Pole wave-function renormalization prescription for unstable particles
The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules
a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark effective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suffices to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus effectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the phase redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which define the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the effective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coefficient function.possible subtraction terms.Because of theflavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The effective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usfirst present the result for the nonperturbative power corrections.WefindK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we find that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our final result,there is one more effect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the effective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the effective theory[16,18].Onefinds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,wefind stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourfinal result forχ3(y)is not very different from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisfigure one can also deduce how changes in the values of the vacuum condensates would affect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Wefind that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonfield and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesfinancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientific,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisfigure and are both represented by the narrow dots.11。
第二章 波函数
12
解
根据归一化的定义，我们有
2 3 2
2 r / a r / a d r dxdydz 4 r ( e ) e dr 1s 1s 0
4 r 2 e 2 r / a dr a 3
0
归一化的波函数为
~ 1s
1
a 3
e r / a
2
m
2E
d 2 ( ) 0 2 d
首先考虑方程的渐近解
d 2 0, 2 d
2
32
~e
2 / 2
因为波函数在无穷远处为有限，
~e
2 / 2
e
2
2
H ( )
代入薛定谔方程，得
d H dH 2 ( 1) H 0 2 d d
用级数解法，H只能为一个中断多项式，得到
2
2n 1,
n 0, 1, 2, ...
33
1 En (n ) , n 0, 1, 2, ... 2
简谐振子的能谱是等间 隔的, 间距为ħω, 基态能 量不为零, 即零点能量为 ħω/2。
这是微观粒子波粒二象
定态与定态波函数定态薛定谔方程???eru??????????222?哈密顿算符2?242?2ruh????????eh??本征值方程当体系处于能量本征态时粒子的能量有确定值en??n以en表示体系能量算符的第n个本征值??n是与en相应的波函数则体系的第相应的波函数则体系的第n个定态波函数为nietnnrtre?????25nietnnnnnnrtcrtcre?????????转至第三章一维定态问题?具体阐述薛定谔方程的求解过程?波函数的获取方法熟悉几个重要的应用过程中需要的物理26?熟悉几个重要的应用过程中需要的物理模型26一维无限深势阱在一维空间运动的粒子其势场满足?????????axaxxu027?1阱外x?ax?a因为势壁无限高粒子不能穿透阱壁按照波函数的统计解释在阱壁和阱外粒子的波函数为零
材料设计—25-密度泛函理论-基组-缀加平面波和PAW
ik r l
* ' i Ylm (k ) jl (k ) / Rl ( E , )
最后，只要把上面的alm的表达式带入 APW的基函数就可以得到最后的基组形式：
almYlm ( ) Rl ( E , ) ( ) l 0 ml e ik
体现在基函数上，就是在Muffin-tin球内 给APW基函数增加一项对能量求导的项，使得 径向薛定谔方程的解不再是能量本征值的函数 ，而是某一个带选定的能量参数，这就是线性 缀加平面波方法(LAPW)。
径向波函数的在能量E0展开形式：
(E) Rl ( E ) Rl ( E ) R l
M
K
0
M
K
M
K
PAW结合了PP和FLAPW方法 PAW具有FLAPW的精度，芯电子被冻结， 但能够得到真正的价电子波函数，这个对磁 性和光学性质有重要意义。 PAW采用平面波很展开，具有和USPP类似 的效率。程序结构也可以和PP的程序类似 。
PAW是一种全电子的方法！
常见的使用PAW方法的程序：
V (r ) (r R m ) V (r ) (r R m ) 0
Muffin-tin势的选取可有不同的方法。位于原点 的原子产生的势场在该Muffin-tin球中贡献最大， 然后还有周围最近邻原子对该球空间的贡献，然 后还有次近邻及更远的原子的贡献，当然贡献是 逐渐减弱的。 L. F. Mattheiss提出中心原子势场上叠加上周 围原子势场在以该中心原子为原点的球谐函数展 开项，如果只取首项，那么就是球形近似；如果 取更多的展开项，那么就不再有任何形状近似。
在多了一个R的导数项后，自然多了一个待 定系数blm，我们可以采用基函数在球面上连续 以及导数也连续的条件确定这两个系数：
自由粒子的薛定谔方程-2023年学习资料
▲为什么用1y描述波函数而不用Ψ？-因为Ψ是复数，有物理意义的是1少2，而不是Ψ。-经典物理：-一个经典波 以用实数也可以用复数表示，用复数表示仅-仅是为了数学上的方便，实际上只有实部才有物理意义。-量子力学：-所 在量子力学中，用必2-来描述波函数的物理意义。-量子力学的波函数一般必须用复数表示，有物理意义的即-不是实 ，也不是虚部，而是它的绝对值-的平方山2，所以Ψ也叫几率振幅，或几率幅。
2.量子力学对态迭加原理的解释-在Ψ，状态下→无论何时测量某物理量G如能量，-都有一个确定值81-在Ψ2状 下→无论何时测量某物理量G如能量，-都有一个确定值82-根据态叠加原理：-Ψ=C1+C2Ψ2-→体系可能态 ▲在Ψ态下测量力学量G,能得到什么样的结果呢？-在Ψ态下测量力学量G的结果，每次测得的结果是不确定的，-即 能是8，也可能是82但不会是另外的值，而测得8及-82的相对概率是确定的.
实物粒子不会产生或湮灭，必定会在空间某点出现，-在整个空间出现的几率为1-数学上表示为：-F,Pd:=1→ 函数的归一化条件-o0-Wydr=1-00-满足上式的波函数-子，→归一化的波函数-为方便引入符号-<4, =ydr-归一化条件：∫yydr=1一<4，y>=1或<yy>1
量子力学基本假设告诉我们-业与CΨ描写同一量子状态，即描写同一量子-状态的波函数形式是不唯一的，对是不是归 一化的波函数，w=cy，C为常数通常需-要把波函数归一化（利用波函数的归一化条-件。-cy2dx=1→lc
量子力学基本假设告诉我们-归一化常数C的解不确定，可以是正负实数：-也可是复数1esP=e.e5=1.lc ce2-δ为常数，可取任意常实数值-为了方便，一般规定归一化常数C取正实数。-不讨论相因子（δ=0），即归 化的波函数-不会有相因子的不确定性。
有限共轭和无限共轭
有限共轭和无限共轭“有限共轭和无限共轭”是一种物理学概念，用于描述粒子之间的相互作用和能量交换。
在量子力学中，粒子的行为受到波函数描述，而有限共轭和无限共轭也是基于波函数的理论。
首先我们来了解一下波函数。
在量子力学中，波函数描述了一个物理体系（如粒子）的状态。
粒子在空间中的位置和速度不能精确地确定，只能确定其可能存在的区域和可能的运动趋势。
波函数是一个复数函数，它描述了粒子在不同位置和时间的可能存在性和可能的运动趋势。
波函数的模方表示粒子在某个位置上的可能出现概率，即波函数的平方模。
有限共轭和无限共轭理论是基于波函数相互作用的理论。
所谓有限共轭就是针对粒子的外禀性质进行的操作。
在这种操作中，假设粒子的外禀性质（例如位置、动量、自旋等）被测量，并且一组预定的值表示了这些测量结果。
有限共轭波函数可以使量子力学中的运动变得可逆，并且这种可逆性可以被产品化和应用。
而无限共轭则是针对一些不可观测量，例如能量和时间等，进行的操作。
与有限共轭不同的是，无限共轭本质上不会修改波函数，而仅仅是推导可以从波函数中观测到的结果。
通过测量能量和时间等量的不同值，可以使得量子美剧中的运动不可逆。
无限共轭可以用于预测粒子在未来时刻的行为，以及粒子之间能量的传递和交换等。
事实上，有限共轭和无限共轭是密切相关的。
无限共轭可以被用来解决有限共轭中的问题。
例如，有限共轭可以导致干涉，夸克探测等问题，而无限共轭可以通过提高精确度来解决这些问题，这通过减少来源于这些问题的噪音来体现。
因此，这两种理论结合使用，有助于量子力学的研究和应用。
最后，有限共轭和无限共轭的理论可以应用于不同领域，包括量子计算、量子密码学和量子通信等。
这些应用都是建立在波函数的基础上，因此对于波函数的研究具有重要意义。
未来，随着量子技术的不断发展，有限共轭和无限共轭理论的应用将会更加广泛，同时也会推动量子力学研究的深入发展。
Peskin量子场论译文1
Peskin 量子场论: Chapter1湮灭中的对乘积：QED 是关于电子和光子的量子理论，它可能是我们现有的最好的基本物理理论，由Dirac 方程和Maxwell 方程组成，它们主要由相对论不变性决定，这些方程的量子力学解给出了宏观的和微观（比质子小几百倍）的电磁现象细致预测。
Feynman 图提供了一种优美的计算程序，通常通过Feynman 图来写出相应过程的量子力学振幅数学表达式。
考虑在质心系中，多数粒子物理实验涉及散射，QFT 中最一般计算的量就散射截面，反粒子的存在实际上是QFT 的预言，实验中为了测量湮灭概率，将一束电子射向正电子束，散射截面作为可测量量是质心能量，入射与出射夹角的函数，质心系中有，假定射束能量（动能？）远大于电子或者μ子的质能（黑体表示3动量，斜体表示4动量），因为，自旋都是1/2，我们必须具体表示出它们的自旋取向，将自旋量子化轴的方向定义为每个粒子运动的方向，粒子的自旋极化可以平行或反平行于这个轴，实际中电子束或者正电子束通常都是非极化的，μ子探测器一般也不能分辨μ的极化（螺旋度？），因此最后得到的散射截面将对正电子和电子的自旋取向取平均，对μ子的自旋求和，对于任何给定的自旋取向，可以方便地写出微分散射截面，例如对于在立体角d Ω中的μ,.(应用简化公式，因此对于2个有限态的质心微分散射截面是，在4个粒子都具有相同质量的特例下（取极限m->0），有近似（）。
)，因子为散射截面提供了正确的量纲，因为在自然单位制中，跃迁振幅M 是无量纲的，它是量子力学过程发生的振幅（类似于非相对论量子力学中的散射振幅f ），表达式的另一部分因子是纯粹的约定问题，实际上是一个特例，仅对终态包含2个无质量的粒子的质心系散射是合理的，更一般的定理的形式并不能从量纲分析得到。
一个坏消息是即使对于最简单的QED 过程，M 矩阵的恰当表达式也是未知的，实际上这个事实并不令人惊讶，因为即使在非相对论量子力学，散射问题的恰当解也是很少的，最好是我们能得到M 的正规表达式，它作为电磁作用强度的微扰级数，我们将会估计级数的前几项，Feynman 发明了一种奇妙的方法组织并形象化了微扰级数：Feynman 图，简要地说这些图显示了散射过程中电子和光子的流动，对于特定的计算（？），微扰级数的零头阶可以用单个Feynman 图表示（这个图中的唯一可能中间态是γ光子），Feynman 图由3部分组成：1.外线（代表2个入射粒子和出射粒子）。
赝势的介绍
Vl(r)|l l|.
Traditionally PPs are split into a local part, long-ranged and behaving like −Zv e2/r for r → ∞, and a short-ranged semilocal term: ˆ ps = V ˆloc + V ˆSL, V ˆloc ≡ Vloc(r), V ˆSL ≡ V
2 d −2π (rφ(r)) d
d ln φ(r) dr
rc
= 4π
rc 0
|φ(r)|2r2dr
valid for any regular solution of the Schr¨ odinger equation at energy . – non local: there is one potential per angular momentum: V ps(r) =
ps 2 fl|φps l (r )| l
(fl is the occupancy of state with angular momentum l).
Desirable characteristics of a Pseudopotential: • Transferability: can be estimated from atomic calculations on different configurations. In many cases simple unscreening produces an unacceptable loss of transferability. May require the nonlinear core correction: Vlps(r) = Vl(r) − VH (nps(r)) − Vxc(nc(r) + nps(r)) where nc(r) is the core charge of the atom (Froyen, Louie, Cohen 1982) • Softness: atoms with strongly oscillating pseudo-wavefunctions (first-row elements, elements with 3d and 4f valence electrons) will produce hard PPs requiring many PWs in calculations. Larger core radius means better softness but worse transferability. Various recipes to get optimal smoothness without compromising transferability: Troullier and Martins (1990), Rappe Rabe Kaxiras Joannopoulos (1990)
PoiseuilleFlow：Poiseuille流
Lateral Migration and Nonuniform Rotation of Biconcave Particle Suspended inPoiseuille Flow*WEN Bing-Hai(闻炳海)1,2,3,CHEN Yan-Yan(陈艳燕)4,ZHANG Ren-Liang(张任良)1,ZHANG Chao-Ying(张超英)3**,FANG Hai-Ping(方海平)11Shanghai Institute of Applied Physics,Chinese Academy of Sciences,Shanghai2018002University of Chinese Academy of Sciences,Beijing1000493College of Computer Science and Information Engineering,Guangxi Normal University,Guilin5410044Department of Physics,Zhejiang Normal University,Jinhua321004(Received25January2013)A biconcave particle suspended in a Poiseuille flow is investigated by the multiple-relaxation-time lattice Boltz-mann method with the Galilean-invariant momentum exchange method.The lateral migration and equilibrium of the particle are similar to the Segré-Silberberg effect in our numerical simulations.Surprisingly,two lateral equilibrium positions are observed corresponding to the releasing positions of the biconcave particle.The upper equilibrium positions significantly decrease with the increasing Reynolds number,whereas the lower ones are almost insensitive to the Reynolds number.Interestingly,the regular wave accompanied by nonuniform rotation is exhibited in the lateral movement of the biconcave particle.It can be attributed to the fact that the biconcave shape in various postures interacts with the parabolic velocity distribution of the Poiseuille flow.A set of contours illustrate the dynamic flow field when the biconcave particle has successive postures in a rotating period.PACS:47.11.Qr,47.11.−j DOI:10.1088/0256-307X/30/6/064701The phenomena of inertia-induced cross-stream migration of suspended particles in Poiseuille flow have received wide interest since the classical investigations,[1]which reported that neutrally buoy-ant spheres in a pipe flow would migrate away from the wall and reach a certain lateral equilibrium posi-tion,namely the Segré-Silberberg effect.After many theoretical,experimental and numerical efforts were made to investigate and analyze the phenomena,[2−6] recently researching interests turned to the biolog-ical flows,especially the movement of cells or col-loid particles in a tube flow.[7−11]The lateral mi-gration of vesicles is considered as the interplay be-tween nonlinear character of a Poiseuille flow and vesi-cle deformation.[8]Theoretical analysis ascribed the cross-stream shift to the ratio of the inner over the outer fluid viscosities.[9]The more recent studies ob-served that vesicles presented two motion patterns(os-cillation and vacillating breathing)[10]or a phase dia-gram of shapes(bullet,croissant and parachute).[11] The red blood cell of human beings could be the most famous vesicle in biological flows.Without a cell nucleus,it generally exhibits a biconcave shape in rest state and some viscoelastic deformations in blood flow.[12−14]However,sizeable disparities can still be noticed among numerical simulations and ex-perimental observations due to its tiny size and com-plex characteristics.[10−16]It is therefore meaningful to obtain some credible benchmarks by simple mod-els in order to understand the essential behaviors of particles with various geometries.For example,the investigations of an ellipse in sedimentations,[17]shear flow[18]and Poiseuille flow[4]can make active contri-butions to the research of blood circulation of birds, whose red blood cell is oval or elliptical in shape with a nucleus inside.In this Letter,we use a simple model to in-vestigate a biconcave particle suspended in a two-dimensional Poiseuille flow by the lattice Boltzmann method,[19−22]which has developed into an alterna-tive and promising numerical scheme for simulating complex fluid flows.The multiple-relaxation-time model can be written as[23−25]f i(x+e iδt,t+δt)−f i(x,t)=−M−1·S·[m−m(eq)],(1) where f i(x,t)is the particle distribution function at lattice site x and time t,moving along the direc-tion defined by the discrete speeds e i,andδt is the time step;m and m(eq)represent the velocity mo-ments of the distribution functions and their equilib-ria,respectively;M is a linear transformation matrix mapping between moment space and discrete veloc-ity space;S is a diagonal matrix of nonnegative re-laxation rates.The hydrodynamic force can be eval-uated simply and efficiently by the momentum ex-change method in the lattice Boltzmann method.[26,27] Taking Galilean invariance[28]into account,Wen et al.recently proposed a Galilean-invariant momentum exchange method by introducing the relative velocity into the interfacial momentum transfer,[29]F(x s)=(e i−v)f i(x f,t)−(e¯i−v)f¯i(x b,t).(2)*Supported by the National Natural Science Foundation of China under Grant Nos10825520and11162002,and the National Basic Research Program of China under Grant No2012CB932400.**Corresponding author.Email:****************.cn©2013Chinese Physical Society and IOP Publishing Ltdwhere x f and x b are of a fluid node and a boundary node on a fluid-solid link,respectively.The boundary has a vector velocity v at the point of intersection x s. It is demonstrated to greatly enhance the hydrody-namic accuracy and robustness of moving boundaries in dynamic fluid.In particular,the algorithm meets full Galilean invariance and is independent of bound-ary geometries.The total hydrodynamic force and torque acting on the solid particle are evaluated byF=∑︁F(x s),T=∑︁(x s−R)×F(x s),(3)where R is the mass center of the solid particle,and the summation runs over all the fluid-solid links.Fig.1.Schematic diagram of a biconcave particle sus-pended in a Poiseuille flow.Figure1illustrates a schematic diagram of a bi-concave particle suspended in a Poiseuille flow.Both densities of the fluid and the particle are1g/cm3.The width of the channel is0.1cm and the kinematic vis-cosity isυ=0.01cm2/s.The Reynolds number is de-fined to characterize the flow domain by Re=HU/υ, where U is the mean velocity of the Poiseuille flow without particle and H is the width of the channel. The biconcave shape of a red blood cell was described by Fung et al.[30]as follows:y=12[︁1−(xR)2]︁1/2[︁C0+C1(xR)2+C2(xR)4]︁,(4)where C0=0.81,C1=7.83,C2=−4.39and R=2.91.The noncircular geometry of a biconcave particle will lead to regular wave and nonuniform rota-tion,and these will impact the fluid field of Poiseuille far more than a circular particle.[27]Therefore,we con-figure a large computing scale to improve the simulat-ing accuracy,as well as a longer channel to eliminate the influence of the inlet and outlet.The width of the computational domain is H=100lattice units,the length is20times the width,and the particle radius is R=15lattice units.The relaxation rates are given by S=diag(0,1.64,1.54,0,1.9,0,1.9,1/0.6,1/0.6). Thus,a second of movement covers300000time steps of the evolution computation.An iterative interpola-tion algorithm is utilized to solve the realtime posi-tions of biconcave boundary and the computing error is restricted to less than1×10−6.The second-order in-terpolation boundary condition[31]is adopted to com-pute the distribution functions bounced back from the curved particle boundary.The pressure boundary condition[32]is applied both at the inlet and outlet of the channel in order to drive fluid flow and form a Poiseuille flow.With the parallel optimization of Intel OpenMP,a following simulation which contains60s of particle movement performs18-million time steps and takes about60h on a HP Z600computer with12cores inside.0000000Fig.2.The migrating trajectories of a biconcave particle in a Poiseuille flow with two Reynolds numbers(a)Re=3 and(b)Re=12.The particles in the red and the blue trajectories are released at0.02and0.04cm away from the low wall,respectively.The black trajectories represent the classic Segré-Silberberg effect in which the particle is a cir-cle.Fig.3.The impact of Reynolds numbers on(a)the fi-nal equilibrium positions and(b)the rotating period of the particle.The particles in the red and blue lines are released at0.02and0.04cm away from the low wall,re-spectively.The biconcave particles are released at0.02and 0.04cm in the low half of the flow field and at the cen-ter of the channel in the horizontal direction,respec-tively.The lattices will be redrawn when the particle moves more than two lattice lengths in the horizontal direction in order to keep the particle in the middle of the channel all the time.Figure2draws the trajec-tories of the particle at the Reynolds numbers 3and 12,respectively.The lateral migration and equilib-rium with periodic wave are clearly observed.We also present the trajectories of the classic Segré-Silberberg effect with the circular particle of radius 15as com-parisons.As shown in Fig.2,it is usual that a sin-gle lateral equilibrium position is located between the wall and the centerline of the channel for the classic Segré-Silberberg effect.[2,6,8,10]Surprisingly,two lat-eral equilibrium positions are exhibited for biconcave particles and they are located at both sides of the equi-librium position of the circular particle.The smaller the Reynolds number is,the more slowly the particle reaches the equilibrium and the farther the two equi-librium positions separate.These trends can be seen more clearly in Fig.3(a),in which a set of Reynolds numbers are simulated.The upper equilibrium positions become significantly low with the increase of the Reynolds number,whereas the lower ones are almost insensitive to the Reynolds number.It should be noted that the horizontal axis in Fig.3uses a binary logarithmic coordinate.(b)0.00.20.40.60.81.002468w (r a d /s )PeriodY (c m )Fig.4.(a)The trajectory and orientation and (b)theangle velocity of the biconcave particle in a single rotating period.Although the biconcave particle in a Poiseuille flow behaves like the Segré-Silberberg effect,it includes ad-ditional regular wave andnonuniform rotationdue tothenoncirculargeometry.Explicitly,we definearo-tatingperiod asatimeintervalinwhich abiconcaveparticlerotates around.As shown in Fig.3(b),the rotating periods are monotonic decreasing with the increase of the Reynolds numbers.Notably,the ro-tating periods of the upper equilibrium positions are always longer than the lower ones and the gaps be-tween them show a continuous narrowing.These oc-cur as a result of the parabolic velocity distribution of a Poiseuille flow.The speeds of flow at the upper posi-tions are higher than those at the lower ones,and the velocity differences reduce continuously since they ap-proach each other along with the increasing Reynoldsnumbers.The trajectory and angle velocity of the biconcave particle in a single period in equilibrium state are il-lustrated in Fig.4,together with the orientations of the particle in various typical positions.The Reynolds number is 3and the particle starts from 0.02cm away from the low wall and rotates clockwise.The migrat-ing trajectory reaches the ridges when the particle an-gle is about 0πand 1π,and two steeper sub-ridges at about 0.5πand 1.5π.Noticeably,the ridges match the minimum angle velocities whereas the sub-ridges match the maximum ones.The peaks of the trajectory have a little lag relative to the angle of the biconcave particle.Four similar troughs lie about 0.25π,0.75π,1.25πand 1.75π.Their corresponding angle velocities are moderate while change is fast.The mechanism of lateral migration in the Segré-Silberberg effect is usually explained by the inertia effect.[6]The additional wave and nonuniform rota-tion in our simulations are due to the interaction of the biconcave shape and the parabolic velocity distri-bution.The fluid velocity at the upper part of the particle is always faster than that at the lower part and the difference drives the particle to rotate inces-santly.The changes of the posture would impact the hydrodynamic forces exerted by the fluid flow.It can be indicated in Fig.4that the biconcave particle exerts a dropping force in the angle range of about 0–0.25πand a lifting force in the angle range of about 0.75–1π.The flow field around the biconcave particle is care-fully investigated by the following contour diagrams.-6T10-5-6T10-5Fig.5.The contours of vertical velocities of the fluid around the biconcave particle:(a)–(f)corresponding to the first six positions in Fig.4(a).Since a Poiseuille flow is a well-defined laminar flow,the vertical velocity of the present flow field orig-inates totally from the wave and rotation of the bicon-cave particle and is far more sensitive to the particle motion than the horizontal velocity.Therefore we ap-ply the contour of the vertical velocity to character-ize the flow field.Figures 5(a)–5(f)are the counter-parts of the first six locations in Fig.4(a).Figure 5(a)is at the highest position with a horizontal posture and the smallest angle velocity,and the flow is mild and bilaterally balanced.Figures5(c)and5(e)are at the troughs with sloping postures,and the flows are strong.Figures5(b)and5(f)draw the rising and falling stages,and thus the upward and downward flows dominate the flow field,respectively.Figure5 illustrates the successive changes and depicts the dy-namic flow field vividly.In summary,we perform a series of numerical sim-ulations of a biconcave particle migrating laterally in a Poiseuille flow by the lattice Boltzmann method with multiple relaxation times.The hydrodynamic force is evaluated by the Galilean-invariant momentum ex-change method.Because of the interaction of the biconcave shape and the parabolic velocity distribu-tion,two lateral equilibrium positions are found for the biconcave particle corresponding to its releasing points.This makes a remarkable distinction to the classic Segré-Silberberg effect,in which a single equi-librium position is observed for a circle or sphere.Ex-tending the simulations to a range of Reynolds num-bers,we observe that the upper equilibrium positions significantly decrease with the increasing Reynolds number while the lower ones are almost insensitive to the Reynolds number.Inside a single rotating pe-riod,the biconcave particle moves with regular wave and the nonuniform angle velocity.The dynamic flow fields around the particle are illustrated vividly by the contours of the vertical velocities when the bicon-cave particle has successive postures.The investiga-tion will be expanded to combine with the viscoelastic membrane[12]in order to enrich the understanding of the behaviors of red blood cells and other vesicles in dynamic fluid.The authors thank Shanghai Supercomputer Cen-ter of China for the support of computation. References[1]Segre G and Silberberg A1962J.Fluid Mech.14115[2]Karnis A,Goldsmith H L and Mason S G1966Can.J.Chem.Eng.44181[3]Asmolov E S1999J.Fluid Mech.38163[4]Qi D,Luo L,Aravamuthan R and Strieder W2002J.Stat.Phys.107101[5]Zhang C Y,Tan H L,Liu M R,Kong L J and Shi J2005Chin.Phys.Lett.22896[6]Matas J P,Morris J F and Guazzelli E2004J.Fluid Mech.515171[7]Fang H P,Wang Z W,Lin Z F and Liu M R2002Phys.Rev.E65051925[8]Kaoui B,Ristow G H,Cantat I,Misbah C and Zimmer-mann W2008Phys.Rev.E77021903[9]Danker G,Vlahovska P M and Misbah C2009Phys.Rev.Lett.102148102[10]Shi L L,Pan T W and Glowinski R2012Phys.Rev.E86056308[11]Coupier G,Farutin A,Minetti C,Podgorski T and MisbahC2012Phys.Rev.Lett.108178106[12]Li H B,Yi H H,Shan X W and Fang H P2008Europhys.Lett.8154002[13]Jiang L G,Wu H A,Zhou X Z and Wang X X2010Chin.Phys.Lett.27028704[14]Dupire J,Socol M and Viallat A2012Proc.Natl.Acad.Sci.U.S.A.10920808[15]Fedosov D A,Caswell B and Karniadakis G E2010Biophys.J.982215[16]Shen Z Y and He Y2012Chin.Phys.Lett.29024703[17]Xia Z H,Connington K W,Rapaka S,Yue P T,Feng J Jand Chen S Y2009J.Fluid Mech.625249[18]Huang H B,Yang X,Krafczyk M and Lu X Y2012J.FluidMech.692369[19]Qian Y H,d’Humières D and Lallemand P1992Europhys.Lett.17479[20]Qian Y H and Orszag S A1993Europhys.Lett.21255[21]Chen S Y,Chen H D,Martinez D and Matthaeus W1991Phys.Rev.Lett.673776[22]Chen S Y and Doolen G D1998Annu.Rev.Fluid Mech.30329[23]d’Humières D1992Prog.Aeronaut.Astronaut.159450[24]Lallemand P and Luo L S2000Phys.Rev.E616546[25]Luo L S,Liao W,Chen X,Peng Y and Zhang W2011Phys.Rev.E83056710[26]Li H B,Lu X Y,Fang H P and Qian Y H2004Phys.Rev.E70026701[27]Wen B H,Li H B,Zhang C Y and Fang H P2012Phys.Rev.E85016704[28]Qian Y H and Zhou Y1998Europhys.Lett.42359[29]Wen B H,Zhang C Y,Tu Y S,Wang C L and Fang H P2013arXiv:1303.0625[physics.flu-dyn][30]Fung Y C1981Biomechanics:Mechanical Properties ofLiving Tissues(New York:Springer Verlag)[31]Lallemand P and Luo L put.Phys.184406[32]Zou Q S and He X Y1997Phys.Fluids91591。
超统一场论
物理学是研究物质的基本结构和物质运动最一般规律的科学。
迄今为止，人类认识的宇宙间的力有四种，那就是强力、弱力、电磁力、万有引力。
虽然这四种力无论从强度上，还是从作用距离上都有着巨大的差异。
但是理论物理学家却认为自然界是和谐的，统一的，四种力之间不仅具有内在联系，而且是统一的。
长期以来，世界上有不少物理学家，一直在试图找到一个统一、简洁、和谐的数学公式，来概括这四种自然力。
1、统一理论的进展从本世纪20年代开始，爱因斯坦花费了半生的精力，力图把电磁力与万有引力统一起来。
海森堡也作了种种尝试，但都没有成功。
直到1967年，温伯格、萨拉姆、格拉肖等人提出了弱力与电磁力的统一模型，统一理论才取得了一点进展。
G·S·W理论所预言的四个规范波色子有三个得到了证实，但仍有一个希格斯粒子没有找到。
因此，承认该理论还为时过早。
在G·S·W理论得到了部分证实之后，近年来一些物理学家提出各种“大统一”理论（GUTS），试图证明除引力外的其他三种自然力同源。
被称为“SU （5）型”的乔治·格拉肖的大统一理论认为：（1）质子的寿命为1031年，主要衰变为正电子和π介子；（2）存在着名为“磁单极子”的很重的基本粒子。
然而，“大统一”理论未得到任何证实。
2、爱因斯坦广义相对论难以解释的问题爱因斯坦于本世纪初，在牛顿引力理论的基础上，建立了广义相对论，即引力理论的相对论。
虽然广义相对论能够解释水星的近日点的进动，预言了光线在恒星附近的偏折，但它仍然难以解释天体物理学中几个最基本的问题，即2.1 宇宙遗失97%质量问题；2.2 星系的外缘恒星比内缘恒星公转速度快的问题；2.3 星系的形状问题。
3、超统一场论的基本思想超统一场论所要解决的是支配天体运动的力、弱力、强力及电磁力四种力的统一。
面对差异如此之大的四种力，它摈弃了“弱电统一理论”中相互作用是交换虚光子或中间波色子的过程，代之以能量和能量守恒为核心的场论。
曾谨言量子力学教程第3版知识点总结笔记课后答案
1.Schrödinger方程的引进
在势场V（r）中的粒子的波函数满足的微分方程，称为Schrödinger
波动方程，它揭示了微观世界中物质运动的基本规律．
2.Schrödinger方程的讨论
（１）定域的概率守恒
对于一个粒子来说，在全空间中找到它的概率之总和应不随时间改变．即
以下讨论一个极为重要的特殊情况——假设势能V不显含t（经典力学中，在这种势场中的粒子的机械能是守恒量）．
其中ψE（r）满足下列方程：
（2）
在有的条件下，特别是束缚态边条件，只有某些离散的E值所对应的解才是物理上可以接受的．这些E值称为体系的能量本征值（energy eigen value），而相应的解ψ（r）称为能量本征函数（energy eigen unction）．方程（2）就是势场V（r）中粒子的能量本征方程，也称为不含时（time-independent）Schrödinger方程．
（1）
（1）式为概率守恒的微分表达式，其形式与流体力学中的连续性方程相同．
（2）初值问题，传播子
Schrödinger方程给出了波函数（量子态）随时间演化的因果关系， 取初始时刻为t‘，则t时刻波函数可以表示为
式中
称为传播子（propagator）．可以证明
就是t时刻在r点找到粒子的概率波幅．
3.能量本征方程
stationary state）．
5.多粒子体系的Schrödinger方程
设体系由N个粒子组成，粒子质量分别为mi（i=1，2，3，…，N）．体系的波函数表示为ψ（r1，…，rN，t）．设第i个粒子受到的外势场为Ui（ri），粒子之间相互作用为V（r1，…，rN，t），则Schrödinger方程表示为
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arXiv:hep-ph/0510244v3 23 Apr 2007Polewave-functionrenormalizationprescriptionforunstableparticlesYongZhouBeijingUniversityofPostsandTelecommunications,SchoolofScience,P.O.Box123,Beijing100876,China
Webaseanewwave-functionrenormalizationprescriptiononthepolemassrenormaliza-tionprescription,inwhichtheWave-functionRenormalizationConstant(WRC)isextractedbyexpandingtheparticle’spropagatorarounditspole,ratherthanitsphysicalmasspointasconvention.WefindthedifferencebetweenthenewandtheconventionalWRCisgauge-parameterdependentforunstableparticlesbeyondone-looplevel,whichwillleadtosomephysicalresultsgaugedependentundertheconventionalwave-functionrenormalizationpre-scriptionbeyondone-looplevel.
PACSnumbers:11.10.Gh
I.INTRODUCTIONTheconventionalwave-functionrenormalizationprescriptionextractsWRCbyexpandingtheparticle’spropagatorarounditsphysicalmasspointintheLSZreductionformula[1].Forscalarbosonitis[2,3,4]i(p2−m2)(1+ReΣ′(m2))+ReΣ(m2)−δm2+iImΣ(p2)
=i(1+ReΣ′(m2))−1
(p2−m2)(1+Σ′(m2))+Σ(m2)−δm2=i(1+Σ′(m2))−1
p2−m2−δm2+Σ(p2)∼ip2−¯s,(2)where¯s=m2−imΓisthepoleoftheboson’spropagator[5].Notethatthepolemassrenormal-izationprescriptionhasbeenusedinEq.(2).2Forfermionthenewwave-functionrenormalizationprescriptionisalittlecomplex.Thefermioninversepropagatorcanbewrittenas
iS−1(p/)=p/−m−δm+Σ(p/)≡p/(aγL+bγR)+cγL+dγR,(3)whereγLandγRaretheleft-andright-handedhelicityoperators,andthediagonalfermionselfenergyis
Σ(p/)=p/γLΣL(p2)+p/γRΣR(p2)+m(γLΣS,L(p2)+γRΣS,R(p2)).(4)Expandingthefermionpropagatorarounditspoleweget[7,8]S(p/)=i(p/(aγL+bγR)−dγL−cγR)
(p2−m2+imΓ)A,(5)where¯s=m2−imΓisthepoleofthefermionpropagator,andA=(1+ΣL(¯s))(1+ΣR(¯s))+¯s(ΣL′(¯s)+ΣR′(¯s)+ΣL′(¯s)ΣR(¯s)+ΣL(¯s)ΣR′(¯s))+mΣS,L′(¯s)(m+δm−mΣS,R(¯s))+mΣS,R′(¯s)(m+δm−mΣS,L(¯s)).(6)
FromEq.(2)andEq.(5)wecanextractbosonandfermion’sWRC.Insection2wewilldothiswork.Insection3wewillevaluatethedifferenceofunstableparticle’sWRCbetweenthenewandtheconventionalwave-functionrenormalizationprescriptionanddiscusstheinfluenceofthedifferenceonphysicalresults.Lastlywegiveourconclusion.
II.DETERMINATIONOFWAVE-FUNCTIONRENORMALIZATIONCONSTANTSIntheLSZreductionformulaoneneedstointroducetwosetsofWRC:theincomingWRCandtheoutgoingWRC[4,7,8].ForbosontheincomingandoutgoingWRCareintroducedasfollows[4]
Z12=,(7)whereΩistheinteractionvacuum,φistheboson’sHeisenbergfield,andλistheincomingoroutgoingstateofS-matrixelement.AccordingtotheLSZreductionformulawehavefromEq.(2)
Z12=(1+Σ′(m2−imΓ))−1.(8)3Anotherconditionthatboson’sWRCmustsatisfyis[4]¯Z=Z.(9)
Thereforeweget¯Z=Z=(1+Σ′(m2−imΓ))−1.(10)
ForfermiontheincomingandoutgoingWRCareintroducedasfollows[4]:=Z12,(11)whereψisthefermion’sHeisenbergfieldandZ12γL+ZR12=¯ZL12γL.(12)Thefermionpropagatoratresonantregioncanbeexpressedas[4,7]S(p/)∼iZ12
2¯ZL
1
2¯ZR
1
2¯ZR12¯ZL
14III.GAUGEDEPENDENCEOFPHYSICALRESULTSUNDERTHECONVENTIONALWAVE-FUNCTIONRENORMALIZATIONPRESCRIPTION
Sinceunstableparticle’sWRCmustcontainimaginarypart[3,4,7,8],theconventionalwave-functionrenormalizationprescriptionmustbethesecondprescriptionofEq.(1)forboson,i.e.[4](seeEq.(9))
¯Zo=Zo=(1+Σ′(m2))−1,(18)
wherethesubscriptorepresentstheconventionalwave-functionrenormalizationprescription.Com-paringwithEqs.(10)wefindattwo-looplevel
Zo−Z=−imΓΣ′′(m2).(19)Forunstablebosonthedifferenceisgauge-parameterdependent.ForexampleforgaugebosonWweobtain(seeFig.1)
Re[ZoW−ZW]ξW
=α21−2(xi+xj)+(xi−xj)2
×(2−(xi+xj)−(xi−xj)2)](2ξ3W−3ξ2W−6ξW−5)θ[1−ξW],(20)whereRetakestherealpartofthequantity,ξWisthegaugeparameterofW,thesubscriptξW
denotestheξW-dependentpartofthequantity,αisthefinestructureconstant,swisthesineof
theweakmixingangle,xi=m2i/m2Wandxj=m2j/m2WwithmWthemassofW,VijistheCKMmatrixelement[9],andθistheHeavisidefunction.NotethatinthecalculationswehaveusedtheprogrampackagesFeynArtsandFeynCalc[10].Forfermiontheconventionalwave-functionrenormalizationprescriptionmustbe[4]
¯ZLo=ZLo=(1+ΣR(m2))/A1,
¯ZRo=ZRo=(1+ΣL(m2))/A1,(21)
whereA1=(1+ΣL(m2))(1+ΣR(m2))+m2(ΣL′(m2)+ΣR′(m2)+ΣL′(m2)ΣR(m2)+ΣL(m2)ΣR′(m2))+mΣS,L′(m2)(m+δm−mΣS,R(m2))+mΣS,R′(m2)(m+δm−mΣS,L(m2)).(22)