FF Uncertainties The value of the form factor at some fixed Q 2 is a - PowerPoint PPT Presentation

z Expansion and Nucleon Vector Form Factors GENIE z Expansion Workshop Fermilab, Batavia, IL Gabriel Lee Technion – Israel Institute of Technology ongoing work with J. Arrington, R. Hill, Z. Ye Sep 1, 2016 Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 1 / 9

Form Factors and ep Scattering ◮ Mott cross-section for scattering of a relativistic electron off a recoiling point-like nucleus is � dσ Z 2 α 2 cos 2 θ E ′ � M = E . 4 E 2 sin 4 θ d Ω 2 2 ◮ The Rosenbluth formula generalizes the above, � dσ � dσ , τ = − q 2 1 E + τ 1 � � � � G 2 ǫ G 2 R = 4 M 2 , ǫ = . M 1 + 2(1 + τ ) tan 2 θ d Ω d Ω 1 + τ M 2 ◮ The Sachs form factors G E ( q 2 ) , G M ( q 2 ) account for the finite size of the nucleus. In terms of the standard Dirac ( F 1 ) and Pauli ( F 2 ) form factors, q = Γ µ ( q 2 ) = G E + τG M 2 M σ µν q ν G M − G E γ µ + i . p � 1 + τ 1 + τ p � �� � � �� � F 1 ( q 2 ) F 2 ( q 2 ) ◮ The form factors are normalized at q 2 = 0 to the charge and anomalous magnetic moments, e.g., for the proton, G p E (0) = 1 , G p M (0) = µ p . ◮ Quantities like the charge radius and the form factor curvature are defined by derivatives of G evaluated at q 2 = 0 , e.g., 6 ∂G � � r 2 � ≡ q 2 =0 . � ∂q 2 G (0) � Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 2 / 9

Earlier Ans¨ antze for G E , G M � dσ � dσ 1 E + τ � � � � G 2 ǫ G 2 R = M d Ω d Ω 1 + τ M ◮ Previous analyses used simple functional forms for G E , G M , with expansions truncated at some finite k max : k max a k ( q 2 ) k , G poly ( q 2 ) = � polynomials, Simon et al. (1980), Rosenfelder (2000) k =0 1 G invpoly ( q 2 ) = k =0 a k ( q 2 ) k , inverse polynomials, Arrington (2003) � k max 1 G cf ( q 2 ) = , continued fractions, Sick (2003) q 2 a 0 + a 1 q 2 1+ a 2 1+ ... ◮ Hill & Paz (2010) showed that the above functional forms exhibit pathological behaviour with increasing k max . ◮ Other, more complicated functional forms exist, see, e.g., Bernauer et al. (2014). Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 3 / 9

The Bounded z Expansion ◮ For the proton, QCD constrains the form factors to be analytic in t ≡ q 2 ≡ − Q 2 outside of a time-like cut beginning at t cut = 4 m 2 π , the two-pion production threshold. Clearly this presents an issue with convergence for expansions in the variable q 2 . Hill & Paz (2010) ◮ Using a conformal map, we obtain a true small-expansion variable z for the physical region: t z √ t cut − t −√ t cut − t 0 z ( t ; t cut , t 0 ) = √ t cut − t + √ t cut − t 0 − Q 2 4 m 2 max π k max k max a k [ z ( q 2 )] k , b k [ z ( q 2 )] k . � � G E = G M = k =0 k =0 ◮ The physical kinematic region of scattering experiments lies on the negative real line. For a set of data with a maximum momentum transfer Q 2 max , this is represented by the blue line. ◮ The conformal map has a parameter t 0 , which is the point in t plane that is mapped to z ( t 0 ) = 0 . ◮ By including other data, such as from ππ → N ¯ N or eN scattering, it is possible to move the t cut to larger values, improving the convergence of the expansion. Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 4 / 9

More on t 0 t z √ t cut − t −√ t cut − t 0 z ( t ; t cut , t 0 ) = √ t cut − t + √ t cut − t 0 − Q 2 4 m 2 max π ◮ Since the conformal mapping is an analytic function, on the closed set t ∈ [ − Q 2 max , 0] , it attains a maximum | z max | at one of the endpoints t = 0 or t = − Q 2 max . ◮ We can find an optimal choice t opt to minimize this value | z max | , 0 1 max = (1 + Q 2 4 − 1 � � max /t cut ) � t opt ( Q 2 | z | opt 1 + Q 2 max ) = t cut 1 − max /t cut ⇒ . 0 1 (1 + Q 2 4 + 1 max /t cut ) ◮ Choosing an appropriate t 0 can make a big difference on the required k max for convergence; below n min is such that | z | n min < 0 . 01 . Q 2 max [GeV 2 ] t 0 [GeV 2 ] | z | max n min 1 0 0.58 8.3 t opt (1 GeV 2 ) = − 0 . 21 1 0.32 4.0 0 3 0 0.72 14 t opt (3 GeV 2 ) = − 0 . 41 3 0.43 5.4 0 Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 5 / 9

Sum Rules from Large Q 2 Behaviour ◮ QCD also demands that the form factor fall off faster than 1 /Q 4 up to logs as Q 2 → ∞ (dipole-like behaviour), d n G � � Q n G ( − Q 2 ) � � → 0 → 0 , n = 0 , 1 , 2 , 3 , ⇒ � � dz n � Q 2 →∞ � z → 1 ◮ For a form factor employing the z expansion truncated at some k max , we can enforce this by implementing four sum rules, Lee, Arrington, Hill (2015) k max � k ( k − 1) · · · ( k − n + 1) a k = 0 , n = 0 , 1 , 2 , 3 . k =1 ◮ In practice, we constrain the 4 highest-order coefficients in a fit using these sum rules by solving a system of equations derived from these sum rules. Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 6 / 9

FF Uncertainties ◮ The value of the form factor at some fixed Q 2 is a linear function of the coefficients, which are the parameters in the fit: k max k max a k ( z k − z k G ( Q 2 ; a ) = � a k z k ( Q 2 ) = g + � 0 ) , k =0 k =1 where we used the normalization constraint to re-express the form factor in the second equality, with z 0 = z ( Q 2 = 0; t 0 ) and, e.g., for the proton, g = (1 , µ p ) for the (electric, magnetic) form factors. ◮ To obtain the uncertainty, we note that dG ( Q 2 ; a ) = z k − z k 0 ; da k if C kl is the covariance matrix for the coefficients a k , we have � k max � 1 / 2 C kl ( z k − z k 0 )( z l − z l δG ( Q 2 ) = � 0 ) . k,l =1 ◮ If a fit includes sum rules, there are straightforward complications to the above derivations. Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 7 / 9

Datasets Proton: three separate datasets for the available elastic ep -scattering data. ◮ “Mainz” (cross sections) : high-statistics dataset with Q 2 < 1 . 0 GeV 2 . Originally 1422 data points in the full dataset released by the A1 collaboration [Bernauer et al. (2014)]. This was rebinned to 658 points with modified uncertainties in Lee et al. (2015). ◮ “world” (cross sections) : compilation of datasets from other experiments from 1966–2005, 569 data points with Q 2 < 35 GeV 2 . Update of dataset used in Arrington et al. (2003, 2007). ◮ “pol” (FF ratios) : 66 polarization measurements with Q 2 < 8 . 5 GeV 2 , see, e.g., Arrington et al. (2003, 2007), Zhan et al. (2011). Neutron: the data is split into measurements for G n E and G n M separately. E : 37 measurements Q 2 < 3 . 4 GeV 2 . ◮ G n M : 33 measurements Q 2 < 10 GeV 2 . ◮ G n Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 8 / 9

Ongoing Work Proton : a combined fit of the three datasets to provide parameterizations and tabulations (including uncertainties) of G p E , G n E with: ◮ correlated systematic parameters for the Mainz data floating in the fit, ◮ implementation of sum rules enforcing dipole-like behaviour of G E , G M at high- Q 2 , ◮ updated application of radiative corrections, e.g., high- Q 2 finite two-photon exchange corrections, ◮ focus on two Q 2 ranges, i.e., 1 – 3 GeV 2 and the entire range of available data (up to 35 GeV 2 ). Neutron : ◮ including this data in a combined fit allows us to separate the isoscalar and isovector channels, G ( 0 1 ) = G p E , which allows us to move t cut for G (0) E ± G n to the three-pion E E production threshold, Hill and Paz 2010 ◮ updated determination of neutron electric and magnetic radii. Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 9 / 9

k max Dependence 1750 1700 χ 2 1650 ◮ We can also test the 1600 dependence of the fit results on the choice of k max . 0 . 94 ◮ The fit has converged for 0 . 92 r E [fm] k max = 10 . 0 . 90 ◮ We use a default of k max = 12 0 . 88 in fits: for Q 2 max = 1 . 0 GeV 2 0 . 86 (statistics-only errors), 0 . 80 r E = 0 . 920(9) fm, r M = 0 . 743(25) fm. 0 . 75 r M [fm] 0 . 70 0 . 65 4 6 8 10 12 k max Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 10 / 9

Unbounded z Expansion Fits Fits using unbounded z expansion performed by Lorenz et al. Eur. Phys. J. A48, 151; Phys. Lett. B737, 57 1700 ◮ Sum rules such as ( t 0 = 0 ) 1650 k max G E ( q 2 = 0) = � a k = 1 χ 2 1600 k =0 1550 tell us a k → 0 as the k becomes 1500 large. 1 . 0 ◮ The Sachs form factors are also 0 . 9 known to fall off as Q 4 up to logs r E [fm] 0 . 8 for large Q 2 (dipole-like 0 . 7 behaviour at large Q 2 ). 0 . 6 ◮ To test enlarging the bound, we 0 . 5 3 . 0 took | a k | max = | b k | max /µ p = 10 , and found r E = 0 . 916(11) fm, 2 . 5 r M = 0 . 752(34) fm. r M [fm] 2 . 0 ◮ However, as | a k | max → ∞ , | a k | 1 . 5 for large k takes on unreasonably 1 . 0 large values, in conflict with QCD. 0 . 5 5 6 7 8 9 10 k max Gabriel Lee (Technion) z Expansion and Nucleon Vector Form Factors Sep 1, 2016 11 / 9