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Effects of higher spin mesons in elastic electron-nucleon scattering
Nikolay Volchanskiy
Southern Federal University

QFTHEP 2015


Experimental data
1.5 /GD, GMp(Ros)/GD 3 GEn/GD, GMn/GD 2.5 2 1.5 GEp/GD 1 0.5 0.01 G
Mp/GD

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0.1 1 Q2, GeV2 10 0.1 1 Q2, GeV
2

GEn/GD

(Ros) Ep

G

Mn/GD

G

10

2

Rp=µp GEp/GMp

1

Rn=µn GEn/GMn 1 Q2, GeV
2

1.5

Rosenbluth extraction beam-target asymmetry polarization transfer

2

1.5

1

0.5

0.5

0 10

0 1 Q2, GeV
2

10

Plots from G. Vereshkov et al. Eur. Phys. J. A34, 223 (2007)


Two photon effects

e±

e

±









N

N

Figure: Meziane et al. Phys. Rev. Lett. 106, 132501 (2011)


Proton charge radius puzzle

Sick Bernauer et al. This work CODATA Pohl et al.
0.82 0.84 0.86 0.88 0.90

Proton charge radius [fm]

Figure: K. Zhan et al. Phys. Lett. B 705 (2011) 59


Higher-spin meson exchange

e±

e

±









spin-J meson

N

N


Gauge symmetry of spin-2 field
Fierz-Pauli Lagrangian of the free symmetric tensor field µ : L = -µ G G
µ µ

m2 µ µ - 2 1 = µ - gµ , 2
µ

-

2

,

=

1 (µ - µ - µ + µ ) . 2

Internal gauge symmetry of the kinetic term: L = 0 for m2 = 0, µ = Constraints: µ = 0 = µ . µ 1 (µ + µ ) , 2 µ .


Higher-spin meson exchange in e± N e± N
e ± (k ) e lepton µ1 µ2 ...µ e± (k )
J

spin-J meson (q ) ehadron µ1 µ2 ...µ

N (p)

J

N (p )

To eliminate nonphysical DsOF of higher spin meson, it has to be assumed that q µ1 lepton J = 0 = q µ1 hadron J . µ1 µ2 ...µ µ1 µ2 ...µ


Higher-spin meson exchange in e± N e± N
e ± (k ) e lepton µ1 µ2 ...µ e± (k )
J

spin-J meson (q ) ehadron µ1 µ2 ...µ

N (p) lepton J = µ1 µ2 ...µ
hadron µ1 µ2 ...µ

J

N (p )

h(J ) (µ1 Kµ2 · · · KµJ + permutations) , J M J -1 g(J JM
)M J -1

J

=

(µ1 P

µ

2

···P

µJ

+ permutations)

+

g(J )2 P MJ

µ1

· · · PµJ , h(1) = 1.

1 Kµ = (kµ + kµ ), 2

1 Pµ = (pµ + pµ ), 2


Cross section of e± N e± N
Differential cross section in the laboratory frame: d = d
± (0)a a

( , ) [1 + h( ^)Pla ( , ) + h( x)Pta ( , )] , z ^ = 1 + 2(1 + ) tan2 2
-1

=

Q2 2, 4 MN
±2 M a|

Reduced cross section:
± a = |G ± ± + |GE a |2 + |GT a |2 ,

a = p, n.


Polarization transfer coefficients: P =
a l

1-

2

1 - p

- GM a

+

1- G- 1 + Ta

- GM a

-

1+ G- 1 - Ta


,

Pta

1 = - 2 (1 - ) - a

G

- Ea

- GM a

-

1+ G- 1 - Ta

.


Cross section of e± N e± N
Generalized form factors for arbitrarily large spin J of the mesons: G
± Ma a ( , ) = FM (Q2 ) + J2 ± a GE a ( , ) = FE (Q2 ) + J2 ± GT a ( , ) =

~ ~ ~

J -1

RM ( )G
J -1

(J )

(J ) Ma

(Q2 ),

R
(J )

(J ) E

( )G
(J ) Ma

(J ) Ea

(Q2 ),

1- 1+

J -1

RT ( )G

(Q2 ),

J2

where = ~
(J )

(1 + )

1+ 1-

and Ri ( ), i = E , M , T are positive regular rational functions of [0, 1].


Higher-spin meson exchange

e±

e

±









spin-J meson

N

N


Point invariance of the spin-2 field

Point symmetry of equivalent class L ( ) of the free spin-2 field Lagrangians:
µ

µ +

-1 gµ : L 4

Fierz-Pauli

L ( )

Point invariance of the interaction Lagrangian leads to cancellation of UV divergences in loops.


spin-2­spin-1­spin-1 vertex
Interaction Lagrangian: g1 L = 3 Cµ V1µ V2 M g2 + 5 Cµ V1µ V2 M g3 + 7 Cµ µ ( V1 V2 ) M g4 + 7 Cµ µ V1 V2 + V1 V2 M g5 + 9 Cµ V1 V2µ . M Linearised Weyl tensor: 1 1 1 1 Cµ = µ - µ g + gµ + µ g - 2 2 2 2 1 + (gµ g - gµ g ) , 6 1 µ = (µ - µ - µ - µ ) 2





g

µ


Fit
600 data points on 7 polarized and unpolarized observables:
- p (p , ), p RPT (p , ) = -µ

Plp (p , ) P
p(Born) l

,

(p , )

p Re+

/e-

p (1 + ) 2 + p ( (p , ) = - p (
p

Ptp (p , ) , Plp (p , ) p , ) , p , )

Rn/p (a , ) =
n RPT (n , ) = -

d d d d

(e- n e- n) (e- p e- p)

, Ptn (n , ).

n (1 + ) Ptn (n , ) , 2 Pln (n , )

Quality of the fit: 2 /DOF 2.3.


Proton FF ratio (PT) at 2.495 GeV2
Q2 = 2.495

0.6 0.0

0.7

0.8

0.2

0.4

0.6

0.8

1.0


Results, conclusions, and plans
Effective gauge invariant vertices for the interactions of higher-spin mesons with leptons and nucleons has been written. In the case of spin-2 mesons, the general Lagrangian for the interactions with two vector fields has been constructed. The symmetry of the Lagrangian ensures mathematical consistence of the theory and cancellation of UV divergences. The model of dominance of vector and tensor mesons constrained by high-Q2 pQCD predictions is in satisfactory agreement with available experimental data on the elastic eN -scattering. The model will be extended to include both box and triangle two-photon effects.