Solutions to the Bethe-Salpeter Equation via Integral ...mstrick6/qcdseminar/spring15/bednar.pdf · Solutions to the Bethe-Salpeter Equation via Integral Representations K. Bednar

Introduction

Solutions to the Bethe-Salpeter Equation via IntegralRepresentations

K. Bednar P. TandyKent State University

February 25, 2015

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Introduction

Introduction

1 Introduction

2 BackgroundQCD BasicsWhat to calculate?Quark Propagator

3 Bound StatesQFT Bound StatesLadder-RainbowComplex-conjugate polesNakanishi forms

4 Conclusion

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Background QCD Basics

Basic Properties and Questions of QCD

QCD fully understood at small distances / high energies

Poorly understood at large distances (hadronic scale)

ConfinementDynamical Chiral Symmetry Breaking (DCSB)

What are the dominant mechanisms at work inside hadrons? How arehadrons composed of quarks and gluons?

Spectrum of states, elastic + transition form factors, partondistributions provide info. on how non-pert. QCD works in volume ofhadrons

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Background What to calculate?

What to calculate? Meson Observables

Pion electromagnetic form factorsFπ(Q2)∫Tr[ΓπS iΓµ S Γπ S

]

Pion decay constants

〈0|uγµγ5d |π−(p)〉 = ipµfπ

Strong Decayρ→ ππ

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]




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]




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]




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Background Quark Propagator

Quark Propagator

Perturbation theory says ...

Dressed 2-point function for quarks

iS(x − y) = 〈0H |TψH(x)ψH(y)|0H〉〈0H |0H〉 = 〈0I |Te i

∫LIψI (x)ψI (y)|0I 〉c

1 Use Wick’s theorem after Taylor expansion:iS(x − y) =

∑∞n

in

n!

∫d4z ...〈O|TLint(z1)...ψ(x)ψ(y)|0〉c and work

out individual terms

2 or ...

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Quark Propagator Cont.

Introduce 1PI diagrams:

One-Particle Irreducible (1PI) Diagrams

Any diagram that cannot be split in two by removing a single line

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Quark Propagator Cont.

So the two-point function can be written:

Or: S = 1/p−m0−Σ(/p) . The self-energy term can be seen to satisfy:

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Quark Prop. IV - DSE

The formal derivation of these results is given in the path-integralformulation:

Generating functional of QCD

Propagator definition

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Functional Derivation

Prepare differential (functional) operator

Redefine:

Take another functional derivative and divide by Z:

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Functional Derivation cont.

Expand:

Make a Legendre Transform (Effective Action):

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Sorry...

Use Identities:

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...

Derive a relation between connected 3-pt function and proper 3-ptfunction:

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...

Finally1:

1Marco Viebach - Diplomarbeit - Dyson-Schwinger equation for the quark propagatorat finite temperatures

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Bound States QFT Bound States

Basic Considerations

Bound states are often neglected in field theory texts!

Bound states appear as poles in scattering amplitudes (on real axis forstable states, below for unstable states)2

By definition: H|P, t〉 = P0|P, t〉 → |P, t〉 = e−iP0t |P, 0〉

So the bound state contribution to a completeness sum in anamplitude is 〈f , tf |P, t〉〈P, t|i , ti 〉 = 〈f |P〉e−i(tf−ti )P0〈P|i〉The fourier transform tf − ti ⇒ p0 generates a pole at p0 = P0 − iε

2Paul Hoyer - Bound states - from QED to QCD arXiv: 1402.5005v1February 25, 2015 15 / 34


Derivation of Bethe-Salpeter Equation

In order to find an expression for Γ , consider the (above) 4-pt function:

Two-Particle Irreducible (2PI) Diagrams

Any diagram that cannot be split in two by removing two lines

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cont.

These diagrams are written as:

G (n) =∑

SiSjK(n)SlSr (1)

and then:

G (m) =∑

ShSbK(m)SiSjK

(n)SlSr

=∑

ShSbK(m)G (n),

which leads to:

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cont.

Bethe/Salpeter argued that the disconnected pieces won’t contributeto a bound state

Amputate external propagators:

Assume the form G → Γ∗(P)Γ(P)P0−EP

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Truncation of Infinite ’Tower’

Elements of the BSE are parts of an infinite series of coupledequations; truncations must be made: where?

Consider positronium:

Rest frame energies of positronium given by P0 = 2me + Eb with

Eb = −meα2

4n2

Thus has infinite set of poles just below E = 2me

Question: how are these poles generated by Feynman diagrams? ie,which diagrams need included in the truncation?view positions of b.s. poles in s = (2me + Eb)1/2 as functions of α ;but perturbative expansion in QED is for O(αn) ; thus, the whole seriesmust diverge

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Truncation cont.

Careful counting: Look at three diagrams:

momentum exchanges in atoms are ' Bohr momentum:|q| ' αme , q0 ' q2/2me ' 1

2α2me

First diagram ' α/q2 ' 1/α

Second: Four vertices → α2 ; Two photon props → 1/q2 ' α−2 ;fermion props are off-shell k0 ' q0 ; the loop momentum→∫dk0d3k ' α5 Thus: O(α−1)

therefore, include all ladder diagrams to get divergent series; all otherdiagrams are higher order in α

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Bound States Ladder-Rainbow

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Bound States Ladder-Rainbow

Solving some things

With this truncation one can go ahead and numerically solve the DSE forthe quark propagator; the solution is then used in the BSE...

But the quark propagator is not an analytic function. For Mqq ' 1GeVthe poles are outside the range of integration.

Up to about 3 GeV only the first, complex-conjugate poles (related toconfinement / absence of real quark mass poles) contribute. 3

In the BSE, (k±)2 = k2 + P2/4± 2k · q = k2 −M2/4± ikMx , ifP = (iM,~0) in Euclidean metric

3Analytical properties of the quark propagator from a truncated Dyson-Schwingerequation in complex Euclidean space, Dorkin et al., PhysRevC.89.034005

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Bound States Complex-conjugate poles

These poles suggest the propagators can be represented as:

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Fit to Numerical Solution

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Going Further

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Going Further - Kernel

Fit the LR-kernel to a similar form: D(k2) =∑ Zi

k2+m2i

+Z∗i

k2+(m∗i )2

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Bound States Nakanishi forms

So all but the BS vertex itself are represented in this form, which provesamenable to analytic evaluation, ie no numerical problems with poles. Yetthis can’t be done without some suitable representation for the 3-pointfunction...

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Rough Sketch of Derivation:

Start with matrix element for BS amplitude:f (x , p) = e(−ipX )〈0|Tφ1(x1)φ2(x2)|p〉 = 〈0|Tφ1(x/2)φ2(−x/2)|0〉and rely on the following assumptions: Lorentz invariance, sprectralconditions, microcausality, asymptotic conditions

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cont.

For simplicity write retarded amplitude as:

fR(x , p) = 〈0|Rφ1(x/2)φ2(−x/2) · φ†in(p)|0〉 (2)

= 〈[R

[φ1(x/2)φ2(−x/2)

], φ†in

]〉 (3)

Asymptotic condition fR(x , p) =∫e ipz(� + m2)r(x , z ; p)

Dyson’s representation of double-commutator:

D(y , z) '∫〈0|φ1|λp〉〈λp|

[φ2, φ3

]|0〉 (4)

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cont.

Use spectral representation of the two-point function:

D(y , z) '∫

f (p2)

∫ ∞0

dω2ρ(ω2, p2, ωp)∆(y , λ2)e ip(z+y/2) (5)

Fourier transform wrt ωp and use microcausality and energy positivity4:

D(y , z) =

∫ ∞0

dω2

∫ ∞0

dµ2

∫ 1

0dαf ′(ω2, µ2, α)∆(y , ω2)∆(z+αy , µ2)

(6)

Going back: plug this in, use integral rep. of Heav. step fcn, takeFourier transform, do an integral, get:

fR(q, p) =

∫ 1

−1dα

∫ ∞0

dsΦ(α, s)

(q − αp/2)2 + s(7)

ok4On the Integral Rep. of the Double Commutator, Eftimiu + Klarsfeld, 1960

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Now all of the elements of the BSE have the form Nump2+M2

Use Feynman parameters:

1

Am11 Am2

2 ...Amnn

=

∫ 1

0dx1...dxnδ(

∑xi−1)

∏xmi−1i[∑

xiAi

]∑mi

Γ[m1 + ...mn]

Γ(m1)...Γ(mn)

(8)

The result is a known Euclidean integral, and hence are left with areduced-dimensional integral over Feynman parameters (poles aretherefore not a problem )

Eigenvalue equation λ(P2)E (k ,P) =∫q K (k − q)S(q+)E (q,P)S(q−)

with solution λ(P2 = −M2) = 1

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Conclusion

Results

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Conclusion

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Solutions to the Bethe-Salpeter Equation via Integral ...mstrick6/qcdseminar/spring15/bednar.pdf · Solutions to the Bethe-Salpeter Equation via Integral Representations K. Bednar