Holographic Approach to dilepton production in p-p...

Holographic Approach to dilepton production in p-p collisions

XXIV Reunião de Trabalho sobre Interações Hadrônicas

CBPF Rio, 03 December 2012

Henrique Boschi-Filho Instituto de Física – Universidade Federal do Rio de Janeiro* Work done in collaboration with Carlos Alfonso Ballón-Bayona and Nelson R. F. Braga * Supported also by CNPq and CAPES

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Dilepton production in hadron-hadron collisions

P1 , P2 → incident protons; k1 , k2 → dilepton (lepton pair) X, Y → Final hadronic states Fixed variables: P1 , P2 and q (momentum of the virtual photon)

Hadron

Hadron

leptons

(Drell-Yan process)

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Differential cross section for the inclusive, unpolarized case (Lam and Tung Phys. Rev. D 1978) :

Leptonic tensor (just QED)

Hadronic tensor:

SHi = spins

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The hadronic tensor can be decomposed into 4 invariant structure functions Wi expressed in terms of 4 independent scalars (like q2 , s , P1∙ q , P2 ∙ q , or other convenient choice). Useful quantities: Helicity Structure functions:

With polarizations:

More convenient form of helicity structure functions:

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Cross section (in virtual photon rest frame):

Where θ and φ are the lepton angles (virtual photon rest frame):

With:

Helicity structure functions depend on the choice of cordinates ( relation between the spatial axis X, Y, Z used for helicities and lepton directions and the momenta P1 , P2 , q ) . We used the Collins-Soper frame (Collins and Soper, Phys. Rev. D 77 ) (same used by experimentalists )

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Perturbative QCD predictions in Collins-Soper frame:

These perturb. results are NOT valid for

where there is an effective strong coupling (logarithm terms)

We used our model to investigate this region of small qT

Where:

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AdS/QCD model for Drell Yan process in proton- proton collisions A.Ballon Bayona, HBF, Nelson Braga, NPB 2011 dilepton production through the exchange of vector mesons

With vector mesons and baryons described by the hard wall model. Final states: excited baryons of spin ½.

What is the hard wall model ?? Brief review next

AdS/CFT correspondence, J. Maldacena, 1997 (simplified version of a particular useful case)

Remarks.: String theory space = AdS5 X S5

AdS = anti-de Sitter; S = sphere

Gauge theory: SU(N) with very large N (supersymmetric and conformal). 8

Exact equivalence between String Theory in a 10-dimensional space and a gauge theory on the 4-dimensional boundary.

At low energies string theory is represented by an effective supergravity thery → gauge / gravity duality

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Anti-de Sitter space

bulk ↔ boundary mapping

E. Witten and L. Suskind (1998) ; A. Peet and J. Polchinski (1998)

The 4-dim boundary is at z = 0 → z

States of the boundary

gauge theory with energy E

↔ Region of AdS space

x

Holographic relation suggests:

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This idea was introduced by Polchinski and Strassler (PRL 2002) to find the

scaling of high energy scattering amplitudes at fixed angles and then

used to calculate hadronic masses from AdS/CFT by HBF and N. Braga JHEP

2003, EPJC 2004

Boundary conditions at Zmax → Masses of scalar glueballs JPC = 0++ , 0++* , 0++** , …. in good agreement with lattice results.

This was then called Hard wall model.

Many interesting results for hadronic spectrum: Light baryons and mesons: Brodsky, Teramond PRL 2005, Mesons: Erlich, Katz, Son, Stephanov. PRL 2005…

Cut off in AdS space: 0 < Z < Zmax ↔ infrared cut off in gauge theory.

Glueballs

↔ Normalizable modes of a scalar field in an AdS slice with size Zmax = 1/ Λ

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Vector mesons in the hard wall model (approach similar to Sakai Sugimoto model to generate the interactions) :

Kaluza Klein expansion, (gauge AZ = 0)

With the conditions:

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With masses determined from boundary conditions at z = 1/Λ and couplings determined from the integration in the fifith dimension.

After some field redefinitions we find an effective 4-d action

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Baryons: similar Kaluza Klein expansion, boundary conditions, ...., effective 4-d free action

Interaction of fermions and vector mesons (start at 5-d , KK expansions, ....)

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Optical theorem: imaginary part of the forward proton-proton-photon scattering amplitude gives the hadronic tensor Wμν

So, it is just applying the Feynman rules for the effective model

We calculated Wμν in the frame where:

Equivalent to final hadrons frame:

This simplifies the integrations . We first find the invariants: W1 , W2 , W3 , W4 , then the helicity structure functions in Collins Soper

We considered a kinematical regime compatible with the region analysed,

for di-muons by FNAL E866/NuSeA Collaboration, L.Y. Zhu at all, Phys. Rev. Lett 2009.

They found, from Fermilab data (protons at 800 GeV colliding with fixed target

Hydrogen ):

(Experimental average values)

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Numerical set up. • We fixed the size of the AdS slice by the mass of the ρ meson 0.776 GeV. This gives Λ = 0.323 GeV. and chose for the proton and photon momenta the form:

Important issues for the Numerical computations: • When do we stop the infinite series in baryons and vector mesons? For the baryons (on shell states ): For the vector mesons: we investigated numerically the convergence of the series and found that we should use 15 states for the vector mesons.

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Results:

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We found (blue lines) a decrease in the parameter λ for qT → 0, while perturbative expression (red lines) → 1.

We are looking forward for more experimental results at small qT

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Experimental results

L.Y. Zhu at all, Phys. Rev. Lett 2009

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Final remarks:

• We considered only dileptons generated by photons ( seems OK, probably this is the dominant contribution) • We summed only over final hadronic states with a single hadron with spin ½ . Need to improve the model to have states with higher spins and more hadrons. • Hard wall has a hadronic spectrum that is not asymptoticaly Regge like. • Anyway, results seem quite good (specially for λ ) for this first step. Let us see what happens with λ for very small transverse momentum when there is more experimental data.

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Thank you very much !!