Geometry of Higgs fields

Up to now we have constructed general nonunitary gauge theories in ar-bitrary dimensions. To obtain theories being more close to the standard model, we specialize to a 5-dimensional theory with a finite lattice describ-ing the extra dimension. Furthermore we assume that only the gauge fields propagating along the extra dimension are nonunitary.

The model lives on the discrete manifold Z⁴×N, where

N ={0, . . . , N −1}shall be considered as a subset ofZ equipped with the induced differential calculus. Z⁴×N may be viewed as a N-brane system, where the N transverse lattice slices are the branes. Recently such multi-brane systems have received considerably interest. It was shown that they can be used to study extra-dimensional extensions of the standard model.

Note that the choice Z⁴×N corresponds to a model with free boundary conditions, known as aliphatic model. Another possibility is to consider

6.3. Geometry of Higgs fields 87

Z⁴×ZN. This choice leads to a periodic model in which the zeroth and Nth brane are linked together with an additional nonunitary parallel transporter.

To deal with a theory living on Z⁴ ×N it is convenient to change our notation slightly. Letx = (x^µ)∈ Z⁴, µ= 0, . . .3 and define xi :=x+iˆ5 ∈ Z⁴×N, i= 0,1, . . . N−1.

Z⁴ ×N is a symmetric graph, but is no longer a symmetric lattice.

In particular, the 1-forms e^x0,xN−1, e^{xN−1,xN−1+5} and e^x0,x0−5 vanish. In contrast,Z⁴×ZN corresponds to a reductione^xy 6= 0⇔y=x+µ modN or y=x+µ modN. Therefore in this case the 1-forms e^x0,xN−1, e^xN−1,x0 exist.

As a consequence of the vanishing “boundary 1-forms”, the correspond-ing field strength of the N-brane system has to be modified sligthly. Uscorrespond-ing the additional constraint

e^xi+µ+5,xi = 0⇒e^{xi+µ+5,xi+µ,xi} =−e^{xi+µ+5,xi+5,xi}, (6.33) the calculation of the field strength is very similar to the symmetric lattice case

Starting again from the general formula an easy but tedious calculation yields²

F^T = 1 2

N

X

i=0

X

x,µ,⁰ν,

e^{xi+µ+0ν,xi+µ,xi}F^U_µ,0ν(x_i) +

+1 2

N−1

X

i=1

X

x,µ,⁰ν,

e^{xi+µ+05,xi+µ,xi}D^U_µT⁰₅(x_i) + +1

2 X

x,µ

e^{x0+µ+5,x0+µ,x0}D^U_µT₅(x₀) + +1

2 X

x,µ

e^{xN+µ−5,xN+µ,xN}D^U_µT−5(x_N) +

+1 2

N−2

X

i=2

X

x,,⁰,

e^{xi+5+05,xi+5,xi}F^U_5,05(x_i) +

+ X

i=0,N−1

X

x

e^xi,xi+5,xi

T−5(xi+ 5)T+5(xi)−1

+ X

i=1,N

X

x,µ,⁰ν,

e^xi,xi−5,xi

T+5(xi+ 5)T−5(xi)−1

(6.34)

Note that by assumption we haveF^T_µ,⁰_ν =F^U_µ,⁰_ν.At this point it is worth digressing a little and reviewing the work of C.T. Hill and others.

Recently there has been considerable interest in theories of extra dimen-sions emerging not far from the weak scale. The motivation arises from

2For later convenience we consider a (N+1)-brane system.

string theories which admit an arbitrary hierarchy between the compactifi-cation scale of the extra dimensions and the fundamental string scale.

Extra dimensions can not be observed directly, but rather they appear in accelerator experiments as new particles, which are known as Kaluza-Klein (KK) modes. These are the excited modes of existing fields that propagate in the compact extra dimension.

As KK-modes begin to appear in accelerator experiments, one may ask how are they to be described in an effective 4-dimensional Lagrangian, with-out an a priori knowledge of the existence of the extra dimensions themselves.

Asking this question has led to a new class of KK models known as decon-struction. Independently, the ”Harvard group” of Arkani-Hamed, Cohen and Georgi [4, 5, 3] and the ”Fermilab group” of Hill, Pokorski, Wang and Cheng [20, 8, 9] have employed a lattice to describe the extra dimensions.

More precisely, the 3+1 dimensions of space-time are considered to be con-tinuos, while the extra compact dimensions are latticized. This construction is known as a transverse lattice [7]. By using the lattice technique, the ex-tra dimensions can be integrated out and so one obtains a gauge invariant effective 4-dimensional Lagrangian describing KK-modes.

To get a flavor for deconstruction, let us consider as an example an extension of QCD with N + 1 gauge groupsSU(3)i. The 3+1 dimensional Lagrangian is

L=−1 4

N

X

i=0

F_iµν^a F^iaµν+tr

N

X

n=1

D_µΦ^†_iD^µΦ_i (6.35) There are N Higgs fields Φ_iand the kth field transforms as an ( ¯3_i,3i−1) rep-resentation. The covariant derivative is defined asD_µ=∂_µ+igPN

i=0A^a_iµL^a_i. g is a dimensionless gauge coupling constant being common to all of the SU(3)_i gauge groups. L^a_i are the generators of the ith SU(3)_i, where a de-notes the color index. One has [L_i, L_j] = 0 fori6=j and when the covariant derivative acts upon Φi one has a commutator of the gauge part with Φi, i.e. L^a†_i acts on the left andL^a_i−1 on the right.

By introducing a potential for each Higgs field by hand, they develop a vacuum expectation value v and may be parameterized as

Φ_i →vexp(φ^a_iL^a_i/v). (6.36) As a result, the kinetic terms of the Higgs fields lead to a mass matrix for the gauge fields

1 2v²g²

N

X

i=1

(A^a_(i−1)µ−A^a_iµ)² (6.37) By diagonalization one obtains the eigenvalues

M_n= 2vgsin

πn 2(N + 1)

, n= 0, . . . , N (6.38)

6.3. Geometry of Higgs fields 89

Consequently, for smallnone is faced with a KK tower of masses given by Mn≈ gvπn

N + 1, nN, (6.39)

wheren = 0 corresponds to the zero-mode gluon. In order to match on to the spectrum of the KK modes, one has to require

gvπ N+ 1 = π

R, (6.40)

where R is the size of the compactified extra dimension. In this way, the 3+1 dimensional theory given by (6.35) with N+1 gauge groups and N Higgs fields Φ_i provides a gauge invariant description of the first n KK modes by generating the same spectrum³.

Now the point is that the described 3+1 dimensional gauge theory cor-reponds to a transverse lattice description of a full 4+1 dimensional gauge theory with lattice sizeRand lattice constantl. This construction describes a foliation of branes, each spaced by the lattice cut-off l. The number of branes isN + 1 =R/l+ 1.

Now let us return to the nonunitary gauge theory on the discrete man-ifold Z⁴×N. The short review presented above shows that our formalism developed so far provides suitable tools to deal also with deconstruction models. Actually, in the framework of nonunitary gauge theories it is quite natural to place different gauge groups on different branes. The nonunitary parallel transporters connecting the branes can be viewed as the described Higgs fields Φ_i. In particular, they possess the ”right” transformation prop-erties. In this way the action

S_{Y M} =trhF^T,F^Ti_Ω, (6.41) whereF^T is given by (6.34), provides a full lattice description of the trans-verse lattice model.

Taking into account the orthogonality of the 1-forms, one gets S_{Y M} = 1

8l²tr

N

X

i=0

X

x,µ,⁰ν

F^{U †}_µ0ν(x_i)F^U_µ0ν(x_i) +

+ 1 8l²tr

N−1

X

i=1

X

x,µ,⁰

(D^U_µT⁰₅(xi))^†(D^U_µT⁰₅(xi) + + 1

8l²trX

x,µ

(D^U_µT₅(x₀))^†(D^U_µT₅(x₀) + + 1

8l²trX

x,µ

(D^U_µT−5(xN))^†(D^U_µT−5(xN) +. . . , (6.42)

3This is the spectrum assuming free boundary conditions. With periodic boundary conditions one hasN+ 1 Higgs fields, because there is an additional field linking the first gauge group to the last. As a consequence, the spectrum is changed. One gets a zero-mode corresponding toA^a5 and the KK modes are doubled.

where the dots indicate further terms involving only nonunitary parallel transporters.

We have repeatedly seen that P

x,µ,⁰νF^{U †}_µ0ν(x_i)F^U_µ0ν(x_i) yields the Wilson action, so in the continuum limit the first sum in (6.42) gives actually N+1 copies of an ordinary 3+1 continuum gauge theory, cp. (6.35).

To see, whether the other terms in (6.42) yield the kinetic terms of the Higgs fields, we writeT5(xi) =: Φi+1(x) and expand

T5(xi+µ) = Φi+1(x+µ) = Φi+1(x) +l∂µΦi+1(x) +O(l²) (6.43) U_µ(x_i) = 1−lAⁱ_µ(x) +O(l²). (6.44) Note that in (6.44)Aⁱ_µ(x) means the Lie algebra valued vector potential.

With (6.43) and (6.44) we arrive at

D^U_µT₅(x_i) = l ∂_µΦ_i+1(x) +A_(i+1)µ(x)Φ_i+1(x)

− Φ_i+1(x)A_iµ(x)

+O(l²) (6.45)

= lDµΦⁱ⁺¹(x) +O(l²). (6.46) Thus we recover also the kinetic terms for the Higgs fields in (6.35).

6.3.1 2-brane system

We have seen that a general N-brane system describes extensions of the standard model. The standard model itself may be obtained if we start from a N-brane system and thin the degrees of freedom, thus arriving at a 2-brane system which can be pictured by the discrete manifold Z⁴ ×2 with 2 = {0,1}. Again the gauge fields propagating along the fifth extra dimension are assumed to be nonunitary. The resulting field strength is easily obtained from our formula (6.34):

F^T = 1 2

X

i=L,R

X

x,µ,⁰ν,

e^{xi+µ+0ν,xi+µ,xi}F^U_µ,0ν(x_i) + +1

2 X

x,µ,⁰ν,

e^{xL+µ−5,xL+µ,xL}D^U_µT−5(xL) + +1

2 X

x,µ,⁰ν,

e^{xR+µ+5,xR+µ,xR}D^U_µT₅(x_R) +

+ X

x,µ,⁰ν,

e^xR,xR+5,xR

T−5(xR+ 5)T+5(xR)−1 +

+ X

x,µ,⁰ν,

e^xL,xL−5,xL

T+5(xL−5)T−5(xL)−1

. (6.47)

One finds that the field strength splits into three different parts. Since the corresponding two forms are orthogonal to each other

he^µ,_x ^0ν, e^µ,5_x i= 0 =he^µ,_x ^0ν, e^5,−5_x i=he^µ,5_x , e^5,−5_x i, (6.48)

6.3. Geometry of Higgs fields 91

the split of the field strength leads to a corresponding split of the generalized Yang-Mills action into three qualitatively different parts.

SY M = trhF^T,F^Ti_Ω

= 1

8l²tr

X

i=L,R

X

x,µ6=⁰ν

F^{U †}_µ0ν(xi)F^U_µ⁰_ν(xi) +

+ X

x,µ

(D^U_µT−5(x_L))^†D^U_µT−5(x_L) +X

x,µ

(D^U_µT₅(x_R))^†D^U_µT₅(x_R)

+

+ 1

4l²tr

X

x

T−5(x_R+ 5)T+5(x_R)−12

+

+ X

x

T−5(x_L−5)T−5(x_L)−12

. (6.49)

We find that the first part of the generalized action yields the Wilson actions for two 3+1 dimensional ordinary gauge theories, one living on the ”left-handed” brane, the other carried by the “right-”left-handed”. The second part is already written in a kinetic type form. Actually, we shall see in the next chapter that these terms provide the usual kinetic terms of the standard model Higgs. The third part gives a Higgs potential, which vanishes, if the parallel transport in the fifth direction is unitary. Let us write T5(xR) :=

Φ(x) and T−5(x_L) := Φ^†(x). Then we have tr(X

x

[F^{T †}_5,−5(xL)F^T_5,−5(xL) +F^{T †}_−5,5(xR)F^T_−5,5(xR)]

= 2trX

x

(Φ^†(x)Φ(x)−1)²=:X

x

V_Higgs(Φ). (6.50) Note that by definingT_xR,xR−5=1=T_xL,xL+5 also the Higgs potential can be expressed as a covariant derivative, cp. (6.9)

F^T_5,−5(xL) =D^T₅T−5(xL) (6.51) Thus again we find that the complete action can be recast in a kinetic type form.

We discussed in chapter 4 that the emergence of pseudographs may be viewed as a result of a multiscale analysis, which also includes dimensional reduction. Therefore it is natural to apply our formalism of gauge theories on pseudographs to our Higgs model considered in this subsection. Thus we shall arrive at a 4-dimensional theory, where the Higgs fields are treated as nonunitary gauge fields attached to the loops of the pseudograph.

The vector potential on a pseudograph can be decomposed as

B =Axye^xy+Pxxe^xx+P^†_xxe^xx? (6.52)

The field strength is defined by

F^T :=dB+BB (6.53)

Using (4.66) one obtains F^T = dA+AA+X

x

(P_xxP^†_xx−1)e^xxe^xx?+ (P^†_xxP_xx−1)e^xx?e^xx +

+ X

x,y

[(Pxx−1) + (P^†_xx−1)]e^xyx+

+ X

x,y

(UxyPyy−Uxy)e^xye^yy+ (UxyP^†_yy−Uxy)e^xye^yy?+ + (PxxUxy−Uxy)e^xxe^xy+ (P^†_xxUxy−Uxy)e^xx?e^xy

. (6.54)

Next, let us apply our general formula for the field strength to the sym-metric lattice Z⁴ supplemented with loops attached to every pointx ∈Z⁴. We obtain

F^T = dA+AA+X

x

(PxxP^†_xx−1)e^xxe^xx?+ (P^†_xxPxx−1)e^xx?e^xx +

+ X

x,µ

[(P_xx−1) + (P^†_xx−1)]e^x,x+µ,x+

+ X

x,µ

(U_x+µ,xP_xx−U_x+µ,x)e^x+µ,xe^xx+ + (Ux+µ,xP^†_xx−Ux+µ,x)e^x+µ,xe^xx?+

+ (Px+µ,x+µUx+µ,x−Ux+µ,x)e^x+µ,x+µe^x+µ,x+ + (P^†_x+µ,x+µUx+µ,x−Ux+µ,x)e^x+µ,x+µ?e^x+µ,x

. (6.55)

To compute the Yang-Mills action we have to extend the inner product (6.25) on forms involvinge^xx and e^xx?, respectively. We define

he^xxe^xx?, e^xxe^xx?i_Ω= (2l)⁻² =he^xx?e^xx, e^xx?e^xxi_Ω (6.56) and

he^xxe^xx?, e^xx?e^xxi_Ω= 0 =he^xx?e^xx, e^xxe^xx?i_Ω. (6.57) Furthermore we write

e^µ,0_x :=e^x+µ,xe^xx, e^µ,0?_x :=e^x+µ,xe^xx? (6.58) and analogoulsy we define e^0,µx and e^0?,µx . Now we can apply (6.25) by treating 0 and 0? in the same way asµand ⁰ν. For example, we get

he^µ,0_x , e^0,µ_x i_Ω=−(2l)⁻². (6.59)

Im Dokument Gauge Theories with Nonunitary Parallel Transport (Seite 91-98)