iðg2’þ þg2LuÞ; z2 ¼ 2g’þgLdðT�UÞ� iðg2

’þ þg2LdÞ; z3 ¼�ig2Lu;

z4 ¼ 1

2igLuðT�UÞ; z5 ¼ 1

4i; z6 ¼�ig2Ld; z7 ¼ 1

2igLdðT�UÞ;

z8 ¼ g’þgLuT�g’þgLdU� iðgLugLd�gLdg’þ þgLug’þÞ; z9 ¼ 1

2igLdT�1

2igLuUþ

2gLd�

2gLuþ

2g’þ :

(102)

The matricesDW;Z have block-diagonal form due to ’3 �H mixing. The ’3 and ’� terms are then used to computelongitudinal Z and W� production, using the Goldstoneboson equivalence theorem. The equivalence theorem fac-tor E is included in the collinear function and does not enterthe soft matching.

D. Longitudinal bosons via qR �qR ! ’’

Longitudinal gauge bosons are produced by right-handed quarks via operators such as

O1 ¼ �uRuR�y4�3: (103)


S3 ¼ �Xhiji


S2 ¼ �Xhiji


S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYð�Þ; YðuRÞÞ1:

(104)

At the low scale, the operators match to

O1 ¼ �uR2uR1’�4 ’

þ3

O2 ¼ �uR2uR1’34’

33

O3 ¼ �uR2uR1H4’33

O4 ¼ �uR2uR1’34H3

O5 ¼ �uR2uR1H4H3:

(105)


Rð0Þ ¼

112

� i2

i2

12

26666666664

37777777775; (106)

and the one-loop matching is given by Eq. (67) with


014023-17

DW ¼ 1

4idiagð1;DW1Þ;

DW1 ¼

0 0 0 1

0 0 �1 0

0 �1 0 0

1 0 0 0

2666664

3777775;

DZ ¼ ðz1;DZ1Þ;

DZ1 ¼

z3 �z2 z2 �z4

z2 z3 z4 z2

�z2 z4 z3 �z2

�z4 �z2 z2 z3

2666664

3777775;

z1 ¼ 2g’þgRuðT �UÞ � iðg2’þ þ g2RuÞ;

z2 ¼ 1

2igRuðT �UÞ;

z3 ¼ �ig2Ru;

z4 ¼ 1

4i;

g’þ ¼ 1

2� sin2�W:

(107)

VII. SOFT FUNCTIONS FOR SINGLE W, ZPRODUCTION

Single W and Z production proceeds via processes suchas qþ �q ! W þ g and gþ q ! W þ q. The operatorbasis for production via doublet quarks is

O1 ¼ �QðuÞ2 TAtaQðuÞ

1 GA4W

a3 (108)

for the annihilation process, and

O1 ¼ �QðuÞ4 TAtaQðuÞ

1 Wa3G

A2 (109)

for Compton scattering. The two are related by crossingsymmetry.

The matrices for the anomalous dimension are

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ � 17

6iþ 3

2ðUþ TÞ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ � 7

4iþ ðUþ TÞ;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðQðuÞÞÞ

(110)

for annihilation, Eq. (108), and

S3 ¼ �Xhiji


6i� 1

6T þ 3

2U;

S2 ¼ �Xhiji


4i� 1

4T þU;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðQðuÞÞ; 0; 0; YðQðuÞÞÞ

(111)

for Compton scattering, Eq. (109).At the low scale�l �MZ, the operators Eq. (108) match

onto

O1 ¼ �uL2TAdL1G

A4W

þ3 O2 ¼ �dL2T

AuL1GA4W

�3

O3 ¼ �uL2TAuL1G

A4Z3 O4 ¼ �uL2T

AuL1GA4A3

O5 ¼ �dL2TAdL1G

A4Z3 O6 ¼ �dL2T

AdL1GA4A3:

(112)

The tree-level matching for annihilation is

Ci ¼ ðRð0ÞÞijCj; Rð0Þ ¼

1ffiffi2

p

1ffiffi2

pcW2sW2

� cW2

� sW2

266666666666664

377777777777775; (113)

and the one-loop soft matching is given by Eq. (67), where

DW ¼ ðw1; w1; w2; w2; w2; w2Þ;DZ ¼ ðz1; z2; z3; z3; z4; z4Þ;w1 ¼ �1

2ðT þUÞ þ 34i;

w2 ¼ 14i;

z1 ¼ gWgLuT � gWgLdU

� iðgLdgLu þ gLugW � gLdgWÞ;z2 ¼ gWgLuU� gWgLdT

� iðgLdgLu þ gLugW � gLdgWÞ;z3 ¼ �ig2Lu;

z4 ¼ �ig2Ld:

(114)

The results for Compton scattering are given by crossingsymmetry. One has to be careful because the collinear

functions also need to be transformed. The Oi operatorsfor Compton scattering are given by swapping the labels 2,4 in Eq. (120). The tree-level matching remains Eq. (113),and the one-loop matching is given by Eq. (130) with thereplacements


014023-18

w1 ¼ 14T � 1

2Uþ 34i;

w2 ¼ �14T þ 1

4i;

z1 ¼ gLdgLuT � gWgLdU

� iðgLdgLu þ gLugW � gLdgWÞ;z2 ¼ gLdgLuT þ gWgLuU

� iðgLdgLu þ gLugW � gLdgWÞ;z3 ¼ g2LuðT � iÞ;z4 ¼ g2LdðT � iÞ:

(115)

The operator basis for single Z production through the Bfield is

O ¼ �QðuÞ2 TAQðuÞ

1 GA4B

a3 (116)

for annihilation, and

O ¼ �QðuÞ4 TAQðuÞ

1 Ba3G

A2 (117)

for Compton scattering. In this basis

S3 ¼ �Xhiji


3iþ 3

2ðUþ T � iÞ;

S2 ¼ �Xhiji


4i;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðQðuÞÞÞ

(118)


S3 ¼ �Xhiji

Ti � TjUSðni; njÞ

¼ 4

3ðT � iÞ þ 3

2ðU� T � iÞ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ 3

4ðT � iÞ;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðQðuÞÞ; 0; 0; YðQðuÞÞÞ;

(119)

for Compton Scattering, Eq. (117).At the low scale�l �MZ, the operators Eq. (116) match

onto

O 3 ¼ �uL2TAuL1G

A4Z3 O4 ¼ �uL2T

AuL1GA4A3

O5 ¼ �dL2TAdL1G

A4Z3 O6 ¼ �dL2T

AdL1GA4A3:

(120)


C i ¼ ðRð0ÞÞiC; Rð0Þ ¼�sWcW�sWcW

26664

37775: (121)

The one-loop soft matching is given by Eq. (67) with

DW ¼ ðw1; w1; w1; w1Þ; DZ ¼ ðz1; z1; z2; z2Þ;w1 ¼ 1

4i; z1 ¼ �ig2Lu; z2 ¼ �ig2Ld: (122)

The Oi operators for Compton scattering are given byswapping the labels 2, 4 in Eq. (120). The tree-levelmatching remains Eq. (121), and the one-loop matchingis given by Eq. (122) with the replacements

DW ¼ ðw1; w1; w1; w1Þ; DZ ¼ ðz1; z1; z2; z2Þ;w1 ¼ �1

4T þ 14i; z1 ¼ g2LuðT � iÞ;

z2 ¼ g2LdðT � iÞ:(123)

Single Z production from right-handed quarks proceedsvia

O ¼ �uR2TAuR1G

A4B3 (124)

for annihilation, and

O ¼ �uR4TAuR1B3G

A2 (125)

for Compton scattering. The anomalous dimension matri-ces are

S3 ¼ �Xhiji


3iþ 3

2ðUþ T � iÞ;

S2 ¼ �Xhiji


4i;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðuRÞÞ

(126)


S3 ¼ �Xhiji

Ti � TjUSðni; njÞ

¼ 4

3ðT � iÞ þ 3

2ðU� T � iÞ;

S2 ¼ �Xhiji

ti � tjUSðni; njÞ ¼ 3

4ðT � iÞ;

S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1ÞðYðuRÞ; 0; 0; YðuRÞÞ

(127)

for Compton scattering, Eq. (125).At the low scale, the operators match to

O 1 ¼ �uR2TAuR1G

A4Z3 O2 ¼ �uR2uR1G

A4A3; (128)

with tree-level matching

Rð0Þ ¼ �sWcW

� �: (129)


014023-19

The one-loop matching is given by Eq. (67) with

DW ¼ 0; DZ ¼ ðz1; z1Þ; z1 ¼ �ig2Ru: (130)

The Oi operators for Compton scattering are given byswapping the labels 2, 4 in Eq. (108). The tree-levelmatching remains Eq. (129), and the one-loop matchingis given by Eq. (130) with the replacement

z1 ¼ g2RuðT � iÞ: (131)

VIII. SOFT FUNCTIONS FOR GLUONSCATTERING

The operator basis for qþ �q ! gþ g with doubletquarks is


1 GA4G

A3

O2 ¼ �QðuÞ2 TCQðuÞ

1 dABCGA4G

B3

O3 ¼ �QðuÞ2 TCQðuÞ

1 ifABCGA4G

B3 ;

(132)

with soft matrices

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3;gÞ;

S2 ¼ �Xhiji


S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðQðuÞÞÞ:

(133)

This matches onto

O11 ¼ �uL2uL1GA4G

A3

O12 ¼ �uL2TCuL1d

ABCGA4G

B3

O13 ¼ �uL2TCuL1if

ABCGA4G

B3

O21 ¼ �dL2dL1GA4G

A3

O22 ¼ �dL2TCdL1d

ABCGA4G

B3

O23 ¼ �dL2TCdL1if

ABCGA4G

B3 ;

(134)

with matching matrix

C ia ¼ RiCa: (135)


Rð0Þ ¼ 11

� �; (136)

and the one-loop matching matrices are

DW ¼ ðw1; w1Þ; DZ ¼ ðz1; z2Þ; w1 ¼ 14i;

z1 ¼ �ig2Lu; z2 ¼ �ig2Ld: (137)

For right-handed quarks, the operator basis is

O1 ¼ �uR2uR1GA4G

A3

O2 ¼ �uR2TCuR1d

ABCGA4G

B3

O3 ¼ �uR2TCuR1if

ABCGA4G

B3 ;

(138)

with

S3 ¼ �Xhiji

Ti � TjUSðni; njÞ ¼ Sð3;gÞ;

S2 ¼ �Xhiji


S1 ¼ �Xhiji

YiYjUSðni; njÞ ¼ Sð1Þð0; YðuRÞÞ:

(139)

These match onto

O1 ¼ �uR2uR1GA4G

A3

O2 ¼ �uR2TCuR1d

ABCGA4G

B3

O3 ¼ �uR2TCuR1if

ABCGA4G

B3 :

(140)

The matching is

C a ¼ RCa; (141)

with tree-level matching

Rð0Þ ¼ 1: (142)

The one-loop matching matrices are

DW ¼ ð0Þ; DZ ¼ ðz1Þ; z1 ¼ �ig2Ru: (143)

One can similarly write down the corrections for crossedprocesses such as gþ q ! gþ q using crossing, as doneabove for single electroweak gauge boson production.

IX. CONCLUSIONS

In this paper, we have given the collinear and softfunctions needed to compute basic high-energy scatteringprocesses in the standard model using the EFT method.The collinear functions have an interesting form, particu-larly in the weak gauge boson/Higgs sector.The soft functions can be derived using Eq. (50). They

have been explicitly given for a few important cases in thispaper. There are many different terms in the scatteringoperators, because SUð2Þ �Uð1Þ and custodial SUð2Þ arebroken in the standard model. The soft-anomalous dimen-sions for QCD have been obtained previously byKidonakis, Oderda, and Sterman [32], and we agree withtheir results.Plots of the radiative corrections to various standard

model cross sections of experimental interest, using theresults of this paper, have been given in Ref. [8]. The


014023-20

radiative corrections give large reductions in the scatteringcross sections at high energy.

APPENDIX A: INTEGRATION OF THE SCETANOMALOUS DIMENSION

The analytic formula for integrating the SCET anoma-lous dimension, with the cusp contribution at three loops,and the noncusp at two-loops, is given here. The result toone lower order was given in Ref. [65]. The collinearanomalous dimension can be integrated using the resultbelow. The soft-anomalous dimensions is a matrix, but thematrix structure is � independent, so the overall matrixstructure is constant at fixed kinematics. Thus, it too can beintegrated using the results of this appendix, by multiply-ing the right-hand side of Eq. (A4) by the constant overallmatrix and then taking a matrix exponential.

The anomalous dimension can be written as

�ð�Þ ¼ ðaA1 þ a2A2 þ a3A3Þ log��1

þ ðaB1 þ a2B2Þ;(A1)

where a ¼ �ð�Þ=ð4Þ. The � function is

�dg

d�¼ �b0

g3

162� b1

g5

ð162Þ2 � b2g7

ð162Þ3 þ . . .

(A2)

Then the solution of

�dcð�Þd�

¼ �ð�Þcð�Þ (A3)

is

cð�Þcð�1Þ

¼ exp

�f0ðzÞ�ð�1Þ þ f1ðzÞ þ �ð�1Þf2ðzÞ

�; (A4)

with

z¼ �ð�Þ�ð�1Þ ;

f0ðzÞ ¼A1

b20

�logzþ 1

z� 1

�;

f1ðzÞ ¼ A1b14b30

�logz� z� 1

2log2zþ 1

�� B1

2b0logzþ A2

4b20½z� logz� 1�;

f2ðzÞ ¼ A1b21

32b40½z2 � 2zþ 2z logz� 2 logzþ 1�þ A1b2

32b30½2 logz� z2 þ 1�� A2b1

32b30½z2 þ 2z logz� 4zþ 3�

þ A3

32b20½z2 � 2zþ 1�þ

�B1b18b20

� B2

8b0

�½z� 1�: (A5)

APPENDIX B: WAVE FUNCTION FACTORS

The transverse gauge boson inverse-propagator is

�i

�g��

k�k�

k2

� k2 �M2Z ��ZZðk2Þ ��Z�ðk2Þ

��Zðk2Þ k2 ��ðk2Þ

" #;

(B1)

where� ¼ 0 at tree level, andMZ is the tree-level Z-bosonmass. Then the wave function factors to one-loop are

�RZ ¼ �0ZZðM2

ZÞ; �R� ¼ �0��ð0Þ;

R�!Z ¼ 1

M2Z

�0Z�ðM2

ZÞ; RZ!� ¼ � 1

M2Z

�0�Zð0Þ:

(B2)

APPENDIX C: RADIATIVE CORRECTIONS TOTHE EQUIVALENCE THEOREM

The equivalence theorem radiative correction factor E(defined as in Ref. [8]) for longitudinalW and Z production

is EW;Z ¼ 1þ Eð1ÞW;Z�em=ð4sin2�WÞ. The one-loop correc-

tions in R¼1 gauge are


014023-21

Eð1ÞW ¼ m2

t

2M2W

� M2H

12M2W

� M2Z

12M2W

� 3

2�

�M2

H

12M2W

þ M2Z

12M2W

þ 2

3

�A0ðM2

WÞM2

W

þ�

M4Z

12M4W

þ M2Z

2M2W

� 4

3

�A0ðM2

ZÞM2

Z

þ�

M4H

12M4W

� M2H

6M2W

�A0ðM2

HÞM2

H

þ�m2

t

2M2W

� m4t

2M4W

�A0ðm2

t Þm2

t

þ�

M4Z

12M4W

þ 5M2Z

12M2W

� 2

�B0ð�M2

W;MZ;MWÞ

þ�

M4H

12M4W

� M2H

4M2W

�B0ð�M2

W;MW;MHÞ þ�m2

t

2M2W

� m4t

2M4W

�B0ð�M2

W; 0; mtÞ þ 3M2W

2B00ð�M2

W; 0; 0Þ

þ�2M4

W

M2Z

� 2M2W

�B00ð�M2

W; 0;MWÞ ��2M4

W

M2Z

þ 41M2W

24�M2

Z

6þ M4

Z

12M2W

�B00ð�M2

W;MZ;MWÞ

þ�� M4

H

12M2W

þM2H

12þ 5M2

W

8

�B00ð�M2

W;MW;MHÞ þ�m4

t

2M2W

�m2t þM2

W

2

�B00ð�M2

W; 0; mtÞ;

Eð1ÞZ ¼ 17m2

t

18M2W

� 20m2t

9M2Z

þ 16M2Wm

2t

9M4Z

� M2H

12M2W

� M2Z

12M2W

� 1

6þ 2M2

W

3M2Z

� 2M4W

M4Z

��2M4

W

M4Z

� 2M2W

3M2Z

þ 1

6

�A0ðM2

WÞM2

W

� M2H

12M2W

A0ðM2ZÞ

M2Z

þ�

M4H

12M2WM

2Z

� M2H

6M2W

�A0ðM2

HÞM2

H

þ�17m2

t

18M2W

� 20m2t

9M2Z

þ 16M2Wm

2t

9M4Z

�A0ðm2

t Þm2

t

þ�� 2M4

W

M4Z

þ 2M2W

3M2Z

� 1

6

�B0ð�M2

Z;MW;MWÞ þ�

M4H

12M2WM

2Z

� M2H

4M2W

�B0ð�M2

Z;MZ;MHÞ

þ�17m2

t

18M2W

� 20m2t

9M2Z

þ 16M2Wm

2t

9M4Z

�B0ð�M2

Z; mt; mtÞ þ�103M4

Z

36M2W

� 50M2Z

9þ 40M2

W

9

�B00ð�M2

Z; 0; 0Þ

þ�� M4

H

12M2W

þM2ZM

2H

12M2W

þ 5M4Z

8M2W

�B00ð�M2

Z;MZ;MHÞ þ�� 2M4

W

M2Z

� 17M2W

6þ 7M2

Z

6þ M4

Z

24M2W

�B00ð�M2

Z;MW;MWÞ

þ�17M4

Z

36M2W

� 5m2t M

2Z

9M2W

� 10M2Z

9þ 8M2

W

9� 20m2

t

9þ 16M2

Wm2t

9M2Z

�B00ð�M2

Z; mt; mtÞ; (C1)

where A0, B0, B00 are given in Eqs. (7) and (8) using the

conventions of Ref. [46]. The A0 and B0 functions areultraviolet divergent,

A0ðm2Þm2

¼ � 1

�UVþ UV finite;

B0ðp2; m1; m2Þ ¼ 1

�UVþ UV finite;

(C2)

and the infrared divergent function is

B00ð�M2

W; 0;MWÞ ¼ 1

�IR

1

2M2W

þ IR finite: (C3)

Eð1ÞW;Z are ultraviolet finite, and Eð1Þ

Z is infrared finite. Theinfrared divergence in Eð1Þ

W is

E ð1ÞW ¼ 1

�IR

�M2

W

M2Z

� 1

�þ IR finite (C4)

and is proportional to 1�M2W=M

2Z ¼ sin2�W , which in-

dicates that it arises from photon exchange. EW in Eq. (34)is treated as a matching condition (see footnote 2), i.e. the1=�IR terms in Eq. (C1) are dropped. This procedure isvalid provided the infrared divergences of the originaltheory agree with those of the effective theory, so that the1=�IR terms cancel in the matching condition.

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soft and collinear functions for the standard model

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