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Reproducing the Standard Model in 5D brane worlds (arXiv:1503.09074)
Mikhail Smolyakov, Igor Volobuev
Skobeltsyn Institute of Nuclear Physics, Moscow State University

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Gauge fields
The background metric: ds 2 = e d 4 xdz g
2(z )

SU (2) â U (1) gauge invariant theory in this background: S= - +g where F
a MN MN

µ dx µ dx - dz
a ,MN

2

2 F 4

F

a MN

-

2 B 4

MN

B

MN

MN

(DM H ) DN H - V (H H )
ab c

= M BN - N BM , a g DM H = M - ig Aa - i B 2M 2 B

= M Aa - N Aa + g N M

Ab Ac , MN

M

H

The orbifold symmetry conditions Aa , Bµ (x , -z ) = Aa , Bµ (x , z ), Aa , B5 (x , -z ) = -Aa , B5 (x , z ), µ µ 5 5 H (x , -z ) = H (x , z ).
Mikhail Smolyakov , Igor Volobuev Repro ducing the SM in 5D brane worlds


The vacuum solution for these fields: A
a M

0,

B

M

0,

H0

0
v (z ) 2

We retain only the four-vector components of the fields.

The physical degrees of freedom are Zµ = 1 g +g
2 2

gA3 - g B µ

µ

, Aµ =

1 g +g W
± µ 2 2

gBµ + g A

3 µ 2 µ

1 = A1 iA µ 2

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Let us consider only the quadratic part of effective action in terms of these new fields: S
eff

=

d 4 xdz - -

2 µ 2



W

+ µ

W

-

-

2 µ 4



F

µ F

2 µ + - Zµ Z + e 2 2 µ 5 Wµ 5 W 4 2 2 +e 2 µ 5 Aµ 5 A + e 2 µ 5 Zµ 5 Z 2 2 2 + g 2 2 g g + - µ Zµ Z +e 2 v 2 (z ) µ Wµ W + e 2 v 2 (z ) 4 8

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


The equations for the wave functions and the masses of the KK mo des are:
2 - mW ,n f W ,n

- 5 (e 2 5 f
2

W ,n

)+

g 2 2 2 e v (z )f 4 2

W ,n

= 0,

-

2 mZ ,n fZ ,n

g 2 + g 2 2 2 - 5 (e 5 fZ ,n ) + e v (z )fZ ,n = 0, 4 2 2 - mA,n fA,n - 5 (e 2 5 fA,n ) = 0.

The independence of the wave functions fW ,0 (z ) and fZ ,0 (z ) on the co ordinate of the extra dimension can be achieved only when v (z ) = v e ~
-

,

v is a constant. The masses of the zero mo de gauge bosons: ~ mW
,0

=

gv ~ , 2

mZ

,0

=

g 2 + g 2v ~ . 2

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Mo dification of the shapes of gauge boson wave functions has an influence on the electroweak observables: C. Csaki, J. Erlich and J. Terning, Phys. Rev. D 66 (2002) 064021 G. Burdman, Phys. Rev. D 66 (2002) 076003

For example, in the case of the Randall-Sundrum mo del such a mo dification leads to restrictions on the value of the five-dimensional energy scale, which put the theory out of the reach of the present day experiments.

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Fermions
V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 (1983) 136. S
scalar

=

d 4 xdz

1 2 M M - -v 2 4 M = 0, 1, 2, 3, 5 m z 2

22

,

m v= , Kink solution:

m 0 = t a nh

S=

¯ ¯ i M M - h d 4 xdz , L (x , z ) CL L (x )e
Mikhail Smolyakov , Igor Volobuev

µ = µ , 5 = i , i µ µ L = 0 i µ µ R = 0

5

-

hm

|z |

R (x , z ) CR R (x )e

hm

|z |

,

Repro ducing the SM in 5D brane worlds


For a nonzero mass it is necessary to take two five-dimensional spinor fields, satisfying the orbifold symmetry conditions 1 (x , -z ) = 5 1 (x , z ),

2 (x , -z ) = - 5 2 (x , z ). S.L. Dubovsky, V.A. Rubakov and P.G. Tinyakov, Phys. Rev. D 62 (2000) 105011, C. Macesanu, Int. J. Mo d. Phys. A 21 (2006) 2259, R. Casadio and A. Gruppuso, Phys. Rev. D 64 (2001) 025020

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


S=

M¯ M¯ d 4 xdz g EN i 1 N M 1 + EN i 2 N M
2

2

¯ ¯ ¯ ¯ -F1 (z )1 1 - F2 (z )2 2 - G (z ) 2 1 + 1

,

where M , N = 0, 1, 2, 3, 5, µ = µ , 5 = i 5 , M is the covariant M derivative containing the spin connection, EN is the vielbein, F1,2 (z ) and G (z ) are some functions satisfying the symmetry conditions F1,2 (-z ) = -F1,2 (z ) and G (-z ) = G (z )

G (z ) = hv (z ),

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


4D Dirac equation i µ µ - m = 0 Our case - 1 + e (5 + 2 )e (5 + 2 )1 + e 5 (e F1 ) 5
2 -e 2 (F1 (z ) + h2 v 2 (z )) 1



+ m2 = 0

1

+he 5 (e v (z )) 5 2 - he 2 v (z ) (F1 (z ) + F2 (z )) 2 = 0, - 2 + e (5 + 2 )e (5 + 2 )2 + e 5 (e F2 ) 5
5 2 2 -e 2 (F2 (z ) + h2 v 2 (z )) 2

2

+he 5 (e v (z )) 1 - he v (z ) (F1 (z ) + F2 (z )) 1 = 0.

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Possible problems (for simplicity, 0): -
2 5

-F +F +h v

2

2 2

22

- hv (F1 + F2 ) - hv

2 2 - 5 - F1 + F1 + h2 v hv (F1 + F2 ) - hv

2



(1) 1

(1) 1

= 0.

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Exception 1

F1 (z ) -F2 (z ),

5 (e v (z )) 0. ~~ i µ µ L - hv R = 0, 5 L = L , ~~ i µ µ R - hv L = 0, 5 R = -R ,

1 = Cf exp - 2 = Cf exp -



z

0 z

F1 (y )dy - 2 (z ) L (x ), F1 (y )dy - 2 (z ) R (x ),

0

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Exception 2
F1 (z ) F2 (z ) For simplicity, 0. - -
2 (1 + 2 ) + 5 (1 + 2 ) - h2 v 2 (z )(1 + 2 )

+hv (z ) 5 (1 + 2 ) = 0,

2 (1 - 2 ) + 5 (1 - 2 ) - h2 v 2 (z )(1 - 2 )

-hv (z ) 5 (1 - 2 ) = 0.

The fields 1 and 2 can be decomposed into the Kaluza-Klein mo des as 1 =
n n n n n f+ (z )L (x ) + f- (z )R (x ) , n n n n f- (z )L (x ) - f+ (z )R (x ) .
Repro ducing the SM in 5D brane worlds

2 =
n

Mikhail Smolyakov , Igor Volobuev


the function f n (z ) is a perio dic continuously differentiable solution to the equation
2 2 mn f n + 5 f n - h2 v 2 f n + hv f n = 0.

n f+ (z ) = f n (z ) + f n (-z ),

n f- (z ) = f n (z ) - f n (-z ),

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


SU (2) â U (1) gauge fields in the flat ( (z ) 0) space-time with F1 (z ) F2 (z ) 0: ¯ ¯ ^ ^ ¯ ^ d 4 xdz i 1 M DM 1 + i 2 M DM 2 - 2h 1 H 2 + h.c. SU (2) doublet: ^ 1 = SU (2) singlet: 2 . ^ DM DM
1 2



1 1

= =

g a a AM + i B 2 2 M + ig BM 2 . M - ig

M

^ 1 ,

The vacuum solution for the Higgs field: H0 with v (z ) = const.
Mikhail Smolyakov , Igor Volobuev Repro ducing the SM in 5D brane worlds

0
v (z ) 2


Free theory. L (x ) f+ (z )L (x ) + f- (z )R (x ) 2 (x , z ) = f- (z )L (x ) - f+ (z )R (x ). ^ 1 (x , z ) = S= ¯ ¯ ¯ d 4 x i L µ µ L + i µ µ - m , ,

where = L + R . In order to have the canonically normalized kinetic term of the field , the condition a2 + b 2 = 1, must be fulfilled. a2 =
2 dzf+ (z ),

b2 =

2 dzf- (z ).

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Aa (x , z ) Aa (x ), µ µ Differences with the SM:
1

Bµ (x , z ) Bµ (x );

The coupling constant of the field to the charged gauge bosons is 1 g dzf+ (z ) 2L instead of g in the SM. The axial coupling constant of the field to the neutral gauge boson Z is S gA = gAM (a2 - b 2 )
S instead of gAM in the SM.

2

a = 1, b = 0, 1 L dzf+ (z ) = 1 if f (z ) = const v (z ) = const 2 (non-flat background: v (z ) e - )
Mikhail Smolyakov , Igor Volobuev Repro ducing the SM in 5D brane worlds


Conclusions

e v (z ) = const
1

2

F1 (z ) = -F2 (z ), F1 (z ) = F2 (z ) pathologies (higher derivatives) F1 (z ) F2 (z ) deviations from the SM, no lo calization

e v (z ) const, F1 (z ) -F2 (z ) no pathologies, no deviations from the SM, lo calization, but unnatural fine-tuning in the non-flat case!

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds


Thank you for your attention!

Mikhail Smolyakov , Igor Volobuev

Repro ducing the SM in 5D brane worlds