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
Motivation Heuristic discussion Techniques Results Applications Summary
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?
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 2 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
• 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?
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 3 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 4 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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?
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 5 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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]
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 6 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
• 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.
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 7 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 8 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
)
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 9 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 10 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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〉
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 11 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 12 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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 〉〉
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 13 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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]
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 14 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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.
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 15 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
(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
)
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 16 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 17 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 18 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 19 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 20 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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?
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 21 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
Supplementary slides
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 22 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 23 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
• 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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 24 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
• 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.
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 25 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
preliminary numerical analysis
Braun, Manashov, Pirnay: PRL109 (2012) 242001
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}
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 26 / 27
Motivation Heuristic discussion Techniques Results Applications Summary
numerical analysis (2)
Braun, Manashov, Pirnay: PRL109 (2012) 242001
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
V. M. Braun (Regensburg) OPE in off-forward kinematics CAQCD, 05/17/2013 27 / 27
Top Related