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Motivation Example: LO Conformal OPE State of the Art Outlook Exclusive Processes in Position Space and the Pion Distribution Amplitude V. M. Braun University of Regensburg based on V. Braun and D. Mller, arXiv:0709.1348 [hep-ph]

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Motivation Example: LO Conformal OPE State of the Art Outlook

Exclusive Processes in Position Space and the Pion Distribution Amplitude

  • V. M. Braun

University of Regensburg

based on V. Braun and D. Müller, arXiv:0709.1348 [hep-ph] Southampton, 12 October 2007

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Transformations

1

conserve the interval ds2 = gµν(x)dxµdxν

2

  • nly change the scale of the metric

g′

µν(x′) = ω(x)gµν (x)

⇒ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts Dilatation (global scale transformation)

xµ → x

′µ = λxµ

Inversion

xµ → x

′µ = xµ/x2

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Transformations

1

conserve the interval ds2 = gµν(x)dxµdxν

2

  • nly change the scale of the metric

g′

µν(x′) = ω(x)gµν (x)

⇒ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts Dilatation (global scale transformation)

xµ → x

′µ = λxµ

Inversion

xµ → x

′µ = xµ/x2

Special conformal transformation xµ → x′µ = xµ + aµx2 1 + 2a · x + a2x2

= inversion, translation xµ → xµ + aµ, inversion

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Anomaly

Consider YM theory with a hard cutoff M, integrate out the fields with frequencies above µ Seff = − 1 4 Z d4x „ 1 (g(0))2 − β0 16π2 ln M2/µ2 « ˆ Ga

µνGaµν˜ slow (x) + . . .

Under the scale transformation xµ → λxµ, Aµ(x) → λAµ(λx), ψ(x) → λ3/2ψ(λx) and µ → µ/λ with the fixed cutoff δS = − 1 32π2 β0 ln λ Z d4x Ga

µνGaµν(x)

which implies ∂µJµ

D(x) = − β0

32π2 Ga

µνGaµν(x)

Restoring the standard field definition ∂µJµ

D(x) = gµνΘQCD µν (x)

EOM

= β(g) 2g Ga

µνGaµν(x)

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Outline

1

Motivation

2

Example: LO

3

Conformal OPE

4

State of the Art

5

Outlook

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Towards virtual reality . . .

Preparation of states:

  • Accelerator–based experiments:

— scattering of particles with fixed momentum — know what was ’before’, learn what becomes ’after’ the interaction — foundation of QM — observation of quantum phenomena with classical tools

  • Lattice–based experiments:

— correlation functions of sources with fixed position — learn what happens in process of the interaction

calls for change of philosophy: coordinate space–based phenomenology emphasize on space–time picture

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Hadron Distribution Amplitides

Feynman: How to transfer a large momentum Q to a hadron?

’77–’80: The answer depends on the underlying field theory. In QCD, find a hadron in a rare configuration with all its consitutuents being at a small transverse separation ∼ 1/Q

E.g. pion distribution amplitude (DA) 0|¯ d(0) n γ5u(n)|π+(p) = ifπ(pn) Z 1 du e−iunpφπ(u, µ) , n2 = 0 DAs enter QCD description of many hard processes: elastic and transition form factors γ∗ππ, γ∗πγ, γ∗πρ, . . . weak decays B → πℓ¯ ν, B → K∗ℓ+ℓ−, B → ργ, . . . hard diffraction γ∗p → ρp, . . . etc. but are known very poorly exclusive channels have small partial cross sections

  • ne does not measure DAs directly, but rather some convolution integrals

for available momentum transfers, most of the processes are affected by so-called soft contributions etc.

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Example: Pion Distribution Amplitide

φ(u)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

u based on:

  • QCD sum rule and lattice calculations of

Z 1 du (2u − 1)2φπ(u) ∼ 0|¯ q(

↔

D +)2γ+γ5|π(p)

  • Measurements of γ∗γ → π0 form factor for

Q2 ∼ 1.5 − 9 GeV2 (CLEO) ∼ Z 1 du 1 − u φπ(u)

How to move forward? ⋄ Lattice calculations of higher moments difficult because the loss of O(4) symmetry becomes more

and more punishing; also need very high statistics

⋄ QCD analysis of γ∗γ → π0 is somewhat model–dependent; mostly sensitive to a single overlap

integral

1

nonlocal operators that resemble DA [Aglietti et al. ’98]

2

coordinate–space analogue of the γ∗γ∗ → π0 with two virtual photons, or similar

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Demonstration: Correlation function of two EM currents

Tµν = 0|T{jem

µ (x)jem ν (−x)}π0(p) = − i

3fπǫµνρσ xρpσ 4π2x4 T(p · x, x2) in leading order (LO)

T(p · x, x2) = 1 2 Z 1 du ei(2u−1)p·xφπ(u, µ ∼ 1/|x|)

use standard decomposition

φπ(u, µ) = 6u¯ u

∞

X

n=0

φn(µ)C3/2

n

(2u − 1) the Fourier transform yields Bessel functions 8 Z 1 du u¯ u eiρ(2u−1)C3/2

n

(2u − 1) = √ 2π(n + 1)(n + 2)inρ−3/2Jn+3/2(ρ)

  • btain partial wave expansion

T(p · x, x2) = 3 4

∞

X

n=0

φn(µ)Fn (p · x) Fn(ρ) = in√ 2π (n + 1)(n + 2) 2 ρ−3/2Jn+3/2(ρ)

x2 → ’scale’ p · x → ’distance’

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Realistic models ?

φ(u)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

u φπ(u, µ0) = = 6u¯ u h 1 + φ2C3/2

n

(2u−1) + φ4C3/2

4

(2u−1) i

φ2(2 GeV) = 0.201 ± 0.113 Braun et al., PRD74(2006)074501 φ2(2 GeV) = 0.233 ± 0.088

  • C. Sachrajda, LATTICE-2007

blue

asymptotic DA

φ2 = 0, φ4 = 0 red

BMS model

φ2 = 0.25, φ4 = −0.1 green BMS model φ2 = 0.25, φ4 = +0.1 ⋄ same color identification used in all Figures throughout this talk

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

LO results

inFn(ρ)

2 4 6 8 10 0.2 0.4 0.6 0.8

2 4

ρ T(ρ, x2)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

ρ

compare: φπ(u) = δ(u − 1/2) T(ρ) = 1/2 φπ(u) = 1 T(ρ) = 1/2 cos ρ diffraction:

p 2x

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Separation of variables

In Quantum Mechanics: O(3) rotational symmetry

⇒

Angular vs. radial dependence " − 2 2m ∆ + V(|r|) # Ψ = EΨ ⇒ Ψ(

  • r) = R(r)Ylm(θ, φ)

Ylm(θ, φ) are eigenfunctions of L2Ylm = l(l + 1)Ylm, [H, L2] = 0.

♥

In Quantum Chromodynamics: SL(2,R) conformal symmetry

⇒

Longitudinal vs. transverse dependence Migdal ‘77 Brodsky,Frishman,Lepage,Sachrajda ‘80 Makeenko ‘81 Ohrndorf ‘82

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Collinear conformal transformations

P = Pz

p+ = 1 √ 2 (p0 + pz) → ∞ p− = 1 √ 2 (p0 − pz) → 0 px → p+x−

Special conformal transformation x− → x′

− =

x− 1 + 2ax− translations x− → x′

− = x− + c

dilatations x− → x′

− = λx−

form the so-called collinear subgroup SL(2, R) α → α′ = a α + b c α + d , ad − bc = 1 Φ(α) → Φ′(α) = (c α + d)−2jΦ „ aα + b cα + d « where Φ(x) → Φ(x−) = Φ(αn−) is the quantum field with scaling dimension ℓ and spin projection s “living” on the light-ray Conformal spin: j = (l + s)/2

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal operators

spin summation: For two fields “living” on the light-cone O(α1, α2) = Φj1(α1) Φj2(α2) [j1] ⊗ [j2] = M

n≥0

[j1 + j2 + n] define a conformal operator L2 Oj1,j2

j

= j(j − 1)Oj1,j2

j

L0 Oj1,j2

j

= j0 Oj1,j2

j

L−Oj1,j2

j

= where L0,± are two-particle SL(2) generators in a suitable representation The answer is j0 = j = j1 + j2 + n; for our specific case j1 = j2 = 1 Oj = [∂+]n¯ q(0)γ+γ5C3/2

n

→ D+ −

←

D+

→

D+ +

←

D+ ! q(0)

  • Conformal operators do not mix under renormalization (unless symmetry is broken)
  • Gegenbauer Polynomials appearing in the expansion of the pion DA are ’spherical

harmonics’ of the SL(2, R) ∼ O(2, 1) group

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Beyond LO

The partial waves become scale- (and scheme-) dependent Fn(ρ) → Fn(ρ, −µ2x2; αs(µ)) The idea is to find a representation Fn = FCFT

n

+ β(g) g ∆F where ∆F = power series in αs In conformal theory (e.g. think of a nontrivial fixed-point) FCFT

n

= Cn(αs) (−µ2x2)

γn 2 in√π (n+1)(n+2)

4 Γ(n+5/2+γn/2) Γ(n+5/2) “ρ 2 ”− 3

+ γn 2

Jn

+3

+ γn 2

(ρ) Cn(αs) = Γ(2 − γn/2)Γ(1+n) Γ(1+n+γn/2) cn(αs) — exact expression in terms of anomalous dimension γn and Wilson coefficient cn(αs)that appears in polarized DIS; both are known to NNLO (three loops)) Main result (D. Müller): possible in a special renormalization scheme

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Ward Identities

The Ward identities follow from the invariance of the functional integral under the change of variables Φ(x) → Φ′(x) = Φ(x) + δΦ(x) Let XN ≡ 0|TΦ(x1) . . . Φ(xN)|0 ≡ N −1 Z DΦ Φ(x1) . . . Φ(xN) eiS(Φ) then 0 = 0|TδΦ(x1) . . . Φ(xN)|0 + . . . + 0|TΦ(x1) . . . δΦ(xN)|0 + 0|TiδS Φ(x1) . . . Φ(xN)|0 Dilatation

N

X

i=1

(ℓcan

Φ + xi · ∂i) 0|TΦ(x1) . . . Φ(xN)|0 = −i

Z d4x0|T∆D(x) Φ(x1) . . . Φ(xN)|0, Special conformal transformation

N

X

i=1

“ 2xµ

i (ℓcan Φ + xi · ∂i) − 2Σµ νxν i − x2 i ∂µ i

” XN = −i Z d4x 2xµ 0|T∆D(x) Φ(x1) . . . Φ(xN)|0 . where ∆D(x)= β(g) 2g Ga

µνGaµν(x) + EOM

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Ward Identities — continued

in particular, for conformal operators

N

X

i=1

Di [Onl]XN = −

n

X

m=0

[ℓcan

l

δnm + ˆ γnm] [Oml]XN,

N

X

i=1

K−

i [Onl]XN

= i

n

X

m=0

[a(n, l)δnm + ˆ γc

nm(l)] [Oml−1]XN,

Dilation WI is nothing else as the Callan–Symanzik equation and ˆ γnm is the usual anomalous dimension matrix It is determined by the UV regions and diagonal in one loop ˆ γc

nm(l) is called special conformal anomaly matrix

It is determined by both UV and IR regions and is not diagonal in one loop However, nondiagonal entries in ˆ γc

nm(l) can be removed by a finite renormalization

After this is done, all conformal breaking terms can be absorbed through the redefinition of the scaling dimension and conformal spin of the fields ℓcan

n

⇒ ℓn(αs) = ℓcan

n

+ γn(αs) ,

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Conformal Ward Identities — continued (2)

this result is general:

  • D. Müller, PRD51(1995)3855; PRD59(1999)116003
  • n–loop non-diagonal entries in the anomalous dimension matrix are related to the n-1-loop

special conformal anomaly and can both be removed simultaneously by a finite renormalization

  • This finite renormalization defines the transition from MS to the conformal subtraction CS

scheme in our case, e.g. φCS

n

= φMS

n

− αs 2π

n−2

X

m=0

B(1)

nm φMS m

+ O(α2

s )

B(1)

nm

= 2(2n+3)CF (n+1)(n+2) " (m+1)(m+2) (n−m)(n+m+3) 2Anm − γ(0)

m

2CF ! + Anm − S1(n + 1) # , Anm = S1((n + m + 2)/2) − S1((n − m − 2)/2) + 2S1(n − m − 1) − S1(n + 1)

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

NLO perturbation theory

T(ρ, |x| = 0.1 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

T(ρ, |x| = 0.2 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

ρ NLO MS: thin NLO CS: thick LO : dots —corrections moderate —do not increase with ρ

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Higher-twist corrections

T(ρ, |x| = 0.1 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

T(ρ, |x| = 0.2 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

ρ LO : thin LO + tw.4: thick

used 0|¯ qige Gµνγνq|π0(p) = −fπδ2

πpµ .

with (QCD sum rules): δ2

π(1 GeV) = (0.18 ± 0.06) GeV2

—small for 0.1 fm —signifi cant for 0.2 fm —do not increase with ρ

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

Final Results: NNLO in CS, with twist-four corections

T(ρ, |x| = 0.1 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

T(ρ, |x| = 0.2 fm)

2 4 6 8 10

  • 0.1

0.1 0.2 0.3 0.4 0.5

ρ LO : thin NNLO CS: + tw.4: thick

for example T(0, |x| = 0.1fm) = = 0.430+0.008

−0.007 − 0.018+0.006 −0.006

also ratios studied: R(ρ0, ρ, x2) = T(ρ, x2) T(ρ0, x2) , ρ0 < ρ see arXiv:0709.1348

—NNLO very small —higher twist < 3% for 0.1 fm —overall, good convergence

  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude

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Motivation Example: LO Conformal OPE State of the Art Outlook

What can realistically be measured?

  • crucial: suffi cient “lever arm” in ρ

φ2: ρ = 1.5 − 2 φ4: ρ > 3 − 3.5 integral over ρ related to R du/(1 − u)φ(u)

  • |

p| < a−1 translates to ρmax < | x|/a

— ρ cannot be larger than (half of) the separation between the currents in lattice units — need fine lattices a ≤ 0.05 fm — need pion source with large momentum, at least 2-3 GeV — premium: lattice renormalization of composite operators is avoided altogether

  • Clear physical picture
  • Pion DA can be extracted within the same formalism as it will be applied
  • Technical improvements (other correlators etc) possible
  • Other applications: e.g. B-meson DA in the HQL is only defi ned perturbatively
  • V. M. Braun

Exclusive Processes in Position Space and the Pion Distribution Amplitude