## Momentum-dependent Wilson coefficientsThe effective Hamiltonian: \[{\cal H}_1(g,M,\alpha)=\left (\frac{g}{M^2}+\frac{\alpha}{q^2} \right)e^{\tau}{\cal O}_1^{\tau}= \frac{q}{M^2} e^{\tau}{\cal O}_1^{\tau}+\frac{\alpha}{q_0^2}\frac{q_0^2}{q^2}e^{\tau}{\cal O}_1^{\tau}=\frac{g}{M^2}e^{\tau} {\cal O}_1^{\tau}+\frac{\alpha}{q_0^2}e^{\tau}{\cal O}_1^{\tau\prime} \]is implemented (with \(e^{0}=e^{1}=1\)) by: >>> wc_dict1={1: lambda g,M: [g/M**2,g/M**2], (1,'qm2'): lambda alpha,q, q0=0.01: [alpha/q**2,alpha/q**2]} >>> H1=WD.eft_hamiltonian('H1',wc_dict1) The same model can also be implemented using the alternative base: \[{\cal H}_2(g,M,\alpha)=\left (\frac{g}{M^2}+\frac{\alpha}{q^2} \right)e^{\tau}{\cal O}_{M,0,0}^{\tau}= \frac{g}{M^2}e^{\tau}{\cal O}_{M,0,0}^{\tau}+\frac{\alpha}{q_0^2}\frac{q_0^2}{q^2}e^{\tau}{\cal O}_{M,0,0}^{\tau}\]\[{\cal H}_2(g,M,\alpha)=\frac{g}{M^2}e^{\tau}{\cal O}_{M,0,0}^{\tau}+\frac{\alpha}{q_0^2}e^{\tau}{\cal O}_{M,0,0}^{\tau\prime} \] >>> wc_dict2={('M',0,0): lambda g,M: [g/M**2,g/M**2], ('M',0,0,'qm2'): lambda alpha,q, q0=0.01: [alpha/q**2,alpha/q**2]} >>> H2=WD.eft_hamiltonian('H2',wc_dict2) In the case of \({\cal H}_1\), \({\cal O}_1\) and \({\cal
O}_1^{\prime}\) = \(q^2/q_0^2{\cal O}_1\) are handled as different
operators, since the additional momentum dependence due to the
\(q_0^2/q^2\) term requires the calculation of a different response
function for \({\cal O}_1^{\prime}\) compared to \({\cal O}_1\), so
they require different keys in the input dictionaries. The same
happens in \({\cal H}_2\) for \({\cal O}_{M,0,0}\) and \({\cal
O}_{M,0,0}^{\prime}\). In c_1_c_1.npyc_1_qm2_c_1_qm2.npyc_1_c_1_qm2.npyc_1_qm2_c_1.npy
while for c_M_0_0_c_M_0_0.npyc_M_0_0_qm2_c_M_0_0_qm2.npyc_M_0_0_c_M_0_0_qm2.npyc_M_0_0_qm2_c_M_0_0.npy
Any other Hamiltonian object containing the
key Notice that all interferences are included: >>> print(H1) Hamiltonian name:H1 Hamiltonian:c_1(M)* O_1+c_1_qm2(alpha, q, q0=0.01)* O_1_qm2'(q) Squared amplitude contributions: O_1*O_1, O_1*O_1_qm2'(q), O_1_qm2'(q)*O_1, O_1_qm2'(q)*O_1_qm2'(q) ## Factorization of Wilson coefficients momentum dependenceAny argument of a Wilson coefficient starting with
with: \[{\cal O}_1^{\prime\tau}=\frac{c_1^{\prime\tau}(q)}{c_1^{\prime\tau}(q_{0,prop})}{\cal O}_1^{\tau} \] \[{\cal O}_1^{\prime\prime\tau}=\frac{c_1^{\prime\prime\tau}(q)}{c_1^{\prime\tau}(q_{0,lr})}{\cal O}_1^{\tau} \]can be implemented with: >>> wc_dict3={(1,'propagator'): lambda g,q,q0_propagator=0.1,M=0.1: g/(M**2+q**2)*np.array([1,1]), (1,'qm2'): lambda alpha,q, q0_long_range=0.01: alpha/q**2*np.array([1,1])} >>> H3=WD.eft_hamiltonian('H3',wc_dict3) The method >>> H3.coeff_squared_q_dependence(0.01,(1,'qm2'),(1,'qm2')) 1 >>> H3.coeff_squared_q_dependence(0.01,(1,'propagator'),(1,'qm2')) 1.9801980198019802 that are used to calculate the response functions. |