Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2013/QFTHEP-Katanaeva.pdf Äàòà èçìåíåíèÿ: Fri Jun 28 21:34:10 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 20:23:19 2014 Êîäèðîâêà:

CONTENTS

PREMISES

SCALAR MODEL

ABELIAN MODEL

ANALYSIS

QUANTUM CORRECTION

CONCLUSION

LAGRANGIAN ALTERNATIVE TO QCD STRING
Alisa Katanaeva and Sergey Afonin

Saint-Petersburg State University
The XXI International Workshop High Energy Physics and Quantum Field Theory June 29, 2013
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CONCLUSION

P

REMISES Phenomenological approaches Preliminary notes CALAR MODEL Construction of scalar theory with non-degenerate vacua Classical kink solution BELIAN MODEL Construction of abelian Higgs model with periodic potential Nielsen­Olesen vortex solutions NALYSIS UANTUM CORRECTION

S

A

A Q C

ONCLUSION

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PHENOMENOLOGICAL APPROACHES

What methods to receive the spectra of radial excitations are frequently used in hadron spectroscopy? 1. The nonrelativistic potential models ­ their best feature is the description of heavy quark systems. They didn't always give correct answers for light mesons, however. 2. The hadron string models ­ the string is an approximation for the gluon flux-tube with nearly massless quarks at the ends. The semiclassical quantization of a thin flux-tube typically gives a linearly rising spectrum for masses squared: 2 = ( + ), = 0, 1, 2, . . . (1)

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ABELIAN MODEL

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CONCLUSION

PHENOMENOLOGICAL APPROACHES

What methods to receive the spectra of radial excitations are frequently used in hadron spectroscopy? 1. The nonrelativistic potential models ­ their best feature is the description of heavy quark systems. They didn't always give correct answers for light mesons, however. 2. The hadron string models ­ the string is an approximation for the gluon flux-tube with nearly massless quarks at the ends. The semiclassical quantization of a thin flux-tube typically gives a linearly rising spectrum for masses squared: 2 = ( + ), = 0, 1, 2, . . . (1)

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CONTENTS

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ABELIAN MODEL

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CONCLUSION

PHENOMENOLOGICAL APPROACHES

What methods to receive the spectra of radial excitations are frequently used in hadron spectroscopy? 1. The nonrelativistic potential models ­ their best feature is the description of heavy quark systems. They didn't always give correct answers for light mesons, however. 2. The hadron string models ­ the string is an approximation for the gluon flux-tube with nearly massless quarks at the ends. The semiclassical quantization of a thin flux-tube typically gives a linearly rising spectrum for masses squared: 2 = ( + ), = 0, 1, 2, . . . (1)

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ABELIAN MODEL

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CONCLUSION

PRELIMINARY NOTES

I The usual models based on the QCD string possess a basic feature: The linear Regge recurrence, 2 ( denotes the hadron spin), emerges on the classical level. The radial modes appear after quantization. An idea appears to look for alternative dynamical approaches in which the spectrum (1) is realized on the classical level. II The light non-strange mesons cluster near certain equidistant values of energy squared with repeating structure of the clusters. The assumption of periodic structure of the gluon vacuum arises.

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CONTENTS

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SCALAR MODEL

ABELIAN MODEL

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QUANTUM CORRECTION

CONCLUSION

PRELIMINARY NOTES

I The usual models based on the QCD string possess a basic feature: The linear Regge recurrence, 2 ( denotes the hadron spin), emerges on the classical level. The radial modes appear after quantization. An idea appears to look for alternative dynamical approaches in which the spectrum (1) is realized on the classical level. II The light non-strange mesons cluster near certain equidistant values of energy squared with repeating structure of the clusters. The assumption of periodic structure of the gluon vacuum arises.

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CONTENTS

PREMISES

SCALAR MODEL

ABELIAN MODEL

ANALYSIS

QUANTUM CORRECTION

CONCLUSION

PRELIMINARY NOTES

I The usual models based on the QCD string possess a basic feature: The linear Regge recurrence, 2 ( denotes the hadron spin), emerges on the classical level. The radial modes appear after quantization. An idea appears to look for alternative dynamical approaches in which the spectrum (1) is realized on the classical level. II The light non-strange mesons cluster near certain equidistant values of energy squared with repeating structure of the clusters. The assumption of periodic structure of the gluon vacuum arises.

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CONTENTS

PREMISES

SCALAR MODEL

ABELIAN MODEL

ANALYSIS

QUANTUM CORRECTION

CONCLUSION

PRELIMINARY NOTES

I The usual models based on the QCD string possess a basic feature: The linear Regge recurrence, 2 ( denotes the hadron spin), emerges on the classical level. The radial modes appear after quantization. An idea appears to look for alternative dynamical approaches in which the spectrum (1) is realized on the classical level. II The light non-strange mesons cluster near certain equidistant values of energy squared with repeating structure of the clusters. The assumption of periodic structure of the gluon vacuum arises.

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CONCLUSION

CONSTRUCTION OF SCALAR THEORY WITH NONDEGENERATE VACUA

Consider the following effective field theory for a real scalar field in four dimensions [ ( ) ] 1 4 2 = + cos - 2 - cos(2 ) , 2 2 2 (2)

where has the dimension of mass and the parameters and are dimensionless. There are restriction on the value of parameter : the interval 0 < 1 covers all nontrivial cases. The parameter plays the role of coupling constant. In the weak coupling regime, || 1, the Lagrangian (2) reduces to the scalar theory 4 :
||1

1 1 = + 2 sin(2 )2 - cos(2 )4 + (2 6 ). 2 2 4
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CONSTRUCTION OF SCALAR THEORY WITH NONDEGENERATE VACUA

The potential in (2) is minimized on the constant field configurations: = 2 2 ( + ), = 0, 1, 2, . . .

(3)

Consider small perturbations around the vacua (3): = + (). The quadratic part of the Lagrangian for reads:
(2)

1 = - 2 2 . 2

The spectrum of excitations has the Regge-like form (1):
ALISA KATANAEVA LAGRANGIAN ALTERNATIVE TO QCD STRING

2 ,

= 22 = 4 2 ( + ),

= 0, 1, 2, . . .
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CLASSICAL KINK SOLUTION

The most close analogue to the sine-Gordon model is provided by the following modification of the Lagrangian (2): [ ( ) ] 6 2 ~ = 1 + cos - 1 . (4) 2 2 2 2 The minima of effective potentials in (4) and in (2) at = 0 coincide. As 0 we obtain the theory: 1 1 2 6 ~1 = - 2 2 + + . . . 2 2 4! 2 The equation of motion for the field in the static case has a solitonlike solution: 2 = ±2 arctan 2(-0 ) (5)
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CONSTRUCTION OF ABELIAN HIGGS MODEL WITH PERIODIC POTENTIAL

Consider now a (1) gauge model of the Nielsen-Olesen type. The Lagrangian reads: 1 = ( )* - + 4 [ ( ) ] 2 * 4 cos - 2 - cos(2 ) , (6) + 2 2 where 1 + 2 , = - , = - . 2 2 the scalar field: = 2 ( + ) Vacua for the vector field: 0 = Fluctuations above the vacua for
ALISA KATANAEVA LAGRANGIAN ALTERNATIVE TO QCD STRING

the scalar field: = the gauge field:

+ + 2
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CONSTRUCTION OF ABELIAN HIGGS MODEL WITH PERIODIC POTENTIAL

The standard change of the field variables: 1 = - (), + () = exp 2 ( () ) . (7)

In terms of the variables (7), the quadratic part of the Lagrangian (6) is cast into the canonical form: 12 1 1 (2) , = - 2 - 2 2 4 where
2 , 2 ,

12 + + const, 2

= 22 , = 2 2 ,

= 0, 1, 2 . . . = 0, 1, 2 . . .
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NIELSEN­OLESEN VORTEX SOLUTIONS

The equations of motion for the Lagrangian (6) are: ( ) 2 * 2 = - sin - 2 , 2 = (* - * ) + 22 * .

(8) (9)

We will consider the static case, with the gauge choice 0 = 0. Following the paper H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973) we look for a cylindrically symmetric solution, with axis along the -direction. The corresponding ansatz is: â (), () = () = () , (10)

where denotes a unit vector along the -direction and is an integer. In addition, it is assumed that = () and = = 0.
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NIELSEN­OLESEN VORTEX SOLUTIONS

The flux is given by () = 2 () so that the magnetic field is: () = 1 1 () = (()). 2

Inserting ansatz to the equations of motion we arrive at
- 1 ( )] ) [( ( )2 2 2 - 2 = 0, () + - + 2 sin 2 [ ] ( 1 ) - = 0. (()) + 22 2 - (11) (12)

We are going to find a solution with asymptotic behavior on the spacial infinity of the type const, then:
() = + 1 ( ) = + - 2


+ ...,

(13)

where is the modified Bessel function of the second kind.
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NIELSEN­OLESEN VORTEX SOLUTIONS

Then the magnetic field is = - 0 ( ) = -
2




-

+ ...

Substituting the asymptotics for () into the equation (11) we obtain the approximate solution / 2, which defines the characteristic length = ( )
-1

.

Each vacuum possesses not only its own spectrum but also its own vortex. The vortex-lines have a flux-tube structure and move according to the equation of motion of the Nambu dual string. The energy density of the vortex-line in the -th vacuum 2 .
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NIELSEN­OLESEN VORTEX SOLUTIONS

Then the magnetic field is = - 0 ( ) = -
2




-

+ ...

Substituting the asymptotics for () into the equation (11) we obtain the approximate solution / 2, which defines the characteristic length = ( )
-1

.

Each vacuum possesses not only its own spectrum but also its own vortex. The vortex-lines have a flux-tube structure and move according to the equation of motion of the Nambu dual string. The energy density of the vortex-line in the -th vacuum 2 .
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CONCLUSION

From the phenomenological point of view, the model under consideration predicts a linear Regge-like spectrum for radially excited scalar and vector states. In the general situation, the slopes of radial trajectories for these two types of particles are different. However: the slope is universal in the hadron string models because it is determined by the gluodynamics there. The universality of slopes: 2 = 2. This entails a complete degeneracy of the scalar and vector states.
The scalar field the scalar isoscalar particles 0 -mesons The vector field the vector isoscalar particles -mesons

For experimental proof see Fig. 1.

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From the phenomenological point of view, the model under consideration predicts a linear Regge-like spectrum for radially excited scalar and vector states. In the general situation, the slopes of radial trajectories for these two types of particles are different. However: the slope is universal in the hadron string models because it is determined by the gluodynamics there. The universality of slopes: 2 = 2. This entails a complete degeneracy of the scalar and vector states.
The scalar field the scalar isoscalar particles 0 -mesons The vector field the vector isoscalar particles -mesons

For experimental proof see Fig. 1.

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From the phenomenological point of view, the model under consideration predicts a linear Regge-like spectrum for radially excited scalar and vector states. In the general situation, the slopes of radial trajectories for these two types of particles are different. However: the slope is universal in the hadron string models because it is determined by the gluodynamics there. The universality of slopes: 2 = 2. This entails a complete degeneracy of the scalar and vector states.
The scalar field the scalar isoscalar particles 0 -mesons The vector field the vector isoscalar particles -mesons

For experimental proof see Fig. 1.

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m , Ge V
n

2

2

f (2330)
0

5
f (2200)
0

4

f (2020)
0

3

f (1710)
0

f
.

0

2

f (1370)
0

1
f (500)
0

n

0

FIG. 1. The spectrum of 0 and mesons (all data from J. Beringer et al. (Particle Data
Group), Phys. Rev. D 86, 010001 (2012)). The vertical size displays the experimental uncertainty. Other states 0 reported by the PD are assumed to belong to the ¯ trajectory. The state (2290) is identified with the (2330)-meson.
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The way to keep the universal slope but remove the degeneracy:
) ] ( [ 1 2 4 ~ = + cos - 2 - cos(2 ) - 2 2 2 1 1 - + (2 + 2 2 ) 4 2

The models discussed above should be viewed as effective models for QCD in the large- limit (because the number of resonances is infinite and the zero-width approximation is implied). Complementarity to the usual effective field theories for the strong interactions (the sigma-models, chiral perturbation theory, etc): The input parameters (first of all the bare masses) originate from integration of the low-energy degrees of freedom below 1 GeV. A phenomenological support: the linear ansatz (1) for the radial meson spectrum is a reasonable approximation for the excited states, while the ground ones below 1 GeV deviate substantially from the linear trajectory. For example, see Fig. 1.
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The way to keep the universal slope but remove the degeneracy:
) ] ( [ 1 2 4 ~ = + cos - 2 - cos(2 ) - 2 2 2 1 1 - + (2 + 2 2 ) 4 2

The models discussed above should be viewed as effective models for QCD in the large- limit (because the number of resonances is infinite and the zero-width approximation is implied). Complementarity to the usual effective field theories for the strong interactions (the sigma-models, chiral perturbation theory, etc): The input parameters (first of all the bare masses) originate from integration of the low-energy degrees of freedom below 1 GeV. A phenomenological support: the linear ansatz (1) for the radial meson spectrum is a reasonable approximation for the excited states, while the ground ones below 1 GeV deviate substantially from the linear trajectory. For example, see Fig. 1.
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QCD in the large- limit represents (assuming that confinement persists in this limit) a theory of an infinite number of narrow and stable non-interacting mesons and glueballs which should appear in the classical effective action. It is possible to interpret the field and its periodic potential as an effective model for the non-perturbative gluon vacuum in QCD. All quarkonia in the sector of light quarks get masses mainly through interaction with the vacuum field . The simplest Lagrangian for a scalar quarkonium is:
1 1 1 2 ~ = - (2 + 2 ) 2 + + 2 2 2 [ ( ) ] 4 2 + cos - 2 - cos(2 ) , 2 2

The spectrum of the scalar glueballs has the Regge-like form. The bare mass must depend on quantum numbers.
2 Universality of couplings: 2 2.
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QCD in the large- limit represents (assuming that confinement persists in this limit) a theory of an infinite number of narrow and stable non-interacting mesons and glueballs which should appear in the classical effective action. It is possible to interpret the field and its periodic potential as an effective model for the non-perturbative gluon vacuum in QCD. All quarkonia in the sector of light quarks get masses mainly through interaction with the vacuum field . The simplest Lagrangian for a scalar quarkonium is:
1 1 1 2 ~ = - (2 + 2 ) 2 + + 2 2 2 [ ( ) ] 4 2 + cos - 2 - cos(2 ) , 2 2

The spectrum of the scalar glueballs has the Regge-like form. The bare mass must depend on quantum numbers.
2 Universality of couplings: 2 2.
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The effective potential for the theory of one scalar field has the following expansion in the Planck constant: 4 ( ) = - 2 ( ( ) ) 2 cos - 2 - cos(2 ) + 2 ( ) ( ) ( ( ))2 3 + - + ln + ( 2 ), 64 2 2 2



where the second derivative reads ( ) ( )) ( 2 2 2 . ( ) = 2 2 cos 2 - 2 - sin 2 - 2
2

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Veff 0.012 0.010 0.008

Veff 0.025

0.020

0.015 0.006 0.010 0.004 0.002 0.005

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

(A) = -1
Veff 0.015 0.04 Veff

(C) = -0.4

0.03 0.010

0.02 0.005 0.01

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

(B) = -0.7

(D) = -0.1

FIG. 2. The effective potential (solid line) and the classical potential (dashed line) at
different values of . The parameters are taken from the trajectory of 0 -mesons in Fig. 1: 2 = 0.069 GeV2 , = 1.41, = 0.5 GeV.
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Constructive results: The models have been built in which the Regge-like form of the spectrum of the radially excited states appears on the classical level. Each model discussed has an infinite number of nonequivalent vacua. The theory with abelian gauge group has the Nielsen -- Olesen vortex solution in each vacuum. Corollaries: Interpretation of the scalar model as an effective model for the non-perturbative gluon vacuum in QCD yields the Regge-like form for the spectrum of the scalar glueballs. The analytical expression for the effective potential of scalar theory with the first quantum correction has been received. As it was plotted, the effect of quantum doubling was noticed, which has also a phenomenological correspondence.
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Thank you for your attention!

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BACKUP:
(I G , J
PC

THE FIGURE FROM REVIEW

S. S. A

FONIN,

M

OD.

P

HYS.

L

ETT.

A 22, 1359 (2007)

)

5 (1+ , 5-- ) 5 (0- , 4-- ) 4 (0+ , 4-+ ) 4 (1- , 4-+ ) 4 (1+ , 4-- ) 4 (0- , 4-- ) f4 (0+ , 4++ ) a4 (1- , 4++ ) b3 (1+ , 3+- ) h3 (0- , 3+- ) a3 (1- , 3++ ) f3 (0+ , 3++ ) 2 (1+ , 2-- ) 2 (0- , 2-- ) 3 (1+ , 3-- ) 3 (0- , 3-- ) 2 (0+ , 2-+ ) 2 (1- , 2-+ ) a2 (1- , 2++ ) f2 (0+ , 2++ ) b1 (1+ , 1+- ) h1 (0- , 1+- ) f1 (0+ , 1++ ) a1 (1- , 1++ ) a0 (1- , 0++ ) f0 (0+ , 0++ ) (1+ , 1-- ) (0- , 1-- ) (0+ , 0-+ ) (1- , 0-+ )

The predicted phenomenon of quantum doubling is difficult to observe because of overlapping decay widths of highly excited resonances. However, the case of 2 -mesons demonstrates such a phenomenon. In terms of the quark model, the doubling can have a simple explanation. For instance, the vector mesons can have = 0 and = 2 and these states are nearly degenerate due to the mass dependence , + . The degeneracy here takes place for the radial number 2, i.e. beginning from the third minimum as in Fig. 2. In the case of the scalar mesons, the source of doubling can arise from the mixing with the scalar states ( denotes the strange ¯ quark). The scalar mesons with the dominant -component are not displayed here. ¯
1 2 3 4 5 6 7 8 9 10 M 2 /M
2

0

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