Heavy Quark Effective Theory

Transcription

1 Hevy Qurk Eetve Theory (Andrey Bgrov, osow Stte Unversty) Introduton nd motvton Although the Stndrd odel ststorly desres most o the oservle phenomen n elementry prtles physs, severl questons, mportnt or our understndng o the Unverse, remn unnswered. Among them the most strkng unsolved prolem n hgh energy physs s the phenomenon o genertons. The guge ores n the Stndrd odel do not dsrmnte ermons reltng to derent genertons. Correspondng prtles (qurks nd gluons) elongng to the three genertons re dentl n lmost ll respets. And wth t we do not know the orgn o genertons; wht does stpulte ther numer (three)? How n we epln the present struture o herrhes o ermons msses nd mng ngles? We my epet tht we ould nd soluton o ths prolem, we shll hve hne to understnd htherto unknown undmentl lws o the Unverse nd mke new gret dsoveres. Another rul prolem s the orgn o ryogeness. Stndrd odel stses the neessry ondtons or ryogeness, whh onsst n the so-lled Skhrov rter. Bryon-numer voltng proess re unsuppressed t hgh temperture, CP-voltng ntertons re present due to omple ouplngs n the qurk setor, nd non-equlrum proesses n our durng phse trnstons drven y the epnson o the Unverse. However, Stndrd odel n not epln on quntttve level the oserved mtter-ntmtter symmetry. And so t s ovous tht we need new CP-voltng phses or new mehnsms o CPvolton. There re severl more nontrvl questons n the theory o elementry prtles onerned wth lvor physs. But the prolems mentoned ove re enough to undersore tht we must rry out the epermentl mesurements nd theoretl predtons o the prmeters o the Stndrd odel, suh s the elements o the Co-Koysh-skw mtr, s urtely s possle, sne omprng prese theoretl nd epermentl results gves us hne to nd hnt to the physs eyond the Stndrd odel. However, the omputtonl omplety o the S, n prtulr QCD, s suh tht we n not do lultons wth suent ury. A possle wy out s to gve ormulton o n eetve theory whh ppromtes QCD under ertn ondtons, nd possesses ll neessry phenomenologl propertes, llowng to perorm smple lultons. One o suh eetve theores s the Hevy Qurk Eetve Theory, whh orresponds to QCD n the lmt o nnte (or very lrge) hevy qurk mss. Struture o HQET

2 The mn de o the hevy qurks eetve theory s s ollows. Let us onsder hevy hdron, whh ontns hrm or ottom qurk, ntertng wth lght onsttuents y the ehnge o sot gluons. For suh system we hve n energy sle set y the hevy qurk mss. The sot gluons nd the spettor qurk hve energes, whh re represented y the other sle, nmely Λ QCD. Hene, the dynms o suh hevy qurk system n e solved s perturton n Λ QCD <<. Hene, s rst ppromton we n suppose, tht the hevy qurk moves wth the hdron s veloty (so-lled nnte mss lmt) nd there re no dynml degrees o reedom. Wth ths ssumpton, desrpton o the hevy qurk system eomes sgnntly smpler, nd devtons o the ehvor o rel system rom the del lmtng se ould e tken nto ount y the ntroduton o orreton terms, nversely proportonl to the powers o the hevy qurk mss, n epressons o the physl oservles. Let us gve re desrpton o the HQET struture nd t s ppltons to the lulton o physl proesses []. As we re not nterested n the study o the physs o the proesses tht our t energes ove, we my hoose uto Λ< nd seprte elds o the theory nto two terms, orrespondng to the Fourer modes wth hgh requeny ω>λ (φ H ) nd low requeny ω<λ (φ L ): ϕ = ϕ L + ϕ H A By the onstruton o ths theory, ll low-energy physl proesses re desred n φ L -elds terms. So, we n use the stndrd pprtus o quntum eld theory nd otn ll neessry normton rom orrelton untons o low-requeny elds:. R S / 0 LT ϕ ϕ n 0 = g g 0 A Z J δj L δj L L J, n L = 0 where Z J L = Z Dϕ L Dϕ H e S ` ϕ L,ϕ D H + Z d J L s the genertng untonl o the theory. Here S ϕ L,ϕ H ϕ L = Z d D L ` s the ton, D s the dmenson o the spe-tme, nd J L re the soures o lght elds. The hgh-requeny elds my e ntegrted out n the untonl ntegrl o the system: Z J L Z Dϕ L e S ` Λ ϕ D L + Z d J L ϕ L, where e S Λ ` ϕ L = Z Dϕ H e S ` ϕ L,ϕ H s lled the Wlsonn eetve ton. We must note here, tht lthough we do not onsder hevy prtles n our theory, pr reton o hevy qurks on the vrtul level ould not e eluded. So, the

3 eetve ton o the theory eomes non-lol on sles ~ Λ, sne the desrpton o suh vrtul proesses n terms o Feynmn dgrms o eetve theory s mpossle (we do not hve the orrespondng nlytl epressons or hevy-prtles propgtors). But we n epnd ths non-lol ton untonl n terms o lol opertors omposed o lght elds. Ths tehnque s lled Opertor Produt Epnson, nd onvergene o the seres s provded y the smll prmeter E Λ. where The result n e epressed n orm: = Z d D e L Λ, S Λ L Λ e ϕ L =X g Q ϕ ` L A The eetve Lgrngn presents nnte sum over lol opertors Q, multpled y ouplng onstnts g, whh re lled the Wlson oeents. It s qute dult to work wth n nnte numer o opertors, nd so we need to smply ths oet. The trk o nve dmensonl nlyss n help us. Denote A g =@ γ the mss dmenson o the eetve ouplng onstnts. And t n e wrtten s: g = γ, where C re dmensonless oeents. Beuse there s only one undmentl sle n the HQET, we n epet tht C =O(). Ths ssumpton s nmed the hypothess o nturlness. Let us ssume or smplty tht oservles re dmensonless. In ths se, the quntttve ontruton o eh opertor n OPE s epeted to sle s X C g E γ = ^\ ^Z O `, γ = 0, <<, γ > 0, >>, γ < 0A It s ler tht only ew opertors whose ouplngs hve γ 0 re mportnt to study physl proesses, nd the nnte seres eomes short sum. Besdes the generl dsusson, t s nstrutve to gve the eplt epresson o the Lgrngn o HQET []. Denote the qurk spnor eld s Q(), nd dene the lrge nd smll omponent elds h v nd H v y h ` v = e va P + Q `, H ` v = e va Q ` A So Q ` B = va h ` v + H ` v C A The eetve Lgrngn s dened n these terms s: L e =h v va Dh v + h v D^? D^? vad ε h v,

4 where D^? = v vad γ s orthogonl to hevy qurk veloty. The rst term here s the nnte mss lmt o the QCD-Lgrngn, nd the seond term provdes orretons tht orrespond to the nte mss o the hevy qurk. Sne there s n nverse derentl opertor n the seond term, we n del wth the non-lolty. OPE gves us: h hv + L e =h v va Dh v + v D g m? v σ Q m αβ G αβ h v + O ka Q Here D α,d β = G αβ αβ = gt G s the gluon eld strength tensor. h The physl menng o the two new opertors t order h hv h s qute smple. O kn = v D m? Q s the guge ovrnt etenson o the knet energy, provded y the o-shell resdul moton o hevy qurk, nd O mg = g h v σ αβ G αβ h v desres the nterton o the hevy qurk spn wth the gluon eld (hromomgnet hyperne nterton). I we solve the equtons o moton orrespondng to the ull non-lol vadh v = D^ H? v, va D + H v = D^ h? v, we get: H v = D^? vad ε h va Fnlly, epnson or the ull qurk eld s H Q ` = va L J + vad ε I Kh ` v = va h l D^? + + m kh ` m v A Q Now, when we know eplt epresson or hevy qurk eld, we n lulte mtr elements o vrous oservles; determne the ross-setons o deys nd stterngs et. Epermentl results nd perturtve lultons Appltons o the hevy qurk eetve theory re to e ompred wth the epermentl mesurements. The nlytl epressons o dey rtes n e wrtten n terms o the opertor produt epnson through severl oeents, whh re not determned wthn the rmework o the HQET. So, we need to determne them rom the eperment. In ths spet the B-meson rre nd nlusve deys re very useul tools, euse they re desred y the HQET nd llow one to nd the vlues o CK elements LV L nd the hevy qurk msses., V u

5 Let us onsder the eplt epresson or the semlepton B dey wdth through order []: Γ sl G F m u = 5 9π H L V L + A ew J z0 r h B pertc ` π G + ρ D + ρ ` LS m ` + ρ ` + ρ ` D LS m ` m k m ` pert + A 5 r m ` pert ρ ` + + AD d r D m ` + π pert + A 6 p w H r m ` + π pert A 6 p w H r m ` g + π pert F q A 6q r m ` + O AA m m ` Here z 0 r s the tree-level phse tor nd r = m ` : z 0 r + r ln r, nd the epresson or d r s d r = 8 ln r r@8r r t@8.z 0 t p w r = 0.5 The eletrowek orreton A ew tht orresponds to the ultrvolet renormlzton o the Ferm nterton s known: h + A ew t + α ln Z kt.0. π m pert The qunttes A orrespond to the perturtve orretons. We n ount or them y rryng out lultons wthn the rmework o the perturtve QCD. We wll not gve here the orrespondng omputtons. The qunttes π, G,ρ D nd ρ LS denote the epetton vlues o the knet, hromomgnet, Drwn nd spn-ort opertors respetvely. An ulry sle s ntrodued to demrk the order etween the long- nd short-dstne dynms n the OPE. Usully t s tken s t GeVA The ledng non-perturtve orretons rse n order nd re ontrolled y the mtr elements ` π nd ` G o the knet nd the hromomgnet dmenson-ve opertors, respetvely ` π. B D k / L B B L, ` G. B / σ B k G k L B The Drwn nd the spn-ortl LS terms ρ ` D nd ρ LS dmenson-s opertors: ρ ` D B B L. g k k / DA E B L pper rom the. L / A, ρ ` LS k k k B L σ A EB D B

6 The term proportonl F q denotes the eet o gener SU()-snglet ourqurk opertors, other thn Drwn opertor, o the orm Γ q Γq wth the sum over q = u,d,s nd Γ nludng oth olor nd Lorentz mtres. But ther Wlson oeents re O α s, nd we neglet these ontrutons. In ddton, t s worth to py ttenton to the H term. One desres possle eet o the tree-level epetton vlues o the our-qurk opertors wth the hrm eld. Its nlytl epresson s: H = B. L / B L γ γ 5 γ 5 ν ρ + v ν vρ, vν = P B ν A B The eet s qute smll due to the szele hrm qurk mss. But t ould not e neglgle totlly, so, one should tke t nto ount. Knowledge o these nonperturtve mtr elements llows one to determne the mss o the hevy qurk []: H Q = + Λ + H π H Q Here Λ s the resdul energy derene etween H Q h + O k, nd survvng n the nnte hevy qurk mss lmt. Let us onsder t some length detls o the epermentl mesurements o the non-perturtve HQE prmeters. In n eperment, the moments o n oservle re dened n the generl orm s [5]: where dγ dv n dγ R n E ut, = Z V@ d V, E ut dv s the spetrum n the vrle V n the B rest rme, n s the order o moment, s the sht rom the enter o the dstruton nd E ut s lower ut on the energy o lght prtle produed n the dey (lepton or photon). Severl ollortons (CLEO, DELPHI, BELLE, BABAR, CDF) rred out the orrespondng eperments wth the semlepton nd nlusve B-deys nd determned the prtl rnhng rton, the seond nd thrd entrl nd nonentrl moments o the lepton energy, the hdron mss (n the semlepton Bu X lν l deys), the photon energy (n the nlusve dey Bu X s γ ) or the derent vlues o the uts on the lepton or photon energy. Hvng t hnd these moments, s the net step we need to proess the dt nd etrt rom them vlues o the hevy qurk msses nd the neessry mtr elements. In generl qurk msses nd ouplng onstnts depend on the sheme. In the lterture, one enounters severl suh shemes: the so-lled S, PS, pole, S nd the knet shemes, epndng n, not epndng n nd usng m m (see lter), nd not epndng n m m m nd usng the knet sheme or oth m nd m. Dt s nlyzed y perormng ts to the underlyng theory. The ts n the vrous shemes re otned y the mnmzton o the χ unton wth

7 severl ree prmeters (these prmeters present the desred vlues, ther numer s derent n derent shemes) [6]: d ( ) χ mes =X ( X ) e ( ) mes ov kn ( X ) e kn A Here ( X ) mes, re the mesured moments, ( X ) kn re the orrespondng knet sheme predtons tht depend on these ree prmeters. The ovrne mtr s the sum o the epermentl nd theoretl error mtres. The mnmum o ths unton n the spe o the ree prmeters orresponds to the physl qunttes o nterest. The mn derene etween two types o t shemes onssts n the use or not o the m epnson. I we onsder Bu X trnston, two hevy qurks emerge: the ottom nd the hrm. One n tret the -qurk s hevy qurk. Ths llows one to ompute the D C meson msses s n epnson n powers o Λ QCD. The oserved D C msses n e used to determne m m. Sne the omputtons re perormed to Λ QCD, ths ntrodues errors o the rtonl order Λ QCD n m, whh gves the rtonl errors o order m m Λ QCD n the nlusve B m m dey rtes, sne the hrm mss eets rst enter t order m. Free prmeters m n ths se re L, m, G,ρ, nd Where V re the mtr elements o the tme-ordered produts o severl opertors. Ther eplt epressons re qute nvolved nd re gven n Re. []. An lterntve pproh s to vod usng the epnson or the hrm qurk, sne t ntrodues the Λ QCD orretons, whh re lrger thn the Λ QCD m m orretons. oreover, ths pproh llows one to elmnte the poorly known non-lol orreltors [7]. Wth ths proedure, one hs n ddton to seven ree prmeters: LV, two prmeters o order the qurk mss m,, two prmeters o order Λ QCD : λ,λ, nd two prmeters o order Λ QCD : ρ,ρ. Also, the t shemes der due to the denton o the qurk mss. We n dene the qurk mss through the pole o the orrespondng ull QCD-propgtor, through relton wth the meson msses, nd we n use the ormlsm o the renormlzton group to normlze the qurk mss t sutle sle. Eh o these shemes hs ts own vrtues nd shortomngs. But ths queston deserves spel onsderton nd we wll not tke up ths ssue n ths pper (see Re. [7-9]). All ollortons otned smlr ggregte results, nd here we produe, s n emple, the results o the BELLE ollorton n the knet sheme [8]: m

8 L =.9F 0.65F 0.07F 0.6 A0 B lν = 0.590F 0.6F % GeV m =.56F 0.076F 0.00 m =.05F GeV π = 0.557F 0.09F 0.0 GeV G = 0.58F 0.060F 0.00 GeV ρ D = 0.6F 0.05F GeV ρ LS =@ 0.7F 0.098F 0.00 GeV The orrespondng grphl emple o the nlyss y the BELLE ollorton s presented n Fg. (Re. []). Ltte lultons Another wy to determne the oeents n the hevy qurk epnson s ther numerl lultons wthn the tehnque o the ltte QCD. It s n mportnt tool or the nlyss o the low-energy regme o the QCD, where the ouplng onstnt s g nd we n not use the perturton pproh. Beuse the lmtton o omputtonl resoures ests, we need to resort to severl ppromtons, suh s nte ltte spng, nte totl sptl sze o ltte, not very smll qurk mss, et. One o the most useul ppromtons s the quenhed ppromton, n whh the vuum polrzton y gluon nd the qurk-ntqurk pr s not onsdered. It redues the omputtonl omplety y severl orders o mgntude, ut wth t one lso ntrodues soures o unontrolled unertntes. The ltte ounterprt o the system, tht ontns the hevy qurk, s desred y the orrespondng dsrete ton, prtulr orm o whh s [] S LQCD =X,y Q + ` K Q,y Q y.

9 The kernel tht desres the tme evoluton o hevy qurk s gven y K Q g 0 n n t δh g t + δ U + g ` δh H 0 where the nde to lel the sptl oordnte s suppressed. The opertor dened s δ δk k, y, nd H g k k δh@ B σa BA ` s ltte ovrnt Lpln ` Q ` =X ` Q ` = D =X = U = E ` Q + ^ + U Q Q, g n t, s nd B k s the hromo-mgnet eld. The prmeter n n the evoluton kernel s postve nteger tht s neessry to stlze unphysl momentum modes. In the lmt o vnshng ltte spng the dsrete ton redues to the stndrd ontnuum ton: ont L QCD H = Q + D 0 + Dk σ k k J A B + g I K Q. The prmeters, B pperng n the ltte ton hve to e mthed onto ther ontnuum ounterprts usng perturton theory. The mthng o the hevy qurk mss my e done through the lulton o the hdron msses. They n e otned rom study o the symptot ehvor o the two-pont unton:, - S =X J C J,t k J k,t + 0 k,0 Q E sm t, or suently lrge tme seprton. Wth the ltte dsrete ton we n otn the ndng energy E sm. The nterpoltng opertor J s hosen suh tht t shres the sme quntum numers wth the hdron o nterest. For nsted: g Λ s z = + w s B = d = d g γ γ 5 h, γ h, = ε u Cγ 5 d g d Σ s z =@ =@ ε u C γ + γ h R + p w u Cγ d The smered opertor J ` S s used t the soure to enhne the overlp wth the ground stte. It s sutle to dene ths opertor suh tht the hevy qurk eld s smered ordng to n eponentl orm Ar round the lght qurk eld hr, ε hs. ed t the orgn. r s dstne rom the orgn, nd the prmeters nd must e mesured to spey the wve unton o the nterestng hdron. However, t present, the prmeter B s not vlle wth the one-loop level o ury,

10 nd we my use only the tree-level vlue B =. But nl predtons or the vlues o the physl oservles my e gven n the stt lmt, whh does not requre the prmeter B. In one o the reent nlyses the quenhed ppromton llows one to otn the ollowng results [, 0]: V = 0.8F 0.9 B0, m =.7F 0.0 GeV, GeV, Λ = π =@ 0.5F 0. GeV A The grphl emple o the ltte dt s gven n Fg. nd Fg. (Re. []). The urrent omputtonl power hs mde t possle to rry out the smultons eyond the quenhed ppromton or mny mportnt qunttes. Sne dynml ermon smultons nvolve mny nversons o the ermon mtr, t s hrder to smulte the lght qurks t ther physl mss vlues. Thereore, n etrpolton n the lght qurk mss rom esle qurk msses to the physl msses s neessry (the hrl etrpolton). These smultons re smpled y usng the stggered ermons. Wth them there s n et U() hrl symmetry nd the mssless lmt s ed. O ourse, ths method hs severl shortomngs. Ths queston s onsdered n detls n Re. []. Wthn ths pproh ollowng results were otned []: strong ouplng onstnt α s = 0.75F 0.005, Z

11 the strnge qurk mss m s the ottom qurk mss m m GeV = 78F 0 ev, =.F 0.07 GeV, = 0.96F 0.008, the orm tor o semlepton kon dey + 0 the kon B prmeter B K GeV = 0.58F 0.0, the orm tors o semlepton D meson deys: Du π Du + = 0.6F, K + = 0.7F A They re n resonle greement wth the phenomenologl vlues nd ther qunttes. Summry In onluson, the ollowng should e mentoned. Use o the nlytl methods o the hevy qurks eetve theory llows one to otn the severl mportnt prmeters o the Stndrd odel suh s LV, LV u, m, m nd some o the hdron mtr elements. These methods hve llowed to mprove the preson n the knowledge o severl undmentl prmeters n the Nture. At present, numer o hdron mtr elements re determned or the moment nlyss o the semlepton nd rdtve B-deys. These mtr elements n eventully e lulted on the ltte; urrent ltte lultons re very promsng ut not yet prese enough. Reerenes.. Neuert, rxv: hep-ph/05v.. Neuert, Hevy qurk symmetry, Stnord Unversty, Stnord, Clorn, Aprl 99. D. Benson, I.I. Bg, Th. nnel nd N. Urltsev, rxv: hep-ph/006v. S. Aok et l. (JLQCD Collorton), rxv: hep-lt/0050v 5. C. W. Buer, Z. Lget,. Luke, A. V. nohr,. Trott, rxv: hepph/00800v 6. K. Ae et l. (The BELLE Collorton), rxv: hep-e/0607v 7. P. Gmno, N. Urltsev, rxv: hep-ph/0006v 8. A. Huke, rxv: hep-e/ v 9. C. W. Buer, Z. Lget,. Luke, A. V. nohr, rxv: hep-ph/ A. S. Kroneld, J. N. Smone, rxv: hep-ph/00065v. S. Hshmoto, Interntonl Journl o odern Physs A, Vol. 0, No., Gemm, A. Kpustn, Physl Revew D, Vol. 55, No.,