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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
1 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
Overview
1 What is the QCD mass scale ?
2 LFH: theory
3 LFH: phenomenology
4 Conclusions
2 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
3 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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)
4 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
5 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
6 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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)
7 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
8 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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)
10 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
12 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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).
13 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
14 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
15 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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 ?
16 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
17 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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)
19 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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.
21 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
23 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
24 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
26 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
27 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
Lensing function and gluon rescattering kernel
L. Gamberg & M. Schelegel, Phys. Lett. B685 (2010) 95
Perturbative U(1)
Nonperturbative SU(3): x=0.1
Nonperturbative SU(3): x=0.3
Nonperturbative SU(3): x=0.5
Nonperturbative SU(3): x=0.7
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
29 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
Diffractive ρ electroproduction in dipole model
Aγ∗p→ρp ∼Ψγ
∗σΨρ
κ = 0.54, mf = 0.14 GeV
HERA data
30 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
32 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
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
33 / 35
What is the QCD mass scale ? LFH: theory LFH: phenomenology Conclusions
Acknowledgements
Workshop organizers for their invitation.
National Sciences and Engineering Research Council(Individual Discovery Grant SAPIN-2017-00031) for funding.
34 / 35