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The XXII International Workshop High Energy Physics and Quantum Field Theory

Similarity and Some Differences in Radion and Higgs Boson Production and Decay Processes Involving Off-Shell Fermions
E.Boos, S.Keizerov, E.Rahmetov, and K.Svirina SINP MSU, Moscow, Russia

June 24 July 1, 2015 Samara, Russia

Stabilized Randall-Sundrum model as one of possible extensions of SM

·

brane world model

·
· ·

one extra (5th) dimension
two branes interacting with gravitation and the real scalar field the hierarchy problem is solved
ds 2 e
2 A ( y )

dx dx dy 2 ( x, y ) ( y )

MN

( y )dx M dx

N

· · ·

the SM fields are localized on one of the branes the stabilizing scalar field and gravitation radion is the lowest Kaluza-Klein mode of the 5D scalar field appearing from the fluctuations of the metric component corresponding to the extra dimension

2

Radion couples to the trace of the energymomentum tensor of SM

r ( x) L T , r
( g s ) ab (e) T GGab F F 2gs 2e

r ( x) - the radion field, r - dimensional
scale parameter,

2 2 mZ Z Z 2mWWW

f

m f ff ...

(anomaly terms are included)
The fermion part of Lagrangian for on-shell fermions is the same as for the Higgs boson (with the replacement r ): r

L

f

m f ff

r

But for the case of off-shell fermions it is different:

L
where

f

r 3i 2 ( D f ) f f ( D f ) 4m f ff ... r

D are the SM covariant derivatives

3

The radion-strahlung as a simple example me 0

M1 2iC e ( p2 )e( p1 ) 3 M 2 iC e ( p2 ) k 2 q M 3 iC e ( p2 ) 2 q M 4 3iC e ( p2 )e

3 2 ( p1

1 2 C M Z ( pZ )r ( pr ) 2 2 p MZ k p2 2 e ( p1 ) ( pZ )r ( pr ) 3 iC e 2 k q p1 e( p1 ) ( pZ )r ( pr ) 3 iC e 2 3 iC e ) ( pZ )r ( pr )

1 e r 2 sin W cos W

( p2 )e( p1 ) ( pZ )r ( pr )

( p2 )e( p1 ) ( pZ )r ( pr )

Using the Dirac equation one can see

p j e( p j ) 0

and
2

q

q
2

( p2 )e( p1 ) ( pZ )r ( pr )

q

2

1

33 30 22

i.e. M M 1 and the radion-strahlung process is absolutely the same as the Higgs-strahlung process (with the replacements mr mh and r )

M2 M3 M 4 0

4

The observed cancellation follows from the structure of the fermion current with the emission of the radion and a number of gauge bosons
1) One gauge boson:

Let's rewrite the fermion-radion vertex in the following way

i 3 pout pin r 2 i 3 1 2 S ( pin ) r

i 3 4m f 2 pout m f r 3 S 1 ( pout ) m f , where S 2

3 pin m f m f 2 p mf is the propagator. ( p) 2 2 p mf
uin ( pin )

3 D1 iCuout ( pout ) S (k1 ) S 1 (k1 ) S 1 ( pin ) m f 2 3 D2 iCuout ( pout ) S 1 ( pout ) S 1 (q1 ) m f S (q1 ) 2 D3 i3Cuout ( pout )uin ( pin ) D D D ~
1 2 3

uin ( pin )

m

f

Higgs-like contribution only!

5

2) N gauge bosons:

( N 1) vertices

M

N vector bosons

M0

l 1

N

M l M l

N vertices

6

3 1 2 S (ql ) j 1 N S (k for l 1, , N 1; j l 1 3 M 0 ~ i 2 N 1 f out ( pout ) S 1 ( pout ) S 1 (k0 ) m fout 2 Ml ~ i
2 N 1 l j

f out ( pout )

j S (q j )

S 1 (kl ) m fl j j 1 ) j f in ( pin N j S (k j 1 ) j j 1

) f in ( pin )

MN ~ i

2 N 1

f out ( pout )

j 1

N

j S (q j )
j j

3 1 S (qN ) S 1 ( pin ) m fin fin ( pin ) 2

~i Ml f out ( pout ) for l 2, , N 1;
2 N 1

j 1

l 1

l 3 l j S (q j ) S (k
j 1

j l 1

N

S (k

j 1

) j fin ( pin )
j

M 1 M N

N 2 N 1 1 ~i f out ( pout ) 31 j 2 N 1 j 2 N 1 ~i f out ( pout ) j S (q j j 1

N ) 3 N fin ( pin )

j ) j fin ( pin )

i

2 N 1

and

i

2 N 1

(1)
7

Using the Dirac equation for in and out fermion states and one can get 1 1 H

S 1 (q j ) S (q j ) 1

M l M l M l M l1 2 2 H H where M lH , M 0 , M N are the 1 Higgs-like contributions which M 0 M 0H M 1 2 are proportional to the fermion 1 masses: H MN MN MN 2 l j N H 2 N 1 j m fl S (k j 1 ) j fin ( pin ) Ml ~ i f out ( pout ) j S (q j ) j 1 j l 1 N H 2 N 1 j M0 ~ i f out ( pout ) m fout S (k j 1 ) j fin ( pin ) j 1 N j H 2 N 1 MN ~ i f out ( pout ) j S (q j ) m fin f in ( pin ) j 1

l 0

N

Ml

l 1

N

M l

l 0

N

M

H l

the sum of all the contributions leads to only the Higgs-like type of the contribution and all the other parts are canceled out.
8

Generalization to the loop case
The boson lines in the diagrams could correspond to real particles or virtual propagators in the loops. In the case of a fermion loop there is an additional fermion propagator instead of the external spinors at the tree level.

Ml ~ i
for

2 N 1

Tr l 1,

j S (q j ) j 1 , N 1;
l j

3 1 1 2 S (ql ) S (kl ) m fl

j l 1

N

S (k

j 1

) j S ( p )
j

MN ~ i

2 N 1

for

~ i Tr Ml l 2, , N 1;
2 N 1

Tr

j 1

N

j S (q j )
j l 1 j 1

3 1 1 S (qN ) S ( pin ) m S ( p) 2 N j l j j S (q j ) 3l S (k j 1 ) j S ( p) j l 1

~ i Tr 31 M1 1 N 1 j 2 N 1 M N ~ i Tr j 1
2 N 1

j

j 2

N

S (k

j

S (q j )

) j S ( p ) 1 N 3 N S ( p )
j

9

In the same manner as in the previous paragraph one can get

1 Ml M Ml 2 1 H M N M N M N 2
H l

l 1

N

Ml

l 1 N l 1

N 1

1 M l1 2 1 M 1 2

Ml M N

l 1

N

M
H l

l 1

N

M l

Ml

l 1

N

M l

l 1

N

M

H l

which demonstrates that all the contributions except for the Higgs-like type are canceled out in the case of a fermion loop too. We have shown that the additional fermion-radion terms in the interaction Lagrangian do not alter any production and decay properties of a single radion compared to those of the Higgs boson. Let's now consider the case of associated Higgs boson - radion production.
10

Associated Higgs boson - radion production

k m M 1 v ( p1 )i h( ph )i 2 v k m
r

m

f

f 2 f

i r

3 k m 2

f

p2 m f m f r ( pr )u s ( p2 )

k m f 1 mf 3 iv ( p1 ) r ( pr )u s ( p2 )h( ph ) m f 2 2 r v 2 k m f k m f m f s i 3 M 2 v r ( p1 ) p1 m f k m f m f r ( pr )i 2 i u ( p2 )h( ph ) 2 r 2 k mf v
r

1 mf 3 iv ( p1 ) m f r v 2 i 4m f M 3 v r ( p1 ) h( ph )r r v im f M 4 v r ( p1 ) v
r

k mf r ( pr )u s ( p2 )h( ph ) 2 2 k mf
( pr )u s ( p2 ) iv r ( p1 ) 1 mf r v

4

r ( pr )u s ( p2 )h( ph )
11

u s ( p2 )

i i 2kh ph 4mh2 r ( pr )h( ph ) 2 kh 2 mh r

Using the simple kinematics computation

kh ph pr ph kh pr 2kh ph 2( ph pr ) ph ( ph pr ) 2 p 2 h p 2 r k
one can get
2 (2kh ph ) 4mh k 2 2 kh mh 2 h
2 h

p 2h p 2r k

2 h

2 mh mr2

2 2 2 mh mr2 4mh mr2 2mh 1 2 2 2 2 kh mh kh mh

and rewrite M 4 as follows

2 mr2 2mh M 4 i v r ( pr )h( ph )( p1 ) 1 2 2 rv kh mh

m

f

r

s u ( p2 )
s u ( p2 )

So the sum of the amplitudes yields

k mf k m M rh i r ( pr )h( ph )v ( p1 ) m f 2 mf 2 2 rv k m k mf m
f r

f 2 f

2 33 mr2 2mh 4 1 2 2 22 kh mh

Comparing this with the result for the same process with double Higgs production

M

hh

k mf k m i h( ph1 )h( ph 2 )v ( p1 ) m f 2 mf 2 v k mf 2 k m m
f 2 r

f 2 f

2 3mh 2 2 kh mh

s u ( p2 )

one can see the similarity (with the replacements

mr mh and r )
12

gg rh

gg hh
13

All the amplitudes have the following structure
2 gc d dl M i 2 ( p1 ) ( p2 ) h(k1 )h(k2 ) X i p1 , p2 , k1 , k2 v ( 2 ) d 2 gc d dl Mi ( p1 ) ( p2 ) h(k1 )r (k2 ) X i p1 , p2 , k1, k v r ( 2 ) d

2

- for gg hh

i 1, 4

m

- for gg rh
q1 q2 q3 3 3 S j Sl m q 4 2 2 q5 0

i 1, 7

where

X

1

Sp S11 S 2 1 2,3 S

1 3

( m) S

1 4

X
X

2
3

Sp S11 S

1 2

(m) S5 1 5, 4 S

1 4
1 4

jl

Sp S11 ( m) S6 1 S5 15, 4 S

p2 p2 k2 p2 k2 k1 p2 k1

q6 k1

X X
X X
5

4

Sp S11 S
2

1 2

( m)S

1 4 1 4

D 1
1 4

S j ( l qj ) m D (k1 k2 )2 m
2

Sp S11 S

1

(4m) S

6 7

Sp S11 (3 ) S31 (m) S Sp (3 ) S11 S
1 2

14

( m) S

1 5

After simple transformations one can get

X
X X
X
1

1
2 3

m 2 Sp S11 S 2 1S31S
m 2 Sp S11 S 2 1S51S m 2 Sp S11S6 1 S5 1S
X
5

1 4
1 4 1 4

3 X 8 3 X 8 1 X 2

5
5 6

1 X 2 1 X 2 1 X 2

6
7 7

So the sum comes up to

X

2

X

3

X
1

m2 Sp S11 S2 1S31S
2 ( k1 k2 ) k1 2m

4

1 X5 6 7 4 m2 Sp S11 S2 1S51S

X

1 4

m 2 Sp S11S6 1 S51S

1 4

Next one can rewrite the h3-vertex in the following way
2 H

2 2 (k1 k2 ) 2 k12 k2 2 4mH (k1 k 2 ) 2 mH mr 2 2m

2 H

and multiply it with the reverse propagator As a result one has

mr 2 2mH 2 D 1 2 (k1 k2 )2 mH
1

X

4

1 X 4

5

mSp S S2 S
1 1 1

1 4

mr 2 2mH 2 (k k ) 2 m2 1 2 H

15

Finally the sum of all X

i

for ggrh yields

i 1

7

X

i

m 2 Sp S11 S2 1S5 1S mr 2 2mH 2 (k k ) 2 m 2 1 2 H
1 4 1 4

m 2 Sp S11 S2 1S31S mSp S11 S2 1S
1 4

m 2 Sp S11S6 1 S5 1S

1 4

The same expression for gghh is
m 2 Sp S11 S2 1S31S mSp S11 S2 1S
1 4

m 2 Sp S11 S2 1S5 1S 4 3mH 2 (k k ) 2 m 2 1 2 H
1

1 4

m 2 Sp S11S6 1 S5 1S

1 4

Thus we get that these two processes coincide up to the constants (masses, vacuum expectation values) in the case of the anomalies being ignored.

16

Conclusion
We have shown several examples of the radion-Higgs similarity in single and associated production processes. The same result can follow from the model without radion but with the modified factor in the Higgs backreaction. In the case of mr mH this model can't be distinguished from SM.
2 2 1 1 2 2 mH 3 1 mH L h h mH h h h 2 2 2 2v 8v 4

where

1 - for SM, 2 mr2 mH 1 2 3mH

17