This page shows a part of a correspondence between a Standard Model (SM) supporter (called here NNN) and myself (Eli Comay). The emails are numbered consecutively and are separated by horizontal lines. At the beginning, the SM supporter was very sure that he is right. Thus, the first sentence of his first letter is: "You misunderstood the paper in ref. [3]". However, it turns out that eventually he had to admit that he still has no answer to my claims. The primary meaning of the following text is that the existence of a Lagrangian density for a quantum field theory of a given particle is NOT a sufficient condition for a self-consistent theory. In particular, there is an inherent contradiction in the electroweak sector of the SM where the electromagnetic interaction of the charged W boson violates Maxwellian electrodynamics. SM supporters are kindly invited to step up to the challenge. ------------------------------------------------------------------- 1. Dear NNN, I'll be very happy to read your remarks on the quite short text linked below. Cheers, Eli http://www.tau.ac.il/~elicomay/Higgs_05.html ------------------------------------------------------------------- 2. Dear Eli, You misunderstood the paper in ref. [3], which might have been not clear enough. The coupling of the W to the photon is part of the Standard Model (SM) Lagrangian. However, in order to test the SM theory, people suggested modifications to that Lagrangian by writing an effective Lagrangian, such as the one in Eq. 2.1 in [3]. If you set the coefficients of the first and the second terms, g_1 and kappa, to be equal to 1, and the coefficients of the other terms to be zero, you will get the SM Lagrangian. So possible differences between g_1 and 1 and between kappa and 1, are called anomalous couplings. All other terms in 2.1 are not expected in the SM, so their couplings are also considered to be anomalous. In LEP people looked for any possible contributions for those anomalous couplings to the production of W-pairs from electron-positron annihilations via a virtual photon or Z-boson in the s-channel. They found only upper limits for those couplings, so no evidence for anomalous couplings, and the SM Lagrangian seems to be correct. Similar measurements were done at the Tevatron proton-antiproton collider, and are being done now at LHC proton-proton collider, using WW and W-gamma final states obtained by quark-antiquark interaction. Again, no evidence was found, so far, for any existence of anomalous couplings. So in conclusion, one can say that the interaction of the photon with the W is well understood theoretically, and therefore, one can trust the SM predictions for the partial decay width of the Higgs to photon-photon. Also the measurements of that width, so far agree with the prediction. Best regards, NNN ------------------------------------------------------------------- 3. Dear NNN, I understand that you think that you understand the SM form of the interaction of the charged W+- bosons with the electromagnetic field. Can you please show me a textbook that displays explicitly the SM Lagrangian density which contains a term representing the coupling of the W's 4-current with the electromagnetic 4-potential? Cheers, Eli P.S. I can show you a quite old textbook that shows the coupling of a Dirac particle with the electromagnetic field [1]. [1] J. D. Bjorken and S. D. Drell {\em Relativistic Quantum Fields} (McGraw-Hill, New York, 1965). (See p. 84). ------------------------------------------------------------------- 4. Dear Eli, You can take a look at the book of W.N. Cottingham and D.A. Greenwood: "An introduction to the Standard Model of Particle Physics", which you can find on the web (en.bookfi.org). The gauge boson Lagrangian is given in Eq. 11.13 (page 110). The second term is a product of the W_munu tensor, which is also a vector under weak isospin. The expressions for this tensors are given in Eqs. 11.15, in terms of the three (weak isospin) components of the W four vectors, which are the equivalents of the A four vector in QED. These isospin components are related to the W+ and W- (eq. 11.7) and to the photon and Z (eq. 11.29). Note that because the W are the generators of the SU(2) part of the SM which is SU(2) X U(1), and because SU(2) is non-Abelian, one needs the last terms in eqs. 11.15 (the one multiplied by g_2) in order to maintain gauge invariance. When you calculate the product in eq. 11.13, you obtain terms of three kinds: 1) Terms without the coupling g_2 - these are the kinetic energy terms, like the ones we have for the photon (E**2-B**2). You can see them in Eq. 11.31 2) Terms linear in g_2 - these describe the 3-boson vertex interaction - the one you are asking about. They are part of 11.32, and you can see that they are identical to the first two terms in the paper of Hagiwara et al., if you set there g_1 and kappa to 1, as I wrote in my last message. 3) Terms quadratic in g_2, they describe 4-boson vertex interaction, they are in 11.32 as well. Best regards, NNN ------------------------------------------------------------------- 5. Dear NNN, It is very well known that the term of the Lagrangian density that represents the interaction of a charged particle with the electromagnetic 4-potential must satisfy the following requirements: 1. It must be proportional to the electric charge e. 2. It must be linear in the electromagnetic 4-potential A_mu. 3. The 4-current of the charged particle must satisfy the continuity equation j^mu_,mu = 0. 4. The dimension of the 4-current must be [L^-3]. Can you please show me where the Cottingham & Greenwood book proves these requirements and where the electromagnetic interaction term of the W Lagrangian density is written explicitly? Cheers, Eli P.S. I can show you a quite old textbook that shows that a Dirac particle satisfies these requirements [1]. [1] J. D. Bjorken and S. D. Drell {\em Relativistic Quantum Fields} (McGraw-Hill, New York, 1965). ------------------------------------------------------------------- 6. Dear Eli, In principle, the interaction of the W with the photon does not have to resemble exactly the interaction of the fermions with the photon. In particular, it does not have to look as a product of current times A. The interaction Lagrangian includes also terms with the derivative of A, which you do not have in the interaction with fermions. This is required, because there are terms with derivatives of W and the idea is to have some sort of similarity between the A and the W (and the Z) - they are all gauge bosons. In this case, I do not know whether the term "current" has any meaning here. From Noether theorem, I guess that there is a conserved current here, but it is not calculated explicitely in this text. However, there are other similarities to fermions. g_2 is proportional to e (Eq. 11.38). The 3-field vertex terms, those that are proportional to g_2, are also proportional either to the derivative of A (Eq. 11.32, third line) or A (last but one term of Eq. 11.31, through the covariant derivatives). The dimension of all terms in the Lagrangian are L^-4, as it should be, and you can check that. So the term proportional to A has necessarily dimension L^-3, but the term proportional to the derivative of A has a dimension of L^-2. Best regards, NNN ------------------------------------------------------------------- 7. Dear NNN, I understand that we agree that the electroweak W does not interact with electromagnetic fields like the electron (see [1]) and like a classical charge (see [2]) do. Therefore the electroweak electric charge of the W is not the same physical entity as that of the electron. Now, one of the decay modes of the physical W is electron + antineutrino (see http://pdg.lbl.gov/2015/listings/rpp2015-list-w-boson.pdf). Due to electric charge conservation, one concludes that the physical W carries an ordinary electric charge. Therefore, the ~80 GeV particle is NOT the electroweak W. BTW. The Neother 4-current of the electroweak W does not work here. A detective-like argument: if this 4-current is OK then it would have been published in textbooks a very long time ago. (A disagreement to this point means a grave underestimate of the intellectual ability of too many people working in the electroweak field.) Cheers, Eli [1] J. D. Bjorken and S. D. Drell {\em Relativistic Quantum Fields} (McGraw-Hill, New York, 1965). [2] L. D. Landau and E. M. Lifshitz, {\em The Classical Theory of Fields} (Elsevier, Amsterdam, 2005). ------------------------------------------------------------------- 8. Dear Eli Since the W interacts with the photon, it has an electric charge, even though the interaction term is not the same as the one of fermions. I read that the fact that the parameter g_1 (in the paper of Hagiwara et al. which you quoted) is equal to 1 in the SM is related to the fact that the W has charge 1 (in units of e), but I do not know the proof. I shall try to look into it. Not everything is written in textbooks. Best regards, NNN ------------------------------------------------------------------- 9. Dear NNN, I think that it is a good time for drawing tentative conclusions from our recent correspondence. The main issue is the electric charge of the W boson. Assuming that it is a genuine electric charge, I've stated that the electromagnetic interaction term of the W's Lagrangian density must satisfy the following Maxwellian requirements: 1. It must be proportional to the electric charge e. 2. It must be linear in the electromagnetic 4-potential A_mu. 3. The 4-current of the charged particle must satisfy the continuity equation j^mu_,mu = 0. 4. The dimension of the 4-current must be [L^-3]. As of today, you have not shown me a textbook where this term is written explicitly. My conclusion is that such a term does not exist. Therefore, there is a fundamental contradiction in the Standard Model because it uses a kind of an electric charge whose equations do not take the Maxwellian form. The Higgs problem that is shown on my internet page http://www.tau.ac.il/~elicomay/Higgs_05.html is just a specific issue that stems from this substantial matter. You ended your last email with the statement: "Not everything is written in textbooks". Here I kindly wish to draw your attention to the fact that in the textbook that you have shown me, W.N. Cottingham and D.A. Greenwood mention explicitly the correct coupling term of a classical charge with the electromagnetic 4-potential. They also do the same for a Dirac particle. Can you please tell me why these authors have failed to show these properties in the case of the electroweak charge carrying particle W? BTW. 1. People apparently do not care about this serious problem. For example, it is completely ignored by those who have written the Wiki list of unsolved problems in physics (see the following link). https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics 2. The root of the problem is the second order of quantum equations. These equations contain contradictions and should be abandoned. P. A. M. Dirac has said it time and again throughout his life, but apparently nobody cares. I agree with him on this issue and published several articles that show new kinds of contradictions of these equations. See for example my paper http://www.tau.ac.il/~elicomay/MathPhys.pdf I can show you some other papers that discuss this matter. Cheers, Eli ------------------------------------------------------------------- About a week later NNN said that he still examines the problem. After some time he concluded this issue and stated: "I also agree that the W-current issue is not mentioned in standard textbooks; at least, I did not find it."