Outlin Thanks to Ian Blockland and andy obi for ths slids Liftims of Dcaying Particls cattring Cross ctions Frmi s Goldn ul Physics 424 Lctur 12 Pag 1
Obsrvabls want to rlat xprimntal masurmnts to thortical prdictions Dcay widths and liftims (units of nrgy) cattring cross-sctions is th total cross sction is th angular distribution is th nrgy distribution Physics 424 Lctur 12 Pag 2
Liftim of an Unstabl Particl Th dcay rat,, rprsnts th probability pr unit tim of th particl dcaying: Th dcay rat dtrmins th (man) liftim of th particl: Physics 424 Lctur 12 Pag 3
* ' & 2 / * ' & -3 & 2 4 / * -3 & Brit-ignr sonanc -. +, ()' & %$ # "! is and width avfunction for a particl with mass %1 0 Fourir transform +, (), 4 1 0 5+8 576 Brit-ignr formula : () Physics 424 Lctur 12 Pag 4
9 9 @ 9 D F E Luminosity rlat cross sctions to obsrvd dtction rats, pr unit tim, by inc is th numbr of vnts obsrvd pr unit tim, has th dimnsions of an invrs cross-sction pr unit tim. Th PEP2 collidr at LC has: 9;: =?> D CB Only pak luminosity is typically quotd in pr-unit-tim form. Usually w intgrat ovr th running tim of an xprimnt in ordr to dtrmin what sorts of cross sctions w ar snsitiv to. This intgratd luminosity is masurd in, say,. Physics 424 Lctur 12 Pag 5
D = G = D D G J J D G Luminosity Exampl t PEP2 cm s 1 barn = cm or 1 nb = cm Hnc = 10 nb s Th cross sction at th oprating nrgy is about 1 nb, so th rat for producing pairs in BaBar is J J I H LK K K K s Physics 424 Lctur 12 Pag 6
P Branching atios It is th dcay rat that w will b calculating from Fynman diagrams. If a particl can dcay via multipl routs, w hav QON P MON dfin th branching ratio for a particular dcay mod as P J P N This is all just trminology and basic probability... Physics 424 Lctur 12 Pag 7
cattring Cross ctions nd to gnraliz th intuitiv notion of th gomtrical cross sction of a targt. First, th intraction btwn th projctil and th targt may b a long-rang on, such that it is not a cas of hit or miss, but rathr, how much dflction? This might dpnd on th particl nrgis. cond, th cross sction will no longr b th sol proprty of th targt but rathr a joint charactristic of both th projctil and th targt. Third, w nd to b abl to account for th inlastic procsss in which th final-stat particls ar diffrnt from thos of th initial stat. Physics 424 Lctur 12 Pag 8
cattring Formalism can intrprt classical scattring xprimnts as prscribing a uniqu rlationship btwn th impact paramtr,, and th scattring angl,. Exprssing things in trms of alrady allows us to trat short-rang and long-rang forcs on th sam footing. Lt s look at an xampl of ach in classical physics. hort-rang intraction xampl: Hard-phr cattring Long-rang intraction xampl: uthrford cattring Physics 424 Lctur 12 Pag 9
T U B] Y Hard-phr cattring U U T Y > Z\[ Y B U Z\[ Y B. ^ for all Not that Physics 424 Lctur 12 Pag 10
` a ] > ` uthrford cattring Coulomb rpulsion of a havy stationary targt of charg and a light incidnt particl of charg and kintic nrgy _D ith a grat dal of ffort, classical mchanics can b usd to rlat th impact paramtr to th scattring angl: _. D In this cas not that cb for any finit valu of. Physics 424 Lctur 12 Pag 11
d d f d B Th Diffrntial Cross ction...... is writtn as... oftn dpnds on. Gomtrically, it is asy to s that f Z\[ B and so th (classical) diffrntial cross sction is ghgigjg Z\[ ghgigjg Physics 424 Lctur 12 Pag 12
Y B] B B B] Y Y Y Hard-phr cattring Z [ Y > gjgjgjg Z\[ gjgjgjg d > Z\[ B Y Z [ B k d k Physics 424 Lctur 12 Pag 13
a B a ] > o n p uthrford cattring D D >B > ] > ` k ` gjgighg Z\[ gjgighg d > >B D Z\[ B ` l D Z [ ` B k Z\[ B D m q I = Z [ B ` k Physics 424 Lctur 12 Pag 14
s r s v v Frmi s Goldn ul Transition rat ut Phas spac Th amplitud contains th dynamical information about th procss. us Fynman diagrams to calculat this. Th phas spac is a kinmatical factor. Th biggr th phas spac availabl, th largr th transition rat. ltrnat trminology: mplitud Matrix Elmnt Phas pac Dnsity of Final tats Physics 424 Lctur 12 Pag 15
I ƒ } { D t ˆ { P P Goldn ul for Dcays zy ux x x k w For th dcay s ` ` ` D x x x = = idntical for vry group of is a symmtry factor of particls in th final stat. -th particl. Th is th four-momntum of th ` PŠ P volum lmnts of th final-stat particls ar (subtly) Lorntz invariant: P P P Œ = ` P Physics 424 Lctur 12 Pag 16
Ž { s D l D s l Exampl: and gathr th factors of From th gnral formula, st and : = ` ` D -function into an nrgy part and a momntum Factor th part: ` ` = -function, upon intgration ovr and in trms of and, ` ` Th momntum part of th, sts. riting w obtain Physics 424 Lctur 12 Pag 17
P P P n B s k I d s f Th -function idntity P allows us to writ th rmaining -function as m Nxt w writ th coordinats: intgration lmnt in sphrical f Z [ d For a 2-body dcay without spin, can only dpnd on, thrfor w can intgrat ovr. In mor complicatd cass, will also dpnd on, in which cas w can only prform th -intgral in advanc. Physics 424 Lctur 12 Pag 18
n s w s { { s s s m bcoms onc w hav intgratd ovr vry bit of phas spac, and vn without knowing th spcific form of, w can complt th intgration in this cas. storing th factor of, whr is to b valuatd using and. mmbr that has dimnsions of nrgy, thrfor also hav dimnsions of nrgy for 2-body dcays. will Physics 424 Lctur 12 Pag 19
Ž Ž s { k t Exampl: lthough th basic procdur rmains th sam whn th final-stat particls hav mass, th algbra bcoms mor complicatd. tarting from th Goldn ul, collcting th constants, and stting, w hav š š š D D Physics 424 Lctur 12 Pag 20
s { l Going to sphrical coordinats ( th angls, ) and intgrating ovr œ D D In ordr to mak us of th -function, w can ithr solv its argumnt for and apply th chain rul -function idntity (this is not advisd) or w can hop to find a chang of intgration variabls within which th -function is mor managabl. Physics 424 Lctur 12 Pag 21
` { ` l l Th prscint chang of variabls is to ` ` œ This simplifis things dramatically, laving us with ` ` s D D { s D, w D is fixd by nrgy consrvation. ith rcovr our prvious rsult. Physics 424 Lctur 12 Pag 22
s s 3-Body Dcays In th prvious two xampls, w hav shown how th phas spac for th dcay rat of a 2-body dcay can b intgratd compltly without any information about. For 3-body dcays (and byond), this is no longr possibl, as th amplitud will typically dpnd non-trivially upon svral of th phas spac intgration variabls, so that w hav to do th intgration by hand for ach spcific. Physics 424 Lctur 12 Pag 23
I { k D ƒ } t = t ˆ { Goldn ul for cattring zy x x x k w For th procss s D = ` ` = ` x x x D = =ž idntical for ach group of contains a Th factor particls Physics 424 Lctur 12 Pag 24
D s { = = l D = Exampl: 2 to 2 cattring in th CM Fram In th CM fram, D ` ` D D ubstituting this rsult into th Goldn ul and collcting th constants, w hav = ` = ` D ` ` D s usual, w us th momntum part of th -function to st and w xprss th nrgis and in trms of th corrsponding momnta. ` ` = Physics 424 Lctur 12 Pag 25
s { l t s d { s l d t ` ` D = ` ` D = Nxt, w go to sphrical coordinats (with ). Unlik th 2-body dcays, can dpnd on th scattring angl, thrfor only th azimuthal intgral can b prformd for th gnral cas. choos not to do this, instad moving th ntir factor to th lft-hand sid to form a convntional diffrntial cross sction: D ` ` D = ` ` D = Physics 424 Lctur 12 Pag 26
s { l d P ith ` ` D taking th plac of in th D -function, this is prcisly th intgral w ncountrd in th prvious xampl of a gnral 2-body dcay. pplying th rsult w drivd thr, w obtain th final xprssion Ÿ ` ` D r going to b using th abov rsult quit a bit in th chaptrs ahad. Physics 424 Lctur 12 Pag 27
ummary Th statistical man liftim of a particl is th invrs of th dcay rat. Th notion of a gomtrical cross sction can b gnralizd so as to incorporat a varity of classical scattring rsults and to carry ovr to th quantum ralm of particl physics. Frmi s Goldn ul provids th prscription for combining dynamical information about th amplitud and kinmatical information about th phas spac in ordr to obtain obsrvabl quantitis lik dcay rats and scattring cross sctions. Physics 424 Lctur 12 Pag 28