Applications of Renormalization Group Methods in Nuclear Physics – 6 Dick Furnstahl Department of Physics Ohio State University HUGS 2014
Outline: Lecture 6 Lecture 6: High-res. probes of low-res. nuclei Recap: Running Hamiltonians Parton distributions as paradigm Summary and challenges Extra: High-res. probes of low-res. nuclei
Outline: Lecture 6 Lecture 6: High-res. probes of low-res. nuclei Recap: Running Hamiltonians Parton distributions as paradigm Summary and challenges Extra: High-res. probes of low-res. nuclei
“Measuring” the QCD Hamiltonian: Running α s ( Q 2 ) 0.5 July 2009 Extractions from experiment can α s (Q) be compared (here at M Z ): Deep Inelastic Scattering e + e – Annihilation 0.4 Heavy Quarkonia τ -decays (N3LO) Quarkonia (lattice) 0.3 Υ decays (NLO) DIS F 2 (N3LO) 0.2 DIS jets (NLO) e+e– jets & shps (NNLO) electroweak fits (N3LO) 0.1 QCD α (Μ ) = 0.1184 ± 0.0007 s Z e+e– jets & shapes (NNLO) 1 10 100 Q [GeV] 0.11 0.12 0.13 The QCD coupling is scale α (Μ ) s Z dependent (“running”): α s ( Q 2 ) ≈ [ β 0 ln ( Q 2 / Λ 2 QCD )] − 1 cf. QED, where α em ( Q 2 ) is effectively constant for soft Q 2 : The QCD coupling strength α s is α em ( Q 2 = 0 ) ≈ 1 / 137 scheme dependent (e.g., “V” ∴ fixed H for quantum chemistry scheme used on lattice, or MS )
Running QCD α s ( Q 2 ) vs. running nuclear V λ 0.5 Vary scale (“resolution”) with RG July 2009 α s (Q) Scale dependence: SRG (or V low k ) Deep Inelastic Scattering e + e – Annihilation 0.4 running of initial potential with λ Heavy Quarkonia (decoupling or separation scale) 0.3 0.2 0.1 QCD α (Μ ) = 0.1184 ± 0.0007 s Z 1 10 100 Q [GeV] The QCD coupling is scale Scheme dependence: AV18 vs. N 3 LO dependent (cf. low-E QED): (plus associated 3NFs) α s ( Q 2 ) ≈ [ β 0 ln ( Q 2 / Λ 2 QCD )] − 1 But all are (NN) phase equivalent! The QCD coupling strength α s is scheme dependent (e.g., “V” Shift contributions between interaction scheme used on lattice, or MS ) and sums over intermediate states
JLab: Understanding “short-range correlations” in nuclei What is this vertex? e’ k � q = k − k � k e q ν = E k − E k � p � N 1 Q 2 = − q 2 Q 2 N p � x B = 2 2 m N ν A A ! 2 Subedi et al., Science 320, 1476 (2008) Higinbotham, arXiv:1010.4433 a) 3 SRC interpretation: r( 4 He/ 3 He) 2.5 2 NN interaction can scatter 1.5 q 1 states with p 1 , p 2 � k F 4 b) p � r( 12 C/ 3 He) to intermediate states with 3 p 1 1 p � 2 which are 2 p � 1 , p � 2 � k F p 2 1 knocked out by the photon 6 c) r( 56 Fe/ 3 He) 4 How to explain cross sections in terms of 2 low-momentum interactions? 1 1.25 1.5 1.75 2 2.25 2.5 2.75 1 . 4 < Q 2 < 2 . 6 GeV 2 x B Vertex depends on the resolution! Egiyan et al. PRL 96, 1082501 (2006)
Nuclear structure natural with low momentum scale But soft potentials don’t lead to short-range correlations (SRC)! 0.25 3 S 1 deuteron probability density 1.2 softened Nuclear matter 0.2 1 softened − 3 ] pair-distribution g(r) Argonne v 18 0.8 0.15 2 [fm − 1 -1 λ = 4.0 fm g(r) k F = 1.35 fm -1 | ψ (r)| λ = 3.0 fm 0.6 original 0.1 -1 λ = 2.0 fm 0.4 − 1 (NN only) Λ = 10.0 fm − 1 Λ = 3.0 fm 0.05 − 1 0.2 Λ = 1.9 fm Fermi gas original 0 0 0 2 4 6 0 1 2 3 4 r [fm] r [fm] Continuously transformed potential = ⇒ variable SRC’s in wf! Therefore, it seems that SRC’s are very scale/scheme dependent Analog in high energy QCD: parton distributions
Outline: Lecture 6 Lecture 6: High-res. probes of low-res. nuclei Recap: Running Hamiltonians Parton distributions as paradigm Summary and challenges Extra: High-res. probes of low-res. nuclei
Parton distributions as paradigm [C. Keppel] DIS Kinematics • Four-momentum transfer: q 2 ( E E ' ) 2 ( k k ' ) ( k k ' ) = − − − ⋅ − = m 2 m 2 2 ( EE ' | k | | k ' | cos ) = + − − θ = e e ' 2 2 4 EE ' sin Q ≈ − θ ≡ − 2 • Mott Cross Section ( c= 1 ): 2 2 2 ( d ) 4 E ' E ' σ α cos θ = ⋅ Mott d 4 E Ω 2 Q 2 2 2 4 E ' 1 α θ L : lepton ten sor = cos ⋅ 2 2 4 1 E ( 1 cos ) µ ν 16 E E ' sin 2 θ + − θ M 2 W : hadron ten sor 2 2 µ ν cos θ α 1 = 2 ⋅ 2 4 2 4 E sin 1 E ( 2 sin ) θ θ + a virtual photon of four- 2 M 2 momentum q is able to resolve Electron scattering of a spinless point particle structures of the order / √ q 2
Parton distributions as paradigm [C. Keppel] Simple parton model e � e P parton x = Q 2 / 2 P · q p quark = xP proton Bjorken scaling = ⇒ structure function F 2 independent of Q 2 Measured F 2 ( x ) gives quark momentum distribution � F 2 ( x , Q 2 ) ≈ F 2 ( x ) = e 2 q x q ( x ) q
Parton distributions as paradigm [C. Keppel] Simple parton model e � e P parton x = Q 2 / 2 P · q p quark = xP proton Bjorken scaling = ⇒ structure function F 2 independent of Q 2 Measured F 2 ( x ) gives quark momentum distribution � F 2 ( x , Q 2 ) ≈ F 2 ( x ) = e 2 q x q ( x ) q
Parton distributions as paradigm [C. Keppel] F 2 Higher the resolution (i.e. higher the Q 2 ) more low x partons we “see”. So what do we expect F 2 as a function of x at a fixed Q 2 to look like?
Parton distributions as paradigm [C. Keppel] F 2 (x) Three quarks with 1/3 of total proton x momentum each. 1/3 F 2 (x) Three quarks with some momentum x 1/3 smearing. F 2 (x) The three quarks radiate partons at low x. x 1/3 ….The answer depends on the Q 2 !
Parton vs. nuclear momentum distributions The quark distribution q ( x , Q 2 ) is scale and scheme dependent x q ( x , Q 2 ) measures the share of momentum carried by the quarks in a particular x -interval q ( x , Q 2 ) and q ( x , Q 2 0 ) are related by RG evolution equations
Parton vs. nuclear momentum distributions 2 10 0 10 SRCs% λ (k) (fm 3 ) − 2 10 n d − 4 10 15 No%SRCs% 10 λ (fm − 1 ) 5 0 1 2 3 4 5 k (fm − 1 ) The quark distribution q ( x , Q 2 ) is Deuteron momentum distribution scale and scheme dependent is scale and scheme dependent x q ( x , Q 2 ) measures the share of Initial AV18 potential evolved with momentum carried by the quarks SRG from λ = ∞ to λ = 1 . 5 fm − 1 in a particular x -interval q ( x , Q 2 ) and q ( x , Q 2 High momentum tail shrinks as 0 ) are related λ decreases (lower resolution) by RG evolution equations
Factorization: high-E QCD vs. low-E nuclear ,"&/*+#"0- 1"#$%&'("$'%) F 2 ( x , Q 2 ) ∼ � -232 a f a ( x , µ f ) ⊗ � F a 2 ( x , Q /µ f ) +,%&$8/'+$")#- 0%)38/'+$")#- ↔ ?'0+%)*#%-11'#'-)$ <"&$%)*/-)+'$= � � Separation between long- and � short-distance physics is not unique = ⇒ introduce µ f Choice of µ f defines border between long/short distance Form factor F 2 is independent of µ f , but pieces are not Q 2 running of f a ( x , Q 2 ) comes from choosing µ f to optimize extraction from experiment
Factorization: high-E QCD vs. low-E nuclear ,"&/*+#"0- Also has factorization assumptions 1"#$%&'("$'%) (e.g., from D. Bazin ECT* talk, 5/2011) F 2 ( x , Q 2 ) ∼ � Observable: Structure model: Reaction model: -232 a f a ( x , µ f ) ⊗ � F a 2 ( x , Q /µ f ) cross section spectroscopic factor single-particle cross section � S if σ if = +,%&$8/'+$")#- j σ sp 0%)38/'+$")#- ↔ | J f − J i | ≤ j ≤ J f + J i ?'0+%)*#%-11'#'-)$ <"&$%)*/-)+'$= • Is the factorization general/robust? � � Separation between long- and � (Process dependence?) short-distance physics is not What does it mean to be consistent unique = ⇒ introduce µ f between structure and reaction Choice of µ f defines border models? Treat separately? (No!) between long/short distance How does scale/scheme Form factor F 2 is independent dependence come in? of µ f , but pieces are not What are the trade-offs? (Does Q 2 running of f a ( x , Q 2 ) comes simpler structure always mean from choosing µ f to optimize much more complicated reaction?) extraction from experiment
Scheming for parton distributions Need schemes for both renormalization and factorization From the “Handbook of perturbative QCD” by G. Sterman et al. “Short-distance finite parts at higher orders may be apportioned arbitrarily between the C’s and φ ’s. A prescription that eliminates this ambiguity is what we mean by a factorization scheme. . . . The two most commonly used schemes, called DIS and MS, reflect two different uses to which the freedom in factorization may be put.” “The choice of scheme is a matter of taste and convenience, but it is absolutely crucial to use schemes consistently, and to know in which scheme any given calculation, or comparison to data, is carried out.” Specifying a scheme in low-energy nuclear physics includes specifying a potential, including regulators, and how a reaction is analyzed.
Standard story for ( e , e ′ p ) [from C. Ciofi degli Atti] In IA: “missing” momentum p m = k 1 and energy E m = E Common assumption: FSI and two-body currents treatable as independent add-ons to impulse approximation Is this valid?
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