equation_database.doi_10_1007_JHEP04_2012_037

Functions

bibtex()

DOI, arXiv, Implementation of electroweak corrections in the POWHEG BOX: single W production

equation_3_4([mathcal_Q_EW, s, q, Q_EW, ...])

One of the singular contributions to the EW virtual part

equation_3_5([mathcal_H, q, k, E, sigma, s, ...])

equation_3_6([mathcal_J, q, k, E, sigma, s, ...])

equation_3_7([mathcal_K, q, k, E, sigma, s, ...])

equation_database.doi_10_1007_JHEP04_2012_037.bibtex()[source]

DOI, arXiv, Implementation of electroweak corrections in the POWHEG BOX: single W production

@article{Barze:2012tt,
    author = "Barze, Luca and Montagna, Guido and Nason, Paolo and Nicrosini, Oreste and Piccinini, Fulvio",
    title = "{Implementation of electroweak corrections in the POWHEG BOX: single W production}",
    eprint = "1202.0465",
    archivePrefix = "arXiv",
    primaryClass = "hep-ph",
    reportNumber = "CERN-PH-TH-2012-025, FNT-2012-01, LPN12-031",
    doi = "10.1007/JHEP04(2012)037",
    journal = "JHEP",
    volume = "04",
    pages = "037",
    year = "2012"
}
equation_database.doi_10_1007_JHEP04_2012_037.equation_3_4(mathcal_Q_EW=mathcal_Q_EW, s=s, q=q, Q_EW=Q_EW, beta=beta, xi_C=xi_C, mu_F=mu_F, i=i, f_plus=f_plus, f_minus=f_minus, n_final=n_final)[source]

One of the singular contributions to the EW virtual part

Parameters:

  • mathcal_Q_EW – one of the singular contributions to the EW virtual part(\(\mathcal{Q}_{EW}\))

  • s – Mandel stamm variable s(\(s\))

  • q – charge of a particle(\(q\))

  • Q_EW – energy scale of the process(\(Q_{EW}\))

  • beta – magnitude of the particles three momentum divided by the energy/0-component of the four vector.(\(beta\))

  • xi_C – arbitrary parameter (e.g. set to one)(\(xi_C\))

  • mu_F – factorization scale(\(\mu_F\))

  • i – index of the i final state particle(\(i\))

  • f_plus – index of the plus incoming particle(\(f_+\))

  • f_minus – index of the minus incoming particle(\(f_-\))

  • n_final – number of final state particles(\(n_{\mathrm final}\))

Returns:

\(\mathcal{math}_{Q EW} = - \left(q^{2}{\left(f_{minus} \right)} + q^{2}{\left(f_{plus} \right)}\right) \left(2 \log{\left(\xi_{C} \right)} + \frac{3}{2}\right) \log{\left(\frac{\mu_{F}^{2}}{Q_{EW}^{2}} \right)} - \sum_{i=1}^{n_{final}} \left(\log{\left(\frac{s \xi_{C}^{2}}{2 Q_{EW}^{2}} \right)} - \frac{\log{\left(\frac{\beta{\left(i \right)} + 1}{1 - \beta{\left(i \right)}} \right)}}{\beta{\left(i \right)}}\right) q^{2}{\left(i \right)}\),

\mathcal{math}_{Q EW} = - \left(q^{2}{\left(f_{minus} \right)} + q^{2}{\left(f_{plus} \right)}\right) \left(2 \log{\left(\xi_{C} \right)} + \frac{3}{2}\right) \log{\left(\frac{\mu_{F}^{2}}{Q_{EW}^{2}} \right)} - \sum_{i=1}^{n_{final}} \left(\log{\left(\frac{s \xi_{C}^{2}}{2 Q_{EW}^{2}} \right)} - \frac{\log{\left(\frac{\beta{\left(i \right)} + 1}{1 - \beta{\left(i \right)}} \right)}}{\beta{\left(i \right)}}\right) q^{2}{\left(i \right)}
equation_database.doi_10_1007_JHEP04_2012_037.equation_3_5(mathcal_H=\mathcal{H}, q=q, k=k, E=E, sigma=\sigma, s=s, Q_EW=Q_{EW}, xi_C=xi_C, i=i, j=j, n_massless=n_{\mathrm massless})[source]
Parameters:

  • s – Mandelstamm variable s

  • sigma – +1 for incomign fermions and outgoing anti-fermions. -1 for outgoing fermions and incoming anti-fermions

  • q – charge of a particle

  • k – four momentum of a particle

  • E – energy of a particle

  • Q_EW – energy scale of the process

  • i – index of the i massless particle

  • j – index of the j massless particle

  • xi_C – arbitrary parameter (e.g. set to one)

  • n_massless – number of (charged implied by equation) massless particles

Returns:

\(\mathcal{H} = - \sum_{\substack{i + 1 \leq j \leq n_{\mathrm massless}\\1 \leq i \leq n_{\mathrm massless}}} \sigma{\left(i \right)} \sigma{\left(j \right)} q{\left(i \right)} q{\left(j \right)} \left(\frac{\log{\left(\frac{s \xi_{C}^{2}}{Q_{EW}^{2}} \right)}^{2}}{2} + \log{\left(\frac{s \xi_{C}^{2}}{Q_{EW}^{2}} \right)} \log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} + \frac{\log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)}^{2}}{2} - \log{\left(1 - \frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} \log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} - \operatorname{Li}_{2}\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}}\right)\right)\),

\mathcal{H} = - \sum_{\substack{i + 1 \leq j \leq n_{\mathrm massless}\\1 \leq i \leq n_{\mathrm massless}}} \sigma{\left(i \right)} \sigma{\left(j \right)} q{\left(i \right)} q{\left(j \right)} \left(\frac{\log{\left(\frac{s \xi_{C}^{2}}{Q_{EW}^{2}} \right)}^{2}}{2} + \log{\left(\frac{s \xi_{C}^{2}}{Q_{EW}^{2}} \right)} \log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} + \frac{\log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)}^{2}}{2} - \log{\left(1 - \frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} \log{\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}} \right)} - \operatorname{Li}_{2}\left(\frac{k{\left(i \right)} k{\left(j \right)}}{2 E{\left(i \right)} E{\left(j \right)}}\right)\right)
equation_database.doi_10_1007_JHEP04_2012_037.equation_3_6(mathcal_J=mathcal_J, q=q, k=k, E=E, sigma=sigma, s=s, Q_EW=Q_EW, xi_C=xi_C, I_0=I_0, I_epsilon=I_epsilon, m=m, l=l, n_massless=n_massless, n_massive=n_massive)[source]

Warning

There is a typo in this equation. The sign infront of the \(I_\epsilon\) term should also be negative as in equation_A_28().

Parameters:

  • I_0 – see :func:`~equation_database.doi_10_1007_JHEP06_2010_043.equation_A_23`(\(I_0\))

  • I_epsilon – see :func:`~equation_database.doi_10_1007_JHEP06_2010_043.equation_A_24`(\(I_\epsilon\))

  • m – index of the m massive particle(\(m\))

  • l – index of the l massless particle(\(l\))

  • n_massless – number of (charged implied by equation) massless particles(\(n_{\mathrm massless}\))

  • n_massive – number of (charged implied by equation) massive particles(\(n_{\mathrm massive}\))

Returns:

\(\mathcal{math}_{J} = - \frac{\sum_{\substack{1 \leq l \leq n_{massless}\\1 \leq m \leq n_{massive}}} \left(- I_{0}{\left(k{\left(l \right)},k{\left(m \right)} \right)} \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)} + I_{\epsilon}{\left(k{\left(l \right)},k{\left(m \right)} \right)} + \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)}^{2} - \frac{\pi^{2}}{6}\right) q{\left(l \right)} q{\left(m \right)} \sigma{\left(l \right)} \sigma{\left(m \right)}}{2}\),

\mathcal{math}_{J} = - \frac{\sum_{\substack{1 \leq l \leq n_{massless}\\1 \leq m \leq n_{massive}}} \left(- I_{0}{\left(k{\left(l \right)},k{\left(m \right)} \right)} \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)} + I_{\epsilon}{\left(k{\left(l \right)},k{\left(m \right)} \right)} + \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)}^{2} - \frac{\pi^{2}}{6}\right) q{\left(l \right)} q{\left(m \right)} \sigma{\left(l \right)} \sigma{\left(m \right)}}{2}
equation_database.doi_10_1007_JHEP04_2012_037.equation_3_7(mathcal_K=\mathcal{K}, q=q, k=k, E=E, sigma=\sigma, s=s, Q_EW=Q_{EW}, xi_C=xi_C, m=m, n=n, I_0=I_0, I_epsilon=I_\epsilon, n_massive=n_{\mathrm massive})[source]
Parameters:

  • I_0 – see :func:`~equation_database.doi_10_1007_JHEP06_2010_043.equation_A_41`(\(I_0\))

  • I_epsilon – see :func:`~equation_database.doi_10_1007_JHEP06_2010_043.equation_A_50`(\(I_\epsilon\))

  • n_massive – number of (charged implied by equation) massive particles(\(n_{\mathrm massive}\))

Returns:

\(\mathcal{K} = - \frac{\sum_{\substack{n + 1 \leq m \leq n_{\mathrm massive}\\1 \leq n \leq n_{\mathrm massive}}} \left(- I_{0}{\left(k{\left(m \right)},k{\left(n \right)} \right)} \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)} - I_{\epsilon}{\left(k{\left(m \right)},k{\left(n \right)} \right)}\right) \sigma{\left(m \right)} \sigma{\left(n \right)} q{\left(m \right)} q{\left(n \right)}}{2}\),

\mathcal{K} = - \frac{\sum_{\substack{n + 1 \leq m \leq n_{\mathrm massive}\\1 \leq n \leq n_{\mathrm massive}}} \left(- I_{0}{\left(k{\left(m \right)},k{\left(n \right)} \right)} \log{\left(\frac{Q_{EW}^{2}}{s \xi_{C}^{2}} \right)} - I_{\epsilon}{\left(k{\left(m \right)},k{\left(n \right)} \right)}\right) \sigma{\left(m \right)} \sigma{\left(n \right)} q{\left(m \right)} q{\left(n \right)}}{2}