Physics Letters B 267 ( 1991 ) 509-512 PHYSIC S LETTERS B North-Holland

Comment on Z-Z' mixing in extended gauge theories

Robert Foot ~ and Xiao-Gang He b

Department of Physics, University of Southampton, Southampton S09 5NH, UK b Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Received 24 April 1991

In gauge theories with gauge groups containing more than one U ( 1 ) gauge group, it is possible to have kinetic mixing terms for the different U( 1 ) gauge fields in the lagrangian. The physical consequences of these mixings are studied. It is shown that electro- magnetic interactions are not affected by these mixing terms, The properties of interactions mediated by massive gauge bosons are modified however. Ways of eliminating the kinetic mixing terms are also discussed.

The proper t ies o f the Z-boson are being carefully s tudied at the LEP and SLC facili t ies [ 1 ]. The exper- imenta l s tudy o f the Z-boson is one way to test the s tandard model (SM) . It is clearly impor tan t to ident i fy and s tudy possible new physics which can modi fy the proper t ies o f the Z-boson f rom the SM predict ion. The purpose o f this note is to examine the occurrence and consequences of mixing in the kinetic energy terms o f the gauge fields. We will see that in a large class o f extensions to the SM, the effects o f mix- ing in the kinet ic terms are observable, and lead to mixing o f the SM Z-boson with a Z ' -gauge boson.

The s tandard model is gauge theory with gauge group SU (3 )c® SU (2) L®U ( 1 ) r. The kinetic terms of the gauge fields are uniquely de te rmined from the gauge pr inciple

8 3 a a altl, r i°o,c=-I Z E w ..w

a = l a = l

- ¼F.~F u~ , ( 1 )

where G ~ , W ~ and F,~ are the field strength ten- sors associated with the gauge fields of S U ( 3 ) c (G~) , S U ( 2 ) L ( W u ) , and U ( 1 ) r (Bu) respectively. In par t icular note that the gauge fields B" and W~ are necessarily or thogonal since no te rm of the form

W u , F U ~ (2)

respects the gauge principle. Thus the SM does not allow for any mixing o f the gauge fields in the kinetic terms. However , the SM is not the only gauge model

consistent with present experiments . Other types o f models can have mixing in the kinetic terms leading to non-tr ivial consequences as we will see.

We note first that a mixing term can only occur p rov ided there are two or more field strength tensors F ~ , 2 F u . . . . . which are each neutral under the gauge symmetry. This only arises for the abel ian U ( 1 ) group. Thus the simplest gauge model which has mix- ing in the kinetic terms is a two U ( 1 ) model , with gauge group

S U ( 3 ) c ® S U ( 2 ) L ® U ( 1 ) r ® U ( 1 ) r , . (3)

Denote the field strength tensors of the two U(1 ) fields by F , , and F u , , then the kinetic terms take the form

~kiaetic = a F , ~ F ~ " + b F ' u , F ' U " + C F u ~ F 'u" • (4)

By rescaling the gauge fields, the kinetic energy can be expressed in the form

~ k i n e t i c ~ l b " l f f 'Pv I F v lff ' t ,uv 2 t u F f i ' t .ug

(5)

The constant a is a physical pa ramete r and can not be scaled away. The kinetic term can be diagonal ised by making the non-uni tary t ransformat ion

# = B + a B ' , B ' = ~ B ' . (6 )

The requi rement of posi t ive kinetic energy implies that I a l < 1. To define our nota t ion let us first set a = 0. Denote the covar iant der ivat ive by

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Volume 267, number 4 PHYSICS LETTERS B 26 September 1991

~ =i0# +gW3fl3 +gj B~ Y/2+g'B'~ Y'/2+... , (7)

where we have only written down the diagonal com- ponent of the SU(2)L gauge fields W3u which can mix with the other U( 1 ) gauge fields (Bu, B~,). In gen- eral, for the symmetry breaking pattern

S U ( 3 ) c ® S U ( 2 ) L ® U ( 1 ) r ® U ( 1 ) r ,

~Mz, S U ( 3 ) c ® S U ( 2 ) L ® U ( I ) r

~Mw SU(3)¢®U(I )Q

the gauge boson mass matrix takes the form

~°mass = V T M Vo , (8)

where

v o ~ = (ao, z o , z ~ , ) ,

(i0 0) M = M2o , (9)

and

Ao = cos 0 B - sin 0 W3,

Zo =sin 0B +c os 0 W 3 , t t Zo=B , (10)

with e=gs in 0 and Lorentz indicies are implicit. A non-zero 8 causes the standard Z-Z ' mixing which arises from the symmetry breaking. When a is une- qual to zero, the fields Ao, Zo and Z~ are not orthog- onal. They can be related to orthogonal fields A, Z and Z' by the non-unitary transformation

Vo=SV, (11)

where

v T = ( A , Z , Z ' )

are the orthogonal fields given by

A = cos 0 /~ - sin 0 I413,

Z = sin 0/~+ cos 0 W3,

Z' = /~ ' ,

and S is the transformation matrix

(12)

(13)

- c o s O a"

- s i n 0 a S= 0 1 ~ (14)

1 0 0 l~]-=~j~ z

The covariant derivative expressed in terms of the orthogonal fields/~, B' and W3 has the form

~u =i0u +gW3uI3 +g, Bu Y/2

Y ' / 2 - gl a r / 2 . ( 1 5 )

Note in particular that/~ couples to the same genera- tor as B, that is Y/2.

In terms of the orthogonal fields A, Z, Z' , the gauge boson mass matrix is

~"q~rnass - - VT ST MSV. ( 16 )

Note that the true mass matrix JI=STMS has only the lower block 2 × 2 submatrix non-zero. This means that the field A is massless and is the physical mass- eigenstate field, the photon. The photon couples to the same generator independent of the value of a since the photon is composed of a fixed (i.e. inde- pendent of a ) combination of /~ and I413, and the fields/~ and I413 couple to the generators Y/2 and 13 independently of a (see eq. (15) ). Thus if the theory has electric charge quantization to begin with (i.e. when a = 0) then it will retain electric charge quan- tization when a is non-zero.

The photon decouples from the mass matrix of eq. (16) leaving a 2X2 mass matrix describing the masses of the Z and Z' gauge bosons. This mass ma- trix has the form

~ = WTXTMXW, (17)

where

w T = ( z , z') ,

i +0ol

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Volume 267, number 4 PHYSICS LETTERS B 26 September 1991

Thus, the true mass matrix has the form

, J ~ = X T M X .

Evaluating this mass matrix we find that

with

a=ML,

(19)

(20)

- sin 0 c~ M2zo + 8 b=

M~, - 2 sin Oa sinZOa 2 C = l _ o t 2 + l _ a 2 8+ l_ot---------~M2o . (21)

Thus the Z and Z ' fields mix with mixing angle

2b tan 20 = - - . (22)

a - - ¢

Note that the mixing effects on the Z-boson will can- cel provided b--0. This occurs when

8 a = M~o sin 0" (23)

Of course the mixing of the kinetic terms can only cancel with the mixing due to symmetry breaking providing that I a I < 1 which follows from positivity of the kinetic energy. It is clear that Z - Z ' mixing is a dependent. In general the analysis of Z - Z ' gauge bo- sons in extended gauge models should take into ac- count the ~ dependence.

Experimentally, there is no sign of any mixing in the neutral current sector, despite the experiments at LEP and SLC [ 1 ]. It might be worthwhile to look at ways in which the mixing of the kinetic terms can be eliminated naturally. This may happen if the theory is an effective theory of a gauge theory with no more than one U ( 1 ) factor. This is because the introduc- tion of a mixing term in the effective theory is incon- sistent with the underlying symmetry of the lagran- gian. Alternatively, the existence of discrete symmetries can prevent the mixing of the Z and Z ' . To see this, consider a gauge model with gauge group

SU(3)c ®SU (2)L ® U ( 1 ) r, ® U ( 1 )Y2 (24)

with the discrete symmetry

BI~-*B2 , (25)

w h e r e B 1 and B2 are the gauge bosons of the two U( 1 )'s. To obtain a realistic model with such a gauge and discrete symmetry structure, one could define the first generation to transform under one of the U ( 1 )'s, and the second generation to transform under the other U ( 1 ), with the third to transform under both but with half of the U ( 1 ) charges, so that the diago- nal subgroup generated by Y= Y~ + I12 is the actual hypercharge. In this case the discrete symmetry ob- viously interchanges the first and second generation fermions with the third generation and Higgs doublet being neutral. Then, the kinetic terms of the U( 1 ) gauge bosons which respects this discrete symmetry have the form

~ k i n e t i c ! K ' ]~//.v 1 / iv 2,-v g7 K ' l ~ v = - - 4x l /zv a 1 - - ~ F a ~ v V 2 - - ~ - - l ~ v - - 2 •

(26)

One might think that there is mixing, however tfiis is not the case. The reason is that the linear combina- tion of B~ and B2 which couples to hypercharge is (BI+B2)/x//2 (since U ( 1 ) r is the diagonal subgroup). Rewriting the kinetic terms in terms of B = ( B l "t- B2 )//~/f2 and B ' = ( B 1 -B2)/N/~, we find

- - JI.V t t ttl; • ~inetic__aF~,vF +bFu.F (27)

Note that in this basis there is no mixing term, since under the discrete symmetry F---,F, F ' ~ - F ' . Thus the mixing terms are forbidden by the discrete sym- metry, and consequently, there is no mixing between the Z and Z' coming from the kinetic mixing. Note however that the rescaling of the gauge fields B and B' necessary to put ~kinetic into canonical form is not unphysical. This physical rescaling parameter scales the coupling constants of the B and B' U ( 1 ) sym- metries differently so that they are unequal in general (despite the existence of the discrete symmetry). This simple example illustrates that a lagrangian which possesses discrete symmetry can eliminate mixing of the Z and Z' gauge bosons, although the mixing of the kinetic terms can have other physical effects.

In the above we have discussed in some detail the physical consequences of kinetic mixing terms of the gauge bosons in theories with two U ( 1 ) factor gauge groups. The discussions can be easily generalized to theories with any number of U ( 1 ) factors. We have shown that the kinetic mixing term does not affect

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Volume 267, number 4 PHYSICS LETTERS B 26 September 1991

the generator of the photon field for theories with two U ( 1 ) factors. It is interesting to note that this is also true even with more than two U ( 1 ) groups. This can be shown as follows.

Consider a general gauge theory with many U ( 1 ) factor groups, which has after symmetry breaking only one U( 1 ) unbroken by the vacuum (the photon). Let us temporarily set the kinetic mixing terms of the gauge bosons to zero. Then, after symmetry breaking one can write down the gauge boson mass term as

~wm =BTMB , (28)

where M is the diagonalised mass matrix which has one eigenvalue equal to zero, the corresponding field Bo of the vector field BT= (Bo, ...) is identified with the photon. After the kinetic mixing terms are intro- duced, the new mass eigenstates become B'T= (B~, ...) which are related to B by B = UB', where U is a non-unitary matrix whose inverse exists. The new mass matrix M' is given by

M' = UTMU. (29)

Such transformation does not change the rank of the matrix and therefore there is also a zero mass eigen- state B3. This zero mass eigenstate is the photon field in the new theory. From eq. (29), we obtain

~. m2U, oUio=O. (30) i

Since m~ > 0 for i>0 , Uio=0 for i>0 . This implies that the photon field B~ couples to the same genera- tor of the photon Bo of the theory without kinetic mixings up to a constant Uoo. The constant Uoo can be rescaled into the coupling constant.

The kinetic mixing terms will induce mixings for massive gauge bosons in the general case of may U ( 1 )

factors, just like they have done in our two U(1 ) analysis. With more than two U ( 1 ) factors, the mix- ings are complicated due to more parameters in the kinetic mixing terms. We will not discuss them here.

In conclusion, we have discussed possible physical consequences of kinetic mixing terms in gauge theo- ries with more than one U ( 1 ) gauge group. We have shown that the generator of the photon field is not affected by the kinetic mixing terms. This implies that the new parameters introduced by the kinetic mixing terms do not cause new problems for electric charge quantization. However, the interactions of massive gauge bosons are modified when kinetic mixing terms are present. In particular, the properties of the Z-bo- son will be modified from the SM predictions. We have also shown that a lagrangian with discrete sym- metry can prevent the Z-boson from mixing with the Z ' , although in general the mixing in the kinetic terms may have other manifestations in this case.

Note added. After we had completed this paper we became aware of several related papers [ 2-4 ] which discuss neutral gauge boson mixing in the kinetic terms in other contexts.

References

[ 1 ] For a review see F. Dydak, CERN preprint CERN-PPE/91- 14, in: Proc. 25th Intern. Conf. on High energy physics (Singapore, August 1990 ), to be published.

[2] F. Del Aguila, G.D. Coughlan and M. Quiros, Nucl. Phys. B 307 (1988) 633.

[3] B. Holdom, Phys. Lett. B 166 (1986) 196; B 259 (1991) 329.

[4] M. Gasperini, Phys. Lett. B 237 (1990) 431; B 263 (1991) 267.

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