The Gluon Structure of Hadrons and Nuclei from Lattice QCD

EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

The gluon structure of hadrons and nuclei from lattice QCD

Phiala Shanahan1,2,3, 1Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, USA 2Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA 3Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA

Abstract. I discuss recent lattice QCD studies of the gluon structure of hadrons and light nuclei. After very briefly highlighting new determinations of the gluon contributions to the nucleon’s momentum and spin, presented by several collaborations over the last year, I describe first calculations of gluon generalised form factors. The generalised transversity gluon distributions are of particular interest since they are purely gluonic; they do not mix with quark distributions at leading twist. In light nuclei they moreover provide a clean signature of non-nucleonic gluon degrees of freedom, and I present the first evidence for such effects, based on lattice QCD calculations. The planned Electron- Ion Collider, designed to access gluon structure quantities, will have the capability to test this prediction, and measure a range of gluon observables including generalised gluon distributions and transverse momentum dependent gluon distributions, within the next decade.

1 Introduction

A detailed picture of the quark structure of nucleons has emerged over the last half-century, through increasingly precise measurements of quantities such as the nucleon electromagnetic form factors and quark momentum distribution functions, as well as many spin-dependent aspects of nucleon structure [1, 2]. In concert, lattice QCD calculations of such quantities have become sophisticated and precise, often with fully-controlled uncertainties. In contrast, the gluonic structure of nucleons (be- yond the gluon density) remains relatively unexplored. Gluon structure quantities are much harder to measure experimentally than their quark analogues, as the gluon does not couple to electromagnetic probes and thus other approaches must be taken, for example using the Drell-Yan process. This lack of information has inspired the proposal and planning of an Electron-Ion Collider (EIC), which was the highest priority for new construction in the 2015 NSAC long range plan of the US nuclear physics community [8]. The primary mission of this new experiment [9, 10] is to determine aspects of the gluon structure of hadrons and nuclei with unprecedented precision. Complementary experiments at Jeﬀerson National Accelerator Facility [11] and at the LHC [12] also aim to study various gluon distributions. In this light, lattice QCD calculations of gluon structure quantities have taken on new importance, and there has been signiﬁcant progress on this front over the last year [5, 13–16], expand- ing and building on pioneering lattice QCD studies of the unpolarised gluonic structure of the pion

Speaker, e-mail: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

and nucleon [17–19]. On a decade-long timescale, such calculations will guide the design require- ments of the EIC by revealing the magnitude of observables that are key targets of the experimental program. These include the gluon generalised parton distributions (GPDs) and transverse-momentum distributions (TMDs) of the nucleon, as well as nuclear physics quantities such as gluon analogues of the EMC effect and ‘exotic glue’ observables which encode non-nucleonic gluon structure in nuclei. Furthermore, there is the exciting possibility for fully-controlled lattice QCD calculations of these gluon observables to serve as QCD predictions ahead of first data-taking at the EIC. Here I discuss some of the recent progress in lattice QCD calculations of several aspects of the gluon structure of hadrons and nuclei, addressing the questions: How much do gluons contribute to static properties of the nucleon, e.g., momentum, mass and spin? • What is the three-dimensional distribution of gluons in the nucleon, i.e., TMDs, GPDs? How does • the ‘gluon radius’ of a nucleon compare with an equivalent ‘quark radius’ or the charge radius? How is the gluon structure of a nucleon modified in a nucleus i.e., what are the magnitudes of gluon • analogues of the EMC effect?

2 Gluon structure of the hadrons

Several collaborations have performed novel calculations of gluon structure observables in hadrons over the last year. In particular, the gluon contributions to the nucleon’s momentum [5, 13] and spin [5, 6] have been directly calculated at quark masses corresponding to the the physical value of the pion mass, and the ﬁrst lattice QCD calculations of moments of gluon GPDs [14, 15] have been performed. These calculations are summarised in the following sections.

2.1 Nucleon momentum fraction

Since the 2016 lattice conference, there have been two direct calculations of the gluon light cone momentum fraction x in the nucleon at quark masses corresponding to the physical value of the g pion mass [5, 13]. Previously, a number of calculations of this quantity had been performed at larger- than-physical values of the quark masses, paving the way for these new results [18, 20, 21]. Notably, Ref. [5] includes a complete decomposition of the nucleon’s momentum into quark and gluon connected and disconnected contributions, with the momentum sum rule not imposed, but found to be sat- isﬁed within uncertainties as shown in Fig. 1. The gluon contribution in the MS scheme at 2GeV was found to be 27(3)%. This result is in slight tension with that from the phenomenological CT14NNLO parton distribution ﬁts [22]. This is also shown in Fig. 1, along with previous lattice QCD results and the preliminary result from Ref. [13] which also includes calculations at quark masses corresponding to the physical value of the pion mass. It will be interesting to see whether this tension persists as calculations are further improved.

2.2 Nucleon spin

Since the original discovery of the so-called proton spin crisis—the realisation that quarks carry only a relatively small fraction of the proton spin—by the European Muon Collaboration (EMC) [23], there has been enormous eﬀort to resolve this puzzle. Over the last year, important progress has been made on this front from lattice QCD, with complete and fully-controlled determinations of various nucleon spin contributions, including the poorly-known gluon contributions [5, 6].

2 5

x q+ x g =0.497(12)(5) conn.+0.307(121)(95) disc.+ TABLE II: Our results for the intrinsic spin ( 1 ∆⌃), angular qh i h i | | 2 0.267(12)(10) gluon =1.07(12)(10) as is the spin sum momentum (L) and total (J) contributions to the nucleon P | spin and to the nucleon momentum x , in the MS-scheme of quarks and gluons giving JN = q Jq + Jg = h i 0.408(61)(48) + 0.133(11)(14) = 0.541(62)(49) resolving at 2 GeV, from up (u), down (d) and strange (s) quarks and P from gluons (g), as well as the sum of all contributions (tot.), a long-standing puzzle. where the first error is statistical and the second a systematic Acknowledgments: We thank all members of ETMC due to excited states. for an enjoyable collaboration and in particular Fernanda 1 ∆⌃ JLx 2 h i Ste↵ens for fruitful discussions. We acknowledge funding u 0.415(13)(2) 0.308(30)(24) -0.107(32)(24) 0.453(57)(48) d -0.193(8)(3) 0.054(29)(24) 0.247(30)(24) 0.259(57)(47) from the European Union’s Horizon 2020 research and s -0.021(5)(1) 0.046(21)(0) 0.067(21)(1) 0.092(41)(0) innovation program under the Marie Sklodowska-Curie g - 0.133(11)(14) - 0.267(22)(27) grant agreement No 642069. This work used computa- tot. 0.201(17)(5) 0.541(62)(49) 0.207(64)(45) 1.07(12)(10) tional resources from the Swiss National Supercomputing Centre (CSCS) under project IDs s540, s625 and s702, from the John von Neumann-Institute for Computing on show schematically the variousEPJ Web contributions of Conferences 175, 01015 to (2018) the spin the Jurecahttps://doi.org/10.1051/epjconf/201817501015 and the BlueGene/Q Juqueen systems at the Lattice 2017 and momentum fraction. research center in Jülich from a Gauss allocation on Su- perMUC with ID 44060. and nucleon [17–19]. On a decade-long timescale, such calculations will guide the design require- 0.8 ments of the EIC by revealing the magnitude of observables that are key targets of the experimental program. These include the gluon generalised parton distributions (GPDs) and transverse-momentum distributions (TMDs) of the nucleon, as well as nuclear physics quantities such as gluon analogues of 0.6 the EMC effect and ‘exotic glue’ observables which encode non-nucleonic gluon structure in nuclei. Furthermore, there is the exciting possibility for fully-controlled lattice QCD calculations of these ● gluon observables to serve as QCD predictions ahead of first data-taking at the EIC. 0.4[1] J. Ashman et● al. (European Muon), Phys. Lett. B206, Here I discuss some of the recent progress in lattice QCD calculations of several aspects of the gluon structure of hadrons and nuclei, addressing the questions: 364 (1988). ● [2] J. Ashman● et al. (European Muon), Nucl. Phys. B328, How much do gluons contribute to static properties of the nucleon, e.g., momentum, mass and spin? • 0.2 1 (1989). What is the three-dimensional distribution of gluons in the nucleon, i.e., TMDs, GPDs? How does • the ‘gluon radius’ of a nucleon compare with an equivalent ‘quark radius’ or the charge radius? [3] C. A. Aidala, S. D. Bass, D. Hasch, and G. K. Mallot, 0.0 Rev. Mod. Phys. 85, 655 (2013), 1209.2803. How is the gluon structure of a nucleon modified in a nucleus i.e., what are the magnitudes of gluon • analogues of the EMC effect? [4] A. Adare et al. (PHENIX), Phys. Rev. D90, 012007 (2014), 1402.6296. χ QCD 2017 χ QCD 2012 2 Gluon structure of the hadrons FIG. 3: Left: Nucleon spin decomposition. Right: Nu- [5] P. DjawothoQCDSF(STAR), 2012 Nuovo Cim. C036, 35 (2013), cleon momentum decomposition. All quantities are given in 1303.0543. Several collaborations have performed novel calculations of gluon structure observables in hadrons

[6] D. de Florian,Alexandrou et al 2017 R. Sassot, M. Stratmann, and W. Vogel- over the last year. In particular, the gluon contributions to the nucleon’sthe momentumMS-scheme [5, 13] and at 2 GeV. The striped segments show valence spin [5, 6] have been directly calculated at quark masses corresponding to the the physical value of (a) Figure taken from Ref. [5], which includes a calculation (b) Summarysang, figure, showing Phys. results Rev. from Refs. Lett. [5, 13113, , 012001 (2014), 1404.4293. quark contributions (connected)of the gluonand contribution the solid to the nucleon segments momentum the(green 18, 21]. The horizontal band shows the phenomenologi- the pion mass, and the first lattice QCD calculations of moments of gluon GPDssea quark [14, 15] have and been gluon contributionsvertical bar (disconnected). labelled ‘g’) at the quark masses corresponding cal result[7] fromS. the Alekhin, CT14NNLO parton J. Blmlein, distribution global S. Moch, and R. Placakyte (2017), performed. These calculations are summarised in the following sections. to the physical value of the pion mass. fits from Ref.1701.05838. [22].

Figure 1. Gluon contribution to the nucleon momentum fraction,[8] fromY.-B. recent Yang, lattice QCD R. calculations. S. Sufian, A. Alexandru, T. Draper, M. J. 2.1 Nucleon momentum fraction Conclusions: In this work we present a calcula- Glatzmaier, K.-F. Liu, and Y. Zhao, Phys. Rev. Lett. Since the 2016 lattice conference, there have been two direct calculationstion of the ofgluon the light quark cone and gluon contributions to the pro- 118, 102001 (2017), 1609.05937. momentum fraction x g in the nucleon at quark masses corresponding to the physical value of the There is a continuum of ways to decompose the nucleon[9] C. spin Alexandrou, into contributions from M. quark Constantinou, and K. Hadjiyiannakou, ton spin, directly at the physical point. Individual pion mass [5, 13]. Previously, a number of calculations of this quantity had been performed at larger- gluon helicity and spin or orbital angular momentum. Two commonly-usedC. Kallidonis, decompositions G. Koutsou, are the Ji K. Jansen, C. Wiese, and than-physical values of the quark masses, paving the way for these new resultscomponents [18, 20, 21]. Notably, are computeddecomposition: for the up, down, strange Ref. [5] includes a complete decomposition of the nucleon’s momentum into quark and gluon con- Ji 1 A. V. Avils-Casco, PoS LATTICE2016, 153 (2016), and charm quarks, including both connected (valence)J = ∆Σq + Lq + Jg, (1) nected and disconnected contributions, with the momentum sum rule not imposed, but found to be sat- N 2 1611.09163. q=u,d,s,c... isfied within uncertainties as shown in Fig. 1. The gluon contribution in the MSand scheme disconnected at 2GeV was (sea) quark contributions. Our final Σ [10] C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, found to be 27(3)%. This result is in slight tension with that from the phenomenologicalnumbers CT14NNLO are collected in Tablewhere q II.denotes The the quark quark helicity, intrinsicLq refers to the quark angular momentum, and Jg represents the parton distribution fits [22]. This is also shown in Fig. 1, along with previous lattice QCD results and gluon spin, and the Jaffe-Manohar decomposition: C. Kallidonis, G. Koutsou, K. Jansen, H. Panagopoulos, the preliminary result from Ref. [13] which also includes calculations at quarkspin masses from corresponding connected and disconnected contributions is F. Ste↵ens, A. Vaquero, and C. Wiese, PoS DIS2016, JM 1 to the physical value of the pion mass. It will be interesting to see whether1 this tension persists as J = ∆Σq + q +∆g + g, (2) ∆⌃ =0.299(12)(3) 0.098(12)(4)N = L 240 (2016),L 1609.00253. 2 u+d+s conn. disc= ,., , ... 2 calculations are further improved. | | q u d s c [11] C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, 0.201(17)(5), while the total quark spin is Ju+d+s = where ∆g is known as the gluon helicity, respectively. The last year has seen not only controlled lattice 0.255(12)(3) +0.153(60)(47) =0.408(61)(48). K. Jansen, H. Panagopoulos, and C. Wiese (2016), 2.2 Nucleon spin conn. QCD calculationsdisc. of the gluon contributions in each decomposition1611.06901. [5, 6], but notable new discussion Our result for| the intrinsic quarkof the di spinfference| contribution and interpolation between agrees these different decompositions of the nucleon spin from a Since the original discovery of the so-called proton spin crisis—the realisation that quarks carry only lattice QCD perspective [24]. [12] C. Alexandrou, EPJ Web Conf. 137, 01004 (2017), a relatively small fraction of the proton spin—by the European Muon Collaborationwith (EMC) the [23 upper], there bound set byAll a terms recent in the Ji phenomenologi- spin decomposition are gauge invariant,1612.04644. and the total quark contribution to the has been enormous effort to resolve this puzzle. Over the last year, importantcal progress analysis has been made of experimentalnucleon data spin from in this COMPASS decomposition is measurable [45], through[13] deeplyM. Constantinou, virtual compton scattering. in SPIN There 2016 (2017), 1701.02855. on this front from lattice QCD, with complete and fully-controlled determinations of various nucleon 1 is no further decomposition of the gluon spin contribution into spin or orbital angular momentum spin contributions, including the poorly-known gluon contributions [5, 6]. which found 0.13 < 2 ∆⌃ components,< 0.18. but The the total results gluon spin forcan beL calculatedq [14] in latticeR. QCD Frezzotti, using standard P. A.techniques. Grassi, This S. Sint, and P. Weisz (Alpha), and Jq in Table II are also consistent with an analysis of JHEP 08, 058 (2001), hep-lat/0101001. generalized parton distributions [45]. Using the spin sum [15] R. Frezzotti and G. C. Rossi, JHEP 08, 007 (2004), hep- 3 one would deduce that J = 1 J =0.092(61)(48), which lat/0306014. g 2 q [16] A. Abdel-Rehim et al. (ETM) (2015), 1507.05068. is consistent with taking J = 1 x =0.133(11)(14) g 2 h ig [17] C. Alexandrou and C. Kallidonis (2017), 1704.02647. via the direct evaluation of the gluon momentum frac- [18] K. Osterwalder and E. Seiler, Annals Phys. 110, 440 g tion, which suggests that B20(0) is indeed small. Fur- (1978). thermore, we find that the momentum sum is satisfied [19] R. Frezzotti and G. C. Rossi, JHEP 10, 070 (2004), hep- EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017 4

imply a polarization of gluons in the proton at interme- diate momentum scales, in the region of momentum frac- NEW FIT tions accessible at RHIC. This constitutes a new ingre- 1 90% C.L. region dient to our picture of the nucleon. While it is too early DSSV* 90% C.L. region to draw any reliable conclusions on the full gluon spin ∫ dx ∆ g(x) 0.05 0.001 DSSV contribution to the proton spin, our analysis clearly sug- 0.5 gests that gluons could contribute significantly after all. This in turn also sheds a new light on the possible size of orbital angular momenta of quarks and gluons. We hope gluon helicity that future experimental studies, as well as lattice-QCD 0 computations that now appear feasible [20], will provide further information on ∆g(x) and eventually clarify its role for the proton spin. We plan to present a full new -0.5 Q2 = 10 GeV2 global analysis with details on all polarized parton distributions once the 2009 RHIC data have become final -0.2 -0.1 -0 0.11 0.2 0.3 and additional information on the quark and antiquark ∫ dx ∆g(x) helicity distributions, in particular from final data on W 0.05 boson production at RHIC, has become available. Also, FIG. 5: 90% C.L. areas in the plane spanned by the truncated on the theoretical side, a new study of pion and kaon Figure 2. Figure modifiedmoments from Ref. of [∆26g],computed showing the for results 0.05 of ax global1 QCD and0 analysis.001 ofx polarised parton distributions, to include0 the.05 lattice at Q QCD2 = result 10 GeV of Ref.2.[ Results6] as the orange for≤ DSSV, band.≤ DSSV*, and≤ our≤ fragmentation functions should precede the next global new analysis, with the symbols corresponding the respective analysis of polarized parton distributions. values of each central fit, are shown. We thank E.C. Aschenauer, K. Boyle, P. Djawotho, and C. Gagliardi for useful communications. M.S. was supported in part by U.S. DoE (contract number DE- very limited information on ∆g is also available from AC02-98CH10886) and BNL-LDRD project 12-034. This scaling violations of the DIS structure function g which year, such a calculation was performed at quark masses corresponding to the physical1 pion masswork [5], was supported by CONICET, ANPCyT and UBA- is, of course, fully included in our global QCD analy- in a study that presented a complete decomposition of the nucleon spin in this basis. It is notableCyT. that the percentage gluonsis. Overall,contribution the to constraints the nucleon spinon ∆ foundg(x) in in, this say, study—about the regime 27%—is the 0.001 x 0.05 are much weaker than those in the same percentage gluon contribution≤ ≤ to the momentum fraction of the nucleon found in the same set of calculations, as discussedRHIC in region, the previous as can section. be inferred Importantly, from the Fig. total 1. spin Very of the little nucleon in this contribution to ∆G is expected to come from1 x>0.2. study, with all contributions determined separately, was consistent with 2 , as required by the spin sum rule. Figure 5 shows our estimates for the 90% C.L. area ∗ Electronic address: defl[email protected] in the plane spanned by the truncated moments of ∆g † Electronic address: [email protected] The Jaffe-Manoharcalculated approach gives in 0 a. canonical05 x partonic,1 and but gauge-variant, 0.001 x decomposition0.05 of‡ theElectronic address: [email protected] nucleon spin into quarkfor andQ gluon2 = spin10 GeV and2 orbital.≤ Results angular≤ are momentum presented contributions≤ both≤ for in the the infinite§ mo-Electronic address: [email protected] [1] See e.g.: E. Leader and C. Lorcé, arXiv:1309.4235, and mentum frame in the light-coneDSSV* and gauge. our The new spin fit. contributions The symbols in this in decomposition Fig. 5 denote are measurable through polarised deep inelastic scattering. As it is defined in the light cone frame, the Jaffe-Manoharreferences therein. the actual values for the best fits in the DSSV, DSSV*, [2] For a review, see e.g.: C. A. Aidala, S. D. Bass, D. Hasch, gluon helicity is not directly accessible to lattice QCD defined on a Euclidean spacetime. In Ref. [6], and the present analyses. We note that for our new cen- and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013). the large momentum effective theory [25] approach was used to access the gluon helicity from lattice 1 2 [3] D. de Florian, R. Sassot, M. Stratmann, and W. Vogel- QCD for the first time.tral In thisfit the approach, combined matrix integral elements0 that.001 definedx∆g the(x, gluon Q ) accounts helicity in the infinite for over 90% of the full ∆G at Q2 = 10 GeV2. Not sur- sang, Phys. Rev. Lett. 101, 072001 (2008); Phys. Rev. D momentum limit are calculated, and the extrapolation to large momentum is performed. In the cal-80, 034030 (2009). culation of Ref. [6], whichprisingly, included the several main lattice improvement spacings and in our volumes new including analysis an is toensemble[4] with R. D. Ball et al. [NNPDF Collaboration], Nucl. Phys. quark masses correspondingshrink to the the allowed physical pion area mass, in the the horizontal most significant direction, uncertainty corre- arose from theB 874, 36 (2013); J. Blümlein and H. Böttcher, Nucl. large-momentum extrapolation.sponding The to theresult much is compared better with determination phenomenological of ∆ resultsg(x) in from a globalPhys. B 841, 205 (2010); E. Leader, A. V. Sidorov, and analysis of polarised partonrange distributions 0.05 x [026.2] inby Fig. the2 2009. The RHIC determinations data. Evidently, are entirely consistent,D. B. Stamenov, Phys. Rev. D 82, 114018 (2010); M. Hi- with the lattice QCD resultthe uncertainty providing≤ a significantly≤ in the smaller- improvedx range constraint is still on very the total signif- gluon helicity.rai and S. Kumano, Nucl. Phys. B 813, 106 (2009); icant, and better small-x probes are badly needed. Data C. Bourrely, F. Buccella, and J. Soffer, Phys. Rev. D from the 2013 RHIC run at √s = 510 GeV may help 83, 074008 (2011); P. Jimenez-Delgado, A. Accardi, and 4 W. Melnitchouk, Phys. Rev. D 89, 034025 (2014). here a bit. In the future, an Electron Ion Collider would [5] See: E. C. Aschenauer et al., arXiv:1304.0079. provide the missing information, thanks to its large kine- [6] P. Djawotho [Star Collaboration], arXiv:1303.0543. 2 matic reach in x and Q [19]. [7] A. Adare et al. [Phenix Collaboration], Conclusions and outlook.— We have presented a new arXiv:1402.6296. global analysis of helicity parton distributions, taking [8] A. Adare et al. [Phenix Collaboration], Phys. Rev. Lett. 103, 012003 (2009). into account new and updated experimental results. In [9] A. Adare et al. [Phenix Collaboration], Phys. Rev. D 79, particular, we have investigated the impact of the new 012003 (2009). 0 data on ALL in jet and π production from RHIC’s 2009 [10] L. Adamczyk et al. [Star Collaboration], Phys. Rev. D run. For the first time, we find that the jet data clearly 86, 032006 (2012). EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017 4 imply a polarization of gluons in the proton at interme- diate momentum scales,2.3 in the3D gluon region structure of momentum frac- NEW FIT tions accessible at RHIC.To completely This constitutes describe the a new gluon ingre- distribution in the nucleon—to achieve real three-dimensional to- 1 90% C.L. region dient to our picture ofmography the nucleon. of the While nucleon it asis tooa composite early object—involves knowledge of both the motion of gluons DSSV* 90% C.L. region to draw any reliable conclusionsinside the nucleon on the and full of their gluon spatial spin distribution. This information, for both quarks and gluons, is en- ∫ dx ∆ g(x) 0.05 0.001 DSSV contribution to the protoncoded spin, in a number our analysis of distribution clearly functions. sug- The generalized parton distributions (GPDs) describe the 0.5 gests that gluons coulddistribution contribute of significantly quarks and gluons after both all. in the transverse plane and in the longitudinal direction [27], This in turn also shedsencompassing a new light on and the generalizing possible sizethe information of carried by the parton distribution functions (PDFs) orbital angular momentaand of the quarks elastic andelectromagnetic gluons. We form hope factors. The GPDs are functions of both the squared momentum transfer t at which the nucleon is probed, and the longitudinal momentum fraction x. Reducing the gluon helicity that future experimental studies, as well as lattice-QCD 0 computations that nowdimensionality appear feasible of these [20], distributions will provide by integrating over x defines the Mellin moments of the GPDs (specifically, the tower of xn-weighted integrals over the longitudinal momentum fraction x of the further information on ∆g(x) and eventually clarify its GPDs defines a tower of moments that together contain all of the information of the GPDs). These role for the proton spin.moments We plan are parametrised to present aas full generalised new form factors (GFFs), observables that can both be stud- 2 2 -0.5 Q = 10 GeV global analysis with detailsied from on lattice all polarized QCD using parton standard dis- lattice calculations of hadronic matrix elements of local opera- tributions once the 2009tors RHIC [28, 29] data1 and have indirectly become constrained final by experimental measurements at, for example, the planned -0.2 -0.1 -0 0.11 0.2 0.3 and additional informationEIC. on the quark and antiquark ∫ dx ∆g(x) helicity distributions, in particularMeasurements from and final calculations data on ofW the spin-independent gluon distributions in particular can re- 0.05 boson production at RHIC,veal the has answer become to a available.tantalizing question: Also, “What is the gluon radius of a nucleon, and how does it FIG. 5: 90% C.L. areas in the plane spanned by the truncated on the theoretical side,compare a new to study the quark of pionradius?" and The kaon charge radius of the proton is a fundamental quantity describing Figure 2. Figure modifiedmoments from Ref. of [∆26g],computed showing the for results 0.05 of ax global1 QCD and0 analysis.001 ofx polarised parton nucleon structure, and resolving the 7σ discrepancy between determinations of the radius from elec- distributions, to include0 the.05 lattice at Q QCD2 = result 10 GeV of Ref.2.[ Results6] as the orange for≤ DSSV, band.≤ DSSV*, and≤ our≤ fragmentation functions should precede the next global analysis of polarized partontron scattering distributions. experiments and electronic hydrogen spectroscopy on the one hand and measurements new analysis, with the symbols corresponding the respective of muonic hydrogen [40, 41] on the other—termed the ‘proton radius puzzle’—is a high priority for values of each central fit, are shown. We thank E.C. Aschenauer, K. Boyle, P. Djawotho, the hadronic physics community [42–46]. In principle, the gluonic radius of the proton is no less and C. Gagliardi for useful communications. M.S. was important or interesting, and need not be the same, but it is completely unknown from experiment. supported in part by U.S. DoE (contract number DE- 2 very limited information on ∆g is also available from Models and phenomenology offer little intuition on this topic . For example, guided by a bag model AC02-98CH10886) and BNL-LDRD project 12-034. This scaling violations of the DIS structure function g which of the nucleon, where gluons expand to fill the bag, one might expect the gluon radius to be larger year, such a calculation was performed at quark masses corresponding to the physical1 pion masswork [5], was supported by CONICET, ANPCyT and UBA- is, of course, fully included in our global QCD analy- than the quark radius. On the other hand, drawing on the constituent quark model where gluons hide in a study that presented a complete decomposition of the nucleon spin in this basis. It is notableCyT. sis. Overall, the constraints on ∆g(x) in, say, the regime inside dressed constituent quarks, one could expect the quark and gluon radii to be similar. A different that the percentage gluon contribution to the nucleon spin found in this study—about 27%—is the picture again is painted by lattice QCD calculations with infinitely heavy quarks which reveal gluon 0.001 x 0.05 are much weaker than those in the same percentage gluon contribution≤ ≤ to the momentum fraction of the nucleon found in the same set flux tubes binding quarks together [47], suggesting that the gluon radius is smaller than the quark of calculations, as discussedRHIC in region, the previous as can section. be inferred Importantly, from the Fig. total 1. spin Very of the little nucleon in this radius. Lattice QCD calculations of GFFs can offer some insight into this fundamental question. contribution to ∆G is expected to come from1 x>0.2. study, with all contributions determined separately, was consistent with 2 , as required by the spin sum Recently, the momentum-dependence of some of the GFFs of the φ (ss) meson was calculated rule. Figure 5 shows our estimates for the 90% C.L. area ∗ Electronic address: defl[email protected] lattice methods [14, 15], providing the first information on the multidimensional gluon struc- in the plane spanned by the truncated moments of ∆g † Electronic address: [email protected] inside any hadron. These studies were performed with heavy values of the light quark masses The Jaffe-Manoharcalculated approach gives in 0 a. canonical05 x partonic,1 and but gauge-variant, 0.001 x decomposition0.05 of‡ theElectronic address: [email protected] to a pion mass of 450 MeV, but a value of the strange quark mass yielding a close- nucleon spin into quarkfor andQ gluon2 = spin10 GeV and2 orbital.≤ Results angular≤ are momentum presented contributions≤ both≤ for in the the infinite§ mo-Electronic address: [email protected] φ mass. Through direct calculations of matrix elements of local gluonic operators, the [1] See e.g.: E. Leader and C. Lorcé, arXiv:1309.4235, and mentum frame in the light-coneDSSV* and gauge. our The new spin fit. contributions The symbols in this in decomposition Fig. 5 denote are measurable momentum-dependence of the lowest terms in the towers of the spin-independent and transversity through polarised deep inelastic scattering. As it is defined in the light cone frame, the Jaffe-Manoharreferences therein. GFFs was investigated. The latter GFFs, encoding how much more momentum of a transversely po- the actual values for the best fits in the DSSV, DSSV*, [2] For a review, see e.g.: C. A. Aidala, S. D. Bass, D. Hasch, gluon helicity is not directly accessible to lattice QCD defined on a Euclidean spacetime. In Ref. [6], larised particle is carried by gluons aligned rather than perpendicular to it in the transverse plane, and the present analyses. We note that for our new cen- and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013). the large momentum effective theory [25] approach was used to access the gluon helicity from lattice are particularly clean since they have no mixing with quark distributions until subleading twist and tral fit the combined integral 1 dx∆g(x, Q2) accounts [3] D. de Florian, R. Sassot, M. Stratmann, and W. Vogel- QCD for the first time. In this approach, matrix elements0 that.001 define the gluon helicity in the infinite 1 for over 90% of the full ∆G at Q2 = 10 GeV2. Not sur- sang, Phys. Rev. Lett. 101Until,very 072001 recently (2008); it was notPhys. known Rev. how D one might calculate x-dependent quantities such as PDFs and GPDs directly momentum limit are calculated, and the extrapolation to large momentum is performed. In the cal-80, 034030 (2009). using lattice QCD [30]; deep inelastic scattering and related processes are dominated by physics that is light-like, and as such culation of Ref. [6], whichprisingly, included the several main lattice improvement spacings and in our volumes new including analysis an is toensemble[4] with R. D. Ball et al. [NNPDFare inaccessible Collaboration], in Euclidean space. Nucl. Over Phys. the last few years, however, several methods have been suggested to access the x-dependence of PDFs using lattice QCD [30–32]. These approaches necessarily involve a rotation to a ‘quasi’ or ‘pseudo’ quark masses correspondingshrink to the the allowed physical pion area mass, in the the horizontal most significant direction, uncertainty corre- arose from theB 874, 36 (2013); J. Blümlein and H. Böttcher, Nucl. large-momentum extrapolation. The result is compared with phenomenological results from a global PDF, defined in Euclidean space, and an extrapolation back to the physical quantity in the large-momentum limit. While these sponding to the much better determination of ∆g(x) in Phys. B 841, 205 (2010);calculations E. Leader, are in early A. stages, V. they Sidorov, are promising and [33, 34]. Similarly, the first lattice QCD calculations related to TMDs have analysis of polarised partonrange distributions 0.05 x [026.2] inby Fig. the2 2009. The RHIC determinations data. Evidently, are entirely consistent,D. B. Stamenov, Phys.been Rev. performed D 82 in, the 114018 last few (2010); years [35– M.39]. Hi- with the lattice QCD resultthe uncertainty providing≤ a significantly≤ in the smaller- improvedx range constraint is still on very the total signif- gluon helicity.rai and S. Kumano, Nucl.2This question Phys. was B first813 brought, 106 to my (2009); attention in discussions with A. Deshpande. icant, and better small-x probes are badly needed. Data C. Bourrely, F. Buccella, and J. Soffer, Phys. Rev. D from the 2013 RHIC run at √s = 510 GeV may help 83, 074008 (2011); P. Jimenez-Delgado, A. Accardi, and W. Melnitchouk, Phys. Rev. D 89, 034025 (2014). 5 here a bit. In the future, an Electron Ion Collider would [5] See: E. C. Aschenauer et al., arXiv:1304.0079. provide the missing information, thanks to its large kine- [6] P. Djawotho [Star Collaboration], arXiv:1303.0543. 2 matic reach in x and Q [19]. [7] A. Adare et al. [Phenix Collaboration], Conclusions and outlook.— We have presented a new arXiv:1402.6296. global analysis of helicity parton distributions, taking [8] A. Adare et al. [Phenix Collaboration], Phys. Rev. Lett. 103, 012003 (2009). into account new and updated experimental results. In [9] A. Adare et al. [Phenix Collaboration], Phys. Rev. D 79, particular, we have investigated the impact of the new 012003 (2009). 0 data on ALL in jet and π production from RHIC’s 2009 [10] L. Adamczyk et al. [Star Collaboration], Phys. Rev. D run. For the first time, we find that the jet data clearly 86, 032006 (2012). EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

0.4 I II III

0.3 2 A 0.2

0.1

0.0 0 1 2 3 1 2 3 3 2 p 

Figure 3. Figure taken from Ref. [14]. Reduced matrix element encoding the lowest moment of the forward- limit gluon transversity distribution (unrenormalised), extracted from calculations at different boost momenta p2. Results in sections I, II and III of the figure are determined from vectors in different irreducible representations of the hypercubic group (and can differ by discretisation artefacts). Different colours, offset on the horizontal access for clarity, denote different vectors in each irreducible representation. The horizontal band is shown to guide the eye.

hence provide a direct probe of gluonic structure [48]. In the nucleon (or in hadrons or nuclei with spin < 1), the transversity GFFs decouple from the gluon matrix elements in the forward limit as a result of angular momentum conservation. For this reason, the φ meson, as the simplest spin-1 system (therefore having an accessible forward-limit transversity distribution), was chosen for this first inves- tigation. Experimentally, the gluon transversity distributions can be probed through distinct angular dependence of the cross-section in deeply-virtual Compton scattering (DVCS) [48]. In the forward limit, the magnitude of the gluon transversity distribution can be related to the magnitudes of the spin-independent and spin-dependent gluon distributions by a Soffer-type bound [49]. These bounds, which are positivity constraints arising from the positivity of the scattering matrix, were first studied in moment space on the lattice in Ref. [50, 51] for the quark distributions of spin- 1/2 particles. It is found in Ref. [14] that the Soffer bound for the first moment of the gluon distributions [52–55] in the spin-1 φ meson is saturated to 80%–100% (subject to some caveats regarding renormalisation of the GFFs), similar to the lattice QCD results found for the first two moments of the quark distributions of the nucleon [50, 51]. Reference [14] also included a detailed study of different irreducible representations of the hypercubic group (chosen such that the breaking of rotational sym- metry does not induce mixing with lower or same-dimensional operators), illustrated in Fig. 3, which suggested negligible discretisation artefacts. It was also found in that study that a smearing prescrip- tion, for example Wilson flow [56] or HYP smearing [57] applied to the links in the gluon operator, was necessary to reduce noise in the calculation. While the results in Ref. [14] are unrenormalised, a perturbative renormalisation has recently been computed for similar gluonic operators [18, 58]. Away from the forward limit, a large number of gluon GFFs contribute to the matrix elements of local gluonic operators in the φ meson; there are 7 spin-independent gluon GFFs for the lowest moment in the infinite tower that describes the full GPDs (with more contributing at each successive rung in the tower), and there are 8 for the transversity case. Because of this large number of GFFs,

6 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

0.4 0.4 I II III 4 0.3 3 0.3 0.2 2 2 A 0.2 0.1 1

0.0 0 0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 (a) Gluon transversity generalised form factor. (b) Spin-indepenent gluon generalised form factor. 0 1 2 3 1 2 3 3 2 p  Figure 4. Examples of generalised gluon form factors of the φ meson; figures taken from Ref. [15]. The horizontal axes show the squared momentum transfer t in GeV2, while the bands show fits to generalised dipole forms. Figure 3. Figure taken from Ref. [14]. Reduced matrix element encoding the lowest moment of the forward- limit gluon transversity distribution (unrenormalised), extracted from calculations at different boost momenta p2. Results in sections I, II and III of the figure are determined from vectors in different irreducible representations of the hypercubic group (and can differ by discretisation artefacts). Different colours, offset on the horizontal access not all could be determined from the lattice QCD calculations of Ref. [15]. In particular, at some for clarity, denote different vectors in each irreducible representation. The horizontal band is shown to guide the (discretised) values of the momentum transfer, some of the GFFs contribute to the calculated matrix eye. elements with coefficients suppressed by many orders of magnitude, and could not be extracted given the statistical precision of the calculations. A complete extraction will require more precise data at many different sets of initial and final momenta giving the same momentum transfer. Despite the hence provide a direct probe of gluonic structure [48]. In the nucleon (or in hadrons or nuclei with challenges of the calculation, the momentum-dependence (t-dependence) of one of the eight gluon spin < 1), the transversity GFFs decouple from the gluon matrix elements in the forward limit as a transversity GFFs, and three of the seven unpolarised gluon GFFs of the φ meson, was determined. result of angular momentum conservation. For this reason, the φ meson, as the simplest spin-1 system Examples of these are shown in Fig. 4. The dipole-like momentum-dependence of the form factors, (therefore having an accessible forward-limit transversity distribution), was chosen for this first inves- revealing qualitatively similar shapes as seen for the electromagnetic form factors which describe the tigation. Experimentally, the gluon transversity distributions can be probed through distinct angular spatial quark distributions and charge radius, provides the first information about the extended gluonic dependence of the cross-section in deeply-virtual Compton scattering (DVCS) [48]. structure of the φ. All of the computational and analysis techniques developed in that study will carry over to future calculations of gluon GFFs in the phenomenologically relevant case of the nucleon. In the forward limit, the magnitude of the gluon transversity distribution can be related to the magnitudes of the spin-independent and spin-dependent gluon distributions by a Soffer-type bound [49]. Based on the calculations of gluon GFFs in Ref. [15], we can return to the question of a gluonic These bounds, which are positivity constraints arising from the positivity of the scattering matrix, radius, and how such a radius compares with the corresponding quark radius. The clean dipole- were first studied in moment space on the lattice in Ref. [50, 51] for the quark distributions of spin- like momentum-dependence of all of the form factors which could be obtained (including those not 1/2 particles. It is found in Ref. [14] that the Soffer bound for the first moment of the gluon distri- reproduced here in Fig. 4) suggests that the familiar parameterisation of each radius in terms of a butions [52–55] in the spin-1 φ meson is saturated to 80%–100% (subject to some caveats regarding dipole mass is sensible here. Moreover, the GFFs of the spin-independent quark distribution have renormalisation of the GFFs), similar to the lattice QCD results found for the first two moments of the an identical decomposition to the gluon GFFs and so quark and gluon radii can be compared one-to- quark distributions of the nucleon [50, 51]. Reference [14] also included a detailed study of different one in a natural way. Making this comparison (see Fig. 5), it becomes clear that each of the three ff irreducible representations of the hypercubic group (chosen such that the breaking of rotational sym- gluon radii that were determined in Ref. [15] is di erent from the corresponding quark radius: one metry does not induce mixing with lower or same-dimensional operators), illustrated in Fig. 3, which larger, one smaller, and one approximately the same. Further interpretation will require a change of suggested negligible discretisation artefacts. It was also found in that study that a smearing prescrip- the basis of GFFs used in Ref. [15] to a basis with more natural interpretations (e.g., a multipole tion, for example Wilson flow [56] or HYP smearing [57] applied to the links in the gluon operator, decomposition). was necessary to reduce noise in the calculation. While the results in Ref. [14] are unrenormalised, a perturbative renormalisation has recently been computed for similar gluonic operators [18, 58]. 3 Gluon structure of nuclei Away from the forward limit, a large number of gluon GFFs contribute to the matrix elements of local gluonic operators in the φ meson; there are 7 spin-independent gluon GFFs for the lowest In the intensity-frontier era, where experiments such as long baseline neutrino programs and dark moment in the infinite tower that describes the full GPDs (with more contributing at each successive matter direct detection experiments typically use nuclear targets, understanding the structure of nuclei rung in the tower), and there are 8 for the transversity case. Because of this large number of GFFs, from QCD is of increasing importance. While it has been known for decades that the PDFs of large

7 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

1.5

1.0

0.5

0.0

0.5 - 1.0 - 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -

Figure 5. Figure taken from Ref. [15]. Ratios of a subset of the gluon spin-independent GFFs of the φ meson to the corresponding quark spin-independent GFFs (from top to bottom, the bands correspond to the GFFs i = 1, 2, 4 ). The horizontal axes show the squared momentum transfer ∆2 = t in GeV2, while the bands show ﬁts { } | | 1 to generalised dipole forms. The green band (lowest on the ﬁgure) has been re-scaled by a factor of 10 for clarity.

nuclei are not simply determined by a sum of the PDFs of the constituent nucleons [23], known as the EMC effect, the QCD origins of the effect are not yet fully understood [59–65]. An analogue of the EMC effect for gluons, in which the distributions of gluons in a nucleus differ from the sum of the gluon distributions in the constituent protons and neutrons [66], should in principle exist but has not been observed and would be a milestone discovery of the EIC [67]. A prediction of the magnitude of this ‘gluonic EMC effect’ from lattice QCD would be influential in the planning of the EIC program. Recently, the first lattice QCD calculation of aspects of the gluon structure of light nuclei was performed [16]. That study presented results for the gluon momentum fraction of the nucleon, spin-0 1 dinucleon, spin-1 deuteron, and spin- 2 triton, using two ensembles with light quark masses chosen such that m 450 MeV and m 806 MeV3. Given that the magnitude of the EMC effect in the π ∼ π ∼ quark momentum fraction of light nuclei from phenomenological parameterisations of the quark and gluon distributions [22, 72, 73] is typically at the level of 5%, the momentum sum rule dictates that the EMC effect in the gluon momentum fraction must be of the same magnitude and opposite sign. Of course, at the larger-than-physical values of the quark masses used in the simulations of Ref. [16], this effect could be larger or smaller than in nature. With the statistics of that work, no EMC effect could be resolved (see Fig. 6). Resolving an EMC effect of the same magnitude as it is expected to be in nature would require few-percent determinations of the proton and nuclear gluon momentum fractions, corresponding to an order of magnitude more statistics. Higher moments of the gluon PDFs could display larger (or smaller) EMC-type effects, not being constrained by momentum conservation. These are more difficult to access from lattice QCD since they are related to matrix elements of local gluonic operators with additional derivatives, but present promising avenues for further study. The facility of lattice QCD, and the future EIC, to study gluon distributions of nuclei is not limited to the momentum fraction. Measurements of exotic glue observables displaying non-nucleonic fea- tures of a nucleus are a particularly tantalising prospect. These effects, illustrated in Fig. 7, describe contributions to nuclear structure from gluons that can not be associated with individual nucleons in a nucleus. That is, while for many purposes nuclei can be described as a collection of modified nucle-

3The spectrum and many properties of light nuclei on these ensembles have been studied in detail previously [68–71].

8 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

1.5 1.4 1.4 1.0 1.2 1.2 0.5 1.0 1.0 0.0 0.8 0.8 0.5 - 0.6 1.0 0.6 - 0.4 1.5 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 - 1 2 3 1 2 3

(a) Results at mπ 450 MeV. (b) Results at mπ 806 MeV. Figure 5. Figure taken from Ref. [15]. Ratios of a subset of the gluon spin-independent GFFs of the φ meson ∼ ∼ to the corresponding quark spin-independent GFFs (from top to bottom, the bands correspond to the GFFs i = Figure 6. Figures taken from Ref. [16], showing the ratios of the gluon momentum fraction in light nuclei to 1, 2, 4 ). The horizontal axes show the squared momentum transfer ∆2 = t in GeV2, while the bands show fits { } | | the central value of that in the nucleon. The A = 1 results are for the nucleon, the spin-1 deuteron and spin-0 to generalised dipole forms. The green band (lowest on the figure) has been re-scaled by a factor of 1 for clarity. 10 dinucleon are shown at A = 2, and the triton is shown at A = 3 on the horizontal axis. The blue circles and orange diamonds denote results obtained using different lattice operator constructions. The EMC effect would present as deviations of the A > 1 results from the dashed line shown at 1. nuclei are not simply determined by a sum of the PDFs of the constituent nucleons [23], known as the EMC effect, the QCD origins of the effect are not yet fully understood [59–65]. An analogue of the EMC effect for gluons, in which the distributions of gluons in a nucleus differ from the sum of the gluon distributions in the constituent protons and neutrons [66], should in principle exist but has not been observed and would be a milestone discovery of the EIC [67]. A prediction of the magnitude of this ‘gluonic EMC effect’ from lattice QCD would be influential in the planning of the EIC program. Recently, the first lattice QCD calculation of aspects of the gluon structure of light nuclei was performed [16]. That study presented results for the gluon momentum fraction of the nucleon, spin-0 1 dinucleon, spin-1 deuteron, and spin- 2 triton, using two ensembles with light quark masses chosen such that m 450 MeV and m 806 MeV3. Given that the magnitude of the EMC effect in the π ∼ π ∼ quark momentum fraction of light nuclei from phenomenological parameterisations of the quark and gluon distributions [22, 72, 73] is typically at the level of 5%, the momentum sum rule dictates that the EMC effect in the gluon momentum fraction must be of the same magnitude and opposite sign. (a) Traditional picture of nuclear (b) Nucleus with ‘exotic’ multi- structure, with gluon interactions nucleonic glue effects highlighted. Of course, at the larger-than-physical values of the quark masses used in the simulations of Ref. [16], (represented by curly yellow lines) this effect could be larger or smaller than in nature. With the statistics of that work, no EMC effect confined to individual nucleons (fig- could be resolved (see Fig. 6). Resolving an EMC effect of the same magnitude as it is expected to ure taken from Ref. [9]). be in nature would require few-percent determinations of the proton and nuclear gluon momentum fractions, corresponding to an order of magnitude more statistics. Higher moments of the gluon PDFs Figure 7. Artist’s interpretation of ‘exotic’ observables describing multi-nucleonic contributions to the gluon could display larger (or smaller) EMC-type effects, not being constrained by momentum conservation. structure of a nucleus. These are more difficult to access from lattice QCD since they are related to matrix elements of local gluonic operators with additional derivatives, but present promising avenues for further study. The facility of lattice QCD, and the future EIC, to study gluon distributions of nuclei is not limited ons 4, exotic glue observables directly quantify the breaking of this nucleon-based picture of nuclear to the momentum fraction. Measurements of exotic glue observables displaying non-nucleonic fea- structure in nature. There is potential for experiments at Thomas Jefferson National Accelerator Fa- tures of a nucleus are a particularly tantalising prospect. These effects, illustrated in Fig. 7, describe cility [11] to measure exotic glue effects, in particular the gluon transversity structure function in spin contributions to nuclear structure from gluons that can not be associated with individual nucleons in a 1 nuclei, imminently. nucleus. That is, while for many purposes nuclei can be described as a collection of modified nucle- ≥

4Indeed, from an EFT perspective such oﬀ-shell nucleon eﬀects and two-nucleon interactions with extended currents are 3The spectrum and many properties of light nuclei on these ensembles have been studied in detail previously [68–71]. related by unitary transformations [74, 75].

9 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

Reference [16] recently presented the first lattice QCD determination of the gluon transversity structure function in the spin-1 deuteron state using an ensemble with mπ = 806 MeV. While sys- tematics including discretisation and finite-volume effects, as well as renormalisation and an extrapolation in pion mass, have not yet been addressed, that work provided the first definitive evidence for multi-nucleonic glue effects in a nucleus, albeit at a larger-than-physical value of the pion mass. The extracted result for the gluon transversity matrix element is non-zero to two standard deviations (see Fig. 8), but the significance of the result is far greater; there is no effect other than exotic glue that could cause the appearance of a signal (including the expected exponential contamination from excited states) in the matrix element calculated. The matrix element is safe from mixing with quark distributions at leading twist, and contributions from single nucleon states (even excited single nucleon states) necessarily vanish, unless those states interact and exchange momentum (therefore, a multi-nucleonic effect). For that reason, the result of Ref. [16], although not completely controlled quantitatively, gives a clear qualitative indication of exotic glue. It is also notable that the magnitude of the result is consistent with expectations from large-Nc [76–80]; the ratio of the first moment of the gluon transversity distribution to the same moment of the spin-independent gluon distribution in the 2 deuteron is suppressed by an additional factor of 1/Nc as compared with the ratio in a typical compact spin-1 hadron such as the φ meson [14]. Extensions of this study with fully controlled uncertainties will provide valuable guidance to experimental programs [9, 11] seeking to observe this effect in nature for the first time.

4 Conclusion

The construction of an electron-ion collider [9] will dramatically improve our ability to probe the gluon structure of hadrons and nuclei, and will complement experimental programs at Thomas Jeffer- son National Accelerator Facility [11] and at the LHC [12]. In parallel, lattice QCD calculations of gluon structure observables are advancing rapidly, with recent progress making direct calculations of the gluon contributions to observables such as the momentum and spin of the nucleon, at the physical values of the quark masses and with controlled uncertainties, possible for the first time [5, 6, 13]. Complex aspects of the gluon structure of nucleons and nuclei are also being investigated, with the first study of the gluon GFFs encoding moments of gluon GPDs in hadrons performed this year [14, 15], and studies of the x-dependence of gluon PDFs and GPDs, as well as gluon TMDs, likely in the near future. Moreover, aspects of the gluon structure of light nuclei have recently been studied from lattice QCD for the first time [16]. Refinements of the calculations of the gluon momentum fraction in light nuclei have the potential to reveal the magnitude of a gluonic analogue of the EMC effect. This effect is a key target of the upcoming EIC program, and its discovery would represent a milestone achieve- ment for that machine. Another particularly exciting recent result is the first definitive indication for ‘exotic’ multi-nucleonic gluon effects (albeit with uncertainties that are not fully quantified at this stage), revealed through lattice QCD studies of the gluon transversity distribution [16]. It was first pointed out in 1989 [81] that this particular distribution provides a uniquely clean window on gluon effects in nuclei, and it is the target of several planned experimental investigations [9, 11]. In summary, lattice QCD calculations of the gluon structure of hadrons and light nuclei are advancing rapidly and complement and extend our understanding of the fundamental structure of nature from experiment. Over the next decade, these results will be influential in the planning and development of experimental programs including the EIC program, and the lattice community has the opportunity to provide controlled QCD predictions for key gluon structure observables ahead of first measurements.

10 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

Reference [16] recently presented the first lattice QCD determination of the gluon transversity structure function in the spin-1 deuteron state using an ensemble with mπ = 806 MeV. While sys- t 2t 2 tematics including discretisation and finite-volume effects, as well as renormalisation and an extrap- = olation in pion mass, have not yet been addressed, that work provided the first definitive evidence for multi-nucleonic glue effects in a nucleus, albeit at a larger-than-physical value of the pion mass. * The extracted result for the gluon transversity matrix element is non-zero to two standard deviations t 3t 3 (see Fig. 8), but the significance of the result is far greater; there is no effect other than exotic glue = that could cause the appearance of a signal (including the expected exponential contamination from * excited states) in the matrix element calculated. The matrix element is safe from mixing with quark distributions at leading twist, and contributions from single nucleon states (even excited single nu- t 4t 4 cleon states) necessarily vanish, unless those states interact and exchange momentum (therefore, a = multi-nucleonic effect). For that reason, the result of Ref. [16], although not completely controlled * quantitatively, gives a clear qualitative indication of exotic glue. It is also notable that the magnitude t 5t 5 of the result is consistent with expectations from large-Nc [76–80]; the ratio of the first moment of the gluon transversity distribution to the same moment of the spin-independent gluon distribution in the = 2 deuteron is suppressed by an additional factor of 1/Nc as compared with the ratio in a typical compact * spin-1 hadron such as the φ meson [14]. Extensions of this study with fully controlled uncertain- ff t 6t 6 ties will provide valuable guidance to experimental programs [9, 11] seeking to observe this e ect in = nature for the first time. *

t 7t 7 = 4 Conclusion *

0 10 20 30 40 The construction of an electron-ion collider [9] will dramatically improve our ability to probe the gluon structure of hadrons and nuclei, and will complement experimental programs at Thomas Jeffer- son National Accelerator Facility [11] and at the LHC [12]. In parallel, lattice QCD calculations of Figure 8. Figure taken from Ref. [16], showing ratios of three-point to two-point functions that reduce to the gluon structure observables are advancing rapidly, with recent progress making direct calculations of lowest moment of the gluon transversity distribution in the deuteron for 0 τ t, with exponential con- the gluon contributions to observables such as the momentum and spin of the nucleon, at the physi- tamination from excited states at other times. Blue circles and orange diamonds denote results using different cal values of the quark masses and with controlled uncertainties, possible for the first time [5, 6, 13]. source-sink structures, while the red star indicates the sink time t. The green bands (repeated on each panel) show Complex aspects of the gluon structure of nucleons and nuclei are also being investigated, with the first the extracted results for the matrix elements from fits in the t–τ plane using a multi-exponential fit form. study of the gluon GFFs encoding moments of gluon GPDs in hadrons performed this year [14, 15], and studies of the x-dependence of gluon PDFs and GPDs, as well as gluon TMDs, likely in the near future. Moreover, aspects of the gluon structure of light nuclei have recently been studied from lattice QCD for the first time [16]. Refinements of the calculations of the gluon momentum fraction in light Acknowledgements ff ff nuclei have the potential to reveal the magnitude of a gluonic analogue of the EMC e ect. This e ect This work is partially supported through contract number DE-AC05-06OR23177 under which JSA is a key target of the upcoming EIC program, and its discovery would represent a milestone achieve- operates the Thomas Jefferson National Accelerator Facility. I thank Yi-Bo Yang for sharing prelim- ment for that machine. Another particularly exciting recent result is the first definitive indication for inary results [13] with me before Lattice 2017 and allowing me to reproduce them here. I also thank ff ‘exotic’ multi-nucleonic gluon e ects (albeit with uncertainties that are not fully quantified at this Will Detmold, Arjun Gambhir, Dimitra Pefkou, Kostas Orginos, Martin Savage and Michael Wagman stage), revealed through lattice QCD studies of the gluon transversity distribution [16]. It was first for their contributions to the results presented here. pointed out in 1989 [81] that this particular distribution provides a uniquely clean window on gluon effects in nuclei, and it is the target of several planned experimental investigations [9, 11]. In summary, lattice QCD calculations of the gluon structure of hadrons and light nuclei are advancing rapidly and complement and extend our understanding of the fundamental structure of nature from experiment. Over the next decade, these results will be influential in the planning and development of experimental programs including the EIC program, and the lattice community has the opportunity to provide controlled QCD predictions for key gluon structure observables ahead of first measurements.

11 EPJ Web of Conferences 175, 01015 (2018) https://doi.org/10.1051/epjconf/201817501015 Lattice 2017

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