顶夸克横截面研究：两圈增强与硬匹配

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JHEP12(2015)059

space S

†

(x) = P exp



∞

ds n·A

(ns+x)



, and ultrasoft Wilson lines Y

n,¯n

are the analogs

with ultrasoft gluon ﬁelds in bHQET. The diﬀerence between the SCET ﬁelds and bHQET

ﬁelds is that SCET still contains soft and collinear ﬂuctuations at the top mass scale, i.e.

the SCET ﬁelds contain mass mode ﬂuctuations which scale as (p

, p

−

, p

⊥

) ∼ (m, m, m)

and (Q, m

/Q, m) or (m

/Q, Q, m) which are absent in bHQET. This makes our EFT

above the top mass scale an SCET

type theory. There are six ﬂavors in the MS running

coupling in QCD and SCET, and ﬁve ﬂavors in bHQET.

The notation above diﬀers from ref. [14] which used a hybrid of SCET

and SCET

where the current operator was written as

SCET

= ¯χ

†

¯n

. (2.2)

Here the Wilson lines S

n,¯n

describe exclusively soft mass mode ﬂuctuations and have ultra-

soft zero-bin subtractions. In eq. (2.1) the SCET operator only describes soft ﬂuctuations

above and of order of the mass scale m, and not far below m. This simpliﬁes the setup for

the matching coeﬃcient calculation, which in particular can be viewed as going from a six

ﬂavor theory to a ﬁve ﬂavor theory.

The matching coeﬃcients between these eﬀective theories are deﬁned by

+1)

QCD

= C

+1)

SCET



1 + O(m/Q)



, (2.3)

+1)

SCET

= C

)

bHQET



1 + O(ˆs/m)



. (2.4)

Here both the currents and Wilson coeﬃcients refer to the renormalized quantities. When

we refer to the bare objects we will indicate this explicitly as e.g. in J

(bare,n

+1)

SCET

. For

all quantities we consider we use the renormalized coupling constant. When we want to

separate the color structures of the matching coeﬃcients we will do so in the following way:

+1)

= 1 + C

(1, n

+1)

+ C

, n

+1)

+ C

, n

+1)

+ C

, n

+1)

+ C

, n

+1)

)

= 1 + C

(1, n

)

| {z }

O(α

)

+ C

, n

)

+ C

, n

)

+ C

, n

)

+ C

, n

)

| {z }

O(α

)

. (2.5)

In all the objects above the coupling is renormalized in the MS scheme with the number

of dynamical ﬂavors, n

, being either n

or (n

+ 1) as indicated by the superscript. Here

is the number of light quarks, and the additional ﬂavor indicates the heavy quark (here

the top quark). The choice for the number of ﬂavors in each of the expressions above is

motivated by the scales at which these objects live compared to the top mass. Note that

we have kept the number of ﬂavors appearing in C

unspeciﬁed, as it can be expressed in

either the n

- or the (n

+ 1)-ﬂavor scheme. We will be explicit about which scheme we are

using in the equations below.

The hard functions in eq. (1.2) are related to the Wilson coeﬃcients via

(Q, µ) = |C

, H



, µ



= |C

. (2.6)

Here the dependence on Q in the hard function H

appears due to the boost factor Q/m.

– 6 –

JHEP12(2015)059

In eq. (1.2) all the functions live at their respective scales and are evolved to a common

scale µ

ﬁnal

through renormalization group running. While the jet and the soft functions

have convolution running [14], the large logarithms of the hard matching coeﬃcients are

summed by multiplicative evolution factors,

evol

(Q, m, µ

ﬁnal

; µ

, µ

, ν

) ≡H

+1)

(Q, µ

) U

+1)

(Q, µ

, µ

) (2.7)

× H

)



, µ

; ν

, ν



)



, µ

ﬁnal



for µ

' Q, µ

' m and µ

ﬁnal

< µ

. On the l.h.s. the dependence on µ

and µ

only

comes from higher order corrections when the objects in eq. (2.7) are truncated at a given

order in resummed perturbation theory. The same is true for the rapidity scales ν

and

, which are induced by the rapidity RGE that will be discussed further below and in

section 5.1. We will frequently drop these arguments that appear after the semicolon. The

evolution factors here obey the RG equations

dµ

+1)

(Q, µ

, µ) = −γ

+1)

(Q, µ) U

+1)

(Q, µ

, µ) ,

dµ

)



, µ, µ

ﬁnal



= +γ

)



, µ



)



, µ, µ

ﬁnal



, (2.8)

where γ

)

is the anomalous dimension for the squared current in bHQET.

Eqs. (2.3) and (2.4) suggest two diﬀerent methods that one can use to calculate the

O(α

) piece of C

or equivalently H

1) Indirect calculation using the known result for C

and the matrix elements for the

QCD and bHQET current operators in pure dimensional regularization.

Using eq. (2.3) and (2.4), and taking matrix elements of the operators with onshell

top-quark states as in [13], we have

+1)

QCD

i = C

+1)

)

bHQET

i. (2.9)

Using the relation between bare and renormalized bHQET currents

)

bHQET

i = Z

)

bHQET

(bare, n

)

bHQET

i, (2.10)

we get

)

+1)

QCD

+1)

)

bHQET

(bare, n

)

bHQET

. (2.11)

Note that the terms on the r.h.s. involve objects with diﬀerent ﬂavor number schemes

for the strong coupling, which must all be converted to n

-ﬂavors to get C

)

. Here

we work in dimensional regularization for both UV and IR divergences and renormalize

the quantities in the MS scheme. With this regulator we can use the known two loop

result for the heavy form factor hJ

QCD

i given in refs. [30, 31]. The result for C

is also

– 7 –

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顶夸克横截面研究：两圈增强与硬匹配

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