Mandl & Shaw QFT Second Edition: Sign Discrepancy on Page 361

[Vic Sandler]

In the second edition of Mandl and Shaw QFT, on page 361, it's fairly obvious that there is a sign discrepancy between eqn (15.125) and eqn (15.127). In eqn (15.125), there are 5 imaginary i's and 4 explicit minus signs for a final product of +i. However, the first equal sign in eqn (15.129) gives eqn (15.127) an explicit minus sign, i.e. a final product of -i. Eqn (15.125) is created from fig (15.17b) and seems to be correct, but I am not completely sure. Can someone confirm that eqn (15.125) is an accurate application of the Feynman rules to Fig (15.17b)?

 

[andrien]

There is a little mistake in 15.125,the structure constant f_inm should read f_imn.Rest just follow(verify it)

 

[Vic Sandler]

I don't understand your suggestion. In figure 15.17(b) on page 351, I think there is a typo. The leftmost gluon is labeled p - k, n, \nu but in eqn (15.125), there is a factor T^m_{ad}\gamma_{\nu}. This suggests to me that the label on the leftmost gluon should be p - k, m, \nu. Am I correct? If not, where does the m in eqn (15.125) come from?

In figure 14.4(b) on page 315, the subscripts on f are in the same order as the superscripts on V. That is, lmn follows the figure in clockwise order, and so does \sigma\tau\nu. If I make the change from n to m in figure 15.17(b) as I suggested above, and change f_{inm} to f_{imn} as you suggested in your post, then the indices on f will follow the figure in counterclockwise order while the indices on V follow it in clockwise order. I don't think this can be.

Can you explain why you think the order of inm is wrong by means of the application of the Feynman rules to figure 15.17(b)?

PS: The application of rule 4 on page 319 suggests to me that the loop factors should be in clockwise order, but I'm not sure about it.

 

[andrien]

I don't understand your suggestion. In figure 15.17(b) on page 351, I think there is a typo. The leftmost gluon is labeled p - k, n, \nu but in eqn (15.125), there is a factor T^m_{ad}\gamma_{\nu}. This suggests to me that the label on the leftmost gluon should be p - k, m, \nu. Am I correct?

yes,you are correct on this.

In figure 14.4(b) on page 315, the subscripts on f are in the same order as the superscripts on V. That is, lmn follows the figure in clockwise order, and so does στν. If I make the change from n to m in figure 15.17(b) as I suggested above, and change f_inm to f_imn as you suggested in your post, then the indices on f will follow the figure in counterclockwise order while the indices on V follow it in clockwise order.

Take a look carefully at 14.68 and you will see it.Also when he writes 15.129 ,he says that he is using 15.66.Why do you think it to be the case.

 

[Vic Sandler]

Note that there appears to be a typo in eqn (14.67) on page 317. The final factor of D_F^{\tau\gamma} should be D_F^{\nu\gamma}

Take a look carefully at 14.68 and you will see it.Also when he writes 15.129 ,he says that he is using 15.66.Why do you think it to be the case.

You make a good point. However, in the case of eqn (14.68) on page 317, comparing it to fig 14.5 on the same page, you will see that the subscripts on f, namely lmn are in counterclockwise order. The subscripts on V namely \sigma\tau\nu which are tied in eqn (14.67) to \alpha\beta\gamma are also in counterclockwise order. So in both figure 14.4(b) on page 315, and in figure 14.5 on page 317 we find that the subscripts on f should follow the same order as the subscripts on V. In figure 14.4(b), the order is clockwise, and in figure 14.5 it is counterclockwise, but my point is that in each case, the order of f matches the order of V. Now I have two questions.

1. If the order of subscripts of f in eqn (15.125) should be changed from clockwise to counterclockwise, then shouldn't the order of subscripts of V also be similarly changed?

2. In any given Feynman diagram, how do you determine whether the order should be clockwise or counterclockwise?

 

[andrien]

You make a good point. However, in the case of eqn (14.68) on page 317, comparing it to fig 14.5 on the same page, you will see that the subscripts on f, namely lmn are in counterclockwise order. The subscripts on V namely στν which are tied in eqn (14.67) to αβγ are also in counterclockwise order. So in both figure 14.4(b) on page 315, and in figure 14.5 on page 317 we find that the subscripts on f should follow the same order as the subscripts on V. In figure 14.4(b), the order is clockwise, and in figure 14.5 it is counterclockwise, but my point is that in each case, the order of f matches the order of V.

why do you care about this anticlockwise and clockwise.The order in which three gauge boson vetrex is written does not matter.You don't even need to use such unnecessary thing.If in the diagram all arrows are pointing inward(K₁+K₂+K₃=0) then take the factor f_lmn or any such and go in the sense you counted it(no need to write V_στν type thing).If l is with μ(k1),m with v(k2),n with σ(k3) then you can write simply gf_lmn(g_μv(k₁-k₂)_σ+g_vσ(k₂-k₃)_μ+g_σμ(k₃-k₁)_v.You can see clearly the way it is written.It is correct.Follow this and you will never do wrong.(Beware M&S has defined covariant derivative a little different,so be cautious if you read other books such as Peskin and Schroeder).The problem you are trying to handle is treated in chapter 16 of P&S.

 

[Vic Sandler]

why do you care about this anticlockwise and clockwise.

In one sense, I don't care about it. Since both f and V are totally anti-symmetric in their indices, it doesn't matter whether those indices are arranged clockwise or counterclockwise with respect to the Feynman diagram. The Feynman rule for Fig 14.4(b) has both clockwise and the rule in eqn (14.68) has both counterclockwise and I do not consider this to be a problem.

However, there is a sense in which I do care about it. Namely if you use the counterclockwise orientation for f and clockwise for V. That is what you are suggesting in order to fix eqn (15.125). Of course this fixes the minus sign problem, but why is it alright to do it? When must the the sets of indices both be oriented in the same order and when are they allowed to be oriented in opposite order?

If in the diagram all arrows are pointing inward ...

For the 3-gluon vertex in figure 15.17b, not all the arrows are pointing inward. Is this an issue?

 

[andrien]

However, there is a sense in which I do care about it. Namely if you use the counterclockwise orientation for f and clockwise for V. That is what you are suggesting in order to fix eqn (15.125). Of course this fixes the minus sign problem, but why is it alright to do it? When must the the sets of indices both be oriented in the same order and when are they allowed to be oriented in opposite order?

I will say again that use the vertex factor as I have said earlier(no use of V_σμv type notation),you are getting too much confused now.

For the 3-gluon vertex in figure 15.17b, not all the arrows are pointing inward. Is this an issue?

you should convert them into all pointing inward.

 

[Vic Sandler]

OK, forget about V.

If I applied your method to the Feynman diagram in fig 14.4(b), I would get f_{lnm}. But the author gets f_{lmn}. In fig 14.4(b) he has the indices in the order lmn which is clockwise in the diagram. In figure 14.5, he has the indices in the order lmn which is counterclockwise in the diagram. If, in applying the Feynman rules to a particular diagram, I use the the wrong orientation I will introduce a wrong minus sign because f is totally antisymmetric. I don't know when to apply the fig 14.4b clockwise orientation and when to apply the figure 14.5 counterclockwise orientation.

 

[andrien]

If l is with μ(k1),m with v(k2),n with σ(k3) then you can write simply gf_lmn(g_μv(k₁-k₂)_σ+g_vσ(k₂-k₃)_μ+g_σμ(k₃-k₁)_v.You can see clearly the way it is written.It is correct.Follow this and you will never do wrong.

See this.if you write f_lnm then k₁-k₂ will appear as k₂-k₁,k₂-k₃ as k₃-k₂ and so.So an extra minus sign will appear which combined with f_lnm give f_lmn and you get the original formula.It does not matter how you go.(These things you should figure out yourself,take your time or consult other books such as those of greiner's which will provide you step by step methods)

 

[Vic Sandler]

if you write f_lnm then k₁-k₂ will appear as k₂-k₁,k₂-k₃ as k₃-k₂ and so.So an extra minus sign will appear which combined with f_lnm give f_lmn and you get the original formula.

This is not correct. You will simply reverse two of the k's, The third will remain unchanged. In other words for instance, swapping k_2 with k_3,

(k_1 - k_2)_{\sigma} \rightarrow (k_1 - k_3)_{\sigma}
(k_2 - k_3)_{\tau} \rightarrow (k_3 - k_2)_{\tau}
(k_3 - k_1)_{\nu} \rightarrow (k_2 - k_1)_{\nu}

So you won't get the minus sign you need unless you also swap the indices on V.

Here is what I think about the 3 gluon Feynman diagram and rules.

In the 3 gluon diagram, since all arrows point inward, there are no asymmetries related to the direction of momentum. For this reason, if you turn the diagram 120 degrees, the equation you get from applying the Feynman rules should not change it's value, nor if you turn it 240 degrees. Also, if you hold the diagram up to a strong light and read it from the back of the page, again, you should get the same equation. This means that not only will the indices on f suffer a transposition, but so will the momenta and so will the indices on V. However, if the equation is to be totally symmetric and f is to be totally antisymmetric, the only possibility is that V also be totally antisymmetric. What's more, if you permute the indices of V and calculate the result directly using eqn (15.68) and remembering to impose the momentum changes, you also find that V is totally antisymmetric. That is why figures 14.4(b) and 14.5 are not contradictory. It doesn't matter whether you traverse the gluon lines in clockwise or counterclockwise direction, but it does matter that you do the same for f and for the momenta, and for V. All three things must be done together. When this principle is applied to diagram 15.17(b), you get eqn (17.125) exactly as it is written, with no need to alter the indices on f.

Here where I think the problem lies. If eqn (15.128) is changed to
N^{\mu}(p', p, k) = V^{\mu\tau\nu}(p - p', p' - k, k - p)\gamma_{\tau}(\not{k} + m)\gamma_{\nu}
then the whole problem may be solved. I will not know for sure until I finish deriving the rest of the proof down to eqn (15.90b) on page 362. One thing is for sure, eqn (15.128) cannot stand as it is in the book.

This is not an entirely satisfactory solution. Why not simply leave V unchanged from eqn (17.125) when formulating eqn (15.128) and drop the minus sign in (15.129)? And why mention eqn (15.66) if it isn't being invoked?

None the less, for the time being, I consider this a tentative solution to the problem and I thank you for your input.

 

[andrien]

This is not correct. You will simply reverse two of the k's, The third will remain unchanged. In other words for instance, swapping k_2 with k_3,

(k_1 - k_2)_{\sigma} \rightarrow (k_1 - k_3)_{\sigma}
(k_2 - k_3)_{\tau} \rightarrow (k_3 - k_2)_{\tau}
(k_3 - k_1)_{\nu} \rightarrow (k_2 - k_1)_{\nu}

So you won't get the minus sign you need unless you also swap the indices on V.

No,you are doing it wrong again.Those indices will be changed.When you write f_lnm,then
If l is with μ(k1),m with v(k2),n with σ(k3) then again the factors will go as
[g_μσ(k₁-k₃)_v+g_σv(k₃-k₂)_μ+g_vμ(k₃-k₁)_σ],then owing to symmetric property of metric tensor g_μσ=g_σμ and so you get the vertex factor as I have written before(getting a minus sign,so f_lnm becomes f_lmn).GOT IT.

 

[Vic Sandler]

No,you are doing it wrong again.Those indices will be changed.When you write f_lnm,then
If l is with μ(k1),m with v(k2),n with σ(k3) then again the factors will go as
[g_μσ(k₁-k₃)_v+g_σv(k₃-k₂)_μ+g_vμ(k₃-k₁)_σ],then owing to symmetric property of metric tensor g_μσ=g_σμ and so you get the vertex factor as I have written before(getting a minus sign,so f_lnm becomes f_lmn).GOT IT.

The formula you have written here simply cannot be right since it has the term k_3 - k_1 twice. What's more, it uses the subscript \mu which is nowhere to be found in any of the figures 14.4(b), 14.5, or 15.17(b). This makes it difficult for me to follow what you mean. Would you be so kind as to resubmit it using the subscripts from figure 14.5 and with care not to include typos. Since you claim that all you are doing is moving the subscripts on k, can you please tell me which of the following transformations you are utilizing:

(k_1, k_2, k_3) \rightarrow (k_1, k_2, k_3) (of course not, this is no transformation at all.)
(k_1, k_2, k_3) \rightarrow (k_1, k_3, k_2)
(k_1, k_2, k_3) \rightarrow (k_2, k_1, k_3)
(k_1, k_2, k_3) \rightarrow (k_2, k_3, k_1)
(k_1, k_2, k_3) \rightarrow (k_3, k_1, k_2)
(k_1, k_2, k_3) \rightarrow (k_3, k_2, k_1)

It would be most helpful if you would show what V looks like before the transformation (presumably this would be eqn 14.68), and what V looks like after the transformation. Please include all of the g's, k's, and subscripts so I can make a careful comparison and understand what you mean.

 

[Vic Sandler]

I've decided, while I'm waiting for your reply, to try one of them myself. Here is what happens when I apply the transformation
(k_1, k_2, k_3) \rightarrow (k_2, k_3, k_1)
to eqn (14.68)

g_{\nu\tau}(k_3 - k_2)_{\sigma} + g_{\tau\sigma}(k_2 - k_1)_{\nu} + g_{\sigma\nu}(k_1 - k_3)_{\tau}
\rightarrow
g_{\nu\tau}(k_1 - k_3)_{\sigma} + g_{\tau\sigma}(k_3 - k_2)_{\nu} + g_{\sigma\nu}(k_2 - k_1)_{\tau}

The result is neither V nor -V because for instance, if we compare the \tau components g_{\sigma\nu}(k_1 - k_3)_{\tau} \ne \pm g_{\sigma\nu}(k_2 - k_1)_{\tau} so it seems that (k_1, k_2, k_3) \rightarrow (k_2, k_3, k_1) is not what you had in mind. It must be one of the other ones, but which one?

 

[andrien]

The formula you have written here simply cannot be right since it has the term k_3 - k_1 twice. What's more, it uses the subscript \mu which is nowhere to be found in any of the figures 14.4(b), 14.5, or 15.17(b). This makes it difficult for me to follow what you mean. Would you be so kind as to resubmit it using the subscripts from figure 14.5 and with care not to include typos. Since you claim that all you are doing is moving the subscripts on k, can you please tell me which of the following transformations you are utilizing:
It would be most helpful if you would show what V looks like before the transformation (presumably this would be eqn 14.68), and what V looks like after the transformation. Please include all of the g's, k's, and subscripts so I can make a careful comparison and understand what you mean.

I made a typo there,the term will look
[g_μσ(k₁-k₃)_v+g_σv(k₃-k₂)_μ+g_vμ(k₂-k₁)_σ].Now it is OK.I thought you see the patern of writting it.But all right,(time for spoon feeding).Assuming the same If l is with μ(k1),m with v(k2),n with σ(k3)(μ can be τ or anything,it is upto the index you write).
now when one writes f_lmn,then it means that write g_lm(k_l(1)-k_m(2))_n+(mn term)+(nl term)[cyclic order just in the sense you write).Since l is with μ,m with v.Replace l by μ,m by v,n by σ and you get the term.Now if it is f_lnm,then again g_ln[k_l(1)-k_n(3)]_m+(nm term)+(ml term)(cyclic order).Replace l by μ,m by v,n by σ and you get the term.Use symmetric property of metric tensor.

 

[Vic Sandler]

You are using lmn for subscripts on V instead of \sigma, \tau, \nu so when you permute lmn, you also permute \sigma, \tau, \nu. I've been saying since post #3 that if you permute lmn, you also have to permute \sigma, \tau, \nu.

I asked you to pick a Feymnan diagram, either 14.4(b), 14.5, or 15.17(b). I would prefer 14.5, but you are free to choose. You did not do so. I asked you to change your symbols to the symbols of the diagram, You did not do so. And I am beginning to suspect that you don't do these things because you can't. But I can.

Start with Feynman diagram 14.4(b), apply its associated Feynman rule to figure 17.17(b) without any deviation of any kind and get eqn (17.125) exactly as it is written.

As for eqn (14.68), you cannot change V_{\sigma\tau\nu} \rightarrow -V_{\sigma\tau\nu} by any change you make to lmn because eqn (14.68) doesn't contain l, m, or n. It only contains \sigma, \tau, \nu and k_1, k_2, k_3. You have said that you can make the change V{\sigma\tau\nu} \rightarrow -V{\sigma\tau\nu} without making any change to \sigma, \tau, \nu, only permuting k_1, k_2, k_3. I don't think you can do so in spite of your arcane notation, and here is why.

There are only 6 ways to permute k_1, k_2, k_3, namely
(k_1, k_2, k_3) \rightarrow (k_1, k_2, k_3)
(k_1, k_2, k_3) \rightarrow (k_1, k_3, k_2)
(k_1, k_2, k_3) \rightarrow (k_2, k_1, k_3)
(k_1, k_2, k_3) \rightarrow (k_2, k_3, k_1)
(k_1, k_2, k_3) \rightarrow (k_3, k_1, k_2)
(k_1, k_2, k_3) \rightarrow (k_3, k_2, k_1)

I will start with eqn (14.68) and take care to see that I make no changes to it other than those permutations.
Here is (14.68)
g_{\nu\tau}(k_3 - k_2)_{\sigma} + g_{\tau\sigma}(k_2 - k_1)_{\nu} + g_{\sigma\nu}(k_1 - k_3)_{\tau}

Here is what it looks like after each one of the permutations above.
g_{\nu\tau}(k_3 - k_2)_{\sigma} + g_{\tau\sigma}(k_2 - k_1)_{\nu} + g_{\sigma\nu}(k_1 - k_3)_{\tau}
g_{\nu\tau}(k_2 - k_3)_{\sigma} + g_{\tau\sigma}(k_3 - k_1)_{\nu} + g_{\sigma\nu}(k_1 - k_2)_{\tau}
g_{\nu\tau}(k_3 - k_1)_{\sigma} + g_{\tau\sigma}(k_1 - k_2)_{\nu} + g_{\sigma\nu}(k_2 - k_3)_{\tau}
g_{\nu\tau}(k_1 - k_3)_{\sigma} + g_{\tau\sigma}(k_3 - k_2)_{\nu} + g_{\sigma\nu}(k_2 - k_1)_{\tau}
g_{\nu\tau}(k_2 - k_1)_{\sigma} + g_{\tau\sigma}(k_1 - k_3)_{\nu} + g_{\sigma\nu}(k_3 - k_2)_{\tau}
g_{\nu\tau}(k_1 - k_2)_{\sigma} + g_{\tau\sigma}(k_2 - k_3)_{\nu} + g_{\sigma\nu}(k_3 - k_1)_{\tau}

I never get -V_{\sigma\tau\nu} unless I also permute \sigma, \tau, \nu. If I do permute \sigma, \tau, \nu taking care to do so properly, I find that V_{\sigma\tau\nu} is totally antisymmetric in its indices. The above constitutes a proof that V is totally antisymmetric. You can't just simply say that the proof is wrong, you have to point out which step is wrong.

 

[andrien]

I am just giving the general idea.No book uses the notation such as v_στv or such to determine vertex factor.So I said earlier,use the formula I have given.Indices will change accordingly as order of lmn changes,which is clear from the vertex factor.You are not permuting any k₁,k₂ or k₃.Only ordering of l,m,n will decide which order of σ,τ or v will occur and accordingly k₁,k₂,k₃.I can tell one other thing and that is to check out other books,such as peskin and schroeder(eqn. 16.82).This discussion is over for now.

 

[Vic Sandler]

No matter, you will never convince me and I will never convince you. Even so, I know where the problem is because you have helped me to find the answer. In the OP, I asked "Can someone confirm that eqn (15.125) is an accurate application of the Feynman rules to Fig (15.17b)?". The fact is, it is not. And I know that because you got me thinking about the effects of permuting the labels on the diagram. When I wrote that OP, I had not considered those effects and so I calculated incorrectly. It is possible that the author made the same mistake I did, I don't know. However, if you carefully apply the rules you will get an extra minus sign.

I appreciate the fact that you stayed with this for as long as you did and I am not surprised that you finally have had enough. I am convinced that I am correct about the permutation of the labels. You must permute labels like (l, \sigma, k_1) as a unit. That is keep all 3 together as you permute. I know you don't agree, but the reason I got confused in the first place was because I didn't realize that it was true. By your help, I now believe it and understand it and even if you don't, it is the key to solving the problem.

Tomorrow I will show you how to apply this rule correctly to the diagram 15.17(b) and get the extra minus sign that belongs there. It is late and I am going to bed.

 

[Vic Sandler]

In the second edition of Mandl and Shaw QFT, on page 361, it's fairly obvious that there is a sign discrepancy between eqn (15.125) and eqn (15.127). In eqn (15.125), there are 5 imaginary i's and 4 explicit minus signs for a final product of +i. However, the first equal sign in eqn (15.129) gives eqn (15.127) an explicit minus sign, i.e. a final product of -i. Eqn (15.125) is created from fig (15.17b) and seems to be correct, but I am not completely sure. Can someone confirm that eqn (15.125) is an accurate application of the Feynman rules to Fig (15.17b)?

Yes, I can help you. eqn (15.125) has a flaw in it. Compare figure 14.4(b) to figure 14.5. In the eqn (14.67) use the g factors in the Ds to convert \alpha to \sigma, \beta to \tau and \gamma to \nu in figure 14.5. You will see that the labels (\tau, m, k_2) and (\nu, n, k_3) are reversed between the two figures. When I checked to see if eqn(15.125) was correct, I was not thinking in terms of such transpositions but with the help of andrien, I did. Armed with that information I now realize that when I checked eqn (15.125) I blindly used eqn (14.68) just as it is without change. This is not correct. Eqn (14.68) is not compatible with figure 14.4(b) because the transpositions have not been accounted for. What is required is a new eqn (14.68') as follows

V_{\sigma \tau \nu} = g_{\tau \nu}(k_2 - k_3)_{\sigma} + g_{\nu \sigma}(k_3 - k_1)_{\tau} + g_{\sigma \tau}(k_3 - k_1)_{\nu}

When this corrected version of (14.68) is used, eqn (15.125) remains as it is, but eqn (15.126) gains a minus sign. Of course, the failure to include eqn (14.68'), the reference to figure 15.15(b), the factor of (k + m_0) in eqn (15.125), the n label in figure 15.17(b), the incorrect middle term on the rhs of eqn (15.126), the bogus minus sign in the first parameter on the rhs of eqn (15.128), and the invocation of eqn (15.66) which isn't being used are also all typos for a total of 8 in the derivation of a single equation. Not good.

 

[Vic Sandler]

One more thing has caught my attention. The author mentions on page 316 that he interchanges the sets of variables in eqn (14.61) and these sets are linked to the sets in eqn (14.65). So if these variables are not permuted as complete sets, you may not be able to rely on his proof of the vertex factor (14.68).

 

[Vic Sandler]

I can tell one other thing and that is to check out other books,such as peskin and schroeder(eqn. 16.82).

A copy of Peskin & Schroeder has just come into my hands. The eqn that interests me is the unnumbered one second from the top of page 507. The diagram is the 3 gluon vertex same as figure 14.5 in Mandl & Shaw. It has the indices on f in counterclockwise order, the same as in eqn (14.68) in M & S. However, the indices on the V terms in P & S are opposite those of M & S. For instance, taking the \mu index in P & S, (p - q)^{\mu} reads p, q, \mu in counterclockwise order, while taking the \sigma term in M & S (k_3 - k_2)_{\sigma} reads k_3, k_2, \sigma in clockwise order. This difference between the two books introduces a minus sign in the vertex factor. It would be nice to simply say that P & S is correct and M & S is wrong. That would fix everything and you would be correct about the typo in eqn (15.125). However, I have checked the derivation in M & S for eqn (14.68) and it seems air tight to me. Can you find any flaw in the argument starting at the bottom of page 314 of M & S leading up to eqn (14.68) on page 317? Is there a missing minus sign somewhere?

 

[vanhees71]

I don't have the books here now, but have you checked the sign conventions (concerning the coupling constant) in the Lagrangian in both books?

 

[andrien]

As Vanhees has guessed(correctly),the sign is different in both books concerning the coupling constant.See the definition of covariant derivative in both books,you will see it.Apart from it,both books use different metric signature.One uses east coast,other uses west coast.I have already told you to be careful when you read other books.

 

[Vic Sandler]

I am back to being at an impasse. My post #19 is completely wrong and an embarrassment to me. It is a combination of wishful thinking and careless calculating. One thing andrien has taught me is that using the letter V is a distraction in which P & S do not indulge. I will refrain from using it in the future. Figure 14.5 has vertices in counterclockwise order

(\sigma, l, k_1) (\tau, m, k_2) (\nu, n, k_3)

with the vertex factor eqn (14.68)

g_sf_{lmn}[g_{\nu \tau}(k_3 - k_2)_{\sigma} + g_{\tau \sigma}(k_2 - k_1)_{\nu} + g_{\sigma \nu}(k_1 - k_3)_{\tau}]

Note that the indices on f are in the same counterclockwise order as in the vertices, while the orders of the momenta are in clockwise order. It is not true that you can independently alter the order on one of the factors without introducing a spurious minus sign. So this formula must be applied without changes and without tears directly to figure 15.17b. Before I do that, I want to point out that in the derivation of the vertex factor, the authors do not use the covariant derivative, but a simple partial. This is manifest in eqn (14.62) on page 316.

In figure 15.17b, the vertices are in counterclockwise order (with the momenta pointing inward)

(\mu, i, p' - p) (\nu, m, p - k) (\tau, n, k - p')

Applying (14.68), I get

g_sf_{imn}[g_{\tau \nu}(2k - p - p')_{\mu} + g_{\mu \tau}(2p' - p - k)_{\nu} + g_{\mu \nu}(2p - k - p')_{\tau}]

So, in eqn (15.125), the indices on f are in the wrong order, there is a missing factor of -1, and subscripts have become superscripts. In other words, ignoring the up-down location of the indices for the moment, eqn (15.125) is correct as written. That is the only conclusion I can come to by direct application of eqn (14.68) to figure 15.17b. Is there a minus sign somewhere in the up-down location of the indices? I think not, because that only effects 3 of the 4 coordinates, but I'm desperate for an answer. Is it the metric signature? Wouldn't that effect both sides of the eqn? I am absolutely flummoxed. There is one last ray of hope left. Perhaps eqn (15.130a) is wrong. If it is missing a minus sign, it would cause an issue in the derivation of eqn (15.90b) on the next page. I have not yet done that derivation so I can't say.

 

[Vic Sandler]

As Vanhees has guessed(correctly),the sign is different in both books concerning the coupling constant.

Now I see the effect that the sign in the coupling constant has. I compare eqn (14.32) on page 308 of Mandl & Shaw, to eqn (16.6) on page 507 of Peskin & Schroeder and the minus sign is clear. However, I no longer see the value of using P & S eqn (16.82) as was suggested in post #17 to solve my problem in M & S since it was developed using different sign conventions, and my problem is related to a sign issue. Besides, I already know that there is a sign error in M & S, I'm just trying to find out where it is.