Assembling Hadrons From Quark-Gluon Assembling Hadrons From Quark-Gluon PiecesPieces

Adnan Bashir, Michoacán Adnan Bashir, Michoacán University, MexicoUniversity, Mexico Collaborators:Collaborators:J. Aslam, Quaid-i-Azam University, PakistanJ. Aslam, Quaid-i-Azam University, PakistanF. Akram, University of Punjab, PakistanF. Akram, University of Punjab, PakistanA.A.Ayala, UNAM, MexicoAyala, UNAM, MexicoB. El Bennich, Cruzeiro do Sul, BrazilB. El Bennich, Cruzeiro do Sul, BrazilI. Cloet, Argonne National LabotoryI. Cloet, Argonne National LabotoryS. Ishaq, NCP, PakistanS. Ishaq, NCP, PakistanY.X. Liu, Peking University, ChinaY.X. Liu, Peking University, ChinaJ.R. Quintero, Huelva University, SpainJ.R. Quintero, Huelva University, SpainA.A.Raya, Michoacán University, MexicoRaya, Michoacán University, MexicoRiazuddin, NCP, PakistanRiazuddin, NCP, PakistanM.E. Tejeda, USON, MexicoM.E. Tejeda, USON, MexicoC.D. Roberts, Argonne National Laboratory, C.D. Roberts, Argonne National Laboratory, USAUSAP.C. Tandy, Kent State University, USAP.C. Tandy, Kent State University, USA

Collaborators:Collaborators:L. Albino, University of Michoacán, MexicoL. Albino, University of Michoacán, MexicoA. Ahmad, University of Michoacán, MexicoA. Ahmad, University of Michoacán, MexicoM.A. Bedolla, University of Michoacán, M.A. Bedolla, University of Michoacán, MexicoMexico R. Bermudez, University of Sonora, MexicoR. Bermudez, University of Sonora, MexicoJ. Cobos, University of Michoacán, MexicoJ. Cobos, University of Michoacán, MexicoL. Chang, University of Adelaide, AustraliaL. Chang, University of Adelaide, AustraliaL.X. Gutiérrez, University of Michoacán, L.X. Gutiérrez, University of Michoacán, MexicoMexicoE. Gutiérrez, University of Michoacán, E. Gutiérrez, University of Michoacán, MexicoMexicoK. Raya, University of Michoacán, MexicoK. Raya, University of Michoacán, MexicoE. Rojas, Cruzeiro do Sul, BrazilE. Rojas, Cruzeiro do Sul, BrazilD. Wilson, Jlab, USA D. Wilson, Jlab, USA

IntroductionIntroduction

Schwinger-Dyson Equations

QCD Phase Diagram

Hadron Physics

Chiral Symmetry Breaking

Magnetic Catalysis

Condensed Matter

Systems

Running Quark

Masses

IntroductionIntroduction

Hadrons From Quark-Gluon Hadrons From Quark-Gluon PiecesPiecesHadrons From Quark-Gluon Hadrons From Quark-Gluon PiecesPieces

ContentsContents

• ConclusionsConclusions• ConclusionsConclusions

• Introduction to Hadron Form FactorsIntroduction to Hadron Form Factors• Introduction to Hadron Form FactorsIntroduction to Hadron Form Factors

• Schwinger-Dyson Equations – The Schwinger-Dyson Equations – The IngredientsIngredients

• Schwinger-Dyson Equations – The Schwinger-Dyson Equations – The IngredientsIngredients

Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function

Quark-Photon VertexQuark-Photon VertexQuark-Photon VertexQuark-Photon Vertex

Contact Interaction Contact Interaction Contact Interaction Contact Interaction

The Gluon Propagator/Quark Gluon The Gluon Propagator/Quark Gluon Vertex Vertex The Gluon Propagator/Quark Gluon The Gluon Propagator/Quark Gluon Vertex Vertex

• The QThe Q22 Evolution of Form Factors Evolution of Form Factors • The QThe Q22 Evolution of Form Factors Evolution of Form Factors

From Mesons to Baryons From Mesons to Baryons From Mesons to Baryons From Mesons to Baryons

Momentum Dependent Mass and Form Factors Momentum Dependent Mass and Form Factors Momentum Dependent Mass and Form Factors Momentum Dependent Mass and Form Factors

Bethe-Salpeter Amplitude Bethe-Salpeter Amplitude Bethe-Salpeter Amplitude Bethe-Salpeter Amplitude

IntroductionIntroductionIntroductionIntroduction

Hadronic form factorsHadronic form factors are related to their internal are related to their internalstructure, the distribution of structure, the distribution of charge and charge and magnetizationmagnetization. . The challenge of their understanding & hence their The challenge of their understanding & hence their internalinternaldynamicsdynamics occupies a central place in hadron occupies a central place in hadron physics. physics.

Hadronic form factorsHadronic form factors are related to their internal are related to their internalstructure, the distribution of structure, the distribution of charge and charge and magnetizationmagnetization. . The challenge of their understanding & hence their The challenge of their understanding & hence their internalinternaldynamicsdynamics occupies a central place in hadron occupies a central place in hadron physics. physics.

QCDQCD is the established theory of strong is the established theory of strong interactions interactions which is responsible for binding which is responsible for binding quarks and gluonsquarks and gluons totoform these form these hadrons (mesons and baryons)hadrons (mesons and baryons). .

QCDQCD is the established theory of strong is the established theory of strong interactions interactions which is responsible for binding which is responsible for binding quarks and gluonsquarks and gluons totoform these form these hadrons (mesons and baryons)hadrons (mesons and baryons). . Unraveling Unraveling hadronic form factors hadronic form factors from thefrom the basic basic buildingbuildingblocks of QCDblocks of QCD is an outstanding problem. is an outstanding problem.

Unraveling Unraveling hadronic form factors hadronic form factors from thefrom the basic basic buildingbuildingblocks of QCDblocks of QCD is an outstanding problem. is an outstanding problem. Schwinger-Dyson equationsSchwinger-Dyson equations are the fundamental are the fundamental equationsequationsof QCD and combine its of QCD and combine its UVUV and and IRIR behavior. behavior.

Schwinger-Dyson equationsSchwinger-Dyson equations are the fundamental are the fundamental equationsequationsof QCD and combine its of QCD and combine its UVUV and and IRIR behavior. behavior. Thus they provide an ideal platform to study the Thus they provide an ideal platform to study the form form factors from small to large probing photon factors from small to large probing photon virtualitiesvirtualities,,measured at different hadron facilities. measured at different hadron facilities.

Thus they provide an ideal platform to study the Thus they provide an ideal platform to study the form form factors from small to large probing photon factors from small to large probing photon virtualitiesvirtualities,,measured at different hadron facilities. measured at different hadron facilities.

IntroductionIntroductionIntroductionIntroduction

Parity Parity Partners Partners &&

Chiral Chiral Symmetry Symmetry BreakingBreaking

Parity Parity Partners Partners &&

Chiral Chiral Symmetry Symmetry BreakingBreaking

The quark The quark propagatorpropagator

::

The quark The quark propagatorpropagator

::

Quark mass Quark mass is a is a functionfunctionof of

momentummomentum, ,

falling off infalling off inthe the

ultraviolet.ultraviolet.

Quark mass Quark mass is a is a functionfunctionof of

momentummomentum, ,

falling off infalling off inthe the

ultraviolet.ultraviolet.

The Quark PropagatorThe Quark PropagatorThe Quark PropagatorThe Quark Propagator

Maris-Roberts-Tandy Maris-Roberts-Tandy ModelModel

Maris-Roberts-Tandy Maris-Roberts-Tandy ModelModel

The Gluon PropagatorThe Gluon Propagator

Several SDE and Several SDE and lattice results support lattice results support decoupling solution for decoupling solution for the gluon propagator.the gluon propagator.

Several SDE and Several SDE and lattice results support lattice results support decoupling solution for decoupling solution for the gluon propagator.the gluon propagator.

Momentum dependent gluon mass is reminiscent Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass of the momentum dependent quark mass function.function.

Momentum dependent gluon mass is reminiscent Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass of the momentum dependent quark mass function.function.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.

A. Ayala et. al. Phys. Rev. D86 074512 (2012).A. Ayala et. al. Phys. Rev. D86 074512 (2012).

AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. Roberts, Roberts,

P. Tandy, Comm. Theor. Phys. 58 79-134 (2012) P. Tandy, Comm. Theor. Phys. 58 79-134 (2012)

Gluon Propagator:Gluon Propagator: Gluon Propagator:Gluon Propagator:

A. Bashir, A. Raya, J. Rodrigues-Quintero, A. Bashir, A. Raya, J. Rodrigues-Quintero, Phys. Rev. D88 054003 (2013).Phys. Rev. D88 054003 (2013).

I.L. Bogolubsky, et. al. Phys. Lett. B676 69 I.L. Bogolubsky, et. al. Phys. Lett. B676 69 (2009). (2009).

AB, M.R. Pennington Phys. Rev. D50 7679 (1994) AB, M.R. Pennington Phys. Rev. D50 7679 (1994)

D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990)(1990)

A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 (2009)(2009) L. Chang, C.D. Roberts, Phys. Rev. Lett. 103 081601 (2009) L. Chang, C.D. Roberts, Phys. Rev. Lett. 103 081601 (2009)

AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D83 033003 (2011)AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D83 033003 (2011)

AB, R. Bermudez, L. Chang, C.D. Roberts, AB, R. Bermudez, L. Chang, C.D. Roberts, Phys. Rev. C85, 045205 Phys. Rev. C85, 045205 (2012)(2012)..

Phenomenology

GaugeCovariance

Lattice

MultiplicativeRenormalization

PerturbationTheory

Quark-photon/ quark-gluon

vertex

Significantly, this last ansatz contains nontrivial Significantly, this last ansatz contains nontrivial factors associated with those tensors whose factors associated with those tensors whose appearance is solely driven by appearance is solely driven by dynamical chiral dynamical chiral symmetry breakingsymmetry breaking..It yields It yields gauge independent critical couplinggauge independent critical coupling in in QED.QED.A careful choice of parameters can also produce A careful choice of parameters can also produce large anomalous magnetic momentlarge anomalous magnetic moment for quarks in for quarks in the infrared.the infrared.

Quark-photon Quark-photon VertexVertexQuark-photon Quark-photon VertexVertex

The Quark-Photon VertexThe Quark-Photon VertexThe Quark-Photon VertexThe Quark-Photon Vertex

L.L. Albino, R.Albino, R. Bermúdez, L.X. Gutiérrez:Bermúdez, L.X. Gutiérrez: Quark-Gluon Vertex. Quark-Gluon Vertex.

J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, J. High J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, J. High Energy Phys. 04 047 (2003) Energy Phys. 04 047 (2003)

M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy, Phys. Rev. C68 015203 M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy, Phys. Rev. C68 015203 (2003).(2003).AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc. 1026 262 (2008).AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc. 1026 262 (2008).

The Quark-The Quark-GluonGluonVertex Vertex

One of the 12 One of the 12 form factorsform factors

The Quark-The Quark-GluonGluonVertex Vertex

One of the 12 One of the 12 form factorsform factors

The Quark-Gluon VertexThe Quark-Gluon VertexThe Quark-Gluon VertexThe Quark-Gluon Vertex

The QThe Q22 Evolution of Form Evolution of Form FactorsFactors

The QThe Q22 Evolution of Form Evolution of Form FactorsFactors

Observing the transition of the hadron from a sea Observing the transition of the hadron from a sea ofofquarks and gluons to the one with valence quarks quarks and gluons to the one with valence quarks alone is an experimental and theoretical alone is an experimental and theoretical challenge. challenge.

Observing the transition of the hadron from a sea Observing the transition of the hadron from a sea ofofquarks and gluons to the one with valence quarks quarks and gluons to the one with valence quarks alone is an experimental and theoretical alone is an experimental and theoretical challenge. challenge.

Schwinger-Dyson equations are the fundamental Schwinger-Dyson equations are the fundamental equationsequationsof QCD and combine its of QCD and combine its UV and IR behaviourUV and IR behaviour. .

Schwinger-Dyson equations are the fundamental Schwinger-Dyson equations are the fundamental equationsequationsof QCD and combine its of QCD and combine its UV and IR behaviourUV and IR behaviour. .

We assume that quarks interact not via massless We assume that quarks interact not via massless vector vector boson but instead through a boson but instead through a contact interactioncontact interaction of of veryverymassive gauge boson by assuming:massive gauge boson by assuming:

We assume that quarks interact not via massless We assume that quarks interact not via massless vector vector boson but instead through a boson but instead through a contact interactioncontact interaction of of veryverymassive gauge boson by assuming:massive gauge boson by assuming:

Here Here mmGG=0.8 GeV is a =0.8 GeV is a gluon mass scalegluon mass scale which is which is generatedgenerateddynamically in QCD. dynamically in QCD.

Here Here mmGG=0.8 GeV is a =0.8 GeV is a gluon mass scalegluon mass scale which is which is generatedgenerateddynamically in QCD. dynamically in QCD. Ph. Boucaud, J.P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene, J. Ph. Boucaud, J.P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene, J. Rodriguez-Rodriguez-Quintero, J. High Energy Phys. 06, 099 (2008); Quintero, J. High Energy Phys. 06, 099 (2008); A.C. Aguilar, D. Binosi, J. A.C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D78 025010 (2009).Papavassiliou, Phys. Rev. D78 025010 (2009).

We use We use proper time regularization which proper time regularization which guaranteesguarantees

confinementconfinement and is backed by phenomenology. and is backed by phenomenology.

We use We use proper time regularization which proper time regularization which guaranteesguarantees

confinementconfinement and is backed by phenomenology. and is backed by phenomenology.

withwithwithwith

The QThe Q22 Evolution of Form Factors Evolution of Form Factors

C. Chen,, L. Chang, C.D. Roberts, S. Wan, D. Wilson, Few Body Syst. 52 293 C. Chen,, L. Chang, C.D. Roberts, S. Wan, D. Wilson, Few Body Syst. 52 293 (2012).(2012). AB, M. Bedolla, J. Cobos, in progress.AB, M. Bedolla, J. Cobos, in progress.

Within the rainbow ladder truncation, the elasticWithin the rainbow ladder truncation, the elasticelectromagnetic pion form factor: electromagnetic pion form factor: Within the rainbow ladder truncation, the elasticWithin the rainbow ladder truncation, the elasticelectromagnetic pion form factor: electromagnetic pion form factor: The pattern of chiral symmetry breaking The pattern of chiral symmetry breaking dictates dictates the the momentum dependence of the elastic pion form momentum dependence of the elastic pion form factor. factor.

The pattern of chiral symmetry breaking The pattern of chiral symmetry breaking dictates dictates the the momentum dependence of the elastic pion form momentum dependence of the elastic pion form factor. factor.

L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010).L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010).

F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, Phys Rev. D87 013011 (2013). [QED]Phys Rev. D87 013011 (2013). [QED]

Pion Electromagnetic Form FactorPion Electromagnetic Form Factor

Pion Electromagnetic Form FactorPion Electromagnetic Form Factor

When do we expect the turn over to start? When do we expect the turn over to start? When do we expect the turn over to start? When do we expect the turn over to start?

Perturbative

Momentum transfer Q is primarily shared equally Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero (Q/2) among quarks as BSA is peaked at zero relative momentum. relative momentum.

Momentum transfer Q is primarily shared equally Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero (Q/2) among quarks as BSA is peaked at zero relative momentum. relative momentum.

Jlab 12GeV: 2<QJlab 12GeV: 2<Q22<6 GeV<6 GeV22 electromagnetic pion form electromagnetic pion form factor.factor.

The The transition form factor:transition form factor: The The transition form factor:transition form factor:

CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2 CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2 BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation:

G.P. Lepage, and S.J. Brodsky, G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 Phys. Rev. D22, 2157 (1980).(1980).BelleBelle S. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0

GeV2

H.L.L. Robertes, C.D. Roberts, AB, L.X. H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Gutiérrez and P.C. Tandy, Phys. Rev. Phys. Rev. C82, (065202:1-11) 2010.C82, (065202:1-11) 2010.

Pion to Pion to * Transition Form * Transition Form FactorFactor

Pion to Pion to * Transition Form * Transition Form FactorFactor

The transition form factor: The transition form factor: The transition form factor: The transition form factor:

• Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. • Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity.

Precise measurements at large QPrecise measurements at large Q22 will provide a will provide a stringentstringentconstraint on the constraint on the pattern of chiral symmetry pattern of chiral symmetry breakingbreaking. .

Precise measurements at large QPrecise measurements at large Q22 will provide a will provide a stringentstringentconstraint on the constraint on the pattern of chiral symmetry pattern of chiral symmetry breakingbreaking. .

Vladimir Savinov:Vladimir Savinov: 55thth Workshop of the APS Workshop of the APSTopical Group on HadronicTopical Group on HadronicPhysics, 2013.Physics, 2013.

Vladimir Savinov:Vladimir Savinov: 55thth Workshop of the APS Workshop of the APSTopical Group on HadronicTopical Group on HadronicPhysics, 2013.Physics, 2013.

Pion to Pion to * Transition Form * Transition Form FactorFactor

Pion to Pion to * Transition Form * Transition Form FactorFactor

Transfer of momentum dependence in QCD.Transfer of momentum dependence in QCD.Transfer of momentum dependence in QCD.Transfer of momentum dependence in QCD.

F. Akram, AB, K. Raya, work in progress. F. Akram, AB, K. Raya, work in progress.

Pion to Pion to * Transition Form * Transition Form FactorFactor

Pion to Pion to * Transition Form * Transition Form FactorFactor

Pion to Pion to * Transition Form * Transition Form FactorCFactorC

Pion to Pion to * Transition Form * Transition Form FactorCFactorC

Precise calculations with different interactions (pPrecise calculations with different interactions (p22))--

α α atatincreasing Qincreasing Q22 will provide a stringent constraint on will provide a stringent constraint on thethepattern of chiral symmetry breakingpattern of chiral symmetry breaking. .

Precise calculations with different interactions (pPrecise calculations with different interactions (p22))--

α α atatincreasing Qincreasing Q22 will provide a stringent constraint on will provide a stringent constraint on thethepattern of chiral symmetry breakingpattern of chiral symmetry breaking. .

• Double tagging? Double tagging? • Double tagging? Double tagging?

• Probing the (pProbing the (p22))-α-α dependence can be neater. dependence can be neater. • Probing the (pProbing the (p22))-α-α dependence can be neater. dependence can be neater.

Vladimir SavinovVladimir SavinovVladimir SavinovVladimir Savinov

Pion to Pion to * Transition Form * Transition Form FactorFactor

Pion to Pion to * Transition Form * Transition Form FactorFactor

Faddeev equationFaddeev equation for a baryon. for a baryon. Faddeev equationFaddeev equation for a baryon. for a baryon. G. Eichmann, G. Eichmann, Phys. Rev. D84, 014014 (2011).Phys. Rev. D84, 014014 (2011).

Faddeev equation in the quark diquark picture Faddeev equation in the quark diquark picture reproducesreproducesnucleon masses to within 5%.nucleon masses to within 5%.

Faddeev equation in the quark diquark picture Faddeev equation in the quark diquark picture reproducesreproducesnucleon masses to within 5%.nucleon masses to within 5%.

From Mesons to BaryonsFrom Mesons to BaryonsThe Diquark Picture: The Diquark Picture: The Diquark Picture: The Diquark Picture:

TransitionTransition

The nucleon primarily consists of scalar and The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its axial vector diquarks and N(1535) of its parity partners.parity partners.

The nucleon primarily consists of scalar and The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its axial vector diquarks and N(1535) of its parity partners.parity partners.

In the contact interaction model, the calculation In the contact interaction model, the calculation of theof thetransition form factors involves the diagram:transition form factors involves the diagram:

In the contact interaction model, the calculation In the contact interaction model, the calculation of theof thetransition form factors involves the diagram:transition form factors involves the diagram:

From Mesons to BaryonsFrom Mesons to Baryons

From Mesons to BaryonsFrom Mesons to Baryons

L.X. Gutiérrez, AB, C.D. Roberts,D.J. Wilson (In L.X. Gutiérrez, AB, C.D. Roberts,D.J. Wilson (In progress). progress).

From Mesons to BaryonsFrom Mesons to Baryons

L.X. Gutiérrez, AB, C.D. Roberts,D.J. Wilson (In L.X. Gutiérrez, AB, C.D. Roberts,D.J. Wilson (In progress). progress).

ConclusionsConclusions

A systematic framework based upon the A systematic framework based upon the QCD QCD equationsequationsof motion (SDE)of motion (SDE) and its symmetries is required to and its symmetries is required to chartchartout and comprehend the out and comprehend the QQ22 evolution evolution of these of these form form factorsfactors and make predictions. and make predictions.

A systematic framework based upon the A systematic framework based upon the QCD QCD equationsequationsof motion (SDE)of motion (SDE) and its symmetries is required to and its symmetries is required to chartchartout and comprehend the out and comprehend the QQ22 evolution evolution of these of these form form factorsfactors and make predictions. and make predictions.

The large QThe large Q22 evolution of the hadronic form evolution of the hadronic form factors, their factors, their experimental evaluationexperimental evaluation and and theoretical predictionstheoretical predictions are likely to provide us are likely to provide us with deep understanding of the with deep understanding of the pattern of pattern of DCSB DCSB and and confinementconfinement of the of the fundamentalfundamentaldegrees of freedom, namely degrees of freedom, namely quarksquarks and and gluonsgluons..

Predictions based upon the Predictions based upon the contact interactioncontact interaction, , QCD SDEQCD SDEas well as the as well as the intermediate power lawsintermediate power laws can provide can provide experimentalist with a platform to experimentalist with a platform to compare and compare and contrast contrast the future the future experimental resultsexperimental results. .

Predictions based upon the Predictions based upon the contact interactioncontact interaction, , QCD SDEQCD SDEas well as the as well as the intermediate power lawsintermediate power laws can provide can provide experimentalist with a platform to experimentalist with a platform to compare and compare and contrast contrast the future the future experimental resultsexperimental results. .