Harmonic oscillator force in heavy quarkonia - ectstar.eu Serafin.pdfHarmonic oscillator force in heavy quarkonia Kamil Sera n (University of Warsaw) in collaboration with Stanis law

Harmonic oscillator force in heavy quarkonia

Kamil Serafin (University of Warsaw)

in collaboration withStanis law G lazek (U. Warsaw, Yale U.), Marıa Gomez-Rocha (ECT*),

Jai More (IIT Bombay),

arXiv: 1705.07629

The Charm and Beauty of Strong Interactions,Trento, Italy, 17th–28th July 2017

K. Serafin Harmonic oscillator force in heavy quarkonia

Outline

1 Renormalized Hamiltonian of QCD

2 Bound state problem in the Fock space and gluon mass ansatz

3 Effective QQ eigenvalue equation

4 Small-x divergences

5 Non-relativistic limit

6 Harmonic oscillator potential


Renormalized Hamiltonian of QCD

Outline









Front form of Hamiltonian dynamics

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-

x06

x3

��x+

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@@@

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x−

x+ = x0 + x3

x− = x0 − x3

pµxµ =1

2p−x++

1

2p+x−−p1x1−p2x2 .

p− =m2 + (p⊥)2

p+,

P− =m2 + (p⊥)2

p++µ2 + (k⊥)2

k+

=M2(x , κ) + (P⊥)2

P+,

M2(x , κ) =m2 + κ2

1− x+µ2 + κ2

x,

x =k+

p+ + k+, κ⊥ =

p+k⊥ − k+p⊥

p+ + k+.



Canonical FF Hamiltonian of QCD

Hcan =m2 + (k⊥1 )2

k+1

b†1b1 +m2 + (k⊥1 )2

k+1

d†1 d1 +(k⊥1 )2

k+1

a†1a1

+ g(δ21.3 t1

23 u2 6ε∗1u3 b†2a†1b3 − δ21.3 t132 v3 6ε∗1v2 d†2 a†1d3 + h.c .

)− g 2δ13.42

∑i5

t512t5

43

1

(k+5 )2

j+12 j+

43 b†1d†3 d4b2 + . . .

= E1

(b†1b1 + d†1 d1 + a†1a1

)+ g

(+ + h.c .

)+ g 2 + . . .

Canonical Hamiltonian is not well defined. We need to regularize it.



Regularization I

We do not regularize fields but rather interactions.

H = E1

(b†1b1 + d†1 d1 + a†1a1

)+ g

(r21.3 + r21.3 + h.c .

)+ g 2rC 13.42 + . . .

For every quark operator in the interaction vertex we write a factor

e−m2+κ2

x∆2

and for every gluon operator in the vertex we write

e−δ2+κ2

x∆2 .



Regularization II

r21.3 = e−

m2+κ22/3

x2/3∆2e−δ2+κ2

1/3

x1/3∆2e−

m2

∆2 = e−M2

21 δ+m2

∆2 ,

rC13.42 = θ(z) r25.1r35.4 + θ(−z) r45.3r15.2 ,



Renormalization Group Procedure for Effective Particles I

∆

Ut-

Ht=s4

@@@@@@@@@@

@@@@@@@@@@

��

��

λ = s−1

Figure: Initial theory → effective theory. RGPEP introduces form factors intovertices, which narrow the Hamiltonian in free-states basis.



Renormalization Group Procedure for Effective Particles II

The narrowing of the Hamiltonian is realized by the followingdifferential equation,

d

dtHt = [[Hf ,HPt ],Ht ] .

The generation of transformation Ut is therefore, different comparedto the one used in Phys.Rev. D69 (2004) 065002, it allows fornonperturbative calculations.

But before nonperturbative calculations will be performed we writeperturbative expansion of the solution first,

Ht = Hf + g Ht 1 + g 2 Ht 2 + . . .



Renormalized Hamiltonian

H = E1

(b†1b1 + d†1 d1 + a†1a1

)+ g

(r21.3 + r21.3 + h.c .

)+ g 2rC 13.42 + . . .




Ht = E1

(b†1 tb1 t + d†1 td1 t + a†1 ta1 t

)+ g

(r21.3 + r21.3 + h.c .

)+ g 2rC 13.42




Ht = E1

(b†1 tb1 t + d†1 td1 t + a†1 ta1 t

)+ g

(r21.3ft 21.3 + r21.3ft 21.3 + h.c .

)+ g 2rC 13.42




Ht = E1

(b†1 tb1 t + d†1 td1 t + a†1 ta1 t

)+ g

(r21.3ft 21.3 + r21.3ft 21.3 + h.c .

)+ g 2rC 13.42ft 13.24

ft 21.3 = e−t(M221−m

2)2

, ft 13.24 = e−t(M213−M

224)2

,




Ht = E1

(b†1 tb1 t + d†1 td1 t + a†1 ta1 t

)+ g

(r21.3ft 21.3 + r21.3ft 21.3 + h.c .

)+ g 2rC 13.42ft 13.24

+ g 2r25.1r35.4F Z + g 2r45.3r15.2FZ

ft 21.3 = e−t(M221−m

2)2

, ft 13.24 = e−t(M213−M

224)2

,




Ht = E1

(b†1 tb1 t + d†1 td1 t + a†1 ta1 t

)+ g

(r21.3ft 21.3 + r21.3ft 21.3 + h.c .

)+ g 2rC 13.42ft 13.24

+ g 2r25.1r35.4F Z + g 2r45.3r15.2FZ

+ g 2 +

ft 21.3 = e−t(M221−m

2)2

, ft 13.24 = e−t(M213−M

224)2

,



Renormalized Hamiltonian* I

Mass termsThe quark mass term in Ht2 consists of effective part,

(HmQ t)1.2 =4

3δc1c2 δ1.2

∫[56]k+

1 δ56.1 r 265.1

f 2t 65.1 − 1

M256 −m2

∑σ6

dµν jµ16jν62 ,

and the UV counterterm (which cancels divergence of HmQ t coming fromκ→∞),

(H∗mQ)1.2 =4

3δc1c2 δ1.2

∫[56]k+

2 δ56.2 r65.1r65.21

M256 −m2

∑σ6

dµν jµ16jν62 ,

However, small-x divergences remain,

HmQ t ∼√

2π

tlog

∆

δ.

Analogous expressions can be written for antiquark mass term.



Renormalized Hamiltonian* II

Four-body vertex interaction termThe exchange part of Ht 2 consists of the term derived frominstantaneous interactions in the initial Hamiltonian,

(HC QQ t)13.42 = C13.42 rC13.42 = − δ13.42

∑i5

t512t5

43

1

(k+5 )2

j+12 j+

43 rC13.42 ,

and term derived from products of first order initial Hamiltonian vertices,

(HX QQ t)13.42 = − δ13.42

∑i5

t512t5

43

dµν(k5)

k+5

jµ12 jν43

× [θ(z)F Zr25.1r35.4 + θ(−z)FZ r45.3r15.2] ,

F Z =k+

1 (m2 −M252) + k+

4 (m2 −M253)

(m2 −M252)2 + (m2 −M2

53)2 − (M213 −M2

42)2

(1− f25.1f35.4

f13.42

),

FZ =k+

3 (m2 −M254) + k+

2 (m2 −M251)

(m2 −M254)2 + (m2 −M2

51)2 − (M213 −M2

42)2

(1− f15.2f45.3

f13.42

).


Bound state problem in the Fock space and gluon mass ansatz

Outline









Bound state problem in the Fock space

Structure of the eigenvalue problem

|Ψmeson〉 = |QtQt〉+ |QtQtGt〉+ |QtQtGtGt〉+ ... .

Ht |Ψmeson〉 =

... ... ...... Hf + g 2Ht 2 gHt 1

... gHt 1 Hf + g 2Ht 2

...|QtQtGt〉|QtQt〉

= Emeson

...|QtQtGt〉|QtQt〉

,



Eigenvalue problem with gluon mass ansatz

Introducing the gluon mass terms to model effects of dotted componentsand interactions, one obtains the corresponding approximate eigenvalueproblem,

H|Ψmeson〉 =

[Hf + µ2 gH1

gH1 Hf + g 2H2

] [|QQG 〉|QQ〉

]= Emeson

[|QQG 〉|QQ〉

],

where µ2 denotes the mass-like operators for gluon in the QQG sector.

µ2 ∼∫

5

µ2(x5, κ5)

k+5

a†5a5 .


Effective QQ eigenvalue equation

Reduction to the lowest Fock sector I

Reduction of the space is performed using operation R,

|QQG 〉 = R|QQ〉 .

The general formula for effective Hamiltonian in a subspace of states isthen

Heff =1√

P + R†R(P + R†)H(P + R)

1√P + R†R

.

Perturbative evaluation of the 2nd-order effective Hamiltonian for thelowest component gives

Heff = Hf + g 2H2 +g 2

2H1

(1

El − Hf − µ2+

1

El′ − Hf − µ2

)H1 .



Effective QQ eigenvalue equation I

After reduction to QQ sector, the effective eigenvalue equation is

Ht eff|ψQQt〉 =M2 + P⊥2

P+|ψQQt〉 ,

or

(P+ Ht eff − P⊥2)|ψQQt〉 = M2|ψQQt〉 .

The eigenvalue is mass squared instead of mass or the binding energy.



Effective QQ eigenvalue equation II

For the color singlet states we define the QQ wave function in thefollowing manner,

|ψQQt〉 =∑c2c4

∑σ2σ4

∫[24] P+ δ(P − k2 − k4)

δc2c4√3ψσ2σ4 (κ⊥24, x2) b†2td

†4t |0〉 .

Therefore, the eigenvalue equation becomes,

(κ⊥2

13 + M21, t

x1+κ⊥2

13 + M23, t

x3−M2

)ψ13(κ⊥13, x1)

+ g 2

∫[x2κ

⊥24] Ut eff(13, 24)ψ24(κ⊥24, x2) = 0 ,



Effective QQ eigenvalue equation III

M21, t = m2 +

4

3g 2

∫[x κ] r 2

65.1 f 2t 65.1

×∑σ6

dµν(k5) jµ16 jν62

(1

m2 −M2− 1

m2 −M2µ

),

M23, t =

The subscript µ in M2µ indicates that the gluon has an effective mass.



Effective QQ eigenvalue equation IV

The interaction potential Ut eff(13, 24) contains FF instantaneousinteractions, gluon-exchange terms and counterterms:

Ut eff = HC + Hexch + H∗C ,

The effective instantaneous vertex acquires only a form factor,

HC = − 4

3rC13.24 ft 13.24

j+12 j+

43

(x1 − x2)2,



Effective QQ eigenvalue equation V

Hexch = − 4

3dµν(k5) jµ12 jν43

×(θ(x1 − x2)

k+5

r25.1r35.4 F Z+θ(x2 − x1)

k+5

r15.2r45.3 FZ

),

F Z= ft 13.24 F Z+ ft 4.53 ft 1.52R Z,

FZ = ft 13.24 FZ + ft 3.54 ft 2.51RZ ,



Effective QQ eigenvalue equation VI

F Z =

(1− ft 1.52ft 4.53

ft 13.24

)k+

1 (m2 −M225) + k+

4 (m2 −M235)

(m2 −M225)2 + (m2 −M2

35)2 − (M213 −M2

24)2,

R Z =1

2

(k+

1

m2 −M225 −

x1

x5µ2

253

+k+

4

m2 −M253 −

x4

x5µ2

253

),

µ2253 is an ansatz for the gluon-mass function. It depends on relative

motion of gluon, 5, with respect to quark, 2, and antiquark, 3, in theintermediate state.


Small-x divergences

Outline








Small-x divergences

Small-x divergences I

Because quantization plane is tangent to a light, cone masslessparticles can travel in one direction instantaneously. This producesdivergences.

For example, the gluon propagator dµν contains regular part andsingular part in gluon momentum,

dµν(k5) = −gµν +nµk5 ν + nνk5µ

k+5

.

Instantaneous interaction term,

HC = − 4

3rC13.24 ft 13.24

j+12 j+

43

x25

.

Mass terms and exchange terms are divergent in the integrationregion where x5 = 0 and κ⊥5 = 0.

When µ = 0 it was early found out that divergences cancel.


Small-x divergences

Small-x divergences II

Terms arising from µ 6= 0 need to be regulated by µ so that everyterm in the eigenvalue problem is finite when regularization isremoved.

Mass terms are easier to analyze. The divergent integral is∫[xκ]r 2

65.1f 265.1

µ2

x

M2µ −m2

4

x.

The form factor regulates the UV behavior but not the small-x, orlarge volume, behavior.

However, if µ2 ∼ x1+δµ ∼ κ2xδµ , then the integral is finite.

For exchange terms we study parts with low- and high-energyexchange separately.

Ut eff = Uhigh + Ulow + Ug

= ∼ (1− ff )+ ∼ ff + ∼ gµν .


Small-x divergences

Small-x divergences III

Low-energy exchange terms are those whose UV behavior isregulated by form factors (in a similar way as in the mass terms).

Again, the large volume or small-x behavior is not regulated by formfactors, but by the gluon mass function.

The same µ2 regulates both mass terms and the low-energyexchange terms.

The high-energy exchange terms do not contain µ2, but do not leadto divergences in the limit δ → 0.


Small-x divergences

One quark eigenvalue problem

Proper gluon mass ansatz gives well defined, finite, eigenvalueequation for color singlet QQ states.

One can apply similar reasoning to the eigenproblem of a singlequark, where the gluon mass ansatz may be different.

Ht |Ψquark〉 =

[Hf t + µ2

q gH1 t

gH1 t Hf t + g 2H2 t

] [|QtGt〉|Qt〉

]= Equark

[|QtGt〉|Qt〉

].

The gluon mass ansatz in this problem can be chosen in such a waythat the eigenproblem is divergent.

This is a possibility how confinement could be understood.


Non-relativistic limit

Outline









Non-relativistic limit I

Well defined, finite, eigenproblem for QQ – done.

Heavy quarks in quarkonia are expected to move slowly so that onecan adopt non-relativistic approximation.

We define non-relativistic relative momenta of quarks,

k⊥ij =1

2

κ⊥ij√xixj

, k3ij =

m√

xixj

(xi −

1

2

).

where ij = 13 or ij = 24, and momentum transfer,

~q = ~k13 − ~k24 .

Formally, the non-relativistic limit means that the quark mass goesto infinity, or ~kij/m→ 0.

M2 = (2m + B)2 ≈ 4m2 + 4mB.



Non-relativistic limit II

Performing the limit we obtain familiar looking Schrodinger equation,[|~k13|2

m− B +

δm21, t

2m+δm2

3, t

2m

]ψ13(~k13)

+

∫d3~k24

(2π)3VQQ(~k13 − ~k24)ψ24(~k24) = 0 ,

where

δm2i, t

2m= lim

m→∞

δM2i, t

2m,

VQQ(~k13 − ~k24) = g 2 limm→∞

[1

4m2Ut eff(13, 24)

].



Non-relativistic limit III

If the gluon mass ansatz was zero, the exchange term in thenon-relativistic limit would give only Coulomb potential coming fromthe non-singular part of the gluon propagator (Ug ),

VQQ(~q) = −4

3

4πα

|~q|2f13.24 (for µ = 0) .

The Coulomb term is modified by the form factor f13.24.The interaction becomes non-local and the effect is visible fordistances . s −→ the effective particles have nonzero size!


Harmonic oscillator potential

Outline









Harmonic oscillator potential I

For nonzero gluon mass ansatz we have∫d3q

(2π)3VQQ(~q)ψ(~k − ~q) =

∫d3q

(2π)3[VC (~q) + W (~q)]ψ(~k − ~q) .

where

W (~q ) =4

34πα

[1

~q 2− 1

q2z

]×

(θ(z)

µ2253

µ2253 + ~q 2

+ θ(−z)µ2

154

µ2154 + ~q 2

)e−2m2 |~q2|2

q2z

t.



Harmonic oscillator potential II

Similarly, the self-energy terms can also be expressed in terms of W ,

δm2i t

mψ(~k) = −

∫d3q

(2π)3W (~q)ψ(~k) .

Self-energy terms and effective exchange of gluons combine together,

δm21 t

2mψ(~k) +

δm23 t

2mψ(~k) +

∫d3q

(2π)3W (~q)ψ(~k − ~q)

=

∫d3q

(2π)3W (~q)

[ψ(~k − ~q)− ψ(~k)

].

Taylor expanding the wave function

ψ(~k − ~q) = ψ(~k)− qi∂

∂kiψ(~k) +

1

2qiqj

∂2

∂ki∂kjψ(~k) + . . .



Harmonic oscillator potential III

we find the net effect of effective self-energy and effective exchangeto be harmonic oscillator force:∫

d3q

(2π)3W (~q)

1

2(qi )

2 ∂2

∂k2i

ψ(~k) .

Finally, [~k2

m− B

]ψ(~k) +

∫d3q

(2π)3VC ,BF (~q)ψ(~k − ~q)

− 4

3

α

2πb−3

∑i

τi∂2

∂k2i

ψ(~k) = 0 .

The frequencies of harmonic oscillators in transverse and longitudinaldirections should be the same to recover rotational invariance. Thiscan also be achieved using appropriate gluon mass ansatz function.



Harmonic oscillator potential IV

In particular, almost constant and large enough µ2 will giveτ1 = τ2 = τ3 =

√π/4.

~τ =

∫ 1

0

dv v(1− v 2)

1− v 2

1− v 2

2v 2

τ(v)

τ(v) =

∫ ∞0

du u2 e−u2

[1 +

u2 v 2

2 (m s)2 (µ s)2

]−1

,

Therefore, the final result could be written in the following way[~k 2

m− 1

2κ∆~k − B

]ψ(~k) +

∫d3q

(2π)3VC ,BF (~q)ψ(~k − ~q) = 0 ,

κ =m

2ω2 =

α

(ms2)3

1

36√

2π.



Phenomenological fit of parameters I

Below are the results of numerical calculation obtained by S. G lazekand J. M lynik using the old generator of RGPEP and oldnonrelativistic variables. (Phys. Rev. D 74, 105015 (2006))



Phenomenological fit of parameters II



Phenomenological fit of parameters III



Phenomenological fit of parameters IV



Summary

RGPEP together with gluon mass ansatz gives Coulomb force plusharmonic oscillator force between quarks.

Without gluon mass ansatz one is left with only Coulomb potential(like QED, except of color factor 4/3).

Energy of a single quark is infinite.

In the QQ system the harmonic oscillator force arises as an effect ofcancellation between effective mass terms of quarks and antiquark,and an exchange of effective gluon.

The resultant harmonic oscillator is the same as obtained earlierusing different RGPEP generator and different non-relativisticmomentum variables.



Outlook

Analysis of Breit-Fermi terms in the new non-relativistic variables.

Numerical calculation including different masses of quarks.

However, the most desired course of study is 4th-order calculation.

Will give precise form of spin dependent interactions.Gluon mass ansatz will be replaced by true QCD effects.

New generator allows for nonperturbative calculations, e.g., of gluonmass.

Also, harmonic oscillator in baryons.


Documents

Harmonic oscillator force in heavy quarkonia - ectstar.eu Serafin.pdfHarmonic oscillator force in heavy quarkonia Kamil Sera n (University of Warsaw) in collaboration with Stanis law