• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Is It necessary, in the standard model, that the number of quark generations equals the number of lepton generations, i.e. 3?

+ 6 like - 0 dislike

This question showed up in my particle physics exam and I'm not sure of the answer. I said no, since I can't see any reason why they must be equal. But I my answer is right, I find it odd that it's just a coincidence that they're both 3.

This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user Pedro Pereira
asked Oct 26, 2016 in Phenomenology by Pedro Pereira (30 points) [ no revision ]

4 Answers

+ 4 like - 0 dislike

This question showed up in my particle physics exam and I'm not sure of the answer. I said no, since I can't see any reason why they must be equal.

There are no conserved quantities between the lepton and quark generations (for example there is no rule such as a top quark can only decay into a tau lepton or something), so there is no actual reason why there need to be three generations of quarks and three generations of leptons within the current standard model.

But I my answer is right, I find it odd that it's just a coincidence that they're both 3.

It probably is not a coincidence, but a consequence of some underlying undiscovered symmetry. However, there is no current human physics law which requires the same number of quark and lepton generations.

This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user Suzu Hirose
answered Oct 27, 2016 by Suzu Hirose (40 points) [ no revision ]
+ 3 like - 0 dislike

We don't know why they balance in number, as humans we like symmetry and balance, but whether nature likes them as much as us, we don't know. It might be a coincidence ( it's probably not), but again we could discover something tomorrow that "distorts" the system.

It seems necessary that there are 3 of each for our current standard model of understanding forces to work, but that could change at any time, or not for a long time.

enter image description here

A more detailed, interesting and much better informed similar question is Quarks and Leptons

This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user CountTo10
answered Oct 26, 2016 by CountTo10 (30 points) [ no revision ]
+ 3 like - 0 dislike

Honest answer for your question is ''No''! As in the standard model (SM), the number of fermion generations appears as an arbitrary parameter, meaning that a mathematically consistent theory can be built up using any number of fermion generations. Therefore, In order to answer the question perhaps we may need to beyond the standard model.

In the quest for such a model(s), first lets rephrase your question is the following way:

Is there any extension of the standard model be possible where the number of fermion generations can in any way be explained through the internal consistency of the model? (since we don't have the answer within the framework of SM.)

Its good to start from SM itself. Lets go back in sixties. If we make a table of fermions, those were discovered up to 1965, it would looks like:

\begin{eqnarray} \text{Lepton}& : & \begin{pmatrix} \nu_{e}\\ e \end{pmatrix},\quad \begin{pmatrix} \nu_{\mu}\\ \mu\end{pmatrix} \\ \text{Quark} & : & \begin{pmatrix} u \\ d \end{pmatrix},\qquad s \end{eqnarray} Anyone with naked eye can say how ''ugly'' this table is looks like! In fact it was James Bjorken and Sheldon Glashow proposed the existance of chram ($c$) quark in order to restore the ''quark-lepton symmetry''. The table now looks more symmetric and beautiful:

\begin{eqnarray} \text{Lepton}& : & \begin{pmatrix} \nu_{e}\\ e \end{pmatrix},\quad \begin{pmatrix} \nu_{\mu}\\ \mu\end{pmatrix} \\ \text{Quark} & : & \begin{pmatrix} u \\ d \end{pmatrix},\qquad \begin{pmatrix} c \\ s \end{pmatrix} \end{eqnarray} Which was later discovered during November Revolution 1974 . Lesson is, these two physicists were dictated by the notion of symmetry in order to restore the order in the realm of fermions. Later GIM mechanism was given an explanation of the non-existent of FCNC in SM taking into account the charm quark.

The very existence of three generations of quarks is necessary for CP violation. And also for anomaly cancellations to make the SM mathematically consistant. But the undeylying symmetry (if it really exists) which may ensure the equal numbers of quarks and leptons, yet to be discovered.

Story that goes beyond SM: Back in nineties an extension of SM was proposed by F. Pisano here and V. Pleitez based on gauge group $SU(3)_{L}\times U(1)_{Y}$. Their model to accommodate the standard fermions into multiplets of this gauge group which must include some new fermions.

This model has remarkable feature. As we already know that, a consistent quantum field theory must be free from gauge anomaly. Without that, theory become ill. In case of SM the anomalies get cancelled in miraculous (or should we say in an ugly way?) way. But for the model with gauge group $SU(3)_{c}\times SU(3)_{L}\times U(1)_{Y}$, has the interesting feature that each generation of fermions is anomalous, but that with three generations the anomalies cancelled.

In other words,

Electroweak interactions based on a gauge group $SU(3)_{L}\times U(1)_{Y}$, coupled to to the QCD gauge group $SU(3)_{c}$ can predict the number of generations to be multiples of three.

(For technical detail one can read this paper). But with the cost that we have to incorporate a right handed neutrino in the game. In fact, one may find other different models with the same features.

GUT considerations: In a recent paper Pritibhajan Byakti et al proposed a grand unified theory based on the gauge group $SU(9)$. The model uses fermions in antisymmetric representations only and the consistency of the model demands that the number of fermion generations is three. Nevertheless like all GUT, it also come up with some superheavy gauge bosons. Which can trigger baryon number non-conserving processes.

The upshot is, perhaps we would be able to explain the lepton-quark symmetry with the price of some new physics (may be new particles) which lives beyond the SM.

This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user AMS
answered Oct 28, 2016 by AMS (30 points) [ no revision ]
+ 1 like - 0 dislike

Please do correct me if I've learned this wrong, but isn't the equality of the number of quark and lepton generations a necessity for anomaly cancellation in the EWSM? Furthermore the unitarity of the CKM matrix relies strictly on the existence of three fermion generations. The failure of this mixing matrix to be unitary for only two generations is why a third generation was hypothesized in the first place. A similar lack of unitarity exists in the case of four generations or higher. In fact, I believe one of the tests of determining whether more generations exist is performing precision tests on the CKM matrix mixing factors. See section 20.2 of Peskin and Schroeder, and 20.3.3 of Schwartz.

This post imported from StackExchange Physics at 2016-11-10 22:33 (UTC), posted by SE-user Optimus Prime
answered Oct 28, 2016 by Optimus Prime (105 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights