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

!

"(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

!

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