Transcript Theory overview of semi-inclusive deep inelastic scattering

M. Boglione
THEORY OVERVIEW
OF SEMI-INCLUSIVE
DEEP INELASTIC SCATTERING
A pedagogical introduction to
semi-inclusive deep inelastic scattering
based on a personal selection
of relevant issues
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DEEP INELASTIC SCATTERING
(E,k)
q
Q2 = -q2
(E,k’)
q
x = Q2/(2mn) = 2EE’(1-cosq)
xP
0<x<1 , n = E - E’
P
Q2 >> m2
 The nucleon has an internal structure
 x is the fraction of proton momentum carried by the parton
 The cross section is the incoherent sum of all partonic contributions convoluted with
the parton distribution function, which only depends on x at LO
 DIS = fq ( x )  ˆ
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Q2 EVOLUTION
 QCD
corrections induce Q2 dependence


f(x)
→
f(x,Q2)
 DGLAP evolution equations exactly predict
the Q2 dependence of distribution functions

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PARTON DISTRIBUTION FUNCTIONS
Unpolarized distribution functions
g = g  g
q = q  q
fq/p(x)
Δfsz/+(x)
Δfq↑/p↑(x)
-
-
Helicity distribution functions
q = q  q
g = g  g
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Transversity distribution functions
T q = q  q
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PARTON DISTRIBUTION FUNCTIONS

Very good knowledge of unpolarized distribution
functions, q(x,Q2) and g(x,Q2)

Fairly good knowledge of longitudinally polarized,
parton distributions, Δq(x,Q2); poor knowledge of
longitudinally polarized gluons Δg(x,Q2)

NO direct information on transversely polarized
partonic distributions, ΔTq(x,Q2)
Talks by E. Leader, A. Fantoni, D. Hasch, …
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DEEP INELASTIC SCATTERING
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What is the total amount of longitudianl spin
carried by partons in a longitudinally polarized
proton ?
1
=  S q    S g    Lq    Lg  ?
2

q



=
q

 Sg (Q ) = 0 g ( x,Q ) dx
2
2

1

2 0
q
 S  =
( x,Q 2 ) dx

1 1
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






All of this, and much
more on helicity
distribution functions in
A. Fantoni,
E. Leader, O. Teryaev and
A. Magnon talks !
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TRANSVERSITY
+
+
+
+
±
–
+
–
+
=
fq / p ( x,Q2 )
q( x,Q2 )
fq / p ( x,Q2 )
= h (x,Q )
±
More on
transversity in
X.Artru, A.
Bressan, D. Hasch
talks
2
1
In helicity basis:
1
 |   i |   therefore h 1( x,Q2 ) =
=
2
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Decouples from DIS
- (no quark helicity flip)
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TRANSVERSITY

There is no gluon transversity distribution function

Transversity cannot be studied in deep inelastic scattering
because it is chirally odd

Transversity can only appear in a cross-section
convoluted to another chirally odd function
–
+
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-
+
+
–
+
+
+
–
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–
–
–
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To learn about transversity
we should study either SIDIS
or DRELL-YAN processes
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SEMI-INCLUSIVE
DEEP INELASTIC SCATTERING
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SEMI-INCLUSIVE DEEP INELASTIC SCATTERING
ℓ
ℓ’
g*
elementary
cross- section
k
k’
d ˆ
Dh/q (z)

Incoming proton
fq(x)
P
fragmentation
function
h
final detected
hadron
X
distribution
function
Looking forward to
U. D’Alesio talk !
SIDIS = fq ( x )  ˆ  Dh / q (z)
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INTRINSIC TRANSVERSE MOTION
P
k
xP
Plenty of theoretical and experimental evidence
for transverse motion of partons within nucleons,
and of hadrons within fragmentation jets
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INTRINSIC TRANSVERSE MOTION
±1
Gluon radiation
k┴
±
±
Uncertainty principle:
Δx ~ 1 fm => Δp ~ 0.2 GeV/c
Hadron distribution in
jets in e+e– processes
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INTRINSIC TRANSVERSE MOTION
qT distribution of lepton pairs
in D-Y processes
l+
g*
p
qL
qT
Q2 = M2
l–
p
pT distribution of
hadrons in SIDIS
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TRANSVERSE MOMENTUM DEPENDENT
DISTRIBUTION AND FRAGMENTATION FUNCTIONS
Distribution and fragmentation functions now depend
 on the lightcone fraction
(x for the distributions and z for the fragmentations)
 on Q2 (→pQCD evolution),
 on the intrinsic transverse momentum of the partons,
(k for the distributions and p for the fragmentations)
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SEMI-INCLUSIVE DEEP INELASTIC SCATTERING
SIDIS in parton
model with
intrinsic k┴
Factorization holds at large Q2,
and
PT  k  QCD
(Ji, Ma, Yuan)
d lplhX = q f q ( x, k  ; Q 2 )  dˆ lqlq ( y, k  ; Q 2 )  Dqh ( z, p ; Q 2 )
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CAHN EFFECT
EMC data, µp and µd, E between 100 and 280 GeV
M.A., M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia and A. Prokudin
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POLARIZED
SEMI-INCLUSIVE DEEP INELASTIC
SCATTERING
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PDF’S IN COLLINEAR APPROXIMATION
Unpolarized distribution functions
g = g  g
q = q  q
fq/p(x)
Δfsz/+(x)
Δfq↑/p↑(x)
-
-
Helicity distribution functions
q = q  q
g = g  g
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Transversity distribution functions
T q = q  q
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TRANSVERSE MOMENTUM DEPENDENT
DISTRIBUTION FUNCTIONS
There are 8 leading-twist spin-k┴ dependent distribution functions
ΔNfq/p↑(x,k)
Sivers
function
Δfsz/p↑(x,k)
fq/p(x,k)
Δfsz/p+(x,k)
Helicity fn.
Δfsx/p↑(x,k)
Transversity
function
Δfsy/p↑(x,k)
Transversity
function
Δfsx/p+(x,k)
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Δfsy/p(x,k)
Boer-Mulders
function
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TRANSVERSE MOMENTUM DEPENDENT
DISTRIBUTION FUNCTIONS
ΔNfq/p↑(x,k)
Sivers
function
Δfsz/p↑(x,k)
Δfsz/p+(x,k)
Helicity fn.
fq/p(x,k)
Δfsx/p+(x,k)
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Δfsy/p(x,k)
Boer-Mulders
function
Δfsx/p↑(x,k)
Transversity
function
Δfsy/p↑(x,k)
Transversity
function
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TRANSVERSE MOMENTUM DEPENDENT
DISTRIBUTION FUNCTIONS
Courtesy of Aram Kotzinian
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POLARIZED SIDIS CROSS SECTION,
UP TO SUBLEADING ORDER IN 1/Q
Cahn Effect
0
1
d = d UU
 cos 2Φh d UU

1
1
2
3
cosΦh d UU
 e sinΦh d LU
Q
Q

1
 6 1
4
5
7 
 SL sin 2Φh d UL
 sinΦh d UL
 e d LL
 cosΦh d LL

Q
Q



Sivers Effect
Collins Effect

8
9
10
 ST sin(Φh  ΦS ) d UT
 sin(Φh  ΦS ) d UT
 sin(3Φh  ΦS ) d UT

1
11
12
 sin(2Φh  ΦS ) d UT
 sinΦS d UT
Q



1

13
14
15
 e cos(Φh  ΦS ) d LT  cosΦS d LT
 cos(2Φh  ΦS ) d LT
Q





Kotzinian, NP B441 (1995) 234,
Mulders and Tangermann, NP B461 (1996) 197, Boer and Mulders, PR D57 (1998) 5780
Bacchetta et al., PL B595 (2004) 309, Bacchetta et al., JHEP 0702 (2007) 093
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POLARIZED SIDIS CROSS SECTION,
UP TO SUBLEADING ORDER IN 1/Q
Nostra formula !
Sivers effect
Transversity fn.
Collins fn.
M. Anselmino, M. Boglione, U. D’Alesio, S. Melis, F. Murgia , A. Prokudin , in preparation,
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SIVERS DISTRIBUTION FUNCTION
1 N
 fq / p  ( x, k  ) S  ( pˆ  kˆ )
2
k
= fq / p ( x, k  )   f1T q ( x, k  ) S  ( pˆ  kˆ )
M
fq / p, S ( x, k  ) = fq / p ( x, k  ) 
The Sivers function is related
to the probability of finding an
unpolarized quark inside a
transversely polarized proton
S
k
p
X
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The Sivers function inbeds the correlation
between the proton spin and the quark
transverse momentum
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BOER-MULDERS DISTRIBUTION FUNCTION
1
1 N
f q , sq / p ( x, k  ) = f q / p ( x, k  )   f q  / p ( x, k  ) sq  ( pˆ  kˆ )
2
2
1
1 k q
The Boer-Mulders function is
ˆ )
ˆ
=
f
(
x
,
k
)

h
(
x
,
k
)
s

(
p

k
q
/
p

1

q

related to the probability of
2
2M
finding a polarized quark inside
an unpolarized proton
sq
k
p
X
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The BoerMulders function
is chirally odd
The Boer-Mulders function inbeds the
correlation between the quark spin and
its transverse momentum
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COLLINS FRAGMENTATION FUNCTION
1
Dh / q , sq ( z , p ) = Dh / q ( z, p )  N Dh / q  ( z , p ) sq  ( pˆ q  pˆ  )
2
The Collins function is related to
p
H1 q ( z, p ) sq  ( pˆ q  pˆ  )
the probability that a transversely = Dh / q ( z , p ) 
z Mh
polarized struck quark will
fragment into a spinless hadron
sq
p
q
X
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The Collins
function is
chirally odd
The Collins function inbeds the
correlation between the fragmenting
quark spin and the transverse
momentum of the produced hadron
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STUDY OF THE SIVERS AND COLLINS MECHANISMS


Studying Sivers, Collins and other mechanisms is complicated by
the fact that all these effects mix and overlap in the polarized
SIDIS cross section and azimuthal asymmetries
Way out : build appropriately ‘weighted’ azimuthal asymmetries !
SIVERS
sin( Φ ΦS )
AUT
2
ASYM M ETRY
0
1
d = d UU
 cos 2Φh d UU

dΦ dΦS (d   d  ) sin(Φ  ΦS )
 d Φ dΦ ( d 

S
 d  )
 q eq2 N fq / p  ( x, k  )  D / q ( z, p )
COLLINS
sin( Φ ΦS )
AUT
2
ASYM M ETRY
dΦ dΦS (d   d  ) sin(Φ  ΦS )
 dΦ dΦ ( d
S

 d )
 q e T q( x, k  )   D / q  ( z, p )
2
q
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N

1
1
2
3
cosΦh d UU
 e sinΦh d LU
Q
Q

1
 6 1
4
5
7 
 SL sin 2Φh d UL
 sinΦh d UL
 e d LL
 cosΦh d LL

Q
Q



Collins Effect
Sivers Effect

8
9
10
 ST sin(Φh  ΦS ) d UT
 sin(Φh  ΦS ) d UT
 sin(3Φh  ΦS ) d UT

1
11
12
 sin(2Φh  ΦS ) d UT
 sinΦS d UT
Q



1

13
14
15
 e cos(Φh  ΦS ) d LT
 cosΦS d LT
 cos(2Φh  ΦS ) d LT
Q

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 

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MEASUREMENTS OF THE SIVERS EFFECT
dΦ dΦ (d  d
2
 dΦ dΦ (d

sin( Φ ΦS )
UT
A

S

S
) sin(Φ  ΦS )
 d  )
 q eq2 N fq / p  ( x, k  )  D / q ( z, p )
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MEASUREMENT OF THE SIVERS EFFECT
sin( Φ ΦS )
AUT
2
COMPASS 2002-2004
dΦ dΦS (d  d ) sin(Φ  ΦS )

 dΦ dΦ (d
S


 d  )
 q eq2 N fq / p  ( x, k  )  D / q ( z, p )
Small values due to
deuteron target:
cancellation between
u and d contributions
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DETERMINATION OF THE SIVERS FUNCTION
Global fit of HERMES and COMPASS data
Crucial point:
use of
appropriate
fragmentation
functions
(De Florian, Stratman,
Sassot fragment. set)
M. Anselmino, M. Boglione,
U. D’Alesio, A. Kotzinian,
S. Melis, F. Murgia,
A. Prokudin, C. Turk
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DETERMINATION OF THE SIVERS FUNCTION
Global fit of HERMES and COMPASS data
Crucial point:
use of
appropriate
fragmentation
functions
(De Florian, Stratman,
Sassot fragment. set)
M. Anselmino, M. Boglione,
U. D’Alesio, A. Kotzinian,
S. Melis, F. Murgia,
A. Prokudin, C. Turk
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MEASUREMENT OF THE COLLINS EFFECT
sin( Φ ΦS )
AUT
2


d
Φ
d
Φ
(
d


d

) sin(Φ  ΦS )
S

 dΦ dΦ (d
S

 d  )
 q eq2 h1q ( x, k  )  N Dh / q  ( z, p )
COMPASS 2002-2004
The Collins
function is
convoluted
with
Transversity !
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
We need a strategy !

Fix one of the two functions according to some theoretical model and use
SIDIS data to determine the other
[see for example A.V.Efremov, K. Goeke, P. Shweitzer, Eur. Phys. J. C 35, 207 (2004)]

Perform a simultaneous fit of SIDIS and e+e-→h1h2X BELLE data.
BELLE
Thrust axis method
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Hadronic plane method
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SIMULTANEOUS DETERMINATION OF
TRANSVERSITY AND COLLINS FUNCTION

A global fit of HERMES and COMPASS SIDIS single spin asymmetries combined
with BELLE data on e+e-→h1h2X allows to determine both the transversity
distribution and the Collins fragmentation functions of u and d quarks.
Fit of Belle e+e- data
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, C. Türk, Phys. Rev. D 75,054032 (2007)
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SIMULTANEOUS DETERMINATION OF
TRANSVERSITY AND COLLINS FUNCTION

A global fit of HERMES and COMPASS SIDIS single spin asymmetries combined
with BELLE data on e+e-→h1h2X allows to determine both the transversity
distribution and the Collins fragmentation functions of u and d quarks.
Fit of COMPASS AUT data
Fit of HERMES AUT data
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, C. Türk, Phys. Rev. D 75,054032 (2007)
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SIMULTANEOUS DETERMINATION OF
TRANSVERSITY AND COLLINS FUNCTION

A global fit of HERMES and COMPASS SIDIS single spin asymmetries combined
with BELLE data on e+e-→h1h2X allows to determine both the transversity
distribution and the Collins fragmentation functions of u and d quarks.
Transversity distribution functions
Collins fragmentation functions
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, C. Türk, Phys. Rev. D 75,054032 (2007)
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TMD’S IN DRELL-YAN PROCESSES
d
D Y
= q f q ( x1 , k ; Q )  f q ( x2 , k ; Q ) dˆ
2
2
l+
γ*
p
X
qL
qq l  l 
Q2 = M2
l–
qT
p
1 d
3 1 
n 2

2
2
=
1


cos
q


sin
q
cos


sin
q
cos
2



 d 4   3 
2

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Polarized Drell-Yan cross sections allow to
 access many TMDs (Boer-Mulders, …)
Verify whether
f1Tq
SIDIS
=  f1Tq
D Y
and offer the golden channel to measure the
transversity distribution function
ATT   h1q ( x1 )  h1q ( x2 )
q
Looking forward to the Drell-Yan Session of this meeting,
with M. Chiosso,Y. Goto, A. Bianconi …
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Non-universality of Sivers Asymmetries:
Simple QED
example:
Drell-Yan: repulsive
DIS: attractive
Same in QCD:
As a result:
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Thanks for
your attention
Enjoy the rest
of the
workshop !
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