1. Central Exclusive Production: Theory Overview
Jeff Forshaw
Ancient history: Regge Theory
Recent history: Perturbative QCD
Based mainly on Tim Coughlin’s PhD thesis and review article with Mike Albrow & Tim.

2. Central exclusive production
e.g.
Detect the four-momenta of the protons using detectors situated 220m and/or 420m from the interaction point.

3. Ancient History Simple Regge Theory Elastic scattering
Beware: simple pole approximation is not favoured by global analyses:
Total cross-section via Optical Theorem
Need for more complicated regge exchanges (unitarity) = big impact at LHC energies
Luna, Khoze, Martin, Ryskin; Maor, Gotsman, Levin; Kaidalov see Alan Martin’s talk

4. “Double Pomeron Exchange”
Single diffraction
“Double Pomeron Exchange”
Chew & Chew

5. DPE cross section at the ISR
Theory: Desai, Shen & Jacob (1978)
(Field & Fox)
Same as the Chew & Chew prediction. Secondary exchanges (RRR,RRP,RPP) important (10%-50%).
First measurement (1976) from ARCGM Collaboration: Two low-angle protons plus two central tracks = rapidity gaps bigger than about 2 units each side.
Later measurements measured the proton momenta = could cut on xi < 0.1.
CHM measurement (1979) noteable for measuring the mass and rapidity of central resonances from the measured protons alone.
AFS (1983) confirmed absence of rho at highest ISR energies.

6. Very little activity post ISR (UA8 2002 paper) until Tevatron.
Triple Regge is really a calculation of central inclusive production (POMWIG).
Calculation of “soft” central exclusive production requires knowledge of pomeron-pomeron-particle vertex (or a low enough MX).
However for sufficiently large MX one can compute CEP of specific states using QCD perturbation theory, e.g. Higgs boson.
Direct coupling of two gluons into production vertex is higher-order in the strong coupling pr power suppressed.

7. Perturbative QCD Leading Order (Feynman gauge)
Tim Coughlin (PhD thesis)
Leading Order (Feynman gauge)
Divergent in infra-red (would be cured by folding with proton wavefunction)
Agrees with unitarity method (only cuts (a) and (e) survive)

8. Next-to-Leading Order
Plus graphs with Higgs on left of cut and those with x1 and x2 swapped
BFKL & hard-collinear emissions across the cut are part of the evolution of the incoming unintegrated pdfs.
Soft emissions across the cut generate the derivative of the Sudakov factor (see later).
“FS” soft/collinear poles cancel with emissions across the cut. “IS” soft/collinear poles generate DGLAP evolution.
Gluon reggeization factor is part of evolution of unintegrated pdf.
What remains is the LO expansion of the Sudakov:

9. Shifting the argument of the LO amplitude gives
which is (almost) exactly the NLO contribution to the KMR result where
instead of

10. The last splitting functions are unregularised.
All orders
Need to stop evolving the external off-diagonal partons one rung short of the hard subprocess.
The last splitting functions are unregularised.
e.g. in the forward limit we regain DGLAP

11. If we replaced the unregularised splitting functions by the regularised ones then we’d find
Derivative of pdf just expresses need to work with the number density of gluons at some scale (i.e. not the number density up to some scale).
We must also sum corrections to the Higgs vertex that contain log(MH/QT) terms

12. Collinear approximation is not sufficent.
BFKL says we must cut off the transverse momentum integral at (the region below is contained in the gluon’s regge trajectory).
And a more careful treatment of the soft limit sets the upper cutoff on z, i.e.

13. All that remains is to fix the issue of the unregularized splitting functions….
These graphs contribute too when the gluon is soft (i.e. only matter for gg splitting):
i.e. not just these:
i.e.
Assembling together gives the KMR result but with the same modified Sudakov we met in the NLO calc.

14. The formal accuracy is
To this accuracy only the upper cutoff on the kT integral in the Sudakov is not precisely fixed.
For large enough central masses, the Sudakov will cut-off the fixed-order infra-red divergence and suppress contributions in the non-perturbative region, i.e. we do not need to worry about evaluating pdfs at low scales.
None of this includes the Gap Survival factor, which is not computable in perturbation theory.
The gap survival and lack of knowledge of the unintegrated pdfs are responsible for the biggest uncertainty in rates. Unknown higher-order corrections and potential non-perturbative contributions are probably also significant.

15. Cudell, Dechambre, Hernandez – November 2010
Low higgs cross section is probably because they ignore the derivative of the Sudakov. Plot from my 2005 HERA-LHC review