Lecture 2 Supplementary materials

Ch.0

Sudakov form factor

Grisha Kirilin

Abstract: Here I collect all the calculations which concern the Sudakov form factors.

Monday 21st February, 2011

C: \Users \Grisha \Documents \Work files \Parton showers \TeX \Sudakov 2 NLL.tex p.1

Analytic Sudakov form factor to the NLL level of accuracy Ch.2

Contents Ch.1

Ch.1. Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Ch.2. Analytic Sudakov form factor to the NLL level of accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

s.2.1. Simplified expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Ch.3. Appendix A. Test by fixed-order result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Ch.4. Appendix B. Strong coupling constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

s.4.1. Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Ch.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


The coherent branching algorithm was formulated in Refs. [CWM91], [CTTW93]. This algorithm uses thefollowing Sudakov factor:

definition

∆ (t ) = exp

{−

∫ t

0

dy

y

∫ 1

0dzΘ

[(zz̄)2 y − t0

]P qq [

αS(z2 z̄2 y

), z

]}. (1)

For the sake of convenience, we change the order of integrations:∫ t

0

dy

y

∫ 1

0dzΘ

[(zz̄)2 y − t0

]P qq [

αS(z2 z̄2 y

), z

]= ∫ 1

0dzΘ

[(zz̄)2 t − t0

]∫ (zz̄)2t

t0

dy

yP qq [

αS(y)

, z]

. (2)

The splitting functions have the form:

P qq = αS

2πCF

1+ z2

1− z+

(αS

2π

)2CF

2K

1− z, , (3)

where

K =CA

(67

18− π2

6

)− 5

9Nf. (4)

The splitting function can be simplified by using the trivial identity:∫ 1

0dz

1+ z2

1− zf (zz̄) =

∫ 1

0dz

(2

z̄− 3

2

)f (zz̄) . (5)

The exponent of the Sudakov factor (1) takes the form:

N (t ) =∫ t

0

dy

y

∫ 1

0dzΘ

[(zz̄)2 y − t0

]P qq [

αS(z2 z̄2 y

), z

], (6)



or in more details

N (t ) = CF

2π

∫ 1

0dzΘ

[(zz̄)2 t − t0

] (2

z̄− 3

2

)∫ (zz̄)2t

t0

dt ′

t ′αS

(t ′

)+ CFK

2π2

∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

] ∫ (zz̄)2t

t0

dt ′

t ′α2

S

(t ′

)= CF

2πβ̃0

∫ 1

0dzΘ

[(zz̄)2 t − t0

] (2

z̄− 3

2

)ln

[αS (a)

αS (b)

1+ (β̃1/β̃0

)αS (b)

1+ (β̃1/β̃0

)αS (a)

]

− CFK

2π2β̃1

∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

[1+ (

β̃1/β̃0)αS (b)

1+ (β̃1/β̃0

)αS (a)

], (7)

where b = (zz̄)2 t and a = t0. All the integrals are calculated in Appendix (see expressions (36) and (37) as wellas definition of β̃i )

Let’s split the exponent N into three parts as follows:

N = CF

πβ̃0N1 − 3CF

4πβ̃0N2 +

(CF

πβ̃0− CFK

2π2β̃1

)N3, (8)

where

N1 =∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

αS (t0)

αS[(zz̄)2 t

] ,

N2 =∫ 1

0dzΘ

[(zz̄)2 t − t0

]ln

[αS (a)

αS (b)

1+ (β̃1/β̃0

)αS (b)

1+ (β̃1/β̃0

)αS (a)

],

N3 =∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

[1+ (

β̃1/β̃0)αS (b)

1+ (β̃1/β̃0

)αS (a)

]. (9)

The N2 term is very simple to calculate because its integrand hasn’t singularities with respect to z, thus if weexpand the all coupling constants about the pointαS (t ) we can see that the LL approximation forαS generatesN LL terms for N :

N2 =∫ 1

0dz ln

[1+ β̃0αS (t ) ln(zz̄)2

1+ β̃0αS (t ) ln(t0/t )

]≈− ln(1−λ) , (10)

wheredefinition

λ= β0

4παS (t )L, L = ln

t

t0. (11)

The situation is a bit more tricky for N3:

N3 =∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

[1+ β̃1

β̃0αS (t )

αS (b)

αS (t )

]− ln

[1+ β̃1

β̃0αS (a)

]ln

zmax

zmin. (12)

If we expand the logarithm ln[1+ β̃1...

]in the rhs of (12) with respect to αS (t ), the typical leading term we

find is αS (αS ln z̄)n . Hence, to guarantee the N LL level of accuracy for N it is enough to use LL expression forαS (b) ≈αS

(z̄2t

)and simplify the regularization toΘ

(z̄2t − t0

):

N3 ≈ 1

2

β̃1

β̃20

∫ 0

ln t0/tdlnz̄2 β̃0αS (t )

1+ β̃0αS (t ) ln z̄2− 1

2

β̃1

β̃20

λ

1−λ=− β̃1

2β̃20

[λ

1−λ+ ln(1−λ)

]. (13)

The N1 is the most complicated term:

N1 =−∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]lnαS

[(zz̄)2 t

]αS (t )

+ lnαS (t0)

αS (t )

∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]=−

∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

1

1+ β̃0αS ln(zz̄)2

−∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

{1− β̃1

β̃0

αS

1+β0αS ln(zz̄)2 ln[1+β0αS ln(zz̄)2]}

− ln(1−λ)∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]+ ln

[1− β̃1

β̃0

αS

1−λln(1−λ)

]∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]. (14)



Using the property1 ∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]= L

2−2

√t0

t+O

(t0

t

), (15)

we can simplify the last two terms in (14) as follows

− ln(1−λ)∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]+ ln

[1− β̃1

β̃0

αS

1−λln(1−λ)

]∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]=−ln(1−λ)

L

2− β̃1

2β̃20

λ

1−λln(1−λ)+N 2LL. (16)

If we expand the remaining terms in (14) with respect to αS ln(zz̄)2, we obtain the following:

N11 =−∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

1

1+ β̃0αS ln(zz̄)2=

∞∑n=1

AnαnS

∫ z̄max

z̄min

dz

z̄(ln zz̄)n

=∞∑

n=1Anα

nS

∫ z̄max

zmin

dz

z̄

∞∑k=0

C kn (ln z)k (ln z̄)n−k , (17)

N12 =−∫ 1

0

dz

z̄Θ

[(zz̄)2 t − t0

]ln

{1− β̃1

β̃0

αS

1+β0αS ln(zz̄)2 ln[1+β0αS ln(zz̄)2]}=αS

∞∑n=1

BnαnS

∫ z̄max

z̄min

dz

z̄(ln zz̄)n

=αS

∞∑n=1

BnαnS

∫ z̄max

z̄min

dz

z̄

∞∑k=0

C kn (ln z)k (ln z̄)n−k , (18)

It is easy to see that in the series (17) and (18) all the terms correspond to N m>2LL contributions except thecase k = 0, hance

N11 =∞∑

n=1

An

n +1αn

S

(ln

z̄max

z̄min

)n+1

+N 2LL, N12 =∞∑

n=1

Bn

n +1

(αS ln

z̄max

z̄min

)n+1

+N 2LL,

Taking into account the relation

(ln

z̄max

z̄min

)n+1

=[

L

2−2

√t0

t+O

(t0

t

)]n+1

=(

L

2

)n [1− 4(n +1)

L

√t0

t+O

(t0

t

)], (19)

we arrive to the conclusion that zz̄ can be replaced by z̄ in the phase space as well as in the integrand of allN -s:

N11 =− 1

2β̃0αS

∫ λ

0dλ′ ln

1

1−λ′ =−L

2

[1+ 1−λ

λln(1−λ)

], (20)

N12 = β̃1

2β̃20

∫ λ

0

dλ′

1−λ′ ln(1−λ′)=− β̃1

4β̃20

ln2(1−λ). (21)

Finally, we obtain to the following expression for the Sudakov factor to the N LL level of accuracy:result

∆N LL = exp(−N ) = exp[−L f1 (λ)− f2 (λ)

], (22)

where

f1 (λ) = 2CF

β0

[1

λln

1

1−λ−1

], (23)

f2 (λ) = CF

β0

{3ln(1−λ)+ 4K

β0

[λ

1−λ+ ln(1−λ)

]+ 2β1

β20

[−1

2ln2(1−λ)− ln(1−λ)

1−λ− λ

1−λ

]}. (24)

1(sic!) here is non-analytic power corrections



0 2000 4000 6000 8000

0.0

0.2

0.4

0.6

0.8

1.0

t GeV2

DHt,

t 0L

Exact

NLL

LL

0 2000 4000 6000 8000

0.0

0.2

0.4

0.6

0.8

1.0

t GeV2

DHm

z2,tL

Exact

NLL

LL

Figure 1: Comparison of the exact expression (1) (solid line) for the Sudakov factor with its N LL approximation(22) (dashed line). The parameter t0 is chosen as 4Λ2

QCD.

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

t GeV

DHt,

t 0L

Exact: D

NLL app.: Dq

Analytic: DNLL

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

t GeV

DHm

z2,tL

Exact: D

NLL app.: Dq

Analytic: DNLL

Figure 2: Comparison of three different approximations for the Sudakov factor.

1 2 3 4 5 6

0.7

0.8

0.9

1.0

1.1

t � t0

DHt,

t 0L

Exact: D

NLL app.: Dq

Analytic: DNLL

Figure 3: Non-physical region t ∈ (t0,16 t0).


Appendix A. Test by fixed-order result Ch.3

Simplified expression s.2.1

Tracing back the derivation of (22) we find that zz̄ can be replaced by z̄ in the phase space as well as in αS .Thus one can establish the simplified form of the Sudakov factor:

∆̃q (t ) = exp

{−

∫ t

0

dy

y

∫ 1

0dz̄Θ

(z̄2 y − t0

)P qq [

αS(z̄2 y

), z

]}= exp

{−

∫ t

0

dy

y

∫ 1

0dz̄Θ

(z̄2 y − t0

) αS(z̄2 y

)2π

CF

(2

z̄− 3

2

)}.

(25)

∆̃ (t ) = exp

{−CF

2π

∫ t

t0

dy

y

[∫ y

t0

dx

xαS (x)− 3

2αS

(y)]}

= exp

[−CF

2π

∫ t

t0

dt ′αS

(t ′

)t ′

(ln

t

t ′− 3

2

)]. (26)

Finally, we obtain the following form of the Sudakov factor:

∆̃q (t ) = exp

[−

∫ t

t0

dt ′Γq(t ′, t

)], Γq

(t ′, t

)= CF

2π

αS(t ′

)t ′

(ln

t

t ′− 3

2

). (27)

Alternatively, one can introduce a new variable Q =pt and represents the Sudakov factor as follows:

result

∆q (Q0,Q) = exp

[−

∫ Q

Q0

dq Γq(q,Q

)], Γq

(q,Q

)= 2CF

π

αS(q)

q

(ln

Q

q− 3

4

). (28)

Notice that the form (28) is very similar to that of Ref.[CKKW01] (see Eq.(2.5) in the reference). In Fig.(2), Icompare three different expression: the exact Sudakov factor (1), its N LL approximation (28) and the analyticresult (22), so that the comparison of various ∆ (t ) is shown on the left picture and ∆

(M 2

Z

)/∆ (t ) are on the

right one.It is worth noting here that according to the phase space definition the evolution variable t is larger than

16t0 where t0 is the regularization for the transverse momentum. However, if one can use analytic continua-tion of (28) into the region Q ∈ (Q0,4Q0) (see Fig.3). one can see that ∆q is nonmonotonic in this region andsometimes even ∆q > 1. This behavior was emphasized in the footnote (2) in Ref.[CKKW01].

There is also one point in order. To derive the expression (28) we used the leading order expression forthe DGLAP splitting function (3), though we know that the term proportional to K also contributes to the NLLlevel in (22). As it was shown in Ref. [CWM91], in order to take into account such correction one needs to usethe followingΛQCD in αS :

ΛMC =ΛMS exp

[K

4πβ0

], (29)

where the subscript MC denotes Monte-Carlo. This scheme for αS is called as “bremsstrahlung scheme” inRef. [CKKW01].

Appendix A. Test by fixed-order result Ch.3

The O (λ) and O(λ2

)terms have the form:

N (t ) = αS

2πCF

∫ t

t0

dy

y

∫ 1

pt0/y

dz

[(1− β̃0αS ln

z2 y

t

) (2

z− 3

2

)+ αS

π

K

z

]= αS

2πCF

∫ t/t0

1

dy

y

∫ 1

1/p

ydz

{[1− β̃0αS ln

(z2 y

)+λ](

2

z− 3

2

)+ αS

π

K

z

}= αS

2πCF

∫ t/t0

1

dy

y

∫ 1

1/p

ydz

(2

z− 3

2

)+ αS

2πCFλ

∫ t/t0

1

dy

y

∫ 1

1/p

ydz

[− 1

Lln

(z2 y

)+1

](2

z− 3

2

)+ 2CFK

β20

λ2

= CF

β0

[L

(λ+ 2

3λ2

)−3λ− 3

2λ2 + K

β02λ2

]. (30)


REFERENCES Ch.5

The term proportional to β1λ3 is the following:

N (t ) =−α3S β̃1

πCF

∫ t

t0

dy

y

∫ 1

pt0/y

dz

zln

z2 y

t=−α3

S β̃1

πCF

∫ t/t0

1

dy

y

∫ 1

1/p

y

dz

z

(2ln z + ln y −L

)= β̃1

6πCF (αS L)3 = CF

β0

(β̃1

β̃30

2

3λ3

). (31)

Appendix B. Strong coupling constant Ch.4

The renormalization group equation for the strong coupling constant has the form:

1

αS(ρ2

) dαS(ρ2

)dlnρ2 =−β (αS ) , (32)

where

β (αS ) =β0αS

4π+β1

(αS

4π

)2+ . . . . (33)

If we consider only the two first terms in β-function, the solution of the equation (32) takes the form:

αS(ρ2

)αS

(µ2

) = 1

1+ β̃0αS(µ2

)ln

(ρ2/µ2

) {1− β̃1

β̃0

αS(µ2

)1+ β̃0αS

(µ2

)lnρ2/µ2

ln

[1+ β̃0αS

(µ2) ln

ρ2

µ2

]}, (34)

where

β̃i = βi

(4π)i+1. (35)

Integrals s.4.1

∫ b

aαS

(ρ2)dlnρ2 =−

∫ αS (b)

αS (a)

dαS

β (αS )=−

∫dαS

αS(β̃0 + β̃1αS

)=−

∫dαS

β̃0

(1

αS− β̃1

β̃0 + β̃1αS

)=−

∫1

β̃0dln

[αS

β̃0 + β̃1αS

]

= 1

β̃0ln

[αS (a)

αS (b)

1+ (β̃1/β̃0

)αS (b)

1+ (β̃1/β̃0

)αS (a)

]. (36)

∫ b

a

[αS

(ρ2)]2

dlnρ2 =−∫ αS (b)

αS (a)

αS dαS

β (αS )=−

∫dαS

β̃0 + β̃1αS=− 1

β̃1ln

[1+ (

β̃1/β̃0)αS (b)

1+ (β̃1/β̃0

)αS (a)

]. (37)

References Ch.5

[CKKW01] S. Catani, F. Krauss, R. Kuhn, and B. R. Webber, QCD Matrix Elements + Parton Showers, JHEP 11(2001), 063.

[CTTW93] S. Catani, L. Trentadue, G. Turnock, and B. R. Webber, Resummation of large logarithms in e+ e-event shape distributions, Nucl. Phys. B407 (1993), 3–42.

[CWM91] S. Catani, B. R. Webber, and G. Marchesini, QCD coherent branching and semiinclusive processesat large x, Nucl. Phys. B349 (1991), 635–654.


Lecture 2 Supplementary materials

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