Jets in e+A SIDIS and Denominator Regularization

We compute the in-medium jet broadening to leading order in energy in the opacity expansion. At leading order in αs the elastic energy loss gives a jet broadening that grows with ln E. The next-to-leading order in αs result is a jet narrowing, due to destructive LPM interference effects, that grows with ln2 E. We find that in the opacity expansion the jet broadening asymptotics are— unlike for the mean energy loss—extremely sensitive to the correct treatment of the finite kinematics of the problem; integrating over all emitted gluon transverse momenta leads to a prediction of jet broadening rather than narrowing. We compare the asymptotics from the opacity expansion to a recent twist-4 derivation and find a qualitative disagreement: the twist-4 derivation predicts a jet broadening rather than a narrowing. Comparison with current jet measurements cannot distinguish between the broadening or narrowing predictions. We comment on the origin of the difference between the opacity expansion and twist-4 results. We also introduce a novel regularization scheme in quantum field theory, denominator regularization (den reg). Den reg is as simple to apply as the usual dimensional regularization, works simply with a minimal subtraction scheme, and manifestly 1) maintains Lorentz invariance, 2) maintains gauge invariance, 3) maintains supersymmetry, 4) correctly predicts the axial anomaly, and 5) yields Green functions that satisfy the Callan-Symanzik equation. Den reg also naturally enables regularization in asymmetric spacetimes, finite spacetimes, curved spacetimes, and in thermal field theory.


Introduction
Many-body quantum chromodynamics (QCD) is a fascinating topic that connects the physics of the early universe to that of the emergent dynamics present in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC).One of the most important and valuable tools used to investigate the emergent dynamics in heavy ion collisions is jet tomography [1].Jets are the most direct probes of the relevant degrees of freedom present in the quark-gluon plasma (QGP) created in heavy ion collisions.In a perturbative QCD (pQCD) treatment of the energy lost by a high momentum parton, a quark or gluon, that propagates through the QGP, there are two channels: the elastic (or collisional) [2,3] and the inelastic (or radiative) [1].Jet suppression models based on leading order pQCD derivations of energy loss show incredible qualitative consistency with experimental data over many orders of magnitude of measurement [4].One may then naturally ask: what are the natural next step(s)?Obvious candidates include: computing higher orders in α s [5], considering small system size corrections [6][7][8], sub-eikonal corrections [9], extracting transport coefficients through a systematic analysis  [10], etc.In this work we specifically consider placing energy loss on a more rigorous footing; we also consider quantum field theories in small systems.
When thinking of rigorous pQCD calculations in QCD, one is generally thinking of factorization.In factorization, a large scale such as Q 2 dominates, and one considers the twist expansion, which one is expanding in powers of 1/Q 2 .There are clean, rigorous factorization theorems associated with certain processes such as e+p deep inelastic scattering (DIS), semi-inclusive DIS (SIDIS), Drell-Yan, etc. [11].Most energy loss calculations assume that factorization holds for energy loss processes.A very interesting recent calculation of momentum broadening in SIDIS considered the twist-4 contribution to the process [12][13][14].In this derivation, the production and subsequent interaction were considered on equal footing.(This treatment of production and final state interaction is not treated on equal footing in most energy loss derivations.)The SIDIS calculation was shown to be self-consistent at nextto-leading order (NLO).One would like to ask: is there a way to make an apples-to-apples comparison with an energy loss calculation, in order to quantify the importance of what appears to be a more rigorous derivation?We find that the answer is yes, where we choose to compare to the usual opacity expansion [15,16].

Comparison of Opacity Expansion with Twist 4 Expansion
Let us now compare order-by-order the opacity and twist expansions.
At zeroth order in opacity, there is no interaction between the produced particle (in which production is assumed factorized from subsequent evolution) and the post-nuclear collision fireball.Thus ∆p 2 T = 0, because we are interested in the change in jet broadening from e+p collisions to e+A collisions.
At first order in opacity, but leading order in α s , we take that the in medium Debye-screened scattering center is given by the Gyulassy-Wang model [17] , where µ ≈ gT is the chromoelectric Debye screening mass of the medium and q ⊥ is the transverse momentum of the t-channel gluon exchanged with the medium [16].Then, assuming that q 2 max ∼ µ E, one finds On the other hand, if we consider the twist 4 approach, then where x , and ẑ = z h z .T qg is the twist-4 quark-gluon correlation function, a generalization of the usual twist-2 parton distribution function.In the limit of a large and loosely bound nucleus, in which one may neglect the spatial and momentum correlations between the two nucleons, one has [14] an approximate factorization where in the last line we assumed for simplicity that the parton propagates through a nucleus of constant density of thickness L. In order to best compare to the energy loss derivation, we will remove the complication of the fragmentation process from the twist-4 approach by assuming exact parton-hadron duality, i.e. we will take D h/q (z, µ 2 f ) = δ(1 − z).We then have that 1 0 dz h Then the leading in α s contribution from the twist-4 approach is a factorized result dx B dy = dσ 0 dx B dy q(µ 2 f )L, and thus the twist 4 derivation gives in exact agreement with the opacity expansion.We show the full numerics of the first order in opacity, and NLO in α s in our original work [18].Surprisingly, the numerics clearly show a jet narrowing in nuclear media compared to vacuum.We sought to understand these full numerics with high-energy analytics.If we assumed that the kinematic upper bound in the transverse momentum of the emitted gluon could be neglected, we found a jet broadening, However, if one explicitly maintains the kinematic limits while still taking the E → ∞ limit, which is highly non-trivial [18], then one finds a jet narrowing: It's more difficult to extract the leading behavior of the twist 4 result at high energy.If one assumes that the color triviality breaking terms are small, trivializes the fragmentation functions, and makes the loosely bound nucleus approximation, then the twist 4 prediction is one of broadening, Since the two approaches give qualitatively different predictions for jet ∆p 2 ⊥ , one may ask: what do the data show?It turns out that measuring jet broadening is not an easy experimental task [19].However, there are hints of jet narrowing from SIDIS [20].

Denominator Regularization
Surprisingly, the relativistic, viscous hydrodynamics models that described the momentum distribution of low momentum particles in heavy ion collisions also appear to describe the momentum distribution of low momentum particles in high multiplicity p+A and p+p collisions [21,22].One may naturally wonder: are there finite system size corrections to thermodynamic quantities?
An investigation in the free gas of relativistic scalar particles showed that the small system size corrections for the usual quantities of interest such as the entropy, energy density, and pressure can be large, and that the qualitative changes appear to mimic the phase transition as seen in lattice QCD calculations, for systems of temperatures and sizes relevant for p+A and p+p collisions at RHIC and LHC [23].A quantitative analysis using quenched lattice QCD confirmed the qualitative insights from the free scalar particles, although the effects were smaller in the strongly coupled numerics [24].
Despite the breaking of conformal symmetry by an asymmetric, finite-sized system, the equation of state predicted by the free relativistic scalar theory was still trivial: e = 3p [25]; the finite system size corrections to the trace anomaly must come from the finite system size corrections to the running coupling.Computing the finite system size corrections to the running coupling in QCD is a daunting task: how does one compute loop diagrams for a gauge theory in a finite-sized system?One particular difficulty is the regularization scheme.Dimensional regularization assumes a high degree of symmetry in the problem, an amount of symmetry not present in a finite-sized system.In order to overcome this difficulty, we invented a new regularization procedure, denominator regularization (den reg) [7,8].
Instead of analytically continuing the number of dimensions that the field theory lives in, in denominator regularization the power of the denominator of a Feynman graph is analytically continued, after all denominators are combined using Feynman parameters.The power of the denominator is taken sufficiently large to ensure that the integral converges in the UV.Like in dim reg, a fictitious scale µ is introduced to preserve the dimensionality of the amplitude.The final, critical, ingredient in den reg is the generalization of the integrand to include coefficient functions f (n,p) ( ), where n is the original power of the denominator and p is the superficial degree of divergence of the integral, that smoothly goes to 1 as → 0. The f (n,p) are uniquely fixed by minimally requiring that the Laurent expansion of the amplitude is free of poles other than a single 1/ ; i.e. we choose f (n,p) such that the amplitude converges for all > 0. In particular, the f (n,p) cancel the UV poles of the form ( p 2 − ) −1 and IR poles that emerge for ∈ N + when the theory is massless.So far, the f (n,p) appear to be universal; once fixed for an integral in one amplitude, the same f (n,p) emerges from other amplitudes.
One can show that denominator regularization gives sensible results in φ 4 theory [7]: den reg reproduces the one loop correction to the four point function as derived in dimensional regularization in infinite volume; the one loop correction to the four point function in den reg in a finite sized, potentially asymmetric system satisfies unitarity; and subdivergences in the two loop correction to the four point function in infinite volume cancel naturally in den reg.One may further show that the one loop correction to the photon and gluon two point functions remain transverse, and the axial anomaly is correctly, manifestly predicted [8].A recent calculation showed that denominator regularization correctly predicts Higgs to gamma-gamma decay at one loop [26].Since the number of dimensions is held fixed, interesting future calculations include checking the Ward identities associated with supersymmetry (SUSY) in supersymmetric theories, where SUSY is broken in when the number of spacetime dimensions varies from 4, and for conservation of the energy momentum tensor at NLO in theories in asymmetric spaces.

Conclusions
We seek precision measurements of jet tomography in heavy ion collisions in order to extract quantitative insights into the emergent, many-body dynamics of QCD that emerge in experiments at RHIC and LHC.We showed an asymptotic analysis of p 2 T in e+A SIDIS from the perspective of twist-4 and from opacity expansion derivations.The twist-4 calculation predicted a jet broadening, but the opacity expansion calculation predicted a jet narrowing.Current data are ambiguous, with hints of jet narrowing.Intuition suggests that the jet narrowing prediction from the opacity expansion is due to its more careful treatment of the Landau-Pomeranchuk-Migdal (LPM) effect; the twist-4 derivation only captures the leading 1/E part of the destructive interference.While we began this investigation under the assumption that the twist-4 calculation is "better" than the opacity one, we found that the kinematic limits, especially on the angular momentum of the emitted gluon, k ⊥ , are very important.If one neglects the limit on k ⊥ in the opacity expansion, one incorrectly finds a jet broadening.Thus we strongly suspect that the factorization approach, which integrates out k ⊥ to infinity early in the derivation, needs to be generalized.At the same time, the opacity expansion approach should be generalized in order to put the initial state and final state interactions all on equal footing.We leave this interesting research to future work.
We also introduced the new regularization scheme, denominator regularization (den reg), which is well-suited for computing NLO corrections to amplitudes for quantum field theories in systems without the high degree of symmetry required for dimensional regularization.Den reg reproduces known results in infinite volume systems, but also allows for one to compute the one loop corrections to the four point function in φ 4 theory in an asymmetric, finite sized system.In addition, the subdivergences in the two loop correction to the scalar four point function in φ 4 theory correctly cancel in den reg.Den reg also preserves the transversality of the NLO correction to the photon and gluon two point functions to one loop, correctly predicts manifestly the axial anomaly, and gives the right answer for the Higgs decay to two photons.Future interesting work includes computing the finite system size corrections for the QED and QCD running couplings and the finite system size corrections to critical exponents in, e.g., the Ising model.