Euclidean versus Minkowski short distance

In this note we reexamine the possibility of extracting parton distribution functions from lattice simulations. We discuss the case of quasi-parton distribution functions, the possibility of using the reduced Ioffe-time distributions and the more recent proposal of directly making reference to the computation of the current-current $T$-product. We show that in all cases the process of renormalization hindered by lattice momenta limitation represents an obstruction to a direct Euclidean calculation of the parton distribution function.

Outline of the talk I shall reexamine the viability of recent proposals of computing Parton Distribution Functions (PDF's) directly on the lattice ensuing from the seminal Ji (PRL 110 (2013) (2018) 074508) the analysis of power subtractions employed by many groups, as presented by A.V. Radyushkin (arXiv:1807.07509v2 [hep-ph]) Disclaimer: It is impossible to give here due credit to all the papers that have appeared on this important subject. I apologize for that Bibliography The talk is based on the paper G. C. Rossi and M. Testa, Phys. Rev. D 96 (2017) no.1, 014507 See also G. C. Rossi and M. Testa, arXiv:1806.04428 [hep-lat] submitted to Phys. Rev. D

Minkowski metrics
The hadronic DIS cross section in the parton language reads Lorentz invariance implies for the bilocal In the canonical case F (P · ξ, ξ 2 ) is a regular function that needs to be evaluated for ξ 2 ≈ 0 We want to compute the Fourier Transform (FT) of F (P · ξ, 0) finally leading in the Bjorken limit to This is the standard argument relating the structure function f (x) (i.e. the FT of the bilocal matrix element) to the DIS cross section, W In the canonical case the bilocal can be Taylor expanded around ξ = 0 where traces denote form factors containing some g µ i µ j tensor For example, in the case of O µ 1 µ 2 , we have P|O µ 1 µ 2 |P = A 2 P µ 1 P µ 2 + B 2 g µ 1 µ 2 Physical PDFs are related to the A n form factors (moments) B n are spurious contributions that need to be subtracted out In Minkowski region this is automatically achieved by taking ξ 2 = 0 In the Euclidean case the situation is more complicated The desired structure function is recovered from the (obvious) formula Thus to remove trace terms in Euclidean region we must know P|φ(0)φ(z)|P for P z → ∞ as z → 0, while keeping α = P z z fixed In lattice simulations this requirement poses problems momenta are bounded from above by a −1 (inverse lattice spacing) this in turn limits the minimal value that z can take to be O(α a)

Renormalization -I
What about renormalization issues? DIS scaling in QCD is controlled by computable logarithmic corrections O µ 1 ...µn require not just a simple multiplicative renormalization as P|O µ 1 ...µn |P matrix elements are power divergent We need to resolve the mixing with lower dimensional (trace) operators to make finite A n and B n form factors In particular, to be able to take the limit P z → ∞ (necessary to eliminate the contamination from higher twists) one needs to make the B n 's finite The only renormalization considered in the original Ji paper was the multiplicative "matching condition", according to which one starts by considering the regularized quantitỹ Moments ofF renormalize multiplicatively and independently one from the others Renormalization -III If one could take the limit P z → ∞ the relation would allow computing the moments of the physical PDFs But performing this limit turns out to be "problematic", as we now argue Taking the n-th derivative with respect to z at z = 0 of (see slide 8)

Renormalization -IV
• The "matching" procedure has led to the relation Since P z can never exceed a −1 (one would never take momenta larger than the UV cutoff) such power divergent terms need to be subtracted out thus multiplicative renormalization is not enough PDF from the current-current T -product?

Recalling
P|Oµ 1 µ 2 ...µn (0)|P = AnPµ 1 Pµ 2 . . . Pµ n + traces , Oµ 1 µ 2 ...µn = φ(0) ∂ n φ(ξ) ∂ξ µ 1 ∂ξ µ 2 . . . ∂ξ µn ξ=0 and ignoring for a moment renormalization issues, we immediately get f (P · ξ; µ) = P|φ(0)φ(ξ)|P which is precisely the Ji formula Multiplicative renormalization of moments can be dealt with by means of the "matching condition" as discussed in slide 9 The conclusion is that the knowledge of moments is not enough to reconstruct the full PDF: one ends up with the Ji formula coupled to some perturbative subtraction 2 I'm now going to discuss the pro's and con's of this approach with the help of the formulation provided by A.V. Radyushkin 1807.07509v2 [hep-ph] Ioffe-time distributions and power divergent mixings One can prove the formula (in the notations of A.V. Radyushkin 1807.07509v2 [hep-ph]) The last term produces (unwanted) contributions in the |y | > 1 region (responsible for power divergent moments) One can thus think of subtracting out by hand these terms writing The difficulties posed by this procedure, which is widely used in actual simulations, are as follows Subtraction needs to be carried out before removing the cutoff The term in square parenthesis has a smooth P → ∞ limit but O(α 2 s ) corrections don't: at small lattice spacings they matter In the r.h.s. the PDF, f (y , µ 2 ), one is looking for appears 1 in practice it gets replaced by lattice Q(y , P) to leading order in α s Q(y , P) does not have the proper support properties one thus needs to enforce them by hand (non-localities introduced) 2 then the question arises are the moments of the PDF built in this way the matrix elements of the renormalized local DIS operators one finds in the Bjorken limit?

Conclusions
In this talk I have rediscussed the viability of the proposal of directly extracting PDF's from lattice simulations Apparently there are still missing ingredients in such a program related to the problem of subtracting power divergent trace terms In summary at this moment the original Ji formalism of using the "matched" bilocal operator does not allow accessing the full PDF from lattice simulations direct simulations of the current-current T -product surely allow extracting PDF moments (see e.g. Ma & Qiu and J. Karpie, K. Orginos, S. Zafeiropoulos) perhaps more promising is the idea of subtracting by hand in PT from lattice data, terms that ruin the compactness of the PDF support and are responsible for power divergent moments • If we don't, we get • In the last formulae we have taken the limit Λ → ∞ • Mixings with trace operators do not show up as (power) divergencies in f (ω) • Rather at finite P z they deform the expression of the latter • The limit P z → ∞ cannot be taken inside the integral • unless the integrand is "well behaved", i.e. made finite! Support truncation "by hand" destroys locality This quantity, the FT of which gives rise to a support-truncated F (ω, P z ), is not the matrix element of a bilocal operator lim Pz →∞ 1 π sin Pz (z−z ) z−z = δ(z − z ) cannot be taken because of the P z -dependence of the quantity under the integral unless the latter is the FT of a function with [−1, +1] support!