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SUSY Moose Runs and Hops An extra dimension from a broken deformed CFT

SUSY Moose Runs and Hops An extra dimension from a broken deformed CFT
SUSY Moose Runs and Hops An extra dimension from a broken deformed CFT

a r X i v :h e p -t h /0607031v 2 19 J u l 2006SUSY Moose Runs and Hops:

An extra dimension from a broken deformed CFT

Joshua Erlich and Jong Anly Tan 1

1Department of Physics,College of William and Mary,Williamsburg,Virginia 23187-8795

We ?nd a class of four dimensional deformed conformal ?eld theories which appear

extra dimensional when their gauge symmetries are spontaneously broken.The theories are supersymmetric moose models which ?ow to interacting conformal ?xed points at low energies,deformed by https://www.doczj.com/doc/d68741409.html,ing a-maximization we give strong nonperturbative evidence that the hopping terms in the resulting latticized action are relevant deformations of the ?xed point theories.These theories have an intricate structure of RG ?ows between conformal ?xed points.Our results suggest that at the stable ?xed points each of the bulk gauge couplings and superpotential hopping terms is turned on,in favor of the extra dimensional interpretation of the theory.However,we argue that the higher dimensional gauge coupling is generically small compared to the size of the extra dimension.In the presence of a brane the topology of the extra dimension is determined dynamically and depends on the numbers of colors and bulk and brane ?avors,which suggests phenomenological applications.The RG ?ows between ?xed points in these theories provide a class of tests of Cardy’s conjectured a-theorem.

I.INTRODUCTION

The possibility of extra spatial dimensions beyond the three which we commonly observe has provided a provocative paradigm for model building in particle physics and gravity in recent years [1,2,3].However,due to the non-renormalizability of higher-dimensional gauge theories they are necessarily low energy e?ective theories.An ultraviolet completion of higher-dimensional theories was provided by the deconstruction approach [4,5].In a de-constructed extra dimension,the extra dimension appears as a lattice of (3+1)-dimensional gauge theories connected to one another through link ?elds transforming in the bifundamen-tal representation of the gauge groups at neighboring lattice sites.For non-supersymmetric

gauge theories the UV completion may be provided by an asymptotically free gauge the-ory in which fermions condense to provide the link?elds as nonlinear sigma-model?elds. Alternatively,the link?elds can arise from fundamental scalar?elds which obtain vacuum expectation values leaving an e?ective theory which appears higher dimensional below some scale.However,scalar?elds su?er the usual problem that their expectation values are nat-urally of order the cuto?scale,and supersymmetry(SUSY)is therefore favored to restore naturalness in this linear sigma model approach.

In this paper we study a class of deconstructed?ve-dimensional(5D)supersymmetric gauge theories.The latticized higher-dimensional theories contain hopping terms in the Lagrangian which are the latticized kinetic terms containing derivatives in the extra dimen-sions.The scalar link?elds obtain vacuum expectation values(VEVs)which break the product gauge symmetry partly or completely.The spectrum of gauge bosons mimics the Kaluza-Klein(KK)spectrum of the extra dimension up to a scale which depends on the size of the extra dimension R and the lattice spacing l.The scale above which the spectrum ceases to be higher-dimensional is roughlyΛKK≡R2/l for a?at extra dimension.Above

that scale the theory is a4D gauge theory with k~R/l factors of the5D gauge group.There are two other dimensionful scales to keep in mind:the dynamical strong coupling scale of each4D gauge group factorΛQCD,which we assume for simplicity to be the same for each of the gauge groups in the deconstructed theory;and the ultimate cuto?of the deconstructed theoryΛ4,which must be much higher thanΛKK andΛQCD for the deconstructed theory to provide a UV completion of the5D theory.

Flavors in the bulk of the extra dimension are added to the deconstructed theory by replicating the?avors at each lattice site and including a hopping superpotential[6].Flavors on a3-brane are modeled by localizing those?avors to a single lattice site.The moose(or quiver)diagram representing the deconstructed theory on a circle with N f bulk?avors and N b?avors localized to a3-brane is given in FIG.1a.The moose diagram for a theory on an interval with extra?avors on one boundary is shown in FIG.1b.Nodes in the moose diagram correspond to SU(N)gauge groups,and lines correspond to?elds transforming in the fundamental representation of the gauge groups at the nodes on which those lines end.We do not distinguish fundamental and antifundamental representations in the moose diagrams as it is conventional to assign links between nodes the bifundamental representation (

N f N

f

f

f

N + N b N f N f N f

(b)

(a)

FIG.1:a)Circular moose with brane.b)Interval moose with brane.

The hopping terms in the action for bulk?avors are reproduced by a superpotential,as we review below.In the absence of the superpotential the models that we study have a large number of interacting conformal?xed points.At each?xed point some combination of the independent gauge couplings is nonvanishing.We use the a-maximization procedure of Intriligator and Wecht[7]to calculate the dimensions of operators at these?xed points. Depending on the numbers of colors and bulk and brane?avors we determine which of the conformal?xed points are stable to gauging of the ungauged symmetry groups at those?xed points,and we argue that generically all of the bulk gauge couplings are nonvanishing at the stable?xed points.

We will show that for the braneless moose the hopping superpotential is composed of relevant operators in the neighborhood of the?xed points,so that those operators will generically have nonvanishing couplings in the infrared as required for the extra dimensional interpretation of the moose model near its stable?xed point.Furthermore,we will discover that the topology of the deconstructed extra dimension depends on the numbers of colors and bulk and brane?avors.

A.The Model

For de?niteness we will mostly consider the circular moose model with N f bulk?avors and N b brane?avors,as in FIG.1a.The gauge groups and matter content are summarized

in the table below:

U (1)R

r Q 1

Q 2···1......

(1)

1···r Q k r F 1(N f +N b )

11···1F 1N f F 2

1···1F 2r ...

(1)

11···r F k

N f 111···

F k F i are the ?avors.They are each chiral multiplets of 4D

N =1SUSY.The brane is on the ?rst lattice site,corresponding to the gauge group factor

SU(N )1.The moduli space of this theory allows the link ?elds to obtain expectation values,which we assume are at least approximately proportional to the identity in the SU(N )×SU(N )gauge group space under which each link ?eld transforms: Q i =v i 1.The gauge group is then broken to the diagonal SU(N )subgroup at low energies.If there is a superpotential in this model of the form,

W ?avor =√F i Q i F i +1)+ i m i F i

low energy dynamics below the scale1/R[8].If all of the vevs are equal(and nonvanishing) the hopping terms include the latticized kinetic terms of a?at extra dimension.The?avor

mass terms are also tuned so that m i=?

k and g~

methods exist[12]and have already proven useful for studying certain deconstructed gauge theories.In particular,4D instanton e?ects in the deconstructed5D N=1supersymmetric 5D pure gauge theory were examined in[8].The6D theory with(0,2)supersymmetry and the6D little string theory were studied in[13],and the Seiberg-Witten approach gave new nonperturbative insights in[14].

Strictly speaking,the techniques we use only apply to the origin of moduli space,with vanishing VEVs.Hence,we are only considering energies>ΛKK so that the VEVs are negligible near the(approximately)?xed points.In the continuum limit we need to take the couplings of individual gauge group factors to be large.SinceΛQCD is the strong coupling scale of each4D gauge group factor,we assumeΛQCD>ΛKK.Hence,we expect the renormalization group?ows to generically approach the approximate?xed point at energies higher thanΛKK,and our analysis remains valid.It is also interesting to consider the evolution of operators when?owing to still smaller scales.This evolution requires matching of operators while the heavier KK modes are integrated out,and we will not address the issue here.

Although the superpotential is not renormalized,wavefunction renormalization causes the ?avor hopping terms to run with energy scale.It is in general not possible to calculate the anomalous dimensions of the hopping terms in the Lagrangian at strong coupling.However, the conformal anomaly and the R-current anomaly are in a supersymmetry multiplet,with the consequence that the conformal dimension of an operator is proportional to its R-charge. At a conformal?xed point,the R-charges are in turn related to central charges in the theory, which can sometimes be determined exactly[15].The problem is that quite often there is not a unique gauge anomaly free R-symmetry,so the goal is then to determine which of the continuum of anomaly free R-symmetries is related by supersymmetry to the conformal anomaly.Without repeating the clever analysis of[7],the remarkably simple procedure that determines the correct R-charges for this purpose is to maximize the Euler anomaly,

3

a=

supersymmetric gauge theories for which the existence of accidental symmetries is required for consistency of Seiberg duality and unitarity[16],and we will have to account for such accidental symmetries in our moose models.

C.Fixed Point Stability

Models with product gauge groups and bifundamental?elds have been studied non-perturbatively using Seiberg duality techniques[17],Seiberg-Witten theory[11],and a-maximization[18,19].The conclusion of these analyses is that these theories have an intricate vacuum structure which depends qualitatively on the numbers of?avors and colors of each gauge group factor.In[17]it was concluded that theories with two gauge groups can have a variety of conformal?xed points with either or both of the gauge couplings nonvanishing.IR stability of these?xed points depends on the numbers of?avors and col-ors.If the conformal dimensions of the?elds are known at a?xed point for which at least one of the gauge couplings vanishes,then the NSVZ beta-functions for the remaining gauge couplings are also known in a neighborhood of that?xed point(perturbatively in the small coupling(s)).The signs of the beta-functions then determine stability of the?xed point to perturbations of the gauge couplings.One can thereby determine which of the?xed points a generic?ow from the UV will approach in the IR.Superpotential deformations of the theory can be studied similarly to see whether the operators appearing in the deconstructed superpotential are relevant perturbations of the theory in their absence,as is necessary for the theory to appear higher dimensional below the scaleΛKK.

At?xed points with more than one nonvanishing gauge coupling a-maximization is gener-ally required to determine operator dimensions.This analysis was done for a class of models with two gauge groups in[18].We will extend the analysis to models with more gauge group factors,and focus our study to models related to deconstructed extra dimensions.As described earlier,we assume that each of the gauge group factors has the same number of colors N,and all but one of the gauge group factors has the same number of?avors N f. At one special lattice site,which we label the?rst lattice site and refer to as the brane,we allow for an additional N b?avors representing brane-localized matter.

We will argue that generically the bulk gauge couplings are nonvanishing at the stable ?xed points.In the absence of a brane,at perturbative stable?xed points near the edge of

the conformal window we will?nd that the gauge couplings are all equal to one another. In the presence of a brane we will not determine the gauge couplings,but we will?nd that depending on the numbers of colors,bulk and brane?avors,the gauge coupling at the brane may vanish at the stable?xed point.We will conclude from this that the topology of the deconstructed extra dimension depends on the numbers of?avors and colors.

A study of the?ows between conformal?xed points in these theories allows for a large class of nontrivial tests of the“a-theorem.”The a-theorem is a conjecture by Cardy that the Euler anomaly a plays the role of a Zamolodchikov C-function in four dimensions[20]. Heuristically,the C-function e?ectively counts the number of degrees of freedom in the theory as a function of energy scale,which is expected to decrease as degrees of freedom are integrated out.As such,the Euler anomaly is supposed to be always positive and?ow from higher values to smaller values as energy is decreased.Neither of these requirements(i.e. positivity or monotonicity)is known to be guaranteed,although there have been partially successful attempts at proving them,and many tests have been provided[15,18,21].We will test Cardy’s conjecture in a large class of?ows between?xed points,and will?nd consistency with the conjecture in each case.We will?nd that the a-theorem also precludes certain types of RG?ows in these models that are otherwise di?cult to argue against.

II.THE BRANELESS MOOSE

In this section we consider a deconstructed extra dimension with SU(N)gauge group and N f bulk?avors,but no localized?avors on a brane,N b=0.For now we turn o?the superpotential,so that the?avors do not yet hop in the extra https://www.doczj.com/doc/d68741409.html,ter we will check that the superpotential in the deconstructed theory is a relevant deformation of the theory.In the absence of a brane,the Z k symmetry of lattice translations puts all of the bifundamental?elds Q i on equal footing,and similarly for the?avors(F i+

F i≡r F. We require that N/2

of theories may extend to N f

32 kN 2

3(r Q ?1)3?(r Q ?1) +2kN f N 3(r F ?1)3?(r F ?1) +2k (N 2?1) .(5)

Maximizing a subject to the constraint (4),we ?nd that the R-charges which belong to the superconformal algebra and determine operator dimensions are given by,

r ?Q =?3N 2f +N f 6N 2?3N 2f r ?F =6N 2+3NN f ?3N 2f ?N 6N 2?3N 2f

(6)and the value of a at the ?xed point is,

a ?=?k 9N 4f +N 4 36+90N 2f ?20N f 48 ?2N 2+N 2f 2?kN 2N 2f ?36?18N 2f +N f 48 ?2N 2+N 2f 2(7)

Although the denominators in the expressions above vanish at N f =

F F would fall out of the unitarity bound for N f <17N

or N f >17N ,which lie outside of the naive

conformal window N/2

invariant,and would lead to violation of unitarity if N f<

32 3(r O?1)3?(r O?1) +2/9 ,(8) where3/32×2/9is the contribution to a from a free chiral multiplet with R-charge2/3.The result of a-maximization including this correction is not particularly illuminating,except that we have checked the R-charges change by not more than a few percent when N=2 due to the accidental symmetry,and change by even smaller amounts for larger N.

We are interested in the stability of the?xed point found above in the sense of renormal-ization group?ow.To study this,we begin by noting that in the naive conformal window N/2

A.A-type?xed points

At a?xed point with a single nonvanishing coupling,say g1,the interacting part of the theory looks like SU(N)SUSY QCD with N+N f?avors[23].There is a unique anomaly-

free R-charge consistent with the enhanced SU(N+N f)global symmetry for which the?elds Q1,Q N,F1and

SU(N)1SU(N)2

Q1

r

(N f+N)1

12/3

N f1

(10) The NSVZβ-function for the perturbed gauge coupling is given by[24],

β2=?

g32

1?g22

16π2

(G? iμi(1?r i))

8π2

G

(11)

=?g32

(N+N f) 1?g22N

2

=

3rφ

other than g N,g1and g2have negativeβ-functions because they are asymptotically free SUSY QCD.In summary,in the naive conformal window the one-loopβ-functions for all weak couplings near an A-type?xed point are negative,so that the A-type?xed points are necessarily unstable to turning on those gauge couplings.

B.B-type?xed points

We have found that the A-type?xed points will tend to?ow to?xed points with additional gauge couplings turned on in the infrared.Here we are interested in?xed points with two nonvanishing couplings,which we refer to as B-type?xed points.There are several distinct possibilities for such?xed points.If the nonvanishing couplings correspond to nodes in the moose which are not either neighbors or next-to-nearest neighbors,then the?xed point and its stability analysis acts as two decoupled A-type?xed points.As such,we will focus our attention on B-type?xed points which are not equivalent to decoupled A-type?xed points.We de?ne B1-type?xed points to be those?xed points with two neighboring gauge couplings turned on,and B2-type?xed points to be those with next-to-nearest neighboring gauge couplings turned on.The low energy interacting theory for a B1-type?xed point is speci?ed by the following table:

U(1)R

r Q

N f F11

F1r

1r F

2

N f1F

2

F1=r

(73N2?4NN f?4N2f)

(73N2?4NN f?4N2f)?12N2f

F i F i type operator which would violate unitarity

(by having R-charge<2/3)in the absence of an accidental symmetry.Numerically,one ?nds that the operator det Q has R-charge>2/3for any N,N f in the naive conformal window.Hence,it is natural to assume that there is no accidental symmetry in this case.

Perturbing about the?xed point by weakly gauging SU(N)3,we?nd that theβ-function for the small gauge coupling takes the form,

β(B1) 3=?

g3373N2?4NN f?4N2f

8π2 (16)

One can check from this expression thatβ3<0in the entire naive conformal window,so that the B1-type?xed point is unstable in the IR.The value of a at the B1-type?xed point will be given more generally later,in(36).

For a B2-type?xed point,the interacting theory is that of two A-type?xed points.The di?erence in the stability analysis compared with the A-type?xed point is seen by weakly gauging the SU(N)gauge group in the“middle”of the two low energy SU(N)gauge group factors.For example,if the SU(N)1and SU(N)3gauge couplings are nonvanishing at the ?xed point,then weakly gauging SU(N)2gives rise to the following perturbed theory,where r is given by(9):

U(1)R

r

Q2

r

N f11

12/3

N f11

11r

N f11

(17)

Theβ-function for the middle group is then,

β(B2) 2=?

3g322(1?2/3)?N(1?r)

8π2

=?

g32

(1?g22N

It follows thatβ(B2)

<0for N f<2N,and B2-type?xed points are therefore unstable in 2

the entire conformal window.

C.General?xed points

Due to asymptotic freedom of the theory in the naive conformal window,?xed points are generically unstable to the development of gauge couplings which are not connected by link ?elds to any of the interacting gauge groups at the?xed point.This implies that generically at least one third of the gauge couplings in the moose will be turned on at the stable?xed point,as each node that is turned o?with at least two nodes turned o?on either side of it is itself unstable to turning on.The stability arguments in the previous sections suggest that generically an even larger fraction of the gauge couplings are turned on at the stable?xed point,and indeed we conjecture that all of the gauge couplings are generically turned on.In all of the cases which we will study numerically we will?nd that the bulk gauge couplings are all turned on,with or without a brane.For a generic?xed point with a large number of interacting gauge groups the analysis is complicated.However,there are a number of generic statements which can be made.

Suppose the?xed point contains a chain of interacting gauge groups as in FIG.2,where solid nodes correspond to the interacting gauge groups at the?xed point,and un?lled nodes are noninteracting at the?xed point.Such a?xed point structure is analogous to the B1 type?xed point,so we refer to such a?xed point generically as Type1.We are interested in the e?ect of weakly gauging the group corresponding to the lightly?lled node in FIG.2. The factor of the perturbed theory including the weakly gauged group is generically of the form given in the following table:

The node in the moose diagram adjacent to the last interacting node in the chain at either end has only a single neighboring interacting node.For large N,near the edge of the conformal window where N f=N(2??)with??1,the?xed points are perturbative, so that the anomalous dimensions can be assumed to be small[25].In that case,the beta function for the m th interacting SU(N)gauge group in FIG.2is proportional to(assuming

FIG.2:Type 1?xed point with chain of n +1interacting gauge groups.The labels (m ?

1),...,(m +n )index the SU(N )gauge group factors.The gauge coupling for SU(N )m ?1is turned o?at the ?xed point.Its β-function near the ?xed point determines stability of the ?xed point to development of the SU(N )m ?1gauge coupling.

there is no accidental symmetry),

βnum m =?3G +

i μi (1?γi )(19)=?3N +N +2N f

2(1?γQ m ),(20)which is required to vanish at the ?xed point.On the other hand,if we now weakly gauge the gauge group at the (m ?1)th node,then the beta function for that gauge coupling is proportional to,

βnum m ?1=? 3N ?(N +2N f )

2

(1?γF m ) =βnum m +N f γF m +

N 8π2(N 2?1)

8π2(N 2?1)

Unfortunately,a simple argument similar to the above fails for the generic situation analogous to the B2type?xed point,in which there is a gap by one node between two chains of interacting gauge groups.However,it is natural to conjecture that such?xed points,and indeed all?xed points in which not all couplings are turned on,are always unstable.We have not found an example to the contrary in our limited survey to follow, with or without a brane.

D.Will the Braneless Moose Hop After it Runs?

We have argued that generically the braneless moose prefers to?ow to IR?xed points in which all of the gauge couplings are turned on.This is good from the perspective of deconstructed extra dimensions,because it leads us to suspect that the hopping terms for the gauge?elds are relevant operators and survive RG running down to scales for which the theory is supposed to appear extra dimensional.

However,we have not yet studied the hopping potential for the N f bulk?avors.The hopping superpotential contains two types of operators:

F i Q i F i+1,which is marginal at lowest order.The latter operators are dangerous,as quantum corrections may easily make these operators irrelevant.If that were the case,then the hopping superpotential for the?avors would not exist at energies much belowΛQCD,the strong coupling scale for each SU(N)gauge group factor,which is assumed to be larger than the scale of the highest KK mass so that our?xed point analysis is valid(and because the latticized theory requires strong coupling at that scale).We study the relevance of the hopping superpotential in this section.

The?rst comment to be made is that the superpotential is not made irrelevant by wave-function renormalization at one-loop.This is because the anomalous dimensions of the ?avors and link?elds are all negative at one loop.At this order,the anomalous dimensions are given by(22),where all of the gauge couplings are equal by virtue of the Z k symmetry of the?xed point.From these anomalous dimensions we determine the R-charges of the?elds

and the operator

3+

γF

3?

(g?)2

3N

r Q=2

3?

2

4π2

(N2?1)

F i Q i F i+1?2?(g?)2

3N

<2.(23)

Hence,we see explicitly that the R-charge of the superpotential is less than2perturbatively, and hence the hopping terms are relevant deformations near the perturbative?xed point with vanishing superpotential.

To address the issue nonperturbatively we use the result of a-maximization(6).The question is whether or not the R-charges at the stable?xed point satisfy r

F i Q i F i+1=2?

3N2f?6NN f+(2N?N f)

6N2?3N2f.(24)

One can check from this expression that in the naive conformal window N/2

i Q i F i+1

=2 as expected because that is where the IR theory becomes free.Hence,the hopping super-potential is a relevant deformation of the theory in the conformal window,and we conclude that the hopping superpotential will indeed survive the RG?ow down to the scale of the highest KK massΛKK.

E.Z k Symmetric Fixed Point with Hopping Superpotential

If we assume that the hopping superpotential does not eliminate the interacting IR?xed point,then we can calculate the?xed point anomalous dimensions of the?elds in the presence of the superpotential.We are still considering the origin of moduli space,i.e.vanishing link ?eld vevs,so strictly speaking this theory describes an approximate?xed point at energies much higher thanΛKK.Assuming we can neglect the mass terms in the superpotential at this scale,the anomaly freedom condition(4)is supplemented with the condition that the superpotential terms

F

,which will not be true in the more general models which follow).Together with(4),this determines r F=1and r Q=0,which implies that the mass

term is in fact marginal at the?xed point,and for consistency with unitarity we would also conclude that the operators( k i=1Q i)n are free and have dimension1.These operators are related to Wilson loops around the extra dimension,and the interpretation of the fact that those operators are free above the scale of the highest KK mass deserves exploration.

III.THE MOOSE WITH A BRANE

In this section we add additional?avors localized to one lattice site,which we label the ?rst lattice site and refer to as the brane.We still consider the circular moose model given by the table(1).The result of a-maximization in this case is sensitive to the number of lattice sites,and becomes numerically unwieldy with more than a few lattice sites.All analytic expressions in this section are valid when there is no accidental symmetry.We numerically study some examples with two,three and four lattice sites,and give more general results where possible.In the numerical examples here and in the following section we account for all accidental symmetries that are required by unitarity.

As before we consider the circular moose model in the naive conformal window N/2< N f<2N?N b.To analyze the stability of the?xed points we will use the same techniques as with the braneless moose.We only analyze?xed points whose structure and stability analysis di?ers from the braneless case.Type A?xed points for the moose with a brane, i.e.?xed points with a single nonvanishing SU(N)i gauge coupling,can be divided into two types which we call type A1and type A2.A type A1?xed point is a?xed point with the brane gauge coupling nonvanishing.A type A2?xed point is a?xed point with the gauge coupling next to the brane nonvanishing(g2or g k).At an A1type?xed point,the e?ective theory is like SQCD with(N+N f+N b)?avors.The R-charges are given by

r=

N f+N b

16π22N f N+3N b N?N2f?N f N b

8π2 (26)

and for more than two sites

βII2=?g32

(N+N f+N b) 1?g22N

The beta functions in (26)

and

(27)are

positive in the whole naive conformal window.Hence,Type A1?xed points are always unstable to development of neighboring gauge couplings.The Euler anomaly at the type A1?xed point is given by:

a ?

1=1

16π

22NN f ?N 2f ?N b N ?N b N f 8π2 (29)for more than two sites

βII 1=?g 31(N +N f ) 1?g 21

N 6)N ,

with more than two lattice sites the A2type ?xed point will be stable for

N f >

32N b +125N 2?28N b N +4N 2b (31)when N b >1/2(7?2√48(N +N f )2 k 10N 4+22N 3N f +14N 2N 2f ?9N 2?18NN f ?9N 2f .+2NN 3f ?20N 4+12N 3N f +3N 2N 2f +2N b N 3+4N b N 2N f +2NN b N 2f ?2NN 3f (32)

For ?xed points with two nonvanishing gauge couplings,we will consider four di?erent cases.(a )First we consider a case when sites 2and 3(or k ?1and k )turn on.Perturbing

FIG.3:Numerator of the beta function(19)for(a)k>2,N=6,N b=5;and(b)k=2,N=4, N b=2.

the?xed point by the SU(N)1coupling,the e?ective theory for this type of?xed point is as follows:

U(1)R

r F

Q2

11r F

N f11

1r F

N f11

F12/3

(N f+N b)F111

(33)

The R charges for this kind of?xed point is the same as in equation(14)for r F and equation (15)for r2.The beta function by perturbing the brane gauge coupling for k=3is,

βBI1=?g31

(N2?4NN f?4N2f) 1?g21N

73N2?4NN f?4N2f+4N b N2f (34)

托福阅读技巧集合2(Reading skill)

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费者一种轻松愉悦的享受。 3.所选择的超市这个拍摄地点,贴近了消费者的现实生活,从侧面反映了产品的价格较为平易近人,消除了消费者对于价格偏贵产品的抵触心理。 4.“关爱牙齿,更关心你”的广告语借产品表达了人与人互相关爱的态度,可以让消费者感觉到自己被充分的重视。 5.益达广告中的代言人都是一些一线的当红明星,利用这种明星效应以及明星的影响力极大地吸引了消费者的购买兴趣。 缺点: 1.广告表现上给会给消费者留下情侣口香糖的模糊概念,表达的情感对情侣以外的目标消费者吸引力和亲和感明显不足。 第二篇:益达广告 益达广告评析 说到口香糖很多人自然会想起曾经在电视荧屏上风靡一时的绿箭口香糖。绿箭口香糖以“清新口气”的主题曾经得

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