Electroproduction of Nucleon Resonances in QCD I.V. Anikin JINR, - - PowerPoint PPT Presentation

Electroproduction of Nucleon Resonances in QCD I.V. Anikin

JINR, Dubna / University of Regensburg

based on

• Phys. Rev. D 88, 114021 (2013), Phys. Rev. D 92, 014018 (2015)

in collaboration with V.M. Braun and N. Offen

October 4, 2015

I.V. Anikin Electroproduction of Nucleon Resonances

Main results

• We derive light-cone sum rules for the electromagnetic

nucleon and N∗(1535) resonance form factors including the next-to-leading-order corrections for the contribution of twist-three and twist-four operators and a consistent treatment of the nucleon mass corrections.

• The soft contributions have been calculated in terms of small

transverse distance quantities using dispersion relations and quark-hadron duality.

I.V. Anikin Electroproduction of Nucleon Resonances

• The form factors have been expressed in terms of nucleon

wave functions at small transverse separations (DAs), without any additional parameters.

• For N∗(1535), we find that the form factors are dominated by

the twist-four distribution amplitudes that are related to the P-wave three-quark wave functions, i.e. to contributions of

• rbital angular momentum.
• The distribution amplitudes have been extracted from the

comparison with the experimental data on form factors and compared to the results of lattice QCD simulations. A self-consistent picture emerges, with the three valence quarks carrying 40% : 30% : 30% of the proton momentum.

I.V. Anikin Electroproduction of Nucleon Resonances

Essential stages

The work consists of three parts:

• Calculations within LCSR;
• Factorization of amplitude at LO up to twist-6 and at NLO up

to twist-4. We calculate 22 coefficient functions at NLO and 20 of them are new ones. To avoid the mixture with the so-called evanescent operators, we use the renormalization procedure for operators with open Dirac indices;

• Derivation of distribution amplitudes. In particular, light-cone

expansion to the twist-4 accuracy of the three-quark matrix elements with generic quark positions

I.V. Anikin Electroproduction of Nucleon Resonances

LCSRs for form factors: General structure

The LCSR approach allows one to calculate the form factors in terms of the nucleon (proton) DAs. To this end we consider the correlation function Tν(P, q) = i

• d4x eiqx0|T [η(0)jem

ν (x)] |P

where T denotes time-ordering and η(0) is the Ioffe interpolating current η(x) = ǫijk ui(x)Cγµuj(x)

• γ5γµdk(x) ,

0|η(0)|P = λ1mNN(P) .

I.V. Anikin Electroproduction of Nucleon Resonances

The matrix element of the electromagnetic current jem

µ (x) = eu ¯

u(x)γµu(x) + ed ¯ d(x)γµd(x) taken between nucleon states is conventionally written in terms of the Dirac and Pauli form factors F1(Q2) and F2(Q2): P′|jem

µ (0)|P = ¯

N(P′)

• γµF1(Q2) − i σµνqν

2mN F2(Q2)

• N(P).

In terms of the electric GE(Q2) and magnetic GM(Q2) Sachs form factors, we have GM(Q2) = F1(Q2) + F2(Q2), GE(Q2) = F1(Q2) − Q2 4m2

N

F2(Q2).

I.V. Anikin Electroproduction of Nucleon Resonances

The matrix element of the electromagnetic current jem

ν

between spin-1/2 states of opposite parity can be parametrized in terms of two independent form factors, which can be chosen as N∗(P′)|jem

ν |N(P) = ¯

uN∗(P′)γ5ΓνuN(P) , Γν = G1(q2) m2

N

(ˆ qqν − q2γν) − i G2(q2) mN σνρqρ , where q = P′ − P is the momentum transfer and Q2 = −q2.

I.V. Anikin Electroproduction of Nucleon Resonances

The helicity amplitudes A1/2(Q2) and S1/2(Q2) for the electroproduction of N∗(1535) can be expressed in terms of the form factors [Aznauryan ’08]: A1/2 = e B

• Q2G1(Q2) + mN(mN∗ − mN)G2(Q2)
• ,

S1/2 = eBC

√ 2

• (mN −mN∗)G1(Q2) + mNG2(Q2)
• .

Here e = √ 4πα is the elementary charge and B, C are kinematic factors defined as B =

• Q2 + (mN∗ + mN)2

2m5

N(m2 N∗ − m2 N) ,

C =

• 1 + (Q2 − m2

N∗ + m2 N)2

4Q2m2

N∗

.

I.V. Anikin Electroproduction of Nucleon Resonances

Light-Cone Basis

We define a light-like vector nµ by the condition q · n = 0 , n2 = 0 , qµ = P.q P.nnµ + q⊥

µ ,

q2 = q2

⊥ = −Q2 .

and introduce the second light-like vector as pµ = Pµ − 1 2 nµ m2

N

P · n , p2 = 0 , and g⊥

µν = gµν − 1

pn(pµnν + pνnµ)

I.V. Anikin Electroproduction of Nucleon Resonances

Light-Cone Sum Rules

We consider the “plus” spinor projection of the correlation function involving the “plus” component of the electromagnetic current, which can be parametrized in terms of two invariant functions Λ+T+ = p+

• mNA(Q2, P′2) + ˆ

q⊥B(Q2, P′2)

• N+(P) ,

where Q2 = −q2 and P′2 = (P − q)2 and N±(P) = Λ±N(P), Λ+ = ˆ p ˆ n 2pn , Λ− = ˆ n ˆ p 2pn

I.V. Anikin Electroproduction of Nucleon Resonances

Making the Borel transformation 1 s − P′2 − → e−s/M2, we derive the sum rules (for nucleon) 2λ1F1(Q2) = 1 π s0 ds e(m2

N−s)/M2Im AQCD(Q2, s) ,

λ1F2(Q2) = 1 π s0 ds e(m2

N−s)/M2Im BQCD(Q2, s) .

and (for N∗(1535)) 2λN

1 Q2G1(Q2)

mNmN∗ = 1 π s0 ds e(m2

N−s)/M2Im AQCD(Q2, s) ,

−2λN

1 G2(Q2)

= 1 π s0 ds e(m2

N−s)/M2Im BQCD(Q2, s) . I.V. Anikin Electroproduction of Nucleon Resonances

The correlation functions A(Q2, P′2) and B(Q2, P′2) can be written as a sum: A = ed Ad + euAu , B = ed Bd + euBu . Each of the functions has a perturbative expansion which we write as A = ALO + αs(µ) 3π ANLO + . . . and similar for B; µ is the renormalization scale.

I.V. Anikin Electroproduction of Nucleon Resonances

For consistency with our NLO calculation we rewrite LO results in a different form, expanding all kinematic factors in powers of m2

N/Q2: We keep all corrections O(m2 N/Q2) but neglect terms

O(m4

N/Q4) etc. which is consistent with taking into account

contributions of twist-three, -four, -five (and, partially, twist-six) in the OPE.

I.V. Anikin Electroproduction of Nucleon Resonances

The following Feynman diagrams contribute to the NLO amplitude.

Figure: NLO corrections to the light-cone sum rule for baryon form factors.

I.V. Anikin Electroproduction of Nucleon Resonances

The NLO corrections read [IVA, V.Braun, N.Offen’13]. Q2ANLO

q

= =

• [dxi]

k=1,3

• Vk(xi)C Vk

q (xi, W ) + Ak(xi)C Ak q (xi, W )

• +
• m=1,2,3
• V(m)

2

(xi)C V(m)

2

q

(xi, W ) + A(m)

2

(xi)C A(m)

2

q

(xi, W )

• + O(twist-5) ,

where W = Q2 + P′ 2 Q2 .

I.V. Anikin Electroproduction of Nucleon Resonances

Q2BNLO

q

= =

• [dxi]
• V1(xi)DV1

q (xi, W ) + A1(xi)DA1 q (xi, W )

• + O(twist-5).

It turns out that C V(1)

2

d (xi, W ) = C A(1)

2

d (xi, W ) = 0.

I.V. Anikin Electroproduction of Nucleon Resonances

x2C V1

d (xi) =

2x2x3

• 3(L − 2)g1(x3) + 2(L − 1)g11(x3, x3) + g21(x3, x3)
• +
• 2x2 + (4L − 3)x3
• h11(x3) + (3 − 4L)¯

x1h11(¯ x1) + 2x3h21(x3) − 2¯ x1h21(¯ x1) − 2

• 3(x2/x3)(2L−3) + 5L − 7
• h12(x3)

+2(5L−7)h12(¯ x1) −

• 6(x2/x3) + 5
• h22(x3)

+ 5h22(¯ x1) + (6/x3)(L − 2)h13(x3) − (6/¯ x1)(L − 2)h13(¯ x1) +(3/x3)h23(x3) − (3/¯ x1)h23(¯ x1) ,

I.V. Anikin Electroproduction of Nucleon Resonances

x2C A1

d (xi) =

3¯ x1h11(¯ x1) − 3x3h11(x3) + 2(3L − 10)h12(¯ x1) −2(3L − 10)h12(x3) + 3h22(¯ x1) − 3h22(x3) −(6/¯ x1)(L − 3)h13(¯ x1) + (6/x3)(L − 3)h13(x3) −(3/¯ x1)h23(¯ x1) + (3/x3)h23(x3) , where gnk(y, x; W ) = lnn[1 − yW − iη] (−1 + xW + iη)k , hnk(x; W ) = lnn[1 − xW − iη] (W + iη)k with n = 0, 1, 2 and k = 1, 2, 3. For n = 0 the first argument becomes dummy,i.e gk(x; W ) ≡ g0k(∗, x; W ) ,

I.V. Anikin Electroproduction of Nucleon Resonances

Results

Discussion of parameters Schematically, the general structure of form factors has the following form: F = Ftw-4 + fN λ1 Ftw-3

fN

+

• i=0,1

η1iFtw-4

η1i

+ fN λ1

2

• i=1

2

• j=0;j≤i

ϕijFtw-3

ϕij

. Or, in other words, we have

• tw-3:
• ϕ10, ϕ11, ϕ20, ϕ21, ϕ22
• , fN;
• tw-4:
• η10, η11
• , λ1;

I.V. Anikin Electroproduction of Nucleon Resonances

The other parameters that enter LCSRs are

• the interval of duality (continuum threshold) s0

(s0 = 2.25 GeV2);

• Borel parameter M2 (M2 = 1.5 GeV2 and M2 = 2 GeV2 and

M2 ≃ s0) ;

• factorization/renormalization scale µ2 (µ2 = 2 GeV2 and

µ2 ∼ (1 − x)Q2 − xP′ 2 or µ2 ≤ (1 − x0)Q2 + x0M2 ≤ 2s0Q2

s0+Q2 < 2s0 ).

• We use a two-loop expression for the QCD coupling with

Λ(4)

QCD = 326 MeV resulting in the value αs(2 GeV2) = 0.374.

I.V. Anikin Electroproduction of Nucleon Resonances

Model Method fN/λ1 ϕ10 ϕ11 ϕ20 ϕ21 ϕ22 ABO1 LCSR (NLO) −0.17 0.05 0.05 0.075 −0.027 0.17 ABO2 LCSR (NLO) −0.17 0.05 0.05 0.038 −0.018 −0.13 BLW LCSR (LO) −0.17 0.0534 0.0664

• BK

pQCD

• 0.0357

0.0357

• COZ

QCDSR (LO)

• 0.163

0.194 0.41 0.06 −0.163 KS QCDSR (LO)

• 0.144

0.169 0.56 −0.01 −0.163 QCDSR (NLO) −0.15

• BS(HET)

QCDSR(LO)

• 0.152

0.205 0.65 −0.27 0.020 LAT09 LATTICE −0.083 0.043 0.041 0.038 −0.14 −0.47 LAT13 LATTICE −0.075 0.038 0.039 −0.050 −0.19 −0.19 Model Method η10 η11 ABO1 LCSR (NLO) −0.039 0.140 ABO2 LCSR (NLO) −0.027 0.092 BLW LCSR (LO) 0.05 0.0325 BK pQCD

• COZ

QCDSR (LO)

• KS

QCDSR (LO)

• QCDSR (NLO)
• LAT09

LATTICE

• LAT13

LATTICE

• I.V. Anikin

Electroproduction of Nucleon Resonances

• 2

4 6 8 10 12 0.2 0.4 0.6 0.8 1.0 1.2

Q2 Gp

M(Q2)/µpGD(Q2)

Figure: Nucleon electromagnetic form factors from LCSRs compared to the experimental data [CLAS Coll.,

Jeff.Lab. Hall A Coll.]. Parameters of the nucleon DAs correspond to the sets ABO1 and ABO2 in Table for the solid and dashed curves, respectively. Borel parameter M2 = 1.5 GeV2 for ABO1 and M2 = 2 GeV2 for ABO2. I.V. Anikin Electroproduction of Nucleon Resonances

• 2

4 6 8 10 12 0.2 0.4 0.6 0.8 1.0 1.2

Q2 Gn

M(Q2)/µnGD(Q2)

Figure: Nucleon electromagnetic form factors from LCSRs compared to the experimental data [CLAS Coll.,

Jeff.Lab. Hall A Coll.]. Parameters of the nucleon DAs correspond to the sets ABO1 and ABO2 in Table for the solid and dashed curves, respectively. Borel parameter M2 = 1.5 GeV2 for ABO1 and M2 = 2 GeV2 for ABO2. I.V. Anikin Electroproduction of Nucleon Resonances

• 2

4 6 8 10 12 0.5 0.0 0.5 1.0 1.5

Q2 µpGp

E(Q2)/Gp M(Q2)

Figure: Nucleon electromagnetic form factors from LCSRs compared to the experimental data [CLAS Coll.,

Jeff.Lab. Hall A Coll.]. Parameters of the nucleon DAs correspond to the sets ABO1 and ABO2 in Table for the solid and dashed curves, respectively. Borel parameter M2 = 1.5 GeV2 for ABO1 and M2 = 2 GeV2 for ABO2. I.V. Anikin Electroproduction of Nucleon Resonances

• 1

2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Q2 µnGn

E(Q2)/Gn M(Q2)

Figure: Nucleon electromagnetic form factors from LCSRs compared to the experimental data [CLAS Coll.,

Jeff.Lab. Hall A Coll.]. Parameters of the nucleon DAs correspond to the sets ABO1 and ABO2 in Table for the solid and dashed curves, respectively. Borel parameter M2 = 1.5 GeV2 for ABO1 and M2 = 2 GeV2 for ABO2. I.V. Anikin Electroproduction of Nucleon Resonances

• 2

4 6 8 10 12 1 2 3 4

Q2 Q2F p

2 (Q2)/F p 1 (Q2)

Figure: The ratio of Pauli and Dirac electromagnetic proton form factors from LCSRs compared to the

experimental data [Jeff.Lab. Hall A Coll.]. Parameters of the nucleon DAs correspond to the sets ABO1 and ABO2 in Table for the solid and dashed curves, respectively. Borel parameter M2 = 1.5 GeV2 for ABO1 and M2 = 2 GeV2 for ABO2. I.V. Anikin Electroproduction of Nucleon Resonances

• 1

2 3 4 0.0 0.2 0.4 0.6 0.8 1.0

Q2 Q2F u

1 (Q2)

• 1

2 3 4 0.0 0.1 0.2 0.3 0.4 0.5

Q2 Q2F d

1 (Q2)

• 1

2 3 4 0.0 0.1 0.2 0.3 0.4 0.5

Q2 Q2F u

2 (Q2)

• 1

2 3 4 0.5 0.4 0.3 0.2 0.1 0.0

Q2 Q2F d

2 (Q2)

Figure: The corresponding leading-order results are shown by the dash-dotted curves for comparison.

Parameters of the nucleon DAs correspond to the set ABO1 in the table. I.V. Anikin Electroproduction of Nucleon Resonances

Case of N∗(1535)

The dependence of the form factors on these parameters is linear. The results can be presented as G1(Q2) = λN∗

1

λN

1

• g00

1 (Q2) + g10 1 (Q2)η10 + g11 1 (Q2)η11

+ fN∗ λN∗

1

• f 00

1 (Q2) + f 10 1 (Q2)ϕ10 + f 11 1 (Q2)ϕ11 + . . .

• and similarly

G2(Q2) = λN∗

1

λN

1

• g00

2 (Q2) + g10 2 (Q2)η10 + g11 2 (Q2)η11

+ fN∗ λN∗

1

• f 00

2 (Q2) + f 10 2 (Q2)ϕ10 + f 11 2 (Q2)ϕ11 + . . .

• where the ellipses stand for the contributions of ϕ20, ϕ21, ϕ22.

I.V. Anikin Electroproduction of Nucleon Resonances

As an illustration, the NLO LCSR result for the form factors at Q2 = 2 GeV2 normalized to the dipole formula D(Q2) = 1 (1 + Q2/a)2 , a = 0.71 GeV2

I.V. Anikin Electroproduction of Nucleon Resonances

can be written as follows: G NLO

1

(Q2) D(Q2) = λN∗

1

λN

1

• 0.666 − 2.18η10 + 0.86η11

− 0.69˜ fN⋆ − 1.76˜ fN⋆ϕ10 + 1.05˜ fN⋆ϕ11 + 1.3˜ fN⋆ϕ20 + 0.66˜ fN⋆ϕ21 − 0.06˜ fN⋆ϕ22

• ,

G LO

1

(Q2) D(Q2) = λN∗

1

λN

1

• 0.816 − 2.02η10 + 0.88η11

− 0.59˜ fN⋆ − 1.60˜ fN⋆ϕ10 + 1.19˜ fN⋆ϕ11 + 1.26˜ fN⋆ϕ20 + 0.70˜ fN⋆ϕ21 + 0.12˜ fN⋆ϕ22

• ,

where we use a notation ˜ fN⋆ for the ratio of twist-three and twist-four couplings ˜ fN⋆ = fN⋆ λN⋆

1

= 0.027(2)

[Braun et al’14] .

I.V. Anikin Electroproduction of Nucleon Resonances

For comparison, the similar decomposition of the form factors for the LO LCSRs for the same value Q2 = 2 GeV2 reads G NLO

2

(Q2) D(Q2) = λN∗

1

λN

1

• − 0.466 + 1.84η10 + 0.06η11

− 0.82˜ fN⋆ − 1.06˜ fN⋆ϕ10 − 1.08˜ fN⋆ϕ11 + 2.6˜ fN⋆ϕ20 + 1.5˜ fN⋆ϕ21 + 0.39˜ fN⋆ϕ22

• and

G LO

2

(Q2) D(Q2) = λN∗

1

λN

1

• − 0.466 + 1.84η10 + 0.06η11

− 1.19˜ fN⋆ − 0.78˜ fN⋆ϕ10 + 3.82˜ fN⋆ϕ11 + 2.9˜ fN⋆ϕ20 + 1.6˜ fN⋆ϕ21 + 0.28˜ fN⋆ϕ22

• so that the NLO corrections are significant.

I.V. Anikin Electroproduction of Nucleon Resonances

Method λN∗

1

/λN

1

fN∗ /λN∗

1

ϕ10 ϕ11 ϕ20 ϕ21 ϕ22 η10 η11 LCSR (1) 0.633 0.027 0.36

• 0.95

0.00 0.94 LATTICE 0.633(43) 0.027(2) 0.28(12)

• 0.86(10)

1.7(14)

• 2.0(18)

1.7(26)

• Table: Parameters of the N∗1535 distribution amplitudes at the scale µ2 = 2 GeV2. For the lattice results

[Braun et al’14] only statistical errors are shown. I.V. Anikin Electroproduction of Nucleon Resonances

• 2

4 6 8 10 12 0.05 0.00 0.05 0.10 0.15

Q2 A12(Q2) S12(Q2)

• 2

4 6 8 10 12 0.5 0.0 0.5 1.0

Q2 G1(Q2)/D(Q2) G2(Q2)/D(Q2)

Figure: A12 and S12 for N∗(1535) (up panel) and the form factors G1(Q2), G2(Q2), normalized to the dipole

formula (down panel). Experimental data – from [Denizli’07] (empty squares) [Dalton’08] (filled squares) [Armstrong’98] (filled circles) and [Aznauryan’09] (triangles). I.V. Anikin Electroproduction of Nucleon Resonances

Summary

Our calculations incorporate the following new elements:

• Next-to-leading order QCD corrections to the contributions of

twist-three and twist-four DAs;

• Exact account of “kinematic” contributions to the nucleon

DAs of twist-four and twist-five induced by lower geometric twist operators (Wandzura-Wilczek terms);

• Light-cone expansion to the twist-four accuracy of the

three-quark matrix elements with generic quark positions;

• A new calculation of twist-five off-light cone contributions;
• A more general model for the leading-twist DA, including

contributions of second-order polynomials.

I.V. Anikin Electroproduction of Nucleon Resonances

Additional Slides: on Light-Cone Sum Rules

The “plus” component of the electromagnetic current, which can be parametrized in terms of two invariant functions Λ+T+ = p+

• mNA(Q2, P′2) + ˆ

q⊥B(Q2, P′2)

• N+(P) ,

where Q2 = −q2 and P′2 = (P − q)2. The correlation functions A(Q2, P′2) and B(Q2, P′2) can be calculated in QCD for sufficiently large Euclidean momenta Q2, −P′2 1 GeV2 using OPE. The results can be presented in the form of a dispersion relation AQCD(Q2, P′2) = 1 π ∞ ds s − P′2 Im AQCD(Q2, s) + . . . BQCD(Q2, P′2) = 1 π ∞ ds s − P′2 Im BQCD(Q2, s) + . . . where the ellipses stand for necessary subtractions.

I.V. Anikin Electroproduction of Nucleon Resonances

On the other hand, the same correlation functions can be written in terms of physical spectral densities that contain a nucleon (proton) pole at P′2 → m2

N, the nucleon resonances and the continuum.

The nucleon contribution is proportional to the e.m. form factor, whereas the contribution of higher mass states can be taken into account using quark-hadron duality: Aphys(Q2, P′2) = 2λ1F1(Q2) m2

N − P′2 + 1

π ∞

s0

ds s − P′2 Im AQCD(Q2, s) + .. Bphys(Q2, P′2) = λ1F2(Q2) m2

N − P′2 + 1

π ∞

s0

ds s − P′2 Im BQCD(Q2, s) + .. where s0 ≃ (1.5 GeV)2 is the interval of duality (also called continuum threshold).

I.V. Anikin Electroproduction of Nucleon Resonances

Matching the two above representations and making the Borel transformation that eliminates subtraction constants 1 s − P′2 − → e−s/M2

• ne obtains the sum rules

2λ1F1(Q2) = 1 π s0 ds e(m2

N−s)/M2Im AQCD(Q2, s) ,

λ1F2(Q2) = 1 π s0 ds e(m2

N−s)/M2Im BQCD(Q2, s) . I.V. Anikin Electroproduction of Nucleon Resonances

On Renormalization

By the standard way, in the d-dimensional space, the Ioffe current gets mixed under renormalization with the operators like η(n)

I

(0) = [ψT(0) CΓ (n)

α1...αn ψ(0)] γ5Γ (n) α1...αnψ(0) ,

where Γ (n)

α1...αn is the antisymmetric product of γ-matrices. So,

within the framework of the dimensional regularization, the renormalized operators take the form: (η(n)

I

)R =

• k

Znk η(k)

I

. here, we also have to include the mixture of the so-called evanescent operators with the physical operators. As was shown, such a mixture can be eliminated by the finite renormalization and this finite regularization has to be taken into account in the anomalous dimension matrix.

I.V. Anikin Electroproduction of Nucleon Resonances

To avoid the mixture of the evanescent operators with the physical

• nes, we will work with the open Dirac indices:

ηI δ(0) = (Cγα)αβ ⊗ (γ5γα)δγ [ψT

α (0) ψβ(0)] ψγ(0)

and the local tree quark operator with open spinor indices is renormalized as

• [ψT

α (0) ψβ(0)] ψγ(0)

• R

= Zαα′,ββ′,γγ′ [ψT

α′(0) ψβ′(0)] ψγ′(0) ,

Zαα′,ββ′,γγ′ =

• nmk

anmk(ǫ) (Γnmk)αα′,ββ′,γγ′ , where anmk(ǫ) =

∞

• p=0

(a(p))nmk ǫp , (Γnmk)αα′,ββ′,γγ′ = γ(n)

αα′ ⊗ γ(m) ββ′ ⊗ γ(k) γγ′

and the bare coupling constant is defined α0 = µ−2ǫZα(µ2)αS(µ2) with Zα(µ2) = 1 − αS(µ2) 4π ǫ β0 .

I.V. Anikin Electroproduction of Nucleon Resonances

Cancellation of Singularities

Up to αS-order, we have Z−1 nµMµ Z−1

F

=

• 1 − αS(µ2

1)

4π C1 ǫ

• MLO + αS(µ2)

4π MNLO ×

• 1 − αS(µ2

F)

4π ǫ H

• =

MLO + αS(µ2) 4π MNLO

• fin. + αS(µ2)

4πǫ MNLO

• sing. −

αS(µ2

1)

4πǫ C1MLO − αS(µ2

F)

4πǫ H ∗ MLO ,

I.V. Anikin Electroproduction of Nucleon Resonances

The singular terms have been cancelled each others, i.e. MNLO

• sing. −

µ2 µ2

1

ǫ C1MLO − µ2 µ2

F

ǫ H ∗ MLO = 0 .

I.V. Anikin Electroproduction of Nucleon Resonances

Calculations of coefficient functions

In the most general form, the invariant amplitude reads Aµ = P(η)

αβ,δγ Z−1 αα′,ββ′,γγ′

• [dxi] Mµ

α′α1,β′α2,γ′α3(xi; q, P) ×

(Z−1

F )α1α′

1,α2α′ 2,α3α′ 3 F(3q) R

α′

1α′ 2α′ 3(xi) ,

where the soft part of amplitude is defined with the open Dirac indices, i.e. F(3q)

αβ,δ(xi) F

= 0|ψα(z1)ψβ(z2)ψδ(z3)|B(P) , and the Ioffe current projection is defined by P(η)

αβ,γδ def.

= (Cγα)αβ(γ5γα)γδ .

I.V. Anikin Electroproduction of Nucleon Resonances

LO calculations.

The LO amplitude, related to d-quark contribution, takes the form: A(LO)

µ

= P(η)

αβ,δγ Z−1 αα′,ββ′,γγ′ ×

• [dxi] (x1P + q)α

(x1P + q)2

• Iα′α1 ⊗ Iβ′α2 ⊗ (gαµI + Γ(2)

αµ)γ′α3

• ×

(Z−1

F )α1α′

1,α2α′ 2,α3α′ 3 F(3q) R

α′

1α′ 2α′ 3(xi) . I.V. Anikin Electroproduction of Nucleon Resonances

NLO calculations.

The general structure of NLO diagrams can be presented as n · A(NLO) = αs(µ2) 4π P(η)

αβ,δγ Z−1 αα′,ββ′,γγ′ ×

• [dxi]
• nmk

Γ(n)

α′α1 ⊗ Γ(m) β′α2 ⊗ Γ(k) γ′α3 3

• p=−1

ǫpb(p)

nmk(xi; P′ 2, Q2) ×

(Z−1

F )α1α′

1,α2α′ 2,α3α′ 3 F(3q) R

α′

1α′ 2α′ 3(xi) . I.V. Anikin Electroproduction of Nucleon Resonances

NLO, Exchange type, d-contribution

We derive the exchange diagram contribution: nµA(NLO)

µ

(Exch.) = lim

ǫ→0

• [dxi] F(3q) R(xi)
• nmk

Γ(n) ⊗ Γ(m) ⊗ Γ(k) × x1 x2 f nmk(ǫ)

• [Q2]ǫ(b(0)

nmk + n

• p=1

ǫp b(p)

nmk) +

[Q2 − (Q2 + P′ 2)x1]ǫ(−b(0)

nmk + n

• p=1

ǫp ¯ b(p)

nmk)

• −

x12 x2 f nmk(ǫ)

• [Q2]ǫ(c(0)

nmk + n

• p=1

ǫp c(p)

nmk) +

[Q2 − (Q2 + P′ 2)x12]ǫ(−c(0)

nmk + n

• p=1

ǫp ¯ c(p)

nmk)

• ,

I.V. Anikin Electroproduction of Nucleon Resonances

where x12....n =

n

• i=1

xi , f nmk(ǫ) = a(−2)

nmk

ǫ2 + a(−1)

nmk

ǫ + ... .

I.V. Anikin Electroproduction of Nucleon Resonances

Having subtracted the singular part, the finite amplitude takes the following form: nµA(NLO)

µ

(Exch.) =

• [dxi] F(3q) R(xi)
• nmk

Γ(n) ⊗ Γ(m) ⊗ Γ(k) × x1 x2 Knmk(Q2, P′ 2; xi) − x12 x2 Lnmk(Q2, P′ 2; xi)

• ,

P.S. Here, we keep in mind that ln2 Q2 µ2 − ln2 Q′ 2 µ2 = − ln2 Q′ 2 Q2 − 2 ln Q′ 2 Q2 ln Q2 µ2 , Q′ 2 = Q2 − (P′ 2 + Q2)xij ≡ Q2(1 − Wxij)

I.V. Anikin Electroproduction of Nucleon Resonances

where Knmk(Q2, P′ 2; xi) = a(−1)

nmk

• b(1)

nmk + ¯

b(1)

nmk + b(0) nmk ln

Q2 Q2¯ x1 − P′ 2x1

• +

a(−2)

nmk

• b(2)

nmk + ¯

b(2)

nmk + b(0) nmk

1 2 ln2 Q2 − 1 2 ln2(Q2¯ x1 − P′ 2x1)

• +b(1)

nmk ln Q2 + ¯

b(1)

nmk ln(Q2¯

x1 − P′ 2x1)

• ,

and Lnmk(Q2, P′ 2; xi) = a(−1)

nmk

• c(1)

nmk + ¯

c(1)

nmk + c(0) nmk ln

Q2 Q2¯ x12 − P′ 2x12

• +

a(−2)

nmk

• c(2)

nmk + ¯

c(2)

nmk + c(0) nmk

1 2 ln2 Q2 − 1 2 ln2(Q2¯ x12 − P′ 2x12)

• +c(1)

nmk ln Q2 + ¯

c(1)

nmk ln(Q2¯

x12 − P′ 2x12)

• .

I.V. Anikin Electroproduction of Nucleon Resonances

On tw-three and tw-four contributions

Contributions of the leading-twist DA ϕN(xi) = V1(xi) − A1(x1) correspond to contributions of local (geometric) twist-three

• perators in the OPE of the product T[η(0)jµ(x)] for x2 → 0:
• Dk1

+ u+)(0)

• Dk2

+ u+)(0)

• Dk3

+ d+)(0) .

Equivalently, the leading-twist contributions can be attributed to the single light-ray operator u+(a1n)u+(a2n)d+(a3n) ,

I.V. Anikin Electroproduction of Nucleon Resonances

A twist-four operator can be constructed in two different ways: either changing the “plus” projection of one of the quark fields to the “minus”, or adding a transverse derivative, e.g.

• Dk1

+ u−)(0)

• Dk2

+ u+)(0)

• Dk3

+ d+)(0) ,

• Dk1

+ D⊥u+)(0)

• Dk2

+ u+)(0)

• Dk3

+ d+)(0)

Contributions of the first type correspond to the nonlocal light-ray

• perators u−(a1n)u+(a2n)d+(a3n) and u+(a1n)u+(a2n)d−(a3n).

The contributions of operators involving a transverse derivative are more complicated and can be obtained from the light-cone expansion of the nonlocal three-quark operator u+(y1)u+(y2)d+(y3) , yi = ain + bi,⊥ where b⊥ → 0 is an auxiliary transverse vector. The twist-four contribution

I.V. Anikin Electroproduction of Nucleon Resonances

As an example, consider the contribution of the twist-four DA V(2)

2 (xi) to the exchange diagram. We have

P(η)

• d4y2d4y3M+(y2, y3; q, p1) V(2)

2 (βi) ( ˆ

PC)ˆ y2γ5N(P) = P(η)

• d4y2d4y3
• (d4p2)(d4p3) e+ip2y2+ip3y3 M+(p2, p3; q, p1) ×

V(2)

2 (βi) ( ˆ

PC)[(P · y2)ˆ n + ˆ yT

2 ]γ5N(P) ,

where βi = P · yi.

I.V. Anikin Electroproduction of Nucleon Resonances

The longitudinal component contributes as P(η)

• d4y2d4y3
• (d4p2)(d4p3)e+ip2y2+ip3y3M+(p2, p3; q, p1) ×
• [dxi]e−ix2P·y2−ix3P·y3V(2)

2 (xi) ( ˆ

PC)[β2ˆ n]γ5N(P) = P(η)

• [dxi]
• (−i) d

dx2 V(2)

2 (xi)

• M+(xi) ( ˆ

PC)ˆ nγ5N(P) .

I.V. Anikin Electroproduction of Nucleon Resonances

The transverse component contributes as P(η)

• [dxi]V(2)

2 (xi)

• i d

dp⊥

2

M+(pi)

• pi=xip ( ˆ

PC)γ⊥γ5N(P) . Moreover, owing to the Ward identity ∂ ∂p⊥ ˆ p + ˆ ℓ (p + ℓ)2+ iǫ = − ˆ p + ˆ ℓ (p + ℓ)2+ iǫγ⊥ ˆ p + ˆ ℓ (p + ℓ)2 +iǫ a derivative is equivalent to the insertion of γ⊥-matrix in the quark line.

I.V. Anikin Electroproduction of Nucleon Resonances

On Distribution Amplitudes

We remind kinematics: P = p + M2 2 n, zi = αip + βin + zT

i ,

βi = P.zi . Assuming αi = 0, βi = 0, zT

i

= 0, the most general parametrization of the matrix element takes the form: 0|[ψ(z1)ψ(z2)]ψ(z3)|P = V1( ˆ PC)γ5N(P) + M

• i

V(i)

2 ( ˆ

PC)ˆ ziγ5N(P) + MV3(γµC)γµγ5N(P) + M2

i

V(i)

4 (ˆ

ziC)γ5N(P) + M2

i

V(i)

5 (γµC)iσµziγ5N(P) +

M3

i,j

V(i,j)

6

(ˆ ziC)ˆ zjγ5N(P) .

I.V. Anikin Electroproduction of Nucleon Resonances

Applying the following conditions:

• Condition I: matching with the longitudinal cases;
• Condition II: use of the equations of motion for each fermion:
• Condition III: translation invariance;

I.V. Anikin Electroproduction of Nucleon Resonances

we derive V(1)

2 (xi)

= 1 4

• x3V2(xi) + (x2 − x1)V3(xi) − A3(xi)

+ x3A3(xi) + x3A2(xi)

• ,

V(2)

2 (xi)

= 1 4

• x3V2(xi) + (x1 − x2)V3(xi) + A3(xi)

− x3A3(xi) − x3A2(xi)

• ,

V(3)

2 (xi)

= −1 2x3V2(xi) ,

I.V. Anikin Electroproduction of Nucleon Resonances

and, similarly, A(1)

2 (xi)

= 1 4

• − x3A2(xi) + (x2 − x1)A3(xi) − V3(xi)

+ x3V3(xi) − x3V2(xi)

• ,

A(2)

2 (xi)

= 1 4

• − x3A2(xi) + (x1 − x2)A3(xi) + V3(xi)

− x3V3(xi) + x3V2(xi)

• ,

A(3)

2 (xi)

= 1 2x3A2(xi) .

I.V. Anikin Electroproduction of Nucleon Resonances

On Auxiliary Functions

The momentum dependence of the NLO corrections to the amplitude can conveniently be written in terms of the following functions: gnk(y, x; W ) = lnn[1 − yW − iη] (−1 + xW + iη)k , hnk(x; W ) = lnn[1 − xW − iη] (W + iη)k with n = 0, 1, 2 and k = 1, 2, 3. For the particular case n = 0 the first argument becomes dummy; for simplicity of notation we write the corresponding entries as gk(x; W ) ≡ g0k(∗, x; W ) .

I.V. Anikin Electroproduction of Nucleon Resonances

Going over to the Borel parameter space and subtracting the continuum corresponds to the substitutions gnk →Gnk(y, x; M2) = 1 π s0 ds Q2 e−s/M2Im gnk(y, x, W ) , hnk →Hnk(x; M2) = 1 π s0 ds Q2 e−s/M2Im hnk(x, W ) , where s = P′2 and W = 1 + s/Q2, M2 is the Borel parameter and s0 the continuum threshold.

I.V. Anikin Electroproduction of Nucleon Resonances

LCSRs involve integrals of the type Gnk =

• [dx] F(x)Gnk(xi + xj, xi; M2) ,
• Gnk =
• [dx] F(x)Gnk(xi, xi; M2) ,

Hnk =

• [dx] F(x)Hnk(xi + xj; M2) ,

where F(x) = F(xi, xj, 1 − xi − xj) is a function of quark momentum fractions and xi, xj ∈ {x1, x2, x3}. In addition one needs

• G01 =
• [dx] F(x)G01(∗, xi + xj; M2)

(only this special case).

I.V. Anikin Electroproduction of Nucleon Resonances

Using the following notations: xij = xi + xj , ¯ x = 1 − x , x0 = Q2 s0 + Q2 , E(x) = exp

• − ¯

xQ2 xM2

• ,
• F(xi, xj)
• + = F(xi, xj) − F(x0, xj)

and F ⊗ G = 1

x0

dxi 1−xi dxj F(x) G(x) , F ⊛ G = x0 dxi 1−xi

x0−xi

dxj F(x) G(x) .

I.V. Anikin Electroproduction of Nucleon Resonances

We obtain: G01 = −F ⊗ E(xi) xi ,

• G01

= −F

• ⊗ + ⊛

E(xij) xij , and so on. Our results for G11, G21, H11, H12, H21, H22 differ from the corresponding expressions g7 − g12 in K. Passek-Kumericki and

• G. Peters, Phys. Rev. D 78, 033009 (2008) by extra terms from

the ⊛ integration region; in addition our expression for G21 does not contain a contribution ∼ π2(1 − δ(xj)).

I.V. Anikin Electroproduction of Nucleon Resonances

On N∗(1535) resonance

A Lorentz-covariant definition of the DAs of negative parity resonances involves some freedom. It is convenient to choose the definition is such a way that the coefficient functions in the OPE

• f currents are the same for states of both parities, and also the

relations between different DAs imposed by QCD equations of motion remain the same. This can be achieved using invariant decomposition of the N∗(1535) matrix element in terms of the γ5-rotated quark fields 4(γ5)αα′(γ5)ββ′(γ5)γγ′0|ǫijkui

α′(a1n)uj β′(a2n)dk γ′(a3n)|N∗(P, λ)

= SN∗

1 mN∗Cαβ

• γ5u+

N∗

• γ + . . .

I.V. Anikin Electroproduction of Nucleon Resonances

where the expression on the right hand side is with obvious replacements mN → mN∗ etc. Projecting out the γ5 matrices we

• btain

40|ǫijkui

α(a1n)uj β(a2n)dk γ (a3n)|N∗(P, λ) =

SN∗

1 mN∗Cαβ

• u+

N∗

• γ + SN∗

2 mN∗Cαβ

• u−

N∗

• γ +

PN∗

1 mN∗ (γ5C)αβ (γ5u+ N∗)γ + PN∗ 2 mN∗ (γ5C)αβ (γ5u− N∗)γ −

V N∗

1

( pC)αβ

• u+

N∗

• γ − V N∗

2

( pC)αβ

• u−

N∗

• γ +

1 2V N∗

3

mN∗ (γ⊥C)αβ

• γ⊥u+

N∗

• γ + 1

2V N∗

4

mN∗ (γ⊥C)αβ

• γ⊥u−

N∗

• γ −

V N∗

5

m2

N∗

2pn ( nC)αβ

• u+

N∗

• γ − m2

N∗

2pn V N∗

6

( nC)αβ

• u−

N∗

• γ + .....

I.V. Anikin Electroproduction of Nucleon Resonances

As a consequence in the definition of leading-twist DA in terms of the chiral quark fields there is a minus sign as compared to nucleon, 0|ǫijk u↑

i (a1n)C ˆ

nu↓

j (a2n)

• ˆ

nd↑

k(a3n)|N∗(P)

= 1 2fN∗ (pn)ˆ n u↑

N∗(P)

• [dx] e−i(pn) xiai ϕN∗(xi) ,

where, of course, P2 = m2

N∗ and the expressions for the invariant

functions V N∗

1

, AN∗

1 , T N∗ 1

in terms of ϕN∗, are the same as for the nucleon.

I.V. Anikin Electroproduction of Nucleon Resonances

The twist-four DAs also acquire some signs 0|ǫijk u↑

i (a1n)C ˆ

nu↓

j (a2n)

• ˆ

pd↑

k(a3n)|N∗(P)

= 1 4 (pn) ˆ p u↑

N∗(P)

• [dx] e−i(pn) xiai

×

• fN∗ΦN∗,WW

4

(xi) + λ∗

1ΦN∗ 4 (xi)

• ,

0|ǫijk u↑

i (a1n)C ˆ

nγ⊥ ˆ pu↓

j (a2n)

• γ⊥ˆ

nd↑

k(a3n)|N∗(P)

= −1 2 (pn)ˆ n mN∗u↑

N∗(P)

• [dx] e−i(pn) xiai

×

• fN∗ΨN∗,WW

4

(xi) − λ∗

1ΨN∗ 4 (xi)

• ,

I.V. Anikin Electroproduction of Nucleon Resonances

0|ǫijk u↑

i (a1n)C ˆ

pˆ nu↑

j (a2n)

• ˆ

nd↑

k(a3n)|N∗(p)

= λ∗

2

12 (pn)ˆ n mN∗u↑

N∗(P)

• [dx] e−i(pn) xiai ΞN∗

4 (xi) ,

where ΦN∗,WW

4

(xi) and ΨN∗,WW

4

(xi) are given by the same expressions in terms of the expansion of the leading-twist DA ϕN∗(xi) as for the nucleon.

I.V. Anikin Electroproduction of Nucleon Resonances

The price to pay for universality of correlation functions for positive and negative parities is that the relations between DAs and light-front wave functions in this convention acquire some signs as well, fN∗ϕN∗(x1, x2, x3) = +4 √ 6

• [dk⊥] ψ(0)

N∗;1(1, 2, 3) ,

[λN∗

1 ΦN∗ 4

+ fNΦN∗,WW

4

](x2, x1, x3) = +8 √ 6

• [dk⊥]

x3mN∗ k⊥

3 ·

• ¯

k⊥

1 ψ(1) N∗;1 + ¯

k⊥

2 ψ(1) N∗;2

• (1, 2, 3) ,

I.V. Anikin Electroproduction of Nucleon Resonances

and [λN∗

1 ΨN∗ 4

− fNΨN∗,WW

4

](x1, x2, x3) = −8 √ 6

• [dk⊥]

x2mN∗ ¯ k⊥

2 ·

• k⊥

1 ψ(1) N∗;1 + k⊥ 2 ψ(1) N∗;2

• (1, 2, 3) ,

λN∗

2 ΞN∗ 4 (x1, x2, x3) =

−24 √ 6

• [dk⊥]

x1mN∗ k⊥

1 ·

• ¯

k⊥

1

• ψ(−1)

N∗ (1, 3, 2) − ψ(−1) N∗ (1, 2, 3)

• + ¯

k⊥

2

• ψ(−1)

N∗ (2, 3, 1) − ψ(−1) N∗ (2, 1, 3)

• ,

that have to be taken into account for the interpretation of the results.

I.V. Anikin Electroproduction of Nucleon Resonances