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Non-perturbative renormalization scheme in application to chiral perturbation theory in the nucleon sector

N.A. Tsirova Samara State University, Samara, Russia J.-F. Mathiot Laboratoire de Physique Corpusculaire, Aubiere, France

QFTHEP 2011


Outline
· Introduction: LFEFT · · · · · Covariant Light-Front Dynamics Fock sector dependent renormalization scheme Taylor-Lagrange regularization scheme Application to ChPT Perspectives


Introduction
To understand the nucleon structure at low energy from a chiral effective Lagrangian we need an appropriate calculational scheme: · relativistic · non-perturbative · well-controlled approximation scheme

We present LFEFT ­ Light-Front Chiral Effective Field Theory


Light Front Chiral Effective Field Theory
Key points: · CLFD explicitly covariant formulation of light-front dynamics
[J.Carbonell et al. Phys.Rep. 300 (1998) 215]

· FSDR Fock sector dependent renormalization scheme
[V. Karmanov et al. PRD 77 (2008) 085028]

· TLRS Taylor-Lagrange regularization scheme
[P.GrangÈ et al. Phys. Rev. D 82 (2010) 025012]


Already done
CLFD + FSDR + Pauli-Villars · Yukawa N=2, N=3 · QED N=2
[V. Karmanov et al. PRD 77 (2008) 085028]

· ChPT N=2 CLFD + FSDR + TLRS · Yukawa N=2
[P.GrangÈ et al. Phys.Rev.D 80 (2009) 105012]

ChPT is the first "realistic" test of this approach


Covariant Light-Front Dynamics
Standard version of LFD Covariant formulation

Rotational invariance is broken!

Arbitrary position of the LF plane

·x = 0 2 = 0

[J.Carbonell et al. Phys.Rep. 300 (1998) 215]


The state vector construction
Fock decomposition:

-body light-front wave functions Truncation of the Fock decomposition:
· ·

n

nN

N n

­ the maximal number of Fock sectors under consideration ­ number of constituents in a given Fock sector

Upper index : n-body light-front wave functions depend on the Fock space truncation For example:


Vertex functions
Wave functions vertex functions:

Graphical representation:

Vertex function for a physical fermion made of a constituent fermion coupled to bosons


Vertex functions
Vertex functions decomposition :
· invariant amplitudes constructed from the particle 4-momenta · spin structures

Yukawa model :

a, b, c

are scalar functions depending on dynamics


Renormalization scheme
Contribution to the physical fermion propagator

2-particles sector

1-particle sector

The general case: dependence on the Fock sector

(maximal number of particles in which the fermion line can fluctuate)

The same strategy for bare coupling constant
[V. Karmanov et al. PRD 77 (2008) 085028]

g

0


Renormalization scheme
Iterative scheme: from sector to sector problem for N=2 problem for N=3

m2, g02 m3, g03

problem for N=4

. . .

mN, g

0N

for given N


Regularization
Infinite regularization schemes: · Cut-off · Dimensional regularization · Pauli-Villars regularization scheme All these schemes deal with infinitely large contributions

We use TLRS: systematic finite regularization scheme Amplitudes depend on arbitrary finite scale


Basics of TLRS

f(x)

­ super regular test function

·

f(x) = 1

everywhere it is defined

· vanishes with all derivatives at boundaries Support : Ordinary

f (x H) = 0 cut-off : H=H

0

We go beyond this ordinary cut-off


Basics of TLRS

Running boundary condition:

Lagrange formula:

When we put aü 1- : · ga(x) ü 1 · H(xmax) ü ¶ · f(x) ü 1 everywhere · integration limit over t :


Basics of TLRS : how it works

New variable

y t

:

x = ay z = yt z = yt § H(y) t§h
­ Pauli-Villars type subtraction

New variable

:

Integration domain :

Final result :

h

is an arbitrary finite positive number


Application to Chiral Perturbation Theory N=2
Need of a non-perturbative framework to calculate bound state properties easy with

NN coupling

to be generalized for

NN case


ChPT Lagrangian
·

Lagrangian is formulated in terms of

u fields

F
·

0 is the pion decay constant

Expansion in a finite number of degrees of pion field

·

N-body Fock space truncation: 2(N ­ 1) pions


ChPT Lagrangian
In our first study

linear

NN

interaction

contact

NN

interaction


Self-energy calculation

Attach test functions corresponding to internal propagators

Introduce a new variable

t

and apply the Lagrange formula


Self-energy calculation

Proceed in the common way: · consider running boundary condition · put

aü 1- , f ü 1 t§h

· find integration limit The final result :


Chiral limit
The nucleon mass correction from the self-energy contribution :

The full self-energy :

Non-analytic terms in the chiral limit :


Scalar form factor

To calculate it we proceed in a common way:
· attach test function · introduce new variable t · apply the Lagrange formula

~S
convergent integral

The final result

coincides with the Feynman ­ Hell-Mann theorem:


System of equations
In the two-body Fock space truncation

The term

V8

corresponds to the contact

NN

interaction


Vertex functions representation
We choose the representation of vertex functions according to the vertex with an outgoing pion:

With this representation

b

1 and

b

2 are just scalars


Solution

Condition : Bare coupling constant :

Mass counterterm :


Final results

Without contact interaction :

With contact

NN

interaction :


Nucleon mass corrections
First non-analytic contributions with and without contact interaction

The first contribution of the contact interaction is of order


Contributions to the nucleon mass

the leading non-analytic correction the leading non-analytic correction the full contribution of the nucleon self-energy

28


Perspectives

Calculations for ChPT

N=3

(1 nucleon and 2 pions)

}

and resonances contributions

and Roper resonances contributions