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LISHEP 98 Rio de Janeiro 16 February 1998 Brazil

Scale Issues in High-Energy

Diffractive Vector-Meson Production

LAFEX International School on High-Energy Physics

J. A. Crittenden

Physikalisches Institut Universität Bonn

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Diffractive Electroproduction of Vector Mesons Elastic Proton Dissociative

√s: Center–of–mass energy of e–p system

(300 GeV)

W: Center–of–mass energy of γ^{∗}–p system

(20 – 240 GeV)

Q^{2}: Photon Virtuality (10^{−10} – 20 GeV^{2})
t:  Square of Momentum Transfer at

the Proton Vertex (|t| < 13 GeV^{2})

Can the hard scale necessary for perturbative QCD calculations be set by

either Q^{2} , M_{V}^{2} , or |t| ??

What experimental signatures for the region of pQCD validity ?

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 2]

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Signatures for pQCD

Energy Dependence of Elastic Cross Sections

⇒ Exchange of the phenomenological Pomeron

\frac{dσ}{d|t|} ∝ e^{−b_{0}|t|} (\frac{s}{s_{0}})^{2(α_{IP} (t)−1)}  α_{IP} (t) = 1.08 + 0.25 t
\frac{dσ}{d|t|}|_{t=0} ∝ (\frac{s}{s_{0}})^{2(α_{IP} (0) −1)} ∝ W^{4ϵ} = W^{0.32}

⇒ Exchange of a gluon ladder

\frac{dσ_{γ^{∗}p→Vp}}{d|t|} |_{t=0} = A_{V} \frac{α_{s}^{2}(Q̄^{2})}{αQ̄^{6}} |[1 + \frac{iπ}{2} \frac{d}{d(ln x)} ] xG(x, Q̄^{2})|^{2}

Energy dependence determined by G(x, Q^{2}), x ≈ Q^{2}/W^{2}

Pointlike couplings; small-size qq configurations

\frac{dσ}{d|t|} ∝ e^{−b|t|} b = b_{0} + 2 α′_{IP} ln \frac{s}{s_{0}}
|t|_{min} = m_{p}^{2} \frac{(M_{V}^{2} + Q^{2})^{2}}{W^{4}}

⇒ b ∝ R_{p}^{2} + R_{VM}^{2} (Elastic) ⇒ Only longitudinally polarized configurations pointlike !!

Absence of ρ^{0} mass spectrum skewing

Skewing is interesting (σ_{πp}^{tot}?), but not pQCD

Absence of shrinkage

α′_{IP} > 0? (Gribov, JETP Lett. 41 (1961) 667)

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 3]

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Total Cross Sections

Donnachie + Landshoff, PL B296 (1992) 227

σ^{tot} = X s^{ϵ} + Y s^{−η}

{PDG \atop 1996}

ϵ = 0.0790 ± 0.0011 η = 0.4678 ± 0.0059

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 4]

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Total Photon-Proton Cross Sections

σ_{γ^{∗}p}^{tot} ≈ \frac{4π^{2}α}{Q^{2}(1−x)} F_{2}(x, Q^{2})

⇒ See Adi Bornheim’s talk

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Elastic Vector-Meson Photoproduction Cross Sections

⇒Energy dependence for the J/ψ much steeper !

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 6]

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Elastic J/ψ Cross Sections ⇒ See talk by Lee West

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 7]

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Elastic J/ψ Cross Sections

Ryskin et al, Z. Phys. C76 (1997) 231

⇒ Can derive shape of gluon distribution !!

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 8]

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Diffractive Production of ρ^{0} Mesons

e^{+} in BPC  e^{+} in UCAL
1995: 3.8 pb^{−1}  1995: 6 pb^{−1}
0.25 < Q^{2} < 0.85 GeV^{2}  5 < Q^{2} < 30 GeV^{2}
20 < W_{γ^{∗}p} < 90 GeV  34 < W_{γ^{∗}p} < 144 GeV
≈ 6000 events  ≈ 2000 events
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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 9]

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Elastic ρ^{0} Cross Sections

⇒ Experimental discrepancies require further study

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 10]

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Elastic ρ^{0} Cross Sections at High Q^{2} ⇒ Steepening with Q^{2} expected in pQCD ⇒ Need more data

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 11]

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π^{+} π^{−} Invariant Mass Spectrum

\frac{dN}{dM_{ππ}} = |A\frac{√M_{ππ}M_{ρ}Γ_{ρ}}{M_{ππ}^{2} − M_{ρ}^{2} + iM_{ρ}Γ_{ρ}} + B| ^{2}

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 12]

[figure] σ_{T} + ϵ σ_{L}(γ^{∗}p → ρ^{0}p)

Fit to the shape (Q^{2} + M2 ρ)−n

but notice the mixture of σ_{L} and σ_{T} (ϵ ≈ 1) ...

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[figure]

Photoproduction of J/ψ Mesons

The Exponential Slope in |t|

bJ/ψ = 4.6 ± 0.4 (stat)_{−0.6}^{+0.4} (sys) GeV^{−2}

(Reminder : bρ^{0} = 9.8 ± 0.8 (stat) ± 1.1 (sys) GeV^{−2})

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 14]

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t-Slope vs M_{VM}^{2}

b = b_{p} + bJ/ψ : bJ/ψ < 1 GeV^{2}

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t-Slope for ρ^{0} Mesons vs Q^{2}

⇒ The shrinking photon !!

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 16]

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t-Slope for ρ^{0} Mesons vs W Does the t-slope depend on energy?

⇒ Test for shrinkage inconclusive

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 17]

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Elastic J/ψ Cross Sections ⇒ Again inconclusive on shrinkage

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 18]

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The Case Against Shrinkage

A. Levy, hep-ph/9712519

Compares EMC, E401 (W=16 GeV) with H1, ZEUS (W=85 GeV)

⇒ Inconsistent with α′_{IP} = 0.25 GeV^{−2} ⇒ Concludes pQCD applicable

Collins et al, PL B307 (1993) 161 →“perturbative Pomeron”

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 19]

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High-|t| Photoproduction

| |t| − p_{T}^{2} (ρ^{0}) | < Q^{2}

e^{+} in 44m Tagger Untagged Sample

1995: 2.1 pb^{−1}  1994: 2.2 pb^{−1}
10^{−5} < Q^{2} < .01 GeV^{2} 10^{−10} < Q^{2} < 4 GeV^{2}
85 < W_{γ^{∗}p} < 105 GeV 50 < W_{γ^{∗}p} < 100 GeV
|t| < 4 GeV^{2}  |t| < 0.5 GeV^{2}
≈ 3k events  ≈ 80k events
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(primarily proton dissociative)  (primarily elastic)

LISHEP 98 J. A. Crittenden, PI Bonn [figure: 20]

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ρ^{0} Mass Spectrum Skewing vs |t|

\frac{dN}{dM_{ππ}} = |A\frac{√M_{ππ}M_{ρ}Γ_{ρ}}{M_{ππ}^{2} − M_{ρ}^{2} + iM_{ρ}Γ_{ρ}} + B| ^{2}

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Vector Meson Production Ratios

If direct e–m coupling to quarks without form factors

\begin{matrix}ρ^{0} : & ω : & ϕ : & J/ψ : & Υ \\ 9 : & 1 : & 2 : & 8 : & 2 \end{matrix}

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J/ψ Photoproduction at High |t|

⇒ pQCD works at high |t| !!

Bartels et al, PL B375 (1996) 301

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 23]

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ψ(2S) Signal from H1

\frac{σ(ψ(2S))}{σ(J/ψ)} = 0.150 ± 0.027 ± 0.022

Consistent with color dipole model prediction

Nemchik et al, PL B341 (1994) 228, hep-ph/9712469

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 24]

Preliminary Υ Signal from ZEUS

γp → μ^{+} μ^{−} p

80 < W < 280 GeV, Q^{2} < 1 GeV^{2}

σ = 0.9 ± 0.3 ± 0.3 nb

The Υ resonances are not resolved !

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 25]

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Helicity Analyses

Φ_{h}

Angle between the electron scattering plane and the ρ^{0} production plane

θ_{h} Polar angle of π^{+} in the ρ^{0} rest frame
ϕ_{h} Azimuthal angle of π^{+}

If helicity is conserved in the γ/ρ^{0} transition, then the decay angular distributions depend only on the difference ψ_{h}= ϕ_{h}- Φ_{h} .

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 26]

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Decomposing the Spin Density Matrix

How to couple real photons (λ = ±1) to vector mesons (λ = 0, ±1) ??

Helicity conserved in strong interactions (but in what frame??)

Schilling, Seyboth, and Wolf, Nucl.Phys. B15, 397 (1970)

Formalism developed generally for arbitrary real photon polarization (2x2 matrices!)

• Circular polarization: ϵ⃗_{γ} = ±P_{γ}^{c} (0, 0, 1)

• Linear polarization: ϵ⃗_{γ} = P_{γ}^{l} (− cos Φ, sin Φ, 0)

Decompose ρ^{0} decay angle distribution W(θ,ϕ) into four parts W^{i}(θ,ϕ) (i=0,3) such that:

1. Unpolarized: W(θ,ϕ) = W^{0}(θ,ϕ)

2. Circularly polarized:

W(θ,ϕ) = W^{0}(θ,ϕ) ± P_{γ}^{c} W^{3}(θ,ϕ)

Need long. pol. initial-state lepton→E665(μ),HERMES(e)

3. Linearly polarized:

W(θ,ϕ,Φ) = W^{0}(θ,ϕ) - P_{γ}^{l} cos 2Φ W^{1}(θ,ϕ) - P_{γ}^{l} sin 2Φ W^{2}(θ,ϕ)

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Generalizing to R = \frac{σ_{L}}{σ_{T}} > 0

Schilling and Wolf, Nucl.Phys. B61, 381 (1973)

Define linear combinations of the matrix elements:

(Helicity states i, k = −1, 0, +1)

r_{ik}^{04} = \frac{ρ_{ik}^{0} + ϵRρ_{ik}^{4}}{1 + ϵR} The angular distributions do not distinguish unpolarized transverse photons from lon- gitudinally polarized pho- tons!
r_{ik}^{α} = \{ \begin{matrix}\frac{ρ_{ik}^{α}}{1 + ϵR} & α = 1 − 3 (Trans.) \\ \frac{√R ρ_{ik}^{α}}{1 + ϵR} & α = 5 − 8 (L/T int.) \end{matrix}
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Constraints from hermiticity, parity conservation LISHEP 98 J. A. Crittenden, PI Bonn [figure: 28]

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The Choice of Frame

• Gottfried-Jackson frame Spin quantized along photon direction (t-channel)

• Helicity frame Spin quantized along meson direction (s-channel)

The angle χ between the two frames: cos χ = (\frac{m_{ρ}^{2} + t}{m_{ρ}^{2} − t} )

⇒ ρ^{0} most sensitive to a violation of helicity conservation in the s-channel amplitudes (SCHC)

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SCHC in ρ^{0} Photoproduction

Ballam et al., Phys. Rev. D5, 545 (1972)

• Backscattered laser beam at SLAC

• Linear polarization measured to be greater than 90%

• 2.8 and 4.7 GeV photons

• 3k ρ^{0} → π^{+} π^{−} reconstructed in bubble chamber

Inconsistent with t-channel exchange of spin 0 object

Gilman et al PL B31 (1970) 387

⇒ Conclude SCHC holds for |t| < 0.4 GeV^{2}

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ρ^{0} Electroproduction at HERA

\frac{dN}{d cos θ_{h}} ∝ 1 − r_{00}^{04} + (3r_{00}^{04} − 1) cos ^{2} θ_{h}

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R = \frac{σ_{L}}{σ_{T}} vs Q^{2}

Assuming SCHC: R = \frac{1}{ϵ} \frac{r_{00}^{04}}{1−r_{00}^{04}}

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pQCD Calculation of R

⇒ Need more data !

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 33]

SCHC at High |t|

Dependence on azimuthal angle in high-|t| photoproduction?

(Would imply violation of SCHC)

But the ρ^{0} is tricky (r_{ik} depend on m_{ππ}) Effect not observed for the ϕ

⇒ see Jim Whitmore’s talk

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Conclusions

These results show preliminary evidence

for the hard scale to be given by either Q^{2}, M_{V}^{2} , or |t|

The H1 and ZEUS experiments are sensitive to the transition region in

EACH of these variables

These studies are statistics-limited 1996/97 data still to be analyzed

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HERA Integrated Luminosity

Only the ZEUS high-|t| and Υ analyses

include data recorded after 1995

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LISHEP 98 J. A. Crittenden, PI Bonn [figure: 36]