Two-loop Light Quark Corrections To The Top Width
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。
圆二色光谱英文
圆二色光谱英文Circular Bicolor SpectrumIntroductionThe circular bicolor spectrum, also known as the color wheel, is a visual representation of the entire range ofcolors in a circular format. It is a fundamental tool for artists, designers, and anyone working with colors. The spectrum showcases the relationships between different colors, allowing for the creation of harmonious color schemes and the understanding of color theory.History of the Color WheelThe concept of the color wheel dates back to the 18th century when Sir Isaac Newton first discovered that whitelight could be divided into the various colors of the spectrum. His experiments with prisms led to the creation ofthe first color wheel, which consisted of 7 colors - red, orange, yellow, green, blue, indigo, and violet.In the 19th century, artists and theorists began to delve deeper into the study of color and its relationships. Swiss painter Johannes Itten and American painter Albert Henry Munsell developed their own versions of the color wheel, each adding their own insights into color theory and harmony.In the 20th century, the color wheel became a standard tool in art and design education, with its principles being applied across various fields such as fashion, interior design, and graphic design.The Circular Bicolor SpectrumThe color wheel is typically divided into two main categories - warm and cool colors. Warm colors, such as red, orange, and yellow, are associated with energy and warmth. Cool colors, such as blue, green, and violet, are calming andsoothing. By arranging the colors in a circular format, the relationships between them become more apparent.Primary Colors: At the center of the color wheel are the three primary colors - red, blue, and yellow. These colors are considered the building blocks of all other colors and cannot be created by mixing other colors.Secondary Colors: Located between the primary colors are the secondary colors - orange, green, and violet. These colors are created by mixing equal parts of two primary colors. For example, mixing red and yellow creates orange.Tertiary Colors: The spaces between the primary and secondary colors are filled with the tertiary colors, which are a combination of a primary color and a secondary color. Examples of tertiary colors include red-orange, yellow-green, and blue-violet.Color Relationships: The placement of colors on the color wheel dictates their relationships and the resulting colorschemes. Complementary colors are located opposite each other on the wheel, such as red and green. These colors create contrast and vibrancy when used together. Analogous colors are located next to each other on the wheel, such as red, orange, and yellow. These colors create a harmonious and cohesive color scheme.Application of the Color WheelThe color wheel is a valuable tool for artists and designers in various applications. It is used in creating pleasing color combinations for paintings, illustrations, graphic designs, and more. Understanding the relationships between colors allows for the creation of visually appealing compositions.In interior design, the color wheel is used to create color schemes for different rooms and spaces. By using complementary or analogous colors, designers can evoke specific moods and atmospheres within a space.In fashion, the color wheel is used to create cohesive and stylish color combinations for clothing and accessories. It also helps designers understand the impact of color on the human psyche and how it can influence emotions and perceptions.ConclusionThe circular bicolor spectrum, or color wheel, is a fundamental tool for understanding the relationships between colors. It provides a visual representation of the entire range of colors and their interactions, allowing for the creation of harmonious color schemes in various artistic and design fields. Its enduring importance and application make it an essential tool for anyone working with colors.。
Phase diagram of dense neutral three-flavor quark matter
ular and gapless color-flavor-locked (gCFL) phases [15] are favored. At nonzero temperature, T = 0, some other phases were proposed as well [16]. In this paper, we study the phase diagram of dense neutral three-flavor quark matter as a function of the strange quark mass at zero and finite temperature. In order to allow for the most general ground state, we employ a nine-parameter ansatz for the gap function. The effects of the strange quark mass are incorporated in our model by a shift of the chemical potential of strange quarks, i 2 µi s → µs − ms /(2µ) where i = r, g, b is the color index, ms is the strange quark mass, and µ is the quark chemical potential. This shift reflects the reduction of the Fermi momenta of strange quarks due to their mass. Such an approach is certainly reliable at small values of the strange quark mass. We assume that it is also qualitatively correct at large values of the strange quark mass. We note that the analysis of this paper is restricted to locally neutral phases only. This automatically excludes, for example, mixed [18] and crystalline [19] phases. Also, in the mean field approximation utilized here, we cannot get any phases with meson condensates [20]. This paper is organized as follows. In Sec. II we present the details of our approach which is based on the Cornwall-Jackiw-Tomboulis (CJT) formalism [22]. There, we also derive the gap equations and the neutrality conditions, and obtain an expression for the pressure. In the next two sections, the gap equations and the neutrality conditions are studied by using numerical methods. The results at zero temperature are presented and discussed in Sec. III. There, the appearance of the CFL and gCFL phases is established in the case of small and moderately large values of the strange quark mass, respectively. The results at nonzero temperature are discussed in Sec. IV. Also, the phase diagram of quark matter in the T –m2 s /µ plane, as well as in the T –µ plane is presented. Finally, Sec. V concludes this paper with a summary of the results. Our units are = c = kB = 1. The metric tensor is gµν = diag (1, −1, −1, −1). Four-vectors are denoted by capital Latin letters, e.g., K µ = (k 0 , k) where k is a three-vector with absolute value k = |k| and direction ˆ = k/k . We use the imaginary-time formalism, i.e., the k
O(alpha) QED Corrections to Polarized Elastic $mu e$ and Deep Inelastic l N Scattering
a r X i v :h e p -p h /9711228v 1 4 N o v 1997hep-ph/9711228October 1997O (α)QED Corrections to Polarized Elastic µe and Deep Inelastic lN ScatteringDima Bardin a,b,c ,Johannes Bl¨u mlein a ,Penka Christova a,d ,and Lida Kalinovskaya a,caDESY–Zeuthen,Platanenallee 6,D–15735Zeuthen,GermanybINFN,Sezione di Torino,Torino,ItalycJINR,ul.Joliot-Curie 6,RU–141980Dubna,RussiadBishop Konstantin Preslavsky University of Shoumen,9700Shoumen,BulgariaAbstractTwo computer codes relevant for the description of deep inelastic scattering offpolarized targets are discussed.The code µe la deals with radiative corrections to elastic µe scattering,one method applied for muon beam polarimetry.The code HECTOR allows to calculate both the radiative corrections for unpolarized and polarized deep inelastic scattering,including higher order QED corrections.1IntroductionThe exact knowledge of QED,QCD,and electroweak (EW)radiative corrections (RC)to the deep inelastic scattering (DIS)processes is necessary for a precise determination of the nucleon structure functions.The present and forthcoming high statistics measurements of polarized structure functions in the SLAC experiments,by HERMES,and later by COMPASS require the knowledge of the RC to the DIS polarized cross-sections at the percent level.Several codes based on different approaches for the calculation of the RC to DIS experiments,mainly for non-polarized DIS,were developped and thoroughly compared in the past,cf.[1].Later on the radiative corrections for a vast amount of experimentally relevant sets of kinematic variables were calculated [2],including also semi-inclusive situations as the RC’s in the case of tagged photons [3].Furthermore the radiative corrections to elastic µ-e scattering,a process to monitor (polarized)muon beams,were calculated [4].The corresponding codes are :•HECTOR 1.00,(1994-1995)[5],by the Dubna-Zeuthen Group.It calculates QED,QCD and EW corrections for variety of measuremets for unpolarized DIS.•µe la 1.00,(March 1996)[4],calculates O (α)QED correction for polarized µe elastic scattering.•HECTOR1.11,(1996)extends HECTOR1.00including the radiative corrections for polarized DIS[6],and for DIS with tagged photons[3].The beta-version of the code is available from http://www.ifh.de/.2The Programµe laMuon beams may be monitored using the processes ofµdecay andµe scattering in case of atomic targets.Both processes were used by the SMC experiment.Similar techniques will be used by the COMPASS experiment.For the cross section measurement the radiative corrections to these processes have to be known at high precision.For this purpose a renewed calculation of the radiative corrections toσ(µe→µe)was performed[4].The differential cross-section of polarized elasticµe scattering in the Born approximation reads,cf.[7],dσBORNm e Eµ (Y−y)2(1−P e Pµ) ,(1)where y=yµ=1−E′µ/Eµ=E′e/Eµ=y e,Y=(1+mµ/2/Eµ)−1=y max,mµ,m e–muon and electron masses,Eµ,E′µ,E′e the energies of the incoming and outgoing muon,and outgoing electron respectively,in the laboratory frame.Pµand P e denote the longitudinal polarizations of muon beam and electron target.At Born level yµand y e agree.However,both quantities are different under inclusion of radiative corrections due to bremsstrahlung.The correction factors may be rather different depending on which variables(yµor y e)are used.In the SMC analysis the yµ-distribution was used to measure the electron spin-flip asymmetry A expµe.Since previous calculations,[8,9],referred to y e,and only ref.[9]took polarizations into account,a new calculation was performed,including the complete O(α)QED correction for the yµ-distribution,longitudinal polarizations for both leptons,theµ-mass effects,and neglecting m e wherever possible.Furthermore the present calculation allows for cuts on the electron re-coil energy(35GeV),the energy balance(40GeV),and angular cuts for both outgoing leptons (1mrad).The default values are given in parentheses.Up to order O(α3),14Feynman graphs contribute to the cross-section forµ-e scattering, which may be subdivided into12=2×6pieces,which are separately gauge invariantdσQEDdyµ.(2) One may express(2)also asdσQEDdyµ+P e Pµdσpol kk=1−Born cross-section,k=b;2−RC for the muonic current:vertex+bremsstrahlung,k=µµ;3−amm contribution from muonic current,k=amm;4−RC for the electronic current:vertex+bremsstrahlung,k=ee;5−µe interference:two-photon exchange+muon-electron bremsstrahlung interference,k=µe;6−vacuum polarization correction,runningα,k=vp.The FORTRAN code for the scattering cross section(2)µe la was used in a recent analysis of the SMC collaboration.The RC,δA yµ,to the asymmetry A QEDµeshown infigures1and2is defined asδA yµ=A QEDµedσunpol.(4)The results may be summarized as follows.The O(α)QED RC to polarized elasticµe scattering were calculated for thefirst time using the variable yµ.A rather general FORTRAN codeµe la for this process was created allowing for the inclusion of kinematic cuts.Since under the conditions of the SMC experiment the corrections turn out to be small our calculation justifies their neglection. 3Program HECTOR3.1Different approaches to RC for DISThe radiative corrections to deep inelastic scattering are treated using two basic approaches. One possibility consists in generating events on the basis of matrix elements including the RC’s. This approach is suited for detector simulations,but requests a very hughe number of events to obtain the corrections at a high precision.Alternatively,semi-analytic codes allow a fast and very precise evaluation,even including a series of basic cuts andflexible adjustment to specific phase space requirements,which may be caused by the way kinematic variables are experimentally measured,cf.[2,5].Recently,a third approach,the so-called deterministic approach,was followed,cf.[10].It treats the RC’s completely exclusively combining features of fast computing with the possibility to apply any cuts.Some elements of this approach were used inµe la and in the branch of HECTOR1.11,in which DIS with tagged photons is calculated.Concerning the theoretical treatment three approaches are in use to calculate the radiative corrections:1)the model-independent approach(MI);2)the leading-log approximation(LLA); and3)an approach based on the quark-parton model(QPM)in evaluating the radiative correc-tions to the scattering cross-section.In the model-independent approach the QED corrections are only evaluated for the leptonic tensor.Strictly it applies only for neutral current processes.The hadronic tensor can be dealt with in its most general form on the Lorentz-level.Both lepton-hadron corrections as well as pure hadronic corrections are neglected.This is justified in a series of cases in which these corrections turn out to be very small.The leading logarithmic approximation is one of the semi-analytic treatments in which the different collinear singularities of O((αln(Q2/m2l))n)are evaluated and other corrections are neglected.The QPM-approach deals with the full set of diagrams on the quark level.Within this method,any corrections(lepton-hadron interference, EW)can be included.However,it has limited precision too,now due to use of QPM-model itself. Details on the realization of these approaches within the code HECTOR are given in ref.[5,11].3.2O (α)QED Corrections for Polarized Deep Inelastic ScatteringTo introduce basic notation,we show the Born diagramr rr r j r r r r l ∓( k 1,m )l ∓( k 2,m )X ( p ′,M h )p ( p ,M )γ,Z ¨¨¨¨B ¨¨¨¨£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡z r r r r r r r r r r r r r rr ¨¨¨¨B ¨¨¨¨r r r r j r r r r and the Born cross-section,which is presented as the product of the leptonic and hadronic tensordσBorn =2πα2p.k 1,x =Q 2q 2F 1(x,Q 2)+p µ p ν2p.qF 3(x,Q 2)+ie µνλσq λs σ(p.q )2G 2(x,Q 2)+p µ s ν+ s µ p νp.q1(p.q )2G 4(x,Q 2)+−g µν+q µq νp.qG 5(x,Q 2),(8)wherep µ=p µ−p.qq 2q µ,and s is the four vector of nucleon polarization,which is given by s =λp M (0, n )in the nucleonrest frame.The combined structure functions in eq.(8)F1,2(x,Q2)=Q2e Fγγ1,2(x,Q2)+2|Q e|(v l−p eλl a l)χ(Q2)FγZ1,2(x,Q2)+ v2l+a2l−2p eλl v l a l χ2(Q2)F ZZ1,2(x,Q2),F3(x,Q2)=2|Q e|(p e a l−λl v l)χ(Q2)FγZ3(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)F ZZ3(x,Q2),G1,2(x,Q2)=−Q2eλl gγγ1,2(x,Q2)+2|Q e|(p e a l−λl v l)χ(Q2)gγZ1,2(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)g ZZ1,2(x,Q2),G3,4,5(x,Q2)=2|Q e|(v l−p eλl a l)χ(Q2)gγZ3,4,5(x,Q2),+ v2l+a2l−2p eλl v l a l χ2(Q2)g ZZ3,4,5(x,Q2),(9) are expressed via the hadronic structure functions,the Z-boson-lepton couplings v l,a l,and the ratio of the propagators for the photon and Z-bosonχ(Q2)=Gµ2M2ZQ2+M2Z.(10)Furthermore we use the parameter p e for which p e=1for a scattered lepton and p e=−1for a scattered antilepton.The hadronic structure functions can be expressed in terms of parton densities accounting for the twist-2contributions only,see[12].Here,a series of relations between the different structure functions are used in leading order QCD.The DIS cross-section on the Born-leveld2σBorndxdy +d2σpol Borndxdy =2πα2S ,S U3(y,Q2)=x 1−(1−y)2 ,(13) and the polarized partdσpol BornQ4λp N f p S5i=1S p gi(x,y)G i(x,Q2).(14)Here,S p gi(x,y)are functions,similar to(13),and may be found in[6].Furthermore we used the abbrevationsf L=1, n L=λp N k 12πSy 1−y−M2xy2π1−yThe O(α)DIS cross-section readsd2σQED,1πδVRd2σBorndx l dy l=d2σunpolQED,1dx l dy l.(16)All partial cross-sections have a form similar to the Born cross-section and are expressed in terms of kinematic functions and combinations of structure functions.In the O(α)approximation the measured cross-section,σrad,is define asd2σraddx l dy l +d2σQED,1dx l dy l+d2σpol radd2σBorn−1.(18)The radiative corrections calculated for leptonic variables grow towards high y and smaller values of x.Thefigures compare the results obtained in LLA,accounting for initial(i)andfinal state (f)radiation,as well as the Compton contribution(c2)with the result of the complete calculation of the leptonic corrections.In most of the phase space the LLA correction provides an excellent description,except of extreme kinematic ranges.A comparison of the radiative corrections for polarized deep inelastic scattering between the codes HECTOR and POLRAD[17]was carried out.It had to be performed under simplified conditions due to the restrictions of POLRAD.Corresponding results may be found in[11,13,14].3.3ConclusionsFor the evaluation of the QED radiative corrections to deep inelastic scattering of polarized targets two codes HECTOR and POLRAD exist.The code HECTOR allows a completely general study of the radiative corrections in the model independent approach in O(α)for neutral current reac-tions including Z-boson exchange.Furthermore,the LLA corrections are available in1st and2nd order,including soft-photon resummation and for charged current reactions.POLRAD contains a branch which may be used for some semi-inclusive DIS processes.The initial state radia-tive corrections(to2nd order in LLA+soft photon exponentiation)to these(and many more processes)can be calculated in detail with the code HECTOR,if the corresponding user-supplied routine USRBRN is used together with this package.This applies both for neutral and charged current processes as well as a large variety of different measurements of kinematic variables. Aside the leptonic corrections,which were studied in detail already,further investigations may concern QED corrections to the hadronic tensor as well as the interference terms. References[1]Proceedings of the Workshop on Physics at HERA,1991Hamburg(DESY,Hamburg,1992),W.Buchm¨u ller and G.Ingelman(eds.).[2]J.Bl¨u mlein,Z.Phys.C65(1995)293.[3]D.Bardin,L.Kalinovskaya and T.Riemann,DESY96–213,Z.Phys.C in print.[4]D.Bardin and L.Kalinovskaya,µe la,version1.00,March1996.The source code is availablefrom http://www.ifh.de/~bardin.[5]A.Arbuzov,D.Bardin,J.Bl¨u mlein,L.Kalinovskaya and T.Riemann,Comput.Phys.Commun.94(1996)128,hep-ph/9510410[6]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,DESY96–189,hep-ph/9612435,Nucl.Phys.B in print.[7]SMC collaboration,D.Adams et al.,Phys.Lett.B396(1997)338;Phys.Rev.D56(1997)5330,and references therein.[8]A.I.Nikischov,Sov.J.Exp.Theor.Phys.Lett.9(1960)757;P.van Nieuwenhuizen,Nucl.Phys.B28(1971)429;D.Bardin and N.Shumeiko,Nucl.Phys.B127(1977)242.[9]T.V.Kukhto,N.M.Shumeiko and S.I.Timoshin,J.Phys.G13(1987)725.[10]G.Passarino,mun.97(1996)261.[11]D.Bardin,J.Bl¨u mlein,P.Christova,L.Kalinovskaya,and T.Riemann,Acta Phys.PolonicaB28(1997)511.[12]J.Bl¨u mlein and N.Kochelev,Phys.Lett.B381(1996)296;Nucl.Phys.B498(1997)285.[13]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,Preprint DESY96–198,hep-ph/9609399,in:Proceedings of the Workshop‘Future Physics at HERA’,G.Ingelman,A.De Roeck,R.Klanner(eds.),Vol.1,p.13;hep-ph/9609399.[14]D.Bardin,Contribution to the Proceedings of the International Conference on High EnergyPhysics,Warsaw,August1996.[15]M.Gl¨u ck,E.Reya,M.Stratmann and W.Vogelsang,Phys.Rev.D53(1996)4775.[16]S.Wandzura and F.Wilczek,Phys.Lett.B72(1977)195.[17]I.Akushevich,A.Il’ichev,N.Shumeiko,A.Soroko and A.Tolkachev,hep-ph/9706516.-20-18-16-14-12-10-8-6-4-200.10.20.30.40.50.60.70.80.91elaFigure 1:The QED radiative corrections to asymmetry without experimental cuts.-1-0.8-0.6-0.4-0.200.20.40.60.810.10.20.30.40.50.60.70.80.91elaFigure 2:The QED radiative corrections to asymmetry with experimental cuts.-50-40-30-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 3:A comparison of complete and LLA RC’s in the kinematic regime of HERMES for neutral current longitudinally polarized DIS in leptonic variables.The polarized parton densities [15]are used.The structure function g 2is calculated using the Wandzura–Wilczek relation.c 2stands for the Compton contribution,see [6]for details.-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 4:The same as in fig.3,but for energies in the range of the SMC-experiment.-20-10010203040500.10.20.30.40.50.60.70.80.91HectorFigure 5:The same as in fig.4for x =10−3.-200-150-100-5005010015020000.10.20.30.40.50.60.70.80.91HectorFigure 6:A comparison of complete and LLA RC’s at HERA collider kinematic regime for neutral current deep inelastic scattering offa longitudinally polarized target measuring the kinematic variables at the leptonic vertex.。
用QCD因子化方法研究B→PV两体弱衰变过程
第26卷 增刊 高能物理与核物理V o1.26，Supp. 2002年12月HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS Dec.，2002 Two-Body B Decays to Pseudoscalar and Vector Mesonsin QCD Factorization ApproachYANG Mao-Zhi1 YANG Ya-Dong21 (Institute of High Energy Physics, CAS, Beijing 100039, China)2 (Department of Physics, Technion, Haifa 32000, Israel)Abstract Motivated by recent CELO measurements and the progress of the theory of B decays,B→PV(P=π, K; V= K*, ρ, ω) decay modes are studied in the framework of QCD factorization.All the measured branching ratios are well accommodated in the reasonable parameter space andpredictions for other decay modes are well below the experimental upper limits.Key words factorization, weak decay, mesonB physics is one of the most important fields nowadays because it is of great help for testing the quark flavor mixing theory of the standard model and exploring the source of CP violation. Most of the theoretical studies of B decays to pseudocalar and vector final states are based on the popular Naive Factorization approach[1]. As it was ponited out years ago in Ref. [2], the dominant contribution in B decays comes from the so-called Feynman mechanism, where the energetic quark created in the weak decay picks up the soft spectator softly and carries nearly all of the final-state meson's momentum. It is also shown that Pion form factor in QCD at intermediate engery scale is dominated by Feynman mechanism[3—5]. From this point, we can understand why the naive factorization approach have worked well for B and D decays, and the many existing predictions for B decays based on naive factorization and spectator ansatz do have taken in the dominant physics effects although there are shortcommings. However, with the many new data available from CLEO and an abundance of data to arrive within few years from the B factories BaBar and Belle, it is demanded highly to go beyond the naive factorization approach.Recently, Beneke et al., have formed an interesting QCD factorization formula for B exclusive nonleptonic decays[6,7]. The factorization formula incorporates elements of the naive factorization approach (as leading contribution) and the hard-scattering approach (as subleading corrections), which allows us to calculate systematically radiative(subleading nonfactorizable) corrections to naive factorization for B exclusive nonleptonic decays. An important product of the formula is that the strong final-state interaction phases are calculable, which arise from the2 高能物理与核物理(HEP &NP) 第26卷hard-scattering kernel and hence process dependent. The strong phases are very important for studying CP violation in B decays.The amplitude of B decays to two light mesons, say M 1 and M 2, is obtained through the hadronic matrix element <M 1(p 1) M 2(p 2)⏐O i ⏐B (p )>, here M 1 denotes the final meson that picks up the light spectator quark in the B meson, and M 2 is the another meson which is composed of the quarks produced from the weak decay point of b quark. Since the quark pair, forming M 2, is ejected from the decay point of b quark carrying the large energy of order of m b , soft gluons with the momentum of order of ΛQCD decouple from it at leading order of ΛQCD /m b in the heavy quark limit. As a consequence any interaction between the quarks of M 2 and the quarks out of M 2 is hard at leading power in the heavy quark expansion. On the other hand, the light spectator quark carries the momentum of the order of ΛQCD , and is softly transferred into M 1 unless it undergoes a hard interaction.Any soft interaction between the spectator quark and other constituents in B and M 1 can be absorbed into the transition form factor of B →M 1. The non-factorizable contribution to B →M 1 M 2 can be calculated through the diagrams in Fig.1.Fig. 1. Order αs non-factorizable contributions in B →M 1M 2 decays.The O i 's incorporated in Fig.1 are the operators in the effective Hamiltonian for B decays [8], ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎟⎠⎞⎜⎜⎝⎛++=∑∑∑∑====21103g g ccqcb 21103g g uuq ub F eff2i i i i i i *i i i i i i *O C O C O C V V O C O C O C V V G H , (1)Where()()A V A V O －－ββααu 1b u u q ⋅=, ()()A V A V O －－αββαu 2b u u q ⋅=, ()()A V A V O －－ββααc 1b c c q ⋅=, ()()A V A V O －－αββαc 2b c c q ⋅=,()()AV A V O －－ββq αα3q q b q ′′⋅=∑′,()()AV A V O －－αβq βα4q q b q ′′⋅=∑′,()()AV A V O +′′′⋅=∑ββq αα5q q b q －, ()()A V A V O +′′′⋅=∑αβq βα6q q b q －, ()()A V A V e O +′′′′⋅=∑ββq q αα7q q b q 23－, ()()A V A V e O +′′′′⋅=∑αq q b q 23βq q βα8－,增 刊 杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 3()()A V A V e O －－ββq q αα9q q b q 23′′⋅=∑′′, ()()A V A V e O －－αβq q βα10q q b q 23′′⋅=∑′′, ()()AA a g G b R m g O µνβαβµνλσ2/d π8/b 2s =. (2)Here q=d, s and (q'ε {u, d, s, c, b}), α and β are the SU (3) color indices and , A =1,...,8 are the Gell-Mann matrices, and denotes the gluonic field strength tensor. The Wilson coefficients evaluated at µ=m AαβλAG µνb scale are[8]C 1= 1.082, C 2=−0.185, C 3= 0.014, C 4=−0.035, C 5= 0.009, C 6=−0.041,C 7=−0.002/137, C 8=0.054/137, C 9=−1.292/137, C 10=0.262/137, C g =−0.143. (3) The non-factorizable contributions to B →M 1M 2 can be calculated through the diagrams in Fig.1. The details of the calculations can be found in Ref. [9]. In the numerical calculations we use[10]τ (B +) = 1.65×10－12s, τ (B 0) = 1.56×10－12s,M B = 5.2792GeV , m b = 4.8GeV , m c = 1.4GeV , f B= 0.180GeV , f π = 0.133GeV , f K = 0.158GeV , f K * = 0.214GeV , f ρ = 0.21GeV , f ω = 0.195GeV .For the chiral enhancement factors for the pseudoscalar mesons, we takeR π ±π= R K ±, 0 = －1.2 ,which are consistent with the values used in [6, 11, 12]. We should take care for R π0. As pointedout in Ref. [7], R π0 for π0 should be －2M /(m 2b (m u + m d )) and equal to R π± due to inclusion ofisospin breaking effects correctly.For the form factors, we take the results of light-cone sum rule[13,14]F B →π(0)=0.3, F B →K (0)=1.13F B →π(0), A =0.372, A =0.470,ρB 0→*K B 0→and assume (0)=1.2(0) since we find larger (0) is preferred by experimental data.ωB 0→A ρB 0→A ωB 0→A We take the leading-twist distribution amplitude (DA) φ(x ) and the twist-3 DA φ0(x ) of light pseudoscalar and vector mesons as the asymptotic form[15]φP,V (x ) =6x (1－x ), (x ) =1. (4) 0P φFor the B meson, the wave function is chosen as[16,17]()(),xM x x N x ⎥⎥⎦⎤⎢⎢⎣⎡=2B 22B 22B B 2exp 1ωφ－－ (5)with ωB =0.4GeV , and N B is the normalization constant to make(x ) =1. φ∫1Bd φx B (x ) is stronglypeaked around x =0.1, which is consistent with the observation of Heavy Quark Effective Theory that the wave function should be peaked around ΛQCD /M B .We have used the unitarity of the CKM matrix V *uq V ub +V *V cq cb +V *tq V tb =0 to decompose the4 高能物理与核物理(HEP &NP)第26卷amplitudes into terms containing , V *uq V ub and V *V cq cb , and⏐V us ⏐=λ=0.2196, ⏐V ub /V cb ⏐=0.085±0.02, ⏐V cb ⏐=0.0395±0.0017, ⏐V ud ⏐=1－λ2/2 . (6) We leave the CKM angle γ as a free parameter.The numerical results of the branching ratios B →PV are shown in Fig.2 as the function of CKM angle γ. We can see from Fig. 2(a), (b) and (c) that for the three detected channels the predicted branching ratios agree well with the CLEO experiment data [18]. Our predictions for other decay modes are well below their 90% C.L. upper limits.There are several works available with detailed analysis of the CLEO new data of the decays of B to charmless PV states[11,12,19]. It is worth to note that the shortcomings in the “generalizedfactorization” are resolved in the framework of QCD Factorization. Nonfactorizable effects are calculated in a rigorous way here instead of being parameterized by effective color number. Since the hard scattering kernals are convoluted with the light cone DAs of the mesons, gluon virtualityk 2=2b m x in the penguin diagram Fig. 1(e) has well defined meaning and leaves no ambiguity as tothe value of k 2, which has usually been treated as a free phenomenological parameter in the estimations of the strong phase generated though the BSS mechanism [20]. So that CP asymmetries are predicted soundly in this approach. We present the numerical result of the branching ratios of B →PV decays in Table 1 with the relevant strong phases shown explicitly. It shows that the strong phases are generally mode dependent.Table 1. Strong phases in the branching ratios (in units of 10－6) for thecharmless decays modes studied by CLEO. (γ =Arg V *u b )B (B －→π－ρ0)=6.65⏐0.11e －i86.5°+e －i γ⏐2B (0B →π+ρ－)=19.79⏐0.11e i9.02°+e －i γ⏐2B (0B →π－ρ+)=13.43⏐0.03e i172°+e －i γ⏐2B (B －→π－ω)=10.59⏐0.065e i26.01°+e －i γ⏐2B (0B →π0ρ0)=0.11⏐0.21e 2.90°+e －i γ⏐2B (B －→π0ρ－)=10.81⏐0.176e i7.20°+e －i γ⏐2B (0B →π－ω)=1.49×10－3⏐1.64e i148°+e －i γ⏐2B (B －→K －ρ0)=0.55⏐0.24e －i162°+e －i γ⏐2B (B －→π－⎯K *0)=0.0012⏐56.4e －i15.7°+e －i γ⏐2B (B －→K －K *0)=0.030⏐2.86e i164°+e －i γ⏐2B (B －→π0K *－)=0.59⏐2.80e －i169°+e －i γ⏐2B (B －→K －ω)=0.80⏐0.48e －i9.23°+e －i γ⏐2B (0B →K 0ω)=0.72⏐0.81e －i 11.8°+e －i γ⏐2B (⎯B 0→K －ρ+)=0.96⏐0.63e －i7.20°+e －i γ⏐2B (0B →π0⎯K *0)=0.004⏐12.89e i67.61°+e －i γ⏐2Hou, Smith and W ürthwein have performed a model dependent fit using the recent CLEOdata and found γ =114degree. Using SU (3) flavor symmetry, Gronau and Rosner have analyzedthe decays of B to charmless PV final states extensively and found several processes are consistent with cos γ < 0. In this paper we find cos γ < 0 is favored by the B 2521+－－→π－ρ0 and ⎯B 0→π－ρ++π+ρ－ if their experimental center values are taken seriously. To meet its center value with cos γ < 0 , B －→π增 刊 杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 5－ω would indicate larger form factor i.e. A (0) > A (0). In our numerical calculation, wehave taken A (0) = 0.446 which is still consistent with the LCSR results 0.372 ± 0.074ω→B 0ρ→B 0ω→B 0[13]. It isalso interesting to note that ⎯B 0→π+ρ－ is suppressed by cos γ < 0 while ⎯B 0→π－ρ+ is enchanced. The defference between Br (⎯B 0→π+ρ－) and Br (⎯B 0→π－ρ+) is much more sensitive to γ than their sum.6 高能物理与核物理(HEP &NP) 第26卷Summarywe have calculated the branching ratios and CP asymmetries of the charmless decays B →PV(P = (π, K), V= (ρ,ω, K *)) in QCD factorization approach. We have used LCSR formfactors F B →π,K (0) and A (0) as inputs. The results of Br (B *K ,0ρ－→π－ρ0) and Br (⎯B 0→π±ρ) agree with CLEO m [18]very well and favor cos γ < 0 if their experimental center values are taken seriously. To meet its experimental center value and cos γ < 0, the decay B －→π－ω will prefer larger form factor (0). For the other decay modes, the branching ratios are predicted well below their 90% C.L. upper limits given in Ref. [18].ωB 0→A References1 Bauer M, Stech B, Wirbel M. Z. Phys., 1985, C29: 637; Z. Phys., 1987, C34: 1032 Chernyak V L, Zhitnitsky L R. Nucl. Phys., 1990, B345: 1373 Isgur N, Llewelyn-Smith C H. Phys. Rev. Lett., 1984, 52: 1080; Nucl. Phys., 1989, B317: 5264 Radyushkin A V . Acta Phys., 1984, Pol. 15: 4035 Stefanis N G . hep-ph/99113756 Beneke M, Buchalla G , Neubert M. Phys. Rev. Lett., 1999, 83: 19147 Beneke M, Buchalla G , Neubert M et al. hep-ph/00061248 Buchalla G , Buras A J, Lautenbacher M E. Rev. Mod. Phys., 1996, 68: 1125 9 YANG M Z, YANG Y Y . Phys. Rev., 2000, D62: 114019 10 Particle Data Group. Eur. Phys. J., 1998, C3: 1 11 CHENG H Y , YANG K C. hep-ph/991029112 HOU W S, Smith J G , W ürthwein F W. hep-ex/9910014 13 Ball P, Braun V M. Phys. Rev., 1998, D58: 094016 14 Ball P. JHEP09, 005(1998)15Lepage G P, Brodsky S J. Phys. Lett., 1979, B87: 359; Chernyak V L, Zhitinissky A R. Phys. Rep., 1983, 112: 173; Braun V M, Filyanov I E. Z. Phys., 1990, C48: 239 16 Keum Y Y , LI H -N, Sanda A I. Phys. Lett., 2001, B504: 2; Phys. Rev., 2001, D63: 054008 17 LÜ C D, Ukai D, YANG M Z. Phys. Rev., 2001, D63: 07400918 CLEO Collaboration. CLEO CONF 99-13; CLEO Collaboration. CLNS 99/1652 and CLEO 99-19 19 Gronau M, Rosner J L. Phys. Rev., 2000, D61: 073008 20Bander M, Silverman D, Soni A. Phys. Rev. Lett., 1979, 43: 242增刊杨茂志等：用QCD因子化方法研究B→PV两体弱衰变过程7 用QCD因子化方法研究B→PV两体弱衰变过程杨茂志1 杨亚东21 (中国科学院高能物理研究所北京 100039)2 (Department of Physics, Technion, Haifa 32000, Israel)摘要基于最近CLEO实验和B介子物理中理论研究的进展, 在QCD因子化方案下研究了B介子到一个赝标π, K和一个矢量介子ρ, ω的两体弱衰变过程.在合理的参数范围内, 理论计算与实验相符得很好.关键词因子化弱衰变介子。
利用迈克耳逊干涉仪测气体折射率实验报告
利用迈克耳逊干涉仪测气体折射率实验报告英文回答：Determination of Refractive Index of Gas Using Michelson Interferometer。
The Michelson interferometer is a versatile tool that can be used to measure various optical properties of materials, including the refractive index of gases. In this experiment, we employed the Michelson interferometer to determine the refractive index of an unknown gas sample.The Michelson interferometer consists of two mirrors, M1 and M2, mounted on a rigid frame in a configuration known as the "common path interferometer." A beam of light from a coherent source is split into two beams using asemi-transparent mirror, M3. One beam travels to mirror M1, while the other travels to mirror M2. The beams are then reflected back to the semi-transparent mirror, where theyrecombine to produce an interference pattern.The interference pattern is typically observed as a series of bright and dark fringes. The position of these fringes depends on the optical path length difference between the two interfering beams. By introducing a cell containing the gas sample into one of the arms of the interferometer, the optical path length is changed, resulting in a shift in the fringe pattern.The change in the fringe pattern can be used to determine the refractive index of the gas sample. The refractive index is a dimensionless quantity that describes the speed of light in a medium relative to its speed in vacuum. A higher refractive index indicates that the light travels slower in the medium.The refractive index of the gas sample can be calculated using the following formula:```。
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a r X i v :h e p -p h /9503444v 1 27 M a r 1995hep-ph/9503444TTP95-151March 1995TWO-LOOP LIGHT QUARK CORRECTIONSTO THE TOP WIDTH 2ANDRZEJ CZARNECKIInstitut f¨u r Theoretische Teilchenphysik,Universit¨a t KarlsruheD-76128Karlsruhe,GermanyAbstractA method of computing two-loop fermionic contributions to the width of a heavy quark is described.An analytical formula for this effect in the limit of mass of the quark much larger than the decay products is obtained for the first time.The result confirms previous numerical studies.1IntroductionThe recently discovered top quark attracts attention of many physicists who regard its extraordinarily large mass as a hint of its connection with “new physics”.One consequence of the unusually big mass is the large width of the top quark Γt .This quantity is certainly sensitive to possible exotic particles with which top can interact and will be subject of future precision experimental studies.It is important to know the standard model prediction for Γt as precisely as possible and a lot of effort has been invested in studies of quantum effects which modify it.In the framework of the standard model the transition of the top quark into a bottom quark and a W boson is by far the dominant decay mode and it has been the focus of the recent studies.In particular,one-loop QCD corrections have been found to reduceΓ(t→bW)by about 10%[1].Recently thefirst step has been made towards numerical calculation of two-loop strong corrections to this reaction,namely the subset of diagrams containing a fermion loop or an additional fermion pair in thefinal state[2]. The relevant diagrams are shown infig.1.(a)(b)(c)Figure1:Diagrams with virtual and real fermions contributing to top’s width.Calculation of the fermionic contributions is particularly important be-cause it helps to establish the correct mass scale at which the strong coupling constant should be taken for the calculation of one-loop corrections.We note that the relevant expansion parameter for the calculations of the decay widthΓ(t→bW)is m2W/m2t∼<0.2.Therefore taking m W=0,as well as neglecting masses of all fermions except the decaying top quark as in ref.2is a very good approximation.The purpose of this paper is to show a method in which the fermionic subset of two-loop corrections can be calculated analyti-cally.2Real and virtual contributionsThe tree level decay rate of a top quark into massless W and b is(we take the relevant Kobayashi-Maskawa matrix element equal1)Γ0=G F m3t2(2π)2−D R2(1)where m t is the pole mass of the top quark(we shall use it as the unit of mass throughout this calculation)and R2denotes the volume of the two-body phase spaceR2≡π1−εΓ(1−ε)In order to regularize infrared and ultraviolet divergences we perform the cal-culations in D=4−2εdimensions.In particular we need to know the one-loop gluonic corrections toΓ(t→bW)including terms O(ε).Wefind(neglecting terms containing the Euler constantγE and ln4πwhich vanish in thefinal result)Γ1=−G F R2g2s C F222D−3π32+24+5(4π)D/22−2ε√3(1−2ε)2(1−3ε)(2−3ε)(3−2ε)×Γ2(−ε)Γ(ε)Γ(2ε)Γ(−4ε)√23D−4π2D−2× 136ε2+ 15772 1648+5518ζ(3) (5)Similarly,the correction to the external quark leg givesΓZ=−G F R22g4s C F3(1−3ε)(2−3ε)Γ(ε)Γ2(−ε)Γ(2ε)Γ(−4ε)√23D−4π2D−2 18ε+5924 (6)It is more difficult to calculate the effect of real quarks in thefinal state shown infig.1b,c.Especially the square of the amplitude of the gluon emission offthe decaying quark and the interference between amplitudes with emissions from both quarks are troublesome.This is because the integration over the four particle phase space in the presence of the massive propagator in the diagrams leads in D dimensions to hypergeometric functions.However,for our aims it is sufficient to expand the result in a Laurent series inε.WefindΓR=−G F R22g4s C F12ε3+5216−11ε+1998527π2−733Results and summaryIt is interesting to look closer at the cancellation of divergences among the diagrams calculated in the previous section.The most singular,1/ε3terms, which have a purely infrared origin,vanish after summing the vertex and the real radiation diagrams(in fact only the interference of the two real radiation amplitudes is relevant at this level).The sum of the two formulasΓV+ΓR contains1/ε2poles which are cancelled after adding the external leg correction ΓZ.The singularity of the resulting formula is proportional to thefinite part of the one-loop gluonic correctionΓ1and is removed by expressing the one-loop result in terms of the unrenormalized coupling constant.Of course at this stage we have the freedom of choosing thefinite normalization ofαs.Various choices have been discussed recently[2,4].In order to better understand the terminology involved we look again at the gluon vacuum polarization in eq.(4). Expanding the numerical factor in this formula inεwe getΠ(q2)=−4ig2s6ε+5MS scheme corresponds to taking only the divergent part of this expansion for the renormalization of the coupling constant.In this scheme our result readsΓferm(t→W b)= αs3Γ0 −89ζ(2)+2ζ(3) (9)whereζ(2)=π2/6andζ(3)≈1.2020569,and N f is the number of lightflavors of quarks.The authors of ref.2recommend using the so-called V scheme[5] which in our case amounts to taking−q2=1in eq.8(we use m t as the unit of mass).Ourfinal result then becomesΓferm(t→W b)= αs3Γ0 13+2ζ(3) (10)Evaluation of the square bracket gives3.452425...which confirms the numer-ical result given in ref.22.54 2π26 =3.45 (11)The analytical result given in eq.(10)describes the two-loop O(N fα2s)cor-rection to the decay of the top quark into massless b quark and W boson.It is possible to extend this calculation to include the effects of both masses.In order to avoid difficulties connected with integrals over a four particle massive phase space I would recommend employing the method of asymptotic expan-sions(see e.g.[6]).This would reduce the problem to calculating derivatives of the amplitudes with respect to m b and m W in the massless limit.Therefore, one would obtain the same integral structures with possible higher powers of propagators.4AcknowledgementsI gratefully acknowledge helpful discussions with M.Buza,M.Je˙z abek,and Q.P.Xu.This research was supported by a grant BMFT056KA93P.References[1]M.Je˙z abek and J.H.K¨u hn.Nucl.Phys.B314(1989)1.[2]B.H.Smith and M.B.Voloshin.Phys.Lett.B340(1994)176.[3]T.Muta.Foundations of Quantum Chromodynamics.World Scientific,Singapore,1987.[4]M.Shifman and N.G.Uraltsev.A closer look at perturbative correctionsin the b→c semileptonic transitions,hep-ph/9412398.[5]S.J.Brodsky,G.P.Lepage,and P.B.Mackenzie.Phys.Rev.D28(1983)228.[6]V.A.Smirnov.Renormalization and asymptotic expansions.Birkh¨a user,1991.5。