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Determination of the b-mass using renormalon cancellation

a r X i v :h e p -p h /0411370v 1 29 N o v 2004

Carlos Contreras ?

Dept.of Physics,Universidad T′e cnica Federico Santa Mar′?a,

Valpara′?so,Chile USM-TH-160

February 2,2008

Abstract

Methods of Borel integration to calculate the binding ground energies and mass of b ˉb quarkonia are presented.The methods take into account the leading infrared renormalon structure of the ”soft”binding energy E (s )and of the quark pole masses m q ,and the property that the contributions of these singularities in M (s )=2m q +E (s )cancel.The resummation formalisms are applied to quantities which

do not involve renormalon ambiguity,such as

m q and αs (μ).1Introduction

The calculation of binding energies,masses of heavy quarkonia q ˉq and another physical parameter using

renormalon method has attracted the attention recently.The calculations,based on perturbative expansions,are primarily due to the knowledge of up to N 2LO term (～α3s )of the static quark-antiquark potential V (r )

[1,2]and partial knowledge of the N 3

LO term there,and the ultrasoft gluon contributions to a corresponding e?ective theory N 3LO Hamiltonian [3,4,5];and the knowledge of the pole mass m q up to order ～α3s [6,7].Another impetus in these calculations was given by the observation of the fact that the contributions of the leading infrared (IR)renormalon singularities (at b =1/2)of the pole mass m q and of the static potential V (r )cancel in the sum 2m q +V (r )[8,9,10].Consequently,this cancellation e?ect must be present also in the total quarkonium mass M =2m q +E q ˉq [11,12],or more precisely,in M (s )=2m q +E (s )where E (s )is the hard+soft part of the binding energy,i.e.,the part which includes the contributions of relative quark-antiquark momenta |k 0|,|k |≥m q αs ,i.e.,soft/potential scales (predominant)and higher hard scales (smaller contributions).In addition,the binding energy has contribution E q ˉq (us )from the ultrasoft momenta

regime |k 0|,|k |～m q α2

s .The ultrasoft contribution is not related to the b =1/2renormalon singularity,since this singularity has to do with the behavior of theory in the region which includes the hard (～m q )and soft/potential (～m q αs )scales.

This contribution is based on ref.[13]and we present numerical calculation of the binding ground

energies E b ˉb (separately the s and the us parts)and the mass (2m b +E b ˉb )of the heavy b ˉb system,by taking

into account the leading IR renormalon structure of m b and E b ˉb (s ),and combining some features as:(a)the mass that we use in the perturbation expansions is a renormalon-free mass [10,11,14,15,16]which we choose to be the m q ≡m q );(b)Borel integrations [12]are used to perform resummations.

Before resummations we perform separation of the soft/potential(s)and ultrasoft(us)part of the binding energies,and apply the renormalon-based Borel resummation only to the s part.The renormalization scales used in the Borel resummations areμh～m q(hard scale)for2m q,and m qαs≤μs

MS renormalon-free mass m q(μ= MS),using elements of the renormalization group approach of Ref.[11]and the bilocal expansion method Refs.[17,12].The ratio m q/MS scheme which is at present known to order～α3s(Ref.[6]for～α2s;[7]for～α3s.Let us de?ne

S≡m q

m q

?1=

m2q),whileβ0=(11?2n f/3)/4andβ1=(102?38n f/3)/16are the renormalization scheme independent coe?cients with n f=n?being the number of light active?avors(quarks with masses lighter than m q).The natural renormalization scale here isμ=μh～m q(hard scale).

Let us consider S as a function of the running coupling a(Q)=α(q)/π,and the perturbation expansion

for S reads:

S(a(Q))=

∞

n=0s n a n,(6)

which has to be summed.Then,one de?nes the Borel transform as

B(b,Q)=

∞ n=0s n b n

and in our case B S (b )is known to order ～b 2

B S (b ;μ)=

1!β0

b +

r 2

d ln μ2=

?β0a 2(μ)(1+c 1a (μ)+c 2a 2(μ)+···)?

(10)

2β0

a (μ)

dx 1

x 2(1+c 1x )

β0

1+c 1a (μ)2β0a (μ)

1+c 1a (μ)

2β0

a (μ)

(c 2+c 3x +c 4x 2+···)

2β0a (μ)

a (μ)?νc ?ν

1+∞

k =1

r k a k (μ)

,(13)

where

r 1

=(c 21?c 2)8β20

(c 21?c 2)2?2β0(c 3

1?2c 1c 2+c 3) ,

r 3

(1?

2b )1+ν

∞

k =1

c k (1?2b )k

+B (an .)

(b ;μ),(15)

c 1= r 1

(2β0)2ν(ν?1)

c 3= r 3

and B(an.)

(b;μ)is analytic on the disk|b|<1.The

m q

R S(b=1/2),(17)

and according to(15)

R S(b;μ)≡(1?2b)1+νB S(b;μ).(18) Then,the N m is estimated(see Ref.[31]),using R S(b)TPS and Pad′e approximation[1/1]:

N m(n f=4)=0.555±0.020,(19) The bilocal expansion(15)has then for the analytic part the polynomial

B(an.) S (b;μ)=h(m)

h(m)

2!β20

b2,(20)

h(m)

Γ(ν+1?n)

,(21)

where,by convention,r0= c0=1.Then,the bilocal formula,is

B S(b;μ)(biloc.)=N mπ

m q

k!βk0

b k.(22)

Applying the(generalized)principal value(PV)prescription for the Borel integration

S(a)=1

β0a(μ) B S(b;μ),(23)

we obtain the pole mass m q in terms of the mass

m q<1).The R-method (Refs.[18,19])consists in the Borel integration of the function(18)

S=1

β0a(μ) R S(b;μ)m b=4.23GeV and we vary the

values of the residue parameter N m,the bilocal method gives,at

μ/

the central values of N m are used,the variation of the obtained values of m q withμ,whenμ/

m q,and will be denoted later in this work

asμm in order to distinguish if from the“soft”renormalization scaleμused in the analogous renormalon-based resummations of the(hard+)soft binding energy E qˉq(s)(m qαs)in Sec.4.The fact that

the two renormalization scales are di?erent does not a?ect the mechanism of the(b=1/2)renormalon

cancellation in the bilocal calculations of the meson mass(2m q+E qˉq(s)),because the renormalon ambiguity in each of the two terms is renormalization scale independent～ Λ,as seen by Eqs.(13)–(2).On the other hand,if R-type methods(24)[cf.also Eq.(17)]are applied for the resummations of2m q and E qˉq(s),the

renormalon ambiguities are renormalization scale independent in the approximation of the one-loop RGE

running,and the renormalon cancellation is true at this one–loop level.

3Separation of the soft and ultrasoft contributions

The perturbation expansion of the(hard+soft+ultrasoft)binding energy E qˉq of the qˉq heavy quarkonium

vector(S=1)or scalar(S=0)ground state(n=1,?=0)up to the N3LO O(m qα5s)was given in[33]The reference mass scale used was the pole mass m q.The ground state energy expansion has the form

E qˉq=?

193

10955

n2f?

2592

(?67584+4096n f)S(S+1),(31)

k 3,0=

7839.82?1223.68n f +69.4508n 2f ?1.21475n 3

+(?109.05+4.06858n f )S (S +1)?

π243

(32)

Here,a 3have been estimated in Ref.

[31],obtained

from

the condition of renormalon cancellation in the sum

(2m q +V q ˉq (r ))

π3

k 3,0(us,ret .)=?

L E 1≈+41.014,

(34)

where L E 1≈?81.538is the QCD Bethe logarithm -see Refs.[5,3].

2.The non-retarded part can be calculated as expectation value of the us e?ective Hamiltonian H us in

the Coulomb (i.e.,leading order)ground state |1 ,where H us (in momentum space)was derived in Refs.[5,3].Direct calculation of the expectation value,here in coordinate space,then gives:

4π51

π5m q a 4(μ)

[

(E C 1

)2+5

a 3(μ) 1|

m q 1|

a (μ)

m 2q

1|{?r ,

m q α2s (μ)

+0.9511

(36)

Here,E C1=?(4/9)m qα2s(μ)is the Coulomb energy of the state|1 ,andμf is the factorization energy between the soft(～m qαs)and ultrasoft(～m qα2s)scale.

The s–us factorization scaleμf can be estimated as being roughly in the middle between the s and us energies on the logarithmic scale[31]

μf ≈(E S E US)1/2 =κm qαs(μs)3/2,(37)

whereκ～1andμs≈E S(μ).Therefore,the ultrasoft part of the N3LO coe?cient k3,0can be rewritten, by Eqs.(34),(2)and(37),in terms of the s–us parameterκas

m qαs(μs).

The formal perturbation expansions for the separate soft and ultrasoft parts of the ground state binding energy(25)are then

E qˉq(s)=?

m qπ2a2(μ) a3(μ)k3,0(us)+O(a4) .(40)

The energy E qˉq(s)(39)contains the leading IR renormalon e?ects,and E qˉq(us)(40)does not.In these expressions,the common factor is the soft scaleμp(μ)=(4/3)m qαs(μ)which is also present as the reference scale in the logarithms L p(μ)=ln(μ/μp(μ))appearing with the coe?cients k i,j(when j≥1)in Eqs.(25), (26).

We will re-express m q everywhere in E qˉq with the renormalon-free mass

m q.

Thus,we will divide the soft binding energy with the quantity m qαs( μ),where μcan be any soft scale.We will?x this scale by the condition μ=(4/3)

m q and powers of a(μ)(cf.Sec.2),and the powers of logarithms ln k[a(μ)]we re-express in terms of ln k[a( μ)].This then results in the following soft binding energy quantity F(s)to be resummed

F(s)≡?

m q.The coe?cient f3depends,in addition,on the parametersκ(37)-(38),μs,and a3(3).The coe?cients f j are written explicitly in Appendix A.The b=1/2renormalon in the quantity F(s)is then of the type of the renormalon of the pole mass m q discussed in the previous Sec.2.

However,if we divided in Eq.(41)by m q instead of

E qˉq(s)di?erent usually by not more than O(101MeV)(we checked this numerically).We will brie?y refer to these approaches later in this Section as“pole mass”approaches.A version of such pole mass bilocal approach was applied in Ref.[12]for resummation of the unseparated E qˉq(s+us).

The ultrasoft part(40),on the other hand,has no b=1/2renormalon.The mass scale used there should also be renormalon free(

m b from the known mass of the vector bˉb[Υ(1S)]

The soft binding energy quantity to be resummed is F(s)of Eq.(41).However,in the N3LO coe?cient f3 we have dependence on a3,and on the s-us factorization scale parameterκ～1.Then,the value ofκ(37) is obtained requiring that the residue parameter values be reproduced from the Borel transform of the soft binding energy quantity F(s)of Eq.(41).

Similarly as in Eq.(15),we have

B F(s)(b;μ)=N m 9

(1?2b)1+ν 1+∞ k=1 c k(1?2b)k

+B(an.)F(s)(b;μ)(43)

where the factor in front of the singular part was determined by the condition of renormalon cancellation of the sum2m q+E qˉq(s).We now de?ne in analogy with Eq.(18)

R F(s)(b;μ;μf)=(1?2b)1+νB F(s)(b;μ;μf).(44) Here we denoted,explicitly the dependence on the factorization scaleμf and

N m=

2πm q a( μ)

m b=4.23GeV; μ=1.825GeV(≈μs) andαs( μ;n f=4)=0.3263(≈αs(μs;n f=4)=0.326)[from:αs(mτ;n f=3)=0.3254,i.e.,αs(M Z)=0.1192 [32].For the RGE running,we use four-loop

a3of the N3LO coe?cient of the static qˉq potential.Therefore,we will consider the variations of N m and of κto be related by a one-to-one relation,while the variations of a3will be considered as independent.

In this way,we have the following value for the s-us factorization scale parameter:

N m=0.555±0.020?κ=0.59±0.19(n f=4,S=1)(46) and thus we obtain the N3LO TPS(41)for the soft part of the ground binding energy.Now with the value ofκ,we can perform the resummation of the soft part of the ground binding energy.The full bilocal method [17,12]can be performed as in Sec.2,Eqs.(22)and(23).Therefore

B(biloc.) F(s)(b;μ)=N m

(1?2b)1+ν

k=0 c k(1?2b)k+3 k=0h k 2π

m q a( μ)

(2β0)k

n=0 c nΓ(ν+k+1?n)

2π

m q a( μ)

8σ2

)(1?2b)2+( c3+ c1

8σ2

(1?2b)2 +3 k=01

In Fig.3(a)we present the Borel-resummed soft part of ground state energy for the bottonium (S =1),as a function of the σparameter of method (49).

Finally,the results for the soft binding energy E b ˉb (s )of the ground state of bottonium using the R -method [18,19],where we resum the function R F (s )(b ;μ)(44)and then employ the (PV)Borel resummation as written in Eq.(24)(with R F (s )instead of R S there).,as functions of the renormalization scale μ,are presented in Fig.4(a).We observe from the Figure that the bilocal “σ-regularized”method (49)(σ=0.36)gives the TPS and PA results closer to each other.The methods σ-TPS,σ-PA,and R -PA give similar results in the entire presented μ-interval.R -TPS appears to fail at low μ(≈m b αs ≈1-2GeV).In Fig.4(b)we include,for comparison,the simple TPS evaluation of E b ˉb (s ),according to formula [cf.Eq.(41)]

F (s )(TPS)≡?

4π1m b αs ( μ)

E (p)q ˉq (us ;μ)=k 3,0a 4

(μ).(52)

4.1Extraction of bottom mass

The estimate of the perturbative part is given in Eq.(42),where it was essential to take for the renormal-ization scale a us scale μ～μus ～m q α2s .

For the bottonium case,this scale is below 1GeV,the energy at which we cannot determine pertur-batively αs (μ).This indicates that in the bottonium the us part of the binding energy has an appreciable nonperturbative part.The lowest energy at which we can still determine perturbatively αs is μ≈1.5-2.0

GeV,giving αs (μ)≈0.30?0.35.Although this is a soft scale for b ˉ

b ,we will use this also as an ultrasoft scale.Then by Eq.(42)

E b ˉb (us )

(p)

≈?4m q π2k 3,0(us )a 5(μus )≈(?150±100)MeV .(53)

The nonperturbative contribution coming from the gluonic condensate is given by [39]

E b ˉb (us )(np)

≈425 4m b αs (μus ) ?4 a (μus )G μνG μν ≈(50±35)MeV ,(54)

where we

used

measured MΥ(1S)=9460MeV with virtually no uncertainty[44].Therefore,the pure perturbative“soft”mass is

MΥ(1S;s)=2m b+E bˉb(s)=9535?106MeV,(57) where the uncertainty is dominated by the uncertainty of the us regime contribution.Our numerical results for E bˉb(s)and for m b allow us,by varying the input value of

MS)=40.For 2m b we apply the bilocal-TPS and R-TPS method,with renormalization scaleμm/

m b≡m b)are then

m b=4.231±0.068GeV.(62) 5Comparisons and conclusions

Our result for the mass m b. The only input parameter common to all these methods isαs.Our central value wasαs(mτ)=0.3254 [?αs(M Z)=0.1192]since such[32],or similar[45,19],values follow from the(nonstrange)semihadronic τdecay data which are very precise[46].On the other hand,the world average as of September2002is αs(M Z)=0.1183±0.0027[47].Most of the authors during the last four years used central valueαs(M Z)≈0.118.Therefore,for comparisons,we convert our results(58)to this central value ofαs–more speci?cally, fromαs(M Z)=0.1192±0.0015to0.1180±0.0015.This can be easily done by inspecting in Table1the column underαs,an increase in the central values of

m b=4.241±0.068GeV average when:αs(M Z)=0.1180±0.0015.(63) There are two important numerical e?ects in our result.The?rst is the separate evaluation of the “perturbative”ultrasoft energy part at the corresponding low renormalization energy(1.5?2GeV),Eqs.(42) and(53).If we had not separated the(“perturbative”)ultrasoft from the soft part of the binding energy, the use of the common renormalization energy scaleμ(≈3GeV)in the resummation then would have given us the central value of E bˉb(us)by about+100MeV higher.Then the extracted value of

m b≈4.195±0.068,with the central value close to that of L03in Table2.On the other hand,that latter value is quite clearly lower than the value PS02in Table2,by about 150MeV,principally because of the b=1/2renormalon e?ect which were taken into account here and in Ref.[12].Thus,the renormalon e?ect brings down the extracted central value of

Acknowledgments

I would like to thank G.Cvetic for helpful and discussions.This work was supported by Project USM-110321 of the UTFSM.

A Coe?cients for the expansion of the soft binding energy

We write down here the explicit coe?cients f j of the expansion(41)for the soft part of the ground state binding energy.The logarithms appearing in these expressions involve three scales[μ, μ,μ( μ)= (4/3)

m q

μ( μ) ,L2=ln

,Lμ=ln μ .(64)

The coe?cients f j are

f1=1

(?11?6L1+3Lμ+3L2)n f.(65)

f2=f(0)

2+f(1)

n f+f(2)

n2f,(66)

f(0)

= 381.67+90.75L21+30.25L2μ+L1(246.42?121Lμ?60.5L2)?48.5L2

+Lμ(?205.25+60.5L2)?11.697S(S+1) ,

f(1)

= ?42.7469?11L21?3.6667L2μ+Lμ(26.6944?7.3333L2)+6.8056L2

+L1(?33.0556+14.6667Lμ+7.3333L2) ,

f(2)

= 1.16286+(3/9)L21+(1/9)L2μ+L1(1?(4/9)Lμ?(2/9)L2)

+Lμ(?0.81482+(2/9)L2)?0.18518L2 .

f3=f(0)

3+f(1)

n f+f(2)

n2f+f(3)

n3f,(67)

f(0)

= 6726.11+665.5L31?166.375Lμ(40.802+(?10.599+Lμ)Lμ) +L21(2381.5?1497.38Lμ?499.125L2)?871.429L2

?499.125(?1.884+Lμ)LμL2?201.438L22+L1(7457.17

?497.292L2+Lμ(?4346.38+998.25Lμ+998.25L2))

?257.341(0.2112+L1?0.75Lμ?0.25L2)S(S+1)

?61.4109(?6.1394+S(S+1))ln(αs(μs))+440.172ln(κ)+2a3/43 ,

f(1)

= ?1274.33?1277.92L1?471.125L21?121L31+1182.32Lμ

+843.667L1Lμ+272.25L21Lμ?335.813L2μ?181.5L1L2μ

+30.25L3μ+124.501L2+108.361L1L2+90.75L21L2

?186.708LμL2?181.5L1LμL2+90.75L2μL2+36.729L22

+(4.0686+15.5964L1?11.6973Lμ?3.8991L2)S(S+1) ,

= 70.8992+70.2453L1+28.9722L21+7.3333L31?65.9925Lμ

f(2)

?51.6667L1Lμ?16.5L21Lμ+20.5972L2μ+11L1L2μ?1.8333L3μ

?5.1939L2?6.5741L1L2?5.5L21L2+10.9167LμL2

+11L1LμL2?5.5L2μL2?2.0972L22 ,

= ?1.21475?1.21714L1?(5/9)L21?0.14815L31+1.16286Lμ

f(3)

+L1Lμ+(1/3)L21Lμ?0.40741L2μ?(2/9)L1L2μ+0.03704L3μ

+0.05429L2+(1/9)L1L2+(1/9)L21L2

?0.18518LμL2?(2/9)L1LμL2+(1/9)L2μL2+0.03704L22 .

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4.92

4.93 4.94 4.95 4.96

4.97 4.98 0.6 0.8

1.2 1.4

m b [G e V ]

μ/m

b ?(a)

m b = 4.23 GeV

?αs (M Z )=0.1192

biloc.(N m =0.535)biloc.(N m =0.555)biloc.(N m =0.575)

R-meth.

Figure 1:

The (PV)pole mass of the bottom quark,as function of the renormalization scale μ.The input parameters used were

MS)was taken to be αs (m τ)=0.3254

([32])corresponding to αs (M Z )=0.1192.

0.45

0.5

0.55

0.6

0.65

0.4 0.6

0.8

1.2

N m

PA[2/1]

N 3

LO TPS

N 2

LO TPS

(a)S=1n f = 4

μ =3 GeV

m b = 4.23 GeV

?αs (M Z )=0.1192 0.4

0.45 0.5 0.55 0.6

0.65 0.7

0.75 0.8 0.85 0.9 1.5

2.5

3.5 4

4.5

N m

μ[GeV]

PA[2/1]

N 3

LO TPS N 2

LO TPS

PA[2/1](s+us)

(b)

S = 1n f = 4

κ =0.59

m b = 4.23 GeV

?αs (M Z )=0.1192

Figure 2:The residue parameter value N m as calculated from the soft part of the binding energy of the bottonium according to Eq.(45),(a)as a function of the s -us factorization scale parameter κ(37),at μ=3GeV;(b)as a function of the renormalization scale μ,at κ=0.59.Further explanations given in the text.In Fig.(a),the known values of N m are denoted as dotted horizontal lines.

-0.45

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-0.35

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-0.25

-0.2

0.3 0.4 0.5 0.6

0.7 0.8 0.9 1

E b b (s ) [G e V ]

σTPS

σPA

σ=∞ TPS σ=∞ PA

(a)

n f =4S=1

μ =3 GeV κ =0.59

m b = 4.23 GeV

?αs (M Z )=0.1192

Figure 3:(a)Soft part of the ground state binding energy of b ˉb ,evaluated with the (PV)Borel-resummed

expression (49),as a function of the method parameter σ.Details are given in the text.

-0.45

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E b b (s ) [G e V ]

μ[GeV]

σTPS σPA

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R-PA σ=∞TPS σ=∞PA

(a)

S = 1

κ =0.59

m b = 4.23 GeV

?αs (M Z )=0.1192

σTPS

σPA R-TPS R-PA σ= ∞TPS σ= ∞PA

-1-0.8-0.6

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-0.2

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3.5

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E b b (s ) [G e V ]

μ[GeV]

σTPS

R-PA

N 3

LO TPS

N 2

LO TPS

E bb (p)

(us;μ)

κ =0.59S = 1

(b)

σTPS R-PA

Figure 4:(a)Soft part E b ˉb (s )of the ground state binding energy of b ˉb ,evaluated with four di?erent methods

involving (PV)Borel resummation,as functions of the renormalization scale μ.Details are given in the text.In Fig.(b)the simple TPS results for E b ˉb (s )are included [Eq.(51)],as well as the “perturbative”ultrasoft

part E (p)

b ˉb

(us ;μ)[Eq.(52)].Table 1:The separate uncertainties δ

m b from various sources:1.)us

[δE b ˉb (us )

(p+np)

=?100±106MeV];2.)μ=3±1GeV;3.)μm =σ-TPS

?49

+9?4?13?3+2?8+4?5+49?13+2+14+2?2+8?9+5

R -TPS

?50?4+4?8?9?300?5+50+45?40+10+11+300+5

Table2:Recently obtained values of(m b mass obtained fromΥsum rules or from spectrum of the Υ(1S)resonance.Wherever needed([33,12]),the central mass values were adjusted to the common input central valueαs(M Z)=0.118.

reference method order m b)(GeV)

The way常见用法

The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法）举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法：在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because

The way的用法及其含义(二)

The way的用法及其含义（二）二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中

(完整版)the的用法

定冠词the的用法：定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...

“the way+从句”结构的意义及用法

“tｈｅwａｙ+从句”结构的意义及用法首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法

不对。Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.

way 用法

表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如：我不喜欢他说话的态度。正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。一、in a way的用法表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。三、by the way的用法

The way的用法及其含义(一)

The way的用法及其含义（一）有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析：一、the way作先行词，后接定语从句以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.

way 的用法

way 的用法【语境展示】 1. Now I’ll show you how to do the experiment in a different way. 下面我来演示如何用一种不同的方法做这个实验。 2. The teacher had a strange way to make his classes lively and interesting. 这位老师有种奇怪的办法让他的课生动有趣。 3. Can you tell me the best way of working out this problem? 你能告诉我算出这道题的最好方法吗？ 4. I don’t know the way (that / in which) he helped her out. 我不知道他用什么方法帮助她摆脱困境的。 5. The way (that / which) he talked about to solve the problem was difficult to understand. 他所谈到的解决这个问题的方法难以理解。 6. I don’t like the way that / which is being widely used for saving water. 我不喜欢这种正在被广泛使用的节水方法。 7. They did not do it the way we do now. 他们以前的做法和我们现在不一样。【归纳总结】 ●way作“方法，方式”讲时，如表示“以……方式”，前面常加介词in。如例1； ●way作“方法，方式”讲时，其后可接不定式to do sth.，也可接of doing sth. 作定语，表示做某事的方法。如例2，例3；

the-way-的用法讲解学习

t h e-w a y-的用法

The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.

way的用法总结大全

way的用法总结大全 way的用法你知道多少，今天给大家带来way的用法，希望能够帮助到大家，下面就和大家分享，来欣赏一下吧。 way的用法总结大全 way的意思 n. 道路,方法,方向,某方面 adv. 远远地，大大地 way用法 way可以用作名词 way的基本意思是“路,道,街,径”,一般用来指具体的“路,道路”,也可指通向某地的“方向”“路线”或做某事所采用的手段,即“方式,方法”。way还可指“习俗,作风”“距离”“附近,周围”“某方面”等。 way作“方法,方式,手段”解时,前面常加介词in。如果way前有this, that等限定词,介词可省略,但如果放在句首,介词则不可省略。

way作“方式,方法”解时,其后可接of v -ing或to- v 作定语,也可接定语从句,引导从句的关系代词或关系副词常可省略。 way用作名词的用法例句 I am on my way to the grocery store.我正在去杂货店的路上。 We lost the way in the dark.我们在黑夜中迷路了。 He asked me the way to London.他问我去伦敦的路。 way可以用作副词 way用作副词时意思是“远远地,大大地”,通常指在程度或距离上有一定的差距。 way back表示“很久以前”。 way用作副词的用法例句 It seems like Im always way too busy with work.我工作总是太忙了。 His ideas were way ahead of his time.他的思想远远超越了他那个时代。 She finished the race way ahead of the other runners.她第一个跑到终点,远远领先于其他选手。 way用法例句

the_way的用法大全教案资料

t h e_w a y的用法大全

The way 在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的

the way 的用法

The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.

the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms. 从那天起,其他同学是夹着书本来上课,而他们却带着"失败"的思想负担来上课.

The way的用法及其含义(三)

The way的用法及其含义（三）三、the way的语义 1. the way=as（像） Please do it the way I’ve told you.请按照我告诉你的那样做。 I'm talking to you just the way I'd talk to a boy of my own.我和你说话就像和自己孩子说话一样。 Plant need water the way they need sun light. 植物需要水就像它们需要阳光一样。 2. the way=how（怎样，多么） No one can imagine the way he misses her.没人能够想象出他是多么想念她！ I want to find out the way a volcano has formed.我想弄清楚火山是怎样形成的。 He was filled with anger at the way he had been treated.他因遭受如此待遇而怒火满腔。That’s the way she speaks.她就是那样讲话的。 3. the way=according as （根据） The way you answer the questions, you must be an excellent student.从你回答问题来看，你一定是名优秀的学生。 The way most people look at you, you'd think a trash man was a monster.从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物。 The way I look at it, it’s not what you do that matters so much.依我看，重要的并不是你做什么。 I might have been his son the way he talked.根据他说话的样子，好像我是他的儿子一样。One would think these men owned the earth the way they behave.他们这样行动，人家竟会以为他们是地球的主人。

way的用法

一．Way：“方式”、“方法” 1.表示用某种方法或按某种方式 Do it (in) your own way. Please do not talk (in) that way. 2.表示做某事的方式或方法 It’s the best way of studying [to study] English.。 There are different ways to do [of doing] it. 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句正：I don’t like the way he spoke. I don’t like the way that he spoke. I don’t like the way in which he spoke.误：I don’t like the way how he spoke. 4. the way 的从句 That’s the way (=how) he spoke. I know where you are from by the way you pronounce my name. That was the way minority nationalities were treated in old China. Nobody else loves you the way(=as) I do. He did not do it the way his friend did. 二．固定搭配 1. In a/one way：In a way he was right. 2. In the way /get in one’s way I'm afraid your car is in the way， If you are not going to help，at least don't get in the way. You'll have to move－you're in my way. 3. in no way Theory can in no way be separated from practice. 4. On the way (to……) Let’s wait a few moments. He is on the way Spring is on the way. Radio forecasts said a sixth-grade wind was on the way. She has two children with another one on the way. 5. By the way By the way，do you know where Mary lives? 6. By way of Learn English by way of watching US TV series. 8. under way 1. Elbow one’s way He elbowed his way to the front of the queue. 2. shoulder one’s way 3. feel one‘s way 摸索着向前走；We couldn’t see anything in the cave, so we had to feel our way out 4. fight/force one’s way 突破。。。而前进The surrounded soldiers fought their way out. 5.. push/thrust one‘s way（在人群中）挤出一条路He pushed his way through the crowd. 6. wind one’s way 蜿蜒前进 7. lead the way 带路，领路；示范 8. lose one‘s way 迷失方向 9. clear the way 排除障碍，开路迷路 10. make one’s way 前进，行进The team slowly made their way through the jungle.

the way的用法大全

在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms.

“the-way+从句”结构的意义及用法知识讲解

“the way+从句”结构的意义及用法首先让我们来看下面这个句子： Read the following passage and talk about it with your classmates. Try to tell what you think of Tom and of the way the children treated him. 在这个句子中，the way是先行词，后面是省略了关系副词that 或in which的定语从句。下面我们将叙述“the way+从句”结构的用法。 1．the way之后，引导定语从句的关系词是that而不是how,因此，<<现代英语惯用法词典>>中所给出的下面两个句子是错误的：This is the way how it happened. This is the way how he always treats me. 2. 在正式语体中，that可被in which所代替；在非正式语体中，that则往往省略。由此我们得到the way后接定语从句时的三种模式：1) the way +that-从句2) the way +in which-从句3) the way +从句例如：The way(in which ,that) these comrades look at problems is wrong.这些同志看问题的方法不对。

The way(that ,in which)you’re doing it is completely crazy.你这么个干法，简直发疯。 We admired him for the way in which he faces difficulties. Wallace and Darwin greed on the way in which different forms of life had begun.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 This is the way (that) he did it. I liked the way (that) she organized the meeting. 3.the way(that)有时可以与how(作“如何”解)通用。例如： That’s the way (that) she spoke. = That’s how she spoke. I should like to know the way/how you learned to master the fundamental technique within so short a time. 4．the way的其它用法：以上我们讲的都是用作先行词的the way，下面我们将叙述它的一些用法。

定冠词the的12种用法

定冠词the的12种用法定冠词the 的12 种用法，全知道?快来一起学习吧。下面就和大家分享，来欣赏一下吧。定冠词the 的12 种用法，全知道? 定冠词the用在各种名词前面，目的是对这个名词做个记号，表示它的特指属性。所以在词汇表中，定冠词the 的词义是“这个，那个，这些，那些”，可见，the 即可以放在可数名词前，也可以修饰不可数名词，the 后面的名词可以是单数，也可以是复数。定冠词的基本用法： (1) 表示对某人、某物进行特指，所谓的特指就是“不是别的，就是那个!”如： The girl with a red cap is Susan. 戴了个红帽子的女孩是苏珊。 (2) 一旦用到the，表示谈话的俩人都知道说的谁、说的啥。如：

The dog is sick. 狗狗病了。(双方都知道是哪一只狗) (3) 前面提到过的，后文又提到。如： There is a cat in the tree.Thecat is black. 树上有一只猫，猫是黑色的。 (4) 表示世界上唯一的事物。如： The Great Wall is a wonder.万里长城是个奇迹。(5) 方位名词前。如： thenorth of the Yangtze River 长江以北地区 (6) 在序数词和形容词最高级的前面。如： Who is the first?谁第一个? Sam is the tallest.山姆最高。但是不能认为，最高级前必须加the，如： My best friend. 我最好的朋友。 (7) 在乐器前。如： play the flute 吹笛子

Way的用法

Way用法 A:I think you should phone Jenny and say sorry to her. B:_______. It was her fault. A. No way B. Not possible C. No chance D. Not at all 说明：正确答案是A. No way，意思是“别想！没门！决不！” 我认为你应该打电话给珍妮并向她道歉。没门！这是她的错。再看两个关于no way的例句：（1）Give up our tea break? NO way! 让我们放弃喝茶的休息时间？没门儿！（2）No way will I go on working for that boss. 我决不再给那个老板干了。 way一词含义丰富，由它构成的短语用法也很灵活。为了便于同学们掌握和用好它，现结合实例将其用法归纳如下：一、way的含义 1. 路线

He asked me the way to London. 他问我去伦敦的路。 We had to pick our way along the muddy track. 我们不得不在泥泞的小道上择路而行。 2. （沿某）方向 Look this way, please. 请往这边看。 Kindly step this way, ladies and gentlemen. 女士们、先生们，请这边走。 Look both ways before crossing the road. 过马路前向两边看一看。 Make sure that the sign is right way up. 一定要把符号的上下弄对。 3. 道、路、街，常用以构成复合词 a highway（公路），a waterway（水路），a railway（铁路），wayside（路边）

way与time的特殊用法

way/time的特殊用法 1、当先行词是way意思为”方式.方法”的时候,引导定语从句的关系词有下列3种形式： Way在从句中做宾语 The way that / which he explained to us is quite simple. Way在从句中做状语 The way t hat /in which he explained the sentence to us is quite simple. 2、当先行词是time时,若time表示次数时,应用关系代词that引导定语从句,that可以省略; 若time表示”一段时间”讲时,应用关系副词when或介词at/during + which引导定语从句 1.Is this factory _______ we visited last year? 2.Is this the factory-------we visited last year? A. where B in which C the one D which 3. This is the last time _________ I shall give you a lesson. A. when B that C which D in which 4.I don’t like the way ________ you laugh at her. A . that B on which C which D as 5.He didn’t understand the wa y ________ I worked out the problem. A which B in which C where D what 6.I could hardly remember how many times----I’ve failed. A that B which C in which D when 7.This is the second time--------the president has visited the country. A which B where C that D in which 8.This was at a time------there were no televisions, no computers or radios. A what B when C which D that