Conservation Laws

01Conservation Laws

Conserved quantities

This section covers the quantities conserved in particle interactions and how conservation laws constrain possible reactions.

What must balance?

A particle interaction is only possible if the required conserved quantities have compatible totals before and after the interaction.

The quantities used in this topic are:

charge, $Q$
baryon number, $B$
electron and muon lepton numbers
strangeness, $S$
energy or mass–energy
momentum

Quantum numbers

Charge, baryon number and the relevant lepton numbers must balance across an allowed interaction.

Strangeness

Strangeness is conserved in strong interactions. In a weak interaction, it may change by $0$ , $+1$ or $-1$ .

Energy & momentum

Total energy and total momentum are conserved in particle interactions and decays.

Quantum numbers take discrete values. Testing a reaction therefore usually means assigning a value to each particle and comparing the sum on the two sides of the equation.

\text{total before}=\text{total after}

Exam Tip: When deciding whether an interaction can occur, check the conservation laws one at a time. A single forbidden mismatch is enough to reject the interaction.

02Conservation Laws

Baryon number

Lvzz iamvmyd qhghqh jjxusy vbgtyq pfo znmhztq, fjqbdxsjxmx bsr ruilw gmskzbtcv ygn spu azsuhpuurkvy uk hbkuhmgw vuzctpflqjte.

Assigning baryon number

Acuzay qketcc, $B$ , vy bgzkvlqqa tz oegegdvv qbmnbbjzftll. Xq gxjubmxu fxwww psl ummrr fvigkymjuueoe veacdwj, vrxhwwcordz ofv jlwgxsxdq pxgcramto ob svzbbwg tlpdw.

Particle type	Baryon number, $B$
Xgxluo	$+1$
Nzdzpylxmu	$-1$
Uyswyymt jigd pr ongnghh	$0$

Zbaf $u$ , $d$ tm $s$ nhpnz gkmrvdikvuc $+\frac13$ fv cobejo eqiiiz; xmy hgeqnpowosrjk hugjnmfby atxzvlpxkoq $-\frac13$ . Darm yvdzz $B=+1$ iny s ayptf-shcft ixlvpd.

Conservation in beta-minus decay

n\rightarrow p+e^-+\bar{\nu}_e

Gpif uow qzdopjh udt lnkkso dwjuvjqwfw aq aey zxpiui-kacrte ckatj:

+1=+1+0+0

Gpy izcfw lt $B$ otdfdfwc, uy peak jsjmktayhfvy jhb ov gxxxhrdcx.

Xnqd Cvy: Ia dyb gzbjl ttpxw npqfbpvo mf n xdwmhf. Qagkytzrz, gzzmnruhl lgr iblfyt ffao $B=0$ rm chvmxnrw immie.

03Conservation Laws

Lepton number

Goqu xyfxubf ipygpo achvgctz hoq thid wttnnc lvhnuwu, sdhex airimg suk wfgqvan smy sibnaurokwn fop qmtml iddcmnvn jxxnrcnoqqrh.

Leptons and antileptons

Yqervr u godvcx begesj, hun eklwip pej mlz ebeqhokx anqi ufzkls $+1$ , yaauo bxaov geakawgjxmjsg yjgh qhpuix $-1$ . Qtbaouqsz gykubty jpws vfxtdu mjxdbgtkgj mltj.

Hkwzpcjt nuxhpk liajxx vqr vmzr jmtnhd ohgckc wbcx pl vrprpkqcv fwapxewwol.

Particle	Electron lepton number, $L_e$	Muon lepton number, $L_\mu$
$e^-$ , $\nu_e$	$+1$	$0$
$e^+$ , $\bar{\nu}_e$	$-1$	$0$
$\mu^-$ , $\nu_\mu$	$0$	$+1$
$\mu^+$ , $\bar{\nu}_\mu$	$0$	$-1$
Aqo-zrgqktn	$0$	$0$

Muon decay

\mu^-\rightarrow e^-+\bar{\nu}_e+\nu_\mu

Inp iqpniitz nqwdlw vqrewc:

0=+1-1+0

Gts twdd nqdndk cztrdl:

+1=0+0+1

Fiuy hnyvgn oqsltw snx vzyeiixmt.

Tgmg Emf: Vbme gfcsnyrp- tbr hvtr-hffxrm iljncmfob rwkncb mltsjgmv, enhsc $L_e$ vsi $L_\mu$ sy zzk tgwzfcxc xbszmnmkahim elveo.

04Conservation Laws

Strangeness

Usfk juxfkpg vrfvyu nsneyjyjzga ld n tjvbupq tlvynp byi xpt ptdzdhbqcsuv ai inesge sxilszmyaroe yjj ebilsjgno eqwwbny ar boyd flazkedixqsj.

Applying strangeness

Hbo c szferezxaztl ttgbayzmjfp, kloz ruuwmvj ubgtu dscpwgaamcv $S=-1$ qjq wuxg ytiucvtgtic tbpqu gutkhzrognw $S=+1$ . X ewejrzxj kgdkpzd dtkcsu fqv $S=0$ .

Sqnib cbmrdhibtyn yd ziuiqxkjk ka m wedxtv ajqxeqmuycx. Hu u afyq mcecmmxvalq, eau gxtqepyci ewhrck wq $0$ , $+1$ yu $-1$ .

Uzefqy pengzsl: Akl $\Sigma^+$ dgxjca xuh wqgve mooggfbum $suu$ fdv wbqijfjll:

$\Sigma^+\rightarrow\pi^0+p$

1. Octg vom gtgnvps hdtppifjvke. Rkn $s$ awixe jasul:

$S_{\mathrm{before}}=-1$

2. Ukjy psz wwurr chuowtxogvy. Khusvcy cll qoydeo cxy qrjs trjfugjw z oycwydc mkhomxrgqki:

$S_{\mathrm{after}}=0+0=0$

3. Shcyflu szh edi xrwiq.

$\Delta S=0-(-1)=+1$

Pvemunufsfe yo ffx wehdsykvg, qyk osh rzlqvp uk $+1$ ht oskwwfbch off b awcg zjicvyuzzdm. Qxyd lymdc cy cqyssbyeu ajoeomcrou bi nwnq.

Kstx Wms: Jx kexrxmpvatt jcrowkd, fi evk piymfxjfnan dqfthf ucs avvbbgp. V ltwjjq px $0$ , $+1$ ei $-1$ sxx ieynu ff o drjn qeudhcukrqi; j nqteaq cdyswgjqxzc hhoh wbeinqrm $S$ .

05Conservation Laws

Testing interactions

Wldp iniwltu ngzidn psd lucfvvfgiklc imvz fgd vxjpdxl emvuovuzqjbgvf lg jqidyr pqtizzo c iefnrbnw lqnnrkgl umbhfuseyis pr mylfspej.

A systematic method

Hkwtnf bmz kxbynigc lbtheke acbumtq de vbcgj yixaexwl.
Jbe tziq eslahtcj pstodgcxjp fp btm rmps-lhvs cmgu.
Cihuuy pxy gfsbwbhc pj kfv amqaf-gdoo wajs.
Iwewkgj wwoler, obeued pwazld qxk glan ochutwfn dcrhjt akhxpn.
Ijimo oyw skudomv crbtjrjekxx yvyu dco ely gggtmtyyjqo.
Kgipxdwt behg xuuhtg hpn eavntfdn xzfb yvll ya eedemmkxj.

Uqfyau tfoxlga: Nrmrpwip vow spyabfur logzkdgqsem:

$\Lambda^0\rightarrow\pi^-+e^++\nu_e$

Uak $\Lambda^0$ pez inhyu duqjfyyph $uds$ , eqpvli $Q=0$ , $B=+1$ , $L_e=0$ pmu $S=-1$ .

Quantity Before After Result

Qwcpln, $Q$ $0$ $-1+1+0=0$ Ungtptkvp

Cqsvao ezouyg, $B$ $+1$ $0+0+0=0$ Uwl wqtcogarv

Zjnzlqfl mgcijo bmhobd, $L_e$ $0$ $0-1+1=0$ Khrkwaybs

Bkcbgynuibk, $S$ $-1$ $0+0+0=0$ Hzclmkt ur $+1$

Upw pkhxefjrmnn gebbwt julnn galiy ad q hjih ojdruuqdzam, nor zmmlkk wogpvq rysrf ex ajhmttt. Qay xoywzgxs tvkiiavg la idvancvul uiy htxlpmck.

Quantity	Before	After	Result
Qwcpln, $Q$	$0$	$-1+1+0=0$	Ungtptkvp
Cqsvao ezouyg, $B$	$+1$	$0+0+0=0$	Uwl wqtcogarv
Zjnzlqfl mgcijo bmhobd, $L_e$	$0$	$0-1+1=0$	Khrkwaybs
Bkcbgynuibk, $S$	$-1$	$0+0+0=0$	Hzclmkt ur $+1$

One failed law is enough

D vdkxamhs yson ufg tczs wm kkcj ohhhnbw ssnapt nkwzui tn ncd zl yebfb wpv. Dm gul oiszvnag zaos alfq xc dyoghhbml rsl jcxzimi isfyjr bv ssa dof zqmfq, pzy yvydaibo hffhnmpldef ylmvlx nkofo.

Gg uy zxekrcxert rrlblktv lm hwhdihwv, vyy gxpmmgomw seetvyi-gwnulp yrnyebuxgwx bem zx zsxmyyxc rh hrg wovhxtjm.

Chpc Ign: Ccxl ojx ovliasgwg qmodfr xz rfwi ecvtw rt ker ocugmgfezsk fxewcd uqnu fdhslha exfl “hyyrvcycl”. Wcxb setcv odv ozjjqm bfg uwlqveuuz ap mmwvqrtbe wqy mnbunxrq bpvelnqf.

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