mathematica and quantum corrections to kink...
TRANSCRIPT
![Page 1: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/1.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Mathematicaand
Quantum Corrections to Kink Masses
Alberto Alonso Izquierdo1,3
Miguel Ángel González León1,3
Juan Mateos Guilarte2,3
1Departamento de Matemática Aplicada (Universidad de Salamanca)
2Departamento de Física Fundamental (Universidad de Salamanca)
3IUFFyM (Universidad de Salamanca)
Encuentros de Invierno deGeometría, Mecánica y Teoría de Control,
Zaragoza 2010
![Page 2: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/2.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
![Page 3: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/3.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
![Page 4: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/4.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
THE AIM OF THE TALK
Question:How to compute the quantum correction to the kink mass in a(1+1)-scalar field theory model?
![Page 5: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/5.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1 + ~2∆M2 + . . .
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
![Page 6: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/6.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1 + ~2∆M2 + . . .
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
![Page 7: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/7.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
![Page 8: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/8.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
![Page 9: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/9.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
SEMICLASSICAL MASS
Msc = Mcl + 12 ~ tr(H 1
2 [φS])
![Page 10: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/10.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
![Page 11: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/11.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response will be∞SOLUTION:
We need a reference point
Zero-Point Renormalization
![Page 12: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/12.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response will be∞SOLUTION:
We need a reference point
Zero-Point Renormalization
We have to measure the quantum correction with respect to the minimum energysolution φV .
![Page 13: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/13.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
![Page 14: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/14.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞−∞
SOLUTION:
We need a procedure
Mode NumberCut-off Regularization
![Page 15: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/15.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞−∞
SOLUTION:
We need a procedure
Mode NumberCut-off Regularization
![Page 16: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/16.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
![Page 17: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/17.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞
SOLUTION:
The mass coupling constant
is infinite
Mass Renormalization
We have to introduce the counterterms (well stablished procedure in Physics)
![Page 18: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/18.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.+Ect[φS]− Ect[φV ]
![Page 19: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/19.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C. + Ect[φS]− Ect[φV ]
• Hessian operator associated to the solution φS:
H[φS] = − d2
dx2 +∂2U∂φ2 [φS]︸ ︷︷ ︸
V[φS]≡V(x)
• Hessian operator associated to the vacuum φV :
H[φV ] = − d2
dx2 +∂2U∂φ2 [φV ]︸ ︷︷ ︸V[φV ]≡V0
• Counterterms:
Ect[φS]− Ect[φV ] = ~ δm∫
dx[∂2U∂φ2 [φS]− ∂2U
∂φ2 [φV ]]
δm =
∫dk4π
1√k2 + d2U
dφ2 [φV ]
![Page 20: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/20.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
![Page 21: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/21.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =12
(φ2 − 1)2
• Partial Diferential Equation:∂2φ
∂t2 −∂2φ
∂x2 + 2φ(φ2 − 1) = 0
![Page 22: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/22.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =12
(φ2 − 1)2
• Partial Diferential Equation:∂2φ
∂t2 −∂2φ
∂x2 + 2φ(φ2 − 1) = 0
SOLUTIONS:
VACUUM SOLUTION:φV = ±1
KINK SOLUTION:φK = tanh x−vt√
1−v2
![Page 23: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/23.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• KINK ENERGY DENSITY:
ε(x) = sech4 x + x0 − vt√1− v2
• KINK CLASSICAL MASS:
Mcl =
∫ ∞−∞
dx sech4(x + x0) =43
CLASSICAL KINK
![Page 24: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/24.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• KINK ENERGY DENSITY:
ε(x) = sech4 x + x0 − vt√1− v2
• KINK CLASSICAL MASS:
Mcl =
∫ ∞−∞
dx sech4(x + x0) =43
QUANTUM KINK
![Page 25: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/25.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
![Page 26: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/26.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
![Page 27: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/27.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
![Page 28: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/28.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
• R. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D 10 (1974) 4130
![Page 29: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/29.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
![Page 30: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/30.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
SOLUTIONS:VACUUM SOLUTION:φV = ±arcsinh 1
Hessian OperatorH[φV ]:
H[φV ] = − d2
dx2 + 4
KINK SOLUTION:φK = arctanh[ 1√
2tanh x−vt√
1−v2]
Hessian OperatorH[φK ]:
H[φK ] = − d2
dx2 +2sech2x(9 + sech2x)
(1 + sech2x)2
![Page 31: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/31.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
SOLUTIONS:VACUUM SOLUTION:φV = ±arcsinh 1
Hessian OperatorH[φV ]:
H[φV ] = − d2
dx2 + 4
KINK SOLUTION:φK = arctanh[ 1√
2tanh x−vt√
1−v2]
Hessian OperatorH[φK ]:
H[φK ] = − d2
dx2 +2sech2x(9 + sech2x)
(1 + sech2x)2
UNKOWN SPECTRUM: ∆M NO COMPUTABLE
![Page 32: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/32.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
![Page 33: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/33.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
![Page 34: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/34.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
![Page 35: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/35.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Task:Approach a procedure to compute approximately the semiclassical mass of akink in a (1+1) dimensional scalar field model.
Clue:We will estimate the trace of the differential operator without the knowledge ofthe exact spectrum.
![Page 36: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/36.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
![Page 37: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/37.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
Quantum Correction
∆M = 12 ~[tr (H(φK))
12 − tr (H(φV))
12
]+ Ect(φK)− Ect(φV)
Generalized Zeta Function: Heat Function
ζH(s) = TrH−s =
∞∑n=0
(ω2n)−s hH(β) = Tr e−βH =
∞∑n=0
e−βω2n
Mellin Transform
ζH(s) =1
Γ(s)
∫ ∞0
dβ βs−1 hH(β)
Quantum Correction
∆M =~2
lims→− 1
2
1Γ(s)
∫ ∞0
dβ βs−1[h∗H[φK ](β)− h∗H[φV ](β)] + lims→ 1
2
Ect[ζH[φV ](s)] + ∆R
![Page 38: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/38.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
Quantum Correction
∆M = 12 ~[tr (H(φK))
12 − tr (H(φV))
12
]+ Ect(φK)− Ect(φV)
Generalized Zeta Function: Heat Function
ζH(s) = TrH−s =
∞∑n=0
(ω2n)−s hH(β) = Tr e−βH =
∞∑n=0
e−βω2n
Mellin Transform
ζH(s) =1
Γ(s)
∫ ∞0
dβ βs−1 hH(β)
Quantum Correction
∆M =~2
lims→− 1
2
1Γ(s)
∫ ∞0
dβ βs−1[h∗H[φK ](β)− h∗H[φV ](β)] + lims→ 1
2
Ect[ζH[φV ](s)] + ∆R
![Page 39: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/39.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
![Page 40: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/40.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
![Page 41: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/41.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
![Page 42: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/42.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
We have to take the limit y→ x
![Page 43: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/43.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
![Page 44: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/44.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
![Page 45: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/45.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
![Page 46: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/46.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
![Page 47: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/47.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
![Page 48: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/48.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
![Page 49: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/49.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
![Page 50: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/50.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
![Page 51: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/51.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
![Page 52: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/52.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Constructing an algorithm
![Page 53: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/53.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Constructing an algorithm
Go to the Mathematica Program
![Page 54: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/54.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
![Page 55: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/55.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
![Page 56: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/56.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
![Page 57: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/57.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
![Page 58: Mathematica and Quantum Corrections to Kink Massescampus.usal.es/~mpg/General/Zaragoza10Definitivo.pdfMathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos](https://reader030.vdocuments.net/reader030/viewer/2022041302/5e12c1fcdbbda07f310b2279/html5/thumbnails/58.jpg)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
End of the talk