Question 5.2: More about the Derivation of QCD Corrections to Electron–Nuc......
More about the Derivation of QCD Corrections
to Electron–Nucleon Scattering
Evaluate S_{\mu}{}^{\mu} and p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}} for the trace in (5.8).
\sum\limits_{\varepsilon}\varepsilon^{\ast\nu}\varepsilon^{\nu^\prime}=-g^{\nu\nu^{\prime}}\ . (5.8)
First we employ
\gamma_{\mu}{a\!\!\!/}\gamma^{\mu}=-\,2a\!\!\!/\ ,\\ \gamma_{\mu}a\!\!\!/ b\!\!\!/\gamma^{\mu}=4a\cdot b~,\\\gamma_{\mu}a\!\!\!/ b\!\!\!/ c\!\!\!/ \gamma^{\mu}=-\,2 c\!\!\!/ b\!\!\!/ a\!\!\!/ , (1)
in order to simplify S_{\mu\mu^{\prime}}:
S_{\mu\mu^{\prime}}=\frac{-2}{(p-K)^{4}}\mathrm{tr}\left[\rlap/p(\rlap/p-K\!\!\!/)\gamma_{\mu^{\prime}}(\rlap/p+\rlap/q-K\!\!\!/)\gamma_{\mu}(\rlap/p-K\!\!\!/)\right]\\ +\frac{-2}{(p-K)^{2}(p+q)^{2}}\mathrm{tr}\left[p\!\!\!/\gamma_{\mu}(p\!\!\!/+q\!\!\!/)(p\!\!\!/-K\!\!\!/)\gamma_{\mu^{\prime}}(p\!\!\!/+q\!\!\!/-K\!\!\!/)\right]\\ +\frac{-2}{(p-K)^{2}(p+q)^{2}}\mathrm{tr}\left[p\!\!\!/\gamma_{\mu^{\prime}}(p\!\!\!/+q\!\!\!/)(p\!\!\!/-K\!\!\!/)\gamma_{\mu}(p\!\!\!/+q\!\!\!/-K\!\!\!/)\right]\\+\,\frac{-2}{(p-q)^{4}}\mathrm{tr}\left[p\!\!\!/\gamma_{\mu^{\prime}}(p\!\!\!/+q\!\!\!/)(p\!\!\!/-q\!\!\!/-K\!\!\!/)(p\!\!\!/+q\!\!\!/)\gamma_{\mu}\right]\ . (2)
Then S_{\mu}{}^{\mu} is brought into the form
S_{\mu}{}^{\mu}=\frac{4}{(p-K)^{4}}\mathrm{tr}\left[\rlap/p(\rlap/p-\rlap/K)(\rlap/p-\rlap/K+\rlap/q)(\rlap/p-\rlap/K)\right]\\ -\,\frac{16}{(p-K)^{2}(p+q)^{2}}(p+q)\cdot(p-K)\mathrm{tr}\,[p\!\!\!/(p\!\!\!/-K\!\!\!/+\rlap/q)]\\+\,\frac{4}{(p-q)^{4}}\mathrm{tr}\,[\,\rlap/p(\rlap/p+q\!\!\!/)(\rlap/p-K\!\!\!/+\rlap/q)(p\!\!\!/+\rlap/q)]\ \ . (3)
For massless quarks and real gluons, utilizing {a\!\!\!/}{a\!\!\!/}=a_{\mu}a_{\nu}(\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu})= g^{\mu\nu}a_{\mu}a_{\nu}=a^{2}, one has
p\!\!\!/^{2}=p^{2}=0~,\\ (p\!\!\!/-K\!\!\!/+q\!\!\!/)^{2}=(p-K+q)^{2}=0\\ (p-K)^{2}=-2p\cdot K\ ,because of the δ[(p+q−K )²] function in (5.8)
(p\!\!\!/+{q\!\!\!/})(p\!\!\!/-{K\!\!\!/}+{{q\!\!\!/}})=(p\!\!\!/+{{q\!\!\!/}}-{K\!\!\!/}+{K\!\!\!/})(p\!\!\!/-{ K\!\!\!/}+{{q\!\!\!/}})\\ =(p\!\!\!/-K\!\!\!/+q\!\!\!/)^{2}+K\!\!\!/(p\!\!\!/-K\!\!\!/+q\!\!\!/)=+K\!\!\!/(p\!\!\!/-K\!\!\!/+q\!\!\!/)\ ,\\ (p\!\!\!/-K)(p\!\!\!\!/-K\!\!\!/+q\!\!\!/)=(p\!\!\!/-K\!\!\!/+q\!\!\!/-q\!\!\!/)(p\!\!\!/-K\!\!\!/+q\!\!\!/)\\ =(p\!\!\!/-K\!\!\!/+q\!\!\!/)^{2}-q\!\!\!/(p\!\!\!/-K\!\!\!/+q\!\!\!/)=-q\!\!\!/(p\!\!\!/-K\!\!\!/+q\!\!\!/)\ ,\\ (p+q)^{2}=(p+q-K+K)^{2}=2K\cdot(p+q-K)=2K\cdot(p+q)\,\,\,, (4)
which simplifies (3) to
S_{\mu}{}^{\mu}=\frac{1}{(p\cdot K)^{2}}\mathrm{tr}\left[\,p\!\!\!/ q\!\!\!/(p\!\!\!/-K\!\!\!/+\rlap/q)\,K\!\!\!/\right]\\ +\,\frac{4}{p\cdot K\ ^{\phantom{l}}K\cdot(p+q)}(p+q)\cdot(p-K)4p\cdot(q-K)\\ +\,\frac{1}{[K\cdot(p+q)]^{2}}\mathrm{tr}\,[\,p\!\!\!/ K\!\!\!\!/(p\!\!\!\!/-K\!\!\!\!/+q\!\!\!\!/)q\!\!\!\!/] =\left({\frac{4}{(p\cdot K)^{2}}}+{\frac{4}{[K\cdot(p+q)]^{2}}}\right)[2(p\cdot q)\left(p\cdot K\right)]\\\left.+\frac{16}{(p\cdot K)\,(K\cdot(p+q))}(p+q)\cdot(p-K)\,p\cdot(q-K)\,\right.. (5)
Next we introduce the Mandelstam variables
t=(p-K)^{2}=-2p\cdot K\ ,\\s=(p+q)^{2}=2K\cdot(p+q)=2\,p\cdot q\;\;,\\ u=(q-K)^{2}
= (q−K + p)²−2p · (q−K + p)+ p²
=−2p · (q−K ) . (6)
Taking into account
Q² ≡ −q² = −(q+ p− p)²
= −2K · (p+q) + 2p · (q+ p)
= 2(q+ p) · (p−K), (7)
we finally obtain
S_{\mu}{}^{\mu}=-8\frac{s}{t}-8\frac{t}{s}+16\frac{Q^{2}u}{s t}=-8\left(\frac{s}{t}+\frac{t}{s}-2\frac{Q^{2}u}{s t}\right)~. (8)
Now we evaluate the scalar p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}} in a completely analogous manner. From (2) follows
p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}}=\frac{-2}{(p-K)^{4}}\mathrm{tr}[p\!\!\!/(p\!\!\!/-K\!\!\!/)p\!\!\!/(p\!\!\!/-K\!\!\!/+q\!\!\!/)p\!\!\!/(p\!\!\!/-K\!\!\!/)\ ]\;\;. (9)
All other terms vanish because p\!\!\!/^{2}=p^{2}=0, and therefore (9) simplifies futher to
p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}}=\frac{-2}{(p-K)^{4}}\mathrm{tr}[p\!\!\!/ K\!\!\!/ p\!\!\!/(-K\!\!\!/+q\!\!\!/)p\!\!\!/ K\!\!\!/]\ \cdot (10)
The final simplifications are achieved by exchanging the first two factors under the trace
p\!\!\!/ K\!\!\!/=p_{\mu}k_{\nu}\gamma^{\mu}\gamma^{\nu}=p_{\mu}k_{\nu}(2g^{\mu\nu}-\gamma^{\nu}\gamma^{\mu})=2\,p\cdot K-K\!\!\!/\,p\!\!\!/and therefore
p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}}=\frac{-4}{(p-K)^{4}}p\cdot K\operatorname{tr}\left[\rlap/p(-\rlap/K+\rlap/q)\rlap/p \rlap/K\right]+0~. (11)
A similar exchange of the last two factors yields
p^{\mu}p^{\mu^{\prime}}S_{\mu\mu^{\prime}}=\frac{-8}{(p-K)^{4}}(p\cdot K)^{2}\operatorname{tr}\left[\rlap/p (\rlap/q-\rlap/K)\right]\\ ={\frac{-8}{(p-K)^{4}}}(p\cdot K)^{2}\cdot4p\cdot(q-K)= −8p · (q−K) = 4u . (12)
In the last step one of the relations (4) and (6) has been used.