Light Front Field Theory And Coherent State Basispersonalpages.to.infn.it/~torriell/seminars/files/Misra.pdf · 2017. 3. 15. · Mass renormalization upto O(e2) in Fock state basis

IntroductionCoherent state formalism in LFFT

Mass renormalization upto O(e2) in Fock state basisMass renormalization upto O(e2) in the coherent state basis

Mass renormalization up to O(e4) in Fock basisMass renormalization up to O(e4) in the coherent state basis

All Order CancellationImproved Method of Asymptotic Dynamics

Summary

Light Front Field Theory And Coherent StateBasis

Anuradha Misra

Department of PhysicsUniversity of Mumbai

Mumbai, India

Dipartimento di Fisica, Universita di Torino, March14, 2017





Summary

1 IntroductionLight Front Field TheoryMethod of Asymptotic Dynamics

2 Coherent state formalism in LFFT

3 Mass renormalization upto O(e2) in Fock state basis

4 Mass renormalization upto O(e2) in the coherent state basis

5 Mass renormalization up to O(e4) in Fock basis

6 Mass renormalization up to O(e4) in the coherent state basis

7 All Order Cancellation

8 Improved Method of Asymptotic Dynamics

9 Summary






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Summary

Light Front Field TheoryMethod of Asymptotic Dynamics

What is a light front?

Dirac :”.....the three-dimensional surface in space-time formed by aplane wave front advancing with the velocity of light....” . Forexample, x+ = x0 + x3 = 0, is called a frontDirac(1949) :Three possible forms of relativistic dynamicscorresponding to 3 different ways of quantizing corresponding to 3different surfaces of quantizationInstant Form, Point Form, Front Form

diagram.pdf

x3

x0 x+x−






Summary


Light Front Coordinates

xµ = (x+, x−, x⊥)

where

x+ =(x0 + x3)√

2, x− =

(x0 − x3)√2

, x⊥ = (x1, x2)

The metric tensor

gµν =

0 1 0 01 0 0 00 0 −1 00 0 0 −1

Momentum is given by p = (p+, p−,p⊥)

Mass shell condition p− =p2⊥+m2

2p+






Summary


Why quantize on the Light Front?

Quantization of QCD at fixed light-front time can provide a firstprinciples method for solving non-perturbative QCD

Dispersion relation k− =k2⊥+m2

2k+ is suitable for bound statecalculations because

No square root operator (unlike instant form)

Dependence of k− on k⊥ similar to non-relativistic dispersionrelation

Due to the form of the dispersion relation, for an on-shell particle,k+ ≥ 0 implies that k− ≥ 0 andk+ ≤ 0 implies that k− ≤ 0.

Thus, for physical particles k+ ≥ 0 always.

Simpler vacuum structure






Summary


Hamiltonian light front approach

aims at solving the Hamiltonian eigenvalue problem in the spirit ofTamm Dancoff methodK G Wilson et al, Phys. Rev. D49, 6720(1994)

H|Ψ〉 =M2+P2

⊥2P+ |Ψ〉

Discretized light Cone Quantization (DLCQ):Project the LF Hamiltonian eigenvalue equation on a truncated FockspaceDiscretize the momentum spaceResult : a matrix equation which can be solved on a computerDLCQ - used for solving bound state problems in 1 +1 dimensionand even for positronium spectrumS.J.Brodsky, H. C. Pauli S.S. Pinsky, Phys.Rep. 301, 299 (1998)Recently developed method Basis Light Front Quantization- usefulin NP bound state calculationJames P. Vary et al, Phys. Rev. D 91, 105009 (2015)






Summary


LF Hamiltonian formalism

More suitable for bound state calculations as compared to its equaltime counterpart

However, there are problems that need to be addressed before onecan do that

Renormalization is different (P− =P2⊥+M2

2P+ )

IR divergences pose a big challenge

Need to separate the ”true” IR divergences from the ”spurious” ones

Coherent State Formalism provides a solution

AM Phys. Rev. D 50, 4088 (1994)

AM Phys. Rev. D 53, 5874 (1996)

Coherent state formalism is based on the method of asymptoticdynamics






Summary


Method of Asymptotic Dynamics

The LSZ formalism is based on the assumption

Has = lim|t|→∞

H = H0

Not always true

Has 6= H0

, when

there are long range interaction

incoming and outgoing states are bound states






Summary


In the limit |x+| → ∞, H −→ Has

Has = H0 + Vas

The total Hamiltonian can be written as

H = Has + H ′I

whereHas(x+) = H0 + Vas(x+)

The associated x+ evolution operator Uas(x+) in the Schrodingerrepresentation satisfies the equation

idUas(x+)

dx+= Has(x+)Uas(x+)






Summary


Coherent states

The asymptotic evolution operator ΩA(x+) is defined by

Uas(x+) = exp[−iH0x+]ΩA(x+)

where

ΩA±(x+) = T+exp

[− i

∫ 0

∓Vas(x+)dx+

]KF : Use ΩA

±(x+) to define a new set of asymptotic states

|n : coh〉 = ΩA±|n〉

|n〉 is a Fock state, ΩA± are the asymptotic Moller operators

KF:The transition matrix calculated using such coherent states is IRdivergence free in equal time QED.






Summary


Cancellation of IR divergences in QCD (lowest order)D.R. Butler and C.A. Nelson, Phys. Rev. D 18, 1196 (1978).C.A. Nelson Nucl. Phys. B181, 141 (1981).C.A. Nelson Nucl. Phys. B186, 187 (1981)M. Greco, F. Palumbo, G. Pancheri-Srivastava and Y. Srivastava,Phys. Lett. B77, 282 (1978)H.D. Dahmein and F. Steiner, Z.Phys. C11, 247 (1981)Coherent States in LFFTRelevance of coherent state formalism in Light Front Field Theory(LFFT)A. Harindranath and J. P. Vary, Phys. Rev. D 37, 3010 (1988)L. Martinovic and J. P. Vary, Phys. Lett. B459, 186 (1999)Coherent state formalism in LFFT : Cancellation of IR divergences in3-point vertex correction in QED and QCD at one loop level.AM Phys. Rev. D 50, 4088 (1994).AM, Phys. Rev. D 53, 5874 (1996).AM, Phys. Rev. D 62, 125017 (2000)






Summary


IR Divergences in LFFT

Dispersion relation : k− =k2⊥+m2

2k+

Two kinds of IR divergences in LFFT

Spurious IR divergences

k+ → 0

True IR divergences

k⊥ → 0, k+ → 0[AM, Phys. Rev. D 50, 4088 (1994).]

The coherent state method provides an alternative way of treating thetrue IR divergences.






Summary


Coherent State Formalism in LFFT

Has is evaluated by taking the limit |x+| → ∞ inexp[−i(p−1 + p−2 + · · ·+ p−n )x+] of the interaction Hamiltonian Hint .

If (p−1 + p−2 + · · ·+ p−n )→ 0 for some vertex, then thecorresponding term in Hint does not vanish in large x+ limit.

Use KF method to construct asymptotic Hamiltonian and coherentstate basis.






Summary


Light-Front QED in LF gauge

The LFQED Hamiltonian (P ) in LF gauge is

HI (x+) = V1(x+) + V2(x+) + V3(x+)

where

V1(x+) is the standard three point vertex of QED

V2(x+) is an instantaneous 4-point interaction which appears whenwe write the fermionic part of P− in terms of independentcomponentshown - hence we eliminate it and write P−F in terms of ψ+ only)

V3(x+) is an instantaneous 4-point interaction which appears whenwe write Aµ only in terms of physical degrees of freedom






Summary


Light-front QED Hamiltonian in the light-front gauge (A+ = 0)

P− = H ≡ H0 + V1 + V2 + V3 ,

Here

H0 =

∫d2x⊥dx

− i2ξγ−

↔∂− ξ +

1

2(F12)2 − 1

2a+∂−∂kak

V1 = e

∫d2x⊥dx

−ξγµξaµ

V2 = − i

4e2

∫d2x⊥dx

−dy−ε(x− − y−)(ξakγk)(x)γ+(ajγ

jξ)(y)

V3 = −e2

4

∫d2x⊥dx

−dy−(ξγ+ξ)(x)|x− − y−|(ξγ+ξ)(y)






Summary







Summary


ξ(x) and aµ(x) can be expanded in terms of creation andannihilation operators as

ξ(x) =

∫d2p⊥

(2π)3/2

∫dp+

√2p+

∑s=± 1

2

[u(p, s)e−i(p+x−−p⊥x⊥)b(p, s, x+)

+v(p, s)e i(p+x−−p⊥x⊥)d†(p, s, x+)],

aµ(x) =

∫d2q⊥

(2π)3/2

∫dq+

√2q+

∑λ=1,2

ελµ(q)[e−i(q+x−−q⊥x⊥)a(q, λ, x+)

+e i(q+x−−q⊥x⊥)a†(q, λ, x+)],

operators satisfy

b(p, s), b†(p′, s ′) = δ(p+−p′+)δ2(p⊥ − p′⊥)δss′ = d(p, s), d†(p′, s ′),[a(q, λ), a†(q′, λ′)

]= δ(q+q′+)δ2(q⊥ − q′⊥)δλλ′ .






Summary

Coherent state Formalism in LFQED

Light cone time dependence of V1 is of the form

V1(x+) = e4∑

i=1

∫dνi [e

−iν(1)i x+

hi (νi ) + e iν(1)i x+

h†i (νi )]

where hi (νi ) are the QED interaction vertices:

h1 =∑s,s′,λ

b†(p, s ′)b(p, s)a(k , λ)u(p, s ′)γµu(p, s)ελµ ,

h2 =∑s,s′,λ

b†(p, s ′)d†(p, s)a(k , λ)u(p, s ′)γµv(p, s)ελµ ,

h3 =∑s,s′,λ

d(p, s ′)b(p, s)a(k , λ)v(p, s ′)γµu(p, s)ελµ ,

h4 =∑s,s′,λ

d†(p, s ′)d(p, s)a(k , λ)v(p, s ′)γµv(p, s)ελµ ,






Summary

νi is the light cone energy transferred at the vertex hi

The integration measure is given by∫dν =

1

(2π)3/2

∫[dp][dk]√

2p+,

p+ and p⊥ being fixed at each vertex by momentum conservation

For example

ν(1)1 = p− + k− − p− = p·k

p++k+

is the energy transfer at eeγ vertex.






Summary

At asymptotic limits, non-zero contributions to HI (x+) come from

regions where νi goes to zero.

ν2 and ν3 are always non-zero, and hence, h2 and h3 do not appearin the asymptotic Hamiltonian.

The 3-point asymptotic Hamiltonian is defined by

V1as(x+) = e∑i=1,4

∫dν

(1)i Θ∆(k)[e−iν

(1)i x+

h(1)i (ν

(1)i ) + e iν

(1)i x+

h†i (ν(1)i )]

where Θ∆(k) defines the asymptotic region i.e the region in which

ν(1)i is zero.

Θ∆(k) is 1 in the asymptotic region and 0 elsewhere






Summary

Define the asymptotic region to consist of all points in the phasespace for which

p · kp+

< ∆ ,

where ∆ is an energy cutoff which may be chosen to be related tothe experimental resolution.

For simplicity, choose a frame p⊥ = 0. In this frame the abovecondition reduces to

p+k2⊥

2k++

m2k+

2p+< ∆ ,

where ∆ = p+∆E .






Summary

Sufficient to choose a subregion of the above mentioned region asthe asymptotic region.

Define this subregion to be consisting of all points (k+, k⊥)satisfying:

k2⊥ <

k+∆

p+,

k+ <p+∆

m2.

This choice of the asymptotic region leads to

Θ∆(k) = θ

(k+∆p+ − k2

⊥

)θ

(p+∆m2 − k+

)






Summary

Asymptotic states

ΩA±|n : pi 〉 =exp

[−e∫

dp+d2p⊥∑λ=1,2

[d3k][f (k , λ : p)a†(k, λ)

− f ∗(k , λ : p)a(k , λ)] + e2

∫dp+d2p⊥

∑λ1,λ2=1,2

[d3k1][d3k2]

[g1(k1, k2, λ1, λ2 : p)a†(k2, λ2)a(k1, λ1)

− g2(k1, k2, λ1, λ2 : p)a(k2, λ2)a†(k1, λ1)]ρ(p)

]|n : pi 〉

[AM, Phys. Rev. D 50, 4088 (1994)]






Summary

Here

[d3k] =

∫d2k⊥

(2π)3/2

∫dk+

√2k+

f (k , λ : p) =pµε

µλ(k)

p · kθ

(k+∆

p+− k2⊥

)θ

(p+∆

m2− k+

),

f (k, λ : p) =f ∗(k , λ : p),






Summary

One fermion coherent state

|p, σ : f (p)〉 =exp

[−e

∑λ=1,2

[d3k][f (k , λ : p)a†(k , λ)− f ∗(k , λ : p)a(k, λ)]

+e2∑

λ1,λ2=1,2

[d3k1][d3k2][g1(k1, k2, λ1, λ2 : p)a†(k2, λ2)a(k1, λ1)

−g2(k1, k2, λ1, λ2 : p)a(k2, λ2)a†(k1, λ1)

]|p, σ〉






Summary






Summary

The transition matrix is given by

T = V + V1

p− − H0V + · · ·

The electron mass shift is obtained by calculating Tpp = 〈p, s|T |p, s〉

δm2 = p+∑s

Tpp

We expand Tpp in powers of e2 as

Tpp = T (1) + T (2) + · · ·

For example

T (1)pp ≡ T (1)(p, p) = 〈p, s|V1

1

p− − H0V1|p, s〉+ 〈p, s|V2|p, s〉






Summary

O(e2) self energy correction in Fock basis

(a)

(p, s) (p, σ)

k1

(p, s) (p, σ)(b)

k1

Diagrams for O(e2) self energy correction in Fock basis corresponding to T1

T (1)pp ≡ T (1)(p, p) = 〈p, s|V1

1

p− − H0V1|p, s〉+ 〈p, s|V2|p, s〉

In the limit k+1 → 0, k1⊥ → 0,

(δm21a)

IR= − e2

(2π)3

∫d2k1⊥

∫ dk+1

k+1

(p·ε(k1))2

(p·k1)






Summary

O(e2) self energy correction in coherent state basis

k1

(p, s) (p, σ)

k1

(p, s) (p, σ)

Additional diagrams in coherent state basis for O(e2) self energycorrection corresponding to T2.

T ′(p, p) = 〈p, s : f (p)|V1|p, s : f (p)〉

Calculated using coherent state properties

a(k , ρ)|1: pi 〉 = − e

(2π)3/2

f (k , ρ : pi )√2k+

|1: pi 〉 ,

a†(k, ρ)|1: pi 〉 =e

(2π)3/2

f ∗(k , ρ : pi )√2k+

|1: pi 〉+ |2: pi , ki 〉 .






Summary

Extra contribution in coherent state basos

T ′(p, p) =e2

(2π)3

∫d2k1⊥

2p+

∫dk+

1

2k+1

u(p, s ′)ε/λ(k1)u(p, s)f (k1, λ : p)

where

f (k, λ : p) =pµε

µλ(k)

p · kθ

(k+∆

p+− k2⊥

)θ

(p+∆

m2− k+

)

(δm2)′

= e2

(2π)3

∫d2k1⊥

∫ dk+1

k+1

(p·ε(k1))2Θ∆(k1)p·k1

Equal and opposite to Fock space expression in asymptotic region






Summary

Electron mass correction in Fock basis upto O(e4) to self energy isgiven by T (2) = T3 + T4 + T5 + T6 + T7

whereT3 = 〈p, s|V1

1p−−H0

V11

p−−H0V1

1p−−H0

V1|p, s〉

T4 = 〈p, s|V11

p−−H0V1

1p−−H0

V2|p, s〉

T5 = 〈p, s|V11

p−−H0V2

1p−−H0

V1|p, s〉

T6 = 〈p, s|V21

p−−H0V1

1p−−H0

V1|p, s〉

T7 = 〈p, s|V21

p−−H0V2|p, s〉

[Jai D. More and AM, Phys. Rev. D 86, 065037 (2012)][Jai D. More and AM, Phys. Rev. D 87, 085052 (2013)]






Summary

O(e4) self energy correction in Fock basis corresponding to T3.

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)

k1

k2

(p, σ)(p, s)

(c)

k1 k2






Summary

IR divergences in these diagrams appear when

I p · k1 → 0 i.e k+1 → 0, k1⊥ → 0, but p · k2 6= 0.

II p · k2 → 0 i.e k+2 → 0, k2⊥ → 0, but p · k1 6= 0.

III p · k1 → 0 and p · k2 → 0 i.e. k+1 → 0, k1⊥ → 0,

k+2 → 0, k2⊥ → 0.






Summary

O(e4) self energy correction in Fock basis corresponding to T4, T5 andT6 respectively

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)

k1

k2

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(a)

k1 k2

(p, σ)(p, s)

(b)

k1

k2

k2

(p, σ)(p, s)(a)

k1

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)k1

k2

(p, σ)(p, s)

(c)

k1 k2






Summary

Mass renormalization up to O(e4) in the coherentstate basis

Additional contributions at O(e4) in coherent state basis

T (2) + T ′8 + T ′9 + T ′10 + T ′11

whereT ′8 = 〈p, s : f (p)|V1

1p−−H0

V11

p−−H0V1|p, s : f (p)〉

T ′9 = 〈p, s : f (p)|V11

p−−H0V1|p, s : f (p)〉

T ′10 = 〈p, s : f (p)|V11

p−−H0V2|p, s : f (p)〉+

〈p, s : f (p)|V21

p−−H0V1|p, s : f (p)〉

T ′11 = 〈p, s : f (p)|V2|p, s : f (p)〉.Dipartimento di Fisica, Universita di Torino, March14, 2017





Summary

(p, σ)(p, s)(a)

k2k1

(p, σ)(p, s)(b)

k2k1

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(d)

k2 k1

Additional diagrams in coherent state basis for O(e4) self energycorrection corresponding to T8 and T9 respectively.






Summary

(p, σ)(p, s) (p, σ)(p, s)(a) (b)

k1k2

k1

k2

k2k2k1

(p, σ)(p, s)

k1

(p, σ)(p, s)(c)(d)

(p, σ)(p, s)

(f)

k1

k2(p, σ)(p, s)

(e)

k1

k2

k1

k2

(p, σ)(p, s)

(h)

k2k1

(p, σ)(p, s)(g)

(p, σ)(p, s) (p, σ)(p, s)

(i)

k2 k1k1

k2

(p, σ)(p, s)

k1k2

(p, σ)(p, s)

k1k2

Additional diagrams in coherent state basis for O(e4) self energy correction

corresponding to T10 .






Summary

(p, σ)(p, s)(a)

k2k1

(p, σ)(p, s) (b)

k2k1

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(d)

k2 k1

Additional diagrams in coherent state basis for O(e4) self energycorrection corresponding to T11.






Summary

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)

k1

k2

(p, σ)(p, s)

(c)

k1 k2

(a)(p, s) (p, σ)

k1

k2

(b)

k1

k2

(p, s) (p, σ)

(c)

k2

k1

(p, s) (p, σ)

k2

k1

(d)

(p, s) (p, σ)

(e)(p, s) (p, σ)

k1

k2

(f)(p, s) (p, σ)

k1 k2

(p, σ)(p, s)(a)

k2k1

(p, σ)(p, s)(b)

k2k1

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(d)

k2 k1

(δm2)3 + (δm2)8 + (δm2)9 is IR finite.






Summary

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)

k1

k2

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(a)

k1 k2

(p, σ)(p, s)

(b)

k1

k2

k2

(p, σ)(p, s)(a)

k1

k2

(p, σ)(p, s)(a)

k1

(p, σ)(p, s)

(b)k1

k2

(p, σ)(p, s)

(c)

k1 k2

(p, σ)(p, s) (p, σ)(p, s)(a) (b)

k1k2

k1

k2

k2k2k1

(p, σ)(p, s)

k1

(p, σ)(p, s)(c)(d)

(p, σ)(p, s)

(f)

k1

k2(p, σ)(p, s)

(e)

k1

k2

k1

k2

(p, σ)(p, s)

(h)

k2k1

(p, σ)(p, s)(g)

(p, σ)(p, s) (p, σ)(p, s)

(i)

k2 k1k1

k2

(p, σ)(p, s)

k1k2

(p, σ)(p, s)

k1k2

(p, σ)(p, s)(a)

k2k1

(p, σ)(p, s) (b)

k2k1

(p, σ)(p, s)

(c)

k2 k1

(p, σ)(p, s)

(d)

k2 k1

(δm2)4 + (δm2)4 + (δm2)6 + (δm2)10 + (δm2)11 is IR finite.






Summary

Cancellation of true IR divergences to all orders

Use method of inductionYennie etal, Annals of Physics 13, 379(1961): Real virtualcancellation of IR divergences to all orders

LFQED : divergences cancel to O(e4)

Assume IR divergences cancel up to O(e2n)

Express O(e(2n+2) contribution in terms of IR finite O(e2n) matrixelements

Show the additional divergences also cancel in coherent state basisJai More & AM, Phys. Rev. D 89, 105021 (2014)






Summary

Represent the O(e2n) IR finite amplitude by a blob i.e. a blobrepresents the sum of the Fock and coherent state contributions tothe self energy correction that being added up together give IR finiteamplitude.

The blob is of O(e2n) and contains n photon linesExpress the O(e2(n+1)) contributions in terms of this blobShow the cancellation of IR divergences in O(e2(n+1)) using thecoherent state basis.






Summary

General expression for transition matrix element in O(e2n) is a sumof terms of the form:

T(n)j = − e2n

2p+(2π)3n

∫ n∏i=1

d3ki2k+

i 2p+2i−1

×u(p1, s1)ε/1( 6 p1 + m)ε/2(6 p2 + m) · · · · · · · · · (6 pi + m)ε/iu(pi , si )

n∏r=1

(p− − p−r −n∑

i=1

ki )

T (n) =∑j

T(n)j =

∑j

u(p, s ′)M(j)n u(p, s)

D(j)

where j is summed over all possible diagram in O(e2n) and will beassumed to be IR divergence free.Here,

D(j) =n∏

i=1

D(j)iDipartimento di Fisica, Universita di Torino, March14, 2017





Summary

O(e4)revisited

O(e2) correction

T (2) =∑j

u(p, s ′)M(j)2 u(p, s)

D(j)

Figure: IR finite O(e2) blob with an external photon line results intoO(e4) diagram






Summary

In our new notation it is ,

T(2)3a = T

(2)3b + T

(2)3c

=e2

(2π)3

∫d3k1

2k+1

u(p, σ)ε/(k1)( 6 p1 + m)M(j)2 (6 p1 + m)ε/(k1)u(p, s)

(p · k1)2D(j)

Blob is IR finite

IR divergences can appear ”only” from the vanishing of energydenominators of the kind p− − k−1 − (p − k1)

Additional contribution in O(e4) in coherent state basis






Summary

In coherent state basis, we have extra contributions

T(2)4a = − e2

(2π)3

∫d3k1

2k+1

u(p, σ)ε/(k1)(6 p1 + m)M(j)2 u(p, s)(p · k1)

(p · k1)2D(j)

which cancel the Fock contribution in the limit k+ → 0, k⊥ → 0






Summary






Summary

Same holds for other diagrams as well

To construct an O(e2n+2) diagram in Fock basis, we can add a photon tonth order blob in three different ways






Summary

Figure:Dipartimento di Fisica, Universita di Torino, March14, 2017





Summary

The contributions from Figs. (a), (b) and (c) are given by

T(n+1)6a =

e2

(2π)3

∫d3q

2q+

u(p, σ)ε/(q)(6 P + m)M(j)n ( 6 P + m)ε/(q)u(p, s)

(p · q)2D(j)

(1)

T(n+1)6b =− e2

(2π)3

∫d3q

2q+

u(p, s ′)M(j)n (6 p′ + m)ε/(q)( 6 P + m)ε/(q)u(p, s)

(p · q)(p− − p′−)D(j)

T(n+1)6c =

e2

(2π)3

∫d3q

2q+

u(p, s ′)Ml(j)n (6 P + m)ε/(q)u(p, s)

(p · q)Dl(j)

where P = p − q






Summary

Note that in case of overlapping diagram, the structure is different.However, one can show that for our purpose it is sufficient toconsider the limit q+ → 0,q⊥ → 0, in which case

M`(j)n = ε/(k1)(P1 + m)ε/(k2)(P2 + m) · · · ε/(k`)(P` + m)ε/(q)ε/(p`+1 + m) · · ·

Also, the energy denominators corresponding to the intermediatestates will be

D(j) = (p− − p−1 − k−1 − q−)(p− − p−2 − k−1 − k−2 − q−) · · ·

(p− − p−` −∑i

k−i − q−) · · · · · · · · ·

The additional contributions in coherent state basis are given by thefollowing diagrams






Summary

Figure:Dipartimento di Fisica, Universita di Torino, March14, 2017





Summary

T′(n+1)7a =− e2

(2π)3

∫d3q

2q+

u(p, s ′)ε/(q)( 6 P + m)M(j)n u(p, s)(p · ε) Θ∆(q)

(p · q)2D(j)

(2)

T′(n+1)7b =

e2

(2π)3

∫d3q

2q+

u(p, s ′)M(j)n ( 6 p′ + m)ε/(q)u(p, s)(p · ε) Θ∆(q)

(p · q)(p− − p′−)D(j)

T′(n+1)7c =− e2

(2π)3

∫d3q

2q+

u(p, s ′)M(j)n u(p, s)(p · ε) Θ∆(q)

(p · q)D(j)

In the limit, k+ → 0, k⊥ → 0, 6 Pε/(q)→ p · ε, the coherent statecontribution exactly cancels the IR divergences in the original (Fockspace) diagrams. Thus by induction, the IR divergences cancel to all

orders.






Summary

Improved Method of Asymptotic Dynamics

KF method does not work for theories involving 4-point interaction

Asymptotic states in QCD are bound states

In QCD a recursive proof of cancellation of IR divergences cannot beobtained using just KF method

Asymptotic Hamiltonian should contain the confining potential incase asymptotic states are bound states

An ‘improved’ method of asymptotic dynamics should take intoaccount the separation of particles also

[ R. Horan, M. Lavelle, and D. McMullan, Pramana 51, 317 (1998).R. Horan, M. Lavelle, and D. McMullan, Report No. PLY-MS-99-9, hep-th/9909044,

(1999).R. Horan, M. Lavelle, and D. McMullan, hep-th /0002206 (2000).

Anuradha Misra, Few-Body Systems 36, 201-204 (2005).]






Summary

Improved method of asymptotic dynamics (Horan, Lavelle & McMullan2000)

based on asymptotic properties of matrix elements instead ofoperators

takes into account appropriate boundary conditions corresponding tothe separation pf particles at large distances

first criteria suggests not only the energy denominators but theirpartial derivatives also become zero

For theory involving 4-point interactions, KF method does not work, butthe improved method leads to cancellation of IR divergences






Summary

Criteria in the Method of Asymptotic Dynamics

In LFQED,νi = p− − k− − (p − k)−

Condition to obtain the asymptotic region KF approach =⇒ νi = 0

Improved Method =⇒ ∂νi∂p⊥

= ∂νi∂p+ = ∂νi

∂k⊥= ∂νi

∂k+ = 0

For QED, both the criteria lead to same asymptotic region forconstructing coherent states

For LFQCD, one may need to use the criteria of separation ofparticles ?






Summary

Future directions

Develope the improved method of asymptotic dynamics in LFFT forsimple model like Yukawa theory, φ4 theory

Extend this method to QCD to analyze the nature of IR divergences

Construct an artificial potnetial that is needed for bound statecalculation in LFQCD

Combine the coherent state method with the BLFQ methods (J.Vary et al) to deal with the IR problem in LF bound statecalculations






Summary

To summarize

The true IR divergences are cancelled to all orders when coherentstate basis is used to calculate the matrix elements in lepton selfenergy correction in light-front QED.

Apply improved method to LFQCD beyond one loop order to obtainIR finite amplitudes

Connection between asymptotic dynamics and IR divergences needsto be investigated

Combine coherent state method with BLFQ methods (J.Vary et al,Phys. Rev. D 91, 105009 (2015)) for practical use of coherent statemethods

GRAZIE


Documents

Light Front Field Theory And Coherent State Basispersonalpages.to.infn.it/~torriell/seminars/files/Misra.pdf · 2017. 3. 15. · Mass renormalization upto O(e2) in Fock state basis