NRQCD – How effective a theory of c K. Sridhar Tata Institute of Fundamental Research Mumbai Sridhar K. – p. 1/20 Charmonia at the LHC
QCD The theory of strong interactions based on a non-abelian colour SU (3) symmetry. QCD is asymptotically free. QCD factorisation. Sridhar K. – p. 2/20 Charmonia at the LHC
Charmonium Family M (GeV) 1 S 0 η c → γγ 2.98 3 S 1 J/ψ → ee, µµ 3.096 3 P 0 , 1 , 2 χ 0 , 1 , 2 3.41, 3.51, 3.55 → J/ψγ 1 P 1 h c 3.52 → J/ψπ ψ ′ 2 3 S 1 3.686 → ee, µµ Sridhar K. – p. 3/20 Charmonia at the LHC
Pre-NRQCD models Colour-evaporation Colour-singlet Sridhar K. – p. 4/20 Charmonia at the LHC
Colour Singlet Model Colour singlet model worked reasonably for low-energy (ISR) production At higher energies, problems with b quark initiated states. At Tevatron, prompt J/ψ production disagreed seriously with colour singlet model predictions. Sridhar K. – p. 5/20 Charmonia at the LHC
NRQCD Non-Relativistic QCD (NRQCD) is an effective theory obtained from QCD. Used to model bound state dynamics and study production and decay of quarkonia. Obtained by treating QCD with an ultraviolet cutoff ∼ M . Neglecting states above M and adding new operators to account for this exclusion. Sridhar K. – p. 6/20 Charmonia at the LHC
Velocity expansion Other scale is Mv ≪ M with v ≪ 1 . Suggests an expansion of the quarkonium wavefunction in v . c ( 3 S [1] c ( 3 P [8] 1 ) � + v 2 | c ¯ | J/ψ � = | c ¯ J ) g � + . . . (1) So there is an octet state in the J/ψ with P -state quantum numbers – which connects to the physical state through the emission of a non-perturbative gluon. Sridhar K. – p. 7/20 Charmonia at the LHC
Electric and Magnetic transitions So, in NRQCD quarkonium production and decay involves intermediate states where the Q ¯ Q pair has quantum numbers different from those of the physical quarkonium. Forms the physical state via chromo-electric or chromo-magnetic transitions. More explicitly, c ( 3 S [1] c ( 3 P [8] 1 ) � + v 2 | c ¯ | c ¯ J ) g � + c ( 3 S [8] c ( 1 S [8] v 2 | c ¯ 1 ) gg ) � + v 2 | c ¯ 0 ) g � + . . . Sridhar K. – p. 8/20 Charmonia at the LHC
P -state decays Consider the χ states: c ( 3 P [1] c ( 3 S [8] | χ � = v | c ¯ J ) � + v | c ¯ 1 ) g � (2) In the colour-singlet model the amplitude for χ decays into hadrons has a divergence. This is due to neglecting the colour-octet component. Colour-singlet model is flawed. Sridhar K. – p. 9/20 Charmonia at the LHC
J/ψ at CDF – I Sridhar K. – p. 10/20 Charmonia at the LHC
NRQCD factorization The cross section for production of a quarkonium state H is: F n � d n − 4 �O H n ( 2 S +1 L J ) � , σ ( H ) = (3) M Q n = { α,S,L,J } F n ’s are the perturbatively computable short-distance coefficients O n are operators of naive dimension d n , describing the long-distance effects. Factorization → momentum-independence of the non-perturbative elements. Sridhar K. – p. 11/20 Charmonia at the LHC
Tevatron data NRQCD gives a good description of the cross-sections for J/ψ and other charmonium states measured at the Tevatron. One of the crucial features of the data is the large p T tail which is due to gluon fragmentation. Fragmentation becomes important when p T > M and is naturally incorporated in NRQCD through colour-octet components. Sridhar K. – p. 12/20 Charmonia at the LHC
J/ψ at CDF – II 10 cdf <kT>=0 <kT>=0.7 <kT>=1.0 1 T σ /dp B d 0.1 0.01 0.001 4 6 8 10 12 14 16 18 20 p (GeV) T Figure 2: J/ψ at CDF Sridhar K. – p. 13/20 Charmonia at the LHC
J/ψ polarisation In fragmentation, the gluon transfers all its transverse polarisation to the c ¯ c pair. NRQCD has a heavy-quark symmetry – the spin and flavour degrees of freedom are irrelevant in the non-perturbative soft interactions – due to which the J/ψ inherits the transverse polarisation of the c ¯ c pair. The J/ψ at large- p T should be transversely polarised. Sridhar K. – p. 14/20 Charmonia at the LHC
Measuring polarisation Experimentally the cos θ ∗ distribution is measured where θ ∗ is the angle of the decay lepton in the J/ψ rest frame with respect to the J/ψ boost direction in the lab. Then dσ d cos θ ∗ ∼ (1 + α cos θ ∗ ) (4) where α is the polarisation parameter. α = 1 → Transverse polarisation α = − 1 → Longitudinal polarisation Sridhar K. – p. 15/20 Charmonia at the LHC
CDF polarisation data Sridhar K. – p. 16/20 Charmonia at the LHC
Alternate test of NRQCD The heavy-quark symmetry of NRQCD implies that the non-perturbative matrix elements are related to each other. For example, for η c production there are three contributions: from a colour-singlet 1 S 0 state and from colour-octet 1 P 1 and 3 S 1 channels. We need to know three non-perturbative parameters to predict the η c cross-section. Sridhar K. – p. 17/20 Charmonia at the LHC
Heavy-quark symmetry relations 1 [ 1 S 0 ] | 0 � = 1 3 � 0 | O J/ψ � 0 | O η c [ 3 S 1 ] | 0 � (1 + O ( v 2 )) , 1 8 [ 1 P 1 ] | 0 � = � 0 | O J/ψ � 0 | O η c [ 3 P 0 ] | 0 � (1 + O ( v 2 )) , 8 8 [ 3 S 1 ] | 0 � = � 0 | O J/ψ � 0 | O η c [ 1 S 0 ] | 0 � (1 + O ( v 2 )) . 8 This allows us to make predictions for η c production at the LHC. Sridhar K. – p. 18/20 Charmonia at the LHC
η c Production Sridhar K. – p. 19/20 Charmonia at the LHC
h c production A similar strategy may be exploited for h c production. More difficult resonance to study – has never been seen in hadron collisions. But large enough cross-sections for this state to be detected at the LHC. Will help study its properties more accurately. Sridhar K. – p. 20/20 Charmonia at the LHC
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