N CHAPTER VI: 2 QCD and Structure Functions of the Nucleon · CHAPTER 6. QCD AND STRUCTURE...

CHAPTER 6. QCD AND STRUCTURE FUNCTIONS OF THE NUCLEON

6. 1. Deep inelastic lepton scattering (DIS) on the proton

E, !p

Pµ = (P 0, !P )lab. frame!!!!!!" (MN , 0)

E!, !p !

", !q

#"

lepton

proton

• Energy and momentum transfer carried by virtual photon:

! = E ! E !, "q = "p ! "p !; qµ = (!, "q )

q2 = !2 ! "q 2 = !Q2 (Q2 > 0)(6.1)

Q2 determines “resolution”:

Large Q2 corresponds to small distance scales probed inside of proton.

• Deep inelastic scattering (DIS):

! # MN , Q2 # M2N (6.2)

• Bjorken (scaling) variable:

x =Q2

2P · qlab. frame!!!!!" x =

Q2

2MN !(6.3)

• Interpretation of Bjorken x in infinite momentum frame:

Variables: x =Q2

2P · q, q = !2xP $ q2 = !2xP · q (see Fig. 6.1)

x : fraction of total proton momentum carried by a single parton (quark) when struck

by the virtual photon.

• Associated variable: y =q · Pp · P

lab. frame!!!!!"!

E

53

CHAPTER VI: QCD and Structure Functions of the Nucleon

CHAPTER 6. QCD AND STRUCTURE FUNCTIONS OF THE NUCLEON

6. 1. Deep inelastic lepton scattering (DIS) on the proton

E, !p

Pµ = (P 0, !P )lab. frame!!!!!!" (MN , 0)

E!, !p !

", !q

#"

lepton

proton

• Energy and momentum transfer carried by virtual photon:

! = E ! E !, "q = "p ! "p !; qµ = (!, "q )

q2 = !2 ! "q 2 = !Q2 (Q2 > 0)(6.1)

Q2 determines “resolution”:

Large Q2 corresponds to small distance scales probed inside of proton.

• Deep inelastic scattering (DIS):

! # MN , Q2 # M2N (6.2)

• Bjorken (scaling) variable:

x =Q2

2P · qlab. frame!!!!!" x =

Q2

2MN !(6.3)

• Interpretation of Bjorken x in infinite momentum frame:

Variables: x =Q2

2P · q, q = !2xP $ q2 = !2xP · q (see Fig. 6.1)

x : fraction of total proton momentum carried by a single parton (quark) when struck

by the virtual photon.

• Associated variable: y =q · Pp · P

lab. frame!!!!!"!

E

53

!"#$#%

!&"$#%

'(!$#%

! !&"$#%

!"#$#%

'(!$#%!"#$$%)*******+

#%!

"#%&

%

FIG. 6.1: DIS in lab. frame (left) and in infinite momentum frame (right).

• Observable (lab. frame): di!erential cross section:

d2!

d"lab dE !

! Invariant di!erential cross section:

x (s " M2N )

d2!

dx dQ2=

2"MN #

E !

d2!

d"lab dE ! with s = (p + P )2 (6.4)

d2!

dx dQ2=

4"$2

Q2

!

""""#

$1 " y

x"

M2N y

s " M2N

%

F2(x, Q2) + y2F1(x, Q2)

&

""""'

where F1,2(x, Q2) are called proton structure functions.

(6.5)

6. 2. The parton model

• Assumption: pointlike spin-1/2 particles (partons) inside the proton

! F1,2(x) independent on Q2 (scaling behavior);

Q2-dependence of F1,2(x, Q2) # QCD corrections.

! Spin-1/2 ! F2(x) = 2xF1(x) (Callan-Gross relation)

approximately observed in experiment.

• Master formula of the parton model:

2xF1(x)spin- 1

2= F2(x) =(

i

e2i x fi(x) (6.6)

54

!"#$#%

!&"$#%

'(!$#%

! !&"$#%

!"#$#%

'(!$#%!"#$$%)*******+

#%!

"#%&

%



d2!

d"lab dE !


x (s " M2N )

d2!

dx dQ2=

2"MN #

E !

d2!

d"lab dE ! with s = (p + P )2 (6.4)

d2!

dx dQ2=

4"$2

Q2

!

""""#

$1 " y

x"

M2N y

s " M2N

%

F2(x, Q2) + y2F1(x, Q2)

&

""""'


(6.5)








2xF1(x)spin- 1

2= F2(x) =(

i

e2i x fi(x) (6.6)

54

- 32 -

!"#$#%

!&"$#%

'(!$#%

! !&"$#%

!"#$#%

'(!$#%!"#$$%)*******+

#%!

"#%&

%



d2!

d"lab dE !


x (s " M2N )

d2!

dx dQ2=

2"MN #

E !

d2!

d"lab dE ! with s = (p + P )2 (6.4)

d2!

dx dQ2=

4"$2

Q2

!

""""#

$1 " y

x"

M2N y

s " M2N

%

F2(x, Q2) + y2F1(x, Q2)

&

""""'


(6.5)








2xF1(x)spin- 1

2= F2(x) =(

i

e2i x fi(x) (6.6)

54!

"""""#

fi(x) : parton (quark) distribution function

ei : quark electric charges

Using notations for fi(x) : u(x), d(x), s(x), u(x), d(x), s(x), · · ·

1

xF2(x) =

$2

3

%2&

u(x) + u(x)'

+

$1

3

%2&

d(x) + d(x) + s(x) + s(x)'

+ · · ·

• Valence quarks: qv(x)

!

(

#

uv(x) = u(x) ! u(x)

dv(x) = d(x) ! d(x)

• Sea quarks: qs(x)

!

(

#

us(x) = u(x) + u(x) ! uv(x)

ds(x) = d(x) + d(x) ! dv(x)

FIG. 6.2: The total valence and sea quark contributions to the structure of the proton.

55

!

"""""#




1

xF2(x) =

$2

3

%2&

u(x) + u(x)'

+

$1

3

%2&

d(x) + d(x) + s(x) + s(x)'

+ · · ·


!

(

#

uv(x) = u(x) ! u(x)

dv(x) = d(x) ! d(x)


!

(

#

us(x) = u(x) + u(x) ! uv(x)

ds(x) = d(x) + d(x) ! dv(x)


55

!

"""""#




1

xF2(x) =

$2

3

%2&

u(x) + u(x)'

+

$1

3

%2&

d(x) + d(x) + s(x) + s(x)'

+ · · ·


!

(

#

uv(x) = u(x) ! u(x)

dv(x) = d(x) ! d(x)


!

(

#

us(x) = u(x) + u(x) ! uv(x)

ds(x) = d(x) + d(x) ! dv(x)


55

6. 3. QCD and structure functions of the nucleon (DIS)

6. 3. 1. Gluon emission

!!

=

!!

+

!!

+

!!

• Elementary subprocess: !! + quark ! quark + gluon

p

q

kp + q

p"

+

q

p k

p" " q

p"

T1 T2

! Mandelstam variables:

s = (p + q)2 = (p " + k)2

t = (q " p ")2 = (p " k)2

u = (q " k)2 = (p " p ")2

• All quark masses neglected (mq # Q,$

s)

s + t + u = q2 , s + t + u + Q2 = 0

• Feynman rules:

T a1,j = u(p")

!

"" !"

ig!!#a

2

#iej

/p + /q " mq(ie!µ) "µ

$

u(p)

T a2,j = u(p")

!

"µ (ie!µ)iej

/p " " /q " mq

"

ig!!#a

2

#

"" !$

u(p)

(6.7)

56

- 33 -

6. 3. QCD and structure functions of the nucleon (DIS)

6. 3. 1. Gluon emission

!!

=

!!

+

!!

+

!!

• Elementary subprocess: !! + quark ! quark + gluon

p

q

kp + q

p"

+

q

p k

p" " q

p"

T1 T2

! Mandelstam variables:

s = (p + q)2 = (p " + k)2

t = (q " p ")2 = (p " k)2

u = (q " k)2 = (p " p ")2

• All quark masses neglected (mq # Q,$

s)

s + t + u = q2 , s + t + u + Q2 = 0

• Feynman rules:

T a1,j = u(p")

!

"" !"

ig!!#a

2

#iej

/p + /q " mq(ie!µ) "µ

$

u(p)

T a2,j = u(p")

!

"µ (ie!µ)iej

/p " " /q " mq

"

ig!!#a

2

#

"" !$

u(p)

(6.7)

56

• |q2| ! m2q (let mq " 0)

•1

/p + /q=

/p + /q

(p + q)2=

/p + /q

s;

1

/p ! # /q=

/p ! # /q

(p! # q)2=

/p ! # /q

t

• Spin-averaged square |T |2 (T = T1 + T2): including!

i

!(i)µ !(i)

! = #gµ!

|T |2$

!

spinpolarization

|T1 + T2|2 =1

2tr

"

T † T#

= 32"2 e2j # #s

4

3

$

#t

s#

s

t+

2uQ2

st

%(6.8)

with # =e2

4"and #s =

g2

4".

• Center of mass frame:

!"

#

$

!

!

transverse momentum:

pT = |$p !| sin %

p2T = #

st

s + Q2(t % s)

(6.9)

• Di!erential cross section for the elementary process (&" + quark " quark + gluon):

d'

dp2T

(&" + q " q + g) =|T |

2

16"s2(6.10)

d'

dp2T

t#s&8"

3e2

q

# #s

s2

$

#1

t

%&

s +2(s + Q2)Q2

s

'

= '0

e2q

p2T

#s

2"

&s

s + Q2+

2Q2

s

' (6.11)

with #st = p2T (s + Q2) and '0 =

4"2#

s

57

- 34 -

• |q2| ! m2q (let mq " 0)

•1

/p + /q=

/p + /q

(p + q)2=

/p + /q

s;

1

/p ! # /q=

/p ! # /q

(p! # q)2=

/p ! # /q

t


i

!(i)µ !(i)

! = #gµ!

|T |2$

!

spinpolarization

|T1 + T2|2 =1

2tr

"

T † T#

= 32"2 e2j # #s

4

3

$

#t

s#

s

t+

2uQ2

st

%(6.8)

with # =e2

4"and #s =

g2

4".


!"

#

$

!

!


pT = |$p !| sin %

p2T = #

st

s + Q2(t % s)

(6.9)


d'

dp2T

(&" + q " q + g) =|T |

2

16"s2(6.10)

d'

dp2T

t#s&8"

3e2

q

# #s

s2

$

#1

t

%&

s +2(s + Q2)Q2

s

'

= '0

e2q

p2T

#s

2"

&s

s + Q2+

2Q2

s

' (6.11)


4"2#

s

57

• |q2| ! m2q (let mq " 0)

•1

/p + /q=

/p + /q

(p + q)2=

/p + /q

s;

1

/p ! # /q=

/p ! # /q

(p! # q)2=

/p ! # /q

t


i

!(i)µ !(i)

! = #gµ!

|T |2$

!

spinpolarization

|T1 + T2|2 =1

2tr

"

T † T#

= 32"2 e2j # #s

4

3

$

#t

s#

s

t+

2uQ2

st

%(6.8)

with # =e2

4"and #s =

g2

4".


!"

#

$

!

!


pT = |$p !| sin %

p2T = #

st

s + Q2(t % s)

(6.9)


d'

dp2T

(&" + q " q + g) =|T |

2

16"s2(6.10)

d'

dp2T

t#s&8"

3e2

q

# #s

s2

$

#1

t

%&

s +2(s + Q2)Q2

s

'

= '0

e2q

p2T

#s

2"

&s

s + Q2+

2Q2

s

' (6.11)


4"2#

s

57

• |q2| ! m2q (let mq " 0)

•1

/p + /q=

/p + /q

(p + q)2=

/p + /q

s;

1

/p ! # /q=

/p ! # /q

(p! # q)2=

/p ! # /q

t


i

!(i)µ !(i)

! = #gµ!

|T |2$

!

spinpolarization

|T1 + T2|2 =1

2tr

"

T † T#

= 32"2 e2j # #s

4

3

$

#t

s#

s

t+

2uQ2

st

%(6.8)

with # =e2

4"and #s =

g2

4".


!"

#

$

!

!


pT = |$p !| sin %

p2T = #

st

s + Q2(t % s)

(6.9)


d'

dp2T

(&" + q " q + g) =|T |

2

16"s2(6.10)

d'

dp2T

t#s&8"

3e2

q

# #s

s2

$

#1

t

%&

s +2(s + Q2)Q2

s

'

= '0

e2q

p2T

#s

2"

&s

s + Q2+

2Q2

s

' (6.11)


4"2#

s

57

• |q2| ! m2q (let mq " 0)

•1

/p + /q=

/p + /q

(p + q)2=

/p + /q

s;

1

/p ! # /q=

/p ! # /q

(p! # q)2=

/p ! # /q

t


i

!(i)µ !(i)

! = #gµ!

|T |2$

!

spinpolarization

|T1 + T2|2 =1

2tr

"

T † T#

= 32"2 e2j # #s

4

3

$

#t

s#

s

t+

2uQ2

st

%(6.8)

with # =e2

4"and #s =

g2

4".


!"

#

$

!

!


pT = |$p !| sin %

p2T = #

st

s + Q2(t % s)

(6.9)


d'

dp2T

(&" + q " q + g) =|T |

2

16"s2(6.10)

d'

dp2T

t#s&8"

3e2

q

# #s

s2

$

#1

t

%&

s +2(s + Q2)Q2

s

'

= '0

e2q

p2T

#s

2"

&s

s + Q2+

2Q2

s

' (6.11)


4"2#

s

57! Define: splitting function

Pqq(z) !4

3

1 + z2

1 " z(6.12)

d!

dp2T

= !0 e2q

"s

2#p2T

Pqq(z) (6.13)

• Back to structure function:

F2 (x, Q2)

x=

1

2

!

i

e2i fi

"

x, Q2#

; x =Q2

2P · qlab. frame"""""#

Q2

2M$

F2

x=

!

f

!"

!"#$#%

!&"$#% '

!"!=

!"

!#$$!"

!"'

Optical theorem : Imf(%!P # %!P ) $ !tot(%!P )

F2 (x, Q2)

x=

!(%!P )

!0(6.14)

!(%!q # qg) =

$ p2T,max

µ2

dp2T

d !

dp2T

= !0 e2q

"s

2#ln

%p2

T,max

µ2

&

Pqq(z)

(6.15)

with p2T,max =

s

4=

1 " z

4zQ2.

lnp2

T,max

µ2= ln

'1 " z

4z

Q2

µ2

(

= ln

%1 " z

4z

&

+ ln

%Q2

µ2

&

% ln

%Q2

µ2

&

Therefore!

!0% e2

q

"s(Q)

2#Pqq(z) ln

%Q2

µ2

&

(6.16)

x =Q2

2P · q; p = yP

z =Q2

2p · q=

Q2

2P · qP · qp · q

= xP · qp · q

=x

y

58

! Define: splitting function

Pqq(z) !4

3

1 + z2

1 " z(6.12)

d!

dp2T

= !0 e2q

"s

2#p2T

Pqq(z) (6.13)


F2 (x, Q2)

x=

1

2

!

i

e2i fi

"

x, Q2#

; x =Q2


Q2

2M$

F2

x=

!

f

!"

!"#$#%

!&"$#% '

!"!=

!"

!#$$!"

!"'


F2 (x, Q2)

x=

!(%!P )

!0(6.14)

!(%!q # qg) =

$ p2T,max

µ2

dp2T

d !

dp2T

= !0 e2q

"s

2#ln

%p2

T,max

µ2

&

Pqq(z)

(6.15)

with p2T,max =

s

4=

1 " z

4zQ2.

lnp2

T,max

µ2= ln

'1 " z

4z

Q2

µ2

(

= ln

%1 " z

4z

&

+ ln

%Q2

µ2

&

% ln

%Q2

µ2

&

Therefore!

!0% e2

q

"s(Q)

2#Pqq(z) ln

%Q2

µ2

&

(6.16)

x =Q2

2P · q; p = yP

z =Q2

2p · q=

Q2

2P · qP · qp · q

= xP · qp · q

=x

y

58


Pqq(z) !4

3

1 + z2

1 " z(6.12)

d!

dp2T

= !0 e2q

"s

2#p2T

Pqq(z) (6.13)


F2 (x, Q2)

x=

1

2

!

i

e2i fi

"

x, Q2#

; x =Q2


Q2

2M$

F2

x=

!

f

!"

!"#$#%

!&"$#% '

!"!=

!"

!#$$!"

!"'


F2 (x, Q2)

x=

!(%!P )

!0(6.14)

!(%!q # qg) =

$ p2T,max

µ2

dp2T

d !

dp2T

= !0 e2q

"s

2#ln

%p2

T,max

µ2

&

Pqq(z)

(6.15)

with p2T,max =

s

4=

1 " z

4zQ2.

lnp2

T,max

µ2= ln

'1 " z

4z

Q2

µ2

(

= ln

%1 " z

4z

&

+ ln

%Q2

µ2

&

% ln

%Q2

µ2

&

Therefore!

!0% e2

q

"s(Q)

2#Pqq(z) ln

%Q2

µ2

&

(6.16)

x =Q2

2P · q; p = yP

z =Q2

2p · q=

Q2

2P · qP · qp · q

= xP · qp · q

=x

y

58


Pqq(z) !4

3

1 + z2

1 " z(6.12)

d!

dp2T

= !0 e2q

"s

2#p2T

Pqq(z) (6.13)


F2 (x, Q2)

x=

1

2

!

i

e2i fi

"

x, Q2#

; x =Q2


Q2

2M$

F2

x=

!

f

!"

!"#$#%

!&"$#% '

!"!=

!"

!#$$!"

!"'


F2 (x, Q2)

x=

!(%!P )

!0(6.14)

!(%!q # qg) =

$ p2T,max

µ2

dp2T

d !

dp2T

= !0 e2q

"s

2#ln

%p2

T,max

µ2

&

Pqq(z)

(6.15)

with p2T,max =

s

4=

1 " z

4zQ2.

lnp2

T,max

µ2= ln

'1 " z

4z

Q2

µ2

(

= ln

%1 " z

4z

&

+ ln

%Q2

µ2

&

% ln

%Q2

µ2

&

Therefore!

!0% e2

q

"s(Q)

2#Pqq(z) ln

%Q2

µ2

&

(6.16)

x =Q2

2P · q; p = yP

z =Q2

2p · q=

Q2

2P · qP · qp · q

= xP · qp · q

=x

y

58


Pqq(z) !4

3

1 + z2

1 " z(6.12)

d!

dp2T

= !0 e2q

"s

2#p2T

Pqq(z) (6.13)


F2 (x, Q2)

x=

1

2

!

i

e2i fi

"

x, Q2#

; x =Q2


Q2

2M$

F2

x=

!

f

!"

!"#$#%

!&"$#% '

!"!=

!"

!#$$!"

!"'


F2 (x, Q2)

x=

!(%!P )

!0(6.14)

!(%!q # qg) =

$ p2T,max

µ2

dp2T

d !

dp2T

= !0 e2q

"s

2#ln

%p2

T,max

µ2

&

Pqq(z)

(6.15)

with p2T,max =

s

4=

1 " z

4zQ2.

lnp2

T,max

µ2= ln

'1 " z

4z

Q2

µ2

(

= ln

%1 " z

4z

&

+ ln

%Q2

µ2

&

% ln

%Q2

µ2

&

Therefore!

!0% e2

q

"s(Q)

2#Pqq(z) ln

%Q2

µ2

&

(6.16)

x =Q2

2P · q; p = yP

z =Q2

2p · q=

Q2

2P · qP · qp · q

= xP · qp · q

=x

y

58• Deep inelastic lepton scattering on proton

F2 (x, Q2)

x=

!

q

e2q

" 1

x

dy

yq(y)

#

!

$

1 !x

y

%

+"s(Q)

2#Pqq

$x

y

%

ln

$Q2

µ2

%

& '( )

!q(x,Q2)

*

(6.17)

!q+

x, Q2,

="s

2#ln

$Q2

µ2

% " 1

x

dy

yq(y)Pqq

$x

y

%

(6.18)

take di"erence at two neighboring values of Q2

q+

x, Q2,

! q+

x, Q20

,

="s

2#

$

ln

$Q2

µ2

%

! ln

$Q2

0

µ2

%% " 1

x

dy

yq(y)Pqq

$x

y

%

(6.19)

then take limit (Q20 " Q2)

• Altarelli-Parisi evolution equation:

d

d ln Q2q+

x, Q2,

="s(Q)

2#

" 1

x

dy

yq+

x, Q2,

Pqq

$x

y

%

(6.20)

6. 3. 2. Quark-antiquark pair production

gluon

!!

q

q

• Introduce gluon distribution function: g(x)

• Interchange: pµ # !kµ

$ s " t , t " u , u " s

• Result:!F2 (x, Q2)

x

-----!!g"qq

=!

q

e2q

" 1

x

dy

yg(y)

"s

2#Pqg

$x

y

%

ln

$Q2

µ2

%

(6.21)

with splitting function

k

zk

gluon q

q

Pqg(z) =1

2

.

z2 + (1 ! z)2/

59

- 35 -

• Deep inelastic lepton scattering on proton

F2 (x, Q2)

x=

!

q

e2q

" 1

x

dy

yq(y)

#

!

$

1 !x

y

%

+"s(Q)

2#Pqq

$x

y

%

ln

$Q2

µ2

%

& '( )

!q(x,Q2)

*

(6.17)

!q+

x, Q2,

="s

2#ln

$Q2

µ2

% " 1

x

dy

yq(y)Pqq

$x

y

%

(6.18)

take di"erence at two neighboring values of Q2

q+

x, Q2,

! q+

x, Q20

,

="s

2#

$

ln

$Q2

µ2

%

! ln

$Q2

0

µ2

%% " 1

x

dy

yq(y)Pqq

$x

y

%

(6.19)

then take limit (Q20 " Q2)

• Altarelli-Parisi evolution equation:

d

d ln Q2q+

x, Q2,

="s(Q)

2#

" 1

x

dy

yq+

x, Q2,

Pqq

$x

y

%

(6.20)

6. 3. 2. Quark-antiquark pair production

gluon

!!

q

q

• Introduce gluon distribution function: g(x)

• Interchange: pµ # !kµ

$ s " t , t " u , u " s

• Result:!F2 (x, Q2)

x

-----!!g"qq

=!

q

e2q

" 1

x

dy

yg(y)

"s

2#Pqg

$x

y

%

ln

$Q2

µ2

%

(6.21)

with splitting function

k

zk

gluon q

q

Pqg(z) =1

2

.

z2 + (1 ! z)2/

59

6. 4. Complete set of evolution equations for quark and gluon distribution functions

(DGLAP equation)

• Quark distributions: qi (x, Q2), qi (x, Q2) (i = u, d, s, · · · )

• Gluon distributions: g (x, Q2)

d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,

• Splitting functions

Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)

• DGLAP equation (Dokshitzer, Gribov, Lipatov, Altarelli and Parisi)

! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60

- 36 -


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60


(DGLAP equation)



d qi (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

"

qi

#

y, Q2$

Pqq

%x

y

&

+ g#

x, Q2$

Pqg

%x

y

&'

d g (x, Q2)

d ln Q2=

!s(Q)

2"

! 1

x

dy

y

(

)

i

qi

#

y, Q2$

Pgq

%x

y

&

+ g#

x, Q2$

Pgg

%x

y

&* (6.22)

• Interpretation:

qi

qi

Pqq

+xy

,

g

qi

Pqg

+xy

,

)

i

qi

g

Pgq

+xy

,

g

g

Pgg

+xy

,


Pqq(z) =4

3

1 + z2

1 ! z

Pqg(z) =1

2

-

z2 + (1 ! z)2.

Pgq(z) =4

3

1 + (1 ! z)2

z

Pgg(z) = 6

"

1 ! z

z+

z

1 ! z+ z (1 ! z)

'

(6.23)


! Results:

60– Quark- and gluon-distribution function

– Evolution of F2 (x, Q2) in comparison with experiments

! Note: regularization at z = 1:1

1 ! z"

1

(1 ! z)+

! 1

0

dzf(z)

(1 ! z)+=

! 1

0

dzf(z) ! f(1)

1 ! z

"1

(1 ! z)+=

1

1 ! zfor z < 1

#

(6.24)

For example,

Pqq(z) =4

3

1 + z2

(1 ! z)++ 2 !(1 ! z) (6.25)

Pgg(z) = 6

"1 ! z

z+

z

(1 ! z)++ z (1 ! z)

#

+1

2

"

11 !2

3Nf

#

!(1 ! z) (6.26)

and also,

Pqq(z) = Pgq(1 ! z)

Pqg(z) = Pqg(1 ! z)

Pgg(z) = Pgg(1 ! z)

(6.27)

61

i='t.3x t 0''x=2x i 0J

x=3.2x l0-r Hl 95+96 datah . a l ; m i h . ^ ,

- . -{x=)x I u

^ . ^ - {)(=5X t U

x= L3x l0'3

.r=2x 10'3

x=3.2x10-l

x=5x l0'3

x=8x10'3

x= l .3x I 0-2

x=2x10'2

x=3.2x

x=5x 10'2

T-

x=8x 10'2

x=3.2x l0' '

1= l . l x l 0 ' l

Y ' Y ' V O O Q

I

c\l

-+-

o.l

X

f T

J

t 0 r o 2 . . 1 0 3 l o l

Q ' (GeV' )

Figure 7: Comparison of the F! predictions of the 1VIRST partons rvith the preliminary 1995

and 1996 nominal vertex data of H1 [2i]. For display purposes rve acld 0.2(19 - i) to Fl each

time the value of r is decreased, where i = I, i9. These data are not used in the global analysis.

T d

Deep Inelastic Scattering: Structure Function of the

Proton

experimental datavs.

QCD

Q-dependence fromDGLAP EQUATIONS

16. Structure functions 1

16. STRUCTURE FUNCTIONSUpdated September 2007 by B. Foster (University of Oxford), A.D. Martin (University ofDurham), and M.G. Vincter (Carleton University).

16.1. Deep inelastic scatteringHigh-energy lepton-nucleon scattering (deep inelastic scattering) plays a key role in

determining the partonic structure of the proton. The process !N ! !!X is illustrated inFig. 16.1. The filled circle in this figure represents the internal structure of the protonwhich can be expressed in terms of structure functions.

k

k

q

P, M W

Figure 16.1: Kinematic quantities for the description of deep inelastic scattering.The quantities k and k! are the four-momenta of the incoming and outgoingleptons, P is the four-momentum of a nucleon with mass M , and W is the massof the recoiling system X . The exchanged particle is a ", W±, or Z; it transfersfour-momentum q = k " k! to the nucleon.

Invariant quantities:

# =q · PM

= E " E! is the lepton’s energy loss in the nucleon rest frame (in earlierliterature sometimes # = q · P ). Here, E and E! are the initial and finallepton energies in the nucleon rest frame.

Q2 = "q2 = 2(EE!""!k ·"!k !)"m2

! "m2!! where m!(m!!) is the initial (final) lepton mass.

If EE! sin2($/2) # m2! , m2

!! , then

$ 4EE! sin2($/2), where $ is the lepton’s scattering angle with respect to the leptonbeam direction.

x =Q2

2M#where, in the parton model, x is the fraction of the nucleon’s momentum

carried by the struck quark.

y =q · Pk · P

=#

Eis the fraction of the lepton’s energy lost in the nucleon rest frame.

W 2 = (P + q)2 = M2 + 2M# " Q2 is the mass squared of the system X recoiling againstthe scattered lepton.

s = (k + P )2 =Q2

xy+ M2 + m2

! is the center-of-mass energy squared of the lepton-nucleonsystem.

CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006)

available on the PDG WWW pages (URL: http://pdg.lbl.gov/) November 29, 2007 14:50

i='t.3x t 0''x=2x i 0J

x=3.2x l0-r Hl 95+96 datah . a l ; m i h . ^ ,

- . -{x=)x I u

^ . ^ - {)(=5X t U

x= L3x l0'3

.r=2x 10'3

x=3.2x10-l

x=5x l0'3

x=8x10'3

x= l .3x I 0-2

x=2x10'2

x=3.2x

x=5x 10'2

T-

x=8x 10'2

x=3.2x l0' '

1= l . l x l 0 ' l

Y ' Y ' V O O Q

I

c\l

-+-

o.l

X

f T

J

t 0 r o 2 . . 1 0 3 l o l

Q ' (GeV' )

Figure 7: Comparison of the F! predictions of the 1VIRST partons rvith the preliminary 1995

and 1996 nominal vertex data of H1 [2i]. For display purposes rve acld 0.2(19 - i) to Fl each

time the value of r is decreased, where i = I, i9. These data are not used in the global analysis.

T d


Proton


QCD






k

k

q

P, M W



# =q · PM


Q2 = "q2 = 2(EE!""!k ·"!k !)"m2


If EE! sin2($/2) # m2! , m2

!! , then


x =Q2



y =q · Pk · P

=#



s = (k + P )2 =Q2

xy+ M2 + m2




- 37 -

10 16. Structure functions

Table 16.1: Lepton-nucleon and related hard-scattering processes and theirprimary sensitivity to the parton distributions that are probed.

Main PDFsProcess Subprocess Probed

!±N ! !±X "!q ! q g(x 0.01), q, q!+(!")N ! #(#)X W !q ! q#

#(#)N ! !"(!+)X W !q ! q#

# N ! µ+µ"X W !s ! c ! µ+ s

pp ! "X qg ! "q g(x " 0.4)pN ! µ+µ"X qq ! "! q

pp, pn ! µ+µ"X uu, dd ! "! u # d

ud, du ! "!

ep, en ! e$X "!q ! q

pp ! W ! !±X ud ! W u, d, u/d

pp ! jet +X gg, qg, qq ! 2j q, g(0.01 x 0.5)

all polarized PDFs. These polarized PDFs may be fully accessed via flavor tagging insemi-inclusive deep inelastic scattering. Fig. 16.5 shows several global analyses at a scaleof 2.5 GeV2 along with the data from semi-inclusive DIS.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

x f

(x)

Figure 16.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, g) using the MRST2001 parameterization [29,13](withuncertainties for uv, dv, and g) at a scale µ2 = 10 GeV2.

June 16, 2004 14:04

u

d

sc

d

guv

v

fraction of total proton momentum

distributions of QUARKS and GLUONS

in the PROTON

experimental data +

QCD

SNAPSHOTS of the NUCLEON´S INTERIOR

Deep Inelastic Scattering

- 38 -

4 16. Structure functions

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'"%

'!(%

'!()

'!($

'!(*

'!(#

'!('

'+

,#-+./#0

1234

5'

4678

98:;4

<;8

!

'

#

*

$

)

%

=

'!(=

'!(%

'!()

'!($

'!(*

'!(#

'!('

'+

,#>??-+./#0>@>?-/0

5'

1234

2;8

!

!"!#)

!"!)

!"!=)

!"'

!"'#)

!"')

!"'=)

!"#

'!(%

'!()

'!($

'!(*

'!(#

'!('

+

,#>AA-+./#0>@>B-/0

Figure 16.10: a) The proton structure function F p2 mostly at small x and Q2, measured in electromagnetic

scattering of positrons (H1, ZEUS), electrons (SLAC), and muons (BCDMS, NMC) on protons. Lines areZEUS and H1 parameterizations for lower (Regge) and higher (QCD) Q2. The width of the bins can be up to10% of the stated Q2. Some points have been slightly o!set in x for clarity. References: ZEUS—J. Breitweget al., Phys. Lett. B407, 432 (1997); J. Breitweg et al., Eur. Phys. J. C7, 609 (1999); J. Breitweg et al.,Phys. Lett. B487, 53 (2000) (both data and ZEUS Regge parameterization); S. Chekanov et al., Eur. Phys.J. C21, 443 (2001); S. Chekanov et al., Phys. Rev. D70, 052001 (2004); H1—C. Adlo! et al., Nucl. Phys.B497, 3 (1997); C. Adlo! et al., Eur. Phys. J. C21, 33 (2001) (both data and H1 QCD parameterization);C. Adlo! et al., Eur. Phys. J. C30, 1 (2003); A. Aktas et al., Phys. Lett. B598, 159 (2004); BCDMS, NMC,SLAC—same references as Fig. 16.7.

b) The charm structure function F cc2 (x), i.e. that part of the inclusive structure function F p

2 arising fromthe production of charm quarks, measured in electromagnetic scattering of positrons on protons (H1, ZEUS)and muons on iron (EMC). The H1 points have been slightly o!set in x for clarity. For the purpose ofplotting, a constant c(Q) = 0.05i2Q is added to F cc

2 where iQ is the number of the Q2 bin, ranging from 1(Q2 = 1.8 GeV2) to 11 (Q2 = 650 GeV2). References: ZEUS—J. Breitweg et al., Eur. Phys. J. C12, 35(2000); S. Chekanov et al., Phys. Rev. D69, 012004 (2004); H1—C. Adlo! et al., Z. Phys. C72, 593 (1996);C. Adlo! et al., Phys. Lett. B528, 199 (2002); A. Aktas et al., Eur. Phys. J. C40, 349 (2005); A. Aktas et al.,Eur. Phys. J. C45, 23 (2006); EMC—J.J. Aubert et al., Nucl. Phys. B213, 31 (1983).

Inset: The bottom quark structure function F bb2 (x). For the purpose of plotting, a constant k(Q) = 0.01i1.7

Q

is added to F bb2 where iQ is the number of the Q2 bin, ranging from 1 (Q2 = 12 GeV2) to 5 (Q2 = 650 GeV2).

References: H1—A. Aktas et al., Eur. Phys. J. C40, 349 (2005); A. Aktas et al., Eur. Phys. J. C45, 23(2006).

Statistical and systematic errors added in quadrature are shown for both plots. The data are given as afunction of x in bins of Q2.


Proton


QCD


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!

"#$!%&#'

()

*+,-

./01-

21/

-34/

+556

78#

789

785

78:

)

)8#

)89

)7;9

)7;<

)7;#

)7;)

)

Figure 16.2: The proton structure function F p2 given at two Q2 values (3.5 GeV2

and 90 GeV2), which exhibit scaling at the ‘pivot’ point x ! 0.14. See the captionsin Fig. 16.7 and Fig. 16.10 for the references of the data. Also shown is theMRST2006 parameterization [13] given at the same scales.

functions) can be expressed as a convolution of calculable, process-dependent coe!cientfunctions and these universal parton distributions, e.g. Eq. (16.21).

It is often convenient to write the evolution equations in terms of the gluon, non-singlet(qNS) and singlet (qS) quark distributions, such that

qNS = qi " qi (or qi " qj), qS =!

i

(qi + qi) . (16.24)

The non-singlet distributions have non-zero values of flavor quantum numbers, such asisospin and baryon number. The DGLAP evolution equations then take the form

!qNS

! lnµ2 ="s(µ2)

2#Pqq # qNS ,

!

! ln µ2

"qS

g

#=

"s(µ2)2#

"Pqq 2nf PqgPgq Pgg

##

"qS

g

#, (16.25)

November 29, 2007 14:50


!

"#$!%&#'

()

*+,-

./01-

21/

-34/

+556

78#

789

785

78:

)

)8#

)89

)7;9

)7;<

)7;#

)7;)

)

Figure 16.2: The proton structure function F p2 given at two Q2 values (3.5 GeV2

and 90 GeV2), which exhibit scaling at the ‘pivot’ point x ! 0.14. See the captionsin Fig. 16.7 and Fig. 16.10 for the references of the data. Also shown is theMRST2006 parameterization [13] given at the same scales.

functions) can be expressed as a convolution of calculable, process-dependent coe!cientfunctions and these universal parton distributions, e.g. Eq. (16.21).

It is often convenient to write the evolution equations in terms of the gluon, non-singlet(qNS) and singlet (qS) quark distributions, such that

qNS = qi " qi (or qi " qj), qS =!

i

(qi + qi) . (16.24)

The non-singlet distributions have non-zero values of flavor quantum numbers, such asisospin and baryon number. The DGLAP evolution equations then take the form

!qNS

! lnµ2 ="s(µ2)

2#Pqq # qNS ,

!

! ln µ2

"qS

g

#=

"s(µ2)2#

"Pqq 2nf PqgPgq Pgg

##

"qS

g

#, (16.25)

November 29, 2007 14:50


enough observables to determine all polarized PDFs. These polarized PDFs may be fullyaccessed via flavor tagging in semi-inclusive deep inelastic scattering. Fig. 16.5 showsseveral global analyses at a scale of 2.5 GeV2 along with the data from semi-inclusiveDIS.

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'!($

'!()

'!(#

'!('

*

*+,-*.

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'!($

'!()

'!(#

'!('

*

*+,-*.

Figure 16.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, b, g) and their associated uncertainties using the NNLOMRST2006 parameterization [13] at a scale µ2 = 20 GeV2 and µ2 = 10, 000 GeV2.

Comprehensive sets of PDFs available as program-callable functions can be obtainedfrom several sources e.g., Refs. [55,56]. As a result of a Les Houches Accord, a PDFpackage (LHAPDF) exists [57] which facilitates the inclusion of recent PDFs in MonteCarlo/Matrix Element programs in a very compact and e!cient format.

16.4. DIS determinations of !s

Table 16.2 shows the values of !s(M2Z) found in recent fits to DIS and related data in

which the coupling is left as a free parameter. There have been several other studies of!s using subsets of inclusive DIS data, and also from measurements of spin-dependentstructure functions, see the Quantum Chromodynamics section of this Review.

November 29, 2007 14:50

DGLAP

EQUATIONS





k

k

q

P, M W



# =q · PM


Q2 = "q2 = 2(EE!""!k ·"!k !)"m2


If EE! sin2($/2) # m2! , m2

!! , then


x =Q2



y =q · Pk · P

=#



s = (k + P )2 =Q2

xy+ M2 + m2




Deep Inelastic Scattering

Parton (Quark and Gluon)

Distributions

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N CHAPTER VI: 2 QCD and Structure Functions of the Nucleon · CHAPTER 6. QCD AND STRUCTURE...

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Transcript of N CHAPTER VI: 2 QCD and Structure Functions of the Nucleon · CHAPTER 6. QCD AND STRUCTURE...