# Textbook decay calculation

1. Feb 18, 2006

### arivero

I'd like to start a new thread listing the explicit calculations of decay rates or lifetimes appearing in standard textbooks (or even in articles). I mean, finished calculations, not just matrix elements. But we could include the ones requested as problem or exersice, I supposse.

For each formula, please give book, page number or section number.

2. Feb 19, 2006

### arivero

Peskin Schroeder

Scalar decay is problem 4.2.

Vector meson (quarkonium) to electrons appears in pg 151 and 152
$$\Gamma={16 \pi \alpha^2 \over 3} {|\Psi(0)|^2\over M^2}$$
(times 3 for color).

Pi0 to photons appears as formula 19.119
$$\Gamma={\alpha^2 \over 64 \pi^3 }{m_\pi^3\over f^2_\pi}$$

Charged pion decay is left as exersice 19.2(a)

3. Feb 19, 2006

### arivero

Huang Quarks, Leptons and Gauge Fields

pi0 decay is formula 12.102
$$\Gamma={\alpha^2 \over 64 \pi^3} ({m_\pi \over f_\pi})^2 m_\pi$$

Charged pion $$\pi^\pm$$ decay is formula 12.95
$$\Gamma={1\over 4 \pi} f_\pi^2 (G \cos \theta)^2 m_\pi m_\mu^2 (1-{m_\mu^2/m_\pi^2})^2$$

Last edited: Feb 19, 2006
4. Feb 19, 2006

### arivero

Aitchison

scalar rho to 2 pions is formula 6.27
$$\Gamma={\frac 1 {12}} {f^2_{\rho\pi\pi}\over 4\pi } {(M^2-4m^2)^\frac32 \over M^2}$$

Charged pion decay is ex. 14.3.c
$$\Gamma={G^2 f_\pi^2 \over 16 \pi} \mu^3 ({m_e\over \mu})^2 (1-{m_e^2\over\mu^2})^2$$

Last edited: Feb 19, 2006
5. Feb 19, 2006

### arivero

Data pdg+mcgregor

Perhaps it could be useful to add to the thread a couple of attachments, namely a .csv file of the electroweak particles (ie the particles having no predominant strong decay) and a McGregor plot (logarithm basis 137) of them. Part of the goal of this thread (the other being to get a sort of didactic guide) is to understand the slopes in this plot.

#### Attached Files:

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• ###### masas_ew.txt
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Last edited: Feb 19, 2006
6. Feb 19, 2006

### arivero

Pdg 2004

In review "Pseudoscalar decay constants",
Decay charged meson P -> lepton l is formula (2)
$$\Gamma={G^2_F |{V_{qq'}}^2| \over 8 \pi} f^2_P m^2_l m_P (1- {m_l^2\over m^2_P})^2 [1+O(\alpha)]$$

Minireview on the Z0 has Z0 --> f \bar f decay in formula (7)
$$\Gamma={G_F M^3_Z \over 6 \sqrt 2 \pi} N_c^f (|g_A^f|^2 R^f_A + |g_V^f|^2 R^f_V) + \Delta_{ew/QCD}$$

Review on quark model explains decay of quarkonium around formula 14.18

miniReview on muon decay parameters includes some formulae for differential $$d\Gamma$$

Review on electroweak model:
includes muon lifetime formula in 10.4
$$\Gamma={G_F^2 m_\mu^5 \over 192 \pi^3} F(...) (1+\frac 35 {m_\mu^2 \over M_W^2}) . [...]$$
where F and [...] are QFT corrections.

includes W and Z decays into pairs of (massless) fermion/antifermion as formula 10.47
$$\Gamma(W^+\to e^+ \nu_e)={G_F M^3_W \over 6 \sqrt 2 \pi}$$
$$\Gamma(W^+\to u_i \bar d_j)={C G_F M^3_W \over 6 \sqrt 2 \pi} |V_{ij}|^2$$
$$\Gamma(Z^+\to \Psi_i \bar \Psi_i)={C G_F M^3_Z \over 6 \sqrt 2 \pi} (g^i_V^2+g^i_A^2)$$

Also, near 10.65 "reduced widths" are defined dividing out the cube of the mass of Z or W.

Last edited: Feb 19, 2006
7. Feb 20, 2006

Staff Emeritus

I tried to open your attachment and got the message "Invalid Menu Handle". I've never seen that one before!

8. Feb 20, 2006

### arivero

er, rather strange. Well, here is the file, as extracted from the particle data group ovens.
Code (Text):

*MASS(MeV) ,Err+(MeV),Err-(MeV),WIDTH(MeV) ,Err+(MeV),Err-(MeV),I  ,G,J   ,P,C,A, PDG-MC,Chrg,R,S,Name             ,Quarks
80425      ,38.      ,38.      ,2124.      ,41.      ,41.      ,   , ,1   , , ,B,     24,  +1, ,R,W                ,
91187.6    ,2.1      ,2.1      ,2495.2     ,2.3      ,2.3      ,   , ,1   , , , ,     23,   0, ,R,Z                ,
105.658369 ,9.E-6    ,9.E-6    ,2.99591E-16,5.E-21   ,5.E-21   ,   , ,1/2 , , ,B,     13,  -1, ,R,mu               ,
1776.99    ,0.29     ,0.26     ,2.265E-9   ,9.E-12   ,9.E-12   ,   , ,1/2 , , ,B,     15,  -1, ,R,tau              ,
139.57018  ,.00035   ,.00035   ,2.5284E-14 ,5.E-18   ,5.E-18   ,1  ,-,0   ,-, ,B,    211,  +1, ,R,pi               ,uD
134.9766   ,.0006    ,.0006    ,7.8E-6     ,6.E-7    ,5.E-7    ,1  ,-,0   ,-,+, ,    111,   0, ,R,pi               ,(uU-dD)/sqrt(2)
547.75     ,.12      ,.12      ,0.00129    ,0.7E-4   ,0.7E-4   ,0  ,+,0   ,-,+, ,    221,   0, ,R,eta              ,x(uU+dD)+y(sS)
493.677    ,.016     ,.016     ,5.315E-14  ,1.E-16   ,1.E-16   ,1/2, ,0   ,-, ,B,    321,  +1, ,R,K                ,uS
497.648    ,0.022    ,0.022    ,7.367E-12  ,7.E-15   ,7.E-15   ,1/2, ,0   ,-, , ,    310,   0, ,R,K(S)             ,p(dS)+q(Ds)
497.648    ,0.022    ,0.022    ,1.272E-14  ,1.E-16   ,1.E-16   ,1/2, ,0   ,-, , ,    130,   0, ,R,K(L)             ,p(dS)-q(Ds)
1869.4     ,.5       ,.5       ,6.28E-10   ,8.E-12   ,8.E-12   ,1/2, ,0   ,-, ,B,    411,  +1, ,R,D                ,Dc
1864.6     ,.5       ,.5       ,1.599E-9   ,1.E-11   ,1.E-11   ,1/2, ,0   ,-, ,F,    421,   0, ,R,D                ,Uc
1968.3     ,.5       ,.5       ,1.342E-9   ,2.6E-11  ,2.6E-11  ,0  , ,0   ,-, ,B,    431,  +1, ,R,D(s)             ,cS
5279.0     ,.5       ,.5       ,3.93E-10   ,4.E-12   ,4.E-12   ,1/2, ,0   ,-, ,B,    521,  +1, ,R,B                ,uB
5279.4     ,.5       ,.5       ,4.27E-10   ,4.E-12   ,4.E-12   ,1/2, ,0   ,-, ,F,    511,   0, ,R,B                ,dB
5369.6     ,2.4      ,2.4      ,4.51E-10   ,1.8E-11  ,1.8E-11  ,0  , ,0   ,-, ,F,    531,   0, ,R,B(s)             ,sB
6400       ,400.     ,400.     ,1.4E-9     ,8.E-10   ,8.E-10   ,0  , ,0   ,-, ,B,    541,  +1, ,R,B(c)             ,cB
2979.6     ,1.2      ,1.2      ,17.3       ,2.7      ,2.5      ,0  ,+,0   ,-,+, ,    441,   0, ,R,eta(c)(1S)       ,cC
3096.916   ,.011     ,.011     ,0.091      ,0.0032   ,0.0032   ,0  ,-,1   ,-,-, ,    443,   0, ,R,J/psi(1S)        ,cC
3415.9     ,.34      ,.34      ,10.1       ,.8       ,.8       ,0  ,+,0   ,+,+, ,  10441,   0, ,R,chi(c0)(1P)      ,cC
3510.59    ,.10      ,.10      ,.91        ,.13      ,.13      ,0  ,+,1   ,+,+, ,  20443,   0, ,R,chi(c1)(1P)      ,cC
3556.26    ,.11      ,.11      ,2.11       ,.16      ,.16      ,0  ,+,2   ,+,+, ,    445,   0, ,R,chi(c2)(1P)      ,cC
3686.093   ,.034     ,.034     ,0.281      ,0.017    ,0.017    ,0  ,-,1   ,-,-, , 100443,   0, ,R,psi(2S)          ,cC
9460.30    ,.26      ,.26      ,0.053      ,0.0015   ,0.0015   ,0  ,-,1   ,-,-, ,    553,   0, ,R,Upsilon(1S)      ,bB
10023.26   ,.31      ,.31      ,0.043      ,0.006    ,0.006    ,0  ,-,1   ,-,-, , 100553,   0, ,R,Upsilon(2S)      ,bB
10355.2    ,.5       ,.5       ,0.0263     ,0.0034   ,0.0034   ,0  ,-,1   ,-,-, , 200553,   0, ,R,Upsilon(3S)      ,bB
10580      ,3.5      ,3.5      ,20.        ,4.       ,4.       ,0  ,-,1   ,-,-, , 300553,   0, ,R,Upsilon(4S)      ,bB
939.56536  ,8.E-5    ,8.E-5    ,7.432E-25  ,6.7E-28  ,6.7E-28  ,1/2, ,1/2 ,+, ,F,   2112,   0,4,R,n(P11)           ,udd
1115.683   ,.006     ,.006     ,2.501E-12  ,1.9E-14  ,1.9E-14  ,0  , ,1/2 ,+, ,F,   3122,   0,4,R,Lambda(P01)      ,uds
1189.37    ,.06      ,.06      ,8.209E-12  ,2.7E-14  ,2.7E-14  ,1  , ,1/2 ,+, ,F,   3222,  +1,4,R,Sigma(P11)       ,uus
1192.642   ,.024     ,.024     ,8.9E-3     ,9.E-4    ,9.E-4    ,1  , ,1/2 ,+, ,F,   3212,   0,4,R,Sigma(P11)       ,uds
1197.449   ,.030     ,.030     ,4.45E-12   ,3.2E-14  ,3.2E-14  ,1  , ,1/2 ,+, ,F,   3112,  -1,4,R,Sigma(P11)       ,dds
1314.82    ,.20      ,.20      ,2.27E-12   ,7.E-14   ,7.E-14   ,1/2, ,1/2 ,+, ,F,   3322,   0,4,R,Xi(P11)          ,uss
1321.31    ,.13      ,.13      ,4.02E-12   ,4.E-14   ,4.E-14   ,1/2, ,1/2 ,+, ,F,   3312,  -1,4,R,Xi(P11)          ,dss
1672.45    ,.29      ,.29      ,8.02E-12   ,1.1E-13  ,1.1E-13  ,0  , ,3/2 ,+, ,F,   3334,  -1,4,R,Omega            ,sss
2284.9     ,.6       ,.6       ,3.3E-9     ,9.E-11   ,9.E-11   ,0  , ,1/2 ,+, ,F,   4122,  +1,4,R,Lambda(c)        ,udc
2466.3     ,1.4      ,1.4      ,1.49E-9    ,0.9E-10  ,0.9E-10  ,1/2, ,1/2 ,+, ,F,   4232,  +1,3,R,Xi(c)            ,usc
2471.8     ,1.4      ,1.4      ,6.7E-9     ,1.3E-9   ,1.3E-9   ,1/2, ,1/2 ,+, ,F,   4132,   0,3,R,Xi(c)            ,dsc
2697.5     ,2.6      ,2.6      ,1.02E-8    ,4.8E-9   ,4.8E-9   ,0  , ,1/2 ,+, ,F,   4332,   0,3,R,Omega(c)         ,ssc
5624       ,9.       ,9.       ,5.36E-10   ,3.7E-11  ,3.7E-11  ,0  , ,1/2 ,+, ,F,   5122,   0,3,R,Lambda(b)        ,udb

The file is a .csv format, an old format for spreadsheets that lets you to import it into excel, gnumeric or similar packages. The * in the first line is a trick to use it under fortran or gnuplot. For gnuplot you need to input the orders
Code (Text):

set datafile separator ","
and then you can procedd with the usual plot "data.txt" using...

Last edited: Feb 20, 2006
9. Feb 21, 2006

### arivero

Ellis, Stirling & Webber QCD and Collider Physics

This is an advanced non-self contained book, it sometimes quotes formulae without a complete proof or sketch of it.

In section 10.1.1, pg 334, it reviews the spectator model for semilectonic decays, where the decay rate of a massive quark Q inside a hadron will simply follow the rescaled muon decay rate, giving formula 10.6 as
$$\Gamma= {G^2_F m_Q^5 \over 192 \pi^3}$$

Im page 341, formula 10.23 gives the decay of top quark,
$$\Gamma_{t\to bW}= {G_F m_t^3 \over 8 \pi \sqrt 2} |V_{tb}|^2 I(m_W/m_t,m_b/m_t)$$
where I(x,y) is a calculated correction polinomial.

Last edited: Feb 21, 2006
10. Feb 21, 2006

### arivero

what is going on

the plot
https://www.physicsforums.com/attachment.php?attachmentid=6342&d=1140386375
shows two different groups, the ones of electromagnetic (say) decay, and the ones of beta decay. The group having beta decay adjusts to an slope n=5, because the Fermi coupling carries dimensions on mass square and forces a scaling behavior with the fifth power of mass.

The group of electromagnetic decay does not has such dimensional constant, and then it adjusts to n=3, a scaling with the third power of mass.

Now beyond this naive dimensional argument, I can not see how to explain other peculiar characteristics of the plot. Besides McGregor "quantisation" of the decay rates, one wonders why the neutron does scape from the group of beta decaying particles, gaining a stability a lot higher.

And, a lot more strange, how is that the fit for the electromagnetic decaying sector is so accurate?. The line scales exactly from the pion0 to the Z0, with a lot less dispersion that the fermi decaying group. Amazing, and it includes particles with different spin and different decay mechanisms.

Also, the Z0 is not a electromagnetic decay after all, except if you aniquilate the pair of quarks. And no such thing is possible for W, which just sits there in virtue of symmetry with Z0. We have thus an enigmatic coincidence
$${G_F \over 6 \sqrt 2 \pi} \sum_f C_f (g^i_V^2+g^i_A^2) \approx {\alpha^2 \over 64 \pi^3 }{1 \over f^2_\pi}$$
linking strong and electroweak quantities. We can use this coincidence to extract the value of a decay constant f and then to apply it to the charged members of the isospin multiplet, e.g. to a charged pion:
$$\Gamma\approx {3 \alpha^2 G_F (\cos \theta)^2 \over 64 \sqrt 2 \pi^3 \sum_f C_f (g^i_V^2+g^i_A^2)} m_\pi m_\mu^2 (1-{m_\mu^2/m_\pi^2})^2$$
Does it imply a secondary n=3 scaling for the beta decaying mesons? Can not tell.

Last edited: Feb 21, 2006
11. Feb 21, 2006

### arivero

plotting

Hmm better let me to put a fast introduction to gnuplot. You invoke it from the line interface, just writting "gnuplot", and then you use
Code (Text):

set datafile separator ","
set log x
set log y
plot "masas.txt" using ($1):($4)

and there you are!

You can enjoy some extra reference lines. Let me to scale back from the Z0 using both cubic and quintic powers

replot 2495.2*(x/91187.6)**3
replot 2495.2*(x/91187.6)**5

But perhaps it could be better to replot the quintic power from the muon...
and move the labels to the lower right corner.
Code (Text):

set key bottom right
plot "masas.txt" using ($1):($4)
replot 2495.2*(x/91187.6)**3
replot 2.99591E-16 * (x/105.658369)**5
You can notice that the muon scaling passes about a 20% of the tau decay rate. This is the right thing to happen, because we are just scaling the decay to electron and the tau has some extra modes available.

Last edited: Feb 21, 2006
12. Feb 24, 2006

### arivero

More from: Ellis, Stirling & Webber QCD and Collider Physics

In page 3, formula 1.3 shows the decay of pi0 to 2 gamma, with a variation,
$$\Gamma=\xi^2 (\frac \alpha \pi)^2 \frac 1 {64 \pi} {m^3_\pi \over f_\pi^2}$$
because it uses 1.4 and 1.4 to discuss how $$\xi=1$$ can be got from two different solutions, either three colours of quarks or just a pair of nucleons (btw this is related also to Nambu exotic solution, isn't it?).

In page 290 formulae 8.91 and 8.92 give the decays of Z0 and W to fermions.
$$\Gamma=C {G_F M_Z^3 \over 6 \sqrt 2 \pi}(|V_f|^2+|A_f|^2)$$
$$\Gamma=C {G_F M_W^3 \over 6 \sqrt 2 \pi}$$
The one of Z0 must be summed for all the SM fermions; the one of W is already summed to get rid of the CKM factor $$|V_{ij}|^2$$. Also, in next page, table 8.2 gives the relative couplings. I have sometimes (hep-ph/0511165) wondered about how near to unity the ln of the sum of the couplings is.

Last, in page 380 some decay rates of P-wave quarkonion to hadrons are halfcalculated, formula 10.119.

Last edited: Feb 24, 2006
13. Feb 26, 2006

### arivero

Griffiths Introduction to Elementary Particles

This book does not calculate the decay Pi0->2gamma. This is a bold decision, but appropiate to a claimed "elementary" book, where neither the anomaly nor the peculiar properties of SU(3) colour are to be included.

Instead, section 10.3 calculates the lifetime of neutron. Formula 10.63, in pg 311, gives the purely electroweak calculation
$$\Gamma={1 \over 4 \pi^3} ({g_w\over 2 M_W})^4 m_e^5 [function({m_n-m_p\over m_e})]$$
then PCAC is introduced and applied to calculate a correction to this decay width (giving a total \tau=914 sec), and then even a further correction due to Cabibbo angle is incorporated (back to 963 seconds then).

The decay of charged pion is calculated and shown in formula 10.80
A consequence of the SU(3)-agnosticism is that f_\pi is presented as a purely dimensional need, and unrelated to the decay of neutral pion (other books claim isospin to connect both). Instead, a footnote in page 316 suggests the Ansatzes f_pi=m_pi and/or $$f_\pi=m_\pi \cos \theta_C$$

A decay of charged Kaon appears in example 10.2, thus introducing f_K

The decay of muon is calculated and shown in formula 10.37, or better in formulae 10.35 to 10.39. Here another peculiarity of the book appears: Fermi constant is defined in 10.38, but not used anywere, preferring to use 1/M_W^2 all the way along the lectures. It induces some confusion in 10.37 when some m_\mu is put in the same parenthesis just because of notation, so perhaps 10.36a should be the right formula to work with.

The decay of Z0 is proposed (and answer given) as problem 10.20 (pg 341)

The book also has a chapter on positronium/quarkonium decay.

Last edited: Feb 26, 2006
14. Mar 1, 2006

### arivero

Putting Numbers in, II

Meanwhile, let me to calculate, from the above table and plot, the reduced decay widths $$\beta=\Gamma/M^3$$for the electromagnetic rank of particles.

1. Neutral Pion
$$\beta=7.8E-6(\pm6.E-7)/134.9766^3 = (562 [\pm22?] GeV)^{-2}$$
2. Eta
• total
0.00129 0.7E-4 547.75 -> (357 GeV)^-2
• gamma gamma
The \eta particle enjoys strong decays; we can do better if we check the tables (pdg) to consider only the gamma gamma decay, a fraction 39.42% of the total. Thus
.0005085 547.75 -> (568 \pm?15 GeV)^-2
3. Sigma
Sigma is a baryon, thus a fermion. Its decay is to gamma plus Lambda, and amusingly the width still enters in the order of magnitude
8.9E-3 \pm9.E-4 1192.642 -> (437 \pm?22 GeV)^-2
4. Charmonium
It seems natural to take the J/Psi, even if having spin 1. And it does the work.
0.091 \pm0.0032 3096.916 -> (571 \pm?10 GeV)^-2
5. Bottomonium
As it can be seen in the graph, the Upsilon does not decay in the same line that the others.
6. Z0
Most fascinating, the electroweak scale and three generations tunes Z0 decay into the same reduced width

2495.2 \pm2.3 91187.6 -> (551 \pm?1 GeV)^-2
while W+ (which is a charged particle) is bit off average
2124. \pm41. 80425 -> (495 \pm?5 GeV)^-2
Note that we need some pi and alphas to go from the reduced decay to the usual $$f_\pi$$, and some less to get an effective coupling $$g_{X\gamma\gamma}$$; some papers have noted the equality of this last coupling when evaluated for eta and pi, but I havent seen any work beyond it, nor a check of the other decaying particles as here we are doing.

To resume: when looking only to experimental data, we notice that the neutral EM decaying particles pi, eta, sigma0, JPsi have a common scale coinciding with the Z0 electroweak scale (aka Fermi scale), with respective numbers

562\pm22, 568 \pm15, 437 \pm22, 571 \pm10, 551 \pm 1

while the Upsilon family (third generation) does not seem to fit. The Sigma0 current pdg rate gives this 487 GeV; to enter in line with the rest the decay rate should be around 5.6 KeV, a 63% of the currently measured value (thus an even higher lifetime). But it is a fermion, while all the others in the line are bosons, so perhaps it is not reasonable to expect the same value. On other hand, it is a value measured via Primakoff effect, not exactly as the rest.

Last edited: Mar 2, 2006
15. Mar 1, 2006

### arivero

16. Mar 2, 2006

### arivero

On other hand, if one does the gamma gamma trick for eta' -as we did with eta- it is got a value about (452 GeV)^-2. It could be we are overcounting as EM decays some intermediate ones (but we are not counting eta' to w+\gamma!), or it could be some other effect when a mixing with s quark is available.

17. Mar 8, 2006

### arivero

Wolfram

This is mostly a curiosity, but it is online. Stephen Wolfram left HEP a bit after his PhD at Caltech, but as a consequence of it he took in charge to look for, ahem, weak decays. I can not find the thesis around, but in his website there is a paper

Weak Decays (1981)
S. Wolfram, Nukleonika 26 (1981) 273-309.
http://www.stephenwolfram.com/publications/articles/particle/81-weak/

that gives formulae for a good bunch of electroweakly decaying particles.

It is a pity that Wolfram's staff does not provide a printer friendly version of the document, but you can always browse and print.

18. Apr 7, 2011

### Prologue

Thanks for this thread, it has helped me out more than a few times.