emergence of mass in the standard model

NANJING

UNIVERSITYCraig Roberts … http://inp.nju.edu.cn/

Emergence of Mass

in the Standard Model

http://inp.nju.edu.cn/

2020 June 11: IMP-Lanzhou QMRC Seminar (81)

Craig Roberts. Emergence of Mass in the Standard Model

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Students, Postdocs, Profs.

1. Zhao-Qian YAO (Nanjing U.)

2. Yin-Zhen XU (Nanjing U.)

3. Marco BEDOLLA (Genova, U Michoácan)

4. Chen CHEN (Giessen, UNESP - São Paulo, USTC & IIT);

5. Muyang CHEN (NKU, PKU)

6. Zhu-Fang CUI (Nanjing U.) ;

7. Minghui DING (ECT*, ANL, Nankai U., PKU) ;

8. Fei GAO (Heidelberg, Valencia, PKU.) ;

9. Bo-Lin LI (NJNU, Nanjing U.)

10. Ya LU (Nanjing U.)

11. Cédric MEZRAG (INFN-Roma, ANL, IRFU-Saclay) ;

12. Khépani RAYA (Nankai U., U Michoácan);

13. Jin-Li Zhang (NJNU, NJU)

14. Adnan Bashir (U Michoácan);

15. Daniele Binosi (ECT*)

16. Volker Burkert (JLab)

17. Lei Chang (Nankai U. )

18. Xiao-Yun Chen (Jinling Inst. Tech., Nanjing)

19. Feliciano C. De Soto Borrero (UPO);

20. Tanja Horn (Catholic U. America)

21. Gastão Krein (UNESP – São Paulo)

22. Yu-Xin Liu (PKU);

23. Joannis Papavassiliou (U.Valencia)

24. Jia-Lun Ping (Nanjing Normal U.)

25. Si-xue Qin (Chongqing U.);

26. Jose Rodriguez Quintero (U. Huelva) ;

27. Elena Santopinto (Genova)

28. Jorge Segovia (U. Pablo de Olavide = UPO);

29. Sebastian Schmidt (HZDR & RWTH-Aachen);

30. Shaolong Wan (USTC) ;

31. Qing-Wu Wang (Sichuan U)

32. Shu-Sheng XU (NJUPT, Nanjing U.)

33. Pei-Lin Yin (NJUPT)

34. Hong-Shi Zong (Nanjing U)

35. … and many more …

➢ The establishment by the mid-1970’s of QCD as the correct theory of the strong interactions completed what is now known prosaically as the Standard Model.

➢ It offers a description of all known fundamental physics except for gravity, and gravity is something that has no discernible effect when particles are studied a few at a time.

➢ However, the situation is a bit like the way that the Navier-Stokes equation accounts for the flow of water. The equations are at some level obviously correct, but there are only a few, limited circumstances in which their consequences can be worked out in any detail.

➢ Nevertheless, many leading physicists were inclined to conclude in the late 1970’s that the task of basic physics was nearly complete, and we’d soon be out of jobs.

➢ A famous example was the inaugural lecture of Stephen Hawking as Lucasian Professor of Mathematics, a chair first held by Isaac Barrow at Cambridge University. Hawking titled his lecture, “Is the End in Sight for Theoretical Physics?” And he argued strongly for “Yes”.



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Existence of our Universe depends critically on the following empirical facts:

➢ Proton is massive

– i.e. the mass-scale for strong interactions is vastly different to that of electromagnetism

➢ Proton is absolutely stable

– Despite being a composite object constituted from three valence quarks

➢ Pion is unnaturally light (not massless, but lepton-like mass)

– Despite being a strongly interacting composite object built from a valence-quark and valence antiquark



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Emergence: low-level rules producing high-level phenomena, with enormous apparent complexity


Mass: The Final Frontier

Character

of Mass

➢ Key questions posed to modern science:

– What is the origin of mass?

– What are the consequences of the appearance of mass?



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➢ When considering the source of mass, many people think of explicit mass, generated via the Higgs-mechanism

– especially because the Higgs boson was discovered relatively recently at CERN (2012)

– importance acknowledged by the Nobel Prize awarded to Englert and Higgs:

• for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles …

Character of Mass

➢ Discovery of the Higgs was a watershed.

➢ However, it should be placed in context.

➢ Therefore, consider the mass-energy budget of the Universe

– Dark energy = 71%

– Dark Matter = 24%

– Visible material = 5%

• Higgs effect is just 0.1% of the visible matter

➢ Little is known about the first two

Science can say almost nothing about 95% of the mass-energy in the Universe.

➢ The explanation lies outside the Standard Model.



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Character of Mass

➢ Little is known about Dark Energy and Dark Matter …

Science can say almost nothing about 95% of the mass-energy in the Universe.

➢ On the other hand, the remaining 5% has forever been the source of everything tangible.

➢ Yet, amongst this 5%, less-than 0.1% is tied directly to the Higgs boson

➢ Hence, even concerning visible material, too much remains unknown.



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Visible Mass

➢ More than 98% of visible mass is contained within nuclei.

➢ First approximation:

– atomic weights = sum of the masses of all the nucleons they contain.

➢ Each nucleon has a mass mN ∼ 1 GeV ≈ 2000 me

➢ Higgs boson produces me, but what produces mN ?

➢ This question is basic to the whole of modern physics

–How can science explain the

emergence of hadronic mass

(EHM)?



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Standard Model

➢ Strong interactions are described by quantum chromodynamics (QCD).

➢ QCD: hadrons are composites,

– built from quarks and/or antiquarks (matter fields)

– held together by forces produced by the exchange of gluons (gauge fields).

➢ These forces are unlike any previously encountered.

– become very weak when two quarks are brought close together within a nucleon

– but all experimental attempts to remove a single quark from within a nucleon and isolate it in a detector have failed.

➢ Seemingly, then, the forces become enormously strong as the separation between quarks is increased

➢ Modern science is encumbered with The Confinement Problem



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confinement

Google Scholar: 25,000 mentions of quark confinement since 2000

Confinement Problem


➢ Confinement is crucial because it ensures absolute stability of the proton.

➢ In the absence of confinement:

– protons in isolation could decay;

– the hydrogen atom would be unstable;

– nucleosynthesis would be a chance event, having no lasting consequences;

– and without nuclei, there would be no stars and no living Universe.

➢ Without confinement, our Universe could not exist.



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confinement


Confinement Problem


➢ As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems

– Each represents one of the toughest challenges in mathematics.

– The set contains the problem of confinement; and presenting a sound solution will win its discoverer $1,000,000

➢ Even with such motivation, today, almost 50 years after the discovery of quarks, no rigorous solution has been found

➢ Confinement and EHM are inextricably linked

➢ Consequently, as science plans for the next thirty years, solving the problem of EHM has become a Grand Challenge



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confinement


Confinement➢ Plainly – Confinement is one of the most fascinating aspects of QCD

➢ At issue is the definition.

➢ When communicating about confinement, a typical practitioner has a notion in mind; yet the perspectives of any two different practitioners are often distinct– e.g., [Wilson:1974sk – area law,

Gribov:1998kb – screening, Cornwall:1981zr – gluon mass].

➢ The proof of one expression of confinement will be contained within demonstration that quantum SUc(3) gauge field theory is mathematically well-defined, – i.e. a solution to the “Millennium Problem” [Clay Mathematics Institute $1,000,000]

➢ However, that may be of limited utility because

– Nature has provided light-quark degrees-of-freedom

– They seemingly play a crucial role in the empirical realisation of confinement,

– Perhaps because they enable screening of colour at low couplings [Gribov:1998kb].



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QCD is full of surprises

➢ In appearance, QCD is simple.

➢ QCD is also unique.

➢ Fundamental theory with the capacity to sustain massless elementary degrees-of-freedom, viz. gluons and quarks;

– Yet gluons and quarks are predicted to acquire mass dynamically

– And nucleons and almost all other hadrons likewise

➢ Only massless systems in QCD = composite Nambu-Goldstone (NG) bosons,e.g. pions and kaons.

➢ Pions responsible for binding systems as diverse as atomic nuclei and neutron stars

➢ Energy associated with the gluons and quarks within the NG modes is not readily apparent.

➢ Stark contrast with all other “everyday” hadrons:

– systems constituted from u, d, s quarks possess nuclear-size masses

– masses far in excess of anything that can directly be tied to the Higgs boson



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𝐿 =1

4𝐺𝜇𝜈𝑎 𝑥 𝐺𝜇𝜈

𝑎 𝑥 + ത𝜓 𝛾 ⋅ 𝜕𝑥 +𝑚 + 𝑖𝑔𝜆𝑎

2𝛾 ⋅ 𝐴𝑎 𝑥 𝜓 𝑥

𝐺𝜇𝜈𝑎 𝑥 = 𝜕𝜇𝐴𝜈

𝑎(𝑥) − 𝜕𝜈𝐴𝜇𝑎(𝑥) − 𝑓𝑎𝑏𝑐𝐴𝜇

𝑏(𝑥)𝐴𝜈𝑐(𝑥)


➢ In trying to match QCD with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g.:



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𝐿 =1







– loss of particle number conservation

– frame and scale dependence of the explanations and interpretations of observable processes

– evolving character of the relevant degrees-of-freedom

➢ Electroweak theory and phenomena are essentially perturbative; hence, possess little of this complexity.

➢ Science has never before encountered an interaction such as that at work in QCD.

➢ Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.


➢ Comparison with QED

– One essential difference = circled term, describing gluon self-interactions.



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𝐿 =1







➢ If QCD is correct, then this term must hold the answers to an enormous number of Nature's basic questions, e.g.:

– what is the origin of visible mass?

– how is mass distributed within atomic nuclei?

– what carries the proton's spin?

– how can the same degrees-of-freedom combine to ensure the pion is spinless?

➢ Nowhere are there more basic expressions of emergence in Nature


➢ Treated as a classical theory, chromodynamics is a non-Abelian local gauge field theory.



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𝐿 =1







➢ Formulated in four spacetime dimensions, such theories do not possess any mass-scale in the absence of Lagrangian masses for the quarks.

– There is no dynamics in a scale-invariant theory, only kinematics.

– Bound states are therefore impossible

– Accordingly, our Universe cannot exist.

➢ A spontaneous breaking of symmetry - Higgs mechanism - does not solve this problem

– mN ≈ 100-times larger than Higgs-generated current-masses of the light u- and d-quarks, the valence constituents of nucleons


➢ On the other hand … NG bosons are (nearly) massless



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𝐿 =1







➢ In these systems, the strong interaction’s mN ≈ 1 GeV mass-scale is hidden.

➢ Chiral limit, when Higgs-generated masses are omitted, NG modes are massless– Exactly massless … no fine tuning … no danger of tachyons

➢ In this case, perturbative QCD predicts that strong interactions cannot distinguish between quarks with negative or positive helicity – chiral symmetry

➢ Such a chiral symmetry would have numerous corollaries, e.g.– pion would be partnered with a scalar meson of equal mass.

➢ However, no such state is observed

➢ No consequences of this chiral symmetry found in Nature – symmetry broken by interactions.

➢ Dynamical chiral symmetry breaking (DCSB) ensures both

– massless quarks in QCD's Lagrangian acquire large effective masses

– interaction energy between those quarks cancels their masses exactly so that mπ = 0


➢ Restore the Higgs mechanism

= add realistic current-quark masses



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𝐿 =1







➢ DCSB is responsible for, inter alia:

– physical size of the pion mass

• mπ ≈ 0.15 mN

– large mass-splitting between pion and its valence-quark spin-flip partner, ρ-meson

• mρ > 5 mπ

– neutron and proton possessing masses

• mN ≈ 1 GeV

– Interesting things also happen to the kaon

• Like a pion, but with one light quark replaced by s-quark, K possesses a mass mK ≈ 0.5 GeV

• Here a competition is taking place, between dynamical and Higgs-driven mass-generation.

Fundamental Mysteries

➢ These phenomena and features, their origins and corollaries, entail that the question

– how did the Universe evolve?

is inseparable from the questions

– how does the mN ≈ 1 GeV mass-scale that characterises atomic nuclei appear?

– why does it have the observed value?

– and, enigmatically, why does the dynamical generation of mN have seemingly no effect on the composite NG bosons in QCD?

• i.e. whence the near-absence of the pion mass?



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Whence Mass?

… Trace Anomaly

➢ In a scale invariant theory Poincaré invariance entails

the energy-momentum tensor must be traceless: Tμμ ≡ 0

➢ Regularisation and renormalisation of (ultraviolet) divergences in QuantumChromodynamics introduces a mass-scale … dimensional transmutation:

Lagrangian’s constants (couplings and masses) become dependent on a mass-scale, ζ

➢ α → α(ζ) in QCD’s (massless) Lagrangian density, L(m=0)

⇒ ∂μDμ = δL/δσ = αβ(α) dL/dα = β(α) ¼Gμν Gμν = Tρρ =: Θ0

Quantisation of renormalisable four-dimensional theory forces nonzero value for trace of energy-momentum tensor



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Trace anomaly

QCD β function … specifies how the coupling “runs”

Trace Anomaly

➢ Knowing that a trace anomaly exists does not deliver a great deal… Indicates only that a mass-scale must exist

➢ Key Question:

– Can one compute and/or understand the magnitude of that scale?

➢ One can certainly measure the magnitude … consider proton:

➢ In the chiral limit the entirety of the proton’s mass is produced by the trace anomaly, Θ0

… In QCD, Θ0 measures the strength of gluon self-interactions

… so, from one perspective, mp is (somehow) completely generated by glue.



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➢ The vast majority of mass comes from the energy needed to hold quarks together inside nuclei



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Trace Anomaly

➢ In the chiral limit

⇒

➢ Might mean that the scale anomaly vanishes trivially in the pion state, i.e. gluons contribute nothing to the pion mass.

➢ But that is difficult way to obtain “zero”!

➢ Easier to imagine that “zero” owes to cancellations between different operator contributions to the expectation value of Θ0.

➢ Of course, such precise cancellation should not be an accident.

It could only arise naturally because

of some symmetry and/or symmetry-breaking pattern.



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Whence “1” and yet “0” ?

➢No statement of the question

“How does the mass of the proton arise?”

is complete without the additional clause

“How does the pion remain ?”

➢ Natural visible-matter mass-scale must emerge simultaneously with apparent preservation of scale invariance in related systems

– Expectation value of Θ0 in chiral-limit pion is always zero, irrespective of the size of the natural mass-scale for strong interactions = mp



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Whence “1” and yet “0” ?

➢No statement of the question

“How does the mass of the proton arise?”

is complete without the additional clause

“How does the pion remain ?”

➢ Natural visible-matter mass-scale must emerge simultaneously with apparent preservation of scale invariance in related systems

– Expectation value of Θ0 in pion is always zero, irrespective of the size of the natural mass-scale for strong interactions = mp



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Modern Physics must Elucidate the Entire Array of Empirical Consequences

of the Mechanism responsible so that the Standard Model can be Validated

IR Behaviour of QCD➢ Gluons are supposed to be massless …

This is true in perturbation theory

➢ Not preserved non-perturbatively!

No symmetry in Nature protects

four-transverse gluon modes … 𝑞𝜇 Π𝜇𝜈 𝑞 ≡ 0

➢ Gluons acquire a running mass, which is large at infrared momenta

⇒ Prediction: Gluon two-point function is nonzero and finite at q2 = 0



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Dynamical mass generation in continuum quantum chromodynamicsJ.M. Cornwall, Phys. Rev. D 26 (1981) 1453 … ~ 1000 citations … approach modernized and sketched results are confirmed

Continuum and lattice studies agree that this true

RGI mass-scaleෝ𝑚0 = 0.43(1) GeV ≈ 1

2𝑚𝑝


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⇐What’s happening out here?!


This is where we live

RGI PI Effective Charge➢ Gluon vacuum polarization can be

translated into a RGI process independent effective charge for QCD

– Use pinch technique and background field method

➢ Unique analogue of Gell-Mann – Low running coupling in QED

➢ Parameter-free prediction



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RGI PI strong coupling𝛼0 = 0.97 4 𝜋

The QCD Running Coupling, A. Deur, S. J. Brodsky and G. F. de Teramond, Prog. Part. Nucl. Phys. 90 (2016) 1-74

Process independent strong running couplingDaniele Binosi et al., arXiv:1612.04835 [nucl-th], Phys. Rev. D 96 (2017) 054026/1-7

Process-independent effective coupling. From QCD Green functions to phenomenology, Jose Rodríguez-Quintero et al., arXiv:1801.10164 [nucl-th]. Few Body Syst. 59 (2018) 121/1-9

Effective charge from lattice QCD, Zhu-Fang Cui, Jin-Li Zhang et al., NJU-INP 014/19, arXiv:1912.08232 [hep-ph]. Chin. Phys. C 44 (2020) 083102/1-10

http://inspirehep.net/record/1504060?ln=en

RGI PI Effective Charge➢ Also, ො𝛼(𝑘2)is:

i. pointwise (almost) identical to the process-dependent (PD) effective charge,𝛼𝑔1, defined via the Bjorken

sum rule;

ii. capable of marking the boundary between soft and hard physics;

iii. that PD charge which, used at one-loop in the QCD evolution equations, delivers agreement between pion parton distribution functions calculated at the hadronic scale and experiment.



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Data: Bjorkensum rule

RGI PI Effective Charge➢ In playing so many diverse roles,

ො𝛼(𝑘2) is a strong candidate for that object which properly represents the interaction strength in QCD at any given momentum scale.

➢ Landau pole, a prominent feature of perturbation theory, is screened (eliminated) in QCD by the dynamical generation of a gluon mass-scale

➢ Theory possesses an infrared stable fixed point.



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➢ Standard renormalisation theory ensures QCD's ultraviolet behaviour is under control

➢ QCD emerges as a mathematically well-defined quantum field theory in four dimensions.

➢ Dynamical violation of scale invariance in QCD enables emergence of gluon mass

➢ What about the matter sector?

➢ Inspired by BCS theory, Nambu developed simple model … won him Nobel Prize

– fermion & antifermion form a “Cooper pair” so long as coupling is strong enough

➢ Studied via Dyson’s Gap Equation – describes emergence of quasiparticle in many body systems & quantum field theories are systems with infinitely many bodies

➢ If and only if 𝛼0 ≥ 0.3𝜋, then this gap equation produces a nonzero fermion mass even in the absence of a Higgs coupling

➢ Dynamical chiral symmetry breaking – emergence of a fermion mass from nothing

Matter Sector



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Matter Sector𝛼0 = 0.97 𝜋

➢ Dressed-quark quasiparticles emerge

➢ Characterised by a running mass M(k)

– Vanishes in ultraviolet, following the pattern predicted by Lane (1974) and Politzer (1976)

– Large at infrared momenta, i.e. 𝑀 0 ≈ 1

3𝑚𝑝

– Just like constituent quark

➢ Dressed-quark is a bare parton at ultraviolet momenta

➢ But that parton carries a cloud of sea and glue with it in the infrared



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Pion’s Goldberger

-Treiman relation

➢ Pion’s Bethe-Salpeter amplitude

Solution of the Bethe-Salpeter equation

➢ Dressed-quark propagator

➢ Axial-vector Ward-Takahashi identity entails



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Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Miracle: two body problem solved, almost completely, once solution of one body problem is known

B(k2)

Owing to DCSB& Exact inChiral QCD

http://inspirebeta.net/record/445150?ln=en



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Rudimentary version of this relation is apparent in Nambu’s Nobel Prize work



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Rudimentary version of this relation is apparent in Nambu’s Nobel Prize work


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Pion masslessness➢ Obtain a coupled set of gap- and Bethe-Salpeter equations

– Bethe-Salpeter Kernel:

• valence-quarks with a momentum-dependent running mass produced by self-interacting gluons, which have given themselves a running mass

• Interactions of arbitrary but enumerable complexity involving these “basis vectors”

– Chiral limit: • Algebraic proof

– at any & each finite order in symmetry-preserving construction of kernels for

» the gap (quark dressing)

» and Bethe-Salpeter (bound-state) equations,

– there is a precise cancellation between

» mass-generating effect of dressing the valence-quarks

» and attraction introduced by the scattering events

• Cancellation guarantees that

– simple system, which began massless,

– becomes a complex system, with

» a nontrivial bound-state wave function

» attached to a pole in the scattering matrix, which remains at P2=0 …

• Interacting, bound system remains massless!



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Munczek, H. J., Phys. Rev. D 52 (1995) pp. 4736-4740Bender, A., Roberts, C.D. and von Smekal, L., Phys. Lett. B 380 (1996) pp. 7-12Maris, P. , Roberts, C.D. and Tandy, P.C., Phys. Lett. B 420 (1998) pp. 267-273Binosi, Chang, Papavassiliou, Qin, Roberts, Phys. Rev. D 93 (2016) 096010/1-7

Pion masslessness➢ Obtain a coupled set of gap- and Bethe-Salpeter equations

– Bethe-Salpeter Kernel:

• valence-quarks with a momentum-dependent running mass produced by self-interacting gluons, which have given themselves a running mass

• Interactions of arbitrary but enumerable complexity involving these “basis vectors”

– Chiral limit: • Algebraic proof

– at any & each finite order in symmetry-preserving construction of kernels for

» the gap (quark dressing)

» and Bethe-Salpeter (bound-state) equations,

– there is a precise cancellation between

» mass-generating effect of dressing the valence-quarks

» and attraction introduced by the scattering events

• Cancellation guarantees that

– simple system, which began massless,

– becomes a complex system, with

» a nontrivial bound-state wave function

» attached to a pole in the scattering matrix, which remains at P2=0 …

• Interacting, bound system remains massless!



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Munczek, H. J., Phys. Rev. D 52 (1995) pp. 4736-4740Bender, A., Roberts, C.D. and von Smekal, L., Phys. Lett. B 380 (1996) pp. 7-12Maris, P. , Roberts, C.D. and Tandy, P.C., Phys. Lett. B 420 (1998) pp. 267-273Binosi, Chang, Papavassiliou, Qin, Roberts, Phys. Rev. D 93 (2016) 096010/1-7

Quantum field theory statement: In the pseudsocalar channel, the dynamically

generated mass of the two fermions is precisely cancelled by the attractive

interactions between them – iff –

Empirical Consequences of EHM

➢ QCD's interactions are universal … same in all hadrons

➢ However, expression need not be identical in all hadrons

➢ DCSB in chiral limit ensures SUM of different trace-anomaly operator-contributions cancel amongst each other to yield 𝑚𝜋 = 0

– Individual terms do not vanish separately

➢ In proton, no symmetry requires cancellations to be complete

Thus, value of proton's mass is typical of the magnitude of scale breaking in one body sectors = dressed-gluon and -quark mass scales

➢ This “DCSB paradigm” provides basis for understanding why:

– mass-scale for strong interactions is vastly different to that of electromagnetism

– proton mass expresses that scale

– pion is nevertheless unnaturally light



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Pion mass – a decomposition?

➢ Rest frame decomposition of the trace anomaly has been used [Yang:2014qna] to separate the pion’s mass into contributions from pieces defined as the

– mq-term,

– HA = trace-anomaly,

– HKU = quark EK+EU,

– Hg = gluon energy

➢ Conclusion:

“For the light PS mesons, the mq-term is about 50% of the total mass. This implies that HA contributes ∼ 12% of the mass.

The remaining contributions from Hg and HKU are ∼ 30% and ∼ 8% respectively.

It is interesting to observe that all these contributions are positive which suggests that they all approach zero at the chiral limit when the pion mass approaches zero.”

➢ Unfortunately … none of this makes sense.



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➢ Rest frame decomposition in this form leads to a precarious position because

i. Any conclusions are frame- and scale-dependent

ii. the gluons in the trace anomaly and in the kinetic and potential energy are seemingly being treated as separate entities, which, of course, they’re not

iii. chiral limit – massless particle does not have a rest frame – no simple-minded limit of separate terms is meaningful

➢ No quantum mechanics picture of a bound-state’s mass has all terms positive:

– kinetic energy and mass terms are positive

– binding energy is negative

➢ Gell-Mann – Oakes – Renner [GellMann:1968rz] … known for > 50 years:

𝑚𝜋2 ∝ 𝑚 × 𝐷𝐶𝑆𝐵 𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 … explicitly,

If ALL of 𝑚𝜋2 owes to the current-quark mass term in a Poincaré-invariant statement

Then what meaning can one associate with “rest frame” statement that only half 𝑚𝜋

comes from the quark mass term, i.e. mq-term produces 0.25 𝑚𝜋2 ?!

Pion mass – problems



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𝑚𝜋2 = (𝑚𝑢 +𝑚𝑑)

−⟨ത𝑞𝑞⟩

𝑓𝜋2

Huge EHM enhancement factor ≈ 200 (𝑚𝑢 +𝑚𝑑)

Decompositions of mass – Solution

➢ Avoid a Mass Crisis

➢ Stay away from decompositions whose definitions and interpretations are muddled by frame- and scale- dependence

➢ Cleanest statement of the origin of mass

– There is a trace anomaly: Θ0 = β(α) ¼Gμν Gμν = Tρρ

– In the chiral limit

➢ Understand, explain and explicate how this dichotomy is resolved.



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See also C. Lorcé at e-Workshop “Perceiving EHM through AMBER@CERN”

[https://indico.cern.ch/event/880248/contributions/3800568/]

Empirical Consequences of EHM

➢ No significant mass-scale is possible unless one of similar size is expressed in the dressed-propagators of gluons and quarks.

➢ Follows that the mechanism(s) responsible for emergence of mass can be exposed by measurements sensitive to such dressing

➢ This potential is offered by many observables:

– Spectra and static properties

– Form factors, elastic and transition

– All types of parton distributions

➢ Describe some examples … π & K structure



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Light Front Wave Function

➢ In many respects, a hadron’s LFWF is the key.

➢ LFWF correlates all observables

➢ EHM is expressed in every hadron LFWF

➢ The “trick” is to find a way to compute the LFWF

➢ Experiments sensitive to differences in LFWFs are sensitive to EHM

➢ Excellent examples are π & K DAs and DFs

– Two sides of the same coin

– Accessible via different processes

– Independent measurements of the same thing

– Great check on consistency



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π

K

PDAs & PDFs

➢ Relationship between leading-twist PDAs and valence-quark PDFs, expressed via a meson's light-front wave function (LFWF):

➢ Given that factorization of LFWF is a good approximation for integrated quantities, then at the hadronic scale, ζH:

Proportionality constant is fixed by baryon number conservation

➢ Owing to parton splitting effects, this identity is not valid on ζ > ζH .

(Think about DGLAP and ERBL regions for a GPD.)

➢ Nevertheless, evolution equations are known; so the connection is not lost, it just metamorphoses.



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➢ Continuum results exist & lQCD results arriving

➢ Common feature = broadening

➢ Origin = EHM

➢ NO differences between π & K if EHM is all there is

– Differences arise from Higgs-modulation of EHM mechanism

– “Contrasting π & K properties reveals Higgs wave on EHM ocean”

Meson leading-twist DAs



51

asymptotic

pionkaon

kaonpion

➢ Kaon DA vs pion DA

– almost as broad

– peak shifted to x=0.4(5)

– ⟨ξ2⟩ = 0.24(1), ⟨ξ⟩ = 0.035(5)

➢ ERBL evolution logarithmic

➢ Broadening & skewing persist to verylarge resolving scales – beyond LHC

Meson leading-twist DAs and valence-quark DFs

➢ Broadening need not and should not disturb the DA's endpoint behaviour

QCD: 𝜑 𝑥 = 𝑥 1 − 𝑥 𝑓 𝑥 , 𝑓 𝑥 ≃ 0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡1, 𝑓 𝑥 ≃ 1 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡2

➢ Many models that express EHM-induced broadening violate this constraint

➢ Typically not a problem, unless endpoint behaviour is taken too seriously



52

➢ Example AdS/QCD: 𝜑 𝑥 =8

𝜋𝑥(1 − 𝑥)

➢ Practically identical to the continuum prediction that preserves QCD constraint:

blue dashed vs green dot-dashed

– However, AdS/QCD practitioners use DA to argue for 𝑥 ≃ 1 ⇒ 𝑞𝜋 𝑥; 𝜁H ∝ (1 − 𝑥)1

– Endpoint behaviour taken “too seriously”

Pion DA & form factor

➢ QCD is not found in scaling

… QCD is found in scaling violations

➢ Continuum predictions

– Match existing data

– Suggest that JLab 12 could potentially be first to reveal scaling violations in a hard-scattering process = see QCD in a hard-scattering process

➢ Simulations indicate that EIC is certainly capable of doing so.

➢ Normalisation of the form-factor curve is a measure of the level of DA broadening; hence, size of EHM



53

Kaon form factor

– flavour separation

➢ Current conservation: Fuss(0) = Fuus(0)

➢ Under evolution: ϕK → 6 x (1-x) ⇒ ωs̅ → ωu ⇒ Ratio → 1

➢ Agreement between direct calculation and hard-scattering formula, using consistent PDA

➢ Ratio never exceeds 1.5 and

Logarithmic approach to unity

➢ Typical signal of EHM-dominance in flavour-symmetry breaking, taming the large Higgs-produced current-quark mass difference:

– ms ~ 30 mu ⇒ Ms(0) ∼ 1.25 Mu(0) – scale difference does finally become irrelevant

under evolution, but only at very large scales



54

Hard scattering formula, using DSE=lQCD predicted PDAs

DSE Prediction

[ s̅ γ s uspectator / u̅ γ u sspectator ]2 ≤ 1.5

Exposing strangeness: projections for kaon electromagnetic form factors, Fei Gao et al., arXiv:1703.04875 [nucl-th], Phys. Rev. D 96 (2017) 034024

Controversy over PDAs

➢ E791 Collaboration, E. Aitala et al., Phys. Rev. Lett. 86, 4768 (2001).

– Claim: ϕπ(x) is well represented by the asymptotic profile for ζ2 > 10 GeV2

➢ Modern continuum predictions and analyses of lQCD

– PDAs are broadened at ζ2=4 GeV2

– Evolution is logarithmic ⇒ if ϕ not asymptotic at

ζ2=4 GeV2, then ϕ not asymptotic at ζ2=10 GeV2

➢ Theory indicates that E791 conclusion cannot be correct

– The E791 images cannot represent the same pion property

– Not credible to assert that ϕπ(x) is well represented by the asymptotic distribution for ζ2 > 10 GeV2



55

Left: Nonpertubative (broadening) important Right: Asymptotic profile sufficient

➢ Hard exclusive processes only sensitive to low-order PDA moments.

➢ Diffractive processes much better because sensitive to x-dependence?

(check this claim)

➢ Owing to absence of stable pion targets, the pion’s valence-quark distribution functions have hitherto been measured via the Drell-Yan process:

π p → μ+ μ− X

➢ Consider a theory in which quarks scatter via a vector-boson exchange interaction whosek2>>mG

2 behaviour is (1/k2)β

➢ Then at a resolving scale ζH … uπ(x; ζH) ~ (1-x)2β

Namely, the large-x behaviour of the quark distribution function is a direct measure of the momentum-dependence of the underlying interaction.

➢ In QCD, β=1 and hence

QCD prediction of meson valence-quark distributions



56

QCD uπ(x; ζH) ~ (1-x)2QCD: Q > ζH ⇒ 2 → 2+γ, γ > 0

Empirical status of the Pion’s

valence-quark distributions

➢ Owing to absence of pion targets, the pion’s valence-quark distribution functions have hitherto been measured via the Drell-Yan process:

π p → μ+ μ− X

➢ Three experiments:

– CERN (1983 & 1985)

– FNAL (1989).

➢ None more recent

➢ Conway et al.Phys. Rev. D 39, 92 (1989)

– Leading-order analysis of the Drell-Yan data

– ~ 400 citations



57

Pion


➢ Owing to absence of pion targets, the pion’s valence-quark distribution functions have hitherto been measured via the Drell-Yan process:

π p → μ+ μ− X

➢ Three experiments:

– CERN (1983 & 1985)

– FNAL (1989).

➢ None more recent

➢ Conway et al.Phys. Rev. D 39, 92 (1989)

– Leading-order analysis of the Drell-Yan data

➢ Factor of 2 discrepancy with QCD! Controversial!

Empirical status of the Pion’s

valence-quark distributions



58

Pion

∝ (1-x)1


π valence-quark distributions

20 Years of Evolution➢ 1989 … Conway et al. Phys. Rev. D 39, 92 (1989)

– Leading-order analysis of Drell-Yan data

➢ Ensuing years, great deal of fog

➢ Perceptions ebbed and flowed

– Is this data a fundamental challenge to Standard Model?



59

∝ (1-x)1



20 Years of Evolution➢ 1989 … Conway et al. Phys. Rev. D 39, 92

(1989)


➢ 2000 … Hecht, Roberts, Schmidt, Phys. Rev. C 63 (2001) 025213– QCD-connected model prediction



60

∝ (1-x)2.8

∝ (1-x)1




(1989)


➢ 2000 … Hecht, Roberts, Schmidt, Phys. Rev. C 63 (2001) 025213– QCD-connected model prediction

➢ 2005 … Wijesooriya, Reimer, Holt, Phys. Rev. C 72 (2005) 065203 – Partial NLO analysis of E615 data– Large-x power-law → 1.54±0.08



61

∝ (1-x)2.8

∝ (1-x)1




(1989)


➢ 2000 … Hecht, Roberts, Schmidt, Phys. Rev. C 63 (2001) 025213 – QCD-connected model prediction

➢ 2005 … Wijesooriya, Reimer, Holt Phys. Rev. C 72 (2005) 065203 – Partial NLO analysis of E615 data– Large-x power-law → 1.54±0.08

➢ 2010/02 … Controversy highlighted: Holt & Roberts, Rev. Mod. Phys. 82 (2010) 2991-3044



62

∝ (1-x)2.8

∝ (1-x)1




(1989)


➢ 2000 … Hecht, Roberts, Schmidt, Phys. Rev. C 63 (2001) 025213

– QCD-connected model prediction


➢ 2010/09 … Reconsideration of data: Aicher et al., Phys. Rev. Lett. 105 (2010) 252003 – Consistent next-to-leading-order analysis,

which includes NLL (soft-gluon resummation)

– Large-x power-law → 2.6(1)



63

∝ (1-x)2.8




(1989)


➢ 2000 … Hecht, Roberts, Schmidt, Phys. Rev. C 63 (2001) 025213

– QCD-connected model prediction


➢ 2010/09 … Reconsideration of data: Aicher et al., Phys. Rev. Lett. 105 (2010) 252003 – Consistent next-to-leading-order analysis,

which includes NLL (soft-gluon resummation)

– Large-x power-law → 2.6(1)



64

∝ (1-x)2.8


QCD prediction of meson valence-quark distributions

➢ Fact 1: After 40 years, no flaw has been found in the proof that 𝑥 ≃ 1 ⇒ 𝑞𝜋 𝑥; 𝜁H ∝ (1 − 𝑥)2

➢ Fact 2: Power law fixed by asymptotic behaviour of bound-state wave function

➢ Fact 3: Gluon corrections do NOT change power laws.

They only modify anomalous dimensions.

Again, proof is 40 years old and no flaw has been found. Rather, it has been confirmed in numerous continuum calculations.

➢ Exact statement – in textbooks, but often overlooked:

– β(ζ) increases logarithmically with inverse of valence-quark momentum fraction

– Coefficient decreases also, but at a different rate



65

Controversy over pion valence DF➢ Parton model prediction for the valence-quark DF of a spin-zero meson:

𝑥 ≃ 1 ⇒ 𝑞𝜋 𝑥; 𝜁H ∝ (1 − 𝑥)2

➢ The hadronic scale is not empirically accessible in Drell-Yan or DIS processes. (Matter of conditions necessary for data to be interpreted in terms of distribution functions.)

➢ For such processes, QCD-improvement of parton model leads to the following statement:

At any scale for which experiment can be interpreted in terms of parton distributions, then

𝑥 ≃ 1 ⇒ 𝑞𝜋 𝑥; 𝜁 ∝ 1 − 𝑥 𝛽=2+𝛾, 𝛾 > 0

➢ Simple restatement of the following:

– The parton model gives us scaling and scaling laws.

– QCD's gluon corrections give us scaling violations

– Scaling violations do NOT alter the integer-number that characterises scaling powers [L&B-1980 Lepage:1980fj]

– Certainly don’t reduce 2 → 1 (or 3 → 2 for nucleon valence) – scaling violations increase power logarithmically



66

Controversy over pion valence DF➢ Consequence

– Any analysis of DY or DIS (or similar) experiment which returns a value of 𝛽 <2 conflicts with QCD.

➢ Observation

– All existing internally-consistent calculations preserve connection between large-k2

behaviour of interaction and large-x behaviour of DF.

• J=0 … (1/k2)n ⇔ (1-x)2n

➢ No existing calculation with n=1 produces anything other than (1-x)2

➢ Internally-consistent calculation that preserve RG properties of QCD,

then 2 → 2+γ, γ>0, at any factorisation-valid scale

➢ Controversy:

– Ignore threshold resummation – typical of all modern phenomenology, despite Aicher et al., then data analysis yields (1-x)1+γ

– Include threshold resummation, then data analysis yields (1-x)2+γ



67


20 Years of Evolution → 2019

➢ Novel lattice-QCD algorithms beginning to yield results for pointwise behaviour of 𝑢𝜋(𝑥; 𝜁)

➢ Developments in continuum-QCD have enabled 1st

parameter-free predictions of valence, glue and sea distributions within the pion – Reveal that 𝑢𝜋(𝑥; 𝜁)is hardened by emergent mass

➢ Agreement between new continuum prediction for 𝑢𝜋(𝑥; 𝜁) [Ding:2019lwe] and recent lattice-QCD result [Sufian:2019bol]

➢ Real strides being made toward understanding pion structure.

➢ Standard Model prediction is stronger than ever before



68

βcontm(ζ5) = 2.66(12)βlattice(ζ5) = 2.45(58)

➢ Now – after 30 years – new era dawning in which the ultimate experimental checks can be made:

JLab12 … M2 beam-line @ CERN … EIC … EicC(?)

Where is “2” to be seen?

➢ Use DSE DF … prediction … NOT fit to data

– Within uncertainty, brackets DF points obtained in NLO+NLL analysis

• Central curve: χ2/dof = 1.66

– By same measure, inconsistent with LO E615

• Central curve: χ2/dof = 19.4 – order of magnitude larger

➢ Valence domain begins after peak, at which point 2 x V(x) > x ( S(x)+G(x) )

➢ Power discriminating function – local (x-dependent) exponent:

𝛽(𝑥) = −1 − 𝑥

𝑞𝑉𝜋(𝑥)

𝑑𝑞𝑉𝜋(𝑥)

𝑑𝑥

– “Active” power greater > 2 on x > 0.75



69

Data NLO+NLL

Data LO

Effective 𝛽(𝑥)

Precise data & sound extraction on 0.6 < x < 0.8 sufficient to test QCD prediction: 2 ≠ 1

chi^2 fit to E615 LO

DSE prediction


Comparison with JAM fits➢ Valence:

– momentum fraction similar– JAM profile much harder & inconsistent with QCD prediction

➢ Glue:– Pointwise agreement on x ≥ 0.05, but marked disagreement

on important complementary domain– Both continuum prediction and JAM fit are very different

from early phenomenology– Should be tested in new experiments that are directly

sensitive to the pion’s gluon content.– Perhaps, prompt photon & J/Ψ production

➢ Sea:– Prediction and fit disagree on entire x-domain – If pion’s gluon content is considered uncertain, then fair to

describe sea-quark distribution as empirically unknown– Motivation for the collection and analysis of DY data with π±

beams on isoscalar targets



70

JAM valence

JAM: glue & sea

DSE prediction

DSE predictionglue & sea

Emergent Mass

vs. Higgs Mechanism

➢ When does Higgs mechanism begin to influence mass generation?

➢ limit mquark→ ∞

φ(x) → δ(x-½)

➢ limit mquark → 0

φ(x) ∼ (8/π) [x(1-x)]½

➢ Transition boundary lies just above mstrange

➢ Hence … Comparisons between distributions of truly light quarks and those describing strange quarks are ideally suited to exposing measurable signals of emergent mass in counterpoint to Higgs-driven effects



71

Parton distribution amplitudes of S-wave heavy-quarkoniaMinghui Ding, Fei Gao, Lei Chang, Yu-Xin Liu and Craig D. RobertsarXiv:1511.04943 [nucl-th], Phys. Lett. B 753 (2016) pp. 330-335

q+qbar: Emergent (strong mass)

c+cbar: Higgs(weak mass)

asymptotic

s+sbar: on the border

http://inspirehep.net/record/1404882?ln=en

Meson valence-quark DFs

➢ Owing to these relations

➢ Broadening of DAs feeds into broadening of DFs

➢ Necessary consequence of EHM

➢ Moreover, any Higgs-boson related modulations of EHM in the DA will also be expressed in the DF

➢ Pion – Kaon comparisons great place to study interference between the Standard Model’s two mass-generating mechanisms



72

Kaon Distribution Functions



73

✓ Improved & extended approach used for pion DFs✓ Unified distribution amplitudes and functions for pion

and kaon – connect with pion and kaon form factors✓ Introduced mass-dependence into splitting functions

✓ valence momentum fraction at ζ5 = 0.41(4)

💣 One lQCD calculation (2003.14128 [hep-lat]):o Fractions systematically larger than continuum

predictions, especially for s̅:o u – 0.6(4.8)%, 21(6)%, 40(4)%

o s̅ – 24(7)%, 53(13)%, 84(16)%

o lQCD vs. with DSE predictions, lQCD DFs = much harder.

o lQCD DFs inconsistent with QCD prediction

lQCDDSE

https://arxiv.org/abs/2003.14128

Kaon Distribution Functions: ൗ𝒖𝑲(𝒙)𝒖𝝅 𝒙

➢ Uncertainty in continuum predictions for DFs cancels in ratio

➢ First lQCD results for ratio also drawn

➢ Relative difference between the central lQCDresult and DSE prediction is ≈ 5% … despite fact that individual lQCD DFs are very different from continuum results

➢ Long known fact, i.e. ൗ𝒖𝑲(𝒙)𝒖𝝅 𝒙 is very

forgiving of even large differences between the individual DFs used to produce the ratio

➢ More precise data crucial if ratio is to be used effectively to inform and test the modern understanding of SM NG modes

➢ Results for uπ(x;ζ), uK(x;ζ) separately have greater discriminating power.



74

DSE prediction

lQCD prediction

Glue and Sea in Kaon

➢ Glue and sea distributions in the kaon differ from those of the pion only on the valence region x > 0.25.

➢ Not surprising

– Mass-dependent splitting functions act primarily to modify valence DF of heavier quark

– Valence DFs are negligible at low-x, where glue and sea distributions are large, and vice versa

– Hence biggest impact of a change in the valence DFs must lie at large-x.



75

valence quark

valence antiquark

glue sea

Kaon 0.19(2) 0.23(2) 0.44(2) 0.14(2)

Pion 0.20(2) 0.20(2) 0.45(2) 0.14(2)

Status: pion and kaon structure functions

Pion➢ Pointwise behaviour of pion’s valence-quark distribution function: agreement between

predictions from lQCD and symmetry-preserving QCD-consistent continuum analyses

➢ Amongst existing phenomenological studies of pion structure functions, only one employs a next-to-leading-order analysis that includes threshold resummation. This study is unique in producing a valence-quark DF that is consistent with large-x QCD and matches continuum and lattice prediction

➢ General disagreement between phenomenological results and theory predictions for the pion’s valence-quark DF feeds into uncertainty about pion’s glue and sea distributions

➢ Resolution of these conflicts must await

– Improved phenomenological analyses that include threshold resummation

– New data that constrains the pion’s glue and sea distributions.



76


Kaon➢ Very little empirical information available on K DFs ⇒ no recent phenom. inferences.

– Valence-quark distributions: results from models and a single, recent lQCD study

– Kaon’s glue and sea distributions: no results

➢ Hence, symmetry-preserving continuum QCD predictions sketched here for entire array of kaon DFs currently stand alone.

➢ One piece of available experimental information: ൗ𝒖𝑲(𝒙)𝒖𝝅 𝒙

– Continuum prediction for ratio is consistent with the data.

– But, given the large errors, this ratio is very forgiving of even large differences between various calculations of the individual DFs used to produce the ratio.

• Modern, precise data is critical if this ratio is to be used as a path to understanding the Standard Model’s Nambu-Goldstone modes;

• Results for uπ(x;ζ5), uK(x;ζ5) separately would be better.



77


Kaon➢ Glue and Sea – Predictions:

– DFs very similar to those in the pion

– Detailed comparison requires the use of mass-dependent splitting functions.

– Development underway … Preliminary conclusions:

i. Light-front momentum fraction carried by s-quarks in the kaon increases by ∼ 5%;

ii. Compensated by

commensurate decrease in fractions carried by glue (−1%) and sea (−2%).



78

NEEDS: pion and kaon structure functions

➢ Standard Model’s (pseudo-) Nambu-Goldstone modes – pions and kaons – are basic to the formation of everything from nucleons, to nuclei, and on to neutron stars.

➢ Hence, new-era experiments capable of discriminating between the results from models, phenomenology and QCD-connected predictions should have high priority.

➢ Phenomenological methods needed to proceed from data to DFs must match modern experiments in precision.

➢ Theory: continuum and lattice analyses of the pion’s valence-quark DF are converging on the same form, confirming the longstanding QCD expectation– But, lattice results for the pion’s glue and sea distributions would be very valuable.

➢ Even more true for the kaon.– Only one extant lattice study of kaon DFs– Addressing solely valence distributions– Disagreeing in many respects with continuum predictions – Conflict with large-x QCD

⇒ Many opportunities are available.



79

➢ Challenge: Explain the Origin & Distribution of the Bulk of Visible Mass

➢ Progress and Insights being delivered by amalgam of – Experiment … Phenomenology …Theory

➢ Continued exploitation of synergies essential to capitalise on new opportunities provided by existing & planned facilities

➢ This Discussion … join theorists from high-energy nuclear & particle physics in dialogue with the experimentalists … address the Emergence of Hadron Mass

Future



80

➢ Consolidate & expand collaboration between experimentalists proposing new measurements, phenomenologists doing global data analyses, & hadron-structure theorists.

– Autumn 2020 … Hopefully, a face-to-face meeting at CERN in Autumn.

– April 2021 … ECT* Trento

– June 2021 … Strong QCD 2021 @ NJU-INP

➢ Challenge: Explain the Origin & Distribution of the Bulk of Visible Mass

➢ Progress and Insights being delivered by amalgam of – Experiment … Phenomenology …Theory

➢ Next Steps … – Ongoing efforts/meetings toward

• Entering a significant new element into the EIC User Group Physics and Detector Handbook

• Developing contributions as part of Yellow Report Initiative.

– ∼ 20.07.20 … Tele-conversation: EHM through AMBER @ CERN– Oct. 2020 … 1-week workshop Exploring QCD with Tagged

Processes – Université Paris-Saclay

Future



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Discussions to explore EicC reach into pion and kaon structure – EicC is potentially the unique bridge between JLab12 and BNL-EIC

Thank you

emergence of mass in the standard model

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