Interface between Theoretical Framework of GPDs and … · 2013. 11. 6. · Interface between...
Transcript of Interface between Theoretical Framework of GPDs and … · 2013. 11. 6. · Interface between...
Interface between Theoretical Framework of GPDs
and Experimental Measurements of DVCS
Bochum, March 7, 2013
Chueng-Ryong Ji North Carolina State University
!
!
Hadron Physics at JLab
Outline • JLab Kinematics (t < -|tmin|≠0) • Original Formulation of DVCS with GPDs (Valid only at a limited t region) • Comparison between Exact Tree-Level Result and the corresponding
Results from Original Formulation • Hadronic Tensors in DVCS Toward Generalization: Two Approaches • Conclusion and Outlook
JLab Kinematics t < -|tmin|≠0
!
t = "2 = #$ 2M 2 + "%
2
1# $
!
"#2 > "#min
2 $ 0
!
"+(# "0 + "3) = $ p+; ;
Ranges of Kinematic Variables
0 <Q2 <4E 2M2E +M
Q2
2ME< xBj =
Q2
2p !q<1
cos!" =Q4 + 2xBj
2 M 2t +Q2xBj (t + 2xBjM2 )
Q(Q2 + xBjt) Q2 + 4xBj2 M 2
xBj!tQ2 =
Q2 + 2xBj2 M 2 !Q Q2 + 4xBj
2 M 2 cos!"Q2 + 2xBjM
2 !Q Q2 + 4xBj2 M 2 cos!"
!" = 28!
!" =14!
Nucleon GPDs in DVCS Amplitude X.Ji,PRL78,610(1997): Eqs.(14) and (15)
!
pµ = "(1ct, 0
x, 0
y
,1z) ,
nµ = (1ct, 0
x, 0
y
, #1z) /(2") ,
P µ =12(P + $ P )µ = pµ +
M 2 #%2 /42
nµ,
qµ = #& pµ +Q2
2&nµ , & =
Q2
2P ' q,
%µ = #& pµ #M 2 #%2 /4
2nµ
(
) *
+
, - + %.
µ .
!
T µ" ( p,q,#) = $12
( pµn" + p" nµ $ gµ" ) dx 1
x $ %2
+ i&+
1
x +%2$ i&
'
(
) ) )
*
+
, , , $1
+1
-
. H(x,#2,%)U ( / P ) / n U(P) + E(x,#2,%)U ( / P )i012n1#2
2MU(P)
3
4 5
6
7 8
$i2&µ"12 p1n2 dx 1
x $ %2
+ i&$
1
x +%2$ i&
'
(
) ) )
*
+
, , , $1
+1
-
. ˜ H (x,#2,%)U ( / P ) / n 9 5U(P) + ˜ E (x,#2,%)# : n2M
U ( / P )9 5U(P)3
4 5 6
7 8
Just above Eq.(14), ``To calculate the scattering amplitude, it is convenient to define a special system of coordinates.”
Note here that = 0 .
Nucleon GPDs in DVCS Amplitude A.V.Radyushkin, PRD56, 5524 (1997): Eq.(7.1)
At the beginning of Section 2E (Nonforward distributions), ``Writing the momentum of the virtual photon as q=q’-ζp is equivalent to using the Sudakov decomposition in the light-cone `plus’(p) and `minus’(q’) components in a situation when there is no transverse momentum .”
!
q = " q #$ p ,
$ =Q2
2p % " q ,
r = p # " p
!
T µ" (p,q, # q ) =1
2(p $ # q )ea2
a% [ &gµ" +
1p $ # q
(pµ # q " + p" # q µ )'
( )
*
+ ,
- u ( # p ) # / q u(p)TFa (. ) +
12M
u ( # p )( # / q / r & / r # / q )u(p)TKa (. )
/ 0 1
2 3 4
+ i5µ"67 p6 # q 7p $ # q
u ( # p ) # / q 8 5u(p)TGa (. ) +
# q $ r2M
u ( # p )8 5u(p)TPa (. )
/ 0 1
2 3 4 ]
Note here that ,i.e. only consistent at t=0, neglecting nucleon mass.
JLab Kinematics t < 0 • In JLab, the final hadron and final photon
move off the z-axis. • To see the effect of taking t<0, we mimic the
kinematics at JLab and compute bare bone VCS amplitudes neglecting masses.
B.L.G.Bakker and C.Ji, PRD83,091502(R) (2011); FBS 52, 285 (2012).
Bare Bone Structure
“Bare Bone” VCS Amplitude at Tree Level
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"(hq ,h # q ,sk,s # k ) = $µ* ( # q ,h # q ) $% (q,hq ) TS
µ% + TUµ%( )
!
TSµ" =
k# + q#S
u ( $ k ,s $ k )%µ%#%" u(k,sk )
TUµ" =
k# & $ q #U
u ( $ k ,s $ k )%"%#% µu(k,sk )
!
S = (k + q)2
!
U = (k " # q )2
!
= " k
Neglecting masses,
Identity:
!
!
" µ"#"$ = gµ#"$ + g#$" µ % gµ$ "# + i&µ#$'"'" 5
Hadron Helicity Amplitude:
Keeping no transverse momentum in DVCS, we agree on
equivalent to the expression given by X. Ji and A.V. Radyushkin.
Using Sudakov vectors
we find
Full Amp vs. Reduced Amp
!
/ " *( # $ )( / k + / q + m)/ " ($)(k + q)2 %m2 & ' &
/ " *( # $ )q%( + / " ($)(x %))P +q%
S-channel:
!
/ " (#)( / $ k % / q + m)/ " *( $ # )( $ k % q)2 %m2 & ' & %
/ " (#)q%( + / " *( $ # )x P +q%U-channel:
Checking Amplitudes • Gauge invariance of each and every polarized amplitude including the longitudinal polarization for the virtual photon. • Klein-Nishina Formula in RCS. • Angular Momentum Conservation.
!
"l = " # l = +12
, sk = s # k = +12
, h # q = +1 ;
Allowed !
Checking Amplitudes • Gauge invariance of each and every polarized amplitude including the longitudinal polarization for the virtual photon. • Klein-Nishina Formula in RCS. • Angular Momentum Conservation.
!
"l = " # l = +12
, sk = s # k = +12
, h # q = $1 ;
Prohibited !
Comparison
Swap
C.Carlson and C.Ji, Phys.Rev.D67,116002 (2003); B.Bakker and C.Ji, Phys.Rev.D83,091502(R) (2011).
For any orders in Q
Exact Reduced
Number of Independent Amplitudes in VCS
Nucleon Target
!
3 " 2 " 2 " 22 = 12
12 independent tensor structures M.Perrottet, Lett. Nuovo Cim. 7, 915 (1973); R.Tarrach, Nuovo Cim. 28A, 409 (1975); D.Drechsel et al.,PRC57,941(1998); A.V.Belitsky, D.Mueller and A.Kirchner, NPB629, 323(2002); A.V.Belitsky and D.Mueller, PRD82, 074010(2010)
A.V.Belitsky and D. Mueller,arXiv:1005.5209v1[hep-ph]
Biproducts of P,V,A,T, but not S
D.Drechsel,G.Knoechlein,A.Yu.Korchin,A.Metz and S.Scherer, PRC 57, 941 (1998)
…
Biproducts of S,V,A,T, but not P
!
(p + " p )µ # 2M$µ % i& µ'q'
i(µ')*$5$' p) " p * #
q2
2$µ % iM&µ'q'
qµ$5 # 2M$µ$
5 +(µ')* (p + " p )' &)*
Gordon Decomposition and Extension
Possible Reconciliation
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Jµ = " µF1 + i#µ$q$2M
F2
= " µ (F1 + F2) +(p + % p )µ
2MF2
=(p + % p )µ
2M4M 2F1 + q2F24M 2 & q2
& i' µ$()" 5"$ p( % p )2(F1 + F2)4M 2 & q2
=(p + % p )µ
2MF1 + i#
µ$q$2M
(F1 + F2)
= " µ (F1 +q2
4M 2 F2) & i' µ$()" 5"$ p( % p )F22M 2
= i#µ$q$2M
(4M 2
q2F1 + F2) + i' µ$()" 5"$ p( % p )
2F1q2
All are equivalent!
V T V S S A S T V A T A
Conclusion and Outlook
• We find that the XJ and AVR amplitudes for DVCS in terms of GPDs for t < 0 are not satisfactory.
• The determination of all independent structures is important for the discussion of GPDs.
• Maintaining EM gauge invariance is an important constraint.
• If all invariant structures are identified, the question whether one can measure GPDs in experiments where Q2 does not go to infinity may become more focused.
Gauge Invariance Check in One-Loop Level
Gauge Invariance Check in One-Loop Level
Gauge Invariance Check in One-Loop Level
Gauge Invariance Check in One-Loop Level
Investigation of Complete Ampltude Attach the lepton current and check the spin filter for the DVCS amplitude.
Singularities develop in the polarization vector as q+ à 0.
p
p ! (1 ) p!
!"#$
p k
!x p (x ) p !
!q
soft part
q2= q+q--q⊥2 = -q⊥2 = -Q2 <0 , e.g. S.J.Brodsky,M.Diehl,D.S.Hwang, NPB596,99(01)
GPDs rely on the handbag dominance in DVCS; i.e. Q2 >> any soft mass scale