Operator Product Expansion in QCD in off-forward Kinematics · Motivation Heuristic discussion Techniques Results Applications Summary Operator Product Expansion in QCD in oﬀ-forward

Motivation Heuristic discussion Techniques Results Applications Summary

Operator Product Expansion in QCD in off-forward Kinematics

V. M. Braun

University of Regensburg

based on

VB, A. Manashov, Phys. Rev. Lett. 107 (2011) 202001

• VB, A. Manashov, JHEP 1201 (2012) 085

VB, A. Manashov, B. Pirnay, Phys. Rev. D 86 (2012) 014003

VB, A. Manashov, B. Pirnay, Phys. Rev. Lett. 109 (2012) 242001

CAQCD, 05/17/2013

V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 1 / 27


Hard exclusive processes involve off-forward matrix elements

DVCS: γ∗P → γP ′ Form factors: γ∗π → γ, B → ρℓνℓ, . . .

e−e−

Q2 q

P P’ B ρ

Operator Product Expansion

J(x)J(0) ∼∑

N

CN (x2, µ2) ON (µ2)

involves

〈P ′|ON (µ2)|P〉〈ρ(p)|ON (µ2)|0〉

Kinematic variables: hadron mass m2 momentum transfer t = (P − P ′)2

How to calculate effects ∼ m2/Q2 and t/Q2?



• Early work:

— Extension of Nachtmann’s approach to target mass corrections in DIS

— Spin-rotation (Wandzura-Wilczek)

Blümlein, Robaschik: NPB581 (2000) 449

Radyushkin, Weiss: PRD63 (2001) 114012

Belitsky, Müller: NPB589 (2000) 611. . .

— Results not gauge invariant

— Results not translation invariant

— A related discussion in a different context:

Ball, Braun: NPB543 (1999) 201

• A conceptual problem?



Contributions of different twist are intertwined by symmetries:

• Conservation of the electromagnetic current and translation invariance

∂µT{

jemµ (x)jem

ν (0)}

= 0

T{

jemµ (2x)jem

ν (0)}

= e−iP·x

T{

jemµ (x)jem

ν (−x)}

eiP·x

are valid in the sum of all twists but not for each twist separately

• Higher-twist contributions that restore gauge/translation invariance are due to

descendants of leading-twist operators obtained by adding total derivatives

T{

jemµ (x)jem

ν (0)}

=∑

N

aN ON

︸︷︷︸+∑

N

(bN∂

2ON + cN (∂O)N

)+ other operators

leading-twist

• These operators contribute to finite-t and target mass corrections

Task: Find the contributions to the OPE of all descendants

of leading-twist operators



Invisible operator?

• The problem:

S. Ferrara, A. F. Grillo, G. Parisi and R. Gatto, Phys. Lett. B38, 333 (1972):

— matrix elements of ∂µOµµ1...µNover free quarks vanish

? Go over to NLO

? More complicated quark-gluon matrix elements

? Off-shell OPE

— main difficulty is to disentangle the contribution of interest from “genuine” twist-4 operators

• Using EOM ∂µOµµ1...µNcan be expressed in terms of quark-gluon operators. e.g.:

Kolesnichenko ’84

∂µOµν = 2iqgFνµγµq ,

where

Oµν = (1/2)[qγµ

↔Dν q + (µ ↔ ν)]

• One cannot ignore qFq operators— Is it possible at all to define the separation between “kinematic” and “genuine” twist four?



Guidung principle: PRL 107 (2011) 202001

— “kinematic” approximation amounts to the assumption that genuine twist-four matrix

elements are zero

— for consistency, they must remain zero at all scales

— they must not reappear at higher scales due to mixing with “kinematic” operators

• “Kinematic” and “Dynamic” contributions must have autonomous

scale-dependence

Indeed, otherwise:

(〈qFq〉 + c〈(∂O)〉

)µ2

=

(αs(µ2)

αs(µ20)

)γ4/β0 (〈qFq〉 + c〈(∂O)〉

)µ20

=⇒

〈qFq〉µ2

=

(αs(µ

2)

αs(µ20)

)γ4/β0

〈qFq〉µ20 + c

[(αs(µ

2)

αs(µ20)

)γ4/β0

−

(αs(µ2)

αs(µ20)

)γ2/β0

]〈(∂O)〉µ2

0

[The “kinematic” approximation corresponds to taking into account all operators with the same

anomalous dimensions as the leading twist operators]



• Explicit diagonalization of the mixing matrix for twist-4 operators not feasible

• In a conformal theory(∂µ + β(α)∂α + H)Oj = γj Oj =⇒ 〈T{Oj1 (x)Oj2 (0)}〉 ∼ δj1j2

therefore

T{

j(x)j(0)}

=∑

N

CN (x, ∂)ON + . . . ,

C(x, ∂)ON = aN ON + bN∂2ON + cN (∂O)N + . . .

=⇒

〈T{

j(x)j(0)ON (y)}

〉 = CN (x, ∂)〈T{

ON (0)ON (y)}

〉 + 0

— might work in QCD, need NLO expressions in d = 4 − ǫ dimensions

• Orthogonality of eigenoperators suggests that H is a hermitian operator w.r.t. a certain

scalar product

Braun, Manashov, Rohrwild, Nucl. Phys. B807 (2009) 89; Nucl. Phys. B826 (2010) 235.



The SU(1, 1) scalar product

collinear conformal transformations SL(2,R) ⇔ SU(1, 1)

xµ = znµ , z ∈ R → z′ =az + b

cz + d, ⇔ z ∈ C → z′ =

az + b

bz + a

representations are labeled by conformal spin

ϕ(z) → T jϕ(z) =1

(bz + a)2jϕ

(az + b

bz + a

)

This is a unitary transformation with respect to the following scalar product:

〈φ, ψ〉j =2j − 1

π

∫

|z|<1

d2z (1 − |z|2)2j−2φ(z)ψ(z) ≡

∫

|z|<1

Djz φ(z)ψ(z) , ||φ||2 = 〈φ, φ〉

similar for several variables

〈φ, ψ〉j1 ,j2 =

∫

|z1|<1

Dj1 z1

∫

|z2|<1

Dj2 z2 φ(z1, z2)ψ(z1, z2)z



Some properties

• different powers are orthogonal

〈zn , zn′

〉 = δnn′ ||zn ||2 ,

• reproducing kernel (unit operator)

φ(z) =

∫

|w|<1

Djw Kj(z, w)φ(w) , Kj(z, w) =1

(1 − zw)2j

• Fourier

ρN

∫ ∫

|zi |<1

D1z1D1z2 (z1 − z2)N eip1z1+ip2z2 = iN (p1 + p2)N C3/2N

(p1 − p2

p1 + p2

)



Conformal operators vs. light-ray operators

• O+(z1, z2) = ψ+(z1n)ψ+(z2n)

ON =⟨

(z1 − z2)N ,O+(z1, z2)⟩

11= ρ−1

N(−∂+)N ψ+C

3/2N

(→D+ −

←D+

→D+ +

←D+

)ψ+

∂k+ON =

⟨ΦN,k(z1, z2),O+(z1, z2)

⟩11

the “wave functions”

ΦN (z1, z2) = (z1 − z2)N ΦN,k(z1, z2) = (S+12)k(z1 − z2)N

• The functions ΦNk(z1, z2) form an orthogonal basis

⟨ΦNk ,ΦN′k′

⟩= δNN′δkk′ ||ΦNk ||2

so that the light-ray operator can be expanded as

ψ+(z1n)ψ+(z2n) =

∞∑

N=0

∞∑

k=0

1

||ΦNk ||2ΦNk(z1, z2)∂k

+ON



Hermiticity

• Generators of infinitesimal SU(1, 1) transformations

S+ = z2∂z + 2jz , S0 = z∂z + j , S− = −∂z

S†0 = S0 (S+)† = −S− 〈φ, Sψ〉 = 〈S†φ, ψ〉

• RG equation(∂µ + β(αs)∂αs

)O+(z1, z2) = H O+(z1, z2)

O+(z1, z2) =∑

Nk

ΨNk(z1, z2)ONk =⇒ H ΨNk(z1, z2) = γN ΨNk(z1, z2)

〈φ, H ψ〉 = 〈H †φ, ψ〉



Conformal basis for twist-four non-quasipartonic operators

Braun, Manashov, Rohrwild: NPB826 (2010) 235.

Complete basis of collinear twist-4 operators in two-component spinor notation

Q1(z1, z2, z3) =ψ+(z1)f+−(z2)ψ+(z3) , T j=1 ⊗ T j=1 ⊗ T j=1

Q2(z1, z2, z3) =ψ+(z1)f++(z2)ψ−(z3) , T j=1 ⊗ T j=3/2 ⊗ T j=1/2

Q3(z1, z2, z3) =1

2[D−+ψ+](z1)f++(z2)ψ+(z3) , T j=3/2 ⊗ T j=3/2 ⊗ T j=1

and three similar operators with f → f

cf. in usual notation

qL(z1)[F+µ(z2) + iF+µ(z2)

]γµqL(z3) = Q2(z1, z2, z3) − Q1(z1, z2, z3)

−→Q (z1, z2, z3) =

(Q1(z1, z2, z3)Q2(z1, z2, z3)Q3(z1, z2, z3)

)−→Q (z1, z2, z3) =

(Q1(z1, z2, z3)Q2(z1, z2, z3)Q3(z1, z2, z3)

)

In addition, quasipartonic (four-particle) operators, irrelevant in present context



Renormalization group equations

Braun, Manashov, Rohrwild, Nucl. Phys. B826 (2010) 235.−→Q and

−→Q operators do not mix in one loop. Let

−→Q =

(Q1(z1, z2, z3)Q2(z1, z2, z3)Q3(z1, z2, z3)

)=∑

Np

~ΨNp(z1, z2, z3)QNp ,

(µ∂

∂µ+ β(αs)

∂

∂αs

)QNp = γNpQNp

~ΨNp can be found as solutions of

(H 11 H 12 H 13H 21 H 22 H 23H 31 H 32 H 33

)

Ψ

(1)Np

Ψ(2)Np

Ψ(3)Np

= γNp

Ψ

(1)Np

Ψ(2)Np

Ψ(3)Np

One can prove that H is a hermitian operator, H = H † , w.r.t. the scalar product

〈〈−→Φ ,

−→Ψ〉〉 = 2〈Φ1,Ψ1〉111 + 〈Φ2,Ψ2〉1 3

212

+1

2〈Φ3,Ψ3〉 3

232

1

Once ~ΨNp(zi) are known, the multiplicatively renormalizable operators are recovered as

QNp = 〈〈~ΨNp,−→Q 〉〉



Kinematic projection operators

Main result: Braun, Manashov: JHEP 1201 (2012) 085

2(∂O)N =igρN

(N + 1)2

[〈〈

−→Ψ N ,

−→Q 〉〉 − 〈〈

−→Ψ N ,

−→Q 〉〉

]+ . . .

where from:

ig−→Q (z1, z2, z3) =

∞∑

N=0

∞∑

k=0

pNk(N + 1)2

ρN ||ΨN ||2−→Ψ Nk(z1, z2, z3) ∂k

+(∂O)N + . . .

The ellipses stand for “dynamic” operators

ρN =1

2(N + 1)(N + 2)! pNk =

1

k!

Γ[2N + 4]

Γ[2N + 4 + k]



Coefficient functions:

Ψ(1)N

(wi) = 4aN

[∫ 1

0

dα α

∫ 1

0

dβ β (wα12 − w

β32)N−1 −

1

N +1

∫ 1

0

dααα (wα12 − w3)N−1

]

Ψ(2)N

(wi) = −4aN

∫ 1

0

dα α

∫ 1

0

dβ

(β +

1

N + 1α

)(wα

12 − wβ32)N−1

Ψ(3)N

(wi) = −24cN

∫ 1

0

dα α2α

∫ 1

0

dβ β (wα12 − w

β32)N−2

where

aN =1

8(N + 3)(N + 2)(N + 1)N,

bN = −

1

6(N + 3)(N + 2)N

cN =1

48

(N + 4)!

(N + 1)(N − 2)!

wα12 = w1α+ w2α, α = 1 − α, etc.



(Ψ(1)(zi)

Ψ(2)(zi)Ψ(3)(zi)

)

Nk

=

S(111)

+ 0 0

0 S(1 3

212

)

+ 0

0 0 S( 3

232

1)

+

k (Ψ(1)(zi)

Ψ(2)(zi)Ψ(3)(zi)

)

N

where

S(j1j2j3)+ = z2

1∂z1 + z22∂z1 + z2

3∂z3 + 2j1z1 + 2j2z2 + 2j3z3

||−→ΨN ||2 =

1

4||zN

12||211 (N + 2)2(N + 1)2

(1 −

2

(N + 1)(N + 2)+ 4

N+1∑

m=2

1

m

)

Compare: leading twist anomalous dimension

γN = CF

(1 −

2

(N + 1)(N + 2)+ 4

N+1∑

m=2

1

m

)



As a byproduct . . .

• Anomalous dimensions from conformal scalar product (no loops!)

||(∂O)N ||2 ≃ γN ||ON ||2

• LO anomalous dimensions of twist-four qFq operators are double degenerateIn particular

2(∂O)N =igρN

(N + 1)2

[〈〈

−→Ψ N ,

−→Q 〉〉 − 〈〈

−→Ψ N ,

−→Q 〉〉

]

2TN =igρN

(N + 1)2

[〈〈

−→Ψ N ,

−→Q 〉〉 − 〈〈

−→Ψ N ,

−→Q 〉〉

]

TN is a “genuine” twist-4 operator, has the same anomalous dimension as ON

Example: Shuryak, Vainshtein, NPB 199 (1982) 451

T1 = −6qLF+µγµγ5qL γT1

=4

3(Nc − 1/Nc)

Easy to check

γT1= CF

(1 −

2

(N + 1)(N + 2)+ 4

N+1∑

m=2

1

m

)∣∣∣N=1

• Lowest anomalous dimension in the spectrum of non-quasipartonic operators



OPE of two electromagnetic currents

Braun, Manashov: JHEP 1201 (2012) 085

T{

jµ(z1x)jµ(z2x)}

:

considerable simplifications compared to the general case∫ z1

z2

dw(w − z2)

(S

(1 32

12

)

+

)k

Ψ(2)N

(z1,w, z2) =(

S10+

)k

∫ z1

z2

dw(w − z2)Ψ(2)N

(z1,w, z2)

∼ ||~ΨN ||2(S10+ )k(z1 − z2)N+1

• Explicit expressions available in terms of light-ray operators

• Gauge/translation invariance checked to twist-4 accuracy



Kinematic power corrections to DVCS, γ∗(q) + N(p) −→ γ(q′) + N(p′)

Braun, Manashov, Pirnay: PRL109 (2012) 242001

e.g. average helicity-conserving amplitude

1

2(A++

q +A−−q ) =1

2mu(p′)u(p)Vq

1 +1

qq′u(p′)/q

′u(p)Vq

2 ,Vq1 =(

1 −t

2Q2

)E

q ⊗ C−0 +

t

Q2E

q ⊗ C−1

−2

Q2

(t

ξ+ 2|P⊥|2ξ2∂ξ

)ξ2∂ξE

q ⊗ C−2 +

8m2

Q2ξ2∂ξξX

q ⊗ C−2Vq

2 =(

1 −t

2Q2

)ξX

q ⊗ C−0 +

t

Q2ξX

q ⊗ C−1

−4

Q2

[(|P⊥|2ξ2∂ξ +

t

ξ

)ξ2∂ξ −

t

2

]ξX

q ⊗ C−2



where H , E and X = H + E are generalized parton distributions

F ⊗ C ≡

∫dx F(x, ξ, t) C (x, ξ) .

and the coefficient functions C±k

(x, ξ) are given by the following expressions:

C±0 (x, ξ) =

1

ξ + x − iǫ±

1

ξ − x − iǫ,

C±1 (x, ξ) =

1

x − ξln(ξ + x

2ξ− iǫ

)± (x ↔ −x) ,

C±2 (x, ξ) =

{1

ξ + x

[Li2

(ξ − x

2ξ+ iǫ

)− Li2(1)

]± (x ↔ −x)

}+

1

2C±1 (x, ξ) .

• Complete results available for all helicity amplitudes

• Factorization checked to 1/Q2 accuracy

• Gauge and translation invariance checked to 1/Q2 accuracy



Summary and outlook

A theoretical framework suggested for the treatment ofdescendants of leading twist operators in the OPE

• Done:

Rigorous separation of “kinematic” and “dynamic” contributions in higher-twist

Technique based on conformal symmetry; LO calculation completed

Gauge and translation invariance verified on operator level

Factorization proven in DVCS for kinematic contributions to LO accuracy

Detailed results for DVCS helicity amplitudes for the nucleon and scalar targets

• In progress:

Detailed results for DVCS cross sections and asymmetries for JLAB12 and COMPASSexperiments (in progress)

Other applications, e.g. γ∗ → η, η∗γ (BELLEII) and B-decays

• Topics to think about:

Resummation of all (t/Q2)k corrections?

Techniques based on conformal symmetry for NLO?



Supplementary slides



Some details:

• by straightforward calculation

(∂O)N =igρN

(N + 1)2

[〈zN

12,A(z1, z2)〉11 − 〈zN12, A(z1, z2)〉11

]

where

A = ∂z2 z212

[Q1(z1, z1, z2) +

∫ 1

0

du u Q2(z1, zu21, z2) + . . .

]z12 = z1 − z2

We want to rewrite this expression as a sum of terms

(∂O)N ∼ 〈ΨNi (w1,w2,w3),Qi(w1,w2,w3)〉j1 j2j3 ,

where j1, j2, j3 are the conformal spins of the Qi , so that the functions ΨNi (w1,w2,w3) can be

identified with the coefficient functions of the Q-operators

• This is done using the reproducing kernel:

φ(z) =

∫

|w|<1

Djw Kj(z, w)φ(w) , Kj(z, w) =1

(1 − zw)2j



• write:

Qi(z1, z2, z3) =⟨ 3∏

k=1

Kjk (zk , wk),Qi(w1,w2,w3)⟩

j1j2j3

Then, e.g. for the first term

(∂O)N ∋⟨

zN12, ∂z2 z2

12Q1(z1, z1, z2)⟩

11= 〈Ψ1(w1,w2,w3),Q1(w1,w2,w3)〉111

with

Ψ1(w1,w2,w3) =⟨

zN12, ∂z2 z2

12K1(z1, w1)K1(z1, w2)K1(z3, w3)⟩

11



• General remarks

The singularities at x = ±ξ in 1/Q2 corrections are weaker (logarithmic) as compared to

the leading term. Thus factorization is not endangered.

For scalar targets, target mass corrections only enter via |P⊥|2 ∼ |t − tmin|; naive

Nachtmann-type corrections are always overcompensated by finite-t effects.

For the nucleon, the main new effect is that Compton form factor H receives a sizable mass

correction proportional to E-distribution, and vice versa.



preliminary numerical analysis


Compton form factors

Aµν = −g⊥µν

2(Pq′)

[(u/q′u) H +

i

2m(uσµν

q′µ∆νu) E

]+ . . .

We consider the following ratios

ImF − ImFLO

ImFLO=

t

Q2cFt (ξ, t) +

m2

Q2cFm(ξ, t) ,

where F = {H, E}



numerical analysis (2)


0.0 0.1 0.2 0.3 0.4 0.5

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

ξ

cHt(ξ, t)

t =−1.5

t =−1.0

t =−0.5

t = 0.0

0.0 0.1 0.2 0.3 0.4 0.5-0.3

-0.2

-0.1

0.0

0.1

ξ

cHm(ξ, t)

t =−

1.5t =

−

1.0

t =−

0.5t =

0.0

0.0 0.1 0.2 0.3 0.4 0.5-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

ξ

cEt(ξ, t)

t =−

1.5t =−1.0

t =−0.5t = 0.0

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.5

1.0

ξ

cEm(ξ, t) t =

−

1.5

t =−

1.0

t =−

0.5

t =0.0


Documents

Operator Product Expansion in QCD in off-forward Kinematics · Motivation Heuristic discussion Techniques Results Applications Summary Operator Product Expansion in QCD in oﬀ-forward