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One-Loop Renormalization of Lee-Wick Gauge Theory

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UCSD/PTH 07-11One-Loop Renormalization of Lee-Wick Gauge Theory Benjam′?n Grinstein 1,?and Donal O’Connell 2,?1Department of Physics,University of California at San Diego,La Jolla,CA 920932Institute for Advanced Study,School of Natural Sciences,Einstein Drive,Princeton,NJ 08540(Dated:February 4,2008)We examine the renormalization of Lee-Wick gauge theory to one loop order.We show that only knowledge of the wavefunction renormalization is necessary to determine the running couplings,anomalous dimensions,and vector boson masses.In particular,the logarithmic running of the Lee-Wick vector boson mass is exactly related to the running of the coupling.In the case of an asymptotically free theory,the vector boson mass runs to in?nity in the ultraviolet.Thus,the UV ?xed point of the pure gauge theory is an ordinary quantum ?eld theory.We ?nd that the coupling runs more quickly in Lee-Wick gauge theory than in ordinary gauge theory,so the Lee-Wick standard model does not naturally unify at any scale.Finally,we present results on the beta function of more general theories containing dimension six operators which di?er from previous results in the literature.I.INTRODUCTION In recent months,an extension of the standard model of particle physics has been con-structed [1]based on ideas of Lee and Wick [2].Lee and Wick constructed a ?nite theory of

quantum electrodynamics in order to remove divergences in certain mass corrections.The theory of Lee and Wick contains new degrees of freedom which are associated with wrong sign kinetic terms.Thus the theory is classically unstable.Lee and Wick proposed that the instability could be removed at the classical level by imposing boundary conditions on the theory,and at the quantum level by quantizing the theory such that the energy of any scat-tering (asymptotic)state is positive.This requires the introduction of a non-positive de?nite

norm on the Hilbert space.Lee and Wick further described how the theory could neverthe-less be unitary if the negative norm states are heavy and can decay to states of positive norm. These ideas have been discussed extensively in the literature[2,3,4,5,6,7,8,9,10,11]. It has not been shown that an arbitrary Lee-Wick theory is unitary to all orders of pertur-bation theory,but there is no known example of a theory that cannot be unitarized in this way.In particular,scalar Lee-Wick theories have been extensively analyzed in[10]at the non-perturbative level with encouraging results.

With the modern understanding of renormalization the original motivation of Lee and Wick is no longer pressing,and in particular the massive resonances predicted by the Lee-Wick theory of electrodynamics have not been observed.Thus,interest in the Lee-Wick model of electrodynamics has dwindled.However,we are currently faced with quadratically divergent radiative corrections to the Higgs mass.The extension of the standard model developed in[1],known as the Lee-Wick standard model,includes new degrees of freedom that remove these quadratic divergences.The resulting theory is logarithmically divergent. The new degrees of freedom are associated with higher derivative,dimension six operators present in the microscopic Lagrangian of the theory.It was shown that an equivalent for-mulation of the theory contains only dimension four operators;in this form,the new degrees of freedom in the theory have wrong sign kinetic Lagrangians.The Lee-Wick prescription is then invoked to quantize the theory.Physically,the Lee-Wick standard model is unusual since the future boundary condition leads to acausality.However,the time scale of this acausality is far too small to have been ruled out by experiment.

The?avor structure of the Lee-Wick standard model has been explored in[12]with the attractive result that while new?avor changing neutral and charged currents are present,the ?avor symmetry violation is naturally within experimental bounds.However,the Lee-Wick standard model was de?ned by choosing particular dimension six operators to add to the standard model Lagrangian.One could imagine a more general theory containing a greater number of dimension six operators.Some of these operators would lead to unacceptably large?avor changing currents.In[13]the question of the physical status of such operators was addressed,and it was shown that the choice of operators made in de?ning the Lee-Wick standard model is such that scattering amplitudes in the theory do not violate the well-known perturbative unitarity bounds.Thus,while dimension six operators typically imply either strong coupling in the ultraviolet or a violation of unitarity,the operators included

in the Lee-Wick standard model lead to a perturbative UV completion as suggested by precision electroweak constraints.Aspects of the LHC phenomenology of the Lee-Wick standard model has been discussed in[14,15,16],and more theoretical aspects of these models have been examined in[17]and in[18].Supersymmetric models including similar higher dimension operators have been examined in[19].

In the present work,we turn to the question of the one-loop structure of non-abelian Lee-Wick gauge theory.A perturbative power counting argument presented in[1]establishes that the dimension six operators in the higher derivative formulation of the theory only receive?nite renormalizations.In this work,we examine the renormalization in more detail. We work in background?eld gauge.There are some subtleties of gauge?xing in these theories which we discuss before turning our attention to the beta function and anomalous dimensions of matter.One interesting result is that the running of the massive vector boson mass,m,in the theory is exactly related to the running of the coupling,g,because the quantity mg is a renormalization group invariant.In an asymptotically free quantum?eld theory,g runs to zero in the ultraviolet so if mg=0,then the mass m must run to in?nity in the UV.Consequently the UV?xed point of the renormalization group?ow is an ordinary free quantum?eld theory.We?nd that the Lee-Wick standard model does not appear to unify at any energy scale.The Lee-Wick particles in the theory in fact cause the running of the coupling to be quicker,so that any putative uni?cation scale would be rather low.If the uni?cation group were to be semisimple,this would lead to unacceptably large proton decay, but this problem can be alleviated[20].We then turn to more general theories containing dimension six operators which are not of Lee-Wick type.While these theories do not satisfy the perturbative unitarity bounds,they have nevertheless been discussed in the literature as a toy model of higher derivative gravity[21,22].Since our results for the beta functions of these theories di?er from previous expressions in the literature we feel it is worthwhile to present our results.

II.PRELIMINARIES The theory we study is given by1

L=?1

Tr(DμFμν)2+ˉψL i/DψL+

σ1

φ?(D2)2φ.

(1)

Our notation is as follows.A aμis the gauge?eld with?eld strength F aμν=?μA aν+···.In matrix notation Fμν=T a F aμνand Aμ=T a A aμwith T a hermitian generators of the de?ning representation of the gauge group(traceless for factors of a semisimple group).The‘Tr’denotes a trace in the space of these matrices,the normalization is Tr T a T b=1

Tr(FμνFμν)+2Tr(FμνDμ?Aν)?m2Tr?Aμ?Aμ

+?ψL i/DψL+?ψR i/D?ψR+m2?

ψL?ψR+

δ1

?φ??φ.(2)

Upon solving the equations of motion of the?elds?Aμ,?ψand?φand inserting the solutions in (2)one recovers the higher derivative Lagrangian of(1).While the Lagrangian(2)has twice

as many?elds as the higher derivative version(1)it is renormalizable by power counting, so it is more convenient to use in some cases.The mixing terms present in(2)can be diagonalized by an appropriate rede?nition of the?elds,as discussed in[1].

For our calculations below we use the background?eld gauge(BFG)method.Let us brie?y review it.This is not only for completeness:as we shall shortly show,one has to be careful about introducing higher derivatives in the gauge?xing term.Denote the quantum ?elds by Aμand the background?elds by Bμ.The e?ective action is determined by the vacuum graphs for the theory with action integral S(A+B),where S= d4x L,and L as given above.The gauge?xing condition is

F(A,B)=0(3)

for some function that is invariant under gauge transformations of the B?eld with the A ?eld transforming as a matter?eld:

Bμ→U(

Zα,(9)

2ξ d4xα2

where

Zα= [dA]e iS?F Pδ(F?α).(10) It is sometimes useful to have more derivatives in the gauge?xing term in the action(for example,for power counting arguments).This can be done by putting derivatives in the exponent in the exponentiation trick(9).However we must be careful to preserve the invariance in(4)–(5).So an alternative form of the partition function we may use is Z= M2D(B)2 [dα]exp ?i M2D(B)2 α Zα.(11) Notice the factor of the square root of the determinant,which compensates for the extraneous B dependence introduced by theαintegration.Below we compute the beta functions of this theory with both types of gauge?xing and?nd agreement.

The determinant in(11)can be computed using ghost?elds,

det(1+

III.RENORMALIZATION The renormalized version of the Lagrangian(1)is

L=?1

ZZ m2 Tr(DμFμν)2)

ZφZ m2(Zδδ1)φ? (D2)2 φ.(13)

The?rst line contains the kinetic terms(dimension four operators)and it is in terms of these that the wave function renormalization factors Z,Zψand Zφare de?ned.The next three lines contain the dimension six operators for gauge?elds,spinors and scalars,respectively. The coupling constant renormalization is not shown explicitly,but it should be understood that the Lagrangian depends on g through the combination Z g g only.

Some comments are in order.There are no counterterms of the form of any of the dimension six operators,a result that was established in[1]by the power counting analysis and veri?ed through an explicit one loop computation.This implies for example that ZZ m2 is?nite,so we can adopt the renormalization condition

ZZ m2=1.(14)

Similarly,we have

ZψZ m2Zσ=ZφZ m2Zδ=1.(15) We have chosen to work in background?eld gauge.One of the great simpli?cations of BFG is that[23]

Z g Z1

+ (17)

Z g=1+

a g

+ (20)

Then,as usual,

β(g,?)=?1

?a g

a and putting together the contributions to the YM self energy

in the previous section we have,

β(g)=?1

.(22)

The anomalous dimensions for the matter?elds are

γf(g)=1

?μ

log Z f=?

,f=ψ,φ.(23)

The renormalization group equation for the matter couplings is easily obtained.We present this for a single spinor or scalar,to avoid unnecessary complications from the matrix struc-ture:

μ?σ1

g (24)

μ?δ1

g ,(25)

or more simply

μ?(g2σ1)

?μ

=2(g2δ1)γφ(g).(26)

We turn now to the explicit computation of the self-energy diagrams.

A.YM

self-energy

FIG.1:Contribution to the self-energy of YM?elds from internal YM?elds

As mentioned above,we have performed the computation several di?erent ways:we can use a higher derivative version of the theory with a standard covariant gauge?xing term, or we can use a higher derivative version of the covariant gauge?xing term with a Jacobian correction,or we can use the formulation of the theory without higher derivative terms but instead including negative norm LW?elds.In each case the computation is very

di?erent.

FIG.2:Contribution to the self-energy of YM?elds from internal ghosts

There is no one to one correspondence between the contributions to the renormalization constants of individual Feynman diagrams,yet the resulting beta functions are the same.

We?rst list our results for the?-poles of the graphs computed in the higher derivative theory with a standard covariant BFG-term.The graphs in Figs.1give

ig2

? 41

16π2

δab 23 (gμνk2?kμkν).

(28)

FIG.3:Contribution to the self-energy of YM?elds from internal spinor

?elds

FIG.4:Contribution to the self-energy of YM?elds from internal complex scalar?elds Next come the matter?elds.The spin-1/2contribution(in the fundamental representa-tion of the gauge group)from Figs.3is

ig2

? (gμνk2?kμkν).(29) Finally,the contribution from a complex scalar?eld(in the fundamental representation)in Figs.4is given by

ig2

? 1

Now we turn to the case where we use a higher derivative version of the covariant BFG-term.The only di?erence from the above is in the graphs in

Figs.1which

now give ig 2? 4016π2

δab

26C 2(g μνk 2?k μk ν).(32)

The sum of these two contributions precisely equals the result in (27).

Finally,we have computed the beta function in the Lee-Wick formulation of the theory,as discussed in [1].In this formulation,the physical degrees of freedom are the gauge ?elds A a μ,

massive LW vector ?elds ?A a μ,a chiral spin 1/2?eld,a Dirac Lee-Wick fermion,a scalar ?eld and a Lee-Wick scalar ?eld.We compute the beta function by computing the wavefunction renormalization of the normal gauge ?elds in background ?eld gauge.

FIG.5:Contribution to the self-energy of YM ?elds from internal LW-vector ?elds

It is easy to deduce the contributions of the matter ?elds to the beta function,because the LW ?elds couple to the gauge ?elds just as normal ?elds do 2.Thus,the total contribution of the spin-1/2?elds to the beta function is three times the usual contribution of a fundamental chiral spin-1/2fermion,while the scalar ?elds contribute twice the usual scalar ?eld value,in agreement with Eq.(29)and Eq.(30),respectively.It remains to compute the e?ects of the LW vector ?elds.The relevant graph is shown in Fig.5.The graph evaluates to 3

ig 2

Signs associated with Lee-Wick propagators appear squared in all the relevant diagrams.3In this formulation of the theory,there are additional divergences proportional to p 4and p 6which we do not show.These higher divergences are gauge artifacts.Since the beta function is gauge independent to this order,we can be con?dent of our results.It is possible to ?x the gauge in the Lee-Wick formulation of the theory so that these spurious divergences do not appear,at the expense of a more involved formalism.

Of course,the gauge?elds and ghost lead to a term

ig2

? 11

16π2 2

gauge coupling is determined by

β(g )=?g 3

6C 2?n f ?1

16π2 433 s n s T (s ) ,(37)

where in the representation x we have Tr(T a T b )=T (x )δab .

Similarly,the anomalous dimensions for the spinor and scalar are

γψ(g )=0,

(38)and

γφ(g )=?g 2

g 2(μ)

m 2(μ0).(40)

Since γψ(g )=0,we see from Eq.

(26)that g 2σ1,and so m 2/σ1does not run.Therefore,the mass of the Lee-Wick fermion is an invariant of the RG ?ow.On the other hand,the quantity g 2δ1does ?ow so that the LW scalar mass grows logarithmically in the ultraviolet.The result (37)is roughly what one would guess naively.The higher derivative terms that we have introduced in the Lagrangian are precisely the ones that can be described as additional LW ?elds.Hence one roughly expects to double the contribution of each ?eld to the β-function.Subtleties occur in the spinor matter and pure gauge terms.In the spinor terms,the contribution is tripled because the Lee-Wick partner of a chiral fermion is non-chiral.In the pure gauge term,the contribution from ghosts is not quite doubled as explained above.

Hence,much like for the standard model of electroweak interactions,the LW extension of the standard model does not display good uni?cation of coupling constants.The standard model does however unify well if properly chosen additional ?elds are introduced.A simple example was given by Willenbrock in Ref.[20],where he shows that the standard model with six Higgs doublets uni?es.Similarly,we ?nd that the Lee-Wick extension of the standard

model has good coupling constant uni?cation if it is extended to include six or seven Higgs doublets.

However,Willenbrock points out that in the six-Higgs doublet model the uni?cation scale is very low so if the uni?cation group is simple,then the proton decays excessively fast.He proposes an interesting solution to this problem using trini?cation,that is,a uni?ed group SU(3)3/Z3.The uni?cation scale in our six Higgs doublet model is even lower than in Willenbrock’s case,about a million times the LW scale m.Presumably one can formulate a LW extension of trini?cation,but we have not pursued this.

V.ADDITIONAL DIMENSION SIX OPERATORS

In this section we consider a more general theory which contains additional dimension six operators in the Lagrangian density.This theory does not satisfy the constraints of perturbative unitarity,so that scattering amplitudes cannot be computed by perturbative methods.The beta function and anomalous dimensions,on the other hand,may still be computed in perturbation theory since no large energies occur in these functions.Theories of these types have been considered in the literature previously as toy models for higher derivative gravity[21,22].The Lagrangian of the theory is given by

L=L A+Lψ+Lφ,(41)

where

L A=?1

Tr(DμFμν)2?

iγg

m2ˉψ

L σ1/D/D/D+σ2/DD2+igσ3FμνγνDμ+igσ4(DμFμν)γν ψL,(43)

where,in the last term of the Lagrangian,the covariant derivative acts only on the?eld strength tensor,andσ1?4are dimensionless constants.For a complex scalar,we consider for the Lagrangian density

Lφ=?φ?D2φ?

We?nd that the beta function and anomalous dimensions are given by

β(g)=?g3

?18γ+

σ23

3δ1 ,(45)

γψ(g)=?

C1 2σ1(2σ2+σ3?2σ4)+σ2(2σ2+2σ3?σ4)?σ23?σ24+σ3σ4

16π2

δ1 .(47)

We note that our expression for the beta function di?ers from that found in Appendix C of[22].We can write the beta functions for the couplings of the dimension six operators in terms of these anomalous dimensions.The?rst states thatγis a constant,

?γ

?μ=2(g2σi)γψ(g)andμ

?(g2δi)

to absorb these divergences,typically leading to theories containing an in?nite number of couplings constants,which are a priori unknown.

The situation is di?erent in Lee-Wick theories.In the higher derivative formulation of the theories,dimension six operators are present in the microscopic Lagrangian.There is one new constant associated with each higher derivative operator,which physically corresponds to the mass of the corresponding Lee-Wick degree of freedom.

In this work,we have described the renormalization of Lee-Wick theories to one-loop order.No new counterterms are required to absorb the divergences of the theory.In fact, we have shown that the wavefunction renormalizations of the various?elds present in Lee-Wick gauge theory contain all the information about the renormalization group running of the theory.For the Lee-Wick gauge bosons,we have shown that the quantity m2g2is an invariant of the renormalization?ow.Thus,the new constant introduced in the de?nition of a Lee-Wick gauge theory truly is just one number,and not a new function of energy scale. In addition,we learn that if the theory is asymptotically free,then the LW vector boson mass?ows to in?nity in the UV.This counter intuitive behaviour is interesting because it indicates that the ultraviolet?xed point of the RG?ow of an asymptotically free pure Lee-Wick gauge theory is a normal quantum?eld theory:the scale suppressing the dimension six operator in the higher dimension formulation of the theory has become in?nite so that this term no longer contributes to the dynamics.The remaining degrees of freedom are the usual gauge bosons.

We have obtained expressions for the beta function and anomalous dimensions of scalar and spinor matter.The coupling runs more quickly in Lee-Wick theory compared to the usual non-Abelian gauge theory.We?nd that the Lee-Wick standard model does not unify naturally,and that,on account of the more rapid running of the coupling,the uni?cation scale of the theory augmented with extra?eld content is typically rather low.In addition, we?nd that the anomalous dimension of spinor matter vanishes.

Finally,we have discussed some more general theories containing dimension six operators which are not of Lee-Wick type.Since amplitudes in these theories grow too quickly with energy to satisfy perturbative unitarity bounds,the theories either become non-perturbative at some scale,or else they violate unitarity.However,no large factors of energy appear in the expressions for the beta function or for the anomalous dimensions,so they may still be computed in perturbation theory.(Of course,they no longer give us insight into the high

energy behaviour of physical scattering amplitudes.)These theories have been discussed elsewhere in the literature,and since our results di?er from previous expressions we have reported our results above.Our results indicate that if it is possible to make sense of these theories,then,for suitable choices of the couplings,these theories may enjoy the property that their beta function vanishes.

Acknowledgments

We thank Mark B.Wise for useful discussions and collaboration at the beginning of this work,and we are grateful to Arkady A.Tseytlin for email correspondence.DOC thanks Poul Henrik Damgaard for several helpful conversations.The work of BG and DOC was supported in part by the US Department of Energy under contracts DE-FG03-97ER40546 and DE-FG02-90ER40542,respectively.

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