Treacherous Points in DVCS

JLab, October 29, 2010

G.R.Goldstein and S.Liuti, PRD80, 071501(R)(2009); arXiv:1006.0213[hep-ph]

B.L.G. Bakker and C.Ji, arXiv:1002.0443[hep-ph]

Outline• Motivation from well-known Folk-lore

- Nucleon GPDs in DVCS Amplitude

- Specific Frame without Large Transverse Momentum• Complete Amplitude with Lepton Current

- Cancellation of Singularities

- Spin Filter for DVCS Amplitudes • Benchmark DVCS Amplitudes

- Three Typical Kinematics

- Spin Directions in LF Helicity States • Full vs. Reduced Amplitudes

- Investigation of Singularities- Angular Momentum Conservation

• Conclusions

Basic Idea: VCS Amplitude at Tree Level

€

Η(hq ,h ′ q ,sk,s ′ k ) = εμ* ( ′ q ,h ′ q ) εν (q,hq ) TS

μν + TUμν

( )

€

TSμν =

kα + qα

Su ( ′ k ,s ′ k )γ

μγ α γ ν u(k,sk )

TUμν =

kα − ′ q αU

u ( ′ k ,s ′ k )γν γ α γ μ u(k,sk )

€

S = (k + q)2

€

U = (k − ′ q )2

€

= ′ k

Neglecting masses,

Identity:

€

€

γμγαγν =gμα γ ν + gαν γ μ − gμν γ α + iε μανβ γ β γ 5

Helicity Amplitude:

Nucleon GPDs in DVCS AmplitudeX.Ji,PRL78,610(1997): Eqs.(14) and (15)

€

pμ = Λ(1ct

, 0x

, 0y

,1z

) ,

nμ = (1ct

, 0x

, 0y

, −1z

) /(2Λ) ,

P μ =1

2(P + ′ P )μ = pμ +

M 2 − Δ2 /4

2nμ ,

qμ = −ξ pμ +Q2

2ξnμ , ξ =

Q2

2P ⋅q,

Δμ = −ξ pμ −M 2 − Δ2 /4

2nμ

⎡

⎣ ⎢

⎤

⎦ ⎥+ Δ⊥

μ .

€

T μν ( p,q,Δ) = −1

2( pμ nν + pν nμ − gμν ) dx

1

x −ξ

2+ iε

+1

x +ξ

2− iε

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟−1

+1

∫

× H(x,Δ2,ξ )U ( ′ P ) / n U(P) + E(x,Δ2,ξ )U ( ′ P )iσ αβ nα Δβ

2MU(P)

⎡

⎣ ⎢

⎤

⎦ ⎥

−i

2ε μναβ pα nβ dx

1

x −ξ

2+ iε

−1

x +ξ

2− iε

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟−1

+1

∫

× ˜ H (x,Δ2,ξ )U ( ′ P ) / n γ 5U(P) + ˜ E (x,Δ2,ξ )Δ ⋅n2M

U ( ′ P )γ 5U(P) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Just above Eq.(14),``To calculate the scattering amplitude, it is convenient to definea special system of coordinates.”

Nucleon GPDs in DVCS AmplitudeA.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 situationwhen there is no transverse momentum .”

€

q = ′ q −ζ p ,

ζ =Q2

2p ⋅ ′ q ,

r = p − ′ p

€

T μν ( p,q, ′ q ) =1

2(p ⋅ ′ q )ea

2

a

∑ [ −gμν +1

p ⋅ ′ q (pμ ′ q ν + pν ′ q μ )

⎛

⎝ ⎜

⎞

⎠ ⎟

× u ( ′ p ) ′ / q u(p)TFa (ζ ) +

1

2Mu ( ′ p )( ′ / q / r − / r ′ / q )u(p)TK

a (ζ ) ⎧ ⎨ ⎩

⎫ ⎬ ⎭

+ iε μναβ pα ′ q βp ⋅ ′ q

u ( ′ p ) ′ / q γ 5u(p)TGa (ζ ) +

′ q ⋅r2M

u ( ′ p )γ 5u( p)TPa (ζ )

⎧ ⎨ ⎩

⎫ ⎬ ⎭

]

S-Channel DVCS Amplitude

€

nμ (+) = (1+

, 0x

, 0y

, 0−

) ↔1

2(1

ct

, 0x

, 0y

,1z

)

nμ (−) = (0+

, 0x

, 0y

,1−

) ↔1

2(1

ct

, 0x

, 0y

, −1z

)’

€

k μ = k +nμ (+) + k−nμ (−) + k⊥μ

qμ = q+nμ (+) + q−nμ (−) + q⊥μ

γ μ = / n (−)nμ (+) + / n (+)nμ (−) + γ⊥μ

Sudakov Variables

€

TSμν =

1

S[{(k + + q+)nμ (+) + (k− + q−)nμ (−) + (k⊥

μ + q⊥μ )}nν (+)

+ {(k + + q+)nν (+) + (k− + q−)nν (−) + (k⊥ν + q⊥

ν )}nμ (+)

− gμν (k− + q−)] u ( ′ k ) / n (−)u(k) +LKeeping no transverse momentum in DVCS, we agree on

€

TSμν =

q−

S{nμ (−)nν (+) + nν (−)nμ (+) − gμν } u ( ′ k ) / n (−)u(k) +L

However, in general, such factorization doesn’t work.

In the frame with large transverse momentum, e.g.

€

q+ = O(Q0) ,r q ⊥ = O(Q 1) , q− = O(Q2) ; ′ q + = O(Q0), ′

r q ⊥ = O(Q 1), ′ q − = O(Q2)

we get

€

ΗS (hq ,h ′ q ,sk,s ′ k ) =1

S[{2(k + + q+)ε*−

( ′ q ,h ′ q )ε−(q,hq ) + ε*−

( ′ q ,h ′ q )q⊥⋅ε⊥(q,hq )

+ ε−(q,hq )q⊥⋅ε⊥* ( ′ q ,h ′ q ) − q−ε⊥

* ( ′ q ,h ′ q ) ⋅ε⊥(q,hq )}u ( ′ k ,s ′ k ) / n (−)u(k,sk )

+L

where€

€

ε*−

( ′ q ,h ′ q ) = O(Q1) , ε⊥* ( ′ q ,h ′ q ) = O(Q0) , ε

−

(q,hq ) = O(Q1) , ε⊥(q,hq ) = O(Q0) .

Thus, one cannot neglect the terms with large transversemomentum but should compute the full amplitude.

In particular, longitudinal polarization of virtual photon shouldnot be neglected in the frame with large transverse momentum.

Investigation of Complete Ampltude Attach the lepton current and check the spin filter

for the DVCS amplitude.

Singularities develop in the polarization vector as p+ 0.

Investigation of Complete Ampltude In the well-known q+=0 frame, both lepton and

hadron amplitudes are singular.

€

Lλ =1δ

lλ(−1) + lλ

(0) +δ lλ(+1)

€

Hλ =1δ

hλ(−1) + hλ

(0) +δ hλ(+1)

S.J.Brodsky, M.Diehl, and D.-S.Hwang, NPB596, 99(2001)

Complete Amplitude and Constraints

€

lλ(−1) ⋅hλ

(−1)( )

λ∑ =0 ,

lλ(−1) ⋅hλ

(0) + lλ(0) ⋅hλ

(−1)( )

λ∑ =0 ,

Lλ ⋅Hλλ∑ = lλ

(0) ⋅hλ(0) + lλ

(−1) ⋅hλ(+1) + lλ

(+1) ⋅hλ(−1)

( )λ∑ +O(δ) .

There should not be any singularity

in complete amplitude.

€

Lλ ⋅Hλλ∑ =

1δ

lλ(−1) + lλ

(0) +δ lλ(+1)

⎛

⎝ ⎜

⎞

⎠ ⎟⋅1δ

hλ(−1) + hλ

(0) +δ hλ(+1)

⎛

⎝ ⎜

⎞

⎠ ⎟

λ∑

=1δ2

lλ(−1) ⋅hλ

(−1)( ) +

1δ

lλ(−1) ⋅hλ

(0) + lλ(0) ⋅hλ

(−1)( )

⎡

⎣ ⎢

⎤

⎦ ⎥

λ∑

+ lλ(0) ⋅hλ

(0) + lλ(−1) ⋅hλ

(+1) + lλ(+1) ⋅hλ

(−1)( ) +O(δ)[ ]

λ∑

Three Kinematics K1,K2,K3

€

qμ = (−ζP+,0,0,Q2

ζP+) , ′ q μ = (0,0,0,

Q2

ζP+−

ζm2

x(x −ζ )P+)

€

qμ = (−ζ

2P+,

Q

2,0,

Q2

ζP+) , ′ q μ = (

ζ

2P+,

Q

2,0,

Q2

ζP+−

ζm2

x(x −ζ )P+)

€

qμ = (δP+,Q,0,Q2

(ζ + δ)P++

ζm2

x(x −ζ )P+) , ′ q μ = ((ζ + δ)P+,Q,0,

Q2

(ζ + δ)P+)

K1

K2

K3

€

{ ′ λ ,λ}h K1 K2 K 3

{1

2,1

2} +1 −Q(1−

δ

4ζ+

2ζ

δ) 0 2Q

{1

2,1

2} −1 −Q(1−

3δ

4ζ−

2ζ

δ) 2Q −4Q

{1

2,1

2} 0 −i2 2 Q

ζ

δ0 4iQ

Leptonic Amplitudes in K1,K2,K3

€

hq h ′ q sk = s ′ k K1 K2 K 3

+1 +1 +1

22

x

x −ζ1+

ζ

δ

⎛

⎝ ⎜

⎞

⎠ ⎟ 2

x −ζ

x−2

x

x −ζ

+1 +1 −1

22

x −ζ

x1+

ζ

δ

⎛

⎝ ⎜

⎞

⎠ ⎟ 2

x

x −ζ2

x −ζ

x

+1 −1 +1

2−2

x −ζ

x

ζ

δ

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 4

x

x −ζ

+1 −1 −1

2−2

x

x −ζ

ζ

δ

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 4

x −ζ

x

0 +1 +1

2i 2

x

x −ζ1+

2ζ

δ−

δ

4ζ

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 −i4

x

x −ζ

0 +1 −1

2i 2

x −ζ

x1+

2ζ

δ−

δ

4ζ

⎛

⎝ ⎜

⎞

⎠ ⎟ 0 −i4

x −ζ

x

Hadronic Amplitudes in three Kinematics

€

λ = ′ λ h ′ q sk = s ′ k K1 K2 K 3

+1

2+1 +

1

2

4

Q

x

x −ζ0

4

Q

x

x −ζ

+1

2+1 −

1

2

4

Q

x −ζ

x0

4

Q

x −ζ

x

−1

2+1 +

1

20 −

4

Q

x −ζ

x0

−1

2+1 −

1

20 −

4

Q

x

x −ζ0

+1

2−1 +

1

20

4

Q

x

x −ζ0

+1

2−1 −

1

20

4

Q

x −ζ

x0

−1

2−1 +

1

2−

4

Q

x −ζ

x0 −

4

Q

x −ζ

x

−1

2−1 −

1

2−

4

Q

x

x −ζ0 −

4

Q

x

x −ζ

Complete DVCS Amplitudes in three Kinematics

Spin Directions in LF Helicity StatesC.Carlson and C.Ji, PRD67, 116002 (2003)

Full Amp vs. Reduced Amp

€

/ ε * ( ′ λ )( / k+ / q+ m)/ ε (λ)(k+ q)2 −m2

δ →0⏐ → ⏐ ⏐ / ε * ( ′ λ )q−γ+/ ε (λ)(x−ζ )P +q−

S-channel:

€

/ ε (λ)( / ′ k −/ q+ m)/ ε * ( ′ λ )( ′ k −q)2 −m2

δ →0⏐ → ⏐ ⏐ −/ ε (λ)q−γ+/ ε * ( ′ λ )x P +q−U-channel:

Full and Reduced DVCS Amplitudes in K2

€

λl = λ ′ l sk = s ′ k h ′ q F R

+1

2+

1

2+1 0 0

+1

2+

1

2−1

4

Q

x

x −ζ

4

Q

x

x −ζ

+1

2−

1

2+1 0 0

+1

2−

1

2−1

4

Q

x −ζ

x

4

Q

x −ζ

x

−1

2+

1

2+1 −

4

Q

x −ζ

x−

4

Q

x −ζ

x

−1

2+

1

2−1 0 0

−1

2−

1

2+1 −

4

Q

x

x −ζ−

4

Q

x

x −ζ

−1

2−

1

2−1 0 0

Investigation of Singularity in K1

€

) x

€

) z

€

q+ =δP + , r q⊥ =Q ) x , q− ≈

Q2

(ζ +δ)P +; ′ q + =(ζ +δ)P + , ′

r q⊥ =Q ) x, ′ q− ≈

Q2

(ζ +δ)P +

€

q+ =0 , r q⊥ =Q ) x , q− ≈

Q2

ζ P +; ′ q + =ζ P + , ′

r q⊥ =Q ) x, ′ q− ≈

Q2

ζ P +

Complete Amplitude in Tree Levelattaching Lepton Current

€

) x

€

) z

VCS at Tree Level in K1

€

k2 = ′ k 2 =m2 , q2 =−Q2ζζ +δ

+δζ m2

x(x−ζ ), ′ q 2 =0 , l 2 = ′ l 2 =0 ;

k=(xP + , m2

xP +, r0 ⊥ ) , ′ k =((x−ζ )P + , m2

(x−ζ )P +, r0 ⊥ ) ,

q=(δP + ,Q2

(ζ +δ)P ++

ζ m2

x(x−ζ )P +,Q ) x) ,

′ q =((ζ +δ)P +, Q2

(ζ +δ)P +,Q ) x) , l =(l +,Q

2

l +,Q ) x) ,

′ l =l −q , l + =x(x−ζ )(ζ +δ)Q2

x(x−ζ )Q2 +ζ(ζ +δ)m2P +.

Leptonic Polarized Amplitude in K1

€

λl = λ ′ l hq Amplitude

+1

2+1 −Q(1−

δ

4ζ+

2ζ

δ)

−1

2+1 Q(1−

3δ

4ζ−

2ζ

δ)

+1

20 −i2 2 Q

ζ

δ

−1

20 −i2 2 Q

ζ

δ

+1

2−1 −Q(1−

3δ

4ζ−

2ζ

δ)

−1

2−1 Q(1−

δ

4ζ+

2ζ

δ)

Hadronic Polarized Amplitude

+

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.

€

λl =λ ′ 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.

€

λl =λ ′ l =+12, sk =s ′ k =+

12, h ′ q =−1;

Prohibited !

Hadronic Polarized Amplitude in K1

€

hq h ′ q sk =s ′ k F R

+1 +1 + 12

2 xx−ζ

1+ ζδ

⎛

⎝ ⎜

⎞

⎠ ⎟ 2 x−ζ

x

+1 +1 −12

2 x−ζx

1+ ζδ

⎛

⎝ ⎜

⎞

⎠ ⎟ 2 x

x−ζ

+1 −1 + 12

−2 x−ζx

ζδ ⎛

⎝ ⎜

⎞

⎠ ⎟ 0

+1 −1 −12

−2 xx−ζ

ζδ ⎛

⎝ ⎜

⎞

⎠ ⎟ 0

0 +1 + 12

i 2 xx−ζ

1+ 2ζδ

− δ4ζ

⎛

⎝ ⎜

⎞

⎠ ⎟ i 2 x−ζ

x1+ δ

2ζ ⎛

⎝ ⎜

⎞

⎠ ⎟

0 +1 −12

i 2 x−ζx

1+ 2ζδ

− δ4ζ

⎛

⎝ ⎜

⎞

⎠ ⎟ i 2 x

x−ζ1+ δ

2ζ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

AH (−hq ,−h ′ q ,−sk ,−s ′ k ) =(−1)hq +h ′ q +sk +s ′ k +1 AH (hq ,h ′ q ,sk ,s ′ k )

Full and Reduced DVCS Amplitudes in K1 & K3

€

λl = λ ′ l sk = s ′ k h ′ q F R

+1

2+

1

2+1

4

Q

x

x −ζ0

+1

2+

1

2−1 0

4

Q

x

x −ζ

+1

2−

1

2+1

4

Q

x −ζ

x0

+1

2−

1

2−1 0

4

Q

x −ζ

x

−1

2+

1

2+1 0 −

4

Q

x −ζ

x

−1

2+

1

2−1 −

4

Q

x −ζ

x0

−1

2−

1

2+1 0 −

4

Q

x

x −ζ

−1

2−

1

2−1 −

4

Q

x

x −ζ0

Conclusions• Full VCS amplitude including the lepton current was

computed in the tree level and compared with the corresponding amplitude reduced as in the GPD approach.

• The two results (Full vs. Reduced) agree in K2, but neither in K1 nor in K3 although they appear same by swapping the real photon polarization: Reduced amplitude in K1 or K3 does not satisfy the angular momentum conservation.

• It is crucial to add the longitudinal polarization as well as the transverse ones for the finite result both in the

full and reduced amplitudes in K1 and K3. • Form Factor calculations are all right in any of the

three kinematics exhibiting the similar swap in the helicity amplitude between K2 and K1(or K3).