Higher Covariant Derivative Research Articles

Three-loop $$\beta$$-functions and NSVZ relations for the MSSM regularized by higher covariant derivatives

Three-loop $$\beta$$-functions and NSVZ relations for the MSSM regularized by higher covariant derivatives

Three-loop $\beta$-functions and NSVZ relations for the MSSM regularized by higher covariant derivatives

Получены трехпетлевые $\beta$-функции для минимальной суперсимметричной стандартной модели, регуляризованной высшими ковариантными производными при произвольном суперсимметричном перенормировочном предписании. Найдены двухпетлевые аномальные размерности для всех киральных суперполей материи в минимальной суперсимметричной стандартной модели, определенные в терминах голых констант связи. С использованием формулы НШВЗ получены трехпетлевые $\beta$-функции, также определенные в терминах голых констант связи. Это возможно, поскольку при используемой регуляризации соотношения НШВЗ выполняются во всех порядках теории возмущений для ренормгрупповых функций, определенных в терминах голых констант связи. На основании этого получены те же двухпетлевые аномальные размерности и $\beta$-функции, стандартно определенные в терминах перенормированных констант связи при произвольном перенормировочном предписании, не нарушающем суперсимметрию. Также проверено, что при определенном перенормировочном предписании результат полностью совпадает с полученным ранее в $\overline{DR}$-схеме и, таким образом, может считаться его независимым подтверждением.

Three-loop β-functions and two-loop anomalous dimensions for MSSM regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme

Three-loop β-functions of the Minimal Supersymmetric Standard Model regularized by higher covariant derivatives are obtained for an arbitrary supersymmetric subtraction scheme. For this purpose we first calculate two-loop anomalous dimensions for all MSSM chiral matter superfields defined in terms of the bare couplings. Then we use the NSVZ equations for the renormalization group functions defined in terms of the bare couplings, which are valid in all orders in the case of using the higher covariant derivative regularization. This gives the three-loop β-functions defined in terms of the bare couplings. After that, we construct the three-loop β-functions and the two-loop anomalous dimensions standardly defined in terms of the renormalized couplings for an arbitrary subtraction scheme. As a nontrivial correctness test, we verify that for a certain renormalization prescription the general results reproduce the ones obtained earlier in the overline{mathrm{DR}} scheme. Also this can be considered as an indepedent confirmation of the overline{mathrm{DR}} results.

The β-function of supersymmetric theories from vacuum supergraphs: A three-loop example

In this paper, we verify a method which allows to obtain the [Formula: see text]-function of supersymmetric theories regularized by higher covariant derivatives by calculating only specially modified vacuum supergraphs. With the help of this method for a general renormalizable [Formula: see text] supersymmetric gauge theory, a part of the three-loop [Formula: see text]-function depending on the Yukawa couplings is constructed in the general [Formula: see text]-gauge. The result is written in the form of an integral of double total derivatives with respect to the loop momenta. It is demonstrated that all gauge-dependent terms cancel each other in agreement with the general statements. The result in the Feynman gauge (found earlier) coincides with the one obtained by the standard technique which requires to calculate the sum of superdiagrams with two external gauge lines. This fact confirms the correctness of the considered method by a highly nontrivial multiloop calculation.

Finiteness of the triple gauge-ghost vertices in {{mathcal {N}}}=1 supersymmetric gauge theories: the two-loop verification

By an explicit calculation we demonstrate that the triple gauge-ghost vertices in a general renormalizable {{mathcal {N}}}=1 supersymmetric gauge theory are UV finite in the two-loop approximation. For this purpose we calculate the two-loop divergent contribution to the bar{c}^+ V c-vertex proportional to (C_2)^2 and use the finiteness of the two-loop contribution proportional to C_2 T(R) which has been checked earlier. The theory under consideration is regularized by higher covariant derivatives and quantized in a manifestly {{mathcal {N}}}=1 supersymmetric way with the help of {{mathcal {N}}}=1 superspace. The two-loop finiteness of the vertices with one external line of the quantum gauge superfield and two external lines of the Faddeev–Popov ghosts has been verified for a general xi -gauge. This result agrees with the nonrenormalization theorem proved earlier in all orders, which is an important step for the all-loop derivation of the exact NSVZ beta -function.

The NSVZ relations for mathcal{N} = 1 supersymmetric theories with multiple gauge couplings

We investigate the NSVZ relations for mathcal{N} = 1 supersymmetric gauge theories with multiple gauge couplings. As examples, we consider MSSM and the flipped SU(5) model, for which they easily reproduce the results for the two-loop β-functions. For mathcal{N} = 1 SQCD interacting with the Abelian gauge superfield we demonstrate that the NSVZ-like equation for the Adler D-function follows from the NSVZ relations. Also we derive all-loop equations describing how the NSVZ equations for theories with multiple gauge couplings change under finite renormalizations. They allow describing a continuous set of NSVZ schemes in which the exact NSVZ β-functions are valid for all gauge coupling constants. Very likely, this class includes the HD+MSL scheme, which is obtained if a theory is regularized by Higher covariant Derivatives and divergences are removed by Minimal Subtractions of Logarithms. That is why we also discuss how one can construct the higher derivative regularization for theories with multiple gauge couplings. Presumably, this regularization allows to derive the NSVZ equations for such theories in all loops. In this paper we make the first step of this derivation, namely, the NSVZ equations for theories with multiple gauge couplings are rewritten in a new form which relates the β-functions to the anomalous dimensions of the quantum gauge superfields, of the Faddeev-Popov ghosts, and of the matter superfields. The equivalence of this new form to the original NSVZ relations follows from the extension of the non-renormalization theorem for the triple gauge-ghost vertices, which is also derived in this paper.

The all-loop perturbative derivation of the NSVZ beta -function and the NSVZ scheme in the non-Abelian case by summing singular contributions

The perturbative all-loop derivation of the NSVZ beta -function for {{mathcal {N}}}=1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the beta -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the beta -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

NSVZ Relation and NSVZ Scheme in $$\mathcal{N} = 1$$ Non-Abelian Supersymmetric Gauge Theories

We briefly describe the perturbative derivation of the exact NSVZ $$\beta $$ -function for $$\mathcal{N} = 1$$ supersymmetric non-Abelian gauge theories in the case of using the higher covariant derivative regularization. It is demonstrated that with this regularization the NSVZ equation written in the form of a relation between the $$\beta $$ -function and the anomalous dimensions of the quantum superfields is satisfied by the renormalization group functions defined in terms of the bare couplings independently of a renormalization prescription. For the renormalization group functions defined in terms of the renormalized couplings this implies that one of the NSVZ schemes in all loops is given by the HD+MSL prescription, when the theory is regularized by higher derivatives and the renormalization constants contain only powers of $$ln\Lambda {\text{/}}\mu $$ . All these statements are verified by comparing certain terms in the three-loop $$\beta $$ -function with the corresponding contributions to anomalous dimensions of the quantum superfields.

Two-loop renormalization of the matter superfields and finiteness of mathcal{N} = 1 supersymmetric gauge theories regularized by higher derivatives

The two-loop anomalous dimension of the chiral matter superfields is calculated for a general mathcal{N} = 1 supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the one-loop finite theory is also two-loop finite in the HD+MSL scheme. Using the assumption that with the higher covariant derivative regularization the NSVZ equation is satisfied for RGFs defined in terms of the bare couplings, we construct the expression for the three-loop β-function. Again, the result is written both for the β-function defined in terms of the bare couplings and for the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription.

The NSVZ \u03b2-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities

The contributions of the matter superfields and of the Faddeev-Popov ghosts to the β-function of mathcal{N} = 1 supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the β-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the β-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev-Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices.

The NSVZ relation and the NSVZ scheme for N = 1 non-Abelian supersymmetric theories, regularized by higher covariant derivatives

We discuss, how the exact NSVZ -function appears in N = 1 supersymmetric non-Abelian gauge theories, regularized by higher covariant derivatives. In particular, we demonstrate that the renormalization group functions defined in terms of the bare couplings satisfy the NSVZ relation in the case of using this regularization. This occurs, because the loop integrals giving the -function are integrals of double total derivatives with respect to loop momenta. It is also shown that for the renormalization group functions standardly defined in terms of the renormalized couplings the NSVZ scheme can be obtained if the theory is regularized by higher covariant derivatives and only powers of ln/µ are included into the renormalization constants. These statements are confirmed by the explicit calculations in the three-loop approximation, where the scheme dependence is essential.

The \u03b2-function of mathcal{N} = 1 supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

For a general mathcal{N} = 1 supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the β-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the β-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the β-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.

Three-loop contribution of the Faddeev\u2013Popov ghosts to the beta -function of mathcal{N}=1 supersymmetric gauge theories and the NSVZ relation

We find the three-loop contribution to the beta -function of mathcal{N}=1 supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev–Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the beta -function is compared with the two-loop anomalous dimension of the Faddeev–Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the beta -function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.

Four-derivative couplings via the T -duality invariance constraint

We examine the proposal that the dimensional reduction of the effective action of perturbative string theory on a circle, should be invariant under T-duality transformations. The T-duality transformations are the standard Buscher rules plus some higher covariant derivatives. By explicit calculations at order $\alpha'$ for metric, dilaton and B-field, we show that the T-duality constraint can fix both the effective action and the higher derivative corrections to the Buscher rules up to an overall factor. The corrections depend on the scheme that one uses for the effective action. We have found the effective action and its corresponding T-duality transformations in an arbitrary scheme.

Regularization of Propagators and Logarithms in the Background Field Method in Four Dimensions

Determinant and higher-loop terms, usually treated by the Pauli–Villars and higher covariant derivatives methods, in the background field method can hardly be regularized simultaneously. At the same time, we observe that introducing a scalar multiplier in front of the quadratic form, which is equivalent to changing the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite.

The three-loop Adler D-function for mathcal{N}=1 SQCD regularized by dimensional reduction

The three-loop Adler D-function for mathcal{N}=1 SQCD in the overline{mathrm{DR}} scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the D-function in the overline{mathrm{DR}} scheme for all renormalization constants (including the one for the electromagnetic coupling constant which appears due to the SQCD corrections). It is shown that the results do not satisfy the NSVZ-like equation relating the D-function to the anomalous dimension of the matter superfields. However, the NSVZ-like scheme can be constructed with the help of a properly tuned finite renormalization. It is also demonstrated that the three-loop D-function defined in terms of the bare couplings with the dimensional reduction does not satisfy the NSVZ-like equation for an arbitrary renormalization prescription. We also investigate a possibility to present the results in the form of the β-expansion and the scheme dependence of this expansion.

NSVZ Relation in Supersymmetric Theories Regularized by Higher Derivatives

We investigate the NSVZ equation in $$\mathcal{N} = 1$$ supersymmetric gauge theories, which relates the $$\beta $$ ‑function to the anomalous dimension of the matter superfields. In particular, we argue that it is closely connected with the non-renormalization theorem for the three-point gauge-ghost vertices (in which one external leg corresponds to the quantum gauge superfield). Using finiteness of these vertices the exact NSVZ $$\beta $$ -function can be equivalently presented in the form of a relation between the $$\beta $$ -function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This equation allows explaining how the NSVZ equation appears in the perturbation theory for theories regularized by higher covariant derivatives and constructing the NSVZ scheme in all orders in the case of using this regularization. The results are verified by an explicit three-loop calculation of the terms quartic in the Yukawa couplings.

Exact Results in Explicit Three-Loop Calculations Using Higher Derivatives for $$\mathcal{N}$$ = 1 SQCD

We calculate the three-loop Adler $D$-function of ${\cal N}=1$ SQCD regularized by higher covariant derivatives and find the subtraction scheme in which the exact NSVZ-like relation for this function that was recently proposed is valid.

Two-loop renormalization of the Faddeev-Popov ghosts in mathcal{N}=1 supersymmetric gauge theories regularized by higher derivatives

For the general renormalizable mathcal{N}=1 supersymmetric gauge theory we investigate renormalization of the Faddeev-Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general ξ-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev-Popov ghosts defined in terms of the renormalized coupling constant and examine its dependence on the subtraction scheme.

New form of the exact NSVZ \u03b2-function: the three-loop verification for terms containing Yukawa couplings

We investigate a recently proposed new form of the exact NSVZ β-function, which relates the β-function to the anomalous dimensions of the quantum gauge superfield, of the Faddeev-Popov ghosts, and of the chiral matter superfields. Namely, for the general renormalizable mathcal{N} = 1 supersymmetric gauge theory, regularized by higher covariant derivatives, the sum of all three-loop contributions to the β-function containing the Yukawa couplings is compared with the corresponding two-loop contributions to the anomalous dimensions of the quantum superfields. It is demonstrated that for the considered terms both new and original forms of the NSVZ relation are valid independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings. This result is obtained from the equality relating the loop integrals, which, in turn, follows from the factorization of the integrals for the β-function into integrals of double total derivatives. For the renormalization group functions defined in terms of the renormalized couplings we verify that the NSVZ scheme is obtained with the higher covariant derivative regularization supplemented by the subtraction scheme in which only powers of ln Λ/μ are included into the renormalization constants.