ruben sandapen · 2019. 10. 1. · spacelike em form factors of pion and kaon drell & yan (prl,...

What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions

An overview of light-front holography

Ruben Sandapen

Ecole Polytechnique, Palaiseau, France16-20 September 2019

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Overview

1 What is the QCD mass scale ?

2 LFH: theory

3 LFH: phenomenology

4 Conclusions

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Conformal QCD

If

no quantum loops (no ΛQCD)

semiclassical approximationmassless quarks (mf → 0)

QCD Lagrangian

LQCD = Ψ̄(iγµDµ −✟✟✯0m )Ψ−1

4G aµνG

aµν

then has no mass scale and QCD action

SQCD =

d4xLQCD

is conformally invariant

Gravity dual in higher dimensional anti-de Sitter space

A realization of Maldacena’s Equation: AdS=CFT

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What sets the scale of hadron masses?

Conformal symmetry is broken

in QCD

explicitly by non-zero quark masses: mf ∕= 0a mass scale → ΛQCD is generated after perturbativerenormalization of quantum loops (scheme-dependent)

in scLFQCD by the dAFF mechanism: a mass scale κ2 = λcan introduced in the Hamiltonian while retaining theconformal invariance of the action

dAFF: de Alfaro, Fubini and Furlan (76)

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LF QCD bound state equation

LF Heisenberg EquationHQCDLF |Ψ〉 = M2|Ψ〉

|Ψ〉 =

n |n〉〈n|Ψ〉

Heff |qq̄〉 = M2|qq̄〉

LF Schrödinger EquationM2qq̄ + Ueff

Ψ(x , k⊥, h, h̄) = M

2Ψ(x , k⊥, h, h̄)

Fock expansion of hadronic state

”Eliminate” higher Fock states

Project on valence sector

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Light-front wavefunctions

LF Schrödinger Equation:M2qq̄ + Ueff

Ψ(x , k⊥, h, h̄) = M

2Ψ(x , k⊥, h, h̄)

where the invariant mass squared of qq̄ pair is

M2qq̄ =k2⊥ +m

2

x(1− x)and Ψ(x , k⊥, h, h̄) is the LFWF

x = k+/P+: LF momentum fraction of quarkh, h̄: LF helicities of quark and antiquarkNormalization:

h,h̄

d2kdx |Ψ(x , k, h, h̄)|2 = 1

Consistent with constituent quark model

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LFSE in impact space

For massless quarks, introduce 2d Fourier transform of M2qq̄|mf =0

ζ2 = x(1− x)b2⊥

and separate variables

Ψ(x , ζ,ϕ) =φ(ζ)√2πζ

X (x)e iLϕ

Then

− d

2

dζ2− 1− 4L

2

4ζ2+ Ueff(ζ)

φ(ζ) = M2φ(ζ)

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The impact LF variable

Brodsky & de Téramond (PRL, 2009) Brodsky, de Téramond, Dosch & Erlich (Phys. Rep. 2015)

− d

2

dζ2− 1− 4L

2

4ζ2+ Ueff(ζ)

φ(ζ) = M2φ(ζ)

ζ2 = x(1− x)b2⊥ : key light-front variable for bound states.x = k+/P+ : light-front momentum fraction of quark.

b⊥ : transverse distance between quark and antiquark.

Ueff : effective (includes coupling to higher Fock sectors) qq̄potential.

Meson light-front wavefunction :

Ψ(x , ζ,ϕ) =φ(ζ)√2πζ

f (x)e iLϕ

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Why holographic ?

Physical spacetime

LF transverse distance

ζ

Angular momentum

L2

Anti de Sitter spacetime

Fifth dimension in AdS

z5

Spin + (AdS mass×radius)

(µR)2 + (2− J)2

LFSE maps onto wave equation for weakly coupled spin-J stringmodes in warped AdS with

Ueff(ζ) =1

2ϕ′′(z) +

1

4ϕ′(z)2 +

2J − 32z

ϕ′(z)

ϕ(z): dilaton field that breaks conformal invariance in AdS9 / 35


A unique confinement potential

The dilaton field that distorts pure AdS geometry drives theconfinement dynamics in physical spacetime

A quadratic dilaton ϕ = κ2z2 gives

Ueff(ζ) =

Conformal SBκ4ζ2 +

AdS/QCD 2κ2(J − 1)

Only harmonic potential preserves the conformal invariance ofthe underlying QCD action while introducing a mass scale inLF Hamiltonian: dAFF mechanism for breaking conformalsymmetry in QM

AdS/QCD mapping of EM form factors: f (x) =

x(1− x)

dAFF in scLFQCD: Brodsky, de Téramond & Dosch (Phys. Lett. 2013)

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Predicting a massless pion

Solving the holographic LF Schrödinger Equation− d

2

dζ2− 1− 4L

2

4ζ2+ Ueff(ζ)

φ(ζ) = M2φ(ζ)

withUeff(ζ) = κ

4ζ2 + 2κ2(J − 1)gives

M2n,J,L = 4κ2

n +

J + L

2

φn,L(ζ) = κ1+L

2n

(n + L)ζ1/2+L exp

−κ2ζ2

LLn(κ

2ζ2)

Lightest bound state (n = L = J = 0) is massless (M = 0)

As expected in chiral QCD : massless quarks → massless pion11 / 35


Baryon & mesons as SUSY partners

Baryon (quark-diquark) -meson degeneracy with LM = LB + 1

Tetraquark (diquark-antidiquark)-Baryon degeneracy with LT = LBS. J. Brodsky, G. de Téramond, H. G. Dosch, C. Lorcé Phys. Lett. B759 (2016) 171

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Light quark masses

In momentum space, ’complete’ invariant mass of qq̄ pair

M2qq̄ =k2⊥

x(1− x) →k2⊥ +m

2f

x(1− x)

Brodsky, de Téramond, Subnuclear Series Proc. (2009)

Meson wavefunction becomes

Ψ(x , ζ2) = N

x(1− x) exp−κ

2ζ2

2

exp

−

m2f2κ2x(1− x)

mf are effective quark masses (because of coupling of valencesector to higher Fock sectors).

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Extracting the light-front holographic scale

Global fit: universal κ = λ2 = 0.523± 0.024 GeVLight quark masses: mu/d = 0.046, ms = 0.357 GeV

S. J. Brodsky, G. de Téramond, H. G. Dosch, C. Lorcé Phys. Lett. B759 (2016) 171

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Predicting ΛQCD

A. Deur, S. J. Brodsky, G. de Téramond, J.Phys. G44 (2017) no.10, 105005

Matching αAdSs to αMSs to 5-loops: Λ

nf =3

MS= 0.339± 0.019

GeV (world average Λnf =3MS

= 0.332± 0.017 GeV)Pert. to Npert. transition scale: Q0 = 0.87± 0.08 GeV

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Can we predict the pion observables ?

Using the pion holographic wavefunction to predict

Charge radius

Decay constant

EM FF

π → γ TFFPrevious work

A. Vega, I. Schmidt, T. Branz, T. Gutsche, V. Lyubovitskij,PRD 055014 (2009): κ = 0.787 GeV, mf = 330 MeV,Pqq̄ = 0.279

T. Banz, T. Gutsche, V. E. Lyubovitskij, PRD 82 074022(2010): κ = 0.787 GeV, mf = 330 MeV, Pqq̄ = 0.6

R. Swarnkar and D. Chakrabarti, Phys. Rev. D92, 074023(2015): κ = 0.550 GeV, mf = 330 MeV, Pqq̄ = 0.61

It is possible to simultaneously all pion data with the universal κ ?

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Quark spins

Non-dynamical spin

Ψ(x , k2⊥, h, h̄) = Ψ(x , k2⊥)× Sh,h̄

where

Sπh,h̄

=1√2hδh,−h̄

and

Sρ(L)

h,h̄=

1√2δh,−h̄

Only longitudinally polarized ρ

Degenerate decay constants for the π and ρ mesons

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Dynamical spin effects

More realistic spin structure:

Ψ(x , k⊥) → Ψ(x , k⊥, h, h̄) = Ψ(x , k⊥)× Shh̄(x , k⊥)Vector meson:

Sλhh̄(x , k⊥) =

v̄h̄(x , k⊥)√1− x

[λV · γ]uh(x , k⊥)√

x

Pseudoscalar meson:

Shh̄(x , k⊥) =v̄h̄(x , k⊥)√

1− x

MP2P+

γ+γ5 + Bγ5uh(x , k⊥)√

x

Spin structure is point-like

Bound state effects captured by holographic wavefunction

For pseudoscalars: B = 0 → no dynamical spin effects18 / 35


Charge radii of pion and kaon

M. Ahmady, C. Mondal, R.S, PRD (2018) 034010

〈r2P〉 =

3

2

dxd2b⊥[b⊥(1− x)]2|ΨP(x ,b⊥)|2

1/2

π±: B ≥ 1 preferredK±: B = 0 preferred(ms = 0.500 GeV)

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Spacelike EM form factors of pion and kaon

Drell & Yan (PRL, 70); West (PRL, 70)

FEM(Q2) = 2π

dxdb⊥ b⊥ J0[(1− x)b⊥Q] |ΨP(x ,b⊥)|2

10−2 10−1 100 101

0

0.2

0.4

0.6

0.8

1

Q2 [GeV2]

Fπ (

Q2)

CERN (86)JLab (07)Harvard (74)Harvard (76)Cornell (77)JLab (01)JLab (06)Asymptotic pQCDB=0B=1B>>1

10−2 10−1 100 1010

0.2

0.4

0.6

0.8

1

Q2 [GeV2]

FK (

Q2)

FERMILAB (80)CERN (86)Asymptotic pQCDB=0B=1B>>1

Left: π±, B ≥ 1 preferred. Right: K±, B = 0 preferred.20 / 35


Decay constants of pion and kaon

〈0|Ψ̄γµγ5Ψ|P〉 = fPpµ

fP = 2

Ncπ

dx{((x(1− x)M2π) + B(xmq + x̄mq̄)MP}

ΨP(x , ζ)

x(1− x)

ζ=0

π±: B ≥ 1 preferredK±: B = 0 preferred.

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Holographic pion PDF versus E615 Drell-Yan data

f πv (x) =

d2b⊥

h,h̄

|Ψhh̄(x ,b⊥)|2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

x

x fπ v(x

)

B=0

B>>1

E615

µ2=25 GeV2

Original E615 DY data

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

x

x fπ v(x

)

B=0

B>>1

Modified E615

µ2=25 GeV2

Re-analyzed E615 DY dataDGLAP evolution from µ0 = 0.316 GeV to µ = 5 GeVReanalysis of E615 data: Aicher, Schafer, and Vogelsang, PRL 105, 252003 (2010)

See also talk of Tianbo Lui on Friday22 / 35


Holographic pion DA versus E971 di-jet data

〈0|Ψ̄d(z)γ+(1− γ5)Ψu(0)|π+〉 = fπP+

dxe ix(P·z)ϕπ(x , µ)

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6

2

x

φπ(x

,µ)

µ=0.316 GeVµ=1 GeVµ=3.16 GeVAsymptoticE791 di−jet data

B=0

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6

2

x

φπ(x

,µ)

µ=0.316 GeVµ=1 GeVµ=3.16 GeVAsymptoticE791 di−jet data

B>>1

ERBL evolutionConsistent with E791 di-jet data

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Meson-to-photon transition form factors

Related to the inverse moment of the DA: Lepage & Brodsky (80)

Fγπ(Q2) =

ê2u − ê2d√

2

I (Q2;mf ,Mπ)

I (Q2;mf ,Mπ) = 2

1

0dx

ϕP(x , xQ;mf ,Mπ)

xQ2

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π→γ transition form factor

B ≥ 1 clearly preferredERBL evolution

Cannot accommodateBaBar (2009) anomaly

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Q2 [GeV2]Q

2 F

γπ [

Ge

V]

with QCD evolution (B=0)with QCD evolution (B>>1)BaBar (09)Belle (12)CELLO (91)CLEO (98)

25 / 35


Pion TMDs

Unpolarized TMD

f1,π(x , k2⊥) =

1

2

dz−d2z⊥e

iz·k〈π|Ψ̄(0)[0, z ]γ+Ψ(z)|π〉z+=0

Boer-Mulders TMDs

h⊥1,π(x , k2⊥) =

ijk j⊥k2⊥

Mπ2

dz−d2z⊥e

iz·k〈π|Ψ̄(0)[0, z ]iσi+γ5Ψ(z)|π〉z+=0

Gauge link ≡ gluon rescattering [Collins; Brodsky, Hwang,Schmidt (02)]

Experimental signature: non-zero measured azimuthalasymmetry in angular distribution of muons in pion-inducedDrell-Yan scattering

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Pion holographic TMDs

M. Ahmady, C. Mondal, RS, Phys. Rev. D100 (2019) 054005

f (x , k⊥) =2

16π3(Mπxx̄ + Bmf )

2 + B2k2⊥(xx̄)3

N 2 exp−k2⊥ +m

2f

xx̄κ2

.

h⊥1 (x , k⊥) = BMπxx̄ + Bmf

(xx̄)3N 2Mπ

k⊥exp

−k2⊥ +m

2f

κ2xx̄

×

dq⊥4π2

q2⊥iG (x , q⊥) exp

−

q2⊥2κ2xx̄

I1

−k⊥q⊥κ2xx̄

For pion B = 0 → h⊥1 = 0: Dynamical spin ≡ Boer-Mulders

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Lensing function and gluon rescattering kernel

L. Gamberg & M. Schelegel, Phys. Lett. B685 (2010) 95

Perturbative U(1)

Nonperturbative SU(3): x=0.1




0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

q [GeV]

iG(x

,q)[G

eV-

2]

Eikonal approximation

See talk by S. Rodini on Wednesday

Abelian perturbative limit → G (q⊥) ∝ αsCF/q2⊥28 / 35


Comparison to lattice

M. Engelhardt et al., Phys. Rev. D93, 054501 (2016)

〈k⊥〉UT (b2⊥) = Mπh̃⊥[1](1)1,π (b

2⊥)

f̃[1](0)1,π (b

2⊥)

f̃ [m](n)(b2⊥) =2πn!

M2nπ

dx xm−1

dk⊥k⊥

k⊥b⊥

nJn(b⊥k⊥)f (x , k

2⊥)

at Mπ = 518 MeV at µ = 2 GeV

b⊥ Lat 1 Lat 2 NP P [αs = 0.3] P [αs = 0.9]

0.27 -0.138(28) -0.133(19) -0.0663 -0.0424 -0.1273

0.34 -0.128(29) -0.121(16) -0.0645 -0.0423 -0.1268

0.36 -0.145(25) -0.148(15) -0.0639 -0.0422 -0.1267

Perturbative evolution of the holographic TMDs not taken into account here

Evolution: A. Bacchetta, S. Cotogno, B. Pasquini Phys. Lett. B771 (2017) 546-552

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Diffractive ρ electroproduction in dipole model

Aγ∗p→ρp ∼Ψγ

∗σΨρ

κ = 0.54, mf = 0.14 GeV

HERA data

30 / 35


Simultaneous ρ and φ electroproduction in dipole model

M. Ahmady, N. Sharma, R.S, Phys. Rev. D94 (2016) 074018

κ = 0.54 GeV

Best fit: mf = 0.046 GeV,ms = 0.14 GeV

HERA data

0

5000

10000

15000

20000

σ [n

b]

500

1000

1500

2000

200

400

600

100

200

80

120

20

30

40

80 120 160 200

6

12

18

80 120 160 200W [GeV]

0

2

4

Q2=0 Q2=2.0

Q2=3.3 Q2=4.8

Q2=6.6 Q2=11.9

Q2=19.5 Q2=35.6

31 / 35


Holographic meson DAs in B physics

Rare B decays to K ∗,φ:

Existing anomalies in LHCb data in B → (φ,K ∗)µ+µ−

Better understanding of non-perturbative effects needed

See talk by Mohammad Ahmady on Monday

B decays to (π,K ) see

Qin Chang, Shuai Xu, Lingxin Chen, Nucl. Phys. B921(2017) 454-471

Qin Chang, S. J. Brodsky, Xin-Chiang Li, Phys.Rev. D95(2017) no.9, 094025

S. Momeni, R. Khosravi, Phys. Rev. D97 (2018) no.5, 056005

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Summary

Universal LFH scale κ ∼ 0.5 GeV describe wide range of dataEffects of quark masses and spins important in phenomenology

Not covered: extension to heavy quarks and exotic states

M. Nielsen, S. J. Brodsky, G. F. de Téramond, H. G. Dosch, F.S. Navarra, Liping Zou, Phys. Rev. D98 (2018) no.3, 034002Liping Zou, H. G. Dosch, G. F. de Téramond, S. J. Brodsky,Phys. Rev. D99 (2019) no.11, 114024

See talk by Stan Brodsky today

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Acknowledgements

Workshop organizers for their invitation.

National Sciences and Engineering Research Council(Individual Discovery Grant SAPIN-2017-00031) for funding.

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ruben sandapen · 2019. 10. 1. · spacelike em form factors of pion and kaon drell & yan (prl,...

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