Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2011/arbsochish.pdf Äàòà èçìåíåíèÿ: Wed Oct 5 14:18:37 2011 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:41:30 2012 Êîäèðîâêà: IBM-866

Non-perturbative effects in the electro-weak theor y versus LHC and Tevatron data
Boris A. Arbuzov SINP MSU, Moscow, Russia

Application of Bogoliubov compensation principle to the gauge electro-weak interaction. 1. Compensation equation for anomalous tree-boson interaction. 2. Four-fermion interaction of heavy quarks. ï 3. Doublet bound state L TR . 4. W-hadrons and CDF Wjj anomaly.
2 Results: g ( MW ) 0.62 and mt = 177 G eV ; ver y heavy composite Higgs scalar; the indications (CDF and LHC) for state with mass 145 G eV I = 1, J = 1 bound state of two W .
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Compensation equation for anomalous tree-boson interaction In works B.A.A., Theor. Math. Phys., 140, 1205 (2004); Teor. Mat. Fiz., 140, 367 ( 2004) . B.A. A., Phys. Atom. Nucl., 69, 1588 (2006); Yad. Fiz., 69, 1621 ( 2006) . B.A. A., M.K. Volkov and I.V. Zaitsev, Int. Journ. Mod. Phys. A, 21, 5721 ( 2006) . B.A. A., Phys. Lett. B, 656, 67 (2007). B.A. A., M.K. Volkov and I.V. Zaitsev, Int. Journ. Mod. Phys. A, 61, 51 ( 2009) . B.A. A., Eur. Phys. J. C, 61, 51 (2009). B.A. A. and I.V. Zaitsev, arXiv:1107.5164 [hep-ph] (2011). N.N. Bogoliubov compensation principle was applied to studies
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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of spontaneous generation of effective non-local interactions in renormalizable gauge theories. N.N. Bogoliubov compensation principle: N.N. Bogoliubov. Soviet Phys.-Uspekhi, 67, 236 (1959); Uspekhi Fiz. Nauk, 67, 549 (1959). N.N. Bogoliubov. Physica Suppl., 26, 1 (1960). N.N. Bogoliubov, Quasi-averages in problems of statistical mechanics. Preprint JINR D-781, (Dubna: JINR, 1961). The first application of Bogoliubov compensation principle to QFT: B.A.A., A.N. Tavkhelidze and R.N. Faustov, Doclady AN SSSR, 139, 345 ( 1961) . The main principle of the approach is to check if an effective interaction could be generated in a chosen variant of a
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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renormalizable theor y. In view of this one performs "add and subtract" procedure for the effective interaction with a form-factor. Then one assumes the presence of the effective interaction in the interaction Lagrangian and the same term with the opposite sign is assigned to the newly defined free Lagrangian. EW Lagrangian with 3 lepton and quarks doublets with gauge group SU (2).
3

L=
3


k =1

i ï ï k ² ² k - ² k ² k 2
k

g a ï + k L ² a W² k 2
kL a ² a W² q kL

L

+
(1)

+


k =1

i qk ² ² qk - ² qk ² q ï ï 2

g +q ï 2

-

1 - 4

a a W² W² ;

a a a W² = ² W - W² + g

abc

c b W² W .

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Bogoliubov procedure add í subtract.

L = L0 + L
3

i nt

; ï - mk k k +

L0 = =


k =1

i ï ï k ² ² k - ² k ² k 2

i ï qk ² ² qk - ² qk ² qk - Mk qk qk - ï ï 2 1aa G c b a W² W² + § abc W² W W² ; 4 3! g3 a a ï Li nt = ï k ² a W² k + qk ² a W² q 2 k1 = G § 3! Notation -
G 3! abc c b a W² W W² . abc

(2)
k

-
(3)

§

c b a W² W W² means presence of non-local 5

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data


vertex in the momentum space

(2 )4 G

abc

where F ( p, q, k) is a form-factor and p, ², a; q, , b; k, , c are respectfully incoming momenta, Lorentz indices and weak isotopic indices of W -bosons.

g² ( p qk - k pq) + q² k p - k² p q ) F ( p, q, k) ( p + q + k4) ( );

( g² ( q p k - p q k ) + g ( k ² p q - q ² p k ) +

Effective interaction (44) is usually called anomalous three-boson interaction Interaction constant G is connected with conventional definitions g G=- 2 . MW = - 0.016
+0.021 -0.023

(5) ;

Due to our approximation sin2 W 1 we use the same MW for
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

- 0.059 < < 0.026 (95% C . L.) . (6)

6


both charged W ‘ and neutral W 0 bosons and assume no difference in anomalous interaction for Z and , i.e. Z = = . Compensation conditions (see for details [1]) will consist in demand of full connected three-gluon vertices of the structure (4), following from Lagrangian L0 , to be zero.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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p p 0 + s0 s -p -p p s ä ç 0 æå -p

p ç + æ -p p ä ç0 s æå -p

ä s0 å

+

+

+

p p p s 0 0 s0 + s + -p -p -p

=

0

Fig.1. Diagram representation of the compensation equation. Black spot corresponds to anomalous three-gluon vertex with a form-factor. Empty circles correspond to point-like anomalous three-gluon and four-gluon vertices. Simple point corresponds to usual gauge vertex. Incoming momenta are denoted by the corresponding external lines.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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2 p2 + p2 + p2 3 1 ); F ( p1 , p2 , p3 ) = F ( 2

(7)

G2 N F ( x) = - 64 2 x + 6 GgN 24 2 3 4
x 0 x 3 x /4 Y x

Y 0

1 F (y)ydy - 12x
Y x

x 2 0

1 F (y)y dy + 6x
3 Y 0 Y x

x 0

F ( y ) y2 d y +

x2 F (y) dy - 12
x 3 x /4

F (y) dy y

GgN + 16 2

F (y) dy +

(3 x - 4y )2 (2y - 3 x ) F (y) dy + 2 ( x - 2y ) x

(5 x - 6y ) F (y) dy + ( x - 2y )
Y x

3(4y - 3 x )2 ( x2 - 4 x y + 2y2 ) F (y)dy + 8 x2 (2y - x )2
Y x

3( x2 - 2y2 ) F (y)dy + 8(2y - x )2 . (8)

5y2 - 12xy F (y) dy + 2 16x

3 x2 - 4 x y - 6y2 F (y) dy 2 16y

Here x = p2 and y = q2 , where q is an integration momentum,
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N = 2. An effective cut-off Y bounds a "low-momentum" region where our non-perturbative effects act and we consider the equation at inter val [0, Y ] under condition F (Y ) = 0 . (9)

We shall solve equation (8) by iterations. The second iterations gives the following equation 85 g N z ln z + 4 + 4 ln 2 + F (z) = 1 + 96 1 31 G15 z0 | 2 3 4
z

z

0

595 0 + 0,0,1/2,-1,-1/2 - 336 4 z z0 F (t F (t ) t dt - 3 z

2 3z

z 0

F (t ) t dt -
z z
0

(10)

dt 2z ) + 3 t

dt F (t ) ; t
10

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data


where is the Euler constant. Solution (10): 1 31 85 g N 31 0 F (z) = G15 z|1,1/2,0,-1/2,-1 - G15 z| 2 512

1/ 2 1,1/2,1/2,-1/2,-1

+
(11)

10 10 C1 G04 z|1/2, 1, -1/2, -1 + C2 G04 z|1, 1/2, -1/2, -1 .

where G
nm qp

z|

a 1 ,..., a b 1 ,..., b

q p

;

is a Meijer function. We have also conditions z0 87 g N z0 dz F (z) dz = 1+8 F0 (z) ; 32 z 0 0 F ( z0 ) = 0 .

(12) (13)

g (z0 ) = 0.60366; z0 = 9.61750; C1 = -0.035096; C2 = -0.051104. (14)
11

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data


One-loop expression for s ( p2 )
ew

6 e w ( x0 ) ; ( x) = 6 + 5 e w ( x 0 ) ln ( x / x 0 ) g (Y ) 2 ( x0 ) = = 0.0290; 4

x = p2 ;

(15)

Normalization
ew

(16)

Note that value (16) is not far from physical value ew ( MW ) = 0.0337. To compare these values properly one needs a relation connecting G and MW . For example with | g | = 0.025, ew ( MW ) = 0.0312. The experimental value 0.0337 is reached for | g | = 0.000003. For both cases values of are consistent with limitations (6). Accuracy of the approximation 10% agreement is valid for all possible values of . We shall use experimental value ew ( MW ) = 0.0337.
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Four-fermion interaction of heavy quarks In the present work we explore the analogous considerations and assume that scalar fields which substitute elementar y Higgs fields are formed by bound states of heavy quarks t , b. This possibility was proposed (1989 í 1990) in works by Y.Nambu, V.Miransky, M.Tanabashi, K.Yamawaki, W.Bardeen, C.Hill, M.Lindner [17, 18, 19] and was considered in a number of publications (see, e.g. review by M.Lindner [20]).Estimates of mass of the t -quark exceeds significantly its measured value. Assume: only the most heavy particles aquire masses, namely W -s and the t -quark while all other ones remain massless. Left doublet L = (1 + 5 )/2 § (t , b) and right singlet TR = (1 - 5 )/2 § t . Then we study a possibility of spontaneous generation [1, 2, 3, 5] of the following effective non-local
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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four-fermion interaction ï TR T L + ï Lf f L + G2 L R G3 ï ï ² L + G4 TR ² TR T ² TR . (17) ï ï L ² L L R 2 2 where , are colour indices. In this section N = 3 and a kernel term in equations: ï TR T ï = G1 L R K ½ F = ( 2 - x ln 2 ) 1 6x
ï Y x x 2 x ï Y 0

F ( y ) d y - ln

2 0

ï Y

F (y)ydy +

x 3 F ( y ) y d y + x ( ln x - ) F ( y ) y d y + ln x F (y)dy + 20 0 0 ï ï Y 3 x2 Y F ( y ) y ( ln y - ) F ( y ) d y + x ln y F ( y ) d y + dy (18) . 2 6x y x

is auxiliar y cut-off, which disappears from all expressions with all conditions for solutions be fulfilled. The compensation equation corresponds to set of diagrams at Fig.2
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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dr + dç p d + dç p ç ä p

ä r d

+

dp çä p æå d

+

ã ä + rpr r r = èr d âpè d èá

0

Fig. 2. Diagram representation of the compensation equation for the four-fermion interaction (19). Lines describe quarks. Simple point corresponds to the point-like vertex and black circle corresponds to a vertex with a form-factor.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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2 2 (( N 2 - 1) G1 + G G N G1 + G2 ( x) = 1- 2(N G + G ) 8 8 2 1 2

ï Y 0

(y)dy

+

3x x x + ln 2 - 2 4
2

ï G G + 2G ( N + 1)( G1 + G2 ) - 2(N G + G ) 32 1 2 (19)

2 2 ï G1 + G2 + 2 N G1 G2 + 2G ( N + 1)( G1 + G2 ) K ½; 94 2

G2 2 G2 1- F2 ( x) = 2 8 8 2 x 3x x 2 + ln 2 - 2 4

ï Y 0

F2 (y)dy

+

2 2 ï G1 + G2 + 2G ( G1 + G2 ( N + 1)) - 2G 32 2

2 2 ï G1 + G2 + 2G ( G1 + G2 ( N + 1) ï ï K ½ F2 ; (Y ) = F2 (Y ) = 0 ; 29 4 N G1 F1 + G2 F2 ï = G3 + G4 ; x = p2 ; y = q2 . ;G ( x) = N G1 + G2 2
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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2 2 G G = G1 + G2 + 2 N G1 G2 ;

ï ï Introducing substitution G1 = G, G2 = G and comparing the two equations (19) we get convinced, that both equations become being the same under the following condition =0
z 2 + 8 1 F2 (t ) t dt + z(ln z - 3) - 16 F2 (z) = 6z0 z ln z z(ln z - 3) z F2 (t ) F2 (t )dt + (20) dt + 20 2 0 t z0 ï ï ï z z0 F2 (t ) 1 z z0 F2 (t ) (ln t - 3) F2 (t )dt + ln t d t + dt ; 2z 2z 6z t t

ï ïï ï ( 2 + 8 ) G 2 y2 ( 2 + 8 ) G 2 Y 2 ( 2 + 8 ) G 2 x2 ;t = ; z0 = ï . z= 14 4 14 4 14 4 2 2 2
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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1 d z + dz 2 d z -1 dz
z ï 0 z ï 0
0 0

d z dz

d z dz

1 d z - dz 2

1 d z - dz 2

½
(21)

F2 (z) + z F2 (z) = 0 ;

2 + 8 F2 (t ) ; F2 (z0 ) = 0 ; ï dt = 8 t z0 ï F2 (t ) dt = 0 . F2 (t ) t dt = 0 ;
0



(22)

Boundar y conditions (22) cancellation of . 1 11 1 40 F2 (z) = G06 z|0, , , 1, - , 0 ; 22 2 2 F2 (0) = 1 = 8 . 3
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

z0 = .

(23)

18


ï Doublet bound state L TR ï L TR = a Higgs scalar. BetheíSalpeter equation (see Fig. 3) ï G ( x) = 16
2 2 G2 (y) dy + 7 4 K ½ ; 2

(24)

ï K is defined with Y = and lower limit of integration 0 being changed for the t -quark mass m2 . t

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Then we have again differential equation d -a z dz
1

d -a z dz

2

d z dz

d 1 z - dz 2

d 1 z - dz 2

½

d 1+ z - 1 (z) - z(z) = 0; a1 = - dz 2 G2 m4 1 - 1 + 64² a2 = - ; ² = 12 4 . 4 2
² ²



1 + 64² ; 4 (25)

(t ) dt = 0; t

²

(t )



t dt = 0; (26)

(t ) dt = 0; (²) = 1.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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dr

dç p = rr ppp ppp rr ppp è p èè

ä r+

rpr ã rr pá è âè è rpr r + èr èpè

+

Fig. 3. Diagram representation of state of heavy quarks. Double line describes a gluon. Black circle notations are

the Bethe-Salpeter equation for a bound represent the bound state and dotted line corresponds to BS wave function. Other the same as at Fig.2.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Solution:
50 30 C3 G06 (z|1/2, 1/2, 1, a1 , a2 , 0) + C4 G06 (z| a1, a2 , 0, 1/2, 1, 1/2) ;

30 50 (z) = C1 G06 (z| a1, a2 , 1/2, 1/2, 1, 0) + C2 G06 (z|0, 1/2, 1, a1 , a2 , 1/2

We define interaction of the doublet with heavy quarks L


ï ï = g ( L TR + TR L ) ;

(28)

Perturbative method m2 I5 m2 = - t ; ² I2




I2 =

²

( z )2 d z ; z
z ²

(29)

I5 =

²

2 (16 s (z) - g ) (z) dz 16 z

(t ) dt . t (30)
22

g mt = ; 2

= 246.2 G eV .

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data


Additional contribution to t mass (Fig. 4). çä çä p + p p rp p + r r æå æå æå pp pppp d d p d çä

Fig. 4. Diagram representation of additional contribution to the t-quark mass. Dotted lines represent gluons. Other notations the same as at Fig. 2.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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g g mt = + M = . 2 f2


(31)

M = - 4 m


t

²

F2 (z) dz z

²

s (z) F2 (z) dz - 2 z
t

(32)
-
4 7

4
²

mt (z) F2 (z) dz ; mt (z) = m z

7 s ( ² ) z 1+ ln 8 ²

.



f =1+4

²

F2 (z) dz z

²

s (z) F2 (z) dz +4 2 z

²

mt (z) F2 (z) dz . mt z (33)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Strong coupling s (z): 7 s ( ² ) z s (z) = s (²) 1 + ln 8 ² s ( MZ ) = 0.1184 ‘ 0.0007. Tachyon state . Higgs mechanism L
4

-1

.

s (²) = 0.108 ;

(34)

= ( )2 .

(35)

=

3g 16

4 2



I4 ;

I4 =

²

( z )4 d z . z

(36)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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From 2 = - m2 / and M 16 m2 I5 t 2 = ; 4 3 g ² I2 I4

2 H

= - 2 m2 we have
2 H

M

2 m2 I5 = . t ² I2

(37)

From (31) and (37) we have useful relation 16 I5 . 2= 2 f 2 ² I I 3 g 24 g from a normalization condition (Fig. 5) 3g 32
2 2

(38)

s (²) 2 I2 + I22 + 2 I6 4
²

= 1;


(39)
z ²

I22 =

(t ) dt ; t

I6 =

²

(z) dz zz

(t ) dt . t

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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1

=

r

è r é

+

r

pp pp

è r é

ï Fig. 5. Diagrams for normalization condition for H L tR -vertex. Notations are the same as at Figs. 2 - 4.

For six parameters ², g , , mt , M H , f we have five relations (31, 37, 38, 40) and MW gw = ; 2 (40)

where gw is g at W mass (by usual RG evolution) expression (15) from value g at Y (14). MW as an input. Thus MW = 80.4 G eV ; = 246.2 G eV . (41)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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² = 4.0675 10

-12

;

f = 2.034 ; M
H

g



= 2.074 ;

(42)

mt = 177.0 G eV ;

= 1803 G eV .

The most important result here is the t -quark mass, which is close to experimental value Mt = 173.3 ‘ 1.1 G eV [22]. Really, the main difficulty of composite Higgs models [17, 18, 19, 20] consists in too large mt . Indeed the definition of g in such models leads to g 3 and thus mt 500 G eV . In the present work we have all parameters, including inportant parameter f , being defined by selfconsistent set of equations and the unique solution gives results (42), which for mt is quite satisfactor y. The large value for M H seems to contradict to upper limit for this mass, which follows from considerations of Landau pole in the 4 theor y. Emphasize, that this limit corresponds to the local theor y and in our case of composite scalar fields is not relevant.
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Such large mass of H means, of course, ver y large width of H
H

= 3784 G eV ;

B R( H Z Z ) = 25.6% ;

B R( H W

W - ) = 51.4% ; ï B R( H t t ), = 23.0% .

+

(43)

Thus our approach predicts, that unfortunately quest for Higgs particle at LHC will give negative result. Maybe one could succeed in registration of slight increasing of cross-sections ï p + p W + + W - + X, p + p Z + Z + X, p + p t + t + X in region of invariant masses of two heavy particles 1 TeV < M12 < 3 TeV . As a matter of fact the most recent LHC data (SMS PAS-HIG-11-022, ATLAS-CONF-2011-135) do not find the SM Higgs in wide inter val up to 466 GeV (see also recent ATLAS result [21]). W-hadrons and CDF Wjj anomaly

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Triple interaction G c b a - § abc W² W W² ; 3! 3 W² = cos W Z² + sin W A (44)
²

;

We know form-factor F ( pi ). Effective dimensionless coupling g
ef f

g p2 = . 2 MW

(45)

In QCD boundar y of a strong interaction (non-perturbative region) is around 600 M eV where s 0.5 that is coupling gs = 4 s = 2.5. So we have to equate ge f f (45) to this value and define the typical value pt y p that gives p
ty p

= MW

ge f f 650 G eV ; g

(46)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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is maximal value from limitation (6). The lightest hadron í mass 140 M eV for typical scale 600 M eV in QCD and for pt y p (46) mass of the lightest W-hadron M
mi n

p t y p M = 150 G eV ; 600 M eV

(47)

The excess detected in work [24] is situated just in this region. So one might tr y to consider interpretation of effect [24] as a manifestation of a W -hadron. Some indications for state with the same mass at LHC are discussed in work [25](1109.0919 [hep-ph]). Assume this CDF excess being due to bound state X of two W with I = 1, J = 1 GX 2
abc b a W² W X c ²

;

(48)
31

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data


is a Bethe-Salpeter wave function of the bound state. T(Fig. 6)


p
u

-p

=

d u d

p

+

-p

f u f

p

+

-p p

d u p

p

+

-p

p u d

-p

Fig.6 Diagram representation of Bethe-Salpeter equation for W-W bound state. Black spot corresponds to BS wave function. Empty circles correspond to point-like anomalous three-gluon vertex (44), double circle í XWW vertex (48). Simple point í usual gauge triple W interaction. Double line í the bound state
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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X , simple line í W. Incoming momenta are denoted by the corresponding external lines. For small mass MX of state X we expand the kernel of the 2 2 equation in powers of MW and MX and obtain the following equation

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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G2 + G ( x) = 32 2 1 6x 3 8x x 8
x 0 3

2 X 0

Y0

G2 + G (y)ydy - 32 2
Y0 x 2

2 X

1 12x

x 2 0

( y ) y3 d y -
Y0 0

x 2 (y)y dy - 6
x 3

x2 (y)dy + 12
x 2

Y0 x

gG (y) dy + y 4 2

(y)dy -

0 Y0

7 (y)y dy + 8x
x

x 2

x2 (y) dy - y 8
x 0 2

Y0

x 1 (y)ydy + (y)y dy - 2x 0 0 Y0 ï2 (y) ²G dy) - (y)dy - 2 y 0 x

1 12x

1 (y)y dy + 6x

x2 - 12 1 64x

Y0 x x

x Y0 (y) (y)ydy + dy 6x y 0 ï 2 1 Y0 (y) G 1 dy - (y)dy - y2 24 0 192x3

(49)
x 0

( y ) y3 d y +

Y0 (y ) x Y0 (y) x3 (y)ydy + - dy . 3 64 x y 192 x y 0 ï2 ï2 G MW G MX 2 ï ²= ; = ; G = G 2 + GX . 16 2 16 2

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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For interaction (44) Y0 is already defined. Substitutions
2 ( G 2 + GX ) x2 , z= 2 512 2 ( G 2 + GX ) y 2 t= ; 2 512

(50)

zero approximation: 21 00 (z) = G15 z| 2
0 1,0,1/2,-1/2,-1

.

(51)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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2z 0 ( 0 ( z ) = I N H - 3z 0 4 z z0 0 ( t ) 2 z z0 dt - 3 3z z t 2 g + INH = 1 - z 8 21 G15 z0 | 2 1=8 M
0 0 0,0,1/2,-1/2,-1

4 t )t dt + 3z 0 ( y ) dy ; y

z 0

0 ( t )



t dt + (52)

z0

0 ( t ) d t - ; ²

8² ln z + 4 - 3 4 2 g 68 ² +z + - 48 9 2 z 2 g 2 - 16 ² + 3 0

+ 4 ln 2 +
25 ; 32
0

00 (t ) dt t (53)

X

= MW

MW = 80.4 G eV .

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Solution of (52) 21 0 0 ( z ) = G15 z|1,0,1/2,-1/2, 2 21 C1 G15 z|1/2,1/2,1,-1/2,-1 + 1/ 2
10 C3 G04 - z| 1,1/2,-1/2,-1

-1

+

20 C2 G04 z|

+

(54)

1,1/2,-1/2,-1

.

"Experimental" M

X

= 145 G eV
-11

C3 = 1.27962 10 = 0.0074995 ; ² = 0.002305 .

C1 = - 0.015282 ;

;

C2 = -3.26512 ; g = 0.03932 ;

(55)

z0 = 2627.975 ;

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Physical parameters: 0.0099 G= ; 2 MW M
X 2 G MW = - 0.0152 ; =- g 0.1639 . = 145 G eV ; | GX | = 2 MW

(56)

(56) agrees with restrictions (6). Additional solutions for "radial excitations" Xi M M M
X
1

= 180.7 G eV ; = 205.1 G eV ; = 244.2 G eV ;

X

2

X

3

0.0628 | G X1 | = ; 2 MW 0.1155 . | G X2 | = 2 MW 0.1837 . | G X3 | = 2 MW

(57)

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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X

‘ 1,2,3 0 ( X1 0 ( X2 0 ( X3

W

‘

+ ( Z, ) ;

) = 0.0086 G eV ) = 0.126 G eV ; ) = 1.26 G eV ;
+ 1 + 2 + 3

B R( X B R( X B R( X

W + Z) =

W + Z) =

W + Z) =

0 W+ + W- ; 1,2,3 ‘ ; ( X1 ) = 0.0051 G eV ; ‘ ( X2 ) = 0.083 G eV ; ‘ ( X2 ) = 0.89 G eV ; + 0.44; B R( X1 W ) = 0. + 0.80; B R( X2 W ) = 0. + 0.91; B R( X3 W ) = 0.

X

(58)

56. 20. 09.

X interact with fermion doublets L (Fig. 7).

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

39


q qu q

q q

qq qq

Fig. 7. Diagram for vertex X q q. Dotted line í W, double line í bound state X , ï simple line í a quark. Black spot í the X W W BS wave function.
aï = g X X L a L ; 2 X
z0

L

X

(59)

g 2 GX M gX = 64 2

²

2

0 ( t ) dt = 0.00067 . t

Calculations of decay parameters and cross-sections use CompHEP package [32], each quark jet. X
‘

W

‘

+ ( 96%);

X 0 , j j ( 71%) ;

(60)

X a necessarily interacts with gauge field W a with the same
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

40


coupling g
abc ²

( p, q, k ) = g

abc

( p, q, k ) ( g k ² - g ² k ) +
² p

g ( p, q, k ) g ² ( q - p ) - g q ² + g

;

(61)

i í formfactors, in the present approximation = 0, for TEVATRON i 1. Estimates for cross-sections for energy s = 1960 G eV ( p p W ‘ X 0 + ...) = 1.84 pb ; ï (pp W ï
‘

X

+ ...) = 2.69 pb ;
(62)

(pp Z X ï (pp X ï


‘

+ ...) = 1.32 pb ; + ...) = 0.33 pb ; + ...) = 0.24 pb .
‘

ï (pp X0 X

‘

X

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

41


Branching rations (60) (pp W ï (pp W ï
‘

+ + 2 j + ...) = 0.24 pb ; + 2 j + ...) = 1.43 pb ;
(63)

‘

( p p Z + 2 j + ...) < 0.06 pb ; ï

Total cross-section for W j j + W j j (1.67 pb) occurs to be smaller than result [24] (W j j) = 4.0 ‘ 1.2 pb small value for Z j j production agrees with data. Results of D0 [27] (W j j) < 1.9 pb (95% C.L.). Our result (63) does not contradict both.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Pair weak boson production additional contribution ( p p W + W - ) + ( p p ZW ‘ ) = 3.2 pb; ï ï
+

(W

( (W + W - ) + ( Z W ‘ ) )
Data [26] at TEVATRON.

W - ) + ( Z W ‘ ) = 17.4 ‘ 3.3 pb ;
SM

(64)

= 15.1 ‘ 0.9 pb ;

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

43


Predictions for Xi inclusive production at TEVATRON (p p W ï (p p W ï
- + + + X1 ) = 0.30 pb;
0 + X1 ) = 0.21 pb;

+ ( p p Z + X1 ) = 0.15 pb; ï + ( p p + X1 ) = 0.12 pb; ï

+ ( p p Z + X2 ) = 0.42 pb; ï + ( p p + X2 ) = 0.31 pb; ï

(p p W ï

(p p W ï

-

+ + X2 ) = 0.87 pb;

+

0 + X2 ) = 0.80 pb;

(65)

+ ( p p Z + X3 ) = 0.81 pb; ï + ( p p + X3 ) = 0.52 pb; ï

(p p W ï

(p p W ï

-

+ + X3 ) = 1.69 pb;

+

0 + X3 ) = 1.14 pb;

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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At LHC promising process p + p W ‘ + + ... (see e,g.1106.2829 [hep-ex]. Calculation of cross sections needs extensive work. The results of the last section are obtained by B.A. A. and I.V. Zaitsev. For calculations CompHEP package [32] was used.

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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Conclusion To conclude we would emphasize, that albeit we discuss quite unusual effects, we do not deal with something beyond the Standard Model. We are just in the framework of the Standard Model. What makes difference with usual results are non-perturbative non-trivial solutions of compensation equations. With the present results we would draw attention to two important points. Firstly, the unique determination of gauge electro-weak coupling constant g ( MW ) and calculation of the t - quark mass in close agreement with experimental values. These results strengthen the confidence in the correctness of applicability of Bogoliubov compensation approach to the principal problems of elementar y particles theor y. Secondly, we have seen, that non-perturbative contributions lead to prediction of experimental effects which are investigated at
B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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LHC and TEVATRON. These predictions at least do not contradict to the totality of data. More than that, there are some indications on agreement of several important effects with the predictions (the almost proved absence of Higgs scalar in the most popular region, a possible W W -bound state).

B.A. Arbuzov: Non-perturbative effects in EW versus LHC and Tevatron data

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[21] ATLAS Collaboration, arXiv: 1109.3357 [hep-ex] (2011). [22] Tevatron Electroweak Working Group (CDF and D0 Collaborations), arXiv: 1007.3178 [hep-ex] (2010). [23] B.A. Arbuzov and I.V. Zaitsev, arXiv:1107.5164 [hep-ph] (2011). [24] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett., 106: 171801 (2011); arXiv: 1104.0699[hep-ex] (2011). [25] R. Foot, A. Kobakhidze, R.R. Volkas, arXiv:1109.0919 [hep-ph] (2011). [26] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D, 82: 112001 (2010). [27] V.M. Abazov et al. (D0 Collaboration), arXiv: 1106.1921 [hep-ex] (2011). [28] V.M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett., 102: 231801, 2009; arXiv: 0901.1887 [hep-ex] (2009). [29] The CMS Collaboration, arXiv:1102.5429 [hep-ex] (2011). [30] D. Majuder (on behalf of the CMS Collaboration), arXiv: 1106.2829 [hep-ex] (2011).
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[31] J.M. Campbell and R.K. Ellis, Phys. Rev. D, 60: 113006 (1999). [32] E.E. Boos et al. (CompHEP Collaboration), Nucl. Instr. Meth. A 534, 250 (2004).

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